Ill f! ! I : ! BERKELEY LIBRARY or WASHINGTON OBSERVATIONS FOR 1870. APPENDIX III. . ON THE RIGHT ASCENSIONS EQUATORIAL FUNDAMENTAL STARS AND THE CORRECTIONS NKCIWSAKY TO REUUCK THE RIGHT ASCENSIONS OF DIFFERENT CATALOGUES MEAN HOMOGENEOUS SYSTEM BY SIMON NEWCOMB, PROFESSOR OF MATHEMATICS, UNITED STATES NAVY. PREPARED AT THE U. S. NAVAL OBSERVATORY 1!Y ORDER OF REAR-ADMIRAL H. F. SANDS, U. S. N., SUPERINTENDENT. WASHINGTON: GOVERNMENT PRINTING OFFICE. I S 7 2 . TABLE OF CONTENTS. Pape. i I. Introduction 5 S 2. Determination of a system of corrections to the relative right ascensions of ISessel s Tabuliv Kegiomnnlana. , pre sumed to be free from periodic errors $3. Discussion of the probable error of these determinations .... 17 $4. Systematic discordances among the catalogues employed 21 $ 5. Determination of the common correction to the first set of positions on account of equinox 21 $ 6. Corrections for the terms multiplied by the second and third powers of the time 34 $ 7. Corrections to different catalogues to reduce them to the preceding system 39 J 8. Derivation of definitive right ascensions 47 Table of right ascensions from 1750 to 1900 53 Corrections for orbital motion of Sirius and Procyon 69 Hill s formulae for the secular variation of the annual motion of the stars 73 ON THE RIGHT ASCENSIONS EQUATORIAL FUNDAMENTAL STARS THE CORRECTIONS NECESSARY TO REDUCE THE RIGHT ASCENSIONS OF DIFFERENT STAR-CATALOGUES TO A MEAN HOMOGENEOUS SYSTEM. INTRODUCTION. Within the past few years very important additions have been made to the mate rial at our disposal for an accurate determination of the positions and proper motions of the fundamental stars. These comprise Dr. Amvers reduction of Bradley s observa tions, the Pulkowa catalogue of 1845, an( l the continuation of the Greenwich and Washington observations. It is not too much to say that this additional material alone would suffice to furnish results far more accurate than any that could be obtained from the older data. The object of the present work is to do for the right ascensions of the equatorial and zodiacal stars, on which the reductions of lunar and planetary observations depend, what has been done by Dr. Auwers for the declinations, namely, to furnish the data necessary to reduce the principal original catalogues of stars to a homogeneous system by freeing them of their systematic differences. I think this can best be done by com paring the catalogue-positions of the fundamental stars with a uniform set of standard positions as free as possible from systematic error, and to consider any systematic dis cordance between the standard and the catalogue position as due to error of the latter, and correct it accordingly. For the reason already mentioned, it will not be admissible to take any existing catalogue as the standard. The first step of our work is, therefore, the preparation of a system of right ascensions of fundamental stars which shall be as free as possible from the suspicion of systematic error. In the formation of such a catalogue I have been guided by certain considerations respecting the probable presence of systematic errors in the reduced right ascensions of many modern catalogues. It has long been the custom, in the reduction of right- ascension observations, to deduce a correction of the clock from adopted right ascen sions of stars, and to consider the mean correction thus deduced as applicable to all the stars observed, the clock-stai S included. On this system it is clear that if the assumed positions of all the clock-stars used on any one night are affected by a common error, 6 RIGHT ASCENSIONS OE FUNDAMENTAL STARS. that same error will be carried into the concluded positions. Thus any periodic error depending- on the right ascension will tend to perpetuate itself. In a former paper I showed that if the clock-stars observed on each night are distributed uniformly over a period of six hours, and if the assumed positions are affected with the error a sin 11. A. -f- 1 cos K. A. -f- a 1 sin 2 11. A. -f- ?/ cos 2 R. A. the concluded positions will be affected with the error 8 2 ( sin R A. + & cos R A.) + -*, <y , S m 2 R A. + V cos 2 R A.) Hence the error of single period will be cut down very slowly by successive revisions of the catalogue, each revision reducing it by less than one-fifth. On the other hand, the error of double period Avill rapidly disappear. The introduction of these periodic errors is very easy, even where the greatest care is taken in the reductions, owing to the diurnal variation of the conditions to which the instrument, the clock, and the observer are subjected. If the pointing" of the instrument, the rate of the clock, or the personal equation of the observer vary during the twenty-four hours, the right ascensions deduced will be affected by a cor responding error. It is true that if the variation continues uniform during an entire year, and if the observations are uniformly distributed throughout the year, this error will be eliminated from a year s work. But neither of these conditions is completely fulfilled, the diurnal change of temperature being greater in summer than in winter, while the annual period in the length of the night and the degree of cloudiness interferes Avith the continuity of the system of observation. 2. DETERMINATION OE A SYSTEM OF CORRECTIONS TO THE RELATIVE RIGHT ASCENSIONS OF BESSEL S TABULAE REGIOMONTANvE, PRESUMED TO BE FREE FROM PERIODIC ERRORS. For the reasons already mentioned, it seems best to deduce the periodic error of the standard catalogue solely from determinations which are entirely independent of any previous hypothesis respecting the right ascensions of clock-stars, and from those in which there is reason to believe that the periodic error has been entirely eliminated. I have selected the following twelve original catalogues as certainly or probably fulfill ing one of these conditions : 1. Ameers Bradley, 1755. Dr. Auwers has very kindly placed his definitive results for relative right ascensions, which are as yet unpublished, at my disposal. 2. Piazzi, 1800. " Praecipuarum stellarum inerrantium positiones media? ineunte saeculo XIX. Panormi, 1814." 3. Maskclync, 1805. Catalogue found in Maskelyne s Astronomical Observations made at the Royal Observatory, Greenwich, from 1799 to 1810, p. 112 of part 2. 4. Ameers Cacciatore. "Reduction der Beobachtungen der Fundamentalsterne am Passageninstrument der Sternwarte zu Palermo in den Jahren 1803 his l &5i un( l Bestimmung der mittleren Rectascensionen fur 1805. Von Arthur Auwers. Publica tion der astronomischen Gesellschaft, V. Leipzig, 1866." DETERMINATION OF A SYSTEM OF CORRECTIONS. 7 5. Bcssd I. "Bestimmung dor geraden Aufsteigungen der 36 Maskelyneschen Fundamental-Sterne fur 1815, auf Kb nigsberger Beobachtungen gegriindet." Berlin Memoirs for iSiS- ig, p. 19 of part 2. 6. Besscl II. " Neue Untersuchungen iibor die geraden Aufsteigungen der 36 Fundamental-Sterne." Berlin Memoirs for 1825, p. 23 of part 2. 7. Stntcc. Catalogue found in "Positiones media 1 ," p. xxxxi. s. Brinklcy. Catalogue found in the Astronomische Nachrichten, Band IV, S. 102. 9. Argelander. Corrections yireii ly Besscl in Astronomische NacJiricJitcn, No. 5 1 5. These differ from the positions in Argelander s "DLX. Stellartun fixarum positiones media 1 " only in the addition of the third decimal of seconds. 10. Ptilkoica, 1845. "Observations do Poulkova, publiees par Otto Struve, Direc- teur de 1 observatoire central Nicolas. Saint-Petersbourg, 1869." Vol. I, p. 120. ir. Greenwich, 1864. "New seven-year catalogue of 2760 stars deduced from observations extending from 1861 to 1867 at the Royal Observatory, Greenwich, and reduced to the epoch 1864." (Appendix II to the Greenwich observations for 1868.) 1 2. WasJiiiiyton, 1870. "Positions of fundamental stars deduced from observations made at the United States Naval Observatory between the years 1862 and 1867." (Washington observations for 1867, Appendix III.) On this selection I remark that, I find no positive evidence that, the catalogues of Piazzi and Brinkley are really independent of any previous determination, and that 1 have judged them to be so because the modern system of reduction had not, then come into general use. Argelander, Greenwich, and Washington are not rigorously inde pendent, but they arc included, because, in the case of each, there is reason to believe that the systematic errors have been more or less completely eliminated. The follow ing are the reasons in the case of each catalogue: An examination of Argelander s observations shows that they were generally widely distributed throughout the twenty-four hours, his habit being to observe both morning and evening. From such a day s work the error of single period will be almost entirely eliminated. The Memoirs of the Royal Astronomical Society, vol. 34, contain a paper by Mr. Stone, in which the stars y Pegasi, Pollux, Spica, and a. Aquila? are compared, with the express purpose of detecting any inequality depending on the right ascension. The resiilt is that the relative positions of these stars, as given in the seven-year catalogue for 1860, require the following corrections: / Pegasi Pollux, -.001 or .014 Pollux Spica, +.029 or -(-.015 Spica - a. Aquihv, -f- .008 or .005 a Aquihc y Pegasi, +- OI 8 or +-004 The first column shows the differences independently deduced ; the second their values when each is diminished by a constant so as to make their sum zero. The smallness of this result indicates that the periodic error, if it exists at all, must be very small, and it ought to be smaller still in the catalogue of 1864, which includes still another revision. 8 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. In the case of Washington ;i special effort was made to determine the periodic error of Dr. Gould s right ascensions; which was of the form a sin R. A. + b cos R A. the details of which are given in the catalogue cited. The determination was entirely satisfactory except in one point a large diurnal inequality showed itself during the second year. This inequality had to be determined on the supposition that it remained constant throughout the year, an hypothesis which, though probable, was not certain. Having selected our data, the next question will be the mode of combining. It was my original intention to compare each catalogue with Dr. Gould s standard, and thus obtain a correction to the latter of the form a sin a -\-l) cos a-\-c A series of values of a, b, and c being thus found for the epochs of the various catalogues, the true values of these quantities should increase uniformly with the time. We should then determine a , a , 1 , V, c oj c , from the equations a + ft t = n 1> + Vt - I c + c t = c of which we should .have as many sets as there were catalogues compared. Having found the most probable values of a , a , etc., the residuals left after substituting them in the equations given by each catalogue would show the systematic error of that cat alogue. But it has been deemed best to begin by determining each star independently of any catalogue whatever, and of any assumed proper motion. This is done most easily and elegantly by assigning the same weight to all the stars in any one catalogue ; this weight being determined not by the general accuracy of the catalogue, but solely by its probable freedom from systematic errors. It will then be unnecessary to apply any reduction on account of eqitinox in this preliminary determination, because, if all tlie right ascensions of any one catalogue be changed by a constant, that change will affect all the concluded positions by the same amount, and therefore will not change the rel ative positions at all. From the nature of the case, it is clearly impossible to make any numerical calcu lation of the relative weights to be assigned to the different determinations. This is necessarily a matter of individual judgment. Without going into the consideration of each particular case, I- may remark that the following weights assigned were founded on considerations respecting the probable freedom of the instruments from sources of systematic error and the care with which systematic error was avoided in the observa tions and reductions, probable accidental errors not being taken into account, and that they were decided on before any comparison of results. They have, therefore, no necessary analogy with the weights to be assigned in a definitive discussion, after sys tematic error has been allowed for. DETERMINATION OF A SYSTEM OF CORRECTIONS. Weights assigned to observations. No. Catalogue. Year. Weight. I 2 Auwers Bradley . . I iazzi 1755 1800 i 3 iSos 4 5 Auwers Cacciotorc . Bessel I ... 1805 181=; I 6 7 Bessel II Struve . . 1823 iS"i 3 7 8 Brinkley i82a 9 10 Argclander Pulkowa 1828 i8j? 2 6 ii 1864 2 12 Washington l87O 1 The dates here given to the catalogues are generally supposed to be about the means of the times of observation on which they depend. The principal exceptions are Brinkley and Washington. The mean date of the former could not be inferred with certainty, but must be earlier than 1824. The date assigned for the latter is, through inadvertence, that to which the catalogue was reduced, which is five years later than the actual mean of the times of observation. The reduction to the epoch was, however, made with Dr. Gould s proper motions, which arc so near the truth that few or none of the results are altered the thousandth of a second by this error. I have, therefore, not corrected it, The results are all exhibited in the form of corrections to the mean positions and annual variations of the Tabula; Regiomontanre for the sake of convenience, and the principal steps of the computation are exhibited in the following tables. The first column in each table shows the correction to the right ascension of the Tabula; Regio- montanse given by the catalogue. Following one star through the several catalogues, we have a series of corrections applicable at the various epochs. The true correction being of the form x -\- y t, each catalogue result gives an equation of condition between x and y. As an epoch near the mean leads to the most convenient computations, I have chosen the year 1820 as the origin of time. Let us now represent by the corrections given by the several catalogues, the subscript numbers being the same as those of the catalogues to which they correspond in the preceding list. Taking the century as the unit of time, the equations of condition will then be ./ 0.65 // = (\ ./ O.2O1J r 2 x o. 1 5 y r , etc., etc., etc. 2 F s I0 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. The solution of these equations by least squares, using the weights in the preceding list, will give the following values of x and y: -\- 0.04 c. 2 - o.io f 2 + 0.04 f;, - 0.08 C 3 + 0.04 c 4 - 0.08 c 4 + 0.04 r 5 - 0.03 c, + o. 1 1 c 6 - 0.04 c t; + o. 1 1 c, 0.03 c 7 + 0.03 c e o.o i c 8 + 0.08 r,, + o.o i c, + 0.19 c, +0.40 r 10 + 0.06 r n +0.27 c n No account is here taken of the discordant equinoxes of the various catalogues, because, all the stars in the same catalogue having the same weight, a change of equinox will not change the relative positions of the stars. In the selection of stars, attention has been at first confined to the Maskelyne fundamental stars between the limits of declination -f 40 and 30. Several stars have to be omitted, namely, Procyon and Sirius, on account of the irregularity of their proper motions which has been so exhaustively investigated by Auwers that no further discussion of their positions is yet necessary ; Castor, on account of its duplic ity, and the confusion of the observations upon it, which render it unfit for use as a standard star ; /? Virginis, a. Librae, and a. Capricorni, because they have of late dropped out of use. Under each catalogue, in the following table, we have, in column c, the correction to the Tabulae Regiomontanae given by the catalogue, from which the values of x and y are deduced by the preceding formulae. The value of the correction x -f t y is next computed for the epoch of each cata logue, and subtracted from the correction given by observation. The excess of the catalogue result thus obtained is given in the second column of the table. To correct the residuals for discordance of equinoxes, the mean value of the numbers in this column is taken, and subtracted from the individual numbers. The remainders repre sent the discordances between the catalogue and the correction as finally deduced. DETERMINATION OF A SYSTEM OF CORRECTIONS. I I Corrections to rig/it ascensions given by standard catalogues. Star. Bradley, 1755 Piaxzi, 1800. f\ c c" fl <V <V y Pegasi .... + 65 - 31 54 - 65 4- 18 a Arietis .... - 56 + 40 + 9 - lOS - 144 - 61 Ccti. 4- 1$ 4- 6.1 4- T1 76 g(3 <7 ~ j^ i ^4 ^ J J 7 J a Taiiri .... 12 + 51 + 20 - 194 - 172 89 /? Orionis. + 40 + 33 4- 2 54 - 30 + 47 i) Tauri .... + 152 + 31 O 53 - TOO 17 a Orionis. + 30 + 22 9 7. 13 4- 70 ,3 Geminorum . 102 + 41 + 10 - 173 - 133 - 50 a Hydrx .... - 139 4- 23 - 3 - 118 - So + 3 Lconis .... 106 + 43 + 12 - 170 121 38 /? Lconis .... - 76 + 34 + 3 ! - 85 - 68 + 19 Virginis 42 4- 2S 3 - S6 54 + 29 a Bootis .... - 36 + 39 4- 8 - 164 - M3 - 60 a- Libra: .... + 190 + 23 - 8 4- 30 - 33 + 50 a Corona} - 66 + 37 4- 6 - 240 208 - 125 a Serpontis . + 122 + 35 + 4 - 56 - 86 - 3 11 Scorpii .... + 84 + 4 - =7 4- 122 + 98 4- 181 Hcrculis . 16 + 40 + 9 - 225 202 - 119 a Ophiuchi . - 162 + 44 + 13 - If) I - n6 - 33 a Lyrne .... 30 + 23 - 3 - 1 = 7 108 - 25 y Aquilrc. + 163 + 29 2 - 35 - 85 - 2 a Aquilic. 12 + 18 13 - 5 37 4- 46 /} Aquilx. + MI + 26 - 5 - f>7 - -89 6 a 3 Capricorn!. 7 + 7 24 20 4- i + 84 a Aquarii. - I 5 + 16 - 15 74 - 22 4- 61 a Pegasi .... + 7 4- 29 - 2 - i* 92 - 9 a Andromcdx . - 73 + 45 + 14 -187 - 159 - 76 12 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Corrections /<> rig/it ascensions, &c. Continued. Star. Maskclync, 1805. Auwers Cacciotorc. Oi (3 f-.l" ft ft ft" y Pcgasi .... ~ MS - 152 24 + 56 +52 + 29 a Arietis .... - I 4 S - 133 - 5 23 8 - 3i a Ccti .... 228 - Ill - 243 94 - "5 + 34 + 27 +12 + 14 +31 ii + 8 a Tauri .... /} Orionis. - 139 - 159 - 31 + 61 +41 + 18 ft Tauri .... - 62 IOI + 27 + 110 + 71 + 48 n Orionis. 123 - 129 I +112 + 106 + 83 ft Geminoi um (i Hyclnc 189 - 133 i/3 - 161 106 - I3& - 33 + 22 - S + f 5 + 93 + 28 +55 + 25 +62 + /o + 32 + 39 Lconis .... ft Lconis .... - 98 - 88 + 40 + 21 +31 + 8 ti Virginis - 122 93 + 35 29 o - 23 a Bootis .... - 97 - 82 + 46 +22 + 37 + M ft- Librge .... 50 - 101 + 27 2 53 76 Corona; 1 20 96 + 32 74 50 - 73 Serpentis . 125 - 148 20 - 53 - 7G - 99 a Scorpii .... 206 223 - 95 + 69 +52 + 29 a Herculis - in 91 + 37 Si - 61 4 a Ophiuchi . - 163 - I3i 3 28 + 4 | 19 a Lyra .... - 65 51 + 77 - 5 +9 14 y Aquilic .... - 7i - in + 17 + 54 + 14 - 9 <i Aquil;c .... - 162 - 151 23 + 44 +55 + 32 ,3 Aquilse .... a- Capricorni. .. - 1 20 - 148 - 135 - 126 + 2 ,+ 17 +2 2O +2 21 21 a Aquarii. - 208 : - 164 - 36 5 +39 + 16 Pegasi .... - MO - 133 5 + 47 +54 + 31 a Andromedx . 146 125 + 3 +23 +44 + 21 DETERMINATION OF A SYSTEM OF CORRECTIONS. Corrections to right ascensions, & f. Continued. Star. Bessel I, 1815. Bcsscl II, i825. ft Ct ft" fa < <"u" y Pcgasi .... + 140 + 151 + 98 + 22 + 27 a Arietis .... + 123 + I2O + 67 17 12 n Ccti + 126 + IOO + 47 o - 35 - 3" a Tauri .... +20 + 27 26 o o + 5 ,j Orionis. ... S -. 33 - 86 - 30 - 25 ,i Tauri .... + 32 + 10 43 o - 9 4 Orionis. . . . 45 5i - 104 - 3 + 2 /j Geminorum . . 5 J 51 o 17 12 llydne .... 78 8 I3C o 28 00 a Lconis .... 16 2 j j - 55 o 4 ^ J + I ; i Lconis . . . . 1 +34 + 21 - 32 31 26 ( Virginis - 3 + M 39 + 17 + 22 a Bootis .... + 33 + 4 13 - 8 - 3 + 166 4- I3O + 86 o _ s T a Coronx ... + 102 i * Jy + ICQ + 56 6 J a Serpentis . + 83 + 73 + 20 + I + 6 a Scorpii .... + 121 + 118 + 65 + 7 + 12 a Hcrculis + 53 + 66 + 13 o + 7 + 12 a Ophiuchi . + 27 + 20 - 33 - 37 - 32 a Lyric .... + 56 + 61 + 8 2 + 3 j Aquilx .... + 65 + 44 9 o - 6 I a Aquilx .... + 48 + 55 + 2 + 4 + <) t 3 Aquilie .... + 63 + 7i + 18 + 26 + 31 n- Capricorni. + 52 + 76 + 23 + 26 + 31 a Aquarii. + (>4 + 37 16 o + 13 + 18 Pcgasi .... + 112 + 116 + 63 + 2 + 7 Andromeda . + 123 + 123 + 70 o 17 12 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Corrections to right ascensions, &c. Continued. Qt.-l Y Struve. Brinkley. Mar. cr c-, CT <8 <" <" 7 Pegasi . . . . + II + 35 I - 150 126 + 9 a Arictis .... + 57 + 33 + 2 - 93 112 + 23 a Ccti -f- 73 + 37 + I - 150 1 86 51 a T.iuri .... + 22 + 21 - 15 - 145 - 146 it ft Orionis. + 119 + 89 + 53 - 82 - 112 + 23 ,? Tauri .... + 20 + 14 22 - 152 160 25 a Orionis. + 28 + 22 - 14 - 128 - 134 "+ I /? Geminorum + 34 + 13 - 23 - So - 101 + 34 + 03 + 61 + 2 5 - 6 - 3? + Q7 a Lconis .... 1 j j + 39 + 32 - 137 - M4 1 j i - 9 ,:) Leonis .... + 58 + 24 12 - no - 144 9 a Virginis + 4 + 20 16 - 182 - 1 66 3i a Bootis .... + 29 + 19 - 17 - 139 - 149 M 4- 22 + 17 10 126 131 + 4 a Cororue + 73 + 66 j + 30 - 137 j * - 144 9 a Scrpcntis . + 71 + 74 + 33 - 156 - 153 IS a Scorpii .... + 44 + 52 + 16 1 86 - 177 42 a llcrculis + 67 + 74 + 38 - 115 108 + 27 Ophiuchi . + 118 + 76 + 40 - Si 123 + 12 tt L)TDB .... - 6 - 9 - 45 - 101 - 104 + 31 } AquiUe .... + 50 + 46 + 10 - 146 - 150 - 15 Aquilic .... + 27 + 30 6 ,- MS - MS 10 ,? AquiUc .... + 10 + 39 + 3 205 - 176 41 <i~ Capricorn!. + 37 + 63 + 27 189 163 - 28 a Aquarii + 21 + 33 - 3 - 127 - "5 + 20 a. Pegasi .... + 5 + 7 - 29 132 - 130 + 5 a Andromedx . - - 25 - 61 - 87 - 106 + 29 DETERMINATION OF A SYSTEM OF CORRECTION S. Corrections to right ascension, &c. Continued. Star Argelander. I ulkowa. fa f Ct" Cut c fu>" Y Pcgasi .... 1C -4- 11 CQ ii 6 Arietis .... * ;? + 2S 1^ **T + 2 DV + 62 + 8 + 13 n Cell + 26 ic 4- c i o -U c * j 1 J~T ^ D Tauri .... + 12 + 8 t f + 6 16 II ft Orionis. + 9 - 23 . + 3S + 2 + 7 ,1 Tauri .... + 13 + 12 - 7 + 10 + 15 Orionis. + 17 + II . 13 - 17 - 12 fi Gcniinorum . + 44 + 14 t t + 7 + I + 6 n Ilydrx .... + 44 I + 90 O + 5 a Lconis .... + 12 o + 50 3 + 2 /J Leonis .... + 42 I . + 75 + 5 n Virginis - 26 13 + 3 + 2 + 7 a Bootis .... + s - 6 . + 38 + 5 + 10 n 2 Libra; .... 8 4 60 if) ii Corona; + 36 + 22 + 34 7 o Scrpcntis . 2 + 6 . - 37 - 8 - 3 Scorpii .... - 55 41 54 "9 M a Hcrculis . + 6 + 10 - ID - 23 18 n Ophiuchi . + 73 + If> . + 94 21 1 6 a Lyrx .... + 6 O + 34 + 12 + 17 } Aquihe . -1- 7 + 10 31 + 5 + 10 n Aquilffi .... _ 5 - 3 - + 7 + I + f) /} Aquihc. 34 + 4 . 75 - 5 a- Capricorn! . - 5S - 31 32 7 2 Aquarii - 26 21 + 16 10 5 a Pegasi .... + 7 + 8 ii - 15 10 a Andromeda; . + 41 + 13 -t- 63 + I + 6 i6 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Corrections to right ascensions, crV. Continued. Star. Greenwich, 1864. Washington. u CM fti" Cn tii Ca" 7 Pcgasi .... - 124 51 - 30 - 59 + 22 . 12 a. Arictis .... + 53 33 12 + 123 + 27 - 7 a Ceti + 53 13 + 8 + 132 + 5 & + 22 a Tami .... + 40 + 1 +22 + 9i f 48 + 14 /? Orionis. + 24 18 + 3 - 73 + 29 5 /? Tauri .... - 58 o + 21 52 + 16 - 18 a Orionis. + 6 + 3 + 24 + 38 + 35 + i /3 Geminorum . + in - 3 + 18 + 151 + 23 n a Hydrne .... + 129 15 + 6 + 185 + 23 n a. Leonis .... + 83 13 + 8 + 148 + 39 + 5 /? Leonis .... + no 4 + 17 + 160 + 33 i Virginis 22 - 38 17 < + 57 + 37 + 3 a. Bootis .... + 39 - 17 + 4 + 97 + 33 i 03 5 + 16 67 + 34 a Coronx + 48 23 2 + 98 + 17 17 a Serpcntis . - 79 - 25 4 - 32 + 29 - 5 a Scorpii .... - 91 - 32 n - 19 + 48 + M a Ilerculis + 12 10 + II + 62 + 35 + i a Ophiuchi . + 178 - 5 + 16 + 246 + 42 + 8 n Lyric .... + 9 29 g + 68 + 25 9 y Aquilx .... - 117 46 - 25 42 + 40 + 6 Aquilx .... - 32 18 + 3 + 49 + 34 /? Aquihc .... - 148 39 18 - 89 + 33 i n 2 Capricorn i. 54 - 29 - 8 o + 25 9 Aquarii. + 6 - 53 - 32 + "5 + 4<J + 12 Pegasi .... 2O : 30 9 + 60 + 49 + 15 a Andromedx . +78 -22 I + 151 + 40 + 6 DISCUSSION OF PROBABLE ERROR. The following are the values of x and y, in which the units are thousandths of seconds of time: Star. X y } Peijasi - 18 127 ft Arietis + 12 F- 167 Ccti . -J- -12 -t- 89 (i Tauri -|- n-3 ft Orionis .... ft Tauri . . . + 2S + M + 32 lf 5 Orionis .... ft Geminorum llvdnc Lconis . + 5 + 10 + 19 - 5 + 236 + 286 + 225 ft Lconis .... + 2 4 + 206 a Virginis .... B,,otis . if) + 4 + 78 + 121 - Librae -4- IT 211 n Coronic .... n Serpentis . + I + 3 + I f)0 - 129 fi Scorpii . 3 - I2S Keren! is .... " Ophinclii .... ti Lvrx - 9 + 2f) + 72 + 357 + 83 } Aqnil:c + 12 - 1 88 ft Aquilrc ... 5 + 39 ft Aquihc . 10 206 : C apricorni a Aquarii .... n I t^asi - 25 18 3 I + 174 + 29 Andromcdic . + II + I W $3- DISCUSSION OF PROBABLE KRROK. From these discordances we shall next deduce the probable error of the resulting 1 x and y. It is well known that results for probable error generally prove entirely illusory. This does not arise from any defect in the theory of the subject, but from the fact that the probable error generally has to be deduced from the discordances among a series of results affected by some common source of error, which changes ;ill the results in the same way, and is, therefore, not eliminated from the mean. 15ut the twelve star-catalogues here employed may be regarded as absolutely independent of each other. In deducing the probable error from each series of residuals, I take the mean value of the residuals without regard to the sign. In the more elegant method of deducing the probable error from the sum of the squares of the residuals, it seems to me that too much weight is assigned to those large residuals the presence or absence 3 v s jg RIGHT ASCENSIONS OF FUNDAMENTAL STARS. of which is a mutter of chance. The probable error being proportional to the mean error, and only a little smaller, I shall use the latter throughout instead of the former. The mean values of the numbers found in the third columns, taken without regard to the sign, are as follows: 1. Auwers Bradley ... 10.7 2. Piazzi - - 49 3. Maskelyne .... 3 4. Auwers Cacciotore 36 5. Bessol I 47 6. Bessel II ^3-7 7. Struvo - - - - 211 8. Brinkley 23.3 9. Argelander 11.4 10. Pulkowa 8.2 1 1. Greenwich . 13.1 12. Washington 7-9 Had the catalogues all been entirely accordant, these residuals would all have been zero. Let us next see in what manner they are made up. In the case of Bradley the residual is formed by subtracting from each c t the value of x 0.65 y deduced from the expression on page 10, substituting for c i} r 2 , etc., their values found in the preceding table. Substituting in this expression these values of x and y in terms of (\, c y , etc., which have already been given, we find x 0.65 y .68 c t + .10 c., -f .09 c,, -f .09 c 4 -f .06 c~, + .14 C R + .13 r 7 + .04 r 8 + .07 r fl .07 r, .1 2 c n .22 f, 2 Subtracting this from r,, the difference is .32 Ci . i o r., .09 r : , .09 r 4 .06 Cr, . 1 4 c 6 . 1 3 c- .04 r H - .07 c a + .07 r, + .12 Cu + .22 r 12 r; and it is this expression of which the value is given in the second column under each catalogue in the preceding tables. Let us now represent by fo, e, . f 12 tlie probable errors of c \i CM fa .... f 12 or of the star-positions in each catalogue. The square of the mean error of a sum being equal to the sum of the squares of the individual mean errors, the square of the mean error of the last expression, or of c , will be .IO2 i~ + .OTO f., 2 + .OO8 :i 2 -f -OOS ,, 2 -f .OO4 * + .029 ,f -f- .OI 7 -, -(- .OO2 E^ + -005 ,/ + .005 10 2 + .014 U 2 + .048 K " The mean Bradley residual being 10.7, the value of this expression is 114. A similar equation being formed for each of the twelve catalogues, we have twelve equations of the first degree, which suffice completely to determine the twelve values of DISCUSSION OF PROBABLE ERROR. F, and therefore the mean error of each catalogue. These equations are as follows, writing only the co-efficients of the unknown quantities: * E * f 6" * tf , , ^ .IO2 .010 .OOS .008 .004 .O2O .017 .002 .005 1 .005 .014 .048 -- 114 .109 .866 .004 .004 .OO2 .014 .014 .OOI : .OO6 .012 .O .0 2601 .084 .002 . 902 .004 .OO2 .014 .012 .001 .OOO .017 .000 .000 = 900 .084 .036 .002 .002 .OO4 .002 .902 .002. .OO2 .922 .014 .OI2 .012 .OI2 .001 .001 .0 .006 .017 .032 .0 .002 .0 .006 1296 = 2209 .022 .002 .002 .002 .OO2 .792 .OI2 .001 .006 .040 .OO5 .OIO = 196 .020 .OO2 .OO2 .002 .OO2 .012 .792 .001 .006 .044 .OO5 .012 - 441 .020 .002 .002 .002 .002 .OI2 .OI2 .940 .006 .044 .005 .012 = 529 ,OI2 .001 .001 .OOI .002 .012 .OI2 .OOI .846 .048 .OO6 .017 = 121 .OOI .000 .000 .OOO .OOI .OIO .OIO .001 .006 .504 .017 .044 = 6 7 .029 .000 .000 .OOO .OOI .008 .OIO .001 .006 - 37 .672 .090 - 170 .048 . .000 .OOO .OOO .ooS .OIO .OOI .006 .152 .036 .462 = 6 3 Those equations will give the middle scries of probable errors with great, certainty ; but those which lie near the extremes in point of time will be quite uncertain, the necessary uncertainty of the last members of each equation being considerably exag gerated iu the result. If we solve by successive approximation, the equations for determining e* and F,./ will be . 1 02 e i~ zr 1 1 4 74 40 ; f , =. o".O2O .462 F 12 a = 63 55 = 8 ; F,, o*.oo4 We may roughly estimate the uncertainty of the second members of the. preceding equations as equal to one-fifth of their entire amount, so that the numlx?r for which we have found the value 1 14 might have ranged anywhere between 91 and 135, while that which we have found to be 63 might have ranged between 50 and 76. Hence the value of F! may, from this cause alone, range between o s .oi3 and o".O25, and that of F,., between zero and O 8 .oo7. In the latter case we have an independent means of esti mating the probable error. The catalogue result actually employed is the mean of two nearly independent determinations with different instruments. The mean differ ence .of the results was o".oi3. Wo may therefore put o".oo6 as the inferior limit of ,. To make the estimate too great rather than too small, I have supposed e 12 to be o*.oo8, and hence obtained the following scries of probable errors, which nearly satisfy all the equations except the last: X. F! .019 ^ =-054 f. =-037 F,, = .048 F (i .014 F 7 =.023 <H =-023 F,, =.010 F 10 = .OIO t f = 3 6l !./ ZZ29I6 f :1 - =1 9OO ff =1369 f,- 2304 f "ti~ = 96 v =529 529 100 100 f/ =: f ID F,., .OO8 ?n"= 196 ,.;- = 64 20 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. It may be remarked that the only uncertainty among these probable errors which materially affect our final result for the probable error of the concluded positions is that of (. To find the probable error of a computed star-position at the time t, we must sub stitute the values of x and y in the expression x-\- y t, which gives the correction to the Tubitlce licyiomontance at the time t. Calling this correction ^ we have -4 = + (.17 - .789*1 + (.04 .I00<3 + (.04 -.08 f)c 3 + (.04 .03 c 5 + (.n . 040 A; + (. 1 1 .03 c 7 + (.03 .01 Of + (.o8 + .oiQcb + (.19 + .409 CM 4- (.06 + . 279 e u + (-09 + 479 c is The square of the probable error of this expression will be (. 1 7 - .78 O 2 *, s + (-04 - - iotf v + etc. = 36 - 1 25 1 + 303 f t being, it will be remembered, the fraction of a century after 1820. This expression would attain its minimum in 1840, when its value would be ~ = 23 or s o s .oo48 At any other time T its value will be /T-i84oV --- 23 4- 1,01, I r y 100 y Owing to the independence of the twelve determinations on which the concluded positions rest, I conceive that this expression for the probable error is not entirely illu sory, but is very nearly real. Of the exactness of the constant term no serious doubt can be entertained. But the co-efficient 303 multiplied by the square of the time depends very largely on the concluded probable error of Auwers Bradley, and we have seen that there is some uncertainty in this determination. If this probable error is really as great as o s .O25, which we have seen to be quite possible, we should have /T-i8 4 iy t- 23 + 464 ( 1 h y 100 / The first expression will give for the probable error in 1900 zz o s .o r i while the second will give ZZO a .OI4 It must be understood that these results for probable error are true only on the assumption that the proper motion of each star carries it forward on a great circle with a uniform velocity. SYSTEMATIC DISCORDANCES AMONG THE CATALOGUES. 21 $4- SYSTEMATIC DISCORDANCES AMONG THE CATALOGUES. Let us next inquire how far the outstanding residuals of the corrections to each catalogue are of a systematic character depending either upon the right ascension or declination. 1 have already set forth the reasons for believing that the systematic, errors to be looked for are those depending on the sine and cosine of the right ascen sion. Let us then suppose the residuals of each catalogue to be of the form m cos a -f n sin a -f- c If we put >,, r.,, etc., for the residuals corresponding to the several stars Ave shall have twenty-seven equations of the form m cos oc i -f- n sin <*; + c r- t from which to determine -m, 11, and c. If these equations be solved by least squares, the normal equations will be m 2 cos 2 a-\-n 2 sin a cos a -f- c 2 cos a 2 r cos a m 2 sin a cos a -(- n 2 sin 2 a -\- c 2 sin a 2 > sin a m 2 cos ct-\- n 2 sin a. -f- 2 7 c 2 > ( )wing to the minuteness of the quantities 2 cos <x, 2 sin <*, w, and 11, and to the fact that when we take the residuals r ;! in the third column of the table of catalogues we have 2 r o, we may, using these same residuals, determine m and n from the first two equations, putting c = o. Putting for 2 cos 2 a-, etc., their values derived from the right ascensions of the twenty-seven stars in question at or near the mean epoch, these equations become 1 1 . 74 m + 0.93 n 2 > cos a 0.93 m + 15.26 n -=.2 r$\\i a. which give i i .68 m 2 r (cos a .06 sin a) i 5. 1 9 n ^ r (sin a .08 cos a) Computing the values of m and n corresponding to each catalogue from these formula. , we find that the residuals in the second column, which represent the excess of the cat alogue right ascension over that computed, nuiy be represented as follows: No. Catalogue. Systematic residuals. s. s. s. i Auwers Hradley . r = + .031 .003 cos i + .007 sin a IMazxi, (1800) .... r= .087 +.003 cos a .001 sin a 3 Maskclync, (1805) . . . r= .128 .019 cos a .006 sin a 4 Auwers Cacciotore r = + .023 + .010 cos a + .033 sin a 5 Hessel I, (1815). . . . r + .053 4- .037 cos a .037 sin a 6 Hessel II, (1825) . . . ) =; .005 +.006 cos a .010 sin a 7 Slruvc, (1825) .... r = + .036 .007 cos a .005 sin a S Hiinklcy, (1825 . . . . /= .135 .002 cosrt +.oot) sin a g Argelandcr, (1828). . . r= .000 + .001 cos a +.002 sin a 10 1 ulkowa, (1845) . . . r = .005 + .001 cos a + .002 sin a II Greenwich, (1864) . >-= .021 .oil cos a +.010 sin n 12 Washington, (1870) r +.034 + .004 cos .001 sin RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Tlie catalogues in which the periodic character of the residual is most strongly indicated are Auwers Cacciotore, Bessel I, and Greenwich. In the case of the latter the systematic difference, small as it is, makes up the greater part of the residual, since, if it is subtracted, the mean value of the residual is reduced from .0131 to .0064, while the probable error is reduced in an even larger proportion. Taking away the three catalogues just mentioned, the co-efficients m and n are no greater than might result from the accumulation of accidental errors ; it is therefore doubtful whether we ought to regard them as real. To find what discrepancies depending on the declinations arc indicated, I have grouped the twenty-seven stars as follows: 7 stars north of + 20 1 1 stars between -f- 6 and -f- 16 9 stars south of-f 6 The mean residual from the third column corresponding to each group is Catalogue. North group. s. Bradley .... + .006 . 046 Maskelyne + .021 Cacciotore. + .OO5 Bessel I .... + .013 Bessel II .... . OOf) OIQ Brinkley .... + .OIO Argelantlcr + .008 + .008 Greenwich. . + .003 Washington .008 Middle. South. s. s. .000 .006 - .013 + .049 .000 .022 + .002 .006 . 006 - .004 + .001 + . 004 + .004 + .OIO . 002 -005 + .007 - .016 .005 .001 + .004 . 006 + .003 + .003 Notwithstanding their general minuteness, these residuals depending on the decli nation are more certainly indicated than those depending on the right ascension. The catalogues in which they are largest, those of Piazzi and Maskelyne, have received but a small weight in the preceding disciission. In all the others, with the possible exception of Struve, they are so small that they may be treated as accidental errors. Their farther consideration is therefore, for the present, omitted. 5- . DETERMINATION OP THE POSITION AND MOTION OF THE EQUINOX. For the determination of the common correction to the right ascensions of all stars on account of error in the assumed equinoctial point, we are still dependent entirely on observations of the sun. We may ultimately hope to determine some of the quantities on which the absolute positions of the stars depend from observations and theories of the planets, the orbit of each planet, when corrected for perturbations, being a -fixed great circle, and the planets being susceptible of more accurate observation than the sun, but this course is not possible at present. It is well known that the best deter minations of the .equinox exhibit discordances far larger than can result from purely DETERMINATION OF THE POSITION AND MOTION OF THE EQUINOX. accidental errors of observation, and HO indicate the existence of systematic errors in the observations of the sun, the nature and cause of which are obscure. The deter mination of each co-ordinate of the sun seems about equally liable to them. We know that the absolute right ascension of a star depends effectively upon two elements, to be determined from observation, namely: 1 . The moment at which the sun crossed the equinox, from which its absolute rig-lit ascension at any time may be obtained, the elements of its orbit being- known. 2. The difference of right ascension between the sun and a star, from which the absolute right ascension of the latter is obtained from that of the former. Since any error depending on the sun s zenith distance will, in the first determina tion, be eliminated from the mean of the t\vo equinoxes, it might be supposed that there would be a general agreement among different authorities in this determination. Hut such is by no means the case, and I can assign no probable cause for the observed discrepancies. With reference to the second determination, it is well known that the personal equation of an observer is frequently different for the sun and for a star, and this dif ference will appear in the difference of their right ascensions. Diurnal changes in the clock, the instrument, and the personal equation of the observer will also introduce errors into the determination. Under these circumstances the most we can do is to base our determination on the widest range of material at our disposal, depending for accuracy on the number and variety of the determinations employed rather than upon the precision of any one. All the data at my disposal which did not involve a re-reduction of observations are comprised in the following discussion. The different common corrections to the preceding determination of right ascen- siovis of stars are represented by c. We begin with an examination of those published catalogues in which the star-positions are referred directly to the sun. BESSEL S BRADLEY. The right ascensions of the following stars were determined by Bessel by direct comparison with the sun. Opposite each star is given Auwers correction to Bessel s position, and the concluded correction already found, the signs being changed: Star. H. - A. n - (i). + .012 4- 063 /? Orionis .... - .040 .007 ra Orionis .... .030 .008 f) Gcminorum . + .102 + .143 a Lcnnis .... + . 106 + -149 n Virginis .... + .042 + .070 -4- 036 + .075 a Lyrnc + .030 + .058 a Aquilx .... + .012 + .030 Mean .... + .030 + .064 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. We liavo, therefore, from Bessel s determination for 1755 c =z PlAZ/I, l8oO. Of Piazzi s determination of tlie equinox I know nothing-. The mean correction given by his catalogue cannot, therefore, receive much weight, although the geograph ical position of his observatory is very favorable to its determination. The value of c, resulting- from this and from all the other original catalogues employed in the preceding- determination is the same with the constant terms in the systematic residuals, given on page 21, and represented by r. MASKELYNE, 1805. Maskclyne states that his catalogue was "settled with respect to the equinoctial points, from the observed declinations of the sun about the equinoxes in 1804, 1805, and 1806." With the same catalogue he gives one for 1756, deduced from Bradley s observations, the mean of which is o .io less than Bessel s, if we compare the twenty- seven stars under consideration, or o". 1 5 less, if we compare only the stars which Bessel found by direct comparison with the sun. This large difference may arise from Maske- lyne s having used only observations of the sun very near the equinoxes, while Bessel extended his determination to a declination of 14 on each side. An examination of the tables of corrections corresponding to different declinations of the sun, given on page 15 of the Fundamenta, docs, in fact, show a systematic difference depending on the declination, but their source and nature are too obscure to admit of much discus sion. All we can do is to suppose Maskelyne s determination an independent one, entitled only to small weight. BESSEL, 1815 AND 1825. Both Bessel s catalogues are referred directly to the sun. The first depends on three years observations with the old transit and the Cary circle, numbering 290 in all; the second on five years observations with the Reichenbach meridian-circle, num bering about 400 in all. STEUVK, 1825. Struvc s equinox depends on 216 observations of the sun, extending through a period of four years. (Positiones Media 1 , p. xxxx.) ARGELANDER, 1830. From a remark in the introduction to his catalogue, page iii, it would appear that Argelander, from his own observations of the sun, found a correction of +".072 to the right ascensions of the Tabulae llegiomontana?. We have, therefore, on this authority c + o".o64; epoch, 1828. POND, 1830. In the introduction to his "Catalogue of 1112 stars," Pond states that the right ascensions of Maskelyne s last catalogue have been increased by o .2O, being a quantity determined by observations of the sun made about the equinoxes with the present DETERMINATION OF THE POSITION AND MOTION OF THE EQUINOX. 25 transit-instrument since 1816. The mean value of e given by this catalogue is -f-o s .o/3. Hut, as the mean epoch of the observations is about seven years earlier, we may inquire whether the correction is different for this epoch. A correction of -f o s .2O applied to Maskelyne s catalogue for i8o5 would leave I infer, therefore, that the correction should have the same value for the mean epoch, which we may consider to be 1825. AIKV.S GREENWICH CATALOGUES. The five catalogues published by Airy have been so carefully referred to the sun that they do not appear to need any further discussion. The comparison of the indi vidual catalogues with the positions already found give the following values of e: s. First six-year catalogue . epoch 1839, r .052 Second six-year catalogue l &45, e .003 Third six-year catalogue 1 85 1, e -}- .01 7 First seven-year catalogue 1 857, i- .005 Second seven-year catalogue . 1864, c .02 i PULKOWA. The Pulkowa equinox depends on observations of the sun with the transit-instru ment and the vertical circle, extending from the year 1842 to 1850. The copious dis cussion of these observations bv Wagner is found in "Observations de Poulkova," vol. Ill, pp. 41-67. Adding Washington, 1870, this completes the list of original catalogues referred independently to the sun. But we have still a collection of data in the form of obser vations of the sun made and reduced at various observatories, and compared with the tables. I have reduced all these in a sumniarv manner, which, though coarse and rude, will, I am persuaded, lead to results differing from those of the most refined discussion only by quantities much smaller than the unavoidable probable errors of the results. The mean correction to the tabular right ascension and declination was taken for each month through a series of years, all the observations made in any month of a given name January, for instance being combined into a single mean. This mean was that of the entire number of individual results. The correction to the sun s absolute mean right ascension was then deduced from the observed errors of the tabular declination, being given by the approximate formula S R. A. (in time) . . 2 A <5 cos / / being the sun s longitude, and the factor cos / being the meail value of the cosine of the sun s true longitude during each month. The mean correction of the sun s absolute right ascension thus obtained is com pared with the correction obtained by comparison with the stars, and the difference, supposed to be due to error in the adopted right ascensions of the latter, gives the mean correction to the equinox to obtain the absolute right ascension of the stars. 4 F s 26 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. PARIS, i854- 63. (Corrections to CarfniUs tables.} No of No. of Month. A a obs. <5 obs. cos / <5 cos / s. ,, .. January . + .132 56 + 1.07 43 + -44 + 0.47 February . + .154 86 + 2.76 73 + .Si + 2.24 March . + .165 104 4- 2. 68 98 + .98 + 2.63 Apr 1 . . . + .209 116 + 2.25 105 + .89 + 2.OO May . . . + .245 So + 1.73 77 + .56 4- i. oo unc . + -234 Si + 1.20 74 + .10 4- 0.12 July . . . + .207 1 20 + o.So 84 39 0.31 August + .213 126 -t- 0.19 95 - -79 - 0.15 September + -194 98 0.31 76 - .98 + 0.30 October . + -133 93 + 0.50 6S - -92 0.46 November + .196 72 4- o.n 52 - .60 0.07 December + .148 49 + 1-30 40 . 10 0.13 Sum + 2.230 + 7-f>4 We hence deduce s. Mean correction to sun s absolute right ascension -(-0.212 Mean correction. deduced from stars ... +0.186 Therefore star s right ascensions require the correction + 0.026 A comparison of the corrections to the Tabula? Regiomontanre adopted in the reduc tion of the Paris observations with those deduced in this paper show that the latter are greater by o".O24. We may therefore put as the result of this series of Paris obser vations c =r o s .ooo PARIS, i864- 65 (Corrections to Le Verrier s tables.") \ No. of No. of Month. A a obs. ,5 obs. A ! cos / s. January . .072 10 + 0.20 7 + 0.09 February . .060 7 + 1-35 6 + i. II March . + .028 16 4- i.oS 16 + i. 06 April . .030 24 + 1-39 22 + 1.24 May . . . .013 21 0.05 2O 0.03 June . .052 19 0.29 18 0.03 July . . . .064 14 - o.SS 16 + 0.34 August .022 13 + O.II M 0.09 September .OO6 23 + 0.52 25 - 0.51 October . .022 9 + 0.45 1 6 0.41 November + .037 15 1.48 14 + 0.89 December - .017 13 - o.S6 13 + 0.09 Sum - 0.293 % + 3-75 i DETERMINATION OF THE POSITION AND MOTION OF THE EOflNOX. We hence deduce s. Mean correction to sun s absolute right ascension . . +0.104 Mean correction deduced from stars 0.024 Correction to adopted right ascensions of stars +0.128 The Paris concluded right ascensions for 1864 are less than those here deduced by o s .oi3. The result of these two years work at Paris is therefore r = + o s . 1 1 5 In the following two years nearly all the observations of the sun at Paris were made by a single observer, whose results are remarkably different from those of other observers, as may be seen by the very large and uniformly positive corrections to the right ascension and longitude of the sun, given in the collected results of the observa tions. This error evidently arises from the personal equation of the observer not being the same for the sun and for a star. Such differences are known to exist, but they are generally so minute as hardly to admit of positive detection. We have, however, one or two cases of the differences attaining an extraordinary magnitude, and, as they do not follow the usual law of error in their distribution -among different observers, we must reject entirely the cases in question. CAMBK1DGE, (ENGLAND.) The results of the observations of the sun during the five years 1833 37 are as follows: Month. da No. 61 obs. A,i No.of obs. A (! cos / s. ,, ,, January + .203 3i + I . oo 30 + o-44 February . + -I JS 40 ! + 0.96 ; 41 + o.So March . . . . -t- .175 44 + ".18 44 + 0.18 April . . . | + .230 55 + 0-36 56 + 0.32 May . . . i + .215 96 + o.f)8 95 + 0.38 June + .138 71 + 0.35 77 + 0.04 July . . . + .2<>5 59 4- 0.66 53 0.26 August + .182 77 + 0-14 80 O.I I Sepicmber . + . 208 65 , + 0.07 65 0.07 October . . + .180 73 + 0.46 75 <>-43 November . + -237 61 + 0.79 59 0.47 December . . + .175 42 + 0.93 37 0.09 Sum + 2.343 + 0.73 Mean correction to sun s absolute right ascension Correction deduced from stars . Correction to adopted right ascension of stars . Adopted right ascensions f/rctifcr than ours by . Hence c 0".03; epoch, 1835. s. -f- O.O2O + 0.195 -0.1/5 0.146 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. in subsequent years the discussion of the correction for each year in which suffi cient observations of the sun were made is given along with the results of the obser vations. The following are the mean values of c for series, each including five years actual observation: ,1. 1841, r=:-fo.o6 1849, c +0.06 1856, c:zz O.oi OXFOED. Since 1861 the sun has been pretty regularly observed at the Ifadclifte Observa tory, and the apparent errors of the tabular positions are given in the results of each year s observations. They are divided into two series, the first comprising the years i86i- 63, in which the results are compared with Carlini s tables, and the observations are considerably scattered; the second, comprising the years i864~ 68, during which the comparison has been made with Le Terrier s tables. The tabular summary of the results for the years i86i- 63 is as follows: No. of No.of Month. A 1 obs. A(i obs. A ti cos / s. ,, ,, January . . -t- .18 9 + 0.2 12 + O.I February . + .40 4 + 0.6 M + 0.5 March ... + .34 12 + "o . 6 15 + 0.6 April . ! + .32 19 + 1-4 19 + 1.3 May . . . ; + .33 16 + 1.5 II + 0.8 June ... + .16 13 + 0.6 II + O.I July . . .1 + .30 1 6 + 0.5 17 0.2 August . . + .38 8 + 1.4 8 I.I September. . + -35 M 1-3 M + 1.3 October . . + .53 7 + 0.8 9 - -7 November . . + .46 15 + 0.7 15 0.4 December . . + .40 12 + 2.3 12 O.2 Sum . . +4.15 . . + 2.1 8, Mean correction to sun s absolute right ascension . -}- 0.06 The same deduced from stars +-35 Correction to adopted right ascension of stars . 0.29 These adopted right ascensions are the same as those used during the correspond ing year in the reduction of the Greenwich observations, the mean of which corresponds very nearly to our adopted results. We have, therefore, from these Itadcliffe observa tions r o".29; epoch, 1862. The great and improbable magnitude of this result renders the question of its introduction somewhat embarrassing. DETERMINATION OF THE POSITION AND MOTION OF THE EQUINOX. For the live subsequent years, i864~ 6S, we have 29 Month. Au No. of obs. A<1 No. of obs. A cos s. 11 ,, anuary + .IT 36 + 2.0 36 + 0.9 "ebruary . + .10 37 + i.o 37 + o.S larch . + .13 28 . + i.o 27 + i.o Vpril . . . + .17 43 O.I 45 o.i lay + -15 45 o.i 47 o.i une + .o3 56 - 0.7 55 o.i uly . . + .06 55 + 0.3 57 i o.i August + .08 47 + 0.7 47 0.6 eptembcr . + .08 52 + o.i 50 O.I )ctober + .10 46 0.2 47 -i- 0.2 November . + -12 42 + I.o 41 0.6 )ecember . + .10 + 1.4 28 o.i Sum + 1.28 . + I.I 8, Correction to sun s absolute rig-lit ascension -f- 0.03 Correction deduced from stars . -f o. 1 1 Correction to adopted right ascension of stars -0.08 The adopted right ascensions beingstill the same as at Greenwich, this result gives ; O .OQ : epoch, 1 866. The correction is still negative, as in the years i86i- 63, though not so great. A comparison of the Radcliffe results with those derived from the Greenwich observations shows that the difference* is found chiefly in the observed relative right ascensions of the sun and stars. Tims, during the years i86i- 63, the mean correction to the sun s right ascension deduced from the Greenwich observations is -f o". 17, while that from the Radcliffe observations, nsing the same adopted right ascensions of stars, is + O 8 .35, so that the Uadclifi e results are greater by o s . 18. An examination of the Radcliffe observations shows that this great discordance did not exist in the year 1861, but commenced in 1862, -\\ith the introduction of the new transit-circle. It might, therefore, well give rise to the suspicion that the pointing of the circle is affected with a diurnal change. As the clock-error is determined from observations made at night, a constant error would thus be introduced into the results of transits of the sun. 1 ut this explanation seems to be negatived by the fact that the observations of the moon, though made at the same time with the stars, deviate still more widely from the truth. In fact, the menu correction to Hansen s right ascension, given by the observations of 1862, is -j- o".62, while all other observations show that the right ascension was then very nearly correct. We may heilcc conclude that the cause is to be sought for in some personal peculiarities of the observers. Though this ;o RIGHT ASCENSIONS OF FUNDAMENTAL STARS. explanation is made unsatisfactory by the circumstance that the observations of the two principal observers lead to nearly the same result, it is noticeable that in the year 1863 Ave find four observations of the moon signed M., probably made by the director of the observatory, the mean correction from which is -j- o a .O35. This seems, therefore, to be the only admissible explanation. EDINBURGH. In the Edinburgh observations for the years i836- 39 the correction to the equinox is deduced from each year s observations. The mean result is r ^-f- s -4j fpoch, 1838. Observations of the sun were continued till 1843. The published corrections to the tables lead to the following mean results: Month. <irt No. of obs. A 6 No. of obs. A <! COS / j. January + -17 50 + 1-3 32 4- 0.6 February . 4- .12 37 + 1-3 41 + i.i March . + .09 53 + 0.4 60 + 0.4 April + .08 57 + 0.5 50 + 0.4 May. . . . + .13 42 O.I 46 0. I June + .11 51 + 0.8 47 + O.I July. . + .09 37 0.2 43 4- o I August. + .05 43 0.2 44 4- 0.2 September . + .08 44 I.O 43 4- i.o October + .18 32 - 0.3 46 4- 0.3 November . + .05 32 + 0.3 20 0.2 December . + .13 3S - 0.3 41 0.0 Sum + 1.28 + 3-9 s. Mean correction to sun s absolute right ascension -j-- IO 8 Mean correction deduced from stars . . . . . . + o. 106 Correction to adopted right ascensions -f- 0.002 The adopted right ascensions are o s .oi less than those in this paper; therefore - r o s .oi; epoch, 1842. BERLIN. /. The introduction to the first volume of the Berlin observations, published by Encke, gives the results of observations of the sun from 1839 to 1842. The following are the mean results: DETERMINATION OF THE POSITION AND MOTION OF THE EOUINOX. Montli. A<; N o. of obs. . A ! No. of obs. A * cos s. January 14 4 1.4 5 0.6 February . + .04 20 o.o 20 o.o March . . . + .12 5 +0.2 4 +0.2 April 4- .06 IS + 0.9 18 + o.S May . . . + .02 14 0.4 14 0.2 June .00 7 i.i 7 o., July . . . .03 7 + 2.4 6 -0.9 August. + . 12 II +0.1 II 0. I September . + .07 25 o.o 23 o.o October + .04 13 - 1.3 10 + 1.2 November . + .11 14 + 0.7 14 0.4 December . + .11 7 + 0.2 S o.o Sum . . -t- -52 0. I 81 Mean correction to absolute right ascension o.oo? \J Correction from stars -f- 0.043 Correction to right ascension of stars - 0.046 The adopted right ascensions are those of the Tabula; Regiomontanae, which at this time are o".oi4 less than those deduced. We have, therefore, r = o".o6; epoch, 1841. The Berlin observations are distributed among the months with unusual irre>-u- larity. Had the mean by weights been taken, the result would have been ( o s .o; WASHINGTON. The following are the mean results of observations made with the transit-instru ment and mural circle during the four years i862- 65: No. of No. of Month. An obs. A,! obs. A(! cos / January 0.052 42 + 0.17 34 + 0.07 February . - 0.045 42 0.30 30 - 0.24 March . . . 0.022 41 + 0.40 31 + 0.39 April + O.022 41 + 0.70 36 + 0.62 May. . . . O.O2O 53 + 0.22 53 + 0.12 June . [ + 0.008 50 + 0.25 66 + 0.02 July . . . | + 0.015 5i + 0.30 55 - 0.12 August ... - 0.002 57 o.oo 50 o.oo September . . 0.038 42 0.22 47 + 0.22 October . . - 0.033 35 - 0. 10 55 + 0.09 November . . 0.027 35 O. IO 46 + O.o6 December . . 0.052 27 -r 0.57 32 - 0.06 Sum . 1 - 0.238 + I.I7 3 2 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean correction to sun s absolute right ascension -f- 0.032 Correction deduced from stars - 0.020 Correction to adopted rig-lit ascension of stars + 0.052 The adopted right ascensions are those of Dr. Gould, the mean of which, at this epoch, scarcely differs from that of our deduced positions. We have, therefore, from these observations f + o s .O5; epoch, 1864. Since 1866 the observations have been made with the transit-circle, but the instru ment was dismounted a part of the time. The first two years work gave the discordant results .S . 1866 frr-f-0.07 1867 p o.oo Mean c + 0.035 The following are the mean results for i868- 69: Xo.of No. of Month. A a obs. A .5 obs. A 6 cos / s. ,, ,, January . + o. 10 10 +- o.S 8 + 0.4 February . + 0.04 20 + 0.3 9 + 0.2 March . + 0.05 20 + 0.7 18 + 0.7 April . + 0.06 15 + 0.7 T 9 + 0.6 May . . . 4- O.O2 22 o.o 24 0.0 June . 4- o . 06 13 0.2 13 o.o July . . . + 0.07 15 - 0.3 15 + O.I August + 0.06 9 + 0. 5 7 0.4 September + 0.03 9 0.7 9 + 0.7 October . O.O2 13 i.5 14 + 1-4 November + O.OI 12 0.7 12 + 0.4 December + o . 06 10 0.9 IO + O.I Sum -I- o m + 4- 2 <^ " J*t S. Mean correction to sun s absolute right ascension -f- o. 1 1 7 Correction deduced from stars -f- 0.045 Whence c -f- o s .o7 The comparison is with Ilansen s tables, the mean right ascension of which is about o s .o6 or o s .o7 less than that of Le Verrier s tables. The true value of c being of the form .r -f ?/T, every separate determination of its value gives an equation of condition between x and y . The following is the system of equations thus obtained. I have included the mean result of the Greenwich observa tions for 1868, 1869, and 1870, which have been published since the appearance of the last Greenwich seven-year catalogue. The Radcliffe resiilts are omitted. DETERMINATION OF THE POSITION AND MOTION OF THE EQUINOX. 33 The weights have generally been assigned with reference to the entire number of observations on which each result depends, and also on the number of years over which the observations extend. But they have not been supposed to increase propor tionally to either the time or the number of observations, so that, although the weight i would be assigned to the result of a single average year s work with the best instru ments, the weight assigned to seven consecutive years work by the same observers, and with the same instruments, is much less than 7. On the same principle, greater proportional weight has been given to isolated determinations than to determinations founded on lony series of observations. Observations. Year. Equation. Years. Weight. s. Bessel s Uradley . 1756 . . &4_r + .06 7 3 Picizzi I Soo .v . 2OJ = .08 (> 2 V- Maskelyne ... 1805 - .i5/= - - 3 2 I Bessel I .... iSiy y - .03,1- = + .05 3 2 Bessel II . . . . 1822 .v + .02/ .00 5 4 Pond . . 1823 x + .03 v -f- .07 14 A Struve .... 1825 x + ,o$y + .04 5 4 Argelander . . . 1828 x + .08 y = -t- .06 tf . 1 Cambridge . , . 1835 x + . isy = - .03 5 2 Edinburgh ... 1838 x + . isy + .04 4 2 Airy , 1839 x + .ig/ = - .05 (, 3 Berlin . 1841 i.v + . 21 y ~ .07 4 2 Cambridge ... 1841 x -t- .2i/ = + .06 t 5 2 Edinburgh . . . 1842 .- + . 22J- = .01 4 2 Pulkowa .... 1845 x + .25; .00 8 5 Greenwich . . . 1845 x + .25/ -~ .00 6 3 Cambridge . ! 1849 x + .2<)/ = + .06 5 2 Greenwich . . . 1851 X + -3I/ + .02 6 3 Cambridge . . . 1856 .1- + . 3&y = .01 5 2 Greenwich . 1857 x + .37 y - - - oi 7 3 Paris . . 1859 x -4- 10 v .00 10 Greenwich ... 1864 i j7 j x + .44,1 = - .02 7 3 Washington . . . 1864 x + .447 = + .05 4 4 Paris . 1865 x + .45 y = \- . 1 1 2 2 Washington ... 1867 x + .47 y - + .05 4 4 Greenwich . . . 1869 1 x + .4 9 y = - .03 3 3 Treating these equations by the method of least squares, we are led to the normal equations 74 y + 14.9 i/ + i ".02 14.9 x + 7.4 # = 4-0*. 1 7 the solution of which gives x = -f o .oio ]) =. 5 F 34 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. The common correction to be applied to the star-positions first dednced is there fore + o".oi6 .009 T T being the fraction of a century after 1820. Applying- these constant corrections to the values of x and y previously found, we shall have a set of corrections to the Ta bulae Regiomontanae which we may suppose to be free from all systematic error what ever, at least so far as it is possible to free them with our existing data. 6 - CORRECTIONS FOR THE TERMS MULTIPLIED BY THE SECOND AND THIRD POWERS OF THE TIME. We have hitherto proceeded on the supposition that the necessary corrections to the Tabulae Regiomontanse increase uniformly with the time. But, on computing the co-efficients of the second and third powers of the time, in the expression for the mean right ascension of the star at any epoch, and comparing them with those found by induction from the mean places of the TabuLe Regiomontanse, it is found that this sup position is not correct. In fact, it appears that Bessel computed the second differential or secular variation of the annual variation without taking any account of the effect of proper motion, and that he considered it constant, thus neglecting the third derivative entirely. The following table shows, i, the secular variation of the precession actually employed in the Tabulae Regiomontanse ; 2, the adopted secular variation of the annual motion for 1800, computed from the co-efficients of Struve and Peters, using Hill s formulae in the "Star Tables of the American ephemeris," page 19; 3, the resulting correction to the first value; 4, the variation of the secular variation in one century, or the value of pr^g > tne um * ^ r o n ^ ascension being o s .ooi, and the unit Cl X of T one century; 5 and 6, the corrections to # and y necessitated by the addition of these terms to the positions of the Tabula? Regiomontanse. These last terms are derived as follows: Representing, for brevity, by a and b the , d* R. A. , d 3 R.A. ., . , . . , corrections to JJessel s assumed values ot and T-TT-PT . the right ascensions ot dl" d T 3 the Tabulae Reiomontanse should have received the correction for the epoch of each catalogue before the comparison with the catalogue. T being here counted from 1 800, the values of Cj, c. 2 , etc., already given, would have been thus changed by the following amounts: ?! O. lOrt + O.O2& c. 3 to c 4 o. c 5 .01 a c 6 =- .03 a c, .03 a S> T^T_-L _ .03 ft .04 a .IO .026 ?1=- .2Ort .066 CORRECTIONS FOR TERMS MULTIPLIED BY SECOND AND THIRD POWERS OF TIME. 35 Substituting these values of Ci, etc., in the expressions for .rand y, (p. 10,) we find .08 a .o i 6 . i3 .066 the values of which are given in the last two columns. Star. d? R A Bessel s rv-~~ a 1 From Struve, (1800.) h AJ: \y y Pcgasi .... + 97 + 953 12 + 63 O 2 a Arictis .... 4- 2010 + 1998 12 : 4- 41 I a Ccti 4- O7O + QCQ T II ! V> 939 1 1 "J" 1_| 4~ I 41 a Tauri .... 4- 1090 4- 1065 25 - 39 4-2 -h 6 ft Orionis. + 430 + 47 23 14 4-2 +4 ,3 Tauri .... 4- 890 4- 862 28 - 95 + 3 : + 9 a Orionis. + 31 + 303 7 ; 42 + i +4 a Canis Majoris. 4- 40 64 - i Hi ii a Canis Minoris 440 515 75 ~ 20 ft Gcminorum - I22O - 1216 4- ; 64 o 4-4 a Ilydrx .... 150 157 7 + 25 o i Lconis .... - IO2O - 1039 9+37 4-1 o (3 Lconis .... 770 7&S 4- 2 ; 4- 03 I 4 a Virginis 4- 1 1 20 4 1 1 1 r 9 ; + 5<-> 0-2 a Bootis .... 4- 1 20 4- 202 4- )2 4- 44 7 - 3 a- Librae .... + 1550 + 1537 3+17 4-1 +1 a Coronas + 240 4- 228 2 4- 17 4-1 4-1 a Serpcntis . 4- 630 4-609 - : !I 4- 5 4-2 4-2 o Scorpii .... + 1570 + 1549 : !I 56 4- 6 re Hcrculis . + 37 + 353 i 7 i n + 1 +3 1 a Ophiuchi . + 35 + 344 6 13 4-1 4-2 a Lyras .... 4- 1 60 4- 108 i a | - 6 + 4 +7 7 Aquila; .... - So 99 l 9 5 4- 2 4-3 Aquila: .... 140 180 \o 1 + 5 ft Aquilae .... - 150 140 4- 8 i I 2 Capricorn* . 810 834 4 IS 4- 2 ! -4-4 <z Aquarii 430 444 1 4 +35 4- i o a Pcgasi .... + 530 + 523 7+56 2 Andromcdic . + 1770 + 177 1 4- 83 1 i - 5 The values of x and y, or the correction to the right ascension and annual varia tion of the TabuLe Regiomontanae for the epoch 1820, may now be found by adding to the values of those quantities already found (p. i 7) the constants for correction of equinox -f- 1 6 and 9, and the corrections from the above table. After doing this we have, for convenience of application, transferred the epoch to 1800. Adding the co-efficients for the second and third powers of the time, we have the following values of the corrections in thousandths of seconds of time to the right ascensions of the Tabulae Regiomontanee, where we put T fraction of century after 1 800 : . ,6 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. v.7 Corrections to Tabula Rey wmontance. y Pegasi, 4 R. A. - + 26 - 1 3 8T - 6T a + ioT 3 a. Arietis, 3 + T 57 6 +7 Ceti, +33+8i 6 +2 a Tiuiri, 3+90 -12 6 /? Orionis, +41+27 -12 2 /? Tauri, +66165 -14 -16 a Orionis +24-10 4 7 a Canis Majoris, +26+64 -52 2 + j (Auwers) a Canis Minoris, -27+127 -38 3 + q (Auwers) /? Geminorum, 20 + 231 + 2 -11 a Hydra, -20 + 276 4 +4 a Leonis, -29 + 216 -10 + 6 /? Leonis, 0+193 + i +10 a Virginis, -16+67 4 + 8 a Bootis 7+99 +41 +7 a 2 Libra?, +80241 6 +3 a Coronae, -12+152 6 +3 a Serpentis, + 48 J 36 - 10 + i a Scorpii, +41 131 - i o 9 a Herculis, 5+66 8 2 a Ophiuchi, - 2 7 + 35 3 2 a Lyra, +2+86 - 26 i y Aquila?, +69 1 94 - i o i a Aquilo?, +7+35 ~ 2O J 13 Aquila>, +39216+5 i a 2 Capricorni, 6 6 -12 2 a Aquarii, -34+165 7 +6 Tega.si, + 9+ 18 4 +9 a Andromeda 1 , -12 + 185 + H We have here added the stars Sirius and Procyori to the list because Anwers has determined their irregular proper motions with sufficient accuracy to allow of their being used as standards of comparison fora long time to come. The following are the details of the operations by which Auwers results have been applied in the above table. PKOCYON. By his researches on the proper motion of Procyoa, Auwers has found a certain correction, q, to its right ascension, denned as follows: Represent by f/ { , (/ 3 , etc., to q 6 , the several corrections to be applied to the right ascension of Procyon in Wolfers "Tabula; Reductionum," in order that the respective differences between the corrected right ascension and the right ascensions of a Tauri, /3 Orionis, a Orionis, /? Geminorum, a ITydrse, and a Leonis, given in the same tables, may be correct. Then Auwers puts CORRECTIONS FOR TERMS MULTIPLIED BY SECOND AND THIRD POWERS OF TIME. 37 It follows that if we represent by ^, J.,, etc., to J e , the definitive corrections to Wolfers positions of these six stars, the correction to Wolfers position of Procyon will be \24 + q But this result requires a small change arising in this way. Wolfers and Auwers have assumed that the corrections to the mean places of the Tabula? Regiomontana? increase uniformly with the time. In other words, it is assumed by Auwers that q is of the form x + yt + P P being the purely periodic terms depending on the orbital motion. But, on examin ing the values of the terms depending on the second and third powers of the time in the preceding table it will be seen that the correction is really of the form + i s rti 1 i T ^ /// o".03i I -+ P To determine rigorously the correction to q which would have been introduced had Auwers taken account of the term in T s Avould be difficult. Owing to the extreme minuteness of the correction, Ave assume that the effect of changing the term would correspond to the effect of a similar change in the equations used in this paper, or ^lx J R. A. (1820) .08 a + o s .oo5 ^y .i3 = .i3X 62 = + The correction to Auwers will then be T being the fraction of a century after 1 800. The value of this correction will be only .006 in 1/55, and will not exceed o".oo3 during the period 1 800 to 1 860. We therefore conclude that the correction to the position of Procyon given in the Tabula? Reductionum is ^ + 3 + 8T- 3 iT* + / It remains to determine d. Subtracting Wolfers corrections to the positions of the Tabula? Regiomontanae, as found in the Tabula? Reductionum, pp. xxxv, xxxvi, from the corrections given in the preceding table, it seems that our corrections to Wolfers 1 positions are as follows : a Tauri ft Orionis a Orionis ft Gerninorum a Hvdra? */ a Leonis Mean J- -24 - 14 T 7 T 3 T 3 The correction to Wolfers position of Procyon is, therefore, -2i-6T- 3 8T -3T + 2 Adding Wolfers correction to Bessel s position of Procyon, which is -6 + I33T* 25 - 7 T - I 2 r 8 - 6 p r 3 + 2 -61 - I 2 2 + I 99 4 Ij 35 + 5 + 2 - 14 -40 + 28 4 + 4 -47 + 5 - 10 + 6 ^8 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. J we have the result given in the table for the correction to the right ascension of Procyon in the Tabulae Regiomontanae. SIEIUS. Auwers correction to the position of Sirius is given in a form similar to that adopted for Procyon, but the stars of comparison are /? Orionis, a Orionis, and Pro cyon. The positions adopted for comparison are those of the Tabulae Regiomontanpe, except that the right ascension of Procyon is first corrected by the quantity -o s .025 o s .ooi3<86 (t 1825) o a .o;o5 cos 9.oo634 (t 179557) or by the quantity + 10- 139 T + P P being the periodic term. The terms in the correction to the relative motion of Sirius which depend on the second and third powers of the time are . - 34 T 2 +2T 3 and, reasoning as in the case of Procyon, we conclude that Auwers q should be cor rected by the quantity The value of this correction for different dates is s. i 760 .005 1780 +.001 1 800 + .004 1820 +-004 1840 +.002 1860 --003 1880 . -.010 The correction to the right ascension of Sirius in the Tabulee Regiomontanse is, therefore, 4 being the mean correction to the right ascensions of /? Orionis, a Orionis, and Pro cyon, the latter star being, however, first corrected by the expression given above. Comparing this expression with Auwers q, (A. N. No. 1382, p. 62,) the value of which is 39 1 19 T o s .o7O5 cos n (t T) we see that the preliminary correction to the position of Procyon is q 29 20 T CORRECTIONS TO DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. 39 Subtracting this correction from the above correction to Procyon, taking the cor rections to /? Orionis and a Orionis without change, and representing by z/,, J.,, and J-i the individual corrections to Auwers assumed positions of the three stars of com parison, we have ^ = + 41 + 27 T - i 2 T- - 2 T :i - f o + 24- 10 4 -7 ^ 3 =+ 2 + 147 -38 -3 ,4 = + 22 + 55 T--l8T a - 4 T 3 Adding this to the expression which replaces Auwers g, we have the result given in the above table. 7- CORRECTIONS TO DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. The preceding investigations have resulted in giving us a set of 29 right ascen sions, which we may suppose to be free from all systematic errors whatever, and of which we may suppose the mean accidental error to be much less than that of any one catalogue determination. We may, therefore, suppose any systematic differences which exhibit themselves between the right ascensions of any catalogue and those just obtained to be due to errors of the former, and from the entire body of the differences we may obtain the correction necessary to reduce it to the standard. This correction may be considered to consist of three parts 1. A constant, which we may consider as a correction to the adopted equinox. 2. Periodic terms depending on the right ascension, which in all recent catalogues we may suppose to be of the form m cos a. + n sin a and to which terms of double period in cos 2 a + n sin 2 a may be added, if it be deemed necessary. In some of the older catalogues, especially in Bessel I, the terms of double period appear to be quite sensible, but thev ought not to be real in any modem catalogue. 3. A term depending on the declination, which, within the limits in which the present investigations are applicable, may be regarded as proportional to the simple declination. In application, it might be better to suppose it proportional to the sine of the declination. The limits in question may be considered as 40 north declination and 30 south declination, but the correction in question will necessarily become uncertain as we approach either limit. These three corrections have been obtained in the order in which they are num bered, as follows: Represent by RIGHT ASCENSIONS OF FUNDAMENTAL STARS. the several excesses of the standard right ascensions over those of the catalogue to be compared. The mean value of c may then be taken as the correction for equinox, which we may represent by e. Subtract e from each individual c, and represent the residuals by c t , e 2 , etc, The values of m and n may then be obtained in the same manner by which the periodic terms of the residuals of the separate catalogues have been obtained. But the addition of Sinus and Procyon to the list of stars will make a slight change in the equations, so that we shall have i 1.9 in 2 c (cos a .0 sn i 7. i n 2 c (sin a .04 cos a) The values of the coefficients- cos a .03 sin a and sin a .04 cos a for each star are given in the following table. Co-eflicient for Star. cos . sin n .gin 1 7 . 1 72 a Andromcdtc . . + I.oo .00 + i . oo .04 7 Pegasi . + i.oo + .03 + i.oo .01 a Arietis ... + .87 , + .50 + .86 + .47 a Ceti .... | + .73 + .69 + .71 + .66 a Tauri .....+ .39 + .92 + ,36 + .9 P Orionis . . . j + .22 + .97 + .19 4- -9 6 ft Tauri .... + .19 + .98 + .16 + .97 n Orionis ... + .06 ! + i.oo + .03 + i . oo a Canis Majoi is . .16 + .99 .19 + i.oo a Canis Minoris . .38 + .92 .41 + -93 ft Geminorum . . : .40 -f- .91 -43 + -93 a Hydras ... .76 + . 64 - .78 + .67 a Leonis ... .86 + .51 .88 + .54 (3 Leonis ... i.oo + .09 I . 00 + .13 a Virginis ... .95 .32 .94 .28 a Bootis. . . . .85 .53 - .83 - .50 a- Librae .... .77 .6} .75 - .61 a CoronacBorealis. .62 .78 - .60 .76 a Serpentis ... .59 .81 - -57 - -79 a Scorpii ... .43 .90 .40 .88 a Hcrculis . . . .23 .97 . 20 . 96 a Opbiuchi . .16 .99 .13 .98 a Lyric . 4- n on + .16 I . oo 7 Aquilae . j + .41 .91 4- .44 -93 a Aquilse . . . + .43 .90 + .46 ,92 Aquilae . J + .45 .89 + .48 .91 Capricorn! . . 4- .53 .85 + .56 : -37 a Aquarii ... -fr .86 .51 + .88 .54 a Pegasi ... + .96 .28 4- .97 -32 CORRECTIONS TO DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. 41 Subtracting from each c the corresponding value of m cos a -\- n sin a the residuals ought to l>e purely accidental errors, unless they are found to depend on the declination. In this case, if we suppose a term proportional to the simple declina tion, we may represent it in the form .Dec. 10 the zero of reference being 10 north for convenience, because that is nearly the mean declination of all the adopted standard stars, and the degrees of declination being divided by 10 in order to have a more convenient factor. We may then find d by solving the equations of condition formed by putting the above expression equal to c" for each star. The following is a convenient and sufficiently accurate expression for d: 60 d =1 -\- 2 c" of a Andromedac. + i - - a Arietis. i . .a Ceti. + I . - or. Tauri. 2 . . /? Orionis. + 2 /? Tauri. 2 . . a Canis Majoris. + 2 . . /? Geminorum. 2 . . a. Hydra;. -)- i . . /? Leonis. - 2 . . a Virginis. + I . - a Bootis. 2 . .or Libra;. -f- 2 . . a. Corona?. 2 . . a Scorpii. + i - - llerculis. -f- 2 . .a Lyrse. 2 . .or Capricorni. i . . Aquarii. As an example of the method we may discuss the twelve-year series of Paris observations, extending from 1856 to 1867. The different results are shown in the fol- lowin<>- table. First we trive the mean correction to Le Vemer s fundamental cata,- O 5 logue deduced from all the observations, using the mean results given in each volume. Next we have the resulting corrections to Bessel. Subtracting the correction to Bessel for the mean epoch, we have the correction to the standard of this paper, or c. The mean value of c is found to be 32, which is the value of e. Adding this, we have c . 6 F s 42 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. From the values oi c we find m = + 20 n = 23 Adding 20 cos a 23 sin a, we have c". From c" we have f?- + 9 Applying- 9 - to c" we have r, or the definitive corrections to the standard given by the Paris observations. Star. Corrections to - c Computed periodic terms. c" r LeVerrier. Tab. Reg. a Andromeda: . - 15 + 57 47 15 + 20 + 5 + 21 7 Pcgasi - 16 27 - 69 37 + 20 - i? - 13 a Arietis ... 31 + 33 ; - 57 - 25 + 5 20 8 a Ccti .... + 34 + 53 i - 27 + 5 i + 4 2 n Tauri .... 25 + 49 + 3 + 35 13 + 22 + 27 /? Orionis + 52 + 76 + 23 + 55 18 + 37 + 21 ,3 Tauri . . . . a Orionis - 6 + 16 - 59 + 13 - 18 3 + M + 29 19 22 5 + 7 + II + 4 a Canis Majoris (- 228) +39 4 + 28 26 + 2 22 a Canis Minoris . (+ SO + 37 4 + 28 - 29 I 6 ft Geminorum . - 6 + 92 24 + 8 - 29 21 - 5 a Hydra; + 61 + 157 + 12 + 44 - 3 + M - 2 a Lconis + 20 + 98 o + 32 - 29 + 3 + 6 ft Leonis + 4 + 86 - 32 o 22 22 - 17 a Virginis . + 28 o 24 + 8 II - 3 21 o liootis . . , + 7 + 30 - 39 7 4 II 2 a 2 Librx .... + 23 66 + I + 33 + 33 + 10 n Corona 25 + IS 60 - 28 + 6 22 7 n Serpcntis . + 28 74 36 4 + 7 + 3 o a Scorpii + 46 - 36 +6 + 38 + 12 + 50 + 18 n Ilcrculis . ii 27 - 59 27 + IS - 9 4 a Ophiuchi . I + 146 37 5 + 20 + 15 + IS ft Lyne .... - 63 48 -93 - 61 + 25 - 36 10 7 Aquilic - 5 - "3 - 3 - 3i + 29 2 2 a Aquilx . 24 44 - 65 33 + 29 - 4 - 5 ft Aquilse + 6 - 148 60 - 28 + 29 + i 3 i 2 Capricorni + 36 49 - 33 I + 30 + 29 + 8 a Aquarii + 10 + 9 - 55 - 23 + 29 + 6 4 n Pegasi - 42 42 62 - 30 + 25 " 5 1 The systematic corrections obtained in this way for a number of catalogues are given in the following list. The year attached to each catalogue is that to which it is reduced. Systematic corrections much smaller than the accidental errorrs of the cata logue may generally be omitted entirely. CORRECTIONS 1 TO DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. 43 SYSTEMATIC CORRECTIONS TO BE APPLIED TO THE RIGHT ASCENSIONS OF EQUATORIAL AND ZODI ACAL STARS IN DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. AUWERS BRADLEY, (1755.)* 4 rr o s .o I o + O S .OO3 cos a o s .OO7 sin a The mean value of the periodic terms being- less than one-third the probable error of the individual star-places, they may be neglected entirely. BESSEL S FUNDAMENTA, 1755. J zr o a .oi DE ZACII S FIXARUM PRECIPUARUM CATALOGUS Novus EX OHSERVATIONIBUS ASTRONO- 3IICIS IN SPECULA ASTRONOMICA GoTHANA ANNIS 1787, 1788, 1789, 1790, HABITIS ET AD INITIUM ANNI MDCCC. CONSTRUCTS. Dec. i o 4 zz: + o".36 o s .03 - 10 PlA/ZI, l8oO. , Dec. 10 = + o".io-fo a .02S o MASKELYNE, (1805.) Catalogue in vol. 4 of Ms observations, second part, p. 1 12. Dec. 10 -|- o . 145 -|-o .019 cos tx -f- o .006 sin a o .012 - AUWERS CACCIOTORE, (1805.) zz O 8 .oo6 O B .O 10 cos a O 8 .O33 sin a BESSEL S. FIRST FUNDAMENTAL CATALOGUE, (1815.) ^ o".O3 7 o s .O3 7 cos a -f- o".O3 7 sin a . BESSEL S SECOND FUNDAMENTAL CATALOGUE, (1825.) ^ -f o".O2o o".oo6 cos a -\- o".oio sin a STEUVE, (1825.) , Dec. 10 zr O .O2O -f O H .OO7 cos a -{- o .005 sin a -f o .008 - o~ AuworH 1 catalogue being Htill iiiipul)li.Hli( il, tliin correction must lie uiiiU i.stood IIM ap|iliculilu only to tho [iri 1 - limiuury catalogue, in which no correction to BcHM-l s equinox has liecn ili tcrinini-U or 44 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. BRINKLEY, (1824.) ARGELANDER, (1830.) ,- Dec. 10 + o s .oi 7 o .006 - 10 G-ILLISS S FIRST WASHINGTON CATALOGUE, 1840. {From observations made at the old Naval Observatory during the years i838- 42. Wash ington, 1 846. . 8vo.) A -f- O 8 .O3 2 cos a o s .O28 sin a. POND, 1830. J o s .o56 o s .oi5 sin a. (For stars in south declination the additional positive correction O 8 .ooi5 X south declination in degrees is strongly indicated.) AIRY S CAMBRIDGE CATALOGUE. In the Memoirs of the Royal Astronomical Society, vol. XI. 4 o s .O24 o s .oi8 sin 2 a (The term of double period, though a priori somewhat improbable, is strongly indicated by the comparisons. TAYLOR, (1835.) General catalogue of the principal fixed stars from observations made at the honorable the East India Company s observatory at Madras in the years 1830-1843. Dec. 10 ^/ O 8 .O47 + o .023 cos a. -(- o .001 1 The last term is far within the probable error of the catalogue-positions. GREENWICH 12-YEAR CATALOGUE. Catalogue, for 1840. = + o s .o67 -f o s .oi4 cos a 01034 sin a Catalogue for 1845. -|- o".oi9 -|-o 8 .oo7 cos a o s .o25 sin a CORRECTIONS TO DIFFERENT CATALOGUES TO REDUCE THEM TO A MEAN SYSTEM. 45 ROBINSON S ARMAGH CATALOGUE, (1840.) 4 r= -f O".O2O. (No positions of fundamental stars are given in this catalogue. The correction has, therefore, been derived from a comparison of all the stars south of 45 north polar distance which are common to Armagh and the Greenwich 6- Year Catalogue for 1 840, and observed at least five times at each observatory. The differences between these catalogues were corrected for the error of the Greenwich catalogue, and the out standing differences were presumed to be due to error of the Armagh catalogue.) PULKOWA, (1845.) 4 = -f o". RADCLIFFE CATALOGUE, 1845. 4 zz o s .oo9 -f- o".O47 cos a o s .O2o sin a (The positions of the fundamental stars in this catalogue give the correction z/ + o s .ooi + C/.O55 cos a o".O2 1 sin a In the introduction to the catalogue, Mr. Main gives a comparison of this catalogue with Greenwich, 1845, from which I derive Greenwich Radcliffe = o".O3S + 01032 cos a -(- o".oo5 sin a Adding the Greenwich correction, this will give, for Radcliffe, 4 =. o".oi9 + o s .O39 cos a. o".O2O sin a The mean result has been adopted.) GREENWICH 6- YEAR CATALOGUE, (1850.) l o".oo3 -f- o".oi i cos a o".O23 sin a GREENWICH 7- YEAR CATALOGUE, (1860.) J -f o 8 .oi4 -f- o s .oi4 cos a o".oi2 sin a GREENWICH 7-YEAR CATALOGUE, (1864.) d + o".O33 -|-o s -Q I o cos a. o a .oio sin a PARIS ANNUAL CATALOGUES, i856- 67. Dec. 10 -)- o a .O32 -(- o s .O2o cos a o .023 sin a -f o .009 - r~c WASHINGTON "POSITIONS OF FUNDAMENTAL STARS," (1870.) (In Appendix III to the Washington Observations fur 1867.) 4 6 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. WASHINGTON ANNUAL RESULTS OF OBSERVATIONS WITH THE OLD TRANSIT-INSTRUMENT, (i862- 6 5 .) ^/ -)- o s .oo6 -f- o s .O2 1 cos a o s .O24 sin a. WASHINGTON ANNUAL RESULTS OF OBSERVATIONS WITH THE TRANSIT CIRCLE, (1866-72.) A -f- o s .oo6 Those of the above corrections which form a considerable part of the probable error of the catalogue-positions are tabulated in the following- tables. In the corrections depending on the declinations the constant term is included, except for Paris, where it is included in the first table. Corrections to different catalogues depending on the right ascension. R. A. Airy s Cambridge catalogue. Taylor s Madras catalogue. AIRY S GREENWICH CATALOGUES. V Paris annual catalogues, 1855-67. Radclifle catalo ue, 1845. 1840. 1845. 1850, 1860. 1864. h. s. j. s. s. S. s. s. s. s. o .02 .02 + .081 + .026 + .008 + .028 + .043 + .052 + .04 i - -03 .02 + -072 + .020 + .002 + .025 + .040 + -045 + .03 2 - -04 - -03 + .062 + .013 .005 + .O2I -T--037 + .037 + .02 3 - .04 - -03 + .053 + .OO6 .Oil + .015 + -033 + .030 + .01 4 - -04 - .04 + .045 + .001 -.017 + .OIO + .029 + .022 - .01 5 - -03 - .04 + .038 .003 .022 + .005 + .026 + .015 .02 6 .02 - -05 + -033 .006 .026 + .OO2 + .023 + .OOg -03 7 .02 - .05 + .030 .007 .028 .O + .021 + .005 - .04 8 .01 - .66 + .031 .007 .028 .003 + .OIg + .002 -05 9 .01. - .06 + .033 .004 -.027 .004 + .Oig + .002 .06 10 .01 - .07 + .038 + .OOI .024 .004 + .0lg + .003 - .06 II .02 - .07 + .044 + .006 .020 .OO3 + .021 + .007 - .06 12 .02 - .07 + 053 + .OI2 .OI4 .0 + .O23 + .012 - .06 13 - -03 - .07 + .062 + .018 .008 + .003 + .026 + .Dig .05 14 .04 - -07 + .072 + .025 .OOI + .007 + .O29 + -027 - .04 15 - .04 - .06 + .081 + .O32 + .OO5 + .013 "+.033 + .34 - -03 16 - .04 .06 + .089 + 037 + .016 + .018 + .37 + -042 .02 17 - -03 -05 + .096 + .041- + .OI6 + .023 + .040 + .049 .00 " 18 .02 - .05 + .IOI + .044 + .02O + .026 + .043 + .055 + .01 19 .02 - .04 + .104 + .045 + .O22 + .028 + .045 + -059 + .02 20 .01 - .04 + .103 + .045 + .O22 + .031 + .047 + .062 + .03 21 .OI - .03 + .IOI + .042 + .O2I + .032 + .047 + .062 + .04 22 .OI - .03 + . 096 + 37 + .018 + .032 + .047 + .061 + .04 23 .02 .02 + .090 + .032 + .014 + .031 + .045 + .057 + .04 24 .02 .02 + .081 + .O26 + .008 + .028 + .043 + .052 + .04 DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Come/ions depending on the declination. 47 Dec. N. P. D. Piazzi, 1800. Struve, Posit. Med. Argelander, 1830. Paris annual catalogues, i856- 63. s. s. s. s. -t-40 5 + .18 + . oo - .001 + .-027 35 H + -17 .000 + .002 + .023 30 60 + -1C - .004 4- .005 + .018 25 65 + -14 - .008 + .008 + .014 20 7" + -13 .012 + .Oil + .009 15 75 + .11 .016 + .014 + .004 IO So + .10 .020 + .017 .0 + 5 5 + .09 - .024 + .020 - .004 o 90 + .07 .028 + -023 .009 5 95 + .06 .032 + .021 - .014 10 100 4- .04 -036 + .029 .018 -15 105 + .03 - .040 + .032 .022 20 no + .02 - -"44 + -035 - .027 -25 5 .OO - .048 + .038 .032 8.. DEFINITIVE BIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Applying these corrections to the catalogue-positions, we may deduce definitive corrections to the individual right ascensions of the stars under consideration. If, in doing this, we employed only the catalogues used in the first determination, and assigned the same relative weights as in the first discussion, we should reproduce the right ascensions already found, lint the weights were first assigned solely with respect to the probable freedom of the catalogue from periodic errors dependent on the right ascension, while, after elimination of these errors, they should be assigned with respect to the purely accidental errors of the determinations, and this change of weights Avill make a small change in the individual definitive positions. We can also add a number of catalogues not employed in the preliminary determination, particularly the Green wich and Paris catalogues, which will add considerably to the precision of the results. The details of the process are shown in the following tables. Under each star is given, in the first column, the difference of excess of catalogue-place, corrected for sys tematic error over the standard place already deduced. In two cases several catalogues of small weight have been combined, as the exhibition of the individual results of each catalogue would not be of interest. The first set are the catalogues of 1 iazzi 1800, weight i Maskelyne .... 1805, weight 2 Auwers Cacciotore . . 1805, weight 2 Bessel T 8i5, weight i which have been combined into a single result for the mean date 1807. The mean RIGHT ASCENSIONS OF FUNDAMENTAL STARS. systematic correction was so small that only the correction for equinox was actually applied. The next case was Airy s Cambridge catalogue and Pond s general catalogue, which were combined into a single result. No corrections depending on the declination have been applied, except in the case of Paris and Argelander. Under the titles Greenwich, 1869, and Washington, 1869, are given the mean results of the Greenwich and Washington observations for the years 1868, 1869, and 1870, without any correction for periodic error. The weights assigned are constant for each catalogue, except in the case of Bradley, where the weight 2 was assigned to results depending on more than 100 observations, and i to those depending on less. The adopted weights and general solution are as follows, the time T being reckoned from 1845 : No. Catalogue. Date. Weight. T Cx C x C , C"y I Bradley 1755 i or 2 - .90 .035 .054 - .46 - -63 2 Combined catalogue . 1807 I - -33 .029 .024 .20 - .14 -3 Bcssel II .... 1823 2 .22 .055 .048 - .24 - .16 o A Struvc l82( I .21 .027 .023 .11 -7 *f 5 Argelander .... 1828 4 -17 .108 .95 - -39 - -24 6 Pond- and Airy . 1830 i - -15 .026 .023 - .08 - -05 7 Greenwich .... 1839 2 - .06 .050 .048 - .08 - .05 8 Greenwich .... 1845 2 .O .049 .048 .02 .0 l8 45 4 .O .098 .095 .05 J .0 10 Greenwich .... 1851 3 + .06 .072 .071 + .05 + .06 it Greenwich .... 1857 3 + .12 .070 .071 + .14 + .12 12 1862 4 + -17 .092 .095 + -25 + -21 13 Greenwich .... 1864 4 + .19 .090 .095 + -33 + .26 14 Washington .... 1865 5 + .20 .112 .118 + -44 + .35 IS Greenwich .... 1869 2 + -24 .044 .047 + .21 + -17 16 Washington .... 1869 2 + .24 .044 .047 + .21 + -17 The equations given by the several catalogues, when solved by least squares, lead to the result that 4 R. A. for 1 845 = 2 C x X > J Ann. variation 2 C y X r r representing the correction given by any catalogue, and C x , C y , factors constant for each catalogue, except that there will be two systems of these factors, the one corre sponding to the stars for which the results of Bradley s observations receive the weight i, the other to those in which they receive the weight 2. The former are designated by one accent, the latter by two. In the case of C x the differences are so minute that a single mean value has been actually used. In the column /, under each star, is given the outstanding residual after subtract ing the concluded correction 4 x + T 4 y Sinus and Procyon are included only for the sake of comparison. No correction to Auwers places is indicated. DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Residual corrections given by catalogues. 49 a Androm. }- Pegasi. a Arietis. a Ceti. a Tauri. (i Orionis. No. Cutnlou iic Year r / r r r r r r r r r t> i Bradley .... 1755 + 15 + 12 - 29 - 49 + 9 + 4 + 32 + 45 + 24 + 17 + 3 + 6 2 Combined catalogue 1807 + 9 + 4 + 23 + 17 12 II 24 20 G - IS - 7 - 4 3 Bessel II .... 1823 12 - IS + 27 + 25 12 10 - 30 - 29 + 5 - 8 - 25 22 4 1824 - Gi -67 - i - 3 + 2 + 5 + I + 2-15 - 28 + 53 + 56 5 Argelander 1828 + i - 5 + 15 + M - 8 - 5 10 10 + 2 ii IO - 7 6 Pond and Airy 1830 + 27 + 20 + 40 + 40 + 2 + 5 - 28 - 28 + 22 + 8 ii 8 7 Greenwich. 1839 + 26 + 19 o + 2 28 - 24 + 4 + 38 + 42 + 28 - 13 IO 8 Greenwich .... 1845 + 6 I - M II + 23 4- 28 - 23 2G + 44 + 29 - 3 o 9 1845 + 8 + I - 3 + M + 19 + 4 4- I 10 - 25 + 6 + 9 10 Greenwich .... 1851 + 23 + 15 - 7 - 2 + 6 o - 4 + 21 + 6 2 o ii Greenwich .... 1857 + 28 + 20 - 2 + 4 - 9 3 - M - 19 + IS + 2 - 26 - 24 12 Paris 1862 + 21 + J 3 13 _ 8 . i 2 4- 27 + 11 -4- 2T + 21 3 Greenwich. 1864 + 9 + I - 18 10 - 6 + i + 8 7 + 2 i * l + 19 + 3 T^ * * - 4 ^J 2 4 Washington . 1865 + i - 7 - 13 - 4 - 7 o + 21 + 15 + M - 3 - 4 2 15 Greenwich. 1869 + 2 - G 21 II - 30 - 23 IO - 16 + 26 + 9 - M u 16 Washington . 1869 22 - 30 + 9 10 T + 43 + 37 + II - 6 + 6 + 8 No. Catalogue. Year. /3 Tauri . o Orionis. a CanisMaj. a CanisMin. /? Gcmin. a Hydra. I Bradley .... 1755 + 4 + 2 - 9 - 9 o o . + ii + 7 - 5 12 2 Combined catalogue 1807 + 32 + 25 + 27 + 26 - 4 - 6 2 - 4 Bessel II .... 1823 - 4 12 + 2 o . . 12 - 13 - 23 - 24 4 Struvc . ... 1824 22 - 30 - 14 - 16 . . - 23 - 24 + 25 + 24 5 Argelander 1828 + 3 G + M + 12 - 17 . + S - i 4- 10 + 10 6 Pond and Airy 1830 + 15 + G + 32 + 3 + 5 + M + 10 + 9 + 48 4-48 7 Greenwich .... 1839 - 4 *- M + 4 + 2 . + 2 - 24 + II + ii - 15 - 14 8 Greenwich .... 1845 + 6 - 4 - 5 - 7 6 . + 13 + 3 + 3 + 3 + 4 Pulkowa .... 1845 + 14 + 4 13 15 + 6 + 6 -+ 5 + 6 10 Greenwich. 1851 + 17 + 6 - S . + 4 + 2 - 3 - 3 T^ D - 6 - 4 ii Greenwich. . 1857 + 34 + 23 + 3 o + 12 12 + 7 + 7 4 - i 12 1862 + ii i + 4 4- i 1 1 6 , c 2 -f- i 13 Greenwich .... 1864 + 16 + 4 + 16 + 13 * - 17 . + 7 D + G D + 7 - S r 4 Washington . 1865 - M - 2G + i 2 . + If) o ii IO 12 - 9 15 Greenwich .... 1869 + 36 + 24 + 2 I . - 3 + 23 + 9 + 10 4- 26 + 30 16 Washington . 1869 + 19 + 7 + i - 2 + I + 15 - 7 - 6 - 23 - 9 1 F S RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Residual corrections given by catalogues. a Leonis. ft Leonis. a Virginis. a Bootis. a 2 Librae. a Coronae. No. Catalogue. Year. r r r r r r r 1 r r r r i Bradley .... 1755 + 12 + G! + i + ii - 5 - 13 I - 6 - 7 - 8 + 5 II 2 Combined catalogue 1807 o 8 + 10 + 17 2 o + 5 + 3 + 3 i - 28 - 35 3 Bessel II .... 1823 + I - 5 - 26 20 + 22 + 27 - 3 - 3 - 3 - 8 i - 5 4 Struve 1824 - 4 10 12 7 - 16 10 - 17 - 17 - 19 - 24 + 30 + 26 5 Argelandcr 1828 S - 14 2 + 3 o + 6 - 9 - 9 + 19 + 13 + 9 + 6 6 Pond and Airy 1830 9 - 15 - 3 + 2 - 15 - 8 - 6 - C + 26 + 20 + 16 + 13 7 Greenwich .... 1839 + 3S + 32 - 10 - 6 - 15 C + 24 + 25 + 28 + 23 - 5 - 6 8 Greenwich .... 1845 + ii + 5 - 30 - 26 - 15 - 5 + 6 + 7 + 30 + 24 + 9 + 9 Pulkowa .... 1845 + 2 - 4 + G + 10 + 8 + 18 + 10 + ii ii 17 2 10 Greenwich .... 1851 + 6 + i + 4 + S - 16 + 15 + iG + 4 J / 2 + 28 + 28 ii Greenwich .... 1857 + 16 + ii I + 2 -36 - 24 + 8 + 10 - 9 - 15 -t- 15 + 17 12 Paris 1862 + f> + i - 17 - 4 21 - 8 2 o + 10 + 3 - 7 - 5 13 Greenwich .... 1864 5 - 10 + 5 + 8 22 - 9 - 6 4 + 16 + 9 + .3 M Washington . 1865 + 5 - 3 + I + 15 12 10 - 7 - 18 - 15 15 Greenwich .... 1869 + ii + & + 12 + 14 21 - 7 - 19 16 + 10 + 3 10 - 6 16 Washington 1869 + 6 + i - 5 - 3 ~ 14 10 - 7 + 20 + 13 - 25 21 No. Catalogue. Year. a Serpentis. a Scorpii. a Herculis. a Ophiuchi. a Lyrx. y Acjuiloe. i Bradley .... 1755 + 5 + i - 25 - S + ii + 10 + 14 + 16 i + 2 o i - 4 2 Combined catalogue 1807 - 44 -46 + 27 + 36 -38 - 40 - 19 - 23 +14 + M 2 - 4 3 Bessel II . . . . 1823 + 6 + 5 + 12 + IS + 12 + 9 - 32 - 37 4- 3 + 4 I 2 4 Struve 1824 + 38 + 37 + 16 + 22 + 38 J- oe -}- 4O -1- -5C AC ji + 10 + r\ 5 Argclander ^ ~r 1828 I J^ + ii [ J i + 10 - 13 - 7 1 J^ + 6 [ jj + 3 + 10 ~ JO 4 J + 421 f-f 20 + M y + 13 6 Pond and Airy 1830 + 23 + 2 3 - 35 - 30 - 29 - 32 + 3 - 6 20 21 7 Greenwich .... IS39 - 24 - 24 + I + 5 - 4 - 7 + 18 + II + 3 + 5 + 3 + 2 S Greenwich .... IS45 + i + I + 28 + 31 - 18 21 + 12 + 5 o + 2 - 5 - 5 9 Pulkowa .... 1845 3 - 3 - 15 12 - 19 22 - 17 - 24 + 18 + 21 + S + 8 10 Greenwich .... 1851 o o - 37 - 35 + 22 + IS + 8 o + 21 + 24 - 18 - 18 ii Greenwich .... 1857 - 4 - 3 10 - 8 + 33 + 29 + 24 +15 + 17 + 2O + 10 + 10 12 Paris 1862 o _j- i + 18 -t- 10 J 8 4- rfi 4- o JO "7 2 2 13 Greenwich .... 1864 + i + 2 - 5 *y - 5 4 + 18 + M i iu ~f y + 22 +13 + 6 / + 9 12 - 12 14 Washington 1865 4 - 3 + if) + 16 + i -- 3 + f> - - 4 - 5 - 2 + 5 + 5 15 Greenwich .... 1869 - 6 - 5 - 13 - 13 + 10 + 6 4-10 026 -23 - 18 - 18 16 Washington . 1869 + 7 + 8 - 18 - IS - 3 - 7 2 12 34 i - 31 + 19 + 19 DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Residual corrections given by catalogues. No. a Aquiloe. /? Aquilsc. - Capricorn!. Aquarii. a Pegasi. r r r r r r r / r r i Bradley .... 1755 - 9 II - 1 ~ 9 21 - 6 - 16 - 5 2 2 2 Combined catalogue 1807 + 15 + 14 - S 10 + 10 + 19 + 4 + 10 + 2O + 19 3 Bessel II .... 1823 + 9 + 9 + 31 + 29 + 31 + 33 + 18 + 23 + 7 + 5 4 Struve 1824 - 6 - 6 - 3 + i + 27 + 34 - 3 4- 2 - 29 3i 5 Argelander 1828 - 3 - 3 -t- II + 9 - 17 10 - 17 12 + 7 + 5 6 Pond and Airy 1830 r / - 7 32 - 34 - 59 - 53 + 4 + 8 2 - 4 7 Greenwich .... 1839 + 23 + 23 7 - 9 - 27 22 - 4 + 10 + S S Greenwich .... 1845 - 5 - 5 + 5 + 3 5 i - 13 - 10 + 2 9 Pulkowa .... 1845 + 7 + 7 + i i i + 4 - 4 I - 9 ii 10 Greenwich .... 1851 + 2 + 3 + 12 + 9 - 28 - 24 2 + I + 4 + i ir Greenwich .... 1857 O + I - 10 - 13 - 19 - 16 - 18 - 16 - 9 - 12 12 1862 f. 4 3 6 + S + II 4 2 i A *t 4 Greenwich .... 1864 - 7 - 6 4 - 7 + 10 + 12 - 18 - If) + 4 + I 14 Washington . 1865 + 4 + 5 - 3 - 6 - 5 3 + ii + 13 + 13 + 10 15 Greenwich .... 1869 - S - 7 + 24 + 21 + 10 + 12 - 3 2 - 4 - 7 16 Washington 1869 - 19 - 18 + 7 + 4 7 - 5 + 25 + 26 + 19 + 16 1 There are still three of Maskelyne .s fundamental stars wliich it is desirable, for fho sake of completeness, to add, namely, a Auriga?, a Cvgni, and a. Piscis Anstralis. Although these stars lie a little outside the limits we have set, they are frequently used as clock-stars. The corrections to the positions of the Tabulae Regiomontanse were obtained on the same general system with that adopted for the other stars. The; tabular positions were corrected for the terms depending on the second and third powers of the time, as given in the following tables, and the catalogue-positions for systematic error given by the formulae of the last section. From the differences the correction proportional to the time was deduced. RIGHT ASCENSIONS OF FUNDAMENTAL STARS. From the residuals r we derive the folloAving definitive corrections to the right ascensions for 1 800 and to the annual variations of the first standard : Definitive corrections to right ascensions of Tabula JKegiomontance. Star. A R. A. &y Definitive correction. a, Andromeda; . + 5 + 5 - 7 + 190 T + o 1- +14 T 3 y Pcgasi .... + 8 - 26 + 34 164 - 6 +10 n Arietis .... o ii - 3 +146 - 6 +7 re Ccti 5 + 17 + 28 +98 6 +2 a Tauri .... + n + 8 + 8 +98 12 6 Aurigx - 2 +26 -31 -34 ft Orionis. - 3 + i + 38 + 28 12 - 2 ft Tauri .... + 6 + 9 + 72 156 14 16 re Orionis. + I + 3 + 25 7 - 4 - 7 re Canis Majoris . + 155 + 29 -52 - 2 + P re Canis Minoris. -15 + 8 -38 - 3 + P ft Gcminorum + 2 4 18 +227 + 2 ii + 1 IO 17 +266 4 +4 a Lconis .... i J + 6 i 23 +215 10 + 6 ft Leonis .... 7 + 7 7 +200 + i +10 re Virginis . i - 19 -17 +48 - 4 +8 a Bootis .... + 2 7 - 5 +92 +41 + 7 a- Librie -4- A + 6 -1-8 A 2^S 6 +T a Coronic i H + & i? I u-f ^"33 J - 4 +135 - 6 +3 a Scrpcntis . + 2 - 5 + 50 141 10 + i re Scorpii .... IO + 15 + 31 116 10 - 9 re Ilerculis + 2 + 3 -3 +69 -8 -2 n Ophiuchi . + 3 + 10 24 +360 - 3 - 2 re Lyric .... 4 + 2 +S2 26 - I }- Aquilac .... + 2 3 + 71 197 10 - i , Aquilx .... + I - 3 + 8 +32 20 - i ft Aquilrc .... + 2 + i +41 215 +5 - i re - Capricorn! . 10 + 12 16 + 6 12 3 re Cygni .... -32 +184 -17 + 3 n Aquarii. 7 + 8 -41 +173 - 7 + 6 n Piscis Australis . 50 152 - 6 +11 re Pcgasi .... + i + 3 + 10 +21 - 4 +9 In the above table P signifies the periodic part of Auwers correction, which is more convenient to use than q. The following table gives the definitive right ascensions for every five years from 1750 to 1900. I conceive that the mean error of these right ascensions, after correction for equinox, will not exceed o s .oio at any time during the nineteenth century. DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. 53 Mean right ascensions for beginning of each fiflli Bcssdlan fictitious year, \ 750-1850. Year. a ANUKOMEDJE. y PKGASI. K. A. Ann. Var. R. A. Ann. Var. //. m. s. s. /i. in. s. s. /So 23 55 31-079 3 0665 o o 23.520 3.0702 i/55 46.413 3.0674 38.872 3.0707 1760 23 56 1.752 3.0683 54- 227 3.0712 1/65 17-095 3-0691 o i 9.584 3.0716 1770 32.443 3.0700 24-943 3.0721 775 47-795 3.0709 40. 305 3.0726 1780 23 57 3- I5 1 3-0717 55-669 3-0730 1785 18.512 3.0726 2 11.035 3-0735 1790 33-876 3-0735 26. 404 3.0740 795 49. 246 3 0744 41-775 3-0745 1800 23 58 4.620 3-0/53 57- 148 3-0749 1805 19.998 3-0761 o 3 12.524 3-0754 1810 35-38i 3-0770 27.902 3-0/59 1815 50. 768 3-0779 43- 283 3.0764 1820 23 59 6. i 60 3.0788 58.666 3.0769 1825 21. 55 6 3-0797 o 4 14.052 3-0774 1830 36. 956 3 . 0806 29.440 3-0778 35 52.361 3-0815 44-830 3-0783 1840 o o 7.771 3.0824 050. 223 3.0788 1845 23- 185 3-0833 15.618 3-0793 1850 38-603 3.0842 31.016 3.0798 1855 54.026 3-0851 46. 416 3 0803 1860 o i 9.454 3 . 0860 o 6 1.819 3. 0808 1865 24.886 3 . 0869 17.224 3-0813 1870 40.323 3.0879 32.632 3.0818 1875 55-764 3.0888 48.042 3-0823 1880 O 2 I I. 2IO 3-0897 -o 7 3-455 3.0828 1885 26. 661 3.0906 18.870 3-0833 1890 42. 1 16 3-0915 34.288 3.0838 1895 57-575- 3-0925 49. 708 3-0843 1900 o 3 13.040 3-0934 o 8 5.131 3.0848 54 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean riyM ascensions, d c. Continued. Year. a ARIKTLS. a. CETI. R. A. Ann. Viir. R, A. Ann. Vav. //. ill. S. S. h. m. s. s. 1750 i 53 8.365 3-3430 2 49 14.423 3- 1170 1755 25-083 3-3440 30. 009 3- H75 1760 41.805 3-3450 45-598 3- H79 1765 58.533 3-346o 2 50 I. 189 3. 1184 1770 i 54 15.266 3-3470 16. 782 3. 1 189 1775 32.003 3-348o 32.377 3-II94 1780 48. 746 3-3490 47-975 3-1198 1785 i 55 5-493 3-3500 2 51 3-576 3. 1203 1790 22.245 3-3510 19. 179 3. 1208 1795 39.003 3-3520 34- 784 3- 1213 1800 55-765 3-3530 50-391 3.1218 1805 i 56 12.533 3-3540 2 52 6.001 3 . 1222 1810 29.305 1815 46.082 3-3550 3-356o 21.614 37.228 3- !227 3- 1232 1820 i 57 2.865 1825 i9- 6 52 1830 36.445 3-3570 3-358o 3-3590 52.846 2 53 8.465 24.087 3- 1237 3- I2 4 2 3. 1246 1835 53-242 3.3600 39.712 3- 1251 1840 i 58 10.045 3.3610 55-338 3- 1256 1845 26.852 3.3620 2 54 10.968 3. 1261 1850 43-665 3-3630 26. 599 3. 1266 1855 i 59 0.482 3-3640 42.233 3- 1271 1860 17-305 3-3650 57-870 3-1275 1865 34- 133 3.3661 2 55 13.509 3. 1280 1870 50.966 3-3671 29- 150 3-1285 1875 2 o 7.803 3-368! 44- 794 3- I2 90 1880 24. 646 3-3691 2 56 0.440 3- 1295 1885 41.494 3-37oi 16.088 3- 1300 1890 58.347 3-37 11 31-739 3- 1304 1895 2 115. 2O6 3-3721 47-393 3- 1309 1900 32. 069 3-3732 2 57 3-049 3- i3H DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean right ascensions, &c. Continued. 55 Year. a TAUKI. a AURIGA. 11. A. Ann. Var. 11, A. Ann. Var. //. III. S. s. li. in. ,s. s. i/50 4 21 36.293 3-4227 4 58 15-977 4.4004 i/55 53-408 3-4232 37-98i 4.4014 i 760 4 22 IO. 526 3-4238 59.990 4.4023 1/65 27. 646 3-4243 4 59 22.004 4.4032 1770 44- 769 3-4248 44-023 4.4042 775 4 23 1.894 3-4254 5 o 6.046 4.4051 i 780 19. 022 3-4259 28.073 4 . 4060 1785 3 6 - S3 3-4265 50. 106 4.4069 1790 53 .286 3.4270 5 i 12. 142 4.4078 1795 4 24 10.423 3-4275 34-184 4.4087 1800 27.562 3.4281 56-230 4.4096 1805 44-704 3-4286 5 218. 280 4.4105 1 8 1 o 4 2 5 i . 848 3-4291 40.335 4.4114 1815 18.995 3.4296 5 3 2.394 4.4122 1820 36.144 3-4302 24-457 4-4I31 825 53-297 3-4307 46.525 4.4140 1830 4 26 10.452 3-4312 5 4 8.597 4.4148 1835 27.609 3.43i8 30.673 4-4 57 1840 44. 769 3.4323 52.754 4.4165 1845 4 27 1.932 3-4328 5 5 14-839 4-4174 1850 19. 097 3-4333 36.928 4.4182 1855 36. 265 3-4339 59-022 4.4190 1860 53-436 3-4344 5 621.119 4.4199 1865 4 28 To. 609 3-4349 43- 221 4.4207 1870 8/5 2 7- 7 8 5 44.963 3-4354 3-4359 5 7 5-326 27.436 4-4215 4.4223 1880 4 29 2. 144 3-4364 49-550 4-4232 1885 9-3 2 7 3-4370 5 8 11.668 4.4240 1890 36.5 4 3-4375 33-790 4-4248 < S 95 53-702 3.438o 55-9I6 4.4256 1900 4 30 10.894 3-4385 5 9 18.045 4.4264 RIGHT ASCENSIONS OF FUNDAMENTAL STARS, Mean riylit ascensions, &c. Continued. /? ORIONIS. /? TAUEI. 1 p ll* -1 Cd 1 R. A. Ann. Var. R. A. Ann. Var. li. in. s. s. h. m. s. 8. 1750 5 2 32.084 2-8757 5 10 30.599 3-7775 1755 46.463 2-8759 49.488 3- 7779 1760 5 3 o. 843 2.8761 5 ii 8.378 3- 7784 1765 15.224 2.8763 27. 272 3-7788 1770 29. 606 2.8765 46. 167 3-7793 775 43.989 2.8767 5 12 5-064 3- 7797 1780 58.373 2. 8769 23.964 3- 7802 1785 5 4 12.758 2.8771 42. 866 3. 7806 1790 27. 144 2.8773 5 U 1-770 3.7811 T 795 4i.53i 2.8775 20. 677 3-7815 1800 55.9I9 2.8777 39.585 3./8.9 1805 5 5 10-309 2.8779 58.496 3- 7824 1810 24.699 2.8781 5 14 17.409 3. 7828 1815 39.090 2.8783 36.324 3-7832 1820 53.482 2.8785 55-24I 3.7836 1825 5 6 7.875 2.8787 5 i5 H. 160 3.7840 1830 22. 269 2. 8789 33-o8i 3- 7845 1835 36.665 2.8791 52.005 3-7849 1840 51.061 2.8793 5 16 10.930 3.7853 1845 5 7 5-458 2.8795 29-858 3-7857 1850 19.856 2.8797 48.787 3.7861 i855 34-256 2-8799 5 17 7.719 3-7865 1860 48.656 2.8801 26. 652 3-7869 1865 5 8 3.057 2.8803 45-588 3.7873 1870 17-459 2.8805 5 i 8 4-526 3.7877 1875 31.862 2.8807 23-465 3.7881 1 880 46. 267 2.8809 42.407 3-7885 1885 5 9 0.672 2.8811 5 19 1.350 3.7889 1890 15.078 2. 8813 2O. 296 3-7893 1895 29.485 2.8815 39-243 3-7897 1900 .89 3 2.8817 58. 193 3- 790, DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean riyJtt ascensions, cCr. Continued. 57 a OKIOXIS. a CANIS MAJOKIS.* Year \{. A. f Ann. Yiir. U. A. Ann. Var. //. m. s. s. //. m. s. s. 1750 5 41 38.680 3.2429 6 34 7.882 2. 6446 i/55 54-895 3-2431 21- 105 2.6445 i 760 5 42 1 1 . 1 1 1 3-2432 34-327 2.6445 1765 27.327 3-2434 47-550 2-6445 !/7o 43-545 3-2435 6 35 0.772 2-6444 775 59-763 3-2437 13-994 2-6444 1780 5 43 15.982 3-2439 27. 216 2.6444 1785 32.201 3.2440 40.438 2 - 6444 1790 48.422 3.2442 53-66o . 2.6443 1795 5 44 4-643 3-2443 6 36 6. 88 i 2-6443 1800 20.865 3-2445 2O. IO2 2-6443 1805 37-088 3- 2446 "> 1 1 *> 1 oo- j-4 2.6442 1810 53-3 11 3.2448 46.545 2. 6442 815. 5 45 9-535 3- 2449 59-766 2.6442 1820 25. 760 3- 245 6 37 12.986 2.6441 1825 41.986 3-2452 26. 207 2.6441 1830 58.212 3-2454 39-427 2.6441 1835 5 46 H-439 3-2455 52.647 2. 6440 1840 30.667 3-2456 6 38 5-867 2 . 6440 1845 46.896 3-2458 19.087 2.6440 1850 5 47 3. 125 o- 2459 32.307 2-6439 1855 9-355 3.2461 45-527 2-6439 1860 35-586 3.2462 58. 746 2.6439 1865 51-817 3.2464 6 39 11-965 2-6438 1870 5 48 8.050 3-2465 25. 184 2-6438 1875 24.282 3.2466 38-403 2.6437 1880 40.516 3.2468 51.622 2-6437 1885 56-750 3.2469 6 40 4. 840 2.6437 1890 5 49 12.985 3.2470 18.058 2-6436 1895 29. 220 *> It** * j- -4/ - 31.276 2.6436 1900 45-457 ^ "J \ *7 ^ o- -4/0 44.494 2-6436 8 F S The corrections, P, on p. Gl>, are to l>o applied to tliese positions of a Canis Mnjoris. RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean right ascensions, c. Continued. Year. a CANIS MINORIS.* /? G-EMINORUM. R. A. Ann. Var. II. A. Ann. Var. /;. m. s. s. n. m. s. . s. 1750 7 26 12.027 3-I507 7 29 58.793 3.6964 J 755 27.780 3- 1505 7 30 17- 274 3-6958 1760 43-532 3- J502 35-75 1 3-6952 1765 59.282 3- H99 54-225 3.6946 1770 7 27 15.031 3- H97 7 3 1 12.697 3.6940 !775 30.779 3- H94 31-165 3-6934 1780 46.525 3- H9 2 49.631 3-6928 1785 7 28 2. 271 3- 1489 7 32 8.093 3.6922 1790 18.015 3- 1487 26.553 3.6916 1795 33-757 3- 1484 45-009 3.6910 1800 49-499 3.1482 7 33 3-462 3 6904 1805 7 29 5.239 3- H79 2*1.913 3.6898 1810 20.978 3- J 47 6 40. 360 3.6892 1815 36-715 3- 1474 58.804 3.6885 1820 52-452 3- i47i 7 34 17-246 3-6879 1825 7 30 8.187 3- H69 35-684 3-6873 1830 23.920 3. 1466 54- H9 3.6867 1835 39-653 3- 1463 7 35 12.551 3.6861 1840 55-384 3- H6i 30.980 3-6855 1845 7 3i ii. H3 3- HSS 49.405 3.6848 1850 26.842 3- H56 7 36 7-828 3.6842 1855 42.569 3- H53 26.248 3.6836 1860 58.295 3- H5o 44. 664 3-6829 1865 7 32 14.019 3- 1448 7 37 3-077 3-6823 1870 29.742 3- 1445 21.487 3.6817 1875 45.464 3- H42 39-894 3.6810 1880 7 33 1-185 3.1440 58.298 3 6804 1885 1 6 . 904 3- H37 7 38 16.699 3-6798 1890 32. 622 3- H34 35-097 3.6791 1895 48.338 3- M32 53-491 3-6785 1900 7 34 4-053 3- H29 7 39 11-882 3.6778 * The corrections, P, on p. 71, arc to bo applied to these positions of a Canis Minoris. DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean riylit ascensions, (fr. Continued. 59 Year. a HYDRJE. a LEONIS. R A. Ann. Var. R. A. Ann. Var. I/. III. S. 8. //. m. s. s. 1750 9 15 17.906 2.9512 9 55 1-762 3- 2148 755 32.662 2.9512 17-835 3-2143 1760 47.418 2 -95 i 33-905 3-2138 1765 9 16 2. 173 2.9510 49-973 3-2133 1770 16.928 2-9509 9 56 6.038 3.2128 1775 31.682 2.9508 22. IOO 3 . 2122 1780 46.43 6 2-9507 38. I 60 3.2II7 1785 917 i. 190 2-9507 54-217 3. 21 12 79 5-943 2.9506 9 57 10.272 3.2107 795 30. 695 2-9505 26.324 3. 2102 1800 45.448 2.9504 42-373 3. 2096 1805 918 O. 2OO 2-9503 58.420 3 . 209 i 1810 4-95 1 2-9503 9 S 8 14-465 3. 2086 .815 29. 702 2- 95O2 30-507 3. 2081 1820 44-453 2.9501 46. 546 3.2076 1825 59-203 2-9500 9 59 2.582 3- 2071 1830 9 19 3-953 2.9500 18.617 3. 2066 835 28. 703 2-9499 34.648 3. 2061 1840 43-452 2.9498 50.677 3.2056 1845 58. 2or 2-9497 10 o 6. 704 3.2050 1850 9 20 12.950 2-9497 22. 728 3-2045 855 27.698 2.9496 38.749 3.2040 1860 42.446 2-9495 54- 768 3-2035 1865 57- 93 2-9495 10 i 10. 784 3.2030 1870 921 ir. 940 2 . 9494 26.798 3.2025 i875 26.687 2-9493 42. Sio 3. 2020 1880 41-433 2.9492 58.818 3- 2015 1885 56- 179 2.9492 IO 2 14.825 3- 2OIO 1890 9 22 IO. 925 2.9491 30.829 3-2005 895 25.670 2 . 9490 46. 830 3. 2OOO I9OO 40.416 2-9490 10 3 2.829 3- 995 6o RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean riff Jit ascensions, dec. Continued, Year. /? LEONLS. a VIEGINIS. R A. Ann. Var. R A. Ann. Var. Ii. m. s. s. /(. In. s. s. 1750 ii 36 17.250 3.0745 13 12 3.448 3-i38i 1755 32.622 3-0741 19. 140 3-*3S6 1760 47-992 3-0737 34-834 3- 1392 1765 ** 37 3-36o 3-0733 50.531 3- *397 1770 18.7-25 3-0729 13 13 6.231 3- H03 1775 34.089 3-0725 21-934 3- 1408 1780 49-45* 3.0722 37-639 3- i4H 1785 ii 38 4.811 3-0718 53-348 3- Hi9 1790 20. 169 3. 0714 13 *4 9-059 3- H25 1795 35-5 2 5 3-07*o 24.772 3-i430 1800 50,879 3.0706 40. 489 3- *436 1805 ii 39 6.231 3. O7O2 56. 2O8 3-144* 1810 21.581 3-0699 13 15 11.930 3-*447 1815 36.929 3-0695 27.655 3- 1452 1820 52.276 3-069 1 43-382 3- H58 1825 ii 40 7.620 3.0687 59-ii3 3- *464 1830 22.963 3-0683 | 13 16 14.846 3- 1469 1835 38.304 3.0680 30.582 3-H75. 1840 53-642 3.0676 46.321 3- 1481 1845 ii 41 8.980 3.0672 13 17 2.063 3:*486 | 1850 24-315 3.0669 17.807 3- 1492 1855 39.648 3.^0665 33- 555 3- *498 1860 1865 54.980 1 1 42 10. 309 3- O66l 3.0658 49-305 3-1503 13 i 8 5.058 3-1509 1870 25.637 3-0654 20.814 3-15*5 1875 40-963 3.0650 36.573 3-1520 1880 56.288 3.0647 52-334 3-1526 1885 ii 43 11.610 3-0643 13 19 8.099 3- iSS 2 1890 26. 931 3.0640 23.866 3.1538 1895 42.250 3.0636 39-637 3-1544 1900 57-567 1 3-0633 55-4io 3-*549 DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean right ascensions, dr. Continued. 61 Year. a Boons. or LIBRAE. K. A. Ann. Var. R. A. Ann. Var. //. 111. .V. s. //. ill. S. s. i/5o 14 4 15.978 2.7320 4 37 5-726 3.2881 755 29-638 2.7321 22. I 68 3.2888 1760 43- 2 99 2.7322 38.614 3.2896 1765 56.960 2.7323 55.064 3.2904 1770 14 5 10.622 2.7324 14 38 11.518 3.2911 1/75 24.284 2.7325 27.976 3- 2919 1780 37.946 2.7326 44-437 3-2927 . i 785 51.609 2.7327 14 39 0.902 3-2934 1790 . 14 6 5.273 2.7328 17-37I 3.2*942 795 * l8 -937 2.7329 33.844 3- 2950 1 800 32.601 2.7330 50-321 3-2957 .805 46.266 2.7331 14 40 6. 802 3-2965 1810 59-932 2.7332 23.286 3-2973 1815 14 7 13.598 2. 7333 39- 775 3- 2980 1820 27.265 2.7334 56.267 3.2988 1825 40.932 2.7335 14 41 12. 763 3.2996 1830 54-599 2.7336 29.263 3-3004 1835 1840 14 8 8.267 2.7337 21.936 2.7338 45- 766 14 42 2.274 3-30ii 3-3QJ9 1^45 35- 6 o5 2-7339 18.785 3.3026 .850 49-275 2. 7340 35-301 3-3034 1855 14 9 2.945 2-7341 51.820 3-3042 1860 16.616 2-7342 14 43 8.343 3-3050 ,865 30. 288 2- 7344 24.870 3-3058 1870 43.960 2.7345 41 . 401 3-3065 i875 57-633 2.7346 57-935 3-3073 1880 14 i o 11. 306 2-7347 14 44 14.474 3-3081 ,885 24. 980 2-7348 31.016 3-3089 1890 38-654 2-7350 47-562 3.3096 895 52.329 2-7351 14 45 4- U3 3-3104 1900 14 ii 6 . 005 2-7352 20. 667 3-3"2 62 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean right ascensions, rfr. Continued. a CORONA. Yo-u- a SERPENTIS. J_ CCliJ. . K A. Ann. Var. R. A. Ann. Var. /;. in. s. s. //. 111. S. s. 1750 15 24 6.587 2.5360 J 5 3i 58.355 2.9431 i/55 19.267 2-5361 15 32 13-071 2-9434 1760 3I-948 2.5363 27.789 2-9437 1765 44-630 2-5364 42.508 2 . 9440 i//o 57-3 12 2-5365 57-229 2-9443 1775 J 5 2 5 9-994 2-5366 15 33 11.952 2.9446 1780 22.678 2.5367 26.675 2.9449 785 35-362 2.5368 41.401 2.9452 1790 48.046 2.5369 56.128 .2-9455 1795 15 26 0.731 2-5370 5 34 10.856 2.9458 i 800 13-416 2-5372 25-586 2. 9462 1805 26. IO2 2-5373 40.318 2.9465 1810 38.789 2-5374 55-05I 2.9468 1815 5^476 2-5375 15 35 9- 785 2.9471 1820 15 27 4. 164 2.5376 24.521 2-9474 1825 16.852 2-5377 39-259 2-9477 1830 29.541 2.5378 53.998 2. 9480 1835 42.231 2-5380 15 36 8.739 2.9483 1840 54-921 2.5381 23.481 2. 9486 1845 5 28 7-612 2-5382 38-225 2.9489 1850 20.303 2-5383 52.970 2.9492 1855 32.995 2-5384 15 37 7-7I7 2-9495 1860 45. 687 2.5386 22.465 2.9498 .865 58.380 2-5387 37-215 2.9501 1870 15 29 11.074 2-5388 5^966 2.9504 1875 23.768 2-5389 i5 38 6.719 2.9507 1880 36.463 2.5390 21.473 2.9510 1885 49-159 2.5392 36. 229 2-95 3 1890 15 30 1.855 2-5393 50-987 2.9516 1895 H.552 2-5394 15 39 5-746 2.9520 1900 27. 249 ! I 2-5395 20.506 2.9523 i DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean rigid ascensions, (C-c. Continued. Year. a ScORPIl. a HERCUIJS. R. A. Ann. Var. R. A. Ann. Var. //. in. t>. s. h. in. s. 6 . 1750 1 6 14 7.423 3.6487 17 3 15.554 2.7286 ! 755 1760 25.669 43.918 3-6495 3-6503 29. 198 42.842 2. 7288 2. 7290 r / 6 5 1615 2. 172 3-6511 56.487 2.7291 1770 2O. 429 3-6519 7 4 10. 133 2.7293 1775 38.690 3-6527 23.780 2- 7295 1780 56.956 3-6534 37-428 2. 7297 1785 16 16 15. 225 3-6542 51-077 2.7299 1790 33-498 3-6550 17 5 4-727 2.7300 i/95 51-775 3-6558 18.378 2.7302 1800 16 i 7 10. 056 3.6566 32.029 2.7304 1805 28.341 3.6573 45.681 2.7306 1810 46. 629 3-6581 59-335 2.7307 1815 16 18 4.922 3-6589 17 6 12.989 2. 7309 1820 23.218 3-6596 26. 644 2.7311 1825 41.518 3.6604 40. 300 2- 73 3 1830 1835 59- 822 16 19 18. 130 3. 66 i 2 3.6619 53.956 17 7 7.614 2.7314 2.7316 1840 36.442 3-6627 21. 273 2.7318 1845 54-757 3-6635 34-932 2.7320 1850 16 20 13.076 3.6642 48.592 2.7321 855 3 -399 3-6650 17 8 2.253 2.7323 1860 49- 726 3-6657 I5-9I5 2.7325 1865 16 21 8.057 3- 666 5 29-578 2.7327 1870 26.391 3-6673 43. 2 42 2.7328 875 44- 730 3.6680 56.906 2.7330 1880 l6 22 3.O72 3.6688 17 9 10.572 2.7332 1885 21.417 3-6695 24.238 2-7333 1890 39.767 3-6703 37-905 2-7335 895 58. 120 3.6710 51-573 2-7337 1900 16 23 16.477 3.6718 17 10 5. 242 2-7339 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean rujld ascensions, <( -c. Continued. Year. a OPHIUCIII. a LYRJE. 11. A. Ann. Var. K. A. Ann. Var. li. m. s. s. li. m. s. ,5. 1750 17 23 20.430 2.7781 18 28 28.585 2. 0298 1755 34-320 2.7783 38.734 2. 0299 i 760 48. 212 2-7784 48.883 2. 0299 1765 17 24 2. 105 2.7786 59-033. 2 . O3OO 1770 15.998 2.7788 18 29 9. 183 2 . O3OO 1775 . 29.893 2.7790 19-334 2.0301 1780 43-788 2. 7791 29.484 2. O3OI 1785 57.684 2-7793 39-635 2.0302 1790 17 25 II.58I 2-7795 49.786 2.0303 1795 25-479 2-7797 59-938 2.0303 1800 39-377 2.7798 18 30 10.089 2 . 0304 1805 53-277 2. 7800 20. 241 2.0304 1810 17 26 7. 177 2.7802 30- 394 2.0305 1815 21.079 2. 7 803 40. 546 2.0305 1820 34.981 2.7805 50.699 2. 0306 1825 48.884 2.7807 18 31 0.852 2. 0306 1830 17 27 2.788 2. 7809 i i . 005 2.0307 1835 16.692 2. 7810 21. 159 2.0307 1840 30-59 8 2. 7812 3I-3I3 2. 0308 1845 44- 504 2.7814 41.467 2. 0308 1850 58.411 2.7815 51 . 621 2.0309 1855 17 28 12.319 2.7817 18 32 1.776 2. O3IO 1860 26. 228 2.7819 11.931 2. O3IO 1865- 40. 138 2. 7820 22. 086 2.03II 1870 54.049 2. 7822 32.241 2.03II 1875 17 29 7.960 2. 7 82 4 42-397 2 . 03 I 2 1880 21.872 2.7825 52.553 2.0312 1885 35-785 2. 7827 i 8 33 2.709 2.0313 1890 49.699 2. 7829 12.865 2.0313 1895 17 30 3.614 2./830 23. 022 2.0314 1900 17-530 2.7832 33- 179 2.03H DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean riyld ascensions, dr. Continued. Yea r. y AQI:IL JE. a AQUIL .. K. A. Ann. Yar. R. A. Ann. Yar. //. HI. S. s. //. 1)1. S. S. 750 19 54 22.396 2-8536 9 38 34-939 2.9502 1755 36.664 2-8536 49-590 2.9501 i 760 50.93 2 2 -8535 19 39 4. 240 2.9500 1/65 19 35 5- 200 2-8535 18.890 2.9299 770 19.467 2-8554 33-539 2. 9298 775 -> t ** -> \ jj>- 734 2-8534 48. i 88 2.9297 1780 48. ooi 2-8534 19 4O 2. 856* 2. 9296 -785 19 36 2. 268 2-8533 17 484 2.9295 1790 16.534 2-8533 32- 132 2.9295 795 30. 800 2-8532 46. 779 2.9294 iSoo 45.066 2-8532 19 41 1.425 2.9295 1805 59-332 2-8551 16. 071 2.9292 i 8 10 9 37 3-597 2-853 30. 7 7 2.9291 1815 27.865 2-8530 45-362 2. 9290 1820 42. 128 2-8530 1942 o. 007 2. 9289 1825 5 6 -392 2-8529 14.652 2.9288 1830 1835 19 38 10.657 24.921 2-8529 2.8528 29. 296 43-939 2.9287 2.9286 1840 39- 185 2.8528 58-582 2.9286 i*45 53-448 2-8527 9 43 13-225 2.9285 1850 19 39 7. 712 2.8527 27.867 2.9284 t855 21-975 2.8526 42.508 2.9285 1860 36. 238 2.8526 57- 150 2. 9282 1865 50. 500 2-8525 i 9 44 i i . 790 2. 9281 1870 19 40 4.763 2.8524 26.450 2. 9280 1^75 19.025 2.8524 41. 070 2.9279 1 880 33-287 2-8523 55-7 Q 2.9278 1885 47-548 2-8523 19 45 10.349 2.9277 1890 1941 i . 810 2.8522 24.987 2. 9276 1895 i 6. 071 2.8522" 39-625 2.9275 1900 ** C\ "> *> T U - JJ 1 2.8521 54.262 2.9274 9 F s 66 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. right ascensions, <fr. Continued. /? AQUILJE. a 2 CAPRICORNI. 1 car. K. A. Ann. Var. H. A. Ann. Var. //. 111. S. s. Ji. m. s. s. 1750 19 43 1-8/4 2. 9490 20 4 9-754 3-3439 1755 16.619 2.9490 26.473 ^> ^ 1 ^ C o J4o 1760 3i-3 6 4 2.9489 43 i 89 ^ ^ 1 "> T o 34.5 l i 765 46. 108 1770 19 44 0.852 2.9488 2.9488 59.902 20 5 16.614 3-3426 3-3422 775 15-595 1780 30.339 2.9487 2.9486 33-324 50.032 3-34I8 3-34I4 1785 45-082 2. 9486 20 6 6. 738 3-3410 1790 59- 82 4 2.9485 23-442 3 34o6 795 T 9 45 U-567 2 . 9484 40 . 144 3-3402 1800 29.308 2.9483 56-844 "* "> ** c\ o- o o9/ 1805 44.050 2.9483 20 7 13.542 "> ^ ^ /i t o oo9o 1810 58. 791 2.9482 30-237 3-3389 1815 19 46 I3-53 2 2 . 948 I 46.931 3-3385 1820 28. 273 2.9481 20 8 3.622 1 1 1<s T o- 33 825 43-oi3 2. 9480 20.311 3-3376 1830 57-75 2 2-9479 36.999 1 -> *7 *> o- o j/ - 835 19 47 12.492 2-9479 53-684 -> 1 1^9 3- oo b 1840 27.231 2.9478 20 910. 367 -^ O () 1 o- oo4 1845 41.970 2-9477 27.048 *> *> ^ /Sr^ 0- JO 00 1850 56. 708 2.9476 43-726 3 3355 1855 19 48 11.446 1860 26 . 184 2.9476 2-9475 20 10 0.403 17.078 *>" ^> I f T O- JO5 * -1 o ^> /I * o- jo4/ 1865 40. 921 2.9474 33-750 ^> -l ^ -1 1 o jo4o 1870 55-658 2-9473 50.421 o ^ o o r\ j ooo9 1875 19 49 10.394 2-9473 20 i i 7. 090 3- 3334 1880 25.131 2.9472 23.756 3-3330 1885 39.866 2.9471 40. 420 3-3326 1890 54.602 2.9471 57.082 i o t o O o o J - - l8 95 19 50 9-337 2.9470 2O 12 13. 742 I o " ^y T *7 D- O J j / 1900 24. 072 2 . 9469 30. 400 *> o I T ^ o- oo j o DEFINITIVE RIGHT ASCENSIONS OF THE FUNDAMENTAL STARS. Mean riyld ((scomions, iCc. Continued. 67 Year. a CYGXI. a AQKAKII. R. A. Ann. Yar. R. A. Ann. Yar. //. III. S. S. //. III. S. s. I750 20 32 54-957 2.0414 21 52 56.049 3.0888 1 755 -o 33 5- H4 2.0415 21 55 11.492 5.0886 i 760 15.352 2.0416 26.935 3-0884 1765 25.561 2.0417 42.376 3.0881 1770 35.770 2.0418 57.816 3.0879 775 i 780 45-979 56. 189 2 . 04 i 9 2. O42O 21 54 3- 255 3-0877 28.693 3-0875 1785 20 34 6.399 2. 042 I 44-130 3-0872 1790 16.610 2. O422 59- 565 3.0870 1 795 26.822 2.0425 21 55 15.000 3 . 0868 1800 37-054 2.0424 30-433 3.0866 1805 47.246 2.0425 45-865 3-0863 18 10 57-459 2. 0426 2 I 56 1 . 296 5. 0861 1815 20 55 7-675 2. 0428 16.726 3-0859 1820 17-887 2.0429 32.155 3-0857 1825 28. 101 2.0450 47- 583 3-0855 1830 38.316 2.0451 21 57 5.010 3-0852 i835 48.532 2.0452 i8.435 5.0850 1840 58.748 2-0433 33-86o 3.0848 1845 20 56 8.965 2-0434 49- 285 3 . 0846 1850 19. l82 2.0435 21 58 4.706 3.0844 1855 29.599 2.0456 20. 127 5.0842 1860 59.618 2-0457 35-548 5 . 0840 1865 49.856 2.0458 50-967 5.0838 1870 20 57 0.056 2.0459 2 59 6.585 3-0835 i8/5 10. 276 2.0440 2 I . 8O2 3-0833 1880 20. 496 2.0441 37-218 3-0831 1885 30-7 7 2-0445 52.633 3.0829 1890 40. 959 2.0444. 22 O 8 . 048 3.0827 1895 51. 161 2.0445 25.461 3-0825 1900 20 58 ,.585 2.0446 38.873 3-0823 68 RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Mean right ascensions, <c. Continued. Year. a PlSCIUM AUSTEALIS. a PEGASI. R A. Ann. Var. R. A. Ann. A T ar. ll. III. S. s. It. 111. S. s. 1/5 22 43 46. 710 3-3551 22 52 19.531 2-9775 J 755 22 44 3.483 3-3540 34-4I9 2-9778 i 760 20. 250 3-3529 49- 309 2.9780 1765 37.012 3- r 35 8 22 53 4- 199 2.9783 7/o 53-768 3-3507 19. 091 2.9785 i/75 22 45 10. 518 3-3496 33-985 2.9788 1780 27. 264 3-3485 48.879 2.9790 1785 44-003 3-3474 22 54 3- 775 2-9793 1790 22 46 o. 737 3-3463 18. 672 2-9795 J 795 17.466 3-3452 33-570 2.9798 1 800 34- 189 3-3441 48.470 2. 9801 1805 50.907 3-3430 22 55 3-37 2.9803 1810 22 47 7.619 3-34 9 18.273 2. 9806 1815 24.326 3-3408 33- 177 2. 9809 1820 41 . 028 3-3397 48.082 2. 981 I 1825 57- 7 2 4 3-3387 22 56 2.988 2.9814 1830 22 48 14-4H 3-3376 17. 896 2.9817 1835 31. ioo "* * i (~\ ^ o- oo5 32.805 2. 9819 1840 47-779 3- 3354 47-715 2. 9822 1845 22 49 4.454 3 3343 22 57 2.627 2.9825 1850 21. 123 i f\ *> -7 *5 O- J JOO 17.540 2.9828 1855 37.786 3-3322 32.455 2.9830 1860 54-445 3-33 1 47-370 2.9833 1865 22 50 II. 097 3-3300 22 58 2.288 2.9836 1870 27-745 3.3290 I 7. 2O6 2.9839 i875 44-387 3-3279 32. 126 2.9841 1880 22 51 I.O24 3-3268 47.048 2.9844 1885 17.656 3-3258 22 59 1.971 2.9847 1890 34.282 3-3247 16.895 2.9850 1895 50- 903 3-3236 31.821 2.9853 1900 22 52 7.518 T, . T.226 \.J ^J 46.748 2.9856 CORRECTIONS FOR ORBITAL MOTION OF SIRIUS AND PROCYOX. Periodic term* to be <ij>]>//crl tu tin riyltt d-yci iisionti of Sir ins and F roc yon. 69 a CAMS MAJOKIS. Year. Year. Year. P s. 1750.6 1800. o 1849.4 4- .026 1751.6 i 80 i . o 20 1850.4 +.006 1752.6 1802.0 1851.4 -.014 1753.6 1803.0 1852.4 --031 ^ i 754. 6 1804. o i853-4 --047 1755.6 1805.0 1854.4 -.062 i 756. 6 1806. o 1855.4 -.076_ 1757.6 1807.0 1856.4 -.088 ,758.6 i 808.0 1857.4 -.099 1/59-6 i 809. o I o 1858. 4 - . 109 , 760. 6 1810. o 1859.4 -.n8_ 1761.6 i 8 1 i . o 1860.4 .126 6 1762.6 1812.0 ,861.4 -.132 1763.6 1813.0 ,862.4 -.138 i 764. 6 1814.0 1863.4 -- 43 1765.6 1815.0 1864. 4 -.147" i 766. 6 1816.0 1865.4 -.149" i 767. 6 1817.0 1866.4 - - 5 1768.6 1818.0 1867.4 ~ 5- i 769. 6 1819.0 1868.4 o 5- , -4- i 1770.6 1820. o 1869. 4 - 5 -4- T 1771. 6 1821 . o 1870.4 - 49 , 1772.6 1822. o [871.4 -.i 4 6 4 "773-6 1823. o ,872.4 .142 4- c 1774.6 1824.0 873-4 > + 6 775-6 1825. o ,874.4 - 3i 1776.6 1826.0 1875-4 + 7 * 1 1777.6 1827.0 1876.4 -" 4 -r- O ,778.6 1828.0 .877.4 -.108^ -t- 10 1779.6 1829. o 1878.4 - .098 ] -r- I 2 1780. 6 1830.0 1879.4 -.086] i 2 i 78 1 . 6 I 83 I . 1880.4 ~~ 74 + I" i 7 o RIGHT ASCENSIONS OF FUNDAMENTAL STARS, Periodic tcnns, d c. Continued. a CANIS MAJOKIS Continued. Year. Year. Year. P s. 1782.6 1832.0 1881.4 .061 + 5 1783.6 1833-0 1882.4 1784.6 1834.0 1883.4 -3 + ]8 1785.6 1835-0 1884.4 .OI 2 + 1 9 .1786.6 1836.0 1885.4 t + .007 + 20 1787.6 1837.0 1886.4 + - 27 +22 1788.6 1838.0 1887.4 + -049 , 1789.6 1839. o 1888.4 _1_ "7 ~ i 790. 6 1840. o 1889.4 + - 96 +24 1791.6 1841 .0 1890. 4 + I2 1 oi 1792.6 1842. o 1891 . 4 + .141 1793.6 1843.0 1892.4 + -i5 2 _ 1794.6 1844. o 1893.4 + -I47 1/95-6 1845.0 1894.4 + -130 i 796. 6 1846.0 1895.4 + - 7 1797.6 1847.0 1896. 4 + .082 1798.6 1848.0 1897.4 + -osS 1 >- 0-1 1799.6 1849.0 1898.4 J_ ~ 1800. 6 1850.0 1899.4 + - OI 4 20 i 80 i . 6 1851.0 1900. 4 .006 18 1802.6 1852.0 1901.4 .024 CORRECTIONS FOR ORBITAL MOTION OF SIRIL S AND PROCYON. Periodic terms, ((<: Continued. a CANIS MINORIS. i Year. Year. Year. Year. p s. 1750.0 i 790. o 1830.0 1870. o -45 8 1751.0 1791.0 1831.0 1871.0 - -53 1752.0 1792.0 1832.0 1872.0 -.060 t; 1 753-o 793-0 1833.0 1873.0 .065 1754-0 1794.0 1834.0 1874.0 .068 ~ 755-0 1 795-o 1835.0 1875.0 .O7O 1756.0 1796.0 1836.0 1876.0 O .070 757-o 1797.0 1837.0 1877.0 - .068 4 _[._ ^> 1758.0 1798.0 1838.0 1878.0 .065 i759-o 1799-0 1839.0 1879.0 - .060 + 1760.0 1 800.0 1840.0 1 880.0 --053^ I i 761 . o 1 80 1. o 1841 . o 1881.0 ~ -045 , 1762.0 1802.0 1842.0 1882.0 .o;6 -4- TO 1763.0 1803. o 1843.0 1883.0 ., TT J U .026 1764.0 i 804 . o i 844 . o 1884.0 /-+ I0 .Ol6 1765.0 1805. o 1845.0 1885.0 .006 1766. o i 806 . o 1846. o 1886.0 + I 1 1767.0 1807.0 1847.0 1887.0 + .oist 10 1768.0 1 808.0 1848.0 1888.0 + .026~f 1769.0 1809. 1849. o 1889.0 . , + 10 + 036 1770.0 1810. o 1850.0 1890. o + -045 , 1771.0 181 1 . o 1851.0 1891 . o + -053 j 1772.0 I 8 I 2 . O 1852.0 1892. o + .o6o 4 773-0 1813.0 1853-0 1893.0 + -065 1774.0 1814.0 1854.0 1894. o _j_ 2 775-o 1815.0 1855-0 1895.0 + -O7 1776.0 1816.0 1856.0 1896. o O + -070 1777.0 1817.0 1857.0 1897.0 + .068 1778.0 1818. o 1858.0 1898.0 + .o6 5 _ * 1779.0 1819.0 1859.0 1899. o + .060 1780.0 1820. o 1860. o 1900. o + -054_ 1781.0 1821.0 i 86 i . o i 90 i . o + .046 9 1783.0 i 784. o 1785.0 1786.0 1787.0 1788.0 i 789. o i 790. o RIGHT ASCENSIONS OF FUNDAMENTAL STARS. Periodic twins, (C c. Continued. Your. Yea 1782. o 1822 a CANIS MIXORIS Continued. 1823.0 1824. o 1825.0 1826.0 1827. o 1828.0 . i 1829. o 1830.0 Year. 1862.0 1863.0 1864. o 1865.0 1866. o 1867.0 1868.0 1869. o 1870. o 1902. o 1903.0 1904. o 1905.0 1906. o 1907. o 1908. o 1909. o 1910. o s. + 037 + .027 + .017 -f- .006 -005 .015 .027 -037 .046 IO 10 - 1 1 1 1 IO 9 HILL S FORMULA FOR SECULAR VARIATION OF ANNUAL MOTION OF STARS. 73 HILL S FORMULA FOR THE SECULAR VARIATION OF THE ANNUAL MOTION OF THE STARS. [From Star-Tables of the American Ephcmeris, p. xix.] These formulae, being more complete than those usually employed, are given here, for convenience of reference, in a slightly modified form. Put- ju, the proper motion in right ascension, expressed in seconds of time; //, the same in declination, expressed in seconds of arc ; p, p , the annual precessions, expressed in the same Avay with /* and // , respectively. We then have p rr m -\- n sin a tan 8 p n cos a da ^ = l + " (16 ~di = P+f * 10 f TT = [7-9878] n cos a. tan 8 -f- [6.81 17] n sin a sec 2 S d u loo- zz [9. 1640] JLI sin a ioo = + . 00322 - [6.6338] j> + [7-9878] (p + 2 /() cos a tan 5 + [6.8117] (y + 2 /O s in -f [6.9866] n // tan 5 fl*fi on / too - /r , - = -[6.6338] p [9.1640] (^ + 2 /<) sin a -[8.7367] >" a *in 2 r5 Struve s values of m rt<7 n. Year. m n log log s. s. 1750 3.06987 I -33767 0.12635 20.0650 1.30244 1800 3.07082 1-33738 o. i 2626 20.0607 I -30235 1850 3.07177 I -33710 o. 12616 20.0564 1.30225 1900 3.07272 1.33681 o . i 2607 20.0521 I .30216 IO S 14 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRONOMY, MATHEMATICS- This book isot d below, or on the date to which renewed. Renewed, books are subject to immediate LD 21-40m-10, 65 (F7763slO)476 , . General library University of California tserkeiey \ o U.C. BERKELEY LIBRARIES C03SS71bDfl AITROJOMY