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Les diagrammes suivants illustrent la mAthode. errata to I pelure, 3n d n 32X 1 2 3 1 2 3 4 S 6 1, THE ELEMENTS OF THE FOUR INISTER PLANETS AND THE FUNDAMENTAL CONSTANTS OF ASTRONOMY BY fV3 S - ij"^ SIMON NEWCOMB Supplement to tbe Americau Ephemeris and Nautical Almanac for 1897 ■-* ♦ »•■ WASHINGTON GOVBRNMENT 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 iin 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 ditticulties which astronomers of the future will meet in utilizing the work of our time. On taking charge of the work of prejjaring 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 v.alues 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 publisL in advance the present brief summary of the work. Ill IV PREPAOE. 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 rt'ii(;hed 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 ofHce not especially organized for the work, will afford a sufticient apology for any defects that may be noticed. Nautical Almanac Office, U. S. Naval Observatory, January 7, 1895. CONTENTS. CHAPTEK I.— GENKIIAL OUTLINK OK THE WOUK OK CUMPAHINU THE OHSEKVATIOXS WITH THEOHY, J 1. Reduction to the standard system of Right Ascensions and Declinations $ 2. Observations nsed $ 3. Seniidiaineters of Mercury and Venus.— Table for defective illumination of Mercury in Right Ascension $ 4. Tabular places from Lkvekrieu's tables.— Reduction for masses used by Lk VEimiKii ^ 5. Coniparisoim of observations and tables $ 6. Equations of condition. — Method of formation $ 7. Method of determining the secular variations and the masses of Venus and Mercury independently $ 8. Method of introducing the results of observations on transits of Venus and Mercury ; separate solutions, A from meridian observations without transits; Ji, including both meridian observations and transits 1 1 6 8 8 10 13 Chaptek II.— Discussion and results ok 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 $ 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 $ 11. Formation of equations froui observed Right Ascensions of Sun $ 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 $ 13. Mass of Venus, derived from observations of the Sun's Right Ascension $ 14. Discussion of corrections to the Right Ascensions of the Sun relative to that of the stars V 15 16 17 20 24 25 VI CONTENTS. Page. $ 15. DiHcussion of corroctious to the eccentricity and perihelion of the Kiirth'a orltit 27 ^ 10. KeHiiltH of ohservod DeclinationH of tlio Sun.— Exhihit of individual correctionH to t\w abNoluto longitude and the obli(|uity of tlio ecliptic at the ditlerent obaervutoriea during different pcrioda 29 ^ 17. DiscuHBion 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. Ktlcctof refraction on the obli(|uity; special investigation of tlie secular change of obli(iuity as derived iVom observa- tiouH of the Sun 35 ^ 20. Concluded results for the obliciuity, and its socniar 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 obsorvatio^js of the Huu alone 41 CiiAPTKH III, — Results of obsehvatio.ns ok thk plankts MEiicrHY, Vknus, and Maks. lit J 22. Elements adopted for correction 43 $ 23. Introduction of the corrections to the masses of Venus and M iry 45 $ 24. Ic jtion of the errors of absolute Right Ascension and jiations 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 qnantities of the equations. — Factors for changing corrections of the unlinown quantities into corrections of the elements 55 $ 28. Table of the values of the principal coefflcients 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. Solntion 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. $ 34. Compariion «tf the resnlts derived from meridian olmerva- tioiiH of Mercury with tliose derived from trannitH over the Sun's disk $ 36. Treatment of meridian observations of Venus ^ 36. lleHults of observed transits of Venus ^ 37. liquations derived from observed transits of Venus ^ 3H. Solutions of the ei|uations from Venus $ 39. Comparison of the results of meridian observations of Venns with those of transits $ 40. Solution of the equations for Mars. — Ine(|nality of long period in the mean longitude and perihelion, indicated by observations $ 41. Reduction from the ok iiypothkskh anp dp,tekmina> TION ((K TIIK MAMSK.S l«Y Wllltll TIIK nKVIATKlNrt OK TIIK SRCn.Alt VAItlAI'IONM KIIOM TIIKIK TIIKOKKTICAI. VALIIKS MAY I»K KXPI,AINKI>. « ^ 55. ('oinpariHon of th« obstrved and thfor(>ti(». IlypothoNiH of iionHpliericity of the <>quiputoiitial surfaces of tJ:o Snii ^ Til. HypotliBHis of an intrain«Tciiriiil ring $ .58. ITypotliesiN of an extended niass of diffused matter, like that which rotlects tho /.odiacal light $ 59. Hypothesis of a ring of planets outside t1i« orhit of Mer- cury. — Kh'Uients of such a ring. — This hypothesis the only o\w which represents the observations, but too im))robable to he accepted ^ 60. Examination of the question whether tiie excess of motion of the perilielion of Mars may be due to the action of the zone of minor planets $ 61. Hypothesis that gravitation towari clie Sun is not exactly as the inverse sq-uaro of the distance $62. Degree of precision with which the theory" of the inverse S(piare is established $ 63. Determination of the masses which will best represent the observed secular variations of the eccentricities, uodes, and inclinations $ 6-t. Preliminnry adjustment of the two sets ot masses. — Result- iug valuta of the solar parallax ClIAPTKK VII. — VaI.T'ES of THE PRINCIPAL CONSTANTS WHICH DK1'KNI> UP«)N THE MOTION OF TIIK EaIMII. ^ 65. The precessional constant ^ 66. The constant of nutation, derived from observations % 67. Relations between the constants of precession and nutation and the quantities on which they depend $ 68. The mass of the Moon from the observed constant of nuta- tion $ 69. The constant of aberration $ 70. The values of this constant, derived from observations ^ 71. The lunar inequality in the P^arth's motion § 72. The solar parallax derived from the lunar inequality $ 73. Values of the solar parallax derived from measurements of Venus on the face of the Sun during the transits of l674 and 1882, with tbe heliometer and photoheliograph $ 74. The solar parallax from observed contacts during transits of Venus lOJ) 111 112 115 116 116 118 119 121 122 124 129 131 132 133 135 139 142 143 145 CONIT-NTS. 'i 7D Soliir |»iiralliix from tlir (>li.s<'ivnl niHMtant ofulHTriition jiml niiMiHiiriMl vt'lix'ity <»!' liylit ^ 7(). Solur parallax from tho parallactic in. Adopteoch 18.">0, as inferred from all the dojta of ohsorvation . ^ 89. Definitive values of the secular vari;i.tioiis ^ .!K). Secular accoleration of the mean motions ^ lU. The measure of time ^ J)2. The "onstant of aberration ^ }t3. The mass of tho Moon ^ 94. The parallactic inequality of the Moon \S i)'>. The centimoter-second system of astrorkondcal uu its ^S 9(5. Masses of tho Karth and Moon in centimoter-second units.. SS 97. Parallax of the Moon ^ 98. Mass and parallax of the Sun (J 99. Constant of nutation, and mechanical ellipticity of the Earth $ 100. Precession ^ 101. Obliqu.ty of the ecliptic ^ 102. Relative jtositions of the eqnator and the eclii»tic at ditter- eut epochs for reduction of places of stars and planets .. IX I'agtt. 117 118 154 15« l.'>9 Uil im 1«>8 173 173 17r. 178 179 182 180 188 188 189 190 190 191 193 194 195 196 196 197 ELEMENTS AND CONSTANTS. CHAPTER 1. GENERAL OUTLINE OF THE WORK OF COMPARING THE OBSERVATIONS WITH THEORY. 1. In logical order, the first step in tlie 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 Right Ascensions was that origi- nally worked out in my paper on the Right 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 Vol. 1 of the Astro- nomical Papers of the American Ephemeris. This system CGi.icides 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 C.OS. 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. Ohserrations used. 2. The following is a general statement of the observati(ms used, and the extent to which t_ ay were corrected, or re-re- duced. Oreentcich. — Dr. Auwers courteously supplied me with the •Qsults 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, Aiby's reductions. The 5690 N ALM 1 1 i* ! GENERAL OUTLINE. 12 data necessary for these observations were discussed in Prof. Saffokd's paper, Vol. ii, pt. ii, wliicli 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 Kight 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. — PiAzzi's observations of the Sun and Planets ware completely re-reduced, the zero point of his instrument being- determined from the observed Declinations. Paris. — LeVerkieb's reduction of the Paris observations from 1801 onward was made use of, applying the corroction necessary to reduce the results to the adopted standard. Kon'ujHberg. — 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 Right Ascensions and Declinations of the funda- mental stars with the stan<><> 002 4885 Earth 354936 =-000 002 8174 Mars 2680337 = '^^^^ ^^ ^^^ ^^'^ The following .ible 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 i;i the introduc- tion to the tables. In the last line of the table is shown the factor used in the present provisional theory. 8 GENERAL OUTLINE. [5 Mercury. Venus. Earth. Mars. In tables of — The Sun I 1.004 I 0.895 Mercury .. . I I 1.0026 I Venus .. . I I Mars 0-97S I Present work I 0. 8657 As in the actual work tlie 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 th« extreme dates of each group was always taken so short that it could be j^resumed that the mean of all the residuals would be the correction for the mean of all the dates. The j^eneral rule was that the interval should not e\ceed four or five days in the case of Mercury, or six or eight days in that ot* 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 meaji 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 1,597 6] EQUATIONS OF CONDITION. 9 III 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. [• the observatory wliero made, and divided into periods rarely exceeding eiglit years in length. The elements are separately derived from the observations of each i)eriod. 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 discore Sun, Mercury, and Mur», and the mass of Mercury in tlie e(iuation8 derived from obser- vations of Venus. Tlie coellicients of tlic masses, liowever, depended wholly upon tlie periodi*- perturbations. Were it quite certain that the secular variations arise wholly from the nuisses 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 throuH;h the mass itself. On this hypothesis the secular variations would not appear in the etpiations, but only the masses, liut it is well known that the periheli(m of Mercury is subject to a secular variation which can not be accounted for by any ad- missible masses of the known disturbin;^' jdanets. The same thing may well be true of the secular variations of the other elements. It is therefore necessary, in the absence of a knowu cause for such deviations, to derive the masses of the i)lauets in only at a certain point of the orbit, a fortiori w«mld it be the case for the observations made at various jmints of the orbit, since any tendency to a systematic ettect of the errors of observation W(Mild 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. Hut they do not apply to the case of the action of the eaith itself Jipon the obser\ed planet, or rice rerna. For ex- am])le, 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 lon}>' run have all values, positive and negative, for these several observations, the systematic error in question will not attect 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, detfTminations of these masses might be aft'ected 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] THANSITH OF VENUS AND MEUCUKY. 13 In tho solution of tlu> equiitions ot condition the method of least srinares Iia8 been nsed tlirou^^hont, the iirninf^enient of th«' work, the choice of (|uantitie8 to bu coiTcctc, the various unknown (piantitics anil show to wliat extent their vahies wore interdependent. Hy no otlier method of eombiinition couhl so hir};:e a number of unknown quantities have been separately determined in a way whicli would have been at all satisfactory. On the other hand, in combining- the tinal results and deci«lin{^ upon tlie values of the i'orrections to be adopte«l, the method has not always been applied, for reasons which will be deveh»ped iu Chapter IV. Introduction, of results of olm-rmtitniH on transitH of Vcnm and Mercury. 8. In the ease 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 i)oint 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, tlie 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 natisfied 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 tlie re- sults by the use of the observed transits over the sun's disk, two sei>arate 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 couside" 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 aftei wards included when approximately determined, and vice versa. In the case of solution B, in Avhich 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 liUjht Ascensions. 9. The meridian observations of the Sun have been treated on a system ditt'erent in some points from that adopted in the case of the planets. It was possible to simplify the treatment by 8upposi:ig 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 IVom an observa- tion of its Kight Ascension. An observed liiglit Ascension being compared with a tabular one, the residual gives rise to an ecpiation of C(mdition between the correction of the long- itude, A, of the obliquity, f, and of the Kight Ascension of the Sun, a\ da = cos 6 sec* 6dX — ^ tan e sin 2ad>;. Tiiis equation may be used to express the error of the longi tilde in terms of the error of tike obliquity and of the Right Ascension as follows : 6\ = sec e cos* 66a -f ^ tan f sin 2Me = s'^c f cos* 66a + 0.21 sin 2\de The elements mainly to be determined from the observations in Kight 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 6e sin 2\ whose coefficient does not amount to O'MO after 1800, and has a period of half a year, will be practically without influence 16 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 Hight Ascension by the formula where 6\ = Fda F = cos^ 6 sec e. h The residual 6^ 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 Right Ascension would give rise to an equation of condition of the form — 61" + Ve"67r" + E6e" = 6X = ¥6a We are here to regard 61" and 671" as corrections to the Kight 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 he case supposed is a function only of the longitude and 0Dli(iuity. The equation ♦or exi)ressing the observed correction in Declination in terms ot *he corrections to these two quantities is /J6 = sin a6e -|- cos a sin £6\ 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 mimm llj FORMATION OF EQUATIONS IN RIGHT A£?CENSIONS. 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 ca^se, 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, Jd„ + cos a sin eSX + sin (xde = JfS /IS being the excess of observed over tabular declination, and zl6„ the common error of all the measured declinations of any one series. ForhMtioii of the equations from Rif/ht Ascensions. 11. The method of treating the observed Kight Ascensions of the Sun was suggested by the fact that they are peculiarly liable to systematic and personal errors; tlie former likely to change with the seasons, au«l to be different for ditterent in- struments ; and the latter to continue through the work of one observer. It is now well understood that the observed Right Ascensions of the mean of the Sun's two limbs relative to the fixed stars are aifected 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 tiie relative mean of the Kight 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 Right Ascensions of the Sun would berattected 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 4- y' sin 2g 5690 N ALM- 18 OBSERVATIONS OF THE SUN. [11 The introduction of the coeiBcieuts x' 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 Right Ascensions was the mass of Venus. The question now arose whether, by a uniform series of cbser- vations, extending through some definite period, the correc- tions to the eccentricity and perigee and the coeflBcients x' and y' could br 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 Venv.s could be determined irrespective of all systematic errors repeating themselves with the season, provided that the observc>,tion8 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 -\-y sin g + x' 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 unknown ones of the equations of condition were as follows: //', the factor of correction of the mass of Venus ; a?, 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 Bight Ascension of the Sun relative to the Stars. The coefficient of c was supposed unity throughout. The reduction of the residual in Bight 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. 4.^ v.^ Ill FORMATION OF EQUATIONS IN BIGHT ASCENSION. 10 Separate tables were constructed for 1802 and ISfiO, 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 expressinfi errors of the Sun's Right Ascension in terms of errors of the elements of the EartWs orbit. Coefficients of — da dl dl da x=o.26e y=o.2e6n x' y Jan. I 1.09 0. 91 + 0.1 — 10. + 0. I -f-io. II 1.07 0-93 1.8 9.8 ^•5 9-4 21 1.04 0.96 3-4 9.4 6.5 7.6 31---- 1. 01 0.98 S-o 8.7 8.7 50 Feb. lO 0.98 1. 01 3.4 7.7 9.8 + 1.8 20 0.96 1.04 -:- 7-6 -6.S + 9-9 — 1.6 Mar. 2 0.94 1.06 8.6 5- 1 8.7 4.9 12 0. 92 1.08 9.4 3-5 6.6 7-S 22 0.92 1.08 9.8 1.9 3-7 9-3 Apr. I 0.93 1.07 10. — 0. I + 0.3 10. II 0.94 1.05 + 9-9 + 1.6 — 31 - 9-S 21 0. 96 1.03 9.5 3-2 6. I 7-9 May I 0.99 1. 01 8.8 4.8 8.4 5-4 II 1.02 0.98 7.8 6.2 9-7 — 2. 2 21 1.05 0-95 6.6 7-S 9.9 1. 2 31--- 1.07 0.93 + 5-3 + 8.S -8.9 - 4-S June lO 1.09 0.91 3-7 9-3 6.9 7-2 20 1. 10 0.91 2. I 9.8 4.1 9.1 so- 1.09 0.91 -1-0.4 10. - 0.7 10. July lo 1.08 0.93 — 1-3 9.9 + 2.7 9.6 20 I. OS 0.9s - 30 + 9-5 + 5-8 — 8.2 30- — 1.03 0.97 4.6 8.9 8.2 5-7 Aug, 9-... 1. 00 1. 00 6.1 8.0 9.6 + 2.7 19.-.. 0.97 1.03 7-3 6.8 10. - 0.8 29.... 0-95 I. OS 8.4 5-4 9.1 4.1 Sept. 8.... 0.93 1.07 - 9.2 + 3-9 + 7-2 -6.9 18.... 0. 92 1.08 9-7 2-3 4-S 8.9 28.... 0.92 1.08 10. -f 0.6 + 1.2 9-9 Oct. 8 0.93 1.07 9.9 — I. I — 2.2 9-7 18 0.9S I. OS 9.6 2.8 5.4 8.4 28.... 0.97 !.02 - 9.0 — 4.4 - 7-9 - 6.1 Nov. 7 I. CO 0.99 8.1 S-9 9-5 - 3-1 17 1.03 0.96 7.0 7.2 10. + 0.3 27 1.06 0.94 5.6 8-3 9-3 3-7 Dec. 7 1.08 0.92 4-1 9.1 7-S 6.6 17 1.09 0.91 - 2-S - 9-7 - 4.9 -f 8.7 27 1.09 0.91 - 0.8 — 10.0 — 1.6 + 99 20 OBSEEVATIONS OF THE SUN. [12 iH 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 nuich 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' 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 bei given. This is shown by the general consistency of the corrections to the eccentricity and i)erigee given by the work at the same or diftercut observatories dur- ing dift'erent periods. In marked contrast to this is the discordance among values of the correction c to the relative Right Ascensions of the Sun and Stars. This quantity it is that is affected by personal error and possibly by the efiect 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. 121 SOLUTION OF THE EQi:ATIONS. 21 The third column shows the value of /a', 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 pi' 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 yu', 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 Right Ascension of the Sun, relative to the assumed Ilight 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. Respecting the weights ultimately assigned to these quanti- ties, and to c, 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 uiust liave beeu att'ected by some extraordiutiry cause of error, unless some mistake bus beeu made hi interpreting or treating tbe observations. Tbe Oxford values of c are unusually discordant. Tbe pre- sumption tbat tbis discordance arises mainly from tbe special personal e9 +.031 8 +0.35 3 +0.33 —0. 10 4 i87i-'74 + 23 +.02 1 8 -fo. 12 3 +0.24 4-0.05 4 1875-78 + 27 —.008 8 — 0. 12 3 4-0.26 4-0.06 4 i879-'82 -f 31 +.017 8 —0.05 3 4-0. 21 4-0.14 4 i883-'88 + 36 -f. 001 «3 —0. 20 3 4-0.18 +0.07 4 i889-'92 + 4« -.025 8 -0.44 2 4-0.24 ■ 4-0. II 3 12J SOLUTION OF THE EQUATIONS. 23 KesultH of ohfiervations of the (S'mm'» h'iffht Anfemion — Continued. PARIS. Years. T -.46 -.025 Ji' 14 t" to •_ 1 0.08 u> 1801-07 II -1.78 i8o8-'is — .^8 +-o«5 17 —0.65 0.5 - 0. 01 f 0. 12 I l8l6-'22 — 3> — . 050 14 +0. 18 0.5 -0. 13 +0. 32 i823-'29 -.24 —.050 10 -f-o. 01 o.s -0.31 —0. 02 1837-44 —.09 -.034 19 +0.33 I —0. 04 +0. 10 "■5 i845-'52 +.01 +.009 15 -f 0. 10 I +0.04 -f 0. 10 1-5 >85.5-'59 + .06 +.014 15 +0.66 I — 0. 04 +0.32 2 1 860- '6s +■13 +■003 10 +0.38 I 4-0.07 f 0. 26 2 i866-'7o -t-.i8 .000 7 +2.29 0.3 +o^ »3 4-0. 40 2 i87i-'79 + •25 +•048 II —0. 26 I — 0.06 4-0. 22 2 1 880-' 89 + ■35 -f-.002 14 4-0.44 I +0.24 -4-0.03 2 PALERMO. i79i-'96 — 56 -.079 —0.07 II - -0. 06 —0.85 i797-'oi — 5' -.116 -2.33 —0. 29 -0. 28 i8o2-'os -.46 —.001 -3^" —0.05 ^0.76 i8o6-'i2 —.41 + ■243 -hs. 92 -1. 17 + •■55 cami5Rii)c;k. 1 828-' 34 —.21 + •007 16 -0. 13 2 -fo. 08 4-0. 12 4 1835-40 — . 12 -•033 14 0. 18 2 -fO.06 —0. 06 4 i8|2-'47 -OS — . 026 9 — 0. 21 2 -fo.o8 —0. 12 4 i85o-'s8 +■04 — . 024 20 —0. U 2 4-0.17 —0.04 4 WASHINGTON. i846-'52 i86i-'6s 1 866-' 73 i874-'8i i882-'9i — .01 -.038 5 — o!'85 2 4-0. 20 1 0.00 +■13 -■ 038 8 -0.53 4 -j-o 01 0.00 -^. 20 — . 004 J3 —o. 22 4 +0.18 —0. 03 +.28 -■033 12 —0.45 4 4-0. 07 —0. i6 ,+.37 —.002 17 —0.79 4 +0. 07 -0.07 3 S 6 5 5 KONIGSBERG. i8i6-'23 —•30 4-.C02 »3 4-0! '30 I 4-o'.'o7 —0.28 3 i824-'^o -■23 — .006 12 4-0. 02 I ~o. 16 4-0. II 3 1 83 1 -'38 — «S — .021 «S 4-0.23 I — 0. 12 +0.03 3 i839-'4S —.08 — .021 12 4-0.77 I + 0.08 -f 0. 20 3 24 0B8EUVAT10NS OF THE SIN. [18 Results of ohHerrationtt of the Sun'ti Ri; + 2"A2 8.1 -4-10.;3i) =+ 3".20; +3''.42 jp=-\- (V'.IO; o".m y=+ 0".23; + 0".33 In the case of the ecceutricity the gv°neral accordance is quite p itisftictory, and for the perigee it is much better than in the case f, the relative liight Ascension. Kestilts of observed ileclinationa of the Sun. 10. The Sun's absolute longitude can be found l" W 6e 1 w A6 &'t \V l842-'4S l846-'49 i86i-'6'; i866-'7o -.06 —.02 + •13 + .18 // +0.82 —0. 10 -'>-53 +0.27 2 2 2 2 // -0.35 —0.48 —0.48 -0.31 // — 0. 01 -fo.07 —0.30 -0.38 // —0. 35 —0.48 —0.48 —0.31 I I I I DORPAT. l823-'28 i829-'32 •833-'38 -.24 -.19 —.14 +0.99 +0.99 + 1.00 2 2 2 -1.26 —0. 76 -0.63 +0-59 +1-34 - +1-34 — I. 41 —0.91 —0.78 1 I I CAPE OF GOOD HOPE. i884-'87 i888-'90 +•36 +•39 —0.51 —0.84 4 +0. 05 4 +0.09 +0. II +0.19 — 0. 07 — 0. 21 2 2 STRASBURG. i884-'88 -!-• 36 -0.57 4 —0.05 —0.77 +0. 12 2 leidi:n. » 1 86/ -'69 i87o-'76 +■17 +•23 +0.14 —0. 23 4 4 — 0. 01 —0.06 +0.27 — 0. 04 — 0. 24 —0. 29 2 2 Correction to the Sun''s absolute longitude 17. So far as mere instrumental measurement is concerned, the correction 6 e should be determined with greater precision than 61" in the ratio 5:2, because the errors in declination have to be divided by the factor sin f = 0.40, in order to form 61". Mlowing for this large increase in the source of error, the values of 61" are more accordant than those of 6€. This is what we should expect. The values of the former quantity depend mainly upon the comparison of observations made OBLIQUITY OF ECLIPTIC. 33 W !.■) I 8 I 8 I I I I I r I { I 2 2 2 2 17, 18] near the opposite e(iuiiioxes, when the sun has the sanie decli- nation, and wlien 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 rfi", as would also in any case any constant error which is a function simply of the Sun's Declination. It is tlierefore to be expected that the actual probable error of this (luantity will conform more nearly to that determined from the residuals than in the case of the other. For these reasons the value of 61" does not give rise to much di8(!ussion. The general result from all the observa- tories is, for 61", when developed in the form .v + y T. J- = -f 0".05 ,/ = - 0".J>7. Obliquity of the ecliptic. 18. The determination of the obliquity rests upon an essen- tially difterent 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 adoi)ted. The following special circumstances affecting the observations are to be taken into consideration : The IJRADLEY Greenwich results for 17i)3-'Cli, 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. Aiwers himself, nor for reduction to the Boss system of standard Declinations, llence arises the large value of J6 given by these Declinations. Consequently the value of 6e is 5690 N ALM 3 !U ] i ' i 11 ^ pi 34 OBSERVATIONS OF THE SUN. fl8 H tbat giveu immediately by the instrument, on tlie system of reduction adoj^ted by Dr. Aitwers, in which 1 have supposed that the I'ulkowa refractions were used. From 17(55 to 181G the Greenwich observations were made with the imperfect qnadrant, the Declinations of which are subjected to an error which is not constant. The neces- sary correctit)ns are derived by Safford in Vol. ii of the Astronomical Papim. The corrections are those necessary to reduce to Boss's system, and they vary with the Declination. Dence 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 J^eclinations 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 varj' somewhat Avith the Declination, and they are different also for different periods. Hence we have here a period during which the instrunu'ntal differences of Declination were cor- rected to reduce them to the standard star- system. If the standard system were subject to no further error than a constant one, common to all J)ec]inations 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 orr) + (4- 0".14 ± 0".23) T (1850) . . (b) Here the mean error is naturally smaller than in the case of tie 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 othci'. The result of combining all Greenwich observa- tions, with the weights as assigned, is — (Ve = - O'Ml + 0".r)0 T (0) In this combination I have used the weak results of JNIaske- LYNE, with the small weights assigned, although they d 4>end wholly upon the standard declinations of stars. In view of the discordance between Bradley's two results, tliis 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 — (Jf =+0".01-0".3GT 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 redetermiuation of the zero points of the instrument from fundamental stars. 19, 20 1 DISCUSSION OF RESULTS OF OBLIQUITY. 39 If we omit the (ireenwich, Paris, and Palermo results, and combine all the others into a sinj^le set of eqnations of condi- tion, we have the eijuationa and resnlts: 3fi.<)j+0.2«J»/= - 14".37 U.LM) + 1.S8 = + 1"-01 y=+ 0".r)9 Here .»• is the valne of rf'f for 18<»0, and y its centenniiil varia- tion. Transferring; the epoch to .1850, as nsual, the result is — 6'f= - o".4.") + (>"..v.rr e?) No reliable mean error can be computed, owing to systematic errors. In view of these, one mode of treatment would be to form eijuations of coiuTition in whic h a possible systematic error at each observatory wouhl appear as one of the unknown quantities. By this process we should j^et the same result for the secular variation as if we made an independent determi- nation from the work of eaciii ol)8ervatory. At most of the observatories the period throuj^h which the observations are made, with one instrument an9 X = - ()".40 y=+ 0".(M («) Concluded results for the oblir'ty. 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 an«l transit circles, ia markedly greater 40 OBSERVATIONS OF THE HUN. (20 h tban that given by otlna- iiioderii obMervatioiis. This may be most ])hiUHibly attributed to the atinusplieric coiKlitioiis within the observing room. (2) The niinnteness of tlie change of tlje correction given by these instrnnionts during nearly eighty years. To this cinnnnstance is due the snuiUness of the centenuial variation, 0".itO, found from tlie totality of the (heenwich observations. A comparison of Bkadley with the mean of the T. C. results only would have given a change of 0'M)7 in 117 years, or a centennial change of about 0".8(). The long periotl, uniformity of plan, and systematic devia- tion of the modern (irecuwich 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 =k 0".23 (2) All other results + 0".(i5 To the latter no reliable mean error can be assigned. To jutlge its reliability we may compare it with the results {a), (c), and (d) — Greenwich T. C, alone, + 0".21 ± ()".46 Greenwich obseivations in general, 4- 0".50 Miscellaneous modern observations, + 0".51) We may, it would seem, fairly give double weight to the result (2), thus obtaining, as the detiuite result from observa- tions of the Sun alone: Correction to Leverrier's centenuial variation of the obliq- uity of the ecliptic ( — ■17".594) ■f 0".48 ± 0".30 the mean error being an estimate from the general discordance of the data. For the coustaut jiart of the correction I take — (y*(1850) = -0".30 21 Sl'MMAEY OP KKSULTS. 41 tSumiiiartf and vAtmpariHon of rcHHltn. 21. From what procodcs wo liiivo tlio lollowiii{i iim tin* values of the unknown quantitien, and of tiieir socular variations, as given by ob-servatiouH of the Sun alone. Value for 1S50. fir" = H- O'MO I (V'.OM c>'{S7T"-\.a) = 0".0() 4-. 0".(>7 8\"-\.a = - 0".(>2 61" = -f 0".(»5 4r 0".12 rff = - 0".aO 4: 0".15 'M)7 1: 0".23 4- 0".48 -Jtz 0".30 + ()".34 No estininte of the probable errors of these (luantities would be useful which did not take account of the Hysteniatic dif- ferences between the results of ditterent observatories. We have therefore formed the mean outstanding residual correc- tions given by the several observatories, us shown in the tables which foH(»w. 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 61" 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 df, 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. Jt is noteworthy that the results are somewhat more accordant when this correction is omitted and jiurely instrumental errors are used for the obli(iuity. The mean errors giveu 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 6e for the reason that they depend on declinations referred to star 11= 42 OBSFiiVATlONS OF THE SUN. [21 places wbi<'li may be att'ected by differences in different Right Ascensions. They are, liowever, retained in the table to show the ivinounts of outstanding discordance. Outstandirnj mean residual corrections to qnantities depending on the tSun '« Bight A scension. 8e" e'Sn" 2w Greenwich + 0".09 - 0".03 o4.5 Paris - 0".09 + o".n 17 (.'anibridge + 0".02 0".00 16 \\'asliington - 0".05 -0".1L' 24 Kbnigsberg - 0".08 + 0".()8 12 Oxford + 0".0•, the radii vectores of the Earth and planet; 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 Right Asceusion of the ascending notle of the orbit on this plane; . U, the argument of helioc entric declination of the planet or its angular heliocentric distance from the node on the etjuator; rf, (J, the geocentric Right Ascension atul Decliimtiou of the planet, e, the obliquity of the ecliptic; We shall then have — .r =/(>'. R. L. J. X. U., K) (a) lAu- the correction to the tabnlar Right Ascension arising from symbolic corrections to t!' >se seven quantities, we have the e(iuation — m 43 ( . 44 MER(!URY, VENUS, AND MABS. [22 I with a similar e(|uation for the declinatiou, formed from this by writing 6 for a. The relations by which these two equations are derived, as well as the expr«»s8iou8 for the difterential coefticieuts they contain, are given very fully in A. P., Vol. II, Part I, to which reference may be made. The corrections dN and 6U are not, however, the most convenient ones to choose. It will be found in the paper faiuded 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 e(|uened 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, ai 1 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. Could all the observations be directly referred to a visible eijuinox and e<|uator, the corrections above enumerated Mould have been the only ones which it was necessary to include in the equaticis of condition. But, as a nmtter of fact, the observations were all referred to an assumed system of Hight Ascensions and Declinations of standard stars — my own system in Hight Ascension and Boss's in Declination. We must therefore introduce two additional unknowns into the equations, which 1 have repn^sented in the following way: a, the common error of tlie adopted Right Ascensions. 6, the common error of Boss's 1 )eclinations. 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 coeflScient being unity in each case. 24] COREECTION OF EQUINOX AND EQUATOR. 47 That the value of 6 found iu 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 tlu^ fact that these absolute Declinations depend upon assumed constants and laws of refraction, which are necessarily aft'ected witli 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 tlie Declina- tions deiived from the work of different observatories, it can not be assumed that these Declinations are free from sys- tematic error. Now, in one circle ot Decimation, say the e(iuator, 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 diftei very greatly from the error at the equator. Even if the difference should be considerable the various values of the error of the different Declinations must have a certain mean value, so that in the case of each particular star, or each region of the lieavens, we may conceive the actual error to be divided into two parts — one the mean value in (juestiou, and the other the deviation from this mean. The latter is probably smaller chixn the former, and in any case can not very well be determined from observations of the l)lanets. 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 this equations of condition. The value of x', y', . . into the original equations of condition, using for t the value r„ which pertains to the mean epoch of the period. Our equation of condition will thus become — a^o + byo + + ar^x' + bT,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 Xo', [aa],a?o -j- [ab], j/o + . . + r, [aa],x' + r, [ab],y' + . . = [an], (4) (1) 25] INTEODUOTION OF SECULAR VARIATIONS. 81 Partial normal equation iu x' j T, [aixljdo + T, [ablj/o + . . 4- r,» [(m],jr' + t,^ [ab],y' + . . = r, [an], (5) We t'oiicliide that the piirtial 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 j?o, yo, . . . is formed from that in x, y, etc., by adjoining to the first member of the equation the member itself multiplied by r and then changing X, y, . . .to Xo, Xu', and, in the products by r, changing ^, y, . . . into x', y', . . . (2) The partial normal equation in x', y', . . . is formed from the partial equation in Xf>, y^, . . . by multiplying all the terms throughout by the factxjr t. The final or complete normal equations in all the unknown quantities being formed by the addition of the partial normals, the formula} for the coefficients are as follow : (2) (3) For the final equation in Xq [aa] = [««],+ [aa]i + [ab] = [a6J, 4- [ab]i + • • . . . . [aa]' = n [fl«J, + T2 [aa]2 + [an] = [an]i+ [«»]« + For the final equation in x' [aa]" = Ti^aa\i + Ti'[aa\2 + [ab]" = ji^[ab]i+T2^ab]2 + • • • • • • « • • t t ■ • t • + T„ [aa]„ • • t + [«w]„ 4- T„* [««]„ [an]" = r, [an]i + tj [an]^ + . . . + r„ [an],, (6) (7) The final equations for all the unknown quantities will then be of the form [aa] Xo + [ab] y„ -f . . + [aa]' x' + , . . = [an] • • • • • • • • • • " • • • • [aa]'xo+[abyyo+ . . . +[aa]"x'+ . . . =[an]" (8) M 52 MEUCUUY, VENUS, AND MARS. [25, 26 Tlio epoch from which we count the time, r, is arbitrary. An obvious sulvantago will be {jfaiued in countiuf^ it from the mid-epocli of all the observations. Then we shall have, by putting K'l, Wt, etc., for the sum of the wei{,'lit8 for the different periods: M'l Ti + Wi Ti + + '»» r„ = (9) If the observations are then equally distributed around the orbits of the planet and of the Earth it may be expected that the coetlicients [aa\', [ah]' (10) will all nearly or quite vanish. Practically we may expect that as observations are continued through successive rev^olutions the ratios of these to the other coetlicients will approach zero as a limit. We may then divide the normal equations into two sets, one containing the ((uantities .<■„, ^d, etc., and the other .r', y', etc. The coetlicients (10) being small, the two sets of normals will be nearly independent, and we may omit the terms (10) in the flrst 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 synunetry will be gained by so choosing it that the mean value of T^ shall not differ greatly from unity. It was found that twenty-tive years was a sutllciently near approximation to be adopted for all three planets. Dates and weights for epocha and periods. 20. As want of space m.akes 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 I i -i « 26] DATES AND WEIGHTS FOR EPOCHS AND PERIODS. 53 of this ineaii period from the mid-ei»och of the ohservationis, which is takeu us follows: For Mercury, 18« iRCURY, VENUS, AND MA US. [26 WeiijhtH, epochH, tnul periods of partial iiornud cquntiom. MERCURY. Period. Right Ascension. Declination. V. Mean year. 1766.60 1784. 22 1799.81 ^ VVt (units of 25;/.) " ' V. i Mean year. r (unitsr)r25.t.) Wt. I 2 -3- 93(io -3- 2312 —2.6076 3-4 18.8 26.1 1765.50 1782.99 —3. 98o<:> —3. 2804 0. 2 4.9 3i 1796.4a 1802.37 1809. 18 1824. 83 -2. 7432 —2.5052 -2. 2328 —1.6068 5.0 'x 39- 9 ■?,! 52. 8 ^a^ 74- I ^^ 1809. 53 -2. 21S8 18.9 i 1818.79 1825.80 i«?S-S6 -1.8484 — I. 5'>8o — I. 1776 0.9 34.5 75. i i 6 6, V833;¥4 1838.26 •843-97 1855.92 1862. 79 1867. 18 1872.64 1877.05 18S2. 17 1886.1:9 l8oi;. 70 — 1 . 2464 — 1 . 0696 —0.841..: -0. 3632 —0. 0884 +0. 0872 +0. 3056 1 0.4820 +0. 6868 -1^0.8516 +0. 9880 ._. 75- ^ ^^ 6, 141.5 281.5 201. 5 189.5 294. : 214.0 204. 5 «7«-5 338. 176.0 7 8 9i 9.' lo, lo. II, lij lij 1 i 1843. 74 1855.90 1863. 10 1867. 12 1872.62 1877.12 1882.24 1886.29 1889. 82 —0. 8504 —0. 3640 —0. 0760 +0. 0848 4-0. 3048 fo. 4848 1 0. 6896 +0.8516 -j-o. 9928 98.8 83.3 99.8 186.0 129.8 129.8 108.2 199.8 109.5 i i i t\ i i i i VKNUS. I '7SS-83 —4. 2868 "•3 * 1759. 69 -4. 1324 7.0 2 1767.92 —3.8032 19.7 * 1770. 18 -3- 7 '28 10.0 3 1781.06 -3- 2776 3-7 * 1793.25 — 2. 7900 •3-5 4 1792.47 —2.8212 »2. 3 * 1S06.73 —2. 2508 65-5 5 1802. 64 -2.4144 23- 3 * '815- 59 — 1.8964 67-5 6 1810.31 —2. 1076 34-0 + •823.75 -1.5700 197.0 7 1816.88 -1.8448 42.7 * 1836.02 — 1.0792 762.0 8 '825.55 —I. 4980 141. i 1844.08 -0. 7568 650.0 9 '835.3I — !. 1076 339-3 * 1854. 24 —0. 3504 333- 10 1843-98 — 0. 7608 259-3 * 1861.43 —0. 0628 749.0 II 1853-51 -0. 3796 205-3 * 1868.06 +0. 2024 815.0 12 1861.60 —0. 0560 353- 7 * •875-32 -j-o. 4928 692.0 13 1868. 12 +0. 2048 466.0 + 1883. 15 +0. 8060 819. «4 1875. 38 +0. 4952 399- 5 i 1888.56 + 1.0224 801.0 15 1883. 09 4-0. 8016 5'4-5 520.5 4 16 1888. 67 +1.0268 i m 2«»,li71 UNKNOWN QUANTITIES OF EC^UATIONH. W'eif/litti, cporhH, omi periods of partial not >.ud ei/initionM. MARS. 05 Kiijht Asci-nsion. 1 ! 1 1 Menu r Wt. F. ■« year. (un tsdf 25 1. I '7S7.43 —J. 942S •!53 li 2 '770.55 —J, 4180 II. 3 17S7.82 -2.7272 U' if 4 •7W. 77 — 2. 2492 20.7 5 ■?<>«. 32 -1.7872 •47 6 1829. 17 -I.07J2 60. 7 I8J7-39 -0. 7444 121. 8 1845- 39 -0. 4244 7t'- 3 9 1853. 36 —0. 1056 90.0 lO 1861.07 -jo. 2028 114. u II 1869. 20 4-0. 5280 124.0 I i 12 1877.71 -f 0. 8684 IJ2.0 '3 iSSjj 27 4 1. 0908 91.0 14 1S8S.8S ^ 1.3 140 HS-S l>(.'clinntion. Mean r ' yrnr. 'lunilsnf 25 r 1 Wt. 1758.82 '773-79 1794.48 |8<.>4. 91 181 ]. 00 1828.04 1 8,57. 18 1S44.95 1853.02 |8()(). 94 1868. 80 1877.38 1 88}. 26 1 888. 48 -3- 8872 3. 2884 — 2. 4608 - -2.043() I, 7200 1. 1184 o. 752'^ o. 4420 -o. 1192 fo. 1976 f o. 5120 +0. 8552 41. 0904 -'I. 2992 8.8 i 8.8 i 13.0 i 47- i i 305 i t 93.0 37'" I 255.0 245.0 306. 197.0 257- 1 60. if)7.o I'lihioirn tjuantities 0/ the eqiidfionn. 27. For convenience in solviii},' the eqnations of I'oiulitiou the coenic'-Mit.s of tlie ('qnation.s were innltiplicd hy .snch nnmerical factors a.s wonhl rednce tlicir jfcneral mean ab.so- Uite value to nnmoers of a|)pro.\iiiMit 1y the .same order of niagnitnde. Hence, the unknown ) the factor by which the unknown (juantity, as tinally found, must be tk^ MERCURY, VENUS, AND MArfS. [27, 28 iMultipIied to obtain the correction as expressed iu the last column. In the case of Venus and Mars these factors are the same. Factor» by which the unknotcn quantitiefi are to be multiplied to obtain corrections of the elements. Symbol of Factor for — Corr. of unknown. Mercury (") (/') Venus. Mars. element. WJ 1 0.1 7 0.3 6m : Wo / ] 40 4 5 2 6\ I J] 30 3 G 2.5 tfj |NJ 30 3 7 2.5 sin JfJN 1 ^' 1 30 3 3 10^7 8e Tt 100 10 439 100^7 dn Tt 100 2.05G 3 1.3323 edn ^ 1 10 1 4 4 dt . «" 6 0.6 2.5 2 6e" n" 6 0.0 2 2 e"6n" [ <^ \ 10 1 1 5 a ^ \ 10 1 5 5 S I" \ 10 1 4 3 61" The secular variation of each unknown in 2~) years is e.tpressed sometimes by a suttixed 1, sometimes by an accent, thus: [/ 1' = |/], = change of [/] in 2.j years. 28. It may also oe useful to give the values of the principal coenicients in each of the normal equations. They are found in tlie followinj; table. Were the other coefficients all zero, tliose numbers would inJicate the weights of the different unknown nuantities as resulting from the solution. Several of them were greatly diminished by the i)roce8S of solution. 28,29] ORDER OP ELIMINATION. 5T Values of the prineipai diagonal coefficienU in the normal equations. Mercury. Venus. Mars. Symbol of From coefficient. From mer. observa- From transits. Sum. 1" rom mer. observa- From transits. .Surn. mer. observa- tions. tions. tions. [ '"•' ] 5488 5488 5868 2929 8797 17887 ^ // 10559 1 1 308 21867 5981 3540 9521 20924 r n 1 15222 1296 16518 13232 7444 20676 28783 - XN ■ 14176 2304 16480 1 795 1 1636 19587 32478 e' t' 19015 5076 24091 5686 3350 9036 201 19 TT TV 8621 8352 '6973 5290 1732 7022 20564 f e IIOOI 196 11197 1 1429 3598 15027 31460 ■,,//,.//■ 9757 508 10265 9586 665 1025 1 15909 \" tt"' 9099 261 9360 5836 1895 7731 14911 ,if ^>t ■ 5242 1 304 1 5242 13583 ' l"l" ' 542 11031 2349 53380 15427 na 13230 13230 .335 335 25138 ■ riJ ■ 24657 24657 15196 15196 53975 ' // ' / 7014 67155 74169 6005 8983 14988 26689 n / 12366 9383 21749 9837 13014 228';i 23440 XN ■ / "035 16682 27717 14724 2874 17598 29494 (' (* / 15437 29647 45084 5743 8610 14353 24364 ff 77 / 6745 493 « 8 56063 4948 4483 9431 27131 f, f / 8488 1418 9906 8458 6306 14764 25675 ' ,•'' e'^ ' / 8409 29o; 1 1346 9805 1682 11487 22947 tt" tt'' / 8410 I5»3 9952 1 5242 4805 10047 17356 ■ ," r"^ / 54)2 S432 , I" I" ' / 1 1629 3126 •4755 10677 5667 16344 2065s a (I / 1 1400 1 1400 i 297 297 33624 [ '''' '. / 18716 18716 10772 10772 42405 NoTK. — The coefficients for Mercury and Venn? in this table are j^iven as they were used in the solution, after droppinii; the units from all tiie terms of the equations, except those from transits of Nieicury. Order of climinafion. 29. Ill dealing with so extonsive a system of uuknown quantities it is impracticable to inveHti}jrate tlio depeiMlence of Oiich upon all the others. It is therefore essentia) to arrange the unknowns in an order partly that of interdependence and partly that of the liability of em'h to subse(|uent change by discussion and adjustment. Hence, tlie mass of the planet, Mercury or Venus, should be first eliminatad, as being that unknown which is least affected by changes in the tinal values of the other unknowns. The secular variations, as derived M MERCUrY, VENUS, AND MARS. [29,30 ! N from meridisiii observations, are nearly indei>endent of the corrections to tlio other elements. The solar elements are to be subswjuently determined by a combination (►f the results of the observations of the Sun and of the three i)lanets. Guided by these considerations, the order of elimination was, with some exceptions, as follows: 1. The mass of the disturbing planet. 2. The live elements of the observed planet. 3. Tlie four elements of the Karth's orbit. 4. The corrections to the star-positions for the mid-epoch. 5. The secular variations of the eleven tpiantities (2), (3), and (4), taken in the same order. Treatment of meridian ohservations of Mercury. 30. In tlie case of ]Mercury the factors of the coetticients of the equations were chosen large enough to admit of the deci- mals being dropi>eer is increased by the ((uautity — rt f J? ± f'y .-t . aoi MBUIDIAN OBSERVATIONS OP MERCURY. s» and tlie only effect n|M)n the precision of the results will be that arising from this increased probable error. Let us esti- mate Its magnitude. From an examination of the tabh'S used in iinding the coellicients I infer that the probable error of the eoet1i<'ient of n was I: 1, and tiiat of all the other coetlicients rli 0.<». The mean value of the unknown «|uantities M-as gener- ally a small fraction of a second. We conclude, therefore, that the probable w mean value of the «'rror A t.r \ >!f \ ... would in any casi^ be only a small frai'tion of a second. More- ov«*r, these errors would be purely accidental and not syst«'n)- atic, since the intervals of time between the e(|uations were generally so long that the coefticicnts for ditlerent equations came from «liti'erent tables, so that no error from omitted deci- mals in anyone e(|uation 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 approxim?'~.iig to this degre<' of a«'(!urary in the secoiul members of the <'qtni- ti(Mis. Were the observations rigorously correc^t and the values oftho unknown (|uantities finally determined affe«'ted by no eiror except that arising in this way, they wjadd be many tinuvs more a<;curate than we can hope to make tln-m. The «'rrors miglit, in fact, be considered unimportant in tl>« present state <>f astroufuny. It has alreai'events this mean err(U" from «'orrespon«ling to the unit of weigiit which would be found from the solutions in the usual way. In the first phu-e, the absolute members were id! multiplied by !(►; in other words, the decimal point was dropped from tenths of sccon«ls. and no further account taken of it. Secon«lly, in (;onse(|ueuce of the probable error in the coefllcients of the normal equations arising from the imperfec- i 60 MKRCUKY, VENUS, AND MARS. tioiis of the decimals, the final values of these coeiTicients would be subject to probable errors ranging between 50 aud ICO 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. Tliu », considering only the efli'ect of this oi)eration, the unit of weight would correspond to a mean error of i 1.0 in units of the absolute term. But in dropping off the last figure from the coeflicients we i)ractically 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 nmltipliers by ten, so that effectively we nmltiplied the coefficients in the ei|uations 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 eftect 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 f, = 1.0x \/l0=i3.1 As the mean error is at best a rather indefinite quantity in a case like the present, we may consider its value as 4 uniif^: 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 unkowns such weights would be entirely illusory. Tlie fact is that in solving so immense a nmss of equations, we must expect systematic errors to vitiate many of the results. The observations of Mercury, especially of its Kight Ascension, are not nmde on a uniform system; sometimes the limb is observed, sometimes the apparent center or the center of light. 30,31 TRANSITS OF MEUCl'RY. Gl All ideally perfect 'ysteiii of reduction would require us to reduce each separate observation with a seniitlianu»ter corre spondinj; to the personal equation of tlie observer. This being entirely impracticable, we nuist reganl the reduction of the observer's seinidianieter to that used in tlie reductions as a probable error. In fact, however, it will be of a systematic character, varying at each point of the relative orbit of ^fercury, and going through a cycle of changes impossible to determine in ii synodic perioil 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 cqitations (h'rireil from ohitcrecft trouxits of Merc or If. M. The relations between the elements of Mercury and the Karth tlerived from this sour<;e are shown in my Dincusniou of Transits of Mercuri/ (A. P.. Vol, I, Fart VI.) (hi page 417 are found e.\i)ressions for those linear functions of the corret'tions to the elements whi<;h are deteiinini'd by the November .md May transits, resjtectively. With a slight change (►f notation to correspond with that «>f the i)resent paper, these functions are as follows: V = 1.487 61 — 0.487 rtV - 1.137 rfc - 1.01 rU" + 1.19 e"dn" + l.uSfV \V = 0.7U; (U + O.L'84 r» + 0.8«<)ch for the transits is taken as 1820, to which the zero vahies correspond. The valnes for 18G5.0, the mid epoch for the meridian observations, are, therefore, from the transits alone — V = - L"'.08 ± 0".41 W = + 1".«7 i 0".37 This, however, is only a tirst approximation to the quantities which shouhl he introduced. Since the meridian observations help to determine the values of V and W, we should not regard the reductions to 1805.0 as final, but retain the results in the form (<»). 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 — a!»d found from all the transits up to 1881, N = - 0".1({ ± 0".27 + (0".28 ± 0".r>2) (T - 1820) (fr) The values of V, 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 valines 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 talcing 1820.0 as the epoch for all three of the quantities, V, W, and N. If we substitute for sin i 6ft its value in terms of 6J, etc., namely, Sin i6f)= - O.fiOlS 6,1 + 0.7»«» sin JdN -f 0.721 6e (c) and then for 6 J, ($N, 6fy their values in terms of the unknowns of the equations of condition, we shall have N = - 1.805 [J] + 2.394 [N] + 0.721 [e] - 0.122 [I"] {d) 311 TRANSITS OF MEBCl RY. 63 Similar expressions will be found for the values (»f V and W by substituting for tlie corrections to tlie elements tlie unknown quantities of the conditional equations, as already given. Taking 181*0.0 as the mid epoch, we may regard the inde- pendent quantities given by the transittj of Mercury to be the six following ones : Vn 1.8 V V, W„-!.8\V,: X„- 1.8 N, W {e) Here Vn, Wq, and N,, indicate values for 1865, the mid-epoch of the meridian ob8ervati years. The six conditional equations thus found from the transits may be written Vo -1.8 V, Wo -1.8W, No - 1.8 N, V, w, N, - 0".9O 1: 0".31 + 0".s4 -J: o".i>ri - 0".lf accursicy 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 <>. 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 ElilATIONS FOU MKUCl'RY. 60 tli« other oh'meiits. Whether this belief is jnstiHee meridian observations. They express what is supposed to [ij equivalent to observations of the tiiree quantities V, VV, and N in 18L'0, when r = — 1.8. Ilenee, fnun the partial nornuils in the si's lu-ini'ipal unknowns, [f], [t-] . . . [r"], the eom- plete normals nuiy be formed by nudtiplication by r and 7* (r = — 1.8) in the way set lorth in §2.">. SolittioitH of the equations/or Mcrcurif. ^2. In the case of Mercury ami Venus, it is desirable to know to what extent the results of the transits diver^je from those of the meridian observations. Hence, as already remarked, two solutions of the equations were made, termed A and li. Solution A is that derived from the meridian observations alone. Solution B is that of the nornnil equations formed from both the meridian observati^dl" -0. 577 — I. 261 \ a \ / —0. 3160 -o- 1973 4- D,a — I. 264 -0.789 Mean ejioch of corrections, 1865.0. DiHCortJance in the ohnerred Right Ancennions of Mercnrti. 33. The most remarkable feature in tlie result i.s the value of the (piantity represented by [/•"]. The unkiiowu quantity introduced with this symbol had as its roetticient the derivative of the geocentric phu^e as to the Earth's rmlius vector, and the result would therefore be an apparent constant correction to that radius vector. Since, however, the position of the planet depends (mly on the ratio of the distances of the I<]arth 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 '^1 ^". , r'.WwHK- ■ t tMWg » WW»M>U jsJ^^1M> 33) DISCOllDANCK OF (»nKERVATI()N8, 67 value of tliis unknown «|uantity hIiouIiI lu' n^i^ardcd an /oro, aiul tliu result as a purely tictitious one, arising from errone- ous eluiui'iits of reduction or systematic' personal errors. It was the possibility of the latter that led to its introduetion. When the planet is i^ast of the Sun, observations are always nnide on or near its west limb, or at h'ast on some point west of tlie true tenter, and vice rcrna. The value <»f 7'>5-'79l 1793-1X15 1817-1839 l8. near approach to constancy hi the vahies of the niunbers in each column, after the secular variation is allowed for, and the ]arp> magni- tude of the corrections. The tuost natural coin-Iusion is that the reduction from the limb of the planet or the observed center of lijyht 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 seeuis iuadmissible. 1 have not yet been able to enter upou the investigation of the source of this anomaly. A very im]>ortant (piestion is that of its influence on the results, i^ixwe a constant error in the radius vector of a planet would have opposite eflects on the elements in different i)oints of the relative orbit, it nniy be inferred that the effect of the error would be nearly elimiiuited »■■" V S^i, m\ CUMPAUiaoN OF OUHKHVaTION.^ OP MKIUTUY. fll) in III! e.\t»MiHiv«' series of obseivation.s '67r" - 1.62 6e" 1 I* ■ 70 MERCURY, VKNI S, AND MARS, we find tlu' follow j:>g results: [34, 35, 36 From meridian observations V = - 2".00 + 0".fi9T Prom November transits - 1 .(iO - 2 .03 T From combined solution 2 .30 T From meridian observations W = -f- 0". — 4".."i5T From May transits alone + 1 .39 + 1 .84 T From combined solution +1 .39 4-0 .42 T We eoin'liide tliat, had no transits ever been observed, tl'O errors of the eh'ments and their secular variations, derived from tlie j-reat mass of meridian observ.atious, would have caused an error of some ."»" 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 solutirincipal points of dirt'erence: 1. The hypothetical (luautity i^r'-' is omitt«'d. Hence no iiulex tt) the consistency of the observations at f p; /?, A, its ecliptic latitude and longitude; L, the Sun's true h)ngitudec 36 1 EQUATIONS OF CONDITION FKOM TRANSITS OF VENUS. 71 Then 6\ = cos i 6v + sin'-' / rf^ •= 0.01W2 6r + iwrm isiii i dH Wo thus have, lor the dates of the observed transits, 1701- 09 ; 6fi=- 0.0ri!)2 (Sr + 0.9982 sin 1 6fi 1 87-4-'b2 ; l and 17(i9 in AstronomUal Paperx, Vol. ii. The tinal results whieh 1 shall use are found on i)age 404 of that volume. Jlere I have put. .r, correction to A — T^; — I/, eorreetion to /^, the Sun's latitude being sui>|M)sed to require no corret'tion. The values of x ami // for 1709 are clistinyuished by an aceent. I have also rei/risented by c.. and ^, the eorreetions to the dif- ferenee of the semidianieters of the Sun and jdanet, tor the respeetive internal eontnets, to which may b»' a".(M1 1709 II;- .04 .!•' - .77 y' + z^ = - 0".27 HI; 4 .SI _ .r,r, 4. .-, = 4- o",(»2 We have here more unknown iiuantities than equations, .so that it is not pratfticable to determine them all separately. What I have dcuie has been llrst to assjinu' z^ = r,. This pre- supposes that the vlistance of centers at the estimated appa- \ 72 MEBCT RY, VENUS, AND MARS. [36 rent coiitatrt tit t>t;r»ss is, in tlio {j^eiuMal incan, the .sanu* as iit injjress. Tlio result of any error in this liypothesis will be almost completely eliminated from the mean latitude at the two transits, but not from the longitude. Still, the values of .i- and //ean not be sejiarately deteimined; I have tiierefore so combined the e<|uations as to obtain mean values of .!• and y for the two ('ontacts, assuming that this would be the result of sui»i»osin}; these i|uantities to havt^ the same viilues at both epochs. Calling; these values .v" and y", we have by addition an luailc nftiT thii corn'ctioii to tlie nMitonnial iiiotion nf tbo mule in dcternuiu'd, wo HhuiiUl put, oil account of thin oorrectiou, ,/ =-_,/' -O'.ll ,/^y +0".U The Holution svoiild tlu-u give y" = + 0'.Ofi X"— . + .14 I huv«) carriiMl through a iiioru * laeful iip|iroxiinatioii in a aii)i8oi|Uont obuptf r. [36 30] El^VATlONS OF CONDITION FROM TRANSITS OF VKNUS. 73 We tlins HihI, for tbc coiToctions to Levekkier's tables at the e|M)ch l?l}.").5, (5 A — rfL ==. + 0".01> 4- 0".2r> = 4- «".34 6fi = - 0".(M> + 2".(H) = + 1".»4 and lieiicc Bin J fJ ^ = 4- 1".!).") + 0".O.VJ rf L A stir farther iiioditicatioii is riMiuired to the taludar loiif;!- ttule on uccoiiiit of the correction to tlie mass of the Harth used by LEVERitiKil, a:id hence U^ the periodic perturbations in ion{;itude. This correction is -f (>".2(). Wo thus have for tlie correction to the orbit W>ngitu«h' of V«'nuH— rfr = + 0".02 + 0".!)<»8 r? I, For the results of the transits of 1S7J and 1882 f have de]>ended entirely on the helioineter measures and photo- jfrajihs njade by the (ierir\an and American expeditions, respectiv«'ly. The «lellnitive results of thetrernnin observa- tions, as \vork«'d up by Dr. AuwEKs, are found in \'ol. V of the (xerman Heports on the Transits.* The American ]ihoto- j^ra',)hic measures of 1S74 have not been otiicially worked up and published, but a preliminary investi};ation from the data contained in the pubhshed nieasures was male by 1). I'. Todd, and publisiied in the Amrrivan 'lournnl of Sru-nw, Vol. 21, 18S1, pa«;e 4!M. Tiie measures of 1.S.S2 have but only the results published. They are tbund in tiu^ report of the Superintendent of the U. 8. Naval Observatory for the year IHUO. Tile corrections to the ;i('ocentric Kijrht Ascension and l>eclinati{t) 1S7I innl IHSi* Ituricia ubiT «lio iX-utmheu ]<0(»l»iicliliiaK<)ii KiUit'tor Mnn«l, Hurliu, 181i:{. 74 MERCURY, VENUS. AND MARS. given ill the following table. In taking the mean the weights are not strictly those which wouhl result from the probable errors as assigned, but, in accordance with a general princi- l)le, independent results have received a weight more near to e(iuality than would be indicated by the mean errors. 1874: German, d K. A. = + 4.77 t 0.28 American, . . . + 4.14 i 0.30 Adopted, ...-}- 4.44 pi " I ; German, 6 Dec. = + 2.28 i 0.10 American, . . -f 2.50 i 0.30 Adopted, . . . + 2.34 1882: (German, rf R. A. = + 0.03 1 0.12 American, . . . +0.101-0.08 Adopted, . . . +0.07 German, d Dec. = + 2.02 ± 0.00 American, . . . + 2.02 ± O.OS Atlopted, . . . +2.02 We change these results successively to geocentric longi* tude and latitude, heliocentric longitude and latitude, and orbital hmgitude and latitude. The results of these several changes are as follow: 1874 1882. Corr. in gcoc. h)ng. + 3''.a53 + 8".077 Corr. in hit' + 2 .724 + 2. 071 Corr. in hel. long. -1 .415 -2 .965 Qorr. in hcl. lat. + I .(M)l + 1 .001 Corr. in orbital long. - I .35 -2 .90 Value of sin i6 6 -I .26 t i 37 J EQUATIONS FROM TRANSITS OF VENUS. 75 Eqwit ions from transits of Venus. 37. The corrections to the heliocentric ])ositioii8 of Venus and the Earth, as thns found, are now to be expressed in terms of corrections to the elements. The results of this expression are shown iu the following equations: Equations f/iren by the corrvctions to the orbital longitude, I. Epovh, 17(m.5; t- -3.90; weight = 20() 0.902 (H+IM e^TT + 1.62 rfe - 0.970 61"- 1.81 ('"d?r"- 0.85 6e" = +0".02 }, 0."15 II. Epoch, 1874.9; r = + 0.48; weight = 400 - 0".88// + \.mt (U - 1.223 <'rtV - 1.590 6e - l.(>30 61" + 1.8(»4 e"6n" + 0.817 6e" =.-. — 1".35 i 0".08 III. Epoch, lS82.th r = + 0.80; weight = 8(M) 0".<;0//+ 1.008 61 - 1.140 e6n - 1.051 6e - 1.028 61" -\- 1.825 e"6n" -f 0.900 6e" =: - 2".!M) i 0."027 Equations (jircn hif the corrections to the orbital latitude. 1. 1765.5; sin i6fi- 0.057 61" -O.ll e"d;r"— 0.05 6e"=z + 1".95 i O'.IO II. 1874.!>; sin^J^-0.06l()7"-H0.110e"fJn'"+0.018f>\"= - 1".08 I 0".04 Ml. 1882.9; sin/rfW-0.061 fy/"-}-0.l07e"rf;r"-f 0.()53rt\'" = -l ".26 i 0".0l9 The weiglits assigned to these three equations are, respec- tively, 200, 600, iiiid l,(i00. Before using these ecinations tlie corrections to tiie elements were transformed into Hie unknown (|naiitities dilincil in i'i', and their secular variations by multiplying the coeOicientf by the factors given on page 56. 70 MEKCl'UY, VENl H, AND MAKS. 138, 39 SolulioHH of the equationM for Venus. 38. The ptirtH of tliu uoriiial eiiiiutiuiiH furiiUHl fVoin the pi-ecediii^: conditional equations were added t4) tlie parts from tlie meridian obHervations, and tlie resulting solution H obtainid. Ah in the case of Mercury, a solution A was made of the nornuil ei|uations derived from the meridian observa- tions alone. The results are as follows: VKMS, HchuUh of MolutiouH It/ the normal equatUnktt. Unknowns. Svml)ol. / e f t" n I .1 N t 15. u. -0.0834 7. — o. 0708 I -o. 143s -o. 1501 5. i +0. 1156 40. IJ40 6. j jo. 0164 f 0.0106 j 7. 1 f o. 0941 10. 1003 I 3. jo. 0628 t o. 0764 I 3. f 0.0246 |-o. 0271 I 4. fo.0336 (-0,0318! 2.5 0.0274 ~0. 02I2 \ 2. 40.4742 1 0.4642 i I. -o. 03S3 -o. 037s 5. —0.0768 -o. 0743 4. — o. 1846 — o. 1983 20. 4-0.0970 jo. 1088 : 34, -0.0561 i 0.0594 28. 4-0. 1472 ! 4-0, 1644 12. 1 0.0555 10.0698 12. f 0.0182 4-0.0202 16. i 0.0283 4-0.0317 10. . (). 0399 i o. 0506 I 8. -n. 0820 o. 0347 4. 0.0020 0.0002 2o. -0.056a -o. o6(»2 16. CniT«ction.t" 4-0.084 40. 080 ." & If" -0. 055 -0. 042 1 -f-o. 474 -i 0. 464 ,\ —0. 192 -0. 18S M" —0. 307 -0. 297 I ), r' / —3. 692 -3- y66 I'. 1 4 2. 328 f 2. Oil sin 1 1 ), N - '57" I.r.63 i>,. -f 1.766 f '••>7J M),r -1-0. 666 40.838 1).« 4-0. 291 +0. 323 I),.-" 4-0. 2S3 (0.317 r"lJ,T" -f 0. 3«9 -f-o. 405 I >, (1 ~o. 328 0. 139 I), a -0. 040 - 0. 004 1 ),.»/" — 0. 899 -1.059 Mean epoch nf correction, 1863.0 CouiixfriHou 0/ tniHHitu 0/ Vt-nHH icith meridian ithHtrrationx, :v,) To .show to what extent Ihf resnUs of the meridian observations ditfei iVom tho.stMtt' th«« observetl transits over the Sun, we {\\\u\ tlie xidnes of the tibsolnte terms of the e«|iiationM of condition, §.'17, tlrst by substituting the values A of the corrections, and tht>n the vallu^s H. We thus have: I J 1 39, 4(>| K«,HATroNS KUOM TUANSITS Oi" VHNUS. lieHulualH in orbital huijitude. (a) From iiu'iidiaii ol»s. aloiu' ( (i) From comhiiu'd Holiitioii {y) From transits alone . . Discordance, (;') — (") • Discorilancu, (v) — (li) . «76S 5 - ".07 -I- WM 4- (Y'SYl -|-(>".01> — ()".(>» lS74'i. - V'M - \".\:\ - 1 '..'{o -f ".01 4- O.OH UixitinalH ill orbital latitihli: {it) I'rom meridian ohs. alone (fi) From coiid>lned solntioii (r) From transits alone . . hisi'oidance, (r) — (<>) . Discordance, ()'j — (fi) . '7"5-5- -I- I '".02 +• J'.OU 4- i".o:» 4- o'".o:{ -(►".11 i.S74.'». -0"'.77 -0'".!U — l".o,s - o"'.;jl -0".17 77 lSS2.n. - Ii"'.."i4 - L'".7H - '2" AH) - {)":M -0'".I2 1S83.9. - o".m} - l"".!:-' - I ".'-m; - o"".;{(» -0".14 It will b«' seen that the comhined solution represents the ohservatioMs of th«> transits mucli lietter here than in the cuku of .Mer«niy. Solution 0/ ihr niuations for Mars. 40. As the formation of the normal eqnations for Mars was a|i|)roachinu its end, a singular discordance anion;; the resid- uals of tlic paiiiiil normal ci|uations for ditVerent perioiKs was noticed. On traeinp; the matter out it appeared that while the correction of riie «eocentri«^ lon^fitude of liKVHititlKK's tahles in 1H4.'> and a^ain in I.S'.rj was i|uit«' small, the correction in ISO12 wan consideral>le. Now theie i.s an iiio<|Uality of lon^' l>eri«»d. about forty years, in the mean m«)tion of Mars, depend - iuffonthe action of the lOiirth, and havin;:: for its ar;;innent ITu/' — 8f/. This coeni4ient isof the st^venth (U'dcr in the eccen- tricities, and the terms of tin' ninth or even of the eh'venth order ndffht he sensible in a dcvclopnu-nt in powers of the eecentriciti«'s and sines or cosines of nndtiples of the mean louj^'itudes. The conclusion which I reached was that the the- oretical \alue coelliricnl u as not determined with suill- dent precision. As the wtak of solvint; the e(|uations could not wait for a lu'w determinati(m and a im*w formation of the absolute terms of the normal c(|uatiiu'h perJoil, uiid the solution wuh thou proceeded with. The (;hiiiiceA se<'in to l>e that by this prm^ess the iiijii- rioiiH el!e(!t of tlie error up(»ii the ehMuents derived from the equations would he inconsideruble; thiH is, however, a point on whieh it is impossible to speak witli certainty. It is the intention of tiie \vrit4>r to recompute the doubtful terms of the peiturbations, and, if ixissible, reconstnu;t the absolute terms of the niuiual eipiations in accordance with the c(U'rected theory. Meanwhile, the present work nei-essarily rests on the imperfect theory with the approximate empirical corrections, which are as follow: 61 = U":60 cos (15f/' - Hfi _ 223^) Mtt = O'Mo cos (1%' - Hji) As the elements of Mars are derived wholly from nu'ridian observatifuis, only one set of cipiations of condition was formed. The results of the solution are shown in the following table: MAKS. rnkiiowns. ^ f'i>rrcction.s nf c]cment>. Symhol. N'iilue. —.02278 03 Symbol, ft •« : m Value. ['"'1 1 -0. 17 // / ] -.44S54 2. .5/ 0.897 ;•' ■ -f. 05479 2-5 'M +0. 137 , ^ '. f-.of>724 2-5 Sin 1. IN +0. 168 ' <■ ' + •4380.? 1 II r fo. 626 T — ,0505() U ij ,\z —0. 722 f f. 07474 4. iSr +0. 299 1 -. 4989S 2, i\e" -0. 998 '_'/' - . 42409 2. <-"<)t" -0. 848 (t + .1S545 5- a -f 0. 927 ■ l\ ' — • 045.?<> 5- i\ —0. 227 '-"'' + .05786 i- ,\l" +0. 174 ■ / ■ -\-. 16605 8. I),<1/ + 1.328 ; J t-. 1340S 10. 1».J 4-I-34I ; N ; —.02265 10. SinJD.N -0. 226 (* -.03180 S« l\c —0. 182 7T — . 00928 lyu D.r -0. 530 t 1 . o6o()7 16. \hr +0. 976 c" -• "2597 s. \h.-" — 1.008 '_//' + .0*^853 8. r'\\j" fo. 068 (1 —.09670 20. l),a -1-934 .) —.01168 20. I),.l -0. 234 /"; ' . 13111 1 2 D,.!/" + >-S73 140 • 411 REFERENCE TO THE ECLIPTIC. (9 HefervHce to the ecliptic. 41. In all the precedinj; (leterriiiiiatioiiH the planes of the orbits are referred to the plane of the ICartli's i>(|iiator, or, to speak more exactly, to a phine through the Snn parallel to tin* lOarth'H e(|uator. Ah in aHtrononiical practice the ecliptic is tai^en as the fiiiulamcntal plane, it is necessary to investiKute the reduution of the elements from one plane to the other. Let us consider the spherical triant;le formed on the celestial sphere by the plane of the orbit, the pl.me of the ecliptic, and the plane of the Karth's equator. For the sides and opposite angles of this triangle we have Sides: }S( B if Opjwjsite angles: / 1S(P — J f When equatorial coordinates are used, the position of the planet is considered as a function of the three (quantities N| Jj (") When ecliptic coordinates are used, the three corresponding quantities are e^ ii (i) Taking the set of quantities (a) as the fundamental parts of the triangle, and expressing the »'orre«!tions of the otiier parts as functions of them, we have Si= ■{■ cos //'«yj + sin »/• sin Jrf X — cos Hfif sin iS6 =s — sin i/^Sf) 4- cos »/' sin JrfN -{- cos / sin Wdf (c) Taking (h) as the fundamental parts, we have for the correc tions to N and J sin ff J = cos yrfj — sin '/' sin /rtV/ -f cos Ndf .IrfX= sin i/'6i -\- cos //• sin iSH — cos .1 sin NfJf «/) The numerical values assigned to the eoeflleients in these eqinitions are those corresponding to the mean epoch ISM). The fact that they change somewhat in the course of a hundred years has not been taken account of. The future astrontuner will meet with a real dilliculty iu that the corrections to a Hi) MKItCUBY, VENl'M, AND MAU8. [41 ii svt of cUMiiciits at Olio eiH)('.h «Io not iiccumtoly corrcHpoiul to Hitniliir (MirrtM'tioiiH at anotlier e|M>c)i. It is iinpossiblo to do away ligoioiiHly with tin' liitliciilty thus arisiiij;, oxoept by iiitrodiu'iiii; a iiioro ^ciicral HyHttMii of oleinuiitH than elliptic onoH. Tlu; error is, happily, not important in the present statu of astronomy. The equations in question for the three planets are as follow : ,U = 4- .71«> fU -I- .«»0L* sin .1 rJ N — A\HH fSf sin jrtvy = - .OOU rf.I -I- .71M» sin J rf N 4- .7L'l 6e VenuH. fj; = + :m:\ (^ .i + .tn's sin .i ti X - .l*.m rf* sin i6& = - .1»28 fif .1 4- .•{T.'t sin .1 f)' N + .JM»7 fJ* 6i = .70.'{ rf J + .IVJ sill .1 rV N - .«iSH + .1M»8 rff siu .1 dorivi'd four Hoparate valiu's of tlu^ six roin'ftioiis, <'iilar variations, which pertain to the orbit and motion of the ICarth rehitivo to th<' stars. \V«' hav«' now to couibinc tlu'so fonr resnits so as to diMive tlic most probable valnos of thf) twi'lvo unknown ijuantities in ipifstion. IhriotionH/rom the imthod of least Hijiuircti. 42. If W(> applii'd witliont inodillcation tht' principles of thu mi't hod of least s<|uares. we sliould first eliminate the elements and secular variations for <'a(;h planet from the normal equa- tions {iiveii by observations of that plaiu't, which would leave us with three sets of nornml equations, eontaininf; only the twelve quantities depending on tln^ motion of the Karth. We should then reduce these uornuil eqtnitinns to equality of weight, by multi|)lying each of them by the appropriate factors, and we should then consider the observi'd corrections to the solar elements j «;), . • • for the possible values of h, and («) Pi, Pi, P3, . - . for the several probabilities that // has these respective values. Then the probability function will become

iu'd is necessarily a result of a judgment based on all the circumstances. Relative pn'vision of the tiro methods of iletermininii the elements of the EartWH 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 tu the case of any of the elements. This, however, is i)roved by the results not to be the case. Attentum has already been (lalled to the extreme consistency of the values iound for the correction to the eccen- tricity and perdielion of the Earth's orbit from observations of the Sun. This consistency insi)ires us with conlidence tliat the probable errors of the corrections to the elements as given do not exceed a few hundredt!>s of a second. But the deter- . mmation 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 ol>servations being mada 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 I'ase nuiy iiave an effect upon the result. Hut ])ersonal e([uation is likely to be smaller in the case of Mercury than in that of Venus, owing to the smallness 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 orbit 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 u; its own plane in a direction at right angles to the line of apsides. The result is that in a series of [41', 43 43,44| SECULAU VARIATIONS OF THE SoLAU ELKMKNTS. 87 observations of ii i»lsinet from the Earth the eonectioiis to the eccentricity and perihehaof the two orbits can not be entirely independent, and we can determine with entire precision only two linear functions expressive of the relati ^ displacements just described. It may be admitted that, w e observations exactly similar in kind made around the entue relative orbit in e(|ual numbers, the etVect of the principle systemati<' errors would b(^ nT; 1 -0.21-0.2:{T; 4 -0.04-0.4;JT; 3 + 0.04- 0.08 T; 4 Mean ; rJ = - 0".09 - 0".42 T Adopted; (S = - .08 - .501 -o. 30 -0.30 Not only observations of the planets but those of the fixed stars are available for the determination of S and of its secular variation. In the discussion of the Declinations derived from observations with the (xreenwich and Washington transit cir- cles {Atitronomical Papern, Vol. II), I have shown that the Greenwich observations iu.iicate, with some uncertainty, a secular variation of the corrections to the standard declina- tions which will give a value of about — 0".o5 for the seen lar variation of rf. But Bradley's Declinations, as reduced by AuwERS, would give a still larger negative vsilue, approxi- mating to an entire second. As the value which we may assume for 6 does not greatly influence the other elements, I have adopted as a convenient probable result, the varia- tion -0".50 T. flO KLHMKNTa Ol' EARTirs OIIIMT. f4(} Hi * 1 1 '■: SP ii If IhJinitiniiec'Klar mriationH of the planvtarij vie men In from ohaer- iHifions alone. Hi. Iliiviii^' «U'(;i(l»Ml upon tlic adoptiMl valucH of tlu- six quiintitios tonnd in tlu^ last artii'le, wr rejjard theiu as known (inantitics, anil siihstitnfc tlicni in tlie cliniinatin;; qniitions aie not tlie corrections themselves, but certain functions of thiMU, we pre pare the following; table, showinj; the formation of the quan- tities which are to be substituted in the several efpiations. The tal)le scarcely seems to need any explanation, except that tfia unknown diiautities j^iven in the three columns on the rijiht are formed by dividinj; the secuhir variations lor twenty- five years by the coetlicients {jjiven in § 21. Ailoptol secular variations of the solar unlcnon'ns, to be substi- tutett ill the eliminatinff equations for the several planets. Mercury. Venus. Mars. 1), rt7" =-\".m; [l" I' =-0.2.")(); -(MMUT); -0.0833; l)^6 =-0.50; [rJ |'=-0.12r); -0.(L'.")0; -0.0250; D^n =-0.30; [rr |' =-0.075; -0.0750; -0.0150; e"DtrJ7r"= + .20; [ .t" ]' =+0.108; +0.03'J5; -f 0.0325; '\(h-" =+0 .21; [e" \' = + 0.087; +0.0210; +0.0202; ^rfi =+0.48: [^ 1' =+0.120; +0.0300; +0.0300. To facilitate a Judgment or rediscussiou of this part of the process, we give on the next three pages the normal eiiuations between all the secular variations which renmin 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 ecpiations which were actually used in the substitution, because they show more fully the relations betweeu the unknown quantities, and can therefore be better used in any ulterior discussion. Regarding the preceding six quantities as known, and substituting them in the normal equations for the secular variations, we derive the detinitive 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. 4<>] NORMAL Kl^UATIONS FOR SECl'LAR VARIATIONS. 91 f? ^« '5 !t •J •^ 1- • • 1 CO .■.-5 s 1 ^ JJ^ ^^ '-t «^ ?i 7 1 ri t) o pH t M ^H •^ 1 1 + + + 4- 1 1 4- 1 7 1 II II II II II II II II II II II II s ^ 1 2 • • 1 + CO 1 1 + 1^- i 1 1- 1 i 1 + X ••• X S ^ Iz v^ § z /5 X 1-^ 'S ?5 1- X ^i n s- ^ JX ** rH -^ ** •>i^ ^ U + 1 + + 1 +■ 1 + 1 4- 4- 8 ^ s* «4< i 1- i ?1 X 1 3S •-H ^ rt 1 + 1 + 1 • 1 1 4- 1 4- ^ 5 ^ o — _ *« ri «A rc ^ 1-4 ^ ^ ^ ^ 7. -t ^ 1- ^H X ?i 1^ 1- Xk ?» 7\ X ^ - ?1 '.-5 S 1 + 1 7 + 1 + 1 4- V t' • ^ ^^ '* '•£ 1 ^ X ^ ^^ -t ^ ^ ■ C^ X ^^ X F^ X ?) 7\ H T. \ ■M^ X r-t t g , •-S U ' *-- > w + + 1 -L 1 1 + 4- ^ ti — * s ^ ::;— ?1 S s i ^-( ^ ^^ •/a s« a -— -t t- A^ r-^ A >s 2 + 1 1 1 1 + 4- ;^ s r? ^ ■M •-i 1- ?i 1 - ^ >-j nj 1^ ^ w^ ^ 73 i^ ;£ ^^ •—1 Ci is 1 1 + 1 1 + V ^ I— r^ _ ^^ t— 2 1 te i^ X i ,2 7 + + 4- + 8 o j^ • A ^ 3? ^ c^ 5 ■«-*i '^ rt ' r^ ?i --mi QO 10 s s 1 + 1 + s » ^ 5>1 ^4) X ?I 3 ;25 ~5 rj '^ s M ^ a 1 1 + K ^ X X •*.* 1^ « •-s X ^ "V + + + 92 NOUMAL Et^UATIONS FOR 8E0ULAB VAUIATIONS. [46 ^ I § i$ s ^ ^ ^^ ^ It U » M I- + I + + I ?1 CO L' "-< i-t 2:1. O »H -H rZ I + I I I I CC ?3 + &s i i- '^ § 3 ^ ^ ^. r ' — 4J rt c: s ^ 5 .^ CO I + II II «5 OS Jl CO I- o I I + + + + 38 5 + I c: r. zr x? »« I- -♦ « -: 2 -^ re f^ s + I + + + ^ 5 S ^: .7 !i ^ '^ Tj S A ^9; X t>. 5 p CO OS • iH I'. 00 I I I + + 02 .f^ 'N Si I jj ^ bo + CO « S= CS + I + to -r ;i_ I- ^ a . — S n 01 c8 ^ ,-$ s ^ CO « o it 00 § s? ?1 (M + + I- CI e ~ V S ? ;:: '^ s 5 »c ».-: CI + 11 + + I + + X CC ri + » + + + d Ci + 40] NORMAL EQUATIONS FoU SErULAIt VABIATIONH. 03 ?2 S iS :5 i' '"^ •- ej v: ri -- * :- I- i» f3^ rv I' -^ i* r-i ^ -5 I + I + I I + 1 r-3 1— • 1-: 5C r. 1- s ^ 1 + 4- 1 + + 1 4- 1 + 4- •A 2 'o :'5 1- =? 3 1^ 1 <. i ?^ S ' — ' rH ^5 I'* ^ -/ 1- i-t in V. ^ '^t e ^ « + 4- + 1 + + 1 1 + 1 s ^ r: i i 1 •H y—l ?I ^ ^ s + 1 + -t- + + 1 4- 4- *■*. «5 ^ X H 1 1 -1 ^ r-! 1— ( i-H *-^ 2 3 1 1 1 + + 4- -f 5! •2 •S 1 r^ 1 . + 00 I- + 1—1 4- I- + f. 1 1 ^ k be a s 2 "•,?^ t ' a ^ o o I 1 g - _ T-l § ^ 1 - ,^ ?" J?i « 1 + ci 1 I- 1 + + o + ;? I- + r-i 1 ri + + 1- 1— • + 1 + 2 + JO 1— I + -JO 94 SECULAR VABIATIONS FROM OBSERVATIONS. [46 •Values of the secular variations as derived from observations only. Uiikuowu. C'orr. Tables. Result. Mercury. DtC -.0091 -0.&1 + 4.19 + 3.3G±0.50 eDtTT +.1577 +1.30 +116.94 +118.24±0.40 Dt* +.0r)93J +0.83 + 6.31 + 7.14 i 0.80 siniDt^ +.08iriN +0.70 - 92.59 - 91.89±0.50 Dtdl -.0967 -1.55 Venus. Dte +.1393 +1.67 - 11.13 - 9.46 ±0.20 eB^TT +.0685 +0.82 - 0.53 + 0.29±0.20 Dt* +.1153 J -0.65 + 4.52 + 3.87 ±0.30 siniDt^ -.0592N -2.73 -102.67 — 105.40±0.12 Dtrfi -.1919 -3.84 Earth. U^e +0.21 - 8.76 - 8.55±0.09 +0.26 + 19.22 + 19.48±0.12 +0.48 - 47.53 - 47.11 ±0.25 Mars. DtC —.1190 -0.68 + 19.68 + 19.00±0.27 +0.29 +149.26 +149.55±0.35 +0.17 - 2.43 - 2.26 ±0.20 siuiDt^ -.0442N -0.76 - 71.84 - 72.60±0.20 J\6l -.0946 -0.76 The first eohiuiu of numbers in this table gives the unknown quantity as found immediately from the eliminating eijuations. These quantities being multiplied by the factors given in §27, Ave have the corrections ts) the tabuLir secular varia- tions, as given in the cclumn "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. l>iit as that of the longitude is the preponderating quantity in its eft'ect on geocentric Dte eDt TT Dtf DtC —.1190 eDt 7r + .0536 Dti + .1136 ;i I 40, 47 CORKECTIOWS TO THE .SOLAR ELEMENTS. 95 places, I luive regardi'd the value <)f the ecceiitr city used in the tables of the equation of the center as the tabular one to be adoi)ted. The numbers in the coluum " rnknown," which are followed by the letttrs J and ^\ are the respective values of [.I], and [N|i, which are changed to (U and siu i6& 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 th« probable magnitude of the sources of error to which they are liable. As in other cases, these ([uantities are very largelv a matter of judgment, because the probable err«ns as determined in the usual way from the eliminating e(|uations would be entirely unreliable. Dcjinitire corrections to the nolnr elements for 18')0. 47. Leaving the above results to be subsequently discussed, we go on with the solution of the equations. By a continuation ux'the process Just described, we regard the preceding secular variations as known (luantities, 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 observgtions of each planet, which are to be redm-ed to ISoO 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 ditliculty that confronted us in the case of the secular \ ariations. With the exception of the Sun's mean longitude, we are t«> regard the results derived from each of the planets as uiiected by obscure sources of systematic error, the probab'e magnitude of which can only be inferred from the general deviation of the (piantitios them- selves. As in the former case, <* is not regarde ( Bd. 13(), 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, trun- scribed from the publication in question, shows the sep-irate results and the conclusions finally reached: Wt Reciprocal of mass of Jupiter from — All observations of the satellites, Action on Faye's comet (Mollek), Action on Themis (Kkiteuer), Action on Saturn (Hill), Action on Polyhymnia, Action on Winnecke's comet (v. Haerdtl), 1047.17 10 m. e. 6690 N ALM- 1047.82 1 1047. 7'J I I047.r>4 .~» 1047.38 7 1(>47.34 L»0 1047.17 10 1047.35 1 0.0 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 ob' ervations 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. Tiie probable error assigned by v. Haeudtl to his result is very much smaller than that which I find for the maav 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 49,50,51] MASSES OF THK KAUTH, MARS, AND JIPITER. 90 be considered as having always been made on the center of gravity of a welldelined mass, moving as if that center were a material point subject to tlie gravitation of the Sun and phiuets. This tlistrust seems to be amply Justified by our general experience of the failure of ccunets to move in exact accordance with their ephemerides. I ))ropose to accept tlie value thus found, Mass of Jupiter = 1 — 1(>47..'?"» as the delinitive one to be used in the i)lanetary theories. Mann of Mt(r.s. 50. In consequence of the minuteness of the mass of Mars, measures of its satellites, esjuHtially the outer one, afford a value of its mass much better '^han 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 jingle 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 lialf a second. It therefore appears to me that the mean error in adopting Prof. Hall s value of the mass does not exceed its tiftietli part. This is a degree of precision much higher than that of any determination through the action of Mars on another jdanet. Prof. Hall's measures of 189i» show a minute increase of the mean distance given by his woik of 1.S77. The result is — ,/'" — = + 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. 1 have adhered to the original value in the work of the jjresent chapter. Mass of the Earth. 51. I have already pointed out the ditticulty in the way of determining the mass of the Earth from its action on the other planets. On the 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- r 100 MASS OF THE EARTH. [51 etary theories can best be derived from it. The theory of the relation between the mass of tlie Earth and its distaiiee from tlie Sun, as )ntact observations, transits of Venus. Aberration and velocity of light. Parallactic e(|uation of the Mocm, Measures of small planets on Gill's plan, 8.8CV ± .007 Leveruieu's method, 8.818 ± .030 Measures of Venus from Sun's center, 8.8.">7 ± .0132 8.780 i .020 8.704 i .018 8.798 1: .00.") 8.799 i .007 Wt. 1 1 16 5 8 0..5 1 Mean result, tt = 8".802 i 0".00o ; f; ■i '. - 1 I 1 - y i. i L 1 : 1 have provisionally taken this mean a., the most probable value of the solar parallax derived from all sources excei)t the m iss of the Earth. The above relation then gives M = 332 040 f51 r»i,r»2| MASS OF VENTS. 101 Taking for the mass of the Moon 1 -i- robal>U' error may be rejjfarded as some- tiiing more tlnin ii thousandth i)art of its whole amount. Mann of Venus. r>L*. The mass of Venus adopted in the provisional theory, to which Levkkrieu's tables were redm-ed, was .000 002 4885 = 1 -^ 401847, which is tiiat of Leveurier's tables of Mer- cury. In the precedinj; discussicuis the foHowing' three factors of ciurection to this mass have been found: From observations of the Sun From observations of Mercury From obs<'rvatious of Mars Mean — .0118 J .0034 - .0121 4 .(K):>0 — .007«i :1 . ( ? ) - .0119 i .0028 rent 1 L(> 5 8 0.5 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 nund^er of completely independent ugh 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 Is. -gely a matter of Judgment. The fact that it was necessary to introduce an empirical correction, with a period of about forty years, into the n»eau longitude of Mars, vitiates the deternjination of the nmss 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 uatiou of the mass independent of this empirical correction. I have therefore assigned no weight to the result. We thus T 102 MASS OF MERCURY. 152,53 I have for the masa of Venus, as derived from tlie periodic per- turbations of Mercury and the Earth produced by its action. !»' = 1 -j- lOfi cm i 1140 Mass of Mercurif. 03. 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 freciuently produces a perturbation of more than one second in the lieliocentric; longitude of Venus. When the latter is near inferior conjunction, the a(;tual perturbation will be more than doubled in the geocentric i)lace, 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 wouUl be obtained from the periodic perturbations of Venus. IJcferring to § -7, it will be seen that the indeterminate nmss of Mercury appears in the equations in the form 1 + 7// - 3 000 000 From the solution B, § 38, the value of jn comes out /< = - 0.0834 corresponding to a mass of INIercury 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 eijuation for /j, we find yu = - 0.0889 which gives m = 1 -j- 7 943 000 The work of the present chapter is based on the former value. m u 08] MASS OF MEKCTRY. 103 A consideration of the inol)jibIe error of tliis rosult in inipor- ttiiit. The fortuitous errors which mostly affect it are of the chiss whieli I have termed semi-Hi/Htemofic. Under this term I itn'Uide tliat larjje class of errors which, extendinj; through or injuriously atfeciinji a limited series of observations, cause the probable error of a result to be larfjer than that ^nven by the solution of the eeati'd, with the perturbation in the opposite direction. If, now, the observa- tions were nnide by the same observer, and under the same circumstances, the personal error would be eliminate:. Hut a probable error which ought to be reliable can be obtaiued by a proj-ews similar to that which 1 have adoi)ted elsewhere in this paper, namely, dividing up the materials into periods, and G. 20l> + 53 1857-'(i4. 4506 - 129 17!MU'0«). 345 + 11) 1865-71. 7736 - 265 180(;-'14. 431) 4- 135 1871-'79. 70(52 - 761 18U-'L5). 1)42 + 2 1879-86. 4958 - 407 1820-'30. 178(5 -330 1885-'92. 9561 -1306 Sum : 49 (5(58 // = — 4095 ^=- 0.0824 ± .019 The difference between this value of ju, 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 £1 = i 4".2 US] MASS OF MEKCVRV. l(>r> >vhich, us aiitieipiitcd, ih niiicli larg<>r than that which woiiUl be given by the iliscordance of the <>rij;iiinl observations. Tliis does not mean that the original observations are atVected by any mucIi mean t-rror as A: 4'MJ, but that the tliseordances between tlie 10 values of /* are as great as w»' hIiouM rxpeet them to be if the origiiuil observaticms were absohitely free from systematie error, but atfeeted by purely aceidental eriors uf this mean amount. The results ot' tin* suhition for the mass of Mercury nuiy be expressed in the form 1 i 0..*i2 , 1 ± 0.35 7 210 000 7 i» l.{ 000 In all researches which have been nnide on the nioti(»n of Encke's comet by Hncke, von Asten, and IJACKLr.MJ, the determination of this mass has been kept in view. The results are, liowever, so discordant that, as already rcmarkcl, scarcely any definitive result can be derived from them. To this statement there is, however, one apparent execption. Ill an appendix to his very careful and elaborate discussion of Winnecke's comet, vox Haeudtl has derived the value of the mass of Mercury from all the return of Encke's conu't as worked up by voN AsjTEN and Hacklund.* The only inter- pretation which 1 can put upon hia result is this: If we regard the acceleration of the c-I8«J8, Ml = 1 4- .") (i 18 (KM) ± 2000 1871-1885, w = 1 -r o 009 700 ± 000 000 He also finds, from the motion of Winnecke's comet, wj = 1 -1- .-) 012 842 ± Ol>7 803 * Denkschrifteii der Kaiaerlichen Akademie der Wisseuschaften, Vol. 56, p. 172-175. Vienna, 1889. ; 100 MASH OF MEUCl'RY. and from four oquatioiis of Leverkikk 1 53, 54 1 4. .-» .->14 700 4 m) (MM) III Tlu' consistnM'y of tlieso results seems to me entirely Weyond what the observjitions art' capable of giving, and I hesitate to ascuihe great weight to them. Moreover, the result implitiitly contained in these numbers, that tlie supposed He<;ula]- accel- eration of the, comet lanet may form a basis for at least a rude estimate of its probable mass. The faet that the Kaith, N'enus, and Mars have ,(MM),000. In view of the fact that the measured diameter is probably somewhat too small, these consider- ations lead us to conclude that the uuiss is probably between 1:0,000,000 and 1:0.000,000. As the v.alue of the mass to be used in investigating the secular variations, I have adoi)ted y = + 0.08 Mass of Mercury = 1.08 7 500 000 Secular variations resulting from theory. 54. In the Astronomical Papers, Vol. V, Part IV, were com- puted the secular variatious of the elements of the orbits in question using, as the basis of the work, the values of tlie \fc_:..,-^ ri4i TiiKoiMrrrcvL skculau variations. to; iiiiiHs»'s wliosr leoiprociils iin- toiiinl in tlir colmnii A 1h»|(>\v. Ill (.'oliimu U aro cited the musses whicU I have decided upon. A B OriKiual AdpohMl rociproiiil rucipriMiil ot' mass. of muss. y Mercury, 7."»(KMMM) <; on 14 1 + .O.S0 Wmiiis, no <)()(» KM J 7.'»0 4-.0 Kurtli + Moon, ;L'7 000 .i'JS 000 -.oo;{or> Mills, .^^^{^(N) .iO!);(.joo Jupiter, 104 7. S8 io47..{r) 4-.ooo.*)<» Saturn, ;{5oi.<>'^+ --Sc'-f 1.1 /'"-O.l ('"' = + 4.21 el)t;r, = + 100.3(» 4-0.00 -|-r.(J.8 4-18.8 4-O.r) =4-100.70 Dt/ =-|- 0.7(1 -0.04 - 0.0 - 1.4 4-0.0 =4- 0.70 sin / D, '^ =- 02.12 —0.3;{ -4!>.;i -12.2 —1.2 =- 02..j0 Veil lift. I>tC =- O.r.8 -1.30J/4- O.Oj''- 4.0i'"— 0.2/'"'=- 0.(»7 eDtar, =-f 0.39-0.81 4-0.0 - 3.0 4-0..") =-|- 0.34 Dtt =-f 3.43 4-0.70 -\- 0.0 -f 0.0 —0.3 =+ 3.49 sin iDt^ =-105.92 4-0.20 —20.2 —43.2 -1.2 =-10<;.00 108 el), TT THEORETICAL SECULAR VARIATIONS Uarfh. :- 8..")7 _o.lL>//_|_ i.;i,,/ = + llUd -0.18 ^ 5 ,s :- 4(}.l _28.3 [54 -1.0,'"'= _ ^.^7 + 1.0 =+ 1JI..39 -0.7 =_ 4(j.80 Mars. --+ 18.71 +0.(13,-+ 0.1,.'+ -J,!,." " _ , ,0';, = + U8.8.+0.0« + 4.«+,.,, Z+'l;^ — .'..« -0.04 +1L>.0 +0.0 +0.0,."'=_ ..o, - 72.43 -0.27 -.".,1 _;.4 I,., ^_ - i CHAPTER Vr. EXAMINATION OF THE HYPOTHESES BY WHICH THE DEVIATIONS OF THE SECULAR VARIATIONS FROM THEIR THEORETICAL VALUES MAY BE EXPLAINED. oo. Tlio inve8tijj:ationH of the present cliapter are founded on a comparison of the secular variations derived purely from observations in Chapter IV, with those resultinj;- 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 dilferenccs are collected in the following table. The mean errors assigned to tlu^ theoretical values are those which result frotn the prob- aWe 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 •I combination of those of the other two colunms. The errors of the observed quantities must not, however, be judged from those of the ditlorences, because subseqiieut changes in the masses of Mercury, Venus, and the Earth nuty produce a general diminution in the discordances. Mercury. Observation. // // T'.R'ory. // // Diff. \/w. Dte + 3..%i 0.50-1- 4.24^.01 -0.88-1 . no -0.80 :i c'DtTT +118.1'4 L0.40 -f-10!>.70 t.lO -f8.18L.43 . . Dt» + 7.14 4,0.80-1- 0.704-.01 -f-0..'J8:t-.80 -f0.;{8 1^ sin iDifi - Ol.80i0.45 _ 0L>..->0i:.10 -f0.01i..5l> -fO.23 2.2 Venus. Dte - 0.40±0.20- 0.07i.24 -|-0.21±.31 -f 0.12 5 el\7c -f 0.20i:0.20 -f- 0.34i.l5 -0.05±.25 . . Dti -f 3.87±0.30-|- 3.40 ±.14 +0.38 i. 33 4-0.44 3^ siniD^e -10.5.40±0.12 -100.{K)i.l2 +0.fiOi.l7 +0.52 8 109 110 COMPARISON or SEC;ULA11 VARIATIONS. [55 Earth. ObBervation. Theory. Ditf. J y,r. // /' DtC - 8.r)54(>.(M> - .S.57i.()4 4-0.02i.10 +0.0L' 10 eDtTT + 19.48-i:0.1li + 10.38i.05 +0.10i.l.'i . . Dtf - 47.11 i0.2;{ - 40.S!>±.01> -0.22±.27 -0.40 4^ Mars. l\e -\- 19.004:0.27 4- 18.71±.01 -|-0.20±.L>7 4-0.29 ;i.7 eDt^r + 149..").") 4- 0.35 +148.804.04 +0.7.") 4. .T) . . Dj/ - 2.2(J40.20 - 2.254.04-0.014.20+0.08 5 siiiiDt'^ - 72.0040.20 - T2.034.09 +0.034.22 -0.17 5 If we umltiply the nieuu errors given by 0.0745, to reduce tliem to probable errors, we shall see that only four of the fifteen ditterences are less than their probable errors. The deviations which call for especial consideration are the follow- ing four : 1. The motion ()f the perihelion of Mercury. The discord- ance i'.i che 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 ]\Iercury. The discordance is more than twice its probable error. It is to be emarked, however, that the ])robrble error of this quantity is very largely a matter of judgment, and that its value may have been under- estim.'ted. The deviations, if not due to erroneous masses, may be explained on two hypotheses. One is that propounded by l*rof. Hall,* that the gravitation of the 8un 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 hy])othesi8 is that of the action of unknown masses or arrangements of matter. Since the latter hypothesis * AaH'onomical Journal, Vol. XIV, p. 7. li 55, 56J NON-SPHERICITY OF THE SUN. HI would account for other motions than those of the perihelia, it might seem that the existence of the other discordaniies tells very strongly in its favor. The hypotheses of possible dis- tributions of unknown niatter, therefore, have tirst to be con- sidered.* Hypothesis of nonsphericity 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 <;enter and passing outward, the lirst question arises wh< ther 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 accepteIer<'ury, 1), tt Venus, Earth, Mars, 40.7; « Dt 7T = 8.;W 4.6 {}.0M 1..5 0.025 o.;j4 o.o;u Owing to the sniallness of the eccentricities the effect is insensible, except in the case of Mercury, so tliat the ring will not account for the observed excess of motion of the perihelion of ]Mars. Such a ring will necessarily ])roduce 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 orb;ts. Let us put ii for its inclination to the ecliptiit, and ^i for the longitude of its node on the ecliptic; and let us put, also, pi = /[ sin ^1 f/i = /, cos ^1 and let y>, />', . . . , r/, -i?,) = 40".7(i>-^,) Expressing the motions of p and q in terms of the motions oft and ^, which is necessary, owing to the very dilferent weights of the determination of the motion of the planes of Mercury and Venus in the dire<;tion of these two coordimites, we have 5<)90 N ALM 8 114 INTRA-MERCrRIAL GROUP. [57 I i'i 5*! '' ;:l I the following expressions for these two motions, which we equate to the observed excesses :* // 4.90 + 26.9 qi + 2SApi 0.27+ 0.8 + 3.0 0.00 + 28.4 - 26.9 0.00+ 3.0 - 0.8 0.00 0.0 — 1.5 + 0.57 ± 0.50 + 0.63 ± 0.12 + 0.50 ± 0.80 : + 0.45 ± 0.30 - 0.25 ± 0.25 Multiplying the conditional ejiuations thus formed by such factors as will make the mean error of each equation nearly dL 0".5, we have the following conditional eciuations for pi and 9i : 27(/, + 2Spi // = + 5.53 3 + 12 = + 3.00 17 -16 == + 0.30 5 - 1 = + 0.77 - 3 = - 0.50 The solution of these equations gives very nearly 9i = + 0.11; i)i = +0.12; ?, = 9° ^1 = 48 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 l)erihelion 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 uoticed that in formiug these equatious I have neither used the tiual 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 toaftect the conclusion. 57, 58] ZODIACAL LIGHT. 115 There would of course be uo dyuaraieal 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 j-TT^ that of the Sun itself, and one-fourth of this if at a distance etpial 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. ler used ct that hanges iou. 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 iuatter which extends out to, or a little beyond, the orbit of the Earth, the density of which diminishes very rapidly as wo recede from the Sun, The question arises whether the total mass of this matter m.ay 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 (piestion 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 diffimilty 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 sUghtly greater than that given by the law of the inverse square. This in(!rease 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 h it" r 1 i 1i 1 1 ! I 110 EXTRA.MEBCUIIIAL QROUP. f.j8, 59, GO Sun is relatively diminished as the planet passes from aphelion to perihelion, and the motion of tuo perihelion would bo 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 thai portion contained between the aphelion and the i)erihelion distantes. The result in erihelion of Mercury would change the jiosition of the planes of the orbitfs, and since observations give a])i)arent indications of such a <'hauge in the i)lane of the orbit of Venus, it might appear that we have here a very good ground for tlie view that all the motions are due to the attraction of unknown masses. But the great diniculty is that the excess of motion of the orbital planes is in the opposite direction from what we should expect. A grou]) of bodies revolving near the phine of the ecliptic would jtroduce a retrograd<^ motion of the nodes. But the observed excess is direct. A rodnce any ai)i)ie('iable etlect. Whether we, can assign to the components of such a group any magnitude so small that they would be individually invis- ible, aud 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 eflects, is a very imp(U'tant (piestion which can not be decided without exa1. 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 /• Force = "JUL the exponent n of r is not exactly 2, but 2 + rf, 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 » = 2.000 000 1574 the excess of motion of each i>erihelion 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 oil LAW OP GRAVITATION. 119 very siiiull eccentruities of their orbitvS, while tlie observed luotiou iu the ease of Mars is very eh)sely re|>resente(l. Dt^r «I)t;r // // Merniry, 42..^4 8.70 Venus, l^.-'iS 0.11 Kartli, lO.L'O 0.17 Mars, oA2 0.51 All iiidepeinlent test of this hyi)othesis in the case of otiier bodies is very desirable. The only case in which there is any hope of deterniininji: such an excess is that of the Moon, where the excess wouhl amount to about 140" per century. This is very nearly the hundred-thousandth i)art of the total motion of the perigee. The theoretical motion has not yet been com- puted with (juite this degree of precision. The only determi- nation which aims at it is that made by Hansen.* lie finds Theory. 01)8cr. Diff. // // // Annual mot. of perigee, 140 434.04; IMM.i.l.OO; 4-1.5(3 Annual mot. of node, -00 070.70 -0(m;70.02; -L'.SO The observed excess of motion agrees well with the hy]»oth- esis, but loses all sustaining force from the disagreement in the case of the node. The difterences Hansen attributes (wrongly, I think) to the deviation of the figure of the Moon from mechanical sphericity. Conftifttenvji of HaWs hypothesis with the ff€)i€ral results of the law of gravitation. 02. The law of the inverse scjuare is proven to a high degree of approximation through a wide range of distances. The close agreement between the observed parallax of the ^loon 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 tlie square can be only a small fraetion of the * Darlegung, etc.: Abhandhungen der Math.-Phys. Cluase der lion. Saehai- Bchen (ieselhchaft der Wisienvchaften , vi, p. 348. -^«- ».ju* '■ lao hall's IIYPOTIIESIH. («2 thousaiKltli jmit, <»r, wo iiuiy Huy, u quantity of th«^ or(l(>r of maKiiitixU' of tli(> livc-tlioiiHuiMllli ))art. Coiiiiiij^ down to snialU'i- distaniM's, w« llnd that the <;lo8e ajjieemcnt h'-twocn the dt-naity of the Earth as derived troni the attraction of small masses, at distances of a fraction of a meter, with th«^ density whi«h we niijjht /< priori snpposo the Earth to hav*-, shows that within a ranjje of distance extend- ing from less than one meter to more than six million meters, the acenmiilated deviation from the law can scarcely amount to its third part. The coincidence of the disturbin}^ force of the Snn ui)on the Moon with that computed upon the theory of jjravitation, extends the coincidence from the distance of the Moon to that of the Snn, while Ivhplkr's third law extends it t(> the outer planets of the system. Here, however, the result of observations so far nuide is relatively less pre- cise. We may therefore say, with entire coulidence, as a result of accurate measurement, that the law of the inverse square holds true within its live-thousandth part from a dis- tance e(|ual to the I^arth's radius to the distance of the Sun, a rauf^e of twenty lour thousand times; that it holds true within a third of its whole amount throu;;h the ranye 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 Franus, in which the multiplicatiou is tweutyfold. J ni all's hypothesis contradicted these couclusious it would be untenable, liut a very simple computation will show that, assuming*' the force to vary as r- - ' * , 6 beinj^ a minute cou- stant sullicieut to acccmnt for the motion of the perihelion of INIercury, the effect would be entirelj' 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 mate'ial point on a si)herical 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 suiiHce to include the visible universe would be very small. ■/ m\ roilKKl TION OF MASSES. 121 MasNrs it/ the planets irliivli rrpt'i'Hviit the neoular nd'hitionn of other eh'inentx thmt the perihelia. 1ia will rciiiaiii iiiicliaii^«Ml. Oiir iM'xt i)r(»l)U'm is to coiisidrr the possibility of leincsont- iiij^- tin' \ariatioiis of tln^ otluT t'h'iiH'iits l»y admissil)!)' masses ol" tln' known plaiu'ts. in i .Vi 1 have yiviMi a conipaiisoii of tlie scculai' variations as tlu-y result from obsiMvafions. with tlu'ir tht'orotical exjni'ssiims in terms of corrections to a cer tain system of masses. When the e(iiuiti«»ns thus formed are multiplied by the fa<'tors V ir. Mhieh make the mean erroi- of ea«'h e(|uatifMi unity, we have the followinj^' system of equa- turns, in which we jnit /' = lO.r; O.r + {', ,.' 4- 1' ,-" + (» ,' 0—1 — L' (» _ 7 _i(),s - 27 - .". - L' 1 - 1 0-1 -L'.ii -;u(; -i(( -f i;{ -1(5 -iL'U - ;i 0-1-8 -0.) + 2.-i + -'1 -lli — + I -11 + (50 -12(5 = - 1.7 = -I- o.r» = 4- O.a = 4- 0.0 == + i.:> = 4 4.2 = 4- 0.1' = - 2.0 r=. + 1.1 = -I- 0.-1 = - 0.8 - 1.8 + 0..-. -t-1.1 4-0.7 + l.a 0.0 + 0.1 -0.7 + i.;j -0.2 -0.2 The resulting; noruuil ecpiations are .-,70(;,,. _ ir.(5;i;''- 40!>1 /'" 4- 1 10 i'"' = -(- lU - 15G;{ + 101231 4- 88r)5(5 -|- .34.~m = - <570 _ 4901 4. 88.M(5 -I- 122 1(52 -|- IMrA) = - 1 14(5 4. 140 -f .'54r)r) 4- ;}7r)0 4- 101 = _ ;',j' - 1563 v' - 4091 v" + 140j'"' = +134 -1503 + 230831 4- 88556 +3455 = +374 -49!H + 88556 + 122462 +3750 = -1446 + 140 + 3455 + 3750 -i- 3S01 = -26 The solution oftliese equations ftives the following values of the unknown quantities: .1- = + 0.0071 i: .0120 /' = + 0.071 + .120 y' = + 0.0084 i .0024 / " = - 0.0177 ± .0035 u"' — + 0.0027 .01(5 Here aj^ain we note that, the Earth aside, the results for the masses are quite satisfactory. The correction to Prof. Hall's original mass of ^lars is so minute and so much less than its probahki error that we may consider this value of the mass to be confirmed, and may adopt it !•■>> Jefinitive without question. The corrections to the masses of Mercury and Venus are scarcely changed. The mean residu.il is reduced to f = i 0.01 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 Kanh thus derived: 7T = 8".750 L ".010 Tiie critical cvamination and comjiarison of this and other values of the i)arallax is the work of the next two chapters. r ill II;" if ' if Cn AFTER VII. VALUES OK THE PRINCIPAL CONSTANTS WHICH DEFINE THE MOTIONS OF THE EARTH. The Pnvensional Constant. 05; The accurate dctcriiiiiuitioii of the annual or centennial motion of precession is somewhat (lillicult, owing to its depend- ence on several distinct dements, and to the probable system- atic errors of the older observations in Right Ascension and Decimation. What is wanted is the annual motion of the e(|uinox, arising from the combined motions of the eijuator and ilu! eclii>tic, relative to |{M)Lhv, there is frequently such commu- nity of proi»er motion among neighboring stars that we <.'an not be (juite sjire that all rotation is eliminated in the general mean. Tuder these circumstances we liave only to make the best use that we can of existing material. We must also reuu'inber (hat observed Itight Ascensions are not directly referred to the efjuinox, but to the ►Sun, of which the error of absolute nu'an Right Ascension must bo det*^^ mined. This again can be done only from observed declina- tions, since by delinition the equinox is the point at which the Sun crosses the e(iuator. 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 the means include the whole list to l)e used aa absolute points of reference. We therctbre have three separate steps in (letenjiin- iug completely a correction to the adopted annual precession: (1) The correction to tlie Sun's absolute mean Hij;ht Ascen- sion or longitude. (2) The correction to the general mean Ifight 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 indepen. Reducing each determination to the equinox thus defined, we iuive the following results for the general preces- sion in Right Ascension at the epoch 1800: L. Stritve, from the comparison of Auwers Bradley with the modern I'ulkowa Right Ascensions . . . mj = 40".050l ; ?r = 4 Drevkr, from the comparison of LaLande's Right Ascensions with those of Schiellerfp 40 .0011; w = 2 Nyrkn, by the comparison of BesselV, Right Ascensions with those of Schjellerup 40 .0450; w = l Mean 40 .0520 [65 65] THE PREOESSIONAL CONSTANT. 127 The weights here assigned are of course a matter of judgment. Tlie general agreement of the results is as good as we could expect. From observed declinations we have — L. Struve, from the comi)arison of Atwers - Bradley with modern Pulkowa catalogues « = 20".0495; »r = 2 BoLTE, from the comparison of La- lande's Declinations with those of Scnj.ELLERUP 20 .0537 ; w = 1 Mean 20 .0500 We have now to combine these independent results. I ]>ro- pose to call Precessional Constant that function of the masses of the Sun, Earth, and Moon, and of the elements of the orbits of the Earth and Moon, which, being multiplied by half the sine of twice the obliquity, will give the annual or centennial motion of the pole on a great -circle, and being nndtii)lied by the cosine of the 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 Right Ascension and Declination — m = P cos'^ e — « sin L cosec 6 M = P sin t cos 5 L being the longitude of the instantaneous axis of rotation of the ecliptic, and u its annual or centennial motion. From the definitive obliipiity and masses of the planets ad(q>ted hereafter, we find the following values of «, L, and e, for 1800 and 1850: 1800. 1H50. . log«= 1.07372; 1.07341 L = 1730 2'.31 ; 1730 29'.08 f= 23 27.92; 23 27.53 128 THE PRECESSIONAL CONSTANT. We thus ttiul the following values of r, the unit of time being 100 solar yeais: From IkigUt Ascensions, From Declinations, p = r)490.12; w =2 r = 5489.44; n' = l Mean, P = 5480".89 As the data used in Strttve'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 Stbuve's work alone. They are // From liight Ascensions, Fi'oni Declinations, P = 5489.83 P = 5489.00 Giving double weight to the results from the Right Asceu sion^>., the results may be expressed as follows : // From Struve's investigation, p = .">489.57 From the other two works, P = 5490.18 B«'fore 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, 548!>".89, by more than' the probable error of the latter, and diverges fiom it in the direction of the best deter- mination, I have decided to adhere to it as the detinitive value. The centennial value of P is subjected to a secular diminu- tion of 0".00;?64 per ceutnry, owing to the secular diminution of the eccentricity of the Earth's orbit. We therefore adopt // // p = 5489.78 - 0.00304 T for a tropical century. p = 5489.90 - 0.00304 T for a Julian century. In the use of P I at first neglected the secular variation, but have added its efiect to the results developed in powers of the time. 66] THE CONSTANT OF NVTATION. Constant of nntai'm ilerivid from obscrrathnts. 129 <;6. The determination of this constant from observationH is extremely satisfactorj', owing to tlie cutation from the discord- ance of. the sepfirate results, but, to a certain extent, a matter of judgment. It.nurst be understood that the results below are not always those given by the authors who are (pioted, 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, Nvren's ecpmtions have been reconstructed on a system slightly ditterent from his, and have been corrected forCiiANDLER's variaticm of latitude. Peteks's classical work has also been corrected by the introduction of later data, and by a resolution of his equations. The Green- wich and Washington results have been derived from the dis- cussion in Astronomical Papers, Vol. II, Part VI. 5600 N ALM 9 it! W. ' 1.50 THE CONSTANT OF NUTATION. [()(» Valncn offhe constant of nutation derived from ohxerrationH. HuscH, from Bhadlky's ol>.servati<)iis with the zenith .sector ".».2.{L' 1 EoniNSON, from lliveiiwi<;h mnrjil circles . . 9.22 1 Peters, from Right Ascensions of Polaris . 0.214 4 Li'NDAiiL, fnrni Declinsitions of Polaris . . 9.2.'}<) X.'y Nyr6n, from ?» Urs. ^laj 9.2r»4 3 " '' oDracouis 9.242 2.5 »< '' / Draconis 9.240 4 DkBall, from Wagner's Ri{?ht Ascensions of Polaris t>.ir.2 3 De IUll, from Wagner's Declinations of Polaris 9.213 3 DeBall, from Wagner's Rijjht Ascensions ofrilfephei 9.2r>2 3 DeIUll, from Wagner's Declinations of 51 ('e]>hei 9.227 3 DeBall, from Wagner's Right Ascensions offHTrs. Min 9.208 3 De Bael, from Wagner's Declinations of rf Urs. Min 9.203 3 Greenwich XorthPolar Distances of Sonth- ern Stars, Series I 9.110 3 Green\v'ich North- Polar Distances of South- ern Stars, Series II 9.201 3 Greenwich North-Pohir Distances.of North- ern Stars, Series I 9.204 4 Greenwich North- Polar Distances of North- ern Stars, Series TI 9.223 4 Washington Transit Circle, southern stars . 9.217 « " " northern stars . 9.1 77 3 Greenwich, Right Ascensions of Polaris . . 9.1."»3 2 '< Declinations of Polaris .... 9.242 2 " Right Ascensions of 51 Ceilhei . 9.135 2 " Declinations of 51 Cephei . . . 9.102 2 " Right Ascensions of fJ Urs. Min. 9.147 2 " Declinations of 6 l^rs. Min. . . 9.235 2 " Right Ascensions of A Urs. Min. 9.1C1 1 *< Declinations of A Urs. Miu. . . 9.339 1 Mean 9.210; w<.=72 «(;,()7 PRECESSION AND NUTATION. 131 The iin'iin error eorreapondiufj: to weight unity wlieii derived from tlie diHcordance of the results i.s I (>".(M!8. wliile the estimate was i 0".07(). We luuy therefore put, as the resulj of observation — KelatioHs between the constatits of pr eves ft ion anti nutation, and the qxanfit'es on irhirh they depend. (17. The foriuuhe of precession and nutation liave been developed by Oppolzer w'th very great rigor and witli great numerical completeness as regards the elements of the Moon's orbit, in tlie first volume of his Buhnhestimmnng der Komcten and Planeten, second edition, Leipzig, 188L'. What is remarkable about this work 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 dis«u)very since Oppol- zer wrote. His theory however fails to take account of the change in the i)eriod of the Eulerian nutation produced Ivy the mobility of the ocean and the elasticity of the Earth. I'«ut this effect is of no importance in the present discussion. From Oppolzer's developments, 1 have I, 182 MASS OP THE MOON. [67,68 We thus liuve, luni-.s«)lar precossitm = 1* + P' f, the obli(|iiity ()t" the ecliptic; /<, the ratio of the masH of the Moon to tliat of the Earth; A, the nu'iin luomciit of inertia of tlie Earth rehitivo to axes pa.s.siny througli its eijuator; C, the same moment relative to its polar axis. With these iletiiiitions we have, (ieiK^ral viiliio. ."^in'cial viiliio for 1850. /' u ( ' - A N = [r).4(>289| cos 6 , '' y~~ = io.aG.")4l'| , '- " ~ /« C - A ^ J o.»;{75S5 1 J.L. ^' - "" ' ' 1 + /< C ('- A r =[."».!)7r)052]cos f l + ;< C P' = [;5.725()!)] cos f ^ ~ ^^ = |3.(J87(52] C The special values for 1850 are fouiul by putting for the value of the oblicpiity of the ecliptic for 18.^0, £ = 22° 27' 31".7 The )uasfi of the ^foon from the observed constant of nutation. 08. From the two quantities given by observation \ and P + P' =ih\, these equations enable us to determine the two (J A unknown quantities /.4 and - /i •• -^^^ 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 ditt'erent values of these quan- tities in the following table: Reciprocals of the mans of the Moon corresponding to different values of the nutation-constant and J mii-solar precession. /o N:=9'^20 N:=9'^2i N 1 -9''. 22 // . 50. 35 50.36 50. 37 81.81 81.86 81.91 81.53 81.58 81.63 81. 25 81.30 81-35 C8, l»yj THE CONSTANT OF ABERRATION. 133 Taking for the constaut of nutation the value Just found, y = 9".L'10 L "MS and for the luni-Hohu' ))re(!e«sion, p^ = .■iO".3ht Ascensions of close polar stars also lead to a value of this constant. But the same ditlienlty still exists. In this case the tnaxima and minima of aberration occur when the star culminates at noon and mi iixis, nor aro thoso foiiiulod on (h'rlinations of tlioso Hfais, if only tli«^ (U'clinatioiis an* observ*'*! ('(|Hiilly at Ixttli «'iiliniiia- tion.H. I>ur (Ictei'iniiiatioiis fonntion «)f water, ice, and air on its surface — W(»uld appear as an anumd tei-m in the latitude, six times as yreat as its actual amount. n 1)6 lie 111 Valtns of the constant of aberration (terireil from ohner rations. 10. What I hav<' done since this discovery by Ciiam)Li:r has been to rcexanune the detei-minations of the constant of aberration made from time to time, to make sncli conections in their bases as seemed necessary, and more especially to determine the correction to be applied to each sepaiate result on account of the periodic term in the latitude. No attempt was nnule to rework completely the original material, except in the case of the results of the I'ulkowa 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 nmtter of indifference whether we regard them as belonging to his work or to my own. Owing- to the very ditierent estimates placed by the astro- nomical world upon the Pulkowa determinations and those L — WU M 136 THE CONSTANT OF ABERRATION. [70 I m ml Is i : 1' f b It-.! made elsewhere, I have used the former quite apart from the others. The complete discussion of each separate value is too volununous for tlie present publication, and is therefore reserved for a more exteuded future publication. At pres- ent it appears sufHcient to judge the final result by the general discordance of the material on whicli it rests, rather thau by a separate criticism of each i>ai'ticular cmse. In the exhibit of results which follows it is to be remarked that NYUE^'s prime vertical observations do not receive a weigiit 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 wi*h 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 'lecessarily 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 thau 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 tluit if, in consequence of this diU'erence of aspect, the observer notes the passage of the faint image too late, the -effect will be to make the constant oi aberration too large. The existence of this IV. 'in of personal ei 2 Observations with Verticil Circle; 7 niis<"~lia leoiis stars, by I*eters 20.47 2 Observations with Vortical Circle; 18G;J-1870, Po- laris, by Gylden 20. tl 2 Observations with Vertical Circle; 1S71-1S75, Po- laris, by Nyren 20.51 2 Observations with l*riuie Vertical; 1S42-1844, by Struve 20.48 4 Observations with Prime \'ertical; 1.S70-1880 by ^^YRl:N 20.52 Obsc'.vations with Prime Vertical; 187.VIS70, by Nyren 2n.r).{ 1 Observations with ^'ertical (^Urcle; 180;>-IS7.">, by Gylden and Nyren 20.52 2 Wagner: Transits of three polar stars .... 20.48 5 From Pight Ascensions of Polaris; lS42-lcS44, by LiNDiiAaEN and Sciiweizer 20..">0 2 Mean reault: 20".49;{ L 0".011 Tliis residt nniy be regardeii. as identical with that fonnd by Nyren in 1882. B. Other determinations: Ah. e ut. AmvERS, from observations witli the ,, zenith sector at Kew l;(»,.5.'i [.12 0.5 ArwERS, from Wansted observations . 20.4<» .4z.l2 (»,5 Peters, from Pradley's obscrv;itions of y Draconis at Greenwich with zenith sector, 1750-1754 20.07 0.5 Bessel, from Pijiht Ascciisioh i observed by Bradley at Greenwich .... 20.71 L.071 0.5 LiNDENAU, from Pight Ascensions of J*olaris observed at various observa- tories between 1750 afid 1810 . . , . 20.45 ±.05 3 n 138 TJIE CONSTANT OF ABERRATION. tSrjtarate n-nults for the nnisUtnt of dhet't'tttioH J>. Otlier (k't( riiiiiiJitioii.s — Continued. I)RIM\LE\ , from ^.^^sel■viltion.s of thirteen "'• stars at Trinity C<>llej;« , J)ublin, with ^, the S foot ciide 20.40 Peters, I'roni Sira vi:\s J)orpat observa- tions of six jjairsofcircunipohir stars . L'(>..'3(* l{I(•lIARI)so^, i'roni observations witli tlie (1 reel! wieh innral circles L'O.oO PetkK'S, from Kiglit Ascensions of Polaris at Dori.at 20.41 IjUNDAHL, fi'om Declinations of P<>l:iris at Dorpat 20.05 HiiNDHRSoN an«l M<"Lear. from /»' and (\'^ Centauri I'O.o'J Main, from observati(tiis with the; (irreen- wici: zenith tnbe 20.20 J>()WNlN(l, from observations of /\J)ra eonis with reth'x /.eiiith tube .... 20..~)2 Xkwcomp., IVom observations <»f ^fLyra* with th(^ \\ asliiuffton lU'ime vertical transit, 18 CoMSToCK, using- LoKwvV method, sli-ihtly modified 20.44 Ki STNHR, from M Auoi se's observations*, 188!)-1800 20.40 Waxacii, from Pulkowa prime vertical observations 20.40 1 70 — Continued. wt. i.io I -L .07 >> ±.00 3 6 5 1.10 1 i.io I i .05 4 tO.4 ri .05 3 4 i.05 4 J, .04 5 3 -t.018 4 d= .015 4 70, 71j THE LUNAR 1N1:C)UAL1TY. 139 Separate rtsulU for the cnnstont of aberration — Contimied. I'.. OtluT deteriniiiations — (.'(mtimu*(l. .lb. wt. From Greenwich Kight Ascensions of ])oljir stars ,^ with the transit circh^ i'0.;{!) ;5 BEf;ivEK, from ohservations at Strasbnr};' by the Talc'ott method, 1S1M)-1,S!K3 LMU7 (5 Davidson, from simihir observations at Sun Francisco, 1802-1894 20. J8 (» Mean result of IJ: Ab. const. = 20".46;{ -1 0".013 The two results. A and B, dirter by 0".0;{0, a quantity so much f»reater than their mean errors as to leave room for a suspicion <»f constant error in one; or both means. Thr Lunar hutinaliti/ in the Earth\ motion. 71. The source of t- s ine(|uality is the revolution of the center of the Earth ai.'ind the <'enter of mass of the Karth and ^looii. The former center describes an orbit which is similar to tliat of the Moon around the Farth. Since tliis orbit is not a Keplerian eclipse, but is affected by all tlu' per- turbations of thi; ]\Ioon by the Sun. no such element as a semi- major axis can be assigned to it. Instead of this I take as the jtrincipal element of the orltit tin- coettlcient of the sine of the Moon's mean e! )ngation fron -nn in thr expression foi- the Sun's true longitude. This elcinenr is a tnnction (»f th«' -^ular parallax and of the mass of the Moon w hi» li may Ik ('('ii\cd from the foUowing expression. Let us put /<; the ratio of the mass of the ,Moon t>> ihar oi the Earth ; >•, A, /i^; the radius vector, true longitude and l;itiiudc of the Moon ; r',\',fi'', the same coordinates of the Sun; .s; the linear distance of tiie Earth's center frouj the center of mass of the Earth and Moon. ■M I I il 140 THE LVNAB INEQUALITY. I'l We then have, for the perturbations of the Suu'8 geocentric place due to the cause in question : J loy r' = *, cos (i cos {\--X') J\' = * cos /^ain (\-A.') J/3' = ' sin /i r and /< r 1 -f 7< »^' I have developed those exiuessions, putting TTo = .S".H48 /< = 81 and taking for the Moon's coordinates the values found by Delai'NAY. Putting D; the mean value of A— A.' (f, (f ; the mean anomalies of the Moon and Sun, respectively, v'; the Sun's mean elongation from the Moon's ascending node ; the result for JA' is JA' = (».533 sin D + 0.013 sin 3 D + 0.179 sin (D + fj) -0.4L»l)sin (I) -f/) + 0.174 sin (D —g') -0.0(51 ,vin (I) + r/') + 0.030 sill (3D — (/) -O.OU sin (D -(J -.(/') — 0.013 sin 2 «' This value of the lunar inequality is substantially uientical with that computed from the tables and formuhe of Lever- 711 THE LUNAR INEliUALlTY. 141 BIEB's solar tables. The development of the niunbers there given lead to the value (»".5;J4 of the principal eoetticient. We have now to find what valne of the coetlicient is given by observations. The observations I make use of are (1) all the observations of the Sun's Right Ascension from early in the century till 1804; (U) The heliometer observations of Vic- toria made in 1881) on Gill's i)lan and worked up by him. I had intende8; - .047 2.0 Dorpat, 1823-'38; 4- .100 0.3 Pulkowa, 1842-'04; - .058 0.5 Washington 1840-04; .000 0.2 Mean, JP = - 0".048 i- 0".01S Gill's result is given in the Monthly Xofices, lioi/al Astro- uomieal Society, for April, 1894 (Vol. LIV, page 3.")0.) It is derived in the following way. In the solar ephemeris which he usyd for comparison the lunar inequalities were computed rigorously from the coordinates of the Moon, putting 7r = 8".880 ;/ = 1 ^ 83 To the coefficient P thus arising he found a correction, JP = +O".O40 142 THE LUNAR INE(,>rALITY. |71,72 h I The above valiU'S (»f rr aiul /< yiv*', on tlu^ tlifory .just devel- oped, V = (I". K»0 Thus Gill's result is, in etlect, V = (;".44(; whih' Miine, from obfiervations of the Sun, is (;".53;? — ()".04S =<>".48ri I consider that these results are entitled to eijual weij^ht, an«l that we may take, as the result of observation, p = (;".4<;r» rt (►".oir. Soior paralht.r/roin the lunar iiiciiuality. 72. With the mass of the Moon already found from the observed coustant of nutation, ;/ = 1 :S1.5S (1 i .OOlio) we niiiy now derive a value of the solar parallax (piite inde pendent of all other values. The. relation between P, tt, and the mass of the Moon is of the jjfeneral form where k ia a numerical constant, and, for brevity, We have found that the following values <'(M'res]»ond to one theory : TT = 8".848 ; //' = 82 ; V = iV'.b'Sd Hence follows log /.• = 1.78207 BO that we have /y'P= [1.78207] TT The numerical values P = 6".4(m and /«' = 82.58 now give ;r = 8".818±0".030 li 72 73J PAKALLAX FROM TRANSITS OF VENIS. 143 1(1 IV e il Values of the solar paraUa.v derirol from meamrenun ts of Venus on the fare of the iSun dnrUuj the traimits of 1ST4 and 188:^, with the heliometer and photohelioijraph . 73. I put these (leterinination.s into one class because they rest essentiiilly on the same ]M-iiicipIe. lioth consist, in eftect, in measures of tlie distance between the center of Venus and tlie center of the Sun; the latter being defined through the visible limb. The method is therefore subject to this serious drawback : that the parallax depends ui)on the measured (lifter- euce between arcs which nniy 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 <'ondition given by the American photo graphs of 1374 are found in Part I of Observations of the Transit of Venus, December U, 1874; Washington, (jovernmeut Printing Oflice, 1880. A preliminary solution of these ecpia- tions, the only one, however, to which they have yet been sub- jected, was published by D. P. Todd, in the Ameriean Journal of Seience for .lune, 1881. (Vol. XXT, page 4!)0.) The photographs of 1882 have been completely worked up by Professor IIaukness, and the results are found in the Report of the Superintendent of the Naval Observatory for 1880. The ecjuations derived from the German heliometer measures, with a preliminary discussion of their results, are officially published by Dr. AuwEUS, in the Bericht ilber die deutsehen lieobachtuuyen, V, p. 710. The sepiirate results for the parallax, with the probai)le errors assigned by the investigators, are as follows: 1874: n 1882: Photographic distances, Position angles, Measures with heliometer, Photographic dist?,nces. Position angles, Measures with heliometer, 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 ± ".010 ,1 IC. w 8.888 ± 0.040 6 1 8.873 ± 0.060 3 3 8.87() ± 0.042 5 5 8.847 i 0.012 04 6 8.772 ±0.050 4 4 8.871) i 0.025 10 10 lU PARALLAX FROM TRilNSITS OP VENUS. 173 it If I ('(uiceive, liowever, that these rehitive weights lane. From this error the positi«m 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 — rr = 8".857 -1- ".016 This mean error is derived from the individual discordances, and n«)t from i'omparisons with the vahies of tlie parallax otherwise determined. As there may be a fortuitous agree- ment among the separate values, another estimate may bo made on the basis of the total mean error derived by Auwers, which includes all known sources of error. lie finds f = i. ".(►32 for the combined heliometer results, to which I have Pissigued weight 15. Hence, for the total weight 20, we have— e=i 0".02;i The deviation of the above result from the mean of all the other good ones is worthy of special attention. The deviation is more tlian 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 liave been considerably underestimated, or the method is affected with some undiscoverable sourca of systemati*; error, which nuxkes 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 err«»r is a systematic one, and that through some unknown caui^e Venus is always measured too low relatively from the center of the Sun. 1 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 Koaiiy the most probable value of the parallax, namely 8".80, we have — ■ // Mean on ror to weight 1 ; I .1 48 Mean error of result I: .0-*.> Solar itarnUux t'fnm nhserreil coHtnctn ilurinij transitu of VenU8, 74. The contact observations of 17«»1 and 17 are discussed in Astrnii3 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, f, = ± ".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 iuter ae. This is what we should expect from constant errors in the determi- nations of i)arallax 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 roo great or too small. I conclude, therefore, that the mean err^r 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 ; n = 8.776 ± .023 From external contacts ; tt = 8.908 ± .00 These meau errors are those derived from the concluded results and they show that the exteriml 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 meau error, and therefore assign a proportionally smaller weight to the results of external contact, we have, for the concluded result, 7T = 8".791 ± ".022 As these two hypotheses seem about equally probable, I shall adopt the mean result, 7r=:8".794 75] PARALLAX FRO 31 VELOCITY OF LIOUT. 147 Solar parallax from thv obnerved vomttant of aberration and measured velocity of light. 75. The question of tlie souiidnesH of the proposition that the aberratioti i.s equal to the quotient of the veh)cit> of the Earth in its orbit by the velocity of Ii},'ht is too broad a one to he discu8se98G0 Newcomb, including all determinations 299810 I have concluded, Velocity of light in vacuo, = 2998G0 ± 30 k. m. Taking as the etpiatorial radius of the Earth 0378.2 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. = 20.40 7T = 8.8076 20.47 8.8033 20.48 8.7990 20.49 8.794G 20.50 8.7903 20.51 8.7859 20.52 8.7810 20.53 8.7773 20.54 8.7730 148 • VHALLACrnc INE(,>rALITV. 7"), 70 Wi^ tliUH Imv<' for tin' valiU'M siiltiiij{ from tln' two values of tluj coiistiint of aberration alruiuly derived: // From I'nikowa determinations; Al». = L't>.4!».'{. rr = s.793 From miseellan«^ons determinations; Al). = 2t».i(».); tt = S.SOO ! ,: iSolar parnUnx fmm the partdlaclh' invijualii}! of tJir Moon. 7<». I Inive tlorivj'd a valneolllie parallaetie ine(|naiity <»♦' tlie Moon from the nieridnin observations made at (Jreeinvicli and Wasldnjjton sin<'e 1S«Jl*. The deternunation of this ineqnaliuy is i)eenliarly liable to systenuitie error, owin^ to the fact that observations have to be made on one lind> of tho Moon when the ineiinality is ]»ositive, and on the other lind) whou it is nefjative. Hence, if we determine the ineqnality by the eom- parison of its extrem(3 observed effects on tho Moon's longitnde or Rif^lit Ascension, any error in the adopted semidiamet'U" of the Moon will atfoct the result by its fall amonnt. It diiiiinisli the irnulia(iarisoii of tlu' uhsnvatioiis. the 4'oiTi'cti«Mi to Im' appliod in order to ri'iliur oltsnvatioiis made during dayli^lit or t\vili;;lit to what tliry would liave hvm had tile sky not bei-ii ilhiininuted. The reduction was sniaMrr than I had expected, and somewhat wn tjuantitics are — X, a constant, depending- (»n the semidianicter. p«'rsonal eipmtion, etc.; y, the eorreetion to the parallactic inefpiality of the .M(ton after reduction to the value 8".S4« of the solar jtarallax. (JliKKNWIcn. Li nth I, t '" h // 4.(;; .r-f 0.JK3.J/ = -0.53; irt. 0.2 5.6 0.00 - 0.72 o.i; 6.5 0.00 - 0.11 1 7.5 0.02 — 0.50 1 8.3 0.70 — 0.54 1 9.5 0.01 -0.13 1 10.5 o.;{s - 0.00 1 11.5 0.13 - o.ot; 1 L iiiih II. 12.5; x'-0.13y = + 0.20; wt, 1 13.5 - 0.38 -!- O.IG 14.5 - 0.01 + 0.28 1 5.5 - 0.7J» + 0.54 1<>.5 - 0.02 -0.1! 17.5 ^ 0.90 — 0.02 18.4 -0.90 + 0.44 0.5 10.4 — 0.03 + 1.21 0.2 150 PARALLACTIC INEQTTALITY. WASHINClTOIi. TJmb T. [76 4.0; •r + 0.03 y = - 1.02; ict. = 0.2 5.0 0.00 - 1 .2(5 0.4 0.5 (».00 - 0.85 7.5 0.92 - 0.04 8." 0.70 - 0.71 0.5 (KOI - 0.71 10.5 0.38 -0.48 11.5 0.1.i Lim -- 0.23 h JL 12.5; x'- 0.13 y = + 0.?!; irt. = 1 i;i.5 — 0.38 0.43 1 14.5 — 0.01 0.52 1 15.5 — 0.79 0.40 1 10.5 — 0.02 0.72 1 17.5 — 0.00 . 0.00 0.5 18.4 — 0.00 1.32 0.3 10.4 ^ 0.03 J. 50 0.1 With tlu'se <*(|natioii!S wo liavo our choice to deteriitinc the ',5iiranactic ine(|uality by assifjiii'ijif a valu" to the seini.55 — 1.23.r " '^ II; -0.28+ 1.23 .»' Wasliiiiffto I : Iamb T ; »/ = - 0.0!) - 1 .23 ,»■ "II; -0.88+1.20.1' If we choose to ittilize Mie observed diameters we liave the fol- lowing results: From 0(5 transits of the Moon's diameter observed at (Ireenwich ; .!■ - .»•' = - {y.iw 6 76] PARALLACTIC INEQiJALlTY. 151 From 33 transi<^s observed at Washington: .»•-.»•' = - r'.lL* We sliould thus liave, // Front Ureenwicili observations, y=— 0.02 From Wasliington observations, // = — 0.23 If, on he other liaud, we eliminate .r from eaeli pair of normal ecjuations, the final results for y will be // // // ..,f Gieenwich : Limb I ; (M>t // = - 0.4."»; y = — 0.70 4. O.IO " II; (M;4//= 0.00; »/= 0.(M> | 0.30 1* n Washington : Limb I ; 0.04 // = - 0.5l»; y = — O..SI J: O.Ki G " "II; iirtli y = - 0.32 ; y = - O.r.0 [: 0.27 3 The weighted mean of these results is .»/= -0".64 1 0".12 The resulting value ot the solar parallax is ;r = o".S02 I 0".00.S A very careful determination of the solar parallax was made from the same theory by Dr. IJatteuman, by meansof oceulta- tions, and the result is discMissed very fully in the publica- tions of the IJerlin Observatory. J)r. Battkuman's definitive result is TT = 8".704 A, ".010 I have slightly revised this result, by applying a c(U-reetiou to the ('oerticient for the parallax adopted by Dr. Battkrman, with the result T = 8". 789 :L ".010 Accepting this result, and combining it witli that ab*eady found from meridian observations, the parallax from this method will tinally come out 7T = 8".709 i ".007 This mean error may be reganled as belonging to the doubtful class. -^''1 162 SOLAR PARALLAX FROM MINOR PLANETS. [70,77 While tbis work is passing tlnongli the press there appears an important ]»aper by Franz of Konigsborg,* giving the value of the parallactic equation derived from observations on tlie lunar crater Mnsliug A. The correctiim to Hansen's eoetli- cient is found to be - 2".10 i 0".30 Tlie corresponding result for the solar parallax is 8".7G7 ± 0".021 We may combine the three results for the solar parallax tlius : Gret!\wicli and Washington meridian obser- ,, vation.: ;r = 8.802; tr = 6 Battkkmann from oceultations 8.7.S".>; 2 Franz from crater jl/o*///*^/ A 8.7«»7; 1 Mean 8.704 ±".008 Solar imrallax from ohscrraiionft on minor planetn icith the heliometer. 77. The fact that tlie 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 tlie absence of a sensible disk, was long since pointed out by Dr. Galle. In 1870 be pub- lished a discussion of observations on Flora, made at nine northern obscrvat(UMes, and at the Cape, Cordoba, and Mel- bourne in the Southern hemisjdiere.t The result was n = 8".873. An examination of the residuals of the several observatories shows that in tlie case of at least one of the Southern observa- tories there is a systeniatic diil'ercnce of a considerable fraction * Astronoinisolie Xachrlchteii, Vol. 136, 8.351. trdier oiiit) Hcstiiiiiimiig der Soinifn-l'nrullaxo aua corre8ponlan was put into operatir i. Then, in 1889 and 1890, a (!oncerted 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 compaiison itifrr 86 was m.ade at the observatories which took part in the measures. The tabular data for the reductions were supplied by the office of the BerUmr Jahrhnck, aiul the reductions and discu'^sion were made by Gill himself for Victoria and Sappho, and by Dr. Elk!N, on Gill's plans, for Iris. The three results, as comnuinicated in advance of their complete otlicial publication, are // // From Victoria: tt = 8.8(10 p. e. A: 0.006 Iris : 8.8l.'r» p. e. ± 0.(M)S Sappho: 8.7fM) p. e. J^ 0.012 I assign the resjiective weights 4, 2, and 1, thu.-* obtaining, as the final result of this method, 7r = 8".807 ± 0".00G I have included in a separate category Gill's determina- tion by Mars, at Ascension, in 1877, as jmblislM'd by the J r 154 UNCERTAINTY OY PARALLAX PROM MARS. [77, 78 ill lit Royal Astronomical Society {Memoirs Royal Antronomical So- ciety, Vol. XLVI), for the reason that, owing to the disk of Mar^, and its reddish color, determinations made on it are liable to errors peculiar to that planet, or at least dift'erent from those which might come in in the case of the small planets. Remarks on determinations of the parallax tchich 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 equatorial obser- vations on Mars. 1 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 gi-'en 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 st.ar 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 n strong suspicion of being affected by a cause which would cause it to deviate in one direction, it is logical to conclade a posteriori that the cause has not acted J8 0- )f e It II 78J FKCERTAINTY OF PARALLAX FROM MAKS. 155 if the (loviatiou is found *o be in the other direction, may not be a perfectly sonnd one, but I have nevertheless acted upon it. In the next place Gill himself, as a part of his discus- sion, compared the observations wlien Mars was at ditterent altitudes, in order to detennine 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. Vv . L. Elkin, to measure the ettect in . We have, iu what precedes, fouud or collected uiiie separate values of the i>arallax of the Sun, by methods of wliich seven may be rejj:arded 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 servations on the minor planets 8.807 i- .007 20 From the lunar etfuation in the motion of the Fart h 8.825^.030 1 From measurements of the distance of Venus from the 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 tleterinination, it seemed likely might attect them, no allowance being made for mere possibility of systematic error. The weights assigned are convenient snuill integers, generally sticli as to make the weight unity corre- spond to the mean error i 0".30, allowance being made, how- 158 THE 80LAR PARALLAX. ?J ever, for doubt as to what value should be assigned to the incau error and for the difl'ereut liabilities to systeiuatit* error. The mean result is — [:! From all deterniinatious; tt — 8.707 Omitting the first result; tt = .s.800 J: .0038 The last value ditt'era 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 bo materially greater than H"..SOO, 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 uuiy be considered as about 0".0(> or 0".07, which is four times the computed probable error. The prob- ability against this, eveu in the case of one result out of eight or nine, is so suuiU that we must either regard the method as being affecte, It i.s tnu' that results of meridian observations are combined witli them; but no cxpla- nation is thus atlorded of th(^ ditlleulty, bei'ause the results of these observations agree with those of the transits (r. §3J>). What adds to the embarrassment and prevents us fnun whitlly discarding the suspicion that some disturbing cause has acted on the motion of Venus, or tiiat sonu- theoretical error has crept into the work, is that, of all the determinations of the solar ]>arallax this is the one which seems the nu)st free from doubt arising from possible undiscovered sources of error. It is, as we shall i»resently 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 nunc e\a«'t estinuite of the sources «»f error and the nuignitude of the mean error. In this reexamination I have regarded the Sun^s parallax as a known (puuitity e(|ual to 8".71>8, aixl then obtained the results of the ol (.l«^ro- nomieal Paperti, Vol. II, Part V), I introduced a quantity expressive of the error in the observed time of contact arising from imperfections of the telescope aiul atmospheric absorp- tion and dispersion. The constants on which these errors depend are represented by synd)ol8 kt and A:> As 1 have worked up the observations, the ultimate result of each observation of contact is the value of an unknown done is to n-prrsent the undiscoverable minute corre<*tioii to tU' thus arising by the 8ynd)oI c^ for second contact, and ^i for thinl contact. In the present re examination the absolute terms are reduced to the parallax H".7!>H by putting Sn^ - - ".05 and n' = - ".025 in the linal - nuationsof the origiind paper. After each result is given the mean error with which it is aiVected, as deter ndned by the investigation in question. When thus treated, the equations which 1 have given on pages .iiH-.'JOH of the paper referred to give the following muinal equations for rfc, the indeterminates hi ami A*;, being retained as such in order to show their tinal elVect t)U the result. 1701. II; H.5 (Sv III; 41.7 6c 1769. 11; 41.8 6e HI; 12.1 (h- + 0.76 - 18. 1 k. 1 0.78 — 2.81 - 10. 2A-3 :|- 1.30 — S.OO - 104.1 A-, ± 1.05 + o.;m - It Au 1 0.70 In order to vary the i>roceeding as nuich as possible from that of the former investigation, I now express rfc in terms of S\ and Sfi, which, for the time being, I take as the cju-rections to the heliocentric longitude and latitude of Venus referred to the Karth, and these again in terms of 6r and sin i6f), which latter, for brevity, I call u. The lirst transfonnation is made with the coetlicients of \). 71, where we have put .r and — y for S\ and rf/J^, and the last by the «'<|uations // rfA = rfr -I- 0.06 i( 6(i = u — 0.06 y Putting Ml for the value of n in 1765, we have, in coD8e(j[uence of the known change in the motion of the node, // In 1761 ; tt = «, -f 0.11 In 1769; m = m, - O.U W| MOTION OF THE NODE OF VENUS. 101 \V> tlms have the four or|nntion^ whhh follow for «l(>teriniiiiii<; fVr iiiul III, the forim r hiiiiy supposed tlie same at the times of the two transits. // // _ ,84 ,Jr _ .55 »/, -)- :^i =s + 0.15 — L»,L' /... | O.m + .73 ._ .m 4- -. = 4. HM - 0.5 k-. I ().(>;{ _ .«iO + .T.J + :, = - 0.10 - L».;{ A . I (».(M + .81 4. .00 -I- c, = -I- 0.10 — l.;{ /,-, I (MMi Kliniituitiiijr any siihtiaetiii}; tlie first (' (ir + 1.28 «, = _ iK'jr* - 0.1 A. , 0.10 .OS ,Sr + 1.20 M, = + 0.00 - 0.8 A^ ± 0.07 We thus have hu- «; the vahie Wi = _ 0' .04 - 0.08 ,)r - 0.o;i A-, - 0.;{0 A, I ()".05 rfpcan not be determined independtMitly of c. an*' *. Assum- ing these quantities to be equal, wo have already fouml it to be uidy O'.ol'. and may therefore, to detmnine its luobable etfeet up<»n the result by assi|,'nin{>- to it the value 'h' = 0".(K> I 0".22 In tii'3 former paper i have found for A. and A;, the values // // J(i = + 0.040 I 0,040 A;, = _ 0.034 :L 0.040 A preliminary correction of -f 2".02 having been applied to the tabular fubital latitude, we have, for tlie epoch 1705.5 sin ifU* = 4. l".00 I: 0".00 Combininj'- this result with that of the transits of IS74 and 1882, we have the following results, whi|Mi(hs. We nuiy estimate tin* combinetl value of these at I 1", eoirespond- injf to an error »»r (»".(«» in sin / 1>, tin. Tlins we have sill / I >,()■«= _ L>".77 I ".0.S4 I coneeive this mean error to he as real as any that can be determined in aHtiiuiomy. This eonvietinn rests upon the tact (1) that the systen»ati<' eiiors alVeetin^' the lour eontaets are shown ti» be small by the f^eneral minuteness of the fotir values of fVr; (li) that whatever systematic errors may alVe".(>.S4 (a) which gives for r" the value y" = - (M)lll + O.tMMJ r - 0.<»7«i )' rl .nOU» The value of the sidar i)arallux for i" = is 8"..sil. Hence, for the value expressed in terms of the coire<'tions to the assumed masses of Venus and Mercury, this equation gives n = S".778 + 0".020 r - 1".08 i-' We have found from the periodic perturbatious J = - 0.055 i .25 r' = + 0.(i mass of Venus. In making' so lar};e a ehan;;e in this elenn-nt, wo iue<'t with insuperable ditlieiilties. The observed motion of the t'cliptie, which is a fairly well-determined quantity, indi- cat«'s a still further increase of this mass. We may put this ditliculty in another form. The observed motion of the nrxlo of \'enus is a relative one, consistinj; in the conduned etV«'ct of the motion of the e(diptic around an axis at right unifies to the node of N'enus, and an abstdute motion of the orbit of N'enus arouiul nearly the same axis. This motion ol' the ecliptic depends mainly on the mass of Venus; the absolute motion of the orbit of Venus mainly on that of the Karth. 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 ;;reater than we have supposed it to be, and shoidd tluM'efore llnd a yet smaller mass of the IC;irth than that heretofore concluded. The deternunation of the mass of Venus already made from observati(»ns of the Sun and Mercury seems to adndt of no doubt. We can not coiic«'ive that the mean of liftecn deter- minations, nnide duriuju; one hundred ami thirty years, at dif- ferent observattuies, which tleterminations are so separated as to be entirely independent of each (»ther, can be atfected by any considerable common error. The entire acccudance of the result thus reached from the i>eriodie i)erturbations produced by Venus with that from a combination of all the secular variations, as shown in Chapter VI, strengthens the result yet further. Unknowu actions and i)ossible defects of theory m ■ 1 i 104 SYSTEM ATir EUU«>RS OF PARALLAX. •O.M asi)|«'. it st'eins to iii«' tliat tin- valiu' of tlu* solar parallax ilcrivi'd troin tins discussion is h-ss M]H'n to doiibt from any known ciuisi* than anv (lotcrtnitnitioii that can be nnnli*. I'nxsihlr si/sh'tnotir nvnys in (IctcntkiuatioHH n/ Ihr panilhi.r, SI. Wo have now to return to the otln'j- values, in order to see to what i^xtent they may be afl'eetejl by systematic error. I have aheady excused myself from discnssiny the validity of the assumed relation between the c".(>'-• than the result of Nykkn's best dj-terndmitions. fVs for niiscellane«>us determinations of the constant, it is to be remenil>ered that the corre(!tions api)lied to a part of the separate values on a<'«'ount of the ('InnuUeriau inequality <»f latitude arc som<'what doubtfid, and the ^'en<'ral mean uuiv have been affect«'«l by a few humlredths of a se«'ond in conse- queuce. It is not, however, possible to determine the amount of the (orrection, I'xcept by an exhaustive rediscussion of the whole of the orifjinal observations, ami even then the result wouhl still be doubtful. J'ext in the order of weight we ha>e the lesnlts of measures oc. the nunor planets with the heiiometer, on (Jin/s plan. I !iave already remarked upon the possible error in such obser- vations aiisin^i from the probal>h> ditterence of c(dor between the platiet and the star. A hypothetical estinuite of the am«nint «>f this erroi' is worth attemptinj;. Let us assnnu' that in the case of a minor planet the mean of the visdde spec- trum corresponds to tlu- line 1). and that in the ca^e of a star tKe same mean is halfway between the lines 1) and K. 811 SYSTEMATIC EUBOBS OF I'AUALLAX. 1(>5 The imlcx of it'Ciiutioii of air has Immmi (letrniiined iiule ]H>ii(1eiitIy by Kin tlkr aixl Lokkmz for the dirt'eieiit rays. Tlu' mean of their results (or the ravs I> and K is 1m »r I), n = l.(M»OLM >:.>«> For K. )i = 1.(100 IMMO These results are accordant in ;,'i\iiiji a dispersion between these two lines equal to about .(KKST of the total refra<'tion. We have hypothetically taken the extreme possible ditVerence between ])lanet and star to be ohe-iialCof this. At an altitmU^ of l."*"^, where the refraction is about (10", (lie err()r would be 0".1I. At iin altitmle of IW (he error would be ((".I'O. We are thus led tt» the noteworthy conclusion: ll'thi' tli()'nin(«' hi tinrii i'l' .^fHrtra <>/ innl 11. iin ivror of 0".(>S or (>".(>■' 1 111(11/ he proihinit in tJir iijiitnrciit paraHn.r of tlw phiiu't. The (pu'stion tlips urisiii^' may be i-eadily settled by measures with the hcliometer. The distances of pairs of stars ditVcrin;; as widely as possible in ccdoi- sliould be measured at ditVerent altituiies, when one is nearly al>t>ve oi- bdow the other, in order to see what ditlerence of refraction depcndin;^ on the color is iiulicated. A colored doultle star, such as fi (.'ygni, miji'ht also be used for the same purpose. The minor planets are oi' (litfer«Mit <'olors. I am not aware of any evidence that \'i<'t(uia or Sapplm already remarked tliat the valin* oi' tiie s<»lar parallax derived from the paralla<-tic etpnition oftlieMoon is one of which the probable mean ei ror is subject to imcertainty. While it is true that the value may be smaller than that we have assigned, we nnist also admit that it may 1h> much larjjer. The probable error of the determinati or lesH probable sources of systematic error Just pointed out. The nnxlilications we maU(^ in the weights lornu'rly assigned are these: We redu<'e the weight of (llLJ.'.s Ascension result to oiH' half, owing to the unceitainty arising from the color of the planet Mars. We r«'tain the iMilkowa ory that the actual nn'an error nnist be larger than that given by the disi-oidanci' of results. Our combination will then In- as Ibllows: From the iiKidoii of thf uoile of \'riiiis , . . n-sf 8.7<»H 10 Frotn iiiiA.'fi Asccnuioii olmcrratioHs .... S.T.SU I From thr Pidkoirti ionslaiif of alnrr(ttioit . . . S.7!K{ 40 From vontortH of Vnnis iritli llir tSiiii's limh . . S.TiU ."{ From li(liomr(tr olmcrraiions on Victoria ami Sapi»lii> H.71M> .') From thr jtarallartir iiminalitji if tlir Moon . . H.TIM 10 From niiHfi ilannms ilrtvrminationx of tlir con- stant of ahrrration S.SOtJ 10 From the hntar incfinalitii in the motion of the Forth H.818 1 From nicannres on Venus in transit S.H.*)? 1 Mt-an residt, ignoring the llrHt; 8".70«'i."» I .004.") This mean result still ditVers t'rouj that given by the motion of the node < f \enus by nnue than ii\e time.s the i»robable eri*«rof the litter, and is vet farther from the combined result 821 llESULTH FOR THE SOLAR PARALLAX. 107 of all the st'culiir variations, so that no nM-onciliation is broufjht about. The eiiiharrassin^r (pu'stion whith now morts us is whether we have here sonx' uni\iio'.vn einise of tlilfereiire, <»r whether the (liscrepaney arises from iin aceiileiitai aectiniulatioii of I'ortnitous <'rrors in tht; separati* of al>norinal action. The motion of the phmes of the orbits is tluit whieh is h'ast bk«'ly to (h'viate from th«'ory, because it is independent of all forms of action dependin^^ u])on «listanee from the 8nn, or upon the vlocity of t lie i)lanct. An examination and comparison of all the results shows one curious feature: the unanimity with which the secular varia- tions speak auainst the larye value of the solar parallax, or of the mass of the llartli. as tli * one quantity at fault. The adopted motion of tiie node of Veinis is sustained not only by the meridian observations, but l>y th«' external contacts at the transits <»f 17«il and 17(»'.», and, wiakly, by a comparison ol the transits of 1S7I and ISSi*. if we deteriiiiiH' the coireclion of the inassoftlie Marth from other se<-nlar variations than that of the node of Venus, by the e<|uations of ^ (i.'l. we havi', alter eliminating tlu' masses of Mercury und Venus, r" = — n.02!»; p. V. I .(US If, insteivl of eliminating tiiese values, we put I' — + .08 J U' ss + .0080; we have I'" = - n.OLMi: p. e. i .011 In eaeh ca«e the value of tlu' parallax is yet smaller than that found from the motion of the node of N'enus. 1 have already remarked that the observed motion of the ecliptic indicrates an iii'-rease of the nniss of Venus. The (piestion thus lakes the form, whether it is possible that the mean of the ^even determinations of the Nolar [tarallax t TT =s.S".707 I "Am') ir,a DEFINITIVE ADJUST «ENT. [82, 83 can with reasonublc possiljility be in error by an amount the wjrreetiun of wliieli would brin^' it within the ranj;e of adjuHt- inent of the other (|nantities. From what has aheady been said of the systenmtii; errors to wiiieh tnery one ()f the determinations may be Habh', it is eviih'nt that we shouhl liave no dir.ieulty in aeeepting the iieeessiiiy reduction «if v.w.h of tln^ separate values. The improbability which meets us is uot so nuich the anumnt of the individual errors of the ht independent deterinimitions should all be hiifj^ely in error in the same direi'lion.* Still, under the eir- eumstaiu'«'s, we must admit this jjossibility, and nuike what seems to be the best adjustment of all t!:f' '"esults. V- Ih' fin it ire (nljustmnit. 8.S. In niiikintj the delinitive adjustment 1 shall i»roceed on the supposition that no correction is necessary to theadoptetl mass of Mars, I also {^o on the i>rinciple that in> result is to be rejeete*! on account of «loubt or discordance, except when it is alfcctcd witli a wcirestalilished causeof syst«'matic error, and sIkiws a larye deviation in tlu! direction in which thii cause would act. At the sanu' time it will be admissible to diminish the weiylnts in special esult must have been greatly underestimated, l^roceediiiji' on this plan, we mi^ht rt'weijiht the last eij;ht results fui' the snlar parallax, so as to j^et a result slijihtly dilVerent from S' .7!»7. Ibit 1 rration on the gi'onnd that no one nnthod ; hould hav«' so luepoiulerating a weight as this has. li we did so the result might be increased to .S".8(K». Wo For a vi-r\ McartlmiK criticisin nt'tlio syHli'iiiatir i-rrnrHwitli which the 'ffd, n'foi— luc may 1 *e (Iftt'i'iiiinatioiiH <>t' thi> solar parallax may nt iiiatlc t(» the liiHt two arti« Kh 1»,v Dr. David Giu., iu WA. I of The obterva' '"'■.'/■ 831 DEFINITIVK AD.I I STMKNT. Ui\t m'gbt very larffoly increase the relative weiylit assignetl to the helioineter observations on N'ietoria and Sapplut. but no admissible in«-reast' would appreciably ehan;ie the result. We niigiit iflso diminish the relative weight of the largely dis conlant result derived from nn>asures of Venus «luring tninsii. lint as, by throwing out this result altogether, we shoidd only diminish the mean by ".«KH. it is seareely worth while to do so. -Vltogether no rediseussion (»f the relative weights seems necessary. On the other hand, the weight whi<'h we assign to the mean result will enter as a very impoitant factor into tiie final adJiiStni«Mit. This is a point on which ii is impossible to reach a positive numerical conclusion by any mathematical process. If, as one extreme case, we considi'r that the mean error ot" eacii separate result Ci»rres[«)nds to I (»".03 for weight unity, we shall have a imnin error of I ".(KKJo for tlie value s".7J»7. The residt will not be very dillerent if we determine the mean error from the discord'ince of tin' eight sepaiate results. On the other hand, if we include the deviation of tin* result givi-n by the motion of the node of N'enus. the uhmii erriU" for weight unity will be increased to L '.(MM5, The latter is undonbt edly the most logical c<»urs(>, so long as we proceed on the hyp«)thesis that the d«'\iations of the tinal adjustment <'au all be exi»lained as due to fortuitous errors. If we include a cotn l)arison with the results of all the secular variations we shall have a yet larger mean error. To show the lesult of assigning one weight or the other I siiall make tw(» solutions, A and 11, in one of which a h'ss and in the otiier a gieater weight will be assigned. To the value S".7!>7 i .0(»:» or j .007 of I lie solai parallax correspond.'. ]'" = - (MH!» i .(MHI'm.!' .(10". According as \\v assign one weight ur ihe other to this rcMilt, we may take as the corresponding ei|iialion ol condillon )f weight unity or (A); (B); 4()(h " = - 2.0 000-" = - 2.U (") 170 DEFINITIVE ADJISTMENT. 183 The masses of Venus and Mercury, ileterinined by niethoils in(le)>(U(lently of the seeuhir Viiriat ions, also enter as conditions into the adjustment. I have, however, made a revision of the preliminary adjustment ^^iven in § <»4, the hitter heiiifif l{ase4.MI(K» ih) The Viduea of the denonnnator corresi)on«lin^ to the mean limits here assijrned are .^SIKMMMI and 12lM(M)0(> These limits are tut wide as to include all admissible results for the nmss of Mercury. Moreover, we can not dellnitely say that the value (h) of this mass is markcilly {;reater or h'ss than Hint jjivcn by the wei^ihtccl mean of all other results, since we miyht so weight the latter as to jfivc a result {jreater or less without transcending tlu' bounds of judicrious Judf^ment. I conceive, theretbre, that we are Justilled in reducinj,' the mean error to | (».l,M», which will yiv*' i«s the e<|uation of cjondition and hence (>.().").•> I O.L'5 U).v= - ().2'J II {<') When, in the normal «'<|uation for the mass of Venus, given by the observations on Mercury, we sulistifut*^ the values of the secular \ariation.s found from the general condiination of § Mi, the result is )'=-(MHl4 Coudiining this with the result from the Hun, wi> have 1 '= -U.UllT In view of the (iwt that the nmss lU'rivcd ft'oni observations of Mercury may be alVecte NN'ith tlu^se eqiuitions of coiidiiion wr liave to coinbine the eleven e(|uatioiis of «0.i. wiiich we nse unehaii;xe«l, exeept that we double the wei;iht assifjiied to tlu' sixtii ••((nation, that deiived from the motion of tin; node of N'einis, on a('<'onnt of the .smaller i>n>l»able eiior of tin' I'esnit of our precedinji;' redis- eussioii, and use the value of the alisolute teiiii fonml in ^^SO. If we aeeept the view that all the perilielia nn»ve accordiiij; to the sauu^ law of j^ravitation towaid the Snn, nannls , that expressed by llAiJ/s hypothesis, then the \aIneol the (pniu- tity '') in the formida expressiii}; the law of /^navitation is so well determinetionof the perihelion of Mars, it is a <|Uestion whether we should not exclude that motion fiom the equations. The conditional equations ^'iven by the motions of the thieo perihelia in question aie found l»y comparinH" the resnitsof M4t), 54, and (»1. Tliey are 40.r -I-. /' + 20 r" =: -f 1.0 14 4- 40 4-0 = - 0.;; li - l.l + 0! = 4- (►.: {«> Tin' conditional eqinitions to be combined aic the eleven e(|nations of i(i;$, the sixth of which is to have (|ii!itioiis to wliirli \v«» arc thuH UmI arc tli« i'ollowiiig, wliicli sIkiw tlic rcsiiltM of the tour (■oinbiiiatioiis we may inakt> accordiiiK as we iihc (A) or (H) tor tiic 4M|iiati()ii givt'ii by tlic mass of tlio Martli, and omit or iiirliuh' tlic third ('(|iiiitioii (f/), which 18 ;j;ivcii by the motion of the perihelion of Mars. {n.) Iwhiilitiff the motiitn of tliv perihelion of Marx. \) m-j - 7 I 17 1' - 11 .'{.Tm" = + L'LM) - 7 117 +207 171 4-H>«7l-'7 =-587 _ II ;{.{.-» 4-H»S7*J7 + MM».'{0(> =-;{;iSS(A) - 1 1 ;{.'{.'» -I- KiH 7l'7 -f (i(Mi :\m = - i.L's ( \\) {fi.) (hnHtintj the motion of thr prrihelion iff Mot'ft. 1Mi(K5.r - 7 iL'l I'' - 11 157 1" = -f L'lH - 7 121 -fL'(»7 0(»;{ + HIMoL'O =-.'.78 - 11 i:»7 4- n;u:.LM> -i- 4(r.»r»78 = - 'MM (A) - 11 l.'»7 -I- Hl'jr.L'O + ()0L'."i7S = _ l;{71 (M) The lesubs of tlie solutions in tin- f'onr eases are: A a .r f 0.0117 I' +0.117 !'■ -1-0.001 .11 ,/" _ (i.no!» 7.{ I J- „t C'lKOOOO 1 -r in' lOS L'.iO TT 8".7s;s A/i 4- (MM 12 + 0.112 + (UHll ti(» — o.(noo."» i> r»(;7 000 lOS 120 .H".7H2 + 0.01(il + o.n»i + 0.00:1 10 — 0.007 70 100 000 lOH 7.50 .S".7.S1> + 0.015S + o.l."»s + 0.00; { 2."» - 0.007 87 (;J77tMK> 408 070 8".78.S 1 eonceive that il'tlie se<'iilar vaiiations, especially the motion ot the no«le of \cnus, arc not atVceted l>y any unkn(»\vn cansc, some mean between these shcnild be r«';iarded as the most ])robablc solntion. Tiie resnif does not, howevei'. biin;;: alxiut a satisfiictory leconciliation. \V«' still lind onrsi'lvrsecnifronled by this embarrassing dilemma : llilher there is .somethinff abnormal in connection with the node of Venns, due to an nnkin)wn canse aetinj; on tlio planet, to some extraordinary errors in the ob.srrvations or their reduction, or to some error in the theory on which the di.senssion iu based, or the deter- N 8.'i, Sl,H.'»| ADOI'TKI) I'AUALLAX AND MASSKS. 1 7.*t minatioiiM of tli<> solar parallax an> lu'arly all in error in n\ni (lins-tioii by aiiioiiiilH wliicli are, in more than one case, «|uite snrprising. VoHHiblv rauHvH iff the ohs^Tiu il ^liMronhnirt'M. 84. Two possible causes of the hypotheses of ellipticity of the Sun wo should have a discrepancy of about .{".0 between theory and observati(Ui. This iMUidusion aloin' seems fatal to the theory, which otherwise has been shown to be s<'arc«'l\' tenable. The other possible cause is an iiuMpiality of Ion;; period; especisilly one dcpendinj; lui the ar;.;uineiit ]'M" — Sl' which has a period of about two hundred and forty three years. A very simple <'oiiiputation shows that the coelllcient of this term is only of tiie order of magnitude O'^Ol. It is a curious coincidi-nce that if we had neylecled to add the mass of the Moon to that of the ICarth, in eompntiii;: the .secular variati«»ns, the discrepain-y would not Ini'.e i'xisted. Adopfnl rahics of thv doubtful (jKantiti'n, 8.">. The practical ipiestion which has been before the writer in working out tin* pre MASSKS. l» (■<)in|tiii-iHoiis of such talilcH witli obsci vittiuiiH, except after lediiciiijt the tahuhii results to some coiisistiMit thtMiry. Tho iiii|iosilioii of siicli a hiWor upon the future iuvestiuiitoi- is not t«i he tliou^ht of. MoreoNcr, there is no ctMtaiiity that tho tal»h's whii'li wouhl l>est represent past ohservations would also i test lepH'sent future ones. Our tahlus must he fouudetl on sonu> perfectly consistent th«>ory, as simple as possible, the t'lcMU'iits of which shall be so chosen as best to represent tho obs(>rvations. In choosing the theory an position of a planet or ^Moup of planets between Mercury ami Venus. It s(>ems to me that the introdiU'tion of the action of such a {froup into astnuiomical tables wiailtl not be Justiliable. The more I have relUM-tcd upon the subject the uuue stronifly Hcems to me the evidence that no such yroup can exist, and, indeed, that whatexer anomalies exist cannot be due to the action of unknown masses of nuitter. Besides, the six elements of sim'Ii a j;roup wouhl constitute a complication in the tabular theory. On the other hand, it did not seem to me best that we sluudd wlutlly reject the possibility of some abn(u;nnil action ov some defect between the assunu'd relations of the various quanti- ties. What I tlnally dt'cided on doin;; was to increase the theo- retical motion of each periheli(Ui by the same fraction of the mean motion, a cours(> which will represent the observations without committing us to any hypothesis as to the cause of the excess of motion, tlnuigh it accords with the result of Hall's hypothesis of the law of y:ravitation; to reject entirely the ii) p(»thesis of the action of uidvuown nnisses, and to adopt for the elements what we mi^iht call coujpromise values between those reached by the preceding adjustment and thos«' which would exist if there is abnormal action. The exijjency of hav- iuiX to prejtarc the tal)les reipiircd me to reach a conclusion on this subject before the tinal revision of the preceding discus- nr>,m\ Fl tiki: DKTKKMINATIONM. nr> HJoii, so that the iiiimlH'is iisnl an* not wliolly based upon it. The coiirliisioiis I have rcachtMl an* these: SjiM-e, if there is notliiii); altiioriiial in tlie tlieoiy, the sohit paialhix is probably not intieli hir^er tlian H".7HU, anil It there is anything abnonnal it is probalily as hirt;e uh S".7!)."» or even 8".HtMt, we may adopt the vahie H".7!»0 as one whii-h is almost certainly too lar);e ou the one hypothesis and too snnill on the other, and \vhi«h is tlu'ret'ore best adapted to alVord a decision ol tlie qnestion. For the nmss ol Venus I t«M»k, as an intermediate value, IM' = l-;-|O.S(MM> For the uiaas «)f Men-ury I t«M»k 1 - C.tNNMHNi Actually it seems that this masH is larp'r than the most prob- able one on either hypothesis, tliou^di not without the ran^i' ot easy possibility. With tlu'se values the outstanding; dill'erence between theory and observatiiui in the centennial motion ot' the node of Venus 18 Jsin/I>, " = 0".L'5 If this dilVerence arises wlndly from the error of the theory, then between the transits of ISTI and L'(M)4 the accumulated error woidd amount to 0"..'i2 in the heliocentric latitude, and about i)".S in the p>o<'entric latitude. Unless an improvetnent is made in the nu>thod of determining^ the position of N'entis by observation, the twentieth century must approach its end before this dilVeri'iice can l»e dete«ted. liearinfi of futKrt' (ictiriniimtionn on tin- qu(»tisn|uent determinations <»f the principal elements will have upon our Jud^nuMit a.s to the solution of the dilemma. The changes in the second column will, by emphasi/.iiij; the discordance between tin restdts, tend to conlirm the hypothesis of an abnorr ..' .!'>feet in the theory, while the opposite ones, in the latit column, will tend to reconcile theory and observation: IMAGE EVALUATION TEST TARGET (MT-S) k A A I/. fA 1.25 us 1^ IM 2.0 M 11 16 V] y ^> * 4"^ ^ Photographic Sciences Corporatiori 23 WeST MAIN STMIT WEBSTER, N.Y. 14580 (716) 872-4503 V -^^ N> ;\ <^.>^ ^ .,<» ^» louj;- a period for a coiiiplotc observation. 'Die yivai leti<»n 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 tlie ICnropean observatories are Itivorable to the determinatioji of tlie alx-rration-constant. Loewy's method has over all others the great advantaf,'e of being iiulependent of the direction of the vertical. 15ut its ap]>lication, and the reduction of the observations made with it, are laborious in a high degree. So far as practical astronomy has yet develoi)ed, the best nicl .»d of directly measuring planetary jtarallax, and there- fore the only one to be considered, is that of (Jill. It there- fore seems desirable that measures by this method shouhl Ije continued. At the same time it is very necessary that the spectra of the small planets to be used should be carefully studied i)hotometrically, and that the probable inlluence of coloration upon ihe 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 up' to a difference of 0".1lM in the heliocentric lati- tude. This is considerably laiger than the ]>ro])able error of the results of the observations of the transits. It may, there- fore, be <|uestioned whether the latter are not entitled to a greater relative weijjht than that assigned, owin;^' to the prob- able systematic eirors of the meridian observations. A second value [h) has therefore been derived from the observations of the transits alone. In subsecpient investigations we may choose between these two values. Formation of drtinitive elements of the four inner pkmets, for tlii\ epoch 11^50, January 0, Greenwich mean noon. Mercury. Unknown of Corr. of equations. element. Rcl. to 1850, // o.o Tabular element. Concluded element. // // 538 100 (554.40 538 100 053.72 n -.0940 - 0.77 e - .0741 - 0.222 - 0.005 42 400.088 42 408.861 75 7 13.78 75 7 VX37 323 11 23.53 323 11 23.83 7 7.71 7 7.00 40 33 8.03 40 33 12.24 jt + .0703 -I- 5.59 I —.0402 + 0.30 i - .2702 J - 0.04 ~ 0.07 d — .0001N+ 3.88 - 0.27 f » ' ^ J 1 '' i 182 DEFINITIVE QUANTITIES. [88,89 Formation of dejinitive elements, etc. — Continued. Ven lift. Unknowuot C'OIT. of Ked. to T.altiilar Conchuled equations. clt'lMCllt. 1850. element. element. // // // // n - .1783 - 3.57 210 669 165.04 210 669161.47 e + .1403 + 0.43C ) - 0.105 1 411.522 1411.796 71 + .0835 + 36.6 -16.4 129 27 14.3 129 27 34.5 I -.1330 - 0.67 + 0.46 243 57 44.34 243 57 44.13 i + .0008 J + 0.31 + 0.12 3 23 34.83 3 23 35.26 e{a)+ .0120 ]S - 9.39 + 6.63 75 19 52.21 7o 19 49.45 ^(6) - 20.36 +15.56 • Earth. // 75 19 47.41 n -1.10 129 602 767.84 129 002 766.74 e + 0.12 3 459.334 3 459.454 7t -2.4 100 21 43.4 100 21 41.0 e - 0.15 23 27 31.83 23 27 31.68 I + 0.02 // 99 Mars. 48 18.72 ft 99 48 18.74 // n - .1094 -0.88 68 910 105.38 68 910 104.50 e - .1088 - 0.155 + 0.058 19 237.101 19 237.004 TV + .1063 + 2.38 + 0.02 333 17 52.47 333 17 54^87 I - .4029 -0.81 + 0.05 83 9 16.92 83 9 16.16 i - .0507 J + 0.18 - 0.01 1 51 2.28 1 51 2.45 +. 1 135 N+ 0.56 + 1.34 48 23 53.02 48 24 0.92 Definitive values of the secular variations. 89. The definitive values of tiie 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 foun4 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 Dt Ttt, is measured along the plane of the orbit itself. The numbers 891 SECULAR VARIATIONS. 183 given being divided by the corresponding valnes of the eccen- tricity we have the motion of the perilielion itself along the plane. The symbols /„ Ji"d ^o 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;' fi. The transformations to the latter (pumtities are nnide 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 rtxed. If we put i„ the inclination of the fixed orbit of the planet at any epoch To to the moving ecliptic at any time; ^1, the longitude of the corresponding node, Q i ; V, the distance from the node Q i to the instantaneous rota- tion axis of the orbit at the epoch To ; we shall have Dt V = h" cosec i'l sin (L" — di) {a) If we compute vo and h from the ecpiations H sin vo = sin lo D? fti H cos Vo = D? 'alu this correction are // Mercury ; Dt 71 = 43.37 Venus; 10.98 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 Dt 7t" = 7".08 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 iu Astronomical Papers^ Vol. V, Part IV, pp. 337, 338. 891 SECULAR VAlllATIONH. is: Secular rariationn <>/ the ckmcniH of the Join- orbits at flic tlmr epochs, IHOO, 1850, anil 3100, as inferred from the (lefinitivclij adopted masses. Mercury. 1600. 18.iU. I'll 10. // // // Jhe 4- 4.257 + 4.227 + l.lt»0 t'l), Ti + lOO.r.L'4 + I(M.».41>8 + 10!). 175 i);'/n — 21.581 — 21.508 — 21.551 9iir/„U','^o — 54,8! » I — 54.'.>(J0 — 55.04! > D;i — 21.78(i — 21.508 — 21.347 sin i l); H — 54.813 — 54.1 MJ9 — 55.i;{0 d;/ 4- 28.884 + 28.;{33 + 27.785 sill i D; 6 — 37.l!)(i — 37.3J>7 — 37.5!>5 I),/ + 7.()!)8 + 0.705 + 0.438 sin / l>t ^ — \yi.m\) — 1)2.3(J0 — ! 12. 7 25 J I), T — l.OG — 1.00 — 1.00 IX ^ i >5{»3.4l 551 ►8.70 . ')004.O2 D,e 42()2.!>.S 42()(;.12 42(m.24 Venus. // // 1 r Dte — !).95!) — !>.8(50 — !>.772 cDtTT, + 0.384 + 0.219 + 0.000 D't'to — 2.484 — 3.071 — 3.(55(5 sin «o D't' 6q — 59.005 — 59.112 — 5! ►.229 Dli — 3.04!) — 3.071 — 3.091 sin i D'; e — 58.978 — 5!>.112 — 59.2(50 D]i + 0.0!>0 + 0.(5!>5 4- (5.097 sin i D{ 6 — 40.758 — 4(5.582 — 4(5.413 Dt i + 3.(541 + 3.(524 -1- 3.(5(M5 sin i Dt f^ — 105.730 — 105.094 — 105.(573 ^BtTT — 0.3(5 — 0.37 — 0.38 Dt;r )()90.07 5072.44 5054.!>2 Dt^ < 3230.39 t 3237.!)8 . 3245.22 l.S<» DKI'INITIVE llEHl'I/i'S. |81>, DO 8' - 8.4«;7 — 8..59.-) — 8.727 e"]\ n" 4- 1J>.2!»3 + 19.210 4- 19.1.39 I), n" . 0179..58 W87.41 01!»5.08 h" sin L„ 4- 4.370 4- .5..'U1 + 0.305 n" cos L„ - 47.113 — 40.838 — 4(».550 log h" l.(}7.500 1.07340 1.07187 L'u 174O42'.04 1730 2!>'.08 172° 17'.18 L" 171° 12'.83 1730 2!>'.08 175o40'.02 i>o .')034.01 5030.13 5037.30 p 5018.28 3023.82 5029.38 U,f - 40.701 — 40.838 — 40.847 Mars, // // // D, t' + 18,77.5 + 18.706 + 18.623 cDt ;ri 4- 148.033 + 148.707 + 148.702 Dl' /o - 28.994 — 2!>.;{90 — 29.803 sin /oDl'^o - 34.023 — 34.012 — 34.017 I); / - 29.482 — 29.390 — 29.309 sin i D;' ^ - 33.60.J — 34.012 — 34.445 DM + 20.904 + 27.104 + 27.245 sin i D; ^ - 38.800 — 38.551 — 38.247 Dti - 2..J18 — 2.292 — 2.004 sin i Dt (9 - 72.40.5 — 72..5()3 — 72.092 JDtTT + 0.08 + 0.07 + 0.00 Dt;r 0021.51 0023.90 062«».25 Dt^ 2770.39 2770.87 2770.03 ISeciilar acceleration of the mean motions. 90. The mean motious 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 formulie for this acceleration are formed by dift'ereutiatiug the known if^,^. DO il. 90] SECULAR ACCELERATIONS. 187 expressioiiH for the variation of the loiij-itiKh- of the v\»h\i in the theory of thr variation of elements, the notation is that of AstroHomiettl rapcrs, Vol. V, Cart I\'. Wo C'lnpute for the aetion of an onter on an inner phmet: B =i(I)-I)'-2I)')(''," 4 W= i(2-UD + 3I)2 + 41)^)rI' o Then D? k = »i' (* n I>, I A 0-2 -f lif'^ - Ce" + \V(r' com (n- - t') I For the action of au inner on an oi't^" phmet we conipnte A'=-(l + D)rT B' = I (I) + 2 1)2 4- i)J)(''; 4: 0' = J(3D4-r.iy^ + 21)^)rr o W'= ^(10 + 3D-1)I)^- tl)^)c'l' 8 D? /„ = m n' Dt \ A' o' + B'e' + C'e'^ + W'^c' cos (;r - n') \ The symbol Dt indicates the secuhir variation of the expres- sion following it prodnced 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 conipntation: Secular change of the centennial mean motions. Action of— Mercury. // Venus, -0.0426 Earth, -0.0029 Mars, +0.0003 Jupiter, -0.00.39 Saturn, -0.0004 Total, -0.0495 +0.0090 -0.0403 +0.0169 Venna. Earth. Mars. // // // . . . -0.0104 + 0.0010 + 0.0128 • t t + 0.0119 -0.0001 - 0.0012 • • • -0.0046 -0.0308 + 0.0004 + 0.0015 + 0.0021 + 0.0036 II ill Hi ii 188 DEFINITIVE QUANTITIES. [91,92 The measure of iime. 91. The fictitious mean Sun whose transit over any meridian detines tlie moment of mean noon on that meridian is a point on the cehistial ^pliere having a uniform sidereal motion in the phuie of the Eai th's equator, and a Right Ascension as nearly as may be e(iual to the Sun's mean longitude. If we put /< for tills uniform sidereal motion and add to ja the precession of the e(|uinox in llight Ascension we have i'ov the mean Kight Ascen- sion cf this fictitious mean Sun T = To + /< T + 4G0()".;}() T + 1".394 T« From §§ 88, 90, and 100 the expression for the Sun's mean longitude, att'ected by aberration, is found to be L = 279047' o8".2 + 129(;0270()".74 T + l".089 T^ Equalizing the «'oet1icients of T we find, for the mean Right Ascension of the lictitious mean Sun r = 2790 47'oS".2 + 129(»0270()".74 T -|- 1".394T^ This differs from the mean longitude of the actual Sun by the quantity r - L = '.30r> T^ = 0«.020 T^ It This difference is of no importance in the astronomy of our time, but may result in an error of 2* 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 (juestion how best to meet the (juestion thus arising. Chang ing to time the expression for r, the ditterence or mean excess of sidereal over mean time for the meridian of Greenwich becomes T = 18i> :i\V" 11«.880 + 24" 0'" 1«.84449 t + 0«.0929 T« t being time in Julian years after 1850, January 0, Greenwich mean noon. Constant ofaherratiov. 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. I; 92,93] MASS OP THE MOON. 189 From the adopted value of the tiolar parallax, n = S".790, and the adopted velocity of light in kilometers per second, V = 290 800, follows for the constant of aberration the value A = 2U"..">01 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 = 2(V'..'511 as the adjusted value of the constant of aberration. Mass of ike Moon. r3. By means of the e«|uation of § 71 between the lunar inequality P in the motion of the Earth and the mass of the Moon /<'r = [1.78207] ;r we may find a fresh value of the Moon's nuiss from the values of TV and P. We have found from observation P = 0".4()5 i .015 Thus follows, for the mass of the Moon, when n = 8".7no, //= 1 :81.32 4 0.20 Combining this with the value found from the constant of nutation, /< = 1 : 81..58 rj 0.20 we have, as the definitive mass of the Moon, /I r= 1 : 81.45 ± 0.15 !! 190 W m DEFINITIVE QUANTITIES. Parallactic inequality of the Moon. [94, 95 94. From the transformation of Hansen's lunar theory in Astronomical Papers, Vol. I, it may be concluded that the solar parallax and the parallactic inequality are connected by the relation IM. = [1.10242] L:^;r = [1.15176] TT Hence we have, for the coefticient of the parallactic inequality of the Moon, corresponding to tt = 8".790, 124" .60 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. Tlie 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 ditticulty 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 sutticient iirecision 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 t 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 . i 95, 96 1 MASSES OF THE EARTH AND MOON. 191 unity. In this definition the unit of force retains its iihysieal 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 reduceM in the expres- sions for those of the centimeter-gram-second system. Masses of the Earth and Moon in centhiuter second unitn. 90. A fundamental (juantity in the centimeter second system is the mass of the P]arth. This mass will be by definition the force of gravity of the Earth, if concentrated in a i)oint at the distance of one centimeter. Were the Earth a sphere of know n 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 : -y/S, is th«' 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- 102 DEFINITIVE Q UANTITIES. [96 leni, owing to the heterogeneity of the Earth and the absence otMeteriiiinations 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, 1 do not think they are such as to greatly ini])air the astronomical results which are to be derived from them. Investigations in geodesy not being practicable in the present work, ] have, nminly from a study of the work of G. W. Hill,* assumed for the lejigth of the seconds pendulum at the point the sine of whose latitude is 1 : y/'i^ which 1 shall call the mean latitude, L, = 99.2715 With this we may compare IIelmekt's expression for the length of the seconds pendulum in terms of the latitude which gives L = 0"'. 990918 (1 + .005310 sin ^U.\T()R. 109 As tlio value of f, does not eliaii|,'o by {)"A\ from one epoch to anotlier, we may, without apiJieciable error, use f„ for f, in tlie formuhe (b) and (c). To u.se tliese equations, we tlrst obtain A- and /7| 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 f, and i/: in powers of the time by the ec|uations (c). The values of n have no reference to any special coordinates. From the table ot § KM) it will be seen that we may put n = 2004".70 - 2".13 t' t' being counted from IS.")!). To And the value of 77, 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 wc may take for the actual node between the ecliptics of any two epochs Ti and T2 the mean of the instantaneous nodes for those two epochs. For example, let it be required to find the value of 77, for the node of the ecliptic of 2100 on that of 1850. We have For 2100 For ISaO, referred to eq. of 2100 Concluded value of /7i ... / L = 17.") 46.()3 L = 17(; 5!).13 77, = 170 22.9 As the basis of our work we have computed the required quantities for the zero ecliptics of 1000, 1850, and 2100, respectively. The values of k and 77, for the ecliptics of two hundred and fifty years before and after these epochs are as follows : Zero epoch. -2SoY + 250 V i- n, /t n, 1600 1850 2IC0 // -118.48 - 118.07 -117.64 / i6» 20.0 170 36. 7 172 53-4 // 4- 1 18. 07 + 117.64 + "7-23 / 174 5-9 176 22.9 178 39.9 1 200 DEl'INITIVK tiUANTITIES. [102 Chaii)>;iiig tlio unit of time to two liiiiidrcMl and titty years, th« eipiatioHH (a) {h) and (c) give the Ibllowing values of the derivatives of ^i and if". D,e D,^ Zero-opooli. — 250 V + 260Y - 250 V + 250Y // // // // 1(100 - 1.4630 + 0.7400 I2600.;);j 12573.0r. IHfiO - 1.17ti8 + 0.4r)l>7 1200H.44 i2r)7(i.or> 21(K) - 0.8898 + O.KMir) 12000.57 12579.71 lin ^•) 1.3 I: At the respective ei)Oolis Drfi vanishes, and Dr*/' has the value«» of the lunisohir precession in longitude (§ 100). Developing in powers of r we have- the following results: Zero-epoch. c / // // // IGOO; f, = 23 29 28.09- + 0.5509 r« - 0.1200 t'-> 1850; f, = 23 27 31.08 + 0.4074 - 0.1207 2100; f, = 23 25 34.50 + 0.2041 - 0.1200 // // 1000; //; = 12587.00 t - 0.07 t« 1850; ?/' = 12590.05 -0.70 2100; »/• = 12593.14 -0.72 // // 1000; A = 45.28 t - 14.83 r^ 1850; \ = 33.52 - 14.8(5 2100; A = 21.75 - 14.88 These values of Si and tp completely fix the position of the ecjuator at the time t relative to the zero ecliptic and e([uinox. 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 P Eo Po, of which the sides and opposite angles are designated Sides, fo fi ^ Opposite angles, 90° - C 90° - Ci '/> If, in the Gaussian relations between the parts of this triangle, we put sin ^ (ei - €o) = ^ (fi - eo) = J Je lOL'l MOTION OF THE El^lATOU. 201 uiul regard the co-sine of this aii;;le us unity, we hiive tau i (;: 4- ;,) = von A (*i 4- '(.) tan A f tau i C - :,) = 2l4iiTTTf7-f- 7jlaiTT7' If we develop the dift'erenoos between the tangent and the arc we find from these e(|iiationH ; -I- ;i = ^' cos A (f, + f„) (1 + -1^ f .sin' fo) ^ = f.u'dhl.)^'-^^--^'f'^^ where we put ^„ for tiie approximate vahie of t — ^i For the iuiiliniition ^ of the nu^an e(iuator of the epoch t to the zero equator, we have tlie eciuatiou gi„ ^ ^ sin f„ sin V cos * and then, by developing in powers of 8 and »/•, we find i/' sin fo • ^ = "cosr ^ -*'/•' cos' fo) = //- sin fo (1 + ^ ;') (1 - i f cos^ fo) We thus find Zero-epoch. // // // 1(500; C + Ci = 11543.70 r - 6.12 t' + 0.r)7 t' 1850; ll.")49.44 -0.14 -f 0.57 2100; 11555.12 - 0,1(5 -f 0.58 // // ItlOO; C - C, = 45,20 t - 0.02 t' 1850; 33.53 - 9.03 2100; 21.7(5 - JJ.04 // // 1600; e = 5017.30 t - 2.(56 t' - 0.64 t^ 1850; 5011.97 -2.67 -0.64 21C0; 5006.64 - 2.67 - 0.65 To show the significance of the preceding quantities, con sider once more the spherical quadrangle Po Eo EP. Let these 202 DEFINITIVE QUANTITIES. [102 letters represent the positions of the poles on the celesticil sphere at any two epochs. In this quadrangle we shall have Angle Eo Po E = 90° Angle P: P Pn = 90° SidePoP = tt = 90° - C + A Let S be the position of a star on the celestial sphere. Its l)olar distances at the two epochs will be Po S and P S and its Eight Ascensions will be determined by the angles Po 'ind P of the triangle S Po P. Thus, if the Right Ascension and Declination of S are given for one epoch, we can lind it for the other epoch by the solu- tion of the triangle S P Po when we have given the values of the quantities ^, I'l, and Z,-\-\. To ttnd the values of these quantities from the preceding formula, let T be the zero-epoch, expressed in calendar years, and let r 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 coefticients of t 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 i^osition of S from one epoch to the other. O (