1896 
 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA. 
 
 01 FT OR 
 
 Received 
 Accession No. 
 
 
 Class No. 
 
THE ELEMENTS 
 
 OF THE 
 
 FOUB INNER PLANETS 
 
 AND THE 
 
 FUNDAMENTAL CONSTANTS OF ASTRONOMY 
 
 BY 
 
 SIMON NEWCOMB 
 
 Supplement to the American Ephemeris and Nautical 
 Almanac for 1897 
 
 WASHINGTON 
 
 GOVERNMENT PRINTING OFFICE 
 1895 
 
PREFACE. 
 
 THE diversity in the adopted values of the elements and 
 constants of astronomy is productive of inconvenience to all 
 who are engaged in investigations based upon these quanti- 
 ties, and injurious to the precision and symmetry of much of 
 our astronomical work. If any cases exist in which uniform 
 and consistent values of all these quantities are embodied in 
 an extended series of astronomical results, whether in the 
 form of ephemerides or results of observations, they are the 
 exception rather than the rule. The longer this diversity 
 continues the greater the difficulties which astronomers of 
 the future will meet in utilizing the work of our time. 
 
 On taking charge of the work of preparing the American 
 Ephemeris in 1877 the writer was so strongly impressed with 
 the inconvenience arising from this source that he deemed it 
 advisable to devote all the force which he could spare to the 
 work of deriving improved values of the fundamental elements 
 and embodying them in new tables of the celestial motions. 
 It was expected that the work could all be done in ten years. 
 But a number of circumstances, not necessary to describe at 
 present, prevented the fulfillment of this hope. Only now is 
 the work complete so far as regards the fundamental constants 
 and the elements of the planets from Mercury to Jupiter inclu- 
 sive. The construction of tables of the four inner planets is 
 now in progress, those of Jupiter and Saturn having already 
 been completed by Mr. HILL. All these tables will be pub- 
 lished as soon as possible, and the investigations on which 
 they are based are intended, so far as it is practicable to con- 
 dense them, to appear in subsequent volumes of the Astro- 
 nomical Papers of the American Ephemeris. As it will take 
 several years to bring out these volumes, it has been deemed 
 advisable to publish in advance the present brief summary of 
 
 the work. 
 
 HI 
 
IV PREFACE. 
 
 The author feels that critical examination of this monograph 
 may show in many points a want of consistency and conti- 
 nuity. The ground covered is so extensive, the material so 
 diverse as well as voluminous, and the relations to be investi- 
 gated so numerous, that no conclusion could be reached on 
 one point which was not liable to be modified by subsequent 
 decisions upon other points. The author trusts that the diffi- 
 culties growing out of these features of the work, as well as 
 those incident to the administration of an office not especially 
 organized for the work, will afford a sufficient apology for any 
 defects that may be noticed. 
 
 NAUTICAL ALMANAC OFFICE, 
 
 U. 8. Naval Observatory, January 7, 1895. 
 
I^IVWITT 
 
 CONTENTS. 
 
 CHAPTER I. GENERAL OUTLINE OF THE WORK OF COMPARING 
 
 THE OBSERVATIONS WITH THEORY. 
 
 Page. 
 1. Reduction to the standard system of Right Ascensions and 
 
 Declinations 
 
 2. Observations used 
 
 3. Semidiameters of Mercury and Venus. Table for defective 
 
 illumination of Mercury in Right Ascension 3 
 
 4. Tabular places from LEVERRIER'S tables. Reduction for 
 
 masses used by LEVERRIER 6 
 
 5. Comparisons of observations and tables 8 
 
 $ 6. Equations of condition. Method of formation 8 
 
 7. Method of determining the secular variations and the masses 
 
 of Venus and Mercury independently 10 
 
 8. Method of introducing the results of observations on transits 
 of Venus and Mercury ; separate solutions, A from meridian 
 observations without transits ; B, including both meridian 
 observations and transits 13 
 
 CHAPTER II. DISCUSSION AND RESULTS OF OBSERVATIONS OF 
 
 THE SUN. 
 
 $ 9. Method of treating observed Right Ascensions of the Sun. 
 Expression of errors of observed Right Ascension as error 
 of longitude 15 
 
 10. Treatment of observed Declinations of the Sun. Formation 
 of equations of condition for the corrections to the 
 obliquity and to the Sun's absolute longitude 16 
 
 $ 11. Formation of equations from observed Right Ascensions of 
 
 Sun 17 
 
 12. Solution of equations from Right Ascensions of the Sun. 
 Tabular exhibit of results of observations of the Sun's 
 Right Ascensions at various observatories during different 
 periods 20 
 
 13. Mass of Venus, derived from observations of the Sun's Right 
 
 Ascension 24 
 
 $ 14. Discussion of corrections to the Right Ascensions of the Sun 
 
 relative to that of the stars 25 
 
 v 
 
VI CONTENTS. 
 
 Page. 
 
 15. Discussion of corrections to the eccentricity and perihelion 
 
 of the Earth's orbit 27 
 
 16. Results of observed Declinations of the Sun. Exhibit of 
 individual corrections to the absolute longitude and the 
 obliquity of the ecliptic at the different observatories 
 during different periods ^ 29 
 
 $ 17. Discussion of the observed corrections to the Sun's absolute 
 
 longitude 32 
 
 18. Discussion of the observed corrections to the obliquity of 
 
 the ecliptic 33 
 
 $ 19. Effect of refraction on the obliquity ; special investigation of 
 the secular change of obliquity as derived from observa- 
 tions of the Sun 35 
 
 $ 20. Concluded results for the obliquity, and its secular varia- 
 tion 39 
 
 21. Summary of results for the corrections to the elements of the 
 Earth's orbit and their secular variations as derived from 
 observations of the Sun alone . 41 
 
 CHAPTER III. RESULTS OF OBSERVATIONS OF THE PLANETS 
 MERCURY, VENUS, AND MARS. 
 
 22. Elements adopted for correction 43 
 
 $ 23. Introduction .of the corrections to the masses of Venus and 
 
 Mercury 45 
 
 24. Introduction of the errors of absolute Right Ascension and 
 
 Declinations of the standard stars 46 
 
 25. Introduction of the corrections to the secular variations. 
 Method of forming the normal equations by periods so as 
 
 to include the correction to the secular variation 49 
 
 $ 26. Dates and weights of the equations for the various periods. 52 
 
 27. Unknown quantities of the equations. Factors for changing 
 corrections of the unknown quantities into corrections of 
 
 the elements 55 
 
 28. Table of the values of the principal coefficients of the normal 
 
 equations '..'...' 56 
 
 29. Order of elimination 57 
 
 30. Treatment of meridian observations of Mercury. Effect of 
 want of approximation in the coefficients of the equations 
 
 of condition 58 
 
 31. Introduction of the equations derived from observed tran- 
 sits of Mercury 61 
 
 $ 32. Solution of the equations for Mercury 65 
 
 $ 33. Systematic discordances among the observed Right Ascen- 
 sions of Mercury in different points of its relative orbit.. 66 
 
CONTENTS. VII 
 
 Page. 
 
 34. Comparison of the results derived from meridian observa- 
 
 tions of Mercury with those derived from transits over the 
 
 Sun'sdisk 69 
 
 35. Treatment of meridian observations of Venus 70 
 
 36. Results of observed transits of Venus 70 
 
 37. Equations derived from observed transits of Venus 75 
 
 38. Solutions of the equations from Venus 76 
 
 39. Comparison of the results of meridian observations of Venus 
 
 with those of transits 76 
 
 40. Solution of the equations for Mars. Inequality of long 
 
 period in the mean longitude and perihelion, indicated by. 
 
 observations 77 
 
 41. Reduction from the equator to the ecliptic 79 
 
 CHAPTER IV. COMBINATION OF THE PRECEDING RESULTS TO 
 
 OBTAIN THE MOST PROBABLE VALUES OF THE ELEMENTS 
 AND OF THEIR SECULAR VARIATIONS FROM OBSERVA- 
 TIONS ALONE. 
 
 42. Modifications of the canons of least squares 81 
 
 43. Relative precision of the two methods of determining the 
 
 elements of the Earth's orbit 86 
 
 $ 44. Concluded secular variations of the solar elements, as 
 
 derived from observations alone 87 
 
 $ 45. Common error of the standard declinations 89 
 
 $ 46. Definitive secular variations of all the elements from obser- 
 vations alone. Matrices of the normal equations for the 
 
 secular variations. Tabular statement of results 90 
 
 $ 47. Definitive corrections to the solar elements for 1850. . 95 
 
 CHAPTER V. MASSES OF THE PLANETS DERIVED BY METHODS 
 
 INDEPENDENT OF THE SECULAR VARIATIONS, WITH THE 
 RESULTING COMPUTED SECULAR VARIATIONS. 
 
 $ 48. Plan of discussion 97 
 
 49. Mass of Jupiter ; general combination of results 97 
 
 50. Mass of Mars. Prof. HALL'S value adopted 99 
 
 51. Mass of the Earth, derived from the preliminary value of the 
 
 solar parallax 99 
 
 52. Mass of Venus, derived from periodic perturbations 101 
 
 53. Mass of Mercury, from various sources 102 
 
 54. Theoretical values of the secular variations for 1850... 106 
 
VIII CONTENTS. 
 
 Page. 
 
 CHAPTER VI. EXAMINATION OF HYPOTHESES AND DETERMINA- 
 TION OF THE MASSES BY WHICH THE DEVIATIONS OF THE 
 SECULAR VARIATIONS FROM THEIR THEORETICAL VALUES 
 MAY BE EXPLAINED. 
 
 55. Comparison of the observed and theoretical secular varia- 
 tions 109 
 
 $ 56. Hypothesis of nonsphericity of the equipotential surfaces 
 
 of the Sun Ill 
 
 57. Hypothesis of an intraniercurial ring 112 
 
 58. Hypothesis of an extended mass of diffused matter, like that 
 
 Which reflects the zodiacal light 115 
 
 $ 59. Hypothesis of a ring of planets outside the orbit of Mer- 
 cury. Elements of such a ring. This hypothesis the only 
 one which represents the observations, but too improbable 
 to be accepted 116 
 
 $ 60. Examination of the question whether the excess of motion 
 of the perihelion of Mars may be due to the action of the 
 zone of minor planets 116 
 
 61. Hypothesis that gravitation toward the Sun is not exactly 
 
 as the inverse square of the distance 118 
 
 62. Degree of precision with which the theory' of the inverse 
 
 square is established 119 
 
 63. Determination of the masses which will best represent the 
 observed secular variations of the eccentricities, nodes, 
 and inclinations 121 
 
 $ 64. Preliminary adjustment of the two sets of 1 masses. Result- 
 ing value of the solar parallax 122 
 
 CHAPTER VII. VALUES OF THE PRINCIPAL CONSTANTS WHICH 
 
 DEPEND UPON THE MOTION OF THE EARTH. 
 
 65. The processional constant 124 
 
 66. The constant of nutation, derived from observations 129 
 
 67. Relations between the constants of precession and nutation 
 
 and the quantities on which they depend 131 
 
 $ 68. The mass of the Moon from the observed constant of nuta- 
 tion --- 132 
 
 69. The constant of aberration 133 
 
 70. The values of this constant, derived from observations 135 
 
 71. The lunar inequality in the Earth's motion 139 
 
 72. The solar parallax derived from the lunar inequality 142 
 
 $ 73. Values of the solar parallax derived from measurements of 
 Venus on the face of the Sun during the transits of 1874 
 
 and 1882, with the heliometer and photoheliograph 143 
 
 $ 74. The solar parallax from observed contacts during transits of 
 
 Venus.. 145 
 
CONTENTS. IX 
 
 75. Solar parallax from the observed constant of aberration and 
 
 measured velocity of light 147 
 
 76. Solar parallax from the parallactic inequality of the Moon. 148 
 
 i 77. Solar parallax from observations of the minor planets with 
 
 the heliometer 152 
 
 78. Remarks on determinations of the parallax which are not 
 used in the present discussion. Errors arising from dif- 
 ferences of color 154 
 
 CHAPTER VIII. DISCUSSION OF RESULTS FOR THE SOLAR PARAL- 
 LAX AND THE MASSES OF THE THREE INNER PLANETS. 
 
 79. Separate values of the solar parallax, and their general 
 
 mean 156 
 
 80. Rediscussion of the motion of the node of Venus 159 
 
 81. Possible systematic errors in determinations of the parallax. 164 
 
 82. Revised list of determinations 166 
 
 83. Definitive adjustment of the masses of the three inner 
 
 planets 168 
 
 84. Possible causes of the observed discordances 173 
 
 85. Adopted values of the doubtful quantities 173 
 
 86. Bearing of future determinations on the question. 175 
 
 CHAPTER IX. DERIVATION OF RESULTS. 
 
 87. Ulterior corrections to the motions of the perihelion and 
 
 mean longitude of Mercury 178 
 
 88. Definitive elements of the four inner planets for the epoch 
 
 1850, as inferred from all the data of observation 179 
 
 $ 89. Definitive values of the secular variations 182 
 
 90. Secular acceleration of the mean motions 186 
 
 91. The measure of time 188 
 
 92. The constant of aberration 188 
 
 93. The mass of the Moon 189 
 
 94. The parallactic inequality of the Moon 190 
 
 95. The centimeter-second system of astronomical units 190 
 
 \S 96. Masses of the Earth and Moon in centimeter-second units.. 191 
 
 ^S 97. Parallax of the Moon 193 
 
 98. Mass and parallax of the Sun 194 
 
 99. Constant of nutation, and mechanical ellipticity of the 
 
 Earth 195 
 
 100. Precession 196 
 
 $ 101 . Obliquity of the ecliptic 196 
 
 102. Relative positions of the equator and the ecliptic at differ- 
 ent epochs for reduction of places of stars and planets . . 197 
 
ELEMENTTlfFCONSTANTS. 
 
 OHAPTEK 1. 
 
 GENERAL OUTLINE OF THE WORK OF COMPARING THE 
 OBSERVATIONS WITH THEORY. 
 
 1. In logical order, the first step in the work consists in the 
 reduction of observed positions of the Sun and planets to a 
 uniform equinox and system of declinations. 
 
 The adopted standard of Eight Ascensions was that origi- 
 nally worked out in my paper on the Eight Ascensions of the 
 fundamental stars, found in an appendix to the Washington 
 Observations for 1870, and extended to a fundamental system 
 of time stars in the catalogue published in Yol. 1 of the Astro- 
 nomical Papers of the American Ephemeris. This system 
 coincides closely with that of the Astronomische Gesellschaft 
 and the Berliner Jahrbuch, about the epoch 1870, but the cen- 
 tennial proper motion is greater' by about 8 .08. 
 
 In Declinations, the adopted standard was that of Boss, 
 which has been used in the American Ephemeris since 1881, 
 and on which is based the catalogue of zodiacal stars just 
 referred to. But as Declinations generally are not immediately 
 referred to fundamental stars, the method of reducing obser- 
 vations to this system in Declination was not entirely uniform. 
 
 Observations used. 
 
 2. The following is a general statement of the observations 
 used, and the extent to which they were corrected, or re-re- 
 duced. 
 
 Greenwich. Dr. AUWERS courteously supplied me with the 
 results of his re-reduction of BRADLEY'S observations both of 
 the Sun and planets. From the beginning of MASKYLENE'S 
 work until 1835, the Greenwich observations were completely 
 re-reduced, utilizing, so far as possible, AIRY'S reductions. The 
 5690 N ALM 1 i 
 
GENERAL OUTLINE. [2 
 
 data necessary for these observations were discussed in Prof. 
 SAFFORD'S paper, Vol. n, pt. n, which paper was prepared 
 for this purpose. In the case of the Greenwich observations 
 from 1835 onward, it was deemed sufficient to apply constant 
 corrections to the Eight Ascensions, determined from time to 
 time by comparisons of the adopted Eight Ascensions with 
 the standard ones. In the case of the Declinations, Boss's 
 special tables were used, but in the later years it was judged 
 sufficient to apply the constant correction necessary for reduc- 
 tion to Boss's standard. 
 
 Palermo. PIAZZT'S observations of the Sun and Planets were 
 completely re-reduced, the zero point of his instrument being 
 determined from the observed Declinations. 
 
 Paris. LEVERRIER'S reduction of the Paris observations 
 from 1801 onward was made use of, applying the correction 
 necessary to reduce the results to the adopted standard. 
 
 Konigsberg. BESSEL'S clock corrections were individually 
 corrected by the new positions of the fundamental stars, so 
 that practically the Eight Ascensions may be considered as 
 completely re-reduced. 
 
 In the case of the other observatories, it was deemed suffi- 
 cient to determine, by a comparison of the adopted or of the 
 concluded Eight Ascensions and Declinations of the funda- 
 mental stars with the standard catalogue, what common cor- 
 rections were necessary for reduction to the standard. When, 
 however, the period was covered by Boss's tables, the correc- 
 tion which he gives as varying with the Declination was ap- 
 plied. After more mature consideration, I am inclined to think 
 it would have been better to apply a constant correction to the 
 Declinations in every case, except those where the change 
 with the Declination was quite large. 
 
 Although these processes were somewhat heterogeneous, it 
 is believed that the main object of referring the Declinations 
 to a system of which the error would be a uniformly varying 
 quantity was fairly well attained. The subsequent determi- 
 nation of this error both in Eight Ascension and Declination 
 is a necessary part of the work. 
 
3] OBSERVATIONS USED. 3 
 
 The following is a list of the observatories whose observa- 
 tions of the Sun and Planets were included in the work: 
 
 Greenwich 1750-1892 
 
 Palermo 1791-1813 
 
 Paris __-- 1801-1889 
 
 Konigsberg 1814-1845 
 
 Dorpat 1823-1838 
 
 Cambridge . 1828-1844 
 
 Berlin 1838-1842 
 
 Oxford, Radcliffe 1840-1887 
 
 Pulkowa 1842-1875 
 
 Washington _ 1846-1891 
 
 Leiden 1863-1871 
 
 Strassburg 1884-1887 
 
 Cape of Good Hope 1884-1890 
 
 The number of the meridian observations of the Sun, and 
 of the planets Mercury, Venus, and Mars, actually included in 
 the work is approximately as follows: 
 
 The Sun 40,176 
 
 Mercury _' 54 21 
 
 Venus 12, 319 
 
 Mars 4, 114 
 
 Total 62,030 
 
 Semidiameters of Mercury and Venus. 
 
 3. The reduction of the semidiarneter of the planets was a 
 point to which special attention was given. In the case of 
 Mercury, the adopted semidiameter at distance unity was 3".34. 
 The values adopted by the various observatories in reducing 
 their observations varied so little from this that in cases where 
 the original reductions were accepted no correction was applied 
 for the difference. So, also, when the observers applied a cor- 
 rection for reducing the observed center of light to the actual 
 center of the planet, no revision of this reduction was made. 
 Such was supposed to be the case with the Paris observations. 
 When the published Eight Ascension was that of the center 
 of light simply, a reduction to the true center was computed 
 by the empirical formula used in the Washington observations. 
 If we put i for the angle between the Earth and Sun as seen 
 from the planet, then 1 -f- cos i will represent the fraction of 
 
4 GENERAL OUTLINE. [3 
 
 the apparent transverse diameter of the planet that is illu- 
 minated by the Sun. It was assumed that when the illumina- 
 tion was such that the thickness of the crescent approached 
 zero, the point observed would be two-thirds of the way from 
 the center of the planet to the limb, and that when the planet 
 was dichotomized the center of observation would be five- 
 twelfths of the way from the center to the limb. These con- 
 ditions, with the added one that when the planet was fully 
 illuminated the correction should vanish, suggested the em- 
 ployment of the formula 
 
 Correction = seinidiameter x (1-cos ^(5-f cos i) 
 
 This correction was to be multiplied by the sine or cosine of 
 the angle which the line of cusps made with the meridian to 
 reduce it to Right Ascension and Declination respectively. 
 
 The correction being practically the same whenever the 
 Earth and planet return to the same positions in anomaly, it 
 is possible to embody it in a table of two arguments, one 
 depending on the longitude of the Earth, the other on that of 
 the planet. Actually, however, the table was arranged in a 
 more convenient form, in which one argument is the date at 
 which Mercury last passed perihelion, and the other, its mean 
 anomaly. Owing to the importance which this correction may 
 assume, a partial transcript of the table actually employed for 
 the reduction in Right Ascension is given on the next page. 
 Read horizontally, the numbers show the corrections of the 
 argument through one revolution of the planet. Vertically, 
 they may be regarded as giving the successive corrections corre- 
 sponding to any one position of the planet, while the Earth 
 goes through a complete revolution. The table as actually 
 used extended to every 10, but the values for every 00 of 
 mean anomaly will suffice to show the general magnitude of 
 the correction. 
 
 The correction to the Declination was embodied in a similar 
 table, which it is not deemed necessary to print at present. 
 
 In the case of Venus, it seems scarcely possible to decide 
 upoiT a value of the semidiauieter, or a law of its apparent 
 change, which should apply to all parts of the orbit. After a 
 
3] 
 
 SEMIDIAMETERS OF MERCURY AND VENUS. 
 
 careful examination of the data, it was decided to reduce all 
 the observations with the semidiameter 
 
 8 -^- 5 +0".20 
 
 when made with modern instruments, and to use a value 0".3 
 greater in earlier observations. The actual reductions of all 
 
 Correction for defective illumination of Mercury in R. A. 
 Arguments: Date of perihelion passage at side, and mean 
 anomaly "g" at top. 
 
 g= 
 
 * 
 
 60 
 
 120 
 
 1 80 
 
 240 
 
 300 
 
 360 
 
 
 s 
 
 s 
 
 s 
 
 s 
 
 s 
 
 J 
 
 s 
 
 Jan. o __ 
 
 +.19 
 
 .16 
 
 -.07 
 
 03 
 
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 +.03 
 
 10 __ 
 
 .16 
 
 -.18 
 
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 20 .. 
 
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 . II 
 
 05 
 
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 3-- 
 
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 13 
 
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 -3 
 
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 + .01 
 
 Feb. 9__ 
 
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 17 
 
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 19 ._ 
 
 .08 
 
 
 . 18 
 
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 Mar. i .. 
 
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 ii __ 
 
 05 
 
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 03 
 
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 31 __ 
 
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 . 20 
 
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 Apr. 10 _. 
 
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 +.23 
 
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 .15 
 
 97 
 
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 20 __ 
 
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 . 18 
 
 . 09 
 
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 3-- 
 
 + .01 
 
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 X 
 
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 May 10 __ 
 
 .00 
 
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 15 
 
 . 22 
 
 17 
 
 -.13 
 
 03 
 
 20 __ 
 
 .00 
 
 05 
 
 . 12 
 
 . 20 
 
 .12 
 
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 30 __ 
 
 .00 
 
 4 
 
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 . 18 
 
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 June 9 __ 
 
 .00 
 
 3 
 
 .09 
 
 .14 
 
 +.18 
 
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 I9~ 
 
 .00 
 
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 29-- 
 
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 05 
 
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 05 
 
 o 
 . II 
 
 +.16 
 
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 2 9-- 
 
 . 02 
 
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 Aug. 8 
 
 03 
 
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 18. 
 
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 3 
 
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 28.. 
 
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 05 
 
 13 
 
 
 
 Sept. 7 . . 
 
 .06 
 
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 *7 -- 
 
 .07 
 
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 .00 
 
 + .01 
 
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 27 .. 
 
 .09 
 
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 .00 
 
 .00 
 
 .02 
 
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 + .20 
 
 Oct. 7__ 
 
 . II 
 
 . 02 
 
 .00 
 
 .00 
 
 + .01 
 
 05 
 
 .18 
 
 17-. 
 
 . 12 
 
 .02 
 
 .00 
 
 .00 
 
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 27 __ 
 
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 -3 
 
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 3 
 
 r 3 
 
 Nov. 6 . . 
 
 .16 
 
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 26 
 
 
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 Dec 6 
 
 
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 --. OI 
 
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 16 
 
 
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 26.. 
 
 + .20 
 
 '5 
 
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 Jan. 5-- 
 
 +.18 
 
 .17 
 
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 3 
 
 .01 
 
 .00 
 
 +.03 
 
6 GENERAL OUTLINE. [4 
 
 the principal series of observations were corrected to this value 
 of the element in question. 
 
 Observations of the estimated center of Venus, when made 
 more than one hundred days from superior conjunction, were 
 rejected altogether; when made within that limit, the point 
 observed was assumed to be the center of gravity of the illu- 
 minated portion of the disk, considered as a plane figure, and 
 the necessary reduction to the center was always applied. 
 
 A similar correction was applied to observations of the esti- 
 mated center of Mars. The Paris results, after 1830, and the 
 later Greenwich and Washington results, are published with 
 the reduction for center of light already applied, and in these 
 cases the published corrections were not changed. 
 
 Tabular places. 
 
 4. The tabular elements of the planets adopted for correc- 
 tion were those of LEVERRIER'S tables. These tables having 
 been continuously used in Astronomical Ephemerides since 
 1864, it was judged more convenient to adopt the theory on 
 which they were based as the provisional one to be corrected 
 than it was to construct a new provisional theory. As the tables 
 in their original form are extremely cumbrous to use, the 
 theory was partially reconstructed by making manuscript 
 tables of the principal perturbations, which were, however, 
 carried only to tenths of seconds. With these tables the 
 places of the planets were computed for dates previous to 1864. 
 
 As places of the Sun were necessary not only for direct com- 
 parison with observations of the Sun, but also for the geocen- 
 tric places of the planets, an ephemeris of the Sun's longitude 
 and radius vector was prepared for the entire period 1750-1864 
 to every fifth day, the lunar perturbation being omitted and 
 afterward applied for each date when required. 
 
 The method of deriving the final tabular places varied with 
 circumstances. When there was no accurate ephemeris avail- 
 able for comparison, which was the case before 1830, it was 
 necessary to compute a completely independent set of tabular 
 geocentric places. Sometimes these places were computed for 
 the moment of the individual observations, but more generally, 
 when the observations occurred in groups, an ephemeris was 
 
4] TABULAR PLACES. 7 
 
 computed in order that the work might be checked by differ- 
 ences. After 1830 it was common to compute an ephemeris 
 for intervals of three, five, or ten days, thus deriving the cor- 
 rections necessary to reduce the published ephemerides of the 
 Berliner Jahrbuch or of the Nautical Almanac to those derived 
 from LEVERRIER'S tables. 
 
 Until this plan was mapped out, and work well in progress 
 upon it, it was not noticed that the planetary masses adopted in 
 LEVERRIER'S tables were so diverse that corrections to reduce 
 the geocentric places to a uniform system of masses would be 
 necessary. Although theoretically the necessary reductions 
 were very simple, I can not but feel that the application of 
 such corrections involves more or less doubt and uncertainty, 
 and that it would have been better to have constructed pro- 
 visional tables based on uniform masses quite independent of 
 those of LEVERRIER. 
 
 In Annales de V Observatoire de Paris, Vol. n, LEVERRIER 
 gives the following values of the masses used by him as the 
 basis of his provisional theory : 
 
 Mercury . . ^ = .000 000 333 . . 
 
 Venus 40T847 =- 0000024 885 
 
 Earth . QKA . ftQg =.000002 
 
 Mars 2 680 337 = - 000 00 373 
 
 The following table shows the factors by which these masses 
 were multiplied in the cases of the several planets in LEVER- 
 RIER'S final tables. They were controlled by induction from 
 the numbers of the tables themselves, the result of which was 
 found in all cases to agree with the statements in the introduc- 
 tion to the tables. 
 
 In the last line of the table is shown the factor used in the 
 present provisional theory. 
 
GENERAL OUTLINE. 
 
 
 Mercury. 
 
 Venus. 
 
 Earth. 
 
 Mars. 
 
 In tables of 
 The Sun . 
 
 
 i 004. 
 
 
 o SCK 
 
 Mercury 
 
 
 
 j 
 
 
 Venus 
 
 i 
 
 
 j 
 
 j 
 
 Mars 
 
 
 o. 071; 
 
 i. 0026 
 
 
 Present work 
 
 j 
 
 i 
 
 
 o 86s? 
 
 
 
 
 
 
 As in the actual work the masses of Mercury and Venus 
 were to be determined from the observed periodic perturba- 
 tions which they produced, it was necessary that the perturba- 
 tions produced by them should all be carefully reduced to the 
 adopted standard. The reduction was less necessary in the 
 case of Mars, but was carried through all the work relating to 
 the Sun. 
 
 Comparison of observations and tables. 
 
 5. The result of each separate observation of each body was 
 compared with the tabular result thus derived. The residuals 
 were then taken and divided into groups. The interval 
 between the extreme dates of each group was always taken 
 so short that it could be presumed that the mean of all the 
 residuals would be the correction for the mean of all the dates. 
 The general rule was that the interval should not exceed four 
 or five days in the case of Mercury, or six or eight days in 
 that of Venus, and that not more than six or eight observa- 
 tions should be included in a single group. In taking these 
 means, weights were assigned to the results of each observa- 
 tory founded on the discordance of its residuals. Then to each 
 mean a weight was again assigned equal to the sum of the 
 weights of the individual residuals when these were few in 
 number, but not allowed to exceed a certain limit, how great 
 soever might be the sum of the individual weights. 
 
 Equations of condition. 
 
 6. Each mean result thus derived formed the absolute term 
 of an equation of condition for correcting the tabular elements. 
 The number of these equations was as follows: 
 
 Equations. 
 
 The Sun . 11,676 
 
 Mercury ___ 3, 929 
 
 Venus 4,849 
 
 Mars i, 597 
 
6] EQUATIONS OF CONDITION. 9 
 
 In forming the equations of condition from observations of 
 the planets, I adopted the system suggested in the introduc- 
 tion to Vol. i of these publications, namely, the determination 
 of the solar elements not only from observations of the Sun 
 itself, but from observations of each of the planets. The reason 
 for this course is quite simple and obvious. An observation of 
 the position of a planet as seen from the Earth is the exact 
 equivalent of an observation of the Earth as seen from a 
 planet, and thus depends equally upon the elements of both 
 orbits. Hence, whatever elements of the Earth's orbit could 
 be determined by observations made from a planet can equally 
 be determined by observations made upon the planet. A 
 strong reason for proceeding upon this plan was found in the 
 very large errors, both accidental and systematic, to which 
 observations of the Sun are liable. 
 
 The advantages, however, have not proved relatively so 
 great as were anticipated. The eccentricity and perihelion of 
 the. Earth's orbit come out in the solution of the normal equa- 
 tions as functions of those of the planetary orbit to so great an 
 extent that their weight is much less than that which would 
 correspond to independent determinations from the same num- 
 ber of observations. On the other hand, the determination 
 of these elements from observations of the Sun proved to be 
 much more consistent than was expected, thus indicating a 
 high degree of precision. 
 
 The case is different with the Sun's mean longitude referred 
 to the Stars. Here systematic and personal errors enter so 
 largely that the results from Mercury and Venus appear to be 
 rather more reliable than those from the Sun itself. In the 
 case of these planets it fortunately happens that the weight of 
 the result derived for the Sun's mean longitude is not mate- 
 rially diminished by the uncertainty of the corresponding 
 element of the planet, the errors of the two mean longitudes 
 being nearly separated in a series of observations equally dis- 
 tributed around the orbit. 
 
 The systematic errors in observations of the Sun rendered 
 it unadvisable to determine the elements of the Earth's orbit 
 from observations of the Sun by a single system of equations. 
 The solar observations, therefore, were classified according to 
 
10 GENERAL OUTLINE. [6 
 
 the observatory where made, and divided into periods rarely 
 exceeding eight years in length. The elements are separately 
 derived from the observations of each period. This system has 
 the advantage of eliminating to a large extent the injurious 
 effect of systematic and personal error upon the eccentricity 
 and perihelion of the Earth's orbit, and also enabling us to 
 judge of the precision of the corrections to those elements by 
 the discordance among separate results. 
 
 Meridian observations of the Sun and Planets are referred 
 to the fundamental stars, while the Eight Ascensions of the 
 latter are referred to the equinox, the position of which has 
 heretofore depended on observations of the Sun. The adopted 
 position of the fundamental stars therefore comes in, to a cer- 
 tain extent, as the basis of the work, and the constant parts 
 of their systematic corrections are among the results to be 
 derived. 
 
 Thus, in the case of the equations pertaining to the three 
 planets, the following corrections were introduced as unknown 
 quantities : 
 
 Correction of the mass of Mercury or of Venus. 
 
 Corrections to the elements of the orbit of the planet 
 observed. 
 
 Correction of the obliquity of the ecliptic. 
 
 Corrections to the Sun's mean longitude, eccentricity, and 
 longitude of perihelion. 
 
 Common corrections to the adopted Eight Ascensions and 
 Decimations of the fundamental stars. 
 
 In the case of Mercury an adopted hypothetical correction 
 of the ratio of the radius vector of the planet to that of the 
 Earth was also included in the equations, although little doubt 
 could be felt that the true value of such a quantity must be 
 zero. The reason for introducing it will be explained here- 
 after. 
 
 Determinations of the masses and secular variations. 
 
 7. The secular variation of all the preceding elements, the 
 mean distances excepted, was also introduced into the equa- 
 tions from observations of the planets. In addition to the 
 above elements, the mass of Venus appeared in the equations 
 
7] MASSES AND SECULAR VARIATIONS. 11 
 
 derived from observations of the Sun, Mercury, and Mars, and 
 the mass of Mercury in the equations derived from obser- 
 vations of Venus. The coefficients of the masses, however, 
 depended wholly upon the periodic perturbations. 
 
 Were it quite certain that the secular variations arise 
 wholly from the masses of the known planets, the masses 
 could of course be derived from these variations, and the lat- 
 ter would appear in the equations of condition only through 
 the mass itself. On this hypothesis the secular variations 
 would not appear in the equations, but only the masses. But 
 it is well known that the perihelion of Mercury is subject to a 
 secular variation which can not be accounted for by any ad- 
 missible masses of the known disturbing planets. The same 
 thing may well be true of the secular variations of the other 
 elements. It is therefore necessary, in the absence of a known 
 cause for such deviations, to derive the masses of the planets 
 independently of the secular variations. In the case of Mars 
 the mass is obtained with all necessary precision from the sat- 
 ellites. It is, however, different in the case of Mercury and 
 Venus. Here no resource is left us but to determine them 
 from the periodic inequalities. As the inequality produced by 
 Venus in the Earth's longitude is rarely more than eight sec- 
 onds, it might seem that the coefficient would be too small to 
 obtain a sufficiently precise value of the mass. But in the 
 case of observations upon the Sun, Mercury, and Mars the 
 error of the determination of the mass in question may be 
 almost indefinitely reduced by multiplication and extension 
 of the observations without danger of systematic error. 
 
 To illustrate this, let us suppose the Sun's longitude to be 
 determined with a meridian instrument only once a year, say 
 at equal intervals of three hundred and s^xty-five days. Let 
 the longitudes thus observed be compared with an ephemeris 
 in which the elements are affected with only slight errors. 
 Leaving out of consideration the periodic perturbations pro- 
 duced by the planets, the comparison of the observed longi- 
 tudes with the tabular ones through an entire century should 
 be nearly constant. Any error affecting all the longitudes 
 alike would appear as a constant. The errors of mean motion 
 
12 GENERAL OUTLINE. [7 
 
 would vary imiformly with the time. Thus the other elements 
 would be nearly constant, and could be still more approxi- 
 mately represented by a slight apparent secular variation. 
 
 Now let the disturbing action of a planet, say Venus, be in- 
 troduced. We should then have a series of deviations from the 
 law of uniform increase, which would enable us to evaluate 
 the mass of the planet. The value of this mass thus derived 
 would not be affected by any systematic error common to all 
 the observations, nor even by such an error which varied uni- 
 formly with the time. Nor would small errors in the adopted 
 elements of the Sun have any effect upon the result. 
 
 If this would be the case for observations made- only at a 
 certain point of the orbit, a fortiori would it be the case for 
 the observations made at various points of the orbit, since any 
 tendency to a systematic effect of the errors of observation 
 would thereby be ultimately eliminated. 
 
 Considerations almost identical apply to the case of observa- 
 tions upon either of the planets when we consider the action 
 of the other planet upon the planet observed and upon the 
 earth. But they do not apply to the case of the action of the 
 earth itself upon the observed planet, or vice versa. For ex- 
 ample, in the case of observations of Venus, we may suppose 
 that all observations made when Venus is at a certain point 
 of its relative orbit, near inferior conjunction, say one month 
 before inferior conjunction, are affected with a certain error 
 common to all observations made at that point of the orbit. 
 Since the perturbations produced by the third planet will in 
 the long run have all values, positive and negative, for these 
 several observations, the systematic error in question will not 
 affect the ultimate value of its mass. But the perturbations 
 of Venus produced by the Earth, as well as those of the Earth 
 produced by Venus, will not have all values in such a case, but 
 only special ones dependent on the relative position. Hence, 
 determinations of these masses might be affected by errors of 
 the kind in question. We conclude, therefore, that the mass 
 of the Earth can not be satisfactorily determined by the peri 
 odic perturbations which it produces in the motion of any 
 planet, nor that of Venus by observations on Venus through 
 its periodic perturbations of the Earth. 
 
8] TRANSITS OF VENUS AND MERCURY. 13 
 
 In the solution of the equations of condition the method of 
 least squares has been used throughout, the arrangement of 
 the work, the choice of quantities to be corrected, and the 
 accuracy of the coefficients being so chosen as to minimize the 
 great mechanical labor of making the necessary multiplica- 
 tions. The adoption of this method was necessary in order to 
 separate, so far as possible, the various unknown quantities 
 and show to what extent their values were interdependent. 
 By no other method of combination could so large a number 
 of unknown quantities have been separately determined in a 
 way which would have been at all satisfactory. On the other 
 hand, in combining the final results and deciding upon the 
 values of the corrections to be adopted, the method has not 
 always been applied, for reasons which will be developed in 
 Chapter IY. 
 
 Introduction of results of observations on transits of Venus and 
 
 Mercury. 
 
 8. In the case of Mercury and Venus the observed transits 
 over the Sun give relations between the corrections to the 
 elements more accurate than those ordinarily derivable from 
 meridian observations. This is especially the case with Venus. 
 The value of these observations is greatly increased by the 
 fact that they are made when the planet is near inferior con- 
 junction, and therefore nearest to the Earth, and in a point of 
 the relative orbit where meridian observations are necessarily 
 most uncertain. In the case of Venus the error of the helio- 
 centric place will be more than doubled in the case of the geo- 
 centric place during a transit. As, however, the observation 
 of a transit gives no one element, but only an equation of con- 
 dition between the values of all the elements at the epoch, the 
 only way of treating it is to introduce the result as such an 
 equation, with its appropriate weight. The determination of 
 the proper weight is a difficult matter. The systematic errors 
 of meridian observations are such that the theoretical value 
 of the weights assignable to so great a mass as we have dis- 
 cussed would be entirely illusory. In fact so great is the 
 weight assignable to the observed transits of Venus that if 
 we should regard the results of each transit as a condition to 
 
14 GENERAL OUTLINE. [8 
 
 be absolutely satisfied we should not be dangerously in error. 
 I conclude, therefore, that there is more danger of assigning 
 too small than too great a weight to these observations. 
 
 In order to determine what change was produced in the re- 
 sults by the use of the observed transits over the sun's disk, 
 two separate solutions of the equations of condition for Mer 
 cury and Venus were made. In the one, termed solution A, 
 the meridian observations alone were used; in the other, 
 termed solution B, the, combined equations formed by adding 
 the normal equations derived from the transits to those given 
 by the meridian observations were used. 
 
 In the case of solution A it was originally supposed that by 
 using the mean epoch of all the observing in the case of each 
 planet as that from which the time was to be reckoned, the 
 normal equations for the secular variations would be almost 
 completely separated from those for the corrections to the 
 elements themselves. The separation would be complete were 
 the observations at different epochs similarly distributed 
 around the orbit. But, as a matter of fact, it was found that 
 the accidental deviations from this symmetry were so consider 
 able that the separation could not be regarded as complete. 
 The solution was therefore made by successive approximations, 
 the terms depending on the secular variations being in the 
 first approximation dropped from the normal equations for the 
 corrections to the elements, and afterwards included when 
 approximately determined, and vice versa. 
 
 In the case of solution B, in which the transits were included, 
 such a separation did not occur, and the equations were solved 
 in the usual rigorous way for all the unknown quantities. 
 
CHAPTER II. 
 
 DISCUSSION AND RESULTS OF OBSERVATIONS OF THE 
 
 SUN. 
 
 Treatment of the Eight Ascensions. 
 
 9. The meridian observations of the Sun have been treated 
 on a system different in some points from that adopted in the 
 case of the planets. It was possible to simplify the treatment 
 by supposing that the small latitude of the Sun was always a 
 definitely known quantity, so that when the observations were 
 corrected for it the apparent motion of the Sun could be sup- 
 posed to take place along the great circle of the ecliptic. This 
 allowed the correction of the elements to depend on but two 
 quantities the obliquity of the ecliptic and the Sun's true 
 longitude. Assuming the obliquity to be known, the longi- 
 tude of the Sun could always be determined from an observa- 
 tion of its Right Ascension. An observed Eight Ascension 
 being compared with a tabular one, the residual gives rise to 
 an equation of condition between the correction of the long- 
 itude, A, of the obliquity, , and of the Right Ascension of the 
 Sun, a: 
 
 da = cos s sec 2 ddX tan e sin 2ad?. 
 
 This equation may be used to express the error of the longi- 
 tude in terms of the error of the obliquity and of the Right 
 Ascension as follows : 
 
 #A = sec 8 cos 2 dda + J tan e sin 2Xds 
 = sec s cos 2 6$a + 0.21 sin 2Xds 
 
 The elements mainly to be determined from the observations 
 in Right Ascension being the eccentricity and perihelion of 
 the Earth's orbit, each of the coefficients of which go through 
 a period in a year, the effect of the small term 0.21 $s sin 2A 
 whose coefficient does not amount to 0".10 after 1800, and has 
 a period of half a year, will be practically without influence 
 
 15 
 
16 OBSERVATIONS OF THE SUN. [10 
 
 on the result. The system was therefore adopted of deriving 
 the residual in longitude directly from the residual in Eight 
 Ascension by the formula 
 
 where 
 
 F = cos 2 3 sec e . 
 
 The residual #A in true longitude is then to be expressed in 
 terms of the residual 61" in mean longitude and of corrections 
 to the eccentricity and to the longitude of the perigee relative 
 to the Stars. In this expression the coefficient of the residual 
 in mean longitude was always taken as unity, the value of the 
 correction being so small in the case of LEVERRIER'S tables 
 that no appreciable error would result from this supposition. 
 Thus each residual in Eight Ascension would give rise to an 
 equation of condition of the form 
 
 61" + Pe"6n" + E6e" = tfA = ~F6a 
 
 We are here to regard 61" and 6n" as corrections to the 
 Eight Ascensions relative to the clock stars, and not to the 
 Sun's longitude or perigee simply. I shall therefore use the 
 symbol c instead of 61" to express the relative correction here- 
 after. 
 
 Treatment of the Declinations. 
 
 10. The declination of the Sun in the case supposed is a 
 function only of the longitude and obliquity. The equation 
 for expressing the observed correction in Declination in terms 
 of the corrections to these two quantities is 
 
 Ad = sin a6s -f cos a sin sSX 
 
 Thus each observation of the Sun's Declination gives rise to 
 an equation of condition of this form. 
 
 It is however to be supposed that the observations in Decli- 
 nation made at each observatory will be affected by a constant 
 error. If the observations are truly reduced to the standard 
 system of star places, this error will be that of the standard 
 system. As a matter of fact, however, observations made in 
 the daytime, especially on the Sun and at noon, are made 
 under circumstances so different from night observations on 
 
11] FORMATION OF EQUATIONS IN RIGHT ASCENSIONS. 17 
 
 stars that we can not assume the error of the reduced declina- 
 tion to be necessarily the same as that of the star system. 
 We must, therefore, in each case, regard the constant error in 
 declination as something peculiar to the observatory and the 
 instrument, which may or may not be worthy of subsequent 
 discussion. Thus each residual in declination gives rise to 
 an equation of condition, 
 
 j# o 4. cos a sin (U -f sin ade = AS 
 
 Ad being the excess of observed over tabular declination, 
 and Ad Q the common error of all the measured declinations of 
 any one series. 
 
 Formation of the equations from Right Ascensions* 
 
 11. The method of treating the observed Eight Ascensions 
 of the Sun was suggested by the fact that they are peculiarly 
 liable to systematic and personal errors ; the former likely to 
 change with the seasons, and to be different for different in- 
 struments; and the latter to continue through the work of one 
 observer. It is now well understood that the observed Eight 
 Ascensions of the mean of the Sun's two limbs relative to the 
 fixed stars are affected by personal errors, no means of elimi- 
 nating which have yet been tried. In a series of observations 
 made by a. single observer, under uniform conditions, this error 
 would systematically affect only the relative mean of the Eight 
 Ascensions of the Sun and Stars, leaving the eccentricity and 
 perigee derived from the observations substantially correct. 
 
 On taking up the work it was also supposed that, owing to 
 the different effect of the Sun's rays upon the instrument at 
 different seasons, and the different circumstances under which 
 observations were made, the Eight Ascensions of the Sun 
 would be affected by errors varying in a regular way through 
 the year, but not wholly expressible as a term of single annual 
 period. It was therefore deemed best to consider the observa- 
 tions possibly affected by an error of double period, having the 
 form 
 
 x' cos 2g -f y' sin 2g 
 
 5690 N ALM 2 
 
18 OBSERVATIONS OF THE SUN. [11 
 
 The introduction of the coefficients x 1 and y' added two more 
 terms to the equations of condition, which terms, however, did 
 not express any astronomical fact, but only the possible errors 
 of the observations. 
 
 An additional and very important element to be determined 
 from the observed Eight Ascensions was the mass of Venus. 
 The question now arose whether, by a uniform series of obser- 
 vations, extending through some definite period, the correc- 
 tions to the eccentricity and perigee and the coefficients x 1 and 
 y 1 could be completely separated from the coefficients of the 
 correction to the mass of Venus. Examination showed that 
 from such a series of observations, extending through eight 
 years, the mass of Venus could be determined irrespective of 
 all systematic errors repeating themselves with the season, 
 provided that the observations were equally distributed 
 throughout the year, or even that an equal number were made 
 at the same time through successive years. As neither of 
 these conditions are practically fulfilled it was judged best to 
 assume in the beginning that the systematic errors of an un- 
 known kind repeated themselves at each season during an 
 eight-year period, and that they could be expressed in the 
 form 
 
 c + x cos g H- y sin g + x 1 cos. 2g + y' sin 2g 
 
 x and y would appear as errors of eccentricity and perigee 
 which could not be eliminated. 
 
 The quantities actually introduced as the uuknown ones of 
 the equations of condition were as follows: 
 
 X, the factor of correction of the mass of Venus j 
 #, one-fifth the correction to the eccentricity; 
 y, one-fifth the correction e"dn" ; 
 
 x',y', one-tenth the coefficients expressing the .supposed 
 error of double period arising from all causes whatever ; 
 
 c, the constant correction to the Eight Ascension of the 
 Sun relative to the Stars. 
 
 The coefficient of c was supposed unity throughout. The 
 reduction of the residual in Eight Ascension to that in Longi- 
 tude and the other factors were taken from a table like the 
 following, of which the argument was the day of the year. 
 

 
 
 
 
 11] FORMATION OF EQUATIONS IN EIGHT ASCENSION. 19 
 
 Separate tables were constructed for 1802 and 1850, but they 
 were so nearly identical that no distinction need be made 
 between them. Furthermore, the error introduced by sup- 
 posing the mean anomaly to have the same value on the same 
 day of every year is entirely unimportant. 
 
 Table of coefficients for expressing errors of the Sun's Right 
 Ascension in terms of errors of the elements of the Earth's 
 orbit. 
 
 
 da 
 dl 
 
 dl 
 
 Coefficients of 
 
 da 
 
 ,=0^ 
 
 *-** 
 
 f 
 
 y' 
 
 Jan. I 
 
 1.09 
 
 o. 91 
 
 4- O.I 
 
 10. 
 
 -[- o. i 
 
 4-10.0 
 
 II 
 
 1.07 
 
 0-93 
 
 1.8 
 
 9.8 
 
 3-5 
 
 9-4 
 
 21 
 
 1.04 
 
 o. 96 
 
 3-4 
 
 9.4 
 
 6-5 
 
 7.6 
 
 31 
 
 I. OI 
 
 0.98 
 
 
 8.7 
 
 8.7 
 
 
 Feb. 10 
 
 0.98 
 
 I. OI 
 
 6.' 4 
 
 7.7 
 
 9.8 
 
 4- \\ 
 
 20 
 
 o. 96 
 
 .04 
 
 + 7.6 
 
 -6.5 
 
 + 9-9 
 
 - 1.6 
 
 Mar. 2 
 
 0.94 
 
 .06 
 
 8.6 
 
 5.1 
 
 8.7 
 
 4.9 
 
 12 
 
 o. 92 
 
 .08 
 
 9.4 
 
 3-5 
 
 6.6 
 
 7.5 
 
 22 
 
 o. 92 
 
 .08 
 
 9-8 
 
 1.9 
 
 3-7 
 
 9-3 
 
 Apr. I 
 
 -93 
 
 .07 
 
 IO. O 
 
 O. I 
 
 + 0.3 
 
 IO. O 
 
 II 
 
 0.94 
 
 .05 
 
 + 9-9 
 
 4- 1.6 
 
 *> T 
 
 o* * 
 
 - 9-5 
 
 21 
 
 o. 96 
 
 . 03 
 
 9-5 
 
 3-2 
 
 6. i 
 
 7-9 
 
 May i 
 
 0.99 
 
 . OI 
 
 8.8 
 
 4.8 
 
 8.4 
 
 5-4 
 
 ii 
 
 I. 02 
 
 0.98 
 
 7-8 
 
 6.2 
 
 9-7 
 
 2. 2 
 
 21 
 
 1.05 
 
 0.95 
 
 6.6 
 
 7-5 
 
 9.9 
 
 I. 2 
 
 31 
 
 1.07 
 
 0.93 
 
 + 5.3 
 
 + 8.5 
 
 - 8.9 
 
 4-5 
 
 June 10 
 
 1.09 
 
 o. 91 
 
 3-7 
 
 9-3 
 
 6.9 
 
 7.2 
 
 20 
 
 I. 10 
 
 o. 91 
 
 2. I 
 
 9.8 
 
 4.1 
 
 9.1 
 
 30- 
 
 1.09 
 
 0.91 
 
 + 0.4 
 
 IO. O 
 
 - 0.7 
 
 10. 
 
 July 10 
 
 1. 08 
 
 0.93 
 
 
 9.9 
 
 + 2.7 
 
 9.6 
 
 20 
 
 1.05 
 
 0.95 
 
 3.0 
 
 + 9-5 
 
 + 5.8 
 
 - 8.2 
 
 30 
 
 1.03 
 
 0.07 
 
 4.6 
 
 8.9 
 
 8,2 
 
 5-7 
 
 Aug. 9 
 
 I. OO 
 
 . OO 
 
 6. i 
 
 8.0 
 
 9-6 
 
 + 2.7 
 
 19 
 
 0.97 
 
 03 
 
 7-3 
 
 6.8 
 
 10. 
 
 0.8 
 
 29 
 
 -95 
 
 05 
 
 8.4 
 
 5-4 
 
 9.1 
 
 4. i 
 
 Sept. 8 
 
 0-93 
 
 .07 
 
 9.2 
 
 + 39 
 
 + 7.2 
 
 -6.9 
 
 18 
 
 0.92 
 
 .08 
 
 9-7 
 
 2-3 
 
 4.5 
 
 8.9 
 
 28.... 
 
 o. 92 
 
 .08 
 
 10. 
 
 + 0.6 
 
 4- 1.2 
 
 9-9 
 
 Oct. 8... 
 
 0-93 
 
 .07 
 
 9.9 
 
 1. 1 
 
 2. 2 
 
 9-7 
 
 18 
 
 o-9S 
 
 05 
 
 9.6 
 
 2.8 
 
 5-4 
 
 8.4 
 
 28.... 
 
 0.97 
 
 I. 02 
 
 9.0 
 
 4.4 
 
 7.9 
 
 6.1 
 
 Nov. 7 
 
 I. 00 
 
 0.99 
 
 8.1 
 
 5.9 
 
 9-5 
 
 - 3-1 
 
 17 
 
 1.03 
 
 o. 96 
 
 7.o 
 
 7.2 
 
 IO..O 
 
 + 0.3 
 
 Dec. 7~~" 
 
 .06 
 .08 
 
 0.94 
 o. 92 
 
 5.6 
 4- i 
 
 8.3 
 
 9.1 
 
 9-3 
 
 7-5 
 
 3.7 
 6.6 
 
 17 
 
 .09 
 
 o. 91 
 
 - 2. 5 
 
 9-7 
 
 4-9 
 
 + 8.7 
 
 27.... 
 
 .09 
 
 o. 91 
 
 -0.8 
 
 IO. O 
 
 1.6 
 
 + 9-9 
 
20 OBSERVATIONS OF THE SUN. [12 
 
 Finally, throughout the work the equations of condition 
 were expressed only in entire numbers, the decimals being 
 neglected. To lessen the number of equations of condition, 
 the residuals were divided into groups generally covering from 
 ten to fifteen days, the length of the group being determined 
 by the condition that the perturbations of Venus must not 
 change much during the period. 
 
 While the formation and solution of the equations of condi- 
 tion on this system were going on, it was found that the intro- 
 duction of the assumed coefficients x 1 and y' was a refinement 
 productive of little or no good result. In fact, the observa- 
 tions of the Sun proved to be much freer from annual sources 
 of error than I had supposed, as will be seen by the tables of 
 their results soon to be given. This is shown by the general 
 consistency of the corrections to the eccentricity and perigee 
 given by the work at the same or different observatories dur- 
 ing different periods. 
 
 In marked contrast to this is the discordance among values 
 of the correction c to the relative Eight Ascensions of the Sun 
 and Stars. This quantity it is that is affected by personal 
 error and possibly by the effect of the Sun on the instrument. 
 Under a perfect system of discussion it would be advisable to 
 determine it separately for each observer. This however was 
 practically impossible. 
 
 Solution of the equations. 
 
 12. For the purposes of forming and solving the normal 
 equations, the equations of condition were divided into groups 
 of generally from four to eight years, the exact lengths of 
 which will be seen from the following exhibit of results. The 
 equations for each period were solved on the supposition that 
 the corrections were constant during the period. Thus every 
 separate result is independent of every other, except so far as 
 they may depend on the same instrument or the same observer 
 at different times. 
 
 The first column shows the years through which the obser- 
 vations extend. 
 
 The second one shows to the nearest year the value of T 
 that is, the fraction of the century after 1850. 
 
12] SOLUTION OF THE EQUATIONS. 21 
 
 The third column shows the value of //, or that factor which, 
 being multiplied by the adopted mass of Venus, is to be applied 
 as a correction to that mass, to obtain the value given by the 
 observations. 
 
 All systematic errors arising from the instrument and the 
 observer are so completely eliminated from the separate de- 
 terminations of X that they may be regarded as absolutely 
 independent of each other, that is as not affected by any 
 common systematic error. 
 
 We have next the relative weight assigned to each value 
 of //, which is determined in the usual way from the solu- 
 tion, and is, therefore, on a different scale for different ob- 
 servatories. 
 
 Next is given the value of c, or the apparent correction to 
 the Eight Ascension of the Sun, relative to the assumed Eight 
 Ascensions of the Stars, as given by observations during the 
 several periods and expressed in seconds of arc, followed by 
 the weights assigned to the separate results. 
 
 The next two columns, the corrections to the solar eccen- 
 tricity and to the longitude of the perigee, require no further 
 explanation. 
 
 Eespecting the weights ultimately assigned to these quanti- 
 ties, and to GJ it is to be remarked that they are the result of 
 judgment more than of computation. It is only possible to 
 enumerate in a general way with some examples the consider 
 ations on which they are based. 
 
 In assigning the weight of c the number of observers en- 
 gaged is an important factor in determining it. Other factors 
 are the steadiness of the atmosphere and the adaptation of the 
 instrument to this particular work. General consistency is 
 an important factor in the assignment. In this respect the 
 Cambridge observations are quite remarkable ; if their excel- 
 lence corresponds to their consistency they must be the best 
 ones made. 
 
 It will be seen that PIAZZI'S results are thrown out en- 
 tirely. The wide range of his values of c led to the inquiry 
 whether more consistent results would be obtained by taking 
 shorter periods, but it was found that the values of c varied 
 from time to time in such an irregular way that his instrument 
 
22 
 
 OBSERVATIONS OF THE SUN. 
 
 [12 
 
 must have been affected by some extraordinary cause of error, 
 unless some mistake has been made in interpreting or treating 
 the observations. 
 
 The Oxford values of c are unusually discordant. The pre- 
 sumption that this discordance arises mainly from the special 
 personal equation in observations of the Sun, described on 
 page 17, derives additional weight from the greater relative 
 consistency of the values of 6e" and e"dn". I have therefore 
 allowed the values of these quantities to receive a fair weight. 
 
 The value of c for Paris, 1866-'70, has received a much re- 
 duced weight, solely on account of its excessive value. It 
 seems that the work of one observer who made many observa- 
 tions during this period was affected by an unusual system- 
 atic error. 
 
 Results of observations of the Sun's Right Ascension. 
 
 GREENWICH. 
 
 Years. 
 
 T 
 
 P' 
 
 w 
 
 c 
 
 W 
 
 6e" 
 
 *"<Jir" 
 
 za 
 
 !750-'62 
 
 -94 
 
 .027 
 
 20 
 
 +o'-33 
 
 i-S 
 
 a 
 +0.04 
 
 0.42 
 
 2 
 
 I765-'7I 
 
 .82 
 
 .041 
 
 10 
 
 +0-37 
 
 o-5 
 
 0.08 
 
 o. 64 
 
 I 
 
 1772-^8 
 
 -75 
 
 .022 
 
 10 
 
 +0.74 
 
 o-5 
 
 o. 16 
 
 0.49 
 
 I 
 
 '779-'85 
 
 -.68 
 
 -35 
 
 IJ 
 
 +2.89 
 
 0. 2 
 
 o. 1 8 
 
 o. 73 
 
 -5 
 
 ij86-'()2 
 
 -.61 
 
 037 
 
 8 
 
 + i-5i 
 
 0. 2 
 
 0. 12 
 
 0.88 
 
 o 
 
 i793-'97 
 
 -55 
 
 . 114 
 
 5 
 
 + 1.87 
 
 0. 2 
 
 O. 22 
 
 1.27 
 
 
 
 i798-'o2 
 
 So 
 
 +. 060 
 
 5 
 
 + i. 02 
 
 O. 2 
 
 o. 42 
 
 i. 15 
 
 
 
 i8o3-'o6 
 
 45 
 
 . 002 
 
 5 
 
 +0.27 
 
 O. 2 
 
 o. 03 
 
 1.03 
 
 o 
 
 1807-' 10 
 
 .41 
 
 .068 
 
 5 
 
 0.34 
 
 O. 2 
 
 -0.32 
 
 I. 12 
 
 o 
 
 i8ii_'i4 
 
 37 
 
 095 
 
 3 
 
 -3-33 
 
 0. 2 
 
 +0.17 
 
 -I 08 
 
 o 
 
 i8i5-'i8 
 
 -33 
 
 -.052 
 
 6 
 
 1.99 
 
 o-5 
 
 0. 12 
 
 0-34 
 
 
 
 l8l9-'22 
 
 .29 
 
 + .010 
 
 6 
 
 0.51 
 
 
 +0. 22 
 
 o. 19 
 
 I 
 
 1823-^6 
 
 25 
 
 054 
 
 6 
 
 i. 08 
 
 
 +0.05 
 
 o. 17 
 
 I 
 
 i827-'30 
 
 . 21 
 
 .045 
 
 6 
 
 0.42 
 
 
 o. 09 
 
 0-75 
 
 I 
 
 i83i-'34 
 
 17 
 
 +.016 
 
 7 
 
 +0.76 
 
 
 +0.04 
 
 o. 27 
 
 I 
 
 1835-38 
 
 !3 
 
 + . O2O 
 
 8 
 
 4*1. 16 
 
 
 +0.26 
 
 +0.06 
 
 2 
 
 1839-42 
 
 .09 
 
 + .061 
 
 8 
 
 +0.84 
 
 
 +0.32 
 
 +o. 10 
 
 2 
 
 1843-46 
 
 05 
 
 -.008 
 
 8 
 
 +0.15 
 
 2 
 
 +0.25 
 
 +0. 22 
 
 2 
 
 i847-'5o 
 
 .01 
 
 .045 
 
 8 
 
 0. 10 
 
 2 
 
 +0. 28 
 
 +0.02 
 
 3 
 
 i8si-'54 
 
 +.03 
 
 +.024 
 
 8 
 
 +0.40 
 
 3 
 
 +O. 22 
 
 -j-o. 02 
 
 3 
 
 1855-58 
 
 +.07 
 
 .032 
 
 9 
 
 +0.36 
 
 3 
 
 +0.15 
 
 +O. O2 
 
 3 
 
 i859-'62 
 
 +.11 
 
 .043 
 
 
 0.02 
 
 3 
 
 + 0.25 
 
 +0. 22 
 
 4 
 
 i863-'66 
 
 +.15 
 
 .Ol6 
 
 8 
 
 +0-31 
 
 3 
 
 +0.23 
 
 0.05 
 
 4 
 
 i867-'7o 
 
 +.19 
 
 + .031 
 
 8 
 
 +Q-35 
 
 3 
 
 +0-33 
 
 O. IO 
 
 4 
 
 i8 7 i-' 74 
 
 +.23 
 
 + .021 
 
 8 
 
 +0. 12 
 
 3 
 
 +0.24 
 
 +q.os 
 
 4 
 
 1875-78 
 
 +.27 
 
 .008 
 
 8 
 
 . 0. 12 
 
 3 
 
 +0.26 
 
 +0.06 
 
 4 
 
 i879-'82 
 
 +31 
 
 +.017 
 
 8 
 
 o. 05 
 
 3 
 
 +0. 21 
 
 +o. 14 
 
 4 
 
 i883-'88 
 
 +.36 
 
 + . OOI 
 
 13 
 
 0. 20 
 
 3 
 
 +0.18 
 
 +0.07 
 
 4 
 
 i889-'92 
 
 +.41 
 
 .025 
 
 8 
 
 0.44 
 
 2 
 
 +o. 24 
 
 +0. II 
 
 3 
 
12J SOLUTION OF THE EQUATIONS. 23 
 
 Results of observations of the Sun's Right Ascension Continued. 
 
 PARIS. 
 
 Years. 
 
 T 
 
 /"' 
 
 w 
 
 c 
 
 w 
 
 6e" 
 
 e"fa" 
 
 W 
 
 iSoi-'oy 
 
 -.46 
 
 .025 
 
 H 
 
 _i'/ 7 8 
 
 0-5 
 
 +o'.'o8 
 
 n 
 0.23 
 
 
 1 808-' 1 5 
 
 -38 
 
 +.015 
 
 17 
 
 -0.65 
 
 o-5 
 
 O. OI 
 
 +O. 12 
 
 
 1816-22 
 
 -3i 
 
 . 050 
 
 H 
 
 +o. 18 
 
 0-5 
 
 o. 13 
 
 +0.32 
 
 
 i823-'29 
 
 .24 
 
 .050 
 
 10 
 
 -j-O. 01 
 
 0-5 
 
 0.31 
 
 O. O2 
 
 
 i837-'44 
 
 .09 
 
 -.034 
 
 19 
 
 +Q-33 
 
 i 
 
 o. 04 
 
 +O. IO 
 
 5 
 
 i8 4 5-'52 
 
 + .01 
 
 -f-. 009 
 
 15 
 
 +0. 10 
 
 
 +0.04 
 
 +o. 10 
 
 5 
 
 i853-'59 
 
 +.06 
 
 +.014 
 
 15 
 
 +0.66 
 
 
 o. 04 
 
 +0.32 
 
 2 
 
 i86o-'65 
 
 +-I3 
 
 +.003 
 
 10 
 
 +0.38 
 
 
 +0.07 
 
 +o. 26 
 
 2 
 
 i866-'7o 
 
 +. 18 
 
 .000 
 
 7 
 
 +2.29 
 
 . 3 
 
 +- I 3 
 
 +0.40 
 
 2 
 
 i87i-'79 
 
 + 25 
 
 +.048 
 
 ii 
 
 o. 26 
 
 
 o. 06 
 
 +O. 22 
 
 2 
 
 1 880-' 89 
 
 + 35 
 
 -[-. 002 
 
 14 
 
 + 0.44 
 
 
 +0.24 
 
 -i-o. 03 
 
 2 
 
 PALERMO. 
 
 i79i-'96 
 
 -.56 
 
 .079 
 
 
 
 o. 07 
 
 o 
 
 o. 06 
 
 o'.'ss 
 
 
 
 i797-'oi 
 
 -51 
 
 . 116 
 
 
 
 2.33 
 
 o 
 
 o. 29 
 
 o. 28 
 
 
 
 i8o2-'o5 
 
 -.46 
 
 . OOI 
 
 
 
 3-" 
 
 o 
 
 -0.05 
 
 o. 76 
 
 
 
 1 806-' 1 2 
 
 .41 
 
 +243 
 
 o 
 
 4-5-92 
 
 o 
 
 1.17 
 
 + 1-55 
 
 o 
 
 CAMBRIDGE. 
 
 1 828-' 34 
 
 .21 
 
 + .007 
 
 16 
 
 o'.'i3 
 
 2 
 
 +0.08 
 
 +0.12 
 
 4 
 
 i835-'40 
 
 . 12 
 
 -033 
 
 H 
 
 o. 1 8 
 
 2 
 
 +0.06 
 
 0. 06 
 
 4 
 
 i8|2-'47 
 
 -05 
 
 . 026 
 
 9 
 
 O. 21 
 
 2 
 
 + 0.08 
 
 O. 12 
 
 4 
 
 i8so-'58 
 
 + .0 4 
 
 . O24 
 
 20 
 
 0. II 
 
 2 
 
 +0.17 
 
 0.04 
 
 4 
 
 WASHINGTON. 
 
 1 846-' 5 2 
 
 . 01 
 
 -.038 
 
 5 
 
 -o."8 5 
 
 2 
 
 +0. 20 
 
 o. oo 
 
 3 
 
 i86i-'6s 
 
 +-I3 
 
 -.038 
 
 8 
 
 -0.53 
 
 4 
 
 + O OI 
 
 o. oo 
 
 5 
 
 i866-' 73 
 
 +.20 
 
 .004 
 
 13 
 
 0. 22 
 
 4 
 
 +o. 18 
 
 o. 03 
 
 6 
 
 i8 74 -'8i 
 
 + .28 
 
 -033 
 
 12 
 
 -0.45 
 
 4 
 
 +0.07 
 
 o. 16 
 
 5 
 
 i882-'9i 
 
 + 37 
 
 . 002 
 
 17 
 
 0.79 
 
 4 
 
 +0.07 
 
 o. 07 
 
 5 
 
 KONIGSBERG. 
 
 
 
 
 
 
 
 
 II 
 
 
 i8i6-'23 
 
 3 
 
 + .002 
 
 '3 
 
 +0.30 
 
 I 
 
 +0.07 
 
 0.28 
 
 3 
 
 1824-^0 
 
 23 
 
 .006 
 
 12 
 
 +O. O2 
 
 I 
 
 o. 1 6 
 
 + O. II 
 
 3 
 
 i8 3 i-' 3 8 
 
 - 15 
 
 . O2I 
 
 15 
 
 +0.23 
 
 I 
 
 O. 12 
 
 +0.03 
 
 3 
 
 1 839-45 
 
 .08 
 
 . O2 1 
 
 12 
 
 +0.77 
 
 1 
 
 +0.08 
 
 +o. 20 
 
 3 
 
24 OBSERVATIONS OF THE SUN. [13 
 
 Results of observations of the Sun's Right Ascension Continued. 
 
 OXFORD. 
 
 ! 
 
 
 
 j 
 
 Years. 
 
 T S 
 
 w 
 
 e 
 
 w 
 
 fe" 
 
 e"W 
 
 w 
 
 i840-'49 
 
 05 
 
 .043 
 
 12 
 
 a 
 +2.49 
 
 -3 
 
 +0.24 
 
 0/17 
 
 2 
 
 i86o-'68 
 
 +.14 
 
 +.042 
 
 13 
 
 + 1.96 
 
 o-3 
 
 4-0.08 
 
 o. 13 
 
 2 
 
 1 869-' 76 
 
 +.23 
 
 + 054 
 
 IS 
 
 4-0. 92 
 
 0-3 
 
 4-0. 20 
 
 o. 04 
 
 2 
 
 i88o-'87 
 
 +.34 
 
 -.014 
 
 9 
 
 0.31 
 
 o-3 
 
 4-0.27 
 
 4-0.64 
 
 2 
 
 PULKOWA. 
 
 1 842-' 50 
 
 i86i-'7o 
 
 .04 
 4-.i6 
 
 4-. 047 
 4-. 002 
 
 II 
 
 10 
 
 + i!'a 
 0.40 
 
 i 
 
 0. 12 
 
 4-0.05 
 
 4-0. 20 
 4-0.28 
 
 3 
 
 3 
 
 DORPAT. 
 
 I82 3 -' 3 
 
 i8 3 i-' 3 8 
 
 -23 
 -15 
 
 4-. 021 
 
 + .008 
 
 I 
 
 4-0. 36 
 +0.45 
 
 ! 
 
 0. 12 
 
 4-0.02 
 
 ii 
 O. 22 
 +0.03 
 
 2 
 2 
 
 CAPE OF GOOD HOPE. 
 
 1 884-' 90 
 
 +37 
 
 . 026 
 
 12 
 
 o. 36 
 
 3 
 
 -f-o! 02 
 
 +0. OI 
 
 4 
 
 STRASSBURG. 
 
 i883-'88 
 
 +.36 
 
 .014 
 
 12 
 
 -t!'s 
 
 2 
 
 4-0. 23 
 
 -fO/09 
 
 3 
 
 The mass of Venus. 
 
 13. The mean results for the mass of Venus given by the 
 work at the several observatories are shown as follows: 
 
 The probable error, where given at all, is that derived from 
 the discordance of the separate individual results at the par- 
 ticular observatory. In some cases there are only one or two 
 results; here no probable error could be assigned. 
 
 w' is the sum of the weights of the result at each separate 
 observatory, as given by the equations of condition. Were 
 all the observations of equal accuracy, these would be the 
 weights to be assigned to the separate results. Such not be- 
 
14] 
 
 CORRECTIONS OF RELATIVE RIGHT ASCENSIONS. 
 
 25 
 
 ing the case, we choose for the actual weights certain numbers, 
 founded partly on a compromise between the mean errors fol- 
 lowing each result or upon the values of iv 1 , partly on a judg- 
 ment of the accuracy of the observations. 
 
 Values of //' for the mass of Venus. 
 
 
 / 
 
 w 
 
 W 
 
 Greenwich 
 
 . OI5-J-. 006 
 
 226 
 
 II 
 
 Paris ._. _ _ __ _ 
 
 . OO7-4-. OOQ 
 
 146 
 
 c 
 
 Konigsberg ._ 
 
 . OI2-J-. OIO 
 
 C2 
 
 7 
 
 Cambridge 
 
 . 018^. 009 
 
 Co 
 
 6 
 
 Dorpat _ 
 
 +. 016 
 
 1C 
 
 i 
 
 Pulkowa 
 
 +- 02 5 
 
 21 
 
 i 
 
 Oxford _ . . 
 
 4-. oiA-L. 023 
 
 49 
 
 i 
 
 ^^ashington 
 
 . oi84-. OOQ 
 
 cc 
 
 4 
 
 Cape 
 
 . 026 
 
 12 
 
 I 
 
 Strassburg _ 
 
 . 014 
 
 12 
 
 I 
 
 
 
 
 
 Using the weights in the last column, we have for the mean 
 result 
 
 fi = - .0118 .0034. 
 
 The mean error i .0034 is that given by the discordance of 
 the separate results of the preceding table. 
 
 Corrections of relative Right Ascensions. 
 
 14. The true values of the remaining quantities c, 6e", and 
 e"Sn" are to be regarded as increasing uniformly with the 
 time and therefore of the form 
 
 x+Ty. 
 
 Here T is the time, and in the treatment of these particular 
 equations it is counted from 1850 in units of one century, so 
 that x is the value of the correction at this mean epoch. 
 
 The quantity designated by c is the same which, elsewhere 
 in this discussion, is represented by dl + a, so that 
 
 c = dl" -f a 
 
 I shall, however, for convenience, continue to use the designa- 
 tion c, or #+T y. 
 
26 OBSERVATIONS OF THE SUN. [14 
 
 As the observations at Greenwich and Paris extend over 
 longer periods than at any other observatories, I shall first 
 solve them separately. The totality of the Greenwich obser- 
 vations give for c the following normal equations and solution : 
 
 43.4 x + 1.65 y= + 4".23 
 1.65 4- 4.24 = - I". 25 
 
 x = 4 0".ll 
 y = - 0".34 
 
 Those at Paris give the equations and solution 
 
 8.3 x + 0.04 y= + 1".22 
 0.04 4- 0.48 == 4- 0".77 
 
 x = 4- 0".14 
 y = 4 1". 59 
 
 If we combine all the other results into a single set of normal 
 equations, we have 
 
 40.2 # + 4.26 T/ = -10".84 
 4.26 4- 2.20 = - 3". 98 
 
 x=- 0".10 
 
 It will be seen that the results for t/, the secular motion, are 
 markedly discordant. Indeed, if we refer to the exhibit of 
 results, p. 23, we shall see that the values of c are much more 
 discordant than those of the other two quantities. To obtain 
 a definite value, founded on all the observations of the Sun's 
 Eight Ascension, I do not see that any better result can be 
 obtained than that found from a general solution of the com- 
 bined normal equations. The equations and their solution are 
 as follows : 
 
 91.9 #+ 5.95 # = -5".39 
 5.95 + 6.92 = - 4".46 
 
 x = - 0".02 
 y 0".63 
 or 
 
 61" 4- a = - 0".02 - 0".63T 
 
15] CORK. TO THE SOLAR ECCENTRICITY AND PERIGEE. 27 
 
 Corrections to the solar eccentricity and perigee. 
 
 15. 1 have already mentioned the remarkable consistency 
 of the corrections to these elements given by the results at 
 different observatories and at different epochs. The eccen- 
 tricity is more consistent than the perigee. One cause for 
 this, the consideration of which will throw some light on the 
 relative merits of the observations, is that the error of Bight 
 Ascension depending on the Declination of the object observed 
 effects the eccentricity less than the perigee. It is well known, 
 from a comparison of the results, that the systematic differ- 
 ences in the Eight Ascensions of different star catalogues 
 vary somewhat with the Declination. Now, since the Sun's 
 Declination goes through an annual period, it follows that this 
 error will produce a systematic effect on both the eccentricity 
 and the perigee. But the effect will be much larger in the 
 case of the latter element than in the case of the former, 
 because of the nearness of the perigee to the winter solstice, 
 the difference being only some 10 or 12. Consequently the 
 extreme coefficients in the correction to the eccentricity have 
 nearly the same values, with opposite signs, for the same Decli- 
 nations in different seasons of the year. But it is different 
 with the perigee. The coefficient of this quantity is negative 
 from October until March, when the Sun is in south Declina- 
 tion, attaining its maximum value about January 1; while it 
 is positive during the remaining months when the Sun's Decli- 
 nation is north, attaining its maximum value about July 1. 
 A systematic difference in the errors of Eight Ascension will 
 therefore produce its full effect on the longitude of the perigee, 
 while its effect on the eccentricity will be but slight. 
 
 In this connection, the very large negative values of the cor- 
 rection to the perigee during the period when the old Green- 
 wich transit instrument was in use are quite remarkable. 
 The progressive change in the value of c is also remarkable in 
 this connection. It is to be remarked that the new transit was 
 mounted in 1816, but account was not taken of this fact in 
 grouping the equations. Hence it is only from the year 1819 
 that the results of the table are derived wholly from observa- 
 tions with the new instrument. The anomaly alluded to is 
 
28 OBSERVATIONS OF THE SUN. [15 
 
 then seen to disappear. The fact that the abnormally large 
 corrections in c are positive before 1800 and negative after it, 
 while e" dn" is abnormally negative through the doubtful 
 period 1765-1815, complicates the theory of these errors. I 
 have not been able to consider them in detail, but have simply 
 rejected the results for de" and e" 6n" from 1786 to 1818, hav- 
 ing given them a gradually diminishing weight from BRAD- 
 LEY'S observations to the first epoch. 
 
 As in the case of c, I have made a solution for Greenwich 
 alone, Paris alone, the other observatories combined, and all 
 combined. The results are shown as follows : 
 
 1. From Greenwich observations : 
 
 8e" e"8Tt" 
 
 54.5# + 2.73s/ = + HM4; - 0".88 
 
 2.73 + 5.72 = -}- 1".82$ + 2".69 
 x= + 0".19; -0".04 
 y= + 0".22 ; + 0".49 
 
 2. From Paris observations : 
 
 de" e"dn" 
 
 17.0#+ 0.39 y= + 0".30; + 2".95 ' 
 
 0.39 + 0.99 = + 0".29; + 0".33 
 x= + // .01; +OM7 
 y= + 0^.29 ; + 0".27 
 
 3. The equations and results from all the other modern 
 observations are 
 
 de" e"8n" 
 77.0#+ 4.99# = + 5". 58; + 0".35 
 
 4.99 + 3.68 = + 1".09; + 0".40 
 
 y= +0".22; 
 
16] RESULTS OF OBSERVED DECLINATIONS OF THE SUN. 29 
 
 4. Finally, if we combine all the equations, we have 
 
 de" e"d7t" 
 
 US.Zx -f 8.11 y = + 17".02; + 2"A2 
 
 8.1 -4-10.39 =+ 3". 20-, + 3".42 
 
 a? =4. 0".10; 0".00 
 
 y=+ 0".23; + 0".33 
 
 In the case of the eccentricity the general accordance is 
 quite satisfactory, and for the perigee it is much better than 
 in the case c, the relative Eight Ascension. 
 
 Results of observed declinations of the Sun. 
 
 16. The Sun's absolute longitude can be found only from 
 observations of his declination, because this longitude is 
 referred to the equinox, which is defined only by the Sun's 
 crossing of the equator. 
 
 The corrections to the eccentricity and perigee, as just found, 
 are so slight that they may be neglected in determining the 
 correction of the absolute longitude from that of the declina- 
 tion. Thus, as already stated, the unknown quantities of the 
 equations given by the declinations are the corrections of the 
 mean longitude Z", and of the obliquity f, and a constant A6^ 
 peculiar to each observatory, of which we take no further 
 account. The equation of condition given by each observa- 
 tion or group of observations is 
 
 Ad 4- A sin s 61" + Bfo = dd 
 
 where dd is the excess of the observed over the tabular decli 
 nation, and 
 
 A = cosec e = cos a 
 
 B 
 
30 
 
 OBSERVATIONS OF THE SUN. 
 
 [16 
 
 The equations are grouped and solved for periods, as in the 
 case of the Eight Ascensions, with the results shown in the 
 following table: 
 
 Results of observations of the Sun's Declination. 
 
 GREENWICH. 
 
 Years. 
 
 T 
 
 <?/" 
 
 w 
 
 6. 
 
 IV 
 
 * 
 
 ,. 
 
 w 
 
 1753-57 
 i758-'62 
 
 -95 
 .90 
 
 +0.78 
 + I -5 
 
 
 0-34 
 
 1.81 
 
 I 
 I 
 
 2.43 
 1.94 
 
 0-34 
 1.81 
 
 I 
 
 I 
 
 1765-^0 
 
 .82 
 
 0.23 
 
 
 o. 95 
 
 0-5 
 
 +o. 20 
 
 0-95 
 
 o. 5 
 
 i77i-'78 
 
 . 75 
 
 +0.48 
 
 
 0-93 
 
 0-5 
 
 + i. 25 
 
 0-93 
 
 o. 5 
 
 i779-'85 
 
 . 68 
 
 + i. 23 
 
 
 1.09 
 
 
 0.99 
 
 1.09 
 
 o. 5 
 
 i786-'9i 
 
 .61 
 
 +0.48 
 
 
 o. 50 
 
 o. 3 
 
 f o. 15 
 
 o. 50 
 
 0-3 
 
 1792 '97 
 
 -55 
 
 + 1. 12 
 
 
 o. 70 
 
 0. 2 
 
 0.35 
 
 o. 70 
 
 0. 2 
 
 1798-03 
 
 49 
 
 + 0.41 
 
 
 1.02 
 
 0. I 
 
 0. 10 
 
 1.02 
 
 O. I 
 
 1 804-' i o 
 
 43 
 
 +O.I8 
 
 
 I.4I 
 
 O. I 
 
 0.84 
 
 I.4I 
 
 0. I 
 
 i8i2-'i6 
 
 36 
 
 -0.15 
 
 3 
 
 0-53 
 
 3 
 
 +0.48 
 
 0-53 
 
 3 
 
 l8l7-'22 
 
 30 
 
 o. 41 
 
 3 
 
 +o. 03 
 
 3 
 
 +0.40 
 
 +0.03 
 
 3 
 
 i823~'28 
 
 -.24 
 
 +0-43 
 
 3 
 
 0. 10 
 
 3 
 
 +0.08 
 
 0. 10 
 
 3 
 
 1 829- '34 
 
 .18 
 
 0.08 
 
 3 
 
 +0. 21 
 
 3 
 
 4-0.25 
 
 +0.21 
 
 3 
 
 1835 '40 
 
 . 12 
 
 0. 12 
 
 3 
 
 O. 2O 
 
 3 
 
 4-0.37 
 
 o. 13 
 
 3 
 
 1 84 1 -'46 
 
 . 6 
 
 +0. 21 
 
 3 
 
 4-0.13 
 
 3 
 
 4-0.47 
 
 +0. 12 
 
 4 
 
 1847-52 
 
 o 
 
 4-0.25 
 
 4 
 
 O. OO 
 
 4 
 
 o. 24 
 
 o. 15 
 
 4 
 
 1853-58 
 
 +. 6 
 
 -f'55 
 
 5 
 
 +o. 18 
 
 5 
 
 o. 26 
 
 0.05 
 
 5 
 
 1859-64 
 
 + .12 
 
 +0.03 
 
 5 
 
 +0.28 
 
 5 
 
 0.46 
 
 +0. 12 
 
 5 
 
 i865-'7o 
 
 +.18 
 
 -0.23 
 
 5 
 
 0.15 
 
 5 
 
 4-0.05 
 
 0.36 
 
 5 
 
 i87i-'76 
 
 +.24 
 
 o. 15 
 
 5 
 
 + 0.26 
 
 5 
 
 +o. 16 
 
 o. 1 6 
 
 5 
 
 i877-'82 
 
 +.30 
 
 o. 90 
 
 5 
 
 +0.22 
 
 5 
 
 4-0-34 
 
 +0.08 
 
 5 
 
 1 883-' 88 
 
 +.36 
 
 0.27 
 
 5 
 
 4-0-33 
 
 5 
 
 -0.14 
 
 -j-o. 02 
 
 5 
 
 i889-*92 
 
 +.4-1 
 
 0.05 
 
 3 
 
 4-0.19 
 
 3 
 
 4-o. 13 
 
 o. 07 
 
 3 
 
 1 
 
 
 
 
 
 
 
 
 
 PARIS. 
 
 i8oo-'o3 
 
 .48 
 
 -f-o. 01 
 
 
 I. 93 
 
 
 O. 45 
 
 
 
 i8o4-'o7 
 
 .44 
 
 4-0.7 
 
 
 +0.82 
 
 
 2.02 
 
 
 
 1 808-' 10 
 
 .41 
 
 +2.66 
 
 
 + 1.60 
 
 
 . 95 
 
 
 
 i8ii-'i5 
 
 . 37 
 
 o. 92 
 
 
 I. 2O 
 
 
 i. 18 
 
 
 
 i8i6-'2i 
 
 3 1 
 
 4-0.58 
 
 
 + 1.68 
 
 
 i. 42 
 
 
 
 l822-'28 
 
 . 25 
 
 + 1.09 
 
 7 
 
 +o. 39 
 
 7 
 
 O. OI 
 
 
 
 i837-'42 
 
 . 10 
 
 +o. 79 
 
 7 
 
 o. 15 
 
 3 
 
 +0.40 
 
 
 
 1843 '48 
 
 4 
 
 +o. 43 
 
 2 
 
 o. 03 
 
 3 
 
 -f-o. iq 
 
 
 
 1849-^4 
 
 -f. 2 
 
 + i. 19 
 
 2 
 
 O. OI 
 
 2 
 
 + 1. 74 
 
 
 
 i855-'6o 
 
 -f. 8 
 
 +o. 35 
 
 
 o. 02 
 
 7 
 
 + 1. 22 
 
 
 
 1 86 1 -'66 
 
 4-. H 
 
 + 1.35 
 
 3 
 
 o. oo 
 
 T. 
 
 +O. 12 
 
 
 
 i867-'72 
 
 -f-.20 
 
 +o. 31 
 
 2 
 
 o. 67 
 
 2 
 
 +O. IO 
 
 
 
 i873~'77 
 
 -f. 25 
 
 O. Sq 
 
 2 
 
 +o. 04 
 
 2 
 
 4-1. 01 
 
 
 
 i878-'83 
 
 4--3 1 
 
 o. oq 
 
 2 
 
 o. 32 
 
 2 
 
 +o. q8 
 
 
 
 1 884-' 89 
 
 4-. 37 
 
 0.80 
 
 2 
 
 +o. 32 
 
 2 
 
 +o. 78 
 
 
 
 
 
 
 
 
 
 
 
 
16] RESULTS OF OBSERVED DECLINATIONS OF THE SUN. 31 
 
 Results of observations of the Sun's Declination Continued. 
 
 PALERMO. 
 
 Years. 
 
 T 
 
 w 
 
 w 
 
 6e 
 
 W 
 
 J<? 
 
 Vt 
 
 W 
 
 
 r -j 
 
 // 
 I 46 
 
 o 
 
 // 
 
 O Qi 
 
 
 // 
 4-o. 78 
 
 // 
 
 o. o<> 
 
 o 4 
 
 i/y 1 U J 
 i SOA ' 1 1 
 
 - JJ 
 41 
 
 1 I 7O 
 
 o 
 
 O $2 
 
 
 4-O. 42 
 
 O. 52 
 
 O. A 
 
 1 0^4 i j 
 
 
 
 
 
 
 
 
 
 CAMBRIDGE. 
 
 1877 '78 
 
 T A 
 
 O "I 
 
 2 
 
 O. 77 
 
 
 4-o. So 
 
 O. 54 
 
 i 
 
 10 OJ O 
 
 1870 '44 
 
 .08 
 
 -[-o. 31 
 
 2 
 
 O. 2O 
 
 
 +o. 29 
 
 o. 41 
 
 i 
 
 1847 '57 
 
 oo 
 
 4-O. 21 
 
 2 
 
 4-o. 7,1 
 
 
 O. }2 
 
 -fo. 10 
 
 i 
 
 i854-'s8 
 
 +.06 
 
 o. 15 
 
 2 
 
 4-o. 74 
 
 
 o. 42 
 
 -4-o. 13 
 
 i 
 
 
 
 
 
 
 
 
 
 
 WASHINGTON. 
 
 i840-'49 
 
 . 02 
 
 0.28 
 
 4 
 
 o. 73 
 
 
 o. 47 
 
 o. 81 
 
 2 
 
 i86i-'66 
 
 +.14 
 
 O. II 
 
 4 
 
 o. 43 
 
 
 0.45 
 
 o. 25 
 
 2 
 
 i867-'72 
 
 + . 20 
 
 4/-O. 74 
 
 4 
 
 o. 39 
 
 
 +0.28 
 
 o. 51 
 
 2 
 
 i8y3-'78 
 
 + . 26 
 
 o. 58 
 
 4 
 
 o. 32 
 
 
 -(-o. 10 
 
 o. 45 
 
 2 
 
 1 879-' 84 
 
 + 32 
 
 o. 31 
 
 4 
 
 o. 60 
 
 
 0.35 
 
 o. 72 
 
 2 
 
 i88s-'9i 
 
 -f-38 
 
 o. 02 
 
 4 
 
 0.05 
 
 
 o. 20 
 
 o. 18 
 
 2 
 
 
 
 
 
 
 
 
 
 
 KONIGSBERG. 
 
 i8i 
 
 
 
 
 
 
 
 I O7 
 
 
 1 820-' 23 
 
 28 
 
 o. 14 
 
 2 
 
 O. 22 
 
 
 - 59 
 
 O. 47 
 
 I 
 
 
 . 24 
 
 -(-0.65 
 
 2 
 
 4~- 49 
 
 
 o. 60 
 
 -f-o. 24 
 
 I 
 
 i828-'3i 
 
 . 20 
 
 41.08 
 
 2 
 
 +o. 09 
 
 
 o. 64 
 
 o. 1 6 
 
 I 
 
 1 832-' 34 
 
 . 17 
 
 o. 72 
 
 2 
 
 o. 15 
 
 
 I. 72 
 
 o. 40 
 
 I 
 
 1 837-' 44 
 
 . 09 
 
 0.66 
 
 2 
 
 o. 62 
 
 
 2. 24 
 
 o. 87 
 
 I 
 
 
 
 
 
 
 
 
 
 
 OXFORD. 
 
 1 840-' 45 
 1 846-' 5 1 
 i86i-'66 
 
 .07 
 . 01 
 
 + H 
 
 +0-79 
 
 +0.35 
 -j-o. 36 
 
 2 
 2 
 2 
 
 +0.42 
 4-0-40 
 
 o. 81 
 
 
 
 +0.67 
 4-0.89 
 
 4-o. 10 
 
 4-0. 22 
 4-0. 20 
 I. OI 
 
 O. 2 
 0. 2 
 
 O 2 
 
 1 867-' 7 2 
 
 +.20 
 
 o. 1 6 
 
 2 
 
 o. 24 
 
 
 -j-O. 2Q 
 
 O. 44 
 
 O 2 
 
 1873-76 
 i88o-'83 
 
 + .25 
 + 32 
 
 0.38 
 -43 
 
 2 
 2 
 
 0-33 
 
 -f-O. 12 
 
 
 
 +0.29 
 
 o. 17 
 
 o 53 
 o 08 
 
 0. 2 
 O. 2 
 
 i88 4 -'87 
 
 -f-36 
 
 o. 24 
 
 2 
 
 -fo. 23 
 
 
 o. 19 
 
 4-o. 07 
 
 O. 2 
 
 
 
 
 
 
 
 
 
 
OBSERVATIONS OF THE SUN. [16, 17 
 
 Results of observations of the Sun's Declination Continued. 
 
 PULKOWA. 
 
 Years. 
 
 61" 
 
 W 6 e 
 
 \\ 
 
 W 
 
 1842^45 .06 -fO. 82 2 0.35 0.01 0.35 I 
 
 .02 o. 10 2 0.48 +0.07 0.48 .1 
 
 i86i-'65 +.13 0.53 2 0.48 0.30 0.48 
 
 i866-'7o -f. 18 +0.27 2 0.31 0.38 0.31 
 
 DORPAT. 
 
 i823-'28 .24 +0.99 2 1.26 +0.59 1.41 i 
 
 i829-'32 .19 -f- 99 2 o. 7 6 -f I -34 0.91 I 
 
 l8 33~'3 8 .14 4- 1 - 00 2 0.63 -fi-34 0.78 i 
 
 CAPE OF GOOD HOPE. 
 
 i884-'87 +.36 0.51 4 +0.05 +o. ii 0.07 2 
 
 i888-'9O -}- 39 0.84 4 -{-0.09 -f - 1 9 0.21 2 
 
 . STRASBURG. 
 
 i884-'88 +.36 0.57 4 0.05 0.77 4-0.12 2 
 
 LEIDEN. 
 
 i864-'69 -)-. 17 +0.14 4 o. 01 -(-0.27 0.24 2 
 
 i87o-'76 -(-.23 0.23 4 0.06 0.04 0.29 2 
 
 Correction to the Sun's absolute longitude 
 
 17. So far as mere instrumental measurement is concerned, 
 the correction d s should be determined with greater precision 
 than dl" in the ratio 5:2, because the errors in decimation 
 have to be divided by the factor sin s = 0.40, in order to form 
 dl". Allowing for this large increase in the source of error, 
 the values of 6 1" are more accordant than those of 6 8. This 
 is what we should expect. The values of the former quantity 
 depend mainly upon the comparison of observations made 
 
17, 18] OBLIQUITY OF ECLIPTIC. 33 
 
 near the opposite equinoxes, when the snn has the same decli- 
 nation, and when the season is not greatly different. Indeed, 
 if the season changed exactly with the sun's declination, all 
 effects of annual change of temperature would be completely 
 eliminated from 61", as would also in any case any constant 
 error which is a function simply of the Sun's Declination. It 
 is therefore to be expected that the actual probable error of 
 this quantity will conform more nearly to that determined from 
 the residuals than in the case of the other. 
 
 For these reasons the value of dl" does not give rise to 
 much discussion. The general result from all the observa- 
 tories is, for dl", when developed in the form x -f- y T. 
 
 x = + 0".05 
 y = 0".97. 
 
 Obliquity of the ecliptic. 
 
 18. The determination of the obliquity rests upon an essen- 
 tially different basis from that of the absolute longitude, in 
 that it depends upon actual differences of measured Declina- 
 tions, which differences are still further complicated by the 
 fact that they are necessarily made at opposite seasons. A 
 more detailed discussion of them is therefore necessary, and 
 some modification may have to be made in the separate results 
 as adopted. The following special circumstances affecting the 
 observations are to be taken into consideration : 
 
 The BRADLEY Greenwich results for 1753-^2, are derived 
 from a manuscript communicated by Dr. AUWERS, containing 
 the results of his very careful reduction of BRADLEY'S ob- 
 served Declinations of the Sun, which were compared with 
 HANSEN'S tables. The corrections were reduced to those of 
 LEVERRIER'S tables by being computed at intervals suffi- 
 ciently short to permit of the reduction being interpolated with 
 all necessary precision. No reduction was applied either on 
 account of the constant error of the Declinations determined 
 by Dr. AUWERS himself, nor for reduction to the Boss system 
 of standard Declinations. Hence arises the large value of Ad 
 given by these Declinations. Consequently the value of df is 
 5690 N ALM 3 
 
34 OBSERVATIONS OF THE SUN. [18 
 
 that given immediately by the instrument, on the system of 
 reduction adopted by Dr. AUWERS, in which I have supposed 
 that the Pulkowa refractions were used. 
 
 From 17G5 to 1816 the Greenwich observations were made 
 with the imperfect quadrant, the Declinations of which are 
 subjected to an error which is not constant. The neces- 
 sary corrections are derived by S AFFORD in Vol. n of the 
 Astronomical Papers. The corrections are those necessary to 
 reduce to Boss's system, and they vary with the Declination. 
 Hence the arc on which the obliquity depends is not that 
 measured with the instrument itself, but that so corrected as 
 to reproduce as nearly as may be the standard Declinations. 
 
 From 1812 onward the two mural circles were used. Up to 
 1830 no correction except the constant one derived by SAF- 
 FORD was applied to the Declinations as measured with these 
 instruments. Hence the arc of obliquity is that measured 
 with the instrument itself without being corrected by the 
 standard stars. 
 
 After 1830 the Declinations were corrected by the tables for 
 Greenwich given in Boss's paper. These corrections vary 
 somewhat with the Declination, and they are different also 
 for different periods. Hence we have here a period during 
 which the instrumental differences of Declination were cor- 
 rected to reduce them to the standard star- system. 
 
 If the standard system were subject to no farther error than 
 a constant one, common to all Declinations within the zodiac, 
 which common correction would be subject to a uniform change 
 with the time, this system would doubtless be the best one to 
 adopt in order to obtain the secular variation in the obliquity 
 of the ecliptic. But, as a matter of fact, the standard Decli- 
 nations are simply the mean results of Declinations measured 
 with different instruments. It is, therefore, a question whether 
 we shall get any better results by applying reductions to a 
 standard system than we should get by simply taking the 
 mean of the instrumental results, because the system is itself 
 only a mean of such results. It is true that the standard sys- 
 tem depends on more instruments than the obliquity, though 
 not on better ones; but it is also to be considered that the 
 reductions in the case of the Sun may be different from those 
 
18, 19] OBLIQUITY OF ECLIPTIC. 35 
 
 in the case of the stars, owing to the very different conditions 
 in which the observations are made. 
 
 Another troublesome point arises from the refraction used 
 in the reductions. The effect of refraction is always to make 
 the measured obliquity less than the actual one; the correc- 
 tion to the obliquity on account of refraction is therefore a 
 positive quantity, which is a minimum for an observatory at 
 the equator and increase equally towards each pole. Some 
 values of the obliquity were derived from BESSEL'S refractions 
 of the Tabulae Regiomontance, and others from the Pulkowa 
 tables. Since the secular variation of the obliquity is more 
 important than the absolute value of the quantity, it is essen- 
 tial that the standard to which all determinations of the ob- 
 liquity are reduced should be as nearly as possible the same, 
 and therefore that the same refraction should be used. But in 
 reductions to standard star places we meet with the addi- 
 tional complication that the differences in the constant of 
 refraction might be wholly or partially eliminated by the 
 reductions to a standard system. It would therefore be a dif- 
 ficult question how far we should modify the values of 6s on 
 account of the use of different tables of refraction. 
 
 To avoid all these difficulties I have judged it best to make 
 the obliquity depend mainly upon absolute measures, the 
 reductions being made with the Pulkowa refractions. 
 
 Effect of refraction on the obliquity. 
 
 19. The determination of the average or most probable effect 
 on the obliquity produced by using the Pulkowa refractions, 
 instead of those of the Tabulce Regiomontanw, is easily deter- 
 mined. We divide the ecliptic into a number of equal arcs 
 throughout the year, and by equations of condition express 
 differences of refraction in terms of differences of Declination, 
 and hence differences of obliquity. We thus find that at 
 certain latitudes where observations were made, and where 
 BESSEL'S refractions were used in the reduction, the follow- 
 ing corrections are necessary to reduce the obliquity to the 
 ones given by the Pulkowa refractions: 
 
 Pulkowa; y = 59.S; Jf = 0".325 
 Greenwich; <p = 51.o; z/f = 0".20 
 Washington; q> = 3S.9; 4e = - 0".125 
 
36 OBSERVATIONS OF THE SUN. [19 
 
 Hence I conclude that for . 
 
 Dorpat; As 0".29 
 Konigsberg; Js = /7 .26 
 Cambridge; Je = - 0".21 
 Cape Town; At = - 0".12 
 
 The corrections to the obliquity thus derived, depending 
 mainly on direct instrumental measurement, and reduced to the 
 Pulkowa refractions, are designated as 6'f . The results for this 
 quantity are given in the last column of the several tables. 
 
 In the case of BRADLEY'S Greenwich results, I have taken 
 as 6'e Dr. AUWERS'S results unchanged, assuming in the 
 absence of any specific statement that he has used the Pul- 
 towa refraction tables. 
 
 In the case of MASKYLENE'S observations, I have, by excep- 
 tion, used them as reduced to the standard star-system, 
 because we have no other results at these times, and the en or 
 of his instrument is so strongly shown that it would not do to 
 use the results unchanged. It will be seen, however, that 
 small weights are assigned, and that the weights diminish 
 towards the end of the 'series. 
 
 In the case of the Greenwich observations from 1812 to 
 about 1834, no change has to be made, as the results are gen- 
 erally or always purely instrumental, and Pulkowa refractions 
 are used in SAFFORD'S work. 
 
 From 1835 onward I have depended mainly on certain cor- 
 rected Greenwich reductions. First, for tf 7 , I have used the 
 results given by Mr. CHRISTIE in his very valuable paper on 
 the Greenwich Declinations, in M. E. A. S., Vol. XLV, where 
 the Declinations from 1836 to 1879 are reduced on a uniform 
 system. Later, I have adopted the corrected results given in 
 Appendix III to the Greenwich observations for 1887. In 
 each case the result has been reduced to the Pulkowa refrac- 
 tions. 
 
 The Paris results rest on a different basis from the others, 
 in that the zero point of the instrument depends wholly upon 
 LEVERRIER'S Declinations of the stars, and I fear it was not 
 always accurately determined. Observations near the winter 
 solstice are mostly referred to one set of stars; those near the 
 
19J OBLIQUITY OF ECLIPTIC. 37 
 
 summer to another set, the error of which may be systemat- 
 ically different. Certain it is that the results during the early 
 years were very discordant. The weights as given in the table 
 are those assigned a priori, without sufficient reference to the 
 discordance of the older results. I have felt constrained to 
 evade a decision as to their treatment by entirely omitting 
 their results in the final discussion. 
 
 Iii the case of some other observatories it was difficult to 
 determine exactly what refractions had been used in each 
 special case and what reductions should be made. I have, how- 
 ever, determined the corrections in the best way I was able. 
 
 A precise determination of the secular change in the ob- 
 liquity is of more importance for our present object than a 
 precise determination of its amount. Hence a series of obser- 
 vations extending through a long period of time, and made on 
 a uniform system, has an advantage over a number of isolated 
 values, in that any constant error with which it may be 
 affected will be eliminated from the secular variation. Possi- 
 ble constant differences between the determinations of the 
 various observatories at different epochs will vitiate the sec- 
 ular variation, but the probable amount of this error may be 
 diminished by using a number of separate determinations, 
 such as are presented in the preceding table. In the Green- 
 wich transit circle we have a very uniform series, extending 
 over a period of forty years, but giving results systematically 
 different from other determinations. This series gives for the 
 correction to the obliquity : 
 
 Transit Circle, 1847->91 : 
 
 d'e = - 0".ll i 0".06 + (0".21 i 0".46) T . . . (a) 
 
 Here, in view of the uniformity of method and reduction, 
 we may regard the mean error of the centennial variation from 
 the discordance alone as a fair approximation to the probable 
 mean error. It will be seen that I have here included four 
 years (1847-'50) of the Mural Circle results. 
 
 Continuing the Greenwich series backward, the question 
 arises whether we can regard the results of the mural circle 
 from 1812 to 1850 as comparable with those of the transit circle. 
 
38 OBSERVATIONS OF THE SUN. [19 
 
 There is certainly nothing in the table to indicate any system- 
 atic difference. From the combination of the two we have 
 
 M. C. andT. 0., 1812->50: 
 
 <J'6 = - 0".08 0".05 + (+ 0".14 i 0".23) T (1850) . . (b) 
 
 Here the mean error is naturally smaller than in the case of 
 the transit circle alone, but is now more subject to possible 
 systematic difference between the two instruments. 
 
 If we now go back to BRADLEY, we meet with the very diffi- 
 cult question, whether we should regard his results as best 
 comparable with the modern Greenwich observations, or with 
 modern observations in general. If we assume that the differ- 
 ence between the Greenwich and other modern results is due 
 to any cause which has remained unchanged since BRADLEY, 
 we should reach one conclusion; otherwise, we should reach 
 the other. The result of combining all Greenwich observa- 
 tions, with the weights as assigned, is 
 
 6'e = -0".ll + 0".50T (c) 
 
 In this combination I have used the weak results of MASKE- 
 LYNE, with the small weights assigned, although they depend 
 wholly upon the standard declinations of stars. In view of 
 the discordance between BRADLEY'S two results, this seems 
 the only admissible course. 
 
 Next in the length of time which they include come the Paris 
 observations, of which the results, with the. weights assigned, 
 are 
 
 6f= + 0".01 0".36T 
 
 I give this result in order that nothing may be omitted. 
 Undue weight has probably been assigned to the earlier 
 determinations; in any case the method of deriving it from 
 the original observations is so objectionable that no further 
 use is made of it. A satisfactory discussion of the observa- 
 tions would require a complete redetermination of the zero 
 points of the instrument from fundamental stars. 
 
19, 20] DISCUSSION OF RESULTS OF OBLIQUITY. 39 
 
 If we omit the Greenwich, Paris, and Palermo results, and 
 combine all the others into a single set of equations of condi- 
 tion, we have the equations arid results : 
 
 36.9# + 0.26 y = - 14".37 
 0.26 + 1.88 = + 1".01 
 
 x = - 0".39 
 y=+ 0".59 
 
 Here x is the value of 6'e for 1860, and y its centennial varia- 
 tion. Transferring the epoch to 1850, as usual, the result is 
 
 d'e = - 0".45 + 0".59 T ..... (d) 
 
 No reliable mean error can be computed, owing to systematic 
 errors. In view of these, one mode of treatment would be to 
 form equations of condition in which a possible systematic 
 error at each observatory would appear as one of the unknown 
 quantities. By this . process we should get the same result 
 for the secular variation as if we made an independent determi- 
 nation from the work of each observatory. At most of the 
 observatories the period through which the observations are 
 made, with one instrument and on an unchanged plan, is too 
 short to render such a course advisable. 
 
 As a last combination, we shall combine the earlier Green- 
 wich results, up to 1810, with Palermo and with all the modern 
 results except Paris, first dividing the weights of the Green- 
 wich results by 2. We then have the equations 
 
 39.8 a? -1.82 y = - 17 ".12 
 - 1.8 + 3.47 = + 2".99 
 
 x = 0".40 
 
 y=+0".65 ....... () 
 
 Concluded results for the obliquity. 
 
 20. The data on which these various results for the obliquity 
 rest show the following noteworthy features : 
 
 (1) That the correction given by the modern Greenwich 
 instruments, mural and transit circles, is markedly greater 
 
40 OBSERVATIONS OF THE SUN. [20 
 
 than that given by other modern observations. This may be 
 most plausibly attributed to the atmospheric conditions 
 within the observing room. 
 
 (2) The minuteness of the change of the correction given 
 by these instruments during nearly eighty years. To this 
 circumstance is due the smallness of the centennial variation, 
 0".50, found from the totality of the Greenwich observations. 
 A comparison of BRADLEY with the mean of the T. C. results 
 only would have given a change of 0".97 in 117 years, or a 
 centennial change of about 0".80. 
 
 The long period, uniformity of plan, and systematic devia- 
 tion of the modern Greenwich observations lead me to consider 
 them as forming a series distinct from all others. We have 
 therefore the following two completely independent determi- 
 nations of the centennial variation : 
 
 (1) Modern Greenwich results: y = + 0".14 i 0".23 
 
 (2) All other results + 0".6o 
 
 To the latter no reliable mean error can be assigned. To 
 judge its reliability we may compare it with the results (), (c), 
 and (d) 
 
 Greenwich T. C., alone, + 0".21 0".46 
 
 Greenwich observations in general, -f- 0".50 
 Miscellaneous modern observations, + 0".59 
 
 We may, it would seem, fairly give double weight to the 
 result (2), thus obtaining, as the definite result from observa- 
 tions of the Sun alone: 
 
 Correction to LEVERRIER'S centennial variation of the obliq- 
 uity of the ecliptic (- 47".594) 
 
 + 0".48 0".30 
 
 the mean error being an estimate from the general discordance 
 of the data. 
 For the constant part of the correction I take 
 
 tie (1850) = - 0".30 
 
21] SUMMARY OF RESULTS. 41 
 
 Summary and comparison of results. 
 
 21. From what precedes we have the following as the values 
 of the unknown quantities, and of their secular variations, as 
 given by observations of the Sun alone. 
 
 de" = 
 
 Value for 
 1850. 
 
 + 0".10 d 
 
 - 7/ .03 
 
 Cent, 
 var. 
 
 + 0".23 
 
 0^.10 
 
 e"(dn"+a) = 
 
 0".00 J 
 
 r /7 .07 
 
 + C^.33 i 
 
 ;/ .12 
 
 dl"+a = 
 
 - 0".02 
 
 
 - G^.63 
 
 
 dl" = 
 
 + 0^.05 J 
 
 - 0".12 
 
 - /7 .97 i 
 
 0^.23 
 
 ds = 
 
 - /7 .30 = 
 
 t 7/ .15 
 
 + 77 .48 i 
 
 : 0".30 
 
 a = 
 
 -0".07 
 
 
 4- /7 .34 
 
 
 No estimate of the probable errors of these quantities would 
 be useful which did not take account of the systematic dif- 
 ferences between the results of different observatories. We 
 have therefore formed the mean outstanding residual correc- 
 tions given by the several observatories, as shown in the 
 tables which follow. Originally the scale of weights used for 
 the Greenwich observations did not correspond to that for the 
 other observatories; they were, therefore, divided by 2. As 
 used below, however, the change has been made in the case 
 of dl" by multiplying all the weights of the other observatories 
 by 2, and, in the case of 6s, by dividing the Greenwich weights 
 by 2. 
 
 The correction to the obliquity depends solely on 6'e ; but 
 the comparison has also been made with the values of <?, 
 which, it will be remarked, differ from the others in that 
 account is taken of the supposed variation of the systematic 
 correction with the declination. It is noteworthy that the 
 results are somewhat more accordant when this correction is 
 omitted and purely instrumental errors are used for the 
 obliquity. 
 
 The mean errors given in the preceding summary of results 
 are derived from the discordances in question, and may be 
 regarded as substantially real. 
 
 No use was made of the Paris results for 61" and ds for 
 the reason that they depend on decimations referred to star 
 
42 OBSERVATIONS OF THE SUN. [21 
 
 places which may be affected by differences in different Eight 
 Ascensions. They are, however, retained in the table to show 
 the amounts of outstanding discordance. 
 
 Outstanding mean residual corrections to quantities depending 
 on the Sun's Right Ascension. 
 
 Greenwich + 
 
 0".09 - 0".03 
 
 54.5 
 
 Paris 
 
 0".09 + 0".17 
 
 17 
 
 Cambridge 4- 
 
 7/ .02 0".00 
 
 16 
 
 Washington 
 
 0".05 0".12 
 
 24 
 
 Konigsberg 
 
 /7 .08 4- /7 .08 
 
 12 
 
 Oxford . 4- 
 
 0".06 4- 0".02 
 
 8 
 
 Pulkowa 
 
 /7 .15 4- 77 .22 
 
 6 
 
 Dorpat 
 
 /7 .10 - /7 .03 
 
 4 
 
 Cape 
 
 7/ .16 7/ .ll 
 
 4 
 
 Strassburg 4- 
 
 77 .05 (V.03 
 
 3 
 
 Mean errors for 
 
 
 
 weight unity t \ = 
 
 /7 .34 /7 .39 
 
 
 Mean error of x i 
 
 /7 .03 i /7 .03 
 
 
 Mean error of y 
 
 /7 .10 /7 .12 
 
 
 Outstanding mean residual corrections to quantities 
 
 depending 
 
 on the Sunh 
 
 ? Declination. 
 
 
 SI" 
 
 w de w 
 
 d'K 
 
 Greenwich - 7/ .06 
 
 64 4- 77 .31 29.6 
 
 4-0".! 7 
 
 Paris 4- /7 .45 
 
 4- 7/ .31 
 
 
 Palermo - 77 .39 
 
 - /7 .20 0.8 
 
 /7 .20 
 
 Cambridge - 0".05 
 
 8 4- 77 .35 4 
 
 4- /7 .14 
 
 Washington 4- /7 .07 
 
 24 - 7/ .22 12 
 
 - /7 .29 
 
 Konigsberg - 77 .20 
 
 10 + /7 .31 5.5 
 
 /7 .00 
 
 Oxford 4- /7 .14 
 
 14 + /7 .19 1.4 
 
 - 0".01 
 
 Pulkowa + /7 .12 
 
 8 - 77 .13 4 
 
 - /7 .13 
 
 Dorpat 4- 7/ .75 
 
 6 77 .49 3 
 
 - /7 .64 
 
 Cape - 7/ .35 
 
 8 4- /7 .10 4 
 
 - /7 .02 
 
 Leiden + /7 .10 
 
 8 4- 77 .17 2 
 
 - /7 .06 . 
 
 Strassburg - /7 .26 
 
 4 + /7 .08 4 
 
 4- /7 .25 
 
 for weight unity /7 .81 
 
 7/ .74 
 
 /7 .60 
 
CHAPTEE III. 
 
 RESULTS OF OBSERVATIONS OF MERCURY, VENUS, AND 
 
 MARS. 
 
 Elements adopted for correction. 
 
 22. We first give an outline of the method of expressing the 
 observed corrections to the Eight Ascensions and Declinations 
 of each of the planets as linear functions of the corrections to 
 the tabular elements. This linear function forms the first 
 member of the equation of condition in its original form, and 
 the observed correction forms its second member. 
 Let us put 
 
 E, r, the radii vectores of the Earth and planet 5 
 L, the Sun's true longitude; 
 
 J, the inclination of the orbit of the planet to a plane 
 passing through the Sun's center parallel to the 
 plane of the Earth's equator; 
 N, the Eight Ascension of the ascending node of the 
 
 orbit on this plane; 
 
 U, the argument of heliocentric declination of the planet 
 or its angular heliocentric distance from the node 
 on the equator; 
 a, 6, the geocentric Eight Ascension and Declination of 
 
 the planet. 
 , the obliquity of the ecliptic; 
 
 We shall then have 
 
 a =/(r. E. L. J. X. U., *.) . . . . . . . .V (a) 
 
 For the correction to the tabular Eight Ascension arising 
 from symbolic corrections to these seven quantities, we have 
 the equation 
 
 Sa = A 63 + * tfN + %L SU + % Sr + * <fe 
 dJ dN du dr de 
 
 43 
 
44 MERCURY, VENUS, AND MARS. [22 
 
 with a similar equation for the declination, formed from this by 
 writing <5 for a. 
 
 The relations by which these two equations are derived, as 
 well as the expressions for the differential coefficients they 
 contain, are given very fully in A. P., Yol. II, Part I, to which 
 reference may be made. The corrections tfN and tfU are not, 
 however, the most convenient ones to choose. It will be found 
 in the paper alluded to that they have been transformed by 
 measuring the longitude in orbit of the planet and that of the 
 perihelion from an arbitrary point in the orbit. As to this very 
 convenient device in celestial mechanics, it is to be remarked 
 that the "departure point" always disappears from the final 
 equations which determine the position of the planet. We 
 may, in fact, make abstraction of it by considering that its 
 introduction is equivalent to the following simple linear trans- 
 formations. 
 
 We put 
 
 w, the distance from the node to the perihelion ; 
 
 /, the true anomaly ; 
 
 g, the mean anomaly. 
 
 TT, the longitude of the perihelion ; 
 
 I, the mean longitude of the planet; 
 
 v, its true longitude; 
 
 these longitudes being counted from the departure point. 
 Then, we have the relations 
 
 #U = tfw 4- df --= 6v cos 
 6w = dn - cos JtfN (2) 
 
 61 = dn +dg 
 Hence, 
 
 dn = tfU -f cos JtfN df 
 
 (3) 
 
 The elements finally adopted for correction by the equations 
 of condition were 
 
 I. TT. e. J. N. 
 
22, 23] ELEMENTS ADOPTED FOR CORRECTION. 45 
 
 The value of a, the mean distance, is known with such pre- 
 cision that its correction need not enter into the equations of 
 condition. The latter are formed by substituting in (1) 
 
 n+-e + - cos 
 
 dgj de dg (4) 
 
 dr x . dr ~, dr ^ 
 
 dr = -=- de + -=- 61 =- o n 
 de dg dg 
 
 The coefficients of each equation of condition from the Eight 
 Ascension thus become 
 
 Coefficient of $J . . . 
 
 dS 
 
 XT da da 
 
 ... ^-cosJ^- 
 
 de . . . * 
 
 (5> 
 u u rf i <* ^/ _. ^ ^ r 
 
 ro ^"^ ^ ^ 
 
 da .d \dadr 
 
 u it se dtx d f 4- da dr 
 
 Wde + W fo 
 
 In the second members of the equations a is regarded as 
 a function of the seven quantities (a), as is also #, for which 
 a similar equation is to be formed. 
 
 The corrections of the solar eccentricity, perihelion, and 
 mean longitude were also introduced by putting in (1) 
 
 tfL = dl" + de" + dn" 
 
 de" dn" (6) 
 
 ^R = *L de" + ^ dn" 
 
 de" dn" 
 
 Introduction of the masses of Venus and Mercury. 
 
 23. The correction to the mass of Venus was introduced 
 by taking the tabular perturbation produced by Venus on 
 the geocentric place of the planet at the mean date of each 
 equation as the coefficient of the unknown quantity to be 
 determined. In computing these perturbations regard was 
 
46 MERCURY, VENUS, AND MARS. [23, 24 
 
 had to the action of Venus on the Earth as well as ou the 
 planet. On this system the unknown quantity finally found 
 would be the factor by which the adopted mass of the planet 
 must be multiplied in order to give the correction of that mass. 
 
 It has already been remarked that the mass of a planet can 
 not be determined free from systematic error by observations 
 made upon the planet itself. Hence, the mass of Venus can 
 be determined only from observations of Mercury and Mars, 
 and that of Mercury only from observations of Venus and 
 Mars. But the mass of Mercury is so minute that it would be 
 useless to attempt to determine it from observations either of 
 the Sun or Mars. It was therefore determined solely from the 
 periodic perturbations of Venus. 
 
 It has happened that the mass of Venus could not be deter- 
 mined in a reliable way from observations of Mars, owing to 
 a defect in the theory of the latter planet, which I shall men- 
 tion hereafter, and have not yet had time to correct. Practi- 
 cally, therefore, the mass of Venus is determined only from 
 observations of the Sun and of Mercury, and that of Mercury 
 from observations of Venus. 
 
 Correction of equinox and equator. 
 
 24. t Could all the observations be directly referred to a 
 visible equinox and equator, the corrections above enumerated 
 would have been the only ones which it was necessary to 
 include in the equations of condition. But, as a matter of 
 fact, the observations were all referred to an assumed system 
 of Right Ascensions and Decimations of standard stars my 
 own system in Eight Ascension and Boss's in Declination. 
 We must therefore introduce two additional unknowns into 
 the equations, which I have represented in the following way: 
 
 <*, the common error of the adopted Right Ascensions. 
 #, the common error of Boss's Declinations. 
 
 The first quantity will appear only in the equations derived 
 'from observed Right Ascensions and the second only in the 
 equations derived from Declinations, the coefficient being unity 
 in each case. 
 
24] CORRECTION OF EQUINOX AND EQUATOR. 47 
 
 That the value of 6 found in this way should be regarded 
 as a correction to the Declinations of the equatorial stars will 
 appear by the following considerations. The mean heliocen- 
 tric orbit of a planet as projected on the celestial sphere is 
 undoubtedly a great circle. On the other hand, in view of the 
 systematic discordance always found to exist in measures of 
 absolute Declinations near the equator, and of the fact that 
 these absolute Declinations depend upon assumed constants 
 and laws of refraction, which are necessarily affected with 
 greater or less uncertainty, and are otherwise subject to 
 systematic errors, instrumental or personal, of an obscure 
 character, but strongly shown by a comparison ot.the Declina- 
 tions derived from the work of different observatories, it can 
 not be assumed that these Declinations are free from sys- 
 tematic error. JSow, m one circle ot Decimation, say the 
 equator, we may expect that the error will be nearly constant 
 around the sphere, since the causes of error will generally be 
 nearly constant for any one Declination. This conclusion is 
 confirmed by a comparison of the best star catalogues. 
 Moreover, between the zodiacal limits, the error in each par- 
 ticular case is not likely to differ very greatly from the error 
 at the equator. Even if the difference should be considerable 
 the various values of the error of the different Decimations 
 must have a certain mean value, so that in the case of each 
 particular star, or each region of the heavens, we may conceive 
 the actual error to be divided into two parts one the mean 
 value in question, and the other the deviation from this mean. 
 The latter is probably smaller than the former, and in any 
 case can not very well be determined from observations of the 
 planets. But the condition that the planet moves on a great 
 circle of the sphere admits of the mean value being deter- 
 mined with great precision. It should, therefore, be included 
 in the equations of condition. 
 
 The value of <*, the common error of all the Eight Ascen- 
 sions, can obviously not be determined from the equations in 
 .Eight Ascension alone, because the only result that such 
 observations can give us would be the values of the Eight 
 Ascensions referred to some assumed equinox. The coefficient 
 of a would therefore completely disappear from the equations 
 
48 ' MERCURY, VENUS, AND MARS. [24 
 
 of condition in Eight Ascension. But since the same unknown 
 quantities are introduced into the equations of condition in 
 Eight Ascension and in Declination, the requirement that the 
 two sets of equations shall give common values of these 
 quantities does away with this indetermiriatiou and enables 
 determinate values to be found. In fact, this method does not 
 differ in principle from that usually adopted, in deriving the 
 Eight Ascensions of stars from observations of the Sun. The 
 latter consists in deriving the Sun's absolute longitude from 
 observations of its Declination and absolute Eight Ascensions 
 of the stars by comparing them with the Sun. In the same 
 way we may. consider that, in observations of the planet, the 
 Sun's absolute longitude is derived from observations of Decli- 
 nations of the planet, and then a comes out from the observa- 
 tions in Eight Ascension. 
 
 I have deemed it absolutely necessary that all the equations 
 of condition should be solved by the method of least squares. 
 By this method alone can the results of the observations as 
 regards separate values of the elements and constants be prop- 
 erly brought out. But the work of constructing and solving 
 a system of nine thousand equations of condition, each involv- 
 ing twenty unknown quantities, would be extremely laborious, 
 and might even require a century for its completion, if done in 
 the usual way. It was therefore necessary to adopt every 
 device by which the labor could be reduced to a minimum. 
 One device was the dropping of all superfluous decimals in the 
 coefficients of the equations. Since tbe errors thus produced 
 would be purely accidental, it follows that if the sum of the 
 products obtained by multiplying the value of each unknown 
 quantity by the error of its coefficient in the equation of con- 
 dition is but a small fraction of the necessary probable error 
 of the absolute term, no serious harm will result from the 
 errors of the coefficients. 
 
 Another device was the construction of tables for finding 
 the coefficients. Such tables relating to Mercury and Venus 
 are found in Vol. II, Part 1, of the Astronomical Papers. 
 These tables are, however, only given for one mean anomaly in 
 each case, and therefore require computations dependent on 
 the value of the other anomaly. They were therefore extended 
 
24, 25] INTRODUCTION OF SECULAR VARIATIONS. 49 
 
 to tables of double entry, so that the value of the derivatives 
 of the geocentric Eight Ascension or Declination at any epoch 
 could be taken from the tables at sight. The arguments were 
 the mean anomaly of the planet and the day of the year at 
 which the planet last passed through its perihelion. 
 
 Introduction of the secular variations. 
 
 25. When the equations of condition are formed on the plan 
 just set forth, the unknown quantities will be the corrections 
 to the elements or to the mean longitude at the date of each 
 equation. But every one of the unknown quantities which 
 have been enumerated, the correction of the masses excepted^ 
 is subject to a secular variation. Hence, instead of the 
 unknown quantities heretofore denned, we introduce two 
 others, the one the value of this unknown at some assumed 
 mean epoch, which, for reasons already set forth, must first 
 be determined from the observations; the other the secular 
 variation in a unit of time. The unknown quantities which 
 have been enumerated make twelve for each equation of con- 
 dition. Eleven of these are subject to a secular variation, so 
 that if the secular variations were introduced into the original 
 equations of condition they would each have twenty-three 
 unknown quantities. 
 
 The following device was employed to reduce to a minimum 
 the work of introducing and determining the secular variations 
 of the various elements : 
 
 Firstly, the whole time covered by the observations was 
 divided into periods, never exceeding ten years, except when 
 the observations were very few in number, or entitled to but 
 small weight. It was then assumed that no error would arise 
 from supposing the value of the unknown quantity to be the 
 same throughout the period as it was at the mid-epoch of the 
 period. The maximum absolute error thus arising would be 
 the secular variation during half the length of the period, and 
 the mean error the secular variation during one-fourth of the 
 period; but actually the effect of even this error would be 
 almost entirely nullified by the combination of positive and 
 negative coefficients throughout each period. 
 5690 N ALM 4 
 
50 MERCURY, VENUS, AND MARS. [25 
 
 Let us now put 
 
 x,y, 
 
 the corrections to the elements at any epoch, T. 
 
 Let 
 
 a x+ b y + cz -f . . . ^=n 
 
 be an equation of condition between these quantities at this 
 epoch. From a system of such equations, extending through a 
 period numbered i, during which #, #, etc., may be considered 
 as constant, we derive normal equations of the form 
 
 [aa] t x+[db] t y + . . . = [an], 
 
 which I shall call partial normal equations, and which we 
 might solve so as to obtain the values of x, y, etc. This solu- 
 tion is not, however, necessary. The values of the unknown 
 quantities being really of the general form 
 
 
 we may imagine these values substituted in the normal equa- 
 tions (1), the value T, of t for the mean epoch of the period 
 being substituted for t. 
 
 Let us now suppose that we introduce the quantities # , 2/o, > 
 #', y', . . into the original equations of condition, using for t 
 the value r tj which pertains to the mean epoch of the period. 
 Our equation of condition will thus become 
 
 ax Q + fy/o 4- + ar t x f + br t y' + . . = n (3) 
 
 If from a system of conditional equations of this form we 
 form the normal equations for all the unknown quantities, the 
 results will be these : 
 
 Partial normal equation in x ; 
 
 [aa] f # -f- [db] t y + . . + rJaaJX + r^ab^y 1 + . . = [aw], (4) 
 
25] INTRODUCTION OF SECULAR VARIATIONS. 51 
 
 Partial normal equation in x 1 j 
 
 T,[aa],a?o + T,[a&],y + . . + T 2 [aa],#' + rf [ab^y 1 
 
 + . . = r t [an] t (5) 
 
 We conclude that the partial normal equations, when the full 
 number of unknown quantities is included, may be derived 
 from those of the form (1) by the following rules. 
 
 (1) Each partial normal equation in X Q , y , . . . is formed 
 from that in #, y, etc., by adjoining to the first member of the 
 equation the member itself multiplied by r and then changing 
 x, y, . . .to x j XQ'J and, in the products by r, changing 
 x, y, . . . into a?', y', . . . 
 
 (2) The partial normal equation in a?', y', . . . is formed 
 from the partial equation in x 0j y^ . . . by multiplying all 
 the terms throughout by the factor r. 
 
 The final or complete normal equations in all the unknown 
 quantities being formed by the addition of the partial normals, 
 the formulae for the coefficients are as follow : 
 
 For the final equation in X Q 
 
 [aa] = [aa]! + [aa], + . . . + [aa] n 
 
 [ab] = [a&]i+ [a&] 2 + . . . + [ab] n 
 
 [aa]' = n [aa]i + r 2 [aa] 2 + . . . + r n [aa], 
 [an] = [an]^ [an] 2 + . . . + [aw], 
 
 For the final equation in x' 
 
 [aa]" = r, a [aa]! + r, 2 [oa] 2 + . . . +T n [aa] n 
 
 . . . +r n *[ab] n 
 
 [an]" = n [an]i + T 2 [an] 2 + . . . + r n [an] n 
 
 The final equations for all the unknown quantities will then 
 be of the form 
 
 [oa] x + [aft] y + . . + [aa] 1 x' + . . . = [an] 
 
 ... (8) 
 [aa]'xo+[ab] / y () + . . . +[aa]"0 / + . . . = [an]" 
 
52 MERCURY, VENUS, AND MARS. [25,26 
 
 The epoch from which we count the time, r, is arbitrary. 
 An obvious advantage will be gained in counting it from the 
 mid- epoch of all the observations. Then we shall have, by 
 putting w^ w 2 , etc., for the sum of the weights for the different 
 periods : 
 
 MI r\ + w 2 r 2 + + w n r n = (9) 
 
 If the observations are then equally distributed around the 
 orbits of the planet and of the Earth it may be expected that 
 the coefficients 
 
 [.]', [aft]' .... (10) 
 
 will all nearly or quite vanish. Practically we may expect that 
 as observations are continued through successive revolutions 
 the ratios of these to the other coefficients will approach zero 
 as a limit. We may then divide the normal equations into two 
 sets, one containing the quantities x , y^ etc., and the other 
 #', y', etc. The coefficients (10) being small, the two sets of 
 normals will be nearly independent, and we may omit the 
 terms (10) in the first approximation, and introduce them in 
 one or two successive approximations so far as necessary. 
 
 The unit of time is also arbitrary. A certain advantage in 
 symmetry will be gained by so choosing it that the mean value 
 of T 3 shall not differ greatly from unity. It was found that 
 twenty-five years was a sufficiently near approximation to be 
 adopted for all three planets. 
 
 Dates and weights for epochs and periods. 
 
 26. As want of space makes impracticable the present publi- 
 cation of the great mass of material worked up, the following 
 particulars have been selected as those most likely to be use- 
 ful in judging and criticising the work. We give three tables, 
 showing the division of the dates of observation into periods, 
 and the weights for each period. The first column of each 
 table contains the number or designation of the period, as 
 found in the manuscript books. The second contains the 
 mean year of the period. The third column shows the time 
 
26] DATES AND WEIGHTS FOR EPOCHS AND PERIODS. 53 
 
 of this mean period from the mid-epoch of the observations, 
 which is taken as follows : 
 
 For Mercury, 1865.0 
 Venus, 
 
 The next column contains the sum of the weights of the 
 equations in each period, as used 'in forming the normal equa- 
 tions. These were not, however, the weights actually used 
 in multiplying the coefficients of the equations of condition. 
 Owing to the diversity in the quality of the observations at 
 different times it was not found convenient to reduce the 
 equations at once to a uniform system of weights, and so dif- 
 ferent units of weight were selected for the older observations 
 and for the earlier observations. After the partial normal 
 equations were formed they were multiplied by the factor F, 
 necessary to reduce them to a standard in which the unit of 
 weight should correspond to the mean error 
 
 The sums of the weights reduced by these factors are shown 
 in the table. 
 
 In arranging the weights and selecting the factors it should 
 be remarked that a liberal allowance was made at each step 
 for probable constant errors, which results in the given 
 weights being much smaller than they would have been by 
 the theoretical treatment of the original observations. Not- 
 withstanding this allowance the final result seems to show 
 that it was still insufficient, and that the actual weights of 
 the results are less than would follow even from the final ones 
 as given. . 
 
 The partial normal equations for each period after being' 
 multiplied by the factors F, are added to form the final normal 
 equations as derived from meridian observations. 
 
54 MERCURY, VENUS, AND MARS. [26 
 
 WeightSj epochs, and periods of partial normal equations. 
 
 MERCURY. 
 
 1 
 
 1 
 
 Right Ascension. 
 
 Declination. 
 
 Mean 
 year. 
 
 T Wt 
 (units of 257.) 
 
 F - 
 
 Mean 
 
 year. 
 
 T 
 
 (units of 25 y. 
 
 Wt. 
 
 F. 
 
 I 
 
 2 
 
 3 
 3i 
 
 3-2 
 
 4 
 5 
 5i 
 
 8- 
 
 6, 
 6, 
 
 8 
 9i 
 
 9-2 
 10, 
 
 I0 2 
 
 i 
 
 n 2 
 Ii 3 
 
 1766.60 
 
 1784. 22 
 
 1799.81 
 
 -3. 9360 
 -3.2312 
 
 2. 6076 
 
 26! I 
 
 * 
 1 
 
 1765-50 
 
 1782.99 
 
 -3. 9800 
 3. 2804 
 
 
 
 O. 2 
 4.9 
 
 1 
 f 
 
 1796.42 
 1802.37 
 1809. 1 8 
 1824. 83 
 
 2. 7432 
 -2.5052 
 2. 2328 
 I. 6068 
 
 5-0 
 
 39-9 
 52.8 
 
 74-1 
 
 I 
 
 I 
 
 
 
 
 
 1809. 53 
 
 2. 2188 
 
 18.9 
 
 i 
 
 1818. 79 
 1825. 80 
 1835-56 
 
 1.8484 
 1.5680 
 I. 1776 
 
 0.9 
 34.5 
 
 75-o 
 
 i 
 i 
 i 
 
 
 
 
 
 
 
 
 
 1833.84 
 1838. 26 
 1843. 97 
 
 1855-92 
 1862. 79 
 1867. 18 
 1872.64 
 1877.05 
 1882. 17 
 1886. 29 
 1889. 70 
 
 I . 2464 
 
 75-3 
 141.5 
 
 281.5 
 201. 5 
 
 189.5 
 
 294.5 
 
 214.0 
 
 204.5 
 171.5 
 
 338. o 
 176.0 
 
 1% 
 1 
 
 1 
 1 
 
 1 
 1 
 1 
 
 
 
 
 I 0606 
 
 1843. 74 
 1855.90 
 1863. 10 
 1867. 12 
 1872. 62 
 1877.12 
 1882. 24 
 1886.29 
 1889.82 
 
 o. 8504 
 o. 3640 
 o. 0760 
 +o. 0848 
 4-o. 3048 
 
 -j-o. 4848 
 4-o. 6896 
 
 -j-o. 8516 
 -j-o. 99*28 
 
 98.8 
 
 83-3 
 99-8 
 1 86. o 
 129.8 
 129.8 
 108.2 
 199.8 
 109.5 
 
 i 
 
 i 
 i 
 
 * 
 i 
 
 | 
 
 0.8412 
 o. 3632 
 o. 0884 
 -j-o. 0872 
 -l-o. 3056 
 -j-o. 4820 
 -fo. 6868 
 4-0.8516 
 -j-o. 9880 
 
 VENUS. 
 
 I 
 
 1755.83 
 
 4. 2868 
 
 "3 
 
 * 
 
 1759.69 
 
 4. 1324 
 
 7.0 
 
 i 
 
 2 
 
 1767.92 
 
 3. 8032 
 
 19.7 
 
 i 
 
 1770. 18 
 
 -3.7128 
 
 IO. O 
 
 i 
 
 3 
 
 1781.06 
 
 3- 2776 
 
 3-7 
 
 i 
 
 I793.25 
 
 2. 7900 
 
 13.5 
 
 i 
 
 4 
 
 1792.47 
 
 2.8212 
 
 12.3 
 
 i 
 
 1806. 73 
 
 2. 2508 
 
 65.5 
 
 * 
 
 5 
 
 1802. 64 
 
 2.4144 
 
 23-3 
 
 i 
 
 i8i5-59 
 
 1.8964 
 
 67.5 
 
 i 
 
 6 
 
 1810. 31 
 
 2. 1076 
 
 34-0 
 
 i 
 
 1823.75 
 
 1.5700 
 
 197.0 
 
 i 
 
 7 
 
 1816. 88 
 
 1.8448 
 
 42. 7 
 
 i 
 
 1836. 02 
 
 1.0792 
 
 762. o 
 
 
 8 
 
 1825.55 
 
 I.4 9 80 
 
 141.0 
 
 
 1844.08 
 
 o. 7568 
 
 650. o 
 
 
 9 
 
 1835.31 
 
 I. 1076 
 
 339-3 
 
 i 
 
 1854. 24 
 
 o. 354 
 
 333-o 
 
 
 10 
 
 1843.98 
 
 o. 7608 
 
 259.3 
 
 i 
 
 1861.43 
 
 o. 0628 
 
 749.0 
 
 
 li 
 
 1853-51 
 
 o. 3796 
 
 205.3 
 
 i 
 
 1868.06 
 
 -j-o. 2024 
 
 815.0 
 
 
 12 
 
 1861. 60 
 
 o. 0560 
 
 353-7 
 
 * 
 
 1875-32 
 
 -j-o. 4928 
 
 692.0 
 
 
 13 
 
 1868. 12 
 
 +o. 2048 
 
 466. o 
 
 1 
 
 1883. 15 
 
 4-o. 8060 
 
 819. o 
 
 i 
 
 H 
 
 1875- 38 
 
 +o. 4952 
 
 399-5 
 
 1 
 
 1888.56 
 
 4-1.0224 
 
 801.0 
 
 i 
 
 1C 
 
 1883. 09 
 
 -j-o. 8036 
 
 04. c 
 
 i 
 
 
 
 
 
 16 
 
 1888. 67 
 
 -j-I. 0268 
 
 D T^ J 
 
 520. 5 
 
 2 
 
 | 
 
 
 
 
 
 
 
 
 */ *? 
 
 
 
 
 
 
26,27] UNKNOWN QUANTITIES OF EQUATIONS. 55 
 
 Weights, epochs, and periods of partial normal equations. 
 
 MARS. 
 
 
 Right Ascension. 
 
 Declination. 
 
 T3 
 
 o 
 
 1 
 
 Mean 
 year. 
 
 T 
 
 (units of 25 y.} 
 
 Wt. 
 
 F. 
 
 Mean 
 year. 
 
 r 
 
 (units of 25 y.} 
 
 Wt. 
 
 F. 
 
 , 
 
 1757-43 
 
 3- 9428 
 
 25-3 
 
 i 
 
 1758.82 
 
 -3.8872 
 
 8.8 
 
 i 
 
 2 
 
 1770.55 
 
 3- 4i8o 
 
 II. 
 
 * 
 
 11773-79 
 
 -3- 2884 
 
 8.8 
 
 
 3 
 
 1787.82 
 
 -2.7272 
 
 10. 
 
 * 
 
 1794.48 
 
 2. 4608 
 
 13.0 
 
 i 
 
 4 
 
 1799.77 
 
 2. 2492 
 
 20.7 
 
 1 
 
 1804. 91 
 
 -.-2. 0436 
 
 47.0 
 
 i 
 
 5 
 
 1811.32 
 
 -1.7872 
 
 14.7 
 
 * 
 
 ! 1813.00 
 
 I. 7200 
 
 30.5 
 
 1 
 
 6 
 
 1829. 17 
 
 1.0732 
 
 60. o 
 
 i 
 
 1828.04 
 
 I. Il84 
 
 93-o 
 
 
 7 
 
 1837- 39 
 
 o. 7444 
 
 121. O 
 
 i 
 
 1837. 18 
 
 o. 7528 
 
 371.0 
 
 
 8 
 
 1845- 39 
 
 o. 4244 
 
 76.3 
 
 t 
 
 1844.95 
 
 o. 4420 
 
 255-0 
 
 
 9 
 
 1853-36 
 
 o. 1056 
 
 90. o 
 
 * 
 
 1853. 02 
 
 o. 1192 
 
 245.0 
 
 
 10 
 
 1861. 07 
 
 -j-o. 2028 
 
 114. o 
 
 i 
 
 1860. 94 
 
 +0. 1976 
 
 306.0 
 
 
 ii 
 
 1869. 20 
 
 +o. 5280 
 
 124. o 
 
 * 
 
 1868. 80 
 
 -f o. 5120 
 
 197.0 
 
 
 12 
 
 1877.71 
 
 -j-o. 8684 
 
 132. o 
 
 i 
 
 1877. 38 
 
 +o. 8552 
 
 257.0 
 
 
 J 3 
 
 1883. 27 
 
 4-i. 0908 
 
 91. o 
 
 i 
 
 1883.26 
 
 +1.0904 
 
 1 60. o 
 
 
 14 
 
 1888. 85 
 
 + 1.3140 
 
 115.5 
 
 i 
 
 1888. 48 
 
 + 1.2992 
 
 167.0' 
 
 
 Unknown quantities of the equations. 
 
 27. For convenience in solving the equations of condition 
 the coefficients of the equations were multiplied by such 
 numerical factors as would reduce their general mean abso- 
 lute value to numbers of approximately the same order of 
 magnitude. Hence, the unknown quantities themselves are 
 not the corrections to the elements, but these corrections 
 divided by the adopted factors. 
 
 In the case of Mercury the absolute term was also multi- 
 plied by 10, so that effectively the factors in question were 
 reduced to one-tenth part of their value. The unknown 
 quantities of the equations are represented by the symbols 
 of the elements to which they relate inclosed in brackets. 
 
 For convenience of reference the following table is given, 
 showing the factors used in the case of each planet. In the 
 case of Mercury the column (a) shows the factors by which the 
 differential coefficients were actually multiplied; (b) the factor 
 by which the unknown quantity, as finally found, must be 
 
56 MERCURY, VENUS, AND MARS. [27, 28 
 
 multiplied to obtain the correction as expressed in the last 
 column.. In the case of Venus and Mars these factors are the 
 same. 
 
 Factors by which the unknown quantities are to be multiplied to 
 obtain corrections of the elements. 
 
 Symbol of 
 unknown. 
 
 Mercury 
 
 Factor ior 
 Venus. 
 
 . Mars. 
 
 Corr. of 
 element. 
 
 
 w 
 
 (*) 
 
 
 
 
 [ ] 
 
 1 
 
 0.1 
 
 7 
 
 0.3 
 
 dm : m Q 
 
 ( * ] 
 
 40 
 
 4 
 
 5 
 
 2 
 
 dl 
 
 1 JJ 
 
 30 
 
 3 
 
 6 
 
 2.5 
 
 dJ 
 
 [Nj 
 
 30 
 
 3 
 
 7 
 
 2.5 
 
 sin JdR 
 
 I ] 
 
 30 
 
 3 
 
 3 
 
 10^-7 
 
 de 
 
 f * ] 
 
 100 
 
 10 
 
 439 
 
 100-r7 
 
 d?r 
 
 1 * ] 
 
 100 
 
 2.056 
 
 3 
 
 1.3323 
 
 edn 
 
 1 * 1 
 
 10 
 
 1 
 
 4 
 
 4 
 
 de 
 
 ["] 
 
 6 
 
 0.6 
 
 2.5 
 
 2 
 
 de" 
 
 ["-] 
 
 6 
 
 0.6 
 
 2 
 
 2 
 
 e"dn" 
 
 I 1 
 
 10 
 
 1 
 
 1 
 
 5 
 
 Of 
 
 [ * ] 
 
 10 
 
 1 
 
 5 
 
 5 
 
 d 
 
 [I"] 
 
 10 
 
 1 
 
 4 
 
 3 
 
 dl" 
 
 The secular variation of each unknown in 25 years is 
 expressed sometimes by a suffixed 1, sometimes by an accent, 
 thus: 
 
 [1]' = [l]i = change of [I] in 25 years. 
 
 28. It may also be useful to give the values of the principal 
 coefficients in each of the normal equations. They are found 
 in the following table. Were the other coefficients all zero, 
 these numbers would indicate the weights of the different 
 unknown quantities as resulting from the solution. Several 
 of them were greatly diminished by the process of solution. 
 
28, 29] 
 
 ORDER OF ELIMINATION. 
 
 57 
 
 Values of the principal diagonal coefficients in the normal 
 
 equations. 
 
 
 
 I 
 
 rtercury. 
 
 
 
 Venus. 
 
 
 Mars. 
 
 Symbol c 
 
 f 
 
 
 
 
 
 
 
 From 
 
 oefficien 
 
 :. 
 
 From mer. 
 observa- 
 
 From 
 transits. 
 
 Sum. 
 
 From mer. 
 observa- 
 
 From 
 transits. 
 
 Sum. 
 
 mer. 
 observa- 
 
 
 
 tions. 
 
 
 
 tions. 
 
 
 
 tions. 
 
 mm 
 
 
 5488 
 
 o 
 
 5488 
 
 ^868 
 
 2929 
 
 8797 
 
 17887 
 
 11 
 
 
 I0 559 
 
 11308 
 
 21867 
 
 598i 
 
 3540 
 
 9521 
 
 20924 
 
 " JJ 
 
 
 15222 
 
 1296 
 
 16518 
 
 13232 
 
 7444 
 
 20676 
 
 28783 
 
 : NN ; 
 
 
 14176 
 
 2304 
 
 16480 
 
 I795I 
 
 1636 
 
 19587 
 
 32478 
 
 - ee '. 
 
 
 19015 
 
 5076 
 
 24091 
 
 5686 
 
 3350 
 
 9036 
 
 20119 
 
 7T 7T 
 
 
 8621 
 
 8352 
 
 16973 
 
 5290 
 
 1732 
 
 7022 
 
 20564 
 
 
 
 
 IIOOI 
 
 196 
 
 11197 
 
 11429 
 
 3598 
 
 15027 
 
 31460 
 
 '' e" e" '' 
 
 
 9757 
 
 508 
 
 10265 
 
 9586 
 
 665 
 
 10251 
 
 15909 
 
 'IT" TT" 
 
 
 9099 
 
 261 
 
 9360 
 
 5836 
 
 1895 
 
 7731 
 
 14911 
 
 ' r // r // 
 
 
 ^242 
 
 o 
 
 5242 
 
 
 
 
 
 ' /"/" = 
 
 
 Or 
 
 13041 
 
 542 
 
 13583 
 
 11031 
 
 2349 
 
 I338o 
 
 15427 
 
 aa 
 
 
 13230 
 
 
 
 13230 
 
 335 
 
 o 
 
 335 
 
 25138 
 
 r 66 '' 
 
 
 24657 
 
 
 
 24657 
 
 15196 
 
 o 
 
 15196 
 
 53975 
 
 11 ' 
 
 
 7014 
 
 67155 
 
 74169 
 
 6005 
 
 8983 
 
 14988 
 
 26689 
 
 JJ " 
 
 
 12366 
 
 9383 
 
 21749 
 
 9837 
 
 13014 
 
 22851 
 
 23440 
 
 ; NN ; 
 
 
 "35 
 
 16682 
 
 27717 
 
 14724 
 
 2874 
 
 17598 
 
 29494 
 
 ee ' 
 
 
 15437 
 
 29647 
 
 45084 
 
 5743 
 
 8610 
 
 '4353 
 
 24364 
 
 7T7T 
 
 
 6745 
 
 493 i 8 
 
 56063 
 
 4948 
 
 4483 
 
 943i 
 
 27131 
 
 ee 
 
 
 8488 
 
 1418 
 
 9906 
 
 8458 
 
 6306 
 
 14764 
 
 25675 
 
 : e " e ff " 
 
 
 8409 
 
 2937 
 
 11346 
 
 9805 
 
 1682 
 
 11487 
 
 22947 
 
 : TT" TT //= 
 
 
 8439 
 
 1513 
 
 9952 
 
 5242 
 
 4805 
 
 10047 
 
 17356 
 
 V'r'/ n 
 
 
 5432 
 
 o 
 
 5432 
 
 
 
 
 
 '. l " l " ". 
 
 
 11629 
 
 3126 
 
 I47S5 
 
 10677 
 
 5667 
 
 16344 
 
 20655 
 
 aa 
 
 
 11400 
 
 o 
 
 11400 
 
 297 
 
 o 
 
 297 
 
 33624 
 
 ; 66 \ 
 
 
 18716 
 
 o 
 
 18716 
 
 10772 
 
 o 
 
 10772 
 
 42405 
 
 NOTE. The coefficients for Mercury and Venus in this table are given as they 
 were used in the solution, after dropping the units from all the terms of the 
 equations, except those from transits of Mercury. 
 
 Order of elimination. 
 
 29. In dealing with so extensive a system of unknown 
 quantities it is impracticable to investigate the dependence of 
 each upon all the others. It is therefore essential to arrange 
 the unknowns in an order partly that of interdependence and 
 partly that of the liability of each to subsequent change by 
 discussion and adjustment. Hence, the mass of the planet. 
 Mercury or Venus, should be first eliminated, as being that 
 unknown which is least affected by changes in the final values 
 of the other unknowns. The secular variations, as derived 
 
58 MERCURY, VENUS, AND MARS. [29, 30 
 
 from meridian observations, are nearly independent of the 
 corrections to the other elements. The solar elements are to 
 be subsequently determined by a combination of the results 
 of the observations of the Sun and of the three planets. 
 
 Guided by these considerations, the order of elimination 
 was, with some exceptions, as follows : 
 
 1. The mass of the disturbing planet. 
 
 2. The five elements of the observed planet. 
 
 3. The four elements of the Earth's orbit. 
 
 4. The corrections to the star-positions for the mid-epoch. 
 
 5. The secular variations of the eleven quantities (2), (3), 
 and (4), taken in the same order. 
 
 Treatment of meridian observations of Mercury. 
 
 30. In the case of Mercury the factors of the coefficients of 
 the equations were chosen large enough to admit of the deci- 
 mals being dropped from the products without prejudice to 
 the accuracy of the final result. This was done to facilitate 
 the formation of the normal equations. For the same reason 
 the factors were made so small that the absolute numerical 
 values of the coefficients should generally not exceed 13. As 
 this degree of precision is far short of that usually employed 
 for correcting the elements of a planet, it may be well to set 
 forth the considerations on which it is based. 
 
 Let any equation of condition as actually used be 
 
 ax-\- by + cz + . . . =n (a) 
 
 Let the coefficients a, &, etc., be affected by the mean errors 
 e, ', etc., so that the true equation should be 
 
 . . . = n 
 This true equation may be written in the form 
 
 ax + by + . . . = nx e'y . . . (b) 
 
 We may regard (b) as a rigorous equation, in which the error 
 of the second member is increased by the quantity 
 
30] MERIDIAN OBSERVATIONS OF MERCURY. 5 
 
 and the only effect upon tlie precision of the results will be 
 that arising from this increased probable error. Let us esti- 
 mate its magnitude. From an examination of the tables used 
 in finding the coefficients I infer that the probable error of the 
 coefficient of n Avas 1, and that of all the other coefficients 
 0.6. The mean value of the unknown quantities was gener- 
 ally a small fraction of a second. We conclude, therefore, 
 that the probable or mean value of the error 
 
 ex fy i . . . 
 
 would in any case be only a small fraction of a second. More- 
 over, these errors would be purely accidental and not system- 
 atic, since the intervals of time between the equations were 
 generally so long that the coefficients for different equations 
 came from different tables, so that no error from omitted deci- 
 mals in any one equation would enter into the other equations. 
 
 Now, in view of the necessary systematic errors which affect 
 observations of the planets, there is no hope of approximating 
 to this degree of accuracy in the second members of the equa- 
 tions. Were the observations rigorously correct and the 
 values of the unknown quantities finally determined affected 
 by no error except that arising in this way, they would be 
 many times more accurate than we can hope to make them. 
 The errors might, in fact, be considered unimportant in the 
 present state of astronomy. 
 
 It has already been remarked that the scale of weights was 
 so taken that the unit of weight should correspond approx- 
 imately to a supposed mean error i 1".0 in the value of each 
 absolute term of an equation of condition, so far as the error 
 could be determined from the discordance of the original 
 observations. The corresponding probable error would be 
 dt 0".65. In the case of Mercury, however, modifications were 
 made which prevents this mean error from corresponding to 
 the unit of weight which would be found from the solutions in 
 the usual way. In the first place, the absolute members were 
 all multiplied by 10; in other words, the decimal point was 
 dropped from tenths of seconds, and no further account taken 
 of it. Secondly, in consequence of the probable error in the 
 coefficients of the normal equations arising from the irnperfec- 
 
60 MERCURY, VENUS, AND MARS. [30 
 
 tions of tlie decimals, the final values of these coefficients 
 would be subject to probable errors ranging between 50 and 
 100 units. In consequence there would be no advantage in 
 retaining the last figure in the normal equations, and it was 
 dropped in all the subsequent solution and discussion of these 
 equations. 
 
 In dropping the last figure from the absolute term of the 
 normal equations we may consider that we are merely drop- 
 ping the tenths of seconds and that the units are once more 
 expressed in seconds. Thus, considering only the effect of 
 this operation, the unit of weight would correspond to a mean 
 error of 1.0 in units of the absolute term. But in dropping 
 off the last figure from the coefficients we practically reduce 
 the scale of weights, considered as multipliers of the equa- 
 tions, to one-tenth of their former value. On the other hand, 
 in expressing the unknown quantities in terms of the correc- 
 tions to the elements, we divide the multipliers by ten, so that 
 effectively we multiplied the coefficients in the equations of 
 condition, considering the unknown quantities to be defined 
 as on page 56, by 10. Since these coefficients are of the second 
 degree in the normal equations, it follows that the scale of 
 weights has in effect been increased ten fold. Hence the unit 
 of weight for the normal equations between the unknown 
 quantities as finally solved will correspond to the mean error 
 
 l = 1.0 X VI6 = 3.1 
 
 As the mean error is at best a rather indefinite quantity in a 
 case like the present, we may consider its value as 4 units and 
 even then as by no means rigorously determined. 
 
 Up to the time of writing no attempt has been made to 
 derive rigorously the weights of the unknown quantities from 
 the solution, because in the cases of most of the uukowns such 
 weights would be entirely illusory. The fact is that in solving 
 so immense a mass of equations, we must expect systematic 
 errors to vitiate many of the results. The observations of 
 Mercury, especially of its Eight Ascension, are not made on 
 .a uniform system 5 sometimes the limb is observed, sometimes 
 the apparent center or the center of light. 
 
30, 31] TRANSITS OF MERCURY. 61 
 
 An ideally perfect system of reduction would require us to 
 reduce each separate observation with a semidiaineter corre- 
 sponding to the personal equation of the observer. This being 
 entirely impracticable, we must regard the reduction of the 
 observer's semidiameter to that used in the reductions as a 
 probable error. In fact, however, it will be of a systematic 
 character, varying at each point of the relative orbit of 
 Mercury, and going through a cycle of changes impossible to 
 determine in a synodic period of the planet. It is impracti- 
 cable to give even a full discussion of these errors; we shall, 
 however, meet with a proof of their magnitude. 
 
 Introduction of the equations derived from observed transits of 
 
 Mercury. 
 
 31. The relations between the elements of Mercury and the 
 Earth derived from this source are shown in my Discussion of 
 Transits of Mercury (A. P., Vol. I, Part VI.) On page 447 are 
 found expressions for those linear functions of the corrections 
 to the elements which are determined by the November and 
 May transits, respectively. With a slight change of notation 
 to correspond with that of the present paper, these functions 
 are as follows : 
 
 V = 1.487 61 - 0.487 dx - 1.137 tie - 1.01 dl" -f 1.19 e"dn" 
 
 + 1.58 de" 
 W = 0.716 61 + 0.284 drt + 0.896 de - 0.97 61" - 1.11 e"dn" 
 
 - 1.62 de" 
 
 The values of V and W being derived from a series of transits 
 extending from 1677 to the present time, enable us to deter- 
 mine both these quantities at some epoch, and their secular 
 variations. The values derived from the transits, together 
 with their mean errors, are found on page 460 of the work in 
 question. Omitting the doubtful factor fc, introduced on 
 account of a possible variability of the Earth's axial rotation, 
 which was not proved by the transits, the values of V and W 
 were found to be as follows : 
 
 V = 0".90 i 0".31 + ( - 2 // .63 0';.59) (T - 1820) 
 W == + 0".84 0".25 -f (+ 1".84 i 0".60) (T - 1820) 
 
62 MERCURY, VENUS, AND MARS. [31 
 
 The mean epoch for the transits is taken as 1820, to which 
 the zero values correspond. The values for 1865.0, the mid 
 epoch for the meridian observations, are, therefore, from the 
 transits alone 
 
 V = - 2".08 0".41 
 W = + 1".67 0".37 
 
 This, however, is only a first approximation to the quantities 
 which should be introduced. Since the meridian observations 
 help to determine the values of V and W, we should not 
 regard the reductions to 1865.0 as final, but retain the results 
 in the form (a). 
 
 Another element which is determined from the observed 
 transits of Mercury with greater precision than it can be from 
 meridian observations is the longitude of the node of the orbit 
 relatively to the Sun. In the paper quoted we have put 
 
 F = (30 -61"} sin i 
 and found from all the transits up to 1881, 
 
 N = - 0".16 i 0".27 + (0".28 dL 0".62) (T - 1820) (b) 
 
 The values of Y, W, and N, found from the discussion in 
 question, give rise to six conditional equations, which become 
 completely independent when we take as observed values the 
 secular motions and the absolute values at the mid-epoch of 
 observation. This mid-epoch is not the same for the May and 
 November transits. But I have assumed that no serious error 
 would be introduced by taking 1820.0 as the epoch for all three 
 of the quantities, Y, W, and X. 
 
 If we substitute for sin i 66 its value in terms of tfJ, etc., 
 namely, 
 
 Sin idB=- 0.6018 J + 0.796 sin JtfN + 0.721 de (c) 
 
 and then for tf J, tfN, tff, their values in terms of the unknowns 
 of the equations of condition, we shall have 
 
 N = - 1.805 [J] + 2.394 [N] + 0.721 [*] - 0.122 [V] (d) 
 
31] TRANSITS OF MERCURY. 63 
 
 Similar expressions will be found for the values of Y and W 
 by substituting for the corrections to the elements the unknown 
 quantities of the conditional equations, as already given. 
 
 Taking 1820.0 as the mid epoch, we may regard the inde- 
 pendent quantities given by the transits of Mercury to be the 
 six following ones : 
 
 Vo - 1.8 Y x ; W - 1.8 W,; N - 
 
 V, ; W, 5 N, - 
 
 Here Y , W , and N indicate values for 1865, the mid-epoch of 
 the meridian observations; and Y 1? W 1? and Nj. the variations 
 in 25 years. The six conditional equations thus found from 
 the transits may be written 
 
 Yo - 1.8 Yt = - 0".90 0".31 
 W - 1.8 W t = + 0".84 i 0".25 
 
 :N O - 1.8 N! = - 0".16 0".27 
 
 Y, = - 0".66 i 0".15 
 
 Wi = + 0".46 i 0".15 
 
 Ni = + /7 .07 /7 .15 
 
 Substituting for Y , YI, etc., their expressions as linear func- 
 tions of the unknowns of the conditional equations, we find 
 the following six equations, which are to be used as conditional 
 equations additional to those given by the meridian observa- 
 tions : 
 
 5.95 [1] - 4.87 [TT] - 3.41 [e] - 1.01 [l"\ + 0.71 [n"\ + 0.95 [e"] 
 -1.8)6.95[Z]! - 4.87 [ir]i 3.41 [e],- 1.01 [l"]i+ 0.71 [n"^ 
 + 0.95[e // ] 1 J = -O^O 
 
 Weight = 250 
 
 2.86 [1] + 2.84 [it] + 2.69 [e] - 0.97 [I 11 ] - 0.67 [n"\ - 0.97 [e"} 
 -1.8 {2.86 [I], + 2.84 [w-J! + 2.69 (e}, - 0.97 [l"^ - 0.67 [n"}, 
 -C.97^ 7 '],} = + 7/ .84 
 
 Weight = 300 
 
 - 1.8 [J] + 2.4 [N] + 0.7 [f] - 0.12 [I"} 
 - 1.8 { - 1.8 [ J] x + 2.4 [NJi + 0.7 [f ]x - 0.12 [/ // ] l } = - 0".16 
 
 Weight = 400 
 
64 MERCURY, VENUS, AND MARS. [31 
 
 5.95 [1], - 4.87 [TT]! - 3.41 [e], - 1.01 [l"^ + 0.71 [*"]i+ 0.95 
 = - 0".66 
 
 Weight = 700 
 
 2.86 [l]t + 2.84 [TT]! + 2.69 [e]i - 0.97 [Z"]i - 0.67 [TT"], - 0.97 [a' 7 ]! 
 = + 0".46 
 
 Weight = 700 
 
 -1.8 [J], + 2.4 [N] { + 0.7 [e], - 0.12 [I"}, = + 0".07 
 Weight = 1,600 
 
 The weights assigned to these several equations have been 
 determined by the following considerations: 
 
 We have already found that in the equations of condition 
 from the meridian observations as finally reduced, the scale of 
 weights has so come out as to show a practical mean error for 
 weight unity of about 4". Were this error purely accidental, 
 the weights of the conditional equations derived from the 
 transits would be determined in the same way, from the mean 
 errors assigned to them. But, as a matter of fact, the exist- 
 ence of systematic errors in the meridian observations is 
 shown, as will be subsequently explained, by the large value 
 found for the fictitious quantity 6r 2 . Since observations of 
 transits are made at the point of the relative orbits of Mercury 
 and the Earth, near which meridian observations are rarely 
 available, and are of a higher order of accuracy than meridian 
 observations, it follows from the theory of probabilities that 
 we should assign a larger relative weight to the observations 
 of the transits. How much larger does not admit of being 
 determined with numerical precision. Actually I have taken 
 the weights as if the mean error corresponding to weight 
 unity were between 5 and 6. In the case of the motion of the 
 node a still larger weight has been assigned to the secular 
 variation, from the belief that the accuracy of the determina- 
 tion from transits relative to meridian observations is in this 
 case of a yet higher order of magnitude than in the case of 
 
31, 32] SOLUTION OF EQUATIONS FOR MERCURY. 65 
 
 the other elements. Whether this belief is justified or not 
 must be left to the decision of the future astronomer. 
 
 The first three of the preceding six conditional equations 
 may be treated in a way similar to that adopted for the 
 meridian observations. They express what is supposed to be 
 equivalent to observations of the three quantities V, W, and 
 N in 1820, when r 1.8. Hence, from the partial normals 
 in the six principal unknowns, [e], [>]... [>"], the com- 
 plete normals may be formed by multiplication by r and i* 
 (r = 1.8) in the way set forth in 25. 
 
 Solutions of the equations for Mercury. 
 
 32. In the case of Mercury and Venus, it is desirable to 
 know to what extent the results of the transits diverge from 
 those of the meridian observations. Hence, as already 
 remarked, two solutions of the equations were made, termed 
 A and B. 
 
 Solution A is that derived from the meridian observations 
 alone. Solution B is that of the normal equations formed 
 from both the meridian observations and the transits. 
 
 The results of the solutions in the case of Mercury are shown 
 in the following tables. The relation of the unknown quan- 
 tities given in the first columns, A and B, to the corrections 
 of the elements has been shown in a preceding section ( 27). 
 The upper half of the table shows the corrections to the 
 elements; the lower half those of the secular variations. 
 
 It will be seen that all the values, with a single exception, 
 come out less than a unit. In stating the corrections to the 
 elements, it must be remembered that, owing to the proximity 
 of Mercury to the Sun, the errors of geocentric place are much 
 less than those of the heliocentric elements, so that an error 
 in the latter indicates a proportionally smaller error in the 
 actual observations. For the same reason we must expect a 
 less degree of precision in the elements as finally derived than 
 in the case of the other planets. 
 5690 N ALM 5 
 
66 MERCURY, VENUS, AND MARS 
 
 MERCURY. 
 
 Results of solutions of the normal equations. 
 
 [32, 33 
 
 Unknowns. 
 
 EM 
 
 Corrections of elements. 
 
 Symbol. 
 
 A. 
 
 B. 
 
 Symbol. 
 
 A. 
 
 B. 
 
 ["] 
 
 o. 1478 
 
 o. 1207 
 
 O. I 
 
 6 m : m 
 
 o. 0148 
 
 O. OI2I 
 
 
 ' / 
 
 
 o. 1342 
 
 0.0752 
 
 4- 
 
 ii 
 
 o. 537 
 
 0.301 
 
 
 : j 
 
 
 o. 2436 
 
 o. 2299 
 
 3- 
 
 6] 
 
 o. 731 
 
 o. 690 
 
 
 ;N 
 
 
 o. 0227 
 
 0. 0201 
 
 3- 
 
 SinJdN 
 
 -o. 068 
 
 o. 061 
 
 t 
 
 
 
 
 -fo. 2074 
 
 +o. 2194 
 
 i. 
 
 6 
 
 4-o. 207 
 
 4-0.219 
 
 
 e 
 
 
 O. I2O2 
 
 4-0. 4094 
 
 3- 
 
 6e 
 
 o. 361 
 
 41.228 
 
 
 ' TT 
 
 
 4-0. 5209 
 
 4-0. 2688 
 
 10. 
 
 6 7T 
 
 +5- 209 
 
 4-2. 689 
 
 
 ' e" 
 
 
 -f o. 0669 
 
 4-0. 8397 
 
 0.6 
 
 (5 e f/ 
 
 -f-o. 040 
 
 -f o. 504 
 
 
 V' 
 
 
 o. 2248 
 
 o. 7027 
 
 0.6 
 
 e" 6 TT // 
 
 o. 135 
 
 o. 422 
 
 
 >// 
 
 
 4-1. 1240 
 
 4-1.0566 
 
 2. 
 
 6r 
 
 4-2. 248 
 
 4-2. 113 
 
 
 6 
 
 
 o. 2310 
 
 o. 2556 
 
 I. 
 
 6 
 
 0.231 
 
 o. 256 
 
 
 /" 
 
 
 -0-0354 
 
 o. 0897 
 
 I. 
 
 61" 
 
 -o. 035 
 
 o. 090 
 
 
 a 
 
 
 4-0. 4803 
 
 4-0. 4930 
 
 I. 
 
 a 
 
 -fo. 480 
 
 +o- 493 
 
 
 / 
 J 
 
 
 o. 2060 
 o. 0114 
 
 o. 1209 
 -f-o. 0636 
 
 1 6. 
 
 12. 
 
 D t # 
 
 3. 296 
 
 o. 137 
 
 i. 935 
 
 -f o. 764 
 
 
 N; 
 
 
 4~o. looo 
 
 4~o. 0930 
 
 12. 
 
 SInjDt<m 
 
 -|-i. 200 
 
 4-1. 116 
 
 
 " e ~ 
 
 
 4-o. 0681 
 
 4-o. 0966 
 
 4- 
 
 Dt 6 e 
 
 4-0. 272 
 
 4-0. 386 
 
 
 e 
 
 
 o. 1165 
 
 +o. 0987 
 
 12. 
 
 D t 6e 
 
 -i. 398 
 
 4-1.184 
 
 
 7T 
 
 
 o. 2385 
 
 o. 0252 
 
 40. 
 
 D t d7T 
 
 9- 540 
 
 1.008 
 
 
 e" 
 
 
 o. 1968 
 
 +- I 3 I 7 
 
 2.4 
 
 D t 6 e // 
 
 o. 472 
 
 4-0.316 
 
 
 7T // " 
 
 
 o. 1677 
 
 o. 1193 
 
 2.4 
 
 e/s D t 6 TT // 
 
 o. 402 
 
 o. 286 
 
 
 r" 
 
 
 4-o. 1108 
 
 4-o. 0806 
 
 8. 
 
 D t dr" 
 
 4-0. 886 
 
 4-0. 645 
 
 
 6 
 
 
 o. 1826 
 
 o. 1233 
 
 4- 
 
 D t <5 
 
 o. 730 
 
 o. 493 
 
 
 I" 
 
 
 o. 1442 
 
 0.3152 
 
 4- 
 
 D t d/ x/ 
 
 -o. 577 
 
 i. 261 
 
 
 a 
 
 
 o. 3160 
 
 o. 1973 
 
 4- 
 
 D t a 
 
 i. 264 
 
 -o. 789 
 
 Mean epoch of corrections, 1865.0. 
 
 Discordance in the observed Right Ascensions of Mercury. 
 
 33. The most remarkable feature in the result is the value 
 of the quantity represented by [r"\. The unknown quantity 
 introduced with this symbol had as its coefficient the derivative 
 of the geocentric place as to the Earth's radius vector, and the 
 result would therefore be an apparent constant correction to 
 that radius vector. Since, however, the position of the planet 
 depends only on the ratio of the distances of the Earth and 
 Mercury, it follows that the actual correction may be regarded 
 as a correction to the ratio of the mean distances. 
 
 The determination of the mean distances by KEPLER'S 
 third law may be regarded as so unquestionable that the true 
 
33] DISCORDANCE OF OBSERVATIONS. 67 
 
 value of this unknown quantity should be regarded as zero, 
 and the result as a purely fictitious one, arising from errone- 
 ous elements of reduction or systematic personal errors. It 
 was the possibility of the latter that led to its introduction. 
 When the planet is east of the Sun, observations are always 
 made on or near its west limb, or at least on some point west 
 of the true center, and vice versa. The value of dr" therefore 
 indicates that there is a remarkable systematic difference in 
 the observed Eight Ascension according as the planet is east 
 or west of the Sun, and therefore according to the illuminated 
 side. The sign of the result shows that the reduction to the 
 center of the planet was apparently too small. It is there- 
 fore of interest to learn according to what law this error 
 changed as the planet moved around its relative orbit. 
 
 It has up to the present time been impracticable to substi- 
 tute the unknown quantities in the original equations of con- 
 dition, and thus determine the separate residuals, and for the 
 purpose of investigating the present case such a substitution 
 is the less necessary, owing to the sniallness of the unknown 
 quantities. I have therefore simply determined the mean 
 correction to the Right Ascension given by all the observa- 
 tions during the various periods in six segments of the relative 
 orbit, near the elongations, and before and after the two 
 conjunctions. The results are shown in the following table. 
 Commencing with the moment of inferior conjunction, column 
 A contains the mean correction to the tabular Eight Ascension, 
 from observations made within about twenty days following. 
 Column B contains the observations made from twenty days 
 after the inferior conjunction until twenty days before superior 
 conjunction, a period during which the planet was generally 
 near its greatest west elongation. Column C contains the 
 observations made during the twenty days following and up 
 to superior conjunction. Then follow in regular order the 
 corresponding results when the planet was east of the Sun, 
 beginning with the twenty days following superior conjunc- 
 tion and going around to inferior conjunction. 
 
68 
 
 MERCURY, VENUS,- AND MARS. 
 
 [33 
 
 Table showing the mean corrections to the tabular Right Ascen- 
 sion of Mercury in six segments of its relative orbit. 
 
 Epochs. 
 
 A 
 
 B 
 
 C 
 
 1765-1791. 
 
 " wt. 
 
 4-3. 24 4 
 
 " wt. 
 
 4-2. 61 5 
 
 " wt. 
 
 1793-1815. 
 
 -(-2.06 6 
 
 4-1.82 10 
 
 -f o. 97 4 
 
 1817-1839 
 
 +3- 06 6 
 
 4-1.79 24 
 
 -|-i. 13 24 
 
 1840-1849 
 
 + 1.46 6 
 
 + 1.48 18 
 
 o. 38 20 
 
 1850-1859 
 1860-1869. 
 
 4-3-72 4 
 4-1. 18 28 
 
 4-0. 77 20 
 + 1.14 72 
 
 4-o. 08 1 6 
 4-O. 31 44 
 
 1870-1880 
 
 + i. 18 25 
 
 4-o. 74 65 
 
 o. 20 6 1 
 
 1881-1892 
 
 4-1. 19 38 
 
 4-0. 98 63 
 
 o. 15 62 
 
 
 
 
 
 
 D 
 
 E 
 
 F 
 
 1765-1791 
 
 " wt. 
 4-0.92 . i. 5 
 
 " wt. 
 -{-1.30 10 
 
 " wt. 
 
 +o. 81 3 
 
 1793-1815. 
 
 +2. 82 5 
 
 4-i. 10 16 
 
 + 1.85 5 
 
 1817-1830 
 
 4-O. 27 25 
 
 4-3. 76 24 
 
 i. 20 ; 
 
 1 84.0 1 840 
 
 j O 22 22 
 
 o S7 30 
 
 j_o 71; 3 
 
 i8co 18^0 
 
 J-O 60 14. 
 
 O 7Q 28 
 
 O 6< 4. 
 
 1860-1869 
 
 o. 44 5 "> 
 
 o. <u 69 
 
 V. \J^ ^ 
 o. 35 1 6 
 
 1870-1880 
 
 o. 52 57 
 
 jj _? 
 i. 25 67 
 
 o. 30 24 
 
 1881-1892 
 
 o. 84 80 
 
 o. 73 102 
 
 o. 37 26 
 
 
 
 
 
 The remarkable feature of these results is the near approach 
 to constancy in the values of the numbers in each column, 
 after the secular variation is allowed for, and the large magni- 
 tude of the corrections. The most natural conclusion is that 
 the reduction from the limb of the planet or the observed 
 center of light to the true center was too small by an amount 
 which, at the mean distance of the Sun, must have been nearly 
 or quite a second of arc (cf. 3). The adopted semidiameter 
 3".4 seems so well established, both by micrometric measures 
 and by heliometer measures during transits of Mercury, that 
 such a correction to the diameter seems inadmissible. 
 
 I have not yet been able to enter upon the investigation of 
 the source of this anomaly. A very important question is that 
 of its influence on the results. Since a constant error in the 
 radius vector of a planet would have opposite effects on the 
 elements in different points of the relative orbit, it may be 
 inferred that the effect of the error would be nearly eliminated 
 
33, 34] COMPARISON OF OBSERVATIONS OF MERCURY. 69 
 
 in an extensive series of observations distributed equally 
 between the two elongations. Actually, however, there seems 
 to have been an appreciable lack of symmetry in this respect, 
 as the influence of the unknown quantity upon the other 
 unknowns is not inconsiderable. Although the law of change, 
 as shown in the preceding table, does not correspond to the 
 magnitude of the coefficient of 6r", this coefficient being rela- 
 tively too small near inferior conjunction and too large near 
 superior conjunction, it is still probable that through the intro- 
 duction and elimination of dr" a large part of the injurious 
 effect is eliminated. 
 
 Comparison of transits and meridian observations of Mercury. 
 
 34. Another remarkable result which may be associated with 
 this is shown by the difference between the solutions A and B, 
 in the case of the eccentricity and perihelion not only of the 
 planet, but of the Sun. It will be seen that the meridian 
 observations alone give a negative correction to the eccen- 
 tricity of the planet, while, when the transits are included, 
 the correction becomes positive. That this is due to a system- 
 atic cause running through the observations is shown by the 
 fact that the same thing is true of the secular variation of 
 the eccentricity. This relation of the correction to its secular 
 variations holds true for three of the four relative elements, 
 and for the eccentricity and perihelion both of the planet and 
 of the Earth. In the case of the Earth's perihelion, however, 
 there is a nearer approach to conformity between the two 
 results. 
 
 There is yet another anomaly in this connection, which indi- 
 cates a very considerable systematic error in the older meridian 
 observations, which is not completely eliminated from the ele- 
 ments. If we take the values of the unknown quantities and 
 their secular variations, which result from the two solutions, 
 and substitute them in the linear functions of the corrections 
 to the elements derived from the transits alone, namely 
 
 V = 1.487 dl - 0.487 drr - 1.137 de - 1.01 dl" + 1.19 e"dn" 
 
 + 1.58 de" 
 
 W = 0.716 dl + 0.284 dn + 0.896 de - 0.97 dl" - 1.11 e"d7t" 
 
 - 1.62 de" 
 
70 MERCURY, VENUS, AND MARS. [34, 35, 36 
 
 we find the following results : 
 
 From meridian observations V = 2".99 + 0".69T 
 From November transits 1 .69 2 .63 T 
 
 From combined solution 2 .77 2 .30 T 
 
 From meridian observations W = -f- 0".89 4".55 T 
 From May transits alone +1 .39 + 1 .84 T 
 
 From combined solution +1 .39 + .42 T 
 
 We conclude that, had no transits ever been observed, the 
 errors of the elements and their secular variations, derived 
 from the great mass of meridian observations, would have 
 caused an error of some 5" per century in the heliocentric 
 place of the planet at the times of the May transits, and of 
 some 3" at the time of the November transits. 
 
 The fact that the combined solution B satisfies the transits 
 so much better than A, although the total weight of equations 
 A is so much greater than that of the transit equations, shows 
 that the meridian observations give only weak results for the 
 functions in question. 
 
 Meridian observations of Venus. 
 
 35. So far as the meridian observations are concerned, those 
 of Venus were treated on the same general plan as the observa- 
 tions of Mercury. The following are the principal points of 
 difference : 
 
 1. The hypothetical quantity dr" is omitted. Hence no 
 index to the consistency of the observations at different points 
 of the relative orbit can be derived from the solution. 
 
 2. Tenths of a unit were included in the coefficients of the 
 equations, and no modification was made in the units. The 
 units and tenths were, however, dropped in the final solution 
 of the normal equations. 
 
 Results of observed transits of Venus. 
 
 36. We put, at the time of a transit, 
 
 v, the longitude in orbit of Venus ; 
 Z, its mean longitude, or the mean vame of v; 
 fi, A, its ecliptic latitude and longitude; 
 L, the Sun's true longitude. 
 
36 ] EQUATIONS OF CONDITION FROM TRANSITS OF VENUS. 71 
 
 Then 
 
 tf A = cos i 6v + sin 2 i 66 
 
 = 0.99S2 dv + 0.0592 sin i 36 
 
 We thus have, for the dates of the observed transits, 
 
 1761-'69 ; dp = 0.0592 6v + 0.9982 sin i S6 
 1874->82 ;/?= + 0.0592 Sv - 0.9982 sin * dd 
 
 I have discussed very fully the observations of the transits 
 of 1761 and 1769 in Astronomical Papers, Vol. n. The final 
 results which I shall use are found on page 404 of that volume. 
 Here I have put. 
 
 x, correction to A L ; 
 y, correction to /?, 
 
 the Sun's latitude being supposed to require no correction. 
 The values of x and y for 1769 are distinguished by an accent. 
 I have also represented by z 2 and 2 3 the corrections to the dif- 
 ference of the semidiameters of the Sun and planet, for the 
 respective internal contacts, to which may be added the. un- 
 known but probably nearly constant quantity due to personal 
 error in estimating the time of contact. From their very nature 
 these quantities do not admit of accurate determination, and 
 must therefore be eliminated from the equations. From the 
 observations of internal contact are derived the following four 
 equations : 
 
 1761 II; ,87a? + .50 y + z 2 = - 0".07 
 HI ; + .68 + .73 + 2 3 = - 0".06 
 
 1769 II; - .64 a? 7 - .11 y' + z 2 = - 0".27 
 III; + .84 - .55 + 2 3 = + 0".02 
 
 We have here more unknown quantities than equations, so 
 that it is not practicable to determine them all separately. 
 What I have done has been first to assume ^ = 2 3 . This pre- 
 supposes that the distance of centers at the estimated appa- 
 
72 MERCURY, VENUS, AND MARS. [36 
 
 rent contact at egress is, in the general mean, the same as at 
 ingress. The result of any error in this hypothesis will be 
 almost completely eliminated from the mean latitude at the 
 two transits, but not from the longitude. 
 
 Still, the values of x and y can not be separately determined; 
 I have therefore so combined the equations as to obtain mean 
 values of x and y for the two contacts, assuming that this 
 would be the result of supposing these quantities to have the 
 same, values at both epochs. Calling these values x" and y", 
 we have by addition and subtraction, supposing z 2 = 23, 
 
 - 0.39 a?" + 2.55 y" = 0".12 
 3.03 #"4- 0.45 y" = 0".30 
 
 We thus have* 
 
 x" = + 0".09 
 y 1 ' = + 0".06 
 
 These corrections are not applicable to the coordinates from 
 LEVERRTER'S . tables as they stand, but to those quantities 
 as corrected by the following amounts : 
 
 A\ = 4- 0".25 
 Afi= 4-2".00 
 
 * In a second approximation to these quantities, which may be made 
 after the correction to the enteimial motion of the node is determined, 
 we should put, on account of this correction, 
 
 The solution would then give 
 
 I have carried through a more careful approximation in a subsequent 
 chapter. 
 
36] EQUATIONS OF CONDITION FROM TRANSITS OF YENUS. 73 
 
 We thus find, for the corrections to LEVERRIER'S tables at 
 the epoch 1765.5, 
 
 d A - L = + 0".09 + 0".25 = + 0".34 
 3 = _ 0".06 2".00 .= + 1".94 
 
 and hence 
 
 dv = + 0".22 + 0".998 L 
 sin i 6 S = + 1".95 + // .059 6L 
 
 A still farther modification is required to the tabular longi- 
 tude on account of the correction to the mass of the Earth 
 used by LEVERRIER, and hence to the periodic perturbations 
 in longitude. This correction is + 0".20. We thus have for 
 the correction to the orbit longitude of Venus 
 
 dv = + 0".02 4- 0".998 6 L 
 
 For the results of the transits of 1874 and 1882 I have 
 depended entirely on the heliometer measures and photo- 
 graphs made by the German and American expeditions, 
 respectively. The definitive results of the German observa- 
 tions, as worked up by Dr. AUWERS, are found in Vol. V of 
 the German Keports on the Transits.* The American photo- 
 graphic measures of 1874 have not been officially worked up 
 and published, but a preliminary investigation from the data 
 contained in the published measures was made by D. P. TODD, 
 and published in the American Journal of Science, Vol. 21, 
 1881, page 491. The measures of 1882 have been definitively 
 worked up by HARKNESS, but only the results published. 
 They are found in the report of the Superintendent of the 
 TJ. S. Naval Observatory for the year 1890. 
 
 The corrections to the geocentric Eight Ascension and 
 Declination of Venus relative to the Sun thus derived are 
 
 * Die Venus-durchgiinge 1874 und 1882 Bericht uber die Deutsclien 
 Beobachtungen Fuufter Band, Berlin, 1893. 
 
74 MERCURY, VENUS, AND MARS. [36 
 
 given in the following table. In taking the mean the weights 
 are not strictly those which would result from the probable 
 errors as assigned, but, in accordance with a general princi- 
 ple, independent results have received a weight more near to 
 equality than would be indicated by the mean errors. 
 
 1874: German, 6 E. A. = + 4.77 0.28 
 American, . . . + 4.14 0.30 
 
 Adopted, . . . +4.44 
 
 German, 6 Dec. = + 2.28 + 0.10 
 American, . . . -f 2.50 0.30 
 
 Adopted, . . . +2.34 
 
 1882: German, 6 E. A. = + 9.03 + 0.12 
 American, . . . + 9.10 + 0.08 
 
 Adopted, . . . +9.07 
 
 German, d Dec. = + 2.02 i 0.06 
 American, . . . +2.02 + 0.08 
 
 Adopted, . . . +2.02 
 
 We change these results successively to geocentric longi- 
 tude and latitude, heliocentric longitude and latitude, and 
 orbital longitude and latitude. The results of these several 
 changes are as follow: 
 
 Corr. in geoc. long. 
 Corr. in lat. 
 
 1874. 
 + 3''.853 
 
 + 2 .724 
 
 1882. 
 + 8".077 
 
 + 2. 971 
 
 Corr. in hel. long. 
 Corr. in hel. lat. 
 
 -1 .415 
 + 1 .001 
 
 -2 .965 
 + 1 .091 
 
 Corr. in orbital long. 
 Value of sin i 6 
 
 -1 .35 
 -1 .08 
 
 -2 .90 
 
 -1 .26 
 
37] EQUATIONS FROM TRANSITS OF VENUS. 75 
 
 Equations from transits of Venus. 
 
 37. The corrections to the heliocentric positions of Yenus 
 and the Earth, as thus found, are now to be expressed in 
 terms of corrections to the elements. The results of this 
 expression are shown in the following equations: 
 
 Equations given by the corrections to the orbital longitude. 
 
 I. Epoch, 1765.5; T= 3.90 ; weight = 200 
 
 0.992 61 + 1.17 eSn + 1.62 de - 0.976 61" 1.81 e"dn"- 0.85 de" 
 
 = +0".02 O."15 
 
 II. Epoch, 1874.9 ; r = + 0.48; weight = 400 
 
 - 0".88/* + 1.009 61 - 1.223 edn - 1.596 6e - 1.030 61" 
 $*" + 0.817 6e" = - 1".35 0".08 
 
 III. Epoch, 1882.9; T = + 0.80; weight = 800 
 
 0".60,w+ 1.008 61 - 1.146 edn - 1.651 6e - 1.028 <M"+ 1.825 6"$w" 
 
 + 0.900 $6" = 2".90 i 0.' / 027 
 
 Equations given by the corrections to the orbital latitude. 
 
 I. 1765.5; sin i66- 0.057 <"- 0.11 ^^^"-O.OS^^ + 1".95 
 
 i 0".10 
 
 II. 1874.9; sinid9-0. 
 
 /7 .04 
 
 III. 1882.9; sin^^-0. 
 
 i 0".019 
 
 The weights assigned to these three equations are, respec- 
 tively, 200, 600, and 1,600. 
 
 Before using these equations the corrections to the elements 
 were transformed into the unknown quantities denned in 27, 
 and their secular variations by multiplying the coefficients by 
 the factors given on page 56. 
 
76 
 
 MERCURY, VENUS, AND MARS. 
 
 [38, 39 
 
 Solutions of the equations for Venus. 
 
 38. The parts of the normal equations formed from the 
 preceding conditional equations were added to the parts from 
 the meridian observations, and the resulting solution B 
 obtained. As in the case of Mercury, a solution A was made 
 of the normal equations derived from the meridian observa- 
 tions alone. The results are as follows : 
 
 VENUS. 
 Results of solutions of the normal equations. 
 
 Unknowns. 
 
 Factors. 
 
 Corrections of elements. 
 
 Symbol. 
 
 A. 
 
 B. 
 
 Symbol. 
 
 A. 
 
 B. 
 
 [>] 
 
 o. 0708 
 
 o. 0834 
 
 7- 
 
 <5 m : m 
 
 o. 496 
 
 0.^584 
 
 
 ' / ' 
 
 
 o. 1435 
 
 o. 1501 
 
 5- 
 
 61 
 
 -0.718 
 
 0-751 
 
 
 " J ' 
 
 
 +o. 1156 
 
 +o. 1340 
 
 6. 
 
 6 T 
 
 +o. 694 
 
 +o. 804 
 
 
 N; 
 
 
 4-o. 0164 
 
 -j-o. 0106 
 
 7- 
 
 sinJrfN 
 
 +o. 115 
 
 +o. 074 
 
 
 e 
 
 
 +o. 0941 
 
 -j-o. 1003 
 
 3- 
 
 fit 
 
 +o. 282 
 
 -|-o. 301 
 
 
 7T ] 
 
 
 4-o. 0628 
 
 -j-o. 0764 
 
 3- 
 
 e6K 
 
 4-o. 1 88 
 
 +o. 229 
 
 
 e 
 
 
 4-o. 0246 
 
 +0.0271 
 
 4- 
 
 6e 
 
 -j-o. 098 
 
 4-o. 1 08 
 
 
 ~ e "\ 
 
 
 4-o. 0336 
 
 4-o. 0318 
 
 2-5 
 
 5e" 
 
 4-o. 084 
 
 +o. 080 
 
 
 ~ir"' 
 
 
 o. 0274 
 
 0.0212 
 
 2. 
 
 e"tv" 
 
 o. 055 
 
 o. 042 
 
 
 ' a 
 
 
 +o. 4742 
 
 +o. 4642 
 
 I. 
 
 a 
 
 +o. 474 
 
 +o. 464 
 
 
 i 6 '' 
 
 
 o. 0383 
 
 -o. 0375 
 
 5- 
 
 6 
 
 o. 192 
 
 o. 1 88 
 
 
 */// 
 
 
 o. 0768 
 
 o. 0743 
 
 4- 
 
 61" 
 
 o. 307 
 
 -o. 297 
 
 
 ' / ' 
 
 
 o. 1846 
 
 -o. 1983 
 
 20. 
 
 D t d/ 
 
 3. 692 
 
 3. 966 
 
 
 :*: 
 
 
 +o. 0970 
 o. 0561 
 
 +o. 1088 
 o. 0594 
 
 2 4 . 
 
 28. 
 
 DtJ 
 sinJD t N 
 
 +2. 328 
 I-57I 
 
 +2.611 
 -1.663 
 
 
 e 
 
 
 +o. 1472 
 
 +o. 1644 
 
 12. 
 
 Dt e 
 
 + 1.766 
 
 + r -973 
 
 
 TT 
 
 
 +0.0555 
 
 4-o. 0698 
 
 12. 
 
 ^D t 7r 
 
 +o. 666 
 
 +o. 838 
 
 
 
 
 
 -(-o. 0182 
 
 -j-O. O2O2 
 
 1 6. 
 
 Dte 
 
 +o. 291 
 
 +o. 323 
 
 
 "e"\ 
 
 
 4-o. 0283 
 
 +0.0317 
 
 10. 
 
 Dt^ x/ 
 
 +o. 283 
 
 
 
 7T // 
 
 
 +o. 0399 
 
 +o. 0506 
 
 8. 
 
 e" Dt TT // 
 
 +o. 3 J 9 
 
 4-o. 405 
 
 
 a 
 
 
 o. 0820 
 
 -o. 0347 
 
 4- 
 
 D t a 
 
 o. 328 
 
 o. 139 
 
 
 ! <* s 
 
 
 0. 0020 
 
 O. OOO2 
 
 20. 
 
 Dt d 
 
 o. 040 
 
 o. 004 
 
 
 L/// - 
 
 
 --o. 0562 
 
 o. 0662 
 
 1 6. 
 
 EM/" 
 
 o. 899 
 
 i. 059 
 
 Mean epoch of correction, 1863.0 
 
 Comparison of transits of Venus with meridian observations. 
 
 39. To show to what extent the results of the meridian 
 observations differ from those of the observed transits over 
 the Sun, we form the values of the absolute terms of the 
 equations of condition, 37, first by substituting the values 
 A of the corrections, and then the values B. We thus have : 
 
39, 40] EQUATIONS FROX TRANSITS OF VENUS. 77 
 
 Residuals in orbital longitude. 
 
 1765.5. 1874.9. 1882.0. 
 
 (a) From meridian obs. alone . - 0".07 - 1".36 - 2".54 
 
 ( ft) From combined solution . + 0".04 - 1".43 - 2".78 
 
 (y) From transits alone . . . + 0".02 - I". 35 - 2".90 
 
 Discordance, (y)-(tf) . . +0".09 + 0".01 - 0".36 
 
 Discordance, (y)- (ft) . . - 0".04 + 0.08 -0".12 
 
 Residuals in orbital latitude. 
 
 1765.5. 1874.9. 1882.9. 
 
 (a) From meridian obs. alone . + 1".92 - 0".77 - 0".96 
 
 (ft) From combined solution . + 2".06 - 0".91 - 1".12 
 
 (y) From transits alone . . . + 1".95 - 1".08 - 1".26 
 
 Discordance, (y)- () + 0".03 - 0".31 - 0".30 
 
 Discordance, (y) (ft) . . - 0".ll - 0".17 - 0".14 
 
 It will be seen that the combined solution represents the 
 observations of the transits much better here than in the case 
 of Mercury. 
 
 Solution of the equations for Mars. 
 
 40. As the formation of the normal equations for Mars was 
 approaching its end, a singular discordance among the resid- 
 uals of the partial normal equations for different periods was 
 noticed. On tracing the matter out it appeared that while the 
 correction of the geocentric longitude of LEVERRIER'S tables 
 in 1845 and again in 1892 was quite small, the correction in 
 1862 was considerable. Now there is an inequality of long 
 period, about forty years, in the mean motion of Mars, depend- 
 ing on the action of the Earth, and having for its argument 
 15# 7 Sg. This coefficient is of the seventh order in the eccen- 
 tricities, and the terms of the ninth or even of the eleventh 
 order might be sensible in a development in powers of the 
 eccentricities and sines or cosines of multiples of the mean 
 longitudes. The conclusion which I reached was that the the- 
 oretical value of this coefficient was not determined with suffi- 
 cient precision. As the work of solving the equations could 
 not wait for a new determination and a new formation of the 
 absolute terms of the normal equations, it was decided to make 
 an approximate empirical correction to the theory. This was 
 used to correct the absolute terms of the partial normal equa- 
 
78 
 
 MERCURY, VENUS, AND MARS. 
 
 tions for each period, and the solution was then proceeded 
 with. The chances seein to be that by this process the inju- 
 rious effect of the error upon the elements derived from the 
 equations would be inconsiderable; this is, however, a point 
 on which it is impossible to speak with certainty. It is the 
 intention of the writer to recompute the doubtful terms of the 
 perturbations, and, if possible, reconstruct the absolute terms 
 of the normal equations in accordance with the corrected 
 theory. Meanwhile, the present work necessarily rests on the 
 imperfect theory with the approximate empirical corrections, 
 which are as follow : 
 
 M = 0".30 cos (150' - 8-7 - 223) 
 edn = 0".15 cos (150' 80) 
 
 As the elements of Mars are derived wholly from meridian 
 observations, only one set of equations of condition was formed. 
 The results of the solution are shown in the following table : 
 
 MARS. 
 
 Unknowns. 
 
 Factors. 
 
 Corrections of elements. 
 
 Symbol. 
 
 Value. 
 
 Symbol. 
 
 Value. 
 
 [ tn/ ~\ 
 
 . 02278 
 
 0-3 
 
 6 m : m 
 
 -o. 007 
 
 
 1 ] 
 
 
 . 44854 
 
 2. 
 
 61 
 
 -o. 897 
 
 
 N; 
 
 
 + 05479 
 +.06724 
 
 2-5 
 2-5 
 
 SinJJN 
 
 +o. 137 
 +o. 1 68 
 
 
 e 
 
 
 + .43803 
 
 V 
 
 6 e 
 
 +o. 626 
 
 
 TT 
 
 
 . 05056 
 
 1^-Q 
 
 6w 
 
 o. 722 
 
 
 e 
 
 
 +.07474 
 
 4- 
 
 6s 
 
 +o. 299 
 
 
 e"\ 
 
 
 . 49898 
 
 2. 
 
 be" 
 
 o. 998 
 
 
 V /x " 
 
 
 .42409 
 
 2. 
 
 e' f 6 K" 
 
 o. 848 
 
 
 a 
 
 
 + 18545 
 
 5- 
 
 a 
 
 +o. 927 
 
 
 ; 6 ' 
 
 
 . 04536 
 
 5- 
 
 6 
 
 o. 227 
 
 
 
 ///' 
 
 
 + .05786 
 
 3- 
 
 61" 
 
 +o. 174 
 
 
 " / " 
 
 
 +. 16605 
 
 8. 
 
 D t d/ 
 
 +1.328 
 
 
 ; JT 
 
 
 + 13408 
 . 02263 
 
 10. 
 
 10. 
 
 D t J 
 SinJDtN 
 
 +1.341 
 
 o. 226 
 
 
 e 
 
 
 . 03180 
 
 V 1 
 
 D t <? 
 
 o. 182 
 
 
 TC 
 
 
 . 00928 
 
 A^Q 
 
 D t 7T 
 
 -o. 530 
 
 
 e 
 
 
 +. 06097 
 
 1 6. 
 
 Dt 
 
 +0.976 
 
 
 ' // 
 
 
 . 12597 
 
 8. 
 
 l}^e // 
 
 1. 008 
 
 
 "TT //= 
 
 
 +.00853 
 
 8. 
 
 e" D t TT" 
 
 +o. 068 
 
 
 a 
 
 
 . 09670 
 
 20. 
 
 D t a 
 
 i. 934 
 
 
 " 6 ] 
 
 
 .01168 
 
 20. 
 
 Dt^ 
 
 -o. 234 
 
 
 \l>'\ 
 
 
 +. 13111 
 
 12. 
 
 D t rf/" 
 
 +1.573 
 
41] REFERENCE TO THE ECLIPTIC. 79 
 
 Reference to the ecliptic. 
 
 41. In all the preceding determinations the planes of the 
 orbits are referred to the plane of the Earth's equator, or, to 
 speak more exactly, to a plane through the Sun parallel to the 
 Earth's equator. As in astronomical practice the ecliptic is 
 taken as the fundamental plane, it is necessary to investigate 
 the reduction of the elements from one plane to the other. 
 
 Let us consider the spherical triangle formed on the celestial 
 sphere by the plane of the orbit, the plane of the ecliptic, and 
 the plane of the Earth's equator. For the sides and opposite 
 angles of this triangle we have 
 
 Sides: N 6 y 
 
 Opposite angles : i 180 J e 
 
 When equatorial coordinates are used, the position of the 
 planet is considered as a function of the three quantities 
 
 N; J; e (a) 
 
 When ecliptic coordinates are used, the three corresponding 
 quantities are 
 
 0; I-, e (b) 
 
 Taking the set of quantities (a] as the fundamental parts of 
 the triangle, and expressing the corrections of the other parts 
 as functions of them, we have 
 
 6 i = + cos rp6J + sin ip sin JtfN cos Ode 
 sin i d 9 = sin fidJ + cos >/> sin J6N + cos * sin Ode 
 
 Taking (b) as the fundamental parts, we have for the correc- 
 tions to N and J 
 
 d J = cos fi6i sin ip sin i$6 + cos 
 sin J#N= sin 6i + cos sin idd cos J sin Ntfs 
 
 The numerical values assigned to the coefficients in these 
 equations are those corresponding to the mean epoch 1850. 
 The fact that they change somewhat in the course of a hundred 
 years has not been taken account of. The future astronomer 
 will meet with a real difficulty in that the corrections to a 
 
80 MERCURY, VENUS, AND MARS. [41 
 
 set of elements at one epoch do not accurately correspond to 
 similar corrections at another epoch. It is impossible to do 
 away rigorously with the difficulty thus arising, except by 
 introducing a more general system of elements than elliptic 
 ones. The error is, happily, not important in the present state 
 of astronomy. The equations in question for the three planets 
 are as follow: 
 
 Mercury. 
 
 di = + .799 (5 J + .602 sin J fi X .688 fo 
 sin idO = - .602 6J + .799 sin J d N + .721 fa 
 
 Venus. 
 
 di = + .373 6 J + .928 sin J N - .255 fa 
 sin idS .928 d J + .373 sin J d N 4- .967 de 
 
 Mars. 
 
 di = .703 6 J + .712 sin J tf N - .664 6s 
 sin id 6 = - .712 3J + .703 sin J 3X + .747 fa 
 
 For the inverse relations we have 
 
 Mercury. 
 
 3J= .799 <H - .602 sin idti + .983 de 
 sin J SIS = .602 6i + .799 sin id 6 - .162 de 
 
 Venus. 
 
 6 J = .373 di .928 sin idO + .990 de 
 sin J d N = .928 di + .373 sin idB - .125 fa 
 
 Mars. 
 
 6 J = .703 <M - .712 sin id 6 + .998 fa 
 sin J N = .712 < + .703 sin id B - .052 fa 
 
CHAPTER IY. 
 
 COMBINATION OF THE PRECEDING RESULTS TO OBTAIN 
 THE MOST PROBABLE VALUES OF THE ELEMENTS 
 AND OF THEIR SECULAR VARIATIONS FROM OBSER- 
 VATIONS ALONE. 
 
 In the two preceding chapters are derived four separate 
 values of the six corrections, <*, #, 6e, 61", de", and e"6n", and 
 of their secular variations, which pertain to the orbit and 
 motion of the Earth relative to the stars. We have now to 
 combine these four results so as to derive the most probable 
 values of the twelve unknown quantities in question. 
 
 Deviations from the method of least squares. 
 
 42. If we applied without modification the principles of the 
 method of least squares, we should first eliminate the elements 
 and secular variations for each planet from the normal equa- 
 tions given by observations of that planet, which would leave 
 us with three sets of normal equations, containing only the 
 twelve quantities depending on the motion of the Earth. We 
 should then reduce these normal equations to equality of 
 weight, by multiplying each of them by the appropriate 
 factors, and we should then consider the observed corrections 
 to the solar elements derived from observations of the Sun 
 alone as affording equations of condition to be reduced to the 
 adopted system of weights, and then multiplied by their coeffi- 
 cients and added to the normal equations. The solution of 
 the single set of normal equations thus formed would lead to 
 the definitive values of the solar elements and of their secular 
 variations, which, being substituted in the eliminating equa- 
 tions from each planet, would lead to the definitive elements 
 of the planet and of their secular variations. 
 
 This proceeding is not, however, advisable in the present 
 case, because, owing to the immense mass of material worked 
 
 5690 N ALM 6 
 
 81 
 
82 PROBABILITY OF ERRORS. [42 
 
 up, the errors to be principally feared are not the accidental 
 ones, of which alone the method of least squares takes account, 
 but the systematic ones arising principally from personal 
 equation and imperfect reduction of the observations to the 
 actual center of the planet or of the Sun. These errors affect 
 different elements in very different ways and to different 
 amounts; from some they will be almost completely elim- 
 inated and from others they will not. We must therefore pro- 
 ceed by a tentative process, ascertaining at each step, so far 
 as possible, how each result .will come out before we accept it 
 as final, to be combined with other results. In doing this it 
 is necessary to deviate so widely from what are commonly 
 regarded as fundamental principles of the theory of the com- 
 bination of observations that a brief presentation of the prin- 
 ciples involved is appropriate. 
 
 It is frequently accepted as an axiom that when we have 
 several non-accordant determinations of the same quantity, 
 between which we have no reason for choosing, the most prob- 
 able value is the arithmetical mean. The operation of taking 
 the arithmetical mean is, in fact, the simplest application of 
 the method of least squares. The fundamental hypothesis on 
 which this method rests is that the probability of an error of 
 magnitude i x is given by the well-known exponential equa- 
 tion 
 
 h M** 
 <p (Ji, x)dx = e dx (a) 
 
 h, the modulus of precision, being a constant. It was shown by 
 GAUSS that this function for the probability follows rigorously 
 from the principle of the arithmetical mean. It therefore fol- 
 lows that the method of the arithmetical mean, and therefore 
 that of least squares, is rigorously correct only so far as the 
 law of error is expressed by the above exponential function. 
 
 It scarcely needs to be pointed out that, as a matter of fact, 
 the law of error in question is not true. Not only so, but in 
 astronomical experience it deviates from the truth in a way 
 admitting of precise statement. It presupposes that the mod- 
 ulus of precision is a determinate quantity. Were this the 
 case, then, to take a single instance, the probability of an 
 
42] PROBABLE ERRORS AND WEIGHTS. 83 
 
 error five times as great as the probable error would be less 
 than 0.001, and the probability of an error six times as great 
 would be about 0.0001. This is not true, because, taking the 
 function q> (ft, x) as a basis, we may say that the modulus of 
 precision, ft, is nearly always in practice an uncertain quan- 
 tity. Let us then put 
 
 hi, lit, h 3 , . . . 
 for the possible values of ft, and 
 
 for the several probabilities that h has these respective values. 
 Then the probability function will become 
 
 <p(x)=Pi <P(h\, x) +Pz(p'(h*, oo) + ... (6) 
 
 Now this form can not be reduced to the form (a) with any 
 value whatever of the modulus h. If we make the closest rep- 
 resentation possible, we shall have a curve in which small 
 values and large values of x are relatively less probable as 
 compared with the facts than are intermediate values. To 
 show that this is the actual case, let us suppose that we have 
 three determinations of an unknown quantity. If we proceed 
 in the usual way, we should infer the value of ft, the measure 
 of precision, from the discordance of these three values. But 
 it is evident that this determination of ft w.ould be very uncer- 
 tain. Should the three values chance to be fortunately accord- 
 ant, then, proceeding in the usual way, our function would lead 
 to the conclusion that the probability of an error of a certain 
 magnitude in the mean was very small, when, as a matter of 
 fact, it might be very considerable.* The value of ft being 
 
 * To take a simple and quite possible instance, let three observations of 
 a star with a meridian circle give, for the seconds of declination, 0".4, 0".5, 
 and 0".6. By the canons of least squares the mean result would be 
 
 0".50 0".039 
 and the probability of an error as great as 0".l would come out about 0.08. 
 
84 PROBABLE ERRORS AND WEIGHTS. [42 
 
 uncertain, the true form of the function is not (a} but (b). It 
 follows that we may lay down the following general rule : 
 
 The best value from a system of non- accordant determinations 
 is not the arithmetical mean, but a mean in which less iceiglit is 
 assigned to those results which deviate most widely from the 
 mean of the others. 
 
 I have considered the subject from this point of view in the 
 American Journal of Mathematics, Vol. VIII, p. 343, and given 
 tables for determining the weights to be assigned to the results 
 when the law of error is that derived from several hundred 
 observed contacts of the limb of Mercury with that of the Sun 
 during transits of the planet. 
 
 Another well-known defect in the method of least squares 
 is that it does not take any account of systematic errors. The 
 greater the number of observations that are combined, the 
 larger the proportion in which the errors of the results may 
 be due to the systematic errors in the observations or the 
 elements of reduction. Although such errors may elude inves- 
 tigation so far as their determination and elimination is con- 
 cerned, we may yet be able to point out their origin, and to 
 show to what extent they would influence each separate result. 
 Of some results we can say with entire confidence that they 
 are but slightly affected with systematic error 5 of others, that 
 they may be very largely so affected. In the latter case, the 
 weights of the results, as determined from the solution of the 
 normal equations, give no clue whatever to the probable mag- 
 nitude of the error. 
 
 The result of this is that in the following paper we are more 
 than once confronted with the following problem: Among 
 several determinations of a quantity one is known to be free 
 from systematic error and to be affected with a well determined 
 probable mean error, i e. There are also one or more other 
 determinations of which the probable error is unknown and 
 can not be determined, because we have no sufficient knowl- 
 edge of the probable effect of systematic errors upon the result. 
 What shall be the relative weight assigned to two such results 
 in order to obtain the mean? The decision of this question is 
 necessarily a matter of judgment, the grounds for which it 
 might be extremely prolix to state at length. An attempt has 
 
42 1 PROBABLE ERRORS AND WEIGHTS. 85 
 
 been made in these cases to classify the results, so as to give 
 a general idea of what is likely to be their modulus of pre- 
 cision, and weight them accordingly. 
 
 Any attempt at numerical accuracy in such an estimate 
 would be labor thrown away. It has therefore been considered 
 sufficient in such cases to state what the conclusion of the 
 author is, leaving its revision and criticism to the future 
 investigator. Indeed, in some cases, as in that of the correc- 
 tion to the centennial motion of the Sun in longitude, a con- 
 venient round number has been chosen, very near to the result 
 of well-decermined weight. 
 
 We should be carrying the preceding conclusions too far if 
 they led us to a general distrust of the conclusions reached by 
 the method of least squares. The doctrines that there is a 
 necessary limit to the accuracy with which astronomical deter- 
 minations can be made; that systematic errors necessarily 
 affect every such determination 5 and that the canons of least 
 squares necessarily lead to illusory probable errors, are too 
 sweeping. We may lay down the general rule that if we have 
 a sufficient number of really independent determinations of an 
 unknown quantity, of which we individually know nothing 
 except that they are the results of actual measures, and not 
 mere guesses, then the arithmetical mean will be a definite 
 result, the probable deviation from which will actually follow 
 the law given by the canons in question with a closeness 
 which will continually increase with the number of independent 
 determinations. 
 
 If we have such knowledge of the relative values of the 
 various determinations as to assign greater weight to some 
 than to others, the result will be still better when those 
 weights are used, provided always that they are assigned 
 without undue bias in favor of those results which most nearly 
 approach the value supposed to be approximately correct. 
 
 These considerations lead me to a policy which I have 
 always adopted when it was easy to do so in the following 
 discussions, namely, that of so conducting the work as to 
 lead to as many independent determinations of a quantity 
 as possible, arid of always giving a less relative weight to such 
 sets of determinations as might from any cause whatever be 
 
86 ELEMENTS OF EARTH'S ORBIT. [42,43 
 
 supposed affected by au important common source of error. 
 Where the independent determinations are few in number, the 
 computation of a definite probable error is impracticable, and 
 the probable mean error assigned is necessarily a result of a 
 judgment based on all the circumstances. 
 
 Relative precision of the tico methods of determining the elements 
 of the Earth's orbit. 
 
 43. When the system of determining the solar elements from 
 observations of the planets as well as of the Sun was originally 
 decided upon, it was supposed that the two methods would 
 give results not greatly differing in accuracy in the case of any 
 of the elements. This, however, is proved by the results not to 
 be the case. Attention has already been called to the extreme 
 consistency of the values found for the correction to the eccen- 
 tricity and perihelion of the Earth's orbit from observations of 
 the Sun. This consistency inspires us with confidence that 
 the probable errors of the corrections to the elements as given 
 do not exceed a few hundredths of a second. But the deter- 
 mination of these elements from observations of Mercury and 
 Venus may be seriously affected by the form of the visible 
 disks of those planets, which results in observations being 
 made only upon one limb when east of the Sun and the other 
 limb when west of it. Thus personal equation and the uncer- 
 tainty of the semidiameter to be applied in each case may have 
 an effect upon the result. But personal equation is likely to be 
 smaller in the case of Mercury than in that of Venus, owing 
 to the smalluess of its disk. 
 
 There is another circumstance which weakens the inde- 
 pendent determination of the Earth's eccentricity and perigee 
 from observations of the planets. If we define the orbit of a 
 planet, not as a curve, but as the totality of points which the 
 planet occupies at a great number of given equidistant moments 
 during its revolution, then it is easy to see that the general 
 mean effect of an increase of the eccentricity is to displace the 
 entire or.bit toward the point of the celestial sphere marked by 
 its aphelion, while the effect of a change of its perihelion is to 
 move the entire orbit in its own plane in a direction at right 
 angles to the line of apsides. The result is that in a series of 
 
43, 44J SECULAR VARIATIONS OF THE SOLAR ELEMENTS. 87 
 
 observations of a planet from the Earth the corrections to the 
 eccentricity and perihelia of the two orbits can not be entirely 
 independent, and we can determine with entire precision only 
 two linear functions expressive of the relative displacements 
 just described. It may be admitted that, were observations 
 exactly similar in kind made around the entire relative orbit 
 in equal numbers, the effect of the principle systematic errors 
 would be nearly eliminated from the result. But we can not 
 rely upon this being the case, and even were it the case there 
 would probably be a residual effect which would be large in 
 proportion to the interdependence of the two sets of correc- 
 tions. But in this connection the important remark is to be 
 made that, so far as these systematic errors are invariable, 
 they would not affect the secular variations, but only the abso- 
 lute values of the elements. We may therefore assign greater 
 relative weights to the former than to the latter. 
 
 So far as we cau classify the results, I have concluded that 
 in the case of the secular variations of f, e"< and TI" , the weight 
 of the determination from Mercury and Venus might receive a 
 weight one-fifth that from the Sun. But in the case of the 
 absolute values of these quantities, it would seem from the 
 discordance of the results that the relative weight of the 
 planetary results should be much smaller. 
 
 In dealing with the common error, a, of the adopted Right 
 Ascensions of the stars, it is to be remarked that we may 
 regard the observations in Eight Ascension as fitted to give 
 the values of a + 61", while 61" necessarily depends solely 
 upon the observations of declination, in effect if not in form. 
 Hence, although the unknown quantities of the solution are 
 a and 61", I have deemed it best to derive the result by 
 regarding a + 61" as the quantity to be first found, instead of 
 a itself. 
 
 Secular variations of the solar elements. 
 
 44. The following table shows the corrections to the tabular 
 secular variations of the solar elements, as they have been 
 found from observations. In the cases of Mercury and Venus 
 the results of both solutions are given for the sake of compari 
 son, although only solution B is used. The relative weights 
 
88 
 
 ELEMENTS OF EARTH'S ORBIT. 
 
 [44 
 
 have been determined by the considerations already set forth. 
 In the case of Mars, the final determinant of the solution for 
 the solar elements came out so- nearly evanescent as to show 
 that no reliable values could be obtained, a result which we 
 
 Corrections to the secular variations of the solar elements derived 
 from observations only. 
 
 
 Dt* 
 
 Dt<J/" 
 
 Dt#f* 
 
 From observations of 
 The Sun 
 
 " w. 
 
 4-0. 48 5 
 
 " w. 
 
 o. 97 i 
 
 " TC/. 
 +- 2 3 5 
 
 Mercury, solution A . 
 " " B 
 
 -4-0.27 
 
 -4-O 3Q I 
 
 0.58 
 i 26 i 
 
 0.47 
 
 1 O 32 I 
 
 Venus, solution A _ _ 
 
 -f o. 29 
 
 o. 90 
 
 4-0.28 
 
 B 
 
 4-O. 32 I 
 
 i. 06 i 
 
 4-o. 72 i 
 
 Mars 
 
 + 1.03 | 
 
 
 
 
 
 
 
 Mean 
 
 -1-0.48 
 
 I. IO 
 
 4-o. 26 
 
 Adopted _ . _ . 
 
 +0.48 
 
 . I.OO 
 
 4~O. 21 
 
 
 
 
 
 
 *"D t c57r" 
 
 D t (a + d/") 
 
 D t a 
 
 From observations of 
 The Sun 
 
 " IV. 
 
 4-0. 32 5 
 
 " w. 
 
 0. 63 2 
 
 // 
 
 4-0. 34 
 
 Mercury, solution A 
 
 0.40 
 
 -1.84 
 
 1.26 
 
 B 
 
 o. 29 i 
 
 2. 05 3 
 
 o. 79 
 
 Venus, solution A 
 
 4-o. 32 
 
 
 O. ?? 
 
 " " B 
 
 4~o. 46 i 
 
 I. 2O 2 
 
 o. 14 
 
 Mars 
 
 
 
 
 
 
 
 
 Mean 
 
 -fo. 25 
 
 I. 4O 
 
 o. 30 
 
 Adopted 
 
 -l-o. 26 
 
 I. ^O 
 
 o. 30 
 
 
 
 
 
 might expect, because, in order to separate the principal ele- 
 ments of the Earth's orbit from those of the planet, observa- 
 tions should be continued all around the relative orbit, whereas, 
 as a matter of fact, they are generally made only near the time 
 of opposition. I have judged, however, that the correction to 
 the secular variation of the obliquity obtained by putting 
 D t dl" = 1".00 MI the equation for D t tf e might enter with 
 half the weight that it does in the cases of Mercury and 
 Venus. Before the final values and weights of the quanti- 
 ties in the table had been definitively revised, provisional 
 values were used in the subsequent part of the investigation. 
 
44, 45] CORRECTION TO THE STANDARD DECLINATIONS. 89 
 
 These provisional values are given in the last line of the table. 
 Jt is also to be noted that the secular variations of e, e" TT, n" 
 and in the definitive theory and tables are those computed 
 from the adopted masses of the planets. 
 
 Correction to the standard of Declination. 
 
 45. The results for the secular variation of 3, the common 
 error of the standard Declinations within the zodiacal limits, 
 are not given in the table, as other data are available for its 
 determination. The following shows the separate values of 
 6 and its secular variation, derived from observations of the 
 planets to Saturn inclusive. For reasons already stated obser- 
 vations of the Sun are not used for this purpose. 
 
 " " w. 
 
 From observations of Mercury, 6 = 0.18 - 0.49 T ; 2 
 
 Venus, - 0.19- 0.00 T; 1 
 
 Mars, - 0.21- 0.23 T; 4 
 
 Jupiter, - 0.04 - 0.43 T ; 3 
 
 Saturn, + 0.04- 0.68 T; 4 
 
 Mean ; d= 0".09 0".42 T 
 Adopted ; <? = -0 .08-0 .50T 
 
 Not only observations of the planets but those of the fixed 
 stars are available for the determination of 6 and of its secular 
 variation. In the discussion of the Declinations derived from 
 observations with the Greenwich and Washington transit cir- 
 cles (Astronomical Papers, Vol. II), I have shown that the 
 Greenwich observations indicate, with some uncertainty, a 
 secular variation of the corrections to the standard declina- 
 tions which will give a value of about 0".55 for the secu- 
 lar variation of d. But BRADLEY'S Declinations, as reduced 
 by AUWERS, would give a still larger negative value, approxi- 
 mating to an entire second. As the value which we may 
 assume for d does not greatly influence the other elements, 
 I have adopted as a convenient probable result, the varia- 
 tion 0".50 T. 
 
90 ELEMENTS OF EARTH'S ORBIT. [46 
 
 Definitive secular variations of the planetary elements from obser- 
 vations alone. 
 
 46. Having decided upon the adopted values of the six 
 quantities found in the last article, we regard them as known 
 quantities, and substitute them in the eliminating equations, 
 which give the values of the remaining secular variations. 
 As the unknown quantities in these equations are not the 
 corrections themselves, but certain functions of them, we pre 
 pare the following table, showing the formation of the quan- 
 tities which are to be substituted in the several equations. 
 The table scarcely seems to need any explanation, except that 
 the unknown quantities given in the three columns on the 
 right are formed by dividing the secular variations for twenty- 
 five years by the coefficients given in 27. 
 
 Adopted secular variations of the solar unknowns, to be substi- 
 tuted in the eliminating equations for the several planets. 
 
 D t <M" = -T'.OO; 
 D t tf =0 .50; 
 D t a = -0.30; 
 
 e"D t <y7r" = + .26; 
 D t 6e ff =+0 .21; [ e" 
 D t fo =+0.48; 
 
 To facilitate a judgment or rediscussion of this part of the 
 process, we give on the next three pages the normal equations 
 between all the secular variations which remain after the cor- 
 rections to the elements of the Sun and planets are eliminated 
 from the original normal equations. We give these rather 
 than the eliminating equations "which were actually used in 
 the substitution, because they show more fully the relations 
 between the unknown quantities, and can therefore be better 
 used iu any ulterior discussion. Regarding the preceding six 
 quantities as known, and substituting them in the normal 
 equations for the secular variations, we derive the definitive 
 values of the secular variations which relate to the planets. 
 They are shown in the next table. In the latter the values of 
 the solar elemen ts are repeated for the sake of completeness. 
 
 
 Mercury. 
 
 Venus. 
 
 Mars. 
 
 I" 
 
 1' =-0.250; 
 
 -0.0625; 
 
 -0.0833; 
 
 d 
 
 ]' =-0.125; 
 
 -0.0250; 
 
 -0.0250; 
 
 a 
 
 ]' =-0.075; 
 
 -0.0750; 
 
 -0.0150; 
 
 n" 
 
 ]' = + 0.108; 
 
 + 0.0325; 
 
 + 0.0325; 
 
 e" 
 
 ]' = + 0.087; 
 
 + 0.0210; 
 
 + 0.0262; 
 
 
 
 ]/ = +0.120; 
 
 + 0.0300; 
 
 + 0.0300. 
 
46] NORMAL EQUATIONS FOR SECULAR VARIATIONS. 91 
 
 I I 
 
 
 
 8 
 
 CO 
 
 3 
 
 + 1 1 l 
 
 p 
 
 
 
 ^ 
 I 
 
 -T7 CO 
 CS O5 
 
 + 
 
 SI 
 
 ^ o 
 
 r 3! 
 
 5. fl . . 
 
 
 
 co 
 
 <M 
 
 10 
 
 te oo 
 
 5 < 
 
 rH CO 
 
 i 
 
 10 
 
 s 
 
 8- 
 
 CO 
 
 1O 
 
 CO 
 
 I I 
 
 o 
 
 I 
 
 
 
 T* 
 rH 
 
 1 
 
 I 
 
 28 
 
 I 
 
 Oi 
 
 00 
 
 rH 
 
 rH 
 I 
 
 CO 
 
 i 
 
 + + i 
 
 O GO CO 
 
 rH rH CO 
 
 CO rH C5 
 
 3 s 
 
 CM t- 
 
 I I I 
 
 ^ 5 51 
 
 <M CO 
 
 vr 
 
 3 55 
 
 1 1 
 
 3 
 
 I I + I 
 
 3 3 
 
 CO <^ rH CO 
 
 rH rH IO 
 
 I + I + 
 
 i i 1 
 
 CO 
 
 + 
 
 OO 
 
 00 
 rH 
 
 + 
 
 CT *3 
 
 Z co 
 
 __ (N CO CO 
 
 rH <M 
 
 ^$ 
 
 ^L 3 
 
 I I 
 
 00 
 
 o 
 
 00 
 
 -8 
 
NORMAL EQUATIONS FOE SECULAR VARIATIONS. [46 
 
 CO 
 
 CO 
 CO 
 
 CO 
 
 TH <M O 
 
 O CO Tf 
 
 S * 
 
 i i 
 
 ^ S 
 
 v .2 
 
 * 
 
 ^ a 
 
 s I 
 
 CD 
 
 QO 
 
 GO 
 
 t^ rH 00 
 
 1 ^ 
 I I 
 
 4- + I 
 
 rH CO CO' 
 rH <M 
 
 CO 
 
 i 
 
 CO T* rH 
 
 + 4- + I + 
 
 I -I I 
 
 S 
 
 CO 
 
 QQ 
 
 t> 
 
 > 
 
 I I 
 
 CO t*- 
 
 TH CO 
 
 TH 
 
 O 
 
 rH IS 
 
 00 b- 
 CM rH 
 C5 t 
 
 CO CO rH 
 
 C2 - 
 
 
 ^- ^ ^ r< 
 
 * I 
 
 S S 
 
 CO 
 
 ^> 
 
 I I + 
 
 8 g 53 
 
 " S 
 
 I + + 
 
 I 
 
 z 
 
 9 
 
 
 
 1O t~ 
 
 rH 
 
 I I + 
 
 ^ CO 
 S ^ 
 
46] NORMAL EQUATIONS FOR SECULAR VARIATIONS. 
 
 CO 
 
 O O O CO 
 
 O b- JO CO O b- 
 
 H rH rH CM rH rH 
 
 I + I I + I 
 
 1 1 1 + 
 
 b'38 
 
 CO 
 
 + + 
 
 cp CQ 
 
 CO 
 
 + + i + + + 
 
 a ^ 
 
 ^^ co 
 
 rH O5 
 
 tr^" tr** 
 
 CO t- 
 
 Tfl b- CO O O 
 
 C5 SO rH IO O 
 
 lO rH b- rH C5 
 
 b- rH rH 
 
 + + I I + 
 
 rH CO 
 
 CM Ci 
 
 8 a 
 
 ^ 
 
 1 
 
 o 
 
 3 
 I 
 
 tJO 
 
 ti 
 
 R rH 
 
 C^ CO 
 
 2L S 
 
 CO CM CO 
 Ci <M CO 
 b- rH <M 
 
 1 
 
 I I 
 
 (M CO 
 
 CM CO 
 
 CO iO 
 lO b- 
 
 b- rH O CO 
 05 rH CM CO 
 CM 
 
 a s 
 
 b- O CO 
 
 rH CO ^* 
 
 CO 
 
 cc 
 
 ! 
 
 ^ 
 
 I I 
 
 ^a 
 
 KJ T^ 
 
 J5- ^ 
 
 i 
 
 - 8 
 
 b- 
 
 
 i i 
 
 CM C5 
 
 1^- b- 
 
 O O 
 
 CM CO 
 
 ^^ T"H 
 
 co o 
 
 tfHIVBESITT, 
 
 i 
 + 
 
 CO 
 CO 
 
 
94 
 
 SECULAR VARIATIONS FROM OBSERVATIONS. 
 
 [46 
 
 Values of the secular variations as derived from observations 
 
 only. 
 
 Unknown. Corr. 
 
 Tables. 
 
 Result. 
 
 Mercury. D t e 
 
 D t i 
 
 -.0691 -0.83 + 4.19 + 3.36i0.50 
 
 + .1577 +1.30 +116.94 +118.240.40 
 
 + .0593J +0.83 + 6.31 + 7.14i0.80 
 
 + .0815K +0.70 - 92.59 - 91.890.50 
 
 -.0967 -1.55 
 
 Yenus. 
 
 Earth. 
 
 Mars. 
 
 D t e +.1393 +1.67 - 11.13 - 9.46i0.20 
 
 eD t 7T +.0685 +0.82 - 0.53 + 0.290.20 
 
 D t -t +.1153 J -0.65 + 4.52 + 3.87i0.30 
 
 sintD t -.0592K -2.73 -102.67 -105.40+0.12 
 
 D t tfZ -.1919 -3.84 
 
 D t e +0.21 - 8.76 - 8.55i0.09 
 
 eV t 7t +0.26 + 19.22 + 19.480.12 
 
 +0.48 - 47.59 - 47.11 =fc 0.25 
 
 D t f 
 
 D t e 
 
 eD t ;r 
 
 D t i 
 
 siniD t 
 ~D t 6l 
 
 .1190 -0.68 + 19.68 + 19.OOiO.27 
 
 +.0536 +0.29 +149.26 +149.550.35 
 
 +.1136J +0.17 - 2.43 - 2.260.20 
 
 -.0442N -0.76 - 71.84 - 72.600.20 
 -.0946 -0.76 
 
 The first column of numbers in this table gives the unknown 
 quantity as found immediately from the eliminating equations. 
 These quantities being multiplied by the factors given in 
 27, we have the corrections to the tabular secular varia- 
 tions, as given in the column "correction." The next column 
 gives the value of the tabular secular variations, which are 
 in all cases those actually adopted by LEVERRIER. In the 
 case of the Earth, as has been pointed out by STURMER and by 
 INNES, the secular variation of the radius vector does not cor- 
 respond to that of the longitude. But as that of the longitude 
 is the preponderating quantity in its effect on geocentric 
 
46, 47] CORRECTIONS TO THE SOLAK ELEMENTS. 95 
 
 places, I have regarded the value of the eccentr city used in 
 the tables of the equation of the center as the tabular one to 
 be adopted. 
 
 The numbers in the column "Unknown," which are followed 
 by the letters J and N, are the respective values of [J] L and 
 [aST]i, which are changed to <5i and sin id 8 by the equations 
 of 41. 
 
 Finally, we have the results given in the last column for the 
 actual secular variations of the several elements as derived 
 from the preceding discussion of all the observations. 
 
 The result is followed by the probable mean error of each of 
 the quantities as estimated from the probable magnitude of 
 the sources of error to which they are liable. As in other 
 cases, these quantities are very largelv a matter of judgment, 
 because the probable errors as determined in the usual way 
 from the eliminating equations would be entirely unreliable. 
 
 Definitive corrections to the solar elements for 1850. 
 
 47. Leaving the above results to be subsequently discussed, 
 we go on with the solution of the equations. By a continuation 
 of the process just described, we regard the preceding secular 
 variations as known quantities, and substitute them in the 
 eliminating equations for the solar elements which are derived 
 from the normal equations for each planet. By this substitu- 
 tion, we reach three fresh sets of values of the corrections of 
 the solar elements themselves, one set from the observe tions 
 of each planet, which are to be reduced to 1850 and combined 
 with those already found from observations of the Sun, in 
 order to obtain the most probable result. 
 
 Here we meet with the same difficulty that confronted us in 
 the case of the secular variations. With the exception of the 
 Sun's mean longitude, we are to regard the results derived 
 from each of the planets as affected by obscure sources of 
 systematic error, the probable magnitude of which can only 
 be inferred from the general deviation of the quantities them- 
 selves. As in the former case, a is not regarded as a quantity 
 independently determined, but a-\- 61" has been taken instead. 
 The concluded value of a is "then found by subtracting 61' '/ 
 from 61" -f- a. Since the corrections to the solar elements 
 pertain to each separate epoch, those derived from the obser- 
 
96 
 
 ELEMENTS OF EARTH'S ORBIT. 
 
 [47 
 
 vations of the planets are severally reduced to 1850, and the 
 results are shown in the following table : 
 
 Separate values of the corrections to the solar elements for 1850, 
 after the above definitive values of the secular variations are 
 substituted in the eliminating equations from solution B, 
 reduced to 1850. 
 
 
 Je 
 
 *>> 
 
 6f 
 
 e "6K" 
 
 a + <J/" 
 
 a 
 
 From observations of 
 The Sun 
 
 // 
 . 30 
 
 // 
 
 +-5 
 
 (f 
 
 + .10 
 
 // 
 . oo 
 
 // 
 . 02 
 
 // 
 
 .07 
 
 Mercury 
 
 + I 3 
 
 H--7 
 
 +.48 
 
 47 
 
 -f .60 
 
 + -53 
 
 Venus 
 
 -f. 13 
 
 .17 
 
 +.06 
 
 7 
 
 -f .34 
 
 + -5 
 
 Mars 
 
 + 2 5 
 
 +24 
 
 .83 
 
 .82 
 
 + i. 18 
 
 + 94 
 
 
 
 
 
 
 
 
 Adopted 
 
 . 20 
 
 .02 
 
 -f . 12 
 
 .04 
 
 + -46 
 
 + -48 
 
 
 
 
 
 
 
 
 These adopted values are employed in the subsequent stages 
 of the discussion, but are not in all cases regarded as definitive. 
 In the case of f the value 0".20 is that which I have actually 
 used in the subsequent determinations of the elements, but for 
 the final value of the obliquity it will be seen that I have 
 taken O'MS as more probable. 
 
CHAPTER Y. 
 
 MASSES OF THE PLANETS DERIVED BY METHODS INDE- 
 PENDENT OF THE SECULAR VARIATIONS WITH THE 
 RESULTING COMPUTED SECULAR VARIATIONS. 
 
 48. The plan of discussion laid down in Chapter I contem- 
 plates the determination of the masses of each of the planets 
 from all data independent of the secular variations, in order 
 to determine how far the observed secular variations can be 
 reconciled with these masses. The following is a summary of 
 these determinations. The planets outside of Jupiter need no> 
 discussion, as the well-known determinations of their masses 
 are amply accurate for all our present purposes. 
 
 Mass of Jupiter. 
 
 49. One of the works connected with the present subject has 
 been the determination of the mass of Jupiter from the motions 
 of (33), Polyhymnia. My work on this subject has not yet been 
 printed in full, but I have given in Astronomische Nachrichten y 
 No. 3249 (Bd. 136, S. 130), a brief summary of the results. The 
 mass of Jupiter has been derived not only from the motions 
 of Polyhymnia, but from such other sources as seemed best 
 adapted to give a reliable -result. The following table, tran- 
 scribed from the publication in question, shows the separate 
 results and the conclusions finally reached : 
 
 Reciprocal of mass of Jupiter from wt 
 
 All observations of the satellites, 1047.82 1 
 
 Action on FATE'S comet (MOL.LER), 1047.79 1 
 
 Action on Themis (KRUEGKER), 1047.54 5 
 
 Action on Saturn (HiLL), 1047.38 7 
 
 Action on Polyhymnia, 1047.34 20 
 
 Action on WINNECKE'S comet (v. HAERDTL), 1047.17 10 
 
 1047.35 
 
 in. e. 0.065 
 
 5690 N ALM 7 97 
 
98 MASS OF JUPITER. [49 
 
 It will be seen that the result from observations of the- satel- 
 lites has been assigned a very small weight. This course has 
 been indicated by the circumstances. Other conditions being 
 equal, the greater the mass of a planet the less the propor- 
 tionate precision with which that mass can be determined by 
 observations on the satellites. In any case, if the measures of 
 the distances between the satellites and the primary are' in 
 error by a small fraction, or, of their whole amount, then the 
 error of the mass will be in error by the fraction 3 a of its 
 amount. For reasons founded on the construction and use of 
 the heliometer, I doubt whether the absolute measures made 
 with those forms of that instrument which have been used in 
 determining the mass of Jupiter can be relied upon within 
 their three-thousandth part. If so, the determination of the 
 mass of the planet itself would be doubtful by its thousandth 
 part in each separate case. The chance of personal equation 
 between transits of the satellites and the planet vitiates in the 
 same way the results from observed transits of the planet and 
 satellites. Notwithstanding the great refinement of the dis- 
 cussion by KEMPF of observations made at Potsdam, and the 
 care with which he, SCHUR, and others have determined the 
 mass of Jupiter by a discussion of all the^observations of the 
 satellites, I can not conceive that the probable error of any 
 possible result they could derive would be less than 0.3 or 0.4 
 in the denominator. 
 
 In this connection the discordances between" the mass of 
 Saturn, found by Prof. HALL and by other observers from 
 observations of the satellites, are worthy of consideration. 
 They lead us to suspect that perhaps it is through good for- 
 tune rather than by virtue of their absolute reliability that 
 determinations of the mass of Jupiter from observations of the 
 satellites have agreed so well. 
 
 As to the weights assigned to the other results, only the last 
 needs especial mention. The probable error assigned by v. 
 HAERDTL to his result is very much smaller than that which 
 I find for the mean of all the results. But, as remarked in the 
 paper in question, it has received a smaller relative weight 
 than that corresponding to its assigned probable error, because 
 of distrust on my part whether observations on a comet can 
 
49, 50, 51] MASSES OF THE EARTH, MARS, AND JUPITER. 99 
 
 be considered as having always been made 011 the center of 
 gravity of a well-defined mass, moving as if that center were 
 a material point subject to the gravitation of the Sun and 
 planets. This distrust seems to be amply justified by our 
 general experience of the failure of comets to move in exact 
 accordance with their ephemerides. 
 
 I propose to accept the value thus found, 
 
 Mass of Jupiter = 1 4- 1047.35 
 
 as the definitive one to be used in the planetary theories. 
 Mass of Mars. 
 
 50. In consequence of the minuteness of the mass of Mars, 
 measures of its satellites, especially the outer one, afford a 
 value of its mass much better than can be derived by its action 
 on the planets. When nearest the earth, the major axis of the 
 orbit of the outer satellite subtends an angle of 70". I can 
 not think that the systematic error to be feared in the best 
 measures, such as those made by Prof. HALL, can be as great 
 as half a second. It therefore Appears to me that the mean 
 error in adopting Prof. HALL'S value of the mass does not 
 exceed its fiftieth part. This is a degree of precision much 
 higher than that of any determination through the action of 
 Mars on another planet. 
 
 Prof. HALL'S measures of 1892 show a minute increase of 
 the mean distance given by his work of 1877. The result is 
 
 v>" = + 0.014 
 
 These observations, however, were made when the position of 
 the orbit of the satellite was unfavorable to an exact deter- 
 mination of the elements of motion. I have adhered to the 
 original value in the work of the present chapter. 
 
 Mass of the Earth. 
 
 51. I have already pointed out the difficulty in the way of 
 determining the mass of the Earth from its action on the 
 other planets. On tbe other hand, the solar parallax has, in 
 recent years, been determined in various ways with such 
 precision that the mass of the Earth to be used in the plan- 
 
100 MASS OF THE EARTH. [51 
 
 etary theories can best be derived from it. The theory of the 
 relation between the mass of the Earth and its distance from 
 the Sun, as given by observations of the seconds pendulum 
 and the length of the sidereal year, is one of the best estab- 
 lished results of celestial mechanics. It is, in effect, the 
 principle on which the lunar theory is constructed. In this 
 theory the disturbing action of the Sun is necessarily a func- 
 tion of the ratio of the mass of the Sun to that of the Earth. 
 But in the accepted theory this ratio is eliminated through 
 the ratio of the lunar month to the sidereal year. From the 
 well-established ratio between the distance of the Moon and 
 the length of the seconds pendulum, the ratio of the masses 
 of the Sun and Earth come out of this theory with great 
 precision. It need not be developed here; it will suffice to 
 give the numerical result, which is that between the ratio M 
 of the mass of the Sun to that of the Earth and the mean 
 equatorial horizontal parallax of the Sun in seconds of arc 
 there exists the relation 
 
 7r 3 M = [8.35493] 
 
 I have derived seven values of the solar parallax by different 
 methods, of which the following are the preliminary results : 
 
 wt. 
 GILL'S observations of Mars, 1877, 8.780 .020 1 
 
 Contact observations, transits of Venus. 8.704 i .018 1 
 
 Aberration and velocity of light, 8.798 i .005 16 
 
 Parallactic equation of the Moon, 8.799 .007 5 
 
 Measures of small planets on GILL'S plan, 8.807 i .007 8 
 
 LEVERRIER'S method, 8.818 .030 0.5 
 
 Measures of Venus from Sun's center, 8.857 i .022 1 
 
 Mean result, n = 8".802 i 0".005 
 
 I have provisionally taken this mean as the most probable 
 value of the solar parallax derived from all sources except the 
 mass of the Earth. The above relation then gives 
 
 M = 332 040 
 
51,52] MASS OF VENUS. 101 
 
 Taking for the mass of the Moon 1 4- 81.52, we have for the 
 ratio of the combined masses of the Earth and Moon to the 
 mass of the Sun 
 
 m" 
 
 328 016 
 
 a result of which the probable error may be regarded as some- 
 thing more than a thousandth part of its whole amount. 
 
 Mass of Venus. 
 
 52. The mass of Venus adopted in the provisional theory, 
 to which LEVERRIER'S tables were reduced, was .000 002 4885 
 = 1 +- 401847, which is that of LEVERRIER'S tables of Mer- 
 cury. In the preceding discussions the following three factors 
 of correction to this mass have been found : 
 
 From observations of the Sun . . .0118 =t .0034 
 
 From observations of Mercury . . .0121 =t .0050 
 
 From observations of Mars ... .0076 =t . ( ? ) 
 
 Mean - .0119 .0028 
 
 The mean error assigned to the result from observations of 
 the Sun may be regarded as real, because the result is the 
 mean of a great number of completely independent determina- 
 tions, among which no common error is either a priori prob- 
 able or shown by the discordance of the results. In the 
 case of Mercury, however, as already remarked, the effect of 
 systematic errors is such that, although they are almost com- 
 pletely eliminated from the result, the mean error computed 
 in the usual way would be misleading. The weight assigned 
 is therefore largely a matter of judgment. 
 
 The fact that it was necessary to introduce an empirical 
 correction, with a period of about forty years, into the mean 
 longitude of Mars, vitiates the determination of the mass of 
 Venus from its action on that planet, because one of the prin- 
 cipal terms in the action of Venus on Mars has a period which 
 does not differ from forty years enough to make the determi- 
 nation of the mass independent of this empirical correction. 
 I have therefore assigned no weight to the result. We thus 
 
102 MASS OF MERCURY. [52,53 
 
 have for the mass of Venus, as derived from the periodic per- 
 turbations of Mercury and the Earth produced by its action. 
 
 m' = 1 -^ 406 690 i 1140 
 Mass of Mercury. 
 
 53. The mass of Mercury which I have heretofore adopted, 
 1 -=- 7 500 000, was rather a result of general estimate than of 
 exact computation. The fact is that the determinations of 
 this mass have been so discordant, and varied so much with 
 the method of discussion adopted, that it is scarcely possible 
 to fix upon any definite number as expressive of the mass. 
 An examination of LEVERRIER'S tables of Venus shows that 
 with the mass of Mercury there adopted (1:3 000 000) Mercury 
 frequently produces a perturbation of more than one second 
 in the heliocentric longitude of Venus. When the latter is 
 near inferior conjunction, the actual perturbation will be more 
 than doubled in the geocentric place, so that the latter might 
 not infrequently be changed by 1", even if the mass of Mer- 
 cury be less than one-half LEVERRIER'S value. It was there- 
 fore to be expected that a fairly reliable value of the mass of 
 Mercury would be obtained from the periodic perturbations 
 of Venus. 
 
 Eeferriug to 27, it will be seen that the indeterminate mass 
 of Mercury appears in the equations in the form 
 
 1+7;, 
 3000000 
 
 From the solution B, 38, the value of /* comes out 
 
 ^ = _ 0.0834 
 
 corresponding to a mass of Mercury of 1 : 7 210 000. But in 
 a subsequent solution of the equations, when the secular vari- 
 ations are determined from theory and substituted in the 
 normal equation for /v, we find 
 
 ,u = - 0.0889 
 which gives 
 
 m = 1 -4- 7 943 000 
 
 The work of the present chapter is based on the former 
 value. 
 
53] MASS OF MERCURY. 103 
 
 A consideration of the probable error of this result is impor- 
 tant. The fortuitous errors which mostly affect it are of the 
 class which I have termed semi- systematic. Under this term I 
 include that large class of errors which, extending through or 
 injuriously affecting a limited series of observations, cause the 
 probable error of a result to be larger than that given by the 
 solution of the equations, but which, nevertheless, like purely 
 accidental ones, would be eliminated from the mean result of 
 an infinite series of observations. To this class belong the 
 errors arising from personal equation in observing the limb of 
 Venus, or, what is the same thing, a difference between the 
 practical semidiameter corresponding to the observer and that 
 adopted in the reductions. We may suppose that, during a 
 period of several days, when Venus is not far from inferior 
 conjunction, its geocentric position is affected by a perturba- 
 tion produced by Mercury. Through the error alluded to, all 
 the observations made by any one observer, and in fact all 
 that are made anywhere, may be affected by a certain con- 
 stant error in Right Ascension. Near another inferior con- 
 junction the same state of things may be repeated, with the 
 perturbation in the opposite direction. If, now, tne observa- 
 tions were made by the same observer, and under the same 
 circumstances, the personal error would be eliminated from 
 the mean of these two results so far as the mass of Mercury is 
 concerned. But very frequently different observers will have 
 made the observations under the two circumstances, and dif- 
 ferent conditions will have prevailed. Thus, it is only through 
 the general law of averages that we can expect the effect of 
 these fortuitous but systematic errors to be completely elim- 
 inated. That they would be eliminated in the long run is 
 evident from the fact that there can be no permanent rela- 
 tion between the personal equations of the observers and the 
 changes in the action of Mercury upon Venus. Moreover, 
 Venus has been observed with a fair degree of accuracy 
 through more than half a century, and it seems reasonable 
 to suppose that during that time the errors in question would 
 nearly disappear. 
 
 It is clear from these considerations that the probable 
 error derived from the solution of the equations would be 
 
104 
 
 MASS OF MERCURY. 
 
 [53 
 
 entirely misleading. But a probable error which ought to be 
 reliable can be obtained by a process similar to that which I 
 have adopted elsewhere in this paper, namely, dividing up the 
 materials into periods, and determining the probable error from 
 the discordances among the results of the several periods. 
 This probable error will be reliable, because there is no reason 
 why the same error should affect the mass of Mercury through 
 any two periods. I therefore take the partial normal equa- 
 tions in n derived from Eight Ascensions during the several 
 periods, substitute in them the values of the unknown quanti- 
 ties found from solution B, /* excepted, and thus form six- 
 teen partial normal equations in /i. These equations may be 
 changed into the corresponding equations of condition, of 
 weight unity, by dividing each by the square root of the 
 coefficient of the unknown quantity. The residuals then left 
 when the definitive value of the unknown quantity is substi- 
 tuted will be those from whose discordance the probable error 
 may be inferred. 
 
 The partial normal equations thus found from the Eight 
 Ascensions are as follow: 
 
 1750-'62. 
 
 44; 
 
 i= - 38 
 
 1830-'40. 
 
 5649; 
 
 i=- 831 
 
 1765-'74. 
 
 1265 
 
 -165 
 
 1840-'49. 
 
 2913 
 
 - 18 
 
 1775-'86. 
 
 15 
 
 - 5 
 
 1849->56. 
 
 2238 
 
 - 49 
 
 1787-'96. 
 
 209 
 
 4- 53 
 
 1857->64. 
 
 4506 
 
 - 129 
 
 1796->06. 
 
 345 
 
 4- 19 
 
 1865-'71. 
 
 7736 
 
 - 265 
 
 1806-'14. 
 
 439 
 
 4-135 
 
 1871-79. 
 
 7062 
 
 761 
 
 1814->19. 
 
 942 
 
 + 2 
 
 1879-'86. 
 
 4958 
 
 - 407 
 
 1820-'30. 
 
 1786 
 
 330 
 
 1885-'92. 
 
 9561 
 
 -1306 
 
 Sum: 49668/<= -4095 
 
 ^ = _ 0.0824 i .019 
 
 The difference between this value of yw, which is obtained 
 only to find the probable error, and that formerly found, arises 
 principally from the omission of the declination equations. 
 The mean error corresponding to weight unity comes out 
 
 ft = 4".2 
 
53] MASS OF MERCURY. 105 
 
 which, as anticipated, is much larger than that which would 
 be given by the discordance of the original observations. 
 This does not mean that the original observations are affected 
 by any such mean error as 4".2, but that the discordances 
 between the 16 values of /* are as great as we should expect 
 them to be if the original observations were absolutely free 
 from systematic error, but affected by purely accidental errors 
 of this mean amount. 
 
 The results of the solution for the mass of Mercury may be 
 expressed in the form 
 
 . 10.32 d 
 
 ~ ' 
 
 7 210 000 ' 7 043 000 
 
 In all researches which have been made on the motion of 
 ENCKE'S comet by ENCKE, VON ASTEN, and BACKLUND, the 
 determination of this mass has been kept in view. The 
 results are, however, so discordant that, as already remarked, 
 scarcely any definitive result can be derived from them. 
 
 To this statement there is, however, one apparent exeeption. 
 In an appendix to his very careful and elaborate discussion of 
 WINNECKE'S comet, VON HAERDTL has derived the value of 
 the mass of Mercury from all the return of ENCKE'S comet as 
 worked up by VON ASTEN and BACKLTJND.* The only inter- 
 pretation which I can put upon his result is this : If we regard 
 the acceleration of the comet, which it is supposed results 
 from all the observations made upon it, as non-existent, the 
 following two masses of Mercury are derivable from the obser- 
 vations : 
 
 1819-1868, w = 1 4- 5 648 600 i 2000 
 1871-1885, m = 1 +- 5 669 700 600 000 
 
 He also finds, from the motion of WINNECKE'S comet, 
 m = 1 4- 5 012 842 697 863 
 
 * Denkschriften der Kaiserlichen Akadeinie der Wissenschaften, Vol. 
 56, p. 172-175. Vienna, 1889. 
 
106 MASS OF MERCURY. (53,54 
 
 and from four equations of LEVERRIER 
 
 1 4- 5 514 700 100 000 
 
 The consistency of these results seems to me entirely beyond 
 what the observations are capable of giving, and I hesitate to 
 ascribe great weight to them. Moreover, the result implicitly 
 contained in these numbers, that the supposed secular accel- 
 eration of the comet disappears when we attribute the pre- 
 ceding mass to Mercury, merits farther inquiry. 
 
 The probable density of the planet may form a basis for at 
 least a rude estimate of its probable mass. The fact that the 
 Earth, Yenus, and Mars have densities not very different from 
 each other, while that of the Moon is 0.6 the density of the 
 Earth, leads us to suppose that Mercury, being nearest to the 
 Moon in mass, has probably a slightly greater density. Its 
 diameter at distance unity has been repeatedly measured and 
 found to be 6".6, or, roughly speaking, three-eighths that of the 
 Earth. Were its. density 0.7, its mass would therefore be 
 about 1 : 9,000,000. In view of the fact that the measured 
 diameter is probably somewhat too small, these consider- 
 ations lead us to conclude that the mass is probably between 
 1:6,000,000 and 1:9,000,000. 
 
 As the value of the mass to be used in investigating the 
 secular variations, I have adopted 
 
 v = 0.08 
 
 1 08 
 Mass of Mercury = 
 
 7 500 000 
 
 Secular variations resulting from theory. 
 
 54. In the Astronomical Papers, Vol. V, Part IV, were com- 
 puted the secular variations of the elements of the orbits in 
 question using, as the basis of the work, the values of the 
 
THEORETICAL SECULAR VARIATIONS. 
 
 107 
 
 54] 
 
 masses whose reciprocals are found in the column A below. 
 In column B are cited the masses which I have decided upon. 
 
 
 A 
 
 B 
 
 
 
 Original 
 
 Adpofced 
 
 
 
 reciprocal 
 
 reciprocal 
 
 
 
 of mass. 
 
 of mass. 
 
 V 
 
 Mercury, 
 
 7 500 000 
 
 6 944 444 
 
 + .080 
 
 Venus, 
 
 410 000 
 
 406 750 
 
 + .0080 
 
 Earth + Moon, 
 
 327 000 
 
 328 000 
 
 -.00305 
 
 Mars, 
 
 3 093 500 
 
 3093500 
 
 
 
 Jupiter, 
 
 1047.88 
 
 1047.35 
 
 + .00050 
 
 Saturn, 
 
 3501.6 
 
 
 
 
 Uranus, 
 
 22756 
 
 
 
 
 Neptune, 
 
 19 540 
 
 
 
 
 In the case of the Earth we have to add the secular varia- 
 tion of the perihelion produced by the non-sphericity of the 
 system Earth + Moon. For the principal term I have found, 
 
 D t e" d n" = + 0".129 
 
 The resulting values of the secular variations, expressed as 
 functions of v, v 1 , v"j v 1 ", are given in the following section: 
 
 Theoretical secular variations for 1850. 
 
 Mercury. 
 
 // // // // // // 
 
 D t e = + 4.22 +0.00i/+ 2.8 K' + l.lF // -0.1?/ / " = + 4.24 
 
 6D t 7Ti =+109.36+0.00 +56.8 +18.8 +0.5 = + 109.76 
 
 D t ?: =+ 6.76 -0.04 - 0.6 - 1.4 +0.0 =+ 6.76 
 
 sin i Dt ; = 92.12 0.33 -49.3 -12.2 1.2 =92.50 
 
 Venus. 
 
 D t e =- 9.58 1.30^+ 0.0^'- 4.9*/ // 0.2v /// =- 9.67 
 
 eDtTTt =+ 0.39-0.81+0.0 -3.9 +0.5 =+ 0.34 
 
 D t i =+ 3.43+0.76 + 0.0 + 0.0 -0.3 =+ 3.49 
 
 sin iT> t # =-105.92 +0.26 -29.2 -43.2 -1.2 =-106.00 
 
108 THEORETICAL SECULAR VARIATIONS [54 
 
 Earth. 
 
 // // // // // 
 
 T> t e = - 8.57 -0.12F+ 1.3^' -1.6i'"' = - 8.57 
 
 eT> t 7t = + 19.36-0.18 + 5.8 +1.6 =+ 19.39 
 D t * =- 46.65-0.21 -28.3 -0.7 ==-46.89 
 
 Mars. 
 
 // // // // // // 
 
 D t e =+ 18.71 +0.03T/4- O.lv'4- 2.1v 7/ =4- 18.71 
 
 eDtTf! = + 148.824-0.06 + 4.6 +21.4 = + 148.80 
 
 D t i =- 2.34-0.04+12.0 +0.0 +O.OF x// =- 2.25 
 
 intD t <y=- 72.43-0.27 -25.1 - 7.4 -1.0 =- 72.63 
 
CHAPTER VI. 
 
 EXAMINATION OF THE HYPOTHESES BY WHICH THE 
 DEVIATIONS OF THE SECULAR VARIATIONS FROM 
 THEIR THEORETICAL VALUES MAY BE EXPLAINED. 
 
 55. *The investigations of the present chapter are founded 
 on a comparison of the secular variations derived purely from 
 observations in Chapter IV, with those resulting from the 
 values of the masses obtained independently of the secular 
 variations in the last chapter. For the sake of clearness, 
 these two sets of secular variations and their differences are 
 collected in the following table. The mean errors assigned to 
 the theoretical values are those which result from the prob- 
 able mean errors of the respective masses. They are there- 
 fore not to be regarded as independent. The mean errors 
 given in the column of differences are those which result from 
 a combination of those of the other two columns. The errors 
 of the observed quantities must not, however, be judged from 
 those of the differences, because subsequent changes in the 
 masses of Mercury, Venus, and the Earth may produce a 
 general diminution in the discordances. 
 
 Mercury. 
 
 Observation. Theory. Diff. A \/w. 
 
 // // // // // // // 
 
 D t <? + 3.36+0.50 + 4.24 + .01 -0.88i.50 -0.86 2 
 6D t 7r + 118.24+0.40 +109.76+. 16 +8.4S+. 43 . . 
 
 D t i + 7.14+0.80 + 6.76+.01 +0.38^.80 +0.38 1J 
 siniD t # - 91.89+0.45 - 92.50+.16 +0.61+.52 +0.23 2.2 
 
 Venus. 
 
 D t e 9.46+0.20- 9.67+.24 +0.21+.31 +0.12 5 
 
 eD t 7T + 0.29 i 0.20 + 0.34+.15 -0.05+.25 . . 
 
 D t * + 3.87 0.30 + 3.49i.l4 +0.38i.33 +0.44 3J 
 
 siniD t # -105.400.12 -106.00i.12 +0.60i.l7 +0.52 8 
 
 109 
 
110 COMPARISON OF SECULAR VARIATIONS. [55 
 
 JEartli. 
 
 Observation. Theory. Diff. A Vw. 
 
 // // // // // // // 
 
 D t e - 8.55 0.09 - 8.57 .04 +0.02 .10 +0.02 10 
 eD t 7T + 19.48 0.12 + 19.38+ .05 +0.10i. 13 . . 
 D t - 47.lliO.23 - 46.89+.09 0.22+.27 -0.46 4 
 
 Mars. 
 
 D t e + 19.OOiO.27 + 18.71+.01 -fO.29i.27 +0.29 3.7 
 
 0D t 7r 4-149.55iO.35 + 148.80+.04 +0.75+.35 ... 
 
 D t i 2.26+0.20- 2.25 +.04 -0.01 + .20 +0.08 5 
 
 siniD t # - 72.60 + 0.20 - *2.63+.09 +0.03+.22 -0.17 5 
 
 If we multiply the mean errors given by 0.6745, to reduce 
 them to probable errors, we shall see that only four of the 
 fifteen differences are less than their probable errors. The 
 deviations which call for especial consideration are the follow- 
 ing four : 
 
 1. The motion of the perihelion of Mercury. The discord- 
 ance in the secular motion of this element is well known. 
 
 2. The motion of the node of Venus. Here the discordance 
 is more than five times its probable error. 
 
 3. The perihelion of Mars. Here the discordance is three 
 times its probable error. 
 
 4. The eccentricity of Mercury. The discordance is more 
 than twice its probable error. It is to be remarked, however, 
 that the probable error of this quantity is very largely a 
 matter of judgment, and that its value may have been under- 
 estimated. 
 
 The deviations, if not due to erroneous masses, may be 
 explained on two hypotheses. One is that propounded by 
 Prof. HALL,* that the gravitation of the Sun is not exactly as 
 the inverse square, but that the exponent of the distance is a 
 fraction greater than 2 by a certain minute constant. This 
 hypothesis accounts only for the motions of the perihelia, and 
 not for any other discordances. 
 
 The other hypothesis is that of the action of unknown 
 masses or arrangements of matter. Since the latter hypothesis 
 
 * Astronomical Journal, Vol. XIV, p. 7. 
 
55,56] NON-SPHERICITY OF THE SUN. Ill 
 
 would account for other motions than those of the perihelia, it 
 might seem that the existence -of the other discordances 
 tells very strongly in its favor. The hypotheses of possible dis- 
 tributions of unknown matter, therefore, have iirst to be con- 
 sidered.* 
 
 Hypothesis of non- sphericity of the Sun. 
 
 56. In a case where our ignorance is complete, all hypotheses 
 which do not violate known facts are admissible. Beginning 
 at the center and passing outward, the first question arises 
 whether the action may not be due to a non -spherical distri- 
 bution of matter within the body of the Sun, resulting in an 
 excess of its polar over its equatorial moment of inertia. The 
 theory of the Sun which has in recent times been most gener- 
 ally accepted is that its interior may be regarded as gaseous, 
 or rather as a form of matter which combines the elasticity 
 and mobility of a gas with the density of a liquid. Such 
 being the case, we may conceive that vortices of which the 
 axes coincide with that of rotation may exist in the interior 
 in such a way that the surfaces of equal density are non- 
 spherical. A very small inequality of this sort would suffice 
 to account for the motion of the perihelion of Mercury. 
 
 This hypothesis admits of an easy test. Whatever be the 
 nature or amount of the inequality, a simple computation 
 shows that to account for the observed phenomenon it is 
 necessary and sufficient that the equipotential surfaces at the 
 surface of the Sun should have an elliptic! ty of rather more 
 than half a second of arc. It can not, I conceive, be doubted 
 that the visible photosphere is an equipotential surface. We 
 have then to inquire whether there is any such ellipticity of 
 the photosphere as that required by the hypothesis. This 
 question seems completely set at rest by the great mass of 
 heliometer measures made by the German observers in con- 
 nection with the transits of Venus of 1874 and 1882, which 
 have been discussed by Dr. AUWERS. The general result is 
 
 * After carrying out the investigations of this chapter, I find that the 
 subject was studied on similar lines by Dr. P. HARZER nearly three years 
 ago, and that I made certain suggestions on the subject to Dr. BAUSCH- 
 INGER ten years ago. See Astrononiacliv Nachrichten, Vol. 109, p. 32, and 
 Vol. 127, p. 81. 
 
112 INTRA-MERCURIAL GROUP. [56,57 
 
 that the mean of the equatorial measures are slightly less than 
 the mean of the polar measures, the difference, however, being 
 within the probable errors of the results. I conclude that 
 there can be no such n on- symmetrical distribution of matter 
 in the interior of the Sun as would produce the observed effect. 
 This same conclusion seems to apply to matter immediately 
 around the photosphere. An equatorial ring of planetoids, or 
 gaseous substances of the required mass, very near the photo- 
 sphere, would render the equipotential surfaces of the photo- 
 sphere elliptical to a degree which seems precluded by the 
 measures in question. At a very short distance from the sur- 
 face, however, the effect would be inappreciable. 
 
 Hypothesis of an intra-mercurial ring or group of planetoids. 
 
 57. Passing outward, we have next to consider the hypothe- 
 sis of an intra-mercurial ring adequate to produce the observed 
 phenomena. In a first approximation we may suppose the 
 ring circular. Its mass can not be determined, because it will 
 depend upon the distance ; we have to determine a certain 
 function of the mass and distance adequate to produce the 
 observed motion of the perihelion. Then we must inquire what 
 effect the ring will have on the secular variations of the other 
 elements, both of Mercury and of the other planets, and see if 
 these effects can be reconciled with observation. In the com- 
 putations I have assigned to the excess of motion the pro- 
 visional value 40 // .7. If the ring is not very distant from the 
 Sun the motion which it will produce in the perihelion of a 
 planet whose mean motion is n and whose mean distance is a 
 may be represented in the form 
 
 }JL being a function of the mass of the ring and of its radius, 
 which is nearly the same for all of the planets, so long as the 
 radius of the ring is only a small fraction of the distance of 
 Mercury. A first approximation to /* is 
 
 u = . m r 2 
 
57 ) INTRA-MERCTJRIAL 
 
 m being tke ratio of its mass to that of the Sun alSSTFTEs radius. 
 Multiplying these motions in the case of the four planets by 
 their eccentricities, we find that the hypothetical ring will 
 produce the following secular variations : 
 
 Mercury, D t n 40.7; eD t n = 8.38 
 Venus, 4.6 0.031 
 
 Earth, 1.5 0.025 
 
 Mars, 0.34 0.031 
 
 Owing to the sinallness of the eccentricities the effect is 
 insensible, except in the case of Mercury, so that the ring will 
 not account for the observed excess of motion of the perihelion 
 of Mars. 
 
 Such a ring will necessarily produce a motion of the plane 
 of the orbit of Mercury or Venus, or of both, because it can 
 not lie in the plane of both orbits. 
 
 Let us put ii for its inclination to the ecliptic, and 61 for the 
 longitude of its node on the ecliptic; and let us put, also, 
 
 Pi = ii sin #! 
 q l = i, cos 6\ 
 
 and let j?, p', ... , q, q', . . be the corresponding quan 
 tities for the planets. The theory of the secular variations 
 then shows that the ring will produce a motion of the plane of 
 the orbit of Mercury given by the equations 
 
 D t^i = *-ll (9i ~q} = 40".7 (q l - q) 
 
 Expressing the motions of p and q in terms of the motions of i 
 and 0, which is necessary, owing to the very different weights 
 of the determination of the motion of the planes of Mercury 
 and Venus in the direction of these two coordinates, we have 
 5690 N, ALM 8 
 
114 INTRA-MERCURIAL GROUP. [57 
 
 the following expressions for these two motions, which we 
 equate to the observed excesses :* 
 
 - 4.96 4- 26.9 i 4- 28.4_p t = + 0.57 0.50 
 
 - 0.27 + 0.8 4-3.0 = 4- 0.63 0.12 
 0.00 4- 28.4 - 26.9 = + 0.50 0.80 
 0.00 4- 3.0 - 0.8 =4- 0.45 0.30 
 0.00 0.0 1.5 = - 0.25 0.25 
 
 Multiplying the conditional equations thus formed by such 
 factors as will make the mean error of each equation nearly 
 0".5, we have the following conditional equations for p { 
 and #1 : 
 
 27 q, 4- 28^! = + 5.53 
 
 3 + 12 = 4. 3.60 
 17 _ 16 =4- 0.30 
 
 5 - 1 = 4- 0.77 
 
 - 3 = - 0.50 
 
 The solution of these equations gives very nearly 
 
 ^=4-0.12; 0i=4& 
 
 This great inclination seems in the highest degree improbable 
 if not mechanically impossible, since there would be a tend- 
 ency for the planes of the orbits of a ring of planets so 
 situated to scatter themselves around a plane somewhere 
 between that- of the orbit of Mercury and that of the invari- 
 able plane of the planetary system, which is nearly the same 
 as that of the orbit of Jupiter. Moreover, the motion of the 
 perihelion of Mars is still unaccounted for and that of the 
 node of Venus only partially accounted for, as shown by the 
 large residual of the second equation. In fact, the great incli- 
 nation assigned to the ring- comes from the necessity of repre- 
 senting as far as possible the latter motion. 
 
 * It will be noticed that iii forming these equations I have neither used 
 the final values of the absolute terms, nor taken account of the fact that 
 the observed motions of the planes are referred to the ecliptic. Changes 
 thus produced in the equations are too minute to affect the conclusion. 
 
57, 58] ZODIACAL LIGHT. 115 
 
 There would of course be no dynamical impossibility in the 
 hypothesis of a single planet having as great an inclination as 
 that required. But I conceive that a planet of the adequate 
 mass could not have remained so long undiscovered. Whether 
 we regard the matter as a planet or a ring, a simple computa- 
 tion shows that its mass, if at the Sun's surface, would be 
 
 about ri that of the Sun itself, and one-fourth of this if at a 
 
 distance equal to the Sun's radius. We may conceive, if we 
 can not compute, how much light such a mass of matter would 
 reflect. Altogether, it seems to me that the hypothesis is 
 untenable. 
 
 Hypothesis of an extended mass of diffused matter like that which 
 reflects the zodiacal light. 
 
 58. The phenomenon of the zodiacal light seems to show 
 that our Sun is surrounded by a lens of diffused matter which 
 extends out to, or a little beyond, the orbit of the Earth, the 
 density of which diminishes very rapidly as we recede from 
 the Sun. The question arises whether the total mass of this, 
 matter may not be sufficient to cause the observed motion. 
 
 So far as the action of that portion of matter which is near 
 the Sun is concerned, the conclusions just reached respecting 
 a ring surrounding the Sun will apply unchanged, because we 
 may regard such a mass as made up of rings. Observation 
 seems to show that the lens in question is not much inclined 
 to the ecliptic, and if so it would produce a motion of the 
 nodes of Venus and Mercury the opposite of that indicated 
 by the observations. 
 
 There is another serious difficulty in the way of the hypoth- 
 esis. A direct motion of the perihelion of a planet may be 
 taken as indicating the fact that the increase of its gravitation 
 toward the Sun as it passes from aphelion to perihelion is 
 slightly greater than that given by the law of the inverse 
 square. This increase would be produced by a ring of matter 
 either wholly without or wholly within the orbit. But if we 
 suppose that the orbit actually lies in the matter composing 
 such a ring, the effect is the opposite; gravitation toward the 
 
11 6 EXTRA-MERCURIAL GROUP. [58, 59, 60 
 
 Sun is relatively diminished as the planet passes from aphelion 
 to perihelion, and the motion of the perihelion would be retro- 
 grade. 
 
 It can not be supposed that that part of the zodiacal light 
 more distant from the Sun than the aphelion of Mercury is 
 even as dense as that portion contained between the aphelion 
 and the perihelion distances. The result in question must 
 therefore be due wholly to that part of the matter which lies 
 near to the Sun, and we thus have all the difficulties of the 
 intra-mercurial ring theory, with one more added. 
 
 Hypothesis of a ring of planetoids between, the orbits of Mercury 
 
 and Venus. 
 
 59. It appears that any ring or zone of matter adequate 
 to produce the observed effect must lie between the orbits of 
 Mercury and Venus. Its assignment to this position requires 
 a more careful determination of its possible eccentricity. 
 There will be six independent elements to be determined; 
 the mass, the mean distance, the eccentricity, the perihelion, 
 the inclination, and the node. 
 
 I find that the observed excesses of motion of the elements 
 of Mercury and Venus will be approximately represented by 
 elements not differing much from the following: 
 
 Total mass of group 37000000 
 
 Mean distance 0.48 
 
 Eccentricity of orbit 0.04 
 
 Longitude of perihelion . . , . 10 
 
 Longitude of node 35 
 
 Inclination to ecliptic* 7.5 
 
 Probable diameter at distance unity if 
 
 agglomerated into a single planet . 3".5 
 
 Considerations on the admissibility of the hypothesis Possible 
 mass of the minor planets. 
 
 60. Although the preceding hypothesis is that which best 
 represents the observations of Mercury and Venus, we can 
 not, in the present condition of knowledge, regard it as more 
 than a curiosity. True, it is plausible at first sight. Since, 
 
60] POSSIBLE ACTION OF THE MINOR PLANETS. 117 
 
 as already remarked, any disturbing body of sufficient mass 
 to cause the observed excess of motion of the perihelion of 
 Mercury would change the position of the planes of the orbits, 
 and since observations give apparent indications of such a 
 change in the plane of the orbit of Venus, it might appear 
 that we have here a very good ground for the view that all 
 the motions are due to the attraction of unknown masses. 
 But the great difficulty is that the excess of motion of the 
 orbital planes is in the opposite direction from what we should 
 expect. A group of bodies revolving near the plane of the 
 ecliptic would produce a retrograde motion of the nodes. But 
 the observed excess is direct. A direct motion can be pro- 
 duced only in case the orbits are more inclined than those of 
 the disturbed planet. In admitting such orbits we encounter 
 difficulties which, if not absolutely insurmountable, yet tell 
 against the probability of the hypothesis. 
 
 The hypothesis carries with it the probable result that the 
 excess of motion of the perihelion of Mars is produced by the 
 action of the minor planets. I have considered the question 
 of this action in an unpublished investigation. From the prob- 
 able albedo and magnitude of the minor planets and the obser- 
 vations of BARNARD and others on their diameters, I have 
 determined the probable mass of each part of the group having 
 a given opposition magnitude. The result is that the number 
 of these bodies having such a magnitude appears to progress 
 in a fairly uniform manner through several magnitudes. The 
 ratio of progression may lie anywhere between the limits 2 
 and 3. Up to the limit 3 the total mass, if continued on to 
 infinity, could not produce any appreciable effect on the motion 
 of Mars. But if we suppose a larger ratio than 3 to prevail, 
 then the number of planets of smaller magnitude would be so 
 numerous us to form a zone of light across the heavens, as may 
 readily be seen by considering that the total amount of light 
 reflected from the planets of each order of magnitude would 
 form an increasing series, since the ratio between the brillian- 
 cies of two objects of unit difference in magnitude is only 
 about 2.5. We may therefore suppose that the faint band of 
 light which is said to be visible across the entire heavens as 
 a continuation of the zodiacal light, as well as the "gegen- 
 
118 HALL'S HYPOTHESIS. [60,61 
 
 schein," is due to these minute bodies, and yet find their total 
 mass too small to produce any appreciable effect. 
 
 Whether we can assign to the components of such a group 
 any magnitude so small that they would be individually invis- 
 ible, and a number so small that they would not be seen 
 collectively as a band of light brighter than the zodiacal arch, 
 and yet having a total mass so large as to produce the observed 
 effects, is a very important question which can not be decided 
 without exact photometric investigations. It is, however, cer- 
 tain that if we could do so we should have to suppose a very 
 unlikely discontinuity in the law of progression between each 
 magnitude and the number of bodies having that magnitude. 
 It must therefore suffice for our present object that we regard 
 the hypothesis of such bodies as unsatisfactory. 
 
 Hypothesis that gravitation toward the sun is not exactly as the 
 inverse square of the distance. 
 
 61. Prof. HALL'S hypothesis seems to me provisionally not 
 inadmissible. It is, that in the expression for the gravitation 
 between two bodies of masses m and m' at distance r 
 
 Force = 
 
 the exponent n of r is not exactly 2, but 2 + 6, d being a very 
 small fraction. This hypothesis seems to me much more 
 simple and unobjectionable than those which suppose the 
 force to be a more or less complicated function of the relative 
 velocity of the bodies. On this hypothesis the perihelion of 
 each planet will have a direct motion found by multiplying its 
 mean motion by one-half the excess of the exponent of grav- 
 itation. 
 Putting 
 
 n = 2.000 000 1574 
 
 the excess of motion of each perihelion of the four inner 
 planets would be as follows. It will be seen that the evidence 
 in the case of Venus and the Earth is negative, owing to the 
 
01] LAW OF GRAVITATION. 119 
 
 very small eccentricities of their orbits, while the observed 
 motion in the case of Mars is very closely represented. 
 
 Mercury, 
 
 42.34 
 
 8.70 
 
 Venus, 
 
 16.58 
 
 0.11 
 
 Earth, 
 
 10.20 
 
 0.17 
 
 Mars, 
 
 5.42 
 
 0.51 
 
 An independent test of this hypothesis in the case of other 
 bodies is very desirable. The only case in which there is any 
 hope of determining such an excess is that of the Moon, where 
 the excess would amount to about 140" per century. This is 
 very nearly the hundred- thousandth part of the total motion 
 of the perigee. The theoretical motion has not yet been com- 
 puted with quite this degree of precision. The only determi- 
 nation which aims at it is that made by HANSEN.* He finds 
 
 Theory. Obser. Diff. 
 
 // // // 
 
 Annual mot. of perigee, 146 434.04; 146 435.60; ^+1.56 
 
 Annual mot. of node, -69 676.76 69 679.62; -2.86 
 
 The observed excess of motion agrees well with the hypoth- 
 esis, but loses all sustaining force from the disagreement in 
 the case of the node. The differences HANSEN attributes 
 (wrongly, I think) to the deviation of the figure of the Moon 
 from mechanical sphericity. 
 
 Consistency of Hall's hypothesis with the general results of the 
 law of gravitation. 
 
 62. The law of the inverse square is proven to a high degree 
 of approximation through a wide range of distances. The close 
 agreement between the observed parallax of the Moon and 
 that derived from the force of gravitation on the Earth's sur- 
 face shows that between two distances, one the radius of the 
 Earth and the other the distance of the Moon, the deviation 
 from the law of the square can be only a small fraction of the 
 
 *Darlegung, etc.: Abhandhungen der Math.-Phys. Classe der Kon. Sdchsi- 
 Bchen Gesellscltaft der Wissenschaften, vi, p. 348. 
 
120 HALL'S HYPOTHESIS. [62 
 
 thousandth part, or, we may say, a quantity of the order of 
 magnitude of the five- thousandth part. 
 
 Coming down to smaller distances, we find that the close 
 agreement between the density of the Earth as derived Iroin 
 the attraction of small masses, at distances of a fraction of a 
 meter, with the density which we might a priori suppose the 
 Earth to have, shows that within a range of distance extend- 
 ing from less than one meter to more than six million meters, 
 the accumulated deviation from the law can scarcely amount 
 to its third part. The coincidence of the disturbing force of 
 the Sun upon the Moon with that computed upon the theory 
 of gravitation, extends the coincidence from the distance of 
 the Moon to that of the Sun, while KEPLER'S third law 
 extends it to the outer planets of the system. Here, however, 
 the result of observations so far made is relatively less pre- 
 cise. We may therefore say, with entire confidence, as a 
 result of accurate measurement, that the law of the inverse 
 square holds true within its five- thousandth part from a dis- 
 tance equal to the Earth's radius to the distance of the Sun, a 
 range of twenty-four thousand times ; that it holds true within 
 a third of its whole amount through the range of six million 
 times from one meter to the Earth's radius; and within a 
 small but not yet well-defined quantity from the distance of 
 the Sun to that of Uranus, in which the multiplication is 
 twentyfold. 
 
 If HALL'S hypothesis contradicted these conclusions it would 
 be untenable. But a very simple computation will show that, 
 assuming the force to vary as r-(* + 8 \ d being a minute con- 
 stant sufficient to account for the motion of the perihelion of 
 Mercury, the effect would be entirely inappreciable in the ratio 
 of the gravitation of any two bodies at the widest range of 
 distance to which observation has yet extended. Although 
 the total action of a material point on a spherical surface sur- 
 rounding it would converge to zero when the radius became 
 infinite, instead of remaining constant, as in the case of the 
 inverse square, yet the diminution in the action upon a surface 
 no larger than would suffice to include the visible universe 
 would be very small. 
 
63] CORRECTION OF MASSES. 121 
 
 Masses of the planets which represent the secular variations of 
 other elements than the perihelia. 
 
 63. On HALL'S hypothesis the secular variations of all the 
 elements other than the perihelia will remain unchanged. 
 
 Our next problem is to consider the possibility of represent- 
 ing the variations of the other elements by admissible masses 
 of the known planets. In 55 I have given a comparison of 
 the secular variations as they result from observations, with 
 their theoretical expressions in terms of corrections to a cer- 
 tain system of masses. When the equations thus formed are 
 multiplied by the factors Vw, which make the mean error of 
 each equation unity, we have the following system of equa- 
 tions, in which we put v = 10 #: 
 
 Ox 
 
 + QY' 
 
 + 2 v" 
 
 + Ov"' 
 
 = - 1.7 
 
 r = - 1.8 
 
 
 
 - 1 
 
 - 2 
 
 
 
 = +0.5 
 
 + 0.5 
 
 - 7 
 
 -108 
 
 - 27 
 
 - 3 
 
 = +0.5 
 
 + 1.1 
 
 -65 
 
 
 
 - 24 
 
 - 1 
 
 = +0.6 
 
 + 0.7 
 
 +25 
 
 
 
 
 
 - 1 
 
 = + 1.5 
 
 + 1.3 
 
 + 21 
 
 -234 
 
 -346 
 
 -10 
 
 = 4-4.2 
 
 0.0 
 
 -12 
 
 + 13 
 
 
 
 -16 
 
 = + 0.2 
 
 + 0.1 
 
 - 9 
 
 -123 
 
 
 
 - 3 
 
 = -2.0 
 
 -0.7 
 
 + 1 
 
 
 
 + 8 
 
 
 
 = +1.1 
 
 + 1.3 
 
 2 
 
 + 60 
 
 
 
 
 
 = + 0.4 
 
 -0.2 
 
 -14 
 
 -126 
 
 - 37 
 
 - 5 
 
 = -0.8 
 
 -0.2 
 
 The resulting normal equations are 
 
 5766 x 1563 v 1 4991 v" + 140 v 1 " = + 114 
 
 -1563 +101231 + 88556 +3455 670 
 
 - 4991 + 88556 + 122462 + 3750 = - 1446 
 
 + 140 + 3455 + 3750 + 401 39 
 
 Along with the results of the solution of these equations I 
 place, for comparison, the values of Chapter Y, which have 
 been considered most probable. 
 
 From sec. var. From other sources. 
 
 Wx = v = + 0.070 + 0.08 0.20 
 
 v' = + 0.0100 .0056 + 0.0084 0.0028 
 
 v" = - 0.0183 .0052 - 0.00304 i 0.0015 
 
 v" 1 = - 0.0115 .067 + .0037 i 0.018 
 
122 CORRECTION OF MASSES. [63, 64 
 
 By substitution in the conditional equations we find for the 
 mean error corresponding to weight unity 
 
 *i = i 1.14 
 
 In forming these equations they were reduced by multipli- 
 cation to a supposed mean error of 1. Speaking in a 
 general way we may therefore say that the representation. of 
 the secular variations, those of the perihelia being ignored, 
 by these corrections to the masses is satisfactory. Except for 
 the large discordance in the motion of the eccentricity of 
 Mercury the mean error would have been less than unity. 
 
 Comparing the two sets of values we find that the masses 
 of Mercury, Venus, and Mars agree well with those derived 
 from other sources. Very different is it with the mass of the 
 Earth. The discordance is here more than the hundredth 
 part of its whole amount, which involves a discordance of 
 more than the three-hundredth part in the value of the solar 
 parallax. Let us now proceed in the reverse order, and deter- 
 mine the value of the solar parallax from the mass of the Earth, 
 as derived from the preceding data. 
 
 Preliminary adjustment of the two sets of masses. 
 
 64. We make the best adjustment for this purpose by adding 
 to the equations of condition last given the additional ones 
 derived from the values of the masses discussed in Chapter V. 
 Multiplying each value of v by the factor necessary to reduce 
 the mean error of the second member of the equation to unity, 
 we have the following conditional equations : 
 
 50 x =4- 0.4 
 360 v' = + 2.9 
 50 v 1 " = 0.0 
 30 v"> = -f 0.42 
 
 Of the last two equations it may be remarked that the first is 
 that given by Prof. HALL'S original mass of 1877, while the 
 last is derived by Dr. HARSHMAN from HALL'S observations 
 of the outer satellite made during the opposition of 1892. 
 
64] CORRECTION OF MASSES. 123 
 
 When we add to the normal equations already formed the 
 products of these last equations by the factors of the unknown 
 quantities, the system of normal equations is as follows : 
 
 8266 # - 1563 v 1 - 4991 r" + 140 r"' =+134 
 
 -1563 +230831 + 88556 +3455 = +374 
 
 -4991 + 88556 +122462 +3750 = -1446 
 
 + 140 + 3455 + 3750 +3801 == -26 
 
 The solution of these equations gives the following values of 
 the unknown quantities : 
 
 x = + 0.0071 i .0120 
 v = + 0.071 i .120 
 v 1 = + 0.0084 i .0024 
 V n = _ 0.0177 i .0035 
 v'"= + 0.0027 i .016 
 
 Here again we note that, the Earth aside, the results for the 
 masses are quite satisfactory. The correction to Prof. HALL'S 
 original mass of Mars is so minute and so much less than its 
 probable error that we may consider this value of the mass to 
 be confirmed, and may adopt it as definitive without question. 
 The corrections to the masses of Mercury and Veuus are scarcely 
 changed. The mean residual is reduced to 
 
 8 = i 0.91 
 
 which is less than the supposed value. 
 
 We have, therefore, so far as these results go, no reason for 
 distrusting the following value of the solar parallax, which 
 results from that of the mass of the Earth thus derived: 
 
 7i = 8".759 ".010 
 
 The critical examination and comparison of this and other 
 values of the parallax is the" work of the next two chapters. 
 
CHAPTER VII. 
 
 VALUES OF THE PRINCIPAL CONSTANTS WHICH DEFINE 
 THE MOTIONS OF THE EARTH. 
 
 The Precessional Constant. 
 
 65. The accurate determination of the annual or centennial 
 motion of precession is somewhat difficult, owing to its depend- 
 ence on several distinct elements, and to the probable system- 
 atic errors of the older observations in Right Ascension and 
 Declination. What is wanted is the annual motion of the 
 equinox, arising from the combined motions of the equator 
 and the ecliptic, relative to directions absolutely fixed in space. 
 As observations can not be referred to any line or plane which 
 we know to be absolutely fixed, we are obliged to assume that the 
 general mean direction of the fixed stars remains unchanged, 
 or, in other words, that the stellar system in general has no 
 motion of rotation. This is a safe assumption so far as the 
 great mass of stars of smaller magnitude is concerned. But it 
 is not on such stars that we have the earliest accurate obser- 
 vations. Moreover, observed Right Ascensions of these 
 fainter stars relative to the brighter ones are subject to possi- 
 ble systematic errors, arising from the personal equation being 
 different for brighter and fainter stars. In the case of the 
 stars observed by BRA.DLEY, there is frequently such commu- 
 nity of proper motion among neighboring stars that we can 
 noLbe quite sure that all rotation is eliminated in the general 
 mean. Under these circumstances we have only to make the 
 best use that we can of existing material. 
 
 We must also remember that observed Right Ascensions are 
 not directly referred to the equinox, but to the Sun, of which 
 the error of absolute mean Right Ascension must be deter- 
 mined. This again can be done only from observed declina- 
 tions, since by definition the equinox is the point at which 
 the Sun crosses the equator. It is also to be noted that the 
 clock stars which are directly compared with the Sun by no 
 124 
 
65] THE PRECESSIONAL CONSTANT. 125 
 
 means include the whole list to be used as absolute points of 
 reference. We therefore have three separate steps in determin- 
 ing completely a correction to the adopted annual precession : 
 
 (1) The correction to tlie Sun's absolute mean Eight Ascen- 
 sion or longitude. 
 
 (2) The correction to the general mean Eight Ascension of 
 the clock stars relative to the Sun. 
 
 (3) The determination of the clock stars relative to the great 
 mass of stars. 
 
 It goes without saying that the determinations of these three 
 quantities are entirely independent of each other, and that the 
 precision of the result depends on the precision of each sepa- 
 rate determination. 
 
 The motion of the pole of the equator, on which the luni- 
 solar precession depends, may be determined by observed 
 Declinations quite independently of the Eight Ascensions. A 
 determination of the precession from the latter includes the 
 planetary precession, but as this has to be determined from 
 theory independently of observations, we have, in observed 
 Eight Ascensions and Declinations, two independent methods 
 of determining the motion of the equator. 
 
 It fortunately happens that the constant of precession is 
 not so closely connected with other constants that a small 
 error in its determination will seriously affect our general con- 
 clusions, or the reduction of places of the fixed stars, because 
 the effect of an error will be nearly eliminated through the 
 proper motions of the fixed stars, or the motions of the planets 
 in longitude. I have therefore satisfied myself with reviewing 
 and combining ,the four best determinations. 
 
 I pass over in silence the classic determinations of BESSEL 
 and OTTO STRUVE, because the material on which they depend 
 has been incorporated in more recent works. Of these the one 
 which seems entitled to most weight is that of Luowia STRUVE, 
 Bestimmung der Constante der Prcecession, und der eigenen 
 Bewegung des Sonnensy 'stems.* This work was suggested by 
 the completion of AUWERS' re-reduction of BRADLEY'S Obser- 
 vations, and of the Pulkowa standard catalogues for 1845, 
 
 *Me"moires de PAcademie Impe'riale des Sciences de St. Pe'tersbourg. 
 VII e SSrie. Tome xxxv, No. 3. 
 
126 THE PRECESSIONAL CONSTANT. [65 
 
 1855, aiid 1865. It depends entirely on the BRADLEY stars, 
 and the result, when reduced to the most probable equinox, 
 may be regarded as the best now derivable from those stars, 
 or, at least, as not susceptible of any large correction. 
 
 He, of course, includes in his work the determination of the 
 motion of the solar system relative to the mass of the stars. 
 In addition to this, the possibility of a common rotation of 
 the BRADLEY stars around the axis of the Milky Way is con 
 sidered. This rotation I should be disposed to regard as zero 
 for the present. 
 
 In place of considering each of the 2,509 stars singly, he 
 divides the celestial sphere into 120 spherical trapezoids, each 
 covering 15 degrees in Declination, and an arc of Right 
 Ascension equal approximately to one hour of a great circle 
 at the equator. The question might be legitimately raised 
 whether a different system of weighting the trapezoids, founded 
 on a consideration and comparison of the proper motions in 
 Eight Ascension and Declination would not have been advis- 
 able. I am, however, fairly confident that no change in this 
 respect would have materially affected the result. With this 
 work of STRUVE I have combined those of BOLTE, DREYER, 
 and NYREN. 
 
 In the case of the Eight Ascensions it is necessary to reduce 
 all the results to the equinox determined in the last chapter. 
 From this chapter it appears that the standard Eight Ascen- 
 sions with which the reduction of the preceding investigations 
 have been made require a correction to the centennial motion 
 of 4- 0".30. Eeducing each determination to the equinox thus 
 defined, we have the following results for the general preces- 
 sion in Eight Ascension at the epoch 1800 : 
 
 L. STRUVE, from the comparison of 
 AUWERS-BRADLEY with the modern 
 Pulkowa Eight Ascensions . . . m = 46".050l ; w = 4 
 
 DREYER, from the comparison of 
 LALANDE'S Eight Ascensions with 
 those of SCHIELLERUP 46 .0611; w = 2 
 
 NYREN, by the comparison of BESSEL'S 
 Eight Ascensions with those of 
 
 S'CH JELLERUP 46 .0456 ; W = I 
 
 Mean 46 .0526 
 
65] THE PRECESSIONAL CONSTANT. 127 
 
 The weights here assigned are of course a matter of judgment. 
 The general agreement of the results is as good as we could 
 expect. 
 From observed declinations we have 
 
 L. STRUVE, from the comparison of 
 
 AUWERS - BRADLEY with modern 
 
 Pulkowa catalogues w = 20".0495; iv = 2 
 
 BOLTE, from the comparison of LA- 
 
 LANDE'S Declinations with those of 
 
 SCILJELLERUP 20 .0537 ; w = 1 
 
 Mean 20 .0509 
 
 We have now to -combine these independent results. I pro- 
 pose to call Precessional Constant that function of the masses 
 of the SUE, Earth, and Moon, and of the elements of the orbits 
 of the Earth and Moon, which, being multiplied by half the 
 sine o twice the obliquity, will give the annual or centennial 
 motion of the pole on a great -circle, and being multiplied by 
 the cosine of tire obliquity will give the lunisolar precession 
 at any time. It is true that this quantity is not absolutely 
 constant, since it will change in the course of time, through 
 the diminution of the Earth's eccentricity. This change is, 
 however, so slight that it can become appreciable only after 
 several centuries. If, then, we put 
 
 p, the precessional constant, we have, for the annual general 
 precession in Eight Ascension and Declination 
 
 m = p cos 2 e H sin L cosec s 
 n Y sin cos 
 
 L being the longitude of the instantaneous axis of rotation 
 of the ecliptic, and H its annual or centennial motion. From 
 the definitive obliquity and masses of the planets adopted 
 hereafter, we find the following values of #, L, and , for 1800 
 and 1850: 
 
 lOg H = 
 
 1800. 
 1.67372; 
 
 1850. 
 1.67341 
 
 L = 
 
 173 2'.31; 
 
 173 29'.68 
 
 = 
 
 23 27.92; 
 
 23 27 .53 
 
128 THE PRECESSIONAL CONSTANT. [65 
 
 We thus find the following values of p, the unit of time 
 being 100 solar years: 
 
 From Eight Ascensions, P = 5490.12; w = 2 
 From Declinations, p = 5489.44; w = 1 
 
 Mean, p = 5489 // .89 
 
 As the data used in STRUVE'S Investigation may be con- 
 sidered of a more certain kind than those used by the others, 
 we may compare these results with those which follow from 
 STRUVE'S work alone. They are 
 
 From Eight Ascensions, P = 5489.83 
 
 From Declinations, p = 5489.06 
 
 Giving double weight to the results from the Eight Ascen 
 sions, the results may be expressed as follows : 
 
 From STRUVE'S investigation, P = 5489.57 
 From the other two works, p = 5490.18 
 
 Before concluding this investigation, I had adopted as a pre- 
 liminary value 
 
 P = 5489".78 
 
 As this result does not differ from the one I consider most 
 probable, 5489".S9, by more than the probable error of the 
 latter, and diverges from it in the direction of the best deter- 
 mination, I have decided to adhere to it as the definitive 
 value. 
 
 The centennial value of p is subjected to a secular diminu- 
 tion of 0".00364 per century, owing to the secular diminution of 
 the eccentricity of the Earth's orbit. We therefore adopt 
 
 p = 5489.78 0.00364 T for a tropical century. 
 p = 5489.90 - 0.00364 T for a Julian century. 
 
 In the use of p I at first neglected the secular variation, 
 but have- added its effect to the results developed in powers 
 of the time. 
 
66] THE CONSTANT OF NUTATION. 129 
 
 Constant of nutation derived from observations. 
 
 66. The determination of this constant from observations is 
 extremely satisfactory, owing to the completeness with which 
 systematic errors may be eliminated. If, with a meridian 
 instrument, regular observations are made through a draconitic 
 period, on a uniform plan, upon stars equally distributed 
 through the circle of Eight Ascension, the observations being 
 made daily through more than 12 hours of Eight Ascension, 
 all systematic errors in the determination of the nadir point 
 and all having a diurnal or annual period may be completely 
 eliminated from the constant in question. These conditions 
 are so nearly fulfilled in the observations with the Greenwich: 
 transit circle, and, to a less extent, in those with the Wash- 
 ington transit circle, that the results of the Wdrk with those 
 two instruments alone are entitled to greater weight than has 
 hitherto been supposed. I have, however, discussed quite 
 fully all previous determinations of which it seemed that the 
 probable mean error would be less than 0".10. 
 
 Eeferring to the volume on the subject to be hereafter pub- 
 lished, the results of the discussion are presented in the fol- 
 lowing table. The weights are assigned on the supposition 
 that weight unity should correspond to a mean error of about 
 0".07, or to a probable error of /7 .05, this probable value 
 being not entirely a matter of computation from the discord- 
 ance of. the separate results, but, to a certain extent, a matter 
 of judgment. 
 
 It.must be understood that the results below are not always 
 those given by the authors who are quoted, but that their dis- 
 cussion has,, wherever- possible, been subjected to a revision by 
 the introduction of modern data, or by what seemed to me 
 improved combinations. Thus, NYREN'S equations have been 
 reconstructed on a system slightly different from his, and have 
 been corrected for CHANDLER'S variation of latitude. PETERS'S 
 classical work has also been corrected by the introduction of 
 later data, and by a re-solution of his equations. The Green- 
 wich and Washington results have been derived from the dis- 
 cussion in Astronomical Papers, Vol. II, Part VI. 
 5690 N ALM 9 
 
130 THE CONSTANT OF NUTATION. [66 
 
 Values of the constant of nutation derived from observations. 
 
 BUSCH, from BRADLEY'S observations with 
 
 the zenith sector 9.232 1 
 
 ROBINSON, from Greenwich mural circles . . 9.22 1 
 
 PETERS, from Eight Ascensions of Polaris . 9.214 4 
 
 LUND AHL, from Declinations of Polaris . . 9.236 1.5 
 
 NYREN, from v Urs. Maj 9.254 3 
 
 " " oDraconis 9.242 2.5 
 
 " " i Draconis 9.240 4 
 
 DE BALL, from WAGNER'S Eight Ascensions 
 
 of Polaris 9.162 3 
 
 DEBALL, from WAGNER'S Declinations of 
 
 Polaris . 9.213 3 
 
 DEBALL, from WAGNER'S Eight Ascensions 
 
 of51Cephei 9.252 3 
 
 DEBALL, from WAGNER'S Declinations of 
 
 51 Cephei 9.227 3 
 
 DEBALL, from WAGNER'S Eight Ascensions 
 
 of 6 Urs. Miu 9.208 3 
 
 DEBALL, from WAGNER'S Declinations of 
 
 d Urs. Min 9.263 3 
 
 Greenwich North-Polar Distances of South- 
 ern Stars, Series I 9.116 3 
 
 Greenwich North-Polar Distances of South- 
 ern Stars, Series II 9.201 3 
 
 Greenwich North-Polar Distances of North- 
 ern Stars, Series I 9.204 4 
 
 Greenwich North-Polar Distances of North- 
 ern Stars, Series II 9.223 4 
 
 Washington Transit Circle, southern stars . 9.217 6 
 
 " " u northern stars . 9.177 3 
 
 Greenwich, Eight Ascensions of Polaris . . 9.153 2 
 
 " Declinations of Polaris . . . . 9.242 2 
 
 " Eight Ascensions of 51 Cephei . 9.135 2 
 
 " Declinations of 51 Cephei . . . 9.162 2 
 
 " Eight Ascensions of 6 Urs. Min. 9.147 2 
 
 " Decimations of tfUrs. Min. . . 9.235 2 
 
 " Eight Ascensions of A Urs. Min. 9.161 1 
 
 " Declinations of A Urs. Min. 9.339 1 
 
 Mean . 9.210; wt. = 72 
 
66,67] PRECESSION AND NUTATION. 131 
 
 The mean error corresponding to weight unity when derived 
 from the discordance of the results is 0".068, while the 
 estimate was i 0".070. We may therefore put, as the resulj 
 of observation 
 
 N = 9".210 0".008 
 
 Relations betiveen the constants of precession and nutation, and 
 the quantities on which they depend. 
 
 67. The formula of precession and nutation have been 
 developed by OPPOLZER with very great rigor and with 
 great numerical completeness as regards the elements of the 
 Moon's orbit, in the first volume of his Bahnbestimmung der 
 Kometen und Planeten, second edition, Leipzig, 1882. What 
 is remarkable about this Avork is that it constantly takes 
 account of the possible difference between the Earth's axis 
 of rotation and its axis of figure, a distinction which has 
 become emphasized by CHANDLER'S discovery since OPPOL- 
 ZER wrote. His theory however fails to take account of the 
 change in the period of the Eulerian nutation produced by 
 the mobility of the ocean and the elasticity of the Earth. But 
 this effect is of no importance in the present discussion. 
 
 From OPPOLZER'S developments, I have derived the follow- 
 ing expressions, in Avhich the numerical coefficients may be 
 regarded as absolute constants, so accurately determined that 
 no question of their errors need now be considered. These 
 results haA^e been derived quite independently of the similar 
 ones by Mr. HILL in the Astronomical Journal, Yol. XI, which 
 are themselves independent of OPPOLZER'S work. In these 
 formulae we have 
 
 K, the constant of lunar nutation of the obliquity of the 
 ecliptic, as defined by the equation As = ~N cos &, and 
 expressed in seconds of arc; 
 
 P, so much of the precession of the equinox on the fixed 
 ecliptic of the date, in seconds of arc and in a Julian 
 year, as is due to the action of the Moon ; 
 
 P 7 , so much of the same precession as is duejo^j^e^ action 
 of the Sun. 
 
 ^ * ^l - ^- 
 
 Of 
 
132 . MASS OF THE MOON. [67, 68 
 
 We thus have, 
 
 luni-solar precession = P + P 7 
 
 f, the obliquity of the ecliptic; 
 
 yu, the ratio of the mass of the Moon to that of the Earth ; 
 
 A, the mean moment of inertia of the Earth relative to axes 
 
 passing through its equator; 
 C, the same moment relative to its polar axis. 
 
 With these definitions we have, 
 
 General value. Special value for 1850. 
 
 X = [5.40289J cos s 
 P = [5.975052] cos f 
 P' = [3.72509] c 
 
 0-A 
 
 C 
 
 = [5.36542! 
 
 C-A 
 
 
 1 + 
 
 ~ A = [5.937585] _JL_ 
 C J 1 + /i C 
 
 ; "^ = [3.68762] 
 
 C-^. 
 
 
 The special values for 1850 are found by putting for the 
 value of the obliquity of the ecliptic for 1850, 
 
 f = 22 27' 31". 1 
 
 The mass of the Moon from the observed constant of nutation. 
 
 68. From the two quantities given by observation, N and 
 P 4- P' = po, these equations enable us to determine the two 
 
 unknown quantities yu and 
 
 C A 
 
 As the easiest way of 
 
 showing the uncertainty of the Moon's mass, arising from 
 uncertainty of the precession and nutation, I give the value of 
 its reciprocal corresponding to different values of these quan- 
 tities in the following table : 
 
 Reciprocals of the mass of the Moon corresponding to different 
 values of the nutation-constant and luni-solar precession. 
 
 A 
 
 i 
 
 *=" 
 
 50.35 
 50.36 
 50.37 
 
 81.81 
 81.86 
 81. 91 
 
 81-53 
 81.58 
 81.63 
 
 81. 25 
 81.30 
 81-35 
 
68, 69J THE CONSTANT OF ABERRATION. 133 
 
 Taking for the constant of nutation the value just found, 
 
 N = 9 // .210 ".068 
 and for the luni-solar precession, 
 
 lh = 50" .36 ''.006 
 
 we have, for the reciprocal of the mass of 'the Moon and its 
 mean error : 
 
 - = 81.58 0.20 
 p 
 
 The Constant of Aberration. 
 
 69. In the determination of astronomical constants the inves- 
 tigation of the constant of aberration necessarily takes a very 
 important place, not only on its own account but on account of 
 its intimate connection with the solar parallax. A general 
 determination, founded on all the data available, was therefore 
 commenced by me as far back as 1890, before the fact of the 
 variation of terrestrial latitudes had been well established. 
 The successive discoveries of the law of this variation by 
 CHANDLER required such alterations in the work as it went 
 along that much of it is now of too little value for publication 
 in full. Happily the necessity for a new discussion of the best 
 determinations at Pulkowa has been done away with by the 
 papers of CHANDLER himself in the Astronomical Journal. 
 
 Quite apart from the disturbing influence of the revolution 
 of the terrestrial pole upon the determination of the constant 
 of aberration, this constant is itself the one of which the deter- 
 mination is most likely to be affected by systematic errors. 
 In this respect it is at the opposite extreme from the constant 
 of nutation. From the very nature of the case it requires a 
 comparison of observations at opposite seasons of the year, 
 when climatic conditions are different. In most cases the 
 determination must even be made at different times of day. 
 The effect of aberration on a star, for example, is generally at 
 one extreme when the star culminates in the morning, and at 
 the other extreme when it culminates in the evening. The 
 culminations at opposite seasons of the year are necessarily 
 
134 THE CONSTANT OF ABERRATION. [69 
 
 associated with culminations at opposite times of the day. 
 Moreover, in observations to determine the constant of aber- 
 ration from Declination, the stars which give the largest coeffi- 
 cients are, for the northern hemisphere, tbose near 18 h of Eight 
 Ascension. Any error peculiar to the times or seasons at 
 which these stars are observed will therefore affect the result 
 systematically. 
 
 Eight Ascensions of close polar stars also lead to a value of 
 this constant. But the same difficulty still exists. In this 
 case the maxima and minima of aberration occur when the 
 star culminates at noon and midnight. Not only is the aspect 
 of the star different at the two culminations, but the effect of 
 any diurnal change in the instrument will be transferred to the 
 final result for the aberration. 
 
 The prismatic method of LOEWY is free from some of these 
 objections. But its application is extremely laborious, and we 
 have, up to the present time, only two determinations by it, 
 one by LOEWY himself, which is only regarded as preliminary, 
 and one by COMSTOCK, in which a large uncertain correction 
 for personal equation was applied. 
 
 Under these circumstances the seeking of results derived by 
 methods of the greatest possible diversity is yet more strongly 
 recommended than in the case of the other astronomical con- 
 stants. I have therefore used not only the PULKOWA deter- 
 minations, but all those made elsewhere which it seemed worth 
 while to consider. Notwithstanding the great amount of mate- 
 rial added to NYREN'S paper of 1883, it will be seen that the 
 probable error of the final result at which I have arrived is 
 greater than that which he assigns to his result. This is a 
 natural consequence of combining so many separate determi- 
 nations. The advantage is, however, that the assigned prob- 
 able error is more likely to be the real one. It is not to be 
 supposed that any of the systematic errors already indicated 
 would pertain to all observers and to all instruments. The 
 final outcome should be a result in which the discordances of 
 the separate determinations show the probable values of all 
 the actual errors, both accidental and systematic. 
 
 Determinations founded on the Eight Ascensions of circum- 
 polar stars are not affected by the motion of the terrestrial 
 
69, 70] THE CONSTANT OF ABERRATION. 135 
 
 axis, uor are those founded on declinations of these stars, if 
 only the declinations are observed equally at both culmina- 
 tions. But determinations founded on declinations of stars 
 from upper culmination only are necessarily aifected by this 
 cause. If however the stars on which the determination is 
 based extend through the whole circle of Right Ascension the 
 effect of the cause in question may be wholly eliminated by a 
 suitable treatment of the equations of condition. To practically 
 eliminate the injurious effect it is not even necessary to deter- 
 mine the exact law of variation. In fact, if the stars observed 
 are equally scattered in Eight Ascension, the effect of the varia- 
 tion will be partially eliminated without taking account of it. 
 
 CHANDLER has shown that there are two periodic terms in 
 the variation of latitude, one having a period of one year, the 
 other of four hundred and twenty-seven days. I may remark 
 that this combination is in accord with my theory developed in 
 the Monthly Notices of the Royal Astronomical Society for March, 
 1S92. It was there shown that any minute annual change of 
 the position of the principal axis of inertia of the Earth a 
 change which might be produced by the motion of water, ice, 
 and air on its surface would appear as an annual term in the 
 latitude, six times as great as its actual amount, 
 
 Values of the constant of aberration derived from observations. 
 
 70. What I have done since this discovery by CHANDLER 
 has been to reexamine the determinations of the constant of 
 aberration made from time to time, to make such corrections 
 in their bases as seemed necessary, and more especially to 
 determine the correction to be applied to each separate result 
 on account of the periodic term in the latitude. No attempt 
 was made to rework completely the original material, except 
 in the case of the results of the Pulkowa and Washington 
 observations with the prime vertical transit. In the case of 
 the former, however, the preliminary results reached from time 
 to time were so accordant with those of CHANDLER that it is 
 a matter of indifference whether we regard them as belonging 
 to his work or to my own. 
 
 Owing to the very different estimates placed by the astro- 
 nomical world upon the Pulkowa determinations and 'those 
 
136 THE CONSTANT OF ABERRATION. [70 
 
 made elsewhere, I have used the former quite apart from the 
 others. The complete discussion of each separate value is 
 too voluminous for the present publication, and is therefore 
 reserved for a more extended future publication. At pres- 
 ent it appears sufficient to judge the final result by the general 
 discordance of the material on which it rests, rather than by 
 a separate criticism of each particular case. 
 
 In the exhibit of results which follows it is to be remarked 
 that NYREN'S prime vertical observations do not receive a 
 weight as great, relative to the other Pulkowa determinations, 
 as would be given by their assigned probable errors. The 
 reason of this course is that one can not be entirely confident 
 that the results of any one observer with this instrument are 
 free from constant error arising from differences of personal 
 equation in observing a bright and a faint star. Many of the 
 Pulkowa observations are necessarily made in the morning or 
 evening twilight. In the case of an evening observation the 
 star will therefore be much fainter on account of daylight 
 when it transits over the east vertical than it will when it 
 transits over the west vertical one or two hours later. In the 
 case of morning observations the reverse will be true. It is 
 easy to see that if, in consequence of this difference of aspect, 
 the observer notes the passage of the faint image too late, the 
 effect will be to make the constant of aberration too large. 
 The existence of this form of personal equation, when transits 
 are recorded on the chronograph, is so well known that, had 
 NYREN'S observations been made in this way, I should not 
 have hesitated to ascribe the large values of his aberration 
 constant to this cause. Although it has never been shown 
 that any such personal equation exists when observations are 
 made by eye and ear, as KYREN'S were, yet when we consider 
 that we are dealing with quantities amounting only to one or 
 two huudredths of a second of arc, and that a personal equa- 
 tion of this kind, undiscoverable by ordinary investigation, 
 might affect the result by this minute amount, we can not but 
 have at least a suspicion that his values may be slightly too 
 large from this cause. 
 
70! THE CONSTANT OF ABERRATION. 137 
 
 Separate results for the constant of aberration. 
 
 A. Standard Pulkowa determinations : 
 
 A b. wt. 
 
 Observations with Vertical Circle ; Polaris, by n 
 PETERS 20.51 2 
 
 Observations with Vertical Circle ; 7 miscellaneous 
 
 stars, by PETERS 20.47 2 
 
 Observations with Vertical Circle 5 1863-1870, Po- 
 laris, by GYLDEN 20.41 2 
 
 Observations with Vertical Circle; 1871-1875, Po- 
 laris, by XYREN 20.51 2 
 
 Observations with Prime Vertical; 1842-1844, by 
 STRUVE 20.48 4 
 
 Observations with Prime Vertical; 1879-1880 by 
 
 NYREN. . 20.52 6 
 
 Observations with Prime Vertical; 1875-1879, by 
 NYREN 20.53 1 
 
 Observations with Vertical Circle; 1863-1873, by 
 GYLDEN and NYREN . 20.52 2 
 
 WAGNER : Transits of three polar stars .... 20.48 5 
 
 From Eight Ascensions of Polaris; 1842-1844, by 
 
 LINDHAGEN and SCHWEIZER 20.50 2 
 
 Mean result: 20 // .493 0".011 
 
 This result may be regarded as identical with that found by 
 
 NYREN in 1882. 
 
 B. Other determinations: 
 
 Ab. e wt. 
 
 AUWERS, from observations with the n 
 
 zenith sector at Kew 20.53 .12 0.5 
 
 AUWERS, from WANSTED observations . 20.46 .12 0.5 
 
 PETERS, from BRADLEY'S observations 
 of y Draconis at Greenwich with zenith 
 sector, 1750-1754 20.67 0.5 
 
 BESSEL, from Eight Ascensions observed 
 by BRADLEY at Greenwich .... 20.71 i.071 0.5 
 
 LINDENAU, from Eight Ascensions of 
 Polaris observed at various observa- 
 tories between 1750 and 1816 . 20.45 .05 3 
 
138 THE CONSTANT OF ABERRATION. | 70 
 
 Separate results for the constant of aberration Continued. 
 B. Other determinations Continued. 
 
 BRINKLEY, from observations of thirteen --/ft. /. 
 
 stars at Trinity College, Dublin, with 7/ 
 
 the 8-foot circle 20.46 .10 1 
 
 PETERS, from STRUVE'S Dorpat observa- 
 tions of six pairs of circumpolar stars . 20.36 .07 2 
 RICHARDSON, from observations with the 
 
 Greenwich mural circles 20.50 .06 3 
 
 PETERS, from Right Ascensions of Polaris 
 
 at Dorpat . 20.41 6 
 
 LUNDAHL, from Declinations of Polaris 
 
 at Dorpat 20.55 5 
 
 HENDERSON and MCLEAR, from a 1 and 
 
 <* 2 Centauri . . . 20.52 .10 1 
 
 MAIN, from observations with the Green- 
 wich zenith tube . 20.20 .10 1 
 
 DOWNING, from observations of ADra- 
 
 conis with reflex zenith tube . ... . 20.52 .05 4 
 XEWCOMB, from observations of <*Lyra3 
 
 with the 'Washington prime vertical 
 
 transit, 1862-1867 20.46 0.4 6 
 
 NEWCOMB, from Right Ascensions of 
 
 Polaris observed with the Washington 
 
 transit circle, 1866-1867 . 20.55 .05 :J 
 
 KUSTNER, from observations of pairs of 
 
 stars by the TALCOTT method . . . 20.46 4 
 
 PRESTON, from observations with the- 
 
 TALCOTT method at Honolulu, 1891- 
 
 1892 20.43 .05 4 
 
 LOEWY, from his prismatic method . . 20.45 .04 5 
 COMSTOCK, using LOEWY'S method, 
 
 slightly modified 20.44 3 
 
 KiisTNER, from MARCUSE'S observations, 
 
 1889-1890 20.49 .018 4 
 
 WANACH, from Pulkowa prime vertical 
 
 observations . 20.40 .015 4 
 
70, 71] THE LUNAR INEQUALITY. 139 
 
 Separate results for the constant of aberration Continued. 
 B. Other determinations Continued. 
 
 Ab. tvt. 
 
 From Greenwich Eight Ascensions of polar stars 
 
 with the transit circle 20.39 3 
 
 BECKER, from observations at Strasburg by the 
 
 TALCOTT method, 1890-1893 20.47 6 
 
 DAVIDSON, from similar observations at San 
 
 Francisco, 1892-1894 20.48 6 
 
 Mean result of B : Ab. const. = 20".463 0".013 
 
 The two results. A and B, differ by 0".030, a quantity so 
 much greater than their mean errors as to leave room for a 
 suspicion of constant error in one or both means. 
 
 The Lunar Inequality in the Ear til's motion. 
 
 71. The source of this inequality is the revolution of the 
 center of the Earth around the center of mass of the Earth 
 and Moon. The former center describes an orbit which is 
 similar to that of the Moon around the Earth. Since this 
 orbit is not a Keplerian eclipse, but is affected by all the per- 
 turbations of the Moon by the Sun, no such element as a semi- 
 major axis can be assigned to it. Instead of this I take as the 
 principal element of the orbit the coefficient of the sine of the 
 Moon's mean elongation from the sun in the expression for the 
 Sun's true longitude. This element is a function of the solar 
 parallax and of the mass of the Moon, which may be derived 
 from the folio wii>g expression. Let us put 
 
 yw ; the ratio of the mass of the Moon to that of the 
 
 Earth ; 
 r,A, /?; the radius vector, true longitude and latitude of 
 
 the Moon ; 
 r', A',/J'; the same coordinates of the Sun; 
 
 * ; the linear distance of the Earth's center from the 
 center of mass of the Earth and Moon. 
 
140 THE LUNAR INEQUALITY. [71 
 
 We then have, for the perturbations of the Sun's geocentric 
 place due to the cause in question : 
 
 A log r 1 = - x cos ft cos (A-A 7 ) 
 A\' = - t cos ft sin (A A 7 ) 
 
 and 
 
 s 
 
 I have developed these expressions, putting 
 
 7T = 8".848 
 
 -.4 
 
 and taking for the Moon's coordinates the values found by 
 DELAUNAY. Putting 
 
 D; the mean value of A A 7 
 
 g, g 1 } the mean anomalies of the Moon and Sun, respectively, 
 u'-, the Sun's mean elongation from the Moon's ascending- 
 node; 
 
 the result for JA 7 is 
 
 // 
 
 A\' = 6.533 sin D 
 + 0.013 sin 3 D 
 + 0.179 sin (D + g) 
 
 0.429 sin (D g) 
 + 0.174 sin (D g 1 ) 
 - 0.064 sin (D + g') 
 -f 0.039 sin (3 D - g) 
 
 0.014 sin (D g g 1 } 
 
 0.013 sin 2 u 1 
 
 This value of the lunar inequality is substantially identical 
 with that computed from the tables and formula of LEVER- 
 
71] THE LUNAR INEQUALITY. 141 
 
 RIER'S solar tables. The development of the numbers there 
 given lead to the value 6".534 of the principal coefficient. 
 
 We have now to find what value of the coefficient is given 
 by observations. The observations I make use of are (1) all 
 the observations of the Sun's Eight Ascension from early in 
 the century till 1864 ; (2) The heliometer observations of Vic- 
 toria made in 1889 on GILL'S plan and worked up by him. 
 
 I had intended to use all the observations of the Sun up 
 to the present time. 1 found however that those made after 
 1864 gave, by comparison with the published ephemerides, 
 inadmissible positive corrections to the coefficient. This cir- 
 cumstance gives rise to a strong suspicion that in the process 
 of interpolating the Right Ascensions of the Sun during at 
 least some years after 1864, the inequality in question was 
 rounded off to the amount of several hundredths of a second. 
 The results were therefore entirely omitted. 
 
 The results for previous years, when the inequality was 
 computed separately for every day of observation, are: 
 
 Greenwich, 
 
 1820-'64; 
 
 -.068 
 
 3.0 
 
 Paris, 
 
 1801-'64; 
 
 -.050 
 
 0.8 
 
 Konigsburg, 
 
 1820-'45; 
 
 -.054 
 
 1.2 
 
 Cambridge, 
 
 1828->58; 
 
 -.047 
 
 2.0 
 
 Dorpat, 
 
 1823->38; 
 
 + .160 
 
 0.3 
 
 Pulkowa, 
 
 1842-'64; 
 
 -.058 
 
 0.5 
 
 Washington, 
 
 1846-'64: 
 
 .000 
 
 0.2 
 
 Mean, JP = 0".048 i 0".018 
 
 GILL'S result is given in the Monthly Notices, Royal Astro- 
 nomical Society, for April, 1894 (Vol. LIY, page 350.) It is 
 derived in the following way. In the solar ephemeris which 
 he used for comparison the lunar inequalities were computed 
 rigorously from the coordinates of the Moon, putting 
 
 n = 8".880 
 M = 1 -r- 83 
 
 To the coefficient P thus arising he found a correction, 
 
 = + 0".046 
 
142 THE LUNAR INEQUALITY. [71, 72 
 
 The above values of n and // give, on the theory just devel- 
 oped, 
 
 P = 6".400 
 Thus GILL'S result is, in effect, 
 
 P-= 6".446 
 
 while mine, from observations of the Sun, is 
 6".533 0".048 = 6".485 
 
 I consider that these results are entitled to equal weight, and 
 that we may take, as the result of observation, 
 
 P = C>".465 i 0".015 
 
 Solar parallax from the lunar inequality. 
 
 72. With the mass of the Moon already found from the 
 observed constant of nutation, 
 
 // = 1 : 81.58 (1 i .0025) 
 
 we may now derive a value of the solar parallax quite inde 
 pendent of all other values. The relation between P, TT, and 
 the mass of the Moon is of the general form 
 
 // P = A' 7T 
 
 where fc is a numerical constant, and, for brevity, 
 
 We have found that the following values correspond to one 
 theory : 
 
 TT = 8".848; X = 82 5 P = 6". 533 
 Hence follows 
 
 log fc = 1.78207 
 so that we have 
 
 ^P= [1.78207] n 
 
 The numerical values P = 6 /7 .465 and // = 82.58 now give 
 7r = 8 // .818 0".030 
 
73J PARALLAX FROM TRANSITS OF VENUS. 143 
 
 Values of the solar parallax derived from measurements of Venus 
 on the face of the Sun during the transits of 1874 and 1882, 
 with the heliometer and photoheliograph. 
 
 73. I put these determinations into one class because they 
 rest essentially on the same principle. Both consist, in effect, 
 in measures of the distance between the center of Yenus and 
 the center of the Sun j the latter being denned through the 
 visible limb. , The method is therefore subject to this serjous 
 drawback : that the parallax depends upon the measured differ- 
 ence between arcs which may be from thirty to fifty times as 
 great as the parallax itself, the measures being made in 
 different parts of the earth. 
 
 The equations of condition given by the American photo 
 graphs of 1874 are found in Part I of Observations of the 
 Transit of Yenus, December 9, 1874 ; Washington, Government 
 Printing Office, 1880. A preliminary solution of these equa- 
 tions, the only one, however, to which they have yet been sub- 
 jected, was published by D. P. TODD, in the American Journal 
 of Science for June, 1881. (Yol. XXT, page 490.) 
 
 The photographs of 1882 have been completely worked up by 
 Professor HARKNESS, and the results are found in the Eeport 
 of the Superintendent of the Naval Observatory for 1889. The 
 equations derived from the German heliometer measures, with 
 a preliminary discussion of their results, are officially published 
 by Dr. AUWERS, in the Bericht iiber die deutschen Beobachtungen,) 
 Y, p. 710. 
 
 The separate results for the parallax, with the probable 
 errors assigned by the investigators, are as follows: 
 
 // // w. W 
 
 1874 : Photographic distances, n 8.888 0.040 6 1 
 
 Position angles, 8.873 0.060 3 3 
 
 Measures with heliometer, 8.876 i 0.042 5 5 
 
 1882: Photographic distances, 8.847 0.012 64 6 
 
 Position angles, 8.772 i 0.050 4 4 
 
 Measures with heliometer, 8.879 0.025 1C 10 
 
 Under w is given a system of weights proportionally deter- 
 mined from the probable errors as assigned. Using this sys- 
 tem, the mean result is 
 
 n =8".854 i ".016 
 
144 PARALLAX FROM TRANSITS OF VENUS. [73 
 
 I conceive, however, that these relative weights do not cor- 
 respond to the actual precision of the measures. The very 
 small probable error assigned by Prof. HARKNESS to the result 
 of the photographic distances of 1882 does not include the 
 probable error of the angular value of the unit of distance on 
 the plate, which may arise from a number of sources, includ- 
 ing the possible deviation of the mirror of the instrument 
 from a perfect plane. From this error the position angles 
 are entirely free. I have, therefore, assigned another set of 
 weights, w', which seem to me to correspond more nearly to 
 the facts. The result of this system is 
 
 7t = 8".857 i ".016 
 
 This mean error is derived from the individual discordances, 
 and not from comparisons with the values of the parallax 
 otherwise determined. As there may be a fortuitous agree- 
 ment among the separate values, another estimate may be 
 made on the basis of the total mean error derived by AUWERS, 
 which includes all known sources of error. He finds 6 = ".032 
 for the combined heliometer results, to which I have assigned 
 weight 15. Hence, for the total weight 29, we have 
 
 <? = 0".023 
 
 The deviation of the above result from the mean of all the 
 other good ones is worthy of special attention. The deviation 
 is more than three times its mean error, and therefore between 
 four and five times its probable error. We must therefore 
 accept one of two conclusions, either the probable errors have 
 been considerably underestimated, or the method is aifected 
 with some undiscoverable soured of systematic error, which 
 makes it tend to give too large a result. The close accordance 
 of the six separate results, of which only a single one deviates 
 from the adopted mean by more than its probable error, and 
 that by only a little more, would give color to the view that 
 the error is a systematic one, and that through some unknown 
 cause Venus is always measured too low relatively from the 
 center of the Sun. I can not, however, think of any such cause. 
 
 If we determine the mean error from the deviations of the 
 separate results from what we know, in other ways, to be 
 
74) PARALLAX FROM TRANSITS OF VENUS. 145 
 
 nearly the most probable value of the parallax, namely 8".80, 
 we have 
 
 Mean errror to weight 1 ; dz .148 
 Mean error of result dz.029 
 
 Solar parallax from observed contacts during transits of Venus. 
 
 74. The contact observations of 1761 and 1769 are discussed 
 in Astronomical Papers, Vol. III. I have also made a com- 
 plete discussion of those of 1874 and 1882, which, at the date 
 of writing, is unpublished. The separate results from each, 
 contact follow. 
 
 In the case of the second contacts of 1874 and 1882 it was 
 found necessary to divide the observations into two classes: 
 those of mean or true contact, and those of the formation of 
 the thread of light. In the case of the third contact no such 
 division was necessary, as the observations could generally 
 be referred to the same mean phase. The mean error which 
 follows each result is derived from the discordance of the 
 separate observations. 
 
 Values of the solar parallax from observed contacts of the limb 
 of Venus with that of the Sun. 
 
 1761, III; TT = 8. / 78dz / .12; w. = 8 
 IV; 8.75 dz. 20 3 
 
 1769, I; 
 
 9.04 d 
 
 z .17 
 
 4 
 
 II; 
 
 8.55 d 
 
 z .13 
 
 7 
 
 III; 
 
 8.72 d 
 
 - .09 
 
 14 
 
 IV; 
 
 9.01 d 
 
 z .12 
 
 8 
 
 1874, I; 
 
 8.95 d 
 
 z .24 
 
 2 
 
 II; M; 
 
 8.78d 
 
 z .061 
 
 30 
 
 II; L; 
 
 8.75 d 
 
 z .10 
 
 11 
 
 III; 
 
 8.76 d 
 
 z .045 
 
 57 
 
 IV; 
 
 8.74 d 
 
 - .09 
 
 14 
 
 1882, I; 
 
 8.93d 
 
 z .15 
 
 5 
 
 II; M; 
 
 8.76d 
 
 z .042 
 
 64 
 
 II; L; 
 
 8.72d 
 
 - .072 
 
 22 
 
 III; 
 
 8.88 d 
 
 - .042 
 
 64 
 
 IV; 
 
 9.07 d 
 
 - .12 
 
 8 
 
 5690 N ALM 10 
 
 
 
 
146 PARALLAX FROM TRANSITS OF VENUS. [74 
 
 The weights assigned are determined by these mean errors, 
 taken on such a scale that unity is the weight for mean error 
 i ".336. The mean result of the whole series is 
 
 7t = 8".797 i ".023 
 
 This mean error is that resulting from the deviations of the 
 sixteen separate results from the general mean, which give for 
 the mean error corresponding to weight unity, 
 
 1 = ".42. 
 
 The excess of this mean error over that determined from the 
 equations themselves shows that the general discordance of the 
 several contacts is somewhat greater than would be inferred 
 from the individual discordances of the contacts inter se. This 
 is what we should expect from constant errors in the determi- 
 nations of parallax from each separate contact. I conceive, 
 however, that such constant errors are not likely to be large; 
 and we can not conceive that contact observations in general 
 are subject to any constant error tending to make the parallax 
 derived from them always too great or too small. I conclude, 
 therefore, that the mean error determined from the totality of 
 the results may be regarded as real. 
 
 It will be interesting to compare the separate results of 
 internal and external contacts. They are 
 
 // // 
 
 From internal contacts ; TT == 8.776 i .023 
 From external contacts; TT = 8.908 i .06 
 
 These mean errors are those derived from the concluded 
 results and they show that the external contacts are relatively 
 more discordant in proportion to the weights assigned than are 
 the internal ones. If we consider this discordance to indicate 
 a larger mean error, and therefore -assign a proportionally 
 smaller weight to the results of external contact, we have, for 
 the concluded result, 
 
 7t = 8".791 i ".022 
 
 As these two hypotheses seem about equally probable, I shall 
 adopt the mean result, 
 
 n = 8".794 
 
75] PARALLAX FROM VELOCITY OF LIGHT. 147 
 
 Solar parallax from the observed constant of aberration and 
 measured velocity of light. 
 
 75. The question of the soundness of the proposition that 
 the aberration is equal to the quotient of the velocity of the 
 Earth in its orbit by the velocity of light is too broad a one to 
 be discussed here. I can only remark that its simplicity and 
 its general accord with all optical phenomena are such that it 
 seems to me it should be accepted, in the absence of evidence 
 against it. 
 
 In Astronomical Papers, Yol. II, page 202, I have given the 
 following determinations of the velocity of light in vacuo by 
 MICHELSON and myself, expressed in kilometers, per second : 
 
 4 
 
 MICHELSON at Naval Academy in 1879 299910 
 
 MICHELSON at Cleveland, 1882 299853 
 
 NEWCOMB at Washington, 1882, using only results 
 
 supposed to be nearly free from constant errors . 299860 
 
 NEWCOMB, including all determinations . . . , . 299810 
 
 I have concluded, 
 
 Velocity of light in vacuo, = 299860 i 30 k. m. 
 
 Taking as the equatorial radius of the Earth 6378.2 k. m. 
 (CLARK), the following table shows the values of the constant 
 of aberration corresponding to admissible values of the solar 
 parallax when this determination of the velocity of light is 
 accepted. 
 
 Ab. = 20.46 n = 8.8076 
 
 20.47 8.8033 
 
 20.48 8.7990 
 
 20.49 8.7946 
 
 20.50 8.7903 
 20.61 8.7859 
 
 20.52 8.7816 
 
 20.53 8.7773 
 
 20.54 8.7730 
 
148 PAEALLACTIC INEQUALITY. [75, 76 
 
 We thus have for the values of the solar parallax resulting 
 from the two values of the constant of aberration already 
 derived : 
 
 // // 
 
 From Pulkowa determinations; Ab. = 20.493; n 8.793 
 
 From miscellaneous determinations; Ab. = 20.463; n = 8.806 
 
 Solar parallax from the parallactic inequality of the Moon. 
 
 76. I have derived a value of the parallactic inequality of the 
 Moon from the meridian observations made at Greenwich and 
 Washington since 1862. The determination of this inequality 
 is peculiarly liable to systematic error, owing to the fact that 
 observations have to be made on one limb of the Moon when 
 the inequality is positive, and on the other limb when it is 
 negative. Hence, if we determine the inequality by the com- 
 parison of its extreme observed effects on the Moon's longitude 
 or Eight Ascension, any error in the adopted semidiameter of 
 the Moon will affect the result by its full amount. 
 
 It does not seem practicable to make a reliable determina- 
 tion of the Moon's diameter, because it will necessarily be 
 made near the time of full Moon, when the illumination of the 
 extreme limb is less intense than near the quadratures, and 
 when some portions of the limb that might be visible if it were 
 illuminated by a perpendicular Sun will be thrown into shadow 
 by the horizontal one. For these reasons it may be expected 
 that the parallactic inequality determined by using observed 
 semidiameters of the Moon will be too large. I have therefore 
 adopted the plan of determining the inequality from each limb 
 separately. To show in regular progression the errors depend- 
 ing on the elongation from the Sun, I have classified the resid- 
 uals of observations according to the hour of mean time at which 
 the Moon passed the meridian ; and formed equations of con- 
 dition containing two unknown quantities, the one a constant 
 correction depending on the semidiameter, personal equation, 
 etc., and the other the parallactic inequality. The question is 
 further complicated by the fact that the majority of observa- 
 tions near are quadratures made during daylight, when it is 
 to be expected that the illumination of the atmosphere will 
 
76] PAKALLACTIC INEQUALITY. 149 
 
 diminish the irradiation, and thus lead to a smaller apparent 
 sernidiameter. I have therefore sought to determine for the 
 two observatories, by a comparison of the observations, the 
 correction to be applied in order to reduce observations made 
 during daylight or twilight to what they would have been had 
 the sky not been illuminated. The reduction was smaller than 
 I had expected, and somewhat doubtful ; I have assigned pro- 
 portionally less weight to those observations where it was 
 necessary. The following are the equations of condition thus 
 formed. The unknown quantities are 
 
 #, a constant, depending on the semidiaineter, personal 
 
 equation, etc.; 
 2/, the correction to the parallactic inequality of the Moon 
 
 after reduction to the value 8".848 of the solar parallax. 
 
 GEEE^WICH. 
 Limb I. 
 
 4.6; 
 5.6 
 
 x + 0.93 y = 
 0.99 
 
 -0.53; 
 -0.72 
 
 wt. 0.2 
 0.6 
 
 6.5 
 
 0.99 
 
 -0.41 
 
 1 
 
 7.5 
 
 0.92 
 
 -0.59 
 
 1 
 
 8.5 
 
 0.79 
 
 -0.54 
 
 1 
 
 9.5 
 
 0.61 
 
 -0.13 
 
 1 
 
 10.5 
 
 0.38 
 
 -0.09 
 
 1 
 
 11.5 
 
 0.13 
 
 -0.06 
 
 1 
 
 Limb II. 
 // 
 12.5; #'-0.13 y =+0.20; wt. 1 
 
 13.5 
 
 -0.38 
 
 + 0.16 
 
 1 
 
 14.5 
 
 -0.61 
 
 + 0.28 
 
 1 
 
 15.5 
 
 -0.79 
 
 + 0.54 
 
 1 
 
 16.5 
 
 -0.92 
 
 - 0.11 
 
 1 
 
 17.5 
 
 ^0.99 
 
 -0.02 
 
 1 
 
 18.4 
 
 -0.99 
 
 + 0.44 
 
 0.5 
 
 19.4 
 
 0.93 
 
 + 1.21 
 
 0.2 
 
150 PARALLACTIC INEQUALITY. [76 
 
 WASHINGTON. 
 
 . Limb I. 
 
 h 
 4.6 ; x + 0.93 y = 1.62 5 ict. = 0.2 
 
 5.6 
 
 0.99 
 
 - 1.26 
 
 0.4 
 
 6.5 
 
 0.99 
 
 -0.85 
 
 1 
 
 7.5 
 
 0.92 
 
 -0.64 
 
 1 
 
 8.5 
 
 0.79 
 
 -0.71 
 
 1 
 
 9.5 
 
 0.61 
 
 -0.71 
 
 1 
 
 10.5 
 
 0.38 
 
 - 0.48 
 
 1 
 
 11.5 
 
 0.13 
 
 -0.23 
 
 1 
 
 Limb II. 
 
 12.5; 
 
 x'- 0.13 y 
 
 = +0.41; 
 
 wt. = 1 
 
 13.5 
 
 -0,38 
 
 0.43 
 
 1 
 
 14.5 
 
 -0.61 
 
 0.52 
 
 1 
 
 15.5 
 
 -0.79 
 
 0.40 
 
 1 
 
 16.5 
 
 -0.92 
 
 0.72 
 
 1 
 
 17.5 
 
 -0.99 
 
 0.96 
 
 0.5 
 
 18.4 
 
 - 0.99 
 
 1.32 
 
 0.3 
 
 19.4 
 
 - 0.93 
 
 1.50 
 
 0.1 
 
 With these equations we have our choice to determine the 
 parallactic inequality by assigning a value to the semidiaineter, 
 or to eliminate the semidiameter from the normal equations. 
 In each case the equations give the following expressions for y: 
 
 Greenwich : Limb I; y = 0.55 1.23 x 
 " II; 0.28+1.230' 
 
 Washington : Limb I; y = 0.99 1.23.r 
 " II; -0.88 + 1.29 a?' 
 
 If we choose to utilize the observed diameters we have the fol- 
 lowing results: 
 
 From 66 transits of the Moon's diameter observed at Greenwich; 
 
76] PARALLACTIC INEQUALITY. 
 
 From 33 transits observed at Washington : 
 
 We should thus have, 
 
 From Greenwich observations, y = 0. 
 From Washington observations, y = 0.23 
 
 If, on the other hand, we eliminate x from each pair of 
 normal equations, the final results for y will be 
 
 - x/ /x /f wf. 
 
 Greenwich : Limb I; 0.64 y = - 0.45; y = - 0.70 0.16 6 
 u II; 0.64 y = 0.00 ; y = 0.00 i 0.36 2 
 
 Washington : Limb I ; 0.64 y = - 0.52 ; y = - 0.81 i 0.16 6 
 II ; 0.53 y = - 0.32 ; y = - 0.60 0.27 3 
 
 The weighted mean of these results is 
 
 y= - 0".64 0".12 
 The resulting value of the solar parallax is 
 
 7t = 8".802 0".008 
 
 A very careful determination of the solar parallax was made 
 from the same theory by Dr. BATTERMAN, by means of occulta- 
 tions, and the result is discussed very fully in the publica- 
 tions of the Berlin Observatory. Dr. BATTERMAN'S definitive 
 result is 
 
 n = 8".794 ".016 
 
 I have slightly revised this result, by applying a correction 
 to the coefficient for the parallax adopted by Dr. BATTERMAN, 
 with the result 
 
 n = 8".7S9 i ".016 
 
 Accepting this result, and combining it with that already 
 found from meridian observations, the parallax from this 
 method will finally come out 
 
 n = 8".799 i ".007 
 
 This mean error may be regarded as belonging to the doubtful 
 class. 
 
152 SOLAR PARALLAX FROM MINOR PLANETS. [76,77 
 
 While tin's work is passing through the press there appears 
 an important paper by FRANZ of Konigsberg,* giving the value 
 of the parallactic equation derived from observations on the 
 lunar crater Hosting A. The correction to HANSEN'S coeffi- 
 cient is found to be 
 
 - 2".10 i 0".30 
 
 The corresponding result for the solar parallax is 
 8".767 =t 0".021 
 
 We may combine the three results for the solar parallax 
 thus : 
 
 
 
 Greenwich and Washington meridian obser- 
 vations . . ........... n = 8.802; w = 5 
 
 BATTERMANN from occupations ..... 8.789; 2 
 
 FRANZ from crater Hosting A ..... 8.767; 1 
 
 Mean .......... 8.794 i ".008 
 
 Solar parallax from observations on minor planets with the 
 
 neliometer. 
 
 77. The fact that the determination of the parallaxes of the 
 small planets by comparison with neighboring stars is free 
 from the grave uncertainty attaching to similar observations 
 of Venus and Mars, owing to the absence of a sensible disk, 
 was long since pointed out by Dr. GALLE. In 1875 he pub- 
 lished a discussion of observations on Flora, made at nine 
 northern observatories, and at the Cape, Cordoba, and Mel- 
 bourne in the Southern hemisphere.t The result was 
 
 n = 8".873. 
 
 An examination of the residuals Of the several observatories 
 shows that in the case of at least one of the Southern observa- 
 tories there is a systematic difference of a considerable fraction 
 
 * Astronomische Nachrichten, Vol. 136, S. 354. 
 
 tUeber eine Bestimnmng der Sonnen-Parallaxe aus correspondirenden 
 Beobachtuugen des Planeten Flora, in October und November 1873. 
 Breslau, Maruscbke & Berendt, 1875. 
 
77] SOLAR PARALLAX FROM MINOR PLANETS. 153 
 
 of a second. This fact seems to present our assigning any 
 appreciable weight to the final result. 
 
 In 1874, GILL, at Mauritius, made heliometer observations 
 of Juno, east and west of the meridian, with the same object. 
 The result was 8".765, or 8".815 when a discordant observation 
 was rejected. In this connection, only an allusion is necessary 
 to GILL'S expedition to Ascension in 1877, made for the pur- 
 pose of applying the method to Mars at the opposition of that 
 year. 
 
 Shortly afterwards GILL published in the first volume of The 
 Observatory a very exhaustive discussion of the methods of 
 determining the solar parallax, in which he showed that heli- 
 ometer observations of the minor planets, made either at a 
 single station not too far from the equator, or at two stations 
 in different hemispheres, afforded a method of measuring the 
 parallax more precise than any before applied. 
 
 Ten years elapsed before the plan was put into operation. 
 Then, in 1889 and 1890, a concerted system of observations was 
 made on the three minor planets, Victoria, Iris, and Sappho, at 
 a number of observatories in both hemispheres. The observa- 
 tions relating to Victoria were carried out most thoroughly, 
 in that a very careful triangulation of the stars of comparison 
 inter se was made at the observatories which took part in the 
 measures. The tabular data for the reductions were supplied 
 by the office of the Berliner Jahrbuch, and the reductions 
 and discussion were made by GILL himself for Victoria and 
 Sappho, and by Dr. ELKIN, on GILL'S plans, for Iris. The 
 three results, as communicated in advance of their complete 
 official publication, are 
 
 // // 
 
 From Victoria: n = 8.800 p. e. i 0.006 
 Iris: 8.825 p. e. 0.008 
 
 Sappho: 8.796 p. e. J= 0.012 
 
 I assign the respective weights 4, 2, and 1, thus obtaining, 
 as the final result of this method, 
 
 n = 8".807 i 0".006 
 
 I have included in a separate category GILL'S determina- 
 tion by Mars, at Ascension, in 1877, as published by the 
 
154 UNCERTAINTY OF PARALLAX FROM MARS. [77, 78 
 
 Eoyal Astronomical Society (Memoirs Royal Astronomical So- 
 ciety, Vol. XLVI), for the reason that, owing to the disk of 
 Mars, and its reddish color, determinations made on it are 
 liable to errors peculiar to that planet, or at least different 
 from those which might come in in the case of the small 
 planets. 
 
 Remarks on determinations of the parallax which are not used 
 in the present discussion. 
 
 78. In the preceding discussion are given the results of 
 every modern method of determining the solar parallax with 
 which I am acquainted, except meridian and eqiiatorial obser- 
 vations on Mars. I have not used any of the results derived 
 from this source, owing to their large probable error, and 
 the suspicion of systematic error to which they are open. 
 One of these causes of error is to be found in the red color of 
 Mars. This cause will be pointed out and discussed very 
 fully in a subsequent section. Its effect would be to make the 
 observed parallax too large. Since, as a matter of fact, all 
 the determinations of Mars by meridian observations have 
 given a larger parallax than the generality of other methods, 
 color seems to be given to this suspicion. Apart from this, 
 the setting of the threads of a meridian circle upon the appar- 
 ent disk of Mars involves a visual estimate not comparable 
 with that of the bisection of the image of a star by the threads. 
 Hence, there is a chance of systematic personal error arising 
 from this source. The observations generally exhibit large 
 discordances, which may be attributed to one or the other of 
 these causes. 
 
 It may be objected to the inclusion of GILL'S Ascension 
 result that it should be rejected for the same reason, since the 
 color of the planet would affect heliometer observations and 
 meridian observations equally. I have, however, considered 
 it free from the objection in question, for two reasons. In the 
 first place, the result is not too large, but is, on the contrary, 
 the smallest of all the accurate measures. The principle that 
 when a result is open to a strong suspicion of being affected 
 by a cause which would cause it to deviate in one direction, it 
 is logical to conclude a posteriori that the cause has not acted 
 
78] UNCERTAINTY OF PARALLAX FROM MARS. 155 
 
 if the deviation is found to be in the other direction, may not 
 be a perfectly sonnd one, bnt I have nevertheless acted upon 
 it. In the next place G-ILL himself, as a part of his discus- 
 sion, compared the observations when Mars was at different 
 altitudes, in order to determine whether the action of such a 
 cause was indicated, and found a negative result. 
 
 In 1890 an unsuccessful attempt was made, at the writer's 
 request, by Dr. W. L. ELKIN, to measure the effect in question, 
 by placing a refracting prism of very small angle over one of 
 the halves of a heliometer objective, and measuring the refrac- 
 tion thus produced. It was supposed that the dispersing 
 action of the prism would represent that of the atmosphere, 
 greatly magnified. The failure arose from the result that the 
 apparent mean refraction of the star produced by the prism 
 proved to be a function of the star's magnitude, ranging from 
 748".79 for a star of magnitude 2.55 to 751".61 for a star of mag- 
 nitude 6.95. The reason seemed to be that too powerful a prism 
 was used, so that the spectrum was quite sensible; then, in the 
 case of faint stars, the red portion of the spectrum was invis- 
 ible, so that the apparent mean refraction was greater than in 
 the case of the brighter stars. The mean of the observed 
 v displacements of Mars was 748".61, so that it was always less 
 for Mars than for the stars.* 
 
 An investigation of the question whether the same effect is 
 noticeable in meridian observations fails to show any relation 
 between the brightness of a star and its refraction. But this 
 does not disprove the relation between the refraction and the 
 color of a star. 
 
 On the whole it seems to me that, at least in the case of 
 Mars, we have here a cause so mixed up with personal error 
 in making the observations that the objective and subjective 
 effects can not be completely separated. 
 
 * Astronomical Journal, Vol. 10, page 97. 
 
CHAPTER VIII. 
 
 DISCUSSION OF RESULTS FOR THE SOLAR PARALLAX 
 AND THE MASSES OF THE FOUR INNER PLANETS. 
 
 79. We have, in what precedes, found or collected nine 
 separate values of the parallax of the Sun, by methods of 
 which seven may be regarded as completely distinct, in the 
 sense that no one source of error is common to any two. Of 
 these seven the two most nearly associated are those which 
 utilize transits of Venus. These are similar only in the sense 
 of resting upon a determination of the relative parallax of 
 Venus and the Sun during the time of a transit. But the 
 only common elements which enter into the determination are 
 the ratio of the distances of the Sun and Venus, which is 
 determined with such certainty that we can not regard it as 
 subject to error. The methods of determining the parallax in 
 the two cases are . completely distinct from the beginning, 
 there being, I conceive, no common source of error affecting 
 an observation of contact of limbs and one of a distance 
 measured from the center of the Sun while Venus is in transit. 
 
 I have classified as if they were independent the values of 
 the parallax which follow from the Pulkowa determinations 
 of the constant of aberration, and those which follow from all 
 other determinations. Of course whatever doubts may affect 
 the theory of the assumed relation between the constant of 
 aberration and the velocity of light will equally affect both 
 determinations. I do not, however, conceive that there is 
 any source of error which can affect both the Pulkowa deter- 
 minations of the aberration and those made elsewhere. The 
 two could have been combined so as to give a single result 
 of the method ; but as the two values of the constant differ 
 by more than we should expect them to from their probable 
 errors, I have kept them separate, partly not to give a false 
 appearance of agreement of results, and partly to facilitate 
 the inception of any future investigation on the subject. 
 
 156 
 
79] THE SOLAR PARALLAX. 157 
 
 I have also separated tbe result of GILL'S observations on 
 Mars, at Ascension, in 1877, from the determinations made by 
 the same method on the minor planets, because, owing to the 
 color and disk of Mars, the two results may be affected by 
 very different systematic errors. The only common systematic 
 error which seems likely to affect them is that arising from the 
 color of the object, which will be discussed hereafter. 
 
 Results of determinations of the solar parallax arranged in the 
 order of magnitude. 
 
 From the mass of the Earth resulting 
 
 from the secular variations of the wt. 
 
 orbits of the four inner planets . . . 8.759 i .010 9 
 From GILL'S observations of Mars at 
 
 Ascension . . . t 8.780 d= .020 2 
 
 From Pulkoica determinations of the 
 
 constant of aberration 8.793 .0046 40 
 
 From observations of contacts during 
 
 transits of Venus 8.794 i .018 3 
 
 From the parallactic inequality of the 
 
 Moon 8,794 .007 18 
 
 From determinations of the constant of 
 
 aberration made elsewhere than at 
 
 Pulkowa 8.806 .0056 28 
 
 From heliometer observations on the 
 
 minor planets 8.807 .007 20 
 
 From the lunar equation in the motion 
 
 of the Earth 8.825 i .030 1 
 
 From measurements of the distance of 
 
 Venus from t he Sun's center during 
 
 transits 8.857 i .023 2 
 
 The mean errors which follow each value are those which, 
 from a study of the determination, it seemed likely might 
 affect them, no allowance being made for mere possibility of 
 systematic error. The weights assigned are convenient small 
 integers, generally such as to make the weight unity corre- 
 spond to the mean error 0".30, allowance being made, how- 
 
158 THE SOLAR, PARALLAX. [ 7j 
 
 ever, for doubt as to what value should be assigned to the 
 mean error and for the different liabilities to systematic error. 
 The mean result is 
 
 // // 
 
 From all determinations; n 8.797 
 
 Omitting the first result; n = 8.800 .0038 
 
 The last value differs from the preliminary value 8".802 of 
 Chapter V, from a change in the weights. It will be seen 
 that the different values are all as accordant as could be 
 expected, with the exception of the two extreme ones. In the 
 largest value we have a case the principles involved in which 
 have been discussed in Chapter IV. 
 
 We can not suppose the parallax to be materially greater 
 than 8".800, and may take it as probably less than this. Thus 
 the absolute error of the results of measures of Venus on the 
 face of the Sun may be considered as about 0".06 or 0".07, 
 which is four times the computed probable error. The prob- 
 ability against this, even in the case of one result out of eight 
 or nine, is so small that we must either regard the method as 
 being affected by some systematic error, or as affected by 
 an objective probable error larger than that assigned. It 
 seems to me the latter view is not untenable, in view of the 
 very wide range of the possibilities of error which might affect 
 a series of observations with a heliometer exposed to the Sun's 
 rays during a period limited to a few hours. 
 
 Again, in the photographic measures, the value of a second 
 of arc in length on the photographic plate enters as a some- 
 what uncertain element. In this connection it is to be 
 remarked that the measures of position angle on the photo- 
 graphic plates, which are not affected with this uncertainty, 
 although their probable error is quite considerable, give a 
 value of the solar parallax much smaller than the measures of 
 distance. 
 
 Much more embarrassing is the value which results from the 
 mass of the Earth. We here meet in another aspect the same 
 deviation which we encountered in determining the mass of 
 the Earth from the secular variations, and on which we post- 
 poned a conclusion ( 64). This determination rests very 
 
79, 80] MOTION OF THE NODE OF VENUS. 159 
 
 largely on the motion of the node of Venus, as determined 
 from the transits of 1761 and 17G9. It is true that results of 
 meridian observations are combined with them ; but no expla- 
 nation is thus afforded of the difficulty, because the results of 
 these observations agree with those of the transits (v. 39). 
 What adds to the embarrassment and prevents us from wholly 
 discarding the suspicion that some disturbing cause has acted 
 on the motion of Venus, or that some theoretical error has 
 crept into the work, is that, of all the determinations of the 
 solar parallax this is the one which seems the most free from 
 doubt arising from, possible undiscovered sources of error. It 
 is, as we shall presently see, really entitled to twice the relative 
 weight assigned it. As, however, the determination rests 
 mainly on the motion of the node of Venus, and this again 
 mainly rests on the observations of the older transits, I have 
 made a reexamination of the results of these transits with a 
 view of reaching a more exact estimate of the sources of error 
 and the magnitude of the mean error. In this re-examination 
 I have regarded the Sun's parallax as a known quantity equal 
 to 8".798, and then obtained the results of the old observations 
 of the transits on the supposition that the only quantities to 
 be determined were the corrections to the relative heliocentric 
 positions of Venus and the Earth. 
 
 Rediscussion of the motion of the node of Venus. 
 
 80. In discussing the observations of 1761 and 1769 (Astro- 
 nomical Papers, Vol. II, Part V), I introduced a quantity 
 expressive of the error in the observed time of contact arising 
 from imperfections of the telescope and atmospheric absorp- 
 tion and dispersion. The constants on which these errors 
 depend are represented by symbols fc 2 and & 3 . As I have 
 worked up the observations, the ultimate result of each 
 observation of contact is the value of an unknown quantity, 
 dCj which, were there no imperfections of vision and were the 
 radii of the Sun and Venus accurately known, would represent 
 the correction to the tabular distance of centers. As a matter 
 of fact, however, we are to consider 6 c as equal to this correc- 
 tion increased by a rather complex combination of quantities 
 depending on the errors of the assumed semidiameters of 
 
160 MOTION OF THE NODE OF VENUS. [80 
 
 Venus and the S.un, and the thickness of the thread of light 
 when it first became visible at second contact, or vanished at 
 third contact. The observations must be so combined as to 
 eliminate these quantities. What I have done is to represent 
 the undiscoverable minute correction to dc thus arising by 
 the symbol 2 for second contact, and 3 for third contact. In 
 the present re-examination the absolute terms are reduced to 
 the parallax 8".798 by putting 67r Q = - ".05 and n> = - ".025 
 in the final equations of the original paper. After each result 
 is given the mean .error with which it is affected, as deter- 
 mined by the investigation in question. When thus treated, 
 the equations which I have given on pages 391-398 of the 
 paper referred to give the following normal equations for tfc, 
 the indeterminates & 2 and & 3 being retained as such in order to 
 show their final effect on the result. 
 
 // // 
 
 1761. II 5 8.5 do = + 0.76 - 18.5 fc 2 0.78 
 III; 41.7 dc = - 2.81 - 19.2 k, 1.30 
 
 1769. II; 44.8 dc = - 8.00 - 104.1 fc 2 i 1.95 
 III; 12.1 do = + 0.31 - 16.0 fc 3 0.70 
 
 In order to vary the proceeding as much as possible from 
 that of the former investigation, I now express dc in terms of 
 dX and 6fi, which, for the time being, I take as the corrections 
 to the heliocentric longitude and latitude of Venus referred 
 to the Earth, and these again in terms of dv and sin 166, 
 which latter, for brevity, I call u. The first transformation is 
 made with the coefficients of p. 71, where we have put x and 
 y for 6\ and d/3, and the last by the equations 
 
 // 
 
 6X = 6v + 0.06 u 
 8p = u 0.06 v 
 
 Putting Ui for the value of u in 1765, we have, in consequence 
 of the known change in the motion of the node, 
 
 // 
 
 In 1761; u = Ui + 0.11 
 In 1769; u = ^ 0.11 
 
80] MOTION OF THE NODE OF VENUS. 161 
 
 We thus have the four equations which follow for determining 
 6v and HI, the former being supposed the same at the times of 
 the two transits. 
 
 - .84 6v - .55 M! + * a = + 0.15 - 2.2 & 2 i 0.09 
 + .73 - .69 + z 3 = + 0.01 - 0.5 fc 3 0.03 
 
 - .69 + .73 + 2 2 = - 0.10 - 2.3 A" 2 0.04 
 + .81 + .60 + s 3 = + 0.10 - 1.3 & 3 0.06 
 
 Eliminating z. 2 and 3 by subtracting the first equation from 
 the third, and the second from the fourth, we have 
 
 .15 6v + 1.28 % = - o'.25 -~ o'.l A; 2 i 0.10 
 .08 <* + 1.29 ui = + 0.09 - 0.8 fc 3 0.07 
 
 We thus have for Ui the value 
 
 m = - 7/ .04 - 0.08 6v - 0.03 A: 2 - 0.36 fc 3 i O x/ .05 
 
 dv can not be determined independently of z 2 and 3 . Assum- 
 ing these quantities to be equal, we have already found it to 
 be only /7 .02, and may therefore, to determine its probable 
 effect upon the result by assigning to it the value 
 
 In the former paper I have found for k 2 and & 3 the values 
 
 fc 2 = + 0.040 i 0.040 
 
 Jc 3 = - 0.034 0.040 
 
 A preliminary correction of + 2 /7 .02 having been applied to 
 the tabular orbital latitude, we have, for the epoch 1765.5, 
 
 sin id 6 = + 1".99 i 0".06 
 
 Combining this result with that of the transits of 1874 and 
 1882, we have the following results, which are compared with 
 those of meridian observations : 
 
 // 
 Transits of Yenus alone ...... sin i D t $6 = 2.82 
 
 Meridian observations alone .... " 2.45 
 
 Combined solution ........ _ 2.71 
 
 Adjusted with other* results (46) . . . 2.73 
 
 Adopted ........... 2.77 
 
 5690 N ALM - 11 
 
162 MOTION OF THE NODE OF VENUS. [80 
 
 The adopted result is the one which seems the most probable. 
 For the final probable error we are to include that of the pre- 
 cession and of the Sun's longitudes at the two epochs. We 
 may estimate the combined value of these at i 1", correspond- 
 ing to an error of 0".06 in sin i D t 66. Thus we have 
 
 sin i D t 66 = 2". 77 i // .084 
 
 I conceive this mean error to be as real as any that can be 
 determined in astronomy. This conviction rests upon the fact 
 (1) that the systematic errors affecting the four contacts are 
 shown to be small by the general minuteness of the four values 
 of dc; (2) that whatever systematic errors may affect the 
 formation or disappearance of the thread of light are almost 
 completely eliminated from the mean of the transits of 1761 
 and 1769 by the method in which the observations have been 
 combined. The accordance of the observations of external 
 contact made at the same transits strengthens this view. 
 
 The equation thus derived takes the place of the sixth 
 equation of 63 and should have twice the weight there 
 assigned. As the mass of the Earth determined by the secu- 
 lar variations rests mainly on this equation, I shall first con- 
 sider it alone. Expressing the theoretical secular variation of 
 sin i66 in terms of the above observed value, we find that the 
 observed motion of the node of Yenus gives the equation 
 
 0".26 v 29". 2 v 1 43".2 v" = + 0".48 i // .084 (a) 
 which gives for v" the value 
 
 v" = 0.0 ill 4- 0.006 v - 0.676 v 1 i .0019 
 
 The value of the solar parallax for v" is 8" .811. Hence, 
 for the value expressed in terms of the corrections to the 
 assumed masses of Yeuus and Mercury, this equation gives 
 
 n = 8".778 + 0".020 r 1".98V 
 
 We have found from the periodic perturbations 
 
 // // 
 
 v - _ 0.055 i .25 
 v 1 = + 0.0080 i .0025 
 
80] SOLAR PARALLAX. 163 
 
 Whence, 
 
 // // 
 
 Y" = - 0.0168 i .0029 
 n = 8.762 i .0086 
 
 This result of observation, errors and unknown actions aside, 
 Fcan not suppose to be affected by any other mean error than 
 that here assigned. 
 
 We have now to consider how far this result may be recon- 
 ciled with the others by changes in the masses of Mercury 
 and Venus. No admissible change in the former could greatly 
 affect the result. The question then arises whether the dis- 
 crepancy may not be due to an error in the concluded mass 
 of Venus. In making so large a change in this element, we 
 meet with insuperable difficulties. The observed motion of 
 the ecliptic, which is a fairly well-determined quantity, indi- 
 cates a still further increase of this mass. We may put this 
 difficulty in another form. The observed motion of the node 
 of Venus is a relative one, consisting in the combined effect of 
 the motion of the ecliptic around an axis at right angles to the 
 node of Venus, and an absolute motion of the orbit of Venus 
 around nearly the same axis. This motion of the ecliptic 
 depends mainly on the mass of Venus ; the absolute motion 
 Of the orbit of Venus mainly on that of the Earth. If, now, we 
 determine the motion of the ecliptic from observation, we shall 
 find that the relative motion of the orbit of Venus still unac- 
 counted for is yet greater than we have supposed it to be, and 
 should therefore find a yet smaller mass of the Earth than that 
 heretofore concluded. 
 
 The determination of the mass of Venus already made from 
 observations of the Sun and Mercury seems to admit of no 
 doubt. We can not conceive that the mean of fifteen deter- 
 minations, made during one hundred and thirty years, at dif- 
 ferent observatories, which determinations are so separated as 
 to be entirely independent of each other, can be affected by 
 any considerable common error. The entire accordance of the 
 result thus reached from the periodic perturbations produced 
 by Venus with that from a combination of all the secular 
 variations, as shown in Chapter VI, strengthens the result 
 yet further. Unknown actions and possible defects of theory 
 
164 SYSTEMATIC ERRORS OF PARALLAX. [tO, M 
 
 aside, it seems to me that the value of the solar parallax 
 derived from this discussion is less open to doubt from any 
 known cause than any determination that can be made. 
 
 Possible systematic errors in determinations of the parallax. 
 
 81. We have now to return to the other values, in order to 
 see to what extent they may be affected by systematic error. 
 I have already excused myself from discussing the validity of 
 the assumed relation between the constant of aberration and 
 the velocity of light, because there is nothing valuable to be 
 said on the subject, and have alluded to the possible sources 
 of systematic error in the Pulkowa determinations of aberra- 
 tion. It is worthy of attention here that the very best of these 
 determinations, that of NYR^N with the prime vertical transit, 
 in resp,ect to the care with which it was made, and the general 
 accordance of the entire work throughout, gives a result most 
 accordant with that under consideration. In fact, to the value 
 8". 77 of the solar parallax corresponds the value 20 // .55 of 
 the constant of aberration, which is larger by only // .02 than 
 the result of NYREN'S best determinations. 
 
 A.S for miscellaneous determinations of the constant, it is to 
 be remembered that the corrections applied to a part of the 
 separate values on account of the Chandlerian inequality of 
 latitude are somewhat doubtful, and the general mean mav 
 have been affected by a few huudredths of a second in conse- 
 quence. It is not, however, possible to determine the amount 
 of the correction, except by an exhaustive rediscussion of the 
 whole of the original observations, and even then the result 
 would still be doubtful. 
 
 Next in the order of weight we have the results of measures 
 on the minor planets with the heliometer, on GILL'S plan. I 
 have already remarked upon the possible error in such obser- 
 vations arising from the probable difference of color between 
 the planet and the star. A hypothetical estimate of the 
 amount of this error is worth attempting. Let us assume that 
 in the case of a minor planet the mean of the visible spec- 
 trum corresponds to the line D, and that in the case of a star 
 the same mean is halfway between the lines D and E. 
 
81] SYSTEMATIC ERRORS OF PARALLAX. 165 
 
 The index of refraction of air has been determined inde- 
 pendently by KETTLER and LORENTZ for the different rays. 
 The mean of their results for the rays D and E is 
 
 For D, n = 1.000 2920 
 ForE, n = 1.000 2940 
 
 These results are accordant in giving a dispersion between 
 these two lines equal to about .0037 of the total refraction. 
 We have hypothetically taken the extreme possible difference . 
 between planet and star to be one-half of this. At an altitude 
 of 45, where the refraction is about 60", the error would be 
 0".ll. At an altitude of 30 the error would be 0".20. We 
 are thus led to the noteworthy conclusion : 
 
 If the difference between the spectra of a minor planet and a 
 comparison star is such that the means of their respective visible 
 spectra, or the apparent amounts of their respective refractions, 
 differ by one- tenth of the space between D and E, an error of 
 0" .02 or 0" .03 may be produced in the apparent parallax of the 
 planet. 
 
 The question thus arising maybe readily settled by measures 
 with the heliometer. The distances of pairs of stars differing 
 as widely as possible in color should be measured at different 
 altitudes, when one is nearly above or below the other, in 
 order to see what difference of refraction depending on the 
 color is indicated. A colored double star, such as ft Oygni, 
 might also be used for the same purpose. 
 
 The minor planets are of different colors. I am not aware 
 of any evidence that Victoria or Sappho differ in color from 
 the average of the stars, but 1 believe that Iris is somewhat 
 yellow, or reddish. Kow, in this connection, it is a significant 
 fact that the parallax found from observations of Iris, 8".82o, 
 is the largest by GILL'S method. 
 
 I have already remarked that the value of the solar parallax 
 derived from the parallactic equation of the Moon is one of 
 which the probable mean error is subject to uncertainty. 
 While it is true that the value may be smaller than that we 
 have assigned, we must also admit that it may be much larger. 
 
 The probable error of the determination by the lunar equa- 
 tion of the Earth is larger than that of any other method. At 
 
166 RESULTS FOR THE SOLAR PARALLAX. [82 
 
 the same time I do not think that it is liable to systematic 
 error, and we must therefore regard the mean error assigned 
 as real. 
 
 Results for the solar parallax after making allowance for prob- 
 able systematic errors. 
 
 82. Let us now see whether we can reach a satisfactory 
 result by making a liberal allowance for the more or less 
 probable sources of systematic error just pointed out. The 
 modifications we make in the weights formerly assigned are 
 these: We reduce the weight of GILL'S Ascension result to 
 one-half, owing to the uncertainty arising from the color of the 
 planet Mars. We retain the Pulkowa determinations of the 
 constant of aberration with their full weight, but reduce the 
 weight of the miscellaneous determinations. In the case of 
 the parallactic inequality, we reduce the weight for the reasons 
 already given. We omit Iris from the determination from the 
 minor planets. We also reduce to one- half its former value 
 the relative weight assigned to measures of Venus on the Sun, 
 on the theory that the actual mean error must be larger than 
 that given by the discordance of results. Our combination 
 will then be as follows : 
 
 wt. 
 
 From the motion of the node of Venus .... n = 8.708 10 
 From GILL'S Ascension observations .... 8.780 1 
 
 From the Pulkowa constant of aberration . . . 8.793 40 
 
 From contacts of Venus with the Sun's limb . . 8.794 3 
 
 From heliometer observations on Victoria and 
 
 Sappho 8.799 5 
 
 From the parallactic inequality of the Moon . . 8.794 10 
 From miscellaneous determinations of the con- 
 stant of aberration 8.806 10 
 
 From the lunar inequality in the motion of the 
 
 Earth 8.818 1 
 
 From measures on Venus in transit 8.857 1 
 
 Mean result, ignoring the first ; 8".7965i .0045 
 
 This mean result still differs from that given by the motion 
 of the node of Venus by more than five times the probable 
 error of the latter, and is yet farther from the combined result 
 
82] RESULTS FOR THE SOLAR PARALLAX. 167 
 
 of all the secular variations, so that no reconciliation is brought 
 about. 
 
 The embarrassing question which now meets us is whether 
 we have here some unknown cause of difference, or whether 
 the discrepancy arises from an accidental accumulation of 
 fortuitous errors in the separate determinations. We have 
 already discussed the former hypothesis, and have been unable 
 to find any reasonably probable cause of abnormal action. 
 The motion of the planes of the orbits is that which is least 
 likely to deviate from theory, because it is independent of 
 all forms of action depending upon distance from the Sun, 
 or upon the velocity of the planet. 
 
 An examination and comparison of all the results shows one 
 curious feature: the unanimity with which the secular varia- 
 tions speak against the large value of the solar parallax, or 
 of the mass of the Earth, as the one quantity at fault. The 
 adopted motion of the node of Venus is sustained not only by 
 the meridian observations, but by the external contacts at the 
 transits of 1761 and 1769, and, weakly, by a comparison of the 
 transits of 1874 and 1882. 
 
 If we determine the correction of the mass of the Earth from 
 other secular variations than that of the node of Venus, by 
 the equations of 63, we have, after eliminating the masses of 
 Mercury and Venus, 
 
 v" = -0.029; p. e. .018 
 If, instead of eliminating these values, we put 
 
 v = + .08; v 1 = + .0080; 
 we have 
 
 v" = -0.026; p. e. i .014 
 
 In each case the value of the parallax is yet smaller than that 
 found from the motion of the node of Venus. I have already 
 remarked that the observed motion of the ecliptic indicates 
 an increase of the mass of Venus. 
 
 The question thus takes the form, whether it is possible that 
 the mean of the eeven determinations of the solar parallax 
 
 TT = 8".797 i ".0035 
 
168 DEFINITIVE ADJUSTMENT. [82, 83 
 
 can with reasonable possibility be in error by aii amount the 
 correction of which would bring it within the range of adjust- 
 ment of the other quantities. 
 
 From what has already been said of the systematic errors 
 to which every one of the determinations may be liable, it is 
 evident that we should have no difficulty in accepting the 
 necessary reduction of each of the separate values. The 
 improbability which meets us is not so much the amount of 
 the individual errors of the determinations as the fact that 
 seven of the eight independent determinations should all be 
 largely in error in the same direction.* Still, under the cir- 
 cumstances, we must admit this possibility, and make what 
 seems to be the best adjustment of all the results. 
 
 Definitive adjustment. 
 
 83. In making the definitive adjustment I shall proceed on 
 the supposition that no correction is necessary to the adopted 
 mass of Mars. I also go on the principle that no result is to 
 be rejected on account of doubt or discordance, except when 
 it is affected with a well-established cause of systematic error, 
 and shows a large deviation in the direction in which this 
 cause would act. At the same time it will be admissible to 
 diminish the weights in special cases, on account of causes of 
 systematic error which we know to exist, although we can not 
 determine the directions in which they would act ; and also on 
 account of deviations so wide as to show that the probable 
 error of the result must have been greatly underestimated. 
 Proceeding on this plan, we might reweight the last eight 
 results for the solar parallax, so as to get a result slightly 
 different from 8". 797. But 1 doubt whether such a reweight- 
 ing would not involve an objectionable bias. 
 
 We might diminish the weight of the result given by the 
 Pulkowa constant of aberration on the ground that no one 
 method should have so preponderating a weight as this has. 
 If we did so the result might be increased to 8".800. We 
 
 * For a very searching criticism of the systematic errors with which the 
 determinations of the solar parallax may be affected, reference may be 
 made to the first two articles by Dr. DAVID GILL, in Vol. I of The Observa- 
 tory. 
 
83] DEFINITIVE ADJUSTMENT. 169 
 
 might very largely increase the relative weight assigned to 
 the heliometer observations on Victoria and Sappho, but no 
 admissible increase would appreciably change the result. We 
 might also diminish the relative weight of the largely dis- 
 cordant result derived from measures of Venus during transit. 
 But as, by throwing out this result altogether, we should only 
 diminish the mean by ".001, it is scarcely worth while to do 
 so. Altogether no rediscussion of the relative weights seems 
 necessary. 
 
 On the other hand, the weight which we assign to the mean 
 result will enter as a very important factor into the final 
 adjustment. This is a point on which it is impossible to reach 
 a positive numerical conclusion by any mathematical process. 
 
 If, as one extreme case, we consider that the mean error of 
 each separate result corresponds to i0 7/ .03 for weight unity, 
 we shall have a mean error of rt".0035 for the value 8". 797. 
 The result will not be very different if we determine the mean 
 error from the discordance of the eight separate results. On 
 the other hand, if we include the deviation of the result given 
 by the motion of the node of Venus, the mean error for weight 
 unity will be increased to i 0".0045. The latter is undoubt- 
 edly the most logical course, so long as we proceed on the 
 hypothesis that the deviations of the final adjustment can all 
 be explained as due to fortuitous errors. If we include a com- 
 parison with the results of all the secular variations we shall 
 have a yet larger mean error. To show the result of assigning 
 one weight or the other I shall make two solutions, A and B, 
 in one of which a less and in the other a greater weight will 
 be assigned. 
 
 To the value 8".797 i .005 or .007 of the solar parallax 
 corresponds 
 
 r" = - 0.049 i .0016 or .0025 
 
 According as we assign one weight or the other to this result, 
 we may take as the corresponding equation of condition of 
 weight unity 
 
 (A): 400^' =-2.0 
 
 r (BH 600," = -2.9 W 
 
170 DEFINITIVE ADJUSTMENT. [83 
 
 The masses of Venus and Mercury, determined by methods 
 independently of the secular variations, also enter as conditions 
 into the adjustment. I have, however, made a revision of the 
 preliminary adjustment given in 64, the latter being based on 
 the results of 32-38; whereas it is better to use the defini- 
 tive results of the combination used in 46. 
 
 For the mass of Mercury the result found in 53 by the 
 last combination is 
 
 The values of the denominator corresponding to the mean 
 limits here assigned are 
 
 5 890 000 and 12 210 000 
 
 These limits are so wide as to include all admissible results for 
 the mass of Mercury. Moreover, we can not definitely say that 
 the value (6) of this mass is markedly greater or less than that 
 given by the weighted mean of all other results, since we 
 might so weight the latter as to give a result greater or less 
 without transcending the bounds of judicious judgment. I 
 conceive, therefore, that we are justified in reducing the mean 
 error to i 0.26, which will give as the equation of condition 
 
 r= - 0.055 i 0.25 
 and hence 
 
 40 x = - 0.22 i 1 (c) 
 
 When, in the normal equation for the mass of Venus, given 
 by the observations on Mercury, we substitute the values of 
 the secular variations found from the general combination of 
 46, the result is 
 
 v 1 = 0.0114 
 
 Combining this with the result from the Sun, we have 
 
 v 1 = - 0.0117 
 
 In view of the fact that the mass derived from observations of 
 Mercury may be affected by systematic errors of the kind 
 
83] DEFINITIVE ADJUSTMENT. 171 
 
 shown and discussed in 53, the mean error formerly assigned 
 to this result should be somewhat diminished. The result is 
 
 406 600 
 From this we have 
 
 v' = + 0.0084 .0030 
 
 For the equation of condition of weight unity I take 
 
 330 v' = + 2.8 (d) 
 
 With these equations of condition we have to combine the 
 eleven equations of 63, which we use unchanged, except that 
 we double the weight assigned, to the sixth equation, that 
 derived from the motion of the node of Venus, on account of 
 the smaller probable error of the result of our preceding redis- 
 cussion, and use the value of the absolute term found in 80. 
 
 If we accept the view that all the perihelia move according 
 to the same law of gravitation toward the Sun, namely, that 
 expressed by HALL'S hypothesis, then the value of the quan- 
 tity 6 in the formula expressing the law of gravitation is so 
 well determined by the motions of Mercury that it becomes 
 legitimate to use the observed motions of the perihelia of the 
 other three planets as equations of condition. But since it is 
 not impossible that the minor planets between Mars and 
 Jupiter may have an appreciable influence on the motion of 
 the perihelion of Mars, it is a question whether we should not 
 exclude that motion from the equations. 
 
 The conditional equations given by the motions of the three 
 perihelia in question are found by comparing the results of 
 46, 54, and 61. They are 
 
 40 x + v 1 + 20 v" = + 1.0 
 -14+46 +0 = - 0.3 (e) 
 
 2 - 13 +61 = + 0.7 
 
 The conditional equations to be combined are the eleven 
 equations of 63, the sixth of which is to have double weight^ 
 and the six equations (a), (c), (d), and (e). 
 
172 
 
 DEFINITIVE ADJUSTMENT. 
 
 [83 
 
 The normal equations to which we are thus led are the 
 following, which show the results of the four combinations we 
 may make according as we use (A) or (B) for the equation 
 given by the mass of the Earth, and omit or include the third 
 equation (d), which is given by the motion of the perihelion 
 of Mars. 
 
 (a.) Including the motion of the perihelion of Mars. 
 
 9 607,r 7 147 7' ; 11 335j/" = + 220 
 
 = -587 
 = - 3388 (A) 
 = - 4328 (B) 
 
 7 147 + 267 174 + 168 727 
 11 335 + 168 727 + 406 300 
 1 1 335 , + 168 727 + 606 300 
 
 (/?.) Omitting the motion of the perihelion of Mars. 
 
 9 603# - 7 12lv' - 11 457 v" = + 219 
 
 - 7 121 + 267 003 + 169 520 = - 578 
 
 - 11 457 + 169 520 + 402 578 = - 3431 (A) 
 
 - 11 457 + 169 520 + 602 578 = - 4371 (B) 
 
 The results of the solutions in the four cases are: 
 
 
 
 Aa 
 
 A// 
 
 B 
 
 B/? 
 
 
 X -f 
 
 0.0147 
 
 + 0.0142 
 
 + 0.0161 
 
 + 0.0158 
 
 
 V -j- 
 
 0.147 
 
 + 0.142 
 
 -f 0.161 
 
 + 0.158 
 
 
 V 1 + 
 
 0.004 34 
 
 + 0.004 60 
 
 + 0.003 10 
 
 + 0.003 i>.") 
 
 
 v" 
 
 0.009 73 
 
 0.01005 
 
 - 0.007 70 
 
 0.007 87 
 
 1 
 
 -V- m 
 
 6 539 000 
 
 6 567 000 
 
 6 460 000 
 
 6 477 000 
 
 1 
 
 + m' 
 
 408 230 
 
 408 120 
 
 408 730 
 
 408 670 , 
 
 
 7T 
 
 8".783 
 
 8 // .782 
 
 8".789 
 
 8".788 
 
 I conceive that if the secular variations, especially the motion 
 of the node of Venus, are not affected by any unknown cause, 
 some mean between these should be regarded as the most 
 probable solution. The result does not, however, bring about 
 a satisfactory reconciliation. We still find ourselves confronted 
 by this embarrassing dilemma: Either there is something 
 abnormal in connection with the node of Venus, due to an 
 unknown cause acting on the planet, to some extraordinary 
 errors in the observations or their reduction, or to some error 
 in the theory on which the discussion is based, or the deter- 
 
83, 84, 85] ADOPTED PARALLAX AND MASSES. 173 
 
 ruinations of the solar parallax are nearly all in error in one 
 direction by amounts which are, in more than one case, quite 
 surprising. 
 
 Possible causes of the observed discordances. 
 
 84. Two possible causes of discordance may be suggested, 
 one of which has not been touched upon at all in the preceding 
 chapters, and one perhaps inadequately. As to the hypothesis 
 of non-sphericity of the Sun, considered in 56, it may be 
 remarked that Dr. HARTZER shows that an ellipticity of the 
 Sun sufficient to produce the observed motion of the perihelion 
 of Mercury would cause a direct motion of 5".l in the motion 
 of the node of Venus. This would correspond to a change of 
 0".30 in the value siniD t # and would therefore go far toward 
 reconciling the discrepancy. But it is easy to see that this 
 cause would produce a secular motion of 2".6 in the inclina- 
 tion of Mercury. We have seen that the observed motion of 
 the inclination already exceeds the theoretical motion by 0".38; 
 so that introducing the hypothesis of ellipticity of the Sun we 
 should have a discrepancy of about S^.O between theory and 
 observation. This conclusion alone seems fatal to the theory, 
 which otherwise has been shown to be scarcely tenable. 
 
 The other possible cause is an inequality of long period ; 
 especially one depending on the argument \3l" 81' which 
 has a period of about two hundred and forty three years. A 
 very simple computation shows that the coefficient of this term 
 is only of the order of magnitude (V'.Ol. 
 
 It is a curious coincidence that if we had neglected to add 
 the mass of the Moon to that of the Earth, in computing the 
 secular variations, the discrepancy would not have existed. 
 
 Adopted values of the doubtful quantities. 
 
 85. The practical question which has been before the writer 
 in working out the preceding results is : What values of the 
 constants should be used in the tables of the celestial motions 
 of which the results of this discussion are to form the basis ? 
 Should we aim simply at getting the best agreement with obser- 
 vations by corrections more or less empirical to the theory ? 
 It seems to me very clear that this question should be answered 
 in the negative. No conclusions could be drawn from future 
 
174 ADOPTED PARALLAX AND MASSES. [85 
 
 comparisons of such tables with observations, except after 
 reducing the tabular results to some consistent theory. The 
 imposition of such a labor upon the future investigator is not 
 to be thought of. Moreover, there is no certainty that the 
 tables which would best represent past observations would 
 also best represent future ones. Our tables must be founded 
 on some perfectly consistent theory, as simple as possible, the 
 elements of which shall be so chosen as best to represent the 
 observations. 
 
 In choosing the theory and its constants we have again a 
 certain range. If we accept the necessity of assuming the 
 secular variations of the orbits of Mercury and Venus to be 
 affected by the action of unknown masses of matter, then the 
 simplest course to adopt is to construct our theory on the sup- 
 position of a planet or group of planets between Mercury and 
 Venus. 
 
 It seems to me that the introduction of the action of such a 
 group into astronomical tables would not be justifiable. The 
 more I have reflected upon the subject the more strongly 
 seems to me the evidence that no such group can exist, and, 
 indeed, that whatever anomalies exist can not be due to the 
 action of unknown masses of matter. 
 
 Besides, the six elements of such a group would constitute 
 a complication in the tabular theory. 
 
 On the other hand, it did not seem to me best that we should 
 wholly reject the possibility of some abnormal action or some 
 defect between the assumed relations of the various quanti- 
 ties. What I finally decided on doing was to increase the theo- 
 retical motion of each perihelion by the same fraction of the 
 mean motion, a course which will represent the observations 
 without committing- us to any hypothesis as to the cause 
 of the excess of motion, though it accords with the result of 
 HALL'S hypothesis of the law of gravitation ; to reject entirely 
 the hypothesis of the action of unknown masses, and to adopt 
 for the elements what we might call compromise values between 
 those reached by the preceding adjustment and those which 
 would exist if there is abnormal action. The exigency of hav- 
 ing to prepare the tables required me to reach a conclusion on 
 this subject before the final revision of the preceding discus- 
 
85,86] FUTURE DETERMINATIONS. 175 
 
 sion, so that the numbers used are not wholly based upon it. 
 The conclusions I have reached are these: 
 
 Since, if there is nothing abnormal in the theory, the solar 
 parallax is probably not much larger than 8".780, and if there 
 is anything abnormal it is probably as large as 8".795 or even 
 8' '.800, we may adopt the value 8". 790 as one which is almost 
 certainly too large on the one hypothesis and too small on the 
 other, and which is therefore best adapted to afford a decision 
 of the question. 
 
 For the mass of Venus I took, as an intermediate value, 
 
 m ' =1-1-408000 
 For the mass of Mercury I took 
 
 1 4- 6,000,000 
 
 Actually it seems that this mass is larger than the most prob- 
 able one on either hypothesis, though not without the range of 
 easy possibility. 
 
 With these values the outstanding difference between theory 
 and observation in the centennial motion of the node of 
 Venus is 
 
 A sin i D t = 0".25 
 
 If this difference arises wholly from the error of the theory, 
 then between the transits of 1874 and 2004 the accumulated 
 error would amount to 0".32 in the heliocentric latitude, and 
 about 0".8 in the geocentric latitude. Unless an improvement 
 is made in the method of determining the position of Venus 
 by observation, the twentieth century must approach its end 
 before this difference can be detected. 
 
 Bearing of future determinations on the question. 
 
 86. The following shows the influence which subsequent 
 determinations of the principal elements will have upon our 
 judgment as to the solution of the dilemma. The changes in 
 the second column will, by emphasizing the discordance 
 between the results, tend to confirm the hypothesis of an 
 abnormal defect in the theory, while the opposite ones, in the 
 last column, will tend to reconcile theory and observation : 
 
 USUVBRSITf 
 
176 
 
 FUTURE DETERMINATIONS. 
 
 [86 
 
 Element or quantity. 
 
 Change tending to 
 confirm the dis- 
 cordance 
 between theory 
 and observation. 
 
 Change tending to 
 reconcile exist- 
 ing theory with 
 observation. 
 
 The solar parallax. 
 
 Increase. 
 
 Diminution. 
 
 Longitude of the node of Mercury. 
 
 Increase. 
 
 Diminution. 
 
 Longitude of the node of Venus. 
 
 Increase. 
 
 Diminution. 
 
 Constant of aberration. 
 
 Diminution. 
 
 Increase. 
 
 Mass of Venus. 
 
 Increase. 
 
 Diminution. 
 
 Mass of Mercury. 
 
 Diminution. 
 
 Increase. 
 
 Secular diminution of the obliquity. 
 
 Diminution. 
 
 Increase. 
 
 Among these constants are some the values of which can 
 scarcely be decisively obtained except by observations con- 
 tinued through half a century, or even through the whole 
 twentieth century, unless improvements are made in our pres- 
 ent methods of observing. 
 
 The improvement of others, however, is quite within the 
 reach of the astronomy of the present time. Among these 
 the constant of aberration and the solar parallax have the 
 first place. The more accurate determination of these quanti 
 ties thus assumes an importance which may justify some sug- 
 gestions on the subject. 
 
 The observations made on the European continent for the 
 detection and study of the variations of latitude have been 
 executed with such precision that we might look to them for a 
 marked improvement in the determination of the constant of 
 aberration, were it not for a single circumstance. In the gen- 
 eral average few are made after midnight, while the maxima 
 and minima of aberration occur in the morning and evening. 
 The extension of the system into the early morning therefore 
 seems desirable. Although these observations may scarcely 
 equal in accuracy those made by NYREN, with the prime 
 
86] FUTURE DETERMINATIONS. 177 
 
 vertical transit, they have the advantage of not requiring so 
 long a period for a complete observation. The great disad- 
 vantage of the prime vertical instrument is that unless a star 
 culminates within a few minutes of the zenith, an hour, or 
 even several hours, will be required for the completion of a 
 determination, which may thus be made impossible by the 
 'advent of daylight. It may be remarked in this connection 
 that the northern latitudes of the European observatories are 
 favorable to the determination of the aberration-constant. 
 
 LOEWY'S method has over all others the great advantage of 
 being independent of the direction of the vertical. But its 
 application, and the reduction of the observations made with, 
 it, are laborious in a high degree. 
 
 So far as practical astronomy has yet developed, the best 
 method of directly measuring planetary parallax, and there- 
 fore the only one to be considered, is that of GILL. It there- 
 fore seems desirable that measures by this method should be 
 continued. At the same time it is very necessary that the 
 spectra of the small planets to be used should be carefully 
 studied photometrically, and that the probable influence of 
 coloration upon the measures should be investigated. 
 
 The necessity of completing the present work, and of pro- 
 ceeding immediately to the construction of tables founded 
 upon the adopted elements, prevent the author's awaiting the 
 mature judgment of astronomers upon the embarrassing ques- 
 tions thus raised. The regret with which he accepts this 
 necessity is weakened by the consideration that even if the 
 solar parallax which he has adopted requires the largest cor- 
 rection to which it can reasonably be supposed subject, namely, 
 one of 0".015, reducing the value of this constant to 8". 775, 
 the effect of the error will not be prejudicial to the astronomy 
 of the'immediate future. 
 
 More important will be the error /x .035 in the constant of 
 aberration. Yet a long-continued series of observations will 
 be necessary to establish even the existence of such an error, 
 and should it prove detrimental in any astronomical work the 
 evil will be easily remedied by a slight correction. 
 5690 N ALM 12 
 
CHAPTER IX. 
 
 DERIVATION OF RESULTS. 
 
 Ulterior corrections to the motions of the perihelion and mean 
 . longitude of Mercury. 
 
 87. In 32 and 46 we have reached three values of the 
 correction to the tabular motion of the perihelion of Mercury. 
 Of these the first rests on meridian observations alone, the 
 second on the combination of meridian observations with trans- 
 its, and the third is derived by substituting in the eliminating 
 equations the corrections to the solar elements and their secular 
 variations which result from observations. The three values 
 thus reached are 9".54, 1".01, and + 6".34. The pro- 
 gressive divergence of these values, taken in connection with 
 the discrepancy pointed out in 33, leads us to distrust the 
 influence of the meridian observations upon the motion of the 
 perihelion. Under these circumstances I deem it advisable to 
 make such final corrections to the motions in mean longitude 
 and mean anomaly as will best satisfy all the observed transits 
 over the disk of the Sun. In doing this I am enabled to intro- 
 duce the results of a preliminary discussion of the transits of 
 1891 and 1894. By combining the observations of these two 
 transits with those of the older ones I derive the following 
 values of the functions Y and W defined in 31 : 
 
 // // 
 Y = -1.93- 3.03 T 
 
 W= + 1.50 + 2.04 T 
 
 The preliminary theory, so far as yet investigated, gives for 
 the values of this quantity, 
 
 // // 
 Y = - 2.44 3.40 T 
 
 W = + 1.38 + 1.3GT 
 178 
 
87, 88] PERIHELION OF MERCURY. 179 
 
 Equating these values to the corresponding linear functions 
 of the corrections to /, TT, and their secular motions, we have 
 the equations, 
 
 // // 
 
 0.72 SI + 0.28 67f = + 0.12 + 0.68 T 
 + 1 .49 - 0.49 = + 0.51 + 0.37 T 
 
 We find, from these equations, 
 
 // // 
 
 61 = +0.26 + 0.56 T 
 Sn = - 0.24 + 0.97 T 
 
 The preliminary values to which these corrections are appli- 
 cable are 
 
 // // 
 
 61 = +0.04- 1.33 T 
 6jr= + 5.83 + 6.34 T 
 
 The definitive values thus become 
 
 61 = + 0.30- 0.77 T 
 
 tf TT = + 5.59+ 7.31 T 
 
 Definitive elements of the f out inner planets for the epoch 1850, as 
 inferred from all the data of observation. 
 
 88. We have made a fourth solution of the normal equations 
 which give the corrections to the elements of each planet by 
 substituting in those equations the definitive values of all the 
 other quantities, including the values of the secular variations 
 derived from theory. In making this substitution for Mercury, 
 however, the ulterior corrections just found were not applied. 
 The values of the unknowns resulting from this solution are 
 shown in the first column of the next table. From these 
 numbers are derived the definitive elements for 1850, 'by the 
 following processes: 
 
 (a.) By multiplying the unknowns by the appropriate factor 
 given in 27, we have the corrections of the tabular elements 
 at the mid- epoch of observations for each planet. These cor- 
 rections are found in the second column. 
 
 (/?.) The preceding corrections are to be reduced from the 
 respective mid-epochs to 1850. This reduction is found by 
 
180 DEFINITIVE QUANTITIES. [88 
 
 multiplying the definitive correction to the tabular secular 
 variation by the elapsed interval, and is shown in the third 
 column. 
 
 (y) We next have the value of the tabular elements for the 
 fundamental epoch 1850, January 0, Greenwich mean noon. 
 These numbers are those of LEVERRIER'S tables, with the 
 following modifications: 
 
 (d) The reduction from 1850, January 1, Paris noon, to 
 January 0, Greenwich noon 
 
 (f) The corrections to LEVERRIER'S values of the eccen- 
 tricity and perihelion which are necessary to represent those 
 terms in the perturbations of the mean longitude which depend 
 only upon the sine and cosine of the mean anomaly. The 
 theory is more symmetrical in form when all such terms are 
 included with those of the elliptic motion. In LEVERRIER'S 
 tables they have the following values: 
 
 Mercury 5 6v = 4- 0.030 sin I - 0.111 cos I 
 Venus; +0.010 +0.037 
 
 Earth; 0.067 -0.098 
 
 Mars; +1.061 +0.718 
 
 These terms of the longitude may be represented by the follow- 
 ing corrections to the elements: 
 
 Mercury; de = + 0.058 dn r= 0.0 
 
 Venus; -0.012 +2.3 
 
 Earth; +0.054 +1.4 
 
 Mars; +0.613 -1.0 
 
 Applying these corrections d and e to LEVERRIER'S tabular 
 quantities, we have the values of the tabular elements as given 
 in the fourth column. Then applying the preceding correc- 
 tions we have the definitive values given in the last column. 
 
 In some cases this derivation is modified. Instead of using 
 the correction to the perihelion, mean longitude and mean 
 motion of Mercury given by the unknown quantities of the 
 
88] ELEMENTS FOR 1850. 181 
 
 equations, we have used the values for 1850 derived from the 
 discussion of the preceding section. 
 
 The quantities which give the position of the node and 
 inclination have been treated in the same way as their secular 
 variations. The symbols J and N indicate values of the 
 unknown quantities related to the corrections of the elements 
 J and N. These unknowns are then changed to corrections of 
 the elements by the factors of 27, and these again to correc- 
 tion of the inclination and node by the equations of 41. 
 
 In the case of the node of Venus two values are given. The 
 value (a) is that which follows immediately from the normal 
 equations. If we carry forward the position of the node just 
 derived to the mean epoch of the last two transits of Venus, 
 we find a discrepancy amounting to 2".04 in the longitude, 
 corresponding to a difference of 0".121 in the heliocentric lati- 
 tude. This is considerably larger than the probable error of 
 the results of the observations of the transits. It may, there- 
 fore, be questioned whether the latter are not entitled to a 
 greater relative weight than that assigned, owing to the prob- 
 able systematic errors of the meridian observations. A second 
 value (b) has therefore been derived from the observations of 
 the transits alone. In subsequent investigations we may 
 choose between these two values. 
 
 Formation of definitive elements of the four inner planets, for tlit\ 
 epoch 1850 7 January 0, Oreemvich mean noon. 
 
 Mercury. 
 
 Unknown of Corr. of Red. to Tabular Concluded 
 
 equations. element. 1850. element. element. 
 
 // // // // 
 
 n -.0940 - 0.77 0.0 538106654.49 538106653.72 
 
 e - .0741 - 0.222 - 0.005 42 409.088 42 408.861 
 
 n + .6763 -1- 5.59 75 7 13.78 75 7 19'!s7 
 
 t .0402 + 0.30 323 11 23.53 323 11 23.83 
 
 i -.2762J-- 0.64 - 0.07 7 7.71 7 7.00 
 
 d -.0001N+ 3.88 - 0.27 46 33 8.63 46 33 12.24 
 
182 DEFINITIVE QUANTITIES. [88,89 
 
 Formation of definitive elements, etc. Continued, 
 Venus. 
 
 Unknown of Corr. of Ked. to Tabular Concluded 
 
 equations, element. 1850. element. element. 
 
 n - .1783 - 3.57 
 
 210669165.04 210669161.47 
 
 e + .1463 
 
 0.439 - 0.165 
 
 1 411.522 
 
 1 411.796 
 
 129 
 
 7t + .0835 + 36.6 16.4 
 
 z _ .1330 - 0.67 + 0.46. 243 
 
 i + .0968 J + 0.31 + 0.12 3 
 
 0(a)+ .0126 N- 9.39 + 6.63 75 
 
 0(b) -20.36 +15.56 
 
 Earth. 
 
 27 
 57 
 23 
 19 
 
 14.3 129 
 44.34 243 
 34.83 3 
 52.21 75 
 75 
 
 27 
 57 
 23 
 19 
 19 
 
 34.5 
 
 44.13 
 
 35.26 
 
 49.45 
 
 47.41 
 
 1.10 
 0.12 
 
 2.4 
 
 0.15 
 
 0.02 
 
 129 602 767.84 129 602 766.74 
 
 3 459.334 
 
 100 21 43.4 
 23 27 31.83 
 
 99 48 18.72 
 
 Mars. 
 
 - .1094 - 0.88 68 910 105.38 
 
 - .1088 - 0.155 + 0.058 19 237.101 
 
 + .1663 + 2.38 + 0.02 333 
 
 3 459.454 
 
 100 21 41.0 
 23 27 31.68 
 99 48 18.74 
 
 68 910 104.50 
 19 237.004 
 
 - .4029 - 0.81 + 0.05 83 
 
 - .0507 J + 0.18 - 0.01 1 
 + . 1135 N+ 6.56 +1.34 48 
 
 17 
 9 
 
 51 
 23 
 
 52.47 
 16.92 
 2.28 
 53.02 
 
 333 
 
 83 
 
 1 
 
 48 
 
 17 
 9 
 
 51 
 24 
 
 54.87 
 
 16.16 
 
 2.45 
 
 0.92 
 
 Definitive values of the secular variations. 
 
 89. The definitive values of the secular variations, as inferred 
 from the adopted theories and the concluded values of the 
 masses, are shown in the following table, which gives in detail 
 the parts of which each quantity is made up. 
 
 The first four lines of the table give the values of the secular 
 variations as they result from the investigations found in Vol. 
 V, Part IV, of the Astronomical Papers, after correcting the 
 mass of each planet by its appropriate factor. 
 
 The motion of the perihelion first given, denoted by D t n\, 
 is measured along the plane of the orbit itself. The numbers 
 
89 1 SECULAR VARIATIONS. 183 
 
 given being divided by the corresponding values of the eccen- 
 tricity we have the motion of the perihelion itself along the 
 plane. The symbols i and # represent the inclinations and 
 longitudes of the nodes referred at each epoch to the ecliptic 
 and equinox of 1850, regarded as fixed. The motions of these 
 elements are next to be referred to the fixed ecliptic of the 
 date. So referred, they are designated as D? i and D? 6. The 
 transformations to the latter quantities are made by comput- 
 ing an approximate value of the motion of the node due to 
 the motion of the ecliptic alone along the plane of the orbit 
 regarded as fixed. 
 If we put 
 
 ,, the inclination of the fixed orbit of the planet at any epoch 
 TO to the moving ecliptic at any time; 
 
 61, the longitude of the corresponding node, h; 
 
 F, the distance from the node Q t to the instantaneous rota- 
 tion axis of the orbit at the epoch T ; 
 
 we shall have 
 
 D t v = H" cosec i\ sin (L" #1) (a] 
 
 If we compute v and H from the equations 
 
 H sin VQ = sin i D? 
 H cos r = D? i 
 
 and then find Av by integrating the value (a) of D t r from 1850 
 to the date we shall have 
 
 sin i D? # = H sin ( V Q + A v] 
 D i = H cos (v Q -f Av) 
 
 The change of D t ^ between 1850 and the extreme epochs has 
 been found so nearly uniform that it was sufficient to multiply 
 its value at the mid-epoch (1675 or 1975) by 2.5 to obtain Av. 
 
 Next, we have the changes in i and due to the motion of the 
 ecliptic, represented by T>]i and Df0, and computed by the 
 formula 
 
 D l t i= H f/ GOS(r /f -0) 
 sin i D[ 6 = H" cos i sin (v" 0) 
 
184 DEFINITIVE QUANTITIES. [89 
 
 The planetary precession due to the motion of the ecliptic is 
 here omitted, to be afterwards included in the general preces- 
 sion. The sum of the two motions gives the actual variation 
 at each epoch, referred to a fixed equinox. 
 
 The motion of 6 itself thus found is increased by the general 
 precession, which gives the motion of 6 at each epoch. 
 
 The motion of the perihelion to be actually used in the tables 
 is equal to the motion of the node from the mean equinox, plus 
 the increase of the arc of the orbit between the node and 
 perihelion. The adopted value of this quantity is found by 
 increasing the motion of n\ by the following quantities: 
 
 1. The change due to the motion of the plane of the orbit. 
 
 2. The change due to the motion of the ecliptic. 
 The formulae for these two quantities are 
 
 (1) ; d] D t n = tan J i$m i D? d 
 
 (2) 5 <? 2 D t n = H" tan J i sin (L" - 0) 
 
 3. The excess of motion shown by observations in the case 
 of Mercury and Mars, and computed for all four planets as if 
 they gravitated toward the Sun with a force proportional to 
 r~ n where 
 
 n = 2.000 000 16120 
 
 The values of this correction are 
 
 // 
 
 Mercury; D t n 43.37 
 Venus; 16.98 
 
 Earth; 10.45 
 
 Mars; 5.55 
 
 4. The general precession. 
 
 5. In the case of the Earth, the motion arising from the 
 action of the Moon, of which the amount is 
 
 D t n" = 7".68 
 
 But the first two corrections drop out in this case. 
 
 The preceding transformations of the secular variations are 
 made with the original values of the elements e and i, as given 
 in Astronomical Papers, Vol. V, Part IV, pp. 337, 338. 
 
89J 
 
 SECULAR VARIATIONS. 
 
 185 
 
 Secular variations of the elements of the four orbits at the three 
 epochs, 1600, 1850, and 3100, as inferred from the definitively 
 adopted masses. 
 
 Mercury. 
 
 1600. 
 
 1850. 
 
 2100. 
 
 D t e 
 
 + 4.257 
 
 + 4.227 
 
 + 4.196 
 
 tfDtTTi 
 
 + 109.524 
 
 + 109.498 
 
 + 109.475 
 
 DJtp 
 
 21.581 
 
 - 21.568 
 
 - 21.551 
 
 sinioD?0o 
 
 54.891 
 
 - 54.969 
 
 55.049 
 
 D?i 
 
 - 21.786 
 
 - 21.568 
 
 - 21.347 
 
 sintDfd 
 
 54.813 
 
 - 54.969 
 
 - 55.130 
 
 D{i 
 
 + 28.884 
 
 + 28.333 
 
 + 27.785 
 
 sin t DJ B 
 
 - 37.196 
 
 - 37.397 
 
 37.595 
 
 D t * 
 
 + 7.098 
 
 + 6.765 
 
 + 6.438 
 
 sin i D t 
 
 - 92.009 
 
 - 92.366 
 
 - 92.725 
 
 JD t 7T 
 
 1.06 
 
 1.06 
 
 - 1.06 
 
 D t 7T 
 
 5593.41 
 
 5598.70 
 
 5604.02 
 
 D t 
 
 4262.98 
 
 4266.12 
 
 4269.24 
 
 
 
 Venus. 
 
 
 D t e 
 
 - 9.959 
 
 - 9.866 
 
 - 9.772 
 
 eD t 7Ti 
 
 -f 0.384 
 
 + 0.219 
 
 + 0.060 
 
 Dfto 
 
 - 2.484 
 
 - 3.071 
 
 - 3.656 
 
 sin to DJ 
 
 - 59.005 
 
 - 59.112 
 
 - 59.229 
 
 D?t 
 
 - 3.049 
 
 - 3.071 
 
 - 3.091 
 
 sin*D?0 
 
 - 58.978 
 
 - 59.112 
 
 - 59.260 
 
 D<* 
 
 + 6.690 
 
 + 6.695 
 
 + 6.697 
 
 sin i DJ 
 
 - 46.758 
 
 - 46.582 
 
 - 46.413 
 
 D t i 
 
 + 3.641 
 
 + 3.624 
 
 + 3.606 
 
 sin i D t 
 
 - 105.736 
 
 - 105.694 
 
 - 105.673 
 
 JD t 7T 
 
 - 0.36 
 
 - 0.37 
 
 - 0.38 
 
 BtTT 
 
 5090.07 
 
 5072.44 
 
 5054.92 
 
 D t 
 
 3230.39 
 
 3237.98 
 
 3245.22 
 
186 DEFINITIVE RESULTS. |89, 90 
 
 Secular variations of the elements of the four orbits, etc. Cont'd. 
 
 Earth. 
 1600. 1850. 2100. 
 
 D t e" 
 
 - 8.467 
 
 - 8.595 
 
 - 8.727 
 
 e"D t 7r" 
 
 D t 7T" 
 
 + 19.293 
 6179.58 
 
 + 19.210 
 6187.41 
 
 + 19.139 
 6195.68 
 
 H" sin Li' 
 H" cos Li' 
 
 + 4.370 
 47.113 
 
 + 5.341 
 - 46.838 
 
 + 6.305 
 . 46.550 
 
 log H" 
 
 L'o 
 L" 
 
 1.67500 
 1740 42'.04 
 171 12 / .83 
 
 1.67340 
 173 29'.68 
 1730 29 X .68 
 
 1.67187 
 1720 17M8 
 1750 46'.62 
 
 Po 
 
 p 
 
 5034.91 
 
 5018.28 
 - 46.761 
 
 5036.13 
 5023.82 
 - 46.838 
 
 5037.36 
 5029.38 
 - 46.847 
 
 Mars. 
 
 e D t 7T] 
 
 + 18.775 
 + 148.633 
 28.994 
 
 + 18.706 
 + 148.707 
 - 29.396 
 
 + 18.623 
 + 148.762 
 - 29.803 
 
 sin *o D? 
 
 34.023 
 
 34.012 
 
 34.017 
 
 D?* 
 
 - 29.482 
 
 - 29.396 
 
 - 29.309 
 
 sin t D? 
 
 - 33.605 
 
 34.012 
 
 34.445 
 
 DM 
 sin i DJ 
 
 + 26.964 
 
 - 38.860 
 
 + 27.104 
 38.551 
 
 + 27.245 
 38.247 
 
 D t t 
 
 - 2.518 
 
 - 2.292 
 
 2.064 
 
 sin i D t 
 
 72.465 
 
 - 72.563 
 
 - 72.692 
 
 1>^ 
 
 + 0.08 
 6621.51 
 
 + 0.07 
 6623.96 
 
 + 0.06 
 6626.25 
 
 D t 
 
 2776.39 
 
 2776.87 
 
 2776.63 
 
 Secular acceleration of the mean motions. 
 
 90. The mean motions of the planets, like that of the Moon, are 
 subject to a secular acceleration arising from the secular vari- 
 ations of the elements of the orbits. The following formulae 
 for this acceleration are formed by differentiating the known 
 
90] SECULAR ACCELERATIONS. 187 
 
 expressions for the variation of the longitude of the epoch in 
 the theory of the variation of elements. The notation is that 
 of Astronomical Papers, Vol. V, Part IV. 
 We compute for the action of an outer on an inner planet: 
 
 A = D <*\ } 
 
 B = - (D - D 2 2 D 3 ) c ( > 
 
 8 
 
 W- - (2 - 9 D + 3 D 2 + 4 D 3 ) c^ 
 
 8 
 
 Then 
 
 D; = w' a n D t { A <J 2 + Be 2 - Ge' 2 + Wee' cos (n - n')\ 
 For the action of an inner on an outer planet we compute 
 
 A' =-^(l + D)6^ ) 
 
 B 7 = l (D + 2 D 2 + D 3 ) c ( ] 
 4 
 
 8 
 
 W = I (10 + 3 D - 9 D 2 - 4 D 3 ) <t\ } 
 8 
 
 D? 1 = m n' D t j A 7 a 2 + B'e 2 + We' 2 + Wee' cos (n - *') [ 
 
 The symbol D t indicates the secular variation of the expres- 
 sion following it produced by the action of all the planets. The 
 unit of time must be the same one in which n is expressed. 
 
 The following table gives the results of this computation : 
 
 Secular change of the centennial mean motions. 
 Action of Mercury. Venus. Earth. Mars. 
 
 Venus, 0.0426 
 
 . 
 
 -0.0104 
 
 + 0.0010 
 
 Earth, -0.0029 
 
 + 0.0128 
 
 . . . 
 
 + 0.0119 
 
 Mars, + 0.0003 
 
 -0.0001 
 
 - 0.0012 
 
 . 
 
 Jupiter, -0.0039 
 
 0.0046 
 
 -0.0308 
 
 + 0.0004 
 
 Saturn, -0.0004 
 
 + 0.0015 
 
 + 0.0021 
 
 + 0.0036 
 
 Total, -0.0495 +0.0096 -0.0403 +0.0169 
 
188 DEFINITIVE QUANTITIES. [91, 92 
 
 The measure of time. 
 
 91. The fictitious mean Sun whose transit over any meridian 
 defines the moment of mean noon on that meridian is a point 
 on the celestial sphere having a uniform sidereal motion in the 
 plane of the Earth's equator, and a Eight Ascension as nearly 
 as may be equal to the Sun's mean longitude. If we put /* for 
 this uniform sidereal motion and add to JA the precession of the 
 equinox in Eight Ascension we have for the mean Eight Ascen- 
 sion of this fictitious mean Sun 
 
 T = TO + /i T + 4606 // .36 T + 1".394 T 2 
 
 From 88, 90, and 100 the expression for the Sun's mean 
 longitude, affected by aberration, is found to be 
 
 L = 2790 47' 58".2 + 129602766".74 T + 1 // .089 T 2 
 
 Equalizing the coefficients of T we find, for the mean Eight 
 Ascension of the fictitious mean Sun 
 
 r = 279 47' 58".2 + 1296027G6 // .74 T + l // .394 T 2 
 
 This differs from the mean longitude of the actual Sun by the 
 quantity 
 
 r - L = 0".305 T 2 = 8 .020 T 2 
 
 This difference is of no importance in the astronomy of our 
 time, but may result in an error of 2 s in the course of one thou- 
 sand years in the measurement of time by the actual mean 
 sun. We must leave to the astronomers of the future the 
 question how best to meet the question thus arising. Chang- 
 ing to time the expression for r, the difference or mean excess 
 of sidereal over mean time for the meridian of Greenwich 
 becomes 
 
 T = l& 39 ra 11 8 .880 + 24" O m K84449 t + 8 .0929 T 2 
 
 t being time in Julian years after 1850, January 0, Greenwich 
 mean noon. 
 
 Constant of aberration. 
 
 92. We first investigate certain fundamental constants con- 
 nected with the motion of the Sun, Earth, and Moon, on which 
 the precession and nutation depend. 
 
92, 93] MASS OF THE MOON. 189" 
 
 From the adopted value of the solar parallax, 
 
 n = 8 // .790, 
 and the adopted velocity of light in kilometers per second, 
 
 Y = 299 860, 
 
 follows for the constant of aberration the value 
 
 A = 20 // .501 
 
 But if we accept the mean result of the solutions of 83 as 
 giving the most likely value of the solar parallax, we shall 
 have 
 
 n = 8".7854 
 
 Then 75 will give 
 
 A = 20".511 
 
 as the adjusted value of the constant of aberration. 
 Mass of the Moon. 
 
 93. By means of the equation of 71 between the lunar 
 inequality P in the motion of the Earth and the mass of the 
 Moon 
 
 /*'P = [1.78207] n 
 
 we may find a fresh value of the Moon's mass from the values 
 of 7t and P. 
 We have found from observation 
 
 P = 6".465 i .015 
 
 Thus follows, for the mass of the Moon, when 7r = 8".790, 
 yw = 1 : 81.32 i 0.20 
 
 Combining this with the value found from the constant of 
 nutation, 
 
 yw = 1 : 81.58 0.20 
 
 we have, as the definitive mass of the Moon, 
 * = !: 81.45 i 0.1 
 
190 DEFINITIVE QUANTITIES. [94,95 
 
 Parallactic inequality of the Moon. 
 
 94. From the transformation of HANSEN'S lunar theory in 
 Astronomical Papers, Vol. I, it may be concluded that the solar 
 parallax and the parailactic inequality are connected by the 
 relation 
 
 P. I. = [1.16242] ln 7t 
 
 1 + V 
 
 = [1.15176] 7i 
 
 Hence we have, for the coefficient of the parailactic inequality 
 of the Moon, corresponding to n = 8 // .790, 
 
 124".66 
 
 Here the inequality is that in ecliptic longitude. 
 The centimeter -second system of units. 
 
 95. There are certain methods in physics by which the next 
 step in the course of our researches will be guided. The adop- 
 tion of a system of absolute units has simplified the methods 
 and conceptions of physics to such an extent that we may 
 find it advantageous to introduce a similar system into those 
 investigations of astronomy which are closely connected with 
 that science. 
 
 The fundamental units most widely adopted are the centi- 
 meter as the unit of length, the gram as the unit of mass, 
 and the second as the unit of time. There is, however, an 
 insuperable difficulty in the way of introducing the gram, 
 or any other arbitrary terrestrial unit of mass, into astronomy, 
 from the fact that the astronomical masses with which we are 
 concerned can not be determined with sufficient precision in 
 units of terrestrial mass. It is, therefore, quite common in 
 celestial mechanics to regard the unit of mass as arbitrary, 
 and to multiply this arbitrary unit by a factor which will 
 represent its attractive force upon a unit particle at unit dis- 
 tance. The introduction of this factor is, however, needless. 
 It is simpler to adopt the course of DELAUNAY and many other 
 writers, and regard the unit of mass as a derived one, based 
 on the units of time and length, by defining it as that mass 
 which will attract an equal mass at unit distance with force 
 
95, 96] MASSES OF THE EARTH AND MOON. 191 
 
 unity. In this definition the unit of force retains its physical 
 meaning, as that force which, acting on unit mass, will pro- 
 duce a unit of acceleration in a unit of time. 
 
 The number of fundamental units is then reduced to two, 
 those of time and length, and the unit mass becomes a : derived 
 one of dimensions, 
 
 The centimeter as a unit of length wou 
 small for astronomical purposes, if we had to deal mainly r with 
 natural numbers, but it causes no inconvenience in logarith- 
 mic computations, and has the advantage of being assimilated 
 directly to the centimeter-gram-second system in physics. 
 We shall therefore adopt it, expressing our results, however. 
 in terms of other units whenever convenience will thereby be 
 gained. 
 
 I shall make clear this assimilation and the use of the unit 
 of mass as a derived one, by calling this the centimeter- 
 second system. 
 
 In the latter the definitions of units in the centimeter- 
 gram-second system will remain unchanged, except that the 
 derived unit of mass must be substituted for the gram. The 
 dimensions of units in the centimeter-second system will be 
 found by making the above substitution for M in the expres- 
 sions for those of the centimeter- gram -second system. 
 
 Masses of the Earth and Moon in centimeter -second units. 
 
 96. A fundamental quantity in the centimeter- second system 
 is the mass of the Earth. This mass will be by definition the 
 force of gravity of the Earth, if concentrated in a point at the 
 distance of one centimeter. Were the Earth a sphere of known 
 dimensions, it could be readily determined through the force 
 of gravity at any point on its surface. This being not the case, 
 we shall proceed on the accepted approximate theory that the 
 geoid is an ellipsoid of revolution, and that the force of gravity 
 at a point the sine of whose latitude is 1 : -v/3 is the same as 
 if the mass of the Earth were concentrated in its center. 
 
 The determination of this constant with astronomical preci- 
 sion is a difficult and we might say hitherto an insoluble prob- 
 
192 DEFINITIVE Q UANTITIES. [96 
 
 lem, owing to the heterogeneity of the Earth and the absence 
 of determinations of the force of gravity over the surface of the 
 ocean. Although the limits of uncertainty thus arising can 
 not be set with any approach to precision, I do not think they 
 are such as to greatly impair the astronomical results which 
 are to be derived from them. Investigations in geodesy not 
 being practicable in the present work, I have, mainly from a 
 study of the work of G. W. HILL,* assumed for the length of 
 the seconds pendulum at the point the sine of whose latitude 
 is 1 : V3, which I shall call the mean latitude, 
 
 L! = 99.2715 
 
 With this we may compare HELMERT'S expression for the 
 length of the seconds pendulum in terms of the latitude 
 
 L = O m .990918 (1 + .005310 sin 2 tp) 
 which gives 
 
 L! = 99.2688 
 
 From these values of LI we have: 
 
 HILL. HELMERT. 
 
 Gravity at mean latitude, 979.770 979.745 
 
 Correction for centrifugal force, 2.260 2.260 
 
 Attraction of the Earth, 982.030 982.005 
 
 I also accept as the result of CLARKE'S investigation of 1880, 
 
 Equatorial radius of the Earth, 6378249 m 
 Reduction to mean latitude, 7245 
 
 Mean radius of the Earth, 6371004 
 
 From HILL'S and HELMERT'S numbers follows : 
 
 Logarithm mass of Earth expressed in centimeter- second units. 
 
 HILL. HELMERT. 
 
 20.600541. 20.600530. 
 
 * Astronomical Papers, Vol. Ill, p. 339. 
 
96, 97 PARALLAX OF THE MOON. 193 
 
 From the adopted ratio of the mass of the Moon to that of the 
 Earth: 
 
 fji = 1 : 81.45 
 
 follows 
 Logarithm of the mass of the Moon in centimeter-second unite, 
 
 18.68965. 
 Parallax of the Moon. 
 
 97. From these results the distance of the Moon and the 
 relation between the mass and distance of the sun follow in a 
 very simple way. By the formulae of elliptic motion it follows 
 that when we put 
 
 w,w', the masses of any two bodies revolving around each 
 
 other in virtue of their mutual gravitation ; 
 a, the sernimajor axis of the relative orbit, which would 
 
 be the actual distance if the motion were circular; 
 n, their mean angular motion in unit of time; 
 
 we have the relation 
 
 a 3 ri l = m + m' 
 
 This relation is rigorous and independent of the adopted units 
 of length and time, provided we define the unit of mass in the 
 way already done. It follows that if the Moon in its revolu- 
 tion around the Earth were not subject to disturbance, its mean 
 motion in one second, and its distance expressed in centimeters, 
 would be connected by the relation 
 
 Log a 3 n 2 = log m" ( 1 + /*) = 20.605841 
 
 In the theories of DELAUNAY and ADAMS the quantity &, as 
 determined by this equation, is accepted as a fundamental 
 element, and it is shown that in consequence of the perturba- 
 tions produced by the Sun the constant 77 of the Moon's hor- 
 izontal parallax is connected with a by the relation 
 
 a sin 77 = 1. 000907 p 
 
 p being the radius of the Earth corresponding to 77 
 5690 N ALM 13 
 
194 DEFINITIVE QUANTITIES. [97, 98 
 
 From the mean sidereal motion of the Moon in a Julian 
 century 
 
 1336 . 85136 revolutions 
 
 we find, for the co-logarithm of the motion in arc in one 
 second 
 
 log JL= 5.574841 
 
 and thus have for the undisturbed mean distance of the Moon 
 in centimeters 
 
 log a = 10.585174 
 
 and hence 
 
 log sin 77 = 8.219921 
 
 / // 
 
 77 = 57 2.68 
 
 Red. to sine, .16 
 
 Constant of sin n in arc, 57 2.52 
 
 Using HELMERT'S length of the seconds pendulum we 
 should have found for this constant 
 
 3422".55 
 Mass and parallax of the Sun. 
 
 98. In the case of the motion of the center of gravity of the 
 Earth and Moon around the Sun the relation of 97 becomes 
 
 a' 3 n 12 = M] -f m" (1 + ^) 
 
 MI being the mass of the Sun. Replacing a' by TT, the parallax 
 of the Sun, and p the radius of the Earth, we find for the 
 ratio M of the mass of the Sun to the sum of the masses of the 
 Earth and Moon 
 
 M 
 
 m" (I 4- IJL) sin 3 n " 
 
 log M7r 3 = 8.349674 
 
98, 99] SUN'S MASS AND PARALLAX. 195 
 
 The values of M corresponding to certain values of the mean 
 equatorial horizontal parallax of the Sun are as follows : 
 
 M 
 
 8.780 
 
 330514 
 
 8.785 
 
 329951 
 
 8.790 
 
 329388 
 
 8.795 
 
 328827 
 
 8.800 
 
 328266 
 
 Nutation and mechanical ellipticity of the Earth. 
 
 99. Begarding the mass of the Moon as known, we now 
 utilize the equations of 67 to obtain the constant of nutation 
 and the mechanical ellipticity of the Earth. The last two of 
 these equations give, for the absolute precessional constant, 
 when the Julian year is the unit of time, 
 
 p = [[5.975052] j-^ + 5310".o] ~ A 
 
 * 
 We have found, in 66, for a Julian year 
 
 p = 54".8990 
 We then have, for the mechanical ellipticity of the Earth, 
 
 ~ A = 0.0032753 
 
 We also have, from the first equation of 66, for the constant 
 of nutation for 1850 
 
 N = 9".214 
 
 For the parts of the precessional constant which arise from 
 the action of the Sun and of the Moon, respectively, we have 
 
 // 
 
 Action of the Sun 17.3919 
 
 Action of the Moon . 37.5071 
 
196 
 
 DEFINITIVE QUANTITIES. 
 
 Precession. 
 
 [100, 101 
 
 100. In order to develop the terms of the precession and 
 obliquity to higher powers of the time, I have extended their 
 computation one step backward and forward from the three 
 fundamental epochs, by extrapolation of H and L. The results 
 are as follows : 
 
 Year. 
 
 Motion of the ecliptic and equator. 
 
 log. K 
 
 1350 
 
 1.67666 
 
 168 56.13 
 
 - 46.613 
 
 2009.05 
 
 1600 
 
 1.67500 
 
 171 12.84 
 
 - 46.761 
 
 2006.92 
 
 1850 
 
 1.67340 
 
 173 29.68 
 
 - 46.838 
 
 2004.79 
 
 2100 
 
 1.67187 
 
 175 46.63 
 
 - 46.847 
 
 2002.66 
 
 2350 
 
 1.67039 
 
 178 3.50 
 
 - 46.789 
 
 2000.52 
 
 Centennial precessions for tropical centuries. 
 
 Year. 
 
 1350 
 1600 
 1850 
 2100 
 2350 
 
 In longitude 
 
 Lunisolar. 
 
 Planetary. 
 
 General. 
 
 5033.58 * 
 
 - 20.94 
 
 5012.64 
 
 5034.80 
 
 - 16.63 
 
 5018.17 
 
 5036.02 
 
 - 12.31 
 
 5023.71 
 
 5037.25 
 
 - 7.98 
 
 5029.27 
 
 5038.49 
 
 - 3.67 
 
 5034.82 
 
 In Right 
 Ascension. 
 
 4592.41 
 4599.38 
 4606.36 
 4613.35 
 4620.32 
 
 From these values we have the following general expres- 
 sions : 
 
 Annual precession in Eight Ascension; 
 Annual precession in longitude; 
 Centennial precession in longitude; 
 Total precession from 1850; 
 
 46.0636 + 0.0279 T 
 50.2371 + 0.0222 T 
 5023.71 +2.218 T 
 5023.71 T + 1.109 T 2 
 
 Mean obliquity of the ecliptic. 
 
 101. The expression for the mean obliquity when T is counted 
 from 1900 is 
 
 e = 23 27' 8".26 - 46".845T - 0".0059 T 2 + 0".00181 T 3 
 
101, 102] PRECESSION. 197 
 
 Tables of the mean obliquity at different epochs. 
 
 Year. 
 
 Obliquity. 
 
 Year. 
 
 Obliquity. 
 
 
 / // 
 
 
 o / // 
 
 1600 
 
 23 29 28.69 
 
 -2500 
 
 23 58 44.00 
 
 1650 
 
 29 5.31 
 
 -2000 
 
 55 38.99 
 
 1700 
 
 28 41.91 
 
 - J500 
 
 52 23.10 
 
 1750 
 
 28 18.51 
 
 -1000 
 
 48 57.70 
 
 1800 
 
 27 55.10 
 
 - 500 
 
 45 24.14 
 
 1850 
 
 27 31.68 
 
 
 
 41 43.78 
 
 1900 
 
 27 8.26 
 
 + 500 
 
 37 57.97 
 
 1950 
 
 26 44.84 
 
 1000 
 
 34 8.07 
 
 2000 
 
 26 21.41 
 
 1500 
 
 30 15.43 
 
 2050 
 
 25 57.98 
 
 2000 
 
 26 21.41 
 
 2100 
 
 23 25 34.56 
 
 2500 
 
 23 22 27.37 
 
 Relative positions of the equator and ecliptic at different dates. 
 
 102. The motions expressed iii the preceding tables are, for 
 the most part, purely instantaneous ones, referred to the planes 
 of the ecliptic and equator of each separate epoch. For the 
 reduction of the places of the fixed stars from one epoch to 
 another, it is necessary to know the relative position of the 
 planes of the equator or ecliptic at the two epochs. We shall 
 therefore derive the fundamental quantities which express 
 the position of the equator and the ecliptic at any one epoch 
 relatively to their positions at a fundamental epoch taken at 
 pleasure. The latter we shall call zero position. Then, the 
 zero equator and ecliptic are those of the fundamental epoch; 
 the equator and ecliptic simply those of any other varying 
 epoch. So far as convenient, and as conducive to ease in 
 comparing our results with former ones, we shall use the nota- 
 tion of BESSEL. 
 
 To derive the equations for the motions, let us consider the 
 following four points of the celestial sphere: 
 
 E , the pole of the zero ecliptic. 
 
 E, the pole of the actual ecliptic. 
 
 P , the pole of the zero equator. 
 
 P, the pole of the actual equator. 
 
198 DEFINITIVE QUANTITIES. |102 
 
 We put, 
 
 f T = PE , the obliquity of the equator to the zero ecliptic; 
 k = EE , the inclination of the two ecliptics; 
 
 770, the longitude of the node of the ecliptic on the zero 
 
 ecliptic, measured from the zero equinox of the date; 
 
 771, the longitude of the same node, measured from the actual 
 
 equinox; 
 
 A, the arc of the equator intercepted between the two eclip- 
 tics, or the planetary precession on the equator ; 
 
 if? a the total, lunisolar precession on the zero ecliptic from 
 the zero epoch to the actual epoch ; 
 
 w, the rate of motion of the pole of the equator; 
 
 T, the time, expressed in units of 250 years from the zero 
 epoch to any other epoch. 
 
 The position of the variable point E is denned by the quan- 
 tities k and J7 or 77i, which are themselves to be determined 
 through the values of x and L of 100. 
 
 The position of the variable point P is determined by the 
 condition that its motion is constantly at right angles to the 
 arc EP, and its velocity measured on the arc of a great circle 
 is given by the equation 
 
 ds 
 
 = n = P sin s cos s (a) 
 
 ci t 
 
 The positions of the equator and equinox relative to the 
 zero equator and ecliptic are then determined by the quanti- 
 ties 1, ij> and A. The spherical triangle P E E gives the follow- 
 ing equations: 
 
 sin A, sin 77! sin 7I 
 
 sin k " sin e i sin e 
 
 During a period of several centuries the quantities k and A are 
 so small that no distinction is necessary between them and 
 their sines. We may therefore put 
 
 A = k sin 77 t cosec 1 = k sin 77 cosec e (b) 
 
 We also have, from the law of motion of the pole of the 
 
 equator, 
 
 D t t = n sin A 
 
 D t fy = n cos A cosec e\ 
 
102] 
 
 MOTION OF THE EQUATOR. 
 
 199 
 
 As the value of 81 does not change by 0".6 from one epoch to 
 another, we may, without appreciable error, use f for ^ in the 
 formulae (b) and (c). To use these equations, we first obtain k 
 and 77! from the secular motion of the ecliptic, while n is com- 
 puted for any epoch from the formula (a). We then easily 
 develop the values of s-i and ip in powers of the time by the 
 equations (c). The values of n have no reference to any 
 special coordinates. From the table ot 100 it will be seen that 
 we may put 
 
 n = 2004".79 - 2".13 r' 
 
 r' being counted from 1850. 
 
 To find the value of III in each case, we remark that the 
 instantaneous values of L given in 100 show that the instan- 
 taneous node, or intersections of two consecutive ecliptics, 
 moves with so near an approach to uniformity that we may 
 take for the actual node between the ecliptics of any two 
 epochs TI and r 2 the mean of the instantaneous nodes for those 
 two epochs. For example, let it be required to find the value 
 of 77j for the node of the ecliptic of 2100 on that of 1850. We 
 have 
 
 For 2100 
 
 For 1850, referred to eq. of 2100 
 Concluded value of 77i ... 
 
 L = 175 46.63 
 L = 176 59.13 
 77! = 176 22.9 
 
 As the basis of our work we have computed the required 
 quantities for the zero ecliptics of 1600, 1850, and 2100, 
 respectively. The values of k and 77i for the ecliptics of two 
 hundred and fifty years before and after these epochs are as 
 follows : 
 
 Zero epoch. 
 
 1600 
 1850 
 
 2IOO 
 
 250 Y 
 
 + 2 5 OY 
 
 k 
 
 n, 
 
 k 
 
 n, 
 
 ff 
 -118.48 
 118.07 
 -117.64 
 
 / 
 
 1 68 20.0 
 170 36.7 
 172 53-4 
 
 // 
 + 118.07 
 + 117.64 
 
 + 117-23 
 
 / 
 
 174 5-9 
 
 176 22. 9 
 178 39.9 
 
200 DEFINITIVE QUANTITIES. [102 
 
 Changing the. unit of time to two hundred and fifty years, 
 the equations (a) (b) and (c) give the following values of the 
 derivatives of fi and : 
 
 Zero-epoch. 250 Y +250Y 250 Y +250Y 
 
 1600 _ 1.4636 + 0.7400 12600.33 12573.65 
 1850 -1.1768 +0.4527 12603.44 12576.65 
 2100 -0.8898 +0.1665 12606.57 12579.71 
 
 At the respective epochs T> T \ vanishes, and Dr# has the 
 values of the luuisolar precession in longitude ( 100). 
 Developing in powers of r we have- the following results: 
 
 Zero-epoch. o / // // // 
 
 1600; e l = 23 29 28.69-+ 0.5509 r 2 - 0.1206 r 3 
 1850; ei = 23 27 31.68 + 0.4074 - 0.1207 
 2100; t = 23 25 34.56 + 0.2641 - 0.1206 
 
 1600; # = 12587.00 T - 6.67 r 2 
 1850; # = 12590.05 -6.70 
 2100; # = 12593.14 -6.72 
 
 // // 
 
 1600; A = 45.28 r - 14.83 T* 
 1850; A = 33.52 - 14.86 
 2100; A = 21.75 -14.88 
 
 These values of Si and # completely fix the position of the 
 equator at the time T relative to the zero ecliptic and equinox. 
 For the reduction of coordinates from one epoch to another 
 we must express the position of the equator at the time r. We 
 consider the triangle PE P , of which the sides and opposite 
 angles are designated 
 
 Sides, fi 
 
 Opposite angles, 90 - C 90 , # 
 
 If, in the Gaussian relations between the parts of this triangle, 
 we put 
 
 sin A (f, Q } = A (fi e) = A z/ 
 
102] MOTION OF THE EQUATOR. 201 
 
 and regard the cosine of this angle as unity, we have 
 
 tan J (C + Ci) = cos J (e t + e ) tan ip 
 
 If we develop the differences between the tangent and the 
 arc we find from these equations 
 
 r + c, = ^ cos A (fi + f ) (1 - 
 
 where we put z for the approximate value of C Ci 
 
 For the inclination 6 of the mean equator of the epoch r to 
 the zero equator, we have the equation 
 
 sin 9 = 
 
 cos 
 
 and then, by developing in powers of 6 and ^, we find 
 
 = ip sin f (1 + 4 C 2 ) (1 i ^ 2 cos 2 f ) 
 We thus find 
 
 Zero-epoch. // // // 
 
 1600; C + Ci = 11543.79 T - 6.12 r 2 + 0.57 r 3 
 1850; 11549.44 - 6.L4 +0.57 
 
 2100; 11555.12 -6.16 +0.58 
 
 1600; C - Ci = 45.29 r - 9.92 r 2 
 1850; 33.53 -9.93 
 
 2100; 21.76 - 9.94 
 
 1600; e = 5017.30 T - 2.66 r 2 - 0.64 r 3 
 1850; 5011.97 - 2.67 - 0.64 
 
 21CO; 5006.64 -2.67 -0.65 
 
 To show the significance of the preceding quantities, con 
 sider once more the spherical quadrangle P E EP. Let these 
 
202 DEFINITIVE QUANTITIES. [102 
 
 letters represent the positions of the poles on the celestial 
 sphere at any two epochs. In this quadrangle we shall have 
 
 Angle E P E = 90 - Ci 
 
 Angle E P P = 90 - C + A 
 SideP P = # 
 
 Let S be the position of a star on the celestial sphere. Its 
 polar distances at the two epochs will be P S and P S and its 
 Eight Ascensions will be determined by the angles P and P 
 of the triangle S P P. 
 
 Thus, if the Eight Ascension and Declination of S are given 
 for one epoch, we can find it for the other epoch by the solu- 
 tion of the triangle S P P when we have given the values of 
 the quantities 0, Ci, and -f A. 
 
 To find the values of these quantities from the preceding 
 formula, let T be the zero-epoch, expressed in calendar years, 
 and let -c be the interval between the two epochs, taken posi- 
 tively when the zero-epoch is the earlier one, and negatively 
 when it is the later one. We interpolate the coefficients of r 
 and its powers from the preceding formula to the epoch T. 
 Then by substituting the value of r in the formula we shall 
 have the values of the required quantities, and hence the data 
 for reducing the position of S from one epoch to the other. 
 
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