University of California. 
 
 

ELEMENTARY TREATISE 
 
 ON 
 
 ASTRONOMY: 
 
 * <** 
 IN FOUK PAKTS. 
 
 
 
 OONTAINIM.G 
 
 i SYSTEMATIC AND COMPEEHENSIVE EXPOSITION OF THE THEORY, 
 AND THE MOKE IMPORTANT PRACTICAL PROBLEMS ; 
 
 WITH 
 
 SOLAR, LUNAR, AND OTHER ASTRONOMICAL TABLES, 
 
 DESIGNED FOB USE AS A 
 
 TEXT-BOOK IN COLLEGES AND THE HIGHER ACADEMIES. 
 
 - 
 
 BY 
 
 WILLIAM A. NOBTON, A. M,, 
 
 FELLOW OF THE AMERICAN PHILOSOPHICAL SOCIETY, AND OF THE AMERICAN 
 
 ACADEMY OF ARTS AND SCIENCES, AND CORRESPOND**** - : =i==^z 
 MEMBER OF THE NATIONAL INSTITUTE. 
 
 LIBKA i; 
 
 . 
 
 THIBD EDITION. Xl Y |f^ I T V 
 
 CORRECTED, IMPROVED, AND ENLARGED. ( ' \ t V ! 
 
 NEW YORK: 
 JOHN WILEY, 167 BROADWAY. 
 
 .- *;.. v i' " ' 
 
 1853. 
 
 

 Entered according to Act of Congress, in the year 1845. 
 
 By WILLIAM A NORTON, 
 In the Clerk's Office of the District Court for the Southern District of New York. 
 
 Stereotyped by 
 RICHARD C. VALENTINE. 
 
 17 Dnteh-itreet, New YorK. 
 
PREFACE 
 
 TO THE FIRST EDITION. 
 
 THE object for which the present treatise on Astronomy has 
 been written, is to provide a suitable text-book for the use of the 
 students of Colleges and the higher Academies, and at the same 
 time to furnish the practical astronomer with rules, or formulae, 
 and accurate tables for performing the more important astronomi- 
 cal calculations. 
 
 It is divided into four Parts. The first three Parts contain the 
 theory : the First Part treating of the determination of the places 
 and motions of the heavenly bodies ; the Second, of the phenom- 
 ena resulting from the motions of these bodies, and of their ap- 
 pearances, dimensions, and physical constitution ; and the Third, 
 of the theory of Universal Gravitation. The Fourth Part consists 
 of Practical Problems, which are solved with the aid of the Tables 
 appended to the work. An Appendix is added, containing a large 
 collection of useful trigonometrical formulae, and such investiga- 
 tions of astronomical formulae as, from their length, could not, 
 consistently with the plan of the work, be admitted into the text, 
 and which it was still deemed advisable to retain, for the benefit 
 of the few who might wish to pursue them. 
 
 The chief peculiarities of this treatise, as compared with the 
 kindred works now in use in our Colleges, are, 1. The adoption 
 of the Copernican System as an hypothesis at the outset, leaving 
 it to be established by the agreement between the conclusions to 
 which it leads and the results of observation. 2. A connected ex- 
 position of the principles and methods of astronomical observation, 
 embracing the doctrine of the sphere, the construction and use of 
 
.:*/ 
 
 Iv PREFACE TO THE FIRST EDITION. 
 
 the principal astronomical instruments, and the theory of the cor- 
 rections for refraction, parallax, aberration, precession, and nuta- 
 tion. 3. The exhibition of the methods of determining the motions 
 and places of the different classes of the heavenly bodies, in one 
 connection. 4. The explanation of the principles of the construc- 
 tion of astronomical tables. 5. The addition of a chapter on the 
 measurement of time, embracing the explanation of the different 
 kinds of time, the processes by which one is converted into an- 
 other, the methods of determining the time from astronomical obser- 
 vations with the transit instrument and sextant, and the calendar. 
 6. The contemplation of the phenomena of the aspect and appa- 
 rent motion of the heavenly bodies as consequences of their motions 
 in space, and the deduction of the various circumstances of these 
 phenomena from the theory of the orbitual motions previously es- 
 tablished. 7. A comprehensive view of the theory of Universal 
 Gravitation, followed out into its various consequences. 8. An 
 exposition of the operations of the disturbing forces in producing 
 the principal perturbations of the motions of the Solar System. 
 9. The solution of Practical Problems by means of logarithmic 
 formulae instead of rules. 10. The addition of lunar, solar, and 
 other astronomical tables, of peculiar accuracy and improved ar- 
 rangement. 
 
 It may further be mentioned, that many of the investigations 
 have been materially simplified, and that the aim has been to in- 
 troduce into all of them as much simplicity and uniformity of 
 method as possible. Particular attention has also been paid to 
 the diagrams, it being of great importance that they should convey 
 correct notions to the mind of the student. 
 
 The Problems in the Fourth Part /are principally for making 
 calculations relative to the Sun, Moon, and Fixed Stars. The 
 Tables of the Sun and Moon, used in finding the places of these 
 bodies, have, for the most part, been abridged and computed from 
 the tables of Delambre,- as corrected by Bessel, and those of 
 Burckhardt ; and the Tables of Epochs have all been reduced to 
 the meridian of Greenwich. These Tables will give the places 
 
PREFACE TO THE FIRST EDITION. V 
 
 and motions of the Sun and Moon within a fraction of a second of 
 the tables from which they were derived. But as this degree of 
 accuracy will not generally be required, rules are also given in 
 the Fourth Part for obtaining approximate results. The entire set 
 of Tables has been stereotyped, and great pains has been taken, 
 by repeated revisions and verifications, to render them accurate. 
 
 The principal astronomical works which have been consulted, 
 are those of Vince, Gregory, Woodhouse, Delambre, Biot, La- 
 place, Herschel, and Gummere ; also FranccBur's Uranography, 
 Francmur's Practical Astronomy, Encyclopedia Metropolitana, 
 Article " Astronomy," and Baily's Tables and Formula. Free 
 use has been made of the methods of investigation and demonstra- 
 tion pursued in these treatises, such modifications being intro- 
 duced, in those which have been adopted, as the plan of the work 
 required. 
 
 New Yo'*, January, 1839. 
 
PREFACE 
 
 TO THE SECOND EDITION. 
 
 IN preparing a new edition of the present treatise, material al- 
 terations, and, it is hoped, improvements have been made in it. 
 The more abstruse parts are now printed in smaller type, and 
 their connection with the other portions of the book is made such 
 that they can be pursued or omitted at pleasure : by which the 
 opportunity is afforded of making a selection between two courses 
 of study, differing materially in extent, and in the amount of labor 
 and mathematical attainment required for their acquisition. Wood- 
 cuts have also been substituted for the original plates, as more 
 convenient to the student ; and for the sake of more ample illus- 
 tration, nearly fifty new diagrams have been added. Many of 
 these are illustrative of the telescopic appearances of the planets 
 and other heavenly bodies. Considerable additions have been 
 made to several of the Chapters ; especially to the Chapter on 
 Instruments, and those in which the appearances and physical 
 constitution of the heavenly bodies are treated of. These are, for 
 the most part, printed in a small-sized type, as well as the parts 
 above specified. The Chapters on Comets have been rewritten. 
 The Author has also endeavored, in many instances which need 
 not be enumerated, to profit by such criticisms and suggestions of 
 improvement as have been made by others, as well as by his own 
 experience in the use of the work as a text-book. 
 
 The Tables remain unaltered ; with the exception of Tables L, 
 I'L, III., and IV., which have been rendered more accurate. Fre- 
 quent comparisons, since the publication of the first edition, of the 
 Lunar and Solar Tables with the places of the Moon and Sun, as 
 
PREFACE TO THE SECOND EDITION. vii 
 
 given in the Nautical Almanac and the Connaissance des Terns, 
 have furnished additional confirmation of their accuracy. 
 
 Notwithstanding the considerable augmentation which the work 
 has received, the retail price of it is very much reduced. 
 
 The references in the text to the investigations of astronomical 
 formulae in the Appendix, were omitted, in preparing this edition, 
 under the expectation that the new matter to be inserted would 
 render the omission of these investigations necessary. They are, 
 however, retained ; and the articles are designated in which men- 
 tion is made of such formulas. 
 
 In addition to the Astronomical works mentioned in the preface 
 to the first edition, the Author has particularly consulted, in the 
 preparation of this edition, besides periodicals, Littrow's Wonders 
 of the Heavens, KendaWs Uranography, NichoVs Phenomena of 
 the Solar System, Nicholas Architecture of the Heavens, and Ma- 
 son's Introduction to Practical Astronomy. , His acknowledgments 
 are due to Professor Kendall for the copy which he was permitted 
 to take of the delineation of the great comet of 1843, given in his 
 Uranography. 
 
 Where passages have been borrowed entire from any author, 
 credit has been given in the usual way, viz., by references to 
 specifications of title, &c., inserted at the bottom of the page. 
 To these it should be added that the greater portion of the Chap- 
 ter on the Calendar, after the first paragraph, is taken from Wood- 
 house's Astronomy, and most of Art. 463, from Gregory's Astron- 
 omy. Particular assistance has also been derived, in Part IV., 
 from Gummere's Astronomy. It would be idle in every new 
 scientific treatise, to attempt to designate all the instances in which 
 the same forms of expression and the same methods of investiga- 
 tion may have been adopted, that occur in other kindred treatises. 
 DELAWARE COLLEGE, 
 
 Newark, Del, June, 1845. 
 
PREFACE 
 
 TO THE THIRD EDITION. 
 
 SINCE the publication of the previous edition, numerous im- 
 portant and highly interesting astronomical discoveries have 
 been made. These have been introduced into the present 
 edition, by appending*a collection of Notes to the text. The 
 references to these notes, inserted in the text, will bring the 
 different topics of which they treat to the notice of the stu- 
 dent, in the proper connection, while they will collectively 
 form a brief exposition of the progress recently made in astro- 
 nomical science. It has been the intention to make this edi- 
 tion a faithful picture of the present state of the science ; in 
 so far as this end could be attained within the limits which 
 should be observed in the preparation of a college text-book. 
 
 PROVIDENCE, April, 1852. 
 
TABLE OF CONTENTS. 
 
 INTRODUCTION. 
 General Notions General Phenomena of the Heavens 
 
 PART I 
 
 ON THE DETERMINATION OF THE PLACES AND MOTIONS OF 
 THE HEAVENLY BODIES. 
 
 CHAPTER I. 
 
 On the Celestial and Terrestrial Spheres .' ; . . ^ V . n 
 
 CHAPTER II. 
 
 On the Construction and Use of the Principal Astronomical Instruments 23 
 
 Transit Instrument - - - -.-.,* . 26 
 
 Astronomical Clock ^ .- f ,,< : ^%..-.* : /-^ ,* 31 
 
 Astronomical Circle - - - V'_. -> \' ' ' ' '* - 32 
 
 Altitude and Azimuth Instrument , *, - - - - 3$ 
 
 Equatorial - ", - ,..,,V* /.', "- /-;y-^t ;'&. 37 
 
 Sextant > . . . 4 '.!, .... 39 
 
 Errors of Instrumental Admeasurement ----- 43 
 
 Telescope - - .... . . . - #. 
 
 CHAPTER III. 
 
 On the Corrections of the Co-ordinates of the Observed Place of a 
 
 Heavenly Body - - - - - . ,^ . . 43 
 
 Refraction - - - - - * - * , : - 44 
 
 /Parallax .... j--' .; . . . 49 
 
 f Aberration --.-.- . >. - - 55 
 
 f Precession and Nutation ^4>^i.^-%- *' ... 60 
 Remarks on the Corrections. Verification of the Hypothesis 
 
 that the Diurnal Motion of the Stars is Uniform and Circular 66 
 
 2 
 
TABLE OF CONTENTS. 
 
 CHAPTER IV. 
 
 PACK 
 
 Of the Earth its Figure and Dimensions Latitude and Longitude 
 of a Place ..--------66 
 
 Determination of the Latitude and Longitude of a Place 69 
 
 CHAPTER V. 
 Of the Places of the Fixed Stars 72 
 
 CHAPTER VI. 
 Of the apparent motion of the Sun in the Heavens - 77 
 
 CHAPTER VII. 
 
 Of the Motions of the Sun, Moon, and Planets, in their orbits - 82 
 
 Kepler's Laws - - ; '' '< "--;, ,- - - ib. 
 
 Definitions of Terms - - - -- - - - 85 
 
 Elements of the Orbit of a Planet - - - - . - - 87 
 
 Methods of Determining the Elements of the Sun's Apparent 
 
 Orbit, or of the Earth's Real Orbit 88 
 
 Methods of Determining the Elements of the Moon's Orbit - 92 
 
 Methods of Determining the Elements of a Planet's Orbit - 94 
 
 Mean Elements and their Variations - - - - - 102 
 
 CHAPTER VIII. 
 
 On the Determination of the Place of a Planet, or of the Sun, or 
 Moon, for a Given Time, by the Elliptical Theory ; and of the 
 
 Verification of Kepler's Laws ' 105 
 
 Place of a Planet, or of the Sun or Moon in its Orbit - - ib. 
 
 Heliocentric Place of a Planet - - - . - . . V 106 
 
 Geocentric Place of a Planet - - 107 
 
 Places of the Sun and Moon - - - .,- 108 
 
 Verification of Kepler's Laws * -; "-? ib. 
 
 CHAPTER IX. 
 
 On the Inequalities of the Motions of the Planets and of the Moon ; 
 and of the Construction of Tables for finding the Places of these 
 
 Bodies 109 
 
 Construction of Tables - . _ . . . . 114 
 
TABLE OF CONTENTS. 
 
 CHAPTER X. 
 
 PAGE 
 
 Of the Motions of the Comets *V,' . - - - 117 
 
 CHAPTER XL 
 
 Of the Motions of the Satellites - - - .... 134 
 
 CHAPTER XII. 
 
 the Measurement of Time - - - - - - - 137 
 
 Different Kinds of Time ----,. ,. _. - - ib. 
 
 Conversion of one Species of Time into another - - - 128 
 Determination of the Time and Regulation of Clocks by Astro- 
 
 nomical Observations - - - - - . - 130 
 
 ^f the Calendar - - v,, ^ ,^ : , ^ , ^ ,;.... . . 133 
 
 PART II. 
 
 ON THE PHENOMENA RESULTING FROM THE MOTIONS OF 
 THE HEAVENLY BODIES, AND ON THEIR APPEARANCES, 
 DIMENSIONS, AND PHYSICAL CONSTITUTION. 
 
 CHAPTER XIII. 
 
 Of the Sun and the Phenomena attending its Apparent Motions - 137 
 
 Inequality of Days - - -* ." - ib 
 
 Twilight - - - V V 141 
 
 The Seasons * - J^ - , - 144 
 
 Appearance, Dimensions, and Physical Constitution of the Sun 147 
 
 CHAPTER XIV. 
 
 Of the Moon and its Phenomena - - - - *>.* 153 
 
 Phases of the Moon - * * * * * * ** 
 
 Moon's Rising, Setting, and Passage over the Meridian - . 155 
 
 Rotation and Librations of the Moon - . .%- - ' - 158 
 
 Dimensions and Physical Constitution of the Moon * - 159 
 
Hi TABLE OF CONTENTS. 
 
 CHAPTER XV. 
 
 PAS 
 
 Eclipses of the Sun and Moon Occultations of the Fixed Stars - 162 
 
 Eclipses of the Moon - - r *. - .... ib. 
 
 Eclipses of the Sun 171 
 
 Occultations - - '" -, : '.' 183 
 
 CHAPTER XVI. 
 
 Of the Planets and the Phenomena occasioned by their Motions in 
 
 Space * r '- 184 
 
 Apparent Motions of the Planets with respect to the Sun ib. 
 
 Stations and Retrogradations of the Planets - - -- 187 
 
 Phases of the Inferior Planets .... W : : 4' 190 
 
 Transits of the Inferior Planets - - - - * - 191 
 Appearances, Dimensions, Rotation, and Physical Constitution 
 
 of the Planets 192 
 
 CHAPTER XVII. 
 
 Of Comets - - - - - ' - - 201 
 
 Their General Appearance Varieties of Appearance - - ib. 
 
 Form, Structure, and Dimensions of Comets T v - * - 205 
 
 Physical Constitution of Comets - - - . * . 207 
 
 CHAPTER XVIII. 
 
 Of the Fixed Stars *,. *; 211 
 
 Their Number and Distribution over the Heavens -; - ib. 
 
 Annual Parallax and Distance of the Stars .... 213 
 
 Nature and Magnitude of the Stars ..... 216 
 
 Variable Stars - - . 917 
 
 Double Stars V ^ - 219 
 
 Proper Motions of the Stars - *';.- . - 222 
 
 Clusters of Stars Nebulae 223 
 
 Distance and Magnitude of Nebulje ..... 227 
 
 Structure of the Material Universe Nebular Hypothesis - 229 
 
TABLE OF CONTENTS. xiii 
 
 PART III. 
 
 OF THE THEORY OF UNIVERSAL GRAVITATION. 
 CHAPTER XIX. 
 
 ffMP 
 Of the Principle of Universal Gravitation -V,^\y - - 231 
 
 CHAPTER XX. 
 
 Theory of the Elliptic Motion of the Planets - '"' * 234 
 
 CHAPTER XXI. 
 
 Theory of the Perturbations of the Elliptic Motion of the Planets 
 
 and of the Moon ^--...-^ ...... 239 
 
 CHAPTER XXII. 
 
 Of the Relative Masses and Densities of the Sun, Moon, and Planets ; 
 
 and of the Relative Intensity of the Gravity at their surface - 249 
 
 CHAPTER XXIII. 
 
 Of the Figure and Rotation of the Earth ; and of the Precession of 
 
 the Equinoxes and Nutation *? . .>-' - J '. ",-'" '' * 251 
 
 CHAPTER XXIV. 
 
 the Tides *' '"' '"--'., , '-; i." - ' ..* .' .- - - 254 
 
 PART IV. 
 ASTRONOMICAL PROBLEMS. 
 
 EXPLANATIONS OF THE TABLES - - f '^ t " '" " " ^^ 
 
 PROB. 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 266 
 
fclV TABLE OF CONTENTS. 
 
 PAGI 
 
 PBOB. 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 - ' ' * ' ' 267 
 
 PROB. III. To convert Degrees, Minutes, and Seconds of the Equa- 
 tor into Hours, Minutes, &c., of Time v ? - - '- 273 
 PROB. IV. To convert Time into Degrees, Minutes, and Seconds ib. 
 
 PROB. V. The Longitudes of two Places, and the Time at one of 
 
 them being given, to find the corresponding time at the other 274 
 
 PROB. VI. The Apparent time being given, to find the correspond- 
 ing Mean Time ; or, the Mean Time being given, to find the 
 Apparent - ,r - - - - ' ^75 
 
 PROB. VII. To correct the Observed Altitude of a Heavenly Body 
 
 for Refraction - 278 
 
 PROB. VIII. The Apparent Altitude of a Heavenly Body being 
 
 given, to find its True Altitude - - - - - - 279 
 
 PROB. IX. To find the Sun's Longitude, Hourly Motion, and Semi- 
 diameter, for a given Time, from the Tables ... 281 
 
 PROB. X. To find the Apparent Obliquity of the Ecliptic, for a 
 
 given Time, from the Tables - >''' .* ; ./< - ' '-*<*. \- ^ 283 
 
 PROB. XL Given the Sun's Longitude and the Obliquity of the 
 
 Ecliptic, to find his right Ascension and Declination - - 284 
 
 PROB. XII. Given the Sun's Right Ascension and the Obliquity of 
 
 the Ecliptic, to find his Longitude and Declination -jgfc r , .^ 285 
 
 PROB. XIII. The Sun's Longitude and the Obliquity of the Ecliptic- 
 being given, to find the Angle of Position - - - , . _ 285 
 
 PROB. 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 ... 286 
 
 PROB. XV. The Moon's Equatorial Parallax, and the Latitude of 
 
 a Place, being given, to find the Reduced Parallax and Latitude 295 
 
 PROB. XVI. To find the Longitude and Altitude of the Nonagesi- 
 
 mal Degree of the Ecliptic, for a given Time and Place - ib. 
 
 PROB. XVII. To find the Apparent Longitude and Latitude, as 
 affected by Parallax, and the Augmented Semi-diameter of the 
 Moon ; the Moon's True Longitude, Latitude, Horizontal Semi- 
 diameter, and Equatorial Parallax, and the Longitude and Alti- 
 tude of the Nonagesimal Degree of the Ecliptic, being given 298 
 
 PBOB. XVIII. To find the Mean Right Ascension and Declination, 
 or Longitude and Latitude of a Star, for a given Time, from the 
 Tables 302 
 
TABLE OP CONTENTS. XV 
 
 PAOB 
 PROB. XIX. To find the Aberrations of a Star in Right Ascension 
 
 and Declination, for a given Day - .... 393 
 
 PROB. XX. To find the Nutations of a Star in Right Ascension 
 
 and Declination, for a given Day - - - - - 304 
 
 PROB. XXI. To find the Apparent Right Ascension and Declina- 
 tion of a Star, for a given day ...... 306 
 
 PROB. XXII. To find the Aberrations of a Star in Longitude and 
 
 Latitude, for a given Day - 307 
 
 PROB. XXIII. To find the Apparent Longitude and Latitude of a 
 
 Star, for a given Day - ib. 
 
 PROB. XXIV. To compute the Longitude and Latitude of a 
 Heavenly Body from its Right Ascension and Declination, the 
 Obliquity of the Ecliptic being given ----- 308 
 
 PROB. XXV. To compute the Right Ascension and Declination 
 of a Heavenly Body from its Longitude and Latitude, the 
 Obliquity of the Ecliptic being given - 309 
 
 PROB. XXVI. The Longitude and Declination of a Body being 
 given, and also the Obliquity of the Ecliptic, to find the Angle 
 ofPosition 310 
 
 PROB. XXVII. To find from the Tables the Time of New or Full 
 
 Moon, for a given Year and Month - - - - - 311 
 
 PROB. XXVIII. To determine the number of Eclipses of the Sun 
 and Moon that may be expected to occur in any given Year, 
 and the Times nearly at which they will take place - - 314 
 
 PROB. XXIX. To calculate an Eclipse of the Moon 317 
 
 PROB. XXX. To calculate an Eclipse of the Sun, for a given Place 321 
 
 PROB. XXXI. To find the Moon's Longitude, &c., from the Nau- 
 tical Almanac --------- 338 
 
 APPENDIX. 
 
 TRIGONOMETRICAL FORMULAE 
 
 I. Relative to a Single arc or angle a ib. 
 
 II. Relative to Two Arcs a and b, of which a is supposed to be 
 
 the greater - # 
 
 III. Trigonometrical Series 
 
 IV. Differences of Trigonometrical Lines - ib. 
 
 V. Resolution of Right-angled Spherical Triangles ib. 
 
 VI. Resolution of Oblique-angled Spherical Triangles - - 345 
 
XVI TABLE OF CONTENTS. 
 
 PAOB 
 
 INVESTIGATION OP ASTRONOMICAL FORMULJE - 348 
 
 Formulae for the Parallax in Right Ascension and Declination, 
 
 and in Longitude and Latitude - ib. 
 
 Formulas for the Aberration in Longitude and Latitude, and in 
 
 Right Ascension and Declination - 355 
 
 Formulae for the Nutation in Right Ascension and Declination 359 
 Formulae for computing the effects of the Oblateness of the 
 
 Earth's Surface, upon the Apparent Zenith Distance and 
 
 Azimuth of a Star -------- 353 
 
 Solution of Kepler's Problem, by which a Body's Place is found 
 
 in an Elliptical Orbit 364 
 
 NOTES. 
 
 L to XXII - 369-382 
 
CALIFORNIA- 
 
 AN 
 
 ELEMENTARY TREATISE 
 
 ON 
 
 ASTBOIOMY. 
 
 INTRODUCTION. 
 
 GENERAL NOTIONS GENERAL PHENOMENA OF THE HEAVENS.^ 
 
 1 . The space, indefinite in extent, which is exterior to the earth, 
 is called the Heaven or Heavens, or the Firmament. The sun, 
 moon, and stars, the luminous bodies which are posited in this 
 space, are called the Heavenly Bodies. The entire assemblage 
 of these bodies is frequently called the Heavens. 
 
 2. The most casual observation shows us that the heavenly 
 bodies are subject to a variety of motions, as well as to various 
 changes of appearance. The science which treats of the laws 
 and causes of these motions and changes, is called Astronomy ;* 
 or, more particularly, Astronomy is a mixed mathematical science, 
 which treats of the motions, positions, distances, appearances, mag- 
 nitudes, and physical constitution of the heavenly bodies. It has 
 been divided into the two departments of Plane or Pure Astronomy, 
 a.nd Physical Astronomy. Plane Astronomy comprehends, 1st, the 
 description of the motions, appearance^ and: structure of the 
 heavenly bodies, and the description and explanation of their phe- 
 nomena, which may be called Descriptive Astronomy ; 2d, the 
 methods of observation and calculation employed in obtaining a 
 knowledge of the facts embodied in Descriptive Astronomy, and 
 the computation from these of the details of occasional phenome- 
 na, as eclipses of the sun and moon, occupations of the- stars,. &c., 
 which is denominated Practical Astronomy <.. Physical Astronomy 
 investigates inductively the physical causes of the observed mo- 
 tions and constitution of the great bodies of the material universe, 
 and deduces, as a mechanical problem, from the one great cause, 
 the principle of universal gravitation, all the minutiae of the celes- 
 tial mechanism. 
 
 * From Affi^p, a star, and w.w, a law. 
 1 
 
2 INTRODUCTION. 
 
 3. The origin of the science of Astronomy is involved in obscurity ; but it it 
 supposed that its first truths were discovered in the early ages of the world by 
 shepherds, who, at the same time, watched their flocks by night, and followed the 
 motions and noted the varying aspects o'f the heavenly bodies. Each successive 
 age, from that time to the present, has, with occasional interruptions, brought to it 
 its contributions of observations and discoveries. The imposing character of the 
 celestial phenomena, and their intimate relations to the every-day wants of life, as 
 well as the superstitions of the ignorant, have, from time immemorial, conspired 
 to attract to this science the interested attention of mankind, and promote its ad- 
 vancement. From the very nature of things, some of its truths have only unfolded 
 themselves, as century after century has passed away ; while others still await the 
 lapse of future ages. Its history, in a theoretical point of view, presents two prom- 
 inent epochs, viz : 1. The epoch of the discovery of the true system of the world, by 
 Copernicus, towards the middle of the sixteenth century ; soon followed by the 
 discovery of the exact laws of its motions in space, by Kepler, (early in the sev- 
 enteenth century ;) which has so completely changed the whole face of the science, 
 and has been succeeded by such a mass of observations of greatly increased accu- 
 racy, and such an uninterrupted series of important discoveries, that it may 
 almost be said to be the date of its origin, as the science is now taught. 2. The 
 epoch of the discovery of universal gravitation, by Sir Isaac Newton, (1683;) a 
 discovery that has brought Astronomy within the province of Mechanical Philoso- 
 phy, and contributed greatly to its advancement and extension, by making known 
 its physical theory, which has been developed by Laplace and others with great 
 minuteness of detail. Contemplating the science from a practical point of view, 
 we find that its most prominent epoch is that of the discovery of the telescope, a 
 the beginning of the seventeenth century, since which time, by the adaptation of 
 the telescope to instruments for admeasurement, and the improvement of these 
 instruments, its means of research have been gradually perfected and extended, 
 as art and science have advanced hand in hand : until from a few shepherds, un- 
 der the open sky on the plains of Chaldea, with naught but their natural powers 
 of vision, there has come to be a large body of professed Astronomers in charge of 
 permanent observatories erected in almost every civilized country on the globe ; 
 and furnished at the same time with telescopes that bring the heavenly bodies hun- 
 dreds or even thousands of times nearer, and disclose a new world of celestial ob- 
 jects, and with instruments that mark out, with the greatest precision, the ever 
 varying places of all these bodies. 
 
 4. To be able to form correct notions of the phenomena 
 of the heavens, it is necessary to know the form of the earth. 
 We learn from the following circumstances that the earth is 
 a body of a globular form, insulated in space. 1st. When a 
 vessel is receding from the land, an observer stationed upon 
 
 Fig. 1. the coast, first loses sight 
 
 of the hull, then of the 
 lower parts of the sails, and 
 lastly, of the topsails. This 
 is the case whatever is the 
 direction of the course of 
 the vessel, and at whatever 
 part of the earth it is ob- 
 served. That this is a proof 
 of the roundness of the sea, 
 will at once be seen on in- 
 specting Fig. 1. It will 
 readily be perceived that no 
 part of the earth could be- 
 come interposed between the 
 
GENERAL NOTIONS. 
 
 hull and the lower parts 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. 2d.* At sea the visible horizon, or the 
 line bounding the visible portion of the earth's surface, is every- 
 where a circle, of a greater or less extent, according to the altitude 
 of the point of observation, and is on all sides equally depressed. 
 To illustrate this proof, let BOA (Fig. 2) represent a portion 
 
 of the earth's surface supposed to be spherical, P the position of 
 the eye of 'the observer, and DPC a horizontal line. If we 
 conceive lines, such 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, &c., which would 
 be the visible horizon, is a circle. It is also manifest that the an- 
 gles of depression CPA, DPB, &c., of the horizon in different 
 directions, would be equal, and that a greater portion of the earth's 
 surface would be seen, and thus that the horizon would increase in 
 extent, in proportion as the altitude of the point of observation, P, 
 increased. 3d. Navigators, as it is well known, have sailed around 
 the earth in different directions. 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 moun- 
 tains bear an exceedingly small proportion to the dimensions of the 
 whole earth. 
 
 5. If an indefinite number of lines, PA, PB, &c., 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 sufficient light to the eye, are visible. That portion 
 
4 INTRODUCTION. 
 
 of the heavens which is above this surface, presents the appear- 
 ance of a solid vault or canopy, resting upon the earth at the visi- 
 ble 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 distances from the 
 earth, and that the distances of all of them are very great in com- 
 parison 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. 
 
 6. Up and downed! any place on the earth's surface, are from 
 and towards the swfece ; and thus at different places have every 
 variety of absolute direction in space. 
 
 This fact should not merely be acknowledged to be true, but should be dwelt 
 upon until the mind has become familiarized to the conception of it, and divested, 
 as far as possible, of the notion of an absolute up and down in space. 
 
 7. The earth is surrounded with a transparent gaseous medium, 
 called the earth's atmosphere, estimated to be some fifty miles in 
 height ; through which all the heavenly bodies are seen. The at- 
 mosphere is not perfectly transparent, but shines throughout with 
 light received from the heavenly bodies, and reflected from its par- 
 ticles ; and thus forms a luminous 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. 
 
 8. The most conspicuous of the celestial phenomena, is a con 
 tinual 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 recog- 
 nised by all persons. If the heavens be attentively watched on 
 any clear evening, it will soon be seen that the stars have a motion 
 precisely similar to that of the sun and moon. To describe the 
 phenomenon in detail, as witnessed at night : if, on a clear night, 
 we observe the heavens, we shall find that the stars, while they 
 retain the same situations with respect to each other, undergo a 
 continual change of position with respect to the earth. Some will 
 be seen to ascend from a quarter called the East, being replaced 
 by others that come into view, or rise ; others, to descend towards 
 the opposite quarter, the West, and to go out of view, or set : and 
 if our observations be continued throughout the night, with the 
 
GENERAL PHENOMENA OF THE HEAVENS. 5 
 
 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, farther to tne north, that never set, and which appear 
 to revolve in circles, from east to west, about a certain star that 
 seems to remain stationary. This seemingly stationary star is 
 called the Pole Star; and those stars that revolve about it, and never 
 set, are called Circumpolar Stars. It should be remarked, how 
 ever, that the pole star, when accurately observed by means of 
 instruments, is found not to be strictly stationary, but to describe 
 a small circle about a point at a little distance from it as a fixed 
 centre. This point is called the North Pole. It is, in reality, 
 about the north pole, as thus defined, and not the pole star, that 
 the apparent revolutions of the stars at the north are performed. 
 At the corresponding hours of the following night the aspect of the 
 heavens will be the same, from which it appears that the stars re- 
 turn to the same position once in about 24 hours. It would seem, 
 then, that the stars all appear to move from east to west, exactly 
 as if attached to the concave surface of a hollow sphere, which 
 rotates in this direction about an axis passing through the station 
 of the observer and the north pole of the heavens, in a space of 
 time nearly equal to 24 hours. For the sake of simplicity this 
 conception is generally adopted. This motion, common to all the 
 heavenly bodies, is called their Diurnal Motion. It is ascertained, 
 by certain accurate methods of observation and computation, here- 
 after to be exhibited, that the diurnal motion of the stars is strictly 
 uniform and circular. 
 
 9. It is important to observe, that the conception of a single sphere 
 to which the stars are supposed to be attached, will not represent 
 their diurnal motion, as seen from every part of the earth's surface^ 
 unless the sphere be supposed to be of such vast dimensions that 
 the earth is comparatively but a mere point at its centre.* 
 
 10. 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 
 
 * The student should strive to familiarize his mind with this notion of the sphere 
 of the heavens. The disposition, so natural to every one, to conceive the stars to 
 be at no very great distance from the earth, in comparison with the dimensions of 
 so large a body, will, until it is overcome, often give rise to very erroneous concep- 
 tions of the different appearances of the same phenomenon, as viewed from different 
 points of the earth's surface 
 
6 INTRODUCTION. 
 
 and south direction ; and a circle cut out by a plane perpendicular 
 to the axis, has an east and west direction. 
 
 11. The greater number of the stars preserve constantly the 
 same relative position with respect to each other ; and they are 
 therefore called Fixed Stars. There are, however, a few stars, 
 called Planets* which are perpetually changing their places in the 
 heavens. The number of the planets is ten. Each has a distinc- 
 tive name, as follows : Mercury, Venus, Mars, Jupiter, Saturn, 
 Uranus, Ceres, Pallas, Juno, and Vesta. Mercury, Venus, Mars, 
 Jupiter, and Saturn are visible to the naked eye, and have been 
 known from the most ancient times. The other five, namely, Ura- 
 nus, Ceres, Pallas, Juno, and Vesta, cannot be seen without the 
 assistance of the telescope, and were discovered by modern ob- 
 servers.! (See Note I, at the end of the Appendix.) 
 
 1 2. The planets are distinguishable from each other either by a dif- 
 ference 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. Venus never recedes 
 beyond 40 or 50 from the sun, while Jupiter is seen at every va- 
 riety of angular distance from him. Mars is known by the ruddy 
 color of his light. Saturn has a pale, dull aspect. 
 
 13. The apparent motion 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. 
 
 14. The Sun and Moon are also continually changing their 
 places among the fixed stars. 
 
 15. From repeated examinations of the situation of the moon 
 among the stars, it is found that she has with respect to them a 
 progressive circular motion from west to east, and completes a re- 
 volution around the earth in about 27 days. 
 
 16. 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 visi- 
 ble in that part of the heavens where the sun sets. It will be found 
 that those 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 farther to the 
 east. The sun, then, is continually approaching the stars that lie 
 
 * Fom .wAov^rijf, a wanderer. 
 
 t The planet Uranus was discovered in 1781 by Dr. Herschel, who gave it the 
 name of Georgium Sidus. By the European astronomers it was called Herschel. 
 It is now generally known by the name given in the text. Ceres, Pallas, Juno, 
 and Vesta have been discovered since 1800 ; the first by Piazzi, the second and 
 fourth by Olbers, and the third by Harding. 
 
GENERAL PHENOMENA OF THE HEAVENS. 
 
 Fig. 3. 
 
 on the eastern side of 
 him. To make this 
 more evident, let us 
 suppose that the small 
 circle aon (Fig. 3) rep- 
 resents a section of 
 the earth perpendic- 
 ular to the axis of ro- 
 tation of the imagi- 
 nary sphere of the 
 heavens, (8,*) con- 
 ceived to pass through 
 the earth's centre ; the 
 large circle H Z S a 
 section of the heavens 
 perpendicular to the 
 same line, and pass- 
 ing through the sun ; 
 and the right line 
 H o r the plane of the horizon at the 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 dis- 
 tributed that the distances rt, tu, uv, &c. are each equal to S r. 
 Then, at the end of two or three weeks, an hour after sunset the 
 star t will be in the horizon ; at the end of another interval of two 
 or three weeks the star u will be in the same situation at the same 
 hour ; at the end of another interval, the star v, &c. It is plain, 
 then, that the sun must at the ends of these successive intervals be 
 in the successive positions in the heavens, r, t, u, &c. ; otherwise, 
 when he is brought by his 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 he has a mo- 
 tion in the heavens in the direction S r t u v, opposite to that of 
 the diurnal motion ; that is, towards the east. 
 
 Another proof of the progressive motion of the sun among the 
 stars from west to east, is found in the fact that the same stars rise 
 and set earlier each successive night, and week, and month during 
 the year. At the end of six months the same stars rise and set 12 
 hours earlier ; which shows that the sun accomplishes half a revo- 
 lution in this interval. At the end of a year, or of 365 days, the 
 stars rise and set again at the same hours, from which it appears 
 that the sun completes an entire revolution in the heavens in this 
 period of time. 
 
 It is to be observed that the sun does not advance directly to- 
 wards the east. He has also some motion from south to north, and 
 
 * Numbers thus enclosed in a parenthesis refer, in general, to a previous article. 
 
 
8 INTRODUCTION. 
 
 north to south. It is a matter of common observation that the snn 
 is moving towards the north from winter to summer, and towards 
 the south from summer to winter. 
 
 17. When the place of the sun in the heavens is accurately 
 found from day to day by certain methods of observation, hereaf- 
 ter to be explained, it appears that his path is an exact circle, in- 
 clined about 23 to a circle running due east and west, (10.) 
 
 18. The motions of the sun, moon, and planets are for the most 
 part confined to a certain zone, of about 18 in breadth, extending 
 around the heavens from west to east, (or nearly so,) which has 
 received the name of the Zodiac. 
 
 19. There is yet another class of bodies, called Comets* (or 
 hairy Stars,) that have a motion among the fixed stars. They ap- 
 pear only occasionally in the heavens, and continue visible only 
 for a few weeks or months. They shine with a diffusive nebu- 
 lous light, and are commonly accompanied by a fainter divergent 
 stream of similar light, called a tail. 
 
 20. The motions of the comets are not restricted to the zodiac. 
 These bodies are seen in all parts of the heavens, and moving in 
 every variety of direction. 
 
 21. By inspecting the planets with telescopes, it has been dis- 
 covered that some of them are constantly attended by a greater or 
 less number of small stars, whose positions are continually vary- 
 ing. These attendant stars are called Satellites. The planets 
 which have satellites are Jupiter, Saturn, and Uranus. The sat- 
 ellites are sometimes called Secondary Planets ; the planets upon 
 which they attend being denominated Primary Planets. 
 
 22. The sun and moon, the planets, (including the earth,) to- 
 gether with their satellites, and the comets, compose the Solar 
 System. 
 
 23. 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, and which is known 
 by the name of the Copernican System. It is as follows : 
 
 The sun occupies a fixed centre, about which the planets (in- 
 cluding the earth) revolve from west to east,t in planes that are but 
 slightly inclined to each other, and in the following order : Mer- 
 cury, Venus, the Earth, Mars, Vesta, Juno, Ceres, Pallas, Jupiter, 
 
 * From Coma, a head of hair. 
 
 t A motion in space from west to east is a motion from right to left, to a person 
 situated within the orbit described, and on the north side of its plane. To obtain 
 a clear conception of the motions of the solar system, the reader must place him- 
 self, in imagination, in some such situation as this, entirely detached from the 
 earth and all the other bodies of the system. It is customary to take the plane of 
 the earth's orbit as the plane of reference in conceiving of the planetary motions. 
 
 
GENERAL PHENOMENA OF THE HEAVENS. 9 
 
 Saturn, and Uranus. The earth rotates from west to east, about 
 an axis inclined to the plane of its orbit under an angle of about 
 66J, 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. 
 (See the Frontispiece, which is a diagram of the solar 'system in 
 projection.) 
 
 24. We shall here, at the outset, adopt this system as an hypo- 
 thesis, and shall rely upon the simple and complete explanations it 
 affords of the celestial phenomena, as they come to be investi- 
 gated, together with the evidence furnished by Physical Astrono- 
 my, to produce entire conviction of its truth in the mind of the 
 student. 
 
 25. The following are the characters or symbols employed by 
 astronomers for denoting the several planets, and the sun and 
 moon : 
 
 The Sun, Ceres, . . . . ? 
 
 Mercury, Pallas, . . . . $ 
 
 Venus, 5 Jupiter, . . . . ^ 
 
 The Earth, .... Saturn, . . . . ^ 
 
 Mars, ...... $ Uranus, . . . W 
 
 Vesta, fi The Moon, . . V 
 
 Juno, $ 
 
 26. The two planets, Mercury and Venus, whose orbits lie with- 
 in the earth's orbit, are called Inferior Planets. The others are 
 called Superior Planets. 
 
 27. The angular distance between any two fixed stars is found 
 to be the same, from whatever point on 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 these bodies, the whole earth is a mere point. Moreover, the 
 angular distance between any two fixed stars is the same at what- 
 ever period of the year it is measured. Whence, if the earth re- 
 volves around the sun, its entire orbit must be insensible in com- 
 parison with the distance of the stars. 
 
 28. On the hypothesis of the earth's rotation, the diurnal motion 
 of the heavens is a mere illusion, occasioned by the rotation of the 
 earth. To explain this, suppose the axis of the earth prolonged on 
 till it intersects the heavens, considered as concentric with the 
 earth. Conceive a great circle to be tracgd through the two points 
 of intersection and the point directly over head, 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 
 
 2 
 

 10 
 
 INTRODUCTION. 
 
 earth being stationary, the whole heavens rotate about the same 
 axis and at the same rate in the opposite direction. Now, as the 
 motion of the earth is perfectly equable, we are insensible of it, 
 and therefore attribute the changes in the situations of the stars 
 with respect to the earth to an actual motion of these bodies. Ii 
 follows, then, that we must com eive the heavens to rotate as above 
 mentioned, 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 (Art. 8) that the sphere of the heavens ap- 
 pears 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, in appearance, pass through 
 the station of the observer, wherever this may be upon the earth's 
 surface. 
 
 29. We in like manner infer that the observed motion of the 
 sun in the heavens is only an apparent motion, occasioned by the 
 Fig. 4. orbitual motion of the earth. 
 
 Let E, E' (Fig. 4) represent 
 two positions of the earth in 
 its orbit EE'E" about the 
 sun S. When the earth is 
 at E, the observer will refer 
 the sun to that part of the 
 heavens marked s; but when 
 the earth is arrived at E', he 
 will refer it to the part mark- 
 ed s' ; and being in the mean 
 time insensible of his own 
 motion, the sun wall appear 
 to him to have described in 
 the heavens the arc s s', just 
 the same as if it had actu- 
 ally passed over the arc SS' 
 in space, and the earth had, during that time, remained quiescent 
 at E. The motion of the sun from 5 towards s f will be from west 
 to east, since the motion of the earth from E towards E' is in this 
 direction. Moreover, as the axis of the earth is inclined to the 
 plane of its orbit under an angle of 66i, (23,) the plane of the 
 sun's apparent path, which is the same as that of the earth's orbit, 
 will be inclined 23| to a circle perpendicular to the earth's axis, 
 r to a circle directed due east and west. 
 
PART I. 
 
 ON THE DETERMINATION OF THE PLACES AND MOTIONS 
 OF THE HEAVENLY BODIES. 
 
 CHAPTER I. 
 
 ON THE CELESTIAL AND TERRESTRIAL SPHERES. 
 
 30. 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, Astron- 
 omers conceive all these bodies, whatever may be their actual 
 distance from the earth, to be referred to a spherical surface of an 
 indefinitely great radius, having the station of the observer, or 
 what comes to the very same thing, the centre of the earth, for its 
 centre. This imaginary spherical surface is called the Sphere of 
 the Heavens, or the Celestial 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 com- 
 pared to this sphere, is to be considered as a mere point at its 
 centre. 
 
 31. Not only is the size of the earth to be neglected in compari- 
 son with the celestial sphere, but also the size of the earth's orbit. 
 Thus the supposed annual motion of the earth around the sun, 
 does not change the point in which a line conceived to pass from 
 any station upon the earth in any fixed direction into space, pierces 
 the sphere of the heavens ; nor the circle in which a plane cuts the 
 same sphere. 
 
 The fixed stars are so remote from the earth that observers, 
 wherever situated upon the earth, and in the different positions of 
 the earth in its orbit, refer them to the same points of the celestial 
 sphere, (27.) The other heavenly bodies are referred by observ- 
 ers at different stations to points somewhat different. 
 
 32. For the purposes of observation and computation, certain 
 imaginary points, lines, and circles, appertaining to the celestial 
 sphere, are employed, which we shall now proceed to explain. 
 
 (1.) The Vertical Line, at any place on the earth's surface, is 
 
12 
 
 
 ON THE CELESTIAL SPHESE, 
 
 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 Ver- 
 tical Plane. Every plane that is perpendicular to the vertical line, 
 is called a Horizontal 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, (5.) 
 (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 ra- 
 tional horizon is imperceptible, or the two circles appear as one 
 and the same at the distance of the earth, (30.) 
 
 (4.) The Zenith of a place is the point in which the vertical 
 prolonged upward pierces the celestial sphere. The point in 
 which the vertical, when produced downward, intersects the ce- 
 lestial sphere, is called the Nadir. 
 
 The zenith and nadir are the geometrical poles of the horizon. 
 (5.) The Axis of the Heavens is an imaginary right line pass- 
 ing through the north pole (8) 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 South 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 
 n O s (Fig. 5) repre- 
 sent the earth, and the 
 outer circle HZRN 
 the sphere of the 
 heavens ; also let O 
 be a point on the 
 earth's surface, and 
 OZ the vertical line 
 at the station O. 
 Then HOR will be 
 the plane of the sen- 
 sible horizon, HCR 
 the plane of the ra- 
 tional horizon, Z the 
 zenith, and N the na- 
 dir ; and if P be the 
 north pole of the hea- 
 vens, OP, or CP its parallel, will be the axis of the heavens, and 
 P 7 the south pole. 
 
DEFINITIONS 
 
 13 
 
 Fig. 6. 
 
 The lines CP and OP intersect the heavens in the same point, 
 P; and the planes HOR, and HCR, in the same circle, passing 
 through the points H and R. 
 
 Unless we are seeking for the exact apparent place in the heav- 
 ens 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 HOR the same as 
 HCR ; as represented in Fig. 6. 
 In this diagram, the circle of the 
 horizon being supposed to be view- 
 ed from a point above its plane, is 
 represented by the ellipse HARez. 
 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 HARa is pro- 
 jected into its diameter HCR. 
 
 Every different place on the 
 surface of the earth has a different 
 zenith, and, except in the case of diametrically opposite places, a 
 different horizon. To illustrate this, let nesq (Fig. 7) represent 
 the earth, and HZRP' the sphere of the heavens ; then, considering 
 the four stations, e, O, ft, and q, the zenith and horizon of the first 
 
 Fig. 7. 
 
 will be respectively E 
 and PeP' ; of the se- 
 cond Z and HOR ; of 
 the third P and QnE ; 
 of the fourth Q and 
 P'qP. The diametri- 
 cally opposite places 
 and O' have the 
 same rational horizon, 
 viz. HCR. The same 
 is true of the places n 
 and s, and e and q. 
 Their rational hori- 
 zons are respectively 
 QCE and PCP'. 
 
 (7.) Vertical Circles 
 are great circles pass- 
 ing through the zenith 
 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 stai 
 S, called the Vertical Circle of the Star. 
 
 (8.) The Meridian of a place is that vertical circle which con- 
 
 
14 ON THE CELESTIAL SPHERE. 
 
 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 R, m 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 East 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 meas- 
 ured 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 HCM 
 represents the azimuth of the star S. 
 
 The Azimuth and Altitude, or azimuth 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- 
 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 aBS A represents the circle described 
 by the star S in its diurnal motion, ea will be its amplitude al 
 rising, and wA. its amplitude at setting. 
 
DEFINITIONS 15 
 
 (14.) The Celestial Equator, or the Equinoctial, is a great cir- 
 cle of the celestial sphere, the plane of which is perpendicular to 
 the axis of the heavens. The north and south poles of the heav- 
 ens are therefore its geometrical poles. The celestial equator is 
 represented in Fig. 6 by EwQe. 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 an- 
 gles. 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 
 sun appears to describe in the course of the year. 
 
 (18.) The Obliquity of the Ecliptic is the angle under which 
 the ecliptic is inclined to the equator. Its amount is 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 Equi- 
 nox. 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 sum- 
 mer 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 ce- 
 lestial 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 Circle ; the other, about 
 the south pole, is called the Antarctic Circle. 
 
 The polar circles contain the geometrical poles of the ecliptic. 
 
 (23.) The Tropics are parallels of declination at a distance from 
 the equator equal to the obliquity of the ecliptic. That which is 
 
16 
 
 ON THE CELESTIAL SPHERE. 
 
 on the north side of the equator is called the Tropic of Cancer, 
 and the other the Tropic of Capricorn. 
 
 The tropics touch the ecliptic at the solstitial points. 
 
 Fig. 8. 
 
 Let C (Fig. 8) represent the centre of the earth and sphere, 
 PCP' the axis of the heavens, EVQA the equator, -W VTA 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 ; 
 P VP'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 TrcZ, Wrc'Z', the 
 tropics. 
 
 It is important to observe that, agreeably to what has been sta- 
 ted, (Art. 30,) the directions of the equator and ecliptic, of the equi- 
 noctial 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 (18) extends about 9 on each side of the 
 ecliptic. 
 
DEFINITIONS 17 
 
 (25.) The ecliptic and zodiac are divided into twelve equal parts, 
 called Signs. Each sign contains 30. The division commences 
 at the vernal equinox. Setting out from this point, and following 
 around from west to east, the Signs of the Zodiac, with the re- 
 spective characters by which they are designated, are as follows : 
 Aries T, Taurus 8, Gemini n, Cancer Sj, Leo SI, Virgo W, Li- 
 bra =*, Scorpio tn, Sagittarius /, Capricornus V3, Aquarius ss, 
 Pisces }. The first six are called northern signs, being north of 
 the equinoctial. The others are called southern signs. 
 
 The vernal equinox corresponds to the first point of Aries, and 
 the autumnal equinox to the first point of Libra. The summer 
 solstice corresponds to the first point of Cancer, and the winter 
 solstice to the first point of Capricornus. 
 
 The mode of reckoning arcs on the ecliptic is by signs, degrees, 
 minutes, &c. 
 
 A motion in the heavens in the order of the signs, or from west 
 to east, is called a direct motion, and a motion contrary to the or- 
 der of the signs, or from east to west, is called a retrograde mo- 
 tion. 
 
 (26.) The Right Ascension of a heavenly body is the arc of the 
 equator intercepted between the vernal equinox and the declination 
 circle which passes through the centre of the body, as reckoned 
 from the vernal equinox towards the east. It measures the incli- 
 nation of the declination circle of the body to the equinoctial col ure. 
 Thus, PSR being the declination circle of the star S, and V the 
 place of the vernal equinox, VR is the right ascension of the star. 
 It is the measure of the angle VPS. If PS'R' be the declination 
 circle of another star S', SPS', or RR', will be their difference of 
 right ascension. 
 
 (27.) The 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, RS is the declination of the star S.. 
 
 Declination is North or South, according as- the body is north or 
 south of the equator. 
 
 In the use of formulae, a south declination is regarded as nega- 
 tive. 
 
 The right ascension and declination of a heavenly body are two 
 co-ordinates, which, taken together, fix its position in the sphere 
 of the heavens : for they make known its situation with respect to 
 two circles, the equinoctial colure, and the equator. Thus, VR 
 and RS ascertain the position of the star S with respect to the cir- 
 cles PVP'A, and VQAE. 
 
 (28.) The Polar Distance of a heavenly body is the arc of a de- 
 clination circle, intercepted between the centre of the body and the 
 elevated pole. The polar distance is the complement of the decli- 
 nation, and, therefore, when either is known the other may be 
 found. 
 
 3 
 
18 ON THE TERRESTRIAL SPHERE. 
 
 (29.) Circles of Latitude are great circles of the celestial sphere, 
 which pass through the poles of the ecliptic, and therefore cut this 
 circle at right angles. Thus, KSL represents a part of the circle 
 of latitude of the star S. 
 
 (30.) The Longitude of a heavenly body is the arc of the eclip- 
 tic, intercepted between the vernal equinox and the circle of lati- 
 tude 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 eclip- 
 tic. It therefore expresses the distance of the body from the eclip- 
 tic. 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 formulae, a south latitude is affected with the mi- 
 nus sign. 
 
 The longitude and latitude of a heavenly body are another set 
 of co-ordinates, which serve to fix its position in the heavens. They 
 ascertain its situation with respect to the circle of latitude passing 
 through the vernal equinox and the ecliptic. Thus, VL and LS 
 fix the position of the star S, making known its situation with re- 
 spect to the circles KVK'A and VTAW. 
 
 (32.) The Angle of Position of a star, is the angle included at 
 the star between the circles of latitude and declination passing 
 through it. PSK is the angle of position of the star S. 
 
 (33.) The Astronomical Latitude, or the Latitude, of a place, is 
 the arc of the meridian intercepted between the zenith and the ce- 
 lestial equator. It is North or South, according as the zenith is 
 north or south of the equator. ZE (Fig. 7) represents the latitude 
 of the station O ; QOE or QCE being the equator. 
 
 33. The earth's surface, considered as spherical, (which ac- 
 curate admeasurement, upon principles that will be explained in 
 the sequel, proves it to be, very nearly,) is called the Terrestrial 
 Sphere. The following geometrical constructions appertain to the 
 terrestrial sphere, as it is employed for the purposes of astronomy. 
 It will be observed that they correspond to those of the celestial 
 sphere above described, and are used for similar objects. 
 
 (1.) The North and South Poles of the Earth are the two points 
 in which the axis of the heavens intersects the terrestrial sphere. 
 They are also the extremities of the earth's axis of rotation. 
 
 (2.) The Terrestrial Equator is the great circle in which a 
 plane passing through the centre of the earth, and perpendicular to 
 the axis of the heavens and earth, cuts the terrestrial sphere. The 
 terrestrial and the celestial equator, then, lie in the same plane. 
 
DEFINITIONS 
 
 19 
 
 The poles of the earth are the geometrical poles of the terrestrial 
 equator. The two hemispheres into which the terrestrial equator 
 divides the earth, are called, respectively, the Northern Hemi- 
 sphere and the Southern Hemisphere. 
 
 (3.) Terrestrial Meridians are great circles of the terrestrial 
 sphere, passing through the north and south poles of the earth, and 
 cutting the equator at right angles. Every plane that passes through 
 the axis of the heavens, cuts the celestial sphere in a celestial me- 
 ridian^ and the terrestrial sphere in a terrestrial meridian. 
 
 Let PP' (Fig. 9) represent the axis of the heavens, O the centre 
 of the earth, and p andp' its poles. Then, elq will represent the 
 
 Fig. 9. 
 
 terrestrial equator, (ELQ representing the celestial equator;) and 
 pep' andpsp' terrestrial meridians, (PEP' and PSP' representing 
 celestial meridians.) 
 
 (4.) The Reduced Latitude of a place on the earth's surface is 
 the arc of the 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 angle oOe, is the reduced latitude 
 of the place o. Latitude is North 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 Reduction of Latitude. 
 
 (5.) Parallels of Latitude are small circles of the terrestrial 
 
20 ON THE TERRESTRIAL SPHERE. 
 
 sphere parallel to the equator. Every point of a parallel of latitude 
 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 defs. 22, 23, p. 15.) 
 
 (6.) The Longitude of a place on the earth's surface, is the in- 
 clination of its meridian to that of some particular station, fixed 
 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 pqp' be supposed to re- 
 present the first meridian, the angle spq, 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 Royal Observatory at Paris. In the 
 United States the longitude is, for astronomical purposes, reckoned 
 from the meridian of Greenwich or Paris, (generally the former.) 
 
 The longitude and latitude of a place designate its situation on 
 the earth's surface. They are precisely analogous to the right as- 
 cension and decimation of a star in the heavens. 
 
 34. The diagram (see Fig. 6) which we made use of in Art. 32 
 in illustrating our description of the circles of the celestial sphere, 
 represents the aspect of this sphere at a place at which the north 
 
 pole of the heavens is some- 
 where between the zenith arid 
 horizon. Such is the position 
 of the north pole at all places 
 situated between the equator 
 and the north pole of the 
 earth. For, let O (Fig. 10) 
 represent a place on the earth's 
 surface, HOR the horizon, 
 OZ the vertical, HZR the 
 meridian, and ZE the latitude. 
 QOE will then represent the 
 equinoctial, and P, P', 90 dis- 
 tant from E and on the meri- 
 dian, the poles. Now, we have 
 HP = ZH ZP = 90 ZP ; ZE = PE ZP - 90 ZP. 
 
 Whence HP = ZE. 
 
 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 heav- 
 ens will gradually increase from to 90. 
 
 By inspecting Fig. 7, it will be seen that this increase of the al- 
 
ASPECTS OF THE CELESTIAL SPHERE. 
 
 21 
 
 titude of the pole in going north, is owing to the fact that in fol- 
 lowing the curved surface of the earth the horizon, which is con- 
 tinually tangent to the earth, is being 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 spec- 
 tator it will obviously have the same altitude as the north pole in 
 the northern hemisphere. 
 
 35. Let us now inquire minutely into the principal circumstan- 
 ces of the diurnal motion of the stars, as it is seen by a spectator 
 situated somewhere between the equator and the north pole. And 
 in the first place, it is a simple corollary from the proposition just 
 established, that the parallel of declination to the north, whose 
 polar distance is equal to the latitude of the place, will lie entirely 
 above the horizon, and just touch it at the north point. This cir- 
 cle is called the circle of perpetu- Fig. 11. 
 al apparition ; the line #H (Fig. 11) 
 represents its projection on the me- 
 ridian plane. The stars compre- 
 hended between it and the north 
 pole will never set. As the de- 
 pression of the south pole is equal 
 to the altitude of the north pole, H[ 
 the parallel of declination o R, at 
 ,1 distance from the south pole 
 equal to the latitude of the place, 
 will lie entirely below the horizon, 
 and just touch it at the south point. 
 The parallel thus situated is call- 
 ed the circle of perpetual occultation. The stars comprehended 
 between it and the south pole will never rise. 
 
 The celestial equator (which passes through the east and west 
 points) will intersect the meridian at a point E, whose zenith dis- 
 tance ZE is equal to the latitude of the place (Def. 33, Art. 32,) and 
 consequently, whose altitude RE is equal to the co-latitude of the 
 place. Therefore, in the situation of the observer above supposed, 
 the equator QOE, passing to the south of the zenith, will, togeth- 
 er with the diurnal circles nr, st, &c., 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', &c., 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 excep- 
 tion of the equator, will be divided unequally by the horizon. The 
 greater parts of the circles nr, nY, &c., to the north of the equa- 
 tor, will be above the horizon ; and the greater parts of the circles 
 
22 ON THE CELESTIAL SPHERE. 
 
 st y s't', &c., 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 con- 
 trary, a longer time below the horizon than above it. It is also 
 obvious, from the manner in which the horizon cuts the different 
 diurnal circles, that the disparity between the intervals of time that 
 a star remains above and below the horizon, will be the greater the 
 more distant it is from the equator. Again, the stars will all cul- 
 minate, or attain to their greatest altitude, in the meridian : for, 
 since the meridian crosses the diurnal circles at right angles, they 
 will have the least zenith distance when in this circle. Moreover, 
 as the meridian bisects the portions of the diurnal circles which lie 
 above the horizon, the stars will all employ the same length of 
 time in passing from the eastern horizon to the meridian, as in 
 passing from the meridian to the western horizon. The circum- 
 polar stars will pass the meridian twice in 24 hours ; once above, 
 and once below the pole. These meridian passages are called, 
 respectively, Upper and Lower Culminations, or Inferior and Su- 
 perior Transits. 
 
 It will be observed, that in travelling towards the north the cir- 
 cles of perpetual apparition and occultation, together with those 
 portions of the heavens about the poles which are constantly visible 
 and invisible, are continually on the increase. 
 
 It is evident, from what is stated in Art. 34, 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. 
 
 Fig. 12. 36. When the spectator is sit- 
 
 uated on the equator, both the 
 celestial poles will be in his hori- 
 zon, (34,) and therefore the celes- 
 tial 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 
 horizor . 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. 
 
ASTRONOMICAL INSTRUMENTS. 
 
 23 
 
 37. If the observer be at either 
 oi the poles, the elevated pole of 
 the heavens will be in his zenith, 
 (34,) and consequently, the celes- 
 tial 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 continu- 
 ally visible, while the other hem- 
 isphere will be continually invis- 
 ible. This is called a Parallel 
 Sphere. It is represented 
 Fig. 13. 
 
 m 
 
 Fig. 13. 
 
 CHAPTER II. 
 
 ON THE CONSTRUCTION AND USE OF THE PRINCIPAL ASTRONOMICAL 
 
 INSTRUMENTS. 
 
 38. ASTRONOMICAL INSTRUMENTS are, for the most part, used for 
 the admeasurement of arcs of the celestial sphere, or of angles cor- 
 responding to such arcs at the earth's surface. They consist, es- 
 sentially, of a refracting telescope turning upon a horizontal axis, 
 and of a vertical graduated limb, (or, in some cases, of both a ver- 
 tical and a horizontal graduated limb,) to indicate the angle passed 
 over by the telescope. At the common focus of the object-glass 
 and eye-glass of the telescope is a diaphragm, or circular plate, at- 
 tached to which are two very fine wires, or spider-lines, crossing 
 each other at right angles in its centre. The place of this dia- 
 phragm may be altered by adjusting screws ; it is by this means 
 brought into such a position that the cross of the wires will lie on 
 the axis of the telescope, (that is, the line joining the centres of the 
 object-glass and eye-glass.) The line joining the centre of the ob- 
 ject-glass and the cross of the wires, is technically termed the Line 
 of Collimation. Bringing the cross of the wires upon the axis of 
 the telescope, is called Adjusting the Line of Collimation. A star 
 is known to be on the line of Collimation when it is bisected by the 
 cross-wires. 
 
 The telescope either turns around the centre of the graduated 
 limb, or, which is more common, the limb and telescope are firmly 
 attached to each other, and turn together. In the first arrange- 
 ment a small steel plate, firmly connected with the telescope, slides 
 along the limb, upon this plate a small mark is drawn, which is 
 called the Index. The required angle is read off by noting the 
 
24 
 
 ASTRONOMICAL INSTRUMENTS. 
 
 angle upon the limb which is pointed out by the index ; the zero 
 on the limb being generally, in practice, the point from which the 
 angle is reckoi.ed. When the telescope and graduated limb are 
 firmly connected, the limb slides past the index, which is now sta- 
 tionary. The limbs of even the largest instruments are not divided 
 into smaller parts than about 5', but, by means of certain subsidi- 
 ary contrivances, the angle may, with some instruments, be read 
 off lo within a fraction of a second. 
 
 39. The principal contrivances for increasing the accuracy of 
 the reading off of angles, are the Vernier, the Micrometer Screw, and 
 the Micrometer Microscope or Reading Microscope. The Vernier 
 is only the index plate, so graduated that a certain number of its 
 divisions occupy the same space as a number one less on the limb. 
 Fig. 14 represents a vernier and a portion of the limb of the instru- 
 ment, 15 divisions on the vernier corresponding to 14 on the limb. 
 If we suppose the smallest divisions of the limb to be 15', and call 
 x the number of minutes in one division of the vernier, then, 
 
 15 x == 14 X 15', and x = 14'. 
 
 Thus, the difference between a division on the vernier and one 
 on the limb, will be 1'. Accordingly, if the index, which is the 
 first. mark on the vernier, should be little past the mark 40 on the 
 limb, and the second mark of the vernier should coincide with the 
 next point of division, marked 15', the angle would be 40 1'. If 
 the third mark on the vernier were coincident with the next division 
 of the limb, marked 30', the angle would be 40 2'. If the fourth 
 with the next division to this, 40 3' ; and so on. 
 
 By making the divisions on the vernier more numerous, the an- 
 Fig. 14. gle can be read off with greater 
 
 precision ; but a better expedi- 
 ent is provided in the Microme- 
 ter Screw. This piece of me- 
 chanism is represented in Fig. 
 14. The part E can be fast- 
 ened to the limb of the instru- 
 ment by means of a screw. FG 
 is a screw, with a milled head at 
 F, working in a collar fixed in 
 the under part of E , and in a nut 
 fixed in the under part of the tel- 
 escope T t. When the part E 
 is fixed or clamped, and the 
 screw is turned around by its 
 milled head at F, it must com- 
 municate a direct motion to the 
 nut, and, consequently, to the 
 
 telescope and vernier in the direction of FG. Attached to the screw, 
 or to the small cylinder on which it is formed, is an index D, move- 
 able together with the screw, and on a thin graduated immoveable 
 
READING MICROSCOPE TIME, ETC. 25 
 
 plate, the profile only of which is shown in the figure. Suppose 
 now that the screw is of such fineness that while, together with the 
 index D, it makes a complete revolution, the vernier moves through 
 an arc of 1'. Then, if the plate be divided into 60 equal parts, a 
 motion of the index over one of these parts would answer to a mo- 
 tion of I" on the limb. This being understood, to show the use of 
 the micrometer screw, suppose that no two marks on the vernier 
 and limb are coincident : bring the two nearest into coincidence by 
 turning the screw, and the number of divisions passed over by the 
 index D will be the seconds to be added to or subtracted from the 
 angle read off with the vernier. In observing the coincidence of 
 the divisions of the limb and vernier, the eye is assisted by a mi- 
 croscope.* 
 
 40. The Reading Microscope is a compound microscope firmly 
 fixed opposite to the limb, and furnished with cross-wires in the focus 
 of the eye-glass, or conjugate focus of the object-glass, moveable by 
 a fine-threaded micrometer screw, that is, a screw (such as was de- 
 scribed in the previous article) provided with an immoveable grad- 
 uated circular plate, and an index turning with the screw, and glid- 
 ing over the plate, to measure the exact distance through which the 
 head of the screw is moved. The observer looks through the 
 microscope at the limb. The centre of the microscope corresponds 
 to the index of a fixed vernier plate. By turning the screw the 
 intersection of the wires is moved over the space which separates 
 it from the nearest line of division on the limb, in the direction of 
 the zero, and the number of turns and parts of a turn of the screw 
 being noted by means of the graduated plate, the number of mi- 
 nutes and seconds in this space becomes known. The minutes 
 and seconds thus found being added to the angle read off from the 
 limb, the result will be the angle sought. 
 
 41. 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 astronomi- 
 cal observatories, are relied upon as furnishing angles to within 1" 
 of the truth. 
 
 42. Time is an essential element in astronomical observation. 
 Three different kinds of time are employed by astronomers : Si- 
 dereal, Apparent or True Solar,' and Mean Solar Time. 
 
 43. Sidereal Time is time as measured by the diurnal motion 
 of the stars, or, more properly, of the vernal equinox. A Sidereal 
 Day is the interval between two successive meridian transits of a 
 star, or, (as it is now most generally considered,) the interval be- 
 tween 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. 
 
 44. Apparent, or True Solar Time, is deduced from observa 
 
 * Weodhouse's Astronomy, vl. i. p. 55. 
 
 4 
 
26 ASTRONOMICAL INSTRUMENTS. 
 
 tions upon the sun. An Apparent Solar Day is the interval be- 
 tween two successive meridian passages of the sun's centre ; com- 
 mencing when the sun is on the superior meridian. It appears 
 from observation that it is a little longer than a sidereal day, and 
 that its length is variable during the year. It is divided into 24 
 Apparent Solar Ifours. 
 
 45. Mean Solar Time is measured by the diurnal motion of an 
 imaginary sun, called the Mean Sun, conceived to move uniformly 
 from west to east in the equator, with the real sun's mean motion 
 in the ecliptic, and to have at all times a right ascension equal to 
 the sun's mean longitude. A Mean Solar Day commences when 
 the mean sun is on the superior meridian, 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. 
 
 46. The Astronomical Day commences at noon, and is divided 
 into 24 hours ; but the Calendar Day commences at midnight, and 
 is divided into two portions of 12 hours each. 
 
 47. Astronomical observations are, for the most part, made in 
 the plane of the meridian. But some of minor importance are 
 made out of this plane. The chief instruments employed for me- 
 ridian observations, are the Astronomical Circle, and the Transit 
 Instrument, used in connection with the Astronomical Clock. 
 These are the capital instruments of an observatory, inasmuch as 
 they serve (as will soon be explained) for the determination of the 
 places of the heavenly bodies, which are the fundamental data of 
 astronomical science. The principal instruments used for making 
 observations out of the meridian plane, are the Altitude and Azi- 
 muth Instrument, the Equatorial, and the Sextant. 
 
 TRANSIT INSTRUMENT. 
 
 48. The Transit Instrument is a meridional instrument, employ- 
 ed in conjunction with a clock or chronometer for observing the 
 passage of celestial objects across the meridian, either for the pur- 
 pose of determining their difference of right ascension, or obtaining 
 the correct time. It is constructed of various dimensions, from a 
 focal length of 20 inches to one of 10 feet. The larger and more 
 perfect instruments are permanently fixed in the meridian plane ; 
 the smaller ones are mounted upon portable stands. Fig. 15 rep- 
 resents a fixed transit instrument. AD is a telescope, fixed, as it 
 is represented in the figure, to a horizontal axis formed of two 
 cones. The two small ends of these cones are ground into two 
 perfectly equal cylinders ; which cylindrical ends are called Pivots. 
 These pivots rest on two angular bearings, in form like the upper 
 part of a Y, and denominated Y's. The Y's are placed in two 
 dove-tailed brass grooves fastened in two stone pillars E and W, so 
 erected as to be perfectly steady. One of the grooves is horizontal, 
 the other vertical, so that, by means of screws, one end of the axis 
 
TRANSIT INSTRUMENT. 27 
 
 may be pushed a little forward or backward, and the other end 
 may be either slightly depressed or elevated : which two small 
 movements are necessary, as it will be soon explained, for two ad 
 justments of the telescope. 
 
 Fig. 15. 
 
 Let E be called the eastern pillar, W the western. On the 
 eastern end of the axis is fixed (so that it revolves with the axis) 
 an index n, the upper part of which, when the telescope revolves, 
 nearly slides along the graduated face of a circle, attached, as it is 
 shown in the figure, to the eastern pillar. The use of this part of 
 the apparatus is to adjust the telescope to the altitude or zenith 
 distance of a star the transit of which is to be observed. Thus, 
 suppose the index n to be at o, in the upper part of the circle, 
 when the telescope is horizontal : then, by elevating the telescope, 
 the index is moved downward. Suppose the position to be that 
 represented in the figure, then the number of degrees between o 
 and the index is the altitude. 
 
 The wire plate placed in the focus of the transit telescope, has 
 attached to it five vertical wires together with one horizontal wire. 
 In order to be seen at night, these wires, or rather the field of view, 
 requires to be illuminated by artificial light. The illumination of 
 the field is effected by making one of the cones hollow, and ad- 
 mitting the light of a lamp placed in the pillar opposite the orifice ; 
 which light is directed to the wires by a reflector placed diagonally 
 in the telescope. The reflector, having a large hole in its centre, 
 
28 ASTRONOMICAL INSTRUMENTS. 
 
 does not interfere with the rays passing down the telescope from 
 the object* 
 
 The wires are seen as dark lines upon a bright ground. In 
 some of the best instruments recently constructed there is a neat 
 contrivance for illuminating the wires directly, so as to make them 
 appear bright upon a dark ground, which is intended to be used in 
 making observations upon faint stars. 
 
 Sometimes the transit instrument is furnished with a meridian graduated circle 
 of large size, designed to be used for the measurement of meridian altitudes or 
 zenith distances. It then takes the name of Meridian Circle or Transit Circle ; 
 and serves for the determination of both the right ascension and declination of a 
 heavenly body. The meridian circle of the observatory recently established at 
 Pulkova, near St. Petersburg, has two meridian limbs, provided each with four 
 reading microscopes. 
 
 49. We will now explain the principal adjustments of the tran- 
 sit. Upon setting the instrument up it should be so placed that 
 the telescope, when turned down to the horizon, should point north 
 and south, as near as can possibly be ascertained. This being 
 done, then 
 
 (1.) To adjust the line of collimation. 
 
 This adjustment consists in bringing the central vertical wire, 
 within the telescope, to intersect the optical axis, which is sup- 
 posed to be fixed by the maker of the instrument perpendicularly 
 to the axis of rotation. There is no occasion with this instrument 
 to have the horizontal wire intersect the . optical axis with exact- 
 ness. Direct the telescope to some small, distant, well-defined 
 object, (the more distant the better,) and bisect it with the middle 
 of the central vertical wire ; then lift the telescope out of its 
 angular bearings, or Y's, and replace it with the axis reversed. 
 Point the telescope again to the same object, and if it be still bi- 
 sected, the collimation adjustment is correct; if not, move the 
 wires one half the angle of deviation, by turning the small screws 
 that hold the wire plate, near the eye-end of the telescope, and the 
 adjustment will be accomplished : but, as half the deviation may 
 not be correctly estimated in moving the wires, it becomes neces- 
 sary to verify the adjustment by moving the telescope the other 
 half, which is done by turning the screw that gives the small azi- 
 muth motion to the Y before spoken of, and consequently to the 
 pivot of the axis which it carries. Having thus again bisected the 
 object, reverse the axis as before, and if half the error was cor- 
 rectly estimated, the object will be bisected upon the telescope 
 being directed to it. If it should not be bisected, the operation of ' 
 reversing and correcting half the error must be gone through again, 
 and until after successive approximations the object is found to be 
 bisected in both positions of the axis ; the adjustment will then be 
 perfect.* 
 
 * Woodhouse's Astronomy, vol. i. pp. 70-72 ; also Simm's Treatise on Mathe* 
 matical Instruments, p. 53. 
 
TRANSIT INSTRUMENT. 29 
 
 It is desirable that the central wire should be truly vertical, as 
 we should then have the power of observing the transit of a star 
 on any part of it, as well as the centre. It may be ascertained 
 whether it is so, by elevating and depressing the telescope, when 
 directed to a distant object : if the object is bisected by every part 
 of the wire, the wire is vertical, (or rather it is perpendicular to 
 the axis of rotation of the telescope, and becomes vertical so soon 
 as the axis of rotation is made horizontal.) If it is not bisected, 
 the wire should be adjusted, by turning the inner tube carrying 
 the wire plate until the above test of its vertically be obtained. 
 
 50. (2.) To set the axis of rotation of the telescope horizon- 
 tal. This adjustment is effected by means of a spirit-level ; either 
 attached to two upright arms bent at their upper extremities, by 
 which it is hung on the two pivots of the axis, or else having two 
 legs and standing upon the axis. In the first position it is called 
 a hanging level, and in the second a riding level. At one end of 
 the level is a vertical adjusting screw, by which that end may be 
 elevated or depressed. Put the level in its place, and observe to 
 which end of the level the bubble runs, which will always be the 
 more elevated end ; bring it back to the middle by the Y screw for 
 vertical motion, and take off the level and hang it on again with 
 the ends reversed. Then, if the bubble is again found in the mid- 
 dle, the level is already parallel to the axis, and the axis horizon- 
 tal ; but if not, adjust one half the error by the adjusting screw of 
 the level, and the other half by the Y screw ; and let the operation 
 of reversing, and adjusting by halves, be repeated until the bubble 
 will remain stationary in either position of the level, in which case 
 both the level and axis will be horizontal. 
 
 51 . (3.) To adjust the line of collimation to the plane of the me- 
 ridian. We have said, that upon setting the instrument up, the 
 telescope is to be brought into the meridian plane, as near as can 
 be ascertained. One mode of establishing it, is to direct the tel- 
 escope to the pole star, and by repeated observations find the 
 position corresponding to its greatest or least altitude. At the 
 present time, we may instead compute by means of existing tables 
 founded on observation, the time of the meridian transit of the 
 pole star, and at that computed time bisect the star by the middle 
 vertical wire. Afterwards the line of collimation may be placed 
 still more exactly in the plane of the meridian in the following 
 manner : Note the times of two successive superior transits of the 
 pole star across the central vertical wire, and the time of the inter- 
 vening inferior transit. If the line of collimation were exactly in 
 the plane of the meridian, as the diurnal circles are bisected by 
 this plane, the interval between the superior and next inferior tran- 
 sit would be precisely equal to the interval between the inferior 
 and next superior transit. Accordingly, if these intervals are not 
 in fact equal, find by repeated trials the position of the telescope 
 
30 ASTRONOMICAL INSTRUMENTS. 
 
 and vertical wire for which they are equal, and the line of collima 
 tion will then be in the plane of the meridian. 
 
 Instead of establishing this equality by a system of trial and 
 error, we may, by means of a formula which has been investiga- 
 ted for the purpose, compute from an observed inequality the 
 amount of the movement in azimuth necessary to correct the error 
 of position of the instrument. 
 
 Another, and generally a more convenient method, is to observe one of the tran- 
 sits of the pole star, and also the transit of some star that crosses the meridian near 
 the zenith, and which follows or precedes the pole star by a known interval, (differ- 
 e.nce of right ascensions of the two stars,) and compare the observed interval with 
 the calculated interval. The difference of the two may be made to disappear by 
 repeated trials : or a formula may easily be investigated, which shall make known 
 the angular movement of the instrument necessary to make the observed and cal- 
 culated intervals precisely equal. 
 
 The method of regulating the clock required in making this ad- 
 justment, will be explained when we come to treat of the astro- 
 nomical clock. 
 
 52. When the transit telescope has once been placed accurately 
 in the meridian plane, in order to avoid the repetition of trouble- 
 some verifications of its position, a meridian mark should be set 
 up, and permanently established, at a distance from the instru- 
 ment ; its place being determined by means of the middle or me- 
 ridional wire. At Greenwich two such marks, one to the north 
 and another to the south, are used ; they are vertical stripes of 
 white paint upon a black ground, on buildings about two miles dis- 
 tant from the observatory. The position of the telescope is verified 
 by sighting at the meridian mark, when it is once established. 
 
 53. The times of the transits of the heavenly bodies are ascer- 
 tained as follows : in the case of a star, the moments of its cross- 
 ing each of the five vertical wires are noted ; as the wires are 
 equally distant from each other, the mean of these times (or their 
 sum divided by 5) will be the time of the star's crossing the mid- 
 dle wire, or of its meridian transit. The utility of having five 
 wires, instead of the central 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 wire 
 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 times of the passage of both the west- 
 ern and eastern limb across each of the five wires are noted, and 
 the mean of the whole taken, which will be the instant of the me- 
 ridian transit of the centre of the body. 
 
 The time of the meridian transit of a body may, in this manner, 
 be ascertained within a few tenths of a second. 
 
 54. When a star is on the meridian, its declination circle (Def 
 
 * Simm's Mathematical Instruments, p. 59. 
 
 '* * 
 
ASTRONOMICAL CLOCK. 31 
 
 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, (see def. 26, p. 17.) 
 It follows, therefore, that in the interval between the transits of any 
 two stars, the arc of the equator which expresses their difference 
 of right ascension will pass across the meridian, the rate of the 
 motion being that of 1 5 to a sidereal hour : hence the difference 
 of the times of transit of two stars, as observed with a sidereal 
 clock, when converted into degrees by allowing 15 a to the hour, 
 will be the difference between the right ascensions of the two stars. 
 We may, then, in this manner, by means of a transit instrument 
 and sidereal clock, find the differences between the right ascension 
 of any one star and the right ascensions of all the others. This 
 being done, as soon as the position of the vernal equinox with re- 
 spect to the same star becomes known, (and we shall show how 
 to find it,) the absolute right ascensions of all the stars will 
 also become known. Thus RR', (Fig. 8,) is the difference of 
 right ascension of the stars S and S', their absolute right as- 
 censions being VR and VR', and VR is the distance of the vernal 
 equinox V from the declination circle of the star S ; and it will at 
 once be seen that if RR' be found, in the manner just explained 
 so soon as VR becomes known, by adding it to RR' we shall have 
 VR' the right ascension of the star S'. In the actually existing 
 state of astronomical science, the right ascensions of all the stars 
 are more or less accurately known, and a right ascension sought 
 is now obtained directly, by noting the time of the transit of the 
 body with a sidereal clock regulated so as to indicate Oh. Om. Os. 
 when the vernal equinox is on the meridian, and converting it into 
 degrees. 
 
 ASTRONOMICAL CLOCK. 
 
 55. The astronomical clock is very similar to the common clock. 
 It has a compensation pendulum ; that is, a pendulum so construct- 
 ed that its length is unaffected by changes of temperature. The 
 hours on the face are marked from 1 to 24. 
 
 56. Astronomers make use of sidereal time (as already stated) 
 in determining the right ascensions of the heavenly bodies, but for 
 all other purposes they generally use mean solar time. 
 
 57. To regulate a sidereal clock. When a clock is used for de 
 termining differences of right ascension, (54,) it is adjusted to side- 
 real time if it goes equally and marks out 24 hours in a sidereal 
 day ; it being altogether immaterial at what time it indicates Oh. 
 Om. Os. To ascertain the daily rate of going of a clock which is 
 to be adjusted to sidereal time for the purpose just mentioned, note 
 by the clock the times of two successive meridian transits of the 
 same star : the difference between the interval of the transits and 
 24 hours will be the daily gain or loss (as the case may be) of the 
 
32 ASTRONOMICAL INSTRUMENTS. 
 
 clock with respect to a perfectly accurate sidereal clock.* If the 
 gain or loss, when found after this manner, proves to be the same 
 each day, then the mean rate of going is the same each day. 
 
 Next, to be able to discover the rate from hour to hour during the day, it is ne. 
 cessary to have obtained beforehand, at various times, and under various states of 
 the circumstances likely to influence the rate of going of the clock, the differences 
 between the times of the transits of a number of different stars, (correcting propor- 
 tionally for the daily rate,) and to take the mean of the several differences found for 
 each pair of stars for the exact difference of their transits. When this lias been 
 done, the rate of the clock may be found at all hours during the day by noting by 
 the clock the differences between the times of the transits of these stars, and com- 
 paring these with the exact differences already found. At the present time, the 
 right ascensions of the stars being known, to ascertain the rate from hour to hour, 
 we have only to compare the intervals of time given by the clock between the 
 transits of different stars taken in the order of their right ascension with their 
 differences of right ascension. 
 
 58. The sidereal clocks now in use are made to indicate Oh, 
 Om. Os. when the vernal equinox is on the superior meridian. For 
 the regulation of such clocks, it is necessary to know not only their 
 rate but also their error. This is found by noting the time of the 
 transit of a star, and comparing this with its right ascension ex- 
 pressed in time. If the two are equal the clock is right, otherwise 
 their difference will be its error. 
 
 If the error of the rate of a clock be considerable, it should be 
 diminished by altering the length of the pendulum ; otherwise, it 
 may be allowed for. The stars best adapted to the regulation of 
 clocks are those in the vicinity of the equator ; for, as their motion 
 is more rapid than that of the stars more distant from the equator, 
 there is less liability to error in noting their transits. 
 
 59. A mean solar clock is usually regulated by observations up- 
 on the sun. The method of regulating it cannot be adequately 
 explained until we have treated of the apparent motion of the sun. 
 It will here suffice to state, that with the instruments we have now 
 described, the sun's motion can be ascertained ; and therefore, as 
 a knowledge of this is all that is necessary in order that we may 
 be able to obtain the mean solar time at any instant, that it is pos- 
 sible to express all intervals of time in mean solar time. 
 
 ASTRONOMICAL CIRCLE. 
 
 60. An Astronomical Circle is an instrument designed for the 
 measurement of the zenith distances or altitudes of the heavenly 
 bodies at the instants of their arrival on the meridian. Its essen- 
 tial parts are a graduated circular limb, a telescope turning upon a 
 horizontal axis which passes through the centre of the limb, and a 
 micrometer microscope, (40,) or other piece of apparatus, for read- 
 ing off the angles upon the limb. It is sometimes mounted 
 upon an upright stem, which either turns upon fixed supports or 
 
 * It is not necessary, in order to obtain the daily rate of a sidereal clock, that 
 the transit instrument should be adjusted to the plane of the meridian. It is only 
 requisite that it should be kept fixed in some one vertical plane. 
 
MURAL CIRCLE. 
 
 33 
 
 rests upon a tripod, and can be turned around in azimuth ; but, in 
 general, the larger circles in the best furnished observatories have 
 their axis let into a massive pier, or wall, of stone, and capable 
 of only such small motions in the horizontal and vertical directions, 
 under the action of screws, as may be necessary for its adjustment 
 to the horizontal position and perpendicular to the meridian plane. 
 These are called Mural Circles. For greater accuracy the angle 
 is read off at six different points of the limb by means of six sta- 
 tionary micrometer microscopes, and the mean of the different 
 readings taken for the angle required. Fig. 16 is a side view of a 
 mural circle. The graduation is on the outer rim, which is per- 
 pendicular to the plane of the wall CDFE. One of the reading 
 
 Fig. 16. 
 
 microscopes is represented at A. The others, which are omitted 
 in the figure, are disposed at equal distances around the rim. The 
 position of the telescope may be changed by unclamping it and 
 clamping it to a different part of the limb. In taking an angle, the 
 telescope is made fast to the limb, and the limb glides past the sta- 
 tionary microscopes. 
 
 The six reading microscopes, together with the power of chang- 
 ing the position of the telescope on the limb, so as to read off the 
 angle from all parts of the limb, when the mean results of a great 
 number of observations are taken, do away with, or at least very 
 considerably lessen the errors of graduation, centring, and une- 
 qual expansion. 
 
 o 
 
34 ASTRONOMICAL INSTRUMENTS. 
 
 61. In place of mural circles, Mural Quadrants have been much used. Since 
 the mural quadrant has its graduated limb only one fourth the size of the limb of 
 the mural circle, it can be made larger than the circle. But the circle is better 
 balanced than the quadrant, and the quadrant does not possess the advantages 
 which have been enumerated as resulting in the case of the circle from the use of 
 a number of reading microscopes, and from the power to change the position of 
 the telescope on the limb. Besides, two mural quadrants, one to observe the stars 
 north of the zenith, and another to observe the stars south of the zenith, are neces, 
 sary to effect the general object, accomplished by one mural circle, of ascertaining 
 the zenith distance or altitude of any heavenly body at the time of its arrival on 
 the meridian. 
 
 62. The largest astronomical circles that have yet been con- 
 structed, are to be found, it is said, in the Dublin and Cambridge 
 Observatories. That in the Dublin Observatory is 8 feet in diame- 
 ter, and has an azimuth motion, (that is, a motion about a vertical 
 axis.) The other is a mural circle. The mural circle in the Na- 
 tional Observatory at Washington has a diameter of a little more 
 than 5 feet. 
 
 The large mural quadrants of the Greenwich Observatory are of 8 feet radius. 
 
 63. There is another modification of the astronomical circle, called the Zenith 
 Sector, which is used to measure the meridian zenith distances of stars that cross 
 the meridian within a few degrees of the zenith. The limb extends only about* 
 10 on each side of the lowest point. It can, accordingly, be made larger than the 
 limb of the circle or quadrant. The zenith sector in the observatory at Greenwich 
 has a radius of 12 feet. 
 
 64. The mural circle, like the transit instrument, requires three 
 adjustments : 1 . Its axis must be made horizontal ; 2. Its line of 
 collimation(38) must be made perpendicular to the horizontal axis; 
 3. The line of collimation must be made to move in the plane of 
 the meridian. 
 
 A simple mechanical contrivance exists for carrying the first of 
 the adjustments into complete effect. When the axis is made ho- 
 rizontal, the line of collimation describes a vertical circle ; but it 
 may describe a small circle of the celestial sphere. To make it ne- 
 cessarily describe a great circle, and a meridional circle, there are 
 no mechanical means. Astronomical ones must be resorted to ; 
 and ejen with those, the two latter adjustments are not accom- 
 plished without great difficulty, We may, on this occasion, use 
 the transit. When a star is on the meridional wire of the 
 transit instrument, so move the mural circle that the star may 
 be on its middle wire. Next, observe by the transit instrument 
 when a star, on, or very near to the zenith, crosses the meridian : 
 if, at that time, the star is on the middle vertical wire of the tele- 
 scope of the mural circle, then its line of collimation is rightly 
 adjusted. If the star is on the middle wires of the two telescopes 
 at different times, note their difference and adjust accordingly.* 
 
 65. The horizontal point of the limb, technically so called, is 
 the place of the index (or centre of the microscope) answering to 
 
 * This adjustment must be conducted by some formula which expresses the re- 
 lation between the difference of the times, and the inclination of the line of collima- 
 tion to the plane of the meridian, (Woodhouse's Astronomy, p. 117.) 
 
MURAL CIRCLE. 3$ 
 
 a horizontal position of the line of collimation of the telescope. 
 Perhaps the simplest method of obtaining this point is the follow- 
 ing : Direct the telescope upon some star at the moment of its 
 culmination, and read off the angle on the limb. Procure an arti- 
 ficial horizon, (see art. 79,) and on the following night direct the 
 telescope upon the image of the same -star, as seen in the artificial 
 horizon. By the laws of reflexion, the angle of depression of this 
 image will be equal to the angle of elevation of the star. Accord- 
 ingly the arc on the limb which passes before the reading micro- 
 scope, in moving the telescope from the star to its image, will be 
 double the altitude of the star, and its point of bisection the hori- 
 zontal point.* This point may also be found by directing the 
 telescope upon a star whose altitude is known. 
 
 66. In the case of the mural quadrant, if there is no altitude that can be relied 
 on as having been obtained with all attainable accuracy, it is necessary to have 
 recourse to the zenith sector. This instrument is so constructed and arranged, 
 that its horizontal axis can be reversed in position. By taking the zenith distance 
 of a star with its face towards the east, and then of the same star with the face to- 
 wards the west, the half sum of the two will be its true zenith distance. With this 
 we may readily find the vertical point, and thence the horizontal point, on the limb 
 of the mural quadrant, by directing the telescope upon the star observed with the 
 sector, when it is on the meridian. 
 
 67. The adjustments of the mural circle having all been effect- 
 ed, and the horizontal point determined, if the instrument be set to 
 this point, and the telescope afterwards directed upon any star in 
 the meridian, the arc of the limb that passes by the reading mi- 
 croscope will be the altitude of the star. In making the observa- 
 tion the telescope must be brought into such a position that the 
 star will be bisected by the horizontal wire, as it passes through 
 the field of view. The altitude of the sun, moon, or any planet, 
 may be ascertained by measuring the altitudes of the upper and 
 lower limbs, and taking their half sum for the altitude of the cen- 
 tre : or, if the apparent semidiameter be known, by adding this to 
 the altitude of the lower limb, or subtracting it from the altitude of 
 the upper limb. 
 
 68. The meridian altitude or zenith distance of a heavenly body 
 having been measured with an astronomical circle, or other similar 
 instrument, at a place the latitude of which is known, its declina- 
 tion may easily be found. For, let s, (Fig. 10,) represent the point 
 of meridian passage of a star, or other heavenly body, which crosses 
 the meridian to the north of the zenith (Z.) Es will be its decima- 
 tion, (Def. 27, p. 17,) Zs its meridian zenith distance, and ZE the 
 latitude of the place of observation (O,) (Def. 33, p. 18 :) and we 
 obviously have 
 
 Es = ZE + Zs . . . (a). 
 
 * The method of using the level for the determination of the horizontal point 
 may be found explained in Herschel's Astronomy, p. 93. Another piece of appa- 
 ratus, used for the same purpose, called the Floating Collimator, is described in the 
 same work, p. 95. 
 
36 ASTRONOMICAL INSTRUMENTS 
 
 If the star cross the meridian at some point s' between the ze- 
 nith (Z) and the equator (E,) we shall have Es' = ZE Zs', (b) , 
 and if its point cf transit be some point s" to the south of the equa- 
 tor (E,) we shall have Es" = Zs" ZE, and Es" = ZE 
 Zs", (c). The three formulae (a), (6), 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 latitude is 
 always positive ; 2. The zenith distance is positive when it is 
 North, that is, when the star is north of the zenith, and negative 
 when it is South ; 3. The declination is North if it comes out posi- 
 tive, and South if it comes out negative. 
 
 If the latitude is South, it must be regarded as negative, and the zenith distance 
 must be affected with the minus sign when it is South, and with the plus sign when 
 it is North. The rule for the declination is the same. In general, North latitude is 
 -f-, South latitude . The zenith distance has the same sign as the latitude when 
 it is of the same name, the contrary sign when it is of a contrary name. North 
 declination is -}-, South declination . 
 
 The latitude which is here supposed to be known, maybe found 
 by measuring (67) the meridian altitudes of a circumpolar star at 
 its inferior and superior transits, 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, (34.) 
 
 69. When the right ascension and declination of a heavenly body 
 have been obtained from observation, with a transit instrument and 
 circle, (54, 68,) its longitude and latitude may be computed. For, 
 let S (Fig. 8) represent the place of the body, VRQE the equa- 
 tor, VLTW the ecliptic, and P, K, the north poles of the equator 
 and ecliptic. In the spherical triangle PKS we shall know PS 
 the complement of SR the declination, and the angle KPS = 
 ER = E V + VR = 90 + right ascension ; and if we suppose the 
 obliquity of the ecliptic to be known, we shall know PK. We 
 may therefore compute KS, and the angle PKS. But KS is the 
 complement of SL, which is the latitude of the body S ; and 
 PKS = 180 EKS = 180 (WV + VL) = 180 (90 + 
 longitude) = 90 longitude. 
 
 The obliquity of the ecliptic, which we have here supposed to 
 be known, is, in practice, easily found ; for it is equal to TQ, the 
 sun's greatest declination. 
 
 ALTITUDE AND AZIMUTH INSTRUMENT. 
 
 70. The Altitude and Azimuth Instrument consists, essentially, 
 of a telescope with two graduated limbs, the one horizontal and the 
 other vertical. The telescope turns about the centre of the verti- 
 cal limb, or turns with the limb about its centre ; and the ver- 
 tical limb turns, with the telescope, about the vertical axis of the 
 horizontal limb. 
 
EQUATORIAL. 
 
 37 
 
 If the telescope be brought into the meridian plane, and after- 
 wards directed upon a star out of this plane, the arc of the hori- 
 zontal limb passed over by the index will be the azimuth of the 
 star. The vertical limb will serve to measure its altitude, 
 
 71. The Meridian Line (Def. 8, p 14) at a place may easily be 
 determined with the altitude and azimuth instrument, by a method 
 
 Fig. 17. 
 
 called the Method of Equal Alti- 
 tudes. Let O (Fig. 17) represent 
 the place of observation, NPZ the 
 meridian, and S, S' two positions 
 of the same star, at which the alti- 
 tude is the same. Now, the spher- 
 ical triangles ZPS and ZPS' have 
 the side ZP common, ZS=ZS 7 , 
 and (allowing the stars to move in 
 circles) PS=PS'. Hence 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 me- 
 ridian 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. 
 
 72. When the time is accurately known, the north and south line maybe found 
 very easily by directing the telescope of any instrument that has a motion in azi- 
 muth upon a star in the vicinity of the pole and at a distance from the zenith, at 
 the moment of its arrival on the meridian, (which, as will be understood in the se- 
 quel, can now easily be determined from existing data.) 
 
 EQUATORIAL. 
 
 73. The Equatorial is similar, in its construction, to the altitude 
 and azimuth instrument. It is so called from the circumstance of 
 one of the limbs being placed in a position parallel to the plane of 
 the equator. The axis of this limb is then parallel to the axis of 
 the heavens ; and the other limb, to the centre of which the tele- 
 scope is attached, is parallel in every one of its positions to the 
 plane of some one celestial meridian. The limb which is parallel 
 to the equator serves for the measurement of differences of right 
 ascension, and the other for the measurement of declinations. The 
 equatorial is regarded as one of the most indispensable instruments 
 of an astronomical observatory. It is particularly useful in the 
 measurement of apparent diameters, and in all observations that 
 require the telescope to be directed upon a body for a considerable 
 period of time ; as, by giving the limb to which the telescope is 
 attached a slow motion from east to west, the body may be follow 
 
ASTRONOMICAL INSTRUMENTS, 
 
 ed in its diurnal motion, and kept continually within the field ot 
 view. This motion is generally produced by clock-work, without 
 the use of the hand. 
 
 It is also frequently used for determining the right ascension and declination of a 
 comet, or other heavenly body, which for some reason cannot, at the time, be ob- 
 served in the meridian ; and for finding and obtaining a protracted view, or fixing 
 more accurately the place of an object invisible to the naked eye, whose place has 
 been approximately calculated from the results of previous observations. Another 
 important object to which it may be applied, is the determination of small differ- 
 ences of right ascension and declination, and thus of the relative positions of con- 
 tiguous objects. Its determinations of declinations and differences of right ascen- 
 sion, in general, are to be deemed less accurate than those effected with the mural 
 circle and transit instrument ; as, from its more complicated structure, and peculiar 
 position, the parts have less stability and are more subject to unequal strains, 
 bendings, and expansions, than those of the instruments just named. 
 
 74. The adjustments of the equatorial are somewhat complica- 
 ted and difficult. They are best performed by following the pole- 
 star round its entire diurnal circle, and by observing, at proper 
 intervals, other considerable stars whose places are well ascertained. 
 (Herschel.) 
 
 Fig. 18. 
 
 
 75. In addition to the instruments that have now been de- 
 scribed, which are designed and used for the measurement 
 of the angular distances of bodies from some fixed point or 
 circle in the heavens, astronomers have found it convenient 
 and important to have another instrument, or piece of appa- 
 ratus, with which to determine directly the relative situation 
 of two stars that are near to each other ; so near as to be 
 seen, at the same time, in the same field of view. The ap- 
 paratus used for this purpose is attached to the telescope of 
 the equatorial, or other instrument, and is called a Micrometer. 
 Another important use to which it is put, is the measurement 
 of the apparent diameters of the heavenly bodies. It has a 
 variety of forms. The simplest is known by the name of the 
 Wire-Micrometer. It is placed in the focus of the telescope. 
 It consists of two forks of brass, bb'b, cc'c, (Fig. 18) sliding 
 one within the other, and having each a very fine wire, or 
 spider-line, e, and d, stretched perpendicularly across from 
 one prong to the other. These forks are placed length- 
 wise in a shallow rectangular box, aa'aa', about 2 inches 
 wide and 4 inches long ; and have each fine-threaded mi- 
 crometer screws, /, /, working against the ends, b', and c'. 
 The graduated heads of these screws are not represented in 
 the figure, but they may be seen in Fig. 19. They pass 
 p. .Q through the ends a', a', of the box, 
 
 and have their graduated heads on 
 the outside of it. Between the ends 
 b', c' of the two forks and the con- 
 tiguous ends a', a' of the box are two 
 spiral springs, h, h, which keep the 
 ends of the forks firmly pressed 
 against the ends of the screws, and 
 draw the forks outward and the 
 wires further apart whenever the 
 screws are loosened. By turning 
 the screws in the opposite direction 
 the forks are pushed forward, and 
 the wires brought nearer to each 
 other. The number of complete turns and parts of a turn made by each screw, as 
 
 a'nri 
 
SEXTANT. 39 
 
 shown by its graduated head, will make known the fraction of an inch through 
 which the end of it and the contiguous wire is moved. The screws can be so del- 
 icately cut that they will measure with accuracy the ym^ of an inch. The linear 
 space thus measured in the focus of the telescope must be converted into the equiv- 
 alent angular space in the heavens. This is effected by fixing upon two contigu- 
 ous stars, whose distance is accurately known, and measuring with the micrometer 
 the linear distance of their images formed in the focus. In this way will be found 
 how many seconds of angular space correspond to a given movement of either of 
 the wires, as measured by the micrometer scale. The micrometer box is fastened 
 perpendicularly across the eye-end of the tube of the telescope. The eye-piece 
 of the telescope screws into the outer face of the box, (see Fig. 19,) and on looking 
 into it, the wires d, e within the box are seen in its focus ; where also the images 
 of the stars, formed by the object-glass, fall. To save the necessity of counting 
 the revolutions of the micrometer screws, a linear scale is placed within the box, 
 and at one side of it, consisting of a series of teeth, with intervening notches. This 
 is represented in the diagram, (Fig. 18.) A motion of the wire from one notch to 
 another answers, say, to one turn of the screw, and to 1' in space. 
 
 To measure the angular distance of two stars, the wires are both brought into 
 coincidence at the zero of this scale, when we will suppose that they fall between 
 the stars. By turning the screws they are moved from this position, and the mo- 
 tion is continued until the one star is accurately bisected by one wire, and the other 
 star by the other wire. The number of notches which the wires have passed will 
 express the number of minutes in the space between the stars ; to these are to be 
 added the seconds answering to the fractional parts of a revolution, as shown by 
 the divided heads of the screws. It will be seen, that in order to obtain the real 
 distance between the two stars, the two wires d and e must be brought into such 
 a position as to be perpendicular to the line of the stars. This is effected by giving 
 to the whole box a revolving motion about the optical axis of the telescope, and 
 bringing the wire /, which is perpendicular to d and e, into such a position as to 
 bisect both the stars. The diameter of a heavenly body is measured in a similar 
 manner ; the wires being brought into contact with the opposite limbs. 
 
 76. To measure the angle made by the line of direction of two stars with a fixed 
 line passing through one of them, it is necessary that the micrometer box should 
 not only have a revolving motion around the axis of the telescope, but also a grad- 
 uated circle to measure its amount. The cross-wire I is brought by this motion 
 into coincidence first with one line and then with the other, and then the an- 
 gle read off. In this way may be found the angle made by the line of direction of 
 two contiguous stars with the meridian, or a line perpendicular to the meridian, 
 at the moment one of them is crossing this circle. This angle is called the Angle 
 of Position of the two stars, and the micrometer that serves to measure it is called 
 a Position Micrometer. The position of the wire I when perpendicular to the meri- 
 dian may be found by turning it until one of the stars runs along the wire, while 
 the telescope of the equatorial is stationary. Fig. 19 represents a position microm- 
 eter. The micrometer box b, with its attached eye-piece e, is connected with the 
 circle a, and is turned around with it by the small milled-head screw s, which 
 works on an interior toothed wheel, and the angle is read off upon the stationary 
 graduated circle above a, by aid of the vernier, moveable with the plate a 
 
 SEXTANT. 
 
 77. The instruments which have now been described are ob- 
 servatory instruments, the chief design of whose construction is tc 
 furnish the places of the heavenly bodies with all attainable exact- 
 ness. That of which we are now to treat is much less exact, 
 though still of great utility in determining the essential data of 
 some of the practical applications of astronomical science ; as 
 finding the latitude and longitude of a place, and the time of day: 
 and is used chiefly by navigators, and astronomical observers on 
 land, who are precluded, by their situation or other circumstances, 
 
40 
 
 ASTRONOMICAL INSTRUMENTS. 
 
 Fig. 20. 
 
 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 adjusted at every new place of observation. Besides, 
 as it is held in the hand, it can be used at sea, where, by reason 
 of the agitations of the vessel, no instrument supported in the or- 
 dinary way is of any service. 
 
 78. The sextant may be defined, in general terms, to be an in- 
 strument which serves for the direct admeasurement of the angu- 
 lar distance between any two visible points. The particular quan- 
 tities that may be measured with it, are, 1st, the altitude of .1 
 heavenly body ; 2d, the angular distance between any two visible 
 objects in the heavens, or on the earth. Its essential parts are a 
 graduated limb BC, (Fig. 20,) 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 moveable in 
 c( .mection with an index G about A the centre of the limb, and 
 
 the other (D) called the Ho- 
 rizon Glass, is permanently 
 fixed parallel to the radius 
 AC drawn to the zero point 
 of the limb, and is only half- 
 silvered, (the upper half be- 
 ing transparent ;) and an im- 
 moveable telescope at E, 
 directed towards the horizon- 
 glass. The principle of the 
 construction and use of the 
 sextant may be understood 
 from what follows : A ray of 
 light SA from a celestial ob- 
 ject 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 part of that glass a direct ray from another point 
 or object S'. Now, if AS' be drawn, directed to the object S', 
 SAS', the angular distance between the two objects S and S', is 
 equal to double the angle CAG measured upon the limb of the 
 instrument, (AC being parallel to the horizon-glass.) For, when 
 the index-glass is parallel to the horizon-glass, and the angle on the 
 limb is zero, AD, the course of the first reflected ray, will make 
 equal angles with the two glasses, and therefore the angle SAD 
 will become the angle S'AD, (= ADE ;) and the observer, look 
 ing through the telescope, will see the same object S' both b} 
 direct and reflected light. Now, if the index-glass be moved from 
 this position through any angle CAG, the angle made by the re- 
 flected ray which follows the direction AD with this glass, will be 
 
SEXTANT. 41 
 
 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 reflexion now pursues the 
 same course ADE, and reaches the eye at E, as it is always equal 
 to that made by the reflected ray, will be diminished by this amount. 
 Consequently, the incident ray in question will, on the whole, that 
 is, by the diminution of its inclination to the mirror by the angle 
 CAG and by the motion of the mirror through the same angle, be 
 displaced towards the right, or upward, 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 reflex- 
 ion from the two mirrors, is equal to twice the angle CAG that 
 the index-glass is moved from the position (AC) of parallelism to 
 the horizon-glass. 
 
 Hence the limb is divided into 1 20 equal parts, which are called 
 degrees ; and to obtain the angular distance between two points, 
 it is only necessary to sight directly at one of them, and then 
 move the index until the reflected image of the other is brought 
 into contact with it ; the angle read off on the limb will be the an- 
 gle sought. 
 
 To obtain the angular distance between two bodies which 
 have a sensible diameter, bring the nearest limbs into contact, and 
 to the angle read off on the limb add the sum of the apparent semi- 
 diameters of the two bodies, or bring the farthest limbs into con- 
 tact, and subtract this sum. 
 
 79. The sextant is also employed to take the altitude of a heav- 
 enly 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 shallow vessel of mercury forms a very good artificial horizon. 
 
 In obtaining tne 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. 
 
 80. 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 meas- 
 uring any angle greater than about 90, while the sextant will 
 measure an angle as great as 120, or even 140, (for the gradua- 
 tion generally extends to 140.) The quadrant is also inferior to 
 ihe sextant in respect to materials and workmanship, and its meas- 
 urements are less accurate. 
 
 6 
 
42 ASTRONOMICAL INSTRUMENTS. 
 
 ERRORS OF INSTRUMENTAL ADMEASUREMENT. 
 
 81. 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 adjust- 
 ment of the instrument may be detected and allowed for. But 
 errors of observation are obviously undiscoverable. Since, how- 
 ever, the chances are that an error committed at one observation 
 will be compensated by an opposite error at another, it is to be 
 expected that a more accurate result will be obtained if a great 
 number of observations, under varied circumstances, be made, 
 instead of one, and the mean of the whole taken for the element 
 sought. And accordingly, it is the uniform practice of astronomi- 
 cal observers to multiply observations as much as is practicable. 
 
 TELESCOPE. 
 
 82. An observatory is not completely furnished unless it is supplied with a large 
 telescope for examining the various classes of objects in the heavens, and one or 
 more smaller ones for exploring the heavens and searching for particular objects 
 invisible to the naked eye, as faint comets, and making observations upon occa- 
 sional celestial phenomena, as eclipses of the sun and moon, occultations of the 
 stars, &c. Telescopes are divided into the two classes of Reflecting and Refract- 
 ing Telescopes. In the former class the image of the object is formed by a con- 
 cave speculum, and in the latter by a converging achromatic lens. This image is 
 viewed and magnified by an eye-glass, or rather by an achromatic eye-piece 
 consisting of two glasses. In the simplest form of the reflecting telescope, the 
 Herschelian, the image formed by the concave speculum is thrown a little to one 
 side, and near the open mouth of the tube, where the observer views it through the 
 eye-glass, with his back turned towards the object. 
 
 83. The magnifying 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 faint stars, nebulae, &c., by intercepting and con- 
 verging to a point a much larger beam of rays. The magnifying power is meas- 
 ured by the ratio of the focal length of the object-glass, or speculum, to that of the 
 eye-piece. The illuminating and space-penetrating power, (for faint objects,) if 
 we leave out of view the amount of light lost by reflexion and absorption, is meas- 
 ured by the proportion which the aperture of the object-glass or speculum bears to 
 the pupil of the eye. Telescopes are provided with several eye-glasses of various 
 powers. The power to be used varies with the object to be viewed, and the purity 
 and degree of tranquillity of the atmosphere. Of two telescopes of the same focal 
 length, that which has the largest aperture will form the brightest image in the 
 focus, and therefore, other things being equal, admit of the use of the most power- 
 ful eye-piece. In this way it happens that the available magnifying power indi- 
 rectly depends materially upon the size of the aperture. In all telescopes there- is 
 a certain fixed ratio between the aperture and focal length, or at least limit to this 
 ratio. In reflecting telescopes it is about one inch of aperture for every foot of focal 
 length, and in refracting, 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 available magnifying 
 power that has yet been obtained is about 6,000 ; but this was applicable only to 
 the faintest stars and nebulous spots. With the best telescopes a magnifying power 
 of a few hundred is the highest that can be applied to the moon and planets. The 
 largest reflecting telescope that has yet been constructed, and directed to the heav- 
 ens, is the celebrated one of Sir William Herschel, of 40 feet focus, and 4 feet 
 aperture. Its illuminating power was about 35,000, which makes its space-pene 
 
CORRECTIONS OF CO-ORDINATES. 43 
 
 trating power nearly 190 times the distance of the faintest star visible to the naked 
 eye ; and its highest magnifying power was 6,450.* The most powerful refractor 
 yet constructed is in the new observatory at Pulkova, near St. Petersburg. It has 
 an aperture of very nearly 15 inches, (14.93 inches,) and a focal length of 22 feet. 
 The best telescope in the United States is the refractor in the new observatory at 
 Cincinnati.! Its aperture is 12 inches, and focal length about 17 feet. The 
 field of view of telescopes diminishes in proportion as the magnifying power in- 
 creases. It 's stated that with a magnifying power of between 100 and 200 it is 
 a circle not as large as the full moon ; and with a power of 600 or 1000 is nearly 
 filled by one of the planets, while a star will pass across it in from two to three 
 seconds. 
 
 84. The diminution of the field of view, and the trepidations of the image oc- 
 casioned 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 tele- 
 scopes. This limit, it is probable, is already nearly attained, for the highest pow- 
 ers of the best telescopes can now be used only in the most favorable states of the 
 weather4 
 
 85. The largo refracting telescopes are equatorially mounted, that they may, as 
 readily as possible, be directed and retained upon an object. 
 
 86. The small telescope, called a comet-seeker, is a refractor of large aperture 
 and wide field. Its power does not exceed 100. 
 
 CHAPTER III. 
 
 ON THE CORRECTIONS OF THE CO-ORDINATES OF THE OBSERVED 
 PLACE OF A HEAVENLY BODY. 
 
 '*" " *' 
 
 87. ANGLES measured at the earth's surface with astronomical 
 instruments, answer to the Apparent Place of a heavenly body, 
 and are termed Apparent elements. In astronomical language, 
 the True Place of a heavenly body is its real place in the heavens, 
 as it would be seen from the centre of the earth. Angles which 
 relate to the true place are denominated True elements. The 
 apparent co-ordinates of a star are reduced to the true, by the ap- 
 plication of certain corrections, called Refraction, Parallax, and 
 Aberration. 
 
 88. Refraction and aberration are corrections for errors' com- 
 
 * A reflecting telescope, inferior to Herschel's in size, the diameter of the spe- 
 culum being 3 feet, and the focal length 26 feet, but pronounced by Dr. Robinson 
 superior to it in defining power, has, within a few years, been constructed by the 
 Earl of Rosse, of Ireland. The same nobleman has just completed the construc- 
 tion of a reflecting telescope of unparalleled dimensions, from the use of which im- 
 portant discoveries may be anticipated. The diameter of the speculum is 6 feet, 
 and it has a focal length of 53 feet. 
 
 | See Note II. 
 
 i The illuminating and space-penetrating power of telescopes may, however, yet 
 be greatly increased, and a greater distinctness and definiteness in the outline of 
 objects may be obtained. Much may perhaps be gained also by setting up an ob- 
 servatory on the top or sides of some lofty mountain above the greater impurities 
 and disturbances of the lower regions of the atmosphere, and under a tropical sky 
 
44 CORRECTIONS OF CO-ORDIXATES. 
 
 mitted in the estimation of a star's place, while parallax serves to 
 transfer the co-ordinates from the earth's surface to its centre. The 
 object of the reduction of observations from the surface to the cen- 
 tre of the earth, is to render observations made at different places 
 on the earth's surface directly comparable with each other. Ob- 
 servers occupying different stations upon the earth refer the same 
 body (unless it be a fixed star) to different points of the celestial 
 sphere. Their observations cannot, therefore, be compared to- 
 gether, unless they be reduced to the same point, and the centre 
 of the earth is the most convenient point of reference that can be 
 chosen. 
 
 89. The co-ordinate planes or circles to which the place of a 
 star is referred, (p. 17,) are not strictly stationary, but, on the con- 
 trary, have a continual slow motion with respect to the stars. 
 Hence, the true co-ordinates of a star's place which have been 
 found for any one epoch, will not answer, without correction, for 
 any other epoch. The reduction from one epoch to another is 
 effected by applying two corrections, called Precession and Nu- 
 tation. 
 
 REFRACTION. 
 
 90. We learn from the principles of Pneumatics, as well as by 
 experiments with the barometer, that the atmosphere gradually 
 decreases in density from the earth's surface upward. 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 surface. From the known pressure and density 
 of the atmosphere at the surface of the earth, it is computed, that 
 by the laws of the equilibrium of fluids, if its density were through- 
 out the same as immediately in contact with the earth, its altitude 
 would be about 5 miles. Certain facts, hereafter to be mentioned, 
 show that its actual altitude is not far from 50 miles. Now, it is 
 an established principle of Optics, that light in passing from a va- 
 cuum into a transparent medium, or from a rarer into a denser 
 medium, is bent, or refracted, towards the perpendicular to the 
 surface at the point of incidence. It follows, therefore, that the 
 light which comes from a star, in passing into the earth's atmo- 
 sphere, or in passing from one stratum of atmosphere into another, 
 is refracted towards the radius drawn from the centre of the earth 
 to the point of incidence. 
 
 91. Let MmnN, NraoO, Oo^Q, (Fig. 21,) represent successive 
 strata of the atmosphere. Any ray Sp will then, instead of pur- 
 suing a straight course Spx, follow the broken line pabc ; ' being 
 bent downward at the points p, a, b, c,,&c., where it enters the 
 different strata. But, since the number of strata is infinite, and 
 the density increases by infinitely small degrees, the deflections 
 apx, bay, &c., as well as the lengths of the lines pa, ab, &c., are 
 
KEFRACTION. 
 
 45 
 
 Fig. 21. 
 
 infinitely small ; and 
 therefore pabc, the 
 path of the ray, is a 
 broken line of an in- 
 finite number of parts, 
 or a curved line con- 
 cave towards the 
 earth's surface, as it 
 is represented in Fig. 
 22. Moreover, it lies 
 in the vertical plane 
 containing the origi- 
 nal direction of the 
 ray ; for, this plane is 
 perpendicular to all 
 the strata of the atmosphere, and therefore the ray will continue 
 in it in passing from one to the other. 
 
 92. The line OS' (Fig. 22) drawn tangent to paO, the curvilinear 
 path of the light, at its lowest point, will represent the direction in 
 which the light enters the eye, and therefore the apparent line of 
 
 Fig. 22. 
 
 direction of the star. If, then, OS be the true direction of the star, 
 the angle SOS' will be the displacement of the star produced by 
 Atmospherical Refraction. This angle is called the Astronomical 
 Refraction, or simply the Refraction. 
 
 Sir ce paO is concave towards the earth, OS' will lie above 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 
 of the observer upon the earth's surface.) Thus, to obtain the 
 true altitude from the apparent, we must subtract the refraction ; 
 .and to obtain the true zenith distance from the apparent, we must 
 add the refraction. As refraction takes effect wholly in a vertical 
 plane, (91,) it does not alter the azimuth of a star. 
 
46 
 
 CORRECTIONS OF CO-ORDINATES. 
 
 Fig. 23. 
 
 s' 
 
 93. The amount of the refraction varies with the apparent ze- 
 nith distance. In the zenith it is zero, since the light passes per- 
 pendicularly 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 en- 
 ter its different strata, and therefore, according to a well-known 
 principle of optics, the greater is the refraction. 
 
 94. To find the amount of the refraction for a given zenith dis 
 tance or altitude. Let us first show a method of resolving this 
 problem by the general theory of refraction. According to this 
 theory, the amount of the refraction, except so far as the convexity 
 of the strata of the atmosphere may have an effect, depends whol- 
 ly upon the absolute density of the air immediately in contact with 
 tne 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 sup 
 pose, then, that the whole 
 atmosphere is brought to the 
 same density as that portion 
 of it which is in contact with 
 the earth, and let bah (Fig. 
 23) represent its surface ; 
 also let O represent the sta- 
 tion of the observer upon the 
 earth's surface, and Sa a ray 
 incident upon the atmosphere 
 at a. Denote the angle of 
 refraction OC by p, and the 
 refraction Oax by r. The 
 angle of incidence 
 Z'aS = Z'aS' + S'aS = OaC 
 
 + Oax =p + r. 
 Now if we represent the in- 
 dex of refraction of the atmosphere by m, we have, by the laws of 
 refraction, 
 
 sin Z'aS = m sin OaC, or sin ( p + r) = m sin p ; 
 developing (App. For. 15,) 
 
 sin p cos r + cos p 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, 1+cotp.r" sinl"=m, or r" _.J" ' x =Atang p; 
 
 sin 
 
REFRACTION. 
 
 47 
 
 putting A = 
 
 Let ZCa = C ; and ZOa = Z. OaC = 
 
 sm 1" 
 
 ZOa ZCa, orp = 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 very small with respect 
 to Z. If we neglect it, we have 
 
 r = A tang Z (3) ; 
 
 which is the expression for the refraction, answering to the suppo- 
 sition that the surface of the earth is a plane, and that the light is 
 transmitted through a stratum of uniformly dense air, parallel to 
 its surface. We perceive, therefore, that the refraction, except in 
 the vicinity of the horizon, varies nearly as the tangent of the ap- 
 varent zenith distance. 
 
 95. It has been ascertained by experiment that m, the index of 
 refraction, (the barometer being = 29.6 inches, and the thermome- 
 ter = 50) = 1.0002803. Substituting in equation (3), after hav- 
 ing restored the value of A, and reducing, there results 
 
 , r = 57". 8 tang Z (4). 
 
 96. With the aid of this formula, or of others purely theoretical, 
 astronomers have sought to determine the precise amount of the 
 refraction at various zenith distances from observation, and by col- 
 lating the results of their observations to obtain empirical formulae 
 that are more exact. 
 
 97. 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 
 
 Fig. 24. 
 
 for all altitudes from observation simply. 
 For, let P (Fig. 24) be the elevated pole, 
 Z the zenith, PZE 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 
 been measured. Now, in the triangle 
 ZPS, ZP the co-latitude and PS the 
 polar distance are known by hypothe- 
 sis, and the angle P is the sidereal time 
 which has elapsed since the star's last 
 meridian transit, (or, if the star be to 
 the east of the meridian, the difference 
 between this interval and 24 sidereal 
 hours,) converted into degrees by allow- 
 ing 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 refraction. For the solution of this problem the polar distance may be 
 found by taking the complement of the declination computed from an ob- 
 served meridian zenith distance, (68 ;) and, since the upper and lower transits 
 of a circumpolar star take place at equal distances from the pole, the co-lati- 
 tude may be found by taking the half sum of the greatest and least zenith dis- 
 tances of the pole star. But it is obvious that neither of these quantities can be 
 accurately determined, unless the measured zenith distances be corrected for re- 
 
48 CORRECTIONS OF CO-ORDINATES 
 
 fraction. When, however, the zenith distances in question differ considerably fron* 
 90, the corresponding refractions may be at first ascertained with considerable 
 accuracy by means of equation (4.) When more correct formulae have been ob- 
 tained by this or any other process, the latitude and polar distance, and therefore 
 the refraction answering to the measured zenith distance, will become more accu- 
 rately known. 
 
 98. The various formulae of refraction having been tested by 
 numerous 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 this 
 the refraction is irregular, or is frequently different in amount 
 when the circumstances upon which it is supposed to depend are 
 the same. 
 
 99. The refractive power of the air varies with its density, and 
 hence the refraction must vary with the height of the barometer 
 and thermometer. 
 
 100. The refractions which have place when the barometer 
 stands at 29.6 inches, (or, according to some astronomers, 30 inch- 
 es,) and the thermometer at 50, are called mean refractions. 
 
 The refractions corresponding to any other height of the barom- 
 eter or thermometer, are obtained by seeking the requisite correc- 
 tions to be applied to the mean refractions, on the hypothesis that 
 the refraction is directly proportional to the density of the atmo- 
 sphere. 
 
 101. To save astronomical observers and computers the trouble 
 of calculating the refraction whenever it is needed, the mean re- 
 fractions corresponding to various zenith distances or altitudes are 
 computed from the formulae, as also the correction's for the barom- 
 eter and thermometer, and inserted in a table. Table VIII is Dr. 
 Young's table of mean refractions, and Table IX his table of cor- 
 rections. The refraction answering to any zenith distance not 
 inserted in the table can be found by simple proportion. (See 
 Prob. VII.)* 
 
 102. On inspecting Table VIII, it will be seen that the refrac- 
 tion amounts to about 34' when a body is in the apparent horizon, 
 and to about 68" when it has an altitude of 45. 
 
 OTHER EFFECTS OF ATMOSPHERICAL REFRACTION. 
 
 103. Atmospherical refraction makes the apparent distance of 
 any two heavenly bodies less than the true ; for it elevates them 
 in vertical circles which continually approach each other from the 
 horizon till they meet in the zenith. 
 
 104. Refraction also makes the discs of the sun and moon ap- 
 pear of an elliptical form when near the horizon. As it increases 
 with an increase of zenith distance, the lower limb of the sun or 
 
 * The tables referred to in the text may be found near the end of the book. Th 
 problems referred to are in Part IV. 
 
PARALLAX. 49 
 
 moon is more refracted than the upper, and thus the vertical diam- 
 eter is shortened, while the horizontal diameter remains the same, 
 or very nearly so. This effect is most observable near the hori- 
 zon, for the reason that the increase of the refraction is there the 
 most rapid. The difference between the vertical and horizontal 
 diameters may amount to i part of the whole diameter. 
 
 105. When a star appears to be in the horizon, it is actually 34' 
 below it, (102 :) 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. 
 
 106. The apparent diameter of the sun is about 32' ; as this is 
 less than the refraction in the horizon, it follows, that when the 
 sun appears to touch the horizon it is actually entirely below it. 
 The same is true of the moon, as its apparent diameter is nearly 
 the same with that of the sun. 
 
 PARALLAX. 
 
 ; 107. The correction for atmospherical refraction having been 
 applied, the zenith distance of a body is reduced from the surface 
 of the earth to its centre, by means of a correction called Parallax. 
 
 108. Parallax is, in its most general sense, the angle made by 
 the lines of direction, or the arc of the celes- Fig. 25. 
 
 tial sphere comprised between the places of 
 an object, as viewed from two different sta- 
 tions. It may also be defined to be the an- 
 gle subtended at an object by a line joining 
 two different places of observation. Let S 
 (Fig. 25) represent a celestial object, and 
 A, B two places from which it is viewed. 
 At A it will be referred to the point s of the 
 celestial sphere, and at B to the point s' ; 
 the angle BSA, or the 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 indefinitely 
 great radius, so that the point S is not sen- 
 sibly removed from its centre. 
 
 109. The term parallax is, however, generally used in astrono- 
 my in a limited sense only, namely, to denote the angle included 
 between 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. 
 26) is the centre of the earth, O a point on its surface, and S a 
 heavenly body, OSC is the parallax of the body. 
 
 110. The parallax of a heavenly body above the horizon is call 
 ed Parallax in Altitude. 
 
 7 
 
50 
 
 CORRECTIONS OF CO-ORDINATES. 
 
 The parallax of a body at the time its apparent altitude is ze- 
 ro, or when it is in the plane of the horizon is called the Horizon* 
 tal Parallax of the body. Thus, if the body S (Fig. 26) be sup* 
 
 Fig. 26. 
 
 posed to cross the plane of the horizon at S', OS'C will be its hori- 
 zontal parallax. OSC is a parallax in altitude of this body. 
 
 111. It is to be observed, that the definition just given of the hori- 
 
 Fig. 27. 
 
 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 the se- 
 quel) is a spheroid, and ac- 
 cordingly the vertical and 
 the radius at any point of 
 its surface are inclined to 
 each other ; as represented 
 in Fig. 27, where OC is the 
 radius, and OC' the verti- 
 cal. The points Z and z, 
 in which the vertical and 
 radius pierce the celestial 
 sphere, are called, respec- 
 tively, the Apparent Ze- 
 nith and the True Zenith. 
 In perfect strictness, the horizontal parallax is the parallax at the 
 time zOS, the apparent distance from tie true zenith, is 90. 
 No material error, however, will be committed in supposing the 
 
PARALLAX IN ALTITUDE, 5-] 
 
 earth to be spherical, except when the question relates to the paral- 
 lax of the moon. 
 
 112. Let the apparent zenith distance ZOS=Z, (Fig. 26,) the 
 true zenith distance ZCS = z, and the parallax OSC =p. Since 
 
 ^the angle ZOS is the exterior angle of the triangle OSC, we have 
 
 ZOS = ZCS + OSC, and hence also ZCS = ZOS OSC ; 
 or, 
 
 Z=z+p, andz = Z -p .... (5). 
 
 Thus, to obtain the true zenith distance from the apparent, we have 
 to subtract the parallax ; and to obtain the apparent zenith distance 
 from the true, to add the parallax. 
 
 Parallax, then, takes effect wholly in a vertical plane, like the 
 refraction, but in the inverse manner ; depressing the star, while 
 the refraction elevates it. Thus, the refraction is added to Z, but 
 the parallax is subtracted from it. 
 
 113. To find an expression for the parallax in altitude. 
 
 (1.) In terms of the apparent zenith distance. In the triangle 
 SOC (Fig. 26) the angle OSC = parallax in altitude =p, OC = ra- 
 dius of the earth = R, 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, 
 
 sin p: sin (180 Z) : : R : D; 
 and 
 
 D sinp = R sin Z ; 
 or, 
 
 R 
 
 ship == :~ 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 pro- 
 portion 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 
 
 sinH = g (7): 
 
 substituting, we have 
 
 sinp = sin H sin Z . . . . (8). 
 
 This last 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 nearest, the 
 value of H does not at any time exceed 62'. We may, therefore, 
 without material error, replace sinp and sin H byp and H. This 
 being done, there results, 
 
 p = HsinZ .... (9). 
 
.-':" 
 
 52 CORRECTIONS OF CO-ORDINATES. 
 
 Wherefore, the parallax in altitude equals the product of the hor 
 izontal 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, (111,) and H the horizontal parallax as it is 
 defined in Art. 111. 
 
 (2.) In terms of the true zenith distance. In the actual state of astronomy, the 
 true co-ordinates of the places of the heavenly bodies are generally known, or may 
 be obtained by computation from the results of observations already made, and 
 from these there is often occasion to deduce the apparent co-ordinates. For this 
 purpose there is required an expression for the parallax in altitude in terms of the 
 true zenith distance. 
 
 If we make Z = z -\-p (112) in equation (8), we shall have 
 
 sin p = sin H sin (z -j- ), or sin H =- - ~ 
 whence, 
 
 and 
 
 Dividing, 
 
 1 -{- sin H _ sin (z -\-p) + sin p ^ 
 
 1 sinH sm(z-+-p) sinp' 
 
 1 sip 
 
 tang' (45 + J H) = - , < A PP . For. 36, 29) ; 
 
 whence, 
 
 tang (i z +J>) = tang $z tang 2 (45 + $ H) . . . (10). 
 
 This equation makes known z -\-p, from which we may obtain p by subtract- 
 ing i z. 
 
 In order to be able to compute the parallax in altitude by means of 
 formula (9) or (10), it is necessary to know H, the horizontal parallax. 
 
 114. To find the horizontal parallax. 
 
 Let 0, O' (Fig. 27) represent two stations upon the same ter- 
 restrial meridian OEO', and remote from each other, Z, Z' their 
 apparent zeniths, and z, z' their true zeniths, QCE the equator, 
 and S the body (supposed to be in the meridian) the parallax of 
 which is to be found. Let the angle OSO' = A, *OS = Z, s'O'S 
 = Z' ; also let CO = R, CO' = R', CS = D, the parallax in alti- 
 tude OSC =p, and the parallax in altitude O'SC =p'. Now, by 
 equation (6), replacing the sine of the parallax by the parallax it- 
 self, (113,) 
 
 T? "R' ' 
 
 p = rj sin Z, and p' ~r sin Z' ; 
 
 whence 
 
 R . -R' . - RsinZ-fR'sinZ' 
 
 g- ^ 
 
 but, (equa. 7,^ 
 
 HR , R 
 = 
 
HORIZONTAL PARALLAX. 53 
 
 Substituting this value of D, and deducing the value of H, we 
 have 
 
 ' ' ' ^ >' 
 
 ___ 
 
 R sin Z + R' sin Z' R sin Z + R' sin Z' 
 It remains now to find an expression for A in terms of measura- 
 ble quantities. Let Os and O's (Fig. 27) be the directions at O 
 and O' of a fixed star ' which crosses the meridian nearly at the 
 same time with the body. Owing to the immense distance of the 
 star, these lines will be sensibly parallel to each other, (27.) 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 O', be represent- 
 ed by d'. Now, 
 
 OSO' = OLO' SQ's = SOs SO'*,orA = d d 1 . 
 If the body be seen on different sides of the star by the two ob- 
 servers, we shall have 
 
 A.=d+d f . 
 Substituting in equation (11), there results, 
 
 H = _W^L_ (12) 
 
 RsinZ+R'sinZ' 
 
 If we regard the earth as a sphere, R=R', and dividing by R, 
 we have 
 
 H _ dd' - 
 
 -"' ---- 
 
 115. To find the parallax by means of these formulae, each of 
 the two observers must 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 them for refraction. The 
 difference of the two will be, respectively, the values of d and d f ; 
 and the corrected zenith distances of the body will be the values 
 of Z and Z', if formula (13) be used; if formula (12) be used, 
 the measured zenith distances of the body must still be corrected 
 for the reduction of latitude, (p. 19, Def. 4.) 
 
 It- is not necessary that the two stations should be on precisely 
 the same meridian ; for if the meridian zenith distance of the body 
 be observed from day to day, its daily variation will become 
 known ; then, knowing also the difference of longitude of the two 
 places, the following simple proportion will give the change of ze- 
 nith 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 successive transits: 
 diff. of long., expressed in time, (*) : : variation of zenith dist. 
 in interval T : its variation in interval t. This result, applied 
 to the zenith distance observed at one of the stations, will re- 
 duce it to what it would have been if the observation had been made 
 tn the same latitude on the meridian of the other station* 
 
 116. The horizontal parallax of a heavenly body may be found 
 
54 
 
 CORRECTIONS OF CO-ORDINATES. 
 
 by the foregoing process, to within 1" or 2" of the truth. No 
 greater degree of accuracy is necessary in the case of the moon. 
 But there are certain important uses made of the horizontal paral- 
 lax of a body that will be noticed hereafter, which require that the 
 parallax of the sun, and of the planets, should be known with much 
 greater precision. The more accurate methods employed to deter- 
 mine the parallaxes of these bodies will be explained (in principle 
 at least) in subsequent parts of the w r ork. 
 
 117. In consequence of the spheroidal form of the earth, the hor- 
 izontal parallax of a body is somewhat different at different places. 
 Let H and H' denote the horizontal parallaxes of the same body, 
 and R and R' the radii of the earth at two different places. Then, 
 by equation (7,) 
 
 whence, 
 
 Thus the parallax at the equator, called the Equatorial Paral- 
 lax, is the greatest, and the parallax at the pole the least. The dif- 
 ference between the parallaxes of the moon at the equator and at 
 the pole may amount to about 12". For the other heavenly bodies 
 the difference is too small to be taken into account. 
 
 118. When the horizontal parallax has been found for any one 
 distance and time from observation, the horizontal parallax for any 
 other distance and time may be approximately computed, by means 
 of the principle that the parallax of a body is directly proportional 
 to its apparent diameter. The truth of this principle appears from 
 the fact, that both the parallax (113) and the apparent diameter are 
 inversely proportional to the same quantity, viz : the distance of 
 the body from the earth. 
 
 In the present condition of astronomical science, when the hori- 
 zontal parallax of either one of the heavenly bodies is required for 
 any particular time, it may be obtained by computation, or from 
 tables. It may also be taken out of the Nautical Almanac.* 
 
 119. The equatorial horizontal parallax of the moon varies from 
 53' 48" to 61' 24", according to the distance of the moon from the 
 earth. The equatorial parallax of the moon answering to the mean 
 distance, is 57' 1". 
 
 The horizontal parallax of the sun varies slightly, from a change 
 of distance. At the mean distance it is 8".6. 
 
 The horizontal parallaxes of the planets are comprised within 
 the limits 31", and 0".4. 
 
 * The Nautical Almanac is a collection of data to be used in nautical and as 
 tronomical calculations, published annually in England, and republished in New 
 York. It may generally be obtained two or three yeare previous to the year for 
 which it is calculated. 
 
ABERRATION. 55 
 
 The fixed stars have no parallax.* 
 
 120. Parallax in right ascension and declination, and in longi- 
 tude and latitude. 
 
 Since the parallax displaces a body in its vertical circle, which 
 is generally oblique to the equator and ecliptic, it will alter 
 its right ascension and declination, as well as its longitude and lat- 
 itude. The difference between the true and apparent right ascen- 
 sion is called the parallax in right ascension ; the like differences 
 for the other co-ordinates are called, respectively, parallax in De- 
 clination^ parallax in longitude, and parallax in latitude. 
 
 ABERRATION. 
 
 121. 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 places of the fixed 
 stars were subject to any direct alteration in consequence of the 
 supposed continual change of the earth's position in space. The 
 observations showed that there had been in reality, during the pe- 
 riod 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 sup- 
 posed orbitual motion of the earth. These phenomena Dr. Brad- 
 ley undertook to examine and reduce to a general law. After 
 repeated trials, he at last succeeded in discovering their true ex- 
 planation. His theory is, that they are different effects of one gen- 
 eral cause, a progressive motion of light in conjunction with an 
 orbitual motion of the earth. 
 
 Fig. 28. 
 
 A A' A" B 
 
 122. Let us conceive the observer to be stationed at the earth's 
 centre ; and let ACB (Fig. 28) be a portion of the earth's orbit, so 
 small that it may be considered a right line, CS the true direction 
 
 * The practical method of correcting for parallax is detailed and exemplified in 
 Problem VIII. 
 
56 CORRECTIONS OF CO-ORDINATES. 
 
 of a fixed star as seen from the point C, AC the distance through 
 which the earth moves in some small portion of time, and aC the 
 distance through which a particle of light moves in the same time. 
 Then, a particle of light, which, coming from the star in the direc- 
 tion SC, is at a at the same time that the earth is at A, will arrive at 
 C at the same time that the earth does. Suppose that Aa is the 
 position of the axis or central line of a telescope, when the earth 
 is at A, and that, continuing parallel to itself, it takes up by virtue 
 
 of the earth's motion, the successive positions A'a', A."a" 
 
 CS'. A particle of light which follows the line SC in space will 
 descend along this axis : for aa 1 is to AA' and aa" is to AA", as 
 aC is to AC, that is, as the velocity of light is to the velocity of 
 the earth ; consequently, when the earth is at A' the particle of 
 light is on the axis at a', and when the earth is at A" the particle 
 of light is on the axis at a", and so on for all the other positions of 
 the axis, until the earth arrives at C. The apparent direction of 
 the star S, as far, at least, as it depends upon the cause under con- 
 sideration- will therefore be CS'. 
 
 The angle SCS', which expresses the change in the apparent 
 place of a star S, produced by the motion of light combined with 
 the motion of the spectator, is called the Aberration of the star ; 
 and the phenomenon of the change of the apparent course of the 
 light coming from a star, thus produced, is called Aberration of 
 Light, or simply Aberration. 
 
 123. The phenomenon of the aberration of light may be famil- 
 iarly illustrated by taking falling drops of rain instead of particles 
 of light, and a vessel in motion at sea instead of the earth moving 
 through space ; and considering what direction must be given to 
 a small tube by a person standing upon the deck of the vessel, so 
 as to permit the drops falling perpendicularly to pass through the 
 tube. It is plain, that if the tube had a precisely vertical position, its 
 forward motion would bring the back part of the tube against the 
 drop ; and that the only way to prevent this is to incline the upper 
 end of the tube forward, or draw the lower end backward, whereby 
 the back part of it would be made to pass through a greater dis- 
 tance before it comes up to the line of descent of the drop. The 
 quantity that it is made to deviate in direction from this line must 
 depend upon the relative velocities of the falling drop and moving 
 tube. To the observer, unconscious of his own motion, the drop 
 will appear to fall in the oblique direction of the tube. 
 
 124. If through the point a (Fig. 29) a line as' be drawn parallel 
 to AC, and terminating in CS', the figure Aas'C will be a parallel- 
 ogram, and therefore as' will be equal to AC. Hence it appears, 
 that if on CS, the line of direction of a star S, a line Ca be laid off, 
 representing the velocity of light, and through a a line as' be drawn, 
 naving the same direction as the earth's motion and equal to its ve- 
 locity, the line joining s' and C will be the apparent line of direc- 
 tion of the star, the point S' its apparent place in the heavens, and 
 
ABERRATION. 57 
 
 the angle aCs' its aberration. We conclude, therefore, that by 
 virtue of aberration a star is seen in advance of its true place, in the 
 plane passing through the line of direction of the star and the !ine 
 of the earth's motion. 
 
 Fig. 29. 
 
 The amount of the aberration of a star is always very small, 
 (never greater than about 20",) because of the very great dispropor- 
 tion between the velocity of light and the velocity of the earth. It 
 is very much exaggerated in Figs. 28 and 29. 
 
 125. The aberration is the same when a star is viewed with the 
 naked eye, as when it is seen through a telescope. For, let C, 
 the velocity of the light, be decomposed into two velocities, of 
 which one, AC, is equal and parallel to the velocity of the earth ; 
 the other will be represented by s'C. Now, since the velocity 
 AC^is equal and parallel to the velocity of the earth, it will pro- 
 duce no change in the relative position of a particle of light and 
 the eye, and therefore the relative motion of the light and the eye 
 will be the same that it would be if the earth were stationary and 
 the light had orjy the velocity s'C ; accordingly, the light entering 
 the eye just as it would do if it actually came in the direction s'C, 
 and the eye were at rest, Cs' will be the apparent direction of the 
 star from which it proceeds. 
 
 126. If we regard the observer as situated upon the earth's 
 surface, instead of being at its centre, the aberration resulting 
 from the earth's motion of revolution will be still the same : for, 
 all points of the earth advance at the same rate and in the same 
 direction with the centre. The motion of rotation will produce an 
 aberration proper to itself, but it is so small that there is no occa- 
 sion to take it into account. 
 
 127. To find a general expression for the aberration. We have 
 by Trigonometry, (Fig. 29,) 
 
 sin A.aC : sin CA : : CA : Ca : : vel. of earth : vel. of light ; 
 whence, 
 
 f 1 A 
 
 sin AaC sin CAez -^-, or, since AaC = SCS', 
 (^a 
 
 . vel. of earth . 
 
 sm aberr. = sin CAa = (14). 
 
 vel. of light 
 
 When CAa is 90, the aberration has its maximum value, and 
 this has been found by observation to be about 20"(20".44) ; whence, 
 
 8 
 
58 CORRECTIONS OF CO-ORDINATES. 
 
 vel. of earth 
 
 sin 20" = . F-r-j . . . (15): 
 vel. of light 
 
 substituting, and taking sin BCa for sin CA<z, to which it is very 
 nearly equal, we have 
 
 sin aberr. = sin BCa sin 20" . . . (16). 
 
 We may conclude from this equation, that the aberration in- 
 creases with the angle BCa made by the direction of the star with 
 the direction of the earth's motion ; that it is equal to zero when 
 this angle is zero, and has its maximum value of 20" (more accu- 
 rately 20". 44) when this angle is 90. 
 
 128. Let us now inquire into the entire effect of aberration in 
 the course of a year. Let S (Fig. 30) be the sun ; E the earth ; 
 ~Efg its orbit ; ZTV that orbit extended to the fixed stars, or the 
 ecliptic, (p. 15, Def. 17 ;) ET a tangent to the earth's orbit at E ; 
 the place of S among the fixed stars or in the ecliptic, as seen 
 Fig. 30. from the earth ; s a fixed star ; 
 
 s/T the arc of a great circle pass- 
 ing through s and T. Then, by 
 what has preceded, (124,) the 
 earth moving in the direction 
 E/g, the apparent place of the 
 star may be represented by s' and 
 the aberration by s~Es'. Thus, 
 the effect of aberration at any one 
 time is to displace the star by a 
 small amount, directly towards 
 the point T of the ecliptic, which 
 is 90 behind the sun. As the 
 earth moves, the position of the point T will vary ; and in the 
 course of a year, while the earth describes its entire orbit in the 
 direction E/g", this point will move in the same direction entirely 
 around the ecliptic. In this period of time, therefore, ss' the small 
 arc of aberration will revolve entirely around s the true position of 
 the star ; from which we conclude, that in consequence of aberra- 
 tion a star appears to describe a closed curve in the heavens around 
 its true place. 
 
 As the inclination of the direction of the star to the direction oi 
 the earth's motion will vary during a revolution of the earth, the 
 aberration will also vary during this period, (127,) and hence the 
 curve in question will not be a circle. It appears upon investiga- 
 tion that it is an ellipse, having the true place of the star for its 
 centre, and of which the semi-major axis is constant and equal to 
 20 '.44, and the semi-minor axis variable and expressed by 20''.44 
 sin X, (X denoting the latitude of the star.) Each star, then, de- 
 scribes an ellipse which is the more eccentric in proportion as the 
 star is the nearer to the ecliptic ; for, the expression for the minoi 
 axis shows that the smaller the latitude the less will be this axis. 
 For a star situated in the ecliptic the minor axis will be zero, and 
 
ABERRATION. 59 
 
 the ellipse will be reduced to a right line. For a star in the pole 
 of the ecliptic the minor axis is equal to the major, and the ellipse 
 therefore becomes a circle. 
 
 When the earth is at two diametrically opposite points of its orbit, as E and r, 
 the direction of its motion, which is the same as that of the tangent to the orbit, 
 will make equal angles with the line of direction of the star, but will be towards 
 diametrically opposite points in the sphere of the heavens, (since the earth's orbit 
 is to be considered as a mere point in the centre of this sphere, (27.) It follows, 
 therefore, that in all such positions of the earth the aberration is the same, but in 
 opposite directions. At E and r, where the angle sET included between the line 
 of direction of the star and that of the earth's motion is 90, the aberration is at 
 its maximum, and the star is at the extremities of the major axis of its elliptic orbit. 
 At / and g, 90 distant from E and r, this angle is at its minimum ; the aberration 
 is the least possible, and the star is at the extremities of the minor axis of its orbit. 
 
 129. Since aberration causes the apparent place of a star to dif- 
 fer slightly from its true place, the true and apparent co-ordinates 
 will, in consequence, differ somewhat from each other. The effects 
 of the aberration of light upon the apparent right ascension and 
 declination of a star, are called, -respectively, the Aberration in 
 Right Ascension and the Aberration in Declination. In like man- 
 ner its effects upon the longitude and latitude are called the Aber- 
 ration in Longitude and the Aberration in Latitude* 
 
 130. Since the motion of the earth is at all times in a direction perpendicular, or 
 nearly so, to the line followed by the light which comes from the sun to the earth, 
 the aberration of the sun, which takes place only in longitude, is continually equal 
 to 20".44, (127.) Thus, the sun's apparent place is always 20" behind its true place. 
 
 131. For a planet, the aberration is different from what it is for a fixed star. 
 As a planet changes its place during the time that the light is passing from it to 
 the earth, it would, if the earth were stationary, appear to be as far behind its true 
 place as it has moved during this interval. This aberration due to the motion of 
 the planet, combined with that due to the earth's motion, will give the real aberra- 
 tion of the planet. 
 
 132. For the moon, the aberration occasioned by its motion around the earth is 
 very small. The earth's motion produces no lunar aberration, for the reason that 
 the moon, and consequently the light emitted from it, partakes of this motion. 
 
 133. If the apparent places of a star, found at various times, be 
 corrected for aberration, the same result for the true place of the 
 star is obtained. Again, the deductions of Art. 128 agree in every 
 particular with the observed phenomena of the apparent displace- 
 ment of the stars, first discovered by Dr. Bradley. These facts 
 show that the aberration of light is the true cause of these phe- 
 nomena, and consequently, at the same time establish the fact of 
 the earth's orbitual motion, as well as that of the progressive mo- 
 tion of light. 
 
 134. It may be worth while to state, that the first discovery of 
 the progressive motion of light preceded the detection and expla- 
 nation by Bradley of the phenomena of aberration. The discov- 
 ery was made by Roemer, a Danish astronomer, in the year 1667, 
 from a comparison of observations upon the eclipses of Jupiter's 
 satellites. 
 
 * For the practical method of determining and applying these corrections, 
 Probs. XIX., XXL, XXII., XXIII. 
 
 
60 
 
 CORRECTIONS OF CO-ORDINATES. 
 
 135. As to the actual velocity of:-; light, we have, by equation 
 (15,) vel. of earth : vel. of light : sin 20" .44 : 1 : : 1 : 10,000, 
 (nearly.)' Taking the velocity of the earth at 68,167 miles per 
 hour, and making the calculation by logarithms, we obtain for the 
 velocity of light 191,000 (191,140) miles per second. As deter- 
 mined from observations upon Jupiter's satellites, it is very nearly 
 the same. The time employed by light in coming from the sun to 
 the earth is 8m. 18s. 
 
 PRECESSION AND NUTATION. 
 
 136. In the investigations that follow, we shall take it for grant- 
 ed that it is possible to find the obliquity of the ecliptic and the 
 place of the equinox. Methods of determining them will be given 
 when we come to treat of the apparent motion of the sun. 
 
 137. By comparing the longitudes and latitudes of the same 
 fixed stars, obtained at different periods, (69,) it is found that their 
 latitudes continue very nearly the same, but that all their longi- 
 tudes increase at the mean rate of about 50" per year. The longi- 
 tude of a star being the arc of the ecliptic, intercepted in the order 
 of the signs between the vernal equinox and a circle of latitude 
 passing through the star, (p. 18, Def. 30,) it follows from the last 
 mentioned circumstancej that the vernal equinox must have a mo- 
 tion along the ecliptic in a direction contrary to the order of the 
 signs, amounting to about 50" in a year. As it has been found 
 that the autumnal equinox is always at the distance of 180 from 
 the vernal, it must have the same motion. This retrograde motion 
 of the equinoctial points, is called the Precession of the Equinoxes. 
 
 138. As the latitude of a star is its distance from the ecliptic, 
 (p. 18, Def. 31,) it follows from the circumstance of the latitudes 
 of all the stars continuing very nearly the same, that the ecliptic 
 remains fixed, or very nearly so, with respect to the situations of 
 the fixed stars. 
 
 139. The ecliptic being stationary, it is plain that the precession 
 
 p- gj of the equinoxes must result from 
 
 a continual slow motion of the 
 equator in one direction. It ap- 
 pears from observation, that the 
 obliquity of the ecliptic, or the 
 inclination of the equator to the 
 ecliptic, remains, in the course of 
 this motion, very nearly the same. 
 140. Since the equator is in 
 motion, its pole must change its 
 place in the heavens. Let VLA 
 (Fig. 31) represent the ecliptic, 
 K its pole, which is stationary, 
 P the position of the north pole 
 
PRECESSION. 61 
 
 of the equator or of the heavens at any given time, and YEA the 
 corresponding position of the line of the equinoxes : KPL re- 
 presents the circle of latitude passing through P, or the solsti- 
 tial 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 fol- 
 lows 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 equi- 
 noxes 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 the 
 revolution of the equinox and of the pole of the equator is some- 
 thing less than 26,000 years. 
 
 141. It is an interesting consequence of this motion of the pole 
 of the equator and heavens, that the pole star, so called, will not 
 always be nearer to the pole than any other star. The pole is at 
 the present time approaching it, and it will continue to approach 
 it until the present distance of 1 1 becomes reduced to less than 
 i, which will happen about the year 2100: after which it will 
 begin to recede from it, and continue to recede, until about the 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 1 3,000 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. 
 
 142. 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 ac- 
 cording to the situations of the stars with respect to the equator 
 and equinoctial points. 
 
 143. The ecliptic, although very nearly stationary, as stated in Art. 138, is not 
 strictly so. By comparing the values of the obliquity of the ecliptic, found at dis- 
 tant periods, it is ascertained that it is subject to a gradual diminution from century 
 to century. A comparison of the results of observations made by Flamstead in 
 1690, and by Dr. Maskelyne in 1769, gives for the mean secular diminution 50" 
 
62 CORRECTIONS OF CO-ORDINATES. 
 
 and for the mean annual diminution 0".50. A more accurate determination of the 
 mean annual diminution is 0".4G. 
 
 It appears from observation, that there are minute secular changes in the lati- 
 tudes of the stars, which establish that the diminution of the obliquity of the eclip- 
 tic arises from a slow displacement of the plane of the ecliptic (or of the earth's 
 orbit) in space. 
 
 144. If the ecliptic slowly changes its position in the heavens, its pole must like- 
 wise ; and since the obliquity of the ecliptic is continually diminishing, its pole 
 must be gradually approaching the pole of the equator. 
 
 145. The motion of the ecliptic alters somewhat the precession of the equinoxes, 
 making it a little less than it would be if the equator only was in motion : for, let 
 
 EL (Fig. 32) represent the position of the eclip- 
 tic, and VQ that of the equator, at any assumed 
 date, and EL', V'Q' the positions of the same 
 circles at some later date ; the obliquity L'V'Q' 
 at the second epoch being less than that (LVQ) 
 at the first epoch : also let r be the physical point 
 of the moveable ecliptic, which at the first epoch 
 coincided with the point V, and n the point an- 
 swering to V ; and Vr, V'n the arcs of small 
 circles described by the points V and V. in the 
 motion of the arc EVL about the point E. Since 
 wV'V is a right angle, and Q'V'V an acute an- 
 gle, the point n must fall to the left of V", and 
 therefore V"r will be less than nr, or its equal 
 W, by the small arc nV". But VV is the 
 precession on the fixed ecliptic, and rV" the ac- 
 tual precession. We learn by the aid of Physi- 
 cal Astronomy, that the amount of annual pro- 
 cession would, if the ecliptic were fixed, be 50".35. As we have already seen, the 
 actual precession on the moveable ecliptic is 50", (more accurately, 50".23.) 
 
 146. It remains for us now to take notice of a minute inequal- 
 ity in the motion of the equator and its pole, which we have thus 
 far overlooked. Dr. Bradley, in observing the polar distance of a 
 certain star, (y Draconis,) with the view of verifying his theory of 
 aberration, discovered that the observed polar distance did not agree 
 with the apparent polar distance, as computed from the results of 
 previous observation, by allowing for precession, aberration, and 
 refraction ; and hence inferred the existence of a new cause of vari- 
 ation in the co-ordinates of a star. On continuing his observations, 
 he found that the polar distance alternately increased and dimin- 
 ished, 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, re- 
 volved around a point conceived to move in this manner. 
 
 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 supposed vibration of the axis of the 
 earth, and consequently of that of the heavens, Dr. Bradley gave 
 the name of Nutation. The term Nutation is also applied to the 
 changes of the co-ordinates of a star's place, which are produced 
 by the nutation of the earth's axis. The point about which the 
 pole was conceived to revolve, is the mean position of the pole, or 
 the Mean Pole. 
 
NUTATION. 
 
 63 
 
 Dr. Bradley discovered, from his observations, that the curve described by the 
 pole must be an ellipse, having its major axis in the solstitial colure ; and esti 
 mated the value of the major axis at about 19", and that of the minor axis at about 
 14". He also discovered that a connection existed between the position of the 
 pole in its ellipse, and the position of the moon at the time its latitude was zero, 
 (69,) and changing from south to north, or of the point in which the moon crossed 
 the plane of the ecliptic in passing from the south to the north side of it, called the 
 
 Fig. 33. 
 
 ascending node of the moon's orbit ; for he 
 found that the pole retrograded in like man- 
 ner with the node ; that it completed its revo- 
 lution in the same time, namely, in about 19 
 years ; and that its position was determinable 
 from the place of the node by a geometrical 
 construction. Let P (Fig. 33) represent the 
 mean pole, and p the true pole ; pfg 1 repre- 
 sents the ellipse described by the true pole 
 around P as a centre ; gg', lying in the sol- 
 etitial colure KPL, being its major axis, and 
 ff its minor axis. It is to be observed that 
 the pole P is not stationary, but revolves in 
 the circle NPP', carrying with it the ellipse 
 pfg 1 . It will be seen that this ellipse is very 
 much exaggerated in the figure : a true de- 
 lineation of it on the scale of the figure would 
 be altogether imperceptible. 
 
 This theory of a nutation of the earth's axis has been verified by subsequent ob- 
 servations, and Physical Astronomy has revealed the cause of the phenomenon. 
 
 147. As the equator must move with the axis of the earth or heavens, nutation 
 will change the position of the equinox and the obliquity of the ecliptic. It is plain 
 that its effect upon the position of the equinox will be to make it oscillate periodi- 
 cally and by equal degrees, from one side to the other of the position which corre- 
 sponds to the mean pole ; and that its effect upon the obliquity of the ecliptic will 
 be to make it alternately greater and less than the obliquity corresponding to the 
 mean pole. The position of the equinox which corresponds to the mean pole, is 
 called the Mean Equinox. The obliquity corresponding to the mean pole, is term- 
 ed the Mean Obliquity. Mean Equator has a like signification. The real equinox 
 and the real equator are called, respectively, the True Equinox and the True 
 Equator. The actual obliquity of the ecliptic is termed the Apparent Obliquity. 
 Right ascension and declination, as estimated from the true equator and true equi- 
 nox, are called, respectively, True Right Ascension and True Declination; and 
 longitude, as reckoned from the true equinox, is called True Longitude. Right 
 ascension, declination, and longitude, reckoned from the mean equinox and mean 
 equator, are called, respectively, Mean Right Ascension, Mean Declination, and 
 Mean Longitude. The true and mean co-ordinates differ by reason of nutation. 
 The effect of nutation upon the right ascension is called the Nutation in Right 
 Ascension ; upon the declination, Nutation in Declination ; and upon the longi- 
 tude, Nn'.ation in Longitude. Its effect upon the obliquity of the ecliptic is called 
 Nutation of Obliquity. The distance of the true from the mean equinox in longi- 
 tude, which is the same as the nutation in longitude, is sometimes termed the 
 Equation of the Equinoxes in Longitude ; and the distance in right ascension, the 
 Equation of the Equinoxes in Right Ascension. The precession of the mean 
 equinox is equal to the Mean Precession of the true equinox, which is 50" .2. 
 
 148. Formulae for computing the nutation in right ascension, declination, &c., 
 at any given time, are investigated in some astronomical'works. These formulae 
 cannot be used without a knowledge of the moon's motions. In practice, the nu- 
 tations in right ascension, &c., are found by the aid of tables. (See Probs. XX., 
 XXIII.) If these be applied to the true co-ordinates, the results will be the mean 
 co-ordinates. If the mean co-ordinates be known, the same corrections will fur- 
 nish the true. 
 
 149. Physical Astronomy has made known the existence of another nutation of 
 the earth's axis, too small to be detected by observation. It is called Solar Nuta- 
 tion. The nutation discovered by Dr. Bradley is frequently called Lunar Nutation 
 
64 
 
 CORRECTIONS OF CO-ORDINATES. 
 
 150. To reduce the co-ordinates of a star from one epoch to 
 another. 
 
 This problem is resolved by first converting the true co-ordi- 
 nates into the mean, then transferring the mean co-ordinates from 
 the one epoch to the other, and finally converting the reduced mean 
 co-ordinates into the true. The mode of performing the first and 
 last mentioned operations has already been considered, (148.) It 
 remains now for us to show how to reduce mean co-ordinates from 
 one epoch to another. 
 
 (I.) When the interval of time between the epochs comprises 
 but a few years. In this case the changes, from precession, of 
 the mean right ascension and declination in the course of a year, 
 called the Annual Variation in right ascension and the Annual 
 Variation in declination, are determined, then multiplied by the 
 number of years in the interval, and applied as corrections to the 
 given right ascension and declination. 
 
 Fig. 34. F r * ms purpose formula? have been in- 
 
 vestigated, in which the annual variations 
 in right ascension and declination are ex- 
 pressed in terms of the right ascension and 
 declination of the star and the obliquity of 
 the ecliptic. Let VLA (Fig. 34) be the 
 ecliptic, K its pole, PFP" the circle de- 
 scribed by the mean pole, P the mean pole 
 and VQA the mean equator at any given 
 time, P' the mean pole arid V'Q'A' the 
 mean equator a year afterwards, and s a 
 star. Draw P'r perpendicular to the decli- 
 nation circle Psa. We have 
 an. var. in dec.=sa' sa = Ps P's = Pr ; 
 but since PFr may be considered as a right- 
 angled plane triangle, 
 Pr = PF cos P'Pr = PF sin QPa .... (17). 
 Regarding KPP' as a right-angled isosceles triangle, we obtain 
 sin KPF or 1 : sin KF : : sin PKP' : sin PF ; 
 whence, 
 
 sin PF = sin PKF sin KF, or PF = PKF sin KF (nearly) .... (18) : 
 substituting in equation (17), there results, 
 
 Pr = PKF gin KF sin QPa. 
 
 PKP' = 50".2 (140) ; KF = obliquity of the ecliptic = a, ; 
 
 QPa = VQ Va = 90 R (R designating the right ascension of the star .) 
 Thus, finally, 
 
 an. var. in dec. = 50".2 sin w cos R . . . . (19). 
 Next, we have 
 
 an. var. in r. asc. = V'a' Va = V'a' mb = V'm + ba' . . 
 but, 
 
 V'm = VV cos V V'm = 50".2 cos w ; 
 and since the right-angled triangles sP'r and sba' are similar, 
 
 sin sr or sin sP 1 (nearly) : sin P'r : : sin sa' : sin ba' ; 
 whence, 
 
 (20) ; 
 
 sin6a'= sin 
 
 or baf = P'r 
 
 (nearly). 
 
 The triangle PP'r gives P'r = PF sin P'Pr = PF cos QPa == PKP' sin KF cog 
 QPa (equa. 18) ; and sin P's = cos sa'. Substituting, we obtain 
 
 ba 1 = PKF sin KF cos QPa ^-^', = PKF sin KF cos QPa tang sa'. 
 
 
 
VARIATIONS OF THE CORRECTIONS. 65 
 
 Replacing PKF, KF, and QPa by their values, as above, and taking the declina- 
 tion sa for sa' and denoting it by D, there results, 
 
 bal == 50".2 sin w sin R tang D. 
 Now, substituting in equation (20) the values of V'ra and ba', we have 
 
 an. var. in r. asc. = 50".2 cos w -f- 50" .2 sin &> sin R tang D . . . (21). 
 
 The results of formulae (19, 21) are to be used with their algebraic signs, if the 
 reduction is from an earlier to a later epoch, otherwise with the contrary signs. 
 The declination is always to be considered positive if North, and negative if South. 
 
 V'm = 50' .2 cos u, = 50".2 cos 23 o 28' = 46".0, 
 is the annual retrograde motion of the equinoctial points along the equator. 
 
 (2.) When the inteival of the epochs is of considerable or great length. If the 
 epochs are separated by an interval of more than 10 or 12 years, the foregoing pro- 
 cess will not answer ; for in a period of ten years the annual variations will have 
 sensibly altered.* In this case we may proceed as follows : Convert the right as- 
 cension and declination into longitude and latitude, add to the longitude (or if the 
 reduction be to an earlier epoch, subtract from it) the precession in longitude, 
 which will be the product of 50".23 by the interval of the epochs, expressed in years 
 and parts of a year, and then with the longitude thus obtained, and the latitude, 
 calculate the right ascension and declination, using the mean obliquity of the ecliptic. 
 
 When the period is of great length, or very great precision is desired, the pre- 
 cession on the fixed ecliptic should be used, which is 50".35 per year, (145) ; anc 
 the right ascension should be corrected for the change of the position of the equi- 
 nox on the equator, produced by the motion of the ecliptic , which correction is 
 0".1313 (per year) for later epochs. 
 
 REMARKS ON THE CORRECTIONS VERIFICATION OF THE 
 HYPOTHESIS THAT THE DIURNAL MOTION OF THE STARS 
 IS UNIFORM AND CIRCULAR. 
 
 151. It appears from what we have stated on the subject of the 
 Corrections : 1 . That Refraction varies during the day with the alti- 
 tude of the body, and changes for all altitudes with the state of the 
 atmosphere ; 2. That Parallax varies, like the refraction, with the 
 altitude of the body, and changes from one day to another with its 
 distance ; 3. That Aberration remains sensibly the same for two 
 or three days, and depends for its absolute value on the time of the 
 year ; 4. That Precession and Nutation do not perceptibly alter 
 the co-ordinates of a star, unless it be a circumpolar star, under 
 several days, and that the former increases uniformly with the time 
 while the latter varies periodically, its effects entirely disappearing 
 in about 19 years ; and, 5. That the absolute value of the Nutation 
 depends entirely upon the longitude of the moon's ascending node. 
 
 152. In the determination of the amount and laws of the cor- 
 rections, it was taken for granted by astronomers, that the diurnal 
 motion of the stars was uniform and circular. This hypothesis 
 may be verified in the following manner : Let the zenith distance 
 and azimuth of the same star be measured at various times during 
 a revolution, and corrected for refraction, (the other corrections be 
 ing insensible, (151.) ) Then, if the latitude of the place be 
 known (68) in the triangle ZPS, (Fig. 17, p. 37,) we shall have ZP 
 
 * It is to be understood that we are here giving methods of obtaining very accu- 
 rate results. The process just explained, except for stars near the pole, will fur 
 nish results sufficiently accurate for most purposes, even when the interval com- 
 prises 20 years or more. 
 
 9 
 
66 OP THE EARTH. 
 
 the co-latitude, ZS the zenith distance of the star, and PZS its azi- 
 muth, whence we may compute PS. If this calculation be made 
 foi the time of each observation, it will be found that the same 
 value for PS is obtained in every instance ; which proves the di- 
 urnal motion to be circular. Again, let the angle ZPS be com- 
 puted for the time of each observation, with the same data, and it 
 will be found that it varies proportionally to the time ; which es- 
 tablishes that the diurnal motion is also uniform, or, at least, sensi- 
 bly so during one revolution. 
 
 153. When the transits of a circumpolar star are observed at 
 intervals of several days, and allowance is made for the error of 
 the rate of the clock, as determined from observations upon stars 
 in the vicinity of the equator, and for the aberration in right ascen- 
 sion, it is found that the sidereal times of the transits differ slightly 
 from each other ; from which it appears that the diurnal motion of 
 the stars is not strictly uniform. When, however, allowance is 
 made for the precession and nutation in right ascension, this dif- 
 ference disappears. We may hence conclude that the motion of 
 rotation of the earth is uniform, and that the motions of the earth 
 and of its axis, which produce the phenomena of precession and 
 nutation, alter the times of the transits of the stars, thereby making 
 the period of the apparent revolution of a star to differ slightly 
 from the period of the earth's rotation. 
 
 It may be observed, that the greatest difference obtains in the 
 case of the pole star, and is half a second. 
 
 CHAPTER IV. 
 
 OF THE EARTH ; ITS FIGURE AND DIMENSIONS I LATITUDE AND 
 
 LONGITUDE OF A PLACE. 
 
 154. ALTHOUGH it is in general sufficient for astronomical pur- 
 poses to regard the earth as a sphere, still it is necessary in some 
 cases of astronomical observation and computation, when accurate 
 results are desired, to take notice of its deviation from the spheri- 
 cal form. No account need, however, be taken of the irregulari- 
 ties 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 corresponded with that of the sea. 
 
 155. 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 verticals at the extremities of the arc. 
 
FIGURE AND DIMENSIONS OP THE EARTH. 
 
 67 
 
 Fig. 35. 
 
 It is also called a Degree of Lat- 
 itude. Thus if QNE (Fig. 35) 
 represent a terrestrial meridian, 
 ab will be a degree of it if it be of 
 such length that the angle aCb 
 between the verticals Z'C, Z6C, 
 is 1. 
 
 156. The length of a degree 
 at any place will serve as a meas- 
 ure of the curvature of the me- 
 ridian at that place ; for it is ob- 
 vious, from considerations already 
 presented, (4,) that the earth, if 
 not strictly spherical, must be 
 nearly so, and therefore that a 
 degree ab (Fig. 35) may, with 
 
 but little if any error, be considered as an arc of 1 of a circle 
 which has its centre at C, the point of intersection of the verticals 
 Ca, C6, at the extremities of the arc. The ( curvature will then 
 decrease in the same proportion as the radius of this circle in- 
 creases, and therefore in the same proportion as the length of a 
 degree increases. Wherefore, the form of a meridian may be de- 
 termined by measuring the length of a degree at various latitudes. 
 
 157. To determine the length of a degree of a terrestrial me- 
 ridian. To accomplish this, we have, 
 
 (1.) To run a meridian line ; an operation which is performed 
 in the following manner. An altitude and azimuth instrument (or 
 some other instrument adapted to meridian observations) is first 
 placed at the point of departure, and accurately adjusted to the 
 meridian. A new station is then established by sighting forward 
 with the telescope. To this station the instrument is removed, 
 and is there adjusted to the meridian by sighting back to the first 
 station. A third station is then established by sighting forward 
 with the telescope as before, to which the instrument is removed. 
 By thus continually establishing new stations, and carrying the 
 instrument forward, the meridian line may be marked out for any 
 required distance. The meridian adjustments may be corrected 
 from time to time by astronomical observations, (51, 71.) 
 
 (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 di- 
 rect measurement, it may be calculated with equal precision, by 
 means of a base line and a chain of triangles the angles of which 
 are measured, 
 
 (3.) To find the inclination of the verticals at the extreme sta- 
 tions. This angle may be obtained by measuring the meridian 
 zenith distances of the same fixed star at the two stations, correct- 
 ing them for refraction if they are observed about the same time, 
 
68 OF THE EARTH. 
 
 and for refraction, aberration, precession, and nutation, if they are 
 observed at different times, and taking their difference. For, let 
 O, 0' (Fig. 35) 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 dif- 
 ference 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 de- 
 note 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 sen- 
 sibly from a sphere, we may state the proportion 
 
 1 x A 
 N : A : : 1 : x ; whence x === . . . (22). 
 
 158. Degrees have been measured with the greatest possible 
 care, at various latitudes and on various meridians. Upon a com- 
 parison of the measured degrees, it appears that the length of a 
 degree increases as we proceed from the equator towards either 
 pole. It follows, therefore, (156,) that the curvature of a meridian 
 is greatest at the equator, and diminishes as we go towards the 
 poles ; and consequently, that the earth is flattened at the poles. 
 
 159. The fact of the decrease of the curvature of a terrestrial 
 meridian from the equator to the poles, leads to the supposition 
 that it is an ellipse, having its major axis in the plane of the equa- 
 tor and its minor axis coincident with the axis of the earth. Ana- 
 lytical investigations, founded on the lengths of a degree in differ- 
 ent 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 investiga- 
 tions 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 equato- 
 rial diameter, which is technically termed the Oblateness. 
 
 160. The form of the earth has also been determined by other 
 methods, which cannot here be explained. All the results, taken 
 
 together, indicate an oblateness of . 
 
 o05 
 
 The following are the dimensions of the earth in miles : 
 
 Radius at the equator 3962.6 miles. 
 
 Radius at the pole 3949.6 " 
 
 Difference of equatorial and polar radii 13.0 
 Mean radius, or at 45 latitude . . . 3956.1 " 
 
 Mean length of a degree 69.05 " 
 
 The fourth part of a meridian . . . 6214.2 " 
 
 161. Owing to the elliptical form of a terrestrial meridian, the 
 
LATITUDE AND LONGITUDE OF A PLACE. 
 
 69 
 
 radius and vertical at a place do Fig. 36. 
 
 not coincide. Let ENQS (Fig. 
 36) represent a terrestrial me- 
 ridian. For any point O situa- 
 ted on this meridian, CO will be 
 the radius, and the normal line 
 ZON the vertical. The posi- 
 tion of the vertical will always 
 be such that the apparent zenith 
 Z will lie between the true ze- 
 nith z and the elevated pole P. 
 The inclination of the radius to 
 the vertical, or the angle CON, 
 called the reduction of latitude, is greatest at the latitude 45, and 
 is there equal to about 11'. 
 
 162. The oblateness of the earth occasions some slight modifications in the 
 effects of parallax, which are in some instances to be taken into account in com- 
 puting the apparent azimuth and zenith distance of a body, from the known co- 
 ordinates of its true place. 
 
 DETERMINATION OF THE LATITUDE AND LONGITUDE OF 
 
 A PLACE. 
 
 163. The latitude and longitude of a place ascertain its situation 
 upon the earth's surface, and are essential elements in many astro- 
 nomical investigations. 
 
 164. To find the latitude of a place. 
 
 (1.) By the zenith distances or altitudes of a circumpolar star 
 at its upper and lower transits. The principle of this method has 
 already been demonstrated, (68,) and shown to be a particular case 
 of a well known principle of Fig. 37. 
 
 arithmetical proportions ; the fol- 
 lowing is a more complete proof 
 of it. Let Z (Fig. 37) represent 
 the zenith, HOR the horizon, P 
 the pole, and S, S' the points at 
 which the upper and lower tran- 
 sits of a circumpolar star take 
 place ; HP will be equal to the 
 latitude, (34,) and ZP will be^equal to the co-latitude. Now, 
 we have 
 
 HP = HS + PS, and HP = HS' PS 7 = HS' PS ; 
 
 TTS I TTS' 
 
 whence, 2HP = HS + HS', or, HP = - - ... (23). 
 
 2 
 
 In like manner we obtain 
 
 Wherefore, let the altitudes of a circumpolar star at its upper and 
 
70 OF THE EARTH. 
 
 lower transits be measured and corrected for refraction, and their 
 half sum will be the latitude ; or, let the zenitk distances be meas- 
 ured, and corrected for refraction, and their half sum subtracted 
 from 90 will be the latitude. Stars should be selected that have 
 a considerable altitude at their inferior transit, for, the greater is 
 the altitude the less is the uncertainty as to the amount of the 
 refraction. On this principle the pole star is to be preferred to all 
 others. 
 
 (2.) By a single meridian altitude or zenith distance. Let 
 5, s', s" (Fig. 10, p. 20) be the points of meridian passage of three 
 different stars, the first to the north of the zenith, the second be- 
 tween the zenith and equator, and the third to the south of the 
 equator : ZE = the latitude, and we have for the three stars, 
 
 ZE = sE Zs, ZE = s'E + Zs', ZE = Zs" s"E. 
 Thus, if the zenith distance be called north or south, according as 
 the zenith is north or south of the star when on the meridian, in 
 case the zenith distance and decimation are of the same name 
 their sum will be equal to the latitude ; but if they are of different 
 names their difference will be the latitude, of the same name with 
 the greater. 
 
 This method supposes the declination of a body 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 English Nautical Alma- 
 nac, or other similar work. If the time of the meridian transit be 
 known, the altitude may be measured by a sextant, (79). The ob- 
 served altitude must be corrected for refraction, and also for paral- 
 lax if the body observed is 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. 
 
 165. The astronomical latitude being known, the reduced lati- 
 tude (p. 19, Def. 4) may be obtained by subtracting from it the 
 reduction of latitude. For, if OC (Fig. 36) represents the radius, 
 and ON the vertical, at any place O, and ECQ represents the ter- 
 restrial equator, ONQ will be the^stronomical latitude, OCQ the 
 reduced latitude, and CON the reduction of latitude ; and we have 
 
 ONQ = OCQ + CON, and OCQ = ONQ CON . . (25). 
 (For the practical method of resolving this problem, see Prob. XV.) 
 ^ 166. 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 times, (whether 
 sidereal or solar,) at any place and on the first meridian, is the 
 longitude of the place, expressed in time ; and consequently, a/so, 
 
LONGITUDE OF A PLACE. 71 
 
 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 celestial 
 equator comprised between the meridian of Greenwich and the 
 meridian of the place. Now, it is Oh. Om. Os. of mean solar time 
 or mean noon at any place, when the mean sun (45) is on the me- 
 ridian of that particular place. Therefore, as the mean sun, mov- 
 ing in the equator, recedes from the meridian towards the west at 
 the rate of 15 per mean solar hour, when it is mean noon at a 
 place to the west of Greenwich, it will be as many hours and parts 
 of an hour past mean noon at Greenwich, as is expressed by the 
 quotient of the division of the arc of the celestial equator, or its 
 equal the longitude, by 15. If the place be to the east, instead of 
 to the west of Greenwich, when it is mean noon there it will be as 
 much before mean noon at Greenwich as is expressed by the lon- 
 gitude of the place converted into time, (as above.) In either situ- 
 ation of the place, then, the principle just stated will be true. 
 
 It is plain that the equality between the differences of the times 
 and of the longitudes will subsist equally if sidereal instead of so- 
 lar time be used. 
 
 167. To find the longitude 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 phe- 
 nomenon which is seen at the same instant at both places ; the 
 difference of the observed times will be the longitude in time. 
 These 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 satellites of 
 Jupiter and of the moon, are phenomena adapted to the purpose in 
 question. However, 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 adjust- 
 ed 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 chro- 
 nometer from any one place to another, their difference of longi- 
 tude may be obtained. The error and rate of the chronometer 
 must be determined at the outset, and as often afterwards as cir- 
 cumstances will admit, that the error at the moment of the obser- 
 vation may be known as accurately as possible. To ensure greater 
 certainty and precision in the knowledge of the time, three or four 
 chronometers are often taken, instead of one only. 
 
72 PLACES OF THE FIXED STARS. 
 
 This method is much used at sea ; the local time being obtained 
 from an observation upon the sun or some other heavenly body, in 
 a manner to be hereafter explained. 
 
 (3.) Let the Greenwich time of the occurrence of some 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. 
 Eclipses of the sun and occultations of the stars furnish the most 
 exact determinations of the longitude, but they cannot be used 
 for this purpose unless the longitude is already approximately 
 known. 
 
 The explanation, in detail, of the method of lunar distances, 
 which is chiefly used at sea, may be found in treatises on Naviga- 
 tion and Nautical Astronomy. 
 
 CHAPTER V. 
 
 OF THE PLACES OF THE FIXED STARS. 
 
 168. THE place of a fixed star in the sphere of the heavens is 
 found by ascertaining its true right ascension and declination, which 
 are the co-ordinates of its place. The process of finding the true 
 right ascension and declination of a heavenly body has already 
 been detailed : the apparent right ascension and declination are 
 found as explained in Arts. 54, 68, and to these are applied the 
 several corrections of refraction, parallax (when sensible,) and 
 aberration, (92, 120, 129.) 
 
 When right ascensions and decimations found at different times 
 are to be compared together, or employed in the same calculations, 
 as often becomes necessary, they are to be reduced to the same 
 epoch by correcting for precession and nutation, (p. 64.) 
 
 169. It is important to observe, however, that the places of the 
 fixed stars, as at present known, were not obtained by the direct 
 process just referred to, that is, by observing the right ascension 
 and declination, and applying to them at once all the corrections 
 of which we have treated. They were arrived at by successive 
 approximations. The respective corrections were applied in suc- 
 cession as they came to be discovered ; and more accurate results 
 were obtained, as, by the improvement of the instruments, the ob 
 
THE CONSTELLATIONS. 73 
 
 servations became more and more exact, and as the amount of the 
 corrections came to be known with greater and greater precision. 
 
 170. In order to distinguish the fixed stars from each other, they 
 are arranged into groups, called Constellations, which are ima- 
 gined to form the outlines of figures of men, animals, or other ob- 
 jects, 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 constella- 
 tions is of very remote antiquity ; and the names given by the an- 
 cients to individual constellations are still retained. 
 
 The resemblance of the figure of a constellation to that of the 
 animal or other object from which it is named, is in most instances 
 altogether fanciful. Still, the prominent stars hold certain definite 
 positions in the figure conceived to be drawn on the sphere of the 
 heavens. Thus, the brightest star in the constellation Leo is placed 
 in the heart of the Lion, and hence it has sometimes been called 
 Cor Leonis, or the Lion's Heart : and the brightest star in the 
 constellation Taurus is situdted in the eye of the Bull, and there- 
 fore sometimes called the Bull's Eye; while that conspicuous 
 cluster of seven stars in this constellation, known by the name of 
 the Pleiades, is placed in the neck of the figure. Again, the line 
 of three bright stars noticed by every observer of the heavens in 
 the beautiful constellation of Orion, is in the belt of the imaginary 
 figure of this bold hunter drawn in the skies. The three larger 
 stars of this constellation are, respectively, in the right shoulder, in 
 the left shoulder, and in the left foot. 
 
 171. The constellations are divided, into three classes : North- 
 ern 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 atten- 
 tively observed. 
 
 172. 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 in the heavens. The signs of the zodiac coincided with the 
 zodiacal constellations of the same name, as now defined, about the 
 year 140 B. C. Since then the equinoctial and solstitial points 
 have retrograded nearly one sign : so that now the vernal equinox, 
 or first point of the sign Aries, is near the beginning of the constel- 
 lation Pisces ; the summer solstice, or first point of Cancer, near 
 the beginning of the constellation Gemini ; the autumnal equinox, 
 or first point of Libra, at the beginning of Virgo ; and the winter 
 solstice, or first point of Capricornus, at the beginning of Sagittarius, 
 
 It follows from this, that when the sun is in the sign Aries, he 
 is in the constellation Pisces, and when in the sign Taurus, in the 
 
 10 
 
74 PLACES OF THE FIXED STARS. 
 
 constellation Aries, &c., &o. For the rest, it should be observed 
 that the constellations and signs of the zodiac have not precisely 
 the same extent. 
 
 173. 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 Roman letters, and the numbers 1, 2, 3, &c. ; 
 the characters, according to their order, denoting the relative mag- 
 nitude of the stars. Thus, a Arietis designates the largest star in 
 the constellation Aries ; (3 Draconis, the second star of the Drag- 
 on, &c. 
 
 Some of the fixed stars have particular names, as Sirius, Aide- 
 baran, Arcturus, Regulus, &c. 
 
 174. The stars are also divided into classes, or magnitudes, ac- 
 cording to the degrees of their apparent brightness. The largest 
 or brightest are said to be of the^rs^ magnitude ; 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 percepti- 
 ble to the naked eye. All of a smaller kind are called telescopic 
 stars, being invisible without the assistance of the telescope. The 
 classification according to apparent magnitude is continued with 
 the telescopic stars down to stars of the twentieth magnitude, (ac- 
 cording to Sir John Herschel,) and the twelfth according to Struve. 
 
 The following are all the stars of the first magnitude that occur 
 in the heavens, viz. Sirius, or the Dog-star, Betelgeux, Rig el, Al- 
 debaran, Capella, Procyon, Regulus, Denebola, Cor. Hydra, 
 Spica Virginis, Arcturus, Antares, Altair, Vega, Deneb or Alpha 
 Cygni, Dubhe or Alpha Ursa Majoris, Alpherat or Alpha Andro- 
 meda, Fomalhaut, Achernar, Canopus, Alpha Crucis, and Alpha 
 Centauri. It is the practice of Astronomers to mark more or less 
 of these stars as intermediate between the first and the second 
 magnitude ; and in some catalogues some of them are assigned to 
 the second magnitude. All of these stars, with the exception of 
 ,the last four, come above the horizon in all parts of the United States. 
 
 175. There are two principal modes of representing the stars ; 
 the one by delineating them on a globe, where each star occupies 
 the spot in which it would appear to an eye placed in the centre 
 of the globe, and where the situations are reversed when we look 
 down upon them ; the other is by a chart or map, where the stars 
 are generally so arranged as to be represented in positions similar 
 to their natural ones, or as they would appear on the internal con- 
 cave surface of the globe.* The construction of a globe or chart 
 is effected by means of the right ascensions and declinations of the 
 stars. Two points diametrically opposite to each other on the 
 surface of an artificial globe are taken to represent the poles of the 
 heavens, and a circle traced 90 distant from these for the equator : 
 another point 23 from one of the poles is then fixed upon for one 
 
 * Encyclopedia Metropolitana, Art. Astronomy, p. 505. 
 
RIGHT ASCENSION AND DECLINATION. 7$ 
 
 of the poles of the ecliptic, and with this point as a geometrical 
 pole a great circle described ; the points of intersection of the 
 two circles will represent the equinoctial points. The point which 
 represents the place of a star is found by marking off the right as- 
 cension and decimation 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 1 2 or 
 18 inches in diameter. 
 
 176. The places of the fixed stars are generally expressed by 
 their right ascensions and declinations, but sometimes also by 
 their longitudes and latitudes. A table containing a list of fixed 
 stars designated by their proper characters, and giving their mean 
 right ascensions and declinations, or their mean longitudes and lati- 
 tudes, is called a Catalogue of those stars.* 
 
 Table XC. is a catalogue of fifty principal fixed stars, and gives 
 their mean right ascensions and declinations for the beginning of 
 the year 1840, as well as their annual variations in right ascension 
 and declination. The annual variations serve to extend the use of 
 the catalogue about 10 years (150) before and after the epoch for 
 which it is constructed. (See Prob. XVIII.) Every ten years, or 
 thereabouts, a new catalogue must be formed. 
 
 177. If the true right ascension and declination of a star at a given time be re- 
 quired, correct the mean right ascension and declination found by the catalogue, 
 for nutation. (See Art. 148.) And if the apparent right ascension and declination 
 be required, correct also for aberration. (See Art. 129.) 
 
 178. The latitude and longitude of a fixed star or other heavenly 
 body are obtained originally by computation from its right ascen- 
 sion and declination. 
 
 To convert the right ascension and declination of a body into 
 its longitude and latitude. Let EQ (Fig. 38) represent the equa- 
 tor, EC the ecliptic,?, K the poles of 
 the equator and ecliptic, E the vernal 
 equinox, PSR a circle of declination 
 and KSL a circle of latitude, both 
 passing through a body S. The 
 right ascension of the body is ER = 
 R ; the declination RS = D ; the 
 longitude EL = L ; and the latitude 
 LS = X. REL = w is the obliqui- 
 ty of the ecliptic, which is one of 
 the essential data of the problem. 
 
 * Various catalogues have at different periods been published. The first was be- 
 gun by Hipparchus, 120 years before the Christian era. Of the modern catalogues, 
 the following may be cited as among the most accurate, although not the most 
 extensive, viz. the Catalogues of Flamstead, Lacaille, Bradley, Maskelyue, Piazzi, 
 and of the Royal Astronomical Society, and of the British Association. 
 
 The Nautical Almanac contains a Catalogue of 100 principal fixed stars, of 
 which 54. are designated as Standard Stars that is, stars whose places are sup- 
 posed to be known with all attainable precision. The largest single catalogue evej 
 published is the Histoire Celeste of Lalande, which gives the places of 50,OQO stars 
 
76 
 
 PLACES OF THE FIXED STARS. 
 
 RES = x and LES = y are employed as auxiliary angles. In the 
 right-angled spherical triangle LES we have by Napier's rules for 
 the solution of right-angled triangles, (see Appendix,) 
 
 sin (co. LES) = tang EL tang (co. ES) ; 
 whence, 
 
 tan EL = cos LES tan ES, or, tan L = cos (RES w) tan ES ; 
 but 
 
 sin (co. RES) =tan ER tan (co. ES,) or, tan ES = 
 
 COS 
 
 thus, 
 
 T /Tj-nci \ tan ER cos (x w) tan R , ^ 
 
 tan L = cos (RES w) -%=^ = - - . . (26): 
 
 cos RES cos x 
 
 and to find a?, we have 
 
 sin ER = tan (co. RES) tan RS, or, cot x = sin R cot D . . (27.) 
 
 Again, 
 
 sin EL = tan (co. LES) tan LS, or tan LS = tan LES sin EL, 
 
 which gives 
 
 tang X = tang (x w) sin L . . . (28.) 
 
 Equation (27) makes known the value of x, with which we de- 
 rive the values of L and X by means of equations (26) and (28.) 
 In resolving the equations attention must be paid to the signs of 
 the quantities, which are determined according to the usual trigo- 
 nometrical rules, it being understood that the declination D is to 
 be regarded as negative when it is south, x is to be taken always 
 less than 180, and greater or less than 90 according as its cotan- 
 gent is negative or positive. L will always be in the same quad- 
 rant with R. The latitude X will be north or south according as 
 tang X comes out positive or negative. 
 
 The apparent or mean obliquity is used, according as the case 
 refers to true or mean co-ordinates. (For exemplifications of this 
 problem see Prob. XXIV.) 
 
 179. It is now frequently necessary to resolve the converse problem, that is, to 
 convert the longitude and latitude of a body into its right ascension and decli- 
 nation. 
 The triangle RES (Fig. 38) gives 
 
 sin (co. RES) = tang ER tang (co. ES) ; 
 whence, 
 
 tan ER = cos RES tan ES, or, tan R = cos (LES -j- ) tan ES ; 
 but 
 
 sin (co. LES) = tang EL tang (co. ES), or tan ES = 
 thus, 
 
 tang R = eis (LES + ) *Sft = <* (y + ") tang L _ , _ _ 2 
 
 ' cos LES cos y 
 
 and to find y, we have 
 
 sin EL = tang (co. LES) tang LS, or cot y = sin L cot X . . (30). 
 For the declination, we have 
 
 sin ER = tan (co. RES) tan RS, or, tan RS = tan RES sin ER ; 
 r, 
 
 tang D = tang (y + w) sin R . . . (31.) 
 
OBLIQUITY OF THE ECLIPTIC. 77 
 
 The value of y being derived from equation (30) and substituted in equations 
 (29) and (31), these equations will then make known the values of R and D. The 
 signs of the quantities are determined by the usual trigonometrical rules, the lati- 
 tude A being taken negative when south, y is always less than 180, and greater 
 or less than 90 according as its cotangent comes out negative or positive. R will 
 be in the same quadrant as L. The declination will be north or south according 
 as its tangent comes out positive or negative. (For exemplifications of this prob- 
 lem see Prob. XXV.) 
 
 180. Table XCII. contains the mean longitudes and latitudes 
 of some of the principal fixed stars for the beginning of the year 
 1840, together with their annual variations, which serve to make 
 known the mean longitudes and latitudes at any other epoch. (See 
 Prob. XVIII.) 
 
 181. The fixed stars, so called, are not all of them, rigorously 
 speaking, fixed or stationary in the heavens. It has been discov- 
 ered that many of them have a very slow motion from year to year. 
 These motions of the stars are called their Proper Motions. The 
 annual variations in right ascension and declination, and in longi- 
 tude and latitude, given in Tables XC. and XCII., are the varia- 
 tions due both to the precession of the equinoxes and the proper 
 motions of the stars. 
 
 CHAPTER VI. 
 
 OF THE APPARENT MOTION OF THE SUN IN THE HEAVENS. 
 
 182. THE sun's declination, and the difference of right ascension 
 of the sun and some fixed star, found from day to day throughout 
 a revolution, are the elements from which the circumstances of the 
 sun's apparent motion are derived. 
 
 The motion of the sun, as at present known, has been arrived at 
 in the same approximative manner as the places of the fixed stars, 
 (169.) It would, in fact, be theoretically impossible to correct the 
 co-ordinates of the sun's apparent place for precession, nutation, 
 and aberration, in the original determination of the sun's motion ; 
 for, the knowledge of these corrections presupposes some know- 
 ledge of the motion of the sun. 
 
 183. The curve on the sphere of the heavens passing through 
 the successive positions determined as above from day to day, 1 is 
 the ecliptic. If we suppose it to be a circle, as it appears to be, 
 its position will result from the position of the equinoctial points 
 and its obliquity to the equator. 
 
 184. To find the obliquity of the ecliptic. Let EQA (Fig. 39) 
 represent the equator, EGA the ecliptic, and OC, OQ lines drawn 
 through the centre of the earth and perpendicular to AGE the 
 
78 
 
 APPARENT MOTION OF THE SUN. 
 
 Fig. 39. 
 
 line of the equinoxes ; then the angle COQ will be the obliquity 
 of the ecliptic. This angle has for its measure the arc CQ, and 
 
 therefore the obliquity of the eclip- 
 tic is equal to the greatest decli- 
 nation of the sun. It can but 
 rarely happen that the time of the 
 greatest declination will coincide 
 with the instant of noon at the 
 place where the observations are 
 made, but it must fall within at 
 least twelve hours of the noon for 
 which the observed declination 
 is the greatest. In this interval 
 the change of declination cannot 
 exceed 4", and therefore the greatest observed declination cannot 
 differ more than 4" from the obliquity. A formula has been in- 
 vestigated, which gives in terms ol determinable quantities the 
 difference between any of the greater declinations and the maxi- 
 mum declination. By reducing by means of this formula a num- 
 ber of the greater declinations to the maximum declination, and 
 taking the mean of the individual results, a very accurate value of 
 the obliquity may be found. 
 
 185. To find the position of the vernal or autumnal equinox. 
 (1.) On inspecting the observed declinations of the sun, it is seen 
 that about the 21st of March the declination changes in the inter- 
 val of two successive noons from south to north. The vernal 
 
 Fig. 40. 
 
 S:RS 
 
 equinox occurs at some moment 
 of this interval. Let RS, R'S' 
 (Fig. 40) represent the declinations 
 at the noons between which the 
 equinox occurs : as one is north 
 and the other south, their sum (S) 
 will be the daily change of declina- 
 tion at the time of the equinox. 
 Denote the time from noon to noon 
 by T. Now, to find the interval 
 (x) between the noon preceding 
 the equinox and the instant of the 
 equinox, state the proportion 
 T . _TxRS 
 
 1 * s 
 
 on the principle that the declination changes for a day or more pro- 
 portionally to the time. Next, take the daily change in right 
 ascension (RR') on the day of the equinox and compute the value 
 of RE, by the proportion 
 
 rp v -po 
 
 T : x, or A : : RR' : RE ; 
 
POSITION OP THE EQUINOX. 79 
 
 Bdd RE to MR, the observed difference of right ascension 
 (182) on the day preceding the equinox, and the sum ME will be 
 the distance of the equinox from the meridian of the star observed 
 in connection with the sun.* 
 
 The position of the autumnal equinox may be found by a simi- 
 lar process, the only difference in the circumstances being that the 
 declination changes from north to south instead of from south to 
 north. 
 
 If the value of x which results from the first proportion be add- 
 ed to the time of noon on the day preceding the equinox, the result 
 will be the time of the equinox. 
 
 (2.) In the triangle RES (Fig. 39) we have the angle RES = u 
 the obliquity of the ecliptic, and RS = D the declination of the 
 sun, both of which we may suppose to be known, and we have by 
 Napier's first rule, 
 
 sin ER = tang (co. RES)tangRS = cot wtang D . . (32 ;) 
 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 posi- 
 tions S, S', S", &c., 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. 
 
 ME 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. 
 
 186. 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 M E deduced from obser- 
 vations upon the sun in different positions S, S', S", &c., estab- 
 lishes the truth of this hypothesis. It is also confirmed by the 
 fact, that the right ascensions of the vernal and autumnal equinox 
 differ by 1 80, since we may infer from this that the line of the 
 equinoxes passes through the centre of the earth. 
 
 187. The mean obliquity of the ecliptic is derived from the apparent obliquity, as 
 well as the mean equinox from the true equinox, by correcting for nutation. 
 
 188. The mean obliquity at any one epoch having been found, its value at any 
 assumed time may be deduced from this by allowing for the annual diminution of 
 0".46, (see Table XXII.) In like manner, the place of the mean equinox^at any 
 given time may be derived from its place once found, by allowing for the annual 
 precession of 50".23. 
 
 The mean obliquity having thus been found for any assumed time, the apparent 
 obliquity at the same time becomes known, by applying the nutation of obliquity. 
 (See Prob. X.) 
 
 189. 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. 39,) ES(=L) represents the longi- 
 
 The star is here supposed to be to the west of the sun. 
 
80 APPARENT MOTION OF THE SUN. 
 
 tilde of the sun supposed to be at S, ER (=R) its right ascension 
 and RS (= D) its declination. Now, by Napier's first rule, 
 
 thus, 
 
 cotL==coswcotR,ortanffL = ^ . . (33). 
 
 cos w 
 
 Also, (Napier's second rule, Appendix,) 
 
 sin RS = cos (co. RES) cos (co. ES); whence, sin ES = 
 
 or, 
 
 . , sinD , . 
 
 smL= . . . (34). 
 sin w 
 
 With these formulae the longitude of the sun may be computed 
 from either its right ascension or declination. (See Prob. XII.) 
 
 Formulae (33) and (34) may be written thus, 
 
 tang R = tang L cos w ; sin D = sin L sin w . . . (35). 
 
 These formulae will make known the right ascension and decli- 
 nation of the sun, when his longitude is given. (See Prob. XL) 
 It will be seen in the sequel that in the present advanced state 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. 
 
 190. The interval between two successive returns of the sun to 
 the same equinox, or to the same longitude, is called a Tropical 
 Year. 
 
 And the interval between two successive returns of the sun to 
 the same position with respect to the fixed stars, is called a Side- 
 real Year. 
 
 191. 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 tropical 
 year is obviously very nearly equal to the mean length of the true 
 tropical year in an interval during which it 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. 
 
 Theory shows that the variation in the length of the tropical year arises from 
 the periodical inequality in the precession of the equinoxes which results from nu- 
 tation, and certain periodical inequalities in the sun's yearly rate of motion ; and 
 thus establishes also, that the mean tropical year, as above defined, is the same as 
 the interval between two successive returns of the sun, supposed to have its mean 
 motion, to the same mean equinox. 
 
 According to the most accurate determinations, the length of the 
 mean tropical year, expressed in mean solar time, is 365d. 5h. 
 48m. 47.58s., (48s. nearly.) 
 
SUN'S DAILY MOTION IN LONGITUDE. 81 
 
 192. In a mean tropical year the sun's mean motion in longi- 
 tude is 360 ; hence, to find his mean daily motion in longitude 
 we have only to state the proportion 
 
 365d. 5h. 48m. 48s. : Id. : : 360 : x = 59' 8".33. 
 
 193. The sidereal year is longer than the tropical. For since 
 the equinox has a retrograde motion of 50". 23 in a year, when the 
 sun has returned to the equinox it will not have accomplished a si- 
 dereal revolution, into 50". 23. The excess of the sidereal over 
 the tropical year results from the proportion 
 
 59' 8".3 : 50".23 : : Id. : x = 20m. 23.1s. 
 
 Thus the length of the mean sidereal year, expressed in mean 
 solar time, is 365d. 6h. 9m. 11s. 
 
 194. If from the right ascensions and declinations of the sun, 
 found on two successive days, the corresponding longitudes be de- 
 duced (equas. 33, 34) and their difference taken, the result will be 
 the sun's daily motion in longitude at the time of the observations. 
 The sun's daily motion in longitude is not the same throughout 
 the year, but, on the contrary, is continually varying. It gradually 
 increases during one half of a revolution, and gradually decreases 
 during the other half, and at the end of the year has -recovered its 
 original value. Thus, the greatest and least daily motions occur 
 at opposite points of the ecliptic. They are, respectively, 61' 10" 
 and 57' 11". 
 
 195. The exact law of the sun's unequable motion can only be 
 obtained 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 mo- 
 tion 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 ap- 
 parent 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. Ac- 
 cordingly, the sun is nearest to us when his daily angular motion 
 is the most rapid, and farthest from us when his daily motion is 
 the slowest. The greatest apparent diameter of the sun is 32' 
 36' ; and the least apparent diameter 31' 31". 
 
 11 
 
MOTIONS OF THE PLANETS IN SPACE. 
 
 CHAPTER VII. 
 
 OP THE MOTIONS OF THE SUN, MOON, AND PLANETS, IN 
 THEIR ORBITS. 
 
 KEPLER'S LAWS. 
 
 196. THE celebrated astronomer Kepler, who flourished early 
 in the seventeenth century, by examining the observations upon 
 the planets that had been made by the renowned Danish observer, 
 Tycho Brahe, discovered that the motions of these bodies, and of 
 the earth, were in conformity with the following laws : 
 
 (1.) The areas described by the radius-vector of a planet [or 
 the line drawn from the sun to the planet] are proportional to the 
 times. 
 
 (2.) The orbit of a planet is an ellipse, of which the sun occu- 
 pies one of the foci. 
 
 (3.) The squares of the times 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 mo- 
 tions. They have since been completely verified by other astron- 
 omers. We shall adopt the first two laws for the present as hy- 
 potheses, 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 times of revolution. 
 
 197. 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 
 
 Fig. 41. two facts, let EE'A (Fig. 
 
 41) represent the elliptic or- 
 bit of the earth, and S the 
 position of the sun in space. 
 If the earth move from E to 
 , any point E', as it seems to 
 remain stationary at E, it is 
 plain that the sun will ap- 
 pear to move from S to a 
 position S', on the line ES' 
 drawn parallel to E'S the 
 actual direction of the sun 
 from the earth, and at a dis- 
 
LAW OF THE ANGULAR MOTION OF A PLANET. $3 
 
 tance ES' equal to E'S the actual distance of the sun from the earth. 
 Thus, for every position of the earth in its orbit, the corresponding 
 apparent position of the sun is obtained by drawing a line parallel to 
 the radius-vector of the earth, and equal to it. It follows, therefore, 
 that the area SES' apparently described by the radius-vector of 
 the sun (or the 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 EE' actually described in the same time by the earth. 
 Whence we conclude, that the apparent motion of the sun in space, 
 and the actual motion of the earth, are the same in every particular. 
 
 198, It has been discovered that the motion of the moon in its 
 revolution around the earth, is subject to the same laws as the mo- 
 tion of a planet in its revolution around the sun. We shall assume 
 this to be a fact, and show that our hypothesis is verified by the 
 results to which it leads. 
 
 1 99, 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, respective 
 ly, 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 el- 
 lipses, the perigee and apogee, or the perihelion and aphelion, are 
 the extremities of the major axis of the orbit, 
 
 200, The law of the angular motion of a planet about the sun 
 may be deduced from Kepler's Fig. 42. 
 
 first law. Let PpAp" (Fig, 42) 
 represent the orbit of a planet, con- 
 sidered as an ellipse, and p, p 1 two 
 positions of the planet at two in- 
 stants separated by a short interval 
 of fime ; and let n be the middle 
 point of the arc pp 1 . With the ra- 
 dius Sn describe the small circular 
 arc Inl', and with the radius Sb 
 equal to unity describe the arc ab. 
 It is plain that the two positions p,p' 
 may be taken so near to each other, that the area Spp' will be 
 sensibly equal to the circular sector Sll'. If we suppose this to 
 be the case, as the measure of the sector is \lriP x Sn = \ab X 
 Sn 2 , (substituting for Inl 1 its value ab x Sn,) we shall have 
 
 area Spp f = db x Sn 2 . 
 When the planet is at any other part of its orbit, as n', if 
 
84 MOTIONS OF THE PLANETS IN SPACE. 
 
 Sp"p" be an area described in the same time as before, we shal 
 have 
 
 area Sp"p'" = a!V x 
 But these areas are equal according to Kepler's first law : hence, 
 
 ab x Sn* = \a!V xJJM 2 ^. . . (36) ; 
 and ab : a'b' : : Sri* : Sn 2 > 
 
 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 ra- 
 dius-vector. 
 
 It results from this that the angular motion is greatest at the pe- 
 rihelion, and least at the aphelion, and the same at corresponding 
 points on either side of the major axis : also, that it decreases pro- 
 gressively from the perihelion to the aphelion, and increases pro- 
 gressively from the aphelion to the perihelion. 
 
 201. 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 planet, and to set out at the same instant 
 Fig. 43. with it from the perihelion. Let 
 
 PMAM' (Fig. 43) represent the 
 elliptic orbit of the planet, and 
 PBaB the circle described by the 
 body. The position B of this fic- 
 titious body at auy time will be the 
 mean place of the planet as seen 
 from the sun. The two bodies 
 will accomplish a semi-revolution 
 in the same period of time, and 
 therefore be, respectively, at A and 
 a at the same instant ; for it is ob- 
 vious 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 revolu- 
 tion. At the outset, the motion of the planet is the most rapid, 
 (200,) 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 pSE from the body will increase until its mo- 
 tion becomes reduced to an equality with the mean motion, 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 pe- 
 rihelion, the angular velocity of the planet will at first be less than 
 that of the body, (200,) but it will continually increase, while 
 that of the body will remain unaltered : thus, the body will now 
 get in advance of the planet, and their angular distance p'SB' will 
 increase, as before, until the motion of the planet again attains to 
 an equality with the mean motion, after which it will decrease, as 
 before, until it again becomes zero at the perihelion. 
 
DEFINITIONS OF TERMS. 85 
 
 It appears, then, that from the perihelion to the aphelion the 
 true place is in advance of the mean place, and that from the aphe- 
 lion to the perihelion, on the contrary, the mean place is in ad- 
 vance 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 Equa- 
 tion of the Centre. Thus, pSE is the equation of the centre cor- 
 responding to the particular position p 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. 
 
 202. As the laws of the motion of the moon (198) and of the 
 apparent motion of the sun (197) are the same as those of a planet, 
 the principles established in the two preceding articles are as ap- 
 plicable to these bodies in their revolution around the earth, as to 
 a planet in its revolution around the sun. 
 
 DEFINITIONS OF TERMS. 
 
 203. (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 he- 
 liocentric place. 
 
 (4.) Two heavenly bodies are said to be in Conjunction when 
 their longitudes are the same, and to be in Opposition when their 
 longitudes differ by 180. When any one heavenly body is in 
 conjunction with the sun, it is, for the sake of brevity, said to be 
 in Conjunction ; and when it is in opposition to the sun, to be in 
 Opposition. 
 
 The planets Mercury and Venus, allowing that their distances 
 from the sun are each less than the earth's distance (23), can never 
 be in opposition. But they may be in conjunction, either by being 
 between the sun and earth, or by being on the opposite side of the 
 sun. In the former situation they are said to be in Inferior Con- 
 junction, 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 conjunctions. 
 
 (6.) The Periodic Time of a planet is the period of time in 
 which it accomplishes a revolution around the sun. 
 
 (7.) The Nodes of a planet's orbit, or of the moon's orbit, are 
 tne points in which the orbit cuts the plane of the ecliptic. The 
 
86 
 
 MOTIONS OF THE PLANETS IN SPACE. 
 
 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 &. The other is called the Descending Node, and 
 is marked t3- 
 
 (8.) The Eccentricity of an elliptic orbit is the ratio which the 
 distance between the centre of the orbit and either focus bears to 
 the semi-major axis. 
 
 Fig. 44 
 
 204. To illustrate these definitions, let EE'E" (Fig. 44) repre- 
 sent the orbit of the earth ; C'DC 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 ; LNP a part of the or- 
 bit of Mars, or of any other planet more distant from the sun S 
 than the earth is ; and ANB a part of the projection of this orbit 
 on the plane of the ecliptic : N or & will represent the ascending 
 node of the orbit ; and the descending node will be diametrically 
 opposite to this in the direction Sn'. Also let SV be the direction 
 of the vernal equinox, as seen from the sun, and E V, E' V the par- 
 allel directions of the same point, as seen from the earth in the two 
 positions E and E' ; and P being supposed to be one position of 
 MarS in his orbit, let p be the projection of that position on the 
 plane of the ecliptic. The heliocentric longitude and latitude of 
 Mars in the position P, are respectively VSp and PSp ; and if the 
 earth be at E, his geocentric longitude and latitude are respec- 
 tively VEp and PEp. If we suppose that when Mars is at P the 
 
ELEMENTS OF THE ORBIT OF A PLANET. 87 
 
 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 con- 
 junction; 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 superior in the sense of higher in place, or farther frort 
 the earth. Since the earth and planets are continually in motion, 
 it is manifest that the positions of conjunction and opposition wiL 
 recur at different parts of the orbit, and in process of time in every 
 variety of position. The time employed by a planet in passing 
 around from one position of conjunction, or opposition, to another, 
 called the synodic revolution, is, for the same reason, longer than 
 the periodic time, or time of passing around from one point of the 
 orbit to the same again. 
 
 if 
 ELEMENTS OF THE ORBIT OF A PLANET. 
 
 205. To have a complete knowledge of the motions of the plan- 
 ets, 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 ad- 
 dition to the laws of its motion discovered by Kepler, the position 
 and dimensions of its orbit, its mean motion, and its place at a spe- 
 cified epoch. These necessary particulars of information are sub- 
 divided 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 planet being at its perihelion, or instead, 
 its mean longitude at a given epoch. 
 
 (7.) The periodic time of the planet. 
 
 The first two ascertain the position of the plane of the planet's 
 orbit ; the third and fourth, the dimensions of the orbit ; the fifth, 
 the position of the orbit in its plane ; the sixth, the place of the 
 planet at a given epoch; and the seventh, its mean rate of motion. 
 
 206. The elements of the earth's orbit, or of the sun's apparent 
 orbit, are but five in number ; the first two of the above -mentioned 
 elements being wanting, as the plane of the orbit is coincident with 
 the plane of the ecliptic. 
 
 207. 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 geo- 
 centric longitude of the perigee arid the geocentric longitude of the 
 
88 MOTIONS OF THE PLANETS IN SPACE. 
 
 node of the moon's orbit answer, respectively, to the heliocentric 
 longitude of the perihelion and the heliocentric longitude of the 
 node of a planet's orbit. 
 
 208. The linear unit adopted, in terms of which the semi-major 
 axes, eccentricities, and radii-vectores of the planetary orbits, are 
 expressed, is the mean distance of the sun from the earth, or the 
 semi-major axis of the earth's orbit. When thus expressed, these 
 lines are readily obtained in known measures whenever the mean 
 distance of the sun becomes known. The lines of the moon's 
 orbit are found in terms of the moon's mean distance from the 
 earth, as unity. 
 
 METHODS OF DETERMINING THE ELEMENTS OF THE SUN'S 
 APPARENT ORBIT, OR OF THE EARTH'S REAL ORBIT. 
 
 MEAN MOTION. 
 
 209. The sun's mean daily motion in longitude results from the 
 length of the mean tropical year obtained from observation, (192.) 
 
 SEMI-MAJOR AXIS. 
 
 210. 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 com- 
 puted from the mean horizontal parallax of the sun. 
 
 211. The horizontal parallax of a body being given, to find its 
 distance from the earth. We have (equation 7, p. 51) 
 
 n R 
 
 ~ sin H ' 
 
 where H represents the horizontal parallax of the body, D its dis- 
 tance from the centre of the earth, and R the radius of the earth. 
 The parallax of all the heavenly bodies, with the exception of the 
 moon, is so small, that it may, without material error, be taken in 
 this equation in place of its sine. Thus, 
 
 Again, since 6.2831853 is the length of the circumference of a 
 circle of which the radius is 1, and 1296000 is the number of 
 seconds in the circumference, we have 6.2831853 : 1 : : 1296000" : 
 x = 206264" .8 = the length of the radius (1) expressed in seconds. 
 Hence, if the value of H be expressed in seconds, 
 2068 
 
 212. In the determination of the sun's parallax, by the process 
 of Arts. 114 and 115, 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 (38) from his parallax 
 found in that manner, may be in error by an amount equal to one- 
 
ECCENTRICITY OF THE SUN's APPARENT ORBIT. 89 
 
 fourth or more of the true distance. There is a much more ac- 
 curate method of obtaining the sun's parallax, which will be no- 
 ticed hereafter. It has been found by the method to which we 
 allude, that the horizontal parallax of the sun at the mean distance 
 is 8".58, which may be relied upon as exact to within a small 
 fraction of a second. We have, then, for the sun's mean distance, 
 or the mean semi-major axis of his orbit, 
 
 D - R .' = 24040.19 R = 95,102,992 miles ; 
 
 o .5o 
 
 taking for R the mean radius of the earth = 3956 miles. 
 
 ECCENTRICITY. 
 
 213. First method. By the greatest and least daily motions 
 in longitude. Wo have already explained (194) the mode of de- 
 riving from observation the sun's motion in longitude from day to 
 day. Now, let v = the greatest daily motion in longitude ; v' = 
 the least daily motion in longitude ; r the least or perigean dis 
 tance of the sun ; and r' the greatest or apogean distance ; and we 
 shall have, by the principle of Art. 200, 
 
 r : r' : : V v' : ^ v ; 
 whence, r' + r : r' r : : ^ v -f- ^ v' : ^ v */ v', 
 
 r' + r W+VI 7 " , / - 7 
 
 or, - : r' r : : -- : v v v v' : 
 
 2 2 
 
 but, 
 
 r' + r 
 
 - = semi-major axis = 1 ; and r' r = 2 (eccentricity) = 2 e; 
 
 \/ i) _i_ \/ <t\i _ _ 
 thus, 1 : 2e : : -- - : V.v *J v f , 
 
 v 7> - "*/ 1) f 
 
 and e = L - = . . . (39). 
 
 v v + vV 
 
 The greatest and least daily motions are, respectively, (at a 
 mean,) 61 '.165 and 57'. 192. Substituting, we have 
 e = 0.016791. 
 
 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 in- 
 versely proportional to his corresponding apparent diameters, (195.) 
 
 214. Second method. By the greatest equation of the centre. 
 
 (1.) To find the greatest equation of the centre. Let L = the true longitude, 
 and M = the mean longitude, at the time the true and mean motions are equal 
 between the perigee and apogee, (201) ; L' = the true longitude and M' = the mean 
 longitude, when the motions are equal between the apogee and perigee ; and E 
 the greatest equation of the centre. Then (201) 
 
 L = M + E, and L' = M' E ; 
 whence, L' - L = M' - M - 2E, 
 
 12 
 
90 MOTIONS OP THE PLANETS IN SPACE. 
 
 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 
 wemay therefore, with but slight error, tako 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 interval, by 59' 8".330, the mean daily mo- 
 tion in longitude. 
 
 For example : from observations upon the sun, made by Dr. Maskelyne, in the 
 year 1775, it is ascertained in the manner just explained that the sun was near its 
 greatest equation at noon, or at Oh. 3m. 35s. mean solar time, on the 2d April, and 
 at noon on the 31st, or at 23h. 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 between the two epochs is 182d. 
 ftm. 
 
 Mean motion in 182d. 14m. . . . 179 22' 41".56 
 Difference of two longitudes .... 175 32 5 .39 
 
 Difference 2 ) 3 50 36 .17 
 
 Greatest equation of centre .... 1 55 18 .08 
 
 More accurate results are obtained by reducing observations made during seve- 
 ral days before and after the epoch of the greatest equation, and taking the mean 
 of the different values of the greatest equation thus obtained. According to M 
 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 : 
 
 _JK 11 K 3 587 K 5 
 " 2 3^ 3.5.216 
 
 Tjl 
 
 in which K stands for the expression (E being the greatest equation 
 
 57 . 
 
 of the centre.) In the case of the sun's orbit, K being a small fraction, all its 
 powers beyond the first may be omitted. Thus, retaining only the first term of the 
 series, and taking E = 1<> 55' 3l".66 the greatest equation in 1775, we have 
 
 K IP 55' 31".66 
 
 6 = 2- = 2X570.2957795 " ' Ol1 ' 8 3 ' 
 
 215. It appears from the law of the angular velocity of a re- 
 volving body, investigated in Art. 200, that the amount of the pro- 
 portional 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 eccentri- 
 city of the orbit, (Def. 8, p. 86.) 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, (201,) 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, then, will become determinate : 
 and formulae have been investigated which make known either one 
 
PERIGEE OP THE SUN*S APPARENT ORBIT. 91 
 
 when the other is given. Equation 41 is the formula for the ec- 
 centricity. 
 
 216. 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, called the secular 
 diminution, is 17". 2. 
 
 LONGITUDE AND EPOCH OF THE PERIGEE, j 
 
 217. As the sun's angular velocity is the greatest at tjie perigee, 
 the longitude of the sun at the time its angular velocity (is greatest, 
 will be the longitude of the perigee. The time of Ae greatest 
 angular velocity may easily be obtained within a fewihours, by 
 means of the daily motions in longitude, derived from observation. 
 
 218. The more 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 orbits whose longitudes 
 differ by 180, in passing from one to the other of which the sun 
 employs just half a year. This principle may be inferred from 
 Kepler'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 
 divides the ellipse into equal parts, and by the law in question 
 equal areas correspond to equal times. 
 
 219. By a comparison of the results of observations made atdis 
 tant epochs, it is discovered that the longitude of the perigee is 
 continually increasing at a mean rate of 61 ".5 per year. As the 
 equinox retrogrades 50". 2 in a year, the perigee must then have a 
 direct motion in space of 11". 3 per year. 
 
 It will be seen, therefore, that 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 inter 
 val of time employed by the sun in moving through an arc of 5". 6, 
 the sidereal motion of the apogee and perigee in half a year. 
 
 220. According to the most exact determinations, the mean Ion 
 gitude of the perigee of the sun's orbit at the beginning of the yeai 
 1800, was 279 30' 8".39 : it is now 280^. 
 
 221. The 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. 41, p. 82,) 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 abc 1 80 = geocentric longitude of the sun's perigee 
 180.* 
 
 * It is plain that the same relation subsists between the heliocentric longitude 
 of the earth and the geocentric longitude of the sun in every other position of the 
 earth in its orbit ; or that each point of the earth's orbit is diametrically opposite tc 
 the corresponding point of the sun's apparent orbit. 
 
92 MOTIONS OF THE PLANETS IN SPACE. 
 
 222. The epoch and mean longitude of the perigee of the sun's 
 orbit being once found, the sun's mean longitude at any assumed 
 epoch is easily obtained by means of the mean motion in longitude. 
 
 METHODS OF DETERMINING THE ELEMENTS OF THE MOON'S 
 
 ORBIT. 
 
 LONGITUDE OF THE NODE. 
 
 223. 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 ; and this may be deduced from the lon- 
 gitudes and latitudes of the moon, derived from observed right as- 
 censions and declinations (69), by methods precisely analogous to 
 those by which the right ascension of the sun, at the time its decli- 
 nation is zero, and it is passing from the south to the north of the 
 equator, or the position of the vernal equinox, is ascertained, (185.) 
 
 INCLINATION OF THE ORBIT. 
 
 224. Among the latitudes computed from the moon's observed 
 eight 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. 
 
 225. 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 f In order to obtain the mean tropical revolution, we 
 must compare two longitudes found at distant epochs. Their dif- 
 ference, augmented by the product of 360 by the number of rev- 
 olutions performed in the interval of the epochs, will be the mean 
 motion in longitude in the interval, from which the mean motion in 
 100 years or 36525 days, called the Secular motion, may be ob- 
 tained 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 mo- 
 tions is known with more or less accuracy. 
 
 226. The longitude of the moon, at certain epochs, maybe very 
 conveniently deduced from observations upon lunar eclipses. For, 
 
MOON'S MEAN MOTION IN LONGITUDE. 93 
 
 the time of the middle of the eclipse is very near the time of oppo- 
 sition, when the longitude of the moon differs 180 from that of the 
 sun, and the longitude of the sun results from the known theory 
 of its motion. The recorded observations of the ancients upon the 
 times of the occurrence of eclipses, are the only observations that 
 can now be made use of for the direct determination of the longi- 
 tude of the moon at an ancient epoch. 
 
 227. The mean tropical revolution of the moon is found to be 
 
 27.321 582d. or 27d. 7h. 43m. 4.7s. (5s. nearly.) 
 Hence, 27.321582d. : Id. : : 360 : 13M7639. = 13 10' 35".0 = 
 moon's mean daily motion in longitude. 
 
 228. Since the equinox has a retrograde motion, the sidereal 
 revolution of the moon must exceed the tropical revolution, as the 
 sidereal year exceeds the tropical year. The excess will be equal 
 to the time employed by the moon in describing the arc of preces- 
 sion answering to a revolution of the moon. Thus, 
 
 365.25d. : 50".2 : : 27.3d. : 3". 75 = arc of precession, 
 and 13. 17 : Id. : : 3".75 : 6.8s. = excess. 
 
 Wherefore, the mean sidereal revolution of the moon is 27d. 7h. 
 43m. 12s. 
 
 229. It has been found, by determining the moon's mean rate of motion for pe- 
 riods of various lengths, that it is subject to a continual slow acceleration. This 
 acceleration will not, however, be indefinitely progressive : Laplace has investiga- 
 ted its physical cause, and shown 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. 227 answers to the commencement of the pres- 
 ent century. 
 
 LONGITUDE OF THE PERIGEE, ECCENTRICITY, AND SEMI-MAJOR AXIS. 
 
 230. The methods of determining these elements of the moon's 
 orbit are similar to those by which the Fig. 45. 
 
 corresponding elements of the sun's 
 orbit are found. 
 
 It is to be observed, however, that for the 
 longitudes of the sun, which are laid off in the 
 plane of the ecliptic, in the case of the moon cor- 
 responding angles are laid off in the plane of its 
 orbit. These angles are reckoried from a line 
 drawn making an angle with the line of nodes 
 equal to the longitude of the ascending node, and 
 are called Orbit Longitudes. The orbit longi- 
 tude is equal to the moon's angular distance 
 from the ascending node plus the longitude of 
 the ascending node. "Thus, let VNC (Fig. 45) 
 represent the plane of the ecliptic, and V'NM a 
 portion of the moon's orbit ; N being the as- 
 cending node : also let EV be the direction of 
 the vernal equinox, and let EV be drawn in the 
 plane of the moon's orbit, making an angle 
 V'EN with the line of {he nodes equal to YEN, 
 the longitude of the ascending node N. The 
 orbit longitudes lie in the plane of the moon's orbit, and are estimated from this 
 line, while the ecliptic longitudes lie in the plane of the ecliptic, and are estimated 
 
94 MOTIONS OF THE PLANETS IN SPACE. 
 
 from the line EV. Thus, V'EM, or its measure V'NM, is the orbit longitude of 
 the moon in the position M ; and VEm is the ecliptic longitude, that is, the longi- 
 tude as it has been hitherto considered. V'NM = V'N -f NM = VN + NM ; that 
 is, orbit long. = long, of ^ + 5)'s distance from ^. 
 
 The orbit longitudes are calculated from the ecliptic longitudes ; these being de- 
 rived from observed right ascensions and declinations. 
 
 231. The ecliptic longitude of the moon at any time being given, to find the 
 orbit longitude. As we may suppose the longitude of the node to be given, (223) 
 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 MNm = cot NM tang Nm ; 
 or, cot NM = cos MNw cot Nm. 
 
 Nm = ecliptic long. long, of node ; and MNm = inclination of orbit. 
 
 232. The horizontal parallax of the moon, like almost every 
 other element of astronomical science, 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, an- 
 swers 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' 0".9 ; 
 from which we find by equation (38) the mean distance of the 
 moon from the earth to be 60.3 radii, or about 240,000 miles. 
 
 The first equation of article 211 would give a more accurate result. 
 The greatest and least parallaxes of the moon are 61' 24" and 53' 48". 
 
 233. The eccentricity of the moon's orbit is more than three 
 times as great as that of the sun's orbit. Its greatest equation ex- 
 ceeds 6 (215). 
 
 MEAN LONGITUDE AT AN ASSIGNED EPOCH. 
 
 234. We have already explained (225) 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. 
 
 METHODS OF DETERMINING THE ELEMENTS OF A PLANET'S 
 
 ORBIT. 
 
 235. The methods of determining the elements of the planetary 
 orbits suppose the possibility of finding the heliocentric longitude 
 and the radius-vector of the earth for any given time. Now, the 
 elements of the earth's orbit having been found by the processes 
 heretofore detailed, the longitude may be computed by means of 
 Kepler's first law, and the radius-vector from the polar equation 
 of the elliptic orbit. (See Davies' Analytical Geometry, p. 137.) 
 The manner of effecting such computation will be considered 
 
LONGITUDE OP THE NODE OP A PLANET*S ORBIT. 
 
 95 
 
 hereafter ; at present the possibility of effecting it will be taken for 
 granted. 
 
 HELIOCENTRIC LONGITUDE OP THE ASCENDING NODE. 
 
 236. When the planet is in either of its nodes, its latitude is zero. It follows, 
 therefore, that the longitude of the planet at the time its latitude is zero, is the 
 geocentric longitude of the node at the time the planet is passing through it. Now 
 if the right ascension and declination of the planet be observed from day to day, 
 about the time it is passing from one side of the ecliptic to the other, and convert- 
 
 ed into longitude and latitude, the time at 
 which the latitude is zero, and the longitude 
 at that time, may be obtained by a proportion. 
 When the planet is again in the same node, 
 the geocentric longitude of the node may 
 again be found in the same manner as be. 
 fore. On account of the different position 
 of the earth in its orbit, this longitude will 
 differ from the former. 
 
 Now, if two geocentric longitudes of the 
 same node be found, its heliocentric longitude 
 may be computed. Let S (Fig. 46) be the 
 sun, N the node, and E one of the positions 
 of the earth for which the geocentric longi- 
 tude of the node (VEN) is known. Denote 
 this angle by G, the sun's longitude VES by 
 S, and the radius-vector SE by r. Also, let 
 E' be the other position of the earth, and 
 denote the corresponding quantities for this 
 position, VE'N, VE'S, and SE', respectively, 
 by G', S', and r 1 . Let the radius-vector of 
 the planet when in its node, or SN = V ; and 
 the heliocentric longitude of the node, or VSN = X. 
 
 Fig. 46. 
 
 but 
 
 and 
 
 hence, 
 
 or, 
 
 In like manner, 
 
 Dividing, 
 
 rsin (S G) 
 r ' r'sinCS' GO 
 whence, 
 
 The triangle 'SNE gives 
 sin SNE : sin SEN : : SE : SN ; 
 SEN = VES VEN = S G, 
 SNE = VAN VSN =* VEN VSN = G X ; 
 
 sin (G X): sin (S G)::r: V, 
 r sin (S G) = V sin (G X) . . . (42). 
 r' sin (S' G') = V sin (G' X.) 
 r sin (S G) sin (G X) 
 
 r' sin (S' G') sin (G' X)' 
 
 sin G cos X sin X cos G sin G cos G tang X 
 
 sin G' cos X sin X cos G' ~~ sin G' cos G'tangX ' 
 
 tangX 
 
 r sin (S G) sin G' r> sin (S' G') sin G 
 
 r sin (S G) cos G'r 1 sin (S' G')cosG 
 
 (43). 
 
 Equation (42) gives 
 
 r sin (S G) 
 X) 
 
 sin 
 
 (44). 
 
 237. The longitude of the node may also be found approximately 
 from observations made upon the planet at the time of conjunction 
 or opposition. It will happen in process of time that some of the 
 conjunctions and oppositions will occur when the planet is near 
 one of its nodes ; the observed longitude of the sun at this con- 
 junction or opposition, will either be approximately the heliocentric 
 
96 
 
 MOTIONS OF THE PLANETS IN SPACE. 
 
 Fig. 47. 
 
 E' 
 
 longitude of the node in question, or will differ 180 from it 
 This will be s^een on inspecting Fig. 47. If at a certain time the 
 
 earth should be at E, crossing the 
 line of nodes, and the planet in 
 conjunction, it will be in the node 
 N, and VES the longitude of the 
 sun will be equal to VSN, the heli- 
 ocentric longitude of the node. If 
 the earth should be 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 lati- 
 tude of the planet should be ob- 
 served about the time of the sup- 
 posed conjunction or opposition 
 near the node, the time when the 
 latitude becomes zero, or the pla- 
 net 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 (241) the longitude of the node 
 could be more accurately determined. 
 
 INCLINATION OF THE ORBIT. 
 
 238. The longitude of the node having been found by the pre- 
 ceding 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 decimation, and deduce the geocentric longitude and 
 latitude. Let ENp (Fig. 47) be the plane of the ecliptic, V the 
 vernal equinox, S the sun, N the node, E the earth on the line of 
 nodes, and P the planet as referred to the celestial sphere, from 
 the earth. Let X denote the geocentric latitude Pp ; E the arc 
 Np = Vp VN = geo. long, of planet long, of node ; and I 
 the inclination PNp. The right-angled triangle PNp gives 
 
 sin Np = tang Pp cot PNp = tang X cot I ; 
 
 . T sin E T tang X 
 
 hence, cot I = -, and tang I = . * . . . (45) : 
 
 tang X' sin E 
 
 or, tang inclination = : 7= . '- 7- y- x . . . (46). 
 
 sin (long. long, of node) 
 
 239. It will be understood, that to obtain an exact result, we must compute tha 
 precise time of the day at which the longitude of the sun is the same as that of 
 the node, and then, by means of their observed daily variations, correct the longi- 
 tude and latitude of the planet for the variations in the interval between the tinm 
 thus ascertained and the time of the observation above mentioned. 
 
REDUCTION OF OBSERVATIONS. 97 
 
 PERIODIC TIME. 
 
 240. The interval from the time the planet is in one of its nodes 
 till its return to the same, gives the periodic time or sidereal revo- 
 lution. 
 
 241. Another and more accurate method is to observe the length 
 of a synodic revolution, (p. 85,) and compute the periodic time 
 from this. If we compare the time of a conjunction which has 
 been observed in modern times, with that of a conjunction observed 
 by the earlier astronomers, and divide the interval between them 
 by the number of synodic revolutions contained in it, we shall 
 have the mean synodic revolution with great exactness, from which 
 the mean periodic time may be deduced.* 
 
 The periodic time being known, the mean daily motion around 
 the sun may be found by dividing 360 by the periodic time ex- 
 pressed in days and parts of a day. 
 
 TO FIND THE HELIOCENTRIC LONGITUDE AND LATITUDE, AND THE 
 RADIUS-VECTOR, FOR A GIVEN TIME. 
 
 242. The earth being in constant motion in its orbit, and being 
 thus at different times very differently situated with regard to the 
 other planets, as well in respect to distance as direction, it is ne- 
 cessary for the purpose of comparing the observations made upon 
 these bodies with each other, to refer them all to one common 
 point of observation. As the sun is the fixed centre about which 
 the revolutions of the planets are performed, it is the point best 
 suited to this purpose, and accordingly it is to the sun that the 
 observations are in reality referred. The reduction of observations 
 from the earth to the sun, as it is actually performed, consists in 
 the deduction of the heliocentric longitude and latitude from the 
 geocentric longitude and latitude, these being derived from the 
 observed right ascension and declination. 
 
 We will now show how to effect this deduction, supposing that the longitude of 
 the node and the inclination of the orbit are known. Let NP (Fig. 48) be part of 
 the orbit of a planet, SNC the plane of the ecliptic, N the ascending node, S the 
 sun, E the earth, and P the planet ; also, let P* be a perpendicular let fall from P 
 upon the plane of the ecliptic, and EV, SV, the direction of the vernal equinox. 
 Let A = PEir the geocentric latitude of the planet ; / = PS* its heliocentric lati- 
 tude ; G = VETT its geocentric longitude ; L = VS* its heliocentric longitude ; 
 S = VES the longitude of the sun ; N = VSN the heliocentric longitude of the 
 node ; I = PNC the inclination of the orbit ; r = SE the radius- vector of the 
 earth ; and v = SP the radius-vector of the planet. 
 
 The point * is called the reduced place of the planet, and S* its curtate distance. 
 All the angles of the triangle SEir have also received particular appellations : SirE 
 the angle subtended at the reduced place of the planet by the radius of the earth's 
 orbit, is called the Annual Parallax, SB* the Elongation, and ESr the Commu- 
 
 * We shall, in the sequel, investigate the equation that expresses the relation be- 
 tween the synodic revolution and the periodic time. (See equation 129, p. 187) : ff 
 the synodic revolution () be given, then, the sidereal year (P) being also known, 
 the value of the sidereal revolution of the planet (j) can be calculated from thia 
 equation. 
 
 13 
 
98 
 
 MOTIONS OF THE PLANETS IN SPACE, 
 Fig. 48. 
 
 / V 
 
 tation. Let A = SirE, E = SE, and C =ESir. Draw Sir parallel to Bi- 
 tten A = *Sir' = VSir VS*' = VSir VEir = L G ; E = VETT VES = 
 G S ; C = VSE VS* = 180 + VSE' VS ff = 180 + VES VSir = 180 
 -f S L = T L (putting T = 180+ S). 
 
 (1.) For the latitude. The triangles EP^, SPa-, give 
 
 tang X Sir 
 
 Eir tang X = PJT = Sir tang / whence . = =- : 
 
 tang / E* ' 
 Sir _ sin E f 
 i 
 
 but, 
 substituting, 
 
 sin C 
 
 Sir : E : : sin E : sin C, or, 
 
 tang X _ sin E 
 tang / "~ sin C ' 
 whence, tang X sin C = tang I sin E . . (46) ; 
 
 or, tang X sin (T L) = tang / sin (G S) . . . (47). 
 
 Again, the triangle NP/j gives, by Napier's first rule, 
 
 sin Np = cot PNp tan Pp, or, sin (L N) = cot I tan I . . (48). 
 Either of the equations (47) and (48) will give the value of Z, when the longi- 
 tude L is known. 
 
 (2.) For the longitude. If we substitute in equation (47) the value of tang /, 
 given by equation (48), and replace (G S) by E, we have 
 
 tang X sin (T L) = sin (L N) tang I sin E ; 
 
 but T L = (T N) (L N) = D (L N), (denoting (T N) by D) ; 
 substituting, and designating L N by a:, 
 
 tang X sin (D x) = sin x tang I sin E ; 
 whence, 
 
 tang X sin D cos x tang X cos D sin x = tang I sin E sin x, 
 or, tang X sin D tang X cos D tang x = tang I sin E tang x, 
 
 which gives 
 
 tang X sin D 
 
 = tang X cos D + tang I sin E ' 
 
 Substituting tho values of x, D, and E, we have, finally, 
 
 tangX sin (T N) 
 
 tang(L N) = 
 
 (50). 
 
 tang X cos (T N) + tang I sin (G S) 
 As N is known, the value of L will result from this equation. 
 
 243. The co-oi linates employed to fix the position of a planet 
 in the plane of its orbit, are its orbit longitude (230) and its radius- 
 vector, both of which result from the heliocentric longitude and 
 
REDUCTION OF OBSERVATIONS. 
 
 99 
 
 atitude, the longitude of the node and the inclination of the orbit 
 being known. 
 
 In Fig. 48, V'NP represents the orbit longitude, and SP (=s t>) the radius-vector 
 for the position P. Now, the triangle PSir gives 
 
 and the triangle ES* gives 
 
 sin A : sin E : : SE : Sir 
 
 whence, by substitution, 
 
 r sin E 
 
 SE sin E 
 
 sin A 
 
 r sin E 
 
 sin A J 
 
 rsin(G S) 
 
 (51). 
 
 sin A cos I sin (L G) cos I 
 
 The orbit longitude L' = NP-f-long. of node . . . (52). 
 And to find NP, the triangle NPp gives 
 
 cos PNp = cot NP tang Np, or tang NP = tang Np . . . (52) ; 
 and Np = long, of planet long, of node . . . (52). 
 
 244. The heliocentric longitude Fig. 49. 
 may be obtained in a very simple 
 
 manner, if the observations be made 
 upon the planet at the time of con- c 
 junction or opposition; for, it will 
 then either be equal to the geocen- 
 tric longitude, or differ 180 from it. 
 
 When the heliocentric longitude is 
 thus found, the latitude for the same 
 time may be obtained by solving the 
 triangle PNp, (Fig. 49.) For, by Na- 
 pier's first rule, 
 
 sin Np cot PNp tang Pp, 
 
 or tang Pp = sin Np tang PNp ; 
 where Pp is the latitude sought, 
 PNp the known inclination of the orbit, and Np = VNp VN = 
 long, of planet long, of node, both of which may be considered 
 as known. 
 
 The radius-vector may be computed for the same time from 
 the triangle ESP ; for the side SE, the radius-vector of the earth, 
 is known, as well as the angle SEP the geocentric latitude of the 
 planet, andthe angle ESP = 180 PSp =180 heliocentric lat. 
 
 245. 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 maximum elongation, de- 
 termined from observation. 
 The elongation which obtains 
 at any time may be found by 
 ascertainingfrom instrumental 
 observations* the places of the 
 
 Fig. 50. 
 

 100 
 
 MOTIONS OF THE PLANETS IN SPACE. 
 
 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. 50) thus formed there will be known the two polar dis- 
 tances PS and Pp, which are the complements of the observed de- 
 cimations, and the angle SPp the difference of their observed right 
 ascensions, from which the angular distance Sp between the two 
 
 Fig. 51. 
 
 bodies may be calculated. The 
 maximum elongation being, 
 then, supposed to be known, 
 let NPP' (Fig. 51) represent 
 the orbit of an inferior planet. 
 The line EP drawn from the 
 earth to the planet will, at the 
 time of maximum elongation, 
 be perpendicular 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 different points of 
 the planet's orbit, and therefore we may find by the foregoing pro- 
 cess different radii-vectores. 
 
 LONGITUDE OF THE PERIHELION, ECCENTRICITY, AND SEMI-MAJOR 
 
 AXIS. 
 
 246. The longitude of the perihelion, 
 the eccentricity, and the semi-major 
 axis, may be derived from the helio- 
 centric orbit longitude (243) and the 
 radius-vector found for three different 
 times. 
 
 Let SP, SP', SP" (Fig. 52) be the 
 three given radii-vectores, V'SP, V'SP', 
 V'SP", the three given longitudes, and 
 AB the line of apsides of the planet's 
 orbit. Let the angles PSF, PSP", 
 which are known, be represented by 
 m, n, and the angle BSP, which is 
 rf unknown, by x\ and let the three ra- 
 dii-vectores SP, SP', SP", be denoted by v, t>', v" ; the semi-major axis AC by 
 a, and the eccentricity by e : then, the three unknown quantities which are to be 
 determined, are a, e, and the angle x, and the general polar equation of the ellipse 
 furnishes for their determination the three equations 
 
 1 -f- e cos x 
 
 (53), 
 
 \-\-t cos (x -f- w) 
 
 1 + e cos (x -f- n) 
 
 . . . (55). 
 
SEMI-MAJOR AXIS OF A PLANERS ORBIT. 101 
 
 Equating the values of a (1 e 2 ) obtained from equations (53) and (54), we havt 
 v-\-ve cos x = t>' -|- v'e cos (x -{- m), 
 
 or, e = -- ; -- - - - . . . (56). 
 
 v cos x v cos (x -f- ) 
 
 In like manner from (53) and (55), 
 
 v" v 
 
 tjcosa; v" cos (x -}- ri) ' 
 
 Let c' v=p, and c" o = q ; then, by equating the second members of 
 equations (56), (57), and transforming, we obtain 
 
 p _ COS X - V f COS (X -f- Ml) 
 
 q v cos x 1>" cos (a; + n) 
 
 t) cos x v' cos m cos x + ' sin m sin a; 
 c cos x v" cos n cos x -\- v" sin n sin a; 
 t> ' cos m -f- 1>' sin m tang x 
 ~ v v" cos n+ 15" sin n tang a; ' 
 
 whence, tang x = ' ( ~ P// , C Sw) ~* (v ~ V ' C Sm) . . . (58). 
 qv sin m pv sin n 
 
 The value of a; being found by this equation, and subtracted from the orbit lon- 
 gitude of the planet in the first position P, the result will be the orbit longitude of 
 the perihelion. Also, x being known, e may be computed from either of the equa- 
 tions (56) and (57) : and hence again, the semi-major axis from equation (53), 
 (54), or (55). 
 
 247. The semi-major axis or mean distance from the sun, may 
 also be had by taking the mean of a great number of radii-vectores 
 found for every variety of position of the planet in its orbit, (244), 
 (245). 
 
 248. Now that Kepler's third law has been established by in- 
 vestigations in Physical Astronomy, it furnishes the most accurate 
 method of finding the mean distance of a planet from the sun. 
 Thus, let P = the periodic time of the planet, and a = its mean 
 distance ; then, the length of the sidereal year being 365.256374 
 days, (193), 
 
 (365.256374d.) 2 :P 2 ::l 3 :o 3 ; 
 
 249. If a great number of radii-vectores in a great variety of po- 
 sitions of the planet in its orbit be found by the method explained 
 in Art. 244, the longitude of the planet at the time it has the least 
 calculated radius-vector will be approximately the longitude of the 
 perihelion : or, if it chances that among the radii-vectores deter- 
 mined 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 
 calculated radii-vectores to the mean of the whole, will be the ap- 
 proximate value of the eccentricity. 
 
 EPOCH OF A PLANET BEING AT THE PERIHELION OF ITS ORBIT. 
 
 250. From several observations upon the planet, about the time 
 
102 MOTIONS OF THE PLANETS IN SPACE. 
 
 it has the same longitude as the perihelion, the correct time of its 
 being at the perihelion may be easily determined by proportion. 
 
 251. The mean longitude at an assigned epoch is obtained up- 
 on the same principles as the mean longitude of the sun or moon, 
 (222, 234.) 
 
 REMARKS. 
 
 252. 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 
 the planet Uranus was discovered by Sir William Herschel, La- 
 place contrived methods by which the elements of its elliptic orbit 
 were determined from four observations within little more than a 
 year from its first discovery by Herschel.* After the discovery of 
 Ceres, Gauss invented another general method of calculating the 
 orbit of a planet from three observations, and applied it to the de- 
 termination of the orbit of Ceres, and, subsequently, to the deter- 
 mination 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 by astronomers. 
 
 MEAN ELEMENTS AND THEIR VARIATIONS. 
 
 253. The elements of the planetary orbits, obtained by the foregoing processes, 
 are the true elements at the periods when the observations are made. Upon deter- 
 mining them at different periods, it appears that they are subject to minute varia- 
 tions. A comparison of the values found at various distant epochs shows that they 
 are slowly changing from century to century, and that the changes experienced 
 during equal long periods of time are very nearly the same. The amount of the 
 variation of an element in a period of 100 years is called its Secular Variation. 
 Upon reducing the elements, found at different times, to the same epoch, by allow- 
 ing 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. 
 
 The mean elements at an assigned epoch may be had by finding the true ele- 
 ments at various times, and reducing them to the given epoch, by making allow- 
 ance for the proportional parts of the secular variations, and then taking for each 
 element the mean of all the particular values obtained for it. 
 
 254. 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. 
 
 255. The elements of the moon's orbit are also subject to continual variations. 
 These are, for the most part, periodic, and are far greater than the variations of 
 the corresponding elements of a planet's orbit. It will be seen, then, that in de- 
 termining 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 progressive motion. Theory shows that the other 
 mean elements, with the exception of the semi-major axis, are subject to secular 
 variations, but their effect has hitherto been very inconsiderable. 
 
 * History of the Inductive Sciences, vol. ii. p. 231. 
 
ELEMENTS OF THE PLANETARY ORBITS. 103 
 
 256. The mean elements, which have been derived as above directly from ob- 
 servation, have subsequently been verified and corrected, by comparing the com- 
 puted with the observed places of the planet ; and for this purpose many thousands 
 of observations have been made. 
 
 257. Tables II. and III. contain the elements of the orbits of 
 the principal planets, and of the moon's orbit, together with their 
 secular variations, for the beginning of the year 1801 ; and also, 
 the elements of the orbits of the four small planets, Vesta, Juno, 
 Ceres, and Pallas, for 1831. (See Note III.) 
 
 If an element be desired for any time different from the epoch 
 of the table, we have only to allow for the proportional part of the 
 secular variation, in the interval between the given time and the 
 epoch of the table. 
 
 258. 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 6788.50982 days, or about 18 years 214 days; and 
 the latter in 3231.4751 days, or in about 8 years 309 days. The 
 mean motion of the node, and the mean motion of the perigee, are 
 both subject to a slow secular diminution. 
 
 259. It will be seen, also, that the orbits of the planets are 
 ellipses of small eccentricity, or which differ but slightly from 
 circles ; and that they are, with the exception of the orbit of 
 Pallas, inclined under small angles to the plane of the ecliptic. 
 The eccentricity is in every instance so small, that if a represen- 
 tation of the orbit were accurately delineated, it would not differ 
 perceptibly from a circle. The most eccentric orbits, among those 
 of the seven principal planets, are those of Mercury and Mars ; 
 and the least eccentric, those of Venus and the earth. The eccen- 
 tricity of Mercury's orbit is 12 times that of the earth's, of Mars' 
 6 times, of Venus' \. The eccentricities of the orbits of Jupiter, 
 Saturn, and Uranus, are each about 3 times greater than that of 
 the earth's orbit. 
 
 The orbit of Mercury is more inclined to the ecliptic than the 
 orbit of any other of the seven principal planets ; and the orbit of 
 Uranus is less inclined than that of any other planet. The in- 
 clination of the latter is , of the former 7. 
 
 The orbits of the four asteroids are more eccentric, and more 
 inclined to the plane of the ecliptic, than those of the other planets 
 in general. 
 
104 MOTIONS OF THE PLANETS IN SPACE. 
 
 260. The mean distances of the planets from the sun are, in 
 round numbers, as follows : Mercury 37 millions of miles, Venus 
 69 millions of miles, the earth 95 millions of miles, Mars 145 mil- 
 lions of miles, Juno 254 millions of miles, Jupiter 495 millions of 
 miles, Saturn 907 millions of miles, Uranus 1824 millions of miles. 
 The range of distance is from 1 to 77. The distance of Uranus 
 is about 19 times the earth's distance: of Neptune 30 times. 
 
 261. The approximate periods of revolution of the planets are 
 as follows : Mercury 3 months, (| of a year,) Venus 7| months, 
 (f of a year,) Mars 1} years, Juno 4f years, Vesta of a year 
 shorter, and Ceres and Pallas j of a year longer than that of 
 Juno, Jupiter 12 years, (11-f years,) Saturn 29| years, Uranus 
 84 years, Neptune 164f years. 
 
 262. A remarkable empirical law, called Bode's Law of the 
 Distances, from its discoverer, the late Professor Bode of Berlin, 
 connects the distances of the planets from the sun. It is as fol- 
 lows. If we take the following numbers, 0, 3, 6, 12, 24, 48, 
 96, 192, and add the number 4 to each one of them, so as to ob- 
 tain 4, 7, 10, 16, 28, 52, 100, 196, this series of numbers will 
 express the order of distance of the planets from the sun. This 
 law embodies the following curious relation between the distances 
 of the orbits from one another, viz.: setting out from Venus, the 
 distance between two contiguous orbits increases nearly in a dupli- 
 cate ratio as we recede from the sun ; that is, the distance from 
 the orbit of the earth to the orbit of Mars, is twice the distance 
 from the orbit of Venus to the orbit of the earth, and one half the 
 distance from the orbit of Mars to the orbits of the asteroids, &c. 
 Professor Challis of Cambridge, England, has recently extended 
 this principle to the distances of the satellites ; so that although 
 still somewhat indefinite, it is unquestionably part of the arrange- 
 ments and mechanism of the solar system.* 
 
 Previous to the discovery of the four asteroids, to complete the 
 above law a planet was wanting between Mars and Jupiter. It 
 was on this account surmised by Bode, that another planet might 
 exist between these two. Instead of one such planet, however, it 
 was subsequently discovered that there were in fact four, revolving 
 at pretty nearly the same distance from the sun, and in conformity 
 with the curious law which had been detected by Bode. (Note IV.) 
 
 263. A better idea of the dimensions of the solar system than 
 is conveyed by the statement of distances above given, may be 
 gained by reducing its scale sufficiently to bring it within the 
 scope of familiar distances. Thus, if we suppose the earth to be 
 represented by a ball only 1 inch in diameter, the distance of Mer 
 cury from the sun will be represented on the same scale by 400 feet, 
 the distance of Venus by 700 feet, that of the earth by 1000 feet, 
 (j of a mile nearly,) that of Mars by 1500 feet, that of Juno by a 
 
 * Nichol's Phenomena of the Solar System, p. 
 
PLACE OF A PLANET IN ITS ORBIT. 105 
 
 mile, that of Jupiter by 1 mile, that of Saturn by % 2 miles, (If 
 miles,) and that of Uranus by 3 miles, (3f miles.) On the same 
 scale, the distance of the moon from the earth would be only 2^ 
 feet : that of Neptune 5| miles. 
 
 CHAPTER VIII. 
 
 OP THE DETERMINATION OF THE PLACE OF A PLANET, OR OF THE 
 SUN, OR MOON, FOR A GIVEN TIME, BY THE ELLIPTICAL THEORY J 
 AND OF THE VERIFICATION OF KEPLER ? S LAWS. 
 
 PLACE OF A PLANET, OR OF THE SUN, OR MOON, IN ITS ORBIT. 
 
 264. THE angle contained between the line of apsides of a 
 planet's orbit and the radius-vector, as reckoned from the peri- 
 helion towards the east, is called the True Anomaly. Thus, 
 let BPAP' (Fig. 53) represent Fig. 53. 
 
 the 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 reck- 
 oned from the perihelion to- A| 
 wards 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, (201.) 
 
 Describe a circle BpA on the line of apsides as a diameter ; 
 through P drawpPD perpendicular to the line of apsides, and join 
 p and C : the angle BCp, which the line thus determined makes 
 with the line of apsides, is called the Eccentric Anomaly. 
 
 The corresponding angles appertaining to the sun's apparent 
 orbit, and to me moon's orbit, have received the same appellations. 
 
 The interval between two consecutive returns of a body to either 
 apsis of its orbit, is called the Anomalistic Revolution. The ano- 
 malistic revolution of the earth, or of the sun in its apparent orbit, 
 is termed, also, the Anomalistic Year. 
 
 265. The periodic time, or the mean motion of a body, and the 
 motion of the apsis of its orbit, being known, the anomalistic Devo- 
 lution may l^^asily computed. Let m = the sidereal motion of 
 
 S* 14 
 
108 DETERMINATION OF THE PLACE OF A PLANET. 
 
 the apsis answering to the periodic time, and M = the mean daily 
 motion of the planet; then, 
 
 M : Id. : : m : 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. 
 
 266. The length of the anomalistic year exceeds that of the 
 sidereal year by 4m. 44s. 
 
 267. From the anomalistic 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 -^ . . . (60). 
 
 268. The place of a body in its elliptical orbit is ascertained 
 by finding its true anomaly. The problem which has for its ob- 
 ject the determination of the true anomaly from the mean, was first 
 resolved by Kepler, and is called Kepler's Problem. Another and 
 more convenient method of obtaining the true anomaly, is to com- 
 pute the equation of the centre from the mean anomaly, and add 
 it to the mean anomaly, or subtract it from it, according to the po- 
 sition of the body in its orbit, (201). 
 
 HELIOCENTRIC PLACE OF A PLANET. 
 
 269. The place of a planet in the plane of its orbit is designated 
 by its orbit longitude (230) and radius-vector. To find the orbit 
 longitude we have the equation V'SP = V'SB + BSP (see Fig. 
 53,) or, long. = long, of perihelion -f true anomaly. 
 
 The orbit longitude may also be deduced from the mean longi- 
 tude, by adding or subtracting the equation of the centre ; for, 
 V'SP - V'SM + MSP, 
 
 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, (235,) viz : 
 
 - e cos x 
 
 in which x denotes the true anomaly, e the eccentricity, and a the 
 semi-major axis. 
 
 270. Now to find the heliocentric longitude and latitude which 
 ascertain the position of the planet with respect to the ecliptic, the 
 triangle NPp (Fig. 48, p. 98) gives 
 
 sin Pp = sin NP sin 
 
GEOCENTRIC *LACE OP A PLANET. 107 
 
 or, sin lat. = sin (orbit long. long, of node) x sin (inclin.) . . (62); 
 and 
 
 cos PNp = tang Np cot NP, or tang Np = tang NP cos PNp, 
 or, 
 
 tang (long. long, of node) tang (orbit long. long, of node) 
 x cos (inclination) . . . (63). 
 
 GEOCENTRIC PLACE OF A PLANET 
 
 271. From the heliocentric longitude and latitude and the radius-vector of a 
 planet, to find the geocentric longitude and latitude. Let S (Fig. 48) be the sun, 
 E the earth, P the planet, ir its reduced place, and V the vernal equinox. Denote 
 the heliocentric longitude VS* by L, the heliocentric latitude PSir by /, and the 
 radius-vector SP by v ; and denote the geocentric longitude by G, and the geo- 
 centric latitude by A. Also let E = SB* the elongation ; C = ES?r the commu- 
 tation ; A = SffE the annual parallax ; and r = SE the radius-vector of the 
 earth. Now, 
 
 VE* = SETT -f VES, 
 or, G = E -j- long, of sun. 
 
 This equation will make known the geocentric longitude when the value of E 
 is found. In the triangle PS* 1 the side SJT = SP cos PSn- = t> cos /, and is there- 
 fore known, the side ES is given by the elliptical theory, (269,) and the angle C 
 may be desived from the following equation : C = VSE VSjr = long, of earth 
 long, of planet ; and to find E we have, by Trigonometry, 
 
 ES + S ff : ES ST : : tan $ (E^S -f- SB*) : tan $ (E W S SE*-,) 
 or r + v cos I : r v cos I : : tang ( A -f- E) : tang ( A E) ; 
 
 whence, 
 
 v cos Z 
 
 tang 4 (A-E) = -= tan g i(A+E) tang 1 ,*+ E) 
 
 Let tang 6 = - - - : then, 
 
 tang J (A - E) = ta g J (A + E, ; 
 
 or, tang J (A E) = tang (45 0) tang (A + E) . . . (64). 
 
 But, A + E = 180 C, and E = (A -f E) (A E.) 
 
 Next, to find the geocentric latitude. 
 
 Sir tang / = PJT = E* tang Aj 
 
 STT tang A 
 
 whence, - = - ? ; 
 
 EJT tang I 
 
 "El 
 
 but. Sw : Eir : : sin E : sin C, or :=- = - , 
 
 En- sin C 
 
 sin E tang A 
 
 and therefore ^ = - ^-* 
 
 sin C tang I 
 
 sin E tang I ,__. 
 
 272. When a planet is in conjunction or opposition, the sines of the angles of 
 elongation and commutation are each nothing. In these cases, then, the geo- 
 centric latitude cannot be found by the preceding formula ; it may, however, be 
 easily determined in a different manner. Suppose the planet to be in conjunction 
 at P, (Fig. 49, p. 99 ;) then, 
 
 PTT PIT 
 
108 DETERMINATION OF THE PLACE OF A PLANET. 
 
 But the triangle SPir gives 
 
 Par = v sin Z and Sir = cos l^ and ES = r ; 
 
 hence, tang A = * si " * . . . (66) * 
 
 r + v cos 2 
 
 273. To find the distance of the planet from the earth, represent the distance by 
 D ; then, from the triangles PwS and EPrr, (Fig. 48, p. 98,) we have 
 
 PIT = EP sin PE?r = D sin X, 
 and PIT = SP sin PSs- = v sin I ; 
 
 whence, D = ^-1 ! . . (67). 
 
 sin A 
 
 274. The distance of a planet being known, its horizontal parallax may be com- 
 puted from the equation 
 
 sin H = -,?... (68.) (Art. 113.) 
 
 PLACES OF THE SUN AND MOON. 
 
 275. 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, (221,) 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 to compute its true anomaly, (268,) and thence tne lon- 
 gitude and radius-vector by the equation 
 
 long. = true anomaly + long, of perigee, 
 and the polar equation of the orbit. 
 
 276. The orbit longitude and the radius-vector of the moon are 
 found by the same process as the longitude and radius-vector of 
 the sun. The orbit longitude being known, the ecliptic longitude 
 and the latitude may be determined by a process precisely similar 
 to that by which the heliocentric longitude and latitude of a plane 
 are found, (270.) 
 
 VERIFICATION OF KEPLER'S LAWS. 
 
 277. If Kepler's first two laws be true, then the geocentric 
 places of the planets, computed by the process that we have 
 described, (271^) which is founded upon them, ought to agree 
 with the true geocentric places as obtained for the same time by 
 direct observation ; or, the heliocentric places computed from the 
 observed geocentric places, (242,) ought to agree with the same 
 as computed by the elliptic theory, (269, 270.) 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 
 
 * For inferior conjunction the sign of cos I must be changed, and for opposition 
 the sign of r must be changed. 
 
 ' 
 
VERIFICATION OP KEPLER ? S LAWS. 109 
 
 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. 
 
 278. Kepler's laws have been verified for the sun and moon, in 
 a similar manner. 
 
 279. The relative distances of the sun, or moon, at different 
 times, result for this purpose, from measurements of the appa- 
 rent diameter, upon the principle that any two distances are in- 
 versely proportional to the corresponding apparent diameters. 
 Let A = semi-diameter corresponding to the mean distance, and 
 $ = semi-diameter corresponding to any distance D : then 
 
 5 : A : : 1 : D ; whence, D =-j- . . . (69) ; 
 
 an equation which, when A has been found, will make known the 
 distance corresponding to any observed semi-diameter <J, in terms 
 of the mean distance as a unit. 
 
 Now, to find A, denote the greatest and least semi-diameters 
 respectively by 5', 6", and the corresponding distances by D' and 
 D", and we have 
 
 D'=A D" = 
 
 6" D <$"' 
 
 and thence, 
 
 , or, != 
 
 2 <5' <$" 
 whence, A = ^.. .(70.) 
 
 280. The distance of the sun or moon in terms of the mean 
 distance as a unit, may be found in this manner ; but it may be 
 had more accurately by means of a principle which has been dis- 
 covered from observation, namely, that the distance is inversely 
 proportional to the square root of the daily angular motion. 
 
 CHAPTER IX. 
 
 OF THE INEQUALITIES OF THE MOTIONS OF THE PLANETS AND OF 
 THE MOON J AND OF THE CONSTRUCTION OF TABLES FOR FINDING 
 THE PLACES OF THESE BODIES. 
 
 281. 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 masses, and inversely pro- 
 portional to the square of their distance. The force which causes 
 one body to gravitate towards another, is supposed to arise from a 
 
110 
 
 
 INEQUALITIES OF THE PLANETARY MOTIONS. 
 
 mutual attraction existing between the particles of the. two bodies 
 and is hence called the Attraction of Gravitation. This force of 
 attraction, common to all the bodies of the Solar System, is the 
 general physical cause of their motions. The sun's attraction re- 
 tains the planets in their orbits, and the planets by their mutual 
 attractions slightly alter each other's motions. The reasoning by 
 which Newton's Theory of Universal Gravitation is established, 
 appertains to Physical Astronomy, and will be presented in another 
 part of the work. 
 
 282. If a planet were acted on by no other force than the 
 attraction of the sun, it is proved that its orbit would be accu- 
 rately an ellipse, and that 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 agree- 
 ment between the observed and computed places noticed in Art. 
 277, 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 Plane Astronomy, Inequalities. 
 Although, as we have just seen, the fact of the existence of ine- 
 qualities in the motions of the planets is discoverable from ob- 
 servation, their laws cannot be determined without the aid of 
 theory. 
 
 283. 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 
 
 ro- 
 
 Fig. 54. 
 
 this motion. This, then, is the disturbing force. 
 
 duces the perturbations the 
 turbing Force. To find the dis- 
 turbing force, let P (Fig. 54) be 
 the planet, S the sun, and M the 
 disturbing body; and let PD 
 represent the attraction of M for 
 the planet. Decompose PD in- 
 to two forces, PE and PF, one 
 of which, PE, is equal arid paral- 
 lel to SG, the attraction of M for 
 the sun ; the other, PF, will be 
 known in position and intensity. 
 The two forces, PE and SG, 
 being equal and parallel, they 
 cannot alter the relative motion 
 of the sun and planet, and ac- 
 cordingly may be left out of ac- 
 count : there remains, therefore, 
 the component PF, which will 
 be wholly effectual in disturbing 
 
PROBLEM OF THE THREE BODIES. HI 
 
 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 paral- 
 lel. 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. 
 
 284. 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 ap- 
 pears, therefore, that the disturbing force produces at the same 
 time perturbations or inequalities of longitude, of latitude, and of 
 radius-vector. 
 
 285. Let us now consider how these inequalities may be deter- 
 mined. In the first 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 disturb- 
 ing forces of the system. 1 he 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 resolved 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 be- 
 come known. 
 
 286. 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 
 resolved 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 an- 
 alytical expressions for the perturbations in longitude, latitude, and 
 radius -vector, involving variables and constants. 
 
 287. The general expression for the whole perturbation in longitude, due to the 
 action of any one disturbing body, is 
 
 C sin (F P) -f C' sin 2 (F P) + C" sin 3 (F P) -f &c. . . (71), 
 
 in which C, C', &c., are constants, P the heliocentric longitude of the body di 
 
' * 
 
 112 INEQUALITIES OF THE PLANETARY MOTIONS. 
 
 turbed, and P' that of the disturbing body. The number of terms is, strictly speak- 
 ing, indefinite, but they form a decreasing series, so that only a small number of 
 the first terms (which will be different in different cases) need to be used. 
 
 288. The constants C, C', &c. are to be determined from observation ; they 
 may, however, be determined in the case of some of the planets from theory alone. 
 The process of finding them from observation is as follows : Suppose that the earth 
 is the body whose perturbations are under consideration, and let D denote the per- 
 turbation in longitude, produced by the joint action of all the disturbing forces. 
 Then, supposing, for the sake of simplicity, that the expression for the perturbation 
 due to each disturbing body consists of but two terms, we have 
 
 D=Csin(F P)-f-C'sin2 (P P)-f csin(F' P)+c'sin2 (P" P)+&c. (72). 
 
 Find, by observation, the heliocentric longitude of the earth, (189, 221,) and take 
 the difference between this and the longitude as computed for the same time by 
 the elliptical theory; (269.) This difference will be the value of D at the time of 
 the observation. P, P', P", &c. the heliocentric longitudes of the earth and of the 
 disturbing bodies, and consequently P' P, P" P, &c., are given by the ellipti- 
 cal theory. Thus, in the above equation all will be known but C, C', c, c', &c. 
 By repetitions of this process as many equations may be obtained as there are con- 
 stants to be determined, and from these the values of the constants may be com- 
 puted. But it is usual to obtain a much greater number of equations than there 
 are constants ; as, by combining them according to certain rules, much more ex- 
 act values of the constants may be derived. 
 
 289. In the expression 
 
 C sin (F P) -f- C' sin 2 (F P) -f &c., 
 
 for the perturbation in longitude, due to the action of a disturbing body, each term, 
 C sin (F P),C' sin 2 (F P), &c., is technically termed an Equation,* and is 
 considered as representing a specific inequality. The angle P' P, or 2 (P' P), 
 or other multiple of P' P, the sine of which enters into the equation of an ine- 
 quality, 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 
 anity, the coefficient is equal to the greatest value of the inequality. 
 
 290. The coefficient being known, the value of the inequality at any particular 
 time will become known if that of the argument be found. Now, the argument 
 is the difference between the longitudes of the disturbing body and disturbed body, 
 or some multiple of this difference, and may be found by the elliptical theory. In 
 practice, the mean longitudes may be taken, without material error, in place of the 
 true, and these are easily deduced from the mean longitudes at a given epoch, by 
 means of the mean motions in longitude of the two bodies. When the values of 
 all the inequalities in longitude have been separately determined, by taking their 
 algebraic sum yve shall have the correction to be applied to the elliptic longitude in 
 order to find the exact longitude. 
 
 291. The general expression for the total perturbation of radius- vector, due to the 
 action of one body, is 
 
 C cos (F P) + C' cos 2 (F P) + C" cos 3 (F P) -f &c (73). 
 
 As in the expression for the perturbation of longitude, each term is called an equa- 
 tion, and represents a distinct inequality, the constant being the coefficient, and 
 the variable angle, the cosine of which enters into the equation, the argument of 
 the inequality. The amounts of the different inequalities, at an assumed time, 
 are computed after the same manner as those of the inequalities of longitude, and 
 being added together with their algebraical signs, will give the correction to be 
 applied to the elliptic radius-vector. 
 
 292. The perturbation in latitude is very minute. The inequalities of latitude, 
 as of longitude and radius- vector, are represented by equations composed of a con- 
 stant coefficient and the sine or cosine of a variable argument, or of the form C 
 sin A or C cos A. 
 
 The term equation is applied in Astronomy to all quantities added to mean 
 tlements in order to find the true. 
 
PERIODIC AND SECULAR INEQUALITIES. 113 
 
 293. The arguments of the inequalities we have been consider- 
 ing, are angles depending upon the configurations 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. 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 em- 
 ployed 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. 
 
 294. It has been stated that the elements of the elliptic orbits of 
 the planets are, for the most part, subject to a slow variation from 
 century to century. Investigations in Physical Astronomy have 
 established that the variations of the elements are due to the action 
 of the disturbing forces of the planets, and that they are not pro- 
 gressive, (except in the cases of the longitude of the node and the 
 longitude of the perihelion,) but are really periodic inequalities 
 whose periods comprise many centuries. From the great lengths 
 of their periods these inequalities are termed Secular Inequalities, 
 in order to distinguish them from the inequalities of the elliptic mo- 
 tion, denominated Periodic Inequalities, the periods of which are 
 comparatively short. 
 
 Physical Astronomy furnishes expressions called Secular Equa- 
 tions, which give the value of an element at any assumed time. 
 
 295. The inequalities of the moon's motion arise from the dis- 
 turbing action of the sun. The attractions of each of the planets 
 for the moon and earth are sensibly equal and parallel. The lunar 
 inequalities are investigated upon the same principle as the plan- 
 etary, and are represented by equations of the same general form, 
 that is, consisting of a constant coeificient and the sine or cosine 
 of a variable argument. They far exceed in number and magni- 
 tude those of any single planet. 
 
 296. There are three lunar inequalities of longitude which are 
 prominent above the rest, and were early discovered by observa-, 
 tion. 
 
 The most considerable is called the Evection, and was discover- 
 ed 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". 
 
 15 
 
1 14 INEQUALITIES OF THE PLANETARY MOTIONS. 
 
 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. When greatest it amounts to 11 
 12". 
 
 297. The discovery of the other lunar inequalities (with the ex- 
 ception of one inequality of latitude) is due to Physical Astronomy. 
 
 The whole number of lunar inequalities of longitude, according 
 to Burckhardt, is 34, (without taking into account the equation of 
 the centre and the reduction from the orbit longitude to the eclip- 
 tic longitude :) and according to Damoiseau, 45. 
 
 298. To present now at one view the entire process of finding 
 the exact heliocentric place of a planet, or 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, (294.) 
 
 (2.) Compute the longitude, latitude, and radius- vector, by the 
 elliptic theory, (269, 270.) 
 
 (3.) Compute the values of the inequalities in longitude and lati- 
 tude, and of radius-vector, by means of their equations (290, 291, 
 292,) and apply them individually with their proper signs, as correc- 
 tions to the elliptic values of the longitude, latitude, and radius-vector. 
 
 299. If we suppose the sun to be in motion instead of the earth, 
 its inequalities will be the same as those to which the motion of 
 the earth is actually subject. 
 
 300. When the heliocentric place of a planet has been found, its 
 geocentric place, if required, may be determined by the process 
 explained in Art. 271. 
 
 CONSTRUCTION OF TABLES. 
 
 301. The determination of the place of the sun or moon, or of a 
 planet, may be greatly facilitated by the use of tables. - The prin- 
 ciple 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 com- 
 putation of the sun's longitude. We have the equation 
 
 True long. = mean long. -\- equa. of centre -f- inequalities -j- nutation. 
 
 If, then, tables can be constructed which will furnish by inspection the mean 
 longitude, the equation of the centre, the amounts of the various inequalities in 
 longitude, and the nutation in longitude, at any assumed time, we may easily find 
 the true longitude at the same time. 
 
 302. (1.) For the mean longitude, The sun's mean motion in longitude in a 
 mean tropical year, is 360. From this we may find by proportion the mean mo- 
 tions in a common year of 365 days and a bissextile year of 366 days. 
 
 With these results, and the mean longitude for the epoch of Jan. 1, 1801, (see 
 Table II,) we may easily derive the mean longitude at the beginning of each of 
 the years prior and subsequent to the year 1801. The second column of TaWe 
 XVIII. contains the mean longitude of the sun at the beginning of each of the 
 years inserted in the first column. The third column of this table contains the 
 
115 
 
 mean longitude 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, which is 61".52.* 
 
 Having the sun's mean daily motion in longitude, (192,) we obtain by propor- 
 tion the motion in any proposed number of months, days, hours, minutes, or sec- 
 onds. 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 tho 
 month, and his motion for hours from 1 to 24 ; and Table XXI, the same for min- 
 utes and seconds from 1 to 60. With these tables the sun's mean motion in lon- 
 gitude in the interval between any given time in any year and the beginning of the 
 year may be had : and if this be added to the epoch for the given year, taken out 
 from Table XVIII, the result will be the mean longitude at the given time. (See 
 Problem IX.) 
 
 303. Tables XIX and XX ateo contain the motions of the sun's perigee, from 
 which and the epoch given by Table XVIII results the longitude of the perigee 
 at any proposed time. The longitude of the perigee is given in the Solar Tables 
 for the purpose of making known the mean anomaly, the mean anomaly being 
 equal to the mean longitude minus the longitude of the perigee. 
 
 304. (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 6 + B sin 20 + C sin 3$ + &c. ; 
 
 in which A, B, C, &c., are constants that rapidly decrease in value, and which 
 may be determined for any particular orbit, and 9 the mean anomaly. Now, by 
 giving to the mean anomaly in this equation a series of values increasing by 
 small equal differences (of 1, for instance,) from zero to 360, and computing the 
 corresponding values of the equation of the centre, then registering in a column 
 the different values assigned to 0, and in another column to the right of this the 
 computed values of the equation of the centre, we shall obtain a table which will 
 give on inspection the equation of the centre corresponding to any particular mean: 
 anomaly. In this manner was constructed Table XXV. In this table, however, 
 for the sake of compactness, the values of the equation, instead of being register- 
 ed in one column, are put in as many different columns as there may be different 
 numbers of signs in the value of the mean anomaly ; each column answering to 
 the particular number of signs placed at the head of it. 
 
 If the equation of the centre at an assumed time be required, find th,e mean 
 anomaly by the tables (303,) and with the value found for it take out the equation 
 of the centre from Table XXV. 
 
 The given quantity with which a quantity is taken from a table, is called the 
 Argument of. that quantity. Accordingly the mean anomaly is the argument of 
 the equation of the centre in Table XXV. 
 
 305. (30 For the inequalities. The equations of the inequalities, as we have 
 already stated, are of the form C sin A, the argument A being the difference be- 
 tween the longitude of the disturbing planet and that of the earth, or some mul- 
 liple 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, (287,) 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 ive 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 
 <jarth and the longitude of the disturbing planet a series of values increasing by a 
 common difference, and computing with each set ef the values of these quanti- 
 ties, the corresponding amount of the perturbation. 
 
 In connection with the tables of the perturbations, we must have tables that 
 make known the values of the arguments at any given time. Now, the mean lon- 
 gitude of the sun may be found by the solar tables (302,) and thence the mean he- 
 
 * The quantities in Table XVIII are called Epochs. The Epoch of a quan*' 
 is its value at some chosen epoch. 
 
116 CONSTRUCTION OF ASTRONOMICAL TABLES. 
 
 liocentric 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, IV, V, VI, VII, contain the arguments of all the pertur- 
 bations, for the beginning of each of the years registered in the first column, ex 
 pressed in thousandth parts of a circle. Tables XIX and XX contain the varia- 
 tions of the arguments for months, days, and hours. Their variations for minutes 
 and seccnds are too small to be taken into account. With these tables, and Table 
 XVIII, the values of the arguments at any given time may be found, and by 
 means of the arguments the perturbations may be taken from Tables XXVIII, 
 XXIX, XXX, XXXI, XXXII, and XXXIII. 
 
 306. (4.) For the nutation. The formula for the lunar nutation in longitude, is 
 17".3 sin N 0".2 sin 2 N, in which N denotes the supplement (to 360) of the 
 longitude of the moon's ascending node. With this formula the second column of 
 Table XXVII was constructed. The value of N, in thousandth parts of a circle, 
 results from Tables XVIII, XIX, and XX. The solar nutation is also given by 
 Table XXVII. 
 
 307. Tables may also be constructed that will facilitate the computation of the 
 radius-vector. We have 
 
 * True rad. vector = elliptic rad. vector + perturbations. 
 
 A table of the elliptic radius-vector may be formed by means of the polar equa- 
 tion of the orbit, and tables of the perturbations from their analytical expressions, 
 (291.) The tables of the perturbations will have the same arguments as the tables 
 of the perturbations of longitude. 
 
 308. 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 longi- 
 tude and radius- vector. But other tables are necessary in the case of thes^ bodies, 
 for the computation of the ecliptic longitude and the latitude. 
 
 309. The difference between the orbit longitude and the ecliptic longitude, is 
 called the Reduction to the ecliptic. A formula for the reduction has been investi- 
 gated, in which the variable is the difference between the orbit longitude and the 
 longitude of the node, (or, what amounts to the same, the orbit longitude plus the 
 supplement of the longitude of the node to 360.) If this formula be reduced to 
 a table, by taking the reduction from the table and adding it to the orbit longitude, 
 we shall have the ecliptic longitude. Table LIII is a table of reduction for the moon. 
 
 310. For the latitude, we have the equation 
 
 True lat. = lat. in orbit + perturbations. 
 We have already seen (270) that 
 
 sin (latin orbit )= sin (orbit long. long, of node) sin ^inclina.) 
 
 A table constructed from this formula will have for its argument the orbit longi- 
 tude minus the longitude of the node, which is also the argument of reduction. 
 (See Table LV.) 
 
 The tables of the perturbations in latitude are constructed upon the same prin- 
 ciples as the tables of the perturbations in longitude and radius-vector. 
 
 311. A table exhibiting the longitude and latitude, right ascen- 
 sion and declination, distance, parallax, semi-diameter, &c., of the 
 sun or other body, at stated periods of time, as at noon of each day 
 throughout the year, is called an Epliemeris of the body. An ephe- 
 meris of the sun, of the moon, and of each of the planets, is pub- 
 lished for each year in advance in the English Nautical Almanac, 
 and in the Connaissance des Terns. 
 
FORM OF THE COMETARY ORBITS. 117 
 
 CHAPTER X. 
 
 OF THE MOTIONS OF THE COMETS. 
 
 312. WHEN first seen, a comet is ordinarily at some distance 
 from the sun in the heavens, and moving towards him. After 
 this it continues to approach the sun for a certain time, and then 
 recedes from him to a greater or less distance, and finally disap- 
 pears. In many instances comets have come so near the sun, as 
 to be for a time lost in his beams. It has sometimes happened 
 that a comet has not made its appearance in the firmament until 
 after the time of its nearest apparent approach to the sun, and 
 when it is receding from him in the heavens. This was the case 
 with the great comet of 1843. It was first seen, in this country, 
 in open day, on the 28th of February, in the immediate vicinity 
 (within 3) of the sun ; and after this moved away from him, and 
 gradually diminishing in brightness, in about a month became 
 invisible. 
 
 313. 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 hea- 
 vens, but traverse indifferently every part. 
 
 314. Sir Isaac Newton, from observations that had been made 
 upon the remarkable comet of 1 680, ascertained that this comet 
 described a parabolic orbit, having the sun at its focus, or an ellip- 
 tic 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 about 180 comets have been 
 computed, and found to be, with a few exceptions, of a parabolic 
 form, or sensibly so. 
 
 315. It was demonstrated by Newton, on the theory of gravi- 
 tation, 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 velo- 
 city it will be an hyperbola. Now, as there is but one velocity 
 from which a parabolic orbit will result, and as any comet, whicn 
 may have originally moved in an hyperbola, must have 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 para- 
 bolic arcs, are in fact ellipses of great eccentricity. This is the 
 theory of the cometary motions proposed by Newton. 
 
118 OF THE MOTIONS OF THE COMETS. 
 
 The orbits of some of the comets are known from observation 
 to be very eccentric ellipses. 
 
 316. The elements of a comet's orbit are the longitude of the 
 ascending node, the inclination of the orbit, the longitude of the 
 perihelion, the perihelion distance, and the epoch of the perihe- 
 lion passage. These make known the position and dimensions of 
 the orbit on the supposition that it is a parabola, and thus apper- 
 tain only to the motions of the comet for the period during which 
 it is visible. 
 
 317. Assuming that the radius-vector of a comet describes 
 areas proportional to the times, the elements of its orbit may be 
 computed from three observed geocentric places. But the prob- 
 lem is one of considerable, difficulty. 
 
 318. Astronomers do not, in general, seek to deduce from the 
 observations made during one appearance of a comet its entire 
 elliptic orbit. It is impossible, from such observations, to com- 
 pute the major-axis of its orbit and its period with any accuracy, 
 inasmuch 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 rs 1483 
 years and the greatest 1952 years ; and that of the great comet 
 seen in 1811 is said to be either 2301 or 3056 years. The un- 
 certainty becomes rmich 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 perihelion 
 passages. A comet cannot be recognised at a second appearance 
 by its aspect, for this is liable to great alterations. But it may be 
 identified by means of the elements of its orbit, as it is extremely 
 improbable that the elements of the orbits of two different comets 
 will agree throughout. This method of identifying a comet on a 
 second appearance may sometimes fail of application, inasmuch 
 as the orbit of a comet may experience great alterations, from the 
 attractions of the planets. 
 
 319. Owing to the great lengths of the periods of most of the 
 comets, and the comparatively short interval during which their 
 motions have been carefully observed, there are but three comets, 
 the periods and entire orbits of which have been determined. 
 These are denominated Encke's Comet, Gambarfs Comet, (some- 
 times called Bieltfs,) and Halley's Comet. The two former have 
 never been seen, except in a very few instances, without the assist- 
 ance of a telescope, but the latter, when near its perihelion, is dis- 
 tinctly visible to the naked eye. 
 
 320. Encke's Comet is so called from Professor Encke, of Ber- 
 lin, who first ascertained its periodical return.. It accomplishes 
 'ts revolution in the short period of 1207 days, or about 3^ years, 
 and moves in an orbit inclined under a small angle (13|) to the 
 
ENCKE S COMET. 
 
 
 119 
 
 plane of the ecliptic, and whose perihelion is at the distance of 
 the planet Mercury, and aphelion nearly at the distance of Jupiter. 
 
 
 Fig. 55. 
 
 (See Fig. 55.) This discovery was made on the occasion of its 
 fourth recorded appearance, in 1819. Since then, it has returned 
 several times to its perihelion, and in every instance very nearly 
 as predicted. Its last return took place in 1848 : its next will be 
 in March, 1852. This comet is also called the comet of short 
 period. 
 
 321. The motions of this comet present the anomalous fact, in the solar system, 
 of a period continually diminishing, and an orbit slowly contracting from some 
 other cause than the disturbing actions of the other bodies of the system. Professor 
 Encke finds, that after allowance has been made for all the perturbations produced 
 by the planets, the actual time of each perihelion passage anticipates the time cal- 
 culated from the duration of the previous revolution about 2 hours ; and that the 
 comet now arrives at its perihelion several days (about 21) 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 cannot be attributed 
 to the disturbing attraction of some unknown body, because this attraction would 
 produce other effects, which have not been noticed. Encke conceives that it can 
 arise from no other cause than the action of a resisting medium, or ether, in 
 space. The immediate effect of the resistance 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. We have a similar phenomenon to this 
 
, 
 
 120 OF THE MOTIONS OF THE COMETS. 
 
 in the familiar fact of the shortening of the arc of vibration, and consequent in- 
 crease of the rapidity of vibration of a pendulum, under the influence of the resist- 
 ance of the air. 
 
 322. Gambart's Comet was first seen by M. Biela, at Joseph- 
 stadt in Bohemia, on the 27th of February, 1826, and ten days af- 
 terwards by M. Gambart, at Marseilles. The latter calculated its 
 parabolic elements from his own observations, and on inspecting 
 a general table of comets discovered that the same comet had pre- 
 viously appeared in 1805 and 1772. Its period is about 6f years, 
 (2460 days.) Its orbit is inclined under an angle of 13 to the 
 plane of the ecliptic, and has its perihelion just within the orbit of 
 the earth, and aphelion -beyond the orbit of Jupiter, (see Fig. 55.) 
 By a remarkable coincidence, the orbit of this comet very nearly 
 intersects the orbit of the earth ; so nearly that if the two bodies 
 should ever chance to arrive at the point of crossing at the same 
 time, the earth would encounter a portion of the filmy mass of the 
 comet. It appeared, according to the prediction, in 1832; pass- 
 ing through its perihelion on the 27th of November. At its next 
 and last return, in 1839, it was not seen, owing to certain unfavor- 
 able circumstances, (see Art. 548.) It is announced that it will 
 again return to its perihelion on the llth of February, 1846, and 
 under favorable circumstances. (See Note V.) 
 
 Gambart's comet and Encke's comet both have a direct motion, 
 or in the order of the signs. 
 
 323. Halley's Comet is so called from Sir Edmund Halley, se- 
 cond Astronomer Royal of England, who ascertained its period, 
 and correctly predicted its return. From a comparison of the ele- 
 ments of the orbits described by the comets of 1531, 1607, and 
 1682, he concluded that the same comet had made its appearance 
 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 as- 
 tronomer, undertook the arduous task of calculating its perturba- 
 tions 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 eifect 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 35.3. Accordingly it approaches the sun to with- 
 in about one half the distance of the earth, and recedes from him 
 to nearly twice the distance of Uranus. (See Fig. 55.) 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 pas- 
 sage took place on the 16th of November, 1835, within a few days 
 
GREAT COMET OP 1843. 
 
 121 
 
 of the predicted time. The next will occur about the year 1911. 
 It is to be expected that the perturbations will now be determined 
 with such increased accuracy that the error in the prediction of 
 its next perihelion passage will be less than one day. 
 
 324. Besides the three comets whose motions have now been described, there are 
 three others, the orbits and periods of which are supposed to be known, but which 
 have not as yet returned to verify the predictions concerning them. These are 
 Olber's Comet of 1815, the Great Comet of 1843, and Faye's Comet or the third- 
 comet of 1843. The first and last are telescopic comets. (See Note VI.) 
 
 Fig. 56. 
 
 325. Olber's Comet is believed to accomplish a revolution around the sun in 75 
 years ; and to be destined to return to its perihelion early in the year 1887. 
 
 326. The astronomers of the High School Observatory in Philadelphia, and other 
 astronomers in Europe, suppose that they have identified the Great Comet of 1843 
 with the comets of 1668 and 1689, and predict its return about the beginning of 
 the year 1865. The probable identity of this comet with that of the year 1668, 
 
 16 
 
122 OF THE MOTIONS OF THE COMETS. 
 
 seems to be generally admitted by astronomers ; but more doubt is felt with re- 
 spect to the comet of 1689. Professor Peirce, of Harvard University, contends 
 that the arguments which have been offered in support of the identity of the com- 
 ets of 1843 and 1689 are insufficient; and finds, after an examination of the dif 
 ferent orbits which have been calculated, that the observations are, on the whole, 
 best satisfied by the elliptic orbit of the French astronomers Laugier and Mauvais, 
 which answers to a revolution of 175 years. 
 
 Fig. 56 shows the parabolic path of this comet, together with various correspond, 
 ing positions of the earth and comet, n is the ascending, and n' the descending 
 node : the perihelion, which is within 520,000 miles of the sun's centre, is not far 
 from midway between n and n 1 . The inclination of the orbit is 36. The comet 
 passed its perihelion on the 27th of February, at about 5 P. M., (Philadelphia 
 time.) On the 28th it was seen by day at various parts of New England, the 
 East and West Indies, and the south of Europe. It was then about 3 distant 
 from the sun, and of a dazzling brightness. After this 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. It was followed with the telescope at the 
 High School Observatory until the 10th of April. 
 
 327. Faye's Comet has a period of only about 7 years. Its perihelion is about 
 60 millions of miles without the earth's orbit, and aphelion somewhat beyond the 
 orbit of Jupiter. In respect to eccentricity, its orbit holds nearly a middle place 
 between those of the two comets of shortest period and the most eccentric planet- 
 ary orbits, (259.) The gradation is nearly as the fractions , , and f . 
 
 328. Of the 180 comets whose paths have been traced, about 
 an equal number have a direct and a retrograde motion. More 
 than two-thirds have the perihelia of their orbits within the orbit 
 of the earth. The aphelia, except in the few instances already 
 cited, are beyond the orbit of Uranus. Some have come into 
 close proximity to the sun. The great comet of 1680, according 
 to the computation of Newton, came 166 times nearer the sun 
 than the earth is. The no less remarkable comet of 1843 seems 
 to have approached still nearer to him. When at its perihelion it 
 was less than 100,000 miles from the sun's surface. Its velocity 
 at this time was 360 miles per second, and it accomplished a semi- 
 revolution (from n to n' in Fig. 56) in the remarkably short inter- 
 val of 2 hours. (See Note VII.) 
 
 There is little reason to doubt that many of the comets recede 
 tens of thousands of millions of miles from the sun before they 
 begin to return to him again. The periods of most of them are 
 told by centuries, and of very many of them by tens of centuries. 
 The planes of the orbits are inclined under every variety of angle 
 to the plane of the ecliptic. 
 
 329. The motions of the comets are liable to great derange- 
 ments, from the attractions of the planets. As their orbits cross 
 the orbits of the planets, they may come into proximity to these 
 bodies, and be strongly attracted by them. Halley's comet has 
 already (323) furnished an illustration of this general fact. The 
 comet of 1770, commonly called 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 Burck. 
 
i GREAT NUMBER OF COMETS 123 
 
 hardt undertook to investigate the cause of this phenomenon, and 
 found that, previous to the year 1767, the comet moved in an orbit 
 which answered to a period of 50 years, and never approached near 
 enough to the earth and sun to become visible. Early in the year 
 1767 it came so near the planet Jupiter that his attraction changed 
 its orbit to one of 5^ years. It thus became visible in 1770, and 
 would have again been seen on its return to the perihelion in 1776, 
 had it not been so situated with regard to the earth and sun as to 
 be continually hid by the sun's rays. In the year 1779 it again 
 met with Jupiter, and passed so near him that his attraction was 
 two hundred times greater than the attraction of the sun. The 
 consequence was that its orbit was greatly enlarged, and its period 
 lengthened to 20 years ; so that it no longer comes near enough to 
 the earth to be visible. 
 
 330. The number of recorded appearances of comets is about 
 500. But the actual number of cometary bodies connected with 
 the solar system is undoubtedly far greater than this. 
 
 This list comprises for the great number of years which precede the time of 
 the invention of the telescope, only those comets which were very conspicuous to 
 the naked eye, giving, for example, only three in the thirteenth and three in the 
 fourteenth century ; and since the heavens have begun to be attentively examined 
 with telescopes, from two to three comets, on an average, have made their appear- 
 ance every year, of which the great majority are telescopic. The periods of these, 
 as well as of the others, are, in general, of such vast length (328) that probably 
 not more than half of the whole number of comets have returned twice to their 
 perihelia during the last two thousand years. From these considerations it appears 
 that had the heavens been attentively surveyed with the telescope during the last 
 two thousand years, as many as 2500 different cometary bodies would have been 
 seen. But if we reflect that there are various causes which may tend to prevent 
 a comet from being seen when present in our firmament ; as unfavorable weather, 
 continued proximity to the sun, too grat distance from the sun and earth, (for all 
 distances seern equally probable, a priori.) want of intrinsic lustre, (for there is every 
 gradation of lustre from the highest to the lowest, and the fainter comets are the 
 most numerous,) &c., we shall see it to be highly probable that there are, in fact, 
 many thousands of these bodies. It is not difficult to perceive, as Arago has 
 shown, that the paucity of observed comets with large perihelion distances, though 
 apparently, is not in fact, opposed to the natural supposition that the perihelia are 
 distributed uniformly throughout the region of space which surrounds the sun, even 
 beyond the orbit of the most distant planet. Taking 30 as the number of comets 
 that come within the orbit of Mercury, this distinguished philosopher finds that 
 upon this supposition with respect to the distribution of the perihelia, the number 
 of comets which come within the precincts of the solar system is no less than 
 three millions and a half. 
 
 If the hypothesis upon which this estimate is based is anywhere near the truth, 
 then by far the greater number of the comets can never be seen from the earth ; 
 Cor uo comet has ever been visible at the distance of the orbit of Jupiter. 
 
124 OF THE MOTIONS OF THE SATELLITES. 
 
 CHAPTER XI. 
 
 OF THE MOTIONS OF THE SATELLITES. 
 
 331. As it has already been remarked, the planets which have 
 satellites are Jupiter, Saturn, and Uranus. The number of Jupi- 
 ter's satellites is four ; of Saturn's, eight ; of Uranus', six. 
 
 332. The satellites of Jupiter are perceptible with a telescope 
 of very moderate 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 
 st>me on each side. They are distinguished from each other by 
 the distance to which they recede from the planet, that which re- 
 cedes 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. 
 
 333. The satellites of Saturn and of Uranus cannot be seen 
 except through excellent telescopes. They experience changes of 
 apparent position, similar to those of Jupiter's satellites. 
 
 334. 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 con- 
 firmed 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 direction : 
 and again, while the satellite is passing from the western to the 
 eastern side, it often disappears, and after remaining for a time 
 invisible, reappears at another place. These phenomena are 
 easily explained, if we suppose that the planet and its satellites 
 are opake bodies illuminated by the sun, and that the satellites re- 
 volve around the planet from west to east. On this hypothesis, 
 the dark spot seen traversing the disc of the planet, is the shadow 
 cast upon it by the satellite on passing between the planet and the 
 sun, and the disappearance of the satellite is an eclipse, occasioned 
 by its entering the shadow of the planet. 
 
 As the transit of the shadow occurs during the passage of the 
 satellite from the eastern to the western side of the planet, and the 
 eclipse of the satellite during its passage from the western to the 
 eastern side, the direction of the motion must be from west to east 
 
 335 Analogous conclusions may be drawn from similar phe- 
 nomena 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. 
 
 336. Let us now examine into the principal circumstances of 
 
 
ECLIPSES OF JUPITER S SATELLITES. 
 
 125 
 
 the eclipses of Jupiter's satellites, and of the transits of their shad- 
 ows across the disc of the primary. Let EE'E" (Fig. 57) repre- 
 sent the orbit of the earth, PP'P" the orbit of Jupiter, and ss's" 
 that of one of its satel- Fig. 57. 
 
 lites. Suppose that E is 
 the. position of the earth, 
 and P that of the planet, 
 and conceive two lines, 
 aa f , bb', to be drawn tan- 
 gent to the sun and plan- 
 et : then, while the satel- 
 lite is moving from s to s' 
 it will be eclipsed, and 
 while it is moving from 
 f to f its shadow will 
 fall upon the planet. 
 Again, if Ee, Ee' repre- 
 sent two lines drawn from 
 the earth tangent to the 
 planet on either side, the 
 satellite will, while mov- 
 ing from g to g', traverse 
 the disc of the planet, 
 and while moving from h 
 to h', be behind the plan- 
 et, 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 
 opposition, to E' that of conjunction, the disappearances or immer~ 
 sions 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 east- 
 ern side of the planet, and that the immersions cannot be visible, 
 unless the line Fs, drawn from the earth to the point of immersion, 
 passes to the east of the planet. It appears from observation that 
 the immersion and emersion are never both visible at the same pe- 
 riod, 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. 
 
 337. The periods and other particulars of the motions of the 
 
126 OP THE MOTIONS OF THE SATELLITES. 
 
 satellites, result from observations upon their eclipses. The mid- 
 dle point of time between the satellite entering and emerging 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 interval be- 
 tween 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 revolu- 
 tion. The synodic revolution, however, being observed, and the 
 period of the primary being known, the periodic time of the satel- 
 lite may be computed. 
 
 338. 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 Ju- 
 piter has a small eccentricity ; that of the fourth a larger. 
 
 339. The distances of the satellites from their primary are de- 
 termined from micrometrical measurements of their apparent dis- 
 tances at the times of their greatest elongations. 
 
 A comparison of the mean distances of Jupiter's satellites with 
 their periodic times, proves that Kepler's third law with respect to 
 the planets applies also to these bodies ; or, that the squares of 
 their sidereal revolutions are as the cubes of their mean distances 
 from the primary. 
 
 The same law also has place with the satellites of Saturn and 
 Uranus. 
 
 340. The computation of the place of a satellite for a given time, 
 is effected upon similar principles with that of the place of a planet. 
 The mutual attractions of Jupiter's satellites occasion sensible per- 
 turbations of their motions, of which account must be taken when 
 it is desired to determine their places with accuracy. 
 
 341. Laplace has shown from the theory of gravitation, that, by 
 reason of the mutual attractions of the first three of Jupiter's satel- 
 lites, 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. 
 
 342. It follows from this last relation, that the longitudes of the 
 three satellites can never be the same at the same time, and conse- 
 quently that they can never be all eclipsed at once. 
 
SOLAR TIME. 127 
 
 CHAPTER XTI. 
 
 ON THE MEASUREMENT OF TIME 
 
 \ 
 
 DIFFERENT KINDS OF TIME. 
 
 843. 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 mo- 
 tion 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. 
 
 344. The sidereal day and the mean solar day are each of uni- 
 form 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 from two causes, viz : 
 
 (1.) The inequality of the sun'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' 11" 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. 39,) will be attended with a variation in 
 the daily arc of right ascension. The tendency of this cause is 
 obviously to make the daily arc of right ascension least at the 
 equinoxes, where the obliquity of the arc of longitude is greatest, 
 and greatest at the solstices, where the obliquity is least. 
 
 345. 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 uni- 
 formly in the equator with the real sun's mean motion, a day was 
 obtained of which the length is invariable, and equal to the mean 
 length of all the apparent solar days in a tropical year ; and by 
 supposing the fight ascension of this fictitious sun to be, at the 
 instant of the sun's arrival at the perigee of his orbit, equal to the 
 sun's true longitude, and consequently at all times equal to the 
 sun's mean longitude, the time deduced from its position with re- 
 
128 MEASUREMENT OF TIME. 
 
 spect to the meridian, was made to correspond very nearly witl. 
 apparent solar time. 
 
 346. 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, 24h. 3m. 56.555s. of sidereal time. Hence, a clock regula- 
 ted to sidereal time will gain 3m. 56.555s. in a mean solar day. 
 
 347. In order to find the expression for the sidereal day in 
 mean solar time, we must use 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, 3m. 55.908s. in a sidereal 
 day, and proportionally in other intervals. This is called the daily 
 acceleration of the fixed stars. 
 
 348. To express any given period of sidereal time in mean solar time, we must 
 subtract for each hour ' - = 9.83s., and for minutes and seconds in the 
 same proportion. And, on the other hand, to express any given period of mean 
 solar time in sidereal time, we must add for each hour = 9.86s., and 
 
 for minutes and seconds in the same proportion. 
 
 349. 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. 
 
 350. Another species of time, called Mean Equinoctial Time, has recently been 
 introduced to some extent into astronomical calculations. Mean equinoctial time 
 signifies the mean time elapsed since the instant of the Mean Vernal Equinox. Its 
 use is to afford a uniform date, which shall be independent of the different me- 
 ridians, and of all inequalities in the sun's motion, and shall thus save the neces- 
 sity, when speaking of the time of any event's happening, of mentioning at the 
 same time the place where it was observed or computed. Thus, it is the same 
 thing to say that a comet passed its perihelion on January 5th, 1837, at 5h. 47m. 
 0.0s., mean time at Greenwich ; at 5h. 5Gm. 21.5s., mean time at Paris ; or at 
 1836y. 289d. 6h. 16rn. 40.96s., equinoctial time ; but the former dates make the 
 localities of Greenwich and Paris enter as elements of the expression ; whereas 
 the latter expresses the period elapsed since an epoch common to all the world, 
 and identifiable independently of all localities. By this means, all ambiguities in 
 the reckoning of time are supposed to be avoided.* 
 
 CONVERSION OF ONE SPECIES OF TIME INTO ANOTHER. 
 
 351. The difference between the apparent and mean time is 
 called the Equation of Time. The equation of time, when known, 
 serves for the conversion of mean time into apparent, and the 
 reverse. 
 
 352. To find the equation of time. The hour angle of the sun 
 
 * (Nautical Almanac for 1837, p. 515.) 
 
CONVERSION OF APPARENT INTO MEAN SOLAR TIME. 129 
 
 (p. 15, def. 16) varies at the rate 
 of 360 in a solar day, or 15 per 
 solar hour. If, therefore, its value 
 at any moment be divided by 15, 
 the quotient will be the apparent 
 time at that moment. In like man- 
 ner, the hour angle of the mean 
 sun, divided by 15, gives the mean 
 time. Now, let the circle VSD 
 (Fig. 58) represent the equator, V 
 the vernal equinox, M the point of 
 the equator which is on the me- 
 ridian, and VS the right ascension 
 of the sun, and we shall have 
 
 MS 
 
 appar. time = 
 
 Fig. 58. 
 
 VM VS 
 
 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 = 
 
 15 
 
 15- 
 
 thus, equa. of time = mean time ap. time = 
 
 VS VS' 
 15 
 
 ..(74); 
 
 or, the equation of time is equal to the difference betiveen the 
 surfs true right ascension and mean longitude, converted into 
 time. 
 
 This rule will require some modification if very great accuracy is desired ; for, 
 in seeking an expression for the mean time, the circle VSD ought properly to be 
 considered as the mean equator, answering to the mean pole, (147), and the mean 
 longitude of the sun is really estimated from the mean equinox V, and ought there- 
 fore to be corrected by the arc W, or the equation of the equinoxes in right as- 
 cension, (147.) 
 
 The value of the equation of time, determined from formula 
 (74), 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.) The value of the equation of time 
 at noon, on any day of the year, is also to be found in the epheni- 
 eris of the sun, published in the Nautical Almanac and othei 
 works. If its value for any other time than noon be desired, il 
 may be obtained by simple proportion. 
 
 353. The equation of time is zero, or mean and true time are 
 the same four times in the year, viz., about the 15th of April, 
 the 15th of June, the 1st of September, and the 24th of Decem- 
 ber. Its greatest additive value (to apparent time) is about 14J 
 minutes, and occurs about the llth of February; and its greatest 
 
 17 
 
130 MEASUREMENT OF TIME. 
 
 subtractive value is about 16 minutes, and occurs about the 3d 
 of November. 
 
 354. To convert sidereal time into mean time, and vice versa. Making use 
 of Fig. 58 already employed, the arc VM, called the Right 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 cor- 
 rected for the equation of the equinoxes in right ascension, (352 ;) and MS' ex- 
 pressed in time, is the mean time. Let the arcs VM, MS', and VS', converted 
 into time, be denoted respectively by S, M, and L. Now, 
 
 VM = MS' -f- VS' ; 
 or, S = M + L.. (75); and M = S L.. (76). 
 
 If M -}- L in equation (75) exceeds 24 hours, 24 hours must be subtracted ; and 
 if L exceeds S in equation (76), 24 hours must be added to S, to render the sub- 
 traction possible. 
 
 This problem may in practice be solved most easily by means f^f an ephemeris 
 of the sun, which gives the value of S, or the sidereal time, at the instant of mean 
 noon of each day, together with a table of the acceleration of sidereal on mean 
 solar time, and the corresponding table of the retardation of mean on sidereal time. 
 
 355. The conversion of apparent time into sidereal, or sidereal time into appa- 
 rent, 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 CLOCKS 
 BY ASTRONOMICAL OBSERVATIONS. 
 
 356. The regulation of a clock consists in finding its error and 
 its rate. 
 
 357. The error of a mean solar clock is most conveniently de- 
 termined 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. 
 
 358. The daily rate of a mean solar clock may be ascertained 
 by finding as above the error at two successive apparent noons. 
 It 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. 
 
 359. To find the error of a sidereal clock, compute the true 
 right ascension of some one of the fixed stars, (see Prob. XXI,) 
 and note the time of its transit ; the difference between the time 
 observed and the right ascension in time will be the error. The 
 error of the daily rate is determined by observing two successive 
 transits of the same star. The variation of the time of the second 
 transit from that of the first will be the error in question. 
 
 The error and rate may be determined more accurately from 
 observations upon several stars, taking a mean of the individual 
 
DETERMINATION OF TIME. 
 
 131 
 
 Fig. 59. 
 
 results. Stars at a distance from the pole are to be selected, foi 
 reasons which have been already assigned, (58). 
 
 360. In default of a transit instrument, the time may be obtain- 
 ed and time-keepers regulated by observations made out of the 
 meridian. There are two methods by which this may be accom- 
 plished, 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 re- 
 fraction and parallax, and also, if 
 the sextant is the instrument used, 
 for the semi-diameter of the sun. 
 Then, if Z (Fig. 59) 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, from which we may compute 
 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 fol- 
 lowing equations, (App., Resolution of oblique-angled spherical 
 triangles, Case 1,) 
 
 2 k = ZP + PS + ZS= co-lat. + co-dec. + co-alt. . . . (77); 
 
 sin-PS) 
 
 ' 
 
 sm ZP sin PS 
 
 or, 
 
 i n 2 ip = sin (k co-lat.) sin (ft- co-dec.) 
 sin (co-lat.) sin (co-dec.) 
 
 sin- 
 
 The value of P being derived from these equations and convert- 
 ed into time, (see Prob. Ill,) 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 regarded 
 as corresponding to the mean of the times. The correspondence will be sufficiently 
 exact if the measurements be all made within the space of 10 or 12 minutes, and 
 when the sun is near the prime vertical. If an even number of altitudes be taken, 
 and alternately of the upper and lower limb, the mean of the whole will give the 
 altitude of the sun's centre, without it being necessary to know his apparent semi- 
 diameter. In practice, the declination of the sun may be taken for the solution of 
 this problem from an ephemeris of the sua. For this purpose the time of the ob 
 lervation and the longitude of the place must be approximately known. 
 
132 
 
 MEASUREMENT OF TIME. 
 
 Example. On the 1st of June, 1838, at about lOh. 45m. A. M. 
 the altitude of the sun's lower limb was measured at New York 
 with a sextant, and found to be 64 55' 5". What was the correct 
 time of the observation ? ' 
 
 Measured alt. of the sun's lower limb, . 64 55' 5" 
 Sun's semi-diam., by Conn, des Terns, . 15 47 
 
 Appar. alt. of sun's centre, ';'.?" . 
 Parallax in alt., (Table X), ., 
 Refraction, (Table VIII), ;> . 
 
 True alt. of sun's centre, 
 *>* 
 
 N. York approx. time of observation, 
 Diff. of long, of Paris and N. York, 
 
 Paris approx. time of obs., '' ".' 
 
 Sun's declin. June 1st, M. noon at Paris, 
 " " June 2d, 
 
 65 
 
 10 52 
 
 + 4 
 
 27 
 
 65 10 29 
 
 lOh. 45m. 
 5 5 
 
 3 50 P. 
 
 22 2' 27" 
 22 10 31 
 
 8 4 
 
 Change of declin. in 24 hours, 
 
 24h.:8'4": : 3h. 50m.: 1' 17". 
 Declin. June 1st, M. noon at Paris, 22 2' 27" 
 
 Change of declin. in 3h. 50m., 
 Declin at time of obs., 
 
 90 0' 0" 
 Lat. of N. York, 40 42 40 
 
 1 17 
 
 Co-lat. . 
 Co-dec. . 
 Co-alt. . 
 
 k . 
 k co-lat. 
 k co-dec. 
 
 . 49 
 . 67 
 . 24 
 
 17 20 
 56 16 
 49 31 
 
 2)142 
 
 3 7 
 
 . 71 
 . 21 
 . 3 
 
 1 33 
 44 13 
 5 17 
 
 22 3 44 
 
 ar. co. sin. 0.12033 
 ar. co. sin. 0.03303 
 
 sin. 9.56861 
 sin. 8.73135 
 
 JP= 9 42 7.5. 
 P = 19 24 15 
 4 
 
 Ih. 17m. 37s. 0'" 
 
 10 42 23 A. M. 
 Equa. of time, 2 34 
 
 M. time of obs. 10 39 49 A. M. 
 
 2)18.45332 
 9.22666 
 
THE CALENDAR. 133 
 
 In case the altitude of a star is taken, the value of P derived from formula (79), 
 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 (in- 
 creased 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 hori- 
 zon, he would have equal altitudes at equal times before and after 
 apparent noon. Hence, if to the time of the first observation one 
 half the interval of time between the two observations should be 
 added, the result would be the time of noon, as shown by the clock 
 or watch employed to note the times of the observations. The 
 deviation from 12 o'clock would be the error of the clock with re- 
 spect 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 chang- 
 ing, equal altitudes will not have place precisely at equal times be- 
 fore and after noon, and it is therefore necessary, in order to obtain 
 an exact result, to apply a correction to the time thus obtained 
 This correction is called the Equation of Equal Altitudes. Tables 
 have been constructed by the aid of which the equation is easily 
 obtained. This is at the same time a very simple and very accu- 
 rate method of finding the time and the error of a clock. 
 
 If equal altitudes of a star should be observed, it is evident that 
 half the interval of time elapsed w r ould give the time of the star 
 passing the meridian, without any correction. From this the error 
 of the clock (if keeping sidereal time) may be found, as explained 
 in Art. 359. 
 
 OF THE CALENDAR. 
 
 361. The apparent motions of the sun, which bring about the 
 regular succession of day and night and the vicissitude of the sea- 
 sons, and the motion of the moon to and from the sun in the heav- 
 ens, attended with conspicuous and regularly recurring changes in 
 her disc, furnish three natural periods for the measurement of the 
 lapse of time, viz. 1, the period of the apparent revolution of the 
 sun with respect to the meridian, comprising the two natural pe- 
 riods 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 her 
 phases and returns to the same position relative to the sun, called 
 a 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. 
 
134 MEASUREMENT OF TIME. 
 
 The tropical year contains 365d. 5h. 48m. 48s. The lunar month 
 consists of about 29| days. The week, consisting of seven days, 
 has its origin in Divine appointment alone. A Calendar is a scheme 
 for taking note of the lapse of time, and fixing the dates of occur- 
 rences, 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 ad- 
 justment or regulation of the calendar by astronomical observa- 
 tions has in all ages and with all nations been an object of the 
 highest importance. We propose, in what follows, to explain only 
 the Julian and Gregorian Calendars. 
 
 362. The Julian calendar divides the year into 12 months, con- 
 taining in all 365 days. Now, it is desirable that the calendar 
 should always denote the same parts of the same season by the 
 same days of the same months : that, for instance, the summer and 
 winter solstices, if once happening on the 21st of June and 21st 
 of December, should ever after be reckoned to happen on the same 
 days ; that the date of the sun's entering the equinox, the natural 
 commencement of spring, should, if once, be always on the 20th 
 of March. For thus the labors of agriculture, which really depend 
 on the situation of the sun in the heavens, would be simply and 
 truly regulated by the calendar. 
 
 This would happen, if the civil year of 365 days were equal to 
 the astronomical ; but the latter is greater ; therefore, if the cal- 
 endar should invariably distribute the year into 365 days, it would 
 fall into this kind of confusion, that in process of time, and suc- 
 cessively, 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 com- 
 plete the astronomical or tropical year. At the completions of a 
 second and a third civil year, the sun would be still more and more 
 remote from the equinoctial point, and would be obliged to move, 
 respectively, for 12 and 18 hours before he could rejoin it and com- 
 plete the astronomical year. 
 
 At the completion ol a fourth civil year the sun would be more 
 distant than on the two preceding ones from the equinoctial point. 
 In order to rejoin it, and to complete the astronomical year, he 
 must move for 24 hours ; that is, for one whole day. In other 
 words, the astronomical year would not be completed till the be- 
 ginning of the next astronomical day ; till, in civil reckoning, the 
 noon of March 21 st. 
 
 At the end of four more common civil years, the sun would be 
 
THE CALENDAR. 135 
 
 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 Septem- 
 ber 20th ; and in a period of about 1508 years, the sun would 
 have been in every sign of the zodiac on the same day of the cal- 
 endar, and in the same sign on every day. 
 
 363. 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 most easily avoided. It would be necessa- 
 ry 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 the 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. 
 
 364. The astronomical year being equal to 365d. 5h. 48m. 47.6s., 
 it is less than the mean Julian by llm. 12.4s. or 0.007782d. The 
 Julian correction, therefore, itself needs correction. The calendar 
 regulated by it would, in process of time, become erroneous, and 
 would require reformation. 
 
 The intercalation of the Julian correction being too great, its 
 effect would be to antedate the happening of the equinox. Thus 
 (to return to the old illustration) the sun, at the completion of the 
 fourth civil year, now the Bissextile, would have passed the equi- 
 noctial point by a time equal to four times 0.007782d. ; at the end 
 of the next Bissextile, by eight times 0.007782d. ; at the end of 
 1 30 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 19th. 
 
 In the lapse of ages this error would continue and be increased. 
 Its accumulation in 1300 years would amount to 10 days, and then 
 the vernal equinox would be reckoned to happen on March 10th 
 
 365. The error into which the calendar had fallen, and would 
 continue to fall, was noticed by Pope Gregory XIII. in 1582. At 
 his time the length of the yearwas known to greater precision than 
 at the time of Julius Caesar. It was supposed equal to 365d. 5h. 
 49m. 16.23s. Gregory, desirous that the vernal equinox should 
 be reckoned on or near March 21st, (on which day it happened in 
 the year 325, when the Council of Nice was held,) ordered that 
 the day succeeding the 4th of October, 1582, instead of being 
 called the 5th, should be called the 15th: thus suppressing 10 
 days, which, in the interval between the years 325 and 1582, 
 
136 MEASUREMENT OF TIME. 
 
 represented nearly the accumulation of error arising from the ex- 
 cessive 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 in- 
 tercalary 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 interca- 
 lary 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 that is di- 
 visible 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. 
 
 366. This is a most simple mode of regulating the calendar. It 
 corrects the insufficiency of the Julian correction, by omitting, in 
 the space of 400 years, 3 intercalary days. And it is easy to esti- 
 mate the degree of its accuracy. For the real error of the Julian 
 correction is 0.007782d. in 1 year, consequently 400 x 0.007782d. 
 or 3.1128d. in 400 years. Consequently, 0.1128d. or 2h. 42m. 
 26s. in 400 years, or 1 day in 3546 years, is the measure of the 
 degree of inaccuracy in the Gregorian correction. 
 
 367. The Gregorian calendar was adopted immediately on its 
 promulgation, in all Catholic countries, but in those where the 
 Protestant religion prevailed, it did not obtain a place till some 
 time after. In England, " the- change of style," as it was called, 
 took place after the 2d of September, 1752, eleven nominal days 
 being then struck out ; so that the last day of Old Style being the 
 2d, the first of New Style (the next day) was called the 14th, in- 
 stead of the 3d. The same legislative enactment which estab- 
 lished the Gregorian calendar in England, changed the time of the 
 beginning of the year from the 25th of March to the 1st of January. 
 Thus the year 1752, which by the old reckoning would have com- 
 menced 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 1 800, there is now, for all dates, 1 2 days 
 difference between the old and new style. 
 
 Russia is at present the only Christian country in which the 
 Gregorian calendar is not used. 
 
 368. The calendar months consist, each of them, of 30 or 31 
 days, except the second month, Februaiy, which, in a common 
 year, contains 28 days, and in a Bissextile, 29 days ; the interca- 
 lary day being added at the last of this month. 
 
 369. To find the number of days comprised in any number of 
 civil years, multiply 365 by the number of years, and add to the 
 product as many days as there are Bissextile years in the period. 
 
PART II. 
 
 ^a iA- 
 
 ON THE PHENOMENA RESULTING FROM THE MOTIONS OF THE 
 
 HEAVENLY BODIES, AND ON THEIR APPEARANCES, DIMEN- 
 SIONS, AND PHYSICAL CONSTITUTION, 
 
 CHAPTER XIII. 
 
 OP THE SUN AND THE PHENOMENA ATTENDING ITS APPARENT 
 MOTIONS. 
 
 INEQUALITY OF DAYS * 
 
 370. WE will first give a. detailed description of the sun's ap- 
 parent motion with respect to the equator, the phenomenon upon 
 which the inequality of days (as well as the vicissitude of the 
 seasons, soon to be treated of) immediately depends. 
 
 Fig. 60. 
 
 Let VEAQ (Fig. 60) represent 
 the equator, VT AW (inclined to 
 VEAQ, under the angle TOE, 
 measured by the arc TE, equal 
 to 23^,) the ecliptic, TnZ and 
 Wn'Z' the two tropics, POP' the 
 axis of the heavens, and PEP'Q 
 the meridian and HVRA the ho- B 
 rizon in one of their various po- 
 sitions with respect to the other 
 circles. About the 21st of March 
 the sun is in the vernal equinox 
 V, crossing the equator in the 
 oblique direction VS, towards the 
 north and east. At this time its diurnal circle is identical with the 
 equator, and it crosses the meridian at the point E, south of the 
 zenith a distance ZE equal to the latitude of the place. Ad- 
 vancing towards the east and north, it takes up the successive 
 positions S, S', S", &c., and from day to day crosses the meridian 
 at r, r', &c., farther and farther to the north. Its diurnal circles 
 will be, respectively, the northern parallels of declination passing 
 through S, S', S", &c., 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 2 1st of 
 June, when he reaches the summer solstice T. At this point he 
 
 * The day, here considered, is the interval between sunrise and sunset. 
 
 18 
 
138 OF THE SUN AND ITS PHENOMENA. 
 
 moves for a short time parallel to the equator : his declinatior 
 changes but slightly for several days, and he crosses the meridian 
 from day 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, 
 TraZ, 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.! 
 The sun is, also, when at the summer solstice, at its point of near- 
 est approach to the zenith of every place whose latitude ZE ex- 
 ceeds the obliquity of the ecliptic TE, equal to 23. The distance 
 ZT = ZE ET = latitude obliquity of ecliptic. During 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 f , &c., 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 meridian at the points s", s', 
 <fec., so that on the 21st of March it is again at the vernal equinox. 
 371. To explain now the phenomenon of the inequality of days 
 which obtains at all places north or south of the equator. At all 
 such places, the observer is in an oblique sphere ; that is, the ce- 
 lestial equator and the parallels of declination are oblique to the 
 horizon. This position of the sphere is represented in Fig. 1 1 , 
 p. 21, where HOR is the horizon, QOE the equator, and ncr, set, 
 &c., parallels of declination ; WOT is the ecliptic. It is also rep- 
 resented in Fig. 60, from which Fig. 1 1 differs chiefly in this, that 
 the horizon, equator, ecliptic, and parallels of declination, which 
 are stereographically represented as ellipses in Fig. 60, are in Fig. 
 1 1 orthographically 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, (35,) will be divided into un- 
 equal parts, and that the disparity between the parts will be greater 
 
 * Fiom Sol, the sun, and sto, to stand. t From rpeirw, to turn. 
 
INEQUALITY OF DAYS. 139 
 
 in proportion as the parallel is more distant from the equator; 
 also, that to the north of the equator the greater parts will lie above 
 the horizon, and to the south of the equator below the horizon. 
 Now, the length of the day is measured by the portion of the 
 parallel to the equator, described by the sun, which lies above the 
 horizon; and if^is evident, from what has just been stated, that 
 (as it is showrWby the Fig.) this increases continually from the 
 .winter solstice W to the summer solstice T, and diminishes con- 
 tinually from the summer 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. 
 
 372. As the equator is bisected by the horizon, at the equinoxes 
 the day and night must be each 12 hours long. 
 
 373. When the sun is north of the equator, the greater part of 
 its diurnal circle lies above the horizon, in northern latitudes ; and, 
 therefore, from the vernal to the autumnal equinox the day is, in 
 the northern hemisphere, more than 12 hours in length. On the 
 other hand, when the sun is south of the equator, the greater part 
 of its circle lies below the horizon, and hence from the autumnal 
 to the vernal equinox the day is less than 1 2 hours in length. 
 
 In the latter interval the nights will obviously, at corresponding 
 periods, be of the same length as the days in the former. 
 
 374. The variation in the length of the day in the course of the 
 year, will increase with the latitude of the place ; for the greater 
 is the latitude, the more oblique are the circles described by the 
 sun to the horizon, and the greater is the disparity between the 
 parts into which they are divided by the horizon. This will be 
 obvious, on referring to Fig. 11, p. 21, where HOR, H'OR', rep- 
 resent the positions of the horizons of two different 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. 
 
 375. At the equator the horizon bisects ail the diurnal circles, 
 (36,) and consequently, the day and night are there each 12 hours 
 in length throughout the year. 
 
 376. At the arctic circle the day will be 24 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 
 circle ; whence it follows, that the diurnal circle of the sun at this 
 epoch, will correspond to the circle of perpetual apparition for the 
 parallel in question. 
 
 On the other hand, when the sun is at the winter solstice, the 
 night will be 24 hours long on the arctic circle. 
 
 377. To the north of the arctic circle, the sun will remain con- 
 
140 OF THE SUN AND ITS PHENOMENA. 
 
 tinually above the horizon during the period, before and after the 
 summer solstice, that his north polar distance is less than the lati- 
 tude of the place, and continually below the horizon during the 
 Eeriod, about the winter solstice, that his south polar distance is 
 jss than the latitude of the place. 
 
 At the north pole, as the horizon is coincident with the equator, 
 (37,) the sun will be above the horizon while passing from the ver- 
 nal 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. % 
 
 378. The circumstances of the duration of light and darkness 
 are obviously the same in the southern hemisphere as in the north- 
 ern, for corresponding latitudes and corresponding declinations of 
 the sun. 
 
 379. The latitude of the place and the declination of the sun 
 being given, to find the times of the surfs rising and setting and 
 the length of the day. 
 
 Fig. 61. ' LetHPR(Fig. 61) 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 converted into 
 time expresses the interval between 
 the rising or setting of the sun and 
 his passage over the meridian, is 
 called the Semi-diurnal Arc. Now, 
 
 E* = EM + M* = 90 + M, 
 
 which gives 
 
 cos E = sin M ; 
 and we have, by Napier's first rule, 
 
 sin Mt =cot tM.s tang ts -- tang PMH tang EB =tang PH tang EB : 
 whence, cos E = tang PH tang EB, 
 
 or, cos (semi-diurnal arc) = -r- tang lat. x tang dec. . . (80). 
 
 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. 
 
 Log. tang lat. (40 42' 40") . .- . . 9.93474 
 Log. tang dec. (23 27' 40") . . 9.63749 
 
 Log. cos (semi-diurnal arc) . . . 9.57223 
 
TWILIGHT. 14) 
 
 Semi-diurnal arc ,-,:> . * $& '?**$ HI 55' 40" 
 Time of sun's setting >, v ..-. ... , ? , ; 7h. 27m. 43s. 
 Time of sun's rising - *. '^.. ; .^ . * : 4 32 17 
 Length of day . . > <( ,i ;,<V N 5 ^ > 14 55 26 
 
 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. 28s. and 8h. 53m. 32s. 
 
 Exam. 3. At what hours did the sun rise and set on May 1st, 
 1837, at Charleston; the latitude of Charleston being 32 47', and 
 the declination of the sun being 15 6' 0" N. ? 
 
 Ans. Time of rising, 5h. 19m. 58s. Time of setting, 6h. 40m. 
 2s. 
 
 380. To find the time of the surfs apparent rising or setting; 
 the latitude of the place and the declination of the sun being given. 
 
 At the time of his apparent rising or setting, the sun as seen from 
 the centre of the earth will be below the horizon a distance sS 
 (Fig. 61) equal to the refraction minus the parallax. The mean 
 difference of these quantities is 33' 42". Let it be denoted by R. 
 Now, to find the hour angle ZPS(=P), the triangle ZPS gives, 
 (see Appendix,) 
 
 f ZP + PS + ZS co-lat. + co-dec. -f (90-f R) 
 k = - ~2~ ~2~ --..(81) 
 
 |1D sin (k -ZP) sin (k -PS) 
 
 and siir |r = : ^^ . po , 
 
 sin ZP sin PS 
 
 . _ sin (k co-lat.) sin (A; co-dec.) /0rtX 
 
 or, sin a JP = --r. r ^^ -j , '- . . . (82). 
 
 sin (co-lat.) sin (co-dec.) 
 
 The value of P (in time) will be the interval between apparent 
 noon and the time of the apparent rising or setting. 
 
 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 33' 42" -f 
 sun's semi-diameter, or 49' 43". 
 
 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. When the greatest precision is required, the times of true 
 rising and setting must be computed by equation (80), and the de- 
 clinations found for these times. 
 
 TWILIGHT. 
 
 381. 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 reflected down upon 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 the 
 air posited above the horizon, that is directly illuminated, becomes 
 less. The same effect, though in a reverse order, takes place in 
 
142 
 
 OF TIJE SUN AND ITS PHENOMENA. 
 
 the morning previous to the sun's rising. The light thus produced 
 is called the Crepusculum, or Twilight. This explanation of twi- 
 light will be better understood on examining Fig. 62, where AON 
 represents a portion of the earth's surface, H/cR the surface of the 
 
 Fig. 62. 
 
 atmosphere above it, and kmS a line drawn touching the earth and 
 passing through the sun. The unshaded portion, kcR, of the body 
 of air which lies above the plane of the horizon HOR, is still illu- 
 minated by the sun, and shines down, by reflection, upon O the 
 station of the observer. As the sun descends this will decrease, 
 until finally when the sun is in the direction RNS' he will illumi- 
 nate directly none of that part of the atmosphere which lies above 
 the horizon, and twilight will be at an end. 
 
 382. The close of the evening twilight is marked by the ap- 
 pearance of faint stars over the western horizon, and the beginning 
 of the morning twilight by the disappearance of faint stars situated 
 in the vicinity of the eastern horizon. It has been ascertained from 
 numerous observations, that, at the beginning of the morning and 
 end of the evening twilight, the sun is about 18 below the horizon. 
 
 383. At this time, then, the angle TRS' is equal to 18. This datum will ena- 
 ble us to calculate the approximate height of the atmosphere. For if the verticals 
 at O, m, and N be produced to the centre of the earth, we shall have the angle 
 OCN equal to TRS', or 18, and therefore OCR equal to 9 ; and thus the height 
 of the atmosphere, mR, equal to CR Cm, equal to secant of 9 radius. Making 
 the calculation, we find the height of the atmosphere to be about 47 miles. It is 
 to be understood that this is only a rough approximation. 
 
 It will be seen, on inspecting Fig. 62, that twilight would continue longer if the 
 atmosphere were higher. 
 
 384. The latitude of the place and the sun's declination being 
 given, to find the time of the beginning or end of twilight. 
 
 The zenith distance of the sun at the beginning of morning or 
 end of evening twilight, is 90 -f 18 : wherefore we may solve this 
 problem by means of equations (81) and (82), 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. 
 
 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 
 
TWILIGHT. 
 
 143 
 
 pole at a place the latitude of which is 49, differs only 18 from 
 the polar distance of the sun at this epoch. This may be illustra- 
 ted by Fig. 60, taking Z as the point of passage of the sun across 
 the inferior meridian, PZ=67, and PH =49. At this latitude, 
 therefore, twilight will 'continue all night, at the summer solstice. 
 This will be true for a still stronger reason at higher latitudes. 
 
 385. The duration of twilight varies with the latitude of the 
 place and with the time of the year. At all places 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, dif- 
 ferent for each latitude.* The summer twilights increase in length 
 from the equator northward. 
 
 These facts are consequences of the different situations with respect to the hori- 
 zon of the centres 6f 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 sun, comprised between the horizon and the crepusculum circle, which, for the 
 sake of brevity, we will call the arc of twilight : and this will vary from the two causes 
 just mentioned. For, let hkr (Fig. 63) represent the equator, and h'k'r' a diurnal 
 
 Fig. 63. 
 
 circle described by the sun when north 
 of the equator ; and let hr, st, and A V, 
 s't'f be the intersections of the equator 
 and diurnal circle, respectively, with the 
 planes of the horizon and crepusculum 
 circle. When the sun is in the equator, 
 the arc of twilight is hs, and when he is 
 on the parallel of declination h'k'r 1 it is 
 AY. Draw the chords hs, AY, mn, and 
 the radii cs, cs 1 , cr 1 , en, cp. The angle /* 
 r'AY is the half of r'cs 1 , and the angle f 
 pmn is the half of pen : but r'cs' is less 
 than pen, and therefore r'AY is less than 
 pmn. Again, chs is the half of res, and 
 therefore greater than pmn, the half of 
 the less angle pen. Whence it appears 
 that the chord AY is more oblique to the 
 horizon, and therefore greater than the 
 chord mn, and this more oblique and greater than the chord As. It follows, there- 
 fore, that the arc AY is greater, and contains a greater number of degrees than the 
 arc mn, arid that this arc is greater than A*. Thus, as the sun recedes from the 
 equator towards the north, the arc of twilight, and therefore the duration of twilight, 
 increases from two causes, viz : 1st. The increase in the distance of the line of in- 
 tersection of the horizon with the diurnal circle from the centre of the circle ; and, 
 2d. The diminution in the size of the circle. The change will manifestly be greater 
 in proportion as the latitude is greater. 
 
 * The duration of shortest twilight is given by the following formula : 
 
 sin 9 
 
 ~ coslat. 
 
 Twice the angle a, converted into time, expresses the duration of shortest twilight 
 To find the sun's declination at the time of shortest twilight, we have 
 
 sin dec. = tang 9 sin lat 
 
 (For the investigation of this and the preceding formula, see Gummere's Astrono- 
 my, pages 87 and 88.) 
 
144 OF THE SUN AND ITS PHENOMENA. 
 
 When the sun is south of the equator twilight will, for the same declination, 
 be shorter than when he is north of the equator, because, although the diurnal cir 
 cle will be of the same size, and its intersection with the horizon at the same dis- 
 tance from its centre, the intersection with the crepusculum 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 will be shorter. 
 
 The shortest twilight occurs when the sun is somewhat to the south of the equa- 
 tor, because the arc of twilight, for a time, decreases by reason of the diminution 
 of its obliquity 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 sun 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 between the horizon 
 and the crepusculum circle ; which will happen when the sun is a certain dis- 
 tance 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 at- 
 mosphere. Twilight also increases in length with the obliquity of the sphere. 
 
 386. At the poles 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 hori- 
 zon at the poles is identical with the celestial equator, the twilight 
 which precedes the long day of six months will begin when the sun 
 in approaching the equator, upon the other side, attains to a decli- 
 nation of 18, and this will be about 50 days before he reaches the 
 equator and rises at the pole. In like manner the evening twilight 
 continues until the sun has descended 18 below the equator. 
 
 THE SEASONS. 
 
 387. The amount of heat received from the sun in the course 
 of 24 hours, depends upon two particulars ; the time of the sun's 
 continuance above the horizon, and the obliquity of his rays at 
 noon. Ey reason of the obliquity of the ecliptic, both of these cir- 
 cumstances vary materially in the course of the year ; whence 
 arises a variation of temperature or a change of seasons. 
 
 388. The tropics and the polar circles divide the earth into five 
 parts, called Zones, throughout each of which the yearly change 
 of the temperature is occasioned by a similar change in the cir- 
 cumstances upon which it depends. 
 
 The 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 circles, are called Frigid Zones. 
 
 389. At all places in the nprth temperate zone the sun will al- 
 ways pass the meridian to the south of the zenith ; for the latitudes 
 of all such places exceed 23^, the greatest decimation of the sun. 
 (See Fig. 60.) The meridian zenith distance will be greatest at 
 the winter solstice, when the sun has its greatest southern decli- 
 nation, and least at the summer solstice, when the sun has* its 
 greatest northern declination ; and it will vary continually between 
 

 THE SEASONS. 145 
 
 the values which obtain at these epochs. The day will be longest 
 at the summer solstice, and the shortest at the winter solstice, and 
 will vary in length progressively from the one date to the other. 
 
 We infer, therefore, thaF throughout the zone in question the 
 greatest amount of heat will be received from the sun at the sum- 
 mer 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 maximum 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 during 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. 
 
 390. Within the torrid zone the length of the day varies after 
 the same manner as in the temperate zone, though in a less de- 
 gree ; but the motion of the sun with respect to the zenith is 
 different. At all places in the torrid zone the sun passes the me- 
 ridian during 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. Throughout 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 winter 
 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. 
 
 391 . At the 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 winter solstice. 
 
 392. The amount of the yearly variation of temperature in- 
 creases with the latitude of the place ; for the greater is the lati- 
 tude 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 approach 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 obliquity of the sun's rays at noon, and this increases 
 with the latitude. 
 
 393. The yearly change in the sun's distance from the earth has 
 but little effect in producing a variation of temperature upon the 
 
 19 
 
146 OF THE SUN AND ITS PHENOMENA. 
 
 earth's surface. The change of its heating power from this cause 
 amounts to no more than / y . 
 
 394. It is important to observe, that, although in the main cli- 
 mate varies with the latitude after ttt% manner explained in the 
 foregoing articles, it is still dependent more or less upon local 
 circumstances, such as the vicinity of lakes, seas, and mountains, 
 prevailing winds of some particular direction, &c. 
 
 395. In the north temperate zone, Spring, Summer, Autumn, 
 and Winter, the four seasons into which the year is divided, are 
 considered as respectively commencing at the times of the Ver- 
 nal Equinox, Summer Solstice, Autumnal Equinox, and Winter 
 Solstice. 
 
 Let V (Fig. 64) represent the vernal, and A the autumnal equi- 
 nox ; S the summer, and W the winter solstice. The perigee of 
 
 Fig. 64. 
 
 the sun's apparent orbit is at present about 10 15' 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, respectively 
 proportional to the 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 8 days longer than autumn and winter united. 
 
 Since the perigee of the sun's orbit has a progressive motion, 
 the relative lengths of the seasons must be subject to a continual 
 variation. 
 
 396. At the beginning of the year 1800, the longitude of the 
 sun's perigee was 279 SO' 8",39. If from this we take 180, the 
 longitude of the autumnal equinox, the remainder, 99 30' 8". 39, 
 is the distance of the perigee from the autumnal equinox at that 
 epoch. The motion of the perigee in longitude is at the rate of 
 61".52 per year. Dividing 99 30' 8".39 by 61".52, the quotient 
 is 5822. Hence it appears that about 5800 years anterior to the 
 
DIMENSIONS OF THE SUN. 147 
 
 year 1800, the perigee coincided with the autumnal equinox, and 
 the apogee with the vernal equinox. 
 
 397. It is important to observe that the primary cause of the 
 phenomenon of change of seasons, as well as of that of the ine- 
 quality of days, is the inclination of the earth's axis of rotation to 
 the perpendicular to the plane of its orbit, since this is the occa- 
 sion of the obliquity of the ecliptic, upon which, as we have seen, 
 these phenomena immediately depend. If the axis of rotation 
 were perpendicular to the plane of the orbit, there would neither 
 be a change of seasons nor any inequality in the length of the days 
 and nights. 
 
 APPEARANCE, DIMENSIONS, AND PHYSICAL CONSTITUTION 
 
 OF THE SUN. 
 
 398. The sun presents the appearance of a luminous circular 
 disc. But it does not necessarily follow from this that its surface 
 is really flat ; for such is the appearance of all globular bodies 
 when viewed at a great distance. It is ascertained from observa- 
 tions with the telescope, that the sun has a rotatory motion : this be- 
 ing the fact, its surface must in reality be of a spherical form ; for 
 otherwise it would not, in presenting all its sides, always appear 
 under the form of a circle. 
 
 399. The sun's real diameter is determined from his apparent 
 diameter and horizontal parallax. Fig. 65. 
 
 Let ACB (Fig. 65) represent the 
 sun or other heavenly body, and 
 E the place of the earth ; and 
 let 5 = AEB the sun's apparent 
 diameter, d 2AS his real di- 
 ameter, D = ES his distance 
 from the earth, and R = the radius of the earth. We have, from 
 the triangle AES, 
 
 AS = ES sin 1AEB, or, 2AS = 2ES sin JAEB ; 
 and thus, d = 2D sin <5 : 
 
 but, (equa. 7,) D== ^TH> 
 
 whence, i^^^O^^^fj^ 
 
 The mean apparent diameter of the sun is 32' 1".8, and his 
 mean horizontal parallax 8".58. Accordingly we have, for the 
 real diameter of the sun, 
 
 OO/ 1 // Q 
 
 ^ = 2R W r^y=2R xi 12 (nearly.) 
 
 Thus the diameter of the sun is about 112 times the diameter 
 of the earth. The volume of the sun then exceeds that of the 
 earth nearly in the proportion 112 3 to I 3 , or 1,404,928 to I. 
 
148 OF THE SUN AND ITS PHENOMENA. 
 
 From equation (83) we may derive the proportion 
 d : 2R : : 5 : 2H. 
 
 Thus, the real diameter of a heavenly body is to the diameter 
 of the earthy as the apparent diameter of the body is to double its 
 horizontal parallax. 
 
 400. When the sun is viewed with a telescope of considerable 
 power, and provided with colored glasses, black spots of an irreg- 
 ular form, surrounded by a dark border of a nearly uniform shade, 
 
 Fig. 66. called a penumbra, are often seen 
 
 on its disc, (see Fig. 66.) Some- 
 times several spots are included 
 i:. within the same penumbra. Their 
 num ber, magnitude, and position 
 - : i;;:;||$f :::: on the disc, are extremely variable. 
 In some years they are very fre- 
 quent, and appear in large numbers ; 
 in others, none whatever are seen. 
 ^ n some instances more than one 
 hundred, of various forms and sizes, 
 have been counted. They usually 
 appear in clusters, composed of various numbers, from two to sixty 
 or a hundred. Their absolute magnitude 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 11" : 
 the breadth of such spots must therefore exceed three times the 
 diameter of the earth, or 24,000 miles. Spots two or three times 
 as large as this, or about three times as great as the entire surface 
 of our globe, have been seen. 
 
 401. 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 forms. When a spot disappears, 
 it always contracts into a point, and vanishes before the penumbra. 
 Some spots disappear almost immediately after they become visi- 
 ble ; others remain for weeks, or even months. 
 
 402. Spots and streaks more luminous than the general body 
 of the sun, and of a mottled appearance, are also frequently per- 
 ceived 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 Fa- 
 culce. They are chiefly to be seen near the margin of the disc. 
 The penumbra which surrounds each black spot is also abruptly 
 terminated by a border of light more brilliant than the rest of the 
 disc. 
 
 According to Sir John Herschel, the part of the sun's disc not 
 occupied b.y spots is far from uniformly bright. Its ground is 
 finely mottled with an appearance of minute dark dots, or pores, 
 
SUN'S SPOTS; AND ROTATION. 149 
 
 which, when attentively watched, arc found to be in a constant 
 state of change. 
 
 403. When the positions of the spots on the disc are observed 
 from day to day, it is perceived that they all have a common mo- 
 tion 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 en- 
 tirely across the disc, disappear from view at the western limb, 
 and re-appear 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 per- 
 pendicular 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 sfreak. 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. 
 
 404. 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 
 rotation ; for during this interval the sun has apparently moved 
 forward nearly a sign in the ecliptic ; the spot will therefore 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. 
 
 405. The apparent position of a spot with respect to the sun's 
 centre may be accurately determined, from day to Jay, by observ- 
 ing, when the sun is crossing the meridian, the right ascensions 
 and declinations 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 time of the sun's rotation on his axis is about 25| days ; 
 the inclination of his equator to the ecliptic 7 30' ; and the helio 
 centric longitude of the ascending node of the equator 80 7'. 
 
 406. It is a curious fact, that the region of the sun's spots is con 
 fined within about 30 of his equator. It is only occasionally that 
 spots are seen in higher latitudes than this : and none are ever seen 
 farther than about 60 from the equator. 
 
 407. The only theories relative to the physical constitution of 
 the sun which deserve notice> are those of Laplace and Herschel 
 
150 OF THE SUN AND ITS PHENOMENA. 
 
 Laplace supposed that the sun was an immense globe of solid mat- 
 ter in a state of ignition, and that the spots upon his disc were large 
 cavities, where there was a temporary intermission in the evolution 
 of luminous matter. Sir W. Herschel was of opinion that the sun 
 was an opake solid body, surrounded by a transparent atmosphere 
 of tens of thousands of miles in height, within which floated at a 
 height of from two to three thousand miles above the solid globe a 
 stratum of self-luminous clouds, which was the source of the sun's 
 light and heat, and beneath this another opake and non-luminous 
 stratum, which shone only with the light received from the upper 
 stratum. On this hypothesis the spots are accounted for by sup- 
 posing that openings occasionally take place in the strata, through 
 which the dark body of the sun is seen. The penumbra is the por- 
 tion of the obscure stratum, situated immediately around the open- 
 ing made in it. This theory seems to account for all the circum- 
 stances of the aspect and variation of the form and magnitude of 
 the spots, which the other does not do. 
 
 408. That the dark spots are depressions below the luminous surface of the sun 
 was first shown by Dr. Alexander Wilson, of Glasgow. He noticed that as a large 
 spot, which was seen on the sun's disc in November, 17 69, came near the western limb, 
 the penumbra on the side towards the centre of the disc contracted and disappeared, 
 and that afterwards the luminous matter on that side seemed to encroach upon the 
 central black nucleus, 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 towards the centre of the disc ; and that 
 when this part made its appearance, after the spot had advanced a short distance 
 upon the disc, it was much narrower than the opposite part. These various ap- 
 pearances of the spot in question are represented in Fig. 67. Dr. Wilson drew 
 from these facts the natural conclusion, that the spots were the dark body of the 
 
 Fig. 67. 
 
 
 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 sur- 
 rounding each black spot he conjectured to be the sloping sides of the opening in 
 the stratum of luminous clouds. But according to this the penumbra should shade 
 off gradually and merge into the central black spot without presenting any defi- 
 nite line of demarcation ; whereas its shade is nearly uniform throughout, and it is 
 abruptly terminated, both without and within. Herschel's theory is more com- 
 plete than this, and differs from it essentially in supposing the existence of an 
 opake non-lurninous cloudy stratum between the luminous medium and the dark 
 solid globe. It was devised, after a long and diligent inspection of all the aspects 
 and phenomena of the sun's spots, to account for these in all their varieties. It 
 
 S'ves a satisfactory explanation of the uniformity of shade of the penumbra, which 
 r. Wilson's theory does not do. 
 
 409. Herschel conceives the luminous surface of the sun to be constantly in a 
 state of violent agitation, and thai in comparatively limited districts it is occasion^- 
 ally forced up into masses or waves of hundreds of miles in height, by powerful 
 
PHYSICAL CONSTITUTION OF THE SUN. 
 
 151 
 
 upward currents, or by the exertion of some sort of explosive energy from beneath. 
 The ridges of these waves constitute the faculae, 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. This upheav- 
 ing force is supposed at times to acquire such intensity as to effect an opening both 
 in the lower and the upper stratum, and disclose to view the dark body of the sun. 
 410. Whatever may be the true physical constitution of the sun, the changes 
 which occur upon its surface take place with a rapidity which betokens the action 
 of the most powerful agents, if not the existence of the most subtle and elastic me- 
 dia. Some of the spots are said to have closed at the rate of nearly a mile per 
 second. The slowest motion noticed is not far from a mile per minute. But these ve- 
 locities of approach of the sides of a spot are vastly exceeded by the rate of motion o.. 
 the spots themselves, which has been sometimes noticed. In two well-established in- 
 stances spots have been seen to break into parts, which have then rapidly receded 
 from each other while the observer v/as viewing them through a telescope. Some 
 notion of the stupendous velocity of these changes may be obtained from the con- 
 sideration that the smallest area that can be distinctly discerned upon the sun, 
 even through telescopes, is a circle of 465 miles in diameter. 
 
 411. There has been observed, in connection with the sun, at 
 certain periods of th.e year, a faint light that is visible before sun- 
 rise and after sunset, to which has been given the name of the Zo- 
 diacal Light, from the circumstance of its being mostly compre- 
 hended within the zodiac. Its color is white, and its apparent fig- 
 ure that of a spindle, the base of which rests on the sun, and the 
 axis of which lies in the plane of the sun's equator ; such as would 
 be the appearance of a body of a lenticular shape, having its centre 
 coincident with the sun and its circular edge lying in the plane of 
 the sun's equator. Its length varies with the season of the year 
 
 j?ig. 
 
 and the state of the atmosphere ; 
 being sometimes more than 100, 
 and at ether times not more than 
 40 or 50. Its breadth near the 
 sun varies from 8 to 30. It is 
 nowhere abruptly terminated, but 
 gradually merges into the gerferal 
 light of the sky. (See Fig. 68.) 
 
 412. No generally received ex- 
 planation of this singular phenom- 
 enon 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 millions 
 of miles from the sun's centre 
 the centrifugal force balanced 
 the gravity, and that therefore 
 the sun's atmosphere could not extend beyond this : but this dis- 
 tance is less than one half ftie distance of Mercury from the sun, 
 whereas the substance of the zodiacal light extends beyond the or- 
 bit of Venus, and even beyond the earth's orbit. 
 
152 OF THE SUN AM) ITS PHENOMENA. 
 
 Several theories have been propounded relative to the cause of the zodiacal light 
 Laplace conceived it to be a ring of nebulous, that is, cloudy and self-luminous, 
 matter, encircling the sun in the plane of his equator. Professor Olmsted, of 
 New Haven, has suggested that it may be a large nebulous body revolving around the 
 sun in a regular orbit ; and the same body as that from which the periodical meteoric 
 showers are supposed to proceed. If we were to venture another suggestion upon 
 this perplexing subject, it would be, that the substance of the zodiacal light may be 
 a certain species of matter continually in the act of flowing away from the sun 
 into free space : being expelled by some repulsive force from perhaps all parts of 
 its surface, but in much the greatest quantity from the region of the spots, which 
 lies about the equator. Cassini, after an attentive examination of the zodiacal light 
 and the sun's spots during a series of years, conceived that he had detected a con- 
 nection between these two phenomena ; that the zodiacal light was fainter in propor- 
 tion as the spots were fewer in number and smaller. Thus, he remarks, that after the 
 year 1688, when the zodiacal light began to grow weaker, no spots appeared upon 
 the sun. He thought that this phenomenon became at times entirely invisible ; and 
 that this was the case in the years 1665, 1672, and 1681. From this apparent 
 connection between the two phenomena he drew the natural conclusion, that the 
 substance of the zodiacal light was some emanation from the oun's spots. The 
 explosive actions, which are the most probable cause of these spots, may perhaps 
 furnish the luminous matter, which may afterwards be driven off to an indefinite 
 distance by some repulsive action of the sun. ' Certainly, if there is at the sun's 
 surface any matter of the same nature'as that of which the tails of comets are com- 
 posed, it must be expelled by the same repulsive force that drives off this species 
 of matter from the heads of comets and forms their tails. (See Art. 557.) 
 
 413. The zodiacal light is seen most distinctly in our northern 
 climates 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 summer solstice it is said to be discernible, in a 
 very pure state of the atmosphere, both in the morning and even- 
 ing. The reason of the variations in the distinctness of the zodia- 
 cal light, is found in the change of its inclination to the horizon at 
 the time of sunset or sunrise, together with the variation in the du- 
 ration of twilight. As its length lis in the plane of the sun's equa- 
 tor, its inclination to the horizon will be different like that of this 
 plane, according to the different positions of the sun in the ecliptic. 
 Since the sun's equator makes but a small angle with the ecliptic, 
 at sunset, the zodiacal light will be most inclined to the horizon, 
 and therefore extend higher up in the heavens, towards the vernal 
 equinox, when the inclination of the ecliptic to the hori/ou at sun- 
 set is at its maximum ; 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 more easily and more frequently perceived in the torrid 
 zone than in these latitudes, because the ecliptic and zodiac make 
 there a larger angle with the horizon, and because twilight is of 
 shorter duration. 
 
PHASES OF THE MOON. 153 
 
 CHAPTER XIV. 
 
 OF THE MOON AND ITS PHENOMENA. 
 PHASES OF THE MOON. 
 
 414 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. 69) rep- 
 resent the position of the earth, ABC, &c., the orbit of the moon, 
 
 Fig. 69. 
 
 which we will suppose for the present to lie in the plane 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 parallel 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 continually be illuminated by him, 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 revolution of the moon. Thus, when 
 the moon is in conjunction, as at A, this hemisphere is turned 
 entirely away from the earth, and she 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 en- 
 lightened and dark hemispheres of the moon, called the Circle of 
 Illumination, it will obviously 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 
 
 20 
 
154 OF THE MOON AND ITS PHENOMENA. 
 
 visible, and thus the crescent enlarges, and the eastern limb be- 
 comes less concave. At the point C, 90 distant from the sun, 
 one half of it is seen, and the disc is a semi-circle, the eastern 
 limb being a right line. Beyond this point, more than half be- 
 comes 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 outward. This phase of the moon is repre- 
 sented at D. When the moon appears under this shape, it is said 
 to be Gibbous. In advancing 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. 
 
 415. 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. 
 
 416. When the moon is in conjunction, it is said to be New 
 Moon; and when in opposition, Full Moon. At the time be- 
 tween 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 moon, 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 Syzigies ; 
 and those of the first and last quarter, Quadratures. The four 
 points midway between the syzigies and quadratures are called 
 Octants. 
 
 417. 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". 33, 
 and that of the moon 13 10' 35".03 ; wherefore the moon sepa- 
 rates from the sun at the mean rate of 12 IT 26". 70 per day; 
 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. 
 
 418. To determine the time of mean new or full moon in any 
 given month. 
 
 Let the mean longitude of the. sun, and also the mean longi- 
 tude of the moon, at the beginning of the year, be found, and let 
 
TIME OF NEW OR FULL MOON. 155 
 
 the former be subtracted from the latter, (adding 360 if neces- 
 sary ;) the remainder, which call R, 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 12 11' 
 
 R 
 
 26". 70 per day, , f . - will express the number of days 
 
 1x2 11 <^o . / U 
 
 and fractions of a day, which at this epoch have elapsed since the 
 last new moon. This interval is called the Astronomical Epact. 
 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 ad- 
 dition 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.) 
 
 419. 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. 
 
 420. The earth, as viewed from the moon, goes through the 
 same phases 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 render the obscure part visible. 
 (See Fig. 69.) 
 
 MOON'S RISING, SETTING, AND PASSAGE OVER THE MERIDIAN. 
 
 421. To find the time of the 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 sun 
 and moon: also let t=ihe required time of the meridian passage. 
 At the time t the right ascensions will be, 
 
 For the moon . . . . M + tm, 
 For the sun . . . . S + ts ; 
 
 and, as the moon is on the meridian, the difference of these arcs 
 will be equal to the hour angle t ; whence, 
 
 * = M- S+t(m s)- 9 
 or, if all the quantities be expressed in seconds, 
 
156 OF THE MOON AND ITS PHENOMENA. 
 
 Thus, we find for the time of the meridian passage, 
 
 3600 (M-S) 
 - 
 
 The quantities M, S, m, s, are v 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 tin 
 meridian of New York, on the 1st of August, 1837 ? 
 
 When it is noon at New York, it is 4h. 56m. 4s. at Greenwich. Now, by the 
 Nautical Almanac, 
 
 Aug. 1st, at 4h. ]) 's R. Ascen. . - . 8h. 58m. 36.7s. 
 " at 5h. " " . ..90 38.3 
 
 Ih. : 56m. 4s. : : 2m. 1.6s. : 1m. 53.6s. 
 
 Aug. 1st, at 4h. D 's R. Ascen. . . . 8h. 58m. 36.7s. 
 Variation of R. Ascen. in 56m. 4s, .. 1 53.6 
 
 D 's R. Ascen. at M. Noon at N. York . 9 30.3 
 Aug. 1st, 0's hourly Variation of R. Ascen. . . . 9.704s 
 Ih. : 4h. 56m. 4s. : : 9.704s. : 47.8s. 
 
 Aug. 1st, M. Nocn at Greenw., 's R. Asc. 8h. 45m. 31.5s. 
 Variation of R. Ascen. in 4h. 56m. 4s. 47.8 
 
 's R. Ascen. at M. Noon at N. York . 8 46 19.3 
 
 Aug. 1st, M. Noon at Greenw., > 's R. Asc. 8h. 50m. 27.7s. 
 Aug. 2d, " : " 9 38 J 8.7 
 
 24)47 5J.O 
 
 Aug. 1st, D 's mean hourly Varia. of R. Asc. 1 59.6 (m) 
 
 's 9.7 0) 
 
 m s== l 49.9 = 109.9s 
 
 By Nautical Almanac, equation of time = 5m. 59s. 
 Ih. : 5m. 59s. : : 1m., 59.6s. : 11.9s. 
 
 ]> 's R. Ascen. at M. Noon at N. York . 9h. Om. 30.3s. 
 Correction for equation of time '... " W 11.9 
 
 D 's R. Ascen. at apparent Noon at N. York 9 18.4 (M) 
 's " " " 8 46 18.3 (S) 
 
 M S = 14 0.1 = 840.1s. 
 3600 . .'.'. . log. 3.55630 
 M 8 = 840.1 .... log. 2.92433 
 3600 (m s) = 3490.1 . . ar. co. log. 6.45716 
 
 Apparent time of meridian passage, 14m. 26.5s. = 866.5s. log. 2.93779 
 Equa. of time at merid. passage, 5 58 
 
 Mean time of meridian passage, Oh. 20m. 24s. 
 
 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 
 following day, and say, as 24h. : the difference : : the longitude of the place : a 
 fourth term. This fourth term, added to the time of the meridian passage at 
 
157 
 
 Greenwich on the given day, will give the time of the meridian passage on the 
 same day at the given place. 
 
 422. Since the moon has a motion with respect to the sun, the 
 time of its rising and setting must vary frorn 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 sunrise and rise about sunset, louring 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. 
 
 423. 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 revolution from about 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 varia- 
 tions 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 retardation fioin 
 day to day. These full moons occur when the sun is in the op- 
 posite 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. 
 
 424. To find the time of the moon's rising or setting on any given day.- Com- 
 pute the moon's semi-diurnal arc from equation (82), or (80), according as it ia 
 the time of the apparent rising 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 inter- 
 val between the moon's passage over the meridian and setting, by the following 
 proportion, 24h. : 24 -f m s (421) : : semi-diurnal arc : corrected semi-diur- 
 nal arc ; and add it to the time of the moon's meridian passage, found as ex- 
 plained in Art. 421. 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 
 it? 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 
 (80), the declination 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 equa- 
 tion (81 \ R = refraction parallax = 33' 51" 57' 1" (at a mean) = 23' 10' 
 
158 OF THE MOON AND ITS PHENOMENA. 
 
 ROTATION AND LIBRATIONS OF THE MOON. 
 
 425. The moon presents continually nearly the same face to- 
 wards 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 thus 
 completes one rotation in the same period of time in which it ac- 
 complishes a revolution in its orbit. 
 
 426. 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 disappear and again become visible. This 
 vibratory motion of the moon's spots is called Libration. 
 
 427. 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 portions 
 on its east and west sides alternately come into sight and disap- 
 pear, in consequence of its unequal motion in its orbit. The 
 periodical oscillation of the spots in an easterly and westerly direc- 
 tion from this cause, is 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 > 
 accounted for by supposing that the moon's axis is not exactly 
 perpendicular to the plane of its orbit, and that it remains contin- 
 ually parallel to itself. On this supposition we ought sometimes 
 to see beyond the north pole of the moon, and sometimes beyond 
 the south pole. 
 
 (3.) Parallax is the cause of a third libration of the moon. The 
 spectator upon the earth's surface being removed from its centra, 
 the point towards which the moon continually presents the same 
 hemisphere, he will see portions of the moon a little different 
 according to its different positions above the horizon. The diur- 
 nal motion of the spots resulting from the parallax, is called the 
 Diurr>sd or Parallactic Libration. 
 
 428. 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 ascertained that the plane of the moon's 
 equator is constantly inclined to the plane of the ecliptic under an 
 angle of 1 30', and Intersects it in a line which is always parallel 
 to the line of the nodes. It follows from the last-mentioned cir- 
 cumstance, that if a plane be supposed to pass through the centre 
 of the moon, parallel to the ecliptic, it will intersect the f>lane of 
 the moon's equator ajiti 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. Tt will 
 
159 
 
 make with the first an angle of 1 30', and with the second an 
 angle cf 5 9'. 
 
 DIMENSIONS AND PHYSICAL CONSTITUTION OF THE MOON. 
 
 429. The phases of the moon prove it to be an opake spherical 
 body. Its diameter is found by means of equation (83), viz : 
 
 where d denotes the diameter sought, R the radius of the earth, 6 
 the apparent diameter of the moon at a given distance, and H its 
 horizontal parallax at the same distance. 
 
 The greatest equatorial horizontal parallax of the moon is 61' 
 24", and the corresponding apparent diameter 33' 31" : thus we 
 have 
 
 33' 31" 3 
 
 = 2R , 48// = 2R ^ (very nearly) - 2161 miles. 
 
 The diameter of the moon being to the diameter of the earth 
 as 3 to 11, the surface of the moon is to the surface of the earth 
 as 3 2 to II 2 , or as 1 to 13 ; and the volume of the moon is to the 
 volume of the earth as 3 3 to II 3 , or as 1 to 49. 
 
 430. When the moon is viewed with a telescope, the edge of 
 the disc, which borders upon the dark portion of the face, is seen 
 to be very irregular and serrated, (see Fig. 70.) It is "hence in 
 
 Fig. 70. 
 
 ferred that the surface of the moon is diversified with mountains 
 and valleys. The truth of this inference is confirmed by the fact 
 that bright 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 it. These bright spots are doubtless the 
 tops of mountains illuminated by the sun, while the surrounding 
 
160 OF THE MOON AND ITS PHENOMENA. 
 
 regions that are less elevated are involved in darkness. The disc 
 is also diversified with spots of different shapes and different de- 
 grees of brightness. The brighter parts are supposed to be ele- 
 vated land, and the dark to be plains, and valleys, or cavities. 
 
 431 . The number of the lunar mountains is very great. Many 
 of them, by their form and grouping, furnish decided indications 
 of a volcanic origin. 
 
 From measurements 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. Accord- 
 ing to Herschel, the altitude of the highest is only about 1| Eng- 
 lish miles. But Schroeter of Lilienthal, a distinguished Seleno- 
 graphist, makes the elevation of some of the lunar mountains to 
 exceed 5 miles: and the more recent measurements of MM. Baer 
 and Madler of Berlin lead to similar results. 
 
 432. There are no seas nor other bodies of water upon the sur- 
 face of the moon. Certain dark and apparently level parts of the 
 moon were for some time supposed to be extended sheets of wa- 
 ter, and, under this idea, were named by Hevelius Mare Imbrium, 
 Mare Crisium, &c. : 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 ; besides, 
 when these dark spots are viewed with good telescopes, they are 
 found to contain a number of cavities, whose shadows are dis- 
 tinctly perceived falling within them. The spots in question are 
 therefore to be regarded as extensive plains diversified by mode- 
 rate elevations and depressions. The entire absence of water also 
 from the farther hemisphere of the moon may be inferred from the 
 fact that the moon's face is never obscured by clouds or mists. 
 
 433. It has long been a question among Astronomers, whether the moon has an 
 atmosphere. It ?s asserted, that, if it has any, it must be exceedingly rare, or 
 very limited in its extent, since it does not sensibly diminish or refract the light 
 of a star seen in contact with the moon's limb ; for when a star experiences an 
 occultalion 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 occupation is as it is com- 
 puted, without making any allowance for the refraction of a lunar atmosphere. 
 But it is maintained, on the other hand, that these facts, if allowed, are not op- 
 posed to the supposition of the existence of an atmosphere of a few miles only in 
 height ; and that certain phenomena which have been observed afford indubitable 
 evidence of the presence of a certain limited body of air upon the moon's surface. 
 Thus the celebrated Schroeter, in the course of some delicate observations made 
 upon the crescent moon, perceived a faint grayish light extending from the horns 
 of the crescent a certain distance into the dark part of the moon's face. This he 
 conceived to be the moon's twilight, and hence inferred the existence of a lunar 
 atmosphere. From the measurements which 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 
 gun, occasioned by the interposition of the moon, the dark body of the moon has 
 been ceen surrounded by a luminous ring, which was at first nost distinct at the 
 
161 
 
 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 po.nt where the junction took place. 
 
 On the whole, it seems most probable that the moon has a smaU atmosphere. 
 
 434. The surface of the moon, like that of the earth, presents the two general 
 varieties of level and mountainous districts ; but it differs from the earth's surface 
 in having no seas, or other bodies of water, upon it, (432,) and in being more rug- 
 ged and mountainous. The comparatively level regions occupy somewhat more 
 than one-third of the nearer half of the moon's surface. These are, in general, 
 the darker parts of the disc. The lunar plains vary in extent from 40 or 50 miles 
 to 700 miles in diameter. The mountainous formations of the other parts of the 
 surface offer three marked varieties, viz : 
 
 (1.) Insulated 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 when placed 
 opposite to the sun.* The perpendicular altitudes of some of them, as deter- 
 mined from the lengths of their shadows, are between four and five miles. Insu- 
 lated mountains frequently occur in the centres of circular plains. They are 
 then called Central Mountains. 
 
 (2.) Ranges of Mountains, extending in- length two or three hundred miles. 
 These ranges bear a distinct resemblance to our Alps, Appenines, and Andes, but 
 they are much less in extent, and do not form a very prominent feature of the 
 lunar surface. Some of them appear very rugged and precipitous, and the highest 
 ranges are, in some places, above four miles in perpendicular altitude. In some 
 instances thev run nearly in a straight line from northeast to southwest, as in that 
 range called the Appenines ; in other cases they assume the form of a semicircle 
 or a crescentt 
 
 (3.) Circular Formations. The general prevalence of this remarkable class of 
 mountainous formations is the great characteristic feature of the topography of 
 the moon's surface. It is subdivided by late selenographists into three orders, viz : 
 Walled Plains, whose diameter varies from one hundred and twenty to forty or 
 fifty miles ; Ring Mountains, the diameter of which descends to ten miles ; and 
 Craters, which are still smaller. The term crater is sometimes extended to all 
 the varieties of circular formations. They are also sometimes called Caverns, be- 
 cause 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 ^hole ring. At other times ranges of concentric mountains encircle 
 the inner toot of the wall, leaving intermediate valleys. Again, we have a few 
 ridges of low mountains stretching through the circle contained by the wall, but 
 oftener isolated conical peaks start up, and very frequently small craters having 
 on an inferior scale every attribute of the large one-t The smaller craters, 
 however, offer some characteristic peculiarities. Most of them are without a 
 flat bottom, and have the appearance of a hollow inverted cone with the sides 
 tapering towards the centre. Some have no perceptible outer edge, their margin 
 being on a level with the surrounding regions : these are called Pits. 
 
 Tho bounding ridge of the lunar craters or caverns is much more precipitous 
 within than without ; and the internal depth of the crater is always much lower 
 than the general surface of the moon. The depth varies from one-third of a mile 
 to three miles and a half. 
 
 These curious circular formations occur at almost every part of the surface, but 
 are most abundant in the southwestern regions. It is the strong reflection of their 
 
 * Dick's Celestial Scenery, p. 256. t Ibid. p. 257. 
 
 t Nichol's Phenomena of the S ;lar System, p. 167. 
 21 
 
162 ECLIPSES OF THE SUN AND MOON. 
 
 mountainous ridges which gives to that part of the moon's surface its superior lus- 
 tre. The smaller craters occupy nearly two-fifths of the moon's visible surface. 
 
 CHAPTER XV. 
 
 ECLIPSES OF THE SUN AND MOON. OCCULTATIONS OF THE 
 
 FIX.ED STARS. 
 
 435. AN eclipse of a heavenly body is a privation 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, 
 Fig. 71. of the moon, and of the satellites, 
 
 (334 ;) and, with respect to circum- 
 stances, into total, partial, annular, 
 d central. A total eclipse is one in 
 which the whole disc of the lumi- 
 nary is darkened ; a partial one is 
 when only a part of the disc is dark- 
 ened. In an annular eclipse the 
 whole is darkened, except a ring or 
 annulus, which appears round the 
 dark part like an illuminated border; 
 the definition of a central eclipse will 
 be given in another place. 
 
 ECLIPSES OF THE MOON. 
 
 436. An eclipse of the moon is oc- 
 casioned 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 shadow of the earth, as pro- 
 jected from the sun. Hence it is ob- 
 vious that lunar eclipses can happen 
 only at the time of full moon, for it 
 is then only that the earth can be be- 
 tween the moon and the sun. 
 
 437. Since the sun is much larger 
 than 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 fn>m~~each other. Let the circles AGB, agb, (Fig. 71,) 
 
163 
 
 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 Ac?, Be, tangents on different sides. The triangular space 
 uCb 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 sur- 
 face of the frustrum 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 penumbra will receive light from a portion of the sun's 
 disc, and from a greater portion the more distant they are from the 
 umbra. 
 
 438. To find the length of the earth's shadow. Let L= the 
 length of the shadow ; R=the radius of the earth ; <5 = sun's ap- 
 parent semi-diameter, and p = sun's parallax. The right-angled 
 triangle EaC (Fig. 71) gives 
 
 sin ECa* 
 andECa = SEA EAC = 5 p; whence, 
 
 . . . (86.) 
 
 p) 
 
 p) is only a 
 its sine, and there 
 
 As the angle (<5 p) is only about 16', it will differ but little from 
 fore, 
 
 (nearly); 
 
 or, if 5 and p be expressed in seconds, 
 
 , ^,. 
 
 L==R r -- (nearly) . . . (87). 
 
 The shadow will obviously be the shortest when the sun is the 
 nearest to the earth. We then have 8 = W 18", andp = 9", which 
 gives L = 213R. The greatest distance of the moon is a little 
 less than 64 R, It appears, then, that the earth's sJiadow always 
 extends to more than three times the distance of the moon. 
 
 439. Let kMh be a circular arc, described about E the centre 
 of the earth, and with a radius equal to the distance between the 
 centres of the earth and moon at the time of opposition. The an- 
 gle MEw, 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 ME A, the 
 apparent semi-diameter of a section of the penumbra, at the same 
 distance, is called the Semi-diameter of the Penumbra. 
 
 440. Were the plane of the moon's orbit coincide with the 
 plane of the ecliptic, there would be a lunar eclipse at every full 
 moon ; out, as it is inclined tc 't, an eclipse can happen only when 
 
164 
 
 ECLIPSES OF THE SUN AND MOON. 
 
 Fig. 72. 
 
 -TNf 
 
 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-di- 
 ameters of the moon and of the earth's 
 shadow. This will be better understood 
 on referring to Fig. 72, in which N'C 
 represents a portion of the ecliptic, and 
 N'M a portion of the moon's orbit, N' 
 the descending node, E the earth, ES, 
 ES', ES" three different directions of 
 the sun, s, s', s" sections of the earth's 
 shadow in the three several positions 
 0\\ corresponding to these directions of the 
 \ N v 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 
 1 unless at the time of opposition it is 
 nearer to the node than the point m', where the latitude rn's' is 
 equal to the sum of the semi-diameters of the moon and shadow. 
 
 441. 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 - ECm (Fig. 71) ; 
 
 but Ema = P and ECm = 6 p (438) ; therefore, 
 MEw=A = P+p-<5 . . . (88). 
 
 The semi-diarneter of the shadow is the least when the moon is 
 in its apogee and the sun is in its perigee, or when P has its mini- 
 mum, and <5 its maximum value. In these positions of the moon 
 and sun, P = 53' 48", d=W 18", and p = 9". Substituting, we 
 obtain for the least semi-diameter of the earth's shadow 37' 39". 
 and for its least diameter 1 15' 18". The greatest apparent diam- 
 eter of the moon is 33' 31". Whence it appears, that the diameter 
 of the earth's shadow is always more than twice the diameter of 
 the moon. 
 
 The mean values of P and <5 are respectively 57' 1", and 16' 1"; 
 which gives for the mean semi-diameter of the earth's shadow 
 41' 9". 
 
 442. If to P +j9 <5, the semi-diameter of the earth's shadow, 
 we add d, the semi-diameter of the moon, the sum P + p + d <5 
 will express the greatest latitude of the moon in opposition, at which 
 an eclipse can happen. 
 
 It is easy for a given value of P -\-p + d 8, and for a given in- 
 clination 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 great- 
 
LUNAR ECLIPTIC LIMITS. 165 
 
 est and least values of P -f p + d <$, and also taking into view the 
 inequalities in the motions of the sun and moon, it has been found, 
 that when at the time of mean full moon the difference of the mean 
 longitudes of the moon and node exceeds 13 21', there cannot be 
 an eclipse ; but when this difference is less than 7 47' there must 
 be one. Between 7 47' and 13 21' the happening of the eclipse 
 is doubtful. These numbers are called the Lunar Ecliptic Limits. 
 
 To determine at what full moons in the course of any one year 
 there will be an eclipse, find the time of each mean full moon, 
 (418) ; and for each of the times obtained find the mean longitude 
 of the sun, and also of the moon's node, and compare the differ- 
 ence of these with the lunar ecliptic limits. Should, however, the 
 difference in any instance fall between the two limits, farther cal- 
 culation 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.) 
 
 443. 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' 9" (441), the mean semi-diameter of the moon 15' 33", 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 3fh. 
 
 444. Since the moon moves from west to east, an eclipse of the 
 moon must commence on the eastern limb, and end on the western. 
 
 445. In the investigations in Arts. 438, 441, we have supposed 
 the cone of the earth's shadow to be formed b}^ lines drawn from 
 the edge of the sun, and touching the earth's surface. This, prob- 
 ably, 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 rase 
 the earth's surface, to be stopped and absorbed by the lower strata 
 of the atmosphere. In such a 'case the conical boundary 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, experiences 
 at the beginning of the eclipse a gradual diminution, and at the end 
 a gradual increase. On this account the times of the beginning 
 and end of the eclipse cannot be noted with precision, and in con- 
 sequence astronomers differ as to the amount of the increase in the 
 
166 
 
 ECLIPSES OF THE SUN AND MOON. 
 
 size of the earth's shadow from the cause above mentioned. It is 
 the practice, however, in computing an eclipse of the moon, to in- 
 crease the semi-diameter of the shadow by a ^\ part ; or, which 
 amounts to the same, to add as many seconds as the semi-diamete~ 
 contains minutes. 
 
 446. It is remarked in total eclipses of the moon, that the mooa 
 is not wholly invisible, but appears with a dull reddish light. 
 
 This phenomenon is doubtless another effect of the earth's at- 
 mosphere, though of a totally different nature from the preceding. 
 Certain of the sun's rays, instead of being stopped and absorbed, 
 are bent from their rectilinear course by the refracting power of 
 the atmosphere, so as to form a cone of faint light, interior to that 
 cone which has been mathematically described as the earth's shad- 
 ow, which falling upon the moon renders it visible. 
 
 447. 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. If will be shown that 
 the case is different with eclipses of the sun. 
 
 CALCULATION OF AN ECLIPSE OF THE MOON. 
 
 448. 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 rnoon in 
 longitude and latitude, as uniform during the eclipse. These sup- 
 positions being made, the calculation of the circumstances of an 
 eclipse of the moon is very simple. 
 
 Fig. 73 
 
 Let NF (Fig. 73) be a part of the ecliptic, N the moon's as 
 cending 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 in 
 stant of opposition : then CC', which is the latitude of the moon 
 in opposition, is the distance of the centres of the shadow and 
 moon at that time. The moon and shadow both have a motion, 
 and in the same direction, as from N towards F and L. It is the 
 
167 
 
 practice, however, to regard the shadow as stationary, and to attri- 
 bute to the moon a motion equal to the relative motion of the moon 
 and shadow. Tfce orbit that would be described by the moon's 
 centre if it had such a motion, is called the Relative Orbit of the 
 moon. Inasmuch as the circumstances of the eclipse depend al- 
 together upon the relative motion of the moon and shadow, this 
 mode of proceeding is obviously allowable. 
 
 As the shadow has no motion in latitude, the relative motion of 
 the moon and shadow in latitude will, be equal to the moon's ac- 
 tual 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 lon- 
 gitude. We obtain, therefore, the relative position of the centres 
 of the moon and shadow at any interval t, following opposition, by 
 laying off Cm equal to the difference of the motions of the sun and 
 moon in longitude in this interval, through m drawing rnM. per- 
 pendicular to NF, and cutting off mM. equal to the latitude at op- 
 position 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, is 
 considered uniform, the ratio of C'm' (= Cm, the difference be^ 
 tween the motions of the sun and moon in longitude) to Mm' the 
 moon's motion in latitude, is the same, whatever may be the length 
 of the interval considered. It follows, therefore, that the relative 
 orbit of the moon N'C'M is a right line. 
 
 449. The relative orbit passes through C', the place of the moon's centre at op- 
 position : its position will therefore be known, if its inclination to the ecliptic be 
 found. Now we have 
 
 Mm' moon's motion in latitude 
 
 tan mc.lma. ~ ; = - : ---- - 
 Cm moon's mot. in long. sun's mot. in long. 
 
 450. The following data arc requisite in the calculation of the circumstances of 
 a lunar eclipse : 
 
 T = time of opposition. 
 
 M = moon's hourly motion in longitude. 
 
 n = moon's hourly motion in latitude. 
 
 m = sun's hourly motion in longitude. 
 
 X = moon's latitude at opposition. 
 
 d = moon's semi-diameter. 
 
 & = sun's semi-diameter. 
 
 P = moon's horizontal parallax 
 
 p = sun's horizontal parallax. 
 
 s = semi-diameter of earth's shadow. 
 
 I = inclination of relative orbit. 
 
 h = moon's hourly motion on relative orbit. 
 
 T, M, n, m, X, d, S, P, and p, are derived from Tables of the sun and moon. 
 (See Problems IX and XIV.) 
 
 The quantities s, I, and h, may be determined from these : 
 
 6) (441 and 445) . . . (89)} 
 
 tang I = (449) . . . (90). 
 
168 
 
 The triangle C'Mm' gives 
 
 ECLIPSES OP THE SUN AND MOON. 
 
 C'M 
 
 cos MOW 
 
 M m 
 
 (91). 
 
 451. The above quantities being supposed to be known, let N'CF (Fig. 74) re- 
 present the ecliptic, and C the stationary centre of the earth's shadow. Let 
 
 Fig. 74. 
 
 _K S 
 
 CC' = X, and let N'C'L' represent the relative orbit of the moon. We here sup- 
 pose the moon to be north of the ecliptic at the time of opposition, and near its 
 ascending node : when it is south of the ecliptic X is to be laid off below N'CF, 
 and when it is approaching either node, the relative orbit is inclined to the right. 
 Let the circle KFK'R, described about the centre C, represent the section of the 
 earth's shadow at the moon ; and let /, /', and g, g 1 , 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. C/ = C/ ' = s -f d, and Cg = Cg' = s d. Draw CM 
 perpendicular to N'C'L', and M will represent the place of the moon's centre when 
 nearest the centre of the shadow : it will also be its place at the middle of the 
 eclipse ; for since C/ = C/, and CM is perpendicular to N'C'/', M/ = M/. 
 
 452. 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 (x) em- 
 ployed by the moon in passing from M to C'. Now 
 
 . MC' 
 
 (expressed in parts of an hour) x = ; 
 
 h 
 
 and in the right-angled triangle CC'M we have CC' = X, and < C'CM = 
 < C'N'C = I, and therefore MC' = X sin I ; whence, by substitution, 
 X sin I X sin I X sin I cos I 
 
 cos I 
 
 or, (expressed in seconds,) x = _-!_. x sin I . . (92). 
 
 M m 
 Hence, if M = time of middle, we have 
 
 3600s. cos 1 
 
 . X sin I ... (93) 
 
 M m 
 
 It is obvious that the upper sign is to be used when the latitude is increasing 
 and the lower sign when it is decreasing. 
 
 The distance of the centre of the moon from the centre of the shadow at the mid 
 die of the eclipse, 
 
 = CM = CC'cos C'CM = X cos I . . . (94). 
 
 453. Beginning and end of the eclipse. Let any point I of the relative orbit be 
 the place of the moon's centre at the time of any given phase of the eclipse. Let 
 t = the interval of time between the given phase and the middle ; and k = C/, 
 
CALCULATION OF A LUNAR ECLIPSE. 169 
 
 the distance of the centres of the moon and shadow. In the interval t the moon's 
 centre will pass over the distance MZ ; hence 
 
 M[= M/.COS! 
 
 h ~~ M m 
 
 but, MZ = \/c/2 clV? = V~Wtf ct> S 2 I (equa. 94), 
 
 C S 
 
 and therefore t = <J k* X2 cos2 I ; 
 
 M m 
 
 3600s. cos I 
 
 or, (in seconds,) t = """ ua -" JS * </ (fc + X cos I) (A; A cos I) ... (95) 
 M m 
 
 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 -f *, or T' = M t , 
 
 according as the phase follows or precedes the middle. 
 Now, at the beginning and end'of the eclipse, we have 
 
 k=CforCf = 
 
 substituting in equation (95) we obtain 
 3600s. cos I 
 
 -V (s + d -h X cos I) (s -f d A cos I) . . . (96). 
 JM m 
 
 t' beir.g found, the time of the beginning (B,; and the time of the end (E,) result 
 from the equations 
 
 B = M *', E=M-H'. 
 
 454. Beginning and end off-he total eclipse. At the beginning and end of the 
 total eclipse, k = Cg = Cg' = s d ; whence, by equation (95,) 
 
 3600s. cos I 
 
 t" = " V (s d -f X cos I) (sd XcosI) . . .(97): 
 
 M m 
 
 and, denoting the time of the beginning by B' and the time of the end by E', we 
 have B' = M t", E' = M + t". 
 
 455. Quantity of the eclipse. In a partial eclipse of the moon the magnitude 
 or quantity of the eclipse is measured by the relative portion of that diameter of 
 the moon, which, if produced, would pass through the centre of the earlh's shad- 
 ow, 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 of a digit in the part immersed. When the moon passes entirely 
 within the shadow, as in a total eclipse, the quantity'of the eclipse is expressed by 
 the number of digits contained in the part of the same diameter prolonged outward, 
 which is comprised between the edge of the shadow and the inner edge of the moon. 
 Thus the number of digits contained in SN (Fig. 74) expresses the quantity of the 
 eclipse represented in the figure. Hence, if Q = the quantity of the eclipse, we 
 shall have 
 
 NS 12NS _ 12(NM + MS) 12 (NM + CS CM) 
 NV = NV NV 
 
 12 (d -f * X cos I) 
 
 If X cos I exceeds (s -{- d) there will be no eclipse. If it is intermediate between 
 (* + d} and (s d) there will be a partial eclipse ; and if it is less than (* d) 
 the eclipse will be total. 
 
 CONSTRUCTION OF AN ECLIPSE OF THE MOON. 
 
 456. 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 
 
 22 
 
170 
 
 ECLIPSES OF THE SUN AND MOON. 
 
 (Fig. 75) 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' = X the moon's latitude in opposition, and RL' = X n, the 
 moon's latitude an hour after opposition. The right line C'L', 
 drawn through C' and L', will represent the moon's relative orbit. 
 It should be observed, that if the latitudes are south they must be 
 laid off below N'F, and that N'C'L' will be inclined to the right 
 when the latitude is decreasing. With a radius CE =s (equation 
 89) describe the circle EKFK', which will represent the section 
 of the earth's shadow. With a radius = s + d, and another radius 
 = s d, describe about the centre C arcs intersecting N'L' in 
 /,/', and g, g' ; /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 
 7h. 23m. 15s. At this time the moon's centre will be at C'. To 
 find its place at 7h., state the proportion, 60m. : 23m. 15s. : : moon's 
 hourly motion on the relative orbit : a fourth term. This fourth 
 term will be the distance of the moon's centre from the point C' at 
 7 o'clock ; and if it be taken in the dividers and laid off on the 
 relative orbit from C' backward to the point 7, it will give the 
 moon's place at that hour. This being found, take in the divi- 
 ders 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/,/', g, g', and M, let 
 
ECLIPSES OF THE SUN LUMINOUS FRUSTUM. 
 
 in 
 
 the hour spaces thus found be divided into quarters, and these 
 subdivided into 5-minute or minute spaces, and the times answer- 
 ing 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 point/' falls between 9 and 10, and thus the end of the eclipse 
 will occur somewhere between 9 and 10 o'clock. To find the num- 
 ber 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 1 5-minute 
 spaces, subdivide the part which contains /' into three equal parts 
 or 5-minute spaces, and again that one of these smaller parts 
 within which/ 7 lies, into five equal parts or minute spaces.* 
 
 ECLIPSES OF THE SUN. 
 
 457. 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 prevented from falling upon certain parts of the 
 earth's surface. 
 
 Let AGB and agb (Fig. 76) be sections of the sun and earth 
 
 Fig. 76. 
 
 by a plane passing through their centres S and E, A#, BZ> tan- 
 gents to the circles AGB and agb on the same side, and Ad, Be 
 tangents to the same on opposite sides. The figure A06B 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 triangular 
 space Fed will be a section of a cone formed by rays tangent on 
 opposite sides. An eclipse of the sun will take place somewhere 
 upon the earth's surface, whenever the moon comes within the 
 frustum AabB, and a total or an annular eclipse whenever the 
 moon comes within the cone Fed. 
 
 458. Let mm'M. (Fig. 76) be a circular arc described about the 
 centre E, and with a radius equal to the distance of the centres 
 of the moon and earth at the time of conjunction. The angle 
 ?nES is the apparent semi-diameter of a section of the frustum, 
 and m'ES the apparent semi-diameter of a section of the cone, at 
 the distance of the moon. To find expressions for these semi- 
 diameters in terms of determinate quantities, let the first be de- 
 noted by A, and the second by A' ; and let P = the parallax of 
 
172 ECLIPSES OF THE SUN AND MOON. 
 
 the moon, p = the parallax of the sun; and = the semi-diameter 
 of the sun. Then we have 
 
 mES = A = mEA. + AES = Ema - EAm + AES ; 
 or, A = P - p + d . . . (99) ; 
 
 and m'ES = m'EE - BES = Em'c - EBm' BES ; 
 or, A' = P p -* . . . (100). 
 
 Taking the mean values of P, p, and <$, (441,) we find for the 
 mean value of A 1 12' 53", and for the mean value of A' 40' 51 ". 
 
 459. 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 of the lu- 
 minous frustum (457) at the moon's orbit. This may be illustrated 
 by means of Fig. 72, already used for a lunar eclipse, by supposing 
 the sun to be in the directions Es, Es', Es", and that s, s', s", are 
 sections of the luminous frustum corresponding to these directions 
 of the sun, also that ra, m- 9 m", represent the moon in the cor- 
 responding positions of conjunction. Thus, denoting the moon's 
 semi-diameter by d, and the greatest latitude of the moon in con- 
 junction, at which an eclipse can take place, by L, we have 
 
 L =P -p + d+d . . . (101). 
 
 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-p-d + d . . . (102). 
 
 In order that an annular eclipse may take place, the apparent 
 s emi-diameter of the moon must be less than that of the sun, and 
 the moon must come at conjunction entirely within the luminous 
 frustum. Whence, if L" = the maximum latitude at which an 
 annular eclipse is possible, we have 
 
 L" = P-p + 8-d . . . (103). 
 
 460. 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 of the mean longitudes of the sun or moon and 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. 
 
 461. In order to discover at what new moons in the course of a 
 year an eclipse of the sun will happen, with its approximate time, 
 we have only to find the mean longitudes of the sun and node at 
 each mean new moon throughout the year, (418,) and take the 
 difference of the longitudes and compare it with the solar ecliptic 
 limits. (For a more direct method of solving this problem, see 
 Prob. XXVIII.) 
 
 462. Eclipses both of the sun and moon recur in nearly the 
 
NUilBEJl OF ECLIPSES IN A YEAR. 173 
 
 sail)'* order and at the same-iutervals 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. 6 1985 Id., 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 this period will be in the same posi- 
 tion with respect to the moon and node as at its commencement. 
 The eclipses which occur during one such period being noted, 
 subsequent eclipses are easily predicted. 
 
 This period was known to the Chaldeans and Egyptians, by 
 whom it was called Saros. 
 
 463. As the solar ecliptic limits are more extended than the lu- 
 nar, eclipses of the sun must occur more frequently than eclipses 
 of the 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 ob- 
 vious. The sun passes by both nodes of the moon's orbit but once 
 in a year, unless he passes by one of them in the beginning of the 
 year, in which case he will pass by the same again a little before 
 the end of the year, as he returns to the same node in a period of 
 346 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 
 (4SO), and at the subsequent opposition the moon will be eclipsec 
 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 after 
 wards, 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 twelfih lunation from the eclipse in the begin 
 ning of the year may give a new moon before the year is ended, 
 which, in consequence of the retrogradation of the nodes, may be 
 within the solar ecliptic limit ; and hence there may be seven 
 eclipses in a year, five of the sun and two of the moon.- But when 
 the moon changes in either of the nodes, she cannot be near enough 
 to the other node, at the next full moon, to be eclipsed, as in the 
 interval the sun will move over an arc of 14 32', whereas the 
 greatest lunar ecliptic limit is but 13 21', and in six lunar months 
 afterwards she will change near the other node ; in this case there 
 cannot be more than two eclipses in a year, both of which will be 
 
 * More exactly, 18 years (of 365 days) plus 15d 7h, 42m. 29s, 
 
174 ECLIPSES OF THE SUN AND MOON. 
 
 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. 
 
 464. 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 cir- 
 cumstance 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 paral- 
 lax, or of 1 54'. Therefore, should an observer situated at one 
 of the points of tangency, refer the centre of the moon to the cen- 
 tre of the sun, an observer at the other would see the centres of 
 these bodies distant from each other at an angle of 1 54', and their 
 nearest limbs separated by an arc of more than 1. 
 
 465. Instead of regarding an eclipse of the sun as produced by 
 an interposition of the moon between the sun and earth, as we have 
 hitherto considered it, we may regard it as occasioned by the moon's 
 shadow falling upon the earth. Fig. 77 represents the moon's 
 shadow, as projected from the sun and covering a portion of the 
 earth's surface. Wherever the umbra falls, there is a total eclipse ; 
 and wherever the penumbra falls, a partial eclipse. 
 
 Fig. 77. 
 
 466. In order to discover the extent of the portion of the earth's 
 surface over which the eclipse is visible at any particular time, 
 we have only to find the breadth of the portion of the earth covered 
 by the penumbral shadow of the moon ; but we will first ascertain 
 the length of the moon's shadow. As seen at the vertex of the 
 moon's shadow, the apparent diameters of the moon and sun are 
 equal. Now, as seen at the centre of the earth, they are nearly 
 equal, sometimes the one being a little greater and sometimes the 
 other. It follows, therefore, that the length of the moon's shadow 
 is about equal to the distance of the earthy being sometimes a little 
 greater and at other times a little less. 
 
175 
 
 afey the 
 
 ECLIPSES OF THE SUN. MOON SiPH^DQW. t> 
 
 When the apparent diameter of the moon is tffespi 
 shadow will extend beyond the earth's centre ; and wfitt^ the -ap- 
 parent diameter of the sun is the greater, it will fall short' o/^t. 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 ortfie 
 earth to the distance of the moon, we shall have 31' 38" far th^ 
 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 sen- 
 sibly the same as at the centre of the earth, or 32' 2", 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. 
 
 467. To find a general expression for the length of the moon's 
 shadow, let AGB, a'g'b', and agb (Fig. 78) be sections of the sun, 
 
 Fig. 78. 
 
 moon, and earth, by a plane passing through their centres S, M, 
 and E, supposed to be in the same right line, and Aa', Bb' tan- 
 gents to the circles AGB, a'g'b' : then a'K6' will represent the 
 moon's shadow. Let L == the length of the shadow ; D = the dis- 
 tance of the mocn ; D' = the distance of the sun ; d = the appa- 
 rent semi-diameter of the moon ; and 5 = 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 se- 
 mi-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 com- 
 parison with the distance of the sun (466), the apparent semi-diam- 
 eter of the sun will always be very nearly the same to an observer 
 situated at K as to one situated at the centre of the earth. Now, 
 since the apparent semi-diameter of the moon is inversely propor- 
 tional to its distance, 
 
 angle MKa' : d : : ME : MK ; 
 and thus, 3 : d : : ME : MK : : D : L (nearly) : 
 
 whence, L=D-| . . . (104). 
 
 If a more accurate result be desired, we have only to repeat the cal- 
 culations, after having diminished 5 in the ratio of D' to (D'-fL D). 
 
 468. Now, to find the breadth of the portion of the earth's surface covered by 
 the penumbral shadow, let the lines Ad', Be' (Fig. 78) be drawn tangent to the 
 circles AGB, a'g'b', on opposite sides, aud prolonged on to the earth. The space 
 
176 ECLIPSES OF THE SUN AND MOON. 
 
 hc'd'k will represent the penumbra of the moon's shadow, and the arc gJi one half 
 the breadth of the portion of the earth's surface covered by it. Let this arc or the 
 angle g-EA = 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 = MAZ AME. 
 
 The angle AME is the moon's parallax in altitude at the station A, and MAZ is 
 its zenith distance at the same station. Denote the former by P' and the latter by Z. 
 Thus, S = Z F . . . (105). 
 
 The triangle AMS gives 
 
 AME = F=MSA+MAS; 
 
 MAS = d-\-t> ; and MSA is the sun's parallax in altitude at the station A: let it 
 be denoted by p'. We have, then, 
 
 F = d + S +p' = d + 6 (nearly) . . . (106); 
 and to find Z we have (equa. 9, p. 51), 
 
 F = P sin Z, or sin Z = - . . . (107). 
 
 F and Z being found by these equations, equa. (105) will then make known the 
 value of S. 
 
 If great accuracy is required, the calculation must be repeated, giving now to 
 p' in equation (106) the value furnished by equation (9) which expresses the rela- 
 tion between the parallax in altitude of a body and its horizontal parallax, instead 
 of neglecting it as before ; and Z must be computed from the following equation : 
 
 *.. 
 
 sin P 
 
 The penumbral shadow will obviously attain to its greatest 
 breadth when the sun is in its perigee and the moon is in its apo- 
 gee. The values of d, <5, and P under these circumstances are re- 
 spectively 14' 41", 16' 18'', and 53' 48". Performing the calcula- 
 tions, we find that the breadth of the greatest portion of the earths 
 surface ever covered by the penumbral shadow is 69 18', or about 
 4800 miles. 
 
 469. The breadth of the spot comprehended within the umbra 
 may be found in a similar manner. 
 
 The arc gh' (Fig. 78) represents one half of it : denote this arc or the equal an- 
 gle ffEA' by S'. 
 
 MEA' == S' = MA'Z' A'ME ; 
 or, S' = Z-F . . . (109). 
 
 A'ME = F = MSA' + MA'S; 
 but MA'S = d 5, and MSA' = p', sun's parallax in altitude at A' ; whence, 
 
 Y' = d $+p' = d S (nearly) . . . (110): 
 and we ha^, as before, 
 
 F = PsinZ,orsinZ= . . . (111). 
 
 The greatest breadth will obtain when the sun is in its apogee 
 and the moon is in its perigee. We shall then have 
 
 <5 = 15' 45", d = 16' 45", P = 61' 24". 
 
 Making use of these numbers, we deduce for the maximum 
 breadth of the portion of the earths surface covered by the moon's 
 shadow, 1 50', or 127 miles. 
 
 470. It should be observed that the deductions of the last two 
 
CALCULATION OF AN ECLIPSE OF THE SUN. 177 
 
 articles answer to the supposition that the moon is in the node, and 
 that the axis of the shadow and penumbra passes through the cen- 
 tre of the earth. In every other case, both the shadow and pe- 
 numbra will be cut obliquely by the earth's surface, and the sec- 
 tions will be ovals, and very nearly true ellipses, the lengths of 
 which may materially exceed the above determinations. 
 
 471. Parallax not only causes the eclipse to be visible at some 
 places and invisible at others, as shown in Art. 464 ; but, by making 
 the distance of the centres of the sun and moon unequal, renders 
 the circumstances of the eclipse at those places where it is visible 
 different at each place. This may also be inferred from the cir- 
 cumstance 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 con- 
 sidered in two points of view: 1st. With respect to the whole 
 earth, or as a general eclipse ; and, 2d. With respect to a particu- 
 lar place. 
 
 472. 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 ex- 
 ceed about 6 hours. 2d. A solar eclipse does not happen at the same time at all 
 places where it is seen : as the motion of the moon beyond the sun, and conse- 
 quently of its shadow, is from west to east, the eclipse must begin earlier at the 
 western parts and later at the eastern. 3d. The moon's shadow, being tangent to 
 the earth at the commencement and end of the eclipse, the sun will be just rising 
 at the place where the eclipse is first seen, and just setting at the place where it is 
 last seen. At the intermediate places, the sun will at the time of the beginning and 
 end of the eclipse have various altitudes. 4th. An eclipse of the sun begins on the 
 western side and ends on the eastern. 5th. When the straight line passing through 
 the centres of the sun and moon passes also through the place of the spectator,. the 
 eclipse is said to be central: a central eclipse may be either annular or total, ac- 
 cording 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. 7th. 
 In most solar eclipses the moon's disc is covered with a faint light, a phenomenon 
 which is attributed to the reflection of the light from the illuminated part of the 
 arth. 
 
 CALCULATION OF AN ECLIPSE OF THE SUN. 
 
 (1.) Of the circumstances of th,e general eclipse. 
 
 473. It is a simple inference from what has been established in Art.. 459, that an 
 eclipse of the sun will begin and end upon the earth, at the times before and after 
 conjunction, when the distance of the centres of the moon and sun is equal to 
 P p-\-S-{-d' t that the total eclipse will begin and end when this distance is 
 equal to P p &-\-d\ and the annular eclipse when the distance is equal to 
 P p + &^td. 
 
 474. The times of the various phases of the general eclipse of the sun may bo 
 obtained by a process precisely analogous to that by which the times of the pknses 
 of an eclipse of the moon are found. Let C (Fig. 79) 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 mo- 
 tion 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 which 
 obtains in the actual state of the case. Let N'C'L' represent the orbit that would 
 be described by the moon if it had such a motion, which is called the Relative Or- 
 bit. Let CM be drawn perpendicular to it ; and let C/= C/ = P ^-4-3-f d t 
 and Cg = Cg' =P p S + d, orP p-\-& d t according as the eclipse is to- 
 
 23 
 
178 
 
 ECLIPSES OF THE SUN AND MOON. 
 
 tal or annular. Then, M will be the place of the moon's centre at the middle of 
 the eclipse ; /and/ the places at the beginning and end of the eclipse ; a.ndg and 
 g' the places at the beginning and end of the total, or of the annular eclipse. We 
 hall thus have, as in eclipses of the moon, 
 
 Fig. 79. 
 
 tan? I = -=5 , CM = X cos I, C'M = X sin I ... (112). 
 
 J*L m 
 
 3600s. X sin I cos I 
 
 Jutervalfrom con. to mid. _ ,_ 
 
 ivi m. 
 
 Interval from middle to beginning or end 
 
 3600s. cos I 
 
 : ~lK m 
 Interval for total eclipse 
 
 3600s. cos I 
 
 . . . (113). 
 
 XcosI) . . . (114). 
 
 = " u :r " V(*" -f X cos I) (k - X cos I) 
 
 IVL ~ fTt 
 
 (115). 
 
 Interval for annular eclipse 
 3600s. cos I 
 
 M m 
 
 Quantity = 
 
 "+X cos I) (A:'" X cos I) ... (116). 
 X cos I) 
 
 .. . (117). 
 
 d . . . (118). 
 
 The letters X, M, m, &c., represent quantities of the same name as in the formulas 
 for a lunar eclipse ; but they designate the values of these quantities at the time of 
 conjunction, instead of opposition. These values are in practice obtained from ta- 
 bles of the sun and moon, as in a lunar eclipse. 
 
 475. 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 pre- 
 cisely similar manner with those of an eclipse of the moon, (456.) 
 
 (2.) Of the phases of the eclipse at a particular place. 
 
 476. The phase of the eclipse, which obtains at any instant at a given place, is 
 indicated by the relation between the apparent distance of the centres of the sun 
 and moon, and the sum, or difference, of their apparent semi-diameters : and the 
 calculation of the time of any given phase of the eclipse, consists in the calculation 
 of the time when the apparent distance of the centres has the value relative to the 
 sum or difference of the semi-diameters, answering to the given phase, ihus, if 
 we wish to find the time of the beginning of the eclipse, we have to seek the time 
 when the apparent distance of the centres of the sun and moon first becomes equal 
 to the sum of their apparent semi-diameters. 
 
 477. The calculation of the different phases of an eclipse of the sun, for a par- 
 ticular place, involves, then, the determination of the apparent distance of the cen- 
 tres 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. 
 
SOLAR ECLIPSE. APPROXIMATE TIMES OF PHASES. 179 
 
 478. According to the celebrated astronomer Dus6jour, in order to make the ob- 
 servations agree with theory, it is necessary to diminish the sun's semi -diameter, as 
 it is given by the tables, 3".5. This circumstance is explained by supposing that 
 the apparent diameter of the sun is amplified, by reason of the very lively impres- 
 sion wlu'ch its light makes upon the eye. This amplification is called Irradiation. 
 He also thinks that the semi-diameter of the moon ought to be diminished 2", to 
 make allowance for an Inflexion of the light which passes near the border of this 
 luminary, supposed to be produced by its atmosphere. It must be observed, how- 
 ever, that the astronomers of the present day do not agree either as to the neces- 
 sity or the amount of the diminutions just spoken of. 
 
 479. 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 ap- 
 parent longitude and latitude of the two bodies have been found. Now, the true 
 longitude of the sun, and the true longitude and latitude of the moon, may be found 
 from the tables, (Probs. IX and XIV) ; and from these the apparent longitudes and 
 latitudes may be deduced by correcting for the parallax. But the customary mode 
 of proceeding is a little different from this : the true ivm jitude and latitude of the 
 sun are employed instead 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.) 
 
 480. 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 
 horizontal parallax from the reduced horizontal parallax of the moon,* and calcu- 
 late 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 
 longitude of the sun, we shall know whether apparent ecliptic conjunction occurs 
 before or after the time T. Let T' denote the time an hour earlier or later than 
 the time T, according as the apparent conjunction is earlier or later. With the 
 sun and moon's longitudes, the moon's latitude, and the hourly motions in longi- 
 tude and latitude, at the time T, calculate the longitudes and the moon's latitude 
 for the time T' ; and for 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 sun in longitude, at this time. Let it be denoted by n. Find, 
 in like manner, the apparent distance of the moon from the sun in longitude at the 
 time T', and denote it by n'. In the same manner as at the time T, we find wheth- 
 er apparent conjunction occurs before or after the tinieT'. If it occurs between 
 the times T and T', the sum of n and n', otherwise their difference, will be the 
 apparent relative motion of the sun and moon in longitude in the interval T' T, 
 or T T' ; from which the relative hourly motion will become known. The dif- 
 ference of the apparent latitudes of the moon, at the times T and T', will make 
 known the apparent relative hourly motion in latitude. With the relative hourly 
 motion in longitude and the 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, with the apparent latitude of the moon at the time 
 T and its hourly motion in latitude, find the apparent latitude at the time of ap- 
 parent conjunction thus determined. Then, knowing the relative hourly motion 
 of the sun and moon in longitude and latitude, together with the time of apparent 
 conjunction, and the apparent latitude at that time, and regarding 1 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 the quantity of the 
 eclipse, may be calculated after the same manner as in the general eclipse ; the 
 disc of the sun answering to the section of the luminous frustum mentioned in Art 
 
 * The reduced horizontal parallax of the moon is its horizontal parallax as re 
 duced from the equator to the given place. (See Prob. XV.) 
 
180 
 
 ECLIPSES OF THE SUN AND MOON. 
 
 457, and the apparent elements 
 answering to the true. Let C 
 (Fig. 80) represent, the centre 
 of the sun supposed stationary, 
 CO' 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 an- 
 gle with the ecliptic NT, or by- 
 its passing through the situa- 
 tions of the moon at the times 
 T and T'. Also, let RKFK' 
 be the border of the sun's disc ; 
 f,f the positions of the moon's 
 
 centre at the beginning aud end of the eclipse, determined by describing a circle 
 
 around C as a centre, with a radius equal to the sum of the apparent semi-diame. 
 
 ters of the sun and moon ; and M (the foot of the perpendicular let fall from C 
 
 upon N'C') its position at the time of greatest obscuration. 
 
 If the eclipse should be total or annular, then g, g' will be the positions of the 
 
 moon's centre at the beginning and end of the total or annular eclipse ; these 
 
 points being determined by describing a circle around C as a centre, and with a 
 
 radius equal to the difference of the apparent semi-diameters of the sun and 
 
 moon. 
 
 The results will be a closer approximation to the truth, if the same calculations 
 
 that are made for the time T' be made also for another time T". 
 
 The various circumstances of the eclipse may also be had by construction, after 
 
 the same manner as in a lunar eclipse, (456.) 
 
 481. In order to be able to observe the beginning or end of a solar eclipse, it is 
 necessary to know the position of the point on the sun's limb where the first or 
 last contact takes place. The situation of these points is designated by the dis- 
 tance on the limb, intercepted between them and the highest point of the limb, call- 
 ed the Vertex. The contacts will take place at the points t, t' , (Fig. 80,) on the 
 lines C/, Cf. To find the position of the vertex, with the sun's longitude found 
 for the beginning of the eclipse, calculate the angle of position of the sun at that 
 time, (see Prob. XIII,) and lay it off to the right of the circle of latitude CK when 
 the sun's longitude is between 90 and 270, and to the left when the longitude is 
 less than 90 or more than 270. Suppose CP to be the circle of declination thus 
 determined. Next, let Z (Fig. 24, p. 47) be the zenith, P the xelevated pole, and S 
 the sun ; then in the triangle 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 derived from the tables, (equa. 35.) These three quanti- 
 ties being known, ZSP, the angle made by the vertical through the sun with its 
 circle of declination, may be computed ; and being laid off in the figure to the 
 right or left of CP, (Fig. 80,) according as the time of beginning is before or after 
 noon, the point Z or Z', as the case may be, in which the vertical intersects the 
 limb RKK', will be the vertex, and the arc Z<, or Z't, on the limb, will ascertain 
 the situation of t, the first point of contact, with respect to it. 
 
 The situation of the last point of contact may be found by the same mode of 
 proceeding. 
 
 482. Let us now show how to find the exact times of the beginning, greatest 
 obscuration, and end of the eclipse, the approximate times being known. Let B 
 designate the approximate time of beginning, taken to the nearest minute. Cal- 
 culate for the time B by means of the tables, the 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 thc 
 relative parallax, calculate the apparent longitude, latitude, and semi-diameter of 
 the moon. Subtract the apparent longitude of the moon from the true longitude 
 of the sun ; the difference will be the apparent distance of the moon from the gun 
 in longitude : let it be denoted by a. Denote the apparent latitude of the moon 
 byX. 
 
SOLAR ECLIPSE TRUE TIMES OF PHASES. 181 
 
 Now, let EC (Fig 81) 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 = A. The right-angled triangle MSra being 
 very small, may be considered as a plane triangle, and 
 we therefore have, to determine A, the equation 
 A 2 = a 2- r -A 2 . . . (119).* 
 
 483. Having computed the value of A, we find, by 
 comparing it with the sum of the apparent semi-diame- 
 ters of the sun and moon, whether the beginning of the 
 eclipse occurs before or after the approximate time B. Fix 
 upon a time some 4 or 5 minutes before or after B, ac- 
 cording as the beginning is before or after, and call it 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 lati- 
 tude at the time B', and compute for this tinae thft apparent longitude, latitude, 
 and semi-diameter of the moon. Subtract the apparent longitude of the moon 
 from the true longitude of the sun, and we shall have the apparent distance of the 
 moon from the sun at the time B'. Take the difference between this and the same 
 distance a at the-time B, and we shall have the apparent relative motion of the 
 sun and moon in longitude during the interval of time between B and B'. Then 
 find, by simple proportion, the apparent relative hourly motion in longitude, and 
 denote it by k. Take the difference between the apparent latitudes of the moon 
 at the times B and B', and it will be the apparent relative motion of the sun and 
 moon in latitude, in the interval; from which deduce the apparent relative hourly 
 motion in latitude, and call it n. Now, put t = the interval between the ap- 
 proximate and true times of the beginning of the eclipse, and suppose S and M 
 (Fig. 81) to be the situations of the sun and moon at the true time of beginning. 
 In the time /. the apparent relative motions in longitude and latitude will be, re- 
 spectively, equal to kt and nt, and accordingly we shall have 
 
 Sm = a kt, MOT = A -|- 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, minus 5".5, (478.) We have then, to find t, the equation 
 
 or, developing and transposing, 
 
 ( n 2_|_ jfc2) *2_ 2 (ak An) * = ^2_ (a 2 + X 2 ) = i// 2 A 2 J 
 making A = ^ A 2 , and B = ak An, (n 2 -f &2) t 2 _ 2B* = A, 
 
 - . . . (120 , 
 
 The negative sign must be prefixed to the radical, for, if we suppose A to be equal 
 to zero, t must be equal to zero. Multiplying the numerator and denominator by 
 B-f. V B2-|-A ( n a + A8), and restoring the value of A, we obtain 
 
 3600s. (A 2 
 (in seconds) t= 
 
 Although this equation has been investigated for the beginning of the eclipse, it 
 is plain that it will answer equally well for the determination of the other phases, 
 
 * In place of equation (119) the following equations may be employed in loga- 
 hmic computation: 
 
 j _ _-j 
 
 rithmic computation 
 where 6 is an auxiliary arc 
 
182 ECJJPSES OP THE SUN AND MOON. 
 
 if we give the proper values and signs to ^, a, A, n, and k. k is positive before 
 conjunction :iml negative after it, and the radical quantity is negative after con- 
 junction ; n \3 negative, when the moon appears to recede from the north pole of 
 the ecliptic ; A hr\ the sign , when it is south ; a is always positive.* 
 The value of t taken with its sign is to be added to the time B. 
 
 484. The values of the quantities a, A, n, and k, are found for the other phases 
 after the same manner a* for the beginning. 
 
 To obtain the value of ^ at the time of greatest obscuration, find the rela- 
 tive motions in longitude and latitude, (k and n,} during some short interval near 
 the middle of the eclipse, which is the approximate time of greatest obscuration j 
 then compute the inclination of the relative orbit by the equation 
 
 tang I = | ... (122.) (See equa. 90) : 
 
 after which ^ will result from the equation 
 
 i/' = A cos I , . . (123.) (See equa. 94). 
 
 A is the moon's latitude at the time of apparent conjunction, which is easily tal. 
 culated, by means of the values of A: and n, and the apparent longitude and lati- 
 tude of the moon, found for some instant near the time of apparent conjunction. 
 
 For the beginning and end of the total eclipse, we have, \p = appar. semi-diam. 
 of moon appsir. semi-diam. of sun -f- 1"-5 ; and for the beginning and end of the 
 annular eclipse, i^/=^ appar. semi-diam. of sun appar. semi-diam. of moon 1".5. 
 
 485. If the value of <//, given by equation (123,) be substituted in equation 
 (121,) tlu's equation will make known the time of greatest obscuration; but this 
 may be found more conveniently by a different process. Let NCF (Fig. 82) repre- 
 
 . 82. sent a portion of the ecliptic, EML a portion 
 
 of the relative orbit passed over about the 
 time of greatest obscuration, C the stationa- 
 ry position of the sun's centre, and M the 
 place of the moon's centre at the instant of 
 its nearest approach to C. Also, let a = CR 
 the apparent distance of the moon from the 
 sun in longitude at the time of the nearest 
 approach of the centres, A' = RM the moon's 
 C B. N apparent latitude at the same time, k = M& 
 
 the apparent relative motion in longitude in some short interval about this time, 
 and n = krt the moon's apparent motion in latitude during the same interval. The 
 right-angled triangles Mnk and CMR are similar, for their sides are respectively 
 perpendicular to each other ; whence, 
 
 M* : MR : : kn : CR ; 
 
 and CR = MR^L or, = A' | . . . (124). 
 
 If the moon's apparent latitude be found for the approximate time of greatest 
 obscuration, and substituted for A' in equation (124,) this equation will give very 
 nearly the apparent distance (a) of the two bodies in longitude at the true time of 
 greatest obscuration. ^Vith 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 A' in equation (124) and then re- 
 peating the calculations. 
 
 486. A simpler, though less accurate method than that already given, of find- 
 ing the times of beginning and end of the total or annular eclipse, is to compute 
 the half duration of the total or annular eclipse, and add it to, and subtract it from, 
 
 * Developing the radical in equation (120,) and neglecting all the terms after 
 tht second, as being very small, we obtain for the beginning and end of the eclipse 
 llie mm* convenient formula 
 
 _ 1800s. (A2 ^2) 
 
 B 
 
OCCULTATIONS. 183 
 
 the time, of greatest obscuration. This interval may easily be determined, if we 
 can find the rate of motion on the relative orbit, and the distance passed over by 
 the moon's centre during the interval. Let g,g' (Fig. 82) be the places of the 
 moon's centr5 at the instants of the two interior contacts, and M/i the distance 
 passed over in some short interval (L). Let 9 = < ~M.nk the complement of the 
 inclination of the relative orbit, k = MAr, k' = Mn, and t = half duration of total 
 or annular eclipse. The triangles MnA:, CRM, give 
 
 JL-. . . (125): 
 sin 
 
 p T/r \ l 
 
 and tang RCM = tang Mw& = -^, or, tang = - ... (126). 
 
 Finding the value of by the last equation, and substituting it in equation (125K 
 we obtain the value of k' ; and then, to find t, we have 
 
 _A2 (Art. 484) ; 
 
 L V 4,2 _ A 2 
 
 whence, t = - ^ - = - -^ -- . . . (127) 
 
 487. The apparent distance of the centres of the two bodies at the time of great- 
 est obscuration 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- 
 diameters, and state the proportion, as the sun's apparent diameter : the remain- 
 der : : 12 digits : the digits eclipsed. (For a more particular description of the 
 method of calculating a solar eclipse, see Prob. XXX.) 
 
 OCCULTATIONS. 
 
 488. At all places upon the earth's surface, which at a given 
 time have the moon in the horizon, its apparent place will differ 
 from its true place, by the amount of its horizontal parallax. It 
 follows, therefore, that a star will be eclipsed by the moon some- 
 where 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 hori- 
 zontal parallax. 
 
 The greatest value of the moon's semi-diameter is 1 6' 45", and 
 that of its horizontal parallax 61' 24". If we add the sum of these 
 numbers to 5 17' 34", the maximum latitude of the moon, we ob- 
 tain as the result 6 35' 43". It is then only the stars which have 
 a latitude less than 6 35' 43" that can experience an occn Ration 
 from the moon. 
 
 489. By considering the various situations of the stars liable to an occultation, 
 taking the greatest and least values of the sum of the moon's semi-diameter and 
 horizontal parallax, and allowing for the inequalities of the motions of the moon, 
 it has been found, that, if at the time of the mean conjunction of the moon and 
 a star, (that is, when the moon's mean longitude is the same with the longitude of 
 the star,) their difference of latitude exceed 1 37', there cannot be an occultation ; 
 if the difference be less than 51', there must be an occultation somewhere on the 
 earth ; and that between these limits there is a doubt, which can only be removed 
 by the calculation of the mo&i's true place. 
 
 490. The calculation of an oecultation 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 latitude of the moon, employed in solar eclipses, the difference between 
 
184 OF THE PLANETS AND THEIR PHENOMENA. 
 
 the latitude of the moon and that of the star, and in place of the difference be- 
 tween 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 (Fig. 81) represent the ecliptic, K its pole, s the situation 
 of the star, M that of the moon, and sm 1 an arc passing through s and parallel to 
 the arc EC, we have in place of mM, m'M = mM mm', and in place of S-m, s-m'. 
 The hourly variation of Sm must also be reduced to the arc sm'. 
 
 Fig. 83. 491 - The reduction of the difference of longitude ef the 
 
 moon and star, to the parallel to the ecliptic, p.issing 
 through the star, is effected by multiplying this difference 
 by the cosine of the latitude of the star. For, let AB (Fig. 
 83) be an arc of the ecliptic, and A'B' the corresponding 
 are of a circle parallel to it ; then, since similar arcs of cir- 
 _,/ cles are proportional to their radii, we have 
 
 BC : B'C' : : AB : A'B' 
 
 c But, B'C' = Co = B'C cos BCB' = BC cos BB' : 
 
 . AB.BCcosBB' 
 
 hence, A'B' = - = AB cos BB'. 
 
 BO 
 
 The reduction of the relative hourly motion in longitude to the parallel in ques- 
 tion, is obviously effected in the same manner. 
 
 CHAPTER XVI. 
 
 OF THE PLANETS, AND THE PHENOMENA OCCASIONED BY THEIR 
 MOTIONS IN SPACE. 
 
 APPARENT MOTIONS OF THE PLANETS WITH RESPECT TO 
 
 THE SUN. 
 
 492. 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 him, and never receding from 
 him beyond a certain distance, while the superior planets are seen 
 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. 
 
 Let CAC'B (Fig. 84) represent the orbit of either one of the in- 
 ferior 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 sphere of the heavens to which all bo 
 dies are referred. Suppose, for the present, that the earth is sta 
 tionary in the position P, and through P draw the lines PA, PB, 
 tangent to the orbit of Venus, and prolong them on till they inter 
 
APPARENT MOTIONS OF THE PLANETS. 
 
 185 
 
 sect the heavens at a and b. When Venus is at C, (the earth be- 
 ing at P,) she will be in superior conjunction, and when at C' in 
 inferior conjunction. 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 
 
 Fig. 84. 
 
 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 him, from C' to B 
 recede from him again, and from B to C approach him again, a 
 and 6 will be her positions in the heavens at the times of her great- 
 est eastern and western elongations. 
 
 When Venus is to the east of the sun, she is seen in the even- 
 ing, and called the Evening Star ; and when to the west, she is 
 seen in the morning, and called the Morning Star. 
 
 493. 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 un- 
 der consideration, as it is slower than that of the planet, is to cause 
 the points A, C', B to advance in the orbit, without altering the 
 nature of the apparent motion of the planet with respect to the sun. 
 The orbits of the earth and planet are also ellipses of small eccen- 
 tricity, and are slightly inclined to each other, instead of being cir- 
 cles and lying in the same plane : on this account, as the greatest 
 elongations will occur in various parts of the orbits, they will differ 
 in value. The greatest elongation of Venus varies from 45 to 47 
 12'. Its mean value is about 46. 
 
 494. Owing to the circumstance of the orbit of Mercury being 
 
 24 
 
186 OF THE PLANETS AND THEIR PHENOMENA. 
 
 within the orbit of Venus, the greatest elongation of this planet is 
 less than that of Venus. It varies between the limits 16 12', and 
 28 48'; and is, at a mean, 22 30'. 
 
 495. Next, suppose PKT (Fig. 84) to be the orbit of a superior 
 planet, and CAC'B that of the earth ; and, as the velocity of the 
 earth is much greater than that of the planet, let us, for the present, 
 regard the planet as stationary in the position P, while the earth 
 describes the circle CAC'. When the earth is at C, the planet, 
 being at P, is in conjunction with the sun. When the earth is at 
 A, SAP, the elongation of the planet is 90. When it arrives at 
 C', the planet is in opposition, or 180 distant from the sun : and 
 when it reaches B, the elongation is again 90. At intermediate 
 points the elongation will have intermediate values. If, now, we re- 
 store to the planet its orbitual motion, we shall manifestly be con- 
 ducted 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 sy- 
 nodic revolution a superior planet will be seen at all angular dis- 
 tances from the sun, both on the east and west side of him. From 
 conjunction to opposition, that is, while the earth is passing from 
 C to C', the planet will be to the right, or to the west of the sun ; 
 and will therefore be below the horizon at sunset, and rise some 
 time in the course of the night. But, from opposition to 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. 
 
 496. To find the 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 in- 
 stance,) Venus will have described an angle m, and the earth an 
 angle M around the sun. Now, m is greater than M ; therefore 
 at the end of a day, the separation of Venus from the earth, (mea- 
 suring the separation by an angle formed by two lines drawn from 
 Venus 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 will be 2 (m M) ; at the end of three days, 3 
 (m M) ; at the end of s days, s (m M). When the angle of 
 separation amounts to 360, jhat is, when s (m M) = 360, the 
 sun, Venus, and the earth must be again in the same right line, 
 and in that case 
 
 . 
 
 In which expression s denotes the mean duration of a synodic 
 revolution, m and M being taken to denote the mean daily motions. 
 
 We may obtain from equation (128) another equation, in which 
 the synodic revolution is expressed in terms of the sidereal periods 
 of the earth and planet. 
 
SYNODIC REVOLUTIONS OF THE PLANETS. 187 
 
 Let P and p denote the sidereal periods in question , then, since 
 
 Id. : M : : P : 360, 
 and 1 :.w : : p : 360 ; 
 
 ,, 360 360 , . . 
 
 M = -, and m = - ; substituting, 
 
 **rjf 
 
 Equations (128), -(129), although investigated for an inferio-j 
 planet, will answer equally well for a superior planet, provided we 
 regard m as standing for the mean daily motion of the earth, M foi 
 that of the planet, p for the sidereal period of the earth, and P foi 
 that of the planet. For the earth holds towards a superior planet. 
 the place of an inferior planet, and a synodic revolution of the earth 
 to an observer on the planet, will obviously be a synodic revolu- 
 tion of the planet to an observer* on the earth. 
 
 497. Equation (128) 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 equation (129), 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 pe- 
 riod of the earth. The actual lengths of the synodic /evolutions 
 of the different planets are given in Table V. 
 
 498. The mean synodic revolution of a planet being known, and 
 also the time of one conjunction or opposition, we may easily as- 
 certain its mean elongation at any given time, and thus approxi- 
 mately the time of its rising, setting, and meridian passage. 
 
 499. A planet will rise and set at the same hours at the end of a 
 synodic revolution ; and will be an evening star, that is, above the 
 horizon at sunset, during half of a synodic revolution, and a morn- 
 ing star, that is, above the horizon at sunrise, during an equal in- 
 terval of time. The inferior planets will be evening stars from 
 superior to inferior conjunction ; and the superior planets from op- 
 position 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 RETROGRADATIONS OF THE PIANETS. 
 
 500. The apparent motions of the planets in the heavens, as has 
 already been stated (13), are not, like those of the sun and moon, 
 
188 
 
 OF THE PLANETS AND THEIR PHENOMENA. 
 
 continually from west to east, or direct, but are sometimes also 
 from east to west, or retrograde. The retrograde motion takes 
 place over arcs of but a small number of degrees ; and in changing 
 the direction of their motions, the planets are for several days sta- 
 tionary in the heavens. These phenomena are called the Stations 
 and Retrogradations of the planets. We now propose to inquire 
 theoretically into the particulars of the motions in question, and to 
 show how the phenomena just mentioned result from the motions 
 of the planets in connection with the motion of the earth. 
 
 Let CAC'B (Fig. 84, p. 185) 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 elonga- 
 tion to A the position of greatest eastern elongation, it would ad- 
 vance in the heavens from 6 to a ; that, while it was moving from 
 A to B, that is, from greatest eastern to greatest western elonga- 
 tion, it would retrograde in the heavens from a to b ; and that, in 
 passing the points A and B, as it would be moving directly towards 
 or from the earth, it would for a time appear stationary in the 
 heavens in the positions a and b. 
 
 But the earth is in fact in motion, and the actual apparent mo- 
 tion of the planet is in consequence materially different from this. 
 Let A, A 7 (Fig. 85) be the positions of the planet and earth at the 
 time of the greatest eastern elongation, C', P their positions at in- 
 
 . 85. 
 
 ferior conjunction, and B, B 7 their positions at the greatest western 
 elongation. At the time of the greatest eastern elongation, while 
 
STATIONS AND RETROGRADAT10NS OF THE PLANETS. 189 
 
 the planet describes a certain distance AD on the line of the cen- 
 tres of the earth and planet, the earth moves forward in its orbit a 
 certain distance A'D' ; so that, instead of appearing stationary at a 
 in the interval, the planet will advance in the heavens from a to d. 
 From the same cause it will have a direct motion about the time 
 of the greatest western elongation. As it advances from A towards 
 C', the direct motion will continue ; but, as the daily arc described 
 by the planet will make a less and less angle with the daily arc de- 
 scribed 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 f , 
 at the beginning and end of the day are parallel, it will be station 
 ary 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 retro- 
 grade. 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 un- 
 til it has passed inferior conjunction and arrived at its western sta- 
 tion ; 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 sta- 
 tion ; C'q = C'p, and Pn = Pm. 
 
 The diminution of the elongation of the planet at its two stations 
 is not the only effect of the earth's motion in the case under con- 
 sideration ; it also accelerates the direct, and retards the retrograde 
 motion of the planet, and gives to the planet along with the sun an 
 apparent motion of revolution around the earth. 
 
 501. Let us now pass to the case of a superior planet. . Sup- 
 pose AC'B (Fig. 85) to be the orbit of the earth, and A'PB' that 
 of the planet. Since the earth is an inferior planet to an observer 
 stationed upon a superior planet, it appears by the foregoing arti- 
 cle that it will, to an observer so situated, have a retrograde mo- 
 tion while it is passing over a certain arc pC'q in the inferior part 
 of its orbit, and a direct motion during the remainder of the sy- 
 nodic revolution. Now, it is plain that the direction of the planet's 
 motion, as seen from the earth, will always be the same as the di- 
 rection of the earth's motion as seen from the planet. When the 
 earth is at C', the middle of the arcpC'^, the planet is in opposi- 
 tion. It follows, therefore, that a superior planet has a retrograde 
 motion during a small portion of its .synodic revolution, about the 
 time of opposition. (See Table V.) 
 
190 
 
 OF THE PLANETS AND THEIR PHENOMENA. 
 
 O 
 
 PHASES OF THE INFERIOR PLANETS. 
 
 502. 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 conjunction. 
 
 Mercury exhibits precisely similar phases, but being smaller, at 
 a greater distance from the earth, and much nearer the sun, its 
 phases are not so easily observed as those of Venus. 
 
 503. The phases of Venus are easily accounted for, by suppos- 
 ing it to be an opake spherical body, and to shine by reflecting the 
 sun's light, and by taking into consideration its motion with respect 
 
 to the sun and earth. The hemi- 
 sphere turned towards the sun is 
 illuminated by him, and the other is 
 in the dark, and as the planet re- 
 volves around the sun, various por- 
 tioiis of the enlightened half are 
 turned towards the earth : in supe- 
 rior conjunction, the whole of it ; at 
 the greatest elongations, one half; 
 and near inferior conjunction, but a 
 small part. This will be abundant- 
 ly evident on inspecting Fig. 86. 
 The phases corresponding to the 
 positions represented are delineated 
 in the figure. 
 
 The phases of Mercury are ob- 
 viously susceptible of a similar ex- 
 planation. 
 
 504. The disc of the planet Mars also undergoes changes of 
 form, but they are of comparatively moderate extent. It is some- 
 times 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 conjunction and opposition, but there would always be more 
 tlinn half of it turned towards the earth, and therefore the disc 
 bhould always be larger than a half circle. 
 
 505. The discs of the other superior planets do not experience 
 any perceptible variation of form, for the reason, doubtless, that 
 their orbits are so large with respect to the orbit of the earth, that 
 all, or very nearly all of their illuminated hemispheres, is con- 
 stantly visible from the earth. 
 
TRANSITS OF THE INFERIOR PLANETS. 191 
 
 TRANSITS OF THE INFERIOR PLANETS. 
 
 506 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 ap- 
 parent semi-diameter of the sun. 
 
 507. The transits of Venus take place alternately at intervals of 
 8 and 1051 O r 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 sy- 
 nodic 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 May 8th, 1845. The next two will take place in 1848, 
 and 1861, in the month of November. The first, which will oc- 
 cur on the 9th, will be visible in this country. 
 
 508. A transit is calculated in a precisely similar manner with' a 
 solar eclipse ; the planet in the one calculation answering to the 
 moon in the other. 
 
 509. A transit is an important phenomenon in a practical point 
 of view, as it furnishes the most exact means we possess of ascer- 
 taining the sun's parallax. In order to understand how this phe- 
 nomenon can be used for this purpose, we have only to consider 
 that, in consequence of the difference of the 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, the transit will take place along different chords, 
 and be accomplished in unequal portions of time. This fact is rep- 
 resented to the eye in Fig. 87. It is then to be expected, that, if the 
 durations of the transit at two different places should be noted, the 
 
 Fig. 87. 
 
 difference of the parallaxes of the sun and Venus, upon which 
 alone the difference of the duration depends, could be computed. 
 This computation is in fact possible. Also, the ratio of the paral 
 sixes being inversely as that of the distances, could be found by the 
 
192 OF THE PLANETS AND THEIR PHENOMENA. 
 
 elliptical theory of -the planetary motions, and thus the parallax both 
 of the sun and Venus would become known. 
 
 510. The parallax of the sun, as it is now known, was deduced 
 from observations upon the transits of Venus in 1769 and 1761. 
 Expeditions were fitted out on the most efficient scale, by the 
 British, French, Russian, and other governments, and sent to va- 
 rious parts of the earth, remote from each other, to observe the 
 transit of 1769, that the parallax of the sun might be computed 
 from the results of the observations. The sun's parallax, as de- 
 termined by Professor Encke from the observations made upon the 
 transit in question, and that of 1761, is 8".5776. ft" it 2. 
 
 APPEARANCES, DIMENSIONS, ROTATION, AND PHYSICAL 
 CONSTITUTION OF THE PLANETS. 
 
 511. It appears from admeasurement with the telescope and 
 micrometer, that the apparent diameter of a planet is subject to 
 sensible variations. The apparent diameter of Venus, as well as 
 of Mercury, is greatest in inferior conjunction, and least in superior 
 conjunction ; while the apparent diameter of each of the other 
 planets is greatest in opposition and least in conjunction. These 
 variations of the apparent diameters of the planets, are necessary 
 consequences of the changes that take place in the distances of 
 the planets from the earth. (See Fig. 84.) 
 
 51 2. The real diameter of a planet is deduced from its apparent 
 diameter and horizontal parallax. (See Art. 429.) When the di- 
 ameters 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. 
 
 513. The order of magnitude of the planets is as follows: 
 1 Jupiter, 2 Saturn, 3 Uranus, 4 the Earth, 5 Venus, 6 Mars, 
 7 Mercury, 8 Pallas, 9 Ceres, 10 Juno, 11 Vesta. The range of 
 magnitude, for the principal planets, is from 1 to about 20,000. 
 (The relative magnitudes of the planets are represented to the eye 
 in the Frontispiece.) (See Note VIII.) 
 
 514. Spots more or less dark have 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 are 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. (See Note IX.) 
 
 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. 
 
 515. The amount of light and heat, which the sun bestows upon 
 
MERCURY VENUS. 193 
 
 the planets, decreases as we recede from the sun, in the same ratio 
 that the square of the distance increases. (See Table IV.) 
 
 516 It will be seen in the sequel that the planets are all opake 
 bodies, like the earth ; and that they are surrounded with an atmo- 
 sphere, after the same manner as the earth. 
 
 MERCURY. 
 
 517. In consequence of its proximity to the sun, Mercury is 
 rarely visible to the naked eye. When seen under the most favor- 
 able circumstances about the time of greatest elongation, it presents 
 the appearance of a star of the 3d or 4th magnitude. Its phases 
 show that it is opake, and illuminated by the sun. Its apparent 
 diameter varies with its distance from 5" to 12". Its real diame- 
 ter is about 3000 miles, or f of that of the earth, and its volume 
 is about T V of the earth's volume.* 
 
 Mercury performs a rotation on its axis in 24h. S^m., and its 
 axis is inclined to the ecliptic under a small angle. 
 
 518. Owing to the dazzling splendor of its rays, and the tremulous motion in- 
 duced 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. Schroeter is the only observer who has ever detected any spots 
 upon it. From the fact that spots are only occasionally seen, it has been inferred 
 that the planet is surrounded with a dense atmosphere, which reflects a strong light, 
 and, except when it is particularly pure, prevents the darker body of the planet 
 from being seen. 
 
 Schroeter, in making observations upon Mercury at the time his disc had the 
 form of a crescent, discovered that one of the horns of the crescent became blunt 
 at 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. 
 
 519. Venus is the most brilliant of all the planets, and generally 
 appears larger and brighter than any of the fixed stars. At times, 
 it emits so much light as to be visible at noonday. 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 of about 
 40. They are about 36 days before and after inferior conjunc- 
 tion. The disc is then considerably less than a semicircle, but the 
 increased proximity to the earth more than compensates for the 
 diminished size of the disc. Venus will besides attain to greater 
 splendor in some revolutions than others, in consequence of being 
 nearer the earth, when in the most favorable position. 
 
 520. As seen through a telescope, Venus presents a disc of 
 nearly uniform brightness, and spots have very rarely been seen 
 upon it. Its phases prove it to be an opake spherical body, 
 shining by reflecting the sun's light. Its apparent diameter varies 
 with its distance from 10" to 61". Its real diameter is about 7800 
 
 * The exact diameters, volumes, times of rotation, &C.,. of the different planets*, 
 as far as known, may be found in Table IV. 
 
 25 
 
194 OF THE PLANETS AND THEIR PHENOMENA. 
 
 miles, and its volume about ^y less than that of the earth. The 
 period of its rotation is 23h. 21m. The inclination of its axis tc 
 the plane of its orbit is not exactly known, but is not far from 18. 
 
 521. From the remarkable vivacity of the light of this planet, which far ex- 
 ceeds that of the light reflected from the moon's surface, as well as the transitory 
 nature of the few darkish spots which have been seen upon its disc, it is inferred 
 that it is surrounded by a dense and highly reflective atmosphere, which in gene- 
 ral screens the whole of the darker body of the planet from our view. The truth 
 
 88. of this inference is confirmed by certain deli- 
 
 cate observations made by Schroeter. This 
 astronomer distinctly discerned a faint bluish 
 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. 88, where the left extremity of the dot- 
 ted line represents the natural terminating point 
 of one of the horns of the crescent. This he 
 considered to be a twilight on the surface of 
 Venus. 
 
 Since the transparency of Venus's atmosphere 
 is variable, becoming occasionally such as to 
 admit of the body of the planet's being seen 
 through it, we must suppose that it contains 
 aqueous vapor and clouds, and therefore that 
 there are bodies of water upon the surface of the planet. It is in fact supposed 
 that isolated clouds have actually been seen. The most natural explanation of 
 the bright spots which have sometimes been noticed on the disc is, that they are 
 clouds more highly reflective than the atmosphere or than the clouds in general. 
 
 522. There are great inequalities on the surface of Venus, and, it would seem, 
 mountains much higher than any upon our globe. Schroeter detected these masses 
 by several infallible marks. In the first place the edge of the enlightened part of 
 Venus is shaded, as seen in Figs. 88, 89, and 90, and as the moon appears when 
 in crescent even to the naked eye. This appearance is doubtless caused by shad- 
 ows 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 often appears rounded 
 at the corners, as in Fig. 89, owing undoubtedly to part of the disc being rendered 
 invisible there by the shadow or interposition of some line of eminences ; and at 
 
 Fig. 89. Fig. 90. 
 
 . 
 
 other times, as in Fig. 90, a single bright point appears detached from the disc 
 the top of a high mountain, illuminated across a dark valley. 
 
 Schroeter found that theke appearances recurred regularly at equal intervals of 
 about 23$ hours ; the same period as that which Cassini had previously found for 
 the completion of a rotation, by observations upon the spots. 
 
 
 
MARS. 195 
 
 MARS. 
 
 523 Mars is of the. apparent size of a star of the first or second 
 magnitude, and is distinguished from the other planets by its red 
 and fiery appearance. Ihe observed variation in' the form of its 
 disc (504) shows that it derives its light from the sun. Its' greatest 
 and least apparent diameters are respectively 4" and 18". Its real 
 diameter is something over 4000 miles, or rather more than | of 
 the diameter of the earth, and its bulk is about | of that of the 
 earth. ^ 
 
 : Mars revoIVes'on'its a'xis'in 24h. 37m. ; and its' axis is incliAed 
 to the ecliptic in an angle of about 60. It appears, from meas- 
 urements 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 Sir W. Herschel, its oblateness (159) 
 is. T V ' according to Arago ^ T . 
 
 524. When the disc of Mars is examined with telescopes of 
 great power it is generally seen to be diversified with spots of dif- 
 ferent shades, which, with occasional variations, retain constantly 
 the same size and form. 
 
 They are conjectured to be continents and seas. In fact, Sir J. F. W. Her- 
 schel has on several occasions, in examining this planet with a good telescope, no- 
 ticed that some of its spots are of a reddish color, while others have a greenish 
 tinge. The former he supposes to be land, and the latter water. Fig. 91 repre- 
 sents Mars in its gibbous state as Yig. 91. 
 seen by Herschel in his 20 feet re- 
 flector, on the 16th of August, 
 1'830. The darker parts are seas. 
 The bright spot at the 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 f been 
 conje6tured with a great deal of 
 probability to be snow ; as they 
 are reduced in size, and sometimes 
 disappear when they have been 
 long exposed to the sun, and are 
 greatest when just emerging from 
 the long night of their polar winter. 
 
 525. The great divisions of the 
 surface of Mars are seen with dif- 
 ferent degrees of distinctness at 
 
 different times, arid sometimes disappear, either partially or entirely : parts oi tun 
 disc also appear at times particularly dark or bright. From these facts it is to 'be 
 inferred that this planet is environed with an atmosphere, and that this contains 
 aqueous vapor which, by varying in quantity and density, renders its transpa- 
 rency variable. ; ^. ... 
 
 526. No mountains have been detected upon Mars. But this is no good reason 
 for supposing that they a,re 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 and 
 Mercury, when of the form of a crescent. The same remarks will apply with still 
 greater force to the other superior planets. 
 
 527. The ruddy color of the light of Mars has generally been attributed to its 
 
196 OF THE PLANETS AND THEIR PHENOMENA. 
 
 atmosphere, but Sir John Herschel finds a sufficient cause for this phenomenon in 
 the ochrey tinge of the general soil of the planet (524.) 
 
 JUPITER AND ITS SATELLITES. 
 
 528. Jupiter is the most brilliant of the planets, except Venus, 
 and sometimes even surpasses Venus in brightness. The eclipses 
 of its satellites prove that it is an opake body, and that it shines 
 by reflecting the light of the sun. Its apparent diameter, when 
 greatest, is 46", and when least, 30". 
 
 Jupiter is the largest of all the planets. Its diameter is about 
 1 1 times the diameter of the earth, or about 87,000 miles, and its 
 bulk is more than 1200 times that of the earth. It turns on an 
 axis nearly perpendicular to the ecliptic, and completes a rotation 
 in 9h. 56m. The polar diameter is about T \ less than the equa- 
 torial. 
 
 529. 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 plane of the equator. 
 
 . 90 . These are called the Belts of 
 
 Jupiter. (See Fig. 92, which 
 represents the appearance of 
 Jupiter as seen by Sir John 
 Herschel in his twenty-feet 
 reflector, on the 23d of Sep- 
 tember, 1832.) They vary 
 somewhat in number, breadth, 
 and situation on the disc, but 
 never in direction. Sometimes 
 only one or two are visible ; on 
 other occasions as many as 
 eigh^ have been seen at the 
 same time. Sir William Her- 
 schel 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 run- 
 ning 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 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 doubt that they 
 are on the body of the planet, and that this turns upon an axis. 
 Bright spots have also been noticed upon the belts. The belts 
 generally retain pretty nearly the same appearance for several 
 months together, but occasionally marked changes of form and 
 size have taken place in the course of an hour or two. 
 
 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 thpy are the body of the planet seen through an atmo. 
 
JUPITER SATURN. 197 
 
 sphere of variable transparency ; but in general having extensive tracts of compar- 
 atively clear sky in a direction parallel to the equator. These are supposed to b 
 determined by currents analogous to our trade winds, but of a much more steady 
 and decided character; as would be the necessary consequence of the superior 
 velocity of rotation of this planet. As remarked by Herschel, that it is the com- 
 paratively darker 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 probably bands of 
 clouds or tracts of less pure air. 
 
 530. The satellites of Jupiter, as it has been already remarked, 
 are visible with telescopes of very moderate power. With the 
 exception of the second, which is a little smaller, they are some- 
 what larger than the moon. The orbits of the satellites lie very 
 nearly in the plane of Jupiter's equator. They are therefore all 
 viewed nearly edgewise from the earth, and in consequence the 
 satellites always appear nearly in a line with each other. 
 
 531. Sir W. Herschel, in examining the satellites of Jupiter 
 with a telescope, perceived that they underwent periodical varia- 
 tions 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 
 a revolution around the primary ; and therefore, like the moon, 
 presented continually the same face to the primary. 
 
 SATURN, WITH ITS SATELLITES AND RING. 
 
 532. Saturn shines with a pale dull light. Its apparent diame- 
 ter varies only 3" or 4" by reason of the change of distance, and 
 is at the mean distance about 16". The eclipses of its satellites 
 prove that it is opake and illuminated by the sun. 
 
 Saturn is the largest of the planets, next to Jupiter. Its diame- 
 ter is about 10 times the diameter of the earth, or 79,000 miles; 
 and its volume is about 900 times that of the earth. The rotation 
 on its axis is performed in lOh. 29m. The inclination of its axis 
 to the ecliptic is about 60. Its oblateness is T V 
 
 533. The disc of Saturn, like that of Jupiter, is frequently 
 crossed with dark bands or belts, in a direction parallel to its equa- 
 tor. Extensive dusky spots are also occasionally seen upon its 
 surface. (See Fig. 93.) 
 
 The cause of Saturn's beltsus doubtless the same as of Jupiter's. They accord- 
 ingly prove the existence of an atmosphere and of aqueous vapor, and thus also of 
 bodies of water, upon the surface of Saturn. 
 
 534. The planet Saturn is distinguished from all the other 
 planets in being surrounded by a broad, thin, luminous ring, situ- 
 ated in the plane of its equator, and entirely detached from the 
 body of the planet. (See Fig. 93.) This ring sometimes casts a 
 shadow upon the planet, and is, in turn, at times partially obscured 
 
198 OF THE PLANETS AND THEIR PHENOMENA. 
 
 by the shadow of the planet ; from which we conclude that it is 
 opake, and receives its light from the sun. 
 
 Fig. 93. It is inclined to the 
 
 plane of the ecliptic in an 
 angle of about 28, and 
 during the motion of Sat- 
 urn in its orbit it remains 
 continually parallel to it- 
 self. The face of the ring 
 is, therefore, never viewed 
 perpendicularly from the 
 earth, and for this reason 
 never appears circular, al- 
 though such is its actual 
 form. Its apparent form 
 is that of an ellipse, more 
 or less eccentric, accord- 
 ing to the obliquity under which it is viewed, which varies with 
 the position of Saturn in its orbit. When it is seen under the 
 larger angles of obliquity, it appears as a luminous band nearly 
 encircling the planet, and is visible in telescopes of small power. 
 Stars can, also be seen between it and the planet in these positions. 
 At other times, when viewed very obliquely, it can be seen only 
 with telescopes of high power. When it is approaching the latter 
 state, it has the appearance of two handles or ansce, one on each 
 side of the planet. 
 
 It is also at times invisible. This is the case whenever the 
 earth and sun are on different sides of the plane of the ring, for 
 the reason that the illuminated face is then turned from the earth. 
 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 
 
 535. 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 ye'ar, that the plane of the ring is passing 
 by the orbit of the earth, the earth will sometimes be on the same 
 side of it as the sun, and sometimes on opposite sides. In the 
 latter case the ring will be invisible, and in the former will be seen 
 so obliquely as to be visible only in telescopes of considerable or 
 great power. All this will perhaps be better understood on con- 
 sulting Fig. 94, where efg represents the orbit of the earth. The 
 appearances of the ring in the different positions of the planet in it| 
 orbit are delineated in the figure 
 
 The plane of the ring will pass through the sun every semi- 
 
SATURN S RING, 
 
 199 
 
 revolution of Saturn, or, at a mean, about every 15 years, and at 
 the epochs at which the longitude of the planet is respectively 
 170 and 350. The ring will then disappear once in about 15 
 years ; but, owing to the different situations of the earth in its or- 
 
 Fig. 94. 
 
 bit, under circumstances oftentimes quite different. And the dis- 
 appearance 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 260. 
 The last disappearance took place in 1833 ; the next will be in 1847. 
 At the present time (1845) the north face of the ring is visible. 
 
 536. From observations made upon bright spots seen on the 
 face of the ring, Herschel discovered that it revolved 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 ro- 
 tation is lOh. 32m. It is remarkable that this is the period in 
 which a satellite assumed to be at a mean distance equal to the 
 mean distance of the particles of the ring, would revolve around 
 the primary according to the third law of Kepler. 
 
 The breadth of the ring is about one-half greater than its dis- 
 tance from the surface of the planet, and is about equal to one- 
 third the diameter of the planet, or 29,000 miles. 
 
 537. What we have called Saturn's ring consists in fact of two 
 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. 
 According to the calculations of Sir John Herschel, from the mi- 
 crometric measures of Professor Struve, the breadth of the interior 
 ring is about 17,200 miles, and of the exterior about 10,600 miles; 
 the interval between the rings is nearly 1800 miles, and the dis- 
 tance from the planet to the inside of the interior ring is a little 
 over 19,000 miles. The thickness of the rings is not well known.; 
 the edge subtends an angle much less than 1", which, at the dis- 
 tance of the planet, answers to about 5000 miles. Herschel makes 
 it less than 250 miles. (See Note X.) 
 
 538. Professor Bessel has shown that the double ring is not bounded by parallel 
 plane surfaces. He infers this to be the case from the fact that at almost every 
 
200 OF THE PLANETS AND THEIR PHENOMENA. 
 
 disappearance or reappearance of the ring, the two ansse have not disappeared 01 
 reappeared at the same time. He has also found, from a discussion of the obser- 
 vations which have been made upon the disappearances and reappearances of the 
 ring, that they cannot be satisfied by supposing the two faces of the ring to be 
 parallel planes. In view of all the facts, it seems most probable that the cross sec- 
 tion of each ring is a very eccentric ellipse, instead of a rectangle, and that it varies 
 somewhat in size from one part of the ring to another. It may have irregularities 
 on its surface as great or greater than those which diversify the surface of the earth. 
 
 539. Whatever may be the form of the rings, their matter is not uniformly dis- 
 tributed. For recent micrometric measurements of great delicacy, made by Pro- 
 fessor 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 gravi- 
 tation, that this circumstance was essential to the stability of the rings. He de- 
 monstrated that if the centre of gravity of either ring were once strictly coincident 
 with the centre of gravity of the planet, the slightest disturbing force, such as the 
 attraction of a satellite, would destroy the equilibrium of the ring, and eventually 
 cause the ring to precipitate itself upon the planet. 
 
 540. 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 orbitual motion, and under the full and free influence of all the acting 
 forces. The latter supposition accords with Laplace's theory of the progressive 
 creation of the universe, hereafter to be noticed. 
 
 541. The satellites of Saturn were discovered, the 6th in the 
 order of distance by Huygens, in 1655, with a telescope of 12 feet 
 focus ; the 3d, 4th, 5th, and 8th, by Dominique Cassini, between 
 the years 1670 and 1685, with refracting telescopes of 100 and 
 136 feet in length ; and the 1st and 2d by Sir William Herschel, 
 in 1789, with his great reflecting telescope of 40 feet focus. All 
 but the 1st and 2d are visible in a telescope of a large aperture, 
 with a magnifying power of 200. (See Note XL) 
 
 They all, with the exception of the 8th, revolve very nearly in 
 the plane of the ring and of the equator of the primary. The or- 
 bit of the 8th is inclined under a considerable angle to this plane. 
 According to Sir John Herschel, the 6th satellite is much the lar- 
 gest, and is estimated to be not much inferior to Mars in size. The 
 others diminish in size as we proceed inward ; until the 1 st and 2d 
 are so small, and so near the ring, that they have never been dis- 
 cerned 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 8th satellite is subject to periodical variations of lustre, 
 which prove its rotation on an axis in the period of a sidereal revo- 
 lution of Saturn. 
 
 URANUS AND ITS SATELLITES. 
 
 642. Uranus is scarcely visible to the naked eye. In a tele- 
 scope it appears as a small round uniformly illuminated disc. Its 
 
URANUS VESTA JUNO CERES PALLAS. 201 
 
 apparent diameter is about 4", from which it never varies much, 
 owing to the smallness of the earth's orbit in comparison with its 
 own. Its real diameter is about 34,500 miles, and its bulk 82 
 times that of the earth. Analogy leads us to believe that this pla- 
 net is opake and turns on an axis, but there is no direct proof that 
 this is the case. 
 
 543. The satellites of Uranus were discovered by Sir W. Her- 
 schel. They are discernible only with telescopes of the highest 
 power. (See Note XII.) 
 
 VESTA JUNO CERES PALLAS. 
 
 544. These four planets, although less distant than several of 
 the others, are so extremely small, that they cannot be seen with- 
 out the aid of a telescope. 
 
 Vesta is the most brilliant, and shines with a white light. In 
 the telescope it appears as a star of about the 6th magnitude. Juno 
 and Ceres have the apparent size of a star of the 8th magnitude ; 
 and together with Pallas have a ruddy aspect and a variable lus- 
 tre, indicative of the presence of atmospheres of variable density 
 and purity. Ceres and Pallas generally shine with a pale dull 
 light, and are seen surrounded with a nebulosity, or haziness of, 
 according to Herschel, from three to six times the extent of the 
 body of the planet. This haziness is sometimes so decided as to 
 conceal the body of the planet from view, and at other times en- 
 tirely disappears, leaving the disc of the planet sharply denned and 
 alone visible. 
 
 545. The actual magnitudes of these planets are not well known. 
 The determinations of different Astronomers are widely different. 
 The following are perhaps the nearest approximations to their true 
 diameters that have yet been obtained : Vesta 270 miles ; Juno 
 460 miles ; Ceres 460 miles ; Pallas 670 miles. 
 
 CHAPTER XVII. 
 
 OF COMETS. 
 THEIR GENERAL APPEARANCE VARIETIES OF APPEARANCE. 
 
 546. THE general appearance of comets is that of a mass of 
 some luminous nebulous substance, to which the name Coma has 
 been given, condensed towards its centre around a brilliant Nucleus 
 that is in general not very distinctly denned, from which proceeds 
 in a direction opposite to the sun a fainter stream or train of simi- 
 lar nebulous matter, called the Tail. The coma and nucleus to- 
 gether form what is called the Head of the Comet. (See Fig. 95.) 
 
 26 
 
202 OF COMETS. 
 
 The tail gradually increases in width, and at the same time di- 
 minishes in distinctness from the head to its extremity, where it is 
 generally many times wider than at the head, and fades away un- 
 
 Fig. 95. 
 
 Great Comet of 18 11. 
 
 til it is lost in the general light of the sky. It is, in general, less 
 bright along its middle than at the borders. From this cause the tail 
 sometimes seems to be divided, along a greater or Less portion of 
 its length, into two separate tails or streams of light, with a com- 
 parative 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. 
 The 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 de- 
 viates more or less from this, the position of exact opposition 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. 
 
 547. 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 
 
GENERAL APPEARANCE OF COMETS. 
 
 203 
 
 the period of its visibility. When a comet first appears, in general, 
 no tail is perceptible, and its light is very faint. As it approaches 
 the sun, it becomes brighter: the tail also after a time shoots out 
 from the coma, and increases from day 10 day in extent and dis- 
 tinctness. As the comet recedes from the sun, the tail 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 de- 
 creases in brightness, till at length the comet resumes nearly its 
 first appearance, and ^finally disappears. (See Fig. 97.) It some- 
 times happens that, owing to peculiar circumstances, Fig. 96. 
 a comet does not make its appearance in the firma- 
 ment until after it has passed the sun in the heavens, 
 and not until it has attained to more or less distinct- 
 ness, and is furnished with a tail of considerable or \ 
 even great length. This was remarkably the case i 
 with the great comet, of 1843. (See Art. 326 ; also 
 Fig. 96.) 
 
 548. The tail of a comet 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 1 
 greatest at this time, as they will depend upon the T 
 distance and position of the earth, as well as the ac- 
 tual size and intrinsic brightness of the comet. To 
 Fig. 97. 
 
 N 
 
 illustrate this, let abed (Fig. 97) represent the orbit j 
 of the earth, and MPN the orbit of a comet, having j 
 its perihelion at P. Now, if the earth should chance 
 to' be at a when the comet, moving towards its peri- 
 helion, is at r, it might very well happen that the 
 comet would appear larger and more distinct than 
 
204 OF COMETS. 
 
 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 c or b soon after the perihelion passage : and 
 it would be the least conspicuous possible if the comet, sup- 
 posed to be moving in the direction NPM, should pass from N 
 around to M, while the earth is moving around from a to b or c, so 
 as to be continually comparatively remote from the comet, and so 
 that the comet will be in conjunction with the sun at the time after 
 the perihelion passage when its actual size and intrinsic lustre are 
 the greatest. It is to be observed that the apparent lustre of a 
 comet is sometimes very much enhanced by the great obliquity of 
 the tail, in some of its positions, to the line of sight. This seems 
 to have been the case with the comet of 1843, on February 28th, 
 (see Fig. 56,) and was doubtless 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 different returns of the same comet to its perihelion, it will 
 be seen, on examining Fig. 97, that the circumstances of the ap- 
 pearance and disappearance of the comet, as well as its size and 
 distinctness, may be very various at its different returns. This 
 has been strikingly true in the case of Halley's Comet. Gambart's 
 Comet was also invisible in 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. 
 
 549. Individual comets offer considerable 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 1744 was pro- 
 vided with six, which were spread out, like an immense fan, 
 through an angle of 1 17 ; and that of 1823 with two, one directed 
 from the sun in the 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 entire- 
 ly destitute of every vestige of a tail. They have the appearance of 
 round masses of luminous vapor, somewhat more dense towards the 
 centre. Such are Encke's and Biela's comets. (See Fig. 98.) The 
 point of greatest condensation is often more or less removed from the 
 centre of figure on the side towards the sun ; and sometimes also 
 on the opposite side. (See Note XIII.) 
 
 550. The comets which have had the longest tails are those of 
 1680, 1769, and 1618. The tail of the great comet of 1680, when 
 apparently the longest, extended to a distance of 70 from the head 
 that of the comet of 1 769, a distance of 97 ; and that of the com- 
 
FORM, STRUCTURE, AND DIMENSIONS OF COMETS. 205 
 
 et of 1618, 104. These are the apparent lengths as seen at cer- 
 tain places. By reason of the different degrees of purity and den- 
 sity 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 
 
 Fig. 98. 
 
 Encke's Comet. 
 
 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 
 tail extends an unknown distance beyond the extremity of the ap- 
 parent tail. 
 
 FORM, STRUCTURE, AND DIMENSIONS OF COMETS. 
 
 551. 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 surface 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 surrounds the nu- 
 cleus, 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, j^owever, is not truly spherical, but approximates 
 to that of an hyperboloid having the nucleus in its focus and its ver- 
 tex turned towards the sun. The tail begins where the nebulosity 
 terminates, and seems, in general, to be merely the continuation 01 
 this in nearly a straight line beyond the nucleus. There is ordina- 
 rily, as has been already intimated, a distinct space containing but 
 little luminous matter between the nucleus and the nebulosity, but 
 this is not always the case. The tail of a comet has the shape of 
 a hollow truncated cone, with its smaller base in the nebulosity of 
 the head ; with this difference, however, that the sides are usually 
 
206 OF COMETS. 
 
 more or less curved, and ordinarily concave towards the axis. Tha' 
 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. 
 
 552. In some instances the nucleus is furnished with several 
 envelopes concentric with it : which are formed in succession as 
 the comet approaches the sun. 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 tfeese sev- 
 eral tails lying one within the other, being hollow, may in conse- 
 quence appear so faint along its middle as to have the aspect of 
 two distinct tails. A. comet which has in reality three separate tails, 
 might thus appear to be supplied with six, as was the comet of 
 1744. If the different envelopes were not distinctly separate from 
 each other, then we should have all the tails appearing to proceed 
 from the same nebulous mass. 
 
 553. Supernumerary tails, shorter and less distinct than the 
 principal tail, are by no means uncommon ; but they generally 
 appear quite suddenly, and as suddenly disappear in a few days, as 
 if the stock of materials from which they were supplied had be- 
 come exhausted. These secondary tails, by their periodical 
 changes of position from the one side of the principal tail to the 
 other, have made known the fact that the comets to which they 
 belonged had a rotatory motion around the axis or central line of. the 
 tail. The same fact has been inferred from other phenomena, in 
 the case of some other comets, as the great comet of 1811, and 
 Halley's comet in 1835. 
 
 554. The general position of the tail of a comet is nearly but 
 Hot exactly in the prolongation of the line of the centres of the 
 sun and head of the comet, or of the radius-vector of the comet. 
 (See Fig. 97.) It deviates from this line on the side of the regions 
 of space which the comet has just left ; and the angle of deviation, 
 'Which, when the comet is first seen at a distance from the sun, is 
 very small or not at all perceptible, increases as the comet a 
 
 'proaches the sun, and attains to its maximum vakie soon after the 
 perihelion passage ; after which it decreases, and finally, at a dis- 
 tance from the sun, becomes insensible. For example, the angle 
 of deviation of the tail of the great comet of 1811 attained tO'ite 
 maximum about ten days after the perihelion passage, and was 
 then about 1 1. In the case of the comet of 1664, the same angle 
 about two weeks after the perihelion passage was 43, and was 
 then, decreasing at the rate of 8 per day.. ^ 
 
 The comet of 1823 might seem to (present aat exoeption to the 
 general fact that the tail of a comet is nearly opposite to the sun ; 
 
PHYSICAL CONSTITUTION OF COMETS. 207 
 
 but Arago has suggested that the probable cause of the singular 
 phenomenon of a secondary tail, apparently directed towards the 
 sun in the heavens, was that the earth was in such a position that 
 the two tails, although in fact inclined to each other under a small 
 angle, were directed towards different sides of the earth, and thus 
 were referred to the heavens so as to appear nearly opposite. 
 
 The same principle will serve to show that the deviation of the 
 tail 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 happening to be within the angle formed by the direction of 
 the tail with the radius-vector prolonged. 
 
 555. Comets are the most voluminous bodies in the solar sys- 
 tem. The tail of the great comet of 1680 was found by Newton 
 to have been, when longest, no less than 123,000,000 miles in 
 length : according to Professor Peirce, the remarkable comet of 
 1843, about three weeks after its perihelion passage, had a tail of 
 over 200,000,000* miles in length. Other comets have had tails 
 of from fifty to a hundred millions of miles in length. The heads 
 of comets are usually many thousand miles in diameter. That of 
 the comet of 181 1 had a diameter of 132,000 miles. Its envelope 
 or nebulosity was 30,000 miles in thickness ; and the inner surface 
 of this was *io less than 36,000 miles distant from the centre of the 
 nucleus. The head of the great comet of 1843 was about 30,000 
 miles in diameter. 
 
 The nuclei of comets are in general only a few hundred miles in 
 diameter : but according to Schroeter the nucleus of the comet of 
 1811 had a diameter of 2600 miles ; and the nucleus of the comet of 
 1843 seems to have been still greater. On the other hand, the 
 comet of 1798 had a nucleus of less than 50 miles in diameter. 
 
 It is important to observe that the dimensions of comets are sub- 
 ject to continual variations. The tail increases as the comet ap- 
 proaches the sun, and attains to its greatest size a certain time after 
 the perihelion passage ; after which it decreases. The head, on 
 the contrary, generally diminishes in size during the approach to 
 the sun, and augments during the recess from him. The changes 
 are often very sudden and rapid. 
 
 PHYSICAL CONSTITUTION OF COMETS. 
 
 556. The quantity of matter which enters into the constitution 
 of a comet is exceedingly small. This is proved by the fact that 
 the comets have had no influence upon the motions of the planets or 
 satellites, although they have in many instances passed near, thegj'e 
 bodies. The comet of 1770, which was quite large and bright, 
 passed through the midst of Jupiter's satellites, without deranging 
 their motions in the least perceptible degree. Moreover, since this 
 small quantity of matter is dispersed over a space of tens of thou- 
 
 * According to later determinations 108,000,000 miles. 
 
208 OF COMETS. 
 
 sands, 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 inconceivably more 
 rare and subtile than the lightest known gas, or the most evanescent 
 film of vapor that ever makes its appearance in our sky ; for faint 
 telescopic stars are distinctly visible through all parts of the comet, 
 with, it may be, the exception of the nucleus in some instances, 
 notwithstanding the great space occupied by the matter of the 
 comet which the light of the star has to traverse. The matter of 
 the tail of a comet is even more attenuated than that of the general 
 mass of the nebulosity of the head ; but is apparently of the same 
 nature, and derived from the head. The nucleus is supposed by 
 some astronomers to be, in some instances, a solid, partially 
 or wholly convertible into vapor, under the influence of the sun ; 
 by others, to be in all cases the same species of matter as is in the 
 nebulosity, only in a more condensed state ; and by others still, to 
 be a solid of permanent dimensions, with a thick stratum of con- 
 densed vapors resting upon its surface. Whichever of these views 
 be adopted, it is a matter of observation that the nebulosity fre- 
 quently receives fresh supplies of nebulous matter from the nu- 
 cleus. It was the opinion of Sir William Herschel, an<J it has been 
 the more generally received notion since his time, that the nucleus 
 of a comet is surrounded with a transparent atmosphere of vast 
 extent, within which the nebulous envelope floats, as do clouds in 
 the earth's atmosphere. But Olbers, and after him Bessel, con- 
 ceives 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 ac- 
 tion of these forces and the attraction of the nucleus. 
 
 It is not yet definitively settled whether the cometic matter is 
 self-luminous, or shines with the light received from the sun ; but 
 it is the general opinion that it derives its light from the sun. 
 
 CONSTITUTION AND MODE OF FORMATION OF THE TAILS OF COMETS. 
 
 557. Upon this topic we may lay down the following postulates. 1. The gen- 
 eral 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 
 him, conduct to the same inference. 2. Since the tail lies in the direction of the 
 radius-vector prolonged beyond the head, the particles of matter of which it is 
 made up must have been driven off by some force exerted in a direction from the 
 sun. 3. This force cannot emanate from the nucleus, for such a force would ex- 
 pel the nebulous matter surrounding the nucleus in all directions, instead of one 
 direction only. It is, however, conceivable that, as Olbers supposes, the nebulous 
 matter is in the first instance expelled from the nucleus by its repulsive action, 
 taking effect chiefly on the side towards the sun, and afterwards driven past the 
 nucleus into the tail by a repulsion from the sun. 4. There seems, then, to be little 
 room to doubt that the matter of the tail is driven off from the head by some force 
 foreign to the comet, and taking effect from the sun outwards. 5. This force, 
 
FORMATION OF THE TAILS OF COMETS. 209 
 
 whatever may be its nature, extends far beyond the earth's orbit. For comets 
 have been seen provided with tails of great length, though their perihelion distance 
 exceeded the radius of the earth's orbit, (e. g. the great comet of 1811.) Nothing 
 can be predicated with certainty with respect to the law of variation of this force, 
 but it is at least probable that, like all known central forces, it varies inversely as 
 the square of the distance. 
 
 558. Whatever may be the nature of the force in question ; whether it consists 
 in an impulsive action of the sun's rays, as Euler imagined, or in a repulsion by 
 the distant mass of the sun, consequent upon a polarity of the cometic particles 
 induced by some action of the sun, as supposed by Olbers and Bessel, we will call 
 it the repulsive force of the sun. Granting its existence, there are two modes in 
 which we may conceive it to operate in forming the tail. We may suppose that 
 it drives off the nebulous matter to greater and greater distances, as its intensity 
 increases, without destroying the original physical connection of the parts ; so that 
 the tail and the head will always be revolving as one connected mass. Or we may 
 conceive that it is continually detaching portions of the nebulosity, or turning them 
 back if repelled by the nucleus, and repelling them to an indefinite distance into 
 free space. The first mentioned conception is the theory which has generally pre- 
 vailed hitherto; but there seem to be good and sufficient reasons for rejecting it, 
 and adopting the other in its stead. 1. There appears to be no satisfactory rea- 
 son to be assigned why the force which expels the nebulous matter to the end of 
 the apparent tail should not urge it still farther ; since the extremities of the tails 
 of some comets are not so far removed from the sun as the heads of others, from 
 which the nebulous matter is expelled by the same force. To account for the sup- 
 posed limited extent of the actual tail, we are forced to suppose that the tendency 
 of the particles to return to the nucleus increases as their distance from the nucleus 
 and from one another increases ; which seems highly improbable. 2. Bessel has 
 found that the nebulous matter of comets has no power to refract the light of a 
 star, passing through it, whence he infers that there can be no molecular connec- 
 tion between the particles. 3. It appears, by calculation, that in the case of the 
 great comet of 1843, we cannot find either in the repulsive action of the sun upon 
 the tail, or in the excess of attraction of the sun for the nearer parts of the comet, 
 a force adequate to keep the tail continually opposite to the sun, and which at the 
 same time will not sensibly alter the orbit, without making improbable suppositions 
 as to the disproportion between the quantity of matter in the nucleus and tail.* 
 4. In the case of such comets as that of 1843, and that of 1680, which come near 
 the sun, the centrifugal force generated by the great velocity of rotation about the 
 time of the perihelion passage would be so great as infallibly to dissipate the great- 
 er part of the tail. At the time of the perihelion passage of either of these comets, 
 the centrifugal force must have exceeded the gravity towards the nucleus at only 
 a few hundred miles from the centre of gravity of the whole mass. 5. Whether we 
 suppose the whole mass of the comet to be kept in rotation about its centre of grav- 
 ity by the repulsion or by the attraction of the sun, the velocity of rotation, as it 
 is constantly nearly equal to the angular velocity of revolution, must be on the in- 
 crease up to the time of the perihelion passage. Now, this will not undergo any 
 diminution after the perihelion passage, as the action of the force would tend to 
 increase rather than diminish it, but the velocity of revolution will continually de- 
 crease : it follows, therefore, that soon after the perihelion passage the velocity of 
 rotation would exceed that of revolution, and continually more and more ; so that 
 ere long the tail would inevitably be thrown forward of the line of the radius-vector 
 prolonged, a situation in which the tail of a comet has never been seen. 
 
 We here suppose the dimensions of the comet to remain the same. In point of 
 fact, the apparent tail increases in length for a certain number of days after the 
 perihelion passage. The tendency of this would be to diminish the velocity of ro- 
 tation ; but the supposed subsequent contraction of the tail to its original dimen- 
 sions would restore the original velocity. 
 
 In view of all that has now been stated, it seems highly probable that the tail 
 and head of a comet do not form one connected body of matter, as has been gen- 
 erally supposed ; but, on the contrary, that the tail is made up of particles of mat- 
 ter continually in the act of flowing away at a very rapid rate from the head into 
 
 * See Silliman's Journal, vol. xlvi, No. I, page 110, &c. 
 27 
 
210 
 
 OF COMETS. 
 
 free space, under the action of the repulsive force of the sun, so called. According 
 to this view, the tail which we see at any instant is the collection of all the parti- 
 cles that have been emitted during a certain previous interval ; and at the end of 
 every such interval we are looking at an entirely new tail. This theory of the con- 
 stitution of the tails of comets is identical with Gibers' ; but,as we have seen (556), 
 Olbers has also given a special theory of the constitution of the nebulosity of the 
 heads of comets, of which nothing is here predicated. 
 
 559. If a theory be true, it must furnish a satisfactory explanation of the facts 
 and phenomena that fall within its scope. Let us examine the present theory from 
 this point of view. In the first place, as respects the form of the tail, it is mani- 
 fest from what has already been stated, (546,) that the sides of the tail must often 
 diverge much more rapidly than the lines of action of the repulsive force of the 
 sun upon the opposite parts of the head. Calculation shows this to have been the 
 case even in the comet of 1843. Now, according to Olbers' theory, this fact is a 
 simple consequence of the supposed repulsive action of the nucleus. If we adopt 
 Herschel's theory of the constitution of the head, we have apparently a sufficient 
 cause for the same fact in the centrifugal force generated by the rotation of the tail, 
 (553,) which we must suppose to be a consequence of a rotation of the head. 
 
 560. In the next place, the increase in the length of the apparent tail as the 
 comet approaches the sun, and until a certain time after the perihelion passage, 
 may be naturally supposed to proceed from the emission of greater quantities of 
 luminous matter in a given time, and a continued augmentation, up to the time of 
 the perihelion passage, in the light received from the sun. The actual tail, it is to 
 be observed, is really indefinite in length, and terminates, to us, where its matter 
 becomes too much dispersed and too distant from the sun, the probable source of 
 its light, to send us a perceptible light. 
 
 561. Let us now see, in the third place, how the theory under consideration ac- 
 counts for the situation and curvature of the tail. Let PCA (Fig. 99) be a portion 
 of a comet's orbit, the sun being at S : and suppose a particle to be expelled in the 
 
 Fig. 99. 
 
 direction SAD, when the head is at A, and another particle to be driven off in the 
 direction SEE, when the head is at B. Each particle will retain the orbitual mo- 
 tion which obtained at the time of its departure, as it moves away from the sun ; 
 and thus, when the comet has reached the point C, instead of being at any points 
 D and E on the lines SAD and SEE, will be respectively at certain points a and b 
 farther forward. The line Cba, which, when the comet is at C, is the locus of all 
 the particles that have been emitted during the interval of time in which the comet 
 has been moving over the arc AC, is the tail. We here suppose the head to be a 
 mere point. If we conceive the particles to be continually emitted from the mar- 
 ginal parts of the head, we shall have the hollow conical tail actually observed. It 
 is easy to see that Cba, the line of the tail, must be a curved line concave towards 
 
% w 
 
 NUMBER AND DISTRIBUTION OP THE FIXED STARS. 211 
 
 * , N tx '* 
 
 the regions of space which the comet has left. Supposing the arc, AC to be so 
 small, or its curvature to be so slight that it may be considered as a straight line, 
 and neglecting the change of the velocity in the orbit, Ca will be parallel to AD, 
 and Ch parallel to BE, whence RCa = CSA, and RCb = CSB. Thus the line 
 joining any particle with the nucleus always makes an angle with the prolonga- 
 tion of the radius-vector, equal 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 mo- 
 tion in anomaly to be 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 Co and Cb. But, 
 in point of fact, the velocity increases with the distance, so that the curvature of 
 the tail will be less than on the supposition just made. 
 
 As to the amount of the deviation of the tail from the line of the radius- vector, 
 it must depend upon the proportion between the velocity of the particles and the 
 velocity of the head in its orbit: and it follows from the principle just established, 
 that unless the velocities of emission augment as rapidly as the velocity of revolu- 
 tion, the deviation in question will increase to the perihelion, and afterwards de- 
 crease ; as it is in fact known to do. 
 
 562. In support of Olbers' theory of a repulsion from the nucleus, it may be sta- 
 ted, that the form of the nebulosity which this theory requires, was found by obser- 
 vation to obtain in the case of the great comet of 1811, and also of Halley's Com- 
 et in 1835.* 
 
 CHAPTER XVIII. 
 
 OP THE FIXED STARS 
 THEIR NUMBER AND DISTRIBUTION OVER THE HEAVENS. 
 
 563. THE number of stars visible to the naked eye, in the entire 
 sphere of the heavens, is from 6000 to 7000 ; of which nearly 
 4000 are in the northern hemisphere ; but not more than 2000 can 
 be seen with the naked eye at any one time at a given place. 
 The telescope brings into view many millions, and every material 
 augmentation of its space-penetrating power greatly increases the 
 number. 
 
 564. As to the number of stars belonging to each different mag- 
 nitude, astronomers assign from 20 to 24 to the first magnitude, 
 from 50 to 60 to the second, about 200 to the third, and so on ; 
 the numbers increasing very rapidly as we descend in the scale 
 of brightness ; the whole number of stars already registered down 
 to the seventh magnitude, inclusive, amounting to 12,000 or 15,000.f 
 
 The reason of this increase in the number of the stars, as we 
 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 * 
 
 * See Silliman's Journal, vol. xlv. No. I, page 206. 
 t Herschel's Outlines of Astronomy, p. 520. 
 
212 
 
 OF THE FIXED STARS. 
 
 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. 
 
 565. It is not to be understood that the classification of the stars 
 into different magnitudes is made according to any fixed definite 
 proportion subsisting between the degrees of apparent brightness 
 of the stars belonging to different classes. Stars of almost every 
 gradation of brightness, between the highest and the lowest, are 
 met with. Those which offer marked differences of lustre, form 
 the basis of the classification ; others, which do not differ very 
 widely from these, are united to them. As a necessary conse- 
 quence, there are some stars of intermediate lustre, which cannot 
 be assigned with certainty to either magnitude. Thus, in the cata- 
 logue of the Astronomical Society of London, 3 stars are marked 
 as intermediate between the first and second magnitudes, and 29 
 betw.een the second and third. 
 
 Different astronomers also not unfrequently assign the same star 
 to different magnitudes. 
 
 As to the proportions of light emitted from the average stars of 
 the different magnitudes, according to the experimental comparisons 
 of Sir Wm. Herschel, they are, from the first to the sixth magni- 
 tude, approximately in the ratio of the numbers, 100, 25, 12, 6, 2, 1 . 
 
 566. With the exception of the three or four brightest classes, 
 the stars are not distributed indiscriminately over the sphere of the 
 heavens, but are accumulated in far greater numbers on the borders 
 of that belt of cloudy light in the heavens, which is called the 
 milky way, and in the milky w r ay itself, which the telescope shows 
 to consist of an immense number of stars of small magnitude in 
 close proximity. 
 
 Herschel found that on a medium estimate a segment of the 
 milky way, 15 long, and 2 broad, contained at least 50,000 stars 
 of sufficient magnitude to be distinguished through his telescope.* 
 According to this, taking its average breadth at 14, the milky way 
 must contain more than eight millions of stars. 
 
 567. This great accumulation of stars in a zone of the heavens, 
 encompassing the earth in the direction of a great circle, suggested 
 to the mind of Herschel the idea that the stars of our firmament 
 are not disseminated indifferently throughout the surrounding re- 
 gions of space, but are for the most part arranged in a stratum, 
 the thickness of which is very small in comparison with its breadth ; 
 
 Fig. 100. 
 
 the sun and solar sys- 
 tem being near the mid- 
 dle of the thickness. If 
 S (Fig. 100) represents 
 the place of the sun, it 
 will be seen that upon 
 this supposition the 
 
 * A Newtonian reflecting telescope of 20 feet focus and nearly 19 in. in aperture. 
 
ANNUAL PARALLAX OF THE STARS, 
 
 213 
 
 number of stars in the direction SC of the thickness of the stratum 
 will be less than in any other direction, and that the greatest num- 
 ber will lie in the direction of the breadth, as SB. On one side 
 of the point S, the stratum is supposed to be divided for a cer- 
 tain distance into two laminae, as Fig. 101. 
 shown in the figure, which repre- 
 sents a section of the supposed stra- 
 tum. This supposition is necessary 
 to account for the two branches, 
 with a dark space between them, 
 into which the milky way is divided 
 for about one-third of its course. 
 
 Herschel undertook to gauge this stratum 
 in various directions, on the principle that 
 the distance through to its borders in any 
 direction was greater in proportion as the 
 number of stars seen in that direction was 
 greater. He thus found that its actual form 
 was very irregular : its section, instead of 
 being truly that of a segment of a sphere 
 divided for a certain distance into two lami- 
 nae, as represented in Fig. 100, having the 
 form represented in Fig. 101. He estimated 
 the thickness of the stratum to be less than 
 160 times the interval between the stars, and 
 the breadth to be nowhere greater than 1000 
 times the same distance. He conceived that 
 it extended in no direction a distance equal 
 to tiie space-penetrating power of his tele- 
 scope for individual stars, and much less for 
 collections of stars seen as nebulous spots. 
 
 568. 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 di- 
 rection of the Southern Cross, and 
 that the central parts are so vacant 
 of stars that the whole approximates 
 to the form of an annulus. 
 
 ANNUAL PARALLAX AND DISTANCE OF THE STARS. 
 
 v 
 
 569. The Annual Parallax of a fixed star is the angle made by 
 two lines conceived to be drawn, the one from the sun and the 
 other from the earth, and meeting at the star, at the time the earth 
 is in such part of its orbit that its radius-vector is perpendicular to 
 the latter line ; or, in other words, it is the greatest angle that can 
 
214 OF THE FIXED STARS. 
 
 be subtended at the star by the radius of the earth's orbit. Thus, 
 let S (Fig. 102) be the sun, s a fixed star, and E the earth, in 
 such a position that the radius-vector SE is perpendicular to Es 
 Fig. 102. _ the line of direction of the star, 
 
 then the angle SsE is the an- 
 nual parallax of the star s. 
 
 570. If the annual parallax 
 of a star was known, we might 
 easily find its distance from the 
 earth ; for in the right-angled tri- 
 angle SEs we would know the 
 angle SsE and the side SE, and 
 we should only have to com- 
 pute the side Es. Now, if any 
 of the fixed stars have a sensi- 
 ble parallax, it could be detected by a comparison of the places of 
 the star, as observed from two positions of the earth in its' orbit, 
 diametrically opposite to each other ; and accordingly, the atten- 
 tion 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 paral- 
 lax. But, after exhausting every refinement of observation, they 
 have not been able to establish that any of them have a measura- 
 ble parallax. 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 and uni- 
 versal recognition. We may then conclude that the annual par- 
 allax of the nearest fixed star is less than \" . 
 
 571. 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 billions of miles. The distance of the nearest fixed star must 
 therefore be greater than this. A juster notion of the immense 
 distance of the fixed. stars, than can be conveyed by figures, may 
 be gained from the. consideration that light, which traverses the 
 distance between the sun and earth in 8m. 18s., and would perform 
 the circuit of our globe in } of a second, employs more than three 
 years in coming from the nearest fixed star to the earth. 
 
 According to Struve, the most probable value of the parallax of a star of the 
 first magnitude is no more than about |" ; which would make its distance 5 times 
 greater than the above determination. 
 
 572. The statement made in Art. 570, that the annual parallax of the fixed stars 
 has hitherto escaped certain detection, although truly representing the result of all 
 the many efforts made to solve the great problem of the distance of the fixed stars, 
 until a very recent date, is at the present time (1845) no longer true. The paral- 
 lax of one of the fixed stars is now believed to have been determined by Bessel. 
 This is the star 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 
 
 * * R. A. 314 52', Dec. N. 37 56'. 
 
DISTANCE OF THE STARS. 215 
 
 stars have a motion of revolution around each other, and the two move together 
 at the same rate, of 5".3 per year, as one star, along the sphere of the heavens. It 
 is hence inferred that they are bound together into one system by the principle of 
 gravitation, and are at pretty nearly the same distance from the earth. The great 
 proper motion of this double star, as compared with other stars, led to the suspicion 
 that it was nearer than any other ; and thus to attempts to determine its parallax. 
 The principle of Bessel's method is to find the difference between the parallaxes of 
 the star 61 Cygni, and some other star of much smaller magnitude, and therefore 
 supposed to be at a much greater distance, seen in as nearly the same direction as 
 possible. -This difference will differ from the absolute parallax of the double star 
 by only a small fraction of its whole amount. It was found by measuring with a 
 position micrometer (76) the annual changes in the distance of the two stars, and 
 in the position of the line joining them. To make it evident that such changes 
 will be an inevitable consequence of any difference of parallax in the two stars, 
 conceive two cones having the earth's orbit for a common base, and their vertices 
 respectively at the two stars, and imagine their surfaces 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. 
 103 ; and they will be apparently described by 
 the stars in the course of one revolution of the 
 earth in its orbit. The two stars will always 
 be similarly situated in their parallactic ellip- 
 ses : thus, if one is at A the other will be at 
 a ; and after the earth has made one-quarter 
 of a revolution, they will be at B and b ; and af- 
 ter another quarter of a revolution at C and c, 
 &c. Now it will be manifest, on inspecting 
 the figure, the ellipses being of unequal size, 
 that the line of the stars will be of unequal 
 lengths, and have different directions in the 
 different situations of the stars. 
 
 A much smaller angle of parallax may be 
 found, with the same degree of certainty, by 
 this indirect method than by the direct process 
 explained in Art. 570 ; for, since the two stars are seen in pretty nearly the same 
 direction, they will be equally affected by refraction and aberration ; and since it 
 is only the relative situations of the two stars that are measured, no allowance haa 
 to be made for precession and nutation, or for errors in the construction or adjust- 
 ment of the instrument. It is therefore independent of the errors that are inevita- 
 bly committed in the determination of these several corrections, when it is at- 
 tempted to find directly the absolute parallax, by observing the right ascension and 
 declination at opposite seasons of the year. The measurements made with the 
 micrometer in the hands of the most accurate observers, may be relied on as exact 
 to within a small fraction of 1". 
 
 For the sake of greater certainty Bessel made the measurements of parallactic 
 changes of relative situation between the star 61 Cygni, and two small stars 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. (See Note XIV.) 
 
 573. The amount of light received from the same body at differ- 
 ent distances varies inversely as the square of the distance. Hence, 
 if we admit the light of a star of each magnitude to be half that of 
 one of the next higher magnitude, a star of the first magnitude 
 would have to be removed to 360 times its distance, to appear no 
 brighter than one of the eighteenth. Accordingly, if the difference 
 
216 OF THE FIXED STARS. 
 
 in the apparent magnitude of the stars arises for the most part 
 from a difference of distance, (which is the more probable suppo- 
 sition,) there must be a multitude of stars visible in telescopes, 
 the light of which has taken at least one thousand years to reach 
 the earth. 
 
 A calculation based upon the power of large telescopes to aug- 
 ment the amount of light received from the stars, in connection with 
 the well-known law of diminution of the light received as the dis- 
 tance increases, conducts to about the same result. 
 
 'NATURE AND MAGNITUDE OF THE STARS. 
 
 574. 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 like our own. If it should be con- 
 jectured that some of the fainter stars might be bodies shining 
 by reflected light, like the planets, the answer is, that if we were 
 to suppose the existence ef opake bodies, at the distance of the 
 stars, so inconceivably vast in their dimensions as to send a sensi- 
 ble light to the eye, if illuminated to the same degree as the plan- 
 ets, the stars of the smaller magnitudes are, with the exception 
 perhaps of the members of some of the double stars, too remote 
 from the brighter ones to receive sufficient light from them ; for, 
 the smallest measurable space in the field of the largest telescopes 
 is, at the distance of the nearest star, as large as or larger than 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. ' ? 
 
 575. To be able to determine the magnitude of a star, we must 
 know its distance, and also its apparent diameter. Now the dis- 
 tance of only one star has, as yet, been found ; and the discs of 
 all the stars, even in the most powerful telescopes, are altogether 
 spurious ; so that in no instance have we the data, nor have we 
 reason to expect that they will be hereafter obtained, for determin- 
 ing with certainty the magnitude of a fixed star. 
 
 We may infer, however, from the intensity of their light, that it 
 is highly probable that some at least of the stars are as large as, 
 or even larger than the sun. It has been calculated from the re- 
 sults of photometrical experiments made by Dr. Wollaston, on the 
 relative quantity of light received from Sirius and the sun, that if 
 the sun were removed to the distance of 20 billions of miles, which 
 is known to be less than the distance of any of the stars, he would 
 not send to us so much as half the quantity of light actually re- 
 ceived from Sirius. 
 
 576. Although there are not sufficient data for calculating the magnitude of the 
 star 61 Cygni, there are for ascertaining its mass. This element results from the 
 distance and the motion of revolution of the two members of the double star about 
 each other. Bessel finds it to be less than half of the sun's mass. According to 
 
VARIABLE STARS. 217 
 
 this result the sun, as seen from this star, should appear as a star of about the fifth 
 magnitude. 
 
 ' * 4 
 
 VARIABLE STARS. 
 
 577. A number of the fixed stars are subject to periodical 
 changes of brightness, and are hence called Variable Stars, or 
 Periodical Stars. One of the most remarkable of the variable 
 stars is the star Omicron, in the constellation Cetus. From being 
 as bright as a star of the second magnitude, it gradually decreases 
 until it entirely disappears ; and, after remaining for a time invisi- 
 ble, reappears, and gradually increasing in lustre, finally recovers 
 its original appearance. The period of these changes is 332 days. 
 It remains at its greatest brightness about two weeks, employs 
 about three months in waning to its disappearance, continues invi- 
 sible for about five months, and during the remaining three months 
 of its period increases to its original lustre. Such is the general 
 course of its phases.. It does not, however, always recover the 
 same degree of brightness, nor increase and dimmish by the same 
 gradations. It is related by Hevelius, that in one instance it re- 
 mained invisible for a period of four years, viz. from October, 1672, 
 to December, 1676.* A similar phenomenon has been noticed in 
 the case of another variable star, viz. the star % Cygni. It is sta- 
 ted by Cassini to have been scarcely visible throughout the years 
 1699, 1700, and 1701, at those times when it ought to have been 
 most conspicuous. On the other hand, a variable star, situated in 
 the Northern Crown, sometimes continues visible for several years 
 without any apparent change, and then resumes its regular varia- 
 tions. 
 
 578. The greater number of variable stars undergo a regular 
 increase and diminution of lustre, without ever, like the star just 
 noticed, becoming entirely invisible. The star Algol, or ft Perseii, 
 is a remarkable variable star of this description. For a period of 
 2d. 14h. it appears as a star of the second magnitude, after which 
 it suddenly begins to diminish in splendor, and in about 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 bright- 
 ness, going through all its changes in 2d. 20h. 48m.f 
 
 579. There are also a number of double stars, one or both of 
 the members of which are variable ; as y Virginis, s Arietis, 
 Bootis, &c. 
 
 580. 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 phase of least light lasts much long 
 er than that of their greatest light. The star Algol, which is white, 
 is said to be the only variable star whose light is not of a reddish 
 
 * Herschel's Treatise on Astronomy, p. 356. t Ibid. 357. 
 
 28 
 
218 OP THE FIXED STARS. 
 
 color. The same star also presents an exception to the other gen- 
 eral fact just noticed. (See Note XV.) 
 
 581. There are also some instances on record of temporary 
 stars having made their appearance in the heavens ; breaking forth 
 suddenly in great splendor, and without changing their positions 
 among the other stars, after a time entirely disappearing. One of 
 the most noted of these is the star which suddenly shone forth with 
 great brilliancy on the llth of November, 1572, between the con- 
 stellations Cepheus and Cassiopeia, and was attentively observed 
 by Tycho Brahe. It was then as bright as any of the permanent 
 stars, and continued to increase in splendor till it surpassed Jupiter 
 when brightest, and was visible at mid-day. It began to diminish 
 in December of the same year, and in March, 1 574, it entirely dis- 
 appeared, 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 yel- 
 low, and then to a very distinct red ; after which it became pale, like Saturn. 
 
 In the years 945 and 1264, brilliant stars appeared in the same 
 region of the heavens. It is conjectured from the tolerably near 
 agreement of the intervals of the appearance of these stars and 
 that of 1572, that the three may be one and the same star, with a 
 period of about 300 years. The places of the stars of 945 and 
 1264 are, however, too imperfectly known to establish this with 
 any degree of certainty. 
 
 Besides these three temporary stars, several others have made their appearance, 
 viz. one in the year 125 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; and a fifth in 1670, in the Swan. 
 
 582. What is no less remarkable than the changes we have no- 
 ticed, several stars, which are mentioned by the ancient astrono- 
 mers, have now ceased to be visible, and some are now visible 
 to the naked eye which are not in the ancient catalogues. 
 
 583. The most probable explanation of the phenomenon of variable stars, is, that 
 they are self-luminous bodi< s rotating upon axes, like the sun, and having like him 
 spots upon their surface, bot vastly larger and more permanent. By the rotation 
 these spots are brought periodically around on the side towards the earth, and ac- 
 cording to their size occasion a diminution of the light of the star, or make it en- 
 tirely to disappear. In the case of the star Algol, however, as suggested by Good- 
 ricke, the phenomena are precisely such as would result from the periodical inter- 
 position of an opake body revolving around it. In those cases in which the period 
 of the diminution of the light is a large fraction of the entire period of the star, 
 (580,) as well as those in which there are occasional interruptions in the regular 
 recurrence of the phenomena, (577,) the supposition of the interposition of an opake 
 body will not answer. (See Note XVL) 
 
 584. Temporary stars are most probably suns which have entirely intermitted 
 the evolution of light for a long period of time, and then burst forth anew with a 
 sudden and peculiar splendor. Laplace conjectured that they might be the confla- 
 grations of distant worlds ; but it seems very questionable whether the conflagra- 
 
 * Herschel's Treatise on Astronomy, p. 359. 
 
DOUBLE STARS. 219 
 
 tion of even an entire system of planets would furnish as much light as the sun at 
 its centre ; and the large and permanent spots on the surface of the variable stars 
 would seem to render it probable that some suns have become, for a time, entirely 
 extinct. In support of this theory, that temporary stars are the temporary revival 
 of extinct suns, we have the fact said to have been recently discovered by Bessel, 
 that there are opake bodies in space of the size of suns. It is stated that this 
 distinguished astronomer has ascertained, from a discussion of the most accurate 
 observations that have been made upon these stars, that the proper motions of the 
 two stars Sirius and Procyon deviate sensibly from uniformity ; whence he infers 
 that they must each be revolving about some large non-luminous body in their vi- 
 cinity, and are thus double stars, one of the members of which is non-luminoue. 
 
 DOUBLE STARS. 
 
 585. Many of the stars which to the naked eye appear single, 
 when examined with telescopes are found to consist of two (in 
 some instances three or more) stars in close proximity to each oth- 
 er. These are called Double Stars, or Multiple Stars. (See Fig. 
 104.) This class of bodies was first attentively observed by Sir 
 William Herschel, who, in the years 1782 and 1785, published 
 
 Fig. 104. 
 
 Castor. y Leonis. Rigel. Pole-star. 11 Monoc. $ Cancri. 
 
 catalogues of a large number of them which he had observed. The 
 list has since been greatly increased by Professor Struve, of Dor- 
 pat, Sir J. F. W. Herschel, and other observers, and now amounts 
 to several thousand. 
 
 586. 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 twojcan only be separated by the most pow- 
 erful telescopes. In other instances, the distance is |' and more, 
 and the separation can be effected with telescopes of very moder- 
 ate power. They are divided into different classes or orders, ac- 
 cording to their distances ; those in which the proximity is the 
 closest forming the first class. 
 
 587. The two members of a double star are generally of quite 
 unequal size. (See Fig. 104.) But in some instances, as that of 
 the star Castor, they are of nearly the same magnitude. Double 
 stars occur of every variety of magnitude. 
 
 588. It is a curious fact, that the two constituents of a double star in numer- 
 ous instances shine with different colors ; and it is still more curious that these 
 colors are in general complementary to each other. Thus, the larger star is 
 usually of a ruddy or orange hue, while the smaller one appears blue or green. 
 This phenomenon has been supposed to be in some cases the effect of contrast ; 
 the larger star inducing the accidental color in the feebler light of the other. "Sir 
 John Herschel cites as probable examples of this effect the two stars t Cancri, and 
 yAndromedoe. But it is maintained by Nichol that this explanation cannot be 
 admitted ; for, if true, it ought to be universal, whereas there are many systems 
 similar in relative magnitudes to the contrasted ones, in which both stars are yel- 
 low, or otherwise belong to the red end of the spectrum. Again, if the blue 01 
 
220 OF THE FIXED STARS. 
 
 violet color were the effect of contrast, it ought to disappear when the yellow star 
 is hid from the eye ; which, however, it does not do. Thus, the star /8 Cygni con- 
 sists of two stars, of which one is yellow, and the other shines with an intensely 
 blue light ; and when one of them is concealed from view by an interposed slip of 
 darkened copper, the other preserves its color unchanged. The color, then, of 
 neither of the stars can be accidental. 
 
 It may be remarked in this connection, that the isolated stars also shine with 
 various colors. For example, among stars of the first magnitude, Sirius, Vega, 
 Altair, Spica are white, Aldebaran, Arcturus, Betelgeux red, Capella and 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 red color, almost as deep as that -of blood, occur in many parts of the 
 heavens, but no decidedly green or blue star has ever been noticed unassociatcd 
 with a companion brighter than itself. 
 
 589. 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 expe- 
 rienced any change in consequence of the annual variation of the 
 parallax of the star. With a micrometer adapted to the purpose, 
 (76,) he measured from time to time the apparent distance of the 
 two stars, and the angle formed by their line of junction with the 
 meridian at the time of the meridian passage, called the Angle of 
 Position. Instead, however, of finding that annual variation of 
 these angles, which the parallax of the earth's annual motion would 
 produce, he observed that, in many instances, they were subject to 
 regular progressive changes, which seemed to indicate a real mo- 
 tion of the stars with respect to each other. After continuing his ob- 
 servations for a period of twenty-five years, he satisfactorily ascer- 
 tained that the changes in question were in reality produced by a 
 motion of revolution of one star around the other, or of both around 
 their common centre of gravity ; and in two papers, published in 
 the Philosophical Transactions for the years 1803 and 1804, he 
 announced the important discovery that* there exist sidereal sys- 
 tems composed of two stars revolving about each other in regular 
 orbits. These stars have received the appellation of Binary Stars t 
 to distinguish them from other double stars which are not thus 
 physically connected, and whose apparent proximity may be occa- 
 sioned by the circumstance of their being situated on nearly the 
 same line of direction from the earth, though at very different dis- 
 tances from it. Similar stars, consisting of more than two consti- 
 tuents, are called Ternary, Quaternary, &c. 
 
 590. Since the time of Sir W. Herschel, the observations upon 
 the binary stars have been continued by Sir John Herschel, Sir 
 James South, Struve, Bessel, Madler, and other astronomers : ac- 
 cording to Madler, the number of known binary and ternary stars 
 is now about 250. 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 exceedingly 
 slow, the actual number of such systems is probably a large frac- 
 tion of the whole number of double stars ; at least, if we confine 
 
DOUBLE STARS. 221 
 
 our attention to double stars whose constituents are within y of 
 each other. This may be inferred from the fact, that the num- 
 ber of such double and multiple stars actually observed, which 
 amounts to over 3000, is at least ten times greater than the num- 
 ber of instances of fortuitous juxtaposition that would obtain on the 
 supposition of a uniform distribution of the stars. Besides, there 
 is a number of double stars not yet discovered to have a motion of 
 revolution, which still give indications 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. 
 
 Prom the observations made upon some of the binary stars, as- 
 tronomers have been enabled to deduce the form of their orbits, 
 and approximately the lengths of their periods. The orbits arc 
 ellipses of considerable eccentricity. The periods are of various 
 lengths, as will be seen from the following enumeration of those 
 which are considered as the best ascertained : tf Coronas 608 years ; 
 61 Cygni 540 years ; a Geminorum 232 years ; 7 Virginis 182 
 years ; 3062 Stmve 95 years ; p Ophiuchi 93 years ; X Ophiuchi 
 88 years ; w Leonis 83 years ; g Ursae Majoris 60 years ; Can- 
 cri 59 years ; n Coronae 43 years ; Herculis 31 years. Fig. 105 
 represents a portion of the apparent orbit of the double star y Vir- 
 ginis, and shows the relative positions of the two members of the 
 
 Fig. 105. 
 
 double star in various years. At the time of their nearest approach, 
 in 1836, the interval between them was a fraction of 1", and they 
 could 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 figure is the stereographic projection of the true orbit on a 
 plane perpendicular to the line of sight. (See Note XVII.) 
 
222 OP THE FIXED STARS. 
 
 The actual distance between the members of a binary star hag 
 been found only for the star 61 Cygni. Bessel makes it for this 
 star about two and a half times the distance of Uranus from the sun. 
 
 591. 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 an- 
 other sun ; of one solar system around another solar system ; or 
 rather of both around their common centre of gravity. We learn 
 from it the important fact, that the fixed stars are endued with the 
 same property of attraction that belongs to the sun and planets. 
 
 PROPER MOTIONS OF THE STARS. 
 
 592. It has already been stated (181) that the fixed stars, so 
 called, are not all of them rigorously stationary. By a careful 
 comparison of their places, found at different times with the accu- 
 rate instruments 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 ve- 
 locity and direction of this motion are uniformly the same for the 
 same star, but different for different stars. The star which has 
 the greatest proper motion of any observed, is the double star 61 
 Cygni. During the last fifty years it has shifted its position in the 
 heavens 4' 23" ; the annual proper motion of each of the individ- 
 ual stars being 5". 3. Among isolated stars, j^ Cassiopeiae has the 
 greatest proper motion. It changes its place 3". 74 every year. 
 The proper motions of some of the stars are either partially or en- 
 tirely attributable to a motion of the sun and the whole solar sys- 
 tem in space ; but the motions of others cannot be reconciled with 
 this hypothesis, and must be regarded as in all probability indica- 
 tive of a real motion of these bodies in space. (See Note XVIII.) 
 
 593. 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 care- 
 ful examination 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 X Herculis 
 towards a point diametrically opposite. Whence he inferred that 
 the sun with its attendant system of planets was moving rapidly 
 through space in a direction towards this constellation. Doubt has 
 since been thrown upon these conclusions by Bessel and other as 
 tronomers ; but they have quite recently been decisively re-estab- 
 lished by M. Argelander, of Abo. The investigations of Arge- 
 lander, which were communicated to the Academy of St. Peters- 
 burgh in 1837, have since been confirmed by M. Otho Struve, of 
 the celebrated Pulkova Observatory. 
 
 Combining the determinations of these two astronomers, we find 
 the most probable situation of the point towards which the sun's 
 
CLUSTERS OF STARS. NEBULA. 
 
 motion is directed to be as follows : R. A. 259 9', Dec. N. 34 
 36'. The point in question is situated in the constellation Hercules, 
 near the star u, (No. 68 in the Catalogue of the Astronomical So- 
 ciety,) and about 10 from the point first supposed by Herschel. 
 
 594. O. Struve finds that for a star situated at right angles to the direction 
 of the sun's motion, and placed at the mean distance of the stars of the first 
 magnitude, the annual angular displacement due to the sun's motion is 0".339, 
 (with a probable error of 0".025.) So that, if we assume, according to the best 
 determinations, 0".211 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 about once and a half (1.606) greater than 
 the radius of the earth 's^orbit: which makes it about 150,000,000 of miles. This 
 is at the rate of about 4j miles per second. 
 
 595. The above angle of 0".339 is about the greatest annual displacement which 
 a star can experience in consequence of the sun's motion. Whence it appears that 
 the whole of the proper motion of any star which is over and above this amount 
 must certainly be due to a real motion in space. Thus, in the case of the star 61 
 Cygni, at least 5" of its annual proper motion (5" .23) results from an actual mo- 
 tion in space. This is 14.3 times greater than the parallax of this star, (0".35.) 
 Accordingly if we suppose the direction of its motion to be perpendicular to its line 
 of direction from the sun or earth, its annual motion is 14.3 times greater than the 
 radius of the earth's orbit, or at the rate of 43 miles per second. As we have 
 no means of ascertaining the actual direction of its motion, it is impossible to 
 discover how much it exceeds this determination. 
 
 596. By comparing the particular motions presented by stars of different class- 
 es 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 lu- 
 minary may be ranked among those stars which have a comparatively slow mo- 
 tion in space 
 
 CLUSTERS OF STARS. NEBULAE. 
 
 597. A great number of spaces are discovered in the heavens 
 which are faintly luminous, and shine with a pale white light. 
 These are called Nebula. Some are visible to the naked eye, but 
 the greater number cannot be seen without the aid of a good tele- 
 scope. On applying to them telescopes of great power, they are 
 found for the most part to consist of a multitude of small stars, dis- 
 tinctly separate, but very near each other, and more or less con- 
 densed towards the centre. 
 
 598. There are also clusters of stars in close proximity, dis- 
 persed here and there over the sphere of the heavens, which are 
 seen to be such with the naked eye, or with telescopes of only 
 moderate power. One of the most conspicuous of these clusters 
 is that called the Pleiades. 
 
 To the unaided sight it appears to consist of six or seven stars, 
 but a telescope even of moderate power exhibits within the space 
 they occupy fifty or sixty conspicuous stars. The constellation 
 called Coma Berenices^ is another group, more diffused, and com- 
 posed of larger stars. 
 
 In the constellation Cancer there is a luminous spot, or nebula, 
 called Prcesepe, or the bee-hive, which a telescope of moderate power 
 resolves entirely into stars. In Perseus is another spot crowded 
 with stars^ which become separately visible with a good telescope. ' 
 
224 
 
 OF THE FIXED STARS. 
 
 599. A large number of nebulae are met with, indifferent parts 
 of the heavens, which offer no appearance of stars, even when ex- 
 amined with telescopes of the highest power. A very great diver- 
 sity of form and aspect obtains among them. One of the most 
 prominent is that near the star v in Andromeda. It is visible to the 
 naked eye, and has often been mistaken for a comet. (See Fig. 106. N 
 
 Fig. 106. 
 
 600. The number of nebulae at present known is about 3000. 
 Although they occur in almost every part of the heavens, they are 
 the most abundant in a zone perpendicular to the milky way, and 
 of about the same breadth, and whose general direction is not very 
 remote from that of the equinoctial colure ; and are particularly 
 numerous where it crosses the constellations Virgo, Coma Bereni- 
 ces, and the Great Bear. They are, for the most part, beyond the 
 reach of any but the most powerful instruments. 
 
 They are divided by Sir William Herschel into six different 
 classes, as follows : 
 
 (1.) Resolved Nebula ; that is, nebulae seen in the telescope to be clusters of 
 stars. Of these some are globular in their form, and others of an irregular figure 
 Fig. 107. ( See Fig. 107.) The latter are 
 
 less rich in stars, and less con- 
 densed towards the centre than 
 the globular clusters. They are 
 also less definite in their outline. 
 It is possible that they may be in 
 the act of condensing, as Herschel 
 supposed, and destined in the pro- 
 cess of ages to form truly globu- 
 lar clusters. This idea seems to 
 be supported by the fact of tho 
 occurrence of a regular gradation 
 of clusters, from one 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 and isolated globular clus- 
 ter highly condensed at the cen 
 
NEBULA. 
 
 225 
 
 trc. Globular clusters appear in telescopes of only moderate power, as small, round, 
 or oval nebulous specks, resembling a comet without a tail. The number of stars 
 which they contain is to be told only by thousands and tens of thousands ; although 
 their apparent size does not exceed the y^th part of the moon's disc. 
 
 (2.) Resolvable Nebulae; or such as give indications that they are clusters of 
 sjars, and that they are in their nature resolvable into stars, although the power 
 of the telescope is not yet sufficient to accomplish this. In telescopes of the high- 
 est power they present the same appearance as the resolved globular clusters in 
 telescopes which do not show their individual stars. Many of them have the as- 
 
 Fig. 108. 
 
 pect of these globular clusters just before they are resolved, and which has been 
 characterized by the phrase star-dust. They are of a round or oval form ; and are 
 doubtless real clusters too distant to show either their irregular edges or their indi- 
 vidual stars. (See Fig. 108.) 
 
 (3.) Nebulae Proper; or which offer no appearance of stars, and are supposed 
 to be actual masses of nebulous matter. Their nebulous constitution is inferred, 
 
 Fig. 109. 
 
 Nebula in Orion. 
 
 1st, from their unique appearance, which is often quite different from that of the 
 resolvable nebulae, and unlike what might be supposed to arise from an accumula- 
 tion of stars. (See Fig. 109.) 2d. From their manifest physical connection with 
 
 29 
 
OF THE FIXED STARS 
 
 individual stars much superior to them in brightness. The evidence of this physi- 
 eal connection is found, in some instances, in an apparent condensation upon the 
 in others in the fact that the substance of the nebula, whatever it may be, 
 
 star; 
 
 Fig. 110. 
 
 has apparently vacated the surrounding regions of space and accumulated about 
 certain stars ; (see Fig. 110 ;) and in others still, in the circumstance of its being 
 apparently drawn towards certain stars. (See Fig, 110.) 3. Another argument to 
 the same point is, that though many of them are seen in telescopes of moderate 
 power, and some with the naked eye, they are not only not resolved into stars by 
 the largest telescopes, as other nebula; of the same brightness are, but do not like 
 these assume a different appearance, farther than that they grow brighter, as the 
 illuminating power of the telescope increases. 
 
 Fig. 111. 
 
 ihey present the greatest variety of forms, and occur in every stage of apparent 
 condensation, from rude amorphous masses of almost equally diffused nebulous 
 matter, to masses in which the condensation has progressed so far that a star is, 
 to all appearance, beginning to be formed at the centre. The latter class have 
 received a distinctive name, and will soon be particularly noticed. The condenaa- 
 
DISTANCE OF NEBULJS. 227 
 
 Jion is often going on in the same mass upon several lines or points ; and 
 are formed, presenting, in a regular gradation, all the varieties of appearance, 
 which a mass, breaking up into parts by condensation upon points or lines, would 
 assume down to the time of complete separation. The nebulas in which the con- 
 densation appears to be upon one point or line, are round or oval in their figure. 
 But some are long and spindle-shaped, while others are perfectly circular. A very 
 few of the round nebulae are annular. (Fig. 111.) A conspicuous example of 
 this singular class of nebulae may be seen with a telescope of moderate power 
 midway between the stars /? and y Lyrae. By far the greater portion of the nebulas 
 proper are round. 
 
 (4.) Planetary Nebula ; or nebulae which have an appearance similar to the plan- 
 ets ; being round, of an equable light throughout, and often perfectly definite in their 
 outline. (See Fig. 112.) The uniformity of their light seems fig. 112, 
 to indicate that it proceeds altogether from the surface of some 
 spherical body ; and therefore that the body, if it be a collec- 
 tion of nebulous matter, is of the form of a spherical shell : or 
 else that it is derived from a bed of stars of uniform thick- 
 ness. The latter supposition seems to be the more probable one, 
 and is moreover now known to be true in some instances ; for 
 Lord Rosse has succeeded in resolving one of the planetary ne- 
 bulae of Sir J. Herschel's catalogue, (viz. Fig. 49 ;) and has discovered another 
 (Fig. 45 of the catalogue) to be an annular nebula. 
 
 The largest planetary nebula occurs in the Swan, and is nearly 15' in diameter. 
 
 (5.) Stellar Nebula; that is, nebulae so much condensed at the centre as to offer 
 the appearance of a star there seen through the surrounding nebulous mass. (See 
 
 Fig. 112. 
 
 Fig. 112.) In some instances the condensation is gradual, in others sudden. A 
 good example of stellar nebulae occurs to the south of the star .3 Ursa? Majoris. 
 
 (6.) Nebulous Stars; or stars distinctly seen to be such, surrounded by their 
 nebulous atmospheres. On the supposition of progressive condensation, they 
 would seem to be stellar nebulae in a more advanced state. (See Fig. Il2. ffl ) 
 
 Among stellar nebulae and nebulous stars there exist particular nebulae in every 
 stage of apparent condensation, from the slightest appearance of a star at the cen- 
 tre to a perfect star surrounded with the faintest nebulous haze. (See Note XIX.) 
 
 DISTANCE AND MAGNITUDE OF NEBULA. 
 
 601. Herschel undertook to estimate 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 ; and thus at least 1000 times 
 the distance of the nearest and brightest stars. The others are 
 distributed at the same variety of distance as the telescopic stars. 
 
 602. As to the actual dimensions of these clusters, if we sup- 
 pose the distance of none of them to be more than 1000 times the 
 distance of a star of the first magnitude, a globular cluster whose 
 
22S OF THE FIXED STARS. 
 
 apparent diameter is 10' cannot have a real diameter of more than 
 three stellar intervals. At a distance some 5 times greater, such 
 a cluster would contain several thousand stars as remote from each 
 other as is the nearest fixed star from our sun. 
 
 603. It is to be supposed that the resolvable nebulae are in gene- 
 ral posited beyond the region of resolved nebulae and visible stars. 
 The nearest of them are on the very confines of this region. We 
 may form some estimate of the probable distance of the most re- 
 mote of these objects, by calculating how much farther a cluster, 
 ascertained as above, (601,) to be at 1000 times the distance of 
 the nearest star, and which is just discerned as a whitish speck by 
 a telescope of the space-penetrating power 20, would have to be 
 removed to have the same appearance in a telescope whose power 
 is 200, (which is less than the power of the largest telescope.) It 
 is plain that it would have to be removed 1 times farther, or to 
 1 0,000 times the distance of the nearest isolated stars. 
 
 This calculation supposes, however, that the number of stars in the most remote 
 resolvable nebulee is no greater than in the most distant resolved nebulae. If we 
 suppose the number to be greater in any ratio, the distance will be increased in the 
 proportion of the square root of the same ratio. Thus suppose the number of stars 
 in the remotest resolved nebulae to be 10,000, and that the most distant of the re- 
 solvable nebulas contains a number 1000 times greater, or 10,000,000, (which is 
 not far from the estimate of the number of stars in the stratum of the milky way,) 
 (566). The distance calculated above will be increased about 30 times, that is, 
 will be no less than 300,000 times a stellar interval a distance so enormous that 
 light would employ 1,000,000 of years in traversing it. Some astronomers make 
 the probable distance of stars of the lowest magnitude about three times less than 
 we have taken it, which would make the distance just calculated less in the same 
 proportion. Herschel, on the other hand, makes it about twice as great. If the 
 bed of stars to which our sun belongs were viewed at this distance, it would sub- 
 tend an angle of about 10', and appear about ^th of the size of the moon's disc. 
 It seems probable, a priori, that other similar beds of stars, to that in which our 
 sun is posited, occur in the profundities of space. If this is the case they must 
 then be visible at the enormous distance just stated, unless there be a limit to -the 
 known law of the propagation of light. 
 
 604. It appears then that clusters of stars are distributed through- 
 out space at every variety of distance, from that of stars of about 
 the 4th magnitude to an unknown limit beyond the reach of the 
 most powerful telescopes : and that the telescope succeeds in dis- 
 tinctly resolving only those which are posited within the region of 
 the isolated stars discernible through it. The more, distant ones 
 appear in it as spots of nebulous light, and occupy the fields of 
 space which extend from say 1000 times the distance of stars of 
 the first magnitude to at least 10,000 times this distance. 
 
 605. As respects the nebulae proper, we may form some esti- 
 mate of the distance of some of them by noting the magnitude of 
 the stars with which they seem to be connected. In this way, for 
 example, it is found that the remarkable nebula in Orion occupies 
 the interval between stars of the 3d and 8th magnitudes. It is 
 probable that some of these objects, which give no indications of a 
 physical connection with stars, lie beyond the region of known 
 
STRUCTURE OF THE MATERIAL UNIVERSE. 229 
 
 stars, but we have no means of obtaining even the remotest ap- 
 proximation to the distance of individuals among them. 
 
 606. A mere speck in the heavens, at the distance of the stars, 
 as viewed through a good telescope, is as large as the earth's or- 
 bit : accordingly the collections of nebulous matter which occur in 
 the heavens, in the regions of the stars, must have, at least, as 
 great a superficial extent as the orbit of the earth. Many of them 
 must be vastly larger. For example, the nebula in Andromeda 
 (599) is two-thirds of the apparent size of the moon's disc. Its 
 actual extent cannot be less than 365,000 times that of the earth's 
 orbit, or 1000 times that of the whole solar system. (See Note XX.) 
 
 607. The matter of the smallest of these nebulae may be ex- 
 ceedingly subtile, and yet be sufficient in quantity to condense into 
 a body as large and as dense as the sun ; for it appears, by calcu- 
 lation, that if the matter of the sun were to expand so as to fill the 
 space enclosed within the earth's orbit, it would be about 45,000 
 times rarer than the air. 
 
 STRUCTURE OF THE MATERIAL UNIVERSE NEBULAR 
 HYPOTHESIS. 
 
 608. In view of the facts which have now been presented, it 
 will be seen that the great prominent feature in the structure of the 
 universe is the arrangement of the stars in detached beds. Thus 
 our starry firmament is one immense bed of stars, in which occur 
 a great number of subordinate clusters or beds, so that, in fact, it 
 appears to be chiefly made up of more or less detached and con- 
 densed groups of stars. Exterior to this stratum, as far as the 
 telescope penetrates into the abyss of space, are seen other beds, 
 apparently similar, for the most part, to those which occur within 
 the stratum itself. But some that are seen, it is not improbable, 
 are other firmaments constructed on the same vast scale as that of 
 the milky way, and at a distance from it of 200 or 300 times its 
 own diameter, (603.) Leaving these out of view, the others, al- 
 though occurring here and there in almost every direction, beyond 
 the stratum of the milky way, seem to be, the great majority of them, 
 disposed in a stratum of unknown extent, crossing the stratum of the 
 milky way nearly at right angles ; or rather, if the milky way was 
 correctly gauged by Herschel, surrounding it, without anywhere 
 touching it. These beds are of a great variety of forms. But the 
 greater number of them are generally supposed to be spherical, or 
 nearly so. Some which have been supposed to have this form, 
 may, perhaps, be circular or elliptical strata, or of the form of 
 spherical segments, more condensed towards the centre, and seen 
 either perpendicularly in their true form, or obliquely, so as to have 
 their longer axis foreshortened. Planetary nebulae may be such 
 strata, which are not condensed towards the centre : and annular 
 nebulae, the same, in which there is a deficiency of stars at the 
 central parts. (See Note XXI.) 
 
230 Of THE FIXED STARS 
 
 609. The discoveries that have been made in the heavens seem 
 then to point to this great truth, viz., that the plan upon which 
 the universe has been fashioned, is that of an ascending scale of 
 systems, from isolated suns with their attendant systems of planets, 
 to the stupendous whole which fills the eye of the Infinite Creator. 
 
 But the indications are that the work of creation is still in progress. Dispersed 
 through the realms of space, as we have seen (p. 226), are immense masses of some 
 sort of nebulous matter, which in their various stages of condensation upon one 
 or more points or lines, seem to present in sibylline leaves the whole history of the 
 progressive creation of existing worlds and systems of worlds, and at the same time 
 to picture forth the accomplishment of a similar destiny on the part of these masses 
 themselves. 
 
 610. The theory that worlds have been and are still being 
 slowly evolved from primordial nebulous masses by the gradual 
 operation of the general forces and properties which the Creator 
 has either permanently imparted to matter, or is incessantly 
 renewing in it, is called the Nebular Hypothesis. Its author is 
 Sir William Herschel. But Laplace, by undertaking to trace in 
 detail the progress of the creation of the solar system, has still 
 more effectually stamped his name upon it than the author him- 
 self. The great arguments which are urged in its support are the 
 following : 
 
 (1.) That there is a multitude of shining nebulas masses now 
 scattered throughout space, each of sufficient extent to furnish the 
 materials of a world, and some perhaps of systems of worlds. 
 
 (2.) That these masses present a long unbroken gradation, from 
 a mass " without form and void" to a perfect star : that is, all the 
 various states in which a single nebulous mass would be during 
 the vast period that it occupies in condensing from its first rude 
 formless state into a finished globe. 
 
 (3.) That the universe, as it is, in both the general and particu- 
 lar features of its structure, may be shown to be a natural me- 
 chanical consequence of the hypothesis in question. 
 
PART III. 
 
 OF THE THEORY OF UNIVERSAL GRAVITATION. 
 
 CHAPTER XIX. 
 
 i 
 
 OF THE PRINCIPLE OF UNIVERSAL GRAVITATION. 
 
 611. IT is demonstrated in treatises on Mechanics, that if a body 
 move in a curve in such a manner that the areas traced by the 
 radius -vector about a fixed point, increase proportionally to the 
 times, it is solicited by an incessant force constantly directed to- 
 wards 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. 113) be a portion of 
 
 Fig. 113. 
 
 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 along each of the sides AB, BC, CD, 
 &c. : and since we may suppose the times of de- 
 scribing 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 lakes effect at B. Upon SB let fall the per- 
 pendiculars Am, en, Cr. Then, since AB = Be, 
 Am = en ; and since the equivalent triangles 
 SAB, SBC, have a common base SB, Am = Cr. 
 It follows, therefore, that en = Cr, and conse- 
 quently, that Cc is parallel to BS. The impulse 
 which the body receives at B is therefore directed 
 from B towards S. In the same manner it may 
 be shown that the impulse which it receives 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 radii- 
 vectores of the planets about the sun, are proportional to the times. 
 It follows therefore from this law, that each planet is acted upon 
 by a force which urges it continually towards the sun,. 
 
 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. 
 
 612. It is also proved by the principles of Mechanics, that if a 
 body, continually urged by a force directed to some point, describe. 
 
232 
 
 OF THE PRINCIPLE OF UNIVERSAL GRAVITATION. 
 
 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. 114) be the 
 supposed elliptic orbit of the body, 
 C A and CB its semi-axes, and S the 
 focus towards which the force is 
 constantly directed. Also let P be 
 one position of the body, PR a tan- 
 gent to the orbit at P ; and draw 
 RQ parallel to PS, Qwu, HI, and 
 CD, parallel to PR, Qx perpendicu- 
 lar to SP, PF perpendicular to CD, 
 and join S and Q. CP and CD are 
 semi-conjugate diameters. Denote 
 them, respectively, by A' and B' ; 
 and denote the semi-axes, CA and 
 CB, by A and B. Since HI is par- 
 allel to PR, and, by a well-known 
 
 G 
 
 property of the ellipse, the angle RPS is equal to the angle HPT, PH is equal to 
 PI : and since HC = SC, and CE is parallel to HI, E is the middle of SI. We 
 have, therefore, 
 
 2 2 
 
 Now the force at P is measured by 2Pu ; and we may state the proportion 
 
 A' 
 PK : PC : : PE : PC : : A : A' ; which gives Pt? = P -^. 
 
 By the equation of the ellipse referred to its centre and conjugate diameters, 
 PG and DL, 
 
 If we regard Q as indefinitely near to P, then QM = Qu, and Go = 2CP = 2A' ; 
 and therefore 
 
 But Qtt:Qx::PE:PF::CA:PF: 
 
 and, by analytical geometry, 
 
 CD X PF = C A X CB, or, CA : PF : : CD : CB : : B' : B. 
 
 Hence Qu : Qx : : B' : B, Qu 2 : Qi 2 : : B'2 : B2, and Q 2 = Q? -2? 
 
 Substituting in equation (a), Q* 2 -~ = T~- 2Ptt 5 whence Qa: 2 = -- . 2Pw. 
 
 U A. A 
 
 Now triangular area SQP = k = SP X -~- ; whence Qx = . Substituting, 
 
 SP2 
 
 there results 
 
 SP 2 
 
 .4*2 1.. CD- 
 B2 gp 2 
 
 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 constant, and therefore the force is inversely proportional 
 to the square of the distance SP. 
 
 It therefore follow? from Kepler's second law, viz. : that the 
 planets describe ellipses having the centre of the sun at one of 
 
233 
 
 their foci ; that the force of gravity of each planet towards the 
 sun varies inversely as the square of the distance from the sun's 
 centre . 
 
 613. By taking into view Kepler's third law, it is proved that it 
 is one and the same force, modified only by distance from the sun, 
 which causes all the planets to gravitate towards him, and retains 
 them in their orbits. This force is conceived to be an attraction 
 of the matter of the sun for the matter of the planets, and is called 
 the Solar Attraction. 
 
 To deduce this consequence from Kepler's third law, let t, t', denote the periodic 
 times of any two planets ; r, r', their distances from the sun at any assumed point 
 of time ; k, k', the areas described by them 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 n-AB, jrA'B'. 
 Thus kt : k't' : : AB : A'B', and kW : k'2f2 : : A2fi2 : A'2B'2. 
 
 But, by Kepler's third law, t? : t'2 : : A3 : A'3. 
 
 B2 B'2 
 
 Dividing, and reducing, ft 2 : A;' 2 : : - : : 
 
 . xx A. 
 
 that is, the squares of the areas described in equal times are as the parameters of 
 the orbits. 
 
 Now, let /, /, denote the forces soliciting the two planets. Then, by equation 
 (I), Art. 612, 
 
 
 From which it appears that the planets are solicited by a force of gravitation 
 towards the sun, which varies from one planet to another according to the law of 
 the inverse square of their distance. 
 
 614. The motions of the satellites are in conformity with Kep 
 ler's laws ; hence, the planets which have satellites are endued 
 with an attractive force of the same nature with that of the sun. 
 
 615. The existence of a similar attractive power in each of the 
 planets that are devoid of satellites, is proved by the fact that the 
 observed inequalities of their motions, and of those of the other 
 planets, may be shown upon this supposition to be necessary con- 
 sequences of the attractions of the planets for each other. 
 
 616. In like manner the inequalities in the motions of the satel- 
 lites and their primaries, show that the satellites possess the same 
 property of attraction as the sun. 
 
 617. We learn from the motions produced by the action of the 
 sun and planets upon each other, that the intensities of their at- 
 tractive forces are, at the same distance, proportional to their 
 masses, and that the whole attraction of the same body for differ- 
 ent bodies, is, at the same distance, proportional to the masses of 
 these bodies. From which we may infer that a mutual attraction 
 exists between the particles of bodies, and that the whole force of 
 attraction of one body for another, is the result of the attractions 
 
 30 
 
234 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 
 
 of its individual particles. Moreover, analysis shows, that in or- 
 der 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. * 
 
 618. 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 Univer- 
 sal Gravitation. The theory of its existence was first promul- 
 gated by Sir Isaac Newton, and is hence often called Newton's 
 Theory of Universal Gravitation. The force which urges the 
 particles of matter towards each other is called the Force of Grav- 
 itation^ or the Attraction of Gravitation. 
 
 619. In the following chapters our object will be to develope the 
 most important effects of the principle of gravitation thus arrived at 
 by induction. The perfect accordance that will be observed to 
 obtain between the deductions from the theory of universal gravi- 
 tation and the results of observation, will afford additional confir- 
 mation of the truth of the theory. 
 
 CHAPTER XX. 
 
 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 
 
 620. 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 result from 
 
 the proportion M : F : : r 2 : 1 2 , which gives F= -j. This, in the lan- 
 guage of Dynamics, is the Accelerating Force soliciting the planet. 
 As 2 expresses the attraction of the sun for a unit of mass of 
 the planet, its attraction for the entire mass m of the planet will be 
 expressed by m -^. This is the moving force of the planet, and 
 since it is, at the same distance, proportional to the mass of the 
 
REVOLUTION ABOUT THE CENTRE OF GRAVITY. 
 
 231 
 
 planet, the velocity due to its action is the same, whatever may be 
 the mass. 
 
 621. 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 -3. 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 
 
 M m 
 
 the planet, that is, as -^ : -^, or as M : m. 
 
 As the attraction of the particles of the sun and planet are mu- 
 tual 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. 
 
 622. The sun and any planet revolve about their common cen- 
 tre of gravity. 
 
 To show this, we would remark, in the first place, that it is a 
 principle of Mechanics that the mutual actions of the different 
 members of a system of bodies cannot affect the state of the centre 
 of gravity of the system. This is called the Principle of the 
 Preservation of the Centre of Gravity. It follows from it that 
 the common centre of gravity of the sun and any planet is at rest, 
 unless it has a motion of translation in common with the two bo- 
 dies, imparted by a force extraneous to the system. As we are 
 concerned at present only with the relative motion of the sun and 
 planet, such motion of translation, if it does exist, may be left 
 out of account. Now, let S (Fig. 115) be the Fig. 115. 
 
 sun, and P any planet, supposed for the mo- 
 ment to be at rest. If neither of the two bo- 
 dies should receive a velocity in a direction 
 oblique to PS, the line of their centres, 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 Pt, inclined to the line PS, it is sus- 
 ceptible 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 
 
 * 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. 
 
236 THEORY OF THE ELLIPTIC MOTION OF THE PLANETS. 
 
 of the planet and sun must continually pass through the stationary 
 position of the centre of gravity ; and therefore, when the planet has 
 advanced to any point p, the sun will have shifted its position to 
 some point s on tne line pC prolonged. Moreover, as the two bo- 
 dies mutually gravitate towards each other, the paths of each in 
 space will be continually concave towards the other body, and 
 therefore also towards the centre of gravity C, which is constantly 
 in the same direction as the other body. Since the planet per- 
 forms 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 begin- 
 ning of the revolution, as well as the ratio of their distances PC 
 and SC from the centre of gravity, they will return to the posi- 
 tions P, S, from which they set out, and will therefore move in 
 continuous curves. 
 
 Moreover, these curves are similar to the apparent orbit described by P around 
 S. For, draw Sp 1 parallel and equal to sp, and join Pp and Ss. Then, since 
 sC : Cp : : SC : CP, Pp is parallel to Ss ; and therefore Pp produced passes 
 through p 1 . Whence, CP : Cp : : SP : Sp'. Moreover, the angle PCp = PS/. It 
 follows, therefore, that the area PCp is similar to the area PS// ; and thus that the 
 orbit 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 is therefore also an ellipse. 
 
 As the distances of the sun and planet from their common cen- 
 tre of gravity are constantly reciprocally proportional to their 
 masses, the orbit of the sun will be exceedingly small in compari- 
 son with the orbit of the planet. 
 
 623. If to both the sun and planet there should be applied a 
 
 force equal to the accelerating force of the sun, -^, (621), but in an 
 
 opposite direction, the sun would be solicited by two forces that 
 would destroy each other, but the planet would now be urged 
 towards the sun remaining stationary, with the accelerating force 
 
 TJ , 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 in- 
 vestigating this motion, we are at liberty to conceive the sun to be 
 stationary, if we suppose the planet to be solicited by the accelera- 
 ting force -g . As the mass of the sun is very much greater 
 
 than that of any planet, but little error will be committed in neg- 
 lecting the attraction of the planet, and taking into account only 
 
 the sun's action . 
 r 
 
 624. Analysis makes known the general laws of the motion of 
 a body, when impelled by a projectile force, and afterwards contin- 
 
GENERAL PROBLEM OF MOTION OF REVOLUTION. 237 
 
 ually attracted towards the sun's centre by a force varying inverse- 
 ly as the square of the distance. We learn by it that the body 
 will necessarily describe some one of the conic sections around 
 the sun situated at one of its foci. We learn, also, that the na- 
 ture of the orbit, as well as the length of the major axis, is wholly 
 dependent, for any given distance of the planet, upon the intensity 
 of the projectile force, but that the position of the axis, and the ec- 
 centricity of the orbit, depend also upon, the angle of projection 
 (that is, the angle included, at the commencement of the motion, 
 between the line of direction of the projectile force and the radius- 
 vector.) As to the relative intensity of the projectile force neces 
 sary to the production of each one of the conic sections, a certain 
 intensity of force will produce a parabola ; any less intensity, an 
 ellipse or circle ; and any greater, an hyperbola. 
 
 625. If the velocity that would at a given distance be imparted 
 by the sun's attraction in a second of time, which is the measure 
 of its intensity at the given distance, be found, and also the dis- 
 tance of a planet at any time, as well as its velocity and the angle 
 made by the direction of its motion with the radius-vector, the form, 
 dimensions, and position of the planet's orbit can be computed. 
 This is to determine the orbit a priori. The practice has been, 
 however, to determine the various elements of a planet's orbit by 
 observation, (as already described, Chap. VII.) 
 
 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 
 investigation of these equations may be found in the Encyclopaedia 
 Metropolitana, Article Physical Astronomy, page 653, in the Me- 
 canique Elementaire de Francoeur, and in many other similar 
 works.)* 
 
 626. 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. 
 
 627. According to the principle of the preservation of the centre 
 of gravity (622), 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 (593) that it has been ascertained 
 from observation, that it is in fact in motion. 
 
 628. 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, com 
 pared with that in the sun, (less than T io-,) the extent of the curve 
 described by the centre of the sun cannot be very great. It is 
 
 * The equations are the same with^hose deduced directly from Kepler's laws ol 
 the planetary motions. 
 
38 THEORY OP THE ELLIPTIC MOTION OF THE PLANETS. 
 
 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. 
 
 629. It is demonstrated in treatises on Mechanics, that if foreign 
 forces act upon a system of bodies, the centre of gravity of the sys- 
 tem 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 cen- 
 tre of gravity will revolve around the sun, just as the whole quan- 
 tity 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 re- 
 volve 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. 
 
 630. It may be worth while here to remark, that the revolution 
 of the earth around the common centre of gravity of the earth and 
 moon, occasions an inequality, both of longitude and latitude, in 
 the apparent motion of the sun. It is, however, exceedingly small, 
 for the reason that the distance of the earth's centre from the cen- 
 tre of gravity is very short, in comparison with the distance of the 
 sun. The mass of the earth is to that of the moon as 80 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. 
 
 631. It appears also from the physical investigation of the ellip- 
 tic motion of the planets, that Kepler's third law is not rigorously 
 true. In consequence of the action of the planets upon the sun, 
 the ratio of the periodic times of the different planets depends upon 
 the masses of the planets, as well as their distances from the sun. 
 If p and p' be the periodic times of any two of the planets, a and a' 
 their mean distances from the sun's centre, and m and m' their 
 quantities of matter, that of the sun being denoted by 1, then, dis- 
 regarding the actions of the other planets, 
 
 . 
 1+w 1+m' 
 
 As m and m' are very small fractions, the error resulting from their 
 omission will be very small. If we omit them, we shall have 
 
 /:p" ::a 3 :a *. 
 which is Kepler's third law. 
 
INVESTIGATION OF THE DISTURBING FORCES. 
 
 239 
 
 CHAPTER XXI. 
 
 THEORY OF THE PERTURBATIONS OF THE ELLIPTIC MOTION OF THE 
 PLANETS AND OF THE MOON. 
 
 116. 
 
 632. 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 pro- 
 pose 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 pro- 
 duced by the action of the sun. 
 
 633. We have already (283) shown 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 eifect 
 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. 116) 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 will be the disturbing force in 
 the direction MS'. The component along 
 ME is another disturbing force. It is called 
 the Addititious Force, because it tends to 
 increase the gravity of the moon towards the 
 earth. The disturbing force along MS' will 
 generally be inclined to the plane of the 
 orbit, and may be decomposed into three 
 forces, one in the direction of the tangent, 
 another in the direction of the radius-vec- 
 tor, and a third in the direction of the per- 
 pendicular to the plane. The first men- 
 tioned component is called the Tangential 
 
 Force ; the second is called the Ablatitious Force ; and the third we shall call 
 the Perpendicular Force. 
 
 The actual disturbing force in the direction of the radius-vector is equal to 
 the difference between the addititious and ablatitious forces, and is called the 
 Radial Force. This and the tangential and perpendicular forces constitute 
 the disturbing forces, the direct operation of which is to be considered. 
 
 634. To obtain general analytical expressions for these forces, let the dis- 
 tance of the sun from the earth (which for the present we shall suppose to 
 be constant) be denoted by a, and the distances of the moon from the earth 
 and sun, respectively, by y and z. Also let F = the force exerted by the 
 earth upon the moon, P = the force exerted by the sun upon the earth, and 
 Q = the force exerted by the sun upon the moon. Then, if we denote the 
 
 SB" 
 
240 PERTURBATIONS OP ELLIPTIC MOTION OP THE MOON. 
 
 mass of the earth by 1, and take m to stand for the mass of the sun, we shall 
 have, (620,) 
 
 F--L P-- Q-- 
 
 JC "" ' ' 4 A J ^t Q * 
 
 y" a* z* 
 
 Let the force Q be represented by the line MS (Fig. 116) ; and let its 
 component parallel to ES, or MS' = R, and its component along the radius- 
 vector, or ME = T. 
 
 Q : T : : MS : ME ; or, -^ : T : : z : y. 
 
 Whence, addititious force T = ^ . . . (130). 
 In a similar manner we obtain 
 
 The disturbing force in the direction of the sun 
 
 Now, let a, & y, denote the angles made by the line MS', respectively, with 
 the tangent, the radius-vector, and the perpendicular to the plane of the orbit, 
 and we shall have for the components of the disturbing force R P, along 
 these lines ; 
 
 cos a 
 
 /I 1 \ 
 
 tangential force = ma I ~^s "^"1 
 
 ablatitious force = ma 
 
 perpendicular force =ma I TF j I cos y 
 
 (132) 
 
 (133); 
 
 (134). 
 
 Fig. 117. 
 
 Combining equation (133) with equation (130) 
 we obtain for the radial force, 
 
 radial force = my - ma I ^ I cos /?. 
 
 635. The obliquity of the orbit of the moon 
 to the plane of the ecliptic, affects but 'very 
 slightly the value of the tangential and radial 
 forces. If we leave it out of account, or sup- 
 pose the moon's orbit to lie in the plane of the 
 ecliptic, we shall have (Fig. 117) j3==S / ML 
 = SEM the elongation of the moon = 0, and 
 a = complement of 0, which gives 
 
 tang, force = ma I - - 
 
 (135); 
 
 rad. force=my mat , ) cos q> (136). 
 
 y z 3 \z* aj 
 
 636. Equation (134) may be transformed 
 .into another, which is better adapted to the 
 purposes we have in view. Let MK (Fig. 
 116) represent the perpendicular to the plane 
 of the moon's orbit, MF the intersection of the plane SMK with the plane 
 of the moon's orbit, and SI, IF the intersections of a plane passing through 
 
INVESTIGATION OF THE DISTURBING FORCES. 241 
 
 S and perpendicular to EN, the line of nodes, with the plane of the ecliptic 
 and the plane of the orbit. SF will be perpendicular to both IF and MF 
 Denote SIF, the inclination of the orbit to the ecliptic, by I, SEN the angu- 
 lar distance of the sun from the node by N, and SE ajid SM by a and z> as 
 before. 
 
 Now, in equation (134) Y stands for the angle S'MK, but S'MK = SMK, 
 (nearly,) and 
 
 SF 
 cos SMK = sin SMF = ^-. 
 
 SF = SI sin SIF, and SI = SE sin SEI ; 
 whence SF = SE sin SEI sin SIF = a sin N sin I : 
 
 substituting. 
 
 a sin N sin I a sin N sin I 
 
 cos y = cos SMK = - - = -- . 
 oM z 
 
 Thus we have 
 
 /I 1 \ a sin N sin I 
 perpen. force = mal -- - I -- ... (137). 
 
 637. The variable z may be eliminated from equations (135), (136), and 
 (137), and other equations obtained, involving only the variables y and 0. Let 
 ML (Fig. 116) be drawn through the place of the moon perpendicular to ES. 
 Then, using the same notation as in the preceding articles, 
 
 LS = z (nearly), EL = EM cos LEM = y cos 0. 
 But LS = SE EL; 
 
 whence z = a y cos 0, and z 9 = a 3 3a 2 y cos : 
 
 neglecting the terms containing the higher powers of y than the first, as they 
 are very minute, y being only about 3-^ a. 
 
 I __ 1 __ 1 3y cos 
 "^"^a 3 3a a ycos0 == "^"~* ~~tf ' 
 neglecting all the terms of the quotient that involve higher powers of y than 
 
 the first. Substituting this value of in equation (135), we obtain, 
 
 my cos A sn d> 
 ; ^ : tangential force = - - -- ^- - ; 
 
 or, (App. For. 13), 
 
 3my sin 20 
 tangential force = - - . . . (138). 
 
 Making the same substitution in equation (136), and neglecting the term con- 
 taining y a , there results, 
 
 ,. . , my (I 3cos'0) 
 
 radial force = -3 ; 
 
 or, (App. For. 9), 
 
 ,. , f .my (1+3 cos 20) 
 
 radial force = -- - - x 3 - . . . (139). 
 
 In equation (137) we have to substitute, besides, the value of z, viz. a y 
 cos ; then dividing and neglecting as before, we have 
 
 3wiy cos 
 perpen. force = - - 3 - sin N sin I ... (140.) 
 
 638. If the disturbing forces retained constantly the same intensity and di- 
 rection, the result would be a continual progressive departure from the ellip- 
 tic place ; but, in point of fact, these forces are subject to periodical changes 
 of intensity and direction from several causes, from which results a compeu- 
 
 31 
 
242 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 
 
 eation 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 in- 
 clination 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 the variations of the forces resulting from the first, and occasion 
 each a distinct set of periodical perturbations. 
 
 639. Let us now investigate, in succession, the effects of eacn of the dis- 
 turbing forces, commencing with the tangential force. The tangential force 
 takes effect directly upon the velocity of the moon in its orbit ; and as its line 
 of direction does not pass through the earth, it disturbs the equable descrip- 
 tion of areas. It also affects the radius-vector indirectly, by changing the 
 centrifugal force. To understand the detail of its action we must inquire in- 
 to the variations which it undergoes. 
 
 If we regard y as constant in the expression for the tangential force, (eqna. 
 138,) which amounts to considering the moon's orbit as circular, the expres- 
 sion will become equal to zero when sin 2<p = 0, and will have its maximum 
 value when sin 20= 1. It will also change its sign with sin 20. 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 
 
 Fig. 118. 
 
 the octants. It will be seen, on inspect 
 ing Fig. 118, that it will be a retarding 
 force in the first quadrant, (AB) Accord- 
 ingly, it will be an accelerating force in 
 the second, a retarding force again in the 
 third, and an accelerating force again in 
 the fourth. 
 
 This will also appear upon considering 
 the direction of the disturbing force par- 
 allel 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 to- 
 wards 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 rela- 
 tive motion will be the same on this supposition. In the part of the orbit 
 supposed, then, the disturbing force under consideration will be directed from 
 the sun, as represented in Fig. 118. 
 
 640. It appears, then, that the tangential force will alternately retard and 
 accelerate the motion of the moon during its passage through the different 
 quadrants, and that the maximum of velocity will occur in the syzigies. A, C, 
 where the accelerating force becomes zero, and the minimum of velocity in 
 the quadratures, B, D, where the retarding force becomes zero. On the sup- 
 position that the orbit is a circle, the arcs AB, BC, CD, and DA, would be 
 equal, and the retardation of the velocity in one quadrant would be compen- 
 sated for by an equal acceleration in the next, and at the close of a synodic 
 revolution the velocity of the moon would be the same as at its commence- 
 ment. 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 menu 
 
EFFECTS OF THE TANGENTIAL FORCE. 243 
 
 motion* must have place in the octants. Now, as the moon moves from the 
 syzigy A with a motion greater than the mean motion, her true place will be 
 in advance of her mean place, and will become more and more so till she 
 reaches the octant, where the true motion is equal to the mean. The dif- 
 ference between the true and mean place will then be the greatest ; for after 
 that, the true motion becoming less than the mean, the mean place will ap- 
 proach nearer to the true, till at the quadrature they coincide. Beyond B, 
 the true motion still continuing less than the mean, the mean place will be in 
 advance of the true, and the separation will increase till at the octant the 
 true motion has attained to an equality with the mean motion, after which, the 
 mean motion being the slowest, the true place will approach the mean till at 
 the syzigy C they again coincide. Corresponding effects will take place in 
 the two remaining quadrants. We perceive, therefore, that the tangential 
 force produces an inequality of longitude, which attains to its maximum posi- 
 tive and negative value in the octants, and is zero in the syzigies. This is the 
 inequality known in Plane Astronomy by the name of Variation, (296.) 
 
 641. Let us now inquire into the modifications of the effects of the tangen- 
 tial 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 
 moon, the sun's disturbing force would be on the increase by reason of the 
 diminution of the sun's distance, and that, in consequence, the retardation in 
 the first quadrant would be less than the acceleration in the second, and the 
 retardation in the third less than the acceleration in the fourth. So that, 
 when the moon had again come round into conjunction, the acceleration would 
 have over-compensated the retardation. This kind of action would go on so 
 long as the sun approached the earth ; but when it had passed the perigee of 
 its orbit, and began to recede from the earth, the reverse effect would take 
 place, and a retardation of the moon's orbitual motion would happen each 
 revolution. If the anomalistic revolution of the sun was an exact multiple of 
 the synodic revolution of the moon, the acceleration in each revolution of the 
 moon during the passage of the sun from the apogee to the perigee of its or- 
 bit, would be compensated for by an equivalent retardation in the revolution 
 of the moon answering to the same distance of the sun in its passage from the 
 apogee to the perigee ; and the velocity of the moon would be the same at 
 the close of an anomalistic revolution of the sun as at its commencement. But 
 as this relation does not, in fact, subsist between the anomalistic revolution 
 of the sun and the synodic revolution of the moon, a compensation between 
 the accelerations and retardations, answering to the different revolutions of 
 the moon, will not be effected until conjunctions shall have occurred at every 
 variety of distance of the sun in each 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 revo- 
 lution 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 cer- 
 tain number of synodic revolutions, while the sun performs a certain number 
 of anomalistic revolutions. Avoiding unnecessary precision, we find it to con- 
 sist of but a moderate number of years. 
 
 642. We have next to consider the consequences of the elliptic form of 
 the moon's orbit. We remark, in the first place, that, the orbit being an 
 ellipse, the areas AEB, BEG, CED, and DEA, (Fig. 118,) 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 retardation 
 
 The expressions, mean motion, true motion, mean place, true place, are here 
 to be understood only in relation to the perturbation under consideration. 
 
244 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON 
 
 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 compen- 
 sated 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 ellip- 
 ticity of its orbit. This variation will be periodic, like that just considered", 
 and for similar reasons. The excess or deficiency of velocity at the close of 
 any one revolution, will in time be compensated by an equal deficiency or 
 excess occurring at the close of another revolution, when the sun has a cer- 
 tain different position with respect to the perigee of the moon's orbit. 
 
 643. 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 discuss the general analytical expression for it, already in- 
 vestigated. It is, 
 
 my (I 3 cos 3 ^) 
 radial force = -^ - 3 . 
 
 We shall have radial force = 0, when 1 3 cos 2 9 = 0, or when cos 
 = V?- This value of cos ^ answers to four points lying on either side 
 of the quadratures, and about 35 distant from them. When cos ^ is numeri- 
 cally greater than ^/ the result will be negative, and when it is less than 
 /^ the result will be positive. It follows, therefore, that the radial force 
 increases the gravity of the moon in the quadratures, and for about 35 on 
 each side of them, and that during the remainder of a synodic revolution it 
 diminishes it. 
 
 When the moon is in quadratures, cos 0=0, and 
 
 radial force =.... (141). 
 In the syzigies, we have cos # = i 1, which gives 
 radial force = *jr . . . (142). 
 
 It appears, then, that the diminution of the moon's gravity in the syzigies 
 is double of its increase in the quadratures. 
 
 We learn also from equations (141) and (142), that the radial force in the 
 quadratures and syzigies varies directly as the distance ; from which we con- 
 clude that the gravity of the moon varies at these points by a different law 
 from that of the inverse squares. In the quadratures the gravity will be in- 
 creased most at the greatest distance, where it is the least ; and thus it will 
 vary in a less rapid ratio than the square of the distance. In the syzigies it 
 will be diminished most at the greatest distance, or where it is the least ; and 
 accordingly, at these points it will vary in a more rapid ratio than the square 
 of the distance. 
 
 644. An easy investigation, with the aid of the differential calculus, proves 
 
 that the mean diminution of the moon's gravity from the sun's action is -^-j ; 
 
 r representing in this case the mean distance of the moon from the earth. 
 The value of this expression is readily found to be equal to about the 360th 
 part of the whole gravity of the moon to the earth. 
 
 In consequence of this diminution, the moon must describe her orbit at a 
 greater distance from the earth, with a less angular velocity, and in a longei 
 time, than if she were acted on only by the attraction of the earth. 
 
 645. The radial force of the sun alters the eccentricity of the moon's orbit 
 
EFFECTS OF THE RADIAL FORCE. 245 
 
 and differently in different revolutions of the moon, according to the position 
 of the line of syzigies with respect to the line of apsides. When these lines 
 are coincident the eccentricity is increased. 
 For, suppose PMAN (Fig. 119) to be the 
 elliptic orbit of the moon that would be 
 described under the influence of a force 
 varying inversely as the square of the dis- 
 tance. In going from the apogee to the 
 perigee, the gravity will increase in a 
 greater ratio than that of the inverse 
 square of the distance ; the true orbit will 
 therefore fall within the ellipse, and the 
 perigean distance (EP') will be less than 
 for the ellipse. Consequently, the eccen- 
 tricity will increase so much the more as 
 the major axis diminishes. On the other hand, in going from the perigee to 
 the apogee, the gravity will decrease in a greater ratio than the inverse square 
 of the distance, and the moon will consequently recede farther from the earth 
 than if the orbit described was an ellipse. Therefore, in this half of the or- 
 bit the eccentricity will also be increased. When the apsides are in quadra- 
 tures the eccentricity will be diminished ; for the gravity will then vary from 
 the apogee to the perigee, and from the perigee to the apogee, in a less ratio 
 than that of the inverse squares ; and therefore the results will be contrary 
 to those just obtained. The eccentricity will have its maximum value when 
 the apsides are in syzigies, and its minimum when they are in quadratures ; 
 for, in every other position of the line of apsides with respect to the line of 
 syzigies, the radial force in the apogee and perigee will be less than in these 
 positions, (equa. 139.) and therefore alter less the proportional gravity of the 
 moon in the apogee and perigee. It is evident, from the gradual decrease ot 
 the radial force as we recede from the syzigies and quadratures, that the ec- 
 centricity will continually diminish in the progress of the apsides from the 
 syzigies to the quadratures, and that it will continually increase from the 
 quadratures to the syzigies. 
 
 The change in the eccentricity of the moon's orbit, thus produced, will be 
 attended with a corresponding change in the equation of the centre, and thus 
 of the longitude. And this change is the conspicuous inequality of the moon, 
 known by the name of Evection, (296.) 
 
 646. The radial force also produces a motion of the line of apsides. If the 
 moon was only acted upon by the attraction of the earth its orbit would be an 
 ellipse, and the motion from one apsis to another, or, in other words, from 
 one point where the orbit cuts the radius-vector at right angles to the other, 
 would be 180. In point of fact, however, the gravity due to the earth's 
 attraction is constantly either diminished or increased by the radial disturbing 
 force of the sun, and therefore its true orbit must continually deviate from 
 the ellipse that would be described under the sole action of the earth's attrac- 
 tion. When from the action of this force there is a diminution of the moon's 
 gravity, she will continually recede from the ellipse in question, her path will 
 be less bent, and she must therefore move through a greater angular distance 
 before the central force will have deflected her course into a direction at right 
 angles to the radius-vector. Accordingly, she will move through a greater 
 angular distance than 180 in going from one apsis to another, and thus the 
 apsides will advance. On the other hand, when the same force increases the 
 moon's gravity, her path will fall within the ellipse, its curvature will be in- 
 creased, and therefore it will be brought to intersect the radius-vector at right 
 angles at a less angular distance. In this case, therefore, the apsides will 
 move backward. Now, we have shown (643) that the radial disturbing force 
 of the sun alternately diminishes and increases the moon's gravity to the earth. 
 It follows, therefore, that the motion of the apsides will be alternately direct 
 
246 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 
 
 and retrograde ; but since, as has been shown, (643,) the diminution subsists 
 during a longer part of the moon's revolution, and is moreover greater than 
 the increase, the direct motion will exceed the retrograde, and therefore in 
 an entire revolution the apsides wjll advance. 
 
 647. The observed motion of the apsides of the moon's orbit is not, how- 
 ever, wholly produced by the radial disturbing force. It is in part due to the 
 action of the tangential force. This force alters the centrifugal force of the 
 moon, and thus changes its gravity towards the earth, at the same time with 
 the radial force. 
 
 648. The elliptic form of the sun's orbit is the occasion of a change in 
 the radial force, from which results a perturbation of longitude called the An- 
 nual Equation, (296.) The mean diminution of the moon's gravity, arising 
 
 from the action of the sun, or the mean radial force, is equal to - 8 , (644.) 
 
 Hence this diminution is inversely proportional to the cube of the sun's dis- 
 tance from the earth. Therefore, as the sun approaches the perigee of its 
 orbit, its distance from the earth diminishing, the mean diminution of the 
 moon's gravity to the earth will increase, and consequently the moon's dis- 
 tance from the earth will become greater, and its motion slower, than it other- 
 wise would be. The contrary will take place while the sun is moving from 
 the perigee to the apogee. 
 
 649. The disturbing force perpendicular to the plane of the moon's orbit, 
 produces a tendency in the moon to quit that plane, from which there results 
 a change in the position of the line of the nodes, and a change in the inclina- 
 tion of the plane of the orbit to that of the ecliptic. If we examine the gene- 
 ral expression for this force, viz : 
 
 3my cos 
 perpen. force = -- 3 
 
 . 
 sin N sm I, 
 
 Fig. 120. 
 
 we see that for any given values of N and I, it will be zero in the quadra- 
 tures, and have its greatest value in the syzigies ; and that it will change its 
 direction in the quadratures, lying, in the nearer half of the orbit, on the 
 same side of its plane as the sun, and in the more remote half, on the opposite 
 side. We perceive also that it will be zero for every value of <p, or for every 
 elongation of the moon, when the angle N is zero, that is, when the sun is in 
 the plane of the orbit ; and will attain its maximum, for any given elongation, 
 when the line of direction of the sun is perpendicular to the line of nodes. 
 It will also be the less, other things being the same, the smaller is the incli- 
 nation I. 
 
 650. Now let NM'R (Fig. 120) repre 
 sent the orbit of the moon, and S the sun, 
 supposed stationary, the line of the nodes 
 being in quadratures ; and let L, L' be the 
 points of the orbit 90 distant from the 
 nodes. The direction of the force, in the 
 various points of the orbit, is indicated by 
 the arrows drawn in the figure. When the 
 moon is at any point M' between L and the 
 descending node N', she will be drawn out 
 of the plane in which she is moving by the 
 disturbing force. MTC', and compelled to 
 move in such a line as M'J'. The node N' 
 will therefore retrograde to some point n'. 
 When she is at any point M, moving from 
 the ascending node N towards L, her course 
 will be changed to the line Mt, lying, like 
 the line M'Z', below the orbit, which being produced backward, meets the 
 plane of the ecliptic in some point n, behind N. The nodes, therefore, retro- 
 
EFFECTS OF THE PERPENDICULAR- FORCE. 
 
 247 
 
 121 
 
 grade in this position of the moon, as well as in the former. When the moon 
 is in Ihe half N'L'N of the orbit, lying below the ecliptic, the absolute direc- 
 tion 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 ot 
 the nodes will be retrograde. Accordingly, when the nodes are in quadra- 
 tures, or 90 distant from the sun, they will retrograde during every part of 
 the revolution of the moon. 
 
 651. 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, (649,) and 
 therefore there will be no disturbance of the plane of the moon's orbit. 
 
 652. Next, let the situation of the sun be intermediate between the two 
 just considered, as represented in Figs. 120 and 121. The eifect of the dis 
 turbing force will be the same as in the first situation from the quadrature q 
 (Fig. 120) to the node N', and from the quadrature q 1 to the node N. But 
 throughout the arcs Ny, N'y', the direction of the force, and therefore the 
 effects, will be reversed. The node will then retrograde, as before, while the 
 moon moves over the arcs ^N' and y'N, and advance while she is in the arcs 
 N^, N'^'. But as the force is greatest over the arcs ^N', <?'N, which con- 
 tain the syzigies, (649,) and as these arcs are also longer than the arcs Ny, 
 N'^', the node will, on the whole, retrograde each revolution. The velocity 
 of retrogradation will, however, be less than when the nodes are in quadra- 
 tures, and proportionably less as the distance of the sun from this position is 
 greater. 
 
 In the position represented in Fig. 121, 
 a direct motion will take place over the 
 arcs ^'N' and q~N ; but as Ny' and N'y, the 
 arcs of retrograde motion, are of greater 
 extent than ^'N' and j-N, and moreover 
 contain the syzigies, the retrograde motion 
 in each revolution 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 appears, then, that the line of the 
 nodes has a retrograde motion in every 
 possible position of the sun. 
 
 653. We have thus far supposed the sun 
 to remain stationary in the various posi- 
 tions in which we have supposed it, during the revolution of 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. 120,) the 
 arcs Ny, NV will increase, and the arcs <?N' and <?'N diminish ; from which 
 it appears, that, during the advance of the sun from the point 90 behind the 
 descending node to this node, its motion in the course of each revolution of 
 the moon will cause the retrograde motion of the node to be slower than it 
 otherwise would be. While the sun moves from the ascending node to the 
 90 from it, the effect of its motion will obviously be just the reverse of this. 
 During its passage from the descending to the ascending node, the effect will 
 be the same in either quadrant as in that diametrically opposite. 
 
 The variation in the intensity of the perpendicular force conspires with 
 the difference of situation of the sun and its motion during a revolution of the 
 moon in diminishing or increasing, as the case may be, the velocity of retro- 
 gradation of the nodes. 
 
 654. Let us now treat of the change of the inclination of the orbit, result 
 
248 PERTURBATIONS OF ELLIPTIC MOTION OF THE MOON. 
 
 ing from the disturbing action of the sun. And first, if we refer to Fig. 120 
 we shall see that when the nodes are in quadrature the inclination will dimin- 
 ish while the moon is moving from the ascending node N to the point L 90 
 distant from it, and increase while she 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, (650 ;) and therefore while the moon is moving from N' to L' the in- 
 clination will diminish, and while she is moving from L' to N it will increase. 
 The diminutions and increments will compensate each other, and the original 
 inclination will be regained at the close of the revolution. 
 
 When the nodes are in syzigies there will be no change of inclination, 
 (649.) 
 
 655. In the situations of the sun represented in Figs. 120 and 121 the 
 inclination will decrease from q to L and from q' to L', and increase from L 
 to q' and from L' to q, the effects being the same as when the nodes are in 
 quadratures over the arcs qL and LN' in Fig. 120, and NL and Lq' in Fig. 
 
 Fig. 120. 
 
 Fig. 121. 
 
 121, and being reversed over the arcs Ng and Ny in Fig. 120, and <?N and 
 y'N' in Fig. 121. When the sun has the position represented in Fig. 120, 
 the arcs of increase Lq' and L'q will be greater than the arcs of diminution 
 qL and q'L'. The disturbing force will also be greater in the former arcs 
 than in the latter. In the position supposed, therefore, there will be, on the 
 whole, an increase of inclination eVery revolution. When the sun is in the 
 position represented in Fig. 121, the arcs of diminution qL and q'L will be 
 the greater ; and the force in them will also be the greater. In this case, 
 therefore, there will be a diminution of the inclination each revolution of the 
 moon. 
 
 When the sun is on the other side of the line of nodes, the results will be 
 the same as in the positions diametrically opposite. 
 
 656. To inquire now into the consequences of the sun's motion during the 
 revolution of the moon. As the sun moves from S towards N' (Fig. 120) 
 the arcs Lq', L'q, over which there is an increase of the inclination, will in- 
 crease ; and the arcs qL, q'L', over which there is a diminution, will diminish. 
 The motion of the sun will, therefore, in approaching the descending node, 
 render the increase of the inclination each revolution of the moon greater than 
 it otherwise would be. When the sun is receding from the ascending node, 
 the corresponding arcs will experience corresponding changes, and therefore 
 the diminution will now be less than if the sun were stationary. 
 
 The results will be similar for the opposite quadrants on the other side of 
 the line of nodes. 
 
 657. Since the inclination diminishes as the sun recedes from either node, 
 
PLANETARY PERTURBATIONS. 249 
 
 and increases as it approaches either node, it will be the least when the nodes 
 are in quadratures, and the greatest when they are in syzigies. 
 
 It is important to observe that the change of inclination which we have 
 been considering is modified by the retrograde motion of the node ; and thus, 
 that, besides the variations of this element connected with the motions of the 
 moon and sun, there is another extending through the period employed by the 
 node in completing a revolution with respect to both the sun and moon. 
 
 658. The perturbations of the elliptic motion of the moon, comprising ine- 
 qualities of orbit longitude, and variations in the form and position of the orbit, 
 which have now been under consideration, depend upon the configurations of 
 the sun and moon, with respect to each other, the perigee of each orbit, and 
 the node of the moon's orbit. Their effects will disappear when the configu- 
 rations upon which they depend become the same. They are therefore pe- 
 riodical. 
 
 659. The perturbations of the motions 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 
 disturbing force of the sun. From, this cause, together with the slow rela- 
 tive motion of the disturbing and disturbed body, the motion of the apsides and 
 nodes, and the accompanying variations of eccentricity and inclination, are 
 very much more gradual in the case of the planets than in the case of the 
 moon. Their periods comprise many thousands of years, and on this account 
 they are called Secular Motions or Variations. In consequence of the greater 
 feebleness of the disturbing forces, the periodical inequalities are also much 
 less in amount. Moreover, as the motion of a planet is much slower than 
 that of the moon, and as the variations of its orbit are more gradual than 
 those of the lunar orbit, the compensations produced by a change of configu- 
 rations are much more slowly effected, and thus the periods of the inequali- 
 ties are much longer. 
 
 660. The motions of the moon would be subject to no secular variations if 
 the apparent orbit of the sun were unchangeable ; but the secular variation 
 of the eccentricity of the sun's orbit, which answers to an equal variation of 
 the eccentricity of the earth's orbit, that is produced by the action of the 
 planets, gives rise to a secular inequality in the motion of the moon, called 
 the Acceleration of the Moon. This inequality was discovered from observa- 
 tion. Its physical cause was first made known by Laplace. 
 
 CHAPTER XXII. 
 
 OF THE RELATIVE MASSES AND DENSITIES OP THE SUN, MOON, AND 
 PLANETS j AND OF THE RELATIVE INTENSITY OF THE GRAVITY 
 AT THEIR SURFACE. 
 
 661. THE perturbations which a planet produces in the motions 
 of the other planets, depend for their amount chiefly upon the ra- 
 tio of the mass of the planet to the mass of the sun, and the ratio 
 of the distance of the planet from the sun to the distance of the 
 planet disturbed from the same body. Now, the ratio of the dis- 
 
 32 
 
250 RELATIVE MASSES OF THE SUN, MOON, AND PLANETS. 
 
 tances is known by the methods of Plane Astronomy ; conse- 
 quently, the observed amount of the perturbations ought to make 
 known the ratio of the masses, the only unknown element upon 
 which it depends: 
 
 This is one method of determining the masses of the planets. 
 The masses of those planets which have satellites may be found 
 by another and simpler method, viz. : by comparing the attractive 
 force of the planet for either one of its satellites with the attract- 
 ive force of the sun for the planet. These forces are to each other 
 directly as the masses of the planet and sun, and inversely as the 
 squares of the distances of the satellite from the primary and of 
 the primary from the sun. Thus, calling the forces j, F, the 
 masses m, M, and the distances d, D, we have 
 
 f F m M 
 
 / # - D 2 
 
 whence we obtain m : M : : fd? : FD 2 . If we regard the orbits as 
 circles, then d and D will be the mean distances, respectively, of 
 the satellite from the primary, and of the primary from the sun, 
 and are given in tables II, III, and VI. The ratio of / to F is 
 equal to the ratio of the versed sines of the arcs described by the 
 satellite and primary, in some short interval of time ;* since these 
 are sensibly equal to the distances that the two bodies are deflect- 
 ed 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 satellites are 
 known, these arcs and their versed sines are easily determined. 
 
 662. The second column of Table IV exhibits the relative 
 masses of the sun, moon, and planets, according to the most re- 
 ceived determinations, that of the sun being denoted by 1 . 
 
 663. 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. The third column of 
 Table IV contains the densities of the sun, moon, and planets, that 
 of the earth being denoted by 1 . It will be seen on inspecting it, 
 that, for the most part, the densities of the planets decrease as we 
 recede from the sun. 
 
 664. The relative intensity of the gravity at the surface of the 
 sun, moon, and planets, may also readily be found, when the 
 masses and bulks of these bodies are known. For supposing 
 them to be spherical, and not to rotate on their axes, the 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 
 
 * It is to be observed that the versed sines here mentioned relate to the actual 
 arcs described in the two unequal orbits. 
 
EXPLANATION OF SPHEROIDAL FORM OF THE EARTH. 251 
 
 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 Courtenay's Mechanics, pages 250 
 and 251.) 
 
 The fourth column of Table IV exhibits the relative intensity 
 of the gravity at the surface of the sun, moon, and planets, that at 
 the surface of the earth being denoted by 1 . 
 
 CHAPTER XXIII. 
 
 OP THE FIGURE AND ROTATION OF THE EARTH ; AND OF THE PRE- 
 CESSION OF THE EQUINOXES AND NUTATION. 
 
 665. WE have already seen (159) 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 about j. 
 
 666. From the amount and law of the variation of the force of 
 gravity upon the earth's surface, ascertained by observations upon 
 the length of the seconds' pendulum, it is proved that the matter 
 of the earth is not homogeneous, but denser towards the centre, 
 and that it is arranged in concentric strata of nearly an elliptical 
 form and uniform density. 
 
 The fact of the greater density of the earth towards its centre 
 has also been established by observations upon the deviation of a 
 plumb-line from the vertical, produced by the attraction of a moun- 
 tain ; the amount of the deviation being ascertained by observing 
 the difference in the zenith distance of the same star, as measured 
 with a zenith-sector on opposite sides of the mountain. To the 
 north of the mountain the plummet was drawn towards the south 
 and the zenith distance of a star to the north of the zenith was 
 diminished ; while to the south of the mountain the plummet was 
 drawn towards the north, and the zenith distance of the same star 
 was increased by an equal amount : and thus the difference of the 
 two measured zenith distances was equal to twice the deviation of 
 the plumb-line from the true vertical in either of the positions of 
 the instrument ; (allowance being made for the difference of lati- 
 tude of the two stations, as determined from the distance between 
 them and the known length of a degree.) 
 
 Such observations were made for the purpose of determining the 
 mean density of the earth by Dr. Maskelyne, in 1774, on the sides 
 of the mountain Schehallien in Scotland. The observed deviation 
 of the plumb-line made known the ratio of the attraction of the 
 mountain to that of the whole earth, and thus the relative quanti- 
 ties of matter in the mountain and earth. These being ascertained 
 
252 OF THE FIGURE AND ROTATION OF THE EARTH, ETC. 
 
 and the figure and bulk of the mountain having been determined 
 by a survey, the relative density of the earth and mountain became 
 known by the principle mentioned in Art. 663, and thence the ac- 
 tual 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, the density of water being 1 . 
 
 667. 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 imme- 
 diately on the axis, would be animated by a centrifugal force in- 
 creasing with their distance from the axis. If, therefore, we con- 
 ceive of two columns of fluid extending to the earth's centre, one 
 from near the equator, and the other from near either pole, the 
 weight of the former would by reason of the centrifugal force be 
 less than that of the latter. In order, then, that they may sustain 
 each other in equilibrio, that near the equator must increase in 
 length, and that near the pole diminish. As this would be true at 
 the same time for every pair of columns situated as we have sup- 
 posed, the surface of the whole body of fluid about the poles must 
 fall, and that of the fluid about the equator rise. In this manner 
 the earth would become flattened at the poles and protuberant at 
 the equator. 
 
 668. Upon a strict investigation it appears that a homogeneous 
 fluid of the same mean density with the earth, and rotating on its 
 axis at the same rate that the earth does, would be in equilibrium, 
 if it had the figure of an oblate spheroid, of which the axis was 
 to the equatorial diameter as 229 to 230, or of which the oblate- 
 ness was | o If the fluid mass supposed to rotate on its axis be 
 not homogeneous, but be composed of strata that increase in den- 
 sity 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. 
 
 669. The time of the earth's rotation, as well as the position of 
 its axis, would change if any variation should take place in the 
 distribution of the matter of the earth, or in case of the impact of 
 a 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 
 oortion 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 suf- 
 ficient ever to produce any sensible alteration in the circumstances 
 of the earth's rotation on its axis. 
 
PHYSICAL THEORY OF PRECESSION AND NUTATION. 25.1 
 
 670. It is ascertained from direct observation, that there has in 
 reality been no perceptible change in the period of the earth's ro 
 tation since the time of Hipparchus, 120 years before the begin- 
 ning of the present era. We may therefore conclude, a posteriori, 
 that there has been no material change in the form and dimensions 
 of the earth in this interval. 
 
 671. Were the axis of the earth to experience any change of 
 position with respect to the matter of the earth, the latitudes of 
 places would be altered. A motion of 200 feet might increase or 
 diminish the latitude of a place to the amount of 2", an angle which 
 can be measured by modern instruments. Now, in point of fact, 
 the latitudes of places have not sensibly varied since their first de- 
 termination with accurate instruments ; therefore, in this interval 
 the axis of the earth cannot have materially changed. Indeed, 
 since the earth's surface and its internal strata are arranged sym- 
 metrically with respect to the present axis of rotation, it is to be in- 
 ferred that this axis is the same as that which obtained at the epoch 
 when the matter of the earth changed from a fluid to a solid state. 
 
 672. The motions of the earth's axis, along with the whole body 
 of the earth, which give rise to the Precession of the Equinoxes 
 and Nutation, are consequences of the spheroidal form of the 
 earth, inasmuch as they are produced by the actions of the sun 
 and moon upon that portion of the matter of the earth which lies 
 on the outside of a sphere conceived to be described about the 
 earth's axis. The physical theory of the phenomena in question 
 is analogous to that of the retrogradation of the moon's nodes. The 
 sun produces a retrograde movement of the points in which the 
 circle described by each particle of the protuberant mass cuts the 
 plane of the ecliptic, as it does of the moon's nodes ; the effect 
 produced is, however, exceedingly small, by reason of the inertia 
 of the interior spherical mass connected with the external mass 
 upon which the action takes place. The moon, in like manner, 
 occasions a retrograde movement of the nodes of the same parti- 
 cles 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, (649 ;) and that of the moon will vary during each 
 month with the distance of the moon from the node, and also 
 during a revolution of the nodes of the moon's orbit by reason of 
 the change in the inclination of the orbit to the equator. The 
 mean effect of both bodies is the precession; the inequality re- 
 sulting from the change in the sun's action during the year is the 
 solar nutation ; and the inequality consequent upon the retrogra- 
 dation of the moon's nodes is the lunar nutation, or the chief 
 part of it : the change in the position of the equinox occasioned by 
 the moon's revolution, never exceeds \ of a second of an arc; and the 
 change of the obliquity of the ecliptic from this cause is still less- 
 
254 OP THE TIDES. 
 
 CHAPTER XXIV. 
 
 OF THE TIDES. 
 
 673. 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 Ebb Tide. 
 
 674. The interval between one high water and the next is, at a 
 mean, half a mean lunar day, or 12h. 25m. 14s. Low water has 
 place nearly, but not exactly, at the middle of this interval ; the 
 tide, in general, employing nine or ten minutes more in ebbing than 
 in flowing. As the interval between one period of high water and 
 the second following one is a lunar day, or Id. Oh. 50m. 28s., the 
 retardation in the time of high water from one day to another is 
 50m. 28s., in its mean state. 
 
 675. The time of high water is mainly dependent upon the po- 
 sition of the moon, being always, at any given place, about the 
 same length of time after the moon's passage over the superior or 
 inferior meridian. As to the length of the interval between the 
 two periods, at different places, in the open sea it is only from two 
 to three hours ; but on the shores of continents, and in rivers, 
 where the water meets with obstructions, it is very different at 
 different places, and in some instances is of such length that the 
 time of high water seems to precede the moon's passage. 
 
 676. 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 at the 
 syzigies ; after which it diminishes and becomes the least at the 
 quadratures.* 
 
 677. 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 like manner the lowest of the neap tides is the third or 
 fourth tide after the quadrature. 
 
 The spring tides are, in general, about twice the height of the 
 neap tides. At Brest, in France, the former rises to the height of 
 19.3 feet, and the latter only to 9.2 feet. In the Pacific Ocean the 
 highest of the tides of the syzigies is 5 feet, and the lowest of the 
 tides of the quadratures is between 2 and 2.5 feet. 
 
 678. The tides are also affected by the declinations of the sun 
 and moon : thus, the highest spring tides in the course of the year 
 
 * Sally's Astronomical Tables and Formulae, p. 25. 
 
'* 
 
 PHENOMENA OF THE TIDES. < >> 255 ^ < 
 
 ft /" -/' 
 
 are those which occur near the equinoxes. The djjctraor^narily hi^h / . 
 tides which frequently occur at the equinoxes are, ho^ypr, in f ' 
 part attributable to the equinoctial gales. Also, when the moda'or, 
 the sun is out of the equator, the evening and morning tides diflEw,.\* 
 somewhat in height. At Brest, in the syzigies of the summer sol- 
 stice, 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 sol- 
 stice.* 
 
 679. The distance of the moon from the earth has also a sensi- 
 ble influence upon the tides. In general, they increase and dimin- 
 ish as the distance increases and diminishes, but in a more rapid 
 ratio. 
 
 680. The daily retardation of the time of high water varies with 
 the phases of the moon. It is at its minimum towards the syzigies, 
 when the tides are at their maximum ; and it is then about 40m. 
 But, towards the quadratures, when the tides are at their minimum, 
 the retardation is the greatest possible ; and amounts to about Ih. 
 15m. 
 
 The variation in the distance of the sun and moon from the earth, 
 (and particularly the moon,) has an influence also on this retarda- 
 tion. 
 
 The daily retardation of the tides varies likewise with the decli- 
 nation of the sun and moon.t 
 
 681. The facts which have been detailed indicate that the tides 
 are produced by the actions of the sun and moon upon the waters 
 of the ocean ; but in a greater degree by the action of the moon. 
 To explain them, let us suppose at first that the whole surface of 
 the earth is covered with water. We remark, in the first place, 
 that it is not the whole attractive force of the moon or sun which 
 is^effective in raising the waters of the ocean, but the difference in 
 the actions of each body upon the different parts of the earth ; or, 
 more precisely, that the phenomenon of the tides is a consequence 
 of the inequality and non-parallelism of the attractive forces exert- 
 ed by the moon, as well as by the sun, upon the different particles 
 of the earth's mass. From this cause there results a diminution 
 in the gravity of the particles of water at the surface, for a certain 
 distance about the point immediately under the moon, and the point 
 diametrically opposite to this, and an augmentation for a certain 
 distance on the one side and the other of the circle 90 distant from 
 these points, or of which they are the geometrical poles : in con- 
 sequence 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 
 
 * Laplace's System of the World. t Daily's Tables and Formal, p. 26. 
 
256 
 
 OF THE TIDES. 
 
 show that the inequality will give rise to the results just noted, let 
 us suppose that the circle acbd (Fig. 122) represents the earth, and 
 M the place of the moon ; then a will be the point of the earth's 
 
 surface directly under the moon, b the 
 point diametrically opposite to this, and 
 the right line dc perpendicular to MO 
 will represent the circle traced on the 
 earth's surface 90 distant from a and b. 
 Now, the attraction of the moon for the 
 general mass of the earth is the same as 
 if the whole mass were concentrated at 
 the centre O. But the centre of the 
 earth is more distant from the moon 
 than the point a at the surface. It fol- 
 lows, therefore, that a particle of matter 
 situated at a will be drawn towards the 
 moon with a proportionally greater force 
 than the centre, or than the general mass 
 of the earth. Its gravity or tendency 
 towards the earth's centre will therefore 
 be diminished by the amount of this ex- 
 cess. 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 b. The excess will be a 
 force tending to draw the centre away 
 from the particle ; and the effect will 
 be the same as if the particle were drawn away from the 
 centre by the same force acting in the opposite direction. The 
 result then is, that this particle has its gravity towards the earth's 
 centre diminished, as well as the particle at 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 
 tk parallel to OM, and partially in the direction of the tangent or 
 horizontal line ts. The component (ts) 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 obvi- 
 ously be less than the actual force of attraction tr ; and the dif- 
 ference will be greater in proportion as the particle is more remote 
 from a. But this component will decrease gradually from a, while 
 the attraction for the centre is less than for a by a certain finite differ- 
 ence : it is plain, therefore, 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 particles will therefore be diminished for a certain distance 
 from this point. In a similar manner it may be shown that it will 
 also be diminished for a certain distance from the point b. Let us 
 now consider a particle at c, 90 from the points a and b. The at- 
 
PHYSICAL THEORY OF THE TIDES. 257 
 
 traction of the moon for it will take effect in the two directions cl 
 and cO. The force in the latter direction alone will alter the grav- 
 ity 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 augmented for a distance of 35 on each side of the circle 
 dc, 90 distant from the points a and b. These distances are rep- 
 resented in the Figure. 
 
 This may be easily made out by means of the expression for the radial disturb, 
 ing force of the sun in its action upon the moon, (643,) viz. y (1 3 cos* 0). If 
 
 we consider m as denoting the mass of the moon, a the moon's distance from the 
 earth's centre, y the distance of a particle of matter at some point t of the earth's 
 surface from the earth's centre, and the angular distance or elongation (MO<) 
 of the same particle from the moon, as seen from the centre of the earth, it will ex- 
 press the change in the gravity of a particle at the earth's surface, produced by the 
 moon's action. The points a and b will answer to conjunction and opposition, and 
 Ihe points c and d to the quadratures. Now we have already seen (643) that the 
 gravity of the moon is increased at the quadratures, and for 35 on each side of 
 them ; and diminished at the syzigies, and 55 from them in both directions. It fol- 
 lows, therefore, that the same is true for particles of matter at the earth's surface. 
 
 In consequence of the earth's diurnal rotation, the parts of the 
 surface, at which the rise and fall of the water will take place, will 
 be continually changing. Were the entire rise and fall produced 
 instantaneously, the points of highest water would constantly be the 
 precise points in which the line of the centres of the moon and 
 earth intersects the surface, and it would always be high water on 
 the meridian passing through these points, both in the hemisphere 
 where the moon is, and in the opposite one. On the west side of 
 this meridian, the tide would be flowing ; on the east side of it, it 
 would be ebbing ; and on the meridian at right angles to the same, 
 it would be low water. Bat it is plain that the effects of the moon's 
 action will not be instantaneously produced, and therefore that the 
 points of highest water will fall behind the moon. It appears from 
 observation, that in the open sea the meridian of high water is about 
 30 to the east of the moon. 
 
 The great tide wave thus raised by the moon, and which follows 
 it in its diurnal motion, will be a mere undulation, or alternate rise 
 and fall of the water, without any progressive motion, if, as we have 
 supposed, it is nowhere obstructed by shallows, islands, or the 
 shores of continents. 
 
 682. It is evident that the sun will produce precisely similar 
 effects with the moon, and will raise a tide wave similar to the 
 lunar tide wave, which will follow it in its diurnal motion. 
 
 683. To show that the effects of the sun are less in degree than those of the 
 moon, let us take the general expression for the change of the moon's gravity, 
 arising from the action of the sun, namely, 
 
 ...(a), 
 33 
 
258 OF THE TIDES 
 
 in which -m denotes the mass of the sun, a its distance, (the mean distance of the 
 moon being taken as 1,) y the distance of the moon in its given position, and its 
 elongation from the sun, as seen from the earth's centre. This formula will serve 
 to express the change* in the gravity of a particle of matter upon the earth's sur- 
 face, produced by the sun's action, if we take m = the mass of the sun-, as before, 
 a = its distance expressed in terms of the radius of the earth as unity, y = the 
 distance of the particle from the centre of the earth, and <f> = its elongation from 
 the sun, as seen from the earth's centre. If we designate the corresponding quan- 
 tities for the moon by m', a', y, <f>, we shall have for the change of the gravity of 
 a particle, produced by the moon's action, 
 
 ^Xy(l-3cos2^} ... (6). 
 
 For particles at equal elongations from the sun and moon, we shall have the 
 same in expressions (a) and (6), and y may be regarded as the same without ma- 
 terial error. For such particles, then, the alterations of the gravity, produced 
 by the sun and moon, will bear the same ratio to each other as the quantities 
 
 -5- and i. Now, if we give to m,m'. a, a', their values, we shall find that the 
 o*> a' 3 
 
 latter quantity is nearly three times greater than the former. Accordingly, the 
 effect of the moon's action, at corresponding elongations of the particles, and there- 
 fore generally, is nearly three times greater than that of the sun. 
 
 684. The actual tide will be produced by the joint action of the 
 sun and moon, or it may be regarded as the result of the combina- 
 tion of the lunar and solar tide waves. 
 
 At the time of the syzigies, the action of the sun and moon will 
 be combined in producing the tides, both bodies tending to produce 
 high as well as low water at the same places. But at the quadra- 
 tures they will be in opposition to each other, the one tending to 
 raise the surface of the water where the other tends to depress it, 
 and vice versa. The tides should, therefore, be much higher at 
 the syzigies than at the quadratures. 
 
 Between the syzigies and the quadratures the two bodies will 
 neither directly conspire with each other, nor directly oppose each 
 other, and tides of intermediate height will have place. The points 
 of highest water will also, in the configuration supposed, neither 
 be the vertices of the lunar nor of the solar tide wave, but certain 
 points between them. This circumstance will occasion a variation 
 in the length of the interval between the time of the moon's pas- 
 sage and the time of high water. 
 
 685. The effect of the moon's action being to that of the sun's 
 nearly as 3 to 1, (683,) the spring tides will be to the neap tides 
 nearly as 2 to 1 . For, let x = the effect of the moon, and y 
 the effect of the sun : then the ratio of x + y to x y will be the 
 ratio of the heights of the spring and neap tides. Now, 
 
 * = 3y, and thus ^=2 = 2. 
 x-y 3y-y 
 
 This result is conformable to observation. 
 
 686. The height of the tide, as well as the interval between the 
 time of high water and that of the moon's meridian passage, will 
 vary not only with the elongation of the moon from the sun, but 
 
MODIFICATIONS OF THE GENERAL PHYSICAL THEORY. 259 
 
 also with the distance and declination of the moon and sun. For, 
 expressions (a) and (b) show that the intensities of the moon's and 
 sun's actions vary inversely as the cube of their distance ; and the 
 changes of the declinations of the two bodies must be attended 
 with a change both in the absolute and relative situation of the 
 vertices of the lunar and solar tide waves. 
 
 687. The laws of the tides, which would obtain on the hypothe- 
 sis of the earth being covered entirely with water, are found to 
 correspond only partially with those of the actual tides. The 
 continents have a material influence upon the formation and pro- 
 pagation of the tide wave. 
 
 688. Professor Whewell infers, from a careful discussion of a 
 great number of observations upon the tides, that the tide of the 
 Atlantic Ocean is, for the most part, produced by a derivative tide 
 wave, sent off from the great wave which in the Southern Ocean 
 follows the moon in its diurnal motion around the earth. This 
 wave advances more rapidly in the open sea than along the coasts, 
 where it meets with obstructions. 
 
 Where portions of the tide wave, extending from one point of 
 the coast to another, become detached, and advance into a narrow 
 space, particularly high tides will occur. In this way (as it is sup 
 posed) it happens that the tide rises at certain places in the Bay 
 of Fundy, to the height of 60 or 70 feet. 
 
 689. In channels peculiar tides occur in consequence of the 
 meeting of the waves which enter the channels at their two ex- 
 tremities. Where the two waves meet in the same state, unusually 
 high tides occur. This is observed to be the case at some points 
 in the Irish Channel. In the port of Batsha, in Tonquin, the tides 
 arrive by two channels, of such lengths that the two waves meet 
 in opposite states, or that the flood tide arrives by one channel just 
 as the ebb tide begins to leave by the other, and the consequence 
 is that there is neither high nor low water. 
 
 This is the case when the moon is in the equator. When she 
 has a northern or southern declination, there is a small rise and 
 fall of the water once in a lunar day, owing to the inequality of the 
 morning and evening tides of the open sea. 
 
 690. Lakes and inland seas have no perceptible 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. 
 
 691. The tides experienced in rivers and seas communicating 
 with the ocean, are not produced by the direct actions of the sun 
 and moon, but are waves propagated from the great wave of the 
 open sea. 
 
 In rivers of considerable length, the ascending tides are encoun- 
 tered by those which are returning, so that a great variety of tides 
 occur along their shores. 
 
 692. The mean interval between noon and the time of high 
 water at any port, on the day of new or full moon, is called the 
 
260 OF THE TIDES. 
 
 Establishment of that port. It will be, approximately, the inter- 
 val between the time of the meridian passage of the moon and the 
 time of high water on any day of the month. To obtain this in- 
 terval for a given day more nearly, it is necessary to correct the 
 establishment for the effects of the change of the distance and de- 
 clination of the sun and moon, and of the change in the elongation 
 of the moon from the sun. When it has been determined, by add- 
 ing it to the time of the meridian passage of the moon, we have the 
 time of the next high water. 
 
PART IV. 
 
 ASTRONOMICAL 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- 
 bles 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. The mean longitudes of the sun and of his perigee for the 
 beginning 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 Terns of 1810, which is 
 more accurate than Delambre's Table, this being in some instances 
 liable to an error of 2 seconds. The Table of Nutation (Table 
 XXVII) was extracted from Francceur's Practical Astronomy. 
 The maximum of nutation of obliquity is taken at 9". 25. The 
 Tables of the Sun will give the sun's longitude within a 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 Burckhardt'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- 
 sults of Burckhardt's Tables more than 0".5, and never as much 
 as 1 ' . The error of the latitude and hourly motion in latitude will 
 be still less. The other tables have been taken from some of the 
 most approved modern Astronomical Works. (For the principles 
 of the construction ofjthe Tables, see Chap. IX.) 
 
 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 table, 
 is called the A rgument of this quantity. 
 
62 ASTR01SOMIC.A.L PROBLEMS. 
 
 The 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., parts. 
 
 Tables are of Single or Double Entry, according as they con- 
 tain one or two arguments. The Epoch of a table is the instant 
 of time for which 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. 
 
 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 
 -f-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 
 Time, 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 given date 
 and the epoch of Table XII is easily derived. 
 
EXPLANATION OF THE TABLES. 263 
 
 Table XIV, contains certain other Corrections to be applied to 
 the 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 their 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 
 variations of the arguments from the beginning of any month to the 
 Beginning of any day of the month ; also the same for Hours. 
 
 Table XXI, gives the Sun's Motions for Minutes and Seconds. 
 Tables XVIII to XXI, inclusive, make known the mean longitude 
 of the Sun from the mean equinox, at any moment of time. 
 
 Table XXII, Mean Obliquity of the Ecliptic for the beginning 
 
264 ASTRONOMICAL PROBLEMS. 
 
 of each year contained in the table. It is found for any interme- 
 diate time 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 equation 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's Longitude, due respectively to the attractions of the 
 Moon, Venus, Jupiter, Mars, and Saturn. 
 
 Table XXXIV, is for the variable part of the Sun's Aberration. 
 The numbers have all been rendered positive by the addition of 
 the constant 0".3. 
 
 Table XXXV, contains the Epochs of the Moon's Mean 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 XLI to LIII, inclusive, give the various Equations of 
 the Moon's Longitude. It is to be observed with respect to Table 
 XLI, that the right hand figure of the argument is supposed to be 
 dropped. But when the greatest attainable accuracy is desired, it 
 
EXPLANATION OF THE TABLES. 265 
 
 can be retained, and a cipher conceived to be written after the 
 numbers in the columns of Arguments in the table. In Tables 
 L, LI, LII, and LV, the degrees will be found by referring to the 
 head or foot of the column. (See Problem II., note 2.) 
 
 Table LIV is for the Nutation of the Moon's 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^-g. 
 
 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. 10m. to equation II. ; 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 
 Differences. These tables are given to facilitate 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. 
 
 34 
 
266 ASTRONOMICAL PROBLEMS. 
 
 To find the logistical logarithm of a number composed of min- 
 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. 26*7 
 
 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. When the moon's hourly motion is 30' 12", what is 
 its motion in 16m. 24s. ? 
 
 As 60m. ,''..- -/" . ; '^ 
 
 : 30' 12" . '.' r/ v ; . 2981 
 
 : : 16m. 24s. . 5633 
 
 : 8' 15" . . ' V 4 . 8614 
 
 2. If the moon's declination change 1 31' in 12 hours, what 
 will be the change in 7h. 42m. ? 
 
 f As 12h. . ''* . ar. co. 9.3010 
 
 : 1 31' **?'.' . & 1.5973 
 
 :: 7h. 42m. y . ^ 'Y ./' ./ 8917 
 
 : 58 ; ] 'I [ . . 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 
 
 As 60m. . *,,- 
 1468 . V : 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. 11s. ? Ans. 2 ; 1". 
 
 5. If the sun's decimation change 16' 33" in 24 hours, what 
 will be the change in 14h. 18m. ? Ans. 9' 52". 
 
 6. If the moon's declination change 54".7 in one hour, what will 
 be the change in 52m. 18s. ? Ans. 47".7. 
 
 PROBLEM II. 
 
 To take from a table the quantity corresponding to a given value 
 of the argument, or to given values of the arguments of the 
 table. 
 
268 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, LII, 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 s - 6 24' 36", to find the corre- 
 sponding quantity in Table L. 
 
 7 s - 6 gives 43' 17" .4. 
 
 The difference between 43' 17" .4 and the next following quan- 
 tity in the table is 1' 7".3. 
 
 60' : 1' 7".3 : : 24' 36" : 27".6.* 
 
 * The student can work the proportion, either by the common method, or by lo- 
 gistical logarithms, as he may prefer. In working this and all similar proportions 
 by the arithmetical method, the seconds of the argument may be converted into 
 the equivalent decimal part of a minute by means of Table XVII, (using the sec- 
 onds as if they were minutes.) It will be sufficient to take the fraction to the 
 nearest tenth. 
 
TO TAKE OUT A QUANTITY FROM A TABLE. 269 
 
 From 43' 17" A 
 Take 27 .6 
 
 42 49 .8 
 
 2. Given the argument 2 s - 18 41' 20", to find the corresponding 
 quantity in Table XXV. 
 
 2 s - 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 s - 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 s - 8 14' 52", to find the corresponding 
 quantity in Table LII. Ans. 12' 36".0. 
 
 5. Given the argument 11 8> 11 23' 10", to find the correspond- 
 ing quantity in Table LVI. Ans. 1 1 7 48' .0. 
 
 6. Given the argument s - 26 20', to find the corresponding 
 quantity in Table XII. Ans. 4 P.O. 
 
 Case 2. 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 term. 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 difference of 10' in the argu- 
 ment. In obtaining quantities from these tables, it will be found 
 more convenient to take for the first and second terms of the pro- 
 
270 ASTRONOMICAL PROBLEMS. 
 
 portion, respectively, 10', and the difference furnished by the table, 
 and work the proportion by the arithmetical method. (See note at 
 bottom of page 268.) 
 
 Exam. 1. Given the argument s - 24 42' 15", to find the cor- 
 responding quantity in Table LI. 
 
 s - 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 s - 24 42' 15" over s - 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 quan- 
 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 s - 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 
 taken 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, mul 
 tiply as above, and remove the decimal point two figures to the left. 
 When the common difference of the arguments is 5, proceed as if 
 it were 10, and double the result. In like manner, when the com- 
 mon difference is 50, proceed as if it were 100, and double the 
 result. 
 
TO TAKE OUT A QUANTITY FROM A TABLE. 271 
 
 Exam. 1. Given the argument 973, to find the corresponding 
 quantity in Table XLV column headed 13. 
 
 970 gives 23".5. 
 The difference is 1". 2, and the excess 3. 
 
 1".2 From 23".5 
 
 3 Take .4 
 
 Corr. .36 23 .1 
 
 2. Given the argument 4834, to find the corresponding quantity 
 m 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 corresponding, 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- 
 
272 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 v 1 . Given the argument 64 at the top and 77 at the side, 
 to find the corresponding quantity in Table LXXXI. 
 
 50 and 70 give 47".7. 
 
 The difference between 47".7 and the next quantity below it 
 is I" A. The excess of 77 over 70 is 7, and the argument at the 
 side changes by 10. 
 
 I" A 
 7 
 
 From 47".7 
 Corr. due excess 7, .98, or 1".0. Take 1.0 
 
 Quantity corresponding to 50 and 77, 46 .7 
 The difference between 47". 7 and the adjacent quantity in the 
 next column on the right is 3". 3. The excess of 64 over 50 is 14, 
 and the argument at the top changes by 50. 
 3".3 
 14 
 
 .462 
 2 
 
 From 46".7 
 
 Corr. due excess 14, .924 Take 0.9 
 
 45 .8 
 
 2. Given the argument 223 at the top and 448 at the side, to 
 find the corresponding quantity in Table XXX. 
 
 220 and 440 give 16".0. 
 
 The difference between 16".0 and the quantity next below it 
 is 2".2. 
 
 2".2 
 8 
 
 2) 1.76 
 
 From 16 /; .0 
 Corr. for excess 8, .88, or 0".9. Take .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. 
 ;/ .7 
 3 
 
 To 15".l 
 
 Corr. for excess 3, 1 Add .2 
 
 15.3 
 
TO CONVERT DEGREES, MINUTES, ETC., INTO TIME. 273 
 
 3. Given the argument 472 at the top and 786 at the side, to 
 find the corresponding quantity in Table XXXI. 
 
 AJIS. 9".7. 
 
 4. Given the argument 620 at the top and 367 at the side, to 
 find the corresponding quantity in Table LXXXI. 
 
 Ans. 55".2. 
 
 5. Given the argument 348 at the top and 932 at the side, to 
 find (by the rule given in Note 2) the corresponding quantity in 
 Table XXXII. Ans. 15".4. 
 
 PROBLEM III. 
 
 To convert Degrees, Minutes, and Seconds of the Equator into 
 Hours, Minutes, fyc., of Time. 
 
 Multiply the quantity by 4, and call the product of the seconds, 
 thirds ; of the minutes, seconds j and of the degrees, minutes. 
 Exam. 1. Convert 83 II 7 52" into time. 
 83 11' 52" 
 4 
 
 5 h - 32 m - 47 B - 28'" 
 2. Convert 34 57' 46" into time. 
 
 Ans. 2h. I9m. 51sec. 4"'. 
 
 PROBLEM IV. 
 
 To convert Hours, Minutes, and Seconds of Time into Degrees, 
 Minutes, and Seconds of the Equator. 
 
 Reduce the hours and minutes to minutes : divide by 4, and 
 call 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. 
 7 h. 9 m. 34 . 
 
 60 
 
 4 ) 429 34 
 
 107 23' 30" 
 2. Convert 1 Ih. 24m. 45s. into degrees, &c. 
 
 Ans. 171 11' 
 35 
 
274 ASTRONOMICAL 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 d> 15 h - 13 m - 22 8 - 
 Diff. of Long. ... 4 56 4 
 
 Time at New York . . 24 10 17 18P.M. 
 
 2. When it is June 9th, 5h. 25m. lOsec. P. M. at Washington, 
 what is the corresponding time at Greenwich ? 
 
 Time at Washington, June, 9 d - 5 h - 25 m - 10 8 - 
 Diff. of Long. ... 586 
 
 Time at Greenwich . . 9 10 33 16P.M. 
 
 3. When it is January 15th, 2h. 44m. 23sec. P. M. at Paris, 
 what is the time at Philadelphia ? 
 
 Longitude of Paris . ^f 1 O h - 9 m - 21 fl .6 E. 
 Do. of Philadelphia, . 5 39 .6 W. 
 
 5 10 1.2 
 
 Time at Paris, January, . 15 d 2 h - 44 m - 23'- 
 Diff. of Long. . . ".] 5 10 1 
 
 Time at Philadelphia, "V 14 21 34 22 
 Or January 15th, 9h. 34m. 22sec. A. M. 
 
 4. When it is Marches 1st, 8h. 4m. 21 sec. P. M. at New Haven, 
 what is the corresponding time at Berlin ? 
 
 Ans. April 1st, Ih. 49m. 43sec. A. M. 
 
TO CONVERT APPARENT INTO MEAN TIME. 275 
 
 5. When it is August 10th, lOh. 32m. Msec. A. M. at Boston, 
 what is the time at New Orleans ? 
 
 Ans. Aug. 10th, 9h. 16m. 4sec. A. M. 
 
 6. When it is noon of the 23d of December at Greenwich, what 
 is the time at New York ? 
 
 Ans, Dec. 23d, 7h. 3m. 55sec. A. M 
 
 PROBLEM VI. 
 
 The Apparent Time being given, tojind the corresponding Mean 
 Time ; or the Mean Time being given tojind 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, take 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 within a 
 small fraction of a second is demanded, take the numbers in the 
 columns entitled I, II, III, IV, V, N, in Tables, XVIII, XIX, 
 XX, answering respectively to the year, month, days, and hours, 
 of the given time. With the respective sums of the numbers 
 taken from each column, as arguments, enter Table XIV, and 
 take out the corresponding quantities. These quantities added to 
 the equation of time as given by Tables XII and XIII, and the 
 
276 ASTRONOMICAL PROBLEMS. 
 
 constant 3.0s. subtracted, will give the true Equation of Time, if 
 the given time is Mean Time. When Apparent Time is given, it 
 will be farther necessary to correct the equation of time as 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 
 same place ? 
 
 Time at New York, July, 1840, 16 d - 9 h - 35 m - 22 s - 
 Diff. of Long. ... 4 56 4 
 
 Time at Greenwich, July, 1840, 16 14 31 26 
 
 M. Long. 
 
 1840 .J . . . . 9 s - 10 12' 49" 
 July L'L . . . . 5 29 23 16 
 16d. '. , .... 14 47 5 
 
 I4h. . . ..; : : *';..'.. . 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 m - 44 s - 
 
 Mean Time at New York, July, 1840, 16 d - 9 h - 35 m - 22 s - 
 Equation of time, sign changed, . 5 44 
 
 Apparent Time, . . . . 16 9 29 38P.M. 
 
 2. On the 9th of May, 1842, when it is 4h. 15m. 21sec. A. M. 
 
 apparent time at New Y ork, what is the mean time at the same 
 
 place, and also at Greenwich ? 
 
 Time at New York, May, 1842, 8 d - 16 h - 15 m - 21'- 
 Diff. of Long. . 4 56 4 
 
 Time at Greenwich, . ^ , 8 21 11 25 
 
 M. Long. 
 
 1842 . -V 9* 10 43' 18" 
 May . . 3 28 16 40 
 8d. 6 53 58 
 
 21h. . 51 45 
 
 llm. 27 
 
 M. Long >;- 1 16 46 8. Equa. of time, 3m. 45s, 
 

 TO CONVERT APPARENT INTO MEAN TIME. 
 
 277 
 
 Apparent Time at Greenwich, May, 1842, 8 d - 21 h - ll m - 25' 
 Equation of Time, *-. . 
 
 -3 45 
 
 Mean Time at Greenwich, 
 Diff. of Long. . 
 
 8 21 7 40 
 4 56 4 
 
 8 16 11 36 
 
 Mean Time at New York, 
 
 Or, May 9th, 4h. llm. 36s. A. M. 
 
 3. On the 3d of February, 1855, when it is 2h, 43m. 36s. appa- 
 rent time at Greenwich, what is the exact mean time at the same 
 place ? 
 
 Appar. Time at Greenwich, Feb., 1855, 3d. 2h. 43m. 36s. 
 
 
 M, Long. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 V. 
 
 N. 
 
 1855 
 
 9' 10 34' 30" 
 
 433 
 
 279 
 
 806 
 
 889 
 
 866 
 
 863 
 
 Feb. 
 
 1 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 d - 2 h - 43 m - 36 s - 
 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 T 
 long.) : : 14m. : O.ls. 
 
 II. Ill 
 
 II. IV. .... 
 
 II. V 
 
 I 
 
 N 
 
 Constant. 
 
 of 
 
 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 the 
 same place ? Ans. 6h. 46m. 39s. P. M. 
 
 6. On the 23d of September, 1850, when it is 9h. 10m. 12sec. 
 mean time at Boston, what is the exact apparent time at the same 
 place? Ans. 9h. 18m. 1.0s. 
 
278 ASTRONOMICAL PROBLEMS. 
 
 PROBLEM VII. 
 
 To correct the Observed Altitude of a Heavenly Body for Re - 
 
 fraction. 
 
 With the given altitude take the corresponding refraction from 
 Table VIII. Subtract the refraction from the given altitude, and 
 the result will be the true altitude of the body at the given station. 
 
 This rule will give exact results if the barometer stands at 30 
 inches, and Fahrenheit's thermometer at 50, and results 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. 2042'11".0 
 
 Refrac. (Table VIII), 2' 33".0; Bar.+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 llth 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 42, required the true altitude. Ans. 48 34' 58".8. 
 
* 
 
 TO DEDUCE THE TRUE FROM THE APPARENT ALTITUDE. 279 
 
 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.) -Hog. 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 wil] also be more accurate to make use of 
 equa. (10), p. 52, 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 ? 
 
280 
 
 ASTRONOMICAL PROBLEMS. 
 
 Obs. alt. 
 
 Refraction . \-^ 9 r . 
 
 True alt. at the station, 
 Parallax in alt. (Table X), 
 
 26 J 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 the 
 17th of March, 1837, 8h. P. M., be 66 10' 44" ; the barometer 
 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 
 
 True refrac. 
 
 25.4 
 
 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 
 Par. in alt. 
 
 logarithms, 
 cos. 9.60637 
 
 sin. 8.19731 
 
 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 
 parallax being taken at 1".9, as given by the Nautical Almanac ? 
 
 True alt. . . 56 54' 57' ; . cos. 9.7371 
 
 Hor. par. 1".9 . . . . .... log. 0.2787 
 
 Par. in alt. 
 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 tht 
 true altitude. Ans. 30 22' 41". 
 
 6. Given 15 24' 23" the true altitude of Venus, and 8" its hori- 
 zontal parallax, to find the apparent altitude Ans. 15 27' 41". 
 
TO FIND THE SUN*S LONGITUDE, ETC , FROM TABLES. 281 
 
 PROBLEM IX. 
 
 To find the Sun's Longitude, Hourly Motion, and Semi-diameter, 
 for a given time, from the Tables. 
 
 For the Longitude. 
 
 When the given time is not for the meridian of Greenwich, 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 longi- 
 tude, adding 12 signs when necessary to render the subtraction 
 possible ; the remainder will be the sun's mean anomaly. With 
 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 
 signs whenever the sum exceeds either of these quantities. In adding arguments 
 expressed in 100 or 1000, &c. parts of the circle, when they consist of two figures, 
 reject the hundreds from the sum; when of three figures, the thousands; antf 
 when of four figures, the ten thousands. 
 
 36 
 
282 
 
 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" ; ir 
 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 XXIX 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- 
 mi-diameter, on the 25th October, 1837, at llh. 27m. 38s. A. M 
 mean time at New York. 
 
 Mean time at N. York, Oct. 1837, 24 d - 23 h - 27 m - 38 s - 
 Diif. of Long 4 56 4 
 
 Mean time at Greenwich, 
 
 25 4 23 42 
 
 1837 . 
 
 October 
 25d. . 
 4h. . 
 23m. . 
 42s. 
 
 Eq. Sun's Cent. 
 
 II. III. 
 II. IV. 
 II. V. 
 
 Const. . 
 
 Lunar Nutation 
 Solar Nutation 
 
 Sun'strue long. 
 
 M. Long. 
 
 9 10 55 47.2 
 
 8 29 4 54.1 
 
 23 39 19,9 
 
 9 51.4 
 
 56.7 
 
 1.7 
 
 7 3 50 51.0 
 
 11 28 12 43.5 
 
 2.5 
 
 9.0 
 
 7.7 
 
 19.3 
 
 3.0 
 
 2 4 16.0 
 
 6.3 
 
 1.2 
 
 7 2 4 8.5 
 
 Long. Perigee. I. 
 
 II. 
 
 9 10 8 5816280 
 
 46250748215 
 4810J 66 107 
 
 8 
 
 9 10 8 55882 94 
 7 3 50 51 
 
 III. IV. 
 
 549 
 
 872 
 
 321 
 
 397 
 
 35 
 
 753 
 
 v. IN. 
 
 348895 
 
 63 
 
 40 
 
 416 939 
 
 9 23 41 56 Mean Anomaly. 
 Sun's Hourly Motion, . . 2' 29". 7 
 Sun's Semi-diameter 16' 17".2 
 
 2. Required the sun's longitude, hourly motion, and semi-diam 
 eter, on the 15th of July, 1837, at 8h. 20m. 40s. P. M. mean time 
 at Greenwich. 
 
TO FIND THE APPARENT OBLIQUITY OF THE ECLIPTIC. 
 
 1837 
 July 
 15d. 
 
 8h. 
 20m. 
 40s. . 
 
 Eq. Sun's Cent. 
 
 ii. in! 
 
 II. IV. 
 II. V. 
 I. VI. 
 II. VII. 
 
 Aber. . 
 
 Lunar Nutation 
 Solar Nutation 
 
 Sun's true long. 
 
 M. Long. 
 
 Long. Peri. 
 
 I. 
 
 816 
 129 
 473 
 11 
 
 II. I III 
 
 IV. V. 
 
 N. VI. 
 
 VII. 
 
 so/ // 
 
 9 10 55 47.2 
 5 28 24 7.8 
 13 47 56.6 
 19 42.8 
 49.3 
 1.6 
 
 8 / 
 
 9 10 8 5 
 31 
 
 2 
 
 280549 
 496,806 
 
 38 62 
 1 1 
 
 321 348 
 263 41 
 20 3 
 
 604392 
 
 895 787 
 27569 
 2508 
 11 
 
 600 
 17 
 2 
 
 9 10 8 38 
 3 23 28 25 
 
 429l8is|418 
 
 924875 619 
 
 2' 23".l 
 . 15' 45".4 
 
 3 23 28 25.3 
 11 29 33 10.3 
 10.7 
 6.6 
 5.0 
 7.7 
 1.8 
 0.2 
 0.6 
 
 6 13 19 47 Mean Anomaly. 
 Sun's Hourly Motion, 
 Sun's Semi-diameter, 
 
 3 23 2 8.2 
 
 7.8 
 + 0.8 
 
 
 
 
 
 3 23 2 1.2 
 
 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 long* 
 tude.) 
 
 Ans. Sun's longitude, 2 s - 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 s - 13 I 7 44". 6 ; hourly motion, 2f 
 32".l ; semi-diameter, 16' 14".7. 
 
 PROBLEM X. 
 
 To find the Apparent Obliquity of the Ecliptic, for a given time, 
 from the Tables. 
 
 Take the mean obliquity for the given year from Table XXII. 
 Then with the argument N, found as in the foregoing problem, 
 and the given date, take from Table XXVII the lunar and solar 
 nutations of obliquity. Apply these according to their signs to the 
 mean obliguity, and the result will be the apparent obliquity. 
 
 Exam. 1 . Required the apparent obliquity of the ecliptic on the 
 15th of March, 1839. 
 
284 ASTRONOMICAL 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 the 12th 
 of July, 1845. Ans. 23 27' 28". 2. 
 
 PROBLEM XL 
 
 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 
 declination. 
 
 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 52 S. . '.-- . 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 longitude 
 is known, is found by the foregoing Problem. 
 
TO FIND THE SUN S LONGITUDE AND DECLINATION. 285 
 
 the sun's right ascension and decimation when his longitude if 
 44 18' 25". 
 
 Aiis. Right ascension 41 50' 30", and declination 16 8' 40" N. 
 
 PROBLEM XII. 
 
 Given the Surfs Right Ascension and the Obliquity of the Eclip 
 tic, to find his Longitude and Declination. 
 
 Using the same notation as in the last problem, we have, to find 
 the longitude and declination, 
 
 log. tang L = log. tang R + ar. co. log. cos w, 
 log. tang D = log. sin R + log. tang w 10. 
 
 Exam. 1. What is the longitude and declination of the sun, 
 when his right ascension is 142 11' 34", and the obliquity of the 
 ecliptic 23 27' 40" ? 
 
 R = 142 II 7 34" . /Vv . tan. 9.88979 
 
 w = 23 27 40 '-...i'*. .',... ar. co. cos. 0.03747 
 
 L = 139 46 30 . . . tan. 9.92726 
 
 R=142 11 34 . r .v . sin. 9.78746 
 w= 23 27 40 . . . tan. 9.63750 
 
 D= 14 53 55 N. . . ..,;.,- tan. 9.42496 
 
 2. Given the sun's right ascension 310 25' 11", and the obli- 
 quity of the ecliptic 23 27' 35", to find the longitude and declina- 
 tion. 
 
 Ans. Longitude 307 59' 57", and declination 18 17' 0" S. 
 
 PROBLEM XIII. 
 
 The Surfs 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 w 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 the ecliptic 23 27' 32", required the angle of position. 
 
ASTRONOMICAL PROBLEMS. 
 
 L = 24 15' 20" . . cos 9.9598G 
 w = 23 27 32 . . tan. 9.63745 
 
 p=2l 35 10 . . tan. 9.59731 
 
 The northern part of the circle of latitude is to the west of the 
 circle of declination. 
 
 2. When the sun's longitude is 120 18' 55", and the obliquity 
 of the ecliptic 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 thousands 
 in the next eight, and the thousands in the others. The resulting 
 quantities will be the arguments for the first twenty equations of 
 longitude. 
 
 With the same time, take from the same tables the remaining 
 arguments with their variations, entitled Evection, Anomaly, 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 XLVI, 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 Eqs. of Long. ; and write 
 beneath them the constant 55". Find the sum of the whole, and 
 place it in the column of Evection. Then the sum of the quanti- 
 ties in this column will be the corrected argument of Evection. 
 
 With the corrected argument of Evection, take the Evection 
 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. 
 
TO FIND THE MOON'S LONGITUDE, ETC. 287 
 
 With the corrected Anomaly, take the Equation of the Centre 
 from Table LI, and add it to the last sum in the column of Eqs. 
 of Long. Place the resulting sum in the column of Variation. 
 Then the sum of the quantities in this column will be the corrected 
 argument of Variation. 
 
 With the corrected argument of Variation, take the variation 
 from Table LII, and add it to the last sum in the column of Eqs. 
 of Long. ; the result will be the sum of the principal equations 
 of the Orbit Longitude, amounting in all to twenty-four, and the 
 constants subtracted for the other equations. Place this sum in 
 the column of Longitude. Then the sum of the quantities in this 
 column will be the Orbit Longitude of the Moon, reckoned from 
 the mean equinox. 
 
 Add the orbit longitude to the supplement of the node, and the 
 resulting sum will be the argument of Reduction. 
 
 With the argument of Reduction, take the Reduction from 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 
 LI V, and apply it, according to its sign, to the longitude from- the 
 mean equinox. The result will be the Moon's True Longitude 
 from the Apparent Equinox. 
 
 For the Latitude. 
 
 The argument of the Reduction is also the 1st argument of Lat- 
 itude. Place the sum of the first twenty-four equations of Longi- 
 tude, taken to the nearest minute, in the column of Arg. II. Find 
 the sum of the quantities in this column, and it will be the Arg. II 
 of Latitude, corrected. The Moon's true Longitude is the 3d 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 or 
 South) according as the distance is less or greater than 90. 
 
 For the Equatorial Parallax. 
 With the corrected arguments, Evection, Anomaly, and Varia 
 
288 ASTRONOMICAL PROBLEMS. 
 
 tion, take out the corresponding quantities from Tables LXI, 
 LXII, and LXIII. Their sum, increased by 7", will be the Equa- 
 torial Parallax. 
 
 For the Semi-diameter. 
 
 With the Equatorial Parallax as an argument, take out the 
 moon's semi-diameter from Table LXV. 
 
 For the Hourly Motion in Longitude. 
 
 With the arguments 2, 3, 4, 5, and 6 of Longitude, rejecting the 
 two right-hand figures in each, take the corresponding equations 
 of the hourly motion in longitude from 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 
 anomaly take the 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- 
 sulting sum at the top, and the argument of the Reduction at the 
 side, take the corresponding equation from Table LXXIV. Also, 
 with the argument of the Reduction take the corresponding 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 eouations^of the hourly motion in longitude, except 
 the last two, at tne 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 these four equations and the 
 
TO FIND THE MOON's LONGITUDE, ETC. 289 
 
 constant 1". To the resulting sum apply the constant 237".2. 
 The difference will be the Moon's true Hourly Motion in Latitude. 
 *The moon will be tending North or South, according as the sign 
 is positive or negative. 
 
 Note. The errors of the results obtained by the foregoing rules, 
 occasioned by the neglect of the smaller equations, cannot exceed 
 for the longitude 15", for the latitude 8", for the parallax 7", for 
 the hourly motion in longitude 5", and for the hourly motion in 
 latitude 3" ; and they will generally be very much less. When 
 greater accuracy is required, take from Tables XXXV to XXXIX 
 the arguments from 21 to 31, along with those from 1 to 20, and 
 their variations. The sums of the numbers for these different 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 T 
 tween the sum and 90 will be the more exact latitude. 
 
 For the Equatorial Parallax. With the arguments 1, 2, 4, 5, 
 
 6, 8, 9, 12, 13, take the corresponding equations from Table LX. 
 Find the sum of these and the other equations, omitting the 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- 
 responding equations froti. Table LXVII. Find the sum of the 
 
 37 
 
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 precede! 
 or follows the given time, with the arguments of Tables LXX, 
 LXXII, and LXXIV, take the equations from Tables LXXVI 
 and LXX VII. Also, with the arguments of Evection, Anomaly, 
 Variation, and Reduction, take the equations from Table LXX VIII. 
 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 equation 
 and the sum of all the equations of the hourly motion in longitude 
 except the last two, take the equation from Table LXXXV. Find 
 the sum of these two equations. Then, for the hour which follows 
 the given time, add this sum to the Hourly Motion in Latitude 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. mean 
 time at New York. 
 
 Mean time at New York, October, 14 d - 6 h< 54 m - 34 s - 
 Diff. of Long. .... 4 56 4 
 
 Mean time at Greenwich, October, 14 11 50 38 
 
TO FIND THE MOON 8 LONGITUDE, ETC. 
 
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ASTRONOMICAL PROBLEMS 
 
 
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TO FIND THE MOON S LONGITUDE, ETC. 
 
 293 
 
 
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294 
 
 ASTRONOMICAL PROBLEMS. 
 
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TO FIND THE MOON ? S REDUCED PARALLAX, ETC. 295 
 
 Exam. 2. Required the moon's longitude, latitude, equatorial 
 parallax, semi-diameter, and hourly motions in longitude and lati- 
 tude, on the 9th of April, 1838, at 8h. 58m. 19s. P. M. mean time 
 at Washington. 
 
 Ans. Long. 6 s - 19 45' 31".2; lat. 36' 21".9 S. ; equat. par. 
 54' 36".3 ; semi-diameter 14' 52" .7 ; hor. mot. in Ion*. 30' 15".2 ; 
 and hor. mot. in lat. 2' 47".0, tending south.* 
 
 PROBLEM XV. 
 
 The Moon's Equatorial Parallax, and the Latitude of a Place, 
 being given, to find the Reduced Parallax and Latitude. 
 
 With the latitude of the place, take the reductions from Table 
 LXIV, and subtract them from the Parallax and Latitude. 
 
 Exam. 1. Given the equatorial parallax 55' 15", and the lati- 
 tude of New York 40 42' 40" N., to find the reduced parallax and 
 latitude. 
 
 Equatorial parallax, ... 55' 15" 
 Reduction, . . , . 5 
 
 Reduced parallax, . . . 55 10 
 
 Latitude of New York, . 40 42' 40 ;/ N. 
 
 Reduction, . . . . 11 20 
 
 Reduced Lat. of New York, 40 31 20 
 
 2. Given the equatorial parallax 60' 36", and the latitude of 
 Baltimore 39 17' 23" N., to find the reduced parallax and latitude. 
 
 Ans. Reduced par. 60' 32", and reduced lat. 39 6' 9". 
 
 3. Given the equatorial parallax 57' 22", and the latitude of 
 New Orleans 29 57' 45" N., to find the reduced parallax and lat- 
 itude. 
 
 Ans. Reduced par. 57' 19", and reduced lat. 29 47' 50". 
 
 PROBLEM XVI. 
 
 To find the Longitude and Altitude of the Nonagesimal Degree 
 of the Ecliptic, for a given time and place. 
 
 For the given time, reduced to mean time at Greenwich, find the 
 sun's mean longitude and the argument N from Tables XVIII, 
 XIX, XX, and XXI. To the sun's mean longitude, apply accord- 
 ing to its sign the nutation in right ascension, taken from Table 
 
 * The smaller equations were omitted in working this example. 
 
296 ASTRONOMICAL PROBLEMS. 
 
 XXVII with argument N ; and the result will be the right ascen- 
 sion of the mean sun, (see Art. 45,) reckoned from the true equi- 
 nox. 
 
 Reduce the mean time of day at the given place, expressed as 
 tronomically, to degrees, &c., and add it to the right ascension of 
 the mean sun from the true equinox. The sum, rejecting 360, 
 when it exceeds that quantity, will be the right ascension of the 
 midheaven, or the sidereal time in degrees. 
 
 Next, find the reduced latitude of the place by Problem XV ; 
 and when it is north, subtract it from 90 ; but when it is south, 
 add it to 90. The sum or difference will be the reduced distance 
 of the place from the north pole. 
 
 Also, take the obliquity of the ecliptic for the given year from 
 Table XXII.* 
 
 These three quantities having been found, the longitude and alti- 
 tude of the nonagesimal degree may be computed from the follow- 
 ing formulae : 
 
 log. cos J (H - ) log. cos i (H + ") = A . . . (1); 
 log. tang J (H w) + 10 - log. tang | (H + ) = B . . . (2) ; 
 log. tang E = A + log. tang (S 90 ) ( 3 ) J 
 log. tang F = log. tang E + B . . . (4) ; 
 N = E+F + 90 . . . (5); 
 
 log. tang h = log. cos E + log- tang 4- (H -f w) + ar. co. log. 
 
 cos F - 20 ... (6). 
 in which 
 
 H = the reduced distance of the place from the north pole ; 
 
 w = the Obliquity of the Ecliptic ; 
 
 S = the Sidereal Time converted into degrees ; 
 
 N = the required Longitude of the Nonagesimal ; 
 
 h = the required Altitude of the Nonagesimal ; 
 
 E and F are auxiliary angles. 
 
 We first find the logarithmic sums A and B. With these we de- 
 termine the angles E and F by formulae (3) and (4), and with these 
 again N and h by formulae (5) and (6). 
 
 The angles E, F, are to be taken Jess than 180; and less or 
 greater than 90, according as the sign of their tangent proves to be 
 positive or negative. 
 
 Note 1 . In case the given place lies within the arctic circle, we 
 must take, in place of formula (5), the following : 
 N = E - F + 90. 
 
 * If great precision is required, the apparent obliquity is to be used in place of the 
 mean. (See Prob. X.) 
 
TO FIND THE LONG. AND ALT. OF NONAGESIMAL DEGREE. 297 
 
 Note 2. As the obliquity of the ecliptic varies but slowly from 
 year to year, the values which have once been found for the loga- 
 rithms A, B, and log. tang i (H + w) (C), will answer for several 
 years from the date of their determination, unless very great accu- 
 racy is required. 
 
 Note 3. The angle h derived from formula (6), is the dis- 
 tance of the zenith of the given place from the north pole of the 
 ecliptic. This is not always equal to the altitude of the nonagesi- 
 mal. Throughout the southern hemisphere, and frequently in the 
 northern near the equator, it is the supplement of the altitude. In 
 employing this angle in the following Problem, it is, however, for the 
 sake of simplicity, called the altitude of the nonagesimal in all cases. 
 
 Exam. 1 . Required the longitude and altitude of the nonagesi- 
 mal degree of the ecliptic at New York, on the 1 8th of September, 
 1838, at 3h. 52m. 56s. P. M. mean time. 
 
 The sun's mean longitude taken from the tables, for the given 
 time, is 5 s - 27 19' 17", and the argument N is 987. The nutation 
 taken from Table XXVII with argument N is I". Hence, the 
 right ascension of the mean sun, reckoned from the true equinox, 
 is 5 s - 27 19' 16". The given time of day, expressed astronomi- 
 cally, is 3h. 52m. 56sec. ; which in degrees is 58 14' 0". 
 
 The reduced latitude of New York, found by Problem XV, is 
 40 31' 20", and this taken from 90 leaves the polar distance 49 
 28' 40". The obliquity of the ecliptic, derived from Table XXII, 
 is 23 27' 37". 
 
 Given time in degrees, . . . 58 14' 0" 
 R. Asc. of mean sun, . . . 177 19 16 
 
 Sidereal time in degrees (S), . . 235 33 16 
 
 90 
 
 2)145 33 16 
 H . . 49 28' 40" 
 u . . 23 27 37 i (S 90) 72 46 38 
 
 Diff . . 26 1 3 
 Sum . 72 56 17 
 
 Jdiff. . .13 031 . cos. 9.98870 . tan. + 10,19.36366 
 sum. . 36 28 8 cos. 9.90535 . tan. C. 9.86871 
 
 A. 0.08335 B. 9.49495 
 
 i(S - 90) 72 46 38 . tan. 0.50866 
 
 E . 75 38 55 . tan. 0.59201 .' cos. 9.39422 
 
 B. 9.49495 C. 9.86871 
 
 F . 50 41 55 . tan 0.08696 . ar. co. cos. 0.19832 
 90 
 
 alt. non. 16 7' 54" . tan. 9.46126 
 
 long. non. 216 20 50 
 
 alt. non. 32 15 48 
 
 38 
 
298 ASTRONOMICAL PROBLEMS. 
 
 2. Required the longitude and altitude of the nonagesimal de- 
 gree of the ecliptic at New York, on the 10th of May, 1838, at 
 llh. 33m. 56sec. P. M. mean time. 
 
 Ans. Long. 200 12' 23", and alt. 37 0' 34". 
 
 PROBLEM XVII. 
 
 To find the Apparent Longitude and Latitude, as affected by 
 Parallax, and the Augmented Semi-diameter of the Moon ; the 
 Moon's True Longitude, Latitude, Horizontal Semi-diameter, 
 and Equatorial Parallax, and the Longitude and Altitude of 
 the Nonagesimal Degree of the Ecliptic, being given. 
 
 We have for the resolution of this Problem the following for- 
 mulae : 
 
 log. x = log.P-f log. cos^+ar. co.log.cos X 10 ... (1); 
 
 c = log. x + log. tang h 10 ... (2) ; 
 log. u = c + log. sin K 10 ... (3) ; 
 log.w' = c+log. sin(K + w)- 10 . . . (4); 
 log.jp = c+log. sin (K + O- 10 . . . (5); 
 Appar. long. = true long. +p . . . (6) ; 
 
 log. tang X' = log. p + ar. co. log. cos X + ar. co. log. u + log. 
 
 sin(X-#)- 10* ... (7); 
 
 log. v = log. P + log. cos h + log. cos X 7 10 ... (8) ; 
 log. z log. v -f log. tang h + log. tang X 7 -f log. cos 
 
 (K + fc>)-30 . . . (9); 
 if =v z . . . (10); 
 Appar. lat. = true lat. . . . (11); 
 
 log.R' = leg. p + ar. co. log. cos X + ar. co. log. u + log. 
 
 cos X 7 -flog. R 10 ... (12) : 
 in which 
 
 P = the Reduced Parallax of the Moon ; 
 h = the Altitude of the Nonagesimal ; 
 X = the True Latitude of the Moon (minus when south) ; 
 K = the Longitude of the Moon, minus the longitude of the No- 
 nagesimal ; 
 
 p = the required Parallax in Longitude ; 
 X' = the approximate Apparent Latitude of the Moon ; 
 if = the required Parallax in Latitude ; 
 R = the True Semi-diameter of the Moon ; 
 R ; = the Augmented Semi-diameter of the Moon ; 
 x, u, u', v, z, are auxiliary arcs. 
 
 * Formula (7) will be rendered more accurate by adding to it the ar. co. cos 
 * 10, and will generally give the apparent latitude with sufficient accuracy; 
 thus rendering formulae (8), (9), (10), and (11) unnecessary. 
 
TO FIND THE MOON ? S APPARENT LONG. AND LAT. 299 
 
 Formulae (1), (2), (3), (4), and (5), being resolved in succession, 
 we derive the apparent longitude from formula (6) ; then the appa- 
 rent latitude from equations (7), (8), (9), (10), (11); and lastly, 
 the augmented semi-diameter from equation (12.) 
 
 The latitude of the moon must be affected with the negative 
 sign when south ; and the apparent latitude will be south when it 
 comes out negative. In performing the operations, it is to be re- 
 membered that the cosine of a negative arc has the same sign as 
 the cosine of a positive arc of an equal number of degrees ; but 
 that the sine or tangent of a negative arc has the opposite sign from 
 the sine or tangent of an equal positive arc. Attention must also 
 be paid to the signs in the addition and subtraction of arcs. Thus, 
 two arcs affected with essential signs, which are to be added to 
 each other, are to be added arithmetically when they have like 
 signs, but subtracted if they have unlike signs ; and when one arc 
 is to be taken from another, its sign is to be changed, and the two 
 united according to their signs. An arithmetical sum, when taken, 
 will have the same sign as each of the arcs ; and an arithmetical 
 difference the same sign as the greater arc. 
 
 The use of negative arcs may be avoided, though the calculation 
 would be somewhat longer, by using the true polar distance d, and 
 the approximate apparent polar distance d', in place of X and X', 
 substituting sin d for cos X, cos (d }- x) for sin (X oc\ sin d' for 
 cos X', log. co-tang d' for log. tang X' ; and observing that p is 
 to be subtracted from the true longitude in case the longitude of 
 the nonagesimal exceeds the longitude of the moon ; that z, when 
 it comes out negative, is to be added to v, which is always positive 
 to the north of the tropic, otherwise subtracted ; and that the par- 
 allax in latitude is to be applied according to its sign to the true 
 polar distance. 
 
 In seeking for the logarithms of the trigonometrical lines, it will 
 be sufficient to take those answering to the nearest tens of seconds. 
 
 Note 1 . When great accuracy is not desired, u' may be taken 
 for p, from which it can never differ more than a fraction of a 
 second. 
 
 Note 2. In solar eclipses the moon's latitude is very small, and 
 formula (7) may be changed into the following : 
 
 log. X' = log. p +ar. co. log. cos X+ar. co. log. u -Hog. (X a?) 10 
 
 and cos X' omitted in formula (12) without material error. 
 
 Formulae (8), (9), (10), and (11), may also now be dispensed 
 with, unless very great precision is desired, and the value of X' 
 given by the above formula taken for the apparent latitude. 
 
 It is to be observed also, that in eclipses of the sun P is taken 
 equal to the reduced parallax of the moon minus the sun's horizon- 
 tal parallax. By this the parallax of the sun in longitude and lati- 
 tude is referred to the moon, and the relative apparent places of 
 the sun and moon are correctly obtained, without the necessity of 
 
300 ASTRONOMICAL PROBLEMS. 
 
 a separate computation of the sun's parallax in longitude and 
 latitude. 
 
 Exam. 1 . About the time of the middle of the occultation of the 
 star Antares, on the 10th of May, 1838, the moon's longitude, by 
 the Connaissance des Terns, was 247 37' 6".7; latitude 4 14' 
 14".7 S. ; semi-diameter 15' 24".2 ; and equatorial parallax 56' 
 31 ".7 ; and the longitude of the nonage simal at New York was 
 200 12' 23" ; the altitude 37 0' 34" ; required the apparent Ion- 
 gitude and latitude, and the augmented semi-diameter of the moon 
 at New York, at the time in question. 
 
 Equat. par. 56' 31".7 Moon's long. 247 37' 7" 
 
 Reduction 4 .6 Long, nonag. 200 12 23 
 
 P = 56 27 .1 K = 47 24 44 
 
 h = 37 34 
 X = -4 14 14.7 
 
 P 3387".! . . log. 3.52983 
 
 h . . .37 0'34" . . cos. 9.90230 
 
 a. 3.43213 
 X . . 41415 ar. co. cos. 0.00119 
 
 x 45 12 . 2712" . log. 3.43332 
 
 h . . . 37 34 . . . tan. 9.87725 
 
 c. 3.31057 
 K . . . 47 24 44 . . sin. 9.86701 
 
 tt ; . . 25 5 . 1505" . log. 3.17758 
 
 c. 3.31057 
 47 49 49 sin. 9.86991 
 
 u' 25 15 . 1515".2 . log. 3.18048 
 
 c. 3.31057 
 K + u' . . 47 49 59 . . . sin. 9.86993 
 
 p 25 15.3 . 1515".3 . log. 3.18050 
 
 True long. . . 247 37 6.7 
 
 Appar. long. . 248 2 22.0 
 
 p log. 3.18050 
 
 x x . . . 4 59 27 . . . sin. 8.93957- 
 X . . /%> . . . . ar. co. cos. 0.00119 
 tt . . ^. . ar. co. log. 6.82242 
 
 X' ,51 10 tan. 8.94368- 
 
301 
 
 5 1' 10" . . . cos. 9.99833 
 
 a. 3.43213 
 
 V 44 54.4 . 2694".4 . log. 3.43046 
 
 h ......... tan. 9.87725 
 
 X' ........ tan. 8.94368 
 
 K + | . 47 37 22 . . . cos. 9.82867 
 
 z . . . 2 0.2 . 120".2 . log. 2.08006 - 
 v-z . . . 46 54.6 
 
 vz (sign changed) 46 54.6 
 Truelat. . 4 14 14.7 
 
 Appar. lat. ..51 9.3 S. 
 
 p log. 3.18050 
 
 X ar. co. cos. 0.00119 
 
 u ar. co. log. 6.82242 
 
 X 7 cos. 9.99833 
 
 R 15 24.2 . 924".2 . log. 2.96577 
 
 Augm. semi-diam. 15 29.4 . 929".4 . log. 2.96821 
 
 Exam. 2. About the middle of the eclipse of the sun on the 18th 
 of September, 1838, the moon's longitude was 175 29' 19".0, 
 latitude 47' 47".5, equatorial parallax 53' 53".5, and semi-diame- 
 ter 14' 41 ".1 ; and the longitude of the nonagesimal at New York 
 was 216 20' 50", the altitude 32 15' 48": required the apparent 
 longitude and latitude, and the augmented semi-diameter of the 
 moon. 
 
 Equat. paral. 53' 53".5 Moon's long. 175 29' 19'- 
 
 Reduction, 4 .4 . Long, nonag. 216 20 50 
 
 Sun's 
 
 53 49 .1 
 
 paral. 8 .6 
 
 K = -40 51 31 
 h = 32 15 48 
 
 X = 47 47.5 
 
 P = 53 40 .5 
 
 P 
 
 h 
 X 
 
 . 3220' 
 32 15' 48" . 
 47 47.5 . 
 
 '.5 . . log. 3.50792 
 cos. 9.92716 
 ar. co. cos. 0.00004 
 
 X 
 
 h 
 
 45 23.5. . 
 32 15 48 . 
 
 2723".5 . log. 3.43512 
 tan. 9.80023 
 
 K 
 
 . -40 51 31 . 
 
 c. 3.23535 
 
 sin. 9.81570- 
 
 1845 .1126" . log. 3.05 L05- 
 
302 
 
 ASTRONOMICAL PROBLEMS. 
 
 True lon. 
 
 -41 10' 16" . . 
 
 18 52.9 . 1132 ;/ .9 . 
 
 c. 
 
 sin. 
 
 3.23535 
 9.81844- 
 
 log. 3.05379- 
 
 41 10 24 
 
 18 52.9 
 
 175 29 19.0 
 
 175 10 26.1 
 
 c. 
 
 sin. 
 
 3.23535 
 
 9.81844- 
 
 long. 
 
 Appar. long. 
 
 P 
 
 X ...... 
 
 u ...... 
 
 X-a? . . 2 / 24 // .0 . 144".0 
 
 Appar. latitude 2' 24".9 N. 144".9 
 
 P 
 X 
 
 u 
 R 
 
 1132".9 . log. 3.05379 
 
 ar. co. 
 ar. co, 
 
 ar. co, 
 ar. co. 
 
 Augm. semi-diam. 14 46 .7 . 886".7 . 
 
 log. 
 cos. 
 log. 
 log. 
 
 3.05379 
 0.00004 
 6.94895 
 2.15836 
 
 log. 2.16114 
 
 log. 
 cos. 
 log. 
 log. 
 
 3.05379 
 0.00004 
 6.94895 
 2.94502 
 
 log. 2.94780 
 
 PROBLEM XVIII. 
 
 To find the Mean Right Ascension and Declination, or Longitude 
 and Latitude of a Star, for a given time, from the Tables. 
 
 Take the difference between the given year and 1 840. Then 
 seek in Table XV for the fraction of the year answering to the 
 given month and days, and add it to this difference, if the given 
 time is after the beginning of the year 1840; but if it is before, 
 subcract it. Multiply the sum or difference by the annual variation 
 given in the catalogue, (Table XC, or XCII,) and the product will 
 be the variation in the interval between the given time and the 
 epoch of the catalogue. Apply this product to the quantity given 
 in the catalogue, according to its sign, if the given time is after 
 the beginning of the year 1840, but with the opposite sign if it is 
 before, and the result will be the quantity sought. 
 
 Exam. 1. Required the mean right ascension and declination of 
 the star Sirius on the 15th of August, 1842. 
 
 Interval between given time and beginn. of 1840, (t,) 2.619yrs, 
 Annual variation of right ascension ; _ 2.646s 
 
 Variation of right ascension for interval f, . . 6.93s. 
 
TO FIND A STAR'S ABERR. IN RIGHT ASCENSION, ETC. 
 
 A similar operation gives for the variation of declination in the 
 same interval, 11". 65. 
 
 Mean right ascen., beginning of 1840, Table XC, 6 h - 38- 5.76 s - 
 Variation for interval t, . . . . . +6.93 
 
 Mean right ascension required, . . 6 38 12.69 
 
 Mean declination,beginning of 1840, . . 16 30' 4".79S, 
 Variation for interval t, + 1 1 .65 
 
 Mean declination required, . , . . 16 30 16 .44 S. 
 
 2. Required the mean longitude and latitude of Aldebaran on 
 the 20th of October, 1838. 
 
 Interval between given time and begin, of 1840, (t) 1.200yrs. 
 Annual variation of longitude, .... 50".210 
 
 Variation of longitude for interval t, 60". 2 
 
 A similar operation gives for the variation of latitude in the same 
 interval 0".4. 
 
 Mean longitude, beginning of 1840, . 2 s - 7 33' 5".9 
 Variation for interval t, ... 1 .2 
 
 Mean longitude required, . . . 2 7 32 5 .7 
 
 Mean latitude, beginning of 1840, . . 5 28' 38".0 S. 
 
 Variation for interval t, . + .4 
 
 Mean latitude required, . . . . 5 28 38 .4 S. 
 
 3. Required the mean right ascension and declination of Capella 
 on the 9th of February, 1839 ? 
 
 Ans. Mean right ascension 5 h - 4 ra - 48.74"-, and mean declination 
 45 49' 38".53 N. 
 
 4. Required the mean longitude and latitude of Aldebaran on 
 the 1 6th of April, 1845? 
 
 Ans. Mean longitude 2 s - 7 37' 31".4, and mean latitude 5 28' 
 36".2. 
 
 PROBLEM XIX. 
 
 To find the. Aberrations of a Star in Right Ascension and Decli 
 nation, for a given Day. 
 
 This problem may be resolved for any of the stars in the 
 logue of Table XC by means of the following formulae : 
 
304 ASTRONOMICAL PROBLEMS. 
 
 log. (aber. in right ascen.) == M + log. sin (O + 9j 10. 
 log. (aber. in declin.) = N + log. sin (O + 6) 10, 
 
 in which M, N, are constant logarithms, O the longitude of the sun 
 on the given day, and 9, 0, auxiliary angles. M, N, and the an- 
 gles 9, 6 9 are given for each of the stars in the catalogue, in 
 Table XCI. O may be derived from an ephemeris of the sun, 
 or it may be computed from the solar tables by Problem IX. 
 
 Exam. 1. What was the amount of aberration, in right ascen- 
 sion and declination, of a Orionis on the 20th of December, 1 837, 
 the sun's longitude on that day being 8 s - 28 28' ? 
 
 Right Ascension. 
 Table XCI, 9 . 6 s - 3 13' M . . 0.1361 
 
 O . 8 28 28 
 
 O+9.3 1 41 . .sin. 9.9998 
 
 Aberration = 1".37 . . . .log. 0.1359 
 
 Declination. 
 Table XCI, 6 . 8 8 - 28 23' N. . 0.7521 
 
 O 8 28 28 
 
 O-H . 5 26 51 . .sin. 8.7399 
 
 Aberration = 0".31 .... log. 1.4920 
 
 2. Required the aberrations in right ascension and declination 
 of a Andromedae on*the 1st of May, 1838, the sun's longitude be- 
 ing I 8 - 10 38'. 
 
 Ans. Aberr. in right ascension 1".07, and aberr. in declina- 
 tion - 11".69. 
 
 PROBLEM XX. 
 
 To find the Nutations of a Star in Right Ascension and Declines 
 tion, for a given Day. 
 
 This Problem may be solved by means of the formulae, 
 log. (nuta. in right asc.) = M' +log. sin (ft 4- 9') 10 ; 
 log. (nuta. in declin.) = N' -flog, sin (ft +6') 10; 
 
 in which M', N', are constant logarithms, ft the mean longitude of 
 the moon's ascending node, and 9', &', auxiliary angles. M', N', 
 and the angles 9', ', are given for each of the stars in the cata- 
 logue, in Table XCI. The mean longitude of the moon's ascend- 
 ing node is given for every tenth day of the year in the Nautical 
 Almanac, page 266, and may be easily found for any intermediate 
 
TO FIND A STAR'S NOTATION IN RIGHT ASCEN., ETC. 305 
 
 day from the daily motion inserted at the foot of the column of 
 longitudes. It may also be had by finding the supplement of the 
 moon's node, for the given time, from the lunar tables, and sub- 
 tracting it from 12 s - 7'. 
 
 Exam. 1. What was the amount of the nutation, in right ascen- 
 sion and declination, of a Orionis on the 20th of December, 1 837, 
 the mean longitude of the moon's node on that day being 18 54' ? 
 
 Right Ascension. 
 
 TableXCI, 9' . 6 s - 15' M' . . 0.0481 
 ft . 18 54 
 
 +9'. 6 19 9 . . sin. 9.5159 
 
 Nutation = - 0".37 . log. 1.5640- 
 
 Declination. 
 
 Table XCI, *' . 3 s - 2 37' N' . . 0.9657 
 ft . 18 54 
 
 &+*'. S 21 31 . . sin. 9.9686 
 
 Nutation = 8".60 . log. 0.9343 
 
 2. Required the nutations in right ascension and declination of 
 a Andromedae on the 1st of May, 1838. 
 
 Ans. Nutation in right ascension 0".54, and nutation in de- 
 cimation I" A3. 
 
 Note. When the apparent place of a star is desired with great 
 accuracy, the solar nutations must also be estimated and allowed 
 for. These may be determined by repeating the process for find- 
 ing the lunar nutations, only using twice the sun's longitude in 
 place of the longitude of the moon's node, and multiplying the re- 
 sults by the decimal .075. 
 
 The calculation of the solar nutations in Example 1st, is as fol- 
 lows : 
 
 Right Ascension. 
 
 Table XCI, 9' - - 6 s - 15' M' . . 0.0481 
 2O . 5 26 56 
 
 2O + 9'. 11 27 11 f -^ . sin. 8.6914 
 
 /; .05 . log.~2".7395- 
 .075 
 
 Solar Nutat. = - 0".00 
 39 
 
306 ASTRONOMICAL PROBLEMS* 
 
 Declination. 
 
 Table XCI,d' . . 3 s - 2 37' N' . . 0.9657 
 20 5 26 56 
 
 2O+4'. 8 29 33 . . sin. 10.0000 
 
 9".24 . *^ 0.9657 
 .075 
 
 Solar Nutat. = - 0".69 
 
 In Example 2d, we find for the solar nutation in right ascension, 
 0".08, and for the solar nutation in declination, 0".51. 
 
 PROBLEM XXI. 
 
 To find the Apparent Right Ascension and Declination of a Star, 
 on a given Day. 
 
 Find the mean right ascension and decimation for the given day 
 by Problem XVIII ; then compute the aberrations in right ascen- 
 sion and declination by Problem XIX, and the lunar and solar nu- 
 tations in right ascension and declination by Problem XX. Apply 
 the aberrations and nutations according to their signs, to the mean 
 right ascension and declination on the given day, observing that the 
 declination when south is to be marked negative, and the results 
 will be the apparent right ascension and declination sought. 
 
 Exam. 1 . What was the apparent right ascension and declina- 
 tion of a Orionis on the 20th of December, 1837 ? 
 
 h. m. s. ' " 
 
 Table XC, M. right ascen. 5 46 30.71 M. dec. 7 22 17.14N. 
 Variations - 6.59 2.42 
 
 5 46 24.12 7 22 14.72 
 
 Aberr. . [ iv * +1.37 wU4^ +0.31 
 
 Lun. nutat. . 0.37 . . +8.60 
 
 Sol. nutat. 0.00 . - 0.69 
 
 App. right asc. 5 46 25.12 App.dec.7 22 22.94N. 
 
 2. Required the apparent right ascension and declination of 
 a Andromedae on the 1st of May, 1838. 
 
 Ans. Appar. right ascen. Oh. Om. 0.90s., and appar. dec. 
 28 11' 39".92. 
 
TO FIND A STAR'S ABERRATION IN LONGITUDE, ETC. 307 
 
 PROBLEM XXII. 
 
 To find the Aberrations of a Star in Longitude and Latitude, for 
 a given Day, 
 
 The formulae for the computation are, 
 
 log. (aber. in long.) = 1.30880 + log. cos (6s. + O L) + ar. 
 
 co. log. cos X 10 ; 
 
 log. (aber. in lat.) = 1.30880 -f log. sin (6s. + O - L) + log. 
 
 sin X 20 ; 
 
 in which O = longitude of the sun on the given day ; L = mean 
 longitude of the star ; and X = mean latitude of the star. 
 
 Exam. 1 . Required the aberrations in longitude and latitude of 
 Antares on the 26th of February, 1838, the sun's longitude on that 
 day being 11 s - 7 29'. 
 
 By Prob. XVIII, L = 8 s - 7 30', and X = 4 32' S. 
 6s. + O . 17 7 29 Const, log. 1.3088 
 
 6s. + O - L 8 29 59 . ' -.' cos. 6.4637 
 X . . 4 32 . ar. co. cos. 0.0014 
 
 Aberr. in long. = -0".00 . log. 1.7739 
 
 Const, log. 1.3088 
 
 6s. + O - L 8* 29 59' . . sin. 10.0000 
 X ... 4 32 . . sin. 8.8978 
 
 Aberr. in lat. = - 1 ;/ .61 . log, 0.2066 - 
 
 2. Required the aberrations in longitude and latitude of Arc- 
 turus on the 5th of October, 1838, the sun's longitude being 
 6"- 11 47'. 
 
 Ans. Aberr. in long. 23".34, and aberr. in lat. 1".85. 
 Note. The nutation in longitude of a fixed star may be found 
 after the same manner as the nutation in longitude of the sun. 
 See Problem IX.) 
 
 PROBLEM XXIII. 
 
 To find the Apparent Longitude and Latitude of a Star, for a 
 
 given Day. 
 
 Find the mean longitude and latitude on the given day by Prob- 
 lem XVIIL Find also the aberrations in longitude and latitude by 
 Problem XXII, and the nutation in longitude, as in Problem lA, 
 Apply the aberration and nutation in longitude, according to their 
 
308 ASTRONOMICAL PROBLEMS. 
 
 signs, to the mean longitude, and the result will be the apparent 
 longitude ; and apply the aberration in latitude according to its 
 sign, to the mean latitude, and the result will be the apparent 
 latitude. 
 
 Exam. 1. Required the apparent longitude and latitude of An. 
 tares on the 26th of February, 1838. 
 
 Table XC, M. long. 8 a ' 7 31' 45".2 M. lat. 4 32' 51".6 S. 
 Var. 1 32 .57 .78 
 
 8 7 30 12 .63 . .4 32 50 .82 
 Aberr. . .00 . 1 .61 
 
 Nutat. - 4 .40 
 
 App.long. 8 7 30 8 .23 App. lat. 4 32 49 .21 S. 
 
 2. Required the apparent longitude and latitude of Arcturus on 
 the 5th of October, 1838. 
 
 Ans. Appar.long. 6 s - 21 58' 37" .4, andappar. lat. 30 51' 19". 1. 
 
 PROBLEM XXIV. 
 
 To compute the Longitude and Latitude of a Heavenly Body from 
 its Right Ascension and Declination, the Obliquity of the Eclip- 
 tic being given. 
 
 This Problem may be solved by means of the following for- 
 mulae : 
 
 log. tang x log. tang D + ar. co. log. sin R ; 
 
 log. tang L=log. cos (x w) + log. tang R + ar. co. log. cos x 10; 
 
 log. tang X = log. tang (x w) + log. sin L 10 ; 
 
 in which 
 
 R = the Right Ascension ; 
 
 D = the Declination (minus when South) ; 
 
 L = the Longitude ; 
 
 X = the Latitude ; 
 
 eo = the Obliquity of the ecliptic ; 
 
 x is an auxiliary arc. It must be taken according to the sign of 
 its tangent, but always less than 180. The longitude will always 
 be in the same quadrant as the right ascension. The latitude must 
 be taken less than 90, and will be north or south, according as the 
 sign is positive or negative. 
 
 Note. When the mean longitude and latitude are to be derived 
 from the mean right ascension and decimation, the mean obliquity 
 of the ecliptic is taken. When the apparent longitude and latitude 
 are to be derived from the apparent right ascension and declina- 
 tion, found as in Problem XAl, the apparent obliquity is taken. 
 
TO COMPUTE THE RIGHT ASCEN. AND DEC. OF A BODY. 309 
 
 The mean obliquity of the ecliptic at any assumed time is easily 
 deduced from Table XXII. The apparent obliquity is found by 
 Problem X. 
 
 Exam. 1. On the 20th of June, 1838, the right ascension of 
 Capella was 76 11' 29", the declination 45 49' 35" N., and the 
 obliquity of the ecliptic 23 27' 37" ; required the longitude and 
 latitude. 
 
 D = 45 49' 35" . , . tan. 0.0125295 
 R = 76 11 29 ar. co. sin. 0.0127367 
 
 x = 46 39 56 . . . tan. 0.0252662 
 
 w = 23 27 37 
 
 x u = 23 12 19 . . . cos. 9.9633623 
 R= 76 11 29 . . . tan. 0.6094483 
 x = 46 39 56 ar. co. cos. 0.1635240 
 
 Long. = 79 36 4 . . tan. 0.7363346 
 
 L = 79 36 4 . . . sin. 9.9928075 
 x u = 23 12 19 . tan. 9.6321632 
 
 Lat. = 22 51 49 . . . tan. 9.6249707 
 
 2. Given the right ascension of Spica 199 IT 35", and decli- 
 nation 10 19' 24" S., and the obliquity of the ecliptic 23 27' 36", 
 on the 1st of January, 1840, to find the longitude and latitude. 
 Ans. Long. 201 36' 32", and lat. 2 2' 30" S. 
 
 PROBLEM XXV. 
 
 To compute the Right Ascension and Declination of a Heavenly 
 Body from its Longitude and Latitude, the Obliquity of the 
 Ecliptic being given. 
 
 The formulae for the solution of this problem are, 
 log. tang y = log. tang X -f ar. co. log. sin L ; 
 log. tang R =log.cos(y-r-w) + log. tang L + ar. co. log. cos y 10; 
 log. tang D = log. tang (y + w) + log. sin R 10 ; 
 in which 
 
 L = the Longitude ; 
 
 X = the Latitude (minus when South) ; 
 
 R = the Right Ascension ; 
 
 D the Declination ; 
 
 w = the Obliquity of the ecliptic ; 
 
 y is an auxiliary arc. It must be taken according to the sign of 
 its tangent, but always less than 180. The right ascension will 
 
310 ASTRONOMICAL PROBLEMS. 
 
 always be in the same quadrant with the longitude. The declina- 
 tion must be taken less than 90, and will be north or south, ac- 
 cording as the sign is positive or negative. 
 
 Note. The mean or apparent obliquity of the ecliptic is taken, 
 according as the given and required elements are mean or apparent. 
 
 Exam. 1. On the 1st of January, 1830, the longitude of Sirius 
 was 3 s - 11 44' 18", the latitude 39 34' 1" S., and the obliquity 
 of the ecliptic 23 27' 41 " : required the right ascension and de- 
 clination. 
 
 X = 39 34' 1" . . tan. 9.9171381 - 
 L = 101 44 18 ar. co. sin. 0.0091788 
 
 y = 139 50 14 . . tan. 9.9263169 
 w = 23 27 41 
 
 w = 163 17 55 . . cos. 9.9812819 
 L = 101 44 18 . . tan. 0.6823798- 
 y = 139 50 14 . ar. co. cos. 0.1 167843 
 
 Right ascen = 99 24 48 tan. 0.7804460- 
 
 R = 992448 . . sin. 9.9941121 
 -hw = 163 17 55 . . tan. 9.477 1803- 
 
 Dec.= 16 29 20 S. . ';. tan. 9. 47 12924 
 
 2. Given the longitude of Aldebaran 67 33' 5", and latitude 
 5 28' 38" S., and the obliquity of the ecliptic 23 27' 36", on the 
 1st of January, 1840, to find the right ascension and declination. 
 
 Ans. Right ascension 66 41' 4", and declination 16 10'57"N. 
 
 PROBLEM XXVI. 
 
 The Longitude and Declination of a Body being given, and also 
 the Obliquity of the Ecliptic, to find the Angle of Position. 
 
 The formula is 
 
 log. ship = log. sin w + log. cos L + ar. co. log. cos D 10 : 
 
 p = Angle of Position (required) ; ^ 
 
 L = Longitude ; 
 
 D = Declination ; 
 
 w = Obliquity of the ecliptic. 
 
 The angle of position p must be taken less than 90. It is to be 
 observed also that when the longitude is less than 90, or more 
 than 270, the northern part of the circle of latitude lies to the west 
 of the circle of declination, but that when the longitude is between 
 90 and 270, it lies to the east. 
 
 Note. The angle of position may also be computed from the 
 
TO FIND THE TIME OF NEW OR FULL MOON 311 
 
 right ascension and latitude, by means of a formula similar to thaf 
 just given, namely, 
 
 log. ship = log. sin w + log. cos R + ar. co. log. cos X 10; 
 Exam. 1. Given the longitude of Regulus 147 27' 54", and 
 declination 12 47' 45" N., and the obliquity of the ecliptic 23 
 27' 41", to find the angle of position. 
 
 u = 23 27' 41" . IjSfc sin. 9.6000260 
 L=147 27 54 V y,* cos.9.9258601 
 D= 12 4745 ar. co.cos.0.0109217 
 
 Angle of pos. = 20 7 58 [? ; ; <Ji sin. 9.5368078 
 
 The circle of latitude lies to the east of the circle of declination. 
 2. Given the longitude of Fomalhaut 331 27' 56", and declina- 
 tion 30 31' 14" S., and the obliquity of the ecliptic 23 27' 41", 
 to find the angle of position. Ans. 23 57' 20". 
 
 The circle of latitude lies to the west of the circle of declination. 
 
 PROBLEM XXVII. 
 
 To find from the Tables the Time of New or Full Moon, for a 
 given Year and Month. 
 
 For New Moon. 
 
 Take from Table LXXXVI, the time of mean new moon in 
 January, and the Arguments I, II, III, and IV, for the given year. 
 Take Irom Table LXXXVII, as many lunations with the corre- 
 sponding variations of Arguments I, II, III, and IV, as the given 
 month is months past January, and add these quantities to the for- 
 mer, rejecting the ten thousands from the sums in the columns of 
 the first two arguments, and the hundreds from the sums in the 
 columns of the other two. Seek the number of days from the first 
 of January to the first of the given month, in the second or third 
 column of Table LXXXVIII, according as the given year is a 
 common or bissextile year, and subtract it from the sum in the col- 
 umn of mean new moon : the remainder will be tabular time of 
 mean new moon for the given month. It will sometimes happen 
 that the number of days taken from Table LXXXVIII, will ex- 
 ceed the number of days of the sum in the column of mean new 
 moon : in this case one lunation more, with the corresponding ar- 
 guments, must be added. 
 
 With the sums in the columns I, II, III, and IV, as arguments, 
 take the corresponding equations from Table LXXXIX, and add 
 them to the time of mean new moon : the sum will be the Approxi- 
 mate time of new moon for the given month, expressed in mean 
 time at Greenwich. 
 
 Next, for the approximate time of new moon calculate the true 
 longitudes and hourly motions in longitude of the sun and moon ; 
 
312 
 
 ASTRONOMICAL PROBLEMS. 
 
 subtract the less longitude from the greater, and the hourly mo- 
 tion of the sun from the hourly motion of the moon ; and say, as 
 the difference between the hourly motions : the difference between 
 the longitudes : : 60 minutes : the correction of the approximate 
 time. The correction added to the approximate time, when the 
 sun's longitude is greater than the moon's, but subtracted, when 
 it is less, will give the true time of new moon required, in mean 
 time at Greenwich. This time may be reduced to the meridian 
 of any given place by Problem V. 
 
 For Full Moon. 
 
 Take from Table LXXXVI, the time of mean new moon, and 
 the corresponding Arguments I, II, III, and IV, for January of the 
 given year, and from Table LXXXVII, a half lunation with the 
 corresponding changes of the arguments. Then, when the time 
 of mean new moon for January is on or after the 16th, subtract the 
 latter quantities from the former, increasing, when necessary to 
 render the subtraction possible, either or both of the first two argu 
 ments by 10,000, and of the last two by 100 ; but add them when 
 the time is before the 16th. The result will be the tabular time 
 of mean full moon and the corresponding arguments, for January. 
 Proceed to find the approximate time of full moon after the same 
 manner as directed for the new moon. 
 
 For the approximate time of full moon calculate the true longi- 
 tudes and hourly motions in longitude of the sun and moon. Sub- 
 tract the sun's longitude from the moon's, adding 360 to the latter 
 if necessary. Take the difference between the remainder and VI 
 signs, and call the result R. Also subtract the hourly motion of 
 the sun from the hourly motion of the moon. Then say, as the 
 difference between the hourly motions : R : : 60m. : the correction 
 of the approximate time. The correction added to the approxi- 
 mate time of full moon, when the excess of the moon's longitude 
 over the sun's is less than VI signs, but subtracted when it is 
 greater, will give the true time of full moon. 
 
 Exam. 1. Required the time of new moon in September, 1838, 
 expressed in mean time at New York. 
 
 1838, 
 81un. 
 
 M. New Moon. 
 
 I. 
 
 n. 
 
 m. 
 
 IV. 
 
 d. h. m. 
 
 24 16 53 
 236 5 52 
 
 0681 
 6468 
 
 9175 
 5737 
 
 99 
 22 
 
 85 
 93 
 
 Dap, 
 
 260 22 45 
 243 
 
 7149 4912 21 
 Approximate time. 
 
 78 
 
 Sept'r, 
 
 II. 
 III. 
 IV. 
 
 17 22 45 
 16 
 9 35 
 3 
 10 
 
 Sept'r. 
 
 18 8 49 
 
TO FIND THE TIME OF NEW OR FULL MOON. 
 
 313 
 
 Moon's true long, found for approx. time, is 5 s - 25 29' 19" 
 Sun's do. do. do. 5 25 27 27 
 
 Difference, . y^ ~ : ' '' '&.' 
 
 5 25 27 27 
 1 52 
 
 Moon's hourly motion in long, is 29 28 
 
 Sun's do. do. /.; '.,-*' . 2 27 
 
 Difference, ;''* :. ." , . ' . '*./"' 27 1 
 
 As 27' I" : I' 52" : : 60 m - : 4 m - 9 8 -, the correction. 
 
 Approx. time of new moon, September, . 18 d - 8 h> 49 m - 0" 
 Correction, . " "'*f* . *V 1j . . 4 9 
 
 True time, in mean time at Greenwich, . 18 8 44 51 
 Diff. of meridians, . . .,.-*<. 4 56 4 
 
 True time, in mean time at New York, . 18 3 48 47 
 
 Exam. 2. Required the time of full moon in April, 1838, ex 
 pressed in mean time at New York. 
 
 1838, 
 
 ilun. 
 
 M. Full Moon. 
 
 I. 
 
 II. 
 
 III. 
 
 rv. 
 
 d. h. m. 
 24 16 53 
 14 18 22 
 
 0681 
 404 
 
 9175 
 5359 
 
 99 
 
 58 
 
 85 
 50 
 
 3 lun. 
 
 9 22 31 
 
 88 14 12 
 
 0277 
 2425 
 
 3816 
 2151 
 
 41 
 46 
 
 35 
 
 97 
 
 Days, 
 
 98 12 43 
 90 
 
 2702 5967 87 
 Approximate time. 
 
 32 
 
 April, 
 
 II. 
 III. 
 IV. 
 
 8 12 43 
 8 29 
 16 7 
 15 
 30 
 
 April, 
 
 9 14 4 
 
 Moon's true long, found for approx. time, is 6 s - 19 44' 17 ;/ 
 
 Sun's 
 
 do. 
 
 do. 
 
 do. 
 
 19 45 22 
 
 29 58 55 
 000 
 
 R. . 1 5 
 
 Moon's hourly motion in long, is . 30 15 
 
 Sun's do. do. . 2 27 
 
 Difference . 27 48 
 
 As 27' 48" : 1' 5" : : 60 m - : 2 m - 20 8 -, the correction. 
 
 40 
 
314 ASTRONOMICAL PROBLEMS. 
 
 Approximate time of full moon, April, '* ''?!** 9 d - 14 ht 4 m> s - 
 Correction, % . *P . . . + 2 20 
 
 True time, in mean time at Greenwich, . 9 14 6 20 
 Diff. of meridians, ..... 4 56 4 
 
 True time, in mean time at New York, . 9 91016 
 
 3. Required the time of new moon in September, 1837, ex- 
 pressed in mean time at Philadelphia ; taking the longitudes for 
 the approximate time from the Nautical Almanac. 
 
 Ans. 29d. 3h. Om. 5s. 
 
 4. Required the time of full moon, in October, 1837, expressed 
 in mean time at Boston. Ans. 13d. 6h. 30m. 25s. 
 
 PROBLEM 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. 
 
 For the Eclipses of the Sun. 
 
 Take, for the given year, from Table LXXXVI the time of 
 mean new moon in January, the arguments and the number N. 
 If the number N differs less than 37 from either 0, 500, or 1000, 
 an eclipse must occur at that new moon. If the difference is be- 
 tween 37 and 53, there may be an eclipse, but it is doubtful, and 
 the doubt can only be removed by a calculation of the true places 
 of the moon and sun. If the difference exceeds 53, an eclipse is 
 impossible. 
 
 If an eclipse may or must occur at the new moon in January, 
 calculate the approximate time of new moon by Problem XXVII, 
 and it will be the time nearly of the middle of the eclipse, express- 
 ed in mean time at Greenwich. This may be reduced to the 
 meridian of any other place by Problem V. 
 
 To find the first new moon after January, at which an eclipse 
 of the sun maybe expected, seek in column N of Table LXXXVII 
 the. first number after that answering to the half lunation, that, 
 added to the number N for the given year, will make the sum come 
 within 53 of 0, 500, or 1000. Take the corresponding lunations, 
 changes of the arguments, and the number N, and add them, re- 
 spectively, to the mean new moon in January, the arguments, and 
 the number N, for the given year. Take from the second or third 
 column of Table LXXXVIII, according as the given year is a 
 common or bissextile year, the number of days next less than the 
 days of the sum in the column of mean new moon, and subtract it 
 from this sum ; the remainder will be the tabular time of mean 
 new moon in the month corresponding to the days taken from Ta- 
 
TO FIND THE NUMBER OF ECLIPSES IN A YEAR. 
 
 315 
 
 ble LXXXVIII. At this new moon there may be an eclipse of 
 the sun ; and if the sum in the column N is within 37 of the num- 
 bers mentioned above, there must be one. Find the approximate 
 time of new moon, and it will be the time nearly of the middle of 
 the eclipse. 
 
 If any of the other numbers in the last column of Table 
 LXXXvII are found, when added to the number N of the given 
 year, to give a sum that falls within the limit 53, proceed in a simi- 
 lar manner to find the approximate times of the eclipses. 
 
 Note. When the sum of the numbers N, or the number N itself, 
 in case the eclipse happens in January, is a little above 0, or a 
 little less than 500, the moon will be to the north of the sun, and 
 there is a probability that the eclipse will be visible at any given 
 place in north latitude at which the approximate time of the eclipse, 
 found as just explained and reduced to the meridian of the place, 
 comes during the day-time. When the number N found for the 
 eclipse is more than 500, the moon will be to the south of the sun, 
 and the eclipse will seldom be visible in the northern hemisphere, 
 except near the equator. 
 
 For the Eclipses of the Moon. 
 
 Find the time of full moon and the corresponding arguments and 
 number N, for January of the given year, as explained in Problem 
 XXVII. Then proceed to find the times at which eclipses of the 
 moon may or must occur, after the same manner as for eclipses of 
 the sun, only making use of the limits 35 and 25, instead of 53 
 and 37.* 
 
 Note. An eclipse of the moon will be visible at a given place, 
 if the time of the eclipse thus found nearly, and reduced to the 
 meridian of the place, comes in the night. 
 
 Exam. 1 . Required the eclipses that may be expected in the 
 year 1840, and the times nearly at which they will take place. 
 
 For the Eclipses of the Sun. 
 
 1840, 
 21un. 
 
 M. New Moon. 
 
 I. 
 
 II. 
 
 III. 
 
 rv. 
 
 N. 
 
 d. h. m. 
 3 10 30 
 59 1 28 
 
 0085 
 1617 
 
 6386 
 1434 
 
 65 
 31 
 
 63 
 
 98 
 
 844 
 170 
 
 
 62 11 58 
 60 
 
 1702 7820 96 61 014 
 
 As the sum of the numbers N 
 comes within 37 of 0, there must be 
 an eclipse. 
 
 
 Mean time at Greenwich. 
 
 March, 
 I. 
 II. 
 III. 
 IV. 
 
 2 11 58 
 8 3 
 19 38 
 12 
 13 
 
 March, 
 
 3 16 4 
 
 * The numbers 53, 37, and 35, 25, are the lunar and solar ecliptic limits, as 
 determined by Delambre. The limits given in the text, converted into thousandth 
 parts of the circle, are 55, 37, and 37, 21. 
 
816 
 
 ASTRONOMICAL PROBLEMS. 
 
 1840, 
 8 Ion. 
 
 M. New Moon. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 N. 
 
 d. h. m. 
 3 10 30 
 236 5 52 
 
 0085 
 6468 
 
 6386 
 5737 
 
 65 
 22 
 
 63 
 93 
 
 844 
 682 
 
 
 239 16 22 
 213 
 
 6553 2123 87 56 526 
 
 As the sum of the numbers N 
 comes within 37 of 500, there must 
 be an eclipse. 
 
 Mean time at Greenwich 
 
 August, 
 
 II*. 
 III. 
 IV. 
 
 26 16 22 
 54 
 49 
 15 
 16 
 
 August, 
 
 26 18 36 
 
 For the Eclipses of the Moon. 
 
 1840, 
 
 ilun. 
 
 M. Full Moon. 
 
 I. 
 
 II. 
 
 m. 
 
 IV. 
 
 N. 
 
 3 10 30 
 14 18 22 
 
 0085 
 404 
 
 6386 
 5359 
 
 65 
 
 58 
 
 63 
 
 50 
 
 844 
 43 
 
 llun. 
 
 18 4 52 
 29 12 44 
 
 489 
 808 
 
 1745 
 717 
 
 23 
 15 
 
 13 
 99 
 
 887 
 85 
 
 
 47 17 36 
 31 
 
 1297 2462 38 12 972 
 
 As the sum of the numbers N, al- 
 though it comes within 35 of 1000, 
 does not come within 25, the eclipse 
 may be considered doubtful. 
 
 Mean time at Greenwich. 
 
 Febr. 
 I. 
 II. 
 III. 
 IV. 
 
 16 17 36 
 7 27 
 23 
 5 
 27 
 
 Febr. 
 
 17 1 58 
 
 1840, 
 7 lun. 
 
 M. Full Moon. 
 
 I. 
 
 H. 
 
 III. 
 
 IV. 
 
 N. 
 
 d. h. m. 
 18 4 52 
 206 17 8 
 
 489 
 5659 
 
 1745 
 
 5020 
 
 23 
 
 7 
 
 13 
 
 94 
 
 887 
 596 
 
 
 224 22 
 213 
 
 6148 6765 30 07 483 
 
 As the sum of the numbers N 
 comes within 25 of 500, there must 
 be an eclipse. 
 
 Mean time at Greenwich. 
 
 August, 
 
 IL 
 III. 
 IV. 
 
 11 22 
 1 37 
 19 16 
 3 
 25 
 
 August, 
 
 12 19 21 
 
 2. Required the eclipses that may be expected in the year 1839, 
 and the times nearly at which they will take place, expressed in 
 mean civil time at New York. 
 
TO CALCULATE A LUNAR ECLIPSE. 317 
 
 Ans. One of the sun on the 15th of March, at 9h. 20m. A. M. ; 
 and one of the sun on the 7th of September, at 5h. 24m. P. M. 
 
 3. Required the eclipses that may be expected in the year 1841, 
 and the times nearly at which they will take place, expressed in 
 mean civil time at New York. 
 
 Ans. Four of the sun, namely, one on the 22d of January, at 
 12h. 18m. P. M. ; one on the 21st of February, at6h. 17m. A.M. ; 
 one on the 18th of July, at 9h. 24m. A. M. ; and one on the 16th 
 of August, at 4h. 28m. P.M.: and two of the moon, namely, one 
 on the 5th of February, at 9h. 10m. P. M. ; and one on the 2d of 
 August, at 5h. 5m. A. M. 
 
 The eclipses of the sun in January and August may be con- 
 sidered as doubtful. 
 
 PROBLEM XXIX. 
 
 To calculate an Eclipse of the Moon. 
 
 The calculation of the circumstances of a lunar eclipse is effect- 
 ed with the following fundamental data, derived from the tables of 
 the sun and moon : 
 
 Approximate Time of Full Moon (at Greenwich), T 
 
 Sun's Longitude at that time, L 
 
 Do. Hourly Motion, s 
 
 Do. Semi-diameter, <5 
 
 Do. Parallax, p 
 
 Moon's Longitude, I 
 
 Do. Latitude, X 
 
 Do. Equatorial Parallax, P 
 
 Do. Semi-diameter, d 
 
 Do. Hourly Motion in longitude, m 
 
 Do. Hourly Motion in latitude, n 
 
 We obtain the time T by Problem XXVII ; the quantities ap- 
 pertaining to the sun, namely, L, s, and 5, by Problem IX ;* and 
 those which have relation to the moon, namely, /, X, P, d, m, and 
 n, by Problem XIV. 
 
 From these quantities we derive the following : 
 
 True Time of Full Moon, (at given place,) . T' 
 
 Moon's Latitude at that time, . . . X' 
 
 Semi-diameter of earth's shadow, . . S 
 
 Inclination of Moon's relative orbit, . . ; r I 
 
 T being known, T' is found as explained in Problem XXVII. 
 To obtain X', we state the following proportion, 
 
 1 hour : correction for the time of full moon :::*; 
 * p may be taken = 9". 
 
318 ASTRONOMICAL PROBLEMS. 
 
 from this we deduce the value of x ; and thence find X by the 
 equation 
 
 V = X x. 
 
 When the true time of full moon, expressed in mean time at 
 Greenwich, is later than the approximate time, the upper sign is 
 to be used, if the latitude is increasing, the lower if it is decreas- 
 ing ; but when the true time is earlier than the approximate time, 
 the lower sign is to be used if the latitude is increasing ; the upper 
 if it is decreasing. 
 
 The value of S is derived from the equation 
 
 and the angle I from the formula 
 
 log. tang I = log. n + ar. co. log. (m s). 
 The foregoing quantities having all been determined, the various 
 circumstances of the eclipse may be calculated by the following 
 formulae : 
 
 For the Time of the Middle of the Eclipse. 
 
 3.55630 + log. cos I + ar. co. log. (m s) 20 = R ; 
 
 log. t = R -f log. X' + log. sin I 10 ; 
 
 M = T' t : 
 
 t = interval between time of middle of eclipse and time of full 
 moon ; M = time of middle of the eclipse. 
 
 The upper sign is to be taken in the last equation when the lati- 
 tude is decreasing; the lower, when it is increasing. 
 
 For the Times of Beginning and End. 
 
 log. c = log X' + log. cos I 10 ; 
 log >0 ^log.(S + *+c) + log.(S+d--c) { R . 
 
 B = M v, and E = M + v : 
 
 v half duration of the eclipse ; B = time of beginning ; and E = 
 time of end. 
 
 Note. If c is equal to or greater than S + d, there cannot be an 
 eclipse. 
 
 For the Times of Beginning and End of the Total Eclipse. 
 
 R . 
 
 B' = M v', and E' = M + v 1 : 
 
 v' = half duration of the total eclipse ; B' = time of beginning of 
 total eclipse ; and E' = time of end of total eclipse. 
 
 Note. When c is greater than S d , the eclipse cannot be total. 
 
 For the Quantity of the Eclipse. 
 
 log. Q = 0.77815 + log. (S + d - c) + ar. co. log. d - 10 ; 
 Q = the quantity of the eclipse in digits. 
 
TO CALCULATE A LUNAR ECLIPSE. 319 
 
 Note 1 . An eclipse of the moon begins on the eastern limb, and 
 ends on the western. In partial eclipses the southern part of the 
 moon is eclipsed when the latitude is north, and the northern part 
 when the latitude is south. 
 
 Note 2. When the eclipse commences before sunset, and ends 
 after sunset, the moon will rise more or less eclipsed." To obtain 
 the quantity of the eclipse at the time of the moon's rising, find 
 the moon's hourly motion on the relative orbit by the equation 
 
 log. h log. (m s) + ar. co. log. cos I ; 
 
 in which h = hourly motion on relative orbit. Also find the inter- 
 val between the time of sunset and the time of the middle of the 
 eclipse, which call i. Then, 
 
 1 hour : i : : h : x. 
 
 Deduce the value of x from this proportion, and substitute it in 
 the equation 
 
 in which c designates the same quantity as in previous formulae. 
 Find the value of c', and use it in place of c in the above formula 
 for the quantity of the eclipse, and it will give the quantity of the 
 eclipse at the time of the- moon's rising. When the eclipse begins 
 before and ends after sunrise, the quantity of the eclipse at the 
 time of the moon's setting may be found in the same manner, only 
 using sunrise instead of sunset. 
 
 Example. Required to calculate, for the meridian of New York, 
 the eclipse of the moon in October, 1837. 
 
 Elements. 
 
 Approximate time of full moon, 
 Sun's longitude at that time, . 
 
 Do. hourly motion 
 
 Do. semi-diameter 
 
 Do. parallax, 
 Moon's longitude, 
 
 Do. latitude, 
 
 Do. equatorial parallax, 
 
 Do. semi-diameter, 
 
 Do. hourly motion in long. 
 
 T =ll h - 10 m -(0ct. 13) 
 
 L == 6 s - 20 24' 28 /; 
 s = 2 29 
 
 * = 16 4 
 
 = 9 
 
 = 20 21 51 
 X = 11 28 S. 
 
 P = 59 32 
 
 d = 16 13 
 
 m = 35 54 
 
 Do. hourly motion in lat. (tending north), n = 3 19 
 
 Approx. time of full moon, October, . 13 d< ll h - 10 m - 00"- 
 Correction found by Prob. XXVII, . +4 42 
 
 True time, in mean time at Greenwich, . 13 11 14 42 
 Diff. of meridians, 4 56 4 
 
 True time, in mean time at New York, T' = 13 6 18 38 
 
320 
 
 ASTRONOMICAL PROBLEMS. 
 
 60 m - : 4 1 
 Moon's lat. at approx. time, 
 Correction, . 4 *& 
 
 Moon's lat. at true time, 
 
 Moon's equatorial parallax, 
 Sun's do 
 
 Sum, 
 
 Sun's semi-diameter, . 
 
 Diff. . 
 
 Add 
 
 Semi-diameter of earth's shadow, . . . 
 
 Moon's hor. mot. less sun's (m s) = 2005" . ar. co 
 Moon's hor. motion in latitude, n = 199 . . 
 
 Inclination of rel. orbit, I = 5 40' . . . 
 Time of Middle. 
 
 = 11' 28" S. 
 = 16 
 
 59 41 
 
 5=164 
 
 P+p 
 
 S = 44 21 
 
 log. 6.69789 
 log. 2.29885 
 
 tan. 8.99674 
 
 3.5563C 
 
 5 40' . . cos. 9.99787 
 2005" ar. co. log. 6.69789 
 
 V . 
 I 
 
 t 
 
 T' . 
 
 Middle, 
 
 X' 
 I 
 
 S-fd 
 
 672" 
 
 5 40' 
 
 O h - l ra - 58 8 - = 
 6 18 38P.M. 
 
 R. 0.25206 
 log. 2.82737 
 sin. 8.99450 
 
 log. 2.07393 
 
 . 6.20 36P.M. 
 
 Times of Beginning and End. 
 
 11' 9" = 669" 
 
 4303" 
 2965 
 
 l h -46 m 22'- = 6382* 
 
 log. 2.82737 
 cos. 9.99787 
 
 log. 2.82524 
 
 log. 3.63377 
 log. 3.47202 
 
 2 ) 7.10579 
 
 3.55289 
 R. 0.25206 
 
 log. 3.80495 
 
Middle, 
 
 Beginning, . 
 Ei 
 
 TO CALCULATE A SOLAR ECLIPSE 321 
 
 l h - 46 m - 22 s - = 6382 s - . log. 3.80495 
 6 20 36 
 
 4 34 14 P. M. 
 8 6 58 P. M. 
 
 S -d-c 
 
 Middle, 
 
 2357" 
 1019 
 
 O h. 46 m. 9.= 2769 s - 
 6 20 36 
 
 Beg. of total eclipse, 5 34 27 P. M. 
 End of total eclipse, 7 6 45 P. M 
 
 S+d-c 
 d 
 
 Quantity, 
 
 log. 3.37236 
 log. 3.00817 
 
 2 ) 6.38053 
 
 3.19026 
 R 0.25206 
 
 log. 3.44232 
 
 0.77815 
 
 log. 3.47202 
 
 973" . ar. co. log. 7.01189 
 
 18.3 digits, 
 
 log. 1.26206 
 
 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-diameter ; 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 
 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 6 : 
 41 
 
ASTRONOMICAL PROBLEMS. 
 
 in which 
 
 A = appar. distance of centres ; 
 X' = appar. Lat. of Moon ; 
 
 a = Dift*. of appar. Long, of Moon and Sun = diff. of appar 
 long, of Moon (found as above) and true long, of Sua 
 
 6 is an auxiliary arc. The value of 6 being derived from th 
 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 sum of their ap- 
 parent semi-diameters, there is an eclipse at this time. But if it 
 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 + lh., 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 = V a* 4- A'2. 
 
 If the logarithmic formulae are used, it will be sufficient here to take out the angle 
 to the nearest minute. When we have occasion to obtain the distance of the 
 centres exact to within a small fraction of a second, must be taken to the nearest 
 tens of seconds, if it exceeds 20 or 30. 
 
TO CALCULATE A SOLAR ECLIPSE. 323 
 
 hour following T. Find in like manner the variation of the dis 
 tance during the hour preceding T. Then, if the apparent distance 
 of the centres at the times (T lh.), (T + lh.) is less than the 
 sum of the apparent semi-diameters, deduce 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 ; their 
 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 -f- 2h.) or 
 (T 2h.), find by the same method the distance of the centres at 
 the time' (T + 3h.) or (T - 3h.) These calculations being effect- 
 ed, the times of the beginning, greatest obscuration, and end of the 
 eclipse, will fall between some of the instants T,(T lh.),(T + lh.), 
 &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 + lh.), (T + 2h.), &c. 
 Find the apparent distance of the centres of the sun and moon for 
 the times (T + lh.), (T + 2h.), by the same method as that by 
 which it is found for the times (T + lh.), (T lh.), 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 + lh.), (T + 2h.) ; as 
 in the preceding case the distance and latitude for the time 
 (T+2h.) were deduced from the same at the times (T lh.), T, 
 (T-j-lh.) With the results obtained compute the apparent dis- 
 tance of the centres of the two bodies at the time (T -f 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 - lh.), (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 m - - t (nearly.) 
 
324 \STRONOMICAL PROBLEMS. 
 
 Find the value of t given by this proportion, and add it to the 
 time at the first instant, and the result will be a first approximation 
 to the time of the beginning of the eclipse, which call b. Find, 
 by interpolation,* the distance of the moon from the sun in appa- 
 rent longitude (a\ and the moon's apparent latitude (V), 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-diameters, 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 in the interval B b, 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 pfepara- 
 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 6. 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 b and the nearer of the two instants, (in 
 
 fc' 
 
 minutes.; Then, if we put /= , c = , and v = var. of a in interval z, 
 
 o ob 
 
 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 1 , from 10m. to 10m. to increase or decrease by equal degrees.) 
 
 The general formula for interpolation is Q = q -f- - d' -\ -- - ^ d" + 
 
 - ^^ "73 - d'" -f- &c., in which q is the first of a series of values, found at 
 
 equal intervals, of the quantity whose value Q at the time t is sought, t is reck- 
 oned from the time for which q is found, h is one of the equal intervals, d', d" f 
 d'", &c., are the first, second, third, &c., differences. If we make h = 1, we have 
 
 . ,+. *+ 
 
 t In effecting the reductions of the quantities a and V to the first approximate 
 time of end, k 1 must stand for the variation of a during the hour preceding that 
 comprised between the last two instants, and the last instant must be substituted 
 for the first. (See Note above.) 
 
TO CALCULATE A SOLAR ECLIPSE. 
 
 325 
 
 The middle of the interval between the approximate times of 
 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 observation, 
 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. 
 123,) to represent the ecliptic, assume on it some point C for the 
 
 Fig. 123. 
 
 position of the -sun at the instant of apparent conjunction, and lay 
 off CA, CA', equal to the two differences of apparent longitude ; 
 and to the right or left, according as the moon is to the west or 
 east of the sun at the instants for which the calculations have been 
 made. Erect the perpendiculars Ap, A'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 /, f, for the beginning and end of the eclipse, and by 
 means of a square mark on EF the points 6, e, which answer to 
 the beginning and end. If the eclipse be total or annular, mark 
 the points of immersion and emersion, g, g' y with an opening in 
 the dividers equal to the difference of the semi-diameters, and find 
 the corresponding points ', 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 points 1, 2, &c., answering to lh., 2h., &c. before 
 
326 ASTRONOMICAL PROBLEMS. 
 
 and after the times for which the calculations are made. If the 
 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^, the 
 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. 24, p. 47, for which we have in the triangle PSZ 
 the side PS = co-declination, the side PZ co-latitude, and the 
 included angle ZPS. (The requisite formulae 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 
 the 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 Nautical Almanac. 
 
TO CALCULATE A SCLAR ECLIPSE. 327 
 
 and augmented semi-diameter, for these several times, (making use 
 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 321 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 /c. 
 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 applied to the ap- 
 proximate time of beginning by means of the following formulae : 
 log. b = log. a 4" log. k + ar. co. log. n 10 ; 
 
 c =\' -b,S = d + 5 5"; 
 log t = log. (S + A) + log. (S - A) + ar. co. log. n + ar. 
 
 co. log. c 4- log. L -f 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. motion in lat. in same interval ; 
 
 X'= 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 
 
328 ASTRONOMICAL PROBLEMS. 
 
 and third equations, and substitute them in the fourth, and it will 
 make known the value of t, which is to be applied 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 
 the eclipse. Let some instant be fixed upon, some five or ten 
 minutes before or after the approximate time of the beginning of 
 the 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 determined. 
 
 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' ft, the following : 
 
 * It will be somewhat more accurate to use in place of k and n, as above de- 
 
 . ' _ k k k' _ k 
 
 fined, the values of the following expressions : -^ -- 2 or 3 , 
 
 ^L -- 2* H ~ n or -^ -- 3* " ~ H . The first of each of these pairs of expressions 
 6 3o 6 36 
 
 is to be used in case the true time of beginning is after the approximate time ; 
 the second in the other case, k' 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, k and n will answer to the last half 
 of L, and k 1 and n' to the first half. 
 
TO CALCULATE A SOLAR ECLIPSE. 329 
 
 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 eclipse, and call it k. Take the difference of 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 1 = the distance of the 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 : : X' : 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- 
 parejit 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 for 
 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 321, the apparent distance of the centres of 
 the sun and moon at the time of greatest obscuration. Subtract 
 this distance from the sum of the apparent semi-diameters of the 
 
 42 
 
330 ASTRONOMICAL PROBLEMS. 
 
 two bodies, diminished by 5", and denote the remainder by R 
 Then, 
 
 Sun's semi-diam. (diminished by 3") : R : : 6 digits : number of 
 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 eclipse in question, and will 
 not differ, at most, more than five or eight minutes (according 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 6 log. V +ar. co. log. a, log. k'=\og. k -f- ar. co. log. sin ; 
 S=$-d- 1", orS=d <* + !"; 
 
 loo - - . (S-A) 
 
 log. c - 2 > 
 
 log. t = ar. co. log. k' + log. c + log. D + 1 .77815 10 ; 
 
 Time of Begin. = M t, Time of End = M + 1 : 
 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- 
 
 tive mot. in appar. long, during this interval ; 
 k' = Moon's appar. mot. on relative orbit for same interval ; 
 & = Inclination of relative orbit ; 
 8 = 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 first 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. 331 
 
 For the Point of the Surfs 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 of the 
 eclipse, as explained at page 326. Let the former be denoted by 
 p, and the latter by v. Give to each the negative sign, if laid off 
 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 the 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. X' + ar. co. log. a ; 
 
 taking it always less than 90, and positive or negative according 
 to the sign of its tangent. X 7 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 
 
 p, t;, 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 Art. 490, page 183.) 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- 
 lae 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 
 
332 
 
 ASTRONOMICAL PROBLEMS. 
 
 latitude and meridian of New York, that will occur on the 18th o 1 
 September, 1838. 
 
 For the Approximate Times of the Phases. 
 
 Approximate time of New Moon. 
 
 Sept. 18 d - 8 h - 49 m - 
 
 
 175 27' SI" A 
 
 Do. hourly motion, 
 
 2 26 .7 
 
 Do. semi-diameter, 
 
 15 57 .0 
 
 Moon's longitude, 
 
 . 175 29 19 
 
 Do. latitude, .... 
 
 47 47 
 
 Do. equatorial parallax, 
 
 53 53 
 
 Do. semi-diameter, 
 
 14 41 
 
 Do. hor. mot. in long. 
 
 29 29 
 
 Do. hor. mot. in lat. . 
 
 2 41 
 
 Do. appar. long. (Prob. XVII), . 
 
 . 175 10 26 
 
 Do. appar. lat. (X 7 ), 
 
 2 25 N. 
 
 Do. augm. semi-diameter, . 
 
 14 47 
 
 Diff. of appar. long, (a), 
 
 17 5 
 
 Appar. dist. of cen. (A), 
 
 17 15 
 
 Sum of semi-diameters, 
 
 30 44 
 
 7 h. 4 g m . 
 
 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, 
 
 9 h. 49 t 
 
 Sun's longitude, . 
 Moon's appar. long. . 
 Do. appar. lat. (X'), 
 Do. augm. semi-diameter, 
 Oiif. of appar. long. (), 
 A.ppar. dist. of cen. (A), 
 Sum of semi-diameters, 
 
 175 25' 
 174 47 
 
 8 
 
 4" 
 3 
 
 12 N. 
 
 14 49 
 38 1 
 38 53 
 30 46 
 
 175 
 175 
 
 29' 58" 
 36 15 
 
 2 18 S. 
 14 44 
 
 6 17 
 
 6 42 
 30 41 
 
 7 h. ^ *. 
 
 8 4v 
 9 49 
 1049 
 
 a 
 
 diff. or k. 
 
 X' 
 
 diff. or . 
 
 A 
 
 diff. 
 
 sum semi-d. 
 
 2281" 
 1025 
 377 
 1925 
 
 1256". 
 1402 
 1548 
 
 492" N 
 145 N 
 138 S 
 357 S 
 
 347" 
 283 
 219 
 
 2333" 
 10(35 
 402 
 1958 
 
 1298" 
 1556 
 
 1846" 
 1844 
 1841 
 1839 
 
TO CALCULATE A SOLAR ECLIPSE. 333 
 
 For the Approximate Time of Beginning. 
 
 h = 1298", d = 2333" 1846" = 487" ; 
 
 1298" : 487" : : 60 m - : * = 22 m -.5 
 
 7h. 49111. 
 
 22 
 
 1st Approxi. 8 h - ll m - 
 
 7 h. 49. > a = 2281" . X'=492"N. 
 
 Corrections for 22 m - 447 . 133 (See Note, p> 324) 
 
 8 h - ll m - . a = 1834 . X'=359 N. 
 
 a = 1834" ar. co. log. 6.73660 . . log. 3.26340 
 V = 359 . log. 2.55509 
 
 6 = 11 4' 30" . tan. 9.29169 ar. co. cos. 0.00817 
 
 Appar. dist. of cen. A = 1869" . . log. 3.27157 
 Sum of semi-diam. 1846 
 
 487" : 23" : : 22 m - : t l m - 2 s - 
 8 h. llm . 
 
 + 1 
 
 2d Approxi. 8 h - 12 m - 
 For the Approximate Time of the End. 
 
 h = 1556", d = 1958" - 1839" = 119". 
 
 1556" : 119" : : 60 m - : t = 4 m -.6. 
 10 h. 
 
 -5 
 
 1st Approxi. 10 h - 44 m - 
 
 10 h - 49 m - . a = 1925" . . . V = 357 /; S. 
 Corrections for 5 m> 132 17 
 
 10 h. 44m . a = 1793 ^ ^ . V = 340 S. 
 
 a = 1793" . ar. co. log. 6.74642 . log. 3.25358 
 X'= 340 . . log. 2.53148 
 
 d = ... tan. 9.27790 . ar. co. cos. 0.00767 
 
 Appar. dist. of cen. A = 1825" . 3.26125 
 1839 
 
 133": 14" : : 5 m - : * = O m -.5 
 
334 
 
 ASTRONOMICAL PROBLEMS. 
 10 h. 44 m. 
 
 .5 
 
 2dApproxi. 10 h - 44 m -.5 
 
 For the Approximate Time of Greatest Obscuration. 
 
 Approx. time of begin. . 8 h * 12 m - 
 Approx. time of end, . 10 44 
 
 2 ) 18 56 
 
 IstApproxi. . 9 28 
 For the True Times of the Phases. 
 
 Approx. time of 
 
 Approx. time of 
 Greatest Obscur. 
 
 Approx. time of 
 End. 
 
 pprox 
 Beginning. 
 
 gh. i2 m - 9 h - 28 m< 10 h> 44 m - 
 
 Sun's longitude, 175 26' 1".0 175 29' 6". 8 17532' 12".6 
 
 Do. semi-diam., 15 57 .0 15 57 .0 15 57 .0 
 
 Moon's app.lon. 174 55 36 .7 175 27 7 .7 176 2 17 .2 
 
 Do. app. lat. 5 45 .3N. 43 .58. 5 32 .4 S 
 
 Do.augm.semid. 14 48 .0 14 45 .1 14 41 .7 
 
 1856".7 1840".0 
 1835 .0 1833 .7 
 
 For the True Time of Beginning. 
 
 gh. 12 m. 
 
 9 28 
 10 44 
 
 a 
 
 k 
 
 V I 
 
 n 
 
 1824".3 
 119 .1 
 1804 .6 
 
 1705".2 
 1923 .7 
 
 345".3 N! r 
 43 .58,* 
 332 .4 S j 
 
 88".8 
 88 .9 
 
 a 
 A 
 n 
 
 b = 
 V 
 
 A 
 
 . 1824".3 . 
 . 1705 .2 ... 
 
 . 388 .8 ... 
 
 - 8001 .1 ... 
 . 345 .3 
 
 . log. 3.26109 
 . log. 3.23178 
 ar. co. log. 7.41028 
 
 . log. 3.90315 
 
 ; = 8346 .4 ... 
 . 3696 .7 ... 
 
 ar. co. log. 6.07850 
 . loff. 3.56781 
 
 V 
 8 
 S - A . -16 .7 . 
 
 n . . 
 
 L 76m. 
 
 Corr. of approx. time, + 43 s - .4 
 
 . log. 1.22272- 
 ar. co. log. 7.41028 
 
 . log. 1.88081 
 Const, log. 1.47712 
 
 . log. 1.63724 -f 
 
TO CALCULATE A SOLAR ECLIPSE. 
 
 Corr. of approx. time, + 43 s - .4 
 
 Approx. time, . 8 h - 12 m - .0 
 
 True time of begin. 8 12 43 .4, in Greenwich time. 
 Diff. 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 .... log. 3.25638 
 
 k . . 1923 .7 .... log. 3.28414 
 
 n . . 288.9 . . ar. co. log. 7.53925 
 
 b = - 12016 .3 . . . . log. 4.07977 
 
 X' 332 .4 
 
 X' +b=c= -12348 .7 . . ar. co. log. 5.90838 
 S-fA . . 3668.7 . . .- . log. 3.56451 
 S-A . , 1.3 . . . . log. 0.11394 
 
 n ar. co. log. 7.53925 
 
 L . . . 76m log. 1.88081 
 
 Const, log. 1.47712 
 
 Corr. of approx. time, 3 9 - . log. 0.48401 
 
 Approx. time, . 10 b - 44 m - .0 
 
 True time of end, . 10 43 57 .0, in Greenwich time. 
 Diff. of merid. 4 56 4 
 
 True time of end, . 5 47 53, in New York time. 
 For the True Time of Greatest Obscuration. 
 
 True time of beginning, . . 8 h - 12 m -43 8 -.4 
 
 Do. of end, . . . . 10 43 57 .0 
 
 2) 18 56 40 .4 
 
 2d Approx. 9 28 20 .2 
 
 9 h - 49 m - . . X' = 138" S. 
 9 28 X' = 43 .5 S. 
 
 Diff. 21 Diff. 94 .5 
 
 21 m - : 20 s - : : 94".5 : 1".5 
 43 .5 
 
 . .' X'=45. 
 
336 ASTRONOMICAL PROBLEMS. 
 
 1705".2 388".8 
 
 1923 .7 288 .9 
 
 k = 3628 .9 : n = 677 .7 : : X' = 45".0 : a 1 = 8".4 
 
 Time of beginn. 8 h - 12 m - 43 s - .4, at 9 h - 28 m - a = 119". I 
 Time of end, 10 43 57 .0 a'= 8 .4 
 
 D= 2 31 13 .6 m = - 110 .7 
 
 3628".9 : 110".7 : : 2 h - 31 ra - 13 s - .6 : 4 m - 36 8 - .8 
 
 9 h -28 .0 
 
 True time (nearly) 9 32 36 .8 
 
 21 m -: 4 m - 37 s - : : 94".5 : 20".8 
 43 .5 
 
 At 9 h - 32 m - 37 s -, V = 64 .3 
 
 3628".9 : 677".7 : : 64" .4 : 12" .0 ; at 9 h< 32 ra - 37 s -, a = 8" .4 
 
 a' =12 .0 
 
 m= 3 .6 
 
 3628".9 : 3".6 : : 2 h - 31 m - 13 s -.6 : 9 s - .0 
 9 h - 32 m - 36 .8 
 
 9 32 27 .8 
 
 True time of greatest obscur. . 9 h - 32 m - 27 s -. 8, in Greenw. time, 
 Diflf. of mend. 4 56 4 
 
 True time of greatest obscur. . 4 36 23 .8, in N. Y. time. 
 For the Quantity of the Eclipse. 
 
 9 h - 32 m - 37 s - . X' = 64".3 
 21 m - : 9 s - : : 94".5 : .6 
 
 At nearest approach of centres, . X' = 63 .7 
 " " " . . a = 12 .0 
 
 a . 12".0 . ar. co. log. 8.92082, . . log. 1.07918 
 V , 63 .7 . . 1.80414 
 
 tan. 0.72496, . ar. co. cos. 0.73253 
 
 Shortest distance of centres, 64".8 . . log. 1.81171 
 Sum of semi-diameters, 1837 .0 
 
 1772 .2 
 15' 54" : 1772".2 : : 6 : 11.14 digits eclipsed. 
 
TO CALCULATE A SOLAR ECLIPSE. 337 
 
 For the Situation of the Point at which the Obscuration com- 
 mences. 
 8 h - 12 m - . . a =1824", . . V = 345".3N. 
 
 76 m. . 438. . . 17Q5 // . 16 76 m. . 433. . 
 
 Atthebeginn. . a =1808, . 
 a . 1808 . ar. co. log. 6.74280 
 V . 341.6 . . log. 2.53352 
 
 & = 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 pos. 23 23' 50". 
 Meantime of begin. 3 h - 16 m - 39 s -, Lat. 40 42' 40", Dec. 1 49' 0" 
 Equa. of time, 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" . cos. 9.80210 
 Co. lat. PZ = 49 17 20 tan. 0.06526 
 
 m = 3623' 0" . . tan. 9.86736 
 Co. dec. PS =88 11 
 
 w'=51 48 . ar. co. sin. 0.10466 
 m = 36 23 . . sin. 9.77320 
 
 P= 50 39 15 . tan. 0.08627 
 
 S= 42 38 10 . . tan. 9.96413 
 
 Angle of position, . . 23 23' 50" 
 
 Angle from eclip. (&), . . 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 h> 32 m - 27 S \8 =true time of greatest obscur. 
 
 At this time, a = 12".2,\' =63".7. 
 
 a = 12".2 . ar. co. log. 8.91364 . . log. 1.08636 
 V=63 .7 . . log. 1.80414 
 
 6 = 79 9' 30" . tan. 0.71778 . ar. co. cos. 0.72564 
 
 A = 64".9 . , log. 1.81200 
 43 
 
ASTRONOMICAL PROBLEMS. 
 
 = 135".8 .log. 2. 13290,4 =79 9 ; 30" . ar. co. sin. 0.00783 
 * log. 3.55977 
 
 S - A= 6 .2 . log. 0.79239, &=3628".9 
 2 ) 2.92529, k 1 
 
 1.46264 
 D=152 m - 
 
 Time of greatest obscur. . 4 36 23 .8 
 Formation of ring, . 
 Rupture of do. 
 
 ar. co. log. 6.43240 
 
 . . 1.46264 
 . .log. 2.18184 
 Const, log. 1.77815 
 
 . log. 1.85503 
 
 . 4 35 12 .2, New York time. 
 
 a 
 
 4 37 35 .4 
 
 PROBLEM XXXI. 
 
 To find the Mootfs 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 the 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, 1 2 h * : excess of given 
 time above time of second longitude : : second first difference : 
 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 for 
 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. 339 
 
 For the Latitude. 
 
 Prefix to north latitudes the positive sign, but to south latitudes 
 the 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 is 
 north or south. 
 
 For the Hourly Motion in Longitude. 
 
 Solve the proportion, 1 2 h - : 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 for 
 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. 
 
840 ASTRONOMICAL PROBLEMS. 
 
 With the given time, and half sum of fourth differences, take 
 from Table XCV the correction for fourth differences, giving it 
 always the same sign as the half sum. 
 
 The sum of the third longitude or latitude, the proportional part 
 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, will be 
 the longitude or latitude required. 
 
APPENDIX. 
 
 TRIGONOMETRICAL FORMULAE.* 
 
 I. RELATIVE TO A SINGLE ARC OR ANGLE <z. 
 
 1. sin 2 a + cos 2 a = I 
 
 2. sin a = tan a cos a 
 
 tana 
 
 3. sm a = ; 
 
 cos a 
 
 I + tan 2 a 
 
 sin a 
 
 5. tan a = 
 
 cos a 
 
 I cos a 
 
 6. cot a = = 
 
 tan a sm a 
 
 7. sin a = 2 sin | a cos | 
 
 8. cos a = 1 2 sin 2 J a 
 
 9. cos a = 2 cos 2 a 1 
 
 10. tani 
 
 11. cot 
 
 . . 
 
 12. tan 2 i a = ; 
 
 1 + 
 
 COS 
 
 sin a 
 
 1 cos a 
 1 cos a 
 
 cos a 
 
 13. sin 2 a = 2 sin a cos a 
 
 14. cos 2 a = 2 cos 2 a 1 = 1 2 sin* a 
 
 II. RELATIVE TO Two ARCS a AND 6, op WHICH a is SUPPOSED 
 
 TO BE THE GREATER. 
 
 15. sin (a + b) sin a cos 6 + sin b cos a 
 
 16. sin (a b) sin a cos b sin 6 cos a 
 
 17. cos (a + b) = cos a cos b sin a sin b 
 
 9 The radius is supposed to be equal to unity in all of the formulae. 
 
APPENDIX. 
 
 16. cos (a b) = cos a cos b + sin a sin 
 tan a -\- tan 6 
 
 1Q f-,^ l n _L M 
 
 6 
 
 (-*) 
 ( + *) 
 
 (a -6) 
 (a -6) 
 
 (a + 6) 
 
 ly. Idll IW T t/J , 7 
 
 1 tan a tan 6 
 , x tan a tan 6 
 
 OA tan 1/7 n 1 
 
 <6U. Idll It* Wl 
 
 l+tanatano 
 21 . sin a -h sin 6 = 2 sin | (a + b) cos 
 22. sin a sin 6 = 2 sin | (a &) cos J 
 23. cos a -|-cos& = 2 cos (a + &) cos 
 24. cos b cosa = 2 sin \ (a + b) sin | 
 , sin (a + b) 
 
 cos a cos b 
 , sin (a b) 
 
 9fi tin rr tin n 
 
 cos a cos b 
 "7 rota ^rot/>- Sin(fl + 6) 
 
 sm a sm o 
 
 sin a sin 6 
 on sin a + sin 6 tan % (a + 6) 
 
 sin a sin 6 tan | (a 6) 
 cos b + cos a cot |(a -f 6) 
 
 cos b cos a tan (a 6) 
 tan a -f tan 6 cot b + cot a sin 
 
 tan a tan b cotb cot a sin (a 6) 
 oo cot b tan^z cot a tan b cos (a + 6) 
 
 cot b -|- tan a cot a + tan b cos 
 33. sin 2 a sin 2 6 = sin (a + 6) sin (a - 
 34. cos 2 a sin 2 b = cos (a + 6) cos (a 
 35. 1 sin a = 2 sin 2 (45 i a) 
 
 (-*) 
 
 -6) 
 
 -6) 
 
 1 =F sin a 
 
 37. L.?L? = tan (45 i a) 
 cosa 
 
 00 1 sin a sin 2 (45 i a) 
 
 38. - . a - 
 
 1 cos a sin 2 a 
 a 1 + sin b _ sin 2 (45 +46) 
 
 1 + cosa cos 2 ^ fl 
 
 1 tan b _ . . . 
 
TRIGONOMETRICAL FORMULAE. 343 
 
 42. sin a cos b = sin (a -\-b) + I sin (a b) 
 
 43. cos a sin b = 1 sin (a + b) \ sin (a b) 
 
 44. sin a sin b I cos (a b) J cos (a + b) 
 
 45. cos a cosb cos (a + b) + cos (a b) 
 
 III. TRIGONOMETRICAL SERIES. 
 
 46. ^ 
 
 f . a 3 
 
 + 
 + 
 + 
 
 
 5 
 
 A 
 
 a 6 
 
 - 4- Ar 
 
 "2.3 
 2 
 
 2. 
 
 3.4.5 
 a 4 
 
 
 _3 
 
 2.3.4 
 20 5 
 
 2. 
 
 4_ A 
 
 3.4. 
 7a 7 
 
 5.6 f 
 
 + &C. 
 
 &c. 
 
 3 
 
 1 a 
 
 C0i a ~~ a 3 " 
 
 
 
 3.5 
 
 1 3 2 . 
 
 5.7 
 
 3 2 . 
 
 5 
 
 3 3 .5. 
 
 7 
 
 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 
 nave the formula 
 
 47. a = (a") sin 1" ; log. sin I" =^"6.685574867. 
 
 IV. DIFFERENCES OF TRIGONOMETRICAL LINES. 
 
 48. A sin x + 2 sin \ A x. cos (x + | A x) 
 
 49. A cos x = 2 sin A x. sin (x + A x) 
 
 sin A x 
 
 50. A tan x = 
 51. 
 
 cos x. cos (x + A x) 
 
 sin A x 
 sin x. sin f (x + A x) 
 
 V. RESOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES.* 
 
 Table of Solutions. 
 
 Given. , Required. Solution. 
 
 Hypothen. ( side op. giv. ang. 52 sin x = sin h . sin a 
 and < side adj. giv. ang. 53 tan x tan h . cos a 
 an angle (. the other angle 54 cot x = cos h . tan a 
 
 cos h 
 
 Hypothen 
 
 and 
 a side 
 
 the other side 55 cos x = 
 
 cos s 
 
 ang. adj. giv. side 56 cos x = tan s . cot h 
 
 ., . si 
 
 ang. op. giv. side 57 sin x = 
 
 sin s 
 
 sn 
 
 * Baily's Astronomical Tables and Formal. 
 
344 APPENDIX. 
 
 , the hypothen. 58 sin x 
 A side and I sin a 
 
 the angle J the other side 59 sin x = tan s . cot a ^ | 
 
 opposite __ . cos a 
 
 the other angle 60 sin x = 
 
 I cos s ) | 
 
 A side and f the hypothen. 61 cot x = cos a . cot s 
 the angle < the other side 62 tana: tan a . sin s 
 adjacent [the other angle 63 cos x sin a . cos s 
 
 {the hypothen. 64 cos a? = rectang. cos. of the 
 giv. sides 
 an angle 65 cot x = sin adj. side x cot. 
 
 op. side 
 
 {the hypothen. 66 cos a? = rectang. cot. of the 
 given angles 
 cos. opp. ang. 
 a side 67 cos x = -. *- 
 
 sin. adj. ang. 
 
 In these formulae, x denotes the quantity sought. 
 a = the given angle 
 * = the given side 
 h = the hypothenuse. 
 
 The formulae for the resolution of right-angled spherical trian- 
 gles are all embraced in two rules discovered by Lord Napier, and 
 called Napier 1 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, and 
 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. 345 
 
 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 : 
 
 cos(B + C) 
 68. 
 
 cos (B C) 
 
 sin (a c) 
 
 69. tan 2 IB = . ; , : 
 
 sin (a -f c) 
 
 70. tan 2 ic = tan (a + b) tan 1 (a b) 
 
 71. tan (45 16) = ^ tan (45- #), tan a? = sin a sin B 
 
 72. tan 2 16 -tan -^+45 tan 
 
 a is the hypothenuse, B, C, the acute angles, and b, c, the sides 
 opposite the acute angles. 
 
 VI. RESOLUTION OF OBLIQUE-ANGLED SPHERICAL TRIANGLES. 
 General Formula. 
 
 Let A, B, C, denote the three angles of a spherical triangle, and 
 a, b y c, the sides which are opposite to them respectively, 
 sin A _ sin B _ 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 b + sin a sin b cos C 
 
 75. cos c = cos (a b) 2 sin a sin b sin 2 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 b 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 b cos c cos A 
 
 Case i. Given the three sides, a, b y c. 
 To find one of the angles. 
 
 2 i A sin (It 6) sin (k c) 
 
 80. sin iA = : : : 
 
 sin b sin c 
 
 or 
 
 81 . COS 2 I A = : ji : 
 
 sm b sin c 
 
 82 . A = 1|. ^Ill^pl^lj*^ 
 
 Case ii. Given the three angles. A, B, C 
 
 To find one of the sides. 
 
 cos K cos (K A) 
 
 83. sin a tf = . p .' ' 
 
 sm B sin C 
 
 44 
 
346 
 
 APPENDIX. 
 
 or, 
 
 84. 
 
 85 . K= 
 
 sm B sm C 
 
 Case in. Given two sides a and &, and the included angle C. 
 1. To find the two other angles A and B. 
 
 Napier's Analogies. 
 
 87. 
 
 2. To find the third side c. 
 
 88. 
 
 or, 
 
 tan c= tan (a + b). 
 
 cosi(A+B) 
 
 cos*(A~B) 
 or equa. 73. 
 
 Case iv. Given ft#o angles A awe? B, and the adjacent side c. 
 1. To find the other two sides, a and 6. 
 
 
 sin i (A 
 
 Napier's Analogies. 
 
 90. tan (a b) =tai. 2 ^. -. , , . . ^v , 
 
 sm i (A+ B) J 
 
 2. To find the third angle C. 
 
 91. 
 
 * sin (a b) 
 
 or 
 
 cos $(a+b) 
 ' cos 2 (ab) 
 or equa. 73. 
 
 Case y. Given two sides a, b, and an opposite angle A, 
 
 To find the other opposite angle B ; take equation 73, or the 
 proportion ; sines of the angles are as sines of the opposite sides. 
 (For the methods of determining the remaining angle and side, see 
 page 348, Case 3.) 
 
 Case vi. Given two angles A, B, and an opposite side a. 
 To find the other opposite side b sines of the angle are proper- 
 
RESOLUTION OF SPHERICAL TRIANGLES. 347 
 
 tional to the sines of the opposite sides. (For the methods of de- 
 termining the remaining side and angle, see page 348, Case 4.) 
 
 OTHER METHODS OF RESOLVING OBLIQUE-ANGLED SPHERICAL 
 TRIANGLES.* 
 
 Except when three sides or three angles are given, the data 
 always include an angle A, and the adjacent side &, besides a third 
 nart. The required parts in the different cases may be found by 
 *e following formulae, and formula 73. 
 
 cot n = tan A cos b 
 
 98 
 
 tan B 
 
 From the angle C (Fig. 124) a perpendicular CD is let fall upon 
 the opposite side c, which divides the 
 triangle into two right-angled trian- 
 gles, that are resolved separately. In 
 the one, ACD, A and b are known, 
 and it is easy to find the other parts, 
 which, joined to the third given part, 
 serve to resolve the second right-an- 
 gled triangle BCD, and determine the 
 unknown quantity required, m, m' A 
 denote the two segments of the base ; n, n' the two parts of the 
 angle C ; and k the perpendicular arc CD. 
 
 It must be observed, that if the perpendicular CD fell without 
 the triangle, m and m r , n and n 1 would have contrary signs ; this 
 happens when the angles A and B at the base are of different kinds, 
 (the one Z_, the other >90). When it is not known whether this 
 circumstance has place or not, the problem is susceptible of two 
 solutions. 
 
 The detail of the different cases is as follows : the data are A, 
 6, and another arc or angle. 
 
 Case 1. Given two sides and the included angle ; or 6, 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 
 345,) 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 345) gives c, which is known in kind. 
 
 FranecBur's Practical Astronomy. 
 
3 8 APPENDIX. 
 
 Case 3. Given two sides and an opposite angle; crb, o, A. 
 
 Equation 92 gives m, 96 m', 94 c, 98 and 73 B and C ; 
 
 or else, 93 gives w, 99 ft', 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 -f or ; there are then two values, for c, and also for C. m' 
 and n' enter into equations 97 and 98 by their sines, whence result 
 therefore also two values of B. 
 
 Case 4. Given two angles, and an opposite side; or A, B, b. 
 
 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 1 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 A sin b 
 
 102. tan \ a = tan b cos B 
 
 103. cos b = cot B cot A 
 
 104. cos $ A = cos a sin B 
 
 The knowledge of two of the four elements A, B, #, b, which 
 form the isosceles triangle, is sufficient for the determination of the 
 two others. 
 
 INVESTIGATION OF ASTRONOMICAL FORMULAE. 
 
 Formula for the Parallax in Right Ascension and Declination, 
 and in Longitude and Latitude. (See Article 120, page 55.) 
 
 Let s (Fig. 125) 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 
 in altitude, which takes effect in the vertical circle zs' ; p is the 
 
PARALLAX IN RIGHT ASCENSION AND DECLINATION. 349 
 
 pole ; the hour angle zps = q is changed into zps', and sps 1 = 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 8 of these arcs is the parallax in declination or of polar dis- 
 tance.* We have, (For. 73, p. 345,) 
 
 sin s' : ships (d) : : sin sps' (a) : sin ss f (p), 
 
 sin zps' (q -|-a) : sin zs' (Z) : : sin s' : sinpz (90 I). 
 
 Multiplying, term by term, we obtain 
 
 sin s r sin (<? + ): sin d sin Z : : sin a sin s' : sin p cos / ; 
 
 sin p cosl . . 
 
 whence, sin a = . * . ^ sin (q + a) . 
 
 sin c? sin Z 
 
 Or, substituting forp its value given by equa. (8,) p. 51, and 
 replacing H by P, 
 
 sin P cos / . , x ,.. 
 
 sin a = : - sin (q + a) . . . (A). 
 
 sin d 
 
 This equation makes known a when the apparent hour angle 
 zps q + a , seen from the earth's surface, is given ; but if we 
 know the true hour angle zps q, seen from the centre of the 
 earth, developing sin (q + a )> (For. 15, p. 341), and putting 
 sin P cos I _ 
 
 sin d 
 
 sin a = ?7i (sin q cos a -f- sin a cos q), 
 or, dividing by sin a, 
 
 1 = 77i (sin q cot a + cos q) ; 
 whence, by transformation, 
 
 ffi sin <7 . . . , v 
 
 tan a = 2 = m sm # -f- m * sm <TT CO s g (very nearly.) 
 
 1 m cos q 
 
 Restoring the value of m, 
 
 sin P cos Z . . /sinPcosZXa . 
 
 tan a = : - sm a + I : ; I sm q cos . 
 
 sm d \ smd / 
 
 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. 343,) there 
 results, 
 
 Pcos/ 
 
 . /P COS /\2 . . _ 
 
 a = : j- sin q + I : -j~ ] sin q cos q sm 1" ... (B). 
 smd V smd / 
 
 The parallax in declination (8) is the difference of the arcs ps 
 (= d) and X (= d + 8.) Let *s = z, and zs' = Z. The trian- 
 gles zjos and zps' give (For. 74 and 73), 
 
 cos d sin / cos z cos (<? + ) sin / cos Z 
 
 1. cos pzs = =-T = 3 % . _ , 
 
 cos / sin z cos I sm Z 
 
 * Francceur'a Uranography, p. 418. 
 
350 APPENDIX. 
 
 _ sin d sin q _ sin (d + #) sin (? + ) 
 ^ sin z sin Z 
 
 From the first equation we derive 
 
 , cos d sin Z sin I cos z sin Z . ' 
 cos (a -f- o) = --- : ----- h sin / cos Z 
 
 smz 
 
 _ cos 6? sin Z sin / (cos z sin Z sin z cos Z) 
 
 sin z 
 _ cos d sin Z sin Z sin (Z z) 
 
 sin z 
 or, (equ. 8, p. 51,) 
 
 sinZ , , r 7v 
 = - (cos a sin P sin /) : 
 sm z v 
 
 from the second, 
 
 sinZ _ sin (d -f 5) sin (q -f a) 
 
 sin z sin d sin q 
 
 substituting, 
 
 , _ ~ sin (d + 5) sin (7 -f ) / , r A 
 cos (d -h <5) = - ^ 5^ . - 4^ - - (cos rf sm P sin /) 
 sin d sm q 
 
 cos (df + ^) _ sin (q + a) /cos d sin P sin l\ 
 sin (c? + ^) sin q Vsin d sin d ) 
 
 * /^ i *\ s ^ n (7 + a ) ^ , ^ sin P s i n ^ /r\ 
 cot (rf -f 5) = -- ^ - - I cot d -- : j I ... (C). 
 sm q \ sin d / 
 
 ~ sin P sin / 
 
 Put tan x = - : -j ; 
 sm d 
 
 then, cot (d + 6) = ^^-^ (cot d - tan x) 
 
 sm (] 
 
 __sin (g + a) /cosd sin a?\ 
 sin^ Vsin 6? cosx/ 
 
 _ sin (# + a) cos 6? cos a? sin d sin a: 
 
 sin q sin rf cos a? 
 
 sin + a c 
 
 sin ^ sin d cos 
 
 
 The apparent polar distance (d + 5) being computed by either 
 of the formulae (C) and (D), we have 8 = (d + 5) d. 
 
 Formulas may be obtained that will give the parallax in declina . 
 tion without first finding the apparent declination, (except approx 
 iinately.) 
 
 From equa. (C) we obtain 
 
 sin P sin / sin q cot (d + 5) 
 
 - : -- j ~~ COt rf - : 7 - , - r - , 
 
 sm d sm (q -f ) 
 
PARALLAX IN RIGHT ASCENSION AND DECLINATION. 351 
 
 and we also have 
 
 rx cos d cos (d + <$) sin <S 
 
 cot d cot (d + <5) = - -, . , , , ' = . , . , , , -v ; 
 
 sm rf sin (rf 4- <$) sm d sm (d + d) 
 
 the sum of these equations gives 
 
 sin P sin / /. sin q \ sin d 
 
 sin d! V sin (q + <*)' sin d sin (d + )' 
 
 sin (/ __ sin (y -f ) - sin q 
 
 w, l ~ : ~. . x : ~ . r 
 
 sm (q + a) sm (q + a) 
 
 _ 2 sin | a cos (q + j a) _ sin a cos (y + j a) 
 
 : "7 r : 7 : . : \ OF. ^&/tf, 1 u I 
 
 sin (^ + a) sin (q -f" a) cos ^ a 
 
 cos (7 -f i ) sin P cos I , % 
 
 = . , -T , by equa. (A). 
 
 sm c? cos i a 
 
 sin d sin (d + <*)' 
 or, sin <5 = sin P sin Z sin 
 
 cos (d + 8) cos (<7+a) sin P cos 
 
 cos a 
 = sin P sin / [sin (d + 5) tan y cos (rf -f- $)], 
 
 . . coil cos ( -f ^ ) 
 
 making tan y = - ~ ; 
 
 cosia 
 
 , . . sin P sin I . . , . . ._. 
 
 whence, sin a = -- sm (d + 5 y) . . . (F). 
 cosy 
 
 To facilitate the calculation, the sines of 5 and P in eqs. (E) 
 and (F), may be replaced bv the arcs. 
 
 To obtain an expression for the parallax in declination in terms 
 of the true declination, develope ain (d + <5 y) in equation (F) 
 which gives 
 
 . ,, sin P sin / r . , , , 
 sm d -- [sm (d + o) cos y sin y cos (d + ^)] ; 
 
 developing sin (d + <5) and cos (d + 5), and reducing, we have 
 
 . . sinP 
 sm 5 = 
 
 co 
 
 dividing by cos <J, 
 
 . . sinPsin/ r . 
 
 sm 5 = - , -- [sm (d y) cos o + cos (a y) sin 
 cos y 
 
 ~" y) + COS (rf ~ 
 
352 APPENDIX. 
 
 cos y 
 whence tan 6- -- 
 
 sin P sin / . , , x 
 sm (dy) 
 
 1 --- cos (a v) 
 cosy 
 
 sin P sin/ . ., v , /sin P sin A 2 
 = --- sin (dy) -f I --- I x 
 cosy V cosy / 
 
 sin (d y) cos (dy) (very nearly ;) 
 
 or, replacing tan 8 and sin P by 5 and P, expressing these arcs in 
 seconds, (For. 47, p. 343), and reducing by For. 13, p. 341, 
 
 PsinJ . x , /P sin /\ 2 sin I" . 
 
 J = - sm (d y) + [ -- I - - sm2(d y) . . . (G.) 
 cos y V 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. 126. Let K (Fig. 126) be the 
 
 pole of the ecliptic/ft, p the 
 pole of the equator fa ;/is 
 the vernal equinox, the ori- 
 gin of longitudes and of 
 right ascensions ; hbs 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 = w the obliquity of the ecliptic. The circle 
 Kznv is at the same time perpendicular at n to the ecliptic fb, and 
 at v to the horizon Jib ; 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,^ its longitude ; the point n is the nonagesimal, since bn = 
 90 ; nv is the altitude of this point, and the complement of zn ; 
 nv measures the inclination of the ecliptic to the horizon at the 
 given instant, or the angle b, so that b nv = Kz ; thus/ra =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.* 
 
 * Francoeur's Uranography, p. 421. 
 
LONGITUDE AND ALTITUDE OF THE NONAGESIMAL. 353 
 
 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 aicfi = 90, since the plane Kpi, passing 
 through the poles K and p, is at the same time perpendicular to 
 the ecliptic and to the equator; the arc mi =fi fm 90 s ; 
 then the angle zpK = I80zpi = 180 mi = 90+ s* 
 
 Now, in the spherical triangle pKz we know the sides Kp = u t 
 zp = 9Ql H, and the included angle zpK 90 + s ; and 
 may therefore find Kz = h the altitude of the nonagesimal, and the 
 angle pKz = nc fcfn = 90 N = complement of the longi- 
 tude N of the nonagesimal. Let S = sum of the angles Kzp and 
 zKp, then, (For. 86, page 346,) 
 
 but, 
 
 tan iS =-tan (180-S), and tan (90--s) =-tan i (s-90) ; 
 
 substituting, and denoting (180 58) by E, we have 
 
 Again, letD = zKp Kzp, then, (For. 87,) 
 
 whence, by transforming as above, and denoting (180 D) by F, 
 we have 
 
 Now, iS -f sD =pKz = 90 N ; 
 
 whence, N = 90 (iS + D), 
 
 or, 
 
 N = 360 4-90- (IS 4- D) = 180-S +180-iD +90; 
 
 consequently, N E -f F -f 90 . . . (J), 
 
 rejecting 360 when the sum exceeds that number. 
 
 Next, for the altitude of the nonagesimal, we have, (For. 88,) 
 
 N and h being known, to obtain the formula for the parallax 
 in longitude and latitude, we have only to replace in the formula 
 
 * Francoeur*s Uranography, p. 421. 
 45 
 
354 APPENDIX. 
 
 for the parallax in right ascension and decimation, the altitude I oi 
 the pole of the equator by that 90 h 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), / into 
 90 A, and q into fc fn L N, L being the longitude fc 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 h , . T , , , x /T . 
 
 smn = . -j (smL N-f-n) . . . (L), 
 
 sin (L N + n)/ sin P cos h \ 
 
 - -(cold 
 ) V 
 sin P cos h 
 
 ,, , v sn n/ sn cos \ 
 
 col(d + *)= - ^^ ^r- -(cold -- r-3 1 . . . (M), 
 sm(L N) V smd / 
 
 . (N), 
 
 sine? 
 
 sin * = sin P cos h sin 
 cos (d + if) cos (L N + in) sin P sin h 
 
 >. 
 
 sin P cos /*.,,, v /rix 
 
 sui if = -- sin (d -f y) . . . (R); 
 cos y 
 
 and in terms of the true longitude and latitude, 
 
 _ PsinA . /T _ _ ' /Psin^V 
 n = . sm (L - N) + I . . I X 
 sind \ smd / 
 
 sin(L-N)cos(L-N)sinl // . . , (S), 
 
 Pcosh . ./PcosAV 
 
 * = -- sm (d-y) + i I -- I x 
 cos y \ cos y / 
 
 sm2(dy)sml" . . . (T), 
 . tan A cos (L N+n) 
 cosn 
 
 To facilitate the computation, sin IT, sin tf, and sin P, in formu- 
 lae (L), (P), and (R), maybe replaced by the arcs themselves. 
 
 The distance d of the star from the pole of the ecliptic enters 
 into these formulae in place of the latitude X. 
 
 To find the apparent distance d', we have 
 
MOON'S AUGMENTED SEMI-DIAMETER. 356 
 
 for the apparent latitude X', 
 
 V=X ; 
 
 for the apparent longitude I/, 
 
 L'=L + II. 
 
 The logarithmic formulae given on page 298, were derived from 
 equations (L), (O), and (P), and the logarithmic formula on page 
 299 from equa. (0). 
 
 To determine now the effect of parallax upon the apparent di- 
 ameter of the moon. 
 
 Let ACB (Fig. 65, p. 147) represent the moon, and E the sta- 
 tion of an observer ; also let R = apparent semi-diameter of the 
 moon, and D =its distance. The triangle AES gives 
 
 i -noi AS . -r, AS 
 
 sm AES = ^TFT, or sin R = -fr- 
 
 HjO LJ 
 
 At any other distance D' we should have for the apparent semi- 
 diameter R', 
 
 sin R' D 
 
 whence - sTnir =s>- 
 
 Thus, if "R/ = moon's apparent semi-diameter to an observer at the 
 earth's surface, as at O (Fig. 26, p. 50), R =the same as it would 
 be seen from the centre C, and S represents the situation of the 
 moon, 
 
 sin R' _ CS _ sin ZOS _ sin Z 
 sinR ~OS~sinZCS~sin* ' 
 But we have, (see page 350,) 
 
 sinZ _ (sine? + 5) sin (q -fa) 
 sin z sin d sin q 
 
 or, in terms of the apparent longitude and latitude, (see page 354,) 
 sinZ *m(d + *) sin (L N + n) 
 
 sin z sin d sin (L N) 
 
 Hence, sin R' = *" R sin (<* + *) sin (L - N + n) ^ 
 
 sin d sin (L N) 
 
 Aberration in Longitude and Latitude, and in Right Ascension 
 and Declination* (See Art. 129, page 59.) 
 
 Aberration is caused by the motion of light in conjunction with 
 the motion of the earth. Light comes to us from the sun in 8 m - 
 17"-.8, during which time the earth describes an arc a =20 r .44, 
 
 * Francoeur's Uranography, p. 442, &c. 
 
356 
 
 APPENDIX. 
 
 of its orbilpbdin (Fig. 127,) supposed circular : p is the place of 
 the earth. Let us take any plane whatsoever, which we will call 
 relative, passing through the star and the 
 sun, and let dd' be the intersection of this 
 plane and the ecliptic, with which it makes 
 an angle k : let us seek the quantity 9 by 
 which the aberration displaces the star in 
 the direction perpendicular to this plane. 
 The question is to project on to a line per- 
 pendicular to the relative plane, the small 
 constant arc a which the earth describes, 
 this being the quantity that the star is dis- 
 placed from its line of direction in a direction parallel to the line 
 of the earth's motion, (see Art. 124 of the text:) this projection 
 is 9, variable according to the position of the relative plane in rela- 
 tion to which it is estimated. The velocity along the tangent at 
 p, makes with ph an angle 6 =pch = the arc pd' ; a cos & is then 
 the projection of this velocity on the line ph. The angle of our 
 two planes being k, this projection will be reduced to a cos 6 sin 
 k, when it is taken perpendicularly to the relative plane. Thus, 
 
 9 = a sin k cos 6 . . . (V). 
 
 The aberration displaces the star from the relative plane by this 
 quantity 9, k designating the inclination of this plane to the eclip- 
 tic, and 6 the arc pd', reckoned from JP the place of the earth to d' 
 the point of intersection of these two planes. Let us give to the 
 relative plane the positions which are met with in applications. 
 
 Let us suppose at first that k = 90, or sin k ~ 1 ; the relative 
 plane will then be perpendicular to the ecliptic. Let n be the ver- 
 nal equinox ; we have pd' = np nd' ; np is the longitude of the 
 earth, or 180 -f that O of the sun ; nd' is the longitude / of the 
 star ; whence 
 
 9 = a cos (O I). 
 
 Now, let M (Fig. 126) be the true place 
 of the star, M' the star as displaced by 
 aberration, KM is the circle 01 true lati- 
 tude, KM' the circle of apparent latitude, 
 and MM' 9 : this arc has its centre C 
 on the axis which passes through the pole 
 K of the ecliptic ; the longitude of the 
 star is then altered by the part OO' of the 
 ecliptic comprised between these two 
 planes ; and since OO' is to the arc MM' 
 as the radius 1 is to the radius CM = sin KM = cos latitude X of 
 the star, we have 
 
 aberr. in long. = cos (O 1) . . . (W). 
 
 cos X 
 
 If the relative plane is kc, (Fig. 129,) perpendicular to the circle 
 
 Fig. 128. 
 
ABERRATION IN RIGHT ASCENSION AND DECLINATION. 357 
 
 of latitude Kcd, the aberration <p 
 perpendicularly to it, will be the 
 aberration in latitude. Let kd be 
 the ecliptic, and o the earth ; the 
 angle k is measured by the arc cd 
 = X ; the arc ok = & = O long, 
 of k ; and as kd = 90, long, of 
 point k = I 90 : substituting in equation (V), we find 
 aberr. in lat. = a sin X sin (O 1) - . . (X). 
 These aberrations of the star produce a small apparent orbit, 
 which is confounded with its projection on the tangent plane to 
 the celestial sphere. Let us suppose the orbit to be referred to 
 two co-ordinate axes passing through the true place of the star and 
 lying in the tangent plane, of which one is parallel to the plane of 
 the ecliptic, and the other perpendicular to this, or tangent to the. 
 
 circle of latitude at the star ; and let = aberr, in long., and 
 
 cos X 
 
 y = aberr. in lat. ; y will be the ordinate, and x (the aberr. in long., 
 reduced to the parallel through the star) the abscissa : we have 
 
 ~= ^- cos (0-0, 
 
 cos X cos X 
 
 y = - a sin X sin ( O I) ; 
 or, = cos (O I), 
 
 -? = -sin(0-J). 
 flsmX 
 
 Squaring the last two equations, and adding them together, O dis- 
 appears, and we find 
 
 y 2 + z 3 sm 2 X=:a 2 sin a X . . . (Y), 
 
 whatever may be the place of the earth. Such is the equation of 
 the apparent orbit, which, as we perceive, is an ellipse of which 
 the semi-axes are a and a sin X, and whose centre is the true place 
 of the star. When the star is at the pole of the ecliptic, X = 90, 
 and the ellipse becomes a circle of which the radius is a. When 
 X = 0, this ellipse is reduced to an arc 2a of the ecliptic. 
 
 To find the aberration in right ascension, the relative plane must 
 be perpendicular to the equator. Let kc be the equator, (Fig. 129,) 
 p its pole, psd the relative plane, which is the circle of declination 
 of the star s ; kd the ecliptic, o the earth, k the vernal equinox, 
 kc = R, sc = D. Aberration carries the star s out of the plane 
 pcd a distance <p, which it is the question to determine. Equa. 
 (V) is here 
 
 <p = a sin d cos do a sin d cos (Jed ko) 
 = a sin d (cos kd cos ko -f- sin kd sin ko) 
 -= a sin d cos kd cos ko + a sin d sin kd sin ko 
 

 358 APPENDIX. 
 
 but ko long, of earth = 180 + O ; we have also the angle k = 
 the obliquity w of the ecliptic, and the right-angled spherical trian- 
 gle kcd gives, by Napier's rules, 
 
 cot kd = cot R cos w, sin d sin kd = sin R. 
 The 1st equa. multiplied by the 2d, gives 
 
 sin d cos kd = cos R cos w, 
 whence 9 = a (cos R cos w cos O + sir R sin O). 
 
 The displacement from M to M'(Fig. 128) conducts, as before, 
 to the division of 9 by cos D, to have the conesponding arc of the 
 equator : thus the aberration in right ascension is, 
 
 u = a sin R sec D sin O a cos u cos R sec D cos O (Z). 
 
 Taking the relative plane perpendicular to the circle of declina- 
 tion, we find for the aberration in declination, 
 
 v = a sin D cos R sin O a cos w (tan w cos D sin R sin D) 
 cos O . . . (a). 
 
 These formulae may easily be adapted to logarithmic computa- 
 tion : 
 
 In formula (Z) let a sin R sec D = A, and a cos w cos R sec 
 D = B; then, 
 
 u = - A (sin O +-T- cos O) . . . (Z'). 
 A. 
 
 ~ B a cos u cos R sec D _ 
 
 Put tan 9 = -r- = - fr = cos w cot R ... (6), 
 
 A a sin R sec D 
 
 and we shall have 
 
 u A (sin O -f ^-^ cos O) 
 cos 9 
 
 . sin O cos 9 + sin 9 cos 
 cos 9 
 
 5= sin (O -I- 9). 
 
 cos 9 
 
 Restoring the value of A, and taking ^ for sec D, we obtain 
 
 asinR 
 
 w = __ sm(O-f-<p) . . . (c). 
 
 cos D cos 9 
 
 The auxiliary arc 9 is given by equation (b) ; it must be substi 
 tuted in equation (c), with its sign, and we then obta n u. Tan 
 9, and the co-efficient of sin (O H- 9) are constant, for the same star, 
 for a long period of time, since these quantities vary very slowly 
 with w and the precession. Moreover, the co-efficient of sin 
 (0+9) is the maximum value of u, since it answers to sin 
 (0 + 9) = 1. Thus we shall be able to calculate in advance, for 
 
NUTATION IN RIGHT ASCENSION AND DECLINATION. 359 
 
 any designated star, the values of 9 and of the maximum of the aber- 
 ration in right ascension, or of the logarithm of this maximum. 
 
 The results of these calculations for 50 principal stars are given 
 in Table XCI, columns entitled M and 9. 
 
 If in equation (a) we make a sin D cos R = A', and a cos 
 (tan u cos D sin R sin D) = B', we shall have the equation 
 
 B' 
 
 v =- A' (sin O + ^7 cos O), 
 
 in which A' and B' are constants. This equation is of the same 
 form with equa.(Z'). We therefore have, in the same manner as 
 for the right ascension, 
 
 B' _ cos oj (tan w cos D sin R sinD) 
 "A?" a sin D cos R 
 
 a sin w cos D a cos w sin R sin D 
 
 a sin D cos R 
 sin w cot D 
 
 cos w tan R . . . (d\ 
 
 A' a sin D cos R 
 
 v = -- - sin ( O + o ) = --- - -- x 
 
 cos B cos 6 
 
 sin(O-H) . . . ( e ). 
 
 6 is given by equation (d), and being substituted in equation (e), 
 we shall have v. 6 and the co-efficient of sin O -M are constant 
 
 for the same star, and we can therefore calculate in advance the 
 value of this arc, and of the co-efficient, which is the maximum 
 of the aberration in declination. Columns entitled 6 and N, Table 
 XCI, contain the^ quantities 6 and the logarithms of the maxima of 
 the aberration in declination for 50 principal stars. 
 
 For convenience in calculation, the angles 9, 6, and the maxima, 
 M, N, in Table XCI, have been rendered positive in all cases >. 
 This has been accomplished by adding 12 s - to 9 and 6 whenever 
 the calculation conducted to a negative value, and by adding 6*- to 
 O + 9, or O + d, whenever the co-efficient had the sign , (this 
 sign being changed to + ;) in this manner the sign of each of the 
 two factors is changed, which does not alter the sign of the pro- 
 duct. 
 
 Formula for the Nutation in Right Ascension and Declination* 
 (Referred to in Article 148, p. 63.) 
 
 In deriving these formulae, we must begin with borrowing cer- 
 tain results established by Physical Astronomy. It has been 
 proved, in confirmation of Bradley's conjectures, that the phenom- 
 ena of nutation are explicable on the hypothesis of the pole of the 
 earth describing around its mean place (that place which, see pago 
 
 * Wood house's Astronomy, p. 357, &c. 
 
360 
 
 APPENDIX. 
 
 61, it would hold in the small circle described around the pole of 
 the ecliptic, were there no inequality of precession) an ellipse, in 
 a period equal to the revolution of the moon's nodes. The major 
 axis of this ellipse is situated in the solstitial colure and equal to 
 18".50 ; it bears that proportion to the minor axis (such are the 
 results of theory) which the cosine of the obliquity bears to the 
 cosine of twice the obliquity : consequently, the minor axis will be 
 13".77. 
 
 Let CdA. (Fig. 130) represent such an ellipse, P being the mean 
 place of the pole, K the pole of the ecliptic. CDOA is a circle 
 
 Fig. 130. 
 
 described with the centre P and radius CP. VL is the ecliptic, 
 Vw the equator, KPL the solstitial colure. In order to determine 
 the true place of the pole, take the angle APO equal to the retro- 
 gradation of the moon's ascending node from V : draw Oi perpen- 
 dicular to PA, and the point in the ellipse, through which Oi 
 passes, is the true place of the pole. This construction being ad- 
 mitted, the nutations in right ascension and north polar distance 
 may, Pp being very small, be thus easily computed. 
 
 Nutation in North Polar Distance. 
 Nutation in N. P. D. = Ptf ptf. = Pr = Pp cospPtf, nearly, 
 
 = Pp cos ( APp + R 90) 
 = Ppsm 
 R denoting the right ascension. 
 
NUTATION IN RIGHT ASCENSION AND DECLINATION. 361 
 
 Nutation in Right Ascension. 
 
 The right ascension of the star tf is, by the effect of nutation, 
 changed from Vw into Vts. Now, 
 
 V'fc = V't> -f VM> + ts, nearly, 
 whence, Vw - V'ts = - V'v ts 
 
 = - W cos VV'u - P^ sin Pp<f ^s 
 
 smP<r 
 
 in which expression V'w (= VV cos VV'r) is, as in the case of pre- 
 cession, common to all stars. 
 
 In order to reduce farther the above expression, we have 
 pPa = APp + APrf = APp + R - 90, 
 
 whence, V'v ts = Pp sin APp cot w 
 
 - Pp sin (APp + R - 90) cot N. P. D. 
 = Pp sin APp cot co -f Pp cos (APp + R) cot <*, 
 
 5 representing the north polar distance, and w the obliquity of the 
 ecliptic. 
 
 But these forms are not convenient for computation. In order 
 to render them convenient, we must, from the properties of the el- 
 lipse, deduce the values of Pp, and of the tangent of APp, and 
 then substitute such values in the above expressions : thus, 
 Pp _ sec APp _ cos APO _ cos(12 8 ft) _ cos ft 
 PO ~ sec APO ~~ cos APp cos APp cos APp' 
 ft designating the longitude of the moon's ascending node ; 
 
 , -n PO COS ft 
 
 whence Pp = - r-=: . 
 
 cos APp 
 
 A tan APp pi _ Pd __ Pd 
 
 tanAPO~Oi"PD"PO ; 
 
 nence, tan APp= tan APO = tan (12 8 - - ft) 
 
 Now substitute, and there will result 
 
 The Nutation in North Polar Distance 
 
 = P C ^ (sin APp cos R + cos APp sin R) 
 cos APp 
 
 = PO (tan APp cos R cos ft + cos ft sin R) 
 = Pd cos R sin ft + PO cos ft sin R 
 = - 6".887 cos R sin ft + 9".250 cos ft sin R . . . (/) ; 
 46 
 
362 APPENDIX. 
 
 which is the difference, as far as nutation is concerned, between 
 the mean and apparent north polar distance. The apparent north 
 polar distance, therefore, must be had by adding the preceding 
 quantity, with its sign changed, to the mean. 
 
 Nutation in right ascension = Pd sin & cot w 
 + PO cos ft cos R cot 5 + Pd sin ft sin R cot 5, 
 which, as far as nutation is concerned, is the difference of the mean 
 and apparent right ascensions : and, consequently, the above ex- 
 pression must be subtracted from the mean, in order to obtain the 
 apparent right ascension ; or, which is the same, must be added 
 after a negative sign has been prefixed ; in which case, we have, 
 substituting for PO, Pd their numerical values, 
 
 The Nutation in Right Ascension 
 
 = 6". 887 sin ft cot w 
 9".250 cos ft cos R cot 5-6".887 sin ft sin R cot 5 . . . (g). 
 
 Formulae (/) and (g) are of the same form with (Z) and (a) for 
 the aberrations in right ascension and declination, and therefore 
 formulae may be derived from them similar to (c) and (e), adapted 
 to logarithmic computation. The quantities corresponding to <p, 
 M, , N, have been calculated for the stars in the catalogue of 
 Table XC, and inserted in Table XCI, in the columns entitled 
 <p', M', 0', N'. 
 
 The Solar Nutation arises from like causes as the Lunar, and 
 admits of similar formulae. As an ellipse, made the locus of the 
 true place of the pole, served to exhibit the effects of the lunar 
 nutation, so an ellipse, of different, and much smaller dimensions, 
 may be made to represent the path which the true pole of the 
 equator would, by reason of the sun's inequality of force in caus- 
 ing precession, describe about the mean place of the pole. Thus, 
 in Figure 130, the ellipse AdC will serve to represent the locus 
 of the pele, when AP = 0".545, Pd = 0".500, and APO, instead 
 of being = ft, is equal to 2 0, or twice the sun's longitude, 
 taken in the order of the signs ; the equations, therefore, for the 
 solar nutation in north polar distance, and right ascension, analo- 
 gous to eqs.jf and g will be 
 
 The Solar Nutation in North Polar Distance 
 _ 0".500 cos R sin 2 + 0".545 sin R cos 2 . . . (h). 
 
 The Solar Nutation in Right Ascension 
 = 0".500 sin 2 cot w 
 
 0".545 cos 2 cos R cot <5 0".500 sin 2 sin R cot 5 . . (*). 
 
 If the apparent place of a star should be required with great 
 
 precision, it would be necessary to compute the solar nutations 
 
 from these formulae, and apply them as corrections to the mear. 
 
EFFECTS OF OBLATENESS OF THE EARTH*S SURFACE. 363 
 
 right ascension and declination. The calculation would be per- 
 formed after the same manner as for the lunar nutation ; but it is 
 much abridged by remarking that the form of the equations is the 
 same as that of the equations for the lunar nutation, and that the 
 co-efficients are very nearly the 0.075 of those of the latter equa- 
 tions. Thus we can make use of the same arcs <p', 6', and log. 
 maxima, M', N', repeat the calculation for the lunar nutation, 
 taking 2 O instead of &, arid multiply the nutations in right ascen- 
 sion and declination thus obtained by 0.075. The results will be 
 the solar nutations required. (See Prob. XX.) 
 
 F ' trmul&for computing the effects of the Oblateness of the Earth's 
 Surface upon the Apparent Zenith Distance and Azimuth of a 
 Star* (See Article 162, page 69.) 
 
 From the centre of the earth, an observer would see a star at I, 
 Fig. 131. (Fig. 131,) and would have V for his 
 
 zenith : fftm the surface his zenith is 
 Z, and he sees this star at B ; IB =p 
 is the parallax in altitude ; the azi- 
 p muth VZI is changed into VZB. If 
 for a given time, we wish to calculate 
 the apparent zenith distance BZ, and 
 the apparent azimuth VZB, we hav^ 
 first to resolve the spherical triangle IZP, in which we kp- 
 two sides ZP = co-latitude and IP = co-declination, ana 
 eluded hour angle P ; the azimuth VZI (= A), and the a* 
 (= n) will thus be known. But from the earth's surface, the v 
 is seen at B : the azimuth VZB = VZI +IZB = A + a ; the zeniu 
 distance BZ = n +p, since, VZ (= i) being very small, we have 
 sensibly IB -f- IZ = BZ. By reason of the want of sphericity of 
 the earth, parallax then increases the true azimuth and zenith 
 distance of a star by small quantities, a and p, which it is neces- 
 sary to calculate. In the triangle VIZ we have 
 
 cos IV = cos i cos n + sin i sin n cos A = cos n + k sin n ; 
 
 making cos i = 1, sin i = i, and i cos A = k. Now, k . i, and 
 & fortiori cos k = 1, sin A; = k ; whence 
 
 cos IV = cos n cos k }- sin n sin k = cos (n A;), 
 and IV = n k =n i cos A. 
 
 Thus we correct the calculated arc n by the quantity i cos 
 A, to have 
 
 IV = z n i cos A ... (j). 
 If this value of z be introduced into equation (10), page 52, we 
 
 * Francceur's Uranography, p. 426, &c. 
 
564 
 
 APPENDIX. 
 
 shall have p, and thence the apparent zenith distance Z = n + p 
 = BZ. 
 
 Afterwards, to obtain IZB = a, or the parallax in azimuth, the 
 triangles ZBV, ZBI give 
 
 sin ZBV __ sin (A + a) sin ZBV ^sina 
 
 sin i sin (z +p) ' sin TI sinp ' 
 whence, by equating the values of sin ZBV, 
 
 sin n sin a sin i sin (A + a) 
 
 sin p sin (z + p) 
 
 substituting for sin p its value sin H sin (z +jp) = sin H sin Z, 
 (equa. 8, page 51,) and reducing, we have 
 
 sin a _ sin (A + a) 
 sin H sin i sin n ' 
 
 and as i is very small, sin i sin (A + a) does not differ sensibly 
 from i sin A, and we thus have in seconds, (For. 47, page 343,) 
 
 Hisjn A sin 1" 
 sin n 
 
 Solution of Kepler's Problem, by which a Body's Place is found 
 in an Elliptical Orbit* (See Art. 268, p. 106.) 
 
 Let APB (Fig. 132) be an ellipse, E the focus occupied by the 
 sun, round which P the earth or any other planet is supposed to 
 revolve. Let the time and planet's motion be dated from the ap- 
 
 Fig. 132. 
 
 M 
 
 side or aphelion A. The condition given is the time elapsed from 
 the planet's quitting A; the result sought is the place P ; to be 
 determined either by finding the value of the angle AEP, or by 
 
 * Woodhouse's Astronomy, p. 457, &c. 
 
365 
 
 cutting off, from the whole ellipse, an area AEP bearing the same 
 proportion to the area of the ellipse which the given time bears to 
 the periodic time. 
 
 There are some technical terms used in this problem which we 
 will now explain. 
 
 Let a circle AMB be described on AB as its diameter, and sup- 
 pose a point to describe this circle uniformly, and the whole of it 
 in the same time as the planet describes the ellipse ; let also t de- 
 note the time elapsed during P's motion from A to P ; then if AM = 
 
 r-j x 2 AMB, M will be the place of the point that moves 
 
 uniformly, while P is that of the planet; the angle ACM is 
 called the Mean Anomaly, and the angle AEP is called the True 
 Anomaly. 
 
 Hence, since the time (t) being given, the angle ACM can al- 
 ways be immediately found, (see Art. 267, p. 106,) we may vary 
 the enunciation of Kepler's problem, and state its object to be the 
 finding of the true anomaly in terms of the mean. 
 
 Besides the mean and true anomalies, there is a third called the 
 Eccentric Anomaly, which is expounded by the angle DC A, and 
 which is always to be found (geometrically) by producing the ordi- 
 nate NP of the ellipse to the circumference of the circle. This 
 eccentric anomaly has been devised by mathematicians for the 
 purposes of expediting calculation. It holds a mean place between 
 the two other anomalies, and mathematically connects them. There 
 is one equation by which the mean anomaly is expressed in terms 
 of the eccentric ; and another equation by which the true anomaly 
 is expressed in terms of the eccentric. 
 
 We will now deduce the two equations by which the eccentric 
 is expressed, respectively, in terms of the true and mean anomalies. 
 Let t = time of describing, AP, 
 
 P = periodic time in the ellipse, 
 
 a = CA, 
 
 ae = EC, 
 
 v = L PEA, 
 
 u = L DCA ; (whence, ET, perpendicular to DT, = EC 
 x sin M,) 
 
 P =PE, 
 
 if = 3.14159, &c. ; 
 
 then, by Kepler's law of the equable description of areas, 
 
 px _ =px 
 
 area of elhp. area circle *a? v 
 
 P /ET.DC AD.DC 
 
 P Pi 
 
 = (e sin u + u) : hence, if we put = -, 
 
 *flt tiflf 71 
 
366 APPENDIX. 
 
 we have 
 
 nt = e sin u + u . . . (/), 
 
 an equation connecting the mean anomaly nt, and the eccentric u. 
 In order to find the other equation, that subsists between the 
 true and eccentric anomaly, we must investigate, and equate, two 
 values of the radius-vector p, or EP. 
 
 First value of p, in terms of v the true anomaly, 
 
 == 
 
 1 e cost; 
 Second, in terms of u the eccentric anomaly, 
 
 p = a (1 + e cos u) . . . (2). 
 For, P 2 = EN 2 + PN 2 
 
 = EN 2 + DN 2 x (1 - e 2 ) 
 
 (ae + a cos u) 2 + a 2 sin 2 u (1 e 2 ) 
 
 = a 2 |e*+ 2e cos u + cos 2 u} + a 2 (1 e 2 ) sin 2 u 
 
 = a 2 \ I + 2e cos u + e 2 cos 2 M^ . 
 
 Hence, extracting the square rooj, 
 
 p = ( 1 + e cos #). 
 Equating the expressions (1), (2), we have 
 
 (1 e 2 ) = (1 e cos v) (1 -he cos M), whence, 
 
 e + cos u 
 
 cosv = 7: -- , an expression for v in terms of u ; 
 1 + e cos u 
 
 but, in order to obtain a formula fitted to logarithmic computation, 
 we must find an expression for tan - : now, (see For. 12, p. 341,) 
 
 v _ //I cos v\ _ / /(I e) (I cos u)\ 
 111 2 " V VI + oo* */ = " V V(l +e)l +COSM/ 
 
 These two expressions (Z) and (m), that is, 
 
 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. 
 
 MO is drawn parallel to DC. (1.) Find the half difference of 
 
367 
 
 the angles at the base EM of the triangle ECM, from this ex- 
 pression, 
 
 tan i (CEM - CME) = tan J (CEM + CME) x j-=-|, 
 
 in which, CEM + CME = ACM, the mean anomaly. 
 
 (2.) Find CEM by adding J (CEM + CME) and J (CEM 
 CME) and use this angle as an approximate value to the ec- 
 centric anomaly DC A, from which, however, it really differs by 
 L EMO. 
 
 (3.) Use this approximate value of L DC A = L ECT in 
 computing ET which equals the arc DM ; for, since (see p. 365), 
 
 t = : ;- x DE A, and (the body being supposed to revolve in 
 
 area circle J 
 
 P 
 
 the circle ADM) = ; j- x ACM 5 area AED = area ACM, 
 area circle 
 
 or, the area DEC + area ACD = area DCM + area ACD ; con- 
 sequently the area DEC =the area DCM, and, expressing their 
 values, 
 
 ET x DC DM x DC , , __, ^ 
 
 = , and thus, ET = DM. 
 
 2 
 
 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 EO. 
 
 (4.) Use EO in computing the angle EMO, the real difference 
 between the eccentric anomaly DC A and the /. MEC ; add the 
 computed L EMO to L MEC, in order to obtain L DCA. The 
 result, however, is not the exact value of L DCA, since L EMO 
 has been computed only approximately ; that is, by a process which 
 commenced by assuming L MEC for the value of the L 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 
 L DCA = L ECT, ET, EO, and L EMO must be again com 
 puted. 
 

369 
 
 NOTE 1. 
 
 The number of planets known at the present date (January 1st, 1852), is 
 twenty- two. During the last seven years twelve new planets have been dis- 
 covered. The following table contains the names of these planets, together 
 with the date and place of discovery, and the name of the discoverer. 
 
 Names. 
 
 When discovered. 
 
 By ^hom. 
 
 Where. 
 
 Astraea 
 
 Dec. 8, 1845 
 
 Hencke 
 
 Driessen. 
 
 Neptune 
 Hebe 
 
 Sept. 23, 1846 
 July 1, 1847 
 
 Galle 
 Hencke 
 
 Berlin. 
 Driessen. 
 
 Iris 
 
 Aug. 13, 1847 
 
 Hind 
 
 London. 
 
 Flora 
 
 Oct. 18, 1847 
 
 Hind 
 
 London. 
 
 Metis 
 
 April 25, 1848 
 
 Graham 
 
 Markree. 
 
 Hygeia 
 
 April 12, 1849 
 
 Gasparis 
 
 Naples, 
 
 Parthenope 
 Clio 
 
 May 13, 1850 
 Sept. 13, 1850 
 
 Gasparis 
 Hind 
 
 Naples. 
 London. 
 
 Egeria 
 
 Nov. 2, 1850 
 
 Gasparis 
 
 Naples. 
 
 Irene 
 
 May 20, 1851 
 
 Hind 
 
 London. 
 
 Eunomia 
 
 July 29, 1851 
 
 Gasparis 
 
 Naples. 
 
 Although Neptune was first seen by Galle, at Berlin, the honor of the discov- 
 ery of this planet is generally awarded to Leverrier, a French astronomer. 
 Leverrier ascertained, from a careful examination of the motions of Uranus, that 
 that planet must be subject to the disturbing action of an unknown planet more 
 remote from the sun. He investigated the probable orbit and mass of this 
 unknown planet, that is, the orbit and mass that would serve to account for the 
 previously unexplained irregularities observed in the motions of Uranus, and 
 assigned its probable place in the heavens. At his request Galle, of the Berlin 
 Observatory, undertook the search for it ; and on directing his telescope to the 
 part of the heavens designated by Leverrier, detected the supposed planet 
 within 1 of the place which had been assigned by that astronomer. 
 
 The same investigation was undertaken about the same time, and with very 
 nearly the same results, by a young English mathematician by the name of 
 Adams, who is therefore entitled to a share of the honor of this wonderful 
 discovery. 
 
 The planets Ceres, Pallas, Juno, and Vesta, on account of their diminutive 
 size and certain other peculiarities, have received the appellation of Asteroids. 
 All the newly-discovered planets, with the exception of Neptune, are also classed 
 among the asteroids. The number of asteroids at present known is, accordingly, 
 fifteen. " Besides these fifteen, others yet undiscovered may exist ; and it ia 
 extremely probable that such is the case, the multitude of telescopic stars 
 being so great that only a small fraction of their number has been sufficiently 
 noticed to ascertain whether they retain the same place or not," and from one 
 to three new asteroids having been discovered every year since 1846. 
 
 47 
 
370 
 
 NOTES n. m. 
 
 II. 
 
 At the present date (Jan., 1852), the largest and best telescope in the United 
 States is the great refractor at the Cambridge Observatory, manufactured by 
 Merz and Mahler, of Munich, Bavaria. The aperture of the object-glass is 15 
 inches, and its focal length is 22 feet. It has 18 different powers, varying from 
 180 to 2,000. Its dimensions are a trifle greater than those of the Pulkova 
 refractor, and it is generally conceded to be superior to it in its performance. 
 It is, accordingly, the best refracting telescope in the world. It was erected in 
 June, 1847, and in the hands of Messrs. W. C. and G. P. Bond has already 
 enriched astronomy with many valuable observations and discoveries. 
 
 The accuracy of transit observations has recently been greatly increased by 
 the introduction of the Electro-Chronograph ; by which, with the adaptation of a 
 proper electro-magnetic recording apparatus, the seconds measured off by the pen- 
 dulum of a clock are designated by a series of equally distant dots or breaks in a 
 continuous line, upon a fillet or 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 on the fillet, that designate the duration of the successive seconds. 
 In this way it is believed that 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 multiplied some seven-fold. 
 
 This method of observation has been introduced at the Cambridge Observa- 
 tory, and also at the National Observatory. 
 
 NOTE III. 
 
 ELEMENTS OF THE ORBITS OF THE ASTEROIDS, 
 Arranged in the Order of their Mean Distance from the Sun. 
 
 
 Name. 
 
 Distance. 
 
 Period in 
 
 days. 
 
 Eccentricity. 
 
 Inclination. 
 
 Longitude of 
 Ascending Node. 
 
 
 
 
 
 
 o / // 
 
 Q 1 11 
 
 1 
 
 Flora 
 
 2.201687 
 
 1193.249 
 
 .156557 
 
 5 43 4.8 
 
 110 18 12.0 
 
 2 
 
 Clio 
 
 2.334876 
 
 1303.127 
 
 .217922 
 
 8 23 1.9 
 
 235 19 49.8 
 
 3 
 
 Vesta 
 
 2.361081 
 
 1325.147 
 
 .089569 
 
 7 8 29.7 
 
 103 23 31.6 
 
 4 
 
 Iris 
 
 2.380624 
 
 1341.636 
 
 .229942 
 
 5 28 15.9 
 
 259 48 10.2 
 
 5 
 
 Metis 
 
 2.385607 
 
 1345.850 
 
 .120253 
 
 5 34 27.8 
 
 68 32 17.4 
 
 6 
 
 Eunomia 
 
 2.399440 
 
 1357.573 
 
 .136504 
 
 13 18.5 
 
 292 51 1.8 
 
 7 
 
 Hebe 
 
 2.425786 
 
 1379.994 
 
 .200180 
 
 14 47 56.0 
 
 138 29 42.6 
 
 8 
 
 Parthenope 
 
 2.450833 
 
 1401.000 
 
 .099466 
 
 4 36 56.7 
 
 124 57 55.8 
 
 9 
 
 Irene 
 
 2.552303 
 
 1518.943 
 
 .170022 
 
 8 37 35.7 
 
 87 47 46.2 
 
 10 
 
 Egeria 
 
 2.560070 
 
 1492.230 
 
 .096180 
 
 15 57 59.8 
 
 43 85 24.4 
 
 11 
 
 Astraea 
 
 2.577047 
 
 1511.095 
 
 .188058 
 
 5 19 22.7 
 
 141 25 14.6 
 
 12 
 
 Juno 
 
 2.670837 
 
 1594.296 
 
 .254884 
 
 13 3 22.1 
 
 170 54 45.6 
 
 18 
 
 Ceres 
 
 2.768051 
 
 1682.125 
 
 .076652 
 
 10 37 4.4 
 
 80 48 66.6 
 
 14 
 
 Pallas 
 
 2.772858 
 
 1686.510 
 
 .239815 
 
 34 37 33.0 
 
 172 43 59.7 
 
 15 
 
 Hygeia 
 
 8.150060 
 
 2042.101 
 
 .010103 
 
 3 47 15.5 
 
 287 37 8.6 
 
NOTE IV. 
 
 371 
 
 
 Name. 
 
 Longitude of 
 Perihelion. 
 
 Mean Anomaly 
 at Epoch. 
 
 Epoch in Mean Time. 
 
 
 
 O / // 
 
 o / // 
 
 d. h. 
 
 1 
 
 Flora 
 
 33 40.8 
 
 35 48 7.0 
 
 Berlin M. T. 1848, Jan. 1 
 
 2 
 
 Clio |302 55 1.5 
 
 65 47 23. 
 
 1851, Jan. 
 
 3 
 
 Vesta 
 
 250 46 32.2 
 
 225 44 18.8 
 
 " 1850, Jan. 9 
 
 4 
 
 Iris 
 
 41 41 13.5 
 
 330 41 54. 
 
 " 1848, Jan. 1 
 
 5 
 
 Metis 
 
 70 33 42.8 
 
 146 30 18.5 
 
 1848, May 5 12 
 
 6 
 
 Eunomia 
 
 112 18 15.6 
 
 172 10 21.6 
 
 " 1851, Aug. 5 
 
 7 
 
 Hebe 
 
 14 50 50.3 
 
 275 8 51.3 
 
 1847, Jan. 1 
 
 8 
 9 
 
 Parthenope 
 Irene 
 
 316 49 61.8 
 191 8 27.5 
 
 288 40 43.2 
 41 57 9.5 
 
 " 1850, May 25 
 Greenwich ] 851, June 10 
 
 10 
 
 Egeria 
 
 116 26 49.4 
 
 288 37 17. 
 
 1850, Nov. 2 
 
 11 
 
 Astraea 
 
 135 20 47. 
 
 318 45 3.3 
 
 Berlin 1846, Jan. 1 
 
 12 
 
 Juno 
 
 54 24 12.8 
 
 124 31 10.8 
 
 1850, April 8 
 
 13 
 
 Ceres 
 
 147 46 12.4 
 
 219 6 29.5 
 
 " 1850, Sept. 25 
 
 14 
 
 Pallas 
 
 121 21 48.5 
 
 217 31 10.6 
 
 1850, Aug. 23 
 
 15 
 
 
 Hygeia 
 
 227 49 54.2 
 
 330 52 8.5 
 
 1849, April 15 
 
 NOTE IV. 
 
 The number of planets which are now known to have the situations men- 
 tioned in the text is no less than fifteen. It is a remarkable fact, with respect 
 to these asteroids, as they are called, that their orbits, if we except those ot 
 Iris and Hygeia, have approximately two common points of reunion in opposite 
 regions of the heavens. This singular fact is in accordance with a theory pro- 
 pounded by Dr. Olbers nearly fifty years ago (1802), after the discovery of 
 Ceres and Pallas, that " these small bodies were merely the fragments of a larger 
 planet, which had exploded from some internal convulsion, and that several more 
 might yet be discovered." For, since the supposed fragments must have origi- 
 nally diverged from the same point, their paths must, agreeably to the laws ot 
 planetary motions, have two common points of reunion ; viz., the place occupied 
 by the primitive planet at the time when the convulsion occurred, and the point 
 in the heavens diametrically opposite to this. It is true that, as a matter of 
 fact, the intersection is only approximate, the deviations from a common point 
 being in some instances as much as 4, and in the case of the planetoids Iris and 
 Hygeia no less than 9, but this discrepancy is ascribed, by the advocates of 
 Giber's theory, to the disturbing actions of the planets, and the consequent sec- 
 ular displacement of the orbits of the asteroids, and it is accordingly conjectured 
 that if the secular motion of the node of each orbit were known, we might, by 
 calculating back, find that at some period in the past the orbits all had truly a 
 common point of intersection, and thus determine the date of the supposed ex- 
 plosion of the single primeval planet. On this point Professor Loomis remarks 
 that " we may safely assume that the nodes of all the asteroids have not coin- 
 cided within a period of many thousand years ; and therefore that, if these bodies 
 are the fragments of a larger planet which has exploded, this explosion must 
 have taken place at a very remote epoch. 
 
 " It should also be observed, that not only must the nodes of all the asteroids 
 coincide, but the distance of the planets from the sun must be the same at that 
 instant. Now the distance of these planets from the sun when at their nodes, 
 varies by nearly the radius of the earth's orbit ; so that to bring them all to- 
 gether, we must suppose a corresponding change in the place of their perihe- 
 lia. This also would require the lapse of many centuries ; and when we con- 
 sider the necessity of a coincidence at the same instant, both in distance and di- 
 rection, we can easily suppose that such a result could not have taken place 
 within a million of years." 
 
372 NOTE V. 
 
 4 
 
 NOTE V. 
 
 Gambart's or Biela's comet, at its return in 1846, exhibited a phenomenon al- 
 together unprecedented in the annals of astronomy. On the 13th of January, at 
 the National Observatory in Washington, and on the 15th and subsequently, at 
 all the principal observatories in this country and Europe it was distinctly seen 
 to have become double ; a very small and faint cometic body, having a nucleus ot 
 its own, being observed appended to it at a distance of about 2' from its centre. 
 The two comets moved on side by side, for a period of two months, and through 
 an arc of more than 70, when the companion, after undergoing remarkable 
 changes of magnitude and luminosity, disappeared. During the whole of this 
 interval the apparent distance between the two bodies gradually increased, but 
 the apparent direction of the line of junction remained nearly the same. On 
 the 30th of January, the distance of separation had increased to 3', on the 13th 
 of February to 5', and so until on the 5th of March it was over 9'. " Both bodies 
 had nuclei, both had short tails, parallel in direction, and nearly perpendicular 
 to the line of junction; but whereas, at its first observation on January 13th, 
 the new comet was extremely small and faint, in comparison with the old, the 
 difference, both in point of light and apparent magnitude, diminished. On the 
 10th of February, they were nearly equal, although the day before the moon- 
 light had effaced the new one, leaving the other bright enough to be well ob- 
 served. On the 14th and 16th, however, the new comet had gained a decided 
 superiority of light over the old, presenting at the same time a sharp and star- 
 like nucleus, compared by Lieut. Maury to a diamond spark. But this state of 
 things was not to continue. Already, on the 18th, the old comet had regained 
 its superiority, being nearly twice as bright as its companion, and offering an'un- 
 usually bright and starlike nucleus. From this period the new companion began 
 to fade away," but continued visible until after the middle of March. As seen 
 by the author on the 17th of March in a reflecting telescope of 14 ft. focus, 
 with a low power, the cometic mass had two points of maximum brightness, but 
 the twin comets were not distinctly separate. On March 21 it appeared in the 
 same telescope as one nebulous mass, with a single point of concentration. On 
 the 22d of April this had disappeared. 
 
 " While this singular interchange of light was going forward, indications of 
 some sort of communication between the comets were exhibited. The new or 
 companion comet, besides its tail, extending in a direction parallel to that of the 
 other, threw out a faint arc of light which extended as a kind of bridge from 
 the one to the other ; and after the restoration of the original comet to its former 
 pre-eminence, it, on its part, threw forth additional rays, so as to present (on the 
 22d and 23d of February, as seen by Lieut. Maury, of the National Observatory) 
 the appearance of a comet with three faint tails forming angles of about 120 
 with each other, one of which extended towards its companion." 
 
 What was the relation of these two bodies ? Was the original comet actually 
 divided into two, as appearances seemed to indicate? Professor Plantamour, 
 director of the observatory of Geneva, has furnished a partial answer to these 
 questions. He has found that all the observations are very well represented 
 by supposing that each nucleus described an independent ellipse around the 
 sun. He has computed the orbits of the two bodies upon this supposition, from 
 the extensive and careful series of observations made upon them, and taking 
 into account the disturbing influence of Jupiter, Mars, the Earth, and Venus ; 
 and concludes that " the disturbing action of one nucleus upon the other must 
 have been extremely small, and that it is doubtful whether the observations 
 were sufficiently precise to render this influence in any degree sensible. He 
 has also shown that the increase of distance between the two nuclei, at least 
 during the interval from February IQth to March 22, was simply apparent, be- 
 ing due to the variation of distance from the earth and to the angle under which 
 their line of junction presented itself to the visual ray ; the real distance during 
 all that interval (neglecting small fractions) having been on an average about 
 thirty -nine times the semi-diameter of the earth, or less than two-thirds the dis- 
 tance of the moon from the earth's centre." 
 
 If it be true that the two bodies are in no sensible degree disturbed by their 
 mutual actions, as M. Plantamour infers from bis investigations, and as we should 
 
NOTES VI-IX. 
 
 naturally suppose from the probable minuteness of the two oometary masses, it 
 has been calculsted by Sir John Herschel, from Plantamour's elements, that 
 there will be an interval of 16 d - 4 between their next perihelion passages; " and 
 it will be therefore necessary, at their next reappearance, to look out for each 
 comet as a separate and independent body." " Nevertheless," as remarked by 
 Herschel, " as it is still perfectly possible that some link of connection may sub- 
 sist between them, it will not be advisable to rely on this calculation to the 
 neglect of a meet vigilant search throughout the whole neighborhood of the 
 more conspicuous one, lest the opportunity should be lost of pursuing to its con- 
 clusion the history of this strange occurrence." 
 
 The investigations of M. Plantamour have served to establish that the actual 
 separation of the two bodies did not occur at the time of the apparent separa- 
 tion in 1846. At what point of time anterior to that epoch it took place, it 
 would seem to be impossible to determine. In fact, it is quite possible that the 
 two bodies have been revolving independently of each other for an indefinite 
 time, and that the supposed division of one comet into two was really the chance 
 approach of two independent cometary bodies. Plantamour remarks that " the 
 extraordinary changes which the companion exhibited within the period of 
 few days, and which have often been noticed in other comets, seem to indicate 
 that the brightness of these objects does not depend merely upon their distance 
 from the earth and sun, but upon other unknown causes. These causes might 
 have developed sufficient brightness in the companion at its late return to the 
 sun to render it visible to us ; while at its former returns, on account of its un- 
 favorable position, the companion was too faint to be noticed," 
 
 NOTE VI. 
 
 The list given in the text has recently been increased by the addition of several 
 other comets, viz., De Vice's comet, period 6^- years, perihelion passage Sept. 2d, 
 1844 ; Brorsen's comet, period, according to Hind, 5| years, perihelion passage 
 Feb. 25th, 1847 ; Peters' comet, period nearly 16 years, perihelioa passage June 
 
 NOTE VII. 
 
 The reader will find a <somplete catalogue of all comets whose orbits have been 
 determined, up to 1846, m the American Almanac for 184*7. 
 
 NOTE VIII 
 
 The new planet, Neptune, proves to be the third planet in the order of mag- 
 nitude, being a little larger than Uranus. The newly-discovered asteroids 
 are probably of a more diminutive size than the other four. 
 
 NOTE IX. 
 
 A remarkable analogy in the periods of rotation of the primary planets was 
 discovered a few years since (1848) by Daniel Kirkwood, of Pottsville, Pennsyl- 
 vania. This analogy is now generally known by the name of JKirkwood's LOAD, 
 and is as follows : 
 
 " Let P be the point of equal attraction between any planet and the one next 
 
374 NOTE x. 
 
 interior, the two being in conjunction : P' that between the same and the on* 
 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 7 =the diameter of any other planet's sphere of attraction found in like 
 manner : 
 
 n = the number of sidereal rotations peformed by the former during one side- 
 real revolution round the sun ; 
 
 n'= the number performed by the latter ; then it will be found that 
 
 n 2 : w' 2 : : D 3 : D' 3 ; or n = n' (p 
 
 That is, the square of the number of rotations made by a planet during one revo- 
 lution 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 ex- 
 amination by Mr. Sears C. Walker, with the following result : " We may there- 
 fore conclude," says he, "that whether Kirkwood's Analogy is or is not the ex- 
 pression of a physical law, it is at least that of a physical fact in the mechanism 
 of the universe." (See the American Journal of Science, New Series, vol. x. 
 pp. 19-26.) 
 
 There are but three planets, viz., Venus, the Earth, and Saturn, for which all 
 the elements embraced in this law are known. The diameters of the spheres of 
 attraction of Mercury and Neptune are, from the nature of the case, incapable 
 of determination. The mass of the one planet into which the asteroids are sup- 
 posed once to have been united is not known with certainty, as there may be 
 asteroids yet undiscovered, and its period of rotation is hypothetical only. The 
 diameters of the spheres of attraction of Mars and Jupiter can only be approx- 
 imately determined; and the period of rotation of Uranus is unknown. Pro- 
 fessor Loomis, in a recent article, argues with a good deal of plausibility, that 
 " Uranus and the asteroids cannot be reconciled with Kirkwood's Law by any 
 admissible assumption with regard to the value of their elements." (See Silll- 
 man's Journal, vol. xl p. 217.) 
 
 The objections urged by Professor Loomis have been answered by Professor 
 Kirkwood. (See the Journal of Science, Second Series, vol. XL p. 394.) The 
 considerations adduced by him have served materially to weaken the force of 
 these objections. 
 
 The interest naturally awakened by the announcement of so important a dis- 
 covery was heightened by the fact, that it was at once perceived that it furnished 
 a new and powerful argument in support of the nebular hypothesis (or cosmog- 
 ony) devised by Laplace. (See a paper on this subject by Dr. B. A. Gould, Jr., 
 in the Journal of Science, New Series, vol. x. p. 26, <fcc.) 
 
 NOTE X. 
 
 A new ring of Saturn, interior to the other two, was discovered by Mr. G. P. 
 Bond, assistant at the observatory of Harvard University, on the llth of 
 November, 1850. It was subsequently observed by the Messrs. Bond on re- 
 peated occasions, from that date to the 7th of January, 1851. It shone with a 
 pale dusky light. Its inner edge was sharply defined, but the side next the old 
 ring was not so definite ; so that it was impossible to make out with certainty 
 whether the new was connected with the old ring or not. According to Mr. 
 Bond's measurements the breadth of the new ring is 1".5. 
 
 " The same appearances were noticed by the Rev. W. R. Dawes, at his ob- 
 servatory, near Maidstone, in England, on the 25th and 29th of November, and 
 subsequently by Mr. Lassell, of Starfield, near Liverpool." 
 
NOTES xi, xn. 375 
 
 Mr. G. P. Bond has propounded a bold and ingenious theory relative to the 
 physical constitution of Saturn's rings ; which is, that " they are in a fluid state. 
 and within certain limits change their form and position in obedience to the 
 laws of equilibrium of rotating bodies." He conceives, also, that under peculiar 
 circumstances of disturbance several subdivisions of the two fluid rings may 
 take place, and continue for a short time until the sources of disturbance are 
 removed, when the parts thrown off would again reunite. He supports his 
 theory by arguments drawn from the results of observation, and by certain 
 physical considerations. The chief argument derived from observation is, that 
 several apparent subdivisions of the double ring have been noticed by different 
 observers from time to time, and that these have in general been invisible to the 
 same observers with the same telescopes, and under equally favorable circum- 
 stances, and have also entirely escaped the observation of many other observers 
 provided with equally good telescopes. It is supposed that these facts admit 
 of explanation only on the hypothesis that the ring is a fluid mass, capable of 
 occasional subdivision. (See Mr. Bond's original paper on this subject, published 
 in Nos. 25 and 26 of the Astronomical Journal.) 
 
 Professor Peirce, of Harvard University, has followed up the speculations of 
 Mr. Bond, by undertaking to demonstrate, from purely mechanical considera- 
 tions, that Saturn's ring cannot be solid. " I maintain, unconditionally," says he, 
 " that there is no conceivable form of irregularity and no combination of irregu- 
 larities, consistent with an actual ring, which would serve to retain it perma- 
 nently about the primary if it were solid." 
 
 He is led by his investigations to the curious result, that Saturn's ring it 
 sustained in a position of stable equilibrium about the planet solely by the 
 attractive power of his satellites ; and that " no planet can have a ring unless it 
 is surrounded by a sufficient number of properly arranged satellites." (See 
 Astronomical Journal for June 16th, 1851.) 
 
 NOTE XI. 
 
 The seventh satellite of Saturn, in the order of distance from the primary, 
 was discovered by the Messrs. Bond, with the great refractor of the Cambridge 
 Observatory, on the 16th of September, 1848; and observed two days after- 
 wards by Mr. Lassell, at Starfield, near Liverpool, with his large reflector. In 
 fact, it appears to have been distinctly made out to be a satellite by these two 
 observers on the same night, viz., that of the 19th of September. 
 
 " The orbit of the new satellite serves to fill up a large chasm before existing 
 between the 6th and 8th satellites (see Table VI). It is fainter than either of the 
 two interior satellites discovered by Sir William Herschel. Its time of revolur 
 tion is about 21.18 days, the semi-axis of its orbit, at the mean distance of 
 Saturn, 214", and Messrs. Bond and Lassell have concurred in giving it the 
 name of Hyperion." 
 
 The periods of revolution, and the mean distances of the satellites of Saturn 
 from their primary, together with the mythological names proposed for them by 
 Sir John Herschel, are given in Table VI. 
 
 NOTE XII. 
 
 " Two of the satellites of Uranus are much more conspicuous than the rest. 
 and their periods and distances from the planet have been ascertained with 
 tolerable certainty. They are the second and fourth of those set down in the 
 synoptic table (Table VI). Of the remaining four, whose existence, though an- 
 nounced with considerable confidence by their original discoverer, could hardly 
 be regarded as fully demonstrated, two only have been, hitherto re-observed ; 
 
376 NOTE xm. 
 
 viz., the first of our table, interior to the two larger ones, by the independent 
 observations of Mr. Lassell, and M. Otto Struve, and the third, intermediate be- 
 tween the larger ones, by the former of these astronomers. The remaining two, 
 if future observation should satisfactorily establish their real existence, will 
 probably be found to revolve in orbits exterior to all these." (Herschel's Out- 
 lines of Astronomy, Art. 551.) 
 
 It is just announced (Nov. 28th, 1851), that Mr. Lassell has discovered two 
 new satellites attending upon Uranus. The following information is communi- 
 cated with respect to them : " They are interior to the innermost of the two 
 bright satellites first discovered by Sir William Herschel, and generally known 
 as the second and fourth. It would appear that they are also interior to Sir 
 William's first satellite, to which he assigned a period of revolution of about five 
 days and twenty-one hours, but which satellite I have as yet been unable to 
 recognize. I first saw these two of which I now communicate the discovery, on 
 the 24th of last month, and had then little doubt that they would prove satel- 
 lites. I obtained further observations of them on the 28th and 30th of October, 
 and also last night (Nov. 2d), and find that for so short an interval the observa- 
 tions are well satisfied by a period of revolution of almost exactly four days for 
 the outermost, and two days and a half for the closest. They are very faint ob- 
 jects ; certainly not half the brightness of the two conspicuous ones ; but all the 
 four were last night steadily visible, in the quieter moments of the air, with a 
 magnifying power of 778 on the 20 ft. equatorial." 
 
 This discovery would seem to confirm the inference drawn by Mr. Dawes, from 
 a discussion of the observations formerly made by Lassell and Struve upon the 
 nearest satellite. He considers these observations incompatible with each other. 
 " While Struve's observations indicate a period of three days and twenty hours, 
 LasselFs observations indicate a period of only two days and two hours. He 
 therefore infers that there must be, at least, two satellites interior to that which 
 Herschel denominates the second." He also considers it doubtful whether the 
 other satellite discovered by Lassell is really Herschel s third satellite, as stated 
 above. 
 
 It would seem, therefore, that at least two, and perhaps three, of the Her- 
 echelian satellites have been seen by later observers, and that two new satellites 
 have probably been discovered by Lassell. Accordingly Uranus has certainly 
 three satellites, and probably as many as eight. 
 
 Neptune. 
 
 The apparent diameter of Neptune is nearly 3", and its actual diameter is 
 41,500 miles. " To two observers it has afforded strong suspicion of being sur- 
 rounded with a ring very highly inclined ; and from the observations of Mr. Las- 
 eell, M. Otto Struve, and Mr. Bond, it appears to be attended certainly by one, 
 and very probably by two satellites, though the existence of the second can hardly 
 yet be considered as quite demonstrated." (For the details of the interesting 
 history of the discovery of this planet, see Herschel's Outlines of Astronomy, or 
 Loomis's Progress of Astronomy.) 
 
 THE NEW ASTEROIDS, 
 
 Astrcea, Hebe, Iris, Flora, Metis, Hygeia, Parthenope, Clio, Egeria, Irene, 
 
 Eunomia. 
 
 Of the dimensions and other physical peculiarities of these planetary bodies, 
 no knowledge has as yet been obtained, further than that they are very small 
 bodies, and probably inferior in size to the other four asteroids. They are all 
 of about the ninth apparent magnitude, except Metis, which is of the tenth or 
 eleventh. 
 
 NOTE XIII. 
 
 Certain remarkable phenomena were exhibited by Biela's comet at its hut 
 return (in 1846), an account of which will be found in Note V. 
 
NOTES XIV, XV. 37? 
 
 NOTE XIV. 
 
 The great problem of the determination of the parallax and distance of a 
 fixed star, first solved by Bessel, has since been undertaken, with success by 
 other astronomers. The following is a list of the most reliable determinations 
 which have been hitherto obtained : 
 
 a Centauri 0".913 (Henderson). 
 
 61 Cygni .348 (Bessel). 
 
 aLyrae .261 (Struve). 
 
 Sirius .230 (Henderson). 
 
 ' n lnfi j Peters, Struve, Preuss, 
 
 Polar18 ; - 106 i andLindenau. 
 
 In the case of the Pole Star, the estimated error to which the result obtained 
 is liable, is l / 9 of the parallax. For the other stars it is a still smaller fraction. 
 The parallax of the pole star indicates a distance which light would require 
 more than 80 years to traverse. 
 
 The measurements for a Lyrse, as well as for 61 Cygni, were made with a 
 micrometer. Professor Henderson determined the parallax of a Centauri, from 
 a discussion of a series of observations upon that star made by him, with a large 
 mural circle, in the years 1832 and 1838, at the Royal Observatory of the Cape 
 of Good Hope. Subsequent observations with a similar but more efficient in- 
 strument by Mr. Maclear, have conducted to very nearly the same result. The 
 observations by M. Peters were made with the great vertical circle of the Pul- 
 kova Observatory. His observations with this instrument upon 61 Cygni gave 
 a parallax almost identical with that found by Bessel. This same observer has 
 also undertaken to determine the parallax of several other stars, with the fol- 
 lowing results: Arcturus (0".127), Iota Ursse Majoris (0".138), 1830 Groom- 
 bridge (0".226), Capella (0".046), a Cygni (no measurable parallax). But the 
 probable errors are one-half, or more, of the parallaxes found. 
 
 NOTE XV. 
 
 It is an interesting fact, ascertained by M. Argelander, of Bonn, that the 
 periods of several of the variable stars are subject to a slow alteration. The 
 two stars, Omicron Ceti and Algol, may be cited as examples. It is conjectured 
 that these variations of period are periodical 
 
 Sir John Herschel, in his " Outlines of Astronomy," gives a list of thirty -four 
 variable stars whose periods have been approximately or roughly determined, 
 but each year adds to the number. There are many other stars known to be 
 variable, but whose periods and limits of variation of brightness are unknown. 
 
 The statement made in the text of the second general fact noticed with 
 respect to variable stars should read thus : they pass from their epoch of least 
 light to that of their greatest in considerably less time than from their greatest 
 to their least. 
 
 " The alterations of brightness in the southern star n Argus, which have been 
 recorded, are very singular and surprising. In the time of Halley (1677) it 
 appeared as a star of the fourth magnitude. Lacaille, in 1751, observed it of the 
 second; in the interval from 1811 to 1815 it was again of the fourth; and 
 again, from 1822 to 1826, of the second. On the 1st of February, 1827, it was 
 noticed by Mr. Burchell to have increased to the first magnitude, and to equal 
 a Crucis. Thence again it receded to the second ; and so continued until the end 
 of 1837. All at once, in the beginning of 1838, it suddenly increased in lustre 
 so as to surpass all the stars of the first magnitude, except Sirius, Canopus, and 
 a Centauri, which last star it nearly equalled. Thence it again diminished, but 
 this time not below the first magnitude, until April, 1843, when it had again 
 increased so as to surpass Canopus, and nearly equal Sirius in splendor. A 
 
 48 
 
/ 
 *v 
 
 378 NOTES xvi-xvni. 
 
 strange field of speculation is opened by this phenomenon. The temporary 
 stars heretofore recorded have all become totally extinct. Variable stars, so far 
 as they have been carefully attended to, have exhibited periodical alternations, 
 in some degree, at least, regular, of splendor and comparative obscurity. But 
 here we have a star fitfully variable to an astonishing extent, and whose fluctu- 
 ations are spread over centuries, apparently in no settled period, and with no 
 regularity of progression. "What origin can we ascribe to these sudden flashes 
 and relapses ? What conclusions are we to draw as to the comfort or habita- 
 bility of a system depending for its supply of light and heat on so uncertain a 
 source." (Herschel's Outlines.) 
 
 NOTE XVI. 
 
 It must be conceded that the change in the length of the periods of the vari- 
 able stars, noticed in the previous note, is apparently at variance with the 
 theory given in the text, since all analogy teaches that the periods of rotation 
 should be uniform. Argelander, who has studied the phenomena of variable 
 stars more attentively than any other observer, is of the opinion that " the time 
 has not come in which we should prepare to frame a theory. The minute 
 changes characterizing the phenomena have been too little studied and dis- 
 cussed." 
 
 NOTE XVII. 
 
 "Among the most remarkable triple, quadruple, or multiple stars, may be 
 enumerated, 
 
 a Andromedae. 
 s Lyrae. 
 ?; Cancri. 
 
 Orionis. 
 H Lupi. 
 H Bootis. 
 
 Scorpii. 
 
 11 Monocerotis. 
 
 12 Lyncis. 
 
 Of these a Andromedae, // Bootis, and /a Lupi, appear in telescopes even of 
 considerable optical power, only as ordinary double stars ; and it is only when 
 excellent instruments are used that their smaller companions are subdivided 
 and found to be in fact extremely close double stars, t Lyrae offers the remark- 
 able combination of a double-double star. Viewed with a telescope of low 
 power, it appears as a close and easily divided double star ; but on increasing 
 the magnifying power, each individual is perceived to be beautifully and closely 
 double, the one pair being about 2", the other about 3" asunder. Each of the 
 stars, Cancri, Scorpii, 11 Monocerotis, and 12 Lyncis, consists of a principal 
 star, closely double, and a smaller and more distant attendant, while Orionis 
 presents the phenomenon of four brilliant principal stars, of the respective 4th, 
 6th, 7th, and 8th magnitudes, forming a trapezium, the longest diagonal of which 
 is 21".4, and accompanied by two excessively minute and very close companions, 
 to perceive both of which is one of the severest tests which can be applied to a 
 telescope." (Herschel's Outlines.) 
 
 NOTE XVIII, 
 
 Later observations have led to the discovery that the star Indi has a greater 
 motion than any other star, the amount of its annual displacement being 
 
NOTE XIX. 
 
 379 
 
 An interesting confirmation of the solar motion mentioned in Art. 593 has 
 recently been obtained by Mr. Galloway, from a discussion* of certain observa- 
 tions made at different epochs and by different observers upon eighty-one stars 
 of the southern hemisphere. He concludes from his discussion, that the point 
 towards which the sun's motion is directed, is situated in R. A. 260 1' and N. 
 Dec. 34 23' ; " a result so nearly identical with that afforded by the northern 
 hemisphere as to afford a full conviction of its near approach to truth, and what 
 may fairly be considered a demonstration of the physical cause assigned." 
 
 NOTE XIX. 
 
 The following, according to Herschel, are the places, for 1830, of the principal 
 globular clusters, as specimens of their class : 
 
 E. A. 
 
 N. P. D. 
 
 B.A. 
 
 N. P. D. 
 
 B.A. 
 
 N. P. D. 
 
 h. m. i. 
 
 o / 
 
 h. m. a. 
 
 o / 
 
 h. m. s. 
 
 o / 
 
 16 25 
 
 163 2 
 
 15 9 56 
 
 87 16 
 
 17 26 51 
 
 143 34 
 
 9 8 33 
 
 154 10 
 
 15 34 56 
 
 127 13 
 
 17 28 42 
 
 93 8 
 
 12 47 41 
 
 159 57 
 
 16 6 55 
 
 112 33 
 
 18 26 4 
 
 114 2 
 
 13 4 30 
 
 70 55 
 
 16 23 2 
 
 102 40 
 
 18 55 49 
 
 150 14 
 
 18 16 38 
 
 136 35 
 
 16 35 37 
 
 53 13 
 
 21 21 43 
 
 78 34 
 
 13 34 10 
 
 60 46 
 
 16 50 24 
 
 119 61 
 
 21 24 40 
 
 91 34 
 
 Many of the nebulous objects in the heavens hitherto classed among resolvable 
 nebulae, have lately been resolved by the magnificent reflecting telescope con- 
 structed by Lord Rosse ; and many nebulae which have offered no appearance 
 of stars to all previous observers, and which were supposed by the elder Her- 
 schel to be collections of nebulous matter, have either been partially resolved 
 by this telescope, or have assumed in it the appearance of resolv ability. In 
 view of these facts it must be conceded, that " although nebulae do exist, which 
 even in this powerful telescope appear as nebulae, without any sign of resolu- 
 tion, it may very reasonably be doubted whether there be really any essential 
 physical distinction between nebulae and clusters of stars, at least in the nature 
 of the matter of which they consist, and whether the distinction between such 
 nebulae as are easily resolved, barely resolvable with excellent telescopes, and 
 altogether irresolvable with the best, be any thing else than one of degree, 
 arising merely from the excessive minuteness and multitude of the stars, of 
 which the latter, as compared with the former, consist." 
 
 Sir James South, who made a trial of Lord Rosse's monster telescope in 
 March, 1845, gives the following account of his observations: "Never before hi 
 my life did I see such glorious sidereal pictures as this instrument afforded us. 
 The most popularly known nebulae observed were the ring nebula in the Canes 
 Venatici, which was resolved into stars with a magnifying power of 548, and 
 the 94th of Messier, which is in the same constellation, and which was resolved 
 into a large globular cluster of stars, not much unlike the well-known cluster in 
 Hercules. On subsequent nights observations of other nebulae, amounting to 
 some thirty or more, removed most of these from the list of nebulae, where they 
 had long figured, to that of clusters ; while some of these latter exhibited a 
 sidereal picture in the telescope such as man before had never seen, and which, 
 for its magnificence, baffles all description." 
 
 The following are some of the nebulae which have assumed a new and re- 
 markable appearance when viewed through Lorjl Rosse's telescopes, of 3 ft. and 
 6 ft. aperture : 
 
 1. The Crab-nebula. To previous observers this curious object presented the 
 appearance of an oval resolvable nebula. " Lord Rosse's three feet reflector 
 exhibits it with resolvable filaments singularly disposed, springing principally 
 from its southern extremity, and not, as is usual, in clusters, irregularly in all 
 
380 NOTE XIX. 
 
 directions. It is studded with stars, mixed, however, with a nebulosity, probably 
 consisting of stars too minute to be recognized." 
 
 2. The Dumb-bell nebula, so named from its resemblance to a dumb-bell, as 
 shown by Sir John Herschel's drawing (see Nichol's Architecture of the Heavens), 
 in Lord Rosse's 8 ft, telescope, has quite a different appearance, and is seen to 
 consist of innumerable stars mixed with nebulosity. 
 
 3. The nebula in the Dog's Ear was formerly described as having the form of 
 a ring, divided through about one-third of its course into two separate branches 
 or streams, and thus regarded as presenting a singular counterpart to our own 
 Milky Way. In Lord Rosse's six feet reflector "the former simple shape is 
 transformed into a scroll, apparently 'unwinding with numerous filaments, and a 
 mottled appearance, which looks like the breaking up of a cluster." It has 
 accordingly received the designation of the Scroll or Spiral nebula. 
 
 4. The great nebula in Orion has also been divested of the mystery in which 
 it has so long remained enshrouded, by the same telescope. Lord Rosse says : 
 " I may safely say that there can be little if any doubt as to the resolvability of 
 this nebula. We can plainly see that all about the trapezium is a mass of stars ; 
 the rest of the nebula also abounding with stars, and exhibiting the characteris- 
 tics of resolvability strongly marked" 
 
 Mr. Bond, with the great refractor at Cambridge, has also succeeded in resolv- 
 ing the brighter portion of this nebula immediately adjacent to the trapezium, 
 or the sextuple star 8. 
 
 The great nebula in Andromeda, mentioned in the text, has also been care- 
 fully observed with the Cambridge refractor, and decisive evidence obtained of 
 its resolvability. 
 
 Detailed descriptions of these two nebulae, as seen with the Cambridge tele- 
 scope, accompanied with accurate drawings, have been published by the Messrs. 
 Bond (Transactions of the American Academy of Arts and Sciences, vol. iii). 
 
 In the southern hemisphere there are two remarkable nebulous masses of 
 light, conspicuously visible to the naked eye, which are known by the name of 
 Magellanic Clouds, or Nubeculce (major and minor). Sir John Herschel de- 
 scribes them as being in the appearance and brightness of their light not unlike 
 portions of the Milky Way of the same apparent size, and round or oval in their 
 general form. 
 
 " When examined through powerful telescopes, the constitution of the nube- 
 culae, and especially of the nubecula major, is found to be of astonishing com- 
 plexity. The general ground of both consists of large tracts and patches of 
 nebulosity, in every stage of resolution, from light irresolvable with 18 inches of 
 reflecting aperture, up to perfectly separated stars like the Milky Way, and 
 clustering groups sufficiently insulated and condensed to come under the desig- 
 nation of irregular, and in some cases pretty rich clusters. But, besides these, 
 there are also nebulae in abundance, both regular and irregular ; globular clus- 
 ters in every state of condensation ; and objects of a nebulous character quite 
 peculiar, and which have no analogue in any other region of the heavens. Such 
 is the concentration of these objects, that in the area occupied by the nubecula 
 major, not fewer than 278 nebulae and clusters have been enumerated, besides 
 60 or 60 outliers, which (considering the general barrenness in such objects of 
 the immediate neighborhood) ought certainly to be reckoned as its appendages, 
 being about 6 per square degree, which very far exceeds the average of any 
 other, even the most crowded parts of the nebulous heavens. In the nubecula 
 minor the concentration of such objects is less, though still very striking, 37 hav- 
 ing been observed within its area, and 6 adjacent but outlying. The nubeculae, 
 then, combine, each within its own area, characters which in the rest of the 
 heavens are no less strikingly separated ; viz., those of the galactic and the 
 nebular system. Globular clusters (except in one region of small extent) and 
 nebulae of regular elliptic forms are comparatively rare in the Milky Way, and 
 are found congregated in the greatest abundance in a part of the heavens the 
 most remote possible from thai circle ; whereas, in the nubeculae they are indis- 
 criminately mixed with the general starry ground, and with irregular though 
 email nebulae." (Uerschel's Outlines of Astronomy.) 
 
NOTES XX. XXI. 381 
 
 NOTE XX. 
 
 According to Struve, the nebula in Andromeda is 1$ long by 16' broad, and 
 thus nearly one-half greater than the moon's disk. Mr. G. P. Bond describes it 
 as extending nearly 2 in length, and upwards of 1 in breadth. 
 
 Since, as stated in Note XIX, many of the nebulae, which were supposed by 
 Sir William Herschel to be masses of nebulous matter, have recently been found 
 to consist of stars, it must now be regarded as exceedingly doubtful whether 
 any such supposed nebulous masses really exist in space ; and, on the other 
 hand, highly probable that all the irresolvable nebulae are only vast beds of 
 stars either too remote, or composed of too small or too closely compacted stars, 
 to appear otherwise than one general mass of cloudy light in the best tele- 
 scopes. 
 
 NOTE XXI. 
 
 Struve, of the Pulkova Observatory, in a recent work entitled 6tudes d' As- 
 tronomic Stellaire, has undertaken to establish that the stratum of the Milky 
 Way is really fathomless (at least in every direction except, perhaps, at right 
 angles to the stratum), and shows, by quotations from his later papers en the 
 Milky Way, that Sir William Herschel was led finally to entertain the* same 
 opinion, in opposition to the views he had at first expressed (in 1785). Accord- 
 ingly, by comparing the number of stars seen in the field of view of a telescope 
 when pointed in two different directions into space, we do not obtain the rela- 
 tive distances through to the boundaries of the stratum of the Milky Way, but 
 only the relative condensation of the stars, or relative density of the starry 
 stratum in the two directions. Every augmentation in the power of the tele- 
 scope brings into view, in these directions, other stars before invisible. 
 
 Struve remarks : " It may be asked why astronomers have generally main- 
 tained the old theory concerning the Milky Way, propounded in 1785,. although 
 it had been entirely abandoned by the author himself, as we have demonstrated. 
 I believe that the explanation must be sought in two circumstances. It was a 
 complete system, imposing from the boldness and geometric precision of its 
 construction, and which the author has never revoked as a whole. In his trea- 
 tises, published since 1802, we meet with only partial views, but which are suf- 
 ficient, when they are compared together, to exhibit the final idea of the great 
 astronomer." 
 
 Sir John Herschel does not give his assent to the opinion expressed by Struve. 
 He remarks : " Throughout by far the larger portion of the extent of the Milky 
 Way in both hemispheres, the general blackness of the ground of the heavens 
 on which its stars are projected, and the absence of that innumerable multitude 
 and excessive crowding of the smallest visible magnitudes, and of glare pro- 
 duced by the aggregate light of multitudes too small to affect the eye singly, 
 which the contrary supposition would seem to necessitate, must, we think, be 
 considered unequivocal indications that its dimensions in directions where these 
 conditions obtain, are not only not infinite, but that the space-penetrating power 
 of our telescopes suffices fairly to pierce through and beyond it." 
 
 If it be true that the stratum of the Milky Way is really fathomless that 
 infinite space is occupied by an infinite number of shining stars, the central suns 
 of planetary systems clustered around them, ay first suggested by Kant, then it 
 has been shown by Olbers that the aspect of the sky should be that of a vault 
 shining in all directions with a lustre similar to that of the sun. The conclusion, 
 therefore, is inevitable, either that the bed of stars in which our sun is posited 
 is not infinite in extent, or that space is not perfectly transparent ; in other 
 words, that the light coming from the stars suffers a partial extinction, propor- 
 tional in amount to the distance traversed by it. The latter view was advo- 
 cated by Olbers, and is also adopted by Struve, who by means of this concep- 
 tion endeavors to reconcile his views of the boundless extent of our firmament 
 
382 NOTE xxn. 
 
 with the feeble luminosity of the sky. He conceives, upon a detailed investiga 
 tion, that the actual luminosity of the sky in different directions is adequately 
 explained, in accordance with his theory of the unlimited extent of the stratum 
 of the Milky Way, if it be allowed that the light of the stars suffers an extinc- 
 tion of only Vioo in traversing a distance equal to that of a star of the first mag 
 nitude. Upon this supposition the extinction for the most distant stars visible 
 in telescopes would amount to 88 per cent. 
 
 Herschel urges, in opposition to this theory, that " if applicable to any, it is 
 equally so to every part of the galaxy. We are not at liberty to argue that at 
 one part of its circumference our view is limited by this sort of cosmical veil 
 which extinguishes the smaller magnitudes, cuts off the nebulous light of distant 
 masses, and closes our view in impenetrable darkness ; while at another we are 
 compelled by the clearest evidence telescopes can afford to believe that star- 
 strewn vistas lie open, exhausting their powers and stretching out beyond their 
 utmost reach, as is proved by that very phenomenon which the existence of 
 such a veil would render impossible, viz., infinite increase of number and dimi- 
 nution of magnitude, terminating in complete irresolvable nebulosity." 
 
 NOTE XXII. 
 
 Or Bather, when the planets are compared with respect to density, it will be 
 seen that they may be divided into two classes, viz. : one class, comprising Mer- 
 cury, Venus, the Earth, and Mars, each of which has a density nearly equal to 
 unity ; and a second class, consisting of Jupiter, Saturn, Uranus, and Neptune, 
 whose density is between 0.13 and 0.23. 
 
 It is a curious fact that the same classification holds with respect to magni- 
 tude and period of rotation. 
 

* : * 
 
 
TABLE I. 
 
 Latitudes and Longitudes from the Meridian of Greenwich, of 
 some cities t and other conspicuous places. 
 
 Names of Places. 
 
 Latitude. 
 
 Longitude 
 in Degrees. 
 
 Longitude 
 in Time. 
 
 Albany, Capitol, New York, 
 Altona, Obs., Denmark, 
 Baltimore, Bait. Mon't, } Maryland, 
 Berlin, Obs , Germany, 
 Boston, State House, Massach'ts, 
 
 / // 
 
 42 39 3 N 
 53 32 45 N 
 39 17 23 N 
 52 31 13 N 
 42 21 28 N 
 
 o / // 
 73 44 49 W 
 9 56 89 E 
 76 37 30 W 
 13 23 52 E 
 71 4 9W 
 
 4 54 59.3 
 39 46 6 
 5 6 30 
 53 a5 5 
 4 44 16.6 
 
 Bremen, Obs., Germany, 
 Brunswick, Eowdoin Coll., Maine, 
 Canton, China, 
 Cape of Good Hope, Obs., Africa, 
 Cape Horn, S. America, 
 
 53 436 N 
 43 53 ON 
 23 8 9N 
 33 56 3S 
 55 58 41 S 
 
 8 48 58E 
 69 55 1 W 
 113 16 54 E 
 18 23 45 E 
 67 1053 W 
 
 35 15.9 
 4 39 40 
 7 33 8 
 1 13 o5.0 
 4 23 43 
 
 Charleston, St. Mich'* Ch., S. Carolina,- 
 Charlottesville, Univers., Virginia, 
 Cincinnati, Fort Wash , Ohio, 
 Copenhagen, Obs., Denmark, 
 Dorpat, Obs., Russia, 
 
 32 46 33N 
 38 2 3N 
 39 5 54N 
 55 40 53 N 
 
 53 22 47 N 
 
 79 57 27 W 
 78 31 29 W 
 84 27 W 
 12 34 57 E 
 26 43 45E 
 
 5 19 49 8 
 5 14 6 
 5 37 43 
 50 19 8 
 1 4655 
 
 Dublin, Obs., Ireland, 
 Edinburgh, Obs., Scotland, 
 Gptha, Obs., Germany, 
 Gottingen. Obs., Germany, 
 Greenwich, Obs., England, 
 
 53 23 13 N 
 55 57 23 N 
 
 50 56 5N 
 51 31 48 N 
 51 28 39 N 
 
 6 20 SOW 
 3 10 54 W 
 10 44 6 E 
 9 5637E 
 000 
 
 25 22 
 12 43 6 
 42 56.4 
 39 46.5 
 000 
 
 Konigsberg, Obs., Prussia, 
 London, St. Paul's Ch., England, 
 Marseilles, Obs., France, 
 Milan, Obs., Italy, 
 Naples, Obs., Italy, 
 
 54 42 SON 
 51 30 49 N 
 43 17 50 N 
 4528 IN 
 40 51 47 N 
 
 20 30 7E 
 5 48W 
 5 22 15 E 
 9 11 48 E 
 14 Id 4 E 
 
 1 22 *5 
 
 23 
 21 29 
 36 47.2 
 057 0.3 
 
 New Haven. College, Connecticut, 
 New Orleans, City Hall, Louisiana, 
 New York, City Hall, New York, 
 Palermo, Obs., Italy, 
 Paramatta, Obs., New Holl'd, 
 
 41 18 30 N 
 29 57 45 N 
 40 42 40N 
 38 6 44N 
 
 33 48 SOS 
 
 72 56 45 W 
 90 6 49 W 
 74 1 8W 
 13 21 24 E 
 151 1 34 E 
 
 4 51 47 
 6 027 
 4 56 45 
 5325.6 
 10 4 6.3 
 
 Paris, Obs., France, 
 Petersburgh, Obs., Russia, 
 Philadelphia, IncTce Hall, Pennsylv'a, 
 Point Venus, Otaheite, 
 Princeton, College, New Jersey, 
 
 4850 13 N 
 59 56 31 N 
 39 56 59 N 
 17 29 21 S 
 40 20 41 N 
 
 2 20 24 E 
 
 30 18 57 E 
 75 9 54 W 
 149 23 55 W 
 74 39 33 W 
 
 921.6 
 2 1 15.8 
 5 39.6 
 957 56 
 4 58 38.2 
 
 Providence, University, Rhode Isl'd, 
 Quebec, Castle, L. Canada, 
 Richmond, Capitol, Virginia, 
 Rome, Roman College, Italy, 
 Savannah, Exchange, Georgia, 
 
 41 49 22 N 
 46 49 12 N 
 37 32 17 N 
 41 53 52 N 
 32 456N 
 
 71 24 48 W 
 71 16 W 
 77 27 2SW 
 12 28 40 E 
 81 8 18 W 
 
 4 45 39 2 
 4 45 4 
 5 50 
 49 54.7 
 52433 
 
 Schenectady, New York, 
 Stockholm, Obs., Sweden, 
 Turin, Obs., Italy, 
 Vienna, Obs., Austria, 
 Wardhus, Lapland, 
 "Washington, Capitol, Diet. Colum. 
 
 4248 N 
 59 2031 N 
 45 4 6N 
 48 1235 N 
 70 22 36 N 
 38 53 84 N 
 
 73 55 W 
 18 3 44 E 
 7 42 6E 
 1623 OE 
 31 7 54 E 
 77 1 30 \V 
 
 4 55 40 
 1 12 15 
 30 48.4 
 1 5 32 
 2 4 32 
 586 
 
2 TABLE II. Elements of the, Planetary Orbits. 
 
 Epoch for Vesta, Juno, Ceres, and Pallas, July 23d, 1831, mean noon at 
 Berlin : for the other planets, Jan. 1, 1801, mean noon at Greenwich.* 
 
 Planet's Inclination to q^ v Longitude of As- q.- vr Longitude of 
 Name. the Ecliptic. Sec. Var. cen ding Node. bec ' Var ' Perihelion. 
 
 Mercury 
 
 Venus 
 
 Earth 
 
 Mars 
 
 Vesta 
 
 Juno 
 
 Ceres 
 
 Pallas 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 7 9.1 
 3 23 285 
 
 1 51 62 
 
 7 7 57.3 
 13 2 10 
 10 36 55 7 
 34 35 49.1 
 
 1 18 51.3 
 
 2 29 35.7 
 46 28.4 
 
 + 18.2 
 
 46 
 
 0.2 
 
 12 
 
 44 
 
 22.6 
 
 15.5 
 + 3.1 
 
 45 57 30 9 
 74 51 55 
 
 48 35 
 103 20 28.0 
 170 52 34.5 
 
 80 53 49.7 
 172 38 29 8 
 
 98 26 18 9 
 111 56 37.4 
 
 72 59 35.3 
 
 70.44 
 51.10 
 
 4167 
 26 
 
 + 25 
 
 57.18 
 , 51.12 
 
 + 23.58 
 
 
 
 74 21 46 9 
 128 43 53 1 
 
 99 31 9.9 
 332 23 56 6 
 249 11 37. 
 
 54 17 12.7 
 147 41 23.5 
 121 5 0.5 
 
 11 8 34.6 
 
 89 9 29.8 
 167 31 16.1 
 
 Sec. Var. 
 
 + 9322 
 + 78.30 
 -j- 103.15 
 
 109.71 
 
 157 
 
 
 + 202 
 
 + 9459 
 + 115.68 
 + 87.44 
 
 Ptonot'a ivr n m. Mean Distance from Mean Distance from Eccentricity in Parts 
 flanet s JMame. g un . or Semi-axis. Sun in Miles. of the Semi-axis. 
 
 Sec. Variation. 
 
 Mercury 
 
 Venus 
 
 Earth 
 
 Mars 
 
 Vesta 
 
 Juno 
 
 Ceres 
 
 Pallas 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 03S70981 
 0.7233316 
 1.0000000 
 1.5236923 
 2.3614800 
 2.6694600 
 2.7709100 
 2.7726300 
 5.2027760 
 9.5387861 
 191823900 
 
 36814000 
 
 68787000 
 
 95103000 
 
 144908000 
 
 224584000 
 
 253874000 
 
 263522000 
 
 263685000 
 
 494797000 
 
 907162000 
 
 1824290000 
 
 020551494 
 000686074 
 001678357 
 0.09330700 
 0.08856000 
 0.25556000 
 0.07673780 
 0.24199800 
 004816210 
 0.05615050 
 0.04661080 
 
 + .000003866 
 
 .000062711 
 
 .000041630 
 + .000090176 
 + .000004009 
 
 .000005830 
 
 + .000159350 
 
 .000312402 
 
 .000025072 
 
 'Janet's Name. 
 
 Mean Longitude at 
 the Epoch. 
 
 Mean Sidereal Period 
 Mean Solar Days. 
 
 n Motion in mean Lon. 
 in 1 yr. of 365 days. 
 
 Mercury 
 
 Venus 
 
 Earth 
 
 Mars 
 
 Vesta 
 
 Juno 
 
 Ceres 
 
 Pallas 
 
 Jupiter 
 
 Saturn 
 
 Uranus 
 
 166 .0 48.6 
 
 11 33 30 
 
 100 39 13 3 
 
 64 22 55.5 
 
 84 47 3,2 
 
 74 39 43 6 
 
 307 3 25.6 
 
 290 38 11.8 
 
 112 15 23.0 
 
 135 20 6.5 
 
 177 48 23.0 
 
 87.9692580 
 
 224.7007869 
 
 365.2563770 
 
 6869796458 
 
 1325.4850000 
 
 1593 0670000 
 
 16847350000 
 
 1686.3050000 
 
 4332.5848212 
 
 10759.2198174 
 
 30686.8208296 
 
 53 43 3.6 
 224 47 29.7 
 14 19.5 
 191 17 9.1 
 
 30 20 31.9 
 
 12 13 36.1 
 
 4 17 45.1 
 
 Mean Daily Motion 
 in Longitude. 
 
 4 5 32.6 
 1 36 7.8 
 59 8.3 
 31 26.7 
 16 17.9 
 13 33.7 
 12 49.4 
 12 48.7 
 4 59.3 
 2 0.6 
 42.4 
 
 TABLE III. Elements of Moon's Orbit. Epoch, Jan. 1, 1801. 
 
 Mean inclination of orbit ... 
 Mean longitude of node at epoch 
 Mean longitude of perigee at epoch - 
 Mean longitude of moon at epoch 
 Mean distance from earth, or semi-axis 
 Eccentricity in parts of serai-axis 
 
 Mean sidereal revolution - 
 
 Mean tropical do. ... 
 
 Mean synodical do. .... 
 
 Mean anomalistic do. - 
 
 Mean nodical do. - 
 
 Mean revolution of nodes ; aider. 
 
 Mean revolution of perigee ; sider. - 
 
 d A m 
 
 5 8 47.9 
 13 53 17.7 
 266 10 7.5 
 118 17 8.3 
 59r. 964350 
 0.0548442 
 
 27 7 43 11.5 = 27.32166142 
 27 7 43 4.7 = 27 32158242 
 29 12 44 2.9 = 29.53058872 
 27 13 18 37.4 = 27.55459950 
 27 5 5 36.0 = 27.21222222 
 : 6793d.279 ; trop. = 6798d.l7707 
 i ; trop. = 3231d.4751 
 
 Element* of Meptune. Mean distance, 30.0368000; Period, 60126-V7100000; Eccentricity, 
 0087195 : Inclination of orbit, 1 46' 55T.O ; Long, of Node, 130 5' ll".0 ; Long, of Perihelion, 
 470 IV 5G".7; M.Long. at Epoch, 330 44' 41".8 ; Epoch, 1848, Jan. 1, Oh. G. T. 
 
 * For an accurate table of the Elements of all the Asteroids, see Note HI. 
 
TABLE IV. 3 
 
 Diameters, Volumes, Masses, dec,, of Sun, Moon, and Planets. 
 
 
 Apparent Diameter. 
 
 Equatorial 
 Diameter.* 
 
 Equatorial 
 Diameter, 
 in Mites.* 
 
 Volume. 
 
 Least. 
 
 At Mean 
 Distance. 
 
 Greatest. 
 
 Mercury 
 
 5.0 
 
 it 
 
 6.5 
 
 12.0 
 
 0.396 
 
 3140 
 
 0.062 
 
 Venus 
 
 9.6 
 
 16.5 
 
 61.2 
 
 0.984 
 
 7800 
 
 0.952 
 
 Earth 
 
 
 
 
 1.000 
 
 7926 
 
 1.000 
 
 Mars 
 
 3.6 
 
 5.8 
 
 18.3 
 
 0.517 
 
 4100 
 
 0.138 
 
 Jupiter 
 
 30.0 
 
 36.9 
 
 45.9 
 
 10.976 
 
 87000 
 
 1233.412 
 
 Saturn 
 
 
 16.2 
 
 
 9.987 
 
 79160 
 
 900.000 
 
 Uranus 
 
 
 3,9 
 
 
 4.353 
 
 34500 
 
 82.759 
 
 Neptune 
 
 / // 
 
 3.0 
 
 / n 
 
 / // 
 
 5.236 
 
 4J500 
 
 144.008 
 
 Sun 
 
 31 31.0 
 
 32 1.8 
 
 32 35.6 
 
 112.020 
 
 887870 
 
 1410366.376 
 
 Moon 
 
 29 21.9 
 
 31 7.0 
 
 33 31.1 | 0.273 
 
 2163 
 
 0.020 
 
 
 Mass.f 
 
 Density4 
 
 Gravity. 
 
 Sidereal Rotation4 
 
 L tir- 
 
 Mercury 
 
 TraSTST 
 
 1.12 
 
 0.47 
 
 7i. m. s. 
 24 5 28.3 
 
 6.680 
 
 Venus 
 
 nmrjv 
 
 0.92 
 
 0.93 
 
 23 21 21.9 
 
 1.911 
 
 Earth 
 
 TTsfjTT 
 
 1.00 
 
 1.00 
 
 33 56 4.1 
 
 1.000 
 
 Mars 
 
 _ I _ 
 
 0.95 
 
 0.50 
 
 24 37 20.4 
 
 .431 
 
 Jupiter 
 
 HTIT^TT 
 
 0.24 
 
 2.85 
 
 9 55 26.6 
 
 .037 
 
 Saturn 
 
 sTffT-tfTTtf 
 
 0.14 
 
 1.03 
 
 10 29 16.8 
 
 .011 
 
 Uranus 
 
 _ l 
 
 0.24 
 
 0.76 
 
 
 .003 
 
 Neptune 
 
 T 
 
 0.14 
 
 0.69 
 
 
 .001 
 
 Sun 
 
 1 
 
 0.25 
 
 28.65 
 
 607 48 
 
 
 Moon 
 
 rtnWw 
 
 0.57 
 
 0.15 
 
 27 7 43 
 
 
 TABLE V. 
 Elements of the Retrograde Motion of the Planets. 
 
 Planets. 
 
 Mercury 
 Venus 
 Mars 
 Jupiter 
 Saturn 
 Uranus 
 
 Are of 
 Retrogradation. 
 
 Duration of 
 Retrogradation. 
 
 Elongation at the 
 Stations. 
 
 Synodic 
 Revolution. 
 
 ' 
 
 9 22 to 15 44 
 14 35 to 17 12 
 10 6 to 19 35 
 9 51 to 9 59 
 6 41 to 6 55 
 336 
 
 d k d h 
 23 12 to 21 12 
 40 21 to 43 12 
 60 18 to 80 15 
 116 18 to 122 12 
 138 18 to 135 9 
 151 
 
 o ' o " 
 14 49 to 20 51 
 27 40 to 29 41 
 128 44 to 146 37 
 113 35 to 11642 
 107 25 to 110 46 
 103 30 
 
 days 
 116 
 584 
 780 
 399 
 378 
 370 
 
 Satellites of Neptune. " One only has certainly been observed its approximate 
 period being 5d. 20h. 50m. 45s. ; distance about 12 radii of the planet." 
 
 * According to Herschel, except the diameters of the Sun and Moon. 
 
 f According to Encke, with the exception of the mass of Neptune, which ia Professor Peirce'a 
 determination from Bond's and Lassel's observations of the satellite. By Leverrier's second 
 determination the mass of Mercury is siro^innr* 
 
 $ According to Hansen and Midler, in the case of the planets. 
 
TABLE VI. 
 
 Elements of the Orbits of the Satellites. 
 
 The distances are expressed in equatorial radii of the primaries, 
 periods are expressed in mean solar days. 
 
 I. Satellites of Jupiter. 
 
 Th 
 
 Sat. 
 
 Mean Distance, 
 
 Sidereal 
 Revolution. 
 
 Inclination of 
 Orbit to that 
 of Jupiter. 
 
 Epoch 
 of Ele- 
 ments. 
 
 Mass ; that of 
 Jupiter being 
 1,000,000,000. 
 
 1 
 2 
 3 
 4 
 
 6.04853 
 9.62347 
 15.35024 
 26.99835 
 
 d h m s 
 
 1 18 27 33.506 
 3 13 14 36.393 
 7 3 42 33.362 
 16 16 31 49.702 
 
 O ' " 
 
 3 5 30 
 
 Variable. 
 Variable. 
 2 58 48 
 
 Jan. 1, 
 
 1801. 
 
 17328 
 23235 
 
 88497 
 42659 
 
 II. Satellites of Saturn. 
 
 Name and 
 Order of 
 Satellite. 
 
 Mean 
 Distance. 
 
 Sidereal 
 Revolution. 
 
 M, Long, at the 
 Epoch. 
 
 Eccentricity 
 and Perisatur- 
 nium. 
 
 Epoch 
 of Ele- 
 ments. 
 
 
 
 d A m s 
 
 O ' " 
 
 
 
 1. Mimas 
 
 3.3607 
 
 22 37 22.9 
 
 256 58 48 
 
 
 1790.0 
 
 2. Enceladus 
 
 4.3125 
 
 1 8 53 6.7 
 
 67 41 36 
 
 
 1836.0 
 
 3. Tethys 
 
 5.3396 
 
 1 21 18 25.7 
 
 313 43 48 
 
 0.04(?) 54(?) 
 
 Ditto 
 
 4. Dione 
 
 6.8398 
 
 2 17 41 8.9 
 
 327 40 48 
 
 0.02(?) 42(8) 
 
 Ditto 
 
 5. Rhea 
 
 9.5528 
 
 4 12 25 10.8 
 
 353 44 
 
 0.02(?) 95(?) 
 
 Ditto 
 
 6. Titan 
 
 22.1450 
 
 15 22 41 255 
 
 137 21 24 
 
 ) .029314 
 f 256 38' 
 
 1830.0 
 
 7. Hyperion 
 
 28. 
 
 22 12 ? 
 
 
 
 
 8. lapetus 
 
 64.3590 
 
 79 7 53 40.4 
 
 269 37 48 
 
 
 1790.0 
 
 The longitudes are reckoned in the plane of the ring from its descending node with 
 the ecliptic. The first seven satellites move in or very nearly in its plane ; that of the 
 8th lies about half-way between the planes of the ring and of the planet's orbit. The 
 apsides of Titan have a direct motion of 30' 28" per annum in longitude (on the ecliptic). 
 
 III. Satellites of Uranus. 
 
 Sat. 
 
 Mean 
 Distance. 
 
 Sidereal 
 Revolution. 
 
 Epochs of Passage 
 through Ascending 
 Node of Orbits. Q. T. 
 
 Inclination to Ecliptic. 
 
 
 
 d h m s 
 
 
 The orbits are inclined at 
 
 1 
 2 
 3 
 
 4 
 
 17.0 
 
 19.8 (?) 
 22.8 
 
 4(0 
 8 16 56 31.3 
 10 23(?) 
 13 11 7 12.6 
 
 1787, Feb. 1 6th, 10 
 1787, Jan. 7th, 28 
 
 an angle of about 78 58' 
 to the ecliptic in a plane 
 whose ascending node is 
 in long. 165 30' (Equinox 
 
 5 
 6 
 
 45.5 (?) 
 91.0 
 
 38 2(?) 
 107 12(?) 
 
 
 of 1798). Their motion 
 is retrograde. The orbits 
 are nearly circular. 
 
 TABLE VII. Saturn s Ring. 
 
 Exterior diameter of exterior ring 176,418 miles. 
 
 Interior ditto 155,272 " 
 
 Exterior diameter of interior ring 151,690 " 
 
 Interior ditto 117,339 " 
 
 Equatorial diameter of the body 79,160 " 
 
 Interval between the planet and interior ring 19,090 " 
 
 Interval of the ring* -. 1,791 " 
 
 Thickness of the rings not exceeding 250 
 
 Ditto, according to Professor Bond, not exceeding 50 " 
 
Mean Astronomical Refractions. 
 Barometer 30 in. Thermometer, Fah. 50. 
 
 Ap.Alt. 
 
 Refr. 
 
 Ap. Alt. 
 
 Refr. 
 
 Ap. Alt. 
 
 Refr. 
 
 Alt. 
 
 Refr. 
 
 0' 
 
 33' 51" 
 
 4 0' 
 
 11' 52" 
 
 12 0' 
 
 4' 28.1" 
 
 42 
 
 l'.4.6" 
 
 5 
 
 32 53 
 
 10 
 
 11 30 
 
 10 
 
 4 24.4 
 
 43 
 
 1 2.4 
 
 10 
 
 31 58 
 
 20 
 
 11 10 
 
 20 
 
 4 20.8 
 
 44 
 
 1 0.3 
 
 15 
 
 31 5 
 
 30 
 
 10 50 
 
 30 
 
 4 17.3 
 
 45 
 
 0.58.1 
 
 20 
 
 30 13 
 
 40 
 
 10 32 
 
 40 
 
 4 13.9 
 
 46 
 
 56.1 
 
 25 
 
 29 24 
 
 50 
 
 10 15 
 
 50 
 
 4 10.7 
 
 47 
 
 54.2 
 
 30 
 
 28 37 
 
 5 
 
 9 58 
 
 13 
 
 4 7.5 
 
 48 
 
 52.3 
 
 35 
 
 27 51 
 
 10 
 
 9 42 
 
 10 
 
 4 4.4 
 
 49 
 
 50.5 
 
 40 
 
 27 6 
 
 20 
 
 9 27 
 
 20 
 
 4 1.4 
 
 50 
 
 48.8 
 
 45 
 
 26 24 
 
 30 
 
 9 11 
 
 30 
 
 3 58.4 
 
 51 
 
 47.1 
 
 50 
 
 25 43 
 
 40 
 
 8 58 
 
 40 
 
 3 55.5 
 
 52 
 
 45.4 
 
 55 
 
 25 3 
 
 50 
 
 . 8 45 
 
 50 
 
 3 52.6 
 
 53 
 
 43.8 
 
 1 
 
 24 25 
 
 6 
 
 8 32 
 
 14 
 
 3 49.9 
 
 54 
 
 42.2 
 
 5 
 
 23 48 
 
 10 
 
 8 20 
 
 10 
 
 3 47.1 
 
 55 
 
 40.8 
 
 10 
 
 23 13 
 
 20 
 
 8 9 
 
 20 
 
 3 44.4 
 
 56 
 
 39.3 
 
 15 
 
 22 40 
 
 30 
 
 7 58 
 
 30 
 
 3 41.8 
 
 57 
 
 37.8 
 
 20 
 
 22 8 
 
 40 
 
 7 47 
 
 40 
 
 3 39.2 
 
 58 
 
 36.4 
 
 25 
 
 21 37 
 
 50 
 
 7 37 
 
 50 
 
 3 36.7 
 
 59 
 
 35.0 
 
 30 
 
 21 7 
 
 7 
 
 7 27 
 
 15 
 
 3 34.3 
 
 60 
 
 33.6 
 
 35 
 
 20 38 
 
 10 
 
 7 17 
 
 35 30 
 
 3 27.3 
 
 61 
 
 32.3 
 
 40 
 
 20 10 
 
 20 
 
 7 8 
 
 16 
 
 3 20.6 
 
 62 
 
 31.0 
 
 N 45 
 
 19 43 
 
 30 
 
 6 59 
 
 16 30 
 
 3 14.4 
 
 63 
 
 29.7 
 
 50 
 
 19 17 
 
 40 
 
 6 51 
 
 17 
 
 3 8.5 
 
 64 
 
 28.4 
 
 55 
 
 18 52 
 
 50 
 
 6 43 
 
 17 30 
 
 3 2.9 
 
 65 
 
 27.2 
 
 2 
 
 18 29 
 
 8 
 
 6 35 
 
 18 
 
 2 57.6 
 
 66 
 
 25.9 
 
 5 
 
 18 5 
 
 10 
 
 6 28 
 
 19 
 
 2 47.7 
 
 67 
 
 24.7 
 
 10 
 
 17 43 
 
 20 
 
 6 21 
 
 20 
 
 2 38.7 
 
 68 
 
 23.5 
 
 15 
 
 17 21 
 
 30 
 
 6 14 
 
 21 
 
 2 30.5 
 
 69 
 
 22.4 
 
 20 
 
 17 
 
 40 
 
 6 7 
 
 22 
 
 2 23.2 
 
 70 
 
 21.2 
 
 25 
 
 16 40 
 
 50 
 
 6 
 
 23 
 
 2 16.5 
 
 71 
 
 19.9 
 
 30 
 
 16 21 
 
 9 
 
 5 54 
 
 24 
 
 2 10.1 
 
 72 
 
 18.8 
 
 35 
 
 16 2 
 
 10 
 
 5 47 
 
 25 
 
 2 4.2 
 
 73 
 
 17.7 
 
 40 
 
 15 43 
 
 20 
 
 5 41 
 
 26 
 
 1 58.8 
 
 74 
 
 16.6 
 
 45 
 
 15 25 
 
 30 
 
 5 36 
 
 27 
 
 53.8 
 
 75 
 
 15.5 
 
 50 
 
 15 8 
 
 40 
 
 5 30 
 
 28 
 
 49.1 
 
 76 
 
 14.4 
 
 55 
 
 14 51 
 
 50 
 
 5 25 
 
 29 
 
 44.7 
 
 77 
 
 13.4 
 
 3 
 
 14 35 
 
 10 
 
 5 ^0 
 
 30 
 
 40.5 
 
 78 
 
 12.3 
 
 5 
 
 14 19 
 
 10 
 
 5 15 
 
 31 
 
 36.6 
 
 79 
 
 11.2 
 
 10 
 
 14 4 
 
 20 
 
 5 10 
 
 32 
 
 33.0 
 
 80 
 
 10.2 
 
 15 
 
 13 50 
 
 30 
 
 5 5 
 
 33 
 
 29.5 
 
 81 
 
 9.2 
 
 20 
 
 13 35 
 
 40 
 
 5 (/ 
 
 34 
 
 26.1 
 
 82 
 
 8.2 
 
 25 
 
 13 21 
 
 50 
 
 4 56 
 
 35 
 
 23.0 
 
 83 
 
 7.1 
 
 30 
 
 13 7 
 
 11 
 
 4 51 
 
 36 
 
 20.0 
 
 84 
 
 6.1 
 
 35 
 
 12 53 
 
 10 
 
 4 47 
 
 37 
 
 17.1 
 
 85 
 
 5.1 
 
 40 
 
 12 41 
 
 20 
 
 4 43 
 
 38 
 
 14.4 
 
 86 
 
 4.1 
 
 45 
 
 12 28 
 
 30 
 
 4 39 
 
 39 
 
 11.8 
 
 87 
 
 3.1 
 
 50 
 
 12 16 
 
 40 
 
 4 35 
 
 40 
 
 9.3 
 
 88 
 
 2.0 
 
 if 
 
 12 3 I 
 
 50 
 
 4 31 
 
 41 
 
 6.9 
 
 89 
 
 1.0 
 
TABLE IX 
 
 Corrections of Mean Refractions. 
 
 Ap.Alt. 
 
 [diffo 
 -4-1 B 
 
 dif. fo 
 1F 
 
 1 Ap.Alt. 
 
 Dif. for 
 4-1 B. 
 
 Dif. for 
 1F 
 
 Ap. Alt. 
 
 Dif. fo 
 + IB. 
 
 Dif. for 
 1F 
 
 Alt. 
 
 Dif. for 
 +1B. 
 
 Dif. for 
 
 - -1 F. 
 
 O ' 
 
 
 
 74 
 
 8.1 
 
 4 
 
 24.1 
 
 1.70 
 
 12 
 
 9.00 
 
 0.556 
 
 42 
 
 2.16 
 
 0.130 
 
 5 
 
 71 
 
 7.6 
 
 10 
 
 23.4 
 
 1.64 
 
 10 
 
 8.86 
 
 .548 
 
 43 
 
 2.09 
 
 .125 
 
 10 
 
 69 
 
 7.3 
 
 20 
 
 22.7 
 
 1.58 
 
 20 
 
 8.74 
 
 .541 
 
 44 
 
 2.02 
 
 .120 
 
 15 
 
 67 
 
 7.0 
 
 30 
 
 22.0 
 
 1.53 
 
 30 
 
 8.63 
 
 .533 
 
 45 
 
 1.95 
 
 .116 
 
 20 
 
 65 
 
 6.7 
 
 40 
 
 21.3 
 
 1.48 
 
 40 
 
 8.51 
 
 .524 
 
 46 
 
 1.88 
 
 .112 
 
 25 
 
 63 
 
 6.4 
 
 50 
 
 20.7 
 
 1.43 
 
 50 
 
 8.41 
 
 .517 
 
 47 
 
 1.81 
 
 .108 
 
 30 
 
 61 
 
 6.1 
 
 5 
 
 20.1 
 
 1.38 
 
 13 
 
 8.30 
 
 .509 
 
 48 
 
 1.75 
 
 .104 
 
 35 
 
 59 
 
 5.9 
 
 10 
 
 19.6 
 
 1.34 
 
 10 
 
 8.20 
 
 .503 
 
 49 
 
 1.69 
 
 .101 
 
 40 
 
 58 
 
 5.6 
 
 20 
 
 19.1 
 
 1.30 
 
 20 
 
 8.10 
 
 .496 
 
 50 
 
 1.63 
 
 .097 
 
 45 
 
 56 
 
 5.4 
 
 30 
 
 18.6 
 
 1.26 
 
 30 
 
 8.00 
 
 .490 
 
 51 
 
 1.58 
 
 .094 
 
 50 
 
 55 
 
 5.1 
 
 40 
 
 18.1 
 
 1.22 
 
 40 
 
 7.89 
 
 .482 
 
 52 
 
 1.52 
 
 .090 
 
 55 
 
 53 
 
 4.9 
 
 50 
 
 17.6 
 
 1.19 
 
 50 
 
 7.79 
 
 .476 
 
 53 
 
 1.47 
 
 .088 
 
 1 
 
 52 
 
 4.7 
 
 6 
 
 17.2 
 
 1.15 
 
 14 
 
 7.70 
 
 .469 
 
 54 
 
 1.41 
 
 .085 
 
 5 
 
 50 
 
 4.6 
 
 10 
 
 16.8 
 
 1.11 
 
 10 
 
 7.61 
 
 .464 
 
 55 
 
 1.36 
 
 .082 
 
 10 
 
 49 
 
 4.5 
 
 20 
 
 16.4 
 
 1.09 
 
 20 
 
 7.52 
 
 .458 
 
 56 
 
 1.31 
 
 .079 
 
 15 
 
 48 
 
 4.4 
 
 30 
 
 16.0 
 
 1.06 
 
 30 
 
 7.43 
 
 .453 
 
 57 
 
 1.26 
 
 .076 
 
 20 
 
 46 
 
 4.2 
 
 40 
 
 15.7 
 
 1.03 
 
 40 
 
 7.34 
 
 .448 
 
 58 
 
 1.22 
 
 .073 
 
 25 
 
 45 
 
 4.0 
 
 50 
 
 15.3 
 
 1.00 
 
 50 
 
 7.26 
 
 .444 
 
 59 
 
 1.17 
 
 .070 
 
 30 
 
 44 
 
 3.9 
 
 7 
 
 150 
 
 0.98 
 
 15 
 
 7.18 
 
 .439 
 
 60 
 
 1.12 
 
 .067 
 
 35 
 
 43 
 
 3.8 
 
 10 
 
 14.6 
 
 .95 
 
 15 30 
 
 6.95 
 
 .424 
 
 61 
 
 1.08 
 
 .065 
 
 40 
 
 42 
 
 3.6 
 
 20 
 
 14.3 
 
 .93 
 
 16 
 
 6.73 
 
 .411 
 
 62 
 
 1.04 
 
 .062 
 
 45 
 
 40 
 
 3.5 
 
 30 
 
 14.1 
 
 .91 
 
 16 30 
 
 6.51 
 
 .399 
 
 63 
 
 .99 
 
 .060 
 
 50 
 
 39 
 
 3.4 
 
 40 
 
 13.8 
 
 .89 
 
 17 
 
 6.31 
 
 .386 
 
 64 
 
 .95 
 
 .057 
 
 55 
 
 39 
 
 3.3 
 
 50 
 
 13.5 
 
 .87 
 
 17 30 
 
 6.12 
 
 .374 
 
 65 
 
 .91 
 
 .055 
 
 2 
 
 38 
 
 3.2 
 
 8 
 
 13.3 
 
 .85 
 
 18 
 
 5.94 
 
 .362 
 
 66 
 
 .87 
 
 .052 
 
 5 
 
 37 
 
 3.1 
 
 10 
 
 13.1 
 
 .83 
 
 19 
 
 5.61 
 
 .340 
 
 67 
 
 .83 
 
 .050 
 
 10 
 
 36 
 
 3.0 
 
 20 
 
 12.8 
 
 .82 
 
 20 
 
 5.31 
 
 .322 
 
 68 
 
 .79 
 
 .047 
 
 15 
 
 36 
 
 2.9 
 
 30 
 
 12.6 
 
 .80 
 
 21 
 
 5.04 
 
 .305 
 
 69 
 
 .75 
 
 .045 
 
 20 
 
 35 
 
 2.8 
 
 40 
 
 12.3 
 
 .79 
 
 22 
 
 4.79 
 
 .290 
 
 70 
 
 .71 
 
 .043 
 
 25 
 
 34 
 
 2.8 
 
 50 
 
 12.1 
 
 .77 
 
 23 
 
 4.57 
 
 .276 
 
 71 
 
 .67 
 
 .040 
 
 30 
 
 33 
 
 2.7 
 
 9 
 
 11.9 
 
 .76 
 
 24 
 
 4.35 
 
 .284 
 
 72 
 
 .63 
 
 .038 
 
 35 
 
 33 
 
 2.7 
 
 10 
 
 11.7 
 
 .74 
 
 25 
 
 4.16 
 
 .252 
 
 73 
 
 .59 
 
 .036 
 
 40 
 
 32 
 
 2.6 
 
 20 
 
 11.5 
 
 .73 
 
 26 
 
 3.97 
 
 .241 
 
 74 
 
 .56 
 
 033 
 
 45 
 
 32 
 
 2.5 
 
 30 
 
 11.3 
 
 .72 
 
 27 
 
 3.81 
 
 .230 
 
 75 
 
 .52 
 
 .031 
 
 50 
 
 31 
 
 2.4 
 
 40 
 
 11.1 
 
 .71 
 
 28 
 
 3.65 
 
 .219 
 
 76 
 
 .48 
 
 .029 
 
 55 
 
 30 
 
 2.3 
 
 50 
 
 11.0 
 
 .70 
 
 29 
 
 3.50 
 
 .209 
 
 77 
 
 45 
 
 .027 
 
 3 
 
 30 
 
 2.3 
 
 10 
 
 10.8 
 
 .69 
 
 30 
 
 3.36 
 
 .201 
 
 78 
 
 .41 
 
 .025 
 
 5 
 
 29 
 
 2.2 
 
 10 
 
 10.6 
 
 .67 
 
 31 
 
 3.23 
 
 .193 
 
 79 
 
 .38 
 
 .023 
 
 10 
 
 29 
 
 2.2 
 
 20 
 
 10.4 
 
 .65 
 
 32 
 
 3.11 
 
 .186 
 
 80 
 
 .34 
 
 .021 
 
 15 
 
 28 
 
 2.1 
 
 30 
 
 10.2 
 
 .64 
 
 33 
 
 2.99 
 
 .179 
 
 81 
 
 .31 
 
 .018 
 
 20 
 
 28 
 
 2.1 
 
 40 
 
 10.1 
 
 .63 
 
 34 
 
 2.88 
 
 .173 
 
 82 
 
 .27 
 
 .016 
 
 25 
 
 27 
 
 2.0 
 
 50 
 
 9.9 
 
 .62 
 
 35 
 
 2.78 
 
 .167 
 
 83 
 
 .24 
 
 .014 
 
 30 
 
 27 
 
 2.0 
 
 11 
 
 9.8 
 
 .60 
 
 36 
 
 2.68 
 
 .161 
 
 84 
 
 .20 
 
 .012 
 
 35 
 
 26 
 
 2.0 
 
 10 
 
 9.6 
 
 .59 
 
 37 
 
 2.58 
 
 .155 
 
 85 
 
 .17 
 
 .010 
 
 40 
 
 26 
 
 1.9 
 
 20 
 
 9.5 
 
 .58 
 
 38 
 
 2.49 
 
 .149 
 
 86 
 
 .14 
 
 .008 
 
 45 
 
 25 
 
 1.9 
 
 30 
 
 9.4 
 
 .57 
 
 39 
 
 2.40 
 
 .144 
 
 87 
 
 .10 
 
 .006 
 
 50 
 
 25 
 
 1.9 
 
 40 
 
 9.2 
 
 .56 
 
 40 
 
 2.32 
 
 .139 
 
 88 
 
 .07 
 
 .004 
 
 55 
 
 25 
 
 1.8 
 
 50 
 
 9.1 
 
 .55 
 
 41 
 
 2.24 
 
 .134 
 
 89 
 
 .03 
 
 .002 
 
TABLE X. 
 
 Parallax of the Sun, on the first day of each Month: the mean 
 horizontal Parallax being assumed = 8 ",60. 
 
 Alti- 
 tude. 
 
 Jam 
 
 Feb. 
 Dec. 
 
 March. 
 
 Nov. 
 
 April. 
 Oct. 
 
 May. 
 Sept. 
 
 June. 
 Aug. 
 
 July. 
 
 o 
 
 
 8.75 
 
 8.73 
 
 8.67 
 
 8.60 
 
 8.53 
 
 8.48 
 
 8.46 
 
 5 
 
 8.73 
 
 8.69 
 
 8.64 
 
 8.56 
 
 8.50 
 
 8.44 
 
 8.42 
 
 10 
 
 8.62 
 
 8.59 
 
 8.54 
 
 8.47 
 
 8.40 
 
 8.35 
 
 8.33 
 
 15 
 
 8.45 
 
 8.43 
 
 8.38 
 
 8.30 
 
 8.24 
 
 8.19 
 
 8.17 
 
 20 
 
 8.22 
 
 8.20 
 
 8.15 
 
 8.08 
 
 8.01 
 
 7.97 
 
 7.95 
 
 25 
 
 7.93 
 
 7.91 
 
 7.86 
 
 7.79 
 
 7.73 
 
 7.68 
 
 7.67 
 
 30 
 
 7.58 
 
 7.56 
 
 7.51 
 
 7.45 
 
 7.39 
 
 7.34 
 
 7.33 
 
 35 
 
 7.17 
 
 7.15 
 
 7.11 
 
 7.04 
 
 6.99 
 
 6.94 
 
 6.93 
 
 40 
 
 6.70 
 
 6.68 
 
 6.64 
 
 6.59 
 
 -&S3 
 
 6.49 
 
 6.48 
 
 45 
 
 6.19 
 
 6.17 
 
 6.13 
 
 6.08 
 
 6.03 
 
 5.99 
 
 5.98 
 
 50 
 
 5.62 
 
 5.61 
 
 5.58 
 
 5.53 
 
 5.48 
 
 5.45 
 
 5.44 
 
 55 
 
 5.02 
 
 5.01 
 
 4.98 
 
 4.93 
 
 4.89 
 
 4.86 
 
 4.85 
 
 60 
 
 4.37 
 
 4.36 
 
 4.34 
 
 4.30 
 
 4.26 
 
 4.24 
 
 4.23 
 
 65 
 
 3.70 
 
 3.69 
 
 3.67 
 
 3.63 
 
 3.60 
 
 3.58 
 
 3.57 
 
 70 
 
 2.99 
 
 2.98 
 
 2.97 
 
 2.94 
 
 2.92 
 
 2.90 
 
 2.89 
 
 75 
 
 2.26 
 
 2.26 
 
 2.25 
 
 2.23 
 
 2.21 
 
 2.19 
 
 2.19 
 
 SO 
 
 1.52 
 
 1.52 
 
 1.51 
 
 1.49 
 
 1.48 
 
 1.47 
 
 1.47 
 
 85 
 
 0.76 
 
 0.76 
 
 0.76 
 
 0.75 
 
 0.74 
 
 0.74 
 
 0.74 
 
 90 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 000 
 
 0.00 
 
 0.00 
 
 TABLE XL 
 
 Semi-diurnal Arcs* 
 
 
 Declination. 
 
 Lat. 
 
 1 
 
 5 
 
 10 
 
 15 
 
 20 
 
 25o 
 
 30o 
 
 o 
 
 h m 
 
 h in 
 
 h m 
 
 A m 
 
 h m 
 
 h m 
 
 h m 
 
 5 
 
 6 
 
 6 2 
 
 6 4 
 
 6 5 
 
 6 7 
 
 6 9 
 
 6 12 
 
 10 
 
 6 1 
 
 6 4 
 
 6 7 
 
 6 11 
 
 6 15 
 
 6 19 
 
 6 24 
 
 15 
 
 6 1 
 
 6 5 
 
 6 11 
 
 6 16 
 
 6 22 
 
 6 29 
 
 6 36 
 
 20 
 
 6 1 
 
 6 7 
 
 6 15 
 
 6 22 
 
 6 30 
 
 6 39 
 
 6 49 
 
 25 
 
 6 2 
 
 6 9 
 
 6 19 
 
 6 29 
 
 6 39 
 
 6 50 
 
 7 2 
 
 30 
 
 6 2 
 
 6 12 
 
 6 23 
 
 6 36 
 
 6 49 
 
 7 2 
 
 7 18 
 
 35 
 
 6 3 
 
 6 14 
 
 6 28 
 
 6 43 
 
 6 59 
 
 7 16 
 
 7 35 
 
 40 
 
 6 3 
 
 6 17 
 
 6 34 
 
 6 52 
 
 7 11 
 
 7 32 
 
 7 56 
 
 45 
 
 6 4 
 
 6 20 
 
 6 41 
 
 7 2 
 
 7 25 
 
 7 51 
 
 8 21 
 
 50 
 
 6 5 
 
 6 24 
 
 6 49 
 
 7 14 
 
 7 43 
 
 8 15 
 
 8 r,4 
 
 55 
 
 6 6 
 
 6 29 
 
 6 58 
 
 7 30 
 
 8 5 
 
 8 47 
 
 9 42 
 
 60 
 
 6 7 
 
 6 35 
 
 7 11 
 
 7 51 
 
 8 36 
 
 9 35 
 
 12 
 
 65 
 
 6 9 
 
 6 43 
 
 7 29 
 
 8 20 
 
 9 25 
 
 12 
 
 
TABLE XII. 
 
 Equation of Time, to convert Apparent Time into Mean Time 
 Argument, Mean Longitude of the Sun. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V 
 
 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 
 
 + 6 58.4 
 
 1 29.7 
 
 338.7 
 
 f 1 27.0 
 
 + 6 4.1 
 
 -f 2 49.7 
 
 1 
 
 639.7 
 
 142.0 
 
 334.2 
 
 140.1 
 
 6 6.3 
 
 2 34.5 
 
 2 
 
 620.9 
 
 1 53.8 
 
 329.1 
 
 1 53.1 
 
 6 8.0 
 
 218.9 
 
 3 
 
 6 2.1 
 
 2 5,2 
 
 323.5 
 
 2 6.0 
 
 6 9.1 
 
 2 2.8 
 
 4 
 
 543.3 
 
 2 15.9 
 
 3 17.3 
 
 2 18.9 
 
 6 9.5 
 
 1 46.4 
 
 5 
 
 524.5 
 
 226.1 
 
 3 10.7 
 
 231.7 
 
 6 9.3 
 
 1 29.5 
 
 6 
 
 5 5.7 
 
 235.9 
 
 3 3.5 
 
 244.3 
 
 6 8.5 
 
 1 12.3 
 
 7 
 
 446.9 
 
 245.0 
 
 2 56.0 
 
 256.7 
 
 6 7.2 
 
 054.6 
 
 8 
 
 428.2 
 
 253.6 
 
 247.9 
 
 3 8.9 
 
 6 5.2 
 
 036.6 
 
 9 
 
 4 9.6 
 
 3 1.8 
 
 239.5 
 
 320.8 
 
 6 2.5 
 
 -f-0 18.2 
 
 10 
 
 351.1 
 
 3 9.3 
 
 230.5 
 
 332.5 
 
 559.3 
 
 0.4 
 
 11 
 
 332.6 
 
 3 16.3 
 
 221.2 
 
 343.9 
 
 555.4 
 
 019.5 
 
 12. 
 
 3 14.3 
 
 322.8 
 
 211.5 
 
 355.0 
 
 551.0 
 
 038.8 
 
 13 
 
 2 56.2 
 
 328.6 
 
 2 1.4 
 
 4 5.8 
 
 545.8 
 
 058.4 
 
 14 
 
 238.3 
 
 333.9 
 
 1 51.0 
 
 416.3 
 
 540.1 
 
 1 18.2 
 
 15 
 
 220.5 
 
 338.6 
 
 140.1 
 
 4 26.5 
 
 533.7 
 
 1 38.3 
 
 16 
 
 2 3.0 
 
 342.7 
 
 1 29.0 
 
 436.3 
 
 5 26.7 
 
 1 58.5 
 
 17 
 
 145.7 
 
 346.3 
 
 1 17.6 
 
 445.7 
 
 519.2 
 
 2 19.1 
 
 18 
 
 1 28.6 
 
 349.2 
 
 1 5.9 
 
 454.7 
 
 5 11.1 
 
 239.8 
 
 19 
 
 I 11.7 
 
 351.5 
 
 054.1 
 
 5 3.3 
 
 5 2.3 
 
 3 0.7 
 
 20 
 
 055.2 
 
 353.3 
 
 042.0 
 
 511.3 
 
 453.0 
 
 321.6 
 
 21 
 
 039.1 
 
 354.4 
 
 029.6 
 
 5 18.9 
 
 443.1 
 
 342.8 
 
 22 
 
 023.3 
 
 355.0 
 
 17.1 
 
 526.0 
 
 432.7 
 
 4 4.0 
 
 23 
 
 + 7.8 
 
 3 55.0 
 
 4.4 
 
 532.6 
 
 421.6 
 
 425.3 
 
 24 
 
 7.3 
 
 3 54.5 
 
 -1-0 8.4 
 
 538.6 
 
 4 10.1 
 
 446.6 
 
 25 
 
 022.0 
 
 353.3 
 
 021.5 
 
 544.2 
 
 357.9 
 
 5 8.1 
 
 26 
 
 036.3 
 
 351.5 
 
 034.5 
 
 5 49.3 
 
 345.3 
 
 529.5 
 
 27 
 
 050.3 
 
 349.2 
 
 047.6 
 
 553.9 
 
 332.1 
 
 551.0 
 
 28 
 
 1 3.8 
 
 346.2 
 
 1 0.7 
 
 557.8 
 
 3 18.5 
 
 6 12.3 
 
 29 
 
 1 16.9 
 
 342.8 
 
 1 13.8 
 
 6 1.2 
 
 3 4.3 
 
 6 33.7 
 
 30 
 
 1 29.7 
 
 338.7 
 
 + 1 27.0 
 
 + 6 4.1 
 
 + 2 49.7 
 
 654.9 
 
 TABLE XIII. 
 
 Secular Variation of Equation of Time. 
 Argument, Sun's Mean Longitude. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V* 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 
 
 3 
 
 + 4 
 
 + 11 
 
 + 14 
 
 f 13 
 
 + 9 
 
 3 
 
 2 
 
 5 
 
 11 
 
 14 
 
 13 
 
 8 
 
 6 
 
 1 
 
 6 
 
 12 
 
 14 
 
 12 
 
 8 
 
 9 
 
 1 
 
 6 
 
 12 
 
 15 
 
 12 
 
 7 
 
 12 
 
 
 
 7 
 
 12 
 
 14 
 
 12 
 
 W i 
 
 15 
 
 + 1 
 
 8 
 
 13 
 
 14 
 
 11 
 
 I 
 
 18 
 
 2 
 
 8 
 
 13 
 
 14 
 
 11 
 
 6 
 
 21 
 
 2 
 
 9 
 
 14 
 
 14 
 
 10 
 
 5 
 
 24 
 
 3 
 
 9 
 
 14 
 
 14 
 
 10 
 
 5 
 
 27 
 
 4 
 
 10 
 
 14 
 
 14 
 
 9 
 
 4 
 
 30 
 
 + 4 
 
 -f 11 
 
 + 14 
 
 -r 13 
 
 + 9 
 
 + 4 
 
TABLE XII 
 
 Equation of Time, to convert Apparent Time into Mean Time. 
 Argument, Mean Longitude of the Sun. 
 
 
 VI* 
 
 VII* 
 
 VIII* 
 
 IX* 
 
 X* 
 
 XI* 
 
 o 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. sec. 
 
 min. see. 
 
 
 
 654.9 
 
 15 18.9 
 
 13 58.7 
 
 1 30.6 
 
 + 1130.0 
 
 + 14 3.1 
 
 1 
 
 7 16.1 
 
 15 27.9 
 
 13 43.0 
 
 1 0.2 
 
 1147.0 
 
 13 56.0 
 
 2 
 
 737.2 
 
 1536.1 
 
 13 26.3 
 
 029.8 
 
 12 3.3 
 
 13 48.4 
 
 3 
 
 758.3 
 
 15 43.7 
 
 13 8.9 
 
 + 0.6- 
 
 12 18.7 
 
 1340.1 
 
 4 
 
 8 19.1 
 
 15 50.5 
 
 12 50.5 
 
 031.0 
 
 12 33.4 
 
 1331.1 
 
 5 
 
 839.8 
 
 15 56.5 
 
 1231.4 
 
 1 1.3 
 
 12 47.2 
 
 13 21.6 
 
 6 
 
 9 0.2 
 
 16 1.8 
 
 1211.6 
 
 131.4 
 
 13 0.1 
 
 13 11.4 
 
 7 
 
 920.5 
 
 16 6.3 
 
 1151.1 
 
 v 2 1.3 
 
 13 12.2 
 
 13 0.7 
 
 8 
 
 940.6 
 
 16 9.9 
 
 11 29.9 
 
 231.0 
 
 13 23.5 
 
 12 49.4 
 
 9 
 
 10 0.3 
 
 16 12.9 
 
 11 7.9 
 
 3 0.5 
 
 13 33.9 
 
 12 37.4 
 
 10 
 
 10 19.8 
 
 16 15.1 
 
 10 45.4 
 
 329.7 
 
 13 43.6 
 
 12 25.0 
 
 11 
 
 10 38.9 
 
 16 16.5 
 
 10 22.0 
 
 358.6 
 
 13 52.3 
 
 12 12.2 
 
 12 
 
 10 57.8 
 
 16 17.0 
 
 958.1 
 
 427.1 
 
 14 0.2 
 
 11 58.9 
 
 13 
 
 IT 16.2 
 
 16 16.6 
 
 933.5 
 
 455.2 
 
 14 7.3 
 
 11 45.1 
 
 14 
 
 11 34.4 
 
 16 15.4 
 
 9 8.4 
 
 5 22.9 
 
 14 13.5 
 
 11 30.9 
 
 15 
 
 1152.1 
 
 16 13.4 
 
 8 42.6 
 
 550.2 
 
 14 18.9 
 
 11 16.3 
 
 16 
 
 12 9.5 
 
 16 10.4 
 
 816.4 
 
 6 17.1 
 
 14 23.4 
 
 11 1.1 X 
 
 17 
 
 12 26.5 
 
 16 6.7 
 
 749.6 
 
 643.5 
 
 14 27.2 
 
 10 45.6 
 
 18 
 
 12 42.9 
 
 16 2.1 
 
 722.5 
 
 7 9.3 
 
 14 30.0 
 
 10 29.7 
 
 19 
 
 12 58.9 
 
 15 56.6 
 
 6 54.9 
 
 734.6 
 
 14 32. 1 
 
 10 13.5 
 
 20 
 
 13 14.4 
 
 1550.1 
 
 627.0 
 
 759.3 
 
 14 33.3 
 
 956.9 
 
 21 
 
 13 29.5 
 
 15 42.9 
 
 558.5 
 
 823.4 
 
 14 33.7 
 
 940.1 
 
 22 
 
 1344.1 
 
 15 34.8 
 
 529.7 
 
 "8 46.9 
 
 14 33.3 
 
 923.0 
 
 23 
 
 13 58.0 
 
 15 25.8 
 
 5 0.5 
 
 9 9.8 
 
 14 32.2 
 
 9 5.7 ; 
 
 24 
 
 1411.4 
 
 15 16.0 
 
 431.0 
 
 932.0 
 
 14 30.2 
 
 848.0 j 
 
 25 
 
 1424.1 
 
 15 5.2 
 
 4 1.4 
 
 9 53.5 
 
 14 27.5 
 
 830.2 
 
 26 
 
 14 36.3 
 
 14 53.6 
 
 331.6 
 
 10 14.3 
 
 14 24.0 
 
 8 12.2 
 
 27 
 
 14 47.9 
 
 1441.1 
 
 3 1.5 
 
 " 10 34.4 
 
 14 19.9 
 
 754.0 
 
 28 
 
 14 58.8 
 
 14 27.7 
 
 231.3 
 
 10 53.8 
 
 14 15.0 
 
 735.5 
 
 29 
 
 15 9.2 
 
 14 13.6 
 
 2 1.0 
 
 11 12.3 
 
 14 9.4 
 
 7 17.0 
 
 30 
 
 15 18.9 
 
 1358.7 
 
 1 30.6 
 
 + 1130.0 
 
 + 14 3.1 
 
 + 658.4 
 
 TABLE XIII. 
 
 Secular Variation of Equation of Time. 
 Argument, Sun's Mean Longitude. 
 
 
 VI* 
 
 VII* 
 
 VIII* 
 
 IX* 
 
 X* 
 
 XI* 
 
 o 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 
 
 +4 
 
 2 
 
 10 
 
 15 
 
 15 
 
 10 
 
 3 
 
 3 
 
 3 
 
 10 
 
 15 
 
 14 
 
 10 
 
 6 
 
 3 
 
 4 
 
 11 
 
 15 
 
 14 
 
 9 
 
 9 
 
 2 
 
 4 
 
 12 
 
 15 
 
 14 
 
 8 
 
 12 
 
 1 
 
 5 
 
 12 
 
 15 
 
 13 
 
 8 
 
 15 
 
 + 1 
 
 6 
 
 13 
 
 15 
 
 13 
 
 7 
 
 18 
 
 
 
 7 
 
 13 
 
 15 
 
 12 
 
 6 
 
 21 
 
 
 
 7 
 
 14 
 
 15 
 
 12 
 
 5 
 
 24 
 
 1 
 
 8 
 
 14 
 
 15 
 
 11 
 
 5 
 
 27 
 
 2 
 
 9 
 
 15 
 
 15 
 
 11 
 
 4 
 
 30 
 
 2 
 
 10 
 
 15 
 
 15 | 10 
 
 9 
 
10 
 
 TABLE XIV. 
 
 Perturbations of Equation of Time. 
 
 III. 
 
 II. 
 
 
 
 100 
 
 200 300 
 
 400 
 
 50C 
 
 600 
 
 700 
 
 800 
 
 900 
 
 1000 
 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 sec. 
 
 tec. 
 
 sec. 
 
 
 
 1.4 
 
 0.8 
 
 .0 
 
 1.7 
 
 1.7 
 
 1.2 
 
 0.7 
 
 0.4 
 
 0.6 
 
 1.4 
 
 1.4 
 
 100 
 
 1.2 
 
 1.4 
 
 .1 
 
 1.0 
 
 1.6 
 
 1.8 
 
 1.1 
 
 0.7 
 
 0.6 
 
 0.7 
 
 1.2 
 
 300 
 
 0.9 
 
 1.0 
 
 .2 
 
 1.2 
 
 1.2 
 
 1.5 
 
 1.7 
 
 1.1 
 
 0.5 
 
 0.7 
 
 0.9 
 
 300 
 
 0.7 
 
 1.1 
 
 .1 
 
 0.9 
 
 1.2 
 
 1.4 
 
 1.5 
 
 1.6 
 
 1.2 
 
 0.5 
 
 0.7 
 
 400 
 
 0.5 
 
 0.6 
 
 .2 
 
 1.2 
 
 0.8 
 
 1.0 
 
 1.6 
 
 1.7 
 
 1.5 
 
 1.2 
 
 0.5 
 
 500 
 
 1.0 
 
 0.5 
 
 0.6 
 
 1.2 
 
 1.4 
 
 0.8 
 
 0.8 
 
 1.5 
 
 1.9 
 
 1.5 
 
 1.0 
 
 600 
 
 L7 
 
 1.0 
 
 0.4 
 
 0.5 
 
 1.2 
 
 1.4 
 
 0.9 
 
 0.6 
 
 1.3 
 
 2.0 
 
 1.7 
 
 700 
 
 1.9 
 
 1.8 
 
 .1 
 
 0.4 
 
 0.4 
 
 1.1 
 
 1.6 
 
 1.1 
 
 0.7 
 
 1.2 
 
 1.9 
 
 800 
 
 1.2 
 
 1.8 
 
 .8 
 
 1.2 
 
 0.4 
 
 0.3 
 
 1.0 
 
 1.6 
 
 1.2 
 
 0.7 
 
 1.2 
 
 900 
 
 0.7 
 
 1.1 
 
 .7 
 
 1.8 
 
 1.2 
 
 0.6 
 
 0.2 
 
 0.8 
 
 1.6 
 
 1.3 
 
 0.7 
 
 1000 
 
 1.4 
 
 0.8 
 
 1.0 
 
 1.7 
 
 1.7 
 
 1.2 
 
 0.7 
 
 0.4 
 
 0.6 
 
 1.4 
 
 1.4 
 
 II. 
 
 rv. 
 
 
 
 sec. 
 0.6 
 
 sec. 
 0.7 
 
 sec. 
 0.5 
 
 sec. 
 0.3 
 
 sec. 
 0.2 
 
 sec. 
 0.6 
 
 sec. 
 0.7 
 
 sec. 
 0.5 
 
 sec. 
 0.2 
 
 sec. 
 0.1 
 
 sec. 
 0.6 
 
 100 
 
 0.2 
 
 0.7 
 
 0.6 
 
 0.5 
 
 0.2 
 
 0.3 
 
 0.6 
 
 0.9 
 
 0.5 
 
 0.2 
 
 0.2 
 
 200 
 
 0.2 
 
 0.4 
 
 0.6 
 
 0.5 
 
 0.4 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.5 
 
 0.5 
 
 0.2 
 
 300 
 
 0.4 
 
 0.2 
 
 0.5 
 
 0.5 
 
 0.5 
 
 0.4 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.4 
 
 400 
 
 0.5 
 
 0.4 
 
 0.4 
 
 0.4 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.4 
 
 0.4 
 
 0.5 
 
 500 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.5 
 
 0.4 
 
 0.4 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.3 
 
 0.4 
 
 600 
 
 0.3 
 
 0.3 
 
 0.5 
 
 0.6 
 
 0.4 
 
 0.4 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.4 
 
 0.3 
 
 700 
 
 0.4 
 
 0.2 
 
 0.3 
 
 0.6 
 
 0.6 
 
 0.4 
 
 0.2 
 
 0.2 
 
 0.7 
 
 0.7 
 
 0.4 
 
 800 
 
 0.6 
 
 0.3 
 
 0.2" 
 
 0.3 
 
 0.7 
 
 0.6 
 
 0.3 
 
 02 
 
 0.3 
 
 0.8 
 
 0.6 
 
 900 
 
 0.8 
 
 0.5 
 
 0.3 
 
 0.1 
 
 0.4 
 
 0.7 
 
 0.5 
 
 0.3 
 
 0.1 
 
 0.5 
 
 0.8 
 
 1000 
 
 0.6 
 
 0.7 
 
 0.5 
 
 0.3 
 
 0.2 
 
 0.6 
 
 0.7 
 
 0.5 
 
 0.2 
 
 0.1 
 
 0.6 
 
 II. 
 
 V, 
 
 
 
 sec. 
 1.0 
 
 sec. 
 1.0 
 
 sec. 
 1.1 
 
 sec. 
 1.2 
 
 sec. 
 1.1 
 
 sec. 
 1.0 
 
 sec. 
 0.7 
 
 sec. 
 0.4 
 
 sec. 
 0.6 
 
 sec. 
 0.9 
 
 sec. 
 1.0 
 
 100 
 
 0.9 
 
 0.9 
 
 0.8 
 
 1.0 
 
 1.3 
 
 1.3 
 
 1.0 
 
 0.7 
 
 0.4 
 
 0.5 
 
 0.9 
 
 200 
 
 0.5 
 
 0.7 
 
 0.7 
 
 0.8 
 
 1.0 
 
 1.0 
 
 1.1 
 
 1.2 
 
 0.9 
 
 0.3 
 
 0.5 
 
 300 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.7 
 
 0.8 
 
 1.2 
 
 1.5 
 
 1.5 
 
 1.1 
 
 0.5 
 
 0.2 
 
 400 
 
 0.3 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.7 
 
 0.9 
 
 1.3 
 
 1.4 
 
 1.4 
 
 1.0 
 
 0.3 
 
 500 
 
 0.8 
 
 0.3 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.7 
 
 1.0 
 
 1.4 
 
 1.4 
 
 1.4 
 
 0.8 
 
 600 
 
 1.3 
 
 0.7 
 
 0.3 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.4 
 
 1.6 
 
 1.3 
 
 700 
 
 1.5 
 
 1.1 
 
 0.7 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.8 
 
 1.0 
 
 1.2 
 
 1.4 
 
 1.5 
 
 800 
 
 1.3 
 
 1.3 
 
 1.0 
 
 0.7 
 
 0.4 
 
 0.4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 1.2 
 
 1.3 
 
 900 
 
 1.1 
 
 1.2 
 
 1.2 
 
 1.0 
 
 0.8 
 
 0.6 
 
 0.5 
 
 0.6 
 
 0.9 
 
 1.1 
 
 1.1 
 
 1000 
 
 1.0 
 
 1.0 
 
 1.1 
 
 1.2 
 
 1.1 
 
 1.0 
 
 0.7 
 
 0.4 
 
 0.6 
 
 0.9 
 
 1.0 
 
 Moon and Nutation. 
 
 I. 
 
 tec. 
 0.5 
 
 sec. 
 0.8 
 
 sec. 5 sec. 
 1.0 1.0 
 
 sec. 
 08 
 
 sec. 
 0.5 
 
 *ec. 
 0.2 
 
 sec. 
 0.0 
 
 sec. 
 0.0 
 
 sec. ] sec. 
 0.2 0.5 
 
 N. 
 
 0.1 
 
 0.1 0.2 0.2 
 
 0.2 
 
 0.2 
 
 0.2 
 
 0.2 
 
 0.2 
 
 0.1 10.1 
 
 Constant 3.0 
 
TABLE XV. 
 
 11 
 
 For converting any given day into the decimal part of a year 
 of 365 days. 
 
 Day 
 
 Jan. 
 
 Feb. 
 
 March 
 
 April 
 
 May 
 
 June 
 
 1 
 
 .000 
 
 .085 
 
 .162 
 
 .247 
 
 .329 
 
 .414 
 
 2 
 
 .003 
 
 .088 
 
 .164 
 
 .249 
 
 .331 
 
 .416 
 
 3 
 
 .006 
 
 .090 
 
 .167 
 
 .252 
 
 .334 
 
 .419 
 
 4 
 
 .008 
 
 .093 
 
 .170 
 
 .255 
 
 .337 
 
 .422 
 
 5 
 
 .011 
 
 .096 
 
 .173 
 
 .258 
 
 .340 
 
 .425 
 
 6 
 
 .014 
 
 099 
 
 .175 
 
 .260 
 
 .342 
 
 .427 
 
 7 
 
 .016 
 
 .101 
 
 .178 
 
 .263 
 
 .345 
 
 .430 
 
 8 
 
 .019 
 
 .104 
 
 .181 
 
 .266 
 
 .348 
 
 .433 
 
 9 
 
 .022 
 
 .107 
 
 .184 
 
 .268 
 
 .351 
 
 .436 
 
 10 
 
 .025 
 
 .110 
 
 .186 
 
 .271 
 
 .353 * 
 
 .438 
 
 11 
 
 .027 
 
 .112 
 
 .189 
 
 274 
 
 .356 
 
 .441 
 
 12 
 
 .030 
 
 .115 
 
 .192 
 
 .277 
 
 .359 
 
 .444 
 
 13 
 
 .033 
 
 .118 
 
 .195 
 
 .279 
 
 .362 
 
 .446 
 
 14 
 
 .036 
 
 .121 
 
 .197 
 
 .282 
 
 .364 
 
 .449 
 
 15 
 
 .038 
 
 .123 
 
 .200 
 
 .285 
 
 .367 
 
 .452 
 
 16 
 
 .041 
 
 .126 
 
 .203 
 
 .288 
 
 .370 
 
 .455 
 
 17 
 
 .044 
 
 .129 
 
 .205 
 
 .290 
 
 .373 
 
 .458 
 
 18 
 
 046 
 
 .132 
 
 .208 
 
 .293 
 
 .375 
 
 .460 
 
 19 
 
 .049 
 
 .134 
 
 .211 
 
 .296 
 
 .378 
 
 .463 
 
 20 
 
 .052 
 
 .137 
 
 .214 
 
 .299 
 
 .381 
 
 .466 
 
 21 
 
 .055 
 
 .140 
 
 .216 
 
 .301 
 
 .384 
 
 .468 
 
 22 
 
 .058 
 
 .142 
 
 .219 
 
 .304 , 
 
 .386 
 
 .471 
 
 23 
 
 .060 
 
 .145 
 
 .222 
 
 .30? 
 
 .389 
 
 .474 
 
 24 
 
 .063" 
 
 .148 
 
 .225 
 
 .310 
 
 .392 
 
 .477 
 
 25 
 
 .066 
 
 .151 
 
 .227 
 
 .312 
 
 .395 
 
 .479 
 
 26 
 
 .068 
 
 .153 
 
 .230 
 
 .315 
 
 .397 
 
 .483 
 
 27 
 
 .071 
 
 .156 
 
 .233 
 
 .318 
 
 .400 
 
 .485 
 
 28 
 
 .074 
 
 .159 
 
 .236 
 
 .321 
 
 .403 
 
 .488 
 
 29 
 
 .077 
 
 
 .238 
 
 .323 
 
 .405 
 
 .490 
 
 30 
 
 .079 
 
 
 .241 
 
 .326 
 
 .408 
 
 .493 
 
 31 
 
 .082 
 
 
 .244 
 
 
 .411 
 
 
12 
 
 TABLE XV., Continued. 
 
 For converting any given day into the decimal part of a year 
 of 365 days. 
 
 Day 
 
 July 
 
 August 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 1 
 
 .496 
 
 .581 
 
 .666 
 
 .748 
 
 .833 
 
 .915 
 
 2 
 
 .499 
 
 .584 
 
 .668 
 
 .751 
 
 .836 
 
 .918 
 
 3 
 
 .501 
 
 .586 
 
 671 
 
 .753 
 
 .838 
 
 .921 
 
 4 
 
 .504 
 
 .589 
 
 .674 
 
 .756 
 
 .841 
 
 .923 
 
 5 
 
 .507 
 
 .592 
 
 .677 
 
 .759 
 
 .844 
 
 .926 
 
 6 
 
 .510 
 
 .595 
 
 .679 
 
 .762 
 
 846 
 
 .929 
 
 7 
 
 .512 
 
 .597 
 
 .682 
 
 .764 
 
 .849 
 
 .931 
 
 8 
 
 .515 
 
 .600 
 
 .685 
 
 .767 
 
 .852 
 
 .934 
 
 9 
 
 .518 
 
 .603 
 
 .688 
 
 .770 
 
 855 
 
 .937 
 
 10 
 
 .521 
 
 .605 
 
 .690 
 
 .773 
 
 .858 
 
 .940 
 
 11 
 
 .'523 
 
 .608 
 
 .693 
 
 .775 
 
 .860 
 
 .942 
 
 12 
 
 526 
 
 611 
 
 .696 
 
 .778 
 
 .863 
 
 .945 
 
 13 
 
 .529 
 
 614 
 
 .699 
 
 .781 
 
 .866 
 
 .948 
 
 14 
 
 .532 
 
 .616 
 
 .701 
 
 .784 
 
 .868 
 
 .951 
 
 15 
 
 .534 
 
 .619 
 
 .704 
 
 .786 
 
 .871 
 
 .953 
 
 16 
 
 .537 
 
 .622 
 
 .707 
 
 .789 
 
 .874 
 
 .956 
 
 17 
 
 .540 
 
 .625 
 
 .710 
 
 .792 
 
 877 
 
 .959 
 
 18 
 
 .542 
 
 .627 
 
 .712 
 
 .795 
 
 879 
 
 .962 
 
 19 
 
 .545 
 
 .630 
 
 .715 
 
 .797 
 
 .882 
 
 .964 
 
 20 
 
 .548 
 
 .633 
 
 .718 
 
 .800 
 
 885 
 
 .967 
 
 21 
 
 .551 
 
 .636 
 
 .721 
 
 .803 
 
 888 
 
 .970 
 
 22 
 
 .553 
 
 .638 
 
 .723 
 
 805 
 
 890 
 
 .973 
 
 23 
 
 .556 
 
 .641 
 
 .726 
 
 .808 
 
 893 
 
 .975 
 
 24 
 
 .559 
 
 .644 
 
 .729 
 
 .811 
 
 896 
 
 .978 
 
 25 
 
 .562 
 
 .647 
 
 .731 
 
 .814 
 
 .899 
 
 .981 
 
 26 
 
 .564 
 
 .649 
 
 .734 
 
 .816 
 
 .901 
 
 .984 
 
 27 
 
 .567 
 
 .652 
 
 .737 
 
 .819 
 
 .904 
 
 .986 
 
 28 
 
 .570 
 
 .655 
 
 .740 
 
 .822 
 
 .907 
 
 .989 
 
 29 
 
 .573 
 
 .658 
 
 .742 
 
 .825 
 
 .910 
 
 .992 
 
 30 
 
 .575 
 
 .660 
 
 .745 
 
 .827 
 
 .912 
 
 .995 
 
 31 
 
 .578 
 
 .663 
 
 
 .830 
 
 
 .997 
 
TABLE XVI. 
 
 13 
 
 For converting time into decimal parts of a day. 
 
 Hours 
 
 Minutes 
 
 Seconds 
 
 h. 
 
 m. m. 
 
 s. s. 
 
 1 
 
 .04167 
 
 1 
 
 .00069 
 
 31 
 
 .02153 
 
 1 
 
 .00001 D 31 
 
 .00036 
 
 2 
 
 .08333 
 
 2 
 
 .00139 
 
 32 
 
 .02222 
 
 2 
 
 .00002 1 32 
 
 .00037 
 
 3 
 
 .12500 
 
 3 
 
 .00208 
 
 33 
 
 .02292 
 
 3 
 
 .00003 I 33 
 
 .00038 
 
 4 
 
 .16667 
 
 4 
 
 .00278 
 
 34 
 
 .02301 
 
 4 
 
 .00005 | 34 
 
 .00039 
 
 5 
 
 .20833 
 
 5 
 
 .00347 
 
 35 
 
 .02430 
 
 5 
 
 .00006 
 
 35 
 
 .00040 
 
 6 
 
 .25000 
 
 6 
 
 .00417 
 
 36 
 
 .02500 
 
 6 
 
 .00007 
 
 36 
 
 .00042 
 
 7 
 
 .29167 
 
 7 
 
 .00486 
 
 37 
 
 .02569 
 
 7 
 
 .00008 
 
 37 
 
 .00043 
 
 8 
 
 .33333 
 
 8 
 
 .00556 
 
 38 
 
 .02639 
 
 8 
 
 .00009 
 
 38 
 
 .00044 
 
 9 
 
 .37500 
 
 9 .00625 
 
 39 
 
 .02708 
 
 9 
 
 .00010 
 
 39 
 
 .00045 
 
 10 
 
 .41667 
 
 10 
 
 .00694 
 
 40 
 
 .02778 
 
 10 
 
 .0001 a 
 
 \ 40 
 
 .00046 
 
 11 
 
 .45833 
 
 11 
 
 .00764 
 
 41 
 
 .02847 
 
 11 
 
 .00013 
 
 41 
 
 .00047 
 
 12 
 
 .50000 
 
 12 
 
 .00833 
 
 42 
 
 .02917 
 
 12 
 
 .00014 
 
 42 
 
 .00049 
 
 13 
 
 .54167 
 
 13 ! .00903 
 
 43 
 
 .02986 
 
 13 
 
 .00015 
 
 43 
 
 .00050 
 
 14 
 
 .58333 
 
 14 .00972 
 
 44 
 
 .03056 
 
 14 
 
 .00016 
 
 44 
 
 .00051 
 
 15 
 
 .62500 
 
 15 
 
 01042 
 
 45 
 
 .03125 
 
 15 
 
 .00017 
 
 45 
 
 .00052 
 
 16 
 
 .66667 
 
 16 
 
 .01111 
 
 46 
 
 .03194 
 
 16 
 
 .00018 
 
 46 
 
 .00053 
 
 17 
 
 .70833 
 
 17 
 
 .01180 
 
 47 
 
 .03264 
 
 17 
 
 .00020 
 
 47 
 
 .00054 
 
 18 
 
 .75000 
 
 18 
 
 .01250 
 
 48 
 
 .03333 
 
 18 
 
 .00021 
 
 48 
 
 .00056 
 
 19 
 
 .79167 
 
 19 
 
 .01319 
 
 49 
 
 .03403 
 
 19 
 
 .00022 
 
 49 
 
 .00057 
 
 20 
 
 .83333 
 
 20 
 
 .01389 
 
 50 
 
 .03472 
 
 20 
 
 .00023 
 
 50 
 
 .00058 
 
 21 
 
 .87500 
 
 21 
 
 01458 
 
 51 
 
 .03542 
 
 21 
 
 .00024 
 
 51 
 
 .00059 
 
 22 
 
 .91667 
 
 22 
 
 .01528 
 
 52 
 
 .03611 
 
 22 
 
 .00025 
 
 52 
 
 .00060 
 
 23 
 
 .95833 
 
 23 
 
 01597 
 
 53 
 
 .03680 
 
 23 
 
 .00027 
 
 53 
 
 .00061 
 
 24 
 
 1.00000 
 
 24 
 
 .01667 
 
 54 
 
 .03750 
 
 24 
 
 .00028 
 
 54 
 
 .00062 
 
 
 
 25 
 
 .01736 
 
 55 
 
 .03819 
 
 25 
 
 .00029 
 
 55 
 
 .00064 
 
 
 
 26 
 
 .01805 
 
 56 
 
 .03889 ' 
 
 26 
 
 .00030 
 
 56 
 
 .00065 
 
 
 
 27 
 
 .01875 
 
 57 
 
 .03958 
 
 27 
 
 .00031 
 
 57 
 
 .00066 
 
 
 
 28 
 
 .01944 
 
 58 
 
 .04028 
 
 28 
 
 .00032 
 
 58 
 
 .00067 
 
 
 
 29 
 
 .02014 
 
 59 
 
 .04097 
 
 29 
 
 .00034 
 
 59 
 
 .00068 
 
 
 
 30 
 
 .02083 
 
 60 
 
 .04167 
 
 30 
 
 .00035 
 
 60 
 
 .00069 
 
14 
 
 TABLE XVII. 
 
 For converting Minutes and Seconds of a degree, into the 
 decimal division of the same. 
 
 Minutes 
 
 Seconds 
 
 1 
 
 .OJ667 
 
 31 
 
 .51667 
 
 // 
 1 
 
 .00028 
 
 31 
 
 .00861 
 
 2 
 
 .03333 
 
 32 
 
 .53333 
 
 2 
 
 .00056 
 
 32 
 
 .00889 
 
 3 
 
 .05000 
 
 33 
 
 .55000 
 
 3 
 
 .00083 
 
 33 
 
 .00917 
 
 4 
 
 .06667 
 
 34 
 
 .6*6667 
 
 4 
 
 .00111 
 
 34 
 
 .00944 
 
 5 
 
 .08333 
 
 35 
 
 .58333 
 
 5 
 
 .00139 
 
 35 
 
 .00972 
 
 6 
 
 .10000 
 
 36 
 
 .60000 
 
 6 
 
 .00167 
 
 36 
 
 .01000 
 
 7 
 
 .11667 
 
 37 
 
 .61667 
 
 7 
 
 .00194 
 
 37 
 
 .01028 
 
 8 
 
 .13333 
 
 38 
 
 .63333 
 
 8 
 
 .00222 
 
 38 
 
 .01056 
 
 9 
 
 .15000 
 
 39 
 
 .65000 
 
 9 
 
 .00250 
 
 39 
 
 .01083 
 
 10 
 
 .16667 
 
 40 
 
 .66667 - 
 
 10 
 
 .00278 
 
 40 
 
 .01111 
 
 11 
 
 .18333 
 
 41 
 
 .68333 
 
 11 
 
 .00306 
 
 41 
 
 .01139 
 
 12 
 
 .20000 
 
 42 
 
 .70000 
 
 12 
 
 .00333 
 
 42 
 
 .01167 
 
 13 
 
 .21667 
 
 43 
 
 .71667 
 
 13 
 
 .00361 
 
 43 
 
 .01194 
 
 14 
 
 .23333 
 
 44 
 
 .73333 
 
 14 
 
 .00389 
 
 44 
 
 .01222 
 
 15 
 
 .25000 
 
 45 
 
 .75000 
 
 15 
 
 .00417 
 
 45 
 
 .01250 
 
 16 
 
 .26667 
 
 46 
 
 .76667 
 
 16 
 
 .00444 
 
 46 
 
 .01278 
 
 17 
 
 .28333 
 
 47 
 
 .78333 
 
 17 
 
 .00472 
 
 47 
 
 .01306 
 
 18 
 
 .30000 
 
 48 
 
 .80000 
 
 18 
 
 .00500 
 
 48 
 
 .01333 
 
 19 
 
 .31667 
 
 49 
 
 .81667 
 
 19 
 
 .00528 
 
 49 
 
 .01361 
 
 20 
 
 .33333 
 
 50 
 
 .83333 
 
 20 
 
 .00556 
 
 50 
 
 .01389 
 
 21 
 
 .35000 
 
 51 
 
 .85000 
 
 21 
 
 .00583 
 
 51 
 
 .01417 
 
 22 
 
 .36667 
 
 52 
 
 .86667 
 
 22 
 
 .00611 
 
 52 
 
 .01444 
 
 23 
 
 .38333 
 
 53 
 
 .88333 
 
 23 
 
 .00639 
 
 53 
 
 .01472 
 
 24 
 
 .40000 
 
 54 
 
 .90000 
 
 24 
 
 .00667 
 
 54 
 
 .01500 
 
 25 
 
 .41667 
 
 55 
 
 .91667 
 
 25 
 
 .00694 
 
 55 
 
 .01528 
 
 26 
 
 .43333 
 
 56 
 
 .93333 
 
 26 
 
 .00722 
 
 56 
 
 .01556 
 
 27 
 
 .45000 
 
 57 
 
 .95000 
 
 27 
 
 .00750 
 
 57 
 
 .01583 
 
 28 
 
 .46667 
 
 58 
 
 .96667 
 
 28 
 
 .00778 
 
 58 
 
 .01611 
 
 29 
 
 .48333 
 
 59 
 
 .98333 
 
 29 
 
 .00806 
 
 59 
 
 .01639 
 
 30 
 
 .50000 
 
 60 
 
 1.00000 
 
 30 
 
 .00833 
 
 60 
 
 .01667 
 
TABLE XVIII. 
 Sun's Epochs. 
 
 15 
 
 Years. 
 
 M. Long. 
 
 Long.Peri. 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 N 
 
 VI 
 
 VII 
 
 
 a o / " 
 
 a o / " 
 
 
 
 
 
 
 
 
 
 1830 
 
 9 10 37 46.9 
 
 9 10 54 
 
 228 
 
 279 
 
 169 
 
 598 
 
 758 
 
 519 
 
 989 
 
 362! 
 
 1831 
 
 9 10 23 27.4 
 
 9 10 1 55 
 
 588 
 
 278 
 
 793 
 
 130 
 
 842 
 
 573 
 
 235 
 
 396 j 
 
 1832B. 
 
 9 10 9 7.9 
 
 9 10 2 57 I 948 
 
 278 
 
 418 
 
 661 
 
 926 
 
 627 
 
 482 
 
 430 . 
 
 1833 
 
 9 10 53 56.8 
 
 910 359 
 
 342 
 
 280 
 
 47 
 
 194 
 
 11 
 
 681 
 
 764 
 
 464 
 
 1834 
 
 9 10 39 37.3 
 
 910 5 
 
 702 
 
 279 
 
 671 
 
 725 
 
 95 
 
 734 
 
 11 
 
 498 
 
 1835 
 
 9 10 25 17.8 
 
 910 6 2 
 
 62- 
 
 279 
 
 296 
 
 256 
 
 179 
 
 788 
 
 257 
 
 532 
 
 1836B. 
 
 9 10 10 58.4 
 
 910 7 3 
 
 422 
 
 278 
 
 920 
 
 788 
 
 264 
 
 842 
 
 504 
 
 566 
 
 1837 
 
 9 10 55 47.2 
 
 910 8 5 
 
 816 
 
 280 
 
 549 
 
 321 
 
 348 
 
 895 
 
 787 
 
 600 
 
 1838 
 
 9 10 41 27.8 
 
 9 10 9 6 
 
 176 
 
 279 
 
 173 
 
 852 
 
 432 
 
 949 
 
 33 
 
 634 
 
 1839 
 
 91027 8.3 
 
 9 10 10 8 
 
 536 
 
 279 
 
 798 
 
 383 
 
 511 
 
 3 
 
 279 
 
 668 
 
 1840B. 
 
 9 10 1248.8 
 
 91011 9 
 
 896 
 
 278 
 
 422 
 
 915 
 
 601 
 
 56 
 
 526 
 
 702 
 
 1841 
 
 9 10 57 37.7 
 
 9 10 12 11 
 
 290 
 
 280 
 
 51 
 
 447 
 
 685 
 
 110 
 
 809 
 
 736 
 
 1842 
 
 9 10 43 18.2 
 
 9 10 13 12 
 
 650 
 
 279 
 
 676 
 
 979 
 
 770 
 
 164 
 
 55 
 
 770 
 
 1843 
 
 9 10 28 58.8 
 
 9 10 14 14 
 
 10 
 
 279 
 
 300 
 
 510 
 
 854 
 
 218 
 
 301 
 
 804 
 
 1844B. 
 
 9 10 14 39.3 
 
 9 10 15 15 
 
 370 
 
 278 
 
 924 
 
 41 
 
 938 
 
 272 
 
 548 
 
 838 
 
 1845 
 
 9 10 59 28.2 
 
 9 10 16 17 
 
 764 
 
 280 
 
 553 
 
 574 
 
 23 
 
 325 
 
 831 
 
 872 
 
 1846 
 
 9 10 45 8.7 
 
 9 10 17 19 
 
 124 
 
 280 
 
 177 
 
 106 
 
 107 
 
 379 
 
 77 
 
 906 
 
 1847 
 
 9 10 30 49.2 
 
 9 10 18 20 
 
 484 
 
 279 
 
 802 
 
 637 
 
 191 
 
 433 
 
 324 
 
 940 
 
 1848B. 
 
 9 10 *6 29.8 
 
 9 10 19 22 
 
 844 
 
 278 
 
 427 
 
 168 
 
 276 
 
 487 
 
 570 
 
 974 
 
 1849 
 
 911 118.6 
 
 9 10 20 23 
 
 238 
 
 280 
 
 55 
 
 700 
 
 360 
 
 540 
 
 853 
 
 8 
 
 1850 
 
 9 10 46 59.2 
 
 9 10 21 25 
 
 598 
 
 280 
 
 680 
 
 231 
 
 444 
 
 594 
 
 99 
 
 41 
 
 1851 
 
 9 10 32 39.7 
 
 9 10 22 26 
 
 958 
 
 279 
 
 304 
 
 762 
 
 529 
 
 648 
 
 346 
 
 75 
 
 1852B 
 
 9 10 18 20.2 
 
 9 10 23 28 
 
 319 
 
 278 
 
 929 
 
 294 
 
 613 
 
 701 
 
 592 
 
 109 
 
 1853 
 
 911 3 9.1 
 
 9 10 24 29 
 
 713 
 
 280 
 
 557 
 
 827 
 
 697 
 
 755 
 
 875 
 
 143 
 
 1854 
 
 9 10 48 49.6 
 
 9 10 25 31 
 
 73 
 
 280 
 
 182 
 
 358 
 
 782 
 
 809 
 
 121 
 
 177 
 
 1855 
 
 9 10 34 30.2 
 
 9 10 26 32 
 
 133 
 
 279 
 
 806 
 
 889 
 
 866 
 
 863 
 
 368 
 
 211 
 
 1856B 
 
 9 10 20 10.7 
 
 9 10 27 34 
 
 793 
 
 279 
 
 430 
 
 421 
 
 950 
 
 916 
 
 614 
 
 245 
 
 1857 
 
 911 459.6 
 
 9 10 28 35 
 
 187 
 
 281 
 
 60 
 
 953 
 
 35 
 
 970 
 
 897 
 
 279 
 
 -1858 
 
 9 105040.1 
 
 9 10 29 37 
 
 547 
 
 280 
 
 684 
 
 485 
 
 119 
 
 24 
 
 144 
 
 313 
 
 t !859 
 
 9 10 36 20.7 
 
 9 10 30 39 
 
 907 
 
 279 
 
 308 
 
 16 
 
 203 
 
 78 
 
 390 
 
 347 
 
 1860B 
 
 9 10 22 1.2 
 
 9 10 31 40 
 
 267 
 
 279 
 
 933 
 
 547 
 
 288 
 
 131 
 
 636 
 
 381 
 
 1861 
 
 911 650.1 
 
 9 10 32 42 
 
 661 
 
 281 
 
 562 
 
 80 
 
 372 
 
 185 
 
 919 
 
 415 
 
 1862 
 
 9 10 52 30.6 
 
 9 10 33 43 
 
 21 
 
 280 
 
 186 
 
 612 
 
 456 
 
 239 
 
 166 
 
 449 
 
 1863 
 
 9 1038 11.1 
 
 9 10 34 45 
 
 381 
 
 280 
 
 810 
 
 143 
 
 541 
 
 292 
 
 412 
 
 483 
 
 1864B 
 
 9 10 23 51.7 
 
 9 10 35 46 
 
 741 
 
 279 
 
 435 
 
 674 
 
 625 
 
 346 
 
 659 
 
 517 
 
 1865 
 
 9118 40.5 
 
 9 10 36 48 
 
 135 
 
 281 
 
 64 
 
 207 
 
 709 
 
 400 
 
 941 
 
 551 
 
 1866 
 
 9 105421.1 
 
 9 10 37 49 
 
 495 
 
 280 
 
 688 
 
 738 
 
 794 
 
 453 
 
 188 
 
 585 
 
 1867 
 
 9 10 40 1.6 
 
 9 10 38 51 
 
 855 
 
 280 
 
 313 
 
 270 
 
 878 
 
 507 
 
 434 
 
 619 
 
 1868B 
 
 9 10 25 42.2 
 
 9 10 39 52 
 
 215 
 
 279 
 
 937 
 
 801 
 
 962 
 
 56 
 
 681 
 
 653 
 
 1969 
 
 9 11 1031.0 
 
 9 10 40 54 
 
 609 
 
 281 
 
 566 
 
 334 
 
 47 
 
 615 
 
 963 
 
 687 
 
 1870 
 
 9 1050 11.6 
 
 9 10 41 56 
 
 969 
 
 280 
 
 190 
 
 865 
 
 131 
 
 668 
 
 210 
 
 721 
 
16 
 
 
 TABLE XIX. 
 
 Sun's Motions for Months. 
 
 Months 
 
 M. Long. 
 
 Per. 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 N 
 
 VI 
 
 VII 
 
 
 e o fr 
 
 
 
 
 
 
 
 
 
 
 
 January 
 
 0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 February 
 
 1 33 18.2 
 
 5 
 
 47 
 
 85 
 
 138 
 
 45 
 
 7 
 
 5 
 
 125 
 
 3 
 
 Maivh 5 ^ om- 
 
 1 28 9 11.4 
 
 10 
 
 993 
 
 162 
 
 263 
 
 86 
 
 14 
 
 9 
 
 141 
 
 6 
 
 March ^ Big 
 
 1 2& 8 19.8 
 
 10 
 
 27 
 
 164 
 
 267 
 
 87 
 
 14 
 
 9 
 
 178 
 
 6 
 
 . .. ( Com. 
 
 2 28 42 29.7 
 
 15 
 
 42 
 
 246 
 
 401 
 
 131 
 
 21 
 
 13 
 
 266 
 
 8 
 
 A P rU i Bis. 
 
 2 29 41 38.0 
 
 15 
 
 76 
 
 249 
 
 405 
 
 132 
 
 21 
 
 13 
 
 302 
 
 8 
 
 myr ( Com. 
 
 3 28 16 39.6 
 
 20 
 
 59 
 
 329 
 
 534 
 
 175 
 
 28 
 
 18 
 
 355 
 
 11 
 
 Ma y i Bis. 
 
 3 29 15 47.9 
 
 20 
 
 92 
 
 331 
 
 538 
 
 176 
 
 28 
 
 18 
 
 391 
 
 11 
 
 Tll _ A j Com. 
 
 4 28 49 57.9 
 
 26 
 
 110 
 
 414 
 
 672 
 
 220 
 
 35 
 
 22 
 
 480 
 
 14 
 
 June i Bis. 
 
 4 29 49 6.2 
 
 26 
 
 144 
 
 416 
 
 676 
 
 221 
 
 35 
 
 23 
 
 516 
 
 14 
 
 T , ( Com. 
 
 5 28 24 7.8 
 
 31 
 
 1-29 
 
 496 
 
 806 
 
 263 
 
 41 
 
 27 
 
 569 
 
 17 
 
 Jul y \ Bis. 
 
 % 29 23 16.1 
 
 31 
 
 163 
 
 499 
 
 810 
 
 265 
 
 42 
 
 27 
 
 605 
 
 17 
 
 . ( Com. 
 
 6 28 57 26.1 
 
 36 
 
 182 
 
 580 
 
 943 
 
 309 
 
 49 
 
 31 
 
 694 
 
 20 
 
 Au S' i Bis. 
 
 6 29 56 34.4 
 
 36 
 
 216 
 
 583 
 
 948 
 
 310 
 
 49 
 
 31 
 
 730 
 
 20 
 
 n ( Com. 
 
 7 29 30 44.2 
 
 41 
 
 233 
 
 665 
 
 81 
 
 354 
 
 56 
 
 36 
 
 819 
 
 23 
 
 Se P' i Bis. 
 
 8 29 52.6 
 
 41 
 
 268 
 
 668 
 
 86 
 
 355 
 
 56 
 
 36 
 
 855 
 
 23 
 
 o ( Com. 
 
 8 29 4 54.1 
 
 46 
 
 250 
 
 748 
 
 215 
 
 397 
 
 63 
 
 40 
 
 908 
 
 25 
 
 ifiis. 
 
 9 4 2.5 
 
 46 
 
 284 
 
 750 
 
 219 
 
 399 
 
 63 
 
 40 
 
 944 
 
 25 
 
 Nov 5 Com ' 
 
 9 29 38 12.5 
 
 51 
 
 300 
 
 832 
 
 353 
 
 443 
 
 70 
 
 45 
 
 33 
 
 28 
 
 Nov ' i Bis. 
 
 10 37 20.7 
 
 51 
 
 333 
 
 835 
 
 357 
 
 444 
 
 70 
 
 45 
 
 69 
 
 28 
 
 D < Com. 
 
 10 29 12 22.3 
 
 56 
 
 313 
 
 915 
 
 486 
 
 486 
 
 77 
 
 49 
 
 121 
 
 31 
 
 }Bis. 
 
 11 11 30.6 
 
 56 
 
 347 
 
 917 
 
 491 
 
 488 
 
 77 
 
 49 
 
 158 
 
 31 
 
 TABLE XX. 
 Surfs Motions for Days and Hours. 
 
 ^)ays 
 
 M. Long. 
 
 Per. 
 
 I 
 
 II 
 
 HI 
 
 IV 
 
 V 
 
 N 
 
 VI 
 
 VII 
 
 Hrs. 
 
 Long. 
 
 I 
 VI 
 
 II 
 
 III 
 
 
 o / " 
 
 // 
 
 
 
 
 
 
 
 
 
 
 / // 
 
 
 
 
 1 
 
 0.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 2 27.8 
 
 1 
 
 
 
 
 
 2 
 
 59 8.3 
 
 
 
 34 
 
 3 
 
 4 
 
 1 
 
 
 
 
 
 36 
 
 
 
 2 
 
 4 55.7 
 
 3 
 
 
 
 
 
 3 
 
 1 58 16.7 
 
 
 
 68 
 
 5 
 
 9 
 
 3 
 
 
 
 
 
 73 
 
 
 
 3 
 
 7 23.5 
 
 4 
 
 
 
 1 
 
 4 
 
 2 5725.0 
 
 
 
 101 
 
 8 
 
 13 
 
 4 
 
 1 
 
 
 
 109 
 
 
 
 4 
 
 9 51.4 
 
 6 
 
 
 
 
 5 
 
 3 5633.3 
 
 1 
 
 135 
 
 11 
 
 18 
 
 6 
 
 1 
 
 1 
 
 145 
 
 
 
 5 
 
 12 19.2 
 
 7 
 
 1 
 
 1 
 
 6 
 
 4 5541.6 
 
 1 
 
 169 
 
 14 
 
 22 
 
 r 
 
 1 
 
 1 
 
 181 
 
 
 
 6 
 
 14 47.1 
 
 8 
 
 I 
 
 1 
 
 7 
 
 5 5450.0 
 
 1 
 
 203 
 
 16 
 
 27 
 
 9 
 
 1 
 
 1 
 
 218 
 
 1 
 
 7 
 
 17 14.9 
 
 10 
 
 1 
 
 1 
 
 8 
 
 6 5358.3 
 
 1 
 
 236 
 
 19 
 
 31 
 
 10 
 
 2 
 
 1 
 
 254 
 
 1 
 
 8 
 
 19 42.8 
 
 11 
 
 1 
 
 1 
 
 9 
 
 7 53 6.6 
 
 1 
 
 270 
 
 22 
 
 36 
 
 12 
 
 2 
 
 1 
 
 290 
 
 1 
 
 9 
 
 22 10.6 
 
 13 
 
 1 
 
 2 
 
 10 
 
 8 52 15.0 
 
 1 
 
 304 
 
 25 
 
 40 
 
 13 
 
 2 
 
 1 
 
 327 
 
 1 
 
 10 
 
 24 38.5 
 
 14 
 
 1 
 
 2 
 
 11 
 
 9 51 23.3 
 
 2 
 
 338 
 
 27 
 
 44 
 
 15 
 
 2 
 
 1 
 
 363 
 
 1 
 
 11 
 
 27 6.3 
 
 16 
 
 1 
 
 2 
 
 12 
 
 10 5031.6 
 
 2 
 
 371 
 
 30 
 
 49 
 
 16 
 
 2 
 
 2 
 
 399 
 
 1 
 
 12 
 
 29 34.2 
 
 17 
 
 1 
 
 2 
 
 13 
 
 11 4940.0 
 
 2 
 
 405 
 
 33 
 
 53 
 
 17 
 
 3 
 
 2 
 
 435 
 
 1 
 
 13 
 
 32 2.0 
 
 18 
 
 1 
 
 2 
 
 14 
 
 12 4848.3 
 
 2 
 
 439 
 
 36 
 
 58 
 
 19 
 
 3 
 
 2 
 
 472 
 
 1 
 
 14 
 
 34 29.9 
 
 20 
 
 2 
 
 3 
 
 15 
 
 13 4756.6 
 
 2 
 
 473 
 
 38 
 
 62 
 
 20 
 
 3 
 
 2 
 
 508 
 
 2 
 
 15 
 
 36 57.7 
 
 21 
 
 2 
 
 3 
 
 16 
 
 14 47 4.9 
 
 2 
 
 506 
 
 41 
 
 67 
 
 22 
 
 3 
 
 2 
 
 544 
 
 X* 
 
 16 
 
 39 25.6 
 
 23 
 
 2 
 
 3 
 
 17 
 
 15 46 13.3 
 
 3 
 
 540 
 
 44 
 
 71 
 
 23 
 
 4 
 
 2 
 
 581 
 
 2 
 
 17 
 
 41 53.4 
 
 24 
 
 2 
 
 3 
 
 18 
 
 16 4521.6 
 
 3 
 
 574 
 
 47 
 
 76 
 
 25 
 
 4 
 
 2 
 
 617 
 
 2 
 
 18 
 
 44 212 
 
 25 
 
 2 
 
 3 
 
 19 
 
 17 4429.9 
 
 3 
 
 608 
 
 49 
 
 80 
 
 26 
 
 4 
 
 3 
 
 653 
 
 2 
 
 19 
 
 46 49.1 
 
 27 
 
 2 
 
 4 
 
 20 
 
 18 43 38.3 
 
 3 
 
 641 
 
 52 
 
 85 
 
 28 
 
 4 
 
 3 
 
 690 
 
 2 
 
 20 
 
 49 16.9 
 
 28 
 
 2 
 
 4 
 
 21 
 
 19 4246.6 
 
 3 
 
 675 
 
 55 
 
 89 
 
 29 
 
 5 
 
 3 
 
 726 
 
 2 
 
 21 
 
 51 44.8 
 
 30 
 
 2 
 
 4 
 
 22 
 
 20 41 54.9 
 
 4 
 
 709 
 
 58 
 
 93 
 
 31 
 
 5 
 
 3 
 
 762 
 
 2 
 
 22 
 
 54 12.6 
 
 31 
 
 2 
 
 4 
 
 23 
 
 21 41 3.3 
 
 4 
 
 743 
 
 60 
 
 98 
 
 32 
 
 5 
 
 3 
 
 798 
 
 2 
 
 23 
 
 56 40.5 
 
 32 
 
 3 
 
 4 
 
 24 
 
 22 40 11.6 
 
 4 
 
 777 
 
 63 
 
 102 
 
 33 
 
 5 
 
 3 
 
 835 
 
 2 
 
 24 
 
 59 8.3 
 
 34 
 
 3 
 
 4 
 
 85 
 
 23 39 19.9 
 
 4 
 
 810 
 
 66 
 
 107 
 
 35 
 
 5 
 
 4 
 
 871 
 
 2 
 
 
 
 
 
 
 86 
 
 24 38 28.2 
 
 4 
 
 844 
 
 68 
 
 111 
 
 36 
 
 6 
 
 4 
 
 907 
 
 2 
 
 
 
 
 
 
 27 
 
 25 3736.6 
 
 4 878 
 
 71 
 
 116 
 
 38 
 
 6 
 
 4 
 
 943 
 
 2 
 
 
 
 
 
 
 38 
 
 26 3644.9 
 
 5 
 
 912 
 
 74 
 
 120 
 
 39 
 
 6 
 
 4 
 
 980 
 
 2 
 
 
 
 
 
 
 29 
 
 27 35 53.2 
 
 5 
 
 945 
 
 77 
 
 125 
 
 41 
 
 6 
 
 4 
 
 16 
 
 3 
 
 
 
 
 
 
 30 
 
 28 35 1.6 
 
 5 
 
 979 
 
 79 
 
 129 42 
 
 7 
 
 * 
 
 52 
 
 3 
 
 
 1 
 
 
 
 I 
 
 f ? , 
 
 29 34 9.9 
 
 5 ] H ..''' 
 
 7 
 
 :fl ..-. . ' 
 
 i 
 
TABLE XXI. 
 
 Sun'* Motions for Minutes and Second,. 
 
 TABLE XXII. 17 
 
 Me 
 
 Min. 
 
 Long. 
 
 Min. 
 
 Long. 
 
 Sec. 
 
 Lon 
 
 Sec. Lon. 
 
 I 
 
 1 
 
 2.5 
 
 31 
 
 1 16.4 
 
 1 
 
 0.0 
 
 31 
 
 1.3 
 
 2 
 
 4.9 
 
 32 
 
 1 18.8 
 
 2 
 
 0.1 
 
 32 
 
 1.3 
 
 3 
 
 7.4 
 
 33 
 
 1 21.3 
 
 3 
 
 0.1 
 
 33 
 
 .4 
 
 4 
 
 9.9 
 
 34 
 
 1 23.8 
 
 4 
 
 0.2 
 
 34 
 
 .4 
 
 5 
 
 12.3 
 
 35 
 
 1 26.2 
 
 5 
 
 0.2 
 
 35 
 
 .4 
 
 6 
 
 14.8 
 
 36 
 
 1 28.7 
 
 6 
 
 0.2 
 
 36 
 
 .5 
 
 7 
 
 17.2 
 
 37 
 
 1 31.2 
 
 7 
 
 0.3 
 
 37 
 
 .5 
 
 8 
 
 19.7 
 
 38 
 
 1 33.6 
 
 8 
 
 0.3 
 
 38 
 
 .6 
 
 9 
 
 22.2 
 
 39 
 
 1 36.1 
 
 9 
 
 0.4 
 
 39 
 
 .6 
 
 10 
 
 24.6 
 
 40 
 
 1 38.6 
 
 10 
 
 0.4 
 
 40 
 
 .6 
 
 11 
 
 27.1 
 
 41 
 
 1 41.0 
 
 11 
 
 0.5 
 
 41 
 
 .7 
 
 12 
 
 29.6 
 
 42 
 
 1 43.5 
 
 12 
 
 0.5 
 
 42 
 
 .7 
 
 13 
 
 32.0 
 
 43 
 
 1 46.0 
 
 13 
 
 0.5 
 
 43 
 
 .8 
 
 14 
 
 34.5 
 
 44 
 
 1 48.4 
 
 14 
 
 0.6 
 
 44 
 
 1.8 
 
 15 
 
 37.0 
 
 45 
 
 1 50.9 
 
 15 
 
 0.6 
 
 45 
 
 1.8 
 
 16 
 
 39.4 
 
 46 
 
 1 53.3 
 
 16 
 
 0.7 
 
 46 
 
 1.9 
 
 17 
 
 41.9 
 
 47 
 
 1 55.8 
 
 17 
 
 0.7 
 
 47 
 
 1.9 
 
 18 
 
 44.4 
 
 48 
 
 1 58.3 
 
 18 
 
 0.7 
 
 48 
 
 2.0 
 
 19 
 
 46.8 
 
 49 
 
 2 0.7 
 
 19 
 
 0.8 
 
 49 
 
 2.0 
 
 20 
 
 49.3 
 
 50 
 
 2 3.2 
 
 20 
 
 0.8 
 
 50 
 
 2.0 
 
 21 
 
 51.7 
 
 51 
 
 2 5.7 
 
 21 
 
 0.9 
 
 51 
 
 2.1 
 
 22 
 
 54.2 
 
 52 
 
 2 8.1 
 
 22 
 
 0.9 
 
 52 
 
 2.1 
 
 23 
 
 56.7 
 
 53 
 
 2 10.6 
 
 23 
 
 0.9 
 
 53 
 
 2.2 
 
 24 
 
 59.1 
 
 54 
 
 2 13.1 
 
 24 
 
 1.0 
 
 54 
 
 2.2 
 
 25 
 
 1 1.6 
 
 55 
 
 2 15.5 
 
 25 
 
 1.0 
 
 55 
 
 2.3 
 
 26 
 
 1 4.1 
 
 56 
 
 2 18.0 
 
 26 
 
 1.1 
 
 56 
 
 2.3 
 
 27 
 
 1 6.5 
 
 57 
 
 2 20.5 
 
 27 
 
 1.1 
 
 57 
 
 2.3 
 
 28 
 
 1 9.0 
 
 58 
 
 2 22.9 
 
 28 
 
 1.1 
 
 58 
 
 2.4 
 
 29 
 
 1 11.5 
 
 59 
 
 2 25.4 
 
 29 
 
 1.2 
 
 59 
 
 2.4 
 
 30 
 
 1 13.9 
 
 60 
 
 2 27.8 
 
 30 
 
 1.2 
 
 60 
 
 2.5 
 
 Years 2327 
 
 1835 
 
 3880 
 
 1836 
 
 38.35 
 
 1837 
 
 37.89 
 
 1838 
 
 37.43 
 
 1839 
 
 36.98 
 
 1840 
 
 36.52 
 
 1841 
 
 36.06 
 
 1842 
 
 35.61 
 
 1843 
 
 35.15 
 
 1844 
 
 34.69 
 
 1845 
 
 34.23 
 
 1846 
 
 33.78 
 
 1847 
 
 33.32 
 
 1848 
 
 32.86 
 
 1849 
 
 32.41 
 
 1850 
 
 31.95 
 
 1851 
 
 31.49 
 
 1852 
 
 31.04 
 
 1853 
 
 30.58 
 
 1854 
 
 30.12 
 
 1855 
 
 29.66 
 
 1856 
 
 29.21 
 
 1857 
 
 2875 
 
 1858 
 
 28'29 
 
 1859 
 
 2784 
 
 1860 
 
 27.38 
 
 1861 
 
 2692 
 
 1862 
 
 26.47 
 
 1863 
 
 26.01 
 
 1864 
 
 25.55 
 
 TABLE XXIII. 
 
 Sun's Hourly Motion. 
 
 Argument. Sim's Mean Anomaly. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V, 
 
 
 
 
 
 10 
 20 
 30 
 
 2 32.92 
 2 32.84 
 2 32.59 
 2 32.20 
 
 2 32.20 
 2 31.67 
 2 31.02 
 2 30.28 
 
 2 30.23 
 2 29.46 
 2 28.61 
 2 27.74 
 
 2 27.74 
 2 26.89 
 2 26.07 
 2 25.32 
 
 2 25.32 
 2 24.64 
 2 24.06 
 2 23.60 
 
 2 23.60 
 2 23.26 
 2 23.05 
 2 22.99 
 
 
 
 30 
 20 
 10 
 
 
 
 XI* 
 
 X* 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI* 
 
 
 TABLE XXIV. 
 
 Sun's Semi-diameter. 
 Argument. Sim's Mean Anomaly. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V 
 
 
 
 
 
 10 
 20 
 30 
 
 16 17.3 
 16 17.0 
 16 16.2 
 16 15.0 
 
 16 15.0 
 16 13.3 
 16 11.2 
 16 8.8 
 
 16 8.8 
 16 6.2 
 16 3.4 
 16 0.6 
 
 16 0.6 
 
 15 57.8 
 15 55.1 
 15 52.7 
 
 15 52.7 
 15 50.5 
 15 48.6 
 15 47.0 
 
 15 47.0 
 15 45.9 
 15 45.2 
 15 45.0 
 
 o 
 30 
 20 
 10 
 
 
 
 XI, 
 
 X* 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI, 
 
 
 
 C 
 
 
 
 
 
 
 
18 
 
 TABLE XXV. 
 
 Equation of the Surfs Centre. 
 Argument. Sun's Mean Anomaly. 
 
 
 
 
 i 
 
 Us 
 
 III* 
 
 IV* 
 
 V 
 
 
 
 
 
 8 ' " 
 
 11 29 59 13.9 
 
 5758.5 
 
 1 40 10.7 
 
 1 54 34.1 
 
 1 38 4.8 
 
 55 52.6 
 
 1 
 
 1 1Y.3 
 
 59 43.9 
 
 1 41 8.9 
 
 1 54 30.5 
 
 1 37 2.4 
 
 54 8.7 
 
 2 
 
 3 20.6 
 
 1 1 28.0 
 
 1 42 5.1 
 
 1 54 24.8 
 
 1 3558.1 
 
 5224.0 
 
 3 
 
 5 23.9 
 
 1 3 10.9 
 
 1 42 59.3 
 
 1 54 17.0 
 
 1 3452.2 
 
 50 38.2 
 
 4 
 
 7 27.0 
 
 1 4 52.6 
 
 1 43 51.8 
 
 1 54 7.1 
 
 1 3344.6 
 
 04851.6 
 
 5 
 
 9 30.0 
 
 1 6 33.0 
 
 1 44 42.1 
 
 1 53 55.2 
 
 1 3235.4 
 
 047 4.2 
 
 6 
 
 11 32.8 
 
 1 8 12.3 
 
 1 45 30.4 
 
 1 53 41.0 
 
 1 31 24.4 
 
 45 16.0 
 
 7 
 
 13 35.4 
 
 1 950.1 
 
 1 46 16.8 
 
 1 53 24.9 
 
 1 30 11.9 
 
 43 26.9 
 
 8 
 
 15 37.7 
 
 1 11 26.5 
 
 1 47 1.2 
 
 1 53 6.7 
 
 1 28 57.7 
 
 41 37.0 
 
 9 
 
 17 39.6 
 
 1 13 1.7 
 
 1 47 43.5 
 
 1 52 46.5 
 
 1 2742.0 
 
 03946.5 
 
 10 
 
 19 41.2 
 
 1 14 35.3 
 
 1 48 23.9 
 
 1 52 24.2 
 
 1 26 24.8 
 
 3755.3 
 
 11 
 
 21 42.4 
 
 1 16 7.5 
 
 1 49 2.2 
 
 1 51 59.8 
 
 1 25 5.9 
 
 36 3.3 
 
 12 
 
 23 43.1 
 
 1 1738.2 
 
 1 49 38.4 
 
 1 51 33.4 
 
 1 2345.7 
 
 3410.8 
 
 13 
 
 25 43.4 
 
 1 19 7.5 
 
 1 50 12.6 
 
 1 51 5.0 
 
 1 22 23.8 
 
 32 17.7 
 
 14 
 
 27 43.2 
 
 1 2035.2 
 
 1 30 44.7 
 
 1 50 34.5 
 
 1 21 0.6 
 
 3023.8 
 
 15 
 
 29 42.3 
 
 1 22 1.5 
 
 1 51 14.9 
 
 1 50 - 2.2 
 
 1 1936.0 
 
 2829.6 
 
 16 
 
 31 40.9 
 
 1 2326.0 
 
 1 51 42.9 
 
 1 49 27.7 
 
 1 18 9.9 
 
 26 34.8 
 
 17 
 
 33 38.9 
 
 1 2449.9 
 
 1 52 8.7 
 
 1 48 51.3 
 
 1 1642.4 
 
 2439.6 
 
 19 
 
 35 36.2 
 
 1 26 10.3 
 
 1 52 32.5 
 
 1 48 13.0 
 
 1 15 13.7 
 
 2243.9 
 
 19 
 
 37 32.9 
 
 1 2730.0 
 
 1 52 54.3 
 
 1 47 32.7 
 
 1 1343.5 
 
 2047.9 
 
 20 
 
 39 28.8 
 
 1 2848.0 
 
 1 53 13.9 
 
 1 46 50.4 
 
 1 12 12.1 
 
 1851.4 
 
 21 
 
 41 23.9 
 
 1 30 4.2 
 
 1 53 31.4 
 
 1 46 6.3 
 
 1 1039.3 
 
 1654.6 
 
 22 
 
 43 18.1 
 
 1 31 18.8 
 
 1 53 46.8 
 
 1 45 20.3 
 
 1 9 5.4 
 
 1457.5 
 
 23 
 
 045 11.5 
 
 1 3231.7 
 
 1 54 0.1 
 
 1 44 32.2 
 
 1 7 30.3 
 
 13 0.1 
 
 24 
 
 47 4.0 
 
 1 3342.7 
 
 1 54 11.2 
 
 1 43 42.4 
 
 1 5 54.0 
 
 11 2.6 
 
 25 
 
 48 55.6 
 
 1 3452.0 
 
 1 54 20.4 
 
 1 42 50.7 
 
 1 4 16.5 
 
 9 4.8 
 
 26 
 
 50 46.3 
 
 1 3559.4 
 
 1 54 27.2 
 
 1 41 57.1 
 
 1 2 37.8 
 
 7 6.9 
 
 27 
 
 52 36.0 
 
 1 37 5.1 
 
 1 54 32.1 
 
 1 41 1.7 
 
 1 58.0 
 
 5 87 
 
 28 
 
 54 24.6 
 
 1 38 8.8 
 
 1 54 34.9 
 
 1 40 4.5 
 
 59 17.3 
 
 3105 
 
 29 
 
 56 12.1 
 
 1 39 10.8 
 
 1 54 35.4 
 
 1 39 5.6 
 
 57 35.4 
 
 1 12.2 
 
 L~. 
 
 57 58.5 
 
 1 40 10.7 
 
 1 54 34.1 
 
 1 38 4.8 
 
 55 52.6 
 
 1 
 
 TABLE XXVI. 
 
 Secular Variation of Equation of Sun's Centre. 
 Argument. Sun's Mean Anomaly. 
 
 
 0* 
 
 * 
 
 II* 
 
 III* 
 
 IV* 
 
 V* 
 
 
 
 
 
 
 
 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 
 
 5 
 
 16 
 
 5 
 
 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 
 
 2 
 
 24 
 
 7 
 
 14 
 
 17 
 
 15 
 
 10 
 
 2 
 
 26 
 
 8 
 
 15 
 
 17 
 
 15 
 
 9 
 
 1 
 
 28 
 
 8 
 
 15 
 
 17 
 
 15 
 
 9 
 
 1 
 
 30 
 
 9 
 
 15 
 
 17 
 
 15 
 
 8 
 
 
 
TABLE XXV. 
 
 Equation of the Sun's Centre- 
 Argument. Sun's Mean Anomaly. 
 
 
 VI* 
 
 VII* 
 
 VIII S 
 
 IX* 
 
 X 
 
 s 
 
 
 11* 
 
 11* 
 
 11* 
 
 11* 
 
 11* 
 
 11, 
 
 
 
 
 
 29 59 13.9 
 
 29 235.2 
 
 28 20 23.0 
 
 28 353.7 
 
 28 18 17.1 
 
 29 029.3 
 
 1 
 
 29 57 15.6 
 
 29 052.4 
 
 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 1726.1 
 
 28 355.6 
 
 28 21 22.7 
 
 29 5 51.8 
 
 4 
 
 29 51 20.9 
 
 28 55 50.0 
 
 28 16 30.7 
 
 28 4 0.5 
 
 28 22 28.4 
 
 29 741.5 
 
 5 
 
 29 49 23.0 
 
 28 54 11.4 
 
 28 1537.1 
 
 28 4 7.4 
 
 28 23 35.8 
 
 29 932.2 
 
 6 
 
 29 47 25.2 
 
 28 52 33.8 
 
 28 14 45.4 
 
 28 4 16.6 
 
 282445.1 
 
 29 11 23.8 
 
 7 
 
 29 45 27.7 
 
 28 50 57.5 
 
 28 13 55.6 
 
 28 427.7 
 
 28 25 56.1 
 
 29 13 16.3 
 
 8 
 
 29 43 30.3 
 
 28 49 22.4 
 
 28 13 7.5 
 
 28 441.0 
 
 28 27 9.0 
 
 29 15 9.7 
 
 9 
 
 29 41 33.2 
 
 28 47 48.5 
 
 28 12 21.5 
 
 28 456.4 
 
 28 28 23.6 
 
 29 17 3.9 
 
 10 
 
 29 39 36.4 
 
 28 46 15.7 
 
 28 11 37.4 
 
 28 5 13.9 
 
 28 29 39.8 
 
 29 18 59.0 
 
 11 
 
 29 37 39.9 
 
 28 44 44.3 
 
 28 10 55.1 
 
 28 533.5 
 
 28 30 57.8 
 
 29 20 54.9 
 
 12 
 
 29 35 43.9 
 
 28 43 14.1 
 
 28 10 14.8 
 
 28 555.3 
 
 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 
 
 28 644.9 
 
 28 5 1J8 
 
 29 26 46.9 
 
 15 
 
 29 29 58.2 
 
 28 38 51.8 
 
 28 8 25.6 
 
 28 7 12.9 
 
 28 36 26.3 
 
 29 28 45.5 
 
 16 
 
 2928 4.0 
 
 28 37 27.2 
 
 28 753.2 
 
 28 743.1 
 
 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 
 
 283442.1 
 
 28 654.4 
 
 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 
 
 28 925.6 
 
 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 
 
 284526.1 
 
 29 40 48.2 
 
 22 
 
 29 16 50.8 
 
 28 29 30.1 
 
 28 521.1 
 
 28 11 26.6 
 
 28 47 1.3 
 
 294250.1 
 
 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 
 
 28 446.8 
 
 28 12 57.4 
 
 28 50 15.5 
 
 29 46 55.0 
 
 25 
 
 29 11 23.6 
 
 28 25 52.4 
 
 28 432.6 
 
 28 13 45.7 
 
 28 51 54.8 
 
 29 48 57.8 
 
 26 
 
 29 9 36.2 
 
 28 24 43.2 
 
 28 420.7 
 
 28 14 36.0 
 
 28 53 35.3 
 
 2951 0.8 
 
 27 
 
 29 749.5 
 
 28 23 35.6 
 
 28 4 10.8 
 
 28 15 28.5 
 
 28 55 16.9 
 
 2953 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 
 
 28 353.7 
 
 28 18 17.1 
 
 29 029.3 
 
 29 59 13.9 
 
 TABLE XXVI. 
 
 Secular Variation of Equation of Surfs Centre. 
 Argument. Sun's Mean Anomaly. 
 
 
 VI* 
 
 VII* 
 
 VIII* 
 
 IX* 
 
 X* 
 
 XI* 
 
 o 
 
 
 + 
 
 + 8 
 
 + 15 
 
 tr 
 
 + 17 
 
 + 15 
 
 -f 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 
 
 ft 
 
 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 
 
 22 
 
 6 
 
 13 
 
 17 
 
 16 
 
 11 
 
 2 
 
 24 
 
 7 
 
 14 
 
 17 
 
 16 
 
 10 
 
 *> 
 
 m 
 
 26 
 
 7 
 
 14 
 
 17 
 
 16 
 
 10 
 
 i 
 
 28 
 
 8 
 
 14 
 
 17 
 
 15 
 
 9 
 
 i 
 
 30 
 
 + 8 
 
 + 15 
 
 + 17 
 
 + 15 
 
 + 9 
 
 + 
 
TABLE XXVII. 
 
 Nutations. 
 Argument. Supplement of the Node, or N. 
 
 Solar Nutation, 
 
 N. 
 
 Long. 
 
 R. Asc. 
 
 Obliq. 
 
 N. 
 
 Long. 
 
 R. Asc. 
 
 Obliq. 
 
 
 Long. Obliq. 
 
 
 
 + 0.0 
 
 +0.0 
 
 + 9.2 
 
 500 
 
 0.0 
 
 0.0 
 
 9.3 
 
 Jan. 
 
 ff 
 
 fr 
 
 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 
 
 1.1 
 
 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 
 
 1.2 
 
 + 0.1 
 
 70 
 
 7.2 
 
 6.9 
 
 8.3 
 
 570 
 
 7.5 
 
 6.6 
 
 8.4 
 
 20 
 
 1.0 
 
 0.3 
 
 80 
 
 8.2 
 
 7.8 
 
 8.1 
 
 580 
 
 8.5 
 
 7.5 
 
 8.1 
 
 
 
 
 90 
 
 9.1 
 
 8.7 
 
 7.8 
 
 590 
 
 9.5 
 
 8.4 
 
 7.8 
 
 March. 
 
 
 
 100 
 
 + 10.0 
 
 + 9.4 
 
 4- 7.5 
 
 600 
 
 10.4 
 
 9.1 
 
 7.5 
 
 2 
 12 
 
 0.7 
 + 0.3 
 
 0.5 
 
 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 
 
 
 
 
 130 
 
 12.4 
 
 11.7 
 
 6.3 
 
 630 
 
 12.8 
 
 11.4 
 
 6.3 
 
 April. 
 i 
 
 OC 
 
 A C 
 
 140 
 150 
 
 13.1 
 + 13.8 
 
 12.4 
 + 13.0 1 
 
 5.9 
 
 + 5.5 
 
 640 
 650 
 
 13.5 
 
 14.2 
 
 12.0 
 12.6 
 
 5.9 
 5.4 
 
 i 
 11 
 21 
 
 .*J 
 
 0.8 
 1.1 
 
 u.o 
 
 0.2 
 0.2 
 
 160 
 
 14.4 
 
 13.6 
 
 5.0 
 
 660 
 
 14.8 
 
 13.2 
 
 4.9 
 
 TVT 
 
 
 
 170 
 
 15.0 
 
 14.1 
 
 4.5 
 
 670 
 
 15.3 
 
 13.8 
 
 44 
 
 May. 
 
 
 
 180 
 190 
 
 15.5 
 15.9 
 
 14.5 
 
 14.8 
 
 4.0 
 3.5 
 
 680 
 690 
 
 15.8 
 16.2 
 
 14.2 
 14.7 
 
 3.9 
 3.3 
 
 1 
 11 
 
 91 
 
 1.2 
 1.2 
 
 I 1 
 
 + 0.1 
 0.1 
 ^ 
 
 200 
 
 + 16.3 
 
 + 15.1 
 
 + 2.9 
 
 700 
 
 16.6 
 
 15.0 
 
 2.8 
 
 1 
 
 31 
 
 1 . 1 
 
 0.8 
 
 U.O 
 
 0.4 
 
 210 
 
 16.6 
 
 15.4 
 
 2.4 
 
 710 
 
 16.9 
 
 15.3 
 
 2.2 
 
 
 
 
 220 
 230 
 240 
 250 
 
 16.9 
 17.1 
 17.2 
 + 17.3 
 
 15.6 
 15.7 
 15.9 
 + 15.9 
 
 1.8 
 1.2 
 0.7 
 
 + 0.1 
 
 720 
 730 
 740 
 750 
 
 17.1 
 
 17.2 
 17.3 
 17.3 
 
 15.4 
 15.7 
 15.9 
 15.9 
 
 1.6 
 1.1 
 0.5 
 
 + 0.1 
 
 June. 
 10 
 20 
 30 
 
 0.4 
 0.0 
 + 0.4 
 
 0.5 
 0.5 
 0.5 
 
 260 
 270 
 
 17.3 
 17.2 
 
 15.9 
 15.7 
 
 0.5 
 
 l.l 
 
 760 
 770 
 
 17.2 
 17.1 
 
 15.9 
 15.7 
 
 0.7 
 
 1.2 
 
 July. 
 10 
 
 0.7 
 1 
 
 0.4 
 
 *-l 
 
 280 
 
 17.1 
 
 15.6 
 
 1.6 
 
 780 
 
 16.9 
 
 15.4 
 
 J.8 
 
 J 
 
 
 U.O 
 
 01 
 
 290 
 
 16.9 
 
 15.4 
 
 2.2 
 
 790 
 
 16.6 
 
 15.3 
 
 2.4 
 
 30 
 
 1.2 
 
 .1 
 
 300 
 
 + 16.6 
 
 + 15.1 
 
 _ 2.8 
 
 800 
 
 _16.3 
 
 -15.0 
 
 + 2.9 
 
 Aug. 
 
 
 
 310 
 
 16.2 
 
 14.8 
 
 3.3 
 
 810 
 
 15.9 
 
 14.7 
 
 3.^ 
 
 9 
 
 1.3 
 
 1 9 
 
 + 0.0 
 
 ft 4. 
 
 320 
 330 
 
 15.8 
 15.3 
 
 14.5 
 14.1 
 
 3.9 
 4.4 
 
 820 
 830 
 
 15.5 
 15.0 
 
 14.2 
 
 13.8 
 
 4.0 
 
 4.5 
 
 29 
 
 i . At 
 
 0.9 
 
 U.fc 
 
 0.4 
 
 340 
 
 14.8 
 
 13.6 
 
 4.9 
 
 840 
 
 14.4 
 
 13.2 
 
 5.0 
 
 Sept. 
 
 
 
 350 
 
 + 14.2 
 
 4. 13.0 
 
 5.4 
 
 850 
 
 13.8 
 
 12.6 
 
 + 5.5 
 
 8 
 
 0.6 
 
 0.5 
 
 360 
 370 
 
 13.5 
 
 12.8 
 
 12.4 
 11.7 
 
 5.9 
 6.3 
 
 860 
 870 
 
 13.1 
 12.4 
 
 12.0 
 11.4 
 
 5.9 
 6.3 
 
 18 
 28 
 
 + 0.2 
 0.2 
 
 0.5 
 0.5 
 
 380 
 
 12.0 
 
 11.1 
 
 6.7 
 
 880 
 
 11.6 
 
 10.6 
 
 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 
 
 
 
 
 
 
 
 
 
 OQ 
 
 1.2 
 
 0.2 
 
 410 
 
 9.5 
 
 8.7 
 
 7.8 
 
 910 
 
 9.1 
 
 8.4 
 
 7.8 
 
 ivo 
 
 
 
 420 
 
 8.5 
 
 7.8 
 
 8.1 
 
 920 
 
 8.2 
 
 7.5 
 
 8.1 
 
 Nov. 
 
 
 
 430 
 
 7.5 
 
 6.9 
 
 8.4 
 
 930 
 
 7.2 
 
 6.6 
 
 8.3 
 
 7 
 
 1.2 
 
 + 0.0 
 
 440 
 
 6.5 
 
 6.0 
 
 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 
 
 3.0 
 
 9.2 
 
 970 
 
 3.2 
 
 2.9 
 
 9.0 
 
 7 
 
 0.6 
 
 0.5 
 
 480 
 
 2.2 
 
 2.1 
 
 9.3 
 
 980 
 
 2.1 
 
 2.0 
 
 9.1 
 
 17 
 
 0.2 
 
 0.5 
 
 490 
 
 i.r 
 
 1.0 
 
 9.3 
 
 990 
 
 1.0 
 
 1.0 9.1 
 
 27 
 
 + 0.3 
 
 0.5 
 
 500 
 
 + 0.0 
 
 + 0-0 
 
 9.3 
 
 1000 
 
 0.0 
 
 0.01+ 2.2 
 
 37 
 
 + 0.6 
 
 0.5,' 
 
TABLE XXVIII. 
 
 TABLE XXIX, 
 
 2J 
 
 Lunar Equation, 1st part. 
 Argument I. 
 
 Lunar Equation, 2d part. 
 Arguments I. and VI. 
 I. 
 
 I 
 
 Eq U 
 
 I 
 
 Equ 
 
 
 
 7. 
 
 500 
 
 7.5 
 
 10 
 
 8. 
 
 510 
 
 7.0 
 
 20 
 
 8. 
 
 520 
 
 6.6 
 
 30 
 
 8. 
 
 530 
 
 6.1 
 
 40 
 
 9.4 
 
 540 
 
 5.6 
 
 50 
 
 9.8 
 
 550 
 
 5.2 
 
 60 
 
 10.3 
 
 560 
 
 4.7 
 
 70 
 
 10.7 
 
 570 
 
 4.3 
 
 80 
 
 11. 
 
 580 
 
 3.9 
 
 90 
 
 11.5 
 
 590 
 
 3.5 
 
 100 
 
 11.9 
 
 600 
 
 3.1 
 
 110 
 
 12.3 
 
 610 
 
 2.7 
 
 120 
 
 12.6 
 
 620 
 
 2.4 
 
 130 
 
 13.0 
 
 630 
 
 2.0 
 
 140 
 
 13.3 
 
 640 
 
 1.7 
 
 150 
 
 13.6 
 
 650 
 
 1.4 
 
 160 
 
 13.8 
 
 660 
 
 1.2 
 
 170 
 
 14.1 
 
 670 
 
 0.9 
 
 180 
 
 14.3 
 
 680 
 
 0.7 
 
 190 
 
 14.5 
 
 690 
 
 0.5 
 
 200 
 
 14.6 
 
 700 
 
 0.4 
 
 210 
 
 14.8 
 
 710 
 
 0.2 
 
 220 
 
 14.9 
 
 720 
 
 0.1 
 
 230 
 
 14.9 
 
 730 
 
 0.1 
 
 240 
 
 15.0 
 
 740 
 
 0.0 
 
 250 
 
 15.0 
 
 750 
 
 0.0 
 
 260 
 
 15.0 
 
 760 
 
 0.0 
 
 270 
 
 14.9 
 
 770 
 
 0.1 
 
 280 
 
 14.9 
 
 780 
 
 0.1 
 
 290 
 
 14.8 
 
 790 
 
 0.2 
 
 300 
 
 14.6 
 
 800 
 
 0.4 
 
 310 
 
 14.5 
 
 810 
 
 0.5 
 
 320 
 
 14.2 
 
 820 
 
 0.7 
 
 330 
 
 14.1 
 
 830 
 
 0.9 
 
 340 
 
 13.8 
 
 840 
 
 1.2 
 
 350 
 
 13.6 
 
 850 
 
 1.4 
 
 360 
 
 13.3 
 
 860 
 
 1.7 
 
 370 
 
 13.0 
 
 870 
 
 2.0 
 
 380 
 
 12.6 
 
 880 
 
 2.4 
 
 390 
 
 12.3 
 
 890 
 
 2.7 
 
 400 
 
 11.9 
 
 900 
 
 3.1 
 
 410 
 
 11.5 
 
 910 
 
 3.5 
 
 420 
 
 11.1 
 
 920 
 
 3.9 
 
 430 
 
 10.7 
 
 930 
 
 4.3 
 
 440 
 
 10.3 
 
 940 
 
 4.7 
 
 450 
 
 9.8 
 
 950 
 
 5.2 
 
 460 
 
 9.4 
 
 960 
 
 5.6 
 
 470 
 
 8.9 
 
 970 
 
 6.1 
 
 480 
 
 84 
 
 980 
 
 6.6 
 
 490 
 
 8.0 
 
 990 
 
 7.0 
 
 600 7.5 
 
 000 
 
 7.5 
 
 VI 
 
 
 
 50 
 
 100 
 
 150 200 
 
 25C 
 
 30C 
 
 350 
 
 fiOO 
 
 450 
 
 500 
 
 
 
 .3 
 
 1.2 
 
 1.2 
 
 1.1 ! 1-0 
 
 1.0 
 
 1.0 
 
 1.1 
 
 1.2 
 
 1.2 
 
 1.3 
 
 50 
 
 .5 
 
 1.5 
 
 1.5 
 
 1.3 1.1 
 
 1.0 
 
 0.9 
 
 1.0 
 
 1.1 
 
 1.1 
 
 1.1 
 
 100 
 
 .7 
 
 1.8 
 
 1.7 
 
 1.4 1.2 
 
 1.1 
 
 1.0 
 
 0.9 
 
 0.9 
 
 0.9 
 
 0.9 
 
 150 
 
 .9 
 
 1.9 
 
 1.8 
 
 1.6 1.4 
 
 1.3 
 
 .0 
 
 0.8 
 
 0.8 
 
 0.8 
 
 0.7 
 
 200 
 
 .9 
 
 2.0 
 
 2.0 
 
 1.711.5 
 
 1.4 
 
 .0 
 
 0.8 
 
 0.8 
 
 0.8 
 
 0.7 
 
 250 
 
 2.0 
 
 2.0 
 
 2.0 
 
 1.8 i 1.6 
 
 1.5 
 
 .1 
 
 0.9 
 
 0.7 
 
 0.7 
 
 0.6 
 
 300 
 
 1.9 
 
 1.9 
 
 1.9 
 
 1.9 ! 1.7 
 
 1.6 
 
 .2 
 
 1.0 
 
 0.8 
 
 0.7 
 
 0.7 
 
 350 
 
 1.8 
 
 1.9 
 
 1.9 
 
 1.9 1.7 
 
 1.6 
 
 .4 
 
 1.0 
 
 1.0 
 
 0.9 
 
 0.8 
 
 400 
 
 1.6 
 
 1.7 
 
 1.8 
 
 1.9 ' 1.7 
 
 1.6 
 
 .4 
 
 1.2 
 
 1.1 
 
 1.0 
 
 1.0 
 
 450 
 
 1.5 
 
 1.5 
 
 1.6 
 
 1.7 1.7 
 
 1.7 
 
 .6 
 
 1.4 
 
 .2 
 
 1.2 
 
 1.1 
 
 500 
 
 1.3 
 
 1.4 
 
 1.4 
 
 1.5! 1.7 
 
 1.7 
 
 .7 
 
 1.5 
 
 .4 
 
 1.4 
 
 1.3 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 550 
 
 1.1 
 
 1.2 
 
 1.2 
 
 1.4 ' 1.6 
 
 1.7 
 
 f t 
 
 } 7 
 
 .6 
 
 1.5 
 
 1.5 
 
 600 
 
 1.0 
 
 1.0 
 
 1 lil.tjl.4 
 
 1.6 
 
 .8 
 
 1.8 
 
 .8 
 
 1.7 
 
 1.6 
 
 650 
 
 0.8 
 
 0.9 
 
 1.0 
 
 1.1 
 
 1.3 
 
 1.5 
 
 .7 
 
 1.8 
 
 .9 
 
 1.9 
 
 1.8 
 
 700 
 
 0.7 
 
 0.7 
 
 0.8 
 
 1.1 
 
 1.2 
 
 1.4 
 
 .7 
 
 1.9 
 
 .9 
 
 1.9 
 
 1.9 
 
 750 
 
 0.6 
 
 0.6 
 
 0.7 
 
 1.0 
 
 1.1 
 
 T.3 
 
 .6 
 
 1.9 
 
 .9 
 
 2.0 
 
 2.0 
 
 800 
 
 0.7 
 
 0.7 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.2 
 
 .5 
 
 1.8 
 
 2.0 
 
 .9 
 
 1.9 
 
 850 
 
 0.7 
 
 0.8 
 
 0.8 
 
 0.9 
 
 0.9 
 
 1.1 
 
 .4 
 
 1.7 
 
 .8 
 
 .8 
 
 1.9 
 
 900 
 
 0.9 
 
 0.9 
 
 0.9 
 
 0.9 
 
 1.0 
 
 1.1 
 
 .2 
 
 1.5 
 
 .7 
 
 .7 
 
 1.7 
 
 950 
 
 1.1 
 
 1.0 
 
 1.1 
 
 1.0 
 
 1.0 
 
 1.0 
 
 .1 
 
 1.3 
 
 .4 
 
 .6 
 
 1.5 
 
 
 
 1.3 
 
 1.2 
 
 1.2 
 
 1.1 
 
 1.0 
 
 1.0 
 
 .0 
 
 1.1 
 
 1.2 
 
 .2 
 
 1.3 
 
 I. 
 
 VI 
 
 500 
 
 550 
 
 600 
 
 650 
 
 700 
 
 750 
 
 800 
 
 850 
 
 900 
 
 950 
 
 1000 
 
 
 
 1.3 
 
 1.4 
 
 1.4 
 
 1.5 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.5 
 
 1.4 
 
 .4 
 
 1.3 
 
 50 
 
 1.1 
 
 1.1 
 
 1.2 
 
 1.3 
 
 1.5 
 
 1.5 
 
 1.7 
 
 1.6 
 
 1.5 
 
 .5 1.5 
 
 100 
 
 0.9 
 
 0.9 
 
 0.9 
 
 1.1 
 
 1.3 
 
 1.5 
 
 1.6 
 
 1.7 
 
 1.7 
 
 .7 1.7 
 
 150 
 
 0.7 
 
 0.8 
 
 0.810.9 
 
 1.2 
 
 1.4 
 
 1.6 
 
 1.9 
 
 1.8 
 
 .8 1.9 
 
 200 
 
 0.7 
 
 0.7 
 
 0.6 0.8 
 
 1.1 
 
 1.2 
 
 1.6 
 
 1.8 
 
 .8 
 
 .8 
 
 1.9 
 
 250 
 
 0.6 
 
 0.6 
 
 0.7 
 
 0.7 
 
 1.0 
 
 1.1 
 
 1.5 
 
 1.7 
 
 .9 
 
 .9 
 
 2.0 
 
 300 
 
 0.7 
 
 0.7 
 
 0.7 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.4 
 
 1.6 
 
 .8 
 
 .9 
 
 1.9 
 
 350 
 
 0.8 
 
 0.7 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 1.4 
 
 1.6 
 
 .6 
 
 .7 
 
 1.8 
 
 400 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.8 
 
 0.9 
 
 1.0 
 
 1.2 
 
 1.4 
 
 .5 
 
 .6 
 
 1.6 
 
 450 
 
 1.1 
 
 1.1 
 
 1.0 
 
 0.9 
 
 0.9 
 
 0.9 
 
 1.0 
 
 1.2 
 
 .4 
 
 .4 
 
 1.5 
 
 500 
 
 1.3 
 
 1.2 
 
 1.2 
 
 1.1 
 
 0.9 
 
 0.9 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.2 
 
 1.3 
 
 550 
 
 1.5 
 
 1.4 
 
 1.4 
 
 1.2 
 
 1.0 
 
 0.9 
 
 0.9 
 
 0.9 
 
 1.0 
 
 1.1 
 
 i.i 
 
 600 
 
 1.6 
 
 1.6 
 
 1.5 
 
 1.4 
 
 1.2 
 
 1.0 
 
 0.8 
 
 0.8 
 
 0.8 
 
 0.9 
 
 1.0 
 
 650 
 
 1.8 
 
 1.7 
 
 1.6 
 
 1.6 
 
 1.3 
 
 .1 
 
 0.9 
 
 0.8 
 
 0.7 0.7 
 
 0.8 
 
 700 
 
 1.9 
 
 1.8 
 
 1.8 
 
 1.6 
 
 1.4 
 
 .2 
 
 0.9 
 
 0.7 
 
 0.7 
 
 0.7 
 
 0.7 
 
 750 
 
 2.0 
 
 1.9 
 
 1.9 
 
 1.7 
 
 1.5 
 
 .3 
 
 1.0 
 
 0.7 
 
 0.7 
 
 0.6 
 
 0.6 
 
 800 
 
 1.9 
 
 1.8 
 
 1.8 
 
 1.8 
 
 1.6 
 
 .4 
 
 1.1 
 
 0.8 
 
 0.6 
 
 0.7 
 
 0.7 
 
 850 
 
 1.9 
 
 1.8 
 
 1.8 
 
 1.8 
 
 1.6 
 
 .5 
 
 1.2 
 
 0.9 
 
 0.8 
 
 0.8 
 
 0.7 
 
 900 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.7 
 
 1.6 
 
 1.5 
 
 1.3 
 
 1.1 
 
 0.9 
 
 0.9 
 
 0.9 
 
 950 
 
 1.5 
 
 1.5 
 
 1.5 
 
 1.6 
 
 1.7 
 
 1.6 
 
 1.5 
 
 1.3 
 
 1.2 
 
 1.1 
 
 1.1 
 
 
 
 1.3 
 
 1.4 
 
 1.4 
 
 1.5 
 
 1.6 
 
 1.6 
 
 1.6 
 
 1.5 
 
 1.4 
 
 1.4 
 
 1.3 
 
 Constant 1".3. 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 
 Arguments II and III. 
 
 III. 
 
 fll. 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120' 
 
 1 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 " 
 
 
 
 21.6 
 
 20.8 
 
 19.8 
 
 19.0 
 
 17.9 
 
 16.8 
 
 15.9 
 
 14.7 
 
 14.0 
 
 13.2 
 
 128 125 
 
 12.9 
 
 20 J23.1 
 
 22.7 
 
 21.6 
 
 21.0 
 
 20.1 
 
 19.3 
 
 18.4 
 
 17.4 
 
 16.4 
 
 15.5 
 
 14.5 13.8 
 
 13.4 
 
 40 23.5 
 
 23.2 
 
 22.9 
 
 22.7 
 
 22.0 
 
 21.1 
 
 20.4 
 
 19.5 
 
 18.7 
 
 17.9 
 
 16.9 16.1 
 
 15.3 
 
 60 ! 22.2 
 
 22.5 
 
 23.1 
 
 22.7 
 
 22.8 
 
 22.5 
 
 21.9 
 
 21.3 
 
 20.5 
 
 19.9 
 
 19.1 
 
 18.2 
 
 17.4 
 
 80 ! 20.0 
 
 20.7 
 
 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 
 
 15.3 
 
 16.0 
 
 16.9 
 
 17.7 
 
 18.4 
 
 19.2 
 
 19.8 
 
 20.2 
 
 20.7 
 
 20.8 
 
 21.1 
 
 21.1 
 
 20.8 
 
 140 
 
 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 
 
 160 
 
 12.7 
 
 13.2 
 
 13.6 
 
 14.1 
 
 14.6 
 
 15.0 
 
 15.7 
 
 16.4 
 
 17.0 
 
 17.3 
 
 18.1 
 
 18.7 
 
 19.2 
 
 180 
 
 12.7 
 
 12.9 
 
 13.1 
 
 13.5 
 
 13.9 
 
 14.0 
 
 14.5 
 
 14.8 
 
 15.0 
 
 15.8 
 
 16.4 
 
 16.8 
 
 17.2 
 
 200 
 
 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 
 
 14.8 
 
 14.8 
 
 14.9 
 
 15.1 
 
 15.1 
 
 15.1 
 
 14.9 
 
 14.8 
 
 260 
 
 12.8 
 
 13.3 
 
 13.8 
 
 14.2 
 
 14.6 
 
 15.0 
 
 15.3 
 
 15.6 
 
 15.5 
 
 15.5 
 
 15.6 
 
 15.6 
 
 15.6 
 
 280 
 
 11.5 
 
 12.3 
 
 13.0 
 
 13.4 
 
 14.0 
 
 14.6 
 
 15.1 
 
 15.4 
 
 16.0 
 
 16.2 
 
 16.2 
 
 16.3 
 
 16.2 
 
 300 
 
 10.1 
 
 10.9 
 
 11.3 
 
 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 'll.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;i2.8 
 
 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 
 
 9.2 
 
 580 
 
 22.8 j 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 
 
 640 
 
 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 
 
 24.5 
 
 23.4 
 
 22.5 
 
 21.2 
 
 19.8 
 
 680 
 
 29.7 29.6 
 
 29.5 
 
 29.5 
 
 29.1 
 
 28.8 28.2 
 
 27.6 
 
 27.0 
 
 26.0 
 
 25.0 
 
 23.8 
 
 22.8 
 
 700 
 
 28.8 29.2 
 
 29.3 
 
 29.5 
 
 29.5 
 
 29.5 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 
 
 22.2 23.5 
 
 24.3 
 
 25.3 
 
 26.2 
 
 27.0 j 27.6 
 
 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 25.9 
 
 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 
 
 26.2 
 
 26.6 
 
 840 
 
 13.2 14.0 
 
 15.0 
 
 16.0 
 
 17.0 
 
 18.2 
 
 18.8 
 
 20.3 
 
 21.7 
 
 22.7 
 
 23.6 
 
 24.5 
 
 25.3 
 
 860 
 
 11.5 12.2 
 
 13.0 
 
 13.9 
 
 14.9 
 
 15.9 17.1 
 
 18.0 
 
 18.9 
 
 20.3 
 
 21.4 
 
 22.6 
 
 23.5 
 
 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 
 
 900 
 
 11.2 10.2 
 
 10.9 
 
 11.5 
 
 12.5 
 
 12.1 
 
 12.8 
 
 13.7 
 
 14.5 
 
 15.5 
 
 16.6 
 
 17.9 
 
 18.5 
 
 920 
 
 12.1 11.6 
 
 11.5 
 
 11.1 
 
 11.2 
 
 11.3 
 
 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 15.6 
 
 14.6 13.7 
 
 13.1 
 
 12.5 
 
 11.9 
 
 11.7 
 
 11.6 
 
 11.4 
 
 11.7 
 
 12.1 
 
 12.6 
 
 980 
 
 19.5 18.3 
 
 17.3 
 
 16.4 
 
 15.2 
 
 14.2 
 
 13.4 
 
 12.7 
 
 12.2 
 
 12.0 
 
 11.9 
 
 11.8 
 
 11.8 
 
 1000 
 
 21,6 20.8 
 
 19.8 
 
 19.0 
 
 17.9 
 
 16.8 
 
 15.9 
 
 14.7 
 
 14.0 
 
 13.2. 
 
 12.8 
 
 12.5 
 
 12.2 
 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 59 
 
 60 
 
 70 
 
 80 
 
 90 
 
 00 
 
 110 
 
 120 
 
TABLE XXX. 
 
 Perturbations produced by Venus, 
 
 Arguments II and III. 
 
 HI. 
 
 II. 
 
 120 130 
 
 140 
 
 150 i 160 
 
 170 
 
 180 | 190 200 
 
 210 
 
 220 
 
 230 
 
 240 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12.2 | 12.2 
 
 12.3 
 
 12.4 12.8 
 
 13.3 
 
 13.9 
 
 14.7 15.6 
 
 16.5 
 
 17.7 
 
 18.8 
 
 20.1 
 
 20 
 
 13.4 12.9 
 
 12.6 
 
 12.3 : 12.2 
 
 12.4 
 
 12.9 
 
 13.3; 14.0 
 
 14.6 
 
 15.5 
 
 16.4 
 
 17.3 
 
 40 
 
 15.3 14.4 
 
 14.0 
 
 13.5:13.0 12.9 
 
 12.6 
 
 12.6 13.1 
 
 13.5 
 
 14.0 
 
 14.4 
 
 15.4 
 
 60 
 
 17.4 16.7 
 
 16.0 
 
 15.2 14.5 
 
 14.0 
 
 13.6 
 
 13.3 13.2 
 
 13.2 
 
 13.4 
 
 13.5 
 
 14.1 
 
 80 
 
 19.3118.7 
 
 17.7 
 
 17.1 16.4 
 
 15.9 
 
 15.4 
 
 14.6 14.3 
 
 13.9 
 
 13.8 
 
 13.7 
 
 13.6 
 
 100 
 
 20.5 j 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 I 20.7 
 
 20.4 
 
 20.0 
 
 19.7 
 
 19.2 
 
 18.5 
 
 18.0 
 
 17.3 
 
 16.9 
 
 16.5 
 
 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 
 
 14.7 
 
 14.8 
 
 15.0 
 
 15.1 
 
 15.4 
 
 15.7 
 
 15.8 
 
 16.0 
 
 16.1 
 
 16.1 
 
 16.3 
 
 16.4 
 
 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 
 
 14.9 
 
 14.8 
 
 14.7 
 
 15.0 
 
 300 
 
 16.7 
 
 17.0 
 
 17.1 
 
 16.9 
 
 16.9 
 
 1G.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 
 
 IV. 8 
 
 17.9 
 
 18.1 
 
 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 
 
 15.3 
 
 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 
 
 14.1 
 
 15.0 
 
 15.9 
 
 16.9 
 
 17.7 
 
 18.5 
 
 19.0 
 
 440 
 
 7.1 
 
 7.6 
 
 8.4 
 
 9.0 
 
 9.9 
 
 10.9 
 
 11.8 
 
 12.9 
 
 13.8 
 
 14.9 
 
 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 
 
 500 
 
 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 
 
 6.5 
 
 6.3 
 
 6.1 
 
 6.3 
 
 .6 
 
 7.0 
 
 7.5 
 
 8.0 
 
 8.8 
 
 9.3 
 
 540 
 
 8.2 
 
 7.6 
 
 7.2 
 
 6.8 
 
 6.5 
 
 6.3 
 
 6.2 
 
 6.0 
 
 6.2 
 
 6.5 
 
 6.9 
 
 7.4 
 
 7.9 
 
 560 
 
 9.2 
 
 8.6 
 
 7.9 
 
 7.5 
 
 6.8 
 
 6.6 
 
 6.3 
 
 6.1 
 
 6.0 
 
 6.1 
 
 6.2 
 
 6.5 
 
 6.9 
 
 580 
 
 10.6 
 
 9.8 
 
 9.1 
 
 8.4 
 
 7.7 
 
 7.3 
 
 6.6 
 
 6.3 
 
 6.1 
 
 5.9 
 
 5.7 
 
 5.9 
 
 6.0 
 
 ,600 
 
 12.6 
 
 11.4 
 
 10.5 
 
 9.5 
 
 8.7 
 
 8.1 
 
 7.4 
 
 7.0 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.5 
 
 5.6 
 
 620 
 
 14.7 
 
 13.5 
 
 12.4 
 
 11.4 
 
 10.4 
 
 9.5 
 
 8.7 
 
 7.9 
 
 7.3 
 
 6.7 
 
 6.2 
 
 5.6 
 
 5.2 
 
 640 
 
 17..2 
 
 16.2 
 
 14.9 
 
 13.7 
 
 12.5 
 
 11.4 
 
 10.4 
 
 9.5 
 
 8.7 
 
 7.8 
 
 7.0 
 
 6.5 
 
 5.9 
 
 660 
 
 19.8 
 
 19.0 
 
 17.6 
 
 16.5 
 
 15.1 
 
 13.9 
 
 12.8 
 
 11.5 
 
 10.5 
 
 9.6 
 
 8.6 
 
 7.7 
 
 6.9 
 
 680 
 
 22.8 
 
 21.7 
 
 20.4 
 
 19.3 
 
 18.1 
 
 16.8 
 
 15.7 
 
 14.2 
 
 13.0 
 
 11.9 
 
 10.7 
 
 9.6 
 
 8.6 
 
 700 
 
 25.2 
 
 24.3 
 
 23.3 
 
 22.1 
 
 20.7 
 
 19.7 
 
 18.5 
 
 17.3 
 
 16.0 
 
 14.3 
 
 13.4 
 
 12.1 
 
 11.0 
 
 720 
 
 27.3 
 
 26.4 
 
 25.7 
 
 24.5 
 
 23.7 
 
 22.5 
 
 21.1 
 
 20.2 
 
 18.8 
 
 17.7 
 
 16.4 
 
 15.3 
 
 13.9 
 
 740 
 
 28.4 
 
 27.7 
 
 27.4 
 
 26.6 
 
 25.9 
 
 24.9 
 
 24.0- 
 
 22.8 
 
 21.5 
 
 20.6 
 
 19.2 
 
 18.1 
 
 16.8 
 
 760 
 
 29.0 
 
 28.7 
 
 28.3 
 
 27.8 
 
 27.3 
 
 26.8 
 
 25.9 
 
 25.2 
 
 24.3 
 
 23.0 
 
 21.7 
 
 20.7 
 
 19.7 
 
 780 
 
 28.7 
 
 28.7 
 
 28.8 
 
 28.7 
 
 28.3 
 
 28.0 
 
 27.2 
 
 26.1 
 
 26.1 
 
 25.2 
 
 24.3 
 
 23.3 
 
 22.2 
 
 800 
 
 28.1 
 
 28.3 
 
 28.4 
 
 28.5 
 
 28.5 
 
 28.4 
 
 28.2 
 
 27.3 
 
 27.3 
 
 26.7 
 
 25.9 
 
 25.1 
 
 24.4 
 
 820 
 
 26.6 
 
 27.3 
 
 27.8 
 
 26.1 
 
 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 
 
 27.7 
 
 27.5 
 
 880 
 
 21.1 
 
 22.4 
 
 23.3 
 
 24.2 
 
 25.1 
 
 25.8 
 
 26.5 
 
 27.0 
 
 27.3 
 
 27.5 
 
 27.8 
 
 28.0 
 
 27.7 
 
 900 
 
 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 
 
 14.2 
 
 14.9 
 
 16.1 
 
 17.5 
 
 18.2 
 
 19.6 
 
 20.8 
 
 21.9 
 
 23.0 
 
 23.9 
 
 24.7 
 
 25.7 
 
 26.1 
 
 960 
 
 12.6 
 
 13.3 
 
 14.1 
 
 14.4 
 
 15.9 
 
 17.2 
 
 17.9 
 
 19.5 
 
 20.5 
 
 21.7 
 
 22.7 
 
 23.9 
 
 24.7 
 
 980 
 
 11.8 
 
 12.1 
 
 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 
 
 13.9 
 
 14.7 
 
 15.6 
 
 16.5 
 
 17.6 
 
 8.8 
 
 20.1 
 
 
 120 
 
 130 
 
 140 
 
 150 
 
 160 
 
 170 
 
 180 
 
 190 
 
 200 
 
 210 
 
 220 
 
 230 
 
 24) 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 
 Arguments II. and III. 
 
 III. 
 
 II. 
 
 } 240 
 
 250 
 
 260 
 
 270 280 
 
 290 
 
 300 
 
 310 
 
 320 
 
 330 
 
 340 
 
 350 360 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20.1 
 
 21.1 
 
 22.2 
 
 23.4 24.3 
 
 25.2 
 
 25.8 
 
 26.6 
 
 27.2 
 
 27.6 
 
 27.7 
 
 27.G 
 
 27.6 
 
 20 
 
 17.3 
 
 18.6 
 
 19.7 
 
 20.9 21.9 
 
 23.0 
 
 24.2 
 
 24.9 
 
 25.8 
 
 26.6 
 
 27.0 
 
 27.4 ! 27.7 
 
 40 
 
 15.4 
 
 16.5 
 
 17.3 
 
 18.3 19.4 
 
 20.5 
 
 21.6 
 
 22.7 
 
 23.7 
 
 24.9 
 
 25.5 
 
 26.3 26.9 
 
 60 
 
 14.1 
 
 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 
 
 lu.3 
 
 17.3 
 
 18.2 
 
 19.0 
 
 20.0 
 
 21.1 
 
 22.0123.1 
 
 100 
 
 14.3 
 
 14.3 
 
 14.3 
 
 14.4 14.6 
 
 15.0 
 
 15.5 
 
 16.2 
 
 16.9 
 
 17.7 
 
 18.9 
 
 19.8 20.8 
 
 120 
 
 15.6 
 
 15.2 
 
 14.8 
 
 14.8 15.0 
 
 14.9 
 
 15.0 
 
 15.2 
 
 15.9 
 
 16.3 
 
 17.0 
 
 17.7 18.5 
 
 HO 1 17.0 
 
 1G.6 16.4 
 
 15.8 15.5 
 
 15.4 
 
 15.6 
 
 15.6 
 
 15.5 
 
 15.6 
 
 16.1 
 
 16.7 17.1 
 
 160 ! 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 
 
 180 
 
 18.5 
 
 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 
 
 18.3 
 
 18.4 
 
 18.2 
 
 18.2 
 
 18.2 
 
 18.2 
 
 18.1 
 
 18.1 
 
 17.8 
 
 17.7 
 
 17.6 
 
 17.5 
 
 17.7 
 
 220 
 
 17.5 
 
 17.6 
 
 17.8 
 
 17.8 
 
 18.0 
 
 18.0 
 
 18.2 
 
 18.1 
 
 18.1 
 
 18.3 
 
 18.4 
 
 18.3 
 
 18.3 
 
 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 l 15.0 
 
 14.9 
 
 14.9 
 
 14.9 
 
 14.9 
 
 14.7 
 
 15.0 
 
 15.3 
 
 15.5 
 
 15.9 
 
 16.1 
 
 16.4 
 
 16.8 
 
 300 14.8 
 
 14.6 14.6 
 
 14.2 
 
 14.0 
 
 14.0 
 
 13.9 
 
 13.9 
 
 14.2 
 
 14.5 
 
 14.8 
 
 15.0 
 
 15.5 
 
 320 15.6 
 
 15.3 
 
 14.7 
 
 14.5 
 
 14.4 
 
 13.1 
 
 13.6 
 
 13.4 
 
 13.3 
 
 13.1 
 
 13.4 
 
 13.6 
 
 13.8 
 
 340 
 
 16.8 
 
 16.6 
 
 16.0 
 
 15.5 
 
 15.2! 14.5 
 
 14.3 
 
 13.7 
 
 13.1 
 
 13.0 
 
 12.7 
 
 12.6 
 
 12.6 
 
 360 
 
 18.4 
 
 17.9 
 
 17.5 
 
 17.0 
 
 16.5 
 
 15.9 
 
 15.4 
 
 14.9 
 
 14.3 
 
 13.7 
 
 13.0 
 
 12.6 
 
 12.3 
 
 380 
 
 19.5 
 
 19.2 
 
 18.9 
 
 18.5 
 
 17.9 
 
 17.7 
 
 16.9 
 
 16.4 
 
 15.8 
 
 15.0 
 
 14.5 
 
 13.6 
 
 13.1 
 
 400 
 
 19.8 
 
 19.8 
 
 20.1 
 
 19.7 
 
 19.4 
 
 19.1 
 
 18.6 
 
 18.1 
 
 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 
 
 18.G 
 
 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 
 
 14.2 
 
 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 
 
 11.9 
 
 13.1 
 
 14.1 
 
 15.2 
 
 16.2 
 
 17.2 
 
 580 
 
 6.0 
 
 6.3 
 
 6.6 
 
 7.0 
 
 7.6 
 
 8.4 
 
 9.1 
 
 9.9 
 
 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 
 
 5.0 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.l| 
 
 700 
 
 11.0 
 
 10.0 
 
 P.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 
 
 11.7110.5 
 
 9.4 
 
 8.4 
 
 7.2 
 
 6.5 
 
 5.6 
 
 5.0 
 
 43 
 
 760 
 
 19.7 
 
 185 
 
 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 
 
 25.1 
 
 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 
 
 21.6120.5 
 
 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 
 
 26.5 
 
 26.0 
 
 25.5 
 
 24.7 24.1 
 
 23.2 22.0 
 
 21.4 
 
 20.4 
 
 900 
 
 27.6 
 
 27.8 
 
 27.9 
 
 27.6 
 
 27.1 
 
 26.7 
 
 26.5125.7 25.3 
 
 24.6 23.9 
 
 23.0 
 
 22.0 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 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 
 
 27.5 27.1 
 
 26. 6 '' 26.2 
 
 25.6 
 
 25.5 
 
 960 
 
 24.7 
 
 25.4 
 
 26.2 
 
 88 J 
 
 27.2 
 
 27.5 
 
 27.7 
 
 27.7 27.6 
 
 27.4127.1 
 
 27.0 
 
 26.2 
 
 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 1 260 
 
 270 
 
 280 
 
 290 
 
 300 
 
 310 320 
 
 330 340 
 
 350 
 
 360 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 
 Arguments II. and III. 
 
 III. 
 
 II. 
 
 360 
 
 370 
 
 380 
 
 390 
 
 400 
 
 410 
 
 420 
 
 430 
 
 440 
 
 450 
 
 460 
 
 470 
 
 480 
 
 
 
 27.6 
 
 27.7 
 
 27.3 
 
 26.7 
 
 26.2 
 
 25.5 
 
 24.7 
 
 23.8 
 
 23.1 
 
 22.3 
 
 21.3 
 
 20.2 
 
 19.3 
 
 20 
 
 27.7 
 
 27.8 
 
 27.8 
 
 27.6 
 
 27.4 
 
 26.8 
 
 26.2 
 
 25.6 
 
 24.8 
 
 24.0 
 
 23.1 
 
 22.0 
 
 20.9 
 
 40 
 
 26.9 
 
 27.3 
 
 27.6 
 
 27.9 
 
 27.9 
 
 27.7 
 
 27.5 
 
 27.1 
 
 26.3 
 
 25.6 
 
 24.9 
 
 24.0 
 
 23.2 
 
 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 
 
 24.1 
 
 25.1 
 
 25.8 
 
 26.4 
 
 26.9 
 
 27.3 
 
 27.5 
 
 140 
 
 17.1 
 
 17.9 
 
 18.6 
 
 19.3 
 
 20.3 
 
 21.3 
 
 22.0 
 
 22.9 
 
 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 
 
 24.2 
 
 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 
 
 22.2 
 
 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 
 
 18.9 
 
 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 
 
 20.2 
 
 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 
 
 20.8 
 
 20.9 
 
 320 
 
 13.8 
 
 14.2 
 
 14.6 
 
 15.1 
 
 15.6 
 
 16.2 
 
 16.8 
 
 17.7 
 
 18.3 
 
 18.9 
 
 19.5 
 
 20.1 
 
 20.8 
 
 340 
 
 12.6 
 
 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 
 
 14.2 
 
 14.9 
 
 15.7 
 
 16.5 
 
 17.3 
 
 380 
 
 13.1 
 
 12.5 
 
 11.9 
 
 11.6 
 
 11.5 
 
 11.4 
 
 11.6 
 
 11.7 
 
 12.3 
 
 12.7 
 
 13.3 
 
 14.0 
 
 15.0 
 
 400 
 
 14.8 
 
 13.9 
 
 13.1 
 
 12.5 
 
 11.7 
 
 11.2 
 
 11.1 
 
 10.9 
 
 11.0 
 
 11.1 
 
 11.4 
 
 12.0 
 
 12.6 
 
 420 
 
 16.5 
 
 15.7 
 
 15.1 
 
 14.3 
 
 13.4 
 
 12.5 
 
 11.7 
 
 11.1 
 
 10.8 
 
 10.8 
 
 10.5 
 
 10.6 
 
 10.7 
 
 440 
 
 18.6 
 
 17.9 
 
 17.1 
 
 16.1 
 
 15.6 
 
 14.4 
 
 13.5 
 
 12.8 
 
 11.9 
 
 11.1 
 
 10.6 
 
 10.3 
 
 10.3 
 
 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 
 
 21.2 
 
 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 
 
 21.8 
 
 21.7 
 
 21.5 
 
 21.2 
 
 20.9 
 
 20.3 
 
 19.6 
 
 560 
 
 17.2 
 
 18.4 
 
 19.0 
 
 20.0 
 
 20.8 
 
 21.1 
 
 22.7 
 
 21.9 
 
 22.2 
 
 22.1 
 
 21.9 
 
 21.7 
 
 21.1 
 
 580 
 
 15.0 
 
 16.0 
 
 17.3 
 
 18.2 
 
 19.1 
 
 19.9 
 
 20^.8 
 
 21.1 
 
 21.7 
 
 22.0 
 
 22.2 
 
 22.3 
 
 22.1 
 
 600 
 
 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 
 
 4.1 
 
 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 
 
 1C. 7 
 
 12.9 
 
 11.7 
 
 10.6 
 
 9.5 
 
 8.6 
 
 7.5 
 
 6.6 
 
 5.7 
 
 4.9 
 
 4.4 
 
 860 
 
 18.4 
 
 17.5 
 
 16.6 
 
 15.4 
 
 14.3 
 
 13.1 
 
 12.1 
 
 1.1 
 
 0.0 
 
 9.1 
 
 7.9 
 
 7.0 
 
 6.3 
 
 880 
 
 20.4 
 
 19.6 
 
 18.7 
 
 17.5 
 
 16.6 
 
 15.6 
 
 14.5 
 
 3.6 
 
 2.5 
 
 11.5 
 
 0.4 
 
 9.5 
 
 8.6 
 
 900 
 
 22.0 
 
 21.1 
 
 20.2 
 
 19.4 
 
 18.7 
 
 17.7 
 
 16.5 
 
 5.7 i 
 
 14.7 
 
 13.8 
 
 2.5 
 
 1.9 
 
 109 
 
 920 
 
 23.6 
 
 22.7 
 
 21.7 
 
 21.1 
 
 20.1 
 
 19.4 
 
 18.4 
 
 7.5 
 
 16.7 
 
 15.6 
 
 4.8 
 
 3.9 
 
 13.1 
 
 940 
 
 25.5 
 
 24.1 
 
 23.4 
 
 22.4 
 
 21.4 
 
 20.6 
 
 19.9 
 
 9.0 18.2 17.3 
 
 6.6 
 
 5.7 
 
 14.8 
 
 960 
 
 26.2 
 
 25.6 
 
 24.7 
 
 24.1 
 
 23.3 
 
 22.3 
 
 21.3 
 
 20.6 19.0 18.9 
 
 7.9 
 
 7.1 
 
 16.3 
 
 980 
 
 27.1 
 
 26.7 
 
 26.3 
 
 25.5 
 
 24.9 
 
 23.8 
 
 23.4 2.2 '21.0 20.4 
 
 9.4 
 
 8.6 
 
 17.7 
 
 1000 
 
 27.6 
 
 27.7 
 
 27.3 
 
 26.7 
 
 26.2 
 
 25.5 
 
 24.7 23.8 23.1 22.3 
 
 , L3 
 
 0.2 
 
 19.3 
 
 
 360 
 
 370 
 
 380 
 
 390 
 
 400 410 
 
 420 ! 430 440 450 
 
 460 
 
 470 
 
 480 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 Arguments II and III. 
 
 in. 
 
 II. 
 
 480 
 
 490 
 
 500 
 
 510 
 
 520 
 
 530 
 
 540 
 
 550 
 
 560 
 
 570 
 
 580 
 
 690 600 
 
 
 
 19.3 
 
 18.3 
 
 17.4 
 
 16.6 
 
 15.7 
 
 15.0 
 
 14.2 
 
 13.6 
 
 13.1 
 
 12.3 
 
 11.7 
 
 11.3 10.8 
 
 20 
 
 20.9 
 
 20.2 
 
 19. 1 
 
 18.2 
 
 17.1 
 
 16.2 
 
 15.5 
 
 14.7 
 
 14.1 
 
 13.3 
 
 12.7 
 
 12.2 11.5 
 
 40 
 
 23.2 22.0 
 
 20.8 
 
 20.1 
 
 18.9 
 
 17.9 
 
 17.1 
 
 15.9 
 
 15.1 
 
 14.4 
 
 13.7 
 
 13.0 12.3 
 
 60 
 
 25.0 | 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 
 
 27.2 
 
 26.4 
 
 25.6 
 
 24.1 
 
 23.2 
 
 22.1 
 
 20.8 
 
 20.0 
 
 18.7 
 
 17.9 
 
 16.6 
 
 15.6 ! 14.8 
 
 100 
 
 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 
 
 26.8 
 
 26.3 
 
 25.4 
 
 24.6 
 
 23.7 
 
 22.4 
 
 21.0 
 
 20.1 
 
 18.8 
 
 140 
 
 26.7 
 
 27.0 
 
 27.2 27.4 
 
 27.3 
 
 27.4 26.9 
 
 26.2 
 
 25.4 
 
 24.6 
 
 23.9 
 
 22.6 '21.1 
 
 160 
 
 25.1 
 
 25.6 
 
 26.1 26.7 
 
 26.9 
 
 27.3 j 27.1 
 
 27.0 
 
 26.9 
 
 26.4 
 
 25.5 
 
 24.7 
 
 23.9 
 
 180 
 
 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 
 
 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 
 
 231 
 
 23.3 
 
 23.9 
 
 24.2 
 
 24.6 
 
 260 
 
 20.2 
 
 20.3 
 
 20.6 
 
 21.2 
 
 21.4 
 
 21.7 
 
 21.9 
 
 22.2 
 
 223 
 
 22.7 
 
 23.1 
 
 23.3 
 
 23.6 
 
 280 
 
 20.8 
 
 20.8 
 
 21.0 
 
 21.1 
 
 21.3 
 
 21.4 
 
 21.5 
 
 21.8 
 
 22.0 
 
 22.2 
 
 22.7 
 
 23.0 
 
 23.3 
 
 300 
 
 20.9 
 
 21.0 
 
 21.5 
 
 21.7 
 
 21.7 
 
 22.0 
 
 22.0 
 
 22.1 
 
 22 1 
 
 22.2 
 
 224 
 
 22.6 
 
 22.8 
 
 320 
 
 20.8 
 
 21.2 
 
 21.5 
 
 21.6 
 
 22.0 
 
 22.3 
 
 22.5 
 
 22.5 
 
 226 
 
 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 
 
 235 
 
 23.9 
 
 24.5 
 
 400 
 
 12.6 
 
 13.2 
 
 14.2 
 
 15.4 
 
 16.2 
 
 17.3 
 
 18.1 
 
 19.2 
 
 20.3 
 
 21.4 
 
 224 
 
 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 
 
 201 
 
 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 
 
 15.1 
 
 16.5 
 
 480 
 
 12.2 
 
 11.4 
 
 10.7 
 
 10.1 
 
 9.7 
 
 9.5 
 
 9.7 
 
 9.9 
 
 10.2 
 
 10.7 
 
 11.7 
 
 12.5 
 
 13.4 
 
 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 
 
 560 
 
 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 
 
 206 
 
 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 
 
 21 9 
 
 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 
 
 226 
 
 22.4 
 
 22.0. 
 
 21.6 
 
 21.1 
 
 203 
 
 19.6 
 
 660 
 
 19.0 
 
 20.0 
 
 20.8 
 
 21.3 
 
 22.1 
 
 22.3 
 
 226 
 
 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 
 
 20.0 
 
 221 
 
 21.5 
 
 22.2 
 
 22.6 
 
 22.9 
 
 230 
 
 23.2 
 
 720 
 
 11.5 
 
 12.7 
 
 13.9 
 
 15.0 
 
 16.4 
 
 17.9 
 
 18.6 
 
 19.7 
 
 20.8 
 
 21.6 
 
 22.3 
 
 227 
 
 23.0 
 
 740 
 
 8.9 
 
 9.8 
 
 0.9 
 
 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 
 
 11. 1 
 
 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 
 
 8.2 
 
 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 
 
 34 
 
 3.4 
 
 3.6 
 
 3.9 
 
 900 
 
 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 
 
 1.2 
 
 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 
 
 1.0 
 
 0.1 
 
 9.6 
 
 8.8 
 
 8.1 
 
 7.5 
 
 980 
 
 17.7 
 
 16.8 
 
 16.2 
 
 15.2 
 
 14.5 
 
 13.9 13.1 
 
 2.5 
 
 1.8 
 
 11.2 
 
 10.5 
 
 9.7 
 
 9.3 
 
 1000 
 
 19.3 
 
 18.3 
 
 17.4 
 
 16.6 
 
 15.7 
 
 15.0 14.2 
 
 3.6 
 
 3.1 
 
 12.3 
 
 11.7 
 
 1.3 
 
 0.8 
 
 
 
 
 # 
 
 
 
 
 
 
 
 
 
 
 
 480 
 
 490 
 
 500 
 
 510 
 
 520 
 
 530 
 
 540 
 
 550 
 
 560 
 
 570 
 
 580 
 
 500 
 
 000 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 
 Arguments II. and III. 
 
 111. 
 
 27 
 
 11. 
 
 600 610 
 
 620 
 
 630 
 
 640 
 
 650 
 
 660 
 
 670 680 690 
 
 700 
 
 710 
 
 720 
 
 
 
 10.8 
 
 10.2 
 
 9.5 
 
 9.i 
 
 8.4 
 
 7.9 
 
 7.4 
 
 7.0 6.6 6.3 
 
 5.9 
 
 5.5 
 
 5.4 
 
 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 j 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 1 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 
 
 18.8 
 
 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 
 
 9.9 
 
 160 
 
 23.9 
 
 22.9 
 
 21.5 
 
 20.4 
 
 19.2 
 
 17.9 
 
 16.6 
 
 15.3 14.1 
 
 13.1 
 
 12.0 
 
 11.2 
 
 10.5 
 
 180 
 
 25.6 
 
 24.8 
 
 23.9 
 
 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 
 
 236 
 
 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 
 
 24.5 
 
 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 
 
 540 
 
 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 
 
 700 
 
 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 
 
 232 
 
 234 
 
 23.1 
 
 224 
 
 21 9 
 
 21 3 
 
 9,08 
 
 19.5 
 
 18.5 
 
 176 
 
 164 
 
 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 
 
 23.6 
 
 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 
 
 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 
 
 19.5,20.6 
 
 21.7 
 
 22.6 
 
 23.3 
 
 860 
 
 5.6 
 
 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 
 
 3.9 
 
 4.0 
 
 4.3 
 
 4.7 
 
 5.4 
 
 6.4 
 
 7.3 
 
 8.6 
 
 9.8 
 
 11.2 
 
 940 
 
 5.9 
 
 5.1 
 
 4.6 
 
 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 
 
 5.8 
 
 5.3 
 
 4.7 
 
 4.7 
 
 4.6 
 
 4.6 
 
 4.6 
 
 4.9 
 
 5.4 
 
 6.0 
 
 980 
 
 9.3 
 
 8.7 
 
 7.9 
 
 7.4 
 
 6.8 
 
 6.4 
 
 6.0 
 
 5.6 
 
 5.2 
 
 5.0 
 
 4.9 
 
 5.1 
 
 5.1 
 
 1000 
 
 10.8 
 
 10.2 
 
 9.5 
 
 9.1 
 
 8.4 
 
 7.9 
 
 7.4 
 
 7.0 
 
 6.6 
 
 6.3 
 
 5.9 
 
 5.5 
 
 5.4 
 
 
 600 
 
 610 
 
 620 
 
 630 
 
 640 
 
 650 
 
 660 
 
 670 
 
 680 ! 690 
 
 700 
 
 710 
 
 720 
 
28 
 
 TABLE XXX. 
 
 Perturbations produced by Vtnus. 
 Arguments II. and III. 
 
 in- 
 
 11. 720 
 
 730 740 
 
 750 
 
 760 
 
 770 
 
 780 
 
 790 800 
 
 810 
 
 820 
 
 830 
 
 840 
 
 o' 5.4 
 
 5.5 5.8 
 
 6.0 
 
 6.3 
 
 6.8 
 
 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 
 
 11.2 
 
 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 
 
 9.11 9.0 
 
 8.8 
 
 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 9.9 
 
 9.5 
 
 9.6 
 
 9.4 
 
 9.3! 9.3 
 
 9.0 
 
 9.3 
 
 9.5 
 
 9.8 
 
 9.7 
 
 9.8 
 
 10.0 
 
 160 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 
 
 180 12.0 
 
 11.0 10.1 
 
 9.7 
 
 9.1 8.8 
 
 8.7 
 
 8.3 
 
 8.5 
 
 8.7 
 
 8.8 
 
 9.0 
 
 9.1 
 
 200 14.4 
 
 13.3 12.0 
 
 11.0 
 
 10.1 
 
 9.4 
 
 8.9 
 
 8.5 
 
 8.2 
 
 8.0 
 
 8.0 
 
 8.3 
 
 8.5 
 
 220 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 1 21.4 
 
 20.5 
 
 19.8 
 
 18.7 
 
 17.5 
 
 16.1 
 
 15.0 
 
 13.7 
 
 12.4 
 
 320 24.2 
 
 23.9 23.5 
 
 23.1 
 
 22.7 
 
 22.2 
 
 21.2 
 
 20.6 
 
 19.6 
 
 18.6 
 
 17.5 
 
 16.3 
 
 15.1 
 
 340 23.8 
 
 23.9 23.7 
 
 23.5 
 
 23.2 
 
 22.8 
 
 22.3 
 
 21.4 
 
 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 
 
 24.0 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 26.9 
 
 26.6 26.4 
 
 2G.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 
 
 28.1 
 
 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 86.8 
 
 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 i 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 
 
 13.6 
 
 12.8 
 
 11.9 
 
 11.3 
 
 10.7 
 
 10.3 
 
 780 
 
 230 
 
 22.3 
 
 21.5 
 
 20.5 
 
 19.4 
 
 18.4 
 
 17.2 
 
 l&ft 
 
 14.9 
 
 14.0 
 
 13.0 
 
 12.2 
 
 11.3 
 
 800 
 
 23.9 
 
 23.9 
 
 23.4 
 
 22.6 
 
 21.9 
 
 20.7 
 
 19.8 
 
 18.8 
 
 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 
 
 243 
 
 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 
 
 24.5 
 
 24.3 
 
 23.6 
 
 23.1 
 
 21.9 
 
 21.0 
 
 ; 880 
 
 18.2 
 
 19.7 
 
 20.9 
 
 22.0 
 
 22.8 
 
 23.8 
 
 24.1 
 
 24.6 
 
 24.8 
 
 24.7 
 
 24.5 
 
 24.0 | 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 
 
 s 940 
 
 8.0 
 
 9.3 
 
 10.7 
 
 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 
 
 iO.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 
 
 5.5 
 
 6.8 
 
 5.8 
 
 6.3 
 
 6.8 
 
 7.6 
 
 8.4 
 
 9.3 
 
 10.5 
 
 11.7 
 
 12.9 
 
 14.3 
 
 
 720 
 
 730 740 
 
 750 
 
 760 1 770 
 
 780 
 
 790 
 
 800 
 
 810 820 
 
 830 
 
 840 
 
TABLE XXX. 
 
 Perturbations produced by Venus. 
 
 Arguments II. and III. 
 
 III. 
 
 II. 
 
 840 
 
 850 
 
 860 
 
 870 f 880 
 
 890 
 
 900 
 
 910 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
 
 
 14.3 
 
 15.5 
 
 16.9 
 
 18.2 
 
 19.2 
 
 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 
 
 12.9 
 
 14.0 
 
 120 
 
 9.6 
 
 9.5 
 
 9.3 
 
 9.6 
 
 9.6 
 
 9.7 
 
 9.9 
 
 9.8 
 
 10.4 
 
 10.9 
 
 11.3 
 
 11.8 
 
 12.3 
 
 140 
 
 10.0 
 
 10.2 
 
 10.1 
 
 10.2 
 
 10.1 
 
 10.3 
 
 10.4 
 
 10.5 
 
 10.5 
 
 10.6 
 
 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 
 
 6.7 
 
 6.8 
 
 6.8 
 
 7.0 
 
 7.5 
 
 320 
 
 15.1 
 
 13.9 
 
 12.5 
 
 11.4* 
 
 10.5 
 
 9.7 
 
 8.6 
 
 7.8 
 
 7.4 
 
 7.0 
 
 6.6 
 
 6.5 
 
 6.7 
 
 340 
 
 17.4 
 
 16.4 
 
 15.2 
 
 13.9 
 
 12.7 
 
 11.6 
 
 10.6 
 
 9.7 
 
 8.7 
 
 8.0 
 
 7.3 
 
 6.8 
 
 6.6 
 
 360 
 
 18.9 
 
 18.1 
 
 174 
 
 16.3 
 
 15.1 
 
 13.8 
 
 12.8 
 
 11.7 
 
 10.6 
 
 9.8 
 
 8.8 
 
 8.0 
 
 7A 
 
 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 i 27.8 
 
 27.3 
 
 26.8 
 
 26.3 
 
 25.6 
 
 24.7 
 
 23.9 
 
 23.3 
 
 22.6 
 
 21.8 
 
 20.8 
 
 20.1 
 
 540 
 
 29.2 
 
 29.2 
 
 28.9 
 
 28.5 
 
 27.8 
 
 27.4 
 
 26.8 
 
 26.1 
 
 25.3 
 
 24.4 
 
 23.7 
 
 23.0 
 
 22.0 
 
 560 
 
 29.2 
 
 29.3 
 
 29.5 
 
 29.6 
 
 29.3 
 
 29.1 
 
 28.8 
 
 28.0 
 
 27.4 
 
 26.9 
 
 26.1 
 
 25.1 
 
 24.3 
 
 580 
 
 27.8 
 
 28.6 
 
 29.0 
 
 29.4 
 
 29.6 
 
 29.8 
 
 29.8 
 
 29.3 
 
 28.0 
 
 28.7 
 
 27.9 
 
 27.3 
 
 26.6 
 
 600 
 
 25.5 
 
 26.7 
 
 27.6 
 
 28.4 
 
 28.9 
 
 29.2 
 
 29.6 
 
 29.9 
 
 29.9 
 
 29.8 
 
 29.3 
 
 29.0 
 
 28.5 
 
 620 
 
 22.6 
 
 23.8 
 
 25.0 
 
 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 
 
 16.2 
 
 17.5 
 
 18.8 
 
 20.2 
 
 21.1 
 
 22.9 
 
 24.0 
 
 25.1 
 
 26.2 
 
 27.1 
 
 28.2 
 
 28.8 
 
 29.2 
 
 680 
 
 13.8 
 
 14.7 
 
 15.8 
 
 16.9 
 
 18.4 
 
 19.9 
 
 20.6 
 
 22.3 
 
 23.6 
 
 24.9 
 
 25.8 
 
 26.7 
 
 27.5 
 
 700 
 
 11.5 
 
 12.3 
 
 13.4 
 
 14.6 
 
 15.6 
 
 16.7 
 
 18.0 
 
 19.5 
 
 20.7 
 
 22.0 
 
 23.1 
 
 24.2 
 
 25.1 
 
 720 
 
 10.4 
 
 11.0 
 
 11.4 
 
 12.3 
 
 13.3 
 
 14.3 
 
 15.6 
 
 16.4 
 
 17.7 
 
 19.3 
 
 19.9 
 
 21.6 
 
 22.6 
 
 740 
 
 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 
 
 19.9 
 
 19.7 
 
 18.6 
 
 17.2 
 
 16.4 
 
 15.3 
 
 14.1 
 
 13.3 
 
 920 
 
 24.5 
 
 24.8 
 
 24.7 
 
 243 
 
 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 
 
 24.2 
 
 
 840 
 
 850 
 
 860 
 
 870 
 
 880 890 
 
 900 
 
 910 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
30 
 
 TABLE XXX. XXXI. 
 
 Perturbations by Venus. 
 
 Arguments II and III. 
 
 HI. 
 
 Perturbations by Mars. 
 
 Arguments II and IV. 
 
 IV. 
 
 11. 
 
 960 | 
 
 970 
 
 980 
 
 990 
 
 1000 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 
 
 24.2 
 
 23.7 
 
 23.1 
 
 22.5 
 
 21.6 
 
 9.5 
 
 10.2 
 
 10.8 
 
 11.2 
 
 11.5 
 
 11.7 
 
 11.8 
 
 11.5 
 
 20 
 
 23.6 
 
 23.7 
 
 24.0 
 
 23.4 
 
 23.1 
 
 8.3 
 
 9.1 
 
 9.8 
 
 10.5 
 
 10.9 
 
 11.2 
 
 11.5 
 
 11.6 
 
 40 
 
 21.6 
 
 22.4 
 
 22.9 
 
 23.5 
 
 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 
 
 21.5 
 
 22.2 
 
 5.8 
 
 6.7 
 
 7.6 
 
 8.4 
 
 9.1 
 
 9.8 
 
 10.3 
 
 10.5 
 
 80 
 
 16.2 
 
 17.3 
 
 18.4 
 
 19.7 
 
 20.0! 
 
 4.3 
 
 5.3 
 
 6.4 
 
 7.2 
 
 8.0 
 
 8.9 
 
 9.3 
 
 9.9 
 
 100 
 
 14.0 
 
 14.8 
 
 15.6 
 
 16.5 
 
 17.6 
 
 3.3 
 
 4.2 
 
 5.0 
 
 5.9 
 
 6.8 
 
 7.6. 
 
 8.4 
 
 9.1 
 
 120 
 
 12.3 
 
 12.9 
 
 13.7 
 
 14.3 
 
 15.3 
 
 2.4 
 
 3.1 
 
 3.9 
 
 4.8 
 
 5.6 
 
 6.4 
 
 7.3 
 
 8.0 
 
 140 
 
 11.5 
 
 12.0 
 
 12.6 
 
 12.8 
 
 13.6 
 
 2.1 
 
 2.4 
 
 2.9 
 
 3.8 
 
 4.6 
 
 5.5 
 
 6.3 
 
 7.0 
 
 160 
 
 11.6 
 
 11.8 
 
 12.1 
 
 12.3 
 
 12.7 
 
 2.0 
 
 2.2 
 
 2.4 
 
 2.7 
 
 3.5 
 
 4.4 
 
 5.1 
 
 5.9 
 
 180 
 
 12.2 
 
 12.2 
 
 12.3 
 
 12.5 
 
 12.7 
 
 1.9 
 
 2.0 
 
 2.3 
 
 2.6 
 
 2.9 
 
 3.4 
 
 3.9 
 
 4.9 
 
 200 
 
 12.4 
 
 12.7 
 
 12.8 
 
 13.1 
 
 13.2 
 
 2.3 
 
 2.2 
 
 2.2 
 
 2.4 
 
 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 
 
 4.2 
 
 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 
 
 6.2 
 
 5.8 
 
 5.5 
 
 5.1 
 
 360 
 
 7.4 
 
 6.9 
 
 6.5 
 
 6.5 
 
 6.5 
 
 7.5 
 
 7.4 
 
 7.1 
 
 7.0 
 
 6.8 
 
 6.4 
 
 6.2 
 
 5.8 
 
 880 
 
 8.9 
 
 8.2 
 
 7.5 
 
 6.9 
 
 6.8 
 
 7.5 
 
 7.6 
 
 7.3 
 
 7.3 
 
 7.2 
 
 7.1 
 
 6.7 
 
 6.5 
 
 400 
 
 10.9 
 
 10.0 
 
 9.0 
 
 8.3 
 
 7.5 
 
 7.3 
 
 7.3 
 
 7.5 
 
 7.4 
 
 7.4 
 
 7.4 
 
 7.1 
 
 7.0 
 
 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 
 
 440 
 
 14.2 
 
 13.3 
 
 12.5 
 
 11.6 
 
 10.6 
 
 6.5 
 
 6.8 
 
 6.8 
 
 7.1 
 
 7.2 
 
 7.3 
 
 7.3 
 
 7.4 
 
 460 
 
 15.7 
 
 14.8 
 
 13.9 
 
 13.0 
 
 12.1 
 
 6.2 
 
 6.2 
 
 6.5 
 
 6.7 
 
 6.8 
 
 7.1 
 
 7.1 
 
 7.3 
 
 480 
 
 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 
 
 6.1 
 
 5.1 
 
 5.3 
 
 5.4 
 
 5.6 
 
 5.8 
 
 6.0 
 
 540 
 
 22.0 
 
 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 
 
 22.6 
 
 21.5 
 
 20.6 
 
 4.4 
 
 4.5 
 
 4.6 
 
 4.6 
 
 4.7 
 
 4.8 
 
 4.8 
 
 5.0 
 
 580 
 
 26.6 
 
 25.7 
 
 24.9 
 
 23.8 
 
 23.0 
 
 4.2 
 
 4.3 
 
 4.4 
 
 4.3 
 
 4.5 
 
 4.4 
 
 4.4 
 
 4.5 
 
 600 
 
 28.5 
 
 27.8 
 
 27.0 
 
 26.3 
 
 25.4 
 
 4.0 
 
 4.2 
 
 4.3 
 
 4.2 
 
 4.2 
 
 4.2 
 
 4.2 
 
 4.3 
 
 620 
 
 29.6 
 
 29.2 
 
 28.8 
 
 28.2 
 
 27.4 
 
 4.2 
 
 4.0 
 
 4.1 
 
 4.0 
 
 4.0 
 
 4.0 
 
 4,0 
 
 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 
 
 29.5 
 
 29.7 
 
 29.8 
 
 29.9 
 
 4.6 
 
 4.4 
 
 4.3 
 
 4.1 
 
 4.1 
 
 4.1 
 
 4.0 
 
 3.8 
 
 680 
 
 27.5 
 
 28.6 
 
 28.9 
 
 29.2 
 
 29.7 
 
 4.8 
 
 4.6 
 
 4.5 
 
 4.3 
 
 4.2 
 
 4.1 
 
 4.0 
 
 3.9 
 
 700 
 
 25.1 
 
 26.4 
 
 27.3 
 
 27.8 
 
 28.7 
 
 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 
 
 lp 
 
 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 
 
 5.3 
 
 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 
 
 12.0 
 
 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 
 
 11.2 
 
 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 
 
 12.3 
 
 12.3 
 
 12.2 
 
 11.9 
 
 11.6 
 
 11.0 
 
 10.5 
 
 9.9 
 
 940 
 
 18.4 
 
 17.3 
 
 16.2 
 
 14.5 
 
 14.0 
 
 12.1 
 
 12.1 
 
 12.2 
 
 12.2 
 
 11.8 
 
 11.4 
 
 11.0 
 
 10.6 
 
 960 
 
 21.0 
 
 20.0 
 
 189 
 
 17.9 
 
 16.7 
 
 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 
 
 1000 
 
 24.2 
 
 23.7 
 
 23.1 
 
 22.5 
 
 21.6 
 
 9.5 
 
 10.2 
 
 10.8 
 
 ; 11.2 
 
 11.5 
 
 11.7 
 
 11.8 
 
 11.5 
 
 
 960 
 
 970 
 
 980 
 
 990 
 
 1000 
 
 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
TABLE XXXI. 
 
 31 
 
 Perturbations produced by Mars 
 
 Arguments II and IV. 
 
 IV. 
 
 II. 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 130 
 
 140 
 
 150 
 
 160 
 
 170 
 
 180 
 
 190 
 
 200 
 
 
 
 11.5 
 
 11.2 
 
 11.0 
 
 10.6 
 
 10.1 
 
 9.9 
 
 9.5 
 
 9.0 
 
 8.6 
 
 8.2 
 
 8.1 
 
 7.8 
 
 7.6 
 
 7.4 
 
 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 
 
 7.8 
 
 40 
 
 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 
 
 10.5 
 
 10.4 
 
 10.3 
 
 10.1 
 
 9.9 
 
 9.6 
 
 9.3 
 
 9.0 
 
 8.8 
 
 120 
 
 8.0 
 
 8.8 
 
 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 
 
 9.9 
 
 9.9 
 
 9.7 
 
 9.7 
 
 9.4 
 
 9.3 
 
 8.9 
 
 160 
 
 5.9 
 
 6.5 
 
 7.2 
 
 8.0 
 
 8.5 
 
 8.9 
 
 9.2 
 
 9.6 
 
 9.5 
 
 9.6 
 
 9.5 
 
 9.5 
 
 9.3 
 
 9.1 
 
 180 
 
 4.9 
 
 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 
 
 3.5 
 
 3.9 
 
 4.4 
 
 5.1 
 
 5.8 
 
 6.4 
 
 7.1 
 
 7.6 
 
 7.9 
 
 8.4 
 
 8.6 
 
 8.8 
 
 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 
 
 3.2 
 
 3.8 
 
 4.1 
 
 4.5 
 
 4.9 
 
 5.4 
 
 5.9 
 
 6.6 
 
 7.1 
 
 7.5 
 
 7.7 
 
 8.0 
 
 8.2 
 
 280 
 
 3.4 
 
 3.4 
 
 3.5 
 
 3.8 
 
 4.2 
 
 4.5 
 
 4.9 
 
 5.5 
 
 5.6 
 
 6.2 
 
 6.8 
 
 7.1 
 
 7.5 
 
 7.8 
 
 300 
 
 3.8 
 
 3.7 
 
 3.7 
 
 3.7 
 
 3.9 
 
 4.4 
 
 4.7 
 
 4.9 
 
 5.4 
 
 5.7 
 
 6.0 
 
 6.6 
 
 6.9 
 
 7.3 
 
 320 
 
 4.3 
 
 4.2 
 
 4.1 
 
 4.0 
 
 4.1 
 
 4.2 
 
 4.4 
 
 4.7 
 
 5.0 
 
 5.4 
 
 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 
 
 6.0 
 
 6.3 
 
 360 
 
 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 
 
 5.1 
 
 5.1 
 
 5.4 
 
 5.5 
 
 5.7 
 
 5.8 
 
 400 
 
 7.0 
 
 6.7 
 
 6.7 
 
 6.3 
 
 6.1 
 
 5.9 
 
 5.7 
 
 5.6 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.6 
 
 5.7 
 
 5.9 
 
 420 
 
 7.4 
 
 7.2 
 
 6.9 
 
 7.1 
 
 6.7 
 
 6.4 
 
 6.3 
 
 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 
 
 6.9 
 
 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.G 
 
 6.8 
 
 6.9 
 
 7.2 
 
 7.3 
 
 7.5 
 
 7.5 
 
 7.6 
 
 7.8 
 
 7.7 
 
 7.8 
 
 7.7 
 
 7.6 
 
 7.4 
 
 520 
 
 6.0 
 
 '6.3 
 
 6.5 
 
 6.7 
 
 7.1 
 
 7.2 
 
 7.5 
 
 7.5 
 
 7.7 
 
 7.8 
 
 7.9 
 
 7.6 
 
 7.9 
 
 7.9 
 
 540 
 
 5.5 
 
 5.7 
 
 6.0 
 
 6.3 
 
 6.6 
 
 6.9 
 
 7.1 
 
 7.3 
 
 7.4 
 
 7.7 
 
 7.9 
 
 8.0 
 
 8.2 
 
 8.3 
 
 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 
 
 5.4 
 
 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 
 
 5.0 
 
 5.2 
 
 5.8 
 
 6.3 
 
 6.7 
 
 660 
 
 3.8 
 
 3.7 
 
 3.7 
 
 3.6 
 
 3.6 
 
 3.7 
 
 3.8 
 
 3.9 
 
 4.1 
 
 4.2 
 
 4.5 
 
 5.0 
 
 5.3 
 
 6.0 
 
 680 
 
 3.9 
 
 3.8 
 
 3.6 
 
 3.4 
 
 3.5 
 
 3.4 
 
 35 
 
 3.6 
 
 3.6 
 
 3.7 
 
 3.8 
 
 4.2 
 
 4.6 
 
 4.9 
 
 700 
 
 4.1 
 
 3.9 
 
 3.8 
 
 3.6 
 
 3.5 
 
 3.3 
 
 3.3 
 
 3.2 
 
 3.2 
 
 3.2 
 
 3.5 
 
 3.6 
 
 3.8 
 
 4.2 
 
 720 
 
 4.1 
 
 4.1 
 
 4.0 
 
 3.8 
 
 3.6 
 
 3.5 
 
 3.3 
 
 3.2 
 
 3.3 
 
 3.2 
 
 3.0 
 
 3* 
 
 3.4 
 
 3.6 
 
 740 
 
 4.3 
 
 4.3 
 
 4.2 
 
 4.0 
 
 3.8 
 
 3.7 
 
 3.5 
 
 3.2 
 
 3.0 
 
 3.0 
 
 2.9 
 
 2.8 
 
 2.9 
 
 3.1 
 
 760 
 
 5.0 
 
 4.7 
 
 4.4 
 
 4.3 
 
 4.1 
 
 3.8 
 
 3.7 
 
 3.4 
 
 3.1 
 
 3.0 
 
 2.9 
 
 2.7 
 
 2.7 
 
 2.8 
 
 780 
 
 5.3 
 
 5.1 
 
 4.7 
 
 4.6 
 
 4.4 
 
 4.4 
 
 4.0 
 
 3.8 
 
 3.4 
 
 3.2 
 
 2.9 
 
 2.8 
 
 2.7 
 
 2.5 
 
 800 
 
 5.8 
 
 5.5 
 
 5.4 
 
 4.8 
 
 4.7 
 
 4.7 
 
 4.5 
 
 4.2 
 
 3.9 
 
 3.5 
 
 3.3 
 
 2.9 
 
 2.8 
 
 2.7 
 
 820 
 
 6.5 
 
 6.1 
 
 5.S 
 
 5.6 
 
 5.0 
 
 5.0 
 
 4.9 
 
 4.6 
 
 4.3 
 
 4.1 
 
 3.6 
 
 3.3 
 
 3.0 
 
 2.9 
 
 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 
 
 6.6 
 
 6.2 
 
 6.2 
 
 5.5 
 
 5.4 
 
 5.2 
 
 5.0 
 
 4.8 
 
 4.4 
 
 4.1 
 
 3.6 
 
 880 
 
 8.4 
 
 7.9 
 
 7.6 
 
 7.1 
 
 6.9 
 
 6.4 
 
 6.4 
 
 5.8 
 
 5.7 
 
 .5.4 
 
 5.2 
 
 5.0 
 
 4.6 
 
 4.3 
 
 900 
 
 9.3 
 
 8.7 
 
 8.3 
 
 7.7 
 
 7.4 
 
 7.1 
 
 C.7 
 
 6.6 
 
 6.1 
 
 6.0 
 
 5.6 
 
 5.4 
 
 5.2 
 
 4.9 
 
 920 
 
 9.9 
 
 9.3 
 
 .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 
 
 %.5 
 
 8.9 
 
 8.7 
 
 8.2 
 
 7.8 
 
 7.6 
 
 7.2 
 
 7.1 
 
 6.5 
 
 6.5 
 
 6.3 
 
 6.9 
 
 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 
 
 6.6 
 
 980 
 
 11.4 
 
 11.0 
 
 10.6 
 
 10.2 
 
 9.8 
 
 9.2 
 
 8.9 
 
 8.4 
 
 8.1 
 
 8.0 
 
 7.6 
 
 7.3 
 
 7.2 
 
 6.9 
 
 1000 
 
 11.5 
 
 112 
 
 11.0 
 
 10.6 
 
 10.0 
 
 9.9 
 
 9.5 
 
 9.0 
 
 8.6 
 
 8.2 
 
 8.1 
 
 7.4 
 
 7.6 
 
 7.4 
 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 130 
 
 140 
 
 150 
 
 160 
 
 170 
 
 180 
 
 190 
 
 200 
 
32 
 
 TABLE XXXI. 
 
 Perturbations produced by Mars. 
 
 Arguments II. and IV. 
 
 IV. 
 
 II. 
 
 200 
 
 210 
 
 220 
 
 230 
 
 240 
 
 250 
 
 260 
 
 270 
 
 280 
 
 290 
 
 300 
 
 310 
 
 320 
 
 
 
 7.4 
 
 7.2 
 
 7.0 
 
 6.6 
 
 6.4 
 
 6.2 
 
 5.7 
 
 5.3 
 
 4.9 
 
 4.7 
 
 4.1 
 
 3.8 
 
 3.4 
 
 20 
 
 7.8 
 
 7.2 
 
 7.3 
 
 7.2 
 
 7.0 
 
 6.6 
 
 6.3 
 
 6.0 
 
 5.7 
 
 5.3 
 
 5.0 
 
 4.4 
 
 3.9 
 
 40 
 
 8.2 
 
 8.1 
 
 7.6 
 
 7.5 
 
 7.3 
 
 7.2 
 
 6.8 
 
 6.6 
 
 6.2 
 
 5.9 
 
 5.6 
 
 5.2 
 
 4.7 
 
 60 
 
 8.4 
 
 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 
 
 6.6 
 
 { 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 
 
 8.2 
 
 8.1 
 
 8.3 
 
 8.0 
 
 7.9 
 
 7.8 
 
 7.7 
 
 7.5 
 
 7.4 
 
 -' 160 
 
 9.1 
 
 8.9 
 
 8.7 
 
 8.4 
 
 8.3 
 
 8.3 
 
 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 
 
 8.0 
 
 7.8 
 
 200 
 
 8.9 
 
 8.8 
 
 8.6 
 
 8.4 
 
 8.4 
 
 8.3 
 
 8.1 
 
 8.0 
 
 7.9 
 
 7.8 
 
 7.8 
 
 7.9 
 
 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 
 
 7.3 
 
 7.7 
 
 240 
 
 8.5 
 
 8.4 
 
 8.5 
 
 8.3 
 
 8.1 
 
 8.0 
 
 7.8 
 
 7.8 
 
 7.8 
 
 7.8 
 
 7.8 
 
 7.8 
 
 7.6 
 
 260 
 
 8.2 
 
 8.2 
 
 8.1 
 
 8.1 
 
 8.1 
 
 7.8 
 
 7.8 
 
 7.7 
 
 7.6 
 
 7.6 
 
 7.6 
 
 7.5 
 
 7.4 
 
 280 
 
 7.8 
 
 7.8 
 
 8.0 
 
 7.8 
 
 7.9 
 
 7.9 
 
 7.7 
 
 7.5 
 
 7.5 
 
 7.3 
 
 7.3 
 
 7.4 
 
 7.3 
 
 300 
 
 7.3 
 
 7.6 
 
 7.5 
 
 7.6 
 
 7.7 
 
 7.6 
 
 7.6 
 
 7.6 
 
 7.4 
 
 7.3 
 
 7.1 
 
 7.0 
 
 7.1 
 
 320 
 
 6.6 
 
 7.1 
 
 7.3 
 
 7.4 
 
 7.4 
 
 7.3 
 
 7.4 
 
 7.4 
 
 7.3 
 
 7.1 
 
 7.0 
 
 7.0 
 
 6.8 
 
 340 
 
 6.3 
 
 6.4 
 
 6.7 
 
 7.2 
 
 7.1 
 
 7.2 
 
 7.2 
 
 7.1 
 
 7.1 
 
 7.0 
 
 6.9 
 
 6.8 
 
 6.8 
 
 360 
 
 6.1 
 
 6.2 
 
 6.4 
 
 6.5 
 
 6.9 
 
 6.9 
 
 7.0 
 
 7.0 
 
 6.9 
 
 6.8 
 
 6.7 
 
 6.6 
 
 6.5 
 
 380 
 
 5.8 
 
 6.1 
 
 6.3 
 
 6.4 
 
 6.6 
 
 6.7 
 
 6.6 
 
 6.6 
 
 6.7 
 
 6.8 
 
 6.7 
 
 6.6 
 
 6.5 
 
 400 
 
 5.9 
 
 6.0 
 
 6.2 
 
 6.3 
 
 6.4 
 
 6.5 
 
 6.6 
 
 6.6 
 
 6.5 
 
 6.6 
 
 6.6 
 
 6.5 
 
 6.4 
 
 420 
 
 6.1 
 
 6.3 
 
 6.2 
 
 6.4 
 
 6.3 
 
 6.4 
 
 6.5 
 
 6.6 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.4 
 
 440 
 
 6.3 
 
 6.4 
 
 6.4 
 
 6.6 
 
 6.5 
 
 6.6 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.5 
 
 6.3 
 
 6.3 
 
 6.2 
 
 460 
 
 6.7 
 
 6.5 
 
 6.5 
 
 6.6 
 
 6.7 
 
 6.9 
 
 6.7 
 
 6.6 
 
 6.6 
 
 6.6 
 
 6.5 
 
 6.3 
 
 6.2 
 
 480 
 
 7.1 
 
 7.1 
 
 7.0 
 
 6.9 
 
 6.9 
 
 6.9 
 
 7.0 
 
 7.0 
 
 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 
 
 7.0 
 
 7.5 
 
 7.2 
 
 560 
 
 8.2 
 
 8.6 
 
 8.4 
 
 8.6 
 
 8.7 
 
 8.5 
 
 8.5 
 
 8.4 
 
 8.2 
 
 8.3 
 
 8.2 
 
 8.0 
 
 7.6 
 
 580 
 
 8.2 
 
 8.3 
 
 8.6 
 
 8.8 
 
 8.8 
 
 9.0 
 
 8.9 
 
 8.9 
 
 8.7 
 
 8.7 
 
 8.6 
 
 8.4 
 
 8.4 
 
 600 
 
 7.7 
 
 8.1 
 
 8.5 
 
 8.6 
 
 8.9 
 
 9.1 
 
 9.1 
 
 9.2 
 
 9.2 
 
 9.1 
 
 9.0 
 
 8.8 
 
 8.7 
 
 620 
 
 7.4 
 
 7.6 
 
 8.0 
 
 8.5 
 
 8.7 
 
 9.0 
 
 9.2 
 
 9.5 
 
 9.5 
 
 9.5 
 
 9.4 
 
 9.3 
 
 9.2 
 
 640 
 
 6.7 
 
 7.2 
 
 7.5 
 
 7.9 
 
 8.3 
 
 8.7 
 
 9.0 
 
 9.3 
 
 9.5 
 
 9.8 
 
 9.8 
 
 9.7 
 
 9.7 
 
 660 
 
 6.0 
 
 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 
 
 5.8 
 
 6.4 
 
 6.8 
 
 7.4 
 
 8.0 
 
 8.5 
 
 8.9 
 
 9.2 
 
 9.8 
 
 10.1 
 
 720 
 
 r 3.6 
 
 3.9 
 
 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 
 
 3.3 
 
 3.6 
 
 4.0 
 
 4.4 
 
 5.1 
 
 5.8 
 
 6.5 
 
 7.2 
 
 7.8 
 
 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 
 
 6.9 
 
 7.6 
 
 800 
 
 2.7 
 
 2.5 
 
 2.5 
 
 2.5 
 
 2.5 
 
 2.7 
 
 3.0 
 
 3.4 
 
 3.8 
 
 4.4 
 
 5.0 
 
 5.6 
 
 6.6 
 
 820 
 
 2.9 
 
 2.6 
 
 2.4 
 
 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 
 
 3.0 
 
 2.7 
 
 2.4 
 
 2.3 
 
 2.1 
 
 2.2 
 
 2.3 
 
 2.5 
 
 2.7 
 
 3,0 
 
 3.4 
 
 880 
 
 4.3 
 
 3.8 
 
 3.6 
 
 3.2 
 
 2.8 
 
 2.5 
 
 2.3 
 
 2.1 
 
 2.0 
 
 2.2 
 
 2.3 
 
 2.5 
 
 2.6 
 
 900 
 
 4.9 
 
 4.6 
 
 4.2 
 
 3.6 
 
 3.4 
 
 2.9 
 
 2.6 
 
 2.3 
 
 2.2 
 
 2.2 
 
 2.1 
 
 2.2 
 
 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 
 
 4.7 
 
 4.3 
 
 3.8 
 
 3.4 
 
 3.0 
 
 37 
 
 2.4 
 
 2.1 
 
 2.0 
 
 960 
 
 6.5 
 
 6.2 
 
 59 
 
 1.5 
 
 5.1 
 
 4.9 
 
 4.5 
 
 4.0 
 
 3.4 
 
 3.1 
 
 2.8 
 
 2.4 
 
 2.3 
 
 ; 980 
 
 6.9 
 
 6.8 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.4 
 
 5.1 
 
 4.8 
 
 4.3 
 
 3.9 
 
 3.5 
 
 3.0 
 
 2.7 
 
 1000 
 
 7.4 
 
 7.2 
 
 7.0 
 
 6.6 
 
 6.4 
 
 6.2 
 
 5.7 
 
 5.3 
 
 4.9 
 
 4.7 
 
 4.1 
 
 3.8 
 
 34 
 
 
 200 
 
 210 
 
 220 
 
 230 
 
 240 
 
 250 
 
 260 
 
 270 
 
 280 
 
 290 
 
 300 
 
 310 
 
 320 
 
TABLE XXXI. 
 
 33 
 
 Perturbations produced by Mars. 
 
 Arguments II. and IV. 
 
 IV. 
 
 II. 
 
 320 
 
 330 340 
 
 350 
 
 360 
 
 370 
 
 380 
 
 390 
 
 400 
 
 410 
 
 420 
 
 430 
 
 440 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3.4 
 
 2.8 
 
 2.6 
 
 2.4 
 
 2.2 
 
 2.3 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.4 
 
 4.0 
 
 45 
 
 20 
 
 3.9 
 
 3.5 3.1 
 
 2.7 
 
 2.6 
 
 2.4 
 
 2.4 
 
 2.3 
 
 2.5 
 
 2.7 
 
 3.0 
 
 3.3 
 
 3.J 
 
 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 
 
 2.Q 
 
 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 j 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 
 
 7.6 
 
 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 
 
 480 
 
 6.3! 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 
 
 56 
 
 5.3 
 
 5.1 
 
 620 
 
 9.2 
 
 9.1 
 
 8.9 
 
 8.6 
 
 8.4 
 
 8.1 
 
 7.6 
 
 7.2 
 
 6.8 
 
 65 
 
 6.1 
 
 5.7 
 
 5.3 
 
 640 
 
 9.7 
 
 9.6 
 
 9.4 
 
 9.p 
 
 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 
 
 ]0.5 
 
 10.2 
 
 9.9 
 
 9.4 
 
 760 
 
 8.4 
 
 9.0 
 
 9.5 
 
 9.8 
 
 10.2 
 
 10.6 
 
 10.9 
 
 11.0 
 
 11.0 
 
 11.0 
 
 10.7 
 
 10.5 
 
 10.3 
 
 780 
 
 7.6 
 
 8.2 
 
 8.9 
 
 9.4 
 
 9.9 
 
 10.3 
 
 10.& 
 
 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 
 
 iai 
 
 10.6 
 
 10.8 
 
 11.1 
 
 11.3 
 
 11.1 
 
 IkO 
 
 820 
 
 5.4 
 
 6.0 
 
 7.0 
 
 7.6 
 
 8.2 
 
 8.9 
 
 9.6 
 
 10.0 
 
 10.5 
 
 10.8 
 
 11.0 
 
 11.3^ 
 
 11.3 
 
 840 
 
 4.3 
 
 5.0 
 
 5.6 
 
 6.5 
 
 7.2 
 
 7.9 
 
 9.8 
 
 9.2 
 
 9.9 
 
 10.3 
 
 107 
 
 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 
 
 66 
 
 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 
 
 .l 
 
 9.7 
 
 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 
 
 7.7 
 
 960 
 
 2.3 
 
 2.2 
 
 2.2 
 
 2.3 
 
 2.3 
 
 2.5 
 
 2.8 
 
 3.2 
 
 3.9 
 
 4.5 
 
 5.1 
 
 5.7 
 
 6.5 
 
 980 
 
 $.7 
 
 2.4 
 
 2.2 
 
 2.3 
 
 2.3 
 
 2.4 
 
 25 
 
 2.8 
 
 3.0 
 
 3.6 
 
 4,1 
 
 4.7 
 
 5.5 
 
 1000 
 
 3.4 
 
 2.8 
 
 2.6 
 
 2.4 
 
 2.2 
 
 2.3 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 3.4 
 
 4.0 
 
 4.0 
 
 i 
 
 320 
 
 330 
 
 340 
 
 3.:0 | 360 
 
 370 
 
 380 
 
 390 
 
 400 
 
 410 1 420 
 
 430 
 
 44D 
 
34 
 
 TABLE XXXI. 
 
 Perturbations produced by Mars. 
 
 Arguments II and IV. 
 
 IV. 
 
 n. 
 
 440 
 
 450 
 
 460 
 
 470 
 
 480 
 
 490 
 
 500 
 
 510 
 
 520 
 
 530 
 
 540 
 
 550 
 
 560 
 
 
 
 4.5 
 
 5.2 
 
 5.9 
 
 6.6 
 
 7.3 
 
 8.0 
 
 8.5 
 
 9.0 
 
 9.5 
 
 10.0 
 
 10.4 
 
 10.7 
 
 10.9 
 
 20 
 
 3.8 
 
 4.3 
 
 4.9 
 
 5.6 
 
 6.2 
 
 6.9 
 
 7.6 
 
 8.2- 
 
 8.8 
 
 9.3 
 
 9.7 
 
 10.0 
 
 11.4 
 
 40 
 
 3.2 
 
 3.7 
 
 4.2 
 
 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 
 
 6.3 
 
 6.9 
 
 7.5 
 
 8.0 
 
 8.6 
 
 9.1 
 
 80 
 
 2.9 
 
 3.1 
 
 3.3 
 
 3.5 
 
 3.9 
 
 4.4 
 
 4.9 
 
 5.4 
 
 5.9 
 
 6.5 
 
 7.1 
 
 7.7 
 
 8.2 
 
 100 
 
 3.0 
 
 3.1 
 
 3.2 
 
 3.5 
 
 3.6 
 
 3.8 
 
 4.2 
 
 4.8 
 
 5.3 
 
 5.9 
 
 6.4 
 
 6.9 
 
 7.4 
 
 120 
 
 3.4 
 
 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 
 
 3.6 
 
 3.6 
 
 3.6 
 
 3.7 
 
 4.0 
 
 4.0 
 
 4.2 
 
 4.6 
 
 5.0 
 
 5.4 
 
 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 
 
 4.2 
 
 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 
 
 6.2 
 
 5.9 
 
 5.8 
 
 5.6 
 
 5.4 
 
 5.3 
 
 5.2 
 
 5:i 
 
 5.1 
 
 5.1 
 
 5.2 
 
 5.2 
 
 260 
 
 6.7 
 
 6.6 
 
 6.'4 
 
 6.1 
 
 6.0 
 
 5.9 
 
 5.8 
 
 5.7 
 
 5.6 
 
 5.5 
 
 5.4 
 
 5.4 
 
 5.4 
 
 280 
 
 6.9 
 
 6.8 
 
 6.7 
 
 6.5 
 
 6.3 
 
 6.2 
 
 6.1 
 
 6.0 
 
 5.9 
 
 5.9 
 
 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 
 
 6.9 
 
 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 
 
 6.7 
 
 6.8 
 
 6.9 
 
 360 
 
 6.9 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.7 
 
 6.7 
 
 6.6 
 
 6.6 
 
 6.8 
 
 6.8 
 
 6.8 
 
 6.9 
 
 380 
 
 6.7 
 
 6.5 
 
 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 
 
 6.8 
 
 420 
 
 6.0 
 
 6.2 
 
 6.3 
 
 6.3 
 
 6.2 
 
 6.2 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.3 
 
 6.5 
 
 6.6 
 
 6.7 
 
 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 
 
 5.8 
 
 5.8 
 
 5.8 
 
 5.8 
 
 5.8 
 
 5.9 
 
 6.0 
 
 6.1 
 
 480 
 
 53 
 
 5.3 
 
 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 
 
 4.9 
 
 4.8 
 
 5.0 
 
 5.1 
 
 5.1 
 
 5.1 
 
 5.1 
 
 5.1 
 
 5.0 
 
 5.0 
 
 5.1 
 
 540 
 
 4.8 
 
 4.8 
 
 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 
 
 4.6 
 
 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 
 
 4.3 
 
 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 
 
 .4 
 
 6.0 
 
 5.7 
 
 5.4 
 
 5.0 
 
 4.8 
 
 4.7 
 
 4.5 
 
 1.3 
 
 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 
 
 4.0 
 
 3.9 
 
 700 
 
 7.8 
 
 7.2 
 
 6.8 
 
 6.4 
 
 6.0 
 
 5.6 
 
 5.3 
 
 5.0 
 
 4.7 
 
 4.6 
 
 4.6 
 
 4.3 
 
 4.1 
 
 720 
 
 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 
 
 6.3 
 
 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 
 
 7.8 
 
 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 
 
 8.7 
 
 9.3 
 
 9.9 
 
 10.3 
 
 10.8 
 
 11.0 
 
 u.i 
 
 11.2 
 
 11.2 
 
 11.0 
 
 10.7 
 
 10.4 
 
 10.1 
 
 940 
 
 7.7 
 
 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 
 
 6.5 
 
 7.3 
 
 8.1 
 
 8.6 
 
 9.3 
 
 9.8 
 
 10.2 
 
 10.6 
 
 10.8 
 
 11.1 
 
 11.2 
 
 10.9 
 
 10.8 
 
 990 
 
 5.5 
 
 6.2 
 
 7.0 
 
 7.7 
 
 8.3 
 
 8.9 
 
 9.5 
 
 10.0 
 
 10.4 
 
 10.6 
 
 10.8 
 
 11.0 
 
 10.9 
 
 1000 
 
 4.5 
 
 5.2 
 
 5.9 
 
 6.6 
 
 7.3 
 
 8.0 
 
 8.5 
 
 9.0 
 
 9.5 
 
 100 
 
 10.4 
 
 10.7 
 
 10.9 
 
 
 
 440 
 
 450 
 
 460 
 
 470 
 
 480 
 
 490 
 
 500 
 
 510 
 
 620 
 
 530 
 
 540 
 
 550 
 
 560 
 
TABLE XXXI. 
 
 Perturbations produced by Mars. 
 
 Arguments II and IV. 
 
 IV. 
 
 35 
 
 II. 
 
 560 
 
 570 
 
 580 
 
 590 
 
 600 
 
 610 
 
 620 
 
 630 
 
 640 
 
 650 
 
 660 
 
 670 
 
 680 
 
 
 
 109 
 
 10.8 
 
 10.6 
 
 10.4 
 
 10.3 
 
 10.0 
 
 9.7 
 
 9.2 
 
 8.9 
 
 8.5 
 
 8.1 
 
 7.9 
 
 7.7 
 
 20 
 
 11.4 
 
 10.6 
 
 10.7 
 
 10.6 
 
 10.4 
 
 10.2 
 
 9.9 
 
 9.7 
 
 9.3 
 
 9.0 
 
 8.8 
 
 8.5 
 
 8.1 
 
 40 
 
 9.8 
 
 10.1 
 
 10.4 
 
 10.4 
 
 10.5 
 
 10.3 
 
 10.2 
 
 9.9 
 
 9.6 
 
 9.4 
 
 9.1 
 
 8.9 
 
 8.5 
 
 60 
 
 9.1 
 
 9.4 
 
 9.8 
 
 10.2 
 
 j 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 
 
 10.0 
 
 9.9 
 
 9.8 
 
 9.7 
 
 9.5 
 
 9.3 
 
 91 
 
 100 
 
 7.4 
 
 7.9 
 
 8.4 
 
 8.7 
 
 9.0 
 
 9.4 
 
 9.6 
 
 9.7 
 
 9.8 
 
 9.7 
 
 9.7 
 
 9.5 
 
 9.2 
 
 120 
 
 6.6 
 
 6.9 
 
 7.6 
 
 8.1 
 
 8.3 
 
 8.6 
 
 9.0 
 
 9.2 
 
 9.4 
 
 9.5 
 
 9.5 
 
 | 9.4 
 
 9.3 
 
 140 
 
 6.9 
 
 6.3 
 
 .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 
 
 52 
 
 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 
 
 5.2 
 
 5.2 
 
 5.3 
 
 5.3 
 
 5:4 
 
 5.5 
 
 5.7 
 
 5.9 
 
 6.1 
 
 6.4 
 
 6.6 
 
 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 
 
 6.5 
 
 6.5 
 
 6.6 
 
 6.5 
 
 6.5 
 
 6.3 
 
 6.1 
 
 6.0 
 
 6.0 
 
 340 
 
 6.9 
 
 6.9 
 
 6.9 
 
 7.0 
 
 7.0 
 
 6.9 
 
 6.8 
 
 6.9 
 
 6.9 
 
 6.8 
 
 6.6 
 
 6.5 
 
 6.3 
 
 360 
 
 6.9 
 
 7.0 
 
 7.2 
 
 7.3 
 
 7.3 
 
 7.3 
 
 7.4 
 
 7.3 
 
 7.3 
 
 7.1 
 
 7.1 
 
 7.0 
 
 6.7 
 
 380 
 
 6.9 
 
 7.0 
 
 7.2 
 
 7.4 
 
 7.5 
 
 7.6 
 
 7.7 
 
 7.7 
 
 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 
 
 7.9 
 
 7.8 
 
 420 
 
 6.7 
 
 6.9 
 
 7.0 
 
 "7.2 
 
 7.6 
 
 7.8 
 
 8.0 
 
 8.2 
 
 8.3 
 
 8.4 
 
 8.4 
 
 8.5 
 
 8.4 
 
 440 
 
 6.4 
 
 6.6 
 
 6.9 
 
 7.0 
 
 7.3 
 
 7.5 
 
 7.9 
 
 8.2 
 
 8.4 
 
 8.6 
 
 8.8 
 
 8.8 
 
 8.9 
 
 460 
 
 6.1 
 
 6.2 
 
 6.5 
 
 6.9 
 
 7.1 
 
 7.2 
 
 7.6 
 
 8.0 
 
 8.4 
 
 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 
 
 3.8 
 
 3.7 
 
 3.5 
 
 3.5 
 
 3.4 
 
 3.3 
 
 3.3 
 
 3.4 
 
 3.5 
 
 3.7 
 
 680 
 
 3.9 
 
 4.0 
 
 3.9 
 
 3.8 
 
 3.6 
 
 3.5 
 
 3.4 
 
 3.3 
 
 3.2 
 
 3.1 
 
 3.1 
 
 3.1 
 
 3.1 
 
 700 
 
 4.1 
 
 3.9 
 
 3.9 
 
 3.9 
 
 3.7 
 
 3.5 
 
 3.4 
 
 3.3 
 
 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 
 
 2.5 
 
 780 
 
 5.5 
 
 5.1 
 
 4.9 
 
 4.5 
 
 4.4 
 
 4.3 
 
 4.1 
 
 3.9 
 
 3.8 
 
 3.4 
 
 3.2 
 
 3.0 
 
 2.7 
 
 800 
 
 5.8 
 
 5.6 
 
 5.2 
 
 5.0 
 
 4.6 
 
 4.5 
 
 4.4 
 
 4.3 
 
 4.1 
 
 3.8 
 
 3.5 
 
 3.1 
 
 2.8 
 
 820 
 
 6.6 
 
 6.1 
 
 5.8 
 
 5.5 
 
 5.3 
 
 5.0 
 
 4.8 
 
 4.6! 
 
 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 
 
 5.0 
 
 4.7 
 
 900 
 
 9.4 
 
 9.0 
 
 8.5 
 
 8.0 
 
 7.6 
 
 7.2 
 
 6.8 
 
 6.6 
 
 6.3 
 
 5.9 
 
 5.6 
 
 5.4 
 
 5.2 
 
 920 
 
 10.1 
 
 9.8 
 
 9.2 
 
 8.7 
 
 8-3 
 
 7.8 
 
 7.4 
 
 7.0 
 
 6.7 
 
 6.4 
 
 6.0 
 
 5.8 
 
 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 
 
 61 
 
 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 
 
 rooo 
 
 10.9 
 
 10.8 
 
 10.6 
 
 10.4 
 
 10.3 
 
 10.0 
 
 9.7 
 
 9.2 
 
 8.9 
 
 8.5 
 
 8.1 
 
 7.9 
 
 7.7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 560 
 
 570 
 
 580 
 
 590 
 
 600 | 
 
 610 1 
 
 620 
 
 630 
 
 640 
 
 650 
 
 660 
 
 670 
 
 680 
 
TABLE XXXI. 
 
 Perturbations produced by Mars. 
 
 Arguments II. and IV. 
 
 IV. 
 
 II. 
 
 680 
 
 690 
 
 700 
 
 710 
 
 720 
 
 730 
 
 740 
 
 750 
 
 760 
 
 770 
 
 780 
 
 790 
 
 800 ! 
 
 
 
 7.7 
 
 7.4 
 
 6.9 
 
 6.8 
 
 6.7 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.5 
 
 5.2 
 
 4.8 
 
 4.4 
 
 3.7 
 
 20 
 
 8.1 
 
 7.8 
 
 7.4 
 
 7.0 
 
 7.1 
 
 6.9 
 
 6.7 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.5 
 
 5.1 
 
 4.7 
 
 40 
 
 8.5 
 
 8.3 
 
 7.8 
 
 7.5 
 
 7.2 
 
 7.1 
 
 7.0 
 
 6.9 
 
 6.6 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.3 
 
 60 
 
 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 
 
 120 
 
 9.3 
 
 9.2 
 
 9.0 
 
 8.7 
 
 8.6 
 
 8.4 
 
 8.2 
 
 8.1 
 
 7.9 
 
 7.8 
 
 7.7 
 
 7.6 
 
 7.5 
 
 ; HO 
 
 9.3 
 
 9.2 
 
 9.0 
 
 9.0 
 
 8.7 
 
 8.5 
 
 8.4 
 
 8.3 
 
 8.0 
 
 7.8 
 
 7.7 
 
 7.7 
 
 7.7 
 
 160 
 
 9.0 
 
 9.0 
 
 8.9 
 
 8.8 
 
 8.7 
 
 8.6 
 
 8.5 
 
 8.4 
 
 8.2 
 
 8.0 
 
 7.9 
 
 7.8 
 
 7-8 
 
 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 
 
 8.2 
 
 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 
 
 6.7 
 
 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 
 
 6.0 
 
 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 
 
 6.0 
 
 6.0 
 
 6.0 
 
 6.0 
 
 6.1 
 
 6.1 
 
 6.2 
 
 6.2 
 
 6.3 
 
 6.3 
 
 6.4 
 
 360 
 
 6.7 
 
 6.6 
 
 6.4 
 
 6.1 
 
 6.0 
 
 5.9 
 
 6.0 
 
 5.9 
 
 5.9 
 
 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 
 
 8.4 
 
 8.2 
 
 8.0 
 
 7.8 
 
 7.5 
 
 7.2 
 
 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 
 
 460 
 
 9.2 
 
 9.2 
 
 9.2 
 
 9.0 
 
 8.8 
 
 8.5 
 
 8.2 
 
 7.9 
 
 7.5 
 
 6.9 
 
 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 
 
 9.9 
 
 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 
 
 7.4 
 
 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 
 
 6.8 
 
 7.6 
 
 8.4 
 
 9.2 
 
 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 
 
 10.2 
 
 11.0 
 
 700 
 
 2.9 
 
 2.8 
 
 3.0 
 
 3.4 
 
 3.9 
 
 4.5 
 
 5.2 
 
 6.0 
 
 6.7 
 
 7.5 
 
 8.5 
 
 9.4 
 
 10.1 
 
 720 
 
 2.7 
 
 2.6 
 
 2.5 
 
 2.7 
 
 3.1 
 
 3.5 
 
 4.0 
 
 4.8 
 
 5.6 
 
 6.4 
 
 7.3 
 
 S.2 
 
 9.1 
 
 740 
 
 2.5 
 
 2.4 
 
 2.4 
 
 2.4 
 
 25 
 
 2.7 
 
 3.1 
 
 3.6 
 
 4.5 
 
 5.2 
 
 6.1 
 
 6.9 
 
 7.8 
 
 760 
 
 2.5 
 
 2.3 
 
 2.2 
 
 2.1 
 
 2.1 
 
 2.3 
 
 2.4 
 
 2.8 
 
 3.2 
 
 4.1 
 
 4.7 
 
 5.7 
 
 6.6 
 
 780 
 
 2.7 
 
 2.5 
 
 23 
 
 2.1 
 
 2.0 
 
 1.9 
 
 2.1 
 
 2.2 
 
 2.5 
 
 2.9 
 
 3.6 
 
 4.4 
 
 5.2 
 
 800 
 
 2.8 
 
 2.7 
 
 2.4 
 
 2.2 
 
 2.0 
 
 1.8 
 
 1.8 
 
 1.8 
 
 2.0 
 
 2.3 
 
 2.5 
 
 3.2 
 
 4.0 
 
 820 
 
 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 
 
 3.0 
 
 2.6 
 
 2.3 
 
 2.1 
 
 1.9 
 
 1.6 
 
 1.5 
 
 1.5 
 
 1.6 
 
 1.7 
 
 2.2 
 
 860 
 
 4.4 
 
 4.0 
 
 3.5 
 
 3.2 
 
 2.8 
 
 2.3 
 
 1.9 
 
 1.7 
 
 1.4 
 
 1.3 
 
 1.2 
 
 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 
 
 1.2 
 
 1.2 
 
 900 
 
 5.2 
 
 5.0 
 
 4.6 
 
 4.3 
 
 4.0 
 
 3.6 
 
 3.2 
 
 2.7 
 
 2.2 
 
 1.6 
 
 1.3 
 
 1.2 
 
 1.1 
 
 920 
 
 I 5.7 
 
 5.3 
 
 5.1 
 
 5.0 
 
 4.6 
 
 4.2 
 
 3.8 
 
 *3.4 
 
 2.9 
 
 2.3 
 
 1.9 
 
 1.3 
 
 1.1 
 
 940 
 
 6.1 
 
 5.9 
 
 5.6 
 
 5.4 
 
 5.2 
 
 4.8 
 
 4.5 
 
 3.9 
 
 3.5 
 
 3.1 
 
 2.6 
 
 2.1 
 
 1.5 
 
 960 
 
 6.6 
 
 6.4 
 
 6.2 
 
 5.9 
 
 5.6 
 
 5.4 
 
 5.1 
 
 4.7 
 
 4.3 
 
 3.7 
 
 3.2 
 
 2.8 
 
 2.3 
 
 980 
 
 | 7.2 
 
 6.9 
 
 6.6 
 
 6.4 
 
 6.2 
 
 5.9 
 
 5.6 
 
 5.3 
 
 5.0 
 
 4.6 
 
 4.0 
 
 3.5 
 
 3.0 
 
 1000 
 
 7.7 
 
 7.4 
 
 6.9 
 
 e.a 
 
 6.7 
 
 6.4 
 
 6.1 
 
 5.8 
 
 5.5 
 
 5.2 
 
 4.8 
 
 4.4 
 
 3.7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 680 
 
 690 
 
 700 
 
 710 
 
 720 
 
 730 
 
 740 
 
 750 
 
 760 
 
 770 
 
 780 
 
 790 
 
 800 
 
TABLE XXXI. 
 
 Perturbations produced by Mars. 
 
 Arguments II. and IV. 
 
 IV. 
 
 II. 
 
 $00 
 
 810 
 
 820 
 
 830 840 850 i 860 
 
 870 
 
 880 890 
 
 900 910 
 
 920 
 
 
 
 
 
 
 
 // I .- 
 
 
 
 
 
 
 
 
 3.7 
 
 3.2 
 
 2.6 
 
 2.1 
 
 1.7 
 
 'l.3 
 
 0.9 
 
 0.7 
 
 0.7 
 
 1.0 
 
 1.2 
 
 1.6 
 
 22 
 
 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 
 
 1 5 
 
 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 i 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 
 
 2.4 
 
 140 
 
 7.7 
 
 7.7 
 
 7.5 
 
 7.3 
 
 7.0 
 
 6.7 
 
 6.4 
 
 6.0 
 
 5.6 
 
 5.1 
 
 4.5l 3.8 
 
 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 
 
 4.2 
 
 180 
 
 7.9 
 
 7.8 
 
 7.9 
 
 7.9 
 
 7.7 
 
 7.6 
 
 7.5 
 
 7.1 
 
 7.0 
 
 6.6 
 
 6.1 
 
 5.7 
 
 5.2 
 
 200 
 
 7.9 
 
 7.9 
 
 7.8 
 
 7.9 
 
 7.8 
 
 7.7 
 
 7.6 
 
 7.5 
 
 7.5 
 
 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 
 
 7.4 
 
 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 
 
 7.2 
 
 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 
 
 59 
 
 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 
 
 5.6 
 
 5.6 
 
 5.6 
 
 5.7 
 
 5.7 
 
 5.7 
 
 5.8 
 
 5.9 
 
 5.9 
 
 6.0 
 
 6.1 
 
 6.2 
 
 6.4 
 
 420 
 
 54 
 
 5.4 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.5 
 
 5.6 
 
 5.6 
 
 5.6 
 
 5.7 
 
 5.8 
 
 440 
 
 56 
 
 5.3 
 
 5.3 
 
 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 
 
 5.2 
 
 5.1 
 
 5.0 
 
 5.1 
 
 5.2 
 
 5.2 
 
 5.2 
 
 5.3 
 
 480 
 
 6.5 
 
 6.0 
 
 5.7 
 
 5.4 
 
 5.2 
 
 5.2 
 
 5.1 
 
 4.9 
 
 4.9 
 
 4.9 
 
 4.9 
 
 5.0 
 
 5.0 
 
 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 
 
 4.9 
 
 520 
 
 8.0 
 
 7.4 
 
 7.0 
 
 6.5 
 
 6.1 
 
 5.5 
 
 5.4 
 
 5.1 
 
 4.9 
 
 4.7 
 
 4.7 
 
 4.7 
 
 4.8 
 
 540 
 
 9.0 
 
 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 
 
 9.8 
 
 9.3 
 
 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 
 
 11.3 
 
 11.0 
 
 10.4 
 
 9.8 
 
 9.3 
 
 8.5 
 
 7.8 
 
 7.1 
 
 6.6 
 
 660 
 
 n.sUi.e 
 
 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 
 
 720 
 
 9.1 
 
 10.0 
 
 10.6 
 
 11.4 
 
 11.9 
 
 12.4 
 
 12.6 
 
 12.5 
 
 12.4 
 
 12.0 
 
 11.6 11.2 
 
 0.8 
 
 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 
 
 1.5 
 
 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 
 
 2.1 
 
 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 
 
 10.8 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 
 
 9.2 
 
 10.1 
 
 900 
 
 1.1 
 
 1.1 
 
 1.2 
 
 1.3 
 
 1.7 
 
 2.2 
 
 2.9 
 
 3.8 
 
 4.8 
 
 5.7 
 
 6.8 
 
 7.9 
 
 8.8 
 
 920 
 
 1.1 
 
 1.0 
 
 1.0 
 
 1 i 
 
 111 
 
 1.4 
 
 1.9 
 
 2.6 
 
 34 
 
 4.4 
 
 5.3 
 
 6.3 
 
 7.4 
 
 940 
 
 1.5 
 
 1.1 
 
 0.8 
 
 0.9 
 
 1.0 
 
 1.1 
 
 1.3 
 
 1.6 
 
 2.3 
 
 3.1 
 
 3.9 
 
 5.0 
 
 5.9 
 
 960 
 
 23 
 
 1.7 
 
 1.3 
 
 0.9 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.2 
 
 1.4 
 
 2.0 
 
 2.8 
 
 3.5 
 
 4.6 
 
 980 
 
 30 
 
 2.5 
 
 1.9 
 
 1.4 
 
 1.2 
 
 1.0 
 
 0.8 
 
 0.9 
 
 1.2 
 
 1.4 
 
 1.7 
 
 2.4 
 
 3.3 
 
 1000 
 
 37 
 
 3.2 
 
 2.6 
 
 2.1 
 
 1.7 
 
 1.3 
 
 0.9 
 
 0.7 
 
 0.7 
 
 1.0 
 
 1.2 
 
 1.6 
 
 2.2 
 
 
 800 
 
 810 
 
 820 
 
 830 
 
 840 
 
 850 
 
 860 
 
 870 
 
 880 
 
 890 
 
 900 
 
 910 
 
 920 
 
33 
 
 TABLE XXXI. 
 
 Perturbations by Mars. 
 Arguments II. and IV. 
 IV. 
 
 TABLE XXXII. 
 
 Peris, by Jupiter 
 Arg's. II. and V. 
 V. 
 
 II. 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
 970 
 
 980 
 
 990 
 
 1000 
 
 
 
 10 
 
 20 i 
 
 30 
 
 
 
 2.2 
 
 3.0 
 
 3.8 
 
 4.8 
 
 5.8 
 
 6.9 
 
 7.8 
 
 8.4 
 
 9.5 
 
 15.3 
 
 15.1 
 
 15.0 
 
 15.0 
 
 20 
 
 1.5 
 
 2.1 
 
 2.6 
 
 3.4 
 
 4.4 
 
 5.5 
 
 6.5 
 
 7.6 
 
 8.7 
 
 14.9 
 
 14.9 
 
 14.7 
 
 14.8 
 
 40 
 
 1.0 
 
 1.4 
 
 1.8 
 
 2.5 
 
 3.2 
 
 4.0 
 
 5.2 
 
 6.0 
 
 7.1 
 
 14.7 
 
 14.6 
 
 14.6 
 
 14.5 
 
 60 
 
 1.0 
 
 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 
 
 1.2 
 
 1.2 
 
 1.3 
 
 1.6 
 
 2.0 
 
 2.6 
 
 3.3 
 
 13.2 
 
 13.4 
 
 13.6 
 
 13.7 
 
 I 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 
 
 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 
 
 G.I 
 
 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 
 
 53 
 
 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 
 
 7.3 
 
 7.1 
 
 6.7 
 
 6.3 
 
 5.9 
 
 5.5! 
 
 3.9 
 
 4.1 
 
 4.7 
 
 5.2 
 
 300 
 
 7.7 
 
 7.7 
 
 7.7 
 
 7.7 
 
 7.4 
 
 7.2 
 
 7.0 
 
 6.6 
 
 6.1 
 
 3.4 
 
 3.5 
 
 3.9 
 
 4.3 
 
 320 
 
 7.4 
 
 7.4 
 
 7.6 
 
 7.7 
 
 7.6 
 
 7.6 
 
 7.3 
 
 7.1 
 
 6.9 
 
 3.2 
 
 3.1 
 
 3.4 
 
 3.6 
 
 340 
 
 7.2 
 
 7.2 
 
 7.3 
 
 7.5 
 
 7.7 
 
 7.6 
 
 7.6 
 
 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 
 
 6.8 
 
 6.9 
 
 7.0 
 
 7.0 
 
 7.0 
 
 7.1 
 
 7.3 
 
 7.5 
 
 7.5 
 
 4.5 
 
 4.0 
 
 3.4 
 
 3.1 
 
 400 
 
 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 
 
 4.6 
 
 4.1 
 
 440 
 
 5.5 
 
 5.6 
 
 5.7 
 
 5.8 
 
 6.0 
 
 6.1 
 
 6.3 
 
 6.5 
 
 6.5 
 
 7.5 
 
 6.6 
 
 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 
 
 4.9 
 
 4.9 
 
 5.0 
 
 5.0 
 
 5.0 
 
 5.1 
 
 5.2 
 
 5.3 
 
 5.3 
 
 12.3 
 
 11.3 
 
 10.0 
 
 9.1 
 
 520 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.8 
 
 4.7 
 
 4.9 
 
 5.0 
 
 5.l! 
 
 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 
 
 145 
 
 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 
 
 16.1 
 
 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 
 
 600 
 
 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 
 
 6.6 
 
 6.1 
 
 5.6 
 
 5.4 
 
 5.0 
 
 4.7 
 
 4.6 
 
 4.5 
 
 4.3 
 
 21.6 
 
 20.9 
 
 20.2 
 
 19.5 
 
 660 
 
 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 
 
 10.8 
 
 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 
 
 12.5 
 
 12.1 
 
 11.7 
 
 11.2 
 
 10.5 
 
 9.8 
 
 9.2 
 
 19.6 
 
 19.8 
 
 19.9 
 
 20.1 
 
 820 
 
 12.5 
 
 12.7 
 
 12.8 
 
 12.7 
 
 12.2 
 
 11.9 
 
 11.2 
 
 10.7 
 
 10.1 
 
 18.8 
 
 19.0 
 
 19.2 
 
 19.4 
 
 840 
 
 12.0 
 
 12.4 
 
 12.6 
 
 12.8 
 
 12.6 
 
 124 
 
 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| 
 
 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 i 
 
 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 
 
 1000 
 
 f 2.2 
 
 3.0 
 
 3.8 
 
 4.8 
 
 5.8 
 
 6.9 
 
 7.8 
 
 8.7 
 
 9.5 
 
 15.3 
 
 15.1 
 
 15.0 
 
 15.0 
 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
 970 
 
 980 
 
 990 
 
 1000 
 
 
 
 10 
 
 20 
 
 30 
 
TABLE XXXII. 
 
 39 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 ! n. 
 
 30 1 40 
 
 50 
 
 60 | 70 
 
 80 90 
 
 100 110 
 
 1 120 
 
 130 
 
 140 
 
 150 
 
 
 
 15.0 
 
 14.8 
 
 14.7 
 
 14.7 
 
 14.6 
 
 14.5 i 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 I 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 
 
 13.8 
 
 13.7 
 
 60 
 
 14.4 
 
 14.3 14.3 
 
 14.2 
 
 14.1 
 
 13.9 13.8 
 
 13.6 
 
 Iflfl 
 
 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 i 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 
 
 12.7 
 
 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 
 
 9.9 
 
 10.4 
 
 10.8 
 
 11.3 11.5 
 
 11.8 
 
 11.9 
 
 12.0 
 
 12.0 
 
 12.0 
 
 240 
 
 7.2 
 
 7.7 
 
 8.2 
 
 8.9 
 
 9.4 
 
 9.8 
 
 10.3 
 
 10.6 
 
 11.0 
 
 11.3 
 
 11.5 
 
 11.7 
 
 11.8 
 
 260 
 
 6.1 
 
 6.5 
 
 7.1 
 
 7.6 
 
 8.3 
 
 8.8 
 
 9.3 
 
 9.7 
 
 10.1 
 
 10.5 
 
 10.9 
 
 11.0 
 
 11.2 
 
 280 
 
 5.2 
 
 5.5 
 
 6.0 
 
 6.5 
 
 7.1 
 
 7.6 
 
 8.2 
 
 8.7 
 
 9.2 
 
 9.6 
 
 10.0 
 
 10.4 
 
 10.6 
 
 300 
 
 4.3 
 
 4.7 
 
 5.1 
 
 5.5 
 
 6.1 
 
 6.6 
 
 7.1 
 
 7.6 
 
 8.1 
 
 8.7 
 
 9.1 
 
 9.4 
 
 9.9 
 
 320 
 
 3.6 
 
 3.9 
 
 4.3 
 
 4.6 
 
 5.1 
 
 5.4 
 
 6.0 
 
 6.6 
 
 7.2 
 
 7.7 
 
 8.1 
 
 8.5 
 
 8.9 
 
 340 
 
 3.1 
 
 3.3 
 
 3.5 
 
 3.8 
 
 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 
 
 2.7 
 
 2.8 
 
 2.9 
 
 3.0 
 
 3.2 
 
 3.5 
 
 4.1 
 
 4.6 
 
 5.0 
 
 5.6 
 
 400 
 
 35 
 
 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 
 
 4.3 
 
 440 
 
 4.9 
 
 4.4 3.9 
 
 3.4 
 
 3.1 
 
 2.7 
 
 2.8 
 
 2.7 
 
 2.8 
 
 3.1 
 
 3.1 
 
 3.2 
 
 3.5 
 
 460 
 
 6.3 
 
 5.4 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 i 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 
 
 7.7 
 
 6.9 
 
 6.1 
 
 5.5 
 
 4.8 
 
 4.2 
 
 3.8 
 
 3.5 
 
 3.2 
 
 3.1 
 
 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 1 4.1 
 
 580 
 
 15.7 
 
 14.5 
 
 13.6 
 
 12.5 
 
 11.4 
 
 10.4 
 
 9.3 
 
 8.3 
 
 7.7 
 
 6.9 
 
 6.2 
 
 5.5 
 
 5.0 
 
 600 
 
 17.0 
 
 16.0 
 
 15.0 
 
 14.0 
 
 13.1 
 
 12.0 
 
 11.0 
 
 10.1 
 
 9.2 
 
 8.3 
 
 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 
 
 SS 
 
 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 
 
 IT.J 
 
 10.8 
 
 9.4 
 
 8.7 
 
 660 
 
 20.4 
 
 19.7 
 
 18.9 
 
 18.1 
 
 17.4 
 
 16.3 
 
 15.6 
 
 14.6 
 
 13.7 
 
 12.8 
 
 11.9 
 
 li.O 
 
 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 
 
 20.6 
 
 200 
 
 19.3 
 
 18.7 
 
 18.0 
 
 17.1 
 
 16.5 
 
 15.6 
 
 14.7 
 
 13.8 
 
 13.0' 
 
 720 
 
 2L6 
 
 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 
 
 7SO 
 
 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 
 
 8uO 
 
 20.1 
 
 20.2 
 
 20.3 
 
 20.3 
 
 20.4 
 
 20.3 
 
 20.1 
 
 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 
 
 1S.9 
 
 19.0 
 
 19.2 
 
 19.3 
 
 194 
 
 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 
 
 163 
 
 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 
 
 14.5 
 
 14.6 
 
 14.7 
 
 14.8 
 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 80 
 
 90 
 
 100 
 
 110 
 
 120 
 
 130 
 
 140 
 
 150 
 
TABLE XXXII. 
 
 Perturbations produced by Jupiter. 
 Arguments II. and V. 
 V. 
 
 II. 
 
 150 
 
 160 
 
 170 
 
 180 190 
 
 200 
 
 2101 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 
 
 H.8 14.9 
 
 15.2 15.5 
 
 15.7 
 
 15.9 
 
 16.2 
 
 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 
 
 15.5 
 
 15.8 
 
 16.2 
 
 16.4 
 
 60 
 
 13.3 
 
 13.2 
 
 13.4 
 
 13.5 113.6 
 
 13.8 14.1 
 
 14.3 
 
 14.6 
 
 14.8 
 
 15.1 
 
 15.5 
 
 15.8 
 
 80 
 
 13.1 
 
 13.0 
 
 13.0 
 
 13.0 
 
 13.1 
 
 13.1 
 
 13.3 
 
 13.5 
 
 13.8 
 
 14.1 
 
 14.4 
 
 14.5 
 
 15.1 
 
 100 
 
 12.7 
 
 12.7 
 
 12.7 
 
 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 
 
 12.5 
 
 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 
 
 12.0 
 
 12.0 
 
 12.1 
 
 12.1 
 
 12.3 
 
 12.5 
 
 12.8 
 
 160 
 
 12.5 
 
 12.3 
 
 12.2 
 
 12.1 
 
 12.1 
 
 11.9 
 
 11.8 
 
 11.8 
 
 11.8 
 
 11.8 
 
 11.9 
 
 12.0 
 
 12.2 
 
 180 
 
 12.5 
 
 12.3 
 
 12.2 
 
 12.1 
 
 11.9 
 
 11.8 
 
 11.7 
 
 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 
 
 11.2 
 
 11.3 
 
 11.5 
 
 220 
 
 12.0 
 
 12.0 
 
 12.1 
 
 12.0 
 
 11.8 
 
 11.6 
 
 11.6 
 
 11.5 
 
 11.4 
 
 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 
 
 10.9 
 
 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 
 
 10.9 
 
 300 
 
 9.9 
 
 10.1 
 
 10.5 
 
 10.8 
 
 10.9 
 
 11.0 
 
 11.1 
 
 11.0 
 
 11.0 
 
 11.0 
 
 11.0 
 
 11.1 
 
 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 
 
 7.3 
 
 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 
 
 9.4 
 
 9.7 
 
 9.9 
 
 420 
 
 4.3 
 
 4.8 
 
 5.3 
 
 5.8 
 
 6.2 
 
 6.6 
 
 7.1 
 
 7.4 
 
 7.9 
 
 8.4 
 
 8.7 
 
 9.1 
 
 9.4 
 
 440 
 
 3.5 
 
 3.9 
 
 4.4 
 
 4.9 
 
 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 
 
 2.9 
 
 3.2 
 
 3.6 
 
 3.9 
 
 4.3 
 
 4.7 
 
 5.0 
 
 5.4 
 
 5.9 
 
 6.3 
 
 6.8 
 
 7.3 
 
 500 
 
 2.7 
 
 2.7 
 
 2.9 
 
 3.1 
 
 3.4 
 
 3.6 
 
 4.0 
 
 4.4 
 
 4.8 
 
 5.2 
 
 5.7 
 
 5.9 
 
 6.4 
 
 520 
 
 3.1 
 
 2.8 
 
 2.9 
 
 3.0 
 
 3.1 
 
 3.2 
 
 3.5 
 
 3.8 
 
 4.2 
 
 4,7 
 
 4.9 
 
 5.4 
 
 5.7 
 
 540 
 
 3.5 
 
 3.2 
 
 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 
 
 3.3 
 
 3.2 
 
 3.2 j 3.2 
 
 3.3 
 
 3.5 
 
 3.7 
 
 4.0 
 
 4.3 
 
 4,5 
 
 580 
 
 5.0 
 
 4.6 
 
 4.2 
 
 4.0 
 
 3.6 
 
 3.5 ' 3.3 
 
 3.2 
 
 3.4 
 
 35 
 
 3.7 
 
 4.0 
 
 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 
 
 3.9 
 
 3.9 
 
 4.0 
 
 640 
 
 8.7 
 
 7.8 
 
 7.3 
 
 6.6 
 
 6.1 
 
 5.5 
 
 5.2 
 
 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 
 
 6.2 
 
 5.9 
 
 5.3 
 
 4.9 
 
 4.6 
 
 4.5 
 
 4.4 
 
 680 
 
 11.6 
 
 10.8 
 
 10.0 
 
 9.3 
 
 8.5 
 
 7.5 
 
 7.3 
 
 6.7 
 
 6.3 
 
 5.8 
 
 5.5 
 
 5.2 
 
 4.9 
 
 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 
 
 8.7 
 
 8.0 
 
 7.6 
 
 7.0 
 
 6.6 
 
 740 
 
 15.7 
 
 14.9 
 
 14.2 
 
 13.4 
 
 12.7 
 
 12.0 
 
 11.2 
 
 10.5 
 
 9.7 
 
 9.3 
 
 8.9 
 
 8.2 
 
 7.7 
 
 760 
 
 16.7 
 
 15.9 
 
 15.5 
 
 14.7 
 
 13.9 
 
 13.3 
 
 12.6 
 
 11.8 
 
 11.2 
 
 10.5 
 
 10.0 
 
 9.5 
 
 9.0 
 
 780 
 
 17.6 
 
 17.0 
 
 16.4 
 
 15.7 
 
 15.1 
 
 14.6 
 
 13.8 
 
 13.2 | 12.6 
 
 11.9 
 
 11.2 
 
 10.3 
 
 10.S 
 
 800 
 
 18.2 
 
 17.8 
 
 17.3 
 
 16.8 
 
 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 
 
 18.7 
 
 18.6 
 
 18.4 
 
 18.3 
 
 17.9 
 
 17.4 
 
 17.1 
 
 16.7 
 
 16.3 
 
 15.9 
 
 15.4 
 
 15.0 
 
 8SO 
 
 18.7 
 
 18.5 
 
 18.6 
 
 18.5 
 
 18.3 
 
 18.2 
 
 18.0 
 
 17.7 
 
 17.4 
 
 17.1 
 
 16.6 
 
 16.3 
 
 15.9 
 
 900 
 
 18.2 
 
 18.2 
 
 18.3 
 
 18.3 
 
 18.3 
 
 18.1 
 
 18.1 
 
 18.0 
 
 17.8 
 
 17.6 
 
 17.3 
 
 17.0 
 
 16.7 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 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 
 
 1 j.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 
 
 17.3 
 
 17.6 
 
 17.7 
 
 17.9 
 
 1000 
 
 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 
 
 
 150 
 
 160 
 
 170 
 
 180 
 
 190 
 
 200 
 
 210 
 
 220 
 
 230 240 
 
 250 
 
 260 
 
 270 
 
TABLE XXXII. 
 
 41 
 
 Perturbations produced by Jupiter. 
 Arguments II. and V 
 V 
 
 II. 
 
 270 280 290 
 
 300 
 
 310 
 
 320 
 
 330 
 
 340 350 
 
 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 
 
 J6.1 
 
 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 
 
 1.1.5 
 
 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 
 
 10.6 
 
 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 
 
 10.5 
 
 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 
 
 10.5 
 
 10.5 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.4 
 
 10.3 
 
 10.5 
 
 10.6 
 
 10.8 
 
 380 
 
 10.2 
 
 10.3 1 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 
 
 10.0 
 
 10.2 
 
 10.3 
 
 10.2 
 
 10.2 
 
 10.3 
 
 10.4 
 
 10.3 
 
 10.3 
 
 10.3 
 
 10.5 
 
 420 
 
 9.4 
 
 9.6 
 
 9.8 
 
 9.9 
 
 10.1 
 
 10.2 
 
 10.1 
 
 10:2 
 
 10.2 
 
 10.2 
 
 10.3 
 
 103 
 
 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 
 
 10.0 
 
 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.G 
 
 8.9 
 
 9.2 
 
 V 
 
 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 
 
 540 
 
 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 
 
 580 
 
 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 
 
 5.3 
 
 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 
 
 5.9 
 
 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 
 
 6.5 
 
 800 
 
 11.5 
 
 11.0 
 
 10.4 
 
 9.8 
 
 9.4 
 
 8.7 
 
 8.5 
 
 8.3 
 
 8.0 
 
 7.7 
 
 7.6 
 
 7.3 
 
 7.1 
 
 820 
 
 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 
 
 840 
 
 13.8 
 
 13.2 
 
 12.8 
 
 12.3 
 
 11.9 
 
 11.3 
 
 10.9 
 
 10.5 
 
 10.2 
 
 9.6 
 
 9.4 
 
 9.1 
 
 8.9 
 
 860 
 
 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 
 
 15.4 
 
 15.0 
 
 14.4 
 
 14.2 
 
 13.7 
 
 13.4 
 
 129 
 
 12.5 
 
 12.0 
 
 11.5 
 
 113 
 
 11.1 
 
 900 
 
 16.7 
 
 16.4 
 
 15.9 
 
 15.5 
 
 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 
 
 14.3 
 
 14.0 
 
 13.6 
 
 13.3 
 
 940 
 
 17.7 
 
 17.5 
 
 17.3 
 
 17.1 
 
 16.9 
 
 16.6 
 
 16.3 
 
 16.1 
 
 16.0 
 
 15.5 
 
 15.0 
 
 14.7 
 
 14.5 
 
 960 
 
 17.9 
 
 17.8 
 
 17.6 
 
 17.5 
 
 17.4 
 
 17.2 
 
 17.0 
 
 16.9 
 
 1.6.8 
 
 16.4 
 
 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 
 
 .6.6 
 
 1000 
 
 17.5 
 
 17.7 
 
 17.7 
 
 17.8 
 
 17.9 
 
 17.9 18.0 
 
 18.0 
 
 17.9 
 
 17.7 
 
 17.6 
 
 17.5 
 
 17.5 
 
 
 27C | 280 
 
 290 ,300 
 
 310 
 
 320 330 
 
 340 
 
 350 
 
 360 
 
 370 
 
 380 
 
 390 
 
42 
 
 TABLE XXXII. 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 II. 
 
 390 
 
 400 
 
 410 
 
 420 
 
 430 
 
 440 
 
 450 
 
 460 
 
 470 
 
 480 
 
 490 
 
 500 
 
 510 
 
 
 
 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 
 
 19.3 
 
 19.4 
 
 19.3 
 
 19.3 
 
 19.3 
 
 19.3 
 
 19.2 
 
 19.2 
 
 18.9 
 
 18.8 
 
 100 
 
 18.7 
 
 18.9 
 
 19.1 
 
 19.4 
 
 19.7 
 
 19.8 
 
 19.8 
 
 19.8 
 
 19.8 
 
 19.8 
 
 19.9 
 
 19.7 
 
 19.7 
 
 120 
 
 18.3 
 
 18.6 
 
 18.9 
 
 19.2 
 
 19.5 
 
 19.8 
 
 20.0 
 
 20.1 
 
 20.3 
 
 20.3 
 
 204 
 
 20.4 
 
 20.4 
 
 140 
 
 17.7 
 
 18.2 
 
 18.6 
 
 18.9 
 
 19.2 
 
 19.6 
 
 20.0 
 
 20.3 
 
 20.5 
 
 20.6 
 
 20.7 
 
 20.8 
 
 21.0 
 
 160 
 
 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 
 
 180 
 
 16.3 
 
 16.8 
 
 17.3 
 
 17.9 
 
 18.3 
 
 18.8 
 
 19.3 
 
 19.8 
 
 20.3 
 
 20.6 
 
 20.9 
 
 21.1 
 
 21.4 
 
 200 
 
 15.5 
 
 16.0 
 
 16.5 
 
 17.1 
 
 17.7 
 
 18.2 
 
 18.6 
 
 19.1 
 
 19.8 
 
 20.2 
 
 20.7 
 
 21.0 
 
 21.4 
 
 220 
 
 14.5 
 
 15.0 
 
 15.6 
 
 16.1 
 
 16.9 
 
 17.4 
 
 18.0 
 
 18.6 
 
 19.0 
 
 19.7 
 
 20.3 
 
 20.7 
 
 21.1 
 
 240 
 
 13.8 
 
 14.2 
 
 14.7 
 
 15.2 
 
 15.9 
 
 16.5 
 
 17.1 
 
 17.7 
 
 18.4 
 
 18.9 
 
 19.5 
 
 20.1 
 
 20.7 
 
 260 
 
 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 
 
 15.9 
 
 16.7 
 
 17.2 
 
 17.8 
 
 18.4 
 
 19.1 
 
 300 
 
 11.8 
 
 11.9 
 
 12.4 
 
 12.8 
 
 13.3 
 
 13.8 
 
 14.4 
 
 14.9 
 
 15.7 
 
 16.3 
 
 17.0 
 
 17.6 
 
 18.2 
 
 320 
 
 11.2 
 
 11.5 
 
 11.8 
 
 12.2 
 
 12.7 
 
 13.0 
 
 13.6 
 
 14.1 
 
 14.7 
 
 15.3 
 
 16.0 
 
 16.6 
 
 17.4 
 
 340 
 
 10.8 
 
 11.2 
 
 11.4 
 
 11.6 
 
 12.1 
 
 12.4 
 
 12.9 
 
 13.4 
 
 13.9 
 
 14.4 
 
 15.1 
 
 15.7 
 
 16.4 
 
 360 
 
 10.8 
 
 10.8 
 
 11.0 
 
 11.2 
 
 11.6 
 
 11.9 
 
 12.3 
 
 12.6 
 
 13.2 
 
 13.6 
 
 14.2 
 
 14.8 
 
 15.5 
 
 380 
 
 10.6 
 
 10.6 
 
 10.7 
 
 10.9 
 
 11.2 
 
 11.4 
 
 11.9 
 
 12.2 
 
 12.6 
 
 12.9 
 
 13.5 
 
 13.9 
 
 14.5 
 
 400 
 
 10.5 
 
 10.5 
 
 10.6 
 
 10.6 
 
 10.9 
 
 11.1 
 
 11.4 
 
 11.8 
 
 12.2 
 
 12.5 
 
 12.9 
 
 13.3 
 
 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 
 
 IP- 1 
 
 10.2 
 
 10.4 
 
 10.5 
 
 10.7 
 
 10.8 
 
 10.9 
 
 11.0 
 
 11.2 
 
 11.3 
 
 11.7 
 
 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 
 
 560 
 
 8.7 
 
 8.9 
 
 9.1 
 
 9.3 
 
 9.7 
 
 9.8 
 
 10.1 
 
 10.3 
 
 10.5 
 
 10.6 
 
 10.7 
 
 10.8 
 
 11.2 
 
 580 
 
 8.1 
 
 8.5 
 
 8.7 
 
 8.7 
 
 9.2 
 
 9.4 
 
 9.7 
 
 9.9 
 
 10.2 
 
 10.4 
 
 10.6 
 
 10.7 
 
 10.9 
 
 600 
 
 7.6 
 
 7.9 
 
 8.2 
 
 8.5 
 
 8.8 
 
 9.0 
 
 9.3 
 
 9.5 
 
 9.8 
 
 10.0 
 
 10.3 
 
 10.5 
 
 10.7 
 
 620 
 
 7.0 
 
 7.3 
 
 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 
 
 6.2 
 
 6.4 
 
 6.6 
 
 6.9 
 
 7.3 
 
 7.6 
 
 7.9 
 
 8.1 
 
 83 
 
 8.6 
 
 8.9 
 
 9.2 
 
 9.5 
 
 680 
 
 5.8 
 
 6.1 
 
 6.2 
 
 6.5 
 
 6.8 
 
 7.0 
 
 7.4 
 
 7.6 
 
 7.9 
 
 8.1 
 
 8.4 
 
 8.7 
 
 9.0 
 
 700 
 
 5.6 
 
 5.8 
 
 6.0 
 
 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 
 
 6.2 
 
 6.3 
 
 6.5 
 
 6.8 
 
 7.1 
 
 7.2 
 
 7.5 
 
 7.7 
 
 8.0 
 
 740 
 
 5.7 
 
 5.7 
 
 5.7 
 
 5.8 
 
 6.0 
 
 6.1 
 
 6.2 
 
 6.4 
 
 6.7 
 
 6.9 
 
 7.1 
 
 7.2 
 
 7.5 
 
 760 
 
 6.0 
 
 6.0 
 
 6.0 
 
 6.0 
 
 6.0 
 
 6.1 
 
 6.2 
 
 6.3 
 
 6.4 
 
 6.5 
 
 6.7 
 
 6.8 
 
 7.1 
 
 780 
 
 6.5 
 
 6.3 
 
 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 
 
 .7 
 
 820 
 
 7.9 
 
 7.6 
 
 7.5 
 
 7.3 
 
 7.2 
 
 7.0 
 
 7.0 
 
 6.8 
 
 6.8 
 
 6.7 
 
 6.6 
 
 6.6 
 
 6.7 
 
 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 
 
 9.0 
 
 8.7 
 
 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 
 
 13.3 
 
 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 
 
 14.5 
 
 14.1 
 
 13.8 
 
 13.5 
 
 13.2 
 
 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 
 
 16.6 
 
 16.3 
 
 16.0 
 
 15.7 
 
 15.6 
 
 15.2 
 
 14.9 
 
 14.6 
 
 14.2 
 
 13.8 
 
 13.6 
 
 12.9 
 
 12.7 
 
 1000 
 
 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 
 
 
 390 
 
 400 
 
 410 
 
 420 
 
 430 
 
 440 
 
 450 
 
 460 
 
 470 
 
 480 
 
 490 
 
 500 
 
 510 
 
TABLE XXXII. 
 
 43 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 II. 
 
 510 
 
 | 520 
 
 530 
 
 540 
 
 550 
 
 | 560 
 
 570 
 
 580 
 
 590 
 
 600 
 
 610 
 
 620 
 
 630 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 13.9 
 
 13.4 
 
 13.1 
 
 12.7 
 
 12.1 
 
 11.8 
 
 11.3 
 
 10.8 
 
 10.2 
 
 9.9 
 
 9.4 
 
 I 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 
 
 |io.o 
 
 9.4 
 
 40 
 
 16.5 
 
 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 
 
 121.2 
 
 21.0 
 
 20.6 
 
 20.4 
 
 20.1 
 
 19.6 
 
 19.1 
 
 180 
 
 21.4 
 
 21.6 
 
 21.8 
 
 22.0 
 
 22.0 
 
 22.1 
 
 21.9 
 
 21.8 
 
 21.6 
 
 21.4 
 
 21.1 
 
 20.7 
 
 20.3 
 
 200 
 
 21.4 
 
 21.7 
 
 21.9 
 
 22.1 
 
 22.3 
 
 22.5 
 
 22.5 
 
 22.5 
 
 22.4 
 
 22.3 
 
 22.1 
 
 21.8 
 
 21.5 
 
 220 
 
 21.1 
 
 21.5 
 
 21.8 
 
 22.2 
 
 22.5 
 
 22.8 
 
 23.1 
 
 23.1 
 
 22.9 
 
 22.8 
 
 22.9 
 
 22.6 
 
 22.5J 
 
 240 
 
 20.7 
 
 21.1 
 
 21.5 
 
 21. '9 
 
 22.3 
 
 22.7 
 
 23.0 
 
 23.3 
 
 23.4 
 
 23.5 
 
 23.4 
 
 23.3 
 
 23.2J 
 
 260 
 
 20.0 
 
 20.6 
 
 21.0 
 
 21.6 
 
 22.0 
 
 22.4 
 
 22.8 
 
 23.2 
 
 23.5 
 
 23.8 
 
 23.8 
 
 23.8 
 
 23.9 
 
 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 
 
 17.6 
 
 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 
 
 13.0 
 
 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 
 
 560 
 
 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 
 
 7.2 
 
 7.4 
 
 7.6 
 
 7.9 
 
 8.1 
 
 SCO 
 
 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 
 
 6.7 
 
 6.8 
 
 6.8 
 
 7.0 
 
 7.2 
 
 900 
 
 8.5 
 
 8.2 
 
 7.9 
 
 7.7 
 
 7.5 
 
 7.3 
 
 7.2 
 
 7.1 
 
 6.9 
 
 6.9 
 
 6.8 
 
 6.8 
 
 6.8 
 
 920 
 
 9.4 
 
 9.2 
 
 8.7 
 
 8.4 
 
 8.1 
 
 7.9 
 
 7.6 
 
 7.4 
 
 7.1 
 
 7.0 
 
 6.9 
 
 6.8 
 
 6.7 
 
 940 
 
 10.4 
 
 10.0 
 
 9.7 
 
 9.4 
 
 8.9 
 
 8.6 
 
 8.3 
 
 8.1 
 
 7.7 
 
 7.4 
 
 7.1 
 
 6.9 
 
 6.7 
 
 960 
 
 11.5 
 
 11.2 
 
 10.7 
 
 10.4 
 
 9.8 
 
 9.5 
 
 9.1 
 
 8.8 
 
 8.5 
 
 8.1 
 
 7.7 
 
 7.4 
 
 7.1 
 
 980 
 
 12.7 
 
 12.3 
 
 11.8 
 
 11.5 
 
 11.1 
 
 10.6 
 
 10.0 
 
 9.7 
 
 9.2 
 
 8.9 
 
 8.5 
 
 8.1 
 
 7.7 
 
 1000 
 
 13.9 
 
 13.4 
 
 13.1 
 
 12.7 
 
 12.1 
 
 11.8 
 
 11.3 
 
 10.8 
 
 10.2 
 
 9.9 
 
 9.4 
 
 8.9 
 
 8.4 
 
 
 510 
 
 520 
 
 530 
 
 540 
 
 550 
 
 560 
 
 570 
 
 580 
 
 590 
 
 600 
 
 610 
 
 020 
 
 630 1 
 
TABLE XXXII. 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 II. 
 
 630 
 
 640 
 
 650 
 
 660 
 
 670 
 
 680 
 
 690 
 
 700 
 
 710 
 
 720 
 
 730 
 
 740 
 
 750 
 
 
 
 8.4 
 
 8.0 
 
 7.7 
 
 7.3 
 
 6.9 
 
 6.7 
 
 6.5 
 
 6.5 
 
 6.3 
 
 6.2 
 
 6.2 
 
 6.4 
 
 6.5 
 
 20 
 
 9.4 
 
 9.0 
 
 8.4 
 
 8.0 
 
 7.5 
 
 7.1 
 
 6.9 
 
 6.7 
 
 6.4 
 
 6.3 
 
 6.0 
 
 6.1 
 
 6.1 
 
 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 
 
 180 
 
 20.3 
 
 19.9 
 
 19.4 
 
 18.8 
 
 18.0 
 
 17.3 
 
 16.7 
 
 15.8 
 
 15.0 
 
 14.1 
 
 13.2 
 
 12.4 
 
 11.5 
 
 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 
 
 14.7 
 
 240 
 
 232 
 
 23.0 
 
 22.9 
 
 22.5 
 
 22.0 
 
 21.6 
 
 21.1 
 
 20.5 
 
 19.8 
 
 19.1 
 
 18.2 
 
 17.3 
 
 16.4 
 
 260 
 
 23.9 
 
 23.8 
 
 23.7 
 
 23.5 
 
 23.1 
 
 22.7 
 
 22.3 
 
 21.8 
 
 21.2 
 
 20.6 
 
 19.8 
 
 19.1 
 
 18.1 
 
 280 
 
 24.1 
 
 24.3 
 
 24.2 
 
 24.2 
 
 24.0 
 
 23.7 
 
 23.5 
 
 23.1 
 
 22.4 
 
 21.8 
 
 21.2 
 
 20.5 
 
 19.8 
 
 300 
 
 24.3 
 
 24.5 
 
 24.6 
 
 24.6 
 
 24.5 
 
 24.4 
 
 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 
 
 25.1 
 
 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 
 
 25.2 
 
 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 
 
 13.8 
 
 14.2 
 
 14.6 
 
 14.9 
 
 15.1 
 
 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 
 
 15.6 
 
 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 
 
 16.8 
 
 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 
 
 135 
 
 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 
 
 11.8 
 
 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 
 
 8.6 
 
 8.9 
 
 9.4 
 
 9.9 
 
 940 
 
 6.7 
 
 6.7 
 
 6.7 
 
 6.8 
 
 6.7 
 
 6.8 
 
 6.8 
 
 7.1 
 
 7.4 
 
 7.7 
 
 8.1 
 
 8.4 
 
 8.9 
 
 960 
 
 7.1 
 
 7.0 
 
 6.8 
 
 17 
 
 6.5 
 
 6.5 
 
 6.6 
 
 6.7 
 
 6.8 
 
 7.1 
 
 7.3 
 
 7.7 
 
 8.0 
 
 980 
 
 7.7 
 
 7.4 
 
 7.1 
 
 6.9 
 
 6.6 
 
 6.5 
 
 6.4 
 
 6.4 
 
 6.3 
 
 6.5 
 
 6.8 
 
 6.9 
 
 7.3 
 
 1000 
 
 8.4 
 
 8.0 
 
 7.7 
 
 7.3 
 
 6.9 
 
 6.7 
 
 6.5 
 
 6.5 
 
 6.3 
 
 6.2 
 
 6.2 
 
 6.4 
 
 6.5 
 
 
 63) 
 
 140 
 
 650 
 
 660 
 
 670 
 
 680 
 
 690 
 
 700 
 
 710 
 
 720 
 
 730 
 
 740 
 
 750 
 
TABLE XXXII. 
 
 45 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 II. 
 
 750 
 
 760 
 
 ; 770 
 
 780 
 
 790 
 
 800 
 
 810 
 
 820 
 
 830 
 
 840 
 
 850 
 
 860 
 
 870 
 
 0' 
 
 6, 
 
 6.8 
 
 7.2 
 
 7.5 
 
 8.0 
 
 8.4 
 
 8.8 
 
 9.5 
 
 10.1 
 
 10.5 
 
 11.0 
 
 11.6 
 
 12.4 
 
 20 
 
 6.1 
 
 6.2 
 
 6.5 
 
 6.7 
 
 7.0 
 
 7.4 
 
 7.9 
 
 8.4 
 
 9.0 
 
 9.5 
 
 10.0 
 
 10.6 
 
 11.1 
 
 40 
 
 5.8 
 
 5.9 
 
 5.9 
 
 6.2 
 
 6.4 
 
 6.6 
 
 6.9 
 
 7.4 
 
 7.8 
 
 8.2 
 
 8.8 
 
 9.5 
 
 10.0 
 
 60 
 
 5.8 
 
 5.7 
 
 5.7 
 
 5.7 
 
 5.9 
 
 6.1 
 
 6.2 
 
 6.5 
 
 6.9 
 
 7.2 
 
 7.7 
 
 8.3 
 
 8.8 
 
 80 
 
 6.2 
 
 5.8 
 
 5.7 
 
 5.6 
 
 5.4 
 
 5.6 
 
 5.7 
 
 5.9 
 
 6.1 
 
 6.3 
 
 6.7 
 
 7.3 
 
 7.8 
 
 100 
 
 6.8 
 
 6.3 
 
 5.9 
 
 5.6 
 
 5.5 
 
 5.3 
 
 5.3 
 
 5.4 
 
 5.4 
 
 5.6 
 
 5.9 
 
 6.3 
 
 6.8 
 
 120 
 
 7.6 
 
 7.4 
 
 6.5 
 
 6.0 
 
 5.7 
 
 5.5 
 
 5.1 
 
 5.2 
 
 5.1 
 
 5.1 
 
 5.2 
 
 5.5 
 
 5.8 
 
 140 
 
 8.8 
 
 8.1 
 
 7.4 
 
 6.8 
 
 6.2 
 
 5.8 
 
 5.4 
 
 5.2 
 
 5.0 
 
 4.9 
 
 4.8 
 
 5.0 
 
 5.1 
 
 160 
 
 10.0 
 
 9.3 
 
 8.5 
 
 7.8 
 
 7.2 
 
 6.5 
 
 5.9 
 
 5.5 
 
 5.1 
 
 5.9 
 
 4.7 
 
 4.7 
 
 4.7 
 
 180 
 
 11.5 
 
 10.6 
 
 9.7 
 
 9.0 
 
 8.2 
 
 7.5 
 
 6.9 
 
 6.3 
 
 5.8 
 
 5.2 
 
 4.8 
 
 4.7 
 
 4.5 
 
 200 
 
 13.1 
 
 12.2 
 
 11.2 
 
 10.4 
 
 9.5 
 
 8.8 
 
 7.9 
 
 7.1 
 
 6.5 
 
 5.9 
 
 5.3 
 
 5.0 
 
 4.7 
 
 220 
 
 14.7 
 
 13.8 
 
 12.9 
 
 12.0 
 
 11.1 
 
 10.2 
 
 9.3 
 
 8.4 
 
 7.5 
 
 6.7 
 
 6.1 
 
 5.5 
 
 5.2 
 
 240 
 
 16.4 
 
 15.3 
 
 14.5 
 
 13.6 
 
 12.6 
 
 11.7 
 
 10.7 
 
 9.8 
 
 8.8 
 
 7.9 
 
 7.0 
 
 6.5 
 
 5.9 
 
 260 
 
 18.1 
 
 17.2 
 
 16.3 
 
 15.3 
 
 14.3 
 
 13.3 
 
 12.2 
 
 11.4 
 
 10.4 
 
 9.4 
 
 8.3 
 
 7.7 
 
 6.9 
 
 280 
 
 19.8 
 
 18.9 
 
 17.9 
 
 17.0 
 
 16.1 
 
 15.0 
 
 14.0 
 
 13.0 
 
 11.9 
 
 10.9 
 
 9.9 
 
 8.9 
 
 8.0 
 
 300 
 
 21.2 
 
 20.4 
 
 19.6 
 
 18.7 
 
 17.7 
 
 16.8 
 
 15.8 
 
 14.7 
 
 13.7 
 
 12.6 
 
 11.5 
 
 10.5 
 
 9.4 
 
 320 
 
 22.5 
 
 21.9 
 
 21.2 
 
 20.4 
 
 19.4 
 
 18.5 
 
 17.4 
 
 16.5 
 
 15.5 
 
 14.2 
 
 13.2 
 
 12.3 
 
 11.2 
 
 340 
 
 23.7 
 
 23.0 
 
 22.4 
 
 21.8 
 
 21.1 
 
 20.2 
 
 19.2 
 
 183 
 
 17.1 
 
 16.1 
 
 15.0 
 
 13.9 
 
 12.9 
 
 360 
 
 24.5 
 
 24.0 
 
 23.6 
 
 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 
 
 24.5 
 
 24.2 
 
 23.7 
 
 23.2 
 
 22.3 
 
 21.5 
 
 20.7 
 
 198 
 
 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 
 
 500 
 
 23.9 
 
 24.2 
 
 24.7 
 
 25.0 
 
 25.3 
 
 25.4 
 
 25.5 
 
 25.5 
 
 25.4 
 
 25.2 
 
 24.9 
 
 24.7 
 
 24.3 
 
 520 
 
 23.0 
 
 23.6 
 
 23.9 
 
 24.3 
 
 24.7 
 
 24.9 
 
 25.2 
 
 25.4 
 
 25.4 
 
 25.3 
 
 25.2 
 
 25.1 
 
 24.8 
 
 540 
 
 2 
 
 22.G 
 
 23.2 
 
 23.6 
 
 24.0 
 
 24.4 
 
 24.6 
 
 24.9 
 
 25.1 
 
 25.0 
 
 25.1 
 
 25.1 
 
 25.0 
 
 560 
 
 2L3 
 
 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 
 
 207 
 
 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 
 
 18.5 
 
 18.7 
 
 193 
 
 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 
 
 185 
 
 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 
 
 180 
 
 18.2 
 
 780 
 
 14.9 
 
 15.3 
 
 15.6 
 
 15.9 
 
 16.1 
 
 16.3 
 
 16.5 
 
 16.7 
 
 16.9 
 
 17.1 
 
 17.3 
 
 17.6 
 
 17.7 
 
 800 
 
 14.5 
 
 14.7 
 
 15.2 
 
 15.5 
 
 15.8 
 
 15.9 
 
 16.2 
 
 16.5 
 
 16.6 
 
 16.8 
 
 16.9 
 
 17.1 
 
 17.3 
 
 820 
 
 14.0 
 
 14.4 
 
 14.7 
 
 15.1 
 
 15.4 
 
 15.7 
 
 15.8 
 
 16.1 
 
 16.3 
 
 16.4 
 
 16.6 
 
 16.9 
 
 17.0 
 
 840 
 
 13.4 
 
 13.7 
 
 14.1 
 
 14.5 
 
 15.1 
 
 15.4 
 
 15.4 
 
 15.8 
 
 15.9 
 
 16.1 
 
 16.2 
 
 16.6 
 
 16.7 
 
 860 
 
 12.6 
 
 13.1 
 
 13.5 
 
 13.9 
 
 14.3 
 
 14.8 
 
 15.2 
 
 15.5 
 
 15.6 
 
 15.8 
 
 16.0 
 
 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 
 
 15.7 
 
 920 
 
 9.9 
 
 10.3 
 
 10.8 
 
 11.4 
 
 12.0 
 
 12.5 
 
 12.9 
 
 13.4 
 
 14.0 
 
 14.3 
 
 14.7 
 
 15.0 
 
 15.3 
 
 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 
 
 14.7 
 
 960 
 
 8.0 
 
 8.3 
 
 8.8 
 
 94 
 
 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 i 
 
 1000 
 
 6.5 
 
 6.8 
 
 7.2 
 
 7.5 
 
 8.0 
 
 8.4 
 
 8.8 
 
 9.5 
 
 10.0 
 
 10.5 
 
 11.0 
 
 11.6 
 
 12.4 
 
 
 750 
 
 760 
 
 770 
 
 780 
 
 790 
 
 800 
 
 810 
 
 820 
 
 830 
 
 840 
 
 850 
 
 860 
 
 870 
 
46 
 
 TABLE XXXII. 
 
 Perturbations produced by Jupiter. 
 
 Arguments II. and V. 
 
 V. 
 
 II. 
 
 870 
 
 880 
 
 890 
 
 900 
 
 910 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
 970 
 
 980 
 
 990 
 
 1000 
 
 
 
 12.4 
 
 12.9 
 
 13.2 
 
 13.6 
 
 13.9 
 
 14.2 
 
 14.4 
 
 14.8 
 
 15.0 
 
 15.1 
 
 15.1 
 
 15.2 
 
 15.2 
 
 15.3 
 
 20 
 
 11.1 
 
 11.7 
 
 12.2 
 
 12.7 
 
 13.2 
 
 13.6 
 
 13.8 
 
 14.1 
 
 14.4 
 
 14.7 
 
 14.8 
 
 15.0 
 
 14.9 
 
 14.9 
 
 40 
 
 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 
 
 6.6 
 
 7.1 
 
 7.6 
 
 8.1 
 
 8.7 
 
 9.4 
 
 9.9 
 
 10.4 
 
 10.8 
 
 11.4 
 
 11.8 
 
 12.3 
 
 140 
 
 5.1 
 
 5.3 
 
 5.6 
 
 6.0 
 
 6.5 
 
 7.0 
 
 7.5 
 
 8.2 
 
 8.7 
 
 9.3 
 
 9.7 
 
 10.3 
 
 10.8 
 
 11.3 
 
 160 
 
 4.7 
 
 4.8 
 
 4.8 
 
 5.2 
 
 5.6 
 
 5.9 
 
 6.3 
 
 6.8 
 
 7.4 
 
 8.0 
 
 8.6 
 
 9.2 
 
 9.7 
 
 10.2 
 
 180 
 
 4.5 
 
 4.5 
 
 4.4 
 
 4.5 
 
 4.8 
 
 5.1 
 
 5.4 
 
 5.8 
 
 6.2 
 
 6.9 
 
 7.4 
 
 8.0 
 
 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 
 
 4.2 
 
 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 
 
 4.9 
 
 4.3 
 
 3.8 
 
 3.5 
 
 380 
 
 16.5 
 
 15.4 
 
 14.2 
 
 13.0 
 
 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 
 
 9.5 
 
 8.4 
 
 7.5 
 
 460 
 
 22.5 
 
 21.6 
 
 20.6 
 
 19.7 
 
 18.9 
 
 17.9 
 
 16.7 
 
 15.6 
 
 14.3 
 
 13.3 
 
 12.2 
 
 10.9 
 
 10.0 
 
 9.0 
 
 480 
 
 235 
 
 22.7 
 
 22.0 
 
 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 
 
 23.0 
 
 22.3 
 
 21.6 
 
 20.7 
 
 19.7 
 
 18.8 
 
 17.8 
 
 16.7 
 
 15.4 
 
 14.5 
 
 13.4 
 
 12.3 
 
 520 
 
 24.8 
 
 24.3 
 
 23.7 
 
 23.2 
 
 22.7 
 
 21.9 
 
 21.1 
 
 20.2 
 
 19.2 
 
 18.3 
 
 17.2 
 
 16.1 
 
 15.0 
 
 14.0 
 
 540 
 
 25.0 
 
 24.8 
 
 24.3 
 
 23.9 
 
 23.4 
 
 22.8 
 
 22.1 
 
 21.3 
 
 20.6 
 
 19.7 
 
 18.7 
 
 17.6 
 
 16.6 
 
 15.6 
 
 560 
 
 24.9 
 
 24.8 
 
 24.7 
 
 24-4 
 
 24.0 
 
 23.6 
 
 22.9 
 
 22.4 
 
 21.0 
 
 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 
 
 23.5 
 
 23.0 
 
 22.5 
 
 22.0 
 
 21.4 
 
 20.6 
 
 198 
 
 620 
 
 23.6 
 
 23.7 
 
 23.9 
 
 24.0 
 
 24.1 
 
 24.1 
 
 23.9 
 
 23.7 
 
 23.4 
 
 23.1 
 
 22.6 
 
 22.1 
 
 21.4 
 
 20.8 
 
 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 
 
 22.0 
 
 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 
 
 22.0 
 
 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 
 
 1.9.4 
 
 19.6 
 
 19.9 
 
 20.1 
 
 20.3 
 
 20.5 
 
 20.8 
 
 21.0 
 
 21.2 
 
 780 
 
 17.7 
 
 17.8 
 
 18.0 
 
 18.1 
 
 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 
 
 18.1 
 
 860 
 
 16.4 
 
 16.5 
 
 16.5 
 
 16.6 
 
 16.6 
 
 16.7 
 
 16.8 
 
 16.9 
 
 16.9 
 
 17.0 
 
 17.0 
 
 17.1 
 
 17.2 
 
 17.4 
 
 880 
 
 16.1 
 
 16.3 
 
 16.3 
 
 16.5 
 
 16.5 
 
 16.5 
 
 16.6 
 
 16.6 
 
 16.6 
 
 16.6 
 
 16.6 
 
 16.7 
 
 16.7 
 
 16.9 
 
 900 
 
 15.7 
 
 15.9 
 
 16.1 
 
 16.2 
 
 16.3 
 
 16.4 
 
 16.3 
 
 16.3 
 
 16.2 
 
 16.2 
 
 16.2 
 
 16.3 
 
 16.3 
 
 16.3 
 
 920 
 
 15.3 
 
 15.5 
 
 15.6 
 
 15.9 
 
 16.0 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.0 
 
 16.1 
 
 16.1 
 
 16.1 
 
 16.0 
 
 16.0 
 
 940 
 
 14.7 
 
 15.9 
 
 15.2 
 
 15.4 
 
 15.7 
 
 15.8 
 
 15.8 
 
 16.0 
 
 15.9 
 
 15.9 
 
 15.9 
 
 15.8 
 
 15.7 
 
 15.8 
 
 960 
 
 14.1 
 
 14.3 
 
 14.5 
 
 14.8 
 
 15.2 
 
 15.5 
 
 15.5 
 
 15.7 
 
 15.7 
 
 15.7 
 
 15.6 
 
 15.6 
 
 15.5 
 
 15.5 
 
 980 
 
 13.3 
 
 12.7 
 
 13.9 
 
 14.2 
 
 14.5 
 
 14.8 
 
 15.1 
 
 15.3 
 
 15.4 
 
 15.5 
 
 15.4 
 
 15.4 
 
 15.4 
 
 15.3 
 
 1000 
 
 12.4 
 
 12.9 
 
 13.2 
 
 13.6 
 
 13.9 
 
 14.2 
 
 14.4 
 
 14.8 
 
 15.0 
 
 15.1 
 
 15.1 
 
 15.2 
 
 15.2 
 
 15.3 
 
 
 870 
 
 880 
 
 890 
 
 900 
 
 910 
 
 920 
 
 930 
 
 940 
 
 950 
 
 960 
 
 970 
 
 980 
 
 990 
 
 1000 
 
TABLE XXXIII. 
 Perturbations produced by Saturn. 
 
 Arguments II and VII. 
 VII. 
 
 47 
 
 II 
 
 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 600 
 
 700 
 
 800 
 
 900 
 
 1000 
 
 
 
 1.2 
 
 1.5 
 
 1.4 
 
 1.0 
 
 0.7 
 
 0.6 
 
 0.5 
 
 0.5 
 
 0.4 
 
 0.8 
 
 1.2 
 
 100 
 
 0.9 
 
 1.2 
 
 1.3 
 
 1.1 
 
 0.9 
 
 0.8 
 
 0.7 
 
 0.7 
 
 0.6 
 
 0.7 
 
 0.9 
 
 200 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 1.0 
 
 0.9 
 
 0.8 
 
 0.8 
 
 0.9 
 
 0.8 
 
 0.7 
 
 300 
 
 0.9 
 
 0.8 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 1.0 
 
 1.0 
 
 1.0 
 
 .0 
 
 0.9 
 
 400 
 
 .0 
 
 0.9 
 
 0.6 
 
 0.4 
 
 0.6 
 
 0.9 
 
 1.0 
 
 1.1 
 
 1.1 
 
 .1 
 
 1.0 
 
 500 
 
 .1 
 
 1.0 
 
 0.8 
 
 0.4 
 
 0.2 
 
 0.5 
 
 1.0 
 
 1.3 
 
 1.3 
 
 .2 
 
 1.1 
 
 600 
 
 .2 
 
 1.1 
 
 0.9 
 
 0.6 
 
 0.2 
 
 0.2 
 
 0.5 
 
 1.1 
 
 1.5 
 
 .5 
 
 1.2 
 
 700 
 
 .4 
 
 1.1 
 
 1.0 
 
 0.8 
 
 0.4 
 
 0.1 
 
 0.3 
 
 0.8 
 
 1.4 
 
 .7 
 
 1.4 
 
 800 
 
 .6 
 
 1.3 
 
 1.0 
 
 0.8 
 
 0.6 
 
 0.4 
 
 0.1 
 
 0.3 
 
 1.0 
 
 .6 
 
 1.6 
 
 900 
 
 .5 
 
 1.4 
 
 1.1 
 
 0.9 
 
 0.7 
 
 0.6 
 
 0.3 
 
 0.2 
 
 0.6 
 
 .2 
 
 1.5 
 
 1000 
 
 .2 
 
 1.5 
 
 1.4 
 
 1.0 
 
 0.7 
 
 0.6 
 
 0.5 
 
 0.5 
 
 0.4 
 
 0.8 
 
 1.2 
 
 Constant, l."0 
 
 TABLE XXXIV. 
 
 Variable Part of Sun's Aberration. 
 Argument, Sun's Mean Anomaly. 
 
 
 0* 
 
 I* 
 
 Us 
 
 III* 
 
 IV* 
 
 V* 
 
 
 o 
 
 // 
 
 // 
 
 
 
 Z 
 
 // 
 
 
 
 o 
 
 
 
 0.0 
 
 0.0 
 
 0.1 
 
 0.3 
 
 0.5 
 
 0.6 
 
 30 
 
 3 
 
 0.0 
 
 0.0 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.6 
 
 27 
 
 6 
 
 0.0 
 
 0.0 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.6 
 
 24 
 
 9 
 
 0.0 
 
 0.0 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.6 
 
 21 
 
 12 
 
 0.0 
 
 0. 
 
 0.2 
 
 0.4 
 
 05 
 
 0.6 
 
 18 
 
 15 
 
 0.0 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 15 
 
 18 
 
 0.0 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 12 
 
 21 
 
 0.0 
 
 0. 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.6 
 
 9 
 
 24 
 
 0.0 
 
 0. 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.6 
 
 6 
 
 27 
 
 0.0 
 
 0.1 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.6 
 
 3 
 
 30 
 
 0.0 
 
 0.1 
 
 0.3 
 
 0.5 
 
 0.6 
 
 0.6 
 
 
 
 
 XI* 
 
 x* 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI* 
 
 
 Constant, 0."3 
 
48 
 
 TABLE XXXV. 
 Moorfs Epochs. 
 
 Years. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 12 
 
 | 
 
 1830 * 
 
 00174 
 
 4541 
 
 4461 4638 
 
 9885 0635 5979 9921 
 
 7623 
 
 219 
 
 226,458 
 
 468 
 
 1831 
 
 00103 
 
 1749 
 
 41279381 
 
 2357 
 
 64327040 2378 6487 825 
 
 58 7|1 77 
 
 940 
 
 1832 B 
 
 00032 
 
 8957 
 
 3793 ! 4125 
 
 4829 
 
 22298100;48355351 
 
 432 
 
 948 
 
 897 
 
 413 
 
 1833 
 
 00235 
 
 6816 
 
 4499 9156 
 
 7636 
 
 8399 9219 
 
 76834239 
 
 108 
 
 340 
 
 687 
 
 920 
 
 1834 
 
 00164 
 
 4024 
 
 4164 
 
 3900 
 
 0107 
 
 41960279 01403103 
 
 715 
 
 701 
 
 406 
 
 393 
 
 1835 
 
 00093 
 
 1232 
 
 3830 
 
 8644 
 
 2579 
 
 99931340 
 
 2598 
 
 1967 
 
 321 
 
 061 
 
 125 
 
 866 
 
 183613 
 
 00022 
 
 8441 
 
 3496 
 
 3388 
 
 5051 
 
 5791 2400 
 
 50550831 
 
 928 
 
 422 
 
 845 
 
 339 
 
 1837 
 
 00224 
 
 6299 
 
 4202|8419 7858 1960 3518 
 
 7903 9719 
 
 605 
 
 814 
 
 635 
 
 846 
 
 1838 
 
 00153 
 
 3508 
 
 3868 
 
 3163 0329 7757 4579 
 
 03608583 
 
 211 175 
 
 354 
 
 319 
 
 1839 
 
 00082 
 
 0716 
 
 3534 
 
 7907 
 
 2801 
 
 3555 5639 
 
 2818 
 
 7447 
 
 818 536 
 
 074 
 
 792 
 
 1840 B 
 
 00011 
 
 7925 
 
 3199 
 
 2651 5273 
 
 9352^6700 
 
 5275 
 
 6310 
 
 424 896 
 
 793 
 
 265 
 
 1841 
 
 00213 
 
 5783 
 
 390676828080 
 
 55227818 
 
 8123 
 
 5199 
 
 101 288 
 
 583 
 
 772 
 
 1842 
 
 00142 
 
 2991 
 
 3571 2425 0551 
 
 13198879 
 
 0580 
 
 4062 
 
 707 649 
 
 302 
 
 245 
 
 1843 
 
 00071 
 
 0200 
 
 3237 
 
 7169 3023 
 
 71 16, 9939 
 
 3038 
 
 2926 
 
 314010 
 
 022 
 
 718 
 
 1844 B 
 
 00000 
 
 7408 
 
 2903 
 
 19135495 
 
 2914 
 
 1000 
 
 5495 
 
 1790 
 
 920 371 
 
 741 
 
 191 
 
 1845 
 
 00203 
 
 5266 
 
 3609 
 
 6944*8302 
 
 9083 
 
 2118 
 
 8343 
 
 0678 
 
 597 
 
 763 
 
 531 
 
 698 
 
 1846 
 
 00132 
 
 2475 
 
 3275 
 
 1688 0773 
 
 4880 
 
 3179 
 
 0800 9542 
 
 203 
 
 123 
 
 250 
 
 171 
 
 1847 
 
 00061 
 
 9683 
 
 2941 
 
 6432,3245 
 
 0678 
 
 4239 
 
 3257' 8406 
 
 810 
 
 484 
 
 970 
 
 644 
 
 1848 B 
 
 99990 
 
 6892 
 
 2606 
 
 11765717 
 
 6475 
 
 5300 
 
 5715 
 
 7270 
 
 416 
 
 845 
 
 689 
 
 117 
 
 1849 
 
 00192 
 
 4750 
 
 3312 
 
 6207,8524 
 
 2644 
 
 6418 
 
 8563 
 
 6158 
 
 093 
 
 237 
 
 479 
 
 624 
 
 1850 
 
 00121 
 
 1958 
 
 2978 
 
 0951 
 
 0995 
 
 8442 
 
 7479 
 
 1020 
 
 5022 
 
 700 
 
 597 
 
 199 
 
 097 
 
 1851 
 
 00050 
 
 9167 2644 
 
 5695 3467 
 
 4239 
 
 8539 
 
 3477 
 
 3885 
 
 306 
 
 958 
 
 918 
 
 570 
 
 1852 B 
 
 99979 
 
 6375 2310 
 
 04395939 
 
 0036 
 
 9600 
 
 5935 2749 
 
 913 
 
 319 
 
 637 
 
 043 
 
 1853 
 
 00181 
 
 4233 3016 
 
 5469 8746 
 
 6206 
 
 0718 
 
 8782 
 
 1637 
 
 589 
 
 711 
 
 427 
 
 550 
 
 1854 
 
 00110 
 
 14422681 
 
 0213 
 
 1217 
 
 2003 
 
 1778 
 
 1240 0501 
 
 196 
 
 072 
 
 147 
 
 023 
 
 1855 
 
 00039 
 
 8650 '2347 
 
 4957 
 
 3689 
 
 78012839 
 
 3697 
 
 9365 
 
 802 
 
 432 
 
 866 
 
 496 
 
 1856 B 
 
 99968 
 
 5859 2013 
 
 9701 
 
 6160 
 
 35983899 
 
 61558229 
 
 409 
 
 793 
 
 586 
 
 969 
 
 1857 
 
 00171 
 
 37172719 
 
 4732 8968 
 
 976715018 
 
 90027117 
 
 086 
 
 185 
 
 375 
 
 476 
 
 1858 
 
 00100 
 
 09252385 
 
 9476 1439 
 
 5565 
 
 607S 
 
 14605981 
 
 692 
 
 546 
 
 095 
 
 949 
 
 1859 
 
 00029 
 
 8 134 '2051 
 
 42203911 
 
 1362 
 
 7139 
 
 39174845 
 
 299 
 
 907 
 
 814 
 
 422 
 
 1860 B 
 
 99958 5342 1716 
 
 8964 
 
 6383 
 
 7159 
 
 8199 
 
 6374 3709 
 
 905 
 
 267 
 
 534 
 
 895 
 
 1861 
 
 00160 3200 
 
 2423 
 
 3995 
 
 9190 
 
 3329 
 
 9317 
 
 9222,2597 
 
 581 
 
 659 
 
 323 
 
 402 
 
 1862 
 
 00089 0409 
 
 2088 
 
 8739 
 
 1661 
 
 9126 0378 
 
 1679 
 
 1461 
 
 188 
 
 020 
 
 043 
 
 875 
 
 1863 
 
 00018 7617 
 
 1754 
 
 3483 4133 
 
 4923 1438 
 
 4137 0324 
 
 795 
 
 381 
 
 762 
 
 348 
 
 1864 B 
 
 99947 4826 
 
 1420 
 
 8227 6605 
 
 0721 
 
 2499 
 
 65949188 
 
 401 
 
 742 
 
 482 
 
 821 
 
 1865 
 
 0014912684 
 
 2126 
 
 3257 
 
 9412 
 
 6890 
 
 3617 
 
 9442 8076 
 
 078 
 
 134 
 
 272 
 
 328 
 
 1866 
 
 00078 
 
 9893 
 
 1792 
 
 8001 1883 
 
 2687 
 
 4678 1899 6940 
 
 685 
 
 494 
 
 991 
 
 801 
 
 1867 
 
 00007 
 
 7101 
 
 1457 
 
 2745 4355 
 
 8485 
 
 5738 4357 5804 
 
 291 
 
 855 
 
 711 
 
 274 
 
 1868 B 
 
 99936 
 
 4309 1123 
 
 7489 6827 
 
 4282 
 
 6799 68144668 
 
 898 
 
 216 
 
 431 
 
 747 
 
 1869 
 
 00138 
 
 2168 1829 
 
 2520 9634 
 
 0452 
 
 791796623556 
 
 574 
 
 608 
 
 220 
 
 254 
 
 1870 
 
 00067 
 
 93761495 
 
 7264 2105 6249 
 
 8978! 21 19 2420 
 
 181 
 
 968 
 
 940 
 
 727 
 
TABLE XXXV. 
 Moon's Epochs. 
 
 Years. 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 2223 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29|30|31 
 
 1830 
 
 921 
 
 392 
 
 230 
 
 588 
 
 462 
 
 523 
 
 536 
 
 52 
 
 6044 
 
 94 
 
 51 
 
 47 
 
 98 
 
 99 
 
 99 8'9 52 
 
 1831 
 
 115 
 
 532 
 
 589 
 
 940 
 
 937 
 
 296 
 
 703 
 
 30 
 
 7041 
 
 65 
 
 53 
 
 94 
 
 48 
 
 24 
 
 2451 
 
 44 
 
 1832 B 
 
 309 
 
 673 
 
 949 
 
 293 
 
 412 
 
 070 870 
 
 07 
 
 81 38 
 
 36 
 
 55 
 
 42 97 
 
 48 49 14 35 
 
 1833 
 1834 
 
 602 
 796 
 
 844 
 984 
 
 345 
 
 704 
 
 688 
 040 
 
 913 
 
 388 
 
 845 037 
 619 203 
 
 85 
 62 
 
 9245 
 0342 
 
 07 
 
 77 
 
 61 
 63 
 
 9253 
 4003 
 
 77 
 01 
 
 777727 
 0139J18 
 
 1835 
 
 989 
 
 124 
 
 063 
 
 393 
 
 863 
 
 392 370 
 
 39 
 
 1338 
 
 48 
 
 65 
 
 87 
 
 51 
 
 26 
 
 2602 
 
 10 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 1836 B 
 
 183 
 
 265 423 
 
 745 
 
 338 
 
 166 
 
 537 
 
 17 
 
 2435 
 
 19 
 
 67 
 
 34 
 
 01 
 
 50 
 
 51 6401 
 
 1837 
 
 476 
 
 436819 
 
 140 
 
 840 
 
 942 
 
 704 
 
 94 
 
 3542 
 
 90 
 
 73 
 
 8558 
 
 79 
 
 792793 
 
 1838 
 
 670 
 
 576 178 
 
 492 
 
 315 
 
 715 
 
 870 
 
 72 
 
 4638 
 
 60 
 
 75 
 
 32^07 
 
 04 
 
 04 
 
 8984 
 
 1839 
 
 864 
 
 716 537 
 
 845 
 
 790 
 
 489 
 
 037 
 
 49 
 
 5635 
 
 31 
 
 77 
 
 8056 
 
 28 
 
 28 
 
 5276 
 
 1840 B 
 
 058 
 
 857 
 
 8971197 
 
 265 
 
 262 
 
 204 
 
 26 
 
 6732 
 
 02 
 
 79 
 
 27 
 
 06 
 
 53 
 
 52 
 
 1467 
 
 1841 
 
 351 
 
 028 
 
 293592 
 
 766 
 
 038 
 
 371 
 
 04 
 
 7839 
 
 73 
 
 85 
 
 77 
 
 62 
 
 81 
 
 81 
 
 77 
 
 59 
 
 1842 
 
 544 
 
 168 
 
 652944 
 
 241 
 
 811 
 
 537 
 
 81 
 
 89 ! 35 
 
 43 
 
 87 
 
 25 
 
 12 
 
 0606 
 
 40 
 
 51 
 
 1843 
 
 738 
 
 308 
 
 012 297 
 
 716 
 
 585 
 
 704 
 
 58 
 
 9932 
 
 14 
 
 89 
 
 72 
 
 61 
 
 3031 
 
 02 
 
 42 
 
 1844 B 
 
 932 
 
 449 
 
 371 
 
 649 
 
 191 
 
 358 
 
 871 
 
 36 
 
 1029 
 
 85 
 
 01 
 
 19 
 
 10 
 
 55 
 
 55 
 
 65 
 
 34 
 
 1845 
 
 225 
 
 620 
 
 767 
 
 044 
 
 692 
 
 134 
 
 038 
 
 13 
 
 21 
 
 36 
 
 56 
 
 97 
 
 70 
 
 67 
 
 84 
 
 83 
 
 27 
 
 26 
 
 1846 
 
 419 
 
 760 
 
 126 
 
 396 
 
 167 
 
 907 
 
 204 
 
 91 
 
 32 
 
 32 
 
 26 
 
 99 
 
 17 
 
 16 
 
 08 
 
 08 
 
 90 
 
 17 
 
 1847 
 
 613 
 
 901 
 
 486 749 
 
 643 
 
 681 
 
 371 
 
 68 
 
 42 
 
 29 
 
 97 
 
 01 
 
 65 
 
 6533 
 
 33 
 
 52 
 
 09 
 
 1848 B 
 
 806 
 
 041 
 
 845 ' 
 
 101 
 
 118 
 
 454 
 
 538 
 
 45 
 
 53 
 
 26 
 
 68 
 
 03 
 
 12 
 
 15 5758 
 
 15 
 
 00 
 
 1849 
 
 099 
 
 212 
 
 241 : 
 
 496 
 
 619 
 
 230 
 
 705 
 
 23 
 
 64 
 
 33 
 
 39 
 
 09 
 
 03 
 
 71 86 86 
 
 77 
 
 92 
 
 1850 
 
 293 
 
 352 
 
 600 848 
 
 094 
 
 003 
 
 871 
 
 00 
 
 75 
 
 29 
 
 09 
 
 10 
 
 10 
 
 20 10 10 
 
 40 
 
 83 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1851 
 
 487 
 
 493 
 
 960 
 
 201 
 
 569 
 
 777 
 
 038 
 
 78 
 
 85 
 
 26 
 
 80 
 
 12 
 
 57 
 
 70 35 ! 35 
 
 02 
 
 75 
 
 1852 B 
 
 681 
 
 633 
 
 319 
 
 553 
 
 044 
 
 550 
 
 205 
 
 55 
 
 96 
 
 23 
 
 51 
 
 14 
 
 04 
 
 19 59 60 
 
 65 
 
 66 
 
 1853 
 
 974 
 
 04 
 
 715 
 
 948 
 
 545 
 
 326 
 
 372 
 
 33 
 
 07 
 
 30 
 
 22 
 
 20 
 
 55 
 
 7688 
 
 88 
 
 28 
 
 58 
 
 1854 
 
 168 
 
 944 
 
 074 
 
 300 
 
 020 
 
 099 
 
 539 
 
 10 
 
 18 
 
 26 
 
 93 
 
 22 
 
 03 
 
 25 12 
 
 12 
 
 90 
 
 50 
 
 1855 
 
 361 
 
 085 
 
 434653 
 
 495 
 
 873 
 
 705 
 
 87 
 
 28 
 
 23 
 
 63 
 
 24 
 
 50 
 
 7437 
 
 37 
 
 53 
 
 41 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 1856 B 
 
 555 
 
 225 
 
 793005 
 
 970 
 
 64ft 
 
 872 
 
 65 
 
 39 
 
 2034 
 
 26 
 
 97 
 
 23 61 
 
 62 
 
 15 
 
 33 
 
 1857 
 
 848 
 
 396 
 
 189 400 
 
 471 
 
 422 
 
 039 
 
 42 
 
 50 
 
 27105 
 
 32 
 
 48 
 
 80 90 
 
 90 
 
 78 24 
 
 1858 
 
 042 
 
 537 
 
 548 
 
 752 
 
 947 
 
 195 
 
 206 
 
 20 
 
 61 
 
 2476 
 
 34 
 
 95 
 
 29 15 
 
 15 
 
 40 
 
 16 
 
 1859 
 
 236 
 
 677 
 
 908 
 
 105 
 
 422 
 
 969 
 
 372 
 
 97 
 
 71 
 
 2046 
 
 36 
 
 42 
 
 7939 
 
 40 
 
 0307 
 
 1860 B 
 
 430 
 
 817 
 
 267 457 
 
 897 
 
 742 
 
 539 
 
 74 
 
 82 
 
 1717 
 
 38 
 
 89 
 
 28 64 64 
 
 6599 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 1861 
 
 723 
 
 988 
 
 663 
 
 852 
 
 398 
 
 518 
 
 706 
 
 52 
 
 93 
 
 2488 
 
 44 
 
 41 
 
 84 92 92 
 
 28 
 
 91 
 
 1862 
 
 916 
 
 129 
 
 022 
 
 204 
 
 873 
 
 291 
 
 873 
 
 29 
 
 04 
 
 20' 60 
 
 46 
 
 88 
 
 34 17 
 
 17 
 
 91 
 
 82 
 
 1863 
 
 110 
 
 269 
 
 382 
 
 557 
 
 348 
 
 065 
 
 039 
 
 06 
 
 14 
 
 1729 
 
 48 
 
 35 
 
 8241 
 
 42 
 
 53 
 
 74 
 
 1864 B 
 
 304 
 
 409 
 
 741 
 
 909 
 
 823 
 
 838 
 
 206 
 
 84 
 
 25 
 
 1400 
 
 50 
 
 82 
 
 32 66 66 
 
 16 
 
 65 
 
 1865 
 
 597 
 
 580 
 
 137304 
 
 324 
 
 614 
 
 373 
 
 61 
 
 36 
 
 21 71 
 
 56 
 
 33 
 
 89 95 94 
 
 78 
 
 57 
 
 1866 
 
 791 
 
 721 
 
 496 
 
 657 
 
 799 
 
 387 
 
 540 
 
 39 
 
 47 
 
 1742 
 
 58 
 
 80 
 
 38 1919 
 
 41 
 
 49 
 
 1867 
 
 985 
 
 861 
 
 856 009 274 
 
 161 
 
 707 
 
 16 
 
 57 
 
 1412 
 
 60 
 
 28 
 
 87 44J44 
 
 03 
 
 40 
 
 1868 B 
 
 178 
 
 001 
 
 215362 
 
 749 
 
 934 
 
 873 
 
 93 
 
 68 
 
 1183 
 
 62 
 
 75 
 
 37 68 69 
 
 66 
 
 32 
 
 1869 
 
 471 
 
 172 
 
 611; 
 
 756 
 
 251 
 
 710 
 
 040 
 
 71 
 
 79 
 
 1854 
 
 68 
 
 26 
 
 93 97 97 
 
 28 
 
 23 
 
 1870 
 
 665 
 
 313 
 
 970 
 
 109 
 
 726J483 
 
 207148 
 
 90 
 
 1526 
 
 69 
 
 73 
 
 43 21 21 
 
 91 
 
 15 
 
50 
 
 TABLE XXXV. 
 Moon's Epochs. 
 
 Years. 
 
 Evection. 
 
 Anomaly. 
 
 Variation. 
 
 Longitude. 
 
 
 , o / // 
 
 s o f /' 
 
 8 / " 
 
 9 / " 
 
 1830 
 
 5 17 4 12 
 
 11 24 31 4.5 
 
 2 13 2 39 
 
 11 22 55 37.7 
 
 1831 
 
 11 7 35 41 
 
 2 23 14 24.6 
 
 6 22 40 4 
 
 4 2 18 42.8 
 
 1832 B 
 
 4 28 7 11 
 
 5 21 57 44.4 
 
 11 2 17 28 
 
 8 11 41 48.0 
 
 1833 
 
 10 29 57 40 
 
 9 3 44 58.5 
 
 3 24 6 21 
 
 1 4 15 28.4 
 
 1834 
 
 4 20 29 11 
 
 2 28 18.5 
 
 8 3 43 45 
 
 5 13 38 33.6 
 
 1835 
 
 10 11 40 
 
 3 1 U 38.6 
 
 13 21 10 
 
 9 23 1 38.8 
 
 1836 B 
 
 4 1 32 9 
 
 5 29 54 58.7 
 
 4 22 58 34 
 
 2 2 24 44.0 
 
 1837 
 
 10 3 22 39 
 
 9 11 42 12.8 
 
 9 14 47 27 
 
 6 24 58 24.5 
 
 /1838 
 
 3 23 54 9 
 
 10 25 32.9 
 
 1 24 24 51 
 
 11 4 21 29.8 
 
 1839 
 
 9 14 25 38 
 
 39s 53.1 
 
 6 4 2 16 
 
 3 13 44 35.0 
 
 1840 B 
 
 3 4 57 8 
 
 6 7 52 13.2 
 
 10 13 39 42 
 
 7 23 7 40.4 
 
 1841 
 
 9 6 47 37 
 
 9 19 39 27.5 
 
 3 5 28 33 
 
 15 41 20.9 
 
 1842 
 
 2 27 19 7 
 
 18 22 47.6 
 
 7 15 5 58 
 
 4 25 4 26.2 
 
 1843 
 
 8 17 50 37 
 
 3 17 6 7.9 
 
 11 24 43 23 
 
 9 4 27 31.6 
 
 1844 B 
 
 2 8 22 7 
 
 6 15 49 28.1 
 
 4 4 20 48 
 
 1 13 50 37.0 
 
 1845 
 
 8 10 12 36 
 
 9 27 36 42.5 
 
 8 26 9 40 
 
 6 6 24 17.5 
 
 1846 
 
 2 44 6 
 
 26 20 2.8 
 
 1 5 47 5 
 
 10 15 47 23.0 
 
 1847 
 
 7 21 15 35 
 
 3 25 3 23.2 
 
 5 15 24 30 
 
 2 25 10 28.3 
 
 1848 B 
 
 1 11 47 5 
 
 6 23 46 43.5 
 
 9 25 1 55 
 
 7 4 33 33.7 
 
 1849 
 
 7 13 37 35 
 
 10 5 33 57.9 
 
 2 16 50 47 
 
 11 27 7 14.5 
 
 1850 
 
 I 4 9 4 
 
 1 4 17 18.3 
 
 6 26 28 12 
 
 4 6 30 19.9 
 
 1851 
 
 6 24 40 35 
 
 430 38.6 
 
 11 6 5 37 
 
 8 15 53 25.4 
 
 1852 B 
 
 15 12 5 
 
 7 1 43 59.2 
 
 3 15 43 3 
 
 25 16 31.0 
 
 1853 
 
 6 17 2 34 
 
 10 13 31 13.7 
 
 8 7 31 54 
 
 5 17 50 11.6 
 
 1854 
 
 7 34 4 
 
 1 12 14 34.1 
 
 17 9 20 
 
 9 27 13 17.2 
 
 1855 
 
 5 28 5 33 
 
 4 10 57 54.7 
 
 4 26 46 44 
 
 2 6 36 22.7 
 
 1856 B 
 
 11 18 37 3 
 
 7 9 41 15.2 
 
 9 6 24 10 
 
 6 15 59 28.2 
 
 1857 
 
 5 20 27 33 
 
 10 21 28 29.8 
 
 1 28 13 2 
 
 11 8 33 9.1 
 
 1858 
 
 11 10 59 2 
 
 1 20 n 50.3 
 
 6 7 50 27 
 
 3 17 56 14.6 
 
 1859 
 
 5 1 30 33 
 
 4 18 55 10.9 
 
 10 17 27 53 
 
 7 27 19 20.1 
 
 1860 B 
 
 10 22 2 3 
 
 7 17 38 31.4 
 
 2 27 5 18 
 
 6 42 25.8 
 
 1861 
 
 4 23 52 32 
 
 10 29 25 46.1 
 
 7 18 54 10 
 
 4 29 16 6.6 
 
 1862 
 
 10 14 24 2 
 
 1 28 9 6.6 
 
 11 28 31 35 
 
 9 8 39 12.2 
 
 1863 
 
 4 4 55 32 
 
 4 26 52 27.3 
 
 4891 
 
 1 18 2 17.9 
 
 1864 B 
 
 9 25 27 2 
 
 7 25 35 48.0 
 
 8 17 46 25 
 
 5 27 25 23.5 
 
 1865 
 
 3 27 17 31 
 
 11 7 23 2.7 
 
 1 9 35 is 
 
 10 19 59 4.3 
 
 1866 
 
 9 17 49 2 
 
 266 23.3 
 
 5 19 12 43 
 
 2 29 22 10.1 
 
 1867 
 
 3 8 20 31 
 
 5 4 49 44.0 
 
 9 28 50 9 
 
 7 8 45 15.7 
 
 1868 B 
 
 8 28 62 2 
 
 8 3 33 4.7 
 
 2 8 27 34 
 
 11 18 8 21.4 
 
 1869 
 
 3 42 83 
 
 11 15 20 19.6 
 
 7 16 26 
 
 4 10 42 23 
 
 1870 
 
 ai u 2 
 
 2 14 3 40.3 
 
 11 9 53 51 
 
 8 20 5 8.0 
 
TABLE XXXV. 
 Moon's Epochs. 
 
 Years. 
 
 Supp. of Node. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 
 g ' " 
 
 " 
 
 
 
 
 
 
 
 
 
 1830 
 
 677 11.0 
 
 10 24 46 
 
 498 
 
 502 
 
 900 
 
 904 
 
 427 
 
 062 
 
 025 
 
 433 
 
 1831 
 
 6 26 26 53.3 
 
 215 18 
 
 912 
 
 914 
 
 208 
 
 210 
 
 506 
 
 001 
 
 211 
 
 710 
 
 1832 B 
 
 7 15 46 35.5 
 
 6 550 
 
 326 
 
 327 
 
 516 
 
 516 
 
 586 
 
 940 
 
 397 
 
 986 
 
 1833 
 
 8 5 928.4 
 
 10 731 
 
 774 
 
 779 
 
 852 
 
 856 
 
 702 
 
 885 
 
 624 
 
 297 
 
 1834 
 
 8 24 29 10.7 
 
 128 3 
 
 187 
 
 191 
 
 159 
 
 163 
 
 782 
 
 825 
 
 810 
 
 573 
 
 1835 
 
 9 13 48 53.0 
 
 5 18 35 
 
 601 
 
 603 
 
 467 
 
 469 
 
 861 
 
 764 
 
 996 
 
 850 
 
 1836 B 
 
 10 3 8 35.2 
 
 998 
 
 015 
 
 016 
 
 775 
 
 775 
 
 941 
 
 703 
 
 182 
 
 127 
 
 1837 
 
 102231 28.1 
 
 1 1049 
 
 463 
 
 468 
 
 111 
 
 116 
 
 057 
 
 648 
 
 409 
 
 437 
 
 1838 
 
 11 1151 10.4 
 
 5 1 21 
 
 876 
 
 880 
 
 419 
 
 423 
 
 137 
 
 588 
 
 595 
 
 714 
 
 1839 
 
 1 10 52.6 
 
 8 21 53 
 
 290 
 
 292 
 
 726 
 
 729 
 
 217 
 
 527 
 
 781 
 
 991 
 
 1840 B 
 
 20 30 34.9 
 
 1225 
 
 704 
 
 705 
 
 034 
 
 035 
 
 296 
 
 466 
 
 967 
 
 268 
 
 1841 
 
 1 9 53 27.7 
 
 4 14 6 
 
 152 
 
 157 
 
 370 
 
 375 
 
 412 
 
 411 
 
 194 
 
 578 
 
 1842 
 
 1 29 13 10.0 
 
 8 438 
 
 566 
 
 569 
 
 678 
 
 682 
 
 492 
 
 350 
 
 380 
 
 855 
 
 1843 
 
 2 18 32 52.2 
 
 11 25 10 
 
 980 
 
 980 
 
 986 
 
 988 
 
 572 
 
 290 
 
 566 
 
 131 
 
 1844 B 
 
 3 7 52 34.5 
 
 3 1542 
 
 393 
 
 394 
 
 293 
 
 294 
 
 651 
 
 229 
 
 752 
 
 408 
 
 1845 
 
 3 27 15 27.4 
 
 7 1723 
 
 840 
 
 846 
 
 629 
 
 634 
 
 767 
 
 174 
 
 979 
 
 718 
 
 1846 
 
 41635 9.6 
 
 11 7*5 
 
 254 
 
 258 
 
 937 
 
 941 
 
 847 
 
 113 
 
 165 
 
 995 
 
 1847 
 
 5 55451.8 
 
 22827 
 
 668 
 
 670 
 
 245 
 
 247 
 
 927 
 
 053 
 
 351 
 
 272 
 
 1848 B 
 
 525 1434.1 
 
 6 18 59 
 
 082 
 
 083 
 
 553 
 
 553 
 
 006 
 
 992 
 
 537 
 
 549 
 
 1849 
 
 6 14 37 27.0 
 
 10 20 40 
 
 531 
 
 535 
 
 889 
 
 893 
 
 122 
 
 937 
 
 764 
 
 859 
 
 1850 
 
 7 357 9.2 
 
 211 12 
 
 944 
 
 947 
 
 196 
 
 200 
 
 202 
 
 876 
 
 950 
 
 136 
 
 1851 
 
 723 1651.5 
 
 6 144 
 
 358 
 
 359 
 
 504 
 
 506 
 
 282 
 
 816 
 
 136 
 
 413 
 
 1852 B 
 
 8 12 36 33.6 
 
 922 17 
 
 772 
 
 772 
 
 812 
 
 812 
 
 362 
 
 755 
 
 322 
 
 689 
 
 1853 
 
 9 1 59 26.5 
 
 1 2358 
 
 220 
 
 223 
 
 148 
 
 152 
 
 477 
 
 700 
 
 549 
 
 000 
 
 1854 
 
 9 21 19 8.8 
 
 5 1430 
 
 634 
 
 636 
 
 456 
 
 459 
 
 557 
 
 639 
 
 735 
 
 276 
 
 1855 
 
 10 103851.1 
 
 952 
 
 047 
 
 048 
 
 763 
 
 765 
 
 637 
 
 579 
 
 921 
 
 553 
 
 1856 B 
 
 10 29 58 33.3 
 
 02534 
 
 461 
 
 461 
 
 71 
 
 071 
 
 717 
 
 518 
 
 107 
 
 830 
 
 1857 
 
 11 192126.2 
 
 42715 
 
 909 
 
 912 
 
 407 
 
 411 
 
 832 
 
 463 
 
 334 
 
 140 
 
 1858 
 
 841 8.4 
 
 8 1747 
 
 323 
 
 325 
 
 715 
 
 718 
 
 912 
 
 402 
 
 520 
 
 417 
 
 1859 
 
 028 050.7 
 
 8 19 
 
 736 
 
 737 
 
 023 
 
 024 
 
 992 
 
 342 
 
 706 
 
 694 
 
 1860 B 
 
 1 17 20 32.9 
 
 32851 
 
 150 
 
 150 
 
 330 
 
 330 
 
 072 
 
 281 
 
 892 
 
 971 
 
 1861 
 
 2 64325.8 
 
 8 032 
 
 598 
 
 601 
 
 666 
 
 670 
 
 187 
 
 226 
 
 119 
 
 281 
 
 1862 
 
 226 3 8.0 
 
 11 21 4 
 
 012 
 
 014 
 
 974 
 
 977 
 
 267 
 
 165 
 
 305 
 
 558 
 
 1863 
 
 3 15 22 50.1 
 
 3 1136 
 
 426 
 
 426 
 
 282 
 
 283 
 
 347 
 
 105 
 
 491 
 
 834 
 
 1864 B 
 
 4 44232.3 
 
 728 
 
 839 
 
 839 
 
 590 
 
 589 
 
 427 
 
 044 
 
 677 
 
 111 
 
 1865 
 
 424 525.2 
 
 11 349 
 
 287 
 
 291 
 
 926 
 
 929 
 
 542 
 
 989 
 
 904 
 
 422 
 
 1866 
 
 5 13 25 7.3 
 
 22421 
 
 701 
 
 703 
 
 233 
 
 236 
 
 622 
 
 928 
 
 090 
 
 698 
 
 1867 
 
 6 24449.5 
 
 6 1453 
 
 115 
 
 115 
 
 541 
 
 542 
 
 702 
 
 868 
 
 276 
 
 975 
 
 1868 B 
 
 622 431.7 
 
 10 526 
 
 529 
 
 528 
 
 849 
 
 848 
 
 782 
 
 807 
 
 462 
 
 252 
 
 1869 
 
 7112724.6 
 
 277 
 
 977 
 
 980 
 
 185 
 
 188 
 
 897 
 
 752 
 
 689 
 
 562 
 
 1670 
 
 8 047 6.7 
 
 52739 
 
 390 
 
 392 
 
 493 
 
 495 
 
 977 
 
 691 
 
 875 
 
 839 
 
TABLE XXXVI. 
 
 Moon's Motions for Months. 
 
 Months. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 January 
 
 00000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 February 
 
 08487 
 
 0146 
 
 2246 
 
 8896 
 
 0402 
 
 1533 
 
 1789 
 
 2099 
 
 0753 
 
 175 
 
 965 184 
 
 059 
 
 Mo,. V, 5 Com ' 
 
 16153 
 
 8343 
 
 1371 
 
 6931 
 
 9797 
 
 1951 
 
 3404 
 
 3027 
 
 1433 
 
 139 
 
 836 
 
 157 
 
 016 
 
 March \ Bis. 
 
 16427 
 
 8993 
 
 2411 
 
 7218 
 
 0132 
 
 2323 
 
 3462 
 
 3418 
 
 1457 
 
 209 
 
 868 
 
 228 
 
 050 
 
 ., ( Com. 
 
 24640 
 
 8490 
 
 3616 
 
 5827 
 
 0199 
 
 3484 
 
 5193 
 
 5126 
 
 2186 
 
 314 
 
 801 
 
 342 
 
 076 
 
 April < T, 
 r < Bis. 
 
 24914 
 
 9140 
 
 4657 
 
 6114 
 
 0534 
 
 3856 
 
 5251 
 
 5517 
 
 2210 
 
 384 
 
 832 
 
 412 
 
 110 
 
 ,, ( Com. 
 
 32853 
 
 7986 
 
 4822 
 
 4436 
 
 0265 
 
 4646 
 
 6924 
 
 6835 
 
 2914 
 
 419 
 
 735 
 
 456 
 
 101 
 
 M *y isis. 
 
 33127 
 
 8636 
 
 5862'4723 
 
 0600 
 
 5018 
 
 6982 
 
 7226 
 
 2938 
 
 489 
 
 766 
 
 526 
 
 135 
 
 T ( Com. 
 
 41340 
 
 8133 
 
 7067 3332 
 
 0666 
 
 6179 
 
 8713 
 
 8934 
 
 3667 
 
 593 
 
 700 
 
 640 
 
 160 
 
 June < g: g 
 
 41614 
 
 8783 
 
 8107 
 
 3619 
 
 1002 
 
 6551 
 
 8771 
 
 93253691 
 
 663 
 
 731 
 
 710 
 
 194 
 
 T , ( Com. 
 
 49554 
 
 7629 
 
 8273 
 
 1942 
 
 0732 
 
 7341 
 
 0444 
 
 0643 ! 4396 
 
 698 
 
 634 
 
 754 
 
 185 
 
 Ju] y {Bis. 
 
 49828 
 
 8279 
 
 9313 
 
 2228 
 
 1068 
 
 7713 
 
 0502 
 
 10344420 
 
 768 
 
 665 
 
 824 
 
 219 
 
 . 5 Com. 
 
 58041 
 
 7776 
 
 05-18 
 
 0838 
 
 11348874 
 
 2233 
 
 2742 5148 
 
 873 
 
 599 
 
 938 
 
 245 
 
 A g- Us. 
 
 58315 
 
 8426 
 
 1558 
 
 1125 
 
 1470 
 
 9246 
 
 2290 
 
 31335173 
 
 943 
 
 630 
 
 009 
 
 279 
 
 . ( Com. 
 
 66528 
 
 7922 
 
 2764 
 
 9734 
 
 1536 
 
 0408 
 
 4021 
 
 48425901 
 
 048 
 
 563 
 
 123 
 
 304 
 
 S *&' ifiis. 
 
 66802 
 
 8572 
 
 3804 
 
 0021 
 
 1871 
 
 0780 
 
 4079 
 
 523215925 
 
 118 
 
 595 
 
 193 
 
 338 
 
 f*. < Com. 
 
 74741 
 
 7419 
 
 3969 
 
 8343 
 
 1602 
 
 1569 
 
 5752 
 
 6550 6630 
 
 152 
 
 497 
 
 237 
 
 329 
 
 UCt. < T> 
 
 75015 
 
 8069 
 
 5009 
 
 8630 
 
 1938 
 
 1941 
 
 5810 
 
 6941 
 
 6654 
 
 222 
 
 528 
 
 307 
 
 363 
 
 lyr ( Com. 
 
 83228 
 
 7565 
 
 6215 
 
 7239 
 
 2004 
 
 3102 
 
 7541 
 
 8649 
 
 7382 
 
 327 
 
 462 
 
 421 
 
 388 
 
 Nov - }Bis. 
 
 83502 
 
 8215 
 
 7255 
 
 7526 
 
 2339 
 
 3475 
 
 7599 
 
 9040 
 
 7407 
 
 397 
 
 493 
 
 492 
 
 423 
 
 TX < Com. 
 
 91442 
 
 7062 
 
 7420 
 
 5848 
 
 2070 
 
 4264 
 
 9272 
 
 03588111 
 
 432 
 
 396 
 
 535 
 
 414 
 
 \ Bis. 
 
 91716 
 
 7712 
 
 8460 
 
 6135 
 
 2405 
 
 4636 
 
 9330 
 
 0749 
 
 8135 
 
 502 
 
 427 
 
 606 
 
 448 
 
 TABLE XXXVI. 
 
 
 Moon's Motions for Months. 
 
 Months. 
 
 Erection. 
 
 Anomaly. | Variation. 
 
 Longitude. 
 
 January 
 
 0000 
 
 0.0 
 
 000 
 
 
 
 0.0 
 
 February 
 
 11 20 48 42 
 
 1 15 53.1 
 
 17 54 
 
 48 
 
 1 18 28 5.8 
 
 March J Com ' 
 
 10 7 40 26 
 
 1 20 50 4.2 
 
 11 29 15 
 
 15 
 
 1 27 24 26.6 
 
 n \ Bis. 
 
 10 18 59 26 
 
 2 3 53 58.2 
 
 11 26 
 
 42 
 
 2 10 35 1.6 
 
 ... ( Com. 
 
 9 28 29 8 
 
 3 5 50 57.3 
 
 17 10 
 
 3 
 
 3 15 52 32.5 
 
 AP 111 }Bis. 
 
 10 9 48 8 
 
 3 18 54 51.2 
 
 29 21 
 
 29 
 
 3 29 3 7.5 
 
 u J Com - 
 
 9 7 58 51 
 
 4 7 47 56.4 
 
 22 53 
 
 24 
 
 4 21 10 3.3 
 
 y < Bis. 
 
 9 19 17 50 
 
 4 20 51 50.3 
 
 1 5 4 
 
 50 
 
 5 4 20 38.3 
 
 June i Com ' 
 
 8 28 47 33 
 
 5 22 48 49.4 
 
 1 10 48 
 
 11 
 
 6 9 38 9.1 
 
 MlUt < } 
 
 9 10 6 33 
 
 6 5 52 43.4 
 
 1 22 59 
 
 38 
 
 6 22 48 44.1 
 
 T , ( Com. 
 
 8 8 17 16 
 
 6 24 45 48.5 
 
 1 16 31 
 
 32 
 
 7 14 55 39.9 
 
 Jul y iBis. 
 
 8 19 36 15 
 
 7 7 49 42.5 
 
 1 28 42 
 
 59 
 
 7 28 6 15.0 
 
 ( Com. 
 
 7 29 5 59 
 
 8 9 46 41.6 
 
 2 4 26 
 
 20 
 
 9 3 23 45.8 
 
 V S' \ Bis. 
 
 8 10 24 58 
 
 8 22 50 35.5 
 
 2 16 37 
 
 47 
 
 9 16 34 20.8 
 
 Sept $ C ? m- 
 
 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 J Com< 
 
 6 29 24 24 
 
 10 26 44 33.7 
 
 2 28 4 
 
 28 
 
 11 27 9 22.4 
 
 Uct ' \ Bis. 
 
 7 10 43 23 
 
 11 9 48 27.7 
 
 3 10 15 
 
 55 
 
 10 19 57.5 
 
 Nov J Com ' 
 
 6 20 13 6 
 
 11 45 26.8 
 
 3 15 59 
 
 16 
 
 1 15 37 28.3 
 
 llOV N -! 
 
 7 1 32 5 
 
 24 49 20.7 
 
 3 28 10 
 
 43 
 
 1 28 48 3.3 
 
 T* ( Com. 
 
 5 29 42 49 
 
 1 13 42 25.9 
 
 3 21 42 
 
 37 
 
 2 20 54 59.1 
 
 Dec. < T>- 
 
 6 11 1 48 
 
 1 26 46 19.8 
 
 4 3 54 
 
 4 
 
 345 34.1 
 
TABLE XXXVI. 
 Moon's Motions for Months. 
 
 53 
 
 Months. 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 22 
 
 24 
 
 25 
 
 2627 
 
 28 
 
 292 
 
 31 
 
 January 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000000 
 
 000 
 
 00 
 
 00 
 
 OC 
 
 00 
 
 30 
 
 00 00 
 
 00 
 
 00 <] 
 
 00 
 
 February f 
 
 074 
 
 946 
 
 135 
 
 304 
 
 805 066 
 
 014 24 
 
 26 
 
 14 
 
 82 
 
 28 
 
 14 
 
 17 
 
 29 
 
 96 C 
 
 507 
 
 Marrh J C m ' 
 
 851 
 
 801 
 
 159 
 
 482 
 
 532 
 
 125 
 
 |027 
 
 4550 
 
 96 
 
 57 
 
 13 
 
 18 
 
 12 
 
 46 
 
 821 
 
 15 
 
 March i Bis. 
 
 950 
 
 831 
 
 196 
 
 524 
 
 558 
 
 127 
 
 027 
 
 4651 
 
 06 
 
 59 
 
 17 
 
 21 
 
 19 
 
 51 
 
 851 
 
 15 
 
 ., ( Com. 
 A P nl iBis. 
 
 925 
 024 
 
 747 
 
 778 
 
 294 
 331 
 
 786 
 
 828 
 
 336 
 
 362 
 
 191 
 193 
 
 041 
 042 
 
 6877 
 6977 
 
 12 
 
 22 
 
 39 
 
 42 
 
 70 
 
 n 
 
 32 
 
 36 
 
 29 
 36 
 
 76 
 80 
 
 771 
 801 
 
 523 
 6 23 
 
 \/i * Com- 
 
 899 663 
 
 392 
 
 047 
 
 115 
 
 254 
 
 055 
 
 91 02 
 
 15 
 
 19 
 
 H 
 
 43 
 
 38 
 
 01 
 
 702 
 
 1 30 
 
 ivxav s Tj* 
 
 * ( JJlS. 
 
 999 693 
 
 429 
 
 089 
 
 141 256 
 
 055 
 
 92,03 
 
 26 
 
 22 
 
 )S 
 
 47 
 
 45,05 
 
 732 
 
 1 30 
 
 June \ C ? m< 
 
 973 
 
 609 
 
 527 
 
 351 
 
 920320 
 
 069 
 
 15 
 
 28 
 
 29 
 
 01 
 
 21 
 
 57 
 
 55 31 
 
 652 
 
 638 
 
 
 073 
 
 639 
 
 563 
 
 393 
 
 946 
 
 322 
 
 069 
 
 15 
 
 29 
 
 40 
 
 04 
 
 25 
 
 61 
 
 62 
 
 35 
 
 682 
 
 638 
 
 T , ( Com. 
 
 948 
 
 525 
 
 625 
 
 613 
 
 699 
 
 384 083 
 
 37 
 
 54 
 
 33 
 
 81 
 
 15 
 
 68 
 
 6456 
 
 583 
 
 1 45 
 
 y ( Bis. 
 
 047 
 
 555 
 
 661 
 
 655 
 
 725 
 
 386 083 
 
 38 
 
 55 
 
 43 
 
 84 
 
 1:972 
 
 71 
 
 60 
 
 613 
 
 146 
 
 . C Com. 
 
 022 
 
 471 
 
 759 
 
 917 
 
 503 
 
 449 097 
 
 61 
 
 80 
 
 47 
 
 64 
 
 
 82 
 
 81 
 
 85 
 
 533 
 
 653 
 
 Aug. < T,. 
 I DIS. 
 
 121 
 
 501 
 
 796 
 
 959 
 
 529 
 
 451 
 
 097 
 
 62 
 
 81 
 
 5? 
 
 66 
 
 r786 
 
 88 
 
 90 
 
 563 
 
 653 
 
 Q . ( Com. 
 
 096 
 
 417 
 
 894 
 
 221 
 
 308 515 
 
 111 
 
 85 
 
 07 
 
 61 
 
 46 
 
 )097 
 
 97 
 
 15 
 
 494 
 
 261 
 
 oept. < T- 
 
 195 
 
 447 
 
 931 
 
 263 
 
 334517 
 
 111 
 
 85 
 
 08 
 
 71 
 
 49 
 
 )401 
 
 04 
 
 19 
 
 524 
 
 261 
 
 Oct J Com - 
 
 071 
 
 333 
 
 992 
 
 483 
 
 087 578 
 
 125 
 
 07 
 
 32 
 
 65 
 
 26 
 
 23 08 
 
 07 
 
 40 
 
 414 
 
 768 
 
 1 Bis. 
 
 170 
 
 363 
 
 029 
 
 525 
 
 113 
 
 581 
 
 126 
 
 08 
 
 33 
 
 75 
 
 28 
 
 J8 
 
 11 
 
 14 
 
 44 
 
 444 
 
 769 
 
 Nov J Com ' 
 
 145 
 
 279 
 
 127 
 
 787 892 
 
 644 
 
 139 
 
 31 
 
 59 
 
 79 
 
 08 
 
 il 
 
 22 
 
 23 
 
 70 
 
 375 
 
 276 
 
 ( Bis. 
 
 244 
 
 309 
 
 163 
 
 829 
 
 918 646 
 
 140 
 
 32 
 
 60 
 
 89 
 
 11 
 
 )526 
 
 30 
 
 74 
 
 405 
 
 2|76 
 
 Dec ( Com. 
 
 120 
 
 194 
 
 225 
 
 049 
 
 670 708 
 
 153 
 
 54 
 
 85 
 
 83 
 
 88 
 
 r433 
 
 3395 
 
 295 
 
 784 
 
 
 219 
 
 225 261 
 
 091; 
 
 696 710 
 
 153 
 
 54 
 
 86 
 
 93 
 
 90 
 
 r9,37 
 
 40^99 
 
 325 
 
 7^84 
 
 TABLE 
 
 XXXVL 
 
 Moon's Motions for Months. \ ^. 
 
 Months. 
 
 Supp. of Node. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 
 o ' 
 
 s 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 January 
 
 0.0 
 
 
 
 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 February 
 
 1 38 29.7 
 
 11 15 43 
 
 054 
 
 224 
 
 875 
 
 045 
 
 111 
 
 165 
 
 290 
 
 043 
 
 March i Com - 
 
 037 27.5 
 
 9 27 59 
 
 007 
 
 330 
 
 666 
 
 989 
 
 114 
 
 313 
 
 455 
 
 984 
 
 \ Bis. 
 
 3 10 38.2 
 
 10 
 
 9 8 
 
 041 
 
 369 
 
 694 
 
 023 
 
 150 
 
 319 
 
 496 
 
 018 
 
 April j om - 
 
 4 45 57.3 
 
 9 13 42 
 
 061 
 
 554 
 
 542 
 
 034 
 
 225 
 
 478 
 
 745 
 
 027 
 
 
 4 49 7.9 
 
 9 24 51 095 
 
 593 
 
 570 
 
 068 
 
 261 
 
 484 
 
 787 
 
 061 
 
 VTav J Com - 
 
 6 21 16.4 
 
 8 18 15 081 
 
 738 
 
 389 
 
 046 
 
 300 
 
 638 
 
 993 
 
 036 
 
 jviay < T> 
 
 ( JJlS. 
 
 6 24 27.0 
 
 8 29 25 
 
 115 
 
 778 
 
 417 
 
 080 
 
 336 
 
 643 
 
 034 
 
 070 
 
 June \ Com - 
 
 7 59 46.1 
 
 8 
 
 3 58 
 
 136 
 
 962 
 
 264 
 
 091 
 
 411 
 
 802 
 
 282 
 
 079 
 
 
 082 56.7 
 
 8 15 8 
 
 170 
 
 002 
 
 293 
 
 124 
 
 447 
 
 808 
 
 324 
 
 113 
 
 j , ( Com. 
 
 9 35 5.2 
 
 7 
 
 8 32 
 
 156 
 
 147 
 
 112 
 
 103 
 
 486 
 
 962 
 
 531 
 
 088 
 
 July \ Bis. 
 
 9 38 15.9 
 
 7 19 41 190 
 
 186 
 
 140 
 
 136 
 
 522 
 
 967 
 
 572 
 
 122 
 
 Aug. j Com ' 
 
 11 13 35.0 
 
 6 24 15 210 371 
 
 987 
 
 147 
 
 597 
 
 126 
 
 820 
 
 131 
 
 
 11 16 45.6 
 
 7 
 
 5 24 
 
 244 
 
 411 
 
 015 
 
 182 
 
 6 
 
 33 
 
 132 
 
 862 
 
 164 
 
 ge fc ( Com. 
 
 12 52 4.7 
 
 6 
 
 9 58 
 
 265 
 
 595 
 
 862 
 
 193 
 
 708 
 
 291 
 
 110 
 
 173 
 
 ep ' ( Bis. 
 
 12 55 15.4 
 
 6 21 7 
 
 299 
 
 635 
 
 891 
 
 227 
 
 744 
 
 296 
 
 152 
 
 207 
 
 Oct \ Com ' 
 
 14 27 23.8 
 
 5 14 32 
 
 285 
 
 780 
 
 710 
 
 204 
 
 783 
 
 451 
 
 358 
 
 182 
 
 Uct ' \ Bis. 
 
 14 30 34.4 
 
 5 25 41 
 
 319 
 
 819 
 
 738 
 
 238 
 
 819 
 
 456 
 
 400 
 
 216 
 
 Nov $ Com - 
 
 16 5 53 5 
 
 5 
 
 15 
 
 339 
 
 004 
 
 585 
 
 250 
 
 894 
 
 615 
 
 648 
 
 225 
 
 }B; S . 
 
 16 9 4.2 
 
 5 11 24 
 
 373 
 
 043 
 
 613 
 
 283 
 
 930 
 
 621 
 
 690 
 
 259 
 
 j^ ( Com. 
 
 17 41 12.6 
 
 4 
 
 4 49 
 
 359 
 
 188 
 
 432 
 
 261 
 
 969 
 
 775 
 
 896 
 
 234 
 
 
 17 44 23.3 
 
 4 15 58 
 
 393 
 
 228 
 
 461 
 
 295 
 
 005 | 780 
 
 938 
 
 268 
 
TABLE XXXVII. 
 
 Moon's Motions for Days. 
 
 D. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 "1 
 13 
 
 1 
 
 00000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 0000 
 
 000 
 
 000 
 
 000 
 
 000 i 
 
 2 
 
 00274 
 
 0650 
 
 1040 
 
 0287 
 
 0336 
 
 0372 
 
 0058 
 
 0390 
 
 0024 
 
 070 
 
 031 
 
 070 
 
 034; 
 
 3 
 
 00548 
 
 1300 
 
 2080 
 
 0574 
 
 0671 
 
 0744 
 
 0115 
 
 0781 
 
 0049 
 
 140 
 
 062 
 
 141 
 
 068| 
 
 4 
 
 00821 
 
 1950 
 
 3121 
 
 0861 
 
 1007 
 
 1116 
 
 0173 
 
 1171 
 
 0073 
 
 210 
 
 093 
 
 211 
 
 103 
 
 5 
 
 01095 
 
 2600 
 
 4161 
 
 1148 
 
 1342 
 
 1488 
 
 0231 
 
 1561 
 
 0097 
 
 281 
 
 125 
 
 282 
 
 137 
 
 6 
 
 01369 
 
 3249 
 
 5201 
 
 1435 
 
 1678 
 
 1860 
 
 0289 
 
 1952 
 
 0121 
 
 351 
 
 156 
 
 352 
 
 171 
 
 7 
 
 01643 
 
 3899 
 
 6241 
 
 1722 
 
 2013 
 
 2232 
 
 0346 
 
 2342 
 
 0146 
 
 421 
 
 187 
 
 423 
 
 205 
 
 8 
 
 01916 
 
 4549 
 
 7281 
 
 2009 
 
 2349 
 
 2604 
 
 0404 
 
 2732 
 
 0170 
 
 491 
 
 218 
 
 493 
 
 239 
 
 9 
 
 02190 
 
 5199 
 
 8321 
 
 2296 
 
 2684 
 
 2976 
 
 0462 
 
 3122 
 
 0194 
 
 561 
 
 249 
 
 564 
 
 273 
 
 10 
 
 02464 
 
 5849 
 
 9362 
 
 2583 
 
 3020 
 
 3348 
 
 0519 
 
 3513 
 
 0219 
 
 631 
 
 280 
 
 634 
 
 308 
 
 11 
 
 02738 
 
 6499 
 
 0402 
 
 2870 
 
 3355 
 
 3720 
 
 0577 
 
 3903 
 
 0243 
 
 702 
 
 311 
 
 705 
 
 342 
 
 12 
 
 03012 
 
 7149 
 
 1442 
 
 3157 
 
 3691 
 
 4093 
 
 0635 
 
 4293 
 
 0267 
 
 772 
 
 342 
 
 775 
 
 376 
 
 13 
 
 03285 
 
 7799 
 
 2482 
 
 3444 
 
 4026 
 
 4465 
 
 0692 
 
 4684 
 
 0291 
 
 842 
 
 374 
 
 845 
 
 410 
 
 14 
 
 03559 
 
 8449 
 
 3522 
 
 3731 
 
 4362 
 
 4837 
 
 0750 
 
 5074 
 
 03161912 
 
 405 
 
 916 
 
 444 
 
 15 
 
 03833 
 
 9098 
 
 4563 
 
 4018 
 
 4698 
 
 5209 
 
 0808 
 
 5464 
 
 0340 982 
 
 436 
 
 986 
 
 478 
 
 16 
 
 04107 
 
 9748 
 
 5603 
 
 4305 
 
 5033 
 
 5581 
 
 0866 
 
 5854 
 
 0364 052 
 
 467 
 
 057 
 
 513 
 
 17 
 
 04380 
 
 0398 
 
 6643 
 
 4592 
 
 5369 
 
 5953 
 
 0923 
 
 6245 
 
 0389 122 
 
 498 
 
 127 
 
 547 
 
 18 
 
 04654 
 
 1048 
 
 7683 
 
 4878 
 
 5704 
 
 6325 
 
 0981 
 
 6635 
 
 0413 
 
 193 
 
 529 
 
 198 
 
 581 
 
 19 
 
 04928 
 
 1698 
 
 8723 
 
 5165 
 
 6040 
 
 6697 
 
 1039 
 
 7025 
 
 0437 
 
 263 
 
 560 
 
 268 
 
 315 
 
 20 
 
 05202 
 
 2348 
 
 9763 
 
 5452 
 
 6375 
 
 7069 
 
 1096 
 
 7416 
 
 0461 
 
 333 
 
 591 
 
 339 
 
 649 
 
 21 
 
 05476 
 
 2998 
 
 0804 
 
 5739 
 
 6711 
 
 7441 
 
 1154 
 
 7806 
 
 0486 
 
 403 
 
 623 
 
 409 
 
 683 
 
 22 
 
 05749 
 
 3648 
 
 1844 
 
 6026 
 
 7046 
 
 7813 
 
 1212 
 
 8196 
 
 0510 
 
 473 
 
 654 
 
 480 
 
 718 
 
 23 
 
 06023 
 
 4298 
 
 2884 
 
 6313 
 
 7382 
 
 8185 
 
 1269 
 
 8586 
 
 0534 
 
 543 
 
 685 
 
 550 
 
 752 
 
 24 
 
 06297 
 
 4947 
 
 3924 
 
 6600 
 
 7717 
 
 8557 
 
 1327 
 
 8977 
 
 0559 
 
 614 
 
 716 
 
 621 
 
 786 
 
 25 
 
 06571 
 
 5597 
 
 4964 
 
 6887 
 
 8053 
 
 8929 
 
 1385 
 
 9367 
 
 0583 
 
 684 
 
 747 
 
 691 
 
 820 
 
 26 
 
 06844 
 
 6247 
 
 6005 
 
 7174 
 
 8389 
 
 9301 
 
 1443 
 
 9757 
 
 0607 
 
 754 
 
 778 
 
 762 
 
 854 
 
 27 
 
 07118 
 
 6897 
 
 7045 
 
 7461 
 
 8724 
 
 9673 
 
 1500 
 
 0148 
 
 0631 
 
 824 
 
 809 
 
 832 
 
 888 
 
 28 
 
 07392 
 
 7547 8085 
 
 7748 
 
 9060 
 
 0045 
 
 1558 
 
 0538 
 
 0656 
 
 894 
 
 840 
 
 903 
 
 923 
 
 29 
 
 07666 
 
 8 1 47! 9125 
 
 8035 
 
 9395 
 
 0417 
 
 1616 0928 
 
 0680 
 
 964 
 
 872 
 
 973 
 
 957 
 
 30 
 
 07940 
 
 8847^0165 
 
 8322 
 
 9731 
 
 0789 
 
 1673I15T19 
 
 0704 
 
 034 
 
 903 
 
 043 
 
 991 ! 
 
 31 
 
 08213 
 
 9497 1205 8609 
 
 0066 
 
 1161 
 
 1731 [1709 
 
 0729 
 
 105 
 
 934 
 
 114 
 
 025 
 
TABLE XXXVII 
 
 55 
 
 Moon's Motion for Days. 
 
 D. 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 1 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 00 
 
 2 
 
 099 
 
 031 
 
 037 
 
 042 
 
 026 
 
 002 
 
 000 
 
 01 
 
 01 
 
 10 
 
 03 
 
 04 
 
 04 
 
 07 
 
 04 
 
 03 
 
 00 
 
 00 
 
 3 
 
 198 
 
 061 
 
 073 
 
 084 
 
 052 
 
 004 
 
 001 
 
 02 
 
 02 
 
 20 
 
 05 
 
 08 
 
 07 
 
 14 
 
 08 
 
 06 
 
 00 
 
 00 
 
 4 
 
 297 
 
 092 
 
 110 
 
 126 
 
 078 
 
 006 
 
 001 
 
 02 
 
 03 
 
 30 
 
 08 
 
 12 
 
 11 
 
 21 
 
 13 
 
 09 
 
 01 
 
 01 
 
 5 
 
 397 
 
 122 
 
 146 
 
 168 
 
 104 
 
 008 
 
 002 
 
 03 
 
 03 
 
 41 
 
 11 
 
 16 
 
 15 
 
 28 
 
 17 
 
 12 
 
 01 
 
 01 
 
 6 
 
 496 
 
 153 
 
 183 
 
 210 
 
 130 
 
 Oil 
 
 002 
 
 04 
 
 04 
 
 51 
 
 13 
 
 21 
 
 18 
 
 35 
 
 21 
 
 15 
 
 01 
 
 01 
 
 7 
 
 595 
 
 183 
 
 220 252 
 
 156 
 
 013 
 
 003 
 
 05 
 
 05 
 
 61 
 
 16 
 
 25 
 
 22 
 
 42 
 
 25 
 
 18 
 
 01 
 
 01 
 
 8 
 
 694 
 
 214 
 
 256 
 
 294 
 
 182 
 
 015 
 
 003 
 
 05 
 
 06 
 
 71 
 
 19 
 
 29 
 
 26 
 
 49 
 
 29 
 
 22 
 
 01 
 
 02 
 
 9 
 
 793 
 
 244 
 
 293 
 
 336 
 
 208 
 
 017 
 
 004 
 
 06 
 
 07 
 
 81 
 
 21 
 
 33 
 
 30 
 
 56 
 
 33 
 
 25 
 
 01 
 
 02 
 
 10 
 
 892 
 
 275 
 
 329 
 
 379 
 
 234 
 
 019 
 
 004 
 
 07 
 
 08 
 
 91 
 
 24 
 
 37 
 
 33 
 
 63 
 
 38 
 
 28 
 
 02 
 
 02 
 
 11 
 
 992 
 
 305 
 
 366 
 
 421 
 
 260 
 
 021 
 
 005 
 
 08 
 
 09 
 
 01 
 
 27 
 
 41 
 
 37 
 
 70 
 
 42 
 
 31 
 
 02 
 
 02 
 
 12 
 
 091 
 
 336 
 
 403 
 
 463 
 
 286 
 
 023 
 
 005 
 
 08 
 
 09 
 
 11 
 
 29 
 
 45 
 
 41 
 
 77 
 
 46 
 
 34 
 
 02 
 
 03 
 
 13 
 
 190 
 
 366 
 
 439 
 
 505 
 
 312 
 
 025 
 
 005 
 
 09 
 
 10 
 
 22 
 
 32 
 
 49 
 
 44 
 
 84 
 
 50 
 
 37 
 
 02 
 
 03 
 
 14 
 
 289 
 
 397 
 
 476 
 
 547 
 
 337 
 
 028 
 
 006 
 
 10 
 
 11 
 
 32 
 
 34 
 
 53 
 
 48 
 
 91 
 
 54 
 
 40 
 
 02 
 
 03 
 
 15 
 
 388 
 
 427 
 
 512 
 
 589 
 
 363 
 
 030 
 
 006 
 
 11 
 
 12 
 
 42 
 
 37 
 
 58 
 
 52 
 
 98 
 
 58 
 
 43 
 
 02 
 
 03 
 
 16 487 
 
 458 
 
 549 
 
 631 
 
 389 
 
 032 
 
 007 
 
 11 
 
 13 
 
 52 
 
 40 
 
 62 
 
 55 
 
 05 
 
 63 
 
 46 
 
 03 
 
 04 
 
 17 
 
 587 
 
 488 
 
 586 
 
 673 1 
 
 415 
 
 034 
 
 007 
 
 12 
 
 14 
 
 62 
 
 42 
 
 66 
 
 59 
 
 12 
 
 67 
 
 49 
 
 03 
 
 04 
 
 18 
 
 686 
 
 519 
 
 622 
 
 715 441 
 
 036 
 
 008 
 
 13 
 
 14 
 
 72 
 
 45 
 
 70 
 
 63 
 
 19 
 
 71 
 
 52 
 
 03 
 
 04 
 
 19 
 
 785 
 
 549 
 
 659 
 
 757 , 467 
 
 038 
 
 008 
 
 14 
 
 15 
 
 82 
 
 48 
 
 74 
 
 66 
 
 26 
 
 75 
 
 55 
 
 03 
 
 04 
 
 20 
 
 884 
 
 580 
 
 695 
 
 799 , 493 
 
 040 
 
 009 
 
 14 
 
 16 
 
 92 
 
 50 
 
 78 
 
 70 
 
 33 
 
 79 
 
 59 
 
 03 
 
 05 
 
 21 
 
 983 
 
 611 
 
 732 
 
 841 l 519 
 
 042 
 
 009 
 
 15 
 
 17 
 
 03 
 
 53 
 
 82 
 
 74 
 
 40 
 
 84 
 
 62 
 
 03 
 
 05 
 
 22 
 
 082 
 
 641 
 
 769 
 
 883 545 
 
 044 
 
 010 
 
 16 
 
 18 
 
 13 
 
 56 
 
 86 
 
 77 
 
 47 
 
 88 
 
 65 
 
 04 
 
 05 
 
 23 
 
 182 
 
 672 
 
 805 
 
 925 571 
 
 047 
 
 010 
 
 17 
 
 19 
 
 23 
 
 58 
 
 90 
 
 81 
 
 54 
 
 92 
 
 68 
 
 04 
 
 05 
 
 24 
 
 281 
 
 702 
 
 842 
 
 967 j 597 
 
 049 
 
 Oil 
 
 17 
 
 20 
 
 33 
 
 61 
 
 95 
 
 85 
 
 61 
 
 96 
 
 71 
 
 04 
 
 06 
 
 25 
 
 380 
 
 733 
 
 878 
 
 009 
 
 623 
 
 051 
 
 Oil 
 
 18 
 
 20 
 
 43 
 
 64 
 
 99 
 
 89 
 
 68 
 
 00 
 
 74 
 
 04 
 
 06. 
 
 26 
 
 479 
 
 763 
 
 915 
 
 052 
 
 649 
 
 053 
 
 Oil 
 
 19 
 
 21 
 
 53 
 
 66 
 
 03 
 
 92 
 
 75 
 
 04 
 
 77 
 
 04 
 
 06 
 
 27 
 
 578 
 
 794 
 
 952 
 
 094 675 
 
 055 
 
 012 
 
 20 
 
 22 
 
 63 
 
 69 
 
 07 
 
 90 
 
 82 
 
 09 
 
 80 
 
 04 
 
 06 
 
 28 
 
 677 
 
 824 
 
 988 
 
 136 701 
 
 057 
 
 012 
 
 20 
 
 23 
 
 73 
 
 72 
 
 11 
 
 00 
 
 89 
 
 13 
 
 83 
 
 05 
 
 06 
 
 29 
 
 777 
 
 855 
 
 025 
 
 178j727 
 
 059 
 
 013! 
 
 21 
 
 24 
 
 84 
 
 74 
 
 15 
 
 03 
 
 96 
 
 17 
 
 86 
 
 05 
 
 07 
 
 30 
 
 876 
 
 885 
 
 061 
 
 220 ; 753 
 
 061 
 
 013 
 
 22 
 
 25 
 
 94 
 
 77 
 
 19 
 
 07 
 
 03 
 
 21 
 
 89 
 
 05 
 
 07 
 
 31 
 
 975 
 
 916 
 
 098 
 
 262 | 779 
 
 064' 014 
 
 23 
 
 26 
 
 04 
 
 80 
 
 23 
 
 11 
 
 10 
 
 25 
 
 92 
 
 05 
 
 07 
 
56 
 
 TABLE XXXVII. 
 
 Moon's Motions for Days. 
 
 D. 
 
 Evection. 
 
 Anomaly. 
 
 Variation. 
 
 M. Longitude. 
 
 
 * ' 
 
 8 ' " 
 
 8 ' " 
 
 8 ' " 
 
 1 
 
 0000 
 
 00 
 
 0000 
 
 00 
 
 2 
 
 11 18 59 
 
 13 3 54.0 
 
 12 11 27 
 
 13 10 35.0 
 
 3 
 
 22 37 59 
 
 26 7 47.9 
 
 24 22 53 
 
 26 21 10.1 
 
 4 
 
 1 3 56 58 
 
 1 9 11 41.9 
 
 1 6 34 20 
 
 1 9 31 45.1 
 
 5 
 
 1 15 15 58 
 
 1 22 15 35.9 
 
 1 18 45 47 
 
 1 22 42 20.1 
 
 6 
 
 1 26 34 57 
 
 2 5 19 29.8 
 
 2 57 13 
 
 2 5 52 55.1 
 
 7 
 
 2 7 53 57 
 
 2 18 23 23.8 
 
 2 13 8 40 
 
 2 19 3 30.2 
 
 8 
 
 2 19 12 56 
 
 3 1 27 17.8 
 
 2 25 20 7 
 
 3 2 14 5.2 
 
 9 
 
 3 31 55 
 
 3 14 31 11.7 
 
 3 7 31 34 
 
 3 15 24 40.2 
 
 10 
 
 3 11 50 55 
 
 3 27 35 5.7 
 
 3 19 43 
 
 3 28 35 15.2 
 
 11 
 
 3 23 9 54 
 
 4 10 38 59.7 
 
 4 1 54 27 
 
 4 11 45 50.3 
 
 12 
 
 4 4 28 54 
 
 4 23 42 53.7 
 
 4 14 5 54 
 
 4 24 56 25.3 
 
 13 
 
 4 15 47 53 
 
 5 6 46 47.6 
 
 4 26 17 20 
 
 5 8 7 0.3 
 
 14 
 
 4 27 6 53 
 
 5 19 50 41.6 
 
 5 8 28 47 
 
 5 21 17 35.4 
 
 15 
 
 5 8 25 52 
 
 6 2 54 35.6 
 
 5 20 40 14 
 
 6 4 28 10.4 
 
 16 
 
 5 19 44 51 
 
 6 15 58 29.5 
 
 6 2 51 40 
 
 6 17 38 45.4 
 
 17 
 
 6 1 3 51 
 
 6 29 2 23.5 
 
 6 15 3 7 
 
 7 49 20.4 
 
 18 
 
 6 12 22 50 
 
 7 12 6 17.5 
 
 6 27 14 34 
 
 7 13 59 55.5 
 
 19 
 
 6 23 41 50 
 
 7 25 10 11.4 
 
 7 9 26 1 
 
 7 27 10 30.5 
 
 20 
 
 7 5 49 
 
 8 8 14 5.4 
 
 7 21 37 27 
 
 8 10 21 5.5 
 
 21 
 
 7 16 19 49 
 
 8 21 17 59.4 
 
 8 3 48 54 
 
 8 23 31 40.5 
 
 22 
 
 7 27 38 48 
 
 9 4 21 53.4 
 
 8 16 21 
 
 9 6 42 15.6 
 
 23 
 
 8 8 57 47 
 
 9 17 25 47.3 
 
 8 28 11 47 
 
 9 19 52 50.6 
 
 24 
 
 8 20 16 47 
 
 10 29 41.3 
 
 9 10 23 14 
 
 10 3 3 25.6 
 
 25 
 
 9 1 35 46 
 
 10 13 33 35.3 
 
 9 22 34 41 
 
 10 16 14 0.7 
 
 26 
 
 9 12 54 46 
 
 10 26 37 29.2 
 
 10 4 46 7 
 
 10 29 24 35.7 
 
 27 
 
 9 24 13 45 
 
 11 9 41 23.2 
 
 10 16 57 34 
 
 11 12 35 10.7 
 
 28 
 
 10 5 32 45 
 
 11 22 45 17.2 
 
 10 29 9 1 
 
 11 25 45 45.7 
 
 29 
 
 10 16 51 44 
 
 5 49 11.1 
 
 11 11 20 28 
 
 8 56 20.8 
 
 30 
 
 10 28 10 43 
 
 18 53 5.1 
 
 11 23 31 54 
 
 22 6 55.8 
 
 31 
 
 11 9 29 43 
 
 1 1 56 59.1 
 
 5 43 21 
 
 1 6 17 30.8 
 
TABLE. XXXVII. 
 
 57 
 
 Moon's Motions for Days. 
 
 D. 
 
 Supp. of Node. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 
 8 ' " 
 
 * ' 
 
 
 
 
 
 
 
 
 
 1 
 
 00 0.0 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 000 
 
 2 
 
 003 10.6 
 
 11 9 
 
 034 
 
 039 
 
 028 
 
 034 
 
 036 
 
 005 
 
 042 
 
 034 
 
 3 
 
 006 21.3 
 
 22 18 
 
 068 
 
 079 
 
 056 
 
 067 
 
 072 
 
 Oil 
 
 083 
 
 067 
 
 4 
 
 009 31.9 
 
 1 3 27 
 
 102 
 
 118 
 
 085 
 
 101 
 
 108 
 
 016 
 
 125 
 
 101 
 
 5 
 
 12 42.5 
 
 1 14 37 
 
 136 
 
 158 
 
 113 
 
 135 
 
 143 
 
 021 
 
 166 
 
 135 
 
 6 
 
 15 53.2 
 
 1 25 46 
 
 170 
 
 197 
 
 141 
 
 169 
 
 179 
 
 027 
 
 208 
 
 168 
 
 7 
 
 19 3.8 
 
 2 6 55 
 
 204 
 
 237 
 
 169 
 
 202 
 
 215 
 
 032 
 
 250 
 
 202 
 
 8 
 
 22 14.5 
 
 2 18 4 
 
 238 
 
 276 
 
 198 
 
 236 
 
 251 
 
 037 
 
 291 
 
 235 
 
 9 
 
 25 25.1 
 
 2 29 13 
 
 272 
 
 316 
 
 226 
 
 270 
 
 287 
 
 043 
 
 333 
 
 269 
 
 10 
 
 28 35.7 
 
 3 10 22 
 
 306 
 
 355 
 
 254 
 
 303 
 
 323 
 
 048 
 
 374 
 
 303 
 
 11 
 
 31 46.4 
 
 3 21 31 
 
 340 
 
 395 
 
 282 
 
 337 
 
 358 
 
 053 
 
 416 
 
 336 
 
 12 
 
 34 57.0 
 
 4 2 40 
 
 374 
 
 434 
 
 311 
 
 371 
 
 394 
 
 058 
 
 458 
 
 370 
 
 13 
 
 38 7.6 
 
 4 13 50 
 
 408 
 
 474 
 
 339 
 
 405 
 
 430 
 
 064 
 
 499 
 
 404 
 
 14 
 
 41 18.3 
 
 4 24 59 
 
 442 
 
 513 
 
 367 
 
 438 
 
 466 
 
 069 
 
 541 
 
 437 
 
 15 
 
 44 28.9 
 
 568 
 
 476 
 
 553 
 
 395 
 
 472 
 
 502 
 
 074 
 
 583 
 
 471 
 
 16 
 
 47 39.5 
 
 5 17 17 
 
 510 
 
 592 
 
 424 
 
 506 
 
 538 
 
 080 
 
 624 
 
 505 
 
 17 
 
 50 50.2 
 
 5 28 26 
 
 544 
 
 632 
 
 452 
 
 539 
 
 573 
 
 085 
 
 666 
 
 538 
 
 18 
 
 54 0.8 
 
 6 9 35 
 
 578 
 
 671 
 
 480 
 
 573 
 
 609 
 
 090 
 
 707 
 
 572 
 
 19 
 
 57 11.5 
 
 6 20 44 
 
 612 
 
 711 
 
 508 
 
 607 
 
 645 
 
 096 
 
 749 
 
 605 
 
 20 
 
 010 22.1 
 
 7 1 53 
 
 646 
 
 750 
 
 537 
 
 641 
 
 681* 
 
 101 
 
 791 
 
 639 
 
 21 
 
 013 32.7 
 
 7 13 3 
 
 680 
 
 790 
 
 565 
 
 674 
 
 717 
 
 106 
 
 832 
 
 673 
 
 22 
 
 1 6 43.4 
 
 7 24 12 
 
 714 
 
 829 
 
 593 
 
 708 
 
 753 
 
 112 
 
 874 
 
 706 
 
 23 
 
 9 54.0 
 
 8 5 21 
 
 748 
 
 869 
 
 621 
 
 742 
 
 788 
 
 117 
 
 915 
 
 740 
 
 24 
 
 13 4.6 
 
 8 16 30 
 
 782 
 
 908 
 
 650 
 
 775 
 
 824 
 
 122 
 
 957 
 
 774 
 
 25 
 
 16 15.3 
 
 8 27 39 
 
 816 
 
 948 
 
 678 
 
 809 
 
 860 
 
 128 
 
 999 
 
 807 
 
 26 
 
 19 25.9 
 
 9 8 48 
 
 850 
 
 987 
 
 706 
 
 843 
 
 896 
 
 133 
 
 040 
 
 841 
 
 27 
 
 82 36.5 
 
 9 19 57 
 
 884 
 
 027 
 
 734 
 
 877 
 
 932 
 
 138 
 
 082 
 
 875 
 
 28 
 
 25 47.2 
 
 10 1 6 
 
 918 
 
 066 
 
 762 
 
 910 
 
 968 
 
 143 
 
 123 
 
 908 
 
 29 
 
 28 57.8 
 
 10 12 16 
 
 952 
 
 106 
 
 791 
 
 944 
 
 003 
 
 149 
 
 165 
 
 942 
 
 30 
 
 1 32 8.5 
 
 10 23 25 
 
 986 
 
 145 
 
 819 
 
 978 
 
 039 
 
 154 
 
 207 
 
 975 
 
 31 
 
 1 35 19.1 
 
 11 4 34 
 
 020 | 
 
 185 
 
 847 
 
 Oil 
 
 075 
 
 159 
 
 248 
 
 009 / 
 
58 
 
 TABLE XXXVIII. 
 
 Maoris Motions for Hours. 
 
 H. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 1 
 
 11 
 
 27 
 
 43 
 
 12 
 
 14 
 
 16 
 
 2 
 
 16 
 
 1 
 
 3 
 
 1 
 
 3 
 
 1 
 
 2 
 
 23 
 
 54 
 
 87 
 
 24 
 
 28 
 
 31 
 
 5 
 
 33 
 
 2 
 
 6 
 
 3 
 
 6 
 
 3 
 
 3 
 
 34 
 
 81 
 
 130 
 
 36 
 
 42 
 
 47 
 
 7 
 
 49 
 
 3 
 
 9 
 
 4 
 
 9 
 
 4 
 
 4 
 
 46 
 
 108 
 
 173 
 
 48 
 
 56 
 
 62 
 
 10 
 
 65 
 
 4 
 
 12 
 
 5 
 
 12 
 
 6 
 
 5 
 
 57 
 
 135 
 
 217 
 
 60 
 
 70 
 
 78 
 
 12 
 
 81 
 
 5 
 
 15 
 
 6 
 
 15 
 
 7 
 
 6 
 
 68 
 
 162 
 
 260 
 
 72 
 
 84 
 
 93 
 
 14 
 
 98 
 
 6 
 
 18 
 
 8 
 
 18 
 
 9 
 
 7 
 
 80 
 
 190 
 
 303 
 
 84 
 
 98 
 
 109 
 
 17 
 
 114 
 
 7 
 
 20 
 
 9 
 
 20 
 
 10 
 
 8 
 
 91 
 
 217 
 
 347 
 
 96 
 
 112 
 
 124 
 
 19 
 
 130 
 
 8 
 
 23 
 
 10 
 
 23 
 
 11 
 
 9 
 
 103 
 
 244 
 
 390 
 
 108 
 
 126 
 
 140 
 
 22 
 
 146 
 
 9 
 
 26 
 
 12 
 
 26 
 
 13 
 
 10 
 
 114 
 
 271 
 
 433 
 
 120 
 
 140 
 
 155 
 
 24 
 
 163 
 
 10 
 
 29 
 
 13 
 
 29 
 
 14 
 
 11 
 
 125 
 
 298 
 
 477 
 
 131 
 
 154 
 
 171 
 
 26 
 
 179 
 
 11 
 
 32 
 
 14 
 
 00 
 
 16 
 
 12 
 
 137 
 
 325 
 
 520 
 
 143 
 
 168 
 
 186 
 
 29 
 
 195 
 
 12 
 
 35 
 
 16 
 
 35 
 
 17 
 
 13 
 
 148 
 
 352 
 
 563 
 
 155 
 
 182 
 
 202 
 
 31 
 
 211 
 
 13 
 
 38 
 
 17 
 
 38 ' 
 
 18 
 
 14 
 
 160 
 
 379 
 
 607 
 
 167 
 
 196 
 
 217 
 
 34 
 
 228 
 
 14 
 
 41 
 
 18 
 
 41 
 
 20 
 
 15 
 
 171 
 
 406 
 
 650 
 
 179 
 
 210 
 
 233 
 
 36 
 
 244 
 
 15 
 
 44 
 
 19 
 
 44 
 
 21 
 
 16 
 
 182 
 
 433 
 
 693 
 
 191 
 
 224 
 
 248 
 
 38 
 
 260 
 
 16 
 
 47 
 
 21 
 
 47 
 
 23 
 
 17 
 
 194 
 
 460 
 
 737 
 
 203 
 
 238 
 
 264 
 
 41 
 
 276 
 
 17 
 
 50 
 
 22 
 
 50 
 
 24 
 
 18 
 
 205 
 
 487 
 
 780 
 
 215 
 
 252 
 
 279 
 
 43 
 
 293 
 
 18 
 
 53 
 
 23 
 
 53 
 
 25 
 
 19 
 
 217 
 
 515 
 
 823 
 
 227 
 
 266 
 
 295 
 
 46 
 
 309 
 
 19 
 
 56 
 
 25 
 
 56 
 
 27 
 
 20 
 
 228 
 
 542 
 
 867 
 
 239 
 
 280 
 
 310 
 
 48 
 
 325 
 
 20 
 
 58 
 
 26 
 
 53 
 
 28 
 
 21 
 
 239 
 
 569 
 
 910 
 
 251 
 
 294 
 
 326 
 
 50 
 
 341 
 
 21 
 
 61 
 
 27 
 
 61 
 
 30 
 
 22 
 
 251 
 
 596 
 
 953 
 
 263 
 
 308 
 
 341 
 
 53 
 
 358 
 
 22 
 
 64 
 
 28 
 
 64 
 
 31 
 
 23 
 
 262 
 
 623 
 
 997 
 
 275 
 
 322 
 
 357 
 
 55 
 
 374 
 
 23 
 
 67 
 
 30 
 
 67 
 
 33 
 
 24 
 
 274 
 
 650 
 
 1040 
 
 287 
 
 336 
 
 372 
 
 58 
 
 390 
 
 24 
 
 70 
 
 31 
 
 70 
 
 34 
 
 
 
 
 Hours. 
 
 Evection. 
 
 Anomaly. 
 
 Variation. 
 
 Longitude. 
 
 
 
 ' " 
 
 ' // 
 
 O ' " 
 
 ' " * 
 
 
 
 1 
 
 28 17 
 
 32 39.7 
 
 30 29 
 
 32 56.5 
 
 
 
 2 
 
 56 35 
 
 1 5 19.5 
 
 1 57 
 
 1 5 52.9 
 
 
 
 3 
 
 1 24 52 
 
 1 37 59.2 
 
 1 31 26 
 
 1 38 49.4 
 
 
 
 4 
 
 1 53 10 
 
 2 10 39.0 
 
 2 1 54 
 
 2 11 45.8 
 
 
 
 5 
 
 2 21 27 
 
 2 43 18.7 
 
 2 32 23 
 
 2 44 42.3 
 
 
 
 6 
 
 2 49 45 
 
 3 15 58.5 
 
 3 2 52 
 
 3 17 38.8 
 
 
 
 7 
 
 3 18 2 
 
 3 48 38.2 
 
 3 33 20 
 
 3 50 35.2 
 
 
 
 8 
 
 3 46 20 
 
 4 21 18.0 
 
 4 3 49 
 
 4 23 31.7 
 
 
 
 9 
 
 4 14 37 
 
 4 53 57.7 
 
 4 34 17 
 
 4 56 28.1 
 
 
 
 10 
 
 4 42 55 
 
 5 26 37.5 
 
 5 4 46 
 
 5 29 24.6 
 
 
 
 11 
 
 5 11 12 
 
 5 59 17.2 
 
 5 35 15 
 
 6 2 21.0 
 
 
 
 12 
 
 5 39 30 
 
 6 31 57.0 
 
 6 5 43 
 
 6 35 17.5 
 
 
 
 13 
 
 6 7 47 
 
 7 4 36.7 
 
 6 36 12 
 
 7 8 14.0 
 
 
 
 14 
 
 6 36 5 
 
 7 37 16.5 
 
 7 6 40 
 
 7 41 10.4 
 
 
 
 15 
 
 7 4 22 
 
 8 9 56.2 
 
 7 37 9 
 
 8 14 6.9 
 
 
 
 16 
 
 7 32 40 
 
 8 42 36.0 
 
 8 7 38 
 
 8 47 3.4 
 
 
 
 17 
 
 8 57 
 
 9 15 15.7 
 
 8 38 6 
 
 9 19 59.8 
 
 
 
 18 
 
 8 29 15 
 
 9 47 55.5 
 
 9 8 35 
 
 9 52 56.3 
 
 
 
 19 
 
 8 57 32 
 
 10 20 35.2 
 
 9 39 3 
 
 10 25 52.7 
 
 
 
 20 
 
 9 25 50 
 
 10 53 15.0 
 
 10 9 32 
 
 10 58 49.2 
 
 
 
 21 
 
 9 54 7 
 
 11 25 54.7 
 
 10 40 1 
 
 11 31 45.6 
 
 
 
 22 
 
 10 22 24 
 
 11 58 34.5 
 
 11 10 29 
 
 12 4 42.1 
 
 
 
 23 
 
 10 50 42 
 
 12 31 14.2 
 
 11 40 58 
 
 12 37 38.6 
 
 
 
 24 
 
 11 18 59 
 
 13 3 54.0 
 
 12 11 27 
 
 13 10 35.0 
 
 
TABLE. XXXVIII. 
 
 59 
 
 Moon's Motions for Hours. 
 
 H. 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 || 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 1 
 
 4 
 
 1 
 
 2 
 
 2 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 8 
 
 3 
 
 3 
 
 4 
 
 2 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 3 
 
 12 
 
 4 
 
 5 
 
 5 
 
 3 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 4 
 
 16 
 
 5 
 
 6 
 
 7 
 
 4 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 1 
 
 1 
 
 1 
 
 
 1 
 
 5 
 
 21 
 
 6 
 
 8 
 
 9 
 
 5 
 
 
 
 
 
 
 
 
 
 2 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 6 
 
 25 
 
 8 
 
 9 
 
 11 
 
 6 
 
 
 
 
 
 
 
 
 
 3 
 
 1 
 
 1 
 
 1 
 
 2 
 
 
 
 7 
 
 29 
 
 9 
 
 11 
 
 12 
 
 8 
 
 1 
 
 
 
 
 
 
 
 3 
 
 1 
 
 1 
 
 1 
 
 2 
 
 
 
 8 
 
 33 
 
 10 
 
 12 
 
 14 
 
 9 
 
 1 
 
 
 
 
 
 
 
 3 
 
 1 
 
 1 
 
 | 
 
 2 
 
 
 
 9 
 
 37 
 
 11 
 
 14 
 
 16 
 
 10 
 
 j 1 
 
 
 
 
 
 
 
 4 
 
 1 
 
 2 
 
 1 
 
 3 
 
 1 
 
 
 10 
 
 41 
 
 13 
 
 15 
 
 18 
 
 11 
 
 1 
 
 
 
 
 
 
 
 4 
 
 i 
 
 2 
 
 2 
 
 3 
 
 2 
 
 
 11 
 
 45 
 
 14 
 
 17 
 
 19 
 
 12 
 
 1 
 
 
 
 
 
 
 
 5 
 
 i 
 
 2 
 
 2 
 
 3 
 
 2 
 
 
 12 
 
 49 
 
 15 
 
 18 
 
 21 
 
 13 
 
 1 
 
 
 
 
 
 
 
 5 
 
 i 
 
 2 
 
 2 
 
 3 
 
 2 
 
 2 
 
 13 
 
 54 
 
 16 
 
 20 
 
 23 
 
 14 
 
 1 
 
 
 
 
 
 
 
 5 
 
 i 
 
 2 
 
 2 
 
 4 
 
 2 
 
 2 
 
 14 
 
 58 
 
 18 
 
 21 
 
 25 
 
 15 
 
 1 
 
 
 
 
 
 
 
 6 
 
 2 
 
 2 
 
 2 
 
 4 
 
 2 
 
 2 
 
 15 
 
 62 
 
 19 
 
 23 
 
 26 
 
 16 
 
 1 
 
 
 
 
 
 
 
 6 
 
 2 
 
 3 
 
 2 
 
 4 
 
 3 
 
 2 
 
 16 
 
 66 
 
 20 
 
 25 
 
 28 
 
 17 
 
 1 
 
 
 
 1 
 
 1 
 
 7 
 
 2 
 
 3 
 
 2 
 
 5 
 
 3 
 
 2 
 
 17 
 
 70 
 
 21 
 
 26 
 
 30 
 
 18 
 
 1 
 
 
 
 1 
 
 
 7 
 
 2 
 
 3 
 
 1 
 
 5 
 
 3 
 
 2 
 
 18 
 
 74 
 
 23 
 
 28 32 
 
 19 
 
 2 
 
 
 
 1 
 
 
 8 
 
 2 
 
 3 
 
 3 
 
 5 
 
 3 
 
 2 
 
 19 
 
 78 
 
 24 
 
 29 33 
 
 21 
 
 2 
 
 
 
 1 
 
 
 8 
 
 2 
 
 3 
 
 3 
 
 6 
 
 3 
 
 3 
 
 20 
 
 83 
 
 25 
 
 31 35 
 
 22 
 
 2 
 
 
 
 1 
 
 
 8 
 
 2 
 
 3 
 
 3 
 
 6 
 
 3 
 
 3 
 
 21 
 
 87 
 
 26 
 
 32 37 
 
 23 
 
 2 
 
 
 
 1 
 
 
 9 
 
 2 
 
 4 
 
 3 
 
 6 
 
 4 
 
 3 
 
 22 
 
 91 
 
 28 
 
 34 39 
 
 24 
 
 2 
 
 
 
 1 
 
 
 9 
 
 2 
 
 4 
 
 3 
 
 6 
 
 4 
 
 3 
 
 23 
 
 95 29 
 
 35 
 
 40 
 
 25 
 
 2 
 
 
 
 1 
 
 
 10 
 
 3 
 
 4 
 
 4 
 
 7 
 
 4 
 
 3 
 
 24 
 
 99 
 
 31 
 
 37 
 
 42 
 
 26 
 
 2 
 
 1 
 
 
 10 
 
 3 
 
 4 
 
 4 
 
 7 
 
 4 
 
 3 
 
 
 
 
 H. 
 
 Sap. of Nod. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 
 
 , 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 7.9 
 
 28 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 2* 
 
 1 
 
 
 
 2 
 
 15.9 
 
 56 
 
 3 
 
 3 
 
 2 
 
 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 
 
 2 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 
 
 2 
 
 12 
 
 10 
 
 
 
 8 
 
 1 3.6 
 
 3 43 
 
 11 
 
 13 
 
 9 
 
 11 
 
 12 
 
 2 
 
 14 
 
 11 
 
 
 
 9 
 
 11.5 
 
 4 11 
 
 13 
 
 15 
 
 11 
 
 13 
 
 13 
 
 2 
 
 15 
 
 13 
 
 
 
 10 
 
 19.4 
 
 4 39 
 
 14 
 
 16 
 
 12 
 
 14 
 
 15 
 
 2 
 
 17 
 
 14 
 
 
 
 11 
 
 27.4 
 
 5 7 
 
 16 
 
 18 
 
 13 
 
 15 
 
 16 
 
 2 
 
 19 
 
 15 
 
 
 
 12 
 
 35.3 
 
 5 35 
 
 17 
 
 20 
 
 14 
 
 17 
 
 18 
 
 3 
 
 21 
 
 17 
 
 
 
 13 
 
 43.3 
 
 6 2 
 
 18 
 
 21 
 
 15 
 
 18 
 
 19 
 
 3 
 
 23 
 
 18 
 
 
 
 14 
 
 51.2 
 
 6 30 
 
 20 
 
 23 
 
 16 
 
 19 
 
 21 
 
 3 
 
 24 
 
 19 
 
 
 
 15 
 
 59.2 
 
 6 58 
 
 21 
 
 25 
 
 18 
 
 21 
 
 22 
 
 3 
 
 26 
 
 21 
 
 
 
 16 
 
 2 7.1 
 
 7 26 
 
 23 
 
 26 
 
 19 
 
 22 
 
 24 
 
 4 
 
 28 
 
 22 
 
 
 
 17 
 
 2 15.0 
 
 7 54 
 
 24 
 
 28 
 
 20 
 
 24 
 
 25 
 
 4 
 
 29 
 
 24 
 
 
 
 18 
 
 2 23.0 
 
 8 22 
 
 26 
 
 29 
 
 21 
 
 25 
 
 27 
 
 4 
 
 31 
 
 25 
 
 
 
 19 
 
 2 30.9 
 
 8 50 
 
 27 
 
 31 
 
 22 
 
 27 
 
 28 
 
 4 
 
 33 
 
 27 
 
 
 
 20 
 
 2 38.9 
 
 9 18 
 
 28 
 
 32 
 
 24 
 
 28 
 
 30 
 
 4 
 
 35 
 
 28 
 
 
 
 21 
 
 2 46.8 
 
 9 45 
 
 30 
 
 34 
 
 25 
 
 29 
 
 31 
 
 5 
 
 37 
 
 29 
 
 
 
 22 
 
 2 54.8 
 
 10 13 
 
 31 
 
 36 
 
 26 
 
 31 
 
 33 
 
 5 
 
 38 
 
 31 
 
 
 
 23 
 
 3 2.7 
 
 10 41 
 
 33 
 
 38 
 
 27 
 
 32 
 
 34 
 
 5 
 
 40 
 
 32 
 
 
 
 24 
 
 3 10.6 
 
 11 9 
 
 34 
 
 39 
 
 28 1 
 
 34 
 
 36 
 
 5 
 
 42 
 
 34 
 
 
TABLE XXXIX. 
 
 Moon's Motions for Minutes. 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 1 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 
 I 
 
 1 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 1 
 
 2 
 
 3 
 
 1 
 
 1 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 1 
 
 2 
 
 4 
 
 1 
 
 1. 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 1 
 
 3 
 
 4 
 
 1 
 
 1 
 
 2 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 1 
 
 3 
 
 5 
 
 1 
 
 2 
 
 2 
 
 
 
 2 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 2 
 
 4 
 
 6 
 
 2 
 
 2 
 
 2 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 9 
 
 2 
 
 4 
 
 6 
 
 2 
 
 2 
 
 2 
 
 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 2 
 
 5 
 
 7 
 
 2 
 
 2 
 
 3 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11 
 
 2 
 
 5 
 
 8 
 
 2 
 
 3 
 
 3 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12 
 
 2 
 
 5 
 
 9 
 
 2 
 
 3 
 
 3 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 13 
 
 2 
 
 6 
 
 9 
 
 3 
 
 3 
 
 3 
 
 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 14 
 
 3 
 
 6 
 
 10 
 
 3 
 
 3 
 
 4 
 
 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 15 
 
 3 
 
 7 
 
 11 
 
 3 
 
 3 
 
 4 
 
 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 16 
 
 3 
 
 7 
 
 12 
 
 3 
 
 4 
 
 4 
 
 1 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 17 
 
 3 
 
 8 
 
 12 
 
 3 
 
 4 
 
 4 
 
 1 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 18 
 
 3 
 
 8 
 
 13 
 
 4 
 
 4 
 
 5 
 
 1 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 19 
 
 4 
 
 9 
 
 14 
 
 4 
 
 4 
 
 5 
 
 1 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 20 
 
 4 
 
 9 
 
 14 
 
 4 
 
 5 
 
 5 
 
 1 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 21 
 
 4 
 
 10 
 
 15 
 
 4 
 
 5 
 
 5 
 
 1 
 
 6 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 1 
 
 
 
 22 
 
 4 
 
 10 
 
 16 
 
 4 
 
 5 
 
 6 
 
 1 
 
 6 
 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 1 
 
 1 
 
 
 
 23 
 
 4 
 
 10 
 
 17 
 
 5 
 
 5 
 
 6 
 
 1 
 
 6 
 
 
 
 
 
 
 
 1 
 
 2 
 
 
 
 1 
 
 1 
 
 
 
 24 
 
 5 
 
 11 
 
 17 
 
 5 
 
 6 
 
 6 
 
 1 
 
 7 
 
 
 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 25 
 
 5 
 
 11 
 
 18 
 
 5 
 
 6 
 
 6 
 
 1 
 
 7 
 
 
 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 26 
 
 5 
 
 12 
 
 19 
 
 5 
 
 6 
 
 7 
 
 1 
 
 7 
 
 
 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 27 
 
 5 
 
 12 
 
 19 
 
 5 
 
 6 
 
 7 
 
 1 
 
 7 
 
 
 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 28 
 
 5 
 
 13 
 
 20 
 
 6 
 
 7 
 
 7 
 
 1 
 
 8 
 
 
 
 
 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 29 
 
 6 
 
 13 
 
 21 
 
 6 
 
 7 
 
 7 
 
 1 
 
 8 
 
 
 
 1 
 
 
 1 
 
 I 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
 ,30 
 
 6 
 
 14 
 
 22 
 
 6 
 
 7 
 
 8 
 
 1 
 
 8 
 
 
 
 1 
 
 
 1 
 
 1 
 
 2 
 
 1 
 
 1 
 
 1 
 
 
 
TABLE XXXIX. 
 
 61 
 
 Moon's Motions for Minutes. 
 
 
 
 
 
 i 
 
 Sup. 
 
 
 
 
 
 
 
 
 
 Min. 
 
 Evec. 
 
 Anom. 
 
 Varia. 
 
 Long. 
 
 Nod. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 vm 
 
 IX 
 
 XI 
 
 XII 
 
 1 
 
 28 
 
 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 
 
 2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 2 2] 
 
 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 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 19 
 
 8 58 
 
 10 20.6 
 
 9 39 
 
 10 25.9 
 
 2.5 
 
 9 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 20 
 
 9 26 
 
 10 53.2 
 
 10 10 
 
 10 58.8 
 
 2.6 
 
 9 
 
 
 
 1 
 
 
 
 
 
 
 
 1 
 
 
 
 21 
 
 9 54 
 
 11 25.9 
 
 10 40 
 
 11 31.8 
 
 2.8 
 
 10 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 22 
 
 10 22 
 
 11 58.6 
 
 11 11 
 
 12 4.7 
 
 2.9 
 
 10 
 
 1 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 23 
 
 10 51 
 
 12 31.2 
 
 11 41 
 
 12 37.6 
 
 3.0 
 
 11 
 
 1 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 24 
 
 11 19 
 
 13 3.9 
 
 12 12 
 
 13 10.6 
 
 3.2 
 
 11 
 
 1 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 25 
 
 11 47 
 
 13 36.6 
 
 12 42 
 
 13 43.5 
 
 3.3 
 
 12 
 
 1 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 26 
 
 12 16 
 
 14 9.2 
 
 13 13 
 
 14 16.5 
 
 3.4 
 
 12 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 27 
 
 12 44 
 
 14 41.9 
 
 13 43 
 
 14 49.4 
 
 3.6 
 
 13 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 28 
 
 13 12 
 
 15 14.6 
 
 14 13 
 
 15 22.3 
 
 3.7 
 
 13 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 29 
 
 13 40 
 
 15 47.2 
 
 14 44 
 
 15 55.3 
 
 3.8 
 
 13 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
 30 
 
 14 9 
 
 16 19.9 
 
 15 14 
 
 16 28.2 
 
 4.0 
 
 14 
 
 
 
 1 
 
 
 1 
 
 1 
 
 
TABLE XXXIX. 
 
 Moon's Motions for Minutes. 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 ! 
 18 
 
 31 
 
 6 
 
 14 
 
 22 
 
 6 
 
 7 
 
 8 
 
 
 8 
 
 
 
 1 
 
 
 1 
 
 
 2 
 
 
 
 
 1 
 
 32 
 
 6 
 
 14 
 
 23 
 
 6 
 
 7 
 
 8 
 
 
 9 
 
 1 
 
 2 
 
 
 2 
 
 
 2 
 
 
 
 
 
 33 
 
 6 
 
 15 
 
 24 
 
 7 
 
 8 
 
 9 
 
 i 
 
 9 
 
 1 
 
 2 
 
 
 2 
 
 
 2 
 
 
 
 
 
 34 
 
 6 
 
 15 
 
 25 
 
 7 
 
 8 
 
 9 
 
 
 9 
 
 1 
 
 2 
 
 
 2 
 
 
 2 
 
 
 
 
 
 35 
 
 7 
 
 1C 
 
 25 
 
 7 
 
 8 
 
 9 
 
 
 10 
 
 1 
 
 2 
 
 
 2 
 
 
 2 
 
 
 
 
 
 30 
 
 7 
 
 16 
 
 26 
 
 7 
 
 8 
 
 9 
 
 1 
 
 10 
 
 1 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 37 
 
 7 
 
 17 
 
 27 
 
 7 
 
 9 
 
 10 
 
 1 
 
 10 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 38 
 
 7 
 
 17 
 
 27 
 
 8 
 
 9 
 
 10 
 
 2 
 
 10 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 39 
 
 7 
 
 18 
 
 28 
 
 8 
 
 9 
 
 10 
 
 2 
 
 11 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 40 
 
 8 
 
 18 
 
 29 
 
 8 
 
 9 
 
 10 
 
 2 
 
 11 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 41 
 
 8 
 
 19 
 
 30 
 
 8 
 
 10 
 
 11 
 
 2 
 
 11 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 42 
 
 8 
 
 19 
 
 30 
 
 8 
 
 10 
 
 11 
 
 2 
 
 11 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 43 
 
 8 
 
 19 
 
 31 
 
 9 
 
 10 
 
 11 
 
 o 
 
 12 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 44 
 
 8 
 
 20 
 
 32 
 
 9 
 
 10 
 
 11 
 
 2 
 
 12 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 45 
 
 9 
 
 20 
 
 32 
 
 9 
 
 10 
 
 12 
 
 2 
 
 12 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 46 
 
 9 
 
 21 
 
 33 
 
 9 
 
 11 
 
 12 
 
 2 
 
 12 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 47 
 
 9 
 
 21 
 
 34 
 
 9 
 
 11 
 
 12 
 
 2 
 
 13 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 48 
 
 9 
 
 22 
 
 35 
 
 10 
 
 11 
 
 12 
 
 2 
 
 13 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 49 
 
 9 
 
 22 
 
 35 
 
 10 
 
 11 
 
 13 
 
 2 
 
 13 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 50 
 
 9 
 
 23 
 
 36 
 
 10 
 
 11 
 
 13 
 
 2 
 
 13 
 
 
 2 
 
 
 2 
 
 
 3 
 
 
 
 
 
 51 
 
 10 
 
 23 
 
 37 
 
 10 
 
 12 
 
 13 
 
 2 
 
 14 
 
 
 2 
 
 
 2 
 
 
 4 
 
 
 
 
 
 52 
 
 10 
 
 24 
 
 38 
 
 10 
 
 12 
 
 13 
 
 2 
 
 14 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 
 
 
 53 
 
 10 
 
 24 
 
 38 
 
 11 
 
 12 
 
 14 
 
 2 
 
 14 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 
 
 
 54 
 
 10 
 
 24 
 
 39 
 
 11 
 
 12 
 
 14 
 
 2 
 
 14 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 
 2 
 
 
 55 
 
 10 
 
 25 
 
 40 
 
 11 
 
 13 
 
 14 
 
 2 
 
 15 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 1 
 
 2 
 
 
 56 
 
 11 
 
 25 
 
 40 
 
 11 
 
 13 
 
 14 
 
 2 
 
 15 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 1 
 
 2 
 
 
 57 
 
 11 
 
 26 
 
 41 
 
 11 
 
 13 
 
 15 
 
 2 
 
 15 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 1 
 
 2 
 
 
 58 
 
 11 
 
 26 
 
 42 
 
 12 
 
 13 
 
 15 
 
 2 
 
 16 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 2 
 
 2 
 
 
 59 
 
 11 
 
 27 
 
 43 
 
 12 
 
 14 
 
 15 
 
 2 
 
 16 
 
 
 3 
 
 
 3 
 
 
 4 
 
 
 2 
 
 2 
 
 j 
 
 60 
 
 11 
 
 27 
 
 43 
 
 12 
 
 14 
 
 15 
 
 2 
 
 16 
 
 1 
 
 3 
 
 
 3 
 
 
 4 1 
 
 
 2 
 
 2 
 
 
TABLE XXXIX. 
 Moorfs Motions for Minutes. 
 
 63 
 
 
 
 
 
 
 Sup. 
 
 
 
 
 
 
 
 
 
 Min 
 
 Evec. 
 
 Anom. 
 
 Varia. 
 
 Long. 
 
 Nod. 
 
 II 
 
 V 
 
 VI 
 
 VII 
 
 vm 
 
 IX 
 
 XI 
 
 XII 
 
 31 
 
 14 37 
 
 16 52.5 
 
 15 45 
 
 17 1.2 
 
 4.1 
 
 14 
 
 1 
 
 1 
 
 
 i 
 
 
 1 
 
 
 32 
 
 15 5 
 
 17 25.2 
 
 16 15 
 
 17 34.1 
 
 4.2 
 
 15 
 
 1 
 
 1 
 
 
 i 
 
 
 1 
 
 
 33 
 
 15 34 
 
 17 57.9 
 
 16 46 
 
 18 7.1 
 
 4.4 
 
 15 
 
 1 
 
 1 
 
 
 i 
 
 
 1 
 
 
 34 
 
 16 2 
 
 18 30.5 
 
 17 16 
 
 18 40.0 
 
 4.5 
 
 16 
 
 1 
 
 1 
 
 
 i 
 
 
 1 
 
 
 35 
 
 16 30 
 
 19 3.2 
 
 17 47 
 
 19 12.9 
 
 4.7 
 
 16 
 
 1 
 
 1 
 
 
 i 
 
 
 1 
 
 
 36 
 
 16 58 
 
 19 35.8 
 
 18 17 
 
 19 45.9 
 
 4.8 
 
 17 
 
 1 
 
 1 
 
 
 
 i 
 
 
 
 37 
 
 17 27 
 
 20 8.5 
 
 18 48 
 
 20 18.8 
 
 4.9 
 
 17 
 
 1 
 
 1 
 
 
 
 i 
 
 
 
 38 
 
 17 55 
 
 20 41.2 
 
 19 18 
 
 20 51.8 
 
 5.0 
 
 18 
 
 1 
 
 1 
 
 
 
 i 
 
 
 
 39 
 
 18 23 
 
 21 13.8 
 
 19 49 
 
 21 24.7 
 
 5.2 
 
 18 
 
 1 
 
 1 
 
 
 
 i 
 
 
 
 40 
 
 18 52 
 
 21 46.5 
 
 20 19 
 
 21 57.6 
 
 5.3 
 
 19 
 
 1 
 
 1 
 
 
 
 i 
 
 
 
 41 
 
 19 20 
 
 22 19.2 
 
 20 50 
 
 22 30.6 
 
 5.4 
 
 19 
 
 1 
 
 
 
 
 i 
 
 
 
 42 
 
 19 48 
 
 22 51.8 
 
 21 20 
 
 23 3.5 
 
 5.6 
 
 20 
 
 1 
 
 
 
 
 i 
 
 
 
 43 
 
 20 16 
 
 23 24.5 
 
 21 51 
 
 23 36.5 
 
 5.7 
 
 20 
 
 1 
 
 
 
 
 i 
 
 
 
 44 
 
 20 45 
 
 23 57.1 
 
 22 21 
 
 24 9.4 
 
 5.8 
 
 21 
 
 1 
 
 
 
 
 i 
 
 
 
 45 
 
 21 13 
 
 24 29.8 
 
 22 52 
 
 24 42.3 
 
 6.0 
 
 21 
 
 1 
 
 
 
 i 
 
 i 
 
 
 
 46 
 
 21 41 
 
 25 2.5 
 
 23 22 
 
 25 15.3 
 
 6.1 
 
 21 
 
 1 
 
 
 
 i 
 
 
 
 
 47 
 
 22 10 
 
 25 35.1 
 
 23 53 
 
 25 48.2 
 
 6.2 
 
 22 
 
 1 
 
 
 
 i 
 
 
 
 
 48 
 
 22 38 
 
 26 7.8 
 
 24 23 
 
 26 21.2 
 
 6.4 
 
 22 
 
 1 
 
 
 
 i 
 
 
 
 
 49 
 
 23 6 
 
 26 40.5 
 
 24 54 
 
 26 54.1 
 
 6.5 
 
 23 
 
 1 
 
 
 
 i 
 
 
 
 
 50 
 
 23 34 
 
 27 13.1: 
 
 25 24 
 
 27 27.0 
 
 6.6 
 
 23 
 
 1 
 
 
 
 i 
 
 
 
 
 51 
 
 24 3 
 
 27 45.8 
 
 25 55 
 
 28 0.0 
 
 6.8 
 
 24 
 
 1 
 
 
 
 i 
 
 
 
 
 52 
 
 24 31 
 
 28 18.5 
 
 26 25 
 
 28 32.9 
 
 6.9 
 
 24 
 
 1 
 
 
 
 i 
 
 
 
 
 53 
 
 24 59 
 
 28 51.1 
 
 26 56 
 
 29 5.9 
 
 7.0 
 
 25 
 
 1 
 
 
 1 
 
 i 
 
 
 
 
 54 
 
 25 28 
 
 29 23.8 
 
 27 26 
 
 29 38.8 
 
 7.1 
 
 25 
 
 1 
 
 
 1 
 
 i 
 
 
 2 
 
 
 55 
 
 25 56 
 
 29 56.4 
 
 27 56 
 
 30 11.8 
 
 7.3 
 
 26 
 
 1 
 
 
 1 
 
 i 
 
 
 2 
 
 
 56 
 
 2f> 24 
 
 30 29.1 
 
 28 27 
 
 30 44.7 
 
 7.4 
 
 26 
 
 
 1 
 
 1 
 
 i 
 
 
 2 
 
 
 57 
 
 26 52 
 
 31 1.8 
 
 28 57 
 
 31 17.6 
 
 7.5 
 
 27 
 
 
 2 
 
 1 
 
 i 
 
 
 2 
 
 
 58 
 
 27 21 
 
 31 34.4 
 
 29 28 
 
 31 506 
 
 7.7 
 
 27 
 
 
 2 
 
 1 
 
 i 
 
 
 2 
 
 
 59 
 
 27 49 
 
 32 7.1 
 
 29 58 
 
 32 &I..5 
 
 7.8 
 
 28 
 
 
 2 
 
 1 
 
 i 
 
 
 2 
 
 
 60 
 
 28 17 
 
 32 39.8 
 
 SO 29 
 
 32 56.5 
 
 7.9 
 
 28 
 
 
 2 
 
 1 
 
 i 
 
 
 2 
 
 1 
 
; .1 
 64 
 
 I 
 
 TABLE XL. 
 
 Moon's Motions for Seconds. 
 
 Sec. 
 
 Evec. 
 
 Anom. 
 
 Var. 
 
 Long. 
 
 Sec. 
 
 Evec. 
 
 Anom. 
 
 ,Var. 
 
 Long. 
 
 1 
 
 
 
 0.5 
 
 1 
 
 0.5 
 
 31 
 
 15 
 
 16.9 
 
 16 
 
 17.0 
 
 2 
 
 1 
 
 1.1 
 
 1 
 
 1.1 
 
 32 
 
 15 
 
 17.4 
 
 16 
 
 17.6 
 
 3 
 
 1 
 
 1.6 
 
 2 
 
 1.6 
 
 33 
 
 16 
 
 18.0 
 
 17 
 
 J8.1 
 
 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 
 
 5.4 
 
 5 
 
 5.5 
 
 40 
 
 19 
 
 21.8 
 
 20 
 
 22.0 
 
 11 
 
 5 
 
 6.0 
 
 6 
 
 6.0 
 
 41 
 
 19 
 
 22.3 
 
 21 
 
 22.5 
 
 12 
 
 6 
 
 6.5 
 
 6 
 
 6.6 
 
 42 
 
 20 
 
 22.9 
 
 21 
 
 23.1 
 
 13 
 
 6 
 
 7.1 
 
 7 
 
 7.1 
 
 43 
 
 20 
 
 23.4 
 
 22 
 
 23.6 
 
 14 
 
 7 
 
 7.6 
 
 7 
 
 7.7 
 
 44 
 
 21 
 
 24.0 
 
 22 
 
 24.2 
 
 15 
 
 7 
 
 8.2 
 
 8 
 
 8.2 
 
 45 
 
 21 
 
 24.5 
 
 23 
 
 24.7 
 
 16 
 
 8 
 
 8.7 
 
 8 
 
 8.8 
 
 46 
 
 22 
 
 25.0 
 
 23 
 
 25.3 
 
 17 
 
 8 
 
 9.2 
 
 9 
 
 9.3 
 
 47 
 
 22 
 
 25.6 
 
 24 
 
 25.8 
 
 18 
 
 9 
 
 9.8 
 
 9 
 
 9.9 
 
 48 
 
 23 
 
 26.1 
 
 24 
 
 26.4 
 
 19 
 
 9 
 
 10.3 
 
 10 
 
 10.4 
 
 49 
 
 23 
 
 26.7 
 
 25 
 
 26.9 
 
 20 
 
 9 
 
 10.9 
 
 10 
 
 11.0 
 
 50 
 
 24 
 
 27.2 
 
 25 
 
 27.4 
 
 21 
 
 10 
 
 11.4 
 
 11 
 
 11.5 
 
 51 
 
 24 
 
 27.8 
 
 26 
 
 28.0 
 
 22 
 
 10 
 
 12.0 
 
 11 
 
 12.1 
 
 52 
 
 25 
 
 28.3 
 
 26 
 
 28.5 
 
 23 
 
 11 
 
 12.5 
 
 12 
 
 12.6 
 
 53 
 
 25 
 
 28.9 
 
 27 
 
 29.1 
 
 24 
 
 11 
 
 13.1 
 
 12 
 
 13.2 
 
 54 
 
 26 
 
 29.4 
 
 27 
 
 29.6 
 
 25 
 
 12 
 
 13.6 
 
 13 
 
 13.7 
 
 55 
 
 26 
 
 29.9 
 
 28 
 
 30.2 
 
 26 
 
 12 
 
 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 
 
 154 
 
 58 
 
 27 
 
 31.6 
 
 29 
 
 31.8 
 
 29 
 
 14 
 
 15.8 
 
 15 
 
 15.9 
 
 5" 
 
 28 
 
 32.1 
 
 30 
 
 32.4 
 
 SO 
 
 14 
 
 16.3 
 
 15 
 
 16.5 
 
 60 
 
 28 
 
 32.7 
 
 30 
 
 32.9 j 
 
TABLE XLI, 65 
 
 First Equation of Moon's Longitude*^ Argument 1. 
 
 
 
 Diff. 
 
 
 
 Diff. 
 
 Diff. 
 
 
 Diff. 
 
 Arg. 
 
 1 
 
 for 10 
 
 Ar& 
 
 1 
 
 for 10 
 
 Arg, 
 
 1 
 
 for 10 
 
 Arg. 
 
 1 
 
 for 10 
 
 
 50 
 100 
 150 
 200 
 850 
 
 12 40.0 
 12 18.8 
 11 57.7 
 11 36.fi 
 11 15.6 
 10 &4.7 
 
 4.24 
 
 432 
 4.S2 
 480 
 4.1.8 
 4.16 
 
 8500 
 2550 
 SGOO 
 2050 
 2700 
 2750 
 
 1 40.7 
 1 41.5 
 
 1 42.9 
 1 45.0 
 1 47.7 
 1 51.C 
 
 16 
 
 0.28 
 0.42 
 0.54 
 0.66 
 0.80 
 
 5000 
 5050 
 5100 
 r >K>0 
 6200 
 5260 
 
 12 40.0 
 13 0.3 
 13 20.6 
 13 40.7 
 14 0.0 
 14 20.9 
 
 4.06 
 4.04 
 4.04 
 404 
 4.00 
 4.00 
 
 750023 39.3 
 7550 i 23 39.4 
 7BOO'23 38.9 
 765023 37.7 
 770023 35.8 
 7750*23 33.3 
 
 0.02 
 
 0.24 
 0.38 
 0.50 
 0.62 
 
 300 
 350 
 400 
 450 
 500 
 
 10 33.9 
 10 13.2 
 9 55.6 
 9 32.3 
 9 1S.1 
 
 4.14 
 413 
 4.06 
 4.04 
 4,00 
 
 2SOO 
 2850 
 2900 
 ^950 
 3000 
 
 1 55.0 
 1 59.6 
 
 U 4.8 
 2 10.7 
 2 17.1 
 
 0.92 
 1.04 
 
 1.28 
 1.42 
 
 ^300 
 5350 
 
 5450 
 5500 
 
 14 40.9 
 15 0.8 
 15 20.5 
 15 40.1 
 15 59.6 
 
 3.98 
 3.9-4 
 3.92 
 3.90 
 B,84 
 
 780023 30.2 
 785023 26.4 
 790023 22.0 
 795023 16.9 
 8000p 11.2 
 
 O.V6 
 0.88 
 1.02 
 1.14 
 1.26 
 
 550 
 600 
 650 
 
 8 52.1 
 8 32.4 
 8 13.0 
 
 3.94 
 
 3.88 
 
 3050 
 5100 
 
 2 24.2 
 2 3L9 
 2 40.1 
 
 i.M 
 
 1.64 
 
 5550 
 5600 
 
 16 18.8 
 16 56.7 
 
 3.80 
 3,78 
 
 8050 23 4.9 
 810022 57.9 
 815022 50.3 
 
 1.40 
 1.52 
 
 700 
 750 
 
 7 53.8 
 V 34.9 
 
 3.84 
 3.78 
 3.70 
 
 3200 
 
 2 48.9 
 2 58.8 
 
 1.76 
 L.88 
 1,99 
 
 5700 
 
 17 15.3 
 17 33.6 
 
 3.72 
 3.66 
 3.60 
 
 8200 
 
 8250 
 
 22 42.0 
 22 33.2 
 
 1.66 
 1.76 
 1.9.0 
 
 800 
 850 
 
 t 16.4 
 6 58.2 
 
 3.64 
 
 3300 
 1350 
 
 3 8.2 
 3 18.7 
 
 2,10 
 
 5800 
 5850 
 
 17 51.6 
 18 9.4 
 
 3.56 
 
 830022 23.7 
 835022 13.7 
 
 2-.00 
 
 900 
 950 
 1000 
 
 6 40.3 
 6 22.8 
 6 5.7 
 
 3.60 
 3.42 
 3.34 
 
 3400 
 3450 
 3-500 
 
 3 4L3 
 3 53.4 
 
 2.32 
 2.42 
 2.50 
 
 5900 
 5950 
 6000 
 
 18 26.9 
 18 44.0 
 19 0.8 
 
 3.50 
 3.42 
 3.36 
 3.28 
 
 840022 3.1 
 845021 51.9 
 8500 2-1- 40.1, 
 
 2.- 12 
 2.24 
 2.36 
 246 
 
 1050 
 
 150 
 ,200 
 1250 
 
 5 49.0 
 5 32.8 
 5 17.0 
 5 1.6 
 4 46.7 
 
 3.34 
 3,16 
 3.08 
 
 2.98 
 2.88 
 
 3550 
 3600 
 3650 
 3700 
 3750 
 
 4 5.9 
 4 19.0 
 4 32.5 
 4 46.5 
 5 0.9 
 
 2.62 
 2.70 
 
 2.80 
 2.88 
 2.98 
 
 6050 
 6100 
 6150 
 6200 
 6250 
 
 19 17.2 
 19 33.3 
 19 49. 
 20 4.2 
 80* 19.1 
 
 3.22 
 3.14- 
 3.04 
 2.98 
 
 2.88 
 
 8950 
 8600 
 S650 
 1 8700 
 8750 
 
 2J 278 
 21 15.0 
 21 1.6 
 20 47.7 
 20 33.3 
 
 2.56 
 
 2,68 
 2-. 78 
 2.88 
 2.98 
 
 1300 
 I3f)0 
 1400 
 1450 
 1500 
 
 4 32.3 
 4 18.4 
 4 5.0 
 3 52.2 
 3 39.9 
 
 2.78 
 2.68 
 2.56 
 2.46 
 2.36 
 
 3800 
 3850 
 3900 
 3950 
 4000 
 
 5 15.8 
 5 31.0 
 5 46.7 
 6 2.8 
 6 1-9. 
 
 3.04 
 
 3.14- 
 3.22- 
 3.38 
 3.36 
 
 6800 
 6350 
 6-400 
 6450 
 6500 
 
 2Q 33.5 
 20 47.5 
 21 1.0 
 21 14.1 
 21 26.6 
 
 2.80 
 2.70 
 2.62 
 2.50 
 2.42 
 
 880020 18.4 
 8850 20 3.0 
 8900 19 47.2 
 8950 19 31.0 
 9000 19 143 
 
 3.08 
 3.16 
 3.24 
 3.34 
 3.42 
 
 1550 
 
 3 28.1 
 
 2 24 
 
 4050 
 
 6- 36.0 
 
 o 40 
 
 6550 
 
 21 38.7 
 
 sy qo 
 
 9050 18 57.2 
 
 3KA 
 
 1600 
 1650 
 1700 
 1750 
 
 3 16.9 
 3 6.3 
 2 56.3 
 2 46.8 
 
 2.12. 
 2.00 
 1.90 
 1.78 
 
 4150 
 4200 
 4250 
 
 6 53.1 
 7 10.6- 
 
 7 28.4 
 7 46.4 
 
 350 
 3.56 
 3.60 
 3.66 
 
 6600 
 6650 
 16700 
 0750 
 
 21 50.3 
 22 1.3 
 
 22 11.8 
 22 21.7 
 
 2.20 
 2.10 
 1.98 
 1.88 
 
 9100 18 39.7 
 9150 18 21.8 
 9200 18 8.6 
 9250 IT 43: 1 
 
 3.58 
 3.84 
 3.70 
 3.78 
 
 1800 
 
 2 38.0 
 
 1 fift 
 
 430d 8 4.7 
 
 
 6800 
 
 22 31.1 
 
 7ft 
 
 9300 
 
 17 26.2 
 
 q QX 
 
 1850 
 1900 
 1950 
 2000 
 
 2 29.7 
 2 22.1 
 2 15.1 
 2 8.8 
 
 1.52: 
 1.40 
 1.26 
 1.14 
 
 4350 
 4400 
 4450 
 4500 
 
 8 23.3 
 
 8 42.2 
 9 1.2 
 9 20.4 
 
 3.78 
 3.80 
 3.84 
 3.90 
 
 685C 
 690C 
 G95C 
 70QC 
 
 22 39.9 
 
 22 48.1 
 22 55.8 
 
 23 2.9 
 
 1 
 
 .64 
 .54 
 
 .42 
 .28 
 
 9350 
 9400 
 9450 
 9500 
 
 17 7,0 
 16 47.6 
 16 27.9 
 16 7:9 
 
 '3.88 
 3.94 
 4.00 
 4.04 
 
 2050 
 2100 
 2130 
 2200 
 2230 
 
 2 3.1 
 58.0 
 53.6 
 49.8 
 46.7 
 
 1.02 
 0.88 
 0.76 
 0.62 
 0.50 
 
 4550 
 4600 
 4650 
 4700 
 475C 
 
 9 39.9 
 9 59.5 
 10 19.2 
 10 39.1 
 10 59.1 
 
 3.92 
 3.94 
 3.98 
 4.00 
 4.00 
 
 705023 9.3 
 7100J23 15.2 
 7130 23- 2#.4 
 720023 25.0 
 725023 29.0 
 
 .18 
 .04 
 0.92 
 0.80 
 0.66 
 
 9550 
 9600 
 9650 
 9700 
 9750 
 
 15 47:7 
 15,27.4 
 15 6.8 
 14 46.1 
 14 25:3 
 
 406 
 4.12 
 4,14 
 416 
 4.18 
 
 2300 44.2 
 2350! 42.3 
 24001 41.1 
 2450 40 fi 
 
 0.38 
 0.24 
 0.10 
 
 480C 
 485C 
 490C 
 495(] 
 
 11 19.1 
 11 39.3 
 11 59.S 
 12 19 1 
 
 4.04 
 4.04 
 4.04 
 
 7300 23 32.3 
 7350,23 35.0 
 740023 37.1 
 7450 23 38 5 
 
 0.54 
 0.42 
 
 0.28 
 
 9800 
 9850 
 9900 
 9950 
 
 14 4;4 
 13 43-4 
 |13. 2-2-. 3 
 
 1A- 1 9 
 
 4.22-; 
 
 4.22: 
 
 2500 40 7 0.02 
 
 5000 IS 40. ft ^ U U 
 
 750023 39.3 ' lb 
 
 1.0000 12 40,0 
 
 4.1& 
 
66 TABLE XLII. 
 
 Equations 2 to 7 of Moon's Longitude. Arguments 2 to 7 
 
 Arg. 
 
 2 
 
 diff 
 
 3 
 
 diff I 4 
 
 diff 
 
 5 
 
 diff 
 
 6 
 
 diff 
 
 7 
 
 diff 
 
 1 
 Arg. 
 
 
 / ,f 
 
 
 / // 
 
 
 / 
 
 
 , // 
 
 
 / // 
 
 
 / 
 
 
 
 2500 
 2600 
 2700 
 
 457.3 
 
 457.0 
 456.1 
 
 0.3 
 
 0.9 
 
 2.3 
 
 2.4 
 2.8 
 
 0.1 
 
 0.4 
 
 630.3 
 629.9 
 628.8 
 
 0.4 
 1.1 
 
 339.4 
 339.2 
 338.5 
 
 0.2 
 0.7 
 
 6.2 
 
 * 
 
 0.8 
 0.9 
 1.3 
 
 0.1 
 
 0.4 
 
 2500 
 2400 
 2300 
 
 2800 
 
 454.7 
 
 1.4 
 
 3.3 
 
 0.5 
 
 A O 
 
 626.9 
 
 1.9 
 
 3 37.5 
 
 1.0 
 
 j ^lU.O 
 
 1.8! 2200 
 
 2900 
 
 4 52.7 
 
 2.0 
 
 O ft 
 
 4.1 
 
 0.8 
 
 i n 
 
 624.3 
 
 2.6 
 
 q 
 
 336.0 
 
 1.5 
 
 1 o 
 
 8.8 ,'J 
 
 2.7 
 
 ?; 2100 
 
 3000 
 
 450.1 
 
 2 ' 6 5.1 
 
 l.U 
 
 621.0 
 
 o.o 
 
 334,1 
 
 i.y 
 
 010.3 
 
 
 3.7 
 
 IM 2000 
 
 
 
 3.1 
 
 
 1.3 
 
 
 4.1 
 
 
 2.4 
 
 
 1.8 
 
 
 1.3 
 
 
 3100 
 
 447.0 
 
 3 7 
 
 6.4 
 
 J.4 
 
 616.9 
 
 4,7 
 
 331.7 
 
 2 7 
 
 012.1 
 
 2.1 
 
 5.0 
 
 1.4 
 
 1900 
 
 3200 
 3300 
 3400 
 3500 
 
 443.3 
 439.1 
 434.4 
 
 4292 
 
 4.2 
 4.7 
 5.2 
 
 7.8 
 9.4 
 011.3 
 013.3 
 
 1.6 
 1.9 
 2.0 
 
 612.2 
 6 6.8 
 6 0.7 
 5 54,0 
 
 5.4 
 6.1 
 6.7 
 
 3 29.0 
 3 25.9 
 322.4 
 318.5 
 
 31 
 3.'5 
 3.9 
 
 014.2 
 016.6 
 019.2 
 022.2 
 
 2^4 
 2.6 
 3.0 
 
 6.4 
 8.1 
 010.0 
 012.1 
 
 1.7 
 1.9 
 2.1 
 
 1800 
 1700 
 1600 
 1500 
 
 
 
 5.7 
 
 
 2.2 
 
 
 7.4 
 
 
 4.2 
 
 
 3.2 
 
 
 2.3 
 
 
 3600 
 
 4235 
 
 6.1 
 
 15.5 
 
 2.4 
 
 546.6 
 
 7.9 
 
 314.3 
 
 4.6 
 
 025.4 
 
 3.5 
 
 014.4 
 
 2.4 
 
 1400 
 
 3700 
 3800 
 
 417.4 
 410.8 
 
 6.6 
 6.9 
 
 017.9 
 020.5 
 
 2.6 
 
 2.7 
 
 538.7 
 530.3 
 
 8.4 
 9.0 
 
 3 9.7 
 3 4.9 
 
 4.8 
 5.2 
 
 028.9 
 032.7 
 
 3^8 
 3.9 
 
 016.8 
 19.5 
 
 2.7 
 2.8 
 
 1300 
 1200 
 
 3900 
 
 4 3.9 
 
 7.3 
 
 023.2 
 
 
 521.3 
 
 Q 4. 
 
 259.7 
 
 5 4 
 
 36.6 
 
 A. * 
 
 022.3 
 
 2.9 
 
 1100 
 
 4000 
 
 356.6 
 
 7.7 
 
 026.1 
 
 3.0 
 
 511.9 
 
 y.*r 
 9.9 
 
 2543 
 
 5.7 
 
 40.7- 
 
 025.2 
 
 3.1 
 
 1000 
 
 4100 
 
 348.9 
 
 7.9 
 
 029.1 
 
 3.1 
 
 5 2.0 
 
 10.3 
 
 24S.P 
 
 5.9 
 
 045.1 
 
 4.5 
 
 028.3 
 
 3.2 
 
 900 
 
 4200 
 
 341.0 
 
 8.3 
 
 032.2 
 
 3.2 
 
 451.7 
 
 10 7 
 
 242.7 
 
 6.1 
 
 049.6 
 
 4.7 
 
 031 .6 
 
 3.3 
 
 800 
 
 4300 
 
 3 32.7 
 
 8.5 
 
 035.4 
 
 3.4 
 
 441.0 
 
 10.9 
 
 236.6 
 
 6.3 
 
 54.3 
 
 4.9 
 
 034.8 
 
 3.4 
 
 700 
 
 4400 
 
 324.2 
 
 8.7 
 
 038.8 
 
 3.4 
 
 430.1 
 
 Uo 
 
 230.3 
 
 6.5 
 
 9.2 
 
 4.9 
 
 038.2 
 
 3.5 
 
 600 
 
 4500 
 
 3 15.5 
 
 
 042.2 
 
 
 418.8 
 
 . .0 
 
 223.8 
 
 
 1 4.1 
 
 
 041.7 
 
 
 500 
 
 
 
 8.9 
 
 
 3.5 
 
 
 11.5 
 
 
 6.6 
 
 
 5.1 
 
 
 3.6 
 
 I 
 
 4600 
 
 3 6.6 
 
 9.0 
 
 045.7 
 
 3.5 
 
 4 7.3 
 
 11.6 
 
 217.2 
 
 6.7 
 
 1 9.2 
 
 5.1 
 
 45.3 
 
 3.6 
 
 400! 
 
 4700 
 
 257.6 
 
 9.1 
 
 049.2 
 
 3.6 
 
 3 55.7 
 
 11.8 
 
 2 10.5 
 
 3.8 
 
 1 143 
 
 5.2 
 
 048.9 
 
 3.7 
 
 300j 
 
 4800 
 4900 
 
 248.5 
 239.2 
 
 9.3 
 9.2 
 
 052.8 
 56.4 
 
 3.6 
 
 343.9 
 331.9 
 
 12.0 
 11.9 
 
 2 3.7 
 156.9 
 
 6.8 
 6.9 
 
 1 19.5 
 124.7 
 
 5.2 
 5.3 
 
 052.6 
 056.3 
 
 3.7 
 3.7 
 
 100! 
 
 5000 
 
 230.0 
 
 9.2 
 
 1 0.0 
 
 3.6 
 
 320.0 
 
 11.9 
 
 150.0 
 
 6.9 
 
 130.0 
 
 5.3 
 
 1 0.0 
 
 3.7 
 
 
 
 5100 
 
 220.8 
 
 93 
 
 1 3.6 
 
 3.6 
 
 3 8.1 
 
 12.0 
 
 43.1 
 
 6.8 
 
 135.3 
 
 5.2 
 
 1 3.7 
 
 3.7 
 
 9900 
 
 5200 
 
 211.5 
 
 9.1 
 
 1 7.2 
 
 3.6 
 
 256.1 
 
 11.8 
 
 36.3 
 
 6.8 
 
 1 40.5 
 
 5.2 
 
 1 7.4 
 
 3.7 
 
 9800 
 
 5300 
 
 2 2.4 
 
 9.0 
 
 1 10.8 
 
 3.5 
 
 44.3 
 
 11.6 
 
 29.5 
 
 6.7 
 
 1 45.7 
 
 5.1 
 
 1 11.1 
 
 3.6 
 
 9700 
 
 15400 
 !o500 
 
 L53.4 
 
 1 44.5 
 
 8.9 
 
 8.7 
 
 1 14.3 
 I 17.8 
 
 3.5 
 3.4 
 
 232.7 
 
 221.2 
 
 11.0 
 11.3 
 
 22.8 
 16.2 
 
 6.6 
 6.5 
 
 150.8 
 155.9 
 
 5.1 
 4.9 
 
 1 14 7 
 1 183 
 
 3.6 
 3.5 
 
 9600 
 9500 
 
 5600 
 5700 
 5800 
 5900 
 {6000 
 
 135.8 
 127.3 
 1 19.0 
 1 11.1 
 1 3.4 
 
 S.5 
 8.3 
 7.9 
 7.7 
 7.3 
 
 21.2 
 124.6 
 127.8 
 130.9 
 133.9 
 
 3.4 
 3.2 
 3.1 
 
 3.0 
 2.9 
 
 2 9.9 
 1 59.0 
 148.3 
 1 38.0 
 1 28.1 
 
 10,9 
 10.7 
 10.3 
 9.9 
 9.4 
 
 0.7 
 3.4 
 57.3 
 
 045.7 
 
 6.3 
 6.1 
 5.9 
 5.7 
 ,5.4 
 
 2 08 
 2 5.7 
 ,210.4 
 2149 
 2193 
 
 4.9 
 4.7 
 4,5 
 4.4 
 4.1 
 
 151.8 
 125.2 
 128.5 
 131.7 
 
 1 34.8 
 
 3.4 
 3.3 
 3.2 
 3.1 
 2.9 
 
 9400J 
 9300 
 9200 
 9100 
 9000 
 
 6100 
 .6200 
 j6300 
 6400 
 6500 
 
 056.1 
 049.2 
 042.6 
 36.5 
 030.8 
 
 6.9 
 6.6 
 6.1 
 5.7 
 
 136.8 
 139.5 
 142.1 
 144.5 
 146.7 
 
 2.7 
 2.6 
 14 
 
 2.2 
 
 1 18.7 
 1 9.7 
 1 1.3 
 53.4 
 46.0 
 
 9.0 
 
 8.4 
 7.9 
 7.4 
 
 40 3 ; 
 035.1 
 030.3 
 025.7 
 021.5 
 
 5.2 3 f' 4 
 
 1'fl 23L1 
 Jo 34-6 
 
 on 1337.8 
 
 3.9 
 3,8 
 3.5 
 3.2 
 
 1 37.7 
 1 40.5 
 143.2 
 145.6 
 
 147.9 
 
 B'JUU 
 *- 8800 
 
 o I I 8700 
 
 2-4 86 oo 
 23 8500 
 
 
 
 5.2 
 
 
 2.0 
 
 
 6.7 
 
 
 3.9 
 
 
 3.0 
 
 
 2.1 
 
 
 6600 25.6 
 6700 20.9 
 6800 16.7 
 6900 13.0 
 70000 9.9 
 
 47 
 
 4.2 
 37 
 3.1 
 
 148.7 
 1 50.6 
 162.2 
 153.6 
 1549 
 
 1.9 
 1.6 
 1.4 
 1.3 
 
 039.3 
 033.2 
 027.8 
 023.1 
 019.0 
 
 6.1 
 5.4 
 4.7 
 4.1 
 
 017.6 
 014.1 
 011.0 
 8.3 
 5.9 
 
 3.5 
 3.1 
 2.7 
 2.4 
 
 240.8 
 243.4 
 v> 45.8 
 247.9 
 249.7 
 
 2.6 
 2.4 
 
 2.1 
 1.8 
 
 150.0 
 I 51.9 
 153.6 
 155.0 
 1563 
 
 1.9 
 1.7 
 1.4 
 1.3 
 
 8400 
 8300 
 8200 
 8100 
 8000 
 
 
 2.6 
 
 
 1.0 
 
 
 3.3 
 
 
 1.9 
 
 
 1.5 
 
 
 1.0 
 
 
 71000 7.3 
 72000 5.3 
 7300'.0 3.9 
 74000 3.0 
 75000 2.7 
 
 2.0 
 1.4 
 0.9 
 0.3 
 
 155.9 
 
 1 jB7,2 
 157.6 
 1 &7.7 
 
 0.8 
 05 
 
 0.4 
 0.1 
 
 015.7 
 013.1 
 011.2 
 010.1 
 9.7 
 
 2.6 
 1.9 
 1.1 
 0.4 
 
 4.0 
 2.5 
 1.5 
 0.8 
 0.6 
 
 1.5 
 1.0 
 0.7 
 0.2 
 
 251.2 
 2 52.3 
 253.1 
 2 53.6 
 253.8 
 
 1.1 
 0.8 
 0.5 
 0.2 
 
 157.3 
 1 58.2 
 158.7 
 159.1 
 159.2 
 
 0.0 
 0.5 
 0.4 
 0.1 
 
 7900 
 7800 
 7700 
 7600 
 7500 
 
TABLE XLIIL 
 
 TABLE XLIV. C>7 
 
 Equations 8 and 9, 
 
 Equations 10 and 1J. 
 
 Arg. 
 
 8 
 
 9 
 
 Arg, 
 
 8 
 
 9 
 
 
 
 1 20.0 
 
 1 1200 
 
 5000 
 
 1 20.0 
 
 1 20.0 
 
 100 
 
 1 15.5 
 
 1 287 
 
 5100 
 
 1 24.4 
 
 1 25.8 
 
 200 
 
 1 11.1 
 
 1 37.3 
 
 5200 
 
 1 28.8 
 
 1 31.4 
 
 300 
 
 1 6.7 
 
 1 45.7 
 
 5300 
 
 1 33.1 
 
 1 36.9 
 
 400 
 
 1 2.3 
 
 1 53.7 
 
 5400 
 
 1 37.4 
 
 I 42.0 
 
 500 
 
 58.0 
 
 2 1.3 
 
 5500 
 
 1 41.6 
 
 1 46.8 
 
 600 
 
 53.8 
 
 2 8.3 
 
 5600 
 
 1 45.8 
 
 1 51.0 
 
 700 
 
 49.7 
 
 2 14.7 
 
 5700 
 
 1 49.8 
 
 1 54.6 
 
 800 
 
 45.7 
 
 2 20.2 
 
 5800 
 
 1 53.8 
 
 1 57.6 
 
 900 
 
 41.9 
 
 2 25.0 
 
 5900 
 
 1 57.6 
 
 1 59.8 
 
 1000 
 
 38.2 
 
 2 28.9 
 
 6000 
 
 2 1.2 
 
 2 1.3 
 
 1100 
 
 34.7 
 
 2 31.9 
 
 6100 
 
 2 4.7 
 
 2 1.9 
 
 1200 
 
 31.4 
 
 2 33.9 
 
 6200 
 
 2 8.0 
 
 2 1.7 
 
 1300 
 
 28.2 
 
 2 34.9 
 
 6300 
 
 2 11.2 
 
 2 0.7 
 
 1400 
 
 25.3 
 
 2 35.0 
 
 6400 
 
 2 14.1 
 
 1 58.8 
 
 1500 
 
 22.6 
 
 2 34 1 
 
 6500 
 
 2 16.8 
 
 1 56.1 
 
 1600 
 
 20.1 
 
 2 32 2 
 
 6600 
 
 2 19.3 
 
 1 52.5 
 
 1700 
 
 17.9 
 
 2 29.5 
 
 6700 
 
 2 21.6 
 
 1 48.3 
 
 1800 
 
 15.9 
 
 2 25.9 
 
 6800 
 
 2 23.7 
 
 1 43.4 
 
 1900 
 
 14.2 
 
 2 21.5 
 
 6900 
 
 2 25.4 
 
 1 37.8 
 
 2000 
 
 12.7 
 
 2 16.4 
 
 7000 
 
 2 27.0 
 
 1 31.7 
 
 2100 
 
 11.5 
 
 2 10.7 
 
 7100 
 
 2 28.2 
 
 1 25.1 
 
 2200 
 
 10.5 
 
 2 4.4 
 
 7200 
 
 2 29.2 
 
 1 18.2 
 
 2300 
 
 9.9 
 
 1 57.7 
 
 7300 
 
 2 30.0 
 
 1 11. 1 
 
 2400 
 
 9.5 
 
 1 50.7 
 
 7400 
 
 2 30.4 
 
 1 3.8 
 
 2500 
 
 9.4 
 
 1 43.5 
 
 7500 
 
 2 30.6 
 
 56.5 
 
 2600 
 
 9.6 
 
 1 36.2 
 
 7600 
 
 2 30.5 
 
 49.3 
 
 2700 
 
 10.1 
 
 1 28.9 
 
 7700 
 
 2 30.1 
 
 42.3 
 
 2800 
 
 10.8 
 
 1 21.8 
 
 7800 
 
 2 29.5 
 
 35.6 
 
 2900 
 
 11.8 
 
 1 14.9 
 
 7900 
 
 2 28.5 
 
 29.3 
 
 3000 
 
 13.0 
 
 1 8.3 
 
 8000 
 
 2 27.3 
 
 23.6 
 
 3100 
 
 14.6 
 
 1 2.2 
 
 8100 
 
 2 25.8 
 
 18.5 
 
 3200 
 
 16.3 
 
 56.6 
 
 8200 
 
 2 24.1 
 
 14.1 
 
 3300 
 
 18 4 
 
 51.7 
 
 8300 
 
 2 22.1 
 
 10.5 
 
 3400 
 
 20.7 
 
 47.5 
 
 8400 
 
 2 19.9 
 
 7.8 
 
 3500 
 
 23.2 
 
 43.9 
 
 8500 
 
 2 17.4 
 
 5.9 
 
 3600 
 
 25.9 
 
 41.2 
 
 8600 
 
 2 14.7 
 
 5.0 
 
 3700 
 
 28.8 
 
 39.3 
 
 8700 
 
 2 11.8 
 
 5.1 
 
 3800 
 
 32.0 
 
 38.3 
 
 8800 
 
 2 8.6 
 
 6.1 
 
 3900 
 
 35.3 
 
 38.1 
 
 8900 
 
 2 5.3 
 
 8.1 
 
 4000 
 
 38.8 
 
 38.7 
 
 9000 
 
 2 1.8 
 
 11.1 
 
 4100 
 
 42.4 
 
 40.2 
 
 9100 
 
 58.1 
 
 15.0 
 
 4200 
 
 46.2 
 
 42.4 
 
 9200 
 
 54.3 
 
 19.8 
 
 4300 
 
 50.2 
 
 45.4 
 
 9300 
 
 50.3 
 
 25.3 
 
 4400 
 
 54.2 
 
 49.0 
 
 9400 
 
 46.2 
 
 31.7 
 
 4500 
 
 58.4 
 
 53.2 
 
 9500 
 
 42.0 
 
 38.7 
 
 4600 
 
 1 26 
 
 58.0 
 
 9600 
 
 37.7 
 
 46.3 
 
 4700 
 
 1 6.9 
 
 1 3.1 
 
 9700 
 
 33.3 
 
 54.3 
 
 4800 
 
 1 11.2 
 
 1 8.6 
 
 9800 
 
 28.9 
 
 1 2.7 
 
 4900 
 
 1 15.6 
 
 1 14.2 
 
 9900 
 
 24.5 
 
 1 11.3 
 
 5000 
 
 1 20.0 
 
 1 20.0 
 
 10000 
 
 I 20.0 
 
 1 20.0 
 
 Arg. 
 
 10 
 
 11 
 
 Arg. 
 
 10 
 
 11 
 
 
 
 10.0 
 
 10.0 
 
 500 
 
 10.0 
 
 10.0 
 
 10 
 
 9.3 
 
 11.1 i 
 
 510 
 
 9.6 
 
 10.8 
 
 20 
 
 8.6 
 
 12.1 
 
 520 
 
 9.2 
 
 11.5 
 
 30 
 
 8.0 
 
 13.1 
 
 530 
 
 8.9 
 
 12.3 
 
 40 
 
 7.4 
 
 14.1 ! 
 
 540 
 
 8.5 
 
 12.9 
 
 50 
 
 6.8 
 
 15.0 
 
 550 
 
 8.2 
 
 13.6 
 
 60 
 
 6.2 
 
 15.8 
 
 560 
 
 7.9 
 
 14.2 
 
 70 
 
 5.7 
 
 16.6 
 
 570 
 
 7.7 
 
 14.6 
 
 80 
 
 5.3 
 
 17.3 
 
 580 
 
 7.5 
 
 15.0 
 
 90 
 
 4.9 
 
 17.9 
 
 590 
 
 7.4 
 
 15.4 
 
 100 
 
 4.6 
 
 18.3 ' 
 
 600 
 
 7.3 
 
 15.6 
 
 110 
 
 4.3 
 
 18.6 
 
 610 
 
 7.2 
 
 15.7 
 
 120 
 
 4.1 
 
 18.9 
 
 620 
 
 7.3 
 
 15.7 
 
 130 
 
 4.0 
 
 19.0 
 
 630 
 
 7.4 
 
 15.6 
 
 140 
 
 4.0 
 
 18.9 
 
 640 
 
 7.5 
 
 15.4 
 
 150 
 
 4.0 
 
 18.8 
 
 650 
 
 7.8 
 
 15.1 
 
 160 
 
 4.2 
 
 18.6 
 
 660 
 
 8.1 
 
 14.7 
 
 170 
 
 4.4 
 
 18.2 
 
 670 
 
 8.4 
 
 14.2 
 
 180 
 
 46 
 
 17.7 
 
 680 
 
 8.7 
 
 13.5 
 
 190 
 
 4.9 
 
 17.1 
 
 690 
 
 9.2 
 
 12.8 
 
 200 
 
 5.3 
 
 16.5 
 
 700 
 
 9.7 
 
 12.1 
 
 210 
 
 5.7 
 
 15.7 
 
 710 
 
 10.2 
 
 11.3 
 
 220 
 
 6.2 
 
 14.9 
 
 720 
 
 10.7 
 
 10.4 
 
 230 
 
 6.7 
 
 14.1 
 
 730 
 
 11.2 
 
 9.5 
 
 240 
 
 7.2 
 
 13.2 
 
 740 
 
 11.7 
 
 8.6 
 
 250 
 
 7.7 
 
 12.3 
 
 750 
 
 12.3 
 
 7.7 
 
 260 
 
 8.3 
 
 11.4 
 
 760 
 
 12.8 
 
 6.8 
 
 270 
 
 8.8 
 
 10.5 
 
 770 
 
 13.3 
 
 5.9 
 
 280 
 
 9.3 
 
 9.6 
 
 780 
 
 13.8 
 
 5.1 
 
 290 
 
 9.8 
 
 8.7 
 
 790 
 
 14.3 
 
 4.3 
 
 300 
 
 10.3 
 
 7.9 
 
 800 
 
 14.7 
 
 3.5 
 
 310 
 
 10.8 
 
 7.2 
 
 810 
 
 15.1 
 
 2.9 
 
 320 
 
 11.3 
 
 6.5 
 
 920 
 
 15.4 
 
 2.3 
 
 330 
 
 11.6 
 
 3.8 
 
 830 
 
 15.6 
 
 1.8 
 
 340 
 
 11.9 
 
 5.3 
 
 840 
 
 1&.8 
 
 1.4 
 
 350 
 
 12.2 
 
 4.9 
 
 850 
 
 16.0 
 
 1.2 
 
 360 
 
 12.5 
 
 4.6 
 
 860 
 
 16.0 
 
 1.1 
 
 370 
 
 12.6 
 
 4.4 
 
 1 870 
 
 16.0 
 
 1.0 
 
 380 
 
 12.7 
 
 4.3 
 
 880 
 
 15.9 
 
 1.1 
 
 390 
 
 12.8 
 
 4.3 
 
 890 
 
 15.7 
 
 1.4 
 
 400 
 
 12.7 
 
 4.4 
 
 900 
 
 15.4 
 
 1.7 
 
 410 
 
 12.6 
 
 4.6 
 
 910 
 
 15.1 
 
 2.1 
 
 420 
 
 12.5 
 
 5.0 
 
 920 
 
 14.7 
 
 2.7 
 
 430 
 
 12.3 
 
 5.4 
 
 930 
 
 14.3 
 
 3.4 
 
 440 
 
 12.1 
 
 5.8 
 
 940 
 
 13.8 
 
 4.8 
 
 450 
 
 11.8 
 
 6.4 
 
 950 
 
 13.2 
 
 5.0 
 
 460 
 
 11.5 
 
 7.1 
 
 960 
 
 12.6 5.9 
 
 470 
 
 11.1 
 
 7.7 i 970 
 
 12.0 6.9 
 
 480 
 
 10.8 
 
 8.5 i 980 
 
 11.4 7.9 
 
 490 
 
 10.4 
 
 9.2 j 910 
 
 10.7 8.9 
 
 500 
 
 10.0 
 
 10.0 ! 10( 
 
 100 10.0 
 
TABLE XLY. 
 Equations 12 to 19. 
 
 TABLE XLVI. 
 Equation 20. 
 
 Arg. 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 Arg. 
 
 250 
 
 2.3 
 
 1.6 
 
 7.8 
 
 0.0 
 
 33.7 
 
 3.4 
 
 16.7 
 
 0.4 
 
 250 
 
 260 
 
 2.3 
 
 1.6 
 
 7.8 
 
 0.0 
 
 33.7 
 
 3.4 
 
 16.7 
 
 0.4 
 
 240 
 
 270 
 
 2.4 
 
 1.7 
 
 7.9 
 
 0.1 
 
 33,6 
 
 3.5 
 
 16.6 
 
 0.4 
 
 230 
 
 280 
 
 2.6 
 
 1.9 
 
 8.0 
 
 0.2 
 
 33.5 
 
 3. 5 
 
 16.6 
 
 0.5 
 
 220 
 
 290 
 
 2.9 
 
 2.2 
 
 8.2 
 
 0.3 
 
 33v2 
 
 3.6 
 
 16.5 
 
 0.5 
 
 210 
 
 300 
 
 3.2 
 
 2.5 
 
 8.4 
 
 0.5. 
 
 33-. 
 
 3.7 
 
 16.4 
 
 0.6 
 
 200 
 
 310 
 
 3.5 
 
 2.9 
 
 8.7 
 
 0.7 
 
 32.7 
 
 3.9 
 
 16.2 
 
 0.7 
 
 190 
 
 320 
 
 4.0 
 
 3.4 
 
 9.0 
 
 1.0 
 
 32.4 
 
 4.V 
 
 16.1 
 
 0.8 
 
 180 
 
 330 
 
 4.5 
 
 3.9 
 
 9.3 
 
 1.2 
 
 32.0 
 
 4.2' 
 
 15.9 
 
 1.0 
 
 170 
 
 340 
 
 5.1 
 
 4.4 
 
 9.7 
 
 1.6 
 
 31.6 
 
 4A 
 
 1&.7 
 
 1.1 
 
 160 
 
 350 
 
 5.7 
 
 5.1 
 
 10.1 
 
 1.9 
 
 31.1 
 
 4.7 
 
 15.4 
 
 1.3 
 
 150 
 
 360 
 
 6.4 
 
 5.8 
 
 10.6 
 
 2.3 
 
 30.6 
 
 4.9 
 
 15.2 
 
 1.5 
 
 140 
 
 370 
 
 7.1 
 
 6.6 
 
 11.1 
 
 2.7 
 
 30.1 
 
 5.2 
 
 14.9 
 
 1.7 
 
 130 
 
 380 
 
 7.9 
 
 7.4 
 
 11.7 
 
 3.2 
 
 29.4 
 
 5.5 
 
 14.6 
 
 1.9 
 
 120 
 
 390 
 
 8.7 
 
 8.3 
 
 12.2 
 
 3.6 
 
 28.7 
 
 5.8 
 
 14.3 
 
 2.1 
 
 110 
 
 400 
 
 9.6 
 
 9.2 
 
 12.8 
 
 4.1 
 
 28.0 
 
 6.1 
 
 13.9 
 
 2.3 
 
 100 
 
 410 
 
 10.5 
 
 10.1 
 
 13.5 
 
 4.6 
 
 27.3 
 
 6.5 
 
 13.6 
 
 2.5 
 
 90 
 
 420 
 
 11.5 
 
 11.1 
 
 14.1 
 
 5.2 
 
 26.6 
 
 6.8 
 
 13.2 
 
 2.8 
 
 80 
 
 430 
 
 12.5 
 
 12.2 
 
 14.8 
 
 5.7 
 
 25.8 
 
 7.2 
 
 12.9 
 
 3.1 
 
 70 
 
 440 
 
 13.5 
 
 13.2 
 
 15.5 
 
 6.3 
 
 25.0 
 
 7.6 
 
 12.5 
 
 3.3 
 
 60 
 
 450 
 
 14.5 
 
 14.3 
 
 16.2 
 
 6.9 
 
 24.2 
 
 8.0 
 
 12.1 
 
 3.6 
 
 50 
 
 460 
 
 15.6 
 
 15.4 
 
 17.0 
 
 7.5 
 
 23.4 
 
 8.4 
 
 11.7 
 
 3.9 
 
 40 
 
 470 
 
 16.7 
 
 16.5 
 
 17.7 
 
 8.1 
 
 22.6 
 
 8.8 
 
 11.3 
 
 4.1 
 
 30 
 
 480 
 
 17,8 
 
 17.7 
 
 18,5 
 
 8.7 
 
 21.7 
 
 9.2 
 
 10.8 
 
 4.4 
 
 20 
 
 490 
 
 18.-9 
 
 18.8 
 
 19.2 
 
 9.4 
 
 20.9 
 
 9.6 
 
 10.4 
 
 4.7 
 
 10 
 
 500 
 
 20.0 
 
 20.0 
 
 20.0 
 
 10.0 
 
 20.0 
 
 10.0 
 
 10.0 
 
 5.0 
 
 
 
 510 
 
 21.1 
 
 21.2 
 
 20.8 
 
 10.6 
 
 19.1 
 
 10.4 
 
 9.6 
 
 5.3 
 
 990 
 
 520 
 
 22.2 
 
 22.3 
 
 21.5 
 
 11.3 
 
 18.3 
 
 10.8 
 
 9.2 
 
 5.6 
 
 980 
 
 530 
 
 23.3 
 
 23.5 
 
 22.3 
 
 11.9 
 
 17.4 
 
 11.2 
 
 8.7 
 
 5.9 
 
 970 
 
 540 
 
 24.4 
 
 24.6 
 
 23.0 
 
 12.5 
 
 16.6 
 
 11.6 
 
 8.3 
 
 6.1 
 
 960 
 
 550 
 
 25.5 
 
 25.7 
 
 23.8 
 
 13.1 
 
 15.8 
 
 12.0 
 
 7.9 
 
 6.4 
 
 950 
 
 560 
 
 26.5 
 
 26.8 
 
 24.5 
 
 13.7 
 
 15.0 
 
 12.4 
 
 7.5 
 
 6.7 
 
 910 
 
 570 
 
 27.5 
 
 27.8 
 
 25.2 
 
 14.3 
 
 14.2 
 
 12.8 
 
 7.1 
 
 6.9 
 
 930 
 
 580 
 
 28.5 
 
 28.9 
 
 25.9 
 
 14.8 
 
 13.4 
 
 13.2 
 
 6.8 
 
 7.2 
 
 920 
 
 590 
 
 29.5 
 
 29.9 
 
 26.5 
 
 15.4 
 
 12.7 
 
 13.5 
 
 6.4 
 
 7.5 
 
 910 
 
 600 
 
 30.4 
 
 30.8 
 
 27.2 
 
 15.9 
 
 12.0 
 
 13.9 
 
 6.1 
 
 7.7 
 
 900 
 
 610 
 
 31.3 
 
 31.7 
 
 27.8 
 
 16.4 
 
 11.3 
 
 14.2 
 
 5.7 
 
 7.9 
 
 890 
 
 620 
 
 32.1 
 
 32.6 
 
 28.3 
 
 16.8 
 
 10.6 
 
 14.5 
 
 5.4 
 
 8.1 
 
 880 
 
 630 
 
 32.9 
 
 33.4 
 
 28.9 
 
 17.3 
 
 9.9 
 
 14.8 
 
 5.1 
 
 8.3 
 
 870 
 
 640 
 
 33.6 
 
 34.2 
 
 29.4 
 
 17.7 
 
 9.4 
 
 15.1 
 
 4.8 
 
 8.5 
 
 860 
 
 650 
 
 34.3 
 
 34.9 
 
 29.9 
 
 18.1 
 
 8.9 
 
 15.3 
 
 4.6 
 
 8.7 
 
 850 
 
 660 
 
 34.9 
 
 35.6 
 
 30.3 
 
 18.4 
 
 8.4 
 
 15.6 
 
 4.3 
 
 8.9 
 
 840 
 
 670 
 
 355 
 
 36.1 
 
 30.7 
 
 18.8 
 
 8.0 
 
 15.8 
 
 4.1 
 
 9.0 
 
 830 
 
 680 
 
 36.0 
 
 36.6 
 
 31.0 
 
 19.0 
 
 7.6 
 
 16.0 
 
 3.9 
 
 9.2 
 
 820 
 
 690 
 
 36.5 
 
 37.1 
 
 31.3 
 
 19.3 
 
 7.3 
 
 16.1 
 
 3.8 
 
 9.3 
 
 810 
 
 700 
 
 36.8 
 
 37.5 
 
 31.6 
 
 19.5 
 
 7.0 
 
 16.3 
 
 3.6 
 
 9.4 
 
 800 
 
 710 
 
 37.1 
 
 37.8 
 
 31.8 
 
 19.7 
 
 6.8 
 
 16.4 
 
 3.5 
 
 9.5 
 
 790 
 
 720 
 
 37.4 
 
 38.1 
 
 32.0 
 
 19.8 
 
 6.5 
 
 16.5 
 
 3.4 
 
 9.5 
 
 780 
 
 730 
 
 37.6 
 
 38.3 
 
 32.1 
 
 19.9 
 
 6.4 
 
 16.5 
 
 3.4 
 
 9.6 
 
 770 
 
 740 
 
 37.7 
 
 ,38.4 
 
 32.2 
 
 20.0 
 
 6.3 
 
 16.6 
 
 3.3 
 
 9.6 
 
 760 
 
 750 
 
 37.7 
 
 38.4 
 
 32.2 
 
 20.0 
 
 6.3 
 
 16.6 
 
 3.3 
 
 9.6 
 
 750 
 
 Arg. 
 
 20 
 
 Arg. 
 
 
 
 10.0 
 
 500 
 
 10 
 
 10.9 
 
 510 
 
 20 
 
 11.8 
 
 520 
 
 30 
 
 12.7 
 
 530 
 
 40 
 
 13.5 
 
 540 
 
 50 
 
 143 
 
 550 I 
 
 60 
 
 15.0 
 
 560 
 
 70 
 
 15.7 
 
 570 
 
 80 
 
 16.2 
 
 580 
 
 90 
 
 16.7 
 
 590 
 
 100 
 
 17.0 
 
 600 
 
 110 
 
 17.2 
 
 610 
 
 120 
 
 17.4 
 
 620 
 
 130 
 
 17.4 
 
 630 
 
 140 
 
 17.2 
 
 640 
 
 150 
 
 17.0 
 
 650 
 
 160 
 
 16.7 
 
 660 
 
 170 
 
 16,2 
 
 670 
 
 180 
 
 15.7 
 
 680 
 
 190 
 
 15.0 
 
 690 
 
 200 
 
 14.3 
 
 700 
 
 210 
 
 13.5 
 
 710 
 
 220 
 
 12.7 
 
 720 
 
 230 
 
 11.8 
 
 730 
 
 240 
 
 10.9 
 
 740 
 
 250 
 
 10.0 
 
 750 
 
 260 
 
 9.1 
 
 760 
 
 270 
 
 8.2 
 
 770 
 
 280 
 
 7.3 
 
 780 
 
 290 
 
 6.5 
 
 790 
 
 300 
 
 5.7 
 
 800 
 
 310 
 
 5.0 
 
 810 
 
 320 
 
 4.3 
 
 820 
 
 330 
 
 3.8 
 
 830 
 
 340 
 
 3.3 
 
 840 
 
 350 
 
 3.0 
 
 850 
 
 360 
 
 2.8 
 
 860 
 
 370 
 
 2.6 
 
 870 
 
 380 
 
 2.6 
 
 880 
 
 390 
 
 2.8 
 
 890 
 
 400 
 
 3.0 
 
 900 
 
 410 
 
 3.3 
 
 910 
 
 420 
 
 3.8 
 
 920 
 
 430 
 
 4.3 
 
 930 
 
 440 
 
 5.0 
 
 940 
 
 450 
 
 5.7 
 
 950 
 
 460 
 
 6.5 
 
 960 
 
 470 
 
 7.3 
 
 970 
 
 480 
 
 8.2 
 
 980 
 
 490 
 
 9.1 
 
 990 
 
 500 
 
 10.0 
 
 1000 
 
TABLE XLVIL 
 
 TABLE XLVIII. 69 
 
 Equations 21 to 29, 
 
 Equations 30 and 31. 
 
 1 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 1 
 
 25 
 
 7.8 
 
 3.2 
 
 7.1 
 
 6.1 
 
 5.9 
 
 4.1 
 
 5.8 
 
 4.3 
 
 5.7 
 
 25 
 
 127 
 
 7.8 
 
 3.2 
 
 7.1 
 
 6.1 
 
 5.9 
 
 4.1 
 
 5.8 
 
 4.3 
 
 5.7 
 
 23 
 
 29 
 
 7.7 
 
 3.3 
 
 7.0 
 
 6.1 
 
 5.9 
 
 1.1 
 
 5.8 
 
 4.3 
 
 5.7 
 
 21 
 
 31 
 
 7.6 
 
 3.3 
 
 7.0 
 
 6.0 
 
 5.8 
 
 4.2 
 
 5.7 
 
 4.3 
 
 5.7 
 
 19 
 
 33 
 
 7.5 
 
 3.4 
 
 6.8 
 
 6,0 
 
 5.8 
 
 4.2 
 
 5.7 
 
 4.4 
 
 5.6 
 
 17 
 
 35 
 
 7.3 
 
 3.5 
 
 6.7 
 
 5,9 
 
 5.7 
 
 4.3 
 
 5.6 
 
 4.4 
 
 5.6 
 
 15 
 
 37 
 
 7.0 
 
 3.7 
 
 6.5 
 
 5.8 
 
 5.7 
 
 4.3 
 
 5.6 
 
 4.5 
 
 5.5 
 
 13 
 
 39 
 
 6.8 
 
 3.9 
 
 6.3 
 
 5.7 
 
 5.6 
 
 4.4 
 
 5.5 
 
 4.6 
 
 5.4 
 
 11 
 
 41 
 
 6.5 
 
 4.0 
 
 6.1 
 
 5.6 
 
 5.5 
 
 4.5 
 
 5.4 
 
 4.6 
 
 5.4 
 
 09 
 
 43 
 
 6.2 
 
 4.2 
 
 5.9 
 
 5.5 
 
 5.4 
 
 4.6 
 
 5.3 
 
 4.7 
 
 5.3 
 
 07 
 
 45 
 
 5.9 
 
 4.4 
 
 5,6 
 
 5.3 
 
 5.3 
 
 4.7 
 
 5.2 
 
 4.8 
 
 5.2 
 
 05 
 
 47 
 
 5.5 
 
 4.7 
 
 5.4 
 
 5.2 
 
 5.2 
 
 4.8 
 
 5.1 
 
 4.9 
 
 5.1 
 
 03 
 
 49 
 
 5.2 
 
 4,9 
 
 5.1 
 
 5.1 
 
 5.1 
 
 4.9 
 
 5.0 
 
 5.0 
 
 5.0 
 
 01 
 
 51 
 
 4.8 
 
 5.1 
 
 4.9 
 
 4.9 
 
 4.9 
 
 5.1 
 
 5.0 
 
 5.0 
 
 5.0 
 
 99 
 
 53 
 
 4.5 
 
 5.3 
 
 4.6 
 
 4.8 
 
 4.8 
 
 5.2 
 
 4.9 
 
 5.1 
 
 4.9 
 
 97 
 
 55 
 
 4.1 
 
 5.6 
 
 4.4 
 
 4.7 
 
 4.7 
 
 5.3 
 
 4.8 
 
 5.2 
 
 4.8 
 
 95 
 
 57 
 
 3.8 
 
 5.8 
 
 4.1 
 
 4.5 
 
 4.6 
 
 5.4 
 
 4.7 
 
 5.3 
 
 4.7 
 
 93 
 
 59 
 
 3.5 
 
 6.0 
 
 3.9 
 
 4.4 
 
 4.5 
 
 5.5 
 
 4.6 
 
 5.4 
 
 4.6 
 
 91 
 
 61 
 
 3.2 
 
 6.1 
 
 3.7 
 
 4.3 
 
 4.4 
 
 5.6 
 
 4.5 
 
 5.4 
 
 4.6 
 
 89 
 
 63 
 
 3.0 
 
 6.3 
 
 3.5 
 
 4.2 
 
 4.3 
 
 5.7 
 
 4.4 
 
 5.5 
 
 4.5 
 
 87 
 
 65 
 
 2.7 
 
 6.5 
 
 3.3 
 
 4.1 
 
 4.3 
 
 5.7 
 
 4.4 
 
 5.6 
 
 4.4 
 
 85 
 
 67 
 
 2.5 
 
 6.6 
 
 3.2 
 
 4.0 
 
 4.2 
 
 5.8 
 
 4.3 
 
 5.6 
 
 4.4 
 
 83 
 
 69 
 
 2.4 
 
 6.7 
 
 3.0 
 
 4.0 
 
 4.2 
 
 5.8 
 
 4.3 
 
 5.7 
 
 4.3 
 
 81 
 
 71 
 
 2.3 
 
 6.7 
 
 3.0 
 
 3.9 
 
 4.1 
 
 5.9 
 
 4.2 
 
 6.7, 
 
 4.3 
 
 79 
 
 73 
 
 2.2 
 
 6.8 
 
 2.9 
 
 3.9 
 
 4.1 
 
 5.9 
 
 4.2 
 
 5.7 
 
 4.3 
 
 77 
 
 75 
 
 2.2 
 
 6.8 
 
 2.9 
 
 3.9 
 
 4.1 
 
 5.9 
 
 4.2 
 
 5.7 J 
 
 4.3 
 
 75 
 
 TABLE XLIX. 
 Equation 4 32. Argument, Supp. of Node. 
 
 
 III* 
 
 IVs 
 
 V 
 
 VI* 
 
 VII* 
 
 VIII* 
 
 
 
 
 
 
 3.1 
 
 4.0 
 
 6.5 
 
 10.0 
 
 13.5 
 
 16.0 
 
 
 
 30 
 
 2 
 
 3.1 
 
 4.2 
 
 6.8 
 
 10.2 
 
 13.7 
 
 16.1 
 
 28 
 
 4 
 
 3.1 
 
 4.3 
 
 7.0 
 
 10.5 
 
 13.8 
 
 16.2 
 
 26 
 
 6 
 
 3.1 
 
 4.4 
 
 7.2 
 
 10.7 
 
 14.0 
 
 16.3 
 
 24 
 
 8 
 
 3.2 
 
 4.6 
 
 7.4 
 
 11.0 
 
 14.2 
 
 16.4 
 
 22 
 
 10 
 
 3.2 
 
 4.7 
 
 7. 
 
 11.2 
 
 14.4 
 
 16.5 
 
 20 
 
 12 
 
 3.3 
 
 4.9 
 
 7.9 
 
 11.4 
 
 14.6 
 
 16.6 
 
 18 
 
 14 
 
 3.3 
 
 5.0 
 
 8.1 
 
 11.7 
 
 14.8 
 
 16.6 
 
 16 
 
 16 
 
 3.4 
 
 5.2 
 
 8.3 
 
 11.9 
 
 15.0 
 
 16.7 
 
 14 
 
 18 
 
 3.4 
 
 5.4 
 
 8.6 
 
 12.1 
 
 15.1 
 
 16.7 
 
 12 
 
 20 
 
 3.5 
 
 5.6 
 
 8.8 
 
 12.4 
 
 15.3 
 
 16.8 
 
 10 
 
 22 
 
 3.6 
 
 5.8 
 
 9.0 
 
 12.6 
 
 15.4 
 
 16.8 
 
 8 
 
 24 
 
 3.7 
 
 6.0 
 
 9.3 
 
 12.8 
 
 15.6 
 
 16.9 
 
 6 
 
 26 
 
 3.8 
 
 6.2 
 
 9.5 
 
 13.0 
 
 15.7 
 
 16.9 
 
 4 
 
 28 
 
 3.9 
 
 6.3 
 
 9.8 
 
 13.2 
 
 15.8 
 
 16.9 
 
 2 
 
 30 
 
 4.0 
 
 6.5 
 
 10.0 
 
 13.5 
 
 16.0 
 
 16.9 
 
 
 
 
 Us 
 
 I 
 
 O* 
 
 XL? 
 
 X* 
 
 IX* 
 
 
 Arg 
 
 30 
 
 31 
 
 
 
 5.0 
 
 5.0 
 
 2 
 
 5.0 
 
 50 
 
 4 
 
 4.9 
 
 5.1 
 
 6 
 
 4.9 
 
 5.1 
 
 8 
 
 4.8 
 
 5.2 
 
 10 
 
 4.8 
 
 5.2 
 
 12 
 
 4.7 
 
 5.3 
 
 14 
 
 4.6 
 
 5.4 
 
 16 
 
 4.5 
 
 5.5 
 
 18 
 
 4.4 
 
 5.5 
 
 20 
 
 4.2 
 
 5.6 
 
 22 
 
 4.1 
 
 5.7 
 
 24 
 
 4.0 
 
 5.8 
 
 26 
 
 3.9 
 
 5.8 
 
 28 
 
 3.8 
 
 5.9 
 
 30 
 
 3.7 
 
 5.9 
 
 32 
 
 3.7 
 
 5.9 
 
 34 
 
 3.7 
 
 5.9 
 
 36 
 
 3.7 
 
 5.9 
 
 38 
 
 3.8 
 
 5.8 
 
 40 
 
 3.9 
 
 5.7 
 
 42 
 
 4.1 
 
 5.6 
 
 44 
 
 4.3 
 
 5.5 
 
 46 
 
 4.5 
 
 5.3 
 
 48 
 
 4.8 
 
 5.2 
 
 50 
 
 5.0 
 
 5.0 
 
 52 
 
 5.2 
 
 4.8 
 
 54 
 
 5.5 
 
 4.7 
 
 56 
 
 5.7 
 
 4.5 
 
 58 
 
 5.9 
 
 4.4 
 
 60 
 
 6.1 
 
 4.3 
 
 62 
 
 6.2 
 
 4.2 
 
 64 
 
 6.3 
 
 4.1 
 
 66 
 
 6.3 
 
 4.1 
 
 68 
 
 6.3 
 
 4.1 
 
 70 
 
 6.3 
 
 4.1 
 
 72 
 
 6.2 
 
 4.1 
 
 74 
 
 6.2 
 
 4.2 
 
 76 
 
 6.0 
 
 4.2 
 
 78 
 
 5.9 
 
 4.3 
 
 80 
 
 5.8 
 
 4.4 
 
 82 
 
 5.7 
 
 4.5 
 
 84 
 
 5.5 
 
 4.6 
 
 86 
 
 5.4 
 
 4.6 
 
 88 
 
 5.3 
 
 4.7 
 
 90 
 
 5.2 
 
 4.8 
 
 92 
 
 5.1 
 
 4.8 
 
 94 
 
 5.1 
 
 4.9 
 
 96 
 
 5.0 
 
 4.9 
 
 98 
 
 5.0 
 
 5.0 
 
 100 
 
 5.0 
 
 5.0 
 
 Constant 55" 
 
70 
 
 TABLE L. 
 Evection. 
 
 Argument. Evection, corrected. 
 
 
 
 
 
 I* 
 
 
 II* 
 
 
 in* 
 
 
 IV* 
 
 
 i 
 
 1 
 
 'i 
 
 Diff. 2o 
 
 Diff'. ! 2 ; 
 
 Diff 
 
 2 
 
 Diff. 
 
 2 
 
 Diff 
 
 2 
 
 Diff 
 
 
 , .. 
 
 
 / ,/ 
 
 / ,/ 
 
 
 / .. 
 
 M 
 
 , 
 
 
 , 
 
 
 
 
 1 
 9 
 
 3 
 
 4 
 5 
 
 30 00.0 
 31 25.5 
 32 50. y 
 34 16.3 
 3541.6 
 37 6.7 
 
 85.5 
 
 85.4 
 
 J85.4 
 185.3 
 85.1 
 
 1043.5 
 11 56.7 
 13 9.0 
 14 20.6 
 15 31.3 
 1641.1 
 
 73 o 40 9.7 
 L* * 40 50.6 
 
 ';;? 41 30.1 
 
 i*'!; 42 8.3 
 '"' 4245.1 
 4.320.6 
 
 40.9 
 39.5 
 
 138.2 
 (36.8 
 35.5 
 
 50 25.5 
 50 23.5 
 5020.1 
 50 15.2 
 50 8.8 
 50 1.0 
 
 2.0 
 3.4 
 4.9 
 6.4 
 
 7.8 
 
 39 8.3 
 38 24.9 
 37 40.4 
 36 54.6 
 36 7.6 
 35 19.5 
 
 43.4 
 44.5 
 
 45.8 
 47.0 
 48.1 
 
 942.0 
 829.3 
 716.0 
 6 2.0 
 447.4 
 332.2 
 
 72.7 
 73.3 
 74.0 
 74.6 
 75.2 
 
 
 
 85. T 
 
 
 ,69.0 i 
 
 34 1 
 
 
 93 
 
 
 49.3 
 
 
 75.9 
 
 6 
 
 7 
 8 
 9 
 10 
 
 38 SI.S'JMQ 
 39 56*7 
 4l2L4!ti 
 42 45.8 oTo 
 44 10. II 84 ' 3 
 
 1750.1 
 1858.2 
 20 5.3 
 21 11.5 
 22 16.7 
 
 ' rs 43 54.7 
 E'-I 4427.4|ff-J 
 
 66:2 4458 ' 8 29:9 
 
 4951.7 
 4941.0 
 49 28.8 
 49 15.1 
 49 0.2 
 
 10.7 
 12.2 
 13.7 
 14.9 
 
 34 30.2 
 33 39.7 
 3248.1 
 31 55.4 
 31 1.6 
 
 50.5 
 51.6 
 52.7 
 53.8 
 
 2 16.3 
 59.9 
 
 76.4 
 76.9 
 
 77.4 
 78.0 
 
 59 43.0 
 58 25.6 
 57 7.6 
 
 
 83.9 
 
 
 64.3 
 
 
 272 
 
 
 16.7 
 
 
 54.9 
 
 
 78.4 
 
 11 
 12 
 18 
 
 14 
 15 
 
 45 34.0 oo 7 
 4657.7^ 
 
 49441 83 - 
 <tj tt.L g2 6 
 
 2321.0 
 24 24.2 
 25 26.4 
 26 27.6 
 27 27.6 
 
 63.2 
 62.2 
 61.2 
 60.0 
 
 46 24.5 
 46 B*.8gJ'I 
 
 47 37.4 ??'? 
 
 47 58.8 ^ l ' 4 
 
 48 43.5 
 48 25.6 
 48 6.3 
 4745.5 
 17 23.3 
 
 17.9 
 19.3 
 
 20.8 
 
 22.2 
 
 30 6.7 
 
 29 10.7 
 28 13.7 
 27 15.7 
 26 16.6 
 
 56.0 
 57.0 
 58.0 
 59.1 
 
 55 49.2 
 54 30.3 
 53 11.0 
 51 51.3 
 50 31.2 
 
 78.9 f 
 79.3 
 79.7 
 80.1 
 
 
 j82.2 
 
 
 59.0 
 
 200 
 
 
 235 
 
 
 60.0 
 
 
 30.5 
 
 O CO CO -J O) 
 
 522S.9 fl1 
 53 50.7 8 
 55 12.0 SH 
 53 32.9 8 ^ o 
 
 57 53.2 
 
 28 26.6 
 29 24.6 
 30 21.4 
 31 17.0 
 32 11.5 
 
 58.0 
 56.8 
 55.6 
 54.5 
 
 48 18.8 
 48 37.4 
 4854.5 
 4910.1 
 49 24.4 
 
 18.6 
 17.1 
 15.6 
 14.3 
 
 46 59.8 
 
 46 34.8 
 46 8.5 
 45 40.7 
 45 11.6 
 
 25.0 
 26.3 
 27.8 
 29.1 
 
 25 16.6 
 24 15.6 
 23 13.6 
 22 10.7 
 21 6.8 
 
 61.0 
 62.0 
 62.9 
 63.9 
 
 49 10.7 
 47 49.9 
 46 28.8 
 45 7.5 
 43 45.8 
 
 80.8 
 81.1 
 81.3 
 81.7 
 
 
 79.8 
 
 
 53.3 
 
 
 127 
 
 
 30.4 
 
 
 64.7 
 
 
 81.9 
 
 21 
 
 24 
 ,25 
 
 59 13.0 79 ^ 
 "0 32.3 78 ;' 7 
 151.0 78 1 
 3 9.1774 
 426.5, 
 
 33 4.8 
 33 57.0 
 34 47.9 
 35 37.7 
 
 36 26.2 
 
 52.2 
 50.9 
 49.8 
 
 48.5 
 
 4937.1 
 4948.3 
 4958.1 
 50 6.4 
 50 13.3 
 
 11.2 
 9.8 
 8.3 
 6.9 
 
 4441.2 
 44 9.5 
 43 36.4 
 43 1.9 
 
 42 26.2 
 
 31.7 
 33.1 
 34.5 
 35.7 
 
 20 2.1 
 18 56.4 
 1749.9 
 16 42.6 
 15 34.4 
 
 K . 42 23.9 
 
 *i 41 i-s 
 
 X' 39 39.5 
 SI'* 38 17.0 
 OB "* 3654.4 
 
 82.1 
 82.3 
 82.5 
 82.6 
 
 
 76.8 
 
 
 47.2 
 
 
 5.4 
 
 
 37.0 
 
 
 68.9 1 
 
 82.7 
 
 2G 
 27 
 
 30 
 
 5 43.3 
 6 59.4 
 8 14.9 
 9 29.6 
 1043.5 
 
 76.1 1 
 75.5 
 74.7 
 73.9 
 
 37 13.4 
 37 59.4 
 38 44.2 
 39 27.6 
 40 9.7 
 
 46.0 
 44.8 
 43.4 
 42.1 
 
 50 18.7 
 50 22.6 
 50 25.0 
 50 26.0 
 5025.5 
 
 3.9 
 
 2.4 
 
 1.0 
 
 41 49.2 
 41 10.8 
 4031.2 
 39 50.4 
 39 8.3 
 
 38.4 
 39.6 
 40.8 
 42.1 
 
 425.5 
 3 15.7 
 2 5.2 
 10 54.0 
 942.0 
 
 69.8 
 70.5 
 71.2 
 72.0 , 
 
 3531.7 
 34 8.8 
 32 45.9 
 31 23.0 
 30 0.0 
 
 82.9 
 82.9 
 82.9 
 83.0 
 
 
 2 
 
 
 2 
 
 
 2 
 
 ia 
 
 
 2 
 
 
 1 
 
 
TABLE L. 
 Evection. 
 
 Argument. Evection, corrected. 
 
 71 
 
 
 VI* 
 
 | VII* 
 
 
 VIII* 
 
 
 IX* 
 
 
 X* 
 
 
 XI* 
 
 
 s? 
 
 1 
 
 Diff.O 
 
 Diff.'o 
 
 Diff.O 
 
 Diff. 0^ 
 
 Diff. 
 
 <*> 
 
 Diff. 
 
 q 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 3 
 
 4 
 5 
 
 30 00 
 
 23 37.0 
 27 14.1 
 25 51.2 
 24 28.3 
 23 5.6 
 
 83.0 
 82.9 
 82.9 
 829 
 82.7 
 
 50 18.0 
 49 6.0 
 
 47 54.8 
 46 44.3 
 45 34.5 
 44 25.6 
 
 72.0 
 71.2 
 70.5 
 69.8 
 68.9 
 
 2051.7 
 20 9.6 
 19 28.8 
 1849.2 
 18 10.8 
 1733.8 
 
 42.1 
 40.8 
 39.6 
 38.4 
 37.0 
 
 9 34.5 
 9 34.0 J;J 
 935.0 i? 
 937.4 l\ 
 941.3 J.'J 
 946.7 5 ' 4 
 
 19 50.3 
 20 32.4 
 
 21 15.8 
 22 0.6 
 22 46.6 
 23 33.8 
 
 42.1 
 43.4 
 44.8 
 46.0 
 47.2 
 
 49 16.5 
 50 30.4 
 5145.1 
 53 0.6 
 54 16.7 
 55 33.5 
 
 73.9 
 
 74.7 
 75.5 
 76.1 
 76.8 
 
 
 
 82.6 
 
 
 68.2 
 
 
 35.7 
 
 
 6.9 
 
 
 48.5 
 
 
 77.4 
 
 6 
 
 7 
 8 
 9 
 10 
 
 21 43.0 
 20 20.5 
 IS 58.2 
 1738.1 
 16 14.2 
 
 82.5 
 82.3 
 82.1 
 81.9 
 
 43 17.4 
 42 10.1 
 41 3.6 
 39 57.9 
 38 53.2 
 
 67.3 
 66.5 
 65.7 
 64.7 
 
 16 58.1 
 16 23.6 
 15 50.5 
 15 18.8 
 14 48.4 
 
 34.5 
 33.1 
 31.7 
 30.4 
 
 953.6 
 10 1.9 -3 
 1011.7 ^ 
 !*{*; 
 10 35.6 u 
 
 24 22.3 
 
 25 12.1 
 26 3.0 
 26 55.2 
 27 48.5 
 
 49.8 
 50.9 
 52.2 
 53.3 
 
 56 50.9 7fi , 
 58 9.0 'i 
 59 27.7 ?H 
 04770^ 
 2 6.8 
 
 
 
 81 7 
 
 
 639 
 
 
 29.1 
 
 
 143 
 
 
 545 
 
 
 80.3 
 
 11 
 12 
 
 13 
 14 
 15 
 
 14 52.5 
 1331 2 
 12 10.1 
 10 49.3 
 928.8 
 
 81.3 
 81.1 
 
 80.8 
 80.5 
 
 3749.3 
 36 46.4 
 35 44.4 
 34 43.4 
 33 43.4 
 
 62.9 
 62.0 
 61.0 
 60.0 
 
 14 19.3 
 1351.5 
 1325.2 
 13 0.2 
 1236.7 
 
 27.8 
 26.3 
 25.0 
 23.5 
 
 1049.9 
 11 5.5 
 11 22.6 
 1141.2 
 12 1.2 
 
 15.6 
 17.1 
 18.6 
 20.0 
 
 28 43.0 
 29 38.6 
 30 35.4 
 31 33.4 
 32 32.4 
 
 55.6 
 56.8 
 58.0 
 59.0 
 
 327.1 
 448.0 
 6 9.3 
 731.1 
 853.3 
 
 80.9 
 81.3 
 81.8 
 82.2 
 
 
 
 80.1 
 
 
 59.1 
 
 
 22.2 
 
 
 21.4 
 
 
 600 
 
 
 82.6 
 
 16 
 
 17 
 IS 
 19 
 20 
 
 8 8.7 
 649.0 
 529.7 
 4 10.8 
 252.4 
 
 79.7 
 79.3 
 78.9 
 
 78.4 
 
 32 44.3 
 31 46.3 
 30 49.3 
 29 53.3 
 28 58.4 
 
 58.0 
 57.0 
 56.0 
 54.9 
 
 12 14.5 
 11 53.7 
 11 34,4 
 11 16.5 
 10 59.8 
 
 20.8 
 19.3 
 17.9 
 16.7 
 
 1222.6 
 12 45.5 
 13 9.8 
 13 35.5 
 14 2.7 
 
 22.9 
 24.3 
 25.7 
 
 27.2 
 
 33 32.4 
 34 33.6 
 35 35.8 
 36 39.0 
 37 43.3 
 
 61.2 
 62.2 
 63.2 
 64.3 
 
 10 15.9 
 1138.9 
 13 2.3 
 1426.0 
 1549.9 
 
 83.0 
 83.4 
 83.7 
 83.9 
 
 
 
 78.0 
 
 
 53.8 
 
 
 14,9 
 
 
 28.6 
 
 
 65.2 
 
 
 84.3 
 
 21 
 22 
 23 
 24 
 25 
 
 1 34.4 
 17.0 
 
 77.4 
 76.9 
 76.4 
 75.9 
 
 28 4.6 
 27 11.9 
 26 20.3 
 25 29.8 
 24 40.5 
 
 52.7 
 51.6 
 50.5 
 49.3 
 
 10 44.9 
 1031.2 
 10 19.0 
 10 8.3 
 9 59.0 
 
 13.7 
 12.2 
 10.7 
 9.3 
 
 1431.3 
 
 15 1.2 
 15 32.6 
 16 5.3 
 1639.4 
 
 29.9 
 31.4 
 32.7 
 34.1 
 
 38 48.5 
 39 54.7 
 41 1.8 
 42 9.9 
 43 18.9 
 
 66.2 
 67.1 
 68.1 
 69.0 
 
 1714.2 
 1838.eC'* 
 
 20 3.3^ 
 21 28.2 f' 9 
 2253.3i 
 
 59 0.1 
 57 43.7 
 
 56 27.8 
 
 . 
 
 75.2 
 
 
 48.1 
 
 
 7.8 
 
 
 355 
 
 
 698 
 
 
 85.1 
 
 26 
 
 27 
 28 
 29 
 30 
 
 55 12.6 
 53 58.0 
 52 44.0 
 51 30.7 
 50 18.0 
 
 74.6 
 74.0 
 73.3 
 72.7 
 
 23 52.4 
 23 5.4 
 22 19.6 
 21 35.1 
 2051.7 
 
 47.0 
 45.8 
 44.5 
 43.4 
 
 951.2 
 944.8 
 939.9 
 936.5 
 934.5 
 
 6.4 
 4.9 
 3.4 
 2.0 
 
 17 14.9 
 1751.7 
 18 29.9 
 19 9.4 
 19 50.3 
 
 36.8 
 38.2 
 39.5 
 40.9 
 
 4428.7 
 45 39.4 
 4651.0 
 48 3.3 
 49 16.5 
 
 70.7 
 
 71.6 
 723 
 73.2 
 
 24 18.4 
 25 43.7 
 27 9.1 
 28 34.5 
 30 0.0 
 
 85.3 
 
 85.4 
 85.4 
 85,5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
72 
 
 TABLE LI, 
 
 Equation of Moon's Centre. 
 Argument. Anomaly corrected. 
 
 
 0* 
 
 
 I* 
 
 
 II* 
 
 
 III* 
 
 
 IV* 
 
 
 V* 
 
 
 
 7 
 
 Diff 
 for 10 
 
 10 
 
 Diff 
 for 10 
 
 12 
 
 Diff 
 forlO 
 
 13 
 
 Diff 
 for 10 
 
 12 
 
 Diff 
 for 10 
 
 9 
 
 Diff 
 for 10 
 
 ' 
 
 
 30 
 1 
 30 
 2 
 30 
 
 0.0 
 
 332.6 
 7 5.2 
 1037.8 
 1410.3 
 1742.7 
 
 70.9 
 
 709 
 70.9 
 
 70.8 
 70.8 
 
 2057.9 
 2355.6 
 26 52.2 
 2947.7 
 32 42.0 
 3535.2 
 
 59.2 
 
 58:9 
 
 58.5 
 58J 
 57.7 
 
 3843.6 
 40 14.0 
 4142.7 
 43 9.6 
 4434.9 
 4558.4 
 
 30.1 
 29.6 
 29.0 
 
 28.4 
 27.8 
 
 1735.2 
 17209 
 17 4.8 
 1647.1 
 1627.6 
 16 6.5 
 
 K 
 
 5.9 
 6.5 
 7.0 
 
 1620.8 
 1435.3 
 1248.5 
 11 0.4 
 911.1 
 720.5 
 
 // 
 35.2 
 35.6 
 36.0 
 36.4 
 36.9 
 
 5828.9 
 5543.8 
 5258.0 
 5011.6 
 4724.5 
 4436.8 
 
 // 
 55.0 
 55.3 
 55.5 
 55.7 
 55.9 
 
 
 
 70.8 
 
 
 57.3 
 
 
 27.3 
 
 
 7.6 
 
 
 37.3 
 
 
 56.1 
 
 3 
 30 
 4 
 30 
 5 
 
 21 15.0 
 24 47.3 
 2819.4 
 3151.2 
 35 23.0 
 
 70.8 
 70.7 
 70.6 
 70.6 
 
 3827.1 
 41 18.0 
 44 7.6 
 46 56.0 
 49 43.2 
 
 57.0 
 56.5 
 56.1 
 55.7 
 
 47 20.2 
 48 40.3 
 4958.7 
 51 15.3 
 5230.2 
 
 26.7 
 26.1 
 25.5 
 25.0 
 
 1543.7 
 1519.2 
 1453.1 
 1425.2 
 1355.8 
 
 8.2 
 8.7 
 9.3 
 9.8 
 
 528.7 
 335.6 
 141.3 
 
 37.7 
 38.1 
 38.5 
 38.9 
 
 41 48.5 
 3859.5 
 36 10.0 
 33 19.8 
 3029.1 
 
 56.3 
 56.5 
 56.7 
 56.9 
 
 5945.8 
 5749.1 
 
 
 
 70.5 
 
 
 55.3 
 
 
 24.4 
 
 
 10.4 
 
 
 39.3 
 
 
 57.1 
 
 30 
 6 
 30 
 7 
 30 
 
 3854.5 
 42 25.8 
 4556.9 
 4927.7 
 52 58.2 
 
 70.4 
 70.4 
 70.3 
 70.2 
 
 5229.1 
 55 13.8 
 5757.2 
 ~039L3 
 320.1 
 
 54.9 
 54.5 
 54.0 
 53.6 
 
 5343.3 
 5454.7 
 56 4.4 
 57 12.3 
 
 58 18.5 
 
 23.8 
 23.2 
 22.6 
 22.1 
 
 1324.7 
 1251.9 
 1217.4 
 1141.4 
 11 3.7 
 
 10.9 
 11.5 
 12.0 
 12.6 
 
 5551.1 
 53 52.0 
 5151.7 
 49 50.3 
 4747.6 
 
 39.7 
 40.1 
 40.5 
 40.9 
 
 27 37.8 
 24 45.9 
 21 53.5 
 19 0.6 
 16 7.1 
 
 57.3 
 57.5 
 57.G 
 57.8 
 
 
 
 70.1 
 
 
 53.2 
 
 
 21.5 
 
 
 13.1 
 
 
 41.3 
 
 
 58.0 
 
 8 
 30 
 9 
 30 
 10 
 
 56 28.5 
 59 58.4 
 ~3~2lM) 
 657.2 
 1026.0 
 
 70.0 
 69.9 
 69.7 
 69.6 
 
 559.7 
 837.9 
 11 14.8 
 1350.3 
 1624.5 
 
 52.7 
 52.3 
 
 51.8 
 51.4 
 
 59 22.9 
 025.6 
 126.5 
 225.7 
 323.0 
 
 20.9 
 20.3 
 19.7 
 19.1 
 
 1024.3 
 943.4 
 9 0.8 
 816.6 
 730.8 
 
 13.6 
 14.2 
 14.7 
 15.3 
 
 4543.8 
 43 38.9 
 41 32.8 
 39 25.6 
 37 17.3 
 
 41.7 
 42.0 
 42.4 
 42. "8 
 
 1313.1 
 
 1018.6 
 723.6 
 428.1 
 132.2 
 
 58.2 
 58.3 
 58.5 
 58.6 
 
 
 
 69.5 
 
 
 50.9 
 
 
 18.6 
 
 
 15.8 
 
 
 43.1 
 
 
 58.8 
 
 30 
 11 
 30 
 12 
 30 
 
 1354.5 
 
 1722.5 
 2050.1 
 2417.3 
 2744.0 
 
 69.3 
 69.2 
 69.1 
 68.9 
 
 18 57.3 
 21 28.8 
 23 58.8 
 2627.5 
 28 54.7 
 
 50.5 
 50.0 
 49.6 
 49.1 
 
 418.7 
 5 12.5 
 6 4.6 
 654.9 
 743.5 
 
 17.9 
 17.4 
 16.8 
 16.2 
 
 6 43.4 
 554.4 
 5 3.9 
 411.7 
 318.0 
 
 16.3 
 16.8 
 17.4 
 17.9 
 
 35 7.9 
 
 3257.4 
 3045.8 
 2833.1 
 26 19.4 
 
 43.5 
 43.9 
 44.2 
 44.6 
 
 58 35.8 
 5538.9 
 5241.7 
 4943.9 
 4645.8 
 
 59.0 
 59.1 
 59.3 
 59.4 
 
 
 
 68.7 
 
 
 48.6 
 
 
 15.6 
 
 
 18.4 
 
 
 44.9 
 
 ! 59.5 
 
 13 
 30 
 14 
 30 
 15 
 
 31 10.2 
 34 35.8 
 38 1.0 
 41 25.6 
 4449.6 
 
 68.5 
 68.4 
 68.2 
 68.0 
 
 3120.5 
 3344.9 
 36 7.9 
 38 29.4 
 4049.3 
 
 48.1 
 47.7 
 47.2 
 46.6 
 
 830.3 
 915.4 
 958.6 
 1040.1 
 11 19.9 
 
 15.0 
 14.4 
 13.8 
 13.3 
 
 222.7 
 125.8 
 027.4 
 
 19.0 
 19.5 
 20.0 
 20.5 
 
 24 4.6 
 
 2148.8 
 1931.9 
 1714.1 
 1455.2 
 
 45.3 
 45.6 
 45.9 
 46.3 
 
 4347.3 
 4048.4 
 3749.1 
 3449.5 
 31494 
 
 59.6 
 59.8 
 59.9 
 60.0 | 
 
 59 27.4 
 5825.9 
 
 
 8 
 
 
 11 
 
 
 13 
 
 
 12 
 
 
 11 
 
 |8 
 
 
TABLE LI. 
 
 Equation of Moon's Centre. 
 Argument. Anomaly corrected. 
 
 73 
 
 
 VI* 
 
 ' 
 
 VII* 
 
 
 VIII* 
 
 
 IX* 
 
 
 x 
 
 
 XI 
 
 
 
 
 Diff 
 
 
 DifF 
 
 
 Diff 
 
 
 Diff , 
 
 Diff 
 
 oO 
 
 Diff 
 
 
 7 
 
 for 10 
 
 4 
 
 forlO 
 
 1 
 
 for 10 
 
 
 for 10 1 
 
 for 10 
 
 O 
 
 for 10 
 
 O ' 
 
 
 30 
 1 
 30 
 2 
 30 
 
 0.0 
 5654.6 
 5349.2 
 5043.9 
 4738.6 
 4433.4 
 
 61.8 
 61.8 
 61.8 
 61.8 
 61.7 
 
 / rr 
 
 131.1 
 5846.7 
 56 3.0 
 5320.0 
 50 37.7 
 4756.2 
 
 54.8 
 54.6 
 543 
 54.1 
 53.8 
 
 43 39.2 
 41 55.0 
 40 12.0 
 3830.5 
 36 50.3 
 3511.3 
 
 34.7 
 34.3 
 33.8 
 33.4 
 33.0 
 
 / // 
 42 24.8 
 4212.1 
 42 1.2 
 41 52.0 
 41 44.4 
 41 38.7 
 
 4.2 
 3.6 
 ai 
 2.5 
 1.9 
 
 2116.4 
 
 2248.5 
 2422.2 
 2557.7 
 2734.8 
 29 13.7 
 
 30.7 
 31.2 
 31.8 
 32.4 
 33.0 
 
 39 2.1 
 42 0.8 
 45 0.7 
 48 1.7 
 51 3.7 
 54 6.7 
 
 59.6 
 60.0 
 60.3 
 60.7 
 61.0 
 
 
 
 61.8 
 
 
 53.6 
 
 
 32.5 
 
 
 1.4 
 
 
 33.5 
 
 
 61.3 
 
 3 
 30 
 4 
 30 
 5 
 
 4128.1 
 38 23.0 
 35 18.0 
 32 13.0 
 29 8.1 
 
 61.7 
 61.7 
 61.7 
 61.6 
 
 45 15.4 
 4235.3 
 3956.0 
 3717.4 
 3439.6 
 
 53.4 
 53.1 
 52.9 
 52.0 
 
 3333.7 
 31 57.5 
 3022.6 
 2849.0 
 2716.8 
 
 32.1 
 31.6 
 31.2 
 30.7 
 
 41 34.6 
 41 32.2 
 4131.6 
 41 32.7 
 41 35.6 
 
 0.8 
 0.2 
 0.4 
 1.0 
 
 30 54.2 
 3236.3 
 3420.2 
 36 5.6 
 3752.8 
 
 34.0 
 34.6 
 35.1 
 35.7 
 
 57 10.7 
 ~OT5^ 
 321.8 
 628.8 
 936.8 
 
 61.7 
 62.0 
 62.3 
 62.7 
 
 
 
 61.6 
 
 
 52.3 
 
 
 30.2 
 
 
 1.5 
 
 
 36.2 
 
 
 63.0 
 
 30 
 6 
 30 
 7 
 30 
 
 26 3.4 
 
 2258.8 
 1954.3 
 1650.0 
 1345.8 
 
 61.5 
 61.5 
 61.4 
 61.4 
 
 32 2.7 
 2926.5 
 2651.1 
 2416.6 
 2142.9 
 
 52.1 
 51.8 
 51.5 
 51.2 
 
 2546.1 
 2416.7 
 
 2248.7 
 2122.1 
 1956.9 
 
 29.8 
 29.3 
 28.9 
 
 28.4 
 
 4140.1 
 4146.4 
 41 54.5 
 42 4.3 
 4215.9 
 
 2.1 
 2.7 
 3.3 
 3.9 
 
 3941.5 
 4132.0 
 43 24.0 
 45 17.7 
 4712.9 
 
 36.8 
 37.3 
 37.9 
 
 38.4 
 
 1245.7 
 1555.5 
 19 6.2 
 22 17.8 
 25 30.3 
 
 63.3 
 63.6 
 63.9 
 64.2 
 
 
 
 61.3 
 
 
 51.0 
 
 
 27.9 
 
 
 4.4 
 
 
 39.0 
 
 
 64.5 
 
 8 
 30 
 9 
 30 
 10 
 
 1041.9 
 738.0 
 434.4 
 131.0 
 
 58271 
 
 61.3 
 61.2 
 61.1 
 61.1 
 
 1910.0 
 163S.O 
 14 6.9 
 1136.6 
 9 7.3 
 
 50.7 
 50.4 
 50.1 
 49.8 
 
 1833.1 
 1710.8 
 1549.8 
 1430.4 
 1312.5 
 
 27.4 
 27.0 
 265 
 26.0 
 
 4229.2 
 42 44.2 
 43 1.1 
 4319.6 
 4339.9 
 
 5.0 
 5.6 
 6.2 
 
 6.8 
 
 49 9.8 
 51 8.3 
 53 8.4 
 5510.1 
 5713.3 
 
 39.5 
 40.0 
 40.6 
 41.1 
 
 2843.7 
 31 57.8 
 3512.9 
 3828.7 
 41 45.2 
 
 64.7 
 65.0 
 65.3 
 65.5 
 
 
 61.0 
 
 
 49.5 
 
 
 25.5 
 
 
 7.4 
 
 
 41.6 
 
 
 65.8 
 
 30 
 11 
 30 
 12 
 30 
 
 5524.9 
 5222.2 
 4919.7 
 46 17.5 
 43 15.6 
 
 60.9 
 60.8 
 60.7 
 60.6 
 
 638.9 
 411.3 
 144.7 
 
 49.2 
 48.9 
 48.6 
 
 48.2 
 
 1155.9 
 1040.9 
 927.3 
 
 815.2 
 7 4.6 
 
 25.0 
 24.5 
 24.0 
 23.5 
 
 44 2.0 
 
 4425.9 
 4451.5 
 4518.8 
 4548.0 
 
 8.0 
 8.5 
 9.1 
 9.7 
 
 59 18.2 
 124T5 
 332.4 
 541.9 
 752.9 
 
 42.1 
 42.6 
 43.2 
 43.7 
 
 45 2.6 
 
 4820.7 
 5139.6 
 5459.1 
 5819.3 
 
 66.0 
 66.3 
 66.5 
 66.7 
 
 59 18.9 
 56 54.2 
 
 
 
 60.5 
 
 
 47.9 
 
 
 23.1 
 
 
 103 
 
 
 442 
 
 
 
 67.0 
 
 13 
 30 
 14 
 30 
 15 
 
 4014.0 
 3712.6 
 3411.6 
 31 10.9 
 28 10.6 
 
 60.5 
 60.3 
 60.2 
 60.1 
 
 5430.4 
 52 7.5 
 4945.6 
 4724.7 
 45 4.8 
 
 47.6 
 47.3 
 47.0 
 46.6 
 
 555.4 
 447.8 
 341.7 
 237.1 
 134.1 
 
 22.5 
 22.0 
 21.5 
 21.0 
 
 46 18.9 
 4651.5 
 4726.0 
 48 2.2 
 4840.1 
 
 10.9 
 11.5 
 12.1 
 12.6 
 
 10 5.5 
 1219.5 
 1435.1 
 1652.1 
 1910.7 
 
 44.7 
 45.2 
 45.7 
 46.2 
 
 1 40.3 
 5 1.9 
 
 824.1 
 1146.9 
 15104 
 
 67.2 
 67.4 
 67.6 
 67.8 
 
 
 5 3 
 
 
 2 
 
 
 1 
 
 
 
 
 i 
 
 
 5 
 
 
74 
 
 TABLE LI. 
 
 Equation of Moon's Centre. 
 Argument. Anomaly corrected. 
 
 
 0* 
 
 
 I* 
 
 
 II* 
 
 
 III* 
 
 
 IV* 
 
 
 V* 
 
 
 
 8 
 
 Diff 
 forlO 
 
 11 
 
 Diff 
 forlO 
 
 13 
 
 Diff 
 forlO 
 
 12 
 
 Diffl 10 
 forlO n 
 
 Diff 
 forlO 
 
 8 
 
 Diff 
 for 10 
 
 O ' 
 
 / // 
 
 
 / // 
 
 
 / ,' 
 
 
 / // 
 
 
 / ., 
 
 
 , 
 
 
 15 
 30 
 16 
 30 
 17 
 30 
 
 4449.6 
 4813.1 
 5135.9 
 5458.1 
 5819.7 
 
 67.8 
 67.6 
 67.4 
 67.2 
 67.0 
 
 4049.3 
 43 7.9 
 4524.9 
 4740.5 
 4954.5 
 52 7.1 
 
 46.2 
 45.7 
 45.2 
 
 44.7 
 44.2 
 
 1119.9 
 
 1157.8 
 1234.0 
 13 8.5 
 1341.1 
 1412.0 
 
 12.6 
 12.1 
 11.5 
 10.9 
 10.3 
 
 5825.9 
 5722.9 
 56 18.3 
 5512.2 
 54 4.6 
 5255.4 
 
 21.0 
 21.5 
 22.0 
 22.5 
 23.1 
 
 1455.2 
 1235.3 
 1014.4 
 752.5 
 529.6 
 3 5.8 
 
 46.6 
 47.0 
 47.3 
 47.7 
 47.9 
 
 3149.4 
 2849.1 
 2548.4 
 22 47.4 
 1946.0 
 1644.4 
 
 60.1 
 f.0.2 
 60.3 
 60.5 
 60.5 
 
 140 7 
 
 
 
 66.7 
 
 
 437 
 
 
 9.7 
 
 
 23.5 
 
 
 483 
 
 
 60.6 
 
 18 
 30 
 19 
 30 
 20 
 
 5 0.9 
 820.4 
 1139.3 
 1457.4 
 1814.8 
 
 66.5 
 66.3 
 66.0 
 65.8 
 
 5418.1 
 5627.6 
 5835.5 
 ~6~4T8 
 246 7 
 
 43.2 
 
 42.6 
 42.1 
 41.6 
 
 1441.2 
 15 8.5 
 1534.1 
 1558.0 
 1620.1 
 
 9.1 
 
 8.5 
 8.0 
 7.4 
 
 5144.8 
 5032.7 
 4919.1 
 48 4.1 
 46 47.5 
 
 24.0 
 24.5 
 25.0 
 25.5 
 
 041.1 
 58 15.3 
 5548.7 
 5321.1 
 5052.7 
 
 48.6 
 48.9 
 49.2 
 49.5 
 
 1342.5 
 1040.3 
 737.8 
 435.1 
 132.2 
 
 60.7 
 60.8 
 60.9 
 61.0 
 
 
 
 65.5 
 
 
 41.1 
 
 
 6.8 
 
 
 26.0 
 
 
 49.8 
 
 
 61.1 
 
 30 
 21 
 30 
 22 
 30 
 
 2131.3 
 
 24 47. 1 
 28 2.2 
 31 16.3 
 3429.7 
 
 65.3 
 65.0 
 64.7 
 64.5 
 
 449.9 
 651.6 
 851.7 
 1050.2 
 1247.1 
 
 40.6 
 40.0 
 39.5 
 39.0 
 
 1640.4 
 1658.9 
 1715.8 
 1730.8 
 1744.1 
 
 6.2 
 5.6 
 5.0 
 44 
 
 45 29.6 
 4410.2 
 4249.2 
 4126.9 
 40 3.1 
 
 26.5 
 27.0 
 27.4 
 27.9 
 
 48 23.4 
 4553.1 
 4322.0 
 4050.0 
 
 3817.1 
 
 50.1 
 50.4 
 50.7 
 51.0 
 
 58 29.0 
 5525.6 
 5222.0 
 4918.1 
 46 14.2 
 
 61.1 
 61.2 
 61.3 
 61.3 
 
 
 
 64.2 
 
 
 38.4 
 
 
 3.9 
 
 
 28.4 
 
 
 51.2 
 
 
 61.4 
 
 23 
 30 
 24 
 30 
 25 
 
 3742.2 
 4053.8 
 44 4.5 
 4714.3 
 50 23.2 
 
 63.9 
 63.6 
 63.2 
 63.0 
 
 1442.3 
 1636.0 
 1828.0 
 2018.5 
 
 22 7.2 
 
 37.9 
 37.3 
 
 36.8 
 36.2 
 
 1755.7 
 18 5.5 
 1813.6 
 1819.9 
 18 24.4 
 
 33 
 2.7 
 2.1 
 1.5 
 
 33 37.9 
 3711.3 
 35433 
 34 13.9 
 3243.2 
 
 28.9 
 29.3 
 29.8 
 30.2 
 
 3543.4 
 33 8.9 
 3033.5 
 2757.3 
 2520.4 
 
 51.5 
 51.8 
 52.1 
 52.3 
 
 43 10.0 
 40 5.7 
 37 1.2 
 3356.6 
 3051.9 
 
 61.4 
 61.5 
 61.5 
 61.6 
 
 
 
 62.7 
 
 
 35.7 
 
 *' , 
 
 1.0 
 
 
 307 
 
 
 526 
 
 
 61 6 
 
 30 
 26 
 30 
 27 
 30 
 
 5331.2 
 5633.2 
 59 44.2 
 
 62.3 
 R2.0 
 61.7 
 61.3 
 
 23 54.4 
 25 39.8 
 2723.7 
 29 5.8 
 3046.3 
 
 35.1 
 34.6 
 34.1 
 33.5 
 
 1827.3 
 IS 28.4 
 1827.8 
 1825.4 
 1821.3 
 
 0.4 
 0.2 
 0.8 
 1.4 
 
 31 11.0 
 29 37.4 
 28 2.5 
 26 26.3 
 
 2448.7 
 
 31.2 
 31.6 
 32.1 
 32.5 
 
 2242.6 
 20 4.0 
 1724.7 
 1444.7 
 12 3.8 
 
 52.9 
 53.1 
 3.3 
 53.6 
 
 2747.0 
 2442.0 
 21 37.0 
 1831.8 
 1526.6 
 
 61.7 
 61.7 
 61.7 
 61.7 
 
 249.3 
 5 53.3 
 
 
 
 61.0 
 
 
 33.0 
 
 
 1.9 
 
 
 33.0 
 
 
 53.8 
 
 
 61,7 
 
 28 
 30 
 29 
 30 
 30 
 
 856.3 
 1158.3 
 1459.3 
 1759.2 
 2057.9 
 
 60.7 
 60.3 
 60.0 
 59.6 
 
 32 25.2 
 34 2.3 
 35 37.8 
 3711.5 
 3843.6 
 
 32.4 
 31.8 
 31.2 
 30.7 
 
 1815.6 
 18 8.0 
 1758.8 
 1747.9 
 1735.2 
 
 2.5 
 3.1 
 3.6 
 4.2 
 
 23 9.7 
 21 29.5 
 1948.0 
 18 5.0 
 1620.8 
 
 33.4 
 33.8 
 34.3 
 34.7 
 
 922.3 
 640.0 
 357.0 
 1 13.3 
 
 54.1 
 54.3 
 54.6 
 
 54.8 
 
 1221.4 
 916.1 
 610.8 
 3 5.4 
 0.0 
 
 61.8 
 61.8 
 61.8 
 61.8 
 
 5828.9 
 
 ! 10 
 
 
 12 
 
 
 13 
 
 
 12 
 
 
 9 
 
 
 7 
 
 
TABLE LI. 
 
 Equation of Moon's Centre. 
 Argument. Anomaly corrected. 
 
 75 
 
 
 VI* 
 
 
 VII* 
 
 
 VIII* 
 
 
 IX* 
 
 
 X* 
 
 
 XI* 
 
 
 
 
 Diff 
 
 
 Diff o 
 
 Diff 
 
 
 Diff Lo 
 
 Diff 
 
 
 Diff 
 
 
 5 
 
 forlO 
 
 2 
 
 for 10, * 
 
 1 
 
 for 10 
 
 
 
 for 10 
 
 x 
 
 for 10 
 
 5 
 
 forlO 
 
 / 
 
 , ,, 
 
 
 , 
 
 / // 
 
 
 , 
 
 
 , ,, 
 
 
 / 
 
 
 15 
 30 
 16 
 30 
 17 
 30 
 
 28 10.6 
 2510.5 
 2210.9 
 1911.6 
 1612.7 
 1314.2 
 
 60.0 
 59.9 
 59.8 
 59.6 
 59.5 
 
 45 4.8 
 4245.9 
 4028.1 
 3811.2 
 3555.4 
 3340.6 
 
 46.3 
 45.9 
 45.6 
 45.3 
 44.9 
 
 134.1 
 032.6 
 59~32l3 
 5834.2 
 5737.3 
 56 42 
 
 20.5 
 20.0 
 19.5 
 19.0 
 18.4 
 
 4840.1 
 49 19.9 
 50 1.4 
 5044.6 
 5129.7 
 5216.5 
 
 13.3 
 
 13.8 
 14.-4 
 15.0 
 15.6 
 
 1910.7 
 21 30.6 
 2352.1 
 2615.1 
 28 39.5 
 31 5.3 
 
 46.6 
 
 47.2 
 47.7 
 48.1 
 48.6 
 
 15 10.4 
 8 34.4 
 21 59.0 
 2524.2 
 28 49.8 
 3216.0 
 
 68.0 
 68.2 
 68.4 
 68.5 
 68.7 
 
 
 
 594 
 
 
 44.6 
 
 J17.9 
 
 
 16.2 
 
 
 49.1 
 
 
 68.9 
 
 18 
 30 
 19 
 30 
 20 
 
 1016.1 
 718.3 
 421.1 
 124.2 
 
 59.3 
 59.1 
 59.0 
 
 58.8 
 
 31 26.9 
 29 14.2 
 27 2.6 
 2452.1 
 
 2242.7 
 
 44.2 
 43.9 
 43.5 
 43.1 
 
 5548.3 
 5456.1 
 54 5.6 
 5316.6 
 5229.2 
 
 17.4 
 16.8 
 16.3 
 15.8 
 
 53 5.1 
 53 55.4 
 5447.5 
 5541.3 
 56 37.0 
 
 16.8 
 17.4 
 17.9 
 18.6 
 
 3332.5 
 36 1.2 
 
 3831.2 
 41 2.7 
 43 35.5 
 
 ! 49.6 
 50.0 
 50.5 
 50.9 
 
 3542.7 
 39 9.9 
 4237.5 
 46 5.5 
 49 34.0 
 
 69.1 
 69.2 
 69.3 
 69.5 
 
 58278 
 
 
 
 58.6 
 
 
 42.8 
 
 
 15.3 
 
 
 19.1 
 
 
 51.4 
 
 
 69.6 
 
 305531.9 
 21 0,5236.4 
 304941.4 
 22 04646.9 
 304352.9 
 
 58.5 
 58.3 
 
 58.2 
 58.0 
 
 2034.4 
 1827.2 
 1621.1 
 1416.2 
 1212.4 
 
 42.4 
 42.0 
 41.6 
 41.3 
 
 51 43.4 
 5059.2 
 50 16.6 
 4935.7 
 48 56.3 
 
 14.7 
 14.2 
 13.6 
 13.1 
 
 5734.3 
 58 33.5 
 59 34.4 
 03771 
 141 5 
 
 19.7 
 20.3 
 20.9 
 21.5. 
 
 46 9.7 
 48 45.2 
 5122.1 
 54 0.3 
 5639.9 
 
 51.8 
 52.3 
 52.7 
 53.2 
 
 53 2.8 
 5631.9 
 
 1TT76 
 
 331.5 
 7 1.8 
 
 69.7 
 69.9 
 70.0 
 70.1 
 
 
 
 57.8 
 
 
 40.9 
 
 
 12.6 
 
 
 22.1 
 
 
 53.6 
 
 
 70.2 
 
 23 
 30 
 
 24 
 30 
 25 
 
 4059.4 
 38 6.5 
 3514.1 
 
 32 22.2 
 2930.9 
 
 57.6 
 57.5 
 57.3 
 57.1 
 
 10 9.7 
 8 8.3 
 6 8.0 
 4 8.9 
 210.9 
 
 40.5 
 40.1 
 39.7 
 39.3 
 
 4818.6 
 4742.6 
 47 8.1 
 4635.3 
 46 4.2 
 
 12.0 
 11.5 
 10.9 
 10.4 
 
 247.7 
 355.6 
 5 5.3 
 
 616.7 
 
 729.8 
 
 22.6 
 23.2 
 23.8 
 
 24.4 
 
 5920.7 
 
 54.0 
 54.5 
 54.9 
 55.3 
 
 1032.3 
 14 3.1 
 1734.2 
 21 5.5 
 2437.0 
 
 70.3 
 70.4 
 70.4 
 70.5 
 
 2 2.8 
 446.2 
 730.9 
 1016.8 
 
 
 j 56.9 
 
 I 
 
 38.9 
 
 
 9.8 
 
 
 25.0 
 
 
 55.7 
 
 
 70.6 
 
 30 
 26 
 30 
 27 
 30 
 
 2640.2' 
 2350.0 .;?!!' 
 
 21 &5 Sr2 
 
 181L5 i561 
 15 23.2 5fU 
 
 014.2 
 
 38.5 
 38.1 
 37.7 
 37.3 
 
 45 34.8 
 45 6.9 
 4440.8 
 4416.3 
 43 53.5 
 
 9.3 
 
 8.7 
 8.2 
 7.6 
 
 844.7 
 10 1.3 
 11 19.7 
 1239.8 
 14 1.6 
 
 25.5 
 26.1 
 
 26.7 
 27.3 
 
 13 4.0 
 1552.4 
 1842.0 
 2132.9 
 
 2424.8 
 
 56.1 
 56.5 
 57.0 
 57.3 
 
 28 8.8 
 3140.7 
 35 12.8 
 3845.1 
 4217.3 
 
 70.6 
 70.7 
 70.7 
 70.8 
 
 58 18.7 
 5024.4 
 J431.3 
 5239.5 
 
 
 
 55.9 
 
 
 36.9 
 
 
 7.0 
 
 
 27.8 
 
 57.7 
 
 
 70.8 
 
 28 
 30 
 29 
 SO 
 dO 
 
 1235.5 
 948.4 
 7 2.0 
 416.2 
 131.1 
 
 55.7 
 55.5 | 
 55.3 
 55.0 
 
 5048.9 
 48 59.6 
 4711.5 
 45 24.7 
 4339.2 
 
 36.4 
 36.0 
 35.6 
 35.2 
 
 4332.4 
 4312.9 
 4255.2 
 4239.1 
 
 4224.8 
 
 6.5 
 5.9 
 5.4 
 4.8 
 
 1525.1 
 1650.4 
 1817.3 
 1946.0 
 21 16.4 
 
 4 27 18.0 
 
 AiO.rr o/\ -if) o: 
 
 29.0 ! 1 
 
 29 ' 6 36 44 
 3(U 39 il 
 
 58.1 
 58.5 
 58.9 
 59.2 
 
 4549.7 
 49 22.2 
 52 54.8 
 56 27.4 
 0.0 
 
 70.8 
 70.9 
 70.9 
 70.9 
 
 
 4 
 
 
 1 
 
 
 
 
 1 
 
 
 3 
 
 
 7 
 
 
76 
 
 TABLE LII. 
 
 Variation. 
 Argument. Variation, corrected. 
 
 
 0* 
 
 
 I* 
 
 
 II* 
 
 
 111^ 
 
 
 iv 
 
 
 V 
 
 
 fo 
 
 DiffJl 
 
 Diff. 
 
 1 
 
 Diff. 
 
 Diff. 
 
 
 
 Diff. 
 
 
 
 Diff. 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 
 
 I 
 
 
 
 
 L 
 
 
 
 
 
 
 38 0.0 
 39 13.3 
 40 26.5 
 |41 39.5 
 J42 52.2 
 44 4.5 
 
 73.3 
 73.3 
 73.0 
 72.7 
 72.3 
 
 8 1.5 
 835.5 
 9 7.2 
 936.5 
 10 3.4 
 10 27.9 
 
 34.0 
 31.7 
 29.3 
 26.9 
 24.5 
 
 6 57.9 g 
 6 18.0^-J 
 5 35.9 f' 1 
 
 45L7 Ir1 
 
 A K K.46.2 
 4.S 2 
 3 17.3< 48 ^ 
 
 35 54.4 
 34 40.4 
 33 26.6 
 32 13.0 
 30 59.6 
 29 46.7 
 
 74.0 
 
 73.8 
 73.6 
 73.4 
 72.9 
 
 529.5 
 454.2 
 421.3 
 350.6 
 322.3 
 256.5 
 
 35.3 
 
 32.9 
 30.7 
 28.3 
 25.8 
 
 6 1.6 
 641.6 
 723.9 
 8 8.4 
 8 55.0 
 943.7 
 
 // 
 40.0 
 42.3 
 44.5 
 46.6 
 48.7 
 
 
 
 71.9 
 
 
 22.0 
 
 
 50.1 
 
 
 72.4! 
 
 23.4 
 
 
 50.8 
 
 6 
 7 
 8 
 9 
 10 
 
 45 16.4 
 46 27.7 
 47 38.4 
 48 48.3 
 49 57.4 
 
 71.3 
 70.7 
 69.9 
 69.1 
 
 10 49.9 
 11 9.4 
 11 26.4 
 11 40.9 
 11 52.9 
 
 19.5 
 17.0 
 14.5 
 12.0 
 
 2 27.2 - ( 
 
 1 35.3 ^i; 
 JLiLSSs 
 
 5946.1 5?1 
 58490 
 
 28 34.3 
 27 22.4 
 26 11.2 
 25 0.7 
 2351.1 
 
 233.1 
 712 215U 
 
 1 L.& -t eo r/ 
 
 7ft * 1 1 53.7 
 
 /U.O | , 07 Q 
 fiq R j 1 d7.8 
 
 by ' b 124.5 
 
 21.0 
 18.4 
 15.9 
 13.3 
 
 10 34.5 
 1 1 27.3 
 12 22.0 
 13 18.6 
 14 16.9 
 
 52.8 
 54.7 
 56.6 
 58.3 
 
 
 
 68.2 
 
 
 9.3 
 
 58.8 
 
 
 68.8 
 
 
 10.8 
 
 
 60.1 
 
 11 
 12 
 13 
 14 
 15 
 
 51 5.6 
 52 12.8 
 53 18.9 
 54 23.8 
 55 27.5 
 
 67.2 
 66.1 
 64.9 
 63.7 
 
 12 2.2 
 12 9.0 
 12 13.2 
 12 14.8 
 12 13.9 
 
 6.8 
 4.2 
 1.6 
 0.9 
 
 57 50.2 
 56 50.0 
 55 48.3 
 54 45.2 
 53 40.9 
 
 60.2 
 61.7 
 63.1 
 64.3 
 
 22 42.3 
 21 34.5 
 20 27.9 
 19 22.3 
 18 18.0 
 
 67.8 \ 
 
 s-; ' 
 
 643 0570 
 64 3 56.7 
 
 8.2 
 5.5 
 3.0 
 0.3 
 
 15 17.0 
 16 18.7 
 1722.0 
 1826.9 
 19 33.1 
 
 61.7 
 63.3 
 64.9 
 66.2 
 
 
 
 62.3 
 
 
 3.6 
 
 
 65.6 
 
 
 63.0 
 
 
 2.3 
 
 
 67.6 
 
 16 
 
 17 
 18 
 19 
 20 
 
 56 29.8 
 57 30.7 
 5830.1 
 59 28.0 
 
 60.9 
 59.4 
 57.9 
 56.2 
 
 12 10.3 
 12 4.2 
 11 55.5 
 11 44.2 
 11 30.5 
 
 6.1 
 8.7 
 11.3 
 13.7 
 
 52 35.3 
 
 51 28.5 
 50 20.7 
 49 11.9 
 48 2.2 
 
 66.8 
 67.8 
 68.8 
 69.7 
 
 17 15.0 
 16 13.4 
 15 13.2 
 14 14.6 
 13 17.5 
 
 61.6 
 60.2 
 58.6 
 57.1 
 
 059.0 
 1 3.9 
 1 11.5 
 1 21.6 
 134.4 
 
 4.9 
 7.6 
 10.1 
 
 12.8 
 
 20 40.7 
 21 49.6 
 22 59.6 
 24 10.8 
 25 22.9 
 
 68.9/ 
 70.0 
 71.2 
 72.1 
 
 24 2 
 
 
 
 54.5 
 
 
 16.4 
 
 
 70.5 
 
 
 55.3 
 
 
 15.4 
 
 
 73.0 
 
 21 
 22 
 23 
 24 
 25 
 
 1 18.7 
 2 11.4 
 3 2.3 
 351.2 
 438.2 
 
 52.7 
 50.9 
 48.9 
 47.0 
 
 11 14.1 
 10 55.3 
 10 34.0 
 10 10.2 
 944.0 
 
 18.8 
 21.3 
 23.8 
 26.2 
 
 46 51. 7 _ 
 4540.51^ 
 
 4428 - 6! 725 
 481.l{9 
 
 42 3.2i 72 ' 9 
 
 12 22.2 
 11 28.5 
 1036.7 
 9 46.8! 
 
 8 58.81 
 
 53.7 
 51.8 
 49.9 
 48.0 
 
 1 49.8 
 2* 7.8 
 2 28.3 
 251.4 
 3 16.9 
 
 18.0 
 20.5 
 23.1 
 2o.5 
 
 26 35.9 
 27 49.8 
 29 4.5 
 30 19.7 
 31 35.6 
 
 73.9 
 
 74.7 
 75.2 
 
 75.91 
 
 
 
 44.9 
 
 
 28.G ! |73.3 
 
 146.1 
 
 28.1 
 
 
 76.3 
 
 26 
 27 
 28 
 29 
 30 
 
 523.1 
 6 6.0 
 646.7 
 725.2 
 8 1.5 
 
 42.9 
 40.7 
 38.5 
 36.3 
 
 9 15.4 
 844.5 
 8 11.2 
 735.7 
 6 57.9 
 
 30.9 
 33.3 
 35.5 
 
 37.8 
 
 4049.9! 7q7 
 39 36.2 ' 
 38 ftt.4 X A 
 
 o-r o A 74.0 
 
 37 8.4 74Q 
 35 54.4 
 
 8 12.7! 
 728.7 
 646.8 
 6 7.1 
 529.5 
 
 A i n 3 45 - 
 44.0 \ A 1K c 
 
 41 q ! 4 15 6 
 
 39 7 i 4 48 ' 5 
 o? I 5 23.9 
 j7.b /. , c 
 16 1.6 
 
 30.6 
 32.9 
 35.4 
 37.7 
 
 3251.9 
 34 8.6 
 35 25.6 
 36 42.7 
 38 0.0 
 
 76.7 
 77.0 
 77.1 
 77.3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 1 
 
 
 
 
 loo 
 
 1? 
 
 
 
 
 
TABLE LII. 
 Variation. 
 
 Argument. Variation corrected. 
 
 1 VI* 
 
 
 VII* 
 
 . 
 
 VIII* 
 
 
 IX* 
 
 
 X* 
 
 
 XI* 
 
 ' J 
 
 1 
 
 
 
 Diff. 
 
 1 
 
 Diff. 
 
 1 
 
 Diff. 
 
 Oo 
 
 Diff. 
 
 
 
 Diff. 
 
 
 
 Diff; 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 
 38 0.0 
 39 17.3 
 40 34.4 
 4151.4 
 43 8.1 
 44 24.4 
 
 77.3 
 
 77.1 
 77.0 
 76.7 
 76.3 
 
 958.4 
 1036.1 
 11 11.5 
 1144.4 
 1215.0 
 1243.1 
 
 37.7 
 35.4 
 32.9 
 30.6 
 28.1 
 
 1030.5 
 952.9 
 913.2 
 831.3 
 747.3 
 7 1.2 
 
 37.6 
 39.7 
 41.9 
 44.0 
 46.1 
 
 40 5.6 
 3851.6 
 3737.6 
 36 23.8 
 3510.1 
 33 56.8 
 
 74.0 
 74.0 
 73.8 
 73.7 
 73.3 
 
 9 2.1 
 824.3 
 748.8 
 715.5 
 644.6 
 616.0 
 
 37.8 
 35.5 
 33.3 
 30.9 
 28.6 
 
 758.5 
 834.8 
 913.3 
 954.0 
 1036.9 
 1121.8 
 
 36.3 
 
 38.5 
 40.7 
 42.9 
 44.9 
 
 
 
 75.9 
 
 
 25.5 
 
 
 48.0 
 
 
 72.9 
 
 
 26.2 
 
 
 47.0 
 
 6 
 
 7 
 8 
 9 
 10 
 
 4540.3 
 4655.5 
 4810.2 
 4924.1 
 5037.1 
 
 75.2 
 74.7 
 73.9 
 73.0 
 
 13 8.6 
 1331.7 
 1352.2 
 1410.2 
 1425.6 
 
 23.1 
 20.5 
 18.0 
 15.4 
 
 613.2 
 523.3 
 431.5 
 337.8 
 242.5 
 
 49.9 
 51.8 
 53.7 
 55.3 
 
 3243.9 
 3131.4 
 3019.5 
 29 8.3 
 
 2757.8 
 
 72.5 
 71.9 
 71.2 
 70.5 
 
 5 49.8 
 526.0 
 5 4.7 
 445.9 
 429.5 
 
 23.8 
 21.3 
 18.8 
 16.4 
 
 12 8.8 
 12S7.7 
 1348.6 
 1441.3 
 1535.8 
 
 48.9 
 50.9 
 52.7 
 54.5 
 
 
 
 72.1 
 
 
 12.8 
 
 
 57.1 
 
 
 69.7 
 
 
 13.7 
 
 
 56.2 
 
 11 
 12 
 13 
 
 14 
 15 
 
 5149.2 
 53 0.4 
 54 10.4 
 55 19.3 
 5626.9 
 
 71.2 
 70.0 
 68.9 
 67.6 
 
 1438.4 
 1448.5 
 1456.1 
 15 1.0 
 15 3.3 
 
 10.1 
 7.6 
 4.9 
 2.3 
 
 145.4 
 046.8 
 
 58.6 
 60.2 
 61.6 
 63.0 
 
 2648.1 
 2539.3 
 2431.5 
 23 24.7 
 2219.1 
 
 68.8 
 67.8 
 66.8 
 65.6 
 
 415.8 
 4 4.5 
 355.8 
 349.7 
 346.1 
 
 11.3 
 8.7 
 6.1 
 3.6 
 
 1632.0 
 1729.9 
 1829.3 
 19 30.2 
 20 32.5 
 
 57.9 
 59.4 
 60.9 
 62.3 
 
 5946.6 
 58 45.0 
 5742 
 
 
 
 66.2 
 
 
 0.3 
 
 
 64.3 
 
 
 64.3 
 
 
 0.9 
 
 
 63.7 
 
 16 
 17 
 18 
 19 
 20 
 
 5733.1 
 58 38.0 
 5941.3 
 
 64.9 
 63.3 
 61.7 
 60.1 
 
 15 3.0 
 15 0.0 
 
 1454.5 
 1446.3 
 1435.5 
 
 3.0 
 5.5 
 
 8.2 
 10.8 
 
 5637.7 
 5532.1 
 5425.5 
 5317.7 
 52 8.9 
 
 65.6 
 66.6 
 
 67.8 
 68.8 
 
 21 14.8 
 2011.7 
 1910.0 
 18 9.8 
 1711.0 
 
 63.1 
 61.7 
 60.2 
 58.8 
 
 3 45.2 
 346.8 
 351.0 
 357.8 
 4 7.1 
 
 1.6 
 4.2 
 6.8 
 9.3 
 
 2136.2 
 2241.1 
 2347.2 
 2454.4 
 26 2.0 
 
 64.9 
 66.1 
 67.2 
 68.2 
 
 043.0 
 143 1 
 
 
 
 58.3 
 
 
 13.3 
 
 
 69.6 
 
 
 57.1 
 
 
 12.0 
 
 
 69.1 
 
 21 
 22 
 23 
 24 
 25 
 
 241.4 
 38.0 
 432.7 
 525.5 
 616.3 
 
 56.6 
 54.7 
 
 (TO Q 
 
 50.8 
 
 1422.2 
 14 6.3 
 1347.9 
 1326.9 
 13 3.5 
 
 15.9 
 18.4 
 21.0 
 23.4 
 
 5059.3 
 49 48.8 
 48 37.6 
 4725.7 
 46 13.3 
 
 70.5 
 71.2 
 71.9 
 72.4 
 
 1613.9 
 1518.4 
 1424.7 
 1332.8 
 1242.7 
 
 55.5 
 53.7 
 51.9 
 50.1 
 
 419.1 
 433.6 
 450.6 
 510.1 
 532.1 
 
 14.5 
 17.0 
 19.5 
 22.0 
 
 2711.7 
 2821.6 
 29 32.3 
 3043.6 
 31 55.5 
 
 69.9 
 70.7 
 71.3 
 71.9 
 
 
 
 48.7 
 
 
 25.8 
 
 
 72.9 
 
 
 48.2 
 
 
 24.5 
 
 
 72.3 
 
 26 
 
 27 
 28 
 29 
 30 
 
 7 5.0 
 751.6 
 836.1 
 918.4 
 958.4 
 
 46.6 
 44.5 
 42.3 
 40.0 
 
 1237.7 
 12 9.4 
 1138.7 
 11 5.8 
 1030.5 
 
 28.3 
 30.7 
 32.9 
 35.3 
 
 45 0.4 
 4347.0 
 42 33.4 
 4119.6 
 40 5.6 
 
 73.4 
 73.6 
 73.8 
 74.0 
 
 1154.5 
 11 8.3 
 1024.1 
 942.0 
 9 2.1 
 
 46.2 
 44.2 
 42.1 
 39.9 
 
 556.6 
 623.5 
 
 652.8 
 724.5 
 
 758.5 
 
 26.9 
 29.3 
 31.7 
 34.0 
 
 33 7.8 
 3420.5 
 35 33.5 
 36 46.7 
 38 0.0 
 
 72.7 
 73.0 
 73.2 
 73.3 
 
 
 1 
 
 
 1 
 
 
 
 
 
 Oo 
 
 
 
 
 
 
 
 
78 ' TABLE LIII. Reduction. 
 
 Argument. Supplement of Node + Moon's Orbit Longitude. 
 
 
 Oa Vis 
 
 Diff. 
 
 Is VIIsDiff IlaVIHs Diff. 
 
 Ills IXs 
 
 Diff. 
 
 IVs Xs 
 
 Diff. 
 
 VaXIs 
 
 Diff.' 
 
 
 
 , 
 
 
 . /, 
 
 
 / // 
 
 
 / ,/ 
 
 
 , 
 
 
 , // 
 
 
 
 
 1 
 
 2 
 3 
 4 
 
 5 
 
 7 0.0 
 6 45.6 
 6 31.2 
 6 16.9 
 6 2.6 
 5 48.4 
 
 14.4 
 14.4 
 14.3 
 14.3 
 14.2 
 
 1 3.0 ' " 
 
 o se.o ; 
 
 49.5 J;J 
 43.4 J-* 
 37.8 f J 
 3S.7 - 1 
 
 1 3.0 
 1 10.4 
 1 18.3 
 1 26.5 
 1 35.2 
 1 44.2 
 
 7.4 
 7.9 
 
 8.2 
 8.7 
 9.0 
 
 7 0.0 
 
 7 14.4 
 7 28.8 
 743.1 
 7 57.4 
 8 11.6 
 
 14.4 
 14.4 
 14.3 
 14.3 
 14.2 
 
 12 57.0 
 13 4.0 
 13 10.5 
 13 16.6 
 13 22.2 
 13 27.3 
 
 7.0 
 6.5 
 6.1 
 5.6 
 5.1 
 
 12 57.0 
 12 49.6 
 12 41.7 
 12 33.5 
 12 24.8 
 12 15.8 
 
 7.4 
 7.9 
 8.21 
 8.7; 
 
 9.0 
 
 
 
 14.1 
 
 4.5 
 
 
 9.5 
 
 
 14.1 
 
 
 4.5 
 
 
 '9.5; 
 
 6 
 
 7 
 8 
 9 
 10 
 
 5 34.3 
 5 20.3 
 5 6.4 
 
 4 52.6 
 4 39.0 
 
 14.0 
 13.9 
 13.8 
 13.6 
 
 28.2 
 23.9 *3 
 20.0 ,r 
 16.8 ** 
 
 14.1 ^ ' 
 
 1 53.7 
 2 8.5 
 2 13.7 
 2 24.2 
 S 35.0 
 
 9.8 
 10.2 
 10.5 
 10.8 
 
 8 25.7 
 8 39.7 
 8 53.6 
 9 7.4 
 9 21.0 
 
 14.0 
 13.9 
 13.8 
 13.6 
 
 13 31.8 
 13 36.1 
 13 40.0 
 13 43.2 
 13 45.9 
 
 4.3 
 3.9 
 3.2 
 
 2.7 
 
 12 6.3 
 11 56.5 
 11 46.3 
 11 35.8 
 11 25.0 
 
 9.8 
 10.2 
 10.5 
 10.8 
 
 
 
 13.4 
 
 2.3 
 
 
 11.2 
 
 
 13.4 
 
 
 2.3 
 
 
 11.2 
 
 11 
 12 
 13 
 14 
 15 
 
 4 25.6 
 4 12.3 
 3 59.3 
 3 46.5 
 3 33.9* 
 
 13.3 
 13.0 
 12.8 
 12.6 
 
 11.8 . 
 10.1 :{' 
 8-8 i'? 
 8.1 J' 
 
 7.8 ' d 
 
 2 46.2 
 2 57.7 
 3 9.5 
 3 21.6 
 3 33.9 
 
 11.5 
 
 11.8 
 12.1 
 12.3 
 
 9 34.4 
 9 47.7 
 10 0.7 
 10 13.5 
 10 26.1 
 
 13.3 
 13.0 
 
 12.8 
 12.6 
 
 13 48.2 
 13 49.9 
 13 51.2 
 13 51.9 
 13 52.2 
 
 1.7 
 1.3 
 0.7 
 0.3 
 
 11 13.8 
 11 2.3 
 
 10 50.5 
 10 38.4 
 10 26.1 
 
 11.5 
 11.8 
 12.1 
 12.3 
 
 
 
 12.3 ;0.3 
 
 
 12.6 
 
 12.3 
 
 
 0.3 
 
 
 12.6 
 
 16 
 17 
 
 18 
 19 
 
 20 
 
 3 21.6 
 3 9.5 
 2 57.7 
 2 46.2 
 2 35.0 
 
 " Si 
 
 }{$ iai 
 
 }, JO 11.8 
 v -,fl 14.1 
 
 0.7 
 1.3 
 1.7 
 2'.3 
 
 3 46.5 
 3 59.3 
 4 12.3 
 4 25.6 
 4 39.0 
 
 12.8 '}j}j!!-*' 12.1 
 13.0 . go H.8 
 
 13.3 ,:i 11.5 
 
 ,n A 11 10. - , ~ 
 
 13 4 ; 11 25.0 1L2 
 
 13 51.9 
 13 51.2 
 13 49.9 
 13 48.2 
 13 45.9 
 
 0.7 
 1.3 
 1.7 
 2.3 
 
 10 13.5 
 10 0.7 
 9 47.7 
 9 34.4 
 9 21.0 
 
 12.8 
 13.0 
 13.3 
 13.4 
 
 
 
 10.8 
 
 
 2.7 
 
 13.6 1 
 
 10.8 
 
 
 2.7 
 
 
 13.6 
 
 21 
 22 
 23 
 24 
 25 
 
 2 24.2 
 2 13.7 
 2 3.5 
 53.7 
 44.2 
 
 10.5 
 10.2 
 9.8 
 9.5 
 
 16.8 
 20.0 
 23.9 
 28.2 
 32.7 
 
 3.2 
 3.9 
 4.3 
 4.5 
 
 4 52.6 , 
 
 5 6.4 *J; 
 
 5 20.3 JJ'J 
 5 34.3 JJJ 
 5 48.4 
 
 11 35.8 
 11 46.3 
 11 56.5 
 12 6.3 
 12 15.8 
 
 10.5 
 10.2 
 9.8 
 9.5 
 
 13 43.2 
 13 40.0 
 13 36.1 
 13 31.8 
 13 27.3 
 
 3.2 
 3.9 
 4.3 
 4,5 
 
 9 7.4 
 8 53.6 
 8 39.7 
 8 25.7 
 8 11.6 
 
 13.8 
 13.9 
 14.0 
 14.1 
 
 
 
 9.0 
 
 
 5.1 
 
 14.2 
 
 
 9.0 
 
 
 5.1 
 
 
 14.2 
 
 2fi 
 27 
 28 
 29 
 30 
 
 35.2 
 28.5 
 183 
 10.4 
 30 
 
 8.7 
 82 
 7.9 
 7.4 
 
 37.8 
 43.4 
 49.5 
 56.0 
 1 3.0 
 
 56 
 6.1 
 6.5 
 7.0 
 
 6 2.6 
 6 16.9 
 6 31.2 
 6 45.6 
 7 0.0 
 
 1d 12 24.8 
 Jj 31' 12 33.5 
 J! 12 41.7 
 
 14<4 J12 57.0 
 
 8.7 
 8.2 
 7.9 
 7.4 
 
 13 22.2 
 13 16.6 
 13 10.5 
 13 4.0 
 12 57.0 
 
 5.6 
 6.1 
 6.5 
 7.0 
 
 7 57.4 
 7 43.1 
 7 28.8 
 7 14.4 
 7 0.0 
 
 14.3 
 14.3 
 144 
 M4 
 
 TABLE LIV. Lunar Nutation in Longitude. 
 Argument. Supplement .of the Node. 
 
 
 0* 
 
 I* 
 
 !! 
 
 Ills 
 
 IV 
 
 Vs 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 f 
 
 _j_ 
 
 
 o 
 
 
 
 tt 
 
 ,, 
 
 
 
 // 
 
 // 
 
 o 
 
 
 
 0.0 
 
 8.5 
 
 14.8 
 
 17.3 
 
 15.2 
 
 8.8 
 
 30 
 
 2 
 
 0.6 
 
 9.0 
 
 15.1 
 
 17.2 
 
 14.9 
 
 8.1 
 
 28 i 
 
 4 
 
 1.8 
 
 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 
 
 
 
 
 XI 
 
 X* 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI* 
 
 
TABLE LV. 79 
 
 Moon's Distance from the North Pole of the Ecliptic. 
 Argument. Supplement of Node+Moon's Orbit Longitude. 
 
 
 III* 
 
 IV* 
 
 
 V* 
 
 
 VI 
 
 
 VII* 
 
 
 VIII* 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 84 
 
 85 
 
 Diff. 
 for 10 
 
 87 
 
 Diff. 
 for 10 
 
 89 
 
 Diff. 
 
 for 10 
 
 92 
 
 Diff. 
 
 for 10 
 
 94 
 
 
 Q 
 30 
 1 
 30 
 2 
 30 
 
 3916.0 
 3916.7 
 3918.8 
 3922.4 
 3927.3 
 3933.7 
 
 2042.7 
 22 4.2 
 23 27.0 
 2451.0 
 2616.2 
 2742.6 
 
 27.2 
 27.6 
 28.0 
 
 28.4 
 28.8 
 
 1346.6 
 16 6.9 
 1827.8 
 2049.5 
 2311.8 
 25 34.8 
 
 46.8 
 47.0 
 47.2 
 47.4 
 47.7 
 
 48 0.0 
 5041.4 
 5322.9 
 56 4.3 
 
 5845.7 
 1270 
 
 53.8 
 53.8 
 53.8 
 53.8 
 53.8 
 
 / // 
 
 2213.4 
 2433.1 
 2652.2 
 2910.2 
 31 27.5 
 3344.2 
 
 4^.6 
 46.4 
 46.0 
 45.8 
 45.6 
 
 / // 
 1517.3 
 1637.7 
 1756.8 
 1914.6 
 2031.3 
 2146.7 
 
 O ' 
 
 30 
 30 
 29 
 30 
 
 28 
 30 
 
 
 
 
 29 2 
 
 
 47.9 
 
 
 53.8 
 
 
 45.3 
 
 
 
 3 
 
 30 
 4 
 30 
 5 
 
 3941.5 
 39 50.6 
 40 1.2 
 4013.2 
 4026.7 
 
 2910.1 
 3038.9 
 32 8.8 
 3339.9 
 35 12.2 
 
 29.6 
 30.0 
 30.4 
 30.8 
 
 2758.5 
 30 22.8 
 3247.7 
 35 13.2 
 3739.3 
 
 48.1 
 48.3 
 48.5 
 48.7 
 
 4 8.3 
 649.5 
 930.6 
 1211.6 
 1452.5 
 
 53.7 
 53.7 
 53.7 
 53.6 
 
 36 0.2 
 3815.3 
 40 29.7 
 42 43.3 
 4456.2 
 
 45.0 
 44.8 
 44.5 
 44.3 
 
 23 0.8 
 2413.7 
 25 25.3 
 26 35.7 
 2744.8 
 
 27 
 
 do 
 
 26 
 30 
 25 
 
 
 
 
 31.1 
 
 
 48.9 
 
 
 53.6 
 
 
 44.0 
 
 
 
 30 
 6 
 30 
 7 
 30 
 
 4041.5 
 40 57.7 
 41 15.4 
 4134.4 
 41 54.8 
 
 3645.6 
 3820.1 
 39 55.8 
 41 32.7 
 43 10.6 
 
 31.5 
 31.9 
 32.3 
 32.6 
 
 40 6.1 
 42 33.4 
 45 1.2 
 4729.6 
 4958.6 
 
 49.1 
 49.3 
 49.5 
 49.7 
 
 1733.3 
 20 14.0 
 22 54.4 
 2534.8 
 28 14.9 
 
 53.6 
 53.5 
 53.5 
 53.4 
 
 47 8.1 
 49 19.4 
 51 29.7 
 5339.3 
 5548.0 
 
 43.8 
 43.4 
 43.2 
 42.9 
 
 28 52.6 
 29 59.0 
 31 4.3 
 32 8.2 
 33 10.9 
 
 30 
 24 
 30 
 23 
 30 
 
 
 
 
 33.0 
 
 
 49.8 
 
 
 53.3 
 
 
 42.6 
 
 
 
 8 
 30 
 9 
 30 
 10 
 
 42 16.7 
 4239.9 
 43 4.6 
 4330.6 
 4358.1 
 
 4449.7 
 4629.9 
 4811.2 
 4953.5 
 51 37.0 
 
 33.4 
 33.8 
 34.1 
 34.5 
 
 5228.1 
 54 58.2 
 5728.7 
 59 59.8 
 ~231 3 
 
 50.0 
 50.2 
 50.4 
 50.5 
 
 3054.9 
 3334.7 
 3614.3 
 3853.7 
 41 32.8 
 
 53.3 
 53.2 
 53.1 
 53.0 
 
 5755.8 
 2.8 
 2 8.9 
 414.1 
 618.4 
 
 42.3 
 42.0 
 41.7 
 41.5 
 
 3412.2 
 35 12.2 
 36 10.9 
 37 8.3 
 38 4.4 
 
 22 
 30 
 21 
 30 
 20 
 
 
 
 
 34.9 
 
 
 50.7 
 
 53.0 
 
 
 41.1 
 
 
 
 30 
 11 
 30 
 12 
 30 
 
 4426.9 
 4457.1 
 4528.8 
 46 1.8 
 4636.1 
 
 5321.6 
 55 7.1 
 56 53.8 
 5841.6 
 "fj^fjlj 
 
 35.2 
 35.7 
 35.9 
 36.2 
 
 5 3.3 
 735.8 
 10 8.8 
 1242.1 
 1516.0 
 
 508 441L7 
 
 5?'0 4650 ' 4 
 ;} , 14928.7 
 
 "I 52 6.8 
 51<d 5444.6 
 
 52.9 
 52.8 
 52.7 
 52.6 
 
 821.8 
 1024.3 
 1225.9 
 1426.6 
 1626.3 
 
 40.8 
 40.5 
 40.2 
 39.9 
 
 3859.1 
 3952.5 
 4044,6 
 41 35.3 
 4224.7 
 
 30 
 19 
 30 
 18 
 30 
 
 
 
 
 36.6 
 
 
 51.4 
 
 52.5 
 
 39.6 
 
 
 
 13 
 30 
 14 
 30 
 15 
 
 4711.9 
 4749.0 
 48 27.5 
 49 7.4 
 4948.7 
 
 220.1 
 4H.O 
 6 2.9 
 755.7 
 949.6 
 
 37.0 
 37.3 
 37.6 
 38.0 
 
 1750.2 
 20 24.9 
 2259.9 
 2535.3 
 2811.1 
 
 51.6 
 
 51.7 
 51.8 
 51.9 
 
 5722.1 
 59 59.3 
 236.2 
 512.7 
 748.9 
 
 52.4 
 52.3 
 52.2 
 52.1 
 
 1825.0 
 2022.8 
 2219.7 
 24 15.5 
 26 10.4 
 
 39.3 
 38.0 
 38.6 
 38.3 
 
 43 12.7 
 43 59.4 
 4444.7 
 4528.7 
 4611.3 
 
 17 
 30 
 16 
 30 
 15 
 
 
 84 86 
 
 
 88 
 
 
 91 
 
 
 93 
 
 
 94 
 
 
 
 II* 
 
 I* 
 
 
 
 
 
 XI* 
 
 i x ' 
 
 
 IX- 
 
 
80 TABLE LV. 
 
 Moon's Distance from the North Pole of the Ecliptic. 
 Argument. Supplement of Node+tyfoon's Orbit Longitude. 
 
 
 III* 
 
 IV* 
 
 
 V* 
 
 
 VI* 
 
 
 VII* 
 
 
 VIII* 
 
 
 
 84 
 
 86 
 
 Diff. 
 
 for 10 
 
 88 
 
 Dirt, 
 for 10 
 
 91 
 
 Diff. 
 
 for 10 
 
 93 
 
 Ditf: 
 
 for 10 
 
 94 
 
 
 O ' 
 
 / /' 
 
 / /> 
 
 
 , // 
 
 
 , // 
 
 
 / // 
 
 
 / // 
 
 O ' 
 
 15 49 48.7 9 49.6 
 30 50 31.3 ill 44.5 
 16 051 15.3 1 13 40.3 
 3052 0.6|15?72 
 17 052 47.3i 173o.O 
 30 53 35.3 19 33.7 
 
 38.3 
 38.6 
 39.0 
 39.3 
 39.6 
 
 28 11.1 
 30 47.3 
 33 23.8 
 36 0.7 
 38 37.9 
 41 15.4 
 
 52.1 
 52.2 
 52.3 
 
 52.4 
 52.5 
 
 748.9 
 10 24.7 
 13 0.1 
 1535.1 
 18 9.8 
 20 44.0 
 
 51.9 
 51.8 
 51.7 
 51.6 
 51.4 
 
 26 10.4 
 28 4.3 
 29 57.1 
 31 49.0 
 33 39.9 
 35 29.7 
 
 38.0 
 37.6 
 37.3 
 37.0 
 36.6 
 
 46 11.3 
 46 52.6 
 47 32.5 
 48 11.0 
 4848.1 
 49 23.9 
 
 15 
 30 
 14 
 30 
 13 
 30 
 
 
 
 
 39.9 
 
 
 52.6 
 
 
 51.3 
 
 
 36.2 
 
 
 
 18 54 24.7 21 33.4 
 3055 15.4 2334.1 
 19 056 7.5 2535.7 
 3057 0.92738.2 
 20 57 55.6 29 41.6 
 
 40.2 
 40.5 
 40.8 
 41.1 
 
 43 53.2 
 4631.3 
 49 9.6 
 51 48.3 
 54 27.2 
 
 52.7 
 52.8 
 52.9 
 53.0 
 
 23 17.9 
 25 51.2 
 
 28 24.2 
 30 56.7 
 33 28.7 
 
 51.1 
 51.0 
 50.8 
 50.7 
 
 37 18.4 
 39 6.2 
 40 52.9 
 42 38.4 
 44 23.0 
 
 35.9 
 35.6 
 35.2 
 34.9 
 
 49 58.2 
 5031.2 
 51 2.9 
 51 33.1 
 52 1.9 
 
 12 
 30 
 11 
 30 
 10 
 
 
 
 
 41.4 
 
 
 53.0 
 
 
 50.5 
 
 
 34.5 
 
 
 
 305851.7 31 45.9 
 21 59 49.1:3351.1 
 30 047.83557.2 
 22 0' 1 47.8 j 38 4.2 
 30 249.1 40 12.0 
 
 41.7 
 42.0 
 42.3 
 42.6 
 
 42.9 
 
 57 6.3 
 59 45.7 
 T25^3 
 5 5.1 
 745.1 
 
 53.1 
 53.2 
 53.3 
 53.3 
 
 53.4 
 
 36 0.2 
 3831.3 
 41 1.8 
 4331.9 
 46 1.4 
 
 50.4 
 50.2 
 50.0 
 49.8 
 
 49.7 
 
 46 6.5 
 47 48.8 
 49 30.1 
 51 10.3 
 52 49.4 
 
 34.1 
 33.8 
 33.4 
 33.0 
 32.6 
 
 52 29.4 
 52 55.4 
 5320.1 
 53 43.3 
 54 5.2 
 
 30 
 9 
 30 
 8 
 30 
 
 23 
 30 
 24 
 30 
 
 9,5 
 
 351.8 4220.7 
 4 55.7 44 30.3 
 6 1.0 ! 4640.6 
 7 7.4J4S51.9 
 815.251 3.8 
 
 43.2 
 43.4 
 43.6 
 44.0 
 
 10 25.2 
 13 5.6 
 15 46.0 
 
 18 26.7 
 21 7.5 
 
 53.5 
 5-3.5 
 53.6 
 53.6 
 
 48 30.4 
 50 58.8 
 53 26.6 
 55 53.9 
 58 20.7 
 
 49.5 
 49.3 
 49.1 
 
 48.9 
 
 5427.3 
 56 4.2 
 57 39.9 
 59 14.4 
 47 8 
 
 32.3 
 31.9 
 31.5 
 31.1 
 
 5425.6 
 54 44.6 
 55 2.3 
 55 18.5 
 55 33.3 
 
 7 
 30 
 6 
 30 
 5 
 
 
 
 443 
 
 
 53.6 
 
 
 48.7 
 
 
 30.8 
 
 
 
 30 
 
 26 
 30 
 27 
 30 
 
 924.3 
 10 34.7 
 11 46.3 
 12 59.2 
 14 13.3 
 
 53 16.7 
 55 30.3 
 
 57 44.7 
 59 59.8 
 
 44.5 
 44.8 
 45.0 
 45.3 
 
 23 48.4 
 26 29.4 
 29 10.5 
 31 51.7 
 34 33.0 
 
 53.7 
 53.7 
 53.7 
 53.7 
 
 046.8 
 3 12 3 
 
 537.2 
 8 1.5 
 1025.2 
 
 48.5 
 
 48.3 
 48.2 
 47.9 
 
 220.1 
 351.2 
 521.1 
 649.9 
 8 17.4 
 
 30.4 
 30.0 
 29.6 
 
 29.2 
 
 55 46.8 
 55 58.8 
 56 9.4 
 56 18.5 
 56 26.3 
 
 30 
 4 
 30 
 3 
 30 
 
 2158 
 
 
 
 
 45.6 
 
 
 53.8 
 
 
 47.7 
 
 
 28.8 
 
 
 
 28 15 28.7 
 30 16 45.4 
 29 18 3.2 
 30 19 22.3 
 30 20 42.7 
 
 432.5 
 649.8 
 9 7.8 
 11 26.9 
 1346.6 
 
 45.8 
 46.0 
 46.4 
 46.6 
 
 37 14.3 
 39 55.7 
 42 37.1 
 45 18.6 
 48 0.0 
 
 53.8 
 53.8 
 53.8 
 53.8 
 
 1248.2 
 15 10.5 
 17 32.2 
 1953.1 
 22 13.4 
 
 47.4 
 47.2 
 47.0 
 46.7 
 
 943.8 
 11. 9.0 
 12330 
 13 55.5 
 15 17.3 
 
 28.4 
 280 
 27.6 
 27.2 
 
 56 32.7 
 56 37.6 
 5641.2 
 56 43.3 
 56 44.0 
 
 2 
 
 30 
 1 
 30 
 
 
 
 85 |87 
 
 
 89 
 
 j 92 
 
 
 94 
 
 
 94 
 
 
 
 II* 1 I* 
 
 
 O 
 
 I XI* 
 
 
 X* 
 
 
 IX* 
 
 
TABLE LVI. 
 
 Equation II of the Moon's Polar Distance. 
 Argument II, corrected. 
 
 81 
 
 
 III* 
 
 diff. 
 
 IV* 
 
 diff. 
 
 V* 
 
 diff. 
 
 VI* 
 
 diff. 
 
 VII* 
 
 diff. 
 
 VIII* 
 
 diff. 
 
 
 
 
 
 
 
 / // 
 
 
 , ,, 
 
 
 / // 
 
 
 / 
 
 
 / // 
 
 
 o 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 
 13.8 
 13.9 
 14.1 
 14.5 
 15.1 
 15.8 
 
 0.1 
 0.2 
 0.4 
 0.6 
 0.7 
 
 1 24.4 
 1 29.0 
 1 33.8 
 1 38.7 
 143.8 
 149.0 
 
 4.6 
 
 4.8 
 4.9 
 5.1 
 5.2 
 
 436.9 
 444.9 
 4530 
 5 1.1 
 5 9.3 
 517.6 
 
 8.0 
 
 8.1 
 8.1 
 8.2 
 8.3 
 
 9 0.0 
 9 9.2 
 9 18.4 
 927.5 
 936.7 
 9 45:9 
 
 9.2 
 9.2 
 9.1 
 9.2 
 9.2 
 
 13 23.1 
 1331.0 
 13 38.8 
 1346.6 
 13 54.2 
 14 1.8 
 
 7.9 
 
 7.8 
 7.8 
 7.6 
 7.6 
 
 1635.6 
 16 40.2 
 16 44.6 
 16 48.9 
 16 53.0 
 16 56.9 
 
 4.6 
 4.4 
 4.3 
 4.1 
 3.9 
 
 30 
 29 
 28 
 27 
 26 
 25 
 
 
 
 0.9 
 
 
 5.3 
 
 
 8.4 
 
 
 9.1 
 
 
 7.5 
 
 
 3.8 
 
 
 6 
 
 7 
 8 
 9 
 10 
 
 16.7 
 17.7 
 18.9 
 020.3 
 021.8 
 
 1.0 
 1.2 
 1.4 
 1.5 
 
 1 54.3 
 1 59.8 
 2 5.4 
 211.1 
 2 16.9 
 
 55 
 5.6 
 5.7 
 
 5.8 
 
 526.0 
 534.4 
 542.9 
 551.4 
 6 0.0 
 
 8.4 
 8.5 
 8.5 
 8.6 
 
 955.0 
 10 4.1 
 10 13.2 
 10 22.3 
 1031.4 
 
 9.1 
 9.1 
 9.1 
 9.1 
 
 14 9.3 
 1416.7 
 14 24.0 
 1431.2 
 14 38.2 
 
 7.4 
 7.3 
 7.2 
 7.0 
 
 17 0.7 
 17 4.4 
 17 7.9 
 1711.3 
 17 14.5 
 
 3.7 
 3.5 
 
 3.4 
 3.2 
 
 24 
 23 
 22 
 21 
 20 
 
 
 
 1.7 
 
 
 6.0 
 
 
 8.7 
 
 
 9.0 
 
 |7.0 
 
 
 3.0 
 
 
 11 
 
 12 
 13 
 14 
 15 
 
 023.5 
 025.3 
 027.3 
 
 029.4 
 031.7 
 
 1.8 
 2.0 
 2.1 
 2.3 
 
 222.9 
 2 29.0 
 235.2 
 
 241.5 
 247.9 
 
 6.1 
 6.2 
 6.3 
 6.4 
 
 6 8.7 
 6 17.4 
 626.2 
 635.0 
 643.8 
 
 8.7 
 8.8 
 8.8 
 8.8 
 
 10 40.4 
 10 49.4 
 10 58.4 
 11 7.3 
 11 16.2 
 
 9.0 
 9.0 
 8.9 
 8.9 
 
 1445.2 
 1,4 52.1 
 14 58.9 
 15 5.5 
 15 12.1 
 
 6.9 
 6.8 
 6.6 
 6.6 
 
 17 17.5 
 17 20.4 
 17 23.2 
 17 25.8 
 17 28.3 
 
 2.9 
 2.8 
 2.6 
 2.5 
 
 19 
 18 
 17 
 16 
 15 
 
 
 
 2.5 
 
 
 6.6 
 
 
 8.9 
 
 
 8.8 
 
 
 6.4 
 
 
 2.3 
 
 
 16 
 17 
 18 
 19 
 20 
 
 034.2 
 036.8 
 039.6 
 042.5 
 45.5 
 
 2.6 
 2.8 
 2.9 
 3.0 
 
 254.5 
 3 1.1 
 3 7.9 
 3 14.8 
 321.8 
 
 6.6 
 6.8* 
 6.9 
 7.0 
 
 652.7 
 7 1.6 
 7 10.6 
 719.6 
 
 728.6 
 
 8.9 
 9.0 
 9.0 
 9.0 
 
 11 25.0 
 11 33.8 
 11 42.6 
 11 51.3 
 12 0.0 
 
 8.8 
 8.8 
 8.7 
 8.7 
 
 15 18.5 
 15 24.8 
 1531.0 
 15 37.1 
 1543.1 
 
 6.3 
 6.2 
 6.1 
 6.0 
 
 17 30.6 
 17 32.7 
 17 34.7 
 17 36.5 
 1738.2 
 
 2.1 
 2.0 
 1.8 
 1.7 
 
 14 
 13 
 12 
 11 
 10 
 
 
 
 3.2 
 
 7.0 
 
 
 9.1 
 
 
 8.6 
 
 
 5.8 
 
 
 Ii5 
 
 
 21 
 22 
 23 
 24 
 25 
 
 048.7 
 052.1 
 55.6 
 59.3 
 1 3.1 
 
 34 3288 
 
 35 3360 
 37 3 43.3 
 
 j! 3 50.7 
 d ' 8 3 58.2 
 
 7.2 
 7.3 
 
 7.4 
 7.5 
 
 737.7 
 746.8 
 7 55.9 
 8 5.0 
 8 14.1 
 
 9.1 
 9.1 
 9.1 
 9.1 
 
 12 8.6 
 12 17.1 
 12 25.6 
 12 34.0 
 
 12 42.4 
 
 8.5 
 8.5 
 8.4 
 8.4 
 
 15 48.9 
 15 54.6 
 16 0.2 
 16 5.7 
 16 11.0 
 
 5.7 
 5.6 
 5.5 
 5.3 
 
 17 39.7 
 1741.1 
 1742.3 
 1743.3 
 
 17 44.2 
 
 1.4 
 1.2 
 1.0 
 0.9 
 
 9 
 
 8 
 7 
 6 
 5 
 
 
 
 3.9 
 
 7.6 
 
 
 9.2 
 
 
 8.3 
 
 
 5.2 
 
 0.7 
 
 
 26 
 
 27 
 2S 
 29 
 30 
 
 1 7.0 
 1 11.1 
 1 15.4 
 1 19.8 
 1 24.4 
 
 4 4 5.8 
 
 I I'i 4 13 4 
 
 I;? 481.3 
 
 46 4 29 - 
 4 b 4 36.9 
 
 7.6 
 
 7.8 
 7.8 
 7.9 
 
 ,8 23.3 
 8 32.5 
 841.6 
 8 50.8 
 9 0.0 
 
 9 2 12 50.7 
 of 11258.9 
 9 o 13 7.0 
 92 11315 ' 1 
 
 ; 13 23.1 
 
 8.2 
 8.1 
 8.1 
 8.0 
 
 16 16.2 
 1621.3 
 16 26.2 
 16 31.9 
 1635.6 
 
 51 * 744 ' 9 
 AQ 1745.5 
 ? si 17 45.9 
 ~ 1746.1 
 4 ' 6 il7 46.2 
 
 0.6 
 0.4 
 0.2 
 0.1 
 
 4 
 3 
 
 2 
 1 
 
 
 
 II* 
 
 
 I* 
 
 
 0* 
 
 I XI* 
 
 
 X* j IX* 
 
 
 
 TABLE LVII. 
 
 Equation III of Moon's Polar Distance. 
 Argument. Moon's True Longitude. 
 
 
 III* 
 
 IV* 
 
 V* 
 
 VI* 
 
 VII* 
 
 VIII* 
 
 
 
 
 
 
 16.0 
 
 14.9 
 
 12.0 
 
 8.0 
 
 4.0 
 
 1.1 
 
 
 
 30 
 
 6 
 
 16.0 
 
 14.5 
 
 11.3 
 
 7.2 
 
 33 
 
 0.7 
 
 24 
 
 12 
 
 15.8 
 
 13.9 
 
 10.5 
 
 6.3 
 
 2.6 
 
 0.4 
 
 18 
 
 18 
 
 15.6 
 
 13.4 
 
 9.7 
 
 5.5 
 
 2.1 
 
 0.2 
 
 12 
 
 24 
 
 15.3 
 
 12.7 
 
 8.8 
 
 4.7 
 
 1.5 
 
 0.0 
 
 6 
 
 30 
 
 14.9 
 
 12.0 
 
 8.0 
 
 4.0 
 
 1.1 
 
 0.0 
 
 
 
 
 II* 
 
 I* 
 
 O* 
 
 XI* 
 
 X* 
 
 IX* 
 
 : " 1 
 
F2TABLE LVI1I. 
 
 To convert Degrees 
 and Minutes into 
 Decimal Parts. 
 
 TABLE LIX. 
 
 Equations of Moon's Polar Distance. 
 Arguments, Arg. 20 of Long. ; V to IX 
 
 corrected; X not corrected; and XI 
 
 and XII corrected. 
 
 Deg. Dec 
 
 &Mm. parts. 
 
 ' 
 
 
 1 5 
 
 003 
 
 1 26 
 
 4 
 
 148 
 
 5 
 
 2 10 
 
 6 
 
 231 
 
 7 
 
 253 
 
 8 
 
 3 14 
 
 9 
 
 336 
 
 10 
 
 358 
 
 11 
 
 4 i'J 
 
 12 
 
 441 
 
 13 
 
 5 2 
 
 14 
 
 524 
 
 15 
 
 546 
 
 16 
 
 6 7 
 
 17 
 
 629 
 
 18 
 
 650 
 
 19 
 
 7 12 
 
 20 
 
 734 
 
 21 
 
 755 
 
 22 
 
 8 17 
 
 23 
 
 838 
 
 24 
 
 9 
 
 25 
 
 922 
 
 26 
 
 943 
 
 27 
 
 10 5 
 
 28 
 
 1026 
 
 29 
 
 1043 
 
 30 
 
 11 10 31 
 
 11 31 
 
 32 
 
 1153 
 
 33 
 
 12 14 
 
 34 
 
 1236 
 
 3f> 
 
 1258 
 
 36 
 
 1319 
 
 37 
 
 1341 
 
 38 
 
 14 2 
 
 39 
 
 1424 
 
 40 
 
 1446 
 
 41 
 
 15 7 
 
 42 
 
 1529 
 
 43 
 
 1550 
 
 44 
 
 16 12 
 
 45 
 
 1634 
 
 46 
 
 1655 
 
 47 
 
 1717 
 
 48 
 
 1738 
 
 49 
 
 18 
 
 50 
 
 1822! 51 
 
 1843 52 
 
 19 5 53 
 
 Arg 
 
 20 
 
 V 
 
 VI 
 
 VII 
 
 VIE 
 
 IX 
 
 X 
 
 XI 
 
 Arg 
 
 Arg 
 
 XII 
 
 Arg. 
 
 250 
 
 0.3 
 
 55.9 
 
 6.1 
 
 2.6 
 
 25.1 
 
 3.0 
 
 0.7 
 
 0.9 
 
 250 
 
 
 
 4.0 
 
 500 
 
 260 
 
 0.3 
 
 55.8 
 
 6.2 
 
 2.725.1 
 
 3.1 0.7 
 
 0.9 
 
 240 
 
 10 
 
 3.7 
 
 510 
 
 270 
 
 0.4 
 
 55.7 
 
 6.3 
 
 2.8 
 
 25.0 
 
 3.2! 0.8 
 
 1.0 
 
 230 
 
 20 
 
 3.4 
 
 520 
 
 280 
 
 0.6 
 
 55.4 
 
 6.5 
 
 3.0 24.9 
 
 3.5 
 
 1.0 
 
 1.0 
 
 220 
 
 30 
 
 3.1 
 
 530 
 
 290 
 
 0.8 
 
 55.1 
 
 6.9 
 
 3.324.8 
 
 3.8 1.2 
 
 1.1 
 
 210 
 
 40 
 
 2.8 
 
 540 
 
 300 
 
 1.0 
 
 54.6 
 
 7.3 
 
 3.724.7 
 
 4.3 1.5 
 
 1.2 
 
 200 
 
 50 
 
 2.5 
 
 550 
 
 310 
 
 1.3 
 
 54.1 
 
 7.8 
 
 4.2 24.4 
 
 4.9 1.8 
 
 1.3 
 
 190 
 
 60 
 
 2.3 
 
 5GO 
 
 320 
 
 1.7 
 
 53.4 
 
 8.4 
 
 4.7,24.1 
 
 5.6 2.2 
 
 1.4 
 
 180 
 
 70 
 
 2.1 
 
 570 
 
 330 
 
 2.1 
 
 52.7 
 
 9.1 
 
 5.423.8 
 
 6.4 2.7 
 
 1.5 
 
 170 
 
 80 
 
 1.9 
 
 580 
 
 340 
 
 2.6 
 
 51.9 
 
 9.8 
 
 6.1 
 
 23.5 
 
 7.2! 3.2 
 
 1.7 
 
 160 
 
 90 
 
 1.7 
 
 590 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 350 
 
 3.1 
 
 51.0 
 
 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 
 
 8.7J22.4 
 
 10.3 
 
 5.1 
 
 2.3 
 
 130 
 
 120 
 
 1.5 
 
 620 
 
 380 
 
 4.9 
 
 17.7 
 
 13.6 
 
 9.7:21.9 
 
 11.5 
 
 5.8 
 
 2.5 
 
 120 
 
 130 
 
 1.5 
 
 630 
 
 390 
 
 5.6 
 
 46.5 
 
 14.8 
 
 10.721.4 
 
 12.8 
 
 6.6 
 
 2.8 
 
 110 
 
 140 
 
 1.5 
 
 640 
 
 400 
 
 6.4 
 
 45.2 
 
 16.0 
 
 11.820.9 
 
 14.1 
 
 7.4 
 
 3.0 
 
 100 
 
 150 
 
 1.6 
 
 650 
 
 410 
 
 7.1 
 
 43.9 
 
 17.2 
 
 13.020.4 
 
 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 
 
 41.0 
 
 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 
 
 150 
 
 30.0 
 
 30.0 
 
 250 
 
 15.0 
 
 30.0 
 
 17.0 
 
 6.0 
 
 
 
 250 
 
 4.0 
 
 750 
 
 510 159 
 
 284 
 
 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 
 
 25.1 
 
 34.5 
 
 29.2 
 
 13.1 
 
 35.1 
 
 20.1 
 
 7.0 
 
 970 
 
 280 
 
 4.9 
 
 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 
 
 5GO 
 
 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 
 
 5.9 
 
 820 
 
 580 
 
 22.1 
 
 17.5 
 
 41.5 35.8 
 
 10.1 
 
 43.0 
 
 24.9 
 
 8.5 
 
 920 
 
 330 
 
 6.1 
 
 830 
 
 590 
 
 22.9 
 
 16.1 
 
 42.8 37.0 
 
 9.6 
 
 44.5 
 
 25.7 
 
 8.7 
 
 910 
 
 340 
 
 6.3 
 
 840 
 
 600 
 
 23.6 
 
 14.8 
 
 44.0 38.2 
 
 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 
 
 620 
 
 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 
 
 48.4 42.3 
 
 7.2 
 
 50.8 
 
 29.6 
 
 9.9 
 
 860 
 
 390 
 
 6.5 
 
 890 
 
 650 
 
 26.9 
 
 9.0 
 
 49.343.1 
 
 6.8 
 
 51.8 
 
 30.2 
 
 10.1 
 
 850 
 
 400 
 
 6.4 
 
 900 
 
 660 27.4 
 
 8.1 
 
 50.2 43.9 
 
 6.5 
 
 52.8 
 
 30.8 
 
 10.3 
 
 840 
 
 410 
 
 6.3 
 
 910 
 
 670 27.9 
 
 7.3 
 
 50.9144.6 
 
 6.2 
 
 53.6 
 
 31.3 
 
 10.5 
 
 830 
 
 420 
 
 6.1 
 
 920 
 
 680 
 
 28.3 
 
 6.6 
 
 51.645.3 
 
 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 
 
 710 
 
 29.2 
 
 4.9 
 
 53.1 46.7 
 
 5.2 
 
 56.2,32.8 
 
 10.9 
 
 790 
 
 460 
 
 5.2 
 
 960 
 
 720 
 
 29.4 
 
 4.6 
 
 53.5 47.0 
 
 5.1 
 
 56.5 33.0 
 
 11.0 780 
 
 470 
 
 4.9 
 
 970 
 
 730 29.6 
 
 4.3 
 
 53.7 47.2 
 
 5.0 
 
 56.8 33.2 
 
 11.0 770 
 
 480 
 
 4.6 
 
 980 
 
 740 99.7 
 
 4.2 
 
 53.8 47.3 
 
 4.9 
 
 56.933.3 
 
 11. 1 
 
 760 
 
 490 
 
 4.3 
 
 990 
 
 750 J29.7 
 
 4.1 
 
 53.947.4 
 
 4.9 
 
 57.033.3 11.1 
 
 750 
 
 500 
 
 4.0 1000 
 
 Constant 10" 
 
TABLE LX. TABLE LXI. S3 
 
 Small Equations of Moon's Parallax. Moon's Equatorial Parallax. 
 
 \rgs., 1, 2, 4, 5, 6, 8, 9, 12, 13, of Long. Argument. Arg. of Evection. 
 
 A. 
 
 I 
 
 2 
 
 4 
 
 5 
 
 6 
 
 8 
 
 9 
 
 12 
 
 13 
 
 A. 
 
 
 
 0.0 
 
 1.6 
 
 0.6 
 
 1.6 
 
 1.9 
 
 0.0 
 
 3.0 
 
 .4 
 
 2.0 
 
 100 
 
 30.0 
 
 1.6 
 
 0.6 
 
 1.6 
 
 1.9 
 
 0.0 
 
 3.5 
 
 .4 
 
 2.0 
 
 97 
 
 6 
 
 0.0 
 
 1.5 
 
 0.6 
 
 1.5 
 
 1.8 
 
 0.0 
 
 3.1 
 
 .4 
 
 1.9 
 
 94 
 
 9 
 
 0.1 
 
 1.5 
 
 O.G 
 
 .5 
 
 1.8 
 
 0.1 
 
 2.6 
 
 .3 
 
 1.8 
 
 91 
 
 120.1 
 
 1.4 
 
 0.5 
 
 .4 
 
 .7 
 
 0.2 
 
 1.9 
 
 .2 
 
 1.7 
 
 88 
 
 150.1 
 
 1.3 
 
 0.5 
 
 .3 
 
 .6 
 
 0.2 
 
 1.3 
 
 .1 
 
 1.6 
 
 85 
 
 180.2 
 
 1.1 
 
 0.4 
 
 .1 
 
 .4 
 
 0.3 
 
 0.7 
 
 1.0 
 
 1.4 
 
 82 
 
 21 
 
 0.3 
 
 1.0 
 
 0.4 
 
 .0 
 
 .3 
 
 0.5 
 
 0.2 
 
 0.9 
 
 1.2 
 
 79 
 
 240.4 
 
 0.9 
 
 0.3 
 
 0.9 
 
 .2 
 
 0.6 
 
 0.0 
 
 0.7 
 
 1.0 
 
 76 
 
 270.5 
 
 0.7 
 
 0.3 
 
 0.7 
 
 1.0 
 
 0.7 
 
 0.1 
 
 0.6 
 
 0.9 
 
 73 
 
 30 0.5 
 
 0.6 0.2 
 
 0.6 
 
 0.9 
 
 0.8 
 
 0.4 
 
 0.5 
 
 0.7 
 
 70 
 
 330.6 
 
 0.4 0.2 
 
 0.4 
 
 0.7 
 
 0.9 
 
 0.8 
 
 0.4 
 
 0.5 
 
 67 
 
 360.7 
 
 0.3 
 
 0.1 
 
 0.3 
 
 0.6 
 
 1.0 
 
 1.5 
 
 0.3 
 
 0.4 
 
 64 
 
 390.7 
 
 0.2 
 
 0.1 
 
 0.2 
 
 0.5 
 
 1.1 
 
 2.1 0.2 
 
 0.2 
 
 61 
 
 420.8 
 
 0.1 
 
 0.0 
 
 0.1 
 
 0.4 
 
 l.l 
 
 2.8 0.1 
 
 0.1 
 
 58 
 
 45 
 
 0.8 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.3 
 
 1.2 
 
 3.2 
 
 0.0 
 
 0.0 
 
 55 
 
 480.8 
 500.S 
 
 0.0 
 0.0 
 
 0.0 
 0.0 
 
 0.0 
 0.0 
 
 0.3 
 0.3 
 
 1.2 
 1.2 
 
 3.5 0.0 
 
 3.6)0.0 
 
 0.0 
 0.0 
 
 52 
 50 
 
 Constant 7" 
 
 The first two figures only of the Arguments 
 
 are taken. 
 
 
 o 
 
 I* 
 
 II" 
 
 III* 
 
 IV* 
 
 V* 
 
 
 
 
 
 
 120.8 
 
 1 15.6 
 
 1 1.6 
 
 42.6 
 
 24.1 
 
 10.8 
 
 
 
 30 
 
 1 
 
 1 20.8J1 15.2 
 
 1 0.9 
 
 41.9 
 
 23.6 
 
 10.5 
 
 29 
 
 2 
 
 1 20.8 
 
 1 14.9 
 
 1 0.3 
 
 41.3 
 
 23.0 
 
 10.2 
 
 28 
 
 3 
 
 1 20.7 1 14.5 
 
 59.7 
 
 40.6 
 
 22.5 
 
 9.9 
 
 27 
 
 4 
 
 1 20.7 14.2 
 
 59.2 
 
 40.0 
 
 21.9 
 
 9.6 
 
 26 
 
 5 
 
 1 20.6 13.8 
 
 58.6 
 
 39.4 
 
 21.4 
 
 9.4 
 
 25 
 
 6 
 
 1 20.6 13.4 
 
 57.9 
 
 38.7 
 
 20.9 
 
 9.1 
 
 24 
 
 7 
 
 1 20.5; 13.0 
 
 57.3 
 
 38.1 
 
 20.4 
 
 8.8 
 
 23J 
 
 8 
 
 1 20.4! 12.6 
 
 56.7 
 
 37.4 
 
 19.9 
 
 8.6 
 
 221 
 
 9 
 
 1 20.3 12.2 
 
 56.1 
 
 36.8 
 
 19.4 
 
 8.4 
 
 21 
 
 10 
 
 1 20.2; 11.7 
 
 55.5 
 
 36.1 
 
 18.9 
 
 8.2 
 
 20i 
 
 11 
 
 120.1 11.3 
 
 54.9 
 
 35.5 
 
 18.4 
 
 8.0 
 
 19 
 
 12 
 
 19.9 10.8 
 
 54.2 
 
 34.9 
 
 17.9 
 
 7.8 
 
 18 
 
 13 
 
 19.8 10.4 
 
 53.6 
 
 34.2 
 
 17.5 
 
 7.6 
 
 17 
 
 14 
 
 19.6 9.9 
 
 53.0 
 
 33.6 
 
 17.0 
 
 7.4 
 
 16 
 
 15 
 
 19.5 , 9.4 
 
 52.3 
 
 33.0 
 
 16.6 
 
 7.2 
 
 15 
 
 16 
 
 19.3 9.0 
 
 51.7 
 
 32.4 
 
 16.1 
 
 7.1 
 
 14 
 
 17 
 
 19.1 8.5 
 
 51.1 
 
 31.7 
 
 15.7 
 
 6.9 
 
 13 
 
 18 
 
 18.9 8.0 
 
 50.4 
 
 31.1 
 
 15.2 
 
 6.8 
 
 12 
 
 19 
 
 18.7 7.5 
 
 49.8 
 
 30.5 
 
 14.8 
 
 6.7 
 
 11 
 
 20 
 
 18.4 7.0 
 
 49.1 
 
 29.9 
 
 14.4 
 
 6.5 
 
 10 
 
 21 
 
 18.2 6.5 
 
 48.5 
 
 29.3 
 
 14.0 
 
 6.4 
 
 9 
 
 22 
 
 18.0 5.9 
 
 47.8 
 
 28.7 
 
 13.6 
 
 6.3 
 
 8 
 
 23 
 
 17.7, 5.4 
 
 47.2 
 
 28.1 
 
 13.2 
 
 6.3 
 
 7 
 
 24 
 
 17.4 4.8 
 
 46.5 
 
 27.5 
 
 12.9 
 
 6.2 
 
 6 
 
 25 
 
 17.1 
 
 4.3 
 
 45.9 
 
 26.9 
 
 12.5 
 
 6.1 
 
 5 
 
 26 
 
 16.9 
 
 3.8 
 
 45.2 
 
 26.3 
 
 12.1 
 
 6.1 
 
 4 
 
 27 
 
 16.6 
 
 3.2 
 
 44.6 
 
 25.8 
 
 11.8 
 
 6.1 
 
 3 
 
 28 
 
 16.2 
 
 2.6 
 
 43.9 
 
 25.2 
 
 11.5 
 
 6.0 
 
 2 
 
 29 
 
 15.9 
 
 2.1 
 
 43.3124.7 
 
 11.11 6.0 
 
 1 
 
 30 
 
 15.6 
 
 1.5 
 
 42.6!24.l|l0.8l 6.0 
 
 ! 
 
 
 XI* 
 
 X* 
 
 IX* IvTIlJviI* 
 
 VI* 
 
 
84 
 
 TABLE LXII. 
 Moon's Equatorial Parallax. 
 
 Argument. Anomaly. 
 
 
 0* 
 
 diff 
 
 I* 
 
 diff II* 
 
 diff 
 
 III* 
 
 diff 
 
 IV* 
 
 diff V diff 
 
 
 
 
 , 
 
 
 / 
 
 
 , 
 
 
 / 
 
 
 / // 
 
 
 / 
 
 o 
 
 
 1 
 2 
 3 
 4 
 5 
 
 5857.7 
 58 57. 7 1 
 5857.6! 
 58 57.4! 
 5857.1 
 5856.8 
 
 0.0 
 0.1 
 0.2 
 0.3 
 0.3 
 
 58 27-.0 
 58 25.0 
 58 23.0 
 58 20.9 
 5818.7 
 58 16.5 
 
 2.0 
 2.0 
 
 2.1 
 
 2.2 
 2.2 
 
 57 7.9 
 57 4.8 
 57 1.6 
 5658.4 
 5655.2 
 56 52.0 
 
 3.1 
 3.2 
 3.2 
 3.2 
 3.2 
 
 5529.8 
 5526.6 
 5523.4 
 5520.2 
 55 17.0 
 5513.8 
 
 3.2 
 3.2 
 3.2 
 3.2 
 3.2 
 
 54 1.9 
 53 59.4 
 53 56.9 
 53 54.5 
 5352.1 
 5349.7 
 
 2.5 
 2.5 
 
 2.4 
 2.4 
 2.4 
 
 53 3.2 
 53 1.8 
 53 0.5 
 52 59.3 
 5258.1 
 52 57.0 
 
 1.4 
 1.3 
 1.2 
 1.2 
 1.1 
 
 30 
 
 29 
 28 
 27 
 26 
 25 
 
 
 
 0.4 
 
 
 2.2 
 
 
 3.2 
 
 
 3.2 
 
 
 2.3 
 
 
 1 2 
 
 
 6 
 
 7 
 8 
 9 
 10 
 
 58 56.4 
 5856.0 
 5855.4 
 58 54.8 
 58 54.2 
 
 0.4 
 0.6 
 0.6 
 0.6 
 
 58 14.3 
 58 12.0 
 58 9.6 
 58 7.2 
 58 4.8 
 
 2.3 
 
 2.4 
 2.4 
 2.4 
 
 56 48.8 
 5645.5 
 56 42.3 
 56 39.0 
 5635.7 
 
 3.3 
 3.2 
 3.3 
 3.3 
 
 5510.6 
 55 7.5 
 55 4.4 
 55 1.3 
 54 58.2 
 
 3.1 
 3.1 
 3.1 
 3.1 
 
 53 47.4 
 5345.1 
 5342.9 
 5340.6 
 53 38.5 
 
 2.3 
 
 2.2 
 2.3 
 2.1 
 
 52 55.8 
 52 54.8 
 5253.8 
 52 52.8 
 5251.9 
 
 1.0 
 1.0 
 1.0 
 0.9 
 
 24 
 23 
 22 
 21 
 20 
 
 
 
 0.8 
 
 2.5 
 
 
 3.3 
 
 
 3.1 
 
 
 2.2 
 
 
 0.9 
 
 
 11 
 12 
 13 
 14 
 15 
 
 58 53.4 
 5852.6 
 5851.8 
 58 50.8 
 58 49.8 
 
 0.8 
 0.8 
 1.0 
 1.0 
 
 58 23 25 
 5759.8*' 
 
 5757.2 
 5754.6J!! 
 5751.9^' 
 
 5632.4 
 5629.1 
 56 25.8 
 56 22.5 
 56 19.2 
 
 3.3 
 3.3 
 3.3 
 3.3 
 
 5455.1 
 
 5452.1 
 5449.1 
 5446.1 
 5443.1 
 
 3.0 
 3.0 
 3.0 
 3.0 
 
 5336.3 
 53 34.2 
 5332.1 
 5330.1 
 5328.1 
 
 2.1 
 2.1 
 2.0 
 2.0 
 
 5251.0 
 5250.1 
 5249.3 
 5248.6 
 52 47.9 
 
 0.9 
 0.8 
 0.7 
 0.7 
 
 19 
 18 
 17 
 16 
 15 
 
 
 
 1.1 
 
 
 2.7 
 
 
 3.3 
 
 
 2.9 
 
 
 1.9 
 
 
 0.7 
 
 j 
 
 16 
 17 
 18 
 19 
 20 
 
 5848.7 
 5847.6 
 58 46.4 
 5845.1 
 5843.8 
 
 1.1 
 1.2 
 1.3 
 1.3 
 
 5749.2 
 57 46.4 2-8 
 5743.7C 
 5740.8~Q 
 57 38.0 3 ' 8 
 
 56 15.9 
 5612.6 
 56 9.3 
 56 6.0 
 56 2.7 
 
 3.3 
 3.3 
 3.3 
 3.3 
 
 5440.2 
 5437.3 
 5434.4 
 5431.5 
 5428.7 
 
 2.9 
 2.9 
 2.9 
 2.8 
 
 5326.2 
 53 24.3 
 5322.4 
 5320.6 
 53 18.8 
 
 1.9 
 1.9 
 1.8 
 1.8 
 
 5247.2 
 5246.6 
 5246.0 
 5245.5 
 5245.0 
 
 0.6 
 0.6 
 0.5 
 0.5 
 
 14 
 13 
 12 
 11 
 10 
 
 
 
 1.4 
 
 
 2.9 
 
 
 3.4 
 
 
 2.8 
 
 
 1.8 
 
 
 0.4 
 
 
 21 
 22 
 23 
 24 
 25 
 
 58 42.4 
 5840.9 
 58 39.4 
 58 37.8 
 58 36.2 
 
 1.5 
 1.5 
 1.6 
 1.6 
 
 57351 
 5732.2 
 5729.3 
 5726.3 
 5723.3 
 
 2.9 
 2.9 
 3.0 
 3.0 
 
 55 59.3 
 5556.0 
 5552.7 
 5549.4 
 5546.1 
 
 3.3 
 3.3 
 3.3 
 3.3 
 
 5425.9 
 5423.1 
 5420.3 
 5417.6 
 5414.9 
 
 2.8 
 2.8 
 2.7 
 2.7 
 
 53 17.0 
 53 15.3 
 53 13.7 
 53 12.0 
 53 10.4 
 
 1.7 
 1.6 
 1.7 
 1.6 
 
 5244.6 
 5244.2 
 5243.8 
 5243.5 
 5243.3 
 
 0.4 
 0.4 
 0.3 
 0.2 
 
 9 
 8 
 7 
 6 
 5 
 
 
 
 1.8 
 
 
 3.0 
 
 
 3.3 
 
 
 2.7 
 
 
 1.5 
 
 
 0.2 
 
 
 26 
 27 
 28 
 29 
 30 
 
 58 34.4 
 58 32.7 
 58 30.9 
 5829.0 
 58 27.0 
 
 1.7 
 1.8 
 1.9 
 20 
 
 5720.2 
 57 17.2 
 5714.1 
 5711.0 
 57 7.9 
 
 3.0 
 3.1 
 3.1 
 3.1 
 
 5542.8 
 5539.6 
 5536.4 
 5533.1 
 5529.8 
 
 3.2 
 3.2 
 3.3 
 3.3 
 
 54 12.2 
 54 9.6 
 54 7.0 
 54 4.4 
 54 1.9 
 
 2.6 
 2.6 
 2.6 
 2.5 
 
 53 8.9 
 53 7.4 
 53 5.9 
 53 4.5 
 53 3.2 
 
 , , 5243.1 
 
 15242.9 
 
 \l 5242.8 
 
 J'J '5242.7 
 d 5242.7 
 
 0.2 
 0.1 
 0.1 
 0.0 
 
 4 
 3 
 2 
 1 
 
 
 
 XI* 
 
 
 X* 
 
 
 IX* 
 
 
 VIII* 
 
 
 VII* 
 
 
 VI* 
 
 
 
TABLE LXIII. 
 
 85 
 
 Moorfs Equatorial Parallax. 
 Argument. Argument of the Variation, 
 
 
 
 
 Is 
 
 II* 
 
 III* 
 
 IV* 
 
 V 
 
 
 o 
 
 
 
 
 
 
 
 ,/ 
 
 ,/ 
 
 // 
 
 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 
 
 5 
 
 55.2 
 
 38.1 
 
 12.4 
 
 4.3 
 
 21.9 
 
 47.7 
 
 25 
 
 6 
 
 55.0 
 
 37.2 
 
 11.7 
 
 4.5 
 
 22.7 
 
 48.4 
 
 24 
 
 7 
 
 54.8 
 
 36.3 
 
 11.1 
 
 4.7 
 
 23.6 
 
 49.1 
 
 23 
 
 8 
 
 54.6 
 
 35.5 
 
 10.4 
 
 5.0 
 
 24.5 
 
 49.7 
 
 22 
 
 9 
 
 54.3 
 
 34.6 
 
 9.8 
 
 5.3 
 
 25.4 
 
 50.3 
 
 21 
 
 10 
 
 54.0 
 
 33.7 
 
 9.2 
 
 5.6 
 
 26.3 
 
 50.9 
 
 20 
 
 11 
 
 53.7 
 
 32.7 
 
 8.7 
 
 6.0 
 
 27.2 
 
 51.5 
 
 19 
 
 12 
 
 53.3 
 
 31.8 
 
 8.2 
 
 6.3 
 
 28.2 
 
 52.1 
 
 18 
 
 13 
 
 52.9 
 
 30.9 
 
 7.7 
 
 6.8 
 
 29.1 
 
 52.6 
 
 17 
 
 14 
 
 52.5 
 
 30.0 - 
 
 7.2 
 
 7.2 
 
 30.0 
 
 53.1 
 
 16 
 
 15 
 
 52.0 
 
 29.1 
 
 6.7 
 
 7.7 
 
 30.9 
 
 53.5 
 
 15 
 
 16 
 
 51.5 
 
 28.2 
 
 6.3 
 
 8.2 
 
 31.8 
 
 54.0 
 
 14 
 
 17 
 
 51.0 
 
 27.2 
 
 5.9 
 
 8.7 
 
 32.8 
 
 54.4 
 
 13 
 
 18 
 
 50.5 
 
 26.3 
 
 5.6 
 
 9.3 
 
 33.7 
 
 54.8 
 
 12 
 
 19 
 
 49.9 
 
 25.4 
 
 5.3 
 
 9.8 
 
 34.6 
 
 55.1 
 
 11 
 
 20 
 
 49.4 
 
 24.5 
 
 5.0 
 
 10.5 
 
 35.5 
 
 55.4 
 
 10 
 
 21 
 
 48.8 
 
 23.6 
 
 4.7 
 
 11.1 
 
 36.4 
 
 55.7 
 
 9 
 
 22 
 
 48.1 
 
 22.7 
 
 4.5 
 
 11.7 
 
 37.3 
 
 56.0 
 
 8 
 
 23 
 
 47.4 
 
 21.9 
 
 4.3 
 
 12.4 
 
 38.2 
 
 56.2 
 
 7 
 
 24 
 
 46.8 
 
 21.0 
 
 4.1 
 
 13.1 
 
 39.0 
 
 56.4 
 
 6 
 
 25 
 
 46.1 
 
 20.1 
 
 3.9 
 
 13.8 
 
 39.9 
 
 56.6 
 
 5 
 
 26 
 
 45.4 
 
 19.3 
 
 3.8 
 
 14.5 
 
 40.8 
 
 56.8 
 
 4 
 
 27 
 
 44.6 
 
 18.5 
 
 3.7 
 
 15.3 
 
 41.6 
 
 56.9 
 
 3 
 
 28 
 
 43.9 
 
 17.6 
 
 3.7 
 
 16.1 
 
 42.4 
 
 56.9 
 
 2 
 
 29 
 
 43.1 
 
 16.8 
 
 3.7 
 
 16.8 
 
 43.2 
 
 57.0 
 
 1 
 
 30 
 
 42.3 
 
 16.0 
 
 3.7 
 
 17.6 
 
 44.0 
 
 67.0 
 
 
 
 
 55 
 
 X* 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI* 
 
 
S6 TABLE LXIV. 
 
 TABLE LXV. 
 
 Reduction of the Parallax, 
 and also of the Latitude. 
 
 Argument. Latitude. 
 
 Moon's Semi-diameter. 
 Argument. Equatorial Parallax. 
 
 Lat. 
 
 Red. 
 of par 
 
 Red. of 
 Lat. 
 
 Eq.Par Semidia. 
 
 Eq.Par 
 
 Semidia 
 
 Eq.Par 
 
 Semidia. 
 
 sec' Pro. 
 Par. 
 
 ,/ 
 
 , 
 
 
 
 / " 
 
 / // 
 
 / // 
 
 , 
 
 n 
 
 
 ' ,, 
 
 
 
 7T~ 
 
 ~ ~/7~ 
 
 53 
 
 14 
 
 26.5 
 
 56 
 
 15 15.6 
 
 59 
 
 16 
 
 4.6 
 
 1 
 
 0.3 
 
 
 
 0.0 
 
 0.0 
 
 53 10 
 
 14 
 
 29.3 
 
 56 10 
 
 15 18.3 
 
 59 10 
 
 16 
 
 7.4 
 
 2 
 
 0.5 
 
 3 
 
 0.0 
 
 1 11.8 
 
 53 20 
 
 14 
 
 32.0 
 
 56 20 
 
 15 21.0 
 
 59 20 
 
 16 10.1 
 
 3 
 
 0.8 
 
 6 
 
 0.1 
 
 2 22.7 
 
 53 30 
 
 14 
 
 34.7 
 
 56 30 
 
 15 23.8 
 
 59 30 
 
 16 12.8 
 
 4 
 
 1.1 
 
 9 
 
 0.3 
 
 3 32.1 
 
 53 40 
 
 14 
 
 37.4 
 
 56 40 
 
 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 
 54 
 
 14 
 
 14 
 
 40.2 
 42.9 
 
 56 50 
 57 
 
 15 29.2 
 15 31.9 
 
 59 50 
 60 
 
 16 18.3 
 16 21.0 
 
 6 
 
 7 
 
 1.6 
 1.9 
 
 18 
 
 1.0 
 
 6 43.7 
 
 54 10 
 
 14 
 
 45.6 
 
 57 10 
 
 15 34.7 
 
 60 10 
 
 16 23.7 
 
 8 
 
 2.2 
 
 21 
 
 1.4 
 
 7 39.7 
 
 54 20 
 
 14 
 
 48.3 
 
 57 20 
 
 15 37.4 
 
 60 20 
 
 16 26.4 
 
 9 
 
 2.4 
 
 24 
 
 1.8 
 
 8 30.7 
 
 54 30 
 
 14 
 
 51.1 
 
 57 30 
 
 15 40.1 
 
 GO 30 
 
 16 29.2 
 
 10 
 
 2.7 
 
 27 
 30 
 
 2.3 
 
 2.7 
 
 9 16.1 
 9 55.4 
 
 54 40 
 54 50 
 
 14 
 14 
 
 53.8 
 56.5 
 
 57 40 
 57 50 
 
 15 42.8 
 15 45.6 
 
 60 40 
 60 50 
 
 16 31.9 
 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 
 55 40 
 
 15 
 15 
 
 7.4 
 10.1 
 
 58 30 
 58 40 
 
 15 56.5 
 15 59.2 
 
 61 30 
 61 40 
 
 16 45.5 
 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 LXVI. 
 
 
 72 
 
 10.0 
 
 6 45.9 
 
 
 
 
 
 75 
 
 10.3 
 
 5 45.4 
 
 Augmentation of Moon's Semi-diameter. 
 
 78 
 81 
 
 10.6 
 10 8 
 
 4 41.0 
 3 33 5 
 
 
 
 
 //^ 
 
 84 
 
 1LO 
 
 2 23.7 
 
 A li. 
 
 Horizon. Semi-diameter 
 
 Alt 
 
 Horizon. Semi-diameter. 
 
 87 
 90 
 
 11.1 
 11.1 
 
 1 12.3 
 0.0 
 
 Alt. 
 
 14'30" 
 
 15' 
 
 16' 
 
 17 
 
 Alt. 
 
 14' 30" 
 
 15' 
 
 16' 17 
 
 Subsidiary Table. 
 
 
 
 2 
 
 0.6 
 
 0.6 
 
 0.7 
 
 0.8 
 
 42 
 
 9.2 
 
 9.8 
 
 11.2 12.6 
 
 Lat. 
 
 + 3' 
 
 3' 
 
 4 
 
 1 
 
 .0 
 
 1.1 
 
 1.3 
 
 1.5 
 
 45 
 
 9.7 
 
 10.4 
 
 11.8 13.3 
 
 
 
 
 6 
 
 1 
 
 .5 
 
 1.6 
 
 1.9 
 
 2.1 
 
 48 
 
 10.2 
 
 10.9 
 
 12.4 14.0 
 
 o 
 
 >/ 
 
 " 
 
 8 
 
 2.0 
 
 2.1 
 
 2.4 
 
 2.7 
 
 51 
 
 10.6 
 
 11.4 
 
 13.0 14.7 
 
 
 
 + 0.0 
 
 0.0 
 
 10 
 
 2.4 
 
 2.6 
 
 3.0 
 
 3.4 
 
 54 
 
 11.1 
 
 11.8 
 
 13.5 15.2 
 
 $ 
 
 
 
 0.0 
 
 
 
 
 
 
 
 
 
 
 
 12 
 
 v.v 
 
 
 
 0.0 
 
 12 
 
 2.9 
 
 3.1 
 
 3.6 
 
 4.0 
 
 57 
 
 11.5 
 
 12.3 
 
 14.0 15.8 
 
 i& 
 15 
 
 U.v 
 
 0.0 
 
 0.0 
 
 14 
 
 3.4 
 
 3.6 
 
 4.1 
 
 4.7 
 
 60 
 
 11 8 
 
 12.7 
 
 14.4 16.3 
 
 
 A 1 
 
 0.1 
 
 16 
 
 3.8 
 
 4.1 
 
 4.7 
 
 5.3 
 
 63 
 
 12.2 
 
 13.0 
 
 14.9 16.8 
 
 24 
 
 V. 1 
 
 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 
 
 qn 
 
 1 
 
 A 1 
 
 
 
 
 
 
 
 
 
 
 
 ou 
 36 
 
 U. 1 
 
 0.2 
 
 V. 1 
 
 0.2 
 
 24 
 
 5.6 
 
 6.0 
 
 6.8 
 
 7.7 
 
 72 
 
 13.0 
 
 13.9 
 
 15.9 17.9 
 
 42 
 
 A 
 
 0.2 
 
 27 
 
 6.2 ' 
 
 6.7 
 
 7.6 
 
 8.6 
 
 75 
 
 13.2 
 
 14.1 
 
 16.1 18.2 
 
 48 
 
 v. 
 
 0.3 
 
 0.3 
 
 30 
 
 6.9 
 
 7.3 
 
 8.4 
 
 9.5 
 
 78 
 
 13.4 
 
 14.3 
 
 16.3 18.4 
 
 54 
 
 0.3 
 
 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 
 
 Mft 
 
 0.4 
 
 s\ c 
 
 0.4 
 
 39 
 
 8.6 
 
 9.2 
 
 10.5 i 11.9 
 
 90 
 
 13.7 
 
 14.6 
 
 16.7 18.8 
 
 78 
 
 U.5 
 
 0.6 
 
 0.5 
 0.6 
 
 
 84 
 
 0.6 
 
 0.6 
 
 90 
 
 + 0.6 0.6 
 
TABLE LXVII. 
 
 87 
 
 Moon's Horary Motion in Longitude. 
 Arguments. 1 to 18 of Longitude. 
 
 Ar g 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 
 
 7 
 
 8 
 
 9 
 
 Arg. 
 
 
 
 ~ r ~" 
 
 
 
 
 
 
 
 
 
 100 
 
 5.0 
 
 0.0 
 
 2.9 
 
 1.9 
 
 0.0 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.16 
 
 2 
 
 5.0 
 
 0.0 
 
 2.8 
 
 1.9 
 
 0.0 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.15 
 
 98 
 
 4 
 
 4.9 
 
 0.0 
 
 2.8 
 
 1.9 
 
 0.0 
 
 0.01 
 
 0.00 
 
 0.02 
 
 0.15 
 
 96 
 
 6 
 
 4.8 
 
 0.1 
 
 2.8 
 
 1.9 
 
 0.1 
 
 0.03 
 
 0.01 
 
 0.05 
 
 0.14 
 
 94 
 
 8 
 
 4.7 
 
 0.2 
 
 2.7 
 
 1.8 
 
 0.1 
 
 0.06 
 
 0.01 
 
 0.09 
 
 0.12 
 
 92 
 
 10 
 
 4.5 
 
 0.3 
 
 2.6 
 
 1.7 
 
 0.2 
 
 0.09 
 
 0.02 
 
 0.14 
 
 0.10 
 
 90 
 
 12 
 
 4.3 
 
 0.4 
 
 2.5 
 
 1.7 
 
 0.2 
 
 0.13 
 
 0.02 
 
 0.19 
 
 0.09 
 
 88 
 
 14 
 
 4.1 
 
 0.6 
 
 2.3 
 
 1.6 
 
 0.3 
 
 0.18 
 
 0.03 
 
 0.26 
 
 0.07 
 
 86 
 
 16 
 
 3.8 
 
 0.7 
 
 2.2 
 
 1.5 
 
 0.4 
 
 0.23 
 
 0.04 
 
 0.33 
 
 0.05 
 
 84 
 
 18 
 
 3.6 
 
 0.9 
 
 2.0 
 
 1.4 
 
 0.5 
 
 0.28 
 
 0.05 
 
 0.41 
 
 0.03 
 
 82 
 
 20 
 
 3.3 
 
 1.1 
 
 1.9 
 
 1.3 
 
 0.6 
 
 0.34 
 
 0.06 
 
 0.50 
 
 0.02 
 
 80 
 
 22 
 
 3.0 
 
 1.3 
 
 .7 
 
 1.1 
 
 0.7 
 
 0.40 
 
 0.07 
 
 0.58 
 
 0.01 
 
 78 
 
 24 
 
 2.7 
 
 1.5 
 
 .5 
 
 1.0 
 
 0.8 
 
 0.46 
 
 0.08 
 
 0.67 
 
 0.00 
 
 76 
 
 26 
 
 2.3 
 
 1.7 
 
 .3 
 
 0.9 
 
 0.9 
 
 0.52 
 
 0.10 
 
 0.77 
 
 0.00 
 
 74 
 
 28 
 
 2.0 
 
 1.9 
 
 .2 
 
 0.8 
 
 .0 
 
 0.58 
 
 0.11 
 
 0.86 
 
 0.00 
 
 72 
 
 30 
 
 1.7 
 
 2.1 
 
 .0 
 
 0.7 
 
 .1 
 
 0.63 
 
 0.12 
 
 0.94 
 
 0.01 
 
 70 
 
 32 
 
 1.4 
 
 2.2 
 
 0.8 
 
 0.5 
 
 .2 
 
 0.69 
 
 0.13 
 
 1.03 
 
 0.01 
 
 68 
 
 34 
 
 1.2 
 
 2.4 
 
 0.7 
 
 0.4 
 
 .3 
 
 0.74 
 
 0.14 
 
 1.11 
 
 0.03 
 
 66 
 
 36 
 
 0.9 
 
 2.6 
 
 0.5 
 
 0.3 
 
 .3 
 
 0.78 
 
 0.15 
 
 1.18 
 
 0.05 
 
 64 
 
 38 
 
 0.7 
 
 2.7 
 
 0.4 
 
 0.3 
 
 .4 
 
 0.82 
 
 0.16 
 
 1.25 
 
 0.06 
 
 62 
 
 40 
 
 0.5 
 
 2.8 
 
 0.3 
 
 0.2 
 
 .5 
 
 0.86 
 
 0.16 
 
 1.3Q 
 
 0.08 
 
 60 
 
 42 
 
 0.3 
 
 2.9 
 
 0.2 
 
 0.1 
 
 .5 
 
 0.89 
 
 0.17 
 
 1.35 
 
 0.10 
 
 58 
 
 44 
 
 0.2 
 
 3.0 
 
 0.1 
 
 0.1 
 
 .6 
 
 0.91 
 
 0.17 
 
 1.39 
 
 0.11 
 
 56 
 
 46 
 
 0.1 
 
 3.1 
 
 0.0 
 
 0.0 
 
 .6 
 
 0.93 
 
 0.18 
 
 1.42 
 
 0.12 
 
 54 
 
 48 
 
 0.0 
 
 3.1 
 
 0.0 
 
 0.0 
 
 .6 
 
 0.94 
 
 0.18 
 
 1.44 
 
 0.13 
 
 52 
 
 50 
 
 0.0 
 
 3.1 
 
 0.0 
 
 0.0 
 
 1.6 
 
 0.94 
 
 0.18 
 
 1.44 
 
 0.13 
 
 50 
 
 Arg. 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 Alg. 
 
 
 
 
 
 
 
 
 
 
 
 
 100 
 
 0.00 
 
 0.26 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.26 
 
 0.00 
 
 0.21 
 
 2 
 
 0.00 
 
 0.25 0.00 
 
 000 
 
 0.00 
 
 0.00 
 
 0.26 
 
 0.00 
 
 0.20 
 
 98 
 
 4 
 
 0.02 
 
 0.24| 0.01 
 
 0.00 
 
 0.01 
 
 0.00 
 
 0.26 
 
 0.00 
 
 0.20 
 
 96 
 
 6 
 
 0.04 
 
 0.22 
 
 0.03 
 
 0.01 
 
 O.C2 
 
 0.01 
 
 0.25 
 
 0.00 
 
 0.20 
 
 94 
 
 8 
 
 0.08 
 
 0.20 
 
 0.04 
 
 0.02 
 
 0.04 
 
 0.01 
 
 0.25 
 
 0.0 1 
 
 0.20 
 
 92 
 
 10 
 
 0.12 
 
 0.17 
 
 0.07 
 
 0.03 
 
 0.06 
 
 0.02 
 
 0.24 
 
 0.01 
 
 0.20 
 
 90 
 
 12 
 
 0.16 
 
 0.14 
 
 0.09 
 
 0.04 
 
 0.09 
 
 0.02 
 
 0.22 
 
 0.02 
 
 0.19 
 
 88 
 
 14 
 
 0.20 
 
 0.11 
 
 0.12 
 
 0.06 
 
 0.12 
 
 0.03 
 
 0.21 
 
 0.02 
 
 0.19 
 
 86 
 
 16 
 
 0.24 
 
 0.08 
 
 0.16 
 
 0.07 
 
 0.15 
 
 0.04 
 
 0.20 
 
 0.03 
 
 0.18 
 
 84 
 
 18 
 
 0.28 
 
 0.05 
 
 0.19 
 
 0.09 
 
 0.19 
 
 0.05 
 
 0.19 
 
 0.04 
 
 0.18 
 
 82 
 
 20 
 
 0.31 
 
 0.03 
 
 0.23 
 
 0.11 
 
 0.22 
 
 0.06 
 
 0.17 
 
 0.05 
 
 0.17 
 
 80 
 
 22 
 
 0.34 
 
 0.01 
 
 0.27 
 
 0.13 
 
 0.26 
 
 0.07 
 
 0.15 
 
 0.06 
 
 0.17 
 
 78 
 
 24 
 
 0.35 
 
 0.00 
 
 0.31 
 
 0.15 
 
 0.30 
 
 0.08 
 
 0.14 
 
 0.07 
 
 0.16 
 
 76. 
 
 26 
 
 0.36 
 
 0.00 
 
 035 
 
 0.17 
 
 0.34 
 
 0.08 
 
 0.12 
 
 0.07 
 
 0.16 
 
 74 
 
 28 
 
 0.35 
 
 0.01 
 
 0.39 
 
 0.19 
 
 0.38 
 
 0.09 
 
 0.11 
 
 0.08 
 
 0.15 
 
 72 
 
 30 
 
 0.34 
 
 0.02 
 
 0.43 
 
 0.21 
 
 0.42 
 
 0.10 
 
 0.09 
 
 0.09 
 
 0.15 
 
 70 
 
 32 
 
 0.32 
 
 0.04 
 
 0.47 
 
 0.23 
 
 0.45 
 
 0.11 
 
 0.07 
 
 0.10 
 
 0.14 
 
 68 
 
 34 
 
 0.29 
 
 0.06 
 
 0.50 
 
 0.25 
 
 0.49 
 
 0.12 
 
 0.06 
 
 0.11 
 
 0.14 
 
 66 
 
 36 
 
 0.26 
 
 0.09 
 
 0.54 
 
 0.26 
 
 0.52 
 
 0.13 
 
 0.05 
 
 0.12 
 
 0.13 
 
 64 
 
 38 
 
 0.22 
 
 0.11 
 
 0.57 
 
 0.28 
 
 0.55 
 
 0.14 
 
 0.04 
 
 0.12 
 
 0.13 
 
 62 
 
 40 
 
 0.18 
 
 0.14 
 
 0.59 
 
 0.29 
 
 0.58 
 
 0.14 
 
 0.02 
 
 0.13 
 
 0.12 
 
 60 
 
 42 
 
 0.15 
 
 0.16 
 
 0.62 
 
 0.30 
 
 0.60 
 
 0.15 
 
 0.01 
 
 0.13 
 
 0.12 
 
 58 
 
 44 
 
 0.12 
 
 0.19 
 
 0.63 
 
 0.31 ! 0.62 
 
 0.15 
 
 0.01 
 
 0.14 
 
 0.12 
 
 56 
 
 46 
 
 0.10 
 
 021 
 
 0.65 
 
 0.32 ' 0.63 
 
 0.16 
 
 0.00 
 
 0.14 0.12 
 
 54 
 
 48 
 
 0.09 
 
 0.22 
 
 0.6610.32 0.64 
 
 0.16 
 
 0.00 
 
 0.14. '0.12 
 
 52 
 
 50 
 
 0.08 
 
 0.22 0.66 ' 0.32 0.64 
 
 0.16 0.00 
 
 0.14 0.11 
 
 50 
 
TABLE LXVIII. 
 
 Moon's Horary Motion in Longitude. 
 Argument. Argument of the Evection. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V* 
 
 
 
 
 
 80.3 
 
 74.7 
 
 59.6 
 
 39.4 
 
 19.8 
 
 5.9 
 
 o 
 30 
 
 1 
 
 80.3 
 
 74.3 
 
 58.9 
 
 38.7 
 
 19.3 
 
 5.6 
 
 29 
 
 2 
 
 80.3 
 
 73.9 
 
 58.3 
 
 38.0 
 
 18.7 
 
 5.3 
 
 28 
 
 3 
 
 80.2 
 
 73.5 
 
 57.7 
 
 37.3 
 
 18.1 
 
 5.0 
 
 27 
 
 4 
 
 80.2 
 
 73.1 
 
 57.1 
 
 36.6 
 
 17.6 
 
 4.7 
 
 26 
 
 5 
 
 80.1 
 
 72.7 
 
 56.4 
 
 36.0 
 
 17.0 
 
 4.4 
 
 25 . 
 
 6 
 
 80.1 
 
 72.3 
 
 55.8 
 
 35.3 
 
 16.5 
 
 4.1 
 
 24 
 
 7 
 
 80.0 
 
 71.9 
 
 55.1 
 
 34.6 
 
 15.9 
 
 3.8 
 
 23 
 
 8 
 
 79.9 
 
 71.4 
 
 54.5 
 
 33.9 
 
 15.4 
 
 3.6 
 
 22 
 
 9 
 
 79.8 
 
 71.0 
 
 53.8 
 
 33.2 
 
 14.9 
 
 3.4 
 
 21 
 
 10 
 
 79.7 
 
 70.5 
 
 53.1 
 
 32.5 
 
 14.4 
 
 3.1 
 
 20 
 
 11 
 
 79.5 
 
 70.1 
 
 52.5 
 
 31.9 
 
 13.9 
 
 2.9 
 
 19 
 
 12 
 
 79.4 
 
 69.6 
 
 51.8 
 
 31.2 
 
 13.4 
 
 2.7 
 
 18 
 
 13 
 
 79.2 
 
 69.1 
 
 51.1 
 
 30.5 
 
 12.9 
 
 2.5 
 
 17 
 
 14 
 
 79.1 
 
 68.6 
 
 50.5 
 
 29.9 
 
 12.4 
 
 2.3 
 
 16 
 
 15 
 
 78.9 
 
 68.1 
 
 49.8 
 
 29.2 
 
 11.9 
 
 2.1 
 
 15 
 
 16 
 
 78.7 
 
 67.6 
 
 49.1 
 
 28.6 
 
 11.4 
 
 2.0 
 
 14 
 
 17 
 
 78.5 
 
 67.0 
 
 48.4 
 
 27.9 
 
 11.0 
 
 1.8 
 
 13 
 
 18 
 
 78.2 
 
 66.5 
 
 47.7 
 
 27.2 
 
 10.5 
 
 1.7 
 
 12 
 
 19 
 
 78.0 
 
 66.0 
 
 47.0 
 
 26.6 
 
 10.1 
 
 1.6 
 
 11 
 
 20 
 
 77.8 
 
 65.4 
 
 46.4 
 
 26.0 
 
 9.7 
 
 1.4 
 
 10 
 
 21 
 
 77.5 
 
 64.9 
 
 45.7 
 
 25.3 
 
 9.3 
 
 1.3 
 
 9 
 
 "22 
 
 77.2 
 
 64.3 
 
 45.0 
 
 24.7 
 
 8.8 
 
 1.2 
 
 8 
 
 23 
 
 77.0 
 
 63.7 
 
 44.3 
 
 24.1 
 
 8.4 
 
 1.2 
 
 7 
 
 24 
 
 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 
 
 26 
 
 76.1 
 
 62.0 
 
 42.2 
 
 22.2 
 
 7.3 
 
 1.0 
 
 4 
 
 27 
 
 75.7 
 
 61.4 
 
 41.5 
 
 21.6 
 
 6.9 
 
 0.9 
 
 3 
 
 28 
 
 75.4 
 
 60.8 
 
 40.8 
 
 21.0 
 
 6.6 
 
 0.9 
 
 2 
 
 29 
 
 75.0 
 
 60.2 
 
 40.1 
 
 20.4 
 
 6.2 
 
 0.9 
 
 1 
 
 30 
 
 74.7 
 
 59.6 
 
 39.4 
 
 19.8 
 
 5.9 
 
 0.9 
 
 
 
 
 XI* 
 
 X 
 
 IX* 
 
 VIII* 
 
 VII* 
 
 VI* 
 
 
 TABLE LXIX. 
 
 Moon's Horary Motion in Longitude. 
 
 Arguments. Sum of Equations, 2, 3, &c., and Evection corrected 
 { 0" | 10" | 20" | 
 
 * o 
 
 
 
 
 s 
 
 
 
 00 
 
 0.2 
 
 0.5 
 
 XII 
 
 I 
 
 0.0 
 
 0.2 
 
 0.4 
 
 XI 
 
 II 
 
 0.1 
 
 0.2 
 
 0.3 
 
 X 
 
 III 
 
 0.2 
 
 0.2 
 
 0.2 
 
 IX 
 
 IV 
 
 0.3 
 
 0.2 
 
 0.1 
 
 VIII 
 
 V 
 
 0.4 
 
 0.2 
 
 0.0 
 
 VII 
 
 VI 
 
 0.5 
 
 0.2 
 
 0.0 
 
 VI 
 
 0" [ 10" I 20" | 
 
TABLE LXX. 
 
 Moon's Horary Motion in Longitude. 
 Arguments. Sum of preceding equations, and Anomaly corrected. 
 
 
 " 
 
 10" 
 
 20" 
 
 30" 
 
 40" 
 
 50" 
 
 60" 
 
 70" 
 
 80" 
 
 90" 
 
 100" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 
 
 4.1 
 
 5.3 
 
 6.5 
 
 7.6 
 
 8.8 
 
 10.0 
 
 11.2 
 
 12.4 
 
 13.5 
 
 14.7 
 
 15.9 
 
 XII 
 
 5 
 
 4.1 
 
 5.3 
 
 6.5 
 
 7.7 
 
 8.8 
 
 10.0 
 
 11.2 
 
 12.3 
 
 13.5 
 
 14.7 
 
 15.9 
 
 25 
 
 10 
 
 4.2 
 
 5.4 
 
 6.5 
 
 7.7 
 
 8.8 
 
 10.0 
 
 11.2 
 
 12.3 
 
 13.5 
 
 14.6 
 
 15.8 
 
 20 
 
 15 
 
 4.3 
 
 5.5 
 
 6.6 
 
 7.7 
 
 8.9 
 
 10.0 
 
 11.1 
 
 12.3 
 
 13.4 
 
 14.5 
 
 15.7 
 
 15 
 
 20 
 
 4.5 
 
 5.6 
 
 6.7 
 
 7.8 
 
 8.9 
 
 10.0 
 
 11.1 
 
 12.2 
 
 13.3 
 
 14.4 
 
 15.5 
 
 10 
 
 25 
 
 4.8 
 
 5.8 
 
 6.9 
 
 7.9 
 
 9.0 
 
 10.0 
 
 11.0 
 
 12.1 
 
 13.1 
 
 14.2 
 
 15.2 
 
 5 
 
 I -0 
 
 5.1 
 
 6.0 
 
 7.0 
 
 8.0 
 
 9.0 
 
 10.0 
 
 11.0 
 
 12.0 
 
 13.0 
 
 14.0 
 
 14.9 
 
 XI 
 
 5 
 
 5.4 
 
 6.3 
 
 7.2 
 
 8.2 
 
 9.1 
 
 10.0 
 
 10.9 
 
 11.8 
 
 12.8 
 
 13.7 
 
 14.6 
 
 25 
 
 10 
 
 5.7 
 
 6.6 
 
 7.4 
 
 8.3 
 
 9.2 
 
 10.0 
 
 10.8 
 
 11.7 
 
 12.6 
 
 13.4 
 
 14.3 
 
 20 
 
 15 
 
 6.1 
 
 6.9 
 
 7.7 
 
 8.5 
 
 9.2 
 
 10.0 
 
 10.8 
 
 11.5 
 
 12.3 
 
 13.1 
 
 13.9 
 
 15 
 
 20 
 
 6.6 
 
 7.2 
 
 7.9 
 
 8.6 
 
 9.3 
 
 10.0 
 
 10.7 
 
 11.4 
 
 12.1 
 
 12.8 
 
 13.4 
 
 10 
 
 25 
 
 7.0 
 
 7.6 
 
 8.2 
 
 8.8 
 
 9.4 
 
 10.0 
 
 10.6 
 
 11.2 
 
 11.8 
 
 12.4 
 
 13.0 
 
 5 
 
 II 
 
 7.5 
 
 8.0 
 
 8.5 
 
 9.0 
 
 9.5 
 
 10.0 
 
 10.5 
 
 11.0 
 
 11.5 
 
 12.0 
 
 12.5 
 
 X 
 
 5 
 
 , 7.9 
 
 8.4 
 
 8.8 
 
 9.2 
 
 9.6 
 
 10.0 
 
 10.4 
 
 10.8 
 
 11.2 
 
 11.6 
 
 12.1 
 
 25 
 
 10 
 
 8.4 
 
 8.7 
 
 9.1 
 
 9.4 
 
 9.7 
 
 10.0 
 
 10.3 
 
 10.6 
 
 10.9 
 
 11.3 
 
 11.6 
 
 20 
 
 15 
 
 8.9 
 
 9.1 
 
 9.4 
 
 9.6 
 
 9.8 
 
 10.0 
 
 10.2 
 
 10.4 
 
 10.6 
 
 10.9 
 
 11.1 
 
 15 
 
 20 
 
 9.4 
 
 9.5 
 
 9.7 
 
 9.8 
 
 9.9 
 
 10.0 
 
 10.1 
 
 10.2 
 
 10.3 
 
 10.5 
 
 10.6 
 
 10 
 
 25 
 
 9.9 
 
 9.9 
 
 9.9 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.1 
 
 10.1 
 
 10.1 
 
 5 
 
 III 
 
 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 
 
 S5 
 
 10 
 
 11.3 
 
 11.0 
 
 10.8 
 
 10.5 
 
 10.3 
 
 10.0 
 
 9.7 
 
 9.5 
 
 9.2 
 
 9.0 
 
 8.7 
 
 20 
 
 15 
 
 11.7 
 
 11.4 
 
 11.0 
 
 10.7 
 
 10.3 
 
 10.0 
 
 9.7 
 
 9.3 
 
 9.0 
 
 8.6 
 
 8.3 
 
 15 
 
 20 
 
 12.1 
 
 11.7 
 
 11.3 
 
 10.9 
 
 10.4 
 
 10.0 
 
 9.6 
 
 9.1 
 
 8.7 
 
 8.3 
 
 7.9 
 
 10 
 
 25 
 
 12.5 
 
 12.0 
 
 11.5 
 
 11.0 
 
 10.5 
 
 10.0 
 
 9.5 
 
 9.0 
 
 8.5 
 
 8.0 
 
 7.5 
 
 5 
 
 IV 
 
 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 
 
 7.4 
 
 6.7 
 
 25 
 
 10 
 
 13.6 
 
 12.9 
 
 12.1 
 
 11.4 
 
 10.7 
 
 10.0 
 
 9.3 
 
 8.6 
 
 7.9 
 
 7.1 
 
 6.4 
 
 20 
 
 15 
 
 13.9 
 
 13.1 
 
 12.3 
 
 11.5 
 
 10.8 
 
 10.0 
 
 9.2 
 
 8.5 
 
 7.7 
 
 6.9 
 
 6.1 
 
 15 
 
 20 
 
 14.1 
 
 13.3 
 
 12.5 
 
 11.6 
 
 10.8 
 
 10.0 
 
 9.2 
 
 8.4 
 
 7.5 
 
 6.7 
 
 5.9 
 
 10 
 
 25 
 
 14.4 
 
 13.5 
 
 12.6 
 
 11.7 
 
 10.9 
 
 10.0 
 
 9.1 
 
 8.3 
 
 7.4 
 
 6.5 
 
 5.6 
 
 5 
 
 V 
 
 14.6 
 
 13.7 
 
 12.7 
 
 11.8 
 
 10.9 
 
 10.0 
 
 9.1 
 
 8.2 
 
 7.3 
 
 6.3 
 
 5.4 
 
 VII 
 
 5 
 
 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 
 
 11.0 
 
 10.0 
 
 9.0 
 
 8.0 
 
 7.1 
 
 6.1 
 
 5.1 
 
 20 
 
 15 
 
 15.0 
 
 14.0 
 
 13.0 
 
 12.0 
 
 11.0 
 
 10.0 
 
 9.0 
 
 8.0 
 
 7.0 
 
 6.0 
 
 5.0 
 
 15 
 
 20 
 
 15.1 
 
 14.1 
 
 13.0 
 
 12.0 
 
 11.0 
 
 10.0 
 
 9.0 
 
 8.0 
 
 7.0 
 
 5.9 
 
 4.9 
 
 10 
 
 25 
 
 15.1 
 
 14.1 
 
 13.1 
 
 12.0 
 
 11.0 
 
 10.0 
 
 9.0 
 
 8.0 
 
 6.9 
 
 5.9 
 
 4.9 
 
 5 
 
 VI 
 
 15.1 
 
 14.1 
 
 13.1 
 
 12.1 
 
 11.0 
 
 10.0 
 
 9.0 
 
 8.0 
 
 6.9 
 
 5.9 
 
 4.9 
 
 VI 
 
 
 0' 
 
 10" 
 
 20" 
 
 30" 
 
 40" 
 
 50" 
 
 60" 
 
 70" 
 
 80" 
 
 90" 
 
 100" 
 
 
90 TABLE LXXI. 
 
 Moon's Horary Motion in Longitude. 
 Argument. Anomaly corrected. 
 
 
 
 
 diff. 
 
 I* 
 
 diff. II* diff 
 
 III* 
 
 diff. 
 
 IV* 
 
 diff. 
 
 V 
 
 diff. 
 
 
 
 
 
 
 
 
 
 // 
 
 
 ft 
 
 
 /, 
 
 
 
 
 
 
 
 
 
 1 
 
 2 
 3 
 
 4 
 5 
 
 441.5 
 441.5 
 441.3 
 441.1 
 440.8 
 440.4 
 
 0.0 
 0.1 
 0.2 
 0.3 
 0.4 
 
 404.1 
 401.6 
 399.2 
 396.6 
 394.0 
 391.3 
 
 2.5 
 2.4 
 2.6 
 2.6 
 2.7 
 
 309.3 
 305.6 
 301.9 
 298.1 
 294.4 
 290.6 
 
 3.7 
 3.7 
 
 3.8 ' 
 3.7 
 3.8 ! 
 
 195.3 
 191.6 
 
 187.9 
 184.3 
 180.6 
 177.0 
 
 3.7 
 3.7 
 3.6 
 3.7 
 3.6 
 
 95.8 
 93.0 
 90.2 
 87.6 
 84.9 
 82.3 
 
 2.8 
 2.8 
 2.6 
 2.7 
 2.6 
 
 30.6 
 29.2 
 27.8 
 26.4 
 25.1 
 23.8 
 
 1.4 
 1.4 
 1.4 
 1.3 
 1.3 
 
 30 
 29 
 28 
 27 
 26 
 25 
 
 
 
 0.5 
 
 
 2.7 
 
 
 3.8 
 
 
 3.6 
 
 
 2.6 
 
 
 1.2 
 
 
 6 
 7 
 
 8 
 9 
 10 
 
 439.9 
 439.4 
 438.7 
 438.0 
 437.2 
 
 0.5 
 
 0.7 
 0.7 
 0.8 
 
 388.6 
 385.8 
 383.0 
 380.1 
 377.1 
 
 2.8 
 2.8 
 2.9 
 3.0 
 
 286.8 
 283.0 
 279.2 
 275.4 
 271.5 
 
 3.8 
 3.8 
 3.8 
 3.9 
 
 173.4 
 169.8 
 166.3 
 162.8 
 159.3 
 
 3.6 
 3.5 
 3.5 
 3.5 
 
 79.7 
 77.1 
 74.6 
 72.1 
 69.7 
 
 2.6 
 2.5 
 2.5 
 2.4 
 
 22.6 
 21.4 
 20.3 
 19.2 
 18.2 
 
 1.2 
 1.1 
 1.1 
 IrO 
 
 24 
 23 
 22 
 21 
 20 
 
 
 
 0.9 
 
 13.0 
 
 
 3.8 
 
 
 3.5 
 
 
 2.4 
 
 
 1.0 
 
 
 11 
 12 
 13 
 14 
 15 
 
 436.3 
 435.3 
 434.2 
 433.1 
 431.8 
 
 1.0 
 1.1 
 1.1 
 1.3 
 
 374.1 
 371.1 
 368.0 
 364.8 
 361.6 
 
 3.0 
 3.1 
 3.2 
 3.2 
 
 267.7 
 263.8 
 260.0 
 256.2 
 252.3 
 
 3.9 
 3.8 
 3.8 
 3.9 
 
 155.8 
 152.4 
 148.9 
 145.5 
 142.2 
 
 3.4 
 3.5 
 3.4 
 3.3 
 
 67.3 
 65.0 
 62.7 
 60.4 
 
 58.2 
 
 2.3 
 2.3 
 2,3 
 
 2.2 
 
 17.2 
 16.3 
 15.4 
 14.6 
 13.8 
 
 
 0.9 
 0.8 
 0.8 
 
 19 
 18 
 17 
 16 
 15 
 
 
 
 1.3 
 
 
 3.2 
 
 
 3.8 
 
 
 3.3 
 
 
 2.1 
 
 
 0.7 
 
 
 16 
 17 
 
 18 
 19 
 20 
 
 430.5 
 429.1 
 427.6 
 426.1 
 424.5 
 
 1.4 
 1.5 
 1.5 
 1.6 
 
 358.4 
 355.1 
 351.8 
 348.4 
 345.0 
 
 3.3 
 3.3 
 3.4 
 3.4 
 
 248.5 
 244.6 
 240.8 
 236.9 
 233.1 
 
 3.9 
 3.8 
 3.9 
 3.8 
 
 138.S 
 135.6 
 132.3 
 129.1 
 125.9 
 
 3.3 
 3.3 
 3.2 
 3.2 
 
 56.1 
 53.9 
 51.9 
 49.8 
 47.9 
 
 2.2 
 2.0 
 2.1 
 1.9 
 
 13.1 
 12.4 
 11.8 
 11.2 
 10.7 
 
 0.7 
 0.6 
 0.6 
 0.5 
 
 14 
 13 
 12 
 11 
 10 
 
 
 
 1.7 
 
 
 3.4 
 
 
 3.8 
 
 
 3.2 
 
 
 2.0 
 
 
 0.5 
 
 
 21 
 22 
 23 
 24 
 25 
 
 422.7 
 421.0 
 419.1 
 417.2 
 415.2 
 
 1.7 
 1.9 
 1.9 
 2.0 
 
 341.6 
 338.1 
 334.6 
 331.1 
 327.5 
 
 3.5 
 3.5 
 3.5 
 3,6 
 
 229.3 
 225.4 
 221.6 
 217.8 
 214.0 
 
 3.9 
 3.8 
 3.8 
 3.8 
 
 122.7 
 119.6 
 116.5 
 113.4 
 110.4 
 
 3.1 
 3.1 
 3.1 
 3.0 
 
 45.9 
 44.0 
 42.2 
 40.4 
 38.7 
 
 1.9 
 1.8 
 1.8 
 1.7 
 
 10.2 
 9.8 
 9.4- 
 9.1 
 8.8 
 
 0.4 
 0.4 
 0.3 
 0.3 
 
 9 
 8 
 7 
 6 
 5 
 
 
 
 2.1 
 
 
 3.5 
 
 
 3.7 
 
 
 3.0 
 
 
 1.7 
 
 
 0.2 
 
 
 26 
 27 
 28 
 29 
 30 
 
 413.1 
 410.9 
 408.7 
 406.4 
 404.1 
 
 2.2 
 2.2 
 2.3 
 2.3 
 
 324.0 
 320.3 
 316.7 
 313.0 
 309.3 
 
 3.7 
 3.6 
 3.7 
 3.7 
 
 210.3 
 206.5 
 202.8 
 199.0 
 195.3 
 
 3.8 
 3.7 
 3.9 
 3.7 
 
 107.4 
 104.5 
 101.6 
 98.7 
 95.8 
 
 2.9 
 2.9 
 2.9 
 2.9 
 
 37.0 
 35.3 
 33.7 
 32.1 
 30.6 
 
 1.7 
 1.6 
 1.6 
 1.5 
 
 8.6 
 
 8.4 
 8.3 
 8.2 
 8.2 
 
 0.2 
 0.1 
 0.1 
 0.0 
 
 4 
 3 
 2 
 1 
 
 
 
 XI* 
 
 X* 
 
 
 IX* 
 
 VIII* 
 
 
 VII 
 
 
 VI* 
 
 
 
TABLE LXXII. 
 
 Moon's Horary Motion in Longitude. 
 Arguments. Sum of preceding Equations, and Arg. of Variation. 
 
 91 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 100 
 
 150 
 
 200 
 
 250 
 
 300 
 
 350 
 
 400 
 
 450 
 
 500 
 
 550 
 
 600 
 
 a o 
 
 
 4.5 
 
 5.5 
 
 6.5 
 
 7.6 
 
 8.6 
 
 9.6 
 
 10.6 
 
 11.6 
 
 12.6 
 
 13.7 
 
 14.7 
 
 15.7 
 
 16.7 
 
 8 
 
 XII 
 
 5 
 
 4.6 
 
 5.6 
 
 6.6 
 
 7.6 
 
 8.6 
 
 9.6 
 
 10.6 
 
 11.6 
 
 12.6 
 
 13.6 
 
 14.6 
 
 15.6 
 
 16.6 
 
 25 
 
 10 
 
 4.8 
 
 5.8 
 
 6.8 
 
 7.7 
 
 8.7 
 
 9.6 
 
 10.6 
 
 11.5 
 
 12.5 
 
 13.4 
 
 14.4 
 
 15.3 
 
 16.3 
 
 20 
 
 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 
 
 10.5 
 
 11.2 
 
 12.0 
 
 12.8 
 
 13.5 
 
 14.3 
 
 15.1 
 
 10 
 
 25 
 
 6.6 
 
 7.2 
 
 7.8 
 
 8.5 
 
 9.1 
 
 9.7 
 
 10.4 
 
 11.0 
 
 11.7 
 
 12.3 
 
 12.9 
 
 13.6 
 
 14.2 
 
 5 
 
 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 
 
 11.5 
 
 11.8 
 
 12.1 
 
 25 
 
 10 
 
 9.2 
 
 9.3 
 
 9.5 
 
 9.6 
 
 9.8 
 
 9.9 
 
 10.1 
 
 10.2 
 
 10.4 
 
 10.5 
 
 10.7 
 
 10.8 
 
 11.0 
 
 20 
 
 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 
 
 9.5 
 
 9-2 
 
 9.0 
 
 8.8 
 
 8.6 
 
 10 
 
 25 
 
 12.1 
 
 11.7 
 
 11.3 
 
 10.9 
 
 10.5 
 
 10.2 
 
 9.8 
 
 9.4 
 
 9.0 
 
 8.6 
 
 8.3 
 
 7.9 
 
 7.5 
 
 5 
 
 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 
 
 10.3 
 
 9.5 
 
 8.7 
 
 7.9 
 
 7.1 
 
 6.3 
 
 5.5 
 
 4.7 
 
 20 
 
 15 
 
 14.9 
 
 14.0 
 
 13.1 
 
 12.2 
 
 11.3 
 
 10.4 
 
 9.5 
 
 8.6 
 
 7.7 
 
 6.8 
 
 5.8 
 
 4.9 
 
 4.0 
 
 15 
 
 20 
 
 15.3 
 
 14.3 
 
 13.3 
 
 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 
 
 13.5 
 
 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 
 
 5.4 
 
 4.4 
 
 3.3 
 
 25 
 
 10 
 
 15.2 
 
 14.2 
 
 13.3 
 
 12.3 
 
 11.3 
 
 10.4 
 
 9.4 
 
 8.5 
 
 7.5 
 
 6.5 
 
 5.6 
 
 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 
 
 5.7 
 
 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 
 
 YIIIO 
 
 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 
 
 10il0.9 
 
 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 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.0 
 
 10.1 
 
 10.1 10.1 
 
 15 
 
 20 
 
 8.9 
 
 9.1 
 
 9.3 
 
 9.5 
 
 9.7 
 
 9.9 
 
 10.1 
 
 10.3 
 
 10.5 
 
 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 
 
 5 
 
 6.3 
 
 7.0 
 
 7.6 
 
 8.3 
 
 9.0 
 
 9.7 
 
 10.4 
 
 11.1 
 
 11.8 
 
 12.5 
 
 13.2 
 
 13.9 
 
 14.6 
 
 25 
 
 10 
 
 5.6 
 
 6.4 
 
 .7.2 
 
 8.0 
 
 8.8 
 
 9.7 
 
 10.5 
 
 11.3 
 
 12.1 
 
 13.0 
 
 13.8 
 
 14.6 
 
 15.4 
 
 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 
 
 5 
 
 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 
 
 300 350 
 
 400 
 
 450 
 
 500 
 
 550 
 
 600 
 
 
TABLE LXXIII. 
 
 Moon's Horary Motion in Longitude. 
 Argument. Argument of the Variation. 
 
 
 O 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V* 
 
 
 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 
 
 5 
 
 76.6 
 
 51.8 
 
 15.0 
 
 3.1 
 
 27.5 
 
 65.3 
 
 25 
 
 6 
 
 76.4 
 
 50.5 
 
 14.1 
 
 3.3 
 
 28.7 
 
 66.3 
 
 24 
 
 7 
 
 76.1 
 
 49.3 
 
 13.2 
 
 3.7 
 
 30.0 
 
 67.3 
 
 23 
 
 8 
 
 75.7 
 
 48.0 
 
 12.3 
 
 4.0 
 
 31.3 
 
 68.3 
 
 22 
 
 9 
 
 75.3 
 
 46.7 
 
 11.4 
 
 4.4 
 
 32.6 
 
 69.2 
 
 21 
 
 10 
 
 74.9 
 
 45.4 
 
 10.6 
 
 4.9 
 
 33.9 
 
 70.1 
 
 20 
 
 11 
 
 74.4 
 
 44.1 
 
 9.8 
 
 5.3 
 
 35.2 
 
 70.9 
 
 19 
 
 12 
 
 73.9 
 
 42.8 
 
 9.0 
 
 5.9 
 
 36.5 
 
 71.7 
 
 18 
 
 13 
 
 73.3 
 
 41.5 
 
 8.3 
 
 6.4 
 
 37.8 
 
 72.5 
 
 17 
 
 14 
 
 72.7 
 
 40.2 
 
 7.6 
 
 7.0 
 
 39.2 
 
 73.3 
 
 16 
 
 15 
 
 72.0 
 
 38.9 
 
 7.0 
 
 7.7 
 
 40.5 
 
 74.0 
 
 15 
 
 16 
 
 71.3 
 
 37.5 
 
 6.4 
 
 8.3 
 
 41.8 
 
 74.7 
 
 14 
 
 17 
 
 70.6 
 
 38.2 
 
 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 
 
 4.4 
 
 11.5 
 
 47.2 
 
 76.9 
 
 10 
 
 21 
 
 67.2 
 
 31.1 
 
 4.0 
 
 12.3 
 
 48.5 
 
 77.3 
 
 9 
 
 22 
 
 66.3 
 
 29.8 
 
 3.7 
 
 13.2 
 
 49.8 
 
 77.7 
 
 8 
 
 23 
 
 65.3 
 
 28.6 
 
 3.3 
 
 14.2 
 
 51.1 
 
 78.1 
 
 7 
 
 24 
 
 64.4 
 
 27.3 
 
 3.1 
 
 15.1 
 
 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* 
 
 IX 
 
 VIII* 
 
 VII' 
 
 VI* 
 
 
TABLE LXXIV. 03 
 
 Moon's Horary Motion in Longitude. 
 Arguments. Arg. of Reduction and Sum of preceding Equations 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 100 
 
 150 
 
 200 
 
 250 
 
 300 
 
 350 400 
 
 450 
 
 500 550 
 
 600 
 
 650 
 
 o 
 O 
 
 3.3 
 
 3.1 
 
 2.9 
 
 2.7 
 
 2.5 
 
 2.3 
 
 2. 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.3 
 
 1.1 
 
 0.9 
 
 0.7 
 
 XII 
 
 5 13.3 
 
 3.1 
 
 2.9 
 
 2.7 
 
 2.5 
 
 2.3 
 
 2. 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.3 
 
 1.1 
 
 0.9 
 
 0.7 
 
 25 
 
 10 13.2 
 
 3.0 
 
 2.8 
 
 2.6 
 
 2.4 
 
 2.3 
 
 2. 
 
 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. 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.4 
 
 1.2 
 
 1.0 
 
 0.9 
 
 15 
 
 20 
 
 3.0 
 
 2.8 
 
 2.7 
 
 2.5 
 
 2.4 
 
 2.2 
 
 2. 
 
 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: 
 
 1.9 
 
 1.8 
 
 1.7 
 
 1.5 
 
 1.4 
 
 1.3 
 
 1.2 
 
 5 
 
 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 
 
 2.1 
 
 2.0 
 
 2.0 
 
 1.9 
 
 1.8 
 
 1.8 
 
 1.7 
 
 1.6 
 
 1.6 
 
 25 
 
 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 
 
 15 
 
 20 
 
 1.8 
 
 1.8 
 
 1.8 
 
 1.9 
 
 1.9 
 
 1.9 
 
 2.0 
 
 2.0 
 
 2.1 
 
 2.1 
 
 2.1 
 
 2.2 
 
 2.2 
 
 2.2 
 
 10 
 
 25 
 
 1.6 
 
 1.6 
 
 1.7 
 
 1.8 
 
 1.8 
 
 1.9 
 
 2.0 
 
 2.0 
 
 2.1 
 
 2.2 
 
 2.2 
 
 2.3 
 
 2.4 
 
 2.4 
 
 5 
 
 II 
 
 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 
 
 .9 
 
 2. 
 
 2.2 
 
 2.3 
 
 2.5 
 
 2.6 
 
 2.7 
 
 2.8 
 
 25 
 
 10 
 
 1.0 
 
 1.2 
 
 1.3 
 
 1.5 
 
 1.6 
 
 1.8 
 
 .9 
 
 2. 
 
 2.2 
 
 2.4 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.0 
 
 20 
 
 15 
 
 0.9 
 
 1.1 
 
 1.2 
 
 1.4 
 
 1.6 
 
 .8 
 
 .9 
 
 2. 
 
 2.3 
 
 2.5 
 
 2.6 
 
 2.8 
 
 3.0 
 
 3.1 
 
 15 
 
 20 
 
 0.8 
 
 1.0 
 
 1.2 
 
 1.4 
 
 1.6 
 
 .7 
 
 .9 
 
 2. 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.0 
 
 3.2 
 
 10 
 
 25 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.3 
 
 1.5 
 
 .7 
 
 .9 
 
 2. 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.1 
 
 3.3 
 
 5 
 
 III 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.3 
 
 1.5 
 
 .7 
 
 .9 
 
 2. 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.1 
 
 3.3 
 
 IX 
 
 5 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.3 
 
 1.5 
 
 .7 
 
 .9 
 
 2 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.1 
 
 3.3 
 
 25 
 
 10 
 
 0.8 
 
 .0 
 
 1.2 
 
 1.4 
 
 1.6 
 
 .7 
 
 .9 
 
 2 
 
 2.3 
 
 2.5 
 
 2.7 
 
 2.9 
 
 3.0 
 
 3.2 
 
 20 
 
 15 
 
 0.9 
 
 .1 
 
 1.2 
 
 1.4 
 
 1.6 
 
 .8 
 
 .9 
 
 2.1 
 
 2.3 
 
 2.5 
 
 2.6 
 
 2.8 
 
 3.0 
 
 3.1 
 
 15 
 
 20 
 
 1.0 
 
 .2 
 
 1.3 
 
 1.5 
 
 1.6 
 
 .8 
 
 .9 
 
 2.1 
 
 2.2 
 
 2.4 
 
 2.5 
 
 2.7. 
 
 2.9 
 
 3.0 
 
 10 
 
 25 
 
 1.2 
 
 .3 
 
 1.4 
 
 1.6 
 
 1.7 
 
 .8 
 
 .9 
 
 2.1 
 
 2.2 
 
 2.3 
 
 2.5 
 
 2.6 
 
 2.7 2.8 
 
 5 
 
 IV 
 
 1.4 
 
 .5 
 
 1.6 
 
 1.7 
 
 1.8 
 
 .9 
 
 2.0 
 
 2.1 
 
 2.2 
 
 2.3 
 
 2.4 
 
 2.5 
 
 2.6^.7 
 
 VIII 
 
 
 
 
 
 
 
 
 
 
 
 
 
 | 
 
 
 . 5 
 
 1.6 
 
 1.6 
 
 1.7 
 
 1.8 
 
 1.8 
 
 .9 
 
 2.0 
 
 2.0 
 
 2.1 
 
 2 2 
 
 2 
 
 2.3 
 
 2.4 2.4 
 
 25 
 
 10 
 
 1.8 
 
 1.8 
 
 1.8 
 
 1.9 
 
 1.9 
 
 .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 
 
 20 
 
 2.2 
 
 2.2 
 
 2.2 
 
 2.1 
 
 2.1 
 
 2.0 
 
 2.0 
 
 2.0 
 
 1.9 
 
 1.9 
 
 1.9 
 
 .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 
 
 .7 
 
 1.6 
 
 1.6 
 
 5 
 
 V 
 
 2.6 
 
 2.5 
 
 2.4 
 
 2.3 
 
 2.2 
 
 2.1 
 
 2.0 
 
 1.9 
 
 1.8 
 
 1.7 
 
 1.6 
 
 .5 
 
 1.4 
 
 1.3 
 
 Vll 
 
 5 
 
 2.8 
 
 2.7 
 
 2.6 
 
 2.4 
 
 2.3 
 
 2.2 
 
 2.1 
 
 1.9 
 
 1.8 
 
 1.7 
 
 1.5 
 
 .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 
 
 .3 
 
 1.1 
 
 1.0 
 
 20 
 
 15 
 
 3.1 
 
 2 9 
 
 ? 8 
 
 ? 6 
 
 *> 4 
 
 t> 
 
 ?! 1 
 
 1 9 
 
 1 7 
 
 1 *\ 
 
 A 
 
 o 
 
 1 
 
 9 
 
 15 
 
 20 
 
 3.2 
 
 3.0 
 
 2.8 
 
 2.6 
 
 2.4 
 
 2.3 
 
 2.1 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.3 
 
 .1 
 
 1.0 
 
 0.8 
 
 10 
 
 25 
 
 3.3 
 
 3.1 
 
 2.9 
 
 2.7 
 
 2.5 
 
 2.3 
 
 2.1 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.3 
 
 .1 
 
 0.9 
 
 0.7 
 
 5 
 
 VI 
 
 3.3 
 
 3.1 
 
 2.9 
 
 2.7 
 
 2.5 
 
 2.3 
 
 2.1 
 
 1.9 
 
 1.7 
 
 1.5 
 
 1.3 
 
 .1 
 
 0.9 
 
 0.7 
 
 VI 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 50 
 
 100 
 
 150 
 
 200 
 
 250 
 
 300 
 
 350 
 
 400 
 
 450 
 
 500 
 
 550 
 
 600 
 
 650 
 
94 TABLE LXXV. 
 Moon's Horary Motion in Long. 
 Arg. Arg. of Reduction. 
 
 TABLE LXXVI. 
 Moon's Horary Motion in Long. 
 
 (Equation of the second order.) 
 Arguments. Arg's.of Table LXX. 
 
 
 Os Vis 
 
 I* VI!s Us \Ills 
 
 
 o 
 
 
 
 II 
 
 // 
 
 
 
 
 
 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 
 
 14.7 
 
 27 
 
 4 
 
 2.2 
 
 7.0 
 
 14.9 
 
 26 
 
 5 
 
 2.2 
 
 7.3 
 
 15.1 
 
 25 
 
 6 
 
 2.2 
 
 7.5 
 
 15.3 
 
 24 
 
 7 
 
 2.3 
 
 7.8 
 
 15.5 
 
 23 
 
 8 
 
 2.4 
 
 8.1 
 
 15.7 
 
 22 
 
 9 
 
 2.5 
 
 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 
 
 11.1 
 
 17.4 
 
 11 
 
 20 
 
 3.9 
 
 11.4 
 
 17.5 
 
 10 
 
 21 
 
 4.1 
 
 11.6 
 
 17.5 
 
 9 
 
 22 
 
 4.3 
 
 11.9 
 
 17.6 
 
 8 
 
 23 
 
 4.5 
 
 12.2 
 
 17.7 
 
 7 
 
 24 
 
 4.7 
 
 12.5 
 
 17.8 
 
 6 
 
 25 
 
 4.9 
 
 12.7 
 
 17.8 
 
 5 
 
 26 
 
 5.1 
 
 13.0 
 
 17.8 
 
 4 
 
 27 
 
 5.3 
 
 13.2 
 
 17.9 
 
 3 
 
 28 
 
 5.6 
 
 13.5 
 
 17.9 
 
 2 
 
 29 
 
 5.8 
 
 13.7 
 
 17.9 
 
 1 
 
 | 30 
 
 6.0 
 
 14.0 
 
 17.9 
 
 
 
 1 
 
 XTV 
 
 XIV 
 
 IXsIIIs 
 
 
 
 ft 
 
 
 
 
 
 Arg. 
 
 
 
 50 
 
 100 
 
 a 
 
 // 
 
 // 
 
 " 
 
 
 
 0.05 
 
 0.05 
 
 0.05 
 
 I , 
 
 0.08 
 
 0.05 
 
 0.02 
 
 II 
 
 0.10 
 
 0.05 
 
 0.00 
 
 III 
 
 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 
 
 VIII 
 
 0.01 
 
 0.05 
 
 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 
 
 0.05 
 
 
 tt 
 
 
 50 
 
 100 
 
 Constant to be added 27'24".0. 
 
 TABLE LXXVII. 
 Moon's Horary Motion in Longitude. 
 
 (Equations of the second order.) 
 Arguments. Arguments of Tables LXXII and LXXIV. 
 
 
 Variation. 
 
 Reduction. 
 
 
 tt 
 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 600 
 
 
 
 tf 
 600 
 
 
 
 0. VI. 
 
 0.14 
 
 0.14 
 
 0.14 
 
 0.14 
 
 0.14 
 
 0.14 
 
 0.14 
 
 003 
 
 0.03 
 
 I. 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 
 
 n. vin. o 
 
 0.22 
 
 0.19 
 
 0.16 
 
 0.13 
 
 0.10 
 
 0.07 
 
 0.03 
 
 0.01 
 
 0.05 
 
 m. ix. o 
 
 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 
 
 0.15 
 
 0.18 
 
 0.22 
 
 0.26 
 
 0.05 
 
 0.01 
 
 VI. XII. 
 
 0.14 
 
 0.14 
 
 0.14 J0.14 
 
 0.14 
 
 C.14 
 
 0.14 
 
 0.03 
 
 0.03 
 
TABLE LXXVIII. 95 
 
 Moon's Horary Motion in Longitude. 
 
 (Equations of the second order.) 
 Arguments. Args. of Evection, Anomaly, Variation, Reduction. 
 
 
 Evec. 
 
 Anom. 
 
 Var. 
 
 Red. 
 
 Evec. 
 
 Anom. 
 
 Var. 
 
 Red. 
 
 
 
 
 
 
 
 
 
 
 
 
 0* 
 
 0.16 
 
 1.05 
 
 0.34 
 
 0.08 
 
 0.16 
 
 1.05 
 
 0.34 
 
 0.08 
 
 8 
 
 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 
 
 20 
 
 0.10 
 
 0.59 
 
 0.12 
 
 0.12 
 
 0.22 
 
 1.50 
 
 0.56 
 
 0.03 
 
 10 
 
 25 
 
 0.09 
 
 0.49 
 
 0.08 
 
 0.13 
 
 0.24 
 
 1.60 
 
 0.60 
 
 0.02 
 
 5 
 
 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 
 
 0.26 
 
 1.78 
 
 0.66 
 
 0.01 
 
 25 
 
 10 
 
 0.05 
 
 0.24 
 
 0.01 
 
 0.15 
 
 0.27 
 
 1.86 
 
 0.67 
 
 0.00 
 
 20 
 
 15 
 
 0.04 
 
 0.17 
 
 0.01 
 
 0.15 
 
 0.28 
 
 1.92 
 
 0.67 
 
 0.00 
 
 15 
 
 20 
 
 0.03 
 
 0.12 
 
 0.01 
 
 0.15 
 
 0.29 
 
 1.98 
 
 0.67 
 
 0.00 
 
 '* 10 
 
 25 
 
 0.03 
 
 0.07 
 
 0.03 
 
 0.15 
 
 0.30 
 
 2.02 
 
 0.65 
 
 0.01 
 
 5 
 
 II 
 
 0.02 
 
 0.04 
 
 0.06 
 
 0.14 
 
 0.31 
 
 2.05 
 
 0.62 
 
 0.01 
 
 X 
 
 5 
 
 0.01 
 
 0.02 
 
 0.09 
 
 0.13 
 
 0.32 
 
 2.08 
 
 0.59 
 
 0.02 
 
 25 
 
 10 
 
 0.01 
 
 0.00 
 
 0.13 
 
 0.12 
 
 0.32 
 
 2.09 
 
 0.54 
 
 0.03 
 
 20 
 
 15 
 
 0.00 
 
 0.00 
 
 0.18 
 
 0.11 
 
 0.32 
 
 2.10 
 
 0.50 
 
 0.04 
 
 15 
 
 20 
 
 0.00 
 
 0.00 
 
 0.24 
 
 0.10 
 
 0.33 
 
 2.09 
 
 0.44 
 
 0.05 
 
 10 
 
 25 
 
 0.00 
 
 0.02 
 
 0.29 
 
 0.09 
 
 0.33 
 
 2.08 
 
 0.39 
 
 0.06 
 
 5 
 
 III 
 
 0.00 
 
 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 
 
 0.32 
 
 2.00 
 
 0.22 
 
 0.10 
 
 20 
 
 15 
 
 0.01 
 
 0.14 
 
 0.51 
 
 0.04 
 
 0.32 
 
 1.96 
 
 0.17 
 
 .0.11 
 
 15 
 
 20 
 
 0-01 
 
 0.18 
 
 0.56 
 
 0.03 
 
 0.31 
 
 1.91 
 
 0.12 { 
 
 F 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 
 
 lo 
 
 20 
 
 0.14 
 
 0.91 
 
 0.46 
 
 0.05 
 
 0.19 
 
 1.18 
 
 0.22 
 
 0.10 
 
 10 
 
 25 
 
 0.15 
 
 0.98 
 
 0.40 
 
 0.06 
 
 0.18 
 
 1.12 
 
 0.28 
 
 0.09 
 
 5 
 
 VI 
 
 0.16 
 
 1.05 
 
 0.34 
 
 0.08 
 
 0.16 
 
 1.05 
 
 0.34 
 
 0.08 
 
 VI 
 
16 
 
 TABLE LXXIX. 
 
 Moon's Horary Motion in Latitude. 
 Argument. Arg. I of Latitude. 
 
 
 0* 
 
 I* 
 
 II* 
 
 III* 
 
 IV* 
 
 V 
 
 
 
 
 
 // 
 
 // 
 
 
 
 n 
 
 // 
 
 // 
 
 o 
 
 
 
 378.0 
 
 354.3 
 
 289.2 
 
 200.0 
 
 110.8 
 
 45.7 
 
 30 
 
 1 
 
 378.0 
 
 352.7 
 
 286.5 
 
 196.9 
 
 108.1 
 
 44.2 
 
 29 
 
 2 
 
 377.9 
 
 351.1 
 
 283.8 
 
 193.8 
 
 105.4 
 
 42.7 
 
 28 
 
 3 
 
 377.8 
 
 349.4 
 
 281.0 
 
 190.7 
 
 102.8 
 
 41.3 
 
 27 
 
 4 
 
 377.6 
 
 347.7 
 
 278.3 
 
 187.5 
 
 100.2 
 
 39.9 
 
 26 
 
 5 
 
 377.3 
 
 346.0 
 
 275.5 
 
 184.4 
 
 97.7 
 
 38.6 
 
 25 
 
 6 
 
 377.0 
 
 344.2 
 
 272.6 
 
 181.3 
 
 95.1 
 
 37.3 
 
 24 
 
 7 
 
 376.7 
 
 342.3 
 
 269.8 
 
 178.2 
 
 92.6 
 
 36.1 
 
 23 
 
 8 376.3 
 
 340.5 
 
 266.9 
 
 175.1 
 
 90.2 
 
 34.9 
 
 22 
 
 9 
 
 375.8 
 
 338.5 
 
 264.0 
 
 172.1 
 
 87.7 
 
 33.8 
 
 21 
 
 10 
 
 375.3 
 
 336.6 
 
 261.1 
 
 169.0 
 
 85.3 
 
 32.7 
 
 20 
 
 11 
 
 374.7 
 
 334.5 
 
 258.1 
 
 165.9 
 
 83.0 
 
 31.6 
 
 19 
 
 12 
 
 374.1 
 
 332.5 
 
 255.2 
 
 162.9 
 
 80.7 
 
 30.7 
 
 18 
 
 13 
 
 373.5 
 
 330.4 
 
 252.2 
 
 159.8 
 
 78.1 
 
 29.7 
 
 17 
 
 14 
 
 372.7 
 
 328.3 
 
 249.2 
 
 156.8 
 
 76.1 
 
 28.9 
 
 16 
 
 15 
 
 372.0 
 
 326.1 
 
 246.2 
 
 153.8 
 
 73.9 
 
 28.0 
 
 15 
 
 16 
 
 371.1 
 
 323.9 
 
 243.2 
 
 150.8 
 
 71.7 
 
 27.3 
 
 14 
 
 17 
 
 370.3 
 
 321.9 
 
 240.2 
 
 147.8 
 
 69.6 
 
 26.5 
 
 13 
 
 18 
 
 369.3 
 
 319.3 
 
 237.1 
 
 144.8 
 
 67.5 
 
 25.9 
 
 12 
 
 19 
 
 368.4 
 
 317.0 
 
 234.1 
 
 141.9 
 
 65.5 
 
 25.3 
 
 11 
 
 20 
 
 367.3 
 
 314.7 
 
 231.0 
 
 138.9 
 
 63.4 
 
 24.7 
 
 10 
 
 21 
 
 366.2 
 
 312.3 
 
 227.9 
 
 136.0 
 
 61.5 - 
 
 24.2 
 
 9 
 
 22 
 
 365.1 
 
 309.8 
 
 224.9 
 
 133.1 
 
 59.5 
 
 23.7 
 
 8 
 
 23 
 
 363.9 
 
 307.4 
 
 221.8 
 
 130.2 
 
 57.7 
 
 23.3 
 
 7 
 
 24 
 
 362.7 
 
 304.9 
 
 218.7 
 
 127.4 
 
 55.8 
 
 23.0 
 
 6 
 
 25 
 
 361.4 
 
 302.3 
 
 215.6 
 
 124.5 
 
 54.0 
 
 22.7 
 
 5 
 
 26 
 
 360.1% 
 
 299.8 
 
 212.5 
 
 121.7 
 
 52.3 
 
 22.4 
 
 4 
 
 27 
 
 358.7 
 
 297.2 
 
 209.3 
 
 119.0 
 
 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* 
 
 VIII* 
 
 VII* 
 
 Vis 
 
 
 TABLE LXXX. 
 
 Moon's Horary Motion in Latitude. 
 Arguments. Args. V, VI, VII, VIII, IX, X, XI, and XII, of Latitude 
 
 Arg. 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 Arg. 
 
 
 
 0.00 
 
 0.50 
 
 0.34 
 
 0.00 
 
 0.50 
 
 0.04 
 
 0.12 
 
 0.08 
 
 1000 
 
 50 
 
 0.01 
 
 0.49 
 
 0.33 
 
 0.00 
 
 0.49 
 
 0.04 
 
 0.12 
 
 0.07 
 
 950 
 
 100 
 
 0.04 0.45 
 
 0.30 
 
 0.02 
 
 0.45 
 
 0.04 
 
 0.11 
 
 0.05 
 
 900 
 
 150 
 
 0.09 0.40 
 
 0.27 
 
 0.04 
 
 0.40 
 
 0.03 
 
 0.10 
 
 0.03 
 
 850 
 
 200 
 
 0.16 0.33 
 
 0.22 
 
 0.06 
 
 0.33 
 
 0.03 
 
 0.08 
 
 0.01 
 
 800 
 
 
 
 
 
 
 
 
 
 
 250 
 
 0.23 0.25 
 
 0.17 
 
 0.09 
 
 0.25 
 
 0.02 
 
 0.06 
 
 0.00 
 
 750 
 
 300 
 
 0.30 0.17 
 
 0.12 
 
 0.12 
 
 0.17 
 
 0.01 
 
 0.04J0.01 
 
 700 
 
 350 
 
 0.37 0.10 
 
 0.07 
 
 0.14 
 
 0.10 
 
 0.01 
 
 0.02 0.03 
 
 650 
 
 400 
 
 0.42 0.05 
 
 0.04 
 
 6.16 
 
 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 
 
 550 
 
 500 
 
 0.46 0.00' 0.00 
 
 0.18 
 
 0.00 
 
 0.00 
 
 0.00 
 
 0.08 
 
 500 
 
TABLE LXXXI. Moon's Horary Motion in Latitude. 97 
 
 Arguments. Preceding equation, and Sum of equations of Horary 
 
 Motion in Longitude, except the last two. 
 
 Pr. 
 
 eq. 
 
 0" 
 1".6 
 
 50" 
 1".4 
 
 100" 
 
 150" 
 0".9 
 
 200" 
 
 250" 
 0".4 
 
 300" 
 0".l 
 
 350" 
 0''.2 
 
 400" 
 0".4 
 
 A50" 
 
 500' 
 0".9 
 
 550" 
 
 600" 
 
 ^ 
 
 650" 
 
 Dill 
 
 l."l 
 
 0".6 
 
 0".7 
 
 1".2 
 
 1".7 
 ~ 
 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 59.0 
 
 54.5 
 
 50.0 
 
 45.4 
 
 40.9 
 
 36.4 
 
 31.8 
 
 27.3 
 
 22.8 
 
 18.2 
 
 13.7 
 
 9.1 
 
 4.6 
 
 0.1 
 
 4.5 
 
 30 
 
 57.4 
 
 53.1 
 
 48.9 
 
 44.6 
 
 40.3 
 
 36.0 
 
 31.7 
 
 27.4 
 
 23.2 
 
 18.9 
 
 146 
 
 10.3 
 
 6.0 
 
 1.7 
 
 4.3 
 
 40 
 
 55.8151.8 
 
 47.7 
 
 43.7 
 
 39.7 
 
 35.6 
 
 31.6 
 
 27.6 
 
 23.6 
 
 19.5 
 
 15.5 
 
 11.5 
 
 7.4 
 
 3.1 
 
 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 
 
 5.1 
 
 3.8 
 
 60 
 
 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 
 
 21,8 
 
 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 
 
 47.7i45.OJ 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 
 
 28.8 
 
 28. 6| 28.3 
 
 0.3 
 
 200 
 
 30.0 
 
 30.0 
 
 30.0 
 
 30.01 30.0 
 
 30.0 
 
 30.0 
 
 30.0 
 
 30.0 
 
 80.0 
 
 30.0 
 
 30.0 
 
 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 
 
 31.4 31.7 
 
 0.3 
 
 220 
 
 26.3 
 
 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 
 
 23.5 24.6! 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 
 
 23.2 24.5 
 
 25.71 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 J 23.3 
 
 24.9 26.4 
 
 27.9 
 
 29.4 
 
 30.9 
 
 32.4 
 
 33.9 
 
 35.4 
 
 37.0 
 
 38.5 
 
 40.0 
 
 1.5| 
 
 270 
 
 18.7 
 
 20.,*; 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 
 
 19.1J 21.1 23.1 
 
 25.2 
 
 27.2 
 
 29.2 
 
 31.2 
 
 33.2 
 
 35.2 
 
 37.3 
 
 39.3 
 
 41.3 43.3 
 
 2.0 
 
 290 15.5 
 
 17.8 20.0, 22.3 24.6 
 
 26.8 
 
 29.1 
 
 31.4 
 
 33.6 
 
 35.9 
 
 38.2 
 
 40.4 
 
 42.7J 45.0 
 
 2.3 
 
 300 13.9 
 
 16.4 18.9121.4 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 j 
 
 320 10.7|l3.7 16.7; 19.7 22.7 
 
 25.8 
 
 28.8 
 
 31.8 
 
 34.8 
 
 37.9 
 
 40.9 
 
 43.9 
 
 46.9 
 
 50.0 
 
 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 
 
 51.6 
 
 33 
 
 340 
 
 7.4)10.9 
 
 14.5 18.0 
 
 21.5 
 
 25.1 
 
 28.6 32.1 35.6 
 
 39.2 
 
 42.7 
 
 46.2 
 
 49.8 
 
 53.3 
 
 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 3.8 
 
 360 
 
 4.2 
 
 8.2 
 
 12.3 16.3 
 
 20.3 
 
 24.4 
 
 28.4 32.4 
 
 36.4 
 
 40.5 
 
 44.5 
 
 48.5 
 
 52.6 
 
 56.6 
 
 4.0 
 
 370 
 
 2.6i 6.9 
 
 11.1 15.4 
 
 19.7 
 
 24.01 28.3 32.6 
 
 36.8 
 
 41.1 
 
 45.4 
 
 49.7 
 
 54.0 
 
 58.3 
 
 43 
 
 380 
 
 l.OJ 5.5 
 
 10.0 14.6 19.1 
 
 23.6J 28.2 32.7 
 
 37.2 
 
 41.8 
 
 46.3 50.9 
 
 55.4] 59.9 
 
 4.5 
 
 L_ 
 
 0" 
 
 50" 100" 
 
 150" 200" 
 
 1 : 
 
 250" 300" 350" 
 
 400" 450" 
 
 500"!550" 
 
 600" 650" 
 
 
 
 TABLE LXXXI I. Moons Horary Motion in Latitude. 
 Argument. Arg. II. of Latitude. 
 
 
 05 
 
 1.9 
 
 11* 
 
 111* 
 
 IVs 
 
 Vs 
 
 
 
 
 " 
 
 " 
 
 " 
 
 " 
 
 r 
 
 ft 
 
 
 
 
 
 9.3 
 
 8.7 
 
 7.1 
 
 5.0 
 
 2.9 
 
 1.3 
 
 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 
 
 9.2 
 
 8.2 
 
 6.3 
 
 4.1 
 
 2.1 
 
 0.9 
 
 18 
 
 15 
 
 9.1 
 
 8.0 
 
 6.1 
 
 3.9 
 
 2.0 
 
 0.9 
 
 15 
 
 18 
 
 9.1 
 
 7.9 
 
 5.9 
 
 3.7 
 
 1.8 
 
 0.8 
 
 12 
 
 21 
 
 9.0 
 
 7.7 
 
 5.7 
 
 3.5 
 
 1.7 
 
 0.8 
 
 9 
 
 24 
 
 89 
 
 7.5 
 
 5.4 
 
 3.3 
 
 1.5 
 
 0.8 
 
 6 
 
 27 
 
 88 
 
 7.3 
 
 5.2 
 
 3.1 
 
 1.4 
 
 0.7 
 
 . 3 
 
 30 
 
 87 
 
 7.1 
 
 5.0 
 
 2.9 
 
 1.3 
 
 0.7 
 
 
 
 
 XI* 
 
 Xo ; :,\, iVM*:- V!i* 
 
 VI* 
 
 
 
 M 
 
98 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. 
 
 : 
 
 
 
 
 
 
 
 
 
 Prec. 
 
 
 
 
 
 
 
 
 H 
 
 equ. 
 
 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 600 
 
 700 
 
 
 
 2.1 
 
 1.8 
 
 1.5 
 
 1.2 
 
 0.9 
 
 0.6 
 
 0.3 
 
 0.0 
 
 1 
 
 .9 
 
 1.6 
 
 1.4 
 
 1.1 
 
 0.9 
 
 0.7 
 
 0.4 
 
 0.2 
 
 2 
 
 .7 
 
 1.5 
 
 1.3 
 
 1.1 
 
 .0 
 
 0.8 
 
 0.6 
 
 0.3 
 
 3 
 
 .5 
 
 1.4 
 
 1.2 
 
 1.1 
 
 .0 
 
 0.9 
 
 0.8 
 
 0.6 
 
 4 
 
 .3 
 
 1.2 
 
 1.2 
 
 1.1 
 
 .1 
 
 .0 
 
 0.9 
 
 0.9 
 
 5 
 
 .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 
 
 1.1 
 
 .2 
 
 .3 
 
 1.4 
 
 1.6 
 
 8 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.1 
 
 .2 
 
 .4 
 
 1.6 
 
 1.9 
 
 9 
 
 0.3 
 
 0.6 
 
 0.8 
 
 1.1 
 
 1.3 
 
 .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 237" .2. 
 
 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 
 
 " 
 
 
 
 
 
 
 
 
 equ. 
 
 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 600 
 
 700 
 
 0.00 
 
 0.65 
 
 0.57 
 
 0.48 
 
 0.39 
 
 0.31 
 
 0.21 
 
 0.12 
 
 0.00 
 
 0.10 0.62 
 
 055 
 
 0.47 
 
 0.39 
 
 0.31 !o.23 
 
 0.15 
 
 0.04 
 
 0.20 0.69 
 
 0.53 
 
 0.46 
 
 0.39 
 
 0.32 0.25 
 
 0.18 
 
 0.09 
 
 0.30 0.66 
 
 0.51 
 
 0.45 
 
 0.39 
 
 0.33 0.27 
 
 0.21 
 
 0.13 
 
 0.40 0.63 
 
 0.48 
 
 0.44 
 
 0.39 
 
 0.3410.29 
 
 0.24 
 
 0.17 
 
 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 
 
 0.32 
 
 0.30 
 
 0.80 
 
 0.41 ,0.40 
 
 0.39 
 
 0.38 
 
 0.37 
 
 0.36 
 
 0.35 
 
 0.34 
 
 0.90 
 
 0.38 0.38 
 
 0.38 
 
 0.38 
 
 0.38 
 
 0.38 
 
 0.38 
 
 0.38 
 
 1.00 
 
 0.35 
 
 0.36 
 
 0.37 
 
 0.38 
 
 0.39 
 
 0.40 
 
 0.41 
 
 0.42 
 
 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 
 
 0.51 
 
 1.30 
 
 026 
 
 0.30 
 
 033 
 
 0.38 
 
 0.41 
 
 0.46 
 
 0.49 
 
 0.55 
 
 1.40 
 
 0.23 
 
 0.28 
 
 0.32 
 
 0.37 
 
 0.42 
 
 0.47 
 
 0.52 
 
 0.59 
 
 1.50 
 
 0.20 
 
 0.25 
 
 0.31 
 
 037 
 
 0.43 
 
 0.49 
 
 0.55 
 
 0.63 
 
 1.60 
 
 0.17 
 
 0.23 
 
 0.30 
 
 0.37 
 
 0.44 
 
 0.51 
 
 0.58 
 
 0.67 
 
 1.70 
 
 0.14 
 
 0.21 
 
 0.29 
 
 0.37 
 
 0.45 
 
 0.53 
 
 0.61 
 
 0.72 
 
 1.80 
 
 0.11 
 
 0.19 
 
 0.28 
 
 0.37 
 
 0.45 
 
 0.55 
 
 0.64 
 
 0.76 
 
 
 
 
 100 
 
 200 
 
 300 
 
 400 
 
 500 
 
 600 
 
 700 
 
 TABLE LXXXIV. 
 
 Moorfs Hor. Motion in Lat 
 
 (Equa. of second order.) 
 
 Argument. Arg. I of Lat. 
 
 
 I 
 
 I 
 
 
 o 
 
 
 
 // 
 
 
 
 
 
 0.90 
 
 0.90 
 
 XII 
 
 5 
 
 0.83 
 
 0.97 
 
 25 
 
 10 
 
 0.75 
 
 1.05 
 
 20 
 
 15 
 
 0.68 
 
 1.12 
 
 15 
 
 20 
 
 0.61 
 
 .19 
 
 10 
 
 25 
 
 0.54 
 
 .26 
 
 5 
 
 I 
 
 0.47 
 
 .33 
 
 XI 
 
 5 
 
 0.41 
 
 .39 
 
 25 
 
 10 
 
 0.35 
 
 .45 
 
 20 
 
 15 
 
 0.29 
 
 .51 
 
 15 
 
 20 
 
 0.24 
 
 .56 
 
 10 
 
 25 
 
 0.20 
 
 .60 
 
 5 
 
 II 
 
 0.16 
 
 .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 
 
 25 
 
 0.04 
 
 1.76 
 
 5 
 
 III 
 
 0.04 
 
 1.76 
 
 IX 
 
 5 
 
 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 
 
 rv o 
 
 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 
 
 20 
 
 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. 
 Mean New Moons and Arguments, in January. 
 
 Years. 
 
 Mean New 
 Moon in. 
 January. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 N. 
 
 
 d. h m. 
 
 
 
 
 
 
 1821 
 
 2 17 59 
 
 0092 
 
 7859 
 
 80 
 
 78 
 
 823 
 
 1822 
 
 21 15 32 
 
 0602 
 
 7182 
 
 78 
 
 66 
 
 930 
 
 1823 
 
 11 20 
 
 0304 
 
 5787 
 
 61 
 
 55 
 
 953 
 
 1824 B 
 
 29 21 53 
 
 0814 
 
 5110 
 
 59 
 
 43 
 
 060 
 
 1825 
 
 18 6 41 
 
 0516 
 
 3716 
 
 42 
 
 32 
 
 083 
 
 1826 
 
 7 15 30 
 
 0218 
 
 2321 
 
 25 
 
 21 
 
 105 
 
 1827 
 
 26 13 3 
 
 0728 
 
 1644 
 
 24 
 
 09 
 
 213 
 
 1828 B 
 
 15 21 51 
 
 0430 
 
 0250 
 
 07 
 
 98 
 
 235 
 
 1829 
 
 4 6 40 
 
 0131 
 
 8855 
 
 90 
 
 87 
 
 257 
 
 1830 
 
 23 4 12 
 
 0642 
 
 8178 
 
 88 
 
 75 
 
 365 
 
 1831 
 
 12 13 1 
 
 0343 
 
 6784 
 
 71 
 
 64 
 
 387 
 
 1832 B 
 
 1 21 50 
 
 0045 
 
 5389 
 
 54 
 
 53 
 
 409 
 
 1833 
 
 19 19 22 
 
 0555 
 
 4712 
 
 53 
 
 42 
 
 517 
 
 1834 
 
 9 4 11 
 
 0257 
 
 3318 
 
 36 
 
 31 
 
 539 
 
 1835 
 
 28 1 43 
 
 0768 
 
 2641 
 
 34 
 
 19 
 
 647 
 
 1836 B 
 
 17 10 32 
 
 0469 
 
 1246 
 
 17 
 
 08 
 
 669 
 
 1837 
 
 5 19 20 
 
 0171 
 
 9852 
 
 00 
 
 97 
 
 692 
 
 1838 
 
 24 16 53 
 
 0681 
 
 9175 
 
 99 
 
 85 
 
 799 
 
 1839 
 
 14 1 42 
 
 0383 
 
 7780 
 
 82 
 
 74 
 
 822 
 
 1840 B 
 
 3 10 30 
 
 0085 
 
 6386 
 
 65 
 
 63 
 
 844 
 
 1841 
 
 21 8 3 
 
 0595 
 
 5709 
 
 63 
 
 51 
 
 951 
 
 1842 
 
 10 16 51 
 
 0.297 
 
 4314 
 
 46 
 
 40 
 
 974 
 
 i843 
 
 29 14 24 
 
 0807 
 
 3637 
 
 44 
 
 28 
 
 081 
 
 1844 B 
 
 18 23 13 
 
 0509 
 
 2243 
 
 28 
 
 17 
 
 104 
 
 1845 
 
 7 8 1 
 
 0211 
 
 0848 
 
 11 
 
 06 
 
 126 
 
 1846 
 
 26 5 34 
 
 C721 
 
 0171 
 
 09 
 
 94 
 
 234 
 
 1847 
 
 15 14 22 
 
 0423 
 
 8777 
 
 92 
 
 84 
 
 256 
 
 i848B 
 
 4 23 11 
 
 0125 
 
 7382 
 
 75 
 
 73 
 
 278 
 
 184-J 
 
 22 20 43 
 
 0635 
 
 6705 
 
 73 
 
 61 
 
 386 
 
 1850 
 
 12 5 32 
 
 0337 
 
 5311 
 
 56 
 
 50 
 
 408 
 
 1851 
 
 1 14 21 
 
 0038 
 
 3916 
 
 40 
 
 39 
 
 431 
 
 1852 B 
 
 20 11 53 
 
 0549 
 
 3239 
 
 38 
 
 27 
 
 538 
 
 1853 
 
 8 20 42 
 
 0251 
 
 1845 
 
 21 
 
 16 
 
 560 
 
 1854 
 
 27 18 14 
 
 0761 
 
 1168 
 
 19 
 
 04 
 
 6G8 
 
 1855 
 
 17 3 3 
 
 0463 
 
 9773 
 
 02 
 
 93 
 
 690 
 
 1856 B 
 
 6 11 51 
 
 0164 
 
 8379 
 
 85 
 
 82 
 
 713 
 
 1857 
 
 24 9 24 
 
 0675 
 
 7702 
 
 84 
 
 70 
 
 820 
 
 1953 
 
 13 18 13 
 
 0376 
 
 6307 
 
 67 
 
 59 
 
 843 
 
 1859 
 
 3 3 1 
 
 0078 
 
 4913 
 
 50 
 
 48 
 
 865 
 
 18GOB 
 
 22 34 
 
 0588 | 4236 
 
 48 
 
 36 
 
 972 
 
100 
 
 TABLE LXXXVII. 
 
 Mean Lunations and Changes of the Arguments, 
 
 Num 
 
 Lunations. 
 
 I. 
 
 II. 
 
 III. 
 
 IV. 
 
 N. 
 
 
 d. h ni 
 
 
 
 
 
 
 i 
 
 14 18 22 
 
 404 
 
 5359 
 
 58 
 
 50 
 
 43 
 
 1 
 
 29 12 44 
 
 808 
 
 717 
 
 15 
 
 99 
 
 85 
 
 2 
 
 59 1 28 
 
 1617 
 
 1434 
 
 31 
 
 98 
 
 170 
 
 3 
 
 88 14 12 
 
 2425 
 
 2151 
 
 46 
 
 97 
 
 256 
 
 4 
 
 118 2 56 
 
 3234 
 
 2869 
 
 61 
 
 96 
 
 341 
 
 5 
 
 147 15 40 
 
 4042 
 
 3586 
 
 76 
 
 95 
 
 426 
 
 6 
 
 177 4 24 
 
 4851 
 
 4303 
 
 92 
 
 95 
 
 511 
 
 7 
 
 206 17 8 
 
 5659 
 
 5020 
 
 7 
 
 94 
 
 596 
 
 8 
 
 236 5 52 
 
 6468 
 
 5737 
 
 22 
 
 93 
 
 682 
 
 9 
 
 265 18 36 
 
 7276 
 
 6454 
 
 37 
 
 92 
 
 767 
 
 10 
 
 295 7 20 
 
 8085 
 
 7171 
 
 53 
 
 91 
 
 852 
 
 11 
 
 324 20 5 
 
 8893 
 
 7889 
 
 68 
 
 90 
 
 937 
 
 12 
 
 354 8 49 
 
 9702 
 
 8606 
 
 83 
 
 89 
 
 22 
 
 13 
 
 383 21 33 
 
 510 
 
 9323 
 
 98 
 
 88 
 
 108 
 
 TABLE LXXXVIII. 
 
 Number of Days from the commencement of the year 
 to the first of each month- 
 
 Months. 
 
 Com. 
 
 Bis. j 
 
 
 Days. 
 
 Days. 
 
 January 
 
 
 
 
 
 February 
 
 31 
 
 31 
 
 March . 
 
 59 
 
 60 
 
 April . 
 
 90 
 
 91 
 
 May . 
 
 120 
 
 121 
 
 June 
 
 151 
 
 152 
 
 JuJy . 
 
 181 
 
 182 
 
 August . 
 
 212 
 
 213 
 
 September 
 October . 
 
 243 
 273 
 
 244 
 274 
 
 November 
 
 304 
 
 305 
 
 December 
 
 334 
 
 33f 
 
TABLE LXXXIX. 
 
 Equations for New and Fall Moon. 
 
 101 
 
 Arg. 
 
 i | ii 
 
 Arg. 
 
 I 
 
 II 
 
 Arg! Ill 
 
 IV 
 
 Arg 
 
 
 h in 1 h in 
 
 
 h m 
 
 h m 
 
 
 m 
 
 m 
 
 
 
 
 4 20 I 10 10 
 
 5000 
 
 4 20 
 
 10 10 
 
 25 
 
 3 
 
 31 
 
 25 
 
 100 
 
 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 
 
 500 
 
 5 40 
 
 7 22 
 
 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 i 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 
 
 j 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 
 
 7 
 
 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 
 
 5 
 
 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 
 
 7600 
 
 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 
 
 I 49 
 
 15 34 
 
 65 
 
 16 
 
 11 
 
 85 
 
 4100 
 
 6 32 
 
 4 34 
 
 9100 
 
 2 2 
 
 15 5 
 
 66 
 
 16 
 
 11 
 
 84 
 
 4200 
 
 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 
 
 11 52 
 
 72 
 
 17 
 
 10 
 
 78 
 
 4800 
 
 4 51 
 
 8 50 
 
 9SOO 
 
 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 
 
 
 
102 
 
 TABLE XC. 
 
 Mean Right Ascensions and Declinations of 50 principal Fixed 
 Stars, for the beginning of 1840. 
 
 Stars' Name. 
 
 Mag 
 
 light Ascen. 
 
 AnnualVar. 
 
 Declination. 
 
 Ann. Var. 
 
 
 
 ft m a 
 
 8 
 
 ' " 
 
 ~ 
 
 1 Algenib 
 
 2.3 
 
 5 0.31 
 
 + 3.0775 
 
 14 17 38.82 N 
 
 -1- 20.051 
 
 2 /? Andromedae 
 
 2 
 
 1 46.7 
 
 3.309 
 
 34 46 17.2 N 
 
 19.35 
 
 3 Polaris 
 
 2.3 
 
 1 2 10.38 
 
 16.1962 
 
 88 27 21.96N 
 
 19.339 
 
 4 Achernar 
 
 1 
 
 1 31 44.88 
 
 2.2351 
 
 58 3 5.13 S 
 
 18.473 
 
 5 a Arietis 
 
 3 
 
 1 58 9.94 
 
 3.3457 
 
 22 42 11.81 N 
 
 -r- 17.455 
 
 
 
 
 
 
 f 
 
 6 a Ceti 
 
 2.3 
 
 2 53 55.34 
 
 -r- 3.1257 
 
 3 27 30.09 N 
 
 + 14.561 
 
 7 aPersei 
 
 2.3 
 
 3 12 55.97 
 
 4.2280 
 
 49 17 8.74N 
 
 13.371 
 
 8 AldebartM 
 
 1 
 
 4 26 44.77 
 
 3.4264 
 
 16 10 56.82 N 
 
 7.949 
 
 9 Capella 
 
 1 
 
 5 4 52.67 
 
 4.4066 
 
 45 49 42.81 N 
 
 4.793 
 
 10 Rig el 
 
 1 
 
 5 6 51.09 
 
 2.8783 
 
 8 23 29.29 S 
 
 4.620 
 
 11 tfTauri 
 
 2 
 
 5 16 10.96 
 
 4- 3.7820 
 
 28 27 58.20 N 
 
 -r- 3.825 
 
 12 y Orionis 
 
 2 
 
 5 16 33.1 
 
 3.210 
 
 6 11 55.3 N 
 
 + 3.82 
 
 13 a Columbao 
 
 2 
 
 5 33 51.52 
 
 21688 
 
 34 9 47.41 S 
 
 2.291 
 
 14 a Orionis 
 
 1 
 
 5 46 30.71 
 
 3.2430 
 
 7 22 17.14N 
 
 + 1.191 
 
 15 Canopus 
 
 1 
 
 6 20 24.18 
 
 1.3278 
 
 52 36 38.42 S 
 
 1.778 
 
 16 Sirnu 
 
 1 
 
 6 38 5.76 
 
 + 2.6458 
 
 16 30 4.79 S 
 
 -r- 4.449 
 
 17 Castor 
 
 3 
 
 7 24 23.06 
 
 3.8572 
 
 32 13 58.89N 
 
 7.206 
 
 18 Procyon 
 
 1.2 
 
 7 30 55.53 
 
 3.1448 
 
 5 37 48.92 N 
 
 8.720 
 
 19 PoZ/w* 
 
 2 
 
 ,7 35 31.07 
 
 3.6840 
 
 28 24 25.57 N 
 
 8.107 
 
 20 aHydrae 
 
 2 
 
 9 19 43.57 
 
 2.9500 
 
 7 58 4.83 S 
 
 + 15.341 
 
 21 Regulus 
 
 1 
 
 9 59 50.93 
 
 -1- 3.2220 
 
 12 44 49.70 N 
 
 17.356 
 
 22 a Ursae Majoris 
 
 1.2 
 
 10 53 47.98 
 
 3.8077 
 
 62 36 48.93N 
 
 19.221 
 
 23 /JLeonis 
 
 2.3 
 
 11 40 53.69 
 
 3.0660 
 
 15 28 1.16N 
 
 19.985 
 
 24 Virginia 
 
 34 
 
 11 42 21.4 
 
 3.124 
 
 2 40 2 6 N 
 
 19.98 
 
 25 y Ursae Majoris 
 
 2 
 
 11 45 22.93 
 
 3.1914 
 
 54 35 4 67 N 
 
 20.014 
 
 26aCrucis 
 
 2 
 
 12 17 43.7 
 
 + 3.258 
 
 62 12 47. 9S 
 
 -f 19.99 
 
 27 S/>fca 
 28 Centauri 
 
 1 
 2 
 
 13 16 46.36 
 13 57 18.0 
 
 3.1502 
 3.491 
 
 10 19 24.39 S 
 35 34 41.9 S 
 
 18.945 
 17.499 
 
 29 a Draconis 
 
 3.4 
 
 14 2.8 
 
 1.625 
 
 65 8 32.1 N 
 
 17.37 
 
 30 Arcturvs 
 
 1 
 
 14 8 21.96 
 
 2.7335 
 
 20 1 7.67 N 
 
 18.956 
 
 31 a 2 Centauri 
 
 1 
 
 14 28 47.84 
 
 -{- 4.0086 
 
 60 10 6.24 S 
 
 -f 15. 152 
 
 32 a 2 Librae 
 
 3 
 
 14 42 2.44 
 
 3.3088 
 
 15 22 18.25S 
 
 15.256 
 
 33 /> Ursae Minoris 
 
 3 
 
 14 51 14.66 
 
 0.2787 
 
 74 48 34.18N 
 
 14.712 
 
 34 y a Ursae Minoris 
 
 3.4 
 
 15 21 1.3 
 
 0.179 
 
 72 24 14.1 N 
 
 12.81 
 
 35 a Coronae Borealis 
 
 2 
 
 15 27 54.87 
 
 + 2.5277 
 
 27 15 27.71 N 
 
 12.361 
 
 36 a Serpentis 
 
 2.3 
 
 15 36 2343 
 
 + 2.9386 
 
 6 56 2. SON 
 
 11.770 
 
 37 /JScorpii 
 
 2 
 
 15 56 8.68 
 
 3.4729 
 
 19 21 38.82 S 
 
 -1- 10.330 i 
 
 38 Ant&res 
 
 1 
 
 16 19 36.49 
 
 3.6625 
 
 26 4 13.13S 
 
 8.519 
 
 J9 a Herculis 
 
 3.4 
 
 17 7 21.30 
 
 2.7317 
 
 14 34 41.43 N 
 
 4.576 
 
 40 uOphiuchi 
 
 2 
 
 17 27 30 56 
 
 2.7724 
 
 12 40 58.65 N 
 
 2.844 
 
 41 fi Ursae Minoris 
 
 3 
 
 18 23 56.48 
 
 19.2072 
 
 86 35 28.89 N 
 
 + 2.161 
 
 42 Vega 
 
 1 
 
 18 31 31.19 
 
 + 2.0116 
 
 38 38 16.85 N 
 
 2.742 
 
 43 Altair 
 
 1 
 
 19 42 58.61 
 
 2.9255 
 
 8 27 0.21 N 
 
 8.701 
 
 44 a 2 Capricomi 
 
 3 
 
 20 9 10.34 
 
 3.3323 
 
 13 2 5.57 S 
 
 10.705 
 
 45 a Cygni 
 
 1 
 
 20 35 58.80 
 
 2.0416 
 
 44 42 41.38N 
 
 + 12.614 
 
 46 a Aquarii 
 
 3 
 
 21 57 33.93 
 
 + 3.0835 
 
 1 5 38.00 S 
 
 17.256 
 
 47 Fomalhaut 
 
 1 
 
 22 48 47.67 
 
 3.3114 
 
 30 28 4.91 S 
 
 19.092 
 
 48 /?Pegasi 
 
 2 
 
 22 56 1.1 
 
 2.878 
 
 27 13 1.7 N 
 
 -f 19.255 
 
 49 Markal 
 
 2 
 
 22 56 47.75 
 
 2.9771 
 
 14 20 46.92N 
 
 19295 
 
 50 a Andromedae 
 
 1 
 
 24 7.72 
 
 3.0704 
 
 28 12 27.06 N 
 
 20.056 i 
 
TABLE XCI. 
 
 103 
 
 Constants for the Aberration and Nutation in Right Ascension 
 and Declination of the Stars in the preceding Catalogue 
 
 \ 
 
 Aberration. 
 
 Nutation. 
 
 
 
 
 M ! 
 
 t 
 
 N 
 
 
 
 M' 
 
 6' 
 
 N' 
 
 
 * ' 
 
 
 ' 
 
 8 ' 
 
 
 8 ' 
 
 
 1 
 
 8 28 47 
 
 0.1087 
 
 7 27 12,0.9657 
 
 6 8 24 
 
 0.0300 
 
 5 28 30 
 
 0.8381 
 
 2 
 
 8 13 39 
 
 0.1830 
 
 6 19 12 1.0740 
 
 6 19 53 
 
 0.0838 
 
 5 10 8 
 
 0.8496 
 
 3 
 
 8 13 51 
 
 1.6526 
 
 5 16 57 1.3052 
 
 8 16 7 1.3427 
 
 5 10 22 
 
 0.8493 
 
 4 
 
 8 5 20,0.3801 
 
 10 26 46 1.2798 
 
 4 10 12 
 
 0.0775 
 
 5 31 
 
 0.8629 
 
 5 
 
 7 28 26 
 
 0.1397 
 
 702 
 
 0.8972 
 
 6 11 1 
 
 0.0695 
 
 4 22 53 
 
 0.8765 
 
 6 
 
 7 14 11 
 
 0.1149 
 
 8 23 8 
 
 0.8678 
 
 6 1 26 
 
 0.0322 
 
 4 8 16 
 
 0.9078 
 
 7 
 
 7 9 30 
 
 0.3020 
 
 535 
 
 1.0630 
 
 6 18 13 
 
 0.1849 
 
 4 3 47 
 
 0.9179 
 
 8 
 
 6 21 43 
 
 0.1447 
 
 7 23 12 
 
 0.5760 
 
 6 3 27 
 
 0.0726 
 
 3 17 54 
 
 0.9502 
 
 9 
 
 6 12 51 
 
 0.2875 
 
 3 25 37 
 
 0.9112 
 
 6 5 46 
 
 0.1830 
 
 3 10 29 
 
 0.9605 
 
 10 
 
 6 12 20 
 
 0.1355 
 
 9 3 42 
 
 1.0300 
 
 5 28 47 
 
 1.9966 
 
 3 10 4 
 
 0.9608 
 
 11 
 
 6 10 13 
 
 0.1873 
 
 4 19 21 
 
 0.3917 
 
 6 2 52 
 
 0.1008 
 
 3 8 19 
 
 0.9626 
 
 12 
 
 6 10 6 
 
 0.1340 
 
 8 26 4 
 
 0.7851 
 
 6 40 
 
 0.0441 
 
 3 8 14 
 
 0.9626 
 
 13 
 
 665 
 
 0.2145 
 
 9 4 24 
 
 1.2348 
 
 5 26 18 
 
 1.8750 
 
 3 4 57 
 
 0.9648 
 
 14 
 
 6 3 13 
 
 0.1361 
 
 8 28 23 
 
 0.7521 
 
 6 15 
 
 0.0481 
 
 3 2 37 
 
 0.9657 
 
 15 
 
 5 25 22 
 
 0.3491 
 
 8 25 53 
 
 1.2960 
 
 6 8 46 
 
 1.6679 
 
 2 26 15 
 
 0.9657 
 
 16 
 
 5 21 21 
 
 0.1501 
 
 8 25 51 
 
 1.1152 
 
 6 1 51 
 
 1.9658 
 
 2 22 58 
 
 0.9636 
 
 17 
 
 5 10 40 
 
 0.2010 
 
 1 2 17 
 
 0.6620 
 
 5 24 2 
 
 0.1257 
 
 2 14 6 
 
 0.9535 
 
 18 
 
 596 
 
 0.1297 
 
 9 6 54 
 
 0.8071 
 
 5 28 47 
 
 0.0414 
 
 2 12 47 
 
 0.9513 
 
 19 
 
 582 
 
 0.1829 
 
 14 32 
 
 0.6052 
 
 5 24 2 
 
 0.1114 
 
 2 11 53 
 
 0.9499 
 
 20 
 
 4 12 39 
 
 0.1158 
 
 8 17 31 
 
 0.9967 
 
 6 3 41 
 
 0.0081 
 
 1 18 37 
 
 0.9007 
 
 21 
 
 4 2 22 
 
 0.1162 
 
 10 3 47 
 
 0.8457 
 
 5 23 47 
 
 0.0480 
 
 1 7 59 
 
 0.8782 
 
 22 
 
 3 18 7 
 
 0.4366 
 
 3 28 
 
 1.2394 
 
 4 18 58 
 
 0.2407 
 
 21 57 
 
 0.8520 
 
 23 
 
 3 5 21 
 
 0.1117 
 
 10 6 20 
 
 0.9621 
 
 5 20 56 
 
 0.0344 
 
 6 35 
 
 0.8393 
 
 24 
 
 3 4 57 
 
 0.0958 
 
 9 6 51 
 
 0.9075 
 
 5 28 25 
 
 0.0253 
 
 065 
 
 0.8390 
 
 25 
 
 348 
 
 0.3229 
 
 11 17 28 
 
 1.2298 
 
 4 21 46 
 
 0.1465 
 
 055 
 
 0.8383 
 
 26 
 
 2 25 19 
 
 0.4261 
 
 685 
 
 1.2585 
 
 7 16 2 
 
 0.2089 
 
 11 24 14 
 
 0.8390 
 
 27 
 
 2 9 22 
 
 0.1066 
 
 8 3 31 
 
 0.8862 
 
 6 5 51 
 
 0.0154 11 56 
 
 0.8559 
 
 28 
 
 1 28 40 
 
 0.1942 
 
 6 7 12 
 
 1.0176 
 
 6 17 31 
 
 0.1062 10 23 8 i 0.8760 
 
 29 
 
 1 27 53 
 
 0.4824 
 
 10 23 28 
 
 1.2995 
 
 3 25 50 
 
 0.1090 '10 22 16 0.8777 
 
 30 
 
 1 25 46 
 
 0.1336 
 
 9 28 18 
 
 1.0974 
 
 5 18 49 
 
 1.99371 10 20 1 0.8822 
 
 31 
 
 1 20 32 
 
 0.4123 
 
 5 7 54 
 
 1.1820 
 
 6 29 6 
 
 0.2460 il 10 14 36 0.8937 
 
 32 
 
 I 17 26 
 
 0.1273 
 
 7 18 24 
 
 0.6923 
 
 6 6 29 0.0593J 10 11 28 ! 0.9006 
 
 33 
 
 i 14 42 
 
 0.6961 
 
 10 15 5 
 
 1.3087 
 
 2 26 45 
 
 0. 2235 j 10 8 47 0.9066 
 
 34 
 
 1 7 20 0.6386 
 
 10 7 33 
 
 1.3087 
 
 2 27 7 
 
 0.0960 
 
 10 1 45 ' 0.9225 
 
 35 
 
 1 5 45 0.1704 
 
 9 22 28 
 
 1.1785 
 
 5 17 18 
 
 1.9510 
 
 10 18 
 
 0.9257 
 
 36 
 
 1 3 43 0.1237 
 
 9 8 22 
 
 0.9994 
 
 5 27 30 
 
 0.0058 
 
 9 28 26 
 
 0.9298 
 
 37 
 
 28 58 0.1485 
 
 744 
 
 0.6237 
 
 6 5 20 
 
 0.0795 9 24 12)0.9386 
 
 38 
 
 23 24 0.1728 
 
 5 27 59 
 
 0.5816 
 
 6 5 49 
 
 0.1029 9 j9 21 0.9478 
 
 39 
 
 12 13 0.1451 
 
 9 5 25 
 
 1.0962 
 
 5 27 45 
 
 1.9742 9 9 58 0.9610 
 
 40 
 
 7 34 0.1427 
 
 934 
 
 1.0786 
 
 5 28 48 
 
 1.9803 
 
 969 
 
 0.9642 
 
 41 
 
 11 23 47 1.3571 
 
 8 22 49 
 
 1.2821 
 
 11 19 31 
 
 0-8257 
 
 8 24 57 
 
 0.9650 
 
 42 
 
 11 22 50 0.2393 
 
 8 24 29 
 
 1.2545 
 
 6 5 31 
 
 1.8436 i 8 24 10 0.9644 
 
 43 
 
 11 6 15 0.1309 
 
 8 22 59 
 
 1.0237 
 
 2 16 
 
 1.9988 
 
 8 10 21 0.9472 
 
 44 
 
 11 2 0.1341 
 
 9 29 33 
 
 0.6961 
 
 5 26 12 
 
 0061)9 
 
 8 4 55 0.9368 
 
 45 
 
 10 23 29 0.2668 
 
 8 39 
 
 1.2634 
 
 6 28 32 
 
 1.9042 
 
 7 29 0.9242 
 
 46 
 
 10 2 57 0.1057 
 
 9 2 31 
 
 0.8988 
 
 5 29 S6 
 
 0.0264 
 
 7 8 37 0.8794 
 
 47 
 
 9 19 26 0.1638 
 
 11 7 34 1.0271 
 
 5 13 8 
 
 0.0765 
 
 6 23 30 '0.8540 
 
 48 
 
 9 17 29 0.1491 
 
 7 17 1.1171 
 
 6 17 2 
 
 0.0162 
 
 6 21 13! 0.8511 
 
 49 
 
 9 17 17 
 
 0.1120 
 
 825 
 
 1.0138 
 
 6 8 23 
 
 0.0157 
 
 6 20 58 0.8508 
 
 50 
 
 906 
 
 0.1495 
 
 7 6 42 
 
 1.0785 
 
 6 17 20 
 
 0.0444 
 
 6 8 1 0.8380 
 
104 
 
 TABLE XCII. 
 
 Mean Longitudes and Latitudes of some of the principal Fixed 
 Stars for the beginning of 1 840, with their Annual Variations. 
 
 Stars' Name. 
 
 Mag 
 
 Longitude. 
 
 Annual 
 Var. 
 
 Latitude. 
 
 Annual 
 Var. 
 
 a Arietis 
 
 3 
 
 g ' " 
 
 1 5 25 27.6 
 
 50.277 
 
 ' " 
 
 9 57 40.9 N 
 
 + 0.161 
 
 Aldebaran 
 
 1 
 
 2 7 33 5.9 
 
 50.210 
 
 5 28 38.0 s 
 
 0.335 
 
 Capella 
 
 1 
 
 2 19 37 17.8 
 
 50.302 
 
 22 51 44.4 N 
 
 0.052 
 
 Polaris 
 
 2.3 
 
 2 26 19 20.1 
 
 47.959 
 
 66 4 59. 5 N 
 
 + 0.552 
 
 Sirius 
 
 1 
 
 3 11 52 32.9 
 
 49.488 
 
 39 34 4.3 S 
 
 + 0.319 
 
 Canopus 
 
 1 
 
 3 12 44 59.6 
 
 49.366 
 
 75 50 57.6 S 
 
 + 0.459 
 
 Pollux 
 
 2 
 
 3 21 22.0 
 
 49.502 
 
 6 40 20.2 N 
 
 + 0.255 
 
 Regulus 
 
 1 
 
 4 27 36 13.2 
 
 49.946 
 
 27 38. 3 N 
 
 + 0.220 
 
 Spica 
 
 1 
 
 6 21 36 29.2 
 
 50.085 
 
 2 2 29.7 S 
 
 + 0.171 
 
 Arcturus 
 
 1 
 
 6 22 4.7 
 
 50.711 
 
 30 51 17.5 N 
 
 + 0.214 
 
 Antares 
 
 1 
 
 8 7 31 45.2 
 
 50.120 
 
 4 32 51. 6 S 
 
 4 0.424 
 
 Altair 
 
 1.2 
 
 9 29 31 5.9 
 
 50.795 
 
 29 18 37.3 N 
 
 4- 0.080 
 
 Fomalhaut 
 
 1 
 
 11 1 36 22.0 
 
 50.595 
 
 21 6 49. 7 S 
 
 + 0.213 
 
 Achernar 
 
 1 
 
 11 13 2 5.3 
 
 50.346 
 
 17 6 17.3 S 
 
 0.083 
 
 a Pegasi 
 
 2 
 
 11 21 15 24.7 
 
 50.112 
 
 19 24 40.9 N 
 
 + 0.098 
 
 TABLE added to TABLE XC. 
 
 Mean Right Ascensions and Declinations of Polaris and <* Ursae 
 Minoris for 1S30, 1840, 1850, and 1860. 
 
 Stars. 
 
 Years 
 
 Right Asc. 
 
 Ann. Var. 
 
 Declination. 
 
 Ann. Var. 
 
 
 
 O ' ' 
 
 // 
 
 / " 
 
 
 
 
 1830 
 
 59 30.76 
 
 + 15.478 
 
 88 24 8.82 
 
 + 19.371 
 
 Polaris 
 
 1840 
 1850 
 
 1 2 10.32 
 1 5 0.29 
 
 16.470 
 17.567 
 
 88 27 22.43 
 88 30 35.40 
 
 19.309 
 19.240 
 
 
 1860 
 
 1 8 1.79 
 
 18.784 
 
 88 33 47.64 
 
 19.163 
 
 
 1830 
 
 18 27 5.13 
 
 19.167 
 
 86 35 5.70 
 
 + 2.363 
 
 i Ursae Minoris 
 
 1840 
 
 18 23 53.03 
 
 19.241 
 
 86 35 27.93 
 
 2.085 
 
 
 1850 
 
 18 20 40.21 
 
 19.305 
 
 86 35 47.36 
 
 1.805 
 
 
 1860 
 
 18 17 26.77 
 
 19.360 
 
 86 36 3 97 
 
 1.523 
 
 
TABLE XCIII. 
 
 Second Differences. 
 
 105 
 
 Hours & Minutes. 
 
 1' 
 
 2' 
 
 3' 
 
 4' 
 
 5' 
 
 6' 
 
 7' 
 
 8' 
 
 9' 
 
 10' 
 
 ir 
 
 h m 
 
 h m 
 
 
 
 
 
 
 
 
 
 
 
 
 ! 
 
 12 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 10 
 
 11 50 
 
 0.4 
 
 0.8 
 
 1.2 
 
 1.6 
 
 2.0 
 
 2.4 
 
 2.9 
 
 3.3 
 
 3.7 
 
 4.1 
 
 4.5 
 
 20 
 
 11 40 
 
 0.8 
 
 1.6 
 
 2.4 
 
 3.2 
 
 4.1 
 
 4.9 
 
 5.7 
 
 6.5 
 
 7.3 
 
 81 
 
 8.9 
 
 30 
 
 11 30 
 
 1.2 
 
 2.4 
 
 3.6 
 
 4.8 
 
 6.0 
 
 7.2 
 
 8.4 
 
 9.6 
 
 10.8 
 
 12.0 
 
 13.2 
 
 40 
 
 11 20 
 
 1.6 
 
 3.1 
 
 4.7 
 
 6.3 
 
 7.9 
 
 9.4 
 
 11.0 
 
 12.6 
 
 14.2 
 
 15.7 
 
 17.3 
 
 50 
 
 11 10 
 
 1.9 
 
 3.9 
 
 5.8 
 
 7.8 
 
 9.7 
 
 11.6 
 
 13.6 
 
 15.5 
 
 17.4 
 
 19.4 
 
 21.4 
 
 1 
 
 11 
 
 2.3 
 
 4.6 
 
 6.9 
 
 9.2 
 
 11.5 
 
 13.8 
 
 16.0 
 
 18.3 
 
 20.6 
 
 22.9 
 
 25.2 
 
 1 10 
 
 10 50 
 
 2.6 
 
 5.3 
 
 7.9 
 
 10.5 
 
 13.2 
 
 15.8 
 
 18.4 
 
 21.1 
 
 23.7 
 
 26.3 
 
 29.0 
 
 1 20 
 
 10 40 
 
 3.0 
 
 5.9 
 
 8.9 
 
 11.9 
 
 14.8 
 
 17.8 
 
 20.7 
 
 23.7 
 
 26.7 
 
 29.6 
 
 32.6 
 
 1 30 
 
 10 30 
 
 3.3 
 
 6.6 
 
 9.8 
 
 13.1 
 
 16.4 
 
 19.7 
 
 23.0 
 
 26.3 
 
 29.5 
 
 32.8 
 
 36.1 
 
 1 40 
 
 10 20 
 
 3.6 
 
 7.2 
 
 10.8 
 
 14.4 
 
 17.9 21.5 
 
 25.1 
 
 28.7 
 
 32.3 
 
 35.9 
 
 39.5 
 
 1 50 
 
 10 10 
 
 3.9 
 
 7.8 
 
 11.6 
 
 15.5 
 
 19.4 
 
 23.3 
 
 27.2 
 
 31.0 
 
 34.9 
 
 38.8 
 
 42.7 
 
 2 
 
 10 
 
 4.2 
 
 8.3 
 
 12.5 
 
 16.7 
 
 20.8 
 
 25.0 
 
 29.2 
 
 33.3 
 
 37.5 
 
 41.7 
 
 45.8 
 
 2 10 
 
 9 50 
 
 4.4 
 
 8.9 
 
 13.3 
 
 17.8 
 
 22.2 26.6 
 
 31.1 
 
 35.5 
 
 40.0 
 
 44.4 
 
 48.8 
 
 2 20 
 
 9 40 
 
 4.7 
 
 9.4 
 
 14.1 
 
 18.8 
 
 23.5 
 
 28.2 
 
 32.9 
 
 37.6 
 
 42.3 
 
 47.0 
 
 51.7 
 
 2 30 
 
 9 30 
 
 4.9 
 
 9.9 
 
 14.8 
 
 19.8 
 
 24.7 
 
 29.7 
 
 34.6 
 
 39.6 
 
 44.5 
 
 49.5 
 
 54.4 
 
 2 40 
 
 9 20 
 
 5.2 
 
 10.4 
 
 15.6 
 
 20.7 
 
 25.9 
 
 31.1 
 
 36.3 
 
 41.5 
 
 46.7 
 
 51.9 
 
 57.0 
 
 2 50 
 
 9 10 
 
 5.4 
 
 10.8 
 
 16.2 
 
 21.6 
 
 27.1 
 
 32.5 
 
 37.9 
 
 43.3 
 
 48.7 
 
 54.1 
 
 59.5 
 
 3 
 
 9 
 
 5.6 
 
 11.3 
 
 16.9 
 
 22.5 
 
 28.1 
 
 33.8 
 
 39.4 
 
 45.0 
 
 50.6 
 
 56.3 
 
 61.9 
 
 3 10 
 
 8 50 
 
 5.8 
 
 11.7 
 
 17.5 
 
 23.3 
 
 29.1 
 
 35.0 
 
 40.8 
 
 46.6 
 
 52.4 
 
 58.3 
 
 64.1 
 
 3 20 
 
 8 40 
 
 6.0 
 
 12.0 
 
 18.1 
 
 24.1 
 
 30.1 
 
 36.1 
 
 42.1 
 
 48.1 
 
 54.2 
 
 60.2 
 
 66.2 
 
 3 30 
 
 8 30 
 
 6.2 
 
 12.4 
 
 18.6 
 
 24.8 
 
 31.0 
 
 37.2 
 
 43.4 
 
 49.6 
 
 55.8 
 
 62.0 
 
 68.2 
 
 3 40 
 
 8 20 
 
 6.4 
 
 12.7 
 
 19.1 
 
 25.5 
 
 31.8 
 
 38.2 
 
 44.6 
 
 50.9 
 
 57.3 
 
 63.7 
 
 70.0 
 
 3 50 
 
 8 10 
 
 6.5 
 
 13.0 
 
 19.6 
 
 26.1 
 
 32.6 
 
 39.1 
 
 45.7 
 
 52.2 
 
 58.7 
 
 65.2 
 
 71.7 
 
 4 
 
 8 
 
 6.7 
 
 13.3 
 
 20.0 
 
 26.7 
 
 33.3 
 
 40.0 
 
 46.7 
 
 53.3 
 
 60.0 
 
 66.7 
 
 73.3 
 
 4 10 
 
 7 50 
 
 6.8 
 
 13.6 
 
 20.4 
 
 27.2 
 
 34.0 
 
 40.8 
 
 47.6 
 
 54.4 
 
 61.2 
 
 68.0 
 
 74.8 
 
 4 20 
 
 7 40 
 
 6.9 
 
 13.8 
 
 20.8 
 
 27.7 
 
 34.6 
 
 41.5 
 
 48.4 
 
 55.4 
 
 62.3 
 
 69.2 
 
 76.1 
 
 4 30 
 
 7 30 
 
 7.0 
 
 14.1 
 
 21.1 
 
 28.1 
 
 35.2 42.2 
 
 49.2 
 
 56.2 
 
 63.3 
 
 70.3 
 
 77.3 
 
 4 40 
 
 7 20 
 
 7.1 
 
 14.3 
 
 21.4 
 
 28.5 
 
 35.6 
 
 42.8 
 
 49.9 
 
 57.0 64.2 
 
 71.3 
 
 78.4 
 
 4 50 
 
 7 10 
 
 7.2 
 
 14.4 
 
 21.6 
 
 28.9 
 
 36.1 
 
 43.3 
 
 50.5 
 
 57.7 
 
 64.9 
 
 72.2 
 
 79.4 
 
 5 
 
 7 
 
 7.3 
 
 14.6 
 
 21.9 
 
 29.2 
 
 36.5 
 
 43.8 
 
 51.0 
 
 58.3 
 
 65.6 
 
 72.9 
 
 80.2 
 
 5 10 
 
 6 50 
 
 7.4 
 
 14.7 
 
 22.1 
 
 29.4 
 
 36.8 
 
 44.1 
 
 51.5 
 
 58.8 66.2 
 
 73.6 
 
 80.9 
 
 5 20 
 
 6 40 
 
 7.4 
 
 14.8 
 
 22.2 
 
 29.6 
 
 37.0 
 
 44.4 
 
 51.9 
 
 59.3 66.7 
 
 74.1 
 
 81.5 
 
 5 30 
 
 6 30 
 
 7.4 
 
 14.9 
 
 22.3 
 
 29.8 
 
 37.2 
 
 44.7 
 
 52.1 
 
 59.6 67.0 
 
 74.5 
 
 81.9 
 
 5 40 
 
 6 20 
 
 7.5 
 
 15.0 
 
 22.4 
 
 29.9 
 
 37.4 
 
 44.9 
 
 52.3 
 
 59.8 67.3 
 
 74.8 
 
 82.2 
 
 5 50 
 
 6 10 
 
 7.5 
 
 15.0 
 
 22.5 
 
 30.0 
 
 37.5 
 
 45.0 
 
 52.5 
 
 60.0 67.4 74.9 
 
 82.4 
 
 6 
 
 6 
 
 7.5 
 
 15.0 
 
 22.5 
 
 30.0 
 
 37.5 
 
 45.0 
 
 52.5 
 
 60.0 67.5 
 
 75.0 
 
 82.5 
 
 N 
 
106 
 
 TABLE XCIII. 
 
 Second Differences. 
 
 Hours & Mm. 10" 
 
 20" 
 
 30" 40" 
 
 50" 
 
 1" 
 
 2" 
 
 3" 
 
 4'' 
 
 5" 
 
 6" 
 
 7" 
 
 8" 
 
 9" 
 
 h m 
 
 h m " 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 " 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 10 
 
 11 50 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.1 0.1 
 
 020 
 
 11 40 
 
 0.1 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.7 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.1 j 0.1 
 
 30 
 
 11 30 
 
 0.2 0.4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 0.0 
 
 0.0 
 
 0.1 
 
 0.1 
 
 O.I 
 
 0.1 
 
 0.1 
 
 0.2 0.2 
 
 40 11 20 
 
 0.3 
 
 0.5 
 
 0.8 
 
 1.0 
 
 1.3 
 
 0.0 
 
 0. 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.2 0.2 
 
 50 11 10 
 
 0.3 
 
 0.6 
 
 1.0 
 
 1.3 
 
 1.6 
 
 0.0 
 
 0. 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.2 
 
 0.3 0.3 
 
 1 
 
 11 
 
 0.4 
 
 0.8 
 
 1.1 
 
 1.5 
 
 1.9 
 
 0.0 
 
 0. 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 0.3 
 
 10 
 
 10 50 
 
 0.4 
 
 0.9 1.3 
 
 1.8 
 
 2.2 
 
 0.0 
 
 0. 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 0.4 
 
 20 
 
 10 40 
 
 0.5 
 
 1.0 1.5 
 
 2.0 
 
 2.5 
 
 0.0 
 
 0. 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 0.4l 
 
 30 
 
 10 30 
 
 0.5 
 
 1.1 1.6 
 
 2.2 
 
 2.7 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 40 
 
 10 20 
 
 0.6 
 
 1.2 
 
 1.8 
 
 2.4 
 
 3.0 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.5 
 
 50 
 
 10 10 
 
 0.6 
 
 1.3 
 
 1.9 
 
 2.6 
 
 3.2 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 2 
 
 10 
 
 0.7 1.4 
 
 2.1 
 
 2.8 
 
 3.5 
 
 0.1 
 
 0.1 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.6 
 
 2 10 
 
 9 50 
 
 0.7 
 
 1.5 
 
 2.2 
 
 3.0 
 
 3.7 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 2 20 
 
 9 40 
 
 0.8 
 
 1.6 
 
 2.3 
 
 3.1 
 
 3.9 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 2 30 
 
 9 30 
 
 0.8 
 
 1.6 
 
 2.5 
 
 3.3 
 
 4.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.7 
 
 2 40 
 
 9 20 
 
 0.9 
 
 1.7 
 
 2.6|3.5 
 
 4.3 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 2 50 
 
 9 10 
 
 0.9 
 
 1.8 
 
 2.7 ! 3.6 
 
 4.5 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 0.5 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 3 
 
 9 
 
 0.9 
 
 1.9 2.8 
 
 3.8 
 
 4.7 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 0.5 
 
 0.6 
 
 0.7 
 
 0.7 
 
 0.8 
 
 3 10 
 
 8 50 
 
 1.0 1.912.9 
 
 3.9 
 
 4.9 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 3 20 
 
 8 40 
 
 1.0 2.0 3.0 
 
 4.0 
 
 5.0 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 3 30 
 
 8 30 
 
 1.0 2.1 3.1 
 
 4.1 
 
 5.2 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 3 40 
 
 8 20 
 
 1.1 2.1 i3.2 
 
 4.2 5.3 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 | 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 1.0 
 
 3 50 
 
 8 10 
 
 1.1 2.2 
 
 3.3 
 
 4.3 5.4 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 4 
 
 8 
 
 1.1 2.2 
 
 3.3 
 
 4.4 5.6 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 4 10 
 
 7 50 
 
 1.1 2.3 
 
 3.4 
 
 4.5 
 
 5.7 
 
 0. 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 4 20 
 
 7 40 
 
 1.2 
 
 2.3 
 
 3.5 
 
 4.6 
 
 5.8 
 
 0. 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.0 
 
 4 30 
 
 7 30 
 
 1.2 
 
 2.313.5 
 
 4.7 
 
 5.9 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 0.9 
 
 1.1 
 
 4 40 
 
 7 20 
 
 1.2 2.4 j 3.6 
 
 4.8 
 
 5.9 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 1.0 
 
 1.1 
 
 4 50 
 
 7 10 
 
 1.2 
 
 2.4 
 
 3.6 
 
 4.8 
 
 6.0 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.8 
 
 1.0 
 
 1.1 
 
 5 
 
 7 
 
 1.2 
 
 2.4 
 
 3.6 
 
 4.9 
 
 6.1 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 5 10 
 
 6 50 
 
 1.2 
 
 2.5 
 
 3.7 
 
 4.9 
 
 6.1 
 
 0. 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 5 20 
 
 6 40 
 
 1.2 
 
 2.5 
 
 3.7 
 
 4.9 
 
 6.1 
 
 0.1 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 5 30 
 
 6 30 
 
 1.2 
 
 2.5 
 
 3.7 
 
 5.0 
 
 6.2 
 
 0.1 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 LO 
 
 1.1 
 
 5 40 
 
 6 20 
 
 1.2 
 
 2.5 
 
 3.7 
 
 5.0 
 
 6.2 
 
 0.1 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 5 50 
 
 6 10 
 
 1.2 
 
 2.5 
 
 3.7 
 
 5.0 
 
 6.2 
 
 0.1 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
 6 
 
 6 
 
 1.3 
 
 2.6 
 
 3.8 
 
 5.0 
 
 6.3 
 
 0.1 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.0 
 
 1.1 
 
TABLE XCIV. 
 
 107 
 
 Third Differences. 
 
 Time after 
 
 
 
 
 
 
 
 
 
 
 
 Time after 
 
 noon or 
 
 10" 
 
 20" 
 
 30" 
 
 40" 
 
 50" 
 
 1' 
 
 2' 
 
 3' 
 
 4' 
 
 5' 
 
 noon or 
 
 midnight. 
 
 
 
 
 
 
 
 
 
 
 
 midnight. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Oh. Om. 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 12h. Om. 
 
 30 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.4 
 
 0.5 
 
 0.7 
 
 0.9 
 
 11 30 
 
 1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.6 
 
 1.0 
 
 1.3 
 
 1.5 
 
 11 
 
 1 30 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.8 
 
 1.2 
 
 1.6 
 
 2.1 
 
 10 30 
 
 2 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.9 
 
 1.4 
 
 1.9 
 
 2.3 
 
 10 
 
 2 30 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 1.0 
 
 1.4 
 
 1.9 
 
 2.4 
 
 9 30 
 
 3 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.9 
 
 1.4 
 
 1.9 
 
 2.3 
 
 9 
 
 3 30 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.4 
 
 0.9 
 
 1.3 
 
 1.7 
 
 2.2 
 
 8 30 
 
 4 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.7 
 
 1.1 
 
 1.5 
 
 1.9 
 
 8 
 
 4 30 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.3 
 
 0.6 
 
 0.9 
 
 1.2 
 
 1.5 
 
 7 30 
 
 5 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.4 
 
 0.6 
 
 oa 
 
 1.0 
 
 7 
 
 5 30 
 
 0.0 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 6 30 
 
 6 
 
 . J 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 6 
 
 TABLE XCV. 
 
 Fourth Differences. 
 
 Time after 
 noon or 
 
 midnight. 
 
 10" 
 
 20" 
 
 30" 
 
 40" 
 
 50" 
 
 1- 
 
 2' 
 
 3' 
 
 Time after 
 noon or 
 midnight. 
 
 h. m. 
 
 
 
 
 
 
 
 
 
 h. m. 
 
 
 
 0.0 
 
 0.0 
 
 0.0 
 
 00 
 
 0.0 
 
 0.0 
 
 0.0 
 
 0.0 
 
 12 
 
 30 
 
 0.0 
 
 0.1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.2 
 
 0.4 
 
 0.6 
 
 11 30 
 
 1 
 
 0.1 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.3 
 
 0.4 
 
 0.8 
 
 1.2 
 
 11 
 
 1 30 
 
 0.1 
 
 0.2 
 
 0.3 
 
 0.4 
 
 0.5 
 
 0.6 
 
 1.2 
 
 1.7 
 
 10 30 
 
 2 
 
 0.1 
 
 0.2 
 
 04 
 
 0.5 
 
 0.6 
 
 0.7 
 
 1.5 
 
 2.2 
 
 10 
 
 2 30 
 
 0.1 
 
 0.3 
 
 0.4 
 
 0.6 
 
 0.7 
 
 0.9 
 
 1.8 
 
 2.7 
 
 9 30 
 
 3 
 
 0.2 
 
 0.3 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.0 
 
 2.1 
 
 3.1 
 
 9 
 
 3 30 
 
 0.2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 0.9 
 
 1.1 
 
 2.3 
 
 3.4 
 
 8 30 
 
 4 
 
 0.2 
 
 0.4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 1.2 
 
 2.5 
 
 3.7 
 
 8 
 
 4 30 
 
 0.2 
 
 0.4 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.3 
 
 2.6 
 
 3.9 
 
 7 30 
 
 5 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.1 
 
 1.4 
 
 2.7 
 
 4.1 
 
 7 
 
 5 30 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.2 
 
 1.4 
 
 2.8 
 
 4.2 
 
 6 30 
 
 6 
 
 0.2 
 
 0.5 
 
 0.7 
 
 0.9 
 
 1.2 
 
 1.4 
 
 2.8 
 
 4.2 
 
 6 
 
 
108 
 
 TABLE XCVI. Logistical Logarithms. 
 
 f 
 
 
 
 
 1 2 
 
 3 
 180 
 
 4 
 
 5 6 
 
 7 
 
 8 
 
 9 
 
 " 
 
 60 ! 120 
 
 240 
 
 300 
 
 360 
 
 420 
 
 480 
 
 540 
 
 
 
 
 1.7782 
 
 1.4771 
 
 1.3010 
 
 1.1761 
 
 1.0792 
 
 1.0000 
 
 9331 
 
 8751 
 
 8239 
 
 1 
 
 3.5563 
 
 1.7710 
 
 1.4735 
 
 1.2986 
 
 1.1743 
 
 1.0777 
 
 9988 
 
 9320 
 
 8742 
 
 8231 
 
 2 
 
 3.2553 
 
 1.7639 
 
 1.4699 
 
 1.2962 
 
 L1725 
 
 1.0763 
 
 9976 
 
 9310 
 
 8733 
 
 8223 
 
 3 
 
 3.0792 
 
 1.7570 
 
 1.4664 
 
 1.2939 
 
 1.1707 
 
 1.0749 
 
 9964 
 
 9300 
 
 8724 
 
 8215 
 
 4 
 
 2.9542 
 
 1.7501 
 
 1.4629 
 
 1.2915 
 
 1.1689 
 
 1.0734 
 
 9952 
 
 9289 
 
 8715 
 
 8207 
 
 5 
 
 2.8573 
 
 1.7434 
 
 1.4594 
 
 1.2891 
 
 1.1671 
 
 1.0720 
 
 9940 
 
 9279 
 
 8706 
 
 8199 
 
 6 
 
 2.7782 
 
 1.7368 
 
 1.4559 
 
 1.2868 
 
 1.1654 
 
 1.0706 
 
 9928 
 
 9269 
 
 8697 
 
 8191 
 
 7 
 
 2.7112 
 
 1.7302 
 
 1.4525 
 
 1.2845 
 
 1.1636 
 
 1.0692 
 
 9916 
 
 9259 
 
 8688 
 
 8183 
 
 8 
 
 2.6532 
 
 1.7238 
 
 1.4491 
 
 1.2821 
 
 1.1619 
 
 1.0678 
 
 9905 
 
 9249 
 
 8679 
 
 8175 
 
 9 
 
 2.6021 
 
 1.7175 
 
 1.4457 
 
 1.2798 
 
 1.1601 
 
 1.0663 
 
 9893 
 
 9238 
 
 8670 
 
 8167 
 
 10 
 
 2.5563 
 
 1.7112 
 
 1.4424 
 
 12775 
 
 1.1584 
 
 1.0649 
 
 9881 
 
 9228 
 
 8661 
 
 8159 
 
 11 
 
 2.5149 
 
 1.7050 
 
 1.4390 
 
 1.2753 
 
 1.1566 
 
 1.0635 
 
 9869 
 
 9218 
 
 8652 
 
 8152 
 
 12 
 
 2.4771 
 
 1.6990 
 
 1.4357 
 
 1 2730 
 
 1.1549 
 
 1.0621 
 
 9858 
 
 9208 
 
 8643 
 
 8144 
 
 13 
 
 2.4424 
 
 1.6930 
 
 1.4325 
 
 1.2707 
 
 1.1532 
 
 1.0608 
 
 9846 
 
 9198 
 
 8635 
 
 8136 
 
 14 
 
 2.4102 
 
 1.6871 
 
 1.4292 
 
 1.2685 
 
 1.1515 
 
 1.0594 
 
 9834 
 
 9188 
 
 8626 
 
 8128 
 
 15 
 
 2.3802 
 
 1.6812 
 
 1.4260 
 
 1.2663 
 
 1.1498 
 
 1.0580 
 
 9823 
 
 9178 
 
 8617 
 
 8120 
 
 16 
 
 2.3522 
 
 1.6755 
 
 1.4228 
 
 1.2640 
 
 1.1481 
 
 1.0566 
 
 9811 
 
 9168 
 
 8608 
 
 8112 
 
 17 
 
 2.3259 
 
 1.6698 
 
 1.4196 
 
 1.2618 
 
 1.1464 
 
 1.0552 
 
 9800 
 
 9158 
 
 8599 
 
 8104 
 
 18 
 
 2.3010 
 
 1.6642 
 
 1.4165 
 
 1.2596 
 
 1.1447 
 
 1.0539 
 
 9788 
 
 9148 
 
 8591 
 
 8097 
 
 19 
 
 2.2775 
 
 1.6587 
 
 1.4133 
 
 1.2574 
 
 1.1430 
 
 1.0525 
 
 9777 
 
 9138 
 
 8582 
 
 8089 
 
 20 
 
 2.2553 
 
 1.6532 
 
 1.4102 
 
 1.2553 
 
 1.1413 
 
 1.0512 
 
 9765 
 
 9128 
 
 8573 
 
 8081 
 
 21 
 
 22341 
 
 1.6478 
 
 1.4071 
 
 1.2531 
 
 1.1397 
 
 1.0498 
 
 9754 
 
 9119 
 
 8565 
 
 8073 
 
 22 
 
 22139 
 
 1.6425 
 
 1.4040 
 
 1.2510 
 
 1.1380 
 
 1.0484 
 
 9742 
 
 9109 
 
 8556 
 
 8066 
 
 23 
 
 2.1946 
 
 1.6372 
 
 1.4010 
 
 1.2488 
 
 1.1363 
 
 1.0471 
 
 9731 
 
 9099 
 
 8547 
 
 8058 
 
 24 
 
 2.1761 
 
 1.6320 
 
 1.3979 
 
 1.2467 
 
 1.1347 
 
 1.0458 
 
 9720 
 
 9089 
 
 8539 
 
 8050 
 
 25 
 
 2.1584 
 
 1.6269 
 
 1.3949 
 
 1.2445 
 
 1.1331 
 
 1.0444 
 
 9708 
 
 9079 
 
 8530 
 
 8043 
 
 26 
 
 2.1413 
 
 1.6218 
 
 1.3919 
 
 1.2424 
 
 1.1314 
 
 1.0431 
 
 9697 
 
 9070 
 
 8522 
 
 8035 
 
 27 
 
 2.1249 
 
 1.6168 
 
 1.3890 
 
 1.2403 
 
 1.1298 
 
 1.0418 
 
 9686 
 
 9060 
 
 8513 
 
 8027 
 
 28 
 
 2.1091 
 
 1.6118 
 
 1.3860 
 
 1.2382 
 
 1.1282 
 
 1.0404 
 
 9675 
 
 9050 
 
 8504 
 
 8020 
 
 29 
 
 2.0939 
 
 1.6069 
 
 1.3831 
 
 1.2362 
 
 1.1266 
 
 1.0391 
 
 9664 
 
 9041 
 
 8496 
 
 8012 
 
 30 
 
 2.0792 
 
 1.6021 
 
 1.3802 
 
 1.2341 
 
 1.1249 1.0378 
 
 9652 
 
 9031 
 
 8487 
 
 8004 
 
 31 
 
 2.0649 
 
 1.5973 
 
 1.3773 
 
 1.2320 
 
 1.1233 
 
 1.0365 
 
 9641 
 
 9021 
 
 8479 
 
 7997 
 
 32 
 
 2.0512 
 
 1.5925 
 
 1.3745 
 
 1.2300 
 
 1.1217 
 
 1.0352 
 
 9630 
 
 9012 
 
 8470 
 
 7989 
 
 33 
 
 2.0378 
 
 1.5878 
 
 1.3716 
 
 1.2279 
 
 1.1201 
 
 1.0339 
 
 9619 
 
 9002 
 
 8462 
 
 7981 
 
 34 
 
 2.0248 
 
 1.5832 
 
 1.3688 
 
 1.2259 
 
 1.1186 
 
 1.0326 
 
 9608 
 
 8992 
 
 8453 
 
 7974 
 
 35 
 
 2.0122 
 
 1.5786 
 
 1.3660 
 
 1.2239 
 
 1.1170 
 
 1.0313 
 
 9597 
 
 8983 
 
 8445 
 
 7966 
 
 36 
 
 2.0000 
 
 1.5740 
 
 1.3632 
 
 1.2218 
 
 1.1154 
 
 1.0300 
 
 9586 
 
 8973 
 
 8437 
 
 7959 
 
 37 
 
 1.9881 
 
 1.5695 
 
 1.3604 
 
 1.2198 
 
 1.1138 
 
 1.0287 
 
 9575 
 
 8964 
 
 8428 
 
 7951 
 
 38 
 
 1.9765 
 
 1.5651 
 
 1.3576 
 
 1.2178 
 
 1.1123 
 
 1.0274 
 
 9564 
 
 8954 
 
 8420 
 
 7944 
 
 33 
 
 1.9652 
 
 1.5607 
 
 1.3549 
 
 1.2159 
 
 1.1107 
 
 1.0261 
 
 9553 
 
 8945 
 
 8411 
 
 7936 
 
 40 
 
 1.9542 
 
 1.5563 
 
 1.3522 
 
 1.2139 
 
 1.1091 
 
 1.0248 
 
 9542 
 
 8935 
 
 8403 
 
 7929 
 
 41 
 
 1.9435 
 
 1.5520 
 
 1.3495 
 
 1.2119 
 
 1.1076 
 
 1.0235 
 
 9532 
 
 8926 
 
 8395 
 
 '7921 
 
 42 
 
 1.9331 
 
 1.5477 
 
 1.3468 
 
 1.2099 
 
 1.1061 ! 1.0223 
 
 9521 
 
 8917 
 
 8386 
 
 7914 
 
 43 
 
 1.9228 
 
 1.5435 
 
 1.3441 
 
 1.2080 
 
 1.1045 
 
 1.0210 
 
 9510 
 
 8907 
 
 8378 
 
 7906 
 
 44 
 
 1.9128 
 
 1.5393 
 
 1.3415 
 
 1.2061 
 
 1.1030 
 
 1.0197 
 
 9499 
 
 8898 
 
 8370 
 
 7899 
 
 45 
 
 1.9031 
 
 1.5351 
 
 1.3388 
 
 1.2041 
 
 1.1015 
 
 1.0185 
 
 9488 
 
 8888 
 
 8361 
 
 7891 
 
 46 
 
 1.8935 
 
 1.5310 
 
 1.3362 
 
 1.2022 
 
 1.0999 
 
 1.0172 
 
 9478 
 
 8879 
 
 8353 
 
 7884 
 
 47 
 
 1.8842 
 
 1.5269 
 
 1.3336 
 
 1.2003 
 
 1.0984 
 
 1.0160 
 
 9467 
 
 8870 
 
 8345 
 
 7877 
 
 48 
 
 1.8751 
 
 1.5229 
 
 1.3310 
 
 1.1984 
 
 1.0969 
 
 1.0147 
 
 9456 
 
 8861 
 
 8337 
 
 7869 
 
 49 
 
 1.8661 
 
 1.5189 
 
 1.3284 
 
 1.1965 
 
 1.0954 
 
 1.0135 
 
 9446 
 
 8851 
 
 8328 
 
 7862 
 
 50 
 
 1.8573 
 
 1.5149 
 
 1.3259 
 
 1.1946 
 
 1.0939 1.0122 
 
 9435 
 
 8842 
 
 8320 
 
 7855 
 
 51 
 
 1.8487 
 
 1.5110] 1.3233 
 
 1.1927 
 
 1.0924 1.0110 
 
 9425 
 
 8833 
 
 8312 
 
 7847 
 
 52 
 
 1.8403 
 
 1.5071 
 
 1.3208 
 
 1.1908 
 
 1.0909 1.0098 
 
 9414 
 
 8824 
 
 8304 
 
 7840 
 
 53 
 
 1.8320 1.5032 
 
 1.3183 
 
 1.1889 1.0894 1.0085 9404 
 
 8814 
 
 8296 
 
 7832 
 
 54 
 
 1.8239 ! 1.4994 
 
 1.3158 
 
 1.1871 1.0880 1.0073' 9393 
 
 8805 
 
 8288 
 
 7825 
 
 55 
 
 1. 81 59 j 1.4956 1.3133 
 
 1.1852 
 
 1.0865 
 
 1.0061 
 
 9383 
 
 8796 8279 
 
 7818 
 
 56 
 
 1.8081 
 
 1.4918 1.3108 
 
 1.1834 
 
 1.0850 
 
 1.0049 
 
 9372 
 
 8787 
 
 8271 
 
 7811 
 
 67 
 
 1.8004 
 
 1.4881 
 
 1.3083 
 
 1.1816 
 
 1.0835 
 
 1.0036 
 
 9362 
 
 8778 
 
 8263 
 
 7803 
 
 58 
 
 1.7929 1.4844 
 
 1.3059 
 
 1.1797 
 
 1.0821 1.0024 
 
 9351 
 
 8769 
 
 8255 7796 
 
 59 
 
 1 7855 '1.4808 
 
 1.3034 
 
 1.1779 
 
 1.0806 1.0012 
 
 9341 
 
 8760 
 
 8247 7789 
 
 6fl 
 
 1. 7782 ! 1.4771 
 
 1.301C 
 
 I 1761 
 
 1.0792 1.0000 
 
 9331 
 
 8751 
 
 8239 7782 
 

 
 TABLE XCVI. Logistical Logarithms. 
 
 109 
 
 ' 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 " 
 
 600 
 
 t,60 
 
 720 
 
 780 
 
 840 
 
 900 
 
 960 
 
 1020 
 
 1080 
 
 1140 
 
 1200 
 
 1260 
 
 
 
 7782 
 
 7368 
 
 6990 
 
 6642 
 
 6320 
 
 6021 
 
 5740 
 
 5477 
 
 5229 
 
 4994 
 
 4771 
 
 4559 
 
 1 
 
 7774 
 
 7361 
 
 6984 
 
 6637 
 
 6315 
 
 6016 
 
 5736 
 
 5473 
 
 5225 
 
 4990 
 
 4768 
 
 4556 
 
 2 
 
 7767 
 
 7354 
 
 6978 
 
 6631 
 
 6310 
 
 6011 
 
 5731 
 
 5469 
 
 5221 
 
 4986 
 
 4764 
 
 4552 
 
 3 
 
 7760 
 
 7348 
 
 6972 
 
 6625 
 
 6305 
 
 6006 
 
 5727 
 
 5464 
 
 5217 
 
 4983 
 
 4760 
 
 4549 
 
 4 
 
 7753 
 
 7341 
 
 6966 
 
 6620 
 
 6300 
 
 6001 
 
 5722 
 
 5460 
 
 5213 
 
 4979 
 
 4757 
 
 4546 
 
 5 
 
 7745 
 
 7335 
 
 6960 
 
 6614 
 
 6294 
 
 5997 
 
 5718 
 
 5456 
 
 5209 
 
 4975 
 
 4753 
 
 4542 
 
 6 
 
 7738 
 
 7328 
 
 6954 
 
 6609 
 
 6289 
 
 5992 
 
 5713 
 
 5452 
 
 5205 
 
 4971 
 
 4750 
 
 4539 
 
 7 
 
 7731 
 
 7322 
 
 6948 
 
 6603 
 
 6284 
 
 5987 
 
 5709 
 
 5447 
 
 5201 
 
 4967 
 
 4746 
 
 4535 
 
 8 
 
 7724 
 
 731 5 i 6942 
 
 6598 
 
 6279 
 
 5982 
 
 5704 
 
 5443 
 
 5197 
 
 4964 
 
 4742 
 
 4532 
 
 9 
 
 7717 
 
 7309 
 
 6936 
 
 6592 
 
 6274 
 
 5977 
 
 5700 
 
 5439 
 
 5193 
 
 4960 
 
 4739 
 
 4528 
 
 10 
 
 7710 
 
 7002 
 
 6930 
 
 6587 
 
 6269 I 5973 
 
 5695 
 
 5435 
 
 5189 
 
 4956 
 
 4735 
 
 4525 
 
 11 
 
 7703 
 
 /296 
 
 6924 
 
 6581 
 
 6264 
 
 5968 
 
 5691 
 
 5430 
 
 5185 
 
 4952 
 
 4732 
 
 4522 
 
 12 
 
 7696 
 
 7289 
 
 6918 
 
 6576 
 
 6259 
 
 5963 
 
 5686 
 
 5426 
 
 5181 
 
 4949 
 
 4728 
 
 4513 
 
 13 
 
 7688 
 
 7283 
 
 6912 
 
 6570 
 
 6254 
 
 5958 
 
 5682 
 
 5422 
 
 5177 
 
 4945 
 
 4724 
 
 4515 
 
 14 
 
 768i J7276 
 
 6906 
 
 6565 
 
 6248 
 
 5954 
 
 5677 
 
 5418 
 
 5173 
 
 4941 
 
 4721 
 
 4511 
 
 15 
 
 767't 7270 
 
 6900 
 
 6559 
 
 6243 
 
 5949 
 
 5673 
 
 5414 
 
 5169 
 
 4937 
 
 4717 
 
 4508 
 
 16 
 
 7667 
 
 '.''264 
 
 6894 
 
 6554 
 
 6238 
 
 5944 
 
 5669 
 
 5409 
 
 5165 
 
 4933 
 
 4714 
 
 4505 
 
 17 
 
 7660 
 
 7257 
 
 6888 
 
 6548 
 
 6233 
 
 5939 
 
 5664 
 
 5405 
 
 5161 
 
 4930 
 
 4710 
 
 4501 
 
 18 
 
 7653 
 
 7251 
 
 6882 
 
 6543 
 
 6228 
 
 5935 
 
 5660 
 
 5401 
 
 5157 
 
 4926 
 
 4707 
 
 4498 
 
 19 
 
 7646 
 
 7244 
 
 6877 
 
 6538 
 
 b'223 
 
 5930 
 
 5655 
 
 5397 
 
 5153 
 
 4922 
 
 4703 
 
 4494 
 
 20 
 
 7639 
 
 7238 
 
 6871 
 
 6532 
 
 6218 
 
 5925 
 
 5651 
 
 5393 
 
 5149 
 
 4918 
 
 4699 
 
 4491 
 
 21 
 
 7632 
 
 7232 
 
 6865 
 
 6527 
 
 6213 
 
 5920 
 
 5646 
 
 5389 
 
 5145 
 
 4915 
 
 4696 
 
 4488 
 
 22 
 
 7625 
 
 7225 
 
 6859 
 
 6521 
 
 6208 
 
 5916 
 
 5642 
 
 5384 
 
 5141 
 
 4911 
 
 4692 
 
 4484 
 
 23 
 
 7618 
 
 7219 
 
 6853 
 
 6516 
 
 6203 
 
 5911 
 
 5637 
 
 5380 
 
 5137 
 
 4907 
 
 4689 
 
 4481 
 
 24|7611 
 
 7212 
 
 6847 
 
 6510 
 
 6198 
 
 5906 
 
 5633 
 
 5376 
 
 5133 
 
 4903 
 
 4685 
 
 4477 
 
 25 7604 
 
 7206 
 
 6841 
 
 6505 
 
 6193 
 
 5902 
 
 5629 
 
 5372 
 
 5129 
 
 4900 
 
 4682 
 
 4474 
 
 26 7597 
 
 7200 
 
 6836 
 
 6500 
 
 6188 
 
 5897 
 
 5624 
 
 5368 
 
 5125 
 
 4896 
 
 4678 
 
 4471 
 
 27 ! 7590 
 
 7193 
 
 6830 
 
 6494 
 
 6183 
 
 5892 
 
 5620 
 
 5364 
 
 5122 4892 
 
 4675 
 
 4467 
 
 28 ! 7383 
 
 7187 
 
 6824 
 
 6489 
 
 6178 
 
 5888 
 
 5615 
 
 5359 
 
 5118 4889 
 
 4671 
 
 4464 
 
 29 7577 
 
 7181 
 
 6818 
 
 6484 
 
 6173 
 
 5883 
 
 5611 
 
 5355 
 
 511414885 
 
 4668 
 
 4460 
 
 30 7570 
 
 7175 
 
 6812 
 
 6478 
 
 6168 
 
 5878 
 
 5607 
 
 5351 
 
 5110 4881 
 
 4664 
 
 4457 
 
 31 7563 
 
 7168 
 
 6807 
 
 6473 
 
 6163 
 
 5874 
 
 5602 
 
 5347 
 
 5106 4877 
 
 4660 
 
 4454 
 
 32 7556 
 
 7162 
 
 6801 
 
 6467 
 
 6158 
 
 5869 
 
 5598 
 
 5343 
 
 5102 4874 
 
 4657 
 
 4450 
 
 33 7549 
 
 7156 
 
 6795 
 
 6462 
 
 6153 
 
 5864 
 
 5594 
 
 5339 
 
 5098 4870 
 
 4653 
 
 4447 
 
 34 7542 
 
 7149 
 
 6789 
 
 6457 
 
 6148 
 
 5860 
 
 5589 
 
 5335 
 
 5094 4866 
 
 4650 
 
 4444 
 
 35 7535 
 
 7143 
 
 6784 6451 
 
 6143 
 
 5855 
 
 5585 
 
 5331 
 
 5090 4863 
 
 4646 
 
 4440 
 
 36 7528 
 
 7137 
 
 6778 
 
 6446 
 
 6138 
 
 5850 
 
 5580 
 
 5326 
 
 5086 
 
 4859 
 
 4643 
 
 4437 
 
 37 7522 
 
 7131 
 
 6772 
 
 6441 
 
 6133 
 
 5846 
 
 5576 
 
 5322 
 
 5082 
 
 4855 
 
 4639 
 
 4434 
 
 38 7515 
 
 7124 
 
 6766 
 
 6435 
 
 6128 
 
 5841 
 
 5572 
 
 5318 
 
 5079 4852 
 
 4636 
 
 4430 
 
 39 7508 
 
 7118 
 
 6761 
 
 6430 
 
 6123 
 
 5836 
 
 5567 
 
 5314 
 
 5075 ' 4848 
 
 4632 
 
 4427 
 
 40 7501 
 
 7112 
 
 6755 
 
 6425 6118 
 
 5832 
 
 5563 
 
 5310 
 
 5071 
 
 4844 
 
 4629 
 
 4424 
 
 41 ! 7494 
 
 7106 
 
 6749 
 
 6420 6113 
 
 5827 
 
 5559 
 
 5306 
 
 5067 
 
 4841 
 
 4625 
 
 4420 
 
 42 ! 7488 
 
 7100 
 
 6743 
 
 6414 6108 
 
 5823 
 
 5554 
 
 5302 
 
 5063 
 
 4837 
 
 4622 
 
 4417 
 
 43 7481 
 
 7093 
 
 6738 
 
 6409 
 
 6103 
 
 5818 
 
 5550 
 
 5298 
 
 5059 
 
 4833 
 
 4618 
 
 4414 
 
 44 7474 
 
 7087 
 
 6732 
 
 6404 
 
 6099 
 
 5813 
 
 5546 
 
 5294 
 
 5055 
 
 4830 
 
 4615 
 
 4410 
 
 45 7467 
 
 7081 
 
 6726 
 
 6398 
 
 6094 
 
 5809 
 
 5541 
 
 5290 
 
 5051 
 
 4826 
 
 4611 
 
 4407 
 
 46 
 
 7461 
 
 7075 
 
 6721 
 
 6393 
 
 6089 
 
 5804 
 
 5537 
 
 '5285 
 
 5048 
 
 4822 
 
 4608 
 
 4404 
 
 47 
 
 7454 
 
 7069 
 
 6715 
 
 6388 
 
 6084 
 
 5800 
 
 5533 
 
 5281 
 
 5044 
 
 4819 1 4604 
 
 4400 
 
 48 
 
 7447 
 
 7063 
 
 6709 
 
 6383 
 
 6079 
 
 5795 
 
 5528 
 
 5277 
 
 5040 4815 
 
 4601 
 
 4397 
 
 49 
 
 7441 
 
 7057 
 
 6704 
 
 6377 
 
 6074 
 
 5790 
 
 5524 
 
 5273 
 
 5036)4811 
 
 4597 
 
 4394 
 
 50 
 
 7434 
 
 7050 
 
 6698 
 
 6372 
 
 6069 
 
 5786 
 
 5520 
 
 5269 
 
 5032 
 
 4808 
 
 4594 
 
 4390 
 
 51 
 
 7427 
 
 7044 
 
 6692 
 
 6367 
 
 6064 
 
 5781 
 
 5516 
 
 5265 
 
 5028 
 
 4804 
 
 4590 
 
 4387 
 
 52 
 
 7421 
 
 7038 
 
 6687 
 
 6362 
 
 6059 
 
 5777 
 
 5511 
 
 5261 
 
 5025 : 4800 
 
 4587 
 
 4384 
 
 53 
 
 7414 
 
 7032 
 
 6681 
 
 6357 
 
 6055 
 
 5772 
 
 5507 
 
 5257 
 
 5021 
 
 4797 
 
 4584 
 
 4380 
 
 54 
 
 7407 
 
 7026 
 
 6676 
 
 6351 
 
 6050 ! 5768 
 
 5503 
 
 5253 5017 
 
 4793 
 
 4580 
 
 4377 
 
 55 
 
 7401 
 
 7020 
 
 6670 
 
 6346 
 
 6045 I 5763 
 
 5498 
 
 5249 
 
 5013 
 
 4789 '4577 
 
 4374 
 
 56 
 
 7394 
 
 7014 
 
 6664 
 
 6341 
 
 6040 
 
 5758 
 
 5494 
 
 5245 
 
 5009 
 
 4786 , 4573 
 
 4370 
 
 57 
 
 7387 
 
 7008 
 
 6659 
 
 6336 
 
 6035 
 
 5754 
 
 5490 
 
 5241 
 
 5005 
 
 47S2 4570 
 
 4367 
 
 58 
 59 
 
 7:381 
 7374 
 
 7002 
 6996 
 
 6653 
 6648 
 
 6331 
 6325 
 
 6030 j 5749 
 6025 1 5745 
 
 5486 
 5481 
 
 5237 
 5233 
 
 5002 
 4998 
 
 4778 
 4775 
 
 4506 
 4563 
 
 4364 
 4361 
 
 60 
 
 7368 
 
 6990 b42 
 
 6320 
 
 6021 1 5740 
 
 5477 
 
 5229 
 
 4994 
 
 4771 '4559 4357 
 
no 
 
 TABLE XCVI. 
 
 Logistical Logarithms. 
 
 22 23 
 
 24 
 
 25 
 
 2(3 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 32 
 
 33 ^ 
 
 ' f 
 
 1320,1380 
 
 1440 
 
 1500 
 
 1560 
 
 1630 
 
 1680 
 
 1740 
 
 1800 
 
 1860 
 
 1920 
 
 1J80 
 
 
 
 4357 i 4164 
 
 3979 
 
 3802 
 
 3632 
 
 3468 
 
 3310 
 
 3158 
 
 3010 
 
 2868 
 
 2730 
 
 2596 
 
 1 
 
 4354 
 
 4161 
 
 3976 
 
 3799 
 
 3629 
 
 3465 
 
 3307 
 
 3155 
 
 30Q8 
 
 2866 
 
 2728 
 
 2594 
 
 2 
 
 4351 
 
 4158 
 
 3973 
 
 3796 
 
 3626 
 
 3463 
 
 3305 
 
 3153 
 
 3005 
 
 2863 
 
 2725 
 
 2592 
 
 3 
 
 4347 
 
 4155 
 
 3970 
 
 3793 
 
 3623 
 
 3460 
 
 3302 
 
 3150 
 
 3003 
 
 2861 
 
 2723 
 
 2590 
 
 4 
 
 4344 
 
 4152 
 
 3967 
 
 3791 
 
 3621 
 
 3457 
 
 3300 
 
 3148 
 
 3001 
 
 2859 
 
 2721 
 
 2588 
 
 5 
 
 4341 I 4149 
 
 3964 
 
 3788 
 
 3618 
 
 3454 
 
 3297 
 
 3145 
 
 2998 
 
 2856 
 
 2719 
 
 2585 
 
 6 
 
 4338 1 4145 
 
 3961 
 
 3785 
 
 3615 
 
 3452 
 
 3294 
 
 3143 
 
 2996 
 
 2854 
 
 2716 
 
 2583 
 
 7 
 
 4334 
 
 4142 
 
 3958 
 
 3782 3612 
 
 3449 
 
 3292 
 
 3140 
 
 2993 
 
 2852 
 
 2714 
 
 2581 
 
 8 
 
 4331 
 
 4139 
 
 3955 
 
 3779 3610 
 
 3446 
 
 3289 
 
 3138 
 
 2991 
 
 28-' 9 
 
 2712 
 
 2579 
 
 9 
 
 4328 
 
 4136 
 
 3952 
 
 3776 3607 
 
 3444 
 
 3287 
 
 3135 
 
 2989 
 
 284/ 1 2710 
 
 2577 
 
 10 
 
 4325 
 
 4133 
 
 3949 
 
 3773 3604 
 
 3441 
 
 3284 
 
 3133 
 
 2986 
 
 2845 
 
 3707 
 
 2574 
 
 11 
 
 4321 
 
 4130 
 
 3946 
 
 3770 1 3601 
 
 3438 
 
 3282 
 
 3130 
 
 2984 
 
 2842 
 
 2705 
 
 2572 
 
 12 
 
 4318 
 
 4127 
 
 3943 
 
 3768 
 
 3598 
 
 3436 
 
 3279 
 
 3128 
 
 2981 
 
 2840 
 
 2703 
 
 2570 
 
 13 
 
 4315 
 
 4124 
 
 3940 
 
 3765 
 
 3596 
 
 3433 
 
 3276 
 
 3125 
 
 2979 
 
 2838 
 
 2701 
 
 2568 
 
 14 
 
 4311 
 
 4120 
 
 3937 
 
 3762 
 
 3593 
 
 3431 
 
 3274 
 
 3123 
 
 2977 
 
 2835 
 
 2698 
 
 2566 
 
 15 
 
 4308 
 
 4117 
 
 3934 
 
 3759 
 
 3590 
 
 3428 
 
 3271 
 
 3120 
 
 2974 
 
 2833 
 
 2696 
 
 2564 
 
 16 
 
 4305 
 
 4114 
 
 3931 
 
 3756 
 
 3587 
 
 3425 
 
 3269 
 
 3118 
 
 2972 
 
 2831 
 
 2694 
 
 2561 
 
 17 
 
 4302 
 
 4111 
 
 3928 
 
 3753 
 
 3585 
 
 3423 
 
 3266 
 
 3115 
 
 2969 
 
 2828 
 
 2692 
 
 2559 
 
 18 
 
 4298 
 
 4108 
 
 3925 
 
 3750 
 
 3582 
 
 3420 
 
 3264 
 
 3113 
 
 2967 
 
 2826 
 
 2689 
 
 2557 
 
 19 
 
 4295 
 
 4105 
 
 3922 
 
 3747 
 
 3579 
 
 3417 
 
 3261 
 
 3110 
 
 2965 
 
 2824 
 
 2687 
 
 2555 
 
 20 
 
 4292 
 
 4102 
 
 3919 
 
 3745 
 
 3576 
 
 341 
 
 3259 
 
 3108 
 
 2962 
 
 2821 
 
 2685 
 
 2553' 
 
 21 
 
 4289 
 
 4099 
 
 3917 
 
 3742 
 
 3574 
 
 3412 
 
 3256 
 
 3105 
 
 2960 
 
 2819 
 
 2683 
 
 2551 
 
 22 
 
 4285 
 
 4096 
 
 3914 
 
 3739 
 
 3571 
 
 3409 
 
 3253 
 
 3103 
 
 2958 
 
 2817 
 
 2681 
 
 2548 
 
 23 
 
 4282 
 
 4092 
 
 3911 
 
 3736 
 
 3568 
 
 3407 
 
 3251 
 
 3101 
 
 2955 
 
 2815 
 
 2678 
 
 2546 
 
 24 
 
 4279 
 
 4089 
 
 3908 
 
 3733 
 
 3565 
 
 3404 
 
 3248 
 
 3098 
 
 2953 
 
 2812 
 
 2676 
 
 2544 
 
 25 
 
 4276 
 
 4086 
 
 3905 
 
 3730 
 
 3563 
 
 3401 
 
 3246 
 
 3096 
 
 2950 
 
 2810 
 
 2674 
 
 2542 
 
 26 
 
 4273 
 
 4083 
 
 3902 
 
 3727 
 
 3560 
 
 3399 
 
 3243 
 
 3093 
 
 2948 
 
 2808 
 
 2672 
 
 2540 
 
 27 
 
 4269 
 
 4080 
 
 3899 
 
 3725 
 
 3557 
 
 3396 
 
 3241 
 
 3091 
 
 2946 
 
 2805 
 
 2669 
 
 2538 
 
 28 4266 
 
 4077 
 
 3896 
 
 3722 
 
 3555 
 
 3393 
 
 3238 
 
 3088 
 
 2943 
 
 2803 
 
 2667 
 
 2535 
 
 29 4263 
 
 4074 
 
 3893 
 
 3719 
 
 3552 
 
 3391 
 
 3236 
 
 3086 
 
 2941 
 
 2801 
 
 2665 
 
 2533 
 
 30 
 
 4260 
 
 4071 
 
 3890 
 
 3716 
 
 3549 
 
 3388 
 
 3233 
 
 30S3 
 
 2939 
 
 2798 
 
 2663 
 
 2531 
 
 31 
 
 4256 
 
 4068 
 
 3887 
 
 3713 
 
 3546 
 
 3386 
 
 3231 
 
 3081 
 
 2936 
 
 2796 
 
 2660 
 
 2529 
 
 32 
 
 4253 
 
 4065 
 
 3884 
 
 3710 
 
 3544 
 
 3383 
 
 3228 
 
 3078 
 
 2934 
 
 2794 
 
 2658 
 
 2527 
 
 33 
 
 4250 
 
 4062 
 
 3881 
 
 3708 
 
 3541 
 
 3380 
 
 3225 
 
 3076 
 
 2931 
 
 2792 
 
 2656 
 
 2525 
 
 34 
 
 4247 
 
 4059 
 
 3878 
 
 3705 
 
 3538 
 
 3378 
 
 3223 
 
 3073 
 
 2929 
 
 2789 
 
 2654 
 
 2522 
 
 35 
 
 4244 
 
 4055 
 
 3875 
 
 3702 
 
 3535 
 
 3375 
 
 3220 
 
 3071 
 
 2927 
 
 2787 
 
 2652 
 
 2520 
 
 36 
 
 4240 
 
 4052 
 
 3872 
 
 3699 
 
 3533 
 
 3372 
 
 3218 
 
 3069 
 
 2924 
 
 2785 
 
 2649 
 
 2518 
 
 37 
 
 4237 
 
 4049 
 
 3869 
 
 3696 
 
 3530 
 
 3370 
 
 3215 
 
 3066 
 
 2922 
 
 2782 
 
 2647 
 
 2516 
 
 38 
 
 4234 
 
 4046 
 
 3866 
 
 3693 
 
 3527 
 
 3367 
 
 3213 
 
 3064 
 
 2920 
 
 2780 
 
 2645 
 
 2514 
 
 39 
 
 4231 
 
 4043 
 
 3863 
 
 3691 
 
 3525 
 
 3365 
 
 3210 
 
 3061 
 
 2917 
 
 2778 
 
 2643 
 
 2512 
 
 40 
 
 4228 
 
 4040 
 
 3860 
 
 3688 
 
 3522 
 
 3362 
 
 3208 
 
 3059 
 
 2915 
 
 2775 
 
 2640 
 
 2510 
 
 41 
 
 4224 
 
 4037 
 
 3857 
 
 3685 
 
 3519 
 
 3359 
 
 3205 
 
 3056 
 
 2912 
 
 2773 
 
 2638 
 
 2507 
 
 42 
 
 4221 
 
 4034 
 
 3855 
 
 3682 
 
 3516 
 
 3357 
 
 3203 
 
 3054 
 
 2910 
 
 2771 
 
 2636 
 
 2505 
 
 43 
 
 4218 
 
 4031 
 
 3852 
 
 3679 
 
 3514 
 
 3354 
 
 3200 
 
 3052 
 
 2908 
 
 2769 
 
 2634 
 
 2503 
 
 44 
 
 4215 
 
 4028 
 
 3849 
 
 3677 
 
 3511 
 
 3351 
 
 3198 
 
 3049 
 
 2905 
 
 2766 
 
 2632 
 
 2501 
 
 45 
 
 4212 
 
 4025 
 
 3846 
 
 3674 
 
 3508 
 
 3349 
 
 3195 
 
 3047 
 
 2903 
 
 2764 
 
 2629 
 
 2499 
 
 46 
 
 4209 
 
 4022 
 
 3843 
 
 3671 
 
 3506 
 
 3346 
 
 3193 
 
 3044 
 
 2901 
 
 2762 
 
 2627 
 
 2497 
 
 47 
 
 4205 
 
 4019 
 
 3840 
 
 3668 
 
 3503 
 
 3344 
 
 3190 
 
 3042 
 
 2898 
 
 2760 
 
 2625 
 
 2494 
 
 48 
 
 4202 
 
 4016 
 
 3837 
 
 3665 
 
 3500 
 
 3341 
 
 3188 
 
 3039 
 
 2896 
 
 2757 
 
 2623 
 
 2492 
 
 49 
 
 4199 
 
 4013 
 
 3834 
 
 3663 
 
 3497 
 
 3338 
 
 3185 
 
 3037 
 
 2894 
 
 2755 
 
 2621 
 
 2490 
 
 50 
 
 4196 
 
 4010 
 
 3831 
 
 3660 
 
 3495 
 
 3336 
 
 3183 
 
 3034 
 
 2891 
 
 2753 
 
 2618 
 
 2488 
 
 51 
 
 4193 
 
 4007 
 
 3828 
 
 3657 
 
 3492 
 
 3333 
 
 3180 
 
 3032 
 
 2889 
 
 2750 
 
 2616 
 
 2486 
 
 52 
 
 4189 
 
 4004 
 
 3825 
 
 3654 
 
 3489 
 
 3331 
 
 3178 
 
 3030 
 
 2887 
 
 2748 
 
 2614 
 
 2484 
 
 53 
 
 4186 
 
 4001 
 
 3822 3651 
 
 3487 
 
 3328 
 
 3175 
 
 3027 
 
 2884 
 
 2746 
 
 2612 
 
 2482 
 
 54 
 
 4183 
 
 3998 3820 3649 3484 
 
 3325 
 
 3173 
 
 3025 
 
 2882 
 
 2744 
 
 2610 
 
 2480 
 
 55 
 
 4180 
 
 3995 
 
 3817 
 
 3646 
 
 3481 
 
 3323 
 
 3170 
 
 3022 
 
 2880 
 
 2741 
 
 2607 
 
 2477 
 
 56 
 
 4177 
 
 3991 
 
 3814 
 
 36*3 
 
 3479 
 
 3320 
 
 3168 
 
 3020 
 
 2877 
 
 2739 
 
 2605 
 
 2475 
 
 57 
 
 4174 
 
 3988 
 
 3811 
 
 3640 
 
 3476 
 
 3318 
 
 3165 
 
 3018 
 
 2875 
 
 2737 
 
 2603 
 
 2473 
 
 58 
 
 4171 
 
 3985 
 
 3808 
 
 3637 
 
 3473 
 
 3315 
 
 3163 
 
 3015 
 
 2873 
 
 2735 
 
 2601 
 
 2471 
 
 59 
 
 4167 
 
 3982 
 
 3805 
 
 3635 
 
 3471 
 
 3313 
 
 3160 
 
 3013 
 
 2870 
 
 2732 
 
 2599 
 
 2469 
 
 60 
 
 4164 
 
 3979 
 
 3802 
 
 3632 3468 
 
 3310 
 
 3158 
 
 3010 
 
 2868 
 
 2730 
 
 2596 
 
 2467 
 
TABLE XCVI. Logistical Logarithms. 
 
 Ill 
 
 ' 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 42 
 
 43 
 
 44 
 
 45 
 
 
 2040 
 
 2100 
 
 2160 2220 
 
 2280 
 
 2340 
 
 2400 
 
 2460 
 
 2520 
 
 2580 
 
 2640 
 
 2700 
 
 
 
 2467 
 
 2341 
 
 2218 2099 
 
 1984 
 
 1871 
 
 1761 
 
 1654 
 
 1549 
 
 1447 
 
 1347 
 
 1249 
 
 1 
 
 2465 
 
 2339 
 
 2216 
 
 2098 
 
 1982 
 
 1869 
 
 1759 
 
 1652 
 
 1547 
 
 1445 
 
 1345 
 
 1248 
 
 2 
 
 2462 
 
 2337 
 
 2214 
 
 2096 
 
 1980 
 
 1867 
 
 1757 
 
 1650 
 
 1546 
 
 1443 
 
 1344 
 
 1246 
 
 3 
 
 2460 
 
 2335 
 
 2212 
 
 2094 
 
 1978 
 
 1865 
 
 1755 
 
 1648 
 
 1544 
 
 1442 
 
 1342 
 
 1245 
 
 4 
 
 2458 
 
 2333 
 
 2210 
 
 2092 
 
 1976 
 
 1863 
 
 1754 
 
 1647 
 
 1542 
 
 1440 
 
 1340 
 
 1243 
 
 5 
 
 2456 
 
 2331 
 
 2208 
 
 2090 
 
 1974 
 
 1862 
 
 17*2 
 
 1645 
 
 1540 
 
 1438 
 
 1339 
 
 1241 
 
 6 
 
 2454 
 
 2328 
 
 2206 
 
 2088 
 
 1972 
 
 1860 
 
 1750 
 
 1643 
 
 1539 
 
 1437 
 
 1337 
 
 1240 
 
 7 
 
 2452 
 
 2326 
 
 2204 
 
 2086 
 
 1970 
 
 1858 
 
 1748 
 
 1641 
 
 1537 
 
 1435 
 
 1335 
 
 1238 
 
 8 
 
 2450 
 
 2324 
 
 2202 
 
 2084 
 
 1968 
 
 1856 
 
 1746 
 
 1640 
 
 1535 
 
 1433 
 
 1334 
 
 1237 
 
 9 
 
 2448 
 
 2322 
 
 2200 
 
 2082 
 
 1967 
 
 1854 
 
 1745 
 
 1638 
 
 1534 
 
 1432 
 
 1332 
 
 1235 
 
 10 
 
 2445 
 
 2320 
 
 2198 
 
 2080 
 
 1965 
 
 1852 
 
 1743 
 
 1636 
 
 1532 
 
 1430 
 
 1331 
 
 1233 
 
 11 
 
 2443 
 
 2318 
 
 2196 
 
 2078 
 
 1963 
 
 1850 
 
 1741 
 
 1634 
 
 1530 
 
 1428 
 
 1329 
 
 1232 
 
 12 
 
 2441 
 
 2316 
 
 2194 
 
 2076 
 
 1961 
 
 1849 
 
 1739 
 
 1633 
 
 1528 
 
 1427 
 
 1327 
 
 1230 
 
 13 
 
 2439 
 
 2314 
 
 2192 
 
 2074 
 
 1959 
 
 1847 
 
 1737 
 
 1631 
 
 1527 
 
 1425 
 
 1326 
 
 1229 
 
 14 
 
 2437 
 
 2312 
 
 2190 
 
 2072 
 
 1957 
 
 1845 
 
 1736 
 
 1629 
 
 1525 
 
 1423 
 
 1324 
 
 1227 
 
 15 
 
 2435 
 
 2310 
 
 2188 
 
 2070 
 
 1955 
 
 1843 
 
 1734 
 
 1627 
 
 1523 
 
 1422 
 
 1322 
 
 1225 
 
 16 
 
 2433 
 
 2308 
 
 2186 
 
 2068 
 
 1953 
 
 1841 
 
 1732 
 
 1626 
 
 1522 
 
 1420 
 
 1321 
 
 1224 
 
 17 
 
 2431 
 
 2306 
 
 2184 
 
 2066 
 
 1951 
 
 1839 
 
 1730 
 
 1624 
 
 1520 
 
 1418 
 
 1319 
 
 1222 
 
 18 
 
 2429 
 
 2304 
 
 2182 
 
 2064 
 
 1950 
 
 1838 
 
 1728 
 
 1622 
 
 1518 
 
 1417 
 
 1317 
 
 1221 
 
 19 
 
 2426 
 
 2302 
 
 2180 
 
 2062 
 
 1948 
 
 1836 
 
 1727 
 
 1620 
 
 1516 
 
 1415 
 
 1316 
 
 1219 
 
 20 
 
 2424 
 
 2300 
 
 2178 
 
 2061 
 
 1946 
 
 1834 
 
 1725 
 
 1619 
 
 1515 
 
 1413 
 
 1314 
 
 1217 
 
 21 
 
 2422 
 
 2298 
 
 2176 
 
 2059 
 
 1944 
 
 1832 
 
 1723 
 
 1617 
 
 1513 
 
 1412 
 
 1313 
 
 1216 
 
 22 
 
 2420 
 
 2296 
 
 2174 
 
 2057 
 
 1942 
 
 1830 
 
 1721 
 
 1615 
 
 1511 
 
 1410 
 
 1311 
 
 1214 
 
 23 
 
 2418 
 
 2294 
 
 2172 
 
 2055 
 
 1940 
 
 1828 
 
 1719 
 
 1613 
 
 1510 
 
 1408 
 
 1309 
 
 1213 
 
 24 
 
 2416 
 
 2291 
 
 2170 
 
 2053 
 
 1938 
 
 1827 
 
 1718 
 
 1612 
 
 1508 
 
 1407 
 
 1308 
 
 1211 
 
 25 
 
 2414 
 
 2289 
 
 2169 
 
 2051 
 
 1936 
 
 1825 
 
 1716 
 
 1610 
 
 1506 
 
 1405 
 
 1306 
 
 1209 
 
 26 
 
 2412 
 
 2287 
 
 2167 
 
 2049 
 
 1934 
 
 1823 
 
 1714 
 
 1608 
 
 1504 
 
 1403 
 
 1304 
 
 1208 
 
 27 
 
 2410 
 
 2285* 
 
 2165 
 
 2047 
 
 1933 
 
 1821 
 
 1712 
 
 1606 
 
 1503 
 
 1402 
 
 1303 
 
 1206 
 
 28 
 
 2408 
 
 2283 
 
 2163 
 
 2045 
 
 1931 
 
 1819 
 
 1711 
 
 1605 
 
 1501 
 
 1400 
 
 1301 
 
 1205 
 
 29 
 
 2405 
 
 2281 
 
 2161 
 
 2043 
 
 1929 
 
 1817 
 
 1709 
 
 1603 
 
 1499 
 
 1398 
 
 1300 
 
 1203 
 
 30 
 
 2403 
 
 2279 
 
 2159 
 
 2041 
 
 1927 
 
 1816 
 
 1707 
 
 1601 
 
 1498 
 
 1397 
 
 1298 
 
 1201 
 
 31 
 
 2401 
 
 2277 
 
 2157 
 
 2039 
 
 1925 
 
 1814 
 
 1705 
 
 1599 
 
 1496 
 
 1395 
 
 1296 
 
 1200 
 
 32 
 
 2399 
 
 2275 
 
 2155 
 
 2037 
 
 1923 
 
 1812 
 
 1703 
 
 1598 
 
 1494 
 
 1393 
 
 1295 
 
 1198 
 
 33 
 
 2397 
 
 2273 
 
 2153 
 
 2035 
 
 1921 
 
 1810 
 
 1702 
 
 1596 
 
 1493 
 
 1392 
 
 1293 
 
 1197 
 
 34 
 
 2395 
 
 2271 
 
 2151 
 
 2033 
 
 1919 
 
 1808 
 
 1700 
 
 1594 
 
 1491 
 
 1390 
 
 1291 
 
 1195 
 
 35 
 
 2393 
 
 2269 
 
 2149 
 
 2032 
 
 1918 
 
 1806 
 
 1698 
 
 1592 
 
 1489 
 
 1388 
 
 1290 
 
 1193 
 
 36 
 
 2391 
 
 2267 
 
 2147 
 
 2030 
 
 1916 
 
 1805 
 
 1696 
 
 1591 
 
 1487 
 
 1387 
 
 1288 
 
 1192 
 
 37 
 
 2389 
 
 2265 
 
 2145 
 
 2028 
 
 1914 
 
 1803 
 
 1694 
 
 1589 
 
 1486 
 
 1385 
 
 1287 
 
 1190 
 
 38 
 
 2387 
 
 2263 
 
 2143 
 
 2026 
 
 1912 
 
 1801 
 
 1693 
 
 1587 
 
 1484 
 
 1383 
 
 1285 
 
 1189 
 
 39 
 
 2384 
 
 2261 
 
 2141 
 
 2024 
 
 1910 
 
 1799 
 
 1691 
 
 1585 
 
 1482 
 
 1382 
 
 1283 
 
 1187 
 
 40 
 
 2382 
 
 2259 2139 
 
 2022 
 
 1908 
 
 1797 
 
 1C89 
 
 1584 
 
 1481 
 
 1380 
 
 1282 
 
 1186 
 
 41 
 
 2380 
 
 2257 
 
 2137 
 
 2020 
 
 1906 
 
 1795 
 
 1687 
 
 1582 
 
 1479 
 
 1378 
 
 1280 
 
 1184 
 
 42 
 
 2378 
 
 2255 
 
 2135 
 
 2018 
 
 1904 
 
 1794 
 
 1686 
 
 1580 
 
 1477 
 
 1377 
 
 1278 
 
 1182 
 
 43 
 
 2376 
 
 2253 
 
 2133 
 
 2016 
 
 1903 
 
 1792 
 
 1684 
 
 1578 
 
 1476 
 
 1375 
 
 1277 
 
 1181 
 
 44 
 
 2374 
 
 2251 
 
 2131 
 
 2014 
 
 1901 
 
 1790 
 
 1682 
 
 1577 
 
 1474 
 
 1373 
 
 1275 1179 
 
 45 
 
 2372 
 
 2249 
 
 2129 
 
 2012 
 
 1899 
 
 1788 
 
 1680 
 
 1575 
 
 1472 
 
 1372 
 
 1274 1178 
 
 46 
 
 2370 
 
 2247 
 
 2127 
 
 2010 
 
 1897 
 
 1786 
 
 1678 
 
 1573 
 
 1470 
 
 1370 
 
 1272 1176 
 
 47 
 
 2368 
 
 2245 
 
 2125 
 
 2009 
 
 1895 
 
 1785 
 
 1677 
 
 1571 
 
 1469 
 
 1368 
 
 1270 
 
 1174 
 
 48 
 
 2366 
 
 2243 
 
 2123 
 
 2007 
 
 1893 
 
 1783 
 
 1675 
 
 1570 
 
 1467 
 
 1367 
 
 1269 
 
 1173 
 
 49 
 
 2364 
 
 2241 
 
 2121 
 
 2005 
 
 1891 
 
 1781 
 
 1673 
 
 1568 
 
 1465 
 
 1365 
 
 1267! 11 71 
 
 50 
 
 2362 
 
 2239 
 
 2119 
 
 2003 
 
 1889 
 
 1779 
 
 1671 
 
 1566 
 
 1464 
 
 1363 
 
 1266 1170 
 
 51 
 
 2359 
 
 2237 
 
 2117 
 
 2001 
 
 1888 
 
 1777 
 
 1670 
 
 1565 
 
 1462 
 
 1362 
 
 1264 1168 
 
 52 
 
 2357 
 
 2235 
 
 2115 
 
 1999 
 
 1886 
 
 1775 
 
 1668 
 
 1563 
 
 1460 
 
 1360 
 
 1262 ! 1167 
 
 53 
 
 2355 
 
 2233 
 
 2113 
 
 1997 
 
 1884 
 
 1774 
 
 1666 
 
 1561 
 
 1459 
 
 1359 
 
 1261 1165 
 
 54 
 
 2353 
 
 2231 
 
 2111 
 
 1995 
 
 1882 
 
 1772 
 
 1664 
 
 1559 
 
 1457 I 1357 
 
 1259 I 1163 
 
 55 
 
 2351 
 
 2229 
 
 2109 
 
 1993 
 
 1880 
 
 1770 
 
 1663 
 
 1558 1455 
 
 1355 
 
 1257 
 
 1162 
 
 56 
 
 2349 | 2227 
 
 2107 
 
 1991 
 
 1878 
 
 1768 
 
 1661 
 
 1556 
 
 1454 
 
 1354 
 
 1256 
 
 1160 
 
 57 
 
 2347 
 
 2225 
 
 2105 
 
 1989 
 
 1876 
 
 1766 
 
 1659 
 
 1554 
 
 1452 
 
 1352 
 
 1254 
 
 1159 
 
 58 
 
 2-345 
 
 2223 
 
 2103 
 
 1987 
 
 1875 
 
 1765 
 
 1657 
 
 1552 
 
 1450 
 
 1350 
 
 1253 
 
 1157 
 
 59 
 
 2343 
 
 2220 
 
 2101 
 
 1986 
 
 1873 
 
 1763 
 
 1655 
 
 1551 
 
 1449 
 
 1349 
 
 1251 
 
 1156 
 
 60 
 
 2341 
 
 2218 
 
 2099 
 
 1984 
 
 1871 
 
 1761 
 
 1654 
 
 1549 
 
 1447 
 
 1347 
 
 1249 
 
 1154 
 
112 
 
 TABLE XCVI. Logistical Logarithms. 
 
 r 
 
 1 
 
 46 
 
 47 
 
 48 
 
 49 
 
 50 
 
 51 | 52 
 
 53 
 
 54 
 
 55 
 
 56 
 
 57 
 
 58 
 
 59 
 
 " 
 
 2760 
 
 2820 
 
 2880 
 
 2940 
 
 3000 
 
 3060 3120 
 
 3180 
 
 3240 
 
 3300 
 
 3360 
 
 3420 
 
 3480 
 
 3540 
 
 ~0 
 
 1154 
 
 1061 
 
 "0969 
 
 "0880 
 
 "0792" i 0706 
 
 0621 
 
 0539 
 
 0458 
 
 0378 
 
 0300 
 
 0223 
 
 0147 
 
 0073 
 
 1 
 
 1152 
 
 1059 
 
 0968 
 
 0878 
 
 0790 i 0704 
 
 0620 
 
 0537 
 
 0456 
 
 0377 
 
 0298 
 
 0221 
 
 0145 
 
 0072 
 
 2 
 
 1151 
 
 1057 
 
 0966 
 
 OS77 
 
 0789 
 
 0703 
 
 0619 
 
 0536 
 
 0455 
 
 0375 
 
 0297 
 
 0220 
 
 0145 
 
 0071 
 
 3 
 
 1149 
 
 1056 
 
 0965 
 
 0875 
 
 0787 
 
 0702 
 
 0617 
 
 0535 
 
 0454 
 
 0374 
 
 0296 
 
 0219 
 
 0143 
 
 0069 
 
 4 
 
 1148 
 
 1054 
 
 0963 
 
 0874 
 
 0786 
 
 0700 
 
 0616 
 
 0533 
 
 0452 
 
 0373 
 
 0294 
 
 0218 
 
 0142 
 
 0068 
 
 5 
 
 1146 
 
 1053 
 
 0962 
 
 0872 
 
 0785 
 
 0699 
 
 0615 
 
 0532 
 
 0451 
 
 0371 
 
 0293 
 
 0216 
 
 0141 
 
 0067 
 
 6 
 
 1145 
 
 1051 
 
 0960 
 
 0871 
 
 0783 
 
 0697 
 
 0613 
 
 0531 
 
 0450 
 
 0370 
 
 0292 
 
 0215 
 
 0140 
 
 0066 
 
 7 
 
 1143 
 
 1050 
 
 0859 
 
 0869 
 
 0782 
 
 0696 
 
 0612 
 
 0529 
 
 0448 
 
 0369 
 
 0291 
 
 0214 
 
 0139 
 
 0064 i 
 
 8 
 
 1141 
 
 1048 
 
 0957 
 
 0868 
 
 0780 
 
 0694 
 
 0610 
 
 0528 
 
 0447 
 
 0367 
 
 0289 
 
 0213 
 
 0137 
 
 0063 
 
 9 
 
 1140 
 
 1047 
 
 0956 
 
 0866 
 
 0779 
 
 0693 
 
 0609 
 
 0526 
 
 0446 
 
 0366 
 
 0288 
 
 0211 
 
 0136 
 
 0062 
 
 10 
 
 1138 
 
 1045 
 
 0954 
 
 0865 
 
 0777 
 
 0692 
 
 0608 
 
 0525 
 
 0444 
 
 0365 
 
 0287 
 
 0210 
 
 0135 
 
 0061 
 
 11 
 
 1137 
 
 1044 
 
 0953 
 
 0863 
 
 0776 
 
 0690 
 
 0606 
 
 0524 
 
 0443 
 
 0363 
 
 0285 
 
 0209 
 
 0134 
 
 0060 
 
 12 
 
 1135 
 
 1042 
 
 0951 
 
 0862 
 
 0774 
 
 0689 
 
 0605 
 
 0522 
 
 0442 
 
 0362 
 
 0284 
 
 0208 
 
 0132 
 
 0058 
 
 13 
 
 1134 
 
 1041 
 
 0950 
 
 0860 
 
 0773 
 
 0687 
 
 0603 
 
 0521 
 
 0440 
 
 0361 
 
 0283 
 
 0206 
 
 0131 
 
 0057 
 
 14 
 
 1132 
 
 1039 
 
 0948 
 
 0859 
 
 0772 
 
 0686 
 
 0602 
 
 0520 
 
 0439 
 
 0359 
 
 0282 
 
 0205 
 
 0130 
 
 0056 
 
 15 
 
 1130 
 
 1037 
 
 0947 
 
 0857 
 
 0770 
 
 0685 
 
 0601 
 
 0518 
 
 0438 
 
 0358 
 
 0280 
 
 0204 
 
 0129 
 
 0055 
 
 16 
 
 1129 
 
 1036 
 
 0945 
 
 0856 
 
 0769 
 
 0683 
 
 0599 
 
 0517 
 
 0436 
 
 0357 
 
 0279 
 
 0202 
 
 0127 
 
 0053 
 
 17 
 
 1127 
 
 1034 
 
 0944 
 
 0855 
 
 0767 
 
 0682 
 
 0598 
 
 0516 
 
 0435 
 
 0356 
 
 0278 
 
 0201 
 
 0126 
 
 0052 
 
 |18 
 
 1126 
 
 1033 
 
 0942 
 
 0853 
 
 0766 
 
 0680 
 
 0596 
 
 0514 
 
 0434 
 
 0354 
 
 0276 
 
 0200 
 
 0125 
 
 0051 
 
 19 
 
 1124 
 
 1031 
 
 0941 
 
 0852 
 
 0764 
 
 0679 
 
 0595 
 
 0513 
 
 0432 
 
 0353 
 
 0275 
 
 0199 
 
 0124 
 
 0050 
 
 20 
 
 1123 
 
 1030 
 
 0939 
 
 0850 
 
 0763 
 
 0678 
 
 0594 
 
 0512 
 
 0431 
 
 0352 
 
 0274 
 
 0197 
 
 0122 
 
 0049 
 
 21 
 
 1121 
 
 1028 
 
 0938 
 
 0849 0762 
 
 0676 
 
 0592 
 
 0510 
 
 0430 
 
 0350 
 
 0273 
 
 0196 
 
 0121 
 
 0047 
 
 22 
 
 1119 
 
 1027 
 
 0936 
 
 0847 0760 
 
 0675 
 
 0591 
 
 0509 
 
 0428 
 
 0349 
 
 0271 
 
 0195 
 
 0120 
 
 0046 
 
 23 
 
 1118 
 
 1025 
 
 0935 
 
 0846 0759 
 
 0673 
 
 0590 
 
 0507 
 
 0427 
 
 0348 
 
 0270 
 
 0194 
 
 0119 
 
 0045 
 
 24 
 
 1116 
 
 1024 
 
 0933 
 
 0844 0757 
 
 0672 
 
 0588 
 
 0506 
 
 0426 
 
 0346 
 
 0269 
 
 0192 
 
 0117 
 
 0044 
 
 25 
 
 1115 
 
 1022 
 
 0932 
 
 0843 0756 
 
 0670 0587 
 
 0505 
 
 0424 
 
 0345 
 
 0267 
 
 0191 
 
 0116 
 
 0042 
 
 26 
 
 1113 
 
 1021 
 
 0930 
 
 0841 0754 
 
 0669 0585 
 
 0503 
 
 0423 
 
 0344 
 
 0266 
 
 0190 
 
 0115 
 
 0041 
 
 27 
 
 1112 
 
 1019 
 
 0929 
 
 0840 0753 
 
 0668 0584 
 
 0502 
 
 0422 
 
 0342 
 
 0265 
 
 0189 
 
 0114 
 
 0040 
 
 28 
 
 1110 
 
 1018 
 
 0927 
 
 0838 
 
 0751 
 
 0666 
 
 0583 
 
 0501 
 
 0420 
 
 0341 
 
 0264 
 
 0187 
 
 0112 
 
 0039 
 
 29 
 
 1109 
 
 1016 
 
 0926 
 
 0837 
 
 0750 
 
 0665 
 
 0581 
 
 0499 
 
 0419 
 
 0340 
 
 0262 
 
 0186 
 
 0111 
 
 0038 
 
 30 
 
 1107 
 
 1015 
 
 0924 
 
 0835 
 
 0749 
 
 0663 
 
 0580 
 
 0498 
 
 0418 
 
 0339 
 
 0261 
 
 0185 
 
 0110)0036 
 
 31 
 
 1105 
 
 1013 
 
 0923 
 
 0834 0747 
 
 0662 
 
 0579 
 
 0497 
 
 0416 
 
 0337 
 
 0260 
 
 0184 
 
 0109 
 
 0035 
 
 32 
 
 1104 
 
 1012 
 
 0921 
 
 0833 0746 
 
 0661 
 
 0577 
 
 0495 
 
 0415 
 
 0336 
 
 0258 
 
 0182 
 
 0107 
 
 0034 
 
 33 
 
 1102 
 
 1010 
 
 0920 
 
 0831 
 
 0744 
 
 0659 
 
 0576 
 
 0494 
 
 0414 
 
 0335 
 
 0257 
 
 0181 
 
 0106 
 
 0033 | 
 
 34 
 
 1101 
 
 1008 
 
 0918 
 
 0830 
 
 0743 
 
 0658 0574 
 
 0493 
 
 0412 
 
 0333 
 
 0256 
 
 0180 
 
 0105 
 
 0031 
 
 35 
 
 1099 
 
 1007 
 
 0917 
 
 0828 i 0741 
 
 0656 0573 
 
 0491 
 
 0411 
 
 0332 
 
 0255 
 
 0179 
 
 0104 
 
 0030 
 
 36 
 
 1098 
 
 1005 
 
 0915 
 
 0827 0740 
 
 0655 0572 
 
 0490 
 
 0410 
 
 0331 
 
 0253 
 
 0177 
 
 0103 
 
 0029 
 
 37 
 
 1096 
 
 1004 
 
 0914 
 
 0825 0739 
 
 0654 0570 
 
 0489 
 
 0408 
 
 0329 
 
 0252 
 
 0176 
 
 0101 
 
 0028 
 
 38 
 
 1095 
 
 1002 
 
 0912 
 
 0824 ; 0737 
 
 0652 
 
 0569 
 
 0487 
 
 0407 
 
 032S 
 
 0251 
 
 0175 
 
 0100 
 
 0027 
 
 39 
 
 1093 
 
 1001 
 
 0911 
 
 0822 0736 
 
 0651 
 
 0568 
 
 0486 
 
 0406 
 
 0327 
 
 0250 
 
 0174 
 
 0099 
 
 0025 
 
 40 
 
 1091 
 
 0999 
 
 0909 
 
 0821 0734 
 
 0649 
 
 0566 
 
 0484 
 
 0404 
 
 0326 
 
 0248 
 
 0172 
 
 0098 
 
 0024 
 
 41 
 
 1090 
 
 0998 
 
 0908 
 
 0819 0733 
 
 0648 0565 
 
 0483 
 
 0403 
 
 0324 
 
 0247 
 
 0171 
 
 0096 
 
 0023 
 
 42 
 
 1088 
 
 0996 
 
 0906 
 
 0818,0731 
 
 0647 i 0563 
 
 0482 
 
 0402 
 
 0323 
 
 0246 
 
 0170 
 
 0095 
 
 0022 
 
 43 
 
 1087 
 
 0995 
 
 0905 
 
 0816! 0730 
 
 0645 ! 0562 
 
 0480 
 
 0400 
 
 0322 
 
 0244 
 
 0169 
 
 0094 
 
 0021 
 
 44 
 
 1085 
 
 0993 
 
 0903 
 
 0815 
 
 0729 
 
 0644 0561 
 
 0479 
 
 0399 
 
 0320 
 
 0243 
 
 0167 
 
 0093 
 
 0019 
 
 45 
 
 1084 
 
 0992 
 
 0902 
 
 0814 
 
 0727 
 
 0642 0559 
 
 0478 
 
 0398 
 
 0319 
 
 0242 
 
 0166 
 
 009110018 
 
 46 
 
 1082 
 
 0990 
 
 0900 
 
 6812 
 
 0726 
 
 0641 0558 
 
 0476 
 
 0396 
 
 0318 
 
 0241 
 
 0165 
 
 0090 0017 
 
 47 
 
 1081 
 
 0989 
 
 0899 
 
 0811 
 
 0724 
 
 0640 0557 
 
 0475 
 
 0395 
 
 0316 
 
 0239 
 
 0163 
 
 0089 0016 
 
 48 
 
 1079 
 
 0987 
 
 0897 
 
 0809 
 
 0723 
 
 0638 | 0555 
 
 0474 
 
 0394 
 
 0315 
 
 0238 
 
 0162 
 
 OQ88 
 
 0015 
 
 49 
 
 1078 
 
 0986 
 
 0896 
 
 0808 0721 
 
 0637 0554 
 
 0472 - 0392 
 
 0314 
 
 0237 
 
 0161 
 
 0087 
 
 0013 
 
 60 
 
 1076 
 
 0984 
 
 0894 
 
 0806 
 
 0720 
 
 0635 
 
 0552 
 
 0471 
 
 0391 
 
 031310235 
 
 0160 
 
 0085 
 
 0012 
 
 51 
 
 1074 
 
 0983 
 
 0893 
 
 0805 
 
 0719 
 
 0634 
 
 0551 
 
 0470 
 
 0390 
 
 0311 0234 
 
 0158 
 
 0084 0011 
 
 52 
 
 1073 
 
 0981 
 
 0891 
 
 0803 
 
 0717 
 
 0633 
 
 0550 
 
 0468 
 
 0388 
 
 0310 
 
 0233 
 
 0157 
 
 0083 0010 
 
 53 
 
 1071 
 
 0980 
 
 0890 
 
 0802 0716 
 
 0631 
 
 0548 
 
 0467 
 
 0387 
 
 0309 
 
 0232 
 
 0156 
 
 0082 0008 
 
 54 
 
 1070 
 
 0978 
 
 0888 
 
 0801 ! 0714 
 
 0630 
 
 0547 
 
 0466 
 
 0386 
 
 0307 
 
 0230 
 
 0155 
 
 0080(0007 
 
 55 
 
 1068 
 
 0977 , 0887 
 
 0799 | 0713 
 
 0628 
 
 0546 
 
 0464 
 
 0384 0306 j 0229 j 0153 
 
 0079 
 
 0006 
 
 66 
 
 1067 
 
 0975 
 
 0885 0798 0711 
 
 3627 0544 
 
 0463 
 
 0383 : 0305 i 09C8 0152 
 
 0078 
 
 0005 
 
 fi7 
 
 1065 
 
 0974 
 
 0884 
 
 0796 
 
 0710 
 
 0626 0543 
 
 0462 
 
 0382 0304- 0227 
 
 0151 
 
 0077 
 
 0004 
 
 98 
 
 1064 
 
 0972 
 
 0883 
 
 0795 
 
 0709 
 
 0624 ! 0541 1 0460 
 
 0381 
 
 0302 ! 0225 
 
 0150 
 
 0075 0002 
 
 r>9 
 
 1062 
 
 0971 
 
 0881 
 
 0793 
 
 0707 
 
 0623 0540 
 
 0459 
 
 0379 
 
 0301 
 
 0224 
 
 0148 
 
 0074 0001 
 
 no 
 
 1061 
 
 0969 
 
 0880 
 
 0792 
 
 0706 
 
 0621 i0539 
 
 0458 
 
 0378 
 
 0300 
 
 0223 
 
 0147 
 
 0073 0000 
 

 
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 1 JUN 4 1926 
 JUL 26 192? 
 
 
 KP I i8 
 
 \a 
 
 j t 8 8 
 
 W32 
 
 c^ 
 
 \0 
 
 ^v 1 
 
 AUQ 8 
 
 25w-7,'25