SPHERICAL ASTRONOMY.








SPHERICAL ASTRONOMY.
BY
F. BRUNNOW, PH. DR.
4+ —---
TRANSLATED BY THE AUTHOR FROM THE SECOND
GERMAN EDITION.
NEW  YORK:
D. VAN NOSTRAND, 192 BROADWAY.
LONDON: ASHER & Co.
1865.








DEDICATED
TO THE
REV. GEORGE P. WILLIAMS, L. L. D.
PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN
BY THE AUTHOR
AS AN EXPRESSION OF AFFECTION AND GRATITUDE FOR UNVARYING
FRIENDSHIP AND A NEVER CEASING INTEREST IN ALL HIS
SCIENTIFIC PURSUITS.








PREFACE.
During my connection with the University of
Michigan as Professor of Astronomy I felt very much
the want of a book written in the English language,
to which I might refer the students attending my lectures, and it seems that the same want was felt by
other Professors, as I heard very frequently the wish
expressed, that I should publish an English Edition of
my Spherical Astronomy, and thus relieve this want
at least for one important branch of Astronomy. However while I was in America I never found leisure to
undertake this translation, although the arrangements
for it were made with the Publishers already at the time
of the publication of the Second German Edition. In
the mean time an excellent translation of a part of the
book was published in England by the Rev. R. Main; but
still it seemed to me desirable to have the entire work
translated, especially as the Second Edition had been
considerably enlarged. Therefore when I returned to
Germany and was invited by the Publishers to prepare an English translation, I gladly availed myself of
my leisure here to comply with their wishes, and having acted for a number of years as an instructor of




VIII
science in America, it was especially gratifying to me
at the close of my career there to write a work in
the language of the country, which would leave me
in an intellectual connection with it and with those
young men whom I had the pleasure of instructing in
my science.
Still I publish this translation with diffidence, as
I am well aware of its imperfection, and as I fear that,
not to speak of the want of that finish of style which
might have been expected from an English Translator,
there will be found now and then some Gerimanisms,
which are always liable to occur in a translation, especially when made by a German.  I have discovered
some such mistakes myself and have given themn in
the Table of Errors.
I trust therefore that this translation may be received with indulgence and may be found a useful
guide for those who wish to study this particular
branch of science.
JENA, August 1864.
F. BRdNNOW.




TABLES OF CONTENTS.
INTRODUCTION.
A. TRANSFORMATION OF CO-ORDINATES. FORMULAE OF
SPHERICAL TRIGONOMETRY.
Page
1. Formulae for the transformation of co-ordinates.1.....    1
2. Their application to polar co-ordinates..........  2
3. Fundamental formulae of spherical trigonometry.......  3
4.  Other formulae of spherical trigonometry.......            4
5. Gauss's and Napier's formulae.                                    5
6. Introduction of auxiliary angles into the formulae of spherical trigonometry.................... 9
7. On the precision attainable in finding angles by means of tangents
and of sines.................10
S. Formulae for right angled triangles.......... 11
9. The differential formulae of spherical trigonometry......  12
10. Approximate formulae for small angles......... 14
11. Some expansions frequently used in spherical astronomy... 14
B. THE THEORY OF INTERPOLATION.
12. Object of interpolation. Notation of differences.......   18
13. Newton's formula for interpolation.......... 20
14. Other interpolation-formulae............. 22
15. Computation of numerical differential coefficients..     27
C. THEORY OF SEVERAL DEFINITE INTEGRALS USED IN
SPHERICAL ASTRONOMY.
16. The integral fe-t2 dt............... 33
0
17. Various methods for computing the integral  e-t  d t....35
18.  Computation of the integrals............  38
f 3X sin   Adz   and  ex (1- x) sin gdx
c gS   -2   sin2    OsJ f gos 2 +2  sin g2.x




x
D.  THE METHOD OF LEAST SQUARES.
Page
19. Introductory remarks. On the form of the equations of condition
derived from observations..............  40
20. The law of the errors of observation.                            42
21. The measure of precision of observations, the mean error and the
probable error.................. 46
22. Determination of the most probable value of an unknown quantity
and of its probable error from a system of equations....   48
23. Determination of the most probable values of several unknown
quantities from a system of equations.......... 54
24. Determination of the probable error in this case...... 57
25. Example....................                                      60
E. THE DEVELOPMENT OF PERIODICAL FUNCTIONS FROM GIVEN
NUMERICAL VALUES.
26. Several propositions relating to periodical series.              63
27. Determination of the coefficients of a periodical series from given
numerical values............65
28. On the identity of the results obtained by this method with those
obtained by the method of least squares........ 68
SPHERICAL ASTRONOMY.
FIRST SECTION.
THE CELESTIAL SPHERE AND ITS DIURNAL MOTION.
I. THE SEVERAL SYSTEMS OF GREAT CIRCLES OF THE
CELESTIAL SPHERE.
1. The equator and the horizon and their poles........ 71
2. Co-ordinate system of azimuths and altitudes....            73
3. Co-ordinate system of hour angles and declinations...... 74
4. Co-ordinate system of right ascensions and declinations.... 75
5. Co-ordinate system of longitudes and latitudes........ 77
II. THE TRANSFORMATION OF THE DIFFERENT SYSTEMS OF
CO - ORDINATES.
6. Transformation of azimuths and altitudes into hour angles and declinations..................... 79
7. Transformation of hour angles and declinations into azimuths and
altitudes.................... 0
S. Parallactic angle. Differential formulae for the two preceding cases  85
9. Transformation of right ascensions and declinations into longitudes
and latitudes................... 86




XI
Page
10. Transformation of longitudes and latitudes into right ascensions
and declinations................                  88
11. Angle between the circles of declination and latitude. Differential
formulae for the two preceding cases..   89
12. Transformation of azimuths and altitudes into longitudes and latitudes....................    90
III.  THE DIURNAL MOTION AS A MEASURE OF TIME.
SIDEREAL, APPARENT AND MEAN SOLAR TIME.
13. Sidereal time. Sidereal day.............  91
14. Apparent solar time. Apparent solar day.  On the motion of the
earth in her orbit. Equation of the centre. Reduction to the ecliptic  91
15. Mean solar time. Equation of time..........  96
16. Transformation of mean time into sidereal time and vice versa.  98
17. Transformation of apparent time into mean time and vice versa.  99
18. Transformation of apparent time into sidereal time and vice versa  100
IV. PROBLEMS ARISING FROM  TIlE DIURNAL MOTION.
19. Time of culmination of fixed stars and moveable bodies..    101
20. Rising and setting of the fixed stars and moveable bodies..    103
21. Phenomena of the rising and setting of stars at different latitudes  104
22. Amplitudes at rising and setting of stars.......          106
23. Zenith distances of the stars at their culminations..         107
24. Time of the greatest altitude when the declination is variable.. 108
25. Differential formulae of altitude and azimuth with respect to the
hour angle....................  109
26. Transits of stars across the prime vertical.......             109
27. Greatest elongation of circumpolar stars.........  110
28. Time in which the sun and the moon move over a given great circle  111
SECOND SECTION.
ON THE CHANGES OF THE FUNDAMENTAL PLANES TO WHICH
THE PLACES OF THE STARS ARE REFERRED.
I. THE PRECESSION.
1. Annual motion of the equator on the ecliptic and of the ecliptic
on the equator, or annual lunisolar precession and precession produced by the planets.  Secular variation of the obliquity of the
ecliptic                                                        115
2. Annual changes of the stars in longitude and latitude and in right
ascension and declination.............. 119
3. Rigorous formulae for computing the precession in longitude and
latitude and in right ascension and declination.......  124




XII
Page
4. Effect of precession on the appearance of the sphere of the heavens
at a place on the earth at different times. Variation of the length
of the tropical year................ 128
II. THE NUTATION.
5. Nutation in longitude and latitude and in right ascension and declination.................... 130
6. Change of the expression of nutation, when the constant is changed  133
7. Tables for nutation................  134
8. The ellipse of nutation..............  136
THIRD SECTION.
CORRECTIONS  OF  THE  OBSERVATIONS  ARISING  FROM  THE
POSITION  OF  THE  OBSERVER  ON  THE  SURFACE  OF  THE
EARTH AND FROM  CERTAIN PROPERTIES OF LIGHT.
I. THE PARALLAX.
1. Dimensions of the earth. Equatoreal horizontal parallax of the sun  139
2. Geocentric latitude and distance from the centre for different places
on the earth..................  140
3. Parallax in altitude of the heavenly bodies...         144
4. Parallax in right ascension and declination and in longitude and
latitude....................147
5. Example for the moon. Rigorous formulae for the moon.          152
IT. THE REFRACTION.
6. Law of refraction of light. Differential expression of refraction.  154
7. Law  of the decrease of temperature and of the density of the
atmosphere.  Hypotheses by Newton, Bessel and Ivory...    160
S. Integration of the differential expression for Bessel's hypothesis. 163
9. Integration of the differential expression for Ivory's hypothesis  164
10. Computation of the refraction by means of Bessel's and Ivory's
formulae.  Computation of the horizontal refraction..... 66
11. Computation of the true refraction for any indications of the barometer and thermometer.. 169
12. Reduction of the height of the barometer to the normal temperature. Final formula for computing the true refraction. Tables
for refraction..................                              172
13. Probable errors of the tables for refraction.  Simple expressions
for refraction. Formulae of Cassini, Simpson and Bradley.    174
14. Effect of refraction on the rising and setting of the heavenly bodies. Example for computing the time of rising and setting of
the moon, taking account of parallax and refraction.           176
15. On twilight. The shortest twilight........   178




XIII
Page
III. THE ABERRATION.
16. Expressions for the annual aberration in right ascension and declination and in longitude and latitude........  180
17. Tables for aberration..............; I88
18. Formulae for the annual parallax of the stars.......    188
19. Formulae for diurnal aberration............           190
20. Apparent orbits of the stars round their mean places            191
21. Aberration for bodies, which have a proper motion...    192
22. Analytical deduction of the formulae for this case...    194
FOURTH SECTION.
ON THE METHOD BY WHICH THE PLACES OF THE STARS AND
THE VALUES OF THE CONSTANT QUANTITIES NECESSARY FOR
THEIR REDUCTION ARE DETERMINED BY OBSERVATIONS.
I. ON THE REDUCTION OF THE MEAN PLACES OF STARS TO
APPARENT PLACES AND VICE VERSA.
1. Expressions for the apparent place of a star. Auxiliary quantities
for their computation............... 202
2. Tables of Bessel................. 202
3. Other method of computing the apparent place of a star          204
4. Formulae for computing the annual parallax.....           206
II. DETERMINATION OF THE RIGHT ASCENSIONS AND DECLINATIONS
OF THE STARS AND OF THE OBLIQUITY OF THE ECLIPTIC.
5. Determination of the differences of right ascension of the stars. 206
6. Determination of the declinations of the stars......    212
7. Determination of the obliquity of the ecliptic....             214
S. Determination of the absolute right ascension of a star.        218
9. Relative determinations. The use of the standard stars. Observation of zones.......... 223
III.  ON THE METHODS OF DETERMINING  THE MOST PROBABLE
VALUES OF THE CONSTANTS USED FOR THE REDUCTION OF
THE PLACES OF THE STARS.
A. Determination of the constant of refraction.
10. Determination of the constant of refraction and the latitude by upper
and lower culminations of stars. Determination of the coefficient
for the expansion of atmospheric air.........    227
B. Determination of the constants of aberration and nutation and of the
annual parallaxes of stars.
11. Determination of the constants of aberration and nutation from
observed right ascensions and declinations of Polaris   Struve's
method by observing stars on the prime vertical. Determination
of the constant of aberration from the eclipses of Jupiter's satellites  231




XIV
Page
12. Determination of the annual parallaxes of the stars by the changes
of their places relatively to other stars in their neighbourhood  237
C. Determination of the constant of precession and of the proper motions
of the stars.
13. Determination of the lunisolar precession from the mean places of
the stars at two different epochs.                               239
14. On the proper motion of the stars. Determination of the point
towards which the motion of the sun is directed...... 241
15. Attempts made of determining the constant of precession, taking
account of the proper motion of the sun.                        245
16. Reduction of the place of the pole-star from one epoch to another.
On the variability of the proper motions.                       248
FIFTH SECTION.
DETERMINATION  OF TUE POSITION  OF THE FIXED  GREAT
CIRCLES  OF THE CELESTIAL  SPHERE WITH  RESPECT TO
THE HORIZON  OF A PLACE.
I. METHODS OF FINDING THE ZERO OF THE AZIMUTH AND THE
TRUE BEARING OF AN OBJECT.
1. Determination of the zero of the azimuth by observing the greatest elongations of circumpolar stars, by equal altitudes and by
observing the upper and lower culminations of stars.253
2. Determination of the azimuth by observing a star, the declination
and the latitude of the place being known........ 255
3. Determination of the true bearing of a terrestrial object by observing its distance from a heavenly body.2......   257
IT  METHODS OF FINDING THE TIME OR THE LATITUDE BY AN
OBSERVATION OF A SINGLE ALTITUDE.
4. Method of finding the time by observing the altitude of a star. 259
5. Method of computation, when several altitudes of the same body
have been taken..                                  262
6. Method of finding the latitude by observing the altitude of a star  264
7.  Method of finding the latitude by circum-meridian altitudes.    266
S. The same problem, when the declination of the heavenly body is
variable.                                                       269
9. Method of finding the latitude by the pole-star...... 271
10. Method of finding the latitude, given by Gauss....... 275
III  METHODS OF FINDING BOTH THE TIME AND THE LATITUDE
BY COMBINING SEVERAL ALTITUDES.
11  Methods of finding the latitude by upper and lower culminations
of stars, and by observing two stars on different sides of the zenith 278




xv
Page
12. Method of finding the time by equal altitudes. Equation for equal
altitudes....................   279
13  The same, when the time of true midnight is found..... 284
14. Method of finding the time and the latitude by two altitudes of
stars.....................   285
15. Particular case, when the same star is observed twice....  289
16. Method of finding the time and the latitude as well as the azimuths
and altitudes from the difference of azimuths and altitudes and the
interval of time between the observations..            291
17. Indirect solution of the problem, to find the time and the latitude
by observing two altitudes.  Tables of Douwes....           293
18. Method of finding the time, the latitude and the declination by
three altitudes of the same star............  296
19. Method of finding the time, the latitude and the altitude by observing three stars at equal altitudes.  Solution given by Gauss. 296
20. Solution given by Cagnoli..  301
21. Analytical deduction of these formulae........ 303
IV.  METHODS OF FINDING  THE  LATITUDE  AND  THE TIME
BY AZIMUTHS.
22. Method of finding the time by the azimuth of a star...   305
23. Method of finding the time by the disappearance of a star behind
a terrestrial object.............307
24. Method of finding the latitude by the azimuth of a star... 308
25. Method of finding the time by observing two stars on the same
vertical  circle..................    312
V. DETERMINATION  OF THE ANGLE BETWEEN THE MERIDIANS OF
TWO PLACES ON THE SURFACE OF THE EARTH, OR OF THEIR
DIFFERENCE OF LONGITUDE.
26. Determination of the difference of longitude by observing such
phenomena, which are seen at the same instant at both places,
and by chronometers................                             313
27. Determination of the difference of longitude by means of the electric telegraph..       316
28. Determination of the difference of longitude by eclipses. Method
which was formerly used............                   322
29. Method given by Bessel.  Example of the computation of an
eclipse of the sun..............   323
30. Determination  of the difference of longitude by occultations of
stars....................   336
31. Method of calculating an eclipse.........  339
32. Determination of the difference of longitude by lunar distances. 344
33. Determination of the difference of longitude by culminations of
the moon............                                          350




XVI
SIXTH SECTION.
ON  THE DETERMINATION  OF THE DIMENSIONS OF THE EARTH
AND THE HORIZONTAL PARALLAXES OF THE HEAVENLY
BODIES.
I. DETERMINATION OF THE FIGURE AND THE DIMENSIONS OF
THE EARTH.
Page
1. Determination of the figure and the dimensions of the earth from
two arcs of a meridian measured at different places on the earth. 357
2. Determination of the figure and the dimensions of the earth by
any number of arcs................                            360
II. DETERMINATION OF THE HORIZONTAL PARALLAXES OF THE
HEAVENLY BODIES.
3. Determination of the horizontal parallax of a body by observing
its meridian zenith distance at different places on the earth.    366
4. Effect of the parallax on the transits of Venus for different places
on the earth.....                                          375
5. Determination of the horizontal parallax of the sun by the transits
of Venus..........384
SEVENTH SECTION.
THEORY OF THE ASTRONOMICAL INSTRUMENTS.
1. SOME OBJECTS PERTAINING IN GENERAL TO ALL INSTRUMENTS.
A. Use of the spirit-level.
i. Determination of the inclination of an axis by means of the spirit-level............                                390
2. Determination of the value of the unit of its scale..         395
3. Determination of the inequality of the pivots of an instrument    398
B.  The vernier and the reading microscope.
4. Use of the vernier................ 401
5.  Use and adjustments of the reading microscope...... 403
C. Errors arising firom the excentricity of the circle and errors of division.
6. Effect of the excentricity of the circle on the readings. The use
of two verniers opposite each other. Determination of the excentricity by two such verniers.............408
7. On the errors of division and the methods of determining them. 411
D.  On flexure or the action of the force of gravity upon the telescope
and the circle.
S. Methods of arranging the observations so as to eliminate the effect
of flexure. Determination of the flexure.417
E.  On the examination of the micrometer screws.
9. Determination of the periodical errors of the screw. Examination
of the equal length of the threads........... 425




XVII
Page
II. THE ALTITUDE AND AZIMUTH INSTRUMENT.
10. Effect of the errors of the instrument upon the observations.    429
11. Geometrical method for deducing the approximate formulae.    433
12. Determination of the errors of the instrument......    434
13. Observations of altitudes..............             437
14. Formulae for the other instruments deduced from those for the altitude and azimuth instrument..439
III. THE EQUATOREAL.
15. Effect of the errors of the instrument upon the observations.    441
16. Determination of the errors of the instrument......    445
17. Use of the equatoreal for determining the relative places of stars  449
IV. THE TRANSIT INSTRUMENT AND THE MERIDIAN CIRCLE.
18. Effect of the errors of the instrument upon the observations.    451
19. Geometrical method for deducing the approximate formulae.    456
20. Reduction of an observation on a lateral wire to the middle wire.
Determination of the wire-distances...                  457
21. Reduction of the observations, if the observed body has a parallax
and a visible disc................ 461
22. Determination of the errors of the instrument....... 466
23. Reduction of the zenith distances observed at some distance from
the meridian. Effect of the inclination of the wires. The same
for the case when the body has a disc and a parallax.         477
24. Determination of the polar point and the zenith point of the circle.
Use of the nadir horizon and of horizontal collimators....   4S2
V. THE PRIME VERTICAL INSTRUMENT.
25. Effect of the errors of the instrument upon the observations.    484
26. Determination of the latitude by means of this instrument, when
the errors are large. The same for an instrument which is nearly
adjusted......                                               488
27. Reduction of the observations made on a lateral wire to the middle
wire............... 492
28. Determination of the errors of the instrument.....         498
VI. ALTITUDE INSTRUMENTS.
29. Entire circles.............                               499
30. The sextant.  On measuring the angle between two objects. Observations of altitudes by means of an artificial horizon...   500
31. Effect of the errors of the sextant upon the observations and determination of these errors............. 503
Vii. INSTRUMENTS, WHICH SERVE FOR MEASURING THE RELATIVE
PLACE OF TWO HEAVENLY BODIES NEAR EACH OTHER.
(MICROMETER AND HELIOMETER.)
32. The filar micrometer of an equatoreal........ 512
33. Other kinds of filar micrometers.....                    517




XVIII
Page
34. Determination of the relative place of two objects by means of
the ring micrometer................ 518
35. Best way of making observations with this micrometer...   522
36. Reduction of the observations made with the ring micrometer, if
one of the bodies has a proper motion......... 523
37. Reduction of the observations with the ring micrometer, if the objects are near the pole...........   525
38. Various methods for determining the value of the radius of the ring  527
39. The heliometer. Determination of the relative place of two objects
by means of this instrument...........                  532
40. Reduction of the observations, if one of the bodies has a proper
motion....................                                      539
41. Determination of the zero of the position circle and of the value
of one revolution of the micrometer-screw.542
VIII. METHODS OF CORRECTING OBSERVATIONS MADE BY MEANS
OF A MICROMETER FOR REFRACTION.
42. Correction which is to be applied to the difference of two apparent zenith distances in order to find the difference of the true
zenith distances.................    545
43. Computation of the difference of the true right ascensions and declinations, of two stars from the observed apparent differences.    550
44. Effect of refraction for micrometers, by which the difference of
right ascension is found from the observations of transits across
wires which are perpendicular to the daily motion, whilst the difference of declination is found by direct measurement...    551
45. Effect of refraction upon the observations with the ring micrometer  552
46. Effect of refraction upon the micrometers with which angles of
position and distances are observed.......... 555
IX. EFFECT OF PRECESSION, NUTATION AND ABERRATION UPON
THE DISTANCE BETWEEN TWO STARS AND THE ANGLE
OF POSITION.
47. Change of the angle of position by the lunisolar precession and
by nutation. Change of the distance and the angle of position
by aberration...............  556




XIX
E E R A T A.
page 23 line 5 from  bottom   for for a            read a
31 line 5 from bottom   for one               read unity
33 line 13 from top      for formulas         read formulae
48 line 19 from bottom   for as near          read as nearly
57 line 13 from bottom   foSr at last         read finally
and 58 line 17 from top
59 line 12 from bottom   for to               read too
60 line 6 from  bottom   for constant quantity read constant
71 line 17 and 19 from top for North -Pole    read North Pole
-  line20 from top      for South-Pole        read South Pole
-  line 9 from bottom   for Parallel-Circles  read Parallel Circles
73 line 10 and 12 from top for' vertical- circle   read vertical circle
-  line13 from top and             South      read South point
line 2 from bottom       
-  line17 from bottom  for point West        read West point
74 line 14 from top     for polar-distance   read polar distance
75 line 6 from bottom   for right-ascension  read right ascension
76 line 4 from top      for vertical axes    read axes
- line3, 8, 9 fr om top
-  line3,8, 9 from  ftop  -or right- ascension  read right ascension
and 8, 9, 14 from bott. 
-  line 3 from bottom   for multiply          read divide
77 line19 from bottom   for vertical          read perpendicular
78 line 10 from top     for vertical          read perpendicular
81 line 9 from  bottom   for point North     read North point
84 line 17 from bottom   for at last          read finally
93 line 8 from top      for a (1 - cos E)    read a (1 - e cos E)
100 line 6 from bottom   for sidereal time    read of sidereal time
-  line 5 from bottom   for mean time         read of mean time
101 line1 fiom top        for on a             read to the
103 line 6 from bottom   for in                read is
107 line 18 from top      for star             read stars
116 line 18 from top     for against           read to
127 line 3 from bottom   for at last           read finally
135 line12 from bottom   for cos /             read on i
138 line 4 from top      for the light         read light
139 line 1 from top      for the light         read light




xx
page 140 line 16 from bottom
144 linel0 from bottom
147 line 2 from bottom  for at
rfo?~ at           read on
148 line 1 from top
154 line 1l from bottom
155 line 8 from bottom
169 line 9 from top
171 line 4 from top
173 line 1, 2, 18 from topead height
174 line 13 from top
176 linel4, 11 from bott. for the refraction   read refraction
178 line 1 from top     for at              read on
181 line12 from top     for vertical        read perpendicular
190 linell from top     for at              read on
209 line 5 from top     for vertical        read perpendicular
210 line 4 and 5 from top for vertical      read perpendicular
214 line 8 fiom top     for usually         read and as usually
226 line 10 from top    for at last         read finally
232 line 14 from bottom  for Now            read Now let
272 linel3 from bottom  for — p3 sin t cost read — Ip3 sin t cost2
286 line 1S from bottom  for cos 8 sin 2    read cos u' sin 2
x                       x
331 line 9 frpm top     for                 read     -
trag       -tang  tang
397 line18 from top     for a               read an
399 line 1 from bottom  for i and i'        read 2i and 2i'
425 line14 from bottom  for of              read between
450 line 4 from bottom  for of              read between
456 line 16 from top    for form            read from




INTRODUCTION.
A. TRANSFORMATION OF CO-ORDINATES. FORMULAE OF
SPHERICAL TRIGONOMETRY.
1. In Spherical Astronomy we treat of the positions
of the heavenly bodies on the visible sphere of the heavens,
referring them by spherical co-ordinates to certain great circles of the sphere and establishing the relations between the
co-ordinates with respect to various great circles. Instead of
using spherical co-ordinates we can give the positions of the
heavenly bodies also by polar co-ordinates, viz. by the angles,
which straight lines drawn from the bodies to the centre of
the celestial sphere make with certain planes, and by the
distance from this centre itself, which, being the radius of
the celestial sphere, is always taken equal to unity. These
polar co-ordinates can finally be expressed by rectangular
co-ordinates. Hence the whole of Spherical Astronomy can
be reduced to the transformation of rectangular co-ordinates,
for which we shall now find the general formulae.
If we imagine in a plane two axes perpendicular to each
other and denote the abscissa and ordinate of a point by x
and y, the distance of the point from the origin of the co-ordinates by r, the angle, which this line makes with the positive side of the axis of x, by v, we have:
--? Cos v
y  r sin v.
If we further imagine two other axes in the same plane,
which have the same origin as the former two and denote
the co - ordinates of the same point referred to this new sys1




2
tem  by x' and y' and the angle corresponding to v by v',
we have:
--' CO 2s V
-y  r sin v'.
If we denote then the angle, which the positive side of
the axis of x' makes with the positive side of the axis of x,
by w, reckoning all angles in the same direction from  00 to
360", we have in general v   v' - w, hence:
x - r cos V COs w - r sin v' sin w
- sin v' cosw -  cos v' sin to,
or:
x=   x' cos w —' sin w
y  -r sin w +- y' cos w
and likewise:' -   x cos w -+y sin    ()
y' --   sin w- y cos     (
These formulae are true for all positive and negative values
of x and y and for all values of w from  00 to 360".
2. Let x, y, z be the co-ordinates of a point 0 referred
to three axes perpendicular to each, other, let a' be the angle,
which the radius vector makes with its projection on the plane
of xy, B' the angle between this projection and the axis of x
(or the angle between a plane passing through the point 0
and the positive axis of z and a plane passing through the
positive axes of x and z, reckoned from the positive side of
the axis of x towards the positive side of the axis of y from
0~ to 360"), then we have, taking the distance of the point
from the origin of the co-ordinates equal to unity:
x=cosB'cos a', y —sinB'cosa', z  sin'.
But if we denote by a the angle between the radius
vector and the positive side of the axis of z, reckoning it
from the positive side of the axis of z towards the positive
side of the axis of x and y from 0" to 3600, we have:
x=sinacos B', y = sinasinB', z  cosa.
If now we imagine another system of co-ordinates, whose
axis of y' coincides with the axis of y, and whose axes of
x' and z' make with the axis of x and z the angle c and if
we denote the angle between the radius vector and the positive side of the axis of z' by b and by A' the angle between
the plane passing through 0'and the positive axis of z' and the




3
plane passing through the positive axes of x and z, reckoning
both angles in the same direction as a and B', we have:
x' = sin b cos A', y' = sin b sin A', z' = cos b,
and as we have according to the formulae for the transformation of co- ordinates:
z   x'.sin c -4-' cos c
Y = Y
- x cos c - z' sin c,
we find:
cos a -sin b sin c cos A' +- cos b cos c
sin a sin B' — sin b sin A'
sin a cos B'- sin b cos c cos A' - cos b sin c.
3.  If we imagine a sphere, whose centre is the origin
of the co-ordinates' and whose radius is equal to unity and
draw through the point O and the points of intersection of
the axes of z and z' with the surface of this sphere arcs of
a great circle, these arcs form  a spherical triangle, if we use
this term in its most general sense, when its sides as well as
angles may be greater than 180 degrees. The three sides
O Z, OZ' and Z'Z of this spherical triangle are respectively
a, b and c. The spherical angle A at Z' is equal to A', being
the angle between the plane passing through the centre and
the points O and Z' and the plane passing through the centre
and the points Z and Z', while the angle B at Z is generally
equal to 180-B'. Introducing therefore A and B instead
of A' and B' in the equations which we have found in No. 2,
we get the following formulae, which are true for every spherical triangle:
cos a - cos b cos c -- sin b sin c cos A
sin a sin B -sin b sinA
sin a cos B -cos b sin c - sin b cos c cos A.
These are the three principal formulae of spherical trigonometry and express but a simple transformation of co-ordinates.
As we may consider each vertex of the spherical triangle
as the proj'ection of the point 0 on the surface of the sphere
and the two others as the points of intersection of the two
axes z and z' with this surface, it follows, that the above
formulae are true also for any other side and the adjacent
1*




4
angle, if we change the other sides and angles correspondingly.  Hence  we obtain, embracing  all possible  cases:
cos a -- cos b cos c +- sin b sin c cos A
cos b   cos a cos c +- sin a sin c cos B     (2)
cos c -- cos a cos b -+ sin a sin b cos C
sin a sin B = sin b sin A
sin a sin C - sin c sin A              (3)
sin b sin C — sin c sin B
sin a cos B — cos b sin c -- sin b cos c cos A
sin a cos C -- cos c sin b -- sin c cos b cos A
sin b cos A -cos a sin c - sin ca cos c cos B
sin b cos C =  cos c sin a - sin c cos a cos B
sin c cos A —cos a sill b -sin a cos b cos C
sin c cos B    cos b sin a - sin b cos a cos C.
4. We can easily deduce from these formulae all the
other formulae of spherical trigonometry. Dividing the formulae (4) by the corresponding formulae (3), we find:
sin A cotang B = cotang b sin c - cos c cos A
sin A cotang C - cotang c sin b - cos b cos A
sin B cotang A    c- otang a sin c -cos c cos B
sin B cotang C = cotang c sin a - cos a cos B
sin C cotang A =  cotang a sin b - cos b cos C
sin C cotang B -  cotang b sin a -  cos a cos C.
If we write the last of these formulae thus:
cos b sin a sin B i
sin C cos B                      C=  e  cos a sin B cos C,
sin b
we find:
sin C cos B -  cos b sin A - cos a sin B cos C,
or:
sin A cos b -- cos B sin C +- sin B cos C cos a
an equation, which corresponds to the first of the formulae (4),
but contains angles instead of sides and vice versa. By changing  the letters, we find the following  six  equations:
sin A cos b -- cos B sin C -- sin B cos C cos a
sin A cos c -- cos C sin B +- sin C cos B cos a
sin B cos a -- cos A sin C -+ sin A cos C cos b
sin B cos c - cos C sin A +- sin C cos A cos b
sin C cos a -- cos A sin B +- sin A cosB cos c
sin C cos b --- cos B sin A +- sin B cos A cos c
and  dividing  these  equations by  the  corresponding  equations
(3), we have:




sin a cotang b - cotangB sin C-I- cos Ccos a
sin a cotang c - cotang C sinB + cosB cos a
sin b cotang a - cotang A sin C + cos C cos b
sin b cotang c = cotang C sin A + cos A cos b
sin c cotang a - cotang A sinEB + cos B cos c
sin c cotang b = cotangB sin A + cos A cos c.
From  the equations (6) we easily deduce the following:
cos A sin C   sinB cos a - sin A cos C cos b
cos B sin C -- sin A cos b -- sin B cos C cos a.
Multiplying  these  equations by  sin C  and  substituting
the value of sin A sin C cos b taken from the second equation into the first, we find:
cos A — sin B sin C cos a - cos B cos C
and changing the letters we get the following three equations,
which correspond to. the formulae (2), but again contain angles
instead of sides and vice versa:
cos A - sin B sin Ccos a - cos B cos C
cos B =- sin A sin C cos b - cos A cos C  (8)
cos C= sin A sin B cos c - cos A cos B.
5.  If we add the two first of the formulae (3), we find:
sin a [sin B +- sin C] - sin A [sin b -+ sin c],
or:,B   C         B q- C           b -+- c         b -- c
sin4- a cos B-Ces asin         sin  Asin    -.  cos - A cos -2and if we subtract the same equations, we get:
B I -- B-C        B  +- C          b - c           b - c
sin  aC sin.cos. C COS   -  sin A cos.  cos -A sin...
2-' 2                           2  a os        2 e
Likewise we find  by  adding  and  subtracting  the  two
first of the formulae (4):
B -- C B       B+- C            b -- c          b 4- c
sin a cos. os-a cos      =sinsin s2 sn  A  cos -     2
B —C           B B+C            b —c            b- c
sin -asin. cos  i a sin          cos     A. osA cos- -.
2              2                2              2
Each of these formulae is the product of two of Gauss's
equations; but in order to derive from these formulae Gauss's
equations, we must find another formula, in which a different
combination of equations occurs. We may use for this purpose either of the following equations:
B+- C          B - C            b -- c          b -b  c
os 2 co   2. cos  asin       sin -- 2 — n  A cos  -- cos  A cos
B-C.        B- C             bc- c           b —c
sin A a                             A s in  -  cos n  A s in 
2              2                       s 2sin —,  2




6
which we find by adding or subtracting the first two of the
equations (6).
If we take now:
sin   A sin    -- a
sin  A cos   
b —c
cos A sin ---
cos A os  A cos 
and:
B-C
sin  a cos     a'
B-4-C
cos  a cos B+C
B-C
sin  a sin  2  y
COS a sin ---',
we find the following six equations:.a'Sa a',     y, -a'#'  a fi.  7 --,  y'    y' X' - 8, a'y' — ay,
from which we deduce the following:
a,-bS fl,   B        y =,
or:
a  -- C a  -   -, S'- - 
Hence we find the following relations between the angles
and sides of a spherical triangle:
b -4-c           B —C
sin - A sin   c.sin   a cos 
b- c             B Cq-C
sin - A osin  2  si —    a csi   2
(9)
b-c              B —C
cos.- A sin     sin - a sin
2        2        2
b -- c           B  -+C
Cos 2 A cos  2   cos ~ a sin  2
or:
i1n. b-4-c              B-C
sin  A sin  2    - sin    cos
2                   2
b +- c              B -+- C
sin  A cos  2' - Cos a cos 
csAs2               2
b -c                B-C
cos Acsin  2   -- c-sin  a sin B
b - c         I. B+C
COS A cos           cos  a6 siln -
2OH                           2




7
Both systems give us for the unknown quantities, which
may be either two sides anld the included angle or two angles
and the interjacent side, the same value or at least values
differing by 360 degrees. If we wish to find for instance
A, b and c, we should get from the second system of forb -- c   b — c
mulae either for   - and  --  the same values as from  the
2        2
first, but for'A a value which differs 180~, or we should
b c- C    b - c
find for  2  and   2  values which differ 1800 from  those
derived from  the first system, but for I A the same value.
In each case therefore the values of A, b and c as found
from the two systems would differ only by 3600. The four
formulae (9) are therefore generally true and it is indifferent,
whether we use for the computation of A, b and c the quantities a, B, C themselves or add to or subtract from  any of
them  360P ").
The four equations (9) are known as,,Gauss's equations"
and are used, if either one side and the two adjacent angles
of a spherical triangle or two sides and the included angle
are given and it is required to find the other parts. The best
way of computing them  is the following. If a, B and C are
the given parts, we find first the logarithms of the following
quantities:
_-C                     B -- C
(1) cos — \             (4) cos'
~~~~2             2
(2) sin I a             (5) cos I a
B —C                    B+C.
(3) sill                (6) sin  2
2                       2
and from these:
_B-C                       B-C
(7) sin  a cos    C        (9) sin2- a sin
B —.C                     B+-C
(8) cos- acos --         (10) cost asin  2
Subtracting the logarithm of (8) from that of (7) and
the logarithm  of (10) from  that of (9), we find log. tang
(b +- c) and lg. tg.  (b - c), from which we get b and c. Then
we take either log cos - (b - c) or log sin' (b -4-c) and log
cos  (b - c) or log sin  (b - c), whichever is the greater one
*) Gauss, Theoria motus corporum coelestium pag. 50 seq.




8
of the two and subtract the first from the greater one of the
logarithms (7) or (8), the other from the greater one of the
logarithms (9) or (10) and thus find log sin  A and log
cos  A. Subtracting the latter from the first, we get log
tang 2 A, from which we find A. As sin' A as well as
cos a A  must necessarily  give the  same  angle as tang 2 A,
we may use this as a check for our computation.
If for instance we have the following parts given:
a —  110 25' 56."3
B    184   6 55. 4
C —  11  18 40. 3
we have:
2 (B — C)   86o 24' 7."55             2 (B+  C)    970 42'47."85
cos I (B — C) = 8.7976413             cos I (B+ C)   9.1278046,,
sin - a - 8.9982605                   cos E a    9.9978351
sin l (B - C) = 9.9991432             sin 2 (B + C)    9.9960526
sin - a coso (B - C)   7.7959018      sin 4 a sin - (B - C)   8.9974037
cos 2 a cos  (B +- C)   9.1256397,,    cos ~ a sin 2 (B + C)   9.9938877
2 (b + c)  177 19 13.49                 (b - c)    5 45 24.13
coS I (b 4- c)    9.9995248,          cos I (b - c)    9.9978042
sin' A    9.1261149           b —- 1830 4' 37."62
cos' A    9.9960835           c - 171 33 49. 36. A  7~ 40'59."38         A=  15 21 58. 76.
If we had taken B- ---      1750 53'4."6, hence:
2 (B + C)   - -820 17' 12."15
(B -   ) -    93  35 52. 45
we should have found:
2 (b +- c)  --  2~ 40' 46."51
4 (b - c)      185 45 24. 13
hence b -- 1830 4' 37."62 and c   - 1880 26' 10."64.
Dividing Gauss's equations by each other, we find Napier's
equations. Writing A, B, C in place of B, C, A and a, b, c
in place of b, c, a, we find from the equations (9):
a- b
A+.-B         2         C
tang                  cotang
2        a-....
os  2
(9 c0. a-               (9 b
sin.......
A -B          2         C
tang   2  -          cotang
sin
2




9
A-B
COS  2         c
cos 
a - b       2       c
tang       -        tang
2       A-+B tng 2
2
(9 a)
sin  2
tang                tang
2       A- 4-B    2
sin
6. As nearly all the formulae in No. 3 and 4 are under
a form not convenient for logarithmic computation, their second
members consisting of two terms, we must convert them by
the introduction of auxiliary angles into others, which are
free from this inconvenience. Now as any two real, positive
or negative quantities x and y may be taken proportional to
a sine or cosine of an angle we may assume:
z - 7n sin M and y - m cos M
for we find immediately:
tang Mf-= X and  Z1m   + yi,
hence M  and mn expressed by real quantities.  Therefore as
all the above formulas, which consist of several terms, contain in each of these terms the sine and cosine of the same
angle, we can take their factors proportional to the sine and
cosine of an angle and, applying the formulae for the sine
or cosine of a binomial, we  can convert the formulae into
a form convenient for logarithmic computation.
For instance, if we have to compute the three formulae:
cos a — cos b cos c +- sin b sin c cos A
sin a sin B  - sin b sin A
sin a cos B - cos b sin c - sin bcos c cos A,
we may put:
sin b cos A -- m sin II
cos b — in cos I.L
and find:
cos a - in cos (c- M)
sin a sin B -- sin b sin A
sin a cos B =-   sin (c- 2k~).
If we know the quadrant, in which B is situated, we
can also write the formulae in the following manner, subsin b cos A
stituting for m its value  sinM   A We compute first:
tang M-= tang b cos A




10
and then find:
tang B    - tang A sin -1i
sin (c - Ml)
tang a = tang (c —M)
cos B
If we have logarithmic tables, by which we can find
immediately the logarithms of the sum or the difference of
two numbers from  the logarithms of the numbers themselves,
it is easier and at the same time more accurate, to use the
three equations in their original form without introducing the
auxiliary  angle.  Such  tables have been computed for seven
decimals by Zech in Tfibingen. (J. Zech, Tafeln for die Additions- und Subtractions-Logarithmen ftir sieben Stellen.)
KShler's edition of Lalande's logarithmic tables contains
similar tables for five decimals.
7.  It is always best, to  find angles by their tangents;
for as their variation is more rapid than that of the sines
or cosines, we can find the angles more accurately than by
the other functions.
If A x denotes a small increment of an angle, we have:
2iAx
2 (log tang x) =sin 2 x
Now it is cuitomary to express the increments of angles
in seconds of arc; but as the unit of the tangent is the radius, we must express the increment A x also in parts of the
radius, hence we must divide it by the number 206264,8*).
Moreover the logarithms used in the formula are hyperbolic
logarithms; therefore if we wish to introduce common logarithms, we must multiply by  the modulus 0.4342945-  M.
Finally if we wish to find A (log tang x) expressed in units
*) The number 206264.8, whose logarithm is 5.3144251, is always used
in order to convert quantities, which are expressed in parts of the radius,
into seconds of arc and conversely. The number of seconds in the whole
circumference is 1296000, while this circumference if we take the radius as
unit is 2r or 6.2831853. These numbers are in the ratio of 206264,8 to 1.
Hence, if we wish to convert quantities, expressed in parts of the radins'into
seconds of arce, we must multiply them by this number; but if we wish to
convert quantities, which are expressed in seconds of ar/, into parts of the
the radius, we must divide them by this number, which is also equal to the
number of seconds contained in an arc equal to the radius, while its complement is equal to the sine or the tangent of one second.




11
of the last decimal of the logarithms used, we must multiply
by 1QO000000 if we employ logarithms of seven decimals. We
find therefore:
2M     A x"
l (log tang x)  2-'  2. 10000pO0
sin 2x  206264.8
42.1
sin 2x
or:
sin 2x
A x   sin 421  (log tang r).
This equation shows, with what accuracy we may find
an angle by its tangent.
Using logarithms of five decimals we may expect our
computation to be exact within two units of the last decimal.
Hence in this case A (log tang x) being equal to 200, the
error of the angle would be:
200"
A-,, _     sin 2x = 5" sill 2 x.
Therefore if we use logarithms of five decimals, the error
cannot be greater than 5" sin 2x or as the maximum value
of sin 2x is unity, not greater than 5 seconds and an error
of that magnitude can occur only if the angle is near 450.
If we use logarithms of seven decimals, the error must needs
be a hundred times less; hence in that case the greatest error of an angle found by the tangent will be 0."05.
If we find an angle by the sine or cosine, we should
have in the formula for A (log sin x) or A (log cos x) instead
of sin 2x the factor tang x or cotang x which may have any
value up to infinity. Hence as small errors in the logarithm
of the sine or cosine of an angle may produce very great
errors in the angle itself, it is always preferable, to find
the angles by their tangents.
8. Taking one of the angles in the formulae for oblique
triangles equal to 90", we find the formulae for right-angled
triangles. If we denote then the hypothenuse by h, the two
sides by c and c' and the two opposite angles by C and C',
we get from the first of the formulae (2), taking A = 900:
cos 7= -cos  Cos c',
and by the same supposition from the first of the formulae (3):
sin h sin C-= sin c




12
and from the first of the formulae (4):
sin h cos C = cos c sin c'
or dividing this by cos h:tang h cos C= tang c'.
Dividing the same formula by sin h sin C, we find:
cotang C'- cota.Dng c sin c',
or:
tang c = tang C sin c'.
Combining with this the following formula:
tang c'-  tang C' sin c,
we obtain
cos h = cotg C cotg C'.
At last from  the combination of the two equations:
sin h sin C' - sin c'
and sin h cos C   cos c sin c',
we find:
cos C — sin C' cos c.
We have therefore for a right-angled triangle the following six formulae, which embrace all combinations of the five
parts:
COs I = cos c cos c'
sin c  sin h sin C
tang c = tang h cos C'
tang c - tang C sin c'
cos Ih - cotang C cotang C'
cos -= cos c sin C!',
and  these formulae  enable us to find all parts of a rightangled triangle if two of them  are given.
Comparing these formulas with those in No. 6, we easily
see, that by the introduction of the auxiliary quantities m
and lM, we substitute two right-angled triangles for the oblique
triangle. For if we let fall an are of a great circle from the
vertex C of the oblique triangle vertical to the side c, it is
plain, that m is the cosine of this are and M  the part of the
side c between the vertex A and the point, where it is intersected by the vertical are.
9.  For the numerical computation of any quantities in
astronomy we must always take certain data from observations. But as we are not sure of the absolute accuracy
of any of these, on the contrary as we must suppose all of
them to be somewhat erroneous, it is necessary in solving a
problem to investigate, whether a small error of the observed




13
quantity may not produce a large error of the quantity which
is to be found. Now in order to be able easily to make such
an estimate, we  must differentiate the formulae of spherical
trigonometry and in order to embrace all cases we will take
all quantities as variable.
Differentiating thus the first of the equations (2), we get:
- sin ada = db [- sin b cos c - cos b sin c cos A]
-+' de [- cos b sin c + sin b cos c cos A]
sin b sin c sin A. dA.
Here  the  factor of db  is  equal to          sin a cos C and
the factor of dc equal to           sin a cos B; if we write also
-  sin a sin c sin B  instead  of the  factor of A, we find the
differential -formula:
da = — cos Cdb -4- cos Bde +- sin c sin B dA.
Writing  the first of the  equations (3) in a logarithmic
form, we find:
log sin a +- log sin B     log sin b - log sin A
and by differentiating it:
cotang ada -+ cotang Bd B = cotang b db + cotang A dA.
Instead  of the first of the  formulae  (4), we will differentiate the first of the formulae (5), which were found by
the combination of the formulae (3) and (4). Thus we find:
sin A
s —in-  dB  [- dcA [cotang B cos A -sinll A cos c]
sin B 2
sin c
sin   db - dc [cotang b cos c + cos A sin c]
sin 6,2
sin A      cos C          sin c      cos a
or:.    dB-         dA             db+ -— c.
sin B       sin B         sin 2       sin b
Multiplying this equation by sin B, we find:
sin a                    sin C      cos a sin B
- dB - cos CdA  -       -  b —--  db.+.   de,
sin b                     sin b        sin b
or finally:
sin acdB    sin Cib - sin B cos a de - sin b cos CdA.
From  the  first of the formulae (8) we find by similar
reductions as those used for formula (2):
dA = - cos cdB - cos b d C+ — sin b sin Cda.
Hence we have the following differential formulae of trigonometry:
da — cos Cdb -+ cos Bcldc +- sin b sin CdA
cotang a da - cotang B      edB - cotang bdb +- cotang AdA  (1
sin a dB     Sinl Cdb - sin B- cos acde - sin b cos CdA 
dA -- - cos c dB - cos bd C' -- sin b sin Cda.




14
10. As long as the angles are small, we may take their
cosines equal to unity and their sines or tangents equal to
the arcs themselves, or if we wish to have the arc expressed
in seconds we may take 206265 a instead of sin a or tang a.
If the angles are not so small that we can neglect already
the second term of the sine, we may proceed in the following way.
We have:
sin a             1 
aC       6      120
and:
1      1  4
Cos a =       2 + 24 -
hence:
3
i/ cos a- 1   6-   a....
We have therefore, neglecting only the terms higher than
the third power:
sin a   /Vcos
or:
a - sin a V see a
This formula is so accurate that using it for an angle
of 100 we commit only an error less than a second. For we
have:
3
log sin 100~  see 100 = 9.2418864
and adding to this the logarithm 5.3144251 and finding the
number corresponding to it, we get 36000."74 or:
10~ 0' 0."74.
11. As we make frequent use in spherical astronomy
of the developement of formulae in series, we will deduce
those, which are the most important.
If we have an expression of the following form:
a sin x
tangy -  1 —a cosx
we can easily develop y in a series, progressing according
to the sines of the multiples of x. For if we have tang z -',
n dm --?n dn
we find dz =ndn    n.  If we take thus in the formula
n2 + -.-n'




15
for tang y a and y as variable, we find:
dy         sin x
da   1 -2a Cos x-+- a2
and if we develop this expression  by the method  of indeterminate coefficients in a series progressing  according  to  the
powers of a, we find:
sin   - a sin 2 x -t- asin 3 -.... *)
da
Integrating this equation and observing that we have
y -  0 when x =  0, we find the following series for y:
y- = asin x    a2 sin 2x+-    a3 sin 3x +....    (12)
Often we have two equations of the following form:
A sin B — = a sin x
A cos B = 1 -a cos x,
and wish to develop B  and log A  ill a series progressing according to the sines or cosines of the multiples of x.  As in
this case we have:
a sin x
tang B-:1  a cos x
we find for B a series progressing according to the sines of
the multiples of x from the above formula (12). But in order
to develop log A in a similar series, we have:
A =     1 - 2 a cos x - a2.
Now we find the following series by the method of indeterminate coefficients:
a cos x- a2
- a cos x - a2 cos 2 x -I- a3 cos 3x -.. *X *)
1 —2a cos x -- a
Multiplying this by         c  and  integrating  with  respect
a
to a, we find for the left side:
dlog (1-2a cos x - c2)
dcia
and as we have log A  - 0 when a    0, we get:
log/1 —2acosx+a2 =logA —-[acosx+   2 cos2x+ l a3 cos3x+...] (13)
*) It is easily seen, that te first term is sinx, and that the coefficient
of a" is found by the equation:
A,, - 2 A,-_  cos x - A,_-2
**) It is again evident, that the coefficient of a is cos x, while the coefficient of a,, is found by' the equation:
A,, -- 2 A,,- cos x - A,2.




16
If we have the two equations:
A sin B -a sin x
A cos B -- 1 - a cos x
we find by substituting 180 —x instead of x in the equations
(12) and (13):
B-casin x-2 a2 sin 2x' a3 sin3x-.... (14)
log /1 —2a cosx-+-a2 —log A=  cosx — Ia2 cos 2 x -  a3 cos3x-.... (15)
If we have an expression of the following form:
tang y -- n tang x,
a sin x
we can easily reduce it to the form  tang y              -   cos x
For, we have:
tang ( x)   tangy - tang x  (n - 1) tang x
tang (y- x)
1 - tangy tangx       1 +- n tang x2
(n - 1) sin x cos x         (an- 1) sin x cos x
cos X2 --- n sinx2    1   1n
-t - cos 2 x -+ -- -  cos x
2    2         2   2
sin- 2 
(n- 1) sin 2 x       n -4- 1
(n-+-1) —(n1 —I) cos 2x   1  n - 1
I    +   cos 2x
+ t1
Hence, if we have the equation tang y   n tang x, we find:
y    — x- -  Sin      2 —~         s in4x — 1\n-  sin 6xk-...'+" l sin2x +2 (n                       s sin4xl +32(n1- )  (16)
If we take here:
n -- eos a,
n —1
we have:                          - tang  a2.
Hence from  the equation:
tang y -- cos a tang x
we get
-y- x -- tang   2a  2 sin     tang' a4 sin4 x -  tang  a6 a6n6x  x...  (17)
If we have:              n - seca,
we find:                      - tang I a2.
Hence  from  the equation:
tangy --- sec a tang x  or tang x = cos a tangy,
we obtain for y:
y'x+tang  a2 sin2x +- 2                 ] t - a4 sin4x4- tang a6sin6x-+-...  (18)
As we have:
COS t.- COS P
ctang -os           Q - os  a) tang  () tang   (-+-a)
cos a + cos,
alid         sin t   si- n,   tang  (a-  ) cotang-' (a+- 3),
sin a  - sin f




17
we find also from  the equation:
cos a
tang y-   -- tang x
y =    - tang. (a-f) tang  (a +   ) sin 2x
2 tang -(a     2 tang -2  (a — 2 i )2 sin 4x-...
and from:
sin a
tang y       tang x,
sin f
y- x -+ tang 2 (a -) cotang 2 (a +- fl) sin 2 x
-i- I tang " (a -) 2 cotang } (a +- fl)2 sin 4x +-...
By  the  aid  of the  two  last formulae  we  can  develop
Napier's formulae into  a series.  For from  the equation:
sin A-B
a-b          2        c
tang - --            tang 
sin 
2
we find:
a-b    c         B        A              B2       A2
= --   -- tang -  cot;ang 2 sin c 4-  tang 2 cotang   sin 2 c2     2    ag2t2g2                               2
or:
c   a-b         B       A                  B2       A2
2 - -  - -+tang -- cotang 2- sin (a —b) +- 2 tang-2 cotang -2- sin 2 (a-b) -...
and also in the same way from  the equation:
A-B
COS
a+b    0o 2           c
tang                  tang- -
os  2
we find the following two series:
aq-b   c         A      B              _42    B2
2 2   - tang - tang   sin c +-'tang  -tang     sin 2 c+ —.,
c   a —b       A      B                  A2    B2
-2  — 2     tang   tang - sin (a +- b) -    tang - tang — sin 2 (a+ - b) - 
Quite similar series may be obtained from the two other
equations:
sin
A —B           2    180- C
tang.         ang
2. a q- b ang-    -,
sin
2
a — b
A+B           2    180-C
tang ---     - -- =  - tang -  2
2         a..- b      2
os  2
It often happens, that we meet with an equation of the following form:                cos =  os  cos    b
2




18
from  which  we wish to develop y into a series progressing
according to the powers of b.  We  obtain this by applying
Taylor's theorem  to the equation:
y   are cos [cos x + b]
For if we put:
cos x = z and y — f (z - b),
we get:
fy   (z)  b      -cl-f     d 3f b3 +
or as:
(f( )  X f, d-             1- _, dz   d. cos x  sin x
d.
d2f        sin x    dx       cos x
dz2      dx      d. cos x    sin x3
Cos X
d. -
d3 f       sin X3    dx        [1 - 3 cotang x']
dza       dx       d. cosx          sin x3
b              b2                      b
-- x  sinx  cotang  sin —  -,[1 [   cotngx2] s i        (19)
In the same way we find from  the equation:
silly = sin x -+ b
b             b'                      b3
y — x          +- --. - tang x  + -  [- 3 tang 2] — 3+  **) (20)
cos x         cos x                   Cos x
B. THE THEORY OF INTERPOLATION.
12. We continually use in astronomy tables, in which
the numerical values of a function are given for certain numerical values of the  variable  quantity.  But as we often
want to  know  the value of the function for such values of
the variable quantity as are not given in the tables, we must
have means, by which  we may  be able to  compute from
certain numerical values of a function its value for any other
value of the variable quantity or the argument. This is the
object of interpolation.  By it we substitute  for a function,
whose analytical expression is either entirely unknown or at
least inconvenient for numerical computation, another, which
*) Encke, einige Reihenentwickelungen aus der sphirischen Astronomie.
Schumacher's astronomlische Nachrichten No. 562.




19
is derived merely from certain numerical values, but which
may be used instead of the former within certain limits.
We can develop any function by Taylor's theorem  into
a series, progressing according to the powers of the variable
quantity. The only case, which forms an exception, is that,
in which for a certain numerical value of the variable quaintity the value of one of the differential coefficients is infinity,
so that the function ceases to be continuous in the neighbourhood of this value.  The -theory of interpolation being
derived from  the development of functions into series, which
are progressing according to the integral powers of the variable quantity, assumes therefore, that the function is continuous between the limits within which it comes into consideration and can be applied only if this condition is fulfilled.
If we call w the interval or the difference of two following arguments (which we shall consider as constant), we may
denote any argument by  a  - n w, where n is the variable
quantity, and the function corresponding to that argument by
f(a -- nw).  We will denote further the difference of two
consecutive functions f (a -I-  wv) and f(a -4- (n -i 1) w) by
f'(a- n2 — ), writing within the parenthesis the arithmetical
mean of the two arguments, to which the difference belongs,
but omitting the factor tvw ).  Thus f'(a --  ) denotes the
difference of f(a -- w) and f(a), f'(a 4- ) the difference of
f(a-t-2w) and f(a-+-w). In a similar manner we will denote
the higher differences, indicating their order by the accent.
Thus for instance f"(a-V-1) is the difference of the two first
differences f'(a -- 2) and f'(a +-4).
The schedule of the arguments and the corresponding
functions with their differences in thus as follows:
Argument Function   I. Diff.  II. Diff.  III. DiffI  IV. Diff.  V. Diff.
fa (a    
a - 2 wf(a- 2w) f           (a)       (a 
-     J'(a -- w)    (.f-  (( - 1)       f,,(a-   f
ff,~>J" (a f1  (- 
a    f(a)    )(            (a)'+  f"' ( a+   f).\( V (a)
a -w',f (a - )   (,, (a 4- ) f,,, j  l      - (a2)
a-+-2f J'(a~)    fjei)  iJf1(a~9  J             (a
a +3- 3 wf(a+ )'                -- 
c) This convenient notation was introduced by Encke in his paper on
mechanical quadrature in the Berliner Jahrbuch fr 18t37.
2*




20
All differences which have the same quantity as the arguinent of the function, are placed on the same horizontal
line. In differences of an odd order the argument of the
function consists of a-+ a fraction whose denominator is 2.
13. As we may develop any function by Taylor's theorem
into a series progressing according to the integral powers of
the variable quantity, we can assume:
f(a -- n w) - a +- f. n wv -I- r n O w+'  -. n u w3.-..
If the analytical expression of the function f(a) were
known, we might find the coefficients a,,5',   etc., as we
have c =f(a)' —-- etc.  We will suppose however,
that the analytical expression is not given, or at least that
we will not make use of it, even if it is known, but that
we know the numerical values of the function f(a-i-nw) for
certain values of the argument a q — nt w. Then substituting
those different values of the variable n successively in the
equation above, we get as many equations as we know values
of the function and we may therefore find the values of the
coefficients a, Di, y,' etc. from them. It is easily seen, that
we have c - f(a) and that 3'w,'Vw2 etc. are linear functions
of differences, which all may be reduced to a certain series
of differences, so that we may assume f(a-+n w) to be of
the following form:
f (a -- n w) -f(a)  +  ) +-. f (a      B (a -+ - ) +t C ( "'+ ) +-...
where A, B, C... are functions of n, which may be determined
by the introduction of certain values of n. But when n is
an integral number, any function f(a —+nw) is derived from
f(a) and the above differences by merely adding them successively, if we take the higher differences as constant or if we
consider the different values of' the function as forming an
arithmetical series of a higher order. If already the first differences are constant, we have simply f(a+nw) =f(a) +n f'(a+ -),
if the second differences are constant, we must add to the
above value f" (a — 1) multiplied by the sum of the numbers
from  1 to ni-l or by  (n -21); and if only the third differences are constant, we have to add still f" (aI —) multiplied
by the sum  of the numbers I, 1 -- 2, 1    2 + 3 etc. to




21
1 +- 2      +n --- 2 or by n(n-1).(-2.            We have therefore
in general A =n, B           (    )             1  )(n-  2) etc. hence:
1.2        -      1.2.3
f(a + nw)  f(a) +f' nf' (a +) +    1n   f (a 
(n —1) (n —2),  (1)
1.2.3    f
where the law  of progression is obvious ).
This formula is known  as Newton's formula for interpolation.   The  coefficient  of the  difference  of the order n is
equal to the coefficient of x" in the development of (1 -x)".
Example.   According  to  the  Berlin Almanac  for  1850
we have the following heliocentric longitudes of Mercury for
mean noon:
I. Diff.     II. Diff.   III. Diff.
Jan. 0 3030 25' 1". 5
+- 60 41' 50". 0
2 310   6 51.5 2- 18' 48". 0
7  0 38.0                 - +2'44". 4
4 317   7 29 5                     21 32.4    2          - 1 0".   1
72210.4                     254.5
63242939.9                         24926.9                  4.7
7 4637.3             59.2
8 332 16 17.2                      27 26.1
814  3.4
10340 3020.6
If we wish  to  find  now  the  longitude of Mercury for
Jan. 1 at mean noon, we have:
f(a)   3030 25' 1".5 and n= -2:,
further:
f' (a +- I) _- + 60 41' 50". 0, n - 4-    Product:    +- 30 20' 55". 0
f (a- )= +     +  18 48.0,  (-                            2 21.
1.2
*f"' (a +- 3)' +q -   2 44 4,  (n; 6f2           +            10.3
f, (n-I- ) (n-2) (n —3) -.f"V(a q- 2)- --                10        1                 - 0.4
~) We can see this easily by the manner in which the successive functions
are formed by the differences. For if we denote these for the sake of brevity by f', f", f"' etc. we have the following table:
I. Dff.. Dff. III. Diff.
f (a)
f(a) +4f'                   f1
f(a) +   f' +f"               I +f"                     f"ll
f (a) +3f      3f" +" fll,f                   f" +2f"'
f()4f' +  6                   3f"    4f 1'   3f"    3f"-   " +    f"'
f(a) +                            4f' + 10" + 10f""  f"'  " + at'
1(a) - 6f' -+- 1f      20f"' ),  -+ -'1   4f" f
f(a) +- 6f' -  1f." + 350f/' — f 6f"    f"    "    5f"'. (a) - -,7f/ q- 21 f" -+- 35,f"  6




22
Hence we have to add to f(a)
-i 30 18' 43". 9
and we find the longitude of Mercury for Jan. I Oh
3060 43' 45". 4.
We may write Newton's formula in the following more
convenient form, by which we gain the advantage of using
more simple fractions as factors:
f (a + n uw)  f (a) ~+, If' (a    ) --   [f" (a  I) + ---
2               3
A) -     -3 _x-   (a
X [fiv (a +  )+   4[fv (a + 2)]         ( a)
n-  hence
If n is again equal to', we have                    hence,  f'IV(a +- 2) -- 6". 3.  Adding this to f"' (a-+-) and nultiplying the sum  by 3n              we find -     19".0.  Adding this again to f" (a -- 1) and multiplying the sum  by
s2 -    -—, we get - 4'22".2 and if we finally add this to
f' (a-k-) and multiply by n=-, we have to add 30 18'43". 9
to f(a) and thus we find the same value as before, namely
3060 43' 45". 4.
14.  We can find more convenient formulae of interpolation, if we transform  Newton's formula so, that it contains only such differences as are found on the same horizontal line and that for instance starting from f(a) we have to
use only the differences f(a+ —), f"(a) and f"'(a-+-l).  The
two first terms of Newton's formula may therefore be retained.
Now  we have:
f" (a + 1) f=." (a)+f"' (a + 2),
f"' (a + -). =  " (a +- -) -f'i (a + 1)
=-f' (a + l-) +fIv (a) +fv (a + }),
fIv (a +- 2) =fIV(a + 1) +-fv (a +- +)
=fI'(a)+ 2fV (a -- +  ) — fvI (a - 1),
fV (a q- -+) _fV (a -+- +) — +-f  (a +- 2)
-fX7 (a +- ~) +ifvl (a +- 1) +fvI (a - 2),
etc.
We obtain thus as coefficient of f"(a):
n (n- 1)
1.2




23
as coefficient of f"'(a —):
n (n- 1)  n (n - 1) (n - 2)  (ni - 1) n (n - 1)
1.2         1.2.3             1.2.3
as coefficient of fIv (a):
n (n - I) (n - 2)     (n -- I) (n - 2) (n - 3)  ( + 1)  (n - 1) (n -2)
1.2.3              1.2.3.4                 1.2.3.4
at last as coefficient of f'(a-+-1):
n (n - 1) (n- 2) +2. n(n —1) (n-2) (n-3)  n (n - 1) (n - 2) (n - 3) (n - 4)
1.2.3               1.2.3.4                  1.2.3.4.5
(n+2) (n+1) n (n — 1) (n - 2)
1.2.3.4.5
where the law  of progression is obvious.  Hence we have:./(a- +nw)       _ lf(a)t (nf S (a  n)-  +( -- f" () +     f,,i'(a+ t)
f f1.2                  1. 2.33
+ (n+l)n(n —1) (n_ —2) IV z(+2) (?z+l)n(n —l) (?z —2) 
+ i.23.4                   1.2.3.4.5       f (a+ )+'      (2)
If we introduce instead of the differences, whose argument is a+-  those whose argument is a -, we find:
f' (a  -)'- f  (a- ) + f" (a),
f",(a + -) =f"' (a -_ 4) +f"(a),
f' (t +  )./-     (a -  ) +f/VI(a).
Therefore in this case the differences of an odd order
remain the same, but the coefficient of f"(a) is:
n (n- 1)   n (n..-1)
n + —
1.2       1.2
and that of fIv (a):
(n+ 1)n (n — 1)   (n+1)n (- 1) (n -2) _(nn      ( - )n(n —1)(n+2)
1.2.3              1.2.3.4                1.2.3.4
We find therefore:
f(a +n w) =f(a) + nf'(a — )+ n(n( a) ++ (   i) n (+ (a) +  1)
2 l1.2    ()        1.2.3
(n —1)n(n —1)(n+2) f(a)+(n-2) (n-1)n(n+l)(n+2)
1.2.3.4                     1.2.3.4.5               2
where again the law of progression is obvious.
Supposing now, that we have to interpolates   a value,
whose argument lies between a and a   t-w, n will be negative.
But if n shall denote a positive number, we must introduce
-n instead of n in the above formula, which therefore is
changed into the following:




24
f(a —n w) = J(a) - nf'(a -   _  (n-       (a)
1.2
-(n -- 1) n (nq — f1)'"(a —)-t-l)  n(i —1)(n-) I V
(-       ( -+-)2 3               -f (Ca)   (3)
1.2.3      (-+          1.2.3.4
2(n+ 2) (n + ) n (n- ) (n - 2)
1.2.3.4.5
This formula we use therefore if we interpolate backwards. Making the same change with the formulae (2) and
(3) as before made with Newton's formula, we find:
f(a -- n w) =f(c) -+ n [f' (a+ + +- (2)  2+  f"(a)  -3- x
x [f" ( +  ) + n -2    +                 (
n —i        n —1
f(a - nw) =f(a) - n [f' (a -) -  2 [f (a) -  X
x If'" (a -  -  -2 -     () --... (3a)
If we imagine therefore a horizontal line drawn through
the table of the functions and differences near the place which
the value of the function, which we seek, would occupy and
if we use the first formula, when a-i-nw is nearer to a than
to at-w, and the second one, when a-nw  is nearer to a
than to ac- u, we have to use always those differences, which
are situated next to the horizontal line on both sides. It is
then not at all necessary, to pay any attention to the sign
of the differences, but we have only to correct each difference so that it comes nearer to the difference on the other
side of the horizontal line.  For instance if we apply the
first formula, the argument being between a and a +- -w, the
horizontal line would lie between f"(a) and f"(a+- 1). Then
we have to add to f"(a):
+3               3~,
Therefore if f"(c,) is (-  - e) than f "(a - ), the cor\gre ater and hence come nearer f " (a
rected f"(a) will be (greater\) and hence come nearer "(a-1).
\smaller/
A little greater accuracy may be obtained by using instead of the highest difference the arithmetical mean of the
two differences next to the horizontal line on both sides of it.
We shall denote the arithmetical mean of two differences by




25
the sign of the differences, adopted before, but using as the
argument the arithmetical mean of the arguments of the two
differences, so that we have for instance:
f' (a + n)  o        9
As in this case the quantities within the parenthesis are
fractions for differences of an even order and integral numbers for those of an odd order, while in the case of simple
differences they are just the reverse, this notation cannot give
rise to any ambiguity. If we stop for instance at the second
differences, we must use when we interpolate in a forward
direction the arithmetical mean of f"(a) and f"(a -+  1) or
f" (a-+), so that we take now instead of the term
n (n )(a-  )
1.2
the term:' (a+) nr     [2 — (a) f" +2)' f" (a --  )]
1.2                  1.2
Hence while using merely f" (a) we commit an error
equal to the whole third term, the error which we now conmmit, is only:
12 (-i~) (ntl)    n~n-l)'\~.,,,    a (n-i) (n-A-) ~t (a,,
1.2.+3         1i. 2. 2)               1.2.- 23 
If we have n —,      this error, depending on the third
differences, is therefore reduced to nothing, and as it is in
this case indifferent, which of the two formulae (2) or (3)
we use, as we can either start from  the argument a and interpolate in a forward direction or starting from the argument
a --    interpolate in a backward direction, we get the most
convenient formula by the combination of the two. Now for
n2 =  formula (2) becomes:,. ~  1  l,,  1
2 t  z -  F2,'  f":()  2 f,, (a+ ~
f(a~+w) =.(a) +,f' (a+)+         +         123 f"'(a+ 2).3   I   1  3
+2'2  -          fV (_a) v+...
1.2.3.4
while formula (3) becomes, if the argument (a-t-w) is made
the starting point:
1.2
(.+   ) /(a+w) -'1(a+ 4) +   f (a +)
-   12 *2* f(a +)+ 9-*    2' _2ifIV (a+1).




26
If we take the arithmetical mean of these two formulae,
all terms containing  differences of an  odd  order disappear
and we obtain thus for interpolating a value, which lies exactly  in  the  middle  between  two  arguments, the following
very convenient formula, which contains only the arithmetical
mean of' even differences:
1             3               5
f(a +' w)  f(a+  )-       -  (a    + +) + -2 f (a ) - i0]f"(a~-)+...(4)
or:.f(-+~w) =f(a+~) -   [f-"(Ca +.i-)   flv (a              a+) -.., (4a)
where the law of progression is obvious.
Example.  If we wish to find the longitude of Mercury
for Jan. 4 12h, we apply formula (2 a). The differences, which
we have to use, are the following:
I. Diff.    II. Diff.    III. Diff.  IV. Diff.
+ 70 0' 38". 0            + 2' 44". 3
Jan. 4   3170 7' 29". 5              + 21' 32". 4            + 10". 1
7 22 10.4                 2 54.5
6   324 29 39.9                   24 26.9                 4.7
In this case we have ni --    hence:
I I —i 1      3 +    5   n 5   n-2   7
2     8'   3       12'  4       16
taking no account of the signs and we get:
arithmetical mean of the 4t"' differences X  r7,    3". 2
corrected third difference   2 51". 3 X,        1- 1' 1". 4
corrected second difference'22' 43". 8 x   -   8' 31". 4
corrected first difference   70 13' 39". 0 X  1~ 48' 24". 7,
hence the longitude for Jan. 4. 5
3180 55' 54". 2.
If we wish to find the longitude for Jan. 5.5, we have
to apply formula (3a) and to use the differences, which are
on both sides of the lower one of' the two horizontal lines.
Then we find the longitude for Jan. 5. 5
3220 36' 56". 7.
In order to make an application of formula (4a) we will
now  find the longitude for Jan. 5. 0, and get:
arithmetical mean of the 4th" differences X -    -    - - 1. 4arithmetical mean of the 2d differences X -  +     - 2' 52".
arithmetical mean of the functions           - 320~ 48' 34". 7
hence the longitude for Jan. 5.0
3200 45' 42". 4.




27
Computing now the differences of the values found by
interpolation we obtain:
I. Diff.   II. Diff. III. Diff.
Jan. 4.0  3170 7'29".5
~ 10 48)24". 7
4.5  318 55 54.2             -4-1'23".5
1 49 48  2          ~-2".6
5. 0 320 45 42.4         1       26.1
1 5114.3            ~+-2"8.
5.5  322 36 56.7                1 28.9
6.0  324 29 39.9
The regular progression  of the differences shows us,
that the interpolation was accurately made.  This check by
forming the differences we can always employ, when we have
computed a series of values of a function at equal intervals
of the argument.  For supposing that an error x has been
made in computing the value of f(a), the table of the differences will now be as follows:
f (a- 3 w) f (a-            )    f
f(a-2w)'(a)              )
(a — w)      2        (a — x ) (a             v(a —1)-4x
f(a —w)
f(a)+ x                 (a f" (a) -2x         fI(a) )4- 6x
f(a+w) f' (+            (a-   t) —    — x ff,  (a- -)  - 3x
_)f'+(+                              fV((l)
f.l(a-t-2w f)         " (- 2)
f (a  3 w)
Hence an error in the value of a function shows itself
very much increased in the higher differences and the greatest
irregularities occur on the same horizontal line with the erroneous value of the function.
15.  We often have occasion to find the numerical value
of the differential coefficient of a function, whose analytical
expression in not known and of which only a series of numerical values at equal intervals from  each other is given.
In this case we must use the formulae for interpolation in
order to compute these numerical values of the differential
coefficients.
If we develop Newton's formula for interpolation according to the powers of n, we find:
f(a+n w) =-f(a) - n[f' (a ) - -f" (a-)-    + 1 f" (a)- 3
+12 [(f"  +1) — f' (a +  ) +...]
+     _[3f'"' (a +  )-..
but as we have also according to Taylor's theorem;




28
df(a)      d2f(a) n2w w2   d3f(a) n w3,f(a +nwv) — f(a) — a    n to -f-               _        __ _  + -
ia'+cda2   1.2         da3  1.2.3 
we find by comparing the two series:
df (a)    I    11
- f' (a' ~) - {f" (a +- 1) 4 f' -(a +  - - -...]
da        w
d2f(a)      [f"(a+ 1 -f"' (a 3+ 4) +...].
da2    w2
More convenient values of the differential coefficients may
be  deduced  from  formula  (2) in No. 14.   Introducing the
arithmetical mean of the odd differences by the equations:
f' (a + -) =f' (a) ~+ f.t1" (a)
f" (a -- I-).f"'" (a) — + -24fIV(a)
etc.
we find:?I2        (n - %)n (n -I!
f(a -n w).f(a) +-' nf' (a) - 1-  ff (a) + (n + 1) n   f' (a)
1.2       I1.2.3 O
(n - 1)n2 (n )     (a)
1.2.3.4
This formula contains the even differences which are on
the same horizontal line with f(a), and the arithmetical mean
of the odd differences, which are on both sides of the horizontal line. Developing it according to the powers of n we
obtain:
f(a+ nw) =f(a) -+ n [f' (a) - — f'l (a) +,1fv (a) - T4~.( fvn (a) +...]
-  2 If" (a) - -f;Va (a) --           -.(.(a)
+ --. [ If"' (a) _ — _f -   v    (a)    7 f  fv (,) _.]
1.2.3
n4
[fix'  (a)    f` (a)v-4
1.2.3.4. 5
and from this we find:
df   [f (a)- f"' (a) +  i.fv (a) -   1 fVI(a) +...],
da     W'
clc2    w2         i (a) -f  ( a) +  JfV1(a) -..
d3f(a)    1
da3    WU3 [f"'(a) -'fV (a) + Trfl (a) -..  ]
etc.
If we wish to find the differential coefficient of a function,
which is not given itself, for instance of f(a-nwt), we must
substitute in these  formulae  a —n instead of a, so that we
have:




29
df (a + nw)   1
d — [Lf' (a -- I ) - -fa+  (a -+-)      fV(a-I)-. 
da       wv                                               ()
d'f(al -q-w)   I
2([f A"t(a-+-n) - _l f v (a +-2n) -...]
dca2??
etc.
The differences which are to be used now do not occur in
the table of the differences, but must be computed. For the
even differences such as f"(a- +n) for instance this computation is simple, as we find these by the ordinary formulae
of interpolation, considering merely now f" (a), f"(a — +n) etc.
as the functions, the third differences as their first ones etc.
But the odd differences are arithmetical means, hence we must
find a formula for the interpolation of arithmetical means. But
we have:
f' (a+An — ) - f' (ac+, +-1).f' (a   %4-    )   -             _______2
and according to formula (2) in No. 14:
f' (a —  + n) =f' (a-) + - f" (a) +           ( - a)
1.2.3          
(..  +       1) iz(  -2-  1 2
f' (a+ ti+n) =f' (-+ 2) - n.' (ca) A-  (An- 1) A(- )
(n + 1) n (-  1) (a)
1.2.3
therefore taking  the arithmetical mean of both formulae we
find  the following formula for the interpolation of an arithmetical mean:
f' (a+-) =f' (a) A- nf" (a) +-  jt"' (a) -' nflv (a)
(n - 1) n (n - 1)
1.22.3
The two terms:
- -f"' (a) +-' nfIV (a)
1.2
arise from the arithmetical mean of the terms:
n (it - 1)
and
n (n+ - 1)f,,, ( +
-f" (a A)
1':2
which gives:
-1  f' (Ca) A-+ -   f,, (fC -- — ) -f"' ( a — )]-.
1.2          4




30
Combining the two terms, which contain f V(a), we may
write the above formula thus:.f' (a+i) =. f' (a) + nf" (a) +    ~/  (a) --'V ()       (7
The formulae 5, 6 and 7 may be used to find the numerical values of the differential coefficients of a function for
any argument by using the  even differences and the arithmetical means of the odd differences, whenever a series of
numerical values of the function at equal intervals is given.
We can also  deduce other formulae for the differential
coefficients, which contain the simple odd differences and the
arithmetical means of' the even  differences.  For if we introduce in formula (3) in No. 14 the arithmetical means of
the even differences by the aid of the equations:
f(a)   f (a -+ V')-2 f' (a +  2)
f'(a) =f" (a + ) ) — f' (a + - )
flV(a) =fIV(a + ) - lfv (a +- -)
etc.
we find, as we have:
(n + 1) n (e-l)      (n-)       (-1) (n -)
1.2.3        2  1.2         1.2.3
etc.
f(a -+ it w) = f(a -- 2) -+ (n -  ) f' (a -- ) +_ (t- 1 )f (a -- )
1.2.q_ n (n — 1) (n —~-)f,,, (a-{- ~)  (n -- 1) n(n- 1) (n - 2)
+ 1 23  2)a1(a+     )      1.2.3.4    2)'V(a + 
If we write here n — - instead of n, the law  of the coefficients becomes more obvious, as we get:
f [a  4- (,, ) ]   ( )  (   - ) j  ( (a -- 1) -  ) f (a +- 2)
1.2
J fill  (a~ -i-') t?S —)fIv (al  ) q —.
+.2.3                   1.2.3.4         f  (a
Developing this formula according to the powers of n,
we find the terms independent of n:
-f(a   4f (a+) + 8  -- fIV ( +  ) --.. f(a + - W)
hence:
hence:




31
f[a -+ (n +  ) w] -rf(a +  i w)
+n [.f'(a+-         ( -f"'(a ~     + ( +-..]
+      [f(" (a-t   +  ()_f. (a~ +)   + 2- +      ]
1.2            24[f,         576
Comparing this formula with the development of  f(a   w +nw]
~cl~f (a+   _ _ [  a +1)2  +. )1-        37 r V' (ac,+ f  (-.e 1]
da    I [(a+ +) —    7(a+-  ~ v64
1.2d/~4w 1          24 59
fda2          (+ -l _ [f,  fIV(-_ a:,..) +~-; f.(a..]    1
TheseComparing this formula withll be the demost convpmenient in case that+w+n)
accorwe havng  to   ayornd the dierem, we findt
given arguments. For other arguments, for instance a~ (n ~ 6) 
2f/[+- (n+ a) t -    f'  (a -I-  1 ~  - - f"' (a —_  +- a)
+-340 fV(a+ 1+n)..                   (9)
d 2      W      -       24'     - 5760"
etc.
These  formulae  will be the eost convenient in case that
we have to find the differen tial coeffiielnts of a function for
an argument, whicfferenc is thare aritheical mean of two successive
given argby oents.    or other arguments for instance a: -(n-)w
we have again:
CVf~u P (n t ~i-):1,if
f[(a + 2" + )       i~          =./(   +a -- t) (a +   i )+ (-  l )
-    fv (a t  ) -.                     (9)
etc.
Here we can compute the difference f'(aq —+-"n) as well as
all odd differences  by  t    he ordinary forulae  of interpolation.
But as the  followven differencngs  are arithnetical   eans, we m ust
use a'different formula, which we may deduce from  the formula (7) for interpolating an arithmetical mnean of odd differenoes by substuting a -    instead of a and increasing  all
accents by on'e, so that we have for instance:.:.
Example.    According to the Berlin Allanao for 1848
we have the following right-ascensions  of the moon.




32
I. Diff.   II. Diff.  III. Diff. IV. Diff.
Juli 12 01h  16h14m26s.33 +25399
12h     39 30.32               -- 23s 75
2527.74           -is.39
13 01  17  4 58.06                  22.36          — 0OS. 85
25 50.10             2.24
12hi    30 48.16                  20.12   2        0  79
2610.22              3.03
14 Oh      56 58.3S                 17.09             0  67
12h  18 23 25.69   262   31   13.39.70
26 40. 70
15 0'h    50  6  39
If we wish to  find the first differential coefficients for
July 13 101, 111' and 12h and use forimula (9), we must first
compute the first and third  differences for 101, 1111 and  12h.
The third of' the first differences corresponds to the argument
July  13 6" and is f'(a —-), we have therefore for 10t1, 111'
and  121 n  respectively  equal to  L,, ~-  and  -.  Then  interpolating in the ordinary way, we find:
f' (a + e + 2)  f"' (a- — +n)
10h     - 2511157s. 11  - 2s. 51
llh       25 58.81         2.58
12h       26  0.49         2.64
and from  this the differential coefficients:
for 1h    + 2511m57s. 21
111       25 58.92
12h       26  0. 60
where the unit is an interval of 12 hours. If we wish to find
them  so that one hour is the unit, we must divide by 12 and
find thus the following values:
iOh    2m 9s. 77
11h       9.91
12'h     10.05,
which are the hourly velocities of the moon in right-ascension.
If we had employed formula (6), where the arithmetical
means of odd differences are used, taking a -  Juli 13 121,
we would  have found for instance for 1Oh, where n is -,
according to formula (7):
f' (a-  1) —--  -2556s. 77 and f"'(ac -  )   -2s. 51
and  from  these the differential coefficient according to fornula (6) equal to  -1-2"'9s.77.
The second differences are the following:
for 101 o  + 20s. 55
I l     20.34
12h       20. 12.




33
If we add to these the fourth differences multiplied by
-   and divide by 144, we find the second differential coefficients
for 10h    + Os. 1432
111h     0.1417
12h      0.1402.
where again the unit of time is one hour ).
C. THEORY OF SEVERAL DEFINITE INTEGRALS USED IN
SPHERICAL ASTRONOMY.
16.  As the integral fe-'dt, either taken between the
limits 0 and cc or between the limits o and T or T and oo,
is often used in astronomy, the most important theorems regarding it and the formulas used for its numerical computation shall be briefly deduced.
The definite integral fe-'2dt is a transformation of one
of the first class of Euler's integrals known  as the Gamma
functions.  For this class the following notation has been
adopted:
e-x. xa  dx -P(a),                    (1)
0
where a always is a positive quantity, and as we may easily
deduce the following formula:
fe~. x. -I d xfcJe  d X           e —   e — d x
and as the term  without the integral sign becomes equal to
zero after the substitution of the limits, we find:
f x  i  -. xadx
U
or:                      ar(a) -- r(a + 1)                     (2)
But as we have also:
fe-e'  dx  r(1)= 1
*) Encke on interpolation and on mechanical quadrature in,,Berliner
Jahrbuch fiir 1830 und 1837".
3




34
it follows, that when n is an integral number, we have:
-P(n)    (n- 1) (n- 2) (n- 3).....
If we take in the equation (1) x — t2, we find:
2re-t2 t2(a-)+' dt-I(a);
hence for a,=-:
e-2 dt -' r()
In order to find this integral, we will multiply it by a
similar one Je-Y2dy, so that we get:
(Jj e2d t )2 a=fe12d tfeY2 2          (2- y2) dt. d.
e)      o    1)   o         o                    y1
Taking here y =xt, hence dy =  t. dx, we find:
0            t)!1
or as:
e-(l+q-2) t/2 tdt- 
K~- "+''''it~2r  2(1 -2I x")
we find:
co           Go
( - re' A1     x
(f_1;2 dt)2 2J- t~;     -   (are tangc, - are tangO)- 40               o
hence:
-t  t-T  F (I-)                         (3)
From  this follows Z'(-) =  V/, hence from  equation (2):
r(=)2 1 -,, P(-) --  l/r etc.
If we introduce in equation (1) a new  constant quantity
by taking x- ky, where k shall be positive in order that
the limits of'the integral may remain unchanged, we find:
-ky ka-la-ik r1)
hence:
f-ky  a-1 Pd(a)                           (4)
Z,..




35
17.  To find the integral fe-t2dt, various methods are
1I
used. While T is small, we easily obtain by developing
e-' into a series:
C12           T3    I T5    1  T7
e-     -3t2dt    —' 3    2 5    2.3  7
and as we have fe2dt    1'=, we also find from  the above
fe
formula the integral fe-t 2d.
This series must always converge, as the numerators increase only at the ratio of T2, while the denominators are constantly increasing; but only while T is small, does it converge
with sufficient rapidity.  When therefore T is large, another
series is used for computing this integral, which is obtained
by integrating by. parts.  Although  this series is divergent
if continued indefinitely, yet we can find froml it the value of
the integral with sufficient accuracy, as it has the property,
that the sum  of all the terms following  a certain term  is
not greater than this term itself.
We have:
d. d t  t
or integrating by parts:
__2
_2_   l   ft2 dt
s            t2
By the same process we find:
2  t            t
3rit2 dt  3d( e-2   di                     -2
~ t4J ~        di- t23
or finally:
t 22
J  e12 d t_. e
f2  t    e12.11.3'1.3.5
12t    -    52  +(2 t2)2   (2 t2)3
4-.3.5....(2n-1)  1.3.5....(2n+1)  - t2 dt1
(2t2)'             211+1     je   tn+21'
3*




36
or after substituting the limits:
Jt2d e  _  1            1.3  1..3.5
Ject~2TL    2722   (2 T)2   (2 T2)3
1. 3. 5.... (2 n - )   1.3.5....(2+ 1)  - t2 d]  (6)
2),       T         2n+1          t 2ndt 
The factors in the numerator are constantly increasing,
hence they will become greater than 2 T'2; when this happens,
the terms must indefinitely increase, as the numerators increase more than the denominators.  But if we consider the
remainder:
1.O3.5.... (2n-i)  _t2 dt
=F      2n —1 ) f       t++2 d
we can easily prove that it is smaller than the last preceding term.  For the value of the integral is less than
CJ t~,+2
multiplied by the greatest value of e-t2 between the limits T
and c  which is e-'' 2  and as we have:
f dt      1     1'J t~+" L- 2n2 -t-1  T2"'
the remainder must always be less than:
1.3.5... 2n —1 _T2
-     ++ e~
Now this expression is that of the last preceding term
with opposite sign, so that if the last tern is positive, the
remainder is negative and less than it.  In order therefore
to find a very accurate value of the integral, we have only
to see, that the last term which we compute is a very small
one, as the error committed  by neglecting  the remaining
terms is less than this very small term.
Another method for computing this integral, given by
Laplace, consists in converting it into a continued fraction.
If we put:
co
1et 2j  d = U,         (a)
we find:




37
— dU   2 tet2 fe-X2 dx-et2 e2
dt
=2tU- 1.                     (#)
Now  the nth differential coefficient of a product is:
d. xy   d. x        d' -Ix  dy   n (n- 1) d2. x  d 21
dcit'-    dt', Y    -d-t  cdt     1.2   ct'-  clt2 
hence we have:
d'+'l U    cd U      dlt -- U
+    2 t -  -t+ 2 n - 
dk t"S-     cit cdit1dr
If we denote the product 1.2.3.... n by n!, we may write
this equation thus:
(n+1) c1+'. U   2t   U             - U
(n —1)! dt"-'I --    n ddt    (n -  )!dt"-'
or denoting!d "  by U:
(n -t- 1) U,,+l- 2 t U, +- 2 U,,_1.
This equation  is true  for all values of' n froln t-=-1n
when U,, is equal to the function U itself.  We find firom  it:
- 2 -          t - (7, +1) u/
hence:
2  U,,+ I1,2t
U 2 tt- (n -+ )    1- (n+)'2 
or:
U,.            2 t2
(Y)
2  1-(n+l)...1 U+.
2t U,,
But we have from  equation (fi):
U          U
hence:
1    t
2t -        L 
U        2t U
and from equation (y) follows:
1
1  U,       2t2
2t U        1       U2
2t U.




38
If we substitute this value in the former equation and
continue the development, we find:
u   1
U —2t
1   
2t2
+ 2 t2
1+$- 3
2t2
1 +- etc.,
therefore, taking 2  = q
2 TeT2f e-t2 dt — 1
1 -- 2q                         (7)
1 - -3 
1 +4q
1+ etc.
By one of the three formulae (5), (6) or (7) we can
always find the value of the integral e-t2 dt or fe-t2dt, but
0          T
on account of the frequent use of this transcendental function
tables have been constructed for it.  One of such tables is
given in Bessel's Fundamenta Astronomiae for the function:
T2 C2
e J e  dt,
from  which the other forms are easily deduced.  The first
part of this table has the argument T and extends from T= 0
to T=1, the interval of the- argLuments being one hundreth.
But as according to formula (6) the function is the more
nearly inversely proportional to its argument, the greater T
becomes, the common logarithms of T are used as arguments
for values of T greater than 1.  This second part of the
table extends from the logarithm T = 0.000 to log. T — 1.000,
which for most purposes is sufficient. For still greater arguments the computation by formula (6) is very easy.
18. The integral
oD
e    sin    d
COS 2 - 2x sin 52
0j/ -'ixil        ~,




39
can be easily reduced to the one treated above. For if we'
introduce another variable quantity, given by the equation:
1
~ cotg g2 + x    t2
2t
from  which we have       dr    dt,
the above integral is transformled into:
2   2 e —12dt'1'
if we take:            T= cotang ~    - f'
2
If now we introduce the following notation:
e T2fe  dt = Y(r),
1'we have:    J/s 2e        sin2 dx =         (),8)
g cos 2- + 2x sin 52'fr
and also:
If we differentiate the expression e- Vcos - m2   Sill4 x
with respect to z and then integrate the resulting equation
with respect to x between the limits O and w, we easily find:
_____________  dc    = J   2      (  T2)t
where T=cotang       -
And as we have by formula (9)
re    e  s Sill        /2     ) V (1)
we find:
|-x)e~  sin       dr  1/2,B ) - +T2) Y'(1) - - -    (O)
of o fic 
ofwh COS  -forula  w,m
of which formulae we shall also make use hereafsteri




40
D. THE METHOD OF LEAST SQUARES.
19. In astronomy we continually determine quantities
by observations. But when we observe any phenomenon repeatedly, we generally find different results by different observations, as the imperfection of the instruments as well
as that of our organs of sense, also other accidental external causes produce errors in the observations, which render
the result incorrect. It is therefore very important to have
a method, by which notwithstanding the errors of single observations we may obtain a result, which is as nearly correct
as possible.
The errors committed in making an observation are of
two kinds, either constant or accidental. The former are
such errors which are the same in all observations and which
may be caused either by a peculiarity of the instrument used
or by the idiosyncrasy of the observer, which produces the
same error in all observations.  On the contrary accidental
errors are such which as well in sign as in quantity differ
for' different observations and therefore are not produced by
causes which act always in the same sense. These errors
may be eliminated by repeating the observations as often as
possible, as we may expect, that among a very great number
of observations there are as many which give the result too
great as there are such which give it too small. But the final
result must necessarily remain affected by constant errors, if
there are any, when for instance the same observer is observing with the same instrument. In order to eliminate also
these errors, it is therefore necessary, to vary as much as
possible the methods of observation as well as the instruments
and observers themselves, for then also these errors will for
the most part destroy each other in the final result, deduced
from the single results of each method. Here we shall consider all errors as accidental, supposing, that the methods
have been so multiplied as to justify this hypothesis. But
if this is not the case the results deduced according to the
method given hereafter, may still be affected by constant
errors.




41
If we determine a quantity by immediate measurement,
it is natural to adopt the arithmetical mean of all single observations as the most plausible value. But often we do not
determine a single quantity by direct observations, but only
find values, which give us certain relations between several
unknown quantities; we may however always assume, that
these relations between the observed and the unknown quantities have the form  of linear equations. For although in general the function f(  i, ~i,  etc.) which expresses this relation
between the observed quantities and the unknown quantities, 1i,'d, will not be a linear function, we can always procure
approximate values of the unknown quantities from the observations and denoting these by.0, 0,, and'0 and assuming
that the correct values are Z, % -x,?jo -- y, o +,   z etc., we
find from  each observation an equation of the following form:
df    df Y+df
f(~, C, C...) —f(go, r70, g0) +- d~ ox-q c-y q- c z,
provided that the assumed values are sufficiently approximate
as to allow us to neglect the higher powers of x:, y, z etc.
Here f(5,,, C...) is the observed value, f(1o,  201, i0...)
the value computed from  the approximate values, hence
f(o,'io, Co...) -  f(~, V,...) -n is a known quantity.
Denoting then df by a, df by b, df by c etc. and distinguishing these quantities for different observations by different accents, we shall find from  the single observations equations
of the following form:
0     a-=-  -ax+  -- by -+ -..C,
0   --?t   I y -  b'+  cz-,,
etc.,
where x, y, z... are unknown values, which we wish to determine, while n is equal to the computed value of the function
of these unknown quantities minus its observed value. There
must necessarily be as many such equations a's there are observations and their number must be as great as possible,
in order to deduce from them values of x, y, z etc. which
are as free as possible from the errors of observation. We
easily see also, that the coefficients a, b, c.... in the different equations must have different values; for if two of
these coefficients in all the different equations were nearly




42
equal or proportional, we should not be able to separate the
unknown quantities by which they are multiplied.
In order to find from a large number of such equations
the best possible values of the unknown quantities, the following method was formerly employed. First the signs of
all equations were changed so as to give the same sign to
all the terms containing x. Then adding all equations, another equation resulted, in which the factor of x was the
largest possible. In the same way equations were deduced,
in which the coefficient of y and z etc. was the largest possible and thus as many equations were found as there were
unknown quantities, whose solution furnished pretty correct
values of them. But as this method is a little arbitrary, it is
better to solve such equations according to the method of least
squares, which allows also an idea to be formed of the accuracy of the values obtained. If the observations were perfectly right and the number of the unknown quantities three,
to which number we will confine ourselves hereafter, three
such equations would be sufficient, in order to find their true
values. But as each of the values n found by observations
is generally a little erroneous, none of these equations would
be satisfied, even if we should substitute the exact values of
x, y and z; therefore denoting the residual error by A, we
ought to write these equations thus:
An -t-a C+ ax  by   C   1z,
A' n' -- a'x -+- b'y + c'z,
etc.,
and the problem is this: to find from a large number of such
equations those values of x, y and z, which according to
those equations are the most probable.
20. We have a right to assume, that small errors are
more probable than large ones and that observations, which
are nearly correct, occur more frequently than others, also
that errors, surpassing a certain limit, will never occur. There
must exist therefore a certain law depending on the magnitude of the error, which expresses how often any error occurs. If the number of observations is m, and an error of
the magnitude A occurs according to this law p times, P?n




43
expresses the probability of the error A, and shall be denoted by T(A). This function (f,(A) must be therefore zero,
if A surpasses a certain limit and have a maximum for
A = 0, besides it must have equal values for equal, positive
or negative values of A.  As we have p =  m   (A), there
will be among m  observations m'p(A) errors of the imagnitude A, likewise m~(p(A') errors of the magnitude A' etc.; but
as the number of all errors must be equal to the number of
all observations, we have:
n1" 9 (A) +- " 9 (A') -+-.... = 9,
or:
This sum being that of all errors must be taken between
certain limits -- k and  - k, but as according to our hypothesis'p (A) is zero beyond this limit, it will make no difference, if we take instead of the limits — k and -t-k the
limits -oo and A —o.  But as any A between these limits
are possible, as we cannot assign any quantity between the
limits — k and -t-k, which may not possibly be equal to an
error, as therefore the number of possible errors, hence also
the number of the functions p (A) is infinite, each, (A) must
be an infinitely small quantity. The probability that an error
lies between certain limits, is equal to the sum  of all values? (A) which lie between these limits. If these limits are infinitely near to each other, the value'f (A) may be considered
constant, hence'pS(A).dA expresses the chance, that an error lies between the limit A and A -d dA. The probability
that an error lies between the limits a and b, is therefore
expressed by the definite integral
f (A). dA
and we have according to the formula found before:
(A) dA= 1.
According to the theory of probabilities we know, that
when p (A), fp (A') etc. express the probability of the errors
A, A' etc. the probability, that these errors occur together,
is equal to the product of the probabilities of the separate




44
errors. If therefore W denotes the probability, that in a series of observations the errors A, A', A" etc. occur, we have:
w v  () (i) (za').   (")....   (2)
Therefore if for certain assumed values of x, y, z the
errors A, A', A" etc. express the residual errors of the equations (1), W  is the probability that just these errors have
been made and may therefore be used for measuring the probability of these values of x, y and z. Any other system  of
values of x, y and z will give also another system  of residual errors and the most plausible values of x, y and z must
evidently be those, which make the probability that just these
errors have been committed a maximum, for which therefore
the function WV itself is a maximum.  But in order to determine, when (p (A) is a maximum, it is necessary to know the
form  of this flnction.
Now in the case that there is only one unknown quantity, for which the m values n, n', n" etc. have been found
by observations, it is always the rule, to take the mean of
all observation as the most probable value of x.  We have
therefore:
s q-  n' -   n" -....
x -
or:              n- — + — n'- x -f- n — x    0,             (a)
where n - x, n' - x etc. correspond to the errors A, so that
we have n —x, n' — xA' etc. But as W is a maximum
for the most probable value of x, we find differentiating equation (2) in a logarithmic form:
d. log T (A)  dA   d. log p (A') dA'
+__.-t-O —— + = 0,
clA     dx      clA'    dx + 
dA   dA'       __
and as in this case we have d     d          etc.- -1, we find:
dx   dx
cld. log T (n - x)   d. log T (n' - x)
l (n - x)       d (t' - x)
or:
d. log' (n - x)     d.logTq (n'- x)
(n- -)      +         (' —x)          +...o.   (b)
(n - x)(d'. (n — x)   (n'-x) d. (n'- x)
But as according to the hypothesis the arithmetical mean
gives the most probable value of x, the two equations (a)
and (b) must give the same value for x, hence we have:
1    d.log (n - x)    1   d.log p(n'- x)
n - x    d (n - x)    n - c    d (n'- x)




45
where k is a constant quantity.  We have therefore the following equation for determining the function tq(A):
d. log 99 (a).A d — k,
hence
log  (A) --     + -+  log C
and
() -- C. ekA.
The sign of k can easily be determined, for as r (A)
decreases when A is increasing, k mulst be negative; we may
therefore put   k= -h2, so that we have r,(A) —Ce-h-Lr
In order to determine C we use the equation:
+             +
f(A). dA f=c  eA2 dA - 
and as we have fe-x~ 2dx-=   /, we get Je —2A2 dA-1/7,
hence  /-r_ 1 or C- _  and finally:
h  —'A2
(A) —- A/ e.            (3)
The constant quantity h remains the same for a system
of observations, which are all equally good or for which the
probability of a certain error A is the samle.  For such a
system the probability that an error lies between the limits
—' and + —( is:
f      dAj j~e        dx
Now  if i     - another system  of observations the probability' -of an error  h'A  in this sysbility of an error A is expressed by — e          in tlis sys1/ 7r
tem the probability that an error lies between the limits -- 
and -i-', is:
I~ ff    h' h'a} C  -7
fe-  h'' fe  dx
Both integrals become equal when h ) = h'AB. Therefore
if we have h - 2h', it is obvious, that in the second system
an error 2x is as probable as an error x in the first system.




46
The accuracy of the first system  is therefore twice as great
as that of the second and hence the constant quantity h
may be considered as the measure of precision of the observations.
21.  Usually instead of this measure of precision of
observations their probable error is used. In any series of
errors written in the order of their absolute magnitude and
each written as often as it actually occurs, we call that error
which stands exactly in the middle, the probable error.  If
we denote it by r, the probability that an error lies between
the limits — r and +-r, must be equal to 2. Hence we have
the equation:
i fe-th2 LdA_,
or taking hA -- t
hr                  hil
2 —>.t dt —,   therefore  ee-t2dt c
}/ 7CJ                 j           4
1                   0
But as the value of this integral is 4    0.44311, when
hr =  0.47694 4), we find the following relation between r
and h:
0.47694
L h'
The integral /2  et2 d t gives the probability of an error, which is less than n times the probable error and if we
compute for instance the value of this integral for n =,
taking therefore n hr =  0.23847, we find the probability of
an error, which is less than one half of the probable error
equal to 0.264, or among 1.000 observations there ought to
be 264 errors, which are smaller than one half the probable
error. In the same way we find, taking n successively equal
to E~ 2,, 3, 2, 4, 9, 5, that among 1000 observations there
ought to occur:
r) On the computation of this integral see No. 17 of the introduction.




47
688, where the error in less than 3r
823,,,,,,,,,  2r
908,,,,,,,,,,,,
956,,,,,,,,,,,,,  3 r
982,,,r
993,,,,,,,,,,,,,  4r
998......                  9. 
999,,,,, 
and comparing with this a large number of errors of observations, which actually have been made, we may convince
ourselves, that the number of times which errors of a certain
magnitude are met with agrees very nearly with the number
given by this theory.
We will find now the value of h.  Suppose we have a
number of m actual errors of observation, which we denote
by A, A' etc., the probability that these occur together is:
hiL  - h h [iA a' n'+,s' A " +r....]
W, —    e
and if we further suppose, that these errors were actually
committed and hence cannot be altered, the maximum of W
will depend merely on h and that value of h, which gives
the maximum, will be the most probable value of h for these
observations. Denoting now for the sake of brevity the sum
of the squares of the errors A, A' etc. by [LA], we have:
h-, — hh[AA]
and we easily find the following conditional equation for the
maximum:
0  mh"_-e -hh-h[A]  2h.,+ e —-hh [ A]'~-' ~~ —e        -       e         [Akl]
or:                    0 =n -  2 h [  ],
hence follows:
hV2 = m
This square root of the sum of the squares of real errors
of observations divided by their number, is called the mean
error of these observations. If this error had been made in
each observation, it would give the same sum  of the squares
as that of the actual errors. If we denote it by e, or put:?n 1,




48
we have:
1
hV2
and:                   r  0.47694 V 2e
r — 0.674489 s.
22. We will now solve the real problem: To find from
a system  of equations (1), resulting from actual observations,
the most probable values of the unknown quantities x, y and z
and at the same time their probable error as well as that of
the single observations.
If we substitute in the equation (2) instead of, (A),.(AL') etc. their expressions according to equation (3), we
find:
/,    h2 [2+ / 2 + A[l 2+ j +]
w- ~t -- e [
(O/ 7)0
if we suppose that all observations can be considered as
equally good.  Here A, A', A" etc. are not the pure errors
of observations, but depend still on the values of x, y and z.
But as for the most probable values of x, y and z the probability that the then remaining  errors have occurred together, must be as great as possible, as they become as near
as possible equal to the actual errors of observations, which
must be expected among a certain number of observations,
we see that the values of the unknown quantities must be
derived from the equation:
A2 -- A'2 -+ A" -2...  minimum
or the sum of the squares of the residual errors in the equations (1) must be a minimum.  Hence this method to find
the 1most probable values of the unknown quantities from such
equations is called the method of least squares.
If we first consider the most simple case, that the values
of one unknown quantity are found by direct observations,
the arithmetical mean of all observations is the most probable
value.  This of course follows also from  the condition of
the minimum given above. For the residual errors for any
certain value of x are:
A-Z-v, A' —-'it', All x —xn", etc.
We get therefore for the suln of the squares of the residual errors, if we denote




49
the sum of n — n' +-n" t... by [n]
the sum of n2 —-n'2+-n"2 -t... by [nn]
and the number of observations by m:
(x -?z)2  mx 2 - 2x[n] + [nn]
[nz2N]   I       _+(
-'i-R T/? X ~As all terms of the second member are positive, the
sum of the squares will become a minimum, when:
m
and the sum of the squares of the residual errors will be:
[it n] _ [n] _ [n n, J]
2
In order to find the probable error of this result from
the known probable error of a single observation, we must
solve a problem, which on account of an application to be
made hereafter we will state in a more general form, namely:
To find the probable error of a linear function of several
quantities x, x' etc., if the probable errors of the single quantities x, xa' etc. are known.
If r is the probable error of x and we have the simple
function of x:
Xf= ax,
it is evident, that a r is the probable error of X. For if x,,
is the most probable value of x, a x, is the most probable
value of X  and the number of cases, when x lies between
the limits x0 — r and x -- r is equal to the number of cases
in which X lies between a xo,-ar and a xo- a r.
Let X now represent a linear function of two variables
or take:
X; X —- x,
and let a and a' represent the most probable values and r
and r' the probable errors of x and x'. As we must take
then for the errors x and x' respectively h — c and h' —,,
where c is equal to 0.476943 we have the probability of any
value of x:
~2
C    r2 ( -,t)
--- e
rV~




50
and the probability of any value of x':
C2  - -: (xt-a')2
C. er2
hence we — have the probability that any two values x and x'
occur together:
c2    [2 (x-a)2 + -2 (x'- a) 2
rre e
We shall find therefore the probability of two errors x
and x' which satisfy the equation X    x-+x', if we substitute
X   x for x' in the above expression and denoting this prob ability by W, we get:
_ [2 a    2+ C2 (X —r —-l)2]
______  ([ ia)2,t   ( xa2
W " — -'  e
rr T7
If we perform  now  the summation of all cases, in which an
x may unite with an x' to produce X, where of course we
must assign to x all values between the limits -  X and -+  o,
or in other words if we integrate WV between these limits,
we shall embrace all cases, in which X can be produced or
we shall determine the probability of X.
Uniting all terms containing x and giving them the form
of a square, we easily reduce the integral to the following
form:
+o
2     2r2+  r 2       (Xa)+ _ a  22 (-C2
C         r2 err 2    1r 2 a)r,   12+72          cdx
oroX,2+r'2 (X —a- a)2   2  |o2 d
6                 X         du,
J/r2 +r,2 2/7t.
if we put:
c Y/T +tJ 2    r2 (X - a). 2 a?,T!            T.2 -- 2.
and as we have
e0l   du- 
2
c  -. +,-     2




51
But this expression becomes a maximum, when X= a+ -i- a',
hence the most probable value of X is equal to the sum  of
the most probable values of x and x' and the measure of
accuracy for X is   — hence the probable error of X is
/r2p_ _+r'2.  From  this follows in connection with the formula
proved before, that when:
X    x -x + c'x'
the probable error of X is equal to l/ca2r2+-'2r'2.
We may easily extend this theorem to any number of
terms, as in case we have three terms, we can first combine
two of them, afterwards these with the third one and so on.
Hence if we have any linear function:
X= ax -+-      "'x' - t- a"x"....,
and if r, r', r" etc. are the probable errors of x, x', x" etc.
the probable error of X is equal to:
1/a2 r'_  a'- fr 2 r_2 _+ all 2 2.112.
From  this we find immediately the probable error of the
arithmetical mean of m observations, each of which has the
probable error r: for as:
n +)n + n"a -+ no'"....
X —-?n
we have the probable error of the mean equal to  mn. 2
or.
The probable error of the arithmetical mean of m observations is therefore to the probable error of a single observation as I1-:I or its measure of precision to the measure
of a single observation as h  m-: h.  Often the relative accuracy of two quantities is expressed by their weights, which
mean the number of equally accurate observations necessary
in order to find from  their arithmetical mean a value of the
same accuracy as that of the given quantity.  Therefore if
the weight of a single observation is 1, the arithmetical mean
of m observations has the weight nm. Hence the weights of
two quantities are to each other directly as the squares of
4*




52
their measures of precision and inversely as the squares of the
probable errors *).
It remains still to find the probable error r of a single
observation.  If the residual errors x — n A of the original
equations after substituting the most probable value of x were
the real errors of observation, the sum of their squares divided by m  would give the square of the mean error of an
observation according to No. 20, or this error itself would
be  n/;in]. But as the arithmetical mean of the observations
is not the true value, but only the one which according to
the observations made is the most probable, except in case
that the number of observations is infinitely great, the residual errors will not be the real errors of observation and
differ more or less fromu them. Now let xo be the most probable value of x as given by the arithmetical mean, while
xO-1-  may be the true value which is unknown. By substituting the first value in the equations we get the residual
errors x,,- -n,,, —n' etc. which shall be denoted by LA, i'
etc. while the substitution of the true value would give the
errors x0o+ -   - n =-   etc. We have therefore the following
equations:
a +  =  A,
etc.,
and if we take the sum. of their squares observing that the
sum of all A is equal to zero, we find according to the adopted
notation of sums:
[A A] ~?72 2 =   ]
which equation shows that the sum  of the squares of the
residual errors belonging to the arithmetical mnean is always
too small.
As we have [Jc] — me2, when e denotes the mean error
of an observation and further [Al  ] -=Innl], we can write
the equation also in the following form:
[nt 1 _] +  an 2  22 -I 8 2
*) If therefore two quantities have the weights p- 2   and p' - 
the weight of their sum is 2 - 2   p_
+rip p)




v53
Although we cannot compute from  this equation the value of a, as ~ is unknown, still we shall get this value as
near as possible, if we substitute instead of J the mean error
of x and as we have found this to be equal to /8i, we find
thus:
~   - 1
for the mean error of an observation and hence the probable
error:
-- 0.674489 ibn      -n ~
Furthermore we find the mean error of the arithmetical
mean:
C (X)     nz I - l Y...
and the probable error:
0.674489  _/_[__]
Example.  On May 21 1861 the difference of longitude
between the observatory at Ann Arbor and the Lake Survey
Station  at Detroit was determined by means of the electric
telegraph, and  from  31  stars observed at both stations the
following values were obtained:
Difference   Deviation           Difference    Deviation
of longitude. from the mean.     of longitude. from the mean.
Star 1  2m43s. 60   - 0.11       Star 16  2n43s. 50   -0.01
2     43. 49   - 0.00           17    43. 44   +0.05
3     43. 63   -0.14            18    43. 37   +0.12
4     43. 52   - 0.03           19    43. 32  -+- 0.17
5     43. 31    +0.18           20    43. 12   +0.37
6    43. 67   -0.18             21    43. 30   -4- 0.19
7     43. 98   - 0.49           22    43. 72   - 0.23
8    43. 63       0.14          23    43. 25   +0.24
9    43. 83   -0.34             24    43. 13 --  0.36
10     43. 79   — 0.30            25    43. 27   -4-0.22
11    43. 54   -0.00             26    43. 34   +- 0.15
12    43. 18   -0.3'1            27    43. 15   +- 0.34
13     43. 45    +0.04           28    43. 86   - 0.37
14    43. 68   -0.19             29    43. 29   +- 0.20
15     43. 32   +0.17            30    43. 40   -4- 0.09
31    43. 95   - 0.46
Mean 2m43s. 49




54
Here we find the sum of the squares of the residual
errors [nnl] =1.77, and as the number of observations is 31,
we find:
the probable error of a single observation --  Os. 164
hence the probable error of the mean of all observations
-4 Os. 029.
Although we cannot expect that in this case the errors
of observations, the number of observations being so small,
will be distributed according to the law given in No. 21, yet
we shall find, that this is approximately the case. According
to the theory, the number of observations being 31, the number of errors
smaller than ir, r,    r, 2    r, 3r
ought to be 8, 15, 21, 25, 28, 30
while it actually is according to the above table:
6, 12, 22, 24, 29, 30.
The error which stands exactly in the middle of all errors written in the order of their magnitude and which ought
to be equal to the probable error is 0.18.
23. In the general case, when the equations (1) derived
from the observations contain several unknown quantities, the
number of which we will limit here to three, the most probable values of these quantities are again those, which give
the least sum of the squares of the residual errors. As this
sum  must necessarily be a minimum  with respect to x as
well as to y and a, this condition furnishes as many equations as there are unknown  quantities, which therefore can
be determined by their solution.
The equation of the minimum  with respect to x is as
follows:
d A    /' d4  A' " - AllA "
dx     dx      dx
or as we have according to equations (1) d: = 1a, dx -a etc.
we get:
A a q- A' ay - a'l - +  -....  O
If we substitute in this for A, A' etc. their expressions
from (1) and if we adopt a similar notation of the sums as
before, taking:




55
a a + aa' a' -- a" a" +....   [a a]
and a b +- a' b'- - a" b" +. --  [a b] etc.
we get the equation:
[cla]x+ -[ab]y+- [ac]z+ [an]=O;   (A)
and likewise   [a b] x + [b b] y + [b c] z + [b n]  0    (B)
and              [cac] x + [b, c] y + [c c] z + [c --  O    (C)
from  the two  equations of the minimum  with respect to y
and z.  The solution of these tree equations gives the most
probable values of x, y and z.
In  order to solve them  we multiply the first by [ab]
Ia a]
and subtract it from  the second, likewise we multiply the
first by [a] and subtract it from  the third.  Thus we obtain
[a a]
two equations without x, which have the form:
[bbl]y + [bc,] z + [bn,]     (D)
and                [b c,] y+ [cc,] z+ [cn] -O,  (E)
when we take
[bb ]-[bb]  [[ab] [Ct]              a c  ]    c] b [c]
which equations explain the adopted notation.
If we multiply now the equation (D) by [bb,] and subtract it from (E), we find:
[cc2] z + [C,2] 0o  (F),
where we have now:.[cc,]   IC[be]l [be1] [c    ]-[c     [bbc,][bn,]
[cc2]  [C cI] —  [b     [cn~]  [en1] -
From  equation  (F) we find the value of z, while the
equations (D) and (A) give the values of y and x.
If we deduce [A2] from  the equations (1) we find with
the aid of equations (A), (B) and (C) for the sum of the
squares of the residual errors:
[A2] -- [nn] --- [an] x +- [b n] y q- [cn] z.
In order to eliminate here x, y and z, we multiply equatioA.by [a ] and subtract it from the above equation, which
[laa]
gives:
[an]2
[a a][nn]  [a] + [bn,] y + [c n Iz.
If we then multiply the equation (D) by  b] and sub




56
tract it from the last equation, we get:
L[AA] - Enn] -[a ~~ 2  [b- ] -bb]    [Cn2] A
[ an]    [bn]2
and if we here substitute the value of z from (F) we find
at last for the minimum  of the squares of the errors:
[aan]    [bbl.]  [c2   -  nn].
We  can find the equations for the minimum  of the squares
of the errors also without the differential calculus.  For if
we multiply each of the original equations (1) respectively
by ax, by, cz and n and add them, we find:
[Ab    ]  -- [  ] +    [1, a] y +  [A l] z 4+ I[n]y  (a),
where       [aA]    a][a]   --  b] y   + c] z + [an]  (b)
etc.
If we now substitute in (a) instead of x its value taken
from (b), we find:
[a a]2
where
[b,]=  [6b ]y +- [bcn] z -+ [bn,]    (d)
Then substituting in (c) for y its value taken from the first
of the equations (d), we find:
[aa]    [A   ]+[ 2] +n zA]  (e)
[~ aa] =       [-   + Lbb,]
where now
[ca2]: [cc2e]. + [c2,],     (f)
[1n2] = [cn2]2 + [nn2],
and if we finally substitute in (e) for z its value taken from
the first of these last equations,  we have:
[A l=  — ] + [- t ] + cc -   [- -_  n A,3  (g)
Ea a]   [-,] + P-[ Cc2c,
and we easily see that we have [nA]== [nnj].
As the first three terms on the right side of equation (g),
which alone contain x, y, and z, have the form of squares,
we see, that in order to obtain the minimum  of the squares
of the errors, we must satisfy the following equations [a A] =O,
[bAl] =  0 and [c] As —    O, which are identical with those we
found before.  We see also, that [nn] is the minimum  of
the squares of the errors.




57
24. The theorem for the probable error proved in No. 22
will serve us again to find the probable errors of the unknown quantities, as we easily see by the equations A, D
and F  that the most probable values of x, y and z can be
expressed by linear functions of n, n', n" etc.
For in order to find x from  these three equations, we
must multiply each by such a coefficient that taking the sum
of the three equations the coefficients of y and z in the resulting equation become equal to zero.  Therefore if we mul1              A'               A"
tiply (A) by         (D) by [b,'  (F) by             and  add the
la a]                          E[ Cc2]
three equations, we get the following two equations for determining A' and A":
a b] + Al  O,                      (a)
[a al
[ c] A,   [ b c]  All  O
and we have:
[a] - A'     -A" [c]            ()
x [a a]   A[bb-I]   Cc ]B'
In order to find y we multiply (D) by [- -,' (F) by'] and
adding them we get:
[b1c,3
+ B' -  0                (0
[lc b i],                  ()
and'              - [bb,]    [cc2]
At last we have:
[cn23
- icc [ —]'         (C)
Developing the quantities [bn,] and [cn2], we easily find:
[bn,]   A' [an] + [bn]             (1),
[cn2] = A" [an] + B'[ b nJ +  c n]  (9),
and as we may change the letters, the quantities in parenthesis being of a symmetrical form, we find also:
[b b ]: a' [ab] + [Cb]                 (l),
[c c2] - A" [a c] +- B' [b c] + [c c]  (X),
[b c2]= A" [a b] -t B' [b b] + [b c]O  (),
[aC2]= A" [a a] + B' [a b] + [a c] =O  (). *)
*) The two last equations we may easily verify with the aid of the
equations (a), (f) and (a).




58
Now as [an] as well as [bn,] and [cn2] are linear functions of n, we can easily compute their probable errors. First
we have [an] =  an -+- a'n' -4 a"n" +-....  If therefore r denotes the probable  error of one observation, that of [an]
must be:?' ([anl]) =    /a -CtCt +' a' -1 — +aCl a"..' [a a].
Every term  in [bnj] is of' the following form (A' a+-b)n.
In order to find the square of this, we multiply it successively by A'an and bn and find for the coefficient of n2:
A' (A' a -- acb) +- A' ab +b- bb.
This therefore must also be the form  of the coefficients
of each  r'2 in the expression for the square of the probable
error of [bn1] or we have:
(r [b n])2 = [A'(A' [a c] - [a b]) -+- A' [a b] - [b b]] e2,
or:        r (L[b1,]). /[bb,],
as we find immediately by the equations (a) and (6).
At last the coefficient of each n in the expression of
[cn2] is:
A" a -+- B'b -4- c.
Taking the square of this we find:
A" (A"'  a - +- B' cb -4- a c)
+ B' (A" a b +- B'b b +- b c)
- A" a c +- B' b c +- cc.
Now  taking the sum  of all single squares, we find the
coefficient of r2 in the expression of (r [cn])2:
A" (A" [a a] + B'[ca] + [a c])
+-B' (A"[a 1b] -+- B'[) bb] + [b c])
+  " [a c] -+- B' [b c] + [cc],
which according to the equations (z), (A) and (.) is simply
[cc2]; hence we have:
r [cn2]l  r  I [CC2]
We can now  find the probable errors of x, y and z without
any difficulty.  For according  to  equation (,) we have for
the  square  of the  probable  error of x  the following  expression:
[ (x)]'2  [r (n)]2    A A           A" A"
[r()2    [ ]- [a b +  2 (r [b])2    [   2 (7' [cn 2])2
2  i 1    A'A'   A"A 
la a]  [b Ebb]   IC C2]




59
Likewise we find:
I     B" B"
[r(y)12  [bb,] +   C2 -
and                 [r) (z)] 2,2
[C C 2
It remains still to find the probable error of a single
observation.  If we put for x, y and z in the original equationis (1) any determinate values, we may give to the sum
of the squares of the residual errors the following form:
[aA] 2  [b11]2'A-   [3] 2
[AW =       _+ -t           -_  -[ it[nn,]
[a   [a]    [bb,    [IC2] C,
In case that we substitute here for x, y and z the most
probable values resulting from this system of equations, the
quantities [arS], [b A]  and [cA2] become equal to zero and
the sum  of the squares of the residual errors resulting from
these values of x, y and z is equal to [nn,]. But these values will be the true values only in case that thie number of
observations is infinitely great.  Supposing now, that these
true values were known and were substituted in the above
equations, [ A] would  be the sum  of the squares of the
real errors of observation and we should have the following
equation:
2   [r, _] 2  [b    I ]2  [C A21]
-4-      +          I[n n3],
aa]=   [ b 1,]    [ c 2]
where now the quantities [a zi], [bAl] and [cA2] would be a
little different from  zero.  As all these terms are squares,
we see that the sum  of the squares as found from  the most
probable values is to\ small and in order to  come a little
nearer the true value we may substitute for [aA] etc. their
mean errors.  But as in the equations:
axo + biyo + czo  + -2 - 
etc.
no quantity on the left side is affected by errors except n,
A must be affected by the same errors and the mean errors
of [a A], [bAj] and [c A] are equal to those we found for
[an], [b n] and [cn2].  Substituting these in the above equation we find:
2 -  2 +  e2    I ~2 + [22223]
or:                         _ _nn3
m 3




60
Hence the mean  error of an observation is derived from
a finite number of equations between  several unknown quantities by dividing the sum  of the squares of the residual errors, resulting  from   the  condition  of the minimum, by the
number  of- all observations  minus  the  number of unknown
quantities and  extracting  the square root.
Likewise we find for the probable error of all observation:
- 0.674489' [nn3'
Note 1.  We have hitherto always supposed, that all observations, which
we use for the determination of the unknown quantities, may be considered
as equally good.  If this is not the case and if A, h', h"I etc. are the measures of precision for the single observations, the probability of the errors A,
A' etc. of single observations is expressed by:
h  — h2A2   h'    h'2 A'2
— e           _  e, etc.
Hence the function W  becomes in this case:
A_./'./A'..... ( 2'  + /-, zh, 2 +,A, 52,, + 2+...)
T      (-  )   e
and the most probable values of x, y and z will be those, which make
the sum
/2 A2 + h' 2  2   h A"1)2 a-12 ~
a minimum. In order therefore to find these, we must multiply the origfnal
equations respectively by h, h', h" etc. and then computing the sums with
these new coefficients perform the same operations as before.
Note 2.  If we have only one unknown quantity and the original equations have the following form:
O- n -- -ax,
0 =' +- a'x,
O- n"+ a"x, etc.,
we find x   -  [aa] with the probable error r-  -,          where r denotes
Cal a]                            l/[a a]
the probable error of one observation.
25.   This  method  may  be  illustrated  by the following
example, which  is  taken from  Bessel's determination  of the
constant quantity  of refraction, in the seventh volume of the,,Koenigsberger Beobachtungen" pag. XXIII etc.  But of the
52 equations given there only the following 20 have been
selected,  whose  weights  have  been  *taken  as  equal and  in
which  the  numerical term  is a quantity  resulting  from   the
observations  of the stars, while y denotes the  correction  of




61
the constant quantity of refraction and x a constant error
which may be assumed in each observation.
The general form  of the equations of condition in this
case is n —x —by, as the factor denoted before by a is equal
to 1, and the equations derived from  the single stars are:
Residual errors.
a Urs. min.    0=-0". 02-x -+-  0.2 y         -0". 03
i Urs. min.   0 -+0.45 -x +-  8.2y           +0.43
f Cephlei     0= —k0. 10+x  - 20.1y          +0.14
a Urs. maj.   0=-0.14+ —x  +  36.0y         -0.03
a Cephlei     0  -0. 62 +x   - 43.9          -0. 47
8 Cephlei     0=-0.25 + x  +  65.9y           0.00
a Cephei      0 —0.03+x --    74.9y          +0.26
It Cephei     0= -. 24 - x  +  77.8y      — 0.94
a  Cassiop.    0 —  0.59 -+- x  -  75.5 y    + 0.88
y Urs. 2maj.   0 —0.47 —x  -+  79.6y         -0.16
13 Draconis    O=-   0.0 -x +- 104.5 y         0.42
y Draconis    0=- 0.51+- x  + 114.3y         - 0.04
Vr Urs. z aj,   0  --.20+-x  +125.6y          0.68
a Persei      O-+-O.12 - x +-142.1y         +0.72
a Aurigae     0= -1. 31 + x  + 216.81y      -0.37
a Cygni       0= —i.64+x  + 254.8y           — 0.53
e Aurigae     0= -1. 39+- x  4- 280.2y       -O. 16
y Androm.     0 = - -1.24- x  + 393.51y      +0.51
n Aurigae     0= —1.80- x  +419.6y         -+-.06
fi Persei     0    -2.16 - x  + 481.2 y      -0. 01
In  order now  to find from  these the equations for the
most probable values of x and y (equations (A) and (B) in
No. 23), we must first compute all the different sums [aa],
[ab], [an], [bb] and [bn].  In this case, where the number
of unknown  quantities is so small, besides one of the coefficients is constant and equal to one, this computation is very
easy; but if there are more unknown quantities, whose coefficients may be for instance a, b, c, d it is advisable, to
take also the algebraic sum  of the coefficients of each equation, which shall be denoted by s and to compute with these
the sums [as], [bs], [cs] etc., as then the following equations
may be used as checks for the correctness of the computations:
ns] -Cssn] + [szbn] + [cn] + [Cd],
[as]     + [a c     +     [ac] + [ad],
etc.




62
If we compute now  the sums for our example, we find
the following two  equations for determining  the most probable values of x and y:
+- 20.000 x +- 3014.80y - 12.72 = 0,
+ 3014.80 x'+ 844586.1y  - 3700.65 = 0.
The solution of these equations can be made in the following form, which may easily be extended to more unknown
quantities:
[a a]     [a b]              [C an]        [}n n]
+- 20.000  -- 3014.80          -12.72         20.28
1.301030   3.479259           1.104487, -      8.09
[an] --   12.72               [bb]              [b b]         12.19
[a b] x = +  13.78        +844586.1          - 3700.65  [ bn] 8.15
-+  1.06          +454452.0          - 1917.41 [n 2] — 4.04
0.025306,  [[bl] — +-390134.1  [bnl --  1783.24
1.301030                      log [bn] 3.251210
log X  8.724276,,                    log [b bjS 5.591214
x - -0". 053                         log y   7.659996
y  - - 0.0045708
In case that we have computed the quantities [as], [b s] etc.
we may compute also [b s] and use the equation [bbl]  [b s]
as a check.  In the case of 3 unknown quantities we should
use [b b,] -+  [b c]  - [b sl] and [cc2] =  [cs2] and similar equations for a greater number of unknown quantities.
In order to compute the probable errors of x and y,
we use besides [bb,] also the quantity
F[a  1    F       [ [a a]  [b] --     9.2384.
Then  we find the probable  error  of the quantity n  for a
single star:
r- (0.67449  [nn]  - _ 0.3195,
hence the probable errors of x and y:
0".3195
r (x) =-    0_  -- =    0".105,
0"1.3195
r (y)    0. 3195 -4- 0".0005116.
I/ [b b,]
We see therefore, that the determination of x from  the
above equations is very inaccurate, as the probable error is
greater than the resulting value of x; but the probable er



63
ror of the correction of the constant quantity of refraction
is only I of the correction itself.
If we substitute the most probable values of' x and y
in the above equations, we find the residual errors of the
several equations, which have been placed in the table above
at the side of each equation.  Computing the sum  of the
squares of these residual errors, we find 4.04 in accordance
with [nn2], thus proving the accuracy of the computation by
another check.
Note. On the method of least squares coTnsult: Gauss, Theoria motus
corporum coelestium, pag. 205 et seq. Gauss, Theoria combinationis observationunm erroribus minimis obnoxiae. Encke in the appendix to the,,Berliner Jahrhbicher fiur 1834, 1835 und 1836."
E. THE DEVELOPMENT OF PERIODICAL FUNCTIONS FROM GIVEN
NUMERICAL VALUES.
26.  Periodical functions are frequently used in astronomy, as the problem, to find periods in which certain phenomena return, often occurs; but as these are always comprised
within certain limits without becoming infinite, only such periodical functions will come under consideration as contain
the sines and cosines of the variable quantities. Therefore
if X denotes such a function, we may assume the following
form for it:
X-ao +al cosx.+  2 cos2  -+a3 cos3x+..
+-b, sinx-+-b2 sin2x+b3 sin3x+...
Now the case usually occurring is this, that the numerical values of X are given for certain values of x, from
which we must find the coefficients, a problem whose solution
is especially convenient, if the circumference is divided in n
equal parts and the values of X are given for x =0, x  2 
=2 2  etc. to x     (n - 1) 2   as in that case we can make
use of several lemmas, which greatly facilitate the solution.
These lemmas are the following.




64
If A is an aliquot part of the circumference, n A  being
equal to 2s, the sum  of the series
sin A -+- sin 2 A -+- sin 3 A +-...+4- sin (n- 1) A
is always equal to zero; likewise also the sum of the series
cos A +- cos 2A +- cos 3A -+... -4- cos (n - 1) A,
is zero except when A is equal either to 2 a or to a mnultiple of 2 gr, in which case this sum  is equal to n.
The latter case is obvious, as the series then consists
of n terms, each of which is equal to 1. We have therefore to prove only the two other theorems. If we now  put:
cos   --  i sin r -- T
n          n
2I
where we take i   — 1/-  and T= e, we have:
r = nL — 1    =' -= -  1     1 =n — 1
2rc. 214F&7 I          T    Tl1 - I
cos r -- -- i   sin r -     T -          
n             n              T- I
r=0            r'O          r O
As we have now,1 -=  cos 2) -- i sin 2 —=- 1 it follows
that:            r= it -I       r= -i1
cos     - i   sin  -     0,
fZ COS 7'-+j~51I~ri %M  n
r=-O          r=O
r =  -I1
hence:                      sin r     0          (1)
r = O
and this equation is true without any exception, as there is
nothing imaginary on the right side. It follows also, that
we have in general:
r =  — t
Cos aO- -?   0.      (2)
I- -     X
Only when n     0, the expression    -    takes  the form  0
and has the value n, as we can easily see by differentiating it.
From the equations (1) and (2) several others, which
we shall make use of, can be easily deduced. For we find:
=a, -— 1', _, —-1;1     2rc    2 7r           2 7 
sin r - cos -r --      sin 2 -    0,                  (3)
n      ts               n
r=0                    r =O
~' —n -1= t' --- 1
cos rc )                          nn+   cos 2 r   =n in general  (4)
r=O                    r0-O
— n in the exceptional case,




65
finally:
r ='n   I            r —-n —  1
(sin r2)2     n - a-    cos 2r 2    E —n in general     (5)
7'_0                  )'=0
= 0 in the exceptional case.
27.  We will assume now:
X   a,, cos px -4- b,, sin px,
in which equation all integral numbers beginning with zero
must be successively put for p. If now q denotes a certain
number, we have:
X cos q x — ~a, cos (p - q) x -4- a, cos (p - q) x
4 = b,, sin (p +- q) x 4- b, sin (p - q) x,
and if we assign x successively the values 0, A, 2 A.... to
2
(n - 1) A, where A    —, and add the several resulting equations, all terms on the right side will be zero according to
the equations (1) and (2) with the exception of the sum of
the terms of the cosine, in which (p —q) A is equal to 2kn,
which will receive the factor n.  But as A    2=,  we have
for the remaining terms p +-  q =  k n or p -  q -  k n, hence
p —q + kn or =   - q + k n. Therefore denoting the value
of X, which corresponds to the value rA of x by X,.,, we
have:
r= t -1
ZXrA cosqA=2 a-. t + -4-  a.qk
X =O
=  2 — [4-  a. -- a n —q  a,,+ -i a2,, —,    a2,+q.]
But as X does not contain any coefficients whose index
is negative, we must take a_ =  O and get:
= L 1-1
XrA COS qA    2 [aq + a,-q + C,,,,+  a2n-q + a2,,+,  -...]-   (6)
r-=O
Here we have to consider two particular cases. For
when q =  0, we have a_-a,, a,_-,  -a,n+, etc. hence:
r-n-1
XA = n [ao +-,, + c,, 2+...,            (7)
and when n is an even number and q  -ln, a_,  is to be
omitted and a, unites with an_- etc., hence we have also in
this case:
5




66
r= In —
Z XrACOS 2nA  -n [a2CnI a+ C +..]         (8)?0-O
As:       Xsin qx    I       a, sin(psin (p -q) x
+-  bp cos (p - q) x - - bp cos (p -- q) x,
we find in a similar way:
— = HI -- 1
X,rA sin qA =   [bA  -...q + b,+, - be —q + ba+,,q...]. (9)
r =O
If we take now for n a sufficiently large number in proportion to the convergence of the series, so that we can neglect on the right side of the equations (6) to (9) all terms
except the first, we may determine by these equations the
coefficients of the cosines from q =  0 to q   i= n and the coefficients of the sines to q    n -  1, as a larger q gives
only a repetition of the former equations.  The larger we
take n, the more accurate shall we find the values of the
coefficients whose index is small, while those of a high index remain always inaccurate. For instance when n = 12
and q = 4, we have the equation:,yX cos 4 x -- 6 (a4 + - C +...),
hence the value of a, will be incorrect by the quantity as;
but if we had taken n= 24, this coefficient would be only
incorrect by a20.
From the above we find then the following equations:,. =,_  1
acp  --, X,A cos r p A,
r =0
n
with these exceptions, that for pO0 and p=  n we must take
instead of the factor 2
n
It is always of some advantage to take for n a number
divisible by 4, as in this case each quadrant is divided into
a certain number of parts and therefore the same values of
the sines and cosines return only with different signs. As
the cosines of angles, which are the complements to 360~,
are the same, we can then take the sum of the terms, whose
indices are the complements to 3600 and multiply it by the




67
cosine; but the terms of the sine, whose indices are the complements to 360~ must be subtracted' from  each other.  If
we denote then the sum of two such quantities, for instance
XA -+ X(-1,)A by XA, and the difference X  - X(-,_1)  by X,,
we have:                    r=
t   -  XA COS rpA,
n    -
r =0
r = (n —l)
b — 2    X,.i sin rp A.?1
Again denoting here the sum or the difference of two
terms of the cosine, whose indices are the complements to
1800, by  Xr, and X,., and the sum  or difference of two
terms of sines, whose indices are the complements to 180 0,
by X,, and X,,, we have:
r =21 n
a   -2 -S X,.A cos rpA, when p is an even number,  (10)
fl   ++
with the two exceptional cases mentioned before:
a,  -z    X, cos rpA, when p is an odd number,  (11)
r-=
r =  n-i
bp —    X,.A sin rpA, when p is an even number,  (12)
r =1
r=~8 — 1
bp ---      Xi,. sinrpA, when p is an odd number.   (13)
r =1
If for instance n is equal to 12, we find:
a =,    Xo         0    +  XX0 0  o5 
{1 -I~~ X 6   XX+X30cos30~+X6oCeoS 601,
a  Xo+Xcos60- X60cosgO-Xo 
+-q  q+          q-q         qetc.
=b   -i X30 sin 30 + X6  sin 60 -+ X
b2 = -- -t  -
bt2c. X3 0 sin 60+X60 sin 60,
etc.




68
28. If we wish to develop a periodical function up to a
certain multiple of the angle, it is necessary that as many
numerical values are known as we wish to determine coefficients. If then the given values are perfectly correct, we
shall find these coefficients as correct as theory admits, only
the less correct, the higher the index of the coefficient is
compared to the given number of values.  But in case that
the values of the function are the result of observations, it
is advisable in order to eliminate the errors of observation
to use as many observations as possible, therefore to use
many more observations than are necessary for determining the
coefficients. In this case these equations should be treated
according to the method of least squares; but one can easily
see, that this method furnishes the same equations for determining the coefficients as those given in No. 27.  We see
therefore that the values obtained by this method are indeed
the most probable values.
For if the n values X,, XA, X2A... X(-1) A are given,
we should have the following equations, supposing that the
function contains only the sines and cosines of the angle
itself:   0o   - X+a --   +a,,
O =-XA   +- ao -al,cos A  + b, sin A,
O  - X2A+- ao --, cos 2 A - b, sin 2 A,
O  - X(,,-l)A- ito0 -+-a cos(n-l1) A q- b sin(n-1)A,
and according to the method of least squares we should find
for the equations of the minimulnl, when [cos A] again denotes the sum of all the cosines of A, from A   O0 to A     n- 1,
the following:
enao +- [cos A] a,   + [sin A] b,    - [XA]  0 O
[cosA] a + C[cos A 2] a,  -+ [sin A. cos A] b, - [XA cos A] =0,  (14)
[sin A] ac, + [cosAl sinA] a, ~- [sin A'] b,   - [XA sin A] = 0.
But if we take into consideration the equations (3), (4)
and (5) in No. 26 we see. that these equations are reduced
to the following:
ao --- -[XA],
C, -— o [XA cos A],
b    2  XA sin A],




69
which entirely agree with those found in No. 27. What is
shown here for the three first coefficients, is of course true
for any number of them.
We can also find the probable error of an observation
and of a coefficient.  For if [v v] is the sum  of the squares
of the residual errors, which remain after substituting the
most probable values in the equations of condition, the probable error of one observation is
r-0.67449 e/ L3
and that of a= 1j- X
I/ n
r. I/2
1[cosA2]  1/n
a2 J/l[sinA2]    n/2
An  example  will be found in No. 6 of the seventh section.
Note. Consult Encke's Berliner Jahrbuch fiir 1857 pag. 334 and seq.
Leverrier gives in the Annales de l'Observatoire Impdrial, Tome I. another
method for determining the coefficients, which is also given by Encke in the
Jahrbuch for 1860 in a different form.




SPHERICAL ASTRONOMY.
FIRST SECTION.
THE CELESTIAL SPHERE AND ITS DIURNAL MOTION.
In spherical astronomy we consider the positions of the
stars projected on the celestial sphere, referring them by
spherical co-ordinates to certain great circles of the sphere.
Spherical astronomy teaches then the means, to determine the
positions of the stars with respect to these great circles and
the positions of these circles themselves with respect to each
other. We must therefore first make ourselves acquainted
with these great circles, whose planes are the fundamental
planes of the several systems of co-ordinates and with the
means, by which we may reduce the place of a heavenly
body given for one of these fundamental planes to another
system of co-ordinates.
Some of these co-ordinates are independent of the diurnal
motion of the sphere, but others are referred to planes which
do not participate in this motion. The places of the stars
therefore, when referred to one of the latter planes, must continually change and it will be important to study these changes and the phenomena produced by them. As the stars besides the diurnal motion common to all have also other, though
more slow motions, on account of which they change also
their positions with respect to those systems of co-ordinates,
which are independent of the diurnal motion, it is never sufficient, to know merely the place of a heavenly body but it
is also necessary to know the time, to which these places
correspond. We must therefore show, how the daily motion
either alone or combined with the motion of the sun is used
as a measure of time.




71
I. THE SEVERAL SYSTEMS OF GREAT CIRCLES OF THE
CELESTIAL SPHERE.
1. The stars appear projected on the concave surface
of a sphere, which on account of the rotatory motion of the
earth on her axis appears to revolve around us in the opposite direction namely from  east to west.  If we imagine
at any place on the surface of the earth a line drawn parallel to the axis of the earth, it will generate on account of
the rotatory motion of the earth the surface of a cylinder,
whose base is the parallel-circle of the place. But as the
distance of the stars may be regarded as infinite compared
to the diameter of the earth, this line remaining parallel to
itself will appear to pierce the celestial sphere always in the
same points as the axis of the earth. These points which
appear immoveable in the celestial sphere are called the Poles
of the celestial sphere or the Poles of the heavens, and the
one corresponding to the NorthPole of the earth, being therefore visible in the northern hemisphere of the earth is called
the North#Pole of the celestial sphere, while the opposite is
called the South-Pole.  If we now imagine a line parallel to
the equator of the earth, hence vertical to the former, it will
on account of the diurnal motion describe a plane, whose
intersection with the celestial sphere coincides with the great
circle, whose poles are the Poles of the heavens and which
is called the Equcator. Any straight line making an angle
different from 90" with the axis of the earth generates the
surface of a cone, which intersects the celestial sphere in two
small circles, parallel to the equator, whose distance from
the poles is equal to the angle between the generating line
and the axis. Such small circles are called Parallelcircles.
A plane tangent to the surface of the earth at any place
intersects the celestial sphere in a great circle, which separates the visible from  the invisible hemisphere and is called
the Horizon.  The inclination of the axis to this plane is
equal to the latitude of the place.. The straight line tangent to the meridian of a place generates by the rotation of
the earth the surface of a cone, which intersects the celestial sphere in two parallel circles, whose distance from the




72
nearest pole is equal to the latitude of the place and as the plane
of the horizon is revolved in such a manner, that it remains
always tangent to this cone, these two parallel circles must
include two zones, of which the one around the visible pole
remains always above the horizon of the place, while the
other never rises above it. All other stars outside of these
zones rise or set and move from east to west in a parallel
circle making in general an oblique angle with the horizon. A
line vertical to the plane of the horizon points to the highest
point of the visible hemisphere, which is called the Zenith, while
the point directly opposite below the horizon is called the Nadir. The point of intersection of this line with the celestial
sphere describes on account of the rotation a small circle,
whose distance from  the pole is equal to the co-latitude of
the place; hence all stars which are at this distance from
the pole pass through the zenith of the place. As the line
vertical to the horizon as well as the one drawn parallel to
the axis of the earth are in the plane of the meridian of
the place, this plane intersects the celestial sphere in a great
circle, passing through the poles of the heavens and through
the zenith and nadir, which is also called the Meridian. Every
star passes through this plane twice during a revolution of the
sphere. The part of the meridian from the visible pole through
the zenith to the invisible pole corresponds to the meridian of
the place on the terrestrial sphere, while the other half corresponds to the meridian of a place, whose longitude differs
1800 or 12 hours from  that of' the former.  When a star
passes over the first part of the Meridian, it is said to be
in its upper culmination, while when it passes over' the second part it is in its lower culmination.  Hence only those
stars are visible at their upper culmination, whose distance
from the invisible pole is greater than the latitude of the
place, while only those can be seen at their lower culmination, whose distance from the visible pole is less than the
latitude. The arc of the meridian between the pole and the
horizon is called the altitude of the pole and is equal to the
latitude of the place, while the arc between the equator and
the horizon is called the altitude of the equator. One is the
complement of the other to 90 degrees.




73
2. TIn order to define the position of a star on the celestial sphere, we make use of spherical co-ordinates. We
imagine a great circle drawn through the star and the zenith
and hence vertical to the horizon. If we now take the point
of intersection of this great circle with the horizon and count
the number of degrees from this point upwards to the star
and also the number of degrees of the horizon from this point
to the meridian, the position of the star is defined. The great
circle passing through the star and the zenith is called the
vertical/ circle of the star; the arc of this circle between the
horizon and the star is called the altitude, while the arc between
the vertical-circle and the meridian is the azimuth of the star.
The  latter angle  is reckoned from  the point South through
West, North  etc. from  00 to 3600.  Instead of the altitude
of a star its zenith-distance is often used, which is the arc
of the vertical circle between the star and the zenith, hence
equal to the complement of the altitude. Small circles whose
plane is parallel to the horizon are called almucantars.
Instead of using spherical co-ordinates we may also define the position of a star by rectangular co-ordinates, referred  to a system  of axes, of which  that of z is vertical to
the plane of the horizon, while the axes of y and x are situated in its plane, the  axis of x being directed to the origin
of the azimuths, and the positive axis of y towards the azimuth 900 or the point West.  Denoting the azimuth by A,
the altitude by h, we have:
x = cos h cos A, y- cos h sin A, z — = sin h.
Note. For observing these spherical co-ordinates an instrument perfectly
corresponding to them is used, the altitude- and azimuth-instrument. This
consists in its essential parts of a horizontal divided circle, resting on three
screws, by which it can be levelled with the aid of a spirit-level. This circle
represents the plane of the horizon. In its centre stands a vertical column,
which therefore points to the zenith, supporting another circle, which is parallel to the column and hence vertical to the horizon. Round the centre of
this second circle a telescope is moving connected with an index, by which
the direction of the telescope can be measured. The vertical column, which
moves with the vertical circle and the telescope, carries around with it another index, by which one can read its position on the horizontal circle. If.then the points of the two circles, corresponding to the zenith and the point
South,; are known, the azimuth and zenith-distance of any star towards which
the instrument is directed, may be determined.




74
Besides this instrument there are others by which one can observe only
altitudes. These are called altimeters, while instruments, by which azimuths
alone are measured, are called theodolites.
3.  The azimuth and the altitude of a star change on
account of the rotation of the earth and are also at the same
instant different for different places on the earth.  But as it
is necessary for certain purposes to give the places of the
stars by co-ordinates which are the same for different places
and do not depend on the diurnal motion, we must refer the
stars to some great circles, which remain fixed in the celestial sphere. If we lay a great circle through the pole and
the star, the are contained between the star and the equator
is called the declination and the arc between the star and
the pole the polar-distance of the star. The great circle itself
is called the declination-circle of the star. The declination
is positive, when the star is north of the equator and negative, -when it is south of the equator.  The declination
and the polar-distance are the complements of each other.
They correspond to the altitude and the zenith-distance in
the first system of co- ordinates.
The arc of the equator between the declination-circle of
the star and the meridian, or the angle at the pole measured
by it, is called the hour-angle of the star. It is used as the
second co-ordinate and is reckoned in the direction of the
apparent motion of the sphere from east to west from 00
to 360".
The declination-circles correspond to the meridians on
the terrestrial globe and it is evident, that when a star is
on the meridian of a place, it has at the same moment at a
place, whose longitude east is equal to k, the hour-angle k
and in general, when at a certain place a star has the hourangle t, it has at the same instant at another place, whose
longitude is k (positive when east, negative when west) the
hour-angle t — + k.
Instead of using the two spherical co-ordinates, the declination and the hour-angle, we may again introduce rectangular co-ordinates if we refer the place of the star to three
axes, of which the positive axis of z is directed to the Northpole, while the axes of x and y are situated in the plane of




75
the equator, the positive axis of x being directed to the meridian or the origin of the hour-angles while the positive
axis of y is directed towards the hour-angle 900. Denoting
then the declination by S, the hour-angle by t, we have:
x   COs s Cos t, y'  Cos 8 sin t, z'- sin &.
Note. Corresponding to this system of co-ordinates we have a second
class of instruments, which are called parallactic instruments or equatorials.
Here the circle, which in the first class of instruments is parallel to the
horizon, is parallel to the equator, so that the vertical column is parallel to
the axis of the earth. The circle parallel to this column represents therefore
a declination circle. If the points of the circles, corresponding to the meridian, being the origin of the hour-angles, and the pole, are known, the
hour-angle and the declination of a star may be determined by such an instrument.
4.  In  this latter system  of co-ordinates one of them,
the declination, does not change while the hour-angle increases proportional to the time and differs in the same moment at different places on the earth according to the difference  of longitude.  In order to have also the second coordinate invariable, one has chosen a fixed point of the equator
as origin, namely the point in which the equator is intersected
by the great circle, which the centre of the sun seen from
the centre of the earth appears to describe among the stars.
This great circle is called the ecliptic and its inclination to
the equator, which is about 231 degrees, the obliquity of the
ecliptic. The points of intersection between equator and ecliptic  are  called the points of the equinoxes, one that of the
vernal the other that of the autumnal equinox, because day
and night are of equal length all over the earth, when the
sun on the 21st of March and on the 23d of September reaches
those points *). The points of the ecliptic at the distance of
90 degrees from the points of the equinoxes are called solstitial points.
The new  co-ordinate, which is reckoned in the equator
from the point of the vernal equinox, is called the rightascension of the star. It is reckoned from 00 to 3600 from
*) For as the sun is then on the equator, and as equator and horizon
divide each other into equal parts, the sun must remain as long below as
above the horizon,




76
west to east or opposite to the direction of the diurnal motion.
Instead of using the spherical co-ordinates, declination and
right-ascension, we can again introduce rectangular co-ordinates, referring the place of the star to three vertical axes,
of which the positive axis of z is directed towards the Northpole, while the axes of x and y are situated in the plane of
the equator, the positive axis of x being directed towards
the origin of the right-ascensions, the positive axis of y to the
point, whose right-ascension is 900. Denoting then the rightascension by cr, we have:
x" - cos   a    cos a,   / cos  sin a, z"   sin.
The co-ordinates ca and 6 are constant for any star. In
order to find from them the place of a star on the apparent
celestial sphere at any moment, it is necessary to know  the
position of the point of the vernal equinox with regard to
the meridian of the place at that moment, or the hour-angle
of the point of the equinox, which is called the sidereal time,
while the time of the revolution of the celestial sphere is
called a sidereal day and is divided into 24 sidereal hours.
It is Oh sidereal time at any place or the sidereal day commences when the point of the vernal equinox crosses the
meridian, it is 1h when its hour-angle is 15" or Ih etc. For
this reason the equator is divided not only in 3600 but also
into  24 hours.  Denoting the sidereal time by O, we have
always:                    o — =a,
hence   t   - a.
If therefore for instance the right-ascension of a star is
190" 20' and the sidereal time is 4h, we find t-   229" 40' or
1300 20' east.
From  the equation for t follows    -=  a when t =  0.
Therefore every star comes in the meridian or is culminating
at the sidereal time equal to its right-ascension expressed in
time.  Hence when the right-ascension  of a star which is
culminating, is known, the sidereal time at that instant is
also known by it*).
*) The problem to convert an arc into time occurs very often.
If we have to convert an arc into time, we must multiply by 15 and
multiply the remainder of the degrees, minutes and seconds by 4, in order to
convert them into minutes and seconds of time.




77
If the sidereal time at any  place is  0, at the same instant the sidereal time at another place, whose difference of
longitude is hk   must be  0 -+- k  where  k is to  be taken  positive or negative if the second place is East or West of the
first place.
Note. The co-ordinates of the third system can be found by instruments
of the second class, if the sidereal time is known.  In one case these coordinates may be even found by instruments of the first class, namely when
the star is crossing the meridian, for then the right-ascension is determined
by the time of the meridian-passage and the declination by observing the
meridian-altitude of the star, if the latitude of the place is known. For such
observations a meridian-circle is used. If such an instrument is not used for
measuring altitudes but merely for observing the times of the meridian-passages of the stars, if it is therefore a mere azimuth-instrument mounted in
the meridian, it is called a transit-instrument.  If we observe by such an
instrument and a good sidereal clock the times of the meridian-passages we
get thus the differences of the right-ascensions of the stars.  But as the
point from which the right-ascensions are reckoned cannot be observed itself,
it is more difficult, to find the absolute right-ascensions of the stars.
5.   Besides  these  systems  of co -ordinates  a  fourth  is
used, whose fundamental plane is the ecliptic. Great circles
which pass through the poles of the ecliptic and therefore
are vertical to it, are called circles of latitude and the arc
of such a circle between the star and the ecliptic is called
the  latitude  of the star.  It is positive or negative if the star
is North  or South  of the  ecliptic.   The  other co-ordinate,
the longitude, is reckoned in the ecliptic and is the arc between  the circle  of latitude  of the star and the point of the
vernal equinox.  It is reckoned  from  0~ to  3600 in the  same
direction  as the  right-ascension  or  contrary  to  the diurnal
Thus we have 2390 18' 46".75
15 h, 4 X 14 +-  1 minutes, 4 X 3 -- 3 seconds and 0 s. 117
15h 571n 15 s. 117.
If on the contrary we have to convert a quantity expressed in time into
an arc, we must multiply the hours by 15, but divide the minutes and seconds by 4 in order to convert them  into degrees and minutes of arc. The
remainders must again be multiplied by 15.
Thus we have 15h 5711 15s. 117
- 225 ~- 14 degrees, 15 +- 3 minutes and 46.75 seconds
-   390 18' 46". 75.




78
motion of the celestial sphere*). The circle of latitude whose
longitude is zero, is called the colure of the equinoxes and
that, whose longitude is 90", is the colure of the solstices.
The arc of this colure between the equator and the ecliptic,
likewise the arc between the pole of the equator and that
of the ecliptic is equal to the obliquity  of the ecliptic.
The longitude shall always be denoted by i, the latitude
by fi and the obliquity of the ecliptic by e.
If we express  again the spherical co-ordinates (3 and  X
by  rectangular  co-ordinates, referred to  three axes vertical
to  each  other, of which the positive axis of z is vertical to
the ecliptic  and  directed  to  the north-pole  of it, while the
axes of x and y are situated in the plane of the ecliptic, the
positive axis of x being directed to the point of the vernal
equinox, the positive axis of y to the 90th degree of longitude,
we have:
x"cos  cos 2,  "'- cos  sin, z"'- sin fl.
These  co-ordinates  are  never found by  direct observations,
but are only deduced by  computation from  the other systems
of co - ordinates.
Note. As the motion of the sun is merely apparent and the earth really
moving round the sun, it is expedient, to define the meaning of the circles
introduced above also for this case. The centre of the earth moves round
the sun in a plane, which passes through the centre of the sun and intersects the celestial sphere in a great circle called the ecliptic. Hence the longitude of the earth seen from the sun differs always 1800 from that of the
sun seen from the earth. The axis of the earth makes an angle of 661 o
with this plane and as it remains parallel while the earth is revolving round
the sun it describes in the course of a year the surface of an oblique cylinder, whose base is the orbit of the earth. But on account of the infinite
distance of the celestial sphere the axis appears in these different positions
to intersect the sphere in the same two points, whose distance from the poles
of the ecliptic is 232 O. Likewise the equator is carried around the sun parallel to itself and the line of intersection between the equator and the plane
of the ecliptic, although remaining always parallel, changes its position in
the course of the year by the entire diameter of the earth's orbit.  But
the intersections of the equator of the earth with the celestial sphere in all the
different positions to which it is carried appear to coincide on account of the
*) The longitudes of the stars are often given in signs, each of which
has 30. Thus the longitude 6 signs 15 degrees is -  195 o.




79
infinite distance of the stars with the great circle, whose poles are the poles
of the heavens and all the lines of intersections between the plane of the
equator and that of the ecliptic are directed towards the point of intersection
between the two great circles of the equator and the ecliptic.
II. THE TRANSFORMATION OF THE DIFFERENT SYSTEMS OF
CO - ORDINATES.
6.  In  order to find from  the azimuth and altitude of
a star its declination  and hour-angle, we must revolve the
axis of z in the first system of co-ordinates in the plane of
x and z from the positive side of the axis of x to the positive
side of the axis of z through the angle 90- (p (where (p
designates the latitude), as the axes of y of both  systems
coincide. We have therefore according to formula (l a) for
the transformation  of co-ordinates, or according to the formulae of spherical trigonometry in the triangle formed by the
zenith, the pole and the star*):
sin   = sin T sin   cos - cos  1cos h cos A
cos a sin t -- cos hbsin A1
cos                  + cos t = sin h cos? ~ cos h sin  cos A.
In order to render the formulae more convenient for logarithmic computation, we will put:
sin h -- -m cos 11
cos h cos A - m sin M,
and find then:
sin S = m sin (9 - M)
cos a sin t -- cos h sin A
cos a cos t -- m cos (- -MA).
These formulae give the unknown quantities without any
ambiguity. For as all parts are found by the sine and cosine, there can be no doubt about the quadrant, in which they
lie, if proper attention is paid to the signs. The auxiliary
angles, which are introduced for the transformation of such
formulae, have always a geometrical meaning, which in each
case may be easily  discovered.   For the geometrical construction amounts to this, that the oblique spherical triangle
*) The three sides of this triangle are respectively 90~ - h, 90~ -  and
90~ -9 and the opposite angles t, 180 —A and the angle at the star.




80
is either divided into  two  right-angled  triangles or by the
addition of a right-angled triangle is transformed into one.
In the present case we must draw an arc of a great circle
from the star perpendicular to the opposite side 90- -,
and as we have:
tang h - cos A cotang M,
it follows from  the third  of the formulae (10) in No. 8 of
the introduction, that M  is the arc between the zenith and the
perpendicular arc, while m according to the first of the formulae (10) is the cosine of this perpendicular arc itself, since
we have:
sin h --- cos P coS A,
if we denote the perpendicular arc by P.
We will suppose, that we have given:
IT  520 30' 16".0, h = 160 11' 44". 0 and A - 2020 4' 15".5.
Then we have to make the following computation:
cos A 9.9669481,   n m sin AM 9.9493620,,
cos h 9.9824139   m cos M9.4454744
sin A 9.5749045,%    M=- 72 035'54". 61
sin AM 9.9796542,,
p      --   125~ 6' 10". 61
sin (9 —M) 9.9128171  cos S sin t 9.5573184;,   sin A 9.8825249
m    9.9697078  cos S cos t 9.7294114,,  cos a 9.8104999
cos ( — M) 9.7597036,    t — 213 562.22    =-+-4943 46.00
cos t 9.9189115,,.
7. More frequently occurs the reverse problem, to convert the hour-angle and declination  of a star into its azimuth and altitude.  In this case we have again according to
formula (1) for the transformation of co-ordinates:
sin h   sin  sin a - cos oT cos 8 cos t
cos h sinA -- cos A sin t
cos hcos cA = - cos q' sin S - sin n  cos S cos t,
which may be reduced to a more convenient form by introducing an auxiliary angle. For if we take:
cos 8 cos t -- m cos M
sin   — m sin M
we have:
sin h -- m cos (9 - jM)
cos h sin A — cos 8 sin t
cos h cosA - m sin (So - M)




cos l1/tang t
or:                      tang A       (- - -)
cos A
tang h -    COS      *).
tang (9 —121)
When the zenith distance alone is to be found, the following formulae are convenient.  From  the first formula for
sin h we find:
cos z = cos ( -   )- 2 cos 9 cos  sin. t',
or:' z- si n' (9 -   - -  cos )   cos  Asin' 
If we take now:
n  sin4   (9- b)
m -   1/cos c9 cos,
we have:               sin  z2   n2 (q- 2 sin 2 
or taking                     sin 2 t= tang;
sin  z=
Cos A
If sin X should be greater than  cos i, it is more convenient to use the following formula:
sin 2 z —s- sin  t.
sin i
In the formula by which n is found, we must use cp- -,
if the star culminates south of the  zenith, but 83-  p if the
star culminates north of the zenith, as will be afterwards
shown.
Applying  Gauss's  formulae  to  the triangle between the
star, the zenith and the pole, and designating the angle  at
the star by p, we find:
cos 1. sin  (A - p) = sin I4 t  sin 4 (p + AY)
cos i z. cos 2(A -p) -  os 1 t. Cos  (9 — )
sin - z. sin 2 (A +p)   sin I t cos. (9 + 3)
sin  z.              cos -  -- o t   -s 3).
If the azimuth should be reckoned from  the point North,
as it is done sometimes for the polar star, we must introduce
180- A instead' of A in these formulae and obtain now:
COS z. Sin  (p +- A) - cos 4- t. cos 4 (3 —  )
cos } z. cos  (p+ A)   sin 2 t. sin - (-T95)
sin I z. sin  (p- A) = cos - t. sin 2 (3-9-)
sill i z. cos  (p- A)   sin -E t. cos 4  ( +- 9).
~) As the azimuth is always on the same side of the mleridian with the
hour angle, these last formulae leave no doubt as to the quadrant in which it lies.
6




82
Frequently the case occurs, that these computations must
be made very often for the same latitude, when it is desirable
to construct tables for facilitating these computations *). In
this case the following transformation may be used. We had:
(a)           sin h  sin 99sin  il6- cos a  cos 6 cos t
(b)     cos AI sin A   cos 6 sin t
(c)     cos h cos A  -    cos 99 sin 6 +- sin 99 cos  cos t.
If we designate now by Ao and A, those values of A
and A, which substituted in the above equation make h equal
to zero, we have:
(d)              0 - sin 9 sin 6o - cos 9 cos 8o cos t
(e)         sin A0 = cos 0o sin t
(f)          cos Ao = - cos 99 sin 6o -- sin  cos 6 cost.
Multiplying now  (f) by cos T  and  subtracting  from  it
equation (d) after having multiplied it by sin I, further multiplying  equation (f) by sin (p and adding to it equation (d),
after multiplying it by  cos (, we find:
cos Ao cos 99 = - sin 8O
Cos Ao sin -- = cos o8 cos t
sin A  = cos o6 sin t.
Taking then:
sin 9 - sin y cos B
cos 99 cos t - sin y sin B
cos T sin t  cos y,
we find  from  the equation (d) the following:
0 = sin y sin (86 +- B)
or:                            30 - -B
and from (a):
sin Al = sin y sin (8 + B).
Then  subtracting from  the product of equations (b) and
(f) the product of the equations (c) and (e) we get:
cos h sin (A -A 0) -= cos 99 sin t sin (8- So) =cos y sin (+ - B)
and likewise adding to the product of the equations (c) and
(f) the product of the equations (b) and (e) and that of the
equations (a) and (d):
cos h cos (A -A-)   cos 8 cos 80 sin t2 -+ sin S sin 30 +- cos e Cos 80 os tC
-cos (8 — S)  cos (+- B).
*) For instance if one has to set an altitude- and azimuth instrument
at objects, whose place is given by their right ascension and declination. Then
one must first compute the hour angle fiom the right ascension and the sidereal time.




83
Hence the complete system  of formulae is as follows:
sin  -- sin y cos B 
cos T cos t - sin y sinl B   (1)
cos 99 sin t   cos y 
sinB=-cosAO cosT )
cos B    cos t  cos AO sin  (2)
cos B sin t = sin AO
sin h s  sin? sin ( + - B) 
cos / cos (A- A) -- cos (S + B)      (3).
cos h sinl (A - A) = cos y sin ( - +B) )
These formulae by taking  D — sin 7, C-  cos 7 and
A — A- u are changed into the following:
tang B   cotg T cos t
tang Ao sin R tang t
sin h - D sin (B -+  )
tang u =-C tang (B - S)
A — o +t- 
where D and C are the sine and cosine of an angle 7, which
is found from the following equation "):
cotang y - sin B tang t - cotang p sin Ao.
These are the formulae given by Gauss in,,Schumacher's
Hiilfstafeln herausgegeben von Warnstorff pag. 135."  If now
the quantities D, C,- B and A, are brought into tables whose
argument is t, the computation of the altitude and the azimuth from the hour angle and the declination is reduced to
the computation of the following simple formulae:
sin h - D sin (B +- 8)
tang  = C tang (B +- )
A  - Ao -+-u.
Such tables for the latitude of the observatory at Altona
have been published in Warnstorff's collection of tables quoted
above. It is of course only necessary to extend these tables
from  t =  0 to  t   6 L   For it follows from  the equation
tang A,     sin q tang i, that A, lies always in the same quadrant as t, that therefore to the hour angle 12', — t belongs the
azimuth 1800- A. Furthermore it follows from  the equations
for B, that this angle becomes negative, when t:> 6h or > 900,
that therefore if the hour angle is 12' —t the value — B must
be used.  The quantities
*) For we have accotiding to the formunlae (2)
cotang 9R sin A0 = sin B tang t.
g*




84
C=cos cp sin t and D =  /sin   + Cos 2cost2
are not changed if 1800 - t instead of t is substituted in these
expressions.  When t lies between 12 h and 24h, the computation must be carried through with the complement of t to
24h' and afterwards instead  of the resulting  value of A its
complement to 360" must be taken.
It is easy to find the geometrical meaning of the auxiliary angles.  As,, represents that value of (), which substituted in the first of the original equations makes it equal
to zero,'0 is the declination of that point, in which the declination circle of the star intersects the horizon; likewise is
Fig. A.,0 the azimuth of this point. F'urthermore as we have B    -  0, B -+- i
is the are SF Fig. 1 *) of the decliz   nation circle extended to the horizon.
In  the  right angled  triangle  F O If,
which is formed by the horizon, the
A  equator and the side FK — B, we have
_____    daccording to the sixth of the f6rmu/  ae (10) of the introduction, because
a          Bp  X   the angle at 0 is. equal to 900 -(:
sin T - cos B sin 0 F-C.
But as we have also sin (p -D cos B, we see, that D is
the sine of the angle OFK, therefore C its cosine.  At last
we easily see that FIH is equal to A,, and FG equal to iu.
We can find therefore the above formulae from the three
right angled triangles PFH, OFK and SFG.  The first triangle gives:
tang AO -tang t sin T,
the second:
tang B = cotang T cos t
cotang y = sin B tang t = cotg T sinl A0,
and the third:
sin h - sin y sin (B +-`)
tang is = cos y tang (B ~- A).
The same auxiliary  quantities may be used for solving
the inverse problem, given in No. 6, to find the hour angle
*) In this figure P is the pole, Z the zenith, OH the horizon, OA the
equator, and S the star.




85
and the declination of a star from its altitude and azimuth.
For we have in the right angled triangle SKL, designating
LG by B, L K by u, AL by A0 and the cosine and sine of
the angle SLK by C and D:
C tang (z - B) = tang u
D sin ( - B) -= sin A'
and     t- Ao A  i,
where now:
tang B = cotang T9 cos A
tanag Ao - sin p tang A
and where D and C are the sine and cosine of an angle i,
which is found by the equation:
cotang y - sin B tang A.
We use therefore for computing the auxiliary quantities
the same formulae as before only with this difference, that
in these A  occurs in the place of' t; we can use therefore
~also the same tables as before, taking as argument the azimuth converted into time.
8.  The cotangent of the angle;/, which Gauss denotes
by E, can be used to compute the angle at the star in the
triangle between the pole, the zenith andcl the star. This angle
between the vertical circle and the declination circle, which
is called the parallactic ciangle is often made use of., If' we
have tables, such as spoken of before, which give also the
angle E, we find the parallactic angle, which shall be denoted by p, from the following simple formula:
E
tang p-cos(B-)
cos (B q-)
as is easily seen, if the fifth of the formulae (10) in No. 8
of the introduction is applied to the right angled triangle SGF
Fig. 1. But if one has no such tables, the following formulae
which are easily deduced from the triangle S P Z can be used:
cos Ih sin p -cos 5o sin t
cos h cos p   cos 8 sin p - sin 8 cos 9v cos t,
or taking:
cos 50 cos t = n sin N
sin   n cos N,
the following formulae, which are more convenient for logarithmic computation:
cos h sinp - cos q sin t
cos h cos p = n cos (8+- N).




86
The parallactic angle is used, if we wish to compute
the effect which  small increments  of the azimuth and altitude produce in the  declination and the hour angle.  For
we have, applying to the triangle between the pole, the zenith and the star the first and third of the formulae (9) in
No. 11 of the introduction:
d8-cos pch -f- cos td T+q- cos h sin p. dA
cos dt = - sinp dh-+- sin t sin S. dp + cos h cos p. dA
and likewise:
dh - cos pdS  cos A dc  - cos Ssinp. dt
cos 7 dA -- sill p d - sin A sin h d- + cos a cos p dt.
9.  In  order to  convert the right ascension and declination of a star into its latitude and longitude, we must revolve the axis of z" *) in the plane of y"z" through the angle
E equal to the obliquity of the ecliptic in the direction from
the positive axis of y" towards the positive axis of z". As the
axes of x" and x"' of the two systems coincide, we find according to the formulae (la) in No. 1 of the introduction:
cos  Cos    e n COS C cos a
cos fi sin 2 = cos  sin a cos e +- sin a sin ~
sin  = - cos S sin a sin E -+ sin S cos E.
These formulae may be also  derived  from  the triangle
between the pole of the equator, the pole of the ecliptic and
the star, whose three sides are 90"-, 900,8 and E and
the opposite angles respectively  90~ —?, 900-t- c and the
angle at the star.
In order to render these formulae convenient for logarithmic computation, we  introduce  the following auxiliary
quantities:
M sin N — sin            a)
Al] cos N - cos S sin a,
by which the three original equations are changed into the
following:
cos  cos    cos i   cos a
cost  sin     l lcos (N- e)
sin  — 11  sin (N- e),
or if we find all quantities by their tangents and substitute
for M  its value             cos S sin a
cos N
*) See No. 4 of this Section.




87
we get as final equations:
tang N   tang S 
sin a
tang        cos (N —  )         (b).
- cos N    tang a
tang   -— tang (N- ) sin 2
The original formulae give us a and 3 without any ambiguity; but if we use the formulae (b) we may be in doubt
as to  the  quadrant in which we must take 2.  However it
follows from the equation:
cns 8 cos        s  cos a
that 2 must be taken in that quadrant, which corresponds to
the sign  of tang 2 and  at the same time satisfies the condition, that cos ce and cos X lmust have the same sign.
As a check of the computation the following equation
may be used:
cos (N — a)   cos i sin 2
cos N      cos 3 sin a'
which we find by dividing the two equations:
cos i sin 2 -=  Icos (N- a)
cos a sin a = — Mcos 1N.
The geometrical meaning of the auxiliary angles is easily
found. N is the angle which the great circle passing through
the star and the point of the vernal equinox makes with the
equator, and M  is the sine of this arc.
Example. If we have:
a  60 33' 29". 30  S --- 160 22' 35". 45
= 230 27' 31". 72,
the computation of the formulae (b) and (c) stands as follows:
cos 9 9.9820131         tang a    9. 0605604
tangS  9.4681562,        cos(N —     9 0292017,,
cos.N
sin a  9. 0577093           2 — 3590 17' 43". 91
N — - 68 o45' 41". 88
N +23 2- 607 3541.72   tang (N —s) 1. 4114653
e-4- 323 27 31. 72
sin 2. 0897293,,
N-e —92 13 13.60
cos (N —) 8. 5882086,,        1s  -  9.9791948
cos N 9. 5590069
cosfi sin 2 = 8. 0689241,,
cos S sin a - 9. 0397224
9  0292017,,




88
If we  apply  Gauss's formulae  to  the triangle between
the pole of the equator, the pole of the ecliptic and the star
and denote the angle at the star by 90~ —E, we find:
sin (450 --,) sin - (E —A) =  cos (450 4-  a) sin [450 -    i(e+t)]
sin (45 -+-) cos (E-A)   sin (45 +4a) cos[45   A (sE - )
cos (45 — 8) sin  (E+ )-   = sin (45 -4- a) sin [45 --   (E -)]
cos(45 — fi) cos  (E+ 2) - cos(45             45  -      + co s --
These formulae are especially convenient, if we wish to find
besides /? and A also the angle 900 — E.
Note. Encke has given in the Berlin Jahrbuch for 1831 tables, which
are very convenient for an approximate computation of the longitude and latitude from the right ascension and declination. The formulae on which they
are based are deduced by the same transformation of the three fundamental
equations in No. 9 as that used in No. 7 of this section for equations of a
similar form. More accurate tables have been given in the Jahrbuch for 1856.
10. The formulae for the inverse problem, to convert
the  longitude  and  latitude of a star into its right ascension
and declination, are similar.  We get in this case from  the
formulae (1) for the transformation  of co-ordinates  or also
from the same spherical triangle as before:
cos a cos a = cos   cos A
cos A sin a = cos fi sin X cos  -- sin p sin E
sin A = cos, sin A sin E + sin, cos E.
We can find these equations also  by exchanging in the
three  original equations in No. 9 fi and X for 3 and a and
conversely and taking the angle E negative. In the same way
we can deduce from  the formulae (b) the following:
tang -t
sin A
cos (N - ~)
tang a     os       tang 
coS N
tang    a  =  tang (N +- E) sin a
and from (c) the following formula, which may be used as
a check:
cos (N +- e)   cos S sin a
cos N      cos  sin 2
Here is N  the angle, which the great circle passing through
the  star and  the  point of the  vernal  equinox  makes with
the ecliptic.
Finally Gauss's equations give in this case:




89
sin (450 -    ) sin - (E+ a-) =  sin (45~ + — 2) sin [45 " -_  (a +)]
sin (45 --    cos  (E) c os (E - a) — cos(45 -+ -: 2) cos [45 -4- ( — )]
cos (45 - 45c) sin J (E — a) = cos(45 -+ 1 A) sin [45 -   (e —)]
cos (45 - -) cos  (E - a) = sin (45 — 2- i) cos [45 —' (E + -)].
Note. As the sun is always in the ecliptic, the formulae become more
simple in this case. If we designate the longitude of the sun by L, its right
ascension and declination by A and D, we find:
tang A    tang L cos e
sin D -sin L sin E
or:         tang D - tang E sin A.
11.  The  angle  at the star in the triangle between  the
pole  of the  equator, the  pole  of the  ecliptic and the star,
or the angle at the star between its circle of declination and
its circle of latitude, is found  at the same time with A2 and f3,
if Gauss's equations are used for computing them, as, denoting  this  angle  by  A, we  have   j =-  90 -- E.  But if we
wish  to  find  this  angle  without computing  those formulae,
we can obtain it from the following equations:
cos i sin y -  cos a sin e
cosfCos fi C os E Cos s   - sin e sin A sin a
or:
cos d sin? —  cos A sin e
cos S cos 7  cos e cos- sin  sin fi sin i,
or taking:
coS 8 --   coS Ml
sin e sin a --   sin At
or:
coS e — =' COS N
sin e sin 2-n sin I
we may find it from  the equations:
cos fi sin i7 - cos a sin e
cos # cos? =- m coos(A — a)
or:
cos  sin? =- cos A sin E
cos a cos 7   n cos (N+- P).
The angle ii is used to find the effect, which small increments of i) and /3 have on  a and  a  and  conversely.  For
we get by  applying  the first and third  of the formulae (11)
in No. 9 of the introduction to the triangle used before:
df-= cos 0? d5 - cos a sin 7. da - sin 2 de
cos fi c3d= sin 7 d   -4- cos A cos oy. dca - cos 2 sinfclde,
and also:
dS=     cos   d,8 +- cos/  sin 7. d -+ sin a de
cos 8da  -  sin 0? df + cos o  cos 0. dA - cos a sin 5. do.




90
Note.  The supposition made above that the centre of the sun is always
moving in the ecliptic is not rigidly true, as the sun on account of the perturbations produced by the planets has generally a small latitude either north
or south, which however never exceeds one second of arc. Having therefore
computed right ascension and declination by the formulae given in the note
to No. 10, we must correct them  still for this latitude.  If we designate it
by dB, we have the differential formulae:
sin'7
clA --      -.i d B
cos D
dD       cos 7. dB,
or if we substitute the values of sin y and cos 7 from  the formulae for
cos [f cos s and cos a cos i7 after having taken i3 =0, we find:
cos DdA   - - cos A sin E. dB,
dD... dB.
cos                    and azi ths
12.  The formulae   for converting altitudes and azimuths
into  longitudes and latitudes may be briefly  stated, as they
are not made use of:
We have first the co-ordinates with respect to the plane
of the horizon:
x = Cos A cos h,
y =  sin A cos h,
z — sin h.
If we revolve the axis of x in the plane of x and z through
the angle 90  — q  in the direction towards the positive side
of the axis of z, we find the new  co-ordinates:
x'    x sin  +- z cos 9,
z -z sin R -- x cos q.
If we then revolve the axis of x' in the plane of x' and
y', which is the plane of the equator, through the angle  0,
so that the axis of x" is directed towards the point of' the
vernal equinox, we find the following formulae, observing that
the positive side of y" must be directed towards a point whose
right ascension is 90' and that the right ascensions and hour
angle  are reckoned in an opposite direction:
x"    x' cos 0 +y' sin 0
-y"- = y' cos 0 - x' sin 0
If we finally  revolve  the  axis of y" in the plane of y"
and z" through the angle E in the direction towards the positive side of the axis of z", we find:




91
Xrl J   xy~ 
y I,     COS E +-4 z" sin ~
z =-   sinl E -+Z COS E,
and as we also have:
xi I  COS  cos  X
ya  COS Sill a
o"s' cill s,
we can express X and fi directly by A, h  (p, O and E by
eliminating x', y', z' as well as x", y", z".
III. THE DIURNAL MOTION AS A MEASURE OF TIME.
SIDEREAL, APPARENT AND MEAN SOLAR TIME.
13.  The diurnal revolution of the celestial sphere or
rather that of the earth on her axis being perfectly uniform,
it serves as a measure of time. The time of an entire revolution of the earth on its, axis or the time between two successive culminations of the same fixed point of the celestial
sphere, is called a sidereal day. It is reckoned from the moment the point of the vernal equinox is crossing the meridian, when it is Oih sidereal time. Likewise it is lh, 2h, 3h etc.
sidereal time, when the hour angle of the point of the equinox
is ih, 2h, 3h etc. or when the point of the equator whose
right ascension is 1h, 2h', 3h etc. or 15", 30", 450 etc. is on
the meridian.
We shall see hereafter, that the two points of the equinoxes are not fixed points of the celestial sphere, but that
they are moving though slowly on the ecliptic. This motion
is rather the result of two motions, of which one is proportional to the time and therefore unites with the diurnal motion of the sphere, while the other is periodical. This latter,motion has the effect, that the hour angle of' the Foint of
the vernal equinox does not increase uniformly, hence that
sidereal time is not strictly uniform. But this want of uniformity is exceedingly small as it amounts during a period of
nineteen years only to =I- s.
14.  The sun being on the 21th of March at the vernal
equinox it crosses the meridian on that day at nearly Oh si



92
dereal time.  But at it moves in the ecliptic and is at the
point of the autumnal equinox on the 23 1 of September, having the right ascension 12 h, it culminates on this day at
nearly 121' sidereal time.  Thus the time of the culmination
of the sun moves in the course of a year through all hours
of a sidereal day and on account of this inconvenience the
sidereal time would not suit the purposes of society, hence
the motion of the sun is used as the measure of civil time.
The hour angle of the sun is called the apparent solar time
and the time between two successive culminations of the sun
an apparent solar day.  It is 0O1 apparent time when the
centre of the sun passes over the meridian. But as the right
ascension of the sun does not increase uniformly, this time
is also not uniform.  There are two causes which produce
this variable increase of the sun's right ascension, namely the
obliquity of the ecliptic and the variable motion of the sun
in the ecliptic.  This annual motion of the sun is only apparent and produced by the motion of the earth, which according to Kepler's laws moves in an ellipse, whose focus is
occupied by the sun, and in such a manner that the line
joining the centre of the earth and that of the sun (the radius vector of the earth) describes equal areas in equal tiles.
If we denote the length of the sidereal year, in which the earth
performs an entire revolution in her orbit, by r we find for
the areal velocity F of the earth    /        as the area of
the ellipse is eqal to  2l-e2,  or if we take the  emi-a of
the ellipse is equal to a2r I/ 1-e, or if we take the semimajor axis of the ellipse equal to unity and introduce instead
of e the angle of excentricity cp, given by the equation e -sin p,
we find:
7r COS 90.
F -- f cos -
If we call the time, when the earth is nearest to the
sun or at the perihelion T, we find for any other time t
the sector, which the radius vector has described since the time
of the perihelion passage equal to F(t - T). But this sector
is also expr8essed by the definite integral:fr2d, where r des0
ignates the radius vector and V the angle, which the radius




93
vector makes with the major axis, or the true anomaly of the
earth. We have therefore the following equation:
2 F(t —T) J       2o 2v.
As we have for the ellipse r-   c(-e2)   a cos 2'  this
1-+ e cosv  l-ecosv 
integral would become complicated.  We can however introduce another angle for vr; for as the radius vector at the
perihelion is =  a — e, at the aphelion — aq-ae, we may
assume r= Ca (1 4cos E) where E is an angle which is equal
to zero at the same time as v. For we get the following
equation for determining E from the two expressions of' r:
cos ]  -- e v  -F e
cos  E    S,
l + e.cos v
from which we see, that E has always a real value, as the
right side is always less than =- 1.
By a simple transformation we get also:
cos E-e    C      cos sill E
-     cos v and co sin v
I-ecosE            1-ecosE
and differentiating the two expressions for r, we fin"M:
cl}'  a cos 9
dE     r
Introducing now  the variable E into the above definite
integral, we find:
E
2 F(t - T) -C2 cos 9j (1 -e cos E) d E - a 2 cos  (E-e sin E),
hence taking again the semi-major axis equal to unity and
substituting for F its value found before we obtain:
2 r
(t- T) = —E - e sin E,
where 2  is the mean sidereal daily motion of the earth, that
is the daily motion the earth would have if it were performing the whole revolution with uniform velocity in the time r.
The first member of the above equation expresses therefore
the angle, which such a fictitious earth, moving with uniform
velocity, would describe in the time t- T. This angle is
called the mean anomaly and denoting it by M, we can write
the above equation also thus:




94
3f  E-e sinE,
and  having  found  from  this the auxiliary angle E, we get
the true anomaly from  the equation:
cos 9 sin E
tang v -
cos E — e
But in case that the excentricity is small it is more convenient, to develop the difference between the true and mean
anomaly  into a series.  Several elegant methods have been
given for this, whose explanation would lead us too far, but
as we need only a few  terms for our present purpose, we can
easily  find  them  in the following way.  As we have v-= -M
when e —O, we can take:
v —-ll -- v'o. e - 12 v". e2 -+-   v1',. e3 --. 
where v',, v"o etc. designate the first, second etc. differential
coefficient of v with respect to e in case that we take e -0.
cos T sine
If' we differentiate the equation sin v   -— oTs          written
1-!cos E
logarithmically, we find:
cos       dE    os E-e + dd         cos E  e  cos-e -
sill V   sin E  I-  cosE I cosp  1-    E o
n  sin v    sin v     ca Cos T 9   sin v
or:          dv' —_E   dE +-       dT             —: —-dE - dT',
sinlE      cos 99       r         cos T 
and  if we differentiate also the equation'for M, considering
only E  and e as variable, we find:
dE= sinv d p;
dv    sin V                dv    sill v
_  d-vv._si(2 - e cos v) and    =     2 (2  e cos v).
dT   cos 99                ie   cos To
Taking here e =  0, we get'o -  2 sin M.
In  order to  find  also the higher differential coefficients
sin v
we will put P        s= n   and  Q -  2 -- e cos v.  We find then
COS 99
easily, denotiPig the differential coefficients of P and  Q  after
having taken e     0 by P'o, Q'0 etc.
P'o = COS 1/. lo  sin 2 111,
2'0   cos 41,
v"110 -— sin M. Q o0 + 2P' =  sin 2 -1,
P'o = cos 211. v" - sin M. v 2 - 2 sin M= - sin 3M +-  -4 sin 21/,
"    - 2    sin M. vo = -4  sin 1/12,
"V  = sin M. Q"O.   2 Q'0. P'o + -- 2P"=3- sin 3  /1 — sin 3.
Hence we get:
v= [ q- ~+ e. 2 sin 3   e2. sin 2i +l e2  (H+  2 sin 3 2 —'1 sin M3 )
-=  1/1f+ (2e-2 esi ) sin   +       sin 2   +   ein 3  f 
-- M q (2 e - ~ e 3 ) si n   2 q-I e 3 sin 3 A/M  - -   s— e" sin 2 74           -T'




95
The excentricity  of the earth's orbit for the year 1850
is 0.0167712. If we substitute this value for e and multiply
all terms by 206265 in order to get V- M expressed in seconds of are, we find:, -- 1+ 6918". 37 sin Mi+/- 72".52 sin 2M4- 1". 05 sin 3 M1
where the periodical part, which is always to be added to
the mean anomaly in order to get the true anomaly, is called
the equation of the centre.
As the apparent angular motion of the sun is equal to
the angular motion of the earth around the sun, we obtain
the true longitude of the sun by adding to i, the longitude -r
which the sun has when the earth is at the perihelion and
MI -+r  is the longitude of the fictitious mean sun, which is
supposed to move with uniform  velocity in the ecliptic, or
the mean longitude of the sun. Denoting the first by X, the
other by L, we have the following expression for the true
longitude of the sun:
-L -- 6918". 37 sin M +- 72". 52 sin 2M1-+ 1". 05 sin 3M I*),
or if we introduce L instead of /M, as we have Mf= L -
and   -- 2800 21' 41". 0:
L  -+- 1244". 31 sin L    + -   6805". 56 cos L
-   67.82 sin 2L   +-  25.66 cos 2L
-    0. 54 sin 3L    -   0. 90 cos 3L.
In order to deduce the right ascension of the sun from
its longitude, we use the formula:
tang A - tang ). cos a,
which by applying formula (17) in No. 11 of the introduction
is changed into:
A - -  tang - sin 2 +  tang' 4  sin 42  -...
where the periodical part taken with the opposite sign is called the reduction to the ecliptic.
If we substitute in this formula the last formula found
for 2 and develop the sines and cosines of the complex terms
we find after the necessary reductions and after dividing by
15 in order to get the right ascension expressed in seconds
of time:
*) To this the perturbations of the longitude produced by the planets
must be added as well as the small motions of the point of the equinox.




96
A   -- L --  6s. 53 sin L  - 434s. 15 cos L
-596.64 sin 2L  +-   1.69cos2L
-   3.77 sin 3L   -  18. 77 cos 3L
+- 13.23 sin 4 L   -   0. 19 cos 4L
+-  0.16 sin 5L   --  0.82 cos 5L
-   0.36 sin 6L      O0.02 cos 6L
-   0.01 sinl 7 L   -.04 cos 7L.
15.  As the right ascension of the sun does not increase
at a uniform  rate, the apparent solar time, being equal to
the hour angle of the sun, cannot be uniform. Another uniform  time has therefore been introduced, the mean solar time,
which is regulated by the motion of another fictitious sun,
supposed to move with uniform velocity in the equator while
the fictitious sun used before was moving in the ecliptic.
The right ascension of this mean sun is therefore equal to
the longitude L  of the first mean sun.  It is mean noon at
any  place, when  this mean sun  is on the meridian, hence
when the sidereal time is equal to the mean longitude of the
sun  and the hour angle of this mean sun is the mean time
which for astronomical purposes is reckoned from  one noon
to the next fromn 0"1 to 24h1.
According to Hansen the mean right ascension L of the
sun is for 1850 Jan. 0 O0h Paris mean time:
18h 391m 9s. 261,
and as the length  of the tropical year that is the time in
which the sun nmakes an entire revolution with respect to the
vernal equinox is 365.2422008, the mean daily tropical motion of the sun is:
3600
2422008   59 8. 33 or - 31m 56s. 555 in time,
365. 2422005
its motion in 365 days = 231159111 2s. 706 = - 57s. 294,
its motion in 366 days — 24  2 59. 261 = + 2l1 59s 261.
By this we are enabled to compute the sidereal time for
any other time.  In  order to find the sidereal time at noon
for any other meridian, we have the sidereal time at noon
for Jan. 0 1850 equal to:
18h 39',, 9s. 261 -4-   X 311156s. 555,
24
where k denotes the difference of longitude from Paris, taken
positive when West, negative when East ).
*) Here again the small motion of the vernal equinox must be added.




97
The relation between mean and apparent time follows
from the formula for A. The mean sun is sometimes ahead
of the real sun, sometimes behind according to the sign of
the periodical part of the formula for A.
~ If we compute L for mean noon at a certain place, the
value of L — A given by the above formula is the hour angle
of the sun at mean noon, as L is the sidereal time at mean
noon ).  Now  we call equation of time the quantity, which
must be added to the apparent time in order to get the mean
time. In order therefore to find from the expression for L —A
the equation of time x for apparent noon, we must convert
the hour angle L - A into mean time and take it with the
opposite sign. But if n is the mean daily motion of the sun
in time and n+-w the true daily motion on that certain day,
24 hours of mean time are equal to 24 -w hours of apparent
time, hence we have:
x: A - L - 241h: 24h - w'
24h
or x  (A - L)24h
24h _ tFrom the equation for A we can easily see how the
equation of time changes in the course of a year. For if we
take A —L —0   retaining merely the three principal terms,
we have the equation:
0O  86.5 sin L - 596.6 sin 2 L +- 434.1 cos L,
from  which we can find the values of L, for which the equation of time is equal to zero, namely L=230~ 16', L=83~ 26',
L =  1600 15', L =   2730 3', which correspond to the 15t11 of
April, the 14th of June, the 31st of August and the 24th of
December.  Likewise we find the dates, when the equation
of time is a maximum, from the differential equation and we
get the 4 maxima:
-+ 141m 31s,  -- 311 53S    + 61 12s  -- 161n 18s
on Febr. 12,     May 14,     July 26,     Nov. 18.
The apparent solar day is the longest, when the variation
*) The above expression for L —A is only approximate. The true value
must be found from the solar tables and is equal to the mean longitude minus the true right ascension of the sun. The latest solar tables are those
of Hansen and Olufsen (Tables du soleil. Copenhagen 1853.) and Leverrier's
tables in Annales de l'Observatoire Imp6rial Tome IV.
7




98
of the equation  of time in one day is at its maximum  and
positive. This occurs about Dec. 23, when the variation is
30' hence the length of a solar day 24h On' 30s. On the contrary the apparent day is the shortest, when the variation of
the equation of time is negative and again at its maximum.
This happens about the middle of September, when the variation is -21s, hence the length of the apparent day 23 h
59m" 39s.
The transformation of these three different times can now be
performed without any difficulty, but it will be useful, to
treat the several problems separately.
16.  To convert mean solar time into sidereal time and
conversely sidereal into mean time.  As the sun on account
of its motion from West to East fromn one vernal equinox to
the next loses an entire diurnal revolution compared with
the fixed stars, the tropical year must contain exactly one
more sidereal day than there are mean days. We have therefore:
a sidereal day  366242-   mean day,
3(6.242201
=a mean day - 3m 55s. 909 mean time,
366.242201
and a mean day 3624220 sidereal day,
36da(242201
= a sidereal day + 3',56s. 555 sidereal time.
Hence if (9 designates the sidereal time, M the mean
time and  0,, the sidereal time at mean noon, we have:
241h - 3m 55s 909
241
and
241  - 31 56s. 555
241i
The sidereal time at mean noon can be computed by
the formulae given before, or it can be taken from the astronomical almanacs, where it is given for every mean noon.
To facilitate the computation tables have been constructed,
which give the values of
241 - 31155s. 909
24h
and
24h -4- 3111 56s.555
24h




99
for any value of t.  Such tables are published also in the
almanacs and in all collections of astronomical tables.
Example.  Given 1849 Juny 9 14' 16C" 36S. 35 Berlin
sidereal time. To convert it into mean time.
According to the Berlin Almanac for 1849 the sidereal
time at mean noon on that day is
5h 10m11 48s. 30,
hence 915r111 48. 05 sidereal time have elapsed between noon
and the given time and this according to the tables or if
we perform the multiplication by
24h - 3m55s.909
24h
is equal to 9h 41m 18S. 63 mean time. If the mean time had
been given, we should convert it into sidereal hours, minutes
and seconds and add the result to the sidereal time at mean
noon in order to find the sidereal time which corresponds
to the given mean tile.
17.  To convert apparent solcar time into mean time and
mean time into apparent time.  In order to convert apparent
time into mean time, we take simply the equation of time
corresponding to this apparent time from an almanac and add
it algebraically to the given time. According to the Berlin
Almanac we have for the equation of time at the apparent
noon the following values:
I. Diff.    II. Diff.
1849 June 8 - 1l1'20s. 73.
9    1  9.37        +- Os. 27.
11.63
10    0 57.74
Therefore if the apparent time given is June 9 9h 5"li23s. 60,
we find the equation of time equal to -  I.l 4.4s98, hence the
mean time equal to 91 41M 188. 62.
In order to convert mean time into apparent time, the
same equation of time is used.  But as this somletimes is
given for apparent time, we ought to know already the apparent time in order to interpolate the equation of time. But
on account of its small variation, it is sufficient, to take first
an approximate value of the equation of time, find with this
the approximate apparent time and then interpolate with this
a new value of the equation of time. For instance if 9h 411
188. 62 mean time is given, we may take first the equation
7*




100
of time equal to - Imn and then find for 9h 5m 18s.6 apparent
time the equation of time - ln  4. 98, hence the exact apparent time equal to 9h 5m 238. 60.
In the Nautical Almanac we find besides the equation
of time for every apparent noon also the quantity L — A for
every mean noon given, which must be added to the mean
time in order to find the apparent time.  Using then this
quantity, if we have to convert mean time into apparent time,
we perform a similar computation as in the first case.
18. To convert apparent time into sidereal time and conversely sidereal into apparent time. As the apparent time is
equal to the hour angle of the sun, we have only to add the
right ascension of the sun in order to find the sidereal time.
According to the Berlin Almanac we have the following
right ascensions of the sun for the mean noon:
I. Diff.
1849 June 8  511 5m 30s. 79
9    9 38.75 —4m7s.96 +08.27.
10   13 46.9    4 8.23
Now if 9h 5m 23s. 60 apparent time on June 9 is to be
converted into sidereal time, we find the right ascension of
the sun for this time equal to 5h 11m 128. 75, hence the sidereal time equal to 141' 16m 36. 35.
In order to convert sidereal time into apparent time we
must know the apparent time approximately for interpolating
the right ascension of the sun. But if we subtract from the
sidereal time the right ascension at noon, we get the number
of sidereal hours, minutes, etc. which have elapsed since noon.
These sidereal hours, minutes, etc. ought to be converted into
apparent time. But it is sufficient, to convert them into mean
time and to interpolate the right ascension of the sun for this
time.  Subtracting this from the given sidereal time we find
the apparent time.
On June 9 we have the right ascension of the sun at
noon equal to 5h1 9m 38s. 75, hence 9h 61 57s. 60 sidereal
time or 91 511 28..00 mean time have elapsed between noon and
the given sidereal time 14h1 16n 36s. 35.  If we interpolate
for this time the right ascension of the sun, we find again
5h 11m 12s. 75, hence the corresponding apparent time 9h 5m
23. 60.




101
Instead of this we might find from  the sidereal time the
corresponding  mean time and from  this with the aid of the
equation of time the apparent time.
Note. In order to make these computations for the time t of a meridian, whose difference of longitude from the meridian of the almanac is k,
positive if West, negative if East, we must interpolate the quantities from
the almanac, namely the sidereal time at noon, the equation of time and the
right ascension of the sun for the time t + k.
IV. PROBLEMS ARISING FROM THE DIURNAL MOTION.
19.  In consequence of the diurnal motion every star
comes twice on a meridian of a place, namely in its upper
culmination, when  the  sidereal time is equal to its right
ascension and in its lower culmination, when the sidereal time
is greater by 12 hours than its right ascension. The time
of the culmination of a fixed star is therefore immediately
known. But if the body has a proper motion, we ought to
know already the time of culmination in order to be able to
compute the right ascension for that moment.
By the equation of time at the apparent noon, as given
in the almanacs, we find the mean time of the culmination
of the sun for the meridian, for which the ephemeris is published, and the equation of time interpolated for the time k
gives the time of culmination for another meridian, whose
difference of longitude is equal to k.
The places of the sun, the moon and the planets are given
in the almanacs for the mean noon of a certain meridian. Now
let [(a) denote the right ascension of the body at noon, expressed in time, and t the time of culmination, we find the right
ascension at the time of culmination by Newton's formula of
interpolation, neglecting the third differences, as follows:
f(ca) + tf + (a lt to2 f" (ca),
or a little more exact:
(a) -- tf' (a +) --    t(t-l) f" (a - ).
As this must be equal to the sidereal tie  t tht  oAs this must be equal to the sidereal time at that mo



102
ment, we obtain the following equation, where Q0  designates
the sidereal time at mean noon and where the interval of the
arguments of f(a) is assumed to be 24 hours:
00 -+ t (2443m  56s. 5 6) =f(a)+ tf' (       + +  (       )
hence:
t — f(a) - Co
[24h1 311 56s. 56 -f' (ca + 2) - )] 2- f"  ( -- )
The second member of this equation contains it is true t,
but as the second  differences are always small, we can in
computing t from  this formula use for t in the second member the approximate value 24              — Oo,
24 3m 56s.56 -'+ (a+)
The quantity  0o f- [(a) is the hour angle of the body
at noon for the meridian  for which the ephemeris has been
computed;  if k is  the  longitude  of another place,  again
taken  positive if West, the hour angle at this place would
be,-  f(a) -  k, hence the time  of' culmination  for this
place but in time of the first meridian is.;rtn)-  o:C.
24h 311 56s. 56 -.' ( +          -  -  -)
and the local time of' culmination t-t'-k.
Example.  The following  right ascensions of the moon
are given for Berlin mean time:
f(a)
1861 July 14.5   13h 7m 5s.3 +27 17.6
15.0   13 34 22.9    27 58       41.2
15.5   14  2 21.7                 43.5,
28 42.3
16.0   14 31  4.0
and the sidereal time at mean noon on July 15  0,     71 33m
7s. 9. To find the time of the culmination of the moon for
Greenwich.
As the  difference  of longitude in this case is k    53mn
34s. 9, the numerator of the formula for t' becomes 61'54-m 49s. 9,
*) If the interval of the arguments of f(a) were 12 hours instead of
24 hours, the first term of the denominator in the above formula would be 121
1m 58s. 28, and if we start from a value f(a), whose argument is midnight,
we would have to use  0 -t-12hll 58s. 28 instead of O0.




103
the first terms of the  denominator become I lh 33111 59s. 5,
hence the approximate value of t' is 0.59775; with this we
find the correction of the denominator -H- 8s. 5 and the corrected value of t' equal to 0.59762 or 711 10'" 17s. 0O   hence
the local time of -the culmination equal to 6"' 16"' 42s. 1.
For the lower culmination we.have the following equation,
where a again designates the argument nearest to the lower
culmination:
o0 - t (24h 3, 56s. 6) = 121 -f(_a) t- tf' (a _- a)  f"(a -)
1.2
hence the formula for a place whose longitude is k, is:
I1-    I 2h  f' (a) - Oo -- k_
241h 3m 56s. 56 -f' (a -  ) -2f" (a — +)
or in case the interval of the arguments is 12 hours:
12." +fa-     - 19, + k
t', - 1i2, -f  -- 1 —
12"1l' 58. 3 -f' (a     )   --  f" (a -+ )
Example. If we wish to find the time of the lower culmination at Greenwich on July 15, we start from  July 15.5.
Hence the numerator becomes 7h 20"' 50s. 4, the first terms
of the denominator become 111" 33" 16S. 0, hence the aproximate value of t' is equal to 0.6359 and the corrected value
0.63577 or 7" 37m 45s. 1. The lower culmination occurs therefore at 191 3711 45s. 1 Berlin mean time or at 18h 44n' 10s. 2
Greenwich time.
20.  In No. 7 we found the following equation:
sin h- sin 9 sill g     -+- cos f Cos 8 cos t.
If the star is in the horizon, therefore h equal to zero,
we have:
0 - sin c sin  si+ cos 9 cos 8 cos t,.
hence:               cos t, =    tang 99 tang A.
By this formula we find fo  any latitude the hour angle
at rising or setting of a star,'Whose declination in Y.  This
hour angle taken absolute is called the semi-upper diurnal arc
of the star. If we know  the sidereal time at which the star
passes the meridian or its right ascension, we find the time
of the rising or setting of the star, by subtracting the absolute value of to from  or adding it to the right ascension.




104
From  the sidereal time we can find the mean time by the
method given before.
Example.  To find the time when Arcturus rises and
sets at Berlin. For the beginning of the year 1861 we have
the following place of Arcturus:
a = 14h 9m 19s.3       -- + 190 54' 29".
and further we have:
9__ 520 30' 16".
With this we find the semi-diurnal are:
to = 1180 10' 1. 3 -- 7h 521n 40s.        o
Hence Arcturus rises at 6h 16rn 39s and sets at 22h 1m 39s
sidereal time.
In order to find the time of the rising and setting of a
moveable body, we must know its declination at the time of
rising and setting and therefore we have to make the computation twice.  In the case of the sun this is simple.  We
first take an approximate value of the declination and compute with it an approximate value of the hour angle of the
sun or of the apparent time of the rising or setting. As the
declination of the sun is given in the almanacs for every apparent noon, one can easily find by interpolation the declination for the time of the rising or setting and repeat the
computation with this.
In the case of the moon the computation is a little longer.
If we compute the mean time of the upper and lower culminations of the moon, we can find the mean time corresponding to any hour angle of the moon. We then find with
an approximate value of the declination the hour angle at
the time of the rising or setting, find from it an approximate
value of the mean time and after having interpolated the declination of the moon for this time repeat the computation.
An example is found in No. 14 of the third section.
Note. The equation for the hour angle at the time of the rising or setting may be put into another form. For if we subtract it from and add it
to unity, we find by dividing the new equations:
tang2 t- cos (9- A)
cos (q -I- +)
21. The above formula for cos to embraces all the various phenomena, which the rising and setting of stars ac



105
cording to their positions with respect to the equator present
at any place on the surface of the earth.
If 3 is positive or the star is north of the equator, cos to
is negative for all places: which have a northern latitude;
t, therefore in this case is greater than 900 and the star
remains a longer time above than below the horizon. On
the contrary for stars, whose declination is south, to becomes
less than 900, therefore these remain a longer time below
than above the horizon of places in the northern hemisphere.
In the southern hemisphere of the earth, where cp is negative,
it is the reverse, as there the upper diurnal arc of the southern stars is greater than 12 hours. If we have  -= 0, to
is 90o1 for any value of 3; therefore at the equator of the
earth all stars remain as long above as below the horizon.
If we have 3   0, to is also equal to 900 for any value of
t, hence stars on the equator remain as long above the
horizon of any place on the earth as below.
Therefore while the sun is north of the equator, the
days are longer than the nights in the northern hemisphere
of the earth, and the reverse takes place while the sun is
south of the equator.  But when the sun is in the equator,
days and night are equal at all places on the earth.  At
places on the equator this is always the case.
It is obvious that a value of to is only possible while we
have tang 99 tang s< 1.  Therefore if a star rises or sets
at a place whose latitude is b, tang 3 must be less than
cotang so or a < 90 -. If  =90 -r, we find t=-1800
and the star grazes the horizon at the lower culmination.
If we have S > 90 -- cp, the star never sets, and if the
south declination is greater than 90 -p, the star never
rises.
As the declination of the sun lies always between the
limits - E and -- a, those places on the earth, where the sun
does not rise or set at least once during the year, have a
latitude north or south equal to 90- c or 66 0.  These
places are situated on the polar circles.  The places within
these circles have the sun at midsummer the longer above and
in winter the longer below the horizon, the nearer they are
to the pole.




106
Note.  A  point of the equator rises when its honr angle is 6h.  Hence
if we call the right ascension of this point a, we find the stars, which rise
at the same time, if we lay a great circle through this point and the points
of the sphere, whose right ascensions are a- 66h and a-+ 6h and whose declinations are respectively -(90~ — ) and -t-(90~ — ).  Likewise we find
the stars, which set at the same time as this point of the equator, if we lay
the great circle through the points, whose right ascensions are a x+61 and
a - 61  and whose declinations are respectively — (90~- -) and 90~ —o.
The point, which at the time of the rising of the point a was in the horizon
in its lower culmination, is therefore now  in its upper culmination at an
altitude equal to 2 9.  Hence at the latitude of 450 the constellations make
a turn of 900 with respect to the horizon fiom the time of their rising to the
time of setting, as the great circle which is rising at the same time with a
certain point of the equator, is vertical to the horizon, when this point is
setting. On the equator the stars, which rise at the same time, set also at
the same instant.
22.   In  order to  find  the point of the horizon, where
a star rises or sets, we must make in the  equation:
sin  - =  sin T sin h - cos T cos h cos A,
which  was found in  No. 6, hi equal to  zero  and  obtain:
sin 
cos Ao.    -.. (-).
cos 99
The negative value of A,, is the azimuth of the star at its
rising, the  positive  value  that at the time of setting.  The
distance  of the  star, when  rising  or setting, from  the  east
and  west points of the horizon  is called the amplitude of the
star.  Denoting  it by  A,, we have:
Ao = 90 -+ A,
hence:
silln 
sin A, =          (c),
where A, is positive, when  the point where the star rises or
sets, lies  on the  north  of the  east or  west points, negative when it lies towards south.
The formula (c) for the amplitude may be written in a
different shape. For as we have:
1 - sin A,    sin  t +sin S
1 -sin A,   sin V - sin'
when   = 90 - p, we find:
2   tcang'   2
tang 
2




107
For Arcturus we find with the values of 3 and %f, given
before:                   A, -340 0'. 9.
23.  If we write in the equation:
sin h = sin q9 sin A -I- cos g cos a cos t
1 —  2 sinll t  instead of cos t, we get:
sin h -- cos (9-  ) -2 cos  cos a sin - t2.
From  this we see, that equal altitudes correspond to
equal hour angles on both sides of the meridian.  As the
second term of the second member is always negative, h has
its maximum value for t-  0 and the maximum itself is found
from  the equation:
cos Z - cos (90- s)    (d),
from which we get:
z =  p-   or =   -.
If we take therefore in general:
Z -- a-,
we must take the zenith distances towards south as negative,
because for those star, which culminate south of the zenith,
5 is less than cp.
On the contrary h is a minimum  at the lower culmination  or when t   1800, as is seen, when we introduce
180-~ t' instead of t, reckoning therefore t' from that part
of the meridian, which is below  the pole.  For then we
have:
sin It  sin sD sin 8 - cos 99 cos a cos t'.
or introducing again 1 -  2 sin } t'2 instead of cos t':
sin h - cos [1800 == (? +- 8)] +- 2 csos cos sin. t'2
As the second term of the second member is always
positive, h is a minimum  when t' equals zero or at the lower
culmination, when we have:
cos z = cos [1800 ~ (a q- )].
As z is always less than 90~, when the star is visible in
-its lower culmination, we must use the upper sign, when cp
and 8b are positive, and the lower sign for the southern hemisphere, so that we have:
z- = 1800 - (~ + A),
for places in the northern hemisphere, and:
fz -p (lS0O  H- + t-   )
for places in the southern hemisphere.




108
The declination of a Lyrae is 380 39', hence we have
for the latitude of Berlin' —   =  -  130 51'.  The star ca
Lyrae is therefore at its upper culmination at Berlin 130 51'
south of the zenith, and its zenith distance at the lower culmination equal to 180~0 - g-3 is 88" 51'.
24. A body reaches its greatest altitude at the time of'
its culmination only if its declination does not change, and
in case that this is variable, its altitude is a maximum a little
before or after the culmination. If we differentiate the formula:
cos Z  sill 9 sin 3 ~ cos T cos 3 cos t,
taking z, c' and t as variable, we find:
- sin z dz = [sin? cos 3 - cos g9 sin 3 cos t] d3 - cos T cos 3 sin t d t
and from  this we obtain in the case that z is a maximum
or dz -- 0:
sin t  d [tang cp - tangS cos t].
This equation gives the hour angle at the time of the
greatest altitude. d8 is the ratio of the change of the declination to the change of the hour angle, or if d t denotes a
second of are, it is the change of the declination in -L of a
second of time. As this quantity is small for all heavenly
bodies, and as we may take the are itself instead of sin t
and take cos t equal to unity, we get for the hour angle
corresponding to the greatest altitude:
d  206265
t     [tang99 - tang3] —    (g),
dlt               15
where diT is the change of the declination in one second of
time and t is found in seconds of time.  This hour angle
must be added algebraically to the time of the culmination,
in order to find the time of the greatest altitude.
If the body is culminating south of the zenith and apd8
proaching the north pole, so that d/ is positive, the greatest
altitude occurs after the culmination if So is positive; but if
the declination is decreasing, the  greatest altitude  occurs
before the culmination. The reverse takes place, if the body
culminates between the zenith and the pole.




109
25.  If we differentiate the formulae:
cos Ih sin A = cos S sin t,
cos h cos A -  - cos 9' sin 8 +- sin qT cos S cos t,
we find:
d h
sin  h     cos S [sin p cos A sin-t - cos t sin A],
dt
dA
cos h  - cos S [cos A cos t -- sin T sin t sin A],
or:
cl h
-        - cos S sin p     - cos T sin A,
dt
dA
Frequently  we  make use also  of the second  differential
coefficient. For this we find:
d' h                  dA
dt2   -- cos T cos A. dt
cos  cos S cos A cosp
cos h
Likewise we have:
dz
-...= cos 3 sin p --- cos T sin A,
dt
d2 z   cos   co s  cos A cos p
cd t           cos h
Furthermore we find from  the second of the formulae (h):
d 2 A                     dp                   dh
cOS  i2           os h cos 8 sin p   -+- cos n    cos p sin h d
dt2                      d t        dt
But we get also, differentiating the formula:
sin -T  - sin h sin   -+- cos h cos S cos p,
cos  ens  sin        [cos h sin  [cos h  sin h cns h  c os cos ] d
Hence we have:
d2A
cos h2 -dt  -    [cos h sin 8 - 2 cos S sin h cos p] cos S sin p,
or, if we introduce A instead of p:
d2 A
COS 2 dt2     - cos  sin A [cos I sin S +-  2 cos P cos A].
26. As we have:
dh
dt   - cos qT sin A,
dt
we  find  d T    0, or h is a maximum  or minimum, when we
have sin A-  0 or when the star is on the meridian.




110
We find also that c- is a maximum, when sin A   _   1,
hence when A — 90 or -- 270'.
The altitude of a star changes therefore most rapidly, when
it crosses the vertical circle, whose azimuth is 900 or 270".
This vertical circle is called the prime vertical.
In order to find the time of the passage of the star
across the prime vertical as well as its altitude at that time,
we take in the formulae found in No. 6 A — 900 or we consider the right angled triangle between the star, the zenith
and the pole and find:
tang 8
cos t --
tang 9    (I)
sin A
sin h 
sin 99
Finally we havesin  = cos -'
If we have 8 > ~p, cos t would be greater than unity,
therefore the star cannot come then in the prime vertical
but culminates between the zenith and the pole.  If S is
negative, cos t become negative; but as in northern latitudes
the hour angles of the southern stars while above the horizon
are always less than 900, those stars cross the prime vertical
below the horizon.
For Arcturus and the latitude of Berlin we find:
t -730 52'. 1 -- 4h 55mn 28s
— 25  24'.9.
Arcturus reaches therefore the prime vertical before its
culmination at 9"' 13n 519 and after the culmination at 19l,
41 47s.
If the hour angle is near zero, we do not find t very
accurate by its cosine nor h by its sine. But we easily get
from the formula for cos t the following:
sin (99  )
tang  t2  
silln ( +- 8)
and for computing the altitude we may use the formula:
cotang Al - tang t cos 99.
27. As we have:
d A   cos S cos p
dt     cos h




we see that this differential coefficient becomes equal to zero,
or that the star does not change its azimuth for an instant,
when we have cos p=  o, or when the vertical circle is vertical to the declination circle.  But as we.have:
sin 99 - Sin / sinl 
Cos p             —'cos h Cos 
this must occur when sin                  It happens therefore
only to circumpolar stars, whose declination is greater than
the latitude, at the point where the vertical circle is tangent
to the parallel circle. The star is then at its greatest dista(nce from  the meridian and the azimuth at that time is given
by the equation:
cos 
sin A
cos 99
and the hour angle by the equation:
tang 99
cos t 
tang y
For the polar star, whose declination for 1861 is 88'
34' 6" and for the latitude of Berlin, we find:
t   = 88 8' 0"- 5h 52mn 32s,
A = 2o 21' 9" reckoned from the north point, h- 520 31'.7.
28.  Finally we will find the time, in which the discs
of the sun and moon move over a certain great circle.
If A a is the increment of the right ascension between
two consecutive culminations expressed in seconds of time,
we find the number of sidereal seconds x, in which the body
moves through the hour angle t from the following proportion:
x: t -- 86400  A a: 86400
as we may consider the motion of the sun and moon during
the small intervals of time which we here consider, as uniform; hence we have:
86400 - A a
or denoting the second term  of the denominator, which is
equal to the increment of the right ascension expressed in
time in one second of sidereal time, by A:
1 -_.




112
When the western limb of the body is on the meridian,
the hour angle of the centre, is found from the equation:
cos R = sin 82 +- COS 82 COS t
where R designates the apparent radius, or from:
sin 2 R - cos A sin 2 t.
Hence, as t is small, this hour angle expressed in time is:
R
t15 cos S
therefore the sidereal time of the semi-diameter passing the
meridian:
2R     1
15. coS'  - 
When the upper limb of the body is in the horizon, the
depression of the lower limb is equal to 2R, and as we have:
dz
dt = cos  S sin p, the difference of the hour angles of the upper and lower limb in time is:
2 R
15. cos 8 sin p
hence the sidereal time of the diameter rising or setting:
2R        1
15.cosSsinp  1 —i
where p is found from the equation:
sin ~
cos p -'
If we imagine two vertical circles one through the centre,
the other tangent to the limb, the difference of their azimuths
is found from the equation:
sin 2 R -cos h sin 2 a
or, as R is small, fromn the equation:
R -- cos h. a.
But as we have dtcls -  coshdA   we find for the sidereal
cos S cosp
time in which the diameter passes over a vertical circle:
2R        1
15 cos 8. cosp  1 - 
cos   sin 99 - sin a cos 9 cos t
whetre p             h
cos h




SECOND SECTION.
ON THE CHANGES OF THE FUNDAMENTAL PLANES, TO WI-HICH
THE PLACES OF THE STARS ARE REFERRED.
As the two poles do not change their place at the surface of the earth, the angle between the plane of the horizon of a place and the axis of the earth or the plane of the
equator remains constant. Likewise therefore the pole and
the equator of the celestial sphere remain in the same position with respect to the horizon. But as the position of
the axis of the earth in space is changed by the attraction
of the sun and moon, the great circle of the equator and the
poles coincide at different times with different stars, or the
latter appear to change their position with respect to the
equator. Furthermore as the attractions of the planets change
the plane of the orbit of the earth, the apparent orbit of the
sun among the stars must coincide in the course of years
with different stars. Hence the motion of these two planes,
namely that of the earth's equator and that of the earth's
orbit produce a change of the angle between them or of the
obliquity of the ecliptic as well as a change of the points
of intersection of the two corresponding great circles. The
longitudes and latitudes as well as the right ascensions and
declinations of the stars are therefore variable and it is most
important to know the changes of these co-ordinates.
In order to form a clear idea of the mutual motions of
the equator and ecliptic, we must refer them to a fixed place,
for which we take according to Laplace that great circle,
with which the ecliptic coincided at the beginning of the year
1750. Now Physical Astronomy teaches, that the attraction
of the sun and moon on the excess of matter near the equator
8




114
of the spheroid of the earth, creates a motion of the axis of
the earth and hence a motion of the equator of the earth
with respect to the fixed ecliptic, by which the points of intersection have a slow, uniform  and retrograde motion on
this fixed plane and at the same time a periodical motion,
depending on the places of the sun and moon and on the
position  of the moon's nodes viz. of the points in which
the orbit of the moon intersects the ecliptic. The uniform
motion of the equinoxes is called Lutnisolar Precession, the
other periodical motion is called the Nutation or the Equation
of the equinoxes in longitude. Besides this attraction creates
a periodical change of the inclination of the equator to the
fixed plane, dependent on the same quantities, which is called
the Nuttation of' obliquity.
As the mutual attractions of the planets change the inclinations of the orbits with respect to the fixed ecliptic as
well as the position of the line of the nodes, the plane of
the orbit of the earth mnst change its position with respect
to the plane, with which it coincided in the year 1750 or
the fixed ecliptic. This change produces therefore a change
of the ecliptic with respect to the equator, which is called
the Secular variation of the obliquity of the ecliptic and the
motion of' the point of the intersection of the equator with
the apparent ecliptic on the latter, which is called the Genercal
Precession differs from the motion of the equator on the fixed
ecliptic, which is called the luni-solar precession').
But this change of the orbit of the earth has still another effect.  For as;: by it the position of the orbit of the
sun and the moon with respect to the equator of the earth
is changed, though slowly, this must produce a motion of
the equator similar to the nutation only of a period:of great
length, by which the inclination of the equator with respect
to the ecliptic as well as the position of the points of intersection is changed. These changes on account of their long
period can be united with the secular variation of the obliquity of the ecliptic and with the precession.  Hence the
*) The periodical terms, the nutation, are the same for the fixed and
im-oveable ecliptic.




115
motion of the equator, indirectly produced by the perturbations
of the planets, changes a little the lunisolar precession as
well as the general precession and the angle, which the fixed
and the true ecliptic make with the equator *).
I. THE PRECESSION.
1. Laplace has given in ~. 44 of the sixth chapter of
the Mecanique Celeste the expressions for these several slow
motions of the equator and the ecliptic, which can be applied
to a time of 1200 year- before and after the epoch of 1750,
as the secular perturbations of the earth's orbit are taken
into consideration so as to be sufficient for such a space of
time. Bessel has developed these expressions according to
the powers of the time which elapsed since 1750 and has
given in the preface to his Tabulae IRegiomontanae these expressions to the second power.  According to this the annual lunisolar precession at the time 1750  - t is:
dl, = 50". 3 7572 - 0". 000243589 t
dt
or the amount of the precession in the interval of time from
1750 to 1750 -  t:, = t. 50".37572 - t2 0". 0001217945.
This therefore is the are of the fixed ecliptic between
the points of intersection with the equator at the beginning
of the year 1750 and at the time 1750 -- t.
Furthermore the annual general precession is:
-I = 50". 2t1129 +- 0". 0002442966 t
dt
and the general precession in the interval of time from 1750
to 1750 -+- t:
= — t 50". 21129 +- t2 0".0001221483,
and this is the arc of the apparent ecliptic between the points
of intersection with the equator at the beginning of the year
1750 and at the time 1750 -+- t.
*) In the expressions developed in series they change only the terms
dependent on t2.
8*




116
Finally the angle between the equator and the fixed
ecliptic is at the time  1750 — t:
o — 230 28' 18.0 + —I t2 0".0000098423
and the angle between the equator and the ecliptic at the time
1750-+-t (if we neglect as before the periodical terms of nutation), which is called the mean obliquity of the ecliptic, is:
e ~-230 28' 18. 0 - t 0". 48368 - t2 0". 00000272295 *),
so that we have:
d --   _+ 0". 00001968466 t,
cdt
d 
-        0". 48368 - 0". 0000054459 t.
cdt
Now let AA,, Fig. 2 represent the equator and EE,, the
ecliptic both  for the beginning of the year 1750, and let A'A"
and E E' represent the equator and the obliquity of the ecliptic
for 1750i- t; then the arc BD  of the ecliptic, through which
the equator has retrograded on it, is the lunisolar precession
in t years, equal to 1,.  Further are BCE  and A'BE  respectively  the  inclination  of the  true  ecliptic  and  of the  fixed
ecliptic  of 1750  against the equator, equal to  E and  8,.  If
*) Bessel has changed a little the numerical values of the expressions
given in the M/canique Cdleste, as he recomputed the secular perturbations
of the earth with a more correct value of the mass of Venus and determined
the term of the lunisolar precession 1,, which is multiplied by t, from more
recent observations.  The secular variation of the obliquity of the ecliptic
as deduced from  the latest observations differs from the value given above,
as it is 0".4645. But the above value is retained for the computation of the
quantities 7r and 11, which determine the position of the ecliptic with respect
to the fixed plane, as it must be combined for this purpose with the value of
cli
d   based on the same values of the masses.  The terms multiplied by t0,
which depend on the perturbations produced by the planets, are based on
the values of the masses adopted by Laplace and need a more accurate determination.
Peters gives in his work,,Numerus constans nutationis" other v'alues computed with the latest values of the masses.  These are, reduced to the year
1750 and to Bessel's value of the lunisolar precession as follows:
1, = t 50".37572 - t'" 0".0001084
I -- t 50".21484 +- t2 0".0001134
=o =   30 28' 17".9 + 0.00000735 t
-- 23 28' 17".9 - 0".4738 t - 0".00000140 t2.
But as Bessel's values are generally used, they have been retained.




117
Fig. 2.
B     /
-A.
then S represents a star and SL and SL' are drawn vertical
to the fixed and to the true ecliptic, DL is the longitude
of the star for 1750 and CL' the longitude of the star for
1750 — t. If further D' denotes the same point of the true
ecliptic which in the fixed ecliptic was denoted by D, the arc
CD' is the general precession, being the arc of the true
ecliptic between the equinox of 1750 and that of 17504-t.
This portion of the precession is the same for all stars, and in
order to find the complete precession in longitude, we must
add to it D'L' -DL; which portion on account of the slow
change of the obliquity is much less than the other. For
computing this portion we must know  the position of the
true ecliptic with respect to the fixed ecliptic, which is
given by the secular perturbations and may also be deduced
from the expressions given before. For if we denote by I1 the
longitude of the ascending node of the true ecliptic on the
fixed ecliptic (or that point of intersection of the two great
circles setting out from which the true ecliptic has a north
latitude) and if we reckon this angle from the fixed equinox of the year 1750, we have BE =  180~  -  1-1, and
CE = 180"- 11 -1,  as the longitudes are reckoned in the
direction from B towards D and as E is the descending node
of the true ecliptic, hence DE = 180~0 - I.  If we denote
the inclination of the true ecliptic or the angle BE C by n,
we have according to Napier's formulae;




118
tang  Tr. sin  H —  sin             tang  
tang ~~c. Cos Hq4_- t                      2
tn  2-7.co  H             Cos  2   tang -—,
As B  is the same point of the equator which in the year
1750 was at D, B C is the are of the equator, through which
the point of intersection with the ecliptic has moved on the
equator from  west to east during the time t.  If' we denote
this arc, which is the Planetary Precession during the time t,
by a, we find from the same triangle:
8 + —- 80                8    8o
tang  1c. cos  2 -- tang  (, - l) cos 2-2
From  these  equations we can  develop a, as well as ar
and II into a series progressing according to the powers of
t.  From  the last equation, after introducing:
~E -- 4 ( - ~ ) instead of
and  taking  instead  of the  sines and  tangents of the  small
angles 1, — 1, a andl E    e, the arcs themselves, we find:
ai                     sin  8 —
-___ I     (1, —l).      ~-o
COS s      COs ~          206265 
or if we substituLte for 1,,    and E,, their expressions, which
are of the following form, t +-'',t,    t       t2 and    t %-H'  2t
we obtain:
a   -t. t —--    t            (+,- 2   I  sin E0 +
Cos 80       Cos 8,     206265   cos   2 i
or if we substitute the numerical values:
a   t. 0.17926 - t2 0".0002660393,
-   =    0.17926 - t. 0".0005320786.
cdt
In addition we have:
&  -4 — so
tang H q+ —- - -tang 2 n
sin
2
and
1 --      _ -  12    8+ 8o e2o 2
tang~ ----   tang 2tang -               t ang  tang    2  COS - ---
t   2                     2           2
or proceeding in a similar way as before:
tang  11+2(/l + l) } =  &-E  +   2- a e  5
2        2+(8a  isino  + - 2(  COS o)
a sin   oa    - 8)0   206265
7t    a.0)2        ~~~206265




119
Substituting here also for E -      and a the expressions
Ij t -t- 7't  and  e t 4- ct't', we find:
1+ -- -(I+ 1)  are tang a
-tal /ssinle.ia sin
+k a     -il -ft-sin+ ~206265 + _. a cos e  COS c11
sc - t gA 2 Sl~l E o _e U 2 +72                   20 
or substituting the numerical values:
H — 171" 36' 10' — t. 5". 21
r -- t. 0".48892 - t2 0', 0000030715
_t _- 0. 48892 - t. 0". 0000061430.
dt
2.  The mutual changes of the planes, to which the positions of' the stars are referred, having thus been determined,
we can easily find the resulting  changes of the places of
the stars themselves.  If X and fi denote the longitude and
latitude. of a star referred  to the ecliptic of 1750 -- 1, the
co-ordinates of the star with respect to this plane, if we take
the ascending node of the ecliptic  on the fixed ecliptic of
1750 as origin of the longitudes, are as follows:
cos/9 cos ( -  UH- 1), cos  sill (2 - H - 1), sin t9.
If further L and B are the longitude and latitude of the
star referred to the fixed ecliptic of 1750, the three co-ordinates with respect to this plane and the same origin as before are:
cos B cos (L - H), cos B sin (L - H), sin B.
As the fundamental planes of these two systems of coordinates make the angle T with each other, we find by the
formulae (1 a) of the introduction the following equations:
cos F cos (2 -  H - -) =    cos B cos (L -- )
cos bl sin ( -H-  ) =    cos B sin (L - H) cos X + sin B sinl r  (A)
sin,  -- cos B sin (L - TI) sin c +- sin B cos Ar.
If we differentiate these equations, taking L and B as
constant, we find by the differential formulae (11) in No. 9
of the introduction, as we have in this case a — 900~ / l
b  90 - B, c — =, A - 90~4- L - -,  B - 90o~- (A -  I- 1):
d (, - H- l)  -  d-i~ 7tz tang f sin (A - H i- 1) d n
H- tang, cos (2 - H - 1) dcl;
dl- = + 7c cos (A - li — I' dH — sin (r - - H-I) dgr.




120
Dividing by d t and substituting tdt instead of g in the
coefficient of dli, we obtain from these the following formulae for the annual changes of the longitudes and latitudes
of the stars:
dt -=dt + — tangoflicos (A-t- dti     d
dt   c t                         cit i d
cdit  (           cdt   dz
or, as we have  t+d-lt  t=171" 36' 10"-  t.10".42, taking:
d     ci
HT - t d q- 1 =- 171 o 36' 10" - t 39". 79- =  1,
cit
d —  dl                    dc
-  + tang f cos (-) 
dci3    cSill ( - It )   
di                di
where the numerical values for -d  and  dt as given in the
preceding No. must be substituted.
Let L and B  again denote the longitude and latitude
of a star, referred to the fixed ecliptic and the equinox of
1750, then the longitude reckoned from  the point of intersection of the equator of 1750 +  t with the fixed ecliptic, is
equal to L -+ 1, when 1, is the lunisolar precession during
the interval from  1750 to 1750 +- t. IHence the co-ordinates
of the star with respect to the plane of the fixed ecliptic
and the origin of the longitudes adopted last are:
cos B cos (L +-  1), cos B sin (L -+- 1,) and sin B.
If now a and a denote the right ascension and declination of the star, referred to the equator and the true
equinox at the time 1750+- t, the right ascension reckoned
from  the origin adopted before, is equal to a -]- a. We have
therefore the co -ordinates of the star with respect to the
plane of the equator and this origin as follows:
cos 8 cos (a +- a), cos 8 sin (a -+ cc) and sin A.
As the angle between the two planes of co-ordinates is
es, we find from the formulae (1) of the introduction:
cos A cos (a.-  a) cos B cos (L -- 1,)
Cos a sin (a -- a)  cos B sin (L +- 1,) cos so - sin B sin so    (C)
sin 6 = cos B sin (L +- 41) sin so + sin B cos so.




121
If we  differentiate  these equations, taking L  and B  as
constant, we find  from   the differential formulae (11) of the
introduction, as we have in the triangle between the pole of
the  ealiptic, that of the  equator and the star a =  90" - (,
b,,, A900 -(L, B- =,,, A  90  -     +( a+- a)
d (a -- r) = [cos eo +- sin eo tang 8 sin (cL-f- a)] dl, - cos (a -a) tang cdeo
dc    cos (a -q- a) sin o, d l, +- sin (a -- a) dco.
We find therefore for the annual variations of the right
ascensions  and  declinations  of the  stars  the  following forniulae:
dca      dcia                              I,
d — + [cos Eo +- sin Eo tang S sill a] —
d didi d            a
+ | a sin o -- I  lgo tang. cos a,
dc.    cl (,    -    c,   cia,, deo,..... cos asill co     asi n   ct    ct
dit              d      ct sn  — d   -d   sin a
or neglecting  the  last term  of each equation on account of
its being very small ):
t- -          -- + [cos ao + sin a, tang S sin a] d-",
--    n cos t sin eo.  dt
dt                ct
If' we take here:
d,   d cia
cos o -    -    = t
dl
sin ea0
we find silply:
dac
It -  m -+- n tang Ssin a,
d t
dS a(D)
--     cos a,
dt
where the numerical values of m  and n, obtained  by substituting  the numerical values of eo, d-' and  - -, are:
n -- 46".02824 + 0". 0003086450 t
n - 20"." 06442 - 0". 0000970204 t.
In  order to find the precession in longitude and latitude
or  in  right ascension  and  declination  in  the  interval from
di,   da,,
*) The numerical value of the coefficient a sin oa, -I- - -   is only
- 0.0000022471.dt
0.0000022471 t.




122
1750 +  t to  1750 +  t', it would be necessary to take the
integral of' the equations (B) or (D) between the limits t
and t'.  We can find however this quantity to the terms of
the second order inclusively from  the differential coefficient
at the time  + t and from  the interval of time. For if f(t)
and f(t') are two functions, whose difference f(t') - f(t) is
required, (in our case therefore the precession during the time
t'  t), we take:
(t+ t) — X,
(t'- t) =Ax.
Then we have:
f(t) = =f(x --  x)  f(x) - Axf' (x) -+  A x2.f" (x'),
f (t')-f(x -+ A X) x-f(x) +  - Xf' (x) ~-. A 2f"l (x),
where f'(x) and f" (x) denote the first and second differential
coefficient of f(x). From this we find:
f(t') -f(t) = 2 Axf' (x) - (t' - t) f' (    t)
k  2  )'
Hence in order to find the precession during the interval of time t'-  t, it is only necessary to compute the differential coefficient for the time exactly at the middle and
to multiply it by the interval of tilme.  By this process only
terms of the third order are neglected.
For instance if we wish to find the precession in longitude and latitude in the time from 1750 to 1850 for a
star, whose place for the year 1750 is:. -= 2100 0',,= +- 34o 0'
we find the following values of d'l, d  and M for 1800:
d= 50".22350, d  -0".48861, M — 1720 9'20".
cld             cit
With these we find the following place for 1800, computing the precession from  1750 to 1800 only approximately:
2i- 2100 42'.1, f -330 59'.8
from  the formulae (B) we find then the annual variations for
1800:
== -+ 50".48122,  d -- 0". 30447,
d t             I dt
hence the precession in the interval from  1750 to 1850:
in longitude +- 10 24' 8". 12 and in latitude - 30".45.




123
If we wish to find the precession in right ascension and
declination from  1750 to 1850 for a star, whose right ascension and declination for 1750 is:
a 2200 1' 24",  = - 20o 21' 15"
we have for 1800:
m - 46". 04367, a -= 20". 05957,
and the approximate place of the star at that tine:
a = 2200 35'.,   =+- 200 8'. 6
hence we have according to formulae (D):
tang 8 9. 56444    n tang 8 sin a —   4. 78806
sin a 9. 81340,,         m -    46. 04367
da
tang 6 sin a    9. 37784,  41. 2556
cit
-- 1. 30232
cos cc — 9. 88042,,           -  -15. 2314
dt
therefore. the precession  in  the interval of time from  17S50
to 1850
in right ascension 1 8' 45". 56 and in declination - 25' 23". 14.
In the catalogues of stars we find usually for every star
its annual precession in right ascension and declination (variatio  annua) given for the epoch of the catalogue and besides this its variation in one hundred years (variatio saec ularis). If then to denotes the epoch of the catalogue, the
precession of a star according to the above rules equals:
{variatio annua -  200 variatio saecelaris (t -to)
If we differentiate the two formulae:
da
- t  -- -n tang c sin a,
dt
dY
d - =n cos a,
taking  all quantities as variable  and  denoting  the  annual
variations of m  and n by m' and n', we find:
cd' a      n  a               7 n
- -- sin 2 a [2 - }         ta-ang c2 cos a -t- mn'   n' tang 8 sin a,
d t2                          w w
d2 3     n2 2?  n
t2 --- sin a' tang   -  sin a - i  cosa,
dt"      w                to
where wt  signifies the number 206265, and multiplying these
equations by  100  we find the secular variation in right as



124
eension  and  declination.  For the star used before we find
from  this the secular variation:
in right ascension   -- 0".0286,
in declination    = + -0". 2654.
3.   The  differential formulae  given  above cannot be
used if we wish to compute the precession of stars near the
pole. In this case the exact formulae must be'employed.
Let a and {F denote the longitude and the latitude of a
star, referred to the ecliptic and the equinox of 1750+ t,
we find from these the longitude and latitude L and B,
referred to the fixed ecliptic of 1750, from the following
equations, which easily follow from the equations (A) in
No. 2:
cos B cos (L - H) -- cos  cs  (2 -  -- 1)
cos B sin (L - H) = cos f sin (2 - H-  ) cos z - sin f sin m
sin B - cos i sin (2 -  1 - 1) sin  i +- sin fi cos gr.
If we wish  to find now  the longitude and latitude 2'
and gf', referred to the ecliptic and the equinox of 1750q-t',
we get these from  L and B by the following equations, in
which 11', a' and 1' denote the values of 1i, r and I for the
time t':
cos F' cos (2'- H' - I')  cos B cos (L -- H')
cos jf' sin (2' - H' - I') = cos B sin (L - H') cos y' +- sin B' sin A'
sin f3' = - cos B sin (L - H') sin 7r' q- sin B' cos I'.
If we eliminate L and B from these equations, we can
find i' and i3' expressed directly by A and f and the values
of 1, H1  and  r for the times t and t'.
The exact formulae for the right ascension and declination
are similar.  If a and c are the right ascension and declination of a star for 1750 +- t, we find from  them  the longitude  and latitude L and B, referred to the fixed ecliptic of
1750, by the following equations ):
cos B cos (L + I,) - cos  cos (a +- a)
cos B si(L -- ) cos sin (L +- l) c-os a -sinsin a) cos o  sin  sin 
sin B = - cos  sin (a +- a) sin eo -- sin 6 cos so.
If we wish to know now the right ascension and declination a' and 5' for 1750 +- t', we find these from  L and B
*) These equations are easily deduced from the equations (C) in No. 2.




125
by  the  following  equations, in  which  1,, a' and'o  denote the
values  of 1,, a  and  Eo for the  time  t':
cosScos (      - a')  cos  B cos (L  - I',)
coS   sill (a' + — a')   cos B sin (L -- 1',) cos ~' -  sin B sinl'
sin 8'   cos B sin (L   - 1',) sin s'o +-  sin B cos s'
If we eliminate  L  and B  from  the two systems of
equations and observe that we have:
cos B sin L    -cos S cos (a    c) sin 1, +-  cos S sin (a q- a) cos  e COS 1,
-- sin S sin ~ COs 1,
cos B cos L =  cos S cos (a +-  a) cos, +-  cos S sin (a +- C) cos E sin 1,
-I- sin S sin ~ sin 1,
sin B  = -  cos S cos (a A- a) sin s +  sin S cos s,
we  easily  find  the  following  equations:
cos 5' cos (a' +  c') -  cos S cos (a +  a) cos (1'-  1,)
- cos S sin (a -- a) sin (1', - I) cos s,,
sin S sin (1', -- 1) sin -,,
cos 5' sin (a' +-  a')= cos S cos (a +- a) sin (1', -  I,) cos E'o
+- cos Ssin(a + c) [cos (', —,) cos so cos s'o — sin so sin s'o]
-- sin G[cos (', - I,) sin *s cos *o0 - COs so sin 8' 0
sin 5'   cos S cos (a +  a) sin (1,' -  1,) sin s'o'- COs  sin (a —a) [cos (',-I,) cos So sin S' - sin so Cos' o]
+- sin 8 [COS (',-  1,) sin so sin s',-+- cos so cos ~oj.
If we  imagine  a  spherical triangle, whose three  sides are
i1 -,, 90"-z and 90~-+-' whilst the angles opposite those
sides are respectively 0,',o and 180 -,,, we can express
the  coefficients  of the  above  equations,  containing  1',-1, 1,,
and  a', by  0), z  and' and  we  find:
cos S' cos (a' - a')    cos  cos (a +- a) [cos    cos z cos z' - sin z sin z']
-cos S sin (a + — a) [cos o  sin z cos z' I- cos z sin z']
sin S sill O cos Z'
cos S' sin (a' - a') ).=  cos S cos (a +- a) [cos   cos z sin z'   S- sin z cos z']
-  cos S sin (a o- a) [cos O sin z sin z' -  cos z cos z']
-sin   sinl   sin z'
sin a' -   cos a cos (a H- a) sin O cos z
-cos S sin (a +- a) sin O sin z
- sin S cos 0.
Multiplying the first of these equations by sin z', the
second  by  cos z' and  subtracting  the  first,  then  multiplying
the  first by  cos z', the  second  by  sin z' and  adding  the  products we get:
COs t' sin (a' H — a' -  z') -  cos S sin (a H- a +- z)
cos S' cos (a' -  a' -z')= — cOS  cos (a -    -  z) cos O  - sin S sin O   (a).
sin' -  cos S cos (a -I ca — z) sin O +-  sin 8 cos 0.




126
These formulae give ca' and (' expressed by c, Y', a, a'
and the auxiliary quantities z& a' and A. These latter quantities may be found by applying Gauss's formulae to the sphlleriCal triangle considered before, as we have:
sin.l  co s  (z' - z) = sin - (1',- 1,) sin - (~'o -+,)
sin 2 0 sin { (z' -- a)   cos. (1', -C 1) sin ll  (a'o - ao)
cos - 0 sin  (z' -- z)s  -in- (1- l, ) cos Z (E' - E0o)
COS s0 0 Ccos  (z' -z)- eCOS   (1,' -  ) cos 2 (~'o -- ao)
As we may always take here instead of sin.1 (z' --  )
and  sin 2 (Eo -  m0) the arc itself and the corresponding cosines equal to unity, we find the following simple formulae
for computing these three auxiliary quantities:
tang z- (az t.' ) -- ) os i- (a'o +  eo) tang. (1',- 1,)' -  )       (-   ) cota) g, -      (A).
s.in    (S'  -- ao)
tang 4- 0   tang  - (o -+- ao) sin  (ad.- A).
The formulae (a) can be rendered more convenient fbr
computation by the introduction of an auxiliary angle or we
may use instead of them  a different system  of formulae derived  from  Gauss's  equations.  For we arrive at the formulae (a) if we apply  the  three fundamental formulae of
spherical trigonometry to a triangle, whose sides are 90~ —',
90"-.    and 0, whilst the angles opposite the two first sides
are respectively a' — a -1 —    and 180 0 -  c' -  a-' -+-'.  If we
now apply to the same triangle Gauss's formulae and denote
the third angle by c, a- a+z-   by A and a'-+a' —' by A,
we find:
s 4- (900 -+ a') cos - (A' -+ c) = cos, [900 -+ a +   ] cos I- A
cos 2 (90 + A') sin 2 (A' -- c)  cos  [90 -+ --- 0] sinA   (I)
sin i (90 -+ a') os 2 (A' -     sin 2 [90 +4 a +0  ] cos I A
sin  (90 -i-') sin  (A'- c)   sin 2 [90 q-3 — 0] sin 2 A.
As it is even more accurate to find the difference A'- A
instead of the quantity A' itself, we multiply the first of the
equations (a) by  cos A, the  second  by sin A  and subtract
them, then we multiply the first equation by sin A, the second by cos A  and add the products.  We find thus:
cos 5' sin (A' - A) = cos S sin A sin 0 [tang 8 — + tang.- O cos A]
cos A' cos (A' - A)= cos S - cos S cos A sill 0 [tang 8 +- tang' O cos A],
hence:
sin A sin 0 [tang a -f- tang 4- 0 cos A]
1 - cos A sin 0 [tang S -+- tang - O cos A]




127
and from Gauss's equations we find:
Cos   c. sin  (A-A)    sin 0' cos,  (A' -- A)
cos - c. cos  cos (A' - A).
If we put therefore:
p   sin 0 [tang A + tang - 0 cos A]    (B),
we have:
tang (A'- A) )     p Si A
1 - p Cos A
and:                                                     (C).
tang (2' - -) —tang  o                     (-.0'   Cos, (A'- A)
By  the  fornulllae (A), (B) and (C) we  are  enabled to
compute rigorously the right ascension and declination of a star
for the timne 1750 --- t', when the right ascension and declination for the time 1750 -- t are given.
Example.  The right ascension and declination of ax Ursae
minoris at the beginning  of the year 1755 is:
af -10-  55' 44". 955
and                         -  870 59' 41". 12.
If we wish  to  compute from  this the place  referred to
the equator and the equinox of 1850, we have first:
i1  4' 11". 8756' -- 1o 23' 56". 3541
a -- 0". 8897                a' -- 15". 2656
o -- 230 28' 18". 0002       ~0 -= 230 28' 18". 0984.
With this we find from  the formulae (A):
4- (z' -- z)  00 36' 34". 314   - (- z)  10". 6286
hellce:
z - 0~ 36' 23". 685
-— 0~ 36' 44". 943
and:
0 - 00 31' 45". 600
therefore:
A -- a      -  q- z -- II  32'9". 530.
If we comptute then the values of A' — A and A' —   from
the formulae (B) and (C), we find:
log ip - 9,4214471
and:
A' -A  -4" 4' 17". 710,  2 (' _- 0) = 0o 15' 26". 780
hence:
A' --- 150 36' 27". 240
and at last:
a'= 160 12' 56". 917' —88 30 34. 680.




128
4.  As the point of intersection of the equator and the
ecliptic has an annual retrograde motion of 50".2 on the latter, the pole of the ecliptic describes in the course of time
a small circle around the pole of the ecliptic, whose radius
is equal to the obliquity of the ecliptic*). The pole of the
equator coincides therefore with different points of the celestial sphere or different stars will be in its neigbourhood
at different times. At present the extreme star in the tail of the
Lesser Bear (a Ursae minoris) is of all the bright stars nearest
to the north-pole and is called therefore the pole-star. This
star, whose declination is at present 882 0, will approach still
nearer to te pole, until its right ascension, which at present
is 17", has increased to 90'. Then the declination will reach
its maximum 89" 32' and begin to decrease, because the precession in declination of stars whose right ascension lies in
the second quadrant, is negative.
In  order to find the place of the pole for any time t,
we must consider the spherical triangle between the pole of
the ecliptic at a certain time to and the poles of the equator
P and P' at the times to and t. If we denote the right ascension and declination of the pole at the time t referred to the
equator and the equinox at the time t,, by cr and A, and the
obliquity of the ecliptic at the times to and t by s, and a,
we have the sides PP' -  90" -3,  EP              c-E, EP' =-, the
angle at P =90" -c a and the angle at E equal to the general precession in the interval of time t -to; we have therefore according to the fundamental formulae of spherical trigonometry:
cos 8 sin a = sin e cos so cos I - cos e sin so
cos 3 cos a - sin & sin 1
sin 3 = sin s sin so cos I q- cos E Cos So.
This computation does not require any great accuracy,
as we wish to find the place of the pole only approximately
and although the variation of the obliquity of the ecliptic
for short intervals of time is proportional to the time, we
may take e =  eo and get simply:
tang a = - cos oa tang 2 1
*) This radius is strictly speaking not constant, but equal to the actually
existing obliquity of the ecliptic.




129
and:
sin se  sin l
CO S     a
Though c. is found by means of a tangent, we find nevertheless the value of a without ambiguity, as it must satisfy
the condition, that cos cc and cos A have the same sign.
If we wish to find for instance the place of the pole for
the year 14000 but referred to the equinox of 1850, we have
the general precession for 12150 years equal to about 1740,
hence we have:
c -- 2730 16' and  --  43 7'.
This agrees nearly with the place of a Lyrae, whose
right ascension and declination for 1850 is:
a - 277  58' and    3 --- 38  39'.
Hence about the year 14000 this star will be the pole-star.
On account of the change of the declination by the precession stars will rise above the horizon of a place, which
before were always invisible, while other stars now for instance visible at a place in the northern hemisphere, will move
so far south of the equator that they will no longer rise at
this place. Likewise stars, which now always remain above
the horizon of the place, will begin to rise and set, while
other stars will move so far north of the equator that they
become circumpolar stars. The precession changes therefore
essentially the aspect of the celestial sphere at any place on
the earth after long intervals of time.
The latest tables of the sun give the length of the sidereal year, that is, the time, in which the sun describes
exactly 3600 of the celestial sphere or in which it returns to
same fixed star, equal to 365 days 6 hours 9 minutes and
9s.35 or to 365.2563582 mean days. As the points of the
equinoxes have a retrograde motion, opposite to the direction
in which the sun is moving, the time in which the sun returns to the same equinox or the tropical year must be shorter
than the sidereal year by the time in which the sun moves
through the small arc equal to the annual precession.  But
we have for 1850  -- 50".2235 and as the mean motion of
the sun is 59'8".33, we find for this time 0.014154 of a day,
hence the length of the tropical year equal to 365.242204
9




130
days. As the precession is variable and the annual increase
amounts to 0".0002442966, the tropical year is also variable
and the annual change equal to 0.000000068848 of a day. If
we express the decimals in hours, minutes and seconds, we
find the length of the tropical year equal to:
365 days 5h 481" 46s. 42 - Os. 00595 (t - 1800).
II. THE NUTATION.
5. Thus far we have neglected the periodical change
of the equator with respect to the ecliptic, which, as was
stated before, consists of a periodical motion of the point of
intersection of the equator and the ecliptic on the latter as
well as in a periodical change of the obliquity of the ecliptic.
The point in which the equator would intersect the ecliptic,
if there were no nutation, but only the slow changes considered before were taking place, is called the mean equinox
and the obliquity of the ecliptic, which would then occur,
the mean obliquity of the ecliptic. The point however, in
which the equator really intersects the ecliptic at any time
is called the apparent equinox while the actual angle between
the equator and the ecliptic at any time is called the apparent
obliquity of the ecliptic.
The expressions for the equation of the points of the
equinoxes and the nutation of the obliquity are according
to the latest determinations of Peters in his work entitled,,Numerus constans nutationis"':
a - 17". 2405 sin g -+ 0". 2073 sin 2 1
-  1". 2692 sin 2 Q) - 0". 2041 sin 2 (
J+  0". 1279 sin () - P) - 0". 0213 sin (~ + P)
+- 0". 0677 sin ((- P')                  (A)
A   +- 9". 2231 cos  -- 0". 0897 cos 2 5
+  0". 5509 cos 2 () + 0". 0886 cos 2 C
+  0". 0093 cos (() - P),
where S) is the longitude of the ascending node of the moon's
orbit, 0 and ( are the longitudes of the sun and of the
moon and P and P' are the longitudes of the perihelion of
the sun and of the perigee of the moon.  The expressions




131
given above are true for 1800, but the coefficients are a
little variable with the time.and we have for 1900:
= - 17". 2577 sin f) +4- 0". 2073 sin 2 f)
-   1".2693 sin 2 0 - 0". 2041 sin 2 ((
+  0o". 1275 sin (  -P)- o".0213 sin ((-i-P)
+-  0". 0677 sin (  - P')                     (A,)
ia   9". 2240 cos g - 0". 0896 cos 2 5)
+- 0". 5506 cos 2 -- +- 0". 0885 cos 2 ((
- 0". 0092 cos (-A +- P).
In order to find the changes of the right ascensions and
declinations of the stars, arising from this, we must observe,
that we have:
da~t    d         (a  2 )  +(   d )')a+   (_aA  +
and:                                                              (aC)
But we  have  according  to  the  differential formulae in
No. 11  of Section I, if we substitute instead of cos Pi sin j
and cos fi cos Vi their expressions in terms of a, 5 and A:
da                 d8
-cos  +- sin e tang 8 sin a               cosasin
da                                   d8
d -  cos cca tang                 d     in a,
from  which we find by differentiating:
- ),2 =)-:sijn ~2~ [2 sin 2 a + cotang a cos a tang   +- sin 2 a tang 82]
\ d/2-d   - sins  [cos a2 - cotang e tang 8 sin a + tang 32 cos 2a]
(d2  -) -_ _[  sin 2 a --- sin 2 a tang  2]
(d2-sj  — sin a2 sin a [cotang ~ -+- tang a sin a]
- (d..e)   sin a cos a [cotang e +- sin a tang 8]
(de)2   = -   cos a2 tang 8.
If we substitute these expressions in the equations (a)
and introduce instead of A)X and A e their values given before by the equations (A) and take for E the mean obliquity
of the ecliptic at the beginning of the year 1800 =  23" 27' 54". 2,
we find the terms of the first order as follows:
9 




132
a'- a —— 15". 8148 sin f- [6". 8650 sif si sin a - 9". 2231 cos ~) cos a] tang 8
+  0". 1902 sin 2&)+ [0". 0825 sin 2~ sin a +0". 0897 cos2f) cosal tang (
-  1". 1642 sin2(i - [0". 5054 sin 2sinin +0". 5509 cos2( cosca] tang S
-  0".1872 sin2 ( -[0". 0813 sin 2 ( sin a +0". 0886 cos 2(( cosa] tang S
-  0".0195 sin (0(+P)
-  [0".0085 sin(Ob + P) sin a+ 0". 0093 cos (O + P) cos a] tang 8   (B)
+- [0". 0621 -+- 0".0270 sin a tang 5] sin (((- P')
+  [0". 1173 + 0". 0509 sin a tang S] sin (( - P),
8' —        G".8650 sin /~ cos a + 9".2231 cos ) sin a
+  0". 0825 sin 2 /} cos a- 0". 0897 cos 2 f) sin a
-- 0". 5054 sin 2 ( cos a +-  0"..5509 cos 2 () sin a          (C)
-  0".0813 sin 2 (( cos a - 0".0886 cos 2 ( sin a
-  0". 0085 sin ((+) - P) cos cca -- 0". 0093 cos (  -4- P) sin a
- 0".0270 cos a sin ((( - P')
-+   ". 00509 cos a sin (0 - P).
These  expressions are true for 1800; for 1900 they  are
a little different, but the  change is only  of some amount for
the  first terms  depending  on  the  moon's  node.  These  are
for 1900:
in a'- acc:  - 15".8321 sinf) - [6".8683 sin t' sin a+ 9".2240coso Ccos a] tang S
in 8' - 5:  -  6".8683 sin t cosa + 9". 2240 cos   sina.
Of the terms of the second order only those are of
any  amount, which  arise from  the greatest terms in AX  and
BA.  If we put for the sake of brevity:
zAs - 9!'.2231 cos g   a =  C cos /'
and   - sin  2 -  6". 8650 sin --   b sin,
these terms give in right ascension:
bI __a2                        b2
a' — a=-          sin 2 a [tang S2 +  ]      + --  tang 8 cos a cotang e
4                      2~`r    4
a b
- [4 — cotangE sin a tang 8+- tang 82 cos 2 a+ —   cos]2 2l- sin 2a)
b2+a- a2                4 2                   b2 +2.
-}-         tang S2 sin 2 a + -   tang S cos a cotg e -  8    sin 2 a  cos 2 f
and in declination:
(a2+-kb2   a — b2                   b2
S'                   4 8  _ +-  cos 2a tangS  ---   sin a cotang s
ab
- [tang  sin 2 a + 2 cotang E cos a]   sin 2 f'
( 2    a 2              bcos 2 a) tang   --  sin a cotang s cos 2 
Those terms which are independent of ~) change merely




the mean place of the stars and therefore may be neglected.
Another part, namely:
ab.         ab       sin a   2 - - ~b'       cos a
sin 2   k -2 cotang     sin 2   -- - cotang   cos a cos 2   tang g
and
ab                     b2
2 cotang E sin 2 f cos c - --   cotang e sin a cos 2,'
can be united  with the similar terms multiplied by sin 2f&
and cos 2D) of the first order, which then become equal to:
in right ascension
+- 0". 1902 sin 2  +) + [O". 0822 sin 2 &) sin a -+ 0". 0896 cos 2 5 cos a] tang y
and in declination                                        (D)
+ 0".0822 sin 2 ~ cos a - 0". 0896 cos 2 5Q sin a.
The remaining terms of the second order are as follows:
in right ascension
+- 0".0001535 [tang 82 - 4] sin 2 f' cos 2 a
- 0".000160 [tang 82 +- 4] cos 2 f sin 2 a
and in declination                                        (E)
- 0".0000768 tang' sin 2 a sin 2 f'
- [0". 000023 +- 0". 000080 cos 2 a] tang S cos 2 fQ.
But as the first terms amount to 0O. 01 only when the
declination is 880 10' and as the others equal 0".01 only when
the declination is 89~ 26', they are even in the immediate
neighbourhood of the pole of little influence and can be neglected except for stars very near the pole.
6.  We shall hereafter use the changes of the expressions (B) and (C) produced by a change of the constant of
nutation, that is, of the coefficient of cos D in the nutation
of obliquity.  These are different for the terms of the lunar
and solar nutation.  For in the formula of the nutation as
given by theory all terms of the lunar nutation are multiplied by a factor N' which depends on the moments of inertia of the earth as well as on the mass and the mean motion
of the moon, while the terms of the solar nutation are multiplied by a similar factor, which is the same function of the
moments of inertia of the earth and of the mass and mean
motion of' the sun.  But as it is impossible to compute the
moments of inertia of the earth, the numerical values of N
and N' must be determined from  observations. Now  the co



134
efficient of the term of the nutation of obliquity, which is
multiplied  by  sing), is equal to  0.765428N'.  If we take
this equal to 9".2231 (1 —i), where 9".2231 is the value of
the constant of nutation as it follows from the observations,
while 9". 2231 i is its correction, we have therefore:
0.765428 IN'= 9".2231 (1 +- i).
But the lunisolar precession depends on the same quantities N  and N' and  the value determined from  observations
(50".36354 for 1800) gives the following equation between
N and N':
17.469345 = N+ 0.991988 N',
from which we get in connection with the former equation:
N=- 5.516287 (1 - 2 16687 i).
Therefore  if we take  the  constant of nutation equal to
9". 2231  (1 — i) we must multiply  all terms of the lunar
nutation  by  1 -+  i and  all terms of the  solar nutation  by
1 -2. 16687 i.  Taking therefore 9". 2235 i =  dv, we have:
1;= — 1.8702 sinf+0.0225 sin2fl -0.0221 sin2 1+0.0073sin(&-P') d
-t +0.2981 sin 2 (i)-0.0300 sin (  -P) + 0.0050 sin ( - P)    d
dA E  [cos D -0.0097 cos 2 9) + 0.0096 cos 2 (- 0.1294 cos 2 0
-0.0022 cos (O -P)] d!
and from  this we find in the same way as in No. 5:
d (jt' ~a) -- 1.7156 sin ) - [0.7445 sin g Q sin a +- 1 0000 cos g cos a] tang A
dv
- 0.0206 sin 2- +q [0.0090 sin 2) sina+-0.0097 cos2: cosa] tang S
-0,0203 sin2 ( - [0.0088 sinl 2 ( sin a+0.0096 cos2 ( cosa] tangS
+ 0.0067 sin(((- P') - [0.0029 sin ((( -P' ) sin a  ] tang S
~ 0.2735 sin 2G +- [0.1187 sin 2( sina+0..1294 cos2( cosa] tang S
-0.0275 si ((3 -P)-[0.0119 sin (  -P) sina         ] tang o
+- 0.0046 sin ((3 +P) + [0.0020 sin (0 +P) sin a -
+ 0.0022 cos (0(+-P) cosa] tang S
d('-) - ) — 0.7445 sin b cos a +- 1.0000 cos -) sin a
dcv
+ 0.0090 sin 2 t) cos a - 0.0097 cos 2 g sin a
- 0.0088 sin 2 ( cos a -+ 0.0096 cos 2 ( sin a
- 0.0029 sin ((Q - P') cos a
+0.1187 sin 2  cos a - 0.1294 cos 2  sin a
-0.0119 sin (  - P) cos a
-+ 0.0020 sin (( 0- P) sin a — 0.0022 cos (C- + P) sin a.
7.  In order to compute the nutation in right ascension
and declination it is most convenient to find the values of,A. and hem from  the formulae (A) and (Al) and to compute




135
da cda
the numerical values of the differential coefficients d a  dete.
But the labor of computing formulae (B) and (C) has been
greatly reduced by the construction  of tables.  First the
terms:
-15".82 sinf = c and - 1".16sin2() - g
have been brought in tables whose arguments are D) and 20.
The several terms of the nutation in right ascension
multiplied by tang ( are of the following form:
a cos,8  cos ax + b sin, sin a — A [h cos fi cos a +- sin fi sin a].
Now  any expression of this form  may be reduced to
the following form:
x cos [a  a +,
For if we develop the latter expression and compare it
with the former, we find the following equations for determining x and y:
A h cos -- x [cos 8 cos y - sin, sin y]
A sin  - x [sin fB cos y + cos p sin]    (
from  which we find:.2 _ A2 [I _     (L -/2) cos f2]
and:                      (1 - h) sin fi cos f(
tangy   -- 1  -(1 - h) cos f2
where x and y are always real. If we have now tables for
x and y, whose argument is 7], we find the term  of the nutation in right ascension, multiplied by tang' by computing:
x cos [, + y - a]
while:                                     (c),
x sin [f -+-y - a]
gives the term of the nutation in declination depending cos /.
For as these terms have the form:
A [- h cos f sin a +- sin f co's a],
we find taking it equal to x sin ((-F-y - a) the same equations
(b) for determining x and y.
Such tables have been computed by Nicolai and are given in the collection of tables by Warnstorff, mentioned before. These give besides the quantity c the quantities log b
and B  with the argument 0, and with these we find the
terms of the right ascension depending on cos D0 and sin ~)
by computing:
c - b tang S cos (' -+ B - a)




136
and the corresponding terms of the declination by computing:
-b sin (O) + B - a)       (d)
This part of the nutation together with the small terms
depending on 2~, 2 (L and ( —P', is the lunar nutation.
A second table gives the quantities g, log f and F with
the argument 20, by which we find the terms depending on
2C, which for right ascension are:
g — ftang S cos [2 ( -+ F- a]
and for declination:                       (e)
-f sin [2 (+ -- F — a]
This part of the nutation together with the small terms
depending on (-+P and 0 —P is the solar nutation.
No separate tables have been computed for the small
terms depending on 2 (, 2 ~) and () -- P.  For these may
be found from the tables of the solar mutation, using instead
of 2 0 as argument successively 2 (, 180 - 2 &) (because these
terms have the opposite sign) and 0 -P, and multiplying
tle values obtained according to the equations (e) respectively
by  l, y-3 and;,- as these fractions express approximately the
ratio of the coefficients of these terms to that of the solar
nutation.
The form of the terms multiplied by   — P' and (D   P
is different, but analogous to the annual precession in right
ascension and declination; they are therefore obtained by
nmultiplying the annual precession in right ascension and declination by 412 sin ( -.P') and,,1 sin (O-P).
8. If we consider only the largest term  of the nutation
we can render its effect very plain.  We have then:,A i  -- 17".205 sin M2,
A.E- - 9". 22 cos S,
or rather according to theory:
sin E,2-A 10". 05 cos 2 s. sin 5l,:A  - 10". 05 cos e. cos A.
Now  the pole of the equator on account of the lunisolar precession describes a small circle, whose radius is e,
about the pole of the ecliptic. If we imagine now a plane
tangent to the mean pole at any time and in it a system of
axes at right angles to each other so that the axis of x is
tangent to the circle of latitude, we find the co-ordinates of




137
the apparent pole (affected by nutation) y   Sill 8 AX, x'L\ 8
and we have therefore according to the expressions given
above the following equation:                   vi
COS               0 82
The apparent pole describes therefore an ellipse around
the mean pole, whose semi-major axis is C cos E =  9". 22, and
whose semi-conjugate axis is C cos 2  =  6". 86. This ellipse
is called the ellipse of nutation. In order to find the place
of the pole on the circumference of this ellipse, we imagine
a circle described about its centre with the semi-major axis
as radius. Then it is obvious, that a radius of this circle
must move through it in a time equal to the period of the
revolution of the moon's nodes with uniform and retrograde
motion ), so that it coincides with the side of the major axis
nearest to the ecliptic, when the ascending node of the moon's
orbit coincides with the vernal equinox. If we now let fall
from  the extremity of this radius a line perpendicular to the
major axis, the point, in which this line intersects the circumference of the ellipse, gives us the place of the pole.
*) As the motion of the moon's nodes on the ecliptic is retrograde.




THIRD SECTION.
CORRECTIONS OF THE OBSERVATIONS ARISING FROM THE
POSITION OF THE OBSERVER ON THE SURFACE OF THE
EARTH AND FROM CERTAIN PROPERTIES OF THE LIGHT.
The astronomical tables and ephemerides give always the
places of the heavenly bodies as they appear from the centre
of the earth.  For stars at an infinite distance this place
agrees with the place observed from any point on the surface
of the earth. But when the distance of the body has a finite
ratio to the radius of the earth, the place of the body
seen from the centre must differ from the place seen from
any point on the surface. If we wish therefore to compare
any observed place with such tables, we must have means
by which we can reduce the observed place to the place
which we should have seen from  the centre of the earth.
And conversely if we wish to employ the observed place
with respect to the horizon in connection for instance with
its known position with respect to the equator for the computation of other quantities, we must use the apparent place
seen from  the place of observation, and hence we must
convert the place seen from the centre, which is taken from
the ephemeris, into the apparent place.
The angle at the object between the two lines drawn from
the centre of the earth to the body and to the place at the surface is called the parallax of the body. We need therefore
means, by which we can find the parallax of a body at any
time and at any place on the surface of the earth.
Our earth is surrounded by an atmosphere, which has
the property of refracting the light. We therefore do not
see the heavenly bodies in their true places but in the direction which the ray of light after being refracted in the




139
atmosphere has at the moment, when it reaches the eye of
the observer.  The angle between this direction and that,
in which the star would be seen if there was no atmosphere,
is called the refraction. In order therefore to find from observations the true places of the heavenly bodies, we must
have means to determine the refraction for any part of the
sphere and any state of the atmosphere.
If the earth had no proper motion or if the velocity of
light were infinitely greater than that of the earth, the latter
would have no effect upon the apparent place of a star. But
as the velocity of the light has a finite ratio to the velocity
of the earth, an observer on the earth sees all stars a little
ahead of their true places in the direction in which the earth
is moving.  This small change of the places of the stars
caused by the velocities of the earth and of light, is called
the aberration.  In order therefore to find the true places
of the heavenly bodies from  observations, we must have
means, to correct the observed places for aberration.
I. THE PARALLAX.
1. The earth is no perfect sphere, but an oblate spheroid
that is a spheroid generated by the revolution of an ellipse
on its conjugate axis. If a denotes the semi-major axis, b
the semi-minor axis of such a spheroid, and a is their difference expressed in parts of the semi-major axis, we have:
a- b       b
1 — __
a        a
If then s is the excentricity of the generating ellipse or
of the ellipse, in which a plane passing through the minor
axis intersects the surface of the spheroid, also expressed in
parts of the semi-major axis, we have:
a2
therefore:             -    /
and                     --  -1/ 1.2
likewise:                    2 _ —.




140
The ratio -  is for the earth according to Bessel's in295. 1528
vestigations:              291
or we have:               — a 29952
299. 1528
and expressed in toises:
a - 3272077. 14 log a  6 6.5148235
b - 3261139.33 log b - 6.5133693.
However in astronomy we de not use the toise as unit
but the semi-major axis of the earth's orbit. If we denote
then by n the angle at the sun subtended by the equator/al
radius of the earth  and by R  the semi-major axis of the
earth's orbit or the mean distance of the earth from the sun,
we have:
1- R sin 7(
X. =
or~                       a
206265
The angle i or the equatoreal horizontal parallax of the
sun is according to Encke equal to:
8". 57116.
It is the angle at the sun subtended by the radius of a
place on the equator of the earth when the sun at this place
is rising or setting.
2.  In order to compute the parallax of a body for any
place at the surface of the earth, we must refer the place
on the spheroidal earth to the centre by co-ordinates. As the
Fig. 3.               first co-ordinate we use
the sidereal time or the
/       Ace    angle, which a plane passing through the place of
observation andthe minor
L? I-  -/' //             axis ~) makes with the
plane passing through the
sam e axis and the point
of the vernal equinox. If'
/'  \                then OA C Fig. 3 represents the plane through
*) This plane is the plane of the meridian, as it passes through the
poles and the zenith of the place of observation.




141
the axis and the place of observation, we must further know
the distance A 0 -o from  the centre of the earth and the
angle A 0 C, which is called the geocentric latitude. But these
quantities can always be computed from  the latitude A NC
(or the angle which the horizon of A makes with the axis
of the earth or which the normal line AN at the place of
observation makes with the equator) and from the two axes
of the spheroid.
For if x and y are the co-ordinates of A with respect
to the centre 0, the axes of the abscissae and ordinates being
OC and OB, we have the following equation, as A is a point
of an ellipse, whose semi-major and semi-minor axes are a
and b:
a y2 -2- b2 x2   2 b2.
Now we have also, if we denote the geocentric latitude
by A':                           Y
tang T'
dx
and also:              tang T     dx
because the latitude q? is the angle between the normal line
at A and the axis of the abscissae. As we have then from
the differential equation of the ellipse:
y     b2 dx
x     a2 dy
we find the following equation between (q, and (p':
b2
tang 99   - tang 99    (a).
In order to compute o we have:,   /x +       X 2 _o
and as we obtain from the equation of the ellipse:
a                a
1+-/a-g2   1ta +tang. tang99
we find:
_ _    g_   _ tang =    cos 99' Cos (99-) (-)'
If therefore the latitude (~ of a place is given, we can
compute by these formulae the geocentric latitude p,' and the
radius O.




142
For the co-ordinates x and y we easily get the following
formulae, which will be used afterwards:
Xa cos 9
gcos 92 + (I - E2) sin 99 2
a cos q0
-    _ _ _ _ _ _ _ _ _ _ _ _   ( c)
E-,  E2sin 972
and
y   x  tang bpY   x  -- tang 9- x (1 - s2) tang go
a (1 -  2) sin              (d).
Vi-' sin p9
From  the  formula (a) we  can  develop  cp' in  a series
progressing according to the sines of the multiples of,p, for
we obtain by the formula (16) in No. 11 of the introduction:
a2   b2,      a2 - b2(2'  9or  --  2 a b-T.  sin 2  ( q-  2. -b9  sin 4q — etc.  (A)
or taking
a - b
a-I-b
we find:
2 a   s2n 2                       -
99'58 --- 2  sin 2+j  2              ) s2ill  9  etc.      (B).
If we compute the numerical values of the coefficients
from the values of the two axes given above and multiply
them  by 206265 in order to find them  in seconds, we get:
A'._ 9o_ 11' 30". (65 sin 2  V +  1".16 sin 4 A -...  (C),
from  which  we find  for instance  for the  latitude of Berlin
- = 52" 30' 16". 0' -- 520 19' 8".3.
Although e itself cannot be developed into an equally
elegant series, we can  find  one for log qo).   For we get
from formula (b):
t2
os 91   -  tang 
If we substitute here for cos (p'2 its value
a 4
ac4 -+ b4 tang 22
*) Encke in the Berliner Jahrbuch fiir 1852 pag. 326. He gives also
tables, from which the values of p' and log q may be found for any latitude.




143
we find:
a4 Cos P2   4 sin 92
at2 COs 2 2 + b sin T2
a - b -1- (a4 - bV) cos 2 9
a2 -- b' - (a2 - b2) cos 2 S~
(a    b 2)2 + (a2 - b2)2 - 2 (a2 ~ 62) (at 2   b2) cos2 
(a + b)2 -- (a - b)2 + 2 (a + t1) (a - b) cos 2 p
hence:
C         ) + 22.(.    _.cos 2 o]
(a         _    (a 2           a 2_
~(a —))2   a    b
) -      -   -b 2+ 2       C —'- l cos 2 
If we write this formula in a logarithmic  form  and develop  the logarithms of the  square  roots according to formula (15) in No. 11 of the introduction into series progressing according to the cosines of the multiples of 2 (~, we find:
lo'g-b'  I a a    b: a -, b    a
log hyp  = log hyp     a +- - -    
(a2 - b2 )2 (a- b )2
(2__           - "_s-              (D)
(a2- b2)3 _      (a- b\  cos C
-- etc.
or using common logarithms and denoting the quantity
a-b
a+ b
by n, we get:
log =clog (a-j o-) +               -     eMn(s 2 9
- 2 n  co
-   2T1  2+n2s   >?- ~],   cos 4      (E)
-+   -1-x _2n) -n3 Cos 6 9
-   etc.'
where M denotes the modulus of the common logarithms,
hence:
log 11- 9.6377843.
If we compute  again the numerical values of the coefficients and take a — 1, we find:
log e- 9. 9992747 -+ 0.0007271 cos 2 9 -- 0.0000018 cos 4 Tp   (F)
and from this we get for instance for the latitude of Berlin:
log eo = 9.9990880.




144
If we know  therefore the latitude of a place, we call
compute from the two series (C) and (F) the geocentric latitude and the distance of the place from the centre of the
earth and these two quantities in connection with the sidereal
time define the position of the place with respect to the centre
of the earth at any moment.  If we now imagine a system
of rectangular axes passing through the centre of' the earth,
the axis of z being vertical to the plane of the equator, whilst
the axes of x and y are situated in the plane of the equator
so that the positive axis of x is directed towards the point
of the vernal equinox, the positive axis of y to the point
whose right ascension is 90", we can express the position of
the place with respect to the centre by the following three
co-ordinates:
X  e cos T9 Cos 0
y --  cos 9' sin O  (G).
z  9 sin T'
3. The plane in which the lines drawn from the centre
of the earth and from the place of observation to the centre
of the heavenly body are situated, passes through the zenith of the place, if we consider the earth as spherical, and
intersects therefore the celestial sphere in a vertical circle.
Hence it follows that the parallax affects only the altitude
of the heavenly bodies while their azimuth remains unchanged.
If A (Fig. 3) then represents the place of observation, Z
its zenith, S the heavenly body and O the centre of the
earth, Z OS is the true zenith distance z as seen from the
centre of the earth and ZAS the apparent zenith distance za'
seen from the place at the surface. Denoting then the parallax or the angle at S equal to z'  z by p' we have:
sin p'  --  sin z',
where A denotes the distance of the body from the earth,
and as p' is always a very small angle except in the case
of the moon, we can always take the are itself instead of
the sine and have:
p e sinl.206265.
Hence the parallax is proportional to the sine of the apparent zenith distance. It is zero at the zenith, has its max



145
imum in the horizon and has always the effect to decrease
the altitude of the object. The maximum value for a' = 900
p —  206265
is called the horizontal parallax and the quantity
p — 206265,
where a is the radius of the earth's equator, is called the
horizontal equatoreal parallax.
Here the earth has been supposed to be a sphere; but
as it really is a spheroid, the plane of the lines drawn from
the centre of the earth and from the place of observation to
the object does not pass through the zenith of the place,
but through the point, in which the line froml the centre of
the earth to the place intersects the celestial sphere. Hence
the parallax changes a little the azimuth of an object and
the rigorous expression of the parallax in altitude differs a little
from the expression given before.
If we imagine three axes of co-ordinates at right angles
with each other, of which the positive axis of z is directed
towards the zenith of the place, whilst the axes of x and y
are situated in the horizon, so that the positive axis of x
is directed towards the south, the positive axis of y towards
the west, the co-ordinates of the body with respect to these
axes are:
A' sin z' cos A', A' sin z' sin A' and A' cos z',
where A' denotes the distance of the object from the place
and z' and A are the zenith distance and azimuth seen from
the place.
The co-ordinates of the same object with respect to a
system of axes parallel to the others but passing through the
centre of the earth are:
Ai sin z cos A, A sin z sinA and A cos z,
where A denotes the distance of the object from  the centre
and z and A are the zenith distance and the azimuth seen
from the centre. Now as the co-ordinates of the centre of'
the earth with respect to the first system are:
- e sin (p -- p'), 0 and - e cos ( -')
we have the following three equations:
10




146
A' sin z' cos A' = A sin z cos A -- o sin (T - 9T')
A' sin z' sin A' = A sin z sin A' cos'      -  cos z -    cos ( - 99'),
or:          A' sin z' sin (A' - A)-=  sin (T - T') sin A' sin z' cos (A' - A)   A sin z --   sin (5 - p') cos A  (a)
cos Z'                 A cos -- o   cos(- -').
If we  multiply  the first equation by  sin 1 (A'-  A),  the
second by  cos   (A'-  A) and  add the two products, we find:
cos' (A' -I- A)
A' sin.' = A sin z --   sin (59 -- 9')    (A' 
cos I (A' - A)
A' cos' =    cos z -    cos (99 -  99').
Then putting:
cos  (A' +- A)
tangy Y     -  (A  -) tang (59 - )          ()'),
we find:
A' sin z' = A sin z - e cos (T - 99') tang y
A' Cosz' =  A cosz - V c os (9 - T'),
or:
cos y
A',I sin (z'-)=  ens (99-') cos s (  - y) (
A' cos (' -  )    -  - e cos (   )9 - 9 s(  ) 
and besides if we multiply the first equation by sin } (z'- z),
the second by cos  (z'- -) and add the products:
cns (~ -') co [  (e' n  - s]
c o s y C"~. -r.
If we divide the equations (a), (b) and (c) by A and put:
--    ill p,
taking  the  radius  of the earth's equator equal to unity, so
that p  is the horizontal equatoreal parallax, we obtain by the
aid of formulae (12) and (13) in No. 11 of the introduction:
A- A_,o sin p sin (            A +')    A ( sinpsin (9-9'))  i 2 A +
sin z                       sina 
y   cos A (99 - 99')- sin A tang I (A'- A) (go - 99')
sin A sin A' cos  ( (A' +- A)
Cos I (A' - A)3..
*) We have:
cos e  (A' +A)                  1____ A
ncos ] (A' -A) tang (99')   } cos  (A' - A)  3
IV:cos  (Al- A)                      cos I' --  A)'
Substituting here for tang (99 —9') the series
we can easily deduce the expression given above.




147
o sin p cos (5v --')
zI -z-,                  sin (z-y)
cos y
sin p cos (9 9    — (j l 
+ ] I ie o       sin 2 (z - y) +....
log hyp A' = log hyp         sin p cos (  os (z --  )
cos y
~d,(esinpcos(~R-09))2 cos2(z - -y)-.
Cos?
We have therefore neglecting quantities of the order of
sin p (p -  p(') which have little influence on the quantity  7'
-   (99 - I9') cos A
hence the parallax in azimuth is:
A' — A -_? sin p sin (-  ) sin A,
sin z
or its rigorous expression, which  must be used when z is
very small:
e sin p sin ( - ) sin A
sin A
sin z
tang (A' - A) =sin
1 -S-    p sin   (99- -') cos A
sin z
Furthermore as.:
cos (99 -- 9')  cos e (A' -- A) sin (s -- 99')
cos y         cos 2 (A' - A) sin y
is always nearly  equal to unity, the parallax in zenith distance is:
z' - z - e sin p sin [z - (p - I9') cos A],
and the rigorous equations for it are:
Al sin (z' - z) = ~ sin p sin [z - (Rv - 9R') cos A]
A
cos (z' - z)   1 -  sin p cos [z - (9p-') cos A].
Hence if the object is on the meridian, the parallax in
azimuth is zero and the parallax in zenith  distance is:
z'- -e z -   sin p sin [z - ( -')]
4.  In a similar way we obtain the expressions for the
parallax in right ascension and declination.  The co-ordinates
of a body with  respect to the  earth's centre  and the plane
of the equator are:
A cos 3 cos a, A cos a sin a and A sin 3.
The apparent co-ordinates as they appear from the place
at the surface with respect to the same plane are:
A' cos 8' cos a',   cos  sin a' and A' sinl'.
lo*




148
Since the co-ordinates of the place at the surface with respect to the centre referred to the same fundamental plane are:
e co's g' cos 0, o cos 99' sin 0 and g sin 99'
we have the following three equations for determining A', a'
anld':' Cos' Cos a' -  cos 3 cos a - v cos'cos O
zA' cos d sin cc' - A cos 6 sin a - e cos T' sin 0  (a)
A' sin'         ill      - A  sin   sin 9'.
If we multiply  the first equation by sin ce, the second
by cos a and subtract one from  the other, we find:
A' cos S' sin (a' - a)  - q cos p' sin (O - a).
But if we multiply the first equation by cos se, the second by sin a and add them, we find:
cL' COS' cos (a' - a) =, cos  - e cos 90' Cos (o - a).
We have therefore:
tang (a' - a)   cos' sin (a - 0)
tang (o' -car)A cos 3 - e cos T9' cos (a - 0).. sin (a - 0)
/ cos 
i cos S
or developing c'-'a in a series, we find:
a' _     _'X   sin- 0cos +,            sin 2 (a- 0)
Cos cos 3sJ
q-~ -.Acosa/  sin 3 ( -) +...  (A)
In all cases excepting the moon it is sufficiently accurate to take only the first term of the series. Taking then
the radius of the earth's equator. as the unit of ) and writing
in the numerator sin rT as factor (where g is the equatoreal
parallax  of the sun) in order to use the same unit in the
numerator as in the  denominator,  namely the semi-major
axis of the earth's orbit, we get:
Ia   cc= e     sin C cos T  sin (a - 0)
aI ~ ~ ~" —-.............   ~       (B)
cos 8
where a -  O is the east hour angle of the object. The parallax
therefore increases the right ascensions of the stars when east
of the meridian and diminishes them  on the west side of the
meridian. If the object is on the meridian, its parallax in
right ascension is zero.




149
In order to find a similar formula for' —( S. we will
write in the formula for:
A' cos Y' cos (a' - a)
now
1 - 2 sin - (a' - a)2
instead of
cos (a' - c),
and obtain:
CA' c       os A'' A cos  -e    cos ( - a) ~- 2 Sa' cos       (' - a) 2.
If we here multiply and divide the last term by cos " (a' --  )
and make use of the formula:
A' cos Y' sin (a' - a)   - e cos T' sin (0 - a)
we easily find:
cos [-  (a' —  )b)
Introducing  now  the auxiliary quantities A and;, given
by the following equations:
i        sin y  sin'
cos 9' cos [o - E (a' + a)]
f cos -
cos('-_   ) 2
we find from (b):
A' cos' =A cos -, cos y
and from the third of the equations (a):
A' sin' = aisin 8- q   sin y.
From these two equations we easily deduce the following:
A' sin (8' - 8) = -  f sin (y - 8)
A' cos(8' -  )    A - e  cos (y-  ),
or:'  sin (y -  )
tang (l- 8) -   _
1 — ~ cos ( - A)
or according to formula (12) in No. 11 of the introduction:
- sin (y -   )-   A2 sin 2 (y - 8) - etc.   (C)
If we introduce  here instead  of /? its value -sin9   and
sin y
write again Q sin mT instead of o in order to have the same
unit in the numerator as in the denominator, we find, taking
only the first term of the series:
-- c --- Soil  sin      sin (y-)
iA        sin y




150
If we further take in the second of the formulae (c)
cos, (a' - a) equal to unity and write c instead of'  (a'-+ —c),
we have the following approximate formulae for computing
the parallax in right ascension and declination:
arecos _' sin(O-a)
aA       cos 3
tang v99
cos (O - a)
8' — S-  --  d 7 sin n    s in(y  A) * )
A       sin y
If the object has a visible disc, its apparent diameter
must change with the distance.  But we have:
A' Sinl (8' - y)  sin ( -y)
A   A sin (6 - y)
sin (' -y)
and as the semi-diameters, as long as they are small, vary
inversely as the distances, we have:
oR   R - sin (X - y)
sin (8- y)
Exanimple. 1844 Sept. 3 De Vico's comet was observed
at Rome at 20  4111' 38S sidereal time and its right ascension
and declination were found as follows:
--    20~ 35' 55". 5
3= —18 43 21.6.
The logarithm of its distance from the earth was at that
tinme 9.27969 and we have for Roie:?' -41o 42'.5
and
log o  9. 99936.
The computation of the parallax is then performed as
follows:
*) If the object is on the meridian, we find:
--, sin (99'-  )  e   sinl [ -- ( -')],
hence the plarallax in declination is equal to the parallax in altitude.




0 in arc 3100 24'.5
a         2 35.9
0 -a       — 520 11'.4
tang T'  9 94999           y      550 2S'. 6
cos (O - a)  9. 78749      --      -  18 43. 4
sin (O - a)  9.89765,,     y- 8 —  4 ~74  12.0
-_      cos. 5257l'      sin (y - 3) 9. 98327
A  1          ~~*1  ~1.47576,,
secc   0.02362             A
cosec y 0.08413
log (a' - a)    1. 44703   log s' -     1. 04316,
a'  -a    -27". 99     - 8 -- 3 —- 34". 93
Thus the parallax  increases the geocentric right ascension of the comet 28". 0 and diminishes the geocentric declination  34".9.  Hence the place of the comet corrected for
parallax is:
a -    2 0 35' 27". 5
— ~ — -18 42 46.7.
In order to find the parallax of a body for co-ordinates
referred to the plane of' the ecliptic, it is necessary to know
the co-ordinates of the place of observation with respect to
the earth's centre referred  to  the  same fundamental plane.
But if we convert & and,' into longitude and latitude according to No. 9 of the first section and if the values thus
found are 1 and b, these co-ordinates are:
cos b cos 1
o cos b sin 1,o sin b
and we have the following three equations, where  il', 5', A'
are the apparent,,, /J5, A the true longitude and latitude:
A' cos fi cos Z' = A cos f cos Z - o cos b cos 1
A' cos f' sin'- = A cos, sin ) -,o cos b sin I
A' sin fi'    sin  - o sin b,
from  which  we  finally  obtain  similar equations as before,
namely:,Z, —   r cos b sin ( — )'A     cos,
tang b
g  cos (1 --  )
-8, -i   i ~ sinb sin ( —i)
A,     siny
O and ~p' are the right ascension and declination of that point,
in which the radius of the earth intersects the celestial sphere,




152
1 and b are therefore the longitude and latitude of the same
point.  If we consider the earth as a sphere, this point is
the zenith and the longitude of the point of the ecliptic
which is at the zenith is also called the nonagesimal, since
its distance from the points of the ecliptic which are rising
and setting is 90".
5.  As the horizontal equatoreal parallax of the moon
or the angle whose sine is,    being the distance of the
moon from the earth, is always between 54 and 61 minutes,
it is not sufficiently accurate to use only the first term of
the series found for the parallax in right ascension and declination and we must either compute some of the higher
terms or use the rigorous formulae.
If we wish to find the parallax of' the moon in right
ascension  and declination  for Greenwich for 1848 April 10
0lh mean timhe, we have for this time:
a!=  711 43m 20s. 25   1150 50' 3". 75
8 -- -S - 16  27' 22". 9
= — 11h 1711m Os. 02 -- 1690 15' 0".30
and the horizontal equatoreal parallax and the radius of the
moon:                     p -- 56' 57". 5
R= — 15' 31".3.
We have further for Greenwich:'P-= 510 17' 25".4
log o -= 9. 9991134.
If we introduce the horizontal parallax p of the moon
into the two series found for a' c-    and d' -     in No. 4, as
we have sin p=-, we find:
sin2(O-a)
a'-a  -- 206265 ~o cos 99 sin  P sin (O - a)
CossA
+ 9 (C cos  s  sin p  2 (- a)
+   cos. 99 sin p  sin 3 (O - a) -4and:
3       ---  206265  2 sin' sin p sin( - )
i~ cp~4 ssin    n
(e slnssin'  sin p (y-)      )
sin y




153
where we must use the rigorous formula for computing the
auxiliary angle y:
Ceos   (a'- a)
tang y   tang p' cos [-: (a' - a)]
cos [O-E- (a' 4- a)]
If we compute these formulae, we find for a'- a:
fiom the first  term:     -29'45". 71
second,,           -11. 47
third               --  0.03
hence a'- a --         29' 57". 21
and for 8' -:
from the first  term:     - 36' 34". 21
second,,            -20.91
third,,           -  0.12
hence'-         - 36' 55". 24.
The apparent right ascension and declination of the moon
is therefore:
a' -- 115~ 20' 6". 54' -- 150 50' 27". 66.
Finally we find the apparent semi-diameter:
R' - 15' 40". 20.
If we prefer to compute the parallax from the rigorous formulae, we must render them  more convenient for logarithmic
computation.  We had the rigorous formula for tang (a' -  ce):
tang (a' - a)    e cos Tq' sin p sin (a - 0) sec S
1-   cos c' sin p cos (a - 0) see  a
Further from  the two equations:
A' sin 8' = A [sin S -    sin 99' sin p]
and:
A' cos Y' cos (a' - a) = A [cos  -  cos To' sinp cos (a- 0)]
we find:
tang'     [sin  - g sin T' sinkp] cos (a' - a) sec  (
-  -q cos 9' sin p see A cos (a - 0)
Since we have:
A            cos 6' cos (a'- a)' -  cos     e cos T9' sin p cos (a - 0)
we find in addition:
sin 11 — --    cos 8' cos (a' - a) see   sin 
1 - e cos q' sin p see S cos (a - 0)
If we introduce in (a), (b) and (c) the following auxiliary quantities:
cos A = e sin p cos 9T' cos (a -  )
cos 3
and:
sin C= —  sin p sin 9',




154
we find the following formulae which are convenient for logarithmic computation:
tang (ac' - a) _ —   e  cos  i' sin p. sin (a - 0)
cos  sin4l A
tang& b' -sin  ( -C) cos  ( --  C) cos (a' -a)
cos ( sin ~ A2
and:
ail =  2   COS  COS (a' - a)
cos A sin I A 2.
If we compute the values cca'- a, ) and  R' with the
data used before, we find almost exactly as before:
a -a —  29' 57". 21
d' -- Jr- 15  50' 27". 68
R'    15' 40". 2t.
We can find similar formulae for the exact computation
of' the parallax in longitude and latitude and we can deduce
them  immediately from  the above formulae by substituting
x', it, I',:J'g I and b in place of ce', re, X, fi, 06 and cp'.
II. THEIl REFRACTION.
6.  The rays of light from  the stars do not come to us
through a vacuum  but through the atmosphere of the earth.
While in a medium  of uniform  density, the light moves in a
straight line, but when it enters a medium  of a different density, the ray is bent from  its original direction.  If the medium, like our atmosphere, consists of an infinite number of
strata of different density, the ray describes a curve.  But
an observer at the surface of the earth sees the object in the
direction of the tangent of this curve at the point where it
meets the eye and from  this observed direction or the apparent place of the star he must find the true place or the
direction, which  the ray  of light would  have, if it had
undergone no refraction.  The angle between these two directions is called the refraction and as the curve of the ray
of light turns its concave side to the observer, the stars
appear too high on account of refraction.
We will consider the earth as a sphere, as the effect
of the spheroidal form  of the earth upon the refraction is




155
exceedingly small. The atmosphere we shall consider as consisting of concentric strata of an infinitely small thickness,
within which the density and hence the refractive power is
taken as uniform.  In order to determine then the change
of the direction of the ray of light on account of the refraction
at the surface of each  stratum  we must know  the laws
governing the refraction of the light.  These laws are as
follows:
1) If a ray of light meets the surface separating two
media of different density, and we imagine a tangent plane
at the point where the ray meets the surface, and if we draw
the normal and lay a plane through it and through the ray
of light, the ray after its refraction will continue to move
on in the same plane.
2) If we imagine the normal produced beyond the
surface, the sine of the angle between this part of the normnal and the ray of light before entering the medinum (the
angle of incidence) has always a constant ratio to the sine
of the angle between the normal and the refracted ray of
light (the angle of refraction), as long as the density of the
two media is the same.  This ratio is called the index of'
refraction or refractive index.
3) If the index of refraction is given for two media
A and B and also that for two media B and C, the index
of refraction for the two media A  and C is the compound
ratio of the indices between A and B and between B and C.
4) If it is the index of refraction for two media if
the light passes from the medium  A into the medium B, the
index for the same media if the light passes from  the
medium B into the medium A is
Now let O Fig. 4 be a place at the surface of the earth,
C the centre of the earth, S the real place of a star, CJ
the normal at the point J where the ray of light SJ
meets the first stratum of the atmosphere. If we know then
the density of this first stratum, we find the direction of the
ray of light after the refraction according to the laws of
refraction and thus find a new  angle of incidence for the
second stratumn.  If we now consider the nth stratum  taking




156
Fi g. 4.              CN as the line from the
centre  of the earth to
the point in which the
ray  of light meets this
/-i-'astratum, and denoting the
angle of incidence by i,,,
the  angle of refraction
by f,, the index of refraction for the vacuum
and the (n- 1)th stratum
by,,,,, the same for the
nth stratum  by  t.+,,  we
have *):
C                    I    I     I    I      sini,,: sin f,, -,+l:,,.
If further N' is the point in which the ray of light meets
the n -- th stratum, we have in the triangle NCN', denoting
the lines NC and N'C by r,, and r,,t,:
sin: sin i,,+  = r,,+l: r
and combining this formula with the one found before we get:
r,, sin i, i,, -- r,+l sin i,,+l Pt,+1.
Therefore as the product of the distance from the centre
into the index of refraction and the sine of the angle of incidence is constant for all strata of the atmosphere, we may
denote this product by y and we have therefore as the general law of refraction:
ar. g. sin i = y,      (a)
where r, ue and i belong to the same point of the atmosphere.
For the stratum  nearest to the surface of the earth the angle i
or the angle between the last tangent at the curve of the ray
of light and the normal is equal to the apparent zenith distance z of the star. If we therefore denote the radius of the
earth by a, and the index of refraction for the stratum nearest
to the surface of the earth by!,,, we can determine y from
the following equation:
au sin z -y.           (b)
*) These indices are fractions whose numerators are greater than the denominators. For a stratum at the surface of the earth for instance we have
3400
-u = 1.000294 or nearly equal to -
3399




157
If we now assume, that the thickness of the strata, within
which the density is uniform, is infinitely small, the path
of the light through the atmosphere will be a curve whose
equation we can find. Using polar co-ordinates and denoting
the angle, which any r makes with the radius CO by v, we
easily find:              d   tang i. 
dr-tang i.            (c)
The direction of the last tangent at the point where the
curve meets the eye is the apparent zenith distance, but the
true zenith distance C is the angle, which the original direction SJ of the ray of light produced makes with the normal. This i, it is true, has its vertex at a point different
from  the one occupied by the eye of the observer; but as
the height of the atmosphere is small compared with the distance of the heavenly bodies and the refraction itself is a
smalln angle, the angle; differs very little from the true zenith distance seen from the point 0. Even in the case of
the moon, where this difference is the greatest, it does not
amount to a second of arc, when the moon is in the horizon.
We may therefore consider the angle; as the true zenith
distance.
If we now draw a tangent to the ray at the point N, to
which the variable quantities i, r and y, belong and if we
denote the angle between it and the normal CO by 5', we have:'   -i +v.            (d)
Differentiating the general equation (a) written in a logarithmic form, we find:
dr              d/_
rq- cotang i. di -+-  -- 
r               it
and from  this formula in connection with the equations (c)
and (d) we get:       d;'- - tang id-,
or eliminating tang i by the equation:
sill i 
tang i 
1/l  sin i2 / / 2 2 _ y2
and substituting for / its value a So sin z; we find:
- uo sin z ldu
d i"  22  _    2
it  2   Cf o2 sinZ2




158
The integral of this equation taken between the limits
i'=-  and  z' — gives then the refraction.  If we put:
— 1 - s,
we can write the equation in the following form:
d;,_ _            (1 - s) sinz cl(.      (e)
iiJcos2 (z — o    ) +(2s-s2)sinz2
In order to integrate this formula we must know how s
depends upon I,. The latter quantity depends on the density
and we know from  Physics, that the quantity  2 - 1, which
is called the refractive power, is proportional to the density.
If wae introduce now as a new variable quantity the density p,
given by the equation:,2 -     ce,
where c is a constant quantity, we obtain:
1,(- s)sin z. c.      ---
clS' C._
s          * v i(1 —c +    )+ (2s-s2) sinZ2
or taking:
S -Cyo   2=  _ 2a hence c? c           (1- C
1-4- co 0, 2,      t+eo        0
a (1 - s) sin z 
d~' —                        - eo.-    ---                (f
[1 —2a(1 —-']O cosZ2-2a(1 — -+(2s —s',sin. a
The coefficient
1-2 (I    )
is the square of the ratio of the index of refraction for a
stratum  whose radius is r to the index for the stratum  at
the surface of the earth. But as we have I. -1 at the limits
of the atmosphere, and the index of the stratum  at the surface is,,-  3400  the ratio M is always contained between
narrow  limits.  Hence as a is always a small quantity, we
may take instead of the variable factor
I~  a(I-V)




159
its mean value between the two extreme limits 1 and 1 -2 a
or the constant value I -.
If we put for brevity 1 -        to, where w is a function
of s, to be defined hereafter, and if we change the sign of d-',
in order that the formula will give afterwards the quantity,
which is to be added to the apparent place in order to find
the true place, we get:
}    a     (1 -s) sin zdw
1 — a cOS z2-2 at+ -- (2 s -2) sin 2
or as s is always a small quantity, since the greatest value
of s supposing the height of the atmosphere to be 46 miles
is only 0.0115:
a            sin zl dw
-     co sz    2 2o - 2 ssin z2
a   s sin z [cos z2 - 2 a wv] -- s2 sin z2  (g),
1- a6  [cos z2 -2 aw  - 2 s sin zl]2
where already the second term, as we shall see afterwards,
is so small, that it can always be neglected.  In order to
find the refraction from the above equation we must integrate
it with respect to s between the jlimits s =0  and s =H,
where H denotes the height of the atmosphere.
If we now put:
w -F (s)
and introduce the new  variable quantity x, given by the following equation:
-a F(s)
sin z2
or taking:
a F(s)
Sin _2
sin = X -- 9o (s),
s-x +   v (s),
we have according to Lagrange's theorem:
(     F(x)    I  dH(  x)
F (s) =    (A-   (x)    - 99 ()   --- + 12
dx      1.2      x
~      (dx
1.2.3      cldx         +.
hence




160
dP~r)= =L dF(x)           + [(    (x)]   1  d'   )2 dF(xfl
d lF (s)=dF()+'i      dr ) d - 1.2     dx2
d F(x)1.2.3      dX                     ().
In order to find from this the refraction, we must mulct        sin z
tiply each term by 1'COS+.xsi    and integrate be- ay/cos z2 -i-2x sinZ2
tween the limits given above. But in order to perform these
integrations, it is necessary to express w as a function of s
or to find the law, according to which the density of the
atmosphere decreases with the elevation above the surface.
7. Let p, and T0 be the atmospheric pressure and the
temperature at the surface of the earth, p and r- the same
quantities at the elevation x above the surface, m the expansion of atmospheric air for one degree of Fahrenheit's
thermometer; then we have the following equation:
1 4- m. 
1 mto                     (a)
For if we take first a volume of air under the pressure
Po at the temperature To and of the density go and change
the pressure to p, while the temperature remains the same,
the density according to Mariotte's law will change to p 0o.
Po
If then also the temperature increases to r, the resulting density will be:
p  1      +- no
from which we get the equation above. Hence the quantity
_1_     or the quotient: the atmospheric pressure divided by
the density and reduced to a certain fixed temperature, is
always a constant quantity.  Now  if we denote by lo the
height of a column of air of the uniform  density Q( and of
the temperature To, which corresponds to the atmospheric
pressure po, we have, denoting the force of gravity at the
surface of the earth by g,,:
po = (I + dito) e0 gO ~ lo,     (#l
lo is the height which the atmosphere would have if the density and temperature were uniformly the same at any elevation




161
as at the surface of the earth, and if we take for ra, the temperature of 8o" Reaumur _ 10" Celsius = 50 " Fahrenheit, we
have according to Bessel:
o - 4226.05 toises,
equal to the mean height of the barometer at the surface of
the sea multiplied by the density of mercury relatively to
that of air.
If we ascend now in the atmosphere through d r, the
decrease of the pressure is equal to the small column of air
t) d r multiplied by the force of gravity at the distance r, hence
we have:
lp — g o -.o. clr,
and dividing this equation by the equation (/t) and putting
also reckoning the temperature from the temperature ro,, so
that r means the temperature minus 50o Fahrenheit we find:
_cp _  a d-s  IV)
Po      10o
and from the equation (a) we have:             (y)
P -= (1 + si z) (1 -IV ).
Po
If we eliminate p from these two equations, we find 1 - w
and hence the density expressed by s and 1 -rn-r. The latter
quantity is itself a function of s; but as we do not know
the law  according to which the temperature decreases with
the elevation, we are obliged to adopt an hypothesis and to
try whether the refractions computed according to it are in
conformity with the observations. Thus the various theories
of refraction differ from  each other by the hypothesis made
in regard to the decrease of the temperature in the atlmosphere.
If we take the temperature as constant, we have:
P   1- -,  hence  p -   d (1- ),
Po               Po
and we find, combining this with the first of the equation ('):
d(l - w)    a
-
hence   1 -I  - e  lo
1l




162
as the constant quantity which ought to be added to the integral is in this case equal to zero.  This hypothesis was
adopted by Newton, but is represents so little the true state
of the atmosphere that the refractions computed according
to it differ considerably from  the observed refractions.
If we take for 1 I-m   an exponential expression e  h
we arrive at Bessel's form.  We find then by the conmbination of the two equations (y):
a s
cl (l —o  )      - e ] ds
and integrating and determining the constant quantity so that
1 — w is equal to unity when s-  0, we find:
a s
1 -   w 
instead of which we can use the approximate expression:
h- 1)
--   a.s
1 -- wtae =               (a).
Bessel determines the constant quantity h is such a manner that the computed refractions agree as nearly as possible
with the values derived from  observations. But the decrease
as
of the temperature resulting from the formula 1 —~mntre e
for this value of h do not at all agree with the decrease
as observed near the surface of the earth.  For we find
dc      a                                   dr   1
-  for s   0, and as we have also -      fors   0,
ds      km1                                cds   a
we find:
dclr    1
dr      m
at the surface of the earth. Now as m for one degree of
Fahrenheit's thermometer is 0.0020243 and as h according
to Bessel is 116865.8 toises, we find dlt_              There
would be therefore a decrease of the temperature, equal to
10 Fahrenheit if we ascend 237 toises, whilst the observations
show that a decrease of 10 takes place already for a change
of elevation equal to 47 toises.
Ivory therefore in his theory assumes also an exponential
expression for I- m r, but determines it so that it represents




163
the observed decrease of the temperature at the surface of
the earth.  He takes:
1       U — e- t,
where u is a function of s, and further:
1 +w   1 — f ( - e-").
Then we easily get from  the equations (r):
a
cis = (1 -f) cl + 2fe-?'dul,
and  - s=(1 — f) u -+-2f(1 -e').        (o)
Taking r =  a we find from  these two equations:
d-r      I'f
dr     lor0 1 —f
and we see that we must take f equal to 2 in order to make
eu a            1
dr equal to        which value represents the observations at
the surface of the earth.
Several other hypotheses have been adopted by Laplace,
Young, Lubbock and others. Here however we shall confine
ourselves to those of Bessel and Ivory, as the refractions
computed from  their theories are more frequently used, and
the other theories may be treated in a similar manner.
8.  If we put in equation (8):
h — lo
h 1,,
we have for Bessel's hypothesis:
w' = F(s)   - e- sI
we have therefore:
F(x)1  e-Ax
(x)   -a  (e-A —1).
sin z2
and we find:
d F(x)T(x)" s —-(-1)             n.ei      n1,2 
hence as:
dn e-Px
d    (__ ( —1)"p'Le-PX,
dx 
d [dF(x) 9r(x)]  a' n(a+1)en+l)            na'-n.+ (n (1( 1),, e _ (. n",,_),
dx        sin z'( 2 
+ n (n  O)    } e- (n-1).
t.




164
and the general term  of the differential d i' becomes:
a     1  a' "+        sin z      *(n+I)e-(n+l1) 
1-a'  l....n sinz 2'  /cOS2 -+-2xsinz 
n 1 (n -    1       )- 1
- n. n" e-"~3 +    2 1) (n- 1)" e — (".where we have to put for n successively all integral numbers
beginning with zero. All these terms must then be integrated
between the limits s -  O and s -  H, instead of which we
can use also without any sensible error the limits 0 and wc,
as e-AS  is exceedingly small for s -H.  As we have x- O0
when s =  0 and x =c   when s - ce we must integrate the
different terms with respect to x between the limits 0 and cs.
All the integrals which here occur can be reduced to the
functions denoted by ) in No. 18 of the introduction and if
we apply formula (8) of that No., we find the general term
of the expression for the refraction:
aI       2   I            n - 1
1-a'f1 2...n sinz 2
n (n —l)
_t_  _2___ (n - 1) 2 i)(n-1)_...,
or denoting the refraction by SC, we find:
2    sin za     (2) -  (1) 
/ +-.   4 2'2    W  (3)-2  2- s~ ~(s) )  ]
etc.
and as we have:
X2    X3
1-x +              --..  e
1.2   1.2.3'
we can write-this in the following form:
/ _ M
/2   -2i e s                       (
t4 _   2'E e Si1nZ2 (2)
sin __ 2
a2 a 2   3-Sill2
-           32__   sill
1.2. sin4 3e
9.  In Ivory's hypothesis we have:
w: F(u) 1 - e-',




165
and taking 7o    1
1 —f    2 (      -")
If we introduce here the new  variable x, given by the
equation:
sinl2 +s 
the  differential expression  for the  refraction  according to
equation (g) in No. 6 becomes:
d      sin z. cl F (u)
i(x 1//2 + 2sinz-2'
o - - zx  
where x  u —.(1-e-") -fu + 2f (1 - e..).
sin z"
Taking again:
F(x)  1- e-e? (X)  — a  (1 - e- ) +I-'x - 2fJ(1 - e..),
sin Z 2
we find from  the formula (h):
d F(it)  e-1- + ~d [(x) e -x] +  1  d2 [ ()'2 e.-.]
dx      1.2      dx2
As the third term may be already neglected, we have:
e-_t+d         = e — + -—.- [2 e-,-2O: —e-]   +f(- x) e-.-2f [2 e  - 6-e-..].
ar sin      z
If we multiply these terms by  a            saz        and
l V 2 sin z2
I-cosz    Sill2x
integrate them  with respect to x between the limits 0 and o,
we find again according to the formulae (9) and 10) in No. 8
of the introduction:
s5l a z82s!+  a -  l'2 2V(2)-V      -2f  22 (2   _ )
sin  +~w )-f (2-+ T2)  (1) —  T]!
where  T — cotang z / ~
The higher ternms are complicated, but already the next
term  is so small on account of the numerical values of af




166
and f that it can be neglected. For we have for the horizon,
where the term  is the greatest, putting 2f- a —=g
da (p (x) I e....
d2 (p(x2    =f2 x2e-x-(4f 2+2fg)xe-x+S fgxe-2x +(2f 2 +2 +4fg) edx
- (8fg + 8 g2) e —2. + 9g2 e-3.
If we divide each term  by  ~2 and integrate it between
the limits s and oo we find, applying the formulae for FG(),
F(3) etc. given in No. 16 of the introduction:
1    ~/2T 7i/ [2f2 - 3f g (1/2 - 1) + g2 (1 - 2 12 - 3 1/3)]
and if we substitute here the numerical values, which are
given in No. 10, we find that the greatest value of this term,
which occurs in the horizon, is 2". 11. The next term gives
only 0".18. In the differential equation (g) in No. 6 we have
also neglected the second term, as it is small and amounts
to  about half a second in the horizon.   As the sign of
the latter term  is negative, we shall not commit an error
greater than 1". 5 if we compute the horizontal refraction
from formula (1).
10. The numerical computation of the refraction from
formula (k) or (1) can be made without any difficulty, as the
values of the functions Wp can be taken from the tables or
can be computed by the methods given in No. 17 of the introduction.
According to Bessel the constant quantity a at the temperature of 50" Fahrenheit and for the height of the barometer of 29.'6 English inches, reduced to the normal temperature, is
a- 57". 4994, hence log   = 1.759785
and h -- 116865.8 toises.
As we have 1,,   4226.05 toises, we find, if we take
according to Bessel for a the radius of curvature for Greenwich to 3269805 toises:
=-745. 747, hence log    2    3.34729
If we wish to compute for instance the refraction for the
zenith distance 800, we have in this case log T1 = 0.53210
etc. and we find:




167
log n   -   log (n I)! s-n. -2)      log v5 (n)  log e -sinz2
--   1    0.00000            0.00000         9.14983      9.90691
s —- 2    0.15051            9.33113         9.00745      9.81382
on —   3    0.71568         8.36122          8.92228      9.72073
o —  4    1.50515            7.21523         8.86128      9.62763
n -  5    2.44640            5.94430         8,81372     9.53454
n --  6    3.5017           4.57645          8.77473     9.44145
n -- 7    4.6480             3.12943         8.74168      9.34836
--   8    5.8701             1.6155          8.7130       9.2553
---  9    7.157              0.043           8.688        9.162
n -- 10    8.500             8.420           8.665        9.069
The horizontal rows give the  terms within  the parenthesis in  formula (k) and  if we  multiply their sum  by the
constant quantity    a - i/2,, we find 314". 91 exactly in confoimity with Bessel's tables.
Far more simple is the computation of Ivory's formula.
In this case we have:
log af - 9.333826, log 1 _        =/2   = 3.354594, f=.
If we now  compute the refraction according to formula
(1), we have:
log T, = 0.540098  log T, = 0.690613
log 50 (1) - 9.142394 log 50 (2) = 8.999757
and with this the terms independent of f give 315". 32, whilst
the terms multiplied by f give - 0". 12. The refraction is
therefore 315".20 or nearly the same as Bessel's value. The
refractions according to the two formulae continue to agree
about as far as 86" and represent the observed refractions
well.  But nearer to the horizon Bessel's refractions are too
great, while those computed by Ivory's theory are too small.
It is therefore best, to determine the refraction for such great
zenith distances from  observations and to compute tables from
those observed values, as Bessel has done.
We find the horizontal refraction according to Bessel,
as we have in this case:
2 0 --- t d= -  l/- I/
0
an   sbitVti/g Lere. e —22 2fi-{+- 3 e-3- -...]
1 -a2             e                         1.2
and substituting here the numerical values we get 36' 5",




168
According to Ivory we find the horizontal refraction:
V7^               [I -- a  ([1+ (/2 - 1)-f(2 -/2 - )]
= 33' 58",
whilst the observations give 34' 50", a value which is nearly
the mean of the two.
As long as the zenith distance is not too great, it is not
necessary to use the rigorous formulae (k) and (1), but it is more
convenient, to  develop them  into  series.  If we substitute in
formula (1) for qip(1) and  q/(2) the  series found in No. 17
of the introduction and observe that -                1 —-  cotg z2, we
find: *)
I         a              a-9 a
+ O 1 5 fZ I)tng _15   105 a    1575 a  tang7    7
- $ tangzs —-                    +16-tangz+...   ()
or if we substitute the numerical values:
z= [ 1.759845] tag z - [8.821943] tangz3+ [6.383727] tangaz5- [4.180257] tangz7,
where the figures enclosed in brackets are logarithms.
Furthermore the terms multiplied by f give:
a      3             75           1785          46305         i
1-   /                    ta             ta gzp,  -      tang z - a tang' ta
or                                                                    (1.,)
-  [5.506187]tangz5-[3.714510]JtangzT+[1.901468]tangz9-[9.018568]tangz"l
For 75" we find fromi the series   z =  211". 39 and the
part depending on f equal to -  0". 02, hence the refraction
equal to 211". 37 in conformity  with the rigorous formula.
*) For we get:
1 $Z                      15
1 tangza + /  tangz'  - -f - tang z'
/2 # yJ (1) = tang z -  6l   tallg +:_            tang f 7
105
+  -05 tang z -.
__i~           1            3            15
221 /2 f' (2) -tang z -    ng          tag tang -   tang z-    tang z
+ 2/,3 0  tng.z __..
80     tang, a 
Ivory gives in the Phil. Transactions for 1823 another series,, which can be
used for all zenith distances,




169
11. The above formulae give the refraction for any zenith distance but only for a certain density of the air, namely
that, which occurs when the temperature is 500 Fahrenheit and the height of the barometer 29.6 English inches.
The refraction which belongs to this normal state of the
atmosphere is called the mean refraction.  In order to find
from this the refraction for any other temperature r and height
of the barometer b, we must examine, how the refraction is
changed, when the density of the atmosphere or the stand
of the meteorological instruments, upon which it depends,
changes.  Let E be the  expansion  of air for one degree
of Fahrenheit's thermometer, for which Bessel deduced the
following value:
—. 0.0020243
from  astronomical observations.  If we take now a volume
of air at the temperature of 500 as unit, the same volume
att the temperature r will be 1 --  (r —50), hence the density
of the air when the thermometer is r is to the density when
the thermometer is 50 as 1: 1 -- c (r -- 50). WVe know further
from  Mariotte's law, that the density of the air when the
barometer is b is to the density when the barometer is 29.6
as b:2 29.6.  If we therefore denote the density of the air
when the thermometer is r and the barometer is b by ), and
the density in the normal state of the atmosphere by Qo, we
have:
b
) 
o 9.6
o t- + 8(x - 50)
and as the quantity ac which occurs in the fbrmulae for the
refraction may be considered as being proportional to the
density, at least for so small changes of the density as we
take into consideration, we should deduce also the true refraction from the mean refraction by the formula:
AZb
*29.6
1 + E ( - 50)
if a did occur only as a factor, as the quantity 1 - a in the
divisor can be considered as constant on account of the smallness of a. But ce occurs also in the factor of  c-  which
I




170
shall be denoted by Z and the quantity i varies also with
the temperature, as it depends on 1, or when the temperature
is r upon             l- =o [1t + -   (r- 50)]
if we denote the height of an atmosphere of uniform density
at the temperature r by 1.  We find therefore the true refraction from  the following formula:
___ _  b  +  a   dZ           a dZ
8z'      Sz.- b- +a2. d (T-50) +- az  d (b-29.6), (nm)
+  ( — 50)'29. c  1 —a dr         1-a db
but as the influence of the last two terms is small we may
take for the sake of convenience:
y' [~ + E (  - 50)+,'  —. ~         )
But if we develop this we find, neglecting the squares
and higher powers as well as the products of p and q:
4Z.- 520)              29.        6 b
Thus we obtain from  the formulae (m) and (n) the following equations for determining p and q:
a   dZ   1
1 -  a  dc' e7Yz
a   dZ  29.6             (~)
-1_x db z 
if we take in the second member Jz instead of.L  29.6
+ - E (T- 50)'
The moisture diminishes also the density of the atmosphere and hence the refractive power, but, as Laplace has
observed first, this decrease is almost entirely compensated
by the greater refractive power of aqueous vapour.  The
quantity c therefore is hardly changed by the moisture and
as the effect upon the quantities p and q is very small, we
shall pay no regard to the moisture in computing the refraction.
In order to obtain the expressions for p and q, we must
find the differential coeficients    and dZ  but we shall dedt      db'
duce these values only for Ivory's theory, as the deduction
from Bessel's formula is very similar. According to formula (1)
we have:
z =- /2, [(1 -  )   () t+ (   /2 l/ (2) +f Q],




171
taking  X-       -.  From  this we obtain:
sin z2
-. (1 - a) df          +
dZ = 1   (1-)_ — + }"2flA [1/2, (2) - v (1)] d1
+dg2#    J-i    - 2   T(2)   f dv  (P)
as f does not change with the temperature and the stand of
the barometer.
Now we have t (1) =e-T2fe-t2 d t, where T1 - cotg z 
21
= e(2)  -e-'2e-t2 dt, where T2= cotg z /Yf,'2
and as   (1)   2 T  p (1)- I andd ()- 2T2p(2) -I
and  as dT,                          d 2
the last but one term in (p) becomes:
+,BA 1/2, [(1 _  ) (Il 2 V (1)-2 TI ) -4- 2Y2. (T2 2 2  (2)-,!- T2)].
The factor Q consists of two terms, the first of which
having the factor — 2 is equal to the factor of A in the expression of c'z. We therefore embrace this in the latter term
by writing  A-2f instead of A.  There remains then  only
the following term
+/ f 1/2[(l   + T,2) w (1)-  T,]
and as we find differentiating it:
V2 -f 2  T + (2-+ — TY) (T1 li)- T,)y,
the complete expression for d Z becomes:
dp  6z(I- a)  d2
lZ,r d   S   -~ T +  d23 2' [x  /2 v (2) -, (1)]
"+  -     f - — (1 -- -- - if+f T1 ) ( T1,(1) -1)
4             2
+ 1/2 ( - 2f) (T,2   (2) -I T2)As we have:
b
29.6 _           hence       29.6-b              0
a                  fn-  -n=                 — hence 1  - (    50) ],
1 + E iC - 50)'       a           29.6
da   b — 29.6
we find: - --            e (r - 50),
a     29.6




172
and likewise:
3 -t+d (   -a e (- 50),  hence d         E ( 50);
finally we have:
__-'ai        d)   da   dfij   b - 29. 6
X-  ID'  hence   - -+ -- 1. - - 2 E (a- 50).
sinz 2      n    a      i    29.6
We find therefore:
az.q' —1    -         A IV2 ~, (2) - a  (1)]
z. p — - ~z1/2__a. 2 [i2  (2)       (2) -  (1)]  (q)
2     1-a 
+_ 2t2P        K+     -   +( -+2T2)(T,,   (1)- T- )
+ (A -  ) (T22 n (2)- I T2) V2I,
where instead of f its value 2 has been substituted.
If we compute from this p and q for  = 870, Oz being
852".79 we find:
log T1 = 0.0131759 log [V2 ip (2) —  J (1)] =  8.605021,
log (T12,/J (1) - 2 T1) -  9.081168,,, log T2 — 0.163690,
log (T22,p(2) -  - T2) I/ 2 =  9.191771,. and with this
bz.q-   19".71, Jz.p=185".36,
hence:
q-0.0231,
p- 0.2173.
When the zenith distance is not too great, we can find p
and q also by the series given in No. 10. For differentiating
the coefficients of   ain (1,) and (12) with respect to a and A,
we easily find the following series:
q6z = + [7.903991 tang z + [7.90146] tang z'3 - [5.66533] tangz:
-+ [3.54172] tang z --...
p ~z  + _- [7.90399] tang z -- [8.91567] tang z' - [6.70990] tang z5
+- [4.56712] tang z7 -...,
where the coefficients are again logarithms.
For z -  750 for instance we find from  this q-  0o0020
and p=  0.0188.
12.  For the complete computation of the true refraction
from formula (min), we must know the height of the barometer reduced to the normal temperature.  If we take the
length  of the column of mercury at the temperature 500 as
unit and denote the expansion of mercury from  the freezing




173
to the boiling point equal to 5j-5 by q, the stand of the barometer observed at the temperature t*) is to the stand, which
would have been observed if' the temperature had been 500
as 1 -+ 1    (t-50): 1, or the length of the column of mercury reduced to the temperature 50 is:
180
b5 --- bt
180 --- q t —50)
If further s is the expansion of the scale of the barometer from the freezing to the boiling point, s being 0.0018782
if the scale is of brass, we have taking again the length of
the scale at the temperature 500 as unit:
b: bo o-1:1- 1 8 1 -$0 (t - 50).
Hence the height bt of the barometer observed  at the
temperature t, is reduced to 500, taking account of the expansion of the mercury and the scale, by the formula:
180 + s (t- 50)
bso " —bt
180 +-  (t- 50)
The normal length of' an English inch is however not referred to the temperature 500 but to the temperature 620;
hence the stand of the barometer observed at the temperature
50" is measured on a scale which is too small, we must there12s
fore divide the value b,, by 1 -- 10   so that finally we get:
b    80  s (t - s50)    180
180 - q(t - 50) 180~-t12s
If the scale is divided according to Paris lines and the
thermometer is one of' Reaumur, we should get, as the normal temperature of the French inch is 130 R. and we have
500 Fahr. -- 8" Raunm.:
80 -I- s (t - 8)   80
bs -- bt
80 + q (t-8) 80 - 5 s
This embraces every thing necessary for computing fbrmula (mn). If we denote by f the temperature according to
*) The temperature t is observed at a thermometer attached to the barometer, which is called the interior thermometer, whilst the other thermometer
used for observing the temperature of the atmosphere is called the exterior
thermometer.




174
Fahrenheit's thermometer, by r the same according to Reaumur's thermometer, by b~e) and b(') the height of the barometer
expressed in English inches and Paris lines and if we put:
b(4)   180      b(l)   80
29.6 180-12s   333.28 80+-5s
T= 130 + s (f- 50)  80 + s (r - 8)
180+q  (. —50)  80 0+q(r - 8)
1             1
1 + ~-.  (f —50) =1   E (r —S)
and give to the mean refraction the form  S  =z a tang z, we
have:
z'- a tang z. y+P' (B. T)t+q                   (A)
hence log ('z' = log a - log tang z +- (1 +p) log y +- (1 + q) (log B +log T).
If we have then tables, from which we take log a, 1 +-p
and I - q for any zenith distance, and log B, log T and log y
for any stand of the barometer and any height of the interior
and exterior thermometer, the computation of the true refraction for any zenith distance is rendered very easy. This
form, which perhaps is the most convenient, has been adopted
by Bessel for his tables of refraction in his work Tabulae
Regiomontanae.
13. The hypothesis which we have made in deducing
the formulae of refraction, namely that the atmosphere consists of concentric strata, whose density diminishes with the
elevation above the surface according to a certain law, can
never represent the true state of the atmosphere on account
of several causes which continually disturb the state of equilibrium. The values of the refraction as found by theory
must therefore generally deviate from  the observed values
and represent only the mean of a large number of them, as
they are true only for a mean state of the atmosphere. Bessel
has compared the refractions given by his tables with the
observations and has thus determined the probable error of
the refraction for observations made at different zenith distances.  According to the table given in the introduction
to the Tab. Reg. pag. LXIII these probable errors are at
450 4-0".27, at 81"-4-1", at 850~-4-1".7, at 890 30'-4-20". We
thus see, that especially in the neighbourhood of the horizon we can only expect, that a mean obtained from a great
many observations made at very different states of the at



175
mosphere may be considered as free from the effect of refraction.
For zenith distances not exceeding 800 it is almost indifferent, what hypothesis we adopt for the decrease of the
density of the atmosphere with the elevation above the surface of the earth and the real advantage of a theory which
is founded upon the true law consists only in this, that the
refractions very near the horizon as well as the coefficients
1-t — and 1 —-q are found with greater accuracy, hence the
reduction of the mean refraction to the true refraction can
be made more accurately. Even the simple hypothesis, adopted
by Cassini, of an atmosphere of uniform  density, when the
light is refracted once at the upper limit, represents the mean
refractions for zenith distances not exceeding 80" quite well.
In this case we have simply according to the formulae in
No. 6:
sin i -   tto sillf,
or as we have now i —f-''z:
Az = (,, - 1) tangf,
and since we have also, as is easily seen, sin f=   -- sin z, where
I is the height of' the atmosphere, we get:
z_= -(,   o'1)    I n       (ito- 1) tangz(1      )
If we take now for X(,y — 1 the value 57".717, we find
for the refraction at the zenith distances 450, 75" and 800
the values 57".57, 211". 37, 314". 14, whilst according to Ivory
they are 57".45, 211".37 and 315".20.  But beyond this the
error increases very rapidly and the horizontal refraction is
only about 19'.
The equation (f) in No. 6 can be integrated very easily,
if we adopt the following relation between s and r:
For if we introduce a new variable, given by the equation:
l1-2aa (-   )       Sl sinz-w,




176
the equation (f) becomes simply:
cdi
d (2w1)1/)g  1;"
therefore if we integrate and substitute the limits w  - sinll 
2,,- I
and w =  (1 -  2 c) 2 sin z, we find:
2 m -1
Sz -=2 1 Lz   are sin (1-2 a)   sill z
or:
sin [z- (2w  -1) Cz]  (1-2 a)   sinz,
for which we mlay write for brevity:
sin = sin  [z - Nz].
This is Simpson's formula for refraction by which the
refractions  for zenith  distances not exceeding  85" may be
represented very well, if the coefficients 1 and N are suitably
determined.
If we add to the last equation the identical equation
sin z =-sin z and also subtract it, we  easily find two equations from  which we obtain dividing one by the other:
/N   \ 1ll)  iV          N
tang -- - z    1 _+ —_tang z -    C,
(2      1 + ill         2
or   tang (A. Sz) = B tang [z - A. Sz],
which is Bradley's formula for refraction.
14. As the altitude of the stars is increased by the refraction, we can see them  on account ofit, when they rea.llly
are beneath the horizon. The stars rise therefore earlier and
set later on account of the refraction.
We have in general:
cos z = sin o sin  + -- cos cp cos S cos t  (r)
from which follows:
sin z dzcl - cos q cos S sin t. cldt
hence if the object is in the horizon:
dz
cos, 9 cos a sin t
As in this case dz is the horizontal refraction or equal
to  35', we find for the variation  of the hour angle at the
rising or setting:
140s
cos 99 cos a sin t




177
In No. 20 of the first section we found for Arcturus
and the latitude of Berlin:
to = 7h 4211 40s
and as we have'=  190 54'.5,   - =  520 30'.3, we find:
A to - 411 37s.
Arcturus  rises therefore  so much earlier and sets so
much later.  We can compute also directly the hour angle
at the rising or setting with regard to refraction, if we take
in the last formula (r) z-=  90" 35'.  We have then:
cos.z -sin  sin 3
cos t 
cos? cos 3
and adding 1 to both members, we find the following convenient formula:
C_/ coS I (T -  -t ) cos ~ (- -+ a-)
CO   t=   V ~ B ~cos? cos          )
If we subtract both members from  1, we obtain a similar formula:
sin   i/sin (z -- 99 - e) sin - (z   -99)
C os  COsA
In the case of the moon we must take into account besides the refraction her parallax, which increases the zenith
distance and hence makes the time of rising later, that of'
setting earlier. The method of computing them has been
given already in No. 20 of the first section and shall here
only be explained by an example.
For 1861 July 15 we have the following declinations
and horizontal parallaxes of the moon for Greenwich mean
time.
i3                 p
July 15 0l    - 150 32.1   2 19.4   59 13
12h      17 51.5             59 15
2 4.1
16 Oh      19 55.6             59 14
12h      21 42.0             59 13
It is required to find the time of setting for Greenwich.
According to No. 19 of the first section, where the mean time
of the upper and lower culmination was found, we have:
Lunar time   Mean time
Oh       6h 1611. 7
12h     18h 44m. 2
12




178
If we take now an approximate value of the declination
-  170 51'.5 we find with cp = 510 28'.6 and z = 890 35'.8,
t= 4h 211.5 and the mean time corresponding to this lunar
time 10'i 48m. If we interpolate for this time the declination
of the moon, we find - 170 38'.2 and repeating with this
the former computation, we find the hour angle equal to
4'1 22'.9 hence the mean time of setting 101' 49m.6.
15. The effect of the atmosphere on the light produces
besides the refraction the twilight. For as the sun sets later
for the higher strata of the atmosphere than for an observer
at the surface of the earth, these strata are still illuminated
after sunset and the light reflected from  them  causes the
twilight.  According to the observations the sun ceases to
illuminate any portions of the atmosphere which are above
the horizon when he is about 180 below the horizon.  Thus
the moment, when the sun reaches the zenith distance 1080
is the beginning of the morning or the end of the evening
twilight.
If we denote the zenith distance of the sun at the beginning or end of twilight by 90" -- c, by t,, the hour angle
at the time of rising or setting and by r the duration of
twilight, we have:
- sin c = sin 9 sin 8 d- cos go cos  cos (to -+- z)
hence:           os (to       sin  sin -l- sin c
cos (to q- +)cos 99 cos a
or putting H =900 - +Y- _-tsin ~ (to -)    sin -   =c) cos- (H-c)
cos T COS 
from which we can find r after having computed to.
If we call Z' the point of the heavenly sphere, which
at the time of sunset was at the zenith and by Z that point
which is at the zenith at the end of twilight, we easily see
that in the triangle between these two points and the pole
the angle at the pole is equal to r and we have:
cos ZZ' - sin 92 -+- cos 92 cOS Z.
But as we have in the triangle between those two points
and the sun S, ZS- =90+-c, Z'S = 900, we have also calling the angle at the sun S:
cos ZZ' = — cos c cos'




179
and thus we find:
1 - cos c. cos S
S12-    2 cos 992
where S, as is easily seen, is the difference of the parallactic
angles of the sun at the time of sunset and at the end of
twilight.  The equation shows, that r is a minimunm, when
the angle S is zero, or when at the end of twilight the point,
which was at the zenith at sunset, lies in the vertical circle
of the sun. The two parallactic angles are therefore in that
case equal.
The duration of the shortest twilight is thus given by
the equation:
sin - c
sin ~ r =and as we have:
sin  99      sin f + sin c sin
sin' COSp       os c cos 
we find:
sin S  - tang' c sin T,
from which equation we find the declination which the sun
has on the day when the shortest twilight occurs.
If we denote the two azimuths of the sun at the time
of sunset and when it reaches the zenith distance 90-tc by
A and A', we have:
cos 99 sin A - cos S sinp
cos 99 sin A' — cos  sin p'.
Hence we have  at the  time of the shortest twilight
sin A =  sin A' or the two azimuths are then the supplements
of each other to 1800.
From  the two equations:
- sin c = sin 99 sin 8 + cos  cos S cos (to S- )
and
0 =sin 9 sin S + cos 99 cos S cos to
follows also:
cos c sin' c
sin (to -- ) sin 2 - COS  0
cos 6   cos T9
If we take c —180 we find for the latitude q — 810
sin  v -r  1, hence the duration of the shortest twilight for
that latitude is 12 hours. This occurs, when the declination
of the sun is — 9, the sun therefore is then in the horizon
at noon and 180 below at midnight.  But we cannot speak
12 *




180
aly more of the shortest twilight, as the sun only when it
has this certain declination fulfills the two conditions, that it
comes in the horizon  and reaches also a depression of 18'1
below  the horizon; for if the south declination  is greater
the sun remains below  the horizon  and if the south declination is less it never descends 18'' below the horizon.
At still greater latitudes there is no case when we can
speak  of the shortest twilight in the above sense and  lence
the formula for sin l r becomes impossible.
lNote. Consult: on refriaction: Laplace Mecanique Cdleste Livre X.
Bessel Fundamenta Astronomiae pag. 26 et seq. -  Ivory in Philosophical
Transactions for 1823 and 1838. -  Bruhns in his work: Die Astronomische
Strahlenbrechung has given a compilation of all the different theories.
II1. THE ABERRlATION.
16.  As the velocity of the  earth in her orbit round
the sun has a finite ratio to the velocity of light, we do not
see the stars on account of' the motion of the earth  in the
direction, in which they really are, but we see them  a little
displaced in the direction, towards which the earth is moving.
We will distinguish two moments of time t and t' at which
Fia. 5.     the ray of light coming from  an unmoveX1 i    able object (fixed star) strikes in succession the object-glass and the eye-piece of
/   {I'  a telescope (or the lense and the nerve
of' the eye).  The positions of' the objectglass and of the eye-piece in space at the
time t shall be a and b, and  at the time
t' a' and b' Fig. 5.  Then the line a b' represents the real direction  of the ray of
light, whilst a b or a' b', both being parallel
1,4   on account of the infinite distance of the
fixed stars, gives us the direction of the
apparent place, which is observed.  The
angle between the two directions b'Va and
b a is called the annual aberration of the
fixed stars.




181
Let x, y, z be the rectangular co-ordinates of the eyepiece b at the time t, referred to a certain uninoveable point
in space; then:
dx           dy               dz
d+ t  (t  t), y +  (t'- t) and z + -l (t' — t)
cit          cit              cit
are the co-ordinates of the eye-piece at the time t', since durilng
the interval t'- t we may consider the motion of the earth
to be linear.  If the relative co-ordinates of the object-glass
with respect to the eye-piece are denoted by c, C and k, the
co-ordinates of the object-glass at the time t, when the light
enters it, are x3- +, y  - i, z- 4.
If we now take as the plane of the x and y the plane
of' the equator and the other two planes vertical to it, so that
the plane of the x, z passes through the equinoctial, the plane
of y, z through the solstitial points; if we further denote by
a and;S the right ascension and declination of that point in
which the real direction of' the ray of light intersects the celestial sphere and by I. 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:
K (t'- t) cos   s  t't)os,     - t) s sin ax, tc (t'- t) sin 8.
Denoting further the length of the telescope by I and
by a' and c' the right ascension and declination of the point
towards which the telescope is directed, we have for the coordinates of the object-glass with respect to the eye-piece,
which are observed:
— I cos  cos a    cos'sin', Cs  s    sin'.
Now  the true direction of the ray of light is given by
the co-ordinates of the object-glass at the time t:
I cos 8' cos a' + x,
1 cos Y' sin a' +- y,
I sin t'   - z,
and by the co-ordinates of the eye-piece at the time t':
dx
x4+ d- (t'- t),
+   (t'-t),
d z
c it.




182
We have therefore the following equation if we denote
-,     by L:
t -t
cos Cos a  L cos 3' cos a   L cos   d
cos  sin (a  L cos' sin a'dy
dt
Mssin= LsinS'        dz
We easily derive from  these equations the following:
L                         I,         d x
cos' cos (a'- a)    = cos+ —      d sin a     a
IU                       It dt       clt """ dt
- Cos S sin (a'-a)  I-  CCOSa-   sin a
1      dy      rx
see-1 — cosa -  sin a
or:           tang (a' -a)      1   -d           d.dt-   dt
We find a similar equation for tang (3' —3). If we develop both equations into series applying formula (14) in No. 11
of the introduction, we find, if we substitute in the formula
for tang (' —  c) instead of tang l (a' — c) the value derived
from a'- a and omit the terms of the third order:
a  l dxd  d
cX~ ol  -- dtsina — ~-t cosa t see-cS
a-,_  _ ~ isin a - +   s in      cos a
t d         ct       clt     cd
+ f adcos  cos ao -4 a+-  cos 3 sin a +4- -sin 
+  2             cit           dt
X d sin a cos xa - + ysin 3 sin a-.   cos Co
If we now refer the place of the earth to the centre of
the sun by co-ordinates x, y in the plane of the ecliptic,
taking the line from the centre of the sun to the point of
the vernal equinox as the positive axis of x, and the positive axis of y perpendicular to it or directed to the point
of the summer solstice and denoting the geocentric longitude




183
of the sun by 0, its distance from the earth by R, we
have *):
x -- R cos ),
ye-R sin@.
If we refer these co-ordinates to the plane of the equator, retaining as the axis of x the line towards the point of
the vernal equinox and imagining the axis of y in the plane
of y z to be turned through the angle e, equal to the obliquity
of the ecliptic, we get:
x    - R cos,
y - 1R sin A0 cos a,
z  - R sin G sin 8,
and from this we find, since according to the formulae in
No. 14 of the first section we have the longitude of the sun
o =  v --- v   or equal to the true anomaly plus the longitude
of the perihelion:
dx           dRl           dv
-;-l:  - cos 0 -- + R sin   -------
di           dt            dt
dy                cddll             d
y - -sin  cos E   — R coscos e d —
cdit              dt                ct
-dz  sin  sin ed  - -R cos ( sin 8 ddt                dt                dt
But we have also according to the formulae in No. 14
of the first section:
d v   a cosdE and as we have also dE  -addMn
R                               R
we find:                dv   a2 cos p dl3l
dct    R2   dt
Further follows from  the equation R              — 1    j    in con1    +ecos v
nection with the last:
d t-    tang o sin v d
cit              dt
and from this we get:
dx     a   dM10 aC cos   invos
sin Q)  -- - sin 9 sin' cos (D
cos T  dt           R
hence observing that:
an cos _ 1  sin v cos  and  0  v = 2,
dx     a   dill
cd i_ =a    cit [sin (Q)- +sin 9 sin n]  (b)
dct   cos        o d t
*) As the heliocentric longitude of the earth is 180~ - (D).




184
dy       ct      cdll
and           -l -       ---- cos    [COS    +[ sin q cos i]  (b)
dt      cos       cfit
dz        c       d dMd
dz   -        a  S- sin    - [COS ) + sin f cos nC].
d t     cos 99     dt
If we substitute these  expressions in the formulae (a),
the  constant terms dependent on g give in the expressions
for the aberration'also constant terms which change merely
the mean places of the stars and therefore can be neglected.
If we introduce also instead of iR the number k of seconds,
in which the light traverses the semi-major axis of the earth's
orbit, so that we have:
1    k
a
we find, taking only the terms of the first order:'
Ic dMi
a( -c-    -   --- [cos Q cos E Cos a - +sin   Sill a] sec (~
cos  c(I t
k ~ dM
3''       --  -- -.......  [cos 0 (sin a sincos  -   c8 sin- e)- cos a sin 3sinQ].
cos g  dt
The  constant quantity  --    d   is called  the  constant
cos 99 dt
dll
of aberration, and since  it denotes the mean sidereal motion  of the sun  in a second  of time, which is the unit of
k, we are able to compute it, if besides the time in which
the light traverses the semi-major axis of the earth's orbit
is known. Delambre determined this time from the eclipses
of Jupiter's satellites and thus found for the constant of
aberration the value 20". 255. Struve determined this constant latterly from  the observations of the apparent places of
the fixed stars and  found  20".4451  and as we have dtM
dt
59' 81". 19
59s6400        0.0410670  and  cos p —9.999939  we find from
this for the time in which the light traverses the semi-major
axis of the earth's orbit 497s.78 *).
We have therefore the following formulae for the annual aberration of the fixed stars in right ascension and declination:
*) According to Hansen the length of the sidereal year is 365 days 6
hours 9 minutes and 9,35 seconds or 365. 2563582 days, hence the mean
daily sidereal motion of the sun is 59' 8". 193.




185
a' - a = - 20".4451 [cos, cos e cos sa ~ — sin ~ sin a] sec e
a" - S      20". 4451 cos 2 [sin a sin 3 cos E - cos S sin ~]   (A)
- 20".4451 sin (~ cos a sin A.
The terms of the second order are so small, that they
can be neglected nearly in every case. We find these terms
of the right ascension by introducing the values of the differential coefficients (b) into the second term of the formulae
(a), as follows:
4 2 2dt   see S2 [os 2   sin 2 a - os E 2) - 2     cos 2 \a cos -,]
where the small term  multiplied by sin 2 a sin Ea  has been
omitted.  For we find setting aside the constant factor:
2 sin 2 x [cos _2 COS- sin Q2] - 2 sin 2 0 cos ~ [cos a2 -- sin ca']
from which the above expression can be easily deduced. If
we  substitute  the numerical values taking    -=  23~ 28', we
obtain:
-0".0009329 sec a2 sin 2 a cos 20       (
+ 0". 0009295 sec e2 cos 2 a sin 2 (c.
As these terms amount to,  of a second of time only if
the declination  of the star is 851,", they can always be neglected except for stars very near the pole.
The terms of the second order in declination, if we nleglect all terms not multiplied by tang 5, are:
/C2  (M2
cOS - 2  d-ct) tang a [cos 2 ~ (cos 2 a (1 - cos ea) - sinl 2)
- 2 sin 2 0 sin 2 a cos E].
For we find the term  multiplied by tang S, setting aside
the constant factor:
sin G)2 sin aX2 - cos (2 cos 2 a COS a2"   + -   sin 2 0 sin 2 a Cos E
and if we express here the squares of the sines and cosines
by the sines and cosines of twice the angle and omit the
constant terms 1             a- cos a2 as well as the term  cos 2 a sin a2
we easily deduce the above expression.  Substituting again
the numerical values we find:
+ [0". 0000402 - 0". 0004665 cos 2 a] tang 8 cos 2 ( (d
- 0". 0004648 tang 8 sin 2 a sin 2 0.
As these terms also  do not amount to  l of a second
of' arc while the declination is less than 870 6', they are taken
into account only for stars very near the pole.
In  the formulae  (A) for the aberration it is assumed,
that ce, 8 and 0  be referred to the apparent equinox and




186
that E is the apparent. obliquity of the ecliptic.  But in computing the aberration of a star for any long period it is convenient, to neglect the notation and to refer a, cJ and 0 to
the mean equinox and to take for e the mean obliquity.  In
this case however the values of the aberration found in that
way must be corrected. We find the expressions of these
corrections by  differentiating  the  formulae (A) with respect
to a, A, 0  and E and taking da, clS, d(  and de equal to
the  nutation  for these  quantities.   Of course it is only necessary  to  take the largest terms of the nutation and  omiting in the correction of the right ascension all terms, which
are not multiplied by sec J. tang c' and in declination all
terms which are not multiplied by sin 6. tang 6, we easily
see, since the increments dc) and ds do not produce any such
terms, that we need only take the following:
da -- - [6". 867 sin &) sin a +- 9". 223 cos fn cos a] tang ~',
d  = - [6".867 sinl 5 cosa - - 9". 223 cos $f sin a].
Taking here 6". 867 -  b and 9". 223 = -a, we find, if we
substitute these quantities into the differentials of the equations (A):
a'- a   tang8 see  10".2225 - (b + -acos ) sin2acos ((+ A) )
- (bcoss —a)cos2 asin (0)-I- ))
-+ (b - a cos e) sin 2 a cos (( -- ) )
- (b cose-a) cos2 a sin (()- 0 )
3' — S - tang S sin 85". 1112  - (b +- a cos a)cos 2 a os (   -t+&)
- (b cos e -+- a) sin 2 a sin(Q (i)- +)
+ (b- a cos E) cos 2 a cos ( — ~))
- (b cos  - a) sin 2   a sin (()  )
+ (D — acos s) cos (   + ) 
- (b+a a cos ) cos ((:) -)
or if we substitute the numerical values:
a' - a = tang  sec.     0".0007597 sin 2 a cos (  + S)'
- + 0".0007693 cos 2 a sin (  +  ) ) i
- 0".0000790 sin 2 a cos () -  ))(
+- 0".0001449 cos 2 a sin (0 - S))  (e)' - a - tang S sin a.  - 0".0003798 cos 2 a cos (0O  G ) 
- 0".0003847 sin 2 a sin (G + -O))
- 0".0000395 cos 2 a cos (G -  1)
- 0".0000725 sin 2 a sin (0 - Sg)
- 0".0000395 cos (0-t+ $)
- 0".0003798 cos ((0 -  )




187
While the declination is less than 851~, a' —   is less
than!j15 of a second of time and c' --   is greater than ~1
of a second of arc only for declinations exceeding 850 6'.
Hence these terms as well as those given by the equations
(c) and (d) can be neglected except in the case of stars
very the pole.
The equations for the aberration are much more simple,
if we take the ecliptic instead of the equator as the fundamental plane.  For then neglecting again the constant terms
we find:
dx       a        dill
- ---    sin Oi
dt     cosqs      dt
dy       a        dMil
dt     cos     0 dt
dz   0.
dt
and if we substitute these expressions in the formulae (a) and
write X2 and fl in place of a and b, we find for the aberration
of the fixed stars in longitude and latitude:
A' -    = - 20". 4451 cos ( - 0) sec /,   (B)
fi'-     - = - 20".4451 sin (2 —Q) sin fi,
which formulae are not changed if we use the apparent instead of the mean equinox.
The terms of the second order are:
in longitude:   - 0". 0010133 sin 2 (( —i) sec 2,
in latitude:     - 0". 0005067 cos 2 ((-) Z) tang f,
where the numerical factor 0.0010133 is equal to. (20.064451)2
206265
Example.  On the first of April 1849 we have for Arcturus:
a= 14hSm48s- 2120 12'.0,  =+-t- 190 58'.1, 0=11~ 37'.2
= —- 230 27'. 4.
With this we find:
a' - a -  - 18". 88,
8'-a-   -  9". 65,
and as
=- 2020 8',  =  - 300 50',
we find also:
2'- gR  -~- 23". 41,




188
17.  In order to simplify the computation of the aberration in right ascension and declination, tables have been
constructed, the most convenient of which are those given by
Gauss.  He takes:
20". 445 sin 0 -- a sin (0 -+ A),
20". 445 cos ( cos e = a cos (G) +- A),
and thus has simply:
a -   - a see 8 cos ( -  A -a),?' - b  - -- a sin 3 sin ( - A - a)  20". 445 cos ( cos A sill E
- - a sin a sin (0  +  A -  ) - 10".222 sin  cos ((. + — )
-10". 222 sin s cos ((0 - 8).
From  these formulae the tables have been computedd.
The first table gives A and log a, the argument beimng the
longitude of the sun, and with these values the aberration
in right ascension and the first part of the aberration in declination is easily computed.  The second and third part is
found from another table, the angles 0-j-  and O - () being
successively used as arguments. Such tables were first published by Gauss in the Monatliche Correspondenz Band XViI
pag. 312, but the constant there used was that of Delamubre
20". 255.  Latterly they have been recomputed by Nicolai
with the value 20".4451 and have been published in Warnstorff's collection of tables.
For the preceding example we find from those tables:
A --   1i', log a = 1.2748
and with this
ax' -x  --  -   18". 8,
and the first part of the aberration in declination — 2". 15.
For the second and third part we find   3".47 and -4".03,
if we enter the second table with the arguments 31"35' and
8"21. We have therefore:
8' -' —  9". 65.
18. The maximum  and minimum  of' aberration in longitude takes place, when the longitude of the star is either equal to the longitude of the sun or greater by 180",
while the maximum  and minimum in latitude occurs, when
the star is 90" ahead of the sun or follows 90" after. Very
similar to the formulae for the annual aberration are those
for the annual parallax of the stars (that is for the angle




189
which lines drawn froin the sun and from  the earth subtend
att the fixed star) only the maxima and minima in this case
occur at different times.  For if A  be the  distance of the
fixed star from  the sun   X and (? its longitude and latitude
as seen froml  the sun: the co-ordinates of the star with respect to the sun are:
x -A  COS fP Cos 2, y c --- SA COS f sill 2, z --  sinll.
But the co-ordinates of' the star referred to the centre:.of the earth are:
XI= A' COS' cos'2, y  l' -' cos i' sin 2', z' A=' sin ni'
and as the co-ordinates of the sun with respect to the earth are:
X —R  cos () and  Y-  1P sin -
where the semi-major axis of the  earth's orbit is the unit,
we have:
A' cos,f' cos 2' = A cos    COS;    -s - A     cos o
A' cos's i          sin      sin 0
A' sin f' =   sin i8,
from  which we easily deduce:
--    ---- sin (R -  ) see   2. 206265,' —  --   -- cos (2 - 0) sin fl. 206265.
or as y 206265 is equal to the annual parallax n:
2' -- = --   / Sill (b -S in ) see p
fi' — =-f   R cos (A -   )) sin fl.
Hence we see that the formulae are similar to those of
the aberration, only the maximuln and minimum  of the parallax in longitude occurs, when the star is 90" ahead of the
sun  or follows 900 after it, while the maximnmm and mininum
in latitude  occurs, when  the longitude is equal to that of
the sun or is greater by 1800.
For the right ascensions and declinations we have the
following equations:
A' cos 3' cos a'- A cos 3 cos a( + R cos 0
A' cos 3' sin cc' - A cos 3 sin a +t R sin 0) cos e
a'sin'= A sin 3     + — sin  sin e,
from  which we find in a similar way as before:
a' - c   -  VR [cos 0 sil a - sin ( co  s E cos] sec S
A3'-    ---— 7 R [cos E sin a sin A - sin E cos A] sill n   (D)
-   R cos X0 sin 3 cos a.




190
19. The rotation of the earth on her axis produces likewise an aberration which is called the diurnal aberration.
But this is much smaller than the annual aberration, since
the velocity of the rotation of the earth on the axis is much
smaller than the velocity of her orbital motion.
If we imagine three rectangular axes, one of which coincides with the axis of rotation, whilst the two others are situated in the plane of the equator so that the positive axis
of x is directed from  the centre towards the point of the
vernal equinox and the axis of y towards the 90th degree of
right ascension, the co-ordinates of a place at the surface
of the earth are according to No. 2 of this section as follows:
x= e COS c99 COS  O,
y -   cos 99' sin 0,
z  e sin Tp.
We have therefore:
dx                 dO
-  e cos' sin 0., X
dt                 dt
dY =+ q cos 99'cos. d -,
dt                 d't
dz
-0.
dt
If we substitute these expressions in formula (a) in No. 16,
we easily find omitting the terms of the second order:
a - a --       cos I' cos (0 - a) see,
p dt'" --    -"d eO cos 9T' sin (O - a) sin 8.?  dt
If now T be the number of sidereal days in a sidereal
year, the angular motion of a point caused by the rotation
on the axis is T times faster than the angular motion of the
earth in its orbit and we have:
dO     dM
--- T - T
dt     dt
Thus as we have:
- -k  - k sin r
a
where n is the parallax of the sun, k the number of seconds
in which the light traverses the semi-major axis of the earth's
orbit, the constant of diurnal aberration is:
dMi
k.. sin dt. T,
dtt




191
or as we have:
k. -    20". 445,   8". 5712 and T= 366.26 is,
01".3113.
Hence if we take instead of the geocentric latitude (f'
simply the latitude q, we find the diurnal aberration in right
ascension and declination as follows:
a' - a -0"..3113 cos T cos (O - a) see, 
A' - S - 0". 3113 cos 99 sin (O - a) sin S.
The diurnal aberration in declination is therefore zero,
when the stars are on the meridian, whilst the aberration in
right ascension is then at its maximum and equals:
0". 3113. cos 99 see A.
20. We have found the following formulae for the annual aberration of the fixed stars in longitude and latitude:
~' -     k - -c cos (;- ~) sec i,
i' - -- --  k sin ( 2 -  )) sin fi,
where now k denotes the constant 20". 445. If we now imagine
a tangent plane to the celestial sphere at the mean place of
the star and in it two rectangular axes of co-ordinates, the
axes of x and y being the lines of intersection of the parallel
circle and of the circle of latitude with the plane and if' we
refer the apparent place of the star affected with aberration
to the mean place by the co-ordinates:
x     - = (' -  ) cos fi and y = f' —  *),
we easily find by squaring the above equations:
Y/2 _ k2 sin fl2 _- x2 sin f2.
This is the equation of an ellipse, whose semi-major
axis is k and whose semi-minor axis is k sin /f. We see therefore that the stars on account of the annual aberration describe round their mean place an ellipse, whose.semi-major
axis is 20".445 and whose semi-minor axis is equal to the
maximum of the aberration in latitude. Now if the star is
in the ecliptic, # and hence the minor axis is zero.  Such
stars describe therefore in the course of a year a straight
line, moving 20".445 on each side of the mean place. If the
star is at the pole of the ecliptic, fi equals 900 and the mi~) For as the distances from the origin are very small we can suppose
that the tangent plane coincides with that small part of the celestial sphere.




192
nor axis is equal to the major axis.  Such a star describes
therefore in the course of a year about its mnean place a
circle whose radius is 20".445.
In order to find the place which the star occupies at
any time in this ellipse, we imagine round the centre of the
ellipse a circle, whose diamleter is the major axis of the ellipse. Then it is obvious, that the radius must move in the
course of a year over the area of the circle with uniform
velocity so that it coincides with the west side of the major axis, when the longitude of the  sun is equal to  the
longitude of the star, and with the south part of the minor
axis, when the longitude of the sun exceeds the longitude of
the star by 900. If we draw then the radius corresponding
to any time and let fall a perpendicular line from  the extremity of the radius on the major axis, the point, in which
this intersects the ellipse, will be the place of the star.
If the star has also a parallax;r, the expressions for the
two rectangular co-ordinates become:
x = —   cos (-   --  r sin ( - G)
y       in (- ()) sinn sin  - z cos (2- )) sin,
or, taking:
k -- a cos A' -a  sin A/,X- -a cos ( —-A)
3/  +- a sin (2-  A — A) sin fl.
Hence  also  in this case the star describes round  its
nm-ean place all ellipse, whose semi-major  axis is l/tk2_L-n2 ald
whose semi-minor axis is sin fll J/kt -252
The effect of the diurnal aberration is similar. The stars
describe on account of it in the course of a sidereal day
round their mean places an ellipse, whose sem-imajor axis is
0". 3113 cos Tp and whose semi-minor axis is 0". 3113 cos ep sin Si.
If the star is in the equator, this ellipse is changed into a
straight line, while a star exactly at the pole of the heavens
describes a circle.
21.  If the body have a proper motion like the sun, the
moon and the planets, then for such the aberration of the
fixed  stars is not the  complete aberration.   For as such
a body changes its place during the time in which a ray of




193
light travels from it to the earth, the observed direction of
the ray, even if corrected for the aberration of the fixed
stars, does not give the true geocentric place of the object
at the time of observation. We will suppose, that the light,
which reaches the object-glass of the telescope at the time t,
has left the planet at the time T. Let then P Fig. 5 be the
place of the planet at the time T, p its place at the time t,
A the place of the object-glass at the time T, a and b the
places of the object-glass and the eye-piece at the time t and
finally a' and b' their places at the time t', when the light
reaches the eye-piece. Then is:
1) A P the direction towards the place of the body at the
time T: ap that towards the true place at the time t,
2) a b and a' b' the direction towards the apparent place
at the time t or t', the difference of the two being indefinitely small,
3) b'a the direction towards the same apparent place corrected for the aberration of the fixed stars.
Now as P, a, b' are situated in a straight line, we have:
Pa: ab' — t - T: t' - t.
Furthermore as the interval t'-  T is always so small,
that we can suppose, that the earth during the same is moving in a straight line and with a uniform velocity, the points
A, a, a' are also situated in a straight line, so that A a and
aa' are also proportional to the times t —T and t'- t. HIence
it follows that A P is parallel to b'a' or that the apparent
place of the planet at the time t is equal to the true place
at the time T. But the interval between these two times is
the time, in which the light from  the planet reaches the
eye or is equal to the distance of the planet multiplied by
497s.8, that is, by the time in which the light traverses the
semi-major axis of the earth's orbit, which is taken as the unit.
It follows then that we can use three methods, for computing the true place of a planet from its apparent place at
any time t.
I.  We subtract from  the observed time the time in
which the light from the planet reaches the earth; thus we
find the time T and the true place at the time T' is identical with the apparent place at the time t.
13




194
II. We can compute from  the distance of the planet
the reduction of time t —' and from the daily motion of
the planet in right ascension and declination compute the
reduction of the observed apparent place to the time T.
III. We can consider the observed place corrected for
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 used when the distance
of the body is not known, for instance in computing the orbit
of a newly discovered planet or comet.
Since the time in which the light traverses the semimajor axis of the earth's orbit is 497s.8 and the mean daily
motion of the sun is 59' 8". 19, we find the aberration of
the sun in longitude according to rule II. equal to 20". 45,
by which quantity we observe the longitude always too small.
On account of the change of the distance and the velocity
of the sun this value varies a little in the course of a year
but only by some tenths of a second.
22. The aberration for a moveable body, being in fact
the general case, may also be deduced from the fundamental
equations (a) in No. 16. For it is evident, that in this case
we need only substitute instead of the absolute velocity of
the earth its relative velocity with respect to the moveable
body, since this combined with the motion of the light again
determines the angle by which the telescope must be inclined to the real direction of the rays of light emanating
from the body in order that the latter always appear in
the axis of the telescope nothwithstanding the motion of the
earth and the proper motion of the body. If therefore ~, Hi
and ( be the co-ordinates of the body with respect to the
dx  d~
system of axes used there, we must substitute in (a) d- dr'
dy  dgy  dz  d             dx  dy aed  dz
it  -' -d-   instead of      d  and  t But if A is the
distance of the body from the earth, we find the heliocentric
co-ordinates  i, j7, g, since the geocentric co-ordinates are
A cos Y cos a etc., from the formulae:
e = A cos  cos a + x,
a =A cos sin c -y,       (f)
C  A sinS    +z,




195
from  which we easily dedluce the following:
dx - d;) sill a - dy- d) cos a =Acos,
t  c1lt)11       (d-    dit dtJ          ct
dt -- d) stil s Cos a +- d-  (dI  sin S sin a + — t ---  )c-cos - A d
(dit                 \cit dcitdi           \cit  citi        dct
Hence the formulae (a) change into:
A da
It dt
d8
It dt
or as -  equals the time in which the light traverses the distance A, we find, if we denote this by t-  T:
cdt,4 -----  -s- (t — T)'I, 
cd t
which formulae show, that the apparent place is equal to the
true place at the time  T and therefore  correspond  to the
rules I and IL of the preceding number.
But we also find the aberration for this case by adding
to  the  second  memlber of the first formula (a) the term, kll6 sin C a     - cos r] sec J and a similar term  to the second
e 1cll zcl t
member of the second  equation.   We get therefore, if we
denote the aberration of the fixed stars by Da and D:
c'-      -  a t   c si - ll-  c Si a — cos a sec a,
t idt       cit'
S- _ a= =L  +~- d-F   sin a cos  +  t  sill n  silln a -   cos
fL dt            dt            dt
But differentiating the equations (f), taking in the second
member only the geocentric quantities A, ce, Y as variable and
the  co-ordinates of the earth as constant, and denoting the
partial differential coefficients by (d-t) and (- I),  we find the
second members of the above equations respectively equal to:
d (dca ) an  l   A (d8
dtc) and --   d c-t
We therefore have:
aX'-      a - (    T). d /
and S'  D-     - a 8 - (t - T). (d,
13*




196
which formulae correspond to the third rule of the preceding
No.   For  since (           ) and t ) are the differential coefficients
of ce and S, if the heliocentric place of the planet is changed
whilst the place of the earth remains the same, the second
members  of the two  equations give the places of the planet
at the time  T, but as  seen  from  the place which the earth
occupies at the time t.
NVote.  The motion of the earth round the sun and the rotation on the
axis are not the only causes which produce a motion of the points on the
surface of the earth in space, as the sun itself has a motion, of which the
earth as well as the whole solar system participates.  This motion consists
of a progressive motion, as we shall see hereafter, and also of a periodical
one caused by the attractions of the planets.  For if we consider the sun
and one planet, they both describe round their common centre of gravity
ellipses, which are inversely as the masses of the two bodies. The first motion which at present and undoubtedly for long ages may be considered as
going on in a straight line, produces only a permanent and hence. imperceptible change of the places of the stars and the aberration caused by the
second motion is so small that it always can be neglected. For if a and a'
are the radii of the orbits of two planets which are here considered as circular, r and r' their times of revolution, then the angular velocities of the
1 1
two will be as —:     hence their linear velocities as aC6': a',C or as /a': l/a,
qr 7,
since according to the third law of Kepler the squares of the periodic times
of two planets are as the cubes of their semi-major axes.  The constant
of aberration for a planet, the semi-major axis of whose orbit is a, taking
20".45
the radius of the earth's orbit as unit, is therefore        and hence the
constant of aberration caused by the motion of the sun round their common
20". 45
centre of gravity is equal to m. - --, where?i is the mass of the planet
expressed in parts of the mass of the sun. In the case of Jupiter we have
2.-= —  - and a-5.20, hence the constant of aberration caused by the attraction of Jupiter is only 0".0086.
The perturbations of the earth caused by the planets produce also changes
of the aberration, which however are so small, that they can be neglected.
Compare on aberration: The introduction to Bessel's Tabulae Regiomontanae p. XVII et seq.; also Wolfers, Tabulae Reductionum p. XVIII etc.
Gauss, Theoria motus pag. 68 etc.




FOURTH SECTION.
ON THE METHODS BY WHICH THE PLACES OF THE STARS AND
THE VALUES OF THE CONSTANT QUANTITIES NECESSARY FOR
THEIR REDUCTION ARE DETERMINED BY OBSERVATIONS.
The chief problem  of spherical astronomy is the determination of the places of the stars with respect to the fundamental planes and especially the equator, as their longitudes
and latitudes are never determined by observations, but, the
obliquity of the ecliptic being known, are computed from their
right ascensions and declinations. When the observations
are made in such a way as to give immediately the places
of the stars with respect to the equator and the vernal equinox, they are called absolute determinations, whilst relative
determinations are such, which give merely the differences
of the right ascensions and declinations of stars from those
of other stars, which have been determined before.
The observations give us the apparent places of the stars,
that is, the places affected with refraction *) and aberration and
referred to the equator and the apparent equinox at the time
of observation. It is therefore necessary to reduce these
places to mean places by adding the corrections which have
been treated in the two last sections.  But the expressions
of each of these corrections contain a constant quantity, whose
numerical value must at the same time be determined by similar observations as those by which we find the places of
the stars. The values of these constant quantities given in
the last two chapters are those derived from the latest determinations, but they are still liable to small corrections by
future observations.
*) In the case of observations of the sun, the moon and the planets
these places are affected also with parallax.




198
If' we observe the places of the fixed' stars at different
times we ought to find only such differences as can be ascribed to any such errors of the constant quantities and to
errors of observation. However, comparing the places determined at different epochs we find greater or less differences
which cannot be explained by such errors and must be the
effect of proper motions of the stars.  These motions are
partly without any law  and peculiar to the different stars,
partly they are merely of a parallactic character and caused
by the progressive motion of the solar system, that is, by
a proper motion of the sun itself.  So far these proper motions with a few  exceptions can be considered as uniform
and as going on in a great circle.  They must necessarily
be taken into account in order to reduce the mean places
of the stars from one epoch to the other.
The methods for computing the various corrections which
must be applied to the places of the stars have been given
in the two last sections; but as these computations must be
made so very frequently for the reductions of stars, still other
methods are used, which make the reduction of the apparent places of stars to their mean places at the beginning of'
the year as short and easy as possible and which shall be
given now.
I. ON THE REDUCTION OF THE MEAN PLACES OF STARS TO
APPARENT PLACES AND VICE VERSA.
1. If we know  the mean place of' a star for the beginning of a certain year and we wish to find the apparent
place for any given day of another year, we must first reduce
the given place to the mean place at the beginning of' this
other year by applying the precession and if necessary the
proper motion and then add the precession and the proper
mlotion from the beginning of the year to the given day as
well as the nutation and aberration for this day.  Now in
order to make the computation of these three last corrections
easy, tables have been constructed for all of them, which




199
have for argument the  day of the year.  Such tables have
been given by Bessel in his work,,Tabulae Regionontanae" ).
Let a aand     be the mean right ascension and declination
of a star at the beginning of a year, whilst a' and 6' designate
the apparent right ascension  and declination at the time r,
reckoned from the beginning of the year and expressed in
parts of a Julian year. If then,.t und Pi' designate the proper
motion of the star in right ascension and declination, which
is considered to be proportional to the time, we have according to the formulae (D) in No. 2, (B) and (C) in No. 5
of the second section and (A) in No. 16 of the third section
the following expression:
a' - a = -   [7n - 1-n tang S sin a] -+- r t
- [15".8148 +- 6".S650 tangS sin a] sin[}
- 9".2231 tang  ecos a cos 2
- [0".1902 -+ 0".0822 tang 8 sin a] sin 2 9
- 0".0896 tang S cos a cos 2 f)
- [1". 1642 +- 0".5054 tang 8 sin a] sin 2 (
-- 0".5509 tang a cos a cos  2 0
+- [0".1173 -I- 0".0509 tang S sin at] sin (0 -P)
- [0".0195 + 0".0085 tang 8 sin a] sin (- +-P)
- 0".0093 tang S cos a cos (( +- P)
- 20".4451 cos e sec S cos a cos ()
- 20".4451 see 8 sin a sin 0
and:
d' -a  -4- + z a cos a +- r a'
- 6".8650 cos a sin -) + 9".2231 sin a cos
- 0".0822 cos a sin 2 ) -0".0896 sin a cos 2 ~)
- 0".5054 cos a sin 2   + +- 0".5509 sin a cos 2 0
+ 0".0509 cos a sin ( — P)
- 0".0085 cos a sin (- + P) - 0".0093 sin c cos (0 +- P)
+ 20".4451 [sin a sin S cos E - cos S sin e] cos (
-20".4451 cos a sin B sin 0(.
The terms of the nutation, which  depend on twice the
longitude of' the moon 2(( and on the anomaly ( —P' of the
moon have been  omitted  here, as they have a short period
on account of the rapid motion of the moon and therefore
are better tabulated  separately.  Moreover these terms are
only  small and on account of' their short period are nearly
eliminated in the mean of many observations of a star. Hence') For a few stars it is necessary to add also the annual parallax, for
which the most convenient formulae shall be given hereafter,




200
they are only taken into account for stars in the neighbourhood  of the pole, for which  also  the terms depending on the
square and the product of nutation and aberration *) become
significant.  These terms are brought in tables, whose arguments are ((, 0, 0 -+~     and ()- c).
Now  in order to  construct tables for the above expressions for ca'-c     and  &'-, we put:
6".8650 --  i        15".8148- m i - h
0".0822 — ni          0".1902 - m il = I
0".5054 =      zi2    1  ".1642 - mi2 =12
0".0509 - ni3         0".1173 - 2113i = /h3
0".0085 -ni4          0".0195 -n mi4   - 114.
Then  we can write the formulae also in this way:
a - a   [r - i si        isin 2    -i 2sin 2 0+ -+ i3 sin (O — P)
- i4 sin (+ -4- P)] [m -+- n tang  sin a]
- [9".2231 cos t  - 0".0896 cos 2 ) +- 0".5509 cos 2 (D
+- 0".0093 cos ((D 0- P)] tang d'cos a
-20".4451 cos e cos 0. cos a sec
-'20".4451 sin. sin a sec 8
-  sin+ q- 1 sin2  -h2 sin 2 )- 13 sin (( - P) -- 1 sin ( 0+ P)
and:
S' -=[z — i sin n +-i,  sin 2  )-i2 sin 2 (  + i3 sin ((-)-P)
— i4 sin (( Q+ P)]n  cos Ca
+ [9".2231 cos   - 0".0896 cos 2.   -+- 0".5509 cos 2 ()
+ 0".0093 cos ((O + P)] sin a
- 20".4451 cos a cos G [tang a cos 8- sin a sin a]
- 20".4451 sin 0. sin 8 cos ac
If we introduce therefore the following notation:
dA  — i sin -   +1    sin2)-i2 sin2 i n + i3 sinl (0-P) - i4 ill (O +P)
B   - 9"'.2231 cos) -+- 0".0896 cos 2&) - 0".5509 cos 2 —0".0093 cos(-+ —P)'= — 20".4451 cos t cos 
D — 20".4451 sin O(
iE= - Itsin ) -- h,1 sin 2,Q- IA sinin 2 ( + h3 sin () — P) -— h4 sin (  -t+ P)
a- =- t7-  n tang   sin a      Cal  1   COS a
b  tang o cos a                1 -'  - sin a
c - sec  cos a                 c'   tang s coS 8-sin 8 sin a
d-= sec  sin ac                d' - sin 8 cos a,
~) These terms are given by the formulale (E) in No. a of the second
section and (c), (cl) and (e) in No. 16 of the third section.




201
we have simply:
a'- a  A,  B b -  Cc -~- D d + tr  +E- AE
-'   - Aat'+-   b'+4- Cc' -- D d'+ - r',  ()
where the quantities a, b, c, d,',    c', d' depend only on
the place of the star and the obliquity of the ecliptic, while
A, B, (7, D  depend only on 0  and D) and thus being mere
functions of the time may be tabulated with the time for
argument.
The numerical values given in the above formulae are
those for 1800 and we have for this epoch:
i-0.34223  i =0.00410  i2 =0.02519  i3 — 0.00254  i:1 =0.00042
1-=0.0572  hi =0.0016  112 -0.0041   h3 —0.0005   h4 =0.0000.
We see therefore that the quantity E never amounts to
more than a small part of a second, hence it may always
be neglected except when the greatest accuracy should be
required. As several of the coefficients in the above formulae
for t'- -e and'- -   are variable (according to No. 5 of the
second section) and likewise the values of m and n, we have
for the year 1900:
i —0.34256  i,=-0.00410  i —0.02520  i3 - 0.00253  i4 =0.00042
h=-0.0488  h 1 =0.0014  12 —0.0035   h3 =0.0005.
The values of the quantities A, B, C, D, E from the
year 1750 to 1850 have been published by Bessel in his work,,Tabulae Regiomontanae".  But as he has used there a different value of the constants of nutation and of aberration
and also neglected the terms multiplied by  -— P and 0-~ P,
the values given by him  require the following corrections
in order to make them  correspond to the formulae given
above:
For 1750:
dA  -- 0.0090 sin nf +- 0.0001 sin 2  1 +- 0.0013 sin 20,)
+ 0.0025 sill (O - P) - 0.0004 sin (Q + P)
dclB   - 0.2456 cos -i +- 0.0019 cos 2 ) +- 0.0290 cos 2 )
- 0.0093 cos (( + P)
d C  - -0.1744 cos O
dD- - 0.1901 sin 0
dE = - 0.006 sin ) - + 0.001 sin 2 i
For 1850 the value of dB becomes:
dB   - 0.2465 cos ) + 0.0019 cos 2   - +- 0.0291 cos 2 G - 0.0093 cos (0-+P).




202
The values of the quantities A, B etc. fbr the years 1850
to 1860 have been computed by Zech according to Bessel's
formulae, and for the years 1860 to 1880 they have been
given by Wolfers in his work,,Tabulae Reductionunl Observationlum Astronomnicarum", where they have been computed
from  the formulae given above. The values for each year
are published in all astronomical almanacs.
2. The arguments of all these tables are the clays of
the year, the beginning of which is taken at the time, when
the mean longitude of the sun is equal to 280~. 1Hence the
tables are referred to that meridian, for which the beginning
of the civil year occurs when the sun has that mean longitude.  But as the sun performs an entire revolution in 365
days and a fiLaction of a day, it is evident, that in every
year the tables are referred to a different meridian.
Therefore if' we denote the difference of longitude between
Paris and that place, for-which at the beginning of the year
the mean longitude of the sun is 2.80', by k, which we take
positive, when the place is east of Paris, and if further we dclenote by d the difference of longitude between any other place
and Paris, taking it positive, when this place is west of Paris
and if we suppose both k and d to be expressed in time,
we must add to the time of the second place for which we
wish to find the quantities A, B, C, D, E from the tables,
the quantity k-t-d  and for the time thus corrected we must
take the values from  the tables.  The quantity k is found
friom:
L - 2800
where L is the mean longitude of' the sun at the beginning
of the year for the meridian of Paris, while tt is the mean
tropical motion of the sun or 59' 8". 33.  This quantity is
given in the,,Tabulae Regiomontanae" and in Wolfers' Tables
for every year and expressed in parts of a day and the constant quantities A, B, C, D, E are given for the beginning
of the fictitious year or for 18"' 40"' sidereal time of that meridian, for which the sun at the beginning of' the year has
the longitude 280" and then for the same time of every tenth




203
sidereal day ). If now we wish to have these values for any
other sidereal time, for instance for the time of' cunlination
of a star whose right ascension  is eY, we must add to the
argument k — d the quantity:
a - 18h 4011~  a
a'........0.778.
24h       24
Furthermore as on that day, on which the right ascension
of' the  sun is equal to the right ascension of' the star, two
culminations of the star occur, we must after this clday  cldd
a unit to the datum  of the day, so that the complete argument
is always the datum  plus the quantity:
k-  dc -a' + i,
where we have i -           from  the beginning of' the year to the
time, when the right ascension of the sun is equal to ce, while
afterwards we take i —1.
Now  the day, denoted in the tables by Jan. 0, is that9
at the  sidereal time  18 401'" of which the year begins, the
commencement of the days being always reckoned from noon.
Hence  the  cunlmination  of stars,  whose  right ascension  is
<  18h 4011 does not fall on that day, which in the tables is
denoted by 0, but already on the day preceding and therefore
for such stars we must add 1 to the datum  of the day reckoned  froln noon or we must take i =  1 from  the beginning
of the year to the day when the right ascension of the sun
is equal to a and afterwards i   2.
We will find  for instance the  correction  of the mean
place of rc Lyrae for April 1861 and for the time of cuhnination for Berlin. We have for the beginning of' the year:
ta=2780 3'30"  =-+38~ 39'23" E —23"27'22" m'-46".062 logn-=1.30220
and from  this we find:
*) We have therefore to use for computing the tables:
10 a
366. 242201
10n. $60o
Mean longitude of the suin = 2800 + 
366.242201'
where iz must be taken in succession equal to all integral numnbers from 0
to 37. With this we find the true longitude according to I. No. 14. We
have also:
-— 33~ 15' 25". 9 - 19 20'29  (t- 1800)'366 242201 1  20'29". I3,
866,242201




204
log a = 1.47971    log a' = 0.44889
log b =9.04973    log b' - 9.99569
log c- 9.25409    log c'-  9.98106
log d = 0.10309,,  log c' - 8.94233
and besides we have:
log ut - 9.4425   log' = — 9.4564.
Further we have according to Wolfers' Tabulae Reductionum:
log A    log B     log C    log D     loga       E
March 31  9.7494   0.5497,,  1.26G0,,  0.5668,,  9.3905  -+ 0.05
April 10  9.7653   0.5279,,  1.2456,   0.8488,,  9.4362  +- 0.05
20  9.7819   0.4982,,  1.2109,,  1.0089,,  9.4776   +- 0.05
30  9.7995   0.4620,,  1.1596,  1.1155,,  9.5154    - 0.05
and we get according to the formulae (A):
a' -a            6- 6
March 31    - 1 s. 203        - 19". 85
April 10   -+ 1.541          -19.09
20     - 1.871         - 17.79
30       2.185         — 15.97.
a -- 18h 40m11
Now we have    — + 0.124, d --   0.031,   -4i — 0              0.005,
and as here i is equal to 1, because a is less than 181 40"
and  in March  and April the  right ascension  of the sun is
less than 181h40m, the argument in this case is
the datum  -+  1.088.
We find therefore at the time of culmination for Berlin:
March 31    +. 239         - 19". 79
April 10    — 1.577            18.98
20    + 1.906             17.62
30   -+- 2 219             15. 76.
If we subtract these corrections from the apparent place,
we find the mean place at the beginning of the year.
3.  This method of reducing the mean place to the apparent place and vice versa is especially convenient in case,
that we wish to compute an ephemeris for any greater length
of time, for instance if we have to reduce many observations
of the same star.  But in  case that the reduction for only
one day is wanted, the following method may be used with
greater convenience, as it does not require the computation
of the constant quantities a, b, c, etc.
The precession and nutation in right ascension are equal to:
A m? +- A n sin a tang 8 +- B tang 3 cos a + E
and in declination:       An cos a - B sin a.




205
Therefore if we put:          An   g cos G
B = g sin G
A in +- E   f,
the terms for the right ascension become:
f+- g sin (G  - a) tang 3
and those for the declination:
g cos (G 4- a).
Further the aberration in right ascension is:
C see 3 cos a ~- D sec S sin a
and in declination:
- C sin S sin a -+ D sin S cos a -+- C tang s cos 3.
Hence if we put:
C, h sin H   D - h cos H    i =  C tang e,
the aberration in right ascension becomes:'C.           h sin (-II- a) sec 
and in declination:
h cos (It4- a) sin S -H i cos S.
Therefore the complete formulae for the reduction to the
apparent place are:
a'-  ccf-gsil(G- a)tangS+hsin (H — cc) sec) se                 (B)
s'-3 —    gs(G- a)-Acos (H-a) sin -i cos h cos ( ) sin    i cos'.
Here again for the quantities f, g, h, i, G and H tables
may be computed, whose argument is the time. They are
always published in all almanacs for every tenth day and for
nlean noon.
If we wish to find for instance the reduction of a Lyrae
for 1861 April 10 at 171 151"1 mean time, this being the time
of culmination  of a Lyrae  on  that day, we take  from  the
Berlin Jahrbuch for this time:
f=+26".98 g —+1211.20 G-344~ 3' h —+18".98 H=2470 3' i —7".58
hence       G +- a - 2620 6'   H+ a - 1650 6'
cos (G - a)  9.13813,,     9 sin (G + a) 1.08222,,.g      I.08636           tang S      9.90304
sin (G - a)  9.99586,,     l sin (fH- a) 0.68846
cos (H- a)  9.98515,,          cos S      9.89260
ih      1.27830             i        0.87967,
sin (H+- a)  9.41016       h cos (H+ a)  1.26345
sin S     9.79564
f=  + 26".98                   i cos ----  5".92
gsin (G+a) tang 8 -  9".67               g cos (G- + a)= —  1".68
h sin (H —a) sec -= +-  6".25       h cos (H-f-+ a) sin 1S   - 11".46,u   =  +  0".08                   z'   =-    0 (".08
a —-a    + 23".64 - + Is.576' —  6 - 18".98.




206
4.  The formulae (A) and (B) for the reduction to the
apparent place do not contain the daily aberration nor the
annual parallax.  For as the daily aberration depends upon
the latitude of the place, it cannot be included in general
tables; however for meridian observations the daily aberration
in declination is equal to zero and the expression for the
aberration in right ascension being of the same form  as that
of' the correction for the error of collimation, which must be
added to the observations, as we shall see hereafter, it may
in that case always be united with the latter correction.
The annual parallax has been determined only for very
few stars, but for those it must be computed, when the greatest accuracy is required.  Now  the formulae for the annual
parallax are according to No. 18 of the third chapter:
C' - C= --    [COS   sin ca - sil 0 cos - cos a] sec c
3' - 3S X-   [COS s Sill a sin s - sill s cos ] sin C)
— 1 cos 0 sin S cos a.
Therefore if we put:
cos ~ cos a =- k sin K
-sin a - k cos K
sin a sin S cos  -      -c  S sins     sin L
- cos a sin   I cos L,
we have simplyT:
a' —.ex - 7r k cos (KC+ i) see A' --- -  X=r 1 cos (L + ().
But the cases in which this correction must be applied
are rare, for instance when observations of,t Centauri whose
parallax amounts to nearly 1" or those of Polaris are to be
reduced.
II. DETERMINATION OF THE RIGHT ASCENSIONS AND DECLINATIONS
OF THE STARS AND OF THE OBLIQUITY OF THE ECLIPTIC.
5.  If we observe the difference of the time of cullnination of the stars, these are equal to the difference of their
apparent right ascensions expressed in time. We need therefore for these observations only a good clock, that is, one
which for equal arcs of the equator passing across the me



207
ridian gives always an equal number of seconds ) and an
altitude instrument, mounted firmly in the plane of the meridian, that is, a meridian-circle.  This in its essential parts
consists of a horizontal axis, lying  on two firm  Y-pieces,
which carries a vertical circle and a telescope. Attached to
the Y-pieces are verniers or mlicroscopes, which give the arc
passed  over by the telescope by means of the simultaneous
motion of the telescope and the circle round the horizontal axis.
In order to examine the uniform rate of the clock without
knowing  the places of the stars themselves, the interval of
time is observed in which different stars return to the meridian  or to a wire stretched in the focus of the telescope
so that it is always in the plane of the meridian when the
telescope  is turned  round  the  axis ").   Now   the  time
between two successive culminations of the same star is equal
to  241' — z a, where  A C  is the variation  of the apparent
place during those 24 hours.  Therefore if the observations
were right and the instrument at both times exactly in the
plane of the meridian, a condition which we here always assume to be fulfilled, the intervals between two culminations
measured by a perfectly regulated clock would also be found
equal to 241'- Aa.  But on account of the errors of single
observations, we can only assuLme, that the arithmetical mean
of the interval found from  several stars minus the mean of
all A a is equal to 24 hours.  On the contrary if we find,
that this arithmetical mean is not equal to 24 hours but to
24h -  a, we call a the daily rate of the clock and we must
correct all observations on account of it.  In case that for
a certain time all the different stars give so nearly the same
difference 24'1 -- a, that we can ascribe the deviations to possible errors of observation, we take the rate of the clock
during this time as constant and equal to the arithmetical mean
*) It is not necessary to know the error of the clock, as only intervals
of time are observed.
**) Usually there is a cross of wires, one wire being placed parallel to
the daily motion of the stars.  This is effected by letting a star near the
equator run along the wire and by turning the cross by a screw attached to
the apparatus for this purpose, until the star during its passage through the
field does not leave the wire.




208
of all single a and we multiply the observed differences of
24       1
right ascensions by 24 — a           in order to correct theml
24
for the rate of the clock. But if we see that the rate of the
clock is increasing or decreasing with the time and the observations are  sufficiently numerous, we may assume the
hourly rate of the clock at the time t as being of the form
a - b (t - T), where a is the rate at the time T. Multiplying
this by d t and integrating it between the limits t and 24- +t,
we find the rate between two successive culminations of a
star, whose time of culmination is t, equal to:
24a -+- 24b (12+t -- T).
If we compute therefore the coefficient of b for every
star and then take u equal to the rate found from the several
stars, we obtain a number of equations, from  which we can
find the values of a and b by the method of least squares.
The rate during the time t" -t' we find then by means of
the formula:
a (t"- t') + b(t"- t') it -+- T",
and we must correct every interval of time t"- t' according to this.
In case that already the differences of the right ascensions of a number of stars are known, the difference of the
apparent place of each star and of the time U observed by
the clock, gives the error of the clock A U, which ought to
be found the same (at least within the limits of the errors
of observation) from  all the different stars, if the clock is
exactly regulated.  But if it has a rate equal to a at the
time T, each star gives an equation of the following form:
0 -    U-        a (t -- T) -- - (t -- T)2
and from  a great number of stars we may find A U, a and b ).
Now in order to observe the time of culmination of the
stars, it is necessary to rectify the meridian circle in such
*) As we suppose that the right ascensions themselves are not known
yet, at least not with accuracy, the error of the clock U would also be
erroneoulls.




209
a way, that the intersection of the cross wires is in the
plane of the meridian in every position of the telescope or
that at least the deviation from  the meridian is known').
If the line of collimation, that is, the line from  the centre
of the object-glass to the wire-cross is vertical to the axis
of the pivots (the axis of revolution of the instrument), it
describes when the telescope is turned a plane, which intersects the celestial sphere in a great circle. If besides the
axis of the pivots is horizontal, this great circle is at the
same time a vertical circle and if the axis is directed also
to the West and East points, the line of collimation must
always move in the plane of the meridian. Hence the institlment requires those three adjustments.
As will be shown in No. 1 of the last section, we can
always examine with the aid of a spirit-level, whether the
axis of the pivots is horizontal and we may also correct any
error of this kind, since one of the Y-pieces can be raised or
lowered by adjusting screws.  The position  of the line of
collimation with respect to the axis can be examined by reversing the whole instrument and directing the telescope in
each position of the instrument to a distant terrestrial object
or still better to  a small telescope (collimator) placed for
thlis purpose in front of the telescope of the meridian circle
so  that its line  of collimation coincides with that of the
meridian circle. For if there is a wire-cross at the focus of
this small telescope, it can be seen in the telescope of the
meridian circle like any object at an infinitely great distance,
since the rays coming from  the focus of the collimator after
their refraction by its object glass are parallel. Now if the
angle, which the line of collimation makes with the axis of
the meridian circle, differs by x from a right angle, the angles
which the lines of collimation of the two telescopes make
with each other in both positions of the meridian circle, will
differ by 2x or the wire of the collimator as seen in the
*) The complete methods for rectifying the meridian circle and for determining its errors as well as for correcting the observations on account
of them, are given in the seventh section.  Here it is only shown, that
these determinations can be made without the knowledge of the places of
the stars.
14




210
telescope of the meridian circle will appear to have moved
through an angle equal to 2.  Therefore if we move the
wires of the meridian telescope by the adjusting screws in a
plane vertical to the line of collimation through the angle x,
the line of collimation will be vertical to the axis and the
wire of the collimator will remain unchanged with respect
to the wires of the telescope in both positions of the instrument or to speak more correctly it will in both positions
be at the same distance from the middle wire of the telescope.  If this should not be exactly the case, the operation
of reversing the instrument and moving the wires of the telescope must be repeated.
WVhen these corrections have been made, the line of collimation describes a vertical circle. At last in order to direct the horizontal axis exactly from East to West, we must
make use of the observations of stars, but a knowledge of'
their place is not required.  The circumpolar stars, for instance the pole-star, describe an entire circle above the horizon, except at places near the equator.  Therefore if the
telescope moves in a vertical circle which is at least near
the meridian, the line of collimation intersects the parallel
circle twice, and the star can therefore be seen in the telescope twice during one entire revolution. If we observe now
the time of the passage of the star over the wire at first
above and then below the pole and the telescope is accurately in the plane of the meridian, the interval between the
two observations will be 121' -- Ar, where A a designates the
variation of the apparent right ascension of the star in 12
hours; on the contrary, the interval will be greater or less
than 12h ~+ A a  if the plane of the telescope is East or West
of the meridian. Now  as one of the Y-pieces admits always
of a motion in the direction from  North to South, we can
move this until the interval between two observations is ex,
actly 12h — Ace and when this has been accomplished the
telescope is exactly in the plane of the meridian or the axis
is directed from East to West').
*) As the complete adjustment of an instrument would be impracticable
on account of the continuous change of the errors, it is always only approx



211
We can also compare the intervals between three successive culminations with each other, as these must be equal
if the instrument is accurately in the plane of the meridian.
If the intervals are unequal, the telescope is on that side of
the meridian, on which the star remains the shortest time.
If now we observe with an instrument thus adjusted the
times of transit of stars, we find the differences of. the apparent right ascensions and we must apply to these the redulctions to  the  apparent place with the opposite sign  in
order to find the differences of the mean right ascensions
referred to the beginning of the year.  But the computation
of the  formulae  for these corrections requires already an
approximate knowledge of the right ascension and declination, which however can always be taken from former catalogues.
If the observed object has a visible disc, we can only
observe one limb and as such obljects have also a proper
motion, we must compute the time of its semi-diameter passing across the meridian according to No. 28 of the  first
section, and we must add this time to the observed time if
we have observed the first limb or substract it from it, if
we have observed the second limb.  In case of the sun having been observed, where both limbs are usually taken, we
can simply take the arithmetical mean of both times of observation.
The time of culmination of a star may be determined
still by  another method, namely  by  observing  the time,
at which the star arrives at equal altitudes on both sides
of the meridian.  For these observations a circle is required,
which is attached to a vertical column admitting of a motion
round its axis in order that the circle may be brought into
the plane of any vertical circle.  If we observe with such
an instrument the time, when a star arrives at equal altitudes on both sides of the meridian, the arithmetical mean of
both times is the clock-tiime of the culmination of the star.
It is evident, that it is not necessary to know  the altitude
imately adjusted and -the observations are corrected for the remaining errors,
which have been determined by the above methods or by similar ones, which
will be given in the last section.
14t




212
of the star itself, but it is essential, that the telescope in
both  observations  has exactly the same inclination  to the
horizon. If there is a difference of the two inclinations and
this is known, we can easily compute the error of the clocktime of culmination produced by it; for if the zenith distance
on the West side has been observed too great, the star has
been observed in an hour angle which is too great by
hence  we must subtract from  the  arithmetical
cos f sin A 
mean of both times the correction  I oA              Such a corcos go sin A'
rection is always required on account of refraction; for
although  the mean refraction is the same for both observations, yet the different state of the atmosphere, as indicated
by  the  thermometer and barometer, will produce a slight
difference of the refraction, whose effect can be computed
by the above formula.  In case of the sun being observed
the  change  of the declination  during the interval of both
observations will also make a correction necessary.
We see from  the formula c-  cos * sin A, that it is best
to  observe the zenith distances of the stars in  the neighbourhood  of the prime  vertical, because their changes are
then  the most rapid.  It is also  desirable, to make these
observations at a place not too far from the equator, because
then cos (p is also equal to 1, and to observe stars near the
equator.  As the determination of absolute right ascensions
depends upon such  observations, it may be made with advantage by this method at a place near the equator.
6.  If we bring the stars at the time, when they cross
the vertical wire of the meridian circle, on the horizontal
wire and read the circle by a vernier or a microscope, the
differences of these readings for different stars give us the
differences of their apparent meridian altitudes ), and if we
know the zenith point of the circle and subtract this from
*) In the seventh section the corrections will be given, which must be
applied to these readings in order to free them from the errors of the instrument, for instance the errors of division of the circle, or errors produced by the action of the force of gravity upon different parts of the instrument.




213
all readings, we find the apparent zenith distances of the
stars.  This point can be easily determined by observing the
images of the wires reflected from  an artificial horizon.  For
if we turn the telescope towards the nadir, and place a basin
with mercury under the object glas and reflect light from
the outside of the eye-piece towards the mercury, we see in
the light field besides the wires also their reflected images.
Therefore if we turn the telescope until the reflected image
of the horizontal wire coincides with the wire itself; the line
of collimation must be directed exactly to the nadir, hence
we find by the reading of the circle the nadir point or by
adding 1800 the zenith point of the circle.
The apparent zenith distances must first be corrected
for refraction and if the sun, the moon or the planets have
been observed, also for parallax by adding to them the refraction computed according to formula A in No. 12 of the
third section and by subtracting p sin z, where p is the
horizontal parallax ).  If the object has a visible disc, we
must add to or substract from  the zenith distance of the
limb, corrected for refraction and parallax, the radius of the
disc or if in case of observations of the sun, the lower as
well as the upper limb has been observed, we must take the
arithmetical mean of both corrected observations. Since in this
case these observations are made at a little distance from the
meridian, it is still necessary to apply a small correction
(whose expression will be given in the seventh section) because the horizontal wire represents a great circle on the
celestial sphere and therefore differs from the parallel of
the sun.
When the zenith distances at the time of' culmination
are known, the declinations are found according to No. 23
of the first section, if the latitude of the place of observation is known.  But the latter can always easily be determined by observing the zenith distances of any circumpolar
star in its upper and lower cullnination, as the arithmetical mean of these zenith distances corrected for refraction
-i-+A\   is equal to the co-latitude of the place, where AA
*) In the case of the moon the rigorous formula must be used.




214
denotes the variation  of the apparent declination  during
the interval of time.  We may also determine the latitude
by observing any cireumpolar star in its upper and lower
culmination as well direct as reflected from an artificial horizon.  For then the arithmetical mean of the corrected altitudes minus!ASS is equal to the latitude.  But as the reflected observations cannot be made at the same time as the
direct observations, usually also several observations are taken
before and after the time of culmination, we must reduce
first each observation to the meridian by the method given
in the seventh section.
If the place of observation is in the neighbourhood of
the equator, the method of determining the latitude by circumpolar stars cannot be used.  At such a place we must
determine it by observations of the sun as will be shown in
the next number.
When the latitude has been determined we find from
the zenith distances corrected for refraction the apparent declinations of the stars, which are converted into mean declinations for the beginning of the year by applying the reduction to the apparent declination with the opposite sign.
7. If A and D be the right ascension and declination
of the sun, we have:
sin A tangg E - tang D,
hence the observation of the declination of' the sun gives us
either the obliquity of the ecliptic, when the right ascension
is known,  or the right ascension, when the obliquity of' the
ecliptic is known from  other observations. But the differential equation (which we get by differentiating the above equation written in a logarithmic form)
2d a    d A._D
cotang A. cA -+  -
sin 2 E  sin 21D
shows, that it is best, to determine the obliquity of the ecliptic
by observations in the neighbourhood of the solstices and the
right ascension by observations in the neighbourhood of the
equinoxes.  If we determine the declination of the sun exactly at the time, when the right ascension is equal to 90"
or 270" we find immediately by subtracting the latitude of
the sun the obliquity of the ecliptic.  But even if we only




215
observe the declination in the neighbourhood of the solstice
and know approximately the position of the equinox, we can
compute the obliquity of the ecliptic either by the above formula or better by developing it in a series.
If we denote by D' the observed declination, by B the
latitude of the sun, the declination of the sun corrected for
the latitude, which would have been observed, if the centre
of the sun had been in the ecliptic, will be according to
the formulae in the Note to No. 11 of the first Section:
COS 8B
D'...........B —D.
cos D
Moreover if x is the distance of the sun from the solstitial point expressed in right ascension or equal to 90- A,
we have the following equation:
cos x tang -- = tang ),
and as x is a small quantity, we can develop E into a rapidly converging series, for we find according to formula (18)
in No. 11 of the introduction:
E          - D -+-. sin 2 D    tang ~  x4 sin 2 D   tang    (A)
Thus we can easily find the obliquity of the ecliptic
from an observation of the sun in the neighbourhood of the
solstitial points.  It is evident, that the aberration, as it
affects merely the apparent place in the ecliptic, has no influence whatever upon the result, nor is the value of E changed,
if A and D are reduced to another equinox by applying the
precession. But if A and D are the apparent places, affected
with notation, the value of c, which we deduce from them, will
be also the apparent obliquity of the ecliptic, affected with
nutation.
On the 19th of' June 1843 the declination of the sun was
observed at Koenigsberg and after being corrected for refraction and parallax was found equal to -+ 23" 26' 8". 57. At
the same time the right ascension of the sun was 5n 48m 508. 54.
Hence we have in this case x -- 011ml 9s.46 -- 2047'21".90
and as the latitude of the sun was equal to +t0". 70, we have:
D -- +230 26' 7". 87
I. term of the series -- - + 1 29.23
II. term of the series —    + 0.04
— 230 27'37". 14.




216
This is the apparent obliquity of the ecliptic on the 19th
of June 1843, as deduced from this one observation.  If we
compute now the nutation according to the formulae in No. 5
of the second section, taking g) =  272!' 37'.4, O =  870 0',
( -350" 17' and P =  280" 14', we find  a =e  - 0". 05, hence
the mean obliquity on that day according to that one observation is 230 27' 37".09.
We should find the same value only in a more circuitous
way by correcting A and D for nutation according to the formulae in No. 5 and 7 of the second section and computing
the formula (A) with these corrected values. As the nutation
in longitude is equal to  - +17".18, we find  a.- 1. 25,
l a = -- -O0". 39, therefore:
Corrected D — 230 26' 7". 48
I. term    -~- 1 29.57
II. term       -- 0. 04
Mean obliquity - 230 27'37". 09.
In order to free the result from accidental errors of observation, the declination of the sun is observed on as many
days as possible in the neighbourhood of the solstices and
the arithmetical mean taken of all single observations. But
any constant errors, with which x and D are affected, will not
be eliminated in this way.  If we denote the value of the
obliquity of the ecliptic which has been computed from x
and D  according to the above method by e', its true value
by a, the errors of x and D by d x and d D, each observation
gives an equation of the following form:
e  ---   1  tang x sin 2  dx - sin 2D dD,
sin 2D
which is easily deduced from the differential equation given
before and in which dx is expressed in seconds of time. We
have for instance for the above example:
ae -23 27' 37". 09 - 0.212 dx +- 1.001 dl],
from which we see, that an error in x, equal to a second of
time, produces only an error of 0". 21 in the obliquity of'
the ecliptic.  If we assume then a certain value e0, taking
-e,-+- dl   and e,, - a'- it, we find fi'om  each observation
an equation of the following form:
sin 2 e
0- n+ d - d tang x sin e dx -     (lD.::.sin 2 D




217
By applying to them the method of least squares, we
can find de as a function of dx and dD, hence if we should
afterwards be obliged to alter the right ascensions or the declinations of the sun by the constant quantities d A   - d x
and dD, we can easily compute the effect, which these alterations have upon the value of the obliquity of the ecliptic.
Hence we may assume, that the most probable value of the
obliquity of the ecliptic, deduced from observations in the
neighbourhood of a certain solstice, is of the following form:
E' +-adD -+ bdx,
where the coefficient of dD is always nearly equal to unity.
Now  if there are no constant errors in D and x, or if dD
and dx are equal to zero, we ought to find from observations
made in the neighbourhood of the next solstice nearly the
same value of a, the difference being equal to the secular
variation during the interval of time, which amounts to 0". 23.
But since accidental errors committed in taking the single
zenith distances or accidental errors of the refraction are
not entirely eliminated in the arithmetical mean of all observations made in the neighbourhood of the same solstice,
we can only expect to arrive at an accurate value of the
mean obliquity of the ecliptic by reducing the values derived
from a great many solstices to the same epoch and in this
case we may determine at the same time the secular variation. If we have found from observations the mean obliquity
of the ecliptic at the time t equal to e and if we suppose,
that the true value of the obliquity at the time to is equal
to -,,k-de and that the annual variation is A  -- x, we should
have the equation:
E =  o   e C -4  ( e + 2) (t - to)
in case that the observed value were right. Hence if we take:
so- -A (t - to) - e-n,
every determination of the mean obliquity of the ecliptic at
the time of a solstice gives an equation of the following form:'O =  + de + x (t - to)
and if there have been several such determinations made, we
can find from  all equations the most probable values of de
and x according to the method of least squares. In this way
Bessel found from his own observations and those of Brad



218
ley the mean obliquity of the ecliptic for the beginning of the
year 1800 equal to 23" 27' 54".80 and the annual variation
0".457.  Peters comparing Struve's observations with those
of Bradley found:
2930 27' 54". 22 - 0". 4645 (t - 1800)
a value which now generally is considered as more exact.
If a constant error has been committed in observing the
declinations, if for instance the altitude of the pole is only
approximately known, the values of the obliquity derived from
summer or winter solstices will show  constant differences.
Since we have D-=  + --  and if we denote by d Tf the correction which must be applied to the altitude of the pole,
by s the true value of the obliquity of the ecliptic, by a' the
value deduced from observations, we have the following equation from  a summer solstice:
E  E' + adC 9,
and fobr a winter solstice;
hence we have:
El-  + 8 - E
a -t- + 
where E_ —,  is the secular variation during the interval of
time.  This is the correction which must be applied to the
latitude, if a constant error has been committed in observing the zenith distances.  We can find in this way an approximate value of the latitude by observing the zenith distance of the sun on the days of the summer r and winter solstice.  For if a' and z" are those zenith distances corrected
for refraction, parallax and nutation, taken negative if the
sun culminates on the north side of the zenith, we have:
8.  If then the obliquity of the ecliptic be known,  the
absolute right ascension of a star and hence from  the differences of right ascensions that of' all stars may be found
with the utmost accuracy.  For this purpose a bright star
is selected, which can be observed in the daylight as well as
by night and which is in the neighbourhood of the equator,
for instance ca Canis minoris (Procyon) or ct Aquilae (Altair).




219
If then the transit of the, star is observed at the time t, that
of the sun at the time T, the interval t - T, corrected for
the rate of the clock, is equal to the difference of the right
ascensions of' the star and the sun at the time of eulmlination
of the latter.  If now  also the true declination of the sun
has been determined at the time of' culmination, we find the
right ascension of the sun from the following equation:
sin A tang E = tang D,
and we have therefore:
ct -- are sin t -angD   t -- T,
tang E
where strictly the time T must also be corrected for the latitude of the sun by adding +- cos A see' sin E 5.
If now  D and E be in error, we shall on this account
also obtain an erroneous value of t- T, independently of errors of observation in t - T. In order to estimate the effect
of any such errors, we use the differential equation found in
the preceding No.:
2 tang A    2 tangA
dA  d2 --                  D
sin 2    -  sin 2D
and. consequently we obtain from  each observation an equation of the following fornm:
tang D          2 tang A    2 tang A
a.=.areC sin.  --   - +  - T  -   - d+ --  (ID.    (A)
tang a          sin 28      sin 2D
We easily see from this equation, that it is best to make
these observations in the neighbourhood of the equinox, because then the coefficients of de and d D arrive at -their minimnum, that of d C being zero and that of d D being cotang a
or 2.3. Moreover we see that it is possible to combine several observations in such a way, that the effect of an error
in E as well as of any constant error in D is eliminated. For
if in the equation sin A    tan-g  we take the angle A always
tang e
acute, we have, when the right ascension of the sun is 180 - A'
the following equation:
180 -are si ang'         2 tang A'    2 tangA'
w e    10 - a.   asin  agi + t' T' +..... —   da -  h- Cd,
wt'ang E        sin 2ar    sin 2 D'
where t' and T' are again the times of transit of the star




220
and the sun, and if we combine this equation with the former,
we find:
a~-~[(~ —    T) (-      tang D      tangD'      7 8 
a = -[(t- T)-+-(t'- T)] --   arc sin tang    a sin   
+tang A   tang A'-   tang A - tang A'
— +  sn/     2    D I dD —  -sn2    d.  (B)
Lsin 2D  sin 2D'J       sin 2e
If now the acute angle A' —A, then we have also D'=D.
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" - A, the coefficients of d D and de in
equation (B) will be equal to zero 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 will never
be attained with the utmost rigour, as it will never exactly
happen, that, when the sun at one culmination has the right
ascension A, the right ascension 180 —A shall exactly correspond to another culmination.  But if A' be only nearly
equal to 180 ~   A, the remaining errors dependent on dD
and de will be always exceedingly small.
Therefore for the determination of the absolute right
ascension of a star, the difference of right ascensions of the
sun and the star should be observed in the neighbourhood of
the vernal and autumnal equinoxes.  But if one observation
has been made after the vernal equinox, the second must be
made as much before the autumnal equinox and vice veersd.
If we combine any two such observations, the effect of any
constant errors in D and e is eliminated and the result is
only affected with casual errors, which may have been committed in observing the times of transit or the declinations.
These can only be got rid of' in a mass of observations and
hence it is necessary to combine not only two such observations but as great a number as possible of observations
taken before and after the vernal and autumnal equinox, in
which case it is not necessary to confine the observations to
the imlnmediate neighbourhood of the equinox.  Let ac  be an
approximate value and at -e — dclrc the true value of the
right ascension and put:
a0 - i are sin   (t- 7)  n
tang E




221
Then each observation gives an equation of the following
form:
2 tang A   2 tang A
sin 2 e    sin2D
If we treat then all those equations according to the
method of least squares, we can find the most probable values of d a, de and dD or at least d a as a function of de and
d D, so that, if these should be found from other observations
and their values be substituted in the expression for dca, we
get that correction d a which in connection with these determinate values of d E and d D makes the sum of the residual
errors a mlininmm.  In case that the number of observations
is very great and the observations are well distributed about
the equinoxes, the coefficients of de  and d D in the final
equation for dac will always be very small.
If the observations extend to a great distance from the
equinoxes and the observed declinations lie between the limits -FD, it may not be accurate to take dD for the entire
range 2D as constant, for instance, in case that the circlereadings are affected with errors dependent on the zenith distance, or if the constant of refraction should need a correction.  Although even in this case these errors have no effect
upon the result, if the observations are distributed symmetrically around the equinoxes, yet the resulting value of d D
or the term dependent on dD in the final expression of d a
would have no meaning. In this case it is necessary to divide the observations according to the zenith distance into
groups,. within which it is allowable to consider the error
dD as constant and to treat those several groups according
to the method of least squares. Since we have D   9 — z- Q
if the object is south of the zenith, we may take instead of
d D  in the above equation dr  - ddk tang z -3f (z), where
dk denotes the correction of the constant of refraction and
flf(z) the correction which must be applied to the circlereadings. But for determining the values of these quantities,
there are generally other and better methods used.
Bessel observed in 1828 March 24 at Koenigsberg the
declination of the sun's centre, corrected for refraction and
parallax:            D' - +    15' 27". 24




222
and the interval between the transit of the sun and the star
ce Canis minoris, corrected for the rate of the clock:
- T — 7h 19m 29. 86.
As the latitude of the sun was -- 0".21, the correction
of the declination is -  0". 19, whilst that of the time is nothing.  Now  the values D  and T referring  to the sun, need
not be corrected for aberration, since this merely changes
the place of the sun in the ecliptic, but for the star we find
according to formula (A) in No. 16 of the third section, as
the longitude of' the sun is 3" 10' and the approximate place
of the star a -- 112" 46' and   -- --  5" 37':
a'  a -- Os. 42.
This being subtracted from  the time t, we find:
T — 711 19n' 29s.44
D-+ 1   15 27". 05,
both being referred to the apparent equinox at the time of the
observation. If we take now  for the mean obliquity on that
day 23" 27' 35".05, we must add to it the nutation in order
to find the apparent obliquity at the time of observation.
But as:
)-  277o 13'.8, O      =1~4',    283" 5G', P    280 14'
we find by the formula in No. 5 of the second section
L e --- 1". 72, hence:
e  23 o 27' 36". 77.
and with this we find:
A = are sin tag D 2  53'157".44 - 0h 11Hm35s 83.
tang 8
Hence the right ascension referred to the apparent equinox is:
ca - 71k 3 111 5s 27
and adding the mltation in right ascension -  18. 10 and subtracting  the precession and proper motion from  the beginning of the year to March 24 equal to -- 0s.71 (since the
annual variation is -+- 3s. 146) and computing the coefficients
of dc D  and dc, we find according to this observation the
mean right ascension of ce Canis minoris for 1843.0
a      - 7h 311 3s.46 -+- 0. 1539 dD - 0.0092 de,
where dD  and dc  are expressed in seconds of arc.




223
On the 20t1 of September of the same year Bessel observed:
D'=+ --   1 16 29". 22
t' -T' -- -41 171 5s. 82.
As on that clay the latitude of the sun was B  0". 56,
and Q  _ 2670 41'.9, O( —178" 39', ((  135" 41', P-  280" 14'
we find the corrections dependent on B  equal to -  0".51
and -- OOS.01; fuirthermore the aberration is           0' —. 6, the
nutation  of the obliquity is 1-0".27, hence, as the mean
obliquity was on that day 23" 27' 34".82, we find:
D -- -1 16 1t 29". 73
7'  T'- = -4  17m15s.27
-- 23 0 27' 35". 09.
From   this we  get A    2" 56' 22".36 -01" 11" 45-. 49,
hence the right ascension of the suln equal to llh 48"' 14s. 51,
therefore ca - 71h 31"' 9S. 24 and as the nutation was -i- 1. 11,
the precession and proper motion equal to  ~- 2'.27, we find
according to this observation the  mean  right ascension for
1843.0
a — 7h 311' 5s.86 - 0.1539 dD  - 0.0094 de.
Taking the arithmetical mean of both determinations we
find:
a - 711 311 4P. 66 *).
a result which is free from  the constant errors in D and s.
We might have deduced  the  mean right ascension by
subtracting  from  D, T and t the reductions to the apparent
place, neglecting  for the sun the terms dependent on aberration.  Then  using the mean  obliquity for each  day, we
would have found immediately  the right ascension referred
to the mean equinox for the beginning of the year.
9.  When the right ascension of one star has been thus
determined, the right ascensions of all stars, whose differences of right ascension have been observed, are known also
and can be collected in a catalogue together with the dclecli*) According to Bessel's Tabulae Regiomniontanae is a -— 71 31' 4s. 81.
As the arithmetical mean of both observations agrees so nearly with this,
the casual errors on both days must have been also nearly equal. If we
compare the two observed declinations with the solar tables we find the
errors of the declinations equal to +-7".67 and 8".24.




224
nations.  Thus the right ascensions given in the catalogues
of different observers can have a constant difference on account of the errors committed in the determination of the
absolute right ascension.  This can be determined by comparing a large number of stars, contained in the several catalogues, after reducing them  to the same epoch.  Similar
differences may occur in the declinations and can be determined in the same way. But since these errors may be variable, as was stated before, one must form zones of a certain number of degrees and determine the difference for these
several zones.
In order to facilitate the relative determination of the
places of stars as well as of planets and comets, the apparent places of some stars, which have been determined with
great accuracy and are therefore called standard stars, are
given in the astronomical almanacs for the time of culmination for every tenth day of the year.  Thus in order to find
the right ascension and declination of an unknown object,
one compares it with one or several of these standard stars,
determining according to the methods given before the difference of right ascension and declination. In case that the
declination of the unknown object differs little from the standard star, any errors of the instrument will have nearly the
same effect upon both observations and hence their difference
will be nearly free from those errors.
If the unknown object whose difference of right ascension and declination is to be determined, should be very near
the star, one can use for the observation instead of a meridian instrument a telescope furnished with a micrometer (which
will be described in the seventh section).  This method has
this advantage, that the observation can be repeated as often
as one pleases and that it is not necessary to wait for the
culmination of the object, which moreover might happen at
daylight and thus frustrate the observation of a faint object.
This method is therefore always used, if one wishes to observe the relative places of stars very near each other or
the places of new planets and comets.  For this purpose it
is necessary to have a large number of stars determined, so
as to be able to find under all circumstances stars, by which




225
the object can be micrornetrically determined. Therefore on
this account as well as in general for an extensive knowledge
of the fixed stars, large collections of observations of stars
down to the ninth and tenth magnitude have been made and
are still added to. In order to seize as many stars as possible and at the same time to facilitate the reduction of the
stars to their mean places, the observer takes every day only
such stars, which form a narrow zone of a few degrees in
declination and observes the clock-times of transit and the
circle-readings for every star.  Such observations are called
therefore observations of zones.  A table is then computed
for every zone, by which the mean place of every star for
a certain epoch can be easily deduced from  the observed
place and since such tables can be easily recomputed, whenever more accurate means for their computation, for instance
more accurate places of the stars, on which they are based,
are available, the arangement of these observations in zones
is of great advantage.
If now t be the observed transit of a star over the
wire of the instrument, z the circle-reading, it is necessary
to apply corrections to both in order to find the mean right
ascension and declination of the star for a certain epoch.
We must apply to t the error of the clock, the deviation of
the wire from  the meridian, the reduction to the apparent
place with opposite sign, and the precession in the interval
between the time of observation and the epoch, whilst we
must apply to z the polar point of the circle, the errors
of flexure and division, the refraction and, as before, the
reduction to the apparent place with opposite sign and the
precession.  Bessel has introduced a very convenient form
for tabulating these corrections. First a table is constructed,
which gives for every tenth minute of the clock-time t occurring in the zone the values k and d of these corrections
for the declination D corresponding to the middle of the
zone, and besides another table, which gives the variations of
these corrections for a variation of the declination equal
to 100 minutes. The mean right ascension and declination
of any star for the assumed epoch is then found by the formulae:
1,5




226
too'
Z
b'-z + d + qd-i100
where Z denotes the circle-reading corresponding to the middle
of the zone.
If we denote by u and ut' the error of the clock and its
variation in one hour, by e and e' the deviation of the wire
from  the meridian  corresponding  to the position Z and its
variation for 100 minutes, by P  the polar point, by o and
s the refraction and the errors of division and flexure, by o'
and s' their variations; for 100 minutes, at last by A a and
A' Sthe reductions to the apparent place and if we assume,
that the divisions increase in the direction of declination and
that we take as epoch the beginning of the year, we have:
a= t -+- u -+ e +- i' (t - T) - e    ( A -- 
100
Z  z-P+ 900 ~ Fe' -— 1-o  + S +4
But according to. the formulae in No. 3 we have:
f   g           ) g. 
Ac   -+ -- - — ~-'sin (G + a) tang D + -  sin (H — a)-sec D,
15   15                   15
(sinF~ + cI00' - - sin (%I ~4- ct) tQl     - -Z
15  cos2           150sin        cos D      t100o
A 3 —   cos (G - a) -+-  cos (H —- a) sin D -+ i cos D
th cos (Ht+.a) cos D 10'-i sin D100'l 100
hence we find-:
u+ e + t (t - T)-   -     sin ( + c tg D -    sin (H — a) sec D,
1515                    15
V,    g sin (        1- o D2 1 00' - -  sin (1i-t- a)    os 100,
15   os D2         15           cos D
d- - P- 900 =F,o - s -- g cos (G +- +) - h cos (HI+ - a) sin D - i cos D,
d'-=F' - s' - [ cos (H — a) cos D 100' - i sin D 100'].
The error of the clock and the polar point of the
circle are determined  by any known  stars, which occur in
the zone, or by the standard stars, if any of them have been
observed before and after observing the zone-stars and if the
errors of the instrument,  as well as the polar point and
the rate of the clock can either be considered as constant or
be interpolated  from  those  observations.  The values of k,




227
k', d and d' are then tabulated for every tenth minute of
the clock time t and may thus be easily interpolated for any
other value of t.
III. ON THE METHODS OF DETERMINING THE MOST PROBABLE
VALUES OF THE CONSTANTS USED FOR THE REDUCTION OF
THE PLACES OF THE STARS.
A. Determination of the constant of rft'actions.
10. It was shown in No. 6, how the apparent zenith
distances of stars are determined by observations which first
must be cleared from refraction, in order to obtain the true
zenith distances. If the zenith distance of a circulmpolar star
be observed at its upper and lower culmination and corrected
for refraction as well as for the small variations of the aberration, nutation and precession in the interval between the
two observations, the arithmetical mean of the two corrected
zenith distances is equal to the complement of the latitude.
Now if a set of such observations of different stars is made,
all should give the same value for the latitude or at least only
such differences as may be attributed to errors of observation
and casual errors of the refraction as mentioned in No. 13 of
the third section, provided that the adopted formula for the
refraction and especially the adopted value of the constant
of refraction is true.  Hence if there are any differences,
they must enable us to correct the constants on which the
tables of refraction, which are used for the reduction, are
based.
Denoting by z and C the observed zenith distances at
the upper and lower culmination, by r and e the refraction,
we have for any north latitude the equations:
8- 99= Z-t
1800 - 8 — --- =  + e,
where south zenith distances must be taken negative and where
the upper or lower sign must be used, if the star at its upper
culmination be north or south of the zenith.  From  these
equations we find:
900 -= 2   +      2      (a)
-2 —  2




228
If another star be observed at b)oth culminations and the
zenith distances L' and z' be found, we should be able, to
find from the following two equations:
~O+z o- lr4900       -' 
and 900 -   -b 2  - o2
the values of'    o and of that constant which in (, a', r and
r' occurs as factor.  But the values thus found would be
only approximate on account of' the errors of observation;
besides equation (1) in No. 9 of the third section shows, that
the refraction is not strictly proportional to the constant a
but that it contains some other constants, the correct values
of which it is desirable to  determine from  observations.
Ivory's formula contains besides a the constant f, which depends on the decrease of temperature with the elevation above
the surface of the earth, which however shall here be neglected, since its influence, which is always sm11all, is felt only
in the immediate neighbourhood of the horizon; but besides
this, like all other formulae for the refraction, it contains the
coefficient E for the expansion of air by heat, which it is
also best to determine'in this case by a stronomical observations. For since the atmosphere has always a certain degree
of moisture and the expansion of the air depends on its state
of moisture, therefore if we determine this coefficient from
a large number of observed refractions, we shall obtain a
value, which corresponds to a mean state of the atmosphere,
and the refractions computed with this value will give in
the mean of a great many observations as near as possible'
that value which would have been obtained, if the actual
moisture of the atmosphere at the time of each observation
had been taken into account.  Now  denoting the mean and
the true refraction by R and R', we have according to the
formula (12) of the third section:' =1 R [B. T]A [1 + E (r - 50)] i,
where A =- 1 — q and A =  1 -   p.  From this we get:
dR_'= dR'        ( - 50) R'l d
cda      1 +   (, - 50)
or taking:




229
a ~+ cda= a (1 -+ i), E + d =  E(1 + i)
dc'     2e(7:- -0)
d        _R -  a  d   k     ( -- 50) R' i.
d a    I +- (r - 50)
But according to the formula (1) in No. 9 of the third
section we have:
a t-dn -   +2 _-        [2'v (2)T
da~    -I - (l-a) sinz2
The second term  of the second member of this equation
becomes significant only for zenith distances greater than 80'
and if we put:
a dR   (1+ —dca
we can take the values of y from  the following table:
z        Y       I      z         y
800      246            86~       60.5
810      205            870       43.2
820      168            880       29.5
830      135            890       19.0
840      106            890 30'  14.8
850       82
We have therefore:
d R'- R' 1    1 k- -l1   (,  50) It
I 4 - 50)
If we assume therefore, that the values of the refraction,
which have been used for computing formula (a), are erroneous and that the corrections are do and dcr, we get:
90~ - 93       t       t                       -- --
902     2           2              2
if' we denote by m and ~ the values of 1!  (r- 50)  for the
upper and lower culmination. If we also assume an approximate value rf0) for T, the true value being   -  c,, + -dr  p
and take:
~q+z    e4-r
- + —-+ -- + ~o - 90~ -—..
we obtain, combining the result of the upper and lower culmination of each star, an equation of the following form:
O = n  +- dT. 9       k —    (b).




230
Now the observations of the several stars will not have
the same weight, since the accidental errors of observation
are the greater the nearer the star is to the horizon. Hence
the probable error of an observation will generally increase
with the zenith distance of the star. In case that the values
of d c, k and i were already known and were substituted in
the equations, the quantities n would be the real errors of
observation and hence the probable error of one observation
might be determined.  But since these values are unknown,
this can only approximately be found from the deviations of
the single observations from their arithmetical mean. If then
tv and tv' are the probable errors of an observation at the
upper and lower culmination, all equations of the same star
must be divided by 1/t -- w'" in order to give to the equations
of the several stars their true weight. In case that the probable errors should be found very different when the equations have been solved, the whole calculation may be repeated.
Also stars culminating south of the zenith can be used
for determining the correction i of the coefficient e for the
expansion of air. For such stars we have according to the
notation which we used before, taking the zenith distances
positive:
99o   S + d (      z- )- 2' )r  2z +;C —)r  - )7.'1 r,
or taking:
n-   + -- 9- So - Po,
O  n + cl (s-  ) +  (l + -) k-     (c)
If also in this case we multiply the equations of the
several stars by their corresponding weights and deduce the
equations for the minimum  from  all equations of the same
star, we can eliminate the unknown quantities d (-rf,) and
k, so that each star gives finally an equation of the form:
O  N-l- i.        (d)
But a similar equation can be deduced from  every circumpolar star observed at the times of both culminations, if
the equations (b) are treated in a similar way. Hence we
find a number of equations of the form  (d) equal to the
number of observed stars, from which the most probable value




231
of i can  be deduced %).  By this method Bessel determined
the  quantity  i and  thus the  coefficient of the expansion of
air for a mean state of the moisture of the atmosphere from
observations made at Koenigsberg.  (Consult Bessel, Astronomische Beobachtungen, Siebente AbtheiluLng, pag. X) and the
value  found  by him  is the one which was given before namely 0.0020243  for one degree Fahrenheit.
If we  substitute  the  most probable  value  of i in  the
equations (b) or rather in the equations of the minimum, deduced for each  star, we find from  the combination of these
equations corresponding to the several stars, the most probable values of dq, and k "*).
If it should be desirable, to take the correction of' the
quantity f into  account, it would be necessary to add to dR'
dR'
the  term  -.   df or, taking f-t- d f=  f ( 1 -- h), the term
dR',   R
f -_ h- - h, where the values of x  can be taken  from  the
following table:
-        x      I.         x
So       338     |    88      59.3
860      196          890      29.8
870      111          890 30'  20.6.
B.  Determnination of the constants of aberration and nutzation and oJ-' the
annual parallaxes of stars.
11.  The  aberration, nutation  and  annual parallax  are
the periodical terms  contained in the expression for the apparent places of the stars, hence their constants  n-must be deternmined by observing the apparent places of the stars at
different times.  Aberration  and parallax have the period of
*) As a change of temperature has the greatest effect upon low stars, it is
not necessary to take for this purpose stars whose meridian altitude is greater
than 60.'*) The equations given in the example in No. 25 of the introduction are
those, which would have been obtained by giving all observations the same
weight and taking the arithmetical mean of all equations of the same star.
For the form of the equations after the correction of i has been applied, is
0 = n +- d99 -- a k. But Bessel has referred all observations to the polar point
not, as has been assumed here, to the zenith point of the circle, hence the
coefficient a differs from the coefficient of ic in the above equations.




232
a year and therefore may be determined from observations
made during one year.  But the principal term  of nutation
has a period of 18 years and 219 days, the time in which
the moon's nodes perform an entire revolution. Hence the
constant of nutation can be determined only by observations
distribued over a long series of years.
Since the apparent right ascensions of the pole-star are
very much changed by aberration and nutation on account
of the large factors sec o and tang S, their observations afford
the best means for determining these constants; for the same
reason the parallax of the pole-star can be determined in this
way with great advantage. Putting:
-cos ~  cos a     sin A
- sin a = a cos A,
the formulae fbr aberration and parallax in right ascension
in No. 16 and 18 of the third section, can be thus written:
a'- a  -+ kcc sin (- + A) sec  -+ gr a cos (Q + A) sec a + -  (1c2),
where k and n are the constant of aberration and the parallax
and ( (k2) denotes the terms of the second order. If sever:l
observations are taken at the times when sin (( 3- A)  --  1
and hence the maximum  of aberration occurs, an approximate value of k can be found by comparing the right ascensions observed at both times after reducing them to the same
mean equinox. But in order to obtain a more accurate value,
the most probable value must be determined from a great
many observations.  Now  the mean right ascension ce and
the assumed value of the constant k be erroneous by A a and
Ak, the true values being a- +-A   and k +-Ak.  If then a,
denotes that value of the apparent right ascension, which
has been computed from ce with the value k of the constant
of aberration (the computed precession and nutation being
supposed to be the true values) and to which the small terms
dependent on the square of k and on the product of aberration and nutation have also been added, since the effect
of a change of k upon them is very small, and if further cc'
denotes the observed apparent right ascension, we have:
a   a0 -- a +-f Ak a sin (Q  - A) sec -- + 7a cos ( -I- A) sec,
hence, taking:
ao0 -- a,




233
every observation of the right ascension of' Polaris leads to
an equation of the following form:
0 =  + ~ ac +, Ak. a sin (0 + A) see S a- g a cos ((0) + A) see a,
and fromn  all these equations the most probable values of Ace,
A k and jr can be determined according to the methlod of
least squares.
Should these observations embrace a long period of' yars,
the constant of nutation, that is, the coefficient of cos &S in
the expression for the nutation of the obliquity can be determined at the same time. If we denote by Av the correction
of this coefficient, we must add to the above equation the
term d-~ A, where the expression for -   has been given in
No. 6 of the second section. The complete equation for determining the aberration, parallax and nutation from the observation of an apparent right ascension is therefore:
cix,,
0 = n + Ac+ Ak.a sin (0- +A) see A+-sa cos (0 -+A) see K+- -A.
If for this purpose the observations made at different
observatories are used, the probable errors of the observations
of the several observers must be determined and the corresponding weight be given to the different equations.  In
this case also the correction A r may not be the same for
the observations of the several observatories, as the observed
right ascensions may have a constant difference. I-Ience this
difference must be determined and be applied to the observations or the unknown quantities A t A ct' etc. must be eliminated separately by the observations of each observatory.
In this way von Lindenau determined the following values of the constants from  right ascensions of Polaris observed by Bradley, Maskelyne, Pond, Bessel and himself in
the course of 60 years:
k = 20".4486   v = 8". 97707   Xc = 0". 1444,
Peters found later from  observations made by Struve
and Preuss at Dorpat during the years 1822 to 1838 the following values:
k -- 20". 425.5    v - 9". 2361    Z=- 0". 1724.
For the  determination of' these constants by declinations those of Polaris are also very suitable, as their accuracy




234
can be greatly increased by taking several zenith distances
at every culmination of the star.  If we introduce in this
case the following auxiliary quantities:
sin x sin S cos a - cosi    si n   Bsin 
- cos a sin S- b cos B,
the aberration in declination is equal to k b sin (0 -- B), the
parallax equal to r b cos (G-+-B).  Then denoting by SO that
value of the apparent. declinattion which has been computed
from  the mean declination with the constants of aberration
and nutation k and Iv (the computed precession being taken
as accurate) and to which the small terms dependent on the
square of k and on the product of aberration and nutation
have also been added; further denoting the observed apparent
declination by )' and taking c        — )"- n, every observation of
a declination leads to an equation of the following form:
0O =- n q- A a -+ Akb sin (0) +- B) -n-  I) cos (0) -- B) + -— I -A v,
and in case that the observations embrace a sufficiently long
period, the most probable values of AO",  Ak, jT and Av  can
be determined according to the method of least squares ).
It was by such observations that Bradley discovered the aberration. He observed at Kew since the year 1725 principally
the star y Draconis besides 22 other stars, passing nearly
through the zenith of the place, and discovered a periodical
change of the zenith distance, which could not be explained
as being the effect of parallax, for the determination of which
these observations were really intended. The true explanation
of this change as the effect of the motion of the earth combined with *that of light was not given by him  until later.
The instrument, which he used for these observations, was
a zenith sector, that is, a sector of very large radius, with
which he could observe the zenith distances of stars a little
over 12 degrees on each side of the zenith. The star y Draconis, being near the north pole of the ecliptic, was especially suitable for determining the parallax and thus also the
*) If the stars have also proper motions, the terms p(t —to) and q(t —to)
must be added to the equations for right ascensions and declinations, where
p and q are the proper motions in right ascension and declination.




235
aberration, as for this pole we have a = 270~, - = 900-,
hence b = — 1 and B    90" and the maximum  and minimurm
of the aberration and parallax in declination are equal to -4- k
and  -- 7.
By similar observations he discovered also the nutation.
The observations embrace the time from the 19th of August
1727 to the 3d of September 1747, hence all entire period of
the nutation. Busch found frolm their discussion the constant
of aberration equal to 20". 23.  Lundahl found the following
values from the declinations of Polaris observed at Dorpat by
Struve and Preuss:
-- 20". 5508   v -- 9. 2164        — = 0". 1473.
The value of the constant of nutation given in No. 5 of
the second section is taken from Peters's pamphlet,,Numerus
Constans Nutationis".  It was derived from the three determinations made by Peters, Busch and Lundahl, the probable
errors of the single results being taken into account.
But the value of the constant of aberration given in No. 16
of the third section has not been deduced from the values
given above, but has been determined by Struve from the
transits of stars across the prime vertical. For if an instrument is placed exactly in the plane of the prime vertical and
a star is observed on the wire on the east and west side*),
the interval of time divided by 2 is equal to the hour angle
of the star at the transit across the prime vertical. If we denote this by t, we get from the right angled triangle between
the zenith, the pole and the star:
tang S = tang T cos t,
hence we see that the declinations of the stars can be determined by such observations.  Differentiating the formula
in a logarithmic form, we find:
sin 2
sl   i 2 2 8w   -E -2 sin 2 3 tang tdit,
and thus we see that an error in t has the less influence the
smaller t is or the nearer to the zenith the star passes a6ross
the prime vertical. Hence if the zenith distance is very small,
the declination of such a star can be determined by this
*) See No. 26 of the seventh section.




236
method very accurately.  The equations for each star are
in  this case quite similar to those given before and it is
-again preferable to select for these observations stars near
the pole of the ecliptic.  By this method Struve found the
constant of aberration equal to 20".4451, a value which undoubtedly is very. exact.  But his observations embrace too
short a period for determining the constant of nutation, which
however as well as the parallax might also be found by this
method with a great degree of accuracy.
The constant of aberration may also be computed from
the velocity of light and that of the earth according to No. 16
of the third section.  The mean daily motion of the earth
has been determined with great accuracy and is equal to
59' 8".193.  The time in which the light moves through a
distance equal to the semni-diameter of the earth's orbit, was
first determined by Olav Roemer fronm the eclipses of the
satellites of Jupiter.  For he found in the year 1675, that
those eclipses which took place about opposition were observed 8'1 138 earlier and those about conjunction as much
later than an average occurrence *).  Now as the difference
of the distances of Jupiter from  the earth at both times is
equal to the diameter of the earth's orbit, Ri5mer soon found
the true explanation, that the light does not move with an
infinite velocity  and traverses the diameter of the earth's
orbit in 161'" 26'S.  If therefore T' be the time of the beginning or the end of an eclipse computed from the tables, then
must be added to it in order to render it conformable to
the observations, the term
where K is the number of seconds, in which the light traverses the semi-diameter of the earth's orbit and A is the
distance of the satellite from the earth, the semi-major axis
of the earth's orbit being taken as the unit. If then To. is
the time of the eclipse thus corrected, T' the observed time,
every eclipse gives an equation of the form:
0 —- To - T'+- AdK
*) At the opposition the earth stands between Jupiter and the sun, whilst
at conjunction the sun it between Jupiter and the earth.




237
and fromn a large number of such equations the most probable value of d K can. be determined. However the observations of the beginning and the end of an eclipse are always
a little uncertain, since the satellites lose their light only
gradually and  as thus the errors of observation greatly depend upon the quality of the telescope, it is best, to combine only such observations which have been made with
the salte instrument and also to treat the observations of
the beginning  and of the end separately.  Delambre found
by a careful discussion of a large number of observed eclipses
the constant of aberration equal to 20".255, a value which
according to Struve's determination is too small.
12.  The annual parallax  of a star can be determined
still by another method, if the change of the place of the
star relatively to that of another star, which has no parallax,
be observed. This method is even preferable to the former,
because the relative places of two stars near each other can
be measured with great accuracy by means of a micrometer
(as will be shown in the seventh section) and because the
effect of the small corrections upon the places of both stars
is so nearly equal, that any errors in the adopted values of
the constants can have no influence on the difference of the
mean places *).  It is true, this method  gives strictly only,
the difference of the parallaxes of both stars.  But since is
may be taken for granted, that very faint stars are at a great
distance, the parallaxes thus found, when one or several such
faint stars have been chosen as comparison stars, can be
considered as nearly correct.
If the difference of right ascension and declination of
both  stars has been observed, each  observation freed from
the small corrections gives two  equations of the following
form, taking the differences at the time t,, equal to ac'-  a,,C
and 8'0 -    and denoting ce',, a-  -  (a'   cc) and 3o -  3),,
*) In this case, when the stars are near each other, itis preferable, not
to compute the mean place of each star, but to free only the difference of
the apparent places fiom refraction, aberration, precession and notation. The
formulae necessary for this purpose will be given in VIII and IX of the
seventh section.




238
(8'-  A) by n and n' and the errors of the adopted place by
Aa and AS:
0 = n -- A a --- na cos (O- A) see S+p (t - to)
0 — n'+ A 8 q+-   b cos ((+ - B)    + q (t - to).
Usually however instead of the difference of the right
ascensions and declinations of both stars their distance is
observed and besides the angle of position, that is, the angle
which the declination circle of one star makes with the great
circle passing through both  stars.  If' then a and 5 be the
true right ascension and declination  of one star, a' and 5'
their values not freed from  parallax, a" and 5" the right ascension  and  declination of the comparison star, we find the
changes of the differences of the right ascensions and declinations produced by parallax as follows:
d (a" - a) - a   a' - X R [cos () sin a - sin Q( cos s cos a] see A
dl (l8(" —  )' — S'=z- 7t R [Eos e sin a sin  -sin 8 Cos S] sin )
-+ v R sin A cos a cos 0.
If then the true distance and the true angle of position
be denoted by A and P, we have:
A sin P- cos S (a" - a)
A cos P- "- 
hence:
dA = sin P cos 8 d (a"- a) +  cos P d ("- -)
A dP  cos P cos a d (a" - a) - sin P d (8" - 8).
If we substitute here the expressions given before and
take:
m cos M-= sin a sin P + sin S cos a cos P,
m sin Mi  [-cos asinP   sin S sin a cos P] cos s - COS S COS P sin e,
m eos M'  — 1 [sin a cos P - sin S cos a sin P].,
m'1 sin M'-  ~ [- (cos a cos P - sin A sin a sin P) cos E +- cos S sin P sin E],
we easily find:
dcl = z R  en cos (0 - iM)
dP =-  R? m'cos (0 - il').
Therefore if dA0 denotes the correction  of the adopted
distance at the time to,  d q the correction  of the adopted
value of the proper motion in the direction towards the other
star, we find from  the observed  distances equations of the
form:
O —'- do +- (t -to)dq+ R-  Rm cos (  -M)




239
and from the angles of position equations of the form:
0 = v' + dPo +- (t - to) d q' +  rnRm' cos () - M'),
which must be solved according to the method of least squares.
By this method Bessel first determined the parallax  of 61
Cygni.
C. Determination of' the constant of' pr'ecession acnd of' the proper mnotions
of the stars.
13.  We find the change of the right ascension and declination of a star by the precession during the interval t'- t,
if we compute the annual variations:
da                        di   da        dl 
t- =-  m-4- n tg S sin a  cos so- I   -t- sin eo  t tgS sina
d t                       cit  cit        cit
c 2 COS a    sill 8o -- Cos a
(It                dt
c tt
for the time! —-T and then  multiply them  by t'-t.  Now
dac
since the numerical value of-stV is known from  the theory of
the secular perturbations of the planets, we may determine
the lunisolar precession dj either from  the right ascensions
or from the declinations, comparing the difference of the values
found by observations at the time t' and t with the above
formula. Then if the places of the stars were fixed we should
find nearly the same value of the precession front different
stars and the more exactly, the greater the interval is between
the observations, as any errors of observation would have
the less influence. But since not only different stars but also
the right ascensions and declinations of the same star give
different values for the constant of precession, we must attribute these differences to proper motions of the stars. As
they are like the precession proportional to the time, they
cannot be separated from  it and the difficulty is still increased
by the fact, that the proper motions, partly at least, follow
a certain  law  depending on the places of the stars.  Hence
we can eliminate the proper motions only by comparing a
large number of stars distributed over all parts of the heavens
and excluding all those, which on account of their large
proper motion give a very different value for the precession.
The large number will compensate any errors of observation




240
entirely and the effect of the proper motions as much as
possible. As the proper motions are proportional to the time,
the uncertainty of the value of the precession arising from
them  remains the same, however great the interval between
the two comlpa.red catalogues of stars may be, but it will be
most important, that the catalogues are very correct and contain a large number of stars in common and that the interval is long enough so as to make any uncertainty arising
from  errors of observation sufficiently small. If then n,, and
n,0 are the two values of rn and n employed in comparing
the two catalogues, if further cc, 3 and a' and b' are the mean
places of a star for the times t and t', given in the two cat-:l ogues, and A/ce and AA the constant differences of the catalogues for c: and 3 and if we take:
a +- (ino +-  o tg 80 sin a ) (t' - t) -'     (t - t)
and
8 -i- lo cos a o (t' - t)  -'   =' (t' - t)
every star gives two equations of the form:
0 =   t — -   -I- d'mo -+ - dno tg 30 sin ax,,
and
0 - v' —i — t +- dno0 cos a0.
Therefore if we consider the proper motions embraced
in vJ and r' like casual errors of observation, we may find
the most probable values of the unknown quantities from  a
large number of equations by the method of least squares.
This supposition would be justified, if the proper motions
were not following a law depending on the places of the
stars. But as it is very difficult, if not impossible, to introduce
in the above equations a term expressing this law, a matter
which shall be more fully considered afterwards, hardly anything better can be substituted in place of that supposition,
provided that a large number of stars distributed over all
parts of the heavens be used.  We then get from  the right
ascensions a determination of mn and n, from  the declinations a determination of n; but it is evident, that an error of
the absolute right ascensions, which is constant for every
d n rmi   untd wli  da
catalogue, remains united with din and as d-     cos e d
dt         dt  dt




241
there remains also in it any error of the value of dt arising
from incorrect values of the masses of the planets. But the
determination of dn = d, sin %,, from the right ascensions is
independent of any such constant error, and besides the constant difference of the declination may be determined. But
since the supposition, that the latter is constant for all declinations, is not allowable, it is better to divide the stars in
zones of several degrees for instance of 100 of declination
and to solve the equations for the stars of each zone separately, and hence to determine the mean difference Ajb for
each zone. In this way Bessel in his work Fundamenta Astronomiae determined the value of this constant from more than
2000 stars, whose places had been deduced for 1755 and
1800 from Bradley's and Piazzi's observations. He found for
1750 the value 50".340499, which he afterwards changed
according to the observations made at Koenigsberg into
50".37572. (Compare Astron. Nachr. No. 92.)
14. The differences of the places of the stars observed
at two different epochs and the precession in the same interval of time, which has been computed with the value of
the constant determined as before, are then taken as the proper
motions of the stars. In general they may be accounted for
within the limits of possible errors of observation by the supposition, that the single stars are moving on a great circle
with uniform velocity. Halley first discovered in the year
1713 the proper motion of the stars Sirius, Aldebaran and
Arcturus'). Since then the proper motions of a great many
stars have been recognized with certainty and it is inferred,
that all stars are subject to such, although for most stars
these motions have not yet been determined, since they are
small and are still confounded with errors of observation. The
greatest proper motions have 61 Cygni (whose annual change
in right ascension and declination amounts to 5". 1 and 3". 2),
a Centauri (whose annual motion in the direction of the two
*) The last mentioned star has a proper motion of 2" in declination
and has therefore changed its place since the time of Hipparchus more than
one degree.
16




242
co-ordinates is 7".0 and 0".8) and 1830 Groombridge (which
moves 5".2 in right ascension and 5". 7 in declination).
The elder Herschel first discovered a law in the direction
of the proper motions of the stars, when comparing a great
many of them he observed, that in general the stars move
from a point in the neighbourhood of the star 2, Herculis.
Hence he suggested the hypothesis that the proper motions
of the stars are partly at least only apparent and caused by
a motion of the entire solar system towards that point of the
heavens, a hypothesis, which is well confirmed by later investigations on this subject. The proper motions of the fixed
stars are therefore the result of two motions, first of the motion peculiar to each star, by which they really change their
place according to a law hitherto unknown, and secondly of
the apparent or parallactic motion which is the effect of the
motion of the solar system. Now on account of the motion
peculiar to each star, stars in the same region of the celestial
sphere may change their places in any direction whatever,
but the direction of the parallactic motion is at once determined by the place of the star relatively to that towards
which the solar system is moving, and can be easily calculated, if the right ascension and declination A and D of that
point are known. If we compare the direction, computed
for any star, with the direction, which is really observed, we
can etablish for each star the equation between the difference
of the computed and the observed direction and changes of the
right ascension and declination A and D; and since those
portions of these differences, which are caused by the peculiar motions of the stars, follow  no law and can therefore
be treated like casual errors of observation, we can find from
a large number of such equations the most probable values
of dA and dD by the method of least squares.
It is evident that the direction of the parallactic portion
of the proper motion of a star coincides with the great circle,
drawn through the star and the point towards which the
solar system is moving, because the star, supposing of course
that the sun is moving in a straight line, is always seen in
the plane passing through it and the straight line described
by the sun. Now if we denote the motion of the sun during




243
the time t'- t divided by the distance of the star by a, and
then denote the right ascension  and declination  of the star
at the two  epochs t and t' by a, 4 and a', 4', and finally
the ratio of the distances of the star from  the sun at the
same epochs by e, we have the following equations:
e cos' cos a' — = cos  cos a - a cos A cos D
cos A' sin a' n cos 8 sin a - a sin A cos D
e sin' -sinl S - a sin D,
from which we easily deduce:
COs' - CO S - a COS D cos (a - A),
therefore:
cos' (a'- a) = a cos D sin (a - A)
X"    = - ca [cos a sin D - sin 3 cos D cos (a -A)].
But we have also in the spherical triangle between the
pole of the equator, the star and the point, whose right ascension and declination are A and D, denoting the distance of
the star from  that point by A and the angle at the star by P:
sin A, sin P = cos D sin (a - A)
sin A cos P = sin D cos S- cos D sin S cos (a - A).
Now  if we denote the angle, which the direction of the
proper motion of the star makes with the declination circle,
by p, we have:
-  cos  (a' - a)
tang P -  
hence we see, that p     1800" - P or that the star is moving
on a great circle passing through it and the point whose
right ascension and declination is A and D, so that it is mnoving from  the latter point.
From the third of the differential formulae (11) in No. 9
of the introduction, we have:
cos a sin (a- A)    "
clP-,                dD
sin A'
cos D
-+       [sin   cos D- cos  sin D cos (a - A)] dA,
sin l\
hence:~
ldp    en-cos 8 sin (a -A)dD
sin A2
eos D
sin A2 [sin S cos D - cos S sin D cos (a -A)] dA.
Therefore if p' be the observed angle, which the direction
of the proper motion makes with the declination circle, reck16 




244
oned from the north part of it through east from 00 to 3600
so that:
tang p' - cos  (a'-a)
and if further p be the value of 180-P computed according to the formulae (B) with the approximate values A and
D, we have for each star an equation of the form:
0 ~ p - p   cos S sin (a -A)
cos D
s A2 [sin S cosD - cos S sin D cos (a - A)] dA,
or:
cos 8 sin (a - A)
O-(p - p') sin A+   sin  A   dD
sin A
cos D
-    [sin a cos D - cos a sin D cos (a - A)] dA,
sin A
and from a large number of such equations the most probable values of dA and dD can be deduced.
In this way Argelander determined the direction of the
motion of the solar system  ).  Bessel in his work,,Fundamenta Astronomiae" had already derived the proper motions
of a large number of stars by comparing Bradley's observations with those of Piazzi.  Argelander selected from those
all stars, which in the interval of 45 years from 1755 and
1800 exhibited a proper motion greater than 5" and determined their proper motions more accurately by comparing
Bradley's observations with his own made at the observatory
at Abo ~).  For determining the direction of the motion of
the solar system he used then 390 stars, whose annual proper motion amounted to more than 0".1. These were divided into three classes according to the magnitude of the proper motions and the corrections dA and dD determined separately from each class. From those three results, which
well agreed with each other, he finally deduced the following values of A and D, referred to the equator and the equinox of 1800:
A-=-2590~ 51'.8 and DL=+320 29'.1,
) Compare Astronom. Nachrichten No. 363.
**) Argelander, DLX stellarum fixarum positiones mediae ineunte anno
1830. Helsingforsiae 1835.




245
and these agree well with the values adopted by Herschel.
Lundahl determined the position of this point from 147 other
stars, by comparing Bradley's places with Pond's Catalogue
of 1112 stars and found:
A - 2520 24'.4 and D =  - 140 26'.1.
From the mean of both determinations, taking into account their probable errors, Argelander found:
A - 257~ 5b'.7 and D =  - 280 49'.7.
Similar investigations were made by 0. v. Struve and
more recently by Galloway.  Struve comparing 400 stars
which had been observed at Dorpat with Bradley's catalogue,
found:
A -- 2610 23' and D-  370 36'.
Galloway used for his investigations the southern stars,
and comparing the observations made by Johnson on St.
Helena and by Henderson at the Cape of Good Hope with
those of Lacaille, found:
A - 260 1' and DZ  4-+ 340 23'.
Another extensive investigation was made by Maidler,
who found from a very large number of stars:
A - 2610 38'.8 and D -+- 39 53'.9
Since all these values agree well with each other, it seems
that the point towards which the solar system is moving, is
now known with great accuracy, at least as far as it is attainable considering the difficulties of the problem.
15. We may therefore assume, that the direction of the
parallactic proper motion of a star, computed by means of
the formula:
cos D sin (a - A)
tang P -
sin D cos 8 - cosD sin 8 cos (a - A)
with a mean value of A and D, is nearly correct.  If now,
besides, the amount of this portion of the proper motion were
known for every star, we should be able to compute for
every star the annual change of the right ascension and declination, caused by this parallactic motion, and could add
this to the equations given in No. 13 for determining the
constant of precession.  The amount of this parallactic motion must necessarily depend on the distance of the star,
hence if the latter were known, we could determine the par



246
allactic motion  corresponding  to  a  certain distance.  For
since those equations are transformed into the following:
ic cosDo
0 - v - dmo -- dno tg S0 sin (ao -    -os o- sin (ao -A)'A cosY0
and  O  v'+- dno cos ao  -- gsin (G - Do)
where S0 g cos G,
sin o, cos (ao - A)  g sin G,
we could find, if A  were known, from  these equations k,
that is, the motion of the sun as seen from a distance equal
to the adopted unit and expressed in seconds, and besides
we should find the values of dmin and d n, free from this
parallactic proper motion of the stars. Now since the distances of the stars are unknown, 0. v. Struve substituted
for A hypothetical values of the mean distances of the different classes of stars,, which had been deduced by W. v.
Struve in his work, Etudes de l'Astronomie stellaire from the
number of stars in the several classes').  Struve then compared 400 stars which had been observed by W. v. Struve
and Preuss at Dorpat with Bradley's observations and, at first
neglecting the motion of the solar system, he found for the
corrections of the constant of precession from  the right ascensions and declinations two contradicting results, one being
positive, the other negative. But taking the proper motion
of the sun into account he found the corrections -i- 1".16
from  the right ascensions and -+- 0". 66 from the declinations
and hence, taking into account their probable errors, he found
the value of the constant of precession for 1790 equal to
50".23449 or greater than Bessel had found it by 0.01343.
Further he found for the motion of the sun, as seen from a
point at the distance of the stars of the first magnitude,
0".321 from the right ascensions and 0".357 from the declinations.  But although these values of the constant of precession and of the motion of the solar system are apparently
of great weight, it must not be overlooked, that they are
based on the hypothetical ratio of the distances of stars of
*) According to this, the distance of a star of the first magnitude being
1, that of the stars of the second magnitude is 1.71, that of the third 2.57,
the fourth 3.76, the fifth 5.44, the sixth 7.86 and the seventh 11.34.




247
different magnitudes. Besides it cannot be entirely approved
of, that the  number of stars used for this determination,
which are nearly all double stars, is so very small.
If it should be desirable for a more correct determination of the constant of precession, to take the motion of the
solar system  into account, it may be better, not to introduce
the ratios of the distances of stars of different magnitude
according to any adopted hypothesis, but rather to divide
the  stars into  classes according to their magnitude or their
proper motions, and to determine for each class a value of
k and  the  correction  of the constant of precession.   The
values of -  thus found can be considered  as mean values
for these different classes and the values of m and n will
then be independent at least of a portion of the parallactic
motion, which will be the greater, the more nearly equal the
distances of the stars of the  same class are ).  EvenD the
corrections of A and D might be found in this way, since the
equations in this case would be, taking - = a:
cos D
0 -=v'+ dmz, - dno tang 0o sin ao-  os- cos (  -A A) acdA
+ [cos D- sin D dD] sin (   A)
O = i' — dno cos a, - g cos (G —D) adD -- cos D sinYo sin (ao -A) adA
+ aeg sin (G - D)
from   which  the  most probable  values of a, ad A, adD,
d mn  and  dn, can be determined for each  class.  In  case,
that Struve's ratio of the distances be adopted, the unknown  quantity  a after multiplying the factor by  X would
0) The author has undertaken this investigation already many years ago
without being able to finish it.  The proper motions were deduced from a
comparison of Henderson's observations made at Edinborough with those of
Bradley.  The following mean values were found for the annual parallactic
motions of stars of several classes:
for 32 stars of magnitude 4.3. 0".068985 -4- 0.010964,  75,,,,,,       4. 0".069715 =::0.006584
71,,,,,,   4.5. 0".046811 4 0.006925,, 284,,,,,,      5. 0".029043 =4 0.002446.
Stars, whose annual proper motion exceeds 0".3 of arce, were excluded in
making this investigation.




248
be the same for all classes.  (Compare on this subject also
Airy's pamphlet in the Memoirs of the Royal Astronomical
Society Vol. XXVIII.)
16. At present we always assume that the proper motions of the stars are proportional to the time and take place
on a fixed great circle. But the proper motions in right ascension and declination are variable on account of the change
of the fundamental plane to which they are referred, and it
is necessary to take this into account, at least for stars very
near the pole.
The formulae, which express the polar co-ordinates referred to the equinox at the time t' by means of the coordinates referred to another equinox at the time t, are according to No. 3 of the second section:
cos 8' sin (a' - a' - z')   cos a sin (a + a -+ z)
cos 8' cos (a' - a' - z') = cos a cos (a   a +- z) cos O - sin S sin O
sin S' = cos 8 cos (a + a -- +z) sin O +- sin S cos 0,
where a denotes the precession produced by the planets during the time t' - t, and z, z' and O 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 (11) in No. 9 of the introduction, remembering that
the formulae above are derived from a triangle the sides of
which are 900 - A', 900 - a and 0 and the angles of which
are a -4- a 4- z, 1800 -- c'-a'- z' and c:
A a' = cos c A S - sin O sin (a' +- a' - z) A a
cos' a' = sin c As 3- cos 8 cos c  a
or if sin c and cos c be expressed in terms of the other parts
of the triangle:
Aa' — Aa [cos 0 + sin 0 tang S' cos (a' -+ a'- z')] +  As   sin (a-   -)
cos 8        COSs'
(a)
AS' =- Aa sin 6 sin (a'+ a'-') +-   S cos 5' [cos 6 + sin O tang 5' cos (a'+a' —')]
and in the same manner:
A S'    sin (a q- a - z)
Aa= — Aa' [cosO - sin O tang 8 cos (a- c+ az)]-'A sinO  (o  ao
S+Cos 8-   t    cos 6
(b)
AS —Aa' sinO sin(a —+a.-.z) --    cos  [cos O-sinO tang cos(a+a  z)].
Cos.o




249
Example.  The mean right ascension and declination of
Polaris for the beginning of the year 1755 is:
a - 10~ 55' 44". 955      -- 87" 59' 41". 12.
By application of the precession the place of Polaris
was computed in No. 3 of the second section for 1850 Jan. 1,
and found to be:
a' -160 12' 56"..917' =- 88 30'34". 680.
But in Bessel's Tabulae Regiomontanae this place is:
a' - 160 15' 19".530  -        8-t- 88 30'34".898.
The difference between these two values of a' and 6'
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".501189'A a = + 0".002295.
If we wish to find from this, for example, the proper motion of Polaris referred to the equator of 1755, it must be
computed by means of the formulae (b).  But we have:
0 -  0 31' 45". 600
az -- z  110 32' 9". 530
and with this we obtain:
a a- +- 1". 10836   Ac- +- 0".005063.
In the case - of a few stars the assumption of an uniform
proper motion does not satisfy the observations made at
different epochs, since there would remain greater errors,
than can be attributed to errors of observation. Bessel first
discovered this variability of the proper motions in the case
of Sirius and Procyon, comparing their places with those of
stars in their neighbourhood, and he accounted for it by the
attraction of large but invisible bodies of great masses in
the neighbourhood of those stars. Basing his investigations
on this hypothesis, Peters at Altona has determined by means
of the right ascensions of Sirius its orbit round such a central body and has deduced the following formula, which expresses the correction to be applied to the right ascension
of this star:
q -- os. 127 -+- Os.00050 (t - 1800) - O+s.171 sin (u +- 770 44'),




250
where the angle ut is found by means of the equation:
M1- 7. 1865 (t - 1791.431)= u -0.7994 sin u
and  wher"7~0. 1865 is the mean motion of Sirius round the
central body. By the application of the correction computed
according  to  this formula the observed  right ascensions of
Sirius agree well with each other. Safford at Cambridge
has recently shown, that the declinations of Sirius exhibit
the same periodical change, and that the following correction
must be applied to the observed declination:
q' --- 0".56 + 0".0202 (t- 1800) + 1".47 sinll u   0".51 cos tu,
where u is the same as in the formula above *).
*) Of great interest in regard to this matter is the discovery, made recently by A.'\Clarke of Boston, of a faint companion of Sirius at a distance
of about 8 seconds.




FIFTH  SECTION.
DETERMINATION OF THE POSITION OF THE FIXED GREAT
CIRCLES OF THE CELESTIAL SPHERE WITH RESPECT TO
THE'HORIZON OF A PLACE.
It has been already shown in No. 5 and 6 of the preceding section, how the position of the fixed great circles of
the celestial sphere can be determined by means of a meridian instrument.  For if the instrument has been adjusted
so that the line of collimation describes a vertical circle, it
is brought in the plane of the meridian (i. e. the vertical circle
of the pole of the equator is determined) by observing the
circumpolar stars above and below the pole, since the interval between the observations must be equal to 12h of sidereal
time  - Act, where A c is the variation of the apparent place
in the interval of time. Further the observation of the zenith
distances of a star at both culminations gives the co-latitude,
since this is equal to the arithmetical mean of the two zenith
distances corrected for refraction -1- ~', where nA is the variation of the apparent declination during the interval between
the observations.  If the culmination of a star, whose right
ascension is known, be observed, the apparent right ascension
of the star is equal to the hour angle of the vernal equinox
or to the sidereal time at that moment. If a similar observation is made at another place at the same instant, the difference of both times is equal to the difference of the hour
angles of the vernal equinox *at both places or to their difference of longitude, and it remains only to be shown, by
what means the determinations of the time at both places
are made simultaneously or by which at least the difference
of the time of observation at both places becomes known.
These methods, which are the most accurate as well as
the most simple, are used, when the observer can employ afirmly




252
mounted meridian instrument. But the position of the zenith
with respect to the pole and the vernal equinox may also
be determined by observing the co-ordinates of stars, whose
places are known, with respect to the horizon, and thus various methods have been invented, by which travellers or
seamen can make these determinations with more or less advantage according to circumstances and which may be used
on all occasions, when the means necessary for employing the
methods given before are not at hand.
We have the following formulae expressing the relations
between the altitude and azimuth of a star, its right ascension and declination and the sidereal time and the latitude:
sin h - sin q? sin A -+- cos T cos 8 cos (O- a)
cos T tang S
cotangA            ) -  s in (  cotg (O- a).
These equations show, that if the latitude is known, the
time may be determined by the observation of an altitude or
azimuth of a star, whose right ascension and declination are
known, and conversely the latitude can be determined, if the
time is known, therefore by the observations of two altitudes
or azimuths both the latitude and the time can be determined.
The observations used for this purpose must be freed
from refraction and diurnal parallax (if the observed object
is not a fixed star) and the places of the stars must be
apparent places.  The instruments used for these observations are altitude and azimuth instruments, which must be
corrected so that the line of collimation, when the telescope
is turned round the axis, describes a vertical circle (see
No. 12 of the seventh section), or, if only altitudes are taken,
reflecting circles are used, by which the angle between the star
and its image reflected from an artificial horizon, one half of
which is equal to the altitude, can be measured. When an altitude and azimuth instrument is used, the zenith point of the circle
is determined by means of an artificial horizon, or the star is
observed first in one position of the instrument, and again
after it has been turned 1800 round its vertical axis.  For
if f and 5' are the circle-readings in those two positions,
dz     d2 z
corresponding to the times 0 and 0', and if dz and d   are
dO0    dO'2




253
the differential coefficients of the zenith distance (I, 25) corresponding to the time 0,=    2, assuming that in the first
position the divisions increase in the direction of zenith distance  and denoting the zenith point by  Z, then the circlereadings reduced to the arithmetical mean of both times are:
dz   (0          12 _0)   I d~  (O _ 0o)2
zdo + Z      +            2 d ( Oo2
dz               d2z
d (o' —    )+ d Z -2 (o'- 06)2.
Hence the zenith distance z0 corresponding to the arithmetical mean of the times is:
d2z
Z ~'2  -A)-   d 0 2 (0'- 0)2
Finally in  case that the  object is observed direct and
reflected from  an artificial horizon, we have, since the first
member of the second equation is then 180' — zo —Z:
dz
90    0 -  2o -   (~, -) +'~2 (     )  )o'- -,.
In order to observe the azimuth by such an instrument,
the reading of the circle corresponding to the meridian or
the zero of the azimuth must be determined, and this be subtracted from  or added to all circle-readings, if the divisions
increase or decrease in the direction of the azimuth.
I. METHODS OF FINDING THE ZERO OF THE AZIMUTH AND THE
TRUE BEARING OF AN OBJECT.
1. The simplest method of finding the zero of the azimuth consists in observing the time, when a star arrives at
its greatest altitude above the horizon, and for this purpose
one observes the sun with an altitude and azimuth instrument,
*) It is supposed here, that exactly the same point of the circle corresponds to the zenith in both positions. For the sake of examining this, a
spirit level is fastened to the circle, whose bubble changes its position, as soon
as any fixed line of the circle changes its position with respect to the vertical
line.  Such a level indicates therefore any change of the zenith point and
affords at the same time a means for measuring it. (See No. 13 of the seventh section.)




254
and assumes that the sun is on the meridian as soon as it
ceases to change its altitude.  This method is used at sea
to find approximately the moment of apparent noon, but necessarily it is very uncertain, because the altitude of the sun,
being at its maximum, changes very slowly.
Another method is that of observing the greatest distance of the circumpolar stars fromn the meridian. According
to No. 27 of the first section we have for the hour angle of
the star at that time:
_ tang pa  sin (8- T0)
cos t  tg ~ or tang' T t2sin (- 
tang S           sin (S+A)
and the motion of the star is then vertical to the horizon,
since the vertical circle is tangent to the parallel circle.
Therefore if one observes such a star with an azimuth instrument, whose line of collimation describes a vertical circle,
the telescope must in general be moved in a horizontal as
well as a vertical direction in order to keep the star on the
wire-cross, and only at the time of the greatest distance the
vertical motion alone will be sufficient.  If the reading of
the azimuth circle is a in this position of the instrument and
a', when the same observation is made on the other side of the
a -- a'
meridian, a  - is the reading of the circle corresponding to
the zero of the azimuth.  It is best to use the pole-star for
these observations on account of its slow motion.
A third method for determining the zero of the azimuth is that
of taking corresponding altitudes. For as equal hour angles
on both sides of the meridian belong to equal altitudes, it follows, that if a star has been observed at two different times
at the same altitude, then two vertical circles equally distant
from  the meridian are determined by this. Therefore if we
obse(rve a star at the wire-cross of an azimuth instrument,
read the circle and then wait, until the star after the culmination is seen again at the wire-cross, then if the altitude
of the telescope has not been changed but merely its azimuth,
the arithmetical mean of the two readings of the circle is
the zero of the azimuth. If the sun, whose declination changes
in the time between the two observations, is observed, a correction must be applied to the arithmetical mean of the two
readings.  For, differentiating the equation:




255
sin a -- sin 99 sin h - cos 99 cos h cos A,
taking only A and ( as variable, we have:
cos SdS        dS 
dA _..
cos 9 cos h sin A   cos 9 sin t
Therefore if A S  denotes the change of the declination
in the time between the two observations, we must subtract
from  the arithmetical mean of the two readings:
AScos S          Ac
2 cos 9 cos h sin A   2 cos y sin t
if the divisions increase in the direction of the azimuth.
The fourth method is identical with that given in No. 5
of the fourth section for adjusting a meridian circle. For if
we observe the times at which a circumpolar star arrives at the
same azimuth above and below  the pole, the plane of the
telescope coincides with the meridian, if the interval between
the observations is 12h of sidereal time  - A ce, where A a is the
change of the apparent place in the interval of the two times.
But if this is not the case, the azimuth of the telescope is
found in the following way.  If the azimuth be reckoned
from  the north point instead of the south point, we have for
the first observation:
cos hi sin A - cos a sin t
cos  cos A- cos 99 sin S - sin 9T cos S cos t,
and for the second observation below the pole:
cos h' sin A - cos 8 sin t'
cos h' cos A - cos 99 sin 8 - sin 99 cos a cos t'.
Adding the first equation to the third and subtracting
the second equation from  the fourth, and then dividing the
two resulting equations we easily find:
tang A --- cotang     t- ( t) tang  ( ) tang n  (h - h')
sin 9
In case that t'- t is nearly equal to 12 hours of sidereal
time, A as well as 900 -I (t'- t) are small angles, and since
then' (h- +h') and   (h -- h') are nearly equal to S and 900 - 8,
we get:
A    90~ -  (t' -t)
A ==             -
cos 9 tang 8
2.  It is not not necessary for applying any of these methods
to know the latitude of the place or the time, or at least they
need be only very approximately known. But in case they




256
are correctly known, any observation of a star, whose place
is known, with an azimuth instrument, gives the zero of
the azimuth, if the circle-reading is compared with the azimuth computed from the two equations:
cos h sin A = cos 8 sin t
cos h cos A = - cos 9 sin  +- sin 9q cos 8 cos t
In case that a set of such observations has been made,
it is not necessary to compute the azimuth for each observation by means of these formulae, but we can arrive at the
same result by a shorter method. Let 0, 0', O" etc., be the
several times of observation, whose number is n, let O, be
the arithmetical mean of all times and A, the azimuth corresponding to the time 0o, then we have:
dA, dA
A-Ao + dt (o -- O)    d t (o - 0o)2,
dA d(2                A
dt (-)+  d-t2 ('              )
etc.
and since 0 -  0,, +  6 -'0o- -  etc. = O, we find:
Ao   A+A+A  d~A d            (- -  o)' +(-'-O)+...
A        n          2d t2             n
A -- A'-+ A" -4-...  d'A (2 sin  0 ( 0o)2
n       - dt2         n
where 2' 2 sin  (0 -  0,,) denotes the sum  of all;,the quantities 2 sin, (0-,,)2.  These have been introduced instead
of 2    (0-  0 )2 on account of the small difference and because
in  all collections of astronomical tables, for instance in,,Warnstorff's Hiilfstafeln", convenient tables are given, from
which we can take the quantity 2 sin2 2 t expressed in seconds of arc, the argument being t expressed in time. Now
we have according to No. 25 of the first section:
d A _ cos 9 sin Ao [cos ho sin S +- 2 cos T cos A,].
dt2      cos ho 0'
Therefore if we add to the arithmetical mean of all readings of the circle the correction:
cos 95 sin A [cos        os      ] 22 sin 2 (0- _   )2
COS ho 12                              n
we find the value Al, which we must compare with the azimuth computed by means of the formulae (a) for t =  0- a.




257
Differentiating  the equation (a) or using the differential
formulae given in No. 8 of the first section, we find:
dAl-            dt - tang h sin A dp +   csd 
cos h                      cos h
hence we see, that it is especially advisable to observe the
pole-star near the time of its greatest distance from  the meridian, because we have then p    -900 and A is nearly 1800,
except in very high latitudes.  Then  an error of the time
has no influence and an error of the assumed latitude only
a very  small influence on the computed azimuth and hence
on the determination of the zero of the azimuth.
3.  If the zero of' the azimuth has been determined, we
can find the bearing of any terrestrial object ).  This can
also be determined, though with less accuracy, by measuring
the distance of the object from any celestial body, if the time,
the latitude and the altitude of the object above the horizon
are known.
For if the hour angle of the star at the time of the observation is known, we can compute according to No. 7 of
the first section its altitude h and azimuth a, and we have
then in the triangle formed by the star, the zenith and the
terrestrial object:
cos A - sin h sin H +- cos h cos I cos (a - A)
where H and A are the altitude and the azimuth of the object
and A is the observed distance4 ). We find therefore a- A
from the equation
cos A- sill h sin 11
cos (a- -  ) = c       A  sin 11   (A)
cos h cos H
hence also the azimuth of the object A, since a is known.
The  equation  (A) may be changed  into  another form
more convenient for logarithmic computation. For we have:
*) For this a correction is necessary, dependent on the distance of the
object, if the telescope is fastened to one end of the axis. See No. 12 of'
the seventh section.
**) To the computed value of h the refraction must be added, and if the
sun is observed, the parallax must be subtracted froum it. Likewise is H the
apparent altitude of the object, which is found by observation.
17




258
1 + cos (ac - A) - s (I+1 h) + 
cns I cos  cos 
and:
1os (H- ih) - cos A
1-cos (c - A) =
cos 1 cos H
hence:
sin 2 (A - H+ I/) sin 2- (A + I — 7)
tang -        (a ( - A) 2         2_
t  cos -. (Ad -~ 1+ H+iJ cos 3 (H- 11I- a)
or taking:
-2 (A+ -I+ Hih),
ng  ( — A) 2   i  - ) sin (S - h) 
tang 2~ (.,      -- zl)~ =                (.-).
cos S cos (,S -  )
If the terrestrial object is in the horizon, therefore II O,
we have simply:
tang 4- (a -A)2 = tang  (A -- h) tang 1 (A - A).
Differentiating the formula for cos A, taking a — A  and
A a1s variable, we get:
c1-A)-= ~ ~sin A
cos h cos Hsin (a - A)
and from I. No. 8:
cos S cos p
da c  --      clt.
cos h
HIence we see, that the star mUSt not be taken too far
from  the horizon, in order that cos h may not be too small
and errors of the time and distance may not have too great
an influence on A.
If two distances of a star from  a terrestrial object have
been, observed, the hour angle and declination of the latter
can be determined and also its altitude and azimuth.
For if we denote the hour angle and the declination of
the object by T and D, the same for the star by t and 6,
we have in the spherical triangle formed by the pole, the star
and the terrestrial object:
cos A = sill  sin D) -- cos ( cos D cos (t - T).
Then, if'   is the interval of time between both observations, which in case of the sun being observed must be expressed in apparent time, we have for the second distance
a' the equation:
cos A' = sinl  sin D +- cos 8 cos D cos (t - T +- ).
From these equations we can find D and t - T, as will




259
be shown for similar equations in No. 14 of this section. If
then the hour angle t at the time of the first observation be
computed, we can find T and N, and then by means of the
formulae in I. No. 7 A and H.
II. METHODS OF FINDING THE TIME OR THE LATITUDE BY AN
OBSERVATION OF A SINGLE ALTITUDE.
4. If the altitude of a star, whose place is known, is
observed and the latitude of the place is known, we find the
hour angle by means of the equation:
sinl h - sin T sil 8
COS t = —COS? cos 
In order to render this formula convenient for logarithmic computation, we proceed in the same way as in the preceding No. and we find, introducing the zenith distance instead of the altitude:
12  sin ~ (z -   + - ) sin 1' (z +  a-  _ )
tag 2    cos (z + 9 +  ) cos  (99-+-  -z)
or:
t2  sin (S - g) sin (S -) )
tang 2  cos S.cos (S - z)      (A).
where S -  (z+   +A )!
The sign of t is not determined by this formula, but t
must be taken positive or negative, accordingly as the altitude
is taken on the west or onl the east side of' the meridian.
If the right ascension of the star is a, we find the sidereal time of the observation from the equation:
0 -t -+ a,
but if the sun was observed, the computed hour angle is the
apparent solar time.
Example.  Dr. Westphal observed in 1822, Oct. 29, at
Abutidsch in Egypt the altitude of the lower limb of the sun:
h — 330 42' 1 8". 7
at the clock-timle 20"' 16"'" 20s.
The altitude must first be, freed from refraction and parallax; but as the meteorological instruments have not been
observed, only the mean refraction equal to 1' 26".4 can be
used, which is to be subtracted from  the observed altitude.
17 




260
Adding also the parallax in altitude 6".9 and the semi-diameter of the sun 16' 8".7, we find for the altitude of the
centre of the sun:
h -- 33 o 57' 7". 9.
Now  the latitude of Abutidsch is 27o 5' 0" and the declination of the sun was on that day:
13 38' 11".
hence we have:
S  -2 (z + g + a)    + 340 44' 50". 5,.        t-+7~0 39' 50".5,  S- 68 — + 483 23 "., S.  z=- —'- 21~ 0  1 8'. (
and the computation is made as follows:
sin (S - R) 9.1250385     cos S 9.91.46991
sin (S - o) 9.8736752  cos (S - z) 9.9692707
8.9987137
9).8839698
tang. t2 9.1147439    tang -t 9.5573719
t —— 190 50' 37". 98
It- -- 39 41 15.96
t=-  2s 38nl 45s  06.
Hence the apparent time of the observation is 211' 2111
14S.9, and since the equation of time is -  1611" 8.7, the mean
time is 211l 5n 6s.2. The chronometer was therefore 48' 46s. 2
too fast, or -- 48'l 46s.2 lmust be added to the time of the
chronometer in order to get mean time.
Since the declination and the equation of time are variable, we ought to know already the true time, in order to
interpolate, for computing t, the values of the declination, and
afterwards the value of the equation of time, corresponding
to the true time.  But at first we can only use an approximate value for the declination and the equation of time, and
when the true time is approximately known, it is necessary,
to interpolate these values with greater accuracy and to repeat the computation.
The correction which must be applied to the clock-time,
in order to get the true time, is called the error of the clock,
whilst the difference of the errors of the clock at two different times is called the rate of the clock in the interval of
time.   Its sign is always taken  so, that the positive sign
designates, that the clock is losing, and the negative sign,
that the clock is gaining.  If the interval between both tines




261
is equal to 241' -  t and A ul is the rate of the clock in this
time, we find the rate for 24 hours, considering it to be uniform, by means of the formula:
24 A u    A
24- t       t
24
Differentiating the original equation:
sin h  sin 9in T sin   cos +   cos C' cos t,
we find according to I. No. 8:
d = -- cos Acdi - cos S sin p dt,
or since:
cos 8 sin p = cos TF sin A
we get:
1              1i
cl t  s-    _   d Ad -           p.
cos  Sill A   cos T tang A
The value of the coefficients of dh and dmr) is the less,
the nearer A is -4-900. In this case the value of the tangent
is infinity, hence an error of the latitude has no influence
on the hour angle and thus on the time found, if the altitude is
taken on the prime vertical. Since then also sin A is a maximunm, and hence the coefficient of dh is a minimulnm  an error
of the altitude has then also the least influence  on the time.
Therefore, in order to find the time by the observation of an
altitude, it is always advisable, to take this as near as possible
to the prime vertical.
Since the coefficient of d h can also be written-.
cos 8sinpI
it is evident, that one must avoid taking stars of great declination and that it is best to observe equatoreal stars.
If we compute the values of the differential coefficients
for the above example, we find first by means of the formula
sin A A.      A= — 48  25'. 8
cos h
and then
dt - - 1.5013 dlh + 0.9966 dl
or d t expressed in seconds of time:
dt =-q- 0.1001 dh + 0.0664 dp.
Therefore if the error of the altitude be one second of
arc, the error of t would be Os. 10, whilst an error of the
latitude  equal to 1" produces an error of the time equal to
O0. 07.




262
Besides we see from the differential equation, that it is
the less advisable to find the time by an altitude, the less
the value of cos c~, and hence, the less the latitude is. Near
the pole, where cos p is very small, the method cannot be
used at all.
5. In case that several altitudes or zenith distances have
been taken, it is not necessary, to compute the error of the
clock from each observation, unless it is desirable to know
how  far they agree with each other, but the error of the
clock may be found immediately from the arithmetical mean of
all zenith distances. However, since the zenith distances do
not increase proportionally to the time, it is necessary, either
to apply to the arithmetical mean a correction, as was done in
No. 2, in order to find from  this corrected zenith distance
the hour angle corresponding to the arithmetical mean of the
clock-times, or to apply a correction to the hour angle computed from the arithmetical mean of all zenith distances.
Let T, T', z", etc. be the clock-times, at which the zenith
distances, whose number be n, are taken; let T be the arithmetical mean of all, and Z the zenith distance belonging to
the time T, then we have:
z    dZ           d T   Z
-z - Z+  dt (~ —T) +2- (,'-T)2,
dt           dt2
dZ            d2 Z
cit           ct2
etc.,
where t is the hour angle corresponding to the time T', or
since r — T-r'  T — T-" - T —...   O:
z'+- Z"...   d Z (Cr- T)2 + (t,'-T)2.....n        dt2            n
z' - z"...  d2 Z Y2 sin2 (- T)2
n         dt2      n1
c2 z
f' we substitute here the expression for Cdt found in No. 25
of the first section, we finally get:
__z  z' - - z...__   cos  cos  o 2sin') (x-T)2
Z                   -        cos A cosp p I
12         sin Z                  n
With this corrected zenith distance we ought to compute the hour angle and from this the true time, which compared with T gives the error of the clock.  But if we com



263
pute the hour angle with the uncorrected arithmetical mnean
of the zenith distances, we must apply to it the correction:
dt cos c cos 9        22 sin: (- T) 2
dz         1Cos A cos A  cos p,
dt
dcz  sin Z                   a1
or if' we substitute for ddt its value according to No. 25 of
the first section, we find this correction expressed in time:
cos p cos A Y 2 sin 4 ('r - T) 2
(a)
15 sin t       n
where A and p are found by means of the formulae:
sin t
sin A ---   cos 
sin t
and  sin p        sinZcos %.
sin Z
These, it is true, do not determine the sign of cos A and
cos p; but we can easily establish a rule by which we may
always decide about the sign of the correction (a).
If the hour angles are not reckoned in the usual -way,
but on both sides of the meridian from  0" to 180~, the correction is always to be applied to the absolute value of t3
and its sign will depend only upon the sign of the product
cos A cos p, which is positive or negative, if cos p and cos A
have the same or opposite signs.  Now we have:
sin'        sin 9_
Sill (1 -Cosz    ) sin        C- os z)
C08 sp-=    sin — cos     -     sinzcosS
cos A        sin p -si  
sin z cos 8         sin z cos 9
-  t   sin 8 \. cos z sin 99
sin 9~ cos z  -                   -- _
cos A = —--                     -             -
sin z cos?         sin z cos 90
Therefore, if J < q, cos p is always positive,
and cos A is positive, if cos z >.sin 
sin a
negative, if cos   < sin
sin 
and if  Y.>  %, cos A is always negative,
and cos p is negative, if cos' > si s9
sin
positive, if cos z < sin (f
sin ll
Therefore if we take the fraction
sin 8
in, if cp>a
a sin  f if   <  6,
nd  sina~   9




264
the two cosines have the same sign and the correction (a) is
negative, if cos z is greater than this fraction; but they have
opposite  signs and  the correction (a) is positive, if cos z is
less than this fraction. For stars of south declination cos A
and cos p are always positive, hence the sign of the correction is always negative').
Dr. Westphal took on the 29t" of October not only one
zenith distance of the sun but eight in succession, namely:
True zenith distance of
Chronometer-time   the centre of the sun  r - T   2 sin  (r-T)2
201h 161n 20s      560 2' 52". 1       3m 32s      24". 51
17 21           55 52 51.5         2 31         12.43
18 21               42 51.0        1 31         4.52
19 21               32 50.5        0 31          0.52
20 21               22 50.0        0 29          0.46
21 23               12 49.4         1 31         4.52
22 23                2 48.9        2 31         12.43
23 25            4 5248.4         3 33         24.74
20h 19m 51S. 9      550 27' 50". 2                  10". 52.
Now  the arithmetical mean  of the zenith  distances is
550 27'50".2 and the declination of the sun -  130 38' 14". 7,
hence we find the hour angle:
2h 35Ul 3s. 18.
to  which  value  the correction  must be applied.   But we
have:
sinp - 9.83079, sin A = 9.86881,
hence, as the declination is south, the correction is:
-8".32 in are or -Os.55 in time.
With the corrected hour angle -  2" 35nM 12s.63 we find
the mean time  21t' 8m 38. 70, hence the  error of the clock
is equal to:
+ 48m 46s. 8.
6.  If an altitude of a star is taken and the time known,
we can find the latitude of the place.  For we have again
the equation:
sin h = sin T sin S -+- Cos 9 cos S cos t.
*) W~arustorff's Hiilfstafeln pag. 122,




265
Taking now:
sin 8    - sin N,  (A)
cos A cos t = 21 cos N,
we find:
sin h --- M cos (R - N),
and hence:
sin h  sin N
cos ()p - N)   -       sinohi   (B)
M    sins6
The formula leaves it doubtful, whether the positive or
negative value of T -- N must be taken, but it is always easy to
Fig. 6.      decide this in another way. For if in
-    Fig. 6 we draw an arc S Q perpendic/J//  ular to the meridian, we easily see that
/  /  N   90 - F Q or equal to the distance of
Q from the equator, hence that Z Q
cz  - N, whilst M is the cosine of the
arc SQ.  Therefore as long as S Q
~')/o rintersects the meridian south of' the
zenith, we must take the positive value cp - N, but N —- A
is to be taken,  when the point of intersection lies north of
the zenith.  In case that t > 900, the perpendicular are is
below the pole, hence its distance from the equator is > 90"
and the zenith distance of Q equal to N - f. Therefore in
this case the negative value N -  f- of the angle found by
the cosine is to be taken.
If the altitude is taken on the meridian, we find r( by
means of the simple equation
-- = i-4-z,
where the upper or lower sign must be taken, if the star
passes across the meridian south or north of' the zenith. In
case that the star culminates below the pole, we have:
9-= 180 - 8 - z.
Dr. Westphal in 1822 October 19 at Benisuef in Egypt
took the altitude of the centre of the sun at 23' I 1 0Si mean
time and found for it 49" 17' 22". 8. The declination at that
time was - 100 12' 16". 1, the equation of time - 151m 0S.0,
hence the hour angle of the sun 23 16n 1010    l —!0  57' 30".0,
We find therefore:




266
tang a  9. 2552942,
cos t — 9. 9920078
N — 10~ 23' 23". 67
sin N- 9. 2561063,,
sin A   9. 2483695,,
0. 0077368
sin h  9. 8796788
-- N= 390 29 54". 51
hence   -- 29  6 30. 84.
In order to enable us to estimate the effect, which any
errors of h and t can have on p, we differentiate the equation for sin h and find according to 1. No. 8:
dlp = - see Adh - cos (y tang A. dt.
Here the coefficients are at a minimum, when A = 0 or
1 80".  The secant of A is then -= 1,  hence errors of the
altitude are then at least not increased and since tang A is
then equal to zero, errors of' the time have no influcnze at
all.  Therefore in order to find the latitude as correct as
possible by altitudes, they must be taken on the meridian or
at least as near it as possible.
For the example we have A=  -16 "40'. 1, hence we
find:
di= — 1.044 dh + O0.2616 dt,
or if dt be expressed in seconds of time:
d=- 1.044 dh + 3.924 dt.
If several altitudes are taken, we find according to No. 5
the altitude corresponding to the arithmetical mean of the
times by means of the formula:
At_- +A1'_ _ +.   cos 8 cos A  _32 sin' (r- T)'
______1         4-' — +  -..   cos A cosp
7. If the altitude is taken very near the meridian, we
can deduce the latitude from  it in an easier way than by
solving the triangle. For since the altitudes of the stars arrive at a maximum  on the meridian and hence change very
slowly in the neighbourhood of the meridian, we have only
to add a small correction to an altitude taken near the meridian, in order to find the meridian altitude. But this in connection with the- declination gives immediately the latitude.
This method of finding the latitude is called that by
circuinm-meridian altitudes.




267
From:
cos z = sin 99 sin A + cos 9 COS 3 cos t,
we get:
cos z - cos (q9 - 3) - 2 cos 9 cos a sin ~ t2
and from  this according to the formula (19) in No. 11 of
the introduction:
2 cos p cos   2.   2 cos 992 cos 2
sin( - - 99-   sin 2   Sin    ) 2  cotang (t- 3) silln  i t
sin (90 c-)        sin (~ t- -
or denoting cOS  -    by b:
sin ( 3) — b'
99 z + A-b. sin  t2 t   2. cotang ( - ). 2 sin -r t4.    (A)
Therefore if we compute 9p -    and b with an approximate  value of ~p, and  take the  values of 2 sin l   t2 and
2 sinl  t'- from  tables the computation for the latitude is exceedingly simple. Such tables are given for instance in Warnstorff's Huilfstafeln, where for greater convenience also the
logarithms of those quantities are given.  If the value of ~f
should differ considerably from  the assumed value, it is necessary,. to repeat the computation, at least that of the first
term. Stars culminating near the zenith must not be used
for this method, since for these the correction becomes large
on account of the small divisor,f - 35.
Westphal in  1822 October 3 at Cairo took the zenith
distance of the centre of the sun  at 0"h 2m 2. 7 mean time
and found 34" 1' 34". 2. The declination of the sun being
-  3" 48' 51". 2, the equation of time - 1011 48s. 6, and hence
the hour angle  - 12'l" 51. 3, we find from  the tables:
log 2 sin 4 t2 - 2.51105   log 2 sin 4 t4  9.4060.
Taking - = 300 4', we have log b = 0. 19006 and then
the first term  of the correction is -8'22".47, the second
- 0". 91, therefore we have:
Correction -  8' 21". 56
z - - 300 12' 43". 00
-=  300 4'21".44.
A change of 1' in the assumed value of 9f gives in this
case only a change of 0". 30 in the computed value of *, and
the true value, found by repeating the computation, is:
99 - 30" 4' 21". 54.
The formula (A) is true, if the star passes the meridian
south of the zenith.  But if the declination is greater than




268
the latitude and thence the star passes the meridian north of
the zenith, we must use  ) — ( instead of p —f, and we get
in this case:
cos Rcos       12      cos 92 COS 82
Z~_ _+COs oS 2sin'2 t2 CO                cotang' ( - )2sinPt4.
sin (8 -ctn)            (82)sin (i4')2
Finally, if the star be observed near its lower culmination, we have, reckoning t from  the lower culmination:
cos z = cos (180 -- 9 -  ) - 2  cos  A Cossin l t2
and hence:
c=1SO —a-2  os~cos8    " oscp"cos 
lSO 180-S-z-  Csi (z + C) 2 sin   - t 2+ sin(. T C ) cotmng ( +b')2 sin  t4.
If the latitude of a place is determined by this method,
of course not only a single zenith distance but a number of
them  are taken in  succession in  the  neighbourhood of the
meridian. Then the values of 2 sin 1 t2 and 2 sill 1 t4 muslt be
found  for each t and the arithmetical means of all be multiplied by the constant factors.  The correction, found in this
way, is to be added to the arithmetical mean of the zenith
distances *).
The reduction to the meridian can also be made in another form.  For from  the equation:
cos z - cos ( -  )   - 2 cos T cos b' sin - t2
follows:.T - % z si  S -- *' - z
sin          sin           -cos 9p cos a sin. t2.
2           2
Now  if we take the reduction to the meridian:
g - 8      --  = -x,
we find:
99 - a +- Z       +;
hence:
sin  x       C         sinl  t2
sin (T - 8 -- 2 X)
an equation which may be written in this way:
sin z x    cos T cos  i        sin (99 -  ).x            2 sin A t2.x      sin (99 -        sin (90-  + -2 X)
Now it has been proved in No. 10 of the introduction, that
*) In case that the snn is observed, the change of the declination must
be taken into account. See the following No.




269
3
slnl --  /cos a, neglecting terms of' the fourth  order.  If we
apply this and take as a first approximation for x the value c
from  the equation:
-" -- -s —- ) 2  sin  2  (B),
sin (99  -
we find:
sin (9o - =)
13cos X = ~ -i~(~;~_ ~ +~
*  o  Sill 4(   A S _ - +  X) _
or if we find x firom  this equation, write in the second nuimber I instead of x, and denote the new  value of x by;':
sinl (99 - +) see:.
sin (~ - c -+-   see)
This second approximation is in most cases already stfficiently correct. But if this should not be the case, we compute p, from  d', then ~ by means of (B), and find the corrected value:
sin (90 -- 6 ) sec  5
With the data used before, we find:
-- 8' 22". 47
log - 2.70111
sin ((p - S)=9.74620
cosec (9 - ) -+-  )    0.25293
log $'    2.70024,
hence`' - 8' 22".47 and    -- 300 4' 21".53.
8.  If we take circulm-lmeridian altitudes of the sun, we
must take the change of its declination into account, hence
we ought to make the computation for each hour angle with
a different declination. But in order to render the reduction
more convenient, we can proceed in the following way:
We have:
cos 90 Cos 8
f == x + a(it (9o  & 2 si  ti.
Now if D is the declination of the sun at noon, we can
express the declination corresponding to any hour angle t
by D ---  t, where fl is the change of the declination in one
hour and t is expressed  in  parts  of an hour.  Then we
have:
T --  -- D  -  t _.cOs( Co a 2 sin - t 2  (a)
sin (99       2




270
If we take now:
tCosf9Cos?             COS 99COS 
cosin (p- cos   4- 2           2 sin   (t+-y)-,    ()
sill (g, - 6         sin (goT- 6)
we must find y from  the following equation:
2cos T cos [sin 4 (t-+-y)2 - sin - t2] = --  t
sin (99- )
or since:
sin a2 - sin 1 -= sin (a ~- b) sill (a - )
s1 i ll      sin (92 —A)    t
Sin      -  2  cos 99 cos  sin (t +- 2 )
we have:
sin (r- S)   206265
cos c Cos   3600 X 15
where the numerical factor has been added, because we take
sin (t-I-  y)   t, and the unit of t is one hour, whilst the unit
of sin t is the radius or rather unity.   If we  denote the
change of the  declination in 48 hours expressed in seconds
of arc by,t, we have  i=-,  or if we wish to express y in
seconds of time, /i- 70.   We have therefore:
- 1    5[tang 9 -tang c]    (A)
and  then we find the latitude  from  each single observation
by means of' the formula:
c Cos   s2 sill ( +    *)           (X)
sin (~ -  )   2
The quantity y is the hour angle of the greatest altitude,
taken negative.
For in I. No. 24 we found for this the following expression:
de                206265
t =   [tang 9 - tang     15
d 8
where t is expressed in seconds of time and "~ is the change
of the declination in one second of time.  But this is equal
to    3600_  hence the hour angle at the time of the greatest
48 3600o
altitude, expressed in seconds of time, is:
*) To this there ought to be added still the second term dependent on
2  sill. t.




271
FCy~   ~     206265
t i    [tangT - tang 5] 06    15
7230 600 X 5
8-  8   [tang 9 - tang 5],
188.a
which f'ormula is the same as that for y taken with the opposite sign. Hence t -- y is the hour angle of the sun, reckoned not from the time of' the culmlination but from the time
of the greatest altitude.
Therefore if circum-meridian altitudes of a heavenly body
have been taken, whose declination is variable, it is not necessary to use for their reduction the declination corresponding to each observation, but we can use for all the declination. at the time of culmination, if we compute the. hour anoles
so that they are not reckoned from the time'of the culmination but from  the tilme of the greatest altitude.  Then the
computation is as easy as in the former case, when the declination is supposed not to change.
For the observation made at Cairo (No. 7) we have:
log = -- 3.4458,, and D   - 30 48' 38". 57,
with this we get:
y+- 9s. 6, hence t -+-y-13 OS. 9
and hence we find for the first term  of the reduction to the
mer~idian:- - 8' 35". 00.
On account of the second term  multiplied by sinll  t4 we
must add to this -+- 0".91, and we finally find ip =  30' 4'21".54.
In case that only one altitude has been observed, it is
of course easier to interpolate the declination of the sun for
the time of the observation; but if several altitudes have been
taken, the method of reduction just given is more convenient.
9.  Since the polar distance of the pole-star is very
small, it is always in the neighblourhood of the meridian, and
hence its altitude taken at any time may be used with advantage for finding the latitude; but the method given in
No. 7 is not applicable to this case, as the series given there
is converging only as long as the hour angle is small.  In
this case, the polar distance being small, it is convenient to
develop the expression for the correction which is to be applied to the observed altitude according to the powers of
this quantity.




272
Fig. 7.                  If we draw  (Fig. 7) an
arc of a great circle from
the place of the star perpendicular to the meridian,
and denote the arc of the
meridian between the point
of intersection with this arc
and the pole by x, the arc between the same point and the
zenith by z - y, where y is a small quantity, we have:
900 -  -- z - y -- x,
or      9- -900 - z+ -y - x,
and we have in the right angled triangle:
tang x --- tang p cos t
cos z               (a)
cos (.z: —y) 
Cos at
We get immediately from  the first equation:
x — tangp cos t --  tang p3 cos t3,
neglecting the fifth and higher powers of tang p, or neglecting again terms of the same order:
x    p cos t -+- - p3 COS t sin t2.  (b)
If we develop the second equation (a), we find:
1 - cos it
sin y = cotang z    -- -- 2 sin 2 ~ y  cotang z,
Cos It
or neglecting the fifth and higher powers of u:
sin y = cotang z (- u2 + -54 u ) + 2 sin2 2 y cotang z.
But we get from the equation
sin tG = sin p sin t:
t = p sin t -  — p3 sin t cos t,
hence substituting this value in the equation above we find,
again neglecting terms of the fifth order:
y —=2p2 sin t2 cotgs z-2-  p4 sin t2 (4 cos t' —5 sin t2) cotg.z +-  cotg z. y2.  (c)
This formula, it is true, contains still y in the second
member, but on account of the term   eotang. y2 being very
small, it is sufficient, to  substitute in this term  for y the
value computed by means of the first term alone. Thus we
obtain:
T = 900 - z - p COS t -+  - p2 sin t2 cotang z-'3 p3 cos t sin t2
-  p214 sin t2 (5 sin t2 - 4 cos t2) otang z
+ -  p4 sin t-4 cotang z3.          (A)
Since it would be very inconvenient to  compute this




273
formula for every  observation, tables are every  year published in the Nautical Almanac and other astronomical almanacs, which render the computation very easy. They embrace
the largest terms of the above expression, which are always
sufficient,  unless the greatest accuracy should  be required.
If we neglect the terms dependent on the third and fourth
power of' p, we have simply: *)
9  90~.- z —p cos t +- p2 sin t2 cotang z.
If we denote thus a certain value of the right ascension
and polar distance by t,0 and p,o the apparent values at the
time of the observation being
a-  +o t- at,   P - Po + AP
we find substituting these values:
w = 900    z  p, cos to + 1 p 2 cotang z sin t0 2
-/.p cos to - p sin to ia,
where to =     -  a0.
We find now  in the Almanac three tables.  The first
gives the term  -  p0 cos to, the argument being  9, since this
alone is variable.   The second table gives the value of the
term  1 po2 cotang z Sill t0o2 the arguments being z and O. Finally the third  table gives the term  dependent on  O, A a
and Ap
- Ap cos t -p sill to A a,
the arguments being the sidereal time and the days of the
year.
Tables of a different construction  have been published
by Petersen in Warnstorff's ETiilfstafeln pag. 73 and these,embrace all terms and can be used while the polar distance
of the pole-star is between the limits 10 20' and 1" 40'. Let
p, again be a certain value of p, for which Petersen takes
p,,o   10 30', then  the formula (A) can easily be written in
this way:
*) The term multiplied by p3 is at its maximum, when t = 540 44' and
its value, if we take p- 1~ 40', is then only 0". 3). The terms multiplied
by p4 are still less, unless z should be very small. These terms can be
easily embraced in the tables, as the first may be united with p cos t, the
other with 2 p2 sin t2 cotang z.
18




274
q —-90~    - ~     cos ~~[po-  cos tsu'i- F(12O —1 po COol/,
-2- cotang z [ po  sin 1 + _. po:' sin  t' (5 sin t  -4 cos t' )
4~      4
+ o 4 P.o sin 0t cotang 3.
If we put now:
— A
P o
P, COS t +  Pd  COS  Sill 3 2 =,
Aocs  t ( t  _-  1 po 3 CO  in   _ y1  2           Si,) 11 (5   Sill/   2,po2 Sill 2 + m po 4 Sill t2 (5in n2 -- t COS t= i,
A po' sin t0 cotang'  -. A. cotg a,
we obtain:
= 900 -        ~ -A-  2 a -  -+ A82 +  cotagl.a -t- +.
Now  four tables have been constructed,  the first two of
which give rc and ij, the argument being t; a third table gives
the value of the small quantity;,, the arguments being p and t
anld finally  a fourth  table  gives the  quanItity p,  which  is
likewise very  smtall the arguments being y -  A2 i cotang r
and 900 -  z. These tables have been computed from t -  O0
to  t -  61   Therefore  if t >  90", the hour angle must be
reckoned from  the lower culmination, so  that in this case
we have:: =900 - z -+- Ac    A2 -+- y - A   cottang z -- c.
Examnp le.  In 1847 Oct. 12 the altitude of' Polaris was
taken with  a small altitude and  azimuth  instrument at the
observatory  of the late Dr. Htilsmanl  at Dfisseldorf and it
was at 18' 22L' 48g. 8 sidereal time h =  50'' 55' 30". 8, which
is already corrected for refraction.
According  to the Berlin Jahrbuch  the place of Polaris
on that day is:
c -- 115' il 31s. 7,    8 s_= 88O 29' 52". 4.
Hence we have:
p — 1o 30' 7". 6,  t-  1711 17  17s. 1 - 2590 19' IG". 5,
~and:
log A = 0.0006108
and we obtain by means of the tables or the formulae:
ac   1000". 85, i = (68". 28, y = 0". 00, ai _  0'. 02,




275
therefore:
Aa- + - 16' 42". 26
A2afcotang z-    1 24.33
/u=, — +   0.02
sum —I- 18' 6". 61
hence:   -- 50 13'37". 41.
10.  Gauss has also published a method for finding the
latitude from  the arithmietical mean of several zenith distainces, taken  long before or after the  culmination, which  is
especially convenient for the pole-star.
If an approximate value (f, of the latitude  p( is known,
and 06 is the sidereal time9 at which the zenith  distance z
is observed, we can  compute firom  0  and qf, the value of
the zenith distance t by means of the formlulae:
tang x  cos t cotang 3
sin S
cos    sill - sin (0o +x)
COS x
and then we obtain:
d99. --,, —;=I,
hence:
sin S   cos (  ~ x)
cos x    sin.
x is again the arc between the pole and the point in which
an are drawn through the star and perpendicular to the meridian  intersects the latter and since the length of this arc
is always between the limits -4- 90" -  d, we can take in case
sin S
of the pole-star sin   as well as   s (. +.. equal to unity, if
Cos x                sinl     e
the latitude is known within a few  seconds and dq is therefore a small quantity.
If another zenith distance has beell taken at the sidereal
time 0', we have:
tang x' - cos t' tang S
sin 8
cos s'    - sin (op -+ x')
and:
d --
1S*




276
or, if Z denotes the arithmetical mlean of both observed zenith distances equal to   (z, -- z,'):
2 (A +B)'            (
where:
sin s  cos (g90  x)
A --     
cos x    si (S)
sins  cos (To -x')
Cos x'    Silln,
or:                 A -  cotang  c. cotang (0 + x)
(C)
B = cotang c'. cotang (~o + x'),
and finally, if we find d-  from  the original equation:
dcp
cos  =  sin Po sin a + cos C0O cos a cos t
we obtain also:
cos 9p sin b  sin 9 cos (
-(A Jr    C B)         SinZ   cos,,t' + ).   (id)
sin Z      sin Z
In case of the pole-star we have simply:
dcl —-- ~ (p  +') - Z.       ~(e)
If several zenith distances have been observed, we ought
to compute; for each sidereal time separately and we should
then obtain:
I[ -4- g' -4- g"    -J-....1] - 
cl ft-d, (,f)
where Z again denotes the arithmetical mean of all observed
zenith distances. But the following way of proceeding is more
simple.
If we denote by O-, the arithmetical mean of all sidereal
times and put:
0-0 —=', O'-0' Q      r= etc.
and  then  denote by ~0 the zenith distance corresponding to
0,, we obtain in the same way as in No. 5 of this section:
-I- +' t- +- t ".  d'    N o2 sin 2 (0 — o)2
Now  if T is taken from  the following equation:
222 sin I    -(0-o)2
2 sin'  T7 a=      \V - Vo0




277
the zenith distances z and z' at the times N,, - T and 0O,,t-I
are:
d ~0     d2  o
do   d  T  2 - O. ~0 + -dt- 4-+ 7 at~' 7,hence:
2 --    go+  at-  2 sin  T2        __ 
2          d it'   2   -
and we obtain according to the formula (f) simply:
(z' +z)- Z
cl = -,(A'+ B')
if' the values of A  and B  corresponding to z' are denoted
by A' and B'.
Therefore if several zenith distances of a star have been
observed, we take the mean of the observed clock-timnes and
subtract from  it each clock-time without regard to the sign.
These differences converted into sidereal time give the quantities T, for which we find from  the tables the quantities
2 sin  [r'.  From  the same tables we find the argument 7'
corresponding to the arithmetical mean of all these quantities and compute the hour angles:
0     (z -(- T) - t
00 - (a   T= )I
and then z and z' by means of the formulae:
tang x = cos t cotang a
sin 
cos z   --- sin (Tj, + x)
Cos x
and:                  tang x'=- cos t' cotang A
sill n
cos'~ -- - =    sin (po  + x').
In case of the pole-star we then have immediately:
dT, =  (z +- z') - Z,
where Z is now  the arithmetical mean of all observed zenith
distances. For other stars the rigorous formula for de  must
be computed, namely:
-    (A ~ z') Z
where A and B are obtained by means of the formulae (b)>
(c) or (d) after taking; =   and  Z'- =' i).
*) Warnstorff's HJilfstafeln pag. 127.




278
Example.  In  1847 Out. 12 the following ten zenith distances of Polaris were taken at the observatory of Dr. Hiilsmann:
Sidereal time.    Zenith distance.          r           2 sin Q zx
17h5621ts 4        39"  13' 42". 1      13"19s. 75       348. 75
59 54.5            12 17. 6         9 46.65         187.69
18  3 29.7       -          11  6. 8   6 11.45          75.24
6  2.9            10  3. 6         3 38.25        25.98
8 35.0             9 0. 16         I.15            2.39
11   5.1            8 2. 8          1 23.95           3.85
13 32.0             7 7. 6          3 50.85         29.06
16 34.0             6 4.            6 52.85         92. 95
18 28.1             5 15. 3        8 46.95        151.43
22 48.8             342. 7         13  7.65        338.28
0,:-18h 9m41s.15         39" 8'38".39                        125. 56
Refr.     _46".50                  T=7mn59s. 83
Z -390 9'24".89
0, -- (a -  T ) - 16h56m9s.62  00 -(a -  )      -  17h12m119s. 28
- 2540 2' 24".3               - 2580 2' 19". 2.
Now  taking:
5o l0- 51~ 13' 30". 0,
we obtain:
z   390 12' 37". 56    z' -  390 6' 34". 54
2 (z -+ z')   390 9' 36". 05.; (z +   - +' -1 Z    - - 1 ".  6,
hence:
o=51~ 13' 41". 16.!1I. MiETHODS OF FINI)INqG BOTH THE TIME AND THE LATITUDE
BY COMBINING SEVERAL ALTITUDES.
11.  If we observe two altitudes of stars, we  have two
equations:
sin    -= sin T sin S -+ cos   cos   cos t,
sin h'- sin Tp sin X'+- cos T cos c'cos t'.
In these equations, since we always observe stars, whose
[)laces are known, J and J' are known, and further we have:
t'= t + (t' - t)= t + (0' — 0) -- (c'- a).
Now  since ce'   ce and  0' -  0 are likewise known, the latter
being  equal to  the  interval of timue between the two  observations, the two equations contain only two unknown quan



279
tities O  andi q:, which therefore can be found by solving
themn.  Thus the latitude and the time can be found by observing two  altitudes, but the combination of two altitutdes
in some cases is also very convenient for finding either the
latitude or the time alone.
W\Te have seen before that if' two altitudes of the same
star are taken at its upper and lower culmination, their arithmetical mean is equal to the latitude, which thus is determined independently of tile declination.  This is even found
at the same time, since it is equal to half the difference of
the altitudes.
Likewise we can firnd the latitude by the difference of
the meridian zenith distances of two stars, one of which culininates south, the other north of the zenith. For if b is the
declination of the first star, its meridian zenith distance is:
Z = C- b-,
and if Y' is the declination of the other star, north of the zenith, we have:        -,
and therefore we get:
90   d    A- + 1  (z _ z"
12. If two equal altitudes of the same star have been
observed, we have:
sin h1 = sin l sin ll + cos 7) cos S cos t,
sin hI = sin gs sin  -O co  cos S cos t',
firom  which we find t --    t'.  The altitudes therefore are
then taken at equal hour angles on both sides of the meridian.
Now if u is the clock-time of the first, u' that of the second
observation, I (it -- i') is the time, when the star was on the
meridian and since this must be equal to the known right
ascension of the star, we find the error of the clock equal to:
a    (u' -4 — it).
This method of finding the time by equal altitudes is
the most accurate of all methods of finding the time by altitudes.  Since neither the latitude of the place nor the declination of the heavenly  body need be known and  since
for this reason it is also not necessary to know  the longitude of the place  this method is well adapted to find the
time at a place, whose geographical position is entirely unknown.  It is also not all necessary to know  the altitude




280
itself, so that it is possible to obtain by this method accurate
results, even if the quality of the instrument employed does
not admit of any accurate absolute observations. All which is
required for this method is a good clock, which in the interval between the two observations keeps a uniform  rate,
and an altitude instrument, whose circle need not be accurately divided.
We have hitherto supposed, that the declination of the
heavenly body does not change.  But in case that altitudes
of the sun are taken, the arithmetical mean of both times
does not give the time of culmination, for, if the declination
is increasing, that is, if the sun approaches the north pole,
the hour angle corresponding to the same altitude in the
afternoon will be greater than that taken in the forenoon and
hence the arithmetical mean of both times falls a little later
than apparent noon.  The reverse takes place if the declination of the sun is decreasing.  Therefore in case of the
sun a correction dependent on the change of the declination
must be applied to the arithmetical of the two times.  This
is called the equation of equal altitudes.
If a6 is the declination of the sun at noon, A ) the change
of the declination between noon and the time of each observation, we have:
sin h = sin 99 sin (c-A AS) + cosp cos ( - A ) cos t
sin  = sin? sin (O  - +  8) -cos 9 cos (  - A ) cos t'.
Let the clock-time of the observation before noon be denoted by ut, the one in the afternoon by u', then  (u'-+ u)  U
is the time, at which the sun would have been on the meridian, if the declination had not changed.
Then denoting half the interval between the observations 2 (t' -ct) by r, the equation of equal altitudes by x,
the moment of apparent noon is given by U -t-x and we
have:
t =   (i' - u) -+  =   -+ x,
t'=  2 (t' -  u) - x =  - X,
and also:
sin h = sin T sin (8- A 6) +  co    s  cos (-A  ) cos ( +- x)
and;
sin h = sin?p sin (8+A 8) + Cos  os (k —A k) cos ( - x).




281
Fromn these expressions for sin h  we find the following
equation for x:
sill n  cos a sin a- cos 9 sin S sin Ascos X cos x -P cos T cos z a cos Ssin s sin x.
Now  in  case of' the sun x is always so small, that we
call take  eos x equal to  1 and sin x equal to x.   Then we
obtain, taking  also L\c  instead of tang &A(Y:
_  (ta   9 ll z   tang 3a)n8,  *
sin X    tang X
If we  denote  now  by!t the  change of the declination
during  48 hours, which may be considered here to be proportional to the tilue, we have:
A a = B8_ ":44
hence:
X3( (0tang q-i                tang'
or if x is expressed in seconds of time:
X   (    --- tang 7p + ---- tang  ).
-720    sin z        tang T
In  order to  simplify  the  computation  of this  formula5
tables have been  published  by Gauss in Zach's  ionatliche
Correspondenz Vol. XXIII, which  are  also given in Warnstorff's Htilfstafeln.  These tables, whose argument is r, give
the quantities:
_ A
720  sin  -
and:
I     T
720  tang     B
and  thus  the  formula  for the equation  of equal altitudes is
simply:
x_-  A/ utang (+B - -/ tang.         (A)
Differentiating  the  two  formulae (a), taking  5  as constant, we find:
*) We find this also, if we differentiate the original equation for sin h,
cdt
taking 8 and t as variable, since we have x = -    a 8.'*) Since the change of the declination at apparent noon is to be used,
we ought to take the arithmetical mean of the first differences of the declination, preceding and following the day of observation. Instead of this
the almanacs give the quantity p.




'282
clh  - cos Ad  - cos - sinl Adl 
1dh'= --  os A'l, - cos r9 sin A'd/t.
In these equations dt' has been taken  eqtual to dt, since
we can  suppose, that the error committed in takinog the time
of the observation is united with thle errors of the altitudes.
Since we have now A           -  A', we obtain:
d/l = - cos A'dcl, -H cos rp sin A'dt,
dlh' = - cos A' dip - cos cp sin A' dclt
andl:
(dl h- dlh')
cos Tp sin A'
Therefore  we  see, that we must observe the heavenly
body at the time, when its azimuth is as nearly as possible
90" and  -  90"
In  1822 Oct. 8 Dr. Westphal observed  at Cairo the following equal altitudes of the sun:
I)onble the altit(ele of)   Chronometer- time
(Lower limb)      forenoon         afternoon          Mean
730 0'        21_h 7mn 27s      2h 33"', 5 9s  23115011 43s 0
20              8 24          33  3                 43. 5
40              9 23          32  a                 44.0
74  0              10 18          31   9                43.5
20             11  16         30 12                 44.0
40             12 1 1         29 14                 42.
75  0              13 11          28 13                 42.0
20             14   9         27 15                 42.0
40             15 10          26 15                 42.5
76  0              16  6          25 20                 43.0
Hence  we find  for the  arithmetical mean of all observations:
231 50"' 43s. 00.
Now  half the interval  between  the first observation in
the forenoon  and the last in the afternoon is 2"1 43"' 16   l and
that between  the last observation  in  the  forenoon and the
first in the afternoon  2t 341n 37s, hence we take:
-- 2h 382 i 586'. 5 = 211. 649.
If we compute with this A  and B, wve find:
log r  0.42308            0.42308
cosec r  0.19435   cotang, 0.08028
Compl. log 720  7.14267             7.14267
log A  7.7601       log B  7.6460,




283
and as:
-- — 6~7', p' —300 4'
and:
log G- 3 ~ 4391,,,
we obtain:, --- + 1Os. 46.
Therefore the sun was on the meridian or it was apparent noon at the chronometer-time 23'1 50" 53s. 46. Now since
the equation of time was -  121 33s. 18, the sun was on the
meridian at 23h 47126s. 82 mean time, and hence the error
of the chronometer was:
- 3111 2Gs. 64.
If we compute the differential equ~ation and express d t
in seconds of time, we find:
d t  - Os. 048 (cdh' - dl),
and we see, that if an error of 10" was comlnitted in taking
an altitude, the value of the error of the clock would be
0s. 48 wrong.
We can make use of this differential formula in computing the small correction, which must be added to the
arithmetical mean of the times, if the altitudes taken before
and after noon were not exactly but only nearly equal.  For
if h and h' are the altitudes taken before and after noon and
we take h' -       dh dh', we ought to Rapply to h' the correction - dh', and hence the correction of U is:
cl UI30 cos 59 sin A'
dh' cos hA'
30 cos f~ cos A sill t'
In  case that the greatest accuracy is required, sucth a
correction is necessary even if equal altitudes have been taken.
For although the mean refraction is the same for equal apparent altitudes, yet this is not the case with the true refraction, unless the indications of the meteorological instruments
be accidentally the same. Therefore if o is the refraction for
the observation in the forenoon,  -+ cl-, that in the afternoon, the heavenly body has been observed in the afternoon
at a true altitude which is too small by do), and hence we
must add to U the correction:
IUdU=-       q d cos h
30 cos 5 cos c sin t




284
13.  Often the weather does not admit of taking equal
altitudes in the forenoon and afternoon.   But if' we have
obtained  equal altitudes in the afternoon of one day and in
the forenoon of the following day, we can find by them  the
time of midnight.  The expression for the equation of equal
altitudes in this case is of course different.
If' T is half the interval between the observations, the
honr angles are:
- 12, - 7'
and:                          -- I2h -- 7
The case is now  the same as before only with this difference, that if A5r is positive, the sun has the greater declination when the hour angle is -- r, hence the correction, must be taken with the opposite sign and we have in this
case:.(TT
x-j (    tang..- - -tang 5')
s720  sitl  tang       /
=, - ( 2h~ -   tang 7r) -.I2. -  tang 
720   sin,     tango a
If we write instead of' it:
ux  121.... _  T..tang   tang
720    IC   sin r         tang T t
we can use the same tables as before; but besides, the quantity  2_, — must be tabulated, the argument being T or half'
the interval between the observations. This quantity in Warnstorff's Htilfstafeln  is denoted by f, hence we have for the
correction in this case:
x =fuc [A tang 90 - B tang,].
In 1810 Sept. 17 and 18 v. Zach observed at Marseilles
equal altitudes of the sun.  Half the interval of time was
10l 55'1  and as:
l01 5511,' = + 2~ 14' 1lG, t = 430 17' 50"
and:                      log  = -3.4453,,.
We find:
log A = 7.7305  log B - 7.7128,
logf- 1.0033,
ufl1 tang -  - 142s. 33
— tfJ'B tang    -. 67,
hence for the correction:
x -- 136s. 66.




285
ZNote L.  The eqluation for equal altitudes is expressed in apparent solar
time.  If now for these observations a clock adjusted to mean time is used,
we may assume the equation to be expressed in mean time without any
further correction.  But if we use a chronometer adjusted to sidereal time,
366
we must multiply the correction by 35G   a fraction whose logarithm is 0.0012.
Note 2. If the hour angle: is so small, that we may use the arc instead of the sine and the tangent, the equation of equal altitudes becomes:
x —= ~-(  [tang   -- tang 8].
But as the unit of T in the numerator is not the same as in the denonzinator, being in the first case one hour-, in the other the radius or unity,
we nmust multiply the second member of the equation by 206265 and divide
it by 15 X 3600.  Thus we obtain:
= -- S    -- [tang T - tang ],
where now  x is the equation of time for z-  0.  But in this case the two
altitudes are only one, namely the greatest altitude, and hence x is the correction, which must be applied to the time of the greatest altitude in order
to find the time of culmination.
The same expression was found already in No. 8 for the redaction of
circum - meridian altitudes.
14.  If the altitudes  of two  heavenly  bodies have been
observed  as well as the  interval of time  between  the  two
observations, we  can  find the time and  the  latitude  at the
same time.  In  this case we have the two  equations:
sin  -sin p sill  +    s  cos  cos  cos t,
sin lh' - sin 95 sin 3' - cos T' cos S' cos t.
If thern u and  tu' are the clock-times of the first and second  observation, Auz the  error of' the clock on sidereal time,
we have:*)
t -—' -1- Au - a
t'  -'4-' u - a',
where  Au  has  been  taken  the  same  for both  observations,
because the rate of the  clock  must be known  and hence we
can suppose one of the observations to be corrected on account
of it. Then is
*) If the sun is observed and a mean time clock is used, we have, denoting the equation of time for both observations by w and wo':
t         i=    - + %    U- 5U,
t ='- tl- A u - W
hence: A -t u'- iu - (w'- so).




286
a known quantity and we have t'- t +-.  Hence the two
equations contain only the two unknown quantities T~ and t,
which  can  be  found  by  means of them.  For this purpose
we express the three quantities
sin T9 cos q sin t and cos T cos t
by  the parallactic angle, since we have in the triangle between the pole, the zenith and the star:
sin *- = sin h sin 8 + cos h cos 8 cos p,
cos 9 sin t  cos h sin p,                       (a)
cos  Cost -ssin hCo Cos-OS hi sinll Cos p.
Substituting these expressions in the equation for sin h',
we find:
sin h'  [sin 8 sin C' 4- cos 8 cos 8' cos )] sin I/
-+ -cos a sin 5' - sin S cos 3' cos e] cos h cos p
-  COs S sin 2. cos h sin p.
But in the triangle between the two stars and the pole,
denoting  the  distance  of the  stars by D, and the angles at
the stars by s and s', we have:
cos D = sin 6 sin' +- cos S cos 6' cos
sin D cos s - cos S sin S'- sin 6 cos' cos C  (o)
sin D sin s = cos S sin 2,
hence, if we substitute these expressions in the equation for
sin h':
sin h' = cos D sin it - sin D cos /i cos (s -+ p),
sin It' - cos D sin h
hence   os (s — p) =. e s h          (c)
Further if we substitute in
sin It = sin T sin S A- cos   n cos S cos (t' -)
the expressions for sin %(q, cos fq sin t' and cos (q cos t', which
we derive from  the triangle between the pole, the zenith and
the second star, we easily find:
sin h - cos L sin ih'
cos (s' p- ) -   si                       (cl)
sin D cos h''
After the angles p and p' have thus been found by means
of the equations (b) and (c) or (d), the equations (a) or the
corresponding equations for sin t, cos 5p sin t' and cos p cos t'
give finally cpj and t or rp and t'.
The equations (b) give for D  and s the sine and cosine,
the  same is the  case with  the  equations  (ca) for  r., and t,
hence there can never be any doubt, in what quadrant these




287
angles lie.  But the equations (c) and (d) give only the cosine of s - p and s' -  p'; however we have  in the triangle
between the zenith and both  stars:
sill D sin (s -4- ) = cos i' sin (A'- A)
and sin D sin (s' - p')  cos hi sin (A' - A),
hence  we see that sin (s — +p) and sin (s' -  p') have always
the same sign as sin (A'-  A), so that also in this case there
can  never be any  clolbt as to  the quadrant, in which the
angles lie.
The  formulae  (a) and  (b) can  be made  more  convenient by  introducing  auxiliary angles, and  the  formula for
cos (s -- p)  can  be  transformed  into  another  formula  for
tang I (s -+ p)'2 in the same way as in No. 4 of this section.
Thus we obtain the following system  of equations:
sin 1' = sinJl' sin F
cos 8' cos 2- sin fcos F             (e)
cos (' sin  = c.os/,
cos D   sinf cos (F- 6)
sin D cos s   sinJf sin (F —)        (f)
sin D sin s -cosf,
2     cos S. sin (S - 1')
cos (S-D) sin (S- h)'
where S = ~ (D +- h -- h'),
sin g sill G   sin sin 
sin g cos G   cos h cos p       (7,)
cos g       e Cos I sin p,
sinll = sin g cos (G G-  )
cos? sin t  cos                (i)
COs 99 COS t   Sill g sin (G- c).
The Gaussian  formulae  may also be used in  this case.
For -first we have in the triangle between the pole and the
two stars, the sides being D, 90" -  5 and 90"    5' and the
opposite angles 2, s' and s:
sin 2 D. sin. (s' - s) = sin - (- (-) cos  2
sin  D. cos (s')           (       sin         (A)
c  I D  sin -.,I           12      (A)
cos' D. sin- - (s' -t  s)- cos 2 (a'-  ) cos 2  (
COs - D. cos' (s'- s) -- sin' ('+ 8) sin  A 2.
Then we have as before:
tang 2} (s+p)2  _ os S. sin (S - h')-'. (B)
cos S. sin (S -    h)
01 tang 2 (S+  ) -  COS (S -D) sin (S - I)
r tang 4'   cos (S - D) sinl (S - I')




288
Finally we have in the triangle between the zenith, the
pole and the star:
sin (45 -4  99) sin ~ (A +- t) = sin  p Cos. (A -+- )
sin (450 -4) cos 2 (A + t)  cos 2 p sin 7 (   A)  (C)
cos (450 -2 -) sin  (A - t)-= sin - p sin i2 (h - A)
cos (4L50 -' 99) coos  (A - t)  cos * p cos 4 ( -  ),
In case that the other triangle is used, we have similar
equations, in which A', t', p', h' and  )' occur.
Since we find by these formulae also the azimuth, we
have this advantage, that in case the observations have been
made with an altitude and azimuth instrument and the readings
of the azimuth circle have been taken at the same time, the
comparison of these readings with the computed values of
the azimuths gives the zero of the azimuth, which it may
be desirable to know for other observations.
Example. Westphal in 1822 Oct. 29 at Benisuef in Egypt
observed the following altitudes of the centre of the sun:
— 20h1 4S, 48s    1 -- 370 6 5'59". 6
u'  23  7 17    h' — 50 40 55.3,
where u' is already  corrected for the rate of the clock and
h and h' are the true altitudes.  The interval of time converted into  apparent time gives i.    211 18- 28s.66 =  340 37'
9"3.90 and the declination of the sun was for the two observations:
A — 100 10' 50".1 and' —- 100 12' 57".8.
From these data we find by means of the Gaussian formulae:
D    34 o 3' 20". 27
s —  93 12 58.26
s'=   93  6 1.93
Further: s+p=    53 15 41.26
hence:     p —39 57 17.00
and then:  9o=   29  5 39.80
t - -35  24 59.23
A —46 1952.1 7.
It is advisable to compute (p and t' also from the other
triangle as, a verification of the computation, since the values
of (p, must be the same and t'- t = -2.
Now  in order to see, what stars we must select so as
to find the best results by this method, we must resort to
the two differential equations:




289
dh =- cos Adqp - cos w sin Adt
dh'= -  cos A'dT - cos qT sin A'dt
where d t has been supposed to be the same in both equations, because the difference of d t and dt' may be transferred to the error of the altitude.  From  these equations
we obtain, eliminating either d p or d t:
cos A'         cos A
sin (A' - A)   sin (A' - A)
sin A'         sin A   dh'.
sin (A'- A)    sin (A    A) dh'
Hence we see, that if the errors of observation  shall
have no great influence on the values of rf and t, we must
select the stars so that A'- A is as nearly as possible -4- 90",
since, if this condition is fulfilled, we have:
cos Tdt =    cos A'd h - cos A d h'
dl --  sin A'dh -i- sin Adh'.
Then we see, that if A' is -4- 900 and therefore A is 00,
the coefficient of dh in the first equation is 0, that of d h'
equal to -- 1; hence the accuracy of the time depends principally on the altitude taken near the prime vertical. In the
same way we find from  the second equation, that the accuracy of the latitude depends principally on the altitude taken
near the meridian.  For the above example we have, since
A'        1~ 15':
d -9 =  q 0.0308 cl -- 1.0215 d h'
dt -+- 0.1077 dl - 0.0744 dh'.
15. The problem can be greatly simplified, for instance,
by observing the same star twice. Then the declination being
the same and s' =  s, the formulae (A) of the preceding No.
are changed into:
sin  L)D = cos 8 sin  
cos - D sin s - cos 
cos - D cos s = sin a sill  2i.
By means of these we find D and s, and then from  the first
of the equation (B) and the equations (C) pi and t and, if it
should be desirable, A.
In this case we can solve the problem  also in the following way. We find from the formulae:
sin hI = sin qT sin 8 -+- cos n cos S cos /
sin hi' = sin p sin    c -o- os  cos a cos (t i- +)
19




290
by  adding and subtracting them:
cos  sin. cos  sin (t 4 2' )  cos - 4 (h-t-') sin - (h-h')
sin gosin St cos Scos    oscos(t -- 9 2) = sin 4(h-+-h') os' (h-h').  (a)
Therefore if we put:
sin  -- cos b cos B
cos S cos -  -= cos b sin B      (A)
cos 8 Sill' —  sin 1b,
the second of the equations (a) is changed into:
si h 3- h') cos I (/2 - h,')
sin   cos B     cos   cos(t+' 2) sin B    (')  s   (ncos A
and if we finally take:
sin = cos Fcos G
cos R sin (t + -.') = sin G         (B)
COS T COS (t- +2 ) -sin Fe ns G,
we obtain:
os  (a + — a') Silln  (1 - h')
sin G --          sinb
sin b
sin  (h  h') cos i-(h - h')   (
cos G. cos (B -.F) =   -_       -
cos b
Therefore if we first compute the
Fig. 8.         equations (A), we find  G  and F by
means of the equations (C) and then
q~ and t fromz the equations (B). The
geometrical signification  of the  auxiliary  angles  is easily  discovered  by
means of Fig. 8, where PQ  is drawn
perpendicular to the great circle joining the two stars, and Z M  is perpeno %    dclicular to  P Q.   We then  see, that
b  Q S= D, B- P Q, F= PM and
G --   M.
If we use the same data  as in  the  preceding  example,
paying  no  attention  to  the  change  of the  declination  and
taking     --     100 12' 57".8,  we find:
B -- 100~ 41' 23". 1  sin b = 9.466600  cos b = — 9.980534
sin G -  9.432863.    cos G - 9.983445       F= 41~ 1' 53". 3
and hence t = - 350 22'21".0    9-  2 90 5' 42". 7.
In case that the two altitudes are equal, the formulae
(A) or (e) and (f) in No. 14 remain unchanged, but the fornmlae (B) are transformed  into:
= enstgco (h         D)
tallg I (S -1)2 = tang  ('-h p'
cos (h - 5D)




291
and then p being known, (* and t can be computed by means
of the formulae (h) and (i), or %p, t and A by means  of the
formulae (C).
16.  A  similar problem, though  not strictly belonging
to the class of problems we have under consideration at present, is the following: To find the time and the latitude and
at the same time the altitude and the azimuth of the stars
by the differences of their altitudes and azimuths and the
interval of time between the observations.
In this case we must comnpute as before the formulae (A)
in No. 14.
Then we have in the triangle  between the zenith and
both stars, denoting the angles at the two stars by q and q',
the third angle being A'   A and the opposite sides 900-h',
90~- h and D:
cos 2 (hl'- h) cos - (A'-A)
sinll  (q' + q)   S,    
cos  D
Sin  (h'- 1) COS  (A'-A)
sin              2 (q-q)=. -             (B)
sin 4 D
tang I (lh'-+)= cos  (q' +-qg)cotang D.)
cos (q' -q) - 
By meanls of' these equations we find  (h+- h'), thence h
and h' and the angles q and q'.  But since we have according to No. 14 q-  s --- p and q'-  s'-  p', we thus know  p
and p', hence we can compute Tq, t and A by imeans of the
formulae (C) in No. 14 and as a verification of the computation also (f, t' and A'.
In  this case the differential equations are according to
No. 8 of the first section:
dh      os A d - cos S sin p. d -2 — cos  sinl  - 22                2
t' -t- t          - t
dl h' —cosA'cl - cos' sinp'. d.- o- — C    sin p'd 
s 2
cos  cos P  t'-+t  cos  cos    t'-t
dA   - sin A tang h dc  -- cos  d               cos  
cos h    2       cos h     2.,,    cos'cos p' -t'-t  cosY'cosp' t'-t
dA=-sin A' tang    + Aadg              -  o s+ c'    2t —J -
cos h'           co2  s ii'    2
t'+t  t' - t     ti -i- t' - t
where  -t'+t -   anad   - 2+t    -t have been put in place of
2      2         2       2
t' and t occurring in the original formulae.
19 *




292
Subtracting  the first equation from  the second and the
third  from  the fourth, then eliminating first d  2  and then
d,, and remembering that we have:
cos S sinp - cos T9 sin A
cos S cosp
Cos Cs -- sin 9 q- COs 9T tang A cos A
Cos it
we easily find:
A11d =- [tang A cos A - tang t' cos A'] d (h' - h) -+- [sinA - sin A'] d (A' -A)
Fcos 8      S     ens 8     A ]  ('
~Lcos h cosp sin A' —-   cosp' sinA  d('-,
l' ___ t
l/[cos d -— 2-= [tang it sin A- tang h'sin A'] d(h' — )-[cos A- cos A'] d(Al'-A)
-+ [eos  (tgh - tg It') sin2' (A'-i- A)-i- sin 9 (cos A - cos A')] d(t'- t),
where M=  2 [tg t +- tg h') sin2 2 (A'- A).
We see from  this, that it is necessary to select stars for
which the differences of the altitudes and the azimuths are
great, in order that M  be as great as possible. If   (A' -  A)
900, even the coefficient of d (h' -  h) is less than 1.
v. Camphausen has proposed to observe the stars at the
time,  when their altitude is equal to their declination, because then the triangle between the zenith, the pole and the
star is an isosceles triangle and we have t=  180" — A and:
cotg 8 cos t =    cotg S' cos t' = tg (45 -   9T)
- cotg 8 cos A   - cotg 8' cos A'= tg (45 —' T),
by means of which we find:
sin (S -')
tang  ( t'  t) -- sin (,) cotg  ( - t)
or
tang 2 (A'+-A)-  sin (         (A'-A).
sin (8 q- 3')
From  these formulae we obtain t'-+  t or A' -+A  and q.
But since the altitudes are hardly ever taken exactly at the
moment, when they are equal to the declination, the observed
quantities t' - t and A' -A must first be reduced to that
moment.  (Compare Encke, Ueber die Erweiterung des Douwes'schen Problems in the Berlin Jahrbuch for 1859.)
Example.  In  1856 March 30 the following differences
of the altitudes and the azimuths of r Ursae majoris and a
Aurigae were observed at Cologne.




293' -   - - 40 10' 46".0
A'-A  226  2S' 9".9
The interval of time between the observations, expressed in sidereal time,
was Oh 18m 8.70.
The apparent places of the stars were on that day:
-   Ursae majoris a = 13h41m54s. 53  -   +-* 500 1' 45". 9
a Aurigae    ac'   5  6  1.69   8'-=  - 45 5   1.7.
Hence we get X =  1330 30' 23". 1, and we obtain first by
means of the formulae (A) in No. 14:
s --- 31o 22' 33". 1S
s'= -+- 280 41' 50". 20    D — 760 0' 14". 79.
Then we find from  the formulae (B) q'    -  28" 40' 53".44,
q     -- 31~21' 32".80, and since q'= s' -p', q =  s -p   we
find p    -    62" 44' 5". 98, p'- =  - 570 22' 43". 64.  Since we
find - (h' -+ h) -  47" 56' 40". 61, and hence h -  50" 2' 3". 61,
we get by means of the equations (C) in No. 14: q =  50"
55' 55". 57, t =  295~ 2' 56". 70, A =  2440 57' 48". 50.
If we compute also the differential equations we find, if
we express all errors in seconds of arc:
cd    - 0.0342 d (h' - h) - 0.4892 d (A'- A) - 0.2438 d (t'- t)
d --- - - O.S621  d (h' - h)-+ 0.0244 d (A'- A) - 0.0188 d (t'- t).
17.  The method  of finding the latitude and the time
by two  altitudes it often used at sea.  But sailors do not
solve the problem in the direct way which was shown before,
because the computation is too complicate, but they make
use of an indirect method which was proposed by Douwes,
a Dutch seaman.
Since the latitude is always approximately known from
the log-book, they first find an approximate time by the altitude most distant from the meridian, and with this they find
the latitude by the altitude taken near the meridian.  Then
they repeat with this value of the latitude the computation
for finding the time by the first altitude.
Supposing again that the same heavenly body has been
observed twice, we have:
sin h - sin h' - cos T cos a [cos t - cos (t- +  )]
- 2 cos T cos 8 sin (t +-  i) sin' i,
hence:
2 sin (t - --    see T see a cosee' i [sin h - sin A']




294
or, if we write the formula logarithmically:
log. 2 sin (t —'-) — logsec  +- logsec 8-+ log  [sin h - sin h'] - logcosec'.1 (A)
Since an approximate value of qc is known, we find from
this equation t -+  X, and hence also t, and then we find a more
correct latitude by the altitude taken near the meridian by
means of the formula:
cos (-8)  sin h' - cos 9 cos 8. 2 sin I (t +  ) )2 (B)
If the result differs much from the first value of the
latitude, the formulae (A) and (B) must be computed a second
time with the new value of (p.
Douwes has constructed tables for simplifying this computation, which have been published in the,,Tables requisite
to be used with the nautical ephemeris for finding the latitude and longitude at sea" and in all works on navigation.
One table with the heading,,log. half elapsed time" gives the
value of log. cosec  2., -the argument being the hour angle ex —
pressed in time. Another table with the heading,log. middle
time" gives the value of log 2 sin (t --  ), and a third table
with the heading,,log. rising time" gives that of log 2 sin - t2.
The quantity log. sec q! sec' is called log. ratio and we
have therefore according to the equation (A):
Log. middle time = Log. ratio + Log (sin h - sin h')
+ Log. half elapsed time.
By means of the table for middle time we find from
this logarithm immediately t. Then we take from the tables
log. rising  time for the hour angle t -  A  subtract from
it log. ratio and add the number corresponding to it to the
sine of the greater one of the altitudes. Thus we obtain the
sine of the meridian altitude and hence also the latitude.
If we cannot use these tables, we compute:
sin [t  ]  cos 2 (Ih -- h') sin  - (h -  h')
sin  [t - j 2 i]   =     *   ]
cos 99 COS 0 sin 2 A
and:
sin h'
cos (99 - N) - sn 
M
where:                    sin 8 l  sin N
cos 8 cos t -- M cos N.
If we compute the example given in No. 14 according
to Donwests method, we find:
9  29~ 0'




295
log ratio   0. 06512
log (sin h - sin I')   9. 20049,,
log half elapsed time   0. 52645
log middle time   9. 79206,,
t —- -2hlm. 4
hence t'   Oh 211.9
log rising time   5. 90340
log ratio   0. 06512
-+- O. 00007
sin it' -- 0. 77364
cos (, ) —-=   9. 88858
- - 8-=    390 1S'.7
= —-  29  5.7.
In  case that the observations are made at sea, the two
altitudes are taken at two different places on account of the
motion of the ship during the interval of' time between the
observations.  But since the velocity of the motion is known
from  the log and the direction of the course from the needle,
it is very easy to reduce the altitudes to the same place of
observation.
Fig. 9.          The ship at the time of the first ob_,s servation shall be in A (Fig. 9) and at the
time of the second in B.  If we imagine
then a straight line drawn from the centre
/   of the earth to the heavenly body, which
intersects the surface of the earth in S',
then  the side B S' in the  triangle ABS'
will be the zenith distance taken at the place B, and since
BA  is known, we could find, if the angle S'BA were known,
the side A S', that is, the zenith distance which would have been
taken at the place A.  Therefore at the time of the second
observation the azimuth of the object, that is, the angle S'B C
must be observed, and since the angle CBA, which the direction of the course of the ship makes with the meridian,
is known, the angle S'BA  is known also.  Denoting this
angle by a and the distance between the two places A and
B by A, we have:
sin ho0 - sin h cos A ~- sin A cos h cos ea,
where ho is the reduced altitude. If we write instead of this:
sin ho - sin h +- sin A cos h cos a - 2 sin -A2 sin h,




296
and take A instead of sin A, we obtain by means of the formula (20) of the introduction:
h = h o —Acos a -'2 tangh,
where the last term  can in most cases be neglected.
18. If three altitudes of the same star have been observed, we have the three equations:
sin h = sin 99 sin 8 + cos 99 cos 8 cos t
sin h' - sin 9 sin  l- cos 9 cos 8 cos (t -+ v)
sin =" — sin y sin  + cos T cos 8 cos (t +-'),
from  which  we can  find q, t and J.  For if we introduce
the following auxiliary quantities:
x - cos 9 cos  cos t
= cos 99 cos  sin t
z- sin 9T sin 3,
those three formulae are transformed into:
sin It-z -+- x
sin h' =z +- x cos  — y sin,
sin h"  z +- x cos'- y sin 2',
from which we can obtain the three unknown quantities x,
y and z in the usual way. But when these are known, we
find T and t by the equations:
tang t --
x
sin 99 sin 3 - z
COS ~ COS 8 — VXI + y2
This method would be one of the most convenient and
useful, since no further data are required for computing the
quantities sought*). But it is not practical, since the errors of
observation  have a very great effect on the unknown quantities.  But if we do not consider b as constant, that is, if
we observe three different stars, whose declinations are known,
at equal altitudes, the problem  is at once very elegant and
useful.
19. In this case the three equations are:
sin h sin h 9 sin 99   +- cos 9o cos 3 cos t
sin h  sin 9 sin -- cos 99cos' cos (t -+- )    (a)
sin h - sin 99 sin 3"-  cos 99 cos 8" cos (t -+'),
where 2 = (' -  ) - (a' -a)
and    (i"-  ) - (a"a).
) Since three altitudes of the same star have been taken, 2 and;' are
not dependent on the right ascension.




297
If we now  introduce in the two first equations   (`'-+A)
-+  2 (8 -  X') instead of d, and  - (6 ~  3') -  --' (-      ) instead
of (', and subtract the second equation from  the first, we get:
- 9sin T sin')cos     4 (+')- cosf cos t [cos   ( ~+') cos I (( -3')
-sin. (0i —') sin. ((-')]
-cos T cos (t +- ) [cos    - (3' ) cos  - (-3') +- sin - (  3- 8') sin ~ ( -3')]
or:
0 =  Si  sin   (3- 3') cos -, ( + 3')
+- cos T cos E (o  + 3') cos  (-?') sin 2 2 sin (t + -4 2)  
I(a)2 (a)
- cos T sin - (3 3-') sin ~2 (3-3') cos  2 cos(t + 4-).
From  this we find:
tang T9   - sin 2. sin (t + -.) cotang   ( 3-3')
-+- cos 2. cos (t - 2 i) tang 2 ( -+ S').
Introducing  now  the auxiliary quantities A' and B', given
by the formulae:
sin E. cotang 2 (3-')    A' sin B'
cos' 2.  tang  (+' -  )A' cos B'         (A)
B' + -  2 = C',
we obtain:
tang g = A' cos (t - C').          (B)
From   the  first and  third  of the equations (a) we find
in the same way similar equations:
sin 4 2' cotang l (3-  ") - A" sin B"
cos:t 2'  tang'(8-4-") - A" cos B"       (C)
B" + 4 2'    -  C",
tang 9- = A" cos (t ~- C").        (D)
Furthermore we find from  the two formulae (B) and (D):
A' cos (t + C') - A" cos (t + C").
In  order to  find  t from  this  equation,  we will write
it in this way:
A' cos [t + H-+- C' - HI = A" cos [t    H-4- C" - H],
where Ht is an arbitrary  angle, and from  this we easily get:
A' cos (C'-  I) -- A" cos (C"- H)
tang(t ~ H) — A' sin (C' - H) - A" sin (C" - H)
For H we can substitute such a value as gives the formula  the  most convenient form, for instance  0, C' or C".
But we obtain the most elegant form, if we take:
H=-  (C'~ -  C")
for then we have:
tang [t  t- (C' +      C")]      cotang ~  (C'l- C").
2                   cotang W',  -  C"),




298
Introducing  now  an  auxiliary  angle g,  given  by  the
equation:
All
tang    A',              (E)
we find:
A' - A"   1- tang     tang (450
A' l  --   1+- tang 5
hence:
tang It +- ~ (C'+ C')] = tang (450 -) cotang - (C' - C").    (F)
We find therefore first by means of the equations (A)
and (C) the values of the auxiliary quantities A', B', C' and
A", B", C"; then we obtain t by means of the equations (E)
and (F), and finally j:, by either of the equations (B) or (D).
It is not necessary to know  the altitude itself; in order to
find  [f and t, but if we substitute their values in the original equations (a), we find the value of h; hence, if the altitude itself is observed, we can obtain the error of the instrument.
In order to see, how the three stars should be selected
so as to give the most accurate result, we must consider
the differential equations. Since the three altitudes are equal,
we can assume also d h to be the same for the three altitudes, uniting the errors, which may have been committed in
taking the altitudes, with those of the times of observation.
Now since we have:
t = u +-    u - a,
the error d t will we composed of two errors, first of the
error d (Au), thas is, that of the error of the clock, which
may be assumed to be the same for the three observations,
since we suppose the rate of the clock to be known, and
then of the error of the time of observation du which will
be different for the three observations. Hence the three differential equations are:
dclh  - cos A dcl - cos q sin A dc  - cos R sin A d(Au)
dlh  - cos A'dv - cos T sin A'd' - cos q9 sin A' d( u)
dh  - cos A"dcl- cos 9v sin A"du" - cos q' sin A"cd( u).
If we subtract the first two equations from each other,
we find by a simple reduction:




299
A -+-A'           A q- A'               cos  sin A
0- =2 sin   2   d- 2 cos ---   cos Rd(Au)- cos  sin    d;2                2                     A -A'
sin -— 2. —
sin.......sin
2
and in the same way front the first and third equation:
0 A i - A"   2    A  - A"  cos -T sin A
2             02 sin  - - 2 cs  s cd( a) -    A-A"d
sin  2.
2
cos q sin A" d"
A- A" d u"
sin   -
2
From   these  two  equations  we  obtain, eliminating  first
d (L\) and then  d(r:
A'-+A"                      A -4- A"
cos  sin A.s 2               cos g sin A' cos
dT.-  A-A'   A          d. A'-A       A'   "
2 sin       sin    2        2 sin       sin —-9
2         2                 2          2
A -+ A'
cos T sin A" cos 2 —+    A"-A   A"-A' d- "
2 sin   --  sin
2         2
and:
A'-+-A"                     A  - A"
sin A. sin                 sin A' sin
2                           2
A - A''  A -- A"d  + -           --   -A' — A'
2 sin  -    sin A             sin A-  -  -A sin -.
2         2                 2          2
A ~- A'
sin A" sin — +2
+    A-A          A-A'd"
2 sin --  sin
2         2
We  see  fiom  this, that the  stars must be selected  so,
that the differences of the azimuths  of any two of' them  become as great as possible, and hence as nearly as possible equal
to  1200, because  in this case the denominators of the differential coefficients are as great as possible %).
Example. In 1822 Oct. 5 Dr. Westphal observed at
Cairo the following three stars at equal altitudes:
a Ursae minoris at 8h 28m17S
a Herculis            31 21 West of the Meridian
a Arietis   -         47 30 East of the Meridian.
*) This solution of the problem was given by Gauss in Zach's Monatliche Correspondenz Band XVIII pag. 277,




300
The places of the stars were on that day:
a Ursae minoris Oh 58"n 14s. 10 -+ 880 21' 54". 3
a Herculis     17  6 34.26   14 36  2.0
a Arietis       1 57  14.00   22 37 22.7.
Now  we have:
u —  --.- 3mn 49 0'" --  --    -  19 1113s. 0
or expressed in sidereal time:
t'  u --    Oh 3m  4s. 0                  4-. Oh 191n 16s. 16
a' -- --   7 51  39.84         a"- a  -- 0 58 59.90
~ =    7h 5411m44s.34       Z' —   0 39m1143.74
=    118~ 41'  5". 10           =- 9~ 55' 56". 10.
Then we have:
( - 8' - 36" 52' 56". 15
(8 -+' )= 51 28 58. 15
(6 - 6")  32 52 15. 80
( (6 4- ") -=5 29 38. 50,
and from this we obtain:
logA'- 0.1183684            log A"       0.1629829
B'  - 60 48' 11".92         B"   -   50 16' 52". 22
C' -120   8 44.47           "  — 10  14 50.27
1 (C' + C") -    540 56' 57". 10
-l(C' -C")=    65 11 47.37
-=    47 56 16.08
t   -- 56~ 18' 28". 09
--   3 45m 13s. 87
t + C' —   630 50' 16". 38
C" — - 66 33 18.36
and the formulae (B) and (D) give the same value of if:
1 = 300 4' 23". 72.
From t we find the sidereal time:
O = 21h 13m Os. 23,
and since the sidereal time at mean noon was 12h 54m 20.04,
we find the mean time 8h177m36%.44, hence the error of the
chronometer:
lzu --- - 10' 40s. 56.
Computing h from one of the three equations (a) we get:
h = 300 58' 14". 44,
and for the other two hour angles we find:
t'-  620 22' 37". 01
t"  -66 14 24.19.
We then are able to compute the three azimuths:




301
A -- 181~ 35'.2
A'-  89 33. 2
A"-279 50.4;
and finally the three differential equations:
d-  -  0. 329 du -5. 739 du'- 6. 068 du",
d(GAu)  -  0.0018 duz + 0. 468 du' - 0 396 du",
where dcj is expressed in seconds of are, whilst d (Au) and
dut, du', du" are expressed in seconds of' time.
20. Cagnoli has given in his Trigonometry another solution, not of the problem  we have here under consideration,
but of a similar one.  His formulae can be immediately applied to this case, and if it is required, to find the altitude
Fig. 10.               itself besides the latitude and, the time, they are even a little
more convenient.
Let S, S' and S'! (Fig. 10)
be the three stars which are
observed.   In  the  triangle
between the zenith, the pole
and  the  star we have  then
SI' according to Gauss's  or Napier's formulae, denoting the
parallactic angle by p:
tang 4 (p +   - h)   ~       cotang (450 -    )
cos. (t -q P)
cos - (t +P-) tang (450   S)
sin  (t(-p)                     (A)
and:        tang, (1)=-  nz)  - sitang (45t       4 S)
sin  (t -F tAn
sineI (t - p)
-sin s   (t —p) cotang (450+ - 4:
sin  ( (t -Ap)
But in the triangles PS S', P S'S" and PS S" we have also
according to Napier's formulae, putting for the sake of brevity
A  =  [PS "'S'- PS'S"]
A' =  ~[PS"S -PSS"]
A"-.' [PS' S - PSs']:
tang A -— Z sin 2   )cotang 4 ()2'-)  /
t   Cosng A =   o4(3"-k-r' c
tang A'l  (-_ — 3 cotang 2               (B)
CoS 2 (1" 2 +)
tang A"  io 2~   +   )ot  ang2




302
where A and iv' have the salie signification as before.  Now
since we have:
p -.- PS S' - PS'&S - p'
p' - PS' S"    P S" S' — p"
-4- PS S" -- P S" S — p",
we easily find, that:   p   - A' + A" — A
p'  A +A"-A'                 (C
p- "A +- A'- A".
But we also have:
sin t: sin p- cos h: cos 90
sin (t+): sinllp'  cos h: cos 9,
hence:
sin t: sin (t~ 2 ) -- sin p: sin p'
or:
sin t + sin (t + -)   sin [A'+- A"- A] + sin [A +- A"- A']
sin t - sin (t -+ 2)  sin [A'-+- A" - A] - sin [A +- A"C-A']
Fromn this follows:
tang [t +   2j] cotang 4-2 = tang A" cotang (A - A')
or substituting  for tang A" its value taken from  the equations (B):   tang        ]   ('    cotang (A - A').   (D)
Therefore we first find from the equations (B) the values
of A, A' and A", then we find p and t by means of the equations (C) and (D), and then qf and h by means of the equations (A).  An inconvenience connected with these formulae
is the doubt in which we are left in regard to the quadrant
in which the several angles lied all being found by tangents.
However it is indifferent whether we take the angles 1800
wrong, only we must then take 180" -+- t instead of t, if we
should find for rp and h such values, that cos.p and sin h
have oppositive signs. Likewise if we find for (f and h values
greater than 90" we must take the supplement to 180" or to
the nearest multiple of 180". The latitude is north or south,
if sin (p and sin h have either the same sign or opposite signs.
If we compute the example given in No. 19 by means
of these formulae, we have:
E1-2    590 20'32" 55
~2' —=-  4 5758.05
2( S"-')- 40' 40". 35   ( -         3-   -  -  52' 15". 80
-I (a' —A)  -— 360 52' 56". 15
— (S"+') =183642.35   - ("+ 8) =-  55 29 38.50
(8'+8) -=    51 2858.15,




303
and from  this we find:
A —   2 2'1".33, A'  840 49' 4".07, A"  - 29 44' 1.16".52
A - A' —-- 860 51' 5".40
t+a —=       3  2 4.47
t — 56 18 28.08.
Then we find qp and h from  one of the triangles between
the pole, the zenith and one of the stars, and since in the
triangle formed by the first star small angles occur, we choose
the triangle formed by the second star, using the formulae:
cos I (t'  p')
tang  ( +-]- - coo  (t'-p') tang (450 +')
sin,- (t' — p')
tang 4- (p —~)  sin  4 ( t' = -  +) cotang (45 + -4-').
Now  we have:
t'  t + 2 — 62" 22' 37". 02
p' - A - A" - A' = 243 o 24' 38". 08,
therefore we find:
-= 300 4' 23". 73
hi-149  1 45.58
or taking for h the supplement to 180 0:
hi    30 58 14.42,
which  values almost entirely agree with those found in the
preceding No.
21.  We can also find Cagnoli's formulae by an analytical method. According to the fundamental formulae of spherical trigonometry  we  have  for each  of the three stars the
following three equations:
sin a --   sin il sinl   - cos 5 cos A cos t I
cos h sin p-   coS 5 sin t               (a)
cos h cosp   sin 5 cos - cos go sin 8 cos t
sin hs = sin 50 sin S'  cos o  cos 8' cos (t +))
cos h sin p' - cos 5 sin (t -+  )             (b)
cos h cos p'-= sin 9 cos 8' — cos T9 sin S' cos (t-1+.) 
sin h -- sin R sin 8"+ cos, cos 6" cos (t+-2')
cos Ih sinp"= cos5 9 sin (t --')             (c)
cos i cosp" — sin 90 cos I"- cos p sin 8" cos (t+2-')
If we subtract the  first of the  equations (b) from  the
first of the equations (a) and introduce 2 ('"   +) - +-   -'     S')
instead of', and 1 ('-   ) -    (~-  S') instead of'', we find
the equation (ce) in No. 19. By a similar process we deduce
fromn  the third of the equations (a) and (b):




304
cos h sin  (p'+p)   (    )   sin 2(p'p) — sin89 s in  )
cos g sini(n +) cos (8 -3) sin (t+- 2) sin - 
+ cos 9 cos 2 ('~-S) sin " ('-8) cos (t-+- 2) cos -,
and  if we  eliminate  sin My in this equation by means of the
equation (cr), multiplying the first by  cos 2(8' -- ()), the latter
by sin ~2 (Y'q- S), we obtain:
cos h cos 1 (8'+8) sin 4'+p) sin 4(p'-p)  cos'+  sin+) sin  ('- c)ossin ((t+-i-  ) cos  2. (d)
Now if we subtract the second equations (a) and (b),
we find:
cos h cos 4 (p'+-p) sinl  (p'-p)   cos 9 cos (t -+    i) sin 2,
and hence:
tang,1 (pt +p)     s     i   cotang' A - tang A".
We  can  find  similar formulae  by  combining  the  corresponding  equations (a) and (c) and (b) and (c), which we
can write down immediately on account of their symmetrical
form.
tang 2(P"+i)     2-("-+-) cotang   i2'- tang A'
tang(p" —p) 2   Cos  ")
and tang  (p)"+p')' —  o(,,,  cotang:('-) — tang A
If we add finally the second equations (a) and (b), we
find:
cos h sin  (p'+) cos 4 (p'- p) --- cos Tsin (t-+ - 2) cos 2,
and from this in connection with (d) we obtain:
tang (t     2) =   s         cotang  (p-p),
where 4 (p'-p) = A - A'.
When thus p and t for the first star are known, we can
compute  f  and  h by  means  of the  formulae found before,
which were derived by Napier's formulae:
tang  ( + - h) -  os  (t -- ) cotang (450   8)
cos- (t -  )
Cin q  --
tang 2 (R - h)=sn  (t- p tang (450 -    ).
sin  (t +- p)




305
IV. METHODS OF FINDING THE LATITUDE AND THE TIME
BY AZIMUTHS.
22. If we observe the clock-time, when a star, whose
place is known, has a certain azimuth, we can find the error
of the clock, if the latitude is known, because we can compute the hour angle of the star from  its declination, its azilnuth and the latitude. If we-take the observation, when the
star is on the meridian, it is not necessary to know the declination nor the latitude; at the same time, the change of the
azimuth being at its maximum, the observation can be made
with greater accuracy than at other times.
If we differentiate the equation:
cotang A sin t  - cos q) tang c -I- sin yT cos t,
we obtain according to the third formula (11) in No. 9 of
the introduction:
cos h dA  - sin A sin h dcp -1 cos  n cos p. dt.
If the star is on the meridian, we have:
sin A = 0, cosp - 1
and:
h h   90~ -+
at least if the star is south of the zenith, hence we obtain:
sill (T) - )
d cos   dA.
We see therefore, that in order to find the time by the
observation of stars on the meridian, we must select stars
which culminate near the zenith, because there an error of
the azimuth has no influence upon the time.
If a be the right ascension of the star and u the clocktime of observation, we have the error of the clock equal to
a —u, if the clock is a sidereal clock.  But if a mean-time
clock is used, we must convert the sidereal time of the culmination of the star, that is, its right ascension into mean
time.  If we denote this by nm, the error of the clock is
equal to mn - u.
For stars at some distance from the zenith the accuracy
of the determination of the time depends upon the accuracy
of the azimuth or upon the deviation of the instrument from
the meridian. If this error is small, we can easily determine
20




306
it by observing two stars, one of which culminates near the
zenith the other near the horizon, and then we can free the
observation from  that error. For if dA be the deviation from
the meridian, the hour angles O-c  and 0' —t' which the
stars have  at the times of the  observations are also small
and equal to:
cos A'
and-:                      Si1 (99- SA.
uatioence, since O s:    -      we have tle following tnswo
equations:
a. -U  -- - A_ _- AA
ll and:which we  cllli-   d  both'- -   an- Au.    If tte inst    -
from  which  we can find both Au and A A.  If the instrument is so constructed  that we  can  see stars north of the
zenith, we find A A still more accurately if we select two stars,
one of which is near the equator, the other near the pole,
because in this case the coefficient of A A in one of the above
equations is very large and besides has the opposite sign* ).
Example. At the observatory at Bilk the following transits were observed with  the transit-instrument, before it was
well adjusted:
(a Aurigae  ah 61 m 27s. 72
f Orionis  5 8 12.71.
Since the right ascensions of the stars were:
a Aurigae  5h 51)l 33s. 25  + 45~0 50'. 3
pf Orionis  5 7 17.33  -  8 23.1
and the latitude is 51" 12'.5, we have the two equations:
- 54s. 47 A —     - 0.13433 AA
-55. 38 A   - -0.87178 AA,
fromn which we find:
au' --  54s.3()
and:
AA -- - is. 23.
*) It is assumed here, that the instrument be so adjusted, that the line
of collimation describes a vertical circle. If this is not the case, the obsers
vations must be corrected according to the formulae in No. 22 of the seventh
section.




307
23. The time can also be found by a very simple
method, proposed by Olbers, namely by observing the time,
when any fixed star disappears behind a vertical terrestrial
object.  This of course must be a high one and at considerable distance fiom the observer so that it is distinctly seen
in a telescope whose focus is adjusted for objects at an infinite distance.  The telescope used for these observations
must always be placed exactly in the same position, and a
low power ought to be chosen.
Now if for a certain day the sidereal time of the disappearance of the star be known by other methods, we find
by the observation on any other day immediately the error
of the sidereal clock, because the star disappears every day
exactly at the same sidereal time, as long as it does not change
its place.  But if a mean-time clock is used for these observations, the acceleration of the fixed stars must be taken
into account, since the star disappears earlier every day by
Oh 311 555.909 of mean time.
If the right ascension of the star changes, the time of
the disappearance of the star is changed by the sanle quantity, because the star is always observed at the same azimuth
and hence at the same hour angle.  But if the declination
changes, the hour angle of the star, corresponding to this
azimuth, is changed and we have according to the differential
formulae in No. 8 of the first section,  since d A as well as
dpf are in this case equal to zero:
dl- cospdhl
cos d t =- sihin p d h,
hence:
d S. tang p
dt.-.?
cos 
where p denotes the parallactic angle.
Therefore if the change of the star's right ascension and
declination is A f and A, the change of the sidereal time,
at which the star disappears, is:
A a     tang p
15   15 cos 3
Olbers had found from other observations, that in 1800
Sept. 6 the star a Coronae disappeared behind the vertical
wall of a distant spire, whose azimuth was 64" 56' 21". 4, at
20*




308
11h 23m 18". 3 mean time, equal to 22h 26m 21s. 78 sidereal time.
On Sept. 12 he observed the time of the disappearance  of
the star 10I'491 21S.0. Now  since 6x 3m55s.909 is equal to
23n 35S.4, the star ought to have disappeared at 101 591142s.9
mean time, hence the error of the clock on mean time was
equal to  - 10  21s.9.
In 1801 Sept. 6 was:
aO a —  42".0
and:
8 —  13". 2
and since we have:
p =-37' 31'
and:
/~ — = +260 41',
we find:
A,         - tang   11..35
cos 8 
hence the complete correction is -+  53". 35 or 3.56.  Therefore in 1801 Sept. 6 the star O' Coronae disappeared at 22h' 2611
25. 34 sidereal time *).
24.  If we know  the time, we can find the latitude by
observing an azimuth of a star, whose place is known, since
we have:
cotlang A sin t  - cos CT tang 8 -+- sin T cos t.
Differentiating this equation we find:
cos S Cos P  sinp
sill A d99   - cotang hdA +-  dct -+.     dS'.
sin h      sin ih
Hence in order to find the latitude by an azimuth as
accurately as possible, we must observe the star near the
prime vertical, because then sin A  is at a maximum.  Besides we must select a star which passes near the zenith of
the place, since then the coefficients of d A and d t are very
small, as we have:
cos C cos 1) - sin 90 cos h -f- cos V sin h cos A.
Therefore we see that errors of the azimuth and the time
have then no influence, whilst an error of the assumed declination of the star produces the same error of the latitude,
since we have then sinp -1.
If we observe only one star, we must observe the azi) v. Zach, Monatliche Correspondenz Band III. pag. 124.




309
muth itself besides the time. But if we suppose, that two
stars have been observed, we have the two equations:
cotang A sil t - - cos T tang 3 -+- sin p cos t
cotang A' sin t'- - cos 99 tang d' + sin'p cos t'.
Multiplying  the first equation  by  sin t', the second by
sin t, we find:
sin t sin t' i ( A      s 9p [tang 3' sin t - tang'  sin t']
sin A sin A'
- sin'P sin (t'- t),
or as:
cos 3 sin t = cos h sin A,
also:
cos h cos h' sin (A'- A) = cos'p [cos 3 sin 3' sin t - sin a cos 8' sin t']
-- sin'9 sin (t' - t) cos  cos 3'.  (b)
We will introduce now  the following auxiliary quantities:
sin (3' -+- ) sin ~ (t'- t) = m sin M    (h)
sin (3'-3) cos - (t'- t) -   es COS      (A.
If we multiply the first of these equations by cos - (t' +  t),
the  other by  sin   (t'+- t) and  subtract the second equation
from  the first, we get:
m sin [~ (t'-+- t) - M] - sin 3' cos 3 sin t -- cos 8' sin 8 sin t'.
But if we multiply  the first equation  by  cos - (t'- t),
the  second  by  sin  (t' -  t), and  subtract the first equation
from the second, we get:
m sin [I2t' - t)-  M = - sin 3 cos 8' sin (t'  t).
Hence the equation (b) is transformed into the following:
cos h cos h' sin (A'- A)   n  cos'p sin [4- (t'+t) - M]
- m sin P sin [- (t'- t)- 3I] cotang 3.
If' we assume now, that the two stars were observed
either at the  same azimuth or at two azimuths, whose difference is 180", we have in both cases sin (A' — A)   0 and
hence we find:
tang'p = tangsin [ 1        (t' t)-M   (B)
tang  g  sin [ (t'- t) -  (B)
Therefore in  this case it is not necessary to know  the
azirnuth itself, but we find the latitude by the times of observation and by the declination of the star by means of the
formulae (A) and (B).
If the same star was observed both times, the formulae
become still more simple. For since we have in this case
M     90" according to the second formula (A), we find;




310
tang o -= tang a cos -.  (t-  t)C
cos  (t'-t)         (
For the general case, that two stars have been observed
at two different azimuths, the differential equations are:
cos hdA = sinp dr +- cos  cos pdt - sin h sin A df
cos h'lA'- sin p'dS' -- cos' cos p'dt'- sin h'sin A'dT.
If we introduce here also the difference of the azimuths
and therefore multiply the first equation by cos h', the other
by cos h, and subtract them, we get:
cos h cos h'd(A'-A)  -  cos h' cos r cos pdt -+ cos A cos 5' cos p'dt'
- [sin h' cos h sin A' - sin h cos h' sin A] d9T
+ cos h sin p'd S' -- cos h' sin p d?.
Now  since  dt-=  du -+  d (Au) and  dt' =  du' -+  d (Au),
where d u and d u' are the  errors of observation and d(LAu)
that of the error of the clock, we find, if we substitute these
values in place of d t and dt' and take at the same time
A'= 1800  + A ):
sin A d — cos p cos Ad(A u) -    ('     ) [d (A'-A) - sin 99 d (u'-u)1
+es q cosA sin h cos A'    cos h  cos A sin h' cos h
— T   c     s           $ du  s       inh+ os     du'
sin (h' - h)             sin (h' - h)
sin p' cos h d sin p cos h'
sin (h'- h)    sin (h'-+- h)
Hence  we see again that it is best t6 make the observations  on  the  prime  vertical.  For then  the coefficient of
dcp is at a maximum  and those  of the  errors du, du' and
d(LAu) are equal to zero; and only the difference of the two
errors of observation, the errors of the declination and the
quantity, by which the difference of the two azimuths was
greater or less than 1800, will have any effect upon the result. In case that the same star was observed on the prime
vertical in the east and west, we have h= h' and sin p' - sinp,
hence:
dcl     cotang h [d(A'-A)- sin q d(u'- u)] -4-  h da,
~) In order to find the equation given above, we must also substitute
for cos 3 cos p and cos 5' cos p' the following expressions:
cos 8 cosp - sin 7 Cos h -+ cos9  sin h cos A
cos' cos p'  sin T cos h' —- cos T sin!''os A.




and since according to No. 26 of the first section:
sin          - cos E
siln h =  - rand sin p = os
we have:
d ='- cotang h [d(A'- A)- sin Td(u'-u)] +- in   d.
sin 2 
We see again from this eqnation, that it is best to observe stars, which pass near the zenith, because then cotang h
is very large and hence errors in A'-  A  and uW' -u have
only very little influence upon the result. In this case the
coefficient of dO' is equal to 1, since the declination of stars
passing through the zenith is equal to bp, and hence the result
will be affected with the whole error of the declination.  But
if' the difference of latitude  should  be determined  by  this
method  for two  places not far from  each other so that the
same star can be used at each place, this difference will be
entirely free from the error of the declination *).
Example.  The star f  Draconis  passes  very  near the
zenith  of Berlin.  Therefore  this star was observed at the
observatory with a prime vertical instrument.  The interval
between the transits of the star east and west was 34Mn43s.5
hence:
(t' - t)  40 20' 26". 25
and it was
--- 520 25'26". 77.
Now since in case that the observations are taken on
the prime vertical we have - (t'-+-t) = 0, we find from (C)
the following simple formula for finding the latitude:
tang 3    *)
tang 9     t
tang T  --- cos -E (t'-t) )'
and by means of this we obtain:
-- 52 0 30' 13". 04.
Finally the differential equation is:
dq = +- 0.02310 [d(A'-A) - 0.7934 d(u' —u)] +- 0.99925 dS.
*) It is again assumed, that the transit instrument is so far adjusted,
that the line of collimation describes a vertical circle. Compare No. 26 of
the seventh section.
**) This formula is also found simply from the triangle between the pole,
the zenith and the star, which in this case is a right angled triangle.




312
25.  If we observe two stars on the same vertical circle,
we can find the time, if we know  the latitude of the place,
since we have:
sin [2 (t' -+ t) --   ] - tang  [s sin     -    - Al],  (A)
tang Y
where:
t-u -- Au - acc
t' u' -- Au - a'
and
nm sin'M-  sin (A'-+- A) sin 4 (t'- t)
m COS M-  sin (S' - 8) cos 4 (t' - t).
Since t' —t, that is, half the interval of time between
the observations, expressed in sidereal time, is known, we
can find t'-+-t and hence t and t'.
The differential equation given in No. 22 shows, that
for finding the time by azimuths it is best to observe stars
near the meridian, because there the coefficient of dp~ is at
a minimum, that of dt at a maximum.
The azimuth  itself can  also  be found by  such observations.  For we have:
cos a sin t
tang A              --
- cos 9 sin a —- sin T cos 8 cos t
and making use of the equation:
sin [~ (t' -- t) - M]
tang T - tang  si b 2'   l
Sin [I (t' - t) - Yf]
we find:
sin t. sin [I (t'+ t) - M]
sin T tang A                    2
-- sin [- (t'- t) -  1] -- cos t sin [- (t' ~- ) - M]
If we write here
2 (t' -- t) -iM- t instead of - (t'- t) - M,
we easily obtain:
tang [7 (V' + O - 311
tang A     tang [                  (B)
sin T9
If the time of both observations is the same or:
t' - t e a - a',
the formula (A) gives the time, at which two stars are on
the same vertical circle.
The places of c Lyrae  and  ar Aquilae are fbr the bet
ginning of the year 1849:
a Lyrae  a = 18h 31m 47s. 75  --   +  380 38' 52". 2
Aquilae a'-t 9 43  23. 43  3'-t   8 28 30. 5.




313
Therefore we have:
t'  t - - 1i I 1 35s. 68 = - 170 53' 55". 2.
If we take then qj- 52" 30' 16", we find:
M= 1920 55' 53". 0
-(t' -t) — jl - 158  7 9.4
and from  this we get:
2- (t' +- t) — --- 1420 35' 38". 6,
hence:
(t'-~- t)   - 24o 28' 28". 4
-- lh 37m 53s 9
and
t = —11 2m 62 Cs. I,1 t'=  n i341s. 7.
Therefore the sidereal time at which the two stars are
on the same vertical circle is:
0 -- 17h 29m 42s.
Hence if we observe the clock-time when two stars are
on the same vertical circle, if for instance we observe the clocktime when two stars are bisected by a plumb-line, we can find
the error of' the clock at least approximately, when we know
the latitude of the place and compute the time by means of
the formulae given above. It is best to take as one of the
stars always the pole-star, since it changes its place very
slowly, a circumstance which makes the observation more
easy.
V. DETERMINATION OF THE ANGLE BETWEEN THE MERIDIANS OF
TWO PLACES ON THE SURFACE OF THE EARTH, OR OF THEIR
DIFFERENCE OF LONGITUDE.
26.  If the local times, which two different places on
the surface of the earth have at the same absolute instant,
are known, the hour angle of the vernal equinox for each
place is known.  But the difference of these hour angles,
hence the difference of the local times at the same moment,
is equal to the are of the equator between the meridians
passing through the two places and hence equal to their difference of longitude; and since the diurnal motion  of the
heavenly sphere is going on in the direction from  east to
west, it follows, that a place, whose local time at a certain




314
moment is earlier than that of another place, is west of this
place, and that it is east of it, if its local time is later than that
of the other place.  For the first meridian, from which the
longitudes of all other places are reckoned, usually that of a
certain observatory, for instance, that of Paris or Greenwich,
is taken.  But in geographical works the longitudes are more
frequently reckoned from the meridian of Ferro, whose longitude from Paris is 200 0' or 1I 20'2  West.
In order to obtain the local times which exist simlultaneously on two meridians, either artificial signals are observed or such heavenly phenomena as are seen at the same
moment from all places. Such phenomena are first the eclipses of the m1oon.  For since the moon at the time of an
eclipse enters the cone of the shadow of the earth, the beginning and the end of an eclipse as well as the obscurations of different spots are seen from all places on the earth
simultaneously, because the time in which the light traverses
the semi-diameter of the earth is insignificant.  The same is
true for the eclipses of the satellites of Jupiter.
These phenomena therefore would be very convenient
for finding differences of longitude,  since they are simply
equal to the differences of the local times of observations,
if they  could be observed with greater accuracy.   But
since the shadow of the earth on the moon's disc is never
well defined, and thus the errors of observation may amount
to one minute and even more, and since likewise the beginning and end of an eclipse of Jupiter's satellites cannot be
accurately observed, these phenomena are at present hardly
ever used for finding the longitude. If however the eclipses
of Jupiter's satellites should be employed for this purpose, it
is absolutely necessary, that the observers at the two stations
have telescopes of equal power and that each observes the
same number of immersions and emersions and those only of the
first satellite, whose motion round Jupiter is the most rapid.
The arithmetical mean of all these observations will give a
result measurably free of any error, though any very great
accuracy cannot be expected.
Benzenberg has proposed to observe the time of disappearance of shooting stars for this purpose. These can be




315
observed with great accuracy, but since it is not known beforehand, when and in what. region of the heavens a shooting star will appear, it will always be the case, that even if
a great mass of shooting stars have been observed at the two
stations, yet very few, which are identical, will be found
among them; besides the difference of longitude must be
already approximately known, in order to find out these.
Very accurate results can be obtainedl by observing artificial signals, which are given for instance by lighting a quantity
of gunpowder at a place visible from  the two  stations.
Although this method can be used only for places near each
other, yet the difference of longitude of' distant places may
be determined in the following way: Let A and B be the
two places, whose difference of longitude I shall be found, and
let Al, A2, A:3 etc. be other places, lying between those places, whose unknown differences of longitude shall be I,, 1l, 13 etc.
so that 11 is the difference of longitude between A1 and A,
12 that between A2 and A1 etc.  If then signals are given at
the stations A,, A3, A5 etc. at the local times tl, t3, t, etc.,
the signal from  A, is seen  at the place A  at the time
t- 1, - 0, and at the station A2 at the time tl -q-, 12 _.
Further the signal given from  A, is seen at the station A2
at the time t3 - 13  09.,, and at the station A, at the time
t -- l+ 14  03.  But since the difference of longitude of the
places A and B is equal to   -+- 1 -+-... +- 1,,, if the last signal station is A,._1, or since:
-- (O, -  ) +- ( 3-,  )+(Q5- 0-,) etc.,
we find:
/~o....- (o  2.- 3.) —*.. —(o= —o.) —
Therefore at the stations, where the signals are observed,
it is not requisite to know the error of the clocks but only
their rate, and it is only necessary to know the correct time
at the two places, whose difference of longitude is to be
found.
Instead of giving the signals by lighting gunpowder, it
is better to use a heliotrope, an instrument invented by
Gauss, by which the light of' the sun can be reflected in any
direction to great distances.  If the heliotrope is directed to




316
the other station, a signal can be given by covering it suddenly.
The diff'erence of longitude of two places can also be
determined by transporting a good portable chronometer from
one place to the other and finding at each station the error
of the chronometer on local time as well as its rate.  For
if the error found at the first place be At and the dailyrate
be denoted by  dt'  then the error after a days will be
di
Au + a -(-t.  Now if after a days the error of the chronometer at the other place should be found equal to Au', we
have, denoting the longitude of the second place east of the
first by 1:
l d. Aa u
-1 -- A- it —-      u~ -}- A u',
hence
d. Au
1    -- AA +' - Ott.
It is assumed here that the chronometer has kept a uniform rate during the interval between the two observations.
But since this is never strictly the case, it is necessary, to
transport not only one chronometer from  one place to the
other, but as many as possible, and to take the mean of all
the results given by the several chronometers. In this way
the difference of longitude of several observatories, for instance that of Greenwich and that of Pulkova has been determined.  Likewise the longitude at sea is found by this
method, the error of the chronometer as well as its rate
being  determined  at the place from  which the ship sails
and the time at sea being found by altitudes of the sun.
27. The most accurate method of finding the difference
of longitude is that by means of the electric telegraph. Since
telegraphic signals can be observed like any other signals,
the method is of the same nature as some of those mentioned
before, and has no other advantage than perhaps its greater
convenience; but when chronographs are used for recording the
observations at the two stations, it surpasses all other methods by the accuracy of the results. The chronograph is
usually constructed in this way, that a cylinder, about which




317
a sheet of paper is wrapped, is moved around its axis with
uniform  velocity by a clockwork, which at the same time
carries a writing apparatus, resting on the paper, slowly in a
direction parallel to the axis of the cylinder.  Therefore, if
the motion of the cylinder and of the pen is uniform, the
latter markes on the paper a spiral, which when the sheet is
taken from the cylinder, appears as a system of parallel lines
on the paper. Now the writing apparatus is connected with
an electro-magnet so that, every time the current is broken
for an instant and the armature is pulled away from the
magnet by means of a spring attached to it, tlhe pen makes
a plain mark on the paper. If then the pendulum of a clock
breaks the current by some contrivance at every beat, every
second of the clock is thus marked on the sheet of paper,
and since the chronograph is always so arranged that the
cylinder revolves on its axis once in a minute, there will be
on every parallel line sixty marks, corresponding to the seconds of the clock, and the marks corresponding to the same
second in different minutes will also lie in a straight line perpendicular to those parallel lines. We will suppose now, that
at first the current is broken and that the pen is marking an
unbroken line; then if the current be closed just before the
second-hand of the clock reaches the zero-second of a certain
minute, the first second-mark on the paper will correspond
to this certain second, and hence the second corresponding
to any other mark is easily found. If then the current can
also be broken at any time by a break-key in the hand of the
observer, who gives a signal at the instant when a star is seen
on the wire of the instrument, the time of this observation
is also marked on the sheet, and hence it can be found with
great accuracy by measuring the distance of this mark from
the nearest second- mark.
If the current goes to another observatory, whose longitude is to be determined, and passes there also through a
key in the hand of the observer, the signals given by this
observer will be recorded too by the chronograph at the first
station; hence if this observer gives also a signal at the time
when the same star is seen on the wire of his instrument,
the difference of the two times of observation, recorded on




318
the paper and corrected for the deviations of the two instruments from  their respective meridians and for the rate of
the clock in the interval between the two observations, will
be equal to the difference of longitude of the two places.
Since the electrical current, when going to a great distance, is only weak, this main current, which passes through
the keys of the two observers, does not act immediately upon
the electro-magnet of the chronograph, but merely upon a
relay which breaks the local current passing through the
chronograph.
If a chronograph is used at each station and the clocks
are on the local circuits, the signals from each observer and the
seconds of the local clock are recorded by each chronograph,
and hence we get a difference of' longitude by every star
from  the records of each chronograph after being corrected
for the errors of the instruments and the rate of the clock.
But the difference of longitude thus recorded independently
at each station is not exactly the same.  For since the velocity of electricity is not indefinitely great, there will elapse
a very short, but measurable time, at least if the distance
of the two stations is great, till the signal given at the station A, being the farthest east, arrives at the station B.
Hence the time of the signal recorded at the station B corresponds to a time, when the star was, already on the median of a place lying west of A, and the difference of longitude recorded at B is too small by the time, in which the
electricity traverses the distance from A to B. But the same
time will elapse when the signal from B is given, and the
time recorded at the station A will correspond to the tinme
when the star was on the meridian of a place a little west of
B, hence the difference of longitude recorded at the station A
will be too great by the same quantity. Therefore the mean
of the differences of longitude recorded at both stations is
the true difference of longitude and half the difference (subtracting the result obtained at the station B from that obtained at the station A) is equal to the time in which the
electricity traverses the distance from A to B ).
*) The armature-time is also a cause of this difference.




319
A single star, observed in this way, gives already a more
accurate result than a single determination of the longitude
made by any other method, and since the number of stars
can be increased at pleasure, the accuracy can be driven to
a very high degree, provided that also the greatest care is
taken in determining the errors of the two instruments. Since
the same stars are observed at both stations, the difference
of longitude is free from  any errors of the places of the
stars.
In case that the distance between the two stations is
great, sometimes a large number of signals are lost and it
is therefore preferable, to let the main current for a short
time at the beginning and end of the observations pass through
both clocks, so that their beats are recorded by the chronographs at both stations.  If then the current is closed at
each station at a round minute, after having been broken for
a short time, so that the clock-times corresponding to the
records on the chronographs are known, the difference of
the two clocks can be obtained from  every recorded second
or better from the arithmetical mean of all. These differences,
as obtained at both stations, differ again by twice the time,
in which the current passes from one station to the other,
and which in this way can be determined even with greater
accuracy.. A few such comparisons are already sufficient to
give a very accurate result, since the accuracy of one comparison probably surpasses the accuracy with which the errors of the clocks can be obtained from observations.  Certainly the comparisons obtained during a few minutes are
more than sufficient for the purpose so that the telegraphic
part of the operation is limited to a few minutes at the beginning and the end of the observations. After the first set
of comparisons has been made, the clocks as well as the keys
of both observers are put on the local circuit of each observatory and the errors of the clocks determined by each observer. If these errors of the clocks are applied with the
proper signs to the difference of the time of the two clocks,
the difference of longitude of the two stations is found. Also
in this case it is advisable, that the observers use as much
as possible the same stars for finding the errors of their




320
respective clocks, in order to eliminate the influence of any
-errors of the right ascensions of the stars.
Besides errors arising from an inaccurate determination
of the errors of the two instruments, there can remain another
error in the value of the difference of longitude, produced
by the personal equation of the two observers, that is, by
the relative quickness, with which the two observers perceive any impression upon their senses.  But this source of
error is not peculiar to this method, but is common to all
and even of less consequence, when the observations are recorded by the electro-magnetic( method.  In this case the
error depends upon the time, which elapses between the moment, when the eye of the observer receives an impression
and the moment, at which he becomes conscious of this impression and gives the signal by touching the key. If this
time is the same for both observers, the determination of the
difference of the longitude is not at all affected by it; but
if this time is not equal and there exists a personal equation,
the difference of longitude is found wrong by a quantity equal
to it. But the error arising from this source can be entirely
eliminated (at least if the personal equation does not change),
if the same observers determine the difference of longitude
a second time after having exchanged their stations; the difference of the two results is then equal to twice the personal equation, whilst their arithmetical mean is free from it.
The observers can also determine their personal equation,
when they meet at one place and observe the transits of stars
by an instrument furnished with Imany wires, so that one observer takes always the transits over some of the wires and
the other those over the remainder of the wires.  If then
these times of observation are redulced to the middle wire,
(Section VII No. 20) the results for every star obtained by
the two observers will differ by a quantity equal to the personal equation.  The observations are then changed so, that
now the second observer takes the transits over the first set
of wires, and the first one those over the other wires. Then
nearly the same difference between the observers will be obtained and the arithllletical mean of the two values thus found
will be free from  any errors of the wire-distances used for




321
reducing the observations to the middle wire. After the personal equation has thus been found, the value obtained for
the difference of longitude  must be corrected on account of
it.  If the observer whose station is farthest to the east observes later than the other, or if the personal equation is
E-  W=-   a, the value found for the difference of longitude
is too small by the same quantity, and hence -+- a must be
added to it.
Exacmple.  On  the  29t!' of' June 1861  the  difference of
longitude  was determined  between Ann Arbor in the State
of Michigan and Clinton in the State of New  York and from
126 comparisons of the clocks recorded by the chronographs
of the two stations it was found that:
(recorded at A. A.) 13h 591 3s.O Clinton clock-time  191 585' 29s.56 A. A. clock-t.
(recorded at C1.)  13 59 3.0,,,,      19 58 29.40,..
The  clock  at the  observatory  at Clinton  was a mean
time clock and its error on Clinton sidereal time was at the
time 1 3h59's' 3. 0 equal to -+- 6t33"' 46s. 07, while the error of
the clock at Ann Arbor on local sidereal time was ~- Im iS. 87.
From  the records by the chronograph at Ann Arbor we find
therefore:
201b 3211 49s.07 C1. sidereal time — = 191 59m1 31s.43 A. A. sidereal time
and by the chronograph at Clinton:
20h 32'1 49s.07 Cl. sidereal time -- 191 591n 31s. 27 A. A. sidereal time.
Hence we find the difference of longitude by the records
at Ann Arbor equal to
33mi 17s. 64,
and by those at Clinton:
3311 17s. 80,
or the mean  331" 17s. 72.
The personal equation is in this case E - W = -  s.04 *)
hence the corrected difference of longitude is 33"1 17;.76.
Note. The electro-magnetic method for finding the difference of longitude is usually called the American method, since it was proposed by Americans. The idea originated with to Sears C. Walker and W. Bond Esq., to
whom the honour of inventing it must be accorded, although Mitchel of Cincinnati completed the first instrument for recording the observations.
*) Dr. Peters observed at Clinton, the author at Ann Arbor.
21




322
28. Besides the observations of natural or artificial signals, which are seen at the same instant at the two stations,
whose difference of longitude is to be found, we may use
for this purpose also such celestial phenomena, which, though
they are not simultaneous for difierent places, yet can be reduced to the same time; and they afford even this advantage,
that they can be observed with great accuracy, and that they
are visible over a large portion of the surface of the earth
so that it is possible to find the difference of longitude of
places very distant from each other. Such phenomena are the
occultations of fixed stars and planets by the moon, eclipses
of the sun, and transits of the inferior planets Mercury and
Venus.  Since all these heavenly bodies with the exception
of the fixed stars have a parallax, which in the case of the
moon is very considerable, they are seen' at the sanme instant
from different places on the snrface of the earth at different
places on the celestial sphere, and hence the occultations as
well as the other phenomena mentioned before are not simultaneous for different places.  Hence in this case the observations need a correction for parallax, since we must know
the time, when those phenomena would have occurred, if there
had been no parallax or rather, if they had been observed
from the centre of' the *earth.
Therefore we must find first the paralla.xes in longitude
and latitude and the apparent seimi-diamieters of the heavenly
bodies at the time of the beginning and the end of the eclipse
or occultation (or the parallax in right ascension and declination, if it shotld be preferable to use these co-ordinates).
Then in the triangle between the pole of the ecliptic and
the centres of the two bodies the three sides, namely the
complements of' the apparent latitudes and the sum  or the
difference of the apparent semi-diameters, are known; hence
we can compute the angle at the pole, that is, the difference
of the apparent longitudes of the two bodies at the time of
observation and, applying the parallaxes in longitude, we find
the difference of the true longitudes, as seen from the centre
of the earth.  From  this, the relative velocity of the two
bodies being known, we obtain the time of true conjunction,
that is, the time, at which the two bodies have the same




323
geocentric longitude, and expressed in local time of the place
of observation. If the beginning or end of the same eclipse
or occultation has also been observed at another place,
we find in the same way the time of true conjunction expressed in local time of that place.  Hence the difference of
both times is equal to the difference of longitude of the two
places.
If the times of observation, as well as the data used
for the reduction to the centre of the earth were correct,
the difference of longitude thus obtained would also be correct.  But since  they  are  subject to  errors, we must
examine, what influence they have upon the result, and try
to eliminate it by the combination of several observations.
This is the method, which formerly was used for finding the difference of longitude by eclipses. At present a different method is employed. Starting from the equation, which
expresses the condition of the limbs of the two bodies being
in contact with each other and which contains only geocentric quantities, another equation is obtained, in which the
unknown quantity is the time of conjunction or rather the
difference of longitude.
29. The limbs of two heavenly bodies are seen in contact, when the eye is anywhere in the curved surface enveloping the two bodies.   Since the heavenly bodies are so
nearly spherical, that we can entirely disregard the small
deviation from a spherical form, the enveloping surface will
be the surface of a straight cone, and there will always be
two different cones, the vertex being in one case between
the two bodies, while in the other case it lies beyond the
smaller body. If the eye is in the surface of the first cone,
we see an exterior contact, whilst when it is in that of the
second, we see an interior contact.
The equation of a straight cone is the most simple, if
it is referred to a rectangular system  of axes, one of which
coincides with the axis of the cone. If the cone is generated by a right angled triangle revolving about one of its
sides, the equation of its surface is:
x2Z +_ y2 = (c - z)2 tang]f2,
where c is the distance of the vertex from the fundamental
21'




324
plane of the co-ordinates, and f is the vertical angle of the
generating triangle.
We must now find the equation of the cone enveloping
the two bodies and referred to a system of axes one of which
passes through the centres of the two bodies.  If then we
substitute in place of the indeterminate co-ordinates x, y, z
the co-ordinates of a place on the surface of the earth, referred to the same system of axes, we obtain the fundamental equation for eclipses.  For this purpose we must first
determine the position of the line joining the centres of the
two bodies.  But if a and d be the right ascension and declination of that point, in which the centre of the more distant body is seen from  the centre of the nearer body or in
which the line passing through both centres intersects the
sphere of the heavens, and if G denote the distance of the
two centres, further ac, 3 and A  be the geocentric right aseension, declination  and distance  of the nearer body and
a', 8', A' the same quantities for the more distant body, we
have the equations:
G cos d cos a = A' cos' cos a' - A cos 3 cos a
G cos cd sin a = A' cos 3' sin a'- A cos 3 sin a
G sin d —' sin 3' - A sin 3,
or:
G cos d cos (a - a')  A cos' - A cos b cos (a - a')
G cos d sin (c a- a')   - A ('os  sin (a - a')
G sin d= A' sin 3' - A sin 3.
If we take as unit the equatoreal semi-diameter of the
earth, we must take sin' and -instead of A' and A, since
s in. 7r, S il   5r
A and A' are expressed in parts of the semi-major axis of
the earth's orbit, where 7T is the mean horizontal equatoreal
parallax of the nearer body, r' the same for the more distant body; thus we obtain:
sin 7C
sinw 7 G cos d cos (a - a') = A'  Cs'-os  cos (a — a')
sin n'
sin (z G cos d sin (a - a') -      -os 8 sin(a - a')
sin 7r G sin d             ill A     sin'- sin.
sin  n
Now  since we also have:
sin z
sin 7 G os d-' -  cos' cos (a -a') -  os 8 cos (a- a),
sin I'




we find:
sin I' cos 3
tang (a --')'sin' cos a
i' -   sinX co s  cos (a- a')
and:                              sin  /
sin (3 - (Y)
tang (d- 3') - -, -
sin n.......... cos (6 —8')
1 A' sin 7
sin ~
Since in  the case of an eclipse of the sunI sin nr is a
siln 7
small quantity, we obtain from  this by  means of the formula (12) in No. 11 of the introduction:
sin n'  cos S
A' sin'c(a- a')
a-soAinol s   SZ (ta'       (A)
sin  2i'
d=-'-  (' --')
A' sin 7v
G sin 7t'
and putting:               g-     a
sin 7r'
we also find:             g   1'sin-                 (B)
We will imagine now  a rectangular system  of axes of
co-ordinates, whose origin is at the centre of the earth. Let
the axis of y be directed towards the north pole of the equator,
whilst the axes of z and x are situated in the plane of the
equator and directed to points, whose right ascensions are
a and  90 -+a.  Then the co-ordinates of the nearer body
with respect to these axes are:
z'A CoS  Cos (a- Coa),  y'=AsinS, x' =A cos A sin (a -a).
If now we imagine the axes of y and z to be turned in
the plane of y z through the angle -  d ), so that the axis
of z is directed towards the point whose right ascension
and declination are a and d, we find the co-ordinates of the
nearer body with respect to the new system of axes:
sin S sin-d +- cos e cos  cs d cos (a - a)
sin T
sin S cos d - cos S sin d cos (a - a)
sins n
cos a sin (a- a)
x-=
sin Tr
*) The angle d must be taken negative, since the positive side of the
axis of z is turned towards the positive side of the axis of y.




326
Or:
cos (2-d) cos 2 (     -- cc)'2 - COS (QY+ d) sin' (a- a)2
sin T
sin ( d- d) cos 4- (a - a)   sin ( +- d) sin 1 (a - a)2
y3__- (C)
sill 
cos a sin (a - a)
sin 7v
The axis of z is now  parallel to the line joining the
centres of the two bodies.  If we let the axis of z coincide
with this line, the co-ordinates x and y will be the co-ordinates of the centre of the earth with respect to the new
origin but taken negative.
Let cp' be the geocentric latitude of a place on the surface of the earth, ) its sidereal time and ( its distance from
the centre, then the co-ordinates of this place, taking the
origin at the centre of the earth and the axis of i parallel
to the line joining the centres of the two bodies, are:
-- g [sin d sin 99' - cos d cos 9T' cos (0 - a)]
- =  [cos dsin'- sin dcos'os  cos (O — a)]  (D)
=- g cos I' sin (0 - a).
The co-ordinates of this place with respect -to a system
of axes, whose axis of z is the line joining the two centres
itself, are:
-x, r — y and 
and the equation, which expresses, that the place on the surface of the earth, given by o, A' and 0, lies in the surface
of the cone enveloping the two bodies, is:
(x - )2 -t- (y'-7)2 =(C -;)2 tangf2,
where c and f are yet to be expressed by quantities referred
to the centre of the earth. But the angle f is found, as is
easily seen, by the equation:
4- r
sinf    G
where r and r' are the semi-diameters of the two bodies and
where the upper sign must be used for exterior contacts, the
lower one for interior contacts.  Now  since the  unit we
use for G is the semi-diameter of the equator of the earth,
we must refer r and r' to the same unit.  Therefore if k
denotes the semi-diameter of the moon expressed in parts of
the semi-diameter of the equator of the earth and h the ap



327
parent semi-diameter of the sun seen at a distance equal to
the semi-major axis of the earth's orbit, we have, since:
sin Ah
sin''
also:
sinf= G       sin in =4= k sin n']
G sin vr'
or:
sinf      [sin If [ == k sin in'].  (E)
But we have:
log sin n'- 5.6186145,
further  we'have according  to  Burkhardt's  Lunar Tables
k-  0. 2725 and according to Bessel h =  15' 59". 788, hence
we have:
log [sin Ah + — k sin'r] = 7. 6688041 for exterior contacts,
log [sin h - I sin c']- 7. 6666903 for interior contacts.
We must still express the quantity c, that is, the distance of the vertex of the cone from  the plane of xy.  But
we easily see, that:
c -z..-.          (F)
sin f
where again the upper sign is used for an exterior, the lower
one for an interior contact.  If we then denote by  1 the
quantity c tang f, that is, the radius of the circle in which
the plane of xy intersects the cone, and tang f by i,, the general equation for eclipses, which expresses, that the place
on the surface of the earth given by (p', O and e, lies in the
surface of the cone enveloping both bodies, is as follows:
(X —)  + -  _  (     1 --  i;).
Since I is always positive, we must take tang f or X
negative, if we find a negative value of c from the equation (F).
The values of the quantities used for computing x, y,.3
and i$, ), 5 by means of the equations (C) and (D) are taken
from  the tables of the sun and the moon.  Since these are
always a little erroneous, the computed vali,  of x, y etc.
will also differ a little from  the true values.'herefore if,aX, Ay and Al  are the corrections, which must be applied




328
to the computed values x, y and I in order to obtain the
true values, the above equation is transformed into*):
(x -4+ ~Ax - $)2 + (y + Ay -- n)2 -  (I -'1 -- I _  )2.
We will assume now, that the values of e), S, %',t} 61
and a' have been taken from  the tables or almanacs for the
time T of the first meridian.  Then if the unknown time of
the first meridian, at which a phase of the eclipse has been
observed, be T  - T', we have, denoting by x0 and y,, the
values of x  and y corresponding to the time T and by x'
and y' the differential coefficients of x and y:
x - x0 -- x' T' and y  Y==o -+' T'.
In the same way the quantities  i, 1j and, will consist
of two parts.  But since these quantities change only slowly
and an approximate value of the difference of longitude, and
hence of the time of the first meridian corresponding to the
time of observation is always known, we can assume, that
these quantities are known for the time of observation.
Hence the equation is now:
[xO - s -+- x' T' 4_ Ax]2 -4- [LYo  - 7 -+- y' T' q- -Ay]   ( --- (     I - A9)'.
If the changes of x and y were proportional to the -time,
x' and y' would be constant, and therefore it would not be
necessary to know  the time  T — T' for their computation.
Now this is not the case, but since the variations of x' and
y' are very small compared with those of x and y, we can
solve the equation by successive approximations.
If we put:         x' i —        X i' 
y' i -+ x' i' -Ay
and:       m sin M= x, -           n sin N —- x'
m cos l- o — 7          n cos N-y'   (G)
1- - 5 —L,
the above equation is transformed into:
(L    1) z2- = [m cos (Al- N) +- n (T' + i)]2 - [n sin (1- 1V) -n i']',
and we obtain, neglecting the squares of i' and A1, the following equation of the second degree for T' -- i:
i)2   2m                   L —m insin (M- N)a  1122 C   (-Jf- N)
(T +Z)2 ~   C_ co-s (L- N) (T'-os ()
a                            n2                n2
2m                2 L
-sin (] —  N) i' --- A1.
n                n2
*) Errors in a, d and A are here neglected, since they cannot be determined by the observations of eclipses,




..329
Now  since:
1v (a +Ax a >-/ x+ T-V2 / x 
putting:
L sin,y =- i sin (Al- N),  (H)
we find from  this equation:
_              L cos?a
T1 —-   cos (ao —i) e i =   tang,        i — se 
or except in case that ip is very small:
sin (M-N N -4-)               Al
= ~- sin — — i -aF tang i, i'==  see %
-- -'sin i?
Now  since T' for the beginning of the eclipse or any
phase of it must have a less positive or greater negative value
than for the end, the upper sign must be used for the beginning, the lower sign for the end of the eclipse or any
phase, if we take the angle i/I always in the first or fourth
quadrant *).  But if we take,p for the beginning of the
eclipse or any phase in the first or fourth quadrant and for
the end in the second or third quadrant, we have in both
cases:
To,    m  al sin (1f — N +-y) i       i  tangn — sec
1-,-'-'-'....sec'g
Xi sin Ty)                iT
or:
T..  cos ( — N)   _          i - i'tang T  - see p. (J)
The equation (J) is solved by successive approximations.
For this purpose compurte the values of x,  y=, a, d, g, I and
X by means of the formulae (A), (B), (C), (E) and (F) for
several successive hours, so that the values x, and Y, and
their differential coefficients can be interpolated for any time.
Then assume a value of T, as accurately as the approximately known value of the difference of longitude will permit,
interpolate for this time the quantities xo, Y,,, x' and y' and
find an approximate value of' T' by means of the formulae
(D), (G), (H) and (J).  With the value T-+ T' repeat, if
necessary, the whole computation.  If we denote again by
T the value assumed in the last approximation and by T'
the correction found last, we have T -  1"T'   t -  d, where t
is the time of observation and d is the longitude of the place
*) We find this easily from the first expression for 7T',




330
reckoned from  the first meridian, that is, that meridian: for
which the quantities x, y, z etc. have been computed, and
taken positive when the place is east of the first meridian.
Hence we have:...t-T+ —  cos (iI- N) +- - cos   -+ i + i' tang y +-  sec t
-t — T T+- sin (M — N ~- +)            - i i'tang v+  (K)
n sin W                  n
Since the values of x' and y' have one meani hour as
the unit of time, it is assumed, that d in the above formula
is referred to the same unit. Therefore if we wish to find
the difference of longitude expressed in seconds of time, we
must multiply the formula by the number s of seconds contained in one hour of that species of time, in which the observations are expressed.  By this operation t   T is also
expressed in seconds of the same species of time, in which t
is given or T is expressed in the same species of time as t.
Now the equation (K) does not give the longitude of
the place of observation from the first meridian, but only a
relation between this longitude and the errors of the several
elements used for the reduction. But if the same eclipse has
been observed at different places, we obtain for each place
as many equations as phases of the ecliptic have been observed.  By the combination of these equations we can eliminate, as will be shown hereafter, the errors of several of
these elements and thus render the result as independent as
possible of the errors of the tables.
It yet remains to develop the quantities i and i', determined by the equations:
x'ii-  i''- Ax
y'i - x'i' =Ay
or:
n i - sin NA x -+- cos NAy
ni'=- sin NA/ - cos NAx.
The quantities x and y depend upon a - a, S  d and 7r.
Therefore if we suppose these quantities to be erroneous,
we have:
Ax -AA (c - a)+ BA (8- d) -+ CAz
/ y  A'A (a - a) +- B  ( - dcl) + C'A,
where A, B? C are the differential coefficients of x with re



331
spect to a- a,' —d-land  r, and A', B', C' those of y with
respect to the same quantities. N ow since A (c -  a), A (, - d)
and An are always smnall quantities, we can neglect in the
expressions for the differential coefficients the terms containing sin (a —a) and sin (5 - d) as factors, and can write 1 in
place of cos (c - a) and cos (Y - d).  Then we obtain:
cos.        cos 3
A --     cos (a -a)  Cos —
sinll 7          sin 7
sin c sin (a - a)
B —-                  -— =0
sin 7c
cos sin (a -a) cos      x  x
C -- 
sin n2         tang 7r
cos Y sin d sin (a -- a')
Al'    --  -    ------   - -  __.... 0
sin n
cos ('- d) l
sin 7f    sin T
C' =  _
tang X
Now  since i and i', and hence also A(ce —a)), lA( — d)
and A 7 are expressed in part of the radius, we must divide
the differential coefficients by 206265, if we wish to find the
errors of the elements in seconds. Therefore if we put:
S
206265. z sin 7r
we have:
i --  sin Ncos b  (a  - a) +- h cosN   (b'- d) -/2 Cos f 7u [x sin N-y cos N]
i' -hcosNcos    (a  a) -+ hsin 7  (- dcl) -+   cos[ 07 L   xcosN-y sin N],
or multiplying  the upper equation by cos -i/, the lower one
by sin ip and adding them:
[i+-i' tangt] --. sin (N- p) cos A (a - a) + cos (N- V)   (- d)
- cos r Azz [x sin (N —) -ty cos (N- -)].
From this we obtain:
Mn  sin (li — N-f- W)    sin (N -           a)
d= t - T —s   --           --- ---            cos -  (a-a)
n       sin ~,           cos 1
cos (N- y)       d)
cos W
+ / 1  206265 sin   l Al
cos'(p
-h cos    (x sin (N-V) - y Ces (N-o)
Cos I/




332
or putting:
E  sin 2 cos ESA (a - a) -- cos Na ( -d);=  - cos Ncos A A (a - a) H- sin NzA (S - d)
7 - 206265 sin,I                         (L)
O -- cos 7 A/ X
x sin (N - T) - y cos (N-)
cos Z
we finally have:
t - T m  sin (il — N+- ) +  E - 1h  tang T + h  sec T-h E O. (M)
n       sin 1tNow the observation of every phase of an eclipse gives
such an equation and since this contains five unknown quantities, five such equations will be sufficient to find them.
However the quantities t7 and 0 cannot be determined in this
way, unless the observations are made at places which are
at a great distance from each other. Nevertheless the computation of the coefficients will show us the effect, which
errors of g and 1 can have upon the result. Generally it
will only be practicable to free the difference of longitude
from the errors of' and E, but the latter quantity can only
be determined, if the longitude of one place from the first
meridian is already known.  When E and, are known, the
errors of the tables are obtained by means of the equations:
cos  A (a - a) — e sin N-  cos N
a ('- d) - E cos N-+-  sin N.
If we collect all the formulae necessary for computing
the difference of longitude from an eclipse of the sun, they
are as follows:
I   sin A'  cos 
a  = a               (a - a
A'singm  cos (A
sin 7r
d'sinr (-')            (1)
A' sin Zv
sin ir'
gl —
1' sin z'
where a, 1 and a are the right ascension, declination and
horizontal equatoreal parallax of the moon, a':, Y, A' and a'
the right ascension, declination, distance and mean horizontal
equatoreal parallax of the sun.




333
cos' sin (a - a)
si       n         nt
cos (- d) cos (a- a)2 - cos (  -d) sin - (a - a)2
sinf    1
sinf       [sin h =-  k sin h  ]k sin,   (3)
~where:    ~
log [sin h +  c- k sin r']  7. 6 88041
for exterior contacts and
log [sin h - k sin c'] -7. 6666903
for interior contacts.
c- z4-,                      (4)
sinf
where the upper sign is used for exterior contacts, the lower
for interior contacts.
tang f                           (5)
where A has always the same sign as c.
-e  cos T' sin (O - a). = Q [cos d sin I' -- sin d cos T cos (O - a)]  (6),o [sin d sin 9' +- cos d cos q' cos (O - a)],
where qE' and 0 are the geocentric latitude and the distance
of the place from the centre and O is the observed sidereal
time of a phase.
If then we have for the time T:
dx
x o=       dt
dy    I
Y l=Yo._,   YI,
dt
we compute:
m sin M/-  x -  n sinN-x -               -           (7)
I  --  2 - -   (7)
an cos M    yO -         n cos NI-y'
L sin - — n sin (M- N),                 (8)
where for the beginning p must be taken in the first or fourth
quadrant and for the end in the second or third quadrant,
and:
_- sm sin(M —N —)              m                 L cos    (
T'   -- si.                  cos (f —  N)            (9)
n       sin Yt          n                    n
Finally we have:
d=-t- T- T' +- he +-k   h tang'P,          (10)




334
where:
S
206265. it sin 7'
e   sin Ncos  A ( - a) +- cos N1VA ( -  d),
--- cos N cos J A (CL - a) +- sin N, (- d),
hence:
cos 3 A (a - a) -    sin N -   cos N
zA (3- d)  e cos N+-  sin N.
Example.  In  1842  July 7 an eclipse of the sun occurred, which was observed  at Vienna and Pulkova as follows:
Vienna:
Beginning of the total eclipse 18h 49'l 25s. 0 Vienna mean time
End    of the total eclipse 18 51 22.0 O....
Pulkova:
Beginning of the eclipse  19h"  71n 3s. 5 Pulkova mean time
End    of the eclipse  21  12  52. 0,.....
According to  the Berlin Jahrbuch  we have the following
places of the sun and the moon:
Berlin m. t.      a              8               a'          8'
17h   1050 8'49".93 + 230 22'10" 35 106" 50'38".49 -+- 220 33'24'".46
18        47 43.31         15  0.34        53 12.37       33  7.93
19h   106 26 34.14           7 40.45       55 46.24       32 51 —36
20h   107   5 22.32         0 10.75       58 20.09       32 34.75
21h        44 7.75   22  52 31.29 107   053.94           32 18.09
22h   108  2250.34          44 42.13         3!27.78        32  1.40
gf             log A'
171  59' 55". 06    0.0072061
1S8h     56.37             56
19"1     57.65             51
20h      58. 91            46
211  60  0.14             41
22i       1. 35            36.
If we compute first the quantities a, d and g by means
of the formulae (1) we find:
ca                   d               log g
181  106  53' 21". 53  +- 220 33' 2". 04   9.9989808
191        55 50. 33         32 46.47              11
20"h       58 19.10           32 30.87             15
21"  107   0 47.88          32 15.25              19.
Then  we  find  by  means of the formulae  (2), (3), (4)
and (5):




335
x                  jY             log z
17h  - 1. 5632144   -4- 0. 8246864    1. 7585349
18h  - 1. 0061154    +- 0. 7039354    1.7584833
191  - 0. 4489341    - 0. 5827957    1. 7583923
201  -- 0. 1082514    -- 0.4612784    1.7582614
211)  +0.6653785    +- 0.3393985    1.7580909
22h       1. 2224009    +- 0. 2171603    1.7578799.
1 ~log 2
Exterior contact. Interior contact.  Exterior contact. Interior contact.
17h  0. 5362314    0. 0100548       7. 6626222    7. 6605084,
18"h  0.5362001     0.0100860               23             85
1911  0.5361450     0.0101409               25             87
20h  0.5360655    0.0102198                 26             88
211  0.5359622    0.0103227                 27             89'22h  0. 5358345    0. 0104499              29             91.
Now  the time of the beginning of the total eclipse was
observed at Vienna at:
8hll 491m3 25s. o0
or at the sidereal time:
- 11 52m 29s. 8 - 280 7' 27".0;
Further we have:
P  - 48O  12' 35. 5,
hence the geocentric latitude:
T7-480 1' 8". 9
and:
log  = 9. 9991952.
If we take T1'=  18h 3011, we find for this time:
x = -  0. 727530    y0+O -+- 0.643413,
and by mleans of the formulae (6):
-     - 0.654897    r  -+- 0.635482    log g - 9.606857;
moreover by means of the formnulae in No. 15 of the introduction:
x' — 0.557185    y' = — 0.21140,
hence by means of' the formulae (7), (8) and (9):
M-1    2760 13' 54'   log m — 8.863708
N= 102 15 58    log n - 9.756030
t     = 390 57' 10"
T' --- - 61 40s. 85.
Since in this case it is not necessary to repeat the computation, we obtain by means of the formula (10):
d -  - Oh 12,1 44s. 15 - 1.7553 e + 1.4703 g.




336
In the  same way we find from  the observation of the
end of the total eclipse, if we retain the same value of T:
-= - 0. 653763  -  = + 0. 633338   log  - = 9.612367
-1=  277  46' 40"   log   -7.  8. 871874   logL = 8.078638
- = 1500 54' 51". 5
T' = - -Sm 54s. 74,
hence:
d=  + Oh 12n1 27s.26 - 1.7553  - 0. 9764 g.
Likewise from the observations at Pulkova, since:
=590 46' 18". 6,
and hence:'- 590 36' 16". 8
and:
log eo = 9.9989172
we find the following equations:
d'=11h 8$m 26s. 57 -- 1.7559 E - 0. 5064,
d'= 1 8 22.67 -- 1.7541 E - 0.3034.
We have therefore:
d' --   + 5511 42s.42 - 0.9639 C,
d'-cd —-— 55  555.41t0.6730C,
hence:
d'- d — + 55m11 50s. 07
and:
C= - 7". 94.
In order to find the error a, we must assume the longitude of one place reckoned from  the meridian of Berlin as
known.  But the difference of longitude of Vienna and Berlin is:
+0" 1111 56s. 40
and with this we obtain from the first equation for d:
e- -- 20". 55.
Since we have:
cos A A (a - a) = E sin N -   cos N
A ( -dcl)  E cos N+ — sin N,
we find:
cos d A (a - a)  - 21". 78
and:
A ( - d)  - 3". 38.
30. In the case of occultations of' stars by the moon
the formulae  become more simple.  Since then'== 0, we
have a =a', d-= )'.  Hence we need not compute the formulae (1), and the co-ordinates of the place of observation




337
are independent of the place of the moon, since we have
simply:
=- e cos 99' sin (O - a')
7 =-  [sin 99' cos 5' - cos 99' sill' cos (O - a')].
The third co-ordinate ~ is also not used, since we have
in this case f=O and hence 2 —0, so that we have instead
of the enveloping cone a cylinder. The radius 1 of the circle,
in which the plane of the co-ordinates intersects this cylinder, is equal to the semi-diameter of the moon or equal to k.
Hence we need not compute the co-ordinate z and we have
simply:
cos A sin (a - a')
sin X
sin S cos 5' - cos a sin 5' cos (a - O')
sin 7r
Thus the fundamental equation for eclipses is transformed
into the following;
(k + t  7)2 = (X + A x -  )2 - (y - +  y -- v),
which is solved in the same way as before.  Taking again
t - d =T-+-T' and denoting by x0 and y, the values of x
and y for the time T, by x' and y' their differential coefficients, we must compute the auxiliary quantities:
mn sin 1 — xo -      a n sin N   x'
nm cos -l=yo - 77    n cos N-y'
k sin Wa - n sin (M1- N)
and we find:
--- sin (M-A N~ —) +hhptang,
dt- t    T —   s                -    E    tag,
2        siln ty
where h, E and; have the same signification as before.
Example. In 1849 Nov. 29 the immersion and emersion
of a Tauri was observed at Bilk as follows:
Immersion 81h 15n 12s. 1 Bilk mean time
Emersion  9 18 19.8.
The immersion of the same star was observed at Hamburg at
Sh 3311 47s. 2 Hamburg mean time.
The place of the star on that day was according to the
Nautical Almanac:
a'  4h In 16s. 24 - 620 49' 3". 6
S' —- +150 15' 32". 2.
22




338
Further we have for Bilk:
99'  51~ 1' 10". 0
log  -  9.9991201
and for Hamburg:' - 53 o 22' 4". 2
log o - 9.9990624.
Finally  we have the  following  places of the moon according to the Nautical Almanac:
7h1  4h   m 20.35 -   + 150 47' 24". 6    60' 50". S
81"  4   8 35.69        15 54 48.8    60 51.8
9h  4  11  9.31          16  2  6.5      60 52.9.
Hence we find for those three times:
-           I. D.                      1. Diff.  I. Di.
71h - 1.240980       60675      - 0.527577 1 0 118741
8t 3 - 0.634228                - + 0. 646318
+- 0.606864               +- 0.118656.
9h  - 0.027368       6         +- 0.764974
Now  we have for the time of the immersion at Bilk:
0 -- Oh 491m 29s. 93
- a' -- 500 26' 34". 6
hence:
$ — - 0.484015  and   -        +.64 3216.
Taking then T —- 711 50'1, we obtain for this timle:
xo0- -- 0. 251346   0, -           -0. 016682
x' 0=  O. 606789' --   O.ll118713,
hence:
3i- - 2660 12' 10"     N-  + 780 55' 50"
log mi =    9. 401226   logun "-   9. 791194
y'r=- 60 43' 11"
T'  -- - 2' Os. 85.
We find therefore  from  the immersion observed at Bilk
the  following  equation  between  the  difference  of longitude
from  Greenwich and the errors    a;nd a:
d= -27t" 12s.95 -- 1. 5945  -0.  1879 5,
and  in the  same  way  we find  friom  the emersion  observed
at Bilk:      d =+ — 27nm 27S. 10 - 1.5937 S -+O 0.5336 3,
and fron  the einersion observed at Hamburg:
c'- - 4011 3s 76+ - I. 5945 8 - 0. 1362.
We have therefore the two equations:
d'- d-  1211 50s. 81+- 0. 0517 5,
d' —d=- - 12  36. 66 —0.6698,
whence we find:
d' - d - + 12m 49. 80 and 5 =-19". 61.




339
31.  The fundamental equations for eclipses and occultations given in No. 29 and 30 serve also for calculating the
time of their occurrence for any place.  If we take for T
a certain time of the first meridian near the middle of the
eclipse, and compute for this time the quantities XQ,  xO xi y' I
and L, the fundamental equation for eclipses is:
[xo -- +x' T' — ~] I+ [y -o- y' T' -  t'2 --  L 2 *I),
where 5 and ij are the co-ordinates of the place on the earth
at the time T-  T'.  Therefore if we denote by,, the sidereal time corresponding to the time T1, O  -I- do will be the
local sidereal time of the place, for which we calculate the
eclipse9 and if we denote by  ( and  j,, the values of, antcl 
corresponding to the time  o) — d0,, we have:
d (O - a)
= — ro - e COS 9'O Si (Oo - a -+- dC)        ill d.
2:~7=o qI- ~ cos T' sin (  a - do)  d.si
d T
Therefore taking now:
2. Sill    O — yo,  )t sin 1V —- y' -  cos' sin (O-  a +d,).(   an -
sin l p —- sin (M - N),
/ 0
where L, denotes the valute of L corresponding to the time T.,
we find:
(==       _ -- cos'liN)  cosC  J  =- t-  coT-c?,
where ip muinst be taken in the first or fourth quadrlant, and
the upper sign is used for the beginning, the lower for the
end of the eclipse, or if we take:
120L        L
-- cos (i- N) -    cos  -- 
-   cos (i — -N) --   cos  
the time of the beginning expressed in local mean time is:
t = T — d -+-,
and the time of the end:
t'= T -- l+'.
*) For an occultation we have L — k   0. 2725.
22*




340
By the first approximation we find the time of the eclipse
within a couple of minutes, therefore already sufficiently accurate for the convenience of observers.  But if we wish to
find it more accurately, we must repeat the calculation, using
now  T- — r and T-1- T' instead of T.
It is also convenient to know the particular points on
the limb of the sun (or the moon in case of an occultation),
where the contacts take place.  But if we substitute in
xO-  +xT' and 10  -y'7    y  T'
for T' the value:
-- H- cos (if- N) =F -- cos,
we find:
x -    [ m sin li cos N cos Nsin v -  m cos JlMcosNsin Nsin t
= nm sin M cos N sin N cos  -- m cos M sin N sin N cos iP] sin,
or:
m7 sin (l-N) sin (N p)
n: - 5   -           - =  sill (N~= )
sinl 
=-  L sin (N=T,v)
and likewise:
y — = =F L cos (NA:v).
Hence we have for the beginning of the eclipse:
x -      L sin (N-) = L sin (N — 1850~ -)
y -  - - L cos (N — V) -L cos (N+ 1 0~ - w),
and for the end:
x - $-L sin (N +- i'~)
y -  - L cos(N +- ).
Now  we have seen in No. 29 that  x-z and  j1-y are
the co-ordinates of a place on the earth situated in the enveloping surface of the cone and referred to a system of axes,
in which the axis of z is the line joining the centres of the
two heavenly bodies, whilst the axis of x is parallel to the
equator; hence x —-  and y — j are the co-ordinates of that
point, which lies in the straight line drawn from  the place
on the earth to the point of contact of the two bodies, and
whose distance from  the vertex of the cone is equal to that
of the latter point from  the place on the surface of the earth.
Hence x    and Y -    are the sine and cosine of the angle,
which the axis of y or the declination circle passing through




341
the point Z 4) makes with the line drawn from  Z  to  the
point of contact.  But since this point is always very near
the centre of the sun, we can assume without any appreciable error, that       " -    and -    are the sine and the cosine
L         L
of the angle, which the declination circle passing through
the centre of the sun makes with the line from  the centre
of the sun to the point of contact.  Thus this angle is for
the beginning of the eclipse or any phase of the eclipse:
N+ 180~ --   )
and for the end:                         (A)
Therefore the formulae serving for calculating an eclipse
are as follows. We first compute for the time T of the first
meridian to which the tables or ephemerides of' the sun and
the moon are referred (for which we take best a round hour
near the middle of the eclipse) the formulae (1), (2), (3),
(4) and (5) in No. 29 and the differential coefficients x' and
y', and then denoting by 0o the sidereal time corresponding
to the mean time 7' and by do the longitude of the place
reckoned from  the first meridian  and taken positive when
east, we compute the formulae:
-   e cos p9' sin (O0 q- do - a)
7o   [cos d sin 9T' - sin d cos 9' cos (0o +- do - a)]
- o [sin d sin I' + cos d cos 9' cos (0, -+- do - a)].
Computing then the formulae:
m sin M- xo -,, n sin N__x'- e cos' cos (O0+ do- a) d(oa)
mcosM=yo -O,  n cosNy,'-cos,' sin (Oo+do-~ a) d   (o  -a) sin d
Lo - lo Iedo
sin       s — o sin (M- N) (vy always < 4 900)
no
n                n
m               Lo
I = —- cos (M — N)  -- cos,
n                n
*) The point Z is that point, in which the axis of z or the line joining
the centres of the two bodies intersects the sphere of the heavens.




342
we  find  the  time  of the beginning expressed in local mean
time:
t -Tdo +- 
and the time of the end:
t   T- dT Cl+-t'
The  expressions (A) give  then the particular points on
the limb of the sun, where the contact takes place.
For calculating  an  occultation  the formullae are as foLlows.  We compute again for the timne T of the first meridian,
which is near the middle of the occultation:
cos S sin (a - cc')
sin 7C
sin S cos   -cos S siln' cos (Ca - a)
sill 7t
and  the  differential coefficients' andcl?'.  Further we colmpute, denoting by  0, the sidereal time corresponding  to the
mean tilme T:
o -= ecos go sin (O -'+- do)
no =    [sin 9i' cos Y'- cos' sin X cos (O-c a'4- do)].
Then  we compute:
dO
m sin i-=.,x  -  0, n sin N-x'-  cos'cos(0o +do -a')  ctt
d O
M-cosllI-yo -0o, i cosN-'y- cos 99' sin (0 -+- do -c') (-t sin i",
where:
log       9.41916 *)
cit
sin  -= -  sin (M-N-V), l  4. 900,
and:
log k = 9.43537
m                 Ic
cos (ilf-N)-    cos p= —
--    cos (1 — V)  - - cos   -'.
n                 12
A) As one hour is taken as the unit of the differential coefficients, - O
is the change of the honr angle in one mean hour or in 3609s. S8( of sidereal
time. If we multiply by 15 and divide by 20G265 in order to express the
differential coefficient in parts of the radius, we find:
log  -- 9. 41916.
d i




343
Then the immersion takes place at the local mean time:
tI= T- - +  -t- do
and the emersion at the time:
t= -F':' +'  do.
The angle of position of the particular point on the limb,
where the immersion takes place, is found from:
Q - N — 1800 -1 
whilst for the emersion we have:
Exanmple.  If we wish  to calculate the time of the beginning and end of the eclipse of the sun in 1842 July 7
for Pulkova, we take T =19I' Berlin  mean time.  For this
time we have according to No. 29:
x0   - 0.44893,,   +  O.58280, x'- = -- 0.55718, y'  - o. 2133
a - 106~ 55'. 8,  d — 220 32'.8, 1- 0.53614,  log - 7. 66262.
Then we have:
o -— = - h 3m1 Ss,
and since the difference of' longitude between Pulkova and
Berlin is equal to -- 1.1 7fm 43s, we get:
0, - d - a -c   3000 46'. 9,
and with this:, =- 0.43361,,o --- +0.69560, log  -9.75470, log Lo - 9.72716.
Further we find:
=     cos p' cos (0o, t do - a) d(0o -a)  -+ 0.06762 *)
dt                                dt
ed cos'o sin (0,, + do   a) d(O-) sin d= - 0.04352,
dt                               cdt
hence:
x ---- t-c     0.48956 and y        --  0.07781.
ct                         dt
") We have:
d   = - 3609s.86
dt
or01':
- + 57147". 90;
Further we have:
da
--  -- 148".78
dt
hence:
d (0o — a) _ 56999". 12,
e logithn  of  hich  essed in pts of the   
the logarithm of which number expressed in parts of the radius is 9.41796.




344
Then we get:
M —- 187 44'. 1    N — 990 1'. 9
log m = 9.05628    log n -- 9.69522
sj  120 19'.0
hence:
-- -- 1.057     1'  1.046
=      _111 311.4  _ -4 3h 2m. 8,
therefore the beginning and the end of the eclipse occur at
the times:
t = — 19h 4m.3
t' — 2lh Om.5.
These times differ only 3m from  the true times.  If we
repeat the calculation, using T-  18h and T=20h, we should
find the time still more accurately.
The angle of position of the point on the limb of the
sun, where the eclipse begins, is 267" and that of the point,
where it ends, is 11H1~).
32. Another method for finding the longitude is that
by lunar distances, and since this can be used at any time,
whenever the moon is above the horizon, it is one of the
chief methods of finding the longitude at sea.
For this purpose the geocentric distances of the moon
from the sun and the brightest planets and fixed stars are
given in the Nautical Almanacs for every third hour of a
first meridian. If now  at any place the distance of the moon
from one of these stars or planets has been measured, it is
freed from refraction and parallax, in order to get the true
distance, which would have been observed at the centre of
the earth. If then the time of the first meridian, to which
the same computed distance belongs, is taken from  the Almanac, this time compared with the local time of observation
gives the difference of longitude. But since it is assumed
here, that the tables of the moon give its true place, this
method  does  not afford  the  same  accuracy  as that obtained by corresponding observations of eclipses. Besides the
~) Compare on the calculation of eclipses: Bessel, Ueber die Berechnung
der Lange aus Sternbedeckungen. Astr. Nachr. No. 151 and 152, translated
in the Philosophical Magazine Vol. VIII and Bessel's Astronomische Untersuchungen Bd. II pag. 95 etc. W. S. B. Woolhouse, On Eclipses,




345
time of the beginning and end of' an eclipse of' the sun can
be observed with greater accuracy than a lunar distance.
In order to compute the refraction and the parallax of
the two heavenly bodies, their altitudes must be known. Therefore at sea, a little before and after the lunar distance has
been taken, the altitudes of both the moon and the star are
taken, and since their change during a short time can be
supposed to be proportional to the time, the apparent altitudes for the time of observation are easily found and from
these the true altitudes are deduced.
A greater accuracy is obtained by computing the true
and the apparent altitudes of the two bodies. For this purpose the longitude of the place, reckoned from the first meridian, must be approximately known, and then for the approximate time of the first meridian, corresponding to the time
of observation, the places of the moon and tile other body
are taken from the ephemerides. Then the true altitudes are
computed by means of the formulae in No. 7 of the first
Oseetion, and, if the spheroidal shape of the earth be taken
into account, also the azimuths. The parallax in altitude is
then computed by means of the formulae in No. 3 of the
third section, the formulae used for the moon being the rigorous formulae:
sin p' =  sin p sin [z - (- a') cos A]
A- cos p'   1 -   sin p cos [z - (p - 99') cos A],
and finally for the altitudes affected with parallax the refraction is found with regard to the indications of the meteorological instruments.  But since the apparent altitude,
affected with parallax and refraction, ought to be used for
computing the refraction, this computation must be repeated.
The distance of the centres of the two bodies is never
observed, but only the distance of their limbs. Hence we add
to or subtract from  the observed distance the sum  of the
apparent semi-diameters of the two bodies, accordingly as the
contact of the limbs nearest each other or that of the other
limbs has been observed. If r be the horizontal semi-diameter
of the moon, the semi-diameter affected with parallax will be:




346
[rI- r [1 -Vp sin li,
where p is the horizontal parallax expressed in parts of the
radius.
Now  since refraction dininishes the vertical semi-diameter of' the disc, while it leaves the horizontal semi-diameter unchanged, that in the direction of the measured distance
will be the radius vector of an ellipse, whose major and minor axis are the horizontal and the vertical diameter.  The
effect of refraction on the vertical diameter can be computed
by means of the formulae given in VIII of the seventh section, or it cal be taken from  tables which are given in all
Nautical works.  If we denote by T the angle, which the
vertical circle passing through the centre of the moon makes
with the direction towards the other body, by h' the altitude
of the latter and by A the distance between the two bodies,
we have:
sin (A'- A) cos h'
sin g -'
sin A
and:
sin A' - cos A sin h
cos 7: ---
sill A cos h
hence:
I     ctan1 os (  -t- h   1') sill (' - h A    7+-')
tang  7 r'   --  _          1 
2  sin 4 (A +  A'  h) cos,[ (h -( - ) 
Then if we denote the vertical and the horizontal semidiameter by b and a, we find  by means of the equation of
the ellipse:
b
r  V cos 2+ -_ sin r'2
a(2
After the apparent distance of the centres of the bodies
has thus been found, the true geocentric distance is obtained
by means of the apparent and true altitudes of the two bodies.  For if we denote by H', h' and A' the apparent altitudes and the apparent distance of the two bodies and by
E the difference of their azimuths, we have in the triangle
between the zenith and the apparent places of the two bodies:
cos A' — sin HI' sin h/' + cos H' cos h' cos E
= cos (H'-   h') - 2 cos H' cos hA' sin, E.
Likewise we have, denoting by H, h and A their true
altitudes and the true distance:




347
cos A   sin HIsin h  + — cos lIecos h cos E
cos (HI- h) -2 cos Hcos ih sinl  E2,
and if we eliminate 2 sin   E2 we find:
cos H cos hi
cos A   cos (H_- h) +- C eos' 1c o      CO ([OH'S  1' )]-   (a)
cos f~ cos ens
If we take now:
cos II cos li 1
cos H' cos h'   C(A)
we shall have alwcays C>  1I  except when the altitude of the
moon is great and the other body is very nearo the horizon.
If we then take:
H' - A'  d' and I — h  - d          (B)
mand take d' and d positive, we can always put:
cns ci'            cos A'
-col -   c-os d  and    - - cos l"   (C)
C                  C'
because in case that C <1, both cos d' and cos A' are small.
Thus the equation (a) is transformed into:
cos A - cos A"'-  cos d - cos d"
or if we introduce  the  sines of half the sum  and half the
difference of the angles and write instead of sin (A -- A") the
are itself:
A-/\ - A"(dl- d") sin 2 (d + cd")
sin (- (A- + A")
If we take here at first sin  (' i- A") instead of sin 2 (A ~A")
and put:
sin 1 (d ~4 ci")(D
X,(d -d )   2l"   =(D)
sin - (A' + )
we obtain:
a value    is only, but in most cases
a value which is only approximately true: but in most cases
sufficiently accurate. If A should differ considerably from A',
we must repeat the computation and find a new  value of x
by means of' the formula:
x: (c-") sin { (d - cd")
x = (d- )sin  (A+ A)
We have  assumed  here that the angle E  as seen from
the centre of the earth is the same as seen from  a place on
the surface. But we have found in No. 3 of the third section,
*) Bremicker, fiber die Reduction der Monddistanzen. Astronomische
Nachrichten No. 716.




348
that parallax changes also the azimuth of the moon and that,
if we denote by A  and II the true azimuth and altitude, we
have to add to the geocentric azimuth the angle:
\ A --- +- q sin p (9- - 9=') sin A
cos H
in order to find the azimuth as seen from a place on the surface of the earth.  Therefore in the formula for cos A we
ought to use cos (E —AA) instead  of cos E -  cos (A — a),
or we ought to add to A the correction:
d A ~ — cos Hcos h sin (A - a)
sin AL
or:
d    o sinp (c -- T') cos h sin A sin (A -- a)
sin A
Example.  In  1831 June 2 at 23h 8rn 45s apparent time
the distance  of the nearest limbs of the sun and the moon
was observed A' -- 960 47' 10" at a place, whose north latitude was 19~ 31', while the longitude from  Greenwich was
estimated at 81" 50nm. The height of the barometer was 29.6
English inches, the height of the interior thermometer 88"
Fahrenheit, that of the exterior 90" Fahrenheit.
According  to  the Nautical Almanac the places of the
sun and the moon were as follows:
Greenwich m. t.    right asc. (C        decl. (          parallax
June 2   12h     3360 6' 24". 0       10~ 50' 58". 0   56'44". 0
13h          38 4.7              41 48.4         45.9
14h     337   9 45. 7           32 35.0        47.9
15h          41 27.0            23 17. 9       49.9
right asc.      )       decl. 0
June 2   12h   700 5' 23". 2     - 220 11'48". 9
13h       7 56. 9            12  8.4
14h      10 30.5              12 27.9
15h      13  4.1              12 47.3
The time of observation corresponds to 14h 18m 45s Greenwich time and for this time we have:
right asc. (I   3370 19' 39". 6    right asc.    -  700 1l' 18". 5
decl. -- 10 29 41.3           decl. =)- 22  12 33.9
P-        56 48.5                8. —-     8".5.
From  this we find the true altitude and azimuth of the
moon and the sun for the hour angles:
+ 800 2' 53".8




349
and:                     - 120 48' 45". 0:
H- 50 41' 58". 4     h -770 43' 56".7
A  + -+- 760 43'.6  c a — -750 4'.4.
The parallax  of the moon  computed by means of the
rigorous formula:
I    o sin p sin [z - (R - 9o') cos A]'tangp p' =,o sin p cos [z - (?p - T') cos A]
is p'-=56'35".4, hence the apparent altitude I' of the moon
is 4" 45' 23".0.  In order to find the refraction, we first find
an approximate value for it, and applying it to H', we repeat
the computation of the refraction with regard to the indications  of the  meteorological instruments.  We then  find
e-9'3".2  and hence the apparent altitude affected with refraction:
H'- 40 54' 26". 2.
For the sun we find in the same way:
7'  770 44' 6".5.
Further we find the semi-diameter of the moon by multiplying the horizontal parallax by 0.2725 and obtain:
r - 15' 28". 8
and from  this the apparent semi-diameter, as increased by
parallax:
a- 15' 30". 1.
The vertical semi-diameter is diminished  26".0  by the
refraction, and the angle 9/ being 5" 48', the radius of the
moon in the direction towards the sun is:
a'- 15'4". 6,
and since the semi-diameter of the sun was 15'47".0, the
apparent distance of the centres of the sun and the moon is:'  - 970 18'1".6.
Further we find by means of the formulae (A), (B) and (C):
log C — 0.000463
d -- 720 1' 58"
d'-72 49 40
d"   72 50 48
A"= 97 17 33
and at last, computing x twice by means of the formulae (D)
and (E), we find the true distance of the centres of the sun
and the moon:
A - 960 30' 39".




350
Now  we find according to the Almanac the true distance of the centres of the bodies for Greenwich apparent
tilile from the following table:
121" 970 43' 0". 4
13      13 41. 5
14h  96 43 6.5
15hl    13 6.2,
whence we see, that the distance 960 30' 39" corresponds to
the Greenwich apparent time 141 24z1155'. 2  and since the
time of observation was 231 18' 45s.0   the longitude of the
place is:
81 4311 49s. 8 east of Greenwich.
The longitude which we find here is so nearly equal to
that, which was assunled, that the error which we made in
computing the place of the sun anld mroon can only be small.
If the difference had been considerable, it would have been
necessary to repeat the calculation  with the places of the
sun and moon, interpolated for 14' 241m 55' Greenwich timne.
Bessel has given in the Astronomische N;achrichten No. 220
another method'), by which the longitude can be found with
great accuracy by lunar distances.  But the method given
above or a, similar one is always used at sea, and on land
better methods can be employed for finding the longitude.
33.  An excellent way of finding the longitude is that
by lunar culminations.  On account of' the rapid motion of
the moon the sidereal time at the time of its culmination is
very different for different places. Hence if it is known, how
much the righllt ascension of the moon chanoges in a certain
time, the longitude can be determined by observing the difference of the sidereal times at the time of' culmination of
the moon.  Since these observations are made onil the mleridian, neither the parallax nor the refraction will have any
influence on the result. In order to render it also independent of the errors of the instruments, the time of culmination
of' the moon itself is not observed at the two stations, but
rather the interval of time between the time of culmination
of the moon and that of some fixed stars near her parallel.;) The example given above is taken from this paper.




A  list of such stars is always published in the astronomical
almnalacs, in order that the observers may select the same
stars.
The method was proposed already in the last century
by Pigott, but was formerly not much used, because the art
of observing had not reached that high degree of accuracy
which is required for obtaining a good result.
Let a be the right ascension of the moon for the time T
of a certain first imericldian, and the differential coefficients
for the same time be d-;  d/, etc.  We will then suppose,
that at a place whose longitude east of the first meridian
is d, the time of culimination  of the mooln  was observed
at the local time T —t-+-d, corresponding to the time T+-tof the first meridian. Then the right ascension of the mloon
at this time is:
dcn~ 1i'2 Cy~ t2~/3 
c + <4 -t  I- _- t 2 —+' t3 ---- +.
d t      idt2     cit3
If' likewise at another place, whose longitude east from
the first meridian is d', the time of culmination of the moon
was observed at the time T -t' —1- d', corresponding to the
time T - t' of the first meridian,  the right ascension of the
moo00   for this time is:
c r t'.d-+ a t'2  ---  t' I3- +
tld         clt2 dO    c.t3
Now  since these observations are made on the meridian,
the sidereal times of observation are equal to the true right
ascensions of the moon. If we assumle, that the tables, from
which the values of r and the differential coefficients have
been taken, give the right ascension of' the moon too small
by A e and if we put:
T-+- t -+- d = 
and                      1+ t' + d'',
we have the following equations:
=da. +ld +Id  I  C13 t3
a,  + A a+ t    + 2,2 C- 1 +  ct13   +
dx               c da(a
c it  2       dt2
hence:
o' o   = (t't) -d-7 + 2 (t" -t-) d- +...    (a)




352
and since we have also:
d' - d  (' — o) )-(t' - t),        (b)
it is only necessary to find t'- t by means of the equation (a).
In order to do this, we will introduce instead of T the arithmetical mean of the times T —t and T-I-t', that is, the time
T-~s(t-i-t') which we will denote by T'.  Then we must
write T'-  (t'-t) and T' —(t'-t) in place of T —-t and
dea
T-+-t', and if we assume, that the values of a and of dt etc.
belong now  also to the time T", we have the equations:
cia         Cd2 a            d3 a
-= a +-  a -  (t -t)  -a-  (t'  t)  (t'  t)      3
dt           d  2   4        dt3
dt           cit2            dt'
and hence:
d cia         d3 a
9' -     t- O-   d (t'- t)      - (t'-t) 3
From the last equation we can find t'- t, if at first we
neglect the second term of the second member and afterwards
substitute this approximate value of t'- t in that term. Thus
we find:
I   — O,    -[O' 0 3d
t' -- t         -- at        13  (c)
da          d       ct
cit         dt
If the difference of longitude does not exceed two hours,
the last term is always so small, that is may safely be neglected. The solution of the problem is again an indirect
one, since it is necessary to know already the longitude approximately in order to determine the time T'.
For the practical application it is necessary to add a
few remarks.
If 0 and O' are given in sidereal time, 0' — 0 is expressed in sidereal seconds. Thus in order to find also t'- t
expressed in seconds, the same unit must be adopted for
-at or -d must be equal to the change of right ascension in
dtl   dt
one second of time. Therefore if we denote by h the change
of the right ascension expressed in are in one hour sidereal
time, we have:
da,   h
dt  15 3600




353
Now in the ephemerides the places of the moon are not
given for sidereal time but for mean time, and we take from
them the change of the right ascension of the moon in one
hour of mean time. But since 366.24220 sidereal days are
equal to 365.24220 mean days or since we have:
one sidereal day =0.9972693 of a mean day
we find, if h' denotes the change of right ascension expressed
in time in one hour of mean time:
da  0. 9972693             (d)
(d)
dt    3600
15 X 3600' -- 0
hence:            t  t - - 60   00.9972693    h
or from the equation (b):
a' — d=(o' -o) (1 -     9972693'
Now  the second term  within the parenthesis is always
greater than 1, and hence it is better to write the equation
in this way:
d- d' - (0'-  ) ( 0 972693h2 -1',   (e)
and the second place, at which the moon was observed at
the time 6', is west from the other place, if O'- 6 is positive, and east, if 0'- 6 is negative.
Now the time of culmination of the moon's centre cannot be observed, but only that of one limb; hence the latter
must be reduced to the time, at which the culmination of
the centre would have been observed. In the seventh section
the rigorous methods for reducing meridian observations of
the moon will be given, but for the present purpose the following will be sufficient.  We call the first limb the one
whose right ascension is less than that of the centre, the
second limb the one, whose right ascension is greater. Hence
if the first is observed, we must add a correction in order
to find the time of culmination of the centre, and subtract a
correction, if the second limb is observed, and this correction
is equal to the time of the moon's semi-diameter passing
over the meridian, which according to No. 28 of the first
section is equal to Icos  1 —'  where X is equal to the value
of dt as given by the formula (d). Therefore if a9 and Y'
23




354
denote the times at which the moon's limb was observed on
the meridian of the two places, we have:
O1 5 —&COS -A  COS  1 —,  I
0.9972693 h'
where A -- 
3600
and hence we find from formula (e):
15 (COS  cos 6     2(         i)i  (A)
where h' denotes the change of the right ascension of the
moon expressed in time during one hour of mean time and
where the upper sign must be used, if the first limb is observed, whilst the lower one corresponds to the second limb.
If the instrument, by which the transit is observed at
one place, is not exactly in the plane of the meridian of the
place, then the hour angle of the moon at the time of observation is not equal to zero, and if we denote it by s, the
difference of longitude which we find, must be erroneous by
the quantity:
15x3600  _.
0.9972693 it'
Therefore if the instrument is not perfectly adjusted, the
longitude found by this method, can be considerably wrong.
But any error arising from this cause is at least not increased,
if the differences of right ascension of the moon and stars
on the same parallel be observed at both places, since these
are free from any error of the instruments. Nevertheless since
the right ascension of the moon was observed at one place
when its hour angle was s, or when it was culminating at
a place, whose difference of longitude from that place is equal
to s, we find of course the difference of longitude between
the two places wrong by the same quantity. Therefore we
must add to it the hour angle s, if the meridian of the instrument lies between the meridians of the two places, and
subtract s from the difference of longitude, if the meridian of
the instrument corresponds to that of a place which is farther from the other place *). How the hour angle s is found
*) We can add also to the observed difference of right ascension of the
moon and the star the quantity =- 1




355
from  the errors of the instrument, will be shown in No. 18
of the seventh section.
In order that the observers may always use the same
comparison stars, a list of stars under the heading moon-culminating stars is annually published in the Nautical Almanac
and copied in all other Almanacs, for every day, on which
it is possible to observe the moon on the meridian.
Example.  In 1848 July 13 the following clock-times of
the transit of the moon and the moon-culminating stars were
observed at Bilk*):
q Ophiuchi    1711 1n, 52s. 64
v Ophiuchi        12  6. 59
moon's centre       27 34. 60
it' Sagittarii    18  4 52. 99
2 Sagittarii      18 48.12.
On  the same day the following transits were observed
at Hamburg:
v Ophiuchi-  171 Ilm42s. 61
e Ophiuchi -      11 56.91
(( I. Limb =      25 50.43;s' Sagittarii -   18  4 43. 53
2 Sagittarii -    18 38. 56.
The semi-diameter of the moon for the time of culmination at Hamburg was 15'2". 10, the declination - 180 10'. 1,
and the variation of the right ascension in one hour of mean
time equal to 1298.8, hence X-=0.03596. We find therefore:'       R          65s 66,
(1 -2) cos.
hence the time of culmination of the moon's centre:
17h 26m 56s. 09.
Then we find the differences of right ascension of the
stars and the moon's centre:
for Bilk:               for Hamburg:
n Ophiuchi - 25'n 41s. 96         + 25m 13s. 48
e Ophiuchi + 15 28.01           + 14 59.18,u' Sagittarii -37  18.39        -37  47.44
2 Sagittarii — 51  13.52        -51 42.47,
hence the differences of the times of culmination at Bilk and
at Hamburg are:
*) Compare No. 21 of the seventh section.
23*




356
O'-o=  +-28s.48
28. 83
29.05
28.95
mean -t- 28s. 83.
Now we have found in No. 15 of the introduction the
following values of the motion of the moon in one hour for
Berlin time:
10h    +- 2m 9s. 77
11h       2  9.91
12h       2 10.05,
and since the time of observation at Bilk corresponds to
about 101l 30m Berlin time, that at Hamburg to about 10h 161m,
we have:
T' -- 10h 23m
hence:
h'  2m 9s. 82
and we obtain by means of the formula (e):
d - d' — +- 12m 52s. 83 *).
*) Since h is about 30', the value of the coefficient of' —v  in the
equation (A) is about 29, hence the errors of observation have a great influence on the difference of longitude, since an error of Os. 1 in Y'- -   produces an error of 3s in the longitude.




SIXTH SECTION.
ON THE DETERMINATION OF THE DIMENSIONS OF THE EARTH
AND THE HORIZONTAL PARALLAXES OF THE HEAVENLY
BODIES.
In the former section we have frequently made use of
the dimensions of the earth and the angles subtended at the
heavenly bodies by the semi-diameter of the earth or their horizontal parallaxes, and we must show now, by what methods
the values of these constants are determined. Only the horizontal parallax of the sun and the moon is directly found
by observations, since the distances of planets and comets
from the earth, the semi-major axis of the earth's orbit being
the unit of distance, are derived from the theory of their
orbits, which they describe round the sun according to Kepler's laws. Therefore in order to obtain the horizontal parallaxes of those bodies, it is only necessary to know the horizontal parallax of the sun or of one of these planets.
I. DETERMINATION OF THE FIGURE AND THE DIMENSIONS OF
THE EARTH.
1. The figure of the earth is according to theory as
well as actual measurements and observations that of an oblate spheroid, that is, of a spheroid generated by the revolution of an ellipse round the conjugate axis.  It is true,
this would be strictly true only in case that the earth were
a fluid mass, but the surface of an oblate spheroid is that
curved surface which comes nearest to the true figure of the
surface of the earth.




358
The dimensions of this spheroid are found by measuring
the length of a degree, that is, by measuring the linear dimension of an arc of a meridian between two stations by
geodetical operations and obtaining the number of degrees
corresponding to it by observing the latitudes of the two stations. Eratosthenes (about 300 b. Ch.) made use already of
this method, in order to determine the length of the circumference of the earth which he supposed to be of a spherical
form. He found that the cities of Alexandria and Syene in
Egypt were on the same meridian. Further he knew that
on the day of the summer solstice the sun passed through
the zenith of Syene, since no shadows were observed at noon
on that day, whence he knew the latitude of that place. He
observed then at Alexandria the meridian zenith distance of
the sun on the day of the solstice and found it equal to 70 12'.
Hence the arc of the meridian between Syene and Alexandria must be 70 12' or equal to the fiftieth part of the circumference. Thus, since the distance between the two places
was known to him, he could find the length of the entire
circumference. But the result, obtained by him, was very
wrong from several causes. First the two places are not on
the same meridian, their difference of longitude being about
3 degrees; further the latitude of Syene according to recent
determinations is 240 8', whilst the obliquity of the ecliptic at
the time of Eratosthenes was equal to 230 44', *and lastly the
latitude of Alexandria and the distance between the two places was likewise wrong. But Eratosthenes has the merit of
having first attempted this determination and by a method,
which even now is used for this purpose.
Since Newton had proved by theoretical demonstrations,
that the earth is not a sphere but a spheroid, it is not
sufficient to measure the length of a degree at one place on
the surface in order to find the dimensions of the earth, but
it is necessary for this purpose to combine two such determinations made at two distant places so as to determine
the transverse as well as the conjugate axis of the spheroid.
In No. 2 of the third section we found the following
expressions for the co-ordinates of a point on the surface,
referred to a system of axes in the plane of the meridian,




359
the origin of the co-ordinates being at the centre of the earth
and the axis of x being parallel to the equator:
a cos 99
x
l/_s 2 Sin 992
a sin 99 (1 —2)
Y   /1- E2 sin T2
where a and E denote the semi-transverse axis and the excentricity of the ellipse of the meridian, and ~(f is the latitude
of the place on the surface.
Furthermore the radius of curvature for a point of the
ellipse, whose abscissa is x, is:
3
(at2 - _2 x2)a
a b
where b denotes the semi-conjugate axis, or if we substitute
for x the expression given before:
a (l-~2)
r -3
(l-E2 sin 92)
Therefore if G is the length of one degree of a meridian
expressed in some linear measure and q)J is the latitude of
the middle of the degree, we have:
ra (1 - E2)
180 (1 - s sin 992)'
where 7T is the number 3.1415927. If now the length of
another degree, corresponding to the latitude p' has been
measured, so that:
GI       a (1 -a  e)
180 (1 - s2 sinT'2)Y'
we obtain the excentricity of the ellipse by means of the
equation:
1    G )3
sin 99 2_      G )3 sin P2
and when this is known, the semi-transverse axis can be
found by either of the equations for G or G'.
Example.  The distance of the parallel of Tarqui from
that of Cotchesqui in Peru was measured by Bouguer and




360
Condamine and was found to be equal to 176875.5 toises.
The latitudes of the two places were observed as follows:
-30 4' 32". 068
and
+ 0 2' 31". 387.
Furthermore Swanberg determined the distance of the
parallels of Malhrn and Pahtawara in Lappland and found
it to be equal to 92777.981 toises, the latitudes of the two
places being:
650 31' 30".265
and
67 o 8 49". 830.
From  the observations in Peru we obtain the length of
a degree:
-- 56734.01 toises,
corresponding to the latitude
9 -    Io 31' 0". 34,
and from the observations in Lappland we get:
9q - 660 20' 10".05:
G' — 57196.15 toises.
By means of the formulae given above we find from this:.2  0.0064351
a 3271651 toises,
and since the ellipticity of the earth a is equal to 1- 1/1 —2,
we obtain:
1
a  310.29*
In this way the length of a degree has been measured
with the greatest accuracy at different places. But since the
combination of any two of them gives different values for
the dimensions of the earth on account of the errors of observation and especially on account of the deviations of' the
actual shape of the earth from that of a true spheroid, an
osculating spheroid must be found, which corresponds as
nearly as possible to the values of the length of a degree as
measured at all the different places.
2. The length s of an arc of a curve is found by means
of the formula:
I-f   + d/. dx.




361
If we differentiate the expressions of x and y, given in
the preceding No. with respect to p and substitute the values
of dx and dy in the formula for s, we find the expression
for the length of an are of a meridian, extending from the
equator to the place whose latitude is q,:
s- a (l- S2)     fd
(1 — E2 sin Tz2)2
But we have:
3                 35           3 5 7
[1l —saIsin   =_+2] —{' —lq —2s -- 1        
1.2          1.2.8
and if we introduce instead of the powers of sin cp the cosines of the multiples of cp and integrate the terms by means
of the formula:
fcos xx dx   + sin,x
we obtain:
s - a (1 - s2)E [-  a     in2    -sin 4 -f etc.],
where:
3     45      175
E   1   4  64     256
3      15      525
E 8       32 84 1024
154   105
E/-=     ~+         6+...
256     1024 s
If we take here 2 —  180~, we obtain, denoting by g the
average length of a degree:
180 g'  a (1 -_  2) E.. ~7,
and hence:
l80g
s8      [- - a sin 2 Tp - fI sin 4  -..]
Therefore the distance of two parallels whose latitudes
are gp and (5', is:
180g [99s' = s  - - I       -   2 a sin (p' -- T) cos (s' +- p)
- 2 fi sin 2 (p' -- 9) cos 2 (9' +- 99)],
or denoting  t'- - p by I and the arithmetical mean of the
latitudes by L, also expressing 1 in seconds and denoting
206264.8 by w, we find:
(s' - s) =  -2 w a sin 1 cos 2 L + 2 w isin 2 l cos 4 L.
If we substitute here for 1 the difference of the observed
latitudes and for s'- s the measured length of the arc of




362
the meridian, this equation would be satisfied only in case
that we substitute for g and E and hence for g, a and 13
some certain values.  But if we substitute the values, deduced from  the observations at all different places, we can
satisfy these equations only by applying small corrections to
the observed latitudes.  If we write thus c~ +-x and  r,'+- x'
instead of cp and q', where x and x' are small quantities
whose squares and products can be neglected, we obtain,
neglecting also the influence of these corrections upon L:
3600 (s' - s) -- I  2woa sin 1cos 2 L +- 2w  sin 21 cos 4L +- (x'- x) e,
where:
q    - 1 -2 acrcos cos 2 L-+4, cos 2 lcos 4L.
Hence we have:
V g
and a similar equation is obtained from every determination of
the latitudes of two places and of the length of the are of
the meridian between their parallels. Therefore if the number of these equations is greater than that of the unknown
quantities, we must determine the values of g and E so that
the sum  of the squares of the residual errors x' —x etc. is
a minimum.  If we take go and a,, as approximate values of
g and a and take:
g —--- and a —ao ( +- k)
we find, if we neglect the squares and the products of i
and k:
1 3600              2 t
x'-x     -      (s- s)   ) + —[ao sin I cos 2 L-,fo sin 2 1 cos 4 L]
e   go w
1 3600           2w          dfio
I +-.    (s'I- s) i -     [ao sin I cos 2 L - a o d  sin 21 cos 4L] k.
Q         go                          d o
Here rio denotes the value of (3 corresponding to a0,
but in order toget this as well as the differential coefficient
d6 — we must first express fi as a function of a. Now we find:
da,
3 2 +         -15  5256
8    1632    1024
3 e"  45      175
3      3      111
E2- q2 + 4  __    e6 +-...
8      1-   1024




363
and likewise:
15 —  4   156 
256     256
If we reverse the series for a we find:
2   8   32 2       3
2 -— a - -2 + 4   -...
3      9
and if we introduce this in the expression for fi:
5      35
- 5c2+ - a4+...
hence:
d_   5  2   354
a     5 a2 + - a4..
da   6       27
Therefore if we put:
1 (3600 (s'- s)   )
e   go
2 [o(5  s35\2in4Lcos1    (A)
-i-2 zvl [-a, sin I cos 2    ao          sin 2  cos 4 L]    (A)
s2L 2 a0~1080           C
1 3600
a-         (s -s)
and:
=2 wl[a0 sin I cos 2 L - (6  o 2 +   o 4) sin 2 1 cos 4 L],
we obtain the equation:
x'- x - n       a i - b k,            (B)
and a similar equation  is found from  a set of observations
for measuring a degree by combining the station which is
farthest south with one farther north.
If we treat these equations according to the method of
least squares, the equations for the minimum with respect to
x, i and k are for this set of observations, if yu is the number of all observed latitudes:
oxr+ [a] i+  [b] k+ [n] =0
[a] x    + [act] i + [a b] k + [an] = 0
[b] x + [a b] i  [b b] k + [b n]=0,
and if we eliminate x, each set of observations gives the most
probable values of i and k by means of the equations:
0O= [anj] + [a a, ] i + [a b, ] ik
O = [bn,]  [ab,] i-+[bb,] k.
Therefore if we add the different quantities [an,] which
we obtain  from  different sets of observations made in different localities and designate the sum by (an,), likewise




364
the  sum  of all quantities  [aa,] by (aal) etc., we find the
equations:
O   (an,) + (aa,) i -+- (abl) k
O  (bnj) -+- (ab,) i -+- (b bt) kc.
from  which  we derive  the most probable values of i and k
according to all observations made ill different localities.
As an example we choose the following observations:
1) Peruvian arc.
Latitude           I
Tarqui   - 30 4' 32". 068                  Distance of the parallels
Cotchesqui +0 2 31 387    30 7' 3". 45           176875.5 toises
2) East Indian arc.
Trivandeporum  -+ 11 0 44' 52". 59
Paudru       13 19 49.02    1~ 34'56.43          89813.010.
3) Prussian arc.
Truns      540 13' 11". 47
Konigsberg   54 42 50.50    0" 29' 39". 03        28211.629
Memel       55 43 40.45    1 30 28.98          86176.975.
4) Swedish arc.
Mal6rn      650 31' 30". 265
Pahtawara   67  8 49.830   1" 37' 19". 56        92777.981.
Taking now:
57008          1 +- k
g-" 1 + -i and a   — 400
we find:
log a = 7. 39794
log   5 Lao2 +  35 ao]4.41567
log L  ao2 +      ao2- 04 4.71670.
If further we put:
10000 i- y
10  IC-z,
we obtain the following equations for the four arcs:
1) x', - x,   - - 1". 97 +    1.1225 y - 5.6059 z
2) x'2 - x2  +  0. 94 - 0.5697 y -+ 2.5835 z
3) x' - X3  -. 37   0.1779 y - 0.2852 z
X1"3 X- x     3. 79 + 0.5433 y - 0.9157 z
4)  X4- X4 = - 0. 51 - 0.5839y -   1.9711 z




365
and from  these we find:
[n]       [a]        [b]       [an]       [aa]       [a b]
1) -1".97  -+-1.1225  + 5.6059  +- 2.2113  +-1.2600  +-6.2924
2)  +0. 94  + 0.5697  + 2.5835  -+ 0.5355  + 0.3246  + 1.4718
3)  + 3.42  - 0.7212  -1.2009  + 1.9933  + 0.3268  - 0.5482
4)  — 0.51  + 0.5839  -1.9711  — 0.2978  + 0.3409  - 1.1509
[b n]         [b b]
1)  +11.0436    +- 31.4254
2)  +  2.4284        6.6742
3)  -  3.3650         0.9198
4)  +  1.0026         3.8853
and:
[an]             [ aa]            [ab]
1)  +1.1056          +- 0.6300         +- 3.1462
2)  + 0.2678         - 0.1623          + 0.7359
3)   - 1.1711        + 0.1534           -0.2595
4)  — 0.1489          + 0.1705          -0.5755
(an,) —=- 2.3956, (aa,) —-+ 1.1162, (ab,)+ +3.0471,
[bn,]             [bb,]
+ 5.5218           - 15.7127
+- 1.2142        +  3.3371
- 1.9960         +-  0.4391
+- 0.5013         - 1.9426
(bnl)   - 5.2413, (b b,1)4- 21.4315.
Hence the  two  equations by which y and z are found,
are:             o-+ 2.3956 - 1.1162y -   3.0471 z
0 --- 5.2413 -+ 3.0471y  - 21.4315 z,
and we find:
z = +- 0.099012
y- - 2.4165,
hence:
i -   0.00024165 and k = - 0.0099012;
therefore:
57008
1 - 0.00024165   57021.79
and:
1 + 0.0099012
~a   ~ ~ —     -0.002524753.
400
Now since we had before:
8     32
2 -_   --     2-   + 4 a3
3      9
we find:
2 -- 0.006710073,
and the ellipticity of the earth  2971  5
297. 53




366
Moreover we have:
log -   log /1 -2 = 9.9985380,
and since we had:
180g
(1 — e2) E
we find:
log a  - 6.5147884,
and:
log b - 6.5133264.
In this way Bessel*) determined the dimensions of the
earth from 10 arcs, and found the values, which were given
before in No. 1 of the third section:
the ellipticity    a - 299  1528
the semi-transverse axis a —-3272077.14 toises
the semi-conjugate axis b — 3261139.33
log a-  6.5148235
log b  6.5133693.
II. DETERMINATION OF THE HORIZONTAL PARALLAXES OF THE
HEAVENLY BODIES.
3. If we observe the place of a heavenly body, whose
distance from  the earth is not infinitely great, at two places
on the surface of the earth, we can determine its parallax
or its distance expressed in terms of the equatoreal radius
of the earth as unit. Since the length of the latter is known,
we can find then the distance of the body expressed in terms
of any linear measure.
We will suppose, that the two stations are on the same
meridian and on opposite sides of the equator, and that the
zenith distance of the body at the culmination is observed
at both stations. Then the parallax in altitude will be for
one place according to No. 3 of the third section:
sin p' =   sin p sin [z - (p - p')],
where p is the horizontal parallax, z the observed zenith distance cleared from  refraction, q the latitude, T' the geocen*) In Schumacher's Astronomische Nachrichten No. 333 and 438.




367
tric latitude and e the distance of the place from  the centre
of the earth.  Hence we have:
1    g esin [z- (q -')]
sin p       sin p'
We have also, if A' is the latitude of the other place,
q'I and (, the geocentric latitude and the distance from the
centre:
1          sin [zI - (9 1 -9' I)]
sin p        sin p l
If we now  consider the two triangles which are formed
by the place of the heavenly body, the centre of the earth
and the two stations, the angle at the body in one of the
triangles is p', that at the. place of observation 180"   z-   T
- pe', and the angle at the centre cp'-=- 5, where d is the
geocentric declination of the body and where the upper or
the lower sign must be used, if the heavenly body- and the
place of observation are on the same side of the equator or
on different sides. The angles in the other triangle are p'l,
1800 - z,   - - -  q'I and q9', A  J.  We have therefore:
P'I      -- 99'  a
-pi  zI  l- 99 e?
and:
p' I-+pI, = Z' —+ - gl -"'- I'
Therefore if we denote the known quantity p'-+-p', by
g, we have the equation:
e sin [z - (99 -') _    sin [~  I - (99, -[-    -   )]
sin p            sin (c - p')
whence follows:
tang p' -          e sin t sin [z - (9s  - n')]
a, sin [z- (  — 979')] -+  COs  sin [z- (99-')]'
or:
tang, I         sinr sin[z, — (I, -   A',)]
gp l sin [z -  (9i')] +,I cos 7 sin [z, - (, -9',
When either p' or p'1 has been found by means of these
equations, we find p either from:
sin p --   sin Psin p
or from:            sin p        sin P, 
el sin [z, - (99 --  I)
It was assumed, that the two places are on opposite
sides of the equator, a case, which is the most desirable for
determining the parallax.  But if the two places- are on the




368
same side of the equator, the angles at the centre of the
earth in the triangles used before are different, namely q'er
in one triangle and (f', -= ( in the other.  If we put in
this case:
t=ip --   l  Z= -(- 1 Z - ),
we find p' or p'i from the same equations as before.
If the two places are not situated on the same meridian,
the two observations will not be simultaneous, and hence the
change of the declination in the interval of time must be
taken into account.
In this way the parallaxes of the moon and of Mars were
determined in the year 1751 and 1752. For this purpose
Lacaille observed at the Cape of Good Hope the zenith distance of these bodies at their culmination, while corresponding observations were made by Cassini at Paris, Lalande at
Berlin, Zanotti at Bologna and Bradley at Greenwich. These
places are very favorably situated. The greatest difference
in latitude is that between Berlin and the Cape of Good
Hope, being 86 0, whilst the greatest difference in longitude
is that of the Cape and Greenwich, being equal to 1V hour,
a time, for which the change of the declination of the moon
can be accurately taken into account.
By these observations the horizontal parallax of the moon
at its mean distance from the earth was found equal to 57' 5".
A new discussion of these observations was made by Olufsen,
who, taking the ellipticity of the earth equal to 31020-2 found
57' 2".64, while the ellipticity given in the preceding No.,
would give the value 57' 2". 80'). Latterly in 1832 and 1833
Henderson observed at the Cape of Good Hope also the
meridian zenith distances of the moon, from which in connection with simultaneous observations made at Greenwich
he found for the mean parallax the value 57' 1".8 **). The
value adopted in Burkhardt's Tables of the Moon is 57' 0". 52,
while that in Hansen's is 56' 59".59.
The problem of finding the parallax was represented
above in its simplest form, but in the case of the moon it
*) Astron. Nachrichten No. 326.
**) Astron. Nachrichten No. 338.




369
is not quite as simple, since only one limb of the moon can
be observed, and hence it is necessary to know the apparent
semi-diameter, which itself depends upon the parallax.
If r and r' denote the geocentric and the apparent semidiameter, A and A' the distances from the centre of the earth
and from the place of observation, we have:
sin r'  A
sin  A'
Further in the triangle between the centre of the earth,
that of the moon and the place of observation, we have:
A   sin (180~ - z')
A'   sin (' -')
where a' is the angle, which the line drawn from  the place
of observation to the centre of the moon makes with the
radius of the earth produced through the place, and since:
z' = z - (q --') =  r'
where z is the observed zenith distance of the moon's limb
and where the upper sign corresponds to the upper limb, we
have:
A     sin [z - (9 -  I) = r']'  sin [z - (a- p') - p' 4- r']
If we introduce this expression in the equation for s.l.
sinm r
and eliminate p' by means of the equation:
sin p' - e sin p sin [z - (p - S') =t'],
we obtain, writing for the sake of brevity z instead of  -
(r,-  rc') and taking Q =  1:
sin r' -sin r -i- sin r'sin p cos (z 4 - r') 24 — sin' sin p2 sin ( =,
or neglecting terms of the third order:
- r -4- sinl  sin p cos (z -6 r) --  sin r sin p2 sin (z 4- r)2.
Now the geocentric zenith distance Z of the moon, expressed by the zenith distance z of the limb, is:
Z = z -4- r - sin p sin (z -4- r')  sinp3 sin (z 6 r)3
6
or if we substitute for r' its expression found before:
Z   - z   =t    sin -   sinp cos (ezr):I= sin  sinp2 sin ( -)-2 r)  -
sin p3 sin (z -4 r)3
-- sin p sin (z - r)I
6
If we develop this equation and again neglect the terms
of a higher order than the third, we find:
24




370
Z  - z =  rsin sinr  in p sin z -4-  sin r sin p sin z2
sin p 3 sin z3
- sinp cos' sin z + - J sin p sin r2 sin z  - 
or introducing  1 —    sin r2 instead  of cos r and replacing
sin p by e sin p:
Z=-  -- q sip sinz -    sinp sin z      sin    aL   2si 2 sin z2
e3 sin  p3 sin Z3
6
and finally, if we take:
sill r - k sin p,
and hence:
2 _ Sk sinp ~q- 6/c3 sin p 3
and introduce again z -  2 in place of z, where   so -  ~',
we have:
sin p3
Z  - z    - sin p   sin (z -  ) =Fk]   6- -e sin (z -   ) =F= k] 3.
If D is the geocentric declination of the moon's centre,
S the observed declination of the limb, we have also, since
D= cp' -  z and  =-  p' -  (z -):
D       sill p [ sin (z -  )    k]+ — i6  [q sin (z - )  = k] 3.
The quantities ) and a depend on the ellipticity of the
earth, and since it is desirable, to find the parallax of the
moon in such a way, that it can be easily corrected for any
other value of the  ellipticity, we  nmust transform  the  expression  given  above accordingly.  But according to No. 2
of the third section we have:
a 2   b2
a. sin2  -f-+.
b1.
a'
If we introduce here the ellipticity, making use of the
equation':                        2 
and neglect all terms of the order of ct2, we find:
9 - 9'-    Z -c a sin 29.
Moreover we had:
2 = x2 +/+  2   Cos t      (1 -   E)2 sin T2'1O  - ~ 2 sin'1- E2 sin 9sn2
1 - 2E2 Sill 92   81 Sill 2
1   - 2 si + s2




371
If we introduce here also at by means of the equation:
82 = 2 a - 2
and neglect all terms of the order of e2, we find:
--- 1 - a sin T2.
Thus the last expression for D is changed into:
D -   + - [sin z == k] sin p - [sin 992 sin z +- sin 2 9 cos z] a sin p
+- [sin z =: k]3 s6i.
Every observation of the limb of the moon, made at a
place in the northern hemisphere of the earth, leads to such
an equation, in which the upper sign must be taken in case
that the upper limb of the moon has been observed, whilst
the lower sign corresponds to the lower limb of the moon.
Likewise  we find for a place in  the  southern  hemisphere:
D,    -- [sin z, =F: k] sin p, - [sin z, =T 1c]'3 -i P 
+- [sin 9 12 sin z + - sin 2 99T cos z,] sin p.
Now  let t and t1 be the mean times of a certain first
meridian, corresponding to the two times of observation, let
Do be the geocentric declination of the moon for a certain
dD
time T and  d its variation in one hour of mean time and taken
positive, if the moon approaches the north pole, then we find
from the two equations for D and D,:
dD
(t-t) -,   -- [sin z,= k -  (sin 99 2 sinz -- sin 2 9, cos z, )] sin p
- [sin z j= Ik - a (sin 9T2 sin z -- sin 2 9 cos z)] sin p
sin, [13    ]      in p
- [sin, =F= k]3 S          [sin z  s kI] s
6                 6
Moreover if p, is. the parallax for the time T and 7p7 its
change in one hour, we have:
sin    Sill   -~- cos po O (t - T)
sin    sin- inpo - cospo dp (t, - T),
therefore we find the following equation for determining the
parallax for the time T:
24 




372
D                              sin DO3
0     - 1-   +- (t - t1)   - [(sin z, =F k)3 -- sin (z = 7c)3] 
dp
--  cots po [(sin z =F k) (t -T)+- (sin z, == k) (t- T)]
-[sinz, - sin z =k-==] ipsino   sinpo   0    s sin - sin 2    osz )
+ sin, 2sinz x, sin 2, cosz,
If at the two places opposite limbs of the moon are
observed, the coefficient of sin po, is rendered independent
of k, and since this quantity thus only occurs in the small
terms multiplied by sin p,3 and dp, the value of p,, which is
found from the equation, is independent of any error of k.
Since we know the parallaxes from former determinations sufficiently accurately so as to compute the third and the fourth
term of the formula without any appreciable error, we can
consider the first fbur terms of the formula as known, since
all quantities contained in them  have either been observed
or can be taken from  the tables of the moon.  Therefore if
we denote the sum  of these terms by n, the coefficient of
sin p,o by a and that of ac sin po by b, we obtain the equation:
O n-sinp, (a -ba),
from  which P0 can be found  as a function of a.  But instead of the parallax p,, for the time T it is desirable to find
immediately the mean parallax, that is, the horizontal parallax
for the mean distance of the moon from the earth **). Therefore if K is the value of the mean parallax adopted in the
lunar tables, and, the value taken from those tables for the
time T, we have, if we denote the sought mean horizontal
parallax by H:
sin p     sin       p o-  sin sin,
hence the equation found before is transformed into:
0 -- -- sin l (a - b a).
*) If the second differential coefficients are taken into account, we must
add the term:
d2 D
+   [(t - T)2 - (t -T)] dt2
but if we take:           T      (t, + t),
this term vanishes.
**) Namely the distance equal to the semi-major axis of the moon's orbit.




373
Example.  In 1752 February 23 Lalande  observed  at
Berlin the declination of the lower limb of the moon:
8= —- 200 26' 25". 2,
and Lacaille at the Cape of Good Hope the declination of
the upper limb:
8l -+ 210 46' 44". 8.
For the arithmetical mean of the times of observation,
corresponding to the Paris time:
T= 611 40mn
we take from Burkhardt's tables:
dD
--  34". 15
dt
=t - 59' 24". 54
dp _ -+ 0".28;
dt
finally we have:
R- 520 30' 16"
and
m =33 56 3 south.
Since the longitude of the Cape of Good Hope is 20m
198. 5 East of Berlin and the increase of the right ascension
of the moon in one hour was 38' 10", the culmination of the
moon took place 21m 11$ later at Berlin than at the Cape,
hence we have:
dD
t - tl -- -- 21 11llS, hence (t- t) d —- - 12".06
further we have:
-, -a  -- +-1~ 20' 19".6.
The third term, depending on sin p3, we find equal to
0". 12, if we take k = 0.2725; therefore if we omit the
insignificant term  multiplied by d, we find:
n - - 1 20' 7".42
or expressed in parts of the radius:
n -- -+ 0.023307
and since the value of the mean parallax adopted in Burkhardt's tables is:
K — 57' 0". 52
we have:
log  =- 0.01792,
hence:
-- + 0. 022365.
It




374
If we compute the coefficients a and b, we find, since:
z-32~ 3' 51" and z -55 0 42' 48"
the following values:
a   - +1.3571 and b +1.9321
and hence the equation for determining sin HI is:
0 - +- 0.022365 - sin 11(1.3571 - 1.9321 o).
Every combination of two observations gives such an
equation of the form:
0= -- - x (a - ba)
If there is only one equation, we can find from it the
value of x corresponding to a certain value of c.  For instance taking a       I- 991   we find:
log sin 11  8.21901
11 —- 56' 55". 4.
But if there are several equations, we find for the equation of the minimum according to the method of least squares:
[aa] x - [a b] ax -     = o0,
hence:
IX- +a b]
[a a]  la a]
n'      n
[aLa]   -- [ai]'[aa]
[aa]   [aca]  [a a]
Thus Olufsen found for the mean horizontal parallax of
the moon the value 57' 2".80 ).  Since the parallax of the
moon is so large, it may even be determined with some degree of accuracy from  observations made at the same place
by combining observations made near the zenith, for which
the parallax in altitude is small, with observations in the
neighbourhood of the horizon, where the parallax is nearly
at its maximum. In this way the parallax of the moon was
discovered by Hipparchus, since he found an irregularity in
the motion of the moon, depending on its altitude above the
horizon and having the period of a day.
*) Astron. Nachrichten No. 326.




375
4. This method does not afford sufficient accuracy for
determining the horizontal parallax of the sun, but the first
approximate determinations were obtained in this way. In
1671 meridian altitudes of Mars were observed by Richer
in Cayenne and by Picard and Condamine at Paris, and from
these the horizontal parallax  of Mars was found equal to
25".5. But as soon as the parallax of one planet is known,
the parallaxes of all other planets as well as that of the sun
can be found by means of the third law of Kepler, according
to which the cubes of the mean distances of the planets from
the sun are as the squares of the times of revolution. Thus
from  this determination the parallax of the sun was found
equal to 9".5.  Still less accurate was the value found from
the observations of Lacaille and Lalande, namely 10". 25; neither have the observations made latterly in Chili by Gilliss
contributed anything towards a more accurate knowledge of
this important constant.  But allthough all results hitherto
obtained by this method have been insufficient, it is still desirable, that they should be repeated again with the greatest
care, since the great accuracy of modern observations may
lead to more accurate results even by this methodS).
The best method for ascertaining the parallax of the sun
is that by the transits of Venus over the disc of the sun at
her inferior conjunction, which was first proposed by Halley.
The computation of such transits can be made in a similar
way as that given for eclipses in No. 29 and 31 of the preceding section. The following method, originally owing to
Lagrange,  was published by Encke in the Berliner Jahrbuch
for 1842.
If re, (., A and D are the geocentric right ascension and
declination of Venus and the sun for the time T of a certain first meridian, which is not far from the time of conjunction, then we have in the spherical triangle between the
pole of the equator and the centres of Venus and the sun,
denoting the distance of the two centres by m and the angles
at the sun and Venus by Ml and 180~" —1/':
S) Such observations have been made since during the oppositions of
Mars in 1862 and seem to give a greater value of the parallax than the one
considered hitherto as the best.




376
sin - m. sin 2 (M' -+ M)  sin 2 ((- A) cos - (S +- D)
sin,E m. cos (M' -+- 1)- cos I (a - A) sin 2 (8- D)
cos  nz. sin  (M'- M) =sin  (a -A) sin  (8 -- D)
cos M. cos  (    )   ost (- -A) cos ( - D),
or since a -A  and   -D and hence also m and M'- M   are
for the times of contact small quantities:
m sin A31= (a - A) cos 4- ( +-D)   (A)
m cos M  --  - D.
Taking then:
d (a - A)
n sin N=        )cos - ( +- D)
dt
d(6 -D)                   (B)
- Cos N- dN
dt
whered ( — A) and d(!-D) are the relative changes of the
dt           dt
right ascensions and declinationa in the unit of time, and denoting the time of contact of the limbs by T-+-r, we have:
[m sin M-+- rn sin N]J2 + [m cos M l -rn cos N]2 -= [R:f: r]2,
where R and r denote the semi-diameter of the sun and of
Venus, and where the upper sign must be used for an exterior contact, the lower sign for an interior contact.
From  this equation we obtain:
M              Rm' 2 sin (M- N)2
T-   - - cos (1- V)  T- 1           (R=     -  )2
n                            (R -4- r)
Therefore if we put:
m sin (M- N)
adz  4in   =sin y, where   < ==A 90,  (C)
we obtain:?- -- -- cos (1 —N) =F     cos Vd     (D)
n                n
where again the upper sign must be used for the ingress and
the lower for the egress. Therefore at the centre of the earth
the ingress is seen at the time of the first meridian.:
4T — cos (- N). cos.
n                n
and the egress at the time:
T -    cos (M-N)  — +  cos %.
Finally if G is the angle, which the great circle drawn
from  the centre of the sun towards the point of contact ma



377
kes with the declination circle passing through the centre of
the sun, we have:
(R -4- r) cos ()   m cos A q- n cos N.
(R =- r) sin (= m sin M-4- n sin N. r
or:
cos ()   - sin N sin tp =F= cos N cos 5
sin ) -    sin t cos N =F cos yP sin N,
hence for the ingress we have:
) Is800 +N —                    (E)
and for the egress:
0 =N+.                     (F)
These formulae serve for computing the times of the ingress and egress for the centre of the earth. In order to
find from these the times for any place on the surface of the
earth, we must express the distance of the two bodies, seen
at any time at the place, by the distance seen from the centre of the earth.
We have:
cos mn = sin A sin D +- cos 8 cos D cos (a - A).
If a',' A' and D' be the apparent right ascensions and
declinations  of Venus and the sun, seen from  the place on
the surface of the earth, and m' the apparent distance of the
centres of the two bodies, we have also:
cos m'  sin 8' sin D' +- cos d' cos D' cos (a' — A')
and hence:
cos m'  cos is  - (' -  ) [cos s sin D - sin a cos D cos (a - A)]
- (D'- D) [sin  cos D - cos  sin D cos (a - A)]
(a' - a) cos 8 cos D sin (a - A)
+- (A'- A) cos 8 cos D sin (a - A).
But according to the formulae in No. 4 of the third section we have ):
*) We have according to the formulae given there:
t'- a --  sin 9 sin (-Y) =    sin p [sin a cotang y - cos ].
sin y*
but since:
cotang y = cos (a - 0). cotang %,
we have:
5' - a —-  [cos 9F sin 8 cos (a- 0) - sin 7 cos 5].




378
3' - 6 --  r [cos T sin S cos (a - 0) - sin p9 cos 3]
D'- D    p [cos q, sinD  os (a — 0) - sin 9v cos D]
a'- - a= r sec S sin (a- 0) cos 5q
A' - A -p seecD sin (A — 0) cos,
where  a  and p  are  the  horizontal parallaxes of Venus and
the  sun; and if we substitute these  expressions in the equation  for cos n', we obtain:
COS  -- COs m71
+- [cos 8 sin D- sin 8 cos D cos (a —A)] [sT cos 9 sinl  COS (a — 0) - 7c sin  cos C ]
+- [sin  cos D - cos Ssin D cos (a - A)] [p cos qs sinD cos (a- 0) -p sin r cosD]
- cos D sin (a-A). n7 sin (a - 0) cos 99                         (a)
+  cos S sin (a-A). p sin (A —  ) cos 99.
If we develop  this equation, we  find  first for the  coefficient of cos (f:
gr [sin S cos S sin D cos (a -O) - sin 32 COS D cos (a- 6) cos (a- A)
-cos D sin (a- 0) sin (a - A)]
+ p [sin S cos D sin D cos (a — 0) - cos S sin D2 cos.(a -0 ) COS (a - A)
+- cos S sin (a- 0) sin (a- A)]
or since:
sin -2 1-COS S2 and sin D2 — 1 - os D2:
r [(sin 3 sin D - cos  cos D cos ( -A) ) cos  cos (a- 0) — cos D cos (A-0)]
-ip [(sin S sinD + co-s  cos D cos (a-A)) cosD cos (A- 0)-cos S cos (a- 0)],
hence:
7V cos nz cos S cos (a - 0) - 7 Cos D cos (A - 0)
- p cos mn cosD cos (A- 0) -p cos S cos (a - 0).
This we can transform in the following way:
[n cos n CoS  os a -    cos D cos A] cos 0
+  [p COs m7 Cos D COS A - p cos   cos a] cos 0
+ [7 cos in cos S sin a - 7r cos D sin A] sin 0
+- [p cos 1 cos D sin A - p cos   sin a] sin 0,
and hence the term   multiplied  by  cos q~ becomes:
[(n COS m -p) COs S Cos a- (st -p COs iz) Cos D cosA] cos   cos 0   (b
H- [(7 cos ni -p) cos S sin a - (v -p cos m) cos D sin A] cos 9o sin 0.
Further the  coefficient of sin (f, in  the  equation  (a) is:
=r [- cos 2 sin D H- sin S cos S cos D cos (a - A)]
H p [- sin S cos D'2  sin D cosD cos S cos (a - A)],
or since  cos  2           1 -  sin 62 and  cos D2 -     - 1  sin D2:
c [- sinD +- sin S (sin S sin D +- cos S cos D cos (a -A))]
-+-p [- sin S H- sinD (sin S sin I) -+ cos S cos D cos (a -A))].
Therefore  the  term   of the  equation  (a), which is multiplied by sin 6F, is:
(n Cos mn -p) sin S sin 95 -  (z -p cos m) sin D sin 95,




379
and  thus the  equation  (a) is transformed  into  the  following:
COS inf   COS in
+  [(v cos mn-p) coS 0    sa -(T-pC cos - () cosD cosA] cosT  cos O
-+ [(  cos In —p) cosc  sin a -( —p cos im) cosD sin A] cosT  sin O  (c)
+  [(7c cos in-p) sin 3      - (  - p cos in) sin D] sin.
If we take now:
c COs 1in-p  -f sin s              (d)
-  iv sin?n =f cos s,
we have:
c -p cos in:Jf sin (s —mn),
and  hence:
COS min   cos ing
-f [sin s cos 3 cos a -  sin (s - -m) cos D cos A] cosA]   cos cos
-f [sin s cos 3 sin a -  sin (s -in) cos D sin A] cos Tp sin 0   (e)
— f [sin s sin 3     -  sin (s-m) sin D] sin.
Further if we take:
sin s cos 3 cos a -- sin (s -  n) cos D cos A  - P cos 2 cos,
sin s cos 8 sin a -  sin (s-nm) cos D sin A   - P sin 2 cos,3  (f)
sin s sin 3 -  sin (s -  ) sin D      = P sin f,
we  find  by  squaring  these  equations the  following  equation
for P:
p2 =  sin s2 +  sin (s-nim)2 -  2 sin s sin (s - n) cos mi
=sin  -- sin S2 cos m2  - cos s2 sin m2   S  -- sin i2.
Hence we may put:
sin s cos 3 cos a -  sin (s-  n) cos D cosA -  sin in cos2  cos 
sin s cos 3 sin a -  sin (s —i2) cos D sin A    sin m sin 2 cos i
sinr s sin 8 -- sin (s -   ) sin D    -  sin in sin f,
or:
sin in sin (2-A) cos,= -sin s cos 3 sin (a - A)
sin i2 COS (2-A) cos, --- sin s  s   cos (a - A) -  sin (s -   ) cos D    (g)
sin in sin 3 =  sin s sin 3- sin (s —mn) sin D.
But we have:
sin s cos a cos (a-A) - sin (s-n) cos D =  sins [cos 3 cos (a-A) - cos in cos D]
+- cos. sin iz cos D
and:
sin s sin 8 -  sin (s -- in) sin D - sin s [sin 8 -  cos ii sin D]
- cos s. sin in sin D.
Further we  have  in  the  spherical triangle  between  the
pole  of the  equator and  the  geocentric  places  of Venus  and
the  sun, denoting  the  angle  at the  sun  by  M:
sin m sin -11- cos S sin (a -A)
sin im cos 1 —=  sin 8 cos D - cos 8 sin D cos (a - A)      (h)
cos n =- sin 8 sin. D+- cos 3 cos D cos (a- A),




380
hence we have:
cos A cos (o-A) = cos D cos m - sin D sin m cos M
sin A = sin D cos en  +- cos D sin m cos 21,
and the equations (g) are thus transformed into the following:
sin (R -A) cos a = sin s sin M
cos (- A) cos  -cos s cos D - sin s sin D cos Mf   (i)
sin, -~ cos s sin D -+ sin s cos D cos -1g,
where s and M  must be found  by means of the equations
(d) and  (h).  After having  obtained X) and / by the equations (i), rn' is found according to (e) and (f) by means of
the following equation:
cos mn - cos m +-f sin m [cos R c os  cos  cos   sin ) cos  cos 99 sin 0
+  in- sin sin ]
-cos m +-f sin nz [sin T9 sin  q -.cos 9v cos f cos ( - 0)].
Now let T, as before, be that mean time of a certain
first meridian, for which the quantities a., 0, A  and D have
been computed, and L the sidereal time corresponding to it,
further let I be the longitude of the place, to which 0 and?( refer, taken positive when East, we have:
-=1 +L,
therefore:                — e    i,-z.
Hence if we put:
cos  - sin 99 sin? +- cos T costs cos (4 -I/),
we have:                                              (
cos m'- cos m - +f sin mn cos 
All places, for which cos i has the same value, see the
same apparent distance m' simultaneously at the sidereal time
L of the first meridian, or each place at the local mean time
T -4- 1. In order to find the time when these places see the
distance rn, we have:  dc = -f cos,
hence:                    dt      /cos 
dnz
dt
But if m is a small quantity, for instance at the time of
contact of the limbs, we have according to the formulae (A):
-(a - A) cos ~ (8 -+ D) sin  i-+- (8 - D) cos M
d-    d (a-A)   S       D) sin ci+ (d-D)Cos M,
dt     dt       -dt
or according to the formulae (B):
dm
d   t    cos (M — N),
dt




381
f cos
hence:                 dt —    f
n cos (1 - N)
Therefore if an observer at the centre of the earth sees
at the time 7' the angular distance m of the bodies, an observer on the surface of the earth sees the same distance at
the time of the first meridian:
f cos 
in cos (I — N)
or at the local time:
f T+ cos g
n cos (t — N)
Therefore in order to find the times of the ingress and
egress for a place on the surface of the earth from the times
of the ingress and egress for the centre of earth, we need
only use R-t- r and 0) instead of m and M, and since we
have according to the formulae (E) and (F) for the ingress
) = 180" -- N -  1/ and for the egress 0(    N-+q-, we must
add to the times of the ingress and egress for the centre of
the earth:                    fcos 
n cos i
and:                       + f cos
U2 cos 7P
Hence if we collect the formulae for computing a transit
of Venus, they are as follows:
For the centre of the earth.
For. a tinme of a certain first meridian, which is near the
time of conjunction, compute the right ascensions c, A and
the declinations 3, D  of Venus and the sun, likewise their
semi-diameters r and R.  Then compute the formulae:
m silln' - (aA) cos - (a+ D)
m cos 11 —-    - D
n sin N —-       cod (a A)  D)
cdt
d(a-D)
Z cos N —
dt
m sin (If-N) 
—' sin T'T<=4 900
R -4- r
mn             R 4- r
-- Cos ( — - c-    Cos (c'l - 2 Cos (M — N) q4  --- 2cos 5.
n                n




382
Then the time of ingress is:
t - T -i-%,
and we have for this time:
(-= 1800    N — 5,
and the time of egress is:
t' = T-+-',
and for this time
~=N+~.
For a place whose latitude is so and whose east longitude is 1.
Compute for the ingress as well as for the egress, using
the corresponding values of the angle 0, the formulae:
7X cos (R =-4- ) - p =f sin s
- 7tsin (R 4- r) =f Cos s
= g
n Cos 9
sin (R -A) cos 3 = sin s sin 0
cos (A -A) cos = cos s cos D - sin s sin D cos )
sin i 3   cos s sin D - sin s cos D cos 0
cos   sin fi sin 99 +- COs f COs 95 cos (A - I) *),
where L is the sidereal time corresponding to t or t'. Then
the local mean time of the ingress is:
t   - g cos o,
and that of the egress:
t'- I - g cos C.
At those places, for. which the quantity
sin fin n 9 -t + cos, cos T9 cos (A - 1)
is equal 4- 1, the times of contact are the earliest and the
latest. The duration of the transit for a place on the surface may differ by 2g from the duration for the centre, and
since for central transits we have nearly:
X P
the difference of the duration can amount to twice the time,
in which Venus on account of her motion relatively to that
of the sun, describes an are equal to twice the difference of
her parallax and that of the sun.  Now  since the difference
of the parallaxes is 23" and the hourly motion of Venus at
*) I is the angular distance of the point, whose latitude and longitude
are T9 and 1, from the point, whose latitude and longitude are i3 and Al.




383
the time of conjunction is 234", the difference of the duration can amount to 12 minutes, whence we see that the difference of the parallaxes of Venus and the sun, and thus
by Keppler's third law  the parallax of the sun itself can be
determined with great accuracy.
Example.  For the transit of Venus in 1761 June 5 we
have the following places of the sun and of Venus:
Paris m. t.     A             D               a,y
16h    740 17' 1". 8 -+ 220 41' 3". 7  740 25' 50". 3 -+ 220 33' 17". 6
17h        19 36.4       41 19.1      24 13.2        32 32.4
18h        22 10.9       41 34.5      22 36.2        31 47.1
19h        24 45.5       41 49.9      20 59.2        31  1.9
20h        27 20.1       42  5.3      19 22.2        30 16.6,
further:
-- 29". 6068    R - 946". 8
p -  8". 4408    r    29". 0.
In  order to  find the times of exterior contact for the
centre of the earth, we take:
T= 17h
and find:
da   dA
a- A = — 4'36". 8,   -D      8467, dt                411".  6
dt   dt.d.   dD... -60". 65, R+r   975". 8.
dt   dt
From  this we find:
M-=  1540 7'. 2       N-  2550 21'. 9
log _2 -= 2. 76746    log n2  2. 38028
_M — N —  2580 45'. 3
t?=- 36  2.6
-- cos(M —N) =       0. 4756       -- 2h. 8114 = - 2h48m 41s. 0
-4-( R  r ) cs  + 3. 870.7626  ---  3 45 45. 4
n
Therefore the ingress took place for the centre of the
earth:
at 14h 11" 19. 0 Paris mean time,
and it was:
O- 1110~ 24'.5,
and the egress took place at
20h 45m 45s. 4 Paris mean time,
and it was:
=      2190 19'. 3.




384
If we wish to find then the time of the egress for places
on the surface of the earth, we must first compute the constant quantities X,,i and g and find first:
s -- 900 22'. 7, logf= I. 32564, log g = 9. 03764,
and since:
O — 2919'.3, D = 22~ 42'3, A = 740 29'. 3,
we obtain:
2 — 9  15'. 9
and                      P    — 450 44. 4.
Further since 20h 45m 45s. 4 Paris mean time corresponds
to 1l 45mn 34s.6 sidereal time, we have:
A-  170 7'. 7.
If it is required for instance to find the egress for the
Cape of Good Hope, for which:
1 -- +  h 4m 33s. 5
and
--- 330 56' 3",
we find:
log cos  -= 9.94643, g cos    + 5' 47".0,
and hence the local mean time of the egress:
t --  -+- g cos -- 21h 561n 5s. 9.
If we differentiate the equation:
T=t+- 1  g cos ~,
we find, if dT is expressed in seconds:
3600 cos        )
n cos W   p
3600 cos  7X -p
dpo
}X COS  / ]J po
- 40.49 dp  *),
so that an error of the assumed value of the parallax ofthe
sun equal to 0".13  changes the time of the contact of the
limbs by 5s. Conversely any errors of observation will have
only a small effect upon the value of the parallax deduced
from them, and thus this important element can be found
with great accuracy by this method.
5. In order to find the complete equation, to which
any observation of the contact of the limbs leads, we start
from the following equation:
[a' - A']2 COS o 2 + [S' -D']2 [R A  -]2,  (a)') Where Po is the mean horizontal equatoreal parallax.




385
where ca', A',' and D' are the apparent right ascensions and
declinations of the sun and Venus, affected with parallax,
and Y(,, denotes the arithmetical mean'2 +.  But since the
parallaxes of the two bodies are small and likewise the differences of the right ascensions and declinations for the times
of contact of the limbs are small quantities, we can take:
a' - A'    t-A +   (t -p) sec 8o cos T' sin (a0 - 0)' - D' =   - D - (- — p) [cos' sin 8o cos (a -0) - sin p' cos 8o],
where:
ex -- A
If now  we introduce the following auxiliary quantities:
cos' sin (ao - 0)  h sin H
(A)
cos T' sin 8o cos (ao -0O)- sin T' cos 8o  -h I cos T,
the equation (a) is transformed into:
[a- A+ ( -p)  sin Hsec 6] 2 cos 8o2 +[8- D + (7 - p) hcos H]2 =[R=t2
If then a, A,:, D~, %, p, R and r, denote the values which
are taken from  the tables, whilst a -+- dca,  c -+ d8l, A +  dA,
D — +- dD z -4- dt, p -- dp, R -- dR and r -dr are the true
values, and dl is the error in the assumed longitude of the
place of observation, the equation must be written in this way:
a - A +- (s - p) h sin H sec 8 -J- d (a - A)    2
(ad  l - A)    COs 502
L-  d(~ @-p) hi sin H sec Q   -  ) dl 
cd(  -D)
+ [ —D -+ (: — p) h costH+-d(D) -Id() — p) hcoss H-   dl   dl]2
[R +-clR- r + dor]2.
If we develop this equation and neglect the squares and
the products of   — p and the small increments, and put:
a - A -+ (7n - p) h sin Hsec 8, = A'
S - D +- (7c -p) h cosH    -.D',
we find:
A'2 cos jo2 _ D'2 _ (R  4-r)2
-2 2A' cos 0 2 d (a- A) -2 A'h sin H/cos o +- D'h cos H] d ( — p)
— D'd(8 —D)+2 (A d( -A) Cos    +D'              D --   )) dl
+ 2 (R.4- o) d (R 4-?o).
But if we denote:
A'12 COS o 2 _+ D12
by ma2, and since we have approximately:
- 2 - (1R AL ) =) 2 oo [L - (R  )],
25




386
we find:
p. [i — (R4_r)] — A' coS2 d (a - A) - D' d (S-D))
-[A' 7 sin H COs tSo + D' h cos H] d( —p)
+(A' d(a-A) cQs5o2~Dw d(8D))dl+(R4r)d(R)r)
Therefore if we put again:
A' cos So -  n sin Il     (B)
D'      -- rn cos M11
1        d(a -A)
3600coso    d      - ln sin N
(C)
n cos N
3600    dt
the equation becomes:
(R = r -  n)   sin AMcos Sd(a-A)   cos Md (S - D)
n cos (M —  N)     it co (M-N)       n cos (M-N)
h cos (i — H)  -P d         d (R   )   (D)
n cos (M- N) po           n cos (Al - N)
The  difference of longitude dl must be determined by
other observations and thus dl can be taken equal.to 0.  In
this case all the divisors might be omitted, but if we retain
them, R 4- r -  mn is expressed  in  seconds of time, because
we have:
dm
n cos (M — N)
cdt
Example. The interior contact at the egress was observed at the Cape of Good Hope at
21h 38,n 3s 3 mean time.
This time corresponds to
20h 3311 29s. 8 Paris mean time --  h 33m 16S. 2 Paris sidereal time.
We have therefore:
0 - 2h 37 1149s. 7 -- 39"( 271 25".
Moreover we have for that time:
— 740 18' 28". 05        =- 220 29' 51". 32
A —-74  28 46.41        D —22 42 13.90
a-A —- -  10' 18".36    -D —   12' 22".58
ao -= 740 23'37"     ao - O    = 340 56 12"    8o - 220 36'2"
(zr —p) h sin H=- + 10". 07     H — 31 34'. 0  (t-p) h sin H sec SO
(a —p) h cosH   + 16. 39   log I =- 9.95835       =   - 10". 90
A'= - 10' 7". 46
D'- — 12  6.19
M —- 2170 40'. 7      N-  2550 19'. 3
log m = 2.96262       log n = 8.82412.




387
Now since:
R - r -- 917". 80
and:
po  8". 57116,
we find:
- 5.3 - 10.684 d (cc - A) -f- 14.986 d ( - D)
-+- 42.240 dpo + 18.934 d(R -  ).Such an equation of the form:
0 — z+d ( - A) + bd ( -D)+ cdpo + ed(R -r)
is obtained from  each observation of an interior contact and
a similar one containing  d(R-+ r)  from  an  exterior contact, and from a great member of such equations, derived
from observations at different places on the surface of the
earth, the most probable values of dpo,, d (ca —A), d ( —D)
and d (R 4- r) can be found by the method of least squares.
In this way Encke*) found by a careful discussion of
all observations made of the transits of Venus in the years
1761 and 1769 the parallax of the sun equal to 8".5776.
More recently after the discovery of the original manuscript
of Hell's observations of the transit of 1769 made at Wardoe
in Lapland, he has altered this value a little and gives as
the best value
8". 57116
When  the parallax  of the  sun  is known, that of any
other body, whose distance from the earth, expressed in terms
of the semi-major axis of the earth's orbit as unit, is A, is
found by means of the equation:
8". 57116
sinp    -- ~
Note 1. Although a great degree of confidence has always been placed
in the value of the parallax of the sun, as determined by Encke, still not
only the theory of the moon and of Venus, but also the recent observations
for determining the parallax of Mars and a new discussion of the transit of
1769 by Powalky, who used for the longitudes of several places of observa*) Encke, Entfernung der Sonne von der Erde aus dem  Venusdurchgang von. 1761. Gotha 1822.
Encke, Venusdurchgang von 1769. Gotha 1824.
25*




388
tion more correct values than were at Encke's disposal, all seem  to indicate,
that this value must be considerably increased.
Note 2. The transits of Mercury are by far less favourable for determining the parallax  of the sun.  For since the hourly motion of Mercury
at the time of the inferior conjunction is 550", while the difference of the
parallaxes of Mercury and the sun is 9", the coefficient of dpo in the equation (D) in the case of Mercury is to the same coefficient in the case of
Venus as:
23  550
9'  234:
hence 6 times smaller.  Thus an error of observation equal to 5s produces
already an error of 0".8 in the parallax of the sun.  However on account
of the great excentricity of the orbit of Mercury this ratio can become a
little more favourable, if Mercury at the time of the inferior conjunction is in
its aphelion or at its greatest distance from  the sun.




SEVENTH SECTION.
THEORY OF THE ASTRONOMICAL INSTRUMENTS.
Every instrument, with which the position of a heavenly
body with respect to one of the fundamental planes can be
fully determined, represents a system  of rectangular co-ordinates referred to this fundamental plane.  For, such an instrument consists in its essential parts of two circles, one
of which represents the plane of xy of the system of co-ordinates, whilst the other circle perpendicular to it and bearing
the telescope turns around an axis of the instrument perpendicular to the first plane and can thus represent all great
circles which are vertical to the plane of xy. If such an
instrument were perfectly correct, the spherical co-ordinates
of any point, towards which the telescope is directed, could
be read off directly on the circles.  With every instrument,
however, errors must be presupposed, arising partly from the
manner, in which it is mounted, and partly from the imperfect
execution of the same, and which cause, that the circles of
the instrument do not coincide exactly with the planes of the
co-ordinates, but make a small angle with them. The problem  then is, to determine the deviations of the circles of
the instrument from the true planes of co-ordinates, in order
to derive from  the co-ordinates observed on the circles the
true values of these co-ordinates.
Besides other errors occur with instruments, arising partly
from the effect of gravity and temperature on the several
parts of the instrument, partly from the imperfect execution
of particular parts, such as the pivots, the graduation of the
circles etc., and means must be had to determine these errors
as far as possible, so as to find from the indications of the




390
instrument the true co-ordinates of the heavenly bodies with
the greatest possible approximation.
Besides these instruments, with which two co-ordinates
of a body perpendicular to each other can be observed, there
are still others, with which only a single co-ordinate or merely
the relative position of two bodies can be observed. With
regard to these instruments likewise the methods must be
learned, by which the true values of the observed angles can
be obtained from the readings.
I. SOME OBJECTS PERTAINING IN GENERAL TO ALL INSTRUMENTS.
A. Use of the spirit- level.
1. The spirit-level serves to find the inclination of a
line to the horizon.  It consists of a closed glass tube so
nearly filled with a fluid that only a small space filled with
air remains. Since the upper part of this tube is ground out
into a curve, the air-bubble in every position of the level so
places itself as to occupy the highest point in this curve.
The highest point for the horizontal position of the level is
denoted by zero, and on both sides of this point is arranged
a graduated scale marked off in equal intervals and counting
in both directions from the zero of the scale. If the level
could be placed directly on the line, it would only be necessary, in order to render this line horizontal, to change
its inclination to the horizon, until the centre of the bubble
occupy the highest point, that is, the zero of the scale. Since
however this is not practicable, the glass tube for its better
protection is first firmly fixed in a brass tube which leaves
the graduated scale of the level free, and this tube is itself
placed in a wide brass tube of the whole length of the axis
of the instrument.  The upper middle part of this tube is
cut out and covered with a plane glass. In this tube the
other is fastened by means of horizontal and vertical screws
which also serve as adjusting screws, so that the graduated
scale of the level is directly under the plane glass through




391
which it can be read off ). The tube is then provided with
two rectangular supports for placing it upon the pivots or
for the larger instruments with corresponding hooks for suspending it on the axis of the instrument. Generally however
these supports or hooks are not of equal length.  Let AB
Figt. 1.           Fig. 11 be the level, A C and
B D  be the two supports,
whose length is represented
A                              by a and b and suppose
a   the level to be placed on a
line, which makes with the
horizon an angle a, in such
a manner, that BD  shall stand upon the higher side.  Then
will A stand in the height ad- c and B in the height:
b ~- c -+ L tang ac
if L is the length of the level. This is, to be sure, not entirely correct, because the supports A C and BD do not stand
perpendicularly to the horizontal line; since however only
small inclinations of a few minutes, generally of a few seconds,
are always here assumed, this approximation suffices perfectly.
If now we call the angle which the line AB makes with the
horizon x, then we have:
b - a + L tang a
tang xor
b - ac
If we reverse the level so that B  shall stand on the
lower side and call x' the angle, which A B now makes with
the horizon, then we have:
b - a
x == c -  -
L
If furthermore we now assume, that the zero has been
marked erroneously on the level and that it stands nearer
to B  than to A  by ),, then if the level be placed directly
on a horizontal line, we read I-i- t on the side A, if 21 be
*) This arrangement is adopted in order that the level may be in a completely closed place and not liable to be disturbed in reading off by the warmth
of the observer or of the lamp.




392
the length of the bubble, and 1 — A on the side B. Suppose
on the other hand the level to be placed on the line AB,
whose inclination to the horizon is x, then we read on the
side A:
A- 1 +  --  rx,
where r is the radius of the curve A B, in which the level
has been ground out, on the contrary on the higher side B:
B   I - A+rx,
If the level with its supports be reversed in such a
manner that B shall stand upon the lower end, we shall read:
A'= -1-4- +-t rx'
B= -- I --    rx'.
If we now substitute for x and x' the values already
found, we shall find for the four different readings, denoting
the inequality of the supports expressed in units of the scale
of the level by u:
A — 1 - a ~  -ru
B1=+ ra-A +-ru
A' = 1 -+- r a -  -- ru
B' -- 1 - ra - 2 -- r.
It is obvious from the above, that the two quantities A
and r u cannot be separated from each other, and that for
the reading off it is one and the same, whether the zero-point
be not in the centre or whether the supports he of unequal
length.  On the other hand by the combination of these equations we can find A -- rue and a.
If the end B of the bubble is on a particular side of
the axis of an instrument, for instance, on the same side as
the circle, which we will call the circle-end, then after the
reversion of the level we shall read on this side A'.  Now
we have:
B-A
2    -2 + r U + B a
A'- B'
2   = —  ru + r a,
therefore            (B -- A  A'-B'
a-                   206265,
if we wish to have the inclination directly in seconds of arc.06265
The quantity 206265 is then the value of one unit on the
scale expressed in seconds of arc,




393
Therefore, if we wish to determine the inclination of an
axis of an instrument by means of the level, we place it in
two different positions on the axis and read off both ends
of the bubble in each position. We then subtract the reading on the side of the circle from the reading made on the
other side and divide the arithmetical mean of the values
found itn both positions by 2. The result is the elevation
of the circle-end of the axis expressed in units of the scale.
Finally if this number be multiplied by the value of the unit
of the scale in seconds of are, the result will be the elevation of the circle-end in seconds of are.
If we can assume, that the length of the bubble during
the observation does not change, we have also:
(A'- A)
or:
(B-B')
i. e. the inclination would be equal to half the movement
of the bubble on a determined end. If finally the level were
perfectly accurate, then we should have Xt  ru — 0 and it
would not be necessary, to reverse the level, but the inclination could be derived merely from one position by taking
half the difference of the readings on both ends.
Example. On the prime vertical instrument of the Berlin
observatory the following levelings were made:
Circle - end                 Circle - end
17. 0  9   2                6.9    19. 5}
Object glass ast            Object glass East; 16.1 10  
A - 3P. 90              -6. 30
2-A, —.    90, -.- r- — 8P. 80     i -ru -9p. 20
A-     - — 4. 90               -+2.90
-- OP. 50               - 1P.70
Therefore by the mean of both levelings we have b= -- 1". 10,
or since the value of the unit of' the scale was equal to
2" (1 -  -l), b    2". 06.
The above supposes, that a tangent which we imagine
drawn to the zero of the level is in the same plane with
the axis of the instrument.  In order to obtain this result,
the level must first be so rectified,  that this tangent lies in




394
a plane parallel to the axis, which is the case, when A - ru
equals zero.  If this value by the leveling is found to be
equal to zero, then the level is in this sense rectified; if
however, as in the above example, a value different from zero
be found, then the inclination of the level must be so changed
by means of the vertical adjusting screws as to fulfill the
above condition, which will be the case, when A equals
A' and B equals B', or when on the side of the circle-end
as well as on the opposite side, the bubble has the same
position before and after the reversion.  In the above example, where X —rru is 9P.00, it would be necessary to change
the inclination of the level, until the bubble in the last position
for Object glass West indicates 11.6 and 14.8. Then we
should have read on the level so rectified:
12.5  13.7                 lt.4  15.0
Object glass East 12. 7 13.5  Object glass West 11.6 14.8,
whereby we should have found again the inclinations - 0". 50
and - 1".70, and X - ru equal to zero.
If the level has been thus rectified, the tangent to the
zero of the level is in a plane parallel to the axis. If now
the level be turned a little on the axis of the instrument in
such a manner that the hooks always remain closely in contact with the pivots, then will the tangent to the zero, if it
is parallel to the axis, also remain parallel when the level is
turned, and the bubble will not change its position by reason
of this movement.  If however the tangent in the plane parallel to the axis makes an angle with a line parallel to the
axis, then will the inclination to the axis be changed when
the level is turned, and since the bubble always moves towards
the higher end, the end towards which the bubble moves if
the level is turned towards the observer, is too near the observer. This end then must be moved by means of the horizontal adjusting screws, until the bubble preserves its position unaffected, when the level is turned, in which case the
tangent to the zero is parallel to the axis.  By the motion
of the horizontal screws, however, the level is generally somewhat changed in a vertical sense so that ordinarily it will
be necessary to repeat several times both corrections in a




395
horizontal and vertical sense, before the perfect parallelism
of the level with the axis of the instrument can be attained.
2. In order to find the value of the unit of the scale in
seconds, the level must be fixed on a vertical circle of an
instrument provided with an arrangement for that purpose,
and then by means of the simultaneous reading of the level
and of the graduated circle, and by repeating the readings in
a somewhat different position of the circle, the number of
units is found, which corresponds to the number of seconds
which the circle has been turned.  If the bubble passes
through a divisions, whilst the circle revolves through fi
seconds, then is $ the value of the unit of the scale in
seconds.
In making this investigation however it is best, not to
remove the level from the tube, in which it is enclosed, since
it is to be presumed, that the screws which hold it may
produce a somewhat different curve from that which the level
itself would have without them, and since a large level cannot be well fastened on a circle of an instrument, it is best
to use for this purpose a special instrument which consists
in its essential parts of a strong T-shaped supporter, which
rests on three screws and on which the level can be placed
in two rectangular Y-pieces, in such a manner, that the direction of the level passes through one of the screws and is
perpendicular to the line joining the two other screws. The
first screw is intended for measuring and is therefore carefully finished and provided with a graduated head and an
index, by which the parts of a revolution of the screw can
be read' off.  By means of an auxiliary level the apparatus
can be so rectified as to render this screw exactly vertical.
If now the level is read off in one position of the screw
and then again after the screw has been turned a little, the
length of the unit of the scale will be found in parts of
the revolution of the screw. If now we know by exact measurement the distance f of the screw from the line joining the
two other screws and the distance h between the threads of the
screw, then will - be the tangent of the angle, which cor



396
responds to one revolution of the screw or e 206265 be this
angle itself. The perfection of the screw can be easily tested
by observing, whether the bubble always advances an equal
number of units, when the screw is turned the same number
of units of the graduated head. But it is not necessary that
the parts of the scale be really of equal length for the
whole extent of the scale; it is only essential that this equality exists for those parts, which are liable to be used
in leveling and which at least in levels, as they are made
1iow, do not extend far on both sides of the zero. To be
sure the bubble of the level changes its length in heat and
cold on account of the expansion and contraction of the fluid;
but levels are now made so, that there is a small reservoir at
one end of the tube, also partly filled with a fluid, which is
in communication with that in the level through a small
aperture. Then, if the bubble has become too long, the level
can be filled from  the reservoir by inclining it so that the
reservoir stands on the elevated side.  If on the contrary
the bubble is too short, a portion of the fluid can be drawn
off by inclining the level in the opposite direction. In this
manner the bubble can be always kept very nearly of the
same length, and if care be taken, to have the level always
well rectified and the inclination of the axis small, then only
a very few parts will be necessary for all levelings and
their length can be carefully determined.  Besides it would
be well to repeat this determination at very different temperatures in order to ascertain, whether the value of the
unit of the scale changes with the temperature.  If such a
dependence is manifest, then the value of the unit of the
level must be expressed by a formula of the form:
L --  + b (t - to)
where a is the value at a certain temperature t0, and in
which the values of a and b must be determined according
to the method of least squares from the values observed by
different temperatures.
Instead of a special instrument for determining the unit
of' the scale an altitude azimuth and a collimator can also
be used, if the latter be so arranged, that. two rectangular




397
Ys can be fastened to it, in which the level can be placed
so that it is parallel to the axis of the collimator.  If then
this collimator be mounted before an altitude instrument with
a finely graduated circle, and the level be placed in the Ys
and read off and likewise the circle, after the wire-cross of
the instrument is brought in coincidence with the wire-cross
of the collimator, and if this process be repeated after the
inclination of the collimator has been somewhat changed by
means of one of the foot-screws, then will the length of
the unit of the scale be determined by comparing the difference of the two readings of the level with those of the
circle.
Theodolites or altitude and azimuth instruments are
frequently already so arranged, that the length of the unit
of the scale of the level can be determined by means of one
of the foot-screws, which is finely cut for this purpose and is
provided with a graduated head.  These instruments rest
namely on three foot-screws which form a equilateral triangle.
If now the level be set upon the horizontal axis of such an
instrument and the axis be so placed, that the direction of
the level shall pass through the screw a provided with the
graduated head and therefore be perpendicular to the line
joining the two other screws, then can the value of the
unit of the scale be determined from  the readings of the
screw a and the corresponding motion of the bubble of the
level, when the distance between the threads of the screw as
well as the distance of the screw a from the line joining the
two other screws are known. The value of the unit of the
scale for the level attached to the supports of the microscopes or the verniers of the vertical circle is determined by
directing the telescope to the wire-cross of a collimator or
to a distant terrestrial object and then reading off both the
circle and the level. If then the inclination of the telescope
to the object be changed by means of the foot-screws of the
instrument, the amount of the inclination in units of the scale
can be read off on the level, whilst the same can be obtained
in seconds by turning the telescope towards the object and
reading off the circle in the new position.




398
3. The case hitherto considered, to determine by means
of the level the, inclination of a line upon which the level
can be placed, never actually occurs with the instruments,
but the inclination of an axis is always sought which is only
given by a pair of cylindrical pivots on which the level must
be placed. Even if the axis of the cylinders coincides with
the mathematical axis of the instrument, nevertheless the cylinders may be of different diameters, and in that case a level
placed upon them will not give the inclination of the axis of
the instrument. These pivots always rest on Ys, which are
formed  by planes making with each other an angle which
we will denote by 2i. Let the angle of the hooks of the
level, by which it is held on the axis, be 2i' and let the
radius of the pivot on one end (for which here again the
Fig. 12.    circle-end is taklen) be r,, then will b C
(Fig. 12) or the elevation of the centre
of the pivot above the Y  be equal to
/ro cosec i, likewise we have:
a C  ro cosec i',
hence:
C                  a )    [a   r  [coseci'-j-coseci],
and on the other end of the axis we
have:
a'  r,  [cosc i' -4- cosec i],
where r, is the radius of the pivot on
this side. If now the line through the
two Ys makes with the horizon the angle x, then, if the
diameters of the pivots be equal, the same inclination x will
be found by means of the level. If however the pivots are
unequal, then, if x denotes the elevation of the Y of the circleend, we will have for the elevation b of the circle-end:
b  x +-,'  [cosec i' -- cosec i],
where L is the length of the axis. If however the instrument be reversed so that the circle shall now rest on the
lower Y, then will the elevation of the circle-end be:
ib',-x +.j-O-ro -r, [cosec i'  cosec i].
From both equations we derive:




399
b' - b    -r  [cosec i' e  cosec i],
2       L
a quantity which remains constant so long as the thickness
of the pivots does not change.
Now  since we wish to find by means of the level the
inclination of the mathematical axis of both cylinders, we
must subtract from each b the quantity:
L   cosec',
or if rL —  be eliminated, the quantity:
- (b + b') cosec i'
cosec i + cosec i'
or  (b +- b') sin i
orsill i +- sill i
If the correction, as is generally the case, be small,
then we call make i   i' i) and we have therefore to apply
to every result of leveling the quantity - -(b -+b'), in which
b and b' denote the level-errors found in the two different
positions of the instrument.
Example.  On the prime vertical instrument of the Berlin
Observatory the inclination, that is, the elevation of the circleend was found according to No. I. to be b  - 2". 06, when
the circle was south.  After the reversion of the instrument
the leveling was repeated and the inclination found to be
b'- +- 5".02, which value, as before, is the mean of two
levelings by which in one case the object glass of the telescope was directed towards the east and in the other case
towards the west. In this case therefore is:
(b' + b) -- + 0". 74,
hence the inclination of the mathematical axis of the pivots
was:
-2". 80 Circle South
and — + 4". 28 Circle North.
Hitherto it has been assumed, that the sections perpendicular to the axis of the pivots are exactly circular. If this
is the case, then will the level in every inclination of the
telescope give the same inclination of the axis, and the telescope when it is turned round the axis will describe a great
*) Usuallyi,, andi' are equal to about 900.




400
circle.  But if this condition be not fulfilled, then will the
inclination be different for different elevations of the telescope
and the telescope, when it is turned round the axis, will describe a kind of zigzag line instead  of a great circle.  By
means of the level however we can determine the correction
which is to be applied to the inclination in a particular position in order to obtain the inclination for another position.
When, namely, the instrument is so arranged, that the level
by different elevations of the telescope  can  be attached to
the axis, then can the inclination of the axis in different positions of the telescope be found, for instance for every 15t1
or 30th degree of elevation, and only when the telescope is
directed towards the zenith or the nadir will this be impossible. If these observations are also made in the other position of the instrument, then can the inequality of the pivots
or the quantity i (b+ b') be deternmined for the different zenith distances, and if this be subtracted from  the level-error
in the corresponding  positions of the telescope, the inclination of the axis for the different zenith distances will be obtained.  By a comparison  of the same with  the inclination
found for'the horizontal position we can then obtain the corrections, which are to be applied to the inclination in the
horizontal position, in order to obtain the inclination for the
other zenith  distances.  These  corrections can be found by
observations for every tenth or thirtieth degree, and from
these values either a periodical series for the correction may
be found, or more simply by a graphic construction a curve,
the abscissae of the several points being the zenith distances,
and the ordinates the  observed  corrections of the  inclination. Then for those zenith distances, for which the correction has not been found  from  observations, it is taken
equal to the ordinate of this curve  ).
*) The pivots can be examined still better by means of a level, constructed for that purpose, which is placed on the Y in such a manner that
one end rests upon the pivot. If the level is first placed on the pivot at the
circle-end, and read off by different zenith distances of the telescope and then
the mean of the readings in the horizontal position of the telescope is subtracted, it is found, how much higher or lower the highest point of the pivot is
than in the horizontal position. These observed differences shall be uz. Now




401
B.  The vernier and the reading microscope.
4.  The vernier has for its object to read and subdivide
the space between any two divisions on a circle of an instrument, and consists in an are of a circle, which can be
moved round the centre of the graduated circle, and which
is divided into equal parts, the number of which is greater
or less than the number of parts which it covers on the
limb. The ratio of these numbers determines how far the
reading by means of the vernier can be carried.
If we have a scale divided into equal parts, each of
which is a, then the distance of any division from  the zero
can be given by a multiple of a.  If then the  zero of the
vernier or the pointer, which we will denote by y, coincides
exactly with one division of the limb, its distance from the
zero  of the limb  is known.  But if the zero of the vernier
falls between two divisions of the limb, then some one division of the vernier must coincide with a division of the
limb, at least so nearly that the distance from it is less than
the quantity, which can be read off by means of the vernier.
If the distance of this line of the limb from the zero point of
the vernier be equal to p parts of the vernier, each of which
is a', then its distance from the zero of the limb will be:
y -+- p a'.
But it is also qa-4-pa, where q a is that division of the
limb, which precedes the zero of the vernier, hence we have:
y -- p a'- q a — p a,
and therefore the distance of the zero of the vernier from
the zero of the limb is:
f-q qa + p (a- a').
If we have:          n a =(m -    1) a',
that is, if the number of parts on the vernier is greater by
if the same observations are made, when the level is placed on the other
pivot and the values u'z are obtained, then the line through the highest
points of the pivots will have the same inclination in all the different positions
of the instrument, if ua' =-,. But if this is not the case, then the quantity
L   206265, where L is the length of the axis, gives the difference of
the inclination in this position of the telescope from that in the horizontal
position.
26




402
one than the number which it covers on the limb, then we
have:                    a'   11
therefore:  1
therefore:             y= =cu+- P"
The quantity   a  is called the least count of the vernier. Therefore in order to find the distance of the zero of
the vernier from the zero of the limb or to read the instrument by means of a vernier: Read the limb in the direction
of the graduation up to the division-line next preceding the
zero point; this is the reading on the limb: look along the
vernier until a line is found, that coincides with one on the
limb; multiply the number of the line by the least count;
this is the reading on the vernier, and the sum  of these
two readings is the reading of the instrument.
We see that if we take the number mn large enough,
we can make the least count of the vernier as small as we
like. For instance if one degree on the limb of the instrument is divided into 6 equal parts, each being therefore 10
minutes, and we wish to carry the reading by means of the
vernier to 10", we must divide an arc of the vernier whose
length is equal to 590' in 60 parts, because then we have
-  1   10". In order to facilitate the reading of the vernier,
r  -- 1the first line following the zero of the vernier ought to be
marked 10", the second 20" etc., but instead of this only the
minutes are marked so that the sixth line is marked 1, the
twelfth 2 etc.
In general we find m from the equation:
C            a
a - C  -    or rn  -"___ -- 1
mn - I      Ca y- a
taking for a- a' the least count of the vernier and for a the
interval between two divisions of the limb, both expressed in
ternms of the same unit.
Hitherto we have assumed, that:?m a -  (m +- 1) a',
therefore that the number of parts of the vernier is greater
than the number of parts of the limb, which is covered by
the vernier. But we can arrange the vernier also so, that
the number of its parts is less, taking:
(m -+- 1) a  m at'.




403
a
In this case we have:   a' - a — a
and                    y - qa-p- a
m
In this case the vernier must be read in the opposite
direction.
If the length of the vernier is too great or too small by
the quantity A 1, then we have in the first case:
nm cta  (m + 1) a' - A,
therefore using the same notation as before:
y q"pa +-    - A
m11 -- 1   m - - 1
Therefore if the length of the vernier is too great by Al,
we must add to the reading of the vernier the correction:
m — 1 A
where p is the number of the division of the vernier which
coincides with a division of the limb and m-i-1 is the number of parts, into which the vernier is divided. For instance
if we have an instrument, whose circle is divided to 10', and
which we can read to 10" by means of a vernier, so that
59 parts of the circle are equal to 60 parts of the vernier,
and if we find that the length of the vernier is 5" too great, or
A l -- 5", we must add the correction -    5". The length
of the vernier can always be examined by means of the division of the limb.  For this purpose make the zero of the
vernier coincident successively with different divisions on the
limb, and read the minutes and seconds corresponding to the
last division-line on the vernier. Then the arithmetical mean
of these readings will be equal to the length of the vernier.
5. If great accuracy is required for reading the circles,
the instruments, for instance the meridian circles, are furnished
with reading microscopes, which are firmly fastened either
to the piers, or to the plates to which the Ys are attached,
in such a manner, that they stand perpendicular over the graduation of the circles. The reading is accomplished by a moveable wire at the focus of the microscope, which is moved
by means of a micrometer screw whose head is divided into
equal parts, depending upon the extent to which the subdivisions are to be carried. The zero of the screw head is
26*




404
so placed that if the wire coincides with a division-line on
the circle, the reading of the screw head is zero; in this
case the circle is read up to this division-line; hut if the
wire falls between two  division-lines of the circle, it is
moved by turning the screw head until it coincides with the
next preceding line on the circle, in which position the head
of the screw is read, and the reading is then the sum of the
reading on the circle and that on the screw head ). Thus
the zero of the screw head corresponds to the zero of the
vernier, since always the distance of the wire in the position
when the reading of the screw is zero from the next preceding division-line of the circle is measured by means of the
screw head. The value of one revolution of the screw expressed in seconds of arc is determined beforehand, and since
the number of the entire revolutions of the screw can be read
by a stationary comb-scale within the barrel of the microscope, whilst the parts of a revolution are read by means of
the screw head, this distance can always be found. Now it
can always be arranged so that an entire number of revolutions is equal to the interval between two division-lines of the
circle, for the object glass of the microscope can be moved
farther from or nearer to the eye-piece, and thus the image
of the space between two lines can be altered and can be
made equal to the space through which the wire is moved
by an entire number of revolutions of the screw. If the screw
performs more than an entire number of revolutions, when the
wire is moved from one division-line to the next, then the
object glass of the microscopes must be brought nearer to
the eye-piece; but since by this operation the image is thrown
off the plane of the wire, the whole body of the microscope
must be brought nearer to the circle, until the image is again
well defined.
The microscope must be placed so that the wire or the
parallel wires are parallel to the division-lines of the circle,
and that a plane passing through the axis of the microscope
and any radius of the circle is perpendicular to the latter. If
*) It is better to use instead of a single wire two parallel wires and to
bring the division lines of the circle exactly between these wires.




405
it is not rectified in this way, the image of a line moves a little
sideways, when the circle is gently pressed with the hand, and
thus errors would arise in reading off the circle, if it should
not be an exact plane or should not be exactly perpendicular
to the axis. If such a motion of the image arising from the
gentle pressure of the hand be observed, the tube in which
the object glass is fastened must be turned until a position
is found in which such a pressure has no more effect upon
the image.
Since the distance of the microscope from the circle is
subject to small changes, the error of run, that is the difference between an entire number of revolutions and the measured  distance  of two division-lines, must be frequently determined and the reading of the microscope be corrected accordingly'). But it is not indifferent, which two lines of
the circle are chosen for measuring their distance, since this
can slightly vary on account of the errors of division; therefore the exact distance of two certain lines must first be
found and then the run of the microscope always be determined by these two lines.
The micrometer screw itself can be defective so that by
equal parts of a revolution of the screw the wires are not
moved through equal spaces. In order to determine these
errors of the screw, a short auxiliary line (marked so that
it cannot be mistaken for a division-line) is requisite at a
distance from  a division-line, nearly equal to an aliquot part
of the space between two  lines, for instance at a distance
of 10" or 15", in general at the distance a" so that 120=-na.
If now we turn the micrometer screw to its zero and then by
moving the circle bring the line nearest to the auxiliary line
between the wires, we can bring the latter line between the
*) The circle of a meridian instrument is usually divided to 2 minutes,
and two revolutions of the screw are equal to the interval between two division
lines. Hence one revolution of the screw is equal to one minute and the head
being divided into 60 parts, each part is one second, whose decimals can be
estimated. In that position of the wires to which the zero of the screw head
corresponds they bisect a little pointer connected with the comb scale, and if
this pointer should be nearer to the following than to the preceding line, then
one minute must be added to the reading on the screw head.




406
wires by the motion of the screw  and thus measure the distance of the lines by means of the screw. If we leave now
the screw untouched and move the circle, until the first line
is again between the parallel wires, we can again by moving
the screw bring the second line between the wires, and we
can continue this operation, until the screw has made the
two entire revolutions which are always used in reading the
circle ).  If then the different values of the distance of the
two lines as measured by the screw are:
from O to a           a'
from a to 2a          a"
from (n - 1) a to n a  a",
the last reading on the screw will again be nearly zero, and
hence we can assume, that the mean value of all different
a', a" etc. is free from  the errors of the screw.  These observations must be repeated several times and also be changed
so that the intervals are measured in the opposite direction,
starting from 120 instead of 0, and then the means of all the
several values a', a" must be taken.  If we put then:
a' -4- all   a"' +..    a"
n
the correction, which must be added to the reading of the
screw, if also the interval from  -  ce to 0 and that from  n a
to  (n +  1) a is measured  and the corresponding  distances
are denoted by a-' and a"+', will be:
for - a       - a.c -It
0         0
xa         o - a'
2a         2ao - a' - a"
(a — 1) a- (n - 1) a o- a' -..-    a..  -
21 a       0
(n -- 1) a-      ao - a"+.
~) If there is no auxiliary line on the circle, the two parallel wires can
be used for this purpose, if their distance is an aliquot part of 2 minutes.
Then, when the screw is turned to its zero point, the circle is moved until
a line coincides with one wire, and then the other wire is placed on the same
line by moving the screw.




407
By means of these values the correction for every tenth
second can be easily tabulated and then the values for any
intermediate seconds be found by interpolation. The reading
thus corrected is free fromi the errors of the screw and gives
the true distance of the wires in the zero-position from the
next preceding line, expressed in parts of the screw head,
each of which is the sixtieth part of a revolution of the
screw, and hence if two entire revolutions of the screw should
differ from 2 minutes, this distance is not yet the distance
expressed in seconds of arc.
Now  in order to examine this, two lines on the circle
are chosen, whose distance is known and shall be equal to
120 -- y. Then after moving the screw to its zero-point we
move the circle until the following one of' the two lines is
between the wires and then bring by the motion of the screw
the preceding line between the wires %).  If in this position
the corrected reading of the screw is 120 -4- p, then the reading of the screw, if we had moved it from zero through
exactly 120 seconds, would have been 120 -4-p - y; therefore all readings nmust be corrected by multiplying them by:
120
120 4- p — y
It must still be shown, how  the length of an interval
between two certain lines, for instance that between 0~ 0' and
0~ 2', can be found. For this purpose first the length of the
interval in parts of the screw head is found by moving the
circle, after the screw has been turned to its zero, until the
line 0~ 2' is between the wires, and then moving the latter
by means of the screw, until the line 0~ 0' is between them.
The length of the interval expressed in parts of the screw
head shall be from the mean of many observations 120 — x. If
then in the same way a large number of intervals at different places of the circle are measured, we can assume that
there are among them as many too great as there are too
small, so that the arithmetical mean will be the true value
of an interval equal to 120", expressed in parts of the screw
*) The reading of the screw increases, when it is turned in the opposite
direction in which the division runs.




408
head. Now if the mean be 120 - u, the first interval is too
large by x — u   y or is equal to 120 +-y.
The correction, which must be applied to the reading
for this reason, can also be tabulated so that the argument
is the reading on the screw.  As long as the error of the
run remains the same, this table can be united with the one
for the corrections of the screw.
C. Errors arising from an excentricity qf the circle and errors of division.
6. A cause of error which cannot be avoided with all
astronomical instruments is that the centre round which the
circle or the alhidade carrying the vernier revolves is different
from that of the division. We will assume that C Fig. 13
be the centre of the division,
Fig. 13.        C' that of the alhidade and that
the direction C'A' or the angle
-4           0 CA' have been measured equal
/ A' \ j  to A'- O, supposing that the
angles  are  reckoned  from  O.
Then, if the excentricity were
nothing, we- should have read
the angle A CO   A'C'O.  Denoting the radius of the circle
o  CO by r and the angle A CO=
A' C'O by A-0, we have:
A'P - r sin (A' - 0)  A' C' sin (A - 0)
and C'P- r cos (A' - 0) - e  A' C' cos (A -O),
where e denotes the excentricity of the circle.
If we multiply the first equation by cos (A'- 0), the
second by sin (A'-  0) and subtract the second from  the
first, we obtain:
A' C' sin (A - A') = e sin (A' - 0).
But if we multiply the first by sin (A' - 0), the second
by cos (A'- 0) and add them, we find:
A' C' cos (A - A') - r - e cos (A' - 0),
therefore we have:
esin (A' - 0)
tang (A - Al)
1  — cos (A' — 0)




409
or by means of the formula (12) in No. 11 of the introduction:
e              e2
A   A'=-' - - sin (A'- 0)  sin 2 (A'- 0)
r        
+-I 3- sin 3 (A' O) -4.-...
Now  since e is always a very small quantity, the first
term of this series is always sufficient, and hence we find
A - A' expressed in seconds of arc:
A- A'- e sin (A'- 0) 206265,
r
whence we see, that the error A - A' expressed in seconds
can be considerable on account of the large factor 206265,
although e is very small.
r
In order to eliminate this error of the reading caused
by the excentricity, there are always two verniers or microscopes opposite each other used for reading the circle.  For
if the alhidade consists of two stiff arms, each provided with
a vernier, which may make any angle with each other, the
correction for the reading B' by the second vernier would
be similar so that we have:
A   A' +- - sin (A' - 0)
and
B -B' - e sin (B' - 0),
and hence:. (A + B)  I (A' + B') + _  sin [4 (A' +- B') - 0] cos  [A'- B'].
We see therefore, that in case that the angle between the
arms of the alhidade A' -B' is 180", then the arithmetical
mean of the readings by both verniers is equal to the arithmetical mean which we should have found if the excentricity
had been nothing. For this reason all instruments are furnished with two verniers exactly opposite each other, and by
taking the arithmetical mean of the readings, made by these
two verniers, the errors arising from an excentricity of the
circle are entirely avoided.
In order to find the excentricity itself, we will subtract
the two expressions for A and B. Then we get;




410
B - A = B' - A' +- 2   ncos [' (A' +- B') - O] sin  (B' -')
r
or supposing that the angle between the verniers differs from
1800 by the small angle a:
B - A-  180 + a,
B' - A' - 1800   a + 2 - sin (A' - 0)
r
- 1800 + a + 2 --- cos  sin A'- 2 — sin O cos A'.
2'              r
If we take now:
B'-A - 180 =[XA], 2 - cos0-z and 2 - sin O -y,'               r
we obtain:
[XA ]  a +- z sin A' - y cos A',
and hence we can find the unknown quantities a, z and y
by readings at different places of the circle.
Example. With the meridian circle at the Berlin Observatory the following values of B' - A'- 1800 were observed
for two microscopes opposite each other:
Xo  --- 0".3       X0, s    — +1l".5
X30 ==-3.3        X,,,l -0.6
xA0  — +3.8      X,240  +0.7
X90  — 3.1        X270=0.0.   7
X120-+4.8         X300 —2.5
Xl,5, = —+ 6.4    X330 -4.8.
From  this we find the sum  of all these quantities:
+- 16.7 - 12 a
hence:
a -   1 — -". 39.
Moreover we find according to No. 27 of the introduction:
A        XA       XA      XA       XA
+-      +-.     -         +
0~     +0.3   -1.2
300    -1.5   -7.3  -+8.1  -4-15.1
60     +1.3   -4.2  +6.3  +10.4
90       3.8              -2. 4   -+- 2.4
120     +5.5.5           +4.1
150     +5.8              +7.0
180     +- 1.5
and hence:              -2 ny: +  9".62
2  z- +-18. 96,
therefore:       0- =260 54'. 2 and e   1"72.




411
7. If a circle is furnished with several pairs of verniers
or microscopes, as it is generally the case, the arithmetical
mean of the readings by two verniers ought always to differ
from the arithmetical mean of the readings by two other
verniers by the same constant quantity, if there were no other
errors besides the excentricity. However since the graduation
itself is not perfectly accurate, this will never be the case.
But, whatever may be the -nature of these errors of division,
they can always be represented by a periodical series of
the form:
a +- a, cos A + a2 cos 2 A.....
+- b, sin A- b2 sin 2 A +-.....
where A is the reading by a single vernier or microscope.
If now we use i verniers equally distributed over the
circle, then their readings are:
A,A+ -, A+ - 2. 7and
A+(i-1) 2-,
and if we now take the mean of all readings, a large number of terms of the periodical series for the errors of division will be eliminated, as is easily seen, if we develop the
trigonometrical functions of the several angles and make use
of the formulae (1) to (5) in No. 26 of the introduction.
In case that the number of verniers is i, only those
terms remain, which contain itimes the angle. Hence we
see that by using several verniers a large portion of the
errors of division is eliminated, and that therefore it is of
great advantage to use several pairs of verniers or microscopes.
The errors of division are determined by comparing intervals between lines, which are aliquot parts of the circumference, with each other. For instance if the errors of division were to be found for every fifth degree, we should place
two microscopes at a distance of about 5 degrees over the
graduation.  Then we should bring by the motion of the
circle the line marked 00 under one microscope, which we
leave untouched during the entire operation, and measure the
distance of the line marked 50 by the micrometer screw of




412
the second microscope simply by turning this screw  until
that line is between the wires and then reading the head of
the screw.  If now  we turn the circle until the line 50 is
between the wires of the first microscope, the line 10" will
be under the second microscope and its distance from the
line 5" can be measured in the same way, and this operation
can be continued through the entire circumference, so that
we return to the line 0" and measure its distance from the
line 355".  The same operation can be repeated, the circle
being turned in the opposite direction.  If then we take the
arithmetical mean of all readings of the screw and denote
it by ac,, and the readings for the lines 5", 10~ etc. by a',
ca" etc., the error of the line 5", taking that of the line 00 as
nothing, will be ro,,- ac', that of the line 10~, 2 cr0 -'- ce" etc.
But since the circle undergoes during so long a series changes by the change of temperature, it is better, to determine
the errors of the several lines in this way, that first the errors
of a few  lines, for instance those of the lines 00 and 180",
be determined with the utmost accuracy, and then relying
upon these, the errors of the lines 900 and 270" be determined by dividing the arcs of 1800 into two equal parts;
and then by dividing the arcs of 900 again into two or
three equal parts and going on in the same way, the errors
of the intermediate lines are found. Small arcs of 1 degree or
2 degrees may even be divided into five or six equal parts,
but for larger arcs it is always preferable to divide them
only into two equal parts. These operations can be quickly
performed and for the sake of greater accuracy be repeated
several times.
In order to make this examination of the graduation, two
microscopes are requisite which can be placed at any distance from each other over the graduation. For small intervals, for instance of one degree, one microscope with a
divided object glass can be conveniently used.  Before the
operation is begun, the microscopes must of course be rectified according to No. 5, and it is best, to use always the
same microscope for measuring and to arrange the observations even so, that always the same portion of the micrometer screw is used for these measurements.  This end can




413
always be attained, if at the beginning of each series the
screw of that microscope which is merely used as a Zero is
suitably changed.
Example. For the examination of the graduation of the
Ann Arbor meridian circle two microscopes were first placed
at a distance of 1800. When the line 00 was placed under
the first microscope, the reading of the second microscope
after being set at the line 180", was - 17".9; but when the
line 180" was brought under the first microscope, then the reading of the other for the division-line 0" was — 2".7. Hence
the mean is - 10". 3 and the error of the line 180" is 7".60.
The mean of 10 observations gave +-7". 61, which value was
adopted as the error of that line.. In order to find the errors of the lines 90" and 270", the arcs 0O to 180" and 1800
to 0" were divided into two equal parts by placing the two
microscopes at a distance of 90".  If then the line 0" was
brought under the first microscope, the reading of the second
microscope for the line 900 was — 6".5, whilst when the
line 900 was brought under the first microscope, the reading
of the second microscope for the line 180" was -3".5 and,
if this be corrected for the error of that line, -+- 4".11.
The arithmetical mean of - 6". 5 and -- 4". 11 gives - 1". 19,
hence the error of the line 90" is — 5".31. In a like manner the errors of the lines 45", 1350, 2250 and 3150 were
determined by dividing the arcs of 90" into two equal parts.
Then the errors for the arcs of 15 0 might have been determined by dividing the arcs of 45 degrees into three equal
parts. But since the lnicroscopes of the instrument cannot
be placed so near each other, arcs of 315 and 225 were divided into three equal parts. For this purpose the microscopes were first placed at a distance of 105 degrees. When
the lines 0~, 105~ and 210" were in succession brought under
the fixed microscope, the readings of the second microscope
were respectively - 11".9, — 5".6 and +-2".0 or if we add
to the last reading the error of the line 3150, which was
found -0".48, we get — 11".9, — 5".6 and +1".2.  The
arithmetical mean of all is -  5". 33, hence the error of
the line 1050 is +46".57, that of the line 2100 is equal to




414
2ao -a' -  c"= —6". 84.  If the first line which we use is
not the line 0" but another line, whose error has been found
before, the first reading must be corrected also by applying
this error with the opposite sign. For instance when the
first microscope was set in succession at the lines 90", 1950
and 300", the readings of the second microscope for the lines
1950, 300" and 45" were successively -6". 6, -- 2".1 and - 7".9.
Now since the errors of the lines 90" and 450 have been found
to be + 5".46 and +- 3".36, the corrected readings are - 12".06,
-+- 2". 10 and - 4". 54. The mean is - 4". 83, and hence the
error of the line 1950 is -t-7". 23, and that of 300" is - 0". 30.
The errors thus found are the sum of the errors of division and of those caused by the excentricity of the circle
and by the irregularities of the pivots; finally they contain
also the flexure, that is, those changes of' the distance between
the division-lines produced by the action of the force of gravity
on the circle. The errors produced by the latter cause will
change according to the position of a line with respect to
the vertical line, so that the correction which must be applied
to the reading for this reason will be expressed by a series
of the form:
a' cos z - b' sinz -[- " cos 2z +- b"sin 2z ~ a"'cos3z + b'" sin 3z +...
where the coefficients of the sines and cosines are different
for each line and change according to the distance of the line
from a fixed line of the circle. We see therefore, that if a
line is in succession at the distance z and 1800-~z from the
zenith, all odd terms of the series are in those two cases
equal but have opposite signs. Therefore if we measure the
distance between two lines first in a position of the circle, in
which the zenith distance of that line is z and afterwards in
the opposite position, in which its zenith distance is 1800-1 —,
then the mean of the measured distances is nearly free from
flexure and only those terms dependent on 2z, 4z etc. remain in the result. If we repeat the observations in 4 positions of the circle, 900 different from each other, then only
the terms dependent on 4z, 8z remain in the arithmetical
mean., Generally already the second terms will be very small,
and hence the mean of two values for the distance between




415
two lines determined in two opposite positions of the circle
can be considered as free from  flexure *).
The errors arising from the excentricity are destroyed,
if the arithmetical means of the errors of two opposite lines
are taken, and the same is the case with the errors caused
by an imperfect form of the pivots. For such deficiencies
have only this effect, that the error of excentricity is a little
different in different positions of the instrument, since when
the  instrument is turned  round  the axis, the centre of the
division occupies different positions with respect to the Ys**).
If the circle is furnished  with  4 microscopes, as is usually
the case, the arithmetical means of the errors of every four
lines which are at distances of 900 from  each other are taken
and used as the corrections which are to be applied to the
arithmetical mean  of the readings by the 4 microscopes in
order to free it from the errors of division.
By the method given above, the errors of every degree
of the graduation  and  even  of the arcs of 30' may  be determined. If a regularity is perceptible in these corrections,
at least a portion of them can be represented by a series
of the form  a cos 4 z -- b sin 4 z -+ a, cos 8 z --- b sin 8 z etc. and
thus the periodical errors of division are obtained which can
be tabulated. But the accidental errors of the lines must be
found by subdividing the arcs of half a degree into smaller
ones according  to the above method, and  since this would
be an immense labor if excecuted for all lines, Hansen has
proposed a peculiar construction of the circle and the micros*)  Bessel in No. 577, 578, 579 of the Astron. Nachr. has investigated the effect of the force of gravity on a circle in a theoretical way and
has found for the change of the distance between two lines the expression
a' cos z -I- b' sin z. However the case of a perfectly homogoneous circle, which
he considered, wlll hardly ever occur. Usually the higher powers of the expression for flexure will be very small, but it is always advisable, to examine
this by a special investigation.
**) The errors arising from the excentricity of the circle and from the
irregularities of the pivots are of the form:
[e -+- e' cos z -- e" sin z - e'2 cos 2 z +- e", sin 2z] sin (A -O),
where A is the reading of the circle, z the zenith distance of the zero of
the circle, and 07 the direction of the line through the centre of the division
and that of the axis, which is likewise a function of z.




416
copes, for which the number of lines, whose errors must be
determined, is greatly diminished.  (Astron. Nachr. No. 388
and 389.)  The determination of these errors will always be
of great importance for those lines, which are used for the
determination of the latitude, the declination of the standard
stars and the observations of the sun; and after the errors
for arcs of half a degree have been obtained, the errors of
the intermediate lines of any such arc can be found by measuring all intervals of 2 minutes by means of the screw of
the microscope. For this purpose we turn the screw of the
microscope to its zero, then bring by the motion of the circle
the line of a degree between the wires and measure the distance of the next line by means of the screw.  After this
the screw is turned back to its zero and when the same line
has been brought between the wires by turning the circle,
the distance of the following line is measured and so on to
the next line of half a degree. These measurements are also
made in the opposite direction, and the means taken of the
values found for the same intervals by the two series of observation. Then if x and x' are the errors of division of the
first and the last line, and a', a" etc. are the observed intervals between the first and the second, the second and the
third line etc., we have:
a' - a" %- a"' -- ~ ~ ~   x'-x
equal to an interval of 2 minutes as measured by the screw,
and hence the error of the line following the degree line is:
x -i- a, - a
that of the second x  2ao -a' -a"
that of the third  x - 3a -  t - a" - a
and so forth.
Compare on the determination of the errors of division:
Bessel, Knigsberger Beobachtungen Bd. I und VII, also
Astronomische Nachrichten No. 841.  Struve, Astronomische
Nachrichten No. 344 and 345, and Observ. Astron. Dorpat.
Vol. VI sive novae seriae Vol. III; Peters, Bestimmung der
Theilungsfehler des Ertelschen Verticalkreises der Pulkowaer
Sternwarte.




417
D. On flexure or the action of the force of gravity upon the telescope
and the circle.
8. The force of gravity alters the figure of a circle in
a vertical position. If we imagine the point, from which
the division is reckoned, to be directed to the zenith, every
line of the graduation will be a little displaced with respect
to the zero, and for a certain line A the produced displacement shall be denoted by a,,.  If now we turn the circle
so that its zero has the zenith distance z, that is so that
the line z of the graduation is directed towards the zenith,
the displacement of the line A  will be different from  a,.
If we denote by ce. the displacement of the line A, when the
zero has the zenith distance', which shall be reckoned in
the same direction from  00 to 360", then ac5 can be expressed
by a periodical series of the following form:
a' cos - t  a" cos 2  -- a"' cos 3 + —...
+- b' sin  +- b" sin 2 +  b"' sin 3-4-...
But if we take now  another line, the displacement of
it will be expressed by a similar series, in which only the
coefficients a', b' etc. will have different values. These coefficients themselves can thus be expressed by periodical series,
depending on the reading of the circle, so that the displacement of any line u of the graduation, when the zero has
the zenith distance i, can be expressed by a periodical series
of the form:
a,, cos  + a",, cos 2  -- a"',, cos 3 g+...
+ b', sin  -- b",, sin 2 g + b"',, sin 3 --....
where a',U b'5 etc. are periodical functions of u. The sign
of this expression shall be taken so, that the correction given
by the expression is to be applied to the reading of the circle
in order to free it from flexure.
Now a complete reading of the instrument is the arithmetical mean of the readings of the different microscopes,
the number of which is usually 4.  These microscopes we
will suppose to be so placed, that one of them indicates 0~,
when the telescope is directed to the zenith.  The zenith
distance of this microscope which always gives the zenith
distance of the telescope shall be denoted by m. If now the
27




418
telescope is turned so that it is directed to the zenith distance z, the line z will be under this microscope, and since
in this case the zenith distance of the zero is  -i- m, we
have in this case uz —,       = --- n-; hence the correction
which is to be applied to the reading of the microscope, is:
a' cos (z +- rn)t- a"z cos 2 (z - mn) -- a"'11 cos 3 (z H- rm) +...
+- b'z sin (z -- rn) -+ b"z sin 2 (z +- mn) +- b'." sin 3 (z +- nz) -...
For the other microscope, whose reading is 90 — z, we
have u =  90 ~- z, --    -- rm; hence the coefficients in the
expression for flexure become a'9o+: b'9o0+  etc. and thus we
see, that when we use four microscopes at a distance of 90'
from each other, and take the mean of all 4 readings, then
we have to apply to this mean the correction:
a' cos (z +- ) H- a", cos 2 (z ~ i) H- a"', cos 3 (z - z) +...
H- f'  sin (z +-?r) t- p" Sin 2 (z -- n) - pf% sin 3 (z — +  ) --...
where the several ce and [3 are periodical functions of z, but
contain only terms in which 4z, 8  etc. occur, since all the
other terms are elimninated by taking the mean of four readings.  If these terms should be equal to zero, then the force
of gravity has no effect at all on the arithmetical mean of
the readings of four microscopes; otherwise there exists flexure, and since m is constant, the expression for the correction which is to be applied to the mean of the readings of
4 microscopes will have the form:
a' cos z 4- a" cos 2z - a" cos 3  -...
+ b' sin z H- b" sin 2 zH- b"' sil 3 z -+...
But the force of gravity acts also on the tube of the
telescope, bending down both ends of it, except when it is
in a vertical position. If the flexure at both ends is the same
so that the centre of the object glass is lowered exactly as
mtuch as the centre of the wire-cross, it is evident, that it
has no influence at all upon the observations, since in that
case the line joining those two centres (the line of collimation) remains parallel to a certain fixed line of the circle.
But if the flexure at both ends is different, tilhe line of collimation changes its position with respect to a fixed line of
the circle, and hence the angles, through which the line of
collimation moves, do not correspond to the angles as given
by the readings of the circle.  The correction which is to




419
be applied on this account to the readings can again be expressed by a periodical function, and hence we may assume,
that the expression (A) represents these two kinds of flexure,
that of the circle and that of the telescope.
There are two methods of arranging the observations ill
such a manner, that the result is free from  flexure, at least
from  the  greatest portion  of it.  For if we observe a star
at the zenith distance z, its image reflected from an artificial
horizon will be seen at the zenith distance 180 - A, hence
the division-lines corresponding to these zenith distances will
be under that microscope, whose reading  gives the zenith
distance.  Now  if we reverse the instrument, the division of
the circle runs in the opposite direction, and hence the reading for the direct observation is now  360" -    and that for
the  reflected observation 1800~- ~.  Therefore if we denote
the four complete readings, corrected for the errors of division,
for those four observations by z, z', 3" and A"', and by   the
true zenith distance free from flexure, we have the following
four equations, in which N  denotes the nadir point:
Direct          - z -- a' cos z -+ aL cos 2 z -+ a"' cos 3z -I-.. -+- b' sin z
4 — b"sin 2z -t+ b" sin 3z ~-.. - (1I 0~ +N) -- a'-a"-l-a"'*)
Reflected 180~- — z' - a' cos z +- a" cos 2z - a"' cos 3z +-.. + b' sinz
- b"sin 2z -- b"' sin 3z -..- (180~+N) -- a'-a"-+-a'"
Direct  3600 —- - z" +- a' cos z + a" cos 2z -+ a"' cos 3 z +-.. - b' sin z (B
-bsin 2z - b"' sin 3 z —..- (10~+N) — + a' —a"C-a"
Reflected 180~ —+ -z"' - a' cos z ~- a" cos 2z - a"'. cos 3z +.. -- b' sin z
+- b"sin 2- b"' sin 3z -—.. — (180~+0N) - ac' —a"'-a"'.
From  these equations we obtain:
900 --     a2-  ac  Cos z -a" cos 3 z-.. -b" sin 2z —...
900 -  -        -- +  -' cos z +- ac" cos 3 z-.. - b" sin 2z -...,
hence by taking the mean:
90~-0  9       5 — + - -- 2   --  si 2z --.
and we see therefore, that if a star is observed direct and
reflected in both positions of the instrument, only that portion of flexure, which is expressed by the terms b" sin 2 
*) The correction which is to be applied to the nadir point is namely
- a' " +   a?'   -F-. 
27:




420
— b' sin 4z etc. remains in the mean of those four observations.
We obtain also from the mean of the first two equations (B):
90o   --         Cta! cos 2z -..  b' sin z -- b" sin 3 z- -...
2
- (180~ -- N) -- a' - a" -- c"',
likewise:
270" --   - -  -a"cos 2z+. —' sin - h"' sin 3z-...
-- (1800 -N') -+- a' -   a"'-,
fromn which we find:
z q' z'  z" 4- zX l
3G0"  -.-  2 +        2 a" cos 2 z +-..- (N  - N') +- 2 (a' -  "  a- "')
2       2
1800 I         -----   2 1b' sin - - 2 b1" sin 3z.. - N- N'.
2       2
Therefore if we observe different stars direct and reflected in both positions of the instrument, we can find from
those equations the most probable values of the coefficients
ab" acV etc. and b', b"' etc.
Since these observations are made on different days, it
is of course necessary to reduce the zenith distances z, z', z"
and z"' to the same epoch, for instance to the beginning of
the year by applying to the reading of the circle the reduction to the apparent place with the proper sign. Since, besides, the microscopes change continually their position with
respect to the circle, it is also necessary, to determine the
zenith or nadir point after each observation (VII, 24) and
thus to eliminate the change of the microscopes.  Another
correction is required for the reflected observations.  For if
we observe a star reflected, we strictly do not observe the
star from  the place where the instrument stands, but from
that in which the artificial horizon stands, and thus the latitude of the place for those observations is different.  Now
since the artificial horizon is placed in the prolongation of
the axis of the telescope, its distance from the point vertically
below the centre of the telescope will be h tang z, where h
is the height of the axis of the instrument above the artificial
horizon.  Since an arc of the meridian equal to a toise corresponds to a change of latitude equal to 0".063, we must add
to the zenith distance of the reflected image of the star, if h
is expressed in Paris feet, the quantity 0".011 h tang z.




421
A  second method of eliminating the flexure was proposed by Hansen and requires a peculiar construction of the
telescope. The tube of the telescope, namely, is made in such
a manner, that the heads, in which the object glass and the
eye-piece are fastened, can be taken of and their places be
exchanged, without changing the distance off the centres of
gravity of both ends of the tube from  the axis of the instrument. Thus in exchanging the object glass and the eye-piece
the equilibrium  is not at all disturbed and it can be assumed,
that the effect of the force of gravity oil the telescope is the
same in both  cases.  Now  if in one case the line 180" of
the circle is directed to the nadir, and the reading of one
microscope is the zenith distance, then in the other case the
line 0" will correspond to the nadir, and the reading of the
same microscope will be 180" t- the zenith distance. Therefore if g is the zenith distance free fromn flexure, and if the
readings corrected for the errors of division are in the first
case a, and in the other z', we have:
- z +-  a. cos z +- a" cos 2 z   a"' cos 3z +.. +- b' sin z
1- l" sin 2z- b"'sin 3...- (80~ 0 -- N) -  ca'- ct -a"'-.
-- z' -a' cos z q- a" cos 2z - a"' cos 3z +...- b' sin z
- b" sin 2z- l"l sin 3z..  - ( S00 a- V') -a"' - " - c"'-..
Therefore we obtain from  the mean of those two equations, denoting the zenith points 1800-+N  and 180~" —N' by
Z and Z':
Z q- Z' —Z''-, —     — q- at cos 2 z...- +    sin 2 -... -....
whence we see that the arithmletical mean of the zenith distances in the two cases contains only that portion  of flexure, which is expressed by the terms dependent on 2z, 4z etc.
We also obtain by subtracting the above equations:
z'-Z'-(z-Z)
~= 00 ~    2 (    ) -a' cosz-a "'cos3z-...- b'sinz..- b" sin 3z —...
a -a --.,
hence we see, that we can determine the coefficients of the
terms dependent on z, 3z, etc. by observing stars at various
zenith distances or by means of a collimator placed at various zenith distances.
In general we can find these coefficients by placing the
telescope in two positions which differ exactly 180". In order




422
to accomplish this, we mount two collimators so, that their
axes produced pass through the centre of the axis of the instrument, and direct them towards each other through apertures, made for this purpose in the cube of the axis of the
instrument, so that the centres of their wire-crosses coincide.
Then the telescope being directed first to the wire-cross of one
collimator and then to that of the other, will describe exactly
180".  Hence if we read  the circle in the two positions of
the.telescope, and denote the true zenith distance of the collimator by i, we have in one position:
-z +-i a'cosz +- a"cos2z - ac'" cos3z --... +- b' sin z - b" sin 2z
-- b"' sin 3 z  --..Z -Z- aZ    -  a"ll - a"'
and in the other position:
180 +    z  a cos z - a" cos 2 z -- ca"' cos 3 z.. - b' sinz -4- b" sin 2z
- b. sin 3z -... Z -+' - a" - a",
therefore:
z' -z- 180
0          -        --  cos z - a"'cos 3z -... - b'sin z- b"' sin 3 z-...
2
Since we use in reading the circle both times the same
division-lines, the observed quantity A' —z is entirely  free
from  the errors of division.  If we make these observations
by different inclinations of the telescope, that is, at different
zenith distances, we obtain a number of such equations, from
which we can find the most probable values of the coefficients.
There is no difficulty in making these observations when
the telescope is in a horizontal position; but when the inclination is considerable, it would become necessary to place
one of the collimators very high, in which case it might be
difficult to give it a firm stand. However one can use instead of this collimator a plane mirror which is placed at
some distance in front of the object glass or better held by
an arm, which is fastened to the pier of the instrument so
that by turning this arm it may easily be placed in any position'). If then outside of the eye-piece of the lower collimator a plane glass is fastened  at an angle of 45" *), by
*) The mirror must admit of a motion by which it; can be placed so
that a horizontal line in its plane is perpendicular to the axis of the telescope.
**) This plane glass must be fixed so, that one can change its inclination to the eye-piece and that it can be moved around the axis of the




423
means of which, light is reflected into the telescope and which,
while it is not used, can be turned off, and if the telescope
of the collimator is directed to the mirror, then looking into
the telescope through this plane glass we see not only the
wire-cross of the collimator but also its image reflected from
the mirror.  Hence by turning the collimator, until the wirecross and the reflected image coincide, we place its axis perpendicular to the mirror. If then we place by the same means
the telescope of the instrument perpendicular to the mirror,
and afterwards direct it to the wire-cross of the collimator, the
angle, through which the telescope is turned, will be exactly
180, and hence we can find, as before, those terms of the
expression for the flexure, which depend upon z, 3z, etc.
It is best to make these observations in a dark room  and to
reflect the light from  a lamp into the telescope, since then
the reflected images of the wires are better seen.  The only
difficulty will be, to find a plane mirror which will bear a
high magnifying power. But since it need not be larger than
the aperture of the collimator, it will not be impossible, to
excecute such a mirror, especially as it is used only for rays
falling upon it perpendicularly.
The coefficients of the terms dependent upon the cosines
can be determined also by observing the zenith distances of
objects in both positions of the circle, and for this purpose
again either a collimator or the mirror described above can
be used. We find namely from the first and the third of
the equations (B):
1800 =       2      +ca' cos z+act"    c  os2+  3 z+.a.- -   al'-a'-P a"',
where Z —180 —N, Z'=180-+N' and where z and z" are
the readings in both positions, corrected for the errors of
division.
We thus see, that all coefficients can be determined by
simple observations, except those of the sines of even multiples of z.  In order to find these, we must have means to
telescope so as to reflect the light well towards the mirror. It is also better,
to use for these observations an eye-piece with one lens only, since then
the reflected image of the wire-cross is better seen.




424
turn the telescope exactly through certain angles different
firom 900 or 180". There is no contrivance known by which
the telescope may be turned any desired angle; but by means
of the mirror described before and of two collimators the
telescope may be placed at the zenith distance of 450, and
thus at least the coefficient b" may be determined. In order
to do this, the mirror is placed so, that the telescope, when
directed to it, has nearly the zenith distance 1350, and in this
position of the mirror, a small telescope is placed above the
mirror and directed towards the nadir, while a collimator is
placed horizontal in front of it. Both telescopes are placed
so that their axes are directed to the centre of the mirror,
and this can be accomplished by putting covers with a small
hole at the centre over the object glasses, and likewise covering all but the central part of the mirror, and then moving
the two telescopes until the'light from the uncovered portion
of the mirror is reflected into the telescopes. When this is
done, the mirror is turned away, and the line of collimation
of the vertical telescope is made exactly vertical by means
of an artificial horizon, whilst that of the collimator is made
exactly horizontal by means of a level. Then the angle between
the lines of collimation of the two telescopes will be a right
angle. If now the mirror is turned back to its original place,
there is one position of it, in which rays coming from the
wire-cross of one collimator are reflected from  the mirror
into the other telescope so that its image coincides with
the wire-cross of that telescope, and when this is the case,
the angle which the mirror makes with the vertical line is
exactly 45". A small correction is to be applied also in this
case on account of the different latitude of the places of the
collimators. If y is the small angle, which the vertical collimator makes with the vertical line of the instrument, and x
the angle, which the horizontal collimator makes with the
horizon of the instrument, then the angle which the telescope,
when directed to the mirror, makes with the line towards the
nadir is:
450 +  (x - y),
if we assume, that the two collimators are placed on different
sides of the instrument; and if we denote by h and h' the dis



425
tance of the horizontal and the vertical collimator from the
vertical line of the instrument, and if we further denote by b
the inclination of the horizontal collimator as found by means
of the level, taken positive when the side nearer to the instrliment is the higher one, then this angle will be:
450 q+ 0". 0052 (h - h') +-  b.
If we denote this angle by C, and the two readings of
the circle when the telescope is directed to the nadir point
and to the mirror, that is, for the zenith distance 180" and
135~, by z' and z, we have:; -     z -- a' (t -- 2 i/ 2) + a" — a"' (1 +   2) - b1 I/ 2 -+ b - b" 2  2.
If we make now the same observation, when the zenith
distance of the telescope is 225", and if we denote again the
nadir point by a' and by z" the reading of the circle, when
the telescope is directed to the mirror, then we have in this
case:;_ —-+,  (1 —' I/2) -- a+ a"' (1 + -   2) -- E 2 + 1, "  2/2,
therefore we have:
4 ( +')-     - b'' I/2 + - "- b"' 2/2...
provided that the nadir point is the same for both observations.
E. On the examilnation of the micrometer screws.
9. The measurement of the distance of two points by
means of a micrometer screw  presupposes that the linear
motion of the screw and the micrometrical apparatus moved
by it, for instance that of the wire, is proportional to the
indications of the head of the screw  and of the scale, by
which the entire revolutions of the screw are indicated. However this condition is never rigorously fulfilled, since not
only the threads of the screw are not exactly equal for different parts, and hence cause that the amount of the linear
motion  produced  by an entire revolution varies, but also
equal parts of the same revolution move the wire over different spaces. It has been shown already, how the irregularities of the screws of the reading microscopes can be determined, but since in that case only very few threads of the




426
screw  are really used in measuring, the case shall be treated
now, when the entire length of the screw  is employed.
The corrections which  must be applied to the readings
of the screw  head, in order to find from them the true linear
motion of the screw, can again be represented by a periodical series of the form:
*a, cos t -4- b sin t- + a, cos 2u +- b2 sin 2u ~t..
where u is the reading of the screw head. These corrections
will be nearly the same for several successive threads, so
that the coefficients a,, b, etc. can be considered to be equal
for them.  Hence these coefficients are determined from  the
mean of the observations made for several successive threads,
and these determinations are repeated for different portions
of the screw.
If we measure the linear distance between two points,
whose true value is f (forinstance, the distance between two
wires of a collimator) by bisecting each point by the moveable
wire of the micrometer, then, if u and tu' are the indications
of the screw for those positions of the moveable wire, we
have:.f —z'-  A t -- a i (cos U' - COS ) - b I (sin t' - sinu ) -f- a (cos 2u'-cos 2 )
-F b, (sin 2 a' - sin 2 u) 4-...
Now  if the distance is an aliquot part of a revolution,
and we measure the same distance by different parts of the
screw  arranging the observations so, that first we read 0. 00,
when  the moveable wire bisects one point, the next time
01'.10, then 0".20 and so on through one entire revolution
of the screw, then, if these coefficients are small, as is
usually the case, we can assume, that f is equal to the arithmetical mean of all observed values of iu- -u', and we can
take au + f instead of u'.  Therefore if we denote this arithmetical mean by f, every observed value of u'  u gives an
equation of the form:
t'  ut- - f   2 al sin I f sin (f-f) - 2 b I sin fcos ( -I 2 -f)
+2a, sin f sin (2  -}- f) - 2b2 sinf cos (2 Z -f-f)
and since we have ten such equations, because we suppose
that the screw has made one entire revolution, we find the
following equations:




427
10 a, sin ~-f   Y (u' - u -f) sin (u ~- vf)
10 b, sin    -- 2 (' f- u -- f) cos (u + 2t.)
10 a2 sin f-   2 (u' -     ) u -J') sin (2u -+-f)
10 b 2 sin f —- - 2- (7'   G — f) cs (2 tu -f),
from  which we can determine the values of the coefficients.
Example.  Bessel measured by the micrometer screw  of
the heliometer the distance between two objects, which was
nearly equal to half a revolution of a screw, in the way just
described, and found from  the mean of the observations made
on ten successive threads of the screw:-)
Measured distance u' - 
Starting point 0,0    0'. 50045
0,1     0. 49690
0,2      0. 49440
0,3     0. 49240
0,4     0. 49260
0,5    0. 49555
0,6     0. 49905
0,7     0. 50140
0,8      0. 50340
0,9     0. 50350
f-0. 497965 - 179~ 16'.0.
From  this we find:
t -  -f    (u"   - f) sin (u + - f)
O. 0.002485          - 0.002485
-0.001065              0. 000865
-- 0.003565         -0. 001123
-0. 005565            O. 001686
-0.005365            - 0.004320
-0.002415           + 0.002415
- 0. 001085         -0.000882
0. 003435          -0.001083
+~- 0.005435        + 0. 001646
+- 0. 005535        — 0.004457
sunm  - 0.013056,
and since sin f=  1, we have:
10 a, - +- 0. 013056
as:    10 b, — 0. 024874
0. 128 a2 =  - 0. 000147
0. 128 b2=  +- 0. 000337.
*) Astronomische Untersuchungen Bd. 1, pag. 79.




428
Bessel made then a similar series of observations by
measuring a distance, which was nearly equal to one fourth
of one revolution and found:
7. 339 a l _   0. 015915
7.339 b -— 0.016126
9. 970 a,  — 0. 004987
9. 970 b = -~. 000576,
and from these two determinations he obtained according to
Note 2 to No. 24 of the introduction:
at, =-+ 0". 001608
1i,, =-0. 002386
a,  - 0. 000499
lb 2 —0.000057.
These periodical corrections of the screw must be applied to all readings of the screw head. But the observations
can also be arranged in such a manner that these periodical
errors are entirely eliminated. For, if we measure the same
distance first, when the indication of the screw at the bisection of one object is -0"'.25 and then again, when the
reading is -t-0".25 at the bisection of the same object, so
that?u for these two observations is equal to - 90o and + 900,
then in the expression for f the terms al (cos u'o- cos U)
+- b, (sin u'  sin u) will be in one case -j- a, cos u'-+ b' (sin u' —I1)
and in the other case - a, cos t,' - b, (sin u' +l1), and hence
this portion of the correction, dependent on al and b1, will
be eliminated by taking the arithmetical mean of both observwations.  Likewise the result will be free from  that portion of the correction dependent on a, b, a2 and b.2, if we
take the mean of 5 observations,:,rranging them  so that tile
reading of the screw  for the bisection of one object is in
succession -0'. 4, -Or. 2, 0, -- 0". 2 and - 0r.4.
Now  in order to examine, whether the threads of the
screw  are equal, we must measure the same distance, which
is nearly equal to one revolution of the screw  or to a multiple of it, by different parts of the screw, and it will be best
to arrange these observations in the manner just described
in order that the periodical errors may be eliminated.
Bessel measured by the same screw a distance between
two points nearly equal to ten revolutions of the screw, the




429
indications of the scale at the bisection of one point being
in succession 0", 101, 20", etc.  Thus he found:
Reading of the scale at the beginning Or  10. 0142
10      20. 0147
20      30.0131
30      40.0122
40      50.0107
etc.,
where each value is the mean of 5 observations, for instance
the second value that of five observations made when the indications of the scale were 9'.6, 9f'.8, 10, 10.2 and 10.4.
If now  the true distance is 10'" —x,- and the corrections
of the screw  for the readings of the scale 10, 20, etc. are
fit0, f2A, etc. then we have, since we ca.n take f'0-0:
x, - - 0. 0142 t-f, o
x   -0. 0147  -+-f.fl 
x, -. 0131 +.f3, -f2 o
etc.
Likewise he measured a distance, which was equal to
20'-+ x2, in the same way and obtained thus another system
of equations:
X2  a -f-+f2o
x2 = a -f-f4o -f2 o
etc.
Similar systems were obtained by measuring a distance
equal to 30" — +x:,, and from  all these equations he found the
values of x, x2, x,, etc. as well as the corrections of the
screw for the readings 10, 20, etc., that is, f,,, f2,, etc.
II. THE ALTITUDE AND AZIMUTH INSTRUMENT.
10.  One circle of the altitude and azimuth instrument
represents the plane of the horizon and must therefore be
exactly horizontal. Therefore it rests on a tripod by whose
screws its position with respect to the true horizon can be
adjusted by means of a level, as will be shown afterwards.
But since this adjustment is hardly ever perfect, we will
suppose that the circle has still a small inclination to the
horizon. Let therefore P be the pole of this circle of the




430
instrument, whilst the pole of the true horizon is the zenith Z,
and let i be the angle, which the plane of the circle makes
with the plane of the horizon, and whose measure is the arc
of the great circle between P and Z. In the centre of this
circle, which has a graduation, is a short conical axis carrying another circle to which the verniers are attached. On
the circle stand two pillars of equal length, which are furnished at their top with Ys, one of' which can be raised or
lowered by means of a screw. On these Ys rest the pivots
of the horizontal axis supporting the telescope and the vertical circle.  The concentrical circle carrying the verniers
can be firmly connected with the Y, but the telescope and
the graduated circle are turning with the horizontal axis.
Since also the vernier circle turns about a vertical axis, the
telescope can be directed to any object, and the spherical
co-ordinates of it can be obtained from  the indications of
the circles. We will denote by i' the angle, which the line
through both Ys makes with the horizontal circle, and by K
the point, in which this line produced beyond that end on
which the circle is, intersects the celestial sphere. The altitude of this point shall be denoted by b. Now since only
differences of' azimuth are measured by this instrument (if
we set aside at present the observations with the vertical
circle) it will be indifferent, from what point we begin to
reckon the azimuth, and since the points P and Z remain
the same, though K moves through 360 degrees if the vernier
circle is turned on its axis, we can choose as zero of the
azimuth that reading, which corresponds to the position the
instrument has, when K is on the same vertical circle with
P and Z. We will denote this reading by a0. For any other
position we will suppose that we read always that point of
the circle, in which the arc P K intersects the plane of the
circle, and this is allowable, because the difference of this
point and the point indicated by the zero of the vernier is
always constant. The azimuth reckoned in the horizon, but
from the same zero, shall be denoted by A.
If now  we imagine three rectangular axes of co-ordinates, one of which is vertical to the plane of the horizon,
whilst the two others are in the plane of the horizon so that




431
the axis of y is directed to the zero of the azimuth, adopted
ahbove, then the co-ordinates of the point K referred to these
axes will be:
--- sin b, y - cos b cos A
and                     x = cos b sin A.
Moreover the co-ordinates of K referred to three rectangular axes, one of which is perpendicular to the horizontal
plane of the instrument, whilst the two others are situated
in this plane so that the axis of x coincides with the same
axis in the former system, are:
=- sinl i, y - cos i' cos (a - ao), ex  cos i' sin (a-a0a).
Now  since the axis of z in the first system  makes with
the axis of z of the other system the angle i, we have according to the formulae (1) for the transformation of co-ordinates:
sin b- cos i sin i' -    sin i cos i' cos (a - ao)
cos b sin A - cos i' sin (a - ao)
cos 4 cosb ros   sin i sin i' t — cos i cos i co (a - a,,).
We can obtain these equations also from  the triangle
between the zenith Z, the pole of the horizontal circle P and
the point KIf, whose sides PZ, PK and ZK are respectively
i, 900-i' and 90"- b, whilst the angles opposite the sides
PK and ZK are A and 1800-(a -a0).
Now  since b, i and i' are small quantities, if the instrument is nearly adjusted, we can write unity instead of
the cosine and the are instead of the sine, and thus we obtain:
b  - i'- cos (a - a)         (a)
A   a --   a 0.
The telescope is perpendicular to the horizontal axis.
The line of collimation. ought also to be perpendicular to this
axis, but we will assume, that this is not the case, but that
it makes the angle 90"-+-c with the side of the axis towards
the circle.  Tlhe angle c is called the error of collimlation.
It can be corrected by means of screws which move the
wire-cross in a direction perpendicular to the line of collimation.
The telescope shall be directed to the point 0, whose
zenith distance and azimuth are z and e, and whose co-ordinates with respect to the axes of z and y are therefore
cos z and sin z cos e. Now we will suppose that the division




432
increases from the left to the right, that is, in the direction
of the azimuth.  Therefore if the circle-end be on the left
side, the telescope is directed to an azimuth greater than that
of the point K; and hence if we suppose, that the axis of y
is turned so that it lies in the same vertical circle with K,
the co - ordinates will then be: cos z and sin z cos (e  A).
This is true, when the circle is on the left side, whilst we
must take A - e instead of e - A, when the circle is on the
right side. If further we imagine the point O to be referred
to a system  of axes, of which the axes x and y are in the
plane of the instrument, the axis of y being directed to the
point K, then the co-ordinate y of the point O is equal to
-sin c, and since the angle between the axes of z of the
two systems is b, we have according to the formulae for the
transformation of co- ordinates:
- sin c -- cos z sin b +- sin z cos 1b cos (e - A).
We can find this equation also from the triangle between
the zenith Z, the point K and the point 0, towards which
the telescope is directed.  The sides ZO, Z K and OK are
respectively equal to z, 900- b and 90~-+-c, and the angle
KZO is equal to PZO-PZK- e -A.
Since b and c are small quantities, we obtain:
- c - b cos z - sin z cos (e - A),
or finally, substituting for A its value from  the equations (a):
0 = c + b cos z +- sin z cos [e - (a - ao)].
Hence it follows, that
cos [e - (a -- ao)]
is a small quantity of the same order as b and c. Therefore
if we write instead of it:
sin [90~ - e + (a - ao)],
we can take the are instead of the sine and obtain:
0 = c -- b cos z -+ sin z [90 - e + (a  a)].
This formula is true, as was stated before, when the
circle is on the left side. If it is on the right side, we must
take A   e instead of e -A and we obtain then:
c -+ b cos z -+ sin z [900 - (a - a) + e].
Therefore we obtain the true azimuth e by means of
the formulae:




433
e-  a- - a -- 900 + —. -  -) cotang z  Circle left
sin z
and:
e   a-c a - 900.- - b cotang z  Circle right,
sin z
and if we call A the azimuth as indicated by the vernier, and
A A the index error of the vernier, so that A +    A is the
azimuth reckoned on the circle from  the zero of azimuth,
then we have:
e = A + -  A 4- c cosec z 4- b cotang z,
where the upper sign must be used, when the circle is on
the left side and the lower one, when the circle is on the
right side.
Fig. 1i.    11.  We can find these formulae also by a
B ~    geometrical method. Let AB Fig. 14 be the vertical circle of the object and Z the zenith.  If we
assume now that the telescope turns round an axis,
-   whose inclination to the horizon is b, it will describe a vertical circle which passes through the
points A  and B  and the point Z' whose distance
from the zenith is equal to b. Therefore while we
read the azimuth of the vertical circle A Z, the tel-'  escope will be directed to a point on the great
circle AZ'B, say 0, and hence, when the circle
is on the left side, we shall find the azimuth too
A    small.  Now  we have:
sin 0 0' — sin A 0 sin b
cos z. sin b.
But we read the angle at Z subtended by O 0', and therefore the angle OZ O' is the sought correction A A of the azimuth. Now  since:
sin 0 O'-= sin Z 0 sin 5 A,
and hence:
sin A A =- cotang z sin b,
we must add to the reading of the circle on account of the
error b, when the circle is left:
+- b cotang z.
In a similar way we can find the correction for the error of collimation. Let A B again be the vertical circle, which
the line of collimation of the telescope would describe, if
28




434
Pig. 15.  there were no error of collimation.  But if the
angle between this line and the side of the axis
towards the circle be 90 -- c, the line of collimation will describe, when the telescope is turned
around, the surface of a cone, which intersects the
sphere of the heavens in a small circle, whose diszl    i -tance from the great circle A B is equal to c. Fig. 15.
In this case the reading of the circle is again too,-__  small, when the circle is on the left, and if we
denote again the angle AZO  by ASA, we have:
sin c
sin A A s —sin z
or:
A A  - c cosec z.
12. It shall now be shown, how  the errors of the instrument can be determined.
The level-error is found according to the rules given in
No. 1 of this section by placing a spirit-level upon the pivots of the horizontal axis.  But we have according to tile
equation (a) in No. 10:
b = i'- i cos (a - ao),
where i is the inclination of the horizontal circle to the horizon, i' the inclination of the horizontal axis, which carries
the telescope, to the horizontal circle.  This equation contains three unknown quantities, namely i', i and a,,, and hence
three levelings in different positions of the axis will be sufficient for their determination.  We will assume that the inclination b is found by means of the level in a certain position of the axis, when the reading of the circle is a, then
it is best, to find also the inclinations b, and b2 in two other
positions of the  instrument corresponding  to the readings
a-+-120" and a-H-140~. For if we substitute these values in
the above formula, develop the cosines and remember that:
cos 1200- _
and
sin 1200 =- -t 1/ 3,
moreover:
cos 2400 -
and
sin 240~!/3,
we obtain the following three equations:




435
- i' - i cos (a - ao)
b-  i' +-, i cos (a - ao) 4+ 2 i sin (a - ao) / 3
b        i-.i cos (a- a,)'isin (a-O) 1/3
If we add these three equations, we find:
b +- bl +- b2
But if we subtract the third equation from  the second,
we obtain:
i sin (a-ao) - a - 
and if we add the two last equations and subtract the first
after being multiplied by 2, we find:
b1 -+ b2-2b
i cos (a -a) -- 3
Therefore if we level the axis in three positions of the
instrument, which are 1200 apart, we find by means of these
formulae, i, i' and a0, and then we obtain the inclination for
any other position by means of the formula:
b) = i'- i cos (a - a0).
In order to find the collimnation-error, the same distant
terrestrial object mnust be observed both, when the axis is
on the left, as well, when it is on the right, and the circle
be read each time. If the reading in the first case is a, that
in the second case a', we shall have the two equations:
e -- A +-A  -+- b cotang z -F c cosec z
e  A'+ AA - b' cotang z - c cosec z,
from  which we find:
A'-A   b' +- b
c cosec z —     2      cotang z.
2       2
Therefore if the inclinations b and b' in both positions
are known and we get the zenith distance from  the reading
of the vertical circle, we can find the collimation-error by
observing the same object in both positions of the instrument.
It is assumed here, that the telescope is fastened to the
centre of the axis or that, if this is not the case, a very
distant object has been observed. Otherwise we must apply
a correction to the collimation-error, as found by the above
method.  For, if we observe the object 0 Fig. 16 with a
telescope, which is fastened to one extremity of the axis, it
is seen in the direction O F. The angle OF I shall be 90 -- co.
28*




436
ai.~ 16.         Now  if we imagine a telescope
at the centre M of the axis, and
directed to 0, then the angle
0   K will be 90" +t- c. We have
therefore:
C c-o -+ MiOF.
But we have:
tang 11 O FO- d
where d is the distance of the ob_rK r         5~   jeect 0 iM  and o- is half the length
of the axis, and hence, if co is very small, we get:
C = C0 + -'d
Therefore if we observe a terrestrial object with an instrument whose telescope is at one extremity of the axis, the
reading of the circle will be too small by the quantity 7 coseez,
when the circle is on the left, and too large, when the circle
is on the right side. Therefore if these two readings be denoted by A and A', we have the two equations:
e - A -- A A + — b cotang z-+ ( c -+   ) cosec z
e-=A'-+  A-b'cotangz-(c -    -) cosec z,
from which we can find the collimation-error, if d is known.
If the telescope is attached to one extremity of the axis,
its weight can produce a flexure of the axis, which renders
the collimation-error variable with the zenith distance. When
the telescope is horizontal, the flexure has no influence on
the collimation-error, since it merely lowers the line of collimation, but leaves it parallel to the position it would have,
if there were no flexure. But when the telescope is vertical,
the flexure increases the angle, which the line of collimation
makes with the axis.  Hence the collimation-error in this
case can be expressed by the formula cup ca cos z. In order
to find c and a, the error of collimation must be determined
in the vertical as well as in the horizontal position of the
telescope (See No. 22 of this section).




437
If no terrestrial object call be used for finding the collimation-error, it may be determined by observations of the
pole-star.  For, if we observe the pole-star at the time t,
read the circle and then reverse the instrument and observe
the pole-star a second time at the time t', we shall have the
two equations:
e   A - AA   b cotang z -d- c cosec z
and
e'  A' --- 1A  - b' cotang z - c cosec z,
and since we have:
dA
e' =   - e  -  (t'-t)
dt
dA                                                  t-t
where dt denotes the change of the azimuth at the time  2
we obtain:
A'- A   dA  t'- t  b'- -- b
c cosec z     -         2 -        cotang z.
2     cd t   2     2
Finally, in order to find the index error AA, we observe
again a star, whose place is known, for instance the polestar and read the circle. If' then the hour angle of the star
is t, we compute the true azimuth e by means of the formulae:
sin z sin e  cos a sin t
sin z cos e = - cos 9 sin L  sin f  cos  cos t,
and we obtain:
AA = e - A =f b cotang x z= c cosec a,
where A is the reading of the circle and where the upper
sign is used, when the circle is on the left side, the lower
sign, when it is on the right side.
13.  If the instrument serves only for observing the azimuth, it is called a theodolite.  But often the vertical circle
of such an instrument has also a fine graduation so that it
can be used for observing altitudes as well as azimuths. In
this case the vernier-circle is clamped to the Y, whilst the
graduated circle is attached to the horizontal axis and turns
with it. Such an instrument is directed to an object and the
vertical circle having been read in this position, it is turned 1800
in azimuth and again directed to the same object. If then we
subtract the reading in the second position from that in the
first position or conversely, according to the direction in which
the division increases, half the difference of these readings




438
will be the zenith distance of the object or more strictly its
distance from the point denoted before by P.. But this presupposes, that the angles i and i' as well as the error of
cdllimation are equal to 0. Now we can assume again, that
the reading of the circle indicates always the point, where
a plane perpendicular to the circle and passing through the
line of collimation, intersects the circle.  Then the telescope
will be directed to P, when the great circles KO and KP coincide.  (Compare No. 10 of this section.)
When the line of' collimation is turned from here to
point 0, the telescope will describe the angle PKO, but the
side PO will be the'measure of this angle only in case that
OP and PK are 900. On the contrary, if these sides are equal
to 90" -- c and 90" -i', we have, denoting PO  by X and
the reading of the circle, that is, the angle PKO by i':
cos = - sin c sin i' -i cos c cos i' cos 5'
Cos (i' — C) Cos   2 - cos (i' — c) sin 1;'.
If we subtract cos 5' from both members and write (c'- ) sin g'
instead of cos -- cos d', which is allowable, because  —;'
is small, we obtain:
---  -sin /} (c - i') 2 cotg.'-sin    - (i'-c) tang'
or:
c2 +- i'2
5A - -2 g'-t -  cotg C' + i'c cosec';; is then the zenith distance referred to the pole of the instrument P.  But if P does not coincide with the zenith, it
is not yet the true zenith distance. However in this case
all is the same as before, with this difference, that instead
of using the inclination i' of the horizontal axis of the instrument to the horizontal circle, we must talke its inclination
to the horizon, that is:
i'- i cos (a, - a, )- t
and besides, we must subtract from  the reading of the vertical circle the projection of PZ on the circle or the angle
PKZ-= i sin (a- a0).  This angle is always found by means
of a spirit-level attached to the vertical circle. If we denote
by p the reading of the level on that side, on which the division, starting from  the highest point, increases, and that
on the opposite side by n, and finally the point of the circle,




439
corresponding to the middle of the bubble) by Z, then the
zenith point of the circle will be in one position of the instrument Z +-  (p -- in) and in the other Z -K (p'  n'). Therefore if we denote the readings in the two positions by 5' and',x, then the zenith distance in one position will be:
where E expresses the value of one part of the scale of
the level in seconds, and we shall have in the other position:
z-' +    (p' -n'),
and hence we find from  the arithmetical mean the zenith
distance:
- 91 _  (p - n)  -- 2 (p' - n') e
z 2              2
and in order to obtain from  this the true zenith distance,
we must add the correction:
~ sin  (b -1- c) cotg. Z' - sin 2- (b - c)2 tang I Zn
or:
C2 +b2
~-   2   cotg z' -+b c cosec z'.
If we take b   0, since we have it always in our power
to make this error small, we have simply to add:
C2
+- - cotang z'.
If, for instance, c-= 10', we find    = 0".87. Therefore
if z' is a small angle, that is, if the object is near the zenith,
this correction can become very considerable. In case therefore that the zenith distances are less than 45", we must
always take care that we observe the object at the middle
of the field, that is, as near as possible to the wire-cross.
14.  We can deduce the formulae for all other instruments from  the formulae  for the azimuth and altitude instrument. An equatoreal differs from this instrument only
so far as its fundamental plane is that of the equator, whilst
for the other instrument it was that of the horizon.  Therefore if we simply substitute for the quantities which are referred to the horizon, the corresponding quantities with respect to the equator, we find immediately the formulae for
the equatoreal.  The quantity a will then be the reading of
the hour circle, i' will be the inclination of the axis, which




440
carries the telescope, to the hour circle which should be parallel
to the equator. Further i will be the inclination of the hour
circle to the equator, and 90" -- c is again the angle, which
the line of collimation of the telescope makes with the axis.
We can also easily find the formulae for those instruments, which serve for making only observations in a certain
plane. For instance, the transit instrument, is used only in
the plane of the meridian, therefore for this instrument the
quantity a —a, —90o must always be very small.  Denoting
the small quantity by which it differs from zero, by — k, the
formulae given in No. 10 are changed into:
e     — c    b cotang z - c cosec z  Circle left
e -  -    b cotang z - c cosec z  Circle right.
When e is not equal to zero, the body will not be observed exactly in the plane of the meridian, and if e has a
negative value, it will be observed before the culmination.
Now  let r be the time which is to be added to the time of
observation in order to find the time of culmination, then t
is the hour angle of the body at the time of observation,
taken positive on the east side of the meridian. Now since:
sin z
sin    -- sin e.
cos S
sin Z
or:                      x_-e. c
the formulae given above change into:
cos z    sin z
c --- b    - IC    - c sec    Circle left (east)
cos  cos 
and:
Cos sin z
T  -=- b     + k    -s     c sec z  Circle right (west).
cos A    cos a
These are the formulae for the transit instrument. The
quantity b denotes now  the inclination of the horizontal axis
to the horizon, and k is the azimuth of the instrument, taken
positive when east of the meridian.
In a similar way the formulae for the prime vertical instrument are deduced. We have, namely, according to No. 7
of the first section:
cotang A sin t = - cos T tang s + sin T cos t
or, if we reckon the azimuth e from  the prime vertical, so
that A   90" -e:
tang e. sin t = cos R tang  s -in T cos t.




441
Now if e is the time at which the star is on the prime
vertical, we have:
0 = cos cp tang a - sin 9 cos 0
and if we subtract both equations:
tang e sin t -2 sin  sin T  (t -0) sin - (t - 0).
From this we find, if e is small and therefore t is nearly
equal to O:
e = (t --  ) sin T
or:
e
O - t- si -
sin g9
If we substitute here for e the expression found before:
e - - k -4- b cotang z 4- c cosec z,
we obtain the following formulae for the prime vertical instrument:!c    cotang z    cosec z
O ~ t        - -- =:   = b  =    c.
sin        T sin      T sin P
The direct deduction of these formulae will be given for
each instrument in the sequel.
III. THE EQUATOREAL.
15. As the altitude and azimuth instrument corresponds
to the first system of co-ordinates, that of the altitudes and
azimuths, so the equatoreal corresponds to the second system,
that of the hour angles and declinations.  With this instrument therefore that circle, which with the other was horizontal, is parallel to the equator.  Now  let P be the pole of
the heavens, i1 that of the hour circle of the instrument.
Further let A) be the arc of the great circle between those
two points, and h the hour angle of the pole of the instrument.  Finally let i' be the angle, which the axis carrying
the declination circle (the declination axis) makes with the
hour circle, and let K be the point, in which this axis, produced beyond the end on which the circle is, intersects the
sphere of the heavens, and finally let D be the declination
of this point.  As zero of the hour angle we will take again
at first that reading of the hour circle, which we obtain, when
K, P and  // are on the same declination circle.  And wev




442
will assume that every other reading gives us that point of
the circle, in which it is intersected by the great circle passing through P and /I.  This point differs fiom the reading
of the circle only by a constant quantity.  Let the hour
angle reckoned on the true equator, but from the same zero,
be T.
If now we imagine again three rectangular axes of coordinates, of which one is perpendicular to the plane of the
true equator, whilst the other two are situated in the plane
of the equator so, that the axis of y is directed to the adopted
zero of the hour angle, then the three co-ordinates of the
point K, referred to these axes, are:
z- sin D, y = cos D cos T, x = cos D sin T.
Further, the co-ordinates of K, referred to three rectangular axes, one of which is perpendicular to the hour circle
of the instrument, whilst the other two are situated in its
plane, the axis of z coinciding with that of the former system, are:.z- sini', y=cosi'cos(t —t0), x-=osi'sin(t-to).
Now since the axes of z of these two systems make
with each other the angle 2, we have the following equations:
sin D - cos 2 sin i'- sin 2 cos i' cos (t - to)
cos D sin T= cos i' sin (t - to)
cos D cos T- sin 2 sin i' +- cos 2 cos i' cos (t - to).
Since 2, i' and D are small quantities, if the instrument
is nearly rectified, we obtain:
D=~i' — Acos (t -10)
T= t - to.
The telescope is attached to the declination axis and we
will assume, that the part of its line of collimation towards
the object-glass makes with the side of the axis, on which
the circle is, the angle 90" -+  c, c being called the collimation-error. Now if the telescope be directed to a point, whose
declination is () and whose hour angle, reckoned from  the
adopted zero, is Tr, then the co-ordinates of this point will be:
=- sin S, y=cos 8 cos X, and x = cos c sin X,.
We  will assume, that the division  of the circle increases in the direction from south towards west from 0"
to 360" or from  0"' to 24'.  Therefore if the circle-end is




443
west of the telescope, the latter is directed towards a point,
whose hour angle is less than that of the point K.  Therefore if we imagine the axis of y to be turned so that it lies
in the same declination circle with K, if the telescope is directed to the object, then the co-ordinates will be:
z =  sin A, y = cos 8 cos (T — Z1), x -coS S sin (T- r,).
On the contrary, when the circle-end is east of the telescope, these co-ordinates will be:
z- = sin S,   -cos  cos ('1 -  x ),= co S sin (1 - T).
If now we refer the place of the point 0, towards which
the telescope is directed, to a system  of axes, of which the
axis of y is parallel to the declination axis of the instrument
and hence directed to K, whilst the axis of x coincides with
the corresponding axis of the former system, then the three
co-ordinates of the point 0 will be, S' denoting the reading
of the declination circle:
z =sinl' cos c,   =-sin c
and
X = COS 5' cos C.
Now since the axes of z of the two systems make with
each other the angle D, we have:
- sin c = cos 8 cos (r, - T) cos D -I — sin S sin D,
or
- c- cos' cos (  - T) - D. sin S,
and hence, if we substitute for D and T the values found
before:
- c = [i - 2 cos (t - to)] sin a +- cos S cos [-, - (t - to)].
From  this it follows, that:
cos [z, - (t- to)]
is a small quantity.  Therefore if we write:
sin [900 - -, + (t- to)]
instead of
Cos [aI - (t - to)],
we.can take the arc instead of the sine and we find the true
hour angle:
T:  900 +- (t - to) - 2 cos (t - to) tang S -- i' tang S -+ c sec,
when the circle-end is east of the telescope, and:', = (t - to) - 900 ~ 2 cos (t - to) tang  - i' tang SA- c see,
when the circle-end is west of the telescope.
If we acdd h to both members of these equations, we




444
reckon the angles from the meridian.  Then r1 -  h will be
the true hour angle reckoned from  the meridian and:
h -- t - to -- 90"
and hA- t-to -90~
are the hour angles, as given by the instrument in the two
positions. Therefore if we introduce the reading of the circle
and call it t', and the index error At, we have:
It' I+ A t - A   sin [t' +- z t - h] tang 8 -4- c sec  =L4 i' tang 8,
or: q - t' - A t - g sin (r -- h) tang 8 4- c sec  4- i' tang 5,
where the upper sign is used, when the circle-end is west, the
lower one, when it is east.
We can also find these equations and the corresponding
ones for the declination from the spherical triangle between
the pole of the heavens P, the pole of' the instrument 11
and the point 0, towards which the telescope is directed, in
connection with the other triangle formed by 11, 0 and K,
that is, the point in which the declination axis produced intersects the sphere of the heavens.
The sides of the first triangle O P, OH  and PAT  are
respectirely the true polar distance 900 -  of the point towards which the telescope is directed, the distance from the
pole of the instrument 90" —- 5', and i, whilst the angles opposite
the two first sides are 180  -(r'- h) and  - -h, where r —h
is the hour angle, referred to the meridian of the instrument,
and r -  h the hour angle referred to the pole of the instrument and reckoned  from  the meridian  of the instrument.
Hence we have the rigorous equations:
cos S cos (' - h) = sin 8' sin i  -i- cos 8' cos 2 cos (x-' - I)
cos' sin (r - h)- cos' sin (I' - h)
sinS     =sin  sin' cos  - cos 8' Sinll  cos (z' - h),
from  which we obtain in case that t is a small quantity:
r    -  2 tang 8' sin (t' - h)
S - 8' - 2 cos (T' - h).
But T' and (5' are only then equal to the readings of the
circle, when i' and c as well as the index error of the vernier are equal to zero.  First it is evident, that the angle
90" - -" -  As  obtained by the reading of the declination
circle (where /50 is the index error of the declination
circle) is equal to the angle at K in the triangle  IIKO. The
angle SHTO, S being a point on the great circle P11, is




445' — h; the reading  of' the instrument is the angle between
the position of HiK at the time of observation and that, in
which HP coincides with IS. If the above conditions were
fulfilled, this angle would be' -  h, whilst the angle S li-K
would be 90-+-r - h, when the axis is west, and r'- h- 90",
when the axis is east of the telescope. If for the general
case we denote the  latter angle  by  900 +  r"-  h +  At
and  T" -  h -  At -  90", then  the angle Ol.K  will be
equal to 900 ~     " -"- A t -  r', when the axis is west and' — (r"     A t -  90~), when the axis is east of the telescope,
or equal to 90'Q (r' - r" - -At). Now  since the opposite side
in the triangle is 90"-+ c, and since the side ii 0, opposite the
angle 900 -A"- A, is 90" —', and JIK    900- i', we have:
cos X' cos (}r' -  " -   t)    cos c cos (3 -   A 3),
=t= cos X sin (s' - T" - L t) -  sin c cos i' - cos c sin i' sin (Y' -- A),
sin 3'         --- sin c sin i --- cos c cos i' sin (5" +- A 5),
from  which we obtain:
T'  _        t  F t  sec (b" -  ) =f i' tang (' (-[- Lo'),
and in the same way as in No. 13 of this section:
c'  A" +  s -A sin -- (i' - c) tang [450 H-     - (A" -, A)]
-+ sin  (i'- c) cotang [45 O -. 1 (X',*_A I  )],
or also 5' =S' -- A5S- -4 (i ) + c') tang ("   AS) — i c sec (," ~ AS),
and substituting  these expressions in the equations above,
we find:
- z" — + A t - 2 tang S sin (z' - h) =ft c sec 3=t= i' tang S
A  A" H-+ A — 2 cos (' -  ) -   (i' 2 - c2) tang   - i' c sec,
where the upper sign must be taken, when the axis is west,
the lower one, when it is east.  The last equation is true,
when the divison of the circle increases in the direction of
the declination, otherwise we have:
6   3600 -, -   S-a cos (r- /1) -   (i -t- c2) tang  - i'c sec'.
16. It shall now be shown, how the errors of the instrumnent can be determined by observations.  First we find
from  the two last equations for.:,= 1 80~ - (X"I -  I"),
and lhence we see, that the index  error of the declination
circle can be found by directing the telescope in both positions of the instrument to the samle object.  As such we can
choose either a star in the neighbourhood of the meridian, or




446
the pole-star, for then the change of the apparent declination
during the interval between the observations will be insignificant.
The errors i' and c can be determined by observing two
stars, of which one is near the pole, the other near the
equator, each being observed in both positions of the instrument.  We have namely for each star the two equations:
-' T - A - /   sin (z - / h) tang - i' tangS 8 - c sec S
when the circle is east, and:
z, — 1 +-'l      - a- sin (z1 -) /Otang S - i' tang S - c sec,
when the circle is west. Therefore if the interval between
the two  observations is short so that r1    r is a small
quantity, we obtain, denoting the sidereal times of the two
observations by h9 and O1:
i' tang- +csec         [2       ] 
I -' - [0' - -',]
2
and from this equation and the similar one which is deduced
from  the observations of the second star, the values of the
unknown quantities i' and c can be found.
When the errors i' and c have thus been determined
as well as the index error A, then the errors i2 and h as
well as the index error At are found by the observations of
two stars whose places are known. For, if we assume that
the readings are corrected for the errors i' and c and for
the index error A /, we have:
— =' + A t -- 2 sin (w - h) tang o
8 =' -2 cos (- - a),
and likewise for the second star:', = re' + a - A  sin (t, - h) tang',
i= S' - A cos (T - ha).
Fromn  these equations we easily find:
islnL- 2-          -    8 1)
L.2 — -- h ---  2 sinl 
2
cos   -9       a       t...I2 cos.
and from  these the values of h and Xi can be obtained.




447
The index  error At is then found by means of one of
the equations for r or rl.
Since all the quantities obtained by the readings of the
circles are affected with refraction, we must understand by
r, Tr1,    and  Yl also the apparent hour angles and declinations affected with refraction. But if the observations are
not taken very near the horizon, we call use the simple, expression:
d h -- a cotang h,
for computing the refraction, and then we obtain the corresponding  changes  of the hour angle and declination by
means of the formulae:
sin p
dt  - a cotang h.dd     a cotangc h. cos p,
where p is the parallactic angle, which is found by means
of the formulae:
cos T cos t = n sin N
sin  - n cos N
cos 99 sin t
tang P   l ens
tn cos (N -- 6)
or:
cos It sin p - cos 9 sin t
cos h cos p  --  cos (N -I- A).
The altitude h is found by means of the equation:
sinl  = n sin (N +- ).
If *we substitute these values in the expressions for dt
and d', we have also:
t  - a cos 99 sin t
cos 8 sin (N-i- A)
d3 —    - a cotang (N -).
Now  since sin p has always the same sign as sin t, the
hour angle is diminished by refraction in the first and second quadrant, but it is increased, or its absolute value is
diminished also, in the third and fourth quadrant.
If   <  Ap, then  sin Y cos'p is less than cos c sin cp and
hence cosp is always positive. Therefore the declination is
then increased by refraction.  But if 5  > A, then cosp is
always positive when t lies in the second or third quadrant,
therefore then also the declination is always increased by
refraction.  But in the first and the fourth quadrant it may




448
be diminished, and this is the case for all hour angles which
are less than that of the greatest elongation, for which:
tang y 
tang p
When the errors h and AX have been determined and it
is desirable to correct them, this can be accomplished simply
by changing the position of the polar axis of the instrument
in a vertical as well as a horizontal direction.  For if y is
the are of a great circle drawn from  the pole perpendicular
to the meridian, and if s is the distance of the pole from the
point of intersection of this arc with the meridian, then we
have:
tang x = tang it cos is
and:
sin y = sin ), sin h.
Therefore it is only necessary to move the lower end
of the polar axis by the adjusting screws through the distance
y in the horizontal direction and through the distance x in
the vertical direction.
The formulae given above for determining X and h presuppose, that i is a small quantity.  But this condition can
always be fulfilled, since the instrument can very easily be
approximately adjusted. For this purpose the instrument is
set at the declination of a culminating star (the index error
A 3  having been determined before) and then by means of
those foot-screws which act in the plane of the meridian
(or if the instrument is mounted on a stone pier, by the vertical adjusting screws of the plate on which the polar axis
rests) the star is brought to the wire-cross. The same operation is then performed for a star whose hour angle is about
6h, using now those screws which turn the entire instrument
round a horizontal line in the plane of the meridian (or using
the horizontal adjusting screws of the polar axis).
No regard has been paid to the effect of the force of
gravity upon the several parts of the instrument. This produces a flexure of the telescope as well as of the two axes.
Now the flexure of the polar axis need not be taken into
consideration, if the centre of' gravity of all parts of the instrument, which are moveable on this axis, falls within it, and
this must always be the case, at least very nearly, if the in



449
strument is to be in equilibrium  in all different positions.
Only the pole of the instrument will have a different position
on the sphere of the heavens than that which it would have
without flexure, but this position remains constant in whatever position the instrument may be. The flexure of the telescope, which may be assumed equal to y, sin z, can be determined by the method given in No. 8, and since like the
refraction it affects only the zenith distance, the correction
for it can be united with that for refraction by using in the
formulae given above c tang z -q- 2, sin z instead of cr tang z.
The flexure of the declination axis has the effect, that the
angle i' is variable with the zenith distance. Now if the
force of gravity changes the zenith distance of the point K
by fi sin z, then the corresponding change of its declination D
is fi sin z cos p, and that of its hour angle T is - ( sil z sin p
cos D
or since in this case D  is very nearly equal to zero, the
change of declination is /f sin (p and that of the hour angle
tcos (f sin T.  But since we have:
T —- 90~0 - a" if the circle-end is west
and — a" - 900 if the circle-end is east,
we have to take instead of this hour angle:
900 -+  " f-, cos 9o cos a"
or a"- 900 -4- f cos p cos ",
and hence we must use in the formulae given before
r" — f cos (, cos -" instead of r" and i' +-, sin r) instead of i',
since now  TK= -90" -  i' - f sin (f.  Thus we obtain:
- z'"   t — + tg-2Ssin ( -h) = c sec 8= i'tg =Fftg  [sin  -cos  cotg  cos ].
Therefore i' is in this case not constant, but we must take
instead of it:
i' -- bf [sin  - + cos T cotang 8 cos x].
Now the observation of a star in both positions of the
instrument gives an equation of the form:
c sec Am- i' tang A~- fi tang S [sin  -+ cosT  cotg a cos -]  0-  )
and therefore we can determine c, i' and jl by observing three
different stars in both positions of the instrument.
17.  If the equatoreal is well constructed so that the errors can be supposed to remain constant at least for moderate intervals of time, and if the circles have a fine gradua29




450
tion and are furnished with reading microscopes, such an
instrument can be advantageously employed to determine differences of right ascension and declination, and hence to
determine the places of planets and comets. For this purpose the telescope must have two parallel wires which are
a few seconds apart and parallel to the motion of the stars,
and another wire perpendicular to those. The object, which
is observed, is then brought between the parallel wires by
means of the motion of the instrument round the declination
axis, and the transit over the perpendicular wire is observed,
(if there should be several such wires parallel to each other,
then the times of observations are reduced to the middle wire
according to No. 20) and then the two circles of the instrument are read. Then in the same way also the star, whose place
is known, is observed. If the readings of the circle are corrected for the errors of the instrument and for refraction, the
differences of the right ascensions and declinations of the star
and the unknown object are obtained, and if these are added to the apparent right ascension and declination of the
star, the apparent place of the object is found. This method
has this advantage, that one can never be in want of a comparison star and can always choose stars whose places are
well known, even standards stars.  However it is best not
to take the comparison stars at too great a distance from
the object, because otherwise mistakes made in determining
the errors of the instrument would have too much influence
on the results. But when the star is near, those errors will
have very little influence, since both observations will be
nearly equally affected.
Usually however the equatoreal is not perfect enough
for determining the differences of right ascension and declination by it, and these determinations are made by means
of a micrometer connected with the telescope, whilst the parallactic mounting of the instrument serves merely for greater
convenience. Such micrometers, whose theory will be given
in the sequel, are used also to determine the distance of
two objects and the angle of position, that is, the angle,
which the line joining the two objects makes with the declination circle passing through the middle of this line. This




451
angle is obtained from  the reading of the circle of tile micrometer, whose centre is in the line of collimnation of the
telescope.  If the equatoreal is perfectly  adjusted, then in
every position of the instrument the same point of the position circle will correspond to the declination circle of that
object, to which the telescope is directed.  But otherwise
this point varies, and hence the readings of the position circle
must be corrected by the angle, which the great circle passing through the object and the pole of the instrument makes with the declination circle. If we denote this angle by n,
we have in the triangle between the object, the pole and the
pole of the instrument:
cos S sin 9 - sin 2 sin ('' - h)
or  r  2  sin (z'- A) sec.
Therefore we obtain from  the reading of the circle P'
the true angle of position P, reckoned as usually from north
towards east from  00 to 360", by means of the equation:
P - P' +- AP -I- A sin (r'- h) sec 8,
where A P is the index error of the position circle.
Compare on the equatoreal: Hansen, die Theorie des Aequatoreals, Leipzig 1855 and Bessel, Theorie eines mit eineml Heliometer versehenen Aequatoreals. Astronomische Untersuchungen. Bd. I.
IV. THE TRANSIT INSTRUMENT AND TtIE MERIDIAN CIRCLE.
18.  The  transit instrument is an azimuth instrument
which is fixed in the plane of the meridian.  The horizontal
axis of the instrument is therefore perpendicular to the meridian so that the telescope can be turned in the plane of
the meridian.
With portable transit instruments this axis rests again
on two supports which stand on an azimuth circle. But the
large instruments have no such circle and the Ys on which
the pivots of the axis rest are fastened to two insulated stone
piers.  One of the Ys is provided with adjusting screws, by
which it can be raised or lowered in order to rectify the
horizontal axis, whilst the other Y admits of a motion par29*




452
allel to the meridian, by which the azimuth of the instrument can be corrected.
One end of the axis supports the circle, which, if the
instrument is a mere transit, serves only for setting the instrument.  If the circle has a fine graduation, so that the
meridian altitudes can be observed with the instrument, it
is called a meridian circle. The modern instruments of this
kind have all two circles, one on each end of the axis.
Sometimes both  these circles have a fine graduation, but
usually only one of them is finely divided, whilst the other
serves for setting the instrument.  At first we will pay no
regard to the circle of such an instrument and treat it as a
mere transit instrument.
We will suppose that the axis produced beyond the circle
end, which shall be on the west side, intersects the sphere
of the heavens in a point, whose altitude and azimuth are
b and 90" - k, reckoning the azimuths as usually from  the
south point through west etc. from  00 to 3600.  Then we
have the rectangular co-ordinates of this point, referred to
a system, whose axis of z is vertical, whilst the axes of x
and y are situated in the plane of the horizon so that the
positive sides of the axes of x and y are directed respectively to the south and west points:
z -- sin b
y - cos b cos k
x - cos b sin k.
If we denote the declination and the hour angle of this
point by n and 90" - m, then we have the co-ordinates of
this point, referred to a system whose axis of z is perpendicular to the equator, whilst the axis of y coincides with
the corresponding axis of the former system:
z = sin n
y = cos n1 cos m
xz- cos n sin 7m.
Now since the axes of z of the two systems make an
angle equal to 900 -- T with each other, we have:
sin n - sin b sin a -cos b sin k cos T
cos n sin m - sin b cos T +- cos b sin k sin 99
cos ncosos m -- cos b cos k.




453
The same formulae can be deduced from the triangle
between the pole, the zenith and the point Q, towards which
the east end of the axis is directed. For in this triangle we
have  ZP-=90 —(p, ZQ-900 -+ b, P Q   900 +n and
PZQ   900-k, ZPQ = 900 -- m.
If the instrument is nearly adjusted so that b and k as
well as m and n are small quantities, whose sines can be
taken equal to the arcs and whose cosines are equal to unity,
we find the formulae:
n - b sin  - k cos 99
m = b cos +T ~ k sin 9,
or the converse formulae:
b - n sing -9  m cos 9p
k --- n cos -g- m sin 99.
Now  if we assume, that the line of collimation of the
telescope makes with the side of the axis on which the circle
is the angle 900-+cc, and that it is directed to an object,
whose declination is Y and whose east hour angle is r, which
quantity therefore is equal to the interval of time between the
time of observation and the time of culmination of the star,
then the co-ordinates of the star with respect to the equator,
the axis of x being in the plane of the meridian, are:
z sin A, y = - cos 3 sin X
and                    x = cos 6 cos r,
or if we suppose, that the axis of x is perpendicular to the
axis of the instrument:
z --- sin S, y  -cos  sin (z - m)
and                    -- cos 8 cos ('r - m).
Here r - m is the interval between the time of observation and the time at which the star passes over the meridian of the instrument.
If now we imagine another system of co-ordinates, so
that the axis of x coincides with that of the former system,
whilst the axis of y is not in the plane of the equator, but
parallel to the axis of the instrument, then we have:
y  - sin c,
and since the axes of z of these two systems make with each
other the angle n, we have:
sin c - - sin n sin a -+ cos n cos S sin (t - m).




454
In the case of the lower culmination, r -- m is on the
same side of' the meridian, but since then the star is observed after it has passed the meridian of the instrument,
we must take r —mn  negative.  Therefore in this case the
co-ordinates of the point to which the telescope is directed
will be:
z - sin 8, y  +- - cos 8 sin (r -rnV),
and hence we have:
sin c = - sin 2n sin 8 - cos n cos Y sin (r - i2).
Therefore in this case we have only to change the sign
of the second term  in the formula for sin c and we can take:
sin c   - sin  sin sin   cos n cos 8 sin (r - m)
as the general formula, if for lower culminations we use
1800 - a instead of A. These formulae can also be deduced from
the triangle between P, Q and the star 0, of which the sides
are PO    90o -, P Q=  900 -  n,   P    900 -  c, whilst
the angle 0 P Q is equal to 900 -4- mn - T for upper culminations and equal to 900 - m +. z for lower culminations.
From  the above formula we find:
cos n sin (r - n) = sin n tang 8 +- sin c sec 8,
and adding to this the identical equation:
cos n sin m - cos n  sin m,
we obtain:
2cos n sin 2 r cos [-r -ml -= cos n sin m +- sin n tang 8 +- sin c see 8. (a)
Now if we suppose the instrument to be so nearly adjusted that m, n and r are small quantities, we find from this:
T — = in  n tang 3 +- c sec 8 *).
This is Bessel's formula for reducing observations made
with a transit instrument.
If r is known and 7' is the clock-time of observation,
the clock-time of the culmination of the star is T —+T.  If
then At is the error of the clock on sidereal time, then
T-T-r — At will be the sidereal time of the culmination of
the star or be equal to its right ascension a. Hence we have:
a = T+ -/, t +- n + n tang  +- c sec 8.
Therefore if At is' known, the right ascension of the
star can be determined, and conversely, if the right ascension
of' the star is known, the error of' the clock can be found.
*) The same we get immediately from the equation for cos n sin (-r-m).




455
We can express r in terms of b and k, if we substitute
the expressions:
cos n sin m   sin b cos p -+- cos b sin 9 sin k
sin n - sin b sin 9 - cos b cos 99 sin k
in the equation (a).  We find then:
2 sin  - cos an cos [I T -- m] = sin b cos (I- a)
-- cos b sin k sin ( -      see 
cos S 
and from  this:
cos (T -  )    sin (a- A)
X-'- b         -   k  s     -    csec.
cos 3          cos A
This formula is called  Mayer's formula, since Tobias
Mayer used it for reducing his meridian observations.  It is
the same formula which was deduced before from  the formulae for the azimuth instrument.
Hansen has proposed still another form of the equation
for  r, which  is  the  most convenient of all.  For if we
add the two equations:
sin n tang  s sin b    _ cos b sin k sin p
cos T
and
cos n sin m -sin b cos 99 +- cos b sin k sin p,
we find:
cos n sin?n - sin b sec Tp - sin n tang p
and if we substitute this value of cos n sin m  in the equation
(a), we obtain easily:
-- b sec? q- n  [tang S - tang g] +  c sec &.
All these formulae are true, if the circle is on the west
side. But if the circle is east, then the altitude of the west
end  of the  axis is -  b, and the  angle, which the line of
collimation makes with the west end of the axis, will be
900 -c, whilst k remains the same. Therefore in this case
we have only to change the sign of b and c and we have
according to Mayer's formula:
For upper culminations
Circle West a   TA- At A  b ok sin                  c see a
cos (        cos a
Circle East a   T - A t —b cos ( -     k sin (-) se
cos a        cos 6




456
For lower culminations we take 1800 -  J instead of' 
and obtain:
Circle West a +- 12h -- T — At +- b cos ( +-)
cos a
+ k sin ( - c see S
cos 8
Circle East a +- 12h -- T   At - b cos( -   )
cos S
~ ic  ( —-- -)  c secS
cos S
When a large mass of stars is to be reduced, Mayer's
formula is not very convenient, and it is better to employ
then Bessel or Hansen's formula. If we choose Bessel's formula, we must apply to each observation the correction:
n tang S + c sec a
and the error of the clock is then:
a - - m.
If we take Hansen's form we apply the correction:
n [tang S - tang 9] -- c sec e
and obtain the error of the clock form:
a - T- b sec.
19.  These formulae can be deduced easily in the following way: If the circle is West, and b is the altitude of
the point to which the circle-end of the axis is directed, then
the telescope will not move in the plane of the meridian, but
it will describe the great circle A Z'B Fig. 14 pag. 433. If
now the star 0 is observed, we must add to the time of
observation the hour angle:
Fig. 17.                          — OPO'
But we have:
sin 0 0'
sin' 
cos a
and
tang 0 0'= tang b cos O' Z = tang b cos ( a- 5),
therefore:
\\                z -~ b  ~-cos(90-  )
cos S
If the azimuth  of the instrument is k, the
o    o  telescope will describe the vertical circle ZA Fig. 17.
But we have again, if 0 is the star:
sin 0  0'
sin 0 P 0     -- sin   sin O 
XSAS                                      cos 




457
and
tang  0' - tang k sin O'Z,
therefore:
ksin (T — ).
cos 
Finally, if the line of collimation of the telescope makes
with the side of the axis on which the circle is, the angle
90 -- c, it will describe a small circle parallel to the meridian
and we must add to the time of observation the hour angle
(see Fig. 15 pag. 434):
00'
T=-       c sec.
For lower culminations we find the corresponding formulae in the same way.
20. The normal wire of the transit when perfectly adjusted, is a visible representation of the meridian, and the
times are observed, when the stars cross this wire. Now in
order to give a greater weight to these observations, the
transits over several other wires, placed on each side of this
wire (which is called the middle wire) and parallel to it, are
also observed. Then in order that these transits may be taken
always at the same points of the wires, a horizontal wire is
stretched across these wires, in the neighbourhood of which
the transits are always observed. In order to place this wire
perfectly horizontal and thus the other wires perfectly vertical, we let an equatoreal star run along the wire, and turn
the diaphragm, to which the wires are fastened, by means of
two counteracting screws about the axis of the telescope, until the star does not leave the wire during its passage through
the field. If the wires on both sides are equally distant from
the middle wire, the arithmetical mean of all observations will
give the time of the transit over the middle wire. However
usually these distances are not perfectly equal; besides, it has
some interest, to find the time of transit over the middle
wire from the time of observation on each wire, since we
can judge then of the accuracy of the observations by the
deviations of the single results from their mean. Therefore
we must have a method for reducing the time of observation
on any lateral wire to the middle wire, and for this purpose




458
we must know  the distances of the wires from  the middle
wire. This distance f of a wire is the angle at the centre
of the object glass between the line towards the middle wire
and that towards the other wire. But we had:
sin (r -n m) cos n - sin n tang  +- sin c see S.
Now if an observation was taken on a lateral wire whose
distance is f, then the angle which the line from the centre
of the object glass to this wire makes with that side of the
axis on which the circle is, will be:
900  +c f*),
where f is positive, if the star comes to this wire before it
comes to the middle wire. If then r' is the east hour angle
of the star at the time of crossing the wire, we have:
sin (r' - m) cos n = sin n tang a ~- sin (c -f) sec,
and subtracting from this the former equation:
2sin 2 ( -') cos [[ (' +)- )m] cos n = 2 sin ~f cos [c -I- ~f] sec 8.
Now when the instrument is nearly adjusted, so that c,
n and m are small quantities, we find from this the following
formula, if we denote by t the time r -', which is to be
added to the time of observation on a lateral wire in order
to find the time of transit over the middle wire:
sin t = sinf sec 8.
This rigorous formula is used for stars near the pole,
the value of sec' being then very great; but for stars farther from the pole it is sufficient to take:
t  f sec A.
If it is not required to reduce the lateral wires to the
middle wire, we can proceed also in the following way. Let
f', f", f"', etc. be the distances of the lateral wires on the
side towards the circle, and tp', qp", Tp"', etc. those on the
other side, then compute:
f'f -1-f"'  - 9-    -9     a"',
n
where n is the number of wires. Then we must add to the
arithmetical mean of the transits over all the wires the quantity::I: a sec a
*) See Fig. 16 pag. 436, where O is the centre of the object glass,.M
th. 9M iddle wire and F the other wire.




459
where the upper or lower sign is to be used accordingly as
the circle is West or East. For lower culminations the opposite sign is taken.
The equation
sin t - sinf see 
serves also for determining the wire-distances by observing
the transits of a star near the pole and computing:
/f - sin t cos c,
where t is the difference of the transit over the lateral wire
and the middle wire, converted into arc.  In this way the
wire-distances are found very accurately. For the pole-star,
for instance, we have:
cos a -- 0.02609,
and hence we see, that an error of one second of time in
the difference of the times of transit produces only an error
of 0s.03 in the value of the wire-distance.
Gauss has proposed another method for determining the
wire - distances.
Since rays, which strike the object glass of a telescope
parallel, are collected in the focus of the telescope, it follows,
that rays coming from the focus of a telescope are parallel
after being refracted by the object glass. If the rays come
from  different points near the focus, their inclinations to each
other after their refraction are equal to the angles between
the lines drawn from the centre of the object glass to those
different points. Now if another telescope, which is adjusted
for rays coining from an infinite distance, is placed in front
of the first telescope, so that their axes coincide, we can see
through it distinctly any point at the focus of the first telescope. Therefore if there is at the focus of the first telescope a system of' wires, it is seen plainly through the second
telescope, provided that these wires are suitably illuminated.
But this is simply done by directing the eye-piece of the
first telescope towards the sky or any other bright object.
If then the second telescope is that of an azimuth instrument, the apparent distances of the wires can be measured
by it like any other angles.
In order to bring the wires exactly in the focus of the
object glass, the position of the eye-piece with respect to




460
the wires is first changed until they appear perfectly distinct.
Then  the wires are at the focus of the eye-piece.  After
that the telescope is directed to a star, and the entire tube
containing the wires and the eye-piece is moved towards or
from  the object glass, until the star is seen distinctly. When
this is the  case, the wires are at the focus.  In order to
examine this more fully, we direct the telescope to an object
at an infinite distance and bring it on the wire, and then
slighty shifting the eye before the eye-piece we see, whether
the object remains on the wire notwithstanding the motion.
If this should not be the case, it shows, that the wires are
not exactly at the focus, and they are too far from  the object glass, if the eye and the image of the object move towards the same side from  the wire. But if the eye and the
image move to different sides, the wires are too near the object glass *).
In 1850 June 20 Polaris was observed at the lower
culmination with the transit-instrument of the observatory at
Bilk, and the following transits over the wires were obtained:
Circle West.
I           II        III         IV           V
1311 32m 7s    191m 4s    13h 5n 7s    52m 7s    12h 38m 9.
Hence the differences of the times are:
1- III          1 —III       II    IV II- V
271n Os       13'n 57s         13m Os        26m 58s
Since the declination of Polaris on that day was:
880 30' 18". 01
we find by means of the formula:
f = sin t cos a
the following values of the wire-distances:
I —III' —42s.17, II —III — 21s.84, II-IV' -20s.34, IIIV — V=42s.12.
On the same day the star Vn Ursae majoris was observed:
I     II        III        IV     V
7 Ursae maj. Upper culm. 18.  50.3  13h 41m 24S.3  56.0  30.0.
I) It is best to use for this the pole-star. - Since the wire-distances
remain the same only as long as the distance of the wires from the objectglass is not changed, it is necessary to bring the wires exactly in the focus
before determining the wire-distances, and then leave them  always in the
same position.




461
The declination is 50" 4'. Hence the wire-distances are
found by means of the formula:
t =f sec 
I-III- 6s. 70, II —III-34s.02,  -- I- = 31s. 69, III- V —65s. 62.
Since the star was first seen on the first wire, we find
the transits over the middle wire from these wires as follows:.13h 41mn 24s. 20
24.32
24.30
24. 31
24. 38
13h 411, 24s. 30.
The arithmetical mean of all wire-distances, taking them
positive for the wires I and II (these being on the side of
the circle) and negative for the wires IV and V, is:
a -— t- Os. 31.
Now if we take the arithmetical mean of the transits of?j Ursae majoris over the several wires, we find:
1311 41t' 23s8 82,
and adding to it the quantity:
a sec  ~ -=- 8-. 48
taken with the positive sign, because the circle was West,
we find the transit over the middle wire from  the mean of
all wires, as before:
13h 41m 24s8. 30.
21. If the body have a proper motion, this must be
taken into account in reducing the lateral wires to the middle
wire. But since such a body has also a visible disc and a
parallax, we will now consider the general case, that one
limb of such a body has been observed on a lateral wire, and
that we wish to find the time of transit of the centre of the
disc over the middle wire.
We: have found before the following equation, which is
true for circle West:
sin C = - sin n sin a -+- cos 88 cos a sin ( - ni).
Now if the body has been observed on a lateral wire,
whose distance is f, where f is again positive, when the wire
is on the same side from  the middle wire as the circle, then
we must use in this formula c +-f instead of c. But if we
have not observed the centre but only one limb of the body,




462
whose apparent semi-diameter is h', we must take instead of
c now:
c  f -4- h'
where the upper or lower sign must be used accordingly as
the preceding or the following limb has been observed 4). If
then O is the sidereal time of observation, and a' is the apparent right ascension of the body, then its east hour angle is:
and hence we have the following equation, denoting the apparent declination by "':
sin [c  f-4- h'] = - sin n sin' -- cos n cos S' sin [a' - 0 - m],
where the upper or lower sign is to be taken accordingly
as the preceding or the following limb has been observed.
If then A denotes the distance of the body from the earth,
the distance from  the centre of the earth being taken as the
unit, we have also:
A sin [c +f-f  h']  - A sill n sinl 
- A cos n cos m cos 3' sin (O - a')
- A cos n sin m cos 5' cos (O - a'),
and since:
c,, mA, f, h',
and therefore also O — a' are small quantities, their sines
can be taken  equal to the arcs and their cosines equal to
unity, and we obtain:
A cos 5' (a' - 0)   + A. /'4- A. h' + - m A. cos 8' t- n A. sin 5' -+- c A.
The apparent quantities here can be expressed by geocentric quantities.  For we have according to the formulae
(a) in No. 4 of the third section, introducing the horizontal
parallax instead of the distance from the centre of the earth:
A cos 3' cos a'  cos S cos a -'o sin XC cos 9g' cos 0
A cos 8' sin a' = cos a sin a - q sin X cos a' sin d
A sin c'    -sin S      - o sin 7r sin q',
from which we easily obtain:
A cos' cos (O - a') = cos  cos (O -  ) - o sin n coS T'
A cos 5' sin (O -- a') - cos  sin (0 - a)
or in case that  --  a is a small angle*
*) For if the preceding limb is observed on the middle wire, then the
centre would be seen at the same moment on a lateral wire, whose distancef
is equal to ~- h'.




463
A cos A' (O - a')   cos 5 (O - a)
A Cos 3       -= cos 3-  sin 7i cos 9'
A sin 5'      =sin A- osin 7 sin g'.
From the two last equations we find also with sufficient
accuracy:
A   1 - o sin VW cos ('- - 5).
Finally we have, denoting by h the true geocentric semidiameter of the body:
A h' - h.
If we substitute these expressions for the apparent quantities in the above equation for:
A cos 5' (a' - 0),
we find:
cos / (a — )  1=   -        sin 7 cos (cp' - 5)] =t h 
t [cos  -         sin  cosT'] [m -- nt tang S' -- c sec']
or:
h      1     -   sin,t cos (9'- A)
a= 0 4-      +f
cos S           cos                              (a):i [1-o sinX CO si] [m  n tang' + c sec'],
where b' has been retained in the last term  instead  of S,
because it is more convenient in this form.  The apparent
declination 5' is found with sufficient accuracy by the reading ot the small circle for setting the instrument. But if this
is not the case, we must use in the last term also the true
geocentric quantities. Now the last term in the equation for
A cos 5' (c' - &) is:
+- m A cos 5' + n A sin 5' + c A.
If we substitute here for A cos 5', A sin 5' and A the expressions given before, and introduce the following notation:
m   nz - c cos g' e sin g
n   n - c sin 9q' o sin 7t'  C - [nII cos' -+ I- sin 99'] 9 sin 7t,
those three terms are transformed into:
cos  [in' --- n' tang o q- c' sec 5],
and hence we obtain:
h      1 - e sin 7t c     os (Cs  )
a         0- e A  f1-e s- m' - I n' tang -a- c' see S. (b)
cos 8           cos a
Now if the body has a proper motion, we find the time
of culmination from  the time of observation 0 on one of the
lateral wires by adding to 0 the time, in which the body




464
moves through the hour angle cc- O. But this time is equal
to the hour angle itself divided by 1 i-, if X denotes again
the increase of the right ascension expressed in time in one
second of sidereal time. If we put therefore:
1 - B sin 7r cos (T — 5)F
(1-i) cos,
the reduction to the meridian is:
h_            m' - n' tang a +- c' sec
(1-) cos                     - 
or:
A_   h +   + 1 - g sin 2  cos 9' sec  [m  n tang  c see'
If weomfFit              1-           t n tang /'+  sec se.
If we omit the tern /1-see' we find the time of culmination for the observed limb instead for the centre. Moreover, if we omit 1- 2 in the denominator of the last term,
the right ascension of the limb, which is obtained thus, is
not referred to the time of culmination, but to the time of
the transit over the middle wire. Since:
1 -  sin 7: cos S9' sec I'
always differs little from  unity, we can use instead of this
factor unity, if m, n and c are very small quantities *).
Bessel has given a table in his Tabulae Regiomontanae,
which facilitates the computation of the quantity F for the
moon. This table gives the logarithm of
I -q sin gt cos (9I'- 5)
the argument being:
log g sin Xv cos (I'- 5),
and besides it gives the logarithm  of 1-i), the argument
being the change of the right ascension of the moon in 12
hours. Another table gives the logarithm of F and the quanh
tity (1  -)   c for the sun, the arguments being the days of
the year.
If a body, which has a proper motion, has been observed on all the wires, then it is not necessary to know the
quantity F, since we may take again the arithmetical mean
of all the wires and add the small quantity a sec )Y, as was
shown before in No. 20.
*) Compare: Bessel, Tabulae Regiomontanae pag LII.




465
Example.  In 1848 July 13 the transit of the first limb
of the moon was observed with  the  transit instrument at
Bilk, when the circle was West:
I 17h 25m 42s. 9
II     26   5.0
III         28. 8
IV          51.0
V     27  14. 8.
The wire distances were at that time:
I 42s.23    II 21. 96    IV  20. 32    V  42. 30.
Now  in order to reduce the several wires to the middle
wire, we must first compute the quantity F. But on that
day was:
-— 180 10'. 6,
further the increase of the right ascension in one hour of
mean time was:
129s.8, and  = 55'11".0, h   60S.15;
moreover we have for Bilk:
T' - 500 1'. 2, log Q -- 9. 99912.
Now since one hour of mean time is equal to 3609s.86
sidereal, we find:
X- 0~. 03596,
and hence:
F- 0. 03565.
If we multiply the wire-distances by this factor, we find:
45s. 84    23s. 84    22s. 06    45s. 92.
Hence the times of observation reduced to the middle
wire are:
17h 26m 28s. 74
28. 84
28. 80
28. 94
28. 88
mean value 17h 26m 28s. 84.
The term
+ (1 -) cos
is equal to:
+ 65s. 67,
and hence the time of transit of the moon's centre over the
middle wire is:
17h 271m 34s9 51.,
30




466
Now  on that day b and k and therefore also -n and n
were equal to zero, but:
c --  Os. 09.
Therefore taking the factor:
I --  sin 7r cos A' sec s'
1 —2
equal to unity, we find for the time of culmination of the
moon's centre:
1711 271n 34s. 60.
If the parallax of the body is equal to zero or at least
very small, as in case of the sun, the formula for the reduction to the meridian becomes more simple.  For then we
have:
F       1
(1 - Z) cos 
In observing the sun usually the transits of both limbs
over the wires are observed. Then it is only necessary to
take the arithmetical mean of the observations of both limbs,
and thus the computation of the term (1h) e       is avoided
in this case.
22. It shall be shown now, how the errors of the transit instrument are determined by observations.
First the instrument must be nearly adjusted according
to the methods given in No. 5 of the fourth section.  The
level-error can then be accurately determined by means of
the spirit-level according to No. 1 of this section, when the
inequality of the pivots is known from a large number of
observations in both positions of the instrument.  The inclination of the axis can also be found by direct and reflected
observations of a star near the pole, for instance, the polestar.  For if we observe such a star on several wires and
call T the arithmetical mlean- of the times of observation reduced to the middle wire, then we have for the upper culruination the equation:,COS X    sin z
acu T-t- ~At d- ti os        c sec 
cos A   cos A
where i -b, when the circle is West, and i   - b', when
the circle is East, if b and b' denote the elevation of the
circle-end in the two positions.  But if we observe the image




467
of the star reflected from an artificial horizon, in which case
the zenith distance is 1800 - z, we have, denoting now the
arithmetical mean of the times of observation reduced to the
middle wire by T':
acos Z    sin z
a== T' t~ os -     4 - k    =o- c see 3,
cos S   cos A
and hence we find:
T- T cos 8
2    cos z
Since the value of cos J is small, we can find i by such
observations with great accuracy.
Then in order to determine the error c, we observe the
same star in the two positions of the instrument, when the
circle is West and when it is East. For these observations
we must choose again a star near the pole, c, Y or A Ursae
minoris, because for other stars there is no time for reversing the instrument between the observations on the several
wires, and because for these stars the coefficient sec 5 of c
is very great so that errors of observation have only little
influence on the determination of c. If we observe the star
on several wires when the circle is West, and denote by t
the arithmetical mean of the times of observation, reduced
to the middle wire and corrected for the level-error, we have:
sin (95-S)
t - t t-Qt - sin ( a- c sece,
cos 
Then if we reverse the instrument and observe the star
again on several wires, when the circle is East, we have,
denoting now  the arithmetical mean of the times of observation reduced to the middle wire and corrected for the levelerror, by t':
a — t' -- A t   sin (9   - c see S.
cos.
From  the two equations we find therefore:
ti -     t
c-  2  cos.
If there is a very distant terrestrial object in the horizon
in the direction of the meridian (a meridian mark), furnished
with a scale, the value of whose parts is known in seconds,
we can determine the collimation-error by observing this object in the two positions of the instrument, since, if we read
30*




468
the point of the scale in which it is intersected by the middle
wire in the two positions, the collitnation-error is equal to
half the difference of the readings.  Still better is it to use
a collimator for this purpose.  But then the telescope must
have besides the vertical wires, which serve for observing
the transits of the stars, also  a moveable micrometer-wire,
parallel to them, whose position can be easily determined by
means of a scale, which gives the entire revolutions of the
micrometer-screw, and of the divided screw head whose readings give the parts of one revolution of the screw. If the
telescope is furnished with such a wire, it is directed to the
wire-cross of the collimator in both positions, and the moveable wire is moved until it coincides with it each time. Now
if the readings for the moveable wire in the two positions
are a and b, it is easily seen, that 2 (a  - b) corresponds to
that position  of the moveable wire, in which a line drawn
from  it to the centre of the object glass is perpendicular to the
axis of the instrument.  Therefore if the moveable wire is
moved until it coincides with the middle wire, and if the
reading in this position is C, then C- 1 (a+ -b) or 1 (  -jb)   C
is the error of collimation, and its sign is positive, if the
moveable wire in the position     (ac  - b) and the circle-end
of' the axis are on opposite sides of the middle wire.
When there are two collimators opposite each other,
one north, the other south of the telescope, the error of collimation can be determined without reversing the instrument.
For, the two collimators being directed to each other *), one
of them is moved until the two wire-crosses coincide so that
the axes of the two collimators are parallel. Then the telescope is directed in succession to each of the collimators, and
the moveable wire is placed exactly on their wire-crosses. If
the readings for the moveable wire in the two positions be
a and b, then the error of collimation is again   (a + b)-  C
or C-    (a -- b), and we can decide about its sign by the
same rule as was given before.
*) In order that this may be possible if the collimators are on the same
level with the instrument, the cube of the axis of the latter has two apertures opposite each other, through which the two collimators can be directed
to each other, when the telescope of the instrument is in a vertical position.




469
Another method of determining the error of collimation
is that by means of the collimating eye-piece.  For this purpose the telescope  is directed to the nadir and an artificial
horizon placed underneath i).  If then the line of collimation
deviates a little from the vertical line, one sees in the telescope besides the middle wire its reflected image, whose distance from  the wire will be double the deviation of the line
of collimation from the vertical line, which can be easily
measured by means of the moveable wire' ). For this purpose
it is best, to place first the moveable wire so, that the middle
wire is exactly half way between the reflected image and the
moveable wire and afterwards so, that the reflected image
is half way between the middle wire and the moveable wire.
Since there  is also  a reflected image of the moveable wire,
in the first position the two wires and by their side the two
reflected images are seen at equial distances, whilst in the
other position the wires and their images alternately are seen
at' equal distances..  The difference of the two readings for
the moveable wire is equal to three times the distance of the
middle wire from its reflected image.
In order to see the image reflected from the mercury
horizon, it is requisite, that light be so reflected towards the
mercury as to show the wires on a light ground. This is
accomplished  by  placing inside the tube of the eye-piece a
plane  glass inclined  by  an angle of 45" to the axis of the
telescope, an aperture being opposite in the tube, through which
light can be thrown  upon  it.  In  order to  have then  the
*) Usually a mercury horizon, that is, a very flat copper basin filled
with mercury, which is poured into the basin after this has been well rubbed
with cotton dipped into nitric acid. The mercury then dissolves some of the
copper and gives in this impure state a more steady horizontal surface. The
oxyde which is formed on the surface can be easily taken off by means of
the edge of a paper, and thus a perfectly pure reflecting surface is easily
obtained.
**) For all these determinations it is requisite to know  the value of
one revolution of the micrometer-screw of the moveable wire in seconds. But
this can be easily found, if the known interval between two wires is measured also in revolutions of the screw by placing the moveable wire over
each of these wires, and reading the scale and the screw head.




470
whole field uniformely illuminated, it is necessary, as was
first shown by Gauss, that there be no lens between the
wires and the reflector. But since it is always troublesome,
to exchange the common eye-piece so often for this collimating eye-piece, Bessel proposed, to place simply outside upon
the common eye-piece a plane glass in the right inclination
or a small prism, and to reflect by means of it light into
the telescope.  It is true, a small part of the field is then
only illuminated, but there is no difficulty in observing the
reflected image, provided that the glass or the prism is fastened in a frame so that its inclination to the axis can be
changed.
The error of collimation is then determined in the following way. Let b denote the inclination of the line passing
through the Ys, taken positive, when the side on which the
circle is, is the highest; further let uI denote the inequality
of the pivots expressed in seconds and taken positive, when
the pivot on the side of the circle is the thickest one of
the two; finally let c be the error of collimation, taken positive, when the angle, which the end of the axis towards
the circle makes with the part of the line of collimation towards the object glass, is greater than 900; then we have,
denoting by d the distance of the middle wire from its reflected image, and taking it positive, when the reflected image
is on that side of the middle wire, on which the circle is:
-- d — b t-u - c.
Therefore if b-t — is known by means of the spirit-level,
the error of collimation can be found from this equation, and
conversely, if the error of collimation has been determined
by other methods, the inclination of the axis of the pivots
is found. Now if the instrument is reversed, and d' denotes
again the distance of the middle wire from its reflected image,
taken again positive, when it is on the side towards the
circle, we have:
d   - - -b- i - c,
and from  both equations we obtain:
c -  = -   (d -- d')
+ herefo  (by observing  the reflected').
Therefore by observing the reflected.image in both po



471
sitions of the instrument, we can find c as well as the inclination of the axis, if the inequality of the pivots is known.
With small portable instruments, which usually are not
furnished with a moveable wire, we can find the error of
collimation according to the same method but by means of
the spirit-level.  For if one end of the axis is raised or
lowered by means of the adjusting screws, until the reflected
image is made coincident with the middle wire, we have
d  O 0 and hence c- b-+u.  Therefore if' bt u is found by
the spirit-level according to No. 3 of this section, this value
is equal to the error of collimation.
With the meridian circle at Ann Arbor the following
observations were made in the two positions of the instrument.
By means of the level the inclination of the axis of the
pivots was found, when the circle was West, b' = -+ 2".77
and when the circle was East, b'1   - 2".45. The distance
of the middle wire from the reflected image was found in
parts of a revolution of the micrometer-screw:
d -   0 OP. 2260  Circle West
d' —-  0.3107  Circle East.
We have therefore:
c - u - +- 0. 0212 -- -0". 43
b ~-0.1342 -+  2".73;
since one revolution of the screw is equal to 20". 33, and since
u -  0". 17, we have:
c  -- 0". 60,
and the inclination of the axis, when the circle was West,
b' —-4-2". 90, and when the circle was East, b'=- - 2".56.
Then the instrument was directed to one of the collimators, and when the moveable wire was made coincident
with the wire-cross, the reading of the screw was:
21P. 132  Circle West
21.999  Circle East.
We have therefore. (a-I- b)-21.5655; the coincidence
of the wires was 21P.5397, and since we must take 7 (a —b)-C,
in order to find the error of collimation with the right sign,
we obtain:
c =  - 0P. 0258 -  0".52.




472
Finally the two collimators were directed towards each
other and the moveable wire was made coincident with the
wire-crosses. Then the readings of the screw were:
for the south collimator 21". 1190
for the north collimator 22.0127
Hence we have 2 (a~ — b)-  21.5658
C-   21.5397
c =  - OP.0261    0". 53.
The inclination and the error of collimation being thus
determined, it is still necessary, to find the azimuth of the
instrument and the error of the clock.
For this purpose we can combine the observations of
two stars, whose right ascensions are known. But in case
that the rate of the clock is not equal to zero, we must first
reduce the error of the clock to the same time by correcting
one time of observation for the rate of the clock in the interval of time between the two observations.  Then At in
both equations will have the same value.  If then to and t'0
are the two times of transit over the middle wire, corrected
for the level-error, the collimation-error and the rate of the
clock, we have the two equations:
sin (- -)
a = to +At+-I 
cos a
sin (p -')'l- t'of- A t 4k
by means of which we can find the values of the two unknown quantities At and k; for we have:
sin(S - Y')
-' = -- (t'o- to+ )  c   cos  cos  6'
hence  k =      ~
cos T      sin (6 - Y)
After having found k we obtain the error of the clock
from  one of the equations for a or re'.  We see from  the
equation for k, that it is best, when    - 6' is as nearly as
possible 90", and that it is of the greatest advantage, to combine
a star near the pole with an equatoreal star, because then
the divisor sin ( --') is equal to unity and the numerator
is very small. If it is impossible to observe a star near the
pole, we can combine a star culminating near the zenith with
another near the horizon.  But in either case it is always




473
advisable to observe more than two stars, and to find the
most probable values of At and k from  all the observations.
For these determinations the standard stars, whose right
ascensiohs are well known and whose apparent places are
given in the almanacs for every tenth day, are always used.
But these apparent places do not contain the diurnal aberration, since this depends on the latitude of the place. Now
according to No. 19 of the third section the diurnal aberration for culminating stars is:
-4- 0". 3113 cos gp see 3,
where the upper sign corresponds to the upper culmination,
the lower one to the lower culmination.  We see therefore,
that it will be very convenient, to apply this correction with
the opposite sign to the observations, since then it can be
united with the error of collimation.  Therefore the diurlnal
aberration is taken into account, by writing in all the formulae given before c- 0".3113 cos pf instead of c or, expressed
in time, c-08.0208 cos ~, instead of c and - (c- +-0. 0208 cos c)
instead of    c.
The methods given above for determining the azimuth
are generally used for small instruments, which have no very
firm mounting, and they may also be used for larger instruments, especially the first method of the two, when only relative determinations are made. The following may serve as
a complete example for determining the errors of an instrument of the smaller class.
Example.  In  1849 April 5 the following observations
were made with the transit instrument at Bilk.
Circle West.
I       II        III      IV     V       Mean
f, Orionis    54s. 8    15s.3  5h 8r 37s.4  58s.0  20s.1  5h1' 8n37s.44
Polaris U  38m 13s.0 51m14s.O  01h                    t 5 15.25
b — -- Os. 03.
Circle East.
11        11
Polaris U          19m26s.0  lh 51n25s.O              1 5 24.57
b  --  O. 05.
The apparent-places of the two stars were on that day:
Polaris a -- h 4m 17s. 92 a-  880 30' 15". 5,f Orionis a'= 5 7 16.66 6' —8  22. 8.




474
If we reduce the observations to the middle wire and
apply the correction for the level-error, we find:
Circle West, Orionis 5"1 8Um 37s. 42
Polaris 1 5 14.33
Circle East Polaris 1 5 23.05.
From the observations of Polaris in both positions of the
instrument, we find the error of collimation
-  Os. 114,
and since the diurnal aberration for Bilk is equal to 0S.013
sec 3, we must take for c now -4-0'. 101, when the circle is
West, and  - 0. 127, when the circle is East.  If then we
correct the observations in the first position for the error of
collimation, we find:
fi Orionis =t'o -5h 8m 37s. 52
Polaris -to   1 5 18.20.
Hence we have:
t'o - to = 4h 3m 19s. 32   a -- a- 4h2m 58s. 74,
and since:
-51~ 12'. 5
we find:
k   - Os. 85.
Therefore the observation of A Orionis corrected for the
errors of the instrument is:
5h 8m 36s. 78,
and hence:
at: - im 20s. 12.
The methods for determining k, which were given before, have this disadvantage, that they are dependent on the
places of the stars. It is therefore desirable to have another
method, which gives k independent of any errors of the
right ascensions, and which therefore can be employed when
absolute determinations are made with an instrument. For
this purpose the observations of the upper and lower culminations of the same star are used, as has been stated already in No. 5 of the fourth section. In this case we have
a' -c- 12h+-a  and'= 1800 - b, where Act is the change
of the right ascension in the interval between the two culminations, and therefore the formula for k, which was found
before, is transformed into:




475
12It- A a - (t'o - t,) cos 2
cos F       sin 2 a2
121h — A - (t'o - to)
2 cos Tp tang c
Also for this purpose it is best to observe stars very
near the pole at both culminations, because then the divisor
tang Y becomes very great. But the method requires, that
the instrument remains exactly in the same position during
the time between both observations, or at least, if this is not
the case, that any change of the azimuth can be determined
and taken into account.
In order to dispense with frequent determinations of the
azimuth by means of the pole-star, a meridian-mark is usually
erected at a great distance from the instrument. This consists of a stone pillar on a very solid foundation, which bears
a scale on the same level with the instrument. If then by a
great many observations of the pole-star that point of the
scale, which corresponds to the meridian, has been determined, the azimuth of the instrument can be immediately
found by observing the point, in which the scale is intersected by the middle wire, at least, if the scale remains exactly in the same position, and if either the error of collimuation is known or the instrument is reversed and the scale
is observed in the two positions of the instrument; for the
distance of the middle wire from the point of the scale, which
corresponds to the meridian, is in one position equal to k — c
and in the other equal to k - c. But the distance of the
meridian-mark must be great, if great accuracy shall be obtained, since one inch subtends an angle of 1" at a distance
of' 17189 feet, and therefore in this case a displacement
of the scale equal to -L of an inch would produce an error
of the azimuth equal to 0". 1. However such a great distance
is not favorable for making these observations, since the disturbed state of the atmosphere will very seldom admit of an
accurate observation of the scale. And since, besides, the observation of such a meridian-mark is limited to the time of
daylight, Struve has proposed a different kind of meridianmark, which is in use at the observatory at Pulkova.  In
front of the telescope, namely, a lens of great focal length is




476
placed (Struve uses lenses of about 550 feet focal length)
in a very firIn position and so that the axis coincides with
that of the telescope.  The meridian-mark at its focus is
a small hole in a vertical brass plate, which in the telescope
appears like a small and very distinct circle.  The lens is
mounted on an insulated pier and is well protected by suitable coverings against any change. Likewise the meridianmark is placed on a insulated pier in a small house and carefully protected against any external disturbing causes. Since
thus the same care is taken as in the mounting of the instrument itself, it can be supposed, that the changes of the
lens and of the meridian-mark will not be greater that those
of the two Ys of the instrument, and since experience shows,
that the azimuth of a well mounted instrument does not change
more than a second during a day, the probable change of
the line of collimation of the meridian-mark (that is, of the
line from  the centre of the lens to the centre of the small
hole) will be less in the same ratio, as the length of the.axis of the instrument is less than the focal length of the
lens. Therefore if the length of the axis is 3 feet and the
focal length of the lens is 550 feet, this change will not
exceed F}, of a second. The chief advantage of such a meridian-mark is this, that it can be observed at any time of
the day, and thus any change in the position of the instrument can be immediately noticed and taken into account.
When there are two such meridian- marks, one south, the
other north of the telescope, we can find, by observing both,
the change of the error of collimation as well as that of the
azimuth, whilst the observation of one alone gives only the
change of the line of collimation and thus requires, that the
error of collimation has been determined by other methods.
If the readings for the north and south mark are a and b,
and at another time a' and b', and if we take them positive,
when the middle wire appears east of the mark, then we
obtain the changes dc and da of the error of collimation
and of the azimuth by means of the equations:
a' - a -- (b'- b)
dc      --
2
b'- b- (ah- a)




477
where dc must be taken with the opposite sign, when the
circle is East.
23.  If the transit instrument has a divided circle so
that not only the transits but also the meridian zenith distances of the stars can be observed, it is called a meridian
circle.
When a star is placed between the horizontal wires of
such an instrument at some distance from the middle wire,
the angle obtained from the reading of the circle is not the
meridian zenith distance or the declination of the star, because the horizontal wire intersects the celestial sphere in a
great circle, whilst the star describes a small circle. Therefore a correction must be applied on this account to the
reading of the circle.
The co-ordinates of a point of the celestial sphere, referred to a system, whose fundamental plane is the plane of
the equator, whilst the axis of x is perpendicular to the axis
of the instrument, are:
x  cos 8 cos (r - an), =-  cos 8 sin (: -n ) and z — sin 8.
If we imagine now  a second system  of co-ordinates,
whose axis of x coincides with that of the former system,
whilst the axis of y is parallel to the horizontal axis of the
instrument, and if we denote by d' the angle through which
the telescope moves and which is given by the reading of
the circle, and if further we remember, that the telescope
describes an are of a small circle, whose radius is cos c, then
the three co-ordinates of the point, to which the telescope
is directed, are:
x-  Cos c' cos c, y= -sin c, and z = sin d' cos c.
Now since the axes of the two systems make with each
other an angle equal to n, we obtain:
sin a -  sin c sin n +- cos c cos n sin A'
cos / cos (c  -n m) - cos 3' cos c
cos 3 sin (a - m) - sin A' cos c sin n -i-+ sill c cos 
and hence:
cotang a cos (   m) — o 
-sin n sin c -+- cos n cos c sin 8'
This formula can be developed in a series, but since n
is always very small and c, even if the star is observed on




478
the most distant lateral wire, is never more than 15 or 20
minutes, we can write simply:
tang  =- tang d' cos (x - m),
and from  this we obtain  according to formula (17) of the
introduction:
-- _' — tang 2 (4r -- m)2 sin 2 8 -tang ('   - -rn)4 sin 4 8.
This formula is still transformed so that the coefficients
contain the quantities
2 sin I (2  - m)' and 2 sin -- nz)4
because  these quantities can always be taken from  tables.
(V. No. 7).
For this purpose we write instead of
tang - (r - n)22
now:
sin 4- (7; - n) 2
I - COS' (i - n)2
and develop this into the series:
sin'. (xT -  n)2 - sin - (T- --  )4  -...
and since:
- tang 2 ( - n)  = sin 4  (T - nY)4 +...
we obtain:
S   —' - 2 sin 2 (:- S    " )2. I- sin 2 S- 2 sin' (r - -2)4 cos A' sin 2 S,
the first term of which formula is usually sufficient.
The sign of this formula corresponds to the case, when
the division  of the circle increases in the direction  of the
declination  and when the star is observed at its upper culmination.
When the division increases in the opposite direction,
the corrected reading is:' ~- 2 sin 1, (   m- n) 2.  sin 2 S +- 2 sin     ( - -m)4 cos 82 sin 2 a.
Since the circle is numbered in the same direction from
0" to 3600, it follows, that if for upper culminations the division increases in the direction of the declination, the reverse takes place for lower culminations, and hence also
for lower culminations the sign of the formula must be
changed.
We  can  find the formula also in the following way.
Let PO' Fig. 18 represent the meridian and 0 -a star, whose




479
Fig. 18.  hour angle shall be t. If we direct the telescope
rD      to this star and bring it on the horizontal or axial
wire, we observe the polar distance P 0', where
the point O' is found by laying through 0 an arc
of a great circle perpendicular to PS.  Then we
have P O' -  900-' PO -- 900 -  d and hence:
tang' = cos t. tang 1'.
Now  we will further suppose, that the axial
wire is not parallel to the equator, but that it
0o'       makes an angle equal to 900 -- J with the meridian, where J is called the inclination of the wire;
then we observe the polar distance P O", where 0"
S!     is found by laying through 0 a great circle making with the meridian an angle equal to 900 + J.  If we
denote again the observed declination by a', and take 0 0":c,
we have:
sin c sin J  - sin S cos' -1 cos s' silln' cos t
sin c cos J-  cos 8 sin t,
and therefore:
tang 8 -tang?' cos t - sin t'
=tang A' cos (t -+- y),
where:
J
== sin )
When J= 0, the formula gives simply the reduction to
the meridian.  But this reduction plus the correction for the
inclination of the wires is, if we take only the first term of
the series:
-S'- 4 sin 25. 2 sin - (t-  y) 2.
In order to determine the inclination of the wires, a star
near the pole is observed at a great distance from the middle
wire on each side of it.  For, every such observation gives
an equation of the form:
S- =' -9 sin 2 a. 2 sin t2 - cos  sin t. J,
where also the second termr, dependent on sin I t4, can be
added, if it is necessary. Therefore from two such equations we can find d and J, or when more than two observations have been made, we can find the most probable va



480
lues of J and A B, if we assume for 8 the approximate value
o, so that 3-  S, -     ~  A8. The above equation becomes then:
0 =        -   I' +   sill2 S. 2 si la t- + A  -+ cos S sin t. J.
It is also easy to find the correction which must be
applied to the observed declination in case, that a body has
been observed, which has a parallax and a proper motion,
for instance, the moon. If such a body has been observed
on a lateral wire, we have the equations:
cos c cos -= cos  cos (x - m)
cos c sin S' = cos S sin ( - m) sin n - sin a cos n.
Here   - is the apparent declination of the observed point
of the limb, and r is the east hour angle of that point at the
time of observation, whilst 8' is the declination given by the
reading  of the circle.  But if we denote by 8 the apparent
declination of the centre of the moon, and by r its apparent
hour angle, we have:
cos c cos (5': x) - CoS e Cos ( - rn)
cos c sin (' = x) = cos S sin ( —mn) sin n + sin S cos n,
where
sin x cos c -- sin h'
if h' is the apparent semi-diaineter ), and where the upper
or lower sign must be taken accordingly as the upper or
lower limb has been  observed.   If we substitute in these
equations sin h' instead of sin x cos c, eliminate cos  cos c cos x
and multiply the resulting. equation by A, which denotes the
ratio of the distance of' the body from the place of observation to the distance from  the centre of the earth, we find:
-4- A sin h'll-   A COs 8 sin' cos (: — rn)
- A cos S cos 5' sin (T - m) sin n
- A sin S cos 5' cos n,
or since the quantity sill (T - mn) sin n can be neglected and
cos it be taken equal to unity:
=t= A sin It'     A cos. sin S' cos (v —m)
-- A sin S. cos 5'.
If we express now  the apparent quantities in terms of
the geocentric quantities, taking:
*) We find this immediately from the right angled triangle between the
pole of the circle of the instrument, the centre of the moon and the observed point of the limb, the angle at the pole being x and the opposite
side h'.




481
A sin ha' - sill h
-A cos =- cos  -o g sin qt cos Rp~
A Sill   sin   0 -o sill 7C Sin lr,
we easily find:
=I= sin h - q sin 7v sin (9' -')
= sin (' —30) -- os A  sin 8"'   (v- r7I) 2
2062652
Now  if the time of observation is 0, and the time of
culmination of the moon is 0,, we have:
Tr -- - 00.
But when the body has a proper motion and X denotes
the increase of the right ascension in one second, we have:
= (O - 0) (1-  ). 15,
if 0 -  0,, is expressed in *seconds of time.
Now  if we neglect the small quantity m  in (r — m)2
and take:
sin p --  sin sin (9' -- d'),
we have:
152
sin (80 -') ill -p v sin  - I  sin 2 s'(-   ) 2 (- ( 2 206265 
And since:
sin (p =t h)- sin pl  Sin  - 2 siinp I h =F 2 sin h sin,
and hence:
p-4- h           I
sin p -- sin h - sin (p -4- ) -4  --  sin p sin  20625'
we finally obtain:
A,                - 8''F t 1 == - ---- sin p sin h
- 4t         i 20' (  ) ( -o)
This is the formula given by Bessel in the introduction
to the Tabulae Regiomnontanae pag. LV.  The last term  of
this formula corresponds to the first term  of the formula for
the reduction to the meridian, which was found before, multiplied by (1 - ))2.
This true declination of the moon's centre corresponds
to the time 0.  If we wish to have it for the time I', we
must add the term:
-+H  (0' - 0)
dt 
where d/t is the change of the declination in the unit of time.
d t
31




482
24.  In order that the observations with the meridian
circle may give the true declinations or zenith distances, the
readings of the circle must be corrected for the errors of division and for flexure, which must be determined according
to No. 7 and 8 of this section.  Finally the zenith point or
the polar point of the circle must be known.  In order to
find the latter, the pole-star must be observed at the upper'
and lower culmination.  When the readings are freed from
refraction, and from  the errors of division and froln flexure,
the arithmetical mean of the two readings gives the polar
point, provided, that the microscopes have not changed their
position during the interval between the observations.  But
since it is necessary for examining the stability of the microscopes and for determining any change of their position,
to observe the nadir point at the time of the two observations, it is at once the most simple and the most accurate
method, to refer all observations to the zenith point, that is:
to determine the zenith distances of the, stars, and to deduce
from  them  the declinations with the known value of the
latitude.
As has been shown before, the nadir point is determined,
by turning the telescope towards the nadir and observing the
image of the wires reflected from an artificial horizon, which
must be made coincident with the wires themselves. Usually
such an instrument has two axial wires parallel to each other
at a distance of about 10 seconds, and in making an observation the instrumlent is turned, until the star is exactly half
way between these wires.  For determining the nadir point
the reflected images of the two wires are placed in succession half way between the wires, and then the arithmetical mean
of the readings of the circle in these two positions of the
telescope gives the nadir point.  The observations are then
freed from  flexure according to the equations (B) in No. 8
of this section and firom  the errors of division. In order to
obtain the utmost accuracy, it would be necessary to determine the nadir point after every observation of a star; but
since the displacements of the microscopes are only small
and are going on slowly, it is sufficient, to determine it at
intervals, and then to interpolate the value of the nadir point




483
for every observation.  In this way the errors produced by
any changes of the microscopes are entirely eliminated, and
since the observation of the nadir point is so simple and so
accurate, this method for determining zenith distances is the
most recommendable.
Horizontal collimators, of which one is north, the other
south of the telescope, can also be used for determining the
zenith point. For this purpose -the collimattors are constructed
so, that the line of collimation of the telescope is also the
axis of the instrument, the cylindrical tube of the telescope being
provided with two exactly circular rings of bell metal, with
which it lies in the Ys.  These Ys have the usual adjusting
screws for altitude and azimuth, and the wire-cross is likewise furnished with such screws, by which it can be moved
in the plane perpendicular to the axis of the telescope. When the
collimators have been placed so that their line of collimation
coincides nearly with that of the telescope, the line of collimation of the telescope of each collimator is rectified so that
it coincides with the axis of revolution.  This is accomplished by directing one collimator to the other and turning it
1800 about its axis.  If the point of' intersection of the wires
after this motion of' the telescope remains in the same position with respect to that of the other collimator, then the
line of collimation is rectified; if this is not the case, the wirecross is moved by means of the adjusting screws, until the
point of intersection remains exactly in the same position
when the telescope is turned 180".  The inclination of the
axis and hence also of' the line of collilmation is then found
by means of the level, and since the collimator can be reversed so that the object glass is on that side on which the
eye-piece was beforle, the inequality of the pivots can be deternmined and taken into account in the usual way. In order
then to find the horizontal point of' the circle, the collimator
is levelled, and the telescope of' the meridian circle turned
until its wire-cross is coincident with that of the collimator.
In this position the circle is read.  The same operation is
repeated after the collimator has been turned 180~1 about its
axis, to eliminate any error of' the line' of collintlation. Then
the same observations are repeated with the other collimator,
31*




484
and when a and b denote the arithmetical means of the readings of the circle for each collimator,  -5  is the zenith point
of the circle, if the collimators are at equal distances from
the axis of the instrumentS).  If x is the elevation of the
object-end of the collimator, corrected already for the inequality of the pivots, then the zenith distance of the telescope
when it is directed to the wire-cross of the collimator, is
90 ~+- x, taking no account of the angle between the vertical lines of the two instruments, and hence we must subtract x from  the reading or add it, accordingly as the division increases or decreases in the direction of the zenith
distance.
This method being more complicated and therefore probably less accurate than the one mentioned before, the latter
is always preferable.
The latitude is determined best by direct and reflected
observations of the circumpolar stars.  For we obtain from
the observations made at one cuhlmination according to the
equations (B) in No. 8 of this section:
90n- ai- o                     sifn 2    t   l-.c.
and a similar ecquation is found for the lower culmination.
The arithmetical mean of' these two equations gives the latitude independent of the declination of the star, but affected
with those terms of flexure which depend on the sine of
2 z, 4 z etc., the first of which can be determined by the
method given in that No.  The angle between the vertical
lines of the instrument and the artificial horizon must likewise be taken into account, as was shown in the same No.
V. THE PRIME VERTICAL INSTRUMENT.
25. If we observe the transit of a star and its zenith
distance with a transit circle mounted in the plane of the
prime vertical, we can determine two quantities, namely a
*) The readings nmust be corrected for flexure, if there are any terms,
which have an influence upon the mean of the two readings.




485
and (' or 93. But since the observation of zenith distances
in this case is more difficlult, usually only the transits of
stars are observed with such an instrument, in order to find
the latitude or the declinations of the stars.  For this purpose a method is required, by which the true time of passage over the prime vertical can be deduced from  the observed time and the known errors of the instrument.
We will suppose, that the axis of the instrument produced towards north meets the celestial sphere in a point Q,
whose apparent altitude is b and whose azimuth, reckoned
from the north point and positive on the east side of the
meridian, is k. -If we imagine now three axes of co-ordinates,
of which the axis of z is perpendicular to the horizon, whilst
the axes of x and y are situated in the plane of the horizon
so that the positive axis of x is directed to the north point
and the positive axis of y to the east point, then the three
co-ordinates of the point Q are:
z = sin b, y = cos b sin k and x = cos b cos k.
Further if we imagine another system  of co-ordinates,
whose axis of z is parallel to the axis of the heavens, and
whose axis of y coincides with the corresponding axis of the
first system  so that the positive axis of x is directed to the
point in which the equator intersects the meridian below the
horizon, then the three co-ordinates of the point Q, denoting
its hour angle (reckoned in the same way as the azimuth)
by m, and 180o minus its declination by n, are:
z- =sinn,    cos ni sinl m, x - Cos nI Cos n,
and since the axes of z in both systems make with each other
an angle equal to 90"- f., we have the equations:
sin 1b - sin i sin  - cos n cos m cos 9T
cos 1) sin ]k — cos ia sin m2
cos b cos k  cos n cos m sin 9 -+ sin n cos 9
and
sin n - cos b cos c cos 95 +- sin b sin T
cos n sin in  cos b sin k
cos sn cos   cos b) cos Ac sin qT - sin b cos 99.
If we then assume, that the line of collimation of the
telescope makes with the end of the axis towards the circle
an angle equal to 900-  c, and that it is directed to an object, whose declination is 3 and whose hour angle is t, then




486
the three co-ordinates of this point with respect to the equator and supposing the axis of x to be directed towards
north, are:
z - sin 8, y - cos 8 sin t and x  -cos 8 cos t,
and if we take the axis of x in the plane of the equator, but
in the direction of the axis of the instrument:
z = — sin 8
x= - cos 8 cos (t —m).
Now if we imagine another system, of which the axis
of y coincides with that of the former system, whilst the
axis of x coincides with the axis of the instrument, we have:
x   - sin c,
and since the angle between the axes of x in the two systems
is n, we have:
sin c  - sin S sin n +- cos o cos (t-nm) cos n.
WVe can deduce these formulae also from  the triangle
between the pole, the zenith and the point Q, towards which
the side of the axis opposite to that on which the circle is,
is directed.  In this triangle we have, when the circle is
north, P Q — 1800 -    -, n, Z Q — 900+- b and P Z —  900- G,
whilst the angle QPZ  - m and Q Z S =  k. The formula for
sine is deduced from the triangle PS Q, where S is that
point of the sphere of the heavens, to which the telescope
is directed, and in which we have S     900 — c, when S is
west of the meridian and SP-90~ —, PQ=180"- -1 —n,
whilst the angle SPQ   t -m.
From  the last equation we  obtain by substituting for
sin n, cos n cos m and cos n sin mn the values found before, and
taking instead of the sines of b, k and c the arcs themselves
and instead of the cosines unity:
C     Sill S cos T99  cos o sin S   cos s
-[sin a0 sin   - 9   cos a cos e cos t] 
- cos o sill t. I,
and since:
sin S sin T -+ cos +   cos T cos t = Cos z
and
cos 8 sin t = sin z sin A,
or, since A is nearly 900:
cos a sin t = sin z,
we obtain, when the star is west of the meridian:
c H- b cos z - k sin z = - sin a cos 99 +- cos  sin T cos t.




487
If then  O  is tile true sidereal time, at which the star
is on the prime vertical, and if therefore 0   ac is the hour
angle of the star at that moment, we have:
cos (O -a ) =   -tang,
tang T
r':
0   - sin 8 cos   - cos c sin T cos (O- a).
Subtracting this equation from  the other, we obtain:
c - b cos z - k sin z - cos o sin'. 2 sinl -- [0   a-t] sin'- [O —+ t].
Now  since c, b and k are  small quantities and hence
0 —ae and t are nearly equal, we can put:
sin t instead of sin - [9O-a —t]
and'2[0-a-t] instead of sin 2 [0-a-t]and then, remenmbering that
cos l sin t - sill z
we obtain:
c           b         k
O-a -  t -t  -       -.  -
sin z sin T   tang z sin T   sin T
If then a star has been observed on the middle wire of
the instrument at the clock-time T, the true sidereal time
will be l'-+-At, and the hour angle:
-t- At -a -- t.
Therefore we have:
c           b    _    c
O — T —t — A t.      -4sill z si.Sn    tang z sin 9   sin T
This formula is true, when the circle is North and the
star West.  When the star is East, we have:
cos 8 sin t --  sin z.
Therefore, since the signs of the quantities c, b and k
remain the same, we must change in the above formula the
signs of the divisors sinz and tang z and thus we have:
c Tb-                c a  Circle North}
O     -- T — At
sin.Iz sin 9   tangz sin-  sing     Star East
When the circle is South, the quantities b and c have
the opposite sign, and therefore we have:
c          b         k     Circle South
— = T                 -- A t -                     nt
sin z sin T.  tangz sin Rp  sin   t Star West
and
0c zsi n tangb       k -Circle South E
sill Z sin f   tang z sin go  sin 99   Star East




488
If we know 0 and ac, we obtain by means of the formula:
tang 90 cos (O- a) - tang 
either rp, when the declination of the star is known, or the
declination, when the latitude is known.  If 0(  and 0' be
the times, at which the star was on the prime vertical east
and west of the meridian, then 2 (0'- 0) will be the hour
angle of the star at those times, and therefore we have:
tang T cos (0'-  0)  tang C,
so that it is not necessary to know the right ascension of
the star, in order to find ip or J'. When the instrument is
reversed between the two observations, so that one transit
is observed when the circle is North, the other when the
circle is South, then we have:
(0'  )  )   (T'- T),
and hence in that case it is not necessary to know the error
of the clock nor the errors of the instrument except the levelerror.  An example is given in No. 24 of the fifth section.
26. The formulae given before are used, when the instrument is nearly adjusted so that b, c and k are small quantities, whose squares and products can be neglected. But
this method of determining the latitude by observing stars
on the prime vertical is often resorted to by travellers, who
sometimes cannot adjust their instrulment sufficiently and thus
make the observation at a greater distance from the prime
vertical.  In that case the fornmulae given above cannot be
employed.  But we found before the rigorous equation:
sin c  - sin in   n a -I- cos a cos e cos (t - mn),
or if we substitute the values of sin n, cos n cos m and cos n sin m
sin c= - sin b sin S sin p - sin 1 cos 8 cos p cos t -cos b cos k sin A cos e
-1- cos b cos k sin qo cos S cos t I- cos b sin Ic cos S sin t.
Now  if the observation were made on the prime vertical, we should hlave:
sin S  cos z sin 9g,   cos' cos t = cos z cos R
and
cos S sin t - sin z.
But since we assume, that the instrument makes a considerable angle with the prime vertical, we will introduce the
following auxiliary quantities:




489
sin a = cos Z' sin fl
cos a cos t = cos z' cos 99'
cos a sin t = sin z',
by means of which the formula for sin c is transformed into:
sin c = - sin b cos z' cos (9-') -+ cos  cos I COS z! sin (b  - osi')
-- cos b sin k sin z',
so that we obtain:
sin c see z'    tang b  tang k tang a'
tang (T - T')                    - cos    s k  cos (T +
cos b cos kk cos       ( — ) cos
We see from  this formula, that it is best to observe
stars which pass as nearly as possible by the zenith, because
in that case, even if k is not very accurately known, we can
obtain  a good result for the latitude.   And  observing the
star on the east and west side in the two different positions
of the instrument, we can combine the observations so, that
the errors of the instrument are entirely eliminated. For the
above formula is true when the circle is North and the star
West. For the other cases we find the formulae in the same
way as before, taking z negative when the star is East, and
we have:
sin c sec z'   tang' b  tang k tangz' i Circle North
tang (S~-9')cos b csCos kc cos (9-99)   co S  cos (9-99')   Star East 
sin c see z'   tang b  tang ktangz' iCircle Southl
tang (T —9')  -                     ____ —-- c os_ b co kos
cos b cos c cos (-')  cos k    cos (-)   Star West
sin c sec Z'   tang b  tang k tangz  Circle South
tane (T —T9')                tnancet
tcos b) cos i Cos (-')  cos  cos  ( —s')  Star East
Therefore when we reverse the instrument between the
observations, and compute rp -  p' from  each observation, the
arithmetical mean  is free from  all errors of the instrument
except the level-error.  If we cannot observe the same star
east and west of the mleridian, we may observe one star east
and another star west of the meridian after the instrument
has been reversed.  If we choose two stars, whose zenith
distances on the prime vertical are nearly equal, at least a
large portion of the errors of the instrument will be eliminated, and the accuracy of the result for the latitude depends
then merely on the accuracy with which  p~' has been found.
But we have:
tang A
tang 9' = 
cos t




490
therefore  if we  write  the formula logarithmically and differentiate it, we have:
sin 2T
d 9-s   2 A d A+' sin 2 9' tang t dt.
From  this formula we see  again, that it is best to observe stars which pass over the prime vertical near the zenith.
For since we have:
tango z
tang t --,
cos 99
we see that the coefficient- of dt is eclual to sin cp' tang a', and
that it is very small for stars near the zenith, and since for
such  stars' is nearly  equal to (q, an error of the declination is at least non increased.
If the observations have been made on several wires, it
is not even necessary, to reduce thenm  to  the middle wire,
an operation which for this instrument is a little troublesome,
but we  can find a value  of the latitude by combining two
observations made east and west of the meridian, but on the
same wire ).
If we write the formula for tang (rp — (p') in this way:
sin c         tang b
s       in (c  sec Z' -t            cos (9-  9')- tang le tang z',
cos b cos ck      cos ki
then develop sin (f -  a'), and substitute for sin A,' and cos qf?
the values:
sin A sec z' and cos a cos t sec z'
and take cos (p i-') equal to unity, we obtain:
sin c
sil (- 6) =   Ssin )  cos    2 sin. 2 sin  t' - -
cos b cos k
tang b  s 
-F-   cos z - ta ngl sin zx.
When b, c and k are small quantities, we thus find the
following convenient formulae for determining the latitude by
stars near the zenith, writing c — f instead  of c:
t- -= sin T cos. 2 sin   2 t' -4f-f  b + c -    sin z [Circle North, Star West]
-+- b ~- c -+ kc sin z [Circle North, Star East]
- b - c - ic sin z [Circle South, Star West]
- b -   - k sin z [Circle South, Star East].
*) For when we observe on a lateral wire, whose distance is f, it is
the same as if we observe with an instrument whose error of collimation is
c — f.




491
With  the  prime  vertical instrument at the observatory
of Berlin the star f Draconis was observed in 1846 Sept. 10:
Circle North, Star East.
I       II         III         I V         V          VI        VII
19m 9s.0, 17h 10m48s.0, 5m 24s.0, 11116s.5, 16h 55i 6s.3
Circle South, Star West.
1-n 5s.0, 54m 59s.7, 501m47s.8, 17h451M 28s.0, 371 35s.0.
The inclination of the instrument was:
Circle North -+ t4". 64
Circle South --   3. 49.
Further was:
c -- 17h 26m 58S. 59
= —-520 25' 27". 77
/ t:     -  54s. 52,
and the wire-distances expressed in arc were:
I  12' 31". 16
Il   6 43.78
III   3 25.17
V   3 23.14
VI   6 34.21
VII  12 22.32.
Now  in order to compute  p -(, we must know  already
an approximate value of (,.  Assuming:
99 = 520 30' 16",
we have:
log sin T cos  -- 9. 684686,
and we obtain:
Circle North.
III       I V         V         VI          VII
t       8m1144s.1l  17n'5s.11 22Mn29s.11 26m1236S.61 32in46s.81
log2 sin- t2  2. 17552  2. 75807    2. 99648   3'. 14264   3.32351
sing cos82sint t2 1 12.48 4 37.18  7 59.92 11 11.94 16 59.07
f9- -     4 37.65 4 37.18  4 36.78  4 37.73  4 36.75,
and hence from  the mean:
99 -     4' 37". 22 +- 4". 64 +- c ~- k sin z.
Likewise we find  from  the observations made when the
circle was South:
99 - e   4' 53". 53 +- 3". 49 - c - k sin z,
therefore combining these two results, we find:
99 -'= 4'49". 44
9 52" 30' 17". 21
c -+ k sin z = -- 7". 58,




492
This method is the very best for determining the zenith
distance of a star near the zenith with great accuracy, and
it can therefore be used with great advantage to determine the
change of the zenith distance of a star on aecount of aberration, nutation and parallax, and hence to find the constants
of these corrections.  For this purpose is has been used by
Struve with the greatest success.  Since the level-error of
the instrument has a great influence upon the result, because
it remains in the result at its full amount, the instrument
used for such observations must be built so, that it can be
levelled with the greatest accuracy. The instrument built for
the Pulkova observatory according to Struve's directions is
therefore arranged so that the spirit-level remains always on
the axis, even when the instrument is being reversed, so
that any disturbance of the level, which can be produced by
its being placed on the axis, is avoided. When the level is
reversed on the axis and observed in each position, b and b'
are obtained; but it is only necessary to leave it in the same
position when the instrument is reversed, because the two
readings of the level give then immediately b  b', which
quantity alone is used for obtaining the value of p - (A.
A difficulty in making these observations arises from the
oblique motion of the stars with respect to the wires.  A
chronograph is therefore very useful in making these observations, since it is easier to observe the moment when a star
is bisected by the wire, than to estimate the decimal of a
second, at which a star passes over the wire.
If the constant of aberration,  that of nutation, or the
parallax of a star is to be determined by this method, such
stars must be selected, which are near the pole of the ecliptic, because for such the influence of these corrections upon
the declination is the greatest.
27.  The formulae by means of which the observations
on a lateral wire can be reduced to the middle wire, are
found in the same way as for the transit instrument. For
when we have observed on a lateral wire, whose distance is
f, it is the same as if we have observed with an instrument,
whose error of collimation is c   f.  Therefore we have the
equation:




493
sin (c -H-f)   - sin  sin n -+- cos  cos 2 COS (t' -_?),
where t' is the hour angle of the star at the time of the observation  on the lateral wire.  If we subtract from  this the
equation:
sin c = - sin S sin n -- cos S cos n cos (t - 72),
we obtain:
2 sin i.fcos [  A- c] = 2 cos S ns n sin  t') sin  (t - t') - 2n].
Now  since f is only a few  minutes, we can put f instead of the first mlember of the equation and thus we -find:
2 sin'(t - t') =                                           --  -
cos S sin i (t') t Cncos ns C os n -  cons  eCos (t — t) cosn sin  1m'
or if we substitdte for cos in cos m and cos z sin nm the expressions given in the preceding No., we find:
2 sin,' (t - t')
cos S sin R sin    it'  - vbota ng,  ic entang - (t) [ otan    k coang     t) osec ]
Therefore for reducing the observations on a lateral wire
to the middle wire we muust use instead of the wire distance
f the quantity:
-b cotang qT - c COtanLg  (tH.- t') COSC 
and then we have:') sin ll (t-t') -f 
cos asin T siln(+ t') t'
In order to solve this equation we ought to know already
t'.  But we have:
sin n. (t H- t')  sin it -   (t- t')].
If we take then for I (t   t') half the interval of time between
the passages over the lateral wire and over the middle wire,
the second member of the equation is known, and we can
compute t -  t'.  When  the value found differs much frolm
the assumed value, the computation must be repeated with
the new  value.  But this supposes that the value of f' has
been computed before.  Now  in the formula for this the term
b cotaug *p can always be neglected, because b will always
be very small, and likewise if k is small, and the star is not
too near the zenith, the term  dependent on k can also be
neglected, so that then simply f is used instead of f'.  But
when the star is near the zenith, the correction dependent
on k can become considerably large, if k is not very small.
For we have:            tang  cos 9- = tang z,




494
and since f is small, we also have approximately:
tang t' cos *p - tang z'
and hence:
tang I (t + t') cos 9  tang v- (z - z').
Therefore we can write instead of the factor of gk:
cotang Ts cotang +, (z -+ z'),
and thus we see, that the correction can be large, when the
star is near the zenith.
Instead of solving the equation
2 sin  (t - t') =
cos 2  sin r9 sin' (t - t')
by an indirect nlethod, we can develop it in a series.  For
we can write it in this way:
Cos t - -os t -
COS 6 sill T
and from  this we obtain according to formula (19) in No. 11
of the introduction:
-                -       t  C         t
t  -L iK. —-j (1~+3 c otang  tn) 
Cos b' sin 90 Sill tI os   cs sin't; sin t
cos 3 sin T sin t  - 3ong).
Now  when the instrunment is nearly adjusted, we have:
cos:  sin t = sin z,
and hence:
t' - - t- / --' C~til lg' Ibi - / -.-- 6 1+cotang t L -            - -f- 
6si n        Slz siin S
Since this formula  contains also the even powers of f,
we see, that wires, whichll are equally distant from the middle
wire on both sides of it, give different values of t' -  t. For
when f is negative, we have:
t'-t ~4ih-.   -  -I cotang t  i.      
5111 Z sln g E        sin z sin lP
+   6-~ [1  +- 3 cotan, t'2]  - —''"
6           ZI  ~~,,,~.,,,,,a l Csin z sin 9p
In  order to compute this series more conveniently, we
can  construct a table, from  which  we take the  quantities
sin pj sin z,   cotang t, and g (1 -- 3 eotang t2) with the argument )'.
But this series can be used only, when the star is far
from  the zenith, because if' the star is near the zenith these




495
terms of the series would not be sufficient and some higher
ternms would come into consideration.
In this case, when the zenith distance is small, the following method for computing t' can be used with advantage.
We had:
cos t' - cos t --
cos  silln 
If we subtract both members of the equation from unity
and also acid them  to it, we obtain, dividing the two resulting equations:
t' 2 sin  t-I  cos 6 sin 9 — f'
tan   t       s2
2 cos  0 t" cos o sin T +t9Now  since:
tang 8
cos t
tang p
we have:
I - cos t  2 sin t2= t i (- -)
and
I ~- cos t =2 cos   _  t2  Cs;
cos o sin Ep
therefore we get:,,'   sin (99 -  ) — f
tang  t2
-  sin (T 7q- ~) -q-./1'
and if f is negative:
-tng,,>sinl (cp-) A-f'
tang  t' 2 =sinl (p -  ) —'J"
The values of the wire-distances are determined by observing a star near the zenith on all the wires.  If we compute for each observation the quantity:
Sill 9' COS   2 sin + t2,
the differences of these quantities give ts the wire-distances,
because we have for stars near the zenith:'p -  -= sin 9p cos 3. 2 sin, t2  -f-  c +- b +- k sin z.
Thus in the  example of the preceding No. the following  wire-distances would be obtained from  the observations
made when the circle was N orth:
III —  3' 24". 70
V=  3 22.74
VI= —   6 34.76
V'I= — 12 21.89.
In  1838 Oct. 2  a Bootis was observed with the prime
vertical instrument at the Berlin observatory:




496
Circle South, Star West.
I     II    III       I V         V     VI       TVII
r Bootis 44s.7  8s.3  50s.2  19' 211 32S.2  13S.8  55s.4  ln 19s.2.
The wire-distances expressed in time were then:
-- 51. 639
11= 25. 814
III-= 12..lO
V-13. 305
VI= —  26.523
VII-52. 397;
moreover we hlave:
At  — 47.5, c= 1411 111 Gs.5,        -200( 1' 39",p = 52 30' 16"
The quantities b and k were so small, that it *was not
necessary  to  compute the reduced wire-distances f'.  Then
we have:
t - 411 55m1 3s.2 - 730 45' 4S".0, log cos 8 sin t sin -- = 9.85244
and                    log cotang - t = 9. 14552.
Now in order to compute the second term of the series,
we must express sin              in termls  of the radius, that is,
we must multiply it by 15, and divide it by 206265.  Then
we must square it, and in order to express the term  in seconds of time, we must multiply it by 206265 and divide by
15. Thus the factor of:
Lsin  cos s sill t
will be:                  206265  cotangz,
the logarithm  of the numerical factor being 5.00718. Likewise the coefficient of the second term, expressed in seconds
of time, will be:
- L02o0625T [t + 3 cotalng t2J.
But in this case this term  is already insignificant.  Now  if
we compute for instance the reduction for wire I, we have,
since f is negative:. f           _ -— 72s. 533
sin 9T cos  sin t1
~- 20665 *' cotang{t [cos' si t sin  -  0.053,
hence the reduction to the middle wire is:
I-= - 1' 12s. 48.




497
In the same way we find:
II — -36s. 25
III — - 17.71
V —- 18.69
VI= — + 37. 24
VII=-  73.54,
and hence the observations on the several wires reduced to
the middle wire are:
19h 2' g32s. 22
32.05
32.49
32. 20
32.49
32. 64
32.74
mean value 19h 2"1 32s. 40.
In order to give an example for the other method of
reduction, we will take the following observation of ct Persei:
Circle South, Star West.
I         II        III         IV            V
a Persei 4'11 26s. 0  2n 38s. 0  ln 43s. 0  51 Otn 49s. 2  591n 52.0
VI        VII
58' 55s. 2  5711" 2'. 0.
If we compute first:
sin ( -  )
tang    -.....
sin (qP  -c~)
taking:
-- 490 16' 26". 7
and? -520 30' 16". 0
we find:
t  260 58' 58". 88.
If we compute the reduction for the first wire, we have
f negative, and hence we must compute the formula:
tang - t'2 -    (
sin (90  3) -+-f
Now since
f_ 51s. 639 = 12' 54". 585,
or expressed in terms of the radius f  - 0.0037553, we find:
t'- 270~ 53' 6". 72,
hence:
t't —t 00 54' 7". 84
Oh 3-36s. 52.
32




498
Likewise we find for the other wires:
11- lm 49s. 05
IJI     53.48
V     56.85
VI  1 53.85
VII  3 46.77.
However for this star the series is used with greater
convenience, since the influence of the third term for wires
III and V amounts to nothing and for wires 1 and VII it is
only 0O. 12.
28.  It must still be shown, how  the errors of the instrument are determined by observations.
The inclination of the axis is always found by means
of a spirit-level.  The collimation-error can be determined
by observing stars near the zenith east and west of the meridian in the two different positions of the instrument. Or we
can obtain it by combining the observations of the same star
east and west of the meridian, made in the same position of
the instrument.  For we have, when the circle is North:
c       k.
O  7 T+t- LA t               [Star East]
sin z sin      sin ll 
c       kC'='- T}'-+At     -       -  [Star West],
sin z sin f    sin T
if we assume, that the times of' passage over the middle wire
have been corrected for the error of level. Hence we have:
c -sin T sinl  [ (0-  ) - 0) -  (T'- T)].
where the value of 1 (6' — -) is obtained by means of the
equation:
cos, (0-) - tang A
tang 9
or more accurately, taking 2 (' -      -) t, from the equation:
1    sin(99-P)
tang  t   sin (
sin (o t a)
In order that the errors of observation in T and T' may
have as little influence as possible on the determination of c,
we must select such stars which pass over the prime vertical
as near as possible to the zenith.
Adding the two equations for 0 and 0', we find:
-s sin 99 [  (T' + T) -+- at -   (0 + 0')],
2                    2




499
or since 2 ( e 4- 0') = a:
Ic  sin l  [ (T~ T') ~ At -- A].
For the determination of the azimuth k it is best to take
stars, which pass over the prime vertical at a considerable
distance from the zenith, because their transits can be observed  with  greater precision.  With the prime vertical instrument at the Berlin observatory the following observations
were made in 1838:
Circle South:
June 25 a Bootis West 19h 3m  is. 44
26 a Bootis East  9 12 54.49,
these times being the mean  of the  observations  on  seven
wires. On June 25 the level- error was b -- -+ 6".42 and
on June 26 b  -+- 7".98.  If we correct the times by addinog the  correction -1 t, we must add to the first
int~~~~   ~15 tang z si n'
observation -  08.26, and add to the second --- 0.32 so that
we obtain:
T — 191  311 is. 18
T'=  9 12  54.8t.
Hence we have:
(T~+-'7") = 14'- 71m 58s. 00,
and since:
At — - 20. 27 and a = 14h1 Sw 16. 48
we find:
k -- -- 1s. 42.
Note. Compare on the prime vertical instrument: Encke, Bemerkungen
iiber das Durchgangsinstrument von Ost nach West. Berliner astronomisches
Jahrbuch fiir 1843 pag. 300 etc.
VI. ALTITUDE INSTRUMENTS.
29.  The altitude instruments are either entire circles,
quadrants or sextants.  The entire  circle is fastened  to  a
horizontal axis attached to a vertical pillar.  By means of
a spirit-level placed upon the horizontal axis, the vertical
position of the pillar can be examined and corrected by means
322




500
of the three foot-screws.  The adjustment is perfect, when
the bubble of the level remains in the same position while
the pillar is turned  about its axis.  By reversing the level
upon the horizontal axis, the inclination of the latter is found,
which can also be corrected by adjusted screws so that the
circle is vertical.
The  horizontal  axis carries the divided  circle, which
turns at the same time with  the telescope, whilst the concentrie vernier circle is firmly attached to the pillar.  When
the circle is read by means of microscopes, the arm to which
the microscopes are fastened is firmly attached to the pillar
and furnished  with a spirit-level.  By  observing  a star in
two positions of the horizontal axis which differ 180", double
the zenith  distance is determined  in the same way as with
the altitude and azimuth instrument, and everything that was
said  about the observation of zenith distances with that instrument can be immediately applied to this one.
Since the telescope is fastened at one extremity of the
axis, this has the effect, that the error of' collimation is variable with the zenith distance, so that it can be assumed to
be of the form  c -+ a cos z.  With larger instruments of this
kind the  error of collimation in the horizontal position  of
the telescope can be determined by two collimators, and the
error in the vertical position  by  means of the collimating
eye-piece, as was shown in No. 22.  The difference of the
two values obtained gives the quantity a, which however will
always amount only to a few seconds, and hence have no
influence upon the determination of the zenith distances.
Note. The quadrant is similar to the above instrument, but instead of
an entire circle it has only an arc of a circle equal to a quadrant, round
the centre of which the telescope fastened to an alhidade is turning. When
such a quadrant is firmly attached to a vertical wall in the plane of the
meridian, it is called a mural quadrant.  These instruments are now antiquated, since the mural quadrants or mural circles have been replaced by
the meridian circle, and the portable quadrants by the altitude and azimnuth
instruments and by entire circles.
30.   The  most important altitude  instrument is  the
sextant,  or as it is  called  after the  inventor,  Hadley's




501
sextant ).  But this instrument is used not only for measuring altitudes, but for measuring the angle between two objects in any inclination to the horizon; and since it requires
no firm  mounting, but on the contrary the observations are
made, while the instrument is held in the hand, it is especially
useful for making observations at sea, as well for determining the time and the latitude by altitudes of the sun or of
stars, as for determining the longitude by lunar distances.
The sextant consists of a sector of a circle equal to about
one sixth of the entire circle, which is divided and about
the centre of which an alhidade is moving, carrying a planeglass reflector whose plane is perpendicular to the plane of
the sector and passing through its centre. Another smaller
reflector is placed in front of the telescope; its plane is likewise perpendicular to the plane of the sextant and parallel
to the line joining the centre of the divided arc with the
zero of the division.  The two reflectors are parallel when
the index of the albidade is moved to the zero of the division.  Of the small reflector only the lower half is covered
with tinfoil so that through the upper part rays of light from
an object can reach the object glass of the telescope. Now
when the alhidade is turned, until rays of light from another
object are reflected from  the large reflector to the small one
and from that to the object glass of the telescope, then the
images of the two objects are seen in the telescope; and
when the alhidade is turned until these images are coincident,
the angle between the two reflectors, and hence the angle
through which the alhidade has been turned from that position
in which the two reflectors were parallel, is half the angle
subtended at the eye by the line between those two objects.
First it is evident, that when the two reflectors are parallel, the direct ray of light and the ray which is reflected
twice are also parallel.  For if we follow the way of these
rays in the opposite direction, that is, if we consider them
as emanating from the eye of the observer, they will at first
*) In fact Newton is the inventor of this instrument, since after Hadley's
death a copy of the description in Newton's own hand-writing was found
among his papers. But Hadley first made the invention known.




502
coincide. Then one ray passes through the upper uncovered
part of the small reflector to the object A. If cc is the angle,
which the direction of the two rays makes with the small
reflector, then the other ray after being reflected makes the
same angle with it, and since the large reflector is parallel
to the small reflector, the angle of incidence and that of reflection for the large reflector are also equal to a.  HIence
this ray will also reach the object A, if this is at an infinitely great distance so that the distance of the two reflectors is as nothing compared to the distance of the object.
But when the angle between the large and the small
reflector is equal to ), the ray whose angle of reflection from
the small reflector is a, will make a different angle, which
we will denote by fi, with the large reflector.  But in the
triangle formed by the direction of the two reflectors and by
the direction of the reflected ray we have:
180o - x +y+-   180o
or:
The angle of reflection from the large reflector is then
f,i and the direction of this twice reflected ray will make
with the original direction of the ray emanating from the
eye an angle 3, which is equal to the angle subtended by
the line between the two objects, which are seen in the telescope.  But in the triangle formed by the direct ray, the
direction of the ray reflected from  the small reflector and
that of the twice reflected ray, we have:
1 80- - c 2 a  f+ 2p= 1800,
and hence we have:
- 2 a — 2f
or:
= 2y.
The angle between the two objects which are seen
coincident in the telescope is therefore equal to double the
angle, which the two reflectors make with each other and
which is obtained by the reading of the circle. Hence for
greater convenience the arc of measurement is divided into
half-degree spaces, which are numbered as whole degrees,
and thus the reading gives immediately the angle between
the two objects.




503
When altitudes are observed with the sextant, an artificial horizon, usually a mercury horizon, is used, and the
angle between the object and its image reflected from the
mercury is observed, which is double the altitude of the object. But at sea the altitudes of a heavenly body are observed by measuring its distance from the horizon of the sea.
In this case the altitude is measured too great, since
the sensible horizon on account of the elevation of the eye
above the surface of the water is depressed below the rational horizon and is therefore a small circle. It is formed by
the intersection of the surface of a cone, tangent to the surface of the earth and having its vertex at the eye of'the observer, with the sphere of the heavens, whilst the rational
horizon is the great circle in which a horizontal plane passing through the eye intersects the apparent sphere.  If we
denote the zenith distance of the sensible horizon by 900+ —c,
we easily see, that c is the angle at the centre of the earth
between the two radii, one passing through the plane of
observation, the other drawn through a point of the small
circle in which the surface of the cone is tangent to the earth.
Hence if a denotes the radius of the earth, h the elevation
of the eye above the surface of the water, we have:
a
COS ~ C  ----
a +- h
and hence: 2 sin e c2.,
a — h'
I 2h
or: c/ —
By means of this formula the angle c, which is called
the dip of the horizon, can be computed for any elevation
of the eye, and must then be subtracted from the observed
altitude.
31. We will now examine, what influence any errors
of the sextant have upon the observations made with it. If
we imagine the eye to be at the centre of a sphere, the plane
of the sextant will intersect this sphere in a great circle,
which shall be represented by BA C Fig. 19,




504
Fig. 19.
n  wmp
and which at the same time represents the plane in which
the two objects are situated. Let 0 A be the line of vision
towards the object A. When this ray falls upon the small
reflector (which is also called the horizon-glass) it is reflected
to the large reflector, and if p is the pole of the small reflector, that is, the point in which a line perpendicular to
its centre intersects the great circle, the ray after being reflected will intersect the great circle in the point B so that
Bp-pA.
Further if P is the pole of the large reflector (which is also
called the index-glass) the ray after being reflected twice
will intersect the great circle in the point C so that
PC — PB
and in this direction the second observed object will lie. The
angle between the two objects is then measured by A C, the
angle between the two reflectors by pP, and it is again easily
seen that AC is equal to 2pP.
This is the case, if the line of collimation of the telescope is parallel to the plane of the sextant, and both reflectors are perpendicular to this plane. We will now suppose,
that the inclination of the line of collimation to the plane of
the sextant is i. If then B A C represents again the great
circle in which the plane of the sextant intersects the sphere,
the line of collimation will not intersect the sphere in the
point A but in A', the arc AA' being perpendicular to BA C
and equal to i. After the reflexion from the small and the
large reflector the ray will intersect the sphere in the points
B' and C', the arcs BB' and CC' being likewise equal to i
and perpendicular to B A C. If the pole of the great circle
BAC is Q, then the angle QAC is the angle given by the
reading of the sextant, whilst the arc A'C' is equal to the
angle between the two observed objects, and denoting the
first by a, the other by cc', we have in the spherical triangle
A'Q C';




505
cos a'  sin i2 -- cos i cos a
cos a -+ 2 i2 sin  a2,
and hence according to the formula (19) of the introduction:
at = a - i2 tang - a.
Therefore when the telescope is inclined to the plane
of the sextant, all measured angles will be too great. The
amount of the error can be easily found. For in the telescope of the sextant there are two parallel wires, which are
also parallel to the plane of the sextant, and the line from
the centre of the object glass to a point half way between
these wires is taken as the line of collimation.  Now  if
the images of two objects are made coincident near one of
these wires and the sextant is turned so that the images are
seen near the other wire, then the images must still be coincident, if the line of collimation is parallel to the plane of
the sextant, because each time the line of vision was in the
same inclination to the plane of the sextant. But if the two
images are not coincident in the second position of the sextant, it indicates, that the line of collimation is inclined to
the plane of the sextant. Now let the two readings, when
the images are made coincident near each wire, be s and s',
the inclination of the telescope i, the distance of the two
wires a, and the true distance of the objects b, then we
have in one case:
s= b + (-  )tang  s,
and in the other case:
s=b~+(t   +i) tang'so;
therefore putting:
tang 2 s'  tang I s
we have:
i2 —    cotang  s.
It is easily seen that the smaller angle corresponds to that
wire which is nearest to the plane of the sextant, and that a
line parallel to the plane of the sextant would pass through
a point whose distance from  this wire is equal to   -i.
A third wire must then be placed at this distance, and all
observations must be made near it, or, if they are made




506
midways between  the two original wires, the correction
- i2 tang' s must be applied to all measured angles.
It is necessary, that the plane of the horizon-glass be
parallel to that of the index-glass, when the index of the
vernier is at the zero of the scale, and that these two reflectors
be perpendicular to the plane of the sextant. It is easy to
examine whether the first condition is fulfilled, and if there is
any error, it can be easily corrected. For the horizon-glass
has two adjusting screws. One is on the back-side of the
reflector, which by means of it is turned round an axis perpendicular to the plane of the sextant, the other screw serves
to render the plane of the reflector perpendicular to the plane
of the sextant. Now when the index of the vernier is nearly
at the zero of the scale, the telescope is directed to an object at an infinitely great distance, and the direct and reflected images are made coincident. If this is possible, the
two reflectors are parallel and the reading of the circle is
then the index error.  But if it is impossible to make the
two images coincident, and they pass by each other when
the alhidade is turned, it shows, that the planes of the two
reflectors are not parallel. If the images are then placed so
thfiat their distance is as little as possible, then the lines of
intersection of the two reflectors with the plane of the sextant are parallel, and then by means of the second of the
screws mentioned before the horizon-glass can be turned until
the two images coincide and the two glasses are parallel.
The reading in this position is the index error, which must
be subtracted from all readings, in order to find the true
angles between the observed objects.  In order to correct
this error, the alhidade is turned until the index is exactly
at the zero of the scale and then the images of an object
at an infinitely great distance are made coincident by turning
the horizon-glass by means of the screw on its back. Usually
however this error is not corrected, but its amount is determined and subtracted from all readings. For this observation
the sun is mostly used, the reflected image being brought in
contact first with one limb of the direct image and then with
the other.  If the reading the first time is a, the second
at.m         b           b-c-a    a-b.
time b, then   --- is the index-error, and     or -   is the
2                           2        2




507
diameter of the sun, accordingly as a is less or greater than b.
One of these readings will be on the arc of excess, and therefore be an angle in the fourth quadrant; but the readings
on the arc of excess may also be reckoned from the zero
and must then be taken negative.
For observing the sun colored glasses are used to qualify
its light.  When these are not plane glasses, the value of
the index-error found by the sun is wrong. When afterwards
altitudes of the sun are taken, this error has no influence,
as long as the same colored glasses are employed which were
used for finding the index error. But when other observations are made, for instance when lunar distances are taken, the
index-error must be found by a star or by a terrestrial object.
But when a terrestrial object is observed, whose distance
is not infinitely great compared to the distance between the
two reflectors, the index-error c as'found by these observations must be corrected, in order to obtain the true indexerror c0, which would have been found by an object at an
infinitely great distance.  For if A denotes the distance of
the object from the horizon-glass, f the distance between the
two reflectors, f the angle which the line of collimation of
the telescope makes with a line perpendicular to the horizonglass, then we find the angle c, which the direct and the
twice reflected ray make at the object, when the two images
are coincident, from the equation:
f sin 2/5
tang c ---
and hence we have:
c= f sin 2 -  f sin 4 S,
where the second member of the equation must be multiplied
by 206265, in order to find c in seconds. Now if the two
reflectors had been parallel, the ray reflected from the indexglass would have met an object whose distance from the observed object is c, and the true index-error would have been
obtained, if these two objects had been made coincident.
Therefore if the reading was cl, when the object and its
reflected image were coincident, we have:
Co = c 1 -- sin 2  -1A 2 sin 4 /.




508
The angle ji, which was used already before, can be
easily determined, if the sextant is fastened to a stand, and
the index-error cl is found by means of a terrestrial object.
If we then direct a telescope furnished with a wire-cross
to the index-glass, make the wire-cross coincident with the
reflected image of the object, and then measure with the sextant the angle between the object and the wire-cross of the
telescope, we have:.s -o = 2 --  sin 2,
and since:
co = c1 + -  sin 2 S,
we obtain:
2  = s - c.
If the inclination of the horizon-glass to the plane of
the sextant is i, its pole will be at p' (Fig. 20)> the arc pp'
being equal to i and perpendicular to BAC.
Fig.'0.
C
The ray after being reflected from the horizon-glass intersects the sphere in B' and after its reflexion from  the index-glass in C'.  In this case again A C is the angle cr obtained by the reading, while AC' is really the angle a', which
is measured. We have then, as is easily seen:
BB' —- CC'- 2 cos fl. i,
where AS is, as before, the angle between the line of collimation of the telescope and a line perpendicular to the horizonglass, which is equal to Ap.  Moreover we have:
COS a'  COs a cos CC'
cos a- 2 cos A'S i- cos a,
and according to the formula (19) of the introduction:
2 cos 2 i2
CC - a +,_
tang a
If the inclination of the index-glass to the plane of the
sextant were i, and the horizon-glass were parallel to it and
the telescope perpendicular to both, then p', P', A' and likewise B' and C' would lie on a small circle, whose distance




509
from the great circle BA C would be equal to i. Then p'P'
or the.angle   ae' between the two reflectors would be, as in
the former case, when the inclination of the telescope was
equal to i:
a - i2        tang i a,
or:
a'- a - 2 i2 tang + a.
For correcting  this error two metal pieces are used,
which when placed on the sextant, are perpendicular to its
plane. One of these pieces has a small round hole, and the
other piece is cut out and a fine silver-wire is stretched
across the opening so that it is at the same height as the
centre of the hole, when the two pieces are placed on the
sextant.  For correcting the error the sextant is laid horizontal and the piece with the hole is'placed in front of the
index-glass which is turned,  until the image of the piece is
seen through the pole. Then the other piece is likewise placed
before the index-glass so, that the wire is also seen through
the hole. If then the wire passes exactly through the centre
of the reflected ima.,ge of the hole, the index-glass is perpendicular to the plane of the sextant, because then the hole,
its reflected image and the wire lie in a straight line, which
on account of the equal height of the wire and the hole is
parallel to the plane of the sextant. If this is not the case,
the position of the index-glass must be changed by means
of the correcting screws, until the above condition is fulfilled.
The same can be accomplished in this way, though perhtaps not as accurately: If we hold the instrument horizontally with the index-glass towards the eye, and then look
into this glass so that we see the circular are of the sextant as well direct as reflected by it, then, if the index-glass
is perpendicular, the are will appear continuous, and if it
appears broken, the position of the glass must be altered
until this is the case.
It may also be the case, that the two surfaces of the
plane-glas reflectors, which ought to be parallel, make a small
angle with each other so that the reflectors have the form
of prisms.  Let then A B  (Fig. 21) be the ray striking the




510
Fig. 21.        front surface  of the index-glass,
X   a              which will be refracted towards C.
/  / F  After its reflection from  the back
surface it will be refracted at the
front surface and leave this surface in the direction DE.  When
the two surfaces are parallel, the
/F  ~. angle ABF will be equal to G D E,
eC             Aw  but this will not be the case, when
the surfaces are inclined to each other.  Now  if we take
M NP-= -, and denote the angles of incidence A BF and GD E
by a and b, and the angles of refraction by a, and bl, we
have:
a, 4- a - 90  - + 
bl -~ a  900 -,
and hence:
b1 =a1 -2.
Now if' -  is the refractive index for the passage from
m
atmospheric air into glass, we have also:
sin a1 — sin a, sin bI  - sin b;
and hence:
sin a - sin b  -- [sin ca, — sin a, cos 2 -+- cos al sin 2 5]
or:
- b   a2cs —  2  /- see a2 — tang a2
n     cos a       n2...2....see a' -k- 1.
2  2
Now a is the angle, which the line from the eye to the
second object makes with the line perpendicular to the index-glass.  If we denote by if the angle, which the line of
collimation of the telescope makes with the line perpendicular
to the horizon-glass, and by-y the angle between the two
objects, then we have:
a- { (y + a)
and hence:
a - b —2S        - sece (,j    - 12
Now the correction which must be applied to the angle;'
is the difference of the above value and that for  -= 0, be



cause the index-error is also found wrong, when the two
surfaces of the glasses are not parallel. Therefore if we denote this correction by x, we have:
— 2 2P.+   22 2
-x  2 a     2  see  -                1 -2          + 
and we must add this correction, if the side of the glass
towards the direct ray is the thicker one, because then the
reflected ray is less inclined to the line perpendicular to the
glass than the direct ray, and hence the angle read off is
too small.  If the side towards the direct ray is the thinner
one, the correction must be subtracted.
The formula for x can be written more simply thus:
82-   se     Y 1- 2.sin       -Sec se e 1     sine P
71 6 jc2'                 2 2t
or since   is nearly equal to;:
x=-3a tsecs ax-   /i -Y             seeQ  /Il --  si h
Now  in order to find x, we measure after having determrined the index-error the distance of two well defined
objects, for instance, of two fixed stars, which must however
be over 100~.  Then we take the index-glass out of its setting, put it back in the reversed position and determine the
index-error and the same distance a second time. If then A
be the true distance of the stars, we find the second time
A - X  SX
if the first observation gave:
A - ---   S
and hence we have:
b see       -s 
Since rays coining from  the index-glass strike the horizon-glass always at the same angle, it follows, that the error
arising from a prismatic form of this glass is the same for
all positions of the index-glass and hence it has no effect
upon the measured distances.
Finally the sextant may have an excentricity, the centre
on which the alhidade turns being different from that of the




512
graduation.  This error must be determined by measuring
known angles between two objects.  If the angle is c and
the reading of the circle gives s, we have according to No. 6
of this section:
e
a  s  - sin (s- 0) 206265,
or:
- s    - cos  0. sin s - --- sin 2 0 cos s 206265.
Therefore if we measure two such angles, we can find
os 1 0 and -e ill     and hence e and O, and then every
reading must be corrected by the quantity:
+- sin -2 (s- 0) 206265,
Since the error of excentricity is entirely eliminated with
an entire circle, when the readings are made by means of
two verniers which are diametrically opposite, reflecting circles
are for this reason preferable to sextants.  Especially convenient are those invented by Pistor & Martins in Berlin, which
instead of the horizon'glass have a glass-prism. They have
the advantage, that any angles from  0" to 180" can be measured with them.  All that has been said about the sextant
can be immediately applied to these instruments.
NVote. Compare: Encke, Ueber den Spiegelsextanten. Berliner astron.
Jahrbuch fiir 1830.
VII. INSTRUMENTS, WHICH SERVE FOR MEASURING THE RELATIVE
PLACE OF TWO HEAVENLY BODIES NEAR EACH OTHER.
(MICROMETER AND HIELIOMETER).
32.  Filar mnicrodmeter.  For the purpose of measuring
the differences of right ascension and declination of stars,
which are near each other, equatoreals are furnished with a
filar micrometer, which consists of a system  of several parallel wires and one or more normal wires. This system of
wires can be turned about the axis of the telescope so that
the parallel wires can be placed parallel to the diurnal motion of the stars, and this is accomplished, when these wires




513
are turned so that an equatoreal star does not leave the
wire while it is moving through the field of the telescope.
In this position the normal wire represents a declination circle.
Therefore when a known and an unknown star pass through
the field, and the times of transit over this wire are observed,
the difference of these two times is equal to the difference
of the right ascensions of the two stars. In order to measure also the difference of the declinations, the micrometer
is furnished with a moveable wire, which is also parallel to
the diurnal motion of the stars, and which can be moved by
means of a screw so that it is always perpendicular to the
normal wire. The number of entire revolutions of the screw
can be read on a scale, and the parts of one revolution on
the graduated screw-head.  Therefore if the equivalent in
arc of one revolution is known, and the screw is regularly
cut or its irregularities are determined by the methods given
in No. 9 of this section, we can always find, through what
arc of a great circle the wire has been moved by means of
the screw.  Hence if we let a star run through the field
along one of the parallel wires and move the moveable wire,
until it bisects the other star, and then make it coincident
with the wire on which the first star was moving, then the
difference of the readings in these two positions of the moveable wire will be equal to the difference of the declinations
of the two stars. In case that one of the bodies has a proper motion, the difference of the right ascensions belongs to
the time, at which the moveable body crossed the normal
wire, and the difference of the declinations to that time, at
which the moveable body was placed on one of the parallel
wires or bisected by the moveable wire.
The coincidence of the wires is observed so, that the
moveable wire is placed very near the other wire first on
one side and then on the other; it is then equal to the arithmetical mean of the readings in the two positions of the
wire. If this observation is made not only in the middle
of the field, but also on each side near the edge, and the values obtained are the same, it shows, that the moveable wire
is parallel to the others.
The equivalent of one revolution of the screw in sec33




514
onds of are is found in the same way that the wire-distances
of a transit instrument are determined.  The micrometer is
turned so that the normal wire is parallel to the diurnal motion of the stars, and then the times of transit of the polestar over the parallel wires are observed, since these now
represent declination circles. Thus the distances between the
wires are found in seconds of arc, and since they are also
found expressed in revolutions of the screw, if the coincidence
of the moveable wire with each of the parallel wires is observed, the equivalent of one revolution of the screw in seconds of are is easily. deduced.  This method is especially
accurate, when a chronograph is used for these observations.
Another method is that by measuring the distance between the threads of the screw, and the focal length of the
telescope, because if the first is denoted' by mn, the other by
f, we find one revolution of the screw expressed in seconds:
r = —-  206265.
f
We can also find by Gauss's method the distances between
the parallel wires and then the same expressed in revolutions of the screw.  Finally we may measure any known
angle, for instance the distance between two known fixed
stars, by means of the screw; but in either case the accuracy
is limited, in the first by the accuracy with which angles
can be measured with the theodolite, and in the other by the
accuracy of the places of the stars.
Since the focal length of the telescope and likewise the
distance between the threads of the screw vary with the temperature, the equivalent of one revolution of the screw is not
the same for all temperatures.  Hence every determination
of it is true only for that temperature, at which it was made,
and when such determinations have been made at different
temperatures, we may assume r to be of the form:
c= a- b (t - to),
and then determine the values of a and b by means of the
method of least squares.
Usually such a micrometer is arranged so, that it serves
also for measuring the distances and the angles of position
of two objects, that is, the angle, which the great circle




515
joining the two objects makes with the declination circle. In
this case there is a graduated circle (called the position circle)
connected with it, by means of which the angles through
which the micrometer is turned about the axis of the telescope, can be determined.  The distance is then observed
in this way, that the micrometer is turned until the normal
wire bisects both objects, and then one of the objects is
placed on the middle wire while the other is bisected by the
moveable wire.  When afterwards the coincidence of the
wires is observed, the difference of the two readings of the
screw-head is equal to the distance between the two objects.
If another observation is made by placing now the second
object on the middle wire and bisecting the first object by
the moveable wire, then it is not necessary to determine
the coincidence of the wires, since one half of the difference
of the two readings is equal to the distance between the two
objects. If also the position-circle is read, first when the normal wire bisects the two objects, and then, when this wire is
parallel to the diurnal motion of the stars, the difference of
these two readings is the angle of position, but reckoned
from the parallel; however these angles are always reckoned
from the north part of the declination circle towards east
from  0" to 3600, and therefore 90" must be added to the
value found.
In order to make the centre of the micrometer coincident
with the centre of the position angle, we must direct the telescope to a distant object and turn the position circle 180".
If the object remains in the same position with respect to
the parallel wires, this condition is fulfilled; if not, the diaphragm holding the parallel wires must be moved by means
of a screw opposite the micrometer screw, until the error is
corrected. When this second screw is turned, of course the
coincidence of the wires is changed, and hence we must always be careful, that this screw is not touched during a
series of observations, for which the coincidence of the wires
is assumed to be constant.
In order to find from such observations of the distance
and the angle of position the difference of the right ascensions and the declinations of the two bodies, we must find
33*




516
the relations between these quantities.  But in the triangle
between the two stars and the pole of the equator the sides
are equal to A, 900-    and 90~0-  6', whilst the opposite
angles are a', 1800 — p' and p, where p and p' are the
two angles of position and A is the distance, and hence we
have according to the Gaussian formulae:
sill - A sin  (p +- p) = sin I (a' - a) cos t (' -+ 3)
sin    cos cos  ( ) =     - (a' - a) sin  ( -  )
cos 7 A sin -' (p' - p)   sin I (a' - a) sin  (3' — 3)
cos    s  (p' c os     (oa'- a) cos - (3'- ).
In case that cc' - a and 8' - -    are small quantities so
that we can take the are instead of the sines and 1 instead
of the cosines, A is also a small quantity, and since we can
take then p — p', we obtain:
cos - (3 + — 8) [a' - a]  A sinp
3' -!- Acosp.
For observing distances and angles of position it is requisite that the telescope be furnished with a clockwork, by
which it is turned so about the polar axis of the instrument,
that the heavenly body is always kept in the field. But if the
instrument has no clockwork or at least not a perfect one,
the micrometer in connection with a chronograph can still be
advantageously used for such observations, for instance, the
measurement of double stars, without the aid of the screw. For
this purpose the moveable wire is placed at a small, but arbitrary distance from the middle wire, and the position circle
is clamped likewise in an arbitrary position. The transit of the
star A is then observed over the first wire and that of the
star B over the second; let the interval of time be t. Then
the star B is observed on the first wire and the star A on
the second wire, and if the interval of time is t', and if A
denotes the distance between the two stars, p the angle of position, i the inclination of the wires to the parallel circle reckoned from  the west part of the parallel through north, which
is given by the position circle, we have:
aCcos. - (t - t') = A cos (p-i)
sin i
For, a is the arc of the parallel circle of A between A
and a great circle passing through B and making the angle i




517
with the parallel circle.  If we consider the arcs as straight
lines, we have a triangle, in which two sides are A and a,
whilst the opposite angles are i and 900 — p - i.  When
these observations are made in two different positions of the
position circle, we can find from the two values of a the
two unknown quantities A and p, and when the observations
have been made in more than two positions, each observation leads to an equation of the form:
0   Cos (P -    Od  cos (p  i)  dp  sin (p - i) 3600
sin i            sin             sill i  206265
and from  all these equations the values of d/A and dp can
be found by the method of least squares.
At the observatory at Ann Arbor the following observations of E Hydrae were made, where every a is the mean
of ten transits:
i --- 990 24'  500 24'    141~ 40'
ac- - 1".062   — 4".239    +- 2".382.
If we take p - 2070, A - 3".5, we obtain the equations:
0 — - 0".011  -0.306 dA  - 0.590 dp'
0 — +0".070  — 1.191 l     -0.315 dp'
0 —— 0".044  +0.668dA  -0.,089cp',
where p' — =      p.  From  these we find d    - +0". 056,
d p -- + 0". 208, and the residual errors are - 0". 040, - 0". 004
and -~- 0".024.
33. Besides this kind of filar micrometer others were
used formerly, which now however are antiquated and shall
be only briefly mentioned.
Fig.'2.               One is a micrometer, whose
|               wires make angles of 450 with
Av  a /A each other, Fig. 22. If one wire
is placed parallel to the diurnal
motion, we can find from  the
/   time in which a star moves from
A to B,  its distance from  the
centre, for we have:
t'- t
11TC    2   15 cos 6.
2
and since we have for another star:
d' C —  --  15 cos I',




518
the difference of the declinations of the two stars can be
found. The arithmetical mean of the times t and t' is the time
at which the star was on the declination circle CM; if'-+
2
is the same for the second star, the difference is equal to
the difference of the right ascensions.
Fig. 23.         A second micrometer is that invented
-0       by Bradley, whose wires form  a rhombus,
the length of one diagonal being one half
of that of the other, Fig. 23. The shorter
diagonal is placed parallel to the diurnal
motion. If then a star is observed on the
wires at A and B, MD will be equal to the
interval between the observations expressed
in arc and multiplied by cos S, so that:
\MD = 15 (t'- t) cos 3.
And if we have for another star:
Jf'D =- 15 (a' - T) cos 3'.
we easily find the difference of the declinations, whilst the difference of the right ascensions is found
in the same way as with the other micrometer.
Before these micrometers can be used, it must be examined,
whether the wires make the true angles with each other.
They have this inconvenience that the wires must be illurninated, so that they cannot be employed for observing any
very faint objects.  For this reason ring-micrometers are
preferable, since they do not require any illumination, and
besides can be executed with the greatest accuracy.
34. The ring-micrometer consists in a metallic ring,
turned with the-greatest accuracy, which is fastened on a
plane glass at the focus of the telescope, and hence is distinctly
seen in the field of the telescope. If the emersions as well
as the immersions of stars are observed, the arithmetical mean
of the two times is the time at which the star was on the
declination circle passing through the centre of the field.
Therefore the difference of the right ascensions is found in
the same way as with the other micrometers.  And since
the length of the chords can be obtained from the interval
of the times of emersion and immersions the difference of




519
the declinations can be found, if the radius of the ring is
known.
Let t and t' be the times of emersion and of immersion
of a star, whose  declination is A, and let r and r' be the
same for another star, whose declination is &', then we have:
a   G     (Z' + Z)- - (t +t).
If then iL and  t' denote half the chords which the stars
describe, we have:
15
-  (t' - t) cos,
and
15
M     (I'-I-) cos.
Putting:
sin o  -_
r
sin 9,
where r denotes the radius of the ring, we obtain, if we denote by D the declination of the centre of the ring:
S -  D -r cos?
8' - D = r cos 9',
and hence:' -8     [cos 9' -4- cos 99],
accordingly as the stars move through the field on different
sides or on the same side of the centre.
In 1848 April 11 Flora was observed at the observatory
at Bilk with a ring-micrometer, whose radius was 18'46".25.
The declination of Flora was
8' -- 240 5'. 4
and the place of the comparison star was:
a -- 91  12' 59". 01
= —24  1  9.01.
The observations were:
-- 11 1 6m 35s. 0 Sider. time t - llh 171" 53s. 0'-=  17 25.5              t'    19  46.5
I' —  - 508s.' —  t -  1ll 53s. 5.
We have therefore:
logz' —    1.70329       logt'-t 2.05500
log /u' 2. 53878        log   2. 89070
cosT' 9. 97850          cos   9.85941
5' —D   17' 51".9       a —D   13' 34". 8,




520
and since the two bodies passed  through  the field on the
same side of the centre, namely both north of it, we have:
S' -- 8   q+- 4' 17". 1.
The time at which the bodies were on the declination
circle of the centre were:, ('+c ~ )- 11)h 1117m Os. 25  (t'  t)   1l 18m 49s. 75.
Therefore at
1th 171n Os. 25
the difference of the right ascensions and declinations were:
a -a  -   lm 49s. 50'-  = - 4' 17". 1
- 27' 22". 50.
If the exterior edge of such a ring is turned as accurately circular as the other, we can observe the immersions
and  emersions on both edges.  However it is not necessary
in this case to reduce the observations made on each edge
with the radius pertaining to it, but the following shorter
method can be used.
Let It and r be the chord and the radius of the interior ring, and p' and r' the same for the exterior ring, then
we have:
1 cos a (t' -t)    - -r sin99
2 cos A (t' - t )-I'  r' sin 99',
hence:
it -+ it'- (a + b) sin 99 +- (a - b) sin 99'
and:,a- -'  (a +- b) sin T - (a - l) sin 99',
putting:
r - r'        r -r'
-— a and     = — b.
2             2
From  this we find:
g      q - I' 99     -4     99+' Q  9- 9'
lt. --  a sin      cos --   Cbcos+   s    sin   9
2            2        2             2        2
-   a cos 9  sin       - b sin     Cos2     as2             2             2        2
Adding and subtracting the two equations:
a - D   r cos 99
A'- D = r'COS 99'
we further obtain:
(a - b) cos I' - (a +- b) cos 9 -= 0,




521
or:
h+ -'- + i T - 9T
sin      sin
2        2
b —a.
cos — 2 — cosT 2 T
2        2
and
99+99'    99 9-    _  9999      -99
D-  — cos            COS       -   si sin     sin
2        2              2         2
therefore if we substitute the value of b in the expressions for:
Lt' -         -2-' and 8 — D
2       2
we find:
sin 
2
2     a.      t -'
2
- _,     sin -
a. 
and
2  P    99  2   992     /    22
3   D — a.  -
(99991)2
COS      COS -
2        2
cos p9 cos PI
-a   9t9T' TOS9'
c     -Cos - cos
2        2
Therefore if we put:
__    = sin A and   = —= sin B,    (A)
2_a                2a
we obtain:
cos A     --     
COS 2
2
and
cos  Cos'
cos B    -         - 
cos  2
hence:
- D -- a cos A cos B.         (B)
Hence fobr the computation of the distance of the chord
from  the  centre  of the ring  only  the  simple  formulae (A)
and (B) are required.




522
In  1850 June 24 a comet discovered  by Petersen  was
observed with a ring-micrometer at the observatory at Bilk
and compared with a star, whose apparent place was:
a - 2230 22' 41". 30    A = 590 7' 12". 19,
whilst the declination of the comet was assumed to be 59~ 20'.0.
The radius of the exterior ring was 11' 21".09, that of the
interior ring  9'26".29, hence we have:
a - 10' 23". 69.
The observations were as follows:
C. north of the centre                        Star south
Immersion*)    Emersion            Immersion             Emersion
1Sh151n 54s 20s   17m21s 48s     18m 55s 3  13s.0   2 1n20s. 5  37s.5.
With this we obtain:'-Xr  Exterior ring Il 54s        t' - t E. R. 2'11 42s. 2
interior ring  I   1                    2   7. 5
log of the sum 2. 24304                    2. 46195
log of the diff. 1. 72428                 1. 54033
cosA 9.92623                      9.65138
cosB 9. 99418                     9. 99749
9. 92041                   9. 64887
3' -    = — 8' 39".26            - D    -- 4 37". 88,
hence:
$' a- S - - 13' 17". 14,
and the difference of right ascension is found:
ca -a- - 3m 25s. 82 -- - 51' 27". 30.
35.  In  order to  see, how  the  observations  are to be
arranged in the most advantageous manner, we differentiate
the formulae:
r sin 9 —-/,  r sin q' =',  r cos,' =f r cos 9  — S.
Then we obtain:
sin T dr -+- r cos p dp = da
sin'd r + —I- r cos p'd' —   dAt'
[cos p' =F cos p] dr - r sin p'dp' -k- r sin q dp'  d (A'- 8)
or eliminating in the last equation dc ( and d p' by means of
the two first equations:
[cos 9 =!= cos TJ'] dr - sin T9' cos Tdui' =t= sin q cos T'doe
= cos T cos I'd (' -  );
*) For the immersion the first second belongs to the exterior, the second
to the interior ring. The reverse in the case for the emersion.




523
dMu and d1l' are the errors of half the observed intervals of
time. Now the observations made at different points of the
micrometer are not equally accurate, since near the centre
the immersion and emersion of the stars is more sudden than
near the edge. But the observations can always be arranged
so that they are made at similar places with respect to the
centre, and hence we may put d, = ddt' so that we obtain
the equation:
[cos T == cos s'] dr - sin [' == 9] d- = cos T cos p' d (' - ).
Therefore in order to find the difference of the declinations of two stars, we must arrange the observations so
that cos, cos c' is as nearly as possible equal to 1; hence
we must let the stars pass through the field as far as possible from the centre. If the stars are on the same parallel,
in which case the upper sign must be taken and we have
T =   Ic, then an error of r has no influence whatever upon
the determination of the declination. For finding the difference of right ascension as accurately as possible, it is evident, that the stars must pass as nearly as possible through
the centre, since there the immersions and emersions can be
observed best.
36. Frequently the body, whose place is to be determined by means of the ring-micrometer, changes its declination so rapidly that we cannot assume any more, that it
moves through 15" in one sidereal second, and that an arc
perpendicular to the direction of its motion is an are of a
declination circle. In this case we must apply a correction
to the place found simply by the method given before. If
we denote by d the distance of the chord from the centre,
we have:
d2        - (15 t cos 3)2,
where t -=1 (t' - t") is equal to half the interval of time
between the immersion and emersion. Now if we denote by Aa
the increase of the right ascension in one second of time, then
the correction At which we must apply to t on account of it
so that t —A t is half the interval of time which would have
been observed, if L a had been equal to zero, is;
At --— 1- t. Aa.




524
But we have:
152 t cos 62
Ad —-             At,
t2 COS 62 A a
hence:               Ad= 15.    d
or'since we have 15 t cos  -.:
d- — d  15    (A)
Further the tangent of the angle n, which the chord
described by the body makes with the parallel, is:
tang i2t 
(15 -A a) cos'
where AS  is the increase of the declination in one second
of time.
Therefore if we denote by x that portion of the chord
between the declination circle of the centre of the ring and
the arc drawn from  the centre perpendicularly to the chord,
we have:
x = ca tang n 
x =d ttang n  (15 -Aa) cos3'
and  since we must add to the time' -7-i  the  correction
2
x
- or:
cos a
dA      _
15 cos 2 - A a cos S2
we have, neglecting the product of AA and A a:
(B)
In the example given above the change of the right ascension in 241' was - 1015', and that of the declination was
-~10 17', hence we have:
logA a c8.71551 n
and
log a - 8.72694 n;
further we have:
log d-= 2.71538,  log ce  2.52468,
and with this we find:. (8- D) — 0".75 anda (-_a    ) -- - 7". 10.
The change of the right ascension is also taken into
3600 -- A' where 
account, if we multiply the chord by    3~0' where.'a




525
is the hourly change of the right ascension in time, and then
compute with this corrected chord the distance from  the
centre. But we have:
3600 - A'a    M. A'a
log
36;00        3600
where M is the modulus of the commion logarithms, that is,
0.4343. Now since this number is nearly equal 48 times
15 multiplied by 60 and divided by 100000, we have approximately:
MA'a  A' a. 48.15
3600   60. 100000'
therefore we must subtract from  the constant logarithm of
15 cosY'
r - as many units of the fifth decimal as the number of
minutes of arc, by which the right ascension changes in 48
hours.
In the above example the change of the right ascension
in 48 hours is equal to — 2' 30' — - 150', and since the con15 cos'
stant logarithm of 15 2  was 7.48667, we must now take
instead of it 7.48817, and we obtain:
2. 24304
1.72428
cos A 9. 92563
cosB 9. 99415
-- D - 8'38". 50.
37. Thus far we have supposed, that the path which
the body describes while it is passing through the field of
the ring, can be considered to be a straight line. But when
the stars are near the pole, this supposition is not allowable,
and hence we must apply a correction to the difference of
declination computed according to the formulae given before.
But the right ascension needs no correction, since also in
this case the arithmetical mean of the times of immersion
and emersion gives the time at which the body was on the
declination circle of the centre.
In the spherical triangle between the pole of the equator,
the centre of the ring and the point where the body enters
or quits the ring, we have, denoting half the interval of time
between the immersion and emersion by r:




526
cos r - sin D sin S +- cos D cos S cos 15 x,
or:
sinIr2  sin sin(D)"2 — os D cos Ssin
hence:
(3- D)'   - cos 826" (15X)2 - [cos D - cos 6] cos S (15 2)"
r2- COS 82 (15T)9 - (8 - D) sin S cos 3 (15z)2.
If we take the square root of both members and neglect the
higher powers of 6 - D, we have:
- D   [r2 -cos 8 (15 T)2]-_ (8 - D) sinos  (15 
2[r2 _ -COS 2 (15 T) -]'2
The first term is the difference of declination, which is
found, when the body is supposed to move in a  straight
line, the second term is the correction sought. We have
therefore:
a- D=d —  sin Scos (15T)",
where the second term must be divided by 206265, if we
wish to find the correction expressed in seconds.  For the
second star we have likewise:'-D   d' — D    sin X' cos 3' (15,    o,
and hence:
d' - a -- d' - d - 2 [tang A cos AS (15Z)2- tang 6' cos c'2 (15')"],
instead of which we can write without any appreciable error:
5'-   -- d' - d-+-  tang. (8-H- 8') [cos 82 (15r)2 - cos )'2 (15 r')"],
or since:
cOS 62 152 -2 2    r2  d2
and
cOS'2 15"'"=r" -r2   d",
also,'- 8 - d' — d -  tang  (' + ) (d' q- d) (d' - d).
Hence the correction which is to be applied to the difference of declination computed according to the formulae
of No. 34, is:
-+- I tang - (81 -- b') (d' -- d) (d' - d)
2tn(  +0) I V/ 206265
In 1850'May 30 Petersen's comet, whose declination was
740 9' was  compared  with  a  star, whose declination  was
73" 52'.5.  The computation of the formulae of No. 34 gave:
d    - 8' 56". 7,   d' - -+- 7' 36". 9.
With this we find:




527
log (d' -4- d) = 1. 90200,
log (d' — d)- 2. 99721
Compl log 206265 - 4. 68557
Compl log 2- 9. 69897
tang a (S' - 8)- 0. 54286
9. 82661,,
Correct. --   0". 67.
I-ence the corrected difference of declination was:
+- 16' 32". 93;
38.  For  determining  the value of the radius of the
ring, various methods can be used.
If we observe two stars, whose declination is known,
we have:
+- S' -- r [sin 9 +- sin I'] = 2 r sin 2 ( -t t+   ) cos 41 (Sv- ma')
u- 1U' - r [sin  - sin Tg] -2 r cos I (T -~- 9') sin I (T -9)
Further we have:
8'  A 8               8'
cos R - +cos  2 cos 2- (a + a') cos -2 ( -')and hence:
3   8   tang 2 (q9 qT),   --  tang I (P -').
Therefore if we put:
b /,-   = tang A and   --- tang B,
we obtain:
3' S
2 cos A cos B
2 sin A cos B
2 cos A sin B
sin (A - B)
sin (A + B)
The differential equation given  in No. 35  shows, that
the two stars must pass through the field on opposite  sides
of the centre and as near as possible to the edge, because
then the coefficient of dr is a maximum, being nearly equal
to  2, and the  coefficient of d t is very  small.  We must
select therefore such stars, whose difference of declination is
little less than the diameter of the ring.




528
The radius of the interior ring of the micrometer-at the
Bilk observatory was determined by means of the stars Asterope and Merope of the Pleiades, whose declinations are:
8 - 240 4' 24". 26
and'= — 23 28' 6". 85
and half the observed intervals of time were ):
18s.5 and 56s.2.
With this we find:
log (+e - -/') -2  71038
log (It -,')  2.41490
cos A -- 9. 98825
cos B    9. 99693
9.98518
r-18' 46". 5.
The  radius of the ring can also be determined by observing two stars near the pole, but in this case we cannot
use the -above formulae, since the chords of the stars are
not straight lines. But in the triangle between the pole, the
centre of the ring and the point, where the immersion or
emersion  takes place, we have, if we denote half the interval of time between  the two  momlents converted  into  arc,
for one star by r and for the other by T':
cos r - sin 8 sin D -+ cos 8 cos D cos 
cos r   sin 8' sin D  - cos 8' cos D cos z'.
If we write:
q- ~'   3- 3'                   3 +3'   3- 3'
--       instead of 8 and   2        2   instead of 8'
2        2                      2        2
and then subtract the two equations, we obtain:
3-38'.  Z —T'.    -'
tang D - cotang   2   sin      sin
2        2         2
8 -- 8' z''r'
+ tang      cos ----  cos —
2        2         2
Therefore if we put:
*) The stars of the Pleiades are especially convenient for these observations since it is always easy to find among them suitable stars for any ring.
Their places have been determined by Bessel with great accuracy and have
been published in the Astronomische Nachrichten No. 430 and in Bessel's
Astronomische Untersuchungen, Bd. I.




529
8 — i'  Tr -'
otang  2  sin  2 - a cos A
2        2
(A)
qS-'   T- T
tang      os  2    -- sin A,
2        2
we find D from the equation:
tang D — a sin L2  -  ~            (B)
When  thus D  has been found, we can compute r by
means of one of the following equations:
sin - r2 - sin - (8- D)2 - cos 8 cos D sin 2 z2
or
Si - r2- sin  (' D)2 2 — cos' cos D sin 4 712.
If we put here:
tangy-       fl'sin  J /cos  cosD,
sin  (  - D)
or:                                                     (C)
tang y' = -sin  r     cos' cos D,
sin.('-  - D)
we obtain:
sinll r2 - sin - (8 - D)2 secy
sin 4 ('- D)2 see y',
and
6-D   8'- D
cos y    cos               (D)
The solution of the problem is therefore contained in
the formulae (A), (B), (C) and (D).
When the radius of the ring is determined by one of
these methods, the declinations of the stars must be the apparent declinations affected with refraction.  But according
to No. 16 of this section the apparent declinations are, if the
stars are not very near the horizon:
+- 57" cotang (N-.+ 8)
and
a'+- 57" cotang (N-i- a'),
where
tang N — cotg 9q cos t,
and where t is the arithmetical mean of the hour angles of
the two stars.
Hence the difference of the apparent declinations of the
two stars is:
57" sin (' -  )
sin (N4+- 5) sin (N+ 85')
34




530
instead of which we may write:
57" sill (a'- 
sin [N-T +-  ( +  )]::
The  difference  of' declination  thus corrected must be
employed for comuputing the value of the radius of the ring.
These methods of determining the radius of the ring are
entirely  dependent on the declinations of the stars.  Therefore stars of the brighter class, whose places are very accurately  knowl,  ought to be chosen for these observations;
but it is desirable, to  use  also faint stars for determining
the radius of the  ring',  because the objects observed  with
a ring  micrometer are mostly faint, and it lmay be possible
that there is a constant difference between the observtatiols
of bright, a,l  d faint objects; therefore Peters of Clinton has
proposed  another method, by which the radius is found by
observinog  a star passing nearly through the centre  of the
field, and another, which describes only a very small chord
and whose difference  of' declination need not be very aceurately known.
W\Te find narm-ely froml the equation Ju -r sin (:
—,t -q- 2 r sin (-45 o  ) -
Now  if the star passes very nearly through the centre
of the ring, the second term, that is, the correction which
must be applied to a, is very small.  For finding its amount
the observation of the other star is used.  We have namely
according to the equations which where found in No. 38:'   -,r = tang A,   I - -- tang B:; = A -  3B.
Henlce we have:
r,u'- 2r sin [45~ -"i (A -t B)]2,
or because the last term  is very small:
r- =, [I -F 2 sin (450 — 1 (A1 + B))12
-C- [2 - sin (A q — B)].
Since suitable stars for this method can be found anywhere, it is best, to select stars near the meridianl and high
above the horizon so  that the refraction has no influence
upon the result.  In1 case that a cllronograph is used for t he
observations, this method is especially recommendable.




531
We can use also the method proposed by Gauss for
determining the radius of the ring by directing the telescope
of a theodolite to the telescope furnished with the ring micrometer and finding the diameter of the ring by immediate
measurement.
When  solar spots have been observed with the ring
micrometer, it is best to determine the radius of the ring
also by observations of the sun, because the immersions and
emersions of the limb of the sun are usually observed a little
differently from those of stars. For this purpose the exterior
and interior contacts of the limb of the sun with the ring
are employed.  Now  when the first limb of the sun is in
contact with the ring, the distance of the sun's centre from
that of the ring is R  - r, if R denotes the semi-diameter of
the sun and r that of the ring. If we assume the centre of
the sun to describe a straight line while passing through the
field, we have a right angled triangle, whose hypothenuse
is R / — r, whilst one side is equal to the difference of the
declination of the sun's centre and that of the ring, and
the other equal to half the interval of time between the exterior contacts, expressed in arc and multiplied by the cosine of the declination.  Therefore, denoting half this interval of time by t, we have the equation:
(R -- ) -  ( -) - (15 t cos )
For interior contacts we find a similar equation in which
t', i. e. half the interval of time between the interior contacts
occurs instead of t, and R - r instead of R —- r:
(R - r)2 _  (-_ D))2 + (15t' cos) a).
In these two equations the times t and t' must be expressed in apparent solar time in order to account for the
proper motion of the sun. If we eliminate now ( — D)2, we
obtain:
(R +  )" -- (R -)= -(15 cos 8)2 [t -t'2],
and
(15 cos a) [t +- t'] [t - t']
4 R
The sun was observed with one of the ring micrometers
at the Bilk observatory, when its declination was +- 230 14' 50"
and its semi-diameter 15'45".07, as follows:
34*




532
Exterior contact:               Interior contact:
Immersion lOh 31m Ss. 2 Sidereal time     10h 32m 30s. 8
Emersion     34m1 47s. 5                     33 25. 3.
From this we find half the intervals of time expressed
in sidereal time equal to 1'l 49s.65 and Om 27S.25, and these
must be multiplied by 0.99712, in order to be expressed in
apparent time, since the motion of the sun in 24 hours was
equal to 4n 8s. 7.  We have therefore:
t -109s. 33 and t'   27s. 17,
and we find:
r -- 9' 23". 52.
Note. It is evident, that the radius of the ring has the same value only
as long as its distance from the object glass is not changed.  Therefore,
when the radius has been determined by one of the above methods, we must
mark the position in which the tube containing the eye-piece was at the
time of the observation so that we can always place the ring micrometer at
the same distance from the object glass.
01n the ring micrometer compare the papers by Bessel in Zach's Monatliche Correspondenz Bd. 24 and 26.
39.  The Heliometer is a micrometer essentially different
from  those which have been  treated  so far.  It consists of
a telescope whose object glass is cut in two halves, each of
which can be moved by means of a micrometer screw  parallel to the  dividing plane or plane of section and perpendicularly to  the optical axis.  The entire number of revolutions which the screws make in moving the two semi-lenses
can be read on the scales attached to the slides which hold
tile lenses, and the parts of one revolution are obtained by
the readings  of the graduated heads of the screws.  Therefore if the equivalent of one revolution of the screw in seconds of arc is known, we can find the distance through
which the centres of the semi-lenses are moved with respect
to each other. When the semi-lenses are placed so that they
form one entire lens, that is, when their centres coincide,
we shall see in the telescope the image of any object, to
which it is directed, in the direction from the focus of the
lens to its centre.  If then we move one of the semi-lenses
through a certain number of revolutions of the screw, the
image, made by that semi-lens which is not moved, will
remain in the same position, but near it we shall see another




533
image made by the other semi-lens in the direction from its
focus to its centre.  Therefore if there is another object
in the direction from the centre of this semi-lens to the focus
of the fixed lens, then the image of the first object made
by this lens and that of the second object made by the semilens which was moved, will coincide, and the angular distance
between these two objects can be obtained from the number of revolutions of the screw, through which one of the
semi-lenses was moved.
In order that the plane of section may always pass
through the two observed objects, the frame-work supporting the two slides with the semi-lenses is arranged so, that
it can be turned around the optical axis of the telescope.
Therefore if the heliometer has a position circle whose readings indicate the position of the plane of section, then we
can measure with such an instrument angles of position. But
for this purpose it is requisite, that the telescope have a
parallactic mounting.
The eye-piece is also fastened on a slide, whose position is indicated by a scale, and this can likewise be turned
about the axis, and its position be obtained by the readings
of a small position circle whose division increases in the same
direction as that of the position circle of the object glass.
This arrangement serves to bring the focus of the eye-piece
always over the images of the object made by the semi-lenses.
For if one of them is moved so that its centre does not coincide with that of the other, its focus moves also from the
axis of the telescope, and hence the focus of the eye-piece
does not coincide with the image of an object made by this
semi-lens. Therefore in order to see it distinctly, we must
move the eye-piece just as far from the axis of the telescope
and in the right direction, so that its focus and the image
of the object coincide.
Now the plane of section will not pass exactly through
the centre of the position circle. We will call the reading
of the moveable slide +), when the distance of the optical
*) We will assume here, that only one of the slides is moved and that
the other always remains in a fixed position.




534
centre of the lens from  the centre of the circle is a minimum, the zero-point. It can easily be determined, if we find
that position, in which the image of an object seen in the
telescope does not change its place in the direction of the
plane of section, when the object glass is turned 1800. When
this position has been found, the index of the scale of the
slide can be moved so that it is exactly at the middle of'
the scale. In the same way we can find the zero-point of'
the eye-piece, and we will assume, that for this position the
readings of the three scales, namely those on the slides
of the two semi-lenses and that on the slide of the eyepiece, are the same and equal to h. Then the wire-cross
of the telescope must likewise be placed so that its distance
from the axis of revolution is a minimum, and this is accomplished by directing the telescope to a very distant object
and turning both position circles 1800. If the image remains
in the same position with respect to the point of intersection
of the wires, then this condition is fulfilled, but if' it changes its place, the wire-cross must be corrected by means of'
its adjusting screws.
We will assume, that when the image of an object made
by one of the selni-lenses is on the wire-cross, the reading
of the scale is s and that of the position circle, corrected for
the index-error, p; at the same time let the reading of the
scale of the eye-piece be 6, and that of its position circle t.
Let a be the distance of the zero-point from the centre of
the position circle, and t and Y the corrected readings of the
hour-circle and the declination-circle of the instrument; these
belong to that point of the heavens, towards which the axis
of the telescope is directed.  We will imagine then a rectangular system  of axes, the axis of ~ and V being in the
plane of the wire-cross so that the positive axis of' is directed to 0", and the positive axis of 1i directed to 900 of the
position circle, that is, to the east when the telescope is
turned to the zenith.  Finally let the positive axis of' be
perpendicular to the plane of the wire-cross and directed
towards the object glass.  If we put then:
s-  - e and a  h —,e)
and denote by I the focal length of the object glass expressed




535
in  units of the scale, and take a positive, if the zero-point
is on the side where ii is positive, and if the angle of position is either in the first or the fourth quadrant, then the
co - ordinates of the point s are:
e cosp - a sin p, e sinlp - a cosp, 
and those of the point a:
E cos 7t - C sinll r, E sinl  - a cos 7r, 0.
H-Jence the relative co-ordinates of s with respect to a
will be:
= e cos p - e cos tc - a [sin p - sin v]
r = e sin p -   sin p - i    a [cos p - cos ]    (a)
and if' celestial objects are observed, whose distance friom
the focus of the telescope is infinitely great compared to e,
we can assume, that these expressions are also those of' the
co-ordinates of the point s with respect to the focus.
The co-ordinates must now  be changed into such which
are referred to the plane of the equator and the meridian,
the positive axis of x being in -the plane of the meridian and
directed to the zero of the hour- angles, whilst the positive
axis of y is directed to 90~, and the positive axis of z is parailel to the axis of the heavens and directed to the north pole.
For this purpose we first imagine the axis of   to be
turned in the plane of $5  towards the axis of i through the
angle 90~ —; then the new  co-ordinates will be in the plane
of the equator, and we shall have:'  -- ~ sin 3 -  cos 8'= --  sill 6-  cos 3.
Then  we turn  the new  axis of 5' in the plane of ~'4
forwards through the angle 270"' —t, in order that it may
become the positive axis of y, and we obtain:
x   s-' cos t +- 7' sin t
y- =   sin t - 7' cos t
If we eliminate now n', i1', 5' we find:
x-   os 3 cost +   sin 3 cos t -  sin t
y-   cos 8 sin t tsin  sin t -   cos t
z -- 5 sin 3  -  cos 3,
or substituting  the values of  j, 7j,  ~ taken from  the equations (a):




536
x= I cos S cos t -+ [e cosp -e cos] sin 8 cos t -+ [e sinp --  sin ] sin t
- a [sin p -  sin a] sin S cos t + a[cosp -  cos a] sin t
y = I cos 3 sin t + [e cos p - E COs a] sin 3 sin t - [e sin p - e sin a] cos t
- a [sinp -  sin a] sin 8 sin t  a- a[cosp -  cos ] cos t
z   I sin3     - [e Cosp -s Cos ] cos 3    + a[sin p -  sin a] cos.
From  this we find the square of the distance r of the
point s from  the origin of the co-ordinates:
r = 12 -+ [e cosp -  cos  ]2 +  [e sinp -s sin a]2 +  4a2 sin  (p-  )2.
The line  drawn  from  the origin of the co-ordinates to
the point s makes then the following angles with the three
axes of co-ordinates:
x           Y               z
cos a-      cosi-  and cos yr'      r 
But if we  denote  by S' and t' the  declination and the
hour angle of the observed star, that is, of the point, in
which the line joining the wire-cross of the telescope and
the point s intersects the celestial sphere, we have also:
cos a - cos 3' cos t', cos  = -cos 3' sin t', cos y = sin',
therefore if we put:
e       E           a
-    D, -=z    and — d,
1       1           1
and also for the sake of brevity:
1 + [D cos p -   cos nC] 2 + [D sin p -   sin a1] 2 + 4 d2 sin i (p -) 2 _ A
we obtain:
cos Y' cos t'- cos 3 cos t -+ [D cos p - A cos 7] sin S cos t
I/A
[D sin p -A  sin a] sin t
/ A
d [sin p-sin 7] sin 8 cos t -d [os p - cos a] sin t
i/A
cos' sin t' -- cos 8 sin t + [D cos p - A cos ] sin S sin t
/A
[D sin p - A sin a] cos t                 (b)
i/A,  sin — [D cosp -A cos 7] cos 
1/d [sin p - sin ar] cos 3
1/A




537
Now we observe always two objects with the heliometer,
and since thus there will be also the image of another star
made by the second semi-lens on the wire-cross, we shall
have three similar equations, in which
3, 1, A, Tr, dl and p
remain the same, while instead of D, 3' and t' other quantities
referring to this star occur, which shall be denoted by D', ("
and t". We have thus six equations, which however really
correspond only to four, if we find the angles by tangents;
and all quantities occurring in the second members of these
equations will be obtained by the readings of the instrument,
namely 3 and t by the readings of the declination-circle and
the hour-circle, D  and A by the readings of the slides of
the object glass and the eye-piece, and p and T by the readings of the two position circles. Hence we can find by means
of these equations 3', t', 3'" and t". It is true, the instrument does not give the quantities S, t, A and t with the same
accuracy as the other quantities; but since the observed stars
are near each other so that the errors of those quantities
have the same influence upon the places of the two stars,
we shall find the differences 3Y" - 3' and t" - t' perfectly
accurate.
In case that the observed stars are near the pole, we
must find A", 3', t" and t' by means of the rigorous formulae
(b), but in most cases we can use formulae, which give immediately X"- 3' and c"-c a', although they are only approximately true. First we may take d equal to zero. If then we develop the divisor in the equation for sin 3' in a series, and
retain only the first terms, we find:
sin 3- sin 3' - [D cos p - \ cos:] cos S~ +  [D cos p -A cos ]'2 sin 3
- 2 [D sin p -A sin   sin sin,
or according to the formula (20) of the introduction, retaining only the squares of the quantities put in parenthesis:
3' --  = - [D cosp - A cos ] - I [D sinp - A sin ~j]  tang 3.
For the other star we find in the same way:
" —  = - -[D' os p - A cos ] -  [D' sin p - A sin I] 2 tang 3,
and hence we obtain:
3"-' [D - D'] cosp -+-  tang 3[(D -+- D') sinp - 2 A sin kr] [D - D'] sinp, (c)
an equation, by means of which the difference of the decli



538
nations of' the two  stars is found frolm  the readings of the
instrument.
In  order to find also the difference of the right ascensions we multiply the first of thle equations (b) by sin t, the
second by -  cos t and add them.  Then we get:
D sin p - D -   sin Tr
Cos' sill (t -    t')=     ---                           2
1/ -- [D cos p-,cos 2]  - [+Dsin p-  sin ] 2
and in a similar way:
______,D' sin p - A sin 7f
Cos P' sin (t-I ) --  -a)   = =
c/l(l)= D'cosp -A cos ]2 +2 [D' sinp- Asin2]2
If we neglect the squares of D, D' and A, and introduce
the right ascensions instead of the hour angles, these equations are chanlged into:
cos /' (a'  a)  D sinp --  sin 7s
cos Y3(a" (- a) -D' sin p- AL sin 21,
and if we write here instead of' Y' and 6":
I'1 ( + X") + I! A-A)
and  write' -- D" instead  of sinl ('" —  "),  and 1 instead of
cos (A'- "' )  we obtain:
(a' - a,) cOs l (3' -+ oS)   [D sin p2 - A sin A] [1 +-. tang  (" -')]
(al"-a) cos H (s' - A") =- [D'sin p - A sin n] [It +  tang A (A" -')],
and hence:
(a'I - a') cos l  ('-S")  (D'- D) silnp  - -tang S [' D —'] [D'-.]] sinp
- tang 8A/ sin f [S" -'],
and if we substitute instead of 0'" I6' the value found before
(D - D') cos p
we find:
i(a"-a) cos1 (A' +IA)  (D'-D) sinp
-   tang A [(DL'+ D) sinp - 2 A sin d] [D'-D] cosp. (d)
If now we put:
it - -  tang  [(D'q- D) sinp    2'sin ],      (A)
we can write in  the  equations (c) and (d) sin u instead of
the small quantity tu, and add in the first terms of the equations the factor cos iu.  Then we obtain:
"- 3'    - (D' - D) cos (p H-')B
a" - a' =- - (D'- D) sin (p  - s. ) se-e('+'D   (B)
We have assumned  thus far, that simply  the  distance
between the two stars has been measured, and that s is the
reading  of the  slide in that position, in which the images




539
made by the two semi-lenses coincide. But when we have
two objects a' and b' near each other, and we move one of
the semi-lenses, we see in the telescope two new images a'
and b', and we can make the images a and b' coincident.
Then if we turn the screw back beyond the point, at which
the centres of the semli-lenses coincide, we can make also
the images b and a' coincident, and the difference  of the
readings of the slide in those two positions will *be double
the distance.
When the observations have been made in this way, we
munst put  (D'- D) instead of D' — D in the above formulae.
Instead of the angle p -+ u, we obtain from the two observations now p' +-' and p'J- +", and hence we shall have:
PP      -, D' q-D = s-+s'- 2 h, i -- a  h
and
--  tang 8 [(s -+ s' - 2 h) sin p -2 (a- 2 h) sin,c]
A"     (2' -   ('  )) cos (p + -l)
a"  a — -  (D' - D) sin (p - ut) see- (-    \).
If we wish to find 6" —' and a" - a' expressed in seconds and u expressed in minutes, we must multiply    2
by the equivalent of one unit of the scale in seconds of are
206265
and the expression for i by 60265   Now  we can always
arrange the observations so, that we can neglect the term
dependent on p-rT, because we have
A0 -O, when a=  2- andl z -p.
Therefore we must place the eye-piece always, at least
approximately in the position, in which these conditions are
fulfilled, and this is the more necessary, since the images in
this position are seen the most distinctly.
We have assumed thus far, that the coincidence of the
images is observed exactly on the wire-cross. But unless
the stars are very near the pole, it is sufficient, to observe
the coincidence near the middle of the field.
40. If one of the bodies has a proper motion in right
ascension and declination, this must be taken into account
in reducing the observations. If we compute from each ob



540
served distance and the angle of position the differences of
the right ascensions and declinations of the two bodies, then
their arithmetical means will belong to the mean of the times
of observation, since it will be allowable to consider the motion in right ascension and declination to be proportional to
the time. However it is more convenient to calculate the difference of the right ascensions and declinations only once from
the arithmetical mean of all the observed distances and angles
of position. But since these do not change proportionally to
the time, their arithmetical mean will not correspond to the
arithmetical mean of the times of observation, and hence a
correction must be applied similar to that used in No. 5 of
the fifth section for reducing a number of observed zenith
distances to the mean of the times of observation.
Let t, t', t" etc. be the times of observation, and T their
arithmetical mean, and put:
- T  r,  - T=  t',r   - T= -", etc.
Further let p, p', p" etc. be the angles of position corresponding to those times, P that corresponding to the time T?',
and A a and A d' the change of the right ascension and declination in one second of time, assuming that r, r' etc. are
likewise expressed in seconds of time. Then we have:
dP          d2 P   2
p  P — d,  a. a a. z'   d- d  A aa. C1
dP         d2 P         d2 P
+j  5.z-:-    - r d2    ~.   a'    a,..
We shall have as many equations as angles of position
have been observed, and if n is the number of observations,
we obtain:
P  -p +P   __p....
n
2id~P2P dP2 P, da2      da d         2 dS 
where we can take:
2 2 sin ~ r2'_2
instead of -
n               n
if we have tables for these quantities.
Likewise we obtain from the observed distances the distance D corresponding to the arithmetical mean of the times:




541
d - d' - d" -....
D —
d d2D     d2D       3+'t     d2D   __ 2_
d2  M2+ da d              2a  d  S2    n
We must now  find the expressions for the differential
coefficients. But we have:
D sin P-(ao- a') cos 3
D cos P= 38- 3',
a a
or:                    tang P         cos 3
D2   (a - a') cos 82 +- (3 — 31)2,
and we easily find:
dP   cos 3cosP  dP       sin P  dD             dD
da       D'd    -  D   d         c         d 
d2 P     2 cos 2 sinl P cos P  d2 P   2 sin P cos P
da            2D2'   d32       D2
d" P    2 cos 6 sin P2   Cos 3
dad3        D2         D2
d2D   cos 82 cos p2  d2D   sin P2   d2 D     cos 3 sin P cos P
da2        D' d32     D'da.d                D
If we put:
Aa cos 3 - c sin y
3 — c cos 77
we obtain:
p   p  p" -... _ _ sin (P - y) cos (P -- ) C.2
n                  D               n
d_.   d'+d"-. - dsin (P - y)2 2'2
D-        --.       c
n                 D         n
or denoting by M the modulus of the common logarithms:
d - d'+ -    d"+... I  Msin (P-  )2 2 c22
n                             C2
log Dlog          n        -- lD2'         C
It is desirable to find the second term of P expressed
in minutes of arc, and the second term  of log D in units of
the fifth decimal. Therefore, if R is the equivalent of the
unit of the scale in seconds of arc, and if D is expressed in
units of the scale, and A a and A T  denote the changes of
the right ascension and declination in 24 hours, both expressed
in minutes of arc, we must multiply the second term in the
equation for P by
60   206265
864002   R 2
and the term  in the equation for D by:
100000. 602
864002. Re




542
But if we make use of the tables for 2 sin 2 -22 so that
we take:
p  - p' - A-'P...  2sin (P —) cos (P-)   2  sin~2
n                    D 2               n
and
d - d'  d" +...   Msin (P- y))2      2 sin a W
log D=log                                c2
we must multiply these terms respectively by
60. 2062652
S64002. 152. R2
and
100000. 602. 206265
864002. R2. 152
41. It is still to be shown, how the zero of the position circle and the value in arc corresponding to one unit
of' the scale can be determined.
The index of the position circle should be at the zero of
the limb, when the plane of section is perpendicular to the
declination axis.  Therefore, when the two semi-lenses have
been separated considerably, turn the frame of the object
glass so that the index of the position circle is at the zero,
and then make one image of an object coincident with the
point of intersection of the wires X).  If then also the other
image can be brought to this point merely by turning the
telescope round the declination-axis, the plane of section will
be parallel to the plane in which the telescope is moving,
and hence the collimation-error of the position circle will be
zero.  But if this should not be the case, then the object
glass must be turned a little, until both images of an object
pass over the point of intersection of the wires when the
telescope is moved about the declination-axis. Then the reading of' the position circle in this position is its error of collimation.
But this presupposes, that the slides move on a straight
line.  If this is not the case, the error of collimation will
be variable with the distance between the two images.
If the wire-cross is placed so, that an equatoreal star
during its passage through the field moves always on one of the
F) For this purpose it is convcnicnt to have double parallel wires, so
that the mliddlc of the field is indlicated by a small square.




543
wires, this must be parallel to the equator. If then the semilenses are separated, and the object-glass is turned about
the axis of the telescope until the two images of an object
move along this wire, then the reading of the position circle
ought to be 90" or 2700. But if it is in this position 90" - c
or 270" -c, then c is the error of collimation,  which must
be added to all readings.
The equivalent in arc of one unit of the scale can be
found by measuring the known diameter of an object, for
instance, that of the sun, or the distance between two stars,
whose places are accurately known.  For this purpose stars
of the Pleiades may be chosen, as their places have been observed by Bessel with the greatest accuracy.
The method proposed by Gauss can be used also for
this purpose.  For since the axes of the semi-lenses, even
when they are separated, are parallel, it follows, that if' we
direct a telescope, whose eye-piece is adjusted for objects
at an infinite distance, to the object-glass of a heliometer,
we see distinctly the doiuble image of the wire at its focus.
Therefore if one of tile semi-lenses is in that position, in
which the index is exactly at the middle of the scale, while
the other semi-lens is moved so that the index of its scale is
at a considerable distance from the middle, we measure the
distance between the two images of the wire by means of a
theodolite. Comparing then with this angular distance the difference of the readings of the two scales, we can easily find
the equivalent in arc of one unit of the scale. In case that
one of the semi-lenses has no micrometer, the observations
must be made in two different positions of that semi-lens
which is furnished with a graduated screw-head.
Let then S be the reading of the scale of the latter
semi-lens and SO the reading of the scale of' the other semilens which remains always in the same position, finally s
that of the scale of the eye-piece, then we have, if b and. c
are the angles, which straight lines drawn from the points
So and S to the focus make with the axis of the telescope:
(s - So) R - 20G265" tang b
(S - s) R -  206265" tang c,
where R is the value in arce of one unit, of the scale. ~Further




544
let a be the measured angular distance between the two
images of the wire, then we have
a = b ~ c.
If we eliminate b and c by means of the last equation,
we find the following equation of the second degree:
R2              R
(s - So) (S - s) tang a 2062652- (S - So) 206265   tang a,
from which we obtain:
R       (S- S) -  (S  So)   4 (s - SO) (S s)tanga2
206265                2 (s- So) (S- s) tang a
Let then S' be the reading of the scale in the second
position of the semi-lens, s' that of the scale of the eye-piece
and a' the observed angular distance between the two images,
then we shall obtain a similar equation for R, in which S', s'
and a' take the place of S, s and a. Now we can always
arrange the observations in such a way that:
S'-So =S0-S and s - So = So - s'
and then we find from the difference of the two equations:
R       (S' - S) - 1/S(-S)2 -+ 16 (s-So) (S -s) tang 2 (a-+a')2
206265               4 (s - So) (S - s) tang 2 (a + a')
When s -  SO and S - s have the same sign, and if
we put:
tang a  4 tang  (a        - So) (S - s),
Si
we find for R:
[seca - 1]
R - 206265     [se   
tang a /(s - So) (S - s)
tang 1 a
- 206265    tang
/(s- So) (S   S)
But when s -So and S - s have opposite signs, and if
we put:
tang -1 (a + a')
sin fi-_ 4 tang- 2   (s - SO) (S- s)
we find for R:
R   206265       -os
sin f 1/(s- So) (S - s)
tang  I
-- 206265    tang
1(s- So) (s- s)
When s -  S and s' - S', we obtain for R1 instead of
the equations of the second degree the following:




545
(S- S) 20265 — tang a
(So - S') 20    tang a',
hence:
22 tang t    (a -- a')
R -- 206265
S - S'
for which we can also write:
a' + a
S- S'
These formulae can be used also in case, that the diameter of the sun or the distance between two fixed stars is
observed. Then a and a' will be equal to the diameter of
the sun or to the distance between the two stars.
When the heliometer is furnished with a wire-cross, we
can also place one of the wires parallel to the equator and then,
after the two semi-lenses have been separated and turned so
that the two images of a star move along this wire, observe
the transits of the two images over the normal wires.
The' value in are of one revolution of the screw is variable with the temperature and hence it must be assumed
to be of the form:
R =- a - b (t - to).
Hence the value of R must be determined at different
temperatures and the values of a and b be deduced from
all these different determinations.
Note. Compare:
Hansen, Methode mit dem Fraunhofersceen Heliometer Beobachtungen
anzustellen.
and
Bessel, Theorie eines mit einem Heliometer versehenen Aequatoreals.
Astronomische Untersuchungen, Bd. I. K6nigsberger Beobaehtungen
Bd. 15.
VIII. METHODS OF CORRECTING OBSERVATIONS MADE BY MEANS
OF A MICROMETER FOR REFRACTION.
42. The observations made by means of a micrometer
give the differences of the apparent right ascensions and declinations of stars either immediately or so that they can be
35




546
computed from the results of observation. If the refraction were
the same for the two stars, the observed difference of the
apparent places would also be equal to the difference of the
true places.  But since the refraction varies with the altitude
of the objects, the observations made with a micrometer will
need a correction on this account.  Only in case that the
two stars are on the same parallel, there will be no correction, because then the observations are made at the same
point of the micrometer and hence at the same altitude *).
The common tables of refraction, for instance, those published in the Tabulae Regiomontanae give the refraction for
the normal state of the atmosphere (that is, for a certain
height of the barometer and thermometer) in the form:
a tang z,
where z denotes the apparent zenith distance and e is a factor variable with the zenith distance, which for
z-450 is equal to 57".682
and decreases when the zenith distance is increasing so that
for z  850 it is equal to 51"'.310.
By means of these tables others can be calculated, whose
argument is the true zenith distance 5 and by means of which
the refraction is found by the formula:
o - f tang 5,
where # is again a function of 5.  We have therefore:
- z +- fi tang'= -   +- fi' tang i',
hence:' -       -    -4-' I- z' tang i  - i'  tang  tang,
or denoting:' --  -- (z'- z) by A (z'- z)
also:
A (z' —  ) = fi' tang A'- f tang'.  (a)
This is the expression for the correction, which must be
applied to the observed difference of the apparent zenith distances in order to find the difference of the true zenith distances.
*) This remark is not true for micrometers with which distances and
angles of position are measured.




547
If we denote by'3, that value of 35, which corresponds to:
2  -t-o
and which is derived from  the equation:
eo - fo tang go,
we have:
i' tang S' -,, tang'   2-    o tang' (l'-  ) -..
1 dot        _
f tang  = fi, tang  -      t  tang   ( ) +. 
If we write in all terms of the second member, except
the first, tang,o instead of tang i and tang', the teirms containing the second differential coefficients will be the same,
and we have with a considerable degree of accuracy:
j3' tang 9' - f tang    f = lo [tang g' - tang g]
+ dio      tang;   - tang 206265.
do tang         seco 2
Therefore if we put:
dfo tang g 
cl,    d t_206265,
D +dg see go; 
we obtain by means of (a):
A (z' - z) = I- [tang g' - tang ~]
where k must be computed with the value:
o-' gi
and since we can take, neglecting the second power of ~' -:
tang g' - tang     --
Co0 so 2
we have:
A (' -- ) =  (b)
But this formula assumes that the difference of the true
zenith distances is given. If we introduce instead of it the
difference of the apparent zenith distances, we must multiply
the formula by  dlo and we find:
dzO
AX (z' ) — " = 1         ~
ldz   cos g0
or if we put now:
35 *




548
___            tandgo  tang    2 6
-d go   + 2  sin 2 go 206265,        (A)
we finally obtain:
g! - z
A (z' -) -  -.   C,              (B)
COS go 2
The following example will serve to show how accurately the difference of the true zenith distances can be found
from the difference of the apparent zenith distances by means
of this formula:
True zenith distance ~    Apparent zenith distance z    Refraction
870 20'               870 5' 27". 4          14' 32". 6
30                    14 54.8            15  5.2
40                    24 20.7                39.3
50                    33 44.5             16 15.5
88  0                     43  6.4               53.6.
From  this we obtain the following values of fi:
870 20'     40". 6427
30      39. 5209
40      38. 2727
50      36. 9073,
and from these we find by means of the formulae in No. 15
of the introduction the values of lo  that is, the variations
of rio corresponding to a change of so equal to one second:
870 30'    - 0". 0019750
40        0.0021767
50        0.0023967.
If we compute now  the values of k, we find, since the
logarithms of d; are:
870 30'    0. 0271
40     0. 0287
50     0. 0307,
the following values for the logarithms of k:
k
87 o 30'    6. 0505   350
40     6. 0155
50     5.9771
where k is expressed in parts of' the radius.




549
If we take now:
z = 870 10' and z' - 8 0 50',
and hence:
- z = 40',
we have by means of the common tables of refraction:
- 870 24' 47". 8'=-88 7 23.0,
hence:
5' - --- -+ 42' 35". 2
0  870 46' 5". 4.
If we suppose now  that z'- z and'0 are given, and
compute A (z' - z) by means of the formulae (A) and (B),
we find, since the value of log k corresponding to 5o is
5.9925:
A: (I'- z) - — 2' 35". 4,
hence:' --  =- + — 42'35". 4,
which is nearly the same value, which was obtained from
the tables of refraction.
The values of k may be taken from  tables whose argument is the zenith distance.  Such tables have been published in the third volume of the Astronomische Nachrichten
in Bessel's paper,,Ueber die Correction wegen der Strahlenbrechung  bei Micrometerbeobachtungen" and in his work
Astronomische Untersuchungen Bd. I. In the last mentioned
work there are also tables, which give the variations of k
for any change of the height of the thermometer and barometer.
For computing the difference of the true zenith distances;o itself must be known. But since the right ascensions
and declinations of the two stars are known, we can find
this quantity with sufficient accuracy, if we compute it from
the arithmetical mean of the right ascensions and declinations.  For this purpose the following formulae are the most
convenient, since it is also necessary, to know the parallactic
angle:
sin C sin 7 - cos 9 sin to
sin Cos c os 0 - cosio     sin sin   os P9 Cos to
cos   0 i- sin   9 sill  -+ cos8 0 cos 99 cos to.




550
Putting:
cos n = cos T sin t,
sin n sin N-= cos go cos to
sin n cos N   sin 99,
we have:
sill  sin l -l cos n
sin ~ cos nr  sin n cos (Ni - (o)
cos C  sin na sin (N -+ S),
or:
tang          a sin 7 = cotang na. cosec (N-t- o,)
tang 5 cos o? = cotang (N+- S0).
The quantities cotang n and N  can again be tabulated
for any place, the argument being t. In case that the tables,
mentioned in No. 7 of the first section, have been computed,
they  can also  be used for finding the zenith distance and
the  parallactic  angle.   The connection between  the above
formulae and those used for constructing the tables is easily
discovered.
43.  The difference of the true zenith distances having
been found from  that of the apparent zenith distances, the
difference of the true right ascensions and declinations of
two stars is also easily derived from the observed apparent
differences of these co-ordinates. For if ji tang 5 is the refraction for the zenith distance C,. tang  sin   is the refraction in right ascension
and
fi tang ~ cos'1 the refraction in declination.
But we have:
sin                           sin r7         sin si
f3' tangg  cos-  tang         k tang C' si    k tang C
d. tang Co sill           d tang o sin no
cos'So                   cos OYo
-   ('- o) -t k.          (a- a),
dao
and likewise we find:
f' tang C' cos d' — fA tang; cos A- kI   d tang I0 cos o  ( s)
d. tang So cos o (' - a),
where 5' -  S and c' -  ce denote the differences of the apparent right ascensions and declinations.




551
Differentiating the formulae for:
tang g sin  n
os  and tang; cos v
we obtain:
d. tang ~ sin 7
cos S       tang g' sin 7 cos -7 - tang;  sin Cq tang a
d          -                cos
d. tang; sin y
cos d
C- = 1- ttang g cos n tang o +- tang g' sin,r 2
dt
d. tang { cos     -tng;2 COS 2 +
d.* tang C cos g_ tang g2 cos?7 sin V cos  +- tang. sin r sin S,
dct
and these expressions being found we can now treat of the
several micrometers, whose theory was given in No. VII of
this section. But since those -mentioned in No. 33 are at
present entirely out of use, we will omit the corrections
for them.
44. The micrometer, by which the difference of right ascension is found from  the transits over wires perpendicular to
the parallel of the stars, whilst the difference of declination
is found  by direct measurement.   With  these  micrometers
refraction has an influence only at the moment when the two
stars pass over the same declination circle, and hence we need
only to consider the difference of refraction, dependent on
the difference of declination.
Therefore the correction of the apparent right ascension
and declination is for the first star:
tang C sin        --    tang co
cos            -
for the second:
tang g' sin,j',A a-= C-''                   - _ w  tang' sin'     tang' cos 7';
and hence we obtain by means of the formulae in No. 43:
d. tang go sin Go
A ( a'a)   - -k..-    cos S,
d50
A (('  A) -'   k. tang o Co 7o (,  )
( (6' -     d)                q - 8),
dlrd`




552
or substituting the values of the differential coefficients:
~A (a- a) — Ik (/   - I,) tang  2 sin  0 cos I, -tang 50 sin y tang o
A (        k'- A) - k ('-A) [tang  C  s 2 7o2 -  1].
These formulae receive a more convenient form  if we
introduce the auxiliary quantities cotang n and N. For, substituting the values given in No. 42 for;
tang;  sin l  and tang;  cos?7
we obtain:
A (a'- a)  IC (3' 3) cotang n cos.  N -2- 2S0)
sin (N-+- o,) cos 802
and
k (6' — 3)
sin (N+t- c')2
45.  The ring micrometer. If the refdaction were the same
during the passage of the stars through the field of the ring
micrometer, they would describe chords parallel to the equator
and it would only be necessary, to correct the observed differences of right ascension and declination for the difference
of refraction at the moment when the stars pass over the
declination circle of the centre of the ring. Therefore we
would have the same corrections as for the filar micrometer:
tang o2 2sin 7 o cos 7/o - tang go sin n7o tang 8,'A (al - a)         k -A)
cos 
and                                                      (a)
A (3' — 3) —=7k (Y'- - 8) [tang2 COS o 2 +- 1].
But since the refraction really changes while the stars
are passing through the field of the ring, it is the same, as
if the stars have a proper motion in right ascension and declination. Now if h and h' denote the variations of the right
ascension and declination of a star in one second of time,
we must add according to No. 36 of this section the following
correction to the differences of right ascension and declination computed from the observations:
6_ - D
COS 2
ens




553
where D is the declination of the centre of the ring and Pu
is half the chord. Since:
d. tang g sin y
cos S
dt
and
d. tang C cos yn
dt
we have:
tang  2 cos  sin7 z- tang g sin'q tang 8
a aic(-D)
Cos
and likewise for the other star:
tang'a2 cos ~' si l? -f'- itang s' sin n' tang s'
COS 8~
or if we write  in  both  equations  i,, m,, and /0 instead of
a', a;,   and  I',?i', Y', that is, if we neglect terms of the order
of k(8-  D)2, we obtain:
tang C o cos n/ sin nr + tango0 sin /o tang.
COS 3o
If we unite this with the first part of the correction,
which is given by the first of the equations (a), we find:
tang go  sin 2,i7,
A (ca' - a) - k (8' - 8)                   (A)
cos SO
Further we have:
2
1 - D
r_- ('_-D)2    -2,2 _) d
h —-..h.
8 —D            d
If we put'-  D =  d' and denote by ho the value of h
for the centre of the field, we have:
A  S  11 2 d2 - r2 -c12
i'           c i    ho
r2 (d- cl')    dcl' (cl- d')
dc'             d dI
hence:
(3' —         (- -- 8) r —
lA (S    )-   ($ (S-D) (' — D) [1 - tang So cos?7  tang 8o + tang 5o 2 sin o 2]
- k (' —8) [1 —tang g0 cos x?  tang,0 +- tang o" 2 sin o, 2],
and if we unite this with the first part of the correction,
given by the second of the equations (a), we find:




554
a (A'- ) -k (A'-) [tang o0 2 cos 2 7, q- tang go cos 0o tang 30]
72
(( —D) (S' —D)
X [1 - tang go0 2sin rno2 - tang go cos o tang o0]
for the expression of the complete correction of the difference
of declination. Here we can in most cases neglect the terms
multiplied by tang 0o and thus we obtain simply:
~A(a' —  )~ —- kI()    ~5\) tang 0o2 sin 297o
k (a' — a)   k (- 6cB)
cos 80
A ('  3) -  k (' —) tang g, 2 cos 2 7o                 (B)
W2
-c (' - 5) (c- D) (a'r   D) [tang go 2 sin n o - 1].
Example. In 1849 Sept. 9 the planet Metis was observed at Bilk and compared with a star, whose apparent
place was:
a - 221 11 59s. 63,    -  21. 43' 27".08.
The observations corresponding to 23h 23-' 19. 3 sidereal
time, were:
al — a - -  1 9s. 65 =        171 24". 75'  D ---  5' 17". 5,         - D   -         6'34".2
8'-  ---  it' 51".7 and we have r - 9' 26". 29.
Now if we compute ~ and qj with
to =h 201'145s  20~', o0 ----- 21 49'.4 and p   51~ 12'. 5
we obtain:
cotang n   9. 34516    N-370 1'. 9
and
and  =12~ 55'.3    =750 9'. 6.
From  the  tables for k we find for this zenith distance:
log k l-6. 4214,
and then the computation  of the corrections by means of the
formulae (B) is as follows:
log 7k  6. 4214         sin 2 70 9. 6394             0. 0667,,
log (3'- )-  2. 8523,,                0. 4273       cos 3o 9. 9677
tang   1. 1536            cos 2 170 9. 9542    Al(a'-a) — 1".25
0. 4273,, 1 term of A (A' —5)  -2".41
sin  - 28.6990
log (tang g2 sin s2 + 1) - 0. 2335
log r2   5. 5061
k (8' —8)   9. 2737,,
5.0133,,
log (3 —D) (A' —D)   5. 0975,,
II term of A (5'- 5)   +0". 82
A (a'- a)  - - 1". 25
A (8' -  ) --  3". 23.




v555
Hence the  corrected  differences of right ascension and
declination are:
a -- a  --  7' 23". 50' -  8= -        11' 54". 93.
46. The mnicrometer with which angles of position and distances are neasured.  If a'-  r  and ('-' -            denote the differences of right ascension;and declination affected with refraction,  and  a' -  a  and  dl'-  d  the  same differences freed'romn it, we have:
d. tang g sin'i
Cos S
cd3
dc tang  sin s7
cos 3
cd 
where  the  values of the differential coefficients ought to be
computed with the arithmetical means   2j +,               and 3
2  ~  2            2
VVe have therefore:
d. tang ~ sin?7
Cos 3
dl (c^' - a) = - k (c8'-O       ( 
ci3
d. tang 5 sin q7
cos 3
(a — k (a'- a)
dt
and likewise:
d. tankg'  5ic9L Y   - d. tang  cos s
d (8' — (')   k —            - -    + (' - 8)  d. t angcos    k (a' - -a)
d8                       dt
Substituting the values of the differential coefficients found
in No. 43, we get:
d (a'-a) -k ('-  ) tanllg  2sin y7 cos V - tang' sin n? tang 3
cos 3
- k (a'- a) [tang, 2 sin yr - tang  cos q1 tang S-  I]
d ('-3)    k (8' —8) [tang'2 cossy  2- 21]
+ k (c'- a) [tang 2 Cos V sin n cos 3 + tang 5 sin V sin 3].
But, if L  and  T denote  the  apparent distance and the
apparent angle of position, we have:
cos  (a'-a) = A sin g
and' — 3 = A cos a,
hence:
cos 3 (a' - a)
tang       -31
and              A  cos 3 (a'- a) sin t -4- (3'-) cos g,




556
If then A' and n' denote the true distance and the true
angle of position, we have:
cos Xt cos a d (a'- a)- sin 7c d (A'- A)
A' = A -t- sin n cos Sd (a' - a) -- cos g d (' — A).
If now  we substitute  here the values of d (a' —a) and
d(X' —) which  were found  before, and  introduce in them,  and n instead of a' —a  and 8 —A, we obtain:
IC'  7      -!- k tang 2 [sin 9s cos n os 7t COS 7 + Sill r- sin sin  sin 7 cos c
-cos  cos an Cns 7 cos 7c sin z - sin 97 cos s7 sin Xi sin ar]
-k tang; [cos 7 cos 7r sin sy tang  3 — sin z cos x cos s  tang S
4- sin 7c sin 7r sin sy tang A]
-- k sin 7c cos X — k sin -t cos A,
or if we neglect the terms multiplied by tang 5:
A'- = - k tang 2 sin (z - s7) cos (c - s7).
Further we get:'A' = A +  kA tang,2 [sin 7c cos 7c sin a7 cos rl +- sin,22 sin sy2 +- cos e2 cos y2
-+ sin 7r cos 7r sin V cos?7]
- kA tang ~ [cos mt sin m sin sy tang S ~- sin at sin 7c cos s tang S
-sin 7 cos 7v sin s  tang S]
+ k, [sin 7 2 -+- COS X 2],
or if we neglect the terms multiplied by tang o:
A'A +   A [tang 52 cos (      2 -)' + 1].
IX. ON THE EFFECT OF PRECESSION, NUTATION AND ABERRATION
UPON THE DISTANCE BETWEEN TWO STARS AND THE ANGLE
OF POSITION.
47.  The lunisolar precession and the nutation changes
the position of the declination circle and hence the angles
of position of the stars. From the triangle between the pole
of the  ecliptic, that of the  equator and the star we easily
find by means of the formulae in No. 11 of the first section
and the third of the differential equations (11) in No. 9 of
the introduction  the variation of the angle i, which the declination circle makes with the circle of latitude:
cos a d ----  sin e. sin a d  +- cos a dB,
as sin a dB is equal to zero, because the lunisolar precession
and the nutation do not change the latitude of the stars.




557
The sum of this angle iy and of the angle of position p of
another star relatively to this star is equal to the angle, which
the circle of latitude makes with the great circle passing
through the two stars, and since this is not changed by precession and nutation, it follows that the change of p is equal
to that of?i taken with the opposite sign, and that therefore:
cos  cdp - sin e sin a dA - cos a de.  (a)
Since the lunisolar precession does not change the obliquity of the ecliptic, we find the annual change of the angle
of position by precession from the equation
dp           da
cos    - sin  sin E d
dt           dt
or:
--- n sin a see 3
dt
where n --- 20". 06442 - 0". 0000970204 t.
When this formula is employed for computing the change
during a, long interval of time, it is necessary to compute
the values of n, a and 7 for the arithmetical mean of the times, and to multiply the value of d/p  found from  them  by
the interval of time.
In order to find the changes produced by nutation, we
must substitute in (a) instead of d)X and dc the expressions
given in No. 5 of the second section. If we neglect the
small terms, we obtain thus the complete change of p by
precession and nutation from  the formula:
dp -=   20". 0644 sin a see S ~ [- 6".8650 sin n) -1- 0". 0825 sill 2 t)
- 0". 5054 sin 2 (0] sin a sec S
-[9".2231 cos fl - 0".0897 cos 2 ()
+I 0". 5509 cos 2 01 cos a sec,
or if we make use of the notation adopted in No. I of the
fourth section:
dp = A. n sin a sec 8 -+- B cos a sec Y,
which formula gives the difference of' the angle of position
affected with precession and nutation from  that referred to
the mean equinox and the mean equator for the beginning
of the year.
In  order to find the effect of aberration upon the distance and the angle of position we must remember that ac



558
cording  to  the expressions  in  No. 1 of the fourth  section
we have:
for the aberration in right ascension: Cc -- Dd
and for the aberration in declination:           Cc' -J -ld',
where C   - 20".445 cos e cos 0, D - - 20".445 sin C)
C  sec S cos a,           c'  tang E COS - sin 6 sin a
d   see 6 sin a,          d' = sin 6 cos a.
Now if 2 and v denote the differences of the right ascensions and the declinations of the two stars, we find the
changes of these differences by aberration, which are equal
to the difference of the aberration for the two stars, by means
of the equations:,, --  c.,e - - C. A-  d
"A r  C. Ac'+ D. Ad',
where: A c — see   sin a. a,-  see 6 tang 6 cos a.'
A d-   see 8 cos a.  ~- sec 6 tang 8 sinx. v
A'    sin 6 cos a.  - [tang E sin 6+- cos S sin a] r,
Ad'-   - sin 6 sinl.  -,I- cos c cos a. v.
Hence, substituting these expressions we have:
cos S A = C [-sin a. -2 -  tang e8 o  s a. ]   D [cos a. Z +- tang S sin a. r,]
A    -- C [sin 6 cos a. a + (tang a sin 6 + cos 6 sin a) r]
- D [sin 6 sin a. 2 - cos 6 Cos a. v].
But, if we denote the distance and the angle of position
by s and P, we have:
s. sin P --  cos 
S. Cos P-v,
hence:
cos 6
S2   2 COS 82 t- V2,  tang P -.,and therefore:
s. As   co 82 22. A 2V Av - Cos 6 sin  22 (Cc' - Dd').
If we substitute herein the values of A X and Av found
before as well as the values of c' and d', we find after an
easy reduction:
s. As   [22 cs 62 +-, 2] [- C (tang e sin  -i- cns asin ) + D cos 8 cos a]
or:     s   - C. s [tang ~ sin 6 -+- cos  sin a] -D. s cos 6 cos a.
Further we have:
s2 dP= v eos 8. A2 -   ncos SAv - Asin 6v [Cc' - Ddl'],
and if we substitute the values of AX, A v, c' and d', we find
again after a simple reduction:
d P= C tang 6 cos a -+- D tang 6 sin a.




559
Therefore if we introduce the following notation:
a'  - see Ssin ca
60
see a cos ca
60
tang S cos a           s
c                         - [tang a sin 8~3- cos S sin a]
60                IV
tang S sin a        s
d' =                d —-  COS  Cos a,
60              w
I       1       1
where the factors  - and -  =              have  been  added  in
60      Wo   206265
order to find the corrections of the distance and of the angle
of position expressed respectively in seconds of are and minutes of arc, then we have:
Observed distance -- True distance +- c C +- dD
Observed angle of position --- True angle of position for the beginning of the year
- a' A + — b' B -A- c' C +- d' D.
Since c, d, c' and d' are independent of the angle of
position, it follows, that aberration changes the distances,
whatever be their direction, in the same ratio, and all angles
of positions by the same quantity.  Therefore if the circumference of a small circle described round a star is occupied
by stars, such a circle will appear enlarged or diminished
by aberration and at the same time turned a little about its
centre; but it always will remain a circle, and the angles
between the radii of the stars will remain the same.




Berlin, printedI by A. W. ScHnDE, Stallsclreiberstr. 47.