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The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004632240 
 
BRUNNOW'S 
 
 SPHERICAL ASTRONOMY. 
 
ffiambTtBge: 
 
 PKINTED BY 0. J. CLAY, M.A. 
 AT THE UNIVBESITY FKBSS. 
 
BRUNNOWS 
 
 SPHEEICAL ASTRONOMY, 
 
 TEAI^SLATED BY 
 
 The Eev. EGBERT MAIN, M.A. F.E.S. 
 
 FBESIDEKT OF THE EOTAL ASTBOKOMICAL SOCIETY; 
 AUD KABCLirFE OBBEBTEB AT OiFOED. 
 
 PABT I. 
 INCLUDING THE CHAPTEKS ON 
 
 PARALLAX, REFRACTION, ABERRATION, PRECESSION, 
 AND NUTATION. 
 
 CAMBEIDGE: 
 DEIGHTON, BELL, AND CO. 
 
 LONDON: BELL AND DALDY. 
 1860. 
 

 CORNELC 
 
 UNIVERSITYi 
 UBRARY 
 
TRANSLATOR'S PREFACE. 
 
 The want of a Text-book for Spherical Astronomy has 
 been long felt in the University of Cambridge, and indeed 
 in all the Universities of the British Isles in which the 
 Mathematical Sciences are cultivated. The treatise of Maddy, 
 excellent as it was for the time when it was written, has long 
 since ceased to be of much service for the instruction of 
 students who are desirous of knowing the methods, both 
 theoretical and practical, used by the astronomers of the 
 present time; while that of Woodhouse, though it deals 
 more practically with the operations pursued in fixed Ob- 
 servatories, is also comparatively useless, because it describes 
 both instruments and methods which have long been ob- 
 solete. It has been, at least in part, a consequence of the 
 want of a good treatise on practical astronomy, that this 
 science has been far less cultivated in the University of 
 Cambridge than it ought to have been, and that very few 
 indeed of the students acquire any adequate acquaintance 
 with the methods of the present time, as taught in the 
 works of the great German and English astronomers. 
 
 Before the publication of Briinnow's Sphdrischen Astro- 
 nomic these methods were to be sought for either in the 
 PhUosophical Trcmsactions, the Memoirs of the Royal 
 Astronomical Society, or the German Astronomical periodicals, 
 of which the chief and most celebrated is the Astronomische 
 
 B.A. h 
 
VI TRANSLATOR S PEEFACE. 
 
 Nachrichtm. But, even of the comparatively few who could 
 consult with advantage the latter work through their fami- 
 liarity with the German language, only a small number would 
 find the subjects of their research treated in a way at all 
 adapted for academical use, where clear enunciation of 
 methods and concise demonstrations of leading propositions 
 are required rather than the minute refinements introduced by 
 the practical astronomer when treating of special processes. 
 Briinnow's work was, therefore, received in Germany with 
 universal satisfaction, and, even in England, it has been read 
 much more extensively than might have been expected, 
 when we consider that comparatively few of the students 
 of any of our Universities are acquainted with the German 
 language, and still fewer of those who obtain distinction in 
 mathematical acquirements. 
 
 Indeed the excellence of the work was so well recognized, 
 that I was, very soon after its publication, requested by some 
 of the greatest of English astronomers to undertake a trans- 
 lation, and I so far responded to the invitation as to com- 
 plete a considerable portion of it. Want of time, however, 
 and the absorption of a considerable portion of my leisure 
 in works of a more original character, prevented me from 
 completing it, and the Manuscript would stUl have remained 
 neglected, if the increasing desire for some work on astro- 
 nomy which should supply pressing needs had not induced 
 the publishers to undertake the printing of it imperfect as 
 it is. 
 
 It is hoped that it may stiU, although incomplete, be of 
 considerable service, since, with the exception of the detailed 
 accounts, given in the latter portion of the work, of the 
 construction and use of instruments, and of the methods 
 employed by modem astronomers for deducing from obser- 
 vations the most accurate values of the constants employed 
 
TEANSLATOE S PEEPACE. VU 
 
 in astronomy, the portion printed contains nearly the whole 
 of the work which would be valuable to English students. 
 
 In particular I would remark, that the theories of Pa- 
 rallax, Kefraction, Aberration, Precession, and Nutation are 
 given in a more complete form than is to be found in any 
 English work, and that Mr Carrington has, in his admirable 
 Introduction to the Red Hill Catcdogiie of Circumpolar 
 Stars, expressed his obligations to the author by referring 
 to the theory of Precession and Nutation. 
 
 From these considerations, I trust that the imperfect 
 work now offered may meet with a sufficiently favourable 
 reception to repay me by its utility for the generally thank- 
 less labour of translation, and that the lecturers and ex- 
 aminers in our Universities may find it sufficiently well 
 adapted to their purpose to form the basis of an improved 
 system for the teaching of Practical Astronomy. I might 
 then be induced to undertake the labour of translating the 
 remainder of the work, or, at least, of such a portion of it 
 as would be useful to the English student. 
 
 ROBERT MAIN. 
 
 Kadcliffb Obsekvatoky, OxroKD, 
 October 4, i860. 
 
TABLE OF CONTENTS. 
 
 INTRODUCTION. 
 
 A. Transfomudion of Co-ordinates. Formulm of Spherical 
 Trigonometry. 
 
 No. PAGE 
 
 1. Pormnlse for the Transformation of Co-ordinates ... 1 
 
 2. Example 3 
 
 3. Fundamental Formulae of Spherical Trigonometry ... 4 
 
 4. Other Formulae of Spherical Trigonometry . , , , 6 
 
 5. The Gaussian Equations. Napier's Analogies .... 7 
 
 6. Introduction of Auxiliary Angles into the Formulae of Spherical 
 
 Trigonometry . 13 
 
 7. On the advantage due to the finding of Angles by means of the 
 
 Tangents 15 
 
 8. Formulae for right-angled Spherical Triangles ... 17 
 
 9. Differential Formulae of Si^erical Trigonometry . . .18 
 
 10. Approximate Formulae for small Angles .... 19 
 
 11. Some Expansions of frequent occurrence 20 
 
 B. On Interpolation. 
 
 12. Object of Interpolations. Computation of Differences . 26 
 
 13. The Interpolation-Formulae of Newton 27 
 
 14. Other Interpolation-Formulse. 30 
 
 15. Computation of numerical Differential-CoeflScients ... 38 
 
 SPHERICAL ASTRONOMY. 
 
 FIRST SECTION. 
 The Visible Spheeb of the Heavens, and its Daily Motion . 47 
 
 I. On the different Systems ofPUmes and Girdles on the Visile 
 Sphere of the Heavens. 
 
 1. Co-ordinate System of Azimuths and Altitudes ... 48 
 
 I 2. Co-ordinate System of Hour Angles and Decimations . . 50 
 
 ' 3. Co-ordinate System of Right Ascensions and Declinations . 52 
 
 4. Co-ordinate System of Longitudes and Latitudes ... 54 
 
X TABLE OP CONTENTS. 
 
 •■ 
 II. On the Transformation from one to another of the several 
 Systems qf Co-ordinates. 
 
 No. PAGE 
 
 5. Change of Azimuths and Altitudes into Hour Angles and Decli- 
 
 nations 55 
 
 6. Change of Hour Angles and Declinations into Azimuths and 
 
 Altitudes 57 
 
 7. Parallactic Angle. Differential Formulae for the two preceding 
 
 cases 63 
 
 8. Change of Right Ascensions and Declinations into Lon^tudes 
 
 and Latitudes 64 
 
 9. Change of Longitudes and Latitudes into Bight Ascensions and 
 
 Declinations 67 
 
 10. Angle between the Circles of Declination and Latitude. Differen- 
 
 tial Pormulse for the two preceding cases .... 69 
 
 1 1 . Change of Azimuths and Altitudes into Longitudes and Latitudes 70 
 
 III. Particular Phenomena qfthe Daily Motion, 
 
 12. On the Rising and Setting of the Heavenly Bodies ... 71 
 
 13. Amplitudes at Rising and Setting of the Heavenly Bodies . 73 
 
 14. Zenith Distances of the Stars at their Culminations ... 74 
 
 15. Time of the greatest Altitude when the Declination is variable 75 
 
 16. Differential Pormulse of Altitude with respect to the Hour Angle 76 
 
 17. Transits of Stars across the Prime Vertical .... 77 
 
 IV. On the Daily Motion as a Measure of Time. Sidereal Time, 
 Solar Time, Mean Time. 
 
 18. Sidereal Time. Sidereal Day 78 
 
 19. True Solar Time 79 
 
 20. Mean Solar Time 80 
 
 21. Change of Mean Time into Sidereal Time and mce versa . 82 
 
 22. Change of True Time into Mean Time and vice versd . , 83 
 
 23. Change of True Time into Sidereal Time and vice versd . 84 
 
 SECOND SECTION. 
 
 COKRECTIONS OF ObSEEVATIONS, WHICH ARE DEPENDENT ON THE 
 
 Position op the Observer on the Surface op the Earth 
 
 AND oh the PeOPBBTIES OP LiGHT . ... 86 
 
 I. Parallax. 
 
 1. Dimensions of the Earth. BquatoreaJ Parallax of the Sun , 87 
 
 2. Corrected Latitude and Distance from the Centre for different 
 
 Places on the Earth 89 
 
 3. Parallax in Altitude of the Heavenly Bodies ... 93 
 
TABLE OP CONTENTS. xi 
 
 No. _ p^ijj, 
 
 4. Parallax in Right Ascension and Declination, and in Longitude 
 
 and Latitude \ .98 
 
 5. Example for the Moon. Eigorous Pormulse for the Moon . 103 
 
 II. Refraction. 
 
 6. Law of Refraction of Light. Differential Equation of Refraction 106 
 
 7. Integration of this Equation 113 
 
 8. Computation of the Transcendent e'^/'"e-«=^i ... 119 
 
 9. Constant of Refraction. Example of the Computation of Refrac- 
 
 tion from the Pormulse previously found .... 124 
 10. Differential Coefficients of the expression for Refraction in rela- 
 tion to the Thermometer and Barometer. The Tables of 
 
 Bessel 127 
 
 "11. Simpler Expression for Refraction. Pormulse of Simpson and 
 
 Bradley 135 
 
 12. Effect of Refraction on the Phenomena of the Daily Motion . 137 
 
 III. Aberration. 
 
 13. Expression for the Annual Aberration in Right Ascension and 
 
 Declination, and in Longitude and Latitude . . .139 
 
 14. Tables for Aberration in Right Ascension and Declination . 145 
 
 15. Pormulse for the Annual Parallax of the Stars .... 146 
 
 16. Diurnal Aberration 148 
 
 17. Apparent Orbits of the Stars round their Mean Places . . 150 
 
 18. Aberration for the Heavenly Bodies which have a Proper Motion 151 
 
 THIRD SECTION. 
 
 Dbteemination op the Co-ordinates and Ansle of the appa- 
 KENT Sphere op the Heavens independent op the Position 
 OF the Obsebvee on the Surface of the Baeth. Periodical 
 AND Secular Chanoes op these Quantities , . .153 
 
 I. Determination of the Right Ascensions and Declinations of the 
 Stars, and of the OUiquity of the Ecliptic. 
 
 1. Determination of Differences of Right Ascension and Declination 
 
 of the Stars 154 
 
 2. Determination of the Declinations of the Stars. Determination 
 
 of the Absolute Right Ascension of a Star and of the Obli- 
 quity of the Ecliptic by two observations of Difference of 
 Right Ascension of the Sun and the Star in connexion with 
 the Declination of the Sun 158 
 
xii TABLE OF CONTENTS. 
 
 No. PAGE 
 
 3. Determination of the Obliqviiiy of the Ecliptic by observations of 
 
 the Declination of the Sun in the neighbourhood of the 
 Solstices ^61 
 
 4. Determination of the absolute Right Ascension of a Star inde- 
 
 pendently of the constant errors in the Obliquity of the Eclip- 
 tic and the Sun's Declination, by observations of the Differ- 
 ence of Right Ascension of the Star and the Sun, and of the 
 Declination of the latter in the neighbourhood of the two 
 Equinoxes 162 
 
 II. Variationi of the Planes to which the places of the Stars are 
 referred (Precession and Nutation). 
 
 5. Annual motion of the Equator on the BcUptic and of the Ecliptic 
 
 on the Equator, or AnTi nal Lunisolar Precession and Preces- 
 sion produced by the Planets. Secular change of the Obli- 
 quity of the Ecliptic 167 
 
 6. Annual changes of the Stars in Lqpgitude and Latitude, and in 
 
 Right Ascension and Declination. Integration of these Dif- 
 ferential Expressions 172 
 
 7. Rigorous Formulae for the computation of the Precession in Lon- 
 
 gitude and Latitude and in Right Ascension and Declination 177 
 
 8. Effect of Precession on the appearance of the Sphere of the 
 
 Heavens at a place on the Earth at different times. Sidereal 
 and Tropical Revolution of the Sun 183 
 
 9. Nutation 185 
 
 10. Tables for Nutation 190 
 
 11. Determination of the absolute Right Ascension of a Star with 
 
 regard to Precession and Nutation. Star Catalogues. Proper 
 Motions of the Stars 191 
 
 12. Changes of the Proper Motions of the Stars in Right Ascension 
 
 and Declination 195 
 
 III. Mean and Apparent Places of the Fixed Stars. 
 
 13. Expressions for the Apparent Place of a Star .... 198 
 
 14. Tables of Bessel 199 
 
 15. Other methods of computing the Apparent Place of a Star . 202 
 
EREATA. 
 
 Page 11, line 6 from bottom, 
 
 far cos - o . COB n (3 + C) reaA [cos ^ a . cos - (J5 + C)"] 
 
 — 11, line 4 from bottom, 
 
 far cos - a . sin ^ (JS + O) read fcos - o . sin ^ (5 + C)"l 
 
 — 12, middle of page, 
 
 value of - (5 — c), dele the sign mmua before 185<'.45'.24",13 
 
 — 14, line 12 from bottom, /or equations reac2 equation 
 
 — 16, line 2 from bottom of note at foot of page, for complement read 
 
 reciprocal 
 
 — 17, line 24 from top, fm (4) read (6) 
 
 — 17, line 28 from top, for this read the same 
 
 — 19, line 16, 
 
 . sin C „ cos a , , sin C „ cos a .%va.B , 
 
 for :— ^ m + -: — ; . dc read r— , . do H :— 7 — . de 
 
 avrb smo smo smo 
 
 — 22, formula (16), after sin 3a! i/iaert — &e. 
 
 — 24, fifth line from bottom, after sin 2c, for — read + And, in the 
 
 following formula, after sin 2 (a + I), for + read — 
 
 — 25, line 11, In the expansion of y, in the last term of the equation, for - 
 
 read - , and m the expression for — j%^ dele ^ 
 
 — 28, lines 17 and 19, for f.a readf{fl), and, in the line following, "we 
 
 obtain,"/"' /»=» read f{a) = a 
 34, end of line 12, inaert fall 
 
 — 34, line 17,/o»-/"'(a+l) reod/"(a+l) 
 
 3 1 
 
 — 38, line 6 from bottom, for ^f" read ^f" 
 
 — 41, after " The two terms," for - af" (a) read ^ nf" (a) 
 
 — 45, line 10, opposite 12S /or 1,64 read "ifii. 
 
 — 45,laBtUne, /or 25'.56',77 read 26°'. 56',77 
 
 far 2", 51 read 2', 61 
 
 — 46, line 2, far Z. 9-, 77 read 2'°.9',77 
 
 — 65, line 25, for (la) read (2) 
 
 — 61, line 7 from bottom, for A read A„ 
 
 — 70, line 14 from bottom, far y" read —y 
 
70, line 2 from bottom, fm x read id 
 72, line 2 of note, for (d) read (a) 
 78, line 18 far 9". 15^. 5'. read 9". W,5 
 
 for ig''. 3"°. 9'. read ig*. 3"»,9 
 98, line 3 from bottom, after "consequently we have," in the numerator 
 
 of the fraction on the right-hand side of equation, for ft read p 
 100, last paragraph, line 4 from bottom, 
 
 for cos ^ (o-+ o) read ^ (a- + a) 
 
 102, in the Example, log Bin('y — 8), dele the mirms sign 
 
 110, to the second and third expressions for dS', pre/h the sign — 
 
 116, equation (»), for _ — - — t-j- read , „ . . 
 ^ ^ l.i...Bm*e 1.2 8m*z 
 
 116, lapt equation, for (e"^' — 1) read (e^i"— 1) 
 
 117, equation (o), third term in bracket, for 2 . 2'e-^ rea4 2 . 2»,e-''^'' 
 
 118, equation (A), left-hand side, dt should range ynth, e 
 
 120, expression for f e~^dt, insert reference-letter (a) 
 
 127, line i from bottom, for main read ^lean 
 
 131, Une 1, /)r (a) read (o) 
 
 131, line 7 from bottom, for B read b 
 
 136, lines 8 and 10, formulae, prefix the sign -^ 
 
 138, line i from bottom, for 1\ SS"?,? read 1^. SS"?. 7' 
 
 139, last line, for (z - s) read (ir - 1) 
 
 140, line 22, for f rcod f 
 
 144, last line of second paragraph, for 87'.9 read 87°.9' 
 
INTRODUCTION. 
 
 A. TRANSFORMATION OF CO-ORDINATES. FORMULA OF SPHE- 
 RICAL TRIGONOMETRY. 
 
 1. In Spherical Astronomy we treat of the places of the 
 heayenly bodies on the visible sphere of the heavens, by refer- 
 ring them by means of spherical co-ordinates to certain great 
 circles of the sphere, and by investigating the relations between 
 the co-ordinates referred to different great circles. The place of 
 a body can be also given by means of polar co-ordinates as well 
 as by spherical co-ordinates, for instance, by means of the angle 
 which the straight lines drawn from it to the center of the visi- 
 ble sphere make with certain planes and of the distance from that 
 center, which, being the radius of the sphere, is in this case the 
 nnit of distance. These polar co-ordinates can lastly be' with 
 ease expressed in terms of rectangular co-ordinates. The whole 
 of Spherical Astronomy is for these reasons reduced to the 
 transformation of rectangular co-ordinates, for which, first of all, 
 it will be necessary to investigate the ordinary expressions. 
 
 Imagine in a plane two lines crossing each other ai right 
 angles as axes of co-ordinates, and let x and y denote the ab- 
 scissa and ordinate of any point referred to them, and two other 
 similar rectangular axes with reference to which u and 3s are 
 the abscissa and ordinate of the same point, the two pairs of axes 
 having a common point of intersection and making with each 
 other respectively an angle w, then will x and y be frmctions of 
 u and V and of the angle w, so that 
 
 x= ip{u, V, »•), 
 
 B. A. 1 
 
2 TRANSFORMATION OP CO-ORDINATES. 
 
 Denote the co-ordinates of another point referred to the same 
 pairs of axes by x±x', y ± y, and u±u, v ±v' respectively, 
 then will 
 
 a; + 03' = ^ (m + m', V ± v, w), 
 
 y ±y' = i^{u±u, V ± v', w). 
 
 It is however easily seen, that if axes of co-ordinates be 
 drawn through the first point parallel to the others, then 
 
 x±x' =<^{u,v,w) ±(f> {u, v, w), 
 y ±y' — ■^{u,v,w) ±'yjr [u, v',w). 
 
 From these equations it follows that x and y are linear 
 functions of u and v, and since for m = and v = 0, also x = 
 and y = 0, X and y must be of the form 
 
 x=au + ^v\ 
 y ='yu+ Bv) 
 
 («), 
 
 where a, /3, 7, 8 are functions of w only. 
 
 To determine the values of these functions, the following 
 equation must be employed, 
 
 from which, by the help of equations (a), the following equations 
 of condition are obtained, 
 
 (a' + Y'-l) u'+ (/3^ + S=- 1) v'+2 (a)8 + 7S) uv = 0, 
 
 an equation which can in general only be satisfied by making 
 
 ^' + 7^=1 (h), 
 
 i8' + S'=l (c), 
 
 a^ + yS = {d). 
 
 From the equation {b) we obtain 
 
 — + ^=0 
 dio dw ' 
 
 an equation which again is satisfied in general, if, at the same 
 
 time, we make 
 
 dy J dot. 
 
 a = c -J- , and 7 = — c -r- , 
 aw dw 
 
TBANSFOKMATION OP CO-ORDINATES. 
 
 <^7 1 da. 
 
 dw ' dw 
 
 From the first equation then follows 
 
 dw c 
 
 or 
 
 d^ V(i-y)' 
 whence 
 
 7 = sin f— - C\ , and a = cos (^- Cj . 
 
 From the other two equations we should have obtained 
 a = sin( Cj , and 7= cos ( Cj. 
 
 In the same manner it follows from equation (c), that /S and 
 B are the sine and cosine of the angle C. 
 
 Since now for w = 0, x = u and y = v, C must equal 0, and 
 
 thus a = cos — , and 7 = sin — . Again, since for w = 90", y = u, 
 
 and a; = — «, if we reckon the angle w from the positive side of 
 the axis of x towards the positive side of the axis of y, it fol- 
 lows that c = 1, and hence /3 = — sin w, and B = cos w. 
 
 Thus, for the transformation of rectangular co-ordinates, we 
 have the formulae, 
 
 x = u cos w — V sin w 
 y = u sin w ■\- v cos w ) 
 
 (1), 
 
 or " " -- - ■ ^ — - I (2). 
 
 u = x cos w-\- y^va.w 
 V = — xs,m.w + y cosw) 
 
 These formulae serve generally for all positive or negative 
 values of x and y, and for all values of w from 0° to 360°. 
 
 2. Let X, y, z, be the co-ordinates of a point referred to any 
 system of rectangular axes, and let a be the angle which the 
 radius vector makes with its projection on the plane of x, y] E the 
 
 1—2 
 
4 TRANSFORMATION OF CO-ORDINATES, 
 
 angle which this projection makes with the axis of x (that is, the 
 angle which the plane passing through the point. and the posi- 
 tive axis of X makes with the plane passing through the positive 
 axes of X and e, reckoned from the positive side of the axis of 
 X towards the positive side of the axis of y from 0° to 360") ; 
 then, if the distance of the point from the origin of co-ordinates 
 be taken for the unit of distance, 
 
 X = cos B' cos a, y = sin B' cos a, z = sin a. 
 
 Again, if. we denote by a the angle which the radius vector 
 makes with the positive axis of z, reckoned from the positive 
 side of the axis of a towards the positive sides of the axes of 
 X and of y from 0° to 360°, we have 
 
 X = sin a cos B, y = sin a sin B, z = cos a. 
 
 Imagine now a second system of co-ordinates such that the 
 axis of V coincides with the axis of y, and the axes of u and w 
 make an angle c with the axes of x and z, and call b the angle 
 which the radius vector makes with the positive axis of w, A' 
 the angle which the plane passing through O and the positive 
 axis of w makes with the plane which passes through the posi- 
 tive axes of x and z, both angles being reckoned in the same 
 directions as a and B', we have then 
 
 ?«=sin Jcos-d, v = sin 5 sin ^4, ?y = cos5. 
 
 And since, by the formula for the transfonuation of co-ordi- 
 nates, 
 
 s = M sin c + w cos c, 
 
 y=v, 
 
 x = u cos c — w sin c, 
 we obtain 
 
 cos a = sin b sin c cos -4' + cosh cos c 
 
 sin a sin ^' = sin J sin ^' (3). 
 
 sin a cos B' = sin b cos c cos A' — cos b sin c 
 
 3. Imagine now a sphere to be described about the origin 
 of co-ordinates with any radius whatsoevtn- (here taken as the 
 
FORMULA OP SPHERICAL TRIGONOMETRY. 5 
 
 unit of linear measure) and the points of intersection of the axes 
 of z and w to be connected with each other and with the point 
 above mentioned by arcs of great circles, then will these arcs form 
 a spherical triangle, taking the same in its most general signifi- 
 cation, when the angles as well as the sides may be greater than 
 180°. The three sides OZ, OTF,and TFZofthis spherical triangle 
 are respectively equal to a, h, and c. The spherical angle A at 
 the point W, being the angle between the plane passing through 
 and W and that through W and Z and the center of the 
 sphere, will be equal to A'; and in like manner the angle B 
 at the point Z will be equal to 180°- 5'. Put now A and B 
 instead of A! and B in the equations found in (3) and we ob- 
 tain the following formulae applicable to any spherical triangle, 
 
 cos a = cos J cos c + sin J sin c cos .4, 
 sin a sin 5 = sin & sin A, 
 sin a cos B = cos 5 sin c — sin 5 cos c cos A. 
 
 These are the three fundamental formulae of spherical trigo- 
 nometry, which thus expi-ess nothing more than a simple trans- 
 formation of co-ordinates. 
 
 Since now we may look upon any angle of a spherical tri- 
 angle as the projection of the point on the surface of a sphere 
 and the two others as the points of intersection of the axes of 
 s and w with the same, it follows that the foregoing formulae 
 must serve for any other sides and the adjacent angle by simply 
 interchanging the remaining sides and angles amongst each 
 other. Thus we obtain, by collecting all the cases, 
 
 cos a = cos h cos c + sin 5 sin c cos A i 
 
 cosZ> = cosa cos c + sin a sine cos5> (4), 
 
 cos c = cos a cos J -I- sin a sinJ cos Oi 
 
 sin a sin 5 = sin 6 sin ^ i 
 
 sina sinC= sin c sin-4 > , (5), 
 
 sin 5 sin C = sin c sin 5 J 
 
FORMULA OP SPHEEICAL TRIGONOMETEY. 
 
 (6). 
 
 sin a cos B = cos J sin c — sin h cos c cos A 
 sin a cos C = cos c sin S — sin c cos J cos A 
 sin J cos ^ = cos a sin c — sin a cos c cos B 
 sin J cos C= cos c sin a — sin c cos a cos B 
 sta c cos A = cos a sin h — sin a cos I cos C 
 sin c cos B = cos & sin a — sin J cos a cos (7 
 
 4, From these formulse the remaining formulse of spherical 
 trigonometry are easily deduced. By dividing the formula (6) 
 hy the corresponding formidae (5), we obtain, 
 
 sin A cotan B = cotan J sin c — cos c cos A 
 sin A cotan G = cotan c sin J — cos h cos A 
 sin B cotan ^ = cotan a sin c — cos c cos B 
 sin 5 cotan G = cotan c sia a — cos a cos B 
 sin (7 cotan ^ = cotan « sin J — cos J cos G 
 sin C cotan B = cotan J sin a — cos a cos G 
 
 If we write the last of these equations thus, 
 
 cos J sin a sin B 
 
 (7). 
 
 sin C cos B = 
 we obtain 
 
 sin 6 
 
 — cos a sin 5 cos G, 
 
 or 
 
 sin (7 cos B = cos & sin ^ — cos a sin jB cos G, 
 sin ^ cos S = cos 5 sin (7 + sin 5 cos C cos a, 
 
 an equation which corresponds to the first of equations (6), and 
 contains the angles instead of the sides, and vice versd. By chang- 
 ing the letters the following six equations are obtained: 
 
 sin A cos h = cos 5 sin (7 + sin B cos G cos a 
 sin A cos c = cos C sin ^ + sin G cos B cos a 
 sin-Bcosa= cos -4 sin G-\-smA cos CcosJ 
 sin B cos c = cos C sin ^ + sin G cos -4 cos J 
 sin C cos a = cos A sin'jB + sin -4 cos B cos c 
 sin (7 cos & = cos jB sin ^ + sin B cos A cos c 
 
 (8), 
 
 and, dividing these equations by the corresponding ones of 
 equations (5), we obtain 
 
(9). 
 
 FOKMUL^ OP SPHEEICAL TEIGONOMETRY. 
 
 sin a cotan h = cotan jB sin C-V cos G Cos a 
 sin a cotan c = cotan (7 sin 5 + cos B cos a 
 sin & cotan a = cotan ^ sin + cos C cos 5 
 sin h cotan c = cotan (7 sin -4 + cos ^ cos S 
 sin c cotan a = cotan ^ sin 5 + cos B cos c 
 sin c cotan & = cotan B sin ^ + cos ^ cos c 
 
 The equations (8) give in addition 
 
 cos Asm.G= sin 5 cos a — sin A cos G cos 5, 
 cos 5 sin = sin ^ cos 5 — sin B cos G cos a. 
 
 Multiplying both equations by sin G and substituting the 
 value of sin A sin G cos J from the second of the equations in the 
 first, we obtain 
 
 cos A = sin B sin G cos a — cos B cos (7, 
 
 and, by changing the letters, the three equations corresponding 
 to the formula (4), in which angles take the place of sides and 
 vice versd, become 
 
 cos A = s,va.B sin (7 cos a — cos 5 cos C i 
 
 co%B = SVQ.A sin CcosS — cos^ cos G\ (10). 
 
 cos G = smA sin 5 cos c — cos A cos B) 
 
 5, By adding the first two of formulae (5), we obtain 
 
 sin a (sin B + sin G) = sin A (sin b + sin c), 
 or 
 
 . 1 B-G 1 . B+G . 1 . h-c 1 . . b+o 
 
 sin - a cos — — cos - a sm —- =sm-A cos — cos--4 sm — , 
 
 and, by subtracting the same equations, 
 
 . 1 . B-G 1 B+G . 1 , h+c 1 . . b-c 
 
 sm-asin^^— cos-acos-y— =sin-^cos— cos-^sm-g-- 
 
 In the same manner we obtain, by adding and subtracting 
 the first two of formulse (6), 
 
 . 1 B-G 1 B+G . 1 , . b+c . I , b+o 
 
 sm -a cos —-—cos -a cos — — - = sm -u4 sm -^-sm-^cos -^ , 
 
 2 22 2 2 i i i 
 
8 FORMULA OP SPHERICAL TRIGONOMETRY. 
 
 . 1 . B-G 1 . B+G \ . . l-c 1 . b-c 
 
 sm - a sin — - — cos- a sin — - — = cos - A sin -—cos -A cos --- . 
 
 ^ ji A 2. A 22 2 
 
 These four formulae contain the equations of Gauss, multiplied 
 into each other two by two ; we are not able to obtain separately 
 the single equations by the combination of these four formulae, 
 but for that purpose another formula must be contrived in which 
 another combination of these equations is introduced. To this 
 end the following equation is necessary. 
 
 1 B-^-G 1 . 5+C . 1 . i+c 1 . h-c 
 
 1 -a cos — - — cos — asm — - — = sin -^cos-— -cos--4cos-— 
 J Z 2 A 2 222 
 
 which is obtained by adding the first two of the equations (8) 
 together. 
 
 Put now 
 
 and 
 
 . 1 , . h+c 
 sm - A sin -^ = a, 
 
 sin-u4cos-2-=/3, 
 
 1 . . b-c 
 cos - A sm -^ = 7, 
 
 1 . b—c » 
 cos - A cos — — = 6 ; 
 
 2 2 
 
 .1 B-C , 
 
 sin - a cos — - — = a , 
 
 COS 
 
 1 
 2 
 
 a cos 
 
 B+G 
 2 
 
 ■0, 
 
 sin 
 
 1 
 2 
 
 a sin 
 
 B-G 
 2 
 
 y. 
 
 cos 
 
 1 
 2 
 
 a sin 
 
 B+G 
 2 
 
 S', 
 
 and we shall then have the five equations, 
 
 a'S' = aS, y'0' = y^, a'/3' = ay8, y'B' = y&, fi'B' = /2S, 
 from which the following are derived, 
 
 «' = «, /S' = /9, 7' = 7, h' = S, 
 
POEMULiE OP SPHERICAL TEIGONOMETEY. 
 
 Or 
 
 We consequently obtain between the angles and sides of a 
 spherical triangle the following relations, 
 
 . 1 . . h + c . 1 B-C. 
 
 sin - A sin — -- = sm - a cos — - — 
 
 . 1 . h + G 1 B+0 
 
 sm - A cos -— - = cos - a cos — - — 
 
 A ^ 2 2 
 
 . 1 . . b-c . 1 . £-C 
 
 cos - A sm -— — = sm -a sm 
 
 2 2 2 2 
 
 1 . &-e 1 . ^+C 
 cos - A cos — -— = cos - a sm 
 
 2 2 2 2 • 
 
 or also 
 
 (11), 
 
 . 1 , . 5 + c . 1- 5-(7 
 
 sin - A sm -— = - sin - a cos — ^— , 
 
 ■ 1 . & + C 1 B+G 
 
 sm - ^ cos -— — = — cos -f a cos , 
 
 2 2 2 2 
 
 1 . . 5-c . 1. . ^-C 
 
 cos - A sin — — = — sm — a sm — - — , 
 
 Ji A A A 
 
 1 . h-c 1 .B+0 
 cos - A cos — :: — = — cos — a sin — - — . 
 
 2 2 2 2 
 
 From the two systems of equations we obtain for the quantities 
 sought, whether these be two sides and the included angle, or two 
 angles and the adjacent side, either the same values or values 
 differing by 360". For example, if A, b, and c be required, we 
 
 shall find either for ^-r — and — r— the same values from the 
 
 second set of equations as from the first, for - vl a value 
 differing by 180°, or, for ^^ and -^ values differing by 180°, 
 
 and on the contrary for - A the same value. Thus alsd A and b 
 
 will always differ only by 360° from the values formed from the 
 first system of equations. The four formulas (11) are there- 
 fore quite general, and it is indifferent, whether, in the com- 
 
10 FOEMUL^ OF SPHEEICAL TEIGONOMETET. 
 
 putation of A, h, and c the values a, B, C, be employed, or the 
 values of these quantities + 360°.* 
 
 The four equations (11) are known by the name of the 
 Gaussian Equations, and are employed when one side and two 
 adjacent angles of a spherical triangle, or two sides and the in- 
 cluded angle, are given to find the three remaining parts. They 
 are most easily used in the following manner. If a, B, and C are 
 given, we first find the values of 
 
 [1] 
 
 B-G 
 
 cos ^ • 
 
 [4] cos 2 • 
 
 [2] 
 
 . 1 
 
 sm - a. 
 
 [5] cos -a. 
 
 [3] 
 
 . B-G 
 sm 2 . 
 
 m ™^j«. 
 
 and thence 
 
 
 
 [7] sin-, 
 
 B-G 
 
 % cos . 
 
 „ .1 .B-G 
 [9] sm 2 a sm ^ 
 
 [8] cosi. 
 
 B+G 
 a cos — - — . 
 
 r,„T 1 .B+G 
 [10] cos - a sm — - — 
 2 2 
 
 By division of the numbers standing under each other we 
 obtain tan x-(5+c) and tan-(&— c), whence we find h and c. 
 
 We then find the values of cos-(6+c) or sin-{5+c) and 
 
 cos -^(b — c) or sin -^{b — c), accordingly as the sine or the cosine is 
 
 the greater, and subtract the first from the greater of the two loga- 
 rithms [7] or [8], the other from the greater of the two logarithms 
 
 [9] or [10], and then the values of sin -A and cos ~A are 
 
 2 2 
 
 obtained. Combining these to find tan -A, A itself is found. 
 
 Since sin -A and cos -^ must give the same angle as tan -A, 
 there exists a proof of the accuracy of the computation. 
 
 " Gauss, Tlieoria motus Corporum CaUsiium, p. 60 seq. 
 
FOEMULiE OP SPHERICAL TKIGONOMETEY. 11 
 
 Example. 
 Let a= 11°. 25'. 56", 3, 
 
 then we have 
 
 J5=184». 6'. 55", 4, 
 (7= 11M8'.40",3, 
 
 i(5+(7)=97°.42'.47",85, 
 5(5-0) =86". 24'. 7", 55; 
 log cos i {B- C)= 8-7976413, 
 log cos i (5+ C) = - 9-1278046 ; 
 log sin - a = 8-9982605, 
 
 log COS - a = 9-9978351 ; 
 log sin \{B-C) = 9-9991432, 
 log sin \{B+G)= 9-9960526 ; 
 
 Ji 
 
 log[sin5 a cos \ (B-C)] = 7-7959018 
 log[sin 1 a sin 5 (5- (7)] = 8-9974037 ; 
 logfos^acos J (5+ C) = - 9-1256397, 
 
 r 1 1 "" 
 
 log cos i a sin i (5 + CO = 9-9938877 ; 
 
 i(J + c) = 177M9'.13",49, 
 
 l(h-c)=5\ 45'. 24", 13 ; 
 
 log cos 1 (6 + c) = - 9-9995248, 
 
12 FORMULiE OP SPHERICAL TRIGONOMETRY, 
 
 log COS l{b-c) =9-9978042 ; 
 log sin i ^ = 9-1261149, b = 183", 4'. 37",62 ; 
 
 log cos 5 ^ = 9-9960835, c = 17 1°.33'.49",36 ; 
 
 I ^ = 7°. 40'. 59",38, A = 15''.21'.58",76. 
 
 If we had taken 
 
 5=-175''.53'.4",6; 
 then 
 
 i(^+C) = -82M7'.12",15, 
 
 i(5-C')=-93''.35'.52",45, 
 and thus we should have obtained 
 
 i(J + c)=- 2°.40'.46",51, 
 
 1 (& - c) = -f 185. 45'. 24",13, ^. 
 
 and thus 
 
 h = 183°. 4',37",62, and c = - 188°. 26'. 10",64. 
 
 Dividing the Gaussian Equations by each other, we ob- 
 tain Napier's Analogies. Writing A, B, C, in the places of 
 JB, C, A, and a, h, c in the places of I, c, a, we find from 
 
 equations (11) 
 
 a — i 
 
 ^ A + B "°^^- ^ C 
 tan-^-=-_j^cotan-, 
 
 cos-^ 
 
 . a — h 
 
 , A-B ^^"-2- , (7 
 tan-^-=_-__cotan-, 
 
 sin 
 
 2 
 ^-5 
 
 ^ a + S ''' -2~ . c 
 
 *^"-2-=-^4H-B*""2 
 
 cos T 
 
sin 
 
 A-B 
 
 2 
 
 sin 
 
 A+B 
 
 FORMULiE OF SPHERICAL TRIGONOMETRY, 13 
 
 tan — ;;— = ^ . „ tan- , 
 
 6. Since nearly all the formulae in Nos. 3 and 4 consist of 
 two terms, they are unfit for logarithmetical computation; And 
 must be transformed into expressions having only a single term 
 hy means of auxiliary angles. 
 
 Any two possible quantities, whether positive or negative, 
 may be made proportional to a sine or cosine, so that 
 
 x = m sin M, and y = m cos M; 
 
 from which are derived 
 
 tan if =1^ andm = V^T/; 
 
 and thus M and m are expressed in terms of possible quantities 
 ofily. Now all the preceding formulae contain in each of their 
 two terms the sine or cosine of one and the same angle. If then 
 we make the remaining factors of one of the terms proportional 
 to the sine and the other proportional to the cosine of an 
 angle, we shall be enabled to apply the formulae for the sine 
 or cosine of a trigonometrical quantity consisting of iwo terms, 
 and by this means obtain a form convenient for logarithmetical 
 computation. 
 
 Let for example the three formulae for computation be 
 
 cos a =■ cos h cos c + sin & sin c cos A, 
 
 sin a sin 5 = sin 6 sin -4, 
 
 sin a cos B = cos h sin c — sin h cos c cos A, 
 
 and make 
 
 sin h cos A^vn sin M^ 
 
 cos 6 = w cos M, 
 
 We shall have then 
 
 cos a=m cos (c — J/), 
 sin a sin 5= sin h sin A, 
 sin acosB = m sin (c — M). 
 
14 
 
 FORMULA OP SPHERICAL TRIGONOMETRY. 
 
 If the quadrant in which B lies be known, the fonnulse 
 
 may be written in the following manner, putting for m its 
 
 sin h cos A ■^. ^ 
 
 ±irst compute 
 
 value 
 
 sin if 
 
 and then find 
 
 tan M= tan h cos A, 
 
 _ tan A sin M 
 tan 5= 
 
 tana = 
 
 sin (c — ilf) ' 
 
 tan(c — ilf) 
 
 cos 5 
 
 Another transformation may also be effected of the for- 
 mulae 
 
 cos a = cos J cos c + sin J sin c cos -4 (a), 
 
 sin a sin jB= sin J sin -4 (6), 
 
 sin a cos 5 = cos 6 sin c — sin S cos c cos uil (c)) 
 
 which will also be used in the sequel*. 
 
 Denote by B^ and &„ the values of B and h when a = 90°, 
 we have then the three following equations, 
 
 = cos &(, cos c + sin &„ sine cos ^ {d), 
 
 sin 5(1= sin 5„ sin -4. (e). 
 
 cos B^ = cos 6„ sin c — sin b^ cos c cos A, . 
 
 if)' 
 
 Multiply (/) by cos c and subtract from the product equa- 
 tions {d) after multiplying the latter by sine; then multiply 
 equation (/) by sin c, and add to it equation (d) multiplied by 
 cos c, and we obtain 
 
 cos c cos B„ = — sin 6„ cos A 
 
 sin c cos 5„ = cos J„ }■ [A). 
 
 sin B„ = sin b„ sin A 
 
 If then we put 
 
 cos c = sin 7 cos G 
 sin e cos -4 = sin 7 sin G 
 sin c sin A = cos 7 
 
 we obtain from {d) 
 
 = sin 7 cos (5„— G), 
 
 '{£), 
 
 * Enuke, Jahrbuch for 1831. 
 
FORMULA OP SPHEKICAL TEIGOKOMETKY. 15 
 
 also 
 
 and from (a) 
 
 cosa = sin7Cos (G — b). 
 
 If also from the product of the equations (S) and (/) the 
 product of (e) and (c) be subtracted 
 
 sin a sin {B — B^ = sin c sin A sin (6 — J„) , 
 = — cos 7 cos ( (? — 5), 
 
 and, in the same manner, if, to the product of equations (c) 
 and (/) the product of equations (6) and (e) and of (a) and {b) 
 be added, 
 
 slna cos {B—B^)= sin h sin \ sin" ^ + sin b sin5„ cos'^+cosJ cos J„, 
 = cos {b — 5(,) = — sin ( (r — 5). 
 
 The complete system of formula for the computation of a and 
 B is the following, 
 
 sin 7 cos G = cos c, 
 sin 7 sin G = sin c cos A, 
 cos 7 = sin c sin A, 
 cos P„ sin c = — sin G, 
 
 cos jB„ cos C = — cos (r COS ^, 
 
 sin B^ = cos.(t sin A, 
 COS a = sin 7 cos (G — b), 
 sin a sin {B—B^ = — cos 7 cos [G — b), 
 sin acos {B—B„) = — sm{G — b). 
 
 7. In general care must be taken that the angles that are 
 required be found by means of their tangents ; for, since the vari- 
 ations of these functions are most rapid, the value of the angle 
 can be found by means of them with the greatest accuracy. 
 
 Let Ax denote a very small change of an angle, then we have 
 
 A(logtana;)=-j- 
 
 sm2a; 
 
 It is customary to express the variations of an angle in 
 seconds ; since then the tangent is referred to radius as the unit, 
 
16 FORMULA OP SPHERICAL TRIGONOMETRY. 
 
 the variations of Aa; must be also expressed in parts of the radius, 
 and therefore be divided by the number 206264-8 *, The log- 
 arithms are besides understood to be natural or hyperbolic log- 
 arithms ; and, to reduce them to the system of Briggs, the former 
 must be multiplied by the modulus 0-4342945 = M. 
 
 Lastly, if it is desired to express A (log tan a;) in units of the 
 last decimal of the logarithms which are employed, for seven- 
 figure logarithms the expression must be multiplied by 10000000. 
 Thus we obtain 
 
 9 M At-" 
 
 A (logtan.)=^^.^-^^^^x 10000000 
 
 ^^•' Ax", 
 
 sm 2a; 
 
 sin 2a; 
 
 or Ax" = . A (log tan x) . 
 
 From this equation it may now be seen with what precision 
 the value of an angle can be found by means of the tangent. 
 
 If we suppose that five-figure logarithms are employed, since 
 the computation may, in extreme cases, be incorrect to two units 
 of the last decimal, A (log tan x) = 200, and the error in the 
 angle thence arising, will be 
 
 . „ 200" . „ .„ - , 
 A X = -— — sm 2x = 5 sm 2x. 
 42-1 , 
 
 Thus, by the use of five-figure logarithms the error cannot be 
 greater than 5" sin 2a;, or, since sin2x at its maximum is equal 
 to unity, the greatest error may amount to 5", and this error can 
 only be committed when the angle is nearly equal to 45°., With 
 the use of seven-figure logarithms the error must be of necessity 
 
 • The number 206264'8, wliose logarithm is 5-3144251, is always made use of when 
 it is required to transform quantities expressed in parts of the radius into seconds of 
 arc. Tlie number of seconds in the circumference of the circle is 1296000, and the 
 circumference of tbecircle in terms of the radius is 27r =6'2831853. Thesetwo nmnbers 
 are to each ctlier in the proportion of 206264'8 : 1 . If it is required to transform 
 quantities expressed in parts of tire radius into arc, they must be multiplied by this 
 number; on the contrary, to express qoantities which are given in seconds, in parts of 
 the radius, it is necessary to divide by tliis number. The number itself is the number 
 of seconds in an arc equal to the radius, and its oeupleaeut is the sine or the tangent 
 of one second. 
 
FOEMULiE OF SPHEEICAL TRIGONOMETRY. 17 
 
 100 times smaller, and thus the value of the angle obtained by 
 means of the tangent, can at the most be incorrect only to 0"*05. 
 
 If now an angle be given by means of its sine or cosine, then, 
 in the formula for 
 
 A (log sin x) or A (log cos x), 
 
 we should have, instead of the factor sin 2x, tan x or cotan x, 
 each of which may have all values possible up to infinity. We 
 thus see that small errors in the logarithms of the sine or cosine 
 of an angle may produce very large errors in the angle sought by 
 means of these functions, and therefore the values of angles 
 must in preference be always sought by means of their tangents. 
 
 8. Suppose that, in the formulse for oblique-angled spherical 
 triangles, one of the angles is equal to 90°, we then obtain the 
 formulae for right-angled spherical triangles. 
 
 In what follows the hypothenuse will be constantly denoted 
 by Ji, the two sides by c and c', and the angles opposite to them 
 by C and C". 
 
 From the first of the formula (4), we obtain by making 
 
 A = 90°, 
 
 cos h = cos c cos c'. 
 
 Again, from the first of the formulae (5), on the same suppo- 
 sition, we obtain 
 
 sin h sin C = sin c, 
 '/ 
 and from the first of the formulae iA\. [- ) ) 
 
 sin A cos C = cos c sin c, 
 
 or, by dividing this equation by that for the value of cos h, 
 
 tan h cos C = tan c'. 
 
 Again, by dividing 'fhig. formula by the expression for sin h 
 sin G, we shall have 
 
 cotan C = cotan c sin c', 
 or tan c = tan C sin c. 
 
 Both formulae might have been obtained also from the first of 
 equations (7) and the third of equations (9), by making A = 90°, 
 B. A. 2 
 
18 FORMULA OP SPHERICAL TRIGONOMETRY. 
 
 From the first of equations (10) there follows : 
 = cos A sin (7 sin C" — cos (7 cos C, 
 
 or cos h = cotan C cotan C. 
 
 Lastly, by combining the two equations, 
 
 sin h sin C" = sin c', 
 
 and sin A cos C = cos c sin c, 
 
 we obtain 
 
 cos C=sin C" cose. 
 
 We have consequently the following six formulae for right- 
 angled triangles : 
 
 cos h = cos c cos c', 
 sin c = sin A sin C, 
 tan c = tan k cos C", 
 tan c = tan C sin c', 
 cos A = cotan cotan (7', 
 cos (7= cos c sin C", 
 
 (12), 
 
 by means of which, if any two parts of a right-angled triangle be 
 given, the remaining parts may be found. 
 
 9. In astronomy it is necessary in the computation of quan- 
 tities to borrow certain data derived from observation. But since 
 the absolute accuracy of none of these data can be answered for, 
 but in every case a small error must be looked upon as possible, it 
 is necessary to investigate in every problem, whether a small 
 change of the observed quantities can produce no gi-eat error in 
 the quantities to be found. To be able to judge of this in every 
 case, it is necessary to differentiate the formulae of spherical 
 trigonometry, and, that all cases may be included, all the quan- 
 tities must be considered as variable. 
 
 By differentiating the first of equations (4) we obtain 
 — sin ada = db [— sin b cos c + cos b sin c cos A"] 
 + dc [— cos b sin c + sin J cos c cos A] 
 - sin b sin c sin AdA. 
 
poemuljE of spherical teigonometry. 19 
 
 The factor of db is equal to — sin a cos 0, that of dc is equal 
 to — sin a cos B ; writing in addition — sin a sin c sin B instead 
 of the factor of ^, we obtain the differential formula 
 
 da = cos Cdb + cos Bdc + sin c sin BdA. 
 
 Writing the first of equations (5) logarithmically, we obtain 
 
 log sin a + log sin B = log sin b + log sin A ; 
 
 and, differentiating, 
 
 cotan ada + cotan BdB = cotan bdb + cotan Ad A. 
 
 In place of the first of formulae (6), if we differentiate the 
 first of formula (7), which is derived from the combination of 
 (5) and (6), we obtain 
 
 — ^-TTi dB + dA fcotan BcosA — sin A cos c\ 
 sm jD ■- -' 
 
 = — ;— =-7 db + dc Fcotan b cos c + cos A sin c], 
 
 sin^ ,„ cosC 7 , sine „ cos a , 
 or — T-s-T, dB — : — 5 dA= — r-^r db + -. — r dc. 
 sm^B amB sm^b ainb 
 
 By multiplying this equation by sin B, we find ^ ^ 
 
 sin a , J, ^j . sinO y, , cosa»jj 
 
 T— ^ dB- COS CdA = r-n-aoH — = — tOC, 
 
 sm b sivfb sm 6 
 
 or finally ( ; ?• 
 
 sin odB = sin Cdb — sin B cos ode - sin J cos CdA. 
 
 From the first of formulas (10) we obtain, precisely in the 
 same way as from (4), 
 
 dA = — cos cdB — cos bdO+ sin b sin Oda. 
 
 We have thus the following differential equations of spherical 
 trigonometry : 
 
 da = cos Cdb + cos Bdc + sin b sin QdA 
 cotan ada + cotan ^J5 = cotan 5«^J + cotan Ad A 
 sin a J5 = sin Cdb — sin 5 cos adc — sin & cos CdA 
 d A =.— COS cdB — COS bdC + ainb ain Cda 
 
 (13). 
 
 10. For small angles it may be permitted to put unity in 
 place of the cosine, and the arc in place of the sine or the 
 
 2—2 
 
20 FORMULA FOE SMALL ANGLES. 
 
 tangent; thus, if the arc be expressed in seconds we may put 
 206265 a instead of sin a or tan a. 
 
 If the angles are not small enough to allow the second term 
 of the series for the sine to be omitted, we may proceed in the 
 following manner : 
 
 We have 
 
 a 6 120 
 
 and 
 
 thus 
 
 cosa=l-2a'+-a*- 
 
 >/cos a=i ~-xO.'+ &c. 
 
 D 
 
 We have therefore to the third powers inclusive 
 
 sma 
 
 =v< 
 
 cos a, 
 
 a 
 
 or a = sin a 'Jaec a ; 
 
 a formula, which is so accurate, that for an angle of 10° not so 
 much as an error of 1" would arise from its employment. In 
 this case we have 
 
 log (siu 10° ^sec 10°) = 9-2418864, 
 
 and, adding thereto the logarithm 5'3144251 and taking out the 
 corresponding number, we obtain 36000".74, or 10". 0'. 0".74. 
 
 11. In spherical trigonometry very great use is made of 
 developments in series, of which we will here derive the most 
 important. 
 
 If we have an expression of the form 
 
 . a sin X 
 
 tan y = , 
 
 1 — a cos X 
 
 we can easily develope y in a. series, which shall proceed accord- 
 ing to sines of multiples of the angle x. If, namely, tan s = - , 
 
 n 
 
 7 ndm — mdn 
 
 «« = S-; — r-- 
 
TRIGONOMETRICAL DEVELOPMENTS. 21 
 
 Considering then in the formula for tan y both a and y to be 
 variable, we obtain 
 
 dy sin x 
 
 1 — 2a cos 03 + a* 
 
 and, if we develop this expression, by the method of indeter- 
 minate coeflScients, in a seiies proceeding according to the powers 
 of a, 
 
 dy . 
 
 ^ = sma; + asin2a; + a sm3a;+...* 
 
 Integrating this equation, and remarking that for a; = 0, ^ is 
 also = 0, we obtain for y the following series : 
 
 y = asina; + -a''sin2a;+-a'sin3a3+ (14). 
 
 Very frequently we have two equations of the form 
 ^ sin 5= a sin a;, 
 A cosB= 1 — a cos x, 
 
 from which it is proposed to develop B and log^ in a series 
 proceeding according to the sines and cosines of multiples of x. 
 Since here 
 
 a sin 03 
 
 tan-B = 
 
 1 — a cos X ' 
 
 we can, by formula (14), find for B a, series ascending according 
 to sines of multiples of x. To develope log^ in a similar series 
 we have in the first place 
 
 -4 = Vl — 2a cos 03 + a'. 
 
 But, by the method of indeterminate coefficients, the follow- 
 ing series is found : 
 
 a cos x — a' 
 
 1 — 2a cos X 
 
 J = a cos a; + a' cos 2o3 + a' cos 3x + (fecf 
 
 T a 
 
 * It i8 easily seen that the first term is sin x, and that the coefficient of a" will be 
 
 found by the equation 
 
 A, = 2J,-j cos X - A„-2. 
 
 t It is seen immediately that the coefficient of o is equal to cosi, and that the 
 coefficient of a* will be found by the equation 
 
 A^ = 2A^i cos X — A„-i. 
 
22 TRIGONOMETRICAL DEVELOPMENTS, 
 
 By multiplying this expression by and integrating 
 
 with respect to a, we shall hare, since the left side is equal to 
 
 1 d . log (1 — 2a cos 03 + a") 
 
 2 da ' 
 
 and, for a. = 0, log A, also = 0, 
 
 log Vl — 2a cos a; + a'' = log ^ = — [a cos a; + - a" cos 2aj 
 
 + - a' cos 3a5 + &C.J (15). 
 
 In the same manner we obtain, if we have the two equations 
 Awa.B= a sin x, 
 A cos B=\ + a cos x, 
 
 by putting in (14) and (15) 180° — a; in the place of a;, 
 
 5 = a sin a; — - a" sin 2a! + r a' sin 3a;. .T-. (16), 
 
 log Vl + 2a cos x + d? — log J. = a cos a; — - a' cos 2a; 
 
 + -a'cos3a;-&c (17). 
 
 If the expression be of the form 
 
 tan y = n tan x, 
 
 it may easily be reduced to the form' 
 
 a sin X 
 
 tan y = . 
 
 ^ 1 — a cos a; 
 
 We have in fact 
 
 , , tan y — tan x (n — 1) tan x 
 
 tan {y-x)= - — r 1 = V"; — i — w— 
 
 ^^ ' 1 + tan y tan x 1 + n tan x 
 
 _ {n — 1) sinajcosa; (w — 1) sin x cos x 
 
 cos" x+n sin" a; 1 , 1 „ , w w 
 
 - + - cos 2a; + 2 - 2 '^os 2a; 
 
 n-1 . 
 (w-l)sin2a; ^^^■sm2a; 
 
 "~ (»i + 1) — (ra — 1) cos 2a; ~ ^ w — 1 * 
 
 ^ ' ^ ' 1 —cos 2a; 
 
 w + 1 
 
TEIGONOMETElCAIi DEVELOPMENTS. 23 
 
 Therefore if the equation tany = n tan x be given, we obtain 
 
 In this expression put, in the first place, 
 
 n = cos a, 
 
 then 
 
 n—l , . 1 
 
 = — tan - a. 
 
 n + 1 2 
 
 Thus the equation 
 
 tan ^ = cos a tan a? 
 gives 
 
 y = x — tan' - a sin 2a3 + - tan* - a sin 4a! — - tan° - a sin 6aj 
 
 (19). 
 
 
 
 
 
 
 + &C 
 
 If 
 
 
 
 71 = 
 
 = seca 
 
 ■) 
 
 then 
 
 
 
 n-\ 
 
 n + 1 
 
 = tan'' 
 
 1 
 
 2«' 
 
 and we 
 
 thus obtain. 
 
 if 
 
 
 
 
 tan y = sec a tan x, or tan x = cos a tan y, 
 
 y = a; + tan'' - a. sin 2a! + ^tan*- a sin 4a! + -tan'- « sinGo! 
 
 + &C (20). 
 
 „. cos a- cos i8 . 1,- ., 1 //J . N 
 
 Since , , a = **° o (P - «) tan jr (yS + a), 
 
 cosa + cosp 2^ ' 2^ " 
 
 , sin a — sin /3 1 , _. . 1 / , o\ 
 
 and ; — ^-5 = tan-(a-/3) cot-(a + j8), 
 
 cosa+cos/8 2^ "^^ 2^ '^' 
 
 we obtain, if 
 
 cos a , 
 
 tan V = ;i tan x, 
 
 ■^ cos (8 ' 
 
 y = a! — tan- (a-y3) tan - (a + ;8) sin 2a! 
 + itan'i (a -/S) tan'i (a + /S) sin 4a! + ... 
 
24 TKIGONOMETEICAL DEVELOPMENTS. 
 
 and if tan y = —. — 7, tan x, 
 
 " sin /S. 
 
 ^ = a; + tan - (a — /3) cot 5 (a + )S) sin 1x 
 
 + ^ tan" i (a - /3) cot'' i (a + )8) sin 4a; + . . . 
 
 By means of the last two formulse Napier's Analogies may be 
 developed into series. From the equation 
 
 . A-B 
 
 7 sm — - — 
 a—b 2 , c 
 
 *"^^- = — T4-B*"^2' 
 sm-^— 
 
 namely, we obtain 
 
 ——-=- — tan — cot — sine + -tan''— cot' — sm 2c- ... 
 2 2 Z Z 2 2 2 
 
 or 
 
 1 = -^ + tan I cot ^ sin [a-h) + \ tan' f cot" ^ sin2 (a-6) +. . . 
 
 2 2 2 2 2 2^ 
 
 and in the same manner from the equation 
 
 A-B 
 
 , cos 
 
 , a+h 2 ^ c 
 
 ^^"-^^ A + B ^^'^'r 
 
 COS— 2- 
 
 we obtain the following series, 
 
 a + h c , ^ A. B . ,1^ ^A. s-B • „ . 
 = 5 + tan — tan :— sm c + -tan — tan — sm 2c-f»... 
 
 - = ^ tan — tan — sin [a +'J) + - tan" — tan" - sin 2 (a+ 6)4-. . . 
 
 22 2 2^'2 2 2 s/f 
 
 Precisely similar series are obtained from the two other ana- 
 logies 
 
 . a — h 
 A-B ^™~2~, 180°- (7 
 
 sm — -— 
 
TEIGONOMETfilCAL DEVELOPMENTS. 25 
 
 a — h 
 
 , A + B '^^~2'^ 180'- G 
 
 tan — - — = , tan . 
 
 2 a + b 2 
 
 cos-^ 
 
 A case frequently occurs in which a quantity y is given by 
 means of an equation of the form 
 
 cos y = cos x + b, 
 
 and it is necessary to change it into a series proceeding accord- 
 ing to powers of b. For this purpose we develop the equation 
 
 1/ = cos"' (cos x + b), 
 
 hy Taylor's Theorem. Make, namely, 
 
 C08a; = s a.ndy=f{3 + b), 
 
 we have then 
 
 or since 
 
 f(z) = x ^M=_^_= L 
 
 •^ ^ ' ' dz d. (cos a;) sin a; ' 
 
 d ( ^ ) 
 d' (fz) _ ' V sin xj dx cos x 
 
 dz^ dx ' d (cos X) 
 
 - / cosa!\ 
 
 t?°(.^) l~im^j dx ^ y^+^^ 
 
 d^ ~ dx ' d Ccos a;"! m'\ sin' 
 
 d^ dx 'd{cosx) /6'y sin' a; ' 
 
 Thus 
 
 y = x--^ icota;^-J(l + 3cot'a!)^^ (21). 
 
 ^ Bmx 2 sm^'ar 6^ ' sm°a; 
 
 Just in the same way, from the equation 
 
 sin ^ = sin a; + 5, 
 
 we obtain 
 
 « = a;+— + itana;-^+i(l + 3tan=a;)-^ + (22). 
 
 ^ cos a; 2 cos a; 6 ^ • cos a; 
 
 Note. On derelopment of Series compare Bncke, Einige Reihenent- 
 wickdungen am der spharischen Aetronomie. Astronomische Nachrich- 
 ten. No. 562. 
 
26 ON INTERPOLATION. 
 
 B. ON INTERPOLATION. 
 
 12. In astronomy use is constantly made of taUes in which 
 the numerical values of certain functions are given for single 
 numerical values of the variables. Since now in practice there 
 is also need of values of the function for such values of the 
 variables as are not given explicitly in the tables, means must be 
 found for computing the same for any values whatever of the 
 variable or of the argument of the function. For this pur- 
 pose is employed the process of interpolation. The object of it 
 is, to substitute in place of a function whose analytical expression 
 is either altogether unknown or inconvenient for numerical com- 
 putation, andther simpler function formed from given numerical 
 values, which, within the limits of its use, can be employed in- 
 stead of the other. 
 
 By Taylor's Theorem any function can be expanded into a 
 series proceeding according to integral powers of the variable : 
 the only exception to this law being in the case when, for a de- 
 terminate value of the variable, one of the differential coeflBcients 
 becomes infinite but the function in the neighbourhood of this 
 value is discontinuous. Since however the process of interpola- 
 tion is founded on the expansion of functions in series which pro- 
 ceed according to integral powers of the variable, it is thus 
 assumed that the function, within the limits in question, is con- 
 tinuous, and is to be employed only on this supposition. 
 
 Let w be the interval or the difference of two consecutive 
 arguments (which here is always considered to be constant), then 
 any argument whatever may be denoted by a + nw, where n is 
 the variable quantity, and the function, corresponding to this 
 argument, is f{a + nw). The diflference of two consecutive 
 values of the function /(a + mm) and /{«+ (mH- 1) w} may be 
 
 denoted by/'(a + w + -j, where, for the purpose of denoting 
 
 to what values of the function the difference corresponds, the 
 arithmetical mean of the two arguments is put under the expres- 
 sion for the functions and the factor w is omitted*. In the same 
 
 • This very convenient notation is given by Encke in his Treatise On Mechanical 
 Quadratures, in the Jahrbuch for 1837. 
 
On interpolation. 27 
 
 maimer /' (a + -j expresses the difference off (a) and /{a + w), 
 
 f f « + 2) tte difference of /(a + w) and /(a + 2m)). The same 
 
 holds also with respect to higher differences whose order is de- 
 noted by the accents. Thus, for example, /" (a + 2) is the 
 
 difference of the first two differences/' f a + -) and/' fa + -) . 
 
 The scheme of the arguments and of the corresponding values 
 of the function and their differences is therefore as follows : 
 
 Arg. Function. ist Difif. 2nd Diff. 3rd Diff. 4th DifF. 5th Diflf. 
 
 a - 2W /(a - 2w) L f _ f /" (a - 2) .,„ , _ 3. 
 
 a + v, /(« + «')^-J'gtH /';(« + !) f'«i^ /"(«+!) -^^^ 
 a + 2«,/(a + 2«,)^, «tl /"(a + a)-^ ^'' + ''^ 
 
 a + 3w/(a + 3M;) 
 
 All the differences which have the same quantities under the 
 characters of the functions, stand here in the same horizontal line. 
 The differences of odd orders have all of them as the quantities 
 under the function-characters, a + a fraction with the denomi- 
 nator 2. 
 
 13. Since, by Taylor's Theorem, any function can be ex- 
 panded in a series ascending by integral powers of the variable, 
 we may make 
 
 /(a + ww) =a + |8.MW+7.«V+S.wV + &c (a). 
 
 If the analytical expression of the function /(a + wzo) were 
 known, the quantities a, /3, 7, S, might be computed, since 
 
 It is however assumed that this analytical expression is not 
 given, or at least, if even it is known, it is not to be employed, 
 and that the function /(a + nw) is known only for certain values 
 of the argument a + nw. But if, in the above equation, we put 
 
28 ON INTEKPOLATION. 
 
 one after another the diiFerent values of the variaMe n, we 
 obtain as many equations as we know values of the function, 
 and we can calculate just the same number of the coeflScients 
 a, ^, 7, S, &c. 
 
 Let now four numerical values of the function /(a + nw) 
 be given, namely, f{a), f{a + w), f{a + 2w) and f{a + 3w), we 
 have then the four equations : 
 f{a) = a, 
 
 f{a + 2w)=a + 2/3w + 4710" + 8Sw', 
 f{a +3w) = a + 3/8w + 97^" + 27Sw\ 
 But since 
 
 /(« + «,) =/(a)+/' (a +i), 
 /(a + 2w) =f{a) +f (a + ^ +f (a + |) , 
 =y(«)+2/(a+i)+/"(a+l) 
 
 /(a + 3w)=/(a + 2«;)+/'(« + |), 
 
 =/>f3/(a + i)+3/"(a + l)+/"'(a + |); 
 we obtain 
 
 /'(a+i)=/3w + 7w» + S< 
 2/ (a + ^) +/" (a + 1) = 2^w + iyw' + 8Bw\ 
 
 3/ (« + 5) + 3/" (« + 1) +/'" («+!) = 3/8to + 97W' + 27&0', 
 and hence 
 
 /'(«+i)-i/"(a)+|/"'(« + |) = /3«,. 
 
ON INTERPOLATION. 29 
 
 and therefore, if we substitute these values in equation (a) for 
 f{a + nw) and arrange the same in the order of the differences, 
 
 /(a + ««,)=/«+«/'(«+ i) + ^/" (a + 1), 
 
 + 
 
 6 
 or 
 
 This formula is known under the name of Newton's Inter- 
 polation formula. The coefficient of the difference of the order 
 n is the coefficient of x" in the expansion of (1 + a?)". The 
 proof, which is here given for only four values, can he easily 
 extended to any number of values. 
 
 Example. 
 
 In the Berlin Jahrhuch for 1850, there are, for mean noon, 
 the following Heliocentric longitudes of Mercury : 
 
 lat Diff. 2nd DifiF. 3rd Diff. 4th DifiF. 
 
 Jan. 303°.25'. 1",5 
 2 
 4 
 
 8 
 10 340.30.20 ,6 
 
 If from these the longitude be required for mean noon of 
 January 1, we have 
 
 /(a) = 303°. 25'. l",5, and m = i . 
 
 In addition 
 
 fU + ^) = + 6°. 41'. 50",0, n=\. Product = + 3». 20'.55",0, 
 
30 ON INTERPOLATION. 
 
 fia + 1) = + 18'.48",0, 'l!^^ = _ 1 . Product = - 2'.21",0. 
 
 Product = +10,3. 
 
 / (a + 2)-+10,l, 1.2.3.4 " 128- 
 
 Product = - 0,4. 
 
 There is then to be added to /(a), the quantity 
 + 3°. 18'. 43",9, 
 
 and we obtain for the longitude of Mercury, on Jan. 1,0, 
 306°. 43'. 45",4. 
 
 The Newtonian formula may be still more conveniently 
 written in the following manner, which has this advantage, 
 that the multipliers are all small fractions: 
 
 (la). fifl + nw) =f{a) +n [/' (a + i) +^ [fia + l) 
 
 ^-'-f^ [/'"(« + I) +^' [/"(a + 2)+&c. 
 
 ^ "=^ ^ = -|andi^./-(a + 2)=-6",4. 
 
 This, added to/"'fa+-j and the sum multiplied by 
 [= — -), gives — 1'. 19",0. Adding this again to 
 f"{a + 1) and multiplying the sum by — ^ = - - , we obtain 
 
 ^ 4: 
 
 -4'.22",2, and finally, adding this to /'fa + -J and multiplying 
 
 bym = -, we have 3°. 18'.43",9 to add to /(a), and we obtain 
 the same value as before, namely, 306". 43'. 45",4. 
 
 14. More convenient foi-mulse for interpolation are obtained, 
 if Newton's formula of interpolation be so transformed; that in 
 
 n-2 
 
ON INTERPOLATION. 31 
 
 it those differences solely are found, which stand on the same 
 horizontal line, so that, setting out from the value of f{a), 
 
 the differences /' (« + h) > /"(«)> ^^^ /'"(* + o) ' ^- *^® ^ 
 
 be employed. The first two terms of the Newtonian formula 
 can then be taken in. 
 
 But we have 
 
 /"(a+l)-=/"(a)+/"(a + l), 
 
 r(a+l)=r{a+l)+r{a+i), 
 
 r{a + 2)=r{a + l)+f'(a + l), 
 
 /^(«+|)=/^(«+|)+/"(« + 2) 
 
 &c. 
 
 Thus we obtain as the coefficient of/" (a) 
 n(n—l) 
 1.2 ' 
 
 as the coefficient off" (a + -j 
 
 w (w - 1) n(n-l) (w-2) ^ (w+l)w(w-l) 
 ~T:2 ■*■ 1.2.3 1.2.3 ' 
 
 as the coefficient of/" (a) 
 
 w(w-l)(w-2) n.(n-l){n-2){n-3) 
 1.2.3 1.2.3.4 
 
 (n + l)n.{n-l){n-2) 
 1.2.3.4 ' 
 
32 
 
 ON INTERPOLATION. 
 
 and finally, as the coeflScient of/''(a + -] , 
 
 n (w-1) (w-2) n (n-1) (w-2) (n-S) 
 
 1.2.3 "'" ■ 1.2.3.4 
 
 n {n-1) (w-2) (w-3) (w-4) _ (w+2) (w+1) n (w-1) (w-2) 
 1.2.3.4.5 "" 1.2.3.4.5 ' 
 
 where the law of continuation is evident. 
 The whole formula is therefore 
 f{a + nw)=f{a)+nf'(a + ^)+^Lt^f"{a) 
 (w + l)w(w-l) / IN 
 
 ■^ ty::! -^ r+2/ 
 
 (w + l)w(w-l) (w-2) .„ ., 
 "•■ 1.2.3.4 -^ ^''^ 
 
 , (w + 2)(w + l)w(w-l)(w-2) / 1\ 
 + 1.2.3.4.5 -^ l^« + 2; + -i^^ 
 
 If, in place of the differences which have a + -- under the 
 
 sign of the function, we insert those which contain a — - , we 
 have 
 
 /(« + |)=/' («-^)+/"(«), 
 /"'(a + 2^)=/"(«-i)+/-(«.), 
 
 r(«+i)=/^(a-,-)+rW. 
 
 These are the coefficients of the differences of odd orders, 
 and on the other hand the coefficient of/" (a) is 
 
 w (w— 1) m(w+l) 
 
 '^+^-:2 =-S-^' 
 
 and that oif (a) is 
 
 (w+l)w(w-l) (wH-l)w(w-l) (w-2) ^ (w-l)w(w + l)(w + 2) 
 1.2.3 1.2.3.4 1.2.3.4 * 
 
ON INTKKPOLATION. 33 
 
 We obtain therefore 
 
 /(« + nw) =fia) +„/'(«_ 1) +„. 'L^ ./" (a) 
 
 {n-l)n{n+ 1) 
 1.2.3 
 
 ^ '' --'- /'"(a-^) 
 
 , («-l)w(w + l)(« + 2) „v, V 
 "^ 1.2.3.4 -^ ^"^ 
 
 (>i-2)(»-l)n(»+l)(« + 2) ^/ _n , . 
 1.2.3.4.5 ''V 2/"^ 
 
 where again the law of continuation is evident. 
 
 If we assume now that it is required to interpolate a value 
 whose argument lies between a and a— w, m is then negative. But 
 if n always denote a positive value, we must employ — n instead 
 of n in the formula, and the. formula immediately preceding will 
 therefore be for this case : 
 
 f{a - nw) =f{a) - nf (^a - -j + « . ^^/" («) 
 
 (n-l)n(«+l) .,„ / n , (w+l)w(w-l)(n-2) 
 TTiTs ^ r~2J'*' 1.2.3.4 J ^""^ 
 
 (« + 2)(w + l)«(n-l)(w-2) j^(_ 1\ . 
 1.2.3.4.5 J V 2/'^ ^ ^' 
 
 This formula is to be employed if interpolation is to be 
 performed backwards. By writing again the two formulae (2) 
 and (3), as was before done wifh the Newtonian formulse, we 
 obtain 
 
 /(a ^nw) =/(a) + „ |^/' (a + 1) + ^ [/" («), 
 /(«-««,)=/(«)-« [/'(a-i)-^[/"(«), 
 
 B. A. 
 
 3 
 
34 ON INTEEPOLATION. 
 
 Imagine now a horizontal line to be drawn through the 
 scheme of the functions and differences, in the neighbourhood of 
 the place where the function for interpolation happens to lie; 
 then, the first formula being employed i{ a + nw be nearer to a 
 than toa + w, andthesecond,if a— mw lienearer toathanto a— w, 
 those differences must always be used which, on both sides of 
 the horizontal line, lie nearest to the same. No farther attention 
 is necessary to be paid to the signs of the differences, but every 
 difference must be so corrected that it may approximate to that 
 lying on the other side of the horizontal line. 
 
 For example, if the first formula be employed, and the argu- 
 ment therefore lie between a and a+-w, the horizontal line will 
 between/" (a) and/"(a + 1). We have then to add to /"(a) 
 
 + ^/"(a + i) = H-^[/"(«4l)-/"(a)]. 
 
 Thus if /"(a) be {greater} *^''-^"("+^)' *^^ corrected /"(a) 
 
 will be ^^"^^ 
 
 and therefore will always approximate to 
 
 /'"(«+ 1). 
 
 A somewhat greater degree of accuracy is attained if, for the 
 last difference which is employed, the arithmetical mean of the two 
 differences standing next to the horizontal line be taken. The 
 arithmetical mean of the two .differences will be denoted by 
 means of the character of the difference function and the arith- 
 metical mean of the two arguments which stand beneath, so 
 that 
 
 /'(« + «) = 
 
 ^'(^ + ^-|)+/'(^ + >^ + ^) 
 
 As before, there occur alternately fractions in the difference- 
 characters of the even orders and whole numbers in those of the 
 odd orders, so th.at no ambiguity can arise. If, for example, we 
 terminate with the second difference, we must take, in the pro- 
 
ON INTERPOLATION. 35 
 
 cess of interpolation in the forward direction, the arithmetical 
 mean of /" (a) and /"(a + 1), that is, /" (a + ^) • Then, in- 
 
 stead of the term 
 
 w (m — 1) 
 1.2 
 
 we now make use of the term 
 
 /"(«) 
 
 njn-l) ^„/ 1\ 
 
 1,2 
 
 While then, by simply taking /"(a), the omission of the 
 whole of the third term is entailed, in the present case the only 
 omission is that of 
 
 \ in+l)n{n-i) n{n-l) ] .,„/ _^1\ 
 n{n-l)(n- -J 
 
 1.2.3 
 
 /1-J)- 
 
 For n = -, the error, as far as it depends on the third dif- 
 
 ferences, is exactly nothing. For this case, when w = - , and the 
 
 interpolation will therefore be made at the middle term, it is indif- 
 ferent which of the two formulae (2) or (3) is employed, since we 
 may either set out with the argument a and interpolate forwards, 
 or set out with the argument a + w and interpolate backwards. 
 The most convenient formula for this case is however obtained by 
 
 a combination of the two. For n = -, the formula (2) becomes 
 
 1 _1 5 1 _i 
 
 3 113 
 
 2'2' 2- 2 
 1.2.3.4 ■' ^ ^^ 
 
 3—2 
 
36 ON INTERPOLATION. 
 
 On the contrary the formula (3) is, if we set out with the 
 argument a + w, 
 
 1 _1 
 
 /(« + !) =/(« + v>)- If (« + !)+ TTf"^'' + ^^ 
 
 113 3 113 
 
 2*2*2 ,,„/ 1\ 2*2" 2* 2 ,,,, , ,, 
 
 If the arithmetical mean of the two formulae be taken, all 
 the terms in which occur differences of an odd order are got rid 
 of, and we obtain for the interpolation the following very conve- 
 nient formula, in which only the arithmetical means of the even 
 orders appear : 
 
 /(-i-)=/("4)-5/'(-|) + ll8/"(-.-) 
 
 where the law of continuation is evident. 
 
 Example, Let the longitude of Mercury be required for 
 January 4, 12'', then we must employ the formula (2a). The 
 differences to be employed in this case will be the following : 
 
 istDiff, indDifif. 3rd Diff. 4th Diff. 
 
 7° . 0'. 38",0 4- 2'. 44",3 + 
 
 Jan. 4, 317°. 7'. 2d",5_+ 21'. 32",4 + 10",1 
 
 7 . 22 . 10 ,4 2 . 54 ,5 
 
 6,324.29.39,9 24.26,9 4,7 
 
 Here n = -, and so 
 
 M— 1_. 3 n + 1 _ 5 n — 2 7 
 
 2 "~""8' 3 ~12' 4 """16' 
 
ON INTERPOLATION. 37 
 
 paying no farther attention to the signs, we obtain 
 
 arithmetical mean of the 4th differences x 7^ = 0°. 0'. 3",2 
 
 16 
 
 corrected 3rd difference 2'. 51",3 x :^ = 1 . 11 ,4 
 
 a 
 corrected 2nd difference 22'. 43",8 x - = 8 . 31 ,4 
 
 8 
 
 corrected 1st difference 7°. 13'. 39",0 x i = 1 . 48 . 24 ,7 
 
 4 
 
 Thus the true longitude for January 4-5 = 318°. 55'. 54",2. 
 
 If the longitude be required for Jan. 5*5, we must employ 
 the formula (3o), and take the differences which stand on both 
 sides of the lower horizontal line. We find then the longitude 
 for Jan. 5-5 = 322". 36'. 56",7. 
 
 For the purpose of employing the formula (4a), let the lon- 
 gitude be required for Jan. 5'0. 
 
 We obtain then 
 
 3 
 
 arithmetical mean of the 4th differences x — — = — 0°. 0'. 1",4 
 
 lb 
 
 corrected arith. mean of 2nd differences x — - = — 2 . 52 ,3 
 
 o 
 
 arith. mean of functions 320 . 48 . 34 ,7 
 
 whence the true longitude for 
 
 Jan. 5-0 = 320°. 45'. 42",4. 
 
 Exhibiting now the differences of the interpolated values, 
 we obtain 
 
 let Diff. ind Di«f. 3rd Diff. 
 Jan. 4-0 317°. 7'.29",5 , ,0 .q' 91" 7 
 
 4-5 318.55.54,2 +;-;°-;*'' + 1'. 23",5 „ 
 
 5-0 320.45.42,4 itrt^'o 1.26,1+3'^ 
 
 5-5 322 . 36 . 56 ,7 i ^i 4^ 2 ^ • ^^ '» 
 6-0 324 . 29. 39 ,9 ' * ^ ' 
 
 The regular progression of the differences shews the correctness 
 .of the interpolation. This scrutiny by means of differences is 
 
38 ON INTERPOLATION. 
 
 made use of in all computations in which for certain arguments 
 proceeding at small intervals a series of values of the function 
 has heen computed. If namely in any value, f{a) for example, 
 an error x has been committed, the scheme of the diflferences 
 will he as-foUows : 
 
 /(a -3m;) /y„_6i 
 
 f{a - 2w) t)l _ § f"{a - 2) .,„, _ 3^ , „ 
 
 f(a-w) ll_H.^f"{a-l)+x f>_irj^r(«-l)-4r. 
 
 /{a + 2w)Ul+t\ /"{a + 2) /(« + .) "^ 
 
 An error in the value of a function will thus exhibit itself 
 very much magnified in the differences, and the greatest irregu- 
 larity will be found in the horizontal line, in which the erroneous 
 value of the function stands. 
 
 15. A case frequently occurs in which are employed the nu- 
 merical values of the differential coeflScients of a function whose 
 analytical expression is not known, but only a series of nume- 
 rical values of it, which follow each other at equal intervals, 
 are given. In this case recourse must be had to the formula of 
 interpolation for the computation of the numerical values of the 
 differential coefficients. 
 
 By substituting, in the original formula (a) of No. 13, the 
 values found for a, /3, 7, B, or, which is the same thing, deve- 
 loping by the Newtonian interpolation formula by powers of n, 
 we have i- 
 
 + - 
 
 +.4i[/"(-l)- 
 
 Now also since by Taylor's Theorem 
 
 f{a + nw) =f{a) +-iM nw + -j^-j^ 
 
 da^ "1.2.3 
 
 + -^^.7-7^ + 
 
ON INTERPOLATION. 39 
 
 we obtain by comparison of the two series 
 
 '^4[/'(-5)-i/"(«-')-|/1-|)--].' 
 
 More convenient formulae for the differential coefficients will 
 be found from formula (2) in No. 14. Introducing in this formula 
 the arithmetical mean of the odd differences, by making 
 
 /'(« + g =/'(«) +i/"(«), 
 /"(« + |)=/"(«)+i/'^(«), 
 
 we obtain 
 /{a + nw) =f{a) + nf (a) + ^/"(a) + (^±il|l|i:ll/"'(a) 
 
 "^ 1.2.3.4 J^ ^''^' 
 
 a formula which contains the even differences which stand with 
 f{a) on the same horizontal line, and on the contrary the arith- 
 metical means of the odd diflferences, which lie on both sides of 
 the horizontal line. Developing the same according to powers 
 of n, we have 
 
 f{a+nw)=f{a) + n^f (a)- i/"'(a) +^/'(a)- j^/™ W+ -] 
 
 + i^[/"(-)-^/'^W+ro/"W-"] 
 
 +T:^4[/'^w-k"W + -] 
 +t:24^5[^(«)-|-^™(«) + -]- 
 
40 .ON IKTEEPOLATION. 
 
 and from thence 
 
 m 
 
 da 
 
 da' 
 
 (fee, &c. (5). 
 
 If the differential coefficient be required for a function which 
 is not found amongst those given, for example iorf{a + nw), we 
 must put in this formula a + m in place of a, so that 
 
 df(a + nw) If/.;/ \ 1 /./;; / \ 1 y / N ~\ 
 
 l[/"(a+«)-l/'v(«+„) + ...j, 
 
 d'f{a + nw) 
 
 d'a W 
 
 &c. &c (6). 
 
 The differences now to be employed do not appear in the 
 scheme of the formulae above, but must be first computed. For 
 the even differences, for example /"(« + «), this is easy, since 
 they will be obtained by the ordinary interpolation formulae, by 
 considering/" (a),/" (a + w), &c. as the functions, and the third 
 differences as their first differences, &c. But the odd differences 
 are arithmetical means, and a formula must first be developed 
 for the interpolation of arithmetical ineans. 
 
 Now we have 
 
 /'(« + «) = -! 2i.-«J 2/. 
 
 And by the intei-polation formula (2) in No. 14, 
 
 /(»-i+»)./(.-i)+»rw+^^s=il/"(«-i) 
 
ON INTERPOLATIOX, 41 
 
 -^ '"•"i'.°2'i'"V M-Hfe. 
 
 We thus obtain, by taking the arithmetical mean of both 
 formula, the formula for the interpolation of an arithmetical 
 mean, • 
 
 /• (» + «) -/■ (a) + „/" W + ^f („) + L „/.. (a) 
 
 + '"\".i'rV M-^... 
 
 The two terms „ ^ k 
 
 are composed of the arithmetical mean of the terms 
 
 n{n-\) .,„ f 1\ 
 1.2 -^ V~2j' 
 and 
 
 -t:2--^ r+2J' 
 
 which gives 
 
 By combining the two terms which contain /'"{a), the for- 
 mula above may be thus written : 
 
 f{a + n) =f{a) +nf" (a) +^f" (a) +l?^/v(„) + ... (7). 
 
 By means of formulfe (5), (6), and (7), the numerical values 
 of the differential cOeflScients of a function for any argument 
 whatever may be computed from the even differences and the 
 arithmetical means of the odd differences, when there is given a 
 series of numerical values of the function following each other 
 at equal intervals. 
 
 Other formula for the differential coefficients may now be 
 developed, in which the simple differences of odd orders appear 
 instead of the arithmetical means of the even differences. 
 
42 ON INTERPOLATION. 
 
 In fact, if in the interpolation formula (3) the arithmetical 
 mean of the even diflferences he introduced, by making 
 
 we obtain, since 
 
 {n + i)n{n-l) 1 njn-l) ^- ("-^) ('^ " 2 j 
 1.2.3 2' 1.2 1.2.3 
 
 /(<^»»)-/(»4)+(»-l)/(»4)+^^^/"(«4) 
 
 + 
 
 If m + — be written instead of »i) the law of the coefficients 
 will be simpler, since we obtain 
 
 /[a + (« + l)«,]=/(a + i) + n/'(a + i) 
 
 1.2.3.4 -^ l^'^ + a/^"" 
 
 Developing this formula according to powers of w, we obtain, 
 since the terms independent of n are 
 
ON INTERPOLATION. 43 
 
 /{« + («+!) »}-/(. + !») 
 
 -K«-l)-.V"(»4)-4/-(-l)--] 
 
 -ok(-^i/"(-^^„/"(«.|)-...] 
 
 Comparing now this formula with the development of 
 by Taylor's theorem, we find 
 
 — So — ^Kr+2J-2i-^ r+2J 
 — s^ — =^L-^ r+2J-2V r+2J 
 
 &c. &c (8). 
 
 This formula is used most conveniently, when it is required 
 to compute the differential coefiicients of a function for an argu- 
 ment which is the arithmetical mean of two consecutive argu- 
 ments. For other arguments, for example. 
 
 a + 
 
 (n + ^)w, 
 
44 ON INTERPOLATION. 
 
 we have again 
 
 And here again the difference /' ( « + 5 + «) > and in general 
 
 all odd differences, will he computed hy the ordinary interpolation 
 formula. But since the even differences are arithmetical means, 
 we ohtain the formula to be employed for this from formula (7) 
 for the interpolation of an arithmetical mean from the odd differ- 
 ences, by putting a + - instead of a, and, for the purpose of find- 
 
 ing /" ( a + r + w 1, increasing all the accents by unity, so that 
 for example : 
 
 /"(« + ^+-)=/"(« + ^)+«/"'(« + |)+|>(«+-2) 
 
 Example. 
 
 In the Berlin Jahrlmch for 1848, are the, following Right 
 Ascensions of the Moon : 
 
 istDiff. ludDiff. srdDifF. 4th Diff. 
 h. h. m. A. m. a. a. a. s. 
 
 25.27,74 + :„'^^- 1,39 „ „, 
 25.50;i0 . ' 2:24- ' 
 
 26.10,22 :J'i„ 3,03 J!J 
 26.27:31 ' 3:70 «'«^ 
 
 26.40,70 ^'^'^^ 
 
 Supposing that from the above the first differential coefficients 
 be required for July 13, 10", 11", and 32", and that formula (9) 
 
 July 12 
 
 16.14.26,33 
 
 12 
 
 39.30,32 
 
 13 
 
 17. 4.58,06 
 
 12 
 
 30.48,16 
 
 14 
 
 56.58,38 
 
 12 
 
 18.23.25,69 
 
 15 
 
 50. 6,39 
 
ON INTERPOLATION. 45 
 
 be employed, we must first compute the first and third differences 
 for these times. 
 
 The third of the first difierences corresponds to the argument 
 July 13 6", and is /' (« + 1) , and thus, for 10", 11", and 12", n 
 
 is respectively - , - , and - . 
 
 By interpolating in the ordinary way, we obtain 
 
 h. 
 
 for 10 
 ... 11 
 ... 12 
 
 m. s. 
 
 + 25.57,11 
 25.58,81 
 26. 0,49 
 
 -2,51, 
 
 2,58, 
 
 2,64. 
 
 From hence we obtain the differential coefiScients 
 
 h. 
 for 10 
 ... 11 
 ... 12 
 
 the interval w being 12". 
 we must divide by 12, and 
 
 m. 19. 
 + 25.57,21, 
 25.58,92, 
 26. 0,60, 
 
 If the same be required for one hour, 
 L we then obtain the following values : 
 
 h. 
 for 10 
 ... 11 
 ... 12 
 
 
 m. !j. 
 
 + 2. 9,77, 
 
 2. 9,91, 
 
 * 2.10,05, 
 
 which express the hourly velocity of the moon in Right Ascen- 
 sion for those times. 
 
 If we had wished to employ formula (6), in which the 
 arithmetical mean of the odd differences appears, we should have 
 obtained, by taking a = July 13 12" (for 10" for example, in 
 
 which case « = --,) according to formula (7) : 
 
 f'L-iy+ 25*^6",77 and /" (« - ^) = " ^^'^l, - 
 
46 
 
 ON INTEEPOLATION. 
 
 and from thence according to formula (6), for the differential co- 
 efficient, + 2''!9^77. 
 
 The second differences are : 
 
 h. a. 
 
 for 10 +20,55, 
 
 11 20,34, 
 
 12 20,12, 
 
 Applying to these — — part of the 4"' differences and divid- 
 
 ing by 144, we obtain the second differential coefficients for one 
 hour as the unit : 
 
 h. s. 
 
 for 10 +0,1432, 
 
 11 0,1417, 
 
 12 0,1402. 
 
 Note. On the subject of Interpolations, the reader may compare the 
 article by Encke in the Jahrhuch for 1830, and the above-mentioned article 
 on Mechanical Quadratures in the JahrJmch for 1837. 
 
SPHERICAL ASTRONOMY. 
 
 FIEST SECTION. 
 
 THE VISIBLE SPHERE OP THE HEAVENS, AND ITS DAILY 
 
 ■MOTION. 
 
 In Spherical Astronomy are treated of the places of the 
 bodies in the visible sphere of the heavens, by referring them by 
 means of spherical co-ordinates to certain great circles devised 
 on the sphere. Spherical Astronomy gives then the means 
 both of determining the place of a heavenly body in relation to 
 these great circles, and also the relative positions of these circles 
 with regard to each other. It is necessary therefore first to obtain 
 a knowledge of these great circles, whose planes are the ftmda- 
 mental planes of the different systems of co-ordinates, and, at 
 the same time, of the means which have been employed to 
 reduce the place of a body, which is given for one of these 
 fundamental planes, to another system of corordinates. 
 
 Some of these co-ordinates are independ&t of the daily 
 motion of the sphere, others, on the contrary, are referred to 
 planes, which do not partake of this daily motion. The heavenly 
 bodies will therefore, if they are referred to these latter planes, 
 constantly change their places, and it will be of importance to 
 obtain a knowledge of these changes and of the phenomena that 
 are consequent upon them. But, since the heavenly bodies 
 besides these motions which are common to all, exhibit also 
 other, and those much slower motions, by reason of which they 
 change their places also with respect to the systems of co-ordi- 
 nates which ai-e independent of the daily motion, it is not suffi- 
 
48 ALTITUDE AND AZIMUTH. 
 
 cient to detemilne solely the position of a heavenly body, but it 
 is besides necessary that the time corresponding to this position 
 should be given. It is therefore necessary to obtain a knowledge 
 of the methods by which the daily motion of the sphere, partly 
 by itself and partly in connexion with the motion of the sun in 
 the same, is made to serve as a measure of time. 
 
 I. On the different systems of Planes and Circles on the visible 
 Sphere of the Heavens. 
 
 1. The heavens appear to us as the hollow surface of a 
 sphere, on which we see the stars projected and at the centre of 
 which we are situated. For the purpose of determining the 
 positions of the heavenly bodies on this visible sphere various 
 systems of spherical co-ordinates have been devised. The first 
 of these systems is that of Azimuths and Altitudes. In this 
 system the fundamental plane is the plane of the horizon, which 
 is determined by the surface of a fluid at rest, supposed to be 
 extended to infinity. This plane intersects the visible sphere 
 of the heavens in a great circle, which is called the horizon. 
 The plane of the horizon may also be defined as that plane 
 which is perpendicular to the plumbline, that is, to the di- 
 rection of gravity at the surface of the earth. The plumb- 
 line itself cuts the visible sphere of the heavens in two 
 points, which are the poles of the horizon, and of which the one 
 situated above the horizon is called the zenith, and the opposite 
 one the nadir. 
 
 By means of the zenith-point and the horizon, the position 
 of a heavenly body on the sphere of the heavens can now be 
 determined. A great circle for instance is drawn through the 
 zenith and the body whose position is to be determined, and this 
 circle is consequently perpendicular to the horizon. By determin- 
 ing now the point of intersection of this circle and the horizon, 
 and by reckoning from this point upwards along the great circle 
 the number of degrees between the horizon and the body, and 
 in like manner along the horizon the number of degrees to a cer- 
 tain origin or zero of co-ordinates, we have the two spherical 
 co-ordinates, by means of which the position of the body is to be 
 determined. The great circle passing through the zenith and 
 
ALTITUDE AND AZIMUTH INSTRUMENT. 49 
 
 the object is called a vertical circle; the arc of this circle between 
 the horizon and the object is called the altitude of the lody ; and 
 the arc between the object and the zenith is called its zenith- 
 distance. The sum of the altitude and zenith-distance is always 
 equal to 90°. The arc of the horizon between the vertical circle 
 •passing through the object and a point arbitrarily chosen is called 
 its azimuth. This zero-point i« taken arbitrarily, but for sim- 
 plicity let it be chosen so that it may coincide with the zero- 
 point of the second system of co-ordinates under consideration. 
 The direction in which azimuths are reckoned is likewise unim- 
 portant ; let us take therefore, as in the second system of co-ordi- 
 nates, the direction of the diurnal motion, in which we reckon 
 round from left to right from 0° to Set)". Small circles which are 
 parallel to the horizon, are named also horizontal circles or 
 almucantarats. 
 
 We can, instead of using these spherical co-ordinates, deter- 
 mine the place of an object by means of rectangular co-ordinates, 
 refeiTed to a system of axes of which the axis of z is perpen- 
 dicular to the plane of the horizon, whilst the axes of x and y 
 are situated in that plane, and so that the axis of x is drawn 
 towards the zero-point of azimuths, the positive axis of y to- 
 wards an azimuth of 90°. Indicating then the azimuth by A, 
 the altitude by h, we have 
 
 X = cos h cos A, y = cos h sin A, z= sin h. 
 
 Note. In order to observe these co-ordinates we have an 
 instrument completely corresponding to this co-ordinate system^ 
 namely the altitude and azimuth instrument. This consists 
 essentially of an horizontal divided circle, which stands upon 
 three adjusting footscrews, and which can be placed horizontal 
 by means of a spirit-level. This circle represents the plane of 
 the horizon. In the centre of this stands a vertical pillar, which 
 carries a second circle parallel to it, and therefore likewise 
 perpendicular to the horizon. About the centre of this ver- 
 tical circle revolves a telescope which is connected with a 
 pointer, by means of which we can observe the direction of 
 the telescope with reference to the circle. The vertical pillar 
 carries likewise a pointer at right angles to it, which indicates 
 B. A. 4 
 
60 DAILY MOTION OP THE HEAVENS. 
 
 the azimutli upon the horizontal circle. If now it be known 
 what points upon the circles correspond to the zero of the 
 azimuth and to the zenith-point, we are enabled by means of 
 such an instrument, if we direct the telescope to an object, 
 to find its azimuth and its altitude or zenith-distance. Be- 
 sides this there are still other instruments, with which alti- 
 tudes only can be observed. They are called altitude instru- 
 ments ; and are either quadrants, sextants, or entire circles. 
 Instruments, with which we observe azimuths only, are called 
 theodolites. 
 
 2. Heavenly bodies change their places on the visible sphere 
 of the heavens, and indeed every star, in consequence of the daily 
 motion of the earth in that period of time which is called a 
 sidereal day, describes a circle on the sphere which generally 
 is a small circle. Since the planes of all these circles are parallel 
 to one another, they are called parallel circles. The axis about 
 which the daily motion takes place is called the axis of the earth. 
 It meets the visible sphere of the heavens in two points, of which 
 that one which is visible on the northern hemisphere of the 
 earth is called the north pole, while the other is called the 
 south pole. The great circle, the poles of which are those two 
 poles of the earth, is called the equator. The position of the 
 earth's axis can be determined by means of the first system 
 of co-ordinates, by finding the azimuth and the altitude of the 
 pole. This last quantity is called the latitude of the place of 
 observation; and it is equal to the angle between the equator 
 and the zenith of the place. The complement of the elevation of 
 the pole to 90° is the altitude of the equator above the horizon or 
 the altitude of the equator at the place of observation. 
 
 The daily motion of the heavenly bodies now serves for the 
 foundation of a second system of co-ordinates. Great circles 
 which pass through the star and the poles, and are therefore 
 perpendicular to the equator, are called declination-drcles, or 
 hour-circles. The arc of such a great circle which is included 
 between the equator and the star, is called the declination of the 
 star ; on the contrary, the arc between the north pole and the star 
 is called the polar-distance. The declination is said to be posi- 
 tive, when the star lies in that part of the declination-circle which 
 
EEPERENCE OP OBJECTS TO THE EQUATOR. 51 
 
 Is included between the equator and the north pole ; negative, 
 when the star is in that part which is between the equator and 
 the south pole. 
 
 Declination and polar-distance are always complementary 
 to each other, and correspond to the altitude and zenith-dis- 
 tance in the first system of co-ordinates. Analogous to the 
 azimuth there is the hour-angle, that is, the angle formed by the 
 hour-circle passing through the star and a determinate circle 
 taken as the origin, and it is reckoned in the direction of the 
 daily motion from 0° to 360°. For the first hour-circle, that 
 has been chosen which passes through the zenith, and is 
 called the meridian. This meets the horizon in two points, of 
 which that one which lies between the north pole and the. 
 equator is called the north point, and the other for a similar 
 reason the south point. The latter is the zero from which the 
 azimuths are reckoned. Ninety degrees from the north and 
 south points lie the west and east points, which are at the 
 same time the points of intersection of the horizon and equator. 
 
 Instead of determining the place of an object by the two 
 spherical co-ordinates, declination and hour-angle, it can be deter- 
 mined by means of rectangular co-ordinates, by referring it to three 
 co-ordinate axes, of which the positive axis of z is drawn perpen- 
 dicular to the equator and towards the north pole, while the axes 
 of X and y lie in the plane of the equator, and so that the positive 
 axis of X is drawn towards the zero-point, the positive axis of y, 
 on the contrary, to the ninetieth degree of the hour-angle. 
 Designating then the declination by S, the hour-angle by t, we 
 have 
 
 x = cos S cos t, y' = cos 8 sin t, z' = sin S. 
 
 Note. — Corresponding to this second system of co-ordinates, declination 
 and hour-angle, there is another kind of instrument called the parallactic 
 instrument or equatoreal. In this the circle, which, in the first kind of 
 instruments, is parallel to the horizon, is parallel to the equator, so that the 
 pillar at right-angles to it is in the direction of the earth's axis. The circle 
 then which is parallel to this pillar is an hour-circle, and the angles which 
 are read off on the first circle or that parallel to the equator, are hour- 
 angles. Therefore if we know those points of the circle, which correspond 
 to the zero of hour-angle, and to the pole, we can, by means of such an 
 instrument find the declinations and hour-augles of celestial objects. 
 
 4—2 
 
52 EQUINOX OE ZERO OP EIGHT ASCENSIONS. 
 
 3. In this second system, one of the spherical co-ordinates, 
 viz. the declination, is constant, on the other hand the hour-angle 
 changes at every instant, since we begin to reckon it from 
 a point of the heavens which does not partake 6f the daily 
 motion. 
 
 In order now to have the second co-ordinate also constant, 
 we choose as zero-point a determinate point of the equator, and 
 naturally that one in which the equator is cut by the fundamental 
 plane of the following system of co-ordinates, viz. the ecliptic. 
 By virtue of the annual motion of the earth about the sun, the 
 centre of the sun appears to describe in the course of a year 
 a great circle in the heavens, which is called the ecliptic or 
 sun's path. This great circle is inclined to the equator at an 
 angle of about 23^°, which is called the obliquity of ike ecliptic. 
 The points of intersection of the ecliptic with the equator are 
 called respectively the vernal and autumnal equinoxes, because 
 when the sun, on the 21st March and 23rd September in each 
 year, passes through these points, the days and nights at all 
 points of the earth are equal*. The points of the ecliptic which 
 are 90° distant from the equinoxes are called the solstitial points. 
 
 This newly introduced co-ordinate is reckoned on the equator 
 from the vernal equinox, and is called the right ascension of the 
 star. It is reckoned from west to east from 0° to 360°, that is 
 in the direction opposite to that of the daily motion. Instead of 
 the spherical co-ordinates of right ascension and declination we 
 can introduce as before rectangular co-ordinates, by referring the 
 place of a body to three rectangular axes, of which the positive 
 axis of a is drawn perpendicular to the equator and towards the 
 north pole, while the axes of x and y lie in the plane of the 
 equator, and so that the positive axis of x is directed towards the 
 zero-point, the positive axis of y to a point having 90° of riglit 
 ascension. Putting then a for the right ascension, we have 
 
 x" = cos 8 cos a, y" = cos S sin a, z = sin S. 
 
 The co-ordinates a and B are constant for every star, but 
 in order to obtain thereby the place of a star on the visible 
 
 * Since, namely, the sun at that time is in the equator, and since the eqnator and 
 horizon as great circles bisect each other, it follows that the sun on these days is as 
 long abov6 as he is below the horizon. 
 
CELESTIAL, MERIDIAN, AND TEANSIT INSTUUMENT. 53 
 
 sphere of the heavens for any particular instant, we must know 
 the position of the vernal equinox in the heavens for that in- 
 stant. Sidereal time determines the place of the equinox, which 
 is equal to its hour-angle, and of which 24 hours are reckoned 
 to a sidereal day. It is O"" sidereal when the hoxir-angle of the 
 vernal equinox is zero and therefore the vernal equinox is on 
 the meridian ; it is 1" sidereal when the hour-angle of the vernal 
 equinox is one 24* part of the circumference or 15°; &c. This 
 is the reason why the equator besides being divided into 360' 
 is also divided into 2i^. Putting then $ for sidereal time, we 
 have always 
 
 — t = a, and therefore t = 6 — a. 
 
 If, for example, the right ascension = 190*. 20', the sidereal 
 time e = i\ then « = 229°.40'. 
 
 From the equation for t, we have when t = 0, d = a. Every 
 star therefore comes to the meridian or culminates at a sidereal 
 time which expresses also 'its right ascension in time*. If we 
 know therefore, the right ascension of an object, which at a 
 given instant is on the meridian, we have the sidereal time 
 corresponding to that instant. 
 
 Note. The co-ordinates of the third system we can find by 
 means of instruments of the second description, if we know the: 
 sidereal time. In any particular case the co-ordinates may be 
 determined also by means of instruments of the first description, 
 
 • The conrersion of arcs ioto time and vice versS, is of very frequent occnrrence. 
 To convert are into time we must divide by 15 ; we have then 
 
 (15a + 6) degrees, (15 c+<i) minutes, (15e+/>second3i 
 which is in time, 
 
 a hours, ih+c minutes, 4 i + e and J^ seconds. 
 
 For Example 239°. 18' .46" ,76=lSl>,Ux4 + l">, 3x4 + 3«+0,117» 
 = 15^.57 -.IS", 117. 
 On the other hand, to convert time into degrees we must multiply by 15 and after- 
 wards divide the minutes and seconds by 60. We then have 
 
 a hours, 4 6 + c minutes, 4 d^-)-e seconds 
 to convert into degrees, when it becomes 
 
 15 a+6 degrees, 15 c + rf minutes, and 16 e seconds. 
 
 Thus, 
 
 IS"" . 67°' . 15*. 117 = 225 + 14 degrees, 
 
 15+3 minutes and 46,75 seconds, 
 = 23^^18'.16",75. 
 
54 EEPEEENCE TO THE ECLIPTIC. 
 
 namely, by transits of stars over the meridian, since right ascen- 
 sions are determined by observation of transits and declinations 
 by observing the altitude of the star on the meridian, if the 
 elevation of the equator or of the pole at the place of observation 
 be known. The meridian circle which is an altitude circle 
 placed in the plane of the meridian serves for these observations. 
 Should the instrument not serve for determination of altitude, 
 but simply for observations of transits of stars over the meridian, 
 it is then simply an azimuthal instrument which is placed in the 
 plane of the meridian, and is therefore called a transit instrument. 
 By observing with such an instrument, together with a good 
 clock, the transits of stars over the meridian, their differences 
 of right ascension are found. Since the zero-point of right 
 ascensions cannot be observed directly, it is somewhat more diffi- 
 cult to determine absolute right ascensions. 
 
 4. The fourth system of co-ordinates is that one of which 
 the ecliptic is the fundamental plane. Great circles which pass 
 through the poles of the ecliptic, and so are perpendicular to it, 
 are called circles of latitude ; and the arc of such a circle which 
 is included between the ecliptic and the star, is called the lati- 
 tude of the star. This is positive, when the star is in the 
 northern of the two hemispheres formed by the ecliptic, and 
 negative when the star is in the southern hemisphere. The 
 other co-ordinate, the longitude, is reckoned on the ecliptic, and 
 is the arc between the circle of latitude of the star and the 
 vernal equinox. It is reckoned round from 0° to 360° in the 
 same way as the right ascension, and therefore contrary to the 
 daily motion of the heavens*. The circle of latitude, of which 
 the longitude is zero, is called the equinoctial colure. On the 
 other hand, that circle, of which the longitude is 90°, is called the 
 solstitial colure. The arc of this colure which is included be- 
 tween the equator and the ecliptic, is the obliquity of the ecliptic; 
 this is also equal to the arc of the great circle between the poles 
 of the equator and of the ecliptic. 
 
 The longitude is always, in the following pages, represented 
 by X, the latitude by /8, the obliquity by e. 
 
 * The longitudes of stars are also often indicated by tlie signs of the zodiac, each 
 of which includes 30°. 
 
 Thus C signs IS" = 195° of longitude. 
 
TKANSFOEMATION OP CO-OEDINATES. 55 
 
 If we express the spherical co-ordinates ^ and X by means of 
 a system of three rectangular uxes, of which the positive axis of 
 s is drawn perpendicular to the ecliptic and towards the north 
 pole, whilst the axes of x and y lie in the plane of the ecliptic, 
 and so that the positive axis of x is drawn towards the zero-point, 
 the positive axis of y to a point having 90° of longitude, we 
 have: 
 
 «a;"' = cos/8cos\, y" = cos/3sin\, a"' = sin^. 
 
 Note. Formerly there were instruments by which the sphe- 
 rical co-ordinates of the several systems could be observed, and 
 therefore also the longitudes and latitudes. At present these are 
 no longer in use. The co-ordinates of longitude and latitude 
 are never determined by direct observation, but are always ob- 
 tained by means of computation from co-ordinates of the other 
 systems. 
 
 II. On the transformation from one to another of the several 
 systems of Co-ordinates. 
 
 5, In order to reduce the place of a star, which has been 
 referred to the co-ordinate system of azimuth and altitude to the 
 co-ordinate system of hour-angle and declination, we have only 
 to turn the axis of z in the first system in the plane of x and s, 
 reckoning according to the positive directions of x and s, through 
 an angle 90° — <j} (where <f) represents the elevation of the pole)^ 
 since the axis of y is common to both systems, and we then 
 obtain from the formula (J^for the tfansfdrmation of co-ordinates 
 or also by .the formulae of spherical trigonometry, in the tri- 
 angle between the zenith, the pole, and the star* : 
 sin S = sin (/) sin h — cosip cos h cos A, 
 cos S sin < = cos h sin A, 
 cos Sco3t = sin hco3(f) + cos h sin ^ cos A. 
 Should we desire to have the formulae in a form more adapted 
 for logarithmic computation, we must put 
 
 Bmh = m cos M, 
 cos h cos A = msin M, 
 
 * The three sides of this triangle are respectivelyi 
 
 90" -h, 90° -a, 90" -0, 
 and these stand opposite to the angles 
 
 *, 180°- A, and the angle of the star. 
 
56" TEANSFOEMATION OP CO-OEDINATES. 
 
 from which we obtain 
 
 sin 8 = m sin {(f) — M), 
 cos S sin < = cos h sin A, 
 cos 8 cos t = m cos (^ — M). 
 
 These formulae give the required results without any ambi- 
 guity. Since then every part is foimd by means of sines and 
 cosines, we have only to observe the proper sign in order to de- 
 termine the right quadrant for the required part. The subsidiary 
 angles which have been introduced for the transformation of 
 such formulae have always a geometrical meaning, which is 
 found easily in every case. Geometrically considered, the intro- 
 duction of the subsidiary angle depends upon these circum- 
 stances, either that we divide the oblique-angled spherical tri- 
 angle into two right-angled triangles, or compose it out of two 
 right-angled triangles. In the present case we must imagine a 
 perpendicular let fall firom the star on the apposite side 90° — <^, 
 or that side produced, and then 
 
 tan h = cosA cotan M 
 
 where, according to the third of the formulae (12) in No. 8 of 
 the Introduction, M is the arc between the zenith and the foot 
 of the perpendicular; and again, according to the fii-st of the 
 formulae (12), m is the cosine of the same perpendicular, and 
 thus 
 
 sin A = cos Pcos M, 
 
 if we put P for the perpendicular. 
 
 Let ^ (the elevation of the pole) = 52°. 30'. 16",00, 
 h = 16°. U'. 44",0, A = 202°. 4'. 15",5. 
 
 The computation will then be as follows : 
 
 cos ^-9-9669481, »m sin Jf- 9-9493620, 
 
 cos A 9-9824139, mcosM 9-4454744, 
 
 sin A - 9-5749045, 
 
 i/=-72°.35'.54",61, 
 sin J/- 9-9796542, 
 
TRANSFORMATION OP CO-ORDINATES. 57 
 
 ^-ilf=125°.6'.10",61 
 
 sin (^- if) 9-9128171, cos S sin « - 9-5573184, 
 
 m 9-9697078, cosScosi- 9-728dll4, 
 
 sin S 9-8825249, 
 
 cos S 9-8104999, 
 
 < = 223° . 56' . 2",22, S = + 49° . 43' . 46",00 
 
 cos [^-M)- 9-7597036 cos t - 9-9189115. 
 
 6. Far more frequently, however, the converse case to this 
 is used, where it is required to reduce a position which has heen 
 referred to the co-ordinate system of hour-angle and declination 
 to the co-ordinate system of azimuth and altitude. We have 
 then again, according to formula (1), for the transformation of 
 co-ordinates, the following equations : 
 
 sin h = sin ^ sin S + cos ^ cos S cos t, 
 cos Ji sin A = cos S sin t, 
 cos h cos A = — cos^ sin S + sin ^ cos 8 cos t, 
 
 which we can again give in a more convenient form by intro- 
 ducing a subsidiary angle, 
 
 « 
 
 If we put namely ; 
 
 cos 8' cos * = m cos M, 
 sin S = m sin ilf, 
 
 then sin A = m cos {j> — M)f 
 
 cos Jisu\.A = cos S sin t, 
 cos hcosA = m sin (^ — M) ; 
 
 or also 
 
 . cos Mt&n t 
 too. A = 
 
 tan A = 
 
 sin (^- if)' 
 
 cos A ^ 
 tan(^-Jf) ■ 
 
 • Since the azimuth lies always upon the same side of the meridian as the hour- 
 angle, we can also by application of these last formulae never be in doubt as to the 
 quadrant in which it is to be taken. 
 
58 ZENITH-DISTANCE IN TEEMS OP HOUE-ANGLE AND DECLINATION. 
 
 If the zenith-distance alone he required, the following formulffi 
 are convenient, 
 
 cos a = cos ((^ — 8) — 2 cos ^ cos S sin'' - , 
 
 or 
 
 . ,s .„(<5 — o), . <s • ii 
 sin" - = sin ^ + cos ^ cos o sin" - . 
 
 If we now put 
 
 . rf)-S 
 « = sin ^-^— , 
 
 m = Vcos cos 8, 
 then 
 
 sin=-=«'(^l + ^sm-j, 
 
 or, if we put 
 
 m . t , . 
 — sin - = tan \, 
 n 2 
 
 . z n 
 sin-: 
 
 2 cos\ 
 
 If sin X be greater than cos X, then the formula 
 
 . s m . t 
 
 sm- = -^^ sm- 
 
 2 sm \ 2 
 
 is preferable for computation. 
 
 In the formula for computation of w we must moreover, as we 
 shall see further on, use ^ — 8 for stars which culminate south of 
 the zenith, and 8 — (^ for stars which culminate north of the 
 zenith. 
 
 Employing now the Gaussian formulss for the triangle between 
 the zenith, the pole, and the star, we have, if we denote the 
 angle at the star by p, 
 
 s . A-p . t . rf) + 8 
 cos - sm —^ =sm - sm ^^^ , 
 
 z 
 
 A-p t 0-8 
 
 COS - COS — ^ = COS - COS , 
 
 2 Z A A 
 
CALCULATION OF ZENITH-DISTANCE AND AZIMUTH. 59 
 
 Bin - sm— ^ = sin - cos ^^-^ , 
 
 . s A + p t . A-S 
 
 sin - cos — —^ = cos - sm ^ ^ . 
 
 2 2 2 2 
 
 Beckoning now the azimuth from the north point, as would be 
 done in the case of the pole star, we must put 180" — A for A in 
 these formulae, arid we obtain 
 
 a . p+A t S-d> 
 
 cos - sm-t— - — = cos - cos ^ , 
 
 z P + A . t . S + rf) 
 cos - cos^^-y- = sm - sm —^ , 
 
 . s . p-A t . B-4> 
 
 sm - sm-f— - — = cos - sin ^ , 
 
 2 2 2 2' 
 
 . s p-A . t h + d) 
 sm - cos *— - — = sm - cos ——J- . 
 
 The case often occurs in which we have for a ^ven latitude 
 to make a great many such transformations, and for which, for 
 the sake of more ready computation, tables are constructed before- 
 hand*. For this case the second transformation, which is given 
 in No. 6 of the Introduction, for the three fundamental equations, 
 is especially convenient. We easily obtain formulae suitable 
 for the present case, hj putting respectively in the equations 
 there given 
 
 ^0"-%, 90' -S, 90°-^, 180°-^, and t, 
 
 instead of 
 
 a, h, c, B, and A. 
 
 For the sake of perspicuity we will now repeat these trans- 
 formations together with the equations now arrived at: the 
 equations are 
 
 * If, for example, we wish to obtain tbe places of stars of which the positions are 
 given In right ascension and declination, by means of an instrument on which we can 
 only read altitudes and azimnths, it will then be preTionsly necessary to compute the 
 hour-angle from the right ascension and the sidereal time. 
 
60 USE OP AUXILIARY ANGLES. 
 
 (a) sin h = sin<j) sin S + cos ^ cos S cos t, 
 
 (b) cos hsmA = cos S sin t, 
 
 ( c) cos A cos -4 = — cos ^ sin S + sin ^ cos S cos «. 
 
 Denoting then by A^ and' §„ tkese values of -4 and S, which, 
 when substituted in the previous equations, give A = 0, we 
 have 
 
 (d) = sin <j> sin S„ + cos ^ cos 8, cos t^ 
 
 (e ) sin -4„ = cos S„ sin i, 
 
 (/) cos -4„ = — cos ^ sin 8„ + sin ^ cos S„ cos <. 
 
 Multiplying (/) by cos and subtracting from it equation (J) 
 after first multiplying it by sin ^, and, in addition, multiplying 
 equation (/) by sin ^ and adding to it equation (d), previously 
 multiplied by cos 0, we obtain 
 
 cos A^ cos ^ = — sin Sj 1 
 
 co&-4„sin^ = cosS„cos« I {A). 
 
 sin A^ = cos S„ sin < J 
 Put then 
 
 sin ^ = sin 7 cos B, \ 
 
 cos ^ cos < = sin 7 sin B, > (B), 
 
 cos <j)smt = cos 7, J 
 
 we thus obtain from equation (d) 
 
 0=sin7sin(8„ + £) 
 or K = -B, 
 
 and from (a) 
 
 sin h = sin 7 sin (S + J5), 
 
 In addition, we obtain, by subtracting the product of equa- 
 tions (c) and (e) from the product of equations (b) and (/), 
 
 cos h sin {A - A^) = cos ^ sin < sin (S - SJ = cos 7 sin (S + B), 
 
 and similarly, if we add the product of equations (5) and (e) 
 and that of equations (a) and {d) to the product of equations 
 (c) and (/), 
 
 cosAcos {A- A,) = cos8cosS„ sin't + sinS sinS„+cosScosS„ cosV, 
 = cos(S-8„) = cos^8 + J5). 
 
USE OF AUXILIARr ANGLES. 61 
 
 This system of formulae is thus when perfectly formed 
 
 sin ^ = sin 7 cos B ^ 
 
 cos ^ cos « = sin 7 sin 5 \ (1), 
 
 cos ^ sin t = cos 7 J 
 
 sin B = cos A„ cos ^ "j 
 
 cos^cos< = cos^„sin0 > (2), 
 
 cos Bsint= sin ^„ J 
 
 sin h = sin 7 sin (S + 5) | 
 
 cosAcos(^-^J = cos(S + 5) > (3). 
 
 cos h sin {A — A„) = cos 7 sin {B + B)i 
 
 Putting Z> = sin 7, C = cos 7, A — A^=u, 
 
 these formulae become 
 
 tan B = cot <}> cos t, 
 tan -4„ = sin (f> tan <, 
 sin A = i) sin {B+8), 
 tan u = tan (5+ S), 
 
 and Z) and (7 are then the sine and cosine of an angle 7, which 
 is given by the equation 
 
 cot 7 = sin 5 tan t = cot<f) sin AJ^. 
 
 These are the formulas communicated by Gauss in " Schu- 
 macher's Auxiliary Tables lately edited hy Warnstorff, p. 135,' 
 &c." Arranging now the quantities I), G, B and J^m Tables, 
 of which the argument is t, the computation of the altitude 
 and azimuth fi-om the hour-angle and the declination, is 
 reduced to the computation of the preceding formulae: 
 svo.h = D sin {B+S), 
 tanM = Ctan(5+S), 
 A = A„ + u. 
 
 * We have in fact, 
 
 cot (^ sin ^o = — sin 5^ tan t 
 = sin J! tan t. 
 
62 GEOMETRICAL MEANING OP AUXILIAET ANGLES. 
 
 In Warnstorff's Auxiliary Tables we find such a taUe 
 calculated for the latitude of the Altona Observatory. In 
 addition, it is only necessary to calculate these tables from 
 < = to t = &\ Then it follows from the equation tan A^ 
 = sin (f) tan t, that A^ and t lie always in the same quadrant, 
 and that for an hour-angle =12" — « we have only to take 
 180" — ^. It further results from the equation for B that this 
 angle is negative when t is > 6'' or > 90°, and that we must 
 use the value — B for an hour-angle = 12" - 1. The quantities 
 
 C=cos<f} sin t, and D^ = sin' ^ + cos" tj) cos" t, 
 
 are on the contrary not altered, if we put 180" — t instead of t 
 in these expressions. If t lies between 12" and 24", we have 
 to continue the calculation with the complement of f to 24", 
 and to take afterwards for the determination of A its com- 
 plement to 360°. 
 
 It is now easy to find the geometrical meaning of the 
 subsidiary angles. Since S„ is that value of B which makes 
 h = in the first of the primary equations, therefore S^ is 
 the declination of the point in which the hour-circle pass- 
 ing through the star cuts the horizon, and similarly A^ is 
 the azimuth of that point. Further, since B= — S^, B + S is 
 the arc 8F, (fig. 1*) of the hour-circle produced to the horizon. 
 Considering then the right-angled triangle FOK, which is 
 formed by the horizon, the equator, and the side FK=B, 
 we have from the sixth of the formulae (12) in the Introduc- 
 tion, since the angle at is 90" — 0, 
 
 sin ^ = cos B sin OFK. 
 
 But since also sin<^ = Z>cos5, D is the sine and con- 
 sequently G the cosine of the angle OFK. Lastly, as is easy 
 to perceive, the arc FH= A^ and the arc FG = u. 
 
 We thus find the formulae previously given, through the 
 consideration of the three right-angled triangles PFH, OFK, 
 and 8FG. 
 
 * In this figure P is the pole, Z the zenith, OH the horizon, OA the equator, and 
 S tlie star. 
 
FARTHER USE OF AUXILIARY ANGLES. 63 
 
 The first triangle gives 
 
 tan A^ = tan < sin </> ; 
 the second 
 
 tan B=cot([) cos t, 
 cot 7 = sin ^ tan < = cot ^ sin A^ ; 
 and lastly, the third 
 
 sin ^ = sin 7 sin (5 + S), 
 tan M = cos 7 tan {B+S). 
 
 The same subsidiary quantities can now also he employed 
 for the solution of the converse problem treated of in No. 5, 
 to calculate from the altitude and azimuth of a star its hour- 
 angle and declination. We have namely in the right-angled 
 triangle SLK, if we put LG = B, LK=u, AL = A„ and the 
 cosine of the angle SLK= O and the sine = D, 
 
 
 Ctan {Ji—B)= tan u, 
 
 
 Z>sin(A--B)=sinS, 
 
 and 
 
 t = A,-u; 
 
 where now 
 
 
 tan-B = cot (^cos^, 
 tan A„ = sin (f> tan A, 
 
 and I) and O are the sine and cosine respectively of an angle 
 7, which is given by the equation 
 
 cot 7 = sin 5 tan A. ' 
 
 We have also for the calculation of the subsidiary quantities 
 the same formuliB as before, with this difference only, that every- 
 where A occurs instead of t, and hence we can employ the same 
 Auxiliary Tables as before, if we only now take as argument 
 the azimuth converted into time, 
 
 7. The tangent of the angle 6, which Gauss designates by 
 JE, will also serve to calculate the angle at the star in the triangle 
 formed by the pole, the zenith, and the star. This angle, formed 
 
64 . PARALLACTIC ANGLE. 
 
 by the vertical and declination circles, which is called the Paral- 
 lactic angle, is very often required. If we use the Auxiliary 
 Tables mentioned before, in which the quantity E is tabulated, 
 we obtain this angle, which shall be designated by^, by means 
 of the convenient formula, 
 
 E 
 tan p = , p , stN > 
 
 ■^ cos [JD + 0) 
 
 as is immediately seen if we apply to the right-angled triangle 
 SGF, (fig. 1) the fifth of the formulae (12)r in No. 8 of the 
 Introduction. If, on the contrary, we have not the Auxiliary 
 Tables, we obtain by means of the formulae of spherical trigono- 
 metry from the triangle 8PZ, 
 
 cos h sinp = cos ^ sin t, 
 
 cos h cos J? = cos S sin (f> — sin B cos <f> cos t, 
 
 or if we put 
 
 cos <j} cos t = n sin N, 
 s'm<p = n cos N, 
 
 for more convenient logarithmic computation, 
 cos h sin^ = cos <ji sin t, 
 cos h cosp = n cos (S + N). 
 
 The parallactic angle is used, amongst other cases, when 
 we wish to determine the effect which a small change in the 
 azimuth and altitude has upon the hour-angle and declination. 
 We obtain namely, if we apply the first and third of formulse 
 (13), in No. 9 of the Introduction to the triangle between the 
 pole, the zenitlf, and the star, 
 
 dS = cos p .dh + cos t. d^ + cosh sin^ . dA, 
 
 cos S.dt = — sinp.dk + sin t sin 8. d(f) + cos h cos p . dA, 
 and similarly 
 
 dh = cosp .dB — cos A. d<l} — cos B sinp . dt, 
 cos h . dA = sinp. dB — sin A sin h. dj) + cos S cosp. dt. 
 
 8- In order to transform the co-ordinates of right ascension 
 and declination into co-ordinates of longitude and latitude we 
 
TRANSFORMATION OF CO-ORDINATES. 65 
 
 have only to turn the axis of s"* in the plane of y'a" in the posi- 
 tive directions of y" and z" through the angle «, which is equal to 
 the obliquity of the ecliptic. Then we obtain from the formulae 
 (2) in No. 1 of the Introduction, since the .axes of x' and as'" 
 coincide in the two systems : 
 
 cos /8 cos \ = cos 8 cos a, 
 
 cos /3 sin \ = cos 8 sim a cos e + sin 8 sin e, 
 
 sin j8 = — cos 8 sin a sin e + sin 8 cos e. 
 
 These formulae we can also deduce in a different way by 
 consideration of the triangle between the pole of the equator, 
 the pole of the. ecliptic, and the star, in which the three sides 
 are 90°— 8, 90° — yS and e, and the angles respectively opposite to 
 them are 90° — X, 90° + a, and the angle at the star. 
 
 In order to adapt the above formulae for convenient logarithmic 
 computation, we introduce the auxiliary quantities 
 
 Jl/sin^=sin8 1 
 
 JfcoSjJ7'=cos8 sina) 
 
 by which means the three equations are changed into the 
 following, 
 
 cos ^ cos \ = cos 8 cos a, 
 
 cos )8 sin \ = ilf cos {N— e), 
 sin/3 = ifsin(iV"-6), 
 
 or, if we desire to express all the quantities by means of tangents, 
 and substitute for M its value 
 
 cos 8 sin a 
 
 cos.^ ' 
 
 they are changed into the following : 
 tan 8 
 
 tan^=- 
 
 am a 
 
 . ^ cos(^-e) . 
 
 tanX = '^ ,. tana 
 
 cosiV 
 
 tan j8 = tan {N — e) sin X. 
 
 * See No. 3 of this Section 
 
 (^). 
 
 B. A. 
 
66 COMPUTATION OF LONOITUDE AND LATITUDE. 
 
 ■ The primary formtilse give a and 8 without any ambiguity ; 
 hut, if the formulas (SJ are used for computation, it may be 
 doubtful in which quadrant the angle \ is to be taken. But it 
 follows from the equation 
 
 cos /8 cos X = cos B cos a, 
 
 that we must take the angle \ always in that quadrant which at 
 once satisfies the sign of tan \, and besides fulfils the condition 
 that cos a and cos X must have the same sign. 
 
 As a verification of the computation we can besides employ 
 the equation 
 
 cos {N— e) _ cos /3 sin X 
 
 cosJV cosSsina \il>^ 
 
 which arises from the division of the equation 
 
 cos /8 sin \ = Jf cos {N— e) 
 
 by 
 
 cos S sin a = Jf cos N. 
 
 The geometrical signification of the auxiliary quantities is 
 easily discoverable. N is the angle which the great circle con- 
 necting the first point of Aries with the star forms with the 
 equator, and ilf the sine of the arc of the great circle. 
 
 Eocample. 
 
 Let a = 6''.33'.29",30 8 = - 16°.22'.35",45 
 
 e = 22''.27'.31",72. 
 
 Then the computation of the formulfe (SJ and (c,) gives 
 cosS 9-9820131 tana 9"0605604 
 
 tan 8 -9-4681562 cos{N-e) _^.q^^^q 
 
 COSiV 
 
 sin a 9-0577093 X= 359''.17'.43",91 
 N= - 68°.45'.41",88 
 
 e = + 23°.27'.31",72 tan (iV- e) 1-4114653 
 
 N-e = - 92°.13'.13",60 sin \ -8-0897293 
 
 cos {N- e) - 8-5882086 ^S = - 17°.35'.37",53 
 
 cosiV" 9-5590069 cos /3 = 9-9791948 
 
APPLICATION OF THE GAUSSIAN FORMULA. 67 
 
 COS /3 sin \ = - 8-0689241 
 cos S sin a = 9-0397224 
 
 ■9-0292017 
 
 Applying to the triangle between tlie star, the pole of the 
 equator, and the pole of the ecliptic, the Gaussian formulee, we 
 obtain, indicating the angle at the star by 90 — ^*, 
 
 sin (45»- f) sin ^ = cos (45»+ 1) sin (45»-i±i) , 
 sin (450- 1) cos ^ = sin (45»+ 1) cos (^45°- 1^) , 
 
 cos (45°— ^ sin 
 
 = sin(45°+|)sin(45»-l^), 
 
 2/ 2 
 
 cos (45°-f) cos:^ = cos (45»+|) cos (46»-i±^) , 
 
 formulse, which are particularly usefal, if we desire to determine 
 at the same time the quantities X and ^ as well as the angle 
 90" -K 
 
 Note. Encke has in the Jahrbuch for 1831, given tables which, for an 
 approximate computation of the longitude and latitude firom the right 
 ascension and declination, are extremely useful. They depend upon the 
 second of the transformations of the three fundamental equations given in 
 No. 6 of the Introduction, similarly to the tables specified in No. 6 of this 
 section. 
 
 9. For the converse case, when it is required to transform 
 the co-ordinates of a star referred to the ecliptic into co-ordinates 
 referred to the equator, the formulae are exactly similar. We 
 obtain then by means of formulae (1) for the transformation 
 of co-ordinates or by means of the spherical triangle previously 
 treated of, 
 
 cos B cos a = cos /S cos \, 
 cos S sin a = cos ^ sin \ cos e — sin ^ sin e, 
 sin S = cos ;8 sin \ sin e + sin /8 cos e. 
 
 * Gauss, Theoria mains, page 64. 
 
 5—2 
 
68 COMPUTATION OF E.A. AND DECLINATION FROM LONG. AND LAT. 
 
 We obtain also the same equations by replacing, in the 
 three fundamental equations in No. 8, j8 and \ by S and a, 
 and taking the angle e negatively. By this means we find 
 from the formulae (J,), 
 
 tan/3 
 
 tan^= 
 
 sinX 
 
 . cos (N+e) . ^ 
 
 tan a = ^ — ttt-^ tan \ 
 
 cosJv 
 
 tan 8 = tan {N+ e) sin a, 
 
 and, from (Cj), the check equation 
 
 cos (■^"4- e) _ cos S sin a 
 cos^ ~cos;Ssin\' 
 
 where now N signifies the angle which the great circle con- 
 necting the star with the first point of Aries makes with the 
 ecliptic. 
 
 The Gaussian equations give finally for this case : 
 sin (4.5" ^ i S) sin i {E +a) =sin (46" + i \) sin Ld" - 1 (e + /3) ■ , 
 
 in (45»-is) cosi(^+a) = cosf45'' + i\) cos J45''-i(e-;S)|, 
 (45»-ls) sini(^-a)=cos(45'' + ix) sin |45»-1 (e-^)} , 
 
 ^45--^ a) cos i (^-a)=sin (45°+ ^ \) cos |45» - i (e + ;8)| . 
 
 It is not necessary to introduce an example for this case, 
 since the formulae are precisely similar to the former. 
 
 Note. For the sun, which always moves in the ecliptic, the expressions 
 become simpler. Denoting namely the sun's longitude by Z, its right ascen- 
 sion and declination by A and 2), we obtain, 
 
 tan ^= tan Z cose, 
 sin J!5=sinZ sint, 
 or in addition, 
 
 tanZ>=tan esin^. 
 
 sm 
 
 cos 
 
 cos 
 
VABIATIONS OF LONG. AND LAT., E.A, AND DECLINATION. 69 
 
 10. The angle at the star in the triangle formed by great 
 circles joining the pole of the equator, the pole of the ecliptic, 
 and the star, which is contained between the circle of declination 
 and the circle of latitude, is found immediately in terms of \ and 
 ^ or of a and B, by employing the Gaussian formulae for the com- 
 putation of these quantities, since, if this angle be denoted by i], 
 71=90° — E. But if this angle is required, without the necessity 
 of computing by the Gaussian formulae, it is found by means of 
 the equations : 
 
 cos /3 sin 17 = cos a sin e, 
 
 cos j8 cos 1; = cos 6 cos S + sin e sin S sin a, 
 or, 
 
 cos S sin t) = cos \ sin €, 
 
 cos S cos 17 = cos 6 cos ^ — sin e sin /8 sin \ ; 
 
 or, if we put 
 
 cos e = TM cos M, 
 sin e sin a = m sin M, 
 
 or, 
 
 cos e = m cos N, 
 
 sin e sin \ = w sin N, 
 
 by means of the equations 
 
 cos ;8 sin 9? = cos a sin e, 
 
 cos ^ cos ri = m cos {M— S), 
 or, 
 
 cos )8 sin 17 = cos X sin e, 
 
 cos ^ cos »? = w cos {N+ /8). 
 
 This angle again is needed, when it is required to investigate 
 the effect which small changes in the quantities \, /3, and e will 
 produce on a and S, and vice versd. We obtain, namely, by em- 
 ploying for the triangle under consideration the first and third 
 of the formulae (13) in No. 9 of the Introduction, 
 
 cZ/3 = cos ^7 <^ — cos S sin 97 (Za — sin \ de, 
 cos ^ d\ = aiar] dB + coaS cos 17 <Za + cos \ ain^de, 
 and for the converse case, • 
 
 dS = co6r}dfi + CQafi smTjdX + ainade, 
 cos8 da = —ainr]d^ + cos^ cos 17 <?\ - cps a sinSde. 
 
70 FUETHEE TEANSFOEMATION OP CO-OEDINATES. 
 
 11. For the sake of completeness we will in addition give 
 the, formulse for the transformation of the first system of co-ordi- 
 nates into the fourth ; although these are never employed. 
 First with reference to the plane of the horizon : 
 X = cos A cos h, 
 y = smA cos h, 
 z = sin h. 
 
 If the axis of x be turned towards the positive direction, 
 of the axis of z in the plane of xz through an angle 90° — ^, we 
 obtain the new co-ordinates 
 
 as' = £c sin <|) + » cos ^, 
 
 z' = z sin^ — x cos ^. 
 
 If now the axis of x' be turned in the plane of cc'y', which is 
 the plane of the equator, through an angle @, so that the axis 
 of x" is in the direction of the first point of Aries, we obtain, 
 considering that the positive axis of y" is in the direction of the 
 ninetieth degree of right ascension, and that the hour-angle and 
 the right ascension are reckoned in opposite directions, 
 
 x" = x' cos @ + 2^' sin ©, 
 
 ^y" = y' cos @ — «' sin @, 
 
 z" = z'. 
 
 Turn lastly the axis of y" in the plane of y"z" towards the 
 positive direction of the axis of z" through an angle e, and we 
 then obtain 
 
 y'" = y" cos 6 + z" sin e, 
 z'" =—y"sme + z" cos e, 
 
 and, since we have besides, 
 
 x" = Cos /3 cos \, 
 
 y'" = cos /3 sin \, 
 
 a'" = sin A 
 we shall be able, through elimination of x, y', z, and x", y", z", 
 to express \ and yS immediately in terms of A, h, ^, ©, and e. 
 
TIME OP A star's EISINO OR SETTING.^ 71 
 
 III. Particular Phenomena of the Daily Motion. 
 
 12. In No. 6 was found the equation 
 
 sin A = sin <^ sin 8 + cos ^ cos S cos t. 
 
 When the object is in the horizon, A = 0, and hence we 
 have = sin^sinSH-cos^cosScos*o, 
 or cos «„ = — tan <^ tan S (a). 
 
 By means of this formula is found for a given latitude ^ 
 the hour-angle at rising and setting of a star whose declination 
 is S. The value of this hour-angle taken absolutely is termed 
 the semi-diurnal arc of the star. If the sidereal time be known 
 at which the star passes the meridian, or its right-ascension, the 
 sidereal time of its rising or setting can be computed by sub-- 
 tracting from the right ascension or adding to it the absolute 
 value of t^. It follows moreover from this, that, on the equator 
 only, jvhere = and the semi-diuqjal arc for all stars is equal 
 to 90°, all stars which rise at the same time set also at the same 
 time. 
 
 Example. It is required to calculate at what time the star 
 Arcturus rises and sets at Berlin. For the beginning of the 
 year 1848 
 
 a = 14^ 8", 7, S = + 190. 58',5 ; 
 in addition 
 
 = 52". 30'. 16". 
 
 We have therefore 
 
 tan S 9-56048 
 tan^ 0-11.509 
 
 ^„ = 118''. 16',8 
 = 7". bS^. V. 
 
 Arcturus therefore rises at e"". 15"°,6 sidereal time and sets at 
 22^ 1°',8. 
 
 If h be positive, and the star be therefore north of the equa- 
 tor, cos «„ will, for places in north latitudes, be negative ; <„ is 
 therefore greater than 90°, and the star remains a longer time 
 above the horizon than bebw it. For stars with south declina- 
 tion, on the contrary, t^ will be less than 90°, and, for places in 
 the northern hemisphere, these remain for a shorter time above 
 the horizon than below it. 
 
72 TIME OP A stab's EISING OE SETTING. 
 
 In the -southem hemisphere, where <j> has negative values, 
 the converse holds good, since there the diurnal arc for southern 
 stars is greater than 12 hours. If ^ = 0, <„ will, for every value 
 of 8, he equal to 90°; and therefore at the equator all stars remain 
 for an equal time ahove and below the horizon. If 8 = 0, t„ will, 
 for every value of <f), he equal to 90°. Thus, for equatoreal 
 stars, the time during which they are above the horizon is, for 
 all places on the earth, equal to the time during which they are 
 beneath the horizon. 
 
 Thus, if the sun be north of the equator, then, for the north- 
 ern hemisphere of the earth, the daya are longer than the nights, 
 and conversely, when he is south of the equator. If however 
 the sun be in the equator, then, for all places of the earth, 
 day and night are equal. On the equator this is always the 
 case. The value of i„ will besides be possible only as Igng as 
 tan ^ tan S is less than 1. Thus, for a star to set at a place in 
 latitude 0, it is necessary that tan S be less than cot <f), or S less 
 than 90° - ^. If 8 = 90° - 4>, then is i„ = 180°, and the star only 
 touches the horizon at its lower culmination. If S be greater 
 than 90° — ^, the star never sets ; if, on the contrary, the south 
 declination be greater than 90° — <p, the star never ascends above 
 the horizon. 
 
 Since the declinations of the sun lie always between — e 
 and + e, it follows that all places of the earth for which the sun 
 for only one day of the year does not rise or set, must have a 
 north or south latitude equal to 90° — e or to 66^°. These places 
 lie in the two polar circles. Places lying still nearer to the pole 
 of the earth have the sun in summer a still longer time without 
 interruption above the horizon, and in winter beneath it, the 
 nearer they lie to the pole. 
 
 Note. The equation for the hour-angle at rising and setting may still 
 be brought into another form. By subtracting equation 1^ from unity 
 and also adding it to unity, we get, by division of the two new equations, \ 
 
 *™ 2'''~cos(0 + S)- 
 This equation also shews that «„ is only possible as long as 
 
AMPLITUDES AT EISING AND SETTING. 73 
 
 COS {<}> — S) and cos (^ + S) are positive, and that therefore those 
 stars only whose south or north declinations are less than 90° — ^, 
 can for this latitude rise or set. 
 
 13. To find the point of the horizon where a star rises or 
 sets, we have only, in the equation given in No. 5, 
 
 sin S = sin ^ sin h, — cos ^ cos h cos A 
 
 to make k = 0, whereby we obtain 
 
 A sinS 
 C0S^j = T (b). 
 
 The negative value of A^ is the azimuth of the star at its 
 
 rising, the positive value is the azimuth at its setting. The 
 
 distance of the star from the true east or west point is termed 
 
 the morning or evening anvplitude of the star. Denoting this 
 
 by -4j, we have 
 
 ^,, = 90° + ^,; 
 
 and thus 
 
 ■ A sin S , , 
 
 sm^, = T (c), 
 
 ' cos ^ " 
 
 where A^ is positive when the point of rising or setting lies on 
 the north of the east or west point, negative when it lies on the 
 south. 
 
 The formula (c) for the morning and evening amplitude can 
 again be also given in another shape, by writing 
 
 1 + sin A^ _ sin ■<^ + sin S 
 1 — sin J[j sin i|r - sin S ' 
 
 where i^ = 90° — 0. Hence we obtain 
 
 tan-!-^ 
 tan"' 
 
 =(45° -4') = 
 
 tanX__ 
 
 For Arcturus we accordingly obtain, with the preceding 
 values of S and ^, 
 
 A^ = 34°. 8',3. 
 
74 MEEIDIAN ZENITH DISTANCES. 
 
 14. By putting in the equation 
 
 sin h = sin ^ sin 8 + cos <f> cos 8 cos t 
 
 1 — 2 sin" - 1 for cos t, we obtain 
 
 sin h = cos (<^ — 8) — 2 cos ^ cos 8 sin' - 1. 
 
 From this it is seen, in the first place, that, to equal values 
 of t on both sides of the meridian, equal altitudes correspond. 
 In. addition, since the second term is always negative, h has, 
 for < = 0, a maximum value, and this maximum, or the altitude 
 of the star at its upper culmination, is given from the equation 
 
 - sin h = cos {(j> — B) {d). 
 
 For the lower culmination, or for t=180'', h will, on the 
 contrary, be a minimum, as will be most easily seen by intro- 
 ducing 180° + 1' for t, where t' is reckoned fi-om the northern 
 part of the meridian. We shall then have, namely, 
 sin h = sin (p sin 8 — cos <p cos 8 cos t' ; 
 
 or, if again we put 1 — 2 sin'- < for cos t', 
 
 sin h = cos (180° — ^ - 8) + 2 cos ^ cos 8 sin" - i. 
 
 Since both terms of the right side of the equation are now 
 positive, sin li, and consequently A, must, at the lower culmina- 
 tion of the star, be a minimum, and, for its value, 
 
 sinA = cos(180°-^-8) (e). 
 
 From the equation {d) it follows, that 90° — li, or the zenith- 
 distance of the star at its upper culmination, is either ^ — 8, or 
 8 — <f>. But, since the zenith-distance must always be positive, 
 we must, so long as the star culminates on the south side of the 
 zenith, that is so long as 8 is less than <j>, take for the zenith- 
 distance ^ — 8. If however the star culminates on the north 
 side of the zenith, where 8 must be greater than (/>, we must 
 take 8 ^ </> for the zenith-distance. For the zenith-distance at 
 the lower culmination we obtain from equation (e) 
 a = 180° - ^ - 8. 
 
 To bring all the three cases under one algebraical formula, 
 we take as the common expression for the zenith distance of a 
 
GEEATEST ALTITUDES OP PLANETARY BODIES. 75 
 
 star at its passage over the meridian 
 
 ^=^B-^ (/). 
 
 We must then consider south zenith distances as negative, and 
 at the lower culmination take 180° — S instead of S, or we must, 
 in the last case, reckon S from that point of the eq^uator which 
 cuts the visible meridian. 
 
 The declination of a Lyrse is 38°.39', and thus, for the latitude 
 of Berlin, S — <j) = — 13°.51'. The star a. Lyras passes thus at its 
 upper culmination for Berlin south of the zenith at a distance 
 of 13°.51'. In addition 180° — ^ — S, or the zenith distance at the 
 lower culmination, is equal to 88''.51'.* 
 
 15. The greatest altitude of a star takes place on the meri- 
 dian, only when its declination during the time of its continu- 
 ance about the horizon does not change. If, on the contrary, 
 the declination is variable, the body attains its greatest altitude 
 off the meridian. By differentiating the formula 
 
 cos s = sin ^ sin 8 + cos (p cos B cos t, 
 
 regarding z, S, and t as variable, we obtain 
 
 — sin sds = [sin ^ cos S — cos <f) sin S cos t] dS, 
 
 — cos ^ cos S sin t dt, 
 
 and hence, for the case when « is a maximum or dz = 0, 
 
 sin t = -^ [tan ^ — tan S cos t\. 
 
 From this equation is found the hour-angle of the body at 
 the time of its greatest altitude. -^ is the ratio of the change of 
 declination to the change of hour-angle, so that if, for example, 
 dt represents a second of arc, ^ is the change of declination 
 
 in -=• of a second of time. Since this ratio for all the heavenly 
 15 
 
 * [In the preceding remarks zenith-distances are reckoned positively towards the 
 North, but it is more usual as well as more convenient to measure positively towards 
 the South, as well as to measure azimuths from the South point of the horizon — 
 Tbanslatob.] 
 
76' CHANGE OF ALTITUDE OF THE STARS BT THE DIUENAL MOTION. 
 
 bodies is small, we may replace sin t by the arc and put unity 
 for cos t, and we then obtain for the hour-angle corresponding 
 to the greatest altitude 
 
 *. ^S r, . ^ sn 206265 , . 
 
 where -j- is the change of declination in a second of time, and t 
 
 is expressed in seconds of time. This hour-angle t must then 
 always be added algebraically to the time of culmination, to 
 obtain the time of the greatest altitude. 
 
 If the body culminate south of the zenith, and if it be 
 
 approaching the north pole, -3- is positive, and therefore, when <^ 
 
 is positive, the greatest altitude takes place after the culmination ; 
 if, on the contrary, the declination be diminishing, the greatest 
 altitude precedes the culmination. The converse takes place 
 when the body culminates between the zenith and the polef. 
 
 16. Differentiating the formula 
 
 sin ^ = sin ^ sin S + cos ^ cos S cos t 
 
 with respect to h and t, we obtain for the change of altitude 
 of a body, 
 
 cos ^ TT = — cos (^ cos S sin t, 
 
 or 
 
 -^ = — cosS sin^ [h), 
 
 where cos h sin j? = cos sin t, according to No. 7 of this sec- 
 tion. 
 
 In many cases also the second differential coefficient is needed. 
 
 It is 
 
 d^h ~. dp 
 
 ^=-cos8cos^^. 
 
 • [206265 = -;— j7, — Tbanslatob.] 
 
 t [This may take place for European latitudes, but even then but rarely, in the 
 motions of comets. — Tkansiatok.] 
 
PRIME YEETICAL. 77 
 
 Differentiating now the equation 
 
 sin ^ = sin A sin S + cos h cos S cos j?, 
 considering h and^ as variable, we obtain 
 
 = [cos A sin S — sin h cos S cos j>] -^ — cos A cos 8 sinp-^ , 
 
 dp _ cos ^ cos A dh 
 dt cos h cos S sinp ' dt 
 
 _ cos ^ cos ^ 
 cos A 
 
 Substituting this expression for -^ in the equation -^ , we 
 obtain 
 
 or. 
 
 d k cos S cos <6 > , ^ 
 
 ^3-== : ^ cos -4 COS M (t). 
 
 In the same manner we obtain 
 
 ds , s • 
 
 — = + cos o sm^, 
 
 <^a cos S cos (i . ,,, 
 
 -rT = -\ : cos A cosp (k). 
 
 df sms ^ ^ ' 
 
 17. Since cos S sin^ = cos ^ sin A, we have also 
 
 dh 
 dt 
 
 dh . ■ A 
 
 -jT =~ cos ip sm A. 
 
 We shall thus have -^ = 0, and h a maximum or minimum, 
 at 
 
 when A=0, and so the star is on the meridian, and the second 
 
 differential coefficient shews that A is a maximum when -4 = 0, 
 
 and a minimum when A = 180°. 
 
 In addition -j- will ^^ ^ maximum, if sin ^ = + 1, or ^ =90° 
 at 
 
 or 270°. The altitude of a star is thus shewn to change most 
 
 rapidly at the instant when it passes across the vertical circle, 
 
 of which the azimuth is 90° or 270°. This vertical circle is called 
 
 the^Mwe vertical. 
 
 To find the time of passage of the star across the prime 
 vertical, as well as its altitude, we have only, in formula No. 6 
 
78 TIME OF TRANSIT OVEE THE PEIME VEETIOAL. 
 
 of this section, to make A = 90°, or to solve the right-angled 
 triangle between the star, the zenith, and the pole, and we 
 obtain 
 
 tan 6 
 
 cos t = 
 sin A = 
 
 tan (f> 
 
 sin S 
 
 W. 
 
 sm<f> 
 
 If S be greater than <^, cos t will be impossible, and therefore 
 the star cannot transit the prime vertical, but will culminate be- 
 tween the zenith and the pole. If 8 be negative, cos t will be 
 negative, but, since in northern latitudes the hour-angles of 
 southern stars are always less than 90° as long as they are above 
 the horizon, they cannot transit the visible portion of the prime 
 vertical *. 
 
 For Arcturus, with the latitude of Berlin, we obtain 
 
 ^ = 73°.48',5 = 4^55".14' , '" . 
 
 and 
 
 k = 25°.30',2. 
 
 Thus, for Berlin, Arctums transits the prime vertical before 
 the culmination at _ 9''.lf".5', and after it, at 19'*.3™.9^ sidereal 
 time. If the hour-angle be nearly equal to 6, t is found by 
 means of the cosine and h by the sine or tangent very incor- 
 rectly. We obtain in this case, from the formulse for cos t, in 
 the same manner as before, * 
 
 tan^^i sin (0-8) 
 ^'^ 2 ^ - sin (<^ + 8) ' 
 
 andj for the computation of the altitude, we must take the follow- 
 ing formula, 
 
 cotan h = tan t cos (f>. 
 IV. On the Daily Motion as a Measure of Time. 
 
 SIDEREAL TIME, SOLAR TIME, MEAN TIME. 
 
 18, Since the daily revolution of the sphere of the heavens, 
 or, more properly, the revolution of the earth on its axis, is 
 perfectly uniform, it serves us as a measure of time, which we 
 
 * As may bo seen also by the equation for sin^, which then gives a negative value 
 for /(, 
 
SIDEREAL AND SOLAR TIME. 79 
 
 can have an idea of in like manner as Ibeing uniform in its pro- 
 gress. The time occupied by the earth in one revolution on its 
 axis, and consequently the time which elapses between two con- 
 secutive culminations of the same fixed star, is called a sidereal 
 day. The commencement of this day is reckoned, or it is said 
 to be O"" of sidereal time, at the instant when the first point of 
 Aries passes the meridian. In the same manner it is said to be 
 l"", a*", S*", &c. sidereal, when the hour-angle of the first point of 
 Aries is l*", a"", 3"", &c., and therefore when that point of the 
 equator culminates whose right ascension is l*", 2^, 3'^, &c., or 15°, 
 30", 45°, &c. 
 
 We shall see in the sequel, that the equinoctial points, that 
 is, the points of intersection of the ecliptic and the equator, are 
 not fixed points, but that they have a retrograde motion along 
 the equator. This motion is composed of two others, of which 
 one is proportional to the time, and is connected with the daily 
 motion of the heavens, but the other is periodical. Owing to 
 this latter motion, the hour-angle of the first point of Aries does 
 not change with perfect uniformity, and therefore the sidereal 
 time is not a perfectly uniform measure. This want of uniform- 
 ity is however exceedingly small, since the period of nineteen 
 years has only the two maxima — 1' and + 1°. 
 
 19. When the sun on the 21st of March is at the vernal 
 equinox, he passes the meridian very nearly at O*" sidereal. But 
 the sun now moves forward in the ecliptic, and, since on the 
 23rd of September he is in the autumnal equinox, and thus has 
 \2^ right ascension, he culminates on this day at 12'' sidereal. 
 The time of culmination, and, in like manner, the time of rising 
 and setting of the sun runs through therefore in the course of 
 a year all the hours of the sidereal day, and, on account of this 
 inconvenience, sidereal time is not employed in the civil afiairs 
 of life, but the sun himself is used for the purpose of measuring 
 time. The hour-angle of the sun at any time is called true solar 
 time, and the time which elapses between two consecutive culmi- 
 nations of the sun is called a true solar day. At any place it 
 is O** of true time when the sun passes the meridian of that 
 place. 
 
80 MEAN SOLAE TIME. 
 
 This true time has however this inconvenience, that it does 
 not progress uniformly, since the right ascension of the sun does 
 not change uniformly. In the first place, namely, the sun does 
 not move in the equator hut in the ecliptic, and we obtain his 
 right ascension a from his longitude according to the Note to 
 No. 9, by means of the formula 
 
 tan a = tan \ cos e, 
 or, if for this purpose we employ formula (19) in No. 11 of the 
 Introduction, by means of the series 
 
 a = X — tan^ - e sin 2X + - tan* - e sin 4X — &c. 
 
 From this it is seen that the right ascension of the sun 
 increases irregularly, even when the longitude increases uniformly. 
 But besides this the sun moves also irregularly in his orbit, and 
 Theoretical Astronomy teaches that his longitude at any time t is 
 represented by an expression of the form 
 
 where §■ is a periodical function depending upon the longitude 
 of the sun. From both causes then the right ascension of the 
 sun increases irregularly, and consequently also his hour-angle or 
 the true solar time. Since now our clocks have an uniform 
 motion, and so cannot give true solar time, true time cannot 
 be used for the ordinary purposes of life ; an uniform time there- 
 fore is used which is called Mean Solar Time. 
 
 20. Between two successive transits of the sun through the 
 vernal equinox 366"24222 sidereal days elapse, and therefore any 
 particular star will in this time, which is called the tropical year, 
 as often complete its daily revolution on the sphere of the heavens, 
 or as often pass over the meridian. But, since the sun by its 
 proper motion in the ecliptic, has likewise in that time passed 
 through the 24 hours of the equator, so will it during a tropical 
 year pass exactly once less across the meridian than a fixed 
 star, viz. 365"24222 times. The tropical year has been divided 
 into the same number of equal days, which are called Mean 
 Days, and each one of these includes 24 equal hours, so that the 
 tropical year is, when expressed in mean time, equal to 
 365*. 5". 48". 47^,8091. 
 
EQUATION OF TIME. 81 
 
 Imagine then that a fictitious sun moves in the equator 
 with imiform velocity so that the right ascension a for any time 
 t is obtained by means of the expression 
 
 a. = L + /jU, 
 
 where L is the mean longitude of the sun at the commencement 
 of the time t, and 
 
 360° 
 ^ ~ 365-24222 ' 
 
 and therefore if t be expressed in mean days its value = 59'.8",83, 
 and the homf-angle of this mean sun will be the mean time. 
 The mean day begins when the sidereal time is equal to the 
 longitude of the mean sun, or when this fictitious mean sun 
 is on the meridian. In astronomical calculations the hours are 
 reckoned fi-om O"" to 24''. 
 
 Now the mean . sun is sometimes before the true sun and 
 sometimes behind it, according to the sign of the term ^ and 
 
 of the periodical term — tan" - sin 2X + . . . . , which is called 
 
 the Beduction to the Ecliptic. This difierence between mean 
 and true solar time is called the Equation of time, and its alge- 
 braical sign is so taken that it shall be always additive alge- 
 braically to true time to obtain mean time. The equation of 
 time is zero four times in the year, or the mean time is the 
 same as the true time, namely, on the 14th of April, the 14th 
 of June, the 31st of August, and the 23rd of December, or on the 
 day following these. Between the 23rd of December and the 
 14th of April the equation of time reaches its maximum value 
 of 14™.34* about the middle of February, and during this period 
 true time is behind mean time. Between the 14th of April and 
 the 14th of June it reaches a maximum of 3". 54' about the 
 middle of May, and during this period true time is before mean 
 time. Between the 14th of June and the 31st of August, 
 a maximum 6". 11' is attained about the end of July, and during 
 this period tme time is again behind mean time. Lastly, be- 
 tween the 31st of August and the 23rd of December, the 
 maximum 16™. 17' is attained about the middle of November, 
 and here true time is again before mean time. The equation 
 of time is given in astronomical ephemerides, and it is given 
 B. A. 6 
 
82 TBANSFOEMATION OP MEAN TIME INTO SIDEREAL TIME. 
 
 in Encke's Jahrhuch, for every true Berlin noon*. The duration 
 of a true day reaches a maximum at the end of December, when 
 it amounts to 
 
 24". O". 30',0. 
 
 The minimum, which occurs at the end of September, on the 
 contrary, attains the value of 23''. 59". 39°,0. 
 
 There are now three separate kinds of time used in astronomy, 
 and it is therefore necessary to obtain a knowledge of the rules 
 for the mutual transformation of these times. 
 
 21. Transformation of mean time into sidereal time and vice 
 versd. 
 
 Since 365'24222 mean days are equal to 366 '24222 sidereal 
 
 days, 
 
 ., , , 365-24222 , 
 
 one sidereal day = ^^^.^^^^^ mean day 
 
 = a mean day — 3". 55',909 mean time, 
 
 , , 366-24222 ., , - 
 
 and a mean day = 365-24222 ^^^^^^^^ "^^ 
 
 = a sidereal day+ 3™. 56°,555 sidereal time. 
 
 Thus, 6 being the sidereal time, M the mean time and 6^ the 
 sidereal time for M = 0, that is for the commencement of the 
 > mean day or for mean noon, we have 
 
 24'' - 3"". 55°,909 
 
 M=id-0„) 
 and 6 = 6'+ M. 
 
 '0 
 
 24'' 
 
 24'' + 3'°. 56^,555 
 24" 
 
 Thus, in order to transform sidereal time into mean time and 
 vice versS,, it is necessary to know the sidereal time at mean 
 
 • The calculation of the equation of time is deduced from the Solar Tables, 
 from which we can find the mean and true longitudes and likewise the mean and 
 true right ascensions of the sun for any given time. The best Solar Tables are 
 those of Carlinij corrected by Bessel, in the Bffemeridi Astronomiche di Milano per 
 tanno 1844. [Since the publication of Brurmow^s Astronomy, Hansen's and Olnfsen's 
 Solar Tables have been published, as also Le Terrier's. Both are of great excellency, 
 and far superior to Carlini's. — Tkanslator.] 
 
TRANSFORMATION OF TRUE TIME INTO MEAN TIME. 83 
 
 noon, or the right ascension of the mean sun at the beginning 
 of the mean day, and since this increases daily at the rate of 
 3 . 56',555347, it needs be given only for some particular epoch. 
 In astronomical ephemerides this quantity is given for conveni- 
 ence for every mean noon. For further simplification of the 
 calculation there are other tables, which give the values of 
 
 24''-3°'.55^909 
 
 and 24"+3".56»,555 
 
 24" ' 
 
 for different values of the time *. 
 
 Such tables are likewise found in the astronomical epheme- 
 rides and in all collections of astronomical tables. 
 
 Example. To transform 1849, June 9, 14". 16"". 36',35 
 sidereal time for Berlin into mean time. 
 
 The sidereal time at mean noon for this day in Encke's Jahr- 
 huch, is 
 
 5". 10". 48»,30, 
 
 consequently 9". 5". 48%05 sidereal have elapsed between mean 
 noon and the given time, and this, according to the auxiliary 
 tables, or by actually performing the multiplication by the 
 factor 
 
 24" — 3"" 55' 909 
 
 ^ — '- — , produces 9''. 4". 18^63 mean time. 
 
 If the mean time be given it must be transformed by the 
 auxiliary tables into sidereal time, and this must be added to the 
 sidereal time at mean noon for the pui-pose of finding the sidereal 
 time corresponding to the mean time. 
 
 22. Transformation of true time into mean time and vice 
 versd. 
 
 In order to transform true time into mean time we have only 
 to take out from the ephemerides the equation of time for the 
 given true time and to apply this algebraically to the given time. 
 
 6—2 
 
84 TEANSPOEMATION OF TEUE TIME INTO SIDEEEAL TIME. 
 
 According to the Berlin Jahrhuch, we have for the equation of 
 
 time at true noon 
 
 1st Diff. 2nd Diff. 
 1849 June 8 - 1"°. 20',73 , ^s „~ 
 
 9 -1 . 9,37 lit M +0':27 
 10 -0 .57,74 +11'''^ 
 
 If the true time for June 9 be 9". 5". 23',60, we shall find 
 the equation of time to he — 1". 4',98 ; and therefore the mean 
 time will he 9". 4°. 18',62. 
 
 In order to transform mean time into true the same equation 
 of time will serve. But since this is given in the ephemerides 
 for true time, it is necessary to have a knowledge of the true 
 time, in order to be able to interpolate the equation of time. 
 But owing to the small daily change of this quantity it will be 
 sufficient, when we are transforming the given mean time into 
 true, to add to the given time an equation which corresponds 
 only approximately to the given time. The equation of time 
 is then interpolated with this approximate true time. If, for 
 jBxample, the mean time 9''. 4™. 18^62 be given, we take for the 
 equation of time -1". "With the true time 9". 5". 18',6 the 
 equation of time will be found to be — l". 4^,98, and consequently 
 the true time is 9". 5"°. 23',60. 
 
 23. Transformation of true time into sidereal time and vice 
 versa. 
 
 Since true time is nothing else than the hour-angle of the 
 sun, his right ascension A needs only be known in order to 
 obtain the sidereal time from the equation 
 
 e=w+A, 
 
 where TF represents the trae time. 
 
 According to Encke's Jahrhuch, we have the folloTving 
 right ascensions of the sun for true noon at Berlin : 
 
 ist Diff. 2nd Diff. 
 
 1849 June 8 5". 5". 30',79 
 9 
 10 
 
 9 9.38,75 ■--- +0^27 
 
 Should now the tnie time 9". 5°". 23',60 for June 9 be required 
 to be transformed into sidereal time, we have for this time the 
 
TEANSrOEMATION OF SIDEREAL TIME INTO TRUE TIME. 85 
 
 right ascension of tlie sun equal to b^. ll"". 12°,75, and conse- 
 quently the sidereal time equal to 14". 16"", 36°,35. 
 
 In order to transform sidereal time into true time, an ap- 
 proximate knowledge of the true time is required for the inter- 
 polation of the right ascension of the sun. But if from the 
 given sidereal time be subtracted the right ascension of the sun 
 which answers to the commencement of the day, the number of 
 sidereal hours is obtained which have elapsed since that time. 
 These sidereal hours must be transformed into true time. But 
 it is sufficient to transform them into mean time and to inter- 
 polate the right ascension of the sun for this mean time. If 
 then we subtract this from the given sidereal time, we obtain 
 the true time. 
 
 The right ascension of the sun at the beginning of June 9 
 is equal to 5\ 9°°. 38',75, consequently between this and 14''. le"". 
 36',35 sidereal time, 9\ G". 57',60 have elapsed, or 9". 5"°. 28',00 
 of mean time. By interpolating the right ascension of the sun 
 for this time, we obtain again 5\ 11". 12',75, and consequently 
 the true time equal to 9*. 5"". 23'',60. 
 
 These transformations can moreover be performed with equal 
 facility when the mean time is required from the sidereal, and 
 from that the true time by means of the equation of time. 
 
86 COEEECTIONS DEPENDING ON THE POSITION OF OBSEEVEE. 
 
 SECOND SECTION. 
 
 COEEECTIONS OF OBSERVATIONS, WHICH ARE DEPENDENT ON 
 THE POSITION OF THE OBSERVER ON THE SURFACE OF 
 THE EARTH AND ON THE PROPERTIES OF LIGHT. 
 
 The Astronomical Tables and Ephemerides always give the 
 piaces of the heavenly bodies as they would appear from the 
 centre of the earth. For bodies infinitely distant, this place is 
 the same as that which would be observed from any point upon 
 the surface of the earth. But, if the distance of the body bear 
 an appreciable ratio to the radius of the earth, the body will 
 not appear to be in the same position when observed at the 
 centre and at a point on the surface of the earth. Should it 
 therefore be required to compare the observed place of such a 
 body with the tables, it is necessary to devise a method by which 
 the place as seen from the centre of the earth may be calculated 
 from the observed place. Should it be desired, on the other 
 hand, to calculate other quantities from the observed places of 
 such a body referred to the horizon of the observer, for example, 
 in connection with his known position with reference to the 
 equator, the apparent place must then be employed as it ap- 
 pears when seen from the place of observation, and consequently 
 the places as seen from the centre of the earth, which are given 
 in the Ephemerides, must be transformed into apparent places. 
 
 The angle at the star which is included between the two 
 lines drawn respectively from the centre of the earth and the 
 place of observation on the surface to the star, is called the 
 Parallax. Consequently a method is required by which the 
 parallax of a star for any time and place upon the surface of 
 the earth can be calculated. 
 
PAEAXLAX. 87 
 
 Our earth is in addition surrounded, to some considerable 
 altitude, by an atmosphere which possesses the property of re- 
 fracting light. The heavenly bodies therefore are not seen in 
 their true places, but in the direction which the ray of light, 
 after being refracted by the atmosphere, has at the instant when 
 it meets the eye of the observer. 
 
 The difference between this direction and that in which the 
 heavenly body would be seen, supposing no atmosphere to exist, 
 is called the Refraction. In order therefore to obtain the true place 
 of a heavenly body from the observed place, a method is required 
 by which the refraction may be determined for every point of the 
 heavens and for every condition of the atmosphere. 
 
 Had the earth no proper motion, or were the velocity of light 
 infinitely greater than the velocity of the earth, this motion 
 would have no influence upon the apparent places of the heavenly 
 bodies. But, since the velocity of light bears an appreciable 
 ratio to the velocity of the earth, so an observer upon the earth 
 sees all the heavenly bodies In advance of their true places by 
 a small angle which is dependent upon this ratio, and towards 
 that direction in which the earth Is moving. This minute angle 
 by which the places of the heavenly bodies appear to be altered 
 by the motion of the earth and of light, Is called the Aher- 
 ration. In order therefore to obtain the true places of the hea- 
 venly bodies from observation means must be devised for freeing 
 the observed apparent places from this effect of aberration. 
 
 1. Parallax. 
 
 1. Our earth is not a perfect sphere, but an oblate spheroid, 
 that is to say, such as is formed by the revolution of an ellipse 
 about Its minor axis. Denoting by a the semi-major, by h the 
 semi-minor axis, and by a the ellipticity in parts of the semi- 
 major axis, we have 
 
 a— h , b 
 
 a = = 1 . 
 
 a a 
 
 If moreover e be the excentricity of the generating ellipse, 
 that is that ellipse which is formed by the Intersection of the 
 surface of the spheroid and a plane passing through the minor 
 
88 sun's horizontal eqdatoeeal parallax. 
 
 axis, we have, if this excentricity te expressed likewise in parts 
 of the semi-axis major, 
 
 a 
 and therefore - = Jil — e"). 
 
 Hence a = 1 — V(l — e"), 
 
 and e = i\/(^2a — a^. 
 
 The ratio - is, according to Bessel's investigations, in the 
 
 case of the earth, 
 
 298-1528 
 299-1528' 
 
 1 
 
 or a = , 
 
 299-1528' 
 
 and, expressed in toises, 
 
 a = 3272077-14 log a = 6-5148235 
 
 6 = 3261139-33 log J = 6-5133693. 
 
 But in astronomy it is not the toise but the semi-axis major 
 of the orbit of the earth which is taken as the nnit. Denoting 
 then by tt the angle under which the equatoreal radius of the 
 earth, or the semi-axis major of the spheroid, appears when- 
 viewed from the sun, and by B the semi-axis major of the earth's 
 orbit, or the mean distance of the earth from the sun, we have 
 
 a = B, sin tt = Rir sin 1", 
 
 i?7r 
 
 or 
 
 206265 ' 
 
 The angle tt, or the equatoreal horizontal parallax of the sun 
 is, according to Encke, 
 
 8",57116. 
 
 This is the angle under which the radius of the earth's equa- 
 tor is seen from the sun, when the sun for places on the equator 
 is in the horizon. 
 
COEEECTED LATITUDE. 89 
 
 2. In order then to be able to calculate the parallax of a 
 body for any place upon the surface, we must be able by means 
 of co-ordinates to refer to the centre any point upon the spheroid. 
 Take then as the first co-ordinate the sidereal time, that is the 
 angle which a plane* passing through the place of observation 
 and the semi-axis minor makes with a plane passing through the 
 semi-axis minor and the first point of Aries, If then OA (fig. 
 2) be the plane passing through the place of observation A and 
 the semi-axis minor, we must, for the determination of the posi- 
 tion of the place, know in addition the distance A0 = p from 
 the centre of the earth, and the angle AOC, which is called the 
 corrected latitude. 
 
 But these quantities can always be calculated from the astro- 
 nomical latitude ANC, (which is in fact the angle which the 
 horizon at A makes with the axis of the earth, or which the 
 normal AN to the surface at A makes with the equator,) and the 
 two axes of the earth's spheroid. 
 
 Let X and y be the co-ordinates of the point A referred to 
 the centre 0, considering OC as the axis of abscisses and OB 
 as the axis of ordinates ; we have then, since ^ is a point in 
 an ellipse of which a and h are respectively the semi-axes major 
 and minor, the equation 
 
 ay + JV-aV. 
 
 Since now, if we denote by ^' the corrected latitude 
 
 tan </>' = -, 
 
 and also 
 
 dx 
 tan.^ = -^, 
 
 where the latitude ^ is the angle which the normal at A makes 
 with the axis of the abscissae, we obtain, since the difierential 
 equation to the ellipse gives 
 
 y _ V dx 
 
 x~ a^ dy' 
 
 * Since this plane passes throagh the poles of tlie earth and the zenith of the 
 plaee of observation, it is the plane of the meridian. 
 
90 CALCULATIOK ^F THE ANGLE OP THE VERTICAL. 
 
 the following relation between the quantities ^ and ^', 
 tan ^ = -5 . tan <f) 
 In order to calculate p, we have 
 
 tan^'= -j.tan^ , (a). 
 
 p = V(a>' + y) = ; "^ 
 
 cos <!>' ' 
 
 Since now from the equation to the ellipse, 
 a a 
 
 33 = 
 
 Vi'^t 
 
 ~ + 8 A' \ V(l + t^ii ^ tan <^') 
 
 we thence obtain 
 P = 
 
 a sec <}>' /( cos <j) ] ,,, 
 
 VCl + tan^ tan^') ~ V [cos ^' cos (^' — ^) J ^ '' 
 
 By means of these two formulse we can calculate the corrected 
 latitude </>' and the radius p for any place on the surface of the 
 earth of which the latitude ip is known. 
 
 For the co-ordinates x and ^ we obtain the following formulse, 
 which also will be required in the sequel : 
 
 a cos<j) 
 x = — 
 
 v'{cos>+(l-e'')sin''<^} 
 a cos 
 
 and 
 
 V(l-e'sin»' 
 
 52 
 y—x tan <j)' = x-^ tan <j> = x (l — e^) tan (f) 
 
 {0), 
 
 _ g (1 - 6°) sin ^ 
 V(l-e'sin''^) : W- 
 
 From the formula (a) we can express 0' in a series which 
 proceeds according to the sines of the multiples of <^. We 
 obtain, namely from the formulse (18) in No. 11 of the Intro- 
 duction, 
 
 ^, ^ a'-V . „. , l/a'-JV ... „ 
 
 ^ = "^-^q:F'^'^^'^ + 2(?+FJ ^I'^^^-^c (A) 
 
or, putting 
 
 CALCULATION OF EAETH'S KADIUS'VECTOK. 91 
 
 a—h 
 
 a+b' 
 
 ^'=<i>-~.^^^'i>+l(j^:)^^^<i>-&^ (5). 
 
 The angle ^ — <f>' is called tke Angle of the Vertical. 
 If thftj^numerical values of the coefficient for the ellip- 
 ticity of the earth given above be calculated and multiplied 
 
 by 206265 ♦ or -. — -77 ) in order to obtain the result in seconds, 
 \ sml / 
 
 we arrive at 
 
 ^' = ^-ll'.30",65sin2^ + l",16 sin40-&c (C), 
 
 from which, for example, we find for the latitude of Berlin 
 52°. 30'. 16", 0, 
 ^' = 52M9'.8",3. 
 
 Although p itself cannot be expressed in so symmetrical a 
 series as ^', a similar one can be obtained for log/j*. 
 
 The formula (J) ^ves for example : 
 
 , ±_ 
 
 <j>' (l+^tan>) 
 
 and, putting for cos"* <f>' its value ^4 , ^1 ^^^i ! > 
 
 we obtain 
 
 ^ a*cos°<^+ysin''< ^ 
 P " a" cob' <l> + b' sin' if 
 
 -«» + 52^ («"_&») cos 2^ 
 
 (a' + by + (a° - by + 2 (a' + b") (a' - 5') cos 2cj> 
 - ^^a + by+{a-by+2{a+b) {a-b) cos 2<j> 
 
 and consequently, 
 
 • Encke, Jahrhuchfor 1852, p. 326, where tables are likewise given from which 
 log p can be found for any value of tp'. 
 
 P = 
 
92 CALCULATION OP EAETH'S BADIUS-YECTOE, 
 
 If the formula be written logarithmically and the logarithms 
 of the square roots he developed according to the foimula (17) in 
 No. 11 of the Introduction, in series which proceed according to 
 th6 cosines of the multiples of 2^, we obtain: 
 
 , , a^ + V , fa^-V a-h\ „, 
 
 ^ log^ = log«.^:^ + [^:^, - ^cos2^ 
 
 &c (-0), 
 
 or, for the common system of logarithms, 
 , , . a — b 
 
 log p = log (a . ^^) + if {[(il^.) - n^ cos 24> 
 
 4 (^' 
 
 — &c. 
 
 where M denotes the modulus of the common system of loga- 
 rithms, and 
 
 log M= 9-6377843. 
 
 Calculating again the numerical values of the coefficients, we 
 obtain, if a be taken = 1, 
 
 log /3 = 9-9992747 +0-0007271 cos 2^ - 0-0000018 cos 4^... (F), 
 
 from which, for example, for the latitude of Berlin 
 
 log p = 9-9990880. 
 
CALCULATION OF PAEALLAX IN ALTITUDE. 93 
 
 Consequently when the latitude of a place is known, the 
 corrected latitude and the distance of the place from the centre of 
 the earth can be calculated by means of the series ((7) and {F), 
 and by means of these quantities in connexion with the sidereal 
 time the position of the place with reference to the centre of 
 the earth can at any instant be determined. Imagine then a rect- 
 angular system of co-ordinates to be drawn through the centre 
 of the earth, of which the axis of a is perpendicular to the plane 
 of the equator, whilst the axes of x and y lie in the plane of the 
 equator, and so that the positive axis of x is directed towards 
 the first point of Aries and the positive axis of y to a point 
 having 90° of right ascension, then can we also express the 
 position of the place upon the surface with reference to the centre 
 by means of the three rectangular co-ordinates, 
 
 x = p cos (j)' cos j 
 
 y =/j cos 0'sin ^ > (G). 
 
 z=p am <!>' ' 
 
 3. The plane, in which the lines from the centre of the 
 earth and from the place of observation lie, passes, if the earth 
 be considered as a sphere, necessarily through the zenith of the 
 place of observation, and cuts consequently the visible sphere of 
 the heavens in a vertical circle. From" which it follows that the 
 parallaK alters only the altitude of the body, the azimuth re- 
 maining unchanged. Let now A (fig. 2) be the place of obser- 
 vation, Z its zenith, S the star, and the centre of the earth ; 
 then Z08 is the true zenith-distance z as seen from the centre 
 of the earth ; and ZA8 the apparent zenith-distance z' as ob- 
 served firom the place A on the surface. Denoting then the 
 parallax, that is the angle nt S = z' — z, hyp, 
 
 sm^ = -^ sm a , 
 
 where A represents the distance of the body from the earth, and, 
 since p, except in the case of the moon, is a very small angle, 
 the arc may be put in place of the sine, and 
 
 .-. p' = ■£ sin z X 206265. 
 
94 HORIZONTAL EQUATOEEAL PAEALLAX. 
 
 * 
 
 The parallax is consequently proportional to the sine of the 
 apparent zenith-distance. It is nothing at the zenith, attains its 
 maximum value in the horizon, and causes the altitude of every 
 body to appear too small. The maximum value, for z = SO",! 
 
 or 2^ = A 206265, 
 
 is called the Horizontal Parallax, and the value 
 
 P=x 206265, 
 
 where a is the radius of the earth at the equator, is called the 
 Sorizontal-Equatoreal-Parallax, 
 
 Hitherto the earth has been considered as spherical; but 
 since the earth is a spheroid, the plane in which lie the lines 
 drawn from the centre of the earth and from the place of ob- 
 servation to the body does not pass through the zenith of the 
 place of observation but through the point in which the line 
 from the centre of the earth to the place of observation meets the 
 visible sphere of the heavens. On this account will the azimuth 
 of the body also be altered by the parallax, and at the same time 
 the rigorous expression for the parallax in altitude will be differ- 
 ent from that just given*. 
 
 Let us consider then a system of three rectangiilar co-ordi- 
 nate axes, of which the axis of s is drawn towards the zenith 
 of the place of observation, whilst the axes of x and y lie in 
 the plane of the horizon, and so that the positive axis of x is 
 directed towards the south, the positive axis of y towards the 
 west ; then the co-ordinates of a body referred to these axes are 
 
 A' sin z' cos A', A' sin z sin A', and A' cos z', 
 
 where A' represents the distance of the body from the place of 
 observation, z' and A' the apparent zenith-distance and azimuth 
 as seen from the place of observation. 
 
 • The Author is indebted for the following symmetrical investigation to the 
 kindness of Prof. Encke. 
 
CALCULATION OF SPHEROIDAL PARALLAX. 95 
 
 Moreover the co-ordinates of the body referred to a system 
 of axes parallel to the former but which passes through the 
 centre of the earth, are 
 
 Asms cos A, Aainz sin A, and A cos 2, 
 
 if we denote by A the distance of the body from the centre of 
 the earth, and by z and A the zenith-distance and azimuth 
 respectively as seen from the centre of the earth. Since now 
 the co-ordinates of the centre of the earth referred to the first 
 system of axes are respectively 
 
 — p sin (^ — <^'), 0, and — p cos (^ — ^'), 
 we obtain the three equations 
 
 A' sin z cos ^' = A sin a cos ^ — p sin (^ — ^'), 
 
 A' sin z sin ^' = A sin « sin A, 
 
 A' cos z' = A cos a — p cos (0 — <})') ; 
 
 or, 
 
 A' sin z' sin {A' — A)=psm{<j) — <j)') sin A 
 
 A' sins' cos {A' — A) = A sina — psin (0 — ^') cos^ 
 
 A' cos s' = A cos a — p cos (^ — <p') 
 
 (a). 
 
 Multiplying the first equation by sin-(^' — ^), the second 
 by cos- {A' —A), and adding the products, we obtain 
 
 cos-(^' + ^) 
 
 A' sin a' = A sin a — p sin {<j} — ^') 
 
 cos^K-^) 
 
 A' cos a' = A cos z — p cos (^ — ^') ; 
 putting then 
 
 cosi(^' + ^) 
 
 tan 7= = tan(^ — ^') (&), 
 
 we have 
 
 cos|(^'-^) 
 
 A' sin a' = A sin a — p cos (<^ — 4>) tan 7> 
 A' cos a' = A cos a — p cos {<^ — 4') 5 
 
96 
 
 CALCULATION OF SPHEKOIDAL PARALLAX. 
 
 or, 
 
 A sm(g-g)=pcos(0-(^) ^^^^ 
 
 A' cos (/ - «) = A - p cos (<^ - f) ^^^^^ 
 
 ... (c). 
 
 . 1 
 
 And again, if we multiply the first equation by sin-(a' — s), 
 the second by cos - (z' — s), and add the products, 
 
 A' = A-p. 
 
 cos (^ — <f)') cos 
 
 ^- («'+.) -7} 
 
 COS 7 
 
 Dividing equations (a), (J) and (c) by A, and considering 
 the semi-diameter of the earth at the equator as the unit, and 
 putting 
 
 ^ = sm P, 
 
 where P represents the horizontal equatoreal parallax, we obtain, 
 by means of the formulse (14) and (15) in No. 11 of the Intro- 
 duction, 
 
 A —A = I- -—. — '■^ - ^'' sm A 
 
 sms 
 
 , 1 fpsinP8in(<^-<6')r . „ . 
 
 + ]H ^^^ — ^l sm2^+ ..., 
 
 2 ( sms J ' 
 
 7 = cos^ X (^-0') -^sin^ tan-(J.'-^) x {(j} — (f)') 
 
 sin J. sin ^' cos r (^' + -4) 
 ^ CO,' I {A'- A) ^ 
 
 x(<^-,^')3+...* 
 
 We get, namely, 
 
 7 = - 
 
 cos-(.l' + X) 
 
 cos -(A' -A) 
 
 ^cos'-{A'+A) 
 
 tan (,^- ,/,')__ ^ tan»((^ -./.') + ... 
 
 cos»-(^'-^) 
 
 hence putting for tan (tp ~ </>') the series {<p - (^') + - (0 _ 0')a + ... ,ye obtain easily 
 the expression given in the text. 
 
PAEALLAX IN AZIMUTH AND ZENITH-DISTANCE. 97 
 
 , psinPcos{<i — ^') . , . 
 
 Z —Z = - ^i — 21iBui(2_«) 
 
 COS 7 ^ ' 
 
 2 [ cos 7 j ^ ''. 
 
 1 Ai 1 A psinPcos(<i— A') , , 
 
 loge A = log. A - 1- ^2: — ^-^ COS (a - 7) 
 
 cos 7 \ •/ 
 
 1 fpsinPcosfA-rf)')]" „, , 
 
 ^ Iff vr — viZL cos 2 (s- 7)-.... 
 
 2 [ cos 7 J \ " 
 
 Hence, as an approximation aS far as quantities of the order 
 (^ — ^') sin P, which never amount to any significant value in 7, 
 
 7 = (^ — 1^') cos A, 
 
 and we obtain for the azimnthal parallax 
 
 ^,_^^p_sinPsin^^-^^j^^ 
 sms 
 
 or, if s should be very small, accurately 
 
 psinPsin(<i — A') . . 
 ■ ^ — i-^ sm^ 
 
 tan(^-^) = ^ psinPsin( ^-?) 
 
 cos^ 
 sme 
 
 Further, since 
 
 cos 7 ^ I Ai A\ • ' 
 
 ' cos- (-4 —A) sm7 
 
 and is consequently very nearly equal to unity, we obtain for 
 the parallax in zenith distance 
 
 a' - s = p sin P sin {sf — ((^ — <^') cos -4} ; 
 
 or, more strictly, 
 
 A' 
 
 -^ sin (a' — a) = p sin Psin {a — (<^ — <^') cos -4}, 
 
 A' 
 
 -^ cos (a' -a) = 1 -psinPcos{a-(<^-<^')cos^}. 
 
 B.A. 7 
 
98 PARALLAX IN EIGHT ASCENSION AND DECLINATION. 
 
 Consequently, for the meridian, the parallax in azimuth is 
 zero, and the parallax in zenith-distance is 
 
 a' — « = jo sinPsin {a — (^ — ^')}. 
 
 4. Similarly we obtain the parallax for right ascension and 
 declination. Let the co-ordinates of a body referred to the plane 
 of the equator and the centre of the earth be 
 
 A cos B cos a, A cos S sin a, and A sin B. 
 
 Let also the apparent co-ordinates referred to the same planes, 
 as they appear from the place of observation oij the surface of 
 the earth, be 
 
 A' cos 8' cos a'. A' cos S' sin a', and A' sin 8'. 
 
 We have thus, since the co-ordinates of the station on the 
 surface referred to the centre of the earth and to the equator as 
 the fundamental plane, are 
 
 p cos ^' cos @, p cos if>' sin ©, and p sin <f>', 
 
 the three following equations for the determination of A', a', 
 and B' : 
 
 A' cos S' cos a' = A cos 8 cos a — p cos ^' cos @ 
 
 A' cos 8' sin a' = A cos 8 sin a — p cos ^' sin © (a). 
 
 A' sin 8' = A sin 8 —p sin <j>' 
 
 Multiplying the first equation by sin a, the secoild by cos a, 
 and subtracting, 
 
 A' cos 8' sin (a' - a) = - p cos ^' sin (© - a). 
 
 Again, multiplying the first equation by cos a, the second 
 by sin a, and adding, we find 
 
 A' cos 8' cos (a' - a) = A cos 8 - p cos ^' cos (© - a). 
 
 Consequently we have f 
 
 t-infq' n)-,^ fes.^'sin(a-@) 
 
 A cos 8 - p cos (j)' cos (ct - @) 
 
 P cos ^' . 
 
 -7 ^ sm (a — ©) 
 
 A cos 8 ^ ' 
 
 , pcosA' , ' 
 
 '-A^«^«(«-®) 
 
PARALLAX IN EIGHT ASCENSION AND DECLINATION. 99 
 
 or, using the expansion which has been already frequeatly 
 employed, 
 
 a'-a = ^^sin(«-0) + lfa£^ysin2(«-0) 
 
 1 /p cos <A'\' . „ , „, 
 
 . +«S;c (A). 
 
 For all cases, excepting the moon, the first term of this series 
 is sufficient, and if for p we take as unit the radius of the earth's 
 equator, and then introduce into the numerator the factor 
 sin TT (where w is the horizontal equatoreal parallax of the sim) 
 whereby the same unit, namely the semi-major axis of the 
 earth's orbit, is introduced into the numerator and denominator, 
 we get the simple equation 
 
 , p sin TT cos ^' sin (a — ©) ,_. 
 
 "-" A cosS ^^^' 
 
 where a — is the east hour-angle of the body. The parallax 
 consequently increases the right ascensions of stars on the east 
 side of the meridian, and diminishes them on the west side. If the 
 body be on the meridian, its parallax in right ascension is 
 nothing. 
 
 In order to find a similar formula for 8' — S, put, in the for- 
 mula for A' cos S' cos (a'— a), 1 — 2 sin' - (a' — a), in the place of 
 cos (a'— a), by which means We obtain 
 A' cos S' = A cos S - p cos ^' cos (0 - a) + 2A' cos S' sin" - (a' - a). 
 
 Multiplying and dividing the last term by 
 
 cos 2 (a - a), 
 
 we find, by the use of the formula 
 
 A' cos B' sin (a' - a) = - p cos <}>' sin (0 - a), 
 
 cos-0--(a' + a)> 
 A' cos S' = A cos S — p cos <^' ' (S)- 
 
 1 / r \ 
 
 cos - (a' - a) 
 
 7—2 
 
100 PARALLAX IN DECLINATION. 
 
 Introducing now the auxiliary quantities 
 y3 sin 7 = sin 4>, 
 
 cos j>' cos I® - 2 («' + «) f 
 
 j8cos7 = ^j ■■ (c), 
 
 cos -(a -a) 
 
 we obtain from (J) 
 
 A' cos 8" = A cos B — p^ cos 7, 
 and from the third of equations (a) 
 
 A' sin S' = A sin 8 — p/S sin 7. 
 From these two equations we easily obtain : 
 A' sin (8' - S) = - p/3 sin (7-8), 
 A' cos (8' - 8) = A - p/3 cos (7 - 8), 
 
 or tan {8' ~^ =■ 
 
 f sin (7-8) 
 l-^cos(7-S)' 
 
 or also, according to the formula (14) in No. 11 of the Intro- 
 duction, 
 
 S'-S = -^sin(7-8)-l^sin2(7-8)-&c....(C). 
 
 Putting here for B its value . ™ , and p sin tt instead of p, 
 ° sm 7 
 
 in order to have the same unit in the numerator and denominator, 
 
 we obtain, keeping only the first term of the series, 
 
 ^, _ ^ _ p sin TT sin <ji' sin (7 — 8) 
 
 A * sin 7 
 
 Putting, moreover, in the second of formulae (c) (which is 
 
 always allowable) unity instead of cos - (a — a), and a instead of 
 
 cos - (a + a), the complete approximate formuliE for the compu- 
 tation of the parallax in right ascension and declination are the 
 following : 
 
 , _ Trp cos <f>' sin (@ — a) 
 
PAEALLAX COMPUTED FOE AN OBSEEVATION. 101 
 
 tan A' 
 
 tan 7= j^ — r-, 
 
 ' cos (0 — a) 
 
 S' _ S = - "^P ^^^ ^ sin (7 - 8) ^ 
 A ' sin7 
 
 If the body have a visible disk, its apparent semi-diameter 
 will depend on the distance, and a correction will still be neces- 
 sary on this account. Now we have 
 
 A'sin(8'-7)=Asin(S-7); 
 
 • A'-A sin (8 -7) 
 •• ^~'^-sin(S'-7)' 
 
 and, since the semi-diameters, as long as they are small angles, 
 vary inversely as the distances, we have 
 
 sm (o — 7) 
 
 Example. 
 
 1844, Sep. 3, at 20^ 41°'. 38' sidereal, a comet, discovered by 
 de Vice, was observed 
 
 in right ascension 2°.35'. 55",5, 
 and in declination— 18°.43'. 21",6. 
 The logarithm of the distance from the earth was at this time 
 9-28001, and in addition for.Eome 
 
 0' = 41'',42',5, 
 and log p = 9-99936. 
 
 With these data the calculation for the parallax is the 
 
 following : 
 
 ©in arc = 310°. 24',5 
 a = 2 . 35 ,9 
 
 0_a =-52.11,4 
 
 * For the meridian we obtain from this formula, 
 
 i' -a =- ^ sin (.^' -«)= p . ^ . sin [*-((<)- '^')], 
 consequentl; the parallax in declination is equal to the parallax in altitnde. 
 
102 PARALLAX IN LONGITUDE AND LATITUDE. 
 
 log tan ^' = 9-94999 7 = 55''.28',6 
 
 log cos (@- a) = 9-78749 8 = - 18 .43 ,4 
 
 log sin (@- a) =-9-89765 y-S= 74.12,0 
 
 ^\oglI£^^=- 1-52576 log sin (7- 8) =-f 9-98327 
 
 log sec 8= 0-02362 - log '^^^™^ = - 1-47576 
 log (a'- a) = 1-44703 logcosec7= 0-08413 
 
 a'-a = + 27",99 log (8'- 8) = - 1-54316 
 
 8'-B = - 34",93 
 On account of the parallax the observed right ascension of the 
 comet is consequently greater by 28",0, and the declination is 
 less by 34",9, than these quantities would have been if observed 
 from the centre of the earth. The place of the comet freed from 
 the parallax is consequently 
 
 a= 2°. 35' .27",5, 
 S = -18°.42'.46",7- 
 To obtain the parallax of an object for co-ordinates which 
 are referred to the ecliptic as the fandamental plane, it is neces- 
 sary to know the co-ordinates of the place of observation with 
 reference to the centre of the earth, for the same fiindamental 
 plane. But by changing @ and ^' into longitude and latitude 
 according to No. 8 of the first section, and for this purpose ob- 
 taining the values of I and b, these co-ordinates are : 
 p cos b cos I, 
 p cos b sin I, 
 psinb, 
 
 and we have, if X', /S', A' be the apparent and \, /8, A the true 
 quantities, the three equations 
 
 A' cos ;8' cos X' = A cos y8 cos X — p cos b cos I, 
 
 A' cos )S' sin \' = A cos /8 sin X — p cos b sin I, 
 
 A'sin/S' = AsinjS —psinb, 
 
 from whence we obtain formulae similar to those obtained before, 
 namely : 
 
 , Trp cos b sin (I — X) 
 
 X — X 7 . --1 — ' — ^ ' • 
 
 A ' cos (S ' 
 
PARALLAX OF THK MOON. 103 
 
 tan 5 
 
 tan 7 = '■ ■ ., r-r , 
 
 ' cos (Z— X) 
 
 R' -~B= "^P ^^° ^ sin (y-jQ) 
 A ■ siny ' 
 
 and ^' are the right ascension and declination of that point 
 in which the radius of the earth produced meets the sphere of the 
 heavens ; I and b are consequently the longitude and latitude of 
 this same point. If we consider the earth as spherical, this point 
 coincides with the zenith, and its longitude is also called the 
 Nonagesimal, since the point of the ecliptic corresponding to this 
 longitude is distant 90° from that point of it which is in the 
 horizon. 
 
 5. Since the horizontal parallax of the moon or the angle 
 
 "—7 — , where A denotes the distance of the moon from the 
 
 A ' 
 
 earth, lies always between 54' and 61', it follows that, for the 
 computation of the moon's , parallax the first terms of the series 
 for a! — a. and S' — S will not be sufficient, but that it will be 
 necessary either to take account of the terms of higher orders, or 
 to use the rigorous formula. 
 
 Suppose the parallax of the moon in right ascension and 
 declination to be reqxdred for Greenwich on 1848, April 10, 
 at 10". 
 
 For this time we have 
 
 a = 7^ 43". 20%25 = 115°, 50'. 3",75, 
 S = +16°.27',22",9, 
 © = lit. lym^ 0.^02 = 169°. 15'. 0",30, 
 
 the horizontal parallax 
 
 P= 56'. 57",5, 
 
 J? = 15 «31 )3, 
 
 in addition, we have for Greenwich 
 
 f =51°.17'.25",4, 
 log p = 9-9991134, 
 By introducing the horizontal parallax P of the moon into 
 the two series for a- ft, and S'-S, found in No. 4, we shall 
 
 have 
 
 fpcos<i'sinJ' . ,^ V 
 
 «'_« = _ 206265 \^ ^ sm - a) 
 
 [ coso 
 
104 PARALLAX OF THE MOON. 
 
 And 
 
 S' - a = - 206265 fc ^^" ■^' ^^"-^ sin (7 - S) 
 I cosec7 ^ 
 
 1/psln^sinPN^ 
 
 2 V cosec7 / ^' -* 
 
 . 1 /psinA'smPV . „, 5.., 1 
 + o F sin 3 (7- S) + ...K 
 
 3 v. cosec 7 / \ ' / J 
 
 where we must now, for the computation of the auxiliary angle 
 7, employ the rigorous formula 
 
 1/ ' ^ 
 
 cos q (a — a} 
 
 tan 7 = tan ^' . "^^^— — j^— — — . 
 
 Gos{@--(a' + a)} 
 
 Performing then the computations for these formulae, we thus 
 obtain for a' — a 
 
 from the first term - 29'.45",71 
 
 second... — 11,47 
 
 third ... - 0,03 
 
 thus a'-a= - 29 . 57 ,21 
 
 and, for 5' — S ; 
 
 from the first term - 36'. 34",21 
 
 second... - 20 ,91 
 
 third ...- ,12 
 
 thus 8'-S=-36.55 ,24 
 
 The apparent right ascension and declination of the moon is 
 therefore 
 
 a' = 115". 20'. 6",54, S= + 15°. 50'. 27",66. 
 
 Lastly, we obtain for the apparent semi-diameter 
 
 ^' = 15'.40",20. 
 
RIGOROUS COMPUTATION OF PARALLAX. 105 
 
 If it be preferred to compute the parallax by the rigorous 
 formula, it must be rendered more convenient for logarithmic 
 computation. The rigorous formula for tan (a' — a) was 
 
 , , . p cos rf>' sin Psin (a — ©) sec S , . 
 
 tan (a -o) = — i- ^ ., . — „ '■ . ^grr 5 (a). 
 
 ^ ' 1 — p cos sm Pcos (a — @) sec S ^ ' 
 
 In addition, it follows from the two equations 
 
 A' sin S' = sin 8 — p sin tf)' sin P, 
 and 
 
 A' cos S' cos {a! — a) = cos S — p cos ^' sin Pcos (a — @), 
 
 ^T_ i i ^> [sinS — psinrf)'sinPl cos (a' — a)secS „, 
 
 that tanS = *-- <- — ., ■ p c. , — -^r. — (5). 
 
 1 — pcos^ smPsecocos (a— 0) ^' 
 
 Since also 
 
 A. _ cos S* cos (a' — a) 
 
 A' cos S — p cos <f)' sin Pcos (a—©) ' 
 
 we thus obtaia in addition 
 
 . ™ cos 8' cos (a' — a) sec 8 sin ^ , . 
 
 Sm^ =- -rr^. =j-^ = -. ^rr (c>. 
 
 1— pcos^ smPsecocos(a — ©) 
 Introduce now into (a), (&), and (c) the auxiliary quantities 
 
 . _ p sin Pcos <j>' cos (a — @) 
 coso 
 and sin C = p sin P sin <f>', 
 
 and we thus obtain the following convenient formulas for 
 logarithmic computation, 
 
 - p cos <ji' sin Psin (a — 0) 
 tan (a' — a) = ^ , 
 
 cosSsin^--4 
 
 sin- (S— C) cos- (S + C) cos (a — a) 
 tan 8' = ——^—^—^—^—^-m—^-^— ^ 
 cos 8 sin" - A 
 
 and 
 
 - cos 8' cos («' — a) 
 
 E = i J .B. 
 
 cosSsin'-jd 
 
106 
 
 EEFRACTTON. 
 
 If a' — a, S', and E be required for the preceding example 
 according to these formulae, we obtain with precisely the same 
 accuracy as before, 
 
 a'-a = - 29'.57",21, 
 
 S = + 15°. 50'. 27",68, 
 
 B' = 15'. 40",21. 
 
 >' 
 
 For the rigorous computation of the parallaxes in longitude 
 and latitude we obtain precisely similar formulae, with this 
 sole difference that \', X, /3', /8, I, and h appear in place of 
 a', s., S', 8, @, and f . 
 
 II. Eepeaction. 
 
 6. The rays of light do not reach us through empty 
 space, but through the atmosphere of the earth. In empty 
 space the rays proceed in straight lines ; but when they enter 
 into another medium, which refracts the light, they are turned 
 aside from their original direction. If no^ this medium con- 
 sists of innumerable layers whose refractive powers continuously 
 vary as is the case with our atmosphere, the path of the ray 
 of light through it will be a continuous curve. An observer 
 on the earth now sees the object in the direction of the last 
 tangent to the curve, which the ray describes, and he must from 
 this direction, which defines the apparent place of the object, 
 determine that direction of the ray which it would have had 
 in empty space, that is, the true position of the object. The 
 difference between these two directions is called Refraction, and, 
 since the curve formed by the path of the ray in the atmo- 
 sphere turns its concave side towards the observer, it is evident 
 that, on account of refraction, all objects are seen at too great 
 an altitude. 
 
 In what follows the shape of the earth will be supposed 
 to be spherical, since the effect of the spheroidal figure of the 
 earth on the refraction is altogether insignificant. The atmo- 
 sphere will be supposed to consist of concentric layers, withhi 
 which the density, and consequently the refraction which de- 
 pends on it, is constant. Now for the purpose of estimating the 
 change of direction of the rays in each layer on account of 
 
GENERAL LAWS OP EEPRACTION OF LIGHT. 107 
 
 refraction, it is necessary to know the laws of the refraction of 
 light. These are four, namely the following : 
 
 (1) When a ray of light falls on any surface of a body, 
 which separates two media of different refractive powers, if we 
 draw a tangent plane at the point where the ray of light is 
 incident, a normal to this plane, and finally a plane through 
 it and the path of the ray, the ray will not deviate from this 
 plane after its entrance into the body. 
 
 (2) If we suppose the normal to be produced outwards, 
 then for any media whatever and for all angles of incidence, 
 the sine of the angle of incidence (that is, of the angle between 
 the incident ray and the normal) will bear a constant ratio to 
 the sine of the angle of refraction (that is, of the angle be- 
 tween the refracted ray and the normal). This ratio is called 
 the Befractive Index for these two media. 
 
 (3) When the refractive index between two media A and B 
 is given, and also that between two other media B and 0, the 
 refractive index between the media A and G is the compound 
 ratio of the indices between A and B and between B and G. 
 
 (4) If /i be the refractive index for the passage from one 
 
 medium A into another B, then is — the refractive index for 
 
 the passage from the medium B into the medium A. 
 
 Let now (fig. 3) be a place on the surface of the earth, 
 G the earth's centre, 8 the true place of a star, GJ the normal 
 to the point J, at which the ray of light 8J meets the first 
 layer of the atmosphere. Then if the refractive index for this 
 first layer be known, we can by the laws of refraction find the 
 direction of the refracted ray, and we obtain for the second layer 
 a new angle of incidence. Suppose now that there are n layers, 
 and suppose GN to be the line from the centre of the earth 
 to the point, at which the ray of light meets the w"" layer; 
 let also in be the angle of incidence, f^ the angle of refraction, 
 /*„ the refractive index from vacuum to the m*, and fi^^ that to 
 the n 4- 1* layer ; then we have* 
 
 • These reftaetire indices are fractions, whose numerators are greater than unity. 
 For layers at the surface of the earth for example /x = 1 •000294, or nearly —-- . 
 
108 GENERAL LAW OF ATMOSPHERIC EEPEACTION. 
 
 sin 4 ^ /^nti 
 sin/„ //.„ ■ 
 
 If then N' be the point at which the ray of light meets 
 the w + l*" layer, we have in the triangle NCN', denoting by 
 r„ and r^^ the distances of the points ^and N' from the centre 
 of the earth, 
 
 sin/, ^y^, 
 sinvn »•« ' 
 and, combining this equation with the former, 
 »•„ sin 4 /*„ = r^i sin ^^^ /*„,. 
 
 Consequently, since the product of the distance from the 
 centre into the refractive index and the sine of the angle of 
 incidence is the same for all strata of the atmosphere, we thus 
 obtain, if we denote this constant by a^, as the general law 
 of refraction 
 
 r/isin«=a, (a), 
 
 where r, fi and i must belong to the same point of the atmo- 
 sphere. For the surface of the earth, t, (that is the angle which 
 the last tangent of the ray makes with the normal), will be 
 equal to the apparent zenith-distance of the star. Thus, if we 
 call a the radius of the earth, and /i„ the refractive index for a 
 stratum of the atmosphere at its surface, we obtain for the deter- 
 mination of the constant a„ the equation 
 
 a/i„sina = aj (J). 
 
 Assume now that the density of the atmosphere varies con- 
 tinuously, and that consequently the altitude of the stratum 
 within which the density may be regarded as constant is in- 
 definitely small, then the path of the ray through the atmosphere 
 will be a curve, whose equation can be determined. Intro- 
 ducing polar co-ordinates, and calling v the angle which any 
 value of r makes with the radius CO, we easily obtain 
 
 '■^ = tan^ (c). 
 
 The direction of the last tangent is, as has just been seen, 
 the apparent zenith-distance z, whilst the true zenith-distance f 
 
DIFJTEEENTIAL EQUATION FOK EEFEACTION. 109 
 
 is the angle which the original direction 8J of the ray produced, 
 makes with the normal. This ^ has, it is true, its vertex at 
 another point from that at which the eye of the observer is 
 situated, but since the atmosphere is only of small elevation, 
 and on the contrary the luminous bodies ar^ very distant, and 
 especially since the refraction itself is only a small angle, the 
 difference between the angle f and the true zenith-distance, 
 which is to be observed at 0, is quite insignificant. Even in 
 the case of the moon, in which this difierence is most con- 
 spicuous, it amounts only to a very small part of a second. 
 We may therefore assume that the angle ^ is the true zenith- 
 distance. 
 
 At the point N, to which the variable quantities i, r, and 
 fi apply, draw now a tangent to the ray, which makes with the 
 normal CO the angle ?' ; then 
 
 ^' = i+v {d}. 
 
 Diflferentiating then the general equation (a) logarithmically, 
 we obtain 
 
 1- cotan t.di+ — = 0, 
 
 r fj, 
 
 and from this equation, in connection with the equations (e) 
 and ((Z), 
 
 dt = — tan i — ; 
 /* 
 
 or, if we eliminate tan i by the equation 
 
 sin ^ a. 
 
 tan I = ■ 
 
 V(l-sinV)~v'(»-y-0' 
 and put for Oj its value «/*„ sin z, 
 
 — u,„ sins du, 
 The integral of this equation, taken between the limits 
 
110 GENERAL DIFFERENTIAL EQUATION. 
 
 ?' = ? and ?' = «, gives then the amount of refraction. If 
 ■we put 
 
 - = l-s, 
 r 
 
 we can also write* the equation thus, 
 
 ^w ^ (1-s) singt?/! ^ _ .^> 
 
 /. yi cos'^ _ ^1 - -^) + (2* - *') sin^g J 
 
 To integrate this equation, we must now know the value 
 of s as a function of /i. This latter quantity is dependent on the 
 density, and we are taught by the physical sciences that the 
 quantity /*'—!, which is also called the refractive power, is pro- 
 portional to the density. Introducing then the density p as a 
 new variable, given by the equation 
 
 lj^-l=cp, 
 
 where c is a constant, we obtain 
 
 ^- i(l_s)sin«.c.-^ 
 
 or, if we put 
 
 a (1 — s) sin s — 
 |l - 2a (l - -^) j- . a/|cos' s-2a(l~£-^ + {2s -s») sin'' si 
 
 .. ''" ^° ig)' 
 
 The coeflScient 
 
 • Pq is the value of p corresponding to yn = /ug. 
 
 Since also /[»• = 1 + cp, 2 log, ;tt = log, {1+cp); 
 
 p. "-'i+cjo' 
 whence, bj an easy transformation, we get the form of the equation given in the text. 
 [Translatok.] 
 
APPLICATION OP MAEIOTT'S LAW OP PEESSUEE. Ill 
 
 is the square of the ratio of the index of refraction for a stratum 
 at radius r to the index of refraction for a stratum at the surface 
 of the earth. But since, for the limits of the atmosphere, /* = 1, 
 while, on the contrary, for the refraction out of vacuum into strata 
 
 at the surface of the earth u. = — r-— , the ratio — always lies 
 
 '^'' 3900' /i(, •' 
 
 between these narrow limits. Consequently the quantity a is 
 
 small, and instead of the variable factor 
 
 !-.»(.-£), 
 
 we can take its mean value between the two extreme limits 1 and 
 1 — 2a, namely, the constant value 1 — a. 
 
 In order to be able to integrate the equation {g), s must be ex- 
 pressed as a function of p, that is, we must determine the law, accord- 
 ing to which the density of the atmosphere varies with the height 
 above the surface of the earth. Considering first the temperatm-e 
 of the atmosphere as uniform, the density becomes a simple 
 function of the pressure or of the elasticity of the air, and we 
 have according to Mariott's law, if ^ denote the pressure of the 
 air at a point whose distance from the centre of the earth is r, 
 
 If then r be increased by dr the decrement of pressure is 
 equal to the small column pdr, and if this be multiplied into g, 
 the force of gravity corresponding to the distance r, we obtain 
 
 dp = -gpdr, 
 or, since also 
 
 a" 
 
 where g^ represents the force of gravity at the surface of the 
 earth; 
 
 consequently also 
 
 p,^=g,apd{^. 
 
112 LAW OP VAKIATION OF DENSITY OP THE AIR. 
 
 Integrating this equation, and remembering, in the determi- 
 nation of the constant, that for p =/)„, r = a, we obtain 
 
 where e is the base of the natural system of logarithms. Taking 
 then I for the height of a column of air of the density p* which 
 corresponds to the force of gravity ^^^ and to the pressure p^, equili- 
 brium holds when 
 
 we finally obtain, if we again put 
 
 - = l-s, 
 r 
 
 This equation gives thus for every value of s and therefore 
 for every value of r the density of the air on the supposition that 
 the temperature is uniform throughout the whole atmosphere. 
 Since this supposition does not correspond to the case of nature, 
 because the temperature of the atmosphere decreases with the 
 elevation according to an unknown law, it becomes necessary 
 to make some hypothesis concerning the law according to which 
 the density of air varies. 
 
 Bessel, to whom we owe the most accurate Kefraction Tables, 
 adopted for this law the following expression 
 
 ft-l as 
 
 where A is a constant, which must be determined In such a man- 
 ner that the refractions calculated according to this law shall 
 correspond to the observed refractions. Putting then 
 
 h-l 
 
 -JT"-^ W' 
 
 and replacing, in equation (g), p by its value given by the 
 formula 
 
 P = Poe"" ii), 
 
 ' This I is, for the temperature of 8° Reaumur or 50° Fahrenheit, equal to 4226-05 
 toises. It is equal to the mean height of the barometer at the level of the sea multi- 
 plied into the density of mercury relatively to that of air. 
 
INTEGRATION OF THE EQUATION FOE EEFEACTION. 113 
 
 we obtain 
 
 ,o, _ aySe""' sin s {l—s)ds 
 
 ^ ~ (1 - a) Vicos'' « - 2a (1 - e"") + (2s - /) sin'' z} ' 
 
 or, if the quantity under the radical sign be expanded in powers 
 of s, while — s" sin'' z is considered as a small increase of the 
 remaining terms under the radical ; 
 
 a/Se'^' sin s ds 
 
 dt' = 
 
 (1 - a) {cos' a - 2a (1 - e""') + 2s sin' a}* 
 
 a^sdse'^ sin z {cos' a — 2a (1 - e'"') + s sin' s} 
 (1 - a) {cos' a - 2a (1 - e~^') + 2s sin' a}« 
 — &c. 
 
 {k) 
 
 7. Integrating the equation {k) for s between the limits 
 s = H, where H is the height of the atmosphere, and s = 0, we 
 obtain the value of the refraction. In order to obtain however 
 the positive sign, the limits in what follows are to be taken in 
 the inverse order, so that the value then found for the refraction 
 is to be so understood, that we must add it algebraically to the 
 apparent place in order to obtain the mean place. 
 
 Since now, from the height of the atmosphere being very 
 small compared with the radius of the earth, s is always a small 
 quantity, the first term in the series for dl^' will be considerably 
 greater than those which follow, which have only a slight in- 
 fluence, and it can be easily demonstrated that the second term 
 is so small, that it may be always neglected. This term in fact 
 attains its greatest value when a = 90°, and consequently the 
 observed dbject is in the horizon, viz. 
 
 a^sJse~^'{s-2a(l-e"^')} 
 (l-a){2s-2a(l-e-^')P' 
 In order to integrate this expression, we must expand it in 
 a series of powers of s. But it must be borne in mind that the 
 most important part of the integral is that where s is very small, 
 and that thus the first term only is required to be taken, and 
 we obtain then (since 1 — e'^' = ^s) 
 
 a^^/s.dse-^'{l-2o^) 
 (l-a)2«(l-a/8)« • 
 B. A. 8 
 
114 INTEGEATION OP THE EQUATION FOE EEFRACTION. 
 
 We should have now to integrate this expression for par- 
 ticular values from s = to s= IT, hut we may without sensible 
 error take also for these limits and oo , and we shall then find 
 for the integral, observing that 
 
 Jo ^ 
 
 the following value : 
 
 « (1 - 2a^) ^1 
 
 
 4 (1 - a) (1 - a/3)» ■ 
 
 Substituting in this the numerical value of the constant found 
 farther on, we obtain the result 0",55, and since this is the maxi- 
 mum value of the integral of the second term, which besides only 
 applies to the horizon, this term and a fortiori the following 
 terms may be neglected. 
 
 The expression is thus reduced to the first term only of the 
 equation (/c), viz. 
 
 ,^, KjSe"^ sin z ds ,. 
 
 ^ " (1 - a) {cos" s - 2a (1 - e"^) + 2s sin-" zf ^ ^' 
 
 Introducing the new variable 
 
 a ( 1 - e-^) ^ 
 
 s = W^ '- + s, 
 
 sm z 
 
 the denominator becomes simply 
 
 (l-a)(cos'3 + 2s'sin"0)», 
 
 and we have only then to express e'^'ds in terms of s'. But 
 
 since 
 
 1 — e~^' 
 
 s = s' + a.^T-r, — (m), 
 
 sm a ^ ■" 
 
 we can expand the quantity e^% by substituting for s the value 
 from this equation, in powers of «. Putting for instance e^' = u, 
 we have by Maclaurin's theorem 
 
 u=U+aq + o!'q + ... + arq„+ (a), 
 
 • Tills integral is a function represented in analysis bj r, called Euler's integral 
 andr(J). 
 
INVESTI^TION OF LAPLACE'S THEOREM. 115 
 
 where U is the yalue of u when a = 0, or in this case = e"'*', 
 
 We thus need only expand the value -^ when a = firom 
 equation (m). But writing this equation thus*, 
 
 we have 
 
 /ds\ _ (ds 
 
 [daj ~y ^t. 
 and also 
 
 (S)=(S)(i)-e)©=HS) ^- 
 
 If now we integrate ^^cfo with respect to t and afterwards 
 again differentiate with respect to t, whereby the value of ydu 
 will not be altered, we obtain 
 
 /c?m\ _ ldjydu\ 
 \di) " \~drj ' 
 and consequently 
 
 \dai') \ da.dt J' 
 
 But there follows from equation (;S), if we put ^ydu instead 
 of u, 
 
 (djydu\ _ (dJfdiA 
 
 \ da J \ dt J' 
 thus 
 
 fd'u\_ fd'ffdu \^ 
 
 W) V de )'. 
 similarly we obtain 
 
 and generally 
 
 ld''u\_(d''jiy'-du 
 
 ld^u\ _ la 
 
 da") V df j dr"^ 
 
 ''■-■{/(S)} 
 
 ■ b)- 
 
 • What follows is nothing but Laplace's investigation of his Theorem, and might 
 have been omitted. — Tbanslaior. 
 
 8—2 
 
116 INTEGRATION OF THE EQUATION FOE ^EFEACTION. 
 
 And since now for a = 
 
 r-^-m ^^■' 
 
 we at last obtain by combining equations (a), (7), and (S), 
 
 sin" z ^ ' 
 
 e '" = e 
 
 1 . 2 sin* » ds 
 
 — &c. 
 
 («)• 
 
 a"^ t?-'{(i-e-^)"e-^} 
 
 1.2.3...Msin"s' rfs'"-' 
 
 In the equation (Z) however the numerator was to be ex- 
 pressed in terms of the new variable s. But since 
 
 the equation (Q will thus become, if we expand d . e'" by equa- 
 tion (e), 
 
 d^' = 
 
 a^ sin zds' 
 
 (l-a)(cos''s+2s' sin's)* 
 
 
 g d.[{e-^-l)e-^} 
 
 sin' z ds' 
 
 + 
 
 _^ d'{{e-^-iye-^} 
 
 1.2„.sin*s 
 
 _a;; (?"{(e-/^'^-l)''e-/^} 
 
 -1.2.3...wsin*'2; 
 + &c., 
 
 <fe" 
 
 w, 
 
 where in the general term the upper or lower sign must be taken 
 according as n is even or odd. 
 
 But we have 
 
 a 
 
 - 1.2.3...wsm"'2 
 
 <^"{(e-^'-l)"e-'"l 
 
INTEGRATION OP THE EQUATION FOE REFRACTION. 117 
 
 1.2,3...«sin'"s 
 
 + 
 
 w (w — 1) 
 
 -M.w"e""^'' 
 
 1.2 
 
 (n-iye 
 
 ,« .-(n-l)^ 
 
 as is evident, if we expand 
 
 (e--^- 1)" 
 
 in a series, multiply every term by e"^', and then differentiate n 
 times with respect to s' ; accordingly we obtain, if we substitute 
 in equation («) foi the separate terms, this series : 
 
 t?r= 
 
 yS sin sds' 
 
 1— a' (cos''s+2s'sui'»)* 
 
 sm ^ 
 
 + 
 + 
 
 a'^ 
 
 1 , 2 sinz 
 1.2.3 sin's 
 
 (3V"»'-2.2te-'*+e-^'') 
 
 (4V 
 
 ,-*»•'. 
 
 3.3'e-'^'' 
 
 3.2. 
 
 + &C. 
 
 ^1-2 
 
 ..(o); 
 
 and this equation has now to be integrated between the proper 
 limits. And the integral must always be taken from the surface 
 of the earth to the boundary of the atmosphere, whose height 
 we suppose to be IT, and therefore from r = a to r = a + H, 
 But since at the boundary of the atmosphere the density is 
 nothing, and consequently the ray of light, if we consider it as 
 proceeding from the eye of the observer, after reaching that 
 point, experiences no further refraction, we can therefore take for 
 the limit also r = a and r = oo ; and since 
 
 s=l--, 
 r 
 
 the limits, in relation to these variables, will therefore become 
 s = and s = 1, and with reference to s, will be s = 0, and 
 
118 INTEGEATION OP THE EQUATION FOE EEFEACTION. 
 
 sm » 
 
 But, following out equation (t), since /8 is a 
 
 very small number, while at the boundary of the atmosphere 
 the density is nothing, the formula must therefore at least give 
 so small a value for p, that we may neglect it without any sensi- 
 ble error. The quantity e""^'' thus becomes, if we substitute the 
 superior limit for s, collectively very small, and we can therefore 
 take in the integral of equation (o) without sensible error, s' =0 
 and s' = CO as the limits. 
 
 The terms of equation (o) include now, each one of them, a 
 factor of the form 
 
 ^ds'e~ 
 
 •W. 
 
 tion 
 
 \/(cos*«+ 2s' sin's) 
 
 Introducing here a new variable t, obtained from the equa- 
 
 1 cos' s , _ 1 
 2im^"^* ~^' 
 
 s 
 
 the differential expression (rj) becomes 
 
 and consequently if we put 
 
 /r4r~^^°''^W 
 
 where the integral is to be taken between the limits 
 
 t = cotz ,/ f y j , and « = oo , 
 we get 
 
 i„ (cos's + 2s' sin's)* -^^^P;-^. 
 therefore if the refraction be represented by Sz, 
 
 •(A), 
 
 Ss = - 
 
 1-a 
 
 V(2/3) 
 
 
 + 
 
 sm s 
 
 (2*^^(2) -^(1)} 
 
 1.2sin*» 
 
 7-{39^(3)-2.2«^(8)+Vr(l)} 
 
 + L2:3^I5^s ^*^W-3-3^^(3) + 3.2it(2)-f (1)} 
 + &c 
 
INTEGRATION OF THE EQUATION FOB KEFKACTION. 119 
 an expression for which, since 
 
 1 — 3! + ; : 
 
 1.2 1.2.3 
 
 we can also write the following : 
 a 
 
 + ... = e- 
 
 ^^-—^/m 
 
 •(B). 
 
 __o£_ 
 
 „Q Sap 
 
 + -=^2*e~s"E5ii^(2) 
 
 + &C 
 
 The calculation of the amount of refraction is thus made to 
 depend upon the calculation of the transcendent ^Jr (r), or upon 
 
 fe~''dt, taken between the limits of t = * /(^) • cot s, and t=<x> . 
 
 Knowing then this and the numerical values of the constants a 
 and ^, we can find, by the formula (B), the amount of refraction 
 Ss for every apparent zenith-distance s. 
 
 8. Putting 
 
 ^cofa=r', 
 
 we have now therefore to determine the transcendent 
 
 J T 
 
 For calculating this quantity two methods are preferred. 
 The first expands the transcendent in a series, which is obtained 
 by integration by parts, and which is continued to infinity, but 
 froin which we can nevertheless obtain the value to any required 
 accuracy, since it possesses the property that if we stop at any 
 particular term, the following terms do not together amount to 
 more than the one last taken. We have, viz. : 
 
 
 dt 
 
120 INTEGEATION OP THE EQUATION FOE EEPEACTION. 
 or, integrating by parts, 
 
 ~ 2 t 2} f 
 similarly we obtain 
 
 2r f~ 2i"~"Si •' 
 
 T 
 
 _ 1 1 e"'' 1 3 f _pdt 
 ~'^2-2-T'^2-2r t*' 
 
 3 f^pdt 3r\ 2 J dt 3 le-^ „ 
 
 if F = i dt F=-4-2T-'^"- 
 
 U ^-ij — dt — 
 
 consequently, at last 
 
 y 2t I 2e'^ {2ef {2fy "^■•* 
 
 1.3.5...(2re-l) _ 1.3.5. ..(2w + l) /" .^ dt] 
 
 or, putting in the limits, 
 
 r.p, £^f 1 1-3 1.3.5 
 
 )/ '^^ 2T\ 2T^'^ {2Ty {2Ty^'" ' 
 
 1.3.5...(2w-l) _ 1.3 .5... (2w + l) 
 
 {^Ty ^ 2' 
 
 ,n+l 
 
 
 The factors of the numerator constantly increase ; they will 
 therefore at length become greater than 2T', and from this point 
 then every term increases continually, since the quantity put 
 into the numerator is greater than the corresponding quantity 
 put into the denominator. 
 
 But considering now the remainder 
 
 1.3.5... (2w+l) r _p_^ 
 
 it will be easy to shew that this is smaller than the term im- 
 mediately preceding. The value of the integral for instance is 
 evidently smaller than 
 
 r dt 
 
 J T 
 
 j2n+2 ) 
 
■INTEGRATION IN FORM OF A CONTINUED FRACTION. 121 
 
 multiplied into the greatest value of e"'" between the limits T 
 and CO , that is to e"^° ; and since now 
 
 /" 
 
 dt 
 
 an+i 
 
 the remainder will always be smaller than 
 1.3.5 ■..(2>i-l) 
 
 But this expression is nothing more than the previous term 
 with the contrary sign. Consequently, if for example we leave 
 off with a negative term, the remainder will be positive, but 
 smaller than the previous term. Therefore, in order to obtain 
 the correctest value possible of the transcendent by the computa- 
 tion of the series, we require only to continue it up to a term 
 which is very small, and there is then to be feared only an error 
 which is smaller than that previous very small term. 
 
 The second method of calculation consists in transforming, 
 as Laplace was the first to show how, the transcendent into 
 a continued fraction. Putting 
 
 'j\-''dt=U (a), 
 
 we have 
 
 f-<' 
 
 e-'^dt + e'^e-'' 
 r 
 
 = 2tU+l (/3). 
 
 But the w"' differential coefficient of a product ajy is 
 
 d^.xy _ d^x d^-^x dy n{n-l) d'^'^xd^y 
 
 de ~ dt y^'^ df-' dt^ 1.2 df-^ "^+®°--' 
 
 therefore we have also 
 
 = 2t -^-r- + 2m 
 
 in-1 i 
 
 dt'"'- dt ^ de 
 
 an equation which can be written in the following manner, 
 if we represent the product 1 . 2 . 3 . . . w by m ! : 
 
122 INTEGRATION IN FORM OF A CONTINUED FEAQTION. 
 n + 1 d''*'U „^ d^U „ d^-^U 
 
 {n + 1)! dt"^' ~ nldt"^ {n-l)l dr' ' 
 
 d"U 
 or, if be represented by Z7„, 
 
 This equation is to be used from n=\, in whicli case U^ 
 becomes U. We obtain from it 
 
 consequently 
 
 1 p: 1 2« 
 
 <2 TT ~ TT ~ TT ' 
 
 or 
 
 U. 2f 
 
 ^^^^ri_(,^,)^ 
 
 But, referring to equation (yS), 
 
 therefore 
 
 _ 1 
 
 ^* f7 ^ 2^Tr 
 
 but it follows from equation (7), that 
 1 K 2? 
 
 (7). 
 
 2*^ 1-2^ ^^ 
 
 substituting then this in the former equation and continuing the 
 development, we obtain 
 
INTEGRATION IN FORM OP A CONTINUED FRACTION. 123 
 
 
 
 U= 
 
 It 
 1 + 
 
 1 
 
 
 » 
 
 
 H 
 
 ^^he 
 
 
 
 1 + 3 
 
 1 
 
 2i" 
 
 
 
 
 
 
 
 1+&C.; 
 
 and, 
 
 if 
 
 we put 
 
 1 
 
 22T2- 
 
 '2> 
 
 
 
 
 
 2Te'^{ e- 
 
 ^dt = 
 
 1 
 
 
 
 
 
 
 
 1 + 22- 
 
 
 
 
 
 
 
 1+32 
 
 
 
 
 
 
 
 1+49- 
 
 (a) 
 
 1+&C. 
 
 If i is very small, we are enabled advantageously to em- 
 ploy a third method for calculating the transcendent. We have, 
 viz.: 
 
 /•* /•" rT 
 
 e-''dt= e-"dt- e-^dt. 
 
 J T Jo Jo 
 
 But the value of this last integral is easily obtained by ex- 
 panding e-*^ in a series, viz. : 
 
 l> 
 
 ■''dt = 
 
 3 2 5 2.3' 7 
 
 since 
 
 
 />-t. 
 
 value of 
 
 
 f e-^dt 
 
 can be obtained from it, which must further be multiplied by 
 e'"" in order to find the value of the transcendent represented by V^. 
 
124 EEFEACTION FOE ANY STATE OF THE ATMOSPHERE. 
 
 We ^re now always enabled, from what has been shewn, 
 to calculate the value of '^{r). On account of the constant 
 employment of this transcendent it has been brought into a 
 tabular form, and may be found, for example, in Bessel's 
 Fundamenta Astronomioe. The first division of the table thus 
 given has T for the argument, and gives the values for every 
 hundredth part from T= to T=\. But since the transcendent 
 is more nearly inversely proportional to its argument the greater 
 T is, Bessel chooses for the values which have a greater argu- 
 ment than T =\, the common logarithms of T as arguments. 
 This second division of the table then extends from the loga- 
 rithm 0-00 to the logarithm 1-00, or from T= 1 to 7= 10, which 
 will suffice for every requirement. 
 
 9. The formula (B) contains the two constants a and /8, 
 
 whose numerical values must be known, if we wish to determine 
 
 the amount of refraction for any zenith-distance z. If the state 
 
 of the atmosphere were always the same, these constants would 
 
 also always have the same values. But since the density of 
 
 the atmosphere depends upon the readings of the barometer 
 
 and thermometer at the time of observation, a, which indeed 
 
 u? —I 
 is no other than the quantity '^^ „ (where /*„ represents the 
 
 index of refraction from vacuum into the stratum of air at the 
 surface of the earth*), will likewise be a function of the same. 
 
 Similarly /3 or —j-j— • « will depend upon the height of the 
 
 thermometer, since the quantity I or the height of a column 
 of air of .the density p^ , which corresponds to the pressure of 
 the atmosphere, is a function of the temperature of the air. 
 Consequently in order now to find the refraction for a given 
 state of the atmosphere, that is, for given readings of the 
 barometer and thermometer, we must in the calculation of the 
 formula (B) employ the values of the constants really correspond- 
 ing to this state of the atmosphere. 
 
 Now the atmosphere from 0" to 100" of the Centigrade Ther- 
 
 • Or also „,.°f° ■ ■ 
 2(1 + cpo) 
 
REFRACTION FOR ANT STATE OF THE ATMOSPHERE. 125 
 
 3 
 
 mometer is expanded by ^ of its volume or more correctly by 
 
 o 
 
 0'36438. Consequently, if a volume of air of a given temperature, 
 which Bessel has taken 8° Reaumur = 10" Celsius = 50" Fahren- 
 heit*, be equal to unity, the same volume of air at a temperature 
 of t" Fahrenheit is 
 
 1 + (t - 50) ^'^,lf^ = 1 + (t - 50) X 0-0020243. 
 
 If therefore Ig represents the value of the constant I for the 
 temperature t = 50°, then for every other temperature t, 
 
 1=1,{1 + {t- 50) 0-0020243}, 
 
 and similarly, if p^ is the density of the air for t = 50°, for any 
 other temperature 
 
 „^ Po 
 
 ^ 1 + (t- 50) 0-0020243" 
 
 But the density of the air depends also upon the height of the 
 barometer ; and since it also, according to Mariott's law, varies 
 directly as the pressure, we shall have, if we again represent by 
 p„ the density for a given height of the barometer, which 
 Bessel takes at 29*6 English inches, then for any other baro- 
 meter-reading b, 
 
 h 
 
 Since now a, for changes of the density so small as here come 
 under consideration, can be assumed to be proportional to this 
 density, we obtain, if we Call the value of a for the normal con- 
 dition of the atmosphere, Oq, 
 
 b 
 
 "° 29-6 
 
 a = 
 
 1+ (t- 50) 0-0020243 • 
 
 Now Bessel, in his Fundamenfa Astronomioe, has deter- 
 mined the value of the constant a^ from Bradley's observations, 
 and found from thence, that 
 
 a„ = 57",538, 
 
 • According to the thermometer employed by Bradley. 
 
126 EXAMPLE. 
 
 and further for the constant li, 
 
 A = 116865-8 toises. 
 
 From this follows, if we assume the radius of curvature for 
 the Greenwich Observatory to be equal to 3269805 toises, the 
 value of the constant y3 for the normal condition of the atmosphere, 
 
 namely )8„ = 745"747, 
 
 with which we have all the required data, in order to be 
 able to calculate, from the formula (B), the refraction for any 
 zenith-distance and any condition of the atmosphere. If for 
 example there were required the refraction for 80° zenith-distance 
 and for the normal condition of the atmosphere, we have 
 
 log -^— V(2,S) = 3-34688. 
 
 Moreover, we obtain for the logarithms of the separate 
 functions -^^ according to the formulae (a) or (5) in No. 8, and for 
 the logarithms of the factors of these quantities, the following 
 values : 
 
 n=l, 0-00000 0-00000 9-14982 9-90685 
 
 w = 2, 0-15051 9-33142 9-00745 9-81369 
 
 71=3, 0-71568 8-36181 8-92228 9-72054 
 
 m = 4, 1-50515 7-21610 8-86128 9-62738 
 
 n = 5, 2-44640 5-94546 8-81362 9*53423 
 
 w = 6, 3-46568 4-57791 8-77473 9-44108 
 
 w = 7, 4-64804 3-13118 8-74169 9-34792 
 
 From these we obtain then for the separate terms of the 
 formula (B) within the brackets, 
 
 first term = 0-113938, 
 
 second = 0-020094, 
 
 third = 0-005252, 
 
 fourth = 0-001622, 
 
 fifth =0-000549, 
 
EXAMPLE. 127 
 
 sixth term = 0-000182 
 seventh =0-000074 
 
 sum = 0-141711 
 
 log = 9- 15140 
 log const. = 3- 34688 
 refraction = + 5'.15",0 
 
 Bessel has more recently found that the refractions calculated 
 in this manner must be multiplied by 1-003282, in order to re- 
 present the Konigsberg observations. Performing this multi- 
 plication, we obtain for the refraction at 80° zenith-distance 
 and for the normal condition of the atmosphere 
 
 + 5'.16",0. 
 
 If the refraction were required not for the normal condition 
 of the atmosphere but for a temperature of t° Fahrenheit and for 
 a height of h inches of the barometer, we must first seek the 
 values of a and /3 corresponding to these according to the formula 
 given above, and with these perform the calculation of (B). 
 
 But in order to do away with the necessity of this troublesome 
 calculation, the refractions have been reduced to tables which 
 have the apparent zenith-distance for the argument. One table 
 gives the values of the refraction for the normal temperature and 
 the normal barometer height, or of the so-called mean refraction. 
 Another table then gives the corrections, which must be added to 
 this mean refraction, in order to obtain the true refractions 
 corresponding to the existing conditions of the atmosphere. For 
 the calculation of this last table it is necessary to investigate 
 the analytical expressions for the changes in the refraction de- 
 pendent upon the readings of the thermometer and barometer. 
 
 10. Let r denote the true and hz the laaib refraction, then, 
 ^ = S« + ^(r-50) +^^(6-29-6)....; (a). 
 
 But, by formula (B) in No. 8 we have by putting 
 «/3 _ 
 
128 EFFECT OP YAEIATIONS OF TEMPERATURE AND DENSITY. 
 (l-a)S. = sin=../^..X|«' 1.2.3... (.-l) ^-^-^--^W| 
 
 = sin'' a , 
 
 where for n we must substitute all whole numbers beginning 
 with unity. For the sake of brevity let 
 
 2n-l j^n ^-nx 
 
 n' t («) = q„, and t ^^^[^ _^ ^„ = a, 
 then the equation becomes simply 
 
 (l-a)Ss = sin'sy'^. Q^. 
 
 On account of the smallness of a, we may consider the factor 
 1 — a as constant, so that the variables b and t occur only in x 
 and /3, t alone appearing in /8, while in x both t and b appear. 
 Taking now first the differential coefficient of Q„, we have 
 
 But we have also, 
 
 (dQ^\ _ 1-x ^ x''e-^ 
 \dx) X ^ 1.2.3. ..w"^"* 
 
 Or, if we denote the series 2 ^j — ^-— nq„ by Q„', 
 
 i^;=-^^« w; 
 
 moreover 
 
 (dQ„\^^ x\e-^ (dq^\ 
 \d^J ^1.2.3...wU/3/' 
 
 But we have also, by employing the known law for the 
 calculation of the differential coefficient of a definite integral for 
 one of its limits. 
 
 
VARIATIONS OF TEMPERATURE AND DENSITY. 129 
 
 therefore we shall also have 
 
 (dQ^ cof g , _ cotg ^ wVe'"^ ,^ ' 
 
 Moreover, since 
 
 (D-f ©— [i)'"'' 
 
 Consequently from the formula (J), (c), {d), (e), 
 
 Since moreover the variable h only appears in x, we have 
 
 t?g„ _ (dQ^ fdx\ _ 1-x (dx\ 
 ' dh ~\dx)\dh)~ X '^•'{dbj' 
 
 or since 
 
 /dx\ _ X /da\ 
 
 \db)~a\db)' 
 
 (t)=^-«'. w- 
 
 But, by differentiating the expression for (1 — a) Bz, we 
 obtain 
 
 ,, , d.Sz 1. 1 fd^ , . , /^ fdQA 
 
 and 
 
 ,, .d.Bz ,. , /J fdQA 
 
 • By the Formula in No. 9, where £ represents the number 0020243. 
 B. A. ■ 
 
130 VARIATIONS OP TEMPEEATUEE AND DENSITY. 
 
 Substituting in these the values of the differential coefficients 
 of Q„ from the equations (/) and {g), and, in the same manner, 
 
 for (^] its value from the first of equations (e), we find at 
 
 length, 
 
 
 ,n _— nic 
 
 ( T — 7 1 ^ cor » > + e sm' z cot 3 \ 7 -, ] S 
 
 ,A- ;/ 1.2.3... m' 
 
 (A 
 
 , . . d.Bs . 2 /2 1— a; „, 
 
 and (i_„)_^^ = sm«y'^,^g^e„. 
 
 By means of these formulae we could now reduce to a tabular 
 form both the values of the differential coefficients and the values 
 of the mean refraction, the argument being the apparent zenith- 
 distance z, and then compute the true refraction by means of 
 the formula (a). This formula is however not convenient. For 
 more convenient logarithmic computation, make 
 
 r = 
 
 {l+e(7--50)}n29-6J ^^^' 
 
 then are X and A functions of -4 — and ',, , which can them- 
 
 dr do 
 
 selves also be tabulated. 
 
 These functions can be now easily calculated. Since, namely, 
 {1 + e (t - 50)}-'^ = 1 - Xe (t - 50), &c., 
 
 we obtain from equation {£), taking only the first terras of the 
 series, 
 
 b 
 
 r = Sz-\e{T-50)Sz + A(±^-lj 
 
 Sz, 
 
EXAMPLE OP COMPUTATION OP TRUE KEFEACTION. 131 
 
 and, by comparing this equation with the formula (^', o 
 
 1 J.Ss' 
 
 \ = - 
 A = 
 
 0-0020243 &■ dr 
 29-6 d.iz 
 
 dz ' db 
 
 \ {C). 
 
 Example, 
 
 For apparent zenith-distance 80° we obtain, by taking into 
 consideration the terms which contain •^ (8) of which the loga- 
 rithm is 8-71302, 
 
 log ^'„ = 8-58950, 
 
 » « -nr 
 
 and, at the same time, 
 
 log (i.^') = - 7-20207, 
 
 and lastly 
 
 log^ = 5-71441, 
 
 \ = + 1-0428, 
 A = + 1-0042. 
 
 If, for example, we desired to calculate the refraction for 
 T = 15° Reaumur and J^^28'6 English inches, we should obtain, 
 since 15° Reaumur = €5°-7 FsRirenheit, 
 
 ^°g il+0-002024a(x-50)}- = ^-^«^^^' 
 h 
 
 and hence, since 
 
 S« = + 5'.16",0, 
 »• = + 4'. 55",5. 
 
 9—2 
 
132 CONSTRUCTION OF BESSEL'S EEFRACTION TABLES. 
 
 Bessel's Eefraction Tables are constructed according to for- 
 mulse [B) and ((7). The first table gives, with zenith-distance 
 for argument, besides the mean refraction, the quantities A and X. 
 The other tables give, with arguments the temperature observed 
 according to any one of the three scales in use, and the baro- 
 meter-reading either in Paris inches, English inches, or metres, 
 the logarithms of the factors 
 
 and 
 
 {1 + 0-0020243 (t- 50)} 29-6' 
 
 The first factor is the ratio of a volume of air at the tem- 
 perature T = 50° of the Fahrenheit thermometer used by Bradley, 
 to the volume at another temperature. If we denote by 1 the 
 volume of air at the freezing point, then will the volume at any 
 other temperature be 
 
 1 + 0-0020243 (t- 32°) 
 
 _ 0;36438 „ 
 
 Now Bessel has found that the thermometer employed by 
 Bradley gave all temperatures too high by l°-25, wbile the 
 freezing point of the same was too low by the same quantity. 
 Thus the temperature 50° corresponded to true temperature 48°-75. 
 And thus, if we denote by 7 the coefficient 
 
 1 . 
 
 1 + 0-0020248 (t - 50) 
 180 + 16-75 X 0-36438 
 
 {D), 
 
 ' 180 + (/- 32) 0-36438 
 
 where y is the temperature of the air expressed according to the 
 
 scale of a Fahrenheit tliermome^r. If we denote by r and c 
 
 the same temperature according to Reaumur and Celsius, we 
 
 have 
 
 180 + 16-75x0-36438 
 7 = . 
 
 180+ -rx 0-36438 
 4 
 
 180 + 16-75x0-36438 
 
 180 + -ex 0-36438 
 o 
 
 By means of these formulse log 7 has been tabulated. 
 
CONSTKUCTION OF BESSEL'S EEPEACTION TABLES. 133 
 
 For the barometer the reading 29*6 English inches was 
 taken as the normal reading. Since now Bessel has found that 
 this (Bradley's) instrument gave all barometer-heights too small 
 by half of a Paris line, this normal reading becomes 29-644 English 
 inches or 333'78 Paris lines. Barometers are now always divided 
 either according to Paris lines, or English inches, or metres. The 
 lengths of a Paris line, an English inch, and the metre corre- 
 spond to the normal temperature 13° Reaumur, 62° Fahrenheit, 
 and 0° Celsius. Denote then by &"*, ¥'\ and &""' the height of 
 the barometer expressed in terms of a Paris line, an English inch, 
 and the metre, as they are observed at any temperature t, then 
 these will not serve as the true measure, but, if s denote the ex- 
 tension of the scale from the freezing to the boiling point, then 
 will the barometer-height read at the temperature t be to that 
 which would have been read if t had been ec[ual to the normal 
 temperature T, in the proportion of 
 
 l + '-T: !+-«, 
 a a 
 
 if the length of the scale at the freezing point be taken for the 
 unit, and a denote the number of divisions between the freezing 
 and boiling points of the thermometer. Denoting then again 
 by r, f, c the observed temperatures according to E^aumur, 
 Fahrenheit, and Celsius, the height of the barometer referred to 
 the true standard will be 
 
 „ 80 + rg ,„ ].80 + (/-32)5 „, 100 + cs 
 
 80 + 13s ' 180 + 30s ' ■ 100 ' 
 
 where s = 0'0018782 on a scale of brass. 
 
 Since now an English inch = — — — Paris lines, and a 
 
 metre = 443-296 Paris lines, the three preceding barometer- 
 heights are, in Paris lines, 
 
 80+rs ^ „ 12 180 + (/-32)s 
 80 + 13s '1-065765' 180 + 30s 
 
 = &'»'. 443-296. ^^j^ («). 
 
134 CONSTRUCTION OP BESSEL'S KEFEACTION TABLES. 
 
 The normal beigbt of tlie barometer above exbibited or 333'78 
 Paris lines corresponds to the normal temperature 8° Reaumur, 
 or 50° Fahrenheit, or 10° Celsius, and is therefore also measured 
 on a scale of these temperatures. The normal barometer-height 
 reduced to true Paris measure will therefore be 
 
 B - qSS-78 ^Q + ^" 
 ^„-333 78.g^_j^^3^, 
 
 and by this quantity are the observed barometer-heights in (a), 
 when reduced to true Paris measure, to be divided. 
 
 Still however allowance must be made for the expansion of 
 the mercury, which, from the freezing point to the boiling point, 
 
 is equal to -—— part. Denoting this number by c[, the baro- 
 
 meter-height observed at a temperature t will be to that which 
 would have been observed, if t had been equal to the normal 
 temperature T, as 
 
 X + it : l+2r. 
 a a 
 
 We obtain therefore for the three different thermometers the 
 
 following correction-factors, by which the barometer-heights in 
 
 (a) are to be multiplied : 
 
 80 + 8g 180 + 18g , 100 + lOg 
 
 80 + 7-^' 180 + (/-32)2' ^'^ 100 + eg' 
 
 where r,f, and c are the readings of the thermometer attached 
 
 to the barometer. The complete expression for — — will there- 
 
 fore consist of two factors, of which one depends solely on the 
 height of the barometer, the other solely on the temperature of 
 the barometer. Denoting the first by B, the other by T, we 
 have 
 
 B = 
 
 and 
 
 _ 80 + 8g 
 333-78 ■ 80 + 8s 
 
 &'" 12 80 + 13s ''^ 180+18g 
 
 ' 333-78 ■ 1-065765 " 80+ 8.s ' 180 + 30s 
 
 :^^. 443-296. '' + ^'^ 1"" + l^g 
 333-78 80+ 8s 100 
 
 „^ 80 + rs ^ 180 + (/- 32) s ^ 10£+_cs 
 ~m + rq 180 + (/- 32)2 ~ IW + cq 
 
 {E). 
 
SIMPLER FOEMULiE FOE EEFEACTION, 135 
 
 By these formulse (E) and {D) are the tables constructed 
 which give log B with argument the harometer-height accord- 
 ing to the three scales, and log T with argument the height of 
 the thermometer attached to the barometer (the interior ther- 
 mometer) according to the three thermometer scales, and finally 
 log 7 with argument the height of the thermometer suspended 
 in free air (the exterior thermometer) in like manner for all 
 three scales. 
 
 These refraction tables of Bessel are to be found in Bessel's 
 TahulcB Begtomontance, in Schumacher's Hulfstafeln, and also 
 in the Astronomischen Jahrhuchern of Encke. Instead of the 
 quantity hz Bessel gives the quantity «, computed by the equa- 
 tion 
 
 Ss = a tan a, 
 so that the expression for the refraction is the following: 
 log»- = loga + logtana + \log7 + ^ (log 5 + log T)...{F). 
 
 The fundamental constant for the computation of a, is 
 57",538 multiplied by 1-003282 or 57"-727. 
 
 11. The theory of refraction developed in the preceding pages 
 is that given by La Place and Bessel. It corresponds to ob- 
 servations very completely to the greatest possible zenith-dis- 
 tances. There are however still other formulae for refraction which 
 are based on simpler laws for the density of the air, and which 
 therefore give much simpler expressions for the refraction, but 
 which deviate from observations considerably for great zenith- 
 distances. Since however it frequently happens that the simpler 
 analytical expressions are convenient in practice, it is desirable 
 in the following pages to deduce the most important of them. 
 
 In No. 6 was investigated the differential equation [g) 
 _ a — (1 — s) sin z 
 
 £?f = — ^" - _ 
 
 |l- 2a(l--^)[^|cos''s-2a(l--^) + (2s-*=) w^X 
 
 This equation is very easily integrated, if between s and r 
 we assume the law 
 
 -.=|.-.«(.-0-. 
 
a m-l 
 
 a sin a = w. 
 
 136 Simpson's formula foe refraction. 
 
 where m is an unknown quantity to be determined by observa- 
 tion. The equation will then for instance become 
 
 asin«.^ll-2afl-£)r-^' 
 
 or, by introducing another variable, given by the equation 
 
 very simply 
 
 '' (2m - 1) V(l - w'') ' 
 and so by integration 
 
 ' Ss = sin~^ w. 
 
 2m— 1 
 
 Since now at the surface of the earth the density of the 
 air is equal to p„, but is equal to at the boundary of the 
 atmosphere, we must take for the limits of the integral 
 
 2l«-l 
 
 w = sin « and w = (1 — 2a) 2 sin z, 
 and we then obtain 
 
 If r aw- 1 "I) 
 
 ^"'"i^^^^f ~^^''" L^^~^°') ' sins J, 
 or 
 
 2»l-l 
 
 sin [z - {2m - 1) Sz] = (1 - 2a) 2 sin z, 
 
 for which may be written briefly 
 
 ilf sin a = sin (a — ^&) (^^ 
 
 If a = 90°, we have, denoting the corresponding value of 
 Sa, that is the horizontal refraction, by /*, 
 
 M= cos Nh, 
 and thus we have in general 
 
 cos ^A sin a = sin (a — iS^Sa) (J). 
 
BEAbLET'S FOEMTJLA FOE EEFBACTION-. 137 
 
 This is the rule for refraction given by Simpson, which how- 
 ever was not investigated by him analytically but by practical 
 means. If the coefficient be suitably determined, the refraction 
 can be by means of it perfectly well determined to 85° zenith- 
 distance. 
 
 By adding to equation (a) the identical equation 
 
 sin z = sin z, 
 we obtain 
 
 sin 3 (1 + Jf } = 2 sin Ta - - Sz^cos ^ Sa. 
 
 And again, by subtracting the same identical equation, we 
 find 
 
 sin a f 1 — M\ = 2 cos ( a — — & sin — 8z. 
 
 {1 - ilf } = 2 cos (z - jSz) sin^ 
 
 Dividing one equation by the other, we obtain 
 
 *^^2^' = nrF*^°l'-2^^j' 
 
 or, by again introducing the horizontal refraction, 
 
 tan — Sa = tan f^AJ tanfa— — &j '(c). 
 
 This is the rule for refraction proposed by Bradley, which 
 can be briefly written 
 
 tan aBz=^^ tan (a — aSa) . 
 If Sz be small, we may be allowed to put 
 
 Sa = — tan (a — aSa), 
 a 
 
 from which it is seen that, as long as the refraction is small, it 
 may be assumed to be proportional to the tangent of the 
 apparent zenith-distance. 
 
 12. Since refraction causes all stars to appear at a greater 
 altitude than that which they really have, it has also this 
 
138 EFFECTS OF EEPEACTION ON RISING AND SETTING OF STAES. 
 
 effect, that it renders objects visible when they are in fact 
 beneath the horizon. It therefore accelerates the rising, and 
 retards the setting of the heavenly bodies. 
 
 In general we have the equation 
 
 sin A = sin ^ sin S + cos ^ cos S cos t. 
 
 If now a star be seen in the horizon, it is really beneath it 
 by a quantity which is equal to the horizontal refraction. De- 
 noting this by p, we have thus for the rising and setting of 
 a star the equation 
 
 — sin p = sin ^ sin 8 + cos ^ cos 8 cos <„ (a). 
 
 Call now T the hour-angle which the star would have at 
 its rising and setting if the refraction were equal to 0, and 
 we shall thus have 
 
 = sin ^ sin 8 4- cos cos 8 cos T. (J). 
 
 Thus from the two equations (a) and (5) 
 
 cos«„ = cosr(l :r^^^^-^). 
 
 \ COS 9 cos COS iy 
 
 The expression is still simpler, if there be only required the 
 correction AT oi the hour-angle T which is produced by the 
 refraction. This however is found with sufficient accuracy by 
 differentiating the equation for sin A with respect to h and t; we 
 then obtain, if A T be expressed in seconds of time, 
 
 AT-- 
 
 cos (j) . COS 6 sin I' ' 15 
 
 For Arcturus and the latitude of Berlin there was found, in 
 No. 12 of the first section, 
 
 T=7\ 53Vr= 118°. 16',8, 
 
 Computing now the foimula for AT on the assumption that 
 p = 33', we obtain 
 
 A!r=4". 22',0. 
 
ANNUAL ABEEKATION. 139 
 
 By SO much therefore will the rising or the setting of Aro 
 turus at Berlin be accelerated or retarded. 
 
 The zenith-distance of a star culminating south of the zenith 
 is <j>—B, and thus the south declination of a star, which at its 
 upper culmination is exactly in the horizon, is 90°— <^. (Section I. 
 No. 12.) But since now such a star by virtue of the refraction 
 appears at an altitude p, where p again denotes the horizontal 
 refraction, we see that all stars pass above the south horizon whose 
 south declination is less than 90° — ^ + p. In the same manner 
 we see that all stars at their lower culminations appear above the 
 north horizon whose north declination is greater than 90° — ^ — p. 
 
 Remark. On Eefraction compare Laplace, Mec. Oil. 
 Livre x.; Bessel, Fundamenta Astronomwe, page 26 et seq^.; 
 and the Preface to Bessel's Tab, Regiomont, page 59 et seq. 
 
 III. Abeeeation. 
 
 13. Since the velocity of the earth in its annual orbit 
 round the sun has a fixed relation to the velocity of light, we 
 see the stars from the surface of the earth thus moving, not 
 in the direction in which they really are, but always in ad- 
 vance by a small angle in the direction in which the earth is 
 moving. Imagine two separate instants of time t and i at 
 which the ray of light coming from a fixed star arrives suc- 
 cessively at the object-glass and eye-piece of a telescope (or 
 at the lenses and the retina of our eye). Let the positions of 
 the object-glass and the eye-piece in space at the time « be a 
 and h, and at the time t', a and I . Fig. 4. Then is the true 
 direction of the ray of light in space the direction of the 
 straight line ah', and on the contrary, the direction ah or a'V, 
 since the latter, on account of the infinite distance of the stars 
 is parallel to ab, is the direction of the apparent place which 
 is observed. The difference between the direction h'a and ha 
 is called the annual aberration of the fixed stars. 
 
 Let now x, y, z be the rectangular co-ordinates of the eye- 
 piece h at the time t referred to any fixed point in space; 
 then will 
 
 dx , , . dy , , , T dz . , y 
 
 * + T< (* ^ ^)' -^ + J ^^ *)' ^" "*" "^ ^* ~ "^ 
 
140 INVESTIGATION OF FOEMUL^ FOE ABEEEATION. 
 
 he the co-ordinates of tie eye-piece at the time t', since we may- 
 consider the motion of the earth to be linear during the short 
 interval t' — t. Let the co-ordinates of the ohject-glass referred 
 to the eye-piece be ^,17, ?; then are the co-ordinates of the 
 object-glass at the time t at which the light enters it 
 
 as + f y + V, » + 5'- 
 
 Taking now the plane of the equator for the plane of xi/, 
 and the two other co-ordinate planes at right angles to it, so 
 that the plane of xz passes through the equinoctial points, and 
 that of 1/z through the solsticial points, denoting moreover the 
 right ascension and declination of that point in which the true 
 direction of the ray of light meets the visible sphere of the 
 heavens by a and S, and by /j. the velocity of light, then will 
 the latter in the time t' — t describe a space whose projections 
 on the three co-ordinate axes are 
 
 fi (t' — t) cos B cos a, fi it' — t) cos S sin a, fj, [t' — t) sin S. 
 
 In addition, calling the length of the telescope I, and the 
 right ascension and declination of that point in which the ap- 
 parent direction of the ray of light meets the sphere of the 
 heavens, a' and 8' ; the apparent co-ordinates of the object-glass 
 referred to the eye-piece, which are observed, are 
 
 ^ = ? cos S' cos a', 5j = ZcosS'sina', §f=Z sins'.: V" 
 
 ' J 
 
 The true direction of the ray of light is now given by 
 means of the co-ordinates of the object-glass at the time t : 
 
 I cos S' cos a' -I- X, 
 I cos 8' sin a' + y, 
 Z sin S' + » ; 
 
 and by the co-ordinates of the eye-piece at the time t', 
 
INVESTIGATION OP FORMULA FOE ABERRATION;. 141 
 
 We obtain therefore the following equations, denoting by 
 L the quantity -r 
 
 f-t 
 
 fi cos S cos a = i cos S' cos a' — 
 
 , dx 
 
 dt ' 
 
 fi cos S sin 0. = L cos S' sin a! — -^ , 
 
 li sin 8 = i sin S' — =- . 
 
 ^ dt 
 
 From these equations we can easily find 
 L cos S' cos (a — a) = cos S + - i-f. sin a + -^ cos a) , 
 
 i cos S' sin («'-«)= Lgcosa-gsina), 
 
 or 
 
 %, 
 
 - sec S ( -# cos a — -57 sm a ) 
 tan (a - a) = . 
 
 1 + - sec 
 
 » /c?y . dx \ 
 
 We obtain a precisely similar equation for tan (S' — 8). 
 Developing both equations into series by the employment 
 of formula 14 in No. 11 of the Introduction, we obtain, if 
 we substitute in the formula for tan (8' — 8) the value of 
 
 tan. - (a' — a) derivable from a! - a, to terms of the second order 
 
 inclusive, 
 
 , 1 (dx . dy \ e , 1 (dx . dy \ 
 
 X i-j- COS a + ■y: sin a j sec''8 + &c. 
 
 S'_8= (-^sin8cosa + -^sin8sina— -57 cos8 
 
 _ 1 /(j 
 "vie 
 
 !-••■(«) 
 
 1 fdx . dy Vi t 
 
 - sm a — -TT cos a 1 tan 
 
 .dt 
 
 dt 
 
142 INVESTIGATION OP FOEMULiE FOE ABEEEATION. 
 1 [dx - dy ^ . *dz . ^\ 
 
 /dx . 5> dy . ^ . - dz A. 
 
 X ( T- sm cos a + -^ sm 6 sin a — TT cos o 1 . 
 
 Imagine now the place of the earth hy means of co-ordi- 
 nates as, y in the plane of the ecliptic to be referred to the centre 
 of the sun, and take the line from the sun's centre towards the 
 vernal equinox as the positive direction of the axis of x, and, as 
 the positive direction of the axis of y, the perpendicular to it 
 drawn to the summer solsticial colure, then, if we denote the lon- 
 gitude of the sun as seen from the earth by 0, and its distance 
 from the earth by R, we have* 
 
 x = — R cos 0, 
 ^ = — J? sin 0. 
 
 If we refer the co-ordinates to the plane of the equator, 
 taking as the axis of x the line drawn towards the vernal 
 equinox and imagining the co-OTdinate axis of z in the plane 
 yz to be turned through an angle e, equal to the obliquity 
 of the ecliptic, we obtain 
 
 x = — R cos 0, 
 
 ^ = — i2 sin cos e, 
 
 s = — i? sin sin e. 
 
 and from hence, 
 
 dx r, . dQ 
 ^= + ii!sin0^ 
 
 J=-i?COS0COS6^ 
 dz T> r^ • ^Q 
 
 -y- = — M COS sm 6-7— 
 at at 
 
 (b). 
 
 Substituting these values in equations (a) and introducing 
 instead of /* the number of seconds of time required by light 
 to traverse the radius of the earth's orbit, so that 
 
 u= T-, and therefore - = ^i , 
 we have, retaining only the terms of the first order, 
 
 * Since the longitude of the earth as seen from the sun is 180O+©. 
 
ABEEKATION IN EIGHT ASCENSION AND DECLINATION. 143 
 o! — a'^-k— fcos0cosecosa + sin0sina] secS 
 
 J/~\ 
 
 8' — B = + k —f- {cos [sin a sin S cos e — cos S sin e] 
 
 — cos a sin S sin 0}, 
 Now the sun moves* in a mean day 59'. 8",33, so that 
 
 moreover k, or the time required by light to traverse the radius 
 of the earth's orbit is 493^2, and thus we have 
 
 ^f = 20",255. 
 
 We have, therefore, for the annual aberration of the fixed 
 stars in right ascension and declination, the formulae 
 
 a' — a = — 20",255 [cos cos e cos a + sin sin a] sec S -| 
 B' — 8 = + 20", 255 cos [sin a sin S cos e — cos S sin e] }■ ... (A). 
 - 20", 255 cos a sin S sin ^ 
 
 The terms of the second order are so insignificant that they 
 may almost always be neglected. For the right ascension these 
 terms will be, if we introduce into the second term of the formula 
 (a) the values of the differential coefficients (J), 
 
 — J -J- (-J- ] sec'S [cos 20 sin 2a (l+cos'e)— 2 sin 20 cos 2a cos e], 
 
 where the small teim multiplied into sin 2a sin^ e is neglected f. 
 
 If we substitute the numerical values, taking e = 23°. 28', we 
 
 obtain 
 
 -O",0009155 sec'S sin 2a cos 20 
 
 + 0",0009123 sect's cos 2a sin 20. 
 
 • In Btrictnesa, the true motion of the sun in his elliptic orbit should be used for 
 the calculation of the aberration, but for this case the circular orbit may properly 
 be employed instead of the elliptic, since the difference only introduces a constant 
 term into the aberration which remains mixed up with the mean places of the stars. 
 See Bessel's Tahulce Regiomontaruc, page xix. 
 
 -)• We obtain, namely, paying no regard to the factors standing before the brackets 
 2sin2a[cos'0cos'e-sin'0]-2sin2®coac[cos=« -sin^o] 
 
 Since now cos^a-sin'a = cos2a, 003^0=^(1 + cos 2©}, and sin'0= J(l -cos2®), 
 we obtain the expression given above. 
 
144 ABERRATION IN LONGITUDE AND LATITUDE. 
 
 These terms give in the first place for stars whose declination 
 amounts to 85J°, one hundredth of a second of time ; they may 
 therefore be neglected in all cases except for the pole-star. 
 
 For the declination, the tejrms of the second order give, if 
 we neglect the terms which are not multiplied into tan S*, 
 
 ~q'~t[-^) tan S {cos 20 [cos 2a (1 + cos'' e) — sin'' e] 
 
 + 2 sin 2a sin 20 cos e}, 
 or, 
 
 + [0", 000394-0", 0004578 cos 2a] tan S cos 20 
 
 - 0" ,0004561 tan S sin 2a sin 20, 
 
 and these terms also do not amount to more than one hundredth 
 of a second for declinations smaller than 87°"9.' 
 
 If, instead of the equator, we take the ecliptic for the funda- 
 mental plane, the formula (5) will be simpler, namely, 
 
 dx , -o ■ ^dQ 
 ^ = + ii;sm0^, 
 
 _^=_i?cos0^, 
 
 dz 
 d-t=^- 
 
 Substituting these expressions in the formulae (o), and 
 putting \ and ^ in place of a and S, we obtain for the annual 
 aberration of the fixed stars in longitude and latitude the 
 formulse, 
 
 \' _ \ = _ 20", 255 cos (\ - 0) sec /3 1 
 
 /S'-/3 = 4-20 ,255 sin (\-0) sin/3 J ^^'' 
 
 Example. For April 1, 1849, we have for Arcturus, 
 a=14h.8°'.48' = 212M2'.0", S= + 19».58',1, © = 11°.37',2, 
 6 = 23''.27',4, 
 
 * The second term of the expression for S'~Sia equation (a) multiplied into tan« 
 gives, namely, if we disregard the constant factor, 
 
 sin^ ©sin*a + cos20cos''6 cos''a + J sin2 ©sin2a cos e. 
 If now we express the squares of the sine and cosine of and a by cos 2© and cos 2a, 
 and neglect the constant terms 1 + cos'e - cos2 a sin^ e, we obtain the expression given 
 above. 
 
TABULATION OF ABEREATION IN R.A, AND N.P.D, 145 
 
 With these data we find, 
 
 a'-a = + 18",70, 
 S'-S=- 9 ,56, 
 
 and since X = 202°. 8', -S = + 30°. 50', 
 
 also \'-\ = + 23",19, 
 
 ^'-^=- 1 ,89. 
 
 14. For the purpose of simplifying the computation of the 
 aberration in right ascension and declination, which, according 
 to the formulae now given, is somewhat inconvenient, tahles have 
 been constructed. The most convenient are those given by 
 Gauss. 
 
 Gauss assumes 
 
 20",255 sin O = a sin (0 + ^), 
 
 20 ,255 COS0 cos e = a cos (0 + A), 
 
 and obtains then simply : 
 
 a' — a = — a sec 8 cos (0 + ^ — a) 
 
 S'-S=-a sinS sin (0 + J. - a) - 20",255 cos cosS sine 
 = - a sin S sin (0 + ^ - a) - 10",128 sin e cos (0 + S) 
 -10", 128 sine cos (0-S). 
 
 By these formulae the tables are constructed. The first table 
 gives, with the argument sun's longitude, A and log a, from 
 which are obtained the aberration in right ascension and the 
 first part of the aberration in declination. The second and third 
 parts are obtained from the second table which is entered with 
 the arguments + S and Q — S. These tables were first given 
 by Gauss in the Monatliche Correspondent, Vol. xvii, page 312. 
 The constant there used is that which has been previously 
 given, namely, 20",255. More recently the same tables have 
 been recomputed by Nicolai with the constant 20", 4451, and 
 are printed in the Warnstorf Collection of Auxiliary Tables. 
 
 For the preceding example we obtain from the tables last 
 
 mentioned, 
 
 A = l\ 1', log a = 1-2748, 
 
 B.A. 10 
 
146 TABULATION OP ABERRATION IN R.A. AND N.P.D. 
 
 and with these data 
 
 a'-a = + 18",88, 
 
 and the first part of the aberration in declination is — 2",15. In 
 the same manner we find for the second and third parts — 3",47 
 and - 4",03, if the second table be entered with the arguments 
 31».35' and - 8''.21'. Thus we have 
 
 S'_S = _9",65. 
 
 20" 2550 
 Multiplying these values of a' — a and S' — S by „' , , we 
 
 obtain as before 
 
 a'-a = + 18",7, and 8'-S = -9",56. 
 
 Besides these general tables for the aberration there are given 
 in the Astronomische Jahrhuch special tables, which are arranged 
 in the order of the days of the year. There is assumed, namely, 
 
 — 20",255 cos © cos e = ^ sin H, 
 
 — 20 ,255 sin =h cos H, 
 
 — 20 ,255 cos sin e = li tan e sin H= i, 
 
 and we have then 
 
 a' — a = A sin {H+ a) sec S, 
 
 h' — Z = h cos [H+ a) sin B + i cos S. 
 
 Such tables, which give the values of h, H, and i for every 
 tenth or twentieth day, are found in Encke's Jahrhuch. For the 
 example previously employed 
 
 A = + 18",65, 5"= 257". 22', « = -7",90, 
 
 with which we find for a' — a and S' — S the same values as 
 before, 
 
 1.5. The maximum and minimum of aben-ation in longitude 
 take place when the longitude of the star is equal to that of 
 the sun or greater by 180° ; on the contrary the maximum or 
 minimum in latitude occurs when the star precedes the sun by 
 90° or follows it by the same quantity. The formula for the 
 
ANNUAL PARALLAX OP THE STABS. 147 
 
 Annual Parallax of the fixed stars (that is, for the angle which 
 lines drawn from the sun and the earth subtend at the fixed 
 stars) are strictly analogous to those for the annual aherration, 
 with this exception, that in this case the maxima and minima 
 occur at other times. If, for instance, A he the distance of a 
 fixed star from the sun, \ and /3 its longitude and latitude as 
 seen from the sun, then are the co-ordinates of the star referred 
 to the sun : 
 
 03 = A cos ;8 cos \, ^ = A cos /3 sin \, » = A sin 0. 
 
 The co-ordinates of the star referred to the earth will be 
 x' = A' cos yS' cos X', y' = A' cos /8' sin \', z' = A' sin /8', 
 
 and, since the co-ordinates of the sun referred to the earth are 
 
 X=^cos0 and F==i2 sin 0, 
 we have 
 
 A' cos ^' cos \' = A cos /3 cos \ + E cos 0, 
 
 A' cos /3' sin V = A cos yS sin \ + i? sin 0, 
 
 A'sin/3' = Asin/8. 
 
 Hence we easily obtain 
 
 V - \ = - ^ sin (X, - 0) sec 13 x 206265, 
 
 7? 
 
 lS' -/3 = - rrcos (\ - 0) sin/3 x 206265 ; 
 
 73 
 
 or, since -ry x 206265 is the annual parallax ir, 
 
 \' — \ = — TT sin (\ — 0) sec /3 
 /3' - /S = - TT cos (\ - 0) sin ^ 
 
 ■{0). 
 
 The formulae are therefore altogether similar to those for the 
 aberration, only that the maximum and minimum of parallax 
 occur when the star precedes or follows the star by 90° ; on the 
 other hand, the maximum or minimum in latitude occurs when 
 
 10—2 
 
148 DIURNAL ABEEEATION, 
 
 the longitude of the star is equal to that of the sun, or is greater 
 by 180". 
 
 For the right ascension and declination we have the equa- 
 tions 
 
 A' cos S' cos a' = A cos S cos a + i? cos 0, 
 
 A' cos S' sin a' = A cos 8 sin a 4- -S sin cos e, 
 A' sin S' = A sin S + ^ sin sin e, 
 
 from whence, in the same manner as for the aberration, we find 
 
 a' — a = — TT [cos sin a — sin cos e cos a] sec 8 
 8' — S = — TT [cos 6 sin a sin 8 — sin e cos 8] sin \... (D). 
 — TT cos sin 8 cos a 
 
 16. The daily motion of the earth on its axis produces in 
 the same manner as the yearly motion round the sun an Aber- 
 ration, which is called the Diurnal Aberration. This is however 
 much more insignificant than the annual aberration, since the 
 velocity of the motion of the earth on its axis is very much 
 smaller than the velocity of the motion in the annual orbit round 
 the sun. 
 
 The co-ordinates of a place on the surface of the earth re- 
 ferred to three rectangular co-ordinates of which one coincides 
 with the axis of rotation, and the two others lie in the plane of 
 the equator, so that the positive axis of x is drawn from the 
 center towards the vernal equinox, the positive axis of y to- 
 wards the ninetieth degree of right ascension, are, according to 
 No. 2 of this section, 
 
 x = p cos <j)' cos 0, 
 y = p cos (f)' sin 6, 
 z — p sin ^'. 
 
 We have thus 
 
 dx ,, . -dd 
 
 ^ = -pcos</,sme^, 
 
 dy , .• nd6 
 
 s^=^- 
 
DIURNAL ABERRATION. 149 
 
 Substituting these values in the formulse {a) in No. 13, we 
 easily obtain, neglecting the higher powers, 
 
 a — a = - -jT. p cos <f> cos [6 — a) sec o, 
 
 B' — S= - -J-, p cos <^' sin (0 — a.) sin 8. 
 
 Denoting now by T the number of sidereal days contained 
 in the time in which the sun goes through 360° in the heavens, 
 or the so-called sidereal year*, then is the angular motion of a 
 point of the earth's surface on account of its rotation T times 
 greater than the angular motion of the earth in its orbit, so 
 that 
 
 dt dt 
 
 Hence therefore we obtain as the constant of diurnal aber- 
 ration, (since 
 
 - p = ^ -^ = ^ sin TT, 
 
 where -rr is the sun's parallax and k the number of seconds of 
 time required by light to traverse the radius of the earth's 
 orbit,) 
 
 .dQ . rr 
 
 or, (since ^ -J* = 20",255, 7r = 8",5712, and r= 366-26) it is 
 equal to 0",3083. 
 
 If, besides, instead of the corrected latitude tj)' the latitude 
 <l> be simply substituted, we obtain for the diurnal aberration 
 in right ascension and declination, 
 
 a' - a = 0",3083 cos </> cos {0 - a) sec S ] , ™ 
 
 S'-S = ,3083 cos ^ sin ((9 -a) sin S J ^ ''' 
 
 Whence it follows that, on the meridian, the diurnal aberration 
 
 * This time is, as will be seen in the seqnel, somewhat greater than the time 
 which elapses between two passages of the sun through the vernal equinox, since the 
 sidereal ;ear = 36d'25637 mean solar dajs; or is equal to 863 da;s, 6 hours, 9 minutes, 
 and 10'74:96 seconds. 
 
150 ABERRATION — ELLIPSE. 
 
 of the stars in declination is nothing, while, in right ascension, 
 it attains its maximum, namely, 
 
 0",3083 cos ^ sec S. 
 
 17. For the annual aberration of the fixed stars in longi- 
 tude and latitude the following expressions have been already 
 
 found : 
 
 \' — \ = — ^ cos (X — O) sec /3, 
 
 /3' - jS = + ^ sin (\ - O) sin /3, 
 
 where the constant 20",255 is denoted by k. Imagine now at 
 the mean place of the star a tangent plane to the apparent 
 sphere of the heavens, and a rectangular pair of co-ordinates 
 at this point, whose axes of x and y are the lines of intersection 
 of the circles of parallel and of latitude with the tangent plane, 
 and refer now the true place aiFected with aberration to the 
 mean place by the co-ordinates 
 
 x= {\' — \) cos /S, and y = ^' — /3*, 
 
 then we easily obtain, by squaring the above equations, 
 
 y = i'sin''/3-a!''sin'/3. 
 
 Now this is the equation to an ellipse, whose semi-major 
 axis is equal to h and the semi-minor axis to Tc sin /3. Thus, on 
 account of annual aberration, the fixed stars describe round their 
 mean place an ellipse whose semi-major axis is 20",255, and the 
 semi-minor axis is the maximum of the aberration in latitude. 
 For stars which are in the ecliptic /3 = 0, and consequently the 
 semi-minor axis is equal to 0. Such stars therefore describe in 
 the course of a year a straight line, receding along the ecliptic on 
 each side of the mean place, by the quantity 20",255. For a star 
 which is in the pole of the ecliptic, /S = 90°, and consequently the 
 semi-minor axis is equal to the semi-major axis. Such a star will 
 thus in the course of a year describe about its mean place a 
 circle whose radius is 20",255. 
 
 The same reasoning precisely applies to the annual parallax 
 and the diurnal aberration. By means of the latter the stars, in 
 
 * Since for such minute distances from the origin of co-ordinates the tangent 
 plane maj be considered to coincide with this spherical surface. 
 
ABEEEATION OP THE PLANETS. 151 
 
 the course of a sidereal day, describe about their mean place 
 ellipses whose semi-major and semi-minor axes are respectively 
 
 0",3083 cos ^, and 0",3083 cos ^ sin S. 
 
 For stars in the eqUlktor this ellipse becomes a straight line, and 
 for a star at the pole it becomes a circle. 
 
 18. If the body have a proper motion, like 'the sun, the 
 moon, and the planets, then for such the aberration of the fixed 
 stars before treated of, is not the complete aberration. For since 
 such a body changes its place during the time taken by the ray 
 of light to traverse the space between it and the earth, so the 
 observed direction of the ray does not correspond to the true geo- 
 centric place of the body at the time of observation. Let us 
 suppose that the ray of light which reaches the object-glass of 
 the telescope at the time t, set out from the body at the time T. 
 Let also P (fig. 4) be the place of the planet in space at the time 
 T, p the same at the time t, A the position of the object-glass at 
 the time T, a and h the positions of the object-glass and the eye- 
 piece at t, and a and V their positions at the time <', when the 
 ray reaches the eye-piece. Then we have : 
 
 1. AP the direction towards the place of the planet at the 
 time t. 
 
 2. ap the direction towards the true place at the time t. 
 
 3. ap ox dp the direction towards the apparent place at the 
 time t, or at the time i', whose difference is indefinitely small. 
 
 4. h'a the direction towards the same apparent place, freed 
 from the aberration of the fixed stars. 
 
 Since now P, a, and V lie in a straight line, we have 
 Pa:ab' = t- T : t'-t. 
 
 Since moreover the interval of time *' — T is always very 
 small, so that we may assume that within its limits the earth 
 moves in a straight line and with a uniform velocity, the points 
 A, a, and a! also lie in a straight line, so that Aa and ad are 
 also proportional to the times t - T and t' — t. From hence it 
 follows that AP is parallel to h'd, and that thus the apparent 
 place of the planet at the time t is the same as the true place at 
 
152 ABEKRATION OF THE PLANETS. 
 
 * 
 
 the time T. But the difference of the times t' and T is the time 
 in which the light from the planet reaches the eye, or the product 
 of the distance of the planet into 493°,2, that is, into the time in 
 which light traverses the semi-major axis of the earth's orhit, 
 which is taken as the unit. 
 
 Hence there result three methods for computing the true place 
 of a planet frpm the apparent at any time t. 
 
 1. Subtract from the observed time the time in which the 
 light from the planet reaches the earth ; we then obtain the time 
 T, and the true place at the time T is identical with the appa- 
 rent place at the time t. 
 
 2. Compute with the distance of the planet the reduction of 
 time t — T, and with this by help of the daily motion of the 
 body in right ascension and declination the reduction of the 
 observed apparent place to the time T. 
 
 3. Consider the given place freed from the aberration of the 
 fixed stars as the true place at the time T, but as seen from the 
 place which the earth occupies at the time t. 
 
 This last method is to be employed when the distance of the 
 body is not known, for example, in the computation of an orbit 
 of a yet unknown planet or comet. 
 
 Since the time in which the light from the sun reaches the 
 earth is 493', 2, and the mean motion of the sun in a day is 
 59'.8",3, we have (from 2) the aberration of the sun in longitude 
 equal to 20", 25, by which quantity the longitude of the sun is 
 always observed too small. On account of the change of distance 
 and velocity of the sun this value varies in the course of a year 
 by soine tenths of a second. 
 
 Note. On the subject of Aberration compare the Preface to Bessel's 
 ToihuUs RegiomontancB, page xvii. &c., and Gauss, Theoria MotHs, page 
 68, &c. 
 
OBSERVATION OP RIGHT ASCENSION AND DECLINATION. 153 
 
 THIRD SECTION. 
 
 DETERMINATION OF THE CO-ORDINATES AND ANGLE OF THE 
 APPARENT SPHERE OF THE HEAVENS INDEPENDENT OF 
 THE POSITION OF THE OBSERVER ON THE SURFACE OF 
 THE EARTH. PERIODICAL AND SECULAR CHANGES OF 
 THESE QUANTITIES. 
 
 The co-ordinates and angle of the apparent sphere of the 
 heavens which are independent of the position of the ohserver 
 on the surface of the earth are the right ascensions and declina- 
 tions and the longitudes and latitudes of the stars, and finally 
 the angle which the fundamental planes of the two systems of 
 co-ordinates make with one another, or the obliquity of the 
 ecliptic. The spherical co-ordinates of longitude and latitude 
 are never determined immediately by observations, but are 
 always deduced by computation, by means of the formulae for the 
 transformation of co-ordinates, from right ascensions and declina- 
 tions (I. No. 8). There remain therefore to be determined by 
 observation only the right ascensions and declinations and the 
 obliquity of the ecliptic. 
 
 By comparing with each other at different epochs the deter- 
 minations of these quantities, it is found that they are subject to 
 changes of which one part, in intervals of time not very great, 
 is proportional to the time, while the other is periodical. The 
 change proportional to the time of the right ascension and decli- 
 nation as well as of the longitude and latitude is called the Pre- 
 cession; and on the other hand the change proportional to the 
 time of the obliquity of the ecliptic is called the Secular Variation 
 of the Obliquity. The other part of the change whose principal 
 
154 OBSEEVATION OF EIGHT ASCENSION AND DECLINATION. 
 
 terms have a period of 19 years, is denoted by the term Nu- 
 tation. Both changes have their origin in a secular motion 
 of the equator on the ecliptic as well as of the ecliptic on the 
 equator, whereby the inclination of the two planes with respect 
 to each other is altered; and in a periodical oscillation of the 
 intersection of the equator and ecliptic on the latter plane, as 
 well as in a periodical change of the inclination of the ecliptic 
 and equator connected with the same. 
 
 The place of a star at any given time when freed from the 
 periodical part of the change or the Nutation, is called the mean 
 place of the star for that epoch. These mean places of the stars 
 are given in star-catalogues. To obtain from thence the mean 
 places for any other epoch, application must be made of the 
 precession for the difference of the times ; but, if it be required 
 to find the true place of the star referred to the true equinox 
 for this time, it is necessary to add the nutation as well as the 
 precession. It is therefore necessary to find means for knowing 
 the law of the changes of the places of the stars on account of 
 precession and nutation, and at the same time to devise conve- 
 nient methods both for reducing the mean places of the stars to 
 difierent epochs, as also for changing the mean places into true 
 places and vice versd. 
 
 I. Determination of the eight ascensions and de- 
 clinations OF the staes and of the obAquitx op the 
 ecliptic. 
 
 1. If we observe the difi"erence of the times at which stars 
 pass the meridian of a place, these difierences are also the difi'er- 
 ences of right ascension of the stars expressed in time. (I. No. 3. 
 Note.) If also at the same time there be observed the alti- 
 tudes of the stars at their meridian passages, we obtain also the 
 difierences- of their declinations, since every meridian altitude 
 of a star differs firom its declination by a constant quantity. 
 (1. No. 14.) 
 
 For these observations there is necessary a good clock (that 
 is, such a one as for equal arcs of the equator passing across the 
 meridian give an equal number of seconds*) and an altitude 
 
 * It is not necessary to know the absolute time, since only differences of time are 
 observed. 
 
INSTRUMENTS FOK X)BSEEVING E.A. AND N.P.D. 155 
 
 instrument fixed in the plane of the meridian, that is, a meridian 
 circle. This in its essential parts consists of a horizontal axis 
 lying in two fixed T's, which carries a vertical circle and a tele- 
 scope. To one of the supports js fastened an index, which, by 
 means of the simultaneous motion of the telescope and the circle 
 round the horizontal axis, gives on the circle the arc passed over 
 by the telescope. 
 
 To test the uniform rate of going of the clock, consecutive 
 transits are observed of diiferent stars on a vertical wire stretched 
 in the focus of the telescope. If then the instrument have not 
 changed its position in the interval, and if the observation be 
 made on the same part of the vertical wire, then must the clock, 
 if it be adjusted to sidereal time, shew 24 hours exactly between 
 two consecutive transits of the same star. If this be not the case, 
 but the clock gives, for consecutive transits of any star, the time 
 24'' — a, then is a called the daily rate of the clock and must 
 in the observation of diflferences of right ascension be taken into 
 account, by multiplying the observed diiference by 
 
 24 
 24-a' 
 When the uniformity of rate of the clock has been ascer- ■ 
 tained, it is necessary so to adjust the meridian circle that the 
 vertical wire of the telescope may be in the plane of the meridian 
 for every position of the telescope. If the axis of the instrument 
 has been made horizontal by means of a spirit-level, then a star 
 near the equator is allowed when near the meridian* to run 
 along another wire placed at right angles to the first, and the 
 plate carrying the wires is turned till the star as long as it is in 
 the field runs well along the wire. Then is this wire accurately 
 horizontal and the other is accurately vertical. After this has 
 been accomplished the telescope is directed to a distant terrestrial 
 object, and a distinguishable point is noted which is bisected by 
 the vertical wire. Then the instrument must be shifted in its 
 Y's so that the circle which was in the east side before may be 
 now on the west side, and the telescope must be directed in this 
 position to the same object. If then the vertical wire in this 
 
 • The direction of the meridian, as far as this object is concerned, can be found 
 with sufficient accuracy by obserying the time at which the altitudes of the stars 
 do not change. 
 
156 METHOD OF OBSERVINU WITH THIE MERIDIAN CIECLE. 
 
 position also bisect with sufficient accuracy the same point of 
 the object, the line of sight, that is, the line from the center of 
 the object-glass to the wire-cross is sensibly perpendicular to the 
 axis of rotation of the instrj^ment. But if the wire bisect 
 another point, then the wire frame must be shifted by means 
 of screws which give it a motion at right angles to the line of 
 sight, till the vertical wire passes accurately through the point 
 of the object that lies midway between the two points bisected 
 in the reversed positions. The line of sight will be now at right 
 angles to the axis of rotation. If tHs be still not accurately the 
 case, the operation can be repeated till the enror be totally got 
 rid of. 
 
 Finally, to bring the vertical wire into the plane of the 
 meridian, use must be made of the pole star, by observing the 
 transits over the wires at three consecutive and opposite culmi- 
 nations. If for instance the instrument be accurately in the 
 plane of the meridian, the time between the upper culmination 
 and the lower culmination next following must be accurately 
 equal to the time which elapses between the lower and the next 
 following upper culmination. If this be not the case, we know 
 that the vertical circle which the instrument describes deviates 
 from the meridian on that side of it in which the star is 
 for the shorter period of time, and the line of sight can be 
 brought into the plane of the meridian by motion of one of the 
 Y's of the instrument. In this manner can the adjustments of 
 such an instrument be accurately performed, so that the vertical 
 wire of the telescope in each position may be accurately in the 
 plane of the meridian. 
 
 After this has been accomplished, the times of transit of stars 
 over the vertical wires must be observed, and at the same time, a 
 little before or after the meridian passage, they must be brought 
 upon the horizontal wire, and the number must be read off which 
 is shewn by the index on the circle in this position of the tele- 
 scope. We then obtain from the differences of the observed times 
 the differences of the right ascensions, and, from the differences of 
 the readings of the circle, the differences of apparent declinations. 
 To these observed differences are still to be applied the corrections 
 treated of in the preceding sections, for the purpose of deducing 
 the true differences of right ascension and declination of the stars. 
 
ZENITH-POINT OF THE MERIDIAN CIRCLE. 157 
 
 The parallax In right ascension is nothing on the meridian, 
 and therefore is not to be regarded in the observation of the 
 differences of right ascension ; on the contrary the declination, 
 or rather the reading of the circle, must be freed from the 
 parallax, when the body observed is affected by it. If the 
 divisions increase in the direction of the zenith-distance, and thus 
 increase from the zenith towards the horizon, — tt sin a must be 
 .applied to the circle-readings, (II. No. 3) where ir is the hori- 
 zontal parallax and z the apparent zenith-distance of the object*. 
 For the fixed stars this correction is nothing. Since moreover 
 the refraction always acts only in the vertical direction, it does 
 not alter the times of transit of the stars over the meridian; 
 but to all circle results, on the contrary, it is necessary, if the 
 divisions increase in the direction of zenith-distance, to apply 
 the correction + r, where r is to be computed according to for- 
 mula {F) in No. 10 of the second section, and account is to be 
 taken of the readings of the barometer and thermometer at the 
 time of observation. 
 
 Since both corrections require a knowledge of the apparent 
 zenith-distance, it is necessary for the pui-pose of being able to 
 compute them from the circle-readings, to know what point of 
 the same corresponds to the zenith. This point, which is called 
 the zenith point of the circle, is easily found by placing the 
 horizontal wire of the telescope in two different positions of the 
 instrument (that is with the circle East and West) on the same 
 terrestrial object. If ^ be the reading of the circle in one obser- 
 vation and ^' in the other, then is the zenith point -(f+i;'')' 
 
 Instead of a terrestrial object, use may be made of the pole-star 
 at the time when it is very near the meridian, since at this time 
 the zenith-distance changes with extreme slowness. 
 
 In the last place the observed differences of right ascension 
 aild declination are to be freed from aberration, by applying 
 to the observations the expressions {A) given in II, No. 13 ; for 
 right ascensions, to the observed times with signs changed, and 
 on the contrary the correction S' — S with proper sign to the 
 
 * If the divisions increase contrary to the direction of zenith-distance from the 
 horizon to the zenith, all the corrections are to be applied with contrary signs. 
 
158 COERECTIONS FOR EEFRAOTION, PARALLAX, &C. 
 
 observed circle-readings when the stars pass the meridian south 
 of the zenith, and the divisions increase in the direction of the 
 zenith-distances. Since these expressions for a. — a. and S' — S 
 themselves contain the quantities a, S, and e, their computation 
 presupposes an approximate knowledge of them. This we have 
 by means of earlier star-catalogues. In former times the an- 
 cients determined the right ascensions and declinations of the 
 stars naturally without reference to these small corrections, but . 
 by means of a method which was essentially the same as that 
 which is employed at present. Since that time the catalogues 
 have been receiving continual corrections, in part by the superior 
 accuracy of the observations since the invention of the telescope 
 and of the wire-micrometer, and partly by the employment of 
 more accurate values of the small corrections. 
 
 If the object have a visible disk, as for example the sun, 
 then must its semi-diameter be applied to the observation of 
 zenith-distance, or otherwise the upper as well as the lower 
 limb must be observed on the meridian. In these cases the 
 refraction must be applied to the observation of each limb 
 separately, and the mean of the corrected zenith-distances must 
 then be taken. 
 
 After the differences of true right ascension and declination 
 of the stars have been thus found, the only thing still necessary 
 is to determine the true right ascension and declination of a 
 single star, or rather the true right ascension of a star and that 
 point of the meridian circle which corresponds to the pole or to 
 the height of the equator, in order to obtain the right ascensions 
 and declinations of all the other stars. If now these deteimina- 
 tions be made at different times, it is found that, leaving out of 
 account errors of observations, the right ascensions and declina- 
 tions are never the same on account of the changes in the 
 positions of the planes to which the star's places are referred, 
 which cause the stars apparently to change their places with 
 reference to the planes. These changes however cannot be 
 taken into consideration before the place proper for their intro- 
 duction. 
 
 2. The point of the circle, which corresponds to the pole, 
 technically called the polar point, is found by observations of 
 
DETERMINATION OF ABSOLUTE EIGHT ASCENSION, 159 
 
 circumpolar stars at their upper and lower culminations. If for 
 instance ^ be the reading of the circle corrected for refraction 
 for the upper culmination, and f for the lower culmination, then 
 
 is ^ (?'-?)= 90° - S, and i (?'+?) is the point of the circle 
 
 corresponding to the pole. On the other hand, - (f ' + ?) ± 90° 
 
 according to the direction of the divisions of the circle is the 
 point which corresponds to the altitude of the equator, or, to 
 the equatorial point of the circle. If also, the zenith-point 
 Z of the circle he known by the methods before pointed out, 
 then is 
 
 the co-latitude of the place of observation. After the polar point 
 of the circle has been determined, the declinations of all observed 
 objects can be found, and it only remains to deduce from obser- 
 vation, a true right ascension of a single object. 
 
 Since now there is taken for the zero of right ascensions of 
 the stars the point in which the ecliptic (that is, that great 
 circle which the sun in the course of a year appears to describe 
 on the visible sphere of the heavens) cuts the equator, we shall 
 obtain a knowledge of the right ascension of a star by connect- 
 ing the observations of the culmination of the stars with those 
 of the sun. If for instance for several successive days about 
 the times of the equinoxes, besides the culminations of the sun 
 and of a star, there be also observed the declinations of the 
 center of the sun, we obtain for different declinations of the sun 
 the differences of right ascension of the sun and the star, and 
 can therefore compute this difference for the instant when the 
 declination of the sun is equal to 0°, and therefore the right ascen- 
 sion is either 0° or 180°. If then the observations be made at 
 the vernal equinox, the computed difference of right ascension 
 will be the absolute right ascension of the star, and on the 
 other hand, if the observations be made at the autumnal equi- 
 nox, we shall find a value differing from the right ascension by 
 180°. 
 
 The third of the quantities to be determined is the obliquity 
 of the ecliptic, or the angle which the plane of the ecliptic makes 
 
160 DETERMINATION OF THE OBLIQUITY OF THE ECLIPTIC. 
 
 with that of the equator. The measure of this angle is the arc 
 of the solstitial colure (that is, of the circle of latitude passing 
 through the poles of both gi-eat circles) which is comprised be- 
 tween the equator and the ecliptic. The obliquity of the ecliptic 
 is therefore also equal to the greatest declination which the 
 center of the sun has in the course of the year. Therefore if we 
 observe about the time of the summer solstice (June 21) for 
 every day the declination which the sun has at its transit over 
 the meridian, then, if the time of a culmination be coincident 
 with the time of the solstice, the greatest of the observed decli- 
 nations is immediately the obliquity of the ecliptic. But if 
 this be not the case, the greatest declination can be easily de- 
 duced from those observed, by finding the time for which the 
 first difference of the observed declinations is equal to nothing, 
 and by interpolating the declination for this time. 
 
 At the expiration of half a year, at the time of the winter 
 solstice, if the sun were again observed, the same absolute value 
 would be found for the greatest southern declination of the sun 
 if the observations were without error*. In this case, moreover, 
 . when both solstices are observed, the knowledge of the polar point 
 of the circle is not at all requisite, but only the zenith-point, 
 nor is (which amounts to the same thing) a knowledge of the 
 latitude of the place of observation necessary. 
 
 If, for instance, for the least zenith-distance of the sun's 
 center in summer the value a were found, and for the greatest 
 
 zenith-distance in winter the value z, then is - {z' — z) equal to 
 
 the obliquity of the ecliptic and - (s' + z) is equal to the zenith- 
 distance of the equator or the latitude. 
 
 Every two observations of the difference of right ascension 
 of the sun and a star and of the declination of the sun give 
 moreover both the right ascension of the star and the obliquity 
 of the ecliptic. For instance, if a be the right ascension of the 
 star, A the observed difference of right ascension of the sun and 
 
 * With the exception of a trifling difference arising from the secular change of the 
 obliquity and the nutation, , 
 
DETEEMINATION OF THE OBLIQUITY OP THE ECLIPTIC. 161 
 
 star*, D the declination of the sun, and e the obliquity of the 
 ecliptic, we have, according to the Note to No. 9 of Section I, 
 
 sin {A + a) tan e = tan D, 
 
 and, in like manner, from the second observation, 
 
 sin [A' + a) tan e = tan D'. 
 
 From the two observations we find 
 
 . tan D sin A' — tan D' sin A 
 sm a tan e = . , ., jv , 
 
 sin {A! - A) ' 
 
 and 
 
 , tan B' cos A-ta.iaI> cos A' 
 cos a tan e = ; — r^. jr , 
 
 sm {A' - A) ' 
 
 from which a and e can both be computed. 
 
 At the same time it is always desirable to determine the. 
 values of the quantities a and e independently of each other, so 
 that error of the one may not seriously affect the other, and we 
 must for this purpose proceed by a method similar to those 
 previously given. 
 
 3. Assuming that the position of the vernal equinox is ap- 
 proximately known by one of the previous methods, the obliquity 
 of the ecliptic can be rigorously determined from observations 
 of the sun in the neighbourhood of the solstices, in the following 
 manner. If x be the distance of the sun in right ascension from 
 the solstice, which is thus equal to 90° — a, we have the 
 equation 
 
 cos X tan e = tan B. 
 
 Since, by the supposition, a; is a small quantity, e can from 
 this equation be developed in a rapidly converging series, since 
 by formula (20) of the Introduction we obtain 
 
 e = Z> + tan''f sin 2Z> + ^ tan* f sin iD + &c (A). 
 
 • In this manner can the obliquity of the ecliptic be obtained 
 from one observation of the declination of the sun in the 
 neighbourhood of the solstices. 
 
 * So that A + aia the right ascension of the sun. 
 B.A. 11 
 
162 EXAMPLE. 
 
 Bessel observed at Konigsberg, when the right ascension of 
 the sun was 5\51"°.23',5, 
 
 D = 23''.26'.47",83. 
 
 Since the right ascension of the sun at the* time of the 
 solstices is 6 hours, we have here 
 
 a; = 8"'.36',5 = 2".9'.7",5. 
 
 We have thus 
 
 tan'' % sin 2Z) = + 53",13, 
 
 \ tan* I sin 4l> = + 0,01, 
 
 and therefore from this observation we have the obliquity of the 
 ecliptic, 
 
 6 = 23°.27'.40",97. 
 
 Now for the purpose of freeing the result from casual errors 
 of observation, the declination should be observed on several 
 days in the neighbourhood of the solstice, and the means of 
 the separate values of e thus obtained, should be taken. The 
 time of the solstice is necessary to be known only approximately, 
 since an error in x produces only a very small effect on the 
 determination of e. For example, if we take into accoitnt only 
 the first term of the series, 
 
 tan - sin 2Z> 
 
 de. = —^—■^—^— dx, 
 cos^l 
 
 or, from the fundamental equation, 
 
 de = - tan x sin 2edx ; 
 
 SO that, for example, we should have only an error of 1",37 in e, 
 if the assumed value of x were in error to the amount of 100". 
 
 4. If then the obliquity of the ecliptic be known, the abso- 
 lute right ascension of a star can be determined with the utmost 
 accuracy. A bright star is selected, which can be observed in 
 
DETERMINATION OP ABSOLUTE EIGHT ASCENSIONS. 163 
 
 the daylight as well as by night, and which is in the neighbour- 
 hood of the equator. Ordinarily, Altair (a Aquilse) or Procyon 
 (a Canis Minoris) is selected. Then, in the first place, every 
 observation of the sun gives (if A now denote the true right 
 ascension of that body) the equation 
 
 sin A tan e = tan D, 
 or 
 
 . . _, tan D 
 
 A = sm . 
 
 tane 
 
 Now let the star be observed on the meridian at the clock 
 time t, the sun at the clock time T, then is the right ascension 
 a of the star equal to 
 
 „ = sin-^+(e-y)* 
 tan € 
 
 By this equation the right ascension of the star is thus found 
 from the observed difference of right ascension of the star and 
 the sun, the declination of the latter being D, and the obliquity 
 of the ecliptic being e. If therefore D and e be in error f, we 
 shall on this account also obtain an erroneous value of a, inde- 
 pendently of errors of observation in t— T. .But, by differen- 
 tiating logarithmically the equation 
 
 sin A tan e = tan D, 
 we obtain 
 
 cot-drt^H- 
 
 sin26 ~ sin 2i> ' 
 and consequently, if we add these terms to the equation for a, 
 
 ^ „ . _, tanU 2tan^,n 2tan^, 
 tan 6 sm_2i> sm 2e 
 
 Now for the purpose of obtaining a independently of the 
 errors dB and de, 'several observations must be combined in such 
 a way, that these errors shall, eliminate each other's effects, 
 This will be effected by combining an observs^tion near th§ 
 
 " Neglecting fhe rate of tlie clock in the interval t-T. Tkanslatok. 
 
 -|- Properlj speaking it is only a constant error in D that is to he taken into con- 
 sideration here, since casual errors of observations will be got rid of in a mass of 
 observations. 
 
 11-a 
 
16^ DETERMINATION OP ABSOLUTE EIGHT ASCENSIONS. 
 
 vernal equinox with another near the autumnal equinox. For 
 
 instance, if in the equation 
 
 . . tan D 
 
 sin A = 
 
 tan 6 
 
 we take the angle A always acute, we have for the latter obser- 
 vation the equation 
 
 , m, /,„«o • .,tan-D'\ 2tan^' ,„ 2tan^' - 
 a = « -r' + flSO'-sin'- — - — ^--y.r dD -V . „ de, 
 \ . tan e / sin 2Z> sin 2e 
 
 and from the mean of the twQ equations we obtain for a 
 
 Ir — m ~, — TFTi 1/ • _, tani) . _, tanD' ,„„„\ 
 
 ^ a = - [< - r+ i - .T\ + - ( sin-' sm ' + 180° 
 
 2 2 V tan e tan e / 
 
 /tan^ tan^'N ,_, tan -4 — tan ^' , ,„, 
 Vsin2i) sm2i>/ sin2e ^ 
 
 If now the acute angle A! = A, then Z>' is also equal to B ; 
 and if therefore the difference of right ascensions of the sun and 
 the star be observed at the times when the sun has the right 
 ascensions A and 180° — -4, then will the coeflScients of dD and 
 rfe in equation {fi) be nothing, and the constant errors in the 
 declination and the obliquity will thus have no effect on the 
 right ascension of the star. This it is true can never be attained 
 with the utmost rigour, since it will never exactly happen that, 
 when the sun at one culmination has the right ascension A, the 
 right ascension 180° — ^ shall exactly correspond to another cul- 
 mination. But if ^-4 in the first case be oijly nearly equal to 
 180° — ^ in the second, the remaining errors dependent on dD 
 and de will be always exceedingly small. 
 
 Thus, for the determination of the absolute right ascension of 
 a star, the difference of right ascensions of the sun and star 
 should be observed as near as possible to the vernal and autum- 
 iial equinoxes ; but if one observation be made after'the vernal 
 equinox, the second must be made before the Autumnal equinox, 
 and vice versa, that the sun's declination may in each case have 
 the same sign. 
 
 Bessel observed in 1828, March 23, the declination of the 
 sun's center, cleared from refraction and parallax in altitude, 
 Z» = + 1°. 6'. 54",2, 
 
EXAMPLE. 165 
 
 and the transit across the meridian 
 
 T= 0\ ir. 12',57. 
 
 And, on the same day, the transit of a Canis Minoris 
 
 « = 7\3r.l4",62* 
 
 In the same manner he observed on the 20th- of September 
 of the same year • '■■ 
 
 i)' = +lM'.66",8, 
 
 y'=ll\50"'.33',40, 
 
 t' = 7\ SO". 24',82. 
 
 To these observed quantities the. aberration must now be 
 applied. Now for the star we obtain, according to the formula 
 {A) in No. 13 of, the second section, taking a = 112°. 34',3, 
 B = + 5". 39',5, the aberration in right ascension, 
 
 Mar. 23, + 0',42, 
 Sep:2ff, -0",54, . 
 
 where the signs are so to be interpreted, that these corrections 
 are to be applied with changed signs to apparent places, to 
 obtain the mean places. For the sun the aberration is to be 
 computed according to- the prescript in No.. 18 of the second 
 section.' Now- the hourly motion of the sun in right ascen- 
 sion is 
 
 Mar. 23, +9',Q8, ". --"- 
 
 Sep. 30,' + 9, 00 5 
 
 and in declination 
 
 Mar. 23, + 59",08, 
 Sep. 30, -58",38. 
 
 Consequently the aberration of the sun in right ascension 
 and declination is 
 
 Mar. 23, + 1,',24, +8",09, 
 
 Sep. 30, + 1',23, - 8",00. 
 
 • These times are corrected for the rate of the clock, 
 
166 .EXAMPLE. 
 
 And these corrections are to be added algebraically to the ap.- 
 parent places, to obtain the mean places*. 
 
 Taking account of these corrections we find 
 
 « - y= + 7\ 20". 0",39 
 
 (' _ r' = _ 4 . 20 . 9 ,27 
 
 -[t-T+^-T'} = 1.29 .55,56. 
 
 And, if we assume e to be equal to 23°. 27'. 33",4, 
 
 tan D = 8-2901033 
 tan e = 9-6374572 
 
 an-'^^ = 2».34'.32",94 
 tan 6 
 
 = 0\ 10". 18",20, 
 tan D' = 8-2548551 
 tan 6 = 9-6374572 
 
 Bin-^^'=2''.22'.29",63 
 
 tane 
 
 = 0". 9°". 29',98. 
 Thus 
 
 1 fsiu-t^_sin-|5^ + l2A=6^0"'.24Vl, 
 
 2 \ tan e tan e / 
 
 and finally 
 
 a = 7\30"'.19",67. 
 
 In this computation the equinox has been regarded as fixed ; 
 but since this is variable on account of precession and nutation, 
 there is still a correction to be applied to the value above found 
 for the right ascension. The calculation of the example, taking 
 account of this latter correction, is found in No. 11 of this 
 section. 
 
 If the coeflScients of dD and de be computed we obtain 
 
 a = T". 20". 19', 67 + 0=, 000223 dD + 0*, 004406 de. 
 
 * In general it is not necessary to take into account the aberration of the son, 
 since this only alters the time of passage through the equinox, and is eliminated by 
 the combination of the two equations. 
 
PEilOESSION AND NUTATION. 167 
 
 Thus the constant error of the declination and of the assumed 
 ohliquity of the ecliptic are rery nearly eliminated by the comhi- 
 natioii of the two observations. 
 
 II. Variations of the Planes^ to which the places of the 
 Stars are referred. 
 
 PEECESSION AND NUTATION. 
 
 5. If hy the methods above described a series of determina- 
 tions were made of the points of intersection of the ecliptic and 
 equator, it would be found that, with very few exceptions, the 
 right ascensions of the stars increase, and, for intervals of time 
 not very large, leaving out of account small inequalities, that 
 their increase is proportional to the time. For different stars 
 also a different annual change will be observed, without however 
 any conspicuous law being observable. In the same manner 
 if the declinations of stars be observed at different times, there 
 will also be found in this co-ordinate a similar change propor- 
 tional to the time, whose direction is different according to the 
 quadrant in which the right ascension of the star lies. In all 
 these changes there will be immediately discovered a remarkable 
 law, if the stars be no longer referred to the fundamental plane 
 of the equator but to the ecliptic. In this case namely it will be 
 found that the longitudes of all stars increase by nearly equal 
 quantities while their latitudes remain very nearly unchanged. 
 
 This uniform change of the places of the stars with regard to 
 the ecliptic was first discovered by Hipparchus (b.c. 130), who 
 compared his own observations of the stars' places with those of 
 Timocharis, which were made about 160 years earlier. He 
 found from this comparison that the longitudes of all the stars 
 were changed yearly by about 36", and therefore in a hundred 
 years by about 1°. This value is however too small. Hippar- 
 chus foTmd the longitude of Spica Virginis 174''.0'; at the pre- 
 sent time it is 201°.41'. If we take for the interval of time 
 1980 years, and assume the motion to be proportional to the 
 time, we obtain for the annual motion of the equinox in longi- 
 tude 50", 3. 
 
 This change of the stars' places has its origin first in the cir- 
 cumstance that the point of intersection of the equator with the 
 
1"68 PEECESSION, 
 
 ecliptic has a retrograde motion on the latter plane, and, secondly, 
 in the change of inclination of the two planes with respect to each 
 other. The first part of this change is called the precession of 
 the stars, or the retrograde motion of the equinox; the second is 
 called the secular change of the obliquity of the ecliptic. The ex- 
 planation of these phenomena belongs to Physical Astronomy, 
 which teaches that they originate firstly in the attraction of the 
 sun and moon on the spheroidal earth, and secondly on the action 
 of the planets on the position of the plane of the earth's orbit. 
 The attraction of the sun and moon does not change the inclina- 
 tion of the equator to the ecliptic*, but it simply causes the point 
 of intersection of the equator and ecliptic to retrograde on the 
 latter plane. This motion of the equator on the ecliptic is called 
 the Lunisolar Precession. By it the longitudes of all stars are 
 changed while their latitudes remain unchanged. If we take as 
 the fixed plane that great circle of the heavens, with which 
 the ecliptic coincided at the beginning of the year 1750, then, 
 according to Bessel, the lunisolar precession for the year 
 1750 + < is 
 
 ^' = + 50" • 37572 -0"- 000243589 t, 
 
 or the actual change in the interval from 1750 to 1750 + f is 
 Zj = + 50"- 37572 1 - 0"- 000121795 f, 
 
 by which quantity the longitudes of all stars are increased in 
 this interval. 
 
 The mutual attractions of the planets produce in addition a 
 change of inclination of their orbits to each other, and a forward 
 motion of their nodes ; that is, of the intersections of the planes 
 of their orbits. Since now the earth's equator is not changed by 
 these attractions, they produce a change of the obliquity of the 
 ecliptic, and a motion of the point of intersection of the ecliptic 
 and equator on the latter plane. This motion of the equinoctial 
 points is called the Planetary Precession. By its means the right 
 ascensions of all stars are changed, whilst the declinations re- 
 
 * At least the changes of the inclination thereby produced are only periodical 
 -which in the sequel will be treated of under the head of Nutation. 
 
SECULAR DIMINUTION OP THE OBLIQUITY. 169 
 
 main the same, and, according to Bessel, the annual diminution 
 of the right ascension for the time 1750 + t is 
 
 ^ = + 0"-17926 - 0"-0005320788 <* 
 at 
 
 If we denote by a the quantity by which the right ascensions 
 of all stars are diminished in the interval from 1750 to 1750 + *, 
 we have 
 
 a = 0"'17926 t - 0"-0002660394 f. 
 
 At the same time the obliquity of the ecliptic is also changed, 
 and we have for its annual variation by means of the planets for 
 the time 1750 + 1, 
 
 ^ = - 0" -48368 - 0" -0000054459 1 ; 
 
 and, for the obliquity itself for the time 1750 + t, we have 
 
 23°.28'.18",0 - 0"-48368«- 0"-0000027230f. 
 
 But in addition, the position of the ecliptic with respect to the 
 equator being changed, the attraction exerted by the sun and 
 the moon on the spheroidal earth is also changed, and a long in- 
 equality in the inclination of the plane of the equator to the 
 ecliptic is introduced. On this account also there arises a change 
 of the obliquity of the fixed ecliptic for 1750t with respect to 
 the equator, whose annual variation is 
 
 ^ = + 0"-00001968466«, 
 at 
 
 and the obliquity of the fixed ecliptic for the time 1750 + < is 
 
 e = 23°. 28'. 18",0 + 0"- 00000984233 f. 
 
 Let now (fig. 5) AA„ be the equator and EE„ the ecliptic, 
 both for the year 1750; let also A' A" and EE' denote the 
 positions, of the equator and ecliptic for the year 1750 + t; then 
 is BD, the portion of the fixed ecliptic through which the equa- 
 tor has retrograded on the latter, or, the lunisolar precession in t 
 
 • Hence in the year 2087 the motion of the ecliptic on the eqnator, which is now 
 in the contrary direction to that of the equator on the ecliptic, will be in the same 
 direction, 
 
 t Namely, the motion of the equator with reference to the ecliptic with sign 
 changed. 
 
170 CALCULATION OF PEECESSION. 
 
 years = 1^; in addition the portion BG, through which the ecliptic 
 has moved forwards on the equator, or the precession through the 
 planets in t years, is equal to a ; and lastly are BCJE and A' BE 
 respectively, the inclinations of the true and the fixed ecliptic to 
 the equator, equal to e and e^. If then She any star, and SL 
 and SL' be drawn perpendicular to the fixed and the true ecliptic, 
 DL is the longitude of the star for 1750, and GL' is the longi- 
 tude of the star for 1750 + 1. Denote now by B' that point of the 
 moveable ecliptic which on the fixed ecliptic is denoted by I), 
 then is the portion GB', that is, that portion of the true ecliptic 
 between the equinox for 1750 and the equator for the time 
 1750 + 1, called the General Precession in the time t, since this 
 part- of the precession in longitude is the same for all stars. 
 To obtain from, thence the complete precession for a star in 
 longitude, there is still to be added to the general precession 
 the quantity B'L' — BL, But this part is, on account of the 
 slow change of the obliquity, much smaller than the foimer. 
 
 If we denote by 11 the longitude of the ascending node of 
 the true on the fixed ecliptic (that is, the point of intersection 
 of the two great circles, setting out from which the true ecliptic 
 has a north latitude above the second), and reckon this angle 
 from the fixed vernal equinox of the year 1750, then we have, 
 the longitudes being reckoned in the direction from B to B, and 
 E being the descending node of the true ecliptic on the fixed, 
 
 D^=180''-n, BE=lW-Tl-\. 
 
 Moreover, if the general precession CB' be denoted by I, 
 
 EO=U(f-U-l. 
 
 If also we denote by tt the angle BEG, that is, the inclination 
 of the true ecliptic to the fixed, we have by Napier's analogies 
 in the triangle BEG 
 
 tan -J-^ cos —^ = tan - cos -—-' , 
 
 . TT . [„ l^+l\ .1-1 
 tan - sm (IT + — ^ ) = sm ->~ tan 
 
 tan|cos('n + ^i-ti) = 
 
 cos -i— - tan ^ 
 
 2 2 
 
CALCULATION OP PRECESSION. 171 
 
 By these equations I, ir, and 11 can now be developed in 
 series which proceed according to powers of the time t. The 
 first equation gives 
 
 , , cos — — -° 
 
 . li-l . a 2 
 
 tan „ =tan 
 
 2 2 6-6- 
 
 cos — -—2 
 
 or, if we put 6^+ „ ° for ° and replace the sine and- tan- 
 
 gent of the small angles Zj — I, a, and e — e„ by the arcs, 
 
 a , . . 
 -(6-6>ine„ 
 
 ^=^'-«"°'^« + — 206265— (")• 
 
 In addition, 
 or, by proceeding as before, 
 
 • f + ^o 
 
 sm — -— ' 
 
 sin^^ 
 
 a 
 
 ■ cos e„ 
 
 ^ /„ ?, + Z\ a sin 6- 2 
 
 Finally, 
 
 tan» I = (tan'' l^ - tan"^^ + tan' ^°) 
 
 X cos. 
 If we substitute here for tan -i-^ the value found above, we 
 
 obtain (putting again e„+^^ instead of ^-^ and replacing 
 
 the sines of small angles by the arcs, and the cosines of the same 
 by unity) 
 
 , . » / \-> a" sin e: cos e„ (6 — 6.) , . 
 
 ^^a'sm°6,+ (6-6,r-f ;^g^;; ^ (0). 
 
 Putting now in (a), (J), and (c), instead of Z„ a, and e-e^, 
 their expressions, wliich are of the form 
 
172 NUMERICAL VALUES FOE PBECESSION. 
 
 we easily obtain 
 
 |.^sine 
 Z= (X-acose„) t+ f\'- a'cos e, + 206265 j ''' 
 
 V 
 
 + <("'^^^"^->-"^'^^°^°.206265 + gco3 6Jxsec'n, 
 
 1 2 • 
 
 ,, . -a'i7Sine„C0S6 
 ,r = «.V(«=sin=6„ + ^')+-(^««'sin=e. + W + gog^eS J' 
 
 or, if we substitute for X, X', a, a, and 17, 17' the values pre- 
 viously given*, 
 
 I = 50"-21129 t + 0"-0001221483 i", 
 j = + 50"-21129 + 0"-0002442966 t, 
 TT = + 0"-48892 t - 0"-0000030715 f, 
 ^ = + 0"-48892 - 0"-000006143 t, 
 
 n = 171°.36'.10"-5"-21f. 
 
 6. After obtaining a knowledge of the mutual changes of 
 the planes to which the places of the stars are referred, it is easy 
 to determine the resulting changes of the places of the stars' 
 themselves. Denoting by \ and /3 the longitude and latitude of 
 a star referred to the true ecliptic for the epoch 1750 + <, the 
 co-ordinates of the star in relation to this fundamental plane, 
 taking as zero of longitude the ascending node of the true ecliptic 
 on the fixed ecliptic, are 
 
 cos/Scos (\ — 11 — Z), cos/Ssin (\ — 11 — Z), and sin/8. 
 
 If then L and B be the longitude and latitude of the star, 
 referred to the fixed ecliptic for 1750, the three co-ordinates 
 
 * For 1) and »|' the numerical values are to be taken from the following equation ; 
 e-fo=-t. 0",483C8 - <« . 0',00001236528. 
 
PRECESSION IN LONGITUDE AND LATITUDE. 173 
 
 referred to this fundamental plane and reckoned, from the same 
 zero are 
 
 cos^cos (L — n), cos JSsin(i — 11), and sin 5. 
 
 Since now the fundamental planes of the two systems of 
 co-ordinates make with each other the angle tt, we obtain by 
 means of formula (2) of the Introduction 
 
 cosjScos(\ — 11- Z) =cosJ5cos(i — n) 1 
 
 cos /3 sin {\ — n — Z) =cos5sin(L — n)cos7r + sin5sin7r/ (A). 
 sin /3 =— cos5sin(i— n) slnTr+sinjBcosTr) 
 
 Differentiating these equations, considering L and B as con- 
 stant, we obtain by means of the differential formula (13) of the 
 Introduction 
 
 d iX-U-l) = dU -IT tan^ sm{\-U-l) dU 
 
 + tan yS cos (\ — n — I) dv, 
 
 d^ = - IT co3{\-U -I) dU- aha. {X-U-l) dir. 
 
 * dir 
 
 Whence we obtain, after dividing by dt and putting t-j- 
 
 instead of tt in the coefficient of dli, for the annual changes of 
 longitude and latitude of the star, the following formula : 
 
 d^ . / ^ , dU \, 
 
 or, if we make 
 
 dv 
 di' 
 
 n + i . ^ - Z = 171°. 36'. 10" + t . 39"-79 = M, 
 at 
 
 d\ do , rt /-. ii*-\ dir 
 
 i^> 
 
 d3 , ,^ ,,, dir 
 
 where the numerical values for -3- and -5- are given in the pre- 
 ceding No. 
 
174 PRECESSION IN EIGHT ASCENSION AND DECLINATION. 
 
 Denoting again by L and B the longitude and latitude of a 
 star, referred to the fixed ecliptic and the equinox of 1750, then 
 this longitude, reckoned from the intersection of the equator for 
 the year 1750 + 1 with the fixed ecliptic for 1750, will be equal 
 to L + \ where \ is the amount of the lunisolar precession in the 
 interval from 1750 to 1750 + 1. 
 
 The co-ordinates of the star referred to the plane of the 
 ecliptic for 1750 and the above-mentioned point of intersection 
 will thus be 
 
 cos 5 cos (2/ + Zj) , cos 5 sin {L + 1^ , and sin B. 
 
 Denoting then by a and Z the right ascension and declination 
 of the star, referred to the equator and the true equinox for the 
 year 1750 + 1, the right ascension reckoned from the above-men- 
 tioned point of intersection will be « + a. We have thus for 
 the co-ordinates of the star, referred to the plane of the true 
 equator and the assumed point of intersection, 
 
 cos S cos (a + a) , cos S sin (a + a), and sin S. 
 
 Since the two systems of co-ordinates make with each other 
 the angle e„, we obtain by formula {1) of the Intr-oduction 
 
 cosS'cos (a + a) =cos-Bcos (i + ZJ 1 
 
 cos S sin (a + a) =cos5sin(i/ + ZJ cos£^ — sin5sin6„> (C). 
 sin S = cos ^ sin {L + l^ sin e^ + sin 5 cos e„l 
 
 Diflferentiating these formulee again, considering L and B as 
 constant, we obtain by the differential formulae (13) of the 
 Introduction 
 
 <Z (a + a) = [cos ej, + sin e^ tan S sin (a + a)] c?Z, - cos (a+ a) tan 8 Je„ , 
 dZ — cos (a + a) sin e„ d\ + sin (a + a) de^ . 
 
 We have then, for the annual change of right ascension and 
 declination of the stars, the formulae 
 
 clci da ^ y • ■ . ^ ^ . ^-dl 
 ^ = - ^ + [cos e, + sm e„ tan 8 sm a] -^ 
 
 dl de: 
 
 + (asin6„^-3)tanScos«, 
 
PKEOESSION IN EIOHT ASCENSION AND DECLINATION. 175 
 
 dh 
 
 T- = COS a sin e„ 
 
 dl, ( . dl. de,\ . 
 
 or, neglecting the exceedingly small second terms of each 
 equation*, 
 
 da. da ^ J. , . . s ■ i '^h 
 
 _=___ + [cose„+sin6„tanSsm«]^, 
 
 dS . dl, 
 
 ^ = cos asm 6,.^'.. 
 
 And, putting 
 
 dl, da 
 and. sm e^ ^ =n, 
 
 we obtain finally 
 
 -^ = m + w tan b sin a 
 at 
 
 dS 
 
 -7- = w cos a 
 
 {!>)'. 
 
 and, for the numerical values of m and n, if we substitute the 
 values of 6„,^, and ^, 
 
 TO = 46"-02824 + G"-0003086450 t, 
 n = 2O"-0644:2 - 0"-0000970204 t. 
 
 Now to obtain the amount of the precession in longitude 
 
 and latitede or in right ascension and declination in the interval 
 
 from 1750 + < to 1750 + t', it is necessary to take the integrals 
 
 of the equations {B) or {D) between the limits t and t'. In the 
 
 meanwhile we may find this amount also to terms of the second 
 
 t+t' 
 order inclusive from the differential coeflScients for the time -— — 
 
 and the interval. If, for instance, /(^) and/(<') be two functions, 
 
 • The numerical value of the coefBcient xi sin «„ .^ - -^r, ia -0", 0000022472 «. 
 
176 APPEOXIMATE COMPUTATION OF PRECESSION. 
 
 whose difference f{t') -f{t) is- required, as in the present case 
 is wanted the amount of precession in the time t' —t, we put 
 
 Then 
 and 
 
 \{f-t)=^x. 
 
 fit) =f{x - Ax) =f{x) -Axf'ix) + i iAx)y"{x), 
 
 fit') =f{x + Ax) =f{x) + Ax fix) + i {Ax)y"{x) ; 
 
 where /' (a;) and /" (x) denote the first and second differential 
 coefficients of /(a;). Hence we obtain 
 
 f{f) -fit) = 2Axf'ix) = it'- t)f (f±^) . 
 
 Thus to obtain the precession for an interval ^ ' — <, it is only 
 necessary to compute the differential coeflScient corresponding to 
 the arithmetical mean of the times and to multiply it by the 
 interval of time. By this means the terms of the second order 
 are taken into account. 
 
 If, for example, there be required the precession in longitude 
 and latitude in the time from 1750 to 1850 for a star, whose place 
 for 1750 is 
 
 \ = 210°.0'; j8 = + 34°.0', 
 
 then, for 1800, we have for the values oi -j-, -j- , and M, 
 
 ^=50"-22350, ^ = 0"-48861, and Jf = 172°.9'.20". 
 
 Moreover, we obtain for 1800, by reckoning approximately 
 the precession from 1750 to 1800, 
 
 \ = 210°.42',1, /3= + 33°.59',8; 
 
 and, therefore, by formula (5), for 1800, 
 
 |= + 50",48m,f = -0",3044r. 
 
EXAMPLE OP COMPUTATION OF PRECESSION. 177 
 
 and thus, the amount of precession from 1750 to 1850, is 
 in longitude = + 1°. 24'. 8", 12, 
 and in latitude = - 30",45. 
 
 In like mafiner if the amount of precession in right ascension 
 and declination be required from 1750 to 1850, for a star whose 
 right ascension and declination for 1750 are 
 
 a = 220".1'.24", S = + 20''.21'.15", 
 
 then, for 1800, we have 
 
 m = 46",04367, w = 20",05957, 
 
 and, for the approximate place of the star, for this time, 
 
 a = 220°. 35',8, B = + 20". 8',6 ; 
 
 and, from these data, we obtain by formula {D), 
 
 log tan S = 9-56444 log cos a = - 9-88042 
 
 logsina = - 9-81340 logn= 1-30232 
 
 logw= 1-30232 -1-18274 
 
 -"0^68016 ^^_ 15-, 2314 
 
 n sin a tan S = - 4",78806 '^^ 
 
 m = + 46-04367 
 
 -J = + 41-25561 
 
 and therefore the amount of precession from 1750 to 1850 in right 
 ascension = + 1°.8'.45",56, and in declination = - 25'.23",14. 
 
 7. The differential formulse given above do not serve, when 
 it is required to compute the precession for times very far distant 
 from each other, or for stars which are very near the pole. In 
 these cases the exact formulae must be employed. 
 
 Let \ and /8 be the longitude and latitude of a star, referred 
 to the ecliptic and the equinox for the time 1750 + t, then we 
 obtain the longitude and latitude L and B, referred to the fixed 
 ecliptic for 1750, by the following equations, which follow im- 
 mediately from the equations {A) of No. 6 : 
 
 B.A. 12 
 
178 KIGOEOUS COMPUTATION OF PEECESSION IN E.A. AND N.P.D. 
 
 COS B COS (i - n) = COS /3 COS (\ - n - Z), 
 COS jB sin (i - n) = COS y8 sin (X-II-Z) cOS7r-sin^ sinTr, 
 sin B^cos^ sin (X - 11 - Z) sin tt + sin ^ cos ir. 
 If then the longitude and latitude V and ^', referred to the 
 ecliptic and equinox for the epoch 1750 + 1' be -required, these 
 quantities are obtained from L and B by the following equations, 
 if we denote the values of the quantities IT, tt, and I for the time 
 t' by 11', tt', and Z', 
 
 cos /3' cos (V - n' - 1') = cos B cos {L - XT'), 
 cos /3' sin (\' - n' - 1') = cos 5 sin {L — W) cos tt' + sin 5 sin tt', 
 sin /3' = — cos 5 sin (i — 11') sin tt' + sin £ cos ir'. 
 By eliminating 5 and L from these equations we obtain 
 immediately X' and /3' expressed in terms of X and )8, and of Z, IT, 
 and TT for the epochs t and <'. These formulae will however be 
 seldom employed, since, for longitudes and latitudes, the differ- 
 ential formulae previously given, on account of the smallness of the 
 square of tt, serve for very great intervals of time. Thus, in the 
 preceding example, the error of the differential formulae amounts 
 to only 0",02. For right ascension and declination the rigorous 
 formulje will be precisely similar. If the right ascension and 
 declination of a star be a and S for the epoch 1750 + t, then we 
 obtain the longitude and latitude L and B referred to the fixed 
 ecliptic for 1750 by means of the equations*, 
 
 cos B cos {L + ZJ = cos S cos (a + a), 
 
 cos B sin {L + ZJ = cos B sin (a + a) cos e„ + sin S sin e„ , 
 
 sin i? = - cos 8 sin (a + a) sin e„ + sin S cos e„. 
 If the right ascension and declination a and S' for the epoch 
 1750 + <' be required, these are obtained from Z, and B, de- 
 noting the values of l^, a and e„ for the time t' by Zj', a' and e ', 
 by means of the following equations : 
 cos 8' cos {a + a) = cos B cos {L + Z,') , 
 cos S' sin {a! + a) = cos B sin {L + Z/) cos e/ - sin B sin e„', 
 sin 8' = cos B sin {L + Z/) sin e„' + sin B cos e/. 
 
 . These equations are easily fonnd from equations (C) in No. 6, by consideration 
 of the spherical triangle between the star, the pole of the ecliptic for 1750, and tlje 
 pole of the equator for 1750 + i. 
 
EIGOKOUS COMPUTATION OF PRECESSION IN E.A. AND N.P.D. 179 
 
 By eliminating now B and L from the two systems of equa- 
 tions we obtain, (since 
 
 cos B smL = — cos S cos (a + a) sin \ + cos S sin (a + a) cos e cos I^ 
 
 + sin S sin e cos l^ , 
 
 cos B cos i = cos S cos (a + a) cos Z^ 4- cos S sin (a + a) cos e sin Z, 
 
 + sinS sine sin?j, 
 
 sin 5 = - cos S cos (a + a) sin e + sin S cos e,) 
 
 as is easily seen, the following equations : 
 
 cos S cos (a' + a') = cos S cos (a + a) cos (Z/ — ZJ 
 
 — cos S sin (a + a) sin (Z/ — Z,) cos e, 
 
 — sin S sin (Z/ — ZJ sin 6„ , 
 
 cos S' sin («' + a') = cos S cos (a + a) sin (Z/ — ZJ coS ej 
 + cos S sin (a + a) [cos (Z/ — ZJ cos e„ cos e„' + sin 6„ sin 6„'] 
 + sin S [cos (Z,' — Zj) sin e, cos e„' — cos e, sin e„'] 
 
 sin 8' = cos 8 cos (a + a) sin (Z/ — ZJ sin e^' 
 + cos S sin (a + a) [cos (Z/ — Zj) cos e,, sin e„' — sin e^ cos e^,'] 
 + sin S [cos (Z/ — Zj) sin e^ sin e„' + cos c„ cos e^']. 
 
 Imagine now a spherical triangle, whose three sides are 
 l^'-l„ 90° -s, and 90° + a', 
 and whose three opposite angles are respectively®, e^', and 180°— 6„, 
 then the coefficients of the preceding equations which contain 
 l^ — l^, e„, and £„', are expressed by @, z, and s', and we con- 
 sequently find 
 
 cos S' cos (a' + a) = cos S cos (a + a) [cos @ cos z cos a' — sin s sin z'] 
 + cos 8 sin (a + a) [cos © sin s cos z' + cos a sin z'] 
 
 — sin S sin @ cos &' 
 
 cos 8' sin (a' + a') = cos S cos (a + a) [cos © cos z sin z' + sin » cos z] 
 
 — cos S sin (a + a) [cos © sin z sin s' — cos z cos »'J 
 
 — sin S sin © sin z' 
 
 sin S' = cos 8 cos (a + a) sin © cos z 
 
 ' — cos S sin (a + a) sin © sin z 
 
 + sin S cos ©. 
 
 12—2 
 
180 EIGOEOTIS COMPUTATION OF PRECESSION IN E.A. AND N.P.D. 
 
 If we multiply the first of these equations by sin «', tHe 
 second by cos z', and add them together, and again multiply 
 the &st by cos s' and the second by sin«', and add the pro- 
 ducts in the same way, we obtain 
 
 cos S' sin (a' + a' — »') = cos S sin (a + a + a) i 
 
 cos S' cos («' + «'— a') ='^osScos(a+a+») cos©— sinSsin© >..(«). 
 sin 8' = cos S cos {a + a + z). sin © + sin 8 cos © 
 
 These formula serve immediately for expressing a and B' 
 in terms of a, S, a, a and the auxiliary quantities z, z', and ©. 
 Finally we find, by employing upon the above-mentioned spheri- 
 cal triangle the Gaussian formulae, 
 
 sin - © cos - (z' - s) = sin - (Z/ - ZJ sin i (e/ + ej , 
 sin 2 ©sin - {z' -z) = cos - (Z/ - ZJ sin i (e/ - e„) 
 cos i © sin 1 (/ + a) = sin \ {l' - Z,) cos -i {e' + 6„), 
 
 cos^©cosi (s' + s) =cosi(Z;-ZJcosi (6;-e„). 
 
 Now here it is always allowable to replace sin - («' — a) 
 
 and sin - {ej — ej by the arcs, and to assume the corresponding 
 
 cosines to be equal to 1, so that for the computation of the 
 three auxiliary quantities we obtain the following simple 
 formulae: 
 
 tan - {z' + z)= cos - {e,' + e„) tan J (Z/ - ZJ 
 
 1 1 cotan -(Z/-ZJ 
 
 . 1 
 2 
 
 siii^K+eo) 
 
 tan - © = tan - (e,' + e,) sin 1 (z' + z) 
 
 .. {A). 
 
EIGOEOUS COMPUTATION OP PEECESSION IN E.A. AND N.P.D. 181 
 
 The formulse (a^ian be arranged for convenient computation 
 by the introduction of an auxiliary angle, or instead of them 
 another system of equations may be made use of, which are 
 obtained in the same manner as before by the Gaussian formula. 
 In fact we obtain the formulse (a) by employing the three 
 fundamental formulse of spherical trigonometry to a spherjpal 
 triangle whose three sides are 90° -S', 90° -S, and @, and in 
 which the angles a + a + s and 180°- a' -a' + z' are opposite to 
 the first two sides. Employing instead of these the Gaussian 
 formulse, we obtain, if we denote the third angle by c, and for 
 shortness make 
 
 a, + a + z=A, and a' + a' — z' = A', 
 cos^ (90°+8') cos |(^'+ c) =cosi (90°+8+@) cos i A 
 cosi (90°+S') sini (^'+ c) = cosi (90°+S-©) sin i A 
 sin I (90»+S') cos i {A'- c) = sin i (90°+8 +@) cos - A 
 sin I (90°+ 8') sin i {A' - c) =sin i (90°+ 8 - @) sin i ^ 
 
 I. ..(5). 
 
 We arrive at a greater degree of accuracy still if the quantity 
 A' be not required, but only the difference A' - A. We obtain 
 in fact, by multiplying the first of the equations (a) by cos A, 
 and the second by sin J., and subtracting one from the other, 
 and again, by multiplying the first equation by sin A, and the 
 second by cos A and adding, 
 
 cos 8' sin {A! - J.) = cos 8 sin A sin © [tan 8 + tan - @ cos A^ 
 
 cos 
 or 
 
 8' cos {A! -A) = cos 8 - cos 8 cos A sin© [tan8 + tan -@cos^]; 
 
 tan(^'-^) = 
 
 sin A sin © [tan 8 + tan - © cos A'\ 
 1 - cos^ sin © [tan 8 + tan- © cos^] 
 
182 
 
 KXAMPLE. 
 
 and, by the Gaussian formulse, we find 
 
 I e sini (S' - S) = sin ^@cos i {A' + A) 
 
 l(8'-S)=cosi@cosi(4'-^). 
 
 cos ■ 
 
 COS - C COS 
 2 ^ 
 
 Therefore if we put 
 
 ^ = sin @ [tan 8 + tan - © cos A] . 
 
 •{B), 
 
 we have 
 
 tanM--^)^ /^^"^, 
 ^ 1 —p cos A 
 
 tan i (S' - S) = tan 5 @ 
 
 cos|(^' + ^) 
 cosl(^'-^) 
 
 i- (C). 
 
 The rigorous computation of the right-ascension and de- 
 clination of a star for the epoch 1750 + 1' from the right ascen- 
 sion and declination of the same for the epoch 1750 + 1, is by 
 this means referred to the computation of the formulae {A), (B), 
 and (C). 
 
 Sasample. 
 
 The right ascension and declination of Polaris for the 
 beginning of the year 1755 are 
 
 a = 10°. 55'. 44",955, 
 
 8 = 87°. 59'. 41",12. 
 
 If we now compute the place, referred to the equator and 
 equinox of 1850, we have 
 
 I, = 4'. 11",8756 l^' = 1". 23'. 56",3541, 
 
 a = 0",8897, a' = 15",2656, 
 
 e„ = 23°. 28'. 18",0002, €„' = 23°. 28'. 18",0984. 
 With these data we obtain from formula* [A) 
 
 ^(3' + 2)=0°.36'.34",314, 
 
 («' - g) = 10",6286, 
 
 and therefore 
 
 i? = 0°. 36'. 23",685, 
 a' = 0°.36'.44",943, 
 
APPAKENT CHANGE OF POSITION OF POLAEIS. 183 
 
 and 
 
 = 0". 31'. 45",600, 
 and therefore 
 
 A = a + a + z = U\ 32'. 9",530, 
 
 Computing then by formula (B) and (C) the values of 
 A' — A and B' — S, we find 
 
 and 
 
 log^ = 9-4214471, 
 A'-A= 4°. 4'. 17",710, I (S' - S) = 0°. 15'. 26",780, 
 
 and thus 
 and finally 
 
 ^' = 15°.36'.27",240, 
 
 a'=16°.12'.56",917 
 S' = 88'',30'.34",680. 
 
 8. Since the point of intersection of the equator on the 
 ecliptic retrogrades on the latter by about 50",2, the pole of the 
 equator will in course of time describe about the pole of the 
 ecliptic a small circle, of which the radius is equal to the 
 obliquity of the ecliptic*. The pole of the equator is therefore 
 constantly coinciding with other points of the apparent sphere 
 of the heavens, or, at different times, different stars will be in 
 the neighbourhood of it. In our time the extreme star in the 
 tail of the Lesser Bear (a Ursse Minoris) is the nearest to 
 the pole, and is therefore called the pole-star. This star, whose 
 declination is now greater than 88^°, will continually be ap- 
 proaching nearer to the pole, till its right ascension (at present 
 16°) shall have become equal to 90°. The declination will then 
 have reached its maximum, 89°. 32', and from that time will 
 again diminish, since the precession in declination for stars lying 
 in the second quadrant is negative. 
 
 To find now the position of the pole for any time t, consider 
 the spherical triangle between the pole of the ecliptic for a given 
 epoch *„, and the poles of the equator Pand P' for the times <„ 
 and t. Denoting then the right ascension and declination of 
 
 * strictly speaking, tiiis radius is not constant, but is equal to tlie actually existing 
 obliquity of the ecliptic. 
 
184 DETERMINATION OF POSITION OF POLE OF THE EQUATOE. 
 
 the pole at the time t referred to the equator and the equinox of 
 the epoch «^ by a and S, the obliquities of the ecliptic at the times 
 <„ and t by 6„ and e, then the side PP' = 90° -S, EP=e„, 
 EP' = e, the angle at P=90°+a, and the angle at e is equal 
 to the general precession in the interval t — t^; and we have 
 therefore by the three fundamental equations of spherical trigo- 
 nometry, 
 
 cos 8 sin a = sin e cos 6„ cos Z — cos e sin 6„, 
 
 cos S cos a = sin e sin I, 
 
 sin S = sin e sin 6„ cos I + cos e cos 6„. 
 
 Since this computation in general requires no great accuracy, 
 the place of the pole being always required only approximately, 
 and since besides the diminution of the obliquity is only to be 
 looked upon as proportional to the time for short intervals of 
 time, because it has a period of extremely long duration, we may 
 be permitted to make e = e^, and we then obtain the simple 
 equations : 
 
 tan a =— cos 6„tan - 1, 
 
 -. sin 6. sin I 
 
 cos = 5 . 
 
 COS a 
 
 Although a is here found by means of the tangent, its value is 
 still found without any ambiguity, since it must at the same time 
 fulfil the condition, that cos a. and sin I must have the same 
 sign. 
 
 If for example the position of the pole were required for the 
 year 14000 referred to the equinox of 1850, then for the interval 
 12150 years the general precession will amount to about 174°, 
 and therefore we shall have 
 
 a =273°. 16', and S = +43°. 7'. 
 
 This is very near the place of a Lyr^ whose right ascension 
 and declination for 1850 are 
 
 a =277°. 58', and S = + 38°. 39'. 
 
 Thus in the year 14000 this star will have pretensions to the 
 name of the Pole-star. 
 
APPAEENT CHANGES OF POSITION OP THE STAES. 185 
 
 On account of the changes of declination of the stars through 
 the precession, in course of time some will come above the hori- 
 zon of a place which hefore were never visible there, and others 
 which are now, for example, visible at a place in the northern 
 hemisphere of the globe, will on the contrary obtain so southern 
 a declination, that they will no longer rise above the horizon of 
 the place. In like manner will stars which at the present time 
 are always above the horizon of the place, begin to have their 
 risings and settings, while, on the contrary, other stars will 
 attain so large a north declination that they even at their lower 
 culmination will remain above the horizon. The aspect of the 
 heavens at any place on the earth will thus be remarkably 
 changed by means of the precession. 
 
 In the remark appended to No. 16 of the second section, the 
 sidereal year, or the sidereal period of revolution of the sun, that 
 is the time required by the sun to pass over 360° of the sphere of the 
 heavens, or the time in which it again returns to the same fixed 
 star, was estimated at 365 days, 6 hours, 9 minutes, and 10"7496 
 seconds, or at 365'25637 mean days. Since now the equi- 
 noctial point moves backwards, that is contrary to the motion of 
 the sun, the tropical year, that is the time required by the sun 
 to return to the same equinox, will be shorter than the sidereal 
 year by the time taken by the sun to describe the small arc 
 which is equal to the annual precession. Now, for the year 
 1800, ?= 50", 2235, and since the mean daily motion of the sun 
 amounts to 59'. 8", 33, we obtain for this time 0'01415 of a day, 
 and for the length of the tropical year 365'24222 days. But 
 since the precession is variable and its annual increase amounts 
 to 0"- 0002442966, the tropical year is also variable, and the 
 annual change of the same is equal to 0,000000068848 of a day. 
 If the decimal part be expressed in hours, minutes, and seconds, 
 we thus obtain for the length of the tropical year 
 
 365*.5^48".47^8091 - 0^00595 (<- 1800). 
 
 9. The lanisolar precession contains only the terms propor- 
 tional to the time in the motion of the equator on the fixed 
 ecliptic which are produced by the attraction of the sun and moon 
 on the spheroidal earth. But theory teaches that the complete 
 expression of this motion contains, besides those times, others 
 
186 NUTATION. 
 
 periodical, which depend on the places of the sun and moon, but 
 ' especially on the longitude of the moon's node (that is, the longi- 
 tude towards which the line of intersection of the plane of the 
 moon's orhit and the ecliptic is directed)*. This periodical part 
 in the motion of the equator on the fixed ecliptic is denoted by 
 the term Nutation, since it is in a manner produced by a periodi- 
 cal oscillation of the earth's axis about its mean position, and the 
 periodical motion of the point of intersection is called the nutation 
 in longitude, while on the other hand, the periodical part of the 
 change of inclination is called the nutation of the obliquity of the 
 ecliptic. The point in which the equator and the ecliptic at any 
 time really intersect each other is called the true equinox for that 
 time, and on the other hand the point of intersection cleared of 
 the nutation is called the mean equinox. In the same manner by 
 the true obliquity of the ecliptic is meant that inclination of the 
 ecliptic to the equator, which, on account of the secular change 
 and the nutation, really exists, while on the other hand by the 
 mean obliquity is meant the inclination cleared of the nutation. 
 
 The expressions for the change of longitude and obliquity of 
 the ecliptic, AX, and Ae, are at the present time according to 
 Bessel, 
 
 A\ = - 16",78332 sin fl + 0",20209 sin 20 
 
 - 1", 33589 sin 20 - 0", 20128 sin 25 
 and \ (a), 
 
 Ae = + 8", 97707 cos fl - 0"-08773 cos 20 
 
 + 0", 57990 cos 2 + 0", 08738 cos 2]), 
 
 where fl denotes the longitude of the ascending node of the 
 moon's orbit on the ecliptic, and and J denote the longitudes 
 of the sun and moon. Now to compute the amount of the nuta- 
 tion for right ascension and declination, we first obtain, if we 
 denote by a and 8 the mean right ascension and declination, the 
 mean longitude and latitude by the formulae : 
 
 cos ^ cos \ = cos 8 cos a, 
 
 cos j8 sin \ = cos S sin a cos e + sin S sin e, 
 sin ^ = — cos S sin a sin e + sin S cos e. 
 
 * This motion of the moon's node is very rapid, since it amounts to 360° in about 
 19 years. 
 
CALCULATION OF NUTATION. 187 
 
 Augmenting then the longitudes thus found by the nutation 
 A\, and the obliquity of the ecliptic by Ae, we find the apparent 
 right ascension and declination a' and 8' by the equations : 
 cos S' cos a' = cos fi cos (\ + A\), 
 
 cos S' sin a = cos /3 sin (X + A\) cos (e + Ae) — sin jS sin (e + Ae), 
 sin 8' = cos /8 sin (\ + A\) sin (e + Ae) + sin /8 cos (e + Ae) . 
 
 But, since the variations AX. and Ae are but small, differ- 
 ential formulae will be sufficient for the purpose. We shall 
 have, namely, 
 
 and 
 
 But, according to the differential formulae in Section I, No. 
 10, we have, if for cos ^ sin 17 and cos /8 cos 1? we put the expres- 
 sions for them in terms of a and S, 
 
 /'^ j = cos e -I- sin e tan 8 sin a (tt J = cos a sin e, 
 
 (|)=-cos«tanS (?e)=^^^«' 
 
 from which we obtain by differentiation : 
 
 (5). 
 
 d\ 
 
 d\ 
 d\de 
 
 ^ = sin" e \h sin 2a + cotan e cos a tan S + sin 2 a tan^S], 
 d\ 
 
 =— sin e [cos" a — cotan e tan 8 sin a + tan" 8 cos 2a] 
 
 ^ =.- [^ sin 2a + sin 2a tan" 8] . 
 de^ 
 
 — - = — sin'' 6 sin a [cotan e + tan 8 sin a] , 
 dX^ 
 
188 EXPRESSIONS FOE NUTATION IN K.A. AND N.P.D. 
 
 ,, , = sin 6 cos a rcotan e + sin a tan S] 
 aXde 
 
 do H i c» 
 
 -^r^ = — cos a tan 6. 
 
 Substituting these expressions in the equations (J), and 
 putting besides for A\ and Ae the values previously given from 
 equations (a), and for e the mean obliquity for the beginning 
 of the year 1800 = 23°. 27'. 54", we obtain for the terms of the 
 first order, 
 a:-a = - 15", 39537 sin ft- [6", 68299 sin fl sin a 
 
 + 8", 97707 cos fl cos a] tan 8 
 + 0", 18538 sin2fl + [0", 08046 sin 211 sin a 
 
 + 0",08773 cos 212 cos a] tan S 
 
 - 1", 22542 sin20- [0", 53194 sin 20 sin a 
 
 + 0", 57990 cos 20 cos a] tan S 
 
 - 0", 18463 sin 25 - [0", 08015 sin 2 D sin a 
 
 + 0", 08738 cos 2 5 cos a] tan 8 {A) 
 
 ^'-^ = - 6", 68299 sin n cos a + 8", 97707 cos II sin a 
 
 + 0",08046 sin 212 cos a - 0", 08773 cos 211 sin a 
 
 - 0", 53194 sin 20 cos a + 0", 57990 cos 20 sin a 
 
 - 0", 08015 sin 2]) cos a + 0",08738 cos 2 5 sin a. 
 
 Of the terms of the second order those only can be significant 
 which arise from the large terms of AX and Ae. Putting 
 
 Ae = 8", 97707 cos 11 = a cos Q., 
 and 
 
 - sin eA\ = 6", 68299 sin 11 = & sin H, 
 
 these terms will be 
 
 a' — a = — ^ — sin 2a tan'' 8 + - 
 * L 2 
 
 + — tan 8 cos a cotan e 
 4 
 
 + - — cotan e sin a tan 8 + tan^ 8 cos 2a + - cos 2a — sin 2fl 
 
 — - — tan 8 sm 2a + — tan o cos a cotan e 
 4 4 
 
 If + a' 
 
 -I T^ — sm 2a 
 
 cos 211, 
 
TERMS OP NUTATION OF THE SECOND ORDER. 189 
 
 and 
 
 o — 6 = tan o — — sm a cotan e 
 
 4 4 
 
 — (tan S sin 2a + 2 cotan e cos a) -p sin 20 
 
 4 
 
 /a''cos°a-&^sin''a^ s ^^ • 4. \ on 
 
 — I tan 6 — y sm a cotan el cos 211. 
 
 Of these terms those which are independent of D, change 
 merely the mean place of the star, and may therefore he neg- 
 lected. Another portion of the terms, namely, 
 
 — sin 2fl — f — - cotan e sin a sin 2fl + -j cot e cos a cos 20 j tan S, 
 
 and 
 
 - — cotan 6 sin 212 cos oi, + — cotan e sin a cos 2D,, 
 
 are combined with the similar terms multiplied into sin 211 and 
 cos 211 of the first order, so that there wiU be, 
 
 in a, 
 
 + 0"-18545 sin 2fl + (0"-08012 sin 211 sin a 
 
 + 0"-08761 cos 20 cos a) tan S ; 
 and in S, 
 
 + 0"-08012 sin 2f2 cos a - 0"-08761 cos 211 sin a. (B) 
 
 The yet remaining terms of the second order are then the 
 following : 
 
 in right ascension, 
 
 + 0"-0001454 (tan' S + ^ j cos 2a sin 20, 
 
 - 0"-0001518 [tan'' S + -j sin 2a cos 2£1 ; 
 
 and in declination, 
 
 - 0"-0000727 tan 8 sin 2a sin 2D,, 
 
 - (0"-0000217 + 0"-0000759 cos 2a) tan S cos 2fl. 
 
 But since the former terms only attain to the value 0°,01 for 
 declinations equal to 88°. 10', and the others have only the value 
 
190 TABULATION OF NUTATION. 
 
 0",01 for declinations equal to 89°. 26', they may be always 
 safely neglected. 
 
 10. For the purpose of more easily computing the nutation 
 in right ascension and declination, tahles have been constructed. 
 The terms 
 
 -15"-39537sinn = c and - 1",22542 sin20=^ 
 are first tabulated, the arguments of the tables being li and 
 20. 
 
 The 'separate terms for right ascension multiplied into tan 8 
 have always the form 
 
 a cos;8 C0S7 + b sva ^ siay = A {a cosy3cos7 + sin /3 sin 7). 
 
 But to every expression of this form can be given the fol- 
 lowing form, 
 
 a;cos(^-7 + 3^) ....(a), 
 
 by only giving suitable values to the quantities x and y. But 
 by expanding the last expression and comparing it with the 
 former, we obtain for the determination of x and y the equa- 
 tions : 
 
 -4 a cos /Q = 03 (cos ^ cos ^ — sin /3 sin ^), 
 Asin^ = x (sin^S cos^ -1- cos /3 sin?/) ; 
 
 from whence for x and y we obtain the values 
 a;'' = ^'{l-(l-a')cos=;S}, 
 and 
 
 (1 — a) sin B cos B 
 
 *^"y= i_(i-,)cos-^ - 
 
 Tabulating then the values of x and y, whose argument is yS 
 the expression (a) can be easily computed. 
 
 In like manner can tables be constructed for the correspond- 
 ing terms of the nutation in declination, since these are of the 
 form 
 
 A {-a cos ^ sin7 + sin /8 cos 7) ; 
 and, for every such expression we may always put 
 
 xsin{^-y + y), 
 where x and y have the same values as before. 
 
LUNAE NUTATION AND SOLAB NUTATION. 191 
 
 Such a table for the nutation, computed by Nicolai, will be 
 found in Warnstorff's Auxiliary Tables, but the fundamental 
 constants are different from those given above, that is to say, 
 from the Constants of Peters*. In these tables will be found, 
 in addition to the term c, the quantities log h and B, with argu- 
 ment ii, and there will by this means be obtained the terms de- 
 pending on sin O and cos fl, which are, for right ascension, 
 
 c — h tan S cos ifl+B—a), 
 and, for declination, 
 
 -&sin(0-t-£-a). 
 This part of nutation is called Lunar Nutation. 
 
 A second table gives with argument 20 the quantities g, F, 
 and log_^ by means of which are found the terms depending on 
 20, which are, for right ascension, 
 
 g -/tan S cos (20 + i^- a) ; 
 and for declination, 
 
 /sin(20-|-i?'-a). 
 This part of nutation is known by the name of Solar Nutation. 
 
 The terms of nutation depending on the arguments 2]) and 
 211, are then given by the table for solar nutation, if, instead 
 of entering it with 20, it be once entered with 20 and a second 
 time with 20 + 180° (since the last terms have opposite signs); 
 and lastly, if, from the sum of the results corresponding to these 
 two arguments one- sixth part be subtracted, since this is ap- 
 proximately the ratio of its coeflScient to that of solar nutation. 
 
 11. Since we have by this means obtained a knowledge of 
 the changes of the planes to which the places of the stars are 
 referred, the absolute right ascension of a star can be determined 
 with every accuracy by introducing these changes into the com- 
 putation. Therefore in the next place it is necessary to free the 
 observations of the sun and the star from nutation. Taking the 
 example previously given in No. 4 of this section the declina- 
 tions of the sun at the two observations were, for 
 
 " Given in his memoir Numerus Constans Nutatianis in Ascensionibus Reciis Stellce 
 Polaris in Speculd Dorpatensi annis 1822 ad 1838, observatis deductus. Petropoli, 
 1842 Tbawslatoe. 
 
192 
 
 EXAMPLE. 
 
 Mar. 23, Z> = + 1°. 6'. 54", 
 
 Sept. 20, Z>'= + 1". 1'. 57". 
 And the right ascensions of the sun and the star, which are 
 known by the previous determination and the observed difference 
 of right ascension, are 
 
 Mar. 23, A = 2°. 34', 
 
 Sept. 20, ^'=177°. 37', 
 and 
 
 a =112°. 35', 
 
 Moreover the longitude of the ascending node of the lunar 
 orbit was, at the time of the two observations, 
 
 n = 207°.21, Ii'=197».45'; 
 
 and the longitudes of the sun 
 
 = 2°. 49', 0' = 177°. 26'. 
 
 With these values we find : 
 
 Nutation for the Bun. 
 
 Mar. 23, in right ascension = + 0^48, in declination = + 2",8, 
 
 Sept. 20, +0=,34 -. -2",4; 
 
 and the nutation in right ascension for a Canis Minoris (Procy'on), 
 if we assume S = + 5°. 39', 
 
 for Mar. 23, = + 0',47, and Sep. 20 = + 0',30. 
 
 If these values for the nutation be applied with changed 
 signs to the observed times of transit (which differ from the 
 right ascensions only by the amount of clock-error) and to the 
 declinations, we obtain those quantities referred to the mean 
 equinox of the days of observation. Finally, account must be 
 taken of the change of the equinox through precession, or all the 
 observed data must be referred to a fixed equinox. If we take 
 as epoch the beginning of the year 1828, we obtain for the pre- 
 cession for the place of the sun. 
 
 Mar. 23, A^= + 0^,71, AZ) = + 4",6, 
 Sept. 20, A^'= + 2',28, AZ)' = _ li-'^s'. 
 and, for the star. 
 
 Mar. 23, Aa = + 0',73, Sep. 20, Aa' = + 2%32. 
 
PEOPEE MOTIONS OP THE STAES. 193 
 
 Applying these values witli changed signs to the ohserved 
 data, we find, after having taken account of all corrections, 
 
 T= OMr. 12',62, T'= 11\50'°.32',06, 
 <= 7.31.13,00, i= 7.30.22,74, 
 
 i) = + r. 6"."54',9, i)' = + l°. 2". 5',6. 
 
 Finally, we obtain for the obliquity of the ecliptic for both 
 epochs, regard being had to the secular variations and the nuta- 
 tion, 
 
 € = 23°. 27'. 33",9, e' = 23°. 27'. 33",1, 
 and from thence 
 
 1 r . _, tani) . _, tanD'l 
 - sm ' -; sm r 
 
 2 |_ tan 6 tan 6 J 
 
 ^-6'' = 6^ O". 22°,25, 
 
 i(<-r)+i(<'-r') =1.29.55,53; 
 
 and therefore the right ascension of a Canis Minoris referred to 
 the mean equinox of the beginning of the year 1828, 
 
 a = 7". SO-". 17^78. 
 
 If now the absolute right ascensions and declinations of the 
 stars be determined at diiBferent epochs, we obtain from the com- 
 parison of the two positions the amount of the precession in 
 right ascension and declination in the interval, and are able by 
 this means to determine the values of m and n (No. 6) as well 
 as the annual lunisolar precession. But we shall always find 
 that from different stars different values of these constants are 
 obtained, since the stars, besides the apparent motions before 
 treated of, have also proper motions, by virtue of which they 
 will really change their positions in space. Since now these- 
 proper motions, as in the case of. the precessions, at least for 
 intervals of time not very great, appear to be proportional to the 
 time, we can determine the values of both changes in no other 
 way than by computing the values of the lunisolar precession 
 from a very great number of stars atd talking the arithmetical 
 mean of the separate results, assuming thereby that the proper 
 motions, which have different values •and in different directions 
 for different stars, will be eliminated in this mean value. The 
 differences which then exhibit themselves in comparison with 
 B. A. 13 
 
194 SECUIiAR VARIATION, 
 
 observationg made at the time t', when the place of a star for 
 that epoch Is deduced from obserrations made at the time t by 
 means of the value of the lunisolar precession thus determined, 
 is considered to be the proper motion of the star in right ascen- 
 sion and declination during the time t—t. 
 
 On account of the changes of position of the stars through 
 precession and proper motion, catalogues of their places, or star- 
 catalogues, are in all cases serviceable only for one particular 
 epoch. For the purpose then of reducing star-places from one 
 epoch to another more conveniently, it is customary to give for 
 each star in these catalogues the yearly changes in right ascen- 
 sion and declination through precession and proper motion under 
 the title of annual variation and proper motion, and, in addition, 
 the change of the annual variation in a hundred years or the 
 secular variation. If then <„ be the epoch of the catalogue, the 
 change of place of the star during the time < — i„ is equal to 
 
 [annual var. + proper motion + " x secular var.) x (t — <J. 
 
 The computation of the secular variation is made by the 
 following formulae. From No. 6 the annual changes of right 
 ascension and declination are 
 
 — =zm + n tan S sin a, 
 
 -TT = n cos a. 
 at 
 
 By differentiating these equations, treating all the quantities 
 as variable, and denoting the annual variations of m and n 
 by m and n', we easily obtain, 
 
 ^ = n' tan' S sin 2a + w' sin 2a + mn tan S cos a 
 
 + m' + n' tan S sin a, 
 a. tan ^— mn sin a + « 
 
 and, multiplying these second diflferential coefficients by loO 
 
 J2S 
 
 jTj = — «' sin* a tan ^— mn sin a + n' cos a. 
 
EIGOEOUS COMPUTATION OP PKOPEE MOTION. 195 
 
 we obtain the secular variation for right ascension and 
 declination. 
 
 12. As has been remarked, it is always assumed that the 
 proper motions of the stars are proportional to the time, and 
 take place in a fixed great circle. These ti^-o conditions are not 
 strictly coiTect, but on account of the extremely slow motions 
 of the stars no sensible error arises from this assiunption. But 
 since the fundamental planes to which the places of the stars are 
 referred, are variable, on this account the components of the 
 proper motions in the directions of the polar co-ordinates, which 
 are referred to these planes, are also variable. 
 
 The formulse which, for an assumed equinox, express the polar 
 co-ordinates referred to the equator for the time t' by means of 
 the co-ordinates referred to another equinox for the time t, are, 
 by No. 7, 
 
 cos S' sin (a' 4- a' — *') = cos h sin (a + a + a), 
 cos 8' cos (a' + a' -^ z) = cos S cos (a + a + a) cos© — sin S sin ©, 
 sin h' = cos S cos (a + a + «) sin @ + sin S cos ©, 
 where a denotes the precession produced by the planets during 
 the time t' — t, and z, z , and © are auxiliary quantities obtained by 
 means of the formulae (A) of the same No. Since the proper 
 motions are so small, that their squares and products may be 
 neglected, we obtain by the first and third formulae (13) in No. 9 
 of the Introduction, remembering that the formulae above are 
 derived from a triangle of which the sides are 90° — S', 90" — S, 
 and @, and of which the angles are a -H a 4- z, 180° — a' — a' -1- s', 
 and c, 
 
 AS' = cos c AS — sin © sin (a' + a' — a') Aa, 
 cos S' Aa' = sin c AS + cos S cos c Aa, 
 or, if sine and cose be expressed in terms of the other parts of . 
 the triangle, 
 Aa' = Aa [cos © + sin © tan V cos (a' + a' - »')] 
 
 , AS . „ sin (a' + «'-«') 
 
 + 5 sm © i^^ 57 - 
 
 cos 5 cos 
 
 AS' = - Aa sin © sin (a' + a' - «') 
 
 H 5 cos S' [cos © + sin © tan S' cos (a' + a'- »')] 
 
 cos 6 '- ' ^ '■' 
 
 («), 
 
 13—2 
 
196 EXAMPLES. 
 
 and, in the same manner^ 
 Aa = Aa' [cos @ - sin @ tan 8 cos (a + a + z)] 
 
 AS' . ^ sin (a + a + a) 
 cos 6 cos o 
 
 AS = Aa' sin © sin (a + a + z) 
 
 + il£^^ cos S [cos © - sin tan S cos (a + a + a)] 
 coso '- 
 
 (&). 
 
 Example. 
 The mean right ascension and declination of Polaris for the 
 beginning of the year 1755, is 
 
 a = 10°.55'..44",955, S = + 87°.59'.41",12. 
 By application of the precession the place of Polaris was in 
 No. 7 computed for 1850, Jan. 1, and found to be 
 
 a' = 16°.12'.56",917, S' = + 88°.30'.34",680. 
 
 But in Bessel's TahulcB Regiomontance this place is 
 
 'a'=16M5'.19",530, S' = + 88'.30'.34",898. 
 
 This difference arises from the proper motion of Polaris, 
 which thus amounts to + 2'.22",613 in right ascension, and 
 to + 0",218 in declination in the interval from 1755 to 1850. 
 The annual proper motion of Polaris referred to the equator 
 of 1850 is therefore 
 
 Aa' = + 1",501 189, AS' = + 0",002295. 
 
 If we would from this, for example, have the proper motion of 
 
 Polaris referred to the equator of 1755, it must be computed 
 
 from the formulae (6). 
 
 But we have 
 
 © = 0°.31'.45",600, 
 
 a+a + » = ll''.32'.9",530, 
 and by these numbers we obtain 
 
 Aa = + l",10836, AS = + 0",005063. 
 If now the place of Polaris for 1755 and the proper motion 
 were given, and from these data the place for 1850 were to be 
 computed, we should have 
 
 95Aa = + l'.45",294, 
 95AS = + 0,481, 
 
MEAN AND APPARENT PLACES OF THE FIXED STARS. 197 
 
 therefore the place for 1755 + the proper motion to 1850, is 
 
 a = 10°.57'.30",249, 
 
 S = 87°.59'.41",601, 
 
 and, with these values, the compntation in No. 7 must he repeated, 
 when we should find for a' and 8' the values of the Tahulce 
 Regiomontance, 
 
 Note. On Precession and Nutation compare the preface to the To- 
 buloB RegiomontanoB, page in. et seq. 
 
 III. Mean and Apparent Places of the Fixed Stars. 
 
 13. By means of what has preceded we are now in a con- 
 dition to compute the mean place of ^ star for any time what- 
 ever, if this he known for any one epoch. In the same way 
 can we from these mean places find the apparent places, or the 
 places which the stars really appear to occupy on the sphere 
 of the heavens, hy applying to the mean places the separate 
 corrections, nutation, aberration, refraction, and annual parallax. 
 The last of these corrections is for the fixed stars insensible, 
 (excepting a few stars whose annual parallaxes are however 
 extremely small), and since the refraction is always applied to 
 the observations themselves, there remain only the aberration and 
 nutation, whose values can, as has been before seen, be taken out 
 of tables. Since however these reductions from mean to apparent 
 places and vice versd must be performed in large masses, tables 
 still more convenient have been constructed, which have the time 
 for argument, and enable us to perform with greater facility the 
 reduction from the mean places at the beginning of the year to 
 the apparent places for any particular day of the year. Thus if 
 the mean place of a star be known for any epoch, and it is re- 
 quired to find the apparent place for any given day of another 
 year, the mean place for the beginning of this latter year must 
 first be found by application of the precession, andj when neces- 
 sary, of the proper motion, and afterwards the reduction to the 
 apparent place for the given day must be taken out of the tables. 
 These tables are given by Bessel. 
 
 Let a and S denote the mean right ascension and declination 
 of a star for the beginning of i, year, a! and S' the apparent right 
 
. nutation, 
 
 198 COMPLETE EXPEPSSION FOE APPARENT PLACES. 
 
 ascension and declination for the time t, which is reckoned from 
 the beginning of the year and is expressed in parts of it, then 
 a' = a + T [m + w tan S sin a] + r/t, precession and proper motion, 
 
 - [15",39537 + 6",68299 tan S sin a] sinfi 
 
 - 8",97707 tan S cos a cosfl 
 + [0",18538 + 0",08046 tan S sin a] sin 2fl 
 + 0",08773 tan S cos a cos 20 
 
 - [1",22542 + 0",53194 tan 8 sin a] sin 2D, 
 
 - 0",57990 tan S cos a. cos 20 
 
 - 20", 255 cos e sec 8 cos a cos 01, 
 
 f aoerration, 
 
 - 20' ,255 sec sm a sin ) 
 
 and 8' = 8 + T»i cos a + rfjf . . . precession and proper motion, 
 
 - 6",68299 cos a sin fl + 8",97707 sin a cos fl ' 
 
 + 0",08046cosasin2fi-0",08773sinacos2fl • nutation, 
 
 - 0",53194 cos a sin 20+O",5799O sin a. cos 20 
 
 + 20",255 [sin a sin 8 cos e — cos 8 sin e] cos ) , . 
 
 - 20 ,255 cos a sin 8 sin J 
 
 In these formulae those terms of the nutation which depend 
 on twice the moon's longitude are not introduced, since they 
 only alter the place of a star by 0",1, and besides, on account of 
 the rapid motion of the moon, have so short a period, that they 
 are in a great measure eliminated in the mean of several 
 observations. 
 
 To tabulate these expressions for a' — a and B' — B Bessel 
 puts 
 
 6",68299 = to', 15",39537-»w« =h, 
 
 0",08046 = ni', 0",18538 - mi' = h', 
 
 0",53194 = ni", 1",22542 - mi" = h". 
 
 Then the preceding formulae can be written thus : 
 a' = a + [t — t sin O + i' sin 20 — i" sin 2fl] x[m + n tan 8 sin a] 
 
 - [8",97707 cosO-O",O8773cos2n+O",5799cos20] xtan8cosa 
 
 - 20",255 cos 6 cos sec 8 cos « 
 
 - 20",255 sin sec 8 sin a 
 
 + T/4 
 
 -hsina + h' sin 2(1 - h" sin 20, 
 
FOEMUIi^ FOB COMPUTATION. 
 
 199 
 
 («); 
 
 and 
 
 B' = S +"[t - { sin 12 + ^" sin 2fl — *'" sin 20] x w cos a 
 + [8",97707 cos CI - 0",08773 cos 2fi + 0",5799 cos 20] sin a 
 
 — 20",255 cos e cos [tan e cos B - sin'S sin a] 
 
 - 20",255 sin sin 8 cos a 
 
 + T/*'. 
 
 Let now the following notation be introduced : 
 A — r~i&in.il + i' sin 2fl - i" sin 20 
 5 = -8",977O7cosn+O",O8773co»2n-O",5799cos20 
 
 C = -20",255 cose cos 
 
 D=-2O",255sin0 
 
 E = -h, sin ft + A' sin 20- A" sin 20 
 
 a — m-^-n tan S sin a, a' = w cos a 
 
 h =■ tan S cos a, J' = — sin a 
 
 c = sec h cos a, c' = tan e cos S — sin 8 sin a 
 
 <?=sec8sina, «?' = sinScosa 
 
 and thus we have the simple expressions : 
 
 where the quantities a, 5, c, £?, a', 5', c', rtf', depend solely on the 
 place of the star and the obliquity of the ecliptic, while, on the 
 contrary, A, B, G, D, depend solely on the time (since and ft 
 are functions of the time) and so can be tabulated with the time 
 for argimient. Bessel has given the values of the logarithms of 
 A, B, G, and Z) in his Tabulm Begiomontance, for every tenth 
 day from 1750 to 1850. 
 
 The numerical values of the quantities { and h are 
 
 for 1750 » = 0-33308 »' = 0-00401 r = 0-02651, 
 
 ... 1850 0-33324 0-00401 0-02652, 
 and 
 
 for 1750 A = 0-0645 A'= 0-0008 r= 0-0051, 
 
 ... 1850 0-0468 0-0006 0-0037. 
 
 It foUows therefore that the quantity E never amounts to 
 more than a few hundredths of a second, and therefore, in most 
 eases, may be neglected^ 
 
 •(5), 
 
20D EEDU'CTION FOE DIFFEKENT MEEIDIAXS. 
 
 14. The arguments of the tables calculated by Bessel are' the 
 days of the year the beginning of which is taken for the instant 
 when the sun's longitude is 280°. These tables therefore serve 
 for that meridian for which the sun, at the commencement of 
 the civil year, has this longitude. But since the sun does not 
 complete a revolution in a whole number of days, but in 365 days 
 and a fraction, these tables for any other year wiU serve for a dif- 
 ferent meridian. 
 
 Now from the formula (G), in No. 2 of the second section, it 
 follows, that the angle between two meridians on the earth's 
 surface is equal to the difference of the sidereal times at those 
 places, and is thus also equal to the difference of the mean times. 
 Denote then by k, expressed in time, the difference of meri- 
 dians of Paris and of a place for which the sun at the begin- 
 ning of the year has the longitude 280°, and consider le to 
 be positive when the place is east of Paris; denote also by d the 
 meridian difference of any other place from Paris considering it 
 positive when the place is west of Paris, then to the time of this 
 latter place, for which the constants A, B, C, D, are required 
 from the tables, must be added the quantity k + d, and the tables 
 must be entered with this correicted time. The values of h are 
 given by Bessel in the Tdbulm Hegiomontdnce for every year 
 from 1750 to 1850. 
 
 Now in the tables are found the constants A, B, 0, D, for 
 the beginning of the fictitious year, which commences when the 
 longitude of the sun is 280°, and then, for the same time of every 
 tenth sidereal day; they thus apply constantly for 18''.40'" sidereal 
 time of that ftieridian, for which the sun at the beginning of the 
 year has the longitude aforesaid. If now we desire to take from 
 the tables the value for another sidereal time a, which in this 
 instance we have taken equal to the right ascension of the star 
 for the purpose of obtaining the apparent place — say for the time 
 of culmination, then there must be added to the argUiilent k^d 
 the quantity 
 
 , a-18\40'" 
 "=~24- • 
 
 Since, moreover, on that day, on which the right ascension of 
 the sun is equal to the right ascension of the star, two cul^iina- 
 
REDUCTION FOR DIPPEEENT MERIDIANS. 201 
 
 tions of the star must happen, we must at this time a,dd a unit to 
 the datum of the day, so that the complete argument will be 
 equal to the datum + the quantity 
 
 h+d+al + i, 
 
 where i is equalto from the beginning of the year to the time 
 when the right ascension of the sun is equal to a, but afterwards 
 is equal to + 1. 
 
 The day denoted in the tables by Jan. is that, at the 
 sidereal time 18''.40'° of which the year begins, according to the 
 common method of counting days, when the commencement of 
 the same is reckoned from noon. The culmination of those- 
 stars whose right ascensions are less than 18''.40™ does not 
 fall, therefore, on that day denoted by Jan. in the tables, 
 but on the day preceding, and thus to the datum of the day 
 reckoned from noon, must be added a unit, before entering the 
 tables. 
 
 The arguments are consequently the following : 
 
 Xst, if a be < 18^40■", 
 
 from the beginning of the year to the day on which the right 
 ascension of the sun is equal to a, 
 
 , , a-18\40"' , 
 datum + K + a H 5^^ 1- 1 ; 
 
 from that time to the end of the year, 
 
 , , a- 18", 40" „ 
 datum +k'+d-\ ^TH 1" 2- 
 
 2ndly, when a is > IS". 40", 
 
 from the beginning oFthe year to the day on which the right 
 ascension of the sun is equal to a, 
 
 a -IS''. 40" 
 
 datum + i + <? + - 
 
 24" 
 
 and, from that time to the end of the year. 
 
 , , a -18". 40" , , 
 datum + A + a + -^ 1- 1. 
 
202 ANOTHEIl POBM OP CALCULATION. 
 
 15. This kind of computation of apparent places is par- 
 ticularly convenient when it is desired to construct an ephemeris> 
 of stars' places for a considerable length of time. 
 
 If only a single place be required, the following method may 
 be used more conveniently, since thereby the trouble of com- 
 puting the constants a, b, c, &c. is avoided. 
 
 The terms of the precession and nutation are, namely, if the 
 quantity U be neglected : 
 for right ascension. 
 
 Am + An sin a. tan B + Bt&n S cos a ; 
 and, for declination, 
 
 An cos a — 5 sin a. 
 Put now 
 
 An = g cos G, 
 B = ^ sin G, 
 Am=f, 
 then will these terms be, for the right ascension, 
 
 f+gsva. ((? + «) tanS; 
 and, for the declination, 
 
 g cos ((r + a). 
 Moreover we have for the aberration in right ascension and 
 declination, by No. 14 of Section 2, 
 h sin {11+ a) sec 8, 
 
 and 
 
 h cos {11+ a) sin h + { cos S, 
 
 (where A sin ^3"= 0, hcoaS=D, and z= Ctane), 
 
 so that the complete formulas for the apparent place of a star, are 
 
 a.' = a + f+gaisi{G + a.)ta,nS + hBm {H+ a) sec S + t/m, 
 
 S' = S + gcos{G + a.) + h cos {JI+ a.) aiaS + icosS + rfi. 
 
 Note. On the computation of apparent places compare the preface to 
 Bessel'8 Tabtdce Begiomontanm, pp. 24 and 29, &c. 
 
 END OP THE FIRST PART. 
 
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