
REESE LIBRARY
UNIVERSITY OF CALIFORNIA.
Class
SPHERICAL ASTRONOMY.
SPHERICAL ASTRONOMY
BY
F. BRUNNOW, PH. DR.
TRANSLATED BY THE AUTHOR FROM THE SECOND
GERMAN EDITION.
LONDON:
ASHER & CO.
13, BEDFORD STREET, COVENT GARDEN.
1865.
DEDICATED
TO THE
REV. GEORGE P. WILLIAMS, L. L. D.
PROFESSOR OF MATHKMATICS IN THE UNIVERSITY OF MICHIGAN
rt BY THE AUTHOR
AS AN EXPRESSION OF AFFECTION AND GRATITUDE FOR UNVARYING
FRIENDSHIP AND A NEVER CEASING INTEREST IN ALL HIS
SCIENTIFIC PURSUITS.
2 72.
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 lec
tures, 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. How
ever 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 pre
pare an English translation, I gladly availed myself of
my leisure here to comply with their wishes, and hav
ing acted for a number of years as an instructor of
VJII
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 Germanisms,
which are always liable to occur in a translation, espe
cially when made by a German. I have discovered
some such mistakes myself and have given them in
the Table of Errors.
I trust therefore that this translation may be re
ceived with indulgence and may be found a useful
guide for those who wish to study this particular
branch of science.
JENA, August 1864.
F. BRtTNNOW.
TABLES OF CONTENTS.
INTRODUCTION.
A. TRANSFORMATION OF COORDINATES. FORMULAE OF
SPHERICAL TRIGONOMETRY.
Page
1. Formulae for the transformation of coordinates 1
2. Their application to polar coordinates 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 trigo
nometry 9
7. On the precision attainable in finding angles by means of tangents
and of sines 10
8. 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 f e~* dt 33
(/
f**3
17. Various methods for computing the integral I e dt .... 35
T
18. Computation of the integrals 38
(1 x) sin dx
rV^ si n^ and C
J Fcos 2 }2*sin 2 ,
cos 2 h sing 2 .x
P
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. Coordinate system of azimuths and altitudes 73
3. Coordinate system of hour angles and declinations 74
4. Coordinate system of right ascensions and declinations .... 75
5. Coordinate system of longitudes and latitudes 77
II. THE TRANSFORMATION OF THE DIFFERENT SYSTEMS OF
COORDINATES.
6. Transformation of azimuths and altitudes into hour angles and decli
nations 79
7. Transformation of hour angles and declinations into azimuths and
altitudes 80
8. 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 lati
tudes 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 THE 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 pro
duced 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 de
clination 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
II. 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
8. 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 166
11. Computation of the true refraction for any indications of the ba
rometer and thermometer 169
12. Reduction of the height of the barometer to the normal tempera
ture. 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 bo
dies. 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 de
clination and in longitude and latitude . . 180
17. Tables for aberration 188
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
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
8 Determination of the absolute right ascension of a star .... 218
9. Relative determinations. The use of the standard stars. Obser
vation 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 polestar from one epoch to another.
On the variability of the proper motions 248
FIFTH SECTION.
DETERMINATION OF TOE 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 grea
test elongations of circumpolar stars, by equal altitudes and by
observing the upper and lower culminations of stars 253
2. Determination of tfie 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 ob
serving its distance from a heavenly body 257
II 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 circummeridian altitudes . . 266
8. The same problem, when the declination of the heavenly body is
variable . 269
9. Method of finding the latitude by the polestar 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.
1 1 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 ob
serving 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 elec
tric 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.
I. SOME OBJECTS PERTAINING IN GENERAL TO ALL INSTRUMENTS.
A. Use of the spiritlevel.
1. Determination of the inclination of an axis by means of the spi
ritlevel 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
13. The vernier and the reading microscope.
4. Use of the vernier 401
5. Use and adjustments of the reading microscope 403
C. Errors arising from 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 excen
tricity 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.
8. 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 al
titude 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 .... 482
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. Ob
servations of altitudes "by means of an artificial horizon .... 500
31. Effect of the errors of the sextant upon the observations and de
termination 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 ob
jects 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 ap
parent 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 de
clinations 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 dif
ference 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 4
by nutation. Change of the distance and the angle of position
by aberration 556
XIX
ERRATA.
page 23
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formulae
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constant quantity
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constant
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North Pole
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North Pole
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South  Pole
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South Pole
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Parallel Circles
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Parallel Circles
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vertical circle
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South point
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West point
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page 140 line 16 from bottom
144 line 10 from bottom
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148 line 1 from top
154 line 11 from bottom
155 line 8 from bottom
169 line 9 from top
171 line 4 from top /
173 line 1,2, 18 from top I
174 line 13 from top
176 line 14, 11 from bott. for the refraction
178 line 11 from top for at
181 line 12 from top for vertical
190 line 11 from top for at
209 line 5 from top for vertical
210 line 4 and 5 from top for vertical
214 line 8 from top for usually
226 line 10 from top for at last
232 line 14 from bottom for Now
272 line 13 from bottom for ^ p 3 sin t cost
286 line 18 from bottom for cos S sin h
331 line 9 from top for =
tang 7i
397 line 18 from top for a
399 line 1 from bottom for i and {
425 line 14 from bottom for of
450 line 4 from bottom for of
456 line 16 from top for form
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read perpendicular
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INTRODUCTION.
,1. TRANSFORMATION OF COORDINATES. 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 coordinates to certain great cir
cles of the sphere and establishing the relations between the
coordinates with respect to various great circles. Instead of
using spherical coordinates we can give the positions of the
heavenly bodies also by polar coordinates, 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 coordinates can finally be expressed by rectangular
coordinates. Hence the whole of Spherical Astronomy can
be reduced to the transformation of rectangular coordinates,
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 ?/, the distance of the point from the origin of the coor
dinates by r, the angle, which this line makes with the po
sitive side of the axis of a?, by t?, we have:
r cos v
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 coordinates of the same point referred to this new sys
1
tern by x and y and the angle corresponding to by ,
we have:
If we denote then the angle, which the positive side of
the axis of x makes with the positive side of the axis of a?,
by o, reckoning all angles in the same direction from to
360, we have in general v = v \ w, hence :
x = r cos v cos w r sin v sin w
y = r sin v 1 cos w \ r cos v 1 sin w,
or:
x= x cos w y sin w
y = x sin w J y cos w
and likewise:
x = x cos w + y sin w
(1)
y = re sin w f y cos w
These formulae are true for all positive and negative values
of x and y and for all values of w from to 360.
2. Let a;, ?/, z be the co  ordinates of a point 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 a?
(or the angle between a plane passing through the point
and the positive axis of z and a plane passing through the
positive, axes of x and a, 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 coordinates equal to unity:
x = cos B cos , y = sin B cos a , 2 = sin a .
But if we denote by a the angle between the radius
vector and the positive side of the axis of a, reckoning it
from the positive side of the axis of z towards the positive
side of the axis of x and y from to 360, we have:
x = sin a cos B\ y = sin a sin B\ z = cos a.
If now we imagine another system of coordinates, whose
axis of y coincides with the axis of ?/, and whose axes of
x and a make with the axis of x and z the angle c and if
we denote the angle between the radius vector and the posi
tive side of the axis of a 1 by b and by A the angle between
the plane passing through and the positive axis of z and the
plane passing through the positive axes of x and , reckoning
both angles in the same direction as a and B\ we have:
x = sin b cos A\ y = sin b sin A , 2 = cos 6,
and as we have according to the formulae for the transfor
mation of coordinates:
z = x sin c + z cos c
r=*y
# = a cos c z sin c,
we find:
cos a = sin b sin c cos J. H cos 6 cos c
sin a sin .5 = sin 6 sin A
sin a cos B = sin 6 cos c cos A cos b sin c.
3. If we imagine a sphere, whose centre is the origin
of the coordinates and whose radius is equal to unity and
draw through the point and the points of intersection of
the axes of z and * 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
ingles may be greater than 180 degrees. The three sides
Z, Z 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 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
af A 1 and B in the equations which we have found in No. 2,
we get the following formulae, which are true for every spher
ical triangle:
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 6 cos c cos ^4.
These are the three principal formulae of spherical tri
gonometry and express but a simple transformation of coor
dinates.
As we may consider each vertex of the spherical triangle
as the projection of the point 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 correspond
ingly. Hence we obtain, embracing all possible cases:
cos a = cos b cos c H sin b sin c cos A
cos I, = cos a cos c f sin a sin c cos B (2)
CO s c = cos a cos 6 + sin a sin 6 cos C
sin a sin B = sin 6 sin A
sin a sin C = sin c sin vl (3)
sin b sin (7= sin c sin 5
sin a cos B = cos ft sin c sin 6 cos c cos A
sin a cos C = cos c sin b sin c cos b cos 4
sin b cos J. = cos a sin c sin a cos c cos B
sin 6 cos C = cos c sin a sin c cos a cos jB
sin c cos A = cos a sin 6 sin a cos b cos C
sin c cos B = cos 6 sin a sin 6 cos a cos C.
4. We can easily deduce from these formulae all the
other formulae of spherical trigonometry. Dividing the for
mulae (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 = 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 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 sinB
sin C cos J3 = cos a sin 25 cos C,
sm o
we find:
sin C cos .B = cos 6 sin .A cos a sin .B cos C,
or:
sin J. cos b = cos 5 sin C + sin jB 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 chang
ing the letters, we find the following six equations:
sin A cos 6 = cos^B sin (74 sin B cos C cos a
sin A cos c = cos C sin B + sin C cos B cos a
sin 5 cos a = cos A sin C H sin A. cos C cos 6
sin B cos c = cos C sin ^4 f sin C cos J. cos 6
sin C cos a = cos A sin jB f sin A cos J3 cos c
sin (7 cos 6 = cos B sin A { s mB cos J. cos c
and dividing these equations by the corresponding equations
(3), we have:
sin a cotang b = cotang .5 sin C \ cos C cos a
sin a cotang c = cotang C sin B f cos jB cos a
sin 6 cotang a = cotang A sin 6 Y + cos C cos 6
sin b cotang c = cotang C sin J. f cos A cos ft
sin c cotang a = cotang A sinB \ cos .6 cos c
sin c cotang b = cotang B sin A f cos ^4 cos c.
From the equations (6) we easily deduce the following:
cos A sin C = sin .5 cos a sin A cos 6 y cos 6
cos B sin C = sin A cos 6 sin B cos (7 cos a.
Multiplying these equations by sin C and substituting
the value of sin A sin C cos b taken from the second equa
tion 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 C cos a cos B cos C
cosB = sin A sin C cos b cos A cos C (8)
cos C = sin A sin B cos c cos A cos .5.
5. If we add the two first of the formulae (3), we find :
sin a [sin B + sin C] = sin A [sin b f sin c] ,
or:
B C . B+C . 64c 6 c
smj^cos ~ .cos^asm  = sin 5 4 sin . cos ^^4 cos
and if we subtract the same equations, we get:
B C B + C b + c . b c
8in4 a sin  . cos ., a cos ^ =sm^ylcos . cos 4 sin ~  
Likewise we find by adding and subtracting the two
first of the formulae (4):
BC E\C
. . sm.4cos 
2 2i 2
. BC . B + C . b c b c
sm a sin   . cos a sin ^ = cos T M sm cos f A cos ^
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 pur
pose either of the following equations:
B\C . B+C bic b c
cos T a cos ^  .cos^asm   =sin^cos .cos^^lcos n
Z Z 2 Z
. , BC . . B C 6fc b c
sm^acos  .sm^asin =smy^sin .cos 7^4 sin j
* 2
6
which we find by adding or subtracting the first two of the
equations (6).
If we take now :
. 6hc
sin A sm 5 = a
sin? Jcos <r p
. b c
cos j A sin ~ = y
COS 5 .4 COS ~
and:
tf
sm , a cos ~ = a
,
cos a cos  = /a
. BC ,
sin a sm = y
a y =ay,
.  ,,
cos a sm  = o ,
we find the following six equations:
a 8 = a 8, y p =yp, a {3 =a{3, y 8 = y8 t
from which we deduce the following:
= a, /9 = /?, / = y, 3 = ,
or:
= , =   g, / = 7 , 8 = 8.
Hence we find the following relations between the angles
and sides of a spherical triangle:
. b+c BC
sm 5 A sm = sm a cos 
b + c B+C
sm j^. cos ^r = cos .y cos g
(9)
, ,  6 ~ c i BC
cos 5 A sin = = sm 7 a sm ^
6 c .
cos J. cos ^ = cos ijr a sm

or:
. 6+c
sm ^ ^1 sm  = sm 4 a cos
2i
6hc
sm 4 A cos = cos a cos
. 6 c
cos TJ 4 sm r sin 7 a sm
6
c
cos 5 vl cos < = cos j a sn 
Both systems give us for the unknown quantities, which
may be either two sides and 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 for
mulae either for  and ^ the same values as from the
first, but for \A a value which differs 180, or we should
find for c and ~ values which differ 180 from those
derived from the first system , but for A the same value.
In each case therefore the values of 4, b and c as found
from the two systems would differ only by 360. The four
formulae (9) are therefore generally true and it is indifferent,
whether we use for the computation of A, b and c the quan
tities a, B, C themselves or add to or subtract from any of
them 360*).
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 :
BC
(1) cos  (4)
(2) sin ^ a (5) cos I a
(3)
and from these:
,,, . BC . B+C
(3) sm 5^ (6) sin
(7) sin ^ a cos (9) sin ^ a sin
2i 2
(8) cos  a cos  (10) cos \ a sin
Subtracting the logarithm of (8) from that of (7) and
the logarithm of (10) from that of (9), we find log. tang
(b  c) arid Ig. tg. j[ (6 c), from which we get b and c. Then
we take either log cos (6 + c) or log sin i (6 + c) and log
cos ^ (6 c) or log sin (6 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 \ A , from which we find A. As sin \ A as well as
cos  A must necessarily give the same angle as tang \ A,
we may use this as a check for our computation.
If for instance we have the following parts given:
a= 11 25 56."3
. = 184 6 55. 4
C= 11 18 40. 3
we have:
(7) = 86 24 7."55
cos 4 (B C) = 8.7976413
sin ^ a = 8.9982605
sin \ (B (7) = 9.9991432
sin \ a cos \ (B C) 7.7959018
cos 4 a cos  (B f C) 9.1256397.
i(6fc)~ 177 19 13.49^
cos 4 (b h c) _ 9.9995248
sinM 9.1261149
cos ^ A 9.9960835
4 JTTMO 7 59."38~
97 42 47."85
) 9.1278046
cos i a 9.9978351
S i n ^( B + (7) 9.9960526
sin 4 sin ^ ( <7) 8.9974037
cos \ a sin ^(B + (7) 9.9938877
(6 c) 5 45 24. 13
cos^(6 c) 9.9978042
6 = 183 4 37."62
c = 171 33 49. 36
A= 15 21 58. 76.
If we had taken B = 175 53 4.%, hence:
^ ( + C ) = 82 17 12."15
^ (5 C) = 93 35 52. 45
we should have found:
^ (6 _l_c) == _ 240 46."51
7  (i c )= 185 45 24. 13
hence 6 = 183 4 37."62 and c = 188 26 ; 10."64.
Dividing Gauss s equations by each other, we find Napier s
equations. Writing A, B, C in place of 5, C, A and er, 6, c
in place of 6, c, a, we find from the equations (9):
AiB
tang 
tang 
a b
C S ~~
(9 a)
2 C
 cotang
A B
+b > r
2 ~
cos
A B
sin ^
a b 2 c
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:
x = m sin M and y = in cos M
for we find immediately:
tang If = and m = V x" 1 + y* ,
hence M and m expressed by real quantities. Therefore as
all the above formulas, which consist of several terms, con
tain 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 f sin b sin c cos A
sin a sin B = sin 6 sin A
sin a cos B = cos 6 sin c sin b cos c cos A,
we may put:
sin b cos A = m sin M
cos b = m cos M.
and find:
cos a = m cos (c M)
sin a sin B = sin b sin A
sin a cos B = m sin (c M}.
If we know the quadrant, in which B is situated, we
can also write the formulae in the following manner, sub
stituting for m its value S1 : . We compute first:
sin M
tang M= tang b cos A
10
and then find:
tang A sin M
tang= 
sm(c M}
tang(c M)
tang a =
cos ^
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 Tubingen. (J. Zech, Tafeln fur die Ad
ditions und Subtractions Logarithmen fur sieben Stellen.)
Kohler 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 /\x denotes a small increment of an angle, we have:
Now it is customary to express the increments of angles
in seconds of arc ; but as the unit of the tangent is the ra
dius, we must express the increment A & a ls m 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 loga
rithms, 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 129(5000, while this circumference if we take the radius as
unit is 27r 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 radius into
seconds of arc, we must multiply them by this number; but if we wish to
convert quantities, which are expressed in seconds of are, 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 com
plement is equal to the sine or the tangent of one second.
11
of the last decimal of the logarithms used, we must multiply
by 10000000 if we employ logarithms of seven decimals. We
find therefore:
2 M /\x"
A (log tang x} = r  JL , Q 10000000
or:
sin 2,
A (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 a?) being equal to 200, the
error of the angle would be:
900"
A*" = 11 V sin2 * = 5 " sin2 *
4:2,1
Therefore if we use logarithms of five decimals, the error
cannot be greater than 5" sin 2x or as the maximum value
of sin 2 x is unity, not greater than 5 seconds and an error
of that magnitude can occur only if the angle is near 45.
If we use logarithms of seven decimals, the error must needs
be a hundred times less ; hence in that case the greatest er
ror of an angle found by the tangent will be O."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 2 x 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 rightangled
triangles. If we denote then the hypothenuse by /, 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 = 90 :
cos h = cos c 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 = cotang 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 Ccotg C .
At last from the combination of the two equations:
sin h sin C ; = sin c
and sin h cos (7 = cos c sin c ,
we find:
cos = sin C cos c.
We have therefore for a rightangled triangle the follow
ing six formulae, which embrace all combinations of the five
parts :
cos h = cos c cos c
sin c = sin h sin C
tang = tang h cos C"
tang c = tang C sin c
cos h = cotang C cotang C
cos (7= cos r; sin C",
and these formulae enable us to find all parts of a right
angled 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 M, we substitute two rightangled triangles for the oblique
triangle. For if we let fall an arc 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 arc and M the part of the
side c between the vertex A and the point, where it is in
tersected by the vertical arc.
9. For the numerical computation of any quantities in
astronomy we must always take certain data from obser
vations. 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 a da = db [ sin b cos c + cos b sin c cos A]
+ dc [ cos b sin c h 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 E\ if we write also
 sin a sin c sin B instead of the factor of A , we find the
differential formula :
da = cos Cdb J~ cos 13 dc + sin c sin BdA..
Writing the first of the equations (3) in a logarithmic
form, we find:
log sin a + log sin B = log sin b j~ log sin A
and by differentiating it:
cotang a da + cotang Bd.B = cotang bdb \ cotang Ad A.
Instead of the first of the formulae (4), we will dif
ferentiate the first of the formulae (5), which were found by
the combination of the formulae (3) and (4). Thus we find:
dB + dA [cotang B cos A sin A cos c]
sin JD
= , , db + dc [cotang b cos c + cos A sin c]
sm & a
sin A , cos C 7 sin c cos a
or:  dB dA= 72 </6h. : dc.
smB* sm B sin b* sin o
Multiplying this equation by sin B, we find:
sin a sin C cos a sin B
 d B cos CdA = db \ dc,
sm b sin b sm 6
or finally:
sin adB = sin Cdb sin B cos adc 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 bdC + sin b sin Cda.
Hence we have the following differential formulae of tri
gonometry :
da = cos Cdb f cos Bdc H sin b sin CdA
cotang a da + cotang BdB = cotang bdb + cotang A dA
sin adB = sin Cdb sin B cos adc sin b cos CdA
dA = cos cdB cos bdC } 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 fol
lowing way.
We have:
sin a i _ J_ a . _i_ 4 _
a 6 a ^120
and:
cos a= 1 y a 2 + jr a 4
hence :
y cos a = 1 a 2 f
We have therefore, neglecting only the terms higher than
the third power:
sin a \l
= V cos a
a
3
or: i/
a = sin a y sec a
This formula is so accurate that using it for an angle
of 10 we commit only an error less than a second. For we
have :
3
log sin 10 ]/ sec 10 = 9.2418864
and adding to this the logarithm 5.3144251 and finding the
number corresponding to it, we get 36000."74 or:
10 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:
 ,
1 a cos x
we can easily develop y in a series, progressing according
to the sines of the multiples of x. For if we have tang z=,
we find d*= ndm ~ m t . If we take thus in the formula
rf 2
15
for tang y a and y as variable, we find:
dy sin x
  ; 
da 1 2 a cos x + a~
and if we develop this expression by the method of indeter
minate coefficients in a series progressing according to the
powers of , we find:
^ = sinx{asin2xi a 2 sin 3 x + ____ *)
da
Integrating this equation and observing that we have
y = when x = 0, we find the following series for y:
y = a sin x f ^ a 2 sin 2 x + ^ a 3 sin 3 x + ____ (12)
Often we have two equations of the following form:
Asin JB = a sin .r
J. cos B = 1 cos #,
and wish to develop B and log A in a series progressing ac
cording to the sines or cosines of the multiples of x. As in
this case we have :
a sin:r
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 = V I 2acosxia 2 .
Now we find the following series by the method of in
determinate coefficients :
a cos x a 2
~ = a cosx f a cos 2x f a 3 cos ox f .. . )
1 2 a cos x H a 2
Multiplying this by   and integrating with respect
to a, we find for the left side:
2acosa:ta 2 )
<a
and as we have log ^4 = when a = 0, we get :
log ]/l 2acos#a 2 =log^l= [ocosar+^a 2 cos2ar+ a 3 cos3.r + . . .] (13)
*) It is easily seen, that te first term is sin^, and that the coefficient
of a" is found by the equation:
A,, = 2A i cos x Ani
**) It is again evident, that the coefficient of a is cos a:, while the co
efficient of a,, is found by the equation :
A, t = 2 A n \ COS X A n %.
16
If we have the two equations:
A sin B = a sin x
A cos B = 1 + a cos or
we find by substituting 180 x instead ofx in the equations
(12) and (13):
.B = asinar 4 a 2 sin 2*4 j a 3 sin 3* .... (14)
a COS.T .] a 2 cos2:r4 }a 3 cosStf .... (15)
If we have an expression of the following form:
tang y = n tang j?,
we can easily reduce it to the form tang y =
J 1 cos x
For we have:
tang y tang x (n 1 ) tang x
x) = =
1j tang y tango: lfntang* 2
(n 1) sin x cos x (n 1) sin x cos x
x" 1 + n sin x 2 11 n n
2 4 2 cos2*f cos2*
n 1 .
sm 2x
(n 1) sin 2;r
(n4D (M
  cos
n\ 1
Hence, if we have the equation tang y = n tang a?, we find :
y = x} sin 2 x h 4 ( .) sin4a: t4 ( . ) sin6r + ... (16)
nhl Vnflx \njl/
If we take here:
n = cos a,
we have:  = tang 4 a 2 .
nf1
Hence from the equation:
tang^ = cos a tang x
we get
y = x tang^ 2 sin2o:H^tang4a 4 sin4ar ] tang \ a 6 sin6a: + ... (17)
If we have : n = sec ,
we find: ^ = tang $ 2 .
Hence from the equation:
tangy = sec tang a: or tang x = cos a tangj/,
we obtain for y :
^== x _tang^a 2 sin2^+Jtang;a 4 sm4a:hitang^a 6 sinGa:4... (18)
As we have:
cos a cos 8
ioIa Tcos ft
dsin sin /9
sin h sin i
17
we find also from the equation :
cos a
tang y= ^ tang or,
x tang 4 ( /?) tang ( 4 /?) sin
and from:
^ = # h tang ^ ( /?) cotang ^ (a f /9) sin 2 x
+  tang 4 ( ) 2 cotang ^ ( f /9) 2 sin 4or + . . .
By the aid of the two last formulae we can develop
Napier s formulae into a series. For from the equation:
A B
ab Sm 2 c
2 =
s
we find:
ab c B A B 2 A 2
~2~ ~ ~2  tang T cotan g 2 sin c + ^ tan g "^~ cotang sin 2 c ....
or:
c a 6 Z? A B 2 A 2
2 =: ~2 ~ + tan S 2 cotan g 2 sin ( ft ~ 6 )HTtang  cotang y sin 2 (a 6)4 ...
and also in the same way from the equation:
A B
a+ft C S 2
tang 2  = ^ ^tang
cos
we find the following two series :
c A B A 2 B 2
2" tang T tang "2" Sin + tang ~2~ tang T" S
^4 5 ^l 2 B 2
2 ~ ~^  tang 2 tang 2 sin ^ a + ^ + tang 2" tang T sin 2 (l ^)
Quite similar series may be obtained from the two other
equations :
AB sin ~2~ 180 (7
sin  
a~b
~2~ 180C7
cos
It often happens, that we meet with an equation of the fol
lowing form: C os y = cos x H 6
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 = arc cos [cos x f b]
For if we put:
cos x = z and y =/(z f ?>),
we get:
or as:
f f z \ = x d .f= _^.* ... = L
dz d.cosx sin*
d*f_ sin* dx cos x
dz 2 dx d.cosx sin* 3
cos x
d 3 f_ ~ sin x 3 dx __ [1 h 3 cotang**]
dz 3 dx d.cosx sin x 3
y = x ^cotang* , i[lH3cotang* 2 ] ,.... (19)
sin* sin* 2 sin* 3
In the same way we find from the equation:
sin y = sin * f b
y = x\ Ktangs^rH [1 + 3 tang* 2 ] 3 + ...*) (20)
cos * cos * 2 cos * 3
.B. THE THEORY OF INTERPOLATION.
12. We continually use in astronomy tables, in which
the numerical values of a function are given for certain nu
merical 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 spharischen Astronomie.
Schumacher s astronomische 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 quan
tity the value of one of the differential coefficients is infinity,
so that the function ceases to be continuous in the neigh
bourhood 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 va
riable quantity, assumes therefore, that the function is con
tinuous between the limits within which it comes into conside
ration and can be applied only if this condition is fulfilled.
If we call w the interval or the difference of two follow
ing arguments (which we shall consider as constant), we may
denote any argument by a\nw, 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 f nw] and f(a f (n f 1) w) by
/"(ahftfi), writing within the parenthesis the arithmetical
mean of the two arguments, to which the difference belongs,
but omitting the factor w*). Thus /" (a! 5) denotes the
difference of f(a h to) and f(a), f(tfhf) the difference of
f(a l20) and /"(afw?). In a similar manner we will denote
the higher differences, indicating their order by the accent.
Thus for instance f" (a\Y) is the difference of the two first
differences f (aHf) and /"(+).
The schedule of the arguments and the corresponding
functions with their differences in thus as follows:
Argument Function I. Diff. II. Diff. III. Diff. IV. Diff. V. Diff.
a 3w f(a 3 w)
/ (
o3;/(a
) This convenient notation was introduced by Encke in his paper on
mechanical quadrature in the Berliner Jahrbuch fiir 1837.
9*
20
All differences which have the same quantity as the ar
gument 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:
/(a + nw} = a H ft . n w h y . n 2 w" 1 + . n 3 iv 3 H . . .
If the analytical expression of the function f (a) were
known, we might find the coefficients a, ft, 7, 6 etc., as we
have a f(a) /i = ~r 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\nw ) for
certain values of the argument a + nw. 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 , /:?, ; , d etc. from them. It is easily seen, that
we have a f(a) and that pw, /w 2 etc. are linear functions
of differences, which all may be reduced to a certain series
of differences, so that we may assume f(^a\nw) to be of
the following form:
where ^, J5, C... are functions of w, 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 successi
vely, 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 dif
ferences are constant, we have simply f(a}nw) = f(a)+n /"(ajJ),
if the second differences are constant, we must add to the
above value f" (a\Y) multiplied by the sum of the numbers
from 1 to n 1 or by ( y~^; and if only the third diffe
rences are constant, we have to add still /""(aHf) multiplied
by the sum of the numbers 1, l}2, 1 { 2 + 3 etc. to
21
1 + 2 f . . . { 2 or by " (w 7 ^ ( " ~ 2) . We have therefore
1 . J . o
i A n n (>* 1) n n ( n 1) ( n 2) i
in general A = n, B = yg 1 ^ g  etc. hence :
f(a + w ) ==/() 4 n/ (a +*) + ^^/ ( + D
+ ^^ 2) / ( + t)H..., (0
where the law of progression is obvious *).
This formula is known as Newton s formula for interpo
lation. The coefficient of the difference of the order n is
equal to the coefficient of a?" in the development of (1fa?)*.
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. 0303 25 1". 5
2310 651.5 + 6 038 o +18 48 H2 44"4
4317 7 29.5 ! J^ S 21 32 . 4 + * f * h 10". 1
6 324 29 39 9 24 9A 9 2 4 ^ 47
D 3/1 zy oy . j 7 ic 07 q ^ wt> . y 9 _ 9 t . <
8 332 16 17.2 1 27 26 . 1
10340 30 20.6
If we wish to find now the longitude of Mercury for
Jan. 1 at mean noon, we have :
/(a) = 303 25 1". 5 and n = ,
further :
/ ( a f ) = h 6 41 50". 0, n =  Product: h 3 20 55".
/(a + l)= h 18 48.0,^^ =  221.0
1 . Z
+ = + 244.4 n ^=i )2  ) = + s +10.3
*) 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 bre
vity by / , /", / " etc. we have the following table :
I. Diff. II. Diff. III. Diff.
/()
f( \ I O fl I f J J fH i fill J
J(&)~r*J H~/ f \ _, o f n , fin J ~T~ J fin
Q fll I fill J < ^/ ~T" J fll . O f> J
*>j r j ,., Q ,,;; o ,r;/; ./ v /;//
/(a) H 5/ 4 10/ + 10/" f + Yf 1 10^ " " "> 4/ " ^"
/(a) 4 6/ f 15/" + 20/" ^ J fi ,, [T I K> /" + 5/" 7
/(a)47/h21/"435/"" "
22
Hence we have to add to f(cf)
13 18 43". 9
and we find the longitude of Mercury for Jan. 1 O h
300 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:
/(a f nto) =/(a) H n [/ (a + $ + ^ [/" (a+ 1) + ~ X
If n is again equal to , we have  = , hence
/ IV (aH2) = 6". 3. Adding this to f" (4f) and mul
4
tiplying the sum by ? = f, we find  1 19". 0. Ad
ding this again to f" (a f 1) and multiplying the sum by
^~ l  = i, we get 4 22". 2 and if we finally add this to
f (a 4 1) and multiply by n=^ we have to add 3 18 43". 9
to f(d) and thus we find the same value as before, namely
306 43 45". 4.
14. We can find more convenient formulae of inter
polation, if we transform Newton s formula so, that it con
tains only such differences as are found on the same horizon
tal line and that for instance starting from f(a) we have to
use only the differences /X#4), /" GO an( ^ f "(. a ~k~%) The
two first terms of Newton s formula may therefore be re
tained.
Now we have:
/" ( a H 1) = f ()+ f" (a f 1),
/ " ( h ) = f" (a H ) I/ ( a + 1)
/iv ( a + 2) = f lv (a H 1) 4/ v ( + f )
=/ IV ()+2/ v (a + ) f/ v ( + 1),
/v ( a 4 I) ==/% ( + 3 ) + yvi (a + 2 )
=/ v ( 4 i) 4/ VI (a + 1) +/ VI (a + 2),
etc.
We obtain thus as coefficient of f" (a) :
n (n 1)
23
as coefficient of f ^ah^) 
njn 1 ) n (n 1) (n 2) _ (n H 1 )_( w_ _1 )
~T:2 1.2.3 1.2.3~
as coefficient of f lv (a):
n(n l)(n 2) n(n 1) (n 2) (n 3) _ (n + 1) n (n 1) (n 2)
1.2.3 1.2.3.4 1.2.3.4
at last as coefficient of v
n( l)(n 2) n(n l)(n 2)(n 3) n(n l)(n 2)(n 3)(n4)
1.2.3 1.2.3.4 1.2.3.4.5
_ (nf2) (nH1) n (n 1) (n 2)
1 .2.3.4.5
where the law of progression is obvious. Hence we have:
If we introduce instead of the differences, whose argu
ment is aHf those whose argument is a f, we find:
/ (a + i) =./" (a  ) +/" (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 )
1.2 1.2
and that of /" Iv (a) :
(n+l)n(n 1) (n + l)n (n l)(n 2) (n l)n(n + l) (nf2)
1.2.3 1.2.3.4 1.2.3.4
We find therefore:
f" (a) + 1
( n 2)( n l)n(n+l)(nH2)
TTT^IL 4^ ~"i7273 .T.T "
where again the law of progression is obvious.
Supposing now, that we have to interpolate for a value,
whose argument lies between a and a 0, 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
/(a)  n/(a i) + ~^^/ (a)
w ( _ 4) + (n+lnl) 2) /lv
(n42)(n4l)n(nl)(n2)
~lT2T374~5~
This formula we use therefore if we interpolate back
wards. Making the same change with the formulae (2) and
(3) as before made with Newton s formula, we find:
f(a 4 nw) =/() + n [ /" (a K) H ^ [/" (a) + n ~ X
X [/" (a 4) h ^ [/ IV (a) 4 ... (2 a)
/(a _ nw ) =/() _ n [/ ( a  )  ^ 1 [/" (a)  ?^ X
X [/ " (a  $  n ~^ [/ Iv (a)  ... (3 a)
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\nw is nearer to a than
to a\w, and the second one, when a nw is nearer to a
than to a ?, 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 diffe
rence 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/""^) and /" (ahl). Then
we have to add to f" (a):
Therefore if f 00 is ( smaller ) than f"(a hi), the cor
Vgreater/
rected f" (a) will be (f"*^) and hence come nearer f" (a 41).
A little greater accuracy may be obtained by using in
stead 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 :
/ (a + > ,/(+ J)+/(++
2
As in this case the quantities within the parenthesis are
fractions for differences of an even order and integral num
bers 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 /*" (a + 1) or
, so that we take now instead of the term
the term:
?;* f " (a+ * } " "ri (/ " (o) + * / " (a + )! 
Hence while using merely f" (a) we commit an error
equal to the whole third term, the error which we now com
mit, is only:
+> 
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 in
terpolate in a forward direction or starting from the argument
a+w interpolate in a backward direction, we get the most
convenient formula by the combination of the two. Now for
= \ formula (2) becomes :
while formula (3) becomes, if the argument (ofto) is made
the starting point:
" (a t
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 ex
actly in the middle between two arguments, the following
very convenient formula, which contains only the arithmetical
mean of even differences:
 * [/"(aH)  ^ [/ IV (K)  ~ f/ V
where the law of progression is obvious.
Example. If we wish to find the longitude of Mercury
for Jan. 4 12 h , we apply formula (2 a). The differences, which
we have to use, are the following:
I. Diff. II. Diff. III. Diff. IV. Diff.
+ 7 38". H2 44". 3
Jan. 4 317 7 29". 5 _ 21 ^ 2 !jA_ + 10 " l
__ " 7 22 10  4 2 54 . 5
6 324 29 39 ~~9 24 26 . 9~ 4 . 7
In this case we have n = J , hence :
n ~ 1 == A !L] = A n 2 = 7
""2 ~ 8 3 12 4 16
taking no account of the signs and we get:
arithmetical mean of the 4" differences X T 7 g =
corrected third difference 2 51". 3 X ^ = I ll". 4
corrected second difference 22 43". 8 X f = 8 31". 4
corrected first difference 7 13 39". X . , = 1 48 24". 7,
hence the longitude for Jan. 4 . 5
318 55 54". 2.
If we wish to find the longitude for Jan. 5.5, we have
to apply formula (3 a) 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
322 36 56". 7.
In order to make an application of formula (4 a) we will
now find the longitude for Jan. 5 . 0, and get:
arithmetical mean of the 4 th differences X T 3  6 = 1". 4
arithmetical mean of the 2 d differences X ^ = 2 52". 3
arithmetical mean of the functions = 320 48 34". 7
hence the longitude for Jan. 5.0
320 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 SIT" r29 . 5
4.5 318 5554 .2 * hl 23".5 _ _
5.0 3204542.4 126.1 + ,/
5.5 322 3656 .7 128.9 2 8
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 /"(a), the table of the diffe
rences will now be as follows :
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 er
roneous 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 nu
merical 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 ac
cording to the powers of w, we find:
/(oHnuO =/(a) f n[f (a 4^) /" (a 41) + j
+ ^2 [/" Ca H 1) / " (a + f) 4
1.2.3 Ly
but as we have also according to Taylor s theorem:
/v > /v^^/M ,d*f(a)n*w>d f(a)n U ,>
/C + 0=/C) + i_ B , + , i;  + Ta  r 1^3 + ...
we find by comparing the two series:
VQ = JL [/ ( f i)  /" (a + 1)+ I/ " (afi)  ...]
^ = 1 [/ ( + 1) /" (a K) + ...].
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:
etc.
we find:
/(a+nu,) =/() + / (a) 4 ^/ () + ( ^^=^ ) /" (a)
(^D^CnLt)
1.2.3.4 /
This formula contains the even differences which are on
the same horizontal line with /"(a), and the arithmetical mean
of the odd differences, which are on both sides of the hori
zontal line. Developing it according to the powers of n we
obtain :
/(a4nu;)=/(a) + n [/ (a)  J : / "(a) + ^f v (a)  T io/ VI1 (a) + . . .]
H Y~ 2 If" W ~ A / v (o) H F O / VI () ]
+  f/" (a) ~ ^ V (a) + ^ /vn (a) "  ]
and from this we find:
etc.
If we wish to find the differential coefficient of a function,
which is not given itself, for instance of f(a\nw\ we must
substitute in these formulae a\n instead of a, so that we
have:
29
tfI t0 . P ,
, J , /" IV (afn) h .. . ,
. .>
a a z
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 \ ri) for instance this compu
tation is simple, as we find these by the ordinary formulae
of interpolation, considering merely now /" (fl), f"(atri) 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:
/ (0 + ) =
2
and according to formula (2) in No. 14:
/ (a  4 h n) =/ (a  f) + / (a) 4 ^^/" (a
(n+l)(nl)
1 .2.3
/ (aHi) 4 /" (a) H
1.2.3 ~ J
therefore taking the arithmetical mean of both formulae we
find the following formula for the interpolation of an arith
metical mean:
) =/ (a) 4 nf" (a) 4 "/" (a) 4 { nf" (a)
The two terms:
arise from the arithmetical mean of the terms:
n (n 1)
iT^ / ( I)
and
which gives:
l^/" () H ^ f/" (a 4 ) /" (a  ])].
30
Combining the two terms, which contain f lv (a), we may
write the above formula thus:
/ ( aH _ w ) =/ () + / (a) h y / " (a) + ^/^ () H (7)
The formulae 5, 6 and 7 may be used to find the nu
merical values of the differential coefficients of a function for
any argument by using the even differences and the arith
metical 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 in
troduce in formula (3) in No. 14 the arithmetical means of
the even differences by the aid of the equations:
/() = /(a + J) i/(oHj)
etc.
we find, as we have:
(nhl)n(n 1) _ , n (n 1 ) = n (n 1) (n 
1.2.3 1.2 1.2.3
etc.
If we write here w~h instead of w, the law of the co
efficients becomes more obvious, as we get:
/[+ (n hi) w] =f(a H 1) h / ( h D + /" (a + i)
(!^i^^
Developing this formula according to the powers of w,
we find the terms independent of n:
hence :
31
/[a + + 1) w] =/( h { w)
l920 /VII(a+4)   ]
Comparing this formula with the development of f(a\\w+ nw)
according to Taylor s theorem, we find:
(8)
etc.
These formulae will be the most convenient in case that
we have to find the differential coefficients of a function for
an argument, which is the arithmetical mean of two successive
given arguments. For other arguments, for instance a+(n}Qw
we have again:
, 1
=/ ( + 1 *^) / (aH + n)
da
etc.
Here we can compute the difference f (a{\\ri) as well as
all odd differences by the ordinary formulae of interpolation.
But as the even differences are arithmetical means, we must
use a different formula, which we may deduce from the for
mula (7) for interpolating an arithmetical mean of odd diffe
rences by substuting a h \ instead of a and increasing all
accents by one, so that we have for instance:
TZ
/ 1V (a h
Example. According to the Berlin Almanac for 1848
we have the following rightascensions of the moon.
32
I. Diff. II. Diff. III. Diff. IV. DifF.
Juli 12 O h
12h
I6 h 14 ra 26 s
39 30
.33
.32 "
h 25 3s
.99_
j_ 23 s
.75
25 27
13 O h
14 Oh
12"
17
18
4
30
56
23
58
48
58
p
.06
. 16
.38
.69
on
2550
26 10
2627
2640
.22
.31
.70
22
20
17
13
.36~
.12
.09
.39
3
3
.03
.70
15 O h 50 6 .39
If we wish to find the first differential coefficients for
July 13 10 h , II 1 and 12 1 and use formula (9), we must first
compute the first and third differences for 10 h , ll h and 12 h .
The third of the first differences corresponds to the argument
July 13 6 h and is /" (a hi)? we have therefore for 10 h , ll h
and 12 h n respectively equal to *, ^ and \. Then inter
polating in the ordinary way, we find:
10h +25 57s. 11 2s. 51
llh 25 58 .81 2 .58
12h 26 . 49 2 . 64
and from this the differential coefficients:
for 10h +25^573.21
llh 25 58 .92
12h 26 . 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:
10 h 2 99. 77
ll h 9 .91
12h 10 . 05,
which are the hourly velocities of the moon in rightascension.
If we had employed formula (6), where the arithmetical
means of odd differences are used, taking a = Juli 13 12 h ,
we would have found for instance for 10 h , where n is J,
according to formula (7) :
f (a ^) = + 2556s.77 and / "(a ) = 2 . 51
and from these the differential coefficient according to for
mula (6) equal to 42 m 9 s .77.
The second differences are the following:
for 10h j 20s. 55
llh 20 .34
12*> 20 . 12.
33
If we add to these the fourth differences multiplied by
P> and divide by 144, we find the second differential co
efficients
for 1O I s . 1432
lib .1417
12h . 1402.
where again the unit of time is one hour*).
C. THEORY OF SEVERAL DEFINITE INTEGRALS USED IN
SPHERICAL ASTRONOMY.
16. As the integral le ~dt, either taken between the
limits and co or between the limits o and T or T and oo,
is often used in astronomy, the most important theorems re
garding it and the formulas used for its numerical compu
tation shall be briefly deduced.
The definite integral \e~^dt 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 :
le x .x" dx = F(a\ (1)
o
where a always is a positive quantity, and as we may easily
deduce the following formula:
\e x .x" ~ { dx = \e x d(^"^ = e x . *" f * fx a e x dx
and as the term without the integral sign becomes equal to
zero after the substitution of the limits, we find:
CO <X
fir* . x a ~ l dx = fe*. x"
J a J
dx
or: ar(a) = r(a+l} (2)
But as we have also:
*) Encke on interpolation and on mechanical quadrature in Berliner
Jahrbuch fur 1830 und 1837".
3
34
it follows, that when n is an integral number, we have:
F(n} = (n \}(n 2)(n 3).... 1.
If we take in the equation (1) x = J 2 , we find:
o
hence for a = \ :
fe 2 .d/ =
I
In order to find this integral, we will multiply it by a
r
similar one \e~ yl dy, so that we get:
( (>,/, ). = f ," rf , J> d , = Jj>" 2+ " 2) " rf*.
(I I) II tl
Taking here y = x t , hence d/ = t . dx , we find :
or as:
we find:
( I e~ 2 d ty = \ I  = ^ (arc tang GO arc tang 0) = >
(i ii
hence :
From this follows JTQ) = J/TT, hence from equation (2):
r() = /7r, r (I) = 1/7T 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:
hence :
*V ^ = . (4)
35
17. To find the integral le^dt, various methods are
used. While T is small, we easily obtain by developing
< 2 ,, T 3
X
and as we have \e~ *dt= > we also find from the above
formula the integral \e~ li dt.
This series must always converge, as the numerators in
crease only at the ratio of T 2 , while the denominators arc con
stantly 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 from 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:
.
or integrating by parts:
,

By the same process we find:
>~ /2 ) dt ~ rl
j in , , e
or finally
^^=_ e ~ /2 ri l
2t L 2<
1.3.5....(2n + l) f t*
2"+ J e
r e
J
_*2 rf<
3
36
or after substituting the limits:
f
, _e~ T i [ 1 _ l.3_ 1.3.5
= 2 T L 27 12 (2r 2 ) 2 (27 12 ) 3
1.3. 5. ...(2?il) 1.3.5.... (5
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 in
crease more than the denominators. But if we consider the
remainder :
hl) C
J t
we can easily prove that it is smaller than the last pre
ceding term. For the value of the integral is less than
&
,11
multiplied by the greatest value of e~ 2 between the limits T
and OD which is e~ /12 , and as we have:
A = _ L. _1
J /"+ 2n+l T 2 "
r
the remainder must always be less than:
1.3.5...2n 1 _
Now this expression is that of the last preceding term
with opposite sign, so that if the last term 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:
x dx = 7, (a)
J
/
we find :
37
rf7
df<
_ < 2 / X 2 2
= 2te I e dx e
t
= 2* 71. (/?)
Now the n ih differential coefficient of a product is:
d .xy __<*.* d"  * dy , (n 1) e* 8 * *Py ,
n " rf^" 1 rfir " 7 " 1.2 rfr 2 rf^ 2
hence we have:
c/" +1 77 rf 7
If we denote the product 1.2.3  n by w/, we may write
this equation thus:
= 2 o
r = "
or denoting 777 by U n :
(n H 1) 6 7 rt+ i = 2 * / 4 2 7 w _i.
This equation is true for all values of n from n = 1,
when t/ () is equal to the function U itself. We find from it:
hence :
But we have from equation (/9):
~  = 2t ,
hence :
1
2<
o j i
" U
and from equation (; ) follows:
1

2* Z7,
38
If we substitute this value in the former equation and
continue the development, we find:
1 + 3
1 H etc.,
therefore , taking ^^ = g
(7)
143?
144?"
1 4 etc.
By one of the three formulae (5), (6) or (7) we can
always find the value of the integral Ie~ f2 dt or ie~ i2 dt, but
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:
/J.**,
from which the other forms are easily deduced. The first
part of this table has the argument T and extends from T=
to T=l, the interval of the arguments 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 ar
guments the computation by formula (6) is very easy.
18. The integral
 dx
39
can be easily reduced to the one treated above. For if we
introduce another variable quantity, given by the equation:
,
,
the above integral is transformed into:
2 1
from which we have dx= dt,
if we take : T= cotang } ^ .
If now we introduce the following notation
we have : I ^ ^=: dx = } j ^H (8)
and also :
If we diflPerentiate the expression e~ x Vcos^ 2 f^ n x
ft
with respect to x and then integrate the resulting equation
with respect to x between the limits and oo, we easily find :
where T= cotang t
And as we have by formula (9)
o
P
we find:
9
J \l 52 i ^ S111 =>
of which formulae we shall also make use hereafter.
(10)
40
D. THE METHOD OF LEAST SQUARES.
19. In astronomy we continually determine quantities
by observations. But when we observe any phenomenon re
peatedly, we generally find different results by different ob
servations, as the imperfection of the instruments as well
as that of our organs of sense, also other accidental ex
ternal 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 ob
servations 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 ob
serving 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 con
sider 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 ob
servations 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 quan
tities have the form of linear equations. For although in ge
neral the function /"(, ?/, L, etc.) which expresses this relation
between the observed quantities and the unknown quantities
, ?/, C, will not be a linear function, we can always procure
approximate values of the unknown quantities from the ob
servations and denoting these by , ?; , and f and assuming
that the correct values are {.T, ^o4y? Jo ~+" z etc., we
find from each observation an equation of the following form :
,... 9 , ,
provided that the assumed values are sufficiently approximate
as to allow us to neglect the higher powers of ic, ?/, z etc.
Here /"(, r^ ...) is the observed value, /X , >/, ...)
the value computed from the approximate values, hence
tfco o ) f(i Vi f ) = n is a known quantity.
Denoting then ^ by a, f ~ by 6, by c etc. and distinguish
ing these quantities for different observations by different ac
cents, we shall find from the single observations equations
of the following form:
= n  a x + l>y + c z f . . . ,
= n + a x h //y + r z f . . . ,
etc.,
where a?, ?/, a ... are unknown values, which we wish to de
termine, 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 as there are ob
servations and their number must be^as great as possible,,
in order to deduce from them values of a;, */, 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 dif
ferent 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 fol
lowing 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, an
other equation resulted, in which the factor of x was the
largest possible. In the same way equations were deduced,
in which the coefficient oft/ and z etc. was the largest pos
sible 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 ac
curacy of the values obtained. If the observations were per
fectly 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
#, y and z\ therefore denoting the residual error by A^ we
ought to write these equations thus:
A = n 4 ax} byi cz,
/y =,/+ * 4 />V + cX
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 magni
tude of the error, which expresses how often any error oc
curs. If the number of observations is TW, and an error of
the magnitude A occurs according to this law p times,
43
expresses the probability of the error A 5 and shall be de
noted by (/(A). This function </ (A) must be therefore zero,
if A surpasses a certain limit and have a maximum for
/\ = 0, besides it must have equal values for equal, positive
or negative values of A As we have p = m y (A) , there
will be among m observations m<f (A) errors of the magni
tude A? likewise my (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:
. i.
This sum being that of all errors must be taken between
certain limits k and f k , but as according to our hypo
thesis <^(A) is zero beyond this limit, it will make no dif
ference, if we take instead of the limits k and {k the
limits oo and + oo. But as any A between these limits
are possible,, as we cannot assign any quantity between the
limits k and t&, which may not possibly be equal to an
error, as therefore the number of possible errors, hence also
the number of the functions </) (A) is infinite, each cf (A) must
be an infinitely small quantity. The probability that an error
lies between certain limits, is equal to the sum of all values
f(A) which lie between these limits. If these limits are in
finitely near to each other, the value rp (A) may be considered
constant, hence </)(A).dA expresses the chance, that an er
ror lies between the limit A and A H ^A The probability
that an error lies between the limits a and 6, is therefore
expressed by the definite integral
1 9 (A) . </A
and we have according to the formula found before:
According to the theory of probabilities we know, that
when r/>(A), ^ (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 se
ries of observations the errors A? A ) A" etc. occur, we have:
Therefore if for certain assumed values of a?, ?/, z the
errors A? A , A" etc. express the residual errors of the equa
tions (1), W is the probability that just these errors have
been made and may therefore be used for measuring the pro
bability of these values of ,T, y and z. Any other system of
values of x, y and z will give also another system of resi
dual errors and the most plausible values of a?, 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 W itself is a maximum. But in order to deter
mine, when (f (A) is a maximum, it is necessary to know the
form of this function.
Now in the case that there is only one unknown quan
tity, for which the m values w, 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 :
4 n f n" 4 . .
x =
m
or: n _ a .__ n _ ar __ n _ a ..... == o j 0)
where n x, n x etc. correspond to the errors A, so that
we have n x = /\, n x = /\ etc. But as W is a maximum
for the most probable value of a?, we find differentiating equa
tion (2) in a logarithmic form:
dx d{\ dx
rfA = rfA
c?:r JJT
and as in this case we have * = = etc. = 1, we find
.* f/.r
or:
(,) d :]?8fAT^ +(_,) J^2SJ^=^ +....0. W
(n x) d . (n a?) (n a?) d. (n x)
But as according to the hypothesis the arithmetical mean
gives the most probable value of a?, the two equations (a)
and (6) must give the same value for a?, hence we have:
1 c/.logyCn a?) _ 1 ( !_^ o S ( p( n _ x ) _ etc __ ^
n x d(n x) n 1 x d(n x)
45
where k is a constant quantity. We have therefore the fol
lowing equation for determining the function
d_> log y (A)_ _ ,
A.rfA
hence
logy (A) = ?A 2 4logC
and
The sign of k can easily be determined , for as y (A)
decreases when A is increasing, k must be negative; we may
therefore put \k= ft 2 , so that we have q(/\^=Ce **^*.
In order to determine C we use the equation:

and as we have ie~ x * dx = J/TT, we get le~* a ^ a d/\ == ,
00 Of)
hence ^==1 or 0= and finally:
The constant quantity ft remains the same for a system
of observations, which are all equally good or for which the
probability of a certain error /\ is the same. For such ,
system the probability that an error lies between the limits
rV and frV is:
hS
Now if in another system of observations the proba
bility of an error /\ is expressed by  / e~ , in this sys
tem the probability that an error lies between the limits _ <Y
and Hd , is:
+ +h
Both integrals become equal when h <) = h rV. Therefore
if we have h = 2ft , 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 obser
vations.
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 f r, must be equal to \. Hence we have
the equation:
A_ C W* = ^
r
or taking h^ = t
hr
dt = 4, therefore  e~ l dt = 
J
o n
I/ TT
But as the value of this integral is = 0.44311, when
hr = 0.47694 *), we find the following relation between r
and h:
0.47694
nhr
9 r
The integral , Ie~ t2 dt gives the probability of an er
ror, which is less than n times the probable error and if we
compute for instance the value of this integral for n = \,
taking therefore nhr = 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 1000 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 , 2, , 3, J, 4, , 5, that among 1000 observations there
ought to occur:
) On the computation of this integral see No. 17 of the introduction.
47
688, where the error in less than fr
823, 2r
908, . r
956, 3r
982, \r
993, 4r
998, fr
999, 5r,
and comparing with this a large number of errors of obser
vations, 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 etc., the probability that these occur together is:
A AMAA+A A +A"A"+....]
= ^ C
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 ft, 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 [A A]? we have:
**.*"],
and we easily find the following conditional equation for the
maximum :
hence follows : 1
h\/2
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 f, or put:
48
we have:
and: / = 0.47694 / 2 e
r = 0.074489 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 y> (A),
<pGY) etc. their expressions according to equation (3), we
find:
A" A 2 [A 2 +A 2 +A" 2 + ...]
"gF
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 #, y and a.
But as for the most probable values of a?, y and z the pro
bability that the then remaining errors have occurred to
gether, 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:
A 2 H A 2 + A" 2 h = minimum
or the sum of the squares of the residual errors in the equa
tions (1) must be a minimum. Hence this method to find
the most 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 = x ??, i\ ==x n, l \ = x" w", etc.
We get therefore for the sum of the squares of the re
sidual errors, if we denote
49
the sum of n \ri \n" J... by [n]
the sum of w 2  n >2 \ w" 2 {... by [n n]
and the number of observations by m:
nY = mx* 2x[n] + [nr>]
As all terms of the second member are positive, the
sum of the squares will become a minimum, when:
and the sum of the squares of the residual errors will be:
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 a?, x etc., if the probable errors of the single quan
tities a;, x etc. are known.
If r is the probable error of x and we have the simple
function of x:
X = ax,
it is evident, that ar is the probable error of X. For if x
is the most probable value of a?, ax <} is the most probable
value of X and the number of cases, when x lies between
the limits x r and a? Hr is equal to the number of cases
in which X lies between a? ar and aa? +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= and h = c ,,
where c is equal to 0.47694, we have the probability of any
value of x:
50
and the probability of any value of x :
hence we have the probability that any two values x and x
occur together:
We shall find therefore the probability of two errors x
and x whfch satisfy the equation X=*x\x\ if we substitute
X x for x in the above expression and denoting this pro
bability by FT, we get:
W= r e
rr 7t
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 oo and \ oo,
or in other words if we integrate W 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 :
/
"
dx
2 C *
if we put :
~ r*(X a)hr >a a>
rr
and as we have
we find the probability of any value of X:
&&**
51
But this expression becomes a maximum, when X = 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
J/ r 2_j_ r 2 From this follows in connection with the formula
proved before, that when:
the probable error of X is equal to Va z r 2 f a 2 r 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 H a x h a"x" + ....,
and if r, r , r" etc. are the probable errors of re, x\ x" etc.
the probable error of X is equal to:
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:
we have the probable error of the mean equal to j/ m .  a
r
or .
Vm
The probable error of the arithmetical mean of m obser
vations is therefore to the probable error of a single obser
vation as : 1 or its measure of precision to the measure
V m
of a single observation as h]/m:h. Often the relative accu
racy 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 m. Hence the weights of
two quantities are to each other directly as the squares of
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 = & of the original
equations after substituting the most probable value of x were
the real errors of observation, the sum of their squares di
vided by m would give the square of the mean error of an
observation according to No. 20, or this error itself would
be T/fclJ. But as the arithmetical mean of the observations
r m
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 re
sidual errors will not be the real errors of observation and
differ more or less from them. Now let x () be the most pro
bable value of x as given by the arithmetical mean, while
# () { ma y be the true value which is unknown. By substi
tuting the first value in the equations we get the residual
errors o? w, x l} ri etc. which shall be denoted by A? A
etc. while the substitution of the true value would give the
errors a? r 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] 4 >P = [<?<?],
which equation shows that the sum of the squares of the
residual errors belonging to the arithmetical mean is always
too small.
As we have [<)c)] = W 2 , when denotes the mean error
of an observation and further [A A] [n %] , we " can write
the equation also in the following form:
*) If therefore two quantities have the weights p = ^ and p = j^
1 pp
the weight of their sum is = ,^=
2__ a
53
Although we cannot compute from this equation the va
lue of , as 2? is unknown , still we shall get this value as
near as possible, if we substitute instead of g the mean error
of x and as we have found this to be equal to
thus :
,
y m 7
we find
for the mean error of an observation and hence the probable
error :
r 0.674489  1
r m
Furthermore we find the mean error of the arithmetical
mean :
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
2 m 43 s
. 60
0.11
Star 16
2m 43s .
50
0.01
2
43
. 49
0.00
17
43 .
44
hO.05
3
43
. 63
0.14
18
43 .
37
40.12
4
43
. 52
0.03
19
43 .
32
40.17
5
43
. 31
40.18
20
43 .
12
40.37
6
43
. 67
0.18
21
43 .
30
40.19
7
43
. 98
0.49
22
43 .
72
0.23
8
43
. 63
0.14
23
43 .
25
40.24
9
43
. 83
0.34
24
43 .
13
4 0.36
10
43
. 79
0.30
25
43 .
27
40.22
11
43
. 54
0.05
26
43 .
34
40.15
12
43
. 18
40.31
27
43 .
15
4 0.34
13
43
. 45
40.04
28
43 .
86
0.37
14
43
. 68
0.19
29
43 .
29
40.20
15
43
. 32
40.17
30
43 .
40
40.09
31
43 .
95
0.46
Mean 2 m 43 s . 49
* 54
Here we find the sum of the squares of the residual
errors [wJ =1.77, and as the number of observations is 31,
we find:
the probable error of a single observation ==b s . 164
hence the probable error of the mean of all observations
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 num
ber of errors
smaller than r, r, f?*, 2r, fr, 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 er
rors 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 pro
bable 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 3, this condition furnishes as many equa
tions 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 :
... )
ax ax
or as we have according to equations (1) ^=a,  =a etc.
we get:
A + AV + A"a"h... = 0.
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 f a a f a" a" + . . . = [a a]
and a 6 4 a b + a" b" f . . . = [a b] etc.
we get the equation:
[a a] x h [ab] y f [ac] z f [aw] = 0; (4)
and likewise [ a &] x + [bb]y+ [b c] z 4 [6 n] = o (5)
and [rt C ] * j_ [^ c ] y  [ c c ] z 4 [ cw j = 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 3.
In order to solve them we multiply the first by
J
[aa]
and subtract it from the second, likewise we multiply the
first by p and subtract it from the third. Thus we obtain
two equations without #, which have the form:
[66 I ]y + [6c 1 ]+[6i I ] = (D)
when we take
[Ml ] []_ fe^ , [6c,] =[c]  fe^
which equations explain the adopted notation.
If we multiply now the equation (D} by ~p and sub
tract it from (JS), we find:
[cc a l*H[cw a ] = (F),
where we have now:
From equation (F) we find the value of 3, while the
equations (D) and (^4) give the values of y and x.
If we deduce [A 2 ] from the equations (1) we find with
the aid of equations (4), (5) and (C) for the sum of the
squares of the residual errors:
[^2] _ [ ww ] + [ fln ] x _}_ [ 6n ] y __ [ cw ] 2<
In order to eliminate here #, / and 3, we multiply equa
tion ^1 by  ^j and subtract it from the above equation, which
gives :
= [nn]  Cn  + [6m]y H[cn,] *.
If we then multiply the equation (/>) by ~ and sub
56
tract it from the last equation, we get:
and if we here substitute the value of z from (F) we find
at last for the minimum of the squares of the errors :
, , [an] Q..P [cn 2 ] 2
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:
[A A] = [ A] * + [ft A] y + [< A] 4 A] (a),
where [ A] = [a a] x 4 [a 6] y H [a c] 2 4 [a n\ (ft)
etc.
If we now substitute in (a) instead of # its value taken
from (6), we find:
where
Then substituting in (c) for y its value taken from the first
of the equations (d), we find:
[A A] = j^r 4 n^f + t c A 2 ] + [n A 2 ], (c)
where now
and if we finally substitute in (e) for 3 its value taken from
the first of these last equations, we have:
and we easily see that we have [Aa] = [ WW J
As the first three terms on the right side of equation (#),
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 [/\] = 0,
[6/\ 1 ] = and flA 2 l 0, which are identical with those we
found before. We see also, that [w/? 3 ] 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 un
known quantities, as we easily see by the equations A^ D
and F that the most probable values of .T, y and z can be
expressed by linear functions of w, ri, 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 3 in the re
sulting equation become equal to zero. Therefore if we mul
tiply (A} by * , (D) by j , (F) by =4 ] and add the
three equations, we get the following two equations for de
termining A and A":
and we have:
In order to find y we multiply (D) by f , (F) by r ~ and
Lo]J L C>C 2J
adding them we get :
"
and . 
At last we have:
__z J// x<
[aa] ~~^
Developing the quantities [ftwj] and [cw 2 ], we easily find:
[&n,]=4 [an]f[6w] (77),
[cn 2 ] ==^"[aw] f 5 [6n] +[cw] (5 1 ),
and as we may change the letters, the quantities in paren
thesis being of a symmetrical form, we find also:
[&&,]= .4 [&] + [& 6] (0,
[c c 2 ] = A" [a c] f 5 [6 c] f [c c] (x),
[6 c 2 ] = A" [a 6] h B \b 6] + [6 c] = (A),
[a c 2 ] = yl" [ ] + & [ a &] + [ a c ] = Q (^). * )
*) The two last equations we may easily verify with the aid of the
equations (a), (/) and (8).
58
Now as [an] as well as [6%]. and [c 2 ] are linear func
tions of n, we can easily compute their probable errors. First
we have [a n] = a n + a ri h a" n" + If therefore r de
notes the probable error of one observation, that of [an]
must be:
r ([an]) = r J/7?a~4Va 4~ a" a" 4 . . = r V[aa\.
Every term in \bn^\ is of the following form (A 1 r6)w.
In order to find the square of this, we multiply it success
ively by A an and bn and find for the coefficient of ir\
A (A a a 4 a fi) 4 A a b + 1> b.
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 [&wj or we have:
[6 Wl ]) = [_A (A[aa] 4 [aft]) 4 A [ab] 4 [66]] r 2 ,
or: r([6,])=rYp 1 ],
as we find immediately by the equations () and (<.).
At last the coefficient of each n in the expression of
[cn. 2 ] is:
Aa + Bb +
Taking the square of this we find:
A"(A"aa\ B ab
Now taking the sum of all single squares, we find the
coefficient of / in the expression of (r[cw. 2 ]) 2 :
A"(A"[aa] + B [ab] + [ac] )
4 B 1 (A" [a b] 4 B [bb] 4 [6 c])
which according to the equations (x), (A) and (/<) is simply
[cc 2 ]; hence we have:
r[cw 2 ] = /. K[cca]
We can now find the probable errors of x, y and a without
any difficulty. For according to equation (7) we have for
the square of the probable error of x the following ex
pression :
A>A> A " A "\
[66 l ]" + " [cc a ]i*
59
Likewise we find:
K</)] 2 => 2 j
aild [r(z)] 2 =r 2
It remains still to find the probable error of a single
observation. If we put for x,.y and z in the original equa
tions (1) any determinate values, we may give to the sum
of the squares of the residual errors the following form:
In case that we substitute here for #, y and z the most
probable values resulting from this system of equations, the
quantities [a A] 5 [^AJ and [ C A2J become equal to zero and
the sum of the squares of the residual errors resulting from
these values of #, y and z is equal to [wwj. But these val
ues will be the true values only in case that the number of
observations is infinitely great. Supposing now, that these
true values were known and were substituted in the above
equations, [A A] would be the sum of the squares of the
real errors of observation and we should have the following
equation :
[aa] [bb,] [cc 2 ]
where now the quantities [a A] 5 [&AJ and [cA2J 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 [a A] etc. their
mean errors. But as in the equations:
ax 4 by 4 cz f n = A
etc.
no quantity on the left side is affected by errors except ft,
A must be affected by the same errors and the mean errors
of [a A] 5 [&Ai] and [cA 2 ] are equal to those we found for
[aw], [6wj] and [cw 2 ]. Substituting these in the above equa
tion we find:
  3
60
Hence the mean error of an observation is derived from
a finite number of equations between several unknown quan
tities by dividing the sum of the squares of the residual er
rors, 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 an obser
vation :
0.674489
m 3
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" etc. are the mea
sures of precision for the single observations, the probability of the errors A,
A etc. of single observations is expressed by:
h A 2 A 2 h 7/ 2 A 2
V e y/
Hence the function W becomes in this case:
h.h .h"... (/, 2 A 2 +A A 2 +/<" 2 A" 2 + ..0
"orav 1
and the most probable values of or, y nnd z will be those, which make
the sum
7,242 __ 7/2 A 2 fA" 2 A" 2 4....
a minimum. In order therefore to find these, we must multiply the original
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 equa
tions have the following form:
= n t ax,
= n Ho *,
0=w"frt"ar, etc.,
we find x= r  with the probable error r r = , where r denotes
[] V(aa\
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:
Residua]
errors.
a
Urs. min.
=
40
.02 +x 4
0.2?,
&
.03
ft
Urs. min.
=
40
.454
x
4
8.23,
4
.43
ft
Cephei
=
40
. 104
X
4
20.13,
40
.14
a
Urs. maj.
=
0
.144
X
4
36.03,
.03
a
Cephei
=
0
.624
X
4
43.93,
.47
d
Cephei
=
0
.254
X
4
65.9^/
.00
8
Cephei
=
0
.034
x
4
74.93,
4
.26
ft
Cephei
=
 1
.244
X
4
77.83,
.94
a
Cassiop.
=
40
.594
X
4
75.53,
40 .88
y
Urs. maj.
=
0
.474
x
4
79.63,
. 16
ft
Draconis
.004
X
4
104.53,
4
.42
y
Draconis
=
0
.514
X
4
114.33,
.04
y
Urs. maj.
=
 1
.204
X
4
125.63,
.68
a
Persei
=
40
. 12 4
X
4
142.13,
40
.72
a
Aurigae
=

.314
X
4
216.83,
.37
a
Cygni
=
 1
.644
X
4
254.83,
.53
8
Aurigae
=
1
.394
X
4
280.23,
.16
y
Androm.
=

.244
X
4393.53,
4
.51
17
Aurigae
=

.804
X
4419.6^
4
.06
ft
Persei
=
2
.164* 4481.23,
.01
In order now to find from these the equations for the
most probable values of x and y (equations (A) and (/?) in
No. 23), we must first compute all the different sums [a a],
[a 6], [aw], [66] and [few]. In this case, where the number
of unknown quantities is so small, besides one of the coef
ficients is constant and equal to one, this computation is very
easy; but if there are more unknown quantities, whose co
efficients may be for instance a, 6, c, d it is advisable, to
take also the . algebraic sum of the coefficients of each equa
tion, which shall be denoted by s and to compute with these
the sums [as], [6s], [cs] etc., as then the following equations
may be used as checks for the correctness of the compu
tations :
[ns] = [an] 4 [6w] 4 [en] 4 [rfn],
[a^ = [a a] 4 [a 6] 4 [ac] 4 [ad],
etc.
62
If we compute now the sums for our example, we find
the following two equations for determining the most pro
bable values of x and y:
4 20.000 x 4 3014.80 y 12.72 = 0,
4 3014.80 x 4 844586.1y 3700.65 = 0.
The solution of these equations can be made in the fol
lowing form, which may easily be extended to more unknown
quantities :
[a a] [a 6] [an] [wn]
420.000 43014.80 12.72 20.28
1.301030 3.479259 1.104487, ^ 8.09
Ian] =12.72 [66] [6n] 12.19
[a 6]* = 4 13.78 4844586.1 3700.65 ^~ 8.15
[*&J
4 1.06 4454452.0 1917.41 [wn 2 ] = 4.04
0.025306,, [66,] = 4390134.1 [few,] = 1783.24
1.301030 log [6n,] 3.251210
log* = 8.724276,, log [66,] 5.591214
x = 0". 053 log y = 7.659996
y = 4 0.0045708
In case that we have computed the quantities [as], [bs] etc.
we may compute also [6*J and use the equation [66 1 ] = [6sJ
as a check. In the case of 3 unknown quantities we should
use [66 T ] } [6cJ = [6*J and [ecj = [csj and similar equa
tions for a greater number of unknown quantities.
In order to compute the probable errors of x and y,
we use besides [66,] also the quantity
[a a,] = [a a] ^~ = H 9.2384.
Then we find the probable error of the quantity n for a
single star:
,. = 0.67449 / L  " =0.3195,
hence the probable errors of x and y :
^V ^,^
~  = d=0".0005116.
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  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 [wwj, thus proving the accuracy of the computation by
another check.
Note. On the method of least squares consult: Gauss, Theoria motus
corporum coelestium, pag. 205 et seq. Gauss, Theoria combinationis obser
vationum erroribus minimis obnoxiae. Encke in the appendix to the Ber
liner Jahrbucher fur 1834, 1835 und 1836."
E. THE DEVELOPMENT OF PERIODICAL FUNCTIONS FROM GIVEN
NUMERICAL VALUES.
26. Periodical functions are frequently used in astro
nomy, as the problem, to find periods in which certain pheno
mena return, often occurs; but as these are always comprised
within certain limits without becoming infinite, only such pe
riodical 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= a {a, cos a: + a 2 cos2.r + a 3 cosSx h ...
f 6, sin arf 6 2 sin 2x\ b a sin 3 a: H ...
Now the case usually occurring is this, that the nume
rical 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 # = 0, x=?,
x=== 2 ~ etc  to x = ( n 1) ~, 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, nA being
equal to 2?r, the sum of the series
sin A\ sin 2.4 f s mSA h ... + sin (n I) A
is always equal to zero; likewise also the sum of the series
cos A H cos 2 A f cos 3 A+ . . . cos(n 1)^,
is zero except when A is equal either to 2 n or to a mul
tiple of 2 TT, in which case this sum is equal to w.
The latter case is obvious, as the series then consists
of n terms, each of which is equal to 1. We have there
fore to prove only the two other theorems. If we now put:
2?r "27t
cos r h i sin r = 1 r ,
n n
i
where we take i = Vl and T=e n , we have:
r .,_! r= 1 r n 1
2 9 yj. __, J> 1
2 cos T 4 t 2 sin r = ^ T = ^p
r r O f =
As we have now T" = cos2n{i sin 2rc= 1, it follows
that:
7T . ^i .
, cos ?  h t >, sin r = 0,
* n ** n
r=0
hence : ^ sin r =0 (1)
> =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:
,
cos r =0.
n
r i o
Only when n = 0, the expression r _ 1 takes the form ~^
and has the value w, 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:
>, sin r ~  cos r ^   = 4 ^. sin 2 > =0, (3)
* n n " ~* H
r=0 r=0
2n ^ ^  ?^ = n in general (4)
w
= n in the exceptional case,
65
finally:
r=. 1 / =  1
^n / 2?r\ 2 , XT 2?r
>, I sin r ) = i n ^ >, cos 2 r = 4 in general (o)
* V tt /
r = ) =
= in the exceptional case.
27. We will assume now:
X = cip cos p x f bp sin p x,
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 qx = \a p cos (p + 7) a? H / cos (p q} x
+ \ b p sin ( jo 4 9) or + r bp sin (;? f/) x ,
and if we assign x successively the values 0, A^ 2 A to
(n 1) 4, where A = /*, and add the several resulting equa
tions, 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\<f) A is equal to 2/c^r,
which will receive the factor n. But as A = , we have
n
for the remaining terms piq = kn or p q kn, hence
p = qikn or ={~q+kn. Therefore denoting the value
of X, which corresponds to the value rA of a? by X rA , we
have :
2H
XrA COS q A= a  v + A h
f a a
But as X does not contain any coefficients whose index
is negative, we must take a_ 2 = and get:
[<,+ a lt ~
Here we have to consider two particular cases. For
when q = 0, we have a_ ? = a ? , a_j = ct/jfj etc. hence:
and when w is an even number and q =^n, a^ q is to be
omitted and a (J unites with a rt _, y etc., hence we have also in
this case:
5
66
"^XrA cos^nA = n [i n +3 w + ...], (8)
As : X sin q x = J a p sin (p h </) .r 4 ,, sin (p ?) :r
h 6,, cos (p q) x ^ bp cos ( p h r/) .r,
we find in a similar way:
2 ^ sin ^ ^ = IT t b< i ~ bn i + ba+ i ~ b *" i ~*~ >2 " +l 3 C 9 ^
^^ J
If we take now for n a sufficiently large number in pro
portion to the convergence of the series, so that we can ne
glect 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 to q = \n and the co
efficients of the sines to q = \n 1 , as a larger q gives
only a repetition of the former equations. The larger we
take M, the more accurate shall we find the values of the
coefficients whose index is small, while those of a high in
dex remain always inaccurate. For instance when n=l2
and q = 4, we have the equation :
2K cos 4 x = G (a 4 H 8 + ),
hence the value of 4 will be incorrect by the quantity 8 ;
but if we had taken w = 24, this coefficient would be only
incorrect by a M .
From the above we find then the following equations:
2 ^?
a p = >. XrA cos rpA,
n *"
V X,A sin rp A,
~
, = o
with these exceptions, that for /> and p=\n we must take
L instead of the factor
n 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 360 and multiply it by the
67
cosine ; but the terms of the sine, whose indices are the com
plements to 360 must be subtracted from each other. If
we denote then the sum of two such quantities, for instance
X A +X (n i)A by X A , and the difference X A X ln _i M by X A ,
4
we have: 2 r=$
cip = ^ X,A cos rpA,
n *~ +
r =
2 ^j
bp = ^j X, A sin r p A.
n
Again denoting here the sum or the difference of two
terms of the cosine, whose indices are the complements to
180 ft , by X,A and X,.^, and the sum or difference of two
14 4
terms of sines , whose indices are the complements to 180,
by X r _, and X r . 4 , we have:
h
r=in
a p = ^ X,ACOsrpA, when p is an even number, (10)
11 ^j i_
with the two exceptional cases mentioned before:
j^ X,A cos rp J, when p is an odd number, (11)
2 x?
&/, = >j JTr^sinrp^, when /? is an even number, (12)
^, X,^ sin rpA, when p is an odd number. (13)
r=l
If for instance n is equal to 12, we find:
TT *0 ~~ 3 ~~ 6 ~~ 9
a i i \ X f X 3 cos 30 f X 6 cos 60 > ,
"2 = ^ ^C 4 ^ 3 cos GO X 6 cos 60
( + + ++ +4 +
etc.
>i = ff \ X 30 sin 30 h^ 60 sin604X 90 j ,
(4  4 4
etc.
5*
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 coef
ficients. 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 deter
mining 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 () , X A , X^ A ... X (H i)* are given,
we should have the following equations, supposing that the
function contains only the sines and cosines of the angle
itself: = X H +,,
= XA + "+ a \ cos A f&isin^4,
= XZA+ ~+~ i cos 2 A f 6 1 sin 2 A,
= X(i)Ala \a t cos(n 1)^4 + 6, sin(n I) A,
and according to the method of least squares we should find
for the equations of the minimum, when [cos A] again de
notes the sum of all the cosines of A, from A = to A = n 1,
the following:
na f [cos A] a , + [sin A] b t  pG] = 0,
[cos^l]a h[cos^ 2 ]a, f [sin A . cos A] b , [X A cos A] = 0, (14)
[sin A] a j [cos A sin A] a, + [sin^L 2 ] 6, [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:
a, = ACQB A],
2
b , = [X A sin A],
n
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 i>] is the sum of the squares
of the residual errors, which remain after substituting the
most probable values in the equations of condition, the pro
bable error of one observation is
= 0.67449
n  3
and that of a
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 tie 1 Observatoire Imperial, 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 coordinates 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 coordinates 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 coordinates.
Some of these coordinates 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 con
tinually change and it will be important to study these chan
ges and the phenomena produced by them. As the stars be
sides 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 coordinates,
which are independent of the diurnal motion, it is never suf
ficient, to know merely the place of a heavenly body lyt 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 op
posite direction namely from east to west. If we imagine
at any place on the surface of the earth a line drawn par
allel 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 there
fore visible in the northern hemisphere of the earth is called
the NorthPole of the celestial sphere, while the opposite is
called the SouthPole. 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 Equator. 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 sepa
rates 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 tan
gent to the meridian of a place generates by the rotation of
the earth the surface of a cone, which intersects the ce
lestial 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 Na
dir. 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 cor
responds to the meridian of a place, whose longitude differs
180 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 se
cond 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 culmi
nation, 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. In order to define the position of a star on the ce
lestial sphere, we make use of spherical coordinates. 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 to 360. Instead of the altitude
of a star its zenithdistance 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 coordinates we may also de
fine the position of a star by rectangular coordinates, refer
red 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 situa
ted in its plane, the axis of x being directed to the origin
of the azimuths, and the positive axis of y towards the azi
muth 90 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 coordinates 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 spiritlevel. 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 par
allel 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 an
other 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 zenithdistance 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 coordinates 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 ce
lestial sphere. If we lay a great circle through the pole and
the star, the arc contained between the star and the equator
is called the declination and the arc between the star and
the pole the polardistance 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 ne
gative, 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 zenithdistance in
the first system of coordinates.
The arc of the equator between the declinationcircle of
the star and the meridian, or the angle at the pole measured
by it, is called the hourangle of the star. It is used as the
second coordinate and is reckoned in the direction of the
apparent motion of the sphere from east to west from
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 &, the hour angle k
and in general, when at a certain place a star has the hour
angle , it has at the same instant at another place, whose
longitude is k (positive when east, negative when west) the
hour  angle t j k .
Instead of using the two spherical coordinates, the de
clination and the hourangle, we may again introduce rectan
gular coordinates if we refer the place of the star to three
axes, of which the positive axis of z is directed to the North
pole, 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 me
ridian or the origin of the hour angles while the positive
axis of y is directed towards the hourangle 90. Denoting
then the declination by d, the hourangle by , we have:
x = cos cos ?, y = cos sin t, z = sin S.
Note. Corresponding to this system of coordinates 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 me
ridian, 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 in
strument.
4. In this latter system of coordinates one of them,
the declination, does not change while the hour angle in
creases proportional to the time and differs in the same mo
ment at different places on the earth according to the dif
ference of longitude. In order to have also the second co
ordinate 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 23 degrees, the obliquity of the
ecliptic. The points of intersection between equator and eclip
tic 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 21 st of March and on the 23 d of September reaches
those points *). The points of the ecliptic at the distance of
90 degrees from the points of the equinoxes are called sol
stitial points.
The new coordinate, which is reckoned in the equator
from the point of the vernal equinox, is called the right
ascension of the star. It is reckoned from to 360 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 coordinates, declination and
rightascension, we can again introduce rectangular coordi
nates, referring the place of the star to three vertical axes,
of which the positive axis of z is directed towards the North
pole, 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 rightascensions, the positive axis of y to the
point, whose rightascension is 90 . Denoting then the right
ascension by a , we have :
x" = cos S cos , y" = cos sin , z" = sin d.
The coordinates a and d 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 hourangle
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 O h sidereal time at any place or the sidereal day com
mences when the point of the vernal equinox crosses the
meridian, it is P when its hourangle is 15 or P etc. For
this reason the equator is divided not only in 360 but also
into 24 hours. Denoting the sidereal time by 0, we have
always: < = ,
hence / = a.
If therefore for instance the rightascension of a star is
190 20 and the sidereal time is 4 h , we find t = 229 40 or
130 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 rightascension 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 in
stant the sidereal time at another place, whose difference of
longitude is /?, must be f &, where k is to be taken po
sitive or negative if the second place is East or West of the
first place.
Note. The coordinates of the third system can be found by instruments
of the second class, if the sidereal time is known. In one case these co
ordinates 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
meridianaltitude of the star, if the latitude of the place is known. For such
observations a meridiancircle is used. If such an instrument is not used for
measuring altitudes but merely for observing the times of the meridian pas
sages 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 rightascensions are reckoned cannot be observed itself,
it is more difficult, to find the absolute rightascensions of the stars.
5. Besides these systems of coordinates 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 coordinate,
the longitude, is reckoned in the ecliptic and is the arc be
tween the circle of latitude of the star and the point of the
vernal equinox. It is reckoned from to 360 in the same
direction as the right ascension or contrary to the diurnal
Thus we have 239 18 46". 75
= 15 h , 4 X 14 + 1 minutes, 4x343 seconds and s . 117
= 15 h 57m 15s. 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 se
conds by 4 in order to convert them into degrees and minutes of arc. The
remainders must again be multiplied by 15.
Thus we have 15 h 57 m 15 s . 117
= 225 h 14 degrees, 15 f 3 minutes and 46.75 seconds
= 239 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 A, the latitude
by ft and the obliquity of the ecliptic by s.
If we express again the spherical coordinates ft and A
by rectangular coordinates, 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 90 th degree of longitude,
we have:
x " = cos ft cos I , y " = cos /3 sin ^,, z" = sin ft.
These coordinates are never found by direct observations,
but are only deduced by computation from the other systems
of coordinates.
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 inter
sects the celestial sphere in a great circle called the ecliptic. Hence the lon
gitude of the earth seen from the sun differs always 180 from that of the
sun seen from the earth. The axis of the earth makes an angle of 665
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 cy
linder, 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 23^ . Likewise the equator is carried around the sun par
allel 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.
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
COORDINATES.
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 coordinates 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 cp
designates the latitude), as the axes of y of both systems
coincide. We have therefore according to formula (la) for
the transformation of coordinates, or according to the for
mulae of spherical trigonometry in the triangle formed by the
zenith, the pole and the star*):
sin 8 = sin <f> sin k cos <p cos h cos A
cos sin t = cos h sin A
cos 8 cos t = sin h cos y> f cos h sin^P cos A.
Iii order to render the formulae more convenient for lo
garithmic computation, we will put:
sin h = m cos M
cos h cos A = m sin M,
and find then:
sin 8 = m sin (<p M")
cos 8 sin t = cos h sin A
cos 8 cos t = m cos (y> M}.
These formulae give the unknown quantities without any
ambiguity. For as all parts are found by the sine and co
sine, 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 con
struction amounts to this, that the oblique spherical triangle
*) The three sides of this triangle are respectively 90 /?, 90 8 and
90 (f and the opposite angles t, 180 A and the angle at the star.
80
is either divided into two rightangled triangles or by the
addition of a rightangled 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 y,
and as we have:
tang h = cos A cotang 3/,
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 for
mulae (10) is the cosine of this perpendicular arc itself, since
we have:
sin h = cos P cos 3/,
if we denote the perpendicular arc by P.
We will suppose, that we have given:
<p = 52 30 16". 0, A =16 11 44". and A = 202 4 15". 5.
Then we have to make the following computation:
cos ^4 9.9669481,, m sin 3/9.9493620.
cos h 9.9824139 m cos 3/9.4454744
sin A 9.5749045,, 3/= 7^35*54^61
sin 3/9.9796542,,
<p 3/=1256 10".61
sin (y 3/) 9.9128171 cos S sin t 9.5573184,, sin S 9.8825249
m 9.9697078 cos <? cos * 9.7294114,. cos S 9.8104999 _
cos (<p 3/) 9.7597036,, t = 2 1 3 56 2.22 3 = +49 43 46.~00
cos* 9.9189115..
7. More frequently occurs the reverse problem, to con
vert the hour angle and declination of a star into its azi
muth and altitude. In this case we have again according to
formula (1) for the transformation of coordinates:
sin h = sin <p sin 8 + cos <p cos S cos t
cos h sin A = cos S sin t
cos h cos A == cos (p sin S 4 sin y> cos S cos t,
which may be reduced to a more convenient form by introdu
cing an auxiliary angle. For if we take :
cos S cos t = m cos 3/
sin S = m sin 3/
we have:
sin h = m cos (<p 3/)
cos h sin A = cos sin t
cos h cos A = in sin (<p 3/)
81
cos 3/tang t
or : tang A = 
sin (cp M )
cos A
tang h = *).
tang (cp M)
When the zenith distance alone is to be found, the fol
lowing formulae are convenient. From the first formula for
sin h we find :
QOS z = cos (cp 8) 2 cos cp cos 8 sin 2 ,
or : sin T 2 2 = sin^ (cp $) 2 fcos cp cos 8 sin / 2 .
If we take now :
n = sin \ (cp S )
m = YCOS cp cos 8,
we have : sin j z* = n 2 f H , sin j 1 1 * \
or taking  sin t = tang A
sin 4 z =
COS A
If sin A. should be greater than cos A, it is more con
venient to use the following formula:
m
sin .T z = , sin ^ t.
sin A
In the formula by which n is found, we must use (p ,
if the star culminates south of the zenith , but ti qp 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 /?, we find:
cos \ z . sin 4 (A p) = sin 7^ t . sin (cp f 8}
cos j z . COSY (4 p) = cos ,y . cos T (77 $)
sin T 2 . sin  (^4 f p) = sin ^ Z . cos I (9? H 8)
sin 4 2 . cos^ (A H /?) = cos 7 z . sin ^ (9? $).
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 T z . sin { (p\ A) = cos^ t . cos^j (8 cp)
cos 5 z . COSTJ ( p f A) = sin 5 t . sin (Sicp")
sin \ z . sin A (/> 4) = cos \ t . sin  (8 cp)
sin .] z . cos 15 (p A) = sin 5 t . cos A ($49?).
*) As the azimuth is always on the same side of the meridian 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 y sin f cos cp cos cos t
(6) cos h sin A = cos S sin t
(c) cos h cos A = cos y> sin 8 + sin cp cos cos /.
If we designate now by A and d those values of A
and #, which substituted in the above equation make h equal
to zero, we have :
(d) = sin (p sin $ f cos 9? cos S cos t
(e) sin y4 .= cos $ sin 2
(/) cos A o = cos 90 sin $ j sin 9? cos $ (i cos if.
Multiplying now (/") by cos cf and subtracting from it
equation (rf) after having multiplied it by sin <y, further mul
tiplying equation (/*) by sin <f and adding to it equation (c?),
after multiplying it by cos .7, we find:
cos A Q cos 95 = sin S .
cos A sin 95 = cos $ cos t
sin ^4 = cos ^ sin /.
Taking then:
sin (p = sin y cos B
cos 9? cos t = siny sin Z?
cos f sin = cos y,
we find from the equation (d) the following:
= sin y sin (<? f B)
or: <?<, = 
and from (a):
sin A = sin y sin ($ f JB\
Then subtracting from the product of equations (6) and
(/") the product of the equations (c) and (e) we get:
cos h sin ( A A ) = cos <p sin sin (d ^ ) = cos y sin (S + B}
and likewise adding to the product of the equations (c) and
(/") the product of the equations (6) and (e) and that of the
equations (a) and (d):
cos h cos (yl ^1 ) = cos $ cos <? sin t 1 + sin sin t>" + cos S cos $ cos i 2
*) 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 from the right ascension and the side
real time.
83
Hence the complete system of formulae is as follows:
sin cp = sin y cos B \
cosy cos t = sin y sin B (1)
cos fp sin t = cos y
sin B = cos 4 cos gp \
cos 5 cos = cos A sin y (2)
cos .B sin = sin A n
sin A = sin y sin ($ f B) \
cos h sin (4 ^4 ) = cos y sin ($ f B) )
These formulae by taking D = sin y , C = cos / and
,4 ^4 = u are changed into the following:
tang B = cotg cp cos
tang A = sin y tang t
sin 7i = > sin (B f 5)
tang u = C tan
where D and C are the sine and cosine of an angle ; , which
is found from the following equation *) :
cotang y = sin B tang t = cotang cp sin A .
These are the formulae given by Gauss in ,,Schumacher s
Hulfstafeln herausgegeben von Warnstorff pag. 135." If now
the quantities Z>, C, B and A (} are brought into tables whose
argument is f, the computation of the altitude and the azi
muth from the hour angle and the declination is reduced to
the computation of the following simple formulae :
sin/i = Dsin(B h 8)
tang u = C tang (B 4 S)
A = A \ u.
Such tables for the latitude of the observatory at Altona
have been published in WarnstorfFs collection of tables quoted
above. It is of course only necessary to extend these tables
from t = to t = 6 h . For it follows from the equation
tang A () = sin (f tang /, that A () lies always in the same qua
drant as f, that therefore to the hour angle 12 1 t belongs the
azimuth 180 A. Furthermore it follows from the equations
for B, that this angle becomes negative, when t ;> 6 h or ^> 90 ,
that therefore if the hour angle is 12 h t the value B must
be used. The quantities
*) For we have according to the formulae (2)
cotang <p sin A = sin B tang t.
84
C= cosy s mt and D = J/sin y> 2 
are not changed if 180 t instead of t is substituted in these
expressions. When t lies between 12 h and 24 1 , the compu
tation must be carried through with the complement of t to
24 h 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 aux
iliary angles. As r) represents that value of f), which sub
stituted in the first of the original equations makes it equal
to zero, <y o is the declination of that point, in which the de
clination circle of the star intersects the horizon; likewise is
Fig. i. A the azimuth of this point. Further
more as we have B = J , B j ti
is the arc S F Fig. 1 * ) of the decli
nation circle extended to the horizon.
In the right angled triangle FOK^
which is formed by the horizon, the
equator and the side FK = B, we have
according to the sixth of the formu
lae (10) of the introduction, because
the angle at is equal to 90 cf :
sin (p = cos B sin FK.
But as we have ulso sin (f = D cos #, we see, that D is
the sine of the angle OFK. therefore C its cosine. At last
O 7
we easily see that FH is equal to A and FG equal to u.
We can iind therefore the above formulae from the three
right angled triangles PFH, OFK and SFG. The first tri
angle gives :
tang A = tang t sin P,
the second:
tang B = cotang cp cos t
cotang y = sin B tang t = cotg <f> sinA ,
and the third:
sin h = sin y sin (B + S)
tang u = cos y tang (B + 8).
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, A 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 #, LK by ^<, AL by A H and the cosine and sine of
the angle SLK by C and D:
C tang (h B] = tang u
D sin (h ) = sin #
and t = A w,
where now:
tang . = cotang (p cos .4
tang A = sin y tang ^l
and where D and C are the sine and cosine of an angle ;,
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 azi
muth 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 and the star. This angle
between the vertical circle and the declination circle, which
is called the parallactic angle 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 de
noted by p, from the following simple formula:
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 SP Z can be used:
cos h sin p = cos <p sin t
cos h cos p = cos sin <p sin 8 cos (p cos t,
or taking:
cos (p cos t = n sin N
sin (f = n cos N,
the following formulae, which are more convenient for loga
rithmic computation :
cos h sin p = cos (p sin t
cos h cos p = n cos (+N).
86
The parallactic angle is used, if we wish to compute
the effect which small increments of the azimuth and al
titude produce in the declination and the hour angle. For
we have, applying to the triangle between the pole, the ze
nith and the star the first and third of the formulae (9) in
No. 11 of the introduction:
dS = cos p dh H cos t dfp h cos /* sin p . dA
cos Sdt = sin/>c?A+ sin t sin S .dcp f cos h cos p. d A
and likewise:
dh = cos pdS cos A d(p cos S sin/) . dt
cos lid A = sin pd S sin A sin hdcp + cos 8 cospdt.
9. In order to convert the right ascension and decli
nation of a star into its latitude and longitude, we must re
volve the axis ofss" *) in the plane of y" z" through the angle
s 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 ac
cording to the formulae (1 a) in No. 1 of the introduction:
cos /? cos A = cos S cos
cos j3 sin A = cos 8 sin a cos e f sin 8 sin e
sin p = cos 8 sin a sin f H sin 8 cos f .
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 d, 90 ft and s and
the opposite angles respectively 90 A, 90 j a and the
angle at the star.
In order to render these formulae convenient for loga
rithmic computation, we introduce the following auxiliary
quantities :
M sin N= sin 8
TUT AT S> (&)
M cos zV = cos o sin a,
by which the three original equations are changed into the
following:
cos /3 cos A = cos 8 cos a
cos /? sin A = Mcos (N e)
sin {3 = M sin (N s ),
or if we find all quantities by their tangents and substitute
for M its value cos 8 sin
cos N
*) See No. 4 of this Section.
87
we get as final equations :
tang
tang A =
sin
cos (N e)
=! " tanga
tang ft = tang (N e) sin I
The original formulae give us a and d without any am
biguity; but if we use the formulae (6) we may be in doubt
as to the quadrant in which we must take /,. However it
follows from the equation:
cos ft cos k = cos 3 cos a
that I must be taken in that quadrant, which corresponds to
the sign of tang I and at the same time satisfies the con
dition, that cos a and cos h must have the same sign.
As a check of the computation the following equation
may be used:
cos (N e) _ cos {3 sin h .
cos N cos S sin
which we find by dividing the two equations:
cos ft sin /t = Mcos (N e)
cos sin a = Af cos .2V.
The geometrical meaning of the auxiliary angles is easily
found. A 7 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:
fl = 6 33 29". 30 S = 16 22 35". 45
e = 23 27 31". 72,
the computation of the formulae (6) and (c) stands as follows:
cos 9 . 9820131 tang 9 . 0605604
tang<? 9.4681562,,  9 . 0292017,,
cos N
sin a 9 ._057709_3 1 = 359 17 43". 91
jV = 68 45 4 1". 88 , R . Q
27 31 72 tang (#)!. 4114653
sin^ S.OS97293*
 . =  92 13 13 . 60 1^8^37
cos(,Y )8.5882086 n C o S ^ = 9 . 979 1948
cos N 9 . 5590069
cos ft sin;, = 8 .0689241.
cos S sin a = 9. 0397224
9 . 0292017* ^^ ^
TTK , ITY
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 (45  ft) sin i (E A) cos (45+4) sin [45 (eh<?)]
sin (45 4/?) cos^ (E X) = sin (45 +J cos [45 I (s )]
cos(45 $ ft) sin \ (JEM) = sin (45 !) sin [45 $( )]
cos (45 j/5) cos I (JF4) = cos (45 + a) cos [45 ?(e + 8)].
These formulae are especially convenient, if we wish to find
besides ft and A also the angle 90 E.
Note. Encke has given in the Berlin Jahrbuch for 1831 tables, which
are very convenient for an approximate computation of the longitude and la
titude 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 coordinates or also
from the same spherical triangle as before:
cos d cos a = cos ft cos /
cos 8 sin a = cos ft sin A cos E sin ft sin s
sin S = cos ft sin A sin e + sin ft cos e.
We can find these equations also by exchanging in the
three original equations in No. 9 ft and I for $ and a and
conversely and taking the angle s negative. In the same way
we can deduce from the formulae (//) the following:
sn
cos (.TV he)
tang =__ tang I
tang 8 = tang (N+ s) sin a
and from (r) the following formula, which may be used as
a check:
cos (N { s~) _ cos S sin a
cos N cos ft sin I
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 (45 \ } sin \ (E\a] = sin (45 + 4 A) sin [45" (e +/?)]
sin (45 3) cosOEH) = cos(45 MA) cos [45 (,#)]
cos (45 ? <?) sin 4 (E a] cos (45 h \ A) sin [45 (e /?)]
cos (45 4<?) cos 4 (_) = sin (45 H A) cos [45  (s\ft)].
2Vote. 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 I) = sin L sin e
or : tang D = tang e sin ^4.
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 A and /?,
if Gauss s equations are used for computing them, as, de
noting this angle by r\ , we have >/ = 90 E. But if we
wish to find this angle without computing those formulae,
we can obtain it from the following equations:
cos ft sin 77 = cos a. sin e
cos ft cos 77 = cos e cos S + sin e sin sin a
or:
cos S sin 77 = cos A sin e
cos S cos i] = cos e cos ft sin E sin ft sin A,
or taking:
cos = m cos M
sin f sin = m sin /If
or:
cos s = n cos 2V
sin sin A = n sin N
we may find it from the equations :
cos ft sin rj = cos a sin
cos ft cos 77 = w cos (M 8)
or:
cos sin 77 = cos A sin
cos S cos 77 = n cos (2V f /?).
The angle tj is used to find the effect, which small in
crements of A and /> have on a and <) 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:
dft = cos 77 d cos S sin 77 . da sin A de
cos ft o?A = sin 77 d 8 * cos $ cos 77 . da + cos A sin ft de,
and also:
dS= cosr]dft\cosftsmrj.dhtsmad
cos $o? = sin rjdft + cos/? cos 77 . c?A cos sin $ . c/.
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 per
turbations 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 :
<M =  sin y ,. dB ,
COS U
dJj = cos i] . dB,
or if we substitute the values of sin r] and cos 77 from the formulae for
cos ft cos 77 and cos S cos 77 after having taken /?=0, we find:
. cos D dA = cos A sin e . dB,
...
cos D
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 coordinates 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 (f in the direction towards the positive side
of the axis of 3, we find the new coordinates:
x = x sin (f \ z cos (jp,
y =y.
z = z sin (p x cos cp.
If we then revolve the axis of x in the plane of x and
t/, which is the plane of the equator, through the angle &,
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
angles are reckoned in an opposite direction:
x" = x cos & r y sin
y" = y COS x sill
z" = z
If we finally revolve the axis of y" in the plane of y"
and z" through the angle e in the direction towards the pos
itive side of the axis of a", we find:
91
y" ! = y" cos 4 z" sin s
z " = y sin s + z cos ,
and as we also have:
x " = cos p cos I
y" ! = cos fi sin k
z " = sln/3,
we can express A and /? directly by 4, ft, <f , and e by
eliminating x , y , as well as a?", #", a".
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 revo
lution of the earth on its axis or the time between two suc
cessive culminations of the same fixed point of the celestial
sphere, is called a sidereal day. It is reckoned from the mo
ment the point of the vernal equinox is crossing the meri
dian, when it is O h sidereal time. Likewise it is l h , 2 h , 3 h etc.
sidereal time, when the hour angle of the point of the equinox
is l h , 2 h , 3 h etc. or when the point of the equator whose
right ascension is l h , 2 h , 3 h etc. or 15 , 30", 45 etc. is on
the meridian.
We shall see hereafter, that the two points of the equi
noxes 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 propor
tional to the time and therefore unites with the diurnal mo
tion of the sphere, while the other is periodical. This latter
motion has the effect, that the hour angle of the point of
the vernal equinox does not increase uniformly, hence that
sidereal time is not strictly uniform. But this want of uni
formity is exceedingly small as it amounts during a period of
nineteen years only to =1= 1 s . .
14. The sun being on the 21 th of March at the vernal
equinox it crosses the meridian on that day at nearly O h si
92
dereal time. But at it moves in the ecliptic and is at the
point of the autumnal equinox on the 23 d of September, hav
ing the right ascension I2 h , it culminates on this day at
nearly 12 1 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 O h 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 ap
parent and produced by the motion of the earth, which ac
cording 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 ra
dius vector of the earth) describes equal areas in equal times.
If we denote the length of the sidereal year, in which the earth
performs an entire revolution in her orbit, by T we find for
the areal velocity F of the earth  , as the area of
the ellipse is equal to a*nVl e 2 , or if we take the semi
major axis of the ellipse equal to unity and introduce instead
of e the angle of excentricity r/>, given by the equation e = si
we find:
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
V
is also expressed by the definite integral \ Ir 2 e?j/, where r des
o
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:
2F(tT)=j r 
n ,1 IT a (1 e 2 ) a cos y 2 , .
As we have tor the ellipse r =  = , * tnis
Hficos^ l+ecosv
integral would become complicated. We can however in
troduce another angle for r ; for as the radius vector at the
perihelion is a ae, at the aphelion = a\ae, we may
assume r = a(\ icos 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 v + e
cos h =    ,
lje cos v
from which we see, that E has always a real value, as the
right side is always less than =f= 1.
By a simple transformation we get also :
cos E e cos w sin E
 = cos v and  sm v
1 ecosh 1 ecos/t
and differentiating the two expressions for r, we find:
dv a cos cp
r
Introducing now the variable E into the above definite
integral, we find:
E
2 F(t J 7 ) = a 2 cos y 1(1  e cos E} dE a~ cos ip (E e sin E),
o
hence taking again the semi major axis equal to unity and
substituting for F its value found before we obtain:
where w is the mean sidereal daily motion of the earth, that
is the daily motion the earth would have if it were perform
ing the whole revolution with uniform velocity in the time T.
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
M = E e sin E,
and having found from this the auxiliary angle , we get
the true anomaly from the equation:
cos y s mE
tang r=  r ~  .
cos hi e
But in case that the excentricity is small it is more con
venient, 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 = 0, we can take :
v = M+ v\.e + \ v\ .e 2 + l v>\ . e 3 4 . .. ,
where ? , i>" etc. designate the first, second etc. differential
coefficient of v with respect to e in case that we take e = 0.
If we differentiate the equation sin v = c , s  ] written
1 cos E
logarithmically, we find:
cos v _ dE cos E e dy cosE e
sin* sin.E 1 ecosE cosy 1 ecosE
s mr sin v a cos y sin v
or: dv= . ^.dE\ dy = T dEi dy,
sinE . cosy r cosy
and if we differentiate also the equation for M, considering
only E and e as variable, we find:
dE = sin vd<p
dv sin v dv sin v
 = (2 f e cos v) and  =   (2 f e cos v).
dy COS9P de cosy
Taking here e = 0, we get i/ = 2 sin M.
In order to find also the higher differential coefficients
we will put P = ., and Q = 2 h e cos v. We find then
cosy 1
easily, denoting the differential coefficients of P and Q after
having taken e = by P , () etc.
P = cos M . v\ = sin 2 J/,
Q = cos M,
v" ^= sin M. Q H 2P = 4 sin 2 il/,
p" = cos J/. ^" sin M. v\ 2 + 2 sin il/= f sin 3 M h { sin M,
Q" = 2 sin M. v\ = 4 sin Jf 2 ,
v " == S in M. Q" h 2 Q . P + 2P" = V 3 sin 3 If f sin M.
Hence we get:
= 3/h (2 e 1 e 3 ) sin 3/4 ? e 2 sin 2 J/4 [^ e 3 sin 3 J/ 4 ...
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 m order to get v M expressed in sec
onds of arc, we find:
v = M+ G918" . 37 sin M + 72" . 52 sin 2 M f 1" . 05 sin 3M,
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 r the longitude n
which the sun has when the earth is at the perihelion and
M\n 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 A, the
other by L, we have the following expression for the true
longitude of the sun:
I = L f 69 18". 37 sin M + 72". 52 sin 2M+ 1".05 sin 3 M*\
or if we introduce L instead of M , as we have M = L n
and rc = 280 21 41".0:
A = ZM244". 31 sin f 6805". 56 cos L
67. 82 sin 2L + 25. 66 cos 2Z
. 54sin3 . 90 cos 3 L.
In order to deduce the right ascension of the sun from
its longitude, we use the formula:
tang A = tang A . cos e,
which by applying formula (17) in No. 11 of the introduction
is changed into:
A = k tang TT e~ sin 2 1 f ^ tang ^ 4 sin 4^ ...
where the periodical part taken with the opposite sign is cal
led the reduction to the ecliptic.
If we substitute in this formula the last formula found
for / 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 f 86s . 53 s i n L __ 4348 . 15 cos
596 .64sin2L h 1 .69 cos 2 JS
3 .77 sin 3/i  18 . 77cos3L
h 13 . 23 sin 4 L . 19cos4
f 0.16 sin 5 h . 82 cos 5 L
. 36 sin 6 L f . 02 cos 6 L
.01 sin? .04 cosl L.
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 uni
form 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 from O h to 24 h .
According to Hansen the mean right ascension L of the
sun is for 1850 Jan. O h Paris mean time:
18 39 9s. 261,
and as the length of the tropical year that is the time in
which the sun makes an entire revolution with respect to the
vernal equinox is 365 . 2422008, the mean daily tropical mo
tion of the sun is:
9AO
365. 2422008  59 8. 38 o,  8 56 . 555 ta tim.,
its motion in 365 days = 23 h 59 m 2 . 706 = 57 . 294,
its motion in 366 days = 24 2 59 . 261 = 4 2 59 261.
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. 1850 equal to:
18 h 39 " 9s . 261 h X 3 m 56 . 555,
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
o
opposite sign. But if n is the mean daily motion of the sun
in time and ntw 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 == 24 h : 24 h w,
24 h
or x = (AL}~
24 h w
From 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 = , retaining merely the three principal terms,
we have the equation:
= 8G.5 sin L 596.6 sin 2 L + 434.1 cos L,
from which we can find the values of L, for which the equa
tion of time is equal to zero, namely L = 23 16 , L = 83 26 ,
L = 16015 , L = 2733 , which correspond to the 15 th of
April, the 14 th of June, the 31 st of August and the 24 th 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:
H14 m 31s, 3 m 53s, H6 m 12s,  16 IS*
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 mi
nus 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 1 Observatoire Imperial 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 s hence the length of a solar day 24 h O rn 30 s . On the Con
trary 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 va
riation is 21 s , hence the length of the apparent day 23 h
59" 39 s .
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 from 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 there
fore :
365.242201
ay = 366. 242201 mean ^
= a mean day 3 in 55 s .909 mean time,
366.242201
36042201 Sldereal da *
a sidereal day + 3 m 56 s . 555 sidereal time.
366.242201
and a mean day = TTTT^T sidereal day,
J 060. 242201
Hence if (~) designates the sidereal time, M the mean
time and fy, the sidereal time at mean noon, we have :
and
24fa 4 3 50s . 555
0o H "24iT~
The sidereal time at mean noon can be computed by
the formulae given before, or it can be taken from the astro
nomical almanacs, where it is given for every mean noon.
To facilitate the computation tables have been constructed,
which give the values of
24 h 3 " 55s . 9Q9
24 h
and
24 h 4 3 U1 56 s . 555
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 b 16 36 s . 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
5 h 10 " 48 s . 30,
hence 9 1 5 in 48 s . 05 sidereal time have elapsed between noon
and the given time and this according to the tables or if
we perform the multiplication by
24 h 3 m 55s . 909
24*>
is equal to 9 h 4 in 18 s . 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 time.
17. To convert apparent solar 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  1 "20.73 .
9 1 9.37 + S ^+ s.27.
10 57.74
Therefore if the apparent time given is June 9 9 h 5 m 23 s . 60,
we find the equation of time equal to l m . 4 s . 98, hence the
mean time equal to 9 4 m 18 s .62.
In order to convert mean time into apparent time, the
same equation of time is used. But as this sometimes is
given for apparent time, we ought to know already the ap
parent 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 9 h 4 m
18 s . 62 mean time is given, we may take first the equation
7*
100
of time equal to l m and then find for 9 h 5 m 18 s .6 apparent
time the equation of time I m 4 8 .98, hence the exact ap
parent time equal to 9" 5 m 23 s . 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 con
versely 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 :
1849 JuneS 5h 5 m 3Qs,79 ,
9 9 38. 75 + f ^+0s.27.
10 13 46 .98
Now if 9 h 5 m 23 s . 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 5 h 11 "12 s . 75, hence the si
dereal time equal to 14 h 16 m 36 s . 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 5 h 9 m 38 s . 75, hence 9 h 6 m 57 s . 60 sidereal
time or 9 h 5 m 28 s . 00 mean time have elapsed between noon and
the given sidereal time 14 h 16 m 36 s . 35. If we interpolate
for this time the right ascension of the sun, we find again
5 h ll m 12 s . 75, hence the corresponding apparent time 9 h 5 m
23 s . 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 meri
dian, 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 pub
lished, 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 f(a) denote the right ascension of the body at noon, expres
sed 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:
/(a) f tf (a + ) H i~~2~/" ()
or a little more exact:
/(a) H tf (a + ) +  ( {Y  / ( + *)
As this must be equal to the sidereal time at that mo
102
merit, we obtain the following equation, where & designates
the sidereal time at mean noon and where the interval of the
arguments of f(ci) is assumed to be 24 hours:
4 t (24h;> 56s . 56) =/() + // ( + ft H ^^ f" ( h *),
hence :
<== _ _._/M.!?o
._J^3 56". SGrCaH*)] " 1 / (+*)
The second member of this equation contains it is true f,
but as the second differences are always small, we can in
computing t from this formula use for t in the second mem
her the approximate
The quantity 6J 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 O tt f(a) k , hence the time of culmination for this
place but in time of the first meridian is
24 3 " 56s . 5G / ( + ) _ f
2i
and the local time of culmination t=t k.
Example. The following right ascensions of the moon
are given for Berlin mean time:
/()
1861 July 14.5 13" 7 5* . 3
15.0 13 34 22 .9 " Z< V;* +4 i k2
15.5 14 2 21 . 7 ? ^^ 43.5 ;
16.0 1431 4.0
and the sidereal time at mean noon on July 15 r> =7 h 33 m
7 s . 9. To find the time of the culmination of the moon for
Greenwich.
As the difference of longitude in this case is k = 53 m
34 s . 9, the numerator of the formula for t becomes 6 h 54 m 49 s . 9,
*) If the interval of the arguments of / () were 12 hours instead of
24 hours, the first term of the denominator in the above formula would be 12 h
l m 58 s . 28, and if we start from a value /(), whose argument is midnight,
we would have to use H 12 h l m 58 s . 28 instead of 6> .
103
the first terms of the denominator become ll h 33 m 59 s . 5,
hence the approximate value of t is 0.59775; with this we
find the correction of the denominator f 8 s . 5 and the cor
rected value of t equal to 0.59762 or 7 h 10 m 17 s .O, hence
the local time of the culmination equal to 6 h 16" 42 s . 1.
For the lower culmination we have the following equation,
where a again designates the argument nearest to the lower
culmination :
H t (24" 3 56" . G) = 12 H/(a) I */(aH) + ^"^ / (+*),
hence the formula for a place whose longitude is &, is :
24*3 56* . 56/
or in case the interval of the arguments is 1 2 hours :
t , = _ 12 if(a}0 +k
12" 1". 58s . 3 _/ ( + ;) _ < i/ ( a 4. )
Example. If we wish to find the time of the lower cul
mination at Greenwich on July 15, we start from July 15.5.
Hence the numerator becomes 7 h 20 m 50 s .4, the first terms
of the denominator become II 1 33 m 16 s . 0, hence the aproxi
mate value of t is equal to 0.6359 and the corrected value
0.63577 or 7 h 37 m 45 8 .l. The lower culmination occurs there
fore at 19 h 37 m 45 s . 1 Berlin mean time or at 18 h 44 m 10 s .2
Greenwich time.
20. In No. 7^ we found the following equation :
sin h = sin y> sin 8 \ cos cp cos $ cos t. J^j I*
If the star is in the horizon , therefore h equal to zero,
we have:
= sin <f sin f cos cp cos S cos t Q .
hence: cos = tang y tang 8.
By this formula we find for any latitude the hour angle
at rising or setting of a star, whose declination in d. This
hour angle taken absolutejjL^alled the semiupper 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 ab
solute value of t () 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=14 h9m iQs.3 = f 19 54 29".
and further we have:
tf = 52 30 16".
With this we find the semidiurnal arc:
to = Ug 10 1". 3 = ?h 52m 4Qs .
Hence Arcturus rises at 6 h 16 m 39 s and sets at 22 h l m .39 s
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 com
putation twice. In the case of the sun this is simple. We
first take an approximate value of the declination and com
pute 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 ap
parent noon, one can easily find by interpolation the decli
nation 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 cul
minations of the moon, we can find the mean time corres
ponding 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 de
clination 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 set
ting may be put into another form. For if we subtract it from and add it
to unity, we find by dividing the new equations :
, 2 _ cos (90 $)
=
21. The above formula for cos t Q embraces all the va
rious 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 d is positive or the star is north of the equator, cos <
is negative for all places which have a northern latitude;
f therefore in this case is greater than 90 and the star
remains a longer time above than below the horizon. On
the contrary for stars, whose declination is south, t becomes
less than 90, 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 <f< is negative,
it is the reverse, as there the upper diurnal arc of the sou
thern stars is greater than 12 hours. If we have <y/ = 0, t
is 90 for any value of J; therefore at the equator of the
earth all stars remain as long above as below the horizon.
If we have 8 = 0, t (} is also equal to 90 for any value of
, 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 x this is always the case.
It is obvious that a value of t is only possible while we
have tang cp tang d <t 1. Therefore if a star rises or sets
at a place whose latitude is rjp, tang 3 must be less than
cotang y or d < 90 ff. If 8 = 90 r/>, we find t == 180
and the star grazes the horizon at the lower culmination.
If we have d ;> 90 (p , the star never sets , and if the
south declination is greater than 90 rf , the star never
rises.
As the declination of the sun lies always between the
limits s and + e, 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 e or 66^. 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 hour angle is 6 h . 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 6 h and 4O h and whose de
clinations are respectively (90 <p) and 4 (90 tp). 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 46 h and
a G h and whose declinations are respectively (90 90) and 90 <f>.
The point, which at the time of the rising of the point was in the horizon
in its lower culmination, is therefore now in its upper culmination at an
altitude equal to 2<p. Hence at the latitude of 45 the constellations make
a turn of 90 with respect to the horizon from 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 y> sin h cos y> cos h cos A,
which was found in No. 6, h equal to zero and obtain:
COS AQ = (l>).
cos cp
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 n we have:
A =90 4 A
hence :
sin d
sin A t =  (c),
COS (p
where A l is positive, when the point where the star rises or
sets, lies on the north of the east or west points, nega
tive when it lies towards south.
The formula (c) for the amplitude may be written in a
different shape. For as we have:
1 4 sin A { sin t/j 4 sin
1 sin A t sin \p sin 8
when ifj = 90 y, we find :
w 8
tang r ~ 
tang
107
For Arcturus we find with the values of d and r^, given
before: ^1 / = 340 .9.
23. If we write in the equation:
sin h = sin <f> sin S { cos <p cos S cos t
1 2 shir}/ 2 instead of cos f, we get:
sin h = cos (9? 8} *2 cos 9? cos S sin \t^ .
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 = and the maximum itself is found
from the equation:
COS Z = COS (<JT  S) ((/),
from which we get:
z = <p S or = S (f>.
If we take therefore in general:
z = S y>,
we must take the zenith distances towards south as negative,
because for those star, which culminate south of the zenith,
<) is less than (f.
On the contrary /* is a minimum at the lower culmi
nation or when =180, as is seen, when we introduce
180 instead of , reckoning therefore t from that part
of the meridian, which is below the pole. For then we
have :
sin h = sin rp sin S cos rp cos 3 cos t .
or introducing again 1 2 sin \t 2 instead of cos t :
sin h = cos [180 =F (T + 8}] \ 2 cos y cos S sin j* 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 [180 =F (<F 4 S)].
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 c) are positive, and the lower sign for the southern hemi
sphere, so that we have:
for places in the northern hemisphere, and:
z = (180 + <p f 8}
for places in the southern hemisphere.
108
The declination of a Lyrae is 38 39 , hence we have
for the latitude of Berlin d qp = 13 51 . The star a
Lyrae is therefore at its upper culmination at Berlin 13 51
south of the zenith, and its zenith distance at the lower cul
mination equal to 180 cp d 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 for
mula :
cos z = sin cp sin + cos <p cos cos t,
taking , d and t as variable, we find:
sin zdz = [sin <p cos 8 cos y sin cos t] dS cos cp cos S sin tdt
and from this we obtain in the case that z is a maximum
or dz = 0:
d8 r s
sm t =  [tang y tan g " cos *J
This equation gives the hour angle at the time of the
7 ft
greatest altitude. is the ratio of the change of the decli
nation to the change of the hour angle, or if dt denotes a
second of arc, it is the change of the declination in T ^ of a
second of time. As this quantity is small for all heavenly
bodies, and as we may take the arc itself instead of sin t
and take cos t equal to unity, we get for the hour angle
corresponding to the greatest altitude:
dS r ,,206265
t = j [tang <p tang 8] ~^ (g\
7 V<
where 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 ap
7 S>
proaching the north pole, so that is positive, the greatest
altitude occurs after the culmination if y> 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 h sin A = cos 8 sin t,
cos h cos A = cos 90 sin 8 f sin 90 cos cos /,
we find:
sin h = cos 3 [sin cp cos ^4 sin t cos t sin A],
cos A r = cos S [cos ^ cos / f sin cp sin t sin .4],
or:
dh , .
= cos o sm p = cos 90 sin A,
cos A = t cos $ cos p. (A)
a
Frequently we make use also of the second differential
coefficient. For this we find:
d l h t dA
=cosycos^. ,
cos 9? cos S cos J. cos p
cos A
Likewise we have:
t/z ~ .
 = cos o sm p = cos 9? sm ^4,
c? 2 z _ cos cp cos S cos ^4 cos p
~~
Furthermore we find from the second of the formulae (/&) :
d 2 A dp dh
cos /r = cos h cos o sm p f cos o cos p sm h 
c/< 2 * dt dt
But we get also, differentiating the formula:
sin cp = sin h sin S + cos A cos S cos />,
cos h cos $ sin p   = [cos A sin 8 sin h cos 8 cos ] 
dt at
Hence we have:
cos A 2 ^ = + [cos A sin ^ 2 cos 8 sin A cos p] cos # sin p,
or, if we introduce A instead of p:
d* A
cos A 2 2  = cos 95 sin J. [cos A sin 8 f 2 cos 9? cos vlj.
26. As we have :
dh
 = cos 95 sm A,
we find = 0, or A is a maximum or minimum, when we
have sin A = or when the star is on the meridian.
110
We find also that c  1  is a maximum, when sin A = =t 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 90 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 = 90 or we con
sider the right angled triangle between the star, the zenith
and the pole and find:
tang S
cos / =
tang rp ^
. sin 8
sin (f
Finally we have:
COS (f
sin p = ^
cos o
If we have <) ;> <f>, 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 90, those stars cross the prime vertical
below the horizon.
For Arcturus and the latitude of Berlin we find :
t = 73 52 . 1 = 4 h 55 28
h = 25 24 . 9.
Arcturus reaches therefore the prime vertical before its
culmination at 9 b 13 m 51 s and after the culmination at 19 h
4 in 47 s .
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:
, 2 sin (cp $)
sin (y> + S)
and for computing the altitude we may use the formula:
cotang h = tang t cos (p.
27. As we have:
d A cos S cos p
dt cos h
Ill
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 ver
tical to the declination circle. But as we have :
sin <p sin h sin S
cos p =  V
cos h cos d
this must occur when sin (c = & ! n f . It happens therefore
sin d
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 dis
tance from the meridian and the azimuth at that time is given
by the equation:
cos S
sm A = 
cosy
and the hour angle by the equation:
tang (p
cos t h 
tang o
For the polar star, whose declination for 1861 is 88
34 6" and for the latitude of Berlin, we find:
^ = 88 8 0" = 5 52^ 32s
4 = 2 21 9" reckoned from the north point, A = 5231 .7.
28. Finally we will find the time, in which the discs
of the sun and moon move over a certain great circle.
If /\n is the increment of the right ascension between
two consecutive culminations expressed in seconds of time,
we find the number of sidereal seconds #, in which the body
moves through the hour angle t from the following proportion:
x: t = 86400 A: 86400
as we may consider the motion of the sun and moon during
the small intervals of time which we here consider, as uni
form; hence we have:
1
86400 4 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:
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 * f cos S* cos t
where R designates the apparent radius, or from:
sin ^ R = cos 8 sin \ t.
Hence, as t is small, this hour angle expressed in time is:
R
15 cos S
therefore the sidereal time of the semi  diameter passing the
meridian :
2R 1
~15.cos.Tlr
When the upper limb of the body is in the horizon, the
depression of the lower limb is equal to 272, and as we have:
 = cos d sin p, the difference of the hour angles of the up
d t
per and lower limb in time is:
15 . cos d sinp
hence the sidereal time of the diameter rising or setting:
2R_ I
15 . cos S sin p 1 A
where p is found from the equation:
sin (p
cos = 
cos o
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 ^ R = cos h sin  a
or, as R is small, from the equation:
R = cos A . a.
But as we have dt = coshdA ~ we find for the sidereal
cos o cos p
time in which the diameter passes over a vertical circle:
2R J^
15 cosd.cosp 1 A
cos S sin <f sin S cos q> cos t
where =
COS ft
SECOND SECTION.
ON THE CHANGES OF THE FUNDAMENTAL PLANES, TO WHICH
THE PLACES OF THE STARS ARE REFERRED.
As the two poles do not change their place at the sur
face of the earth, the angle between the plane of the hori
zon 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 po
sition 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 coordinates.
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
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 in
tersection 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 Lunisolar 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 Nutation of obliquity.
As the mutual attractions of the planets change the in
clinations 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 must 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 General
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 an
other 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 inter
section is changed. These changes on account of their long
period can be united with the secular variation of the obli
quity of the ecliptic and with the precession. Hence the
*) The periodical terms, the nutation, are the same for the fixed and
moveable 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 Regiomontanae these ex
pressions to the second power. According to this the an
nual lunisolar precession at the time 1750 f t is:
^ = 50". 37572 0". 000243589 t
or the amount of the precession in the interval of time from
1750 to 1750 M:
l t = t. 50". 37572 t 2 0". 0001 2 17945.
This therefore is the arc of the fixed ecliptic between
the points of intersection with the equator at the beginning
of the year 1750 and at the time 1750 M.
Furthermore the annual general precession is :
^j = 50". 21129 + 0". 0002442966 t
and the general precession in the interval of time from 1750
to 1750 M:
l=t 50". 21 129 M 2 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 1 t.
*) In the expressions developed in series they change only the terms
dependent on t 2 .
116
Finally the angle between the equator and the fixed
ecliptic is at the time 1750f:
o = 23 28 18". 4 t* 0". 0000098423
and the angle between the equator and the ecliptic at the time
1750M (if we neglect as before the periodical terms of nu
tation), which is called the mean obliquity of the ecliptic, is :
e = 23 28 18".0 t 0". 48368 z 2 0". 00000272295 *),
so that we have:
dt
d f = 0". 48368 0". 0000054459 t.
dt
Now let AA (} Fig. 2 represent the equator and EE n the
ecliptic both for the beginning of the year 1750, and let A A 1
and E E represent the equator and the obliquity of the ecliptic
for 1750M; then the arc B D of the ecliptic, through which
the equator has retrograded on it, is the lunisolar precession
in t years, equal to /,. Further are BCE and A BE respect
ively the inclination of the true ecliptic and of the fixed
ecliptic of 1750 against the equator, equal to s and . If
*) Bessel has changed a little the numerical values of the expressions
given in the Mecanique Celeste, 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 /,, 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 n and 77, 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
, based on the same values of the masses. The terms multiplied by t~,
dt
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 de
termination.
Peters gives in his work ,,Numerus constans nutationis" other values com
puted 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:
l t = t 50".37572 t" 0".0001084
I = t 50V214S4 h z 2 0".0001134
s = 23 28 17 .9 4 0".00000735 f 2
= 23 28 17".9 0".4738 t 0".00000140 t 2 .
But as Bessel s values are generally used, they have been retained.
117
Fig. 2.
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
1750M. 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 1750 + ?.
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 // 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 equi
nox of the year 1750, we have BE = 180  // /, and
CIS = 180  // /, 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 //. If we denote
the inclination of the true ecliptic or the angle EEC by n,
we have according to Napier s formulae:
118
frr . 4tJi . l t l *f*o
tang 4 7t . sin j II} j = sin    tang  ,
( I,* I \ l t l s
tang ^ 7t . cos j/7f j = cos ^ tang  ,
As 5 is the same point of the equator which in the year
1750 was at Z>, BC is the arc 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 ,
by a, we find from the same triangle:
tang Y a . cos   = tang T  (l t /) cos  
From these equations we can develop a, as well as n
and // into a series progressing according to the powers of
t. From the last equation, after introducing:
o + T ( o) instead of  
and taking instead of the sines and tangents of the small
angles /, /, a and e the arcs themselves, we find:
/, B
206265
or if we substitute for /,, / and s their expressions, which
are of the following form A,f A , 2 , Kt \ K t 2 and
we obtain:
co So ( cos o 8 206265 cos fo 2
or if we substitute the numerical values:
a = t. 0.17926 t 1 0".0002660393,
d " = 0.17926 t . 0".0005320786.
dt
In addition we have:
tang \n+ l l } = tang  . ,J ,
sin ~
and
( I P + 2 S 2 ) /, I 2
tang T} 7T 2 = j tang  L ~^~ tang  h tang j cos ^
or proceeding in a similar way as before :
]
tang \ iJT+lft + Oj =";; + ^^
a 2 sin f o cos o (e )
7T 2 ==a 2 sine 2 + ( o) 2 +
206265
119
Substituting here also for e and a the expression
_ r j j 2 and at \ a f % we find :
sin e
n 4 4 (/ h = arc tang
7?
_
2062boh .cos cos7Z
206265
7i = t \ a? sin 2 H ?7 2 f  \aa sin f ? f rj v/ 
or substituting the numerical values:
77=171 36 10 *.5".21
7t = t.Q". 48892 * a 0". 0000030715
^ = 0". 48892 ^.0". 0000061430.
rf<
2. The mutual changes of the planes, to which the po
sitions of the stars are referred, having thus been determined,
we can easily find the resulting changes of the places of
the stars themselves. If A and ft denote the longitude and
latitude of a star referred to the ecliptic of 1750 + , the
coordinates 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 ft cos (A 77 /), cos ft sin (h 77 J), sin ft.
If further L and B are the longitude and latitude of the
star referred to the fixed ecliptic of 1750, the three coordi
nates with respect to this plane and the same origin as be
fore are:
cos B cos (L 77), cos B sin (L 77), sin B.
As the fundamental planes of these two systems of co
ordinates make the angle n with each other, we find by the
formulae (1 a) of the introduction the following equations :
cos ft cos (A 77 I) = cos B cos (L 77)
cos ft sin (1 77 /) = cos B sin (L 77) cos n + sin B sin n (A)
sin ft = cos B sin (L 77) sin n f sin B cos n.
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 = 90 ft,
6=90 B, c=7r, 4 = 90fL 77, 5 = 90 (I II I}:
d (I 77 /) = flH + n tang ft sin (A 77 /) dll
H tang ft cos (/I 77 /) d n
dft = J n cos (A 77 /) c/77 sin (7 77 I) dn.
120
Dividing by dt and substituting t instead of n in the
coefficient of <///, we obtain from these the following for
mulae for the annual changes of the longitudes and latitudes
of the stars:
dl di t /. dn \d7t
= , f tang B cos (/ II I t\
dt dt \ dt ) dt
dS f . dn \ dn
 = sin I / n I t]
dt \ dt J dt
or, as we have // + d ^t = 171 36 10" MO". 42, taking:
ZTf 1 d ^+ 1= 171 36 10" + t 39".79 = M,
dt
d^ _ dl
dt ~ dt
where the numerical values for and as given in the
dt dt
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 inter
section of the equator of 1750f with the fixed ecliptic, is
equal to L + /,, when /, is the lunisolar precession during
the interval from 1750 to 1750 f 1. Hence the coordinates
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 f /,), cos B sin (L + /,) and sin B.
If now a and 8 denote the right ascension and decli
nation of the star, referred to the equator and the true
equinox at the time 1750f, the right ascension reckoned
from the origin adopted before, is equal to + a. We have
therefore the coordinates of the star with respect to the
plane of the equator and this origin as follows:
cos cos ( f a), cos S sin (a f ) and sin 8.
As the angle between the two planes of coordinates is
c , we find from the formulae (1) of the introduction:
cos 8 cos ( f a) = cos B cos {L \ /,)
cos sin (a \ a) = cos B sin (L + /,) cos e sin B sin e (C)
sin S= cos B sin (L f /,) sin f sin B cos s .
_ UNIVEF
I 1 ^kJ"*. r*. _
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 ecliptic, that of the equator and the star a = 90 <) ,
b = 90 B, c = , A = 90 (L h 0, 5 = 90
d (a 4 ) = [cos f 4 sin e tang sin (a 4 )] ^ cos (a 4 a) tar
dS = cos (a 4 a) sin e dl t 4~ sin (a 4 a) ds .
We find therefore for the annual variations of the right
ascensions and declinations of the stars the following for
mulae :
da da dl
. = h [cos 4 sm tang o sm a]   
( . dl, de \ ~
4 1 a sm e     ? tang o cos ,
rfe
1 sm ,
or neglecting the last term of each equation on account of
its being very small *) :
da da . dl,
, =  r [cos f sin e t) tang o sm 1 ,
at at dt
d
dt
If we take here:
~ = cos sin ,
rfJ, rfa
cos = m.
dt dt
8 rf<
we find simply:
cfa
= m 4 n tang o sin ,
  = n cos ,
where the numerical values of m and w, obtained by substi
tuting the numerical values of g ,  and /tt , are:
w * <Y t
m = 46" . 02824 4 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
*) The numerical value of the coefficient a sin , is only
0.0000022471 t.
122
1750 M to 1750M , it would be necessary to take the
integral of the equations (JB) 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  and from the interval of time. For if
and /"(Y) are two functions, whose difference /"( ) f(f) is
required, (in our case therefore the precession during the time
t ), we take :
( + *) = *,
*(* ) = A*.
Then we have:
/(O =/(*  A*) =/(*)  A*/ GO + 4 A* 2 /" (*),
/(*0=/(* + A*) =/(*) + A */ (*) f IA* 2 /" CO,
where /" (a?) and f" (x) denote the first and second differential
coefficient of f(x). From this we find:
/(O /(O = 2 A*/(aO = (  O
Hence in order to find the precession during the inter
val of time t , it is only necessary to compute the dif
ferential coefficient for the time exactly at the middle and
to multiply it by the interval of time. By this process only
terms of the third order are neglected.
For instance if we wish to find the precession in lon
gitude and latitude in the time from 1750 to 1850 for a
star, whose place for the year 1750 is:
A = 2100 , /? = + 34
we find the following values of  , and M for 1800:
dt dt
=50". 22350, ^=0". 48861, M= 172 9 20".
dt dt
With these we find the following place for 1800, com
puting the precession from 1750 to 1800 only approximately:
/l = 210 42 .l, / 5 = f33 59 .8
from the formulae (5) we find then the annual variations for
1800:
^ = t 50". 48122, ^ = 0". 30447,
dt dt
hence the precession in the interval from 1750 to 1850:
in longitude + 1 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 ascen
sion and declination for 1750 is:
= 220 1 24", ^ = + 20 21 15"
we have for 1800:
m = 46". 04367, n = 20". 05957,
and the approximate place of the star at that time:
== 220 35 . 8, <? = j20 8 . 6
hence we have according to formulae (D):
tang 9 . 56444 n tang sin a = 4 . 78806
sin a 9 . 81340. m = + 46 . 04367
tang 8 sin a = 9 . 37784,, da = + 41 . 25561
n=l. 30232 dt
cos a = 9. 88042,,  = 15 . 2314
at
therefore the precession in the interval of time from 1750
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 (va
riatio annua) given for the epoch of the catalogue and be
sides this its variation in one hundred years (variatio sae
cularis). If then t, denotes the epoch of the catalogue, the
precession of a star according to the above rules equals:
( t t n
variatio annua f ~ OAr r" variatio saecularis (* *)
A(J(J )
If we differentiate the two formulae:
da
= m + n tang o sin a,
dS
 d< =cos,
taking all quantities as variable and denoting the annual
variations of m and n by m and ri, we find:
d * a n 2 . . mn
dt 2 == ^7 Sin " **" tang ^ ~*  tan S ^ cos a H m f n tang 8 sin n,
.
77^ =  sm a 2 tang 8 sin a f n cos a,
where w signifies the number 206265, and multiplying these
equations by 100 we find the secular variation in right as
124
cension and declination. For the star used before we find
from this the secular variation :
in right ascension = f 0". 0286,
in declination = f 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 ft denote the longitude and the latitude of a
star, referred to the ecliptic and the equinox of 1750 + /,
we find from these the longitude and latitude L and #,
referred to the "fixed ecliptic of 1750, from the following
equations, which easily follow from the equations (.4) in
No. 2:
cos B cos {L 77) = cos /9 cos (A II I)
cos B sin (L 77) = cos /? sin (A 77 /) cos n sin /? sin n
sin B = cos /? sin (A 77 f) sin n + sin ft cos 7t.
If we wish to find now the longitude and latitude A
and ft , referred to the ecliptic and the equinox of 1750 \t\
we get these from L and B by the following equations, in
which 77 , n and / denote the values of 77, n and / for the
time t :
cos /? cos (A 77 / ) = cos B cos (L 77 )
cos $ sin (A 77 I ) cos B sin (L 77 ) cos n 1 f sin B sin n
sin /? = cos 73 sin (7L 77 ) sin n + sin B COSTT .
If we eliminate L and B from these equations, we can
find A and /? expressed directly by A and / and the values
of /, 77 and n for the times t and f .
The exact formulae for the right ascension and declination
are similar. If a and 8 are the right ascension and decli
nation of a star for 1750 f f, we find from them the longi
tude and latitude L and J5, referred to the fixed ecliptic of
1750, by the following equations*):
cos B cos {L + Z,) = cos cos (a f a)
cos B sin (L h /,) = cos 8 sin ( + ) cos s + sin S sin
sin 73 = cos $ sin (a + a) sin + sin 8 cos .
If we wish to know now the right ascension and decli
nation a and S for 1750 4 f , we find these from L and 7?
*) These equations are easily deduced from the equations (C) in No. 2.
125
by the following equations, in which l fl a and denote the
values of /,, a and for the time t :
cos 8 cos (a 1 4 ) = cos B cos (X 4 Z ,)
cos <? sin ( 4 ) = cos Z? sin (Z 4 / ,) cos s sin B sin s
sin $ = cos B sin (L 4 Z ,) sin e 4 sin B cos s .
If we eliminate L and 1? from the two systems of
equations and observe that we have:
cos B sin L = cos S cos (a 4 ) sin Z, 4 cos 8 sin (a 4 ) cos cos Z,
4 sin $ sin s cos Z,
cos 7? cos L = cos $ cos ( 4 ) cos Z / 4 cos $ sin ( 4 a) cos e sin Z,
4~ sin $ sin e sin Z,
sin B = cos $ cos (a 4 ) sin e + sin <? cos e,
we easily find the following equations:
cos S cos (a 1 4 ) = cos $ cos (a 4 a) cos (Z , /,)
cos $ sin (a 4 a) sin (Z , Z,) cos e,,
sin $ sin (Z , Z,) sin e
cos $ sin ( 4 ) = cos $ cos (a 4 a) sin (Z , Z,) cos e
4 cos #sin( 4 fi) [cos (Z , Z,) cos e cos e 4sin sin e ]
4 sin$[cos(Z , Z,)sine cose cose sine ]
sin S cos S cos ( 4 a) sin (Z/ Z ( ) sin e
4 cos <?sin(4)[cos(Z / Z,)cose sinf o sine cose ]
4 sin <?[cos(Z , Z,)sine sin 4cos cose ,,].
If we imagine a spherical triangle, whose three sides are
/ , /,, 90 z and 90 f z 1 whilst the angles opposite those
sides are respectively 0, and 180 g , we can express
the coefficients of the above equations, containing / ; /, ()
and e H by 0, ^ and s and we find:
cos 5 cos ( 4 ) = cos 8 cos (a 4 a) [cos cos 2 cos z sin 2 sin z]
cos S sin (a 4 a) [cos sin 2 cos 2 4 cos 2 sin 2 ]
sin 8 sin cos z
cos 5 sin (a 4 a ) = cos 8 cos (a 4 a) [cos cos 2 sin z ] 4 sin 2 cos z 1 ]
cos $ sin (a 4 a) [cos sin z sin 2 cos z cos 2 ]
sin S sin (9 sin 2
sin 5 = cos 8 cos (a 4 a) sin cos 2
cos 8 sin (a 4 ) sin 6> sin 2
4 sin 8 cos <9.
Multiplying the first of these equations by sin * , the
second by cos z and subtracting the first, then multiplying
the first by cos * , the second by sin z and adding the pro
ducts we get:
cos S sin ( 4 a z) = cos 8 sin ( 4 a 4 2)
cos 8 cos ( 4 2 ) = cos S cos (a 4 a 4 2) cos sin ^ sin 6> (a),
sin S = cos ^ cos (a 4 a 4 2) sin 4 sin # cos 0.
126
These formulae give a and if expressed by , #, a, a
and the auxiliary quantities z, z and Q. These latter quanti
ties may be found by applying Gauss s formulae to the spheri
cal triangle considered before, as we have:
sin 4 cos \ (z 1 ~) = sin  (l\ l ( ) sin ^ (e f c () )
sin \ sin ^ (2 2) = cos j (f { I,} sin \ (e\ )
cos sin ^ (2 + 2) = sin ^ (// I,) cos ^ (V + )
cos ^ cos  (2 f 2) = cos ^ (7/ li) cos i (e s )
As we may always take here instead of sin \ (z z)
and sin f (Y ) the arc itself and the corresponding co
sines equal to unity, we find the following simple formulae
for computing these three auxiliary quantities:
tang 4 (z f z) = cos 4 (e + o) tang \ (l t l t )
cotangji/ , l ( )
i u  *) = i c .  .)  iT,v^.r
tang 4 9 = tang .} (e + e ) sin  ( + .2).
The formulae () can be rendered more convenient for
computation by the introduction of an auxiliary angle or we
may use instead of them a different system of formulae de
rived from Gauss s equations. For we arrive at the for
mulae (a) if we apply the three fundamental formulae of
spherical trigonometry to a triangle, whose sides are 90 rV,
90 and 0, whilst the angles opposite the two first sides
are respectively + a f z and 180 a j z . 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 z by A,
we find:
cos (90 4 S ) cos (X I c) = cos J [90 h <? H 0] cos %A
cos (90 I S ) sin  (4 + c) = cos 4 [90 4 8 0] sin 4 4 (ft)
sin 4 (90 4 5 ) cos $ (A c) = sin [90 f <? + 0] cos .4
sin  (90 + <? ) sin (4 c) = sin 4 [90 4 S 0] sin ^ 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 se
cond by cos A and add the products. We find thus:
cos <? sin (A 1 A) = cos 8 sin A sin [tang S f tang cos A]
cos S cos (A 1 A) = cos S cos 8 cos A sin [tang S + tang cos ^L],
hence :
sin ^4 sin [tang S f tang ^ <9 cos 4]
 1 coi 4 sin [teng * H tang * cos 4]
127
and from Gauss s equations we find:
cos 4 c. . sin \ (S 1 ) = sin } cos ^ (A 1 h
COS T} C . COS ? (S S) = COS 4 COS Y (A 
If we put therefore:
p = sin (9 [tang d + tang  cos .4]
we have:
p sin J.
tang (^4 A) =  1
1 p cos ^
and:
By the formulae (A), (5) and (C) we are enabled to
compute rigorously the right ascension and declination of a star
for the time 1750 + t , when the right ascension and decli
nation for the time 1750 + t are given.
Example. The right ascension and declination of a Ursae
minoris at the beginning of the year 1755 is:
= 10 55 44". 955
and #=87 59 41". 12.
If we wish to compute from this the place referred to
the equator and the equinox of 1850, we have first:
I, = 4 11". 8756 / , = 1 23 56". 3541
a = 0". 8897 = 15".2656
o = 23 28 18". 0002 e = 23 28 18". 0984.
With this we find from the formulae (A):
I ( z H ) = o 36 34". 314 J (z z)= 10". 6286
hence:
z = 36 23". 685
2 =0 36 44". 943
and:
= 31 45". 600
therefore:
A=a + a + z = ll Q 32 9". 530.
If we compute then the values of A A and d from
the formulae (#) and (C), we find:
log/; = 9,4214471
and :
A A = 4 4 17". 710, J (? S) = 1 5 26". 780
hence:
4 =153G 27". 240
and at last:
= 16<> 12 56". 917
S = 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 lat
ter, 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 ce
lestial sphere or different stars will be in its neigbourhood
at different times. At present the extreme star in the tail of the
Lesser Bear ( Ursae minoris) is of all the bright stars nearest
to the northpole and is called therefore the polestar. This
star, whose declination is at present 88f , 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 pre
cession 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 ,
we must consider the spherical triangle between the pole of
the ecliptic at a certain time t and the poles of the equator
P and P at the times t and t. If we denote the right ascen
sion and declination of the pole at the time t referred to the
equator and the equinox at the time t (n by a and <?, and the
obliquity of the ecliptic at the times f and t by s and ?,
we have the sides P P = 90" J, EP= , E P = s , the
angle at P = 90 { a and the angle at E equal to the gene
ral precession in the interval of time t 1 ; we have there
fore according to the fundamental formulae of spherical tri
gonometry :
cos 8 sin = sin e cos e cos I cos e sin
cos 8 cos a = sin e sin I
sin S = sin e sin e cos I + cos cos .
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 s = and get simply :
tang a = cos e tang ^ I
*) This radius is strictly speaking not constant, but equal to the actually
existing obliquity of the ecliptic.
129
and:
sin sin I
cos o =
cos a
Though a is found by means of a tangent, we find nev
ertheless the value of a without ambiguity, as it must satisfy
the condition, that cos a and cos I 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 174,
hence we have:
= 27316 and d = H43 7 .
This agrees nearly with the place of a Lyrae, whose
right ascension and declination for 1850 is:
a = 277" 58 and = + 38 39 .
Hence about the year 14000 this star will be the polestar.
On account of the change of the declination by the pre
cession stars will rise above the horizon of a place, which
before were always invisible, while other stars now for in
stance 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 si
dereal year, that is, the time, in which the sun describes
exactly 360 of the celestial sphere or in which it returns to
same fixed star, equal to 365 days 6 hours 9 minutes and
9 s . 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 re
turns 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 5& 48 46 . 42 . 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 consid
ered 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 A = 17". 2405 sin O + 0". 2073 sin 2 O
 1". 2692 sin 2 O 0" . 2041 sin 2 (
4 0" . 1279 sin (0 P) 0". 0213 sin (0 4 P)
4 0".0677 sin (([ P ) (A)
Ae = 4 9". 2231 cos $1 0" 0897 cos 2 Jl
h 0" . 5509 cos 2 4 0" . 0886 cos 2 ([
4 0".0093cos(04P),
where $1 is the longitude of the ascending node of the moon s
orbit, and (L are the longitudes of the sun and of the
moon and P and P are the longitudes of the perihelion of
the sun and of the perigee of the moon. The expressions
131
given above are true for 1800, but the coefficients are a
little variable with the time and we have for 1900:
A A 17" . 2577 sin D + 0". 2073 sin 2 ft
1" . 2693 sin 2 O 0". 2041 sin 2 (C
h 0". 1275 sin (O P) 0".0213 sin
4 0". 0677 sin ((CP )
A = h 9". 2240 cos 41 0". 0896 cos 2 SI
H 0" . 5506 cos 2 h 0" . 0885 cos 2 (
h 0" . 0092 cos (0 h P).
In order to find the changes of the right ascensions and
declinations of the stars, arising from this, we must observe,
that we have :
da , da
and : ()
But we have according to the differential formulae in
No. 11 of Section I, if we substitute instead of cos ft sin 7;
and cos ft cos i] their expressions in terms of <*, 8 and :
rf <*<?
TJ = cos f sm e tang o sin a y = cos a sm e
a/. a A
rfa rf^
7 = cos a tang o  = sm ,
C/ </
from which we find by differentiating:
( 32 ) = sin 2 [5 sin 2 a h cotang e cos a tang f sin 2 tang$ 2 ]
d r* /
( J = sin [cos a 2 cotang s tang sin a + tang 8* cos 2]
(~\ = [% sin 2 H sin 2 a tang ^ 2 ]
f   ;, 2 J = sin f 2 sin a [cotang f tang S sin ]
f  , J = sin e cos a [cotang h sin a tang S]
(v ) = cos a 2 tang $.
c? 2 /
If we substitute these expressions in the equations (a)
and introduce instead of A A and A their values given be
fore by the equations (4) and take for 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
= 15". 8148 sinO [6". 8650 sin O sin a h 9". 2231 cos O cos a] tang 5
+ 0" . 1 902 sin 2O + [0". 0825 sin 2Q sin +0". 0807 cos2^ cosaj tang S
 1 " . 1 642 sin 20  [0". 5054 sin 20 sin +0". 5509 cos20 cos] tan (V
 0".1872sin2([[0".0813sin2((sin+0".0886cos2([cos]tang^
 0".0195sin(04P)
 [0". 0085 sin (0 + P) sin + 0". 0093 cos (0+P) cos ] tang S (B]
4 [0". 0621 4 0".0270 sin tang S] sin (( P )
h [0" .117340". 0509 sin a tang <?] sin (0 P),
<? (?= G". 8650 sin O cos a 4 9". 2231 cos O sin a
H 0".OS25 sin 2 ^ cos a 0".0897 cos 2 f} sin
 0" . 5054 sin 2 cos 4 0" . 5509 cos 2 sin (C)
 0". 0813 sin 2 ([ cos a H 0" . 0886 cos 2 ([ sin
 0" . 0085 sin (0 H P) cos a 4 0" . 0093 cos (0 4 P) sin
4 0". 0270 cos sin ((TP )
4 0" . 0509 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 a:  15".8321 sin^ [6".S683 sin } sin a+9".2240 cos O cos a] tang S
inS :  6^8683 sin O cos a 4 9". 2240 cos 1 sin a.
Of the terms of the second order only those are of
any amount, which arise from the greatest terms in A A and
AC. If we put for the sake of brevity:
Ae = 9" . 2231 cos O = cos }
and  sin s A A = 6" .8650 sin ft = b sin $1 ,
these terms give in right ascension:
a =   sin 2 a [tang S 2 + ^] + tang cos a cotang s
4 [ cotang e sin a tang S\ tang d 2 cos 2 a 4 1 cos 2 a]  sin 2 ft
tang $ 2 sin 2 a 4 ^r tangdcosacotge 4 ~ sin2 a! cos 2i")
and in declination:
a a j .".:*..
cosz( tango sin cotang e
o o / 4
[tango^ sin 2 a 4 2 cotang s cos a] sin 2
U  4  o cos2J tango"  sin a cotang e cos
Those terms which are independent of <O change merely
133
the mean place of the stars and therefore may be neglected.
Another part, namely:
~
and
sin 2 ~ f  cotang e sin a sin 2 ,Q f cotang s cos a cos 2 ,Q J tang
 cotang s sin 2 ") cos a f cotang E sin a cos
can be united with the similar terms multiplied by sin 2O
and cos 2 H of the first order, which then become equal to :
in right ascension
and in declination (/>)
h 0" . 0822 sin 2 f\ cos 0" . 0896 cos 2 ^ sin .
The remaining terms of the second order are as follows:
in right ascension
H 0". 0001 535 [tang <? 2 f ] sin 2 H cos 2
 0". 0001 60 [tang <? 2 + j] cos 2 O sin 2
and in declination (^)
 0" . 0000768 tang 8 sin 2 a sin 2 O
 [0" . 000023 f 0" . 000080 cos 2 a] tang 8 cos 2 O
But as the first terms amount to s . 01 only when the
declination is 88 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 ne
glected except for stars very near the pole.
6. We shall hereafter use the changes of the expres
sions (E) and (C) produced by a change of the constant of
nutation, that is, of the coefficient of cos ,Q 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 multi
plied by a factor N which depends on the moments of in
ertia of the earth as well as on the mass and the mean motion
of the moon, while the terms of the solar nutation are mul
tiplied 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 JV must be determined from observations. Now the co
134
efficient of the term of the nutation of obliquity, which is
multiplied by sinO, is equal to 0. 765428 IV . If we take
this equal to 9". 2231 (1H), 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 N = 9". 2231(1 + 0.
But the lunisolar precession depends on the same quan
tities N and N and the value determined from observations
(50". 36354 for 1800) gives the following equation between
N and IV :
17 .469345 = Nt 0. 991988 JV,
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 f i and all terms of the solar nutation by
1 2. 16687 i. Taking therefore 9". 2235 i = dv, we have:
; _ j 1.8702 sin n+ 0.0225 sin 2O 0.0221 sin 2 (1+0.0073 sin(([P )j
d ^ ~t 4 0.2981 sin 2 0.0300 sin (Q P) + 0.0050 sin (Q + P) i
</A*=[cosO 0.0097 cos 2^10.0096 cos 2 ([ 0.1294 cos 2Q
0.0022 cos (0hP)] dv
and from this we find in the same way as in No. 5:
^.~_ a )_ _i.7t56sinO [0.7445 sin } sin H1 0000 cos O cos ] tang
dv
+ 0.0206 sin 2^ + [0.0090 sin 2^ snuH0.0097 cos2~} cosa] tang
0.0203 sin 2 (L [0.0088 sin 2 ([sin +0.0096cos2 ([ cos]tang<?
h 0.0067 sin ((( P ) h [0.0029 sin (([ P ) sin a } tang 8
40.2735 sin20f[0.1187sin20sina+0.1294cos20 cosa] tang<?
0.0275 sin (0 P) [0.01 19 sin (0 P) sin jtangc?
4 0.0046 sin (0 f P) H [0.0020 sin (Q +P) sin a H
H 0.0022 cos (0hP) cosa] tang 8
^~^= 0.7445 sin O cos a hi. 0000 cos O sin a
dv
i 0.0090 sin 2^^ cos a 0.0097 cos 2O sin a
0.0088 sin 2 ([ cos a + 0.0096 cos 2 ( sin
hO.0029 sin ((I P ) cos a
H0.1187sin20cos 0.1294 cos 2 0sin
0.01 19 sin (0 P)cos
h 0.0020 sin (0 H P ) sin 0.0022 cos (0 h P) sin .
7. In order to compute the nutation in right ascension
and declination it is most convenient to find the values of
A^ and A* from the formulae (4) and (AJ and to compute
135
the numerical values of the differential coefficients ^L A etc.
Cl A d
But the labor of computing formulae (J?) and (C) has been
greatly reduced by the construction of tables. First the
terms :
15".82sinO = c and 1". 16 sin 2 Q = g
have been brought in tables whose arguments are ft and 2 0.
The several terms of the nutation in right ascension
multiplied by tang 5 are of the following form:
a cos ft cos a + b sin ft sin a = A [h cos ft cos a + sin ft sin a].
Now any expression of this form may be reduced to
the following form:
a: cos [ft a\y],
For if we develop the latter expression and compare it
with the former, we find the following equations for determin
ing x and y:
A h cos ft == x [cos ft cos y sin ft sin y]
A sin ft = x [sin ft cos y + cos ft sin #]
from which we find:
x*=A*[l(l ^ 2 ) cos /? 2 ]
and: (1 ft) sin ft cos ff
where x and t/ are always real. If we have now tables for
x and ?/, whose argument is /9, we find the term of the nu
tation in right ascension, multiplied by tang d by computing:
x cos [ft \ y a]
while : ( c ),
gives the term of the nutation in declination depending cos fi.
For as these terms have the form:
A [ h cos ft sin f sin ft cos a] ,
we find taking it equal to x sin (fiy ) the same equations
(6) for determining x and y.
Such tables have been computed by Nicolai and are gi
ven in the collection of tables by Warnstorff, mentioned be
fore. These give besides the quantity c the quantities log b
and B with the argument O, and with these we find the
terms of the right ascension depending on cos 1 and sin O
by computing:
c b tang S cos (ft f B a)
136
and the corresponding terms of the decimation by computing:
 b sin GO + B a) (<0
This part of the nutation together with the small terms
depending on 2O, 2 ([ and d P , is the lunar nutation.
A second table gives the quantities #, log f and F with
the argument 20, by which we find the terms depending on
2O, which for right ascension are:
g /tang S cos [2 Q + F a]
and for declination: ( e )
This part of the nutation together with the small terms
depending on 0fP and P is the solar nutation.
No separate tables have been computed for the small
terms depending on 2 (L , 2 O and f P. For these may
be found from the tables of the solar nutation, using instead
of 20 as argument successively 2d, 180f2,O (because these
terms have the opposite sign) and 0fP, and multiplying
the values obtained according to the equations (e) respectively
by  , 3 6 ~ and i , 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 (I P and P
is different, but analogous to the annual precession in right
ascension and declination; they are therefore obtained by
multiplying the annual precession in right ascension and de
cimation by ji^ sin (<L P ) and ^ sin (0 P).
8. If we consider only the largest term of the nutation
we can render its effect very plain. We have then:
A>1 = 17". 25 sin O,
A = f 9".22cosl,
or rather according to theory:
sineA* = 10". 05 cos 2 f. sin O,
Ae = 10". 05 cos e. cos Jl
Now the pole of the equator on account of the luni
solar precession describes a small circle, whose radius is ,
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 coordinates of
137
the apparent pole (affected by nutation) y = sin s A^? X=&B
and we have therefore according to the expressions given
above the following equation:
?/ 2 = e 2 . cos 2 2 C ~^r x* , where C= 10". 05.
COS 2
The apparent pole describes therefore an ellipse around
the mean pole, whose semimajor axis is C cos e = 9". 22, and
whose semiconjugate axis is C cos 2 e = 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 semimajor 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 cir
cumference 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 com
putation 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 sur
face 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 di
rection 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 ob
servations 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 dif
ference expressed in parts of the semimajor axis, we have:
a_b _ l _b_
a a
If then 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 semimajor axis, we have:
therefore: = V\ e 2
and =1 ^l e
likewise : = ]/% a 2 .
140
The ratio  is for the earth according to BesseFs in
vestigations: " g^g ; /
1
^ ^
and expressed in toises:
a = 3272077. 14 log a = 6. 5148235
6=3201139.33 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 71 the angle at the sun subtended by the equatoreal
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:
a = R sin n
" = 2^265
The angle n 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.
In order to compute the parallax of a body for any
at the surface of the earth, we must refer the place
spheroidal earth to the centre by coordinates. As the
place
on the
Fig. 3.
first coordinate we use
the sidereal time or the
angle, which a plane pas
sing through the place of
observation and the minor
axis *) makes with the
plane passing through the
same axis and the point
of the vernal equinox. If
then OA C Fig. 3 repre
sents 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 = o from the centre of the earth and the
angle AOC, which is called the geocentric latitude. But these
quantities can always be computed from the latitude ANC
(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 coordinates of A with respect
to the centre 0, the axes of the abscissae and ordinates beino
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 6:
fl>H v 6 1 ra*6.
Now we have also, if we denote the geocentric latitude
by </) :
, y
and also : tang y =
dy
because the latitude y 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:
x a" 1 dy
we find the following equation between r/ and r/> :
tang tp } = tang <p (a).
Ill order to compute Q we have:
COS <p
and as we obtain from the equation of the ellipse:
we find:
_ _ = a
cos y
1/1 h tang y tang y cos y cos (y 90)
If therefore the latitude y of a place is given, we can
compute by these formulae the geocentric latitude (f> and the
radius o.
142
For the coordinates x and y we easily get the following
formulae, which will be used afterwards:
_ a cos cp
J/cVs y 2 Kl ) sin 7> 2
a cos 90
and
6 2 ... ^
y x tang y = x j tang 90 = .r (I *) tang 9?
From the formula (a) we can develop y in a series
progressing according to the sines of the multiples of y, for
we obtain by the formula (16) in No. 11 of the introduction:
or taking
a b _
a+ b ~
we find:
2
sin 4 y etc.
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:
(p = y) 11 30". 65 sin 2 yH1". 16 sin 49?... (<?),
from which we find for instance for the latitude of Berlin
<f .== 52" 30 16".
y> = 52 19 8". 3.
Although Q itself cannot be developed into an equally
elegant series, we can find one for log *). For we get
from formula (6):
cos o> 2 1 H 17 tang o> 2
L J
If we substitute here for cos c// 2 its value
a 4
a* f 6 4 tang y 2
*) Encke in the Berliner Jahrbuch fur 1852 pag 326. He gives also
tables, from which the values of 9? and log Q may be found for any latitude.
143
we find:
a 4 cos a> 2 4 b* sin cp 2
+ 6 
a 2 f 6  + (a 2 6 2 ) cos 2 ip
= (a 2 4 6 2 ) 2 H (a 2 6 j ) 2 + 2 (a 2 4 6 2 ) ( 2 6 2 ) cos 2 ?
(a h 6) 2 4 (a 6) 2 4 2 (a 4 b) (a 6) cos 2 y
hence :
_ h ,^ 2 6 2
(o+ft) r./a 6
r./a 6\ 2 _a i HI
^"*"(" ~~r) + 2 T cos 2 OP P
L Va h It/ a + b T _\
If we write this formula in a logarithmic form and de
velop the logarithms of the square roots according to for
mula (15) in No. 11 of the introduction into series progress
ing according to the cosines of the multiples of 2 y, we find :
a a +6 2 , U 2 6 2 a b)
log hyp ? = log hyp j ft +  a . 2  62  ^ cos 2 y
a 6\;
cos49P
6 2 \ 3
 etc.
or using common logarithms and denoting the quantity
a b
a\b
by H, we get:
= log (a } + ;;") + u\ (j ^" n2  )
etc.
where M denotes the modulus of the common logarithms,
hence :
log if =9. 6377843.
If we compute again the numerical values of the coef
ficients and take a = 1, we find:
log q = 9 . 9992747 40.0007271 cos 2 y 0.0000018 cos 4 y> (F)
and from this we get for instance for the latitude of Berlin:
log = 9. 9990880.
144
If we know therefore the latitude of a place, we can
compute from the two series (C) and (F) the geocentric la
titude 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
coordinates :
x = o cos 90 cos
y = $ cos y sin (6?).
2 = (> sin cp
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 ze
nith 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 the centre of the
earth, ZOS is the true zenith distance z as seen from the
centre of the earth and Z AS the apparent zenith distance z
seen from the place at the surface. Denoting then the par
allax or the angle at S equal to z z by p we have:
i C j
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 arc itself instead of
the sine and have :
X = f sin z . 206265.
a
Hence the parallax is proportional to the sine of the ap
parent 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 z = 90
/> = 4 206265
u
is called the horizontal parallax and the quantity
/> =  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 tlie point, in which the line from 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 coordinates 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 coordinates 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 coordinates of the same object with respect to a
system of axes parallel to the others but passing through the
centre of the earth 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 centre
and z and A are the zenith distance and the azimuth seen
from the centre. Now as the coordinates of the centre of
the earth with respect to the first system are:
g sin (9? 9? ), and ^ cos (90 y>~)
we have the following three equations:
10
146
A sin z cos A r = A sin z cos A g sin (9? 95 )
A sin 2 sin A = A sin z sin .4
A cos z = A cos 2 (> cos (90 9? )>
or : A sin z sin (A A) = Q sin (9? 9? ) sin 4
A sin 2 cos (.4 .4) = A sin 2 sin (9? </> ) cos yl (a)
A cose = A cos z Q cos((f> 9? )
If we multiply the first equation by sin (4 4), the
second by cos (X A) and add the two products, we find:
A cos 2 = A cos 2 o cos (9? cp 1 ).
Then putting:
cos 4 (A + A) .. /7N
tang y = ^r, r^ tang (<f> 9? ), (o)
COS l \^* ^*)
we find:
A sin 2 = A sin 2 ^ cos (cp cp ) tang y
A cos 2 = A cos 2 o cos (95 gp )
or:
A sin (2 2) = (> cos (cp cp )
M r \ r ,, cos (2 7) (
A cos (2 2) = A Q cos (cp y>) \
and besides if we multiply the first equation by sin  ( ss),
the second by cos J ( z) and add the products :
, cos (cf cp 1 ) cos [ (2 H z) y]
cos y
If we divide the equations (a), (6) and (c) by A and put:
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:
cos A (cp 9? ) sin A tang 4 (4 4) (y 9? )
, sin A sin ^ cos { (A 1 f 4) , .
 
*.) We have:
Substituting here for tang (95 90 ) the series
( rr y)4{ S p 9P ) 8 ~K
we can easily deduce the expression given above.
sm ^2 y )
cos/
147
(> sin p cos (9? y ]
cos y
Sfsmpcos  (p 9? )\ 2 . 0/ .
4 4 I   ) sin 2 (2 y) H . . . .
\ cos y /
iyp A = log hyp A cos (z y)
( ) cos 2 (c y) ...
V cos y /
We have therefore neglecting quantities of the order of
sin p ((fj (f /) which have little influence on the quantity ; :
y = (99 9? ) cos A
hence the parallax in azimuth is:
or its rigorous expression, which must be used when z is
very small:
o sin p sin (9? cp) .
sin
/ Al Sln Z
tang (A 1 4) = 
_ cos ^
sin 2
Furthermore as:
cos (9? tp) _ cos 4
cos y cos Jr (A 1 A) sin y
is always nearly equal to unity, the parallax in zenith dis
tance is:
2 z = () sin p sin [z (<p 9? ) cos A} ,
and the rigorous equations for it are:
 sin (z z) = (> sin p sin [z (y 9? ) cos A]
cos (z 2) =1 (>sinpcos[2 (cp <f>) cos 4].
Hence if the object is on the meridian, the parallax in
azimuth is zero and the parallax in zenith distance is :
z 2 <) sin p sin [2 (95 9? )]
4. In a similar way we obtain the expressions for the
parallax in right ascension and declination. The coordinates
of a body with respect to the earth s centre and the plane
of the equator are:
A cos 8 cos a, A cos sin a and A sin 8.
The apparent coordinates as they appear from the place
at the surface with respect to the same plane are:
A cos 8 cos , A cos 8 sin and A sin 8 .
10*
148
Since the coordinates of the place at the surface with re
spect to the centre referred to the same fundamental plane are:
^> cos cp cos 0, (> cos cp sin and (> sin cp
we have the following three equations for determining A ?
and 8 :
A cos cos = A cos 8 cos a o cos y cos
A cos d sin = A cos sin a o cos 9? sin (a)
A sin $ = A sin $ Q sin y .
If we multiply the first equation by sin , the second
by cos a and subtract one from the other, we find:
A cos S sin ( ) = (> cos <p sin (0 ).
But if we multiply the first equation by cos , the se
cond by sin a and add them, we find:
A cos cos ( a) = A cos $ (> cos cp cos (0 ).
We have therefore:
, . _ Q cos gp sin (a 6>)
A cos (> cos 90 cos ( )
o cos (f> .
\ ^ sin (a 6>)
A cos o
o cos 90
1  ~ cos (a 0)
A cos o
or developing a a in a series , we find :
? C S sin (,  8) + } ^ rin 2 (  0)
A cos d VAcosd/
In all cases excepting the moon it is sufficiently accu
rate to take only the first term of the series. Taking then
the radius of the earth s equator as the unit of o and writing
in the numerator sin n as factor (where 11 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:
, o sin 7t cos <p sin (a 0}
a a =  .  j . (JB)
A cos o
where a 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 6 #, we will
write in the formula for:
A cos S cos ( )
now
1 2sin(a ) 2
instead of
COS ( a),
and obtain:
A cos = A cos S (> cos <p cos (0 ) + 2 A cos $ sin JS ( ) 2 .
If we here multiply and divide the last term by cos \ (a )
and make use of the formula:
A cos S sin ( ) = Q cos <p sin (6> )
we easily find:
A cos y = A cos ,?  f cos y C 5 j* * ,gffl . ()
Introducing now the auxiliary quantities /? and ; given
by the following equations:
/? sin y = sin y>
cos <p cos [0 I ( H )]
cos y =  V/J , (c)
cos I (a )
we find from (6):
A cos 8 = A cos $ ()f3 cos /
and from the third of the equations (a):
A sin = A sin S ^ /3 sin y.
From these two equations we easily deduce the following:
A sin (S S~) = g ft sin (y $)
A cos (S 1 8) = A f>ft cos (y S),
or:
tang ( S) = }
or according to formula (12) in No. J 1 of the introduction:
S S = s sin (y 8} ^ 3 sin 2 (y $) etc. ((7)
If we introduce here instead of ft its value sm9P and
sm y
write again p sin n 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:
~, o, (} sin n sin cp sin (y 8)
A siny
150
If we further take in the second of the formulae (c)
cos i ( a) equal to unity and write instead of( 4),
we have the following approximate formulae for computing
the parallax in right ascension and declination :
7f(>cos<jp ! sin (0 a)
A cos d
tang cp
tang y
cos (0 a)
> s O *)
A sin/
If the object has a visible disc, its apparent diameter
must change with the distance. But we have:
A sin (8 7) = A sin (8 y)
and as the semi diameters, as long as they are small, vary
inversely as the distances, we have:
. .
sin (o y)
Example. 1844 Sept. 3 De Vice s comet was observed
at Rome at 20 h 41 m 38 s sidereal time and its right ascension
and declination were found as follows :
= 2 35 55". 5
?==_ IS 43 21 .6.
The logarithm of its distance from the earth was at that
time 9.27969 and we have for Rome:
y> = 4142 .5
and
log ? = 9. 99936.
The computation of the parallax is then performed as
follows :
*) If the object is on the meridian, we find :
S 8 = ^ sin (y (?) = $ sin [z (<p y )],
A A
hence the parallax in declination is equal to the parallax in altitude.
151
in arc 310 24 . 5
2 35.9
a 52 11 . 4
tangy 9.94999 y= 55 28 . 6
cos (0 a) 9 . 78749 S= 18 43.4,
sin(6> ) 9. 89765, ~ y =+7412.0
n^cosy ,_ sin(y 5) 9798327
J. O ^ O i u /i . i
A _n 9 sm<p
sec 8 0.02362 A
cosec y . 08413
log (a a) 1 . 44703 log > _ = t ^ 54316/j
a a = + 27". 99 5 5= 34". 93
Thus the parallax increases the geocentric right ascen
sion of the comet 28" . and diminishes the geocentric decli
nation 34". 9. Hence the place of the comet corrected for
parallax is:
a = 2 35 27". 5
<? = IS 42 46 .7.
In order to find the parallax of a body for coordinates
referred to the plane of the ecliptic, it is necessary to know
the coordinates of the place of observation with respect to
the earth s centre referred to the same fundamental plane.
But if we convert and y into longitude and latitude ac
cording to No. 9 of the first section and if the values thus
found are I and 6, these coordinates are:
Q COS b COS I
(> cos b sin I
(> sin b
and we have the following three equations, where A , //, A
are the apparent, A, /?, A the true longitude and latitude:
A cos /? cos A = A cos ft cos A ^ cos b cos I
A cos /? sin A = A cos ft sin 1 $ cos b sin I
A sin ft = A sin ft (> sin 6,
from which we finally obtain similar equations as before,
namely :
, ,, n Q ^ cos b sin (I A)
A cos ft
tang b
^(ii)
, 7t () sin b sin (y ft)
A sin y
& and ff are the right ascension and declination of that point,
in which the radius of the earth intersects the celestial sphere,
152
/ 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 , A 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 de
cimation 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
10 h mean time, we have for this time:
a = 7> 43 fn 2O . 25 = 115 50 3" . 75
= + 16 27 22". 9
6>=llh 17m QS .02 = 169 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:
9, = 51 17 25". 4
log ? = 9. 9991 134.
If we introduce the horizontal parallax p of the moon
into the two series found for a rt and <) j in No. 4, as
we have sin p =  , we find :
_ = _ 206265 P zijpi: sin ( _ a )
cos o
/
K
cos
, , A> cosy sin p\ i
I sin o (^e/ ;(... i
A V cos d /
and: , .
si s i^nnz f>smop smp . .
d d = 206265 sm(y 8)
sin y
153
where we must use the rigorous formula for computing the
auxiliary angle y:
. cos 4 ( )
tang y = tang <p r  .
sy ^ cos[<9 i ( ta)]
If we compute these formulae, we find for a a :
from the first term: 29 45". 71
second 1 1 . 47
third _0 . 03
hence a a = ~~ 29 57". 21
and for S r):
from the first term: 36 34". 21
second 20 . 91
third _0 . 12
hence S ~3Q r 5c) 72l~
The apparent right ascension and declination of the moon
is therefore:
= 115 20 6". 54 5 = 15 50 27". G6.
Finally we find the apparent semi diameter:
# = 15 40". 20.
If we prefer to compute the parallax from the rigorous for
mulae, we must render them more convenient for logarithmic
computation. We had the rigorous formula for tang ( a) :
tang (  ) = ,? C S ?! *?,?. ?.< ~ > ().
1 (> cos (p sm p cos (a 0) sec a
Further from the two equations:
A sin 8 = A [sin S o sin (p 1 sin p]
and:
A cos cos (a a) = A [cos 8 o cos y sinp cos (a &}]
we find:
tang > __ [sin? g sin?/ sin/?] cos ( ) sec d
1 (> cos cp sin /? sec 8 cos (a (9)
Since we have:
A _ cos S cos ( a)
A cos $ (> cos 95 sin /> cos (a (9)
we find in addition:
. , cos cos ( a) sec <?
sin /i =   . 5  sm R (c).
1 (> cos (p smp sec o cos (a 6>)
If we introduce in (a), (6) and (c) the following aux
iliary quantities:
cos A = ? Sin ^ C S ^ ;  cos _^ ~^
cos S
and:
sin (7= $ sin p sin y ,
154
we find the following formulae which are convenient for log
arithmic computation :
*)
tang (  a) =
cos o sin A 2
_ sin ^ (8 C) cos % ($ H (7) cos (a )
cos 8 sin ^ A 2
and:
.
f .4*
If we compute the values a a, 8 and K with the
data used before, we find almost exactly as before:
a = 29 57".21
= 415 50 27". 68
R = 15 40". 21.
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
/t ; , /, ft ) ft, I and b in place of , , <5 , <) , 6> and cp .
II. THE 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 den
sity, the ray is bent from its original direction. If the me
dium, 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 ap
parent 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 di
rections 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 con
sisting 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 nor
mal and the ray of light before entering the medium (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 (7, 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 /LI is the index of refraction for two media if
the light passes from the medium A into the medium #, the
index for the same media if the light passes from the
medium B into the medium A is
f*
Now let 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 stratum. If we now consider the n th stratum taking
156
CJV as the line from the
centre of the earth to
the point in which the
ray of light meets this
stratum, and denoting the
angle of incidence by ,
the angle of refraction
by /", the index of re
fraction for the vacuum
and the (n l) th stratum
by /*, the same for the
w th stratum by #.+ we
have *) :
sin i lt : sin/ n = [i n+ \ . /*.
If further N is the point in which the ray of light meets
the wfl th stratum, we have in the triangle JVC JV , denoting
the lines JVC and JV C by r n and r n+l :
sin/ : sin i,,+i = r+i : r,
and combining this formula with the one found before we get :
r n sin i n fi n = r n +i sin i n+ i /t a+ i.
Therefore as the product of the distance from the centre
into the index of refraction and the sine of the angle of in
cidence is constant for all strata of the atmosphere, we may
denote this product by y and we have therefore as the gene
ral law of refraction:
r . ft . sin i = y, (a)
where r, u 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 dis
tance 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 / from
the following equation:
aju, sin 2 ==/. (6)
*) These indices are fractions whose numerators are greater than the de
nominators. For a stratum at the surface of the earth for instance we have
f) t A A
^=1.000294 or nearly equal to 
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 coordinates and denoting
the angle, which any r makes with the radius CO by 0, we
easily find: r^tehgt. (c)
dr
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 is the angle, which the original di
rection SJ of the ray of light produced makes with the nor
mal. This c, 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 dis
tance of the heavenly bodies and the refraction itself is a
small angle, the angle f differs very little from the true ze
nith 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 JV, to
which the variable quantities i, r and // belong and if we
denote the angle between it and the normal CO by , we have:
= * + . (rf)
Differentiating the general equation (a) written in a log
arithmic form, we find:
dr da
h cotang i.di\  =
r fi
and from this formula in connection with the equations (c)
and (rf) we get: .,., .dp
rf = tang i ,
f 1
or eliminating tang i by the equation:
sin i y
tang i = === =
V 1 sin i 2 yVV 2 / 2
and substituting for y its value a u () sin a; we find:
158
The integral of this equation taken between the limits
= and = gives then the refraction. If we put:
we can write the equation in the following form:
I/
s z z (l 2 )}(2s s 2 )sin2 2
i /
In order to integrate this formula we must know how s
depends upon . 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 we introduce now as a new variable quantity the density p,
given by the equation:
^2 _ i = co ,
where c is a constant quantity, we obtain:
do
^(1 ) sin. c .
(l ^Wc?.? * 2 )sin~ ;
V lic^J
or taking:
co co a A P \
2, hence ^=2a(l 51
1 4 c(> V o /
^ sn
The coefficient
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 u = 1 at the limits
of the atmosphere, and the index of the stratum at the sur
face is /u (} =^ , the ratio is, always contained between
oojy IU.Q
narrow limits. Hence as a is always a small quantity, we
may take instead of the variable factor
159
its mean value between the two extreme limits 1 and 1 2
or the constant value 1 a.
If we put for brevity 1  ^ = ?, where w is a function
of s, to be defined hereafter, and if we change the sign of dC ,
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:
(1 s) sin zdw
z 2 2 aw 4 (2s s 2 )sinz 2
or as s is always a small quantity, since the greatest value
of 5 supposing the height of the atmosphere to be 46 miles
is only 0.0115:
sin zdw
I a ]/ cosz * 2 aw j 2s sin z 2
a s sin z [cos z 2 2 aw] hs 2 sin z 2 *&
[cos* 2 2aw>H2ssins 2 p
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 limits 5 = and 5 = J5T,
where H denotes the height of the atmosphere.
If we now put:
w = F(s)
and introduce the new variable quantity a?, given by the fol
lowing equation:
or taking:
aF(s)
* = x h (p (is),
we have according to Lagrange s theorem:
2
1.2 dx
1.2.3 rfar 5
hence
160
In order to find from this the refraction, we must mul
tiply each term by  . = and integrate be
! J/cos.? 2 42* sins 2
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 r () be the atmospheric pressure and the
temperature at the surface of the earth, p and T the same
quantities at the elevation x above the surface, m the ex
pansion of atmospheric air for one degree of Fahrenheit s
thermometer; then we have the following equation:
Po ()
1 f WT
For if we take first a volume of air under the pressure
p () at the temperature T (} and of the density o {) and change
the pressure to p, while the temperature remains the same,
the density according to Mariotte s law will change to (> .
Po
If then also the temperature increases to r, the resulting den
sity will be:
p 1 h mr
from which we get the equation above. Hence the quantity
~7f^j^~ T ) 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 l () the
height of a column of air of the uniform density o and of
the temperature T O , which corresponds to the atmospheric
pressure p in we have, denoting the force of gravity at the
surface of the earth by </ :
/ is the height which the atmosphere would have if the den
sity and temperature were uniformly the same at any elevation
161
as at the surface of the earth, and if we take for T O the tem
perature of 8 Reaumur = 10 Celsius = 50 Fahrenheit, we
have according to Bessel:
1 =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 dr, the
decrease of the pressure is equal to the small column of air
Qdr multiplied by the force of gravity at the distance r, hence
we have:
, a 2 ,
dp = g ^.Q. dr,
and dividing this equation by the equation (/?) and putting
also reckoning the temperature from the temperature r , so
that r means the temperature minus 50 Fahrenheit we find:
d ? = _/* (!_,)
Po ^o
and from the equation () we have: (y)
? = (l+mr)(l 10).
Po
If we eliminate p from these two equations, we find 1 w
and hence the density expressed by s and l^mr. 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 atmo
sphere.
If we take the temperature as constant, we have:
 = 1 w, hence ? = d (1 w\
Po Po
and we find, combining this with the first of the equation (7) :
d(lw) a ,
= ds,
1 w L
a
T
hence 1 w =
11
162
as the constant quantity which ought to be added to the in
tegral 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.
as
If we take for \\mr an exponential expression e h
we arrive at BesseFs form. We find then by the combi
nation of the two equations (? ):
d(l w) \~ a a h~]
T  = LTr J*
and integrating and determining the constant quantity so that
1 w is equal to unity when 5 = 0, we find:
instead of which we can use the approximate expression :
*=A .. / "
1 lv = e hl (SI
Bessel determines the constant quantity h is such a man
ner 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 \rnr = e h
for this value of h do not at all agree with the decrease
as observed near the surface of the earth. For we find
= = for s = 0, and as we have also = for s = 0,
as hm ds a
we find:
dr_ 1
d r hm
at the surface of the earth. Now as m for one degree of
Fahrenheit s thermometer is . 0020243 and as h according
to Bessel is 116865.8 toises, we find ~=~^ . There
dr "2ot
would be therefore a decrease of the temperature equal to
1 Fahrenheit if we ascend 237 toises, whilst the observations
show that a decrease of 1 takes place already for a change
of elevation equal to 47 toises.
Ivory therefore in his theory assumes also an exponential
expression for 1fmr, but determines it so that it represents
163
the observed decrease of the temperature at the surface of
the earth. He takes:
1 w = e~ " ,
where u is a function of s, and further:
1H WT =1 /(l_ e )
Then we easily get from the equations (; ):
a  ds = (lf)du + 2fe"du,
and  .9 = (1 /) u f 2/(l e "). (*0
o
Taking r = a we find from these two equations :
dr l f
and we see that we must take f equal to  in order to make
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 (d) :
h 1
hi, ~ f
we have for Bessel s hypothesis:
we have therefore :
2 .
sin 2
and we find :
tfF(*)^(^
sin z \ L &
hence as:
dx" 
11
164
and the general term of the differential d becomes:
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 = and s = H, instead of which we
can use also without any sensible error the limits and oo,
as eP* is exceedingly small for 5 = H. As we have x =
when 5 = and x = GO when 5 = GO we must integrate the
different terms with respect to x between the limits and co.
All the integrals which here occur can be reduced to the
functions denoted by ifj in No. 1 8 of the introduction and if
we apply formula (8) of that No., we find the general term
of the expression for the refraction:
(!), .
___(,,_ 1)
y;(n I) ...
or denoting the refraction by <) , we find:
etc.
and as we have :
we can write this in the following form :
*/3
9.
In Ivory s hypothesis we have :
w = .F (it) = 1 e~ " ,
165
and taking = :
If we introduce here the new variable #, given by the
equation :
the differential expression for the refraction according to
equation (g) in No. 6 becomes:
, ,
a 1 /
l/
cosz 2 H
P
where x = u  (1 e) /M + 2/(l e ).
Taking again:
F(^) = l e~ x
<p Or) =  . a/9 a (1  e*) +/*  2/(l  e),
bin 2
we find from the formula (/&):
. .
rfa: 1.2 c/^r 2
As the third term may be already neglected, we have:
e ,+ !M^:: J = e " + 5/1 [2e *_. .]+ / ( 1 _ I )e2/t2e  e ].
t 3? s i n z
If we multiply these terms by   and
* !,/ 2 2 sin, 2
I/ cos s )  a;
^
integrate them with respect to x between the limits and GO,
we find again according to the formulae (9) and 10) in No. 8
of the introduction:
(0
where 7*= cotang 2 l
The higher terms are complicated, but already the next
term is so small on account of the numerical values of a/3
166
and /* that it can be neglected. For we have for the horizon,
where the term is the greatest, putting 2 /*/?=</
* (<(XG
If we divide each term by y ^ and integrate it between
the limits s and oc we find, applying the formulae for /"Q)?
jT() etc. given in No. 16 of the introduction:
1 a ~2 J/f ^f* ~ *f9 ^ ~ 1) + y 2 (1  2 J/2 + 3 /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 (#) 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 (/).
10. The numerical computation of the refraction from
formula (K) or (/) can be made without any difficulty, as the
values of the functions ip can be taken from the tables or
can be computed by the methods given in No. 17 of the in
troduction.
According to Bessel the constant quantity at the tem
perature of 50 Fahrenheit and for the height of the baro
meter of 29 . 6 English inches , reduced to the normal tem
perature, is
= 57". 4994, hence log ," = 1.759785
1 ct
and /* = 116865. 8 toises.
As we have / () = 4226.05 toises, we find, if we take
according to Bessel for a the radius of curvature for Green
wich to 3269805 toises :
^ = 745 . 747, hence log  [/2 /? = 3 . 347295
If we wish to compute for instance the refraction for the
zenith distance 80, we have in this case log 7\ = 0.53210
etc. and we find:
167
H""
logw
n= 1
0.00000
n= 2
0.15051
n= 3
0.71568
n= 4
1.50515
n = 5
2.44640
= 6
3.5017
/i= 7
4.6480
n= 8
5.8701
n= 9
7.157
n = 10
8.500
0.00000
V 8 y v 1
9.14983
9.33113
9.00745
8.36122
8.92228
7.21523
8.86128
5.94430
8.81372
4.57645
8.77473
3.12943
8.74168
1.6155
8.7130
0.043
8.688
8.420
8.665
log
9.90691
9.81382
9.72073
9.62763
9.53454
9.44145
9.34836
9.2553
9.162
9.069
The horizontal rows give the terms within the paren
thesis in formula (&) and if we multiply their sum by the
constant quantity 1 _^ a ^ / 2/?, we find 3 14". 91 exactly in con
foimity with BesseFs tables.
Far more simple is the computation of Ivory s formula.
In this case we have:
log a p = 9.333826, log r  ^2/? = 3.354594, /= *.
1 Ct
If we now compute the refraction according to formula
(/), we have:
log I\ =0.540098 log T 2 = 0690613
log y, (1) == 9.142394 log y (2) = 8.999757
and with this the terms independent of f give 3 15". 32, whilst
the terms multiplied by f give 0".12. The refraction is
therefore 315".2Q or nearly the same as BesseFs 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 BesseFs 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:
and substituting here the numerical values we get 36 5".
168
According to Ivory we find the horizontal refraction:
SZ = 1  a V/7f "[/I U + ^ 0/2 " 1} ~ /(2 1/2 ~ l)]
= 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 (/e) and (/), but it is more
convenient, to develop them into series. If we substitute in
formula (/) for i/^(l) and i//(2) the series found in No. 17
of the introduction and observe that   = 1 4 cote: s 2 , we
sins 2
find: *)
105 n \ /15 105 a 1575 n
or if we substitute the numerical values:
^=[1.759845] tang^ [8.821943] tang2 3 + [6.383727] tangz 5  [4.180257] tang^ 7 ,
where the figures enclosed in brackets are logarithms.
Furthermore the terms multiplied by f give:
75 7 1785 9 46305 M j
" "
or (^,)
 j [5.506187] tangs; 5  [3.714510] tang2 7 f[1.901468]tang2 9 [9.018568]tang2 n 
For 75 we find from the series da = 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 :
P / 2/3v (l) = tang.r tangz 3 f tangz 5 tangz 7
105
H pi tang z
1 ^ 1 **
2* J/27 V (2) = tang z ^ tang a 3 h ^ 2 tangz^ g ^ 3 tang z 1
105
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 ze
nith distance but only for a certain density of the air, namely
that, which occurs when the temperature is 50 Fahren
heit 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 6, 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 s 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 50 as unit, the same volume
at the temperature r will be 1M (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 Hs(r 50). We 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 6:29.6. If we therefore denote the density of the air
when the thermometer is r and the barometer is b by p, and
the density in the normal state of the atmosphere by y (} , we
have :
b
1 4 8 (r 50)
and as the quantity a which occurs in the formulae 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 re
fraction from the mean refraction by the formula:
* 6
,,_ ^ 2976
1 f e (r 50)
if did occur only as a factor, as the quantity 1 a in the
divisor can be considered as constant on account of the small
ness of a. But a occurs also in the factor of " , which
1 cr
170
shall be denoted by Z and the quantity ft varies also with
the temperature, as it depends on / or when the temperature
is T upon / = i [i + e ( r 50)]
if we denote the height of an atmosphere of uniform density
at the temperature T by /. We find therefore the true re
fraction from the following formula:
SJ = . fi = so + rr d ~ (50) + ;  d H (62 J.G), ()
H(T oO; 29.6 1 dr 1 d6
but as the influence of the last two terms is small we may
take for the sake of convenience:
* ,_ U?*_ /_1V + " ( ^
~~ [lf. a <T 50)] +" V29.6/
But if we develop this we find, neglecting the squares
and higher powers as well as the products of p and q:
Thus we obtain from the formulae (m) and (w) the fol
lowing equations for determining p and q:
OQ f
if we take in the second member dz instead of d ~z .  ^.
1 + (r aO)
The moisture diminishes also the density of the atmo
sphere 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 a 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 re
fraction.
In order to obtain the expressions for p and </, we must
rl 7 /I 7
find the differential coefficients  and  , but we shall de
dt db
duce these values only for Ivory s theory, as the deduction
from BesseFs formula is very similar. According to formula (/)
we have:
~ ft? (1) + 1 }/2 y (2) +/ Q],
171
takino a ^= L From this we obtain:
C> C J T1 ~2
: i . ^ (1 ~ a) ^ 4 /2/?/ [/2 y (2)  v CD] y
as f does not change with the temperature and the stand of
the barometer.
Now we have ^(1) = e~ T * fe~ 2 dt, where T^cotg z /,
t~ #2 c? ^, where T 2 = cotg &Vfti
and as ^ =2 T, ./,(!) 1 and ^ = 2 ^02) 1,
dl i dl 2
the last but one term in (/?) becomes:
4 d j Vzp [(i  X) (ir, 2 y a)  1 r, ) 4 A 1/2 . (T 2 2 v (2)  * r 3 )].
The factor () consists of two terms, the first of which
having the factor 2 is equal to the factor of A in the ex
pression of oz. We therefore embrace this in the latter term
by writing / 2f instead of A. There remains then only
the following term
and as we find differentiating it:
the complete expression for dZ becomes:
. rf^ff 8z(\a) dl .
dZijf.   a + T ]/2/3. A [1/2 y, (2)  y, (I)]
I  /2 ~ 4 (1A
As we have:
b
rf /; 29.6
we find: =  ^g   e (r  50),
172
and likewise:
p + dft = 2 2e(T 50) 9 hencc d l = _ E (r _ 50) .
o *o P
finally we have:
/9 </>l rfa dB 6 29.6
*& hence T=^ + f= 29; 6 2.<T50).
We find therefore:
%p . I [1/2 y (2)  y, (I)]

I cc
" 2 A [)/2 y, (2)  y (1)] (ry)
where instead of /" its value f has been substituted.
If we compute from this p and q for 5 = 87, 8z being
852". 79 we find:
log 7\ = 0.013175, log [tf2 V<2) ^ (1)] = 8.605021,
log (I, 2 .//(I) i TO = 9.081 168 /0 log T 2 = 0.163690,
log(T 2 2 i/;(2) 1^)^2 = 9.191771,, and with this
^a.g = 19".71, S*.p = 185". 36,
hence :
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 in (/j) and (/ 2 ) with respect to a and /?,
i  Ct
we easily find the following series:
qSz = f [7.90399] tang z h [7.9014G] tang z^ [5.G6533] tang z :>
+ 1 3.54 172] tang z 7 . . .
p ^ 2 == + [7.90399] tang z + [8.91567] tang 2* [6.70990] tang z 5
4 [4 567 12] tangs 7 ...,
where the coefficients are again logarithms.
For ^ = 75 for instance we find from this = 0.0020
and p= 0.0188.
12. For the complete computation of the true refraction
from formula (m^), we must know the height of the baro
meter reduced to the normal temperature. If we take the
length of the column of mercury at the temperature 50 as
unit and denote the expansion of mercury from the freezing
173
to the boiling point equal to by </, the stand of the baro
Oo.o
meter observed at the temperature *) is to the stand, which
would have been observed if the temperature had been 50
as 1 + g (t 50) : 1, or the length of the column of mer
cury reduced to the temperature 50 is:
180
180 H 7 U 50)
If further s is the expansion of the scale of the baro
meter 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 50 as unit:
Hence the height b, of the barometer observed at the
temperature , is reduced to 50, taking account of the ex
pansion of the mercury and the scale, by the formula:
180 4 s (t 50)
* 50)
The normal length of an English inch is however not re
ferred to the temperature 50 but to the temperature 62;
hence the stand of the barometer observed at the temperature
50 is measured on a scale which is too small, we must there
fore divide the value 6 50 by 1f ^, so that finally we get:
180fs(* 50) 180
180 + q(t 50) 180~4~12s
If the scale is divided according to Paris lines and the
thermometer is one of Reaumur, we should get, as the nor
mal temperature of the French inch is 13 R. and we have
50Fahr. = 8"Reaum.:
80 4 s (t 8) 80
80H7(* 8) 80 + 5*
This embraces every thing necessary for computing for
mula (m^). If we denote by f the temperature according to
*) The temperature t is observed at a thermometer attached to the baro
meter, 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 Reau
mur s thermometer, by b (f} and b (l) the height of the barometer
expressed in English inches and Paris lines and if we put:
3 _ 6(0 180 _^_ 80
""2976 1 80 41 2, s ~~ 333728 8045 .v
_ 180 4 s(f 50) __ 804
180 4 q (/ 50) 80 4 q (r 8)
1_ _1
7 ~~ 1 4 B . (/ 50) 1 4f e (r 8)
and give to the mean refraction the form dz aismgz, we
have :
Sz = a tang z . /+" (B . T^+" (A}
hence log Sz = log a 4 log tang 2 4 (1 4;>) log y 4 (1 4 7) (log B 4 log T).
If we have then tables, from which we take log G, 1 \p
and 1fg for any zenith distance, and log 5, log T and log ;
for any stand of the barometer and any height of the interior
and exterior thermometer, the computation of the true re
fraction 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 con
sists 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 equi
librium. 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 dis
tances. According to the table given in the introduction
to the Tab. Reg. pag. LXIII these probable errors are at
450=1=0". 27, at 81"==1", at 85 + 1". 7, at 89 30 ==20". We
thus see, that especially in the neighbourhood of the hor
izon 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 re
fraction.
For zenith distances not exceeding 80 it is almost in
different, what hypothesis we adopt for the decrease of the
density of the atmosphere with the elevation above the sur
face 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
l\p and l{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 = ^0 sin/,
or as we have now i = f+fizi
Sz = (X, 1) tang/,
and since we have also, as is easily seen, sin f= " sin z, where
/ is the height of the atmosphere, we get:
J^ = = (,,. l)tang z (l? ,).
2 I V a cos z 2 J
,/
I/
If we take now for /< 1 the value 57". 717, we find
for the refraction at the zenith distances 45, 75 and 80
the values 57".57, 211". 37, 314". 14, whilst according to Ivory
they are 57". 45, 21T.37 and 315". 20. But beyond this the
error increases very rapidly and the horizontal refraction is
only about 19 .
The equation (/) 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 equa
tion :
176
the equation (/") becomes simply:
;== _ dw_
(2m 1) Vlw*
therefore if we integrate and substitute the limits w = sin z
and w = (1 2 a) " sin ss, we find:
2 /  1
i
2m 1
or:
2 arc sin (12 a)
<>, 1
sin [2 (2 m I ) Sz] = (1 2 a) " sin z ,
for which we may write for brevity:
If sin z = sin [z NSz].
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 M and N are suitably
determined.
If we add to the last equation the identical equation
sin s = sin* and also subtract it, we easily find two equa
tions from which we obtain dividing one by the other:
N
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 re
fraction, we can see them on account of it, when they really
are beneath the horizon. The stars rise therefore earlier and
set later on account of the refraction.
We have in general:
cos z = sin (f sin + cos y> cos S cos t (r)
from which follows:
sin zdz = cos <p cos S sin t . dt
hence if the object is in the horizon:
______ _ ___
cos y cos S 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:
cos <p cos S sin t
177
In No. 20 of the first section we found for Arcturus
and the latitude of Berlin:
t = 7 h 42 m 40 s
and as we have <?= 19 54 .5, cp = 52 30 . 3, we find:
A/o=437s.
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 ~ sin (p sin 8
C0 st= Zg
COS (p COS
and adding 1 to both members , we find the following con
venient formula:
i _ I/ cos ^s (f ~t~ d ~+~ z) cos TJ (cp + S 2)
COS Cp COS S
If we subtract both members from 1, we obtain a sim
ilar formula:
i / sin i (z j cp <?) sin 4 (z + d OP)
sm * = I/ 2V "
cos y cos ()
In the case of the moon we must take into account be
sides 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.
9 P
July 15 Oh 15 32.1 59 13
12h 17 51.5 . 59 15
16 Oh 19 55.6 59 14
12 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:
Lnnai time Mean time
6hl6 "^ 1227.5.
12
178
If we take now an approximate value of the declination
17 51 . 5 we find with cp = 51 28 . 6 and = 89 35 . 8,
t = k h 21 m .5 and the mean time corresponding to this lunar
time 10 h 48 m . If we interpolate for this time the declination
of the moon, we find 17 38 . 2 and repeating with this
the former computation, we find the hour angle equal to
4 h 22 m .9, hence the mean time of setting 10 h 49 m .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 18 below the horizon. Thus
the moment, when the sun reaches the zenith distance 108
is the beginning of the morning or the end of the evening
twilight.
If we denote the zenith distance of the sun at the be
ginning or end of twilight by 90" + c, by t tt the hour angle
at the time of rising or setting and by T the duration of
twilight, we have:
sin c = sin cp sin \ cos cp cos S cos (t H r)
hell e = COS (* + T) =  >*** **
COS (p COS
or putting H= 90 cf +
i / sin f (H Hhc) cosTf (H ~c)
sin * (< 4 *) = I/
cos cp cos
from which we can find T after having computed t ti .
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 T and we have:
cos ZZ = sin y 2 + cos <p 2 cos r.
But as we have in the triangle between those two points
and the sun S, ZS = 90hc, Z S=90, we have also call
ing the angle at the sun S:
cos ZZ = cos c cos S
179
and thus we find:
1 cos c . cos S
2 COS Q5 2
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 T is a minimum, 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 give.n by
the equation:
sin 4 r =
cos 9?
and as we have:
sin 9? j sin c sin S
. . , cos p ,
sin o cos c cos o
we find:
sin S = tang ^ c sin 95,
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(c by
A and A\ we have:
cos 95 sin A = cos S sinp
cos (f sin A = cos S sinp .
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 180.
From the two equations:
sin c = sin y> sin S f cos y> cos 8 cos (t + 1]
and
= sin 9? sin S f cos 9? cos S cos t
follows also:
cos 4 c sin 4^ c
sm (t f % T) sin 4 r = V >
cos cos y>
If we take c=18 we find for the latitude </>=81
sinr=l, 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 18 below at midnight. But we cannot speak
12*
180
any 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
below the horizon; for if the south declination is greater
the sun remains below the horizon and if the south decli
nation 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 hence
the formula for sin ^ T becomes impossible.
Note. Consult: on refraction: Laplace Mecanique Celeste Livre X. 
Bessel Fundamenta Astronomiae pag. 2G et seq.  Ivory in Philosophical
Transactions for 1823 and 1838. Bruhns in his work: Die Astronomische
Strahlenhrechung has given a compilation of all the different theories.
III. THE ABERRATION.
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
the ray of light coming from an unmove
able object (fixed star) strikes in succes
sion the objectglass and the eyepiece of
a telescope (or the lense and the nerve
of the eye). The positions of the object
glass and of the eyepiece in space at the
time t shall be a and 6, and at the time
t a and b Fig. 5. Then the line a b re
presents the real direction of the ray of
light, whilst a b or a b\ both being parallel
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 a and
b a is called the annual aberration of the
fixed stars.
181
Let #, #, z be the rectangular coordinates of the eye
piece b at the time , referred to a certain unmoveable point
in space; then:
x f ^ (J  t), y + ^ (  and a f (*  )
/ a? ai
are the coordinates of the eyepiece at the time , since during
the interval t t we may consider the motion of the earth
to be linear. If the relative coordinates of the objectglass
with respect to the eyepiece are denoted by , i] and f , the
coordinates of the objectglass at the time , when the light
enters it, are x f , y f ?;, ss f ?.
If we now take as the plane of the x and # 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 #, z through the solstitial points ; if we further denote by
and () the right ascension and declination of that point in
which the real direction of the ray of light intersects the ce
lestial sphere and by u the velocity of light, then will the
latter in the time t t describe a space whose projections
on the three coordinate axes are :
a (t /) cos cos , {u (t t) cos <?sin , t u (t t) sin 8.
Denoting further the length of the telescope by / and
by a and <) the right ascension and declination of the point
towards which the telescope is directed, we have for the co
ordinates of the objectglass with respect to the eye piece,
which are observed:
I = I cos cos n. . // = I cos sin , = / sin d .
Now the true direction of the ray of light is given by
the coordinates of the objectglass at the time t:
I cos cos a + .r,
I cos sin a \y,
I sin <T h z,
and by the coordinates of the eyepiece at the time t :
182
We have therefore the following equation if we denote
u, cos cos a = L cos 8 cos >
a
, cos <? sin = L cos <? sin a ~ ,
{ u sin 8= L sin 8
We easily derive from these equations the following:
cos 8 cos (a a) = cos 8 \ } ^ sin a f  cos [ ,
u, ft at at
L 1 (dy dx
cos 8 sin (a a) = cos sin
p /u dt dt
1 *(dy dx .
sec o ) ~ cos sm
r , . u, \dt dt
or : tang (a ) = 73 :
1 , ! * i ^ , rf;r
H sec o \ ^ sm a +  cos
;W ( rf< (/^
We find a similar equation for tang (d 1 ^). If we de
velop both equations into series applying formula (14) in No. 11
of the introduction, we find, if we substitute in the formula
for tang ((V #) instead of tang( ) the value derived
from a a and omit the terms of the third order:
1 \dx . dy )
a a = { sm a f cos ( sec o
^ rf< dt
dx <
c^ . s> , . e, . e o
o =  sm o cos a H sin o sin a cos o
p ( dt dt dt
(a)
ang ^
1 (dx s dy 9 . c?z . _
cos o cos a H cos () sm a \ sin o
fi 2 (dt dt dt
^(dx . ^ dy . . . </^ )
X )  sin o cos a + sm o sm cos o (
I dt dt dt
If we now refer the place of the earth to the centre of
the sun by coordinates a?, 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 pos
itive 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 O, its distance from the earth by R, we
have *) :
* = .Ecos,
y = R sin Q
If we refer these coordinates to the plane of the equa
tor, 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 g, equal to the obliquity
of the ecliptic, we get:
y = R sin Q cos e.
z = R sin O sir  >
and from this we find, since according to the formulae in
No. 14 of the first section we have the longitude of the sun
= v h 7i or equal to the true anomaly plus the longitude
of the perihelion:
dx * dR dv
__ =s _ co ^_H_* sin0 _
dy dR _^ dv
f = sm (0 cos e  R cos (O cos e
at at dt
dz dR dv
 = sm () sin s   R cos CO sin e _
dt dt dt
But we have also according to the formulae in No. 14
of the first section:
d v =  D dE and as we have also dE = ~ d M
K H
we find : dv _ a 2 cos y dM
~d~t ~ R^ ~dt
Further follows from the equation R = . ^  in con

nection with the last:
dR dM
~ = a tang y sm v 
and from this we get:
dx a dM( . _ a* cosy _.
r =  { sin QO ^ sin fp sm v cos CO
dt cosy dt ( R
hence observing that:
a 7 cosy ^
^ = 1 f sin fp cos v and () v = TT,
it
</^ a dM . __
r = ~ I sm O + sm 9 s sm ^J
dt cos y rf
*) As the heliocentric longitude of the earth is 180 + Q.
184
and   = cos " [cos O H sin or cos n] (fi)
dt cosy dt
dz a dM r
= sin s , I cos CO f sin cp cos TT .
r/i! cosy dt
If we substitute these expressions in the formulae (a),
the constant terms dependent on n 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 /.< the number k of seconds,
in which the light traverses the semimajor axis of the earth s
orbit, so that we have:
1 ___ k
p a
we find, taking only the terms of the first order:
, k dM
 I cos Q cos s cos a f sm M sm a] sec o
cosy dt
^
S 8 = f [cos O (sin sin dcoss cos <?sin e) cos a sin ^sinQl
cos y at
The constant quantity is called the constant
cos y dt
of aberration, and since *  denotes the mean sidereal mo
tion of the sun in a second of time, which is the unit of
A, 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 con
stant latterly from the observations of the apparent places of
the fixed stars and found 20". 4451 and as we have J =
dt
== 0.041 0670 and cos == 9.999939 we find from
this for the time in which the light traverses the semimajor
axis of the earth s orbit 497 s . 78*).
We have therefore the following formulae for the an
nual aberration of the fixed stars in right ascension and de
clination :
*) According to Hansen the length of the sidereal year is 365 days 6
hours minutes and 1), 35 seconds or 3(55.2563582 days, hence the mean
daily sidereal motion of the sun is 59 8". 193.
185
n a = 20" . 4451 [cos cos E cos a + sin sin ] sec S
8 = 4 20". 4451 cos [sin sin S cos cos S sin s] (A)
 20" . 4451 sin cos sin &
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 dif
ferential coefficients (6) into the second term of the formulae
(a), as follows:
& 2 /dJl\ 2
{ a fr J sec<? 2 [cos20sin2(Hcos 2 ) 2 sin 2 cos 2 cose],
where the small term multiplied by sin 2 a sin s 2 has been
omitted. For we find setting aside the constant factor:
2 sin 2 a [cos 2 cos e 2 sin 2 ] 2 sin 2 cos [cos 2 ~ sin 7 ]
from which the above expression can be easily deduced. If
we substitute the numerical values taking s = 23 28 , we
obtain :
 0" . 000932!) sec S 2 sin 2 cos 2
h 0" . 0009295 sec S* cos 2 sin 2
As these terms amount to T( r> of a second of time only if
the declination of the star is 85.]", they can always be ne
glected except for stars very near the pole.
The terms of the second order in declination, if we ne
glect all terms not multiplied by tang r?, are:
 I ~ C ^~~T \~Jl ) tan g S t cos  O ( cos 2 ( 1 h cos f 2 ) sin 2 )
H 2 sin 2 sin 2 a cos t].
For we find the term multiplied by tang J, setting aside
the constant factor:
sin 2 sin a 2 + cos 2 cos 2 cos 2 f ^ sin 2 sin 2 cos
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 f cos 2 as well as the term cos 2 a sin 2
we easily deduce the above expression. Substituting again
the numerical values we find:
h [0". 0000402 0". 0004665 cos 2 a] tang cos 2
 0". 0004648 tang S sin 2 sin 2 0.
As these terms also do not amount to : f j g of a second
of arc while the declination is less than 87 6 , they are taken
into account only for stars very near the pole.
In the formulae (A) for the aberration it is assumed,
that , S and be referred to the apparent equinox and
186
that is the apparent obliquity of the ecliptic. But in com
puting the aberration of a star for any long period it is con
venient, to neglect the nutation and to refer a, 3 and to
the mean equinox and to take for 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, (J, and and taking da, dS, dO and de equal to
the nutation for these quantities. Of course it is only ne
cessary to take the largest terms of the nutation and omit
ing in the correction of the right ascension all terms, which
are not multiplied by sec . tang ti and in declination all
terms which are not multiplied by sin d . tang #, we easily
see, since the increments dQ and ds do not produce any such
terms, that we need only take the following:
da = [6". 867 sin ft sin f 9". 223 cos ft cos ] tang S.
dS= [6" . 867 sin ft cos a h 9" . 223 cos ft sin a].
Taking here 6".867 = & and 9". 223 = , we find, if we
substitute these quantities into the differentials of the equa
tions (A):
a a = tang sec <5 10". 2225 / (&{ cose) sin 2 a cos (Q 4 ft)
} \(b a cos ) sin 2 a cos (0 ft)
\ (b cos a) cos2 a sin (0 ft)
== tang S sin <?5" . 1 1 12 / (b 4 a cos e)cos 2 a cos (0 f ft) \
I (&coseHa)sin2sinCQ4n) I
/ + (b a cose) cos 2 a cos (O O) (
J (b cos a) sin 2 a sin (0 ft) i
}
or if we substitute the numerical values:
a a = tang S sec S . I 0".0007597 sin 2 a cos (0 + ft) ,
) + 0".0007693 cos 2 a sin (0 H ft)
} 0".0000790 sin 2 cos (0 ft) \
( _j_ 0".0001449 cos 2 sin (0 ft) <
== tang S sin 8 . / 0".0003798 cos 2 a cos (0 ift) >
 0".0003847 sin 2 sin (04H) J
 0".0000395 cos 2 a cos (0 ft) (
0".0000725 sin 2 a sin (0 ft)
 0".0000395 cos (04 ft)
\ 0".000379Scos(0 ft)
187
While the decimation is less than 85, a a is less
than T 5Q of a second of time and e) is greater than T J 5
of a second of arc only for declinations exceeding 85 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 funda
mental plane. For then neglecting again the constant terms
we find:
dx a _ d M
7 = H sin W r~ >
at cosy dt
dy a dM
Tt s "cos/ 080 77
*=<>
and if we substitute these expressions in the formulae (a) and
write K and p in place of a and #, we find for the aberration
of the fixed stars in longitude and latitude:
A A = 20". 445 1 cos (/I O) sec ft,
ft /? = + 20". 4451 sin (A 0) sin ft
which formulae are not changed if we use the apparent in
stead of the mean equinox.
The terms of the second order are:
in longitude: = 4 0". 0010133 sin 2 (0 /I) sec /2 2 ,
in latitude : = 0". 0005067 cos 2 (0 A) tang ft,
where the numerical factor 0.0010133 is equal to f . i? ^ 4 ^ 5 !!! .
Example. On the first of April 1849 we have for Arc
turus :
=14h8m48s = 212 12 .0, = 4 19 58 . 1, = 1137 .2
fi = 23 27 . 4.
With this we find:
= 4 18". 88,
S  =  9". 65,
and as
A = 202" 8 , /? = 4 30 50 ,
we find also:
A I = 4 23". 41,
188
17. In order to simplify the computation of the aber
ration in right ascension and declination, tables have been
constructed, the most convenient of which are those given by
Gauss. lie takes:
20" . 445 sin = a sin (Q  A\
20". 445 cos O cos e = a cos (Q f A).
and thus has simply:
= (( sec S cos (044 ) ,
$ <?= sin 8 sin (0 f A a) 20". 445 cos cos t> sin t
= a sin # sin (0 + A a) 10" . 222 sin e cos (0 f <?)
 1 0". 222 sine cos (O #).
From these formulae the tables have been computed.
The iirst table gives A and log a, the argument being the
longitude of the sun, and with these values the aberration
in right ascension and the first part of the aberration in de
clination is easily computed. The second and third part is
found from another table, the angles 0M and 8 being
successively used as arguments. Such tables were first pub
lished by Gauss in the Monatliche Correspondenz Band XVII
pag. 312, but the constant there used was that of Delambre
20". 255. Latterly they have been recomputed by Nicolai
with the value 20". 4451 and have been published in Warn
storff s collection of tables.
For the preceding example we find from those tables:
A = \ 1 , log o = 1.2748
and with this
a = f18". 88
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:
3 1 $=9". 65.
18. The maximum and minimum of aberration in lon
gitude takes place, when the longitude of the star is ei
ther 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 from the sun and from the earth subtend
at the fixed star) only the maxima and minima in this case
occur at different times. For if & be the distance of the
fixed star from the sun, /: and ft its longitude and latitude
as seen from the sun, the coordinates of the star with re
spect to the sun are :
x & cos ft cos A, y = A cos ft sin /, r = A sin ft.
But the coordinates of the star referred to the centre
of the earth are:
x = A cos ft cos A , y A cos ft sin A , == A sin /?
and as the coordinates of the sun with respect to the earth are:
X=RcosQ and r=/2sinQ
where the semimajor axis of the earth s orbit is the unit,
we have:
A cos ft 1 cos ti = A cos /^ cos /I f # cos O
A cos /? sin A = A cos ft sin A j It sin Q
A sin ft = A sin /9,
from which we easily deduce:
A A = * sin (A Q) sec ft . 206265,
u
ft ft = ^ ; cos (/I Q) sin ft . 206265.
or as ^ 206265 is equal to the annual parallax n:
K I = n R S i n (I Q) sec ^
P l3= nR cos (A Q) sin /?.
Hence we see that the formulae are similar to those of
the aberration, only the maximum and minimum of the par
allax in longitude occurs, when the star is 90 ahead of the
sun or follows 90" after it, while the maximum and minimum
in latitude occurs, when the longitude is equal to that of
the sun or is greater by 180.
For the right ascensions and declinations we have the
following equations :
A cos cos a = A cos S cos a + R cos Q
A cos sin = A cos S sin a f R sin Q cos e
A sin 8 = A sin 8 + R sin sin e,+
from which we find in a similar way as before:
a a = TT R [cos sin a sin Q cos s cos ] sec S
$ ^ = T* R [cos sin sin 8 sin cos S] sin (Z>)
nR cos sin S cos .
190
19. The rotation of the earth on her axis produces like
wise 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 coin
cides with the axis of rotation, whilst the two others are sit
uated 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 90 th degree of
right ascension, the coordinates of a place at the surface
of the earth are according to No. 2 of this section as follows :
z gcosy cos 0,
y = q cos 90 sin ,
z = Q sin (f .
We have therefore:
dx

dt
dy
2 = j () COS (p COS 0.
 = o cos (f sin
dt
If we substitute these expressions in formula (a) in No. 16,
we easily find omitting the terms of the second order:
a a = P cos y cos (& a) sec #,
fi dt
8 8=   cos y sin (0 a) sin 8.
ft 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:
d& __ T dM
dt dt
Thus as we have:
 p = k = k sin TT
I
where n is the parallax of the sun, k the number of seconds
in which the light traverses the semimajor axis of the earth s
orbit, the constant of diurnal aberration is:
k . . sin 7t . T,
dt
191
or as we have:
jk. ^"=20".445, 7r==S".5712 and 7 7 =3G6.2G is,
0".3H3.
Hence if we take instead of the geocentric latitude </
simply the latitude <f , we find the diurnal aberration in right
ascension and declination as follows:
a = 0". 31 13 cos y cos (0 a)sceS,
S 8 = 0". 3113 cosy sin (0 ) sin 5.
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 y> sec 8.
20. We have found the following formulae for the an
nual aberration of the fixed stars in longitude and latitude :
A A = k cos (I Q) sec p,
ft p = + k sin (1 0) sin/9,
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 coordinates, 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 coordinates:
x = (A K} cos /9 and y = /? /? *),
we easily find by squaring the above equations:
^ 2 = P sin/? 2 x l sin/5 2 .
This is the equation of an ellipse, whose semi major
axis is k and whose semiminor axis is k sin ft. We see there
fore that the stars on account of the annual aberration de
scribe 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, ft 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, ft equals 90 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 mean 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 diameter is the major axis of the el
lipse. 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 ma
jor 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 90. If we draw then the radius corresponding
to any time and let fall a perpendicular line from the ex
tremity 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 coordinates become:
x k cos (A 0) n sin (A 0)
. y = + k sin (A Q) sin ft n cos (A 0) sin /?
or, taking:
k = a cos A
TC = a sin A
x = a cos (A A )
y = H a sin (/ A) sin /3.
Hence also in this case the star describes round its
mean place an ellipse, whose semimajor axis is Ftf 2 h77 2 and
whose semi minor axis is sin ft V k?\ ^>
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 semimajor axis is
0". 3113 cos (f and whose semiminor axis is 0". 3113 cosy sin 8.
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 objectglass of the telescope at the time ,
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 f,
A the place of the objectglass at the time T, a and b the
places of the objectglass and the eyepiece at the time t and
finally a and b their places at the time , when the light
reaches the eye piece. Then is:
1) AP the direction towards the place of the body at the
time r, ap that towards the true place at the time ,
2) a b and a b the direction towards the apparent place
at the time t or t\ the difference of the two being in
definitely small,
3) b a the direction towards the same apparent place cor
rected for the aberration of the fixed stars.
Now as P, a, b 1 are situated in a straight line, we have:
Pa : a b = 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 mo
ving in a straight line and with a uniform velocity, the points
4, a, a are also situated in a straight line, so that A a and
a a are also proportional to the times t T and t t. Hence
it follows that A P is parallel to 6 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
497 s . 8, that is, by the time in which the light traverses the
semimajor axis of the earth s orbit, which is taken as the unit.
It follows then that we can use three methods, for com
puting 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 ident
ical 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 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 semi
major axis of the earth s orbit is 497 s . 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 in
clined 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 noth withstanding the motion of the
earth and the proper motion of the body. If therefore , ?/
and L, be the coordinates of the body with respect to the
system of axes used there, we must substitute in (a) j  ,
dy_d_n dz_d . d f dx djj an( j dz^ fi if A . h
dt dt dt dt dt dt dt
distance of the body from the earth, we find the heliocentric
coordinates , ?/, f, since the geocentric coordinates are
A cos 8 cos etc. , from the formulae :
f = A cos cos a f x ,
rj = A cos 8 sin f y , (/)
= A sin 8 H z ,
195
from which we easily deduce the following:
(dx dg\ . (dy drj\ da
[ I sm r; r I cos a = A cos o
\dt dt) \dt dtJ dt
(dx dg\ . . (dy dri\ ... (dz d^\ ~ dS
1 sm o cos a + [ I sin o sin a f I J cos o = A r~
\</< c/// W d// Vrf* dt/ dt
Hence the formulae (a) change into:
A da
a a = ,
^ e?
A X * d8
d d ,
ft dt
or as equals the time in which the light traverses the dis
tance A, we find, if we denote this by t 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 II of the preceding number.
But we also find the aberration for this case by adding
to the second member of the first formula (a) the term
^ sin a cos a sec 8 and a similar term to the second
fi [_dt dt J
member of the second equation. We get therefore, if we
denote the aberration of the fixed stars by Da and Dd:
, 1 [~c?! . dr] ~
a = D a \ sm a cos a sec o .
fi \_dt dt J
S 8 = D i sin cos j sin d sin a +  cos 8 .
fi [_dt dt dt J
But differentiating the equations (/*), taking in the second
member only the geocentric quantities A? ? 8 as variable and
the coordinates of the earth as constant, and denoting the
partial differential coefficients by (^) and (V), we find the
second members of the above equations respectively equal to :
A (da\ A /^^\
/u, \dt / /LI \dt /
We therefore have:
and S DS = StT).
13
196
which formulae correspond to the third rule of the preceding
No. For since and are the differential coefficients
of a and cV, 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, buf as seen from the place which the earth
occupies at the time t.
Note. 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 mo
tion which at present and undoubtedly for long ages may be considered as
going on in a straight line, produces only a permanent and hence impercep
tible 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 cir
cular, r and T their times of revolution, then the angular velocities of the
two will be as : 7 , hence their linear velocities as ar : a r or as j/a : J/a,
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
O/\" i **
the radius of the earth s orbit as unit, is therefore   ~ and hence the
ya
constant of aberration caused by the motion of the sun round their common
20 ; .45
centre of gravity is equal to m . ~ r^~ , where m is the mass of the planet
expressed in parts of the mass of the sun. In the case of Jupiter we have
W* = TOTO an d a = 5.20, hence the constant of aberration caused by the at
traction of Jupiter is only 0".OOS6.
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 Regio
montanae p. XVII et seq. ; also Wolfers, Tabulae Reductionum p. XVIII etc.
Gauss, Theoria motus pag. G8 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 deter
mination of the places of the stars with respect to the fun
damental 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 equi
nox, 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 sim
ilar 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 de
terminations, 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 as
cribed to any such errors of the constant quantities and to
errors of observation. However, comparing the places de
termined 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 mo
tions 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 appa
rent 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 be
ginning 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
motion 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 Regiornontanae" *).
Let and d be the mean right ascension and declination
of a star at the beginning of a year, whilst a and $ 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 ( w und . designate the proper
motion of the star in right ascension and declination, which
is considered to be proportional to the time, we have ac
cording to the formulae (/)) in No. 2, (#) and (C) in No. 5
of the second section and (A) in No. 16 of the third section
the following expression:
a = 4 T [mfw tang sin a] + T ft
 [15".8148 + 6".8650 tang S sin ] sin ft
9".2231 tang 8 cos a cos ft
4 [OM902 h 0".OS22 tang S sin ] sin 2 ft
4 0".OS96 tang S cos a cos 2 ft
 [1". 1642 f 0".5054 tang S sin a] sin 2 Q
0".5509 tang S cos a cos 2 Q
H [0".1173 4 0".0509 tang S sin a] sin ( P)
 [0".0195 4 0".0085 tang 5 sin a] sin (0 4P)
 0".0093 tang 8 cos a cos (0 4 P)
20".4451 cos s sec 5 cos a cos
20".4451 sec sin sin
and:
S 8= 4 rn cos f Tp!
 6".8650 cos a sin } H 9".2231 sin a cos O
f 0".0822 cos a sin 2 ft 0".OS96 sin a cos 2 J~)
0".5054 cos a sin 2 40".5509 sin a cos 2
hO".0509cosasin(0 P)
 0".0085 cos a sin (0 4 P) + 0".0093 sin cos (0 4 P)
h 20".4451 [sin a sin 8 cos cos 8 sin e] cos
 20".4451 cos a sin S sin 0.
The terms of the nutation, which depend on twice the
longitude of the moon 2d and on the anomaly (L 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 neighbour
hood 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 argu
ments are ([, 0, OhO and O O.
Now in order to construct tables for the above expres
sions for a a and d , we put:
6".S650 = nz 15".S148 mi = h
0".OS22 = ni, 0".1902 mi l = h l
Q".5054 = ni z 1".1642 mi 2 = fi 2
0".0509 = ni 3 0".1173 m z 3 = / 3
0".0085 = ni 4 0".0195 mil = /* 4 .
Then we can write the formulae also in this way:
n a =[r i sin ft + i l sin 2 } i 2 sin 2 + i 3 sin (0 P)
1 4 sin (0 f P)J [/ + w tang <? sin a]
 [9".2231 cos O 0".0896 cos 2 O f 0".5509 cos 2
H0".0093cos(0+P)] tangtfcosa
20". 4451 cos s cos . cos a sec $
20".4451 sin . sin a sec S
P) 7* 4 s
and:
S S=[r isin^Mi sin 2~} e 2 sin20K 3 sm(0 P)
z 4 sin (0  P)] n cos
+ [9".2231 cos D 0".0896 cos 2^ + 0".5509 cos 20
4 0".0093 cos (0fP)] sin a
20". 4451 cos E cos [tang e cos S sin sin ]
20".4451 sin . sin S cos a
If we introduce therefore the following notation :
A=r { sin H Hi sin 2 i~} l a sin20Hi 3 sin(0 P) / 4 sin (0fP)
,B = 9".223 1 cosO I 0".0896 cos 2^ 0".5509 cos 20 0".0093 cos(0HP)
C == 20".4451 cos cos
/>= 20".4451sin0
^== 7/sin^h^,sin2O A 2 sin20H A 3 sin(0 P) A 4 s
a = w< f n tang $ sin n a! = n cos
ft = tang S cos b = sin
c = sec 8 cos c = tang e cos # sin # sin a
d = sec $ sin a d = sin S cos a,
*) These terms are given by the formulae (E) in No. 5 of the second
section and (c), (d) and (e) in No. 16 of the third section.
201
we have simply:
Aa + Bb f Cc + Dd + r^ f
 Cc
where the quantities a, 6, c, d, a , 6 , c , d depend only on
the place of the star and the obliquity of the ecliptic, while
A, B, (7, D depend only on and H 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 i z =0.02519 i 3 =0.00254 i 4 = 0.00042
A=0.0572 h t =0.0016 A 2 =0.0041 A 3 = 0.0005 A 4 =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 a a and S are variable (according to No. 5 of the
second section) and likewise the values of m and w, we have
for the year 1900:
i=0.34256 i, =0.00410 * = 0.02520 i 3 =0.00253 z 4 =0.00042
A=0.0488 h l =0.0014 h z =0.0035 7*3=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 dif
ferent value of the constants of nutation and of aberration
and also neglected the terms multiplied by P and 0fP,
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 ^ 4 0.0001 sin 2^ + O.OOlo sin 20
H 0.0025 sin (0 P) 0.0004 sin (0+P)
dB= 0.2456 cosO + 0.0019 cos2O + 0.0290 cos 2
0.0093 cos (0 HP)
d C = 0.1744 cos
(/>= 0.1 901 sin
dE = 0.006 sin O + 0.001 sin 2 O
For 1850 the value of dB becomes:
dB= 0.2465 cosiH0.0019cos 2^ H0.0291cos20 0.0093 cos(0fP).
202
The values of the quantities A, B etc. for the years 1850
to 1860 have been computed by Zech according to BesseFs
formulae, and for the years 1860 to 1880 they have been
given by Wolfers in his work Tabulae Reductionum Obser
vationum Astronomicarum", 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 days of
the year, the beginning of which is taken at the time, when
the mean longitude of the sun is equal to 280. Hence the
tables are referred to that meridian, for which the beginning
of the civil year occurs when the sun has that mean longi
tude. But as the sun performs an entire revolution in 365
days and a fraction 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 280, by &, which we take
positive, when the place is east of Paris, and if further we de
note 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 kid and for the time thus corrected we must
take the values from the tables. The quantity k is found
from :
where L is the mean longitude of the sun at the beginning
of the year for the meridian of Paris, while a 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 con
stant quantities A, B, C, D, E are given for the beginning
of the fictitious year or for 18 h 40 m sidereal time of that me
ridian, 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 culmination
of a star whose right ascension is , we must add to the
argument k+d the quantity:
a = 24 h ~ = 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 day add
a unit to the datum of the day, so that the complete argument
is always the datum plus the quantity:
k h d + a + 1,
where we have i = from the beginning of the year to the
time, when the right ascension of the sun is equal to , while
afterwards we take i = 1 .
Now the day, denoted in the tables by Jan. 0, is that,
at the sidereal time 18 h 40 m of which the year begins, the
commencement of the days being always reckoned from noon.
Hence the culmination of stars, whose right ascension is
< 18 h 40 m 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 reck
oned from 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 Lyrae for April 1861 and for the time of culmi
nation for Berlin. We have for the beginning of the year:
a== 2783 30" ^= + 38 39 23" =23"27 22" m = 46".062 logn= 1.30220
and from this we find:
*) We have therefore to use for computing the tables:
= 366 . 242201
Mean longitude of the sun = 280  1  
obb .
where n must be taken in succession equal to all integral numbers from
to 37. With this we find the true longitude according to I. No. 14. We
have also:
^=33 15 25".9 1920 29" 53(t 1800)
204
log a = 1 .4797 1 log a = 0.44889
log 6 = 9.04973 log b 1 = 9.99569
log c = 9.25409 log c = 9.98106
log d = 0.10309,, log d = 8.94233
and besides we have:
log fi = 9.4425 log/* = 9.4564.
Further we have according to Wolfers Tabulae Reductionum
log 4
log.B
log C
logZ;
logr
E
March 31
9.7494
0.5497,,
1.2660,
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)
March 31 + Is . 203  19". 85
April 10 + 1 .541  19 .09
20 +1.871 17.79
30 +2 . 185  15 .97.
Now we have A = + 0.1 24, d= 0.031, ^^ m = 0.005,
and as here i is equal to 1, because a is less than 18 li 40 m
and in March and April the right ascension of the sun is
less than 18 h 40 m , the argument in this case is
the datum + 1.088.
We find therefore at the time of culmination for Berlin :
March 31 + 1.239 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 ap
parent 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, 6, c, etc.
The precession and nutation in right ascension are equal to :
Am {A n sin a tang 8 + B tang S cos a + E
and in declination: An cos a B sin a.
205
Therefore if we put: An = gcosG
B = g sin G
Ami E=f,
the terms for the right ascension become:
ftgsm(G\r ) tang 8
and those for the declination:
g cos (G f a).
Further the aberration in right ascension is:
Csec $ cos a f D sec sin
and in declination:
(7 sin sin a f D sin $ cos a f C tang c cos S.
Hence if we put:
C = h sin // D = h cos /T t = C tang ,
the aberration in right ascension becomes:
h sin (H\ a) sec #
and in declination:
h cos (H+ a) sin $ f i cos $.
Therefore the complete formulae for the reduction to the
apparent place are:
a a=/4 g sin (G + a) tang 8+ h sin (H \ a) sec S \ r/ii
S 8= gcos(G H a) + A cos (//+) sin^ft cos^Hr//.
Here again for the quantities /*, g, h^ i, G and // tables
may be computed, whose argument is the time. They are
always published in all almanacs for every tenth day and for
mean noon.
If we wish to find for instance the reduction of a Lyrae
for 1861 April 10 at 17 h 15 m mean time, this being the time
of culmination of a Lyrae on that day, we take from the
Berlin Jahrbuch for this time:
/==+26".98 <7=+12".20 =3443 A== + 18".98 #=247 3 i= 7".58
hence G \ a = 262 6 7/h = 1656
cos(Gja) 9.13813, g sin (G f a) 1.0S222*
g 1.08636 tang S __M9. 30 L
sin (G + ) 9.99586 a h sin (H+ a) "a68846~
cos (#}) 9.98515 cos^ 9.89260
h 1.27830 i _0^!967_
sin (IT f a) 9.41016 h cos (H+ a) 1.26345
sin 8 9.79564
/=26".98 ;cos$= 5".92
g sin (G + a) tang = 9".67 ^ cos (G + ) = 1".68
sec ^=+ 6".25 h cos (#f a) sin 8= 11".46
r^ =f Q".Q8 r j =
^= 18".98.
206
4. The formulae (A) and (J5) 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 great
est accuracy is required. Now the formulae for the annual
parallax are according to No. 18 of the third chapter:
a a = 7i [cos sin a sin cos cos a] sec d
8 8 = 7t [cos s sin sin d sin e cos 8] sin
TT cos sin 8 cos a.
Therefore if we put:
cos cos a = k sin K
sin a = k cos K
sin a sin 8 cos cos 8 sin e = I sin L
cos a sin 8 = I cos L,
we have simply:
a a = 7tk cos CAT} 0) sec 8
$ 8 = nl cos (L 40).
But the cases in which this correction must be applied
are rare, for instance when observations of 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 culmi
nation of the stars, these are equal to the difference of their
apparent right ascensions expressed in time. We need there
fore 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 me
ridian, 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 Ypieces are verniers or microscopes, 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 me
ridian 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 24 h f/\, where &a 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 as
sume to be fulfilled, the intervals between two culminations
measured by a perfectly regulated clock would also be found
equal to 24 h /\. But on account of the errors of single
observations, we can only assume, that the arithmetical mean
of the interval found from several stars minus the mean of
all A is equal to 24 hours. On the contrary if we find,
that this arithmetical mean is not equal to 24 hours but to
24 h 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 h a, that we can ascribe the deviations to pos
sible 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
right ascensions by ^ , in order to correct them
l ~ii
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 ob
servations are sufficiently numerous, we may assume the
hourly rate of the clock at the time t as being of the form
a~ib(t T), where a is the rate at the time T. Multiplying
this by dt and integrating it between the limits t and 24ff,
we find the rate between two successive culminations of a
star, whose time of culmination is , equal to:
24aH24&(12M T} = u.
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:
t / i /" i
a(t"t ) h b(t"t ) ^P  Fj ,
and we must correct every interval of time t" t accord
ing to this.
In case that already the differences of the right ascen
sions 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 #, 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:
= U a f AZ7 + 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
erroneous.
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 objectglass to the wirecross 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 in
tersects 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 instru
ment requires those three adjustments.
As will be shown in No. 1 of the last section, we can
always examine with the aid of a spiritlevel, whether the
axis of the pivots is horizontal and we may also correct any
error of this kind, since one of the Ypieces can be raised or
lowered by adjusting screws. The position of the line of
collimation with respect to the axis can be examined by re
versing 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
this 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 wirecross 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 de
termining 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 2x. 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 a?,
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 in
strument or to speak more correctly it will in both positions
be at the same distance from the middle wire of the teles
cope. If this should not be exactly the case, the operation
of reversing the instrument and moving the wires of the tele
scope must be repeated.
When these corrections have been made, the line of col
limation describes a vertical circle. At last in order to di
rect 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 in
stance the polestar, describe an entire circle above the hori
zon, 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 tele
scope 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 accu
rately in the plane of the meridian, the interval between the
two observations will be 12 h f &&gt; where j\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 1 2 h  /\ , if the plane of the telescope is East or West
of the meridian. Now as one of the Ypieces admits always
of a motion in the direction from North to South, w r e can
move this until the interval between two observations is ex
actly 12 h fA 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 suc
cessive 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 ap
parent right ascensions and we must apply to these the re
ductions 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 declina
tion, which however can always be taken from former cata
logues.
If the observed object has a visible disc, we can only
observe one limb and as such objects have also a proper
motion, we must compute the time of its semidiameter pass
ing 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 hav
ing been observed, where both limbs are usually taken, we
can simply take the arithmetical mean of both times of ob
servation.
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 alti
tudes on both sides of the meridian, the arithmetical mean of
both times is the clocktime of the culmination of the star.
It is evident, that it is not necessary to know the altitude
imatcly 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.
14*
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 clock
time 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 tp sin A
A *
mean of both times the correction ^ . Such a cor
cos cp sm A
rection is always required on account of refraction; for
although the mean refraction is the same for both observa
tions, 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 ^ = cos (f> sin A^ that it is best
to observe the zenith distances of the stars in the neigh
bourhood 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 (f 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 ad
vantage 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 in
strument, for instance the errors of division of the circle, or errors pro
duced by the action of the force of gravity upon different parts of the in
strument.
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 eyepiece 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 180 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 re
fraction computed according to formula A in No. 12 of the
third section and by subtracting p sin ss, 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) be
cause 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 decimations are found according to No. 23
of the first section, if the latitude of the place of obser
vation is known. But the latter can always easily be deter
mined by observing the zenith distances of any circumpolar
star in its upper and lower culmination, as the arithmet
ical mean of these zenith distances corrected for refraction
rA<? is equal to the co latitude of the place, where A<?
*) 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 circumpolar star in its upper and lower
culmination as well direct as reflected from an artificial ho
rizon. For then the arithmetical mean of the corrected alti
tudes minus A^ is equal to the latitude. But as the re
flected 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 cir
cumpolar 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 de
cimations of the stars, which are converted into mean decli
nations for the beginning of the year by applying the reduc
tion 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 tang = 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 differen
tial equation (which we get by differentiating the above equa
tion written in a logarithmic form)
2de 2dD
cotang A .<lA\ . = = 7777;
sm 2e sm 2Z>
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 ex
actly 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 for
mula 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:
ff^ B^D.
cos/)
Moreover if x is the distance of the sun from the sol
stitial point expressed in right ascension or equal to 90 A^
we have the following equation:
cos x tang e tang D,
and as x is a small quantity, we can develop & into a rap
idly converging series, for we find according to formula (18)
in No. 11 of the introduction:
= /)+ tang ^ x 2 . sin 2 D f ^ tang 4 x* sin 4 D H . . . (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 in
fluence 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 nutation, the value of g, which we deduce from them, will
be also the apparent obliquity of the ecliptic , affected with
nutation.
On the 19 th of June 1843 the declination of the sun was
observed at Koenigsberg and after being corrected for re
fraction and parallax was found equal to + 23 26 8". 57. At
the same time the right ascension of the sun was 5 h 48 m 50 s . 54.
Hence we have in this case x = O h ll m 9 s . 46 = 247 21".90
and as the latitude of the sun was equal to 40". 70, we have:
Z> = 42326 7". 87
I. term of the series = +1 29 . 23
II. term of the series = + . 04
= 23 27 37". 14.
216
This is the apparent obliquity of the ecliptic on the 19 th
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 ft = 272" 37 . 4, = 87 ,
(( = 350 17 and P = 280" 14 , we find A = + 0".05, hence
the mean obliquity on that day according to that one ob
servation is 23 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 for
mulae 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 f 17". 18, we find face = f 1 s . 25,
A = H0".39, therefore:
Corrected D = 23 26 7". 48
I. term h 1 29 . 57
II. term 4^0 . 04
Mean obliquity =23 27 37 77 7o~9^
In order to free the result from accidental errors of ob
servation, the decimation 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 , its true value
by , the errors of x and D by dx and dD, each observation
gives an equation of the following form:
= j V 5 tang j? sin 2 e dx + ^T ^~ dD,
sin Z U
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:
s = 23 27 37". 09 f 0.212 dx f 1.001 dD,
from which we see, that an error in aj, 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 , taking
= re/fi and e () e =n, we find from each observation
an equation of the following form:
sin 2 e ,
= n f as v tang x sin s dx dD.
sin2Z>
217
By applying to them the method of least squares, we
can find de as a function of dx and e?D, hence if we should
afterwards be obliged to alter the right ascensions or the de
clinations of the sun by the constant quantities dA = dx
and dD, we can easily compute the effect, which these al
terations 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 iadD+ bdx,
where the coefficient of (ID is always nearly equal to unity.
Now if there are no constant errors in D and #, 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 , 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 ob
servations 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 varia
tion. 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 t is equal
to e (} \ds and that the annual variation is A^f^ 5 we should
have the equation :
= h tie (A e + ar) (t * )
in case that the observed value were right. Hence if we take :
o
o A (t O e = n,
every determination of the mean obliquity of the ecliptic at
the time of a solstice gives an equation of the following form :
= n f ds f x (t t }
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:
23 27 54". 22 0".4G45 (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 = z 4 cp and if we denote by d <f the cor
rection which must be applied to the altitude of the pole,
by s the true value of the obliquity of the ecliptic, by e the
value deduced from observations, we have the following equa
tion from a summer solstice:
= e + Cfd<f>,
and for a winter solstice:
*, = e" rt rfy
hence we have:
where e s t 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 observ
ing the zenith distances. We can find in this way an ap
proximate value of the latitude by observing the zenith dis
tance of the sun on the days of the summer and winter sol
stice. For if z 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:
~ [ <>
9* = 2
8. If then the obliquity of the ecliptic be known, the
absolute right ascension of a star and hence from the dif
ferences 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 a Canis minoris (Procyon) or a Aquilae (Altair).
219
If then the transit of the star is observed at the time , 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 culmination
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 Z>,
and we have therefore:
. tang D
a = arc sin  h / T,
tang e
where strictly the time T must also be corrected for the lat
itude of the sun by adding J cos A sec d sin s p.
If now D and s be in error, we shall on this account
also obtain an erroneous value oft T, independently of er
rors 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. :
and consequently we obtain from each observation an equa
tion of the following form:
. tang D 2 tang A , 2 tang A
= arcsin H / T ds \   <ID. (A)
tangs sm2f sin 2 Z)
We easily see from this equation, that it is best to make
these observations in the neighbourhood of the equinox, be
cause then the coefficients of ds and dD arrive at their min
imum, that of ds being zero and that of dD being cotang s
or 2.3. Moreover we see that it is possible to combine sev
eral observations in such a way, that the effect of an error
in s as well as of any constant error in I) is eliminated. For
if in the equation sin A = ^? we take the ande A always
tang s J
acute, we have, when the right ascension of the sun is 180 4 ,
the following equation:
=180 arc sin ^ ^f. f_I" +. v "6"</ _"
tang sin 2 e sin 2 D
where i and T are again the times of transit of the star
220
and the sun, and if wo combine this equation with the former,
we find:
( 7 7 )] H i arc sin arc sin f 180
tang e tang e
 tang  1 <*.. ()
sm 2 e
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 dD and ds in
equation (I?) 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 cor
respond to another culmination. But if A be only nearly
equal to 180 A, the remaining errors dependent on dD
and ds 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 versa.
If we combine any two such observations, the effect of any
constant errors in D and 6 is eliminated and the result is
only affected with casual errors, which may have been com
mitted 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 obser
vations but as great a number as possible of observations
taken before and after the \ 7 ernal and autumnal equinox, in
which case it is not necessary to confine the observations to
the immediate neighbourhood of the equinox. Let be an
approximate value and = + d a the true value of the
right ascension and put:
. tang/)
a n arc sin  (t i ) = n.
tangs
221
Then each observation gives an equation of the following
form :
2 tang A 2 tang A
= nha4 da .  rfZ).
sin 2 e sin 2 D
If we treat then all those equations according to the
method of least squares, we can find the most probable val
ues of da, ds and dD or at least da as a function of de, and
dD, so that, if these should be found from other observations
and their values be substituted in the expression for da, we
get that correction da which in connection with these determi
nate values of de and dD makes the sum of the residual
errors a minimum. In case that the number of observations
is very great and the observations are well distributed about
the equinoxes, the coefficients of ds and dD in the final
equation for da will always be very small.
If the observations extend to a great distance from the
equinoxes and the observed declinations lie between the lim
its =p Z>, it may not be accurate to take d D for the entire
range 2D as constant, for instance, in case that the circle
readings are affected with errors dependent on the zenith dis
tance, or if the constant of refraction should need a correc
tion. Although even in this case these errors have no effect
upon the result, if the observations are distributed symmet
rically around the equinoxes, yet the resulting value of dD
or the term dependent on dD in the final expression of da
would have no meaning. In this case it is necessary to di
vide 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 = (p z p,
if the object is south of the zenith, we may take instead of
dD in the above equation dcf> dk tang z fifty, where
dk denotes the correction of the constant of refraction and
fifty the correction which must be applied to the circle
readings. 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 : > = + 1 15 27" . 24
222
and the interval between the transit of the sun and the star
a Canis minoris, corrected for the rate of the clock:
t r=?h 19 " 29*. 86.
As the latitude of the sun was 40". 21, the correction
of the declination is 0".19, whilst that of the time is noth
ing. 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 d = + 5 37 :
a 1 ft = s . 42.
This being subtracted from the time , we find:
t T=l^ 19 " 29 s . 44
Z) = + 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:
^ = 27713 .8, O = l 14 , (1 = 283" 56 , P = 280 14
we find by the formula in No. 5 of the second section
A* = + 1".72, hence:
= 23 27 36". 77.
and with this we find:
A = arc sin ^^ = 2 " 53 57" . 44 = 0" 1 1 35 s . 83.
tang e
Hence the right ascension referred to the apparent equi
nox is:
a = l\> 31 5 S . 27
and adding the nutation in right ascension 4 1 s . 10 and sub
tracting the precession and proper motion from the begin
ning of the year to March 24 equal to f0 s .71 (since the
annual variation is }3 s .146) and computing the coefficients
of dD and de, we find according to this observation the
mean right ascension of a Canis minoris for 1843.0 ,
a = 7 1 31" 3 s .46 h 0. 1539 dD 0. 0092 de,
where dD and de are expressed in seconds of arc.
223
On the 20 th of September of the same year Bessel ob
served :
Z) = +l 16 29". 22
/ T 4 h 17" 5. 82.
As on that day the latitude of the sun was B = 0". 56,
and n = 267 41 . 9, 0=178 39 , (1= 135 41 , P=28014 ,
we find the corrections dependent on B equal to 0".51
and J0 S .01; furthermore the aberration is = 0\l 56, the
nutation of the obliquity is j0".27, hence, as the mean
obliquity was on that day 23" 27 34". 82, we find:
Z> = tl 16 29". 73
t r = 4 h 17 m 5.27
e = 23 27 35". 09.
From this we get A = 2 56 22". 36 = 0" 11 45 s . 49,
hence the right ascension of the sun equal to H h 48 in 14 s . 51,
therefore a = 7 h 31 ni 9 s . 24 and as the nutation was (1 s . 11,
the precession and proper motion equal to f2 s .27, we find
according to this observation the mean right ascension for
1843.0
a = 7 31 5s . 86 0. 1539 dD h .0094 de.
Taking the arithmetical mean of both determinations we
find:
= 7h 31 4 S .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 Z>, T and t the reductions to the apparent
place, neglecting for the sun the terms dependent on aber
ration. 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 differen
ces of right ascension have been observed, are known also
and can be collected in a catalogue together with the decli
*) According to Bessel s Tabulae Regiomontanae is a = 7 h 31 1U 4 8 . 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 ac
count of the errors committed in the determination of the
absolute right ascension. This can be determined by com
paring a large number of stars, contained in the several ca
talogues, after reducing them to the same epoch. Similar
differences may occur in the decimations and can be deter
mined in the same way. But since these errors may be va
riable, as was stated before, one must form zones of a cer
tain 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 appa
rent 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 culmina
tion 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 dif
ference of right ascension and declination. In case that the
declination of the unknown object differs little from the stan
dard 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 ascen
sion and declination is to be determined, should be very near
the star, one can use for the observation instead of a meri
dian 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 ob
serve 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 micrometrically 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 pos
sible 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, when
ever 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 oc
curring 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 for
mulae :
15
226
where Z denotes the circlereading corresponding to the middle
of the zone.
If we denote by u and ri 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
.<? the refraction and the errors of division and flexure, by (>
and s their variations for 100 minutes, at last by A and
&d the 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:
But according to the formulae in No. 3 we have:
A = ~ h p sin ( G + a) tang D + ^ sin ( // + ) sec D,
L
(sin C + ) * $ln ,a,, g D H ^
lo cosZ> 2 la cos /> J 100
& = g cos (6r h a) h /< cos (ff\ ) sin Z) H z cos Z>
h 7i cos (H{ a) cos I> 100 i sin Z) 100 I 
hence we find:
~^ ~s\\\(G{a}tgD ^si
1 1 i
 1QO , + * sin(ff , tang 1* ,
la cos D~ la cos D
d= P4 90 =F (> H * .9 cos (G h a) h cos (f/f ) sin D ? cos Z),
d = =F (/ 4 .s r [A cos (//h ) cos Z> 100 j 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 zonestars and if the
O
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 A 1 ,
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.
ITT. 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. 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 circumpolar star
be observed at its upper and lower culmination and corrected
for refraction as well as for the small variations of the aber
ration, 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 f the observed zenith distances at
the upper and lower culmination, by r and o the refraction,
we have for any north latitude the equations :
S (f = z =t= r
180 8 y> = + (>,
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 :
15*
228
If another star be observed at both culminations and the
zenith distances and z be found, we should be able, to
find from the following two equations :
90. ,_+! + =
and
the values of cp and of that constant which in o , (/, r and
r occurs as factor. But the values thus found would be
only approximate on account of the errors of observation ;
besides equation (/) in No. 9 of the third section shows, that
the refraction is not strictly proportional to the constant r<
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 /", which de
pends on the decrease of temperature with the elevation above
the surface of the earth, which however shall here be ne
glected, since its influence, which is always small, 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 astronomical observa
tions. 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 # , we have according to the
formula (12) of the third section:
R = R[B . T] A [l 4f(r 50)]~ A ,
where A 1 H q and /I = 1 ip. From this we get:
dR A(r50)
dR = . d a    7 R de ,
da 1 f K (T 50)
or taking:
229
a H da a (1
, s { de = e (I + i)
r>7 f ^ *J\rj j..;
7** J7<^56)*
But according to the formula (/) in No. 9 of the third
section we have:
(I a) sins 2
The second term of the second member of this equation
becomes significant only for zenith distances greater than 80
and if we put:
80
y
246
86
81
205
87
82
168
88
83
135
89
84
106
89 30
85
82
da \ y
we can take the values of y from the following table :
y
60.5 ^
43.2
29.5
19.0
14.8
We have therefore:
If we assume therefore, that the values of the refraction,
which have been used for computing formula (a), are erro
neous and that the corrections are do and dr, we get:
f(l
if we denote by m and u the values of    for the
1 h e (T 50)
upper and lower culmination. If we also assume an approx
imate value r/ for (f , the true value being r/> = r/ () f d ff
and take:
we obtain, combining the result of the upper and lower cul
mination of each star, an equation of the following form:
+ dy
(6).
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 y, 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
w and w are the probable errors of an observation at the
upper and lower culmination, all equations of the same star
must be divided by Vw 1 + w ~ in order to give to the equations
o*f the several stars their true weight. In case that the prob
able errors should be found very different when the equa
tions 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 for the
expansion of air. For such stars we have according to the
notation which we used before, taking the zenith distances
positive :
?>o <?o + d (? <?) = ~ H r + r (lt ) k mri,
or taking:
>,. = ~ + r H S <f> ,
= n 4 d (8 y) h r(l + ) k mri. (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 ( J </) and
/e, so that each star gives finally an equation of the form:
= N Mi. (d)
But a similar equation can be deduced from every cir
cumpolar star observed at the times of both culminations, if
the equations (6) 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, Astrono
mische Beobachtungen, Siebente Abtheihmg, pag. X) and the
value found by him is the one which was given before na
mely 0.0020243 for one degree Fahrenheit,
If we substitute the most probable value of i in the
equations (6) or rather in the equations of the minimum, de
duced for each star, we find from the combination of these
equations corresponding to the several stars, the most prob
able values of dy and A**).
If it should be desirable, to take the correction of the
quantity f into account, it would be necessary to add to dR
the term   df or, taking f\d f=f(I j/i), the term
d R R
f h = h, where the values of x can be taken from the
df x
following table:
z
X
z
x
85
338
88
59.3
86
196
S J
29.8
87
111
89 30
20.6.
B. Determination of the constants of aberration and nutation and of the
annual parallaxes of stars.
11. The aberration, nutation and annual parallax are
the periodical terms contained in the expression for the ap
parent places of the stars, hence their constants must be de
termined 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
= n H d(f f 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 k 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 polestar are
very much changed by aberration and nutation on account
of the large factors sec d and tang t) , their observations afford
the best means for determining these constants; for the same
reason the parallax of the polestar can be determined in this
way with great advantage. Putting:
cos cos a = a sin A
sin a = a cos 4,
the formulae for aberration and parallax in right ascension
in No. 16 and 18 of the third section, can be thus written:
a a = t ka sin (0 + A) sec S + n a cos (0 t A) sec h <p (fc 2 ),
where k and n are the constant of aberration and the parallax
and </ (/e 2 ) denotes the terms of the second order. If scvcnil
observations are taken at the times when sin (0 + A) = =t= 1
and hence the maximum of aberration occurs, an approxi
mate value of k can be found by comparing the right ascen
sions 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 a and
the assumed value of the constant k be erroneous by /\a and
A&, the true values being fA and &HA&. If then
denotes that value of the apparent right ascension, which
has been computed from c< 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 aber
ration and nutation have also been added, since the effect
of a change of k upon them is very small, and if further a
denotes the observed apparent right ascension, we have:
a = f AH A&sin (0 + A) sec S + n a cos (0 + A) sec d,
hence, taking:
233
every observation of the right ascension of Polaris leads to
an equation of the following form:
= f f A k . a sin (0 f A) sec 4 TT cos (0 h 4) sec tf,
and from all these equations the most probable values of A?
A/ and TT can be determined according to the method of
least squares.
Should these observations embrace a long period of years,
the constant of nutation, that is, the coefficient of cos <H in
the expression for the nutation of the obliquity can be deter
mined at the same time. If we denote by i\v the correction
of this coefficient, we must add to the above equation the
term   A r, where the expression for , has been given in
No. 6 of the second section. The complete equation for de
termining the aberration, parallax and nutation from the ob
servation of an apparent right ascension is therefore:
= n + Af A& sin (0H4) sec d + na cos (0K4) sec { ( "" 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 cor
responding weight be given to the different equations. In
this case also the correction A** may not be the same for
the observations of the several observatories, as the observed
right ascensions may have a constant difference. Hence this
difference must be determined and be applied to the obser
vations or the unknown quantities A, A etc. must be elim
inated separately by the observations of each observatory.
In this way von Lindenau determined the following va
lues of the constants from right ascensions of Polaris ob
served by Bradley, Maskelyne, Pond, Bessel and himself in
the course of 60 years :
k = 20". 448C v = 8". 97707 TT = 0". 1444,
Peters found later from observations made by Struve
andPreuss at Dorpat during the years 1822 to 1838 the fol
lowing values:
k == 20". 4255 v = 9". 236 1 TT = 0". 1724.
For the determination of these constants by declina
tions 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 a sin 8 cos e cos S sin e. = l> sin B
cos sin S = b cos B,
the aberration in declination is equal to &6 sin (O  #), the
parallax equal to 71 b cos (Oh#). Then denoting by f) that
value of the apparent declination which has been computed
from the mean declination with the constants of aberration
and nutation k and v (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 # d = n, every observation of
a declination leads to an equation of the following form:
7 J5 1
= n + A S f &kb sin (0 + 7?) \ n b cos (Q H B} H A",
<lr
and in case that the observations embrace a sufficiently long
period, the most probable values of /^o, A#, 71 and &v can
be determined according to the method of least squares *).
It was by such observations that Bradley discovered the aber
ration. He observed at Kew since the year 1725 principally
the star ;> 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 com
bined 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 Dra
conis, being near the north pole of the ecliptic, was espe
cially suitable for determining the parallax and thus also the
*) If the stars have also proper motions, the terms p(tt ) and y(t O
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, d = 90 ,
hence 6=1 and 5=90 and the maximum and minimum
of the aberration and parallax in declination are equal to == k
and =t= 7i.
By similar observations he discovered also the nutation.
The observations embrace the time from the 19 th of August
1727 to the 3 d of September 1747, hence an entire period of
the nutation. Busch found from 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 r = 9". 21 04 n = 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 deter
minations 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
o
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 instru
ment is placed exactly in the plane of the prime vertical arid
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 de
note this by , we get from the right angled triangle between
the zenith, the pole and the star:
tang = tang y cos *,
hence we see that the declinations of the stars can be de
termined by such observations. Differentiating the formula
in a logarithmic form, we find:
dd
.
sin 2
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 across
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". 445 J, a value which un
doubtedly 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 semidiameter of the earth s orbit, was
first determined by Olav Koemer from the eclipses of the
satellites of Jupiter. For he found in the year 1675, that
those eclipses which took place about opposition were ob
served 8 13 s 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, Rorner 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 16 111 26 s . If therefore T be the time of the begin
ning 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
4 A A
where K is the number of seconds, in which the light tra
verses 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 2 is
the time of the eclipse thus corrected, T the observed time,
every eclipse gives an equation of the form:
*) At the opposition the earth stands between Jupiter and the sun, whilst
at conjunction the sun it between Jupiter and the earth.
237
and from a large number of such equations the most prob
able value of dK can be determined. However the observa
tions 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 de
pend upon the quality of the telescope, it is best, to com
bine only such observations which have been made with
the same 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 n equal to
and <y o cV and denoting a () ( ) and <) r)
*) In this case, when the stars are near each other, it is preferable, not
to compute the mean place of each star, but to free only the difference of
the apparent places from refraction, aberration, precession and nutation. The
formulae necessary for this purpose will be given in VIII and IX of the
seventh section.
238
(<$ d) by n and w and the errors of the adopted place by
A and &:
Htfa cos lQ 4 4) sec
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 8 be the
true right ascension and declination of one star, and <5
their values not freed from parallax, a" and 8" the right as
cension and declination of the comparison star, we find the
changes of the differences of the right ascensions and decli
nations produced by parallax as follows:
d (" ) = a = TT R [cos Q sin a sin cos E cos a] sec
d (" 8) S 8 = TT R [cos e sin a sin sin e cos S] sin
h 7t R sin S 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 (" )
AcosP=<T S
hence:
d A = sin P cos 8d(a" a) + cos P </ (S" 5)
A rfP = cos Pcosdd (a" a^ smPd (S" S).
If we substitute here the expressions given before and
take :
? cos M= sin a sin P f sin S cos a cos P,
w* sin M = [ cos sin P f sin $ sin cos P] cos f cos S cos P sin e,
m cos j\I = [sin a cos P sin S cos a sin P] ,
A
w sin 3/ = [ (cos a cos Pf sin S sin a sin P) cos e + cos # sin P sin f],
A
we easily find:
d A = n R m cos (0 M)
dP = 7tR m cos (0 J/ ).
Therefore if </A denotes the correction of the adopted
distance at the time f , d(/ 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 :
= v + </Ao H (t <o) d? +7tRm cos (0 M) .
239
and from the angles of position equations of the form:
= f dP 4 (t O dq iTiR m cos (0 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 precession and of the proper motions
of the .stars.
13. We find the change of the right ascension and de
clination of a star by the precession during the interval t ,
if we compute the annual variations:
da dl, da dl. ~
= in f n tg 1 o sin a = cos c    f sm E tg o sin a
d dl
T = n cos a = sm e cos
for the time and then multiply them by t t. Now
since the numerical value of a is known from the theory of
the secular perturbations of the planets, we may determine
the lunisolar precession ( 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 from 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 at
tribute 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 compared catalogues of stars may be, but it will be
most important, that the catalogues are very correct and con
tain a large number of stars in common and that the inter
val is long enough so as to make any uncertainty arising
from errors of observation sufficiently small. If then m () and
M O are the two values of m and n employed in comparing
the two catalogues, if further , c) and a and <) are the mean
places of a star for the times t and t\ given in the two cat
alogues, and A and /\d the constant differences of the cat
alogues for ct and r) and if we take:
a + O 4 w () tg <? sin ) (t /) a = v (t
and
every star gives two equations of the form:
f dm + dn tg sin ,
t t
and
Q = v ,,
t t
Therefore if we consider the proper motions embraced
in v and v 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 any
thing 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 m and n, from the declina
tions a determination of n ; but it is evident, that an error of
the absolute right ascensions, which is constant for every
. , T ,i 7 i dm dl, da
catalogue, remains united with dm and as ^ =cos 
241
there remains also in it any error of the value of  arising
from incorrect values of the masses of the planets. But the
determination of dn dl ( sin from the right ascensions is
independent of any such constant error, and besides the con
stant difference of the declination may be determined. But
since the supposition, that the latter is constant for all decli
nations , is not allowable , it is better to divide the stars in
zones of several degrees for instance of 10 of declination
and to solve the equations for the stars of each zone sep
arately, and hence to determine the mean difference /\J for
each zone. In this way Bessel in his work Fundamenta Astro
nomiae 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 in
terval 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 sup
position, 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
coordinates 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 A Herculis.
Hence v 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 in
vestigations on this subject. The proper motions of the fixed
stars are therefore the result of two motions, first of the mo
tion 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 de
termined by the place of the star relatively to that towards
which the solar system is moving, and can be easily calcu
lated, 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 pecu
liar 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 parsing 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 , 8 and , d , and finally
the ratio of the distances of the star from the sun at the
same epochs by Q, we have the following equations:
Q cos 8 cos a = cos S cos ft a cos A cos D
() cos S 1 sin a = cos S sin a sin A cos D
(> sin S = sin S a sin Z),
from which we easily deduce:
cos S = cos S a cos D cos ( ^4),
therefore :
cos S (a a) = a cos D sin ( ^1)
$ 3= a [cos $sin /> sin $cos /) cos ( yl)].
But we have also in the spherical triangle between the
pole of the equator, the star and the point, whose right ascen
sion and declination are A and P, 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)
sin A cos P = sin Z> cos $ cos /> sin S cos ( A).
Now if we denote the angle, which the direction of the
proper motion of the star makes with the declination circle,
by /?, we have:
cos S (a a)
hence we see, that p = 1 80 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 mov
ing from the latter point.
From the third of the differential formulae (11) in No. 9
of the introduction, we have:
sin A
cos/
sin A
hence :
H . [sin S cos D cos S sin D cos (a A)} dA.
sin A

sin A
 . 2 [sin 8 cos D cos S sin D cos (a A)] dA.
cosD
sin A 5
Therefore if p be the observed angle, which the direction
of the proper motion makes with the declination circle, reck
16*
244
oned from the north part of it through east from to 360
so that:
cos 8 ( a)
and if further p be the value of. \ 80 P computed accord
ing to the formulae (#) with the approximate values A and
D, we have for each star an equation of the form:
( A)
 [sin cosD cos sin D cos (a A)] dA,
or:
cos 8 sin (a A)
.
dD
sin A
[sin <?cos D cos 8 sin D cos ( A}} dA,
sin A
and from a large number of such equations the most prob
able 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 ^Funda
menta Astronomiae" had already derived the proper motions
of a large number of stars by comparing Bradley s observa
tions 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 deter
mined 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 pro
per motion amounted to more than 0" . 1 . These were divi
ded into three classes according to the magnitude of the pro
per motions and the corrections dA and dD determined sep
arately from each class. From those three results , which
well agreed with each other, he finally deduced the follow
ing values of A and D, referred to the equator and the equi
nox of 1800:
4 = 259 51 . 8 and D = + 32 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:
4 = 252 24 . 4 and D 4 14 26 . 1.
From the mean of both determinations, taking into ac
count their probable errors, Argelander found:
.4 = 257 59 . 7 and D = + 28 49 . 7.
Similar investigations were made by O. v. Struve and
more recently by Galloway. Struve comparing 400 stars
which had been observed at Dorpat with Bradley s catalogue,
found :
4 = 261 23 and D = f37 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 D = 4 34 23 .
Another extensive investigation was made by Madler,
who found from a very large number of stars:
4 = 261 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 attain
able 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 4)
sin D cos 8 cos D sin $ cos (a 4)
with a mean value of A and />, 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 de
clination, 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 mo
tion 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:
= v h dm H dn tg 8 sin h ~  sin ( A)
l\ COS 0Q
and O^^ fdn,, cos h # sin ( Z) )
where S = g cos Cr ,
sin $ cos ( A) = g sin G,
we could find, if A were known, from these equations A;,
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 dm and dn t) free from this
parallactic proper motion of the stars. Now since the dis
tances of the stars are unknown, O. v. Struve substituted
for A hypothetical values of the mean distances of the dif
ferent classes of stars, which had been deduced by W. v.
Struve in his work, Etudes de FAstronomie stellaire from the
number of stars in the several classes *). Struve then com
pared 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 as
censions 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 fl".16
from the right ascensions and 40". 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 decli
nations. But although these values of the constant of pre
cession 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 determina
tion 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
and the correction of the constant of precession. The
values of thus found can be considered as mean values
a
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 *). Even the
corrections of A and D might be found in this way, since the
equations in this case would be, taking = a :
= ^4 dm n + dn tang d sin ~ cos ( A) ad A
cos o
f [cos D  sin DdD]
= v idn cos g cos (G D) adD + cos D sin$ sin ( A) ad A
hags m(GD)
from which the most probable values of a, ad A, adD,
dm (t and dn () can be determined for each class. In case,
that Struve s ratio of the distances be adopted, the un
known quantity a after multiplying the factor by would
*) 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".06S9S5 =t= 0.010964
75 4. 0".069715=t= 0.006584
71 4.5. 0".046Sll=t= 0.006925
284 5. 0".029043 0.002446.
Stars, whose annual proper motion exceeds 0".3 of arc, 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 mo
tions of the stars are proportional to the time and take place
on a fixed great circle. But the proper motions in right as
cension 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 coordinates re
ferred to the equinox at the time t by means of the co
ordinates referred to another equinox at the time , are ac
cording to No. 3 of the second section:
cos sin ( j a 2 ) = cos S sin (a f a + z)
cos S cos ( f a z ) = cos S cos (a + a + z) cos sin S sin
sin 8 = cos S cos ( f a f z) sin + sin S cos 0,
where a denotes the precession produced by the planets dur
ing the time t , and 3, z and are auxiliary quantities
obtained by means of the formulae (yl) 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 90 # , 90 8 and S and the angles of which
are a f a + z, 1 80 a a t z and c :
A S = cos c & sin sin ( 4 a z) A
cos $ A = sin c &d + cos S cos c A<*
or if sin c and cos c be expressed in terms of the other parts
of the triangle:
fa = A [cos h sin tang S cos ( ha 2 )] +  sin S1D ^~t a ~ z> }
cos o cos o
(a)
A<9 = A sin sin ( + a z ) h . cos S [cos + sin tang S cos ( + a )]
cos o
and in the same manner:
A = A [cos sin tang 8 cos (a H a 4 z)} s> sin
cos a cos o
(6)
A0 = A sin (9 sin (a f a z) H ^.cosS [cos si
coso
249
Example. The mean right ascension and declination of
Polaris for the beginning of the year 1755 is:
a = 10 55 44". 955 8 = 4 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:
=16 12 56". 9 17 S = 488 30 34". 680.
But in Bessel s Tabulae Regiomontanae this place is:
= 16 15 19". 530 8 = 488 30 34". 898.
The difference between these two values of and S
arises from the proper motion of Polaris, which thus amounts
to { 2 22". 613 in right ascension and to 40". 218 in de
clination in the interval from 1755 to 1850. The annual
proper motion of Polaris referred to the equator of 1850 is
therefore :
A = 41". 501 189 A <? = 40". 002295.
If we wish to find from this, for example, the proper mo
tion of Polaris referred to the equator of 1755, it must be
computed by means of the formulae (6). But we have:
= 31 45". 600
a\a + z=ll 32 9". 530
and with this we obtain :
A = 4 1". 10836 A<? = hO". 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 cen
tral body and has deduced the following formula, which ex
presses the correction to be applied to the right ascension
of this star:
q = Os . 127 4 . 00050 (t 1800) 4 0* . 171 sin ( M 4 77 44 ) ,
250
where the angle u is found by means of the equation:
M 7 . 1865 (* 1791 . 431) = u . 7994 sin u
and where 7. 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:
,? = f0".56hO".0202(* 1 800) r 1". 47 sin w 40". 51 cos M,
where u is the same as in the formula above *).
*) Of great interest in regard to this matter is the discovery, made re
cently 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 prece
ding section, how the position of the fixed great circles of
the celestial sphere can be determined by means of a merid
ian 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 in
terval between the observations must be equal to 12 h of sidereal
time f A 9 where A 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 colatitude,
since this is equal to the arithmetical mean of the two zenith
distances corrected for refraction h A^, where A^ is the varia
tion 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 obser
vation is made at another place at the same instant, the dif
ference of both times is equal to the difference of the hour
angles of the vernal equinox at both places or to their dif
ference 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 a firmly
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 coordinates of stars, whose
places are known, with respect to the horizon, and thus va
rious methods have been invented, by which travellers or
seamen can make these determinations with more or less ad
vantage 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 ascen
sion and declination and the sidereal time and the latitude :
sin h = sin <p sin 8 + cos <f cos S cos (0 a)
cos a> tang S
cotangvl = ~ t sin d cote (0 a),
sm (0 )
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 observa
tions 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 alti
tude 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 180 round its vertical axis. For
if and f are the circle readings in those two positions,
corresponding to the times & and /, and if r^ and   a are
253
the differential coefficients of the zenith distance (I, 25) cor
responding to the time = , assuming that in the first
position the divisions increase in the direction of zenith dis
tance and denoting the zenith point by Z, then the circle
readings reduced to the arithmetical mean of both times are:
* + Z = $ +  (0  0)  1 \ (0  &,) >
.
Hence the zenith distance z (} corresponding to the arith
metical mean of the times is:
Finally in case that the object is observed direct arid
reflected from an artificial horizon, we have, since the first
member of the second equation is then 180" a rZ:
90* = J (5 )HI j^ z a  9 6>) 2 *).
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 sub
tracted from or added to all circle readings, if the divisions
G
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 azi
muth 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 cor
responds 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 se
venth 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 ne
cessarily 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 dis
tance of the circumpolar stars from the meridian. According
to No. 27 of the first section we have for the hour angle of
the star at that time:
tang (f s m(d <p)
cos t  J or tang ^ t 2 = .^r ^ >
tang o sm (o + cp)
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 in
strument, whose line of collimatiou 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
wirecross, 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
meridian, ^~ is the reading of the circle corresponding to
the zero of the azimuth. It is best to use the polestar 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 fol
lows, 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
observe a star at the wire cross of an azimuth instrument,
read the circle and then wait, until the star after the cul
mination is seen again at the wirecross, 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 cor
rection must be applied to the arithmetical mean of the two
readings. For, differentiating the equation:
255
sin 8 = sin 90 sin h cos cp cos h cos A,
taking only A and 8 as variable, we have:
_ cos dS dS
cos (p cos h sin 4 cos 9? sin
Therefore if A^ denotes the change of the declination
in the time between the two observations, we must subtract
from the arithmetical mean of the two readings:
2 cos (p cos h sin A 2 cos <f> 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 12 h of sidereal time fA, where 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 h sin A = cos 8 sin t
cos h cos A = cos rp sin S sin <p cos 8 cos ,
and for the second observation below the pole:
cos h sin A = cos S sin t
cos h cos A = cos rp sin S sin <p cos 8 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) i_ _JLl_> L .
sin <p
In case that t t is nearly equal to 12 hours of sidereal
time, A as well as 90 (* are small angles, and since
then I (7i +/& ) and \ (h h ) are nearly equal to (p and 90 d,
we get:
cos cp tang 8
2. It is 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 azi
muth computed from the two equations:
cos h sin A = cos sin t
7 V> I ^ (fl)
cos h cos A = cos cp sin o + sin (p cos o cos t
In case that a set of such observations has been made,
it is not necessary to compute the azimuth for each obser
vation by means of these formulae, but we can arrive at the
same result by a shorter method. Let 0, (~j\ 0" etc., be the
several times of observation, whose number is w, let be
the arithmetical mean of all times and A l} the azimuth cor
responding to the time , then we have:
A = A* + t (&ej + $ d (6> 6> ) 2 ,
etc.
and since S @ h (") 6> f etc. = 0, we find:
... , d\A [(0 0 ) 2 t(0  ) ? K.."
rf? L~ n J
_ _ 2 2 sinj (0 0J 2
n di* n
where 2 2 sin \{S @,,) 7 denotes the sum of all the quan
tities 2 sin (6> & ) 2 . These have been introduced instead
of ^ (# # o )2 on account of the small difference and because
in all collections of astronomical tables , for instance in
5,Wariistorff s Hulfstafeln", convenient tables are given, from
which we can take the quantity 2 sin 2 \ t expressed in sec
onds of arc, the argument being t expressed in time. Now
we have accordin to No. 25 of the first section:
d l A cos cp sin AQ r . ,
  r [cos A sin o f a cos y> cos A a \.
dr cos A
Therefore if we add to the arithmetical mean of all read
ings of the circle the correction:
cos (p sin A , v . o , ^2 sin(6> 6> ) 2
[cos h sin + 2 cos (f cos ,d t ]  
cos
we find the value 4 19 which we must compare with the azi
muth computed by means of the formulae (a) for t=& () a.
257
Differentiating the equation (a) or using the differential
formulae given in No. 8 of the first section, we find:
cos cos p sin p .
dA =  r  dt tang A sin yJ d<p\ . dS,
cos h cos h
hence we see, that it is especially advisable to observe the
polestar near the time of its greatest distance from the me
ridian, because we have then p = 90 and A is nearly 180,
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 ob
servation is known, w r e 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 A sin H f cos h cos Hcos (a A}
where H and A are the altitude and the azimuth of the object
and A is the observed distance**). We find therefore a A
from the equation
cos A sin h sin H
cos (a A) , (A)
cos h cos H
hence also the azimuth of the object A^ since a is known.
The equation (^4) 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 from it. Likewise is H the
apparent altitude of the object, which is found by observation.
17
and :
hence :
258
, N cos (//h /<) f cos A
1 + cos fa A)== TT ~
cos h cos //
A . cos(H A) cos A
1 cos fa A) = = ^
cos h cos H
. / A ^ sin 4 (A  ^4 A) sin j (A 4 H 7Q
tang 4 (a jl) = TTT ; =7 7i r7zi/~T~r A\
cos 4 (A h //H A] cos (// h A A)
or taking:
sin OS JJ) sin OS 70 , .
tang 4 (a Ay = T^" (*)
cos A cos (S A)
If the terrestrial object is in the horizon, therefore #=0,
we have simply:
tang ,V ( AY = tang ^ (A 4 /O tang 4 (A /<)
Differentiating the formula for cos A? taking a A and
& as variable, we get:
cos A cos 77 sin (17 ^4)
and from I. No. 8:
cos S cos p .
da = at.
cos A
Hence 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 7), the same for the star by t and J,
we have in the spherical triangle formed by the pole, the star
and the terrestrial object:
cbs A = sin d sin L> r cos cos D cos (t J 1 ).
Then, if A is the interval of time between both observa
tions, which in case of the sun being observed must be ex
pressed in apparent time, we have for the second distance
A the equation:
cos A = sin sin D h cos S cos D cos (t T+ /).
From these equations w r e 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 /), 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:
sin h sin a? sin 8
cos t =
cos <p cos o
In order to render this formula convenient for logarith
mic computation, we proceed in the same way as in the pre
ceding No. and we find, introducing the zenith distance in
stead of the altitude:
p. i ,2
__ sin ?( z <P
cos \ (z H (p H 8) cos 4^ (gp H 8 z)
or:
~ sn ~
cos <S . cos (*S z}
where S = \ , (z + <p f $)
The sign of is not determined by this formula, but t
must be taken positive or negative, accordingly as the altitude
is taken on the west or on the east side of the meridian.
If the right ascension of the star is , we find the side
real time of the observation from the equation:
0=*ho,
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 = 33" 42 18". 7
at the clocktime 20 1 16 m 20 s .
The altitude must first be freed from refraction and pa
rallax; 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 semidia
meter of the sun 16 8". 7, we find for the altitude of the
centre of the sun:
h = 33 57 7". 9.
Now the latitude of Abutidsch is 27 5 0" and the de
clination of the sun was on that day:
 13 38 11". 1
hence we have:
,S y = f7 39 50". 5, <? = h48" 23 1".
and the computation is made as follows:
s m(S y>) 9.1250385 cos S 9.9146991
s m(S 8) 9.8736752 cos (S z) 9.9G92707
8.9987137
9.8839698
tang 4 * 2 9.1147439 tang 4* 9.5573719
t = 19 50 37". 98
* = 39 41 15 .96
t = 2s 38 " 45 s . 06.
Hence the apparent time of the observation is 21 h 21 "
14 s . 9, and since the equation of time is 16 m 8 s . 7, the mean
time is 21 h 5 m 6 s . 2. The chronometer was therefore 48 in 46 s . 2
too fast, or f 48 " 46 s . 2 must be added to the time of the
chronometer in order to get mean time.
Since the declination and the equation of time are va
riable, we ought to know already the true time, in order to
interpolate, for computing , 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 approx
imate 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 re
peat the computation.
The correction which must be applied to the clocktime,
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 dif
ferent 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 times
261
is equal to 24 h / and /\ u is the rate of the clock in this
time, wo find the rate for 24 hours, considering it to be uni
form, by means of the formula:
24 A u AM
24 7 ~~ ~^T_
24
Differentiating the original equation:
sin h = sin <f sin 8 H cos <p cos $ cos ,
we find according to I. No. 8:
dh = cos Adcp cos 8 sin p dt<
or since:
cos sin 7> = cos <f> sin A
we get:
clh  A
cos (p sm ^4 cos y tang A
The value of the coefficients of dh and d([> is the less,
the nearer A is =t= 90. 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 max
imum, and hence the coefficient of dh is a minimum, 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 dh can also be written
cos o sin/?
it is evident, that one must avoid taking stars of great de
clination 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
s m^ = 8 * S n( : ^ = 48" 25 . 8
cos h
and then
dt = h 1.5013 dh h 0.9966 cly
or dl expressed in seconds of time:
dt i 0.1001 dh t 0.0664 dtp.
Therefore if the error of the altitude be one second of
arc, the error of t would be s . 10, whilst an error of the
latitude equal to 1" produces an error of the time equal to
s . 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 <^, and hence, the less the latitude is. Near
the pole, where cos cp 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
clocktimes, or to apply a correction to the hour angle com
puted from the arithmetical mean of all zenith distances.
Let r, r , r", etc. be the clocktimes, at which the zenith
distances, whose number be n, are taken ; let T be the arith
metical mean of all, and Z the zenith distance belonging to
the time 7 1 , then we have :
etc.,
where t is the hour angle corresponding to the time 7 T , or
since r Tt r Tfr" Tj.. .=0:
._... _ ^ z _ ,.
n (it* n
If we substitute here the expression for 2 found in No. 25
of the first section, we finally get :
z h z h 2" 4 . . . cos^cosw ^2sin^(r TV
/j =: ^ cos^l cos p .
??. sin Z n
With this corrected zenith distance we ought to com
pute the hour angle and from this the true time, which com
pared with T gives the error of the clock. But if we com
263
pute the hour angle with the uncorrected arithmetical mean
of the zenith distances, we must apply to it the correction:
dt cos cos (p 2 2 sin \ (r 7 1 ) 2
 ^ cos A cos />
dz sin Z n
or if we substitute for ^ its value according to No. 25 of
dz
the first section, we find this correction expressed in time:
cos p cos A JfJ^sin ; [ (r T 7 ) 2 , .
15 sin t n
where A and p are found by means of the formulae:
sin t 2
sin A = . cos o
smZ
sin t
and sin p =  cos if.
smZ
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 ().
If the hour angles are not reckoned in the usual way,
but on both sides of the meridian from 0" to 180", the cor
rection is always to be applied to the absolute value of ,
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 <K , v /sin OP \
sin OP I 1 cos z sm o I cos ~ )
V sin y> \sm o /
cos p = s~ ==: . ja ?
sm z cos o sm z cos o
/ sin $\ , ^ /cos z sin (p \
sin (f I cos z } sin o I ; ^
\ sm (p/ \ sm o /
cos A =   = 
sm z cos (p sin z cos (p
Therefore, if <) <? y, cos p is always positive,
n . . ... .,> sin
and cos A is positive, if cos z > . ,
sm<p
sin o
i
negative, if cos j
siny
and if <) > y, cos A is always negative,
sin (p
sin 8
and cos p is negative, if cos z
... . r, ^ sm (p
positive, it cos z < i,
sin o
Therefore if we take the fraction
sin o .r,
sin
and sin ^, if
sm d 7
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 correc
tion is always negative*).
Dr. Westphal took on the 29 f!i 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 { (rT) 2
20 h 16 m 20 s 56 2 52". 1 3 m 32" 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 31 0.52
20 21 22 50 . 29 . 46
21 23 12 49.4 1 31 4.52
22 23 2 48 . 9 2 31 12 . 43
23 25 54 52 48 . 4 3 33 24 . 74
20 h 19 ra 51 s .9 55 27 50". 2 10". 52.
Now the arithmetical mean of the zenith distances is
55 27 50". 2 and the declination of the sun  13 38 14". 7,
hence we find the hour angle:
2h35 M3s. 18.
to which value the correction must be applied. But we
have :
sin p = 9. 8307 9, sin A = 9 .86881,
hence, as the declination is south, the correction is:
8". 32 in arc or s . 55 in time.
With the corrected hour angle 2 h 35 m 12 s .63 we find
the mean time 21 h 8 m 38 s .70, hence the error of the clock
is equal to :
f_ 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 90 sin 8 f cos y> cos 8 cos t.
*) Warnstorff s Hulfstafeln pag. 122,
265
Taking now:
sin S = M sin N,
cos cos t = Af coslV,
we find :
sin h = M cos (y xV),
and hence:
sin h sin A r .
(H)
The formula leaves it doubtful, whether the positive or
negative value of if N must be taken, but it is always easy to
decide this in another way. For if in
Fig. 6 we draw an arc S Q perpendic
ular to the meridian, we easily see that
JY = 90 F Q or equal to the distance of
Q from the equator, hence that Z Q =
(f N, whilst M is the cosine of the
arc S Q. Therefore as long as S Q
intersects the meridian south of the
zenith, we must take the positive value (p JV, but N tp
is to be taken, when the point of intersection lies north of
the zenith. In case that t ^> 90, the perpendicular arc is
below the pole, hence its distance from the equator is ^> 90"
and the zenith distance of Q equal to N </ . 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 (f by
means of the simple equation
C\ I
9p = d== 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:
Dr. Westphal in 1822 October 19 at Benisuef in Egypt
took the altitude of the centre of the sun at 23 h l m 10 s mean
time and found for it 49 17 22". 8. The decimation at that
time was   10 12 16". 1, the equation of time 15 m O s .O,
hence the hour angle of the sun 23 h 16 m 10 s = 10 n 57 30".0.
We find therefore:
266
tang <5 = 9. 2552942,,
cos t = 9 . 9920078
N= 10 23 23". 67
sin iV= 9. 2561063,,
sin S = 9^2483695,,
"070077368
sin A 9 . 8796788
<p iV = 39 29 54". 51
hence <p = 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 equa
tion for sin h and find according to I. No. 8 :
O
dtp sQvAdh cos ip tang A . dt.
Here the coefficients are at a minimum, when A = or
= 180. The secant of A is then =t= 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 influenze 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 = 1640 .l, hence we
find:
dy> = 1.044 JA + 0. 2616 c//,
or if dt be expressed in seconds of time:
ety= 1.044 dA 43. 924 rf*.
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:
7i4/* 4/i"4... cos S cosy ^2sin4(r T 7 ) 2
//=  h cos^lcosp
n cos H n
1. 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 ar
rive 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 merid
ian, in order to find the meridian altitude. But this in con
nection with the declination gives immediately the latitude.
This method of finding the latitude is called that by
circummeridian altitudes.
267
From:
cos z = sin <p sin 8 f cos <p cos S cos t,
we get:
cos 2 = cos (y $) 2 cos 90 cos sin ^ 2 2
and from this according to the formula (19) in No. 11 of
the introduction:
a , 2 cos OP cos . 2 cosy 2 cos S* . fi
 = <p o h rr^ ~ r sin \t *  cotang (5? S) sin I r .
sin(p o) sin(y> tf) 2
or denoting ?^ by 6:
3 J
6 . sin < 2 4 6 a . cotang (y
Therefore if we compute rp () and b with an approx
imate value of (f y, and take the values of 2 sin  f 2 and
2 sin  ^ from tables, the computation for the latitude is ex
ceedingly simple. Such tables are given for instance in Warn
storfFs Hulfstafeln , where for greater convenience also the
logarithms of those quantities are given. If the value of y
should differ considerably from the assumed value, it is ne
cessary, 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 (p d.
Westphal in 1822 October 3 at Cairo took the zenith
distance of the centre of the sun at O 1 2 2 s . 7 mean time
and found 34 1 34". 2. The declination of the sun being
3 48 51". 2, the equation of time 10 m 48 s . 6, and hence
the hour angle + 12 n 5r s .3, we find from the tables:
log 2 sin 4^ t~ = 2.51 105 log 2 sin 4 t* = 9.4060.
Taking (f = 30 4 , we have log 6 = 0.1 9006 and then
the first term of the correction is 8 22". 47 , the second
+ 0". 91, therefore we have:
Correction 8 21". 56
? + <?= 30 12 43". 00
p= 30 4 21".44.
A change of 1 in the assumed value of (f> gives in this
case only a change of 0". 30 in the computed value of y , and
the true value, found by repeating the computation, is:
(/ ==30 4 21". 54.
The formula (^4) 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 ti y instead of r/> J, and we get
in this case:
v cos (f cos S cos re 2 cos 8 2
<p = d z + TTV 2 sin ^ r  . ^ cotang (8 y) 2 sin It * .
sm(d y) sin (d y) 2
Finally, if the star be observed near its lower culmina
tion, we have, reckoning t from the lower culmination:
cos z = cos (180 (f <?) 4 2 cos y> cos 8 sin ^ t*
and hence :
CO
 1804, 
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 \ 2 and 2 sin \ t 4 must be
found for each t and the arithmetical means of all be mul
tiplied 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 an
other form. For from the equation:
cos z cos ((p 8) = 2 cos y cos 8 sin \ t 1
follows :
. <f> <? h z . ip 8 z
sm  sm^^ ~  = cos (f cos o sin \ t 2 .
Now if we take the reduction to the meridian:
we find:
hence :
COS (f> COS 8
 
  sin
 ;  s  
sin ((f 8 + 1 .r)
an equation which may be written in this way:
sin la: cos rp cos 8 sin (g> 8)
 . x =   ^r ^ sm o t    s~T~~i N "
\x 5111(9 o) sin ((p o\ \.r)
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
a =Vcosa, neglecting terms of the fourth order. If we
apply this and take as a first approximation for x the value
from the equation:
. coso> cos _.
t= . ; v 2sm 4 / 2 (72),
sin (<p d)
we find :
3 / i _ j. sin (<P ^)
sin (cp S + ^ x)
or if we find x from this equation, write in the second num
ber instead of x, and denote the new value of x by :
, sin (tp 8} %
I = I  r 7 7 , jv sec T .
sin (y d H j )
This second approximation is in most cases already suf
ficiently correct. But if this should not be the case, we com
pute (f from , then by means of (5), and find the cor
rected value:
With the data used before, we find:
I = 8 22". 47
log  = 2.701 11
sin (y> 3) = 9.74620
coscc (99 S+ i ) = 0.25293
log I = 2.70024,
hence 8 22". 47 and ff = 30 4 21". 53.
8. If we take circummeridian 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 decimation. But in order to render the reduction
more convenient, we can proceed in the following way:
We have:
, ^ COS OP COS $
<p = z + 8  / 2sin,U 2 .
sm(y> o)
Now if D is the declination of the sun at noon, we can
express the declination corresponding to any hour angle t
by .D/?f, where ft is the change of the declination in one
hour and t is expressed in parts of an hour. Then we
have:
sin (<p
270
If we take now:
COS (f COS .. COS OP COS 8^
ftt . 7*: 2 sm * 2 = . f A 2 sin  ( / + )  , (4)
sin (90 d) sm(r/> 5)
we must find ?/ from the following equation:
or since:
sin a 2 sin b 1 = sin (a f />) sin (a />)
. , P sin (tp 8) t
we have:
2 cosy cos sin
sin (<p 8) 20G265
~ ^ cos y. cos 3600~xl5
where the numerical factor has been added, because we take
sin (}?/) = I, 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 ( , we have fi = , or if we wish to express y in
seconds of time, ft = . We have therefore :
and then we find the latitude from each single observation
by means of the formula:
The quantity y is the hour angle of the greatest altitude,
taken negative.
For in I. No. 24 we found for this the following ex
pression :
dS , ,,,206265
= [tang 90 tang tf] ^
where t is expressed in seconds of time and c is the change
of the declination in one second of time. But this is equal
to ~   , hence the hour angle at the time of the greatest
altitude, expressed in seconds of time, is :
*) To this there ought to be added still the second term dependent on
271
u , 206265
720
which formula is the same as that for y taken with the op
posite sign. Hence t + // is the hour angle of the sun, reck
oned not from the time of the culmination but from the time
of the greatest altitude.
Therefore if circummeridian altitudes of a heavenly body
have been taken, whose declination is variable, it is not ne
cessary to use for their reduction the declination correspond
ing to each observation, but we can use for all the declina
tion at the time of culmination, if we compute the hour angles
so that they are not reckoned from the time of the culmi
nation but from the time of the greatest altitude. Then the
computation is as easy as in the former case, when the de
clination is supposed not to change.
For the observation made at Cairo (No. 7) we have :
100^ = 3.4458,, and D = 3 48 38". 57,
with this we get:
^ = + ys.6, hence t +y = 13 m s . 9
and hence we find for the first term of the reduction to the
meridian: =8 35". 00.
On account of the second term multiplied by sin ~ 4 we
must add to this f 0".91, and we finally find cp = 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 polestar is very
small, it is always in the neighbourhood of the meridian, and
hence its altitude taken at any time may be used with ad
vantage 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 ap
plied 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 per
pendicular to the meridian,
and denote the arc of the
meridian between the point
of intersection with this arc
and the pole by a?, the arc between the same point and the
zenith by z */, where y is a small quantity, we have :
90 <p = z y + x,
or 9?= DO zty x,
and we have in the right angled triangle :
tang x = tang p cos t
. cos 2 (a)
cos (z y) =
cos u
We get immediately from the first equation:
x = tang p cos t ^ tang p 3 cos t 3 ,
neglecting the fifth and higher powers of tang p, or neglect
ing again terms of the same order:
x = p cos t + 3 p 3 cos t sin t z . (6)
If we develop the second equation (a), we find:
1 cos u
sin y = cotang z h "2 sin 2 A y . cotang z,
or neglecting the fifth and higher powers of u:
sin y = cotang z (\ u 1 + , 3 5 T w 1 ) + 2 sin 2 \y cotang z.
But we get from the equation
sin u = sinp sin t :
u = p sin t  p 3 sin t cos t,
hence substituting this value in the equation above we find,
again neglecting terms of the fifth order:
3/~ TP 2 sin if 2 cotg2 ^p 4 sin* 2 (4 cos* 2 Ssin^cotgzh^cotgz.^ 2 . (c)
This formula, it is true, contains still y in the second
member, but on account of the term  cotang z . y 1 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 :
<f> = 90" z p cos t + p* sin t 2 cotang z } p 3 cos t sin t 2
~f~ Ti^ 4 i n t* (5 sin t 1 4 cos* 2 ) cotang z
+ {/>* sin f* cotang 2 3 . (A}
Since it would be very inconvenient to compute this
273
formula for every observation , tables are every year pub
lished in the Nautical Almanac and other astronomical alma
nacs, 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: *)
if = 90 z p cos t +  p 2 sin t 2 cotang z.
If we denote thus a certain value of the right ascension
and polar distance by and p M the apparent values at the
time of the observation being
= H A , p = PO 4 A;>
we find substituting these values:
tp = 90 z p tt cos t h I p 2 cotang z sin / 2
Ap cos / p sin / A,
where t () = .
We find now in the Almanac three tables. The first
gives the term p cos * , the argument being 0, since this
alone is variable. The second table gives the value of the
term  p^ cotang z sin 2 , the arguments being z and &. Fi
nally the third table gives the term dependent on 6>, A
and &p
<Ap cos p sin t 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 Hulfstafeln pag. 73 and these
embrace all terms and can be used while the polar distance
of the polestar is between the limits 1 20 and 1" 40 . Let
p again be a certain value of p, for which Petersen takes
p (] = 1 30 , then the formula (A) can easily be written in
this way:
*) The term multiplied by y/ is at its maximum, when t = 54 44 and
its value, if we take ^ = 140 , is then only 0".G5. The terms multiplied
by p 1 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 /, the
other with 4j 2 sin t 2 cotang z.
18
274
2
<r, = 90 z [p cos / + \p * cos /sin/ 2 ] I f ., 1 )# J cos /sin/"
7>o PoVo
H ^ cotang. z [4;J 2 sin/ 2 h^, P O 4 sin / 2 (5 sin/ 2 4 cos/ 2 )]
;V
f * cotang z 3 .
Po" 
If we put now:
P
p cos / + 3 p
A
^/> 2 sin / 2 f j^Po 4 s i n * 2 & s i n 2 4 cos/ 2 ) ==/?,
* J 4 p 4 sin / 4 cotang c 3 = ^ /I 4 /9 2 . cotang s 3 = //,
we obtain:
tp = 90 ~ Aa y\A*{3 cotang ,~ + u.
Now four tables have been constructed, the first two of
which give and ft, the argument being t , a third table gives
the value of the small quantity ; , the arguments being p and t
and finally a fourth table gives the quantity /, which is
likewise very small, the arguments being y = A^ ft cotang 2
and 90 z. These tables have been computed from t = O h
to t = 6 h . Therefore if t > 90, the hour angle must be
reckoned from the lower culmination, so that in this case
we have:
<p = 90 z h A a h y + A 1 ft cotang z f ft.
Example. In 1847 Oct. 12 the altitude of Polaris was
taken with a small altitude and azimuth instrument at the
observatory of the late Dr. Hulsmann at Diisseldorf and it
was at 18 h 22" 1 48 S .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:
= lh5m3is.7 j 5 = 88 29 52". 4.
Hence we have:
; , = 1 30 7". 6, /=l?h 17 17s. 1 = 259 19 1C". 5,
and:
log A = 0.0006108
and we obtain by means of the tables or the formulae:
275
therefore :
Aa = + 16 42". 26
y! 2 / 3cotangz = t 1 24 . 33
^ = + . 02
sum = 4 18 6". 61
hence: <j> =51 13 37". 41.
10. Gauss has also published a method for finding the
latitude from the arithmetical mean of several zenith distan
ces, taken long before or after the culmination, which is
especially convenient for the polestar.
If an approximate value (f () of the latitude (p is known,
and & is the sidereal time, at which the zenith distance z
is observed, we can compute from ( ) and (f (} the value of
the zenith distance by means of the formulae:
tang x = cos t cotang S
f N
sin UP O f x)
cos.r
and then we obtain:
hence :
u V " :
sm o cos (90
cos;r sin
# is again the arc between the pole and the point in which
an arc drawn through the star, and perpendicular to the me
ridian intersects the latter and since the length of this arc
is always between the limits =t= 90 t), we can take in case
P ,i i sin ,, cos (<p f r) .. ./,
ot the polestar as well as equal to unity, if
cos x sin
the latitude is known within a few seconds and d(f is there
fore a small quantity.
If another^ zenith distance has been taken at the sidereal
time , we have:
tang x cos t tang
; sin o" .
cos = ,sm(<f> n ix)
and:
d(f>
18*
276
or, if Z denotes the arithmetical mean of both observed ze
nith distances equal to * (X { 3, ):
^ ~ . /d d\
M 7 + / )
\dcp da) /
where :
sin 8 cos (OP O f a:)
yl =  .
cos x sm f^\
sin $ cos (9^0 f x}
cosr sin
or: A = cotang . cotang ($>$ + .r) ^ ,
1? = cotang . cotang (9^0 H~ ^ )
and finally, if we find y from the original equation:
eos = sin (p (} sin $ f cos (f> cos ^ cos /
we obtain also:
cos QD sin 8 sin cp cos (5
iCdhB)= r cos 4 (<+/). (^/)
sin Z sin Z
In case of the pole star we have simply:
dy> = i ( h ) Z. (e)
If several zenith distances have been observed, we ought
to compute for each sidereal time separately and we should
then obtain :
i [ + + +... + ,,,]
f j f J
w ^ d c? /
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 () the arithmetical mean of all sidereal
times and put:
i} = r, 6> = T etc. %
and then denote by the zenith distance corresponding to
, we obtain in the same way as in No. 5 of this section:
sn
n
Now if T is taken from the following equation:
277
the zenith distances z and z at the times # T and @ f7
are :
c. d
*= d t
hence :
and we obtain according to the formula (/") simply:
d<f = " ,
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 clocktimes and
subtract from it each clocktime without regard to the sign.
These differences converted into sidereal time give the quan
tities r, for which we find from the tables the quantities
2 sin \ T . From the same tables we find the argument T
corresponding to the arithmetical mean of all these quanti
ties and compute the hour angles :
6> ( t T) = t
(a T) = t
and then z and z by means of the formulae:
tang x = cos t cotang
sin 8
cos z = sin (gpj) + x)
cosx
and tang x cos t cotang
, sin
cos 2 =  , sin (rp a {x).
cosx
In case of the polestar we then have immediately:
where Z is now the arithmetical mean of all observed zenith
distances. For other stars the rigorous formula for d<f must
be computed, namely:
where A and B are obtained by means of the formulae (6),
(c) or (rf) after taking = z and = z *).
*) WarnstorfFs Hulfstafeln pag. 127.
278
Example. In 1847 Oct. 12 the following ten zenith dis
tances of Polaris were taken at the observatory of Dr. Hiils
mann :
Sidereal time. Zenith distance. T 2sin^T 2
17h56 "21s.4 39" 13 42". I 13 n 19.75 348.75
59 54 .5 12 17 . 6 9 46 .65 187.69
18 3 29 .7 11 6 . 8 6 11 .45 75.24
62.9 103.6 3 38 . 25 25 . 98
8 35 .0 90.6 1 6 . 15 2.39
115.1 82.8 123.95 3 . 85
13 32 .0 77.6 3 50 .85 29 .06
16 34 .0 64.8 6 52 .85 92.95
18 28 . 1 5 15 .3 8 46 .95 151 .43
22 48 .8 3 42 . 7 13 7 . 65 __338 . 28
.15 398 38".39 ~~125756
Refr. 46".50 T= 7 59*. 83
Z= 399 r 24".89
= 2542 24".3 =258 2 19". 2.
Now taking:
7> = 51 13 30".0,
we obtain:
z = 39 12 37". 56 z = 39 6 34". 54
(zHy) = 399 36".05
.}0 + 2) = +11". 16,
hence :
= 51 13 41". 16.
III. METHODS OF FINDING BOTH THE TIME AND THE LATITUDE
BY COMBINING SEVERAL ALTITUDES.
11. If we observe two altitudes of stars, we have two
equations :
sin h = sin <p sin 8 + cos <p cos 5 cos t,
sin k = sin y> sin $ + cos <p cos S cos t .
In these equations, since we always observe stars, whose
places are known, <) and d are known, and further we have :
= * + (* f) = t +(& 0) ( ).
Now since a and 6/ B are likewise known, the latter
being equal to the interval of time between the two obser
vations, the two equations contain only two unknown quan
279
titles and f/, which therefore can be found by solving
them. Thus the latitude and the time can be found by ob
serving two altitudes, but the combination of two altitudes
in some cases is also very convenient for finding either the
latitude or the time alone.
We have seen before, that if two altitudes of the same
star are taken at its upper and lower culmination, their arith
metical mean is equal to the latitude, which thus is deter
mined independently of the declination. This is even found
at the same time, since it is equal to half the difference of
the altitudes.
Likewise we can find the latitude by the difference of
the meridian zenith distances of two stars, one of which cul
minates south, the other north of the zenith. For if S is the
declination of the first star, its meridian zenith distance is:
v
and if d is the declination of the other star, north of the ze
nith, we have: , s ,
z =o y,
and therefore we get:
p^tf+tfOM (** )
12. If two equal altitudes of the same star have been
observed, we have:
sin h = sin cp sin S \ cos y cos 8 cos t, . .
sin h = sin <p sin 8 \ cos rp cos 8 cos t ,
from 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 clocktime of the first, u that of the second
observation, J (u { u ) 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 4 <> t M ).
This method of finding the time by equal altitudes is
the most accurate of all methods of finding the time by al
titudes. Since neither the latitude of the place nor the de
clination of the heavenly body need be known and since
for this reason it is also not necessary to know the longi
tude of the place, this method is well adapted to find the
time at a place, whose geographical position is entirely un
known. 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 in
terval between the two observations keeps a uniform rate,
and an altitude instrument, whose circle need not be accu
rately 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 decli
nation 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 S is the declination of the sun at noon, A<) the change
of the declination between noon and the time of each obser
vation, we have:
sin h = sin cp sin (8 A<?) + cos y cos (8 A 8) cos t
sin h = sin y sin (8 f A d) H cos y> cos (d 4 A 8) cos t .
Let the clocktime of the observation before noon be de
noted by M, the one in the afternoon by u\ then (u \ti) U
is the time, at which the sun would have been on the me
ridian, if the declination had not changed.
Then denoting half the interval between the observa
tions (M M) by r, the equation of equal altitudes by x,
the moment of apparent noon is given by U } x and we
have:
t = T (u u) t x = r + x,
t = 4 (11 11) x = T .r,
and also:
sin h = sin (f sin (S A<?) + cos (p cos (8 A<?) cos (T f a:)
and :
sin h = sin <f> sin (8{&8) f cos y cos ($hA$) cos (r #).
281
From these expressions for sin h we find the following
equation for x:
0=singpcos Ssill&S cosy sin $sin A^OSTCOS x \ cosy cos &d cos $sinr sin.r.
Now in case of the sun x is always so small, that we
can take cos x equal to 1 and sin x equal to x. Then we
obtain, taking also &S instead of tang /\r):
r = _/tan g9 ,_tang^\
v sin r tang t /
If we denote now by /< the change of the declination
during 48 hours, which may be considered here to be pro
portional to the time, we have:
A *>.
hence:
U / T T \
x ==  tang a> f tang o }
48 \ smr tang T /
or if x is expressed in seconds of time :
X ~ 7 1A ( ~ tan S 0> +" ~ tall g ^ )
720V smr tang r /
In order to simplify the computation of this formula,
tables have been published by Gauss in Zach s monatliche
Correspondent Vol. XXIII, which are also given in Warn
storTs Hulfstafeln. These tables, whose argument is r, give
the quantities:
720 sin r ~ A
and:
J r
720 tang r
and thus the formula for the equation of equal altitudes is
simply:
x = Au tang y> + J3u tang 8. (A)
Differentiating the two formulae (a), taking d as con
stant, we find:
*) We find this also, if we differentiate the original equation for sin A,
taking 8 and t as variable, since we have x = &.
do
** ) 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 de
clination, preceding and following the day of observation. Instead of this
the almanacs give the quantity fi.
282
d/i = cos A d(p cos <p sin A dt
dh = cos A d(f> cos (p sin A dt.
In these equations dt has been taken equal to dt, since
we can suppose, that the error committed in taking the time
of the observation is united with the errors of the altitudes.
Since we have now A A, we obtain:
dh = cos A drp ( cos rp sin A dt,
dli = cos A d<f cos rp sin A 1 dt,
and :
cos (f sin A
Therefore we see, that we must observe the heavenly
body at the time, when its azimuth is as nearly as possible
490" and 90.
In 1822 Oct. 8 Dr. Westphal observed at Cairo the fol
lowing equal altitudes of the sun:
Double the altitude of Chronometer time_
(Lower limb) forenoon afternoon Mean
73 21 h 7 m 27 2 h 33 m 59 s 23 h 50 m 43 s .O
20 8 24 33 3 43 . 5
40 9 23 32 5 44 .
74 10 18 31 9 43 .5
20 11 16 30 12 44 .0
40 12 11 29 14 42 .5
75 13 11 28 13 42 .0
20 14 9 27 15 42 .0
40 15 10 26 15 42 .5
76 16 6 25 20 43 .
Hence we find for the arithmetical mean of all obser
vations :
23 h 50 " 43 . 00.
Now half the interval between the first observation in
the forenoon and the last in the afternoon is 2 h 43 m 16 s and
that between the last observation in the forenoon and the
first in the afternoon 2 h 34 m 37% hence we take :
T = 9h 38" 56 s . 5 = 2>> . 649.
If we compute with this A and B, we find:
logr 0.42308 0.42308
COSCCT 0.19435 cotang r 0.08028
Compl. log 720 7.14267 7.14267
log 4 "7/7601 logJS 7.6460,
283
and as:
= 6 7 , y> = 304
and:
log <* = 3.4391.,
we obtain:
x = f IQs . 4ft.
Therefore the sun was on the meridian or it was appa
rent noon at the chronometertime 23 h 50 m 53 s . 46. Now since
the equation of time was  12 h 33 s .18, the sun was on the
meridian at 23 h 47 m 26 s .82 mean time, and hence the error
of the chronometer was:
3 26 . 64.
If we compute the differential equation and express dt
in seconds of time, we find:
dt = Qs. 048 (dti dK),
and we see, that if an error of 10" was committed in taking
an altitude, the value of the error of the clock would be
s . 48 wrong.
We can make use of this differential formula in com
puting 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 h=dh\ we ought to apply to h the correc
tion dh\ and hence the correction of U is:
_ _dh _
30 cos <f sin A
dh cos li
30 cos (p cos 8 sin t
In case that the greatest accuracy is required, such a
correction is necessary even if equal altitudes have been taken.
For although the mean refraction is the same for equal ap
parent altitudes, yet this is not the case with the true refrac
tion, unless the indications of the meteorological instruments
be accidentally the same. Therefore if o is the refraction for
the observation in the forenoon, o+dy that in the after
noon, 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:

oO cos
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
hour angles are:
T = 12i> T
and : _ T = i9h + T.
The case is now the same as before only with this dif
ference, that if A# is positive, the sun has the greater de
clination when the hour angle is  r, hence the correction
(i must be taken with the opposite sign and we have in this
case :
X A f ta "g <f> ~ ~~ tail g ^ )
720 \ sin T tang T /
fl ( 12 1 T 12 !l T .A
= rfon I ; tan g ( P ~ tang o \
720 V sin T tang T )
If we write instead of it:
u 12 h r / r r _\
x = foA ~ I " " tan s 9 P ~ tan s ^ )
720 T \ sin r tang r /
we can use the same tables as before ; but besides, the quan
tity  r must be tabulated, the argument being T or half
the interval between the observations. This quantity in Warn
storfTs Htilfstafeln is denoted by /", hence we have for the
correction in this case:
x = ffj, [A tang cp JB tang ].
In 1810 Sept. 17 and 18 v. Zach observed at Marseilles
equal altitudes of the sun. Half the interval of time was
10 h 55 n and as:
10 h 55, <* = H2 14 16", y = 43 17 50"
and: log^ = 3.4453.
We find:
log A = 7.7305 log B = 7.7128,
log/ 1.0033,
ufA tang y = 142* . 33
fifB tang S = + 5 . 67,
hence for the correction:
x = 136s. 66.
285
Note 1. The equation 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,
we must multiply the correction by , a fraction whose logarithm is 0.0012.
obo
Note 2. If the hour angle r is so small, that we may use the arc in
stead of the sine and the tangent, the equation of equal altitudes becomes :
r = [tang y> tang $].
But as the unit of T in the numerator is not the same as in the denom
inator, being in the first case one hour, in the other the radius or unity,
we must multiply the second member of the equation by 206265 and divide
it by 15X3600. Thus we obtain:
x = 18 ^ . [tang ^ tang $\,
where now x is the equation of time for T = 0. But in this case the two
altitudes are only one, namely the greatest altitude, and hence x is the cor
rection, 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 reduction of
circummeridian 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 <f> sin + cos <p cos cos t,
sin h sin cp sin + cos cp cos cos t .
If then u and u are the clocktimes of the first and sec
ond observation, &u the error of the clock on sidereal time,
we have : *)
t U f (\ U 
where AM 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, de
noting the equation of time for both observations by w and w :
t = u + A u w,
hence : A = u u (w w).
286
u it (a ) = A
a known quantity and we have I = t f L Hence the two
equations contain only the two unknown quantities cf and ,
which can be found by means of them. For this purpose
we express the three quantities
sin (p, cos (f> sin t and cos ip cos t
by the parallactic angle, since we have in the triangle bet
ween the pole, the zenith and the star:
sin (p = sin h sin f cos h cos cos p,
cos (f sin t= cos h sin p, (r/)
cos 9? cos t = sin A cos 8 cos h sin cos ;>.
Substituting these expressions in the equation for sin /* ,
we find:
sin h 1 = [sin 8 sin 8 + cos $ cos $ cos 1] sin h
h [cos $ sin sin 8 cos 8 cos 1] cos A cos p
cos $ sin 1 . cos A sin p.
But in the triangle between the two stars and the pole,
denoting the distance of the stars by /), and the angles at
the stars by s and * , we have:
cos D = sin 8 sin 8 f cos 8 cos 8 cos /
sin Z) cos 6 = cos c sin 8 sin 8 cos 8 cos A (/;)
sin D sin s = cos 8 sin A,
hence, if we substitute these expressions in the equation for
sin h :
sin // = cos D sin //. + sin D cos h cos (s t j),
. sin /* cos D sin //
hence cos (. +)= . ( c )
sm Z) cos A,
Further if we substitute in
sin h = sin cp sin 8 + cos y cos 8 cos (Y A)
the expressions for sin r/, cos cj sin < and cos </ cos , which
we derive from the triangle between the pole, the zenith and
the second star, we easily find:
. . .. sin h cos D sin h
cos (s p ) =   , , (</)
sin D cos h
After the angles p and p have thus been found by means
of the equations (6) and (c) or (d), the equations (a) or the
corresponding equations for sin f/, cos (f sin t and cos (f cos <
give finally cp and or <y? and t .
The equations (6) give for D and 5 the sine and cosine,
the same is the case with the equations (a) for (f and ,
hence there can never be any doubt, in what quadrant these
287
angles lie. But the equations (r?) and (rf) give only the co
sine of s + p and s p  however we have in the triangle
between the zenith and both stars:
sin D sin (.<? f p ) = cos // sin {A A)
and sin D sin (.<? p ) = cos h sin (A 1 A),
hence we see that sin (s 4 p) and sin (5 p ) have always
the same sign as sin (A 1  A), so that also in this case there
can never be any doubt as to the quadrant, in which the
angles lie.
The formulae (a) and (6) can be made more conve
nient by introducing auxiliary angles, and the formula for
cos (s  p) can be transformed into another formula for
tang  (sr/?) 2 in the same way as in No. 4 of this section.
Thus we obtain the following system of equations:
sin 8 = sin/ sin F
cos 8 cos^ = sin/cos F (e)
cos 8 sin I cos/,
cos D = sin /cos (F <?)
sin D cos .s = sin/ sin (F 8} (/)
sin D sin s = cos/,
cos . sin (S //)
where 5 = (D f h + /* ),
sin g sin G = sin h
sin <? cos G = cos 7i cos p (//)
cos<7 = cos 7* ship,
sin^ = sin g cos (G (?)
cos (p sin = cos g (?)
cos y cos t = sin # sin (6 S).
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 Z>, 90 d and 90" <V and the
opposite angles A, s and s:
sin ^ Z> . sin ^ (* *) = sin (# 5) cos j A
sin $ D . cosi (* s) = cos4 ( } 8) sin U
cos ] .D . sin (s } .9) = cos 4 (5 S) cos 4 *
cos ^ Z> . cos^ (.9 + s) = sin ^ (5 + <?) sin 4 ^.
Then we have as before:
cos 5. sin (/< )
tang 4 (sf) 2 =  ? ,
D) sin(,S
288
Finally we ha\ 7 e in the triangle between the zenith, the
pole and the star:
sin (45 Ji<p) sin ^ (A + t) = sin ^ p cos ^ (h 4 S)
sin (45 7 <f) cos (A + /) = cos p sin 4 (A 5)
cos (45 %) sin 1, (4 = sin J ;> sin J (A f c?)
cos (45 ^9?) cos \ (A t) = cos .1 p cos 3 (/< 8\
Iii case that the other triangle is used, we have similar
equations, in which A\ t\ p\ ti 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:
u = 20 h 48 " 4S h = 37 56 59". 6
u =23 7 17 7/=50 4055 .3,
where u is already corrected for the rate of the clock and
h and h are the true altitudes. The interval of time con
verted into apparent time gives /. = 2 h 18 in 28 s . 66 = 34 37
9". 90 and the declination of the sun was for the two ob
servations :
^=10 10 50". 1 and S = 10 12 57". 8.
From these data we find by means of the Gaussian formulae:
D= 34 3 20". 27
s= 93 1258.26
s = 93 6 I . 93
Further: * f ;> = 53 1541.26
. hence: p = 39 57 17 .00
and then : (f = 29 5 39 . 80
t = 35 24 59 . 23
.4 = 46 1952.17.
It is advisable to compute (f and t also from the other
triangle as a verification of the computation, since the values
of (fj must be the same and t t = L
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
d/i = cos A d<p cos y sin A dt
dh = cos A dcp cos 9? sin A dt
where dt has been supposed to be the same in both equa
tions, because the difference of dt and dt may be trans
ferred to the error of the altitude. From these equations
we obtain, eliminating either dcp or dt:
cos A cos A
cos ydt = rrT  7\ dh ^~ TT ,   dh
sin (A 1 A} sin (A 1 A)
sin A sin A
dtp =  . dh\ T
.  ^ .
am (A A) am (A 1 A)
Hence we see, that if the errors of observation shall
have no great influence on the values of y> and , we must
select the stars so that A* A is as nearly as possible =t= 90,
since, if this condition is fulfilled, we have :
cosydt= cosA dh cosAdh
dcp = sin A dh + sin Adh .
Then we see, that if A 1 is == 90 and therefore A is 0,
the coefficient of dh in the first equation is 0, that of dh
equal to =t= 1 ; hence the accuracy of the time depends prin
cipally on the altitude taken near the prime vertical. In the
same way we find from the second equation, that the accu
racy of the latitude depends principally on the altitude taken
near the meridian. For the above example we have, since
4 = 115 :
dy> = + 0.0308 dh 1.0215 dh
dt = \ 0.1077 dh 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 TT D = cos sin 4 >l
cos TJ D sin s = cos 4 A
cos ^ D cos s = sin S sin 4 A.
By means of these we find D and 5, and then from the first
of the equation (#) and the equations (C) y and t and, if it
should be desirable, A.
In this case we can solve the problem also in the fol
lowing way. We find from the formulae:
sin h = sin y> sin S f cos cp cos S cos /
sin h = sin (f sin S + cos <p cos 8 cos (t + /)
19
290
by adding and subtracting them:
cos<?sin^/l.cos9Psin(Jf^) = cos.j(//h/i )sin .j (It // )
sin (f sin S\ cos S cos A k . cos(jpcos(t f ^A) = sin (h^h ) cos^ (^ //).
Therefore if we put:
sin = cos 6 cos B
cos $ cos <5 A = cos 6 sin 5 (/I)
cos S sin ^ A = sin 6,
the second of the equations (a) is changed into:
sin (A MO cos 4 (A /< )
sin go cos 5 h cos y> cos (/ + . A) sm /? =
and if we finally take:
sin <f = cos .Fcos G
(B)
in <f = cos .Fcos G
cos y sin (t\\ %) = sin G
cos 9? cos (^ + T^)
we obtain:
sin G =
cos i (A MO
cos(B F) =
sin b
cos 6
ti)
(CO
Fig. 8.
Therefore if we first compute the
equations (4), we find G and F by
means of the equations (C) and then
y and t from the equations (5). The
geometrical signification of the auxi
liary angles is easily discovered by
means of Fig. 8, where PQ is drawn
perpendicular to the great circle join
ing the two stars, and ZM is perpen
dicular to PQ. We then see, that
b=QS = D, B=PQ, F=PM and
G=ZM.
If we use the same data as in the preceding example,
paying no attention to the change of the declination and
taking d =  10 12 57". 8, we find:
jB = 10041 23".l sin b = iUGGGOO cos 6 = 9.980534
sin G = 9.432863. cos G = 9.983445 F=41l 53".3
and hence t = 35 22 21".0 y = 29 5 42". 7.
In case that the two altitudes are equal, the formulae
(A) or (e) and (/") in No. 14 remain unchanged, but the. for
mulae (J5) are transformed into:
cos (h + 4 D)
tang J (s 4y>) 2 = tang
cos (A ^
291
and then p being known, rf and t can be computed by means
of the formulae (ft) and (i), or (p, t and A by means of the
formulae (0).
16. A similar problem, though not strictly belonging
to the class of problems we have under consideration at pres
ent, 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 compute as before the formulae (4)
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 </ ,
the third angle being A A and the opposite sides 90 ft ,
90 h and D:
. , x , . x cos^(// h) cos(A A}
sin 4 (g f 7) = r ~
cos ^ D
. i/i N sin TJT (h li) cos ^ (A 1 A)
By means of these equations we find J (h f ft ), thence ft
and ft and the angles </ and </ . But since we have accord
ing to No. 14 q = s ~f p and q = s ^ , we thus know p
and p , hence we can compute </, Z and ^4 by means of the
formulae (C) in No. 14 and as a verification of the compu
tation also <, t and A .
In this case the differential equations are according to
No. 8 of the first section:
dh = cos A d(f) cos S sin p . d  + cos S sin p d
dh = cos A d(f cos si\i]> . d cos sin pd
cos S cos i) A ] \t cos S cos p t t
dA = sm A tang hdrp\ d d
cos h 2 cos h 2
7 ., ., ,1., . cosS cosn ,t +t cosS cosn t t
dA =BmA tsuagtid<p+ 7 , d  h .,, d ,
cos 7 2 cos h 2
, t \t t  t i t \t t  t , . . n
where 9 h 9 and 9  have been put in place of
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*  and then
dy, and remembering that we have:
cos 8 sin p = cos 9? sin A
cos 8 cos p
= sin CP f cos 9? tang h cos A
cos A
we easily find:
Md<p = [tang h cos J tang ti cos ^4  e/ (ti h) + [sin A sin A ] d (A 1 A)
f  7 cosp sin 4  T cos p sin A\ d(t /),
LCOS h cos A J
Jf cos yrf = [tang A sin A tang A sin A ] d(ti ti) [cos A cos A ] d(A A)
f [cos <f (tg A tg A ) sin 2 ^ (4 + 4) h sin <p (cos J. cos A )] d(t 0
where M = 2 [tg A + tg A ) sin 2  (A 1 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)
= 90, even the coefficient of d (ti Ji) is less than \.
v. Camphausen has proposed to observe the stars at the
time, when their altitude is equal to their declination, be
cause then the triangle between the zenith, the pole and the
star is an isosceles triangle and we have =180 A and:
cotg 8 cos t = cotg 8 cos t = tg (45 4 9?)
cotg 8 cos A = cotg 8 cos A = tg (45 j y>\
by means of which we find:
or
From these formulae we obtain t f t or A + 4 and y.
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 Dou
wes schen Problems in the Berlin Jahrbuch for 1859.)
Example. In 1856 March 30 the following differences
of the altitudes and the azimuths of i] Ursae majoris and a
Aurigae were observed at Cologne.
293
ti h = 410 46".0
A A= 226 28 9".9
The interval of time between the observations, expressed in sidereal time,
was QMS " 8s. 70.
The apparent places of the stars were on that day:
rj Ursae majoris a 13 h 41 m 54 s .53 8 = + 50 1 45". 9
aAurigae = 56 1 . 69 # = + 4551 1 .7.
Hence we get I = 133" 30 23". 1, and we obtain first by
means of the formulae (A) in No. 14:
., = + 31 22 33". 18
., == + 28 41 50". 20 D = 76 14". 79.
Then we find from the formulae (J?) 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 = + 57 22 43". 64. Since we
find  (#4 A) = 47" 56 40". 61, and hence A = 50 2 3". 61,
we get by means of the equations (C) in No. 14: cp = 50"
55 55". 57, / = 295 2 56" .70, A = 244 57 48". 50.
If we compute also the differential equations we find, if
we express all errors in seconds of arc:
dtp = 0.0342 d (/>. A) 0.4892 d(A A] + 0.2438 d(t t)
d~p = 0.8621 rf (A A) f 0.0244 d (A 1 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 w r as proposed by Douwes,
a Dutch seaman.
Since the latitude is always approximately known from
the logbook, they first find an approximate time by the alti
tude 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 <p cos S [cos t cos (t f )]
= 2 cos ^ cos S sin (t + \K) sin A,
hence :
2 sin (t + % A) = sec y> sec 8 cosec } A [sin h sin h ]
294
or, if we write the formula logarithmically:
log . 2 sin (t f \ A j = log sec y H logsec ^h log [sin h sin ti\ + logeosec 5 A. M)
Since an approximate value of (p is known, we find from
this equation t\\ A, and hence also , and then we find a more
correct latitude by the altitude taken near the meridian by
means of the formula:
cos (90 8) = sin /t f cos <p cos 8 . 2 sin 5 (t f A ) 2 . ( J3)
If the result differs much from the first value of the
latitude, the formulae (A) and (#) must be computed a second
time with the new value of (f.
Douwes has constructed tables for simplifying this com
putation, which have been published in the ,,Tables requisite
to be used with the nautical ephemeris for finding the lati
tude 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 f A, 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 + 1 A), and a third table
with the heading r log. rising time" gives that of log 2 sin  2 .
The quantity log. sec f/ sec d is called log. ratio and we
have therefore according to the equation (/I):
Log. middle time = Log. ratio f Log (sin k sin h )
f 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 f / , 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:
. ,, cos ^ (ft + h ) sin (h h )
cos <p cos sin I A
and:
sin
cos ((f 2V) = ,
M
where: sin = J/ sin JV
cos 8 cos t = il/cos 2V.
If we compute the example given in No. 14 according
to Douwes s method, we find:
p = 29
295
log ratio 0.06512
log (sin A sin k ) 9 . 20049*
log half elapsed time . 52645
log middle time 9 . 79206,,
log rising time 5 . 90340
log ratio . 06512
f . 00007
sin ti f . 77364
cos (y <?) = 9 . 88858
<P S= 39 18 .7
0,= 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. . The ship at the time of the first ob
ser^ation 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
O
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
B A 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 di
rection of the course of the ship makes with the meridian,
is known, the angle S BA is known also. Denoting this
angle by and the distance between the two places A and
B by A? we have:
sin h == sin h cos A 4 sin A cos h cos ,
where A is the reduced altitude. If we write instead of this :
sin A = sin h + sin A cos h cos a 2 sin ^ A 2 sin A,
296
and take A instead of sin A, we obtain by means of the for
mula (20) of the introduction:
// = h H A cos .j A 2 tang /<,
where the last term can in most cases be neglected.
18. If three altitudes of the same star have been ob
served, we have the three equations:
sin h = sin y> sin 8 + cos <p cos cos t
sin h = sin tp sin $ h cos y> cos $ cos (t f / )
sin A"= sin 90 sin 8 h cos 90 cos 3 cos (< f A ),
from which we can find </?, t and d. For if we introduce
the following auxiliary quantities:
X = COS (f COS COS
y = cos gp cos S sin ?
z = sin (f sin <?,
those three formulae are transformed into :
sin li = z f x
sin h = z + x cos A y sin A
sin h" z \ x cos 1 y sin A ,
from which we can obtain the three unknown quantities x,
y and z in the usual way. But when these are known, we
find (f and t by the equations:
y
tang t =
x
sin (f sin 3 = z
cos <p cos $ = J/ar 2 + < y 2 .
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 quan
tities. But if we do not consider ci 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 <p sin 8 f cos 95 cos S cos t
sin h = sin cp sin \ cos y cos cos (t 4 A) (a)
sin h = sin y sin S"+ cos <j> cos $"cos (t f A ),
where A = (u 1 it) (a a)
and A =(M"M) (" ).
*) Since three altitudes of the same star have been taken, I and A are
not dependent on the right ascension.
297
If we now introduce in the two first equations \ (o +S)
+. i (<y _ ) instead of <>*, and f (3 + <V) J (<? 5 ) instead
of t) , and subtract the second equation from the first, we get:
= 2 sin T sin  (5 8 ) cos (5 4 8") 4 cos y> cos t [cos ^ (5 4 5 ) cos (5 5 )
 sin  (5H 5 ) sin 4 (5 5 )]
 cos y cos (< } A) [cos (5 + 5 ) cos 4 (8 5 ) 4 sin \ (8 4 5 ) sin .1 (8 5 )J
or:
= sin <f sin 5 (t? 5 ) cos  (5 4 5 )
4 cos y cos (5 H 5 ) cos J[ ( 5 ) sin ^ ^ sin (i! 4 \ A)
 cos <p sin ^ (^ 4 8 ) sin i (55 ) cos 4 I cos (i 4 \ I}.
From this we find:
tang <p = sin ,] A . sin (i! 4  A) cotang ^ (5 5 )
4 cos ^ A . cos (t 4 5 A) tang .1 (5 4 ).
Introducing now the auxiliary quantities A and B\ given
by the formulae:
sin A . cotang  (5 5 ) = .4 sin B
cos 4 A. tang ^(5 4 5 ) = .4 cos Z? (^t)
JB> 4 ^A = C ,
we obtain:
From the first and third of the equations (a) we find
in the same way similar equations:
sin  A cotang \ (5 5") = A" sin " \
cos  A tang (5 4 5") = ^" cos 5" (<7)
fi" 4 ^ = C",
tang 99 = J" cos (< 4 C"). (Z>)
Furthermore we find from the two formulae (B) and (Z>) :
^4 cos ( 4 C Y ) = .4" cos (< 4 C").
In order to find t from this equation, we will write
it in this way:
A cos [t 4 H\ C H] = ^4" cos > 4 T4 C" //J,
where # is an arbitrary angle, and from this we easily get:
ta n g(/ 4 7/)^ ^^ ll^) ~ A " * (C"V)
A sin (C  ff)A r sln~(C f f^
For H we can substitute such a value as gives the for
mula the most convenient form, for instance 0, C or C".
But we obtain the most elegant form, if we take:
H=  (C" 4 C")
for then we have:
tang [t 4 4 (C" 4 C")] = ^r^C cotang * (C" C"),
~
298
Introducing now an auxiliary angle , given by the
equation :
we find:
J
hence :
tang [t + t (C"+ 6 ")] = tang (45  g) cotang  (C C"). (F)
We find therefore first by means of the equations (^4)
and (C) the values of the auxiliary quantities A, /? , C and
A\ /T, C"; then we obtain by means of the equations (E)
and (F), and finally (/ by either of the equations (J5) or (/>).
It is not necessary to know the altitude itself, in order to
find (f and f, but if we substitute their values in the origi
nal equations (a), we find the value of /i; hence, if the alti
tude itself is observed, we can obtain the error of the in
strument.
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 dh to be the same for the three altitu
des, 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 f A 5
the error dt will we composed of two errors, first of the
error 6/(A0, 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 dif
ferential equations are:
dh = cos Ady cos <p sin A du cos (f sin A c?(A M)
dh = cos A d<p cos (f sin A du cos <p sin A d(&u)
dh = cos A"dy cos <p sin A"du" cos y sin A"d(&tt).
If we subtract the first two equations from each other,
we find by a simple reduction:
299
A n . A\rA ^4 + ^4 cos OP sin A
= 2 sm 9 ~ dtp 2 cos vos (f>d(t\n) .,
cos OP sin A
sin 9
sin
&
and in the same way from the first and third equation:
, . A} A" A} A" A . cos OP sin A ,
U=2sm  d<f> 2 cos cos<jprt(/y) r^du
sin ~
From these two equations we obtain, eliminating first
rf (A and then dy:
A +A" A + A"
cos (f sin yi . cos  cos gp sm A cos
2 sin  sin
z z 22
cos p sin A" cos
. ^" A . 4"
2 sm sm
and:
sm ^1 . sm sin .4 sin
2
. A A.A A"
2 sin sm
sm ^ sin
,
sm sm 
We see from this, that the stars must be selected so,
that the differences of the azimuths of any two of them be
come as great as possible, and hence as nearly as possible equal
to 120, because in this case the denominators of the diffe
rential 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 8 h 28 in 17 s
Herculis 31 21 West of the Meridian
_ Arietis 47 30 East of the Meridian.
*) This solution of the problem was given by Gauss in Zach s Monat
liche Correspondenz Band XVIII pag. 277.
300
The places of the stars were on that day:
a Ursae minoris Qh 58 m 14* . 10 + 88 21 54". 3
Herculis 17 6 34 .26 14 36 2.0
Arietis 1 57 14 . 00 22 37 22 . 7.
Now we have:
M _ M = H3m 4s o " M = f. 19m 13s. o
or expressed in sidereal time:
M _ M = l O h 3 m 4s. 50 H()h 19 16*. 16
= 7 51 39 .84 " = hO 58 59 .90
A = 7h 54m 44s . 34 ;/ _ QI> 39 43 . 74
= 118 41 5". 10 = 9055 56". 10.
Then we have:
(# ) = 36" 52 56". 15
i (8 + 8 ) = 51 28 58 .15
i (S 8") = 32 52 15.80
( + ") = 55 29 38.50.
and from this we obtain:
log A = 0. 1183684 log 4" = 0.1629829
B = 60 48 11". 92 B" = 5 16 52". 22
C =120 844.47 C" =10 1450.27
.J (C" H C") = 54 56^ 57". 10
i(C" C" )= 65 11 47 .37
g== 47 56 16 .08
t = 56 18 28". 09
= 3 h 45 13s. 87
t + C = 63 50 16". 38
<HC" = 66 33 18 .36
and the formulae (/?) and (D) give the same value of y :
y = 30 4 23". 72.
From we find the sidereal time:
<9 = 21h 13m o. 23,
and since the sidereal time at mean noon was 12 h 54 m 2 s . 04,
we find the mean time 8 h 17 m 36 8 .44, hence the error of the
chronometer :
A M = 10 40 S .56.
Computing h from one of the three equations (a) we get:
h = 30 58 14". 44,
and for the other two hour angles we find:
= 62 22 37". 01
*= 66 14 24 . 19.
We then are able to compute the three azimuths:
301
A ==181 35 . 2
A = 89 33 .2
.4"= 279 50 .4;
and finally the three differential equations:
d<f= . 329 da 5 . 739 du G . 068 J",
rf(An) = 0.0018 du f . 468 du . 396 du",
where dy is expressed in seconds of arc, whilst t/(/\w) and
du, du\ du" are expressed in seconds of time.
20. Cagnoli has given in his Trigonometry another so
lution, not of the problem we have here under consideration,
but of a similar one. His formulae can be immediately ap
plied to this case, and if it is required, to find the altitude
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
" s " according to Gauss s or Na
pier s formulae, denoting the
parallactic angle by pi
and:
tang % (<JP h h) = V cotang (45
tang J (y> h) = S ] ? tang (45 4 8)
sin 2 ( t f jJ
sin (tp)
cotang (45
sin ] ( t H />)
But in the triangles PSS , PS S" and PSS" we have also
according to Napier s formulae, putting for the sake of brevity
A =1[PS"S PS S"]
A = [PS"S PSS"]
A"=Ji[PS S PSS ]:
tang A =
cos
(B)
302
where /, and // have the same signification as before. Now
since we have:
= p
p +PS S"=PS"S p"
we easily find, that: P = A iA"A
p = A 4 A" A (C)
p"= A 4 A A".
But we also have:
sin t : sin p = cos h : cos cp
sin U4A) : sinp = cos h : cos 9?,
hence :
sin t : sin UfA) = sin 79 : sin>
or:
sin * 4 sin (t + A) __ sin [A 1 4 A" A] + sin [A H A" A ]
Tin"* sin (t +Tf ~~ sin [A f A" A] sin [A h A" A ]
From this follows:
tang [t H 4 A] cotang ^ A = tang .4" cotang (A A )
or substituting for tang A" its value taken from the equa
tions (): sin(S 8)
tang [* H 4 A] = ! cotang U  A ). , (Z
Therefore we first find from the equations (#) the values
of A, yl and A", then we find p and by means of the equa
tions (C) and (D), and then </ and h by means of the equa
tions (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 lie, all being found by tangents.
However it is indifferent whether we take the angles 180
wrong, only we must then take 180 + 1 instead of f, if we
should find for (p and h such values , that cos <f and sin h
have oppositive signs. Likewise if we find for ff 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 ff 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:
,U= 59 20 32". 55
; = 4 57 58 .05
^ (8" ) = 4 O r 40". 35 i (8" S) = 32 52 15 . . 80
; ] ( _) = _ 36 52 56". 15
35 ("})= 55^9 38 .50
= 51 2858 .15,
303
and from this we find:
4 = 2 2 1".33, ^ =84 49 4". 07, A"= 29 44 16". 52
A ^ ==86 51 5". 40
,fl^A= 3 2 4 .47
t = 56 1828 .08.
Then we find y 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:
tang i (pM) = * I y*fy tang (45 h { )
Now we have:
* = < + / = 62 22 37". 02
y = ^t +. ^" A = 243 24 38". 08,
therefore we find:
y,= 30 4 23". 73
A = 149 1 45 .58
or taking for h the supplement to 180:
h = 30 58 14 . 42,
which values almost entirely agree with those found in the
preceding No.
21. We can also find Cagrioli s formulae by an analyt
ical method. According to the fundamental formulae of spher
ical trigonometry w r e have for each of the three stars the
following three equations:
sin h = sin cp sin S j cos cp cos cos t \
cos h sin p = cos y> sin t (a)
cos A cos;? sin rp cos cos y> sin S cos t
sin h = sin <f sin # + cos 90 cos $ cos(i/i) i
cos h sinp = cosy sin (t \r V)  (6)
cos A cos /; = sin 9? cos S cos y sin cos
sin A = sin cp sin ^"4 cos <p cosS" c
cos A sin// cosy sin (< + A ) (c)
cos A cos// = sin gp cos J" cos 9? sin " cos (*HA ) *
If we subtract the first of the equations (6) from the
first of the equations (a) and introduce J (*> f #) f (d <V)
instead of #, and i( ( > 4>^) _J. ( f y <) ) instead of <) , we find
the equation (rr) in No. 19. By a similar process we deduce
from the third of the equations (a) and (6):
304
cos h sin ^ (/> +/>) sin 5 (// p) = sin <f sin \ (8 \8) sin I (8 8)
cos <p sin ^ (<? H<?) cos 4 (8 8) sin (*H A) sin /
h cosy cos ^(<? H?) sin K<? <?) cos(H^)cos4/,
and if we eliminate sin (f in this equation by means of the
equation (), multiplying the first by cos (<) r>), the latter
by smK/VhcT), we obtain:
cos h cos 4 ($ +#) sin ^ (p fp) sin 4(p /) = cos y> sin \ (8 S) cos (H^ A) cos ^ L (o?)
Now if we subtract the second equations (a) and (6),
we find:
cos h cos j (p \p) sin 4 (// />) = cos cp cos (^ + \ /I) sin 5 A,
and hence:
1 X I \ SI 11 K^  ^) Alt
tang J (/> h/>) = l/ cotang ^ / = tang ^ .
We can find similar formulae by combining the cor
responding equations (a) and (c) and (6) and (c), which we
can write down immediately on account of their symmetrical
form :
N siiU ("<?)
+p) = T cotang 4 / = tang A
sin (<?" S")
and tang 5 (/; +;? ;= ,"   cotang (/ /) =
COS^ (.O ~T"O j
If we add finally the second equations (a) and (6), we
find :
cos h sin \ (p ^p} cos ^ (/) p) = cos 9? sin (2 h ^ A) cos ^ A,
and from this in connection with (d) we obtain:
sin ^ (a 1 a)
tang (< H 4 A) = g r^ _{_) cotang f (p p),
where ^ (/ p) = A A .
When thus p and t for the first star are known, we can
compute cf and h by means of the formulae found before,
which were derived by Napier s formulae:
tang * dp H A) = ^r^ cotang (45  * <?)
tang *(?*) = tan ^ < 45  ^ ^
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 com
pute the hour angle of the star from its declination, its azi
muth and the latitude. If we take the observation, when the
star is on the meridian, it is not necessary to know the de
clination 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 (p tang H sin <f> cos t,
we obtain according to the third formula (11) in No. 9 of
the introduction:
cos hdA = sin A sin hdtp + cos cos p . dt.
If the star is on the meridian, we have:
sin A = 0, cos p = 1
and:
A = 90 yf
at least if the star is south of the zenith, hence we obtain:
dt = mr*) dA .
COS
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 clock
time of observation, we have the error of the clock equal to
a ^<, if the clock is a sidereal clock. But if a mean time
clock is used, we must convert the sidereal time of the cul
mination of the star, that is, its right ascension into mean
time. If we denote this by m, the error of the clock is
equal to m 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 ifdA be the deviation from
the meridian, the hour angles (*) a and & a which the
stars have at the times of the observations are also small
and equal to:
si 11(9^ <f)
* A4
cos o
, sin (y S )
and:  s , A A.
COS
Hence, since = u\^u^ we have the following two
equations :
sin 0/5 8)
a = u + A" ^* &A
cos o
and: = + ,i  **=> & A,
COS
from which we can find both &u and &A. If the instru
ment 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 in one of the above
equations is very large and besides has the opposite sign *).
Example. At the observatory at Bilk the following trans
its were observed with the transitinstrument, before it was
well adjusted:
a Aurigae 5 h 6 " 27 s . 72
ft Orionis 5 8 12 . 71.
Since the right ascensions of the stars were :
a Aurigae 5 h 5 ra 33 s .25 445 50 . 3
ft Orionis 57 17 .33  8 23 . 1
and the latitude is 51 12 . 5, we have the two equations:
_ 545 . 47 = A M _ 0.13433 A^
55 . 38 = A" 0.87178 &A,
from which we find:
A u = 54 s . 30
and :
*) 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 obser
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 consid
erable distance from the observer so that it is distinctly seen
in a telescope whose focus is adjusted for objects at an in
finite 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 dis
appearance 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 ob
servations, the acceleration of the fixed stars must be taken
into account, since the star disappears earlier every day by
O h 3 m 55 s .909 of mean time.
If the right ascension of the star changes, the time of
the disappearance of the star is changed by the same quan
tity, 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 dA as well as
d(p are in this case equal to zero:
dS = cos pdh
cos 8dt = sin pdh,
hence :
dS. tang/?
at , >
COS
where p denotes the parallactic angle.
Therefore if the change of the star s right ascension and
declination is A and A (5, the change of the sidereal time,
at which the star disappears, is:
, A A# tang p
15 15 cos<f
Olbers had found from other observations, that in 1800
Sept. 6 the star Coronae disappeared behind the vertical
wall of a distant spire, whose azimuth was 64 56 21". 4, at
20*
308
IP 23 m 18^.3 mean time, equal to 22 h 26 m 21 s . 78 sidereal time.
On Sept. 12 he observed the time of the disappearance of
the star 10"49 m 21 s . 0. Now since 6 x 3 in 55 s .909 is equal to
23 m 35 s .4, the star ought to have disappeared at 10 h 59 " 42 s . 9
mean time, hence the error of the clock on mean time was
equal to + 10 m 21 s . 9.
In 1801 Sept. 6 was:
Aa=5H42".0
and :
A<?= 13". 2,
and since we have:
^ = 37 31 
and :
^ = t2G 41 ,
we find:
. _
A co7 1 " J
hence the complete correction is + 53". 35 or 3 s . 56. There
fore in 1801 Sept. 6 the star d Coronae disappeared at 22 h 26 m
25 s . 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:
cotang A sin t = cos (p tang f sin cp cos t.
Differentiating this equation we find:
cos 8 cos p sin p ~
sin Adtp = cotang lid A \ .  dt f 7 7 do.
sin h sm h
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. Be
sides we must select a star which passes near the zenith of
the place, since then the coefficients of dA and dt are very
small, as we have:
cos S cos p = sin cp cos h h cos y 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 de
clination of the star produces the same error of the latitude,
since we have then sin p = 1 .
If we observe only one star, we must observe the azi
*) v. Zach, Monatliche Correspondent 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 sin t = cos y tang f sin <p cos t .
cotang A sin t = cos <p tang 8 {~ sin (f cos /, .
Multiplying the first equation by sin t\ the second by
sin , we find :
. sin (A A) . .
sin t sin t   ., = cos y tang d sin t tang o sin t J
sm A sin A
h sin (f sin (t 1 *)
or as:
cos 8 sin t = cos A sin A,
also:
cos A cos h sin (^ A) = cos 9? [cos 8 sin 5 sin sin 8 cos 5 sin t ]
h sin 9? sin (t t) cos 8 cos 8 . (&)
We will introduce now the following auxiliary quantities:
sin (8 + 8) sin % (t t~) = ?nsir\M
sin (8 8) cos 5 (< t} = m cos M
If we multiply the first of these equations by eosJ(f Hf),
the other by sin(f M) and subtract the second equation
from the first, we get:
m sin [^ (t \t) M] = sin 8 cos 8 sin t cos 8 sin 8 sin t .
But if we multiply the first equation by cos  (* f),
the second by sin  ( f), and subtract the first equation
from the second, we get:
m sin [ <) IT] = sin 8 cos # sin ( r).
Hence the equation (6) is transformed into the following:
cos A cos k sin (^4 ; A) = m cos 90 sin [\ (< + ifef]
m sin y sin [^ (i t) M] cotang 8.
If we assume now, that the two stars were observed
either at the same azimuth or at two azimuths, whose dif
ference is 180, we have in both cases sin (A A) = and
hence we find:
sin [jfr K) Jf]
tang ? = tang J,^^. (B ]
Therefore in this case it is not necessary to know the
azimuth itself, but we find the latitude by the times of ob
servation and by the declination of the star by means of the
formulae (A) and (5).
If the same star was observed both times, the formulae
become still more simple. For since we have in this case
^=90" according to the second formula (^4), we find:
310
* cos j (YM)
tang f = tang . _ R? _. . (C)
For the general case, that two stars have been observed
at two different azimuths, the differential equations are:
cos h dA = sin p d H cos 8 cos p dt sin h sin A d<p
cos h dA s mp dd + cos S cos p d t sin h s m A dy.
If we introduce here also the difference of the azimuths
and therefore multiply the first equation by cos ft , the other
by cos ft, and subtract them, we get :
cos h cos h d(A A) = cos h cos d cos pdt+ cos h cos S cos p dt
[sin h cos h sin A sin h cos h sin ^1] dy>
\ cos h sin p dS cos h sin pd8.
Now since dt = clu { d (&ii) and c?J = du + r/ (A M),
where du and C/M are the errors of observation and d(&u)
that of the error of the clock, we find, if we substitute these
values in place of dt and dt and take at the same time
4 =180 4 4*):
sin Ad<p cosy cosAd(&u) = 77,, ;>. [d(A ^4) sin cpd(u u)j
sin. \/i r~ fi)
cos (p cos A sin h cos h cos (p cos A sin h cos h ,
^^nr~ ~ii^q^r~
sin /? cos A , sin p cos A _
~ sin (A H A)
Hence we see again that it is best to make the obser
vations on the prime vertical. For then the coefficient of
dcp is at a maximum and those of the errors du, du 1 and
d(u) 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 180", will have any effect upon the re
sult. In case that the same star was observed on the prime
vertical in the east and west, we have ft = ft and sin /? == sin/?,
hence :
h [d(A A) siny>d(u M)] H , d8 t
sin fi
*) In order to find the equation given above, we must also substitute
for cos S cos p and cos 8 cos p the following expressions :
cos d cosp = sin tp cos h H cosy sin h cos A
cos cosp = sin y> cosh cosy sin h cos A,
311
and since according to No. 26 of the first section:
we have:
sin cos fp
sm h = . and sin p =
sm fp cos o
dy> \ cotang h [d(A A) sin <p d(u 11) } f . ^ d &
We see again from this equation, that it is best to ob
serve stars, which pass near the zenith, because then cotang h
is very large and hence errors in A A and u u have
only very little influence upon the result. In this case the
coefficient of d d is equal to 1, since the declination of stars
passing through the zenith is equal to cp, 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 ft 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 34 m 43 8 .5
hence:
{(t t) = 4 20 26". 25
and it was
^ = 52 25 26". 77.
Now since in case that the observations are taken on
the prime vertical we have (Yf) = 0, we mic ^ from ()
the following simple formula for finding the latitude:
and by means of this we obtain:
y, = 5230 13".04.
Finally the differential equation is:
dcf = h 0.02310 [d(A A) 0.7934 d(u u)} 4 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 [i ( +  M] = sin [4 (t 1  t)  M], (A}
where :
t, = u f AW
and
m sin If = sin (d f <?) sin ^ (*
m cos M = sin ($ $) cos ^ (* t).
Since t t , that is , half the interval of time between
the observations, expressed in sidereal time, is known, we
can find J M 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 dcp is at
a minimum, that of dt at a maximum.
The azimuth itself can also be found by such obser
vations. For we have:
cos S sin t
tang A  . 5 * 
cos <f sin o f sm y> cos o cos t
and making use of the equation :
we find:
_ __ _sinjj3in [4 OjO _
"sin ft (?  If] ""
If we write here
^ + M < instead of ^ (i M,
we easily obtain:
sin (f
If the time of both observations is the same or:
t t = a,
the formula (.4) gives the time, at which two stars are on
the same vertical circle.
The places of Lyrae and a Aquilae are for the be
ginning of the year 1849:
a Lyrae a = 18 h 31 47* . 75 S + 38 38 52". 2
ft Aquilae 19 43 23 ,43 8 =+ 8 28 30 .5.
313
Therefore we have:
t t = I 1 1 l m 35* . 68 = 17 53 55". 2.
If we take then f/> = 52 30 16", we find:
3/=19255 53".0
4 ( ^=158 7 0.4
and from this we get :
\ (t 1 + M= 142 35 38" . 6,
hence :
.1 (* M) = 24 28 28". 4
= 1> 37n53 .9
and
* = l h 2 m 6 s . 1 , * = 2 h 13 m 41 s . 7.
Therefore the sidereal time at which the two stars are
on the same vertical circle is:
Hence if we observe the clocktime when two stars are
on the same vertical circle, if for instance we. observe the clock
time when two stars are bisected by a plumbline, 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 polestar, 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 arc of the equator between the meridians
passing through the two places and hence equal to their dif
ference 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 lon
gitude from Paris is 20 or 1" 20 m West.
In order to obtain the local times which exist simulta
neously on two meridians, either artificial signals are ob
served or such heavenly phenomena as are seen at the same
moment from all places. Such phenomena are first the eclip
ses of the moon. For since the moon at the time of an
eclipse enters the cone of the shadow of the earth, the be
ginning and the end of an eclipse as well as the obscura
tions of different spots are seen from all places on the earth
simultaneously, because the time in which the light traverses
the semidiameter 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 begin
ning 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 disap
pearance of shooting stars for this purpose. These can be
315
observed with great accuracy, but since it is not known be
forehand, when and in what region of the heavens a shoot
ing 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 obtained by observing artifi
cial 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 / shall be found, and
let An AM A 3 etc. be other places, lying between those pla
ces, whose unknown differences of longitude shall be / n A 2 , / 3 etc.
so that /! is the difference of longitude between A l and J,
/ 2 that between A z and A l etc. If then signals are given at
the stations 4,, A a , A b etc. at the local times / T , f 3 , /, etc.,
the signal from A is seen at the place A at the time
t l /! = 0, and at the station A^ at the time t l + I, = fc^.
Further the signal given from A. t is seen at the station A^
at the time t 3 / 3 = 6> 2 , and at the station A 4 at the time
^3 f I* = &* But since the difference of longitude of the
places A and B is equal to / f ^ + . . . + /, if the last sig
nal station is A H .\, or since:
/== (0, 0} 4 (6> 3 a ) H (6> 5 4 ) etc.,
we find:
/= 0,, 1 (& 2 0,, a) . . . (6> 2 (9, )
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 sud
denly.
The difference 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 /\u and the daily rate
be denoted by   ", then the error after a days will be
j\u{a u . Now if after a days the error of the chrono
meter at the other place should be found equal to /\ M ? we
have, denoting the longitude of the second place east of the
first by I:
n I h A M H d  d ^ U u = u h AM ,
hence
,= A ,+^  A ..
It is assumed here that the chronometer has kept a uni
form 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 in
stance that of Greenwich and that of Pulkova has been de
termined. 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 me
thods 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 electromagnet 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, the 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 sec
onds of the clock, and the marks corresponding to the same
second in different minutes will also lie in a straight line per
pendicular 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
secondhand of the clock reaches the zerosecond of a certain
minute, the first secondmark 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 breakkey 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 secondmark.
If the current goes to another observatory, whose lon
gitude 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 instru
ments 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 dis
tance, 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 velo
city 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 sta
tion A, being the farthest east, arrives at the station B.
Hence the time of the signal recorded at the station B cor
responds to a time, when the star was already on the me
dian of a place lying west of A, and the difference of longi
tude 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 time
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 (sub
tracting the result obtained at the station B from that ob
tained 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 chrono
graphs 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 clocktimes 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 com
parison probably surpasses the accuracy with which the er
rors of the clocks can be obtained from observations. Cer
tainly 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 be
ginning 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 ob
servatory and the errors of the clocks determined by each ob
server. 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 per
ceive 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 re
corded by the electro magnetic method. In this case the
error depends upon the time, which elapses between the mo
ment, when the eye of the observer receives an impression
and the moment, at which he becomes conscious of this im
pression 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 dif
ference of the two results is then equal to twice the per
sonal 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 many wires, so that one ob
server 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 reduced 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 per
sonal 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 ob
tained and the arithmetical 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 per
sonal 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 ob
serves 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 ~f a must be
added to it.
Example. On the 29 th of June 1861 the difference of
longitude was determined between Ann Arbor in the State
O
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,) 13 59 m 3s.0 Clinton clocktimc=19 b 58 29s .56 A. A. clockt.
(recorded at Cl.) 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 13 h 59 m 3 s .O equal to 4 6" 33 " 46 s . 07, while the error of
the clock at Ann Arbor on local sidereal time was f l m 1 s . 87.
From the records by the chronograph at Ann Arbor we find
therefore :
20 h 32>M9s.07 Cl. sidereal time = 19 h 59 " 31 .43 A. A. sidereal time
and by the chronograph at Clinton:
20 h 32 " 49s. 07 ci. sidereal time = 19 h 59 31 s . 27 A. A. sidereal time.
Hence we find the difference of longitude by the records
at Ann Arbor equal to
33 m 17s.64,
and by those at Clinton:
33 M7s.SO,
or the mean 33 rn 17 s . 72.
The personal equation is in this case E W = f s . 04 *),
hence the corrected difference of longitude is 33 m 17 s .76.
Note. The electro magnetic method for finding the diffeience of lon
gitude is usually called the American method, since it was proposed by Ame
ricans. 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 Cin
cinnati 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 sig
nals, 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 different places, yet can be re
duced 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 same instant
from different places on the surface of the earth at different
places on the celestial sphere, and hence the occultations as
well as the other phenomena mentioned before are not si
multaneous for different places. Hence in this case the ob
servations 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 parallaxes in longitude
and latitude and the apparent semidiameters of the heavenly
bodies at the time of the beginning and the end of the eclipse
or occupation (or the parallax in right ascension and decli
nation, if it should be preferable to use these coordinates).
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 semidiameters, 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 ex
pressed 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 cor
rect. 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 find
ing the difference of longitude by eclipses. At present a dif
ferent 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 geocen
tric 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 con
tact, when the eye is anywhere in the curved surface envel
oping 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 gene
rated by a right angled triangle revolving about one of its
sides, the equation of its surface is:
ar a y 2 = ( c zY tang/ 2 ,
where c is the distance of the vertex from the fundamental
21*
324
plane of the coordinates, 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 coordinates ar, ?/, z
the coordinates of a place on the surface of the earth, re
ferred to the same system of axes, we obtain the fundamen
tal 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 de
clination of that point, in which the centre of the more dis
tant 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 a, d and A be the geocentric right as
cension, declination and distance of the nearer body and
ce i <5 ? A the same quantities for the more distant body, we
have the equations:
G cos d cos a = A cos S cos ft A cos cos #
G cos d sin a = A cos 8 sin A cos S sin ft
sin</=A sin<? A sin <?,
or:
G cos d cos (a a ) = A cos A cos S cos (a )
G cos d sin (a ) = A cos S sin ( )
G sin d = A sin 8 A sin S.
If we take as unit the equatoreal semi diameter of the
earth, we must take  . and instead of A and A, since
sin n sin n
A and A are expressed in parts of the semi major axis of
the earth s orbit, where n is the mean horizontal equatoreal
parallax of the nearer body, n the same for the more dis
tant body; thus w r e obtain:
sin n G cos d cos (a ) = A  cos 8 cos 8 cos (a )
sin n
sin n G cos d sin (a ) = cos 8 sin ( )
. . sin 7t , ,
sin n G sm d = A , sin o sin d.
sin n
Now since we also have :
sin n G cos d = A  f cos 8 cos (a ) cos 8 cos (a ),
sin TF *
325
we find:
sin TC cos
, , sin (ft )
, ,. A SHITT cos d
tang ) = r 5
sin TT cos d
1 771 s? cos (ft a )
A smTT cos o
and: sin n
TJ. sin (o S )
, . c, /N A smn
tang (r/ ) =  
1 .. . cos (()
A
Since in the case of an eclipse of the sun   is a
small quantity, we obtain from this by means of the for
mula (12) in No. 11 of the introduction:
, sin TC cos S
a a . (a )
A S1117T COS . ,
; \A)
and putting: ff = s }
we also find : a = 1 s , in , rm
A sin??
We will imagine now a rectangular system of axes of
coordinates, 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 = & cos S cos (ft ), y = Asin(9, x = A cos S sin (a a).
If now we imagine the axes of y and z to be turned in
the plane of yz 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 coordinates of the
nearer body with respect to the new system of axes:
sin # sin rf + cos 8 cos d cos (a a)
sin n
sin S cos d cos sin d cos (a a)
sin n
cos 8 sin (a a)
sin 7t
*) 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 cos H d) sin (
sin n
sin (fl cQcosi( g) a (sin (j+d)sin^ ( a) 2
_ cos $ sin (a a)
sin TT
The axis of * 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 coordinates x and y will be the coordi
nates of the centre of the earth with respect to the new
origin but taken negative.
Let (f be the geocentric latitude of a place on the sur
face of the earth, its sidereal time and y its distance from
the centre, then the coordinates of this place, taking the
origin at the centre of the earth and the axis of parallel
to the line joining the centres of the two bodies, are:
== C [ g i n d sin <p f cos d cos y cos (0 a)]
*? = (* [ cos d sin tp sin d cos y> cos (0 a)] (Z>)
f C cos 95 sin (0 a).
The coordinates of this place with respect to a system
of axes, whose axis of z is the line joining the two centres
itself, are:
 x, rjy and
and the equation, which expresses, that the place on the sur
face of the earth, given by o, f/ and 6), lies in the surface
of the cone enveloping the two bodies, is:
(x  I) 2 f (y  nY = (c  )" tang/ 2 ,
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:
r =t= r
sin/== ~  ,
Or
where r and r are the semidiameters 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 semidiameter of the moon expressed in parts of
the semidiameter of the equator of the earth and h the ap
327
parent semidiameter of the sun seen at a distance equal to
the semimajor axis of the earth s orbit, we. have, since:
also:
, sin
sin / = r [sin h =t= k sin n }
(JT sm n
or:
sin/= [sin h == k sin n ]. (JE)
A 9
But we have:
log sin n = 5. 6186145,
further we have according to Burkhardt s Lunar Tables
& = 0.2725 and according to Bessel h = 15 59". 788, hence
we have:
log [sin h f k sin 7t ] = 7. 6688041 for exterior contacts,
log [sin h k sin n 1 } = 1 . 6666903 for interior contacts.
We must still express the quantity c, that is, the dis
tance of the vertex of the cone from the plane of xy. But
we easily see, that:
where again the upper sign is used for an exterior, the lower
one for an interior contact. If we then denote by / the
quantity c tang /", that is , the radius of the circle in which
the plane of xy intersects the cone, and tang f by /L, the ge
neral equation for eclipses, which expresses, that the place
on the surface of the earth given by q>\ & and o, lies in the
surface of the cone enveloping both bodies, is as follows :
(x) 2 f( < y7 7 2 ) = (Z^) 2 .
Since / is always positive, we must take tang f or /I
negative, if we find a negative value of c from the equa
tion (F).
The values of the quantities used for computing ic, ?/, z
and , 77, 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 values of x, y etc.
will also differ a little from the true values. Therefore if
A#, A^ an( i A^ are the corrections, which must be applied
328
to the computed values x, y and / in order to obtain the
true values, the above equation is transformed into *) :
(x H A* I)* + (y 4 fry T/) 2 = (I } AZ 1) 2 .
We will assume now, that the values of , , TT, , d
and TI 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 Tf T , we have, denoting by x n 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<> 4 x T and y=y +y T .
In the same way the quantities , r] and J 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:
[x  I + x T + Ar] 2 H [y,  rj f y T + Ay] 2 = (I + A I  A).
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 Tf 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 ?/, we can
solve the equation by successive approximations.
If we put : x i y i> = A*
y i + x i = A#
and : m sin M=x a  n sin N=x }
mcosM=y rj ncosN y (G) i
l )l = L,
the above equation is transformed into:
(L + AO 2 = [m cos (M N} 4 n (T + OP + [m sin (M N] n i J a ,
and we obtain, neglecting the squares of i and /V 5 the fol
lowing equation of the second degree for T ft:
~ sin (M .V) i f 
n n
*) Errors in a, d and k are here neglected, since they cannot be de
termined by the observations of eclipses.
329
Now since :
putting :
L sin y = ?sin(X N\ (//)
we find from this equation:
m L cos yj &l
T = cos (J/ iV) =p i =P tang y ?" =p sec y>,
or except in case that \jj is very small:
m sm(MN==v>) A I
jT =  z =p tang v z =p sec i/>.
n sin \i) n
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 be
ginning, the lower sign for the end of the eclipse or any
phase, if we take the angle /> always in the first or fourth
quadrant *). But if we take ifr 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 :
wsn iv
1 = ? ? tang w sec i/>
11 sin y n
or:
Tit m /*r AT\ L COS W ., A/ f 7N
r = cos (.If N) i ? tang u> sec w. (./)
n n n
The equation (J) is solved by successive approximations.
For this purpose compute the values of x, y, z, a, d, g, I and
/ by means of the formulae (4), (fi), (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 approxima
tely known value of the difference of longitude .will permit,
interpolate for this time the quantities a? , ?/, x and y and
find an approximate value of T by means of the formulae
(D), (6?), (#) and (J). With the value TH 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 + 2 V = t d, where
is the time of observation and d is the longitude of the place
*) We find this easily from the first expression for T ,
330
reckoned from the first meridian, that is, that meridian, for
which the quantities a?, i/, z etc. have been computed, and
taken positive when the place is east of the first meridian.
Hence we have:
d = t T H  cos (M N) \  cos w f i 4 i tang w \  sec W
n n n
TO sin (M N+y) A/ W
= t Ti~ i 1 : + i tang v H sec w.
n sin y n
Since the values of x and y have one mean 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 con
tained in one hour of that species of time, in which the ob
servations 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 (/if) 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 ob
served. By the combination of these equations we can eli
minate, 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 , de
termined by the equations :
or:
ni = sin
ni = sin
The quantities x and y depend upon a cf, d d and n.
Therefore if we suppose these quantities to be erroneous,
we have :
A x = A A ( ) h B A ( S d) h C A n
A y = A & (a a) 4 B b(8d)+ C &Tt,
where A, B, C are the differential coefficients of x with re
331
sped to a, d d and TT, and A , # , C those of y with
respect to the same quantities. Now since A( ), A(<* d)
and A 7 ? are always small quantities, we can neglect in the
expressions for the differential coefficients the terms contain
ing sin (a a) and sin (<) d) as factors, and can write 1 in
place of cos (a a) and cos (JS rf). Then we obtain:
cos S cos
A =  cos (a a) =
sin 7i sin n
_ sin 8 sin (a a) _
sin n
_ cos S sin (a a) cos n ^x
C  ; r =
sin 7i tang n
cos 8 sin d sin ( a)
A=\ =
sin TT
D , cos (8 d) 1
jD =     =
sin n sin TC
Now since i and t , and hence also A( )? A(^ 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:
20G265 . n sin n
we have:
i Asin2v~cos<*A( ) H h cosJVA (S d} hcosn&Ti [x sinN+ycosN]
i h cos NCOS S&(a a)tAsiniVA(<? d) +h COSJC^TT [>coszV y sin A ],
or multiplying the upper equation by cos?/ , the lower one
by sin \\) and adding them :
i fi tangy] = sin (N y;) cos & (a a) f cos (^V ^) A (S d)
cosn&Tt[x sin (2V y/) \y cos (2V y;)].
From this we obtain:
* sin ( M ^+ v) , , sin (^ y) A ,
6 sin y, ~ + h ~ CO s y> COS *A (  )
+ A cosJ2V y ,) M ^_
cos y
M   206265 sin
cos j
332
or putting:
= sin JVcos <?A (a a) H cos 2V A (S d)
= cos 2V cos S A ( a) f sin 2V A (8 d)
^ = 2062 65 sin n A/ ()
(9 = cos n &7t
_ x sin (2V y;) f y cos (2V y>)
cos y
we finally have:
. (Af)
Now T the observation of every phase of an eclipse gives
such an equation and since this contains five unknown quan
tities, five such equations will be sufficient to find them.
However the quantities ?; and cannot be determined in this
way, unless the observations are made at places which are
at a great distance from each other. Nevertheless the com
putation of the coefficients will show us the effect, which
errors of n and I can have upon the .result. Generally it
will only be practicable to free the difference of longitude
from the errors of and , but the latter quantity can only
be determined, if the longitude of one place from the first
meridian is already known. When s and are known, the
errors of the tables are obtained by means of the equations :
cos S A ( ) = sin 2V cos 2V
A (S d) = E cos 2V 7 + sin 2V.
If we collect all the formulae necessary for computing
the difference of longitude from an eclipse of the sun, they
are as follows:
sin 7t cos S .
a = a j, =, (a ) 
A SinTT COS
= "
_
Asinw (
sin n
where , d and n are the right ascension, declination and
horizontal equatoreal parallax of the moon, , r) r , A an( i ^
the right ascension, declination, distance and mean horizontal
equatoreal parallax of the sun.
333
cos S sin (a a)
sin n
sin (S </)cosr(a a) 2 f sin (S\d) sin A (a i*/ v , n ,
y =     ) (2)
SlllTT
cos(^ ef) cos I (a a) ><! cos(S\d~) sin}( a) 2
2 =
sm TT
sin /= j r [sin A =p A; sin TT ], (3)
A 9
where :
log [sin A f fc sin TT ] = 7 . 6588041
for exterior contacts and
log [sin A k sin ?r J = 7 . 6666903
for interior contacts.
c = * A., (4)
sm/
where the upper sign is used for exterior contacts, the lower
for interior contacts.
,
=c.l,
where I has always the same sign as c.
I; = (> cos 90 sin (6> a)
77 = (> [cos rf sin 9? sin d cos 9? cos (<9 a)] (6)
=== ^ [ sm f ^ sm 9 s H~ cos ^ cos 9 cos (^ a )J
where (f and (> are the geocentric latitude and the distance
of the place from the centre and is the observed sidereal
time of a phase.
If then we have for the time T:
dx .
we compute :
m sinM=x  wsin^V=o:
Itf AT I I  Ag = l> (7)
m cos M =y ij ncosN=y
L sin y = m sin (M N) , (8)
where for the beginning i/j must be taken in the first or fourth
quadrant and for the end in the second or third quadrant,
and:
r =  . : = _ . cos _
n sin i/j n n
Finally we have:
d=t T T + AeHA^tangy, (10)
334
where :
206265. n sin TT
E = sin N cos 8 A ( ) 4 cos N &(S d\
= cos 2V cos 5 A ( ) + sin ^V^ (8 c/),
hence :
cos $ A ( ct) = s sin iV cos iV
A (5 rf) = e cos .Vt ^ sin N.
Example. In 1842 July 7 an eclipse of the sun occur
red, which was observed at Vienna and Pulkova as follows:
Vienna :
Beginning of the total eclipse 18 h 49 n 25 s .O Vienna mean time
End of the total eclipse 18 51 22 .
Pulkova:
Beginning of the eclipse 19 h 7 m 3 s . 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 S
a
S
17h 105 8
49".93
42322 10".35
106 50 38
.49 4 22 33
24"
.46
18 47
43.31
15
.34
53 12
.37
33
7
.93
19" 106 26
34.14
7
40
.45
5546
.24
32
51
.36
20 h 107 5
22 .32
10
.75
5820
.09
32
34
.75
21h 44
7 .75
22 5
>
31
.29
107 53
.94
32
is
.09
22h 108 2250.34
44
42
.13
327
.78
32
1
.40
n
log A
17h
59 55"
06
.0072061
IS"
56
37
56
19h
57
65
51
20 h
58
91
46
21h
60
14
41
22 h
1
35
36.
Z^" 1 . OJ OD.
If we compute first the quantities a, d and g by means
of the formulae (1) we find:
a d log g
18 106 53 21". 53 4 22 33 2". 04 9.9989808
19" 55 50 .33 32 46 .47 11
20 h 58 19 . 10 32 30 .87 15
21h 107 47 .88 32 15 .25 19.
Then we find by means of the formulae (2), (3), (4)
and (5):
335
X
17"  1 . 5632144
y
H . 8246864
logs
1 . 7585349
18h 1.0061154
f . 7039354
1 . 7584833
19 h 0.4489341
h . 5827957
1 . 7583923
20 10. 1082514
1 . 4612784
1.7582614
21 f 0.6653785
1 . 3393985
1 . 7580909
22h t 1 . 2224009
+ 0.2171603
1 . 7578799.
17h 0.5362314
. 0100548
7 . 6626222
18h . 5362001
. 0100860
23
19 h 0.5361450
. 0101409
25
20 . 5360655
. 0102198
26
21 h . 5359622
. 0103227
27
22 . 5358345
0.0104499
29
i log;.
Exterior contact. Interior contact. Exterior contact. Interior contact.
7 . 6605084,,
85
87
88
89
91.
Now the time of the beginning of the total eclipse was
observed at Vienna at:
18M9 m 258.0,
or at the sidereal time:
0= lh 52m 29. 8 = 28 7 27".0;
Further we have:
^,==48 12 35". 5,
hence the geocentric latitude:
^ = 48 1 S".9
and:
log? = 9. 999 1952.
If we take T= 18 h 30 11 , we find for this time:
x = 0.727530 # = 4 0.643413,
and by means of the formulae (6):
!= 0.654897 r/ = h . 635482 log g = 9.606857;
moreover by means of the formulae in No. 15 of the intro
duction :
x = H 0.557185 / = 0.121140,
hence by means of the formulae (7), (8) and (9) :
M = 276 13 54" log m = 8 . 863708
^=102 1558 log n = 9. 756030
y; = 39 57 10"
T = 6 40* . 85,
Since in this case it is not necessary to repeat the com
putation, we obtain by means of the formula (10) :
d = + Oh 12 " 44s . 15 H 1 . 7553 e f 1 .4703 .
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. G53763 TI = + . 633338 log = 9 .612367
If =277 46 40" log m = 8. 87 1874 logL= 8. 078638
^=150" 54 51 ".5
T = 8">54".74,
hence :
d = + O h 12 n 27s . 26 H 1 . 7553 s . 9764 .
Likewise from the observations at Pulkova, since:
5^ = 59 46 18". 6,
and hence:
9) = 59 36 16". 8
and:
log o = 9. 9989172
we find the following equations:
d = lh 8 " 26 .57 + 1 .7559 e + 0.5064 ,
d f = 1 8 22 . 67 h 1 . 7541 e 0. 3034 .
We have therefore:
d d = h 55 " 42^ . 42 . 9639 ,
<? <* = + 55 55 .41+0. 6730 ,
hence:
d d= + 55 m 50 8 .07
and:
= 7". 94.
In order to find the error e, we must assume the lon
gitude of one place reckoned from the meridian of Berlin as
known. But the difference of longitude of Vienna and Ber
lin is :
+ h Il n 56.40
and with this we obtain from the first equation for d:
= 20" . 55.
Since we have:
cos S A (a a) = t sin .ZV cos N
&((t) = scosNl sin N,
we find:
cosd(a a) = 21". 78
and:
d) = 3".38.
30. In the case of occupations of stars by the moon
the formulae become more simple. Since then n = , we
have a = , d = d . Hence we need not compute the for
mulae (1), and the coordinates of the place of observation
337
are independent of the place of the moon, since we have
simply :
 = (> cos tp sin (0 )
77 = Q [sin y> cos cos cp sin 8 cos (& )].
The third coordinate is also not used, since we have
in this case fQ and hence A = 0, so that we have instead
of the enveloping cone a cylinder. The radius / of the circle,
in which the plane of the coordinates intersects this cylin
der, is equal to the semidiameter of the moon or equal to k.
Hence we need not compute the coordinate z and we have
simply :
cos 8 sin ( a )
sin S cos 8 cos 8 sin 8 cos (a )
_
sin 7i
Thus the fundamental equation for eclipses is transformed
into the following:
(fc + A / ) 2 = (x 4 A x  ) a 4 (y t \y  i?) a ,
which is solved in the "same way as before. Taking again
t d=T\T and denoting by x lt and y the values of a;
and ?/ for the time 7 , by x and ?/ their difierential coeffi
cients, we must compute the auxiliary quantities:
in sin M= x  n sin jV= x
mcosM*=y, 77 ncosN=i/
k sin y^ = m sin (J/* iV)
and we find:
, m sin (J/ (
^Z = t / H  s  H A H A C tang v>
w sin y
where ft, and J 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 8 h 15 m 12 s . 1 Bilk mean time
Emersion i) 18 10.8.
The immersion of the same star was observed at Ham
burg at
8 h 33 m 47 . 2 Hamburg mean time.
The place of the star on that day was according to the
Nautical Almanac:
= 4h 11". 16s . 24 = 62 49 3". 6
= + 15 15 32". 2.
22
338
Further we have for Bilk:
9? = 51 1 10".0
log == 9.999 1201
and for Hamburg:
^ = 5322 4".2
log Q = 9.9990624.
Finally we have the following places of the moon ac
cording to the Nautical Almanac:
a n
7" 4 1 6" 1 2 . 35 H 15 47 24". G 60 50". 8
S 4 8 35 . 69 15 54 48 . 8 60 51 . 8
9 h 4 11 9 .31 16 2 6 .5 60 52 .9.
Hence we find for those three times:
x I. Diff. y I. Diff.
7h 1.240980 nrnr ~ 9 + 0.527577
8" 0.634228 +0.646318 *
9b 0.027364 +0.764974
Now we have for the time of the immersion at Bilk:
<9 = h 49 29. 93
a = 50 26 34". 6
hence :
I = 0.484015 and rj = \ 0. 643216.
Taking then T=7 h 50 m , we obtain for this time:
TO != 0.251346 yo 77 = 0.016682
x = + . 606789 / = j . 118713,
hence :
J/=266 12 .10" ^= + 78 55 50"
logm= 9.401226 log n = 9.791194
^ = 6 43 11"
T = h 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 s and :
d = h 27 12s . 95 h 1 . 5945 _ Q . 1879 ,
and in the same way we find from the emersion observed
at Bilk : d = H 27 27 . 10 + 1 . 5937 e + . 5336 ^,
and from the emersion observed at Hamburg:
d = + 40 3 . 76 H I . 5945 e 0\. 1362 g.
We have therefore the two equations:
d d= + 12" 50s . 81 I . 0517 ,
d rf = {12 36.66 0.6698^,
whence we find:
d _ rf=H 12m 49s. 80 and = 19". 61.
339
31. The fundamental equations for eclipses and occul
tations 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 a? , ?/ , x\ y
and L, the fundamental equation for eclipses is:
[*o i * T  J a H [y + y T 1 ri*=L* *),
where and i] are the coordinates of the place on the earth
at the time T\ T . Therefore if we denote by the side
real time corresponding to the time T, + d () will be the
local sidereal time of the place, for which we calculate the
eclipse, and if we denote by and v/ the values of and 77
corresponding to the time 6^ +d 05 we have:
 =  + Q cos y cosC^,  a h rf a ) T^ Z"
rj = rj Q j Q cos fp sin (6> fl
U J.
Therefore taking now:
m sin M= x  , n sin N=x (> cos y cos(0 a\d } ~r^r"~
m cosM=y ^ ? n cosN=y g cos y> sin (<9 atd () ) ,  sin d
d J.
sin y = sin (J/ JV),
where L denotes the value of L corresponding to the time T,
we find:
T = cos (M N) =p Z  cosw=tTd,
n n
where ijj must be taken in the first or fourth quadrant, and
the upper sign is used for the beginning, the lower for the
end of the eclipse, or if we take:
cos (M N)  cos w = T
n n
cos (M N} H L  cos w =T
n n
the time of the beginning expressed in local mean time is :
and the time of the end:
*) For an occultation we have L = k = . 2725.
22
340
By the first approximation we find the time of the eclipse
within a couple of minutes, therefore already sufficiently ac
curate for the convenience of observers. But if we wish to
find it more accurately, we must repeat the calculation, using
now T h r and T f 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 occupation),
where the contacts take place. But if we substitute in
aV tha?7" and y Q r]+yT
for T the value:
cos (M JV) =p cos w.
n n
we find:
x = [in sin Mcos NCOS jYsin y m cos M cos N sin Nsin y
=f= m sin M cos N sin N cos u> == m cos M sin N sin N cos w] 
or:
m sin (M N}
sm y
= =p L sin (N=f= y;)
and likewise:
y rj = =p L cos (N=f= y).
Hence we have for the beginning of the eclipse:
x  = L sin (N y/) = L sin (2V+ 180 y)
y n = Lcos (N v) = L cos (iVh 180 y),
and for the end:
x I = L sin (N } y;) v
^ rj = L cos (N\ y).
Sow we have seen in No. 29 that # and ;/ i/ are
the coordinates of a place on the earth situated in the en
veloping 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 i] are the coordinates 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   and ^  are the sine and cosine of the an^le,
L L
which the axis of y or the declination circle passing through
341
the point Z*) 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 appre
ciable error, that  and y n are the sine and the cosine
Lt lj
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:
AThlSO" y )
and for the end: J (A)
A T hy. )
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 6* the sidereal time corresponding
to the mean time T and by d n the longitude of the place
reckoned from the first meridian and taken positive when
east, we compute the formulae :
 = () cos ff sin (6> f d a)
r io Q [cos d sin y> sin d cos y cos (0 f d a)]
So C [ sin d sin y f cos d cos <f cos (0 f d a)].
Computing then the formulae:
m sin M=x Q 1 , n sin N=x (> cosy cos (0 Hd a)
dl,
y *?> ncosN=y ^cosy sin(<9 e? a ) ^ J sin d
dt
sin y = sin (M N) (y; always < == 90)
^o
r = cos (J/ JV)  cos v
n n
r =  cos (MN) + cos y,
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 :
and the time of the end:
;= T+d HT .
The expressions (A) give then the particular points on
the limb of the sun, where the contact takes place.
For calculating an occultation the formulae are as fol
lows. We compute again for the time T of the first meridian,
which is near the middle of the occultation:
cos 3 sin (a a )
_ sin S cos cos S sin cos (a a )
y ~ Bin* ~
and the differential coefficients x and y . Further we com
pute, denoting by the sidereal time corresponding to the
mean time T:
o == C cos T sn
r] = (> [sin 90 cos $ cos 90 sin cos(<9 a h r/ )].
Then we compute:
m sin M=x Q 1 , n sin N=x (>cos9p cos(0 +</ )
7 yQ
mcosM=y ?? , ncosN=y (> cosy sin (6> f(/ a )  sin ,
where :
log ~ = 9. 41016*)
sin ^ =  sin , y;<;==
/J
and:
log jfc = 9. 43537
m f ATN A:
 cos (M N)  COST/>=T
n n
 cos (M N) H  cos t^=T ;
*) As one hour is taken as the unit of the differential coefficients, 
at
is the change of the hour angle in one mean hour or in 3609 s . 86 of sidereal
time. If we multiply by 15 and divide by 206265 in order to express the
differential coefficient in parts of the radius, we find:
log = 9. 41916.
343
Then the immersion takes place at the local mean time:
t=T+
and the emersion at the time:
The angle of position of the particular point on the limb,
where the immersion takes place, is found from :
Q=r2VM80 y
whilst for the emersion we have :
Example. If we wish to calculate the time of the be
ginning and end of the eclipse of the sun in 1842 July 7
for Pulkova, we take T= 19 h Berlin mean time. For this
time we have according to No. 29:
.r = 0.44893, y n =40.58280, x = f 0.55718, / = 0.12133
a = 106 55 . 8, d=j22 32 . 8, 2=0.53614, log A = 7. 66262.
Then we have:
6> = 2 h 3" 1 8 s ,
and since the difference of longitude between Pulkova and
Berlin is equal to fl h 7 m 43 s , we get:
\d a = 300 46 . 9,
and with this:
I = 0.43361, ?= + 0.69560, log = 9.75470, log L H = 9.72716.
Further we find:
^ cosy cos (0 +d a) pL = H 0.06762 *)
f = /, cos y sin (6> + d, a) sin d = 0.04352,
at at
hence:
_ffli = + 0.48956 and y ^ = 0.07781.
*) We have:
^= 3609s. 86
dt
or:
= + 57147". 90;
Further we have:
= + 148" .78
hence:
d(0 a) _ 56999 ^ 12?
dt
the logarithm of which number expressed in parts of the radius is 9.41796.
344
Then we get:
J/=18744 . 1 JV=99"1 .9
log m = 9.05628 log n = 9.69522
v , = 12 19 .
hence:
T = 1.057 T = 1.046
= l h o .4 = hlh2n.8,
therefore the beginning and the end of the eclipse occur at
the times:
*=19h 4m. 3
These times differ only 3 m from the true times. If we
repeat the calculation, using 7 =18 h and T=20 h , 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 111 *).
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 Al
manac, 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 ob
tained by corresponding observations of eclipses. Besides the
*) Compare on the calculation of eclipses: Bessel, Ueber die Berechnung
der Lange aus Stern bedeck nngen. Astr. Nachr. No. 151 and 152, translated
in the Philosophical Magazine Vol. VIII and Bessel s Astronomische Unter
suchungen 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. There
fore 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 alti
tudes 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 pur
pose the longitude of the place, reckoned from the first me
ridian, must be approximately known, and then for the approx
imate time of the first meridian, corresponding to the time
of observation, the places of the moon and the other body
are taken from the ephemerides. Then the true altitudes are
computed by means of the formulae in No. 7 of the first
section, 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 ri
gorous formulae:
v
sin p = (> sin p sin [z (<p y> ) cos A]
/A
cos p = I (> sin p cos [s (<f> y>") cos A],
L\
and finally for the altitudes affected with parallax the re
fraction is found with regard to the indications of the me
teorological 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 tfie observed distance the sum of the
apparent semidiameters 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 semidiameter
of the moon, the semidiameter affected with parallax will be :
346
r = r [1 }/> sin Aj,
where p is the horizontal parallax expressed in parts of the
radius.
Now since refraction diminishes the vertical semi dia
meter of the disc, while it leaves the horizontal semidiame
ter unchanged, that in the direction of the measured distance
will be the radius vector of an ellipse, whose major and mi
nor 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 sec
tion, or it can be taken from tables which are given in all
Nautical works. If we denote by n the angle, Avhich the
vertical circle passing through the centre of the moon makes
with the direction towards the other body, by ti the altitude
of the latter and by A the distance between the two bodies,
we have:
sin (A A) cos ti
sin TI
sin A
and:
sin h cos A sin h
cos n = ,
sin A cos h
hence:
, __ cos 4 (A h h + h ) sin (A H A h }
~ s7nT(l4 ti  K) cos i (h hT A)
Then if we denote the vertical and the horizontal semi
diameter by b and a, we find by means of the equation of
the ellipse:
b
I/ cos 7t 2 H sii
r 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 bod
ies. For if we denote by /T, h and A the apparent alti
tudes 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 H sin h + cos H cos h 1 cos E
= cos (H h } 2 cos H cos h 1 sin 4 E* .
Likewise we have, denoting by #, h and A their true
altitudes and the true distance:
347
cos A = : sin Hsin h f cos Hcos h cos E
= cos (// A) 2 cos Hcos h sin ^
and if we eliminate 2 sin  E 2 we find :
cos A = cos (H A) f f [cos A  cos (JET  h )} (a)
cos
If we take now:
cos If cos h 1 , .v
cos // cos h! G
we shall have always C > 1 , except when the altitude of the
moon is great and the other body is very near the horizon.
If we then take:
H 1 h = d and Hh = d (B)
and take d and d positive, we can always put:
cos d ,,, . cos A .; /^,N
= cos d" and   = cos A (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
arc itself:
,, sii
)
If we take here at first sin  (A h A") instead of sin(AhA")
and put:
we obtain:
A=A"Har, (E)
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:
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, iiber 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 // the true azimuth and altitude, we
have to add to the geocentric azimuth the angle:
o sin p (cp  OP ) sin A
A A = f
cos a
in order to find the azimuth as seen from a place on the sur
face of the earth. Therefore in the formula for cos A we
ought to use cos (E A ^4) instead of cos E = cos (A 0),
or we ought to add to /\ the correction:
cos Hcos h sin {A a)
d A = dA
sm A
or:
o sin p (OP OP ) cos h sin ^ sin (A a)
a = : 7
sm A
Example. In 1831 June 2 at 23 h 8 m 45 s apparent time
the distance of the nearest limbs of the sun and the moon
was observed A = 96 47 10" a ^ a place, whose north lati
tude was 19 3V, while the longitude from Greenwich was
estimated at 8 h 50 m . 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. (( decl. ([ parallax
June 2 12 h 336 6 24".  10 50 58". 56 44".
IS" 38 4.7 41 48.4 45 .9
14h 337 9 45 . 7 32 35 . 47 . 9
15^ 41 27 . 23 17 . 9 49 . 9
right asc. decl.
June 2 12> 70 5 23". 2 f 22 11 48". 9
13 h 7 56 .9 12 8 .4
14" 10 30.5 12 27 .9
15 h 13 4 . 1 12 47 .3
The time of observation corresponds to 14 h 18 m 45 s Green
wich time and for this time we have:
right asc. d = 337 19 39". 6 right asc. = 70 IV 18". 5
decl. (C= 10 2941.3 decl. =H22 1233.9
p= 56 48 .5 TT= 8". 5.
From this we find the true altitude and azimuth of the
moon and the sun for the hour angles:
+ 80" 2 ,56". 8
349
and:  12 48 45". 0:
H== 5 41 58". 4 h = 77 43 56".7
A = h 76 43 . 6 a = 75 4 . 4.
The parallax of the moon computed by means of the
rigorous formula:
. sin p sin [z (a> > ) cos A]
tang/; == .  r  f ^ n
1 n sin p cos [z ((p (f ) cos A\
is // = 56 35".4, hence the apparent altitude // 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 indi
cations of the meteorological instruments. We then find
p = 9 3". 2 and hence the apparent altitude affected with re
fraction :
# = 4 054 96". 2.
For the sun we find in the same way:
A = 77 44 6". 5.
Further we find the semidiameter of the moon by mul
tiplying the horizontal parallax by 0.2725 and obtain:
/= 15 28". 8
and from this the apparent semi diameter, as increased by
parallax:
The vertical semi diameter is diminished 26". by the
refraction, and the angle n being 5 48 , the radius of the
moon in the direction towards the sun is :
r =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:
A = 97 18 1". 6.
Further we find by means of the formulae (4), (#) and (0) :
log C= 0.000463
J=72 1 5S"
of = 72 49 40
d" = 12 50 48
A" =97 17 33
and at last, computing x twice by means of the formulae (#)
and (E), we find the true distance of the centres of the sun
and the moon:
A = 96 30 39".
350
Now we find according to the Almanac the true dis
tance of the centres of the bodies for Greenwich apparent
time from the following table:
12h 97 43 0". 4
13h 13 4 . 5
14 h 96 43 6 . 5
15^ 13 6 .2,
whence we see , that the distance 96 30 39" corresponds to
the Greenwich apparent time 14 h 24 m 55 s . 2, and since the
time of observation was 23 h 8 m 45 s .O, the longitude of the
place is:
gh 43111 498 . 8 east of Greenwich.
The longitude which we find here is so nearly equal to
that, which was assumed, that the error which we made in
computing the place of the sun and moon 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 h 24 m 55 s Greenwich time.
Bessel has given in the Astronomische Nachrichten 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 right ascension of the moon changes in a certain
time, the longitude can be determined by observing the dif
ference of the sidereal times at the time of culmination of
the moon. Since these observations are made on the me
ridian, neither the parallax nor the refraction will have any
influence on the result. In order to render it also independ
ent 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.
351
A list of such stars is always published in the astronomical
almanacs, 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 meridian, and the differential coefficients
for the same time be ^, *, etc, We will then suppose,
that at a place whose longitude east of the first meridian
is d, the time of culmination of the moon was observed
at the local time TMtd?, corresponding to the time T\t
of the first meridian. Then the right ascension of the moon
at this time is:
da , d 2 a d 3 a
H * tSH T <* , 2 + ; t* n *..
dt clr dt*
If likewise at another place, whose longitude east from
the first meridian is eT, the time of culmination of the moon
was observed at the time T + t +</ , corresponding to the
time T f  1 of the first meridian , the right ascension of the
moon for this time is:
,
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 assume, that the tables, from
which the values of a and the differential coefficients have
been taken, give the right ascension of the moon too small
by A ? and if we put:
we have the following equations
dt
hence :
352
and since we have also :
d d=(& 0} (t 0, (6)
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 arith
metical mean of the times TM and T\t\ that is, the time
jli (_!_ ) which we will denote by T . Then we must
wr ite T \(f and T \\(t f) in place of TM and
Tit\ and if we assume, that the values of and of y etc.
belong now also to the time 7", we have the equations:
. [0 @Y d*
" \~da d
L dt J
and hence:
(/ . , c? 3 a
*= O^+^C 1 ^ ^.
From the last equation we can find 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:
 = da
dt
If the difference of longitude does not exceed two hours,
the last term is always so small, that is may safely be ne
glected. The solution of the problem is again an indirect
one, since it is necessary to know already the longitude ap
proximately in order to determine the time T .
For the practical application it is necessary to add a
few remarks.
If and & are given in sidereal time, h 6> is ex
pressed in sidereal seconds. Thus in order to find also t t
expressed in seconds, the same unit must be adopted for
d " or c L a must be equal to the change of right ascension in
dt dt
one second of time. Therefore if we denote by h the change
of the right ascension expressed in arc in one hour sidereal
time, we have:
da h_
dt ~ f5 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 da} 7 =0.9972693 of a mean day
we find, if ti denotes the change of right ascension expressed
in time in one hour of mean time:
da 0. 9972693 ,
r/7 = 3600 "" /i
i ,_ 15x3600 &&
"0.9972693 "~ A ~
or from the equation (6):
. _/ (/> *\(\ l? x ?69()_ \
\ 0. 9972693 A 1 /
Now the second term within the parenthesis is always
greater than 1 , and hence it is better to write the equation
in this way:
,/  <i> = (0>  0} ( 5 _L_^__ _ !) , (e)
and the second place, at which the moon was observed at
the time $ , is west from the other place, if & is pos
itive, and east, if & is negative.
Now the time of culmination of the moon s centre can
not 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 fol
lowing 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
7? 1
section is equal to ~ = . ; , where /I is equal to the value
15 cos o 1 /
of as given by the formula (<f). Therefore if ft and ft
354
denote the times at which the moon s limb was observed on
the meridian of the two places, we have:
R> *
..   . ,
cosd cos dJ 1 A
0.9972693 h
~3600
and hence we find from formula (e) :
where ft 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 ob
served, 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 ob
servation 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 _ \
S VO. 9972693 h /
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 5, 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 in
, O
strument 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 far
ther 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 =*= *
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 mooncul
minating 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 clocktimes of
the transit of the moon and the moonculminating stars were
observed at Bilk *) :
rj Ophiuchi 17 1 l"52s.64
Q Ophiuchi 12 6 .59
moon s centre 27 34 . 60
/t 1 Sagittarii 18 4 52 . 99
I Sagittarii 18 48 . 12.
On the same day the following transits were observed
at Hamburg:
r] Ophiuchi = 17 h 1>" 42 . 61
$ Ophiuchi = 11 56 . 91
([ I. Limb = 25 50 . 43
ft 1 Sagittarii = 18 4 43 . 53
I Sagittarii = 18 38 . 56,
The semi diameter of the moon for the time of culmi
nation at Hamburg was 15 2". 10, the declination 18 10 . 1,
and the variation of the right ascension in one hour of mean
time equal to 129 s . 8, hence A = 0.03596. We find therefore :
TVvT ?;, = 65". 66,
(1 A)cosd
hence the time of culmination of the moon s centre :
Then we find the differences of right ascension of the
stars and the moon s centre:
for Bilk: for Hamburg:
ri Ophiuchi 425 41*. 96 { 25 ra 13^. 48
Q Ophiuchi f 15 28 . 01 f 14 59 . 18
^ Sagittarii 37 18 .39 37 47 .44
I 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
0= }28.48
28 .83
29 .05
28^95_
mean f 28 . 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:
lOb 4 2 m 9 . 77
11" 2 9 .91
12 2 10 .05,
and since the time of observation at Bilk corresponds to
about 10 h 30 111 Berlin time, that at Hamburg to about ID 1 16 111 ,
we have:
T = 10 1 23 m
hence :
/i = 2n9s.S2
and we obtain by means of the formula (e) :
*) Since h is about 30 , the value of the coefficient of # # in the
equation (A) is about 29, hence the errors of observation have a great in
fluence on the difference of longitude, since an error of s . 1 in & & pro
duces ah error of 3 s 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 semidiameter of the earth or their ho
rizontal parallaxes, and we must show now, by what methods
the values of these constants are determined. Only the ho
rizontal parallax of the sun and the moon is directly found
by observations, since the distances of planets and comets
from the earth, the semimajor 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 Kep
ler s laws. Therefore in order to obtain the horizontal par
allaxes of those bodies, it is only necessary to know the ho
rizontal 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 ob
late spheroid, that is, of a spheroid generated by the revo
lution 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 di
mension 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 sta
tions. Eratosthenes (about 300 b. Ch.) made use already of
this method, in order to determine the length of the circum
ference 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 7 12 .
Hence the arc of the meridian between Syene and Alexan
dria must be 7 12 or equal to the fiftieth part of the cir
cumference. 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 24 8 , whilst the obliquity of the ecliptic at
the time of Eratosthenes was equal to 23 44 , and lastly the
latitude of Alexandria and the distance between the two pla
ces 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 de
terminations 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 coordinates of a point on the surface,
referred to a system of axes in the plane of the meridian,
359
the origin of the coordinates being at the centre of the earth
and the axis of x being parallel to the equator:
a cos cp
~ V\
_
~
where a and e denote the semi transverse axis and the ex
centricity of the ellipse of the meridian, and (p is the latitude
of the place on the surface.
Furthermore the radius of curvature for a point of the
ellipse, whose abscissa is #, is:
_ (a 2 2 xrf
~^b~
where b denotes the semiconjugate axis, or if we substitute
for x the expression given before:
(1
Therefore if G is the length of one degree of a meridian
expressed in some linear measure and cp is the latitude of
the middle of the degree, we have:
7ia(l e *)
G =  r ,
180(1 e 2 sin y 2 ) 75
where n is the number 3.1415927. If now the length of
another degree, corresponding to the latitude (p has been
measured, so that:
180(1
we obtain the excentricity of the ellipse by means of the
equation :
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:
3 4 32". 068
and
I 2 31". 387.
Furthermore Swanberg determined the distance of the
parallels of Malorn and Pahtawara in Lappland and found
it to be equal to 92777.981 toises, the latitudes of the two
places being:
65 31 30". 265
and
67 8 49". 830.
From the observations in Peru we obtain the length of
a degree:
G = 56734. 01 toises,
corresponding to the latitude
y = 131 0".34,
and from the observations in Lappland we get:
y/ = 6620 10".05:
= 57196.15 toises.
By means of the formulae given above we find from this :
2=0.0064351
a = 327 1651 toises,
and since the ellipticity of the earth a is equal to 1 j/i_ f 2,
we obtain:
a = 310^9 <
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 ob
servation 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:
Si<
dy l ,
~  dx 
dx 2 
361
If we differentiate the expressions of x and ?/, 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 arc of a meridian, extending from the
equator to the place whose latitude is cf i
s = a(\ t
But we have:
and if we introduce instead of the powers of sin (f the co
sines of the multiples of (f and integrate the terms by means
of the formula:
/I
cos kx dx = z sin hx
A
we obtain:
s = (1 2 ) E [y> sin 2y> f /? sin 4 q> etc.],
where :
If we take here ^ = 180, we obtain, denoting by g the
average length of a degree:
180^ = (1 2 )/i\7r,
and hence:
,y ==. [y, a sin 2 cp f {3 sin 4 cp . . .]
Therefore the distance of two parallels whose latitudes
are (f and <^ ; , is :
ft .9 =    [y cp 2 a sin (y (f) cos (y f y)
+ 2 /? sin 2 <> y) cos 2 fy> + y)],
or denoting r// y by / and the arithmetical mean of the
latitudes by L, also expressing / in seconds and denoting
206264.8 by ?, we find:
3600 , ,
(s ,v) = / 2 ?y a sin / cos 2 Z/ + 2 ?t?/9 sin 2 / cos 4 j&.
If we substitute here for / 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 y , a and ft
some certain values. But if we substitute the values, de
duced 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 cp + x and cp tx
instead of y and ^ , 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 :
r>roo
(* s) = I 2 w a sin / cos 2 L f 2 w 8 sin 2 1 cos 4 L + (x x) o,
9
where :
o = 1 2 cos I cos 2 L h 4 /? cos 2 I cos 4 L.
Hence we have:
x x = (  (s s) (l 2 iva sin I cos 2 L j 2?/;/3 sin 2 / cos 4 LY\ .
V <7 /
and a similar equation is obtained from every determination of
the latitudes of two places and of the length of the arc of
the meridian between their parallels. Therefore if the num
ber of these equations is greater than that of the unknown
quantities, we must determine the values of g and s so that
the sum of the squares of the residual errors x x etc. is
a minimum. If we take g ti and as approximate values of
g and and take :
y = . and = (I f fc)
we find, if we neglect the squares and the products of i
and k:
360
x  x =
*  )  A + 2?0 [ sin /cos 2 L 
sin 2 /cos 4
1 3600 , , 2w r <//?
H  ( s) i H  [ sin I cos 2 L  sm 2 I cos 4LJ fc.
$ go C o
Here /? denotes the value of /? corresponding to ,
but in order to get this as well as the differential coefficient
, , we must first express ft as a function of a. Now we find:
dn
1^ + 15 525 e +
8 * ^ 32 h 1024 ^
363
and likewise:
If we reverse the series for a we find:
f 2 = a  2 +4 3 
and if we introduce this in the expression for ft:
hence :
da 6 27
Therefore if we put:
1 /3GOO , \
n = I (6 s) I )
O \ gr /
H t a o si n I cos 2 ^ f ^n "o 2 H~ in a a o 4 ) s i n 2 / cos 4 L]
1 3600
a = (
and:
2 iv / 5 , , .
6 = sm / cos 2 L I  a n * f^, n 4 sin 2 /cos 4
we obtain the equation:
x x = n + ai + 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
#, i and k are for this set of observations, if u is the num
ber of all observed latitudes:
px+ [a] z+ [b] k+ [n] =0
[a] x h [a a] i{[a b] k f [a n] =
[b] x + [a b] i + [6 b] k H [b n] = 0,
and if we eliminate re, each set of observations gives the most
probable values of i and k by means of the equations:
= [on,] 4 [aa,] if[a&,]fc
*[*,] 4 [aft i] ef[66 l ]Jfc.
Therefore if we add the different quantities [Wj] which
we obtain from different sets of observations made in dif
ferent localities and designate the sum by (an^, likewise
364
the sum of all quantities [aaj by (aa^ etc., we h nd the
equations :
= (an,) f (aa.) z 4 (a M &
from which we derive the most probable values of i and k
according to all observations made in different localities.
As an example we choose the following observations:
1) Peruvian arc.
Latitude /
Tarqui  3 4 32". 068
Cotchesqui +0 2 31 387
3 7 3". 45
Distance of the parallels
176875.5 toises
2) East Indian arc.
Trivandeporum 411 44 52". 59
Paudru 13 19 49 .02 1 34 56. 43
3) Prussian arc.
Trims 54 13 11". 47
Konigsberg 54 4250.50 29 39". 03
55 43 40 . 45 1 30 28 . 98
Memel
Malorn
Pahtawara
4) Swedish arc.
65 31 30". 265
67 8 49 .830 1 37 19". 56
89813.010.
28211.629
86176.975.
92777.981.
Taking now:
57008
i 4 k
we find:
log = 7. 39794
log[yo 2 f 1 3 Q go 4 ] = 4.41567
log[o 2 H ^ <V>] = 4. 71670.
If further we put:
10000 i=y
10 k = z,
we obtain the following equations for the four arcs:
1) x } Xl = 41". 97 4 1.1225^4 5.6059 z
2) x\ ^ 2 =40 . 94 4 0.5697 y 4 2.5835 z
3) x 3 x 3 = Q . 37 4 0.1779 y 0.2852 z
x " 3 X3 == 4 3 . 79 4 0.5433^ 0.9157 z
4) .r 4 xi = .51 + 0.5839^ 1.971 1
365
and from these we find:
[n] [a] [6] [an] [a a]
[a 6]
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
IH [66]
1) +11.0436 +31.4254
2) + 2.4284 6.6742
3)  3.3650 0.9198
4) + 1.0026 3.8853
and:
[an,] [art,] [aft,]
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
(r/ w ,) = + 2.3956, (aa,) = +l.llG2, (aft,) = + 3.0471,
[61,,] [66,]
+ 5.5218 +15.7127
+ 1.2142 + 3.3371
 1.9960 + 0.4391
+ 0.5013 + 1.9426
(ftn,) = + 5.2413, (66,) =+ 2L4315."
and
hence :
therefore
and :
Hence the two equations by which y and z are found,
= + 2.3956 + 1. 1162^+ 3.0471s
= + 5.2413 + 3.0471 y + 21.4315 2,
we find:
2 = + 0.099012
# = 2.4165,
= 0.00024165 and k
0.0099012:
57008
1 0.00024165
 = 57021.79
1 + 0.0099012
0.002524753.
Now since we had before:
32
we find:
I =T"T" H  4
0.006710073,
and the ellipticity of the earth  
366
Moreover we have:
log = log I/I  "e 1 " = 9.9985380,
and since we had:
180$r
(1 e^En
we find:
log = 6.5147884,
and:
log b = 0.5133264.
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 = ^ ^
the seraitransverse axis a = 3272077. 14 toises
the semi conjugate axis fi = 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 /> ==(> sin p sin [z (y> y )],
where p is the horizontal parallax, z the observed zenith dis
tance cleared from refraction, (f the latitude,, (p the geocen
*) In Schumacher s Astronomische Nachrichten No. 333 and 438.
367
trie latitude and (> the distance of the place from the centre
of the earth. Hence we have:
1 _ __ $ sin [z (y> y )]
sin p sin p
We have also, if cp is the latitude of the other place,
and (>j the geocentric latitude and the distance from the
centre :
sin /7 sin/,
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 \ <p
 (p, and the angle at the centre (p =^= <?, where r> 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 19
180 z l j (fi cp\ and <p\ =t= 8. We have therefore:
and:
p > + p > t=g + ~ l VVi
Therefore if we denote the known quantity p f p\ by
TT, we have the equation:
(i > sin [z (y_^J> )] _ (>i sin[g, (y, y ,)]
sin p sin (TT jo )
whence follows:
, _ (> sin TT sin [2 (90 90 )]
lg P $ , sin [2 , (99, 9? , )] H (> cos n sin [s (y 9? )]
or :
tang y __ _ gi sin7Tsin[.g, (y, y ,)]
(> sin [2 (<p <f> )] + $  cos n sin [z , (9? , 9? , )]
When either p or p\ has been found by means of these
equations, we find p either from:
sin
sm ;? = 7   7 
^ sm [z (y 9? )]
or from: sin = r  i3in i )
sin p =
>, sin [2, (95, y> ,)
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 </ =p$
in one triangle and (f\ =p t) in the other. If we put in
this case:
TV = ]> , V .c ,  (y, <p),
we find p or p\ 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
O
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 dis
tance of these bodies at their culmination, while correspond
ing 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 8G, whilst the greatest difference in longitude
is that of the Cape and Greenwich, being equal to 1~ 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 302 Q^ 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 con
nection with simultaneous observations made at Greenwich
he found for the mean parallax the value 57 1". 8**) Tne
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
semidiameter, which itself depends upon the parallax.
If r and r denote the geocentric and the apparent semi
diameter, A and A the distances from the centre of the earth
and from the place of observation, we have:
sin r A
sin r 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(z X)
where z 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(yrt*S
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 (y y ) == /]
A sin [z (yy ^p =fe= r ]
If we introduce this expression in the equation for sin r
sinr
and eliminate p by means of the equation:
sin p 1 = (} sin p sin [z (tp y ) == r ] ,
we obtain, writing for the sake of brevity z instead of z
(ff <^ ) and taking Q = 1 :
sin r = sin r f sin r sin p cos (z == ? ) f \ sin r sin p 2 sin (2 =t r ) 2 ,
or neglecting terms of the third order:
r = r f sin r sin p cos (z == r) f { sin r sin /> 2 sin (z == r) 2 .
Now the geocentric zenith distance Z of the moon, ex
pressed by the zenith distance z of the limb, is:
r , __ i / r ;\ sin 3 sin (2=t=r ) 3
^ = z =t= r sin p Bin (z == r ) ,
6
or if we substitute for r its expression found before:
Z = z =t= r == sin r sin/) cos (2 =t= r) dt= 4 sin r sin/> 2 sin (2 == ?)
... sin n 3 sin (2 ==r) 3
sin p sin (a == r) 
If we develop this equation and again neglect the terms
of a higher order than the third, we find:
370
Z = z == r sin r 2 sin p sin z == 4 sin r sin y> 2 sin z 2
sin/; 3 sin z 3
sm p cos r sin. z + * sin p sin r sin z ,
or introducing 1  sin r 2 instead of cos r and replacing
sin p by y sin p :
Z=z^=i Q sin/? sin z I Q sin;) sin z sin r 2 =i= 7} ^> 2 sin/> 2 sin r sin 2 2
(> 3 sinp 3 sin z 3
"T"
and finally, if we take:
sin r = k sin p ,
and hence:
/ = k sin p + jt A: 3 sin yr 3
and introduce again z A in place of a, where A = ^ </> ,
we have:
Z = z a sm P [f, sin ( s  A) =F A;]  6 fe sin (2 i) =F *] 3 .
If D is the geocentric declination of the moon s centre,
the observed declination of the limb, we have also, since
D = (f Xand d = <f (z A) :
I) = <? 4 sin p [o sin (s A) =j= fc] + ~^^ [Q sin (s A) =f= ^] 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 w r ay, that it can be easily corrected for any
other value of the ellipticity, we must transform the ex
pression given above accordingly. But according to No. 2
of the third section we have:
 r sin 2 y + . . v gf
a 2
If we introduce here the ellipticity, making use of the
equation:
a
and neglect all terms of the order of 2 , we find:
m (fi 1 = K = a sin 2 <p.
Moreover we had:
, __ 2 2 _ cos 9P 2 _ (1 g) 2 siny 2
~ 1 2 "sfn"^ 1 2 sin y 2
_ 1 2 2 sin 9 2 H * sin p 2
1 2 sin "
371
If we introduce here also a by means of the equation:
2 = 2 a a 2
and neglect all terms of the order of 2 , we find:
(> 1 a sin y> 2 .
Thus the last expression for D is changed into:
D = { [sin 2 =p fc] sin p [sin <p 2 sin 2 ~h sin 2 90 cos 2] a sin p
.... sin p 3
f[sms=T=fc] 8  ^.
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 hemi
sphere :
D , = <?! [sin z , =p k\ sin p , [sin z , =p k] 3 ~
b
f [sin tp , 2 sin z, +~ sin 2y>, cos z t ] sin;?,.
Now let t and ^ be the mean times of a certain first
meridian, corresponding to the two times of observation, let
Z) be the geocentric declination of the moon for a certain
time T and c . its variation in one hour of mean time and taken
a t
positive, if the moon approaches the north pole, then we find
from the two equations for D and D 1 :
(*i ^ t = ^j ^ [sin 2, =pl (sin y, 2 sin z t hsin 2^, cos 2,)] ship,
jt [sin .c =p k a (sin y> 2 sin z f sin 2 9? cos 2)] sin p
^fy , 71 , sinp, 3 sin 3
 [sin 2, =f k] 3   [gin 2 =p A;J f .
Moreover if p Q is the parallax for the time T and ^ its
change in one hour, we have:
sin p = sin p f cos p l f (t T}
at
sin p , = sin p + cos p j f (t t T),
therefore we find the following equation for determining the
parallax for the time T:
24*
372
= tf, S H (t /,) [(sins, =f= &) 3 H sin
 . cos p [(sin 2 =f= fc) (/ 7") f (sin c, =p
( sin y 2 sin s + sin 2 OP cos 2 ) ..
 [sm2, fsin2=pA=F/.Jsin;? Hrtsinp J j *).
v 4 sin 09 . sin z . sin z nn . rns 2 . >
If at the two places opposite limbs of the moon are
observed, the coefficient of sin p Q is rendered independent
of /c, and since this quantity thus only occurs in the small
terms multiplied by sinp 3 and j , the value of/> () , which is
found from the equation, is independent of any error of k.
Since we know the parallaxes from former determinations suf
ficiently accurately so as to compute the third and the fourth
term of the formula without any appreciable error, we can
consider the first four 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 ft, the coefficient of
sin p {) by a and that of a sin p by 6, we obtain the equa
tion :
= n sin/> (a b a),
from which p can be found as a function of a. But in
stead 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 **). There
fore if K is the value of the mean parallax adopted in the
lunar tables, and n the value taken from those tables for the
time T, we have, if we denote the sought mean horizontal
parallax by II:
sin p ==~ sin 11= fi sin ZT,
A
hence the equation found before is transformed into:
=   sin 77 (a ba).
ft
*) If the second differential coefficients are taken into account, we must
add the term:
but if we take: T=\ (/,+/),
this term vanishes.
**) Namely the distance equal to the semimajor 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:
S = + 20 26 25". 2,
and Lacaille at the Cape of Good Hope the declination of
the upper limb:
l = + 21 46 44". 8.
For the arithmetical mean of the times of observation,
corresponding to the Paris time:
r=6 h 40,
we take from Burkhardt s tables:
^ = 59 24". 54
^
dt
finally we have:
y = 52 30 16"
and
<p { = 33 56 3 south.
Since the longitude of the Cape of Good Hope is 20 m
19 s . 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 21 m 11 s later at Berlin than at the Cape,
hence we have:
*<, =t21 Ml<S hence (t *,) ~ = 12". 06
at
further we have:
<y, ? = MO 20 19". 6.
The third term, depending on sin p 3 , we find equal to
OM2, if we take ft = 0.2725; therefore if we omit the
insignificant term multiplied by , we find:
n = M<> 20 7". 42
or expressed in parts of the radius:
n = h 0.023307
and since the value of the mean parallax adopted in Burk
hardt s tables is:
^=57 0".52
we have:
log^ = 0. 01792,
hence :
= + 0.022365.
374
If we compute the coefficients a and 6, we find, since:
z = 323 51" and ^=55 42 48"
the following values :
a = 4 1.3571 and /,=+ 1.9321
and hence the equation for determining sin 77 is:
= 4 0.022365 sin 77(1.3571 1.9321 ).
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 nr. For in
stance taking a =  we find :
ij i) 10
log sin 77= 8.21901
II =56 55". 4.
But if there are several equations, we find for the equa
tion of the minimum according to the method of least squares :
[a a] x [a b] a x a = 0,
hence:
.
[a a] [a a]
r n~] r
a a
= L ^J^L
[a a] [a a] [a a]
Thus Olufsen found for the mean horizontal parallax of
the rnoon the value 57 2". 80 *). Since the parallax of the
moon is so large, it may even be determined with some de
gree 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. 32G.
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 Condainine 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 ofLacaille and Lalande, namely 10". 25; nei
ther 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 de
sirable, 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 method *).
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 pre
ceding section. The following method, originally owing to
Lagrange, was published by Encke in the Berliner Jahrbuch
for 1842.
If , <> , A and D are the geocentric right ascension and
declination of Venus and the sun for the time T of a cer
tain first meridian, which is not far from the time of con
junction, 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 M and 180 IT:
*) Such observations luive 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 \ (M 1 + M} = sin \ (
sin  m . cos \ (M 1 f M) = cos ] (a A) sin i (# />)*
cos ^ w . sin ^ ( M 1 M} = sin \ (a .4) sin ^ (8 + D)
cos 4 TO . cos 4 (M M) = cos ^(a A) cos (tf Z>),
or since a A and d D and hence also m and M M are
for the times of contact small quantities:
m sin M (a A) cos ^ (<? +>)
Z).
Taking then:
n cos =
dt
where and are the relative changes of the
dt dt
right ascensions and declinationa in the unit of time, and de
noting the time of contact of the limbs by Tfr, we have:
[m sin M+ r n sin N] 2 H [m cos M f rn cos N] 2 = [R == 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 ex
terior contact, the lower sign for an interior contact.
From this equation we obtain:
Therefore if we put:
m sin (M 2V)
^_^_ r = sin y;, where y < =b 90, (C)
we obtain :
r = cos (M N} =f= cos w. (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:
T  cos (M N} r cos y
n n
and the egress at the time:
T cos (M N) + R= ^ T cos y.
n n
Finally if 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 :
(/2 dt= r) cos = m coe M + n cos N . t
(ft =t= r) sin = m sin M+ n sin N .r
or:
cos = sin N sin y =p cos N cos y
sin = sin y cos .2V =p cos y; sin JV,
hence for the ingress we have:
= 180H2V > (^)
and for the egress :
These formulae serve for computing the times of the in
gress 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 cen
tre of the earth.
We have:
cos m = sin 8 sin D f cos 8 cos I.) cos ( A).
If , <) , 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 sin D f cos 8 cos D cos ( A 1 }
and hence:
cos m = cos m + ( 8 8) [cos 8 sin D sin 8 cos D cos (a A)]
4 (D D) [sin^cosZ* cos # sin Z> cos (a A)]
(a 1 a ) cos 8 cos D sin (a A)
4 (A 1 A) cos 8 cos Z> sin (a 4).
But according to the formulae in No. 4 of the third sec
tion we have *) :
*) We have according to the formulae given there:
w s sin(<? v)
o o Ti sin cp  ; ;= 7t sm cp Ism o cotangy cos ol.
sin y
but since:
cotang Y = cos ( 0} . cotang y>,
we have:
8 8= n [cos cp sin 8 cos (a (9) sin y> cos 8].
378
S S = 7t [cos rp sin $ cos (a 0) sin y cos 8]
// I) = p [cos <p sin D cos (a 0) sin ycos />j
a = rt sec S sin (a 6*) cos ip
A A = p sec D sin (J. 0) cos y,
where n and p are the horizontal parallaxes of Venus and
the sun; and if we substitute these expressions in the equa
tion for cos m , we obtain :
cos m = cos m
f [cos 8 sin/J sin 8 cos D cos ( A}} [TTCOS<JP sin$cos( 0) Trsinycos #]
4 [sin $cos.Z> cos$sin/>cos (a ^1)J [79 cosy sin/>cos( 6>) p sin y cos/)]
cos D sin ( A) . n sin ( 0) cos y ()
+ cos $ sin ( ^4) . /> sin (A 0} cos y.
If we develop this equation, we find first for the coef
ficient of cos tf :
7i [sin S cos S sin D cos ( 6>) sin # 2 cos D cos ( 0) cos ( ^4)
cos Jj sin ( 0) sin ( A)]
\ p [sin $ cos D sin /> cos ( 0) cos S sin JJ* cos ( 0} cos ( ^4)
f cos S sin ( 0~) sin ( vl)J
or since:
sin (V = 1 cos S* and sin D 2 = 1 cos D* :
71 [(sin 8 sin/> + cos #cos Z> cos (a A) ) cos $ cos ( 0} cos D cos (A 0)]
f/>[(sin^sinZ>Hcos^cosZ>cos( ^l))cosDcos(^4 0} cos S cos (a 0)],
hence :
71 COS /ft COS S COS (rt 0) 71 COSZ> COS (A 0)
H /) cos m cos Z> cos (^l 6>) /> cos 8 cos ( 0).
This we can transform in the following way:
?r cos m cos $ cos a n cos Z> cos ^4] cos
f [p cos ?. cos D cos ^1 p cos J cos J cos
f [TT cos M cos $ sin 7t cosD sin^] sin
+ [p cos m cos D sin A p cos 8 sin j sin 6>,
and hence the term multiplied by cos ^ becomes :
[(71 cos m p} cos $cos (n ;) cos m} cos D cos ^4] cos <f cos . .
t [(TT cos /ft p} cos $ sin (it p cos m) cos Z> sin A] cos y sin 0.
Further the coefficient of sin y in the equation (a) is :
7i [ cos 8 * sin D H sin <? cos ^ cos D cos (a ^1)]
+;> [ sin ^ cos // 2 1 sin/^cosjL cos ^ cos ( ^Ijj,
or since cos r) 2 =1 sin <) 2 and cos /> 2 =1 sin D 2 :
TT [ sin D + sin $ (sin 8 sin />+ cos 5 cos Z> cos ( ^4))J
H p [ sin 8 + sin/) (sin 8 sin D f cos ^ cos /) cos ( ^4))J
Therefore the term of the equation (a), which is mul
tiplied by sin y, is :
(?r cos m />) sin ^ sin y (TT jt> cos m) sin Z) sin <p,
379
and thus the equation () is transformed into the following:
cos m = cos in
J [(Vr cos m p) cos S cos a (n p cos TO) cosL> cos ^4] cos (p cos (9
+ [(ft cos TO p) cos S sin (TT y> cos m) cos D sin yl} cos (p sin 6> ( c )
f [O/r cos TO p) sin (V (jt p cos TO.) sin D] sin y.
If we take now:
it cos m  p =f sin s
TT sin m = / cos s,
we have:
7t p cos TO =fsm (s TO),
and henee:
cos in. = cos in.
H/[sin ft cos <? cos a sin (.s m) cos L) cos A] cos y cos
fyfsin s cos $ sin a sin (* in) cosl) sin ^4] cos <f> sin (e)
+/[sin s sin $ sin (s m) sin jDj sin f/>.
Further if we take:
sin s cos 8 cos sin (.s ?//) cos I) cos .4 = P cos A cos /?
sin s cos $ sin a sin (* in) cos D sin .4 = P sin A cos ft (/ )
sin A sin $ sin (,v TO) sin Z> = P sin /^,
we find by squaring these equations the following equation
for P:
P 2 == sin s z H sin (s /) 1<! 2 sin s sin (s m) cos m
= sin A 2 sin .s 2 cos m 2 f cos .$ sin TO  = sin TO 2 .
Hence we may put:
sin s cos $ cos a sin (s TO) cos /) cos A = sin m cos 1 cos (3
sin s cos ^ sin a sin (s m) cos D sin J. = sin TO sin A, cos /9
sin ,v sin ^ sin (.s TO) sin D = sin m sin (3,
or:
sin TO sin (A J) cos ft = sin a cos S sin (a J)
sin // cos (A A) cos p = sin s cos S cos ( ^1) sin (s m) cos/> (</)
sin TO sin /^ = sin s sin S sin (s TO) sin />.
But we have :
sin s cos duos ( J) sin(.s TO) cos L> = sins [cos S cos (a A) cos TO cos D]
H cos ,s . sin TO cos D
and :
sin s sin <? sin (,v TO) sin /> = sin ,s [sin 5 cos w sin />]
+ coss . sin nt 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 TO sin M= cos sin ( A)
sin m cos l/= sin ScosD cos 8 sin D cos (a A) (k)
cos in = sin sin Z) j cos $ cos jD cos ( ^J),
380
hence we have:
cos cos ( A) = cos D cos in sin D sin m cos M
sin $ = sin D cos ?w + cos D sin ? cos 3f,
and the equations (</) are thus transformed into the following:
sin (h A) cos ft = sin s sin 7I/
cos (A ^4) cos /? = cos s cos Z> sin s sin Z) cos M (?)
sin /9 = cos s sin Z) j sin s cos Z) cos M,
where s and M must be found by means of the equations
(d) and (ft). After having obtained A and /? by the equa
tions (i), m is found according to (e) and (/) by means of
the following equation:
cos m = cos m /sin m [cos A cos /? cos y cos f sin A cos /? cos 9? sin
h sin/? sin <p]
= cos m +/sin m [sin <p sin /? + cos y cos /? cos (^ (9)].
Now let T, as before, be that mean time of a certain
first meridian, for which the quantities , r), A and D have
been computed, and L the sidereal time corresponding to it,
further let / be the longitude of the place, to which and
(f refer, taken positive when East, we have:
therefore : I = I L /.
Hence if we put:
A = I L,
cos = sin cp sin 8 + cos <p cos 8 cos (^/ /),
Ti / " N fl \
we have:
COS i s::: 5 COS M ~4~/sin WJ COS
All places, for which cos 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 \ I. In order to find the time when these places see the
distance w, we have: dm = fcos,
hence : dt= 
dm
dt
But if m is a small quantity, for instance at the time of
contact of the limbs, we have according to the formulae (4):
m = (a A) cos ^ (8 + D) sin M\ (S Z>) cos M
dm d(aA) , d(8D) ..
= cos 4 (o 4 D) sin If H cos M.
dt dt at
or according to the formulae (1?) :
381
/cos
hence : dt =  
ncos (M N}
Therefore if an observer at the centre of the earth sees
at the time T the angular distance m of the bodies, an ob
server on the surface of the earth sees the same distance at
the time of the first meridian:
_/co^
ncos (If N)
or at the local time:
ncos(MN)
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=^=r and instead of m and M , and since we
have according to the formulae (E) and (F) for the ingress
O = 180 H N \j) and for the egress O = JVfi//, we must
add to the times of the ingress and egress for the centre of
the earth: _/cos
n cos y
and: + /ll.
n cos y
Hence if we collect the formulae for computing a transit
of Venus, they are as follows:
For the centre of the earth.
For a time of a certain first meridian, which is near the
time of conjunction, compute the right ascensions , A and
the declinations <?, D of Venus and the sun, likewise their
semidiameters r and R. Then compute the formulae:
m sin M= (a A) cos (S + D)
mcosM= S D
n sin N= ^~~y C os i (8 h />)
at
A7 d(8 D}
>tcos N=
.ZV)
T = cos (If N}  cos
n n
r =  cos (M jV) H  cos
n n
382
Then the time of ingress is:
and we have for this time:
= 180 hN ip,
and the time of egress is :
and for this time
For a place whose latitude is y and whose east longitude is I.
Compute for the ingress as well as for the egress, using
the corresponding values of the angle O, the formulae:
7t cos (R =J= r) p = f sin s
7t sin (R =t= /) =/cos *
_/_
n cos y
sin (I A) cos ft = sin s sin
cos (A A) cos ft = cos s cos D sin s sin D cos
sin ft = cos s sin D + sin s cos Z* cos
A = l L
cos = sin ft sin 90 f cos ft cos 90 cos (^/ I) *),
where L is the sidereal time corresponding to t or t . Then
the local mean time of the ingress is:
t 4 I g cos ,
and that of the egress:
t \ I t y cos g.
At those places, for which the quantity
sin ft sin y j cos ft cos 9? cos (A /)
is equal =t= 1, the times of contact are the earliest and the
latest. The duration of the transit for a place on the sur
face may differ by 2g from the duration for the centre, and
since for central transits we have nearly:
n p
> n"
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 arc 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
*) is the angular distance of the point, whose latitude and longitude
are 9? and /, from the point, whose latitude and longitude are ft and A.
383
the time of conjunction is 234", the difference of the dura
tion can amount to 12 minutes, whence we see that the dif
ference 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
16"
17h
IS 1
19 h
20 h
further :
?r = 29". 6068 72 = 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:
7 7 =17h
and find:
=  4 11".6
17 1"
.8
422 41
3".
7
74 25
50".
SH
h22 33
17".
6
1936
.4
41
19
,1
24
13 .
2
32
32 .
4
22 10
.9
41
34
.5
22
36 .
2
31
47 .
1
2445
. 5
41
49
,9
20
59 .
2
31
1 .
9
27 20
.1
42
5 .
3
19
22
2
30
16 .
6,
., ., 
at at
Tt ~ d ft = ~~ 60 " 65 n + r = 975 " 8
From this we find:
M= 154 7 . 2 ^=255 21 . 9
log m = 2 . 76746 log n = 2 . 38028
M N= 258 45 . 3
y = 36 2.6
cos (If A 7 ) = H . 4756 r = 2 h . 8114 = 2 h 4S n 41 .
, = + 3 .7626 = + 3 45 45 .4
Therefore the ingress took place for the centre of the
earth :
at 14 1 11 111 19 S .0 Paris mean time,
and it was:
= 111 24 . 5,
and the egress took place at
20 h 45 ra 45 s . 4 Paris mean time,
and it was :
G = 219 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 con
stant quantities A, ft and g and find first:
s = 90 22 . 7, log/= 1 . 325G4, log# = 9 . 03764,
and since:
O = 219 19 . 3, Z> = 22 42 3, ^ = 74 29 . 3,
we obtain:
1 = 9 15 . 9
and ^ = 45 44 . 4.
Further since 20 h 45 m 45 s . 4 Paris mean time corresponds
to I h 45 m 34 s .6 sidereal time, we have:
A = 17 7 . 7.
If it is required for instance to find the egress for the
Cape of Good Hope, for which:
/= + lh 4m 33s. 5
and
y> = 3356 3",
we find:
log cos = 9 . 94043 , g cos = 4 5 47" . 0,
and hence the local mean time of the egress :
1 + A + g cos = 21h 56 m 5 s . 9.
If we differentiate the equation:
we find, if dT is expressed in seconds:
3600 cos
dT=  d(7C p)
n cos ip
_ 3600 cos np fl
"
n cos iff />
so that an error of the assumed value of the parallax of the
sun equal to 0".13 changes the time of the contact of the
limbs by 5 s . 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:
[  ^l ] 2 cos <? 2 + [S  Z) ] 2 [JR=t r}\ (</)
*) Where ;> is the mean horizontal equatoreal parallax.
385
where , A\ 8 and D are the apparent right ascensions and
declinations of the sun and Venus, affected with parallax,
v; j .,/
and ^ denotes the arithmetical mean . But since the
parallaxes of the two bodies are small and likewise the dif
ferences of the right ascensions and declinations for the times
of contact of the limbs are small quantities, we can take:
ft A = a A+(n p) sec 8 cos cp ! sin ( (9)
8 D = 8 D H (it p} [cosy sin S cos ( 6>) sin y cos <? ],
where :
a + A
. j.
If now we introduce the following auxiliary quantities:
cos (f sin ( 6>) = h sin H
cos cp sin $ cos ( 0} sin y> cos S = h cos //,
the equation (a) is transformed into :
[ A + (n p} h sin //sec # ] 2 cos S 2 f [5 D + (?r p) // cos //J 2 = [7? =fc r ] 2 .
If then , J, J, />, TT, p, /? and r, denote the values which
are taken from the tables, whilst jc/, r) jc?6, ^f^^d,
D ~j c/D, TT + C/TT, p f rf/?, jR j dR and r f 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 f (jc />) h sin //sec <? f d ( J)
h d(n p} h sin //sec 8 <LL^_) rf/
i _
,7^ 7)^
h[5 Di(7t p)/icos/T+rf(5 /))H(/(7ir p)hcosH ~^ J dl]*
If we develop this equation and neglect the squares and
the products of n p and the small increments, and put :
a A+(np)h sin //sec <? = A
L>i(7i;p)hcosH =D\
we find:
yl^cosVh/) 2 CK^r) 2
= 2^ cos <V 2 d(a A) 2 [^ A sin //cos <? H /) A cos H]d(7tp)
^^p^ C o S ^+D d(8 ~
^ at
H 2 CR =J=
But if we denote:
4 l2 C08^ a hZ) a
by m 2 , and since we have approximately:
, M 2 (# d= /) 2 = 2 m [ Mi (R d= r )l,
25
38G
we find:
m [ m (R=r)]= A cos8 *d(a A)D d(8 D )
[A hsmllcos S \ D h cos H] d(n p)
Therefore if we put again:
A cos $ = m
2) = m cos M
1 d(a A}^ \
3600 C S e dt m ( ,
1 d(*Z
3600 ^"^T =ncos^
the equation becomes :
,+ (yj 
n cos (M~) ~ ncos(MN~)
hcs(MH) np d(R^r)
ncos(MN) Po Po ncos(MNY
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==r m is expressed in seconds of time, because
we have:
ncos(Jf 7V) = ~y 
Example. The interior contact at the egress was ob
served at the Cape of Good Hope at
21 h 38 "3 s .3 mean time.
This time corresponds to
20 h 33 m 29 8 .8 Paris mean time = I h 33 16 s . 2 Paris sidereal time.
We have therefore:
= 2 1 37 " 49s . 7 = 39 27 25".
Moreover we have for that time:
= 74 18 28". 05 =22 29 51". 32
A = 74 28 46 . 41  Z) = 22 42 13 .90
a A= 10 18". 36 8 D = 12 22". 58
= 74 23 37" = 34 56 12" <? = 22 36 2"
(7tp) Asin//=h 10". 07 //=3134 . (n p] k sin H sec ^
(n p )k cos /f=h 16 .39 log // = 9.95835 =H10".90
^ = 10 7". 46
D = 12 6 .19
M= 217 40 . 7 N= 255 19 . 3
log m== 2.96262 log n = 8.82412.
* 387
Now since:
R r = 917". 80
and :
/j =8". 57116,
we find:
 5.3 = 10.684 d (a A) + 14.986 d (8 D)
H 42.240 d Po h 18.934 d(R r).
Such an equation of the form:
= n 4 ad (a 4) f 6d (# Z>) H cdp + ed(R r)
is obtained from each observation of an interior contact and
a similar one containing d(Brr) from an exterior con
tact, 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 dp^ d (a A), d (8 D)
and d (/2 =t= 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
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 Venusdurch
gang 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 deter
mining the parallax of the sun. For since the hourly motion of Mercury
at the time of the inferior conjunction is 550", Avhile the difference of the
parallaxes of Mercury and the sun is 9", the coefficient of dp in the equa
tion (Z>) in the case of Mercury is to the same coefficient in the case of
Venus as:
23 550
9 234 :
hence G times smaller. Thus an error of observation equal to 5 s produces
already an error of 0".S 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 coordi
nates referred to this fundamental plane. For, such an in
strument consists in its essential parts of two circles, one
of which represents the plane of xy of the system of coordi
nates, whilst the other circle perpendicular to it and bearing
the telescope turns around an axis of the instrument perpen
dicular 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 coordinates
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
coordinates, but make a small angle with them. The pro
blem then is, to determine the deviations of the circles of
the instrument from the true planes of coordinates, in order
to derive from the coordinates observed on the circles the
true values of these coordinates.
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 coordinates of the heavenly bodies with
the greatest possible approximation."
Besides these instruments, with which two coordinates
of a body perpendicular to each other can be observed, there
are still others, with which only a single coordinate 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 spiritlevel.
1. The spiritlevel serves to find the inclination of a
line to the horizon. It consists of a closed glass tube so
nearly filled with n fluid that only a small space filled with
air remains. Since the upper part of this tube is ground out
into a curve, the airbubble 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 ne
cessary, 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 oft *). The tube is then provided with
two rectangular supports for placing it upon the pivots or
for the larger instruments with corresponding hooks for sus
pending it on the axis of the instrument. Generally however
these supports or hooks are not of equal length. Let AB
Fig. 1 1 be the level, A C and
BD be the two supports,
whose length is represented
by a and b and suppose
the level to be placed on a
line, which makes with the
horizon an angle , in such
a manner, that BD shall stand upon the higher side. Then
will A stand in the height a f c and B in the height:
1> H c + L tang a
if L is the length of the level. This is, to be sure, not enti
rely correct, because the supports AC 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 A B makes with the
horizon a?, then we have:
b a h L tang a
tango: =  >
/ J
or
b a
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:
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 A , then if the level be placed directly
on a horizontal line, we read /  A on the side A, if 21 be
*) This arrangement is adopted in order that the level may be in a com
pletely 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 / I on the side B. Suppose
on the other hand the level to be placed on the line A B,
whose inclination to the horizon is #, then we read on the
side A:
A = l{l rx,
where r is the radius of the curve A #, in which the level
has been ground out, on the contrary on the higher side B:
B = ll\rx,
If the level with its supports be reversed in such a
manner that B shall stand upon the lower end, we shall read :
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 = I ra J A ru
A = I + r a + K ru
It is obvious from the above, that the two quantities A
and ru cannot be separated from each other, and that for
the reading off it is one and the same, whether the zeropoint
be not in the centre or whether the supports be of unequal
length. On the other hand by the combination of these equa
tions we can find A ru 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:
BA
 ( r /  r u f r a
A B
 = / ru i ra,
therefore : , B _ A A > _
* ( 2 + ~2
H \
 206265,
if we wish to have the inclination directly in seconds of arc.
rpi ,. 206265 . , T ,1
The quantity is then the
scale expressed in seconds of arc.
The quantity  is then the value of one unit on the
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 read
ing on the side of the circle from the reading made on the
other side and divide the arithmetical mean of the values
found in both positions by 2. The result is the elevation
of the circleend 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 arc, the result will be the eleva
tion of the circleend in seconds of arc.
If we can assume, that the length of the bubble during
the observation does not change, we have also:
a = U ~ A) ,
T
or:
^^(BB )
r
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 A ru = and it
would not be necessary, to reverse the level, but the incli
nation 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
Object glass East j ; g g 18 Q j 0b J ect S la ss West j , ? ^ !
B ^ = h 3". 90  6". 3(5
A _ B > * ru = 8". 80 I ru = 9". 20
___ = 4 ,90 + 2 . 90
0".50 ~^rp~7o"
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
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 rn
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 J5 , 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 ex
ample, where A ru is 9^. 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 11.4 15.0
Object glass East . Object glass West US
whereby we should have found again the inclinations 0" . 50
and 1".70, and / 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 con
tact 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 pa
rallel 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 ob
server. This end then must be moved by means of the ho
rizontal adjusting screws, until the bubble preserves its posi
tion 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 some
what 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 ft
/?
seconds, then is the value of the unit of the scale in
a
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 can
not be well fastened on a circle of tin instrument, it is best
to use for this purpose a special instrument which consists
in its essential parts of a strong Tshaped supporter, which
rests on three screws and on which the level can be placed
in two rectangular Ypieces, in such a manner, that the di
rection 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 care
fully 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 meas
urement 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 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 equa
lity exists for those parts, which are liable to be used
in leveling and which at least in levels, as they are made
now, 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 tempe
ratures 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 = a +b(t O
where a is the value at a certain temperature , 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 wirecross
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 diffe
rence 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 footscrews, which is finely cut for this purpose and is
provided with a graduated head. These instruments rest
namely on three footscrews 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 micros
copes 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 footscrews 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 cy
linders 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
F ig. 12. circleend is taken) be r , then will b C
(Fig. 12) or the elevation of the centre
of the pivot above the Y be equal to
r cosec i, likewise we have :
a C= r cosec z ,
hence :
a b = r [cosec i + cosec z],
on the other end of the axis we
a 6 = ? I [cosec i f cosec i],
where r l is the radius of the pivot on
this side. If now the line through the
two Ys makes with the horizon the angle #, 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 circle
end, we will have for the elevation 6 of the circle end:
I = x H [cosec i f cosec z],
.Li
where L is the length of the axis. If however the instru
ment be reversed so that the circle shall now rest on the
lower Y, then will the elevation of the circleend be:
b = x h    [cosec i 1 f cosec i].
From both equations we derive :
399
 , r
 =  [cosec i 4 cosec tj,
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:
r o r \ i
 cosec i ,
or if ?0 r be eliminated, the quantity:
(6 + 6 ) cosec i
cosec i 4 cosec i +
or 4: (6 + b ) sin i ^
sin i 4~ sin i
If the correction, as is generally the case, be small,
then we can make 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 circle
end 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
& == 5". 02, which value, as before, is the mean of two
levelings by which in one case the object glass of the teles
cope was directed towards tlie east and in the other case
towards the west. In this case therefore is:
\(b 4 6) = + 0". 74,
hence the inclination of the mathematical axis of the pivots
was:
= 2". 80 Circle South
and = H 4". 28 Circle North!
Hitherto it has been assumed, that the sections perpen
dicular 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 te
lescope when it is turned round the axis will describe a great
*) Usually i and i are equal to about 90.
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 de
scribe 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 posi
tion 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 pos
itions of the telescope be found, for instance for every 15 th
or 30 th degree of elevation, and only when the telescope is
directed towards the zenith or the nadir will this be impos
sible. If these observations are also made in the other posi
tion of the instrument, then can the inequality of the pivots
or the quantity }(b + & ) be determined for the different ze
nith distances, and if this be subtracted from the levelerror
in the corresponding positions of the telescope, the inclina
tion of the axis for the different zenith distances will be ob
tained. By a comparison of the same with the inclination
found for the horizontal position we can then obtain the cor
rections, 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 3, graphic construction a curve,
the abscissae of the several points being the zenith distances,
and the ordinates the observed corrections of the inclina
tion. Then for those zenith distances, for which the cor
rection 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, con
structed 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
circleend, 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 sub
tracted, 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 u z . 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 in
strument, and consists in an arc 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 ?/, 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 di
vision 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 , then its distance from the zero of the limb will be:
y + p a .
But it is also qa\pa, where qa is that division of the
limb, which precedes the zero of the vernier, hence we have :
y + 1> a = q a + p ,
and therefore the distance of the zero of the vernier from
the zero of the limb is:
y = qa}p (a a )
If we have : m a = (m 4 1) ,
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 ,. 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 u x = u/.. But if this is not the case, then the quantity
f 20G265, where L is the length of the axis, gives the difference of
Jj
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 : m
a =  a,
m H 1
therefore : ?/ = H
? 41
The quantity l is called the least count of the ver
nier. Therefore in order to find the distance of th*e zero of
the vernier from the zero of the limb or to read the instru
ment 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 m 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 instru
ment 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
=10". In order to facilitate the reading of the vernier,
m + 1
the 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:
, a a
a a = r or m=  , 1,
m 4 1 a 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
terms of the same unit.
Hitherto we have assumed, that:
ma = (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:
(?>i J 1 ) a = m a .
403
a
In this case we have : a a =
m
and y = q a p
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^? then we have in the first case:
m a = (m f 1 ) a A I ,
therefore using the same notation as before:
pa ^l
Therefore if the length of the vernier is too great by ^/,
we must add to the reading of the vernier the correction :
p  A/
where p is the number of the division of the vernier which
coincides with a division of the limb and mf1 is the num
ber 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 I = + 5", we must add the correction ~ 5". The length
of the vernier can always be examined by means of the di
vision 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 divisionline 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 gra
duation of the circles. The reading is accomplished by a mo
veable 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 sub
divisions 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 prece
ding divisionline of the circle is measured by means of the
screw head. The value of one revolution of the screw ex
pressed 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 micros
cope, 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 revolu
tions is equal to the interval between two divisionlines 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 eyepiece; but since by this operation the image is thrown
oft 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 divisionlines 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 dif
ference between an entire number of revolutions and the meas
ured distance of two division lines, must be frequently de
termined and the reading of the microscope be corrected ac
cordingly *). But it is not indifferent, which two lines of
the circle are chosen for measuring their distance, since this
can slightly vary 911 account of the errors of division ; there
fore the exact distance of two certain lines must first be
found and then the run of the microscope always be deter
mined by these two lines.
The micrometer screw itself can be defective so that by
equal parts of a revolution of the screw the wires arc 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 n a.
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 dis
tance 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 to a a
from a to 2 a a"
from (n 1) to nn 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 ob
servations 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:
the correction, which must be added to the reading of the
screw, if also the interval from a to and that from na
to (n f 1) is measured and the corresponding distances
are denoted by a~ l and o" +l , will be:
for a a + a~ l
a a!
2 2 a a"
(?i 1) = (n 1) a a " ~ l
na=
*) 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 from 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 I y. Then after moving the screw to its zeropoint 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 120jp, then the read
ing of the screw, if we had moved it from zero through
exactly 120 seconds, would have been 120fp y\ there
fore all readings must be corrected by multiplying them by:
120
1204/J y
It must still be shown, how the length of an interval
between two certain lines, for instance that between and
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 2 is between the wires , and then moving the latter
by means of the screw, until the line is between them.
The length of the interval expressed in parts of the screw
head shall be from the mean of many observations 120fic. If
then in the same way a large number of intervals at diffe
rent 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
Fig. 13.
head. Now if the mean be 120fw, the first interval is too
large by x u = y or is equal to 120h?/.
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 of 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,
C that of the alhidade and that
the direction C A or the angle
OCA have been measured equal
to A 0, supposing that the
angles are reckoned from 0.
Then, if the excentricity were
nothing, we should have read
the angle ACO = A C 0. De
noting the radius of the circle
CO by r and the angle ACO =
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 1 O) e = A 1 C cos (A 0) ,
where e denotes the excentricity of the circle.
If we multiply the first equation by cos (A 0), the
second by sin (/! 0) and subtract the second from the
first, we obtain:
A C sin (A 40 = sin (A 1 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 1 0),
therefore we have:
sin (A 1 0)
tang (A  A } = 
1   cos (A 0)
409
or by means of the formula (12) in No. 11 of the intro
duction :
A A = sin (A 0) h 4 ~ sin 2 (A 1 0}
r ~ r*
e 3
+ 1 ^ sin 3 (A 1 0) + . . .
Now since  L 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 = sin (A 1 0) 2062 G5 ,
r
whence we see, that the error A A expressed in seconds
can be considerable on account of the large factor 206265,
although  is very small.
In order to eliminate this error of the reading caused
O
by the excentricity, there are always two verniers or micros
copes 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 1 0)
r
and
B = B +^sin. (B <9),
and hence:
 (A + B) = i (A 1 H B") + 4 sin [ J (A 1 h B ) 0] cos \ [A 1 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 arith
metical mean which we should have found if the excentricity
had been nothing. For this reason all instruments are fur
nished 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 = 13 A 4 2 cos [4 (A 1 4 B ) 0} sin ,1 (B 1 A )
or supposing that the angle between the verniers differs from
180 by the small angle a:
B A = 180 + 4 2 sin (A 1 0)
= 180 4 4 2 cos <9 sin J 2 sin cos A .
r r
and 2 sin = y,
If we take now:
e
r
we obtain:
[XA ] = 4 z sin A y cos A\
and hence we can find the unknown quantities , z and y
by readings at different places of the circle.
Example. With the meridian circle at the Berlin Obser
vatory the following values of B A 180 were observed
for two microscopes opposite each other:
X =40". 3 X,, =41". 5
v i 9 q v (\ n
*TA_ 3 Q """P" O O ^\ 210 ~~~~ U . D
X 90 =43 .1 X a70 =H0 .7
y _ i /tQ "V" O X
^120 * . O ^300 . U
From this we find the sum of all these quantities :
hence :
Moreover we find according to No. 27 of the intro
duction :
A XA XA XA XA
415.1
410.4
42.4 4 2.4
4
.3
1 .
2
30
1
.5
7 .3 4
60
4
1
.3
4.
2
4
90
4
3
.8
4
120
4
5
.5
4
150
4
5
.8
4
180
4
1
.5
and
hence :
t"
y = 4
9"
.62
2 = 4
18
.96,
therefore : = 26 54 . 2 and = 1". 772.
r
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 f a 2 cos 2 A f .....
f b , sin A j 6 2 sin 2 A f .....
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:
and
and if we now take the mean of all readings, a large num
ber of terms of the periodical series for the errors of divi
sion 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 i times 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 micros
copes.
The errors of division are determined by comparing in
tervals between lines, which are aliquot parts of the circum
ference, with each other. For instance if the errors of divi
sion 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 under one microscope, which we
leave untouched during the entire operation, and measure the
distance of the line marked 5 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 5 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 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 and the readings for the lines 5, 10 etc. by ,
" etc., the error of the line 5, taking that of the line as
nothing, will be , that of the line 10, 2a a " etc.
But since the circle undergoes during so long a series chan
ges 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 and 180,
be determined with the utmost accuracy, and then relying
upon these , the errors of the lines 90 and 270 " be deter
mined by dividing the arcs of 180 into two equal parts;
and then by dividing the arcs of 90 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 ancs 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 dis
tance from each other over the graduation. For small in
tervals, 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 rec
tified according to No. 5, and it is best, to use always the
same microscope for measuring and to arrange the observa
tions even so, that always the same portion of the micro
meter 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 180. When the line 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 read
ing of the other for the division line 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 er
rors of the lines 90 and 270", the arcs to 180 and 180
to were divided into two equal parts by placing the two
microscopes at a distance of 90. If then the line was
brought under the first microscope, the reading of the second
microscope for the line 90 was 6". 5, whilst when the
line 90 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, f 4". 11.
The arithmetical mean of 6". 5 and +4". 11 gives 1". 19,
hence the error of the line 90 is f5".31. In a like man
ner the errors of the lines 45, 135, 225 and 315 were
determined by dividing the arcs of 90 into two equal parts.
Then the errors for the arcs of 15 might have been de
termined by dividing the arcs of 45 degrees into three equal
parts. But .since the microscopes of the instrument cannot
be placed so near each other, arcs of 315 and 225 were di
vided into three equal parts. For this purpose the micros
copes 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 j2".0 or if we add
to the last reading the error of the line 315, which was
found 0".48, we get 11". 9, 5". 6 and fl".2. The
arithmetical mean of all is 5 ".33, hence the error of
the line 105 " is +6". 57, that of the line 210 is equal to
414
2cr a " = f6". 84. If the first line which we use is
not the line 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", 195
and 300", the readings of the second microscope for the lines
195, 300 and 45" were successively 6".6, H2".l and 7".9.
Now since the errors of the lines 90" and 45" have been found
to be H5".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 195 is 4 7". 23, and that of 300" is 40".30.
The errors thus found are the sum of the errors of di
vision 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 divisionlines 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 cossh b s\n z \ a" cos 2s + 6" sin 2z + a" cos 3 z h 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 180" tz 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 180f3,
then the mean of the measured distances is nearly free from
flexure and only those terms dependent on 2s, 4z etc. re
main in the result. If we repeat the observations in 4 po
sitions of the circle, 90 different from each other, then only
the terms dependent on 4s, 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 90 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 de
termined. 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 3 f ft sin4^ha 1 cos8s+6 1 sin 8s 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 inves
tigated 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 + b sin z. However the case of a perfectly homogeneous circle, which
he considered, will hardly ever occur. Usually the higher powers of the ex
pression 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 H e cos z + e" sin z + e 2 cos 2z + e" 2 sin 2r] sin (A 0,),
where A is the reading of the circle, z the zenith distance of the zero of
the circle, and O z 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 meas
uring 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 dis
tance 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 ob
servation. Then if x and x are the errors of division of the
first and the last line, and , a" etc. are the observed inter
vals between the first and the second, the second and the
third line etc., we have:
+ a " .+. a > _f_ . . . .+ x > x
15
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 H a
that of the second x + 2 a a"
that of the third x + 3a a a" "
and so forth.
Compare on the determination of the errors of division:
Bessel, Konigsberger Beobachtungen Bd. I und VII, also
Astronomische Nacbrichten No. 841. Struve, Astronomische
Nachrichten No. 344 and 345, and Observ. Astron. Dorpat.
Vol. VI sive novae seriae Vol. Ill; Peters, Bestimmung der
Theilungsfehler des Ertelschen Verticalkreises der Pulkowaer
Stern warte.
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 displa
cement shall be denoted by . If now we turn the circle
so that its zero has the zenith distance a, 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 .
If we denote by a^ the displacement of the line A, when the
zero has the zenith distance , which shall be reckoned in
the same direction from to 360, then ctg can be expressed
by a periodical series of the following form:
a cos h a" cos 2 + a " cos 3 + ...
f // sin + b" sin 2 f b" sin 3 f ...
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 coef
ficients themselves can thus be expressed by periodical series,
depending on the reading of the circle, so that the displa
cement of any line u of the graduation , when the zero has
the zenith distance c, can be expressed by a periodical series
of the form:
a ,, cos f a" u cos 2 f " cos 3 f . . .
H b tl sin 4 6",, sin 2 h & " sin 3 4 . . . ,
where a , b u 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 fret it from flexure.
Now a complete reading of the instrument is the arith
metical 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 dis
tance a, the line z will be under this microscope, and since
in this case the zenith distance of the zero is z + m, we
have in this case u = z, C, = 3fm; hence the correction
which is to be applied to the reading of the microscope, is:
a x cos (z 4 m) 4 a" ,. cos 2 (z + m) + a "* cos 3 (z 4 ni) + . . .
4 //, sin (2 4 ?n) 4 &"* sin 2 (2 H m) 4 &" * sin 3 (2 f ?>0 4 . . .
For the other microscope, whose reading is 90 f a, we
have w = 90  s, c = 3rw; hence the coefficients in the
expression for flexure become a ^^, 690 + 5 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:
. cos (2 4 + " cos 2 (.2 4 m) 4 a ", cos 3 (2 + ;w) 4 . . .
4 , sin (z 4 ?) + ^ ". sin 2 (2 + m) + /? "* sin 3 (2 f + ,
where the several a and /? are periodical functions of a, but
contain only terms in which 4z, 82 etc. occur, since all the
other terms are eliminated by taking the mean of four read
ings. 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 flex
ure, and since m is constant, the expression for the correc
tion which is to be applied to the mean of the readings of
4 microscopes will have the form:
a cos 2 4 a" cos 2 2 + a " cos oz 4 . . .
4 b sin z 4 6" sin 2 z + b" sin 3 z 4 . . .
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
much as the centre of the wirecross, 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 collima
tioii) remains parallel to a certain fixed line of the circle.
But if the flexure at both ends is different, the line of colli
mation 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 ex
pressed 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 in
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 *, its image reflected from an artificial
horizon will be seen at the zenith distance 180 z, 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 read
ing for the direct observation is now 360 z and that for
the reflected observation 180 4 z. Therefore if we denote
the four complete readings, corrected for the errors of division,
for those four observations by 3, , 5" and 3 ", and by the
true zenith distance free from flexure, we have the following
four equations, in which N denotes the nadir point:
Direct = .2 + a cos z f a" cos 2z f a" cos 3z f .. + b sin z
Reflected 180 = * a cos z f a" cos 2 z. a" cos 3 z +..+ b 1 sin z
 &"sin2*h 6 " sin 3z . . (180+iV) ha a"+a "
Direct 360" > = z H cos z 4 a" cos 2zf a " cos 3zf .. b sin z (B
 &"sin2z b "sm3z.. (lSQ+N)ia a"{a"
Reflected 180 +=2" a cos z + a" cos 2. z a" cos 3z f . . b sin z
H b"sm2z b 1 " sin 3z 4 . . (180+^) 4 a a"fa ".
From these equations we obtain:
90 =   a cos s a" cos 3s . . b" sin 2* . . .
+ cos * + " cos 3*  . .  6" sin 2*  . . . ,
hence by taking the mean :
and we see therefore, that if a star is observed direct and
reflected in both positions of the instrument, only that por
tion 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 f a" a " f . .
27*
420
}// v sin4a etc. remains in the mean of those four obser
vations.
We obtain also from the mean of the first two equations (JB):
90 == ~~ ~ h a" cos 2.c f . . 4 6 sin 2 + b " sin 3^ + ...
likewise:
jj . ^/;;
270 = H 1  f " cos 2c + . . V sin z b " sin 3z . . .
 (180 iN ) h a a" + ",
from which we find:
6 sin ~ ~~ 2 6 " sin 3 z + + N ~ N> 
Therefore if we observe different stars direct and re
flected in both positions of the instrument, we can find from
those equations the most probable values of the coefficients
a", a lv etc. and & , b " etc.
Since these observations are made on different days, it
is of course necessary to reduce the zenith distances 3, a , z"
and a " to the same epoch, for instance to the beginning of
the year by applying to the reading of the circle the reduc
tion to the apparent place with the proper sign. Since, be
sides, 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 lat
itude 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 a, 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 cor
responds 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 a.
421
A second method of eliminating the flexure was pro
posed 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 instru
ment. Thus in exchanging the object glass and the eyepiece
the equilibrium is not at all disturbed and it can be assumed,
that the effect of the force of gravity on 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 will correspond to the nadir, and the reading of the
same microscope will be 180f the zenith distance. There
fore if f is the zenith distance free from flexure, and if the
readings corrected for the errors of division are in the first
case 3, and in the other 3 , we have:
= z H a 1 cos z f a" cos 2 z f a " cos 3 z + ...}// sin z
h&"sin2?h&" sin3z. . . (180 h N) + + " ..
= * a cos z h a" cos 2z a " cos 3. c; h ... b sin z
f b"sin2z b "sm3z. . . (180 hiV ) a a " a " ..
Therefore we obtain from the mean of those two equa
tions, denoting the zenith points 180 f IV and 180 f IV by
Z and Z :
Q
whence we see that the arithmetical mean of the zenith dis
tances in the two cases contains only that portion of flex
ure, which is expressed by the terms dependent on 2z, 4 z etc.
We also obtain by subtracting the above equations:
hence we see, that we can determine the coefficients of the
terms dependent on 2, 3 2, etc. by observing stars at various
zenith distances or by means of a collimator placed at va
rious 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 in
strument, and direct them towards each other through aper
tures, made for this purpose in the cube of the axis of the
instrument, so that the centres of their wirecrosses coincide.
Then the telescope being directed first to the wirecross 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 col
limator by , we have in one position:
= 2 4 a cos z + a" cos 2 z + a " cos 3 z f ... f ft sinz 4 b" sin 2z
h b " sin 3z + ... Z + a a" + a "
and in the other position:
180t=2 a cos z+ a" cos 2z a " cos 82 +... b sin z + b" sin 2 2
 b " sin 3z + . . . Z H a a" + a ",
therefore :
= g a cos z a" cos 3. z ... b sin z b" sin 3 2 ...
Since we use in reading the circle both times the same
division lines, the observed quantity * 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 coeffi
cients.
There is no difficulty in making these observations when
the telescope is in a horizontal position; but when the incli
nation 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 in
stead 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 posi
tion *). If then outside of the eyepiece of the lower colli
mator 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 incli
nation 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
wirecross of the collimator but also its image reflected from
the mirror. Hence by turning the collimator, until the wire
cross and the reflected image coincide, we place its axis per
pendicular 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 wirecross 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 3, 3s, 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 (#):
180= Z ~ia coszia"cos2z\a" cos3z+... + a a"fa ",
2i
where Z= 1801 IV, Z =180}/V ; and where z and a" 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 mul
tiples of a. 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
from 90 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 45 , 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 135, 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 co
vering 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 col
limator 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 tjie telescope,
when directed to the mirror, makes with the line towards the
nadir is:
45 HT(* 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 6
the inclination of the horizontal collimator as found by means
of the level, taken positive when the side nearer to the in
strument is the higher one, then this angle will be :
45 f 0".0052 (h //) + j b.
If we denote this angle by f, 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 .3, we have:
= z z a (l 4J/2) f a" a 1 " (I + ]/ 2) & / 2 f b" 6 "^ 1/2.
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 z and by z" the reading of the circle, when
the telescope is directed to the mirror, then we have in this
case:
e=z" s + a (liyya"+a "(l + $V2) b f W2 + b" b "iy2,
therefore we have:
4(: + ) = 2 "~ 2 & ^2H&"& "*l/2...,
provided that the nadir point is the same for both obser
vations.
E. On the examination 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. Ho
wever this condition is never rigorously fulfilled, since not
only the threads of the screw are not exactly equal for dif
ferent 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 dif
ferent spaces. It has been shown already, how the irregu
larities of the screws of the reading microscopes can be deter
mined, 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 perio
dical series of the form:
a, cos u f b l sin u + 2 cos 2u f b 2 sin 2u f .
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 x , b l 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 (for instance, the distance between two
wires of a collimator) by bisecting each point by the moveable
wire of the micrometer, then, if u and u are the indications
of the screw for those positions of the moveable wire, we
have:
/== u u f a, (cos u cosw) f 6, (sinw sin w) { a 2 (cos2w cos2)
H 6 2 (sin 2 u sin 2u) H . . .
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 O r . 00,
when the moveable wire bisects one point, the next time
O r .10, then O r .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 arith
metical mean of all observed values of u M , and we can
take u j f instead of u . Therefore if we denote this arith
metical mean by /", every observed value of u u gives an
equation of the form:
u u /= 2a, sin ^/sin (u + /) 2 6, sin 4/cos (M f /)
+ 2 2 sin / sin (2 u }/) 2 6 2 sin / cos ( 2 u + /)
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 4/= *S(u . u /) sin (u 4 J/)
10 6, .sin 4/= 2(u M /) cos (u 4 1/)
10 a 2 sin /= 2(u u /) sin (2u +/)
10 6 2 sin /= 2 (V M /) cos (2 M 4/) ,
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 u
Starting point 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
. 50045
. 49690
. 49440
. 49240
. 49260
. 49555
. 49905
. 50140
. 50340
. 50350
/== . 497965 = 179 16 . 0.
From this we find :
u u f
4 . 002485
 . 001065
 . 003565
0.005565
 . 005365
 . 002415
40.001085
H . 003435
4 . 005435
4 . 005535
( /) sin (
4 . 002485
 . 000865
0.001123
40.001686
4 . 004320
40.002415
 . 000882
 . 001083
40.001646
4 . 004457
sum 40.013056,
and since sin  f = 1 , we have :
10 ,== 4 0.013056
as: 106, = 0.024874
0. 1 28 2 = 4 0.000147
0.128 6 2 = + 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. 335) a, =  0.015915
7.339 &, = 0.016126
9. 970 a, = 0.004987
9 . 970 b o = . 000576,
and from these two determinations he obtained according to
Note 2 to No. 24 of the introduction:
, =40 . 001608
b i = .002386
2 = .000499
ft a = .000057.
These periodical corrections of the screw must be ap
plied 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 bi
section of one object is O r .25 and then again, when the
reading is 40 . 25 at the bisection of the same object, so
that u for these two observations is equal to 90 and +90,
then in the expression for f the terms a t (cos?/ cos?/)
t6 1 (sin?/ sin M) will be in one casefctj cosw +6 (sin?/+l)
and in the other case a^ cos u b l (sin u + 1), and hence
this portion of the correction, dependent on a l and b l) will
be eliminated by taking the arithmetical mean of both ob
servations. Likewise the result will be free from that por
tion of the correction dependent on r/, 6, nr 2 and 6 2 , if we
take the mean of 5 observations, arranging them so that the
reading of the screw for the bisection of one object is in
succession O r .4, ^O r .2, 0, f0 r .2 and hO r .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 mul
tiple 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 O r , 10 r , 20 r , etc. Thus he found:
Reading of the scale at the beginning (X 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 in
dications of the scale were 9 r .6, 9 r .8, 10, 10.2 and 10.4.
If now the true distance is 10 r \x, and the corrections
of the screw for the readings of the scnle 10, 20, etc. are
AIM Am etc  th en we have, since we can take /" 0:
Xl = H o . 0142 +/i
X} =H0.0147H/ 20 / 10
*, = + 0.013H/ 30 / 20
etc.
Likewise he measured a distance, which was equal to
20 r H# 2 , in the same way and obtained thus another system
of equations:
a: 2 =h/o
x 2 =H/ 40 f., Q
etc.
Similar systems were obtained by measuring a distance
equal to 30 H # :! , and from all these equations he found the
values of #, # 2 , x.^ etc. as well as the corrections of the
screw for the readings 10, 20, etc., that is, /" 10 , /2 , 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 car
rying another circle to which the verniers are attached. On
the circle stand two pillars of equal length, which are fur
nished 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 ver
tical 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
coordinates 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 al
titude of this point shall be denoted by 6. 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 a . For any other
position we will suppose that we read always .that point of
the circle, in which the arc PK 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 w r e imagine three rectangular axes of coordi
nates , 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
above, then the coordinates of the point K referred to these
axes will be :
z = s in b , y = cos b cos A
and x = cos b sin A.
Moreover the coordinates of K referred to three rect
angular 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 :
z = sin i , y == cos i cos (a ) , x = cos i sin (a a ).
Now since the axis of z in the first system makes with
the axis of z of the other system the angle , we have ac
cording to the formulae (1) for the transformation of coor
dinates :
sin b = cos i sin i sin i cos i cos (a )
cos b sin A = cos i sin (a )
cos b cos A = sin i sin i f cos i cos i cos ( ).
We can obtain these equations also from the triangle
between the zenith Z, the pole of the horizontal circle P and
the point /f, whose sides PZ, PK and ZK are respectively
i, 90 i and 90" b , whilst the angles opposite the sides
PK and ZK are A and 180 (a a,,).
Now since 6, i and i are small quantities, if the in
strument is nearly adjusted, we can write unity instead of
the cosine and the arc instead of the sine, and thus we obtain:
b = i cos (a ) (a)
A = a a .
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 he with the side of the axis towards
the circle. The angle c is called the error of collimation.
It can be corrected by means of screws which move the
wire cross in a direction perpendicular to the line of col
limation.
The telescope shall be directed to the point 0, whose
zenith distance and azimuth are z and e, and whose coor
dinates 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 /if; and hence if we suppose, that the axis of y
is turned so that it lies in the same vertical circle with /if,
the coordinates 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 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 coordinate y of the point is equal to
sine, and since the angle between the axes of z of the
two systems is 6, we have according to the formulae for the
transformation of coordinates:
sin c = cos z sin b + sin z cos b 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, ZK and OK are
respectively equal to z, 90 b and 90fc, and the angle
KZO is equal to PZ PZ K= e A.
Since b and c are small quantities, we obtain:
c == b cos z f sin z cos (e J.),
or finally, substituting for A its value from the equations (a) :
= c ( b cos z 4 sin z cos [e (a a )].
Hence it follows, that
cos [e (a a )]
is a small quantity of the same order as b and c. Therefore
if we write instead of it:
sin [1)0 e\(a )],
we can take the arc instead of the sine and obtain:
= c + 6 cos z h sin z [ ( JO e f (a Q )].
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 4 b cos z + sin z [ ( JO (a a ) + c].
Therefore we obtain the true azimuth e by means of
the formulae:
433
e = a a + 1 JO f  4 b cotang z Circle left
sin 2
and:
e = a 90 . 6 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:
c = A + &A^=c cosec z =*= 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.it. 11. We can find these formulae also by a
geometrical method. Let AB Fig. 14 be the vert
ical 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 ft, it will de
scribe 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 A Z B , say 0, and hence, when the circle
is on the left side, we shall find the azimuth too
small. Now we have:
sin O = sin A sin b
= cos z . sin b.
But we read the angle at Z subtended by , and there
fore the angle Z is the sought correction A A of the azi
muth. Now since:
sin = sin Z sin A A,
and hence :
sin A A = cotang z sin b,
we must add to the reading of the circle on account of the
error 6, when the circle is left:
t l> cotang z.
In a similar way we can find the correction for the er
ror of collimation. Let AB again be the vertical circle, which
the line of collimation of the telescope would describe, if
28
434
FL>. is. there were no error of collimation. But if the
angle between this line and the side of the axis
towards the circle be 90 f c, the line of colli
mation will describe, when the telescope is turned
around, the surface of a cone, which intersects the
sphere of the heavens in a small circle, w r hose dis
tance from the great circle AB 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 A .4, we have:
sin c
SIM &A =
sin z
or :
&A = H c cosec ~.
12. It shall now be shown, how the errors of the in
strument can be determined.
The levelerror is found according to the rules given in
No. 1 of this section by placing a spiritlevel upon the pi
vots of the horizontal axis. But we have according to the
equation (a) in No. 10:
b = i i cos (a ),
where i is the inclination of the horizontal circle to the hor
izon, i the inclination of the horizontal axis, which carries
the telescope, to the horizontal circle. This equation con
tains three unknown quantities, namely i , i and (1 , and hence
three levelings in different positions of the axis will be suf
ficient for their determination. We will assume that the in
clination b is found by means of the level in a certain posi
tion of the axis, when the reading of the circle is a, then
it is best, to find also the inclinations b L and 6 2 in two other
positions of the instrument corresponding to the readings
aj120" and af140. For if we substitute these values in
the above formula, develop the cosines and remember that:
cos 120 = ^
and
sin 120 = +
cos 240 =
moreover :
and
sin 240 = 41 o,
we obtain the following three equations:
435
b = i { cos (a a )
b i = i + 4 i cos (a ) + \ i sin (a ) ]/ 3
6 2 = i + ^ i cos (a a n ) 1 1 sin (a a,,) J 7 3.
If we add these three equations, we find:
i _ ?LAI A>
3"
But if we subtract the third equation from the second,
we obtain:
.  f v b l b 9
i sm (a a ) = ,7~^
V "
and if we add the two last equations and subtract the first
after being multiplied by 2, we find:
, , 2b
i cos (a ) =  5
o
Therefore if we level the axis in three positions of the
instrument, which are 120 apart, we find by means of these
formulae, i, i and a , and then we obtain the inclination for
any other position by means of the formula:
b = i i cos (a ).
Iii order to find the collimation error, the same distant
terrestrial object must 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:
G = A H i\A + b cotang z f c coscc z
e = A \ &A b cotang z c cosec z,
from which we find:
A A b + b
c cosec z ~~aT~ 9 cotang z.
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 Fig. 16 with a
telescope, which is fastened to one extremity of the axis, it
is seen in the direction OF. The angle OFK shall be 90Jc y .
28*
436
Now if we imagine a telescope
at the centre M of the axis, and
directed to 0, then the angle
OMK will be 90 he. We have
therefore :
c = , :o hJ/0F.
But we have :
tang 3/0 F = y
where d is the distance of the ob
ject OJH, and o is half the length
of the axis, and hence, if c () is very small, we get:
 cosec c,
Therefore if we observe a terrestrial object with an in
strument whose telescope is at one extremity of the axis, the
reading of the circle will be too small by the quantity^ cosec z,
when the circle is on the left, and too large, when the circle
is on the right side. Therefore if these two readings be de
noted by A and A\ we have the two equations:
e = A + &A \ 1) cotang z 
e = A \ A A 6 cotang z I
from which we can find the collimationerror, 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 collimationerror variable with the zenith distance. When
the telescope is horizontal, the flexure has no influence on
the collimationerror, since it merely lowers the line of col
limation, 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 collimationerror in this
case can be expressed by the formula c h a 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 can be used for finding the col
limation error, it may be determined by observations of the
polestar. For, if we observe the polestar at the time t,
read the circle and then reverse the instrument and observe
the polestar a second time at the time t\ we shall have the
two equations :
e = A f A^4 f b cotang z f c cosec z
and
e = A { &A b cotang z c cosec 2,
and since we have:
where denotes the change of the azimuth at the time  ,
we obtain:
A A dA t t
2 ~~dt ~2~
Finally, in order to find the index error &A, we observe
again a star, whose place is known, for instance the pole
star and read the circle. If then the hour angle of the star
is , we compute the true azimuth e by means of the for
mulae :
sin z sin e = cos sin t
sin z cos e = cos y> sin \ sin cp cos 8 cos t,
and we obtain :
{\A = e A=f= b cotang z =p c cosec z,
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 azi
muth, 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 180
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 pre
supposes, that the angles i and i as well as the error of
collimation 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 K and KP coin
cide. (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 90. On the contrary, if these sides are equal
to 90 + c and 90 i\ we have, denoting PO by and
the reading of the circle, that is, the angle PRO by f:
cos = sin c sin i + cos c cos i cos
= cos (t f c) cos ^  cos (i c) sin 4 2 .
If we subtract cos from both members and write ( C) sin
instead of cos cose , which is allowable, because f
is small, we obtain:
== + sin k (c + i ) 3 cotg 4 % sin \ (i c) 2 tang g
or:
= H 9 cotg I i c cosec ;
C is then the zenith distance referred to the pole of the in
strument 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 in
strument to the horizontal circle, we must take its inclination
to the horizon, that is:
i i cos (a ..) = &
and besides, we must subtract from the reading of the vert
ical circle the projection of PZ on the circle or the angle
PKZ = isin(a a,,). This angle is always found by means
of a spiritlevel attached to the vertical circle. If we denote
by p the reading of the level on that side, on which the di
vision, starting from the highest point, increases, and that
on the opposite side by w, 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 in
strument Zf(/? w) an( l in the other Zi$(p  ). There
fore if we denote the readings in the two positions by and
\, then the zenith distance in one position will be:
Z (p rie,
where e expresses the value of one part of the scale of
the level in seconds, and we shall have in the other position:
and hence we find from the arithmetical mean the zenith
distance :
+
n) e H j (p ~ n) s
_
~ ~~ "2 ~ 2
and in order to obtain from this the true zenith distance,
we must add the correction:
Hh sin I (b + c) 2 cotg 3 sin 4 (b c) 2 tang 4 z
or:
+ cotgz f be cosec 2 .
If we take 6 = 0, since we have it always in our power
to make this error small, we have simply to add:
C "
H Q 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 there
fore 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 instru
ments from the formulae for the azimuth and altitude in
strument. 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. There
fore if we simply substitute for the quantities which are re
ferred to the horizon, the corresponding quantities with re
spect 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 f 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 instru
ments, 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 # f90 () must always be very small. Denoting
the small quantity by which it differs from zero, by &, the
formulae given in No. 10 are changed into:
e = k f b cotang z + c cosec z Circle left
e = k b cotang z c cosec z Circle right.
When e is not equal to zero, the body will not be ob
served 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 r
is the hour angle of the body at the time of observation,
taken positive on the east side of the meridian. Now since :
sins
sin T = sin e . ^
cos o
sins
or: r== e. ,
COS
the formulae given above change into :
and :
cos z sin z _, . , , , N
b 5 FA csectf Circle left (east)
COS O COS
T = 4 6 *\~k ~*+ c sec 3 Circle right (west),
cos o cos o
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 in
strument are deduced. We have, namely, according to No. 7
of the first section:
cotang A sin t = cos y> tang 8 f sin (f cos t
or, if we reckon the azimuth e from the prime vertical, so
that 4 = 90 he:
tang e . sin t = cos (f tang sin <f cos t.
441
Now if (*) is the time at which the star is on the prime
vertical, we have:
= cos y> tang sin (p cos
and if we subtract both equations:
tang e sin t = 2 sin cp sin 4 (t 0} sin \(t\r &)
From this we find, if e is small and therefore t is nearly
equal to 6*:
e = (t 0) sin y
or:
= t .
sm <p
If we substitute here fore the expression found before:
e = k =t= b cotang z == c cosec z,
we obtain the following formulae for the prime vertical in
strument :
k cotaner z cosec z
= +  =p 6 =F c
sin y sin y sm 9?
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 coordinates, that of the altitudes and
azimuths, so the equatoreal corresponds to the second system,
that of the hour angles and declinations. With this instru
ment therefore that circle, which with the other was horizon
tal, is parallel to the equator. Now let P be the pole of
the heavens, /7 that of the hour circle of the instrument.
Further let k be the arc of the great circle between those
two points, and h the hour angle of the pole of the instru
ment. 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, pro
duced 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 w^e obtain, when
/f, P and // are on the same declination circle. And we
442
will assume that every other reading gives us that point of
the circle, in which it is intersected by the great circle pas
sing through P and //. This point differs from 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 co
ordinates, 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 coordinates of the
point /f, referred to these axes, are:
z == sin D, y = cos D cos T, x = cos D sin T.
Further, the coordinates of If, referred to three rect
angular 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 x coinciding with that of the former sys
tem, are:
2 = sini , y = cos i cos (t <), x = cosi sin(i J ).
Now since the axes of z of these two systems make
with each other the angle A, we have the following equations:
sin D = cos A sin i sin A cos i cos (t ? )
cos D sin T cos i sin (t ^ )
cos D cos T sin A sin i .+ cos h cos i cos (t ? ).
Since A, i and D are small quantities, if the instrument
is nearly rectified, we obtain:
D = i I cos (t O
T=tt .
The telescope is attached to the declination axis and we
will assume, that the part of its line of collirnation towards
the objectglass makes with the side of the axis, on which
the circle is, the angle 90 f c, c being called the collima
tionerror. Now if the telescope be directed to a point, whose
declination is <) and whose hour angle, reckoned from the
adopted zero, is r,, then the coordinates of this point will be:
z = sin $, y = cos cos r l and x = cos sin r x .
We will assume, that the division of the circle in
creases in the direction from south towards west from
to 360 or from O h to 24 h . Therefore if the circleend is
443
west of the telescope, the latter is directed towards a point,
whose hour angle is less than that of the point K. There
fore if we imagine the axis of y to be turned so that it lies
in the same declination circle with /if, if the telescope is di
rected to the object, then the coordinates will be:
z = sin , y = cos 8 cos (T T^, x = cos 8 sin ( T TJ).
On the contrary, when the circleend is east of the teles
cope, these coordinates will be :
z sin 8, y = cos S cos (TJ 7"), x = cos 8 sin (T t 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 A , whilst the axis of x coincides with
the corresponding axis of the former system, then the three
coordinates of the point will be, 8 denoting the reading
of the declination circle:
z = sirt 8 cos c, y = sin c
and
X = COS 8 COS C.
Now since the axes of z of the two systems make with
each other the angle J9, we have:
sin c = cos 8 cos (T t T} cos D f sin 8 sin Z),
or
c = cos 8 cos (T ! T} f D . sin 8,
and hence, if we substitute for D and T the values found
before :
c = [i /I cos (t t Q )] sin 8 f cos 8 cos [r x (t )J.
From this it follows, that:
is a small quantity. Therefore if we write:
sin [90 T, +(* * )]
instead of
cos [TI (t Z )J,
we can take the arc instead of the sine and we find the true
hour angle:
r , = 90 {(t < ) A cos (t C tang JM tang 8 + c sec (?,
when the circleend is east of the telescope, and:
Tl =(t Z ) 90 h A cos (< * ) tang <? i tang c sec S,
when the circleend is west of the telescope.
If we add h to both members of these equations, we
444
reckon the angles from the meridian. Then r l j h will be
the true hour angle reckoned from the meridian and:
Ah* * H90"
and AH t t 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 A*, we have:
r = t + A t I sin [t + i\t h] tang 8 == c sec <? =t= { tang ,
or: T = z fA* Asin (T A) tang d== c sec 5 =1= i 1 tang #,
where the upper sign is used, when the circleend 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 //
and the point 0, towards which the telescope is directed, in
connection with the other triangle formed by //, and /if,
that is, the point in which the declination axis produced in
tersects the sphere of the heavens.
The sides of the first triangle OP, OH and P If are
respectirely the true polar distance 90 S of the point to
wards which the telescope is directed, the distance from the
pole of the instrument 90 <) , and /, whilst the angles opposite
the two first sides are 180 (r ti) and r /i, where T h
is the hour angle, referred to the meridian of the instrument,
and TI h the hour angle referred to the pole of the instru
ment and reckoned from the meridian of the instrument.
Hence we have the rigorous equations:
cos cos (r A) = sin 8 sin A j cos S cos A cos (r A)
cos S sin (r A) = cos S sin (r 1 A)
sin S = sin cos A cos sin / cos (T A) ,
from which we obtain in case that A is a small quantity :
T ==T /, tang S sin (T A)
= ;LCOS(T A).
But r and d are only then equal to the readings of the
circle, when i and c as well as the index error of the ver
nier are equal to zero. First it is evident, that the angle
90" d" t\d obtained by the reading of the declination
circle (where A^ is the index error of the declination
circle) is equal to the angle at K in the triangle 77 KO. The
angle S/70, S being a point on the great circle P/7, is
445
T h ; the reading of the instrument is the angle between
the position of UK at the time of observation and that, in
which TIP coincides with IIS. If the above conditions were
fulfilled, this angle would be r A, whilst the angle S/1K
would be 90 r A, when the axis is west, and T h 90,
when the axis is east of the telescope. If for the general
case we denote the latter angle by 90  r"   k + At
and r" /* h &t  90", then the angle ILK will be
equal to 90 J r" + A t *" , when the axis is west and
T (V jA^ 90), when the axis is east of the telescope,
or equal to 90=p(r ?;" AO Now since the opposite side
in the triangle is 90 + c, and since the side // 0, opposite the
angle 90" <T A<?, is90 <* , and ///T=90 i , we have:
cos 8 cos (r T" A i) = cos c cos (" h A #) ,
=J= cos <? sin (T T" A = sin c cos i" cos c sin z sin (8" f A#),
sin $ = sin c sin i ~ cos c cos z sin (8" f A $),
from which we obtain:
T = T" h A =F c sec (S" h A d) =F / tang (<T H A 5),
and in the same way as in No. 13 of this section:
8 = 8" h A 8 sin (i h e) 2 tang [45 H  (" 4 A 8)]
or also <? = 5" f &S 1 (i  1 4 c 2 ) tang (5" h A<?) i c sec (5" f A$),
and substituting these expressions in the equations above,
we find:
T = r" 4 A * ^ tang $ sin (T />) =p c sec $ =^= i tang $
^ = S" 4 A<? /I cos (T ; A) i (t"  h c 2 ) tang 5 z" 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:
<? = 360 8", & I cos (r A) ft 2 f c 2 ) tang 8 i c sec 5.
W. It shall now be shown, how the errors of the in
strument can be determined by observations. First we find
from the two last equations for d:
Afl=lSO (V i +5"),
and hence we see, that the index error of the declination
circle can be found by directing the telescope in both posi
tions of the instrument to the same object. As such we can
choose either a star in the neighbourhood of the meridian, or
446
the polestar, for then the change of the apparent declination
during the interval between the observations will be insigni
ficant.
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 instru
ment. We have namely for each star the two equations:
r =. T h ^r 1 sin (r h) tang f i tang f c sec d,
when the circle is east, and:
T! = T J + AT A sin (T } h} tang i tang S c sec 8,
when the circle is west. Therefore if the interval between
the two observations is short so that r T r is a small
quantity, we obtain, denoting the sidereal times of the two
observations by and 6^:
i tang B \ c. sec 8 =
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 /\ <Y, then the errors A and h as
well as the index error /\ 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:
T = r f A t ^ sin (r K) tang 8
and likewise for the second star:
r t = T ! + i\t Asin(rj //) tang x
From these equations we easily find :
"Vjfr ~1 3 8  (<?i 8 ^
A sin h \ =
. T r ,
A COS  9
* w v cos
2
and from these the values of h and A can be obtained.
447
The index error /\t is then found by means of one of
the equations for r or T I .
Since all the quantities obtained by the readings of the
circles are affected with refraction, we must understand by
r, r 19 d and l also the apparent hour angles and declina
tions affected with refraction. But if the observations are
not taken very near the horizon, we can use the simple ex
pression :
d h = a cotang h,
for computing the refraction, and then we obtain the cor
responding changes of the hour angle and declination by
means of the formulae:
, sin
at= a cotang k .  _
coso
d = + a cotang // . cos p,
where p is the parallactic angle, which is found by means
of the formulae:
cos (p cos t = n sin N
sin cp = n cos N
cos <p sin t
tang = ,
n cos (N h (?)
or:
cos h sin p = cos cp sin t
cos h cos;? = n cos (N \ 8}.
The altitude h is found by means of the equation:
shih = )i sin (N+ ).
If we substitute these values in the expressions for dt
and d<)\ we have also:
. a cos (p sin t
cos 8 sin CZVf )
d8 = H a cotang (A r { 5).
Now since sin p has always the same sign as sin f, the
hour angle is diminished by refraction in the first and sec
ond quadrant, but it is increased, or its absolute value is
diminished also, in the third and fourth quadrant.
If <> <; cp , then sin # cos rp is less than cos d sin cp and
hence cosp is always positive. Therefore the declination is
then increased by refraction. But if <> ></:., then cos p is
always positive when t lies in the second or third quadrant,
therefore then also the decimation 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 cp
cos Z .> 
tang o
When the errors h and A 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 arc of a great circle drawn from the pole perpendicular
to the meridian, and if x is the distance of the pole from the
point of intersection of this arc with the meridian, then we
have :
tang x = tang A cos h
and:
siny = sin k 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 A and h pre
suppose, that /, 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
/\ 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 vert
ical adjusting screws of the plate on which the polar axis
rests) the star is brought to the wirecross. The same ope
ration is then performed for a star whose hour angle is about
6 h , 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 pro
duces 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 in
strument, 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 what
ever position the instrument may be. The flexure of the tel
escope , which may be assumed equal to ; sin z , can be de
termined 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 a tang z f 7 sin z instead of a tang z.
The flexure of the declination axis has the effect, that the
angle * is variable with the zenith distance. Now if the
force of gravity changes the zenith distance of the point K
by ft sin z, then the corresponding change of its declination D
is ft sin z cos p, and that of its hour angle T is ft sin *L^P
cos D
or since in this case D is very nearly equal to zero , the
change of declination is ft sin y and that of the hour angle
ft cos cp sin T. But since we have :
T r =90(T" if the circleend is west
and =r" 90 if the circle end is east,
we have to take instead of this hour angle:
90 HT"^ cosy COST"
or T" 90 H fl cos <p cos T",
and hence we must use in the formulae given before
T"=F/?COS f/ cos T" instead of T" and i 4^siny instead of ,
since now FLK = 90" i ft sm (f. Thus we obtain:
T = r")&t Itgdsin^K) =f=csQc8=f=itgS=i={3tgd[sin(f>l cosy cotg COST].
Therefore i is in this case not constant, but we must take
instead of it:
i + fi [sin (f f cos y> cotang 8 cos r\.
Now the observation of a star in both positions of the
instrument gives an equation of the form:
c sec tff i 1 tang $+ p tang S [sin y> f cos <p cotg S cos r] = T " ^i~~ T> i^
and therefore we can determine c, i and ft by observing three
different stars in both positions of the instrument.
17. If the equatoreal is well constructed so that the er
rors can be supposed to remain constant at least for mod
erate intervals of time, and if the circles have a fine gradua
29
450
tion and are furnished with reading microscopes, such an
instrument can be advantageously employed to determine dif
ferences of right ascension and declination, and hence to
determine the places of planets and comets. For this pur
pose 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 instru
ment are read. Then in the same way also the star, whose place
is known, is observed. If the readings of the circle are cor
rected 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 ad
ded 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 com
parison 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 decli
nation by it, and these determinations are made by means
of a micrometer connected with the telescope, whilst the par
allactic 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 de
clination circle passing through the middle of this line. This
451
angle is obtained from the reading of the circle of the mi
crometer, whose centre is in the line of collimation of the
telescope. If the equatoreal is perfectly adjusted, then in
every position of the instrument the same point of the po
sition 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 pas
sing through the object and the pole of the instrument ma
kes with the declination circle. If we denote this angle by TT,
we have in the triangle between the object, the pole and the
pole of the instrument:
cos S sin ?t = sin 1 sin (i A)
or n = 1 sin (T A) sec 8.
Therefore we obtain from the reading of the circle P 1
the true angle of position P, reckoned as usually from north
towards east from to 360, by means of the equation:
P = p + p 4 I sin (T A) sec 8,
where &P is the index error of the position circle.
Compare on the equatoreal: Hansen, die Tiieorie des Aequatoreals, Leip
zig 1855 and Bessel, Theorie eines mit einem Heliometer versehenen Aequa
toreals. Astronornische Untersuchungen. Ed. 1.
IV. THE TRANSIT INSTRUMENT AND THE 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 me
ridian 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 par
29*
452
allel to the meridian, by which the azimuth of the instru
ment can be corrected.
One end of the axis supports the circle, which, if the
instrument is a mere transit, serves only for setting the in
strument. 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" A;, reckoning the azimuths as usually from the
south point through west etc. from to 360. Then we
have the rectangular coordinates 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 respecti
vely to the south and west points:
z = sin b
y = cos b cos k
x = cos 6 sin k.
If we denote the declination and the hour angle of this
point by n and 90 m, then we have the coordinates of
this point, referred to a system whose axis of z is perpen
dicular to the equator, whilst the axis of y coincides with
the corresponding axis of the former system:
z = sin n
y = cos n cos m
#= cos n sin m.
Now since the axes of z of the two systems make an
angle equal to 90 y> with each other, we have :
sin n = sin b sin 9? cos 6 sin k cos 90
cos n sin m = sin 6 cos y + cos b sin k sin y
cos n cos TO = cos b cos k.
453
The same formulae can be deduced from the triangle
between the pole, the zenith and the point (), towards which
the east end of the axis is directed. For in this triangle we
have ZP = 90 qp, Z = 90 f 6 , P Q = 90 f n and
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 9? k cos <p
m = b cos <p \ k sin 9?,
or the converse formulae:
b = n sin <p + m cos 9?
fc = n cos 99 f m sin 9?.
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 90hc, and that it is directed to an object,
whose declination is d 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 coordinates of the star with respect to the equator,
the axis of x being in the plane of the meridian, are:
z = sin S, y = cos sin r
an d x = cos S cos r,
or if we suppose, that the axis of x is perpendicular to the
axis of the instrument:
z = sin , y = cos S sin (r m)
an u O: = COS#COS(T m).
Here r m is the interval between the time of obser
vation and the time at which the star passes over the meri
dian of the instrument.
If now we imagine another system of coordinates, 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 S f cos n cos sin (r m).
454
In the case of the lower culmination, T m is on the
same side of the meridian, but since then the star is ob
served after it has passed the meridian of the instrument,
we must take r m negative. Therefore in this case the
coordinates of the point to which the telescope is directed
will be:
z = sin 8, y = f~ cos sin (r m),
and hence we have:
sin c = sin n sin 8 cos n cos sin (r m).
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 n sin + cos n cos 8 sin (T ni)
as the general formula, if for lower culminations we use
180 ti instead of J. These formulae can also be deduced from
the triangle between P, Q and the star 0, of which the sides
are P0 = 90 <?, P() = 90 Hrc, OP = 90 c, whilst
the angle P Q is equal to 90 + m r for upper culmina
tions and equal to 90 m + T for lower culminations.
From the above formula we find:
cos n sin (r m) = sin n tang 8 f sin c sec 8,
and adding to this the identical equation:
cos n sin m = cos n sin m,
we obtain:
2 cos n sin ^ r cos [\t m] = cos n sin m f sin n tang 8 + sin c sec 8. (a)
Now if we suppose the instrument to be so nearly ad
justed that m, n and T are small quantities, we find from this:
T = m f n tang 8 j c sec S *).
This is Bessel s formula for reducing observations made
with a transit instrument.
If T is known and T is the clock time of observation,
the clock time of the culmination of the star is Tjr. If
then A* is the error of the clock on sidereal time, then
TtrhA* w iU be the sidereal time of the culmination of
the star or be equal to its right ascension . Hence we have :
a = T 4 A t f m f n tang + c sec 8.
Therefore if A* 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 T in terms of b and &, if we substitute
the expressions:
cos n sin m = sin b cos fp f cos b sin cp sin k
sin ?z = sin b sin 92 cos b cos 9? sin k
in the equation (a). We find then:
COS (cp 0")
2 sin ^ T cos n cos [ t m] = sin 6
cos 8
and from this:
sin (cp $) s
h cos b sin k ~ ( c sec o,
, cos (fp 8) , . sm (fp )
b  iz  f k  =  ( c sec S.
cos o cos o
This formula is called Mayer s formula, since Tobias
Mayer used it for reducing his meridian observations. It is
the same formula which was deduced before from the for
mulae for the azimuth instrument.
Hansen has proposed still another form of the equation
for r, which is the most convenient of all. For if we
add the two equations:
, sin a? 2
sin n tang cp = sm b  cos b sin k sm m
cos cp
and
cos n sin m = sin 6 cos cp f cos 6 sin k sin rp,
we find:
cos n sin m = sin b sec fp sin n tang cp
and if we substitute this value of cos n sin m in the equation
(a), we obtain easily:
t = b sec cp \ n [tang tang cp] f c sec <?.
All these formulae are true, if the circle is on the west
side. But if the circle is east, then the altitude of the west
end of the axis is 6, and the angle, which the line of
collimation makes with the west end of the axis, will be
90 c, whilst A; remains the same. Therefore in this case
we have only to change the sign of b and c and we have
according to Mayer s formula:
For upper culminations
Circle West = T+ A< + 6 5?^$ +t !!?_?$ + c sec ,
COS O COS O
Circle East = T+ A t  b ^~ ^ + k ^rf _ e sec S.
COS COS O
456
For lower culminations we take 180 S instead of 8
and obtain :
Circle West a + 12 h = T\ A* h b 
. sin (ophd)
h k ^ c sec
cos 8
Circle East +12h = Tf A* 6  r  
cos o
. sin (OP h <?)
h A:  f c sec <?.
cos o
W^hen 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 for
mula, we must apply to each observation the correction:
n tang f c sec
and the error of the clock is then :
Tm.
If we take Hansen s form we apply the correction:
n [tang 8 tang (p\ j c sec 8
and obtain the error of the clock form:
a T 6 sec (f.
19. These formulae can be deduced easily in the fol
lowing way: If the circle is West, and 6 is the altitude of
the point to which the circleend 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 is observed, we must add to the time of
observation the hour angle:
Fig. n. r = OPO
But we have:
sin
sin T = sr
cos o
and
tang 00 = tang b cos Z = tang 6 cos (<p 8\
therefore :
If the azimuth of the instrument is &, the
telescope will describe the vertical circle Z A Fig. 17.
But we have again, if is the star:
__, sin 0O
sin OPO = sin T =  ,,
cos
457
and
tang 00 = tang k sin O Z,
therefore :
. sin (<p S)
r = K ~
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
r = ^ = c sec o.
cosS
For lower culminations we find the corresponding for
mulae in the same way.
20. The normal wire of the transit when perfectly ad
justed, 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 vert
ical, 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, un
til 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 in} cos n = sin n tang + sin c sec S.
Now if an observation was taken on a lateral wire whose
distance is /", 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:
90 Hc4/*),
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 (T m) cos n = sin n tang 8 f sin (c (/) sec ,
and subtracting from this the former equation:
2 sin \(t r ~) cos [4 (r { r) m] cos n = 2 sin ^fcos [c f \f\ sec S.
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 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 sin/sec d.
This rigorous formula is used for stars near the pole,
the value of sec d being then very great; but for stars far
ther from the pole it is sufficient to take:
If it is not required to reduce the lateral wires to the
middle wire, we can proceed also in the following way. Let
/", /"", /"" , etc. be the distances of the lateral wires on the
side towards the circle, and (p\ (p", (/> ", etc. those on the
other side, then compute:
where n is the number of wires. Then we must add to the
arithmetical mean of the transits over all the wires the quantity :
=J= a sec S
*) See Fig. 16 pag. 436, where O is the centre of the object glass, M
the middle 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 op
posite sign is taken.
The equation
sin t = sin/sec 8
serves also for determining the wire distances by observing
the transits of a star near the pole and computing:
f = sin t cos S,
where t is the difference of the transit over the lateral wire
and the middle wire, converted into arc. In this way the
wiredistances are found very accurately. For the polestar,
for instance, we have:
cos <? = 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 s . 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 coming 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 tel
escope. Therefore if there is at the focus of the first teles
cope a system of wires, it is seen plainly through the second
telescope, provided that those 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 instru
ment, 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 5 eye piece. After
that the telescope is directed to a star, and the entire tube
containing the wires and the eyepiece 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 eyepiece 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 ob
ject glass, if the eye and the image of the object move to
wards the same side from the wire. But if the eye and the
image move to different sides, the wires are too near the ob
ject glass *).
In 1850 June 20 Polaris was observed at the lower
culmination with the transitinstrument of the observatory at
Bilk, and the following transits over the wires were obtained :
Circle West.
I II III IV V
Hence the differences of the times are:
/ /// II HI III IV IIIV
27 m O s 13 m 57 13 m O 26 m 58 s .
Since the declination of Polaris on that day was:
88 30 18". 01
we find by means of the formula:
/= sin t cos
the following values of the wire distances:
I 111= 42 s.l 7, /////= 2 is. 84, /// /F=20s.34, /// F=42s. 12.
On the same day the star r\ Ursae majoris was observed:
/ // /// IV V
TJ Ursae maj. Upper culm. 18 . 5 50.3 13 h 41 1 * 24<* . 3 56.0 30.0.
*) It is best to use for this the polestar. Since the wire distances
remain the same only as long as the distance of the wires from the object
glass 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 wiredistances are
found by means of the formula:
tfsec 8
I HI 65 s . 70, IT 111= 34s. Q2, 777 7F=31s .69, 777 F=G5 .G2.
Since the star was first seen on the first. wire, we find
the transits over the middle wire from these wires as follows:
13h 41i24*.20
24 .32
24 . 30
24 .31
24 .38
13 h 41 m 24s.30.
The arithmetical mean of all wiredistances, taking them
positive for the wires / and // (these being on the side of
the circle) and negative for the wires IV and F, is :
Now if we take the arithmetical mean of the transits of
?? Ursae majoris over the several wires, we find:
13Ml 23 82,
and adding to it the quantity:
a sec 8 = f . 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:
13 h 41m 24s. 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 8 + cos n cos 8 sin (r rn).
Now if the body has been observed on a lateral wire,
whose distance is /", 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 f instead of c. But if we
have not observed the centre but only one limb of the body,
462
whose apparent semidiameter is ti, we must take instead of
c now:
where the upper or lower sign must be used accordingly as
the preceding or the following limb has been observed*). If
then O is the sidereal time of observation, and a is the ap
parent right ascension of the body, then its east hour angle is:
and hence we have the following equation, denoting the ap
parent declination by d :
sin [c +/=J= h ] = sin n sin f cos n cos 8 sin [ 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 h/== h ] = A sin n sin 8
A cos n cos m cos 8 sin (0 )
A cos n sin m cos 8 cos (0 )>
and since:
c, n, m, /, h ,
and therefore also a are small quantities , their sines
can be taken equal to the arcs and their cosines equal to
unity, and we obtain:
A cos 8 (a 0} = t A /=*= A  h h m A cos 8 h n A . sin 8 + c A.
The apparent quantities here can be expressed by geo
centric 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 8 cos a = cos 8 cos (> sin 7t cos 90 cos
A cos 8 sin a = cos 8 sin a (> sin n cos (p sin
A sin 8 = sin 8 g sin n sin 9? ,
from which we easily obtain:
A cos 8 cos (0 ) = cos 8 cos (0 a) Q sin n cos 9?
A cos 8 sin (0 a ) = cos 8 sin (0 )
or in case that O 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 distance/
is equal to j A .
(a)
eseen
463
A cos 8 (0 ) == cos 8(0 a}
A cos 8 = cos 8 $ sin n cos 9?
A sin 8 = sin 8 (> sin n sin 9? .
From the two last equations we find also with sufficient
accuracy:
A = 1 g sin n cos (9? 8).
Finally we have, denoting by h the true geocentric semi
diameter of the body:
A h = h.
If we substitute these expressions for the apparent quan
tities in the above equation for:
A cos 8 (a 0\
we find:
cos 8 ( 0} ==/[! Q sin n cos (95 8}] =t= k
f [cos 8 (> sin n cos y>] [m f n tang 8 f c sec 8 ]
or:
_/Q_I_ ^ /* 1 (> sin 7t cos (9? $)
COS $ COS $
, fi cosa> ~] r
M 1 P sin n ^j 1 7w + n tang
L cos d J
where 5 has been retained in the last term instead of J,
because it is more convenient in this form. The apparent
declination 8 is found with sufficient accuracy by the read
ing 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 8 ( &) is:
h m A cos 8 f n A sin 8 f c A
If we substitute here for A cos 8 , A sin and A the ex
pressions given before, and introduce the following notation:
m =m c cos <p Q sin n
n = n c sin 9? (t sin 7t
c = c [m cos <f f n sin cp] (> sin n,
those three terms are transformed into :
cos 8 [m f n tang 8 + c sec 8],
and hence we obtain:
h 1 Q sin n cos (9? 8} , , ~ ,
= (9 =t= ^ +/ =^ h m f n tang <? + c sec 8. (6)
cos d cos d
Now if the body has a proper motion, we find the time
of culmination from the time of observation & on one of the
lateral wires by adding to the time, in which the body
464
moves through the hour angle a S. But this time is equal
to the hour angle itself divided by 1 P., if I denotes again
the increase of the right ascension expressed in time in one
second of sidereal time. If we put therefore:
1 $ sin n cos (q> ~)
the reduction to the meridian is:
==1= _ A \fF+ M + H> ta " g S ~*~ S6C 8
(1 *)* y 1 A~
or:
h 1 sinTt cos<jp sec$
=::=t:: 7j TV ^4/F4 z ^ [m 4 n tang 4 e sec ].
/ c
If we omit the term ^. , we find the time of culmi
nation for the observed limb instead for the centre. Moreo
ver, if we ornit 1 I 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 Q sin n cos y> sec
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
1 Q sin n cos (90 $)
the argument being:
log (> sin n cos (95 <?),
and besides it gives the logarithm of 1 A , the argument
being the change of the right ascension of the moon in 12
hours. Another table gives the logarithm of F and the quan
tity  ^ ^ for the sun, the arguments being the days of
the year.
If a body, which has a proper motion, has been ob
served 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 <?, 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:
/ 17h25 m 42s.9
\  II 26 5 .0
/// 28 . 8
IV 51 .0
V 27 14 .8.
The wire distances were at that time:
/ 42*. 23 // 21s. 96 IV 20^.32 F 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:
= 18 10 . 6,
further the increase of the right ascension in one hour of
mean time was :
129s. 8, and 7r = 55 H".0, A = 60s.l5;
moreover we have for Bilk:
y = 50 1 . 2, log ? = 9 . 99912.
Now since one hour of mean time is equal to 3609 s . 86
sidereal, we find:
I = o . 03596,
and hence :
^=0.03565.
If we multiply the wiredistances by this factor, we find:
45 s . 84 23 s . 84 22s . 06 45 . 92.
Hence the times of observation reduced to the middle
wire are:
17h 26m 23s. 74
28 .84
28 .80
28 .94
28 .88
mean value 17 h 26^ 28 s . 84.
The term
is equal to:
h 65 . 67,
and hence the time of transit of the moon s centre over the
middle wire is:
17 b 27 34s. 51.
30
466
Now on that day b and k and therefore also m and n
were equal to zero, but:
c = H s . 09.
Therefore taking the factor:
I (> sin 7f cos cjj sec
~r^r~
equal to unity, we find for the time of culmination of the
moon s centre:
17 h 27n 34" . 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 reduc
tion to the meridian becomes more simple. For then we
have :
F== L_
(1 A)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 ~ is avoided
(1 A) cos o
in this case.
22. It shall be shown now, how the errors of the tran
sit 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
levelerror can then be accurately determined by means of
the spiritlevel 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 incli
nation of the axis can also be found by direct and reflected
observations of a star near the pole, for instance, the pole
star. For if we observe such a star on several wires and
call T the arithmetical mean of the times of observation re
duced to the middle wire, then we have for the upper cul
mination the equation:
= T+ A , + i C ^ + t ^ c sec S,
COS O COS O
where i = b, when the circle is West, and i = & , when
the circle is East, if b and b denote the elevation of the
circleend 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 180 z, we have, denoting now the
arithmetical mean of the times of observation reduced to the
middle wire by T :
and hence we find:
cos
2 cos z
Since the value of cos d 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, , 3 or A Ursae
minoris, because for other stars there is no time for revers
ing the instrument between the observations on the several
wires, and because for these stars the coefficient sec 3 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 levelerror, we have :
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 obser
vation reduced to the middle wire and corrected for the level
error, by t :
From the two equations we find therefore:
t t
c =    cos d.
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 collimationerror by observing this ob
ject 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 collimation 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
micrometerscrew, and of the divided screw head whose read
ings give the parts of one revolution of the screw. If the
telescope is furnished with such a wire, it is directed to the
wirecross of the collimator in both positions, and the move
able 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  (a + />) 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 (af6) or (a}&) C
is the error of collimation , and its sign is positive , if the
moveable wire in the position  (a j 6) 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 col
limation can be determined without reversing the instrument.
For, the two collimators being directed to each other *), one
of them is moved until the two wirecrosses coincide so that
the axes of the two collimators are parallel. Then the teles
cope is directed in succession to each of the collimators, and
the moveable wire is placed exactly on their wirecrosses. If
the readings for the moveable wire in the two positions be
a and 6, then the error of collimation is again ~(a\b) C
or C  (a f 6), 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 aper
tures 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 oollimating eyepiece. For this pur
pose the telescope is directed to the nadir and an artificial
horizon placed underneath *). If then the line of collimation
deviates a little from the vertical line, one sees in the teles
cope besides the middle wire its reflected image, whose dis
tance 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 inoveable 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 equal 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 micrometerscrew of the moveable wire in seconds. But
this can be easily found, if the known interval between two wires is mea
sured 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 eyepiece so often for this collimat
ing eyepiece, 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 fast
ened in a frame so that its inclination to the axis can be
changed.
The error of collimation is then determined in the fol
lowing 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 u 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 pos
itive, when the angle, which the end of the axis towards
the circle makes with the part of the line of collimation to
wards the object glass, is greater than 90; then we have,
denoting by d the distance of the middle wire from its re
flected image, and taking it positive, when the reflected image
is on that side of the middle wire, on which the circle is:
% d = b ( u c.
Therefore if biu is known by means of the spiritlevel,
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:
4 d = b f u c,
and from both equations we obtain:
c t* = J(rfhrf )
l = +\ (dd }.
Therefore by observing the reflected image in both po
471
sitions of the instrument, we can find c as well as the in
clination 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 spiritlevel. 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 = and hence c=b\u. Therefore if b}u is found by
the spiritlevel 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 instru
ment.
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, 6 ! = 2". 45. The distance
of the middle wire from the reflected image was found in
parts of a revolution of the micrometer screw :
d = 4 (K 2260 Circle West
d = .3107 Circle East.
We have therefore:
c u = + 0". 02 12 = f 0". 43
since one revolution of the screw is equal to 20". 33, and since
M = f0". 17, we have:
c = 10". 60,
and the inclination of the axis, when the circle was West,
6 = h2".90, and when the circle was East, b\= 2". 56.
Then the instrument was directed to one of the colli
mators, and when the moveable wire was made coincident
with the wire cross, the reading of the screw was:
21*. 132 Circle West
21 .999 Circle East.
We have therefore \ (at6) = 21 . 5655; the coincidence
of the wires was 21^.5397, and since we must take (046) C,
in order to find the error of collimation with the right sign,
we obtain:
c = f0".025S = }0".52.
472
Finally the two collimators were directed towards each
other and the moveable wire was made coincident with the
wirecrosses. Then the readings of the screw were:
for the south collimator 2 K 1190
for the north collimator 22 .0127
Hence we have (+&) = "TlT5G58"
*C = 21 .5397
c  = h 0^.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 in
terval of time between the two observations. Then &t in
both equations will have the same value. If then and t\ }
are the two times of transit over the middle wire, corrected
for the levelerror, the collimationerror and the rate of the
clock, we have the two equations:
sin (OP )
.,
COS 9
by means of which we can find the values of the two un
known quantities A t and k ; for we have :
. sin (8 9")
a  a = t  t + k 7oslTo  T , COS y,
a a. (t O cos S cos S
hence k = / v we
cosy sin (0 o )
After having found k we obtain the error of the clock
from one of the equations for a or . We see from the
equation for A;, that it is best, when d S 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 /\t and k from all the observations.
For these determinations the standard stars, whose rierht
O
ascensions 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 aber
ration, since this depends on the latitude of the place. Now
according to No. 19 of the third section the diurnal aberra
tion for culminating stars is:
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 diurnal
aberration is taken into account, by writing in all the formu
lae given before c 0". 31 13 cos y instead of cor, expressed
in time, c O s .0208 cosy instead of rand (cf0 s . 0208 cosy)
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 instru
ments, especially the first method of the two, when only re
lative determinations are made. The following may serve as
a complete example for determining the errors of an instru
ment of the smaller class.
Example. In 1849 April 5 the following observations
were made with the transit instrument at Bilk.
Circle West.
/ //
ft Orionis 54.8 15
Polaris U 38m 13s. 5lm 143.0
III
IV
V
Mean
.3 5^8
"378.4
58 s .
20* . 1
5 h 837 8
.44
.0
1 5 15
.25
b =
Os. 03.
Circle
East.
Polaris U 19*268.0 l h 5 25 s .O 1 5 24 .57
The apparent places of the two stars were on that day:
Polaris a = lh 4m HS .92 S= 88 30 15". 5
ft Orionis a = 5 7 16 . 66 <? = S 22 .8.
474
If we reduce the observations to the middle wire and
apply the correction for the level error, we find:
Circle West ft Orionis 5 !l 8 m 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
= h(K 114,
and since the diurnal aberration for Bilk is equal to s . 01 3
sec f) , we must take for c now f s . 101, when the circle is
West, and f s . 127, when the circle is East. If then we
correct the observations in the first position for the error of
collimation, we find:
ft Orionis = t = 5 h 8 m 37* . 52
Polaris =* =1 5 18 .20.
Hence we have:
t t 4 h 3 m 19 .32 a a = 4 h 2 m 5S . 74,
and since:
7> = 51 12 . 5
we find:
k = Os . 85.
Therefore the observation of ft Orionis corrected for the
errors of the instrument is:
5h 8" 36s . 78,
and hence:
&t= 1^208. 12.
The methods for determining k, which were given be
fore, 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 cul
minations of the same star are used, as has been stated al
ready in No. 5 of the fourth section. In this case we have
a = 12 h HA and <J = 180 J, where &a is the change
of the right ascension in the interval between the two cul
minations, and therefore the formula for /?, which was found
before, is transformed into:
475
_ 12 h h A (t o t ] cosS 2
cos <p sin 2 8
2 cos 90 tang $
Also for this purpose it is best to observe stars very
near the pole at both culminations, because then the divisor
tang 8 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 polestar, a meridianmark is usually
erected at a great distance from the instrument. This con
sists 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 polestar that point of the
scale, which corresponds to the meridian, has been deter
mined, the azimuth of the instrument can be immediately
found by observing the point, in which the scale is inter
sected by the middle wire, at least, if the scale remains ex
actly in the same position, and if either the error of colli
mation 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 ob
tained, 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 y 5 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 dis
turbed state of the atmosphere will very seldom admit of an
accurate observation of the scale. And since, besides, the ob
servation of such a meridian mark is limited to the time of
daylight, Struve has proposed a different kind of meridian
mark, 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 firm 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 suit
able coverings against any change. Likewise the meridian
mark is placed on a insulated pier in a small house and care
fully protected against any external disturbing causes. Since
thus the same care is taken as in the mounting of the in
strument itself, it can be supposed, that the changes of the
lens and of the meridianmark 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 T . T of a second. The chief advantage of such a me
ridianmark is this, that it can be observed at any time of
the day, and thus any change in the position of the instru
ment 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 6,
and at another time a and 6 , 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 ah(6 6)
dc^~
da
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 dis
tances 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, be
cause the horizontal wire intersects the celestial sphere in a
great circle, whilst the star describes a small circle. There
fore a correction must be applied on this account to the
reading of the circle.
The coordinates of a point of the celestial sphere, re
ferred 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 S cos (T ?/?), y = cos sin (r in) and z = sin .
If we imagine now a second system of coordinates,
whose axis of x coincides with that of the former system,
whilst the axis of y is parallel te the horizontal axis of the
instrument, and if we denote by # 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 arc of a small circle, whose radius is cos c, then
the three coordinates of the point, to which the telescope
is directed, are:
x = cos 8 J cos c, y = sin c, and z = sin cos c.
Now since the axes of the two systems make with each
other an angle equal to w, we obtain:
sin S = sin c sin n f cos c cos n sin
cos S cos (r ni) = cos d cos c
cos S sin (r ni) = sin S cos c sin n + sin c cos n
and hence:
5, , . COS S COS C
cotang o cos (T m) =
sin n sin c 4 cos n cos c sin S
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 8 = tang cos (r w),
and from this we obtain according to formula (17) of the
introduction :
8 = 8 tang \(r wz) 2 sin 2 8 + ^ tang (r ?w) 4 sin 4 S.
This formula is still transformed so that the coefficients
contain the quantities
2 sin 4 (t w) 2 and 2 sin \(t in)*
because these quantities can always be taken from tables.
(V. No. 7).
For this purpose we write instead of
tang ^ (r m) 2
now:
sin \ (r 7w) 2
1 cos I (r m) 2
and develop this into the series:
sin 4 (r w) 2 H~ sin \ (T wt) 4 "~+~
and since:
\ tang \ (r in) 4 = ? 2 sin ^ (r ni) 4 + . . . ,
we obtain:
8 = 8 2 sin (T mY . sin 2 S 2 sin ^ (r m)* cos 2 sin 2 8,
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 cul
mination.
When the division increases in the opposite direction,
the corrected reading is :
8 + 2 sin };(r m) 2 . ^ sin 2 S + 2 sin \(r ) 4 cos e? 2 sin 2 8.
Since the circle is numbered in the same direction from
to 360, it follows, that if for upper culminations the di
vision increases in the direction of the declination, the re
verse 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 a star, whose
479
Fig. is. hour angle shall be t. If we direct the telescope
to this star and bring it on the horizontal or axial
wire, we observe the polar distance P0\ where
the point is found by laying through an arc
of a great circle perpendicular to PS. Then we
have PO = 90 8 , P0 = 90 8 and hence:
tang = cos t . tang .
Now we will further suppose, that the axial
wire is not parallel to the equator, but that it
makes an angle equal to 90 + J with the merid
ian, where J is called the inclination of the wire;
then we observe the polar distance PO", where 0"
is found by laying through a great circle mak
ing with the meridian an angle equal to 90 + J. If we
denote again the observed declination by <V, and take 00" = c,
we have:
sin c sin .7= sin 8 cos S j cos 8 sin S cos t
sin c cos .7 = cos 8 sin t,
and therefore:
tang S tang S I cos t sin t ~r,
L sin d J
= tang S cos (t{y),
where :
J_
y ~ sin 8
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:
8 8 = l s in2 S.2sml(t+y)*.
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 :
8 = 8 ^ sin 2 8 . 2 sin t 2 cos 8 sin t . J,
where also the second term, dependent on sin  / 4 , can be
added, if it is necessary. Therefore from two such equa
tions we can find 8 and J, or when more than two obser
vations have been made, we can find the most probable va
480
lues of J and AC) , if we assume for S the approximate value
J so that d = c) + A $ The above equation becomes then :
= S S + \ sin 2 <? . 2 siri .U 2 + A S h cos tf sin < . 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 8 = cos S cos (r //?.)
cos c sin = cos S sin (T m) sin w H sin S cos n.
Here c) is the apparent declination of the observed point
of the limb, and T is the east hour angle of that point at the
time of observation, whilst S is the declination given by the
reading of the circle. But if we denote by S the apparent
declination of the centre of the moon, and by T its apparent
hour angle, we have:
cos c cos (S =f= x) = cos S cos (T m)
cos c sin ( =p x) == cos 8 sin (r ni) sin n j sin S cos r?,
where
siri x cos c = sin h
if h is the apparent semidiameter *), 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 c cos x
and multiply the resulting equation by A 5 which denotes the
ratio of the distance of the body from the place of obser
vation to the distance from the centre of the earth, we find:
=t= A sin h = A cos 8 sin S cos (r ni)
A cos S cos 8 sin (r ni) sin n
A sin S cos 8 cos n,
or since the quantity sin (r m) sin n can be neglected and
cos n be taken equal to unity:
=1= A sin h = A cos 8 . sin 8 cos (r ni)
, . c\ c\
A sm . cos .
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 ob
served point of the limb, the angle at the pole being x and the opposite
side h .
481
A sin h j = sin h
A cos S = cos d () sin n cos <p
A sin 8 = sin <? o sin TT sin <p ,
we easily find:
=*= sin h (> sin n sin (90 $ )
= sin (S <T ) cos S sin j (r ) 2 7
Now if the time of observation is 6>, and the time of
culmination of the moon is @ , we have:
r = 6>6> .
But when the body has a proper motion and /, denotes
the increase of the right ascension in one second, we have:
T== 90 )(i;i).i5,
if O is expressed in seconds of time.
Now if we neglect the small quantity m in (r m) 2
and take :
sin p = Q sin n sin (tp $ ),
we have:
sin (* * ) = sin p=Fsin A sin 2<? (6> <9 ) (1 A) a 20 g^
And since:
sin (jo =b A) = sinjw == sin h 2 sin />  A =p 2 sin h sin 1;> 2 ,
and hence:
sin p == sin A = sin (p == A) d= L sin ;^ sin h
we finally obtain:
, = + p =p h =p sin p sin A
This is the formula given by Bessel in the introduction
to the Tabulae Ixegiornontanae 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, mul
tiplied by (1 A) 2 .
This true declination of the moon s centre corresponds
to the time 0. If we wish to have it for the time & , we
must add the term:
7 V
where is the change of the declination in the unit of time.
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 divi
sion 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 polestar must be observed at the upper
and lower culmination. When the readings are freed from
refraction, and from the errors of division and from 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 mi
croscopes and for determining any change of their position,
to observe the nadir point at the time of the two observa
tions, 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 obser
vation the instrument 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 succes
sion 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 (Z?) in No. 8
of this section and from the errors of division. In order to
obtain the utmost accuracy, it would be necessary to deter
mine 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 collimators 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 wirecross is like
wise 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 witli that of the telescope, the line of colli
mation of the telescope of each collimator is rectified so that
it coincides with the axis of revolution. This is accompli
shed by directing one collimator to the other and turning it
180 about its axis. If the point of intersection of the wires
after this motion of the telescope remains in the same posi
tion with respect to that of the other collimator, then the
line of collimation is rectified; if this is not the case, the wire
cross 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 collimation is then found
by means of the level, and since the collimator can be re
versed so that the object glass is on that side on which the
eyepiece was before, the inequality of the pivots can be de
termined 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 wirecross 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 about its
axis, to eliminate any error of the line of collimation. Then
the same observations are repeated with the other collimator,
31*
484
and when a and 6 denote the arithmetical means of the read
ings of the circle for each collimator, ^ is the zenith point
of the circle, if the collimators are at equal distances from
the axis of the instrument *). If x is the elevation of the
objectend of the collimator, corrected already for the inequal
ity of the pivots, then the zenith distance of the telescope
when it is directed to the wire cross of the collimator, is
90 f #, taking no account of the angle between the verti
cal lines of the two instruments, and hence we must sub
tract x from the reading or add it, accordingly as the divi
sion increases or decreases in the direction of the zenith
distance.
This method being more complicated and therefore pro
bably 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 culmination according to the
equations (#) in No. 8 of this section:
and a similar equation is found for the lower culmination.
The arithmetical mean of these two equations gives the lati
tude independent of the declination of the star, but affected
with those terms of flexure which depend on the sine of
2 , 4 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 like
wise 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 must be corrected for flexure, if there are any terms,
which have an influence upon the mean of the two readings.
485
and fi or rp. But since the observation of zenith distances
in this case is more difficult, 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 pur
pose a method is required, by which the true time of pas
sage over the prime vertical can be deduced from the ob
served time and the known errors of the instrument.
We will suppose, that the axis of the instrument pro
duced towards north meets the celestial sphere in a point (),
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 coordinates,
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
coordinates 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 coordinates,
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 coordinates of the point (), denoting
its hour angle (reckoned in the same way as the azimuth)
by M, and 180 minus its declination by ??, are:
z = sin n , y = cos n sin m , x = cos n cos m,
and since the axes of z in both systems make with each other
an angle equal to 90 y, we have the equations:
sin b = sin n sin y> cos n cos m cos y>
cos b sin k = cos n sin m
cos b cos k = cos n cos m sin y + sin n cos cp
and
sin n = cos b cos k cos rp + sin b sin cp
cos n sin m = cos b sin k
cos n cos m = cos b cos k sin cp sin b cos cp.
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 90jG % , and that it is directed to an ob
ject, whose declination is d and whose hour angle is , then
486
the three coordinates of this point with respect to the equa
tor and supposing the axis of x to be directed towards
north , are :
z = sin , y = cos sin t and x = cos S 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
x== cos S cos (t ni).
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 f cos S cos (t m) cos n.
We 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, P0=180 r/5 w, ZQ=W\b and PZ = 90 9,
whilst the angle QPZ = m and QZS=k. The formula for
sine is deduced from the triangle PSQ, where S is that
point of the sphere of the heavens, to which the telescope
is directed, and in which we have 5=90 c, when S is
west of the meridian and SP=90 r> , PQ = 180" cp 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 m the values found before, and
taking instead of the sines of 6, k and c the arcs themselves
and instead of the cosines unity:
c = sin S cos <p + cos sin 90 cos t
[sin sin y> f cos S cos (p cos t] b
( cos sin t . k,
and since:
sin S sin if + cos S cos y cos t = cos z
and
cos S sin t = sin z sin A,
or, since A is nearly 90:
cos sin t = sin z,
we obtain, when the star is west of the meridian :
c\ b cos z k sin z = sin cos (f f cos S sin cp cos t.
487
If then is the true sidereal time, at which the star
is on the prime vertical, and if therefore a is the hour
angle of the star at that moment, we have:
tang
cos (O )= >
tang (p
or:
= sin 8 cos rp j cos sin cp cos (0 a).
Subtracting this equation from the other, we obtain:
c t b cos z k sin z = cos 8 sin <p . 2 sin  [0 t] sin [0 a f t J.
Now since c, 6 and A are small quantities and hence
a and t are nearly equal, we can put :
sin t instead of sin 4 [0 a\t]
and
[0 a t] instead of sin ^[0 t]
and then, remembering that
cos 8 sin t= sin z
we obtain:
c 6 fc
a = t +   :  h   . 
sin z sm </? tang 2 sin 7? smy
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 T h A * ? and the hour angle :
Therefore we have:
sin z sin (p tang 2 sin (p sm<f>
This formula is true, when the circle is North and the
star West. When the star is East, we have:
cos S 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 sin z and tang & and thus we have :
_ c b Jc ( Circle North )
sin z sin rp tangs sin 9? siny Star East *
When the circle is South, the quantities b and c have
the opposite sign, and therefore we have:
<9 =T+A ,_ c _J _____ L jCircle South)
sin z sin (p tang z sin y sin 99 Star West 5
and
^_ c b k ( Circle South j
sin z sin y tang z sin 90 sin 9? Star East >
488
If we know & and a , we obtain by means of the for
mula :
tang <p cos (0 ) = tang
either <jp, when the declination of the star is known, or the
declination, when the latitude is known. If and & be
the times, at which the star was on the prime vertical east
and west of the meridian, then l(@ _ 0) will be the hour
angle of the star at those times, and therefore we have :
tang (p cos Y (0 &) = tang $,
so that it is not necessary to know the right ascension of
the star, in order to find cf or 3. 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:
and hence in that case it is not necessary to know the error
of the clock nor the errors of the instrument except the level
error. An example is given in No. 24 of the fifth section.
26. The formulae given before are used , when the in
strument is nearly adjusted so that 6, c and k are small quan
tities, 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 instrument sufficiently and thus
make the observation at a greater distance from the prime
vertical. In that case the formulae given above cannot be
employed. But we found before the rigorous equation:
sin r, = sin 8 sin n + cos S cos n cos (t m\
or if we substitute the values of sin n, cos n cos m and cos n sin m
sin c = sin !> sin S sin rp sin h cos S cos tf cos t cos t> cos k sin 8 cos <p
f cos b cos k sin y> cos 8 cos t + cos t> sin /, cos S sin t.
Now if the observation were made on the prime vert
ical, we should have:
sin 8 = cos z sin y, cos 8 cos / = cos z cos (f
and
cos 8 sin t = sin z.
But since we assume, that the instrument makes a con
siderable angle with the prime vertical, we will introduce the
following auxiliary quantities:
489
sin S= cos z sin cp
cos 8 cos t = cos 2 cos cp
cos $ sin = sin 2 ,
by means of which the formula for sin c is transformed into :
sin c = sin b cos 2 cos (cp <p } + cos b cos /; cos 2 sin (cp 9 )
f cos b sin A: sin 2 ,
so that we obtain:
_ sin c sec 2 tang b tang fc tang 2
cos 6 cos A; cos (cp y ) cos k cos (<p y )
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 A: tang2 ; ( Circle North)
cos b cos k cos (cp cp} cos k cos (cp cp } Star East )
, sin c sec z tang b tang tang2 ( ^Circle South)
cos ft coskcos((p cp } cos k cos (cp cp } I Star West )
, sine sec z tang b tang k tangs ( Circle South)
cos b cos A: cos (cp cp } cos k cos (cp <f } < Star East
Therefore when we reverse the instrument between the
observations, and compute tp y 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 meridian, 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 elim
inated, and the accuracy of the result for the latitude depends
then merely on the accuracy with which ff has been found.
But we have:
. tanc. S
tang en = ,
" 7 cos t
490
therefore if we write the formula logarithmically and diffe
rentiate it, we have:
dtp 1 = Ts5 dS h J sin 2 OP tang / dt.
sin 20
From this formula we see again, that it is best to ob
serve stars which pass over the prime vertical near the zenith.
For since we have :
tangs
tang t =   ,
COS (f
we see that the coefficient of dt is equal to sin cp tangs , and
that it is very small for stars near the zenith, and since for
such stars # is nearly equal to f/ , an error of the decima
tion is at least non increased.
If the observations have been made on several wires, it
is not even necessary, to reduce them 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 (rf cf ) in this way :
, ,. sin c . tang b
sin (cp g ) =   sec z \ cos (cp on tang k tang z ,
cos 6 cos k cos k
then develop sin (r^ <^ ) 9 and substitute for sin q> and cos cp
the values :
sin sec z and cos S cos t sec z
and take cos (9: <p ) equal to unity, we obtain:
sin (ffo) = cos o sin cp . 2 sin \ t~ f 
cos b cos k
tang b
cos 2; tang k sin z .
cos
When 6, c and A 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:
cp = sin cp cos . 2 sin ^ t 2 =*=/+ b + c k sin ~ [Circle North, Star West]
+ b + c h k sin z [Circle North, Star East]
b c k sin z [Circle South, Star West]
b c f k sin .2 [Circle South, Star East].
*) For when we observe on a lateral wire, whose distance is /, it is
the same as if we observe with an instrument whose error of collimation is
cH/.
491
With the prime vertical instrument at the observatory
of Berlin the star ft Draconis was observed in 1846 Sept. 10:
Circle North, Star East.
/ // /// IV V VI VII
Circle South, Star West.
l 5s.O, 54 " 59s .7^ 50>n47 .8, 17^45 28^ .0, 37 3Ss .0.
The inclination of the instrument was:
Circle North = 4 4" . 64
Circle South = 3 .49.
Further was:
a = 17h26ioSs. 59
=52 25 27". 77
&t=  54*. 52,
and the wire distances expressed in arc were:
/ 12 31". 16
// 6 43 . 78
/// 3 25 .17
V 3 23 . 14
VI 6 34 . 21
VII 12 22 . 32.
Now in order to compute y #, we must know already
an approximate value of cf. Assuming:
y> = 52" 30 16",
we have:
log sin <p cos 8 = 9 . 684686,
and we obtain:
Circle North.
/// IV V VI VII
t 8m44s.ll 17 m 5s.ll 22m 29s. 11 26 ra 36s.61 32 46". 81
log 2 sin 1 1 2 2.17552 2.75807 2.99648 3.14264 3.32351
sin^ cosd 2 sin!* 2 1 12 .48 4 37 .18 7 59 .92 11 11 .94 16 59 .07
<f 4 37 .65 4 37 .18 4 36 .78 4 37 .73 4 36 .75,
and hence from the mean:
7  * = 4 37". 22 + 4". 64 + c + k sin z.
Likewise we find from the observations made when the
circle was South:
<P ~ 8 = 4 53". 53 t 3". 49 c k sin z,
therefore combining these two results, we find:
<p = 4 49". 44
r = 52 30 17". 21
c H 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 account of aber
ration, 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 spiritlevel 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 & , which
quantity alone is used for obtaining the value of y> r?.
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 obser
vations, 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 eclip
tic, 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
/", it is the same as if we have observed with an instrument,
whose error of collimation is c f f. Therefore we have the
equation :
493
sin (c f./O = sin $ sin n f cos S cos ?? cos (t ?n) ,
where t is the hour angle of the star at the time of the ob
servation on the lateral wire. If we subtract from this the
equation :
sin c = sin S sin n f cos 8 cos n cos (/ wz),
we obtain:
2 sin \ /cos [T/+ c] = 2 cos <? cos n sin  (/ t") sin [ (z + 1 ~) m].
Now since f is only a few minutes, we can put f in
stead of the first member of the equation and thus we find:
cos S sin j (*+0 cos n cos m cos S cos \ (<+/ ) cos ?i sin m
or if we substitute for cos n cos m and cos w sin m the ex
pressions given in the preceding No., we find:
2 sin i (<
cos <? sin 9? sin (ff<0 [1 6 cotang y k cotang ( + cosec y]
Therefore for reducing the observations on a lateral wire
to the middle wire we must use instead of the wire distance
f the quantity:
../ . =r
1 b cotang y> k cotang J[ (t\) cosec y
and then we have :
2sinH<0= , . .
cos o sin (p sin ?(t{ t)
In order to solve this equation we ought to know already
t . But we have:
sin 5 (t f = sin [z T (* OJ
If we take then for ^ (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 from
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
6 cotang y> 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 /". 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 t cos ? tang *,
494
and since f is small, we also have approximately :
tang t cos (f> = tang z
and hence :
tang \ (t j t ) cos cp = tang ^ (z + z )
Therefore we can write instead of the factor of k:
cotang (f 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 4 (t ~ = y;
cos sin rp sin r, (t f t )
by an indirect method, we can develop it in a series. For
we can write it in this way:
cos t cos t = ~  1
cos o sm 9?
and from this we obtain according to formula (19) in No. 11
of the introduction:
f r f T 2
t =t Jr cotang t 
cos <) sm 97 sin Z _cos o sin 7 sin t_\
r f i 3
 i v4 (1 h 3 cotang t 2 }.
[_cos o sin (f gmlj
Now when the instrument is nearly adjusted, we have:
cos S sin t = sin z,
and hence:
/ r /"
t = t A cotang /
sm z sm 9? (_sm z sin
[/ is

sin z sin cp J
Since this formula contains also the even powers of /",
we see, that wires, which 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 +  ~ 4 cotang t 
sm z sm 9^ \_sin z sin (p J
r /" i 3
I j r i  *> j 1 I *
_sin z sin 90 J
In order to compute this series more conveniently, we
can construct a table , from which we take the quantities
sin (f sin a, \ cotang i, and ~ (1 f 3 cotang 2 ) with the argu
ment r) .
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
terms would come into consideration.
In this case, when the zenith distance is small, the fol
lowing method for computing t can be used with advantage
We had:
f
cos t = cos t\ .
cos o sin fp
If we subtract both members of the equation from unity
and also add them to it, we obtain, dividing the two result
ing equations:
2 cos i t cos 8 sin y H f 1
Now since:
tang 8
cos t
tang (f
we have:
lcos; = 2sin!^== sin( f^
cos o sin (i)
and
, p co
therefore we get:
^.^^sin^^
sin (9, + 8)
and if f is negative:
v
values of the wiredistances are determined by ob
serving a star near the zenith on all the wires. If we com
pute for each observation the quantity:
sin (f cos 8 . 2 sin f t 2 ,
the differences of these quantities give us the wiredistances,
because we have for stars near the zenith:
<p 8= sin y> cos 8 . 2 sin t 2 ==/f c + h f k sin z.
Thus in the example of the preceding No. the follow
ing wire distances would be obtained from the observations
made when the circle was North:
///== 3 24". 70
r= 3 22 .74
VI= 6 34 .76
r//=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.
7 77 777 7F V VI VII
a Bootis 44. 7 8 s . 3 50 s . 2 19 h 2 32s.2 13 s . 8 55 s . 4 1" 19 S .2.
The wire distances expressed in time were then:
7= 51 s . 639
77=25 .814
777=12 .610
F=13 .305
F7=26 .523
VII =52 .397;
moreover we have:
A* = + 47". 5, = 14 h 8 16s. 5, = + 20 1 39", y> = 52 30 16".
The quantities 6 and k were so small, that it was not
necessary to compute the reduced wire  distances /" . Then
we have:
/ = 4 h 55 m 3s . 2 = 73 45 48". 0, log cos 8 sin t sin 9? = 9 . 85244
and log cotang t = 9 . 14552.
Now in order to compute the second term of the series,
f
we must express  in terms of the radius, that is,
sin <f cos o sm t
we must multiply it by 15, and divide it by 206265. Then
we must square it, and in order to express the term in sec
onds of time, we must multiply it by 206265 and divide by
15. Thus the factor of:
r 1 IT
_sin <f cos sin tj
will be: ,_. cotang 2,
the logarithm of the numerical factor being 5.00718. Like
wise the coefficient of the second term, expressed in seconds
of time, will be:
But in this case this term is already insignificant. Now if
we compute for instance the reduction for wire /, we have,
since f is negative:
72s. 533
sin cp cos o suit
tt.icotang* * =f 0.053,
26o LCOS o sin t sinyj
206265
hence the reduction to the middle wire is:
7= I n 12s.48.
497
In the same way we find:
II = 36*. 25
///= 17 .71
F=H 18 .69
F/=f37 .24
F//=H73 .54,
and hence the observations on the several wires reduced to
the middle wire are:
19 ! 2>32s.22
32 .05
32 .49
32 .20
32 .49
32 . 64
32 .74
mean value 19 h 2 m 32 s . 40.
In order to give an example for the other method of
reduction, we will take the following observation of a Persei :
Circle South, Star West.
/ // III IV V
a Persei 4" 26* . 2 38* . l 43 s .O 5 U " 49 s . 2 59 ni 52 s .
VI VII
58 in 55* . 2 57 2s . Q.
If we compute first:
sin (w )
tang 7 / = .^ ,
sin (y>+~o)
taking :
5 = 40 16 26". 7
and
y> = 52 30 16".
we find :
; = 26 58 58". 88.
If we compute the reduction for the first wire, we have
f negative, and hence we must compute the formula:
. . , sin (OP ~) + /
tang, t = 7 ~ 
sm(y>t)h/
Now since
/= 51s. 639 = 12 54". 585,
or expressed in terms of the radius /"= 0.0037553, we find :
^ = 27 53 G". 72,
hence :
t t 54 7". 84
= O 11 3 36*. 52.
32
498
Likewise we find for the other wires:
// =lm 49s. 05
/// 53 . 48
V 56 .85
VI I 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
/// and V amounts to nothing and for wires / and VII it is
only s . 12.
28. It must still be shown, how the errors of the in
strument are determined by observations.
The inclination of the axis is always found by means
of a spiritlevel. The collimation error can be determined
by observing stars near the zenith east and west of the merid
ian 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:
= rf A t  .  [Star East]
sin z sin (f sin 90
6> =r hA*h C .  . [Star West],
sin z sin (f sin cp
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 <p sin z [, (& &} \ (T 7 1 )].
where the value of \ (6> 6f) is obtained by means of the
equation :
tangy
or more accurately, taking  (6f &) = , from the equation:
sin (cp 8}
tang 1 1 = rrrrjK
sin (y>ho)
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 and 6> , we find:
k = sin y [k(T H T) 4 t % (0 f )],
499
or since f (Q f ) = a :
k = sin <p [i (T{ T") 4 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 ob
served with greater precision. With the prime vertical in
strument at the Berlin observatory the following observations
were made in 1838:
Circle South:
June 25 Bootis West 19 h 3 m 1 s . 44
26 Bootis East 9 12 54 .49,
these times being the mean of the observations on seven
wires. On June 25 the level error was 6 = f6".42 and
on June 26 6 = 4 7". 98. If we correct the times by add
ing the correction + 6 , we must add to the first
10 tang. z smr/> 7
observation s . 26, and add to the second 4 s . 32 so that
we obtain :
T = 19 h 3 Is. 18
T= 9 12 54 .81.
Hence we have:
i(rhr) = 14 h 7 " 58. 00,
and since:
A< = + 20" . 27 and = 14 h S 16* . 48
we find :
^ = his. 42.
Note. Compare on the prime vertical instrument: Encke, Bemerkungen
iiber das Durchgangsinstrument von Ost nach West. Berliner astronomisches
Jahrbuch fur 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 spiritlevel placed upon the horizontal axis, the vertical
position of the pillar can be examined and corrected by means
32*
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 con
centric 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 spiritlevel. 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 in
strument 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 va
riable with the zenith distance, so that it can be assumed to
be of the form c f a cos a. 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 are 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 anti
quated , since the mural quadrants or mural circles have been replaced by
the meridian circle, and the portable quadrants by the altitude and azimuth
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 measur
ing altitudes, but for measuring the angle between two ob
jects 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 determin
ing 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 plane
glass 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 like
wise 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 alhidade is moved to the zero of the divi
sion. 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 par
allel, 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 Hartley 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 a 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 re
flection for the large reflector are also equal to . Hence
this ray will also reach the object A, if this is at an in
finitely great distance so that the distance of the two reflec
tors 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 /^, 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:
180 fyh/? = 180
or:
y = a p.
The angle of reflection from the large reflector is then
/?, and the direction of this twice reflected ray will make
with the original direction of the ray emanating from the
eye an angle , which is equal to the angle subtended by
the line between the two objects, which are seen in the tel
escope. 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:
180 2 a H<? +2/3=180,
and hence we have:
S = 2a 2p
or:
d=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
halfdegree 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 arti
ficial 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 ob
ject. But at sea the altitudes of a heavenly body are ob
served 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 ratio
nal horizon and is therefore a small circle. It is formed by
the intersection of the surface of a cone, tangent to the sur
face of the earth and having its vertex at the eye of the ob
server, with the sphere of the heavens, whilst the rational
horizon is the great circle in which a horizontal plane pass
ing through the eye intersects the apparent sphere. If we
denote the zenith distance of the sensible horizon by 90fc,
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 4 h
and hence: 2 sin \ c~ =
af h
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 BAC Fig. 19,
and which at the same time represents the plane in which
the two objects are situated. Let OA be the line of vision
towards the object A. When this ray falls upon the small
reflector (which is also called the horizonglass) it is reflected
to the large reflector , and if p is the pole of the small re
flector, that is, the point in which a line perpendicular to
its centre intersects the great circle, the ray after being re
flected 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 AC, the
angle between the two reflectors by p P, and it is again easily
seen that A C is equal to 2pP.
This is the case, if the line of collimation of the teles
cope is parallel to the plane of the sextant, and both reflec
tors 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 A A being perpendicular to B A 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 B B 1 and CC being likewise equal to i
and perpendicular to BAC. If the pole of the great circle
BAC is (), then the angle QAC is the angle given by the
reading of the sextant, whilst the arc AC is equal to the
angle between the two observed objects, and denoting the
first by , the other by , we have in the spherical triangle
AQC i
505
cos ft = sin i~ + cos i~ cos ft
= cos f 2 t 2 sin j a ,
and hence according to the formula (19) of the introduction:
a = {  tang 5 .
Therefore when the telescope is inclined to the plane
of fhe sextant, all measured angles will be too great. The
amojint. nf the error can be easily found. For in the teles
cope 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 coin
cident, 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 sex
tant, 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 l
the inclination of the telescope i , the distance of the two
wires J, and the true distance of the objects 6, then we
have in one case:
s=b\ ^  i\ tang I *,
and in the other case:
s = b f ( f i\ tang i s ;
therefore putting:
tang = tang  a
we have :
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
ft
a point whose distance from this wire is equal to  i.
Jj
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
i 2 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 horizonglass
has two adjusting screws. One is on the back side of the
reflector, which by means of it is turned round an axis per
pendicular 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 ob
ject at an infinitely great distance, and the direct and re
flected 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
that their distance is as little as possible, then the lines of
intersection of the two reflectors with the plane of the sex
tant are parallel, and then by means of the second of the
screws mentioned before the horizonglass 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 horizonglass by means of the screw on its back. Usually
however this error is not corrected, but its amount is deter
mined 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
time 6, then a is the indexerror, and ^ or ^ is the
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 there
fore 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 indexerror 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 observa
tions are made, for instance when lunar distances are taken, the
indexerror 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 obser
vations must be corrected, in order to obtain the true index
error c (} , which would have been found by an object at an
infinitely great distance. For if A denotes the distance of
the object from the horizonglass, /"the distance between the
two reflectors, ft the angle which the line of collimation of
the telescope makes with a line perpendicular to the horizon
glass, 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:
/sin 2/9
^ C = ^fcosW
and hence we have:
c = / sin 2/9 4 ^ sin 4/9,
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 index
glass would have met an object whose distance from the ob
served object is c, and the true indexerror would have been
obtained, if these two objects had been made coincident.
Therefore if the reading was c 17 when the object and its
reflected image were coincident, we have:
c =ci h ^sin2/9 ^4r sin 4/9.
a a"
508
The angle /?, which was used already before, can be
easily determined, if the sextant is fastened to a stand, and
the indexerror C T 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 sex
tant the angle between the object and the wirecross of the
telescope, we have :
5 c = 2/? 4 8in M
, . A
and since :
c = c x +^ sin 2 A
we obtain :
If the inclination of the horizon  glass to the plane of
the sextant is , its pole will be at p (Fig. 20), the arc pp
being equal to i and perpendicular to BAC.
Fiji. W.
// C
The ray after being reflected from the horizonglass in
tersects the sphere in B and after its reflexion from the in
dexglass in C . In this case again A C is the angle ob
tained by the reading, while AC is really the angle , which
is measured. We have then, as is easily seen:
BB = CC" = 2 cos^.i,
where ft is, as before, the angle between the line of collima
tiori of the telescope and a line perpendicular to the horizon
glass, which is equal to A p. Moreover we have:
cos a = cos a cos C C
= cos 2 cos /9 2 i cos a,
and according to the formula (19) of the introduction:
. 2 cos ft i 2
a = ft f .
tang a
If the inclination of the index glass to the plane of the
sextant were i, and the horizonglass were parallel to it and
the telescope perpendicular to both, then p , F , A and like
wise B and C would lie on a small circle, whose distance
509
from the great circle BAG would be equal to i. Then p P
or the angle between the two reflectors would be, as in
the former case, when the inclination of the telescope was
equal to i :
j a = 4 a i  tang j ,
or:
a = a 2 i~ 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 hori
zontal and the piece with the hole is placed in front of the
indexglass which is turned, until the image of the piece is
seen through the ^ole. Then the other piece is likewise placed
before the indexglass so, that the wire is also seen through
the hole. If then the wire passes exactly through the centre
of the reflected image of the hole, the index glass is per
pendicular 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 ful
filled.
The same can be accomplished in this way, though per
haps riot as accurately: If we hold the instrument horizon
tally with the index glass towards the eye, and then look
into this glass so that we see the circular arc of the sex
tant as well direct as reflected by it, then, if the indexglass
is perpendicular, the arc 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
planeglas reflectors, which ought to be parallel, make a small
angle with each other so that the reflectors have the form
of prisms. Let then AB (Fig. 21) be the ray striking the
510
front surface of the index glass,
which will be refracted towards C.
After its reflection from the back
surface it will be refracted at the
front surface and leave this sur
face in the direction DE. When
the two surfaces are parallel, the
angle ABF will be equal to GDE,
but this will not be the case, when
the surfaces are inclined to each other. Now if we take
MNP = d, and denote the angles of incidence ABF and GDE
by a and &, and the angles of refraction by t and & t , we
have:
j frt ( JO + 8
b l 4 = DO S,
and hence:
b t = ai 28.
Now if is the refractive index for the passage from
7H
atmospheric air into glass, we have also :
sin a i = sin , sin b t = sin 6 ;
m m
and hence:
sin a sin 6 = [sin a l sin a l cos 2 + cos ci l sin 2 ]
n
or:
= 2 S V , sec a tang a
" 9 I 1
z sec a 2 f 1.
n
Now a is the angle, which the line from the eye to the
second object makes with the line perpendicular to the in
dexglass. If we denote by ft 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:
and hence :
Now the correction which must be applied to the angle ;
is the difference of the above value and that for ; = 0, be
511
cause the index error is also found wrong, when the two
surfaces of the glasses are not parallel. Therefore if we de
note this correction by #, we have:
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:
m \ /? + 7 I/, ~ ~n~~7p~+~y\* ft ./ n*"
x = 2 ) sec r [/ 1  sin   <} sec ~ ]/ 1 
n 1 i m v . * / 2 f in,"
r
or since  is nearly equal to ^ :
m J
Now in order to find #, we measure after having de
termined 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 indexglass out of its set
ting, put it back in the reversed position and determine the
indexerror and the same distance a second time. If then /\
be the true distance of the stars, we find the second time
A x = 6 ,
if the first observation gave :
and hence we have:
,"
S =
Since rays coming from the indexglass strike the hori
zonglass 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 a and
the reading of the circle gives s, we have according to No. 6
of this section:
O) 206265 ,
/
or:
L 1 " c & ~\
cos 4 . sin 4 .s  sin 4 . cos i s 206265.
r J
Therefore if we measure two such angles, we can find
cos * and  sin * 0, and hence and 0, and then every
r r r
reading must be corrected by the quantity :
I sin 4 (* 0) 206265,
r
Since the error of excentricity is entirely eliminated w T ith
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 conve
nient are those invented by Pistor & Martins in Berlin, which
instead of the horizonglass have a glassprism. They have
the advantage, that any angles from to 180 can be mea
sured with them. All that has been said about the sextant
can be immediately applied to these instruments.
Note. Compare: Encke, Ueber den Spiegelsextanten. Berliner astron.
Jahrbuch fur 1830.
VII. INSTRUMENTS, WHICH SERVE FOR MEASURING THE RELATIVE
PLACE OF TWO HEAVENLY BODIES NEAR EACH OTHER.
(MICROMETER AND HEL1OMETER).
32. Filar micrometer. 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 par
allel 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 mo
tion 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 mea
sure 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 mo
veable wire will be equal to the difference of the declinations
of the two stars. In case that one of the bodies has a pro
per 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 arith
metical 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 va
lues obtained are the same, it shows, that the moveable wire
is parallel to the others.
The equivalent of one revolution of the screw in sec
33
514 v
ends of arc is found in the same way that the wiredistances
of a transit instrument are determined. The micrometer is
turned so that the normal wire is parallel to the diurnal mo
tion of the stars , and then the times of transit of the pole
star 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 ob
served, the equivalent of one revolution of the screw in sec
onds of arc is easily deduced. This method is especially
accurate, when a chronograph is used for these observations.
Another method is that by measuring the distance bet
ween the threads of the screw, and the focal length of the
telescope, because if the first is denoted by m, the other by
/", we find one revolution of the screw expressed in seconds :
r = ^ 206265.
We can also find by Gauss s method the distances between
the parallel wires and then the same expressed in revolu
tions 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 tem
perature, 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:
r = a b (t t ) ,
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 decimation 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 tel
escope, 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
screwhead 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 positioncircle is read, first when the nor
mal 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 to 360, 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 tel
escope 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 dia
phragm nolding 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 al
ways 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 ascen
sions 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 , 90 d and 90 , whilst the opposite
angles are a or, 180 p and /?, where p and p are the
two angles of position and A is the distance, and hence we
have according to the Gaussian formulae:
sin A sin (p + p) = sin \ (a! ) cos  ( + <?)
sin I A cos I (p +/>) = cos \ (a! ) sin  (<? )
cos  A sin Y Qo /?) = sin  ( a) sin ^ (<? h 5)
cos Y A cos  (p p] = cos ^ ( a) cos ^ (# d).
In case that a and J d are small quantities so
that we can take the arc 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 (S 1 + S) [a 1 a] = A sin p
Ctl C\
O = A COSjtf.
For observing distances and angles of position it is re
quisite 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 ar
bitrary 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 , and if A
denotes the distance between the two stars, p the angle of posi
tion, i the inclination of the wires to the parallel circle recko
ned from the west part of the parallel through north, which
is given by the position circle, we have:
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 ,
whilst the opposite angles are i and 90 + 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 /?, and when the observations
have been made in more than two positions, each observa
tion leads to an equation of the form:
Acos(p t) cos (p ? ) sin (jo i) 3600
sin.i sin i p sin f 206265
and from all these equations the values of d/\ and dp can
be found by the method of least squares.
At the observatory at Ann Arbor the following obser
vations of 6 Hydrae were made, where every a is the mean
of ten transits:
; = 9924 50 24 141 40
= 1".062 4". 239 H2".382.
If we take p = 207, A = 3". 5, we obtain the equations:
= 0".011  0.306 rf A  0.590 dp
= 40".070 1.191JA  0.315 dp
= 0".044 4 0.668 dA 0.089 d/> ,
where p p. From these we find d A = + 0" . 056,
dp = + 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.
One is a micrometer, whose
wires make angles of 45 with
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 5, its distance from the
centre, for we have:
t
Fig. 42.
15 cos S.
and since we have for another star:
518
the difference of the decimations 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
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
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 J5, MD will be equal to the
interval between the observations expressed
in arc and multiplied by cos d, so that:
And if we have for another star:
M D = 15 (T r) cos d .
we easily find the difference of the decli
nations, 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 illu
minated, 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 immersion, 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 J, and let r and T be the
same for another star, whose declination is J , then we have:
= ! (T f r)  (t H 0
If then u and p denote half the chords which the stars
describe, we have:
fl = j (t t) COS $
and
(A = (T T) cos # .
Putting :
P
sm a? =
r
, /*
sin 9? = >
where r denotes the radius of the ring, we obtain, if we de
note by D the declination of the centre of the ring:
S D = r cos y>
D = r cos 97 ,
and hence:
8 $= r [cos 95 =t= cos 95],
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 ringmicrometer, whose radius was 18 46". 25.
The declination of Flora was
T = 24 5 . 4
and the place of the comparison star was:
= 91 12 59". 01
<?=2.4 1 9 .01.
The observations were:
T = llhi6m35s.o Sider. time t = ll h 17 53* .
T = 17 25 .5 * = 19 46 .5
We have therefore:
log r r 1 . 70329 log t t 2 . 05500
log^ 2.53878 log p 2.89070
cosy 9.97850 cosy 9.85941
> ]) 17 51". 9 S 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:
<? <?=: + 4 17". 1.
The time at which the bodies were on the declination
circle of the centre were:
I (r f T) = Ufa 1? Qs . 25  (* + = Ufa 18m 49 . 75.
Therefore at
Hh 17m Qs. 25
the difference of the right ascensions and declinations were:
. = 1^49*. 50 <? <? = 44 17". 1
= 27 22". 50.
If the exterior edge of such a ring is turned as accu
rately 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 /LI and r be the chord and the radius of the inte
rior ring, and p and r the same for the exterior ring, then
we have:
cos S (t = p = r sin y
<x>sS (t\ t l }=sfi t =r smy> ,
hence :
fi f fi = (a f 6) sin tp\ (a ft) sin y>
and:
ju ft = (a + ft) sin 92 (a ft) sin 9? ,
putting :
r + r 1 rr
^ = a and ^ = 6.
From this we find:
ft I p . <p + QP y OP OP + OP . OP 9?
^ = a sin ^^ cos r  r   ft cos ^^ sin Z_*.
^M w OP f Op . OP 95 . 05 ( OP* OP OP
2 = a cos ^ ~ sin + 6 sm 2~ C S 2
Adding and subtracting the two equations:
S D = r cos 9?
5 Z) = r cos y>
we further obtain:
* ( 6) cos 99 (a f ?;) cos (f = 0,
521
sm 2 2
cos  2  cos
and
d D = a cos T T  cos L ^~ 6 sin 2 Sin 2
therefore if we substitute the value of b in the expressions for;
P\~P p ft ... Tl
~ > and o D
we find:
sin ^
.
sm 
and
/^Hy
C H 2
D = a . :
(D\(p W 
COS fT COS ~~^~
cos y> cos cp
Therefore if we put:
we obtain:
OP  O?
 ^ y 
sin ^4 and ^ _ = sin ^, (A)
2a
V cos
cos 4 = JJ
and
hence :
Hence for the computation of the distance of the chord
from the centre of the ring only the simple formulae (A)
and (#) 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:
rt = 223 22 41". 30 5 = 59 T 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.
Tbe observations were as follows:
C. north of the centre Star south
Immersion*) Emersion Immersion Emersion
18 h 15 m 54s20s 1? 21s 48* 18 m 55.3 13s. 21 20.5 37 . 5.
With this we obtain:
i 1 t Exterior ring l m 54 s t t E.R. 2 m 42s . 2
Interior ring 11 27.5
log of the sum 2 . 24304 2 . 46195
log of the diff. 1 . 72428 1 . 54033
cos ^4 9.92623 4 9:65138
cosJ3 9. 99418 9. 99749
9 . 92041 9 . 64887
8 D = + & 39".26 S D = 4 37". 88,
hence :
a 1 *=hl3 17". 14,
and the difference of right ascension is found:
a a = 3 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 (p = ft , r sin (p = ft , r cos <f> =p r cos cp = S 8.
Then we obtain:
sin (pdr \ r cos <p dtp = dp
sin cp dr \ r cos <f> dy> = dfi
[cos <p =p cos <p\ dr r sin tpdtp ==r sin tpdcp = d (S 1 8}
or eliminating in the last equation dcf and d<p by means of
the two first equations:
[cos (p =f= cos rp] di sin (f 1 cos <pd[* == sin (p cos cp d/u
= cos <p cos cp d (S 8) ;
*) 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
dp and d(.i 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/u = dp! so that we obtain
the equation :
[cos y> =f= cos tp ] dr sin [y> =p <f>] dp = cos <p cos <p d(8 $).
Therefore in order to find the difference of the decli
nations of two stars, we must arrange the observations so
that cos (f cos </ is as nearly as possible equal to 1 ; hence
we must let the stars pass through the field as far as pos
sible from the centre. If the stars are on the same parallel,
in which case the upper sign must be taken and we have
cp = (f, ^ then an error of r has no influence whatever upon
the determination of the declination. For finding the diffe
rence of right ascension as accurately as possible, it is evi
dent, 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 deter
mined by means of the ring micrometer, changes its decli
nation 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 arc 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:
J2=r 2_ (15 ; cog,?) 2 ,
where = ( t") is equal to half the interval of time
between the immersion and emersion. Now if we denote by A
the increase of the right ascension in one second of time, then
the correction A which we must apply to t on account of it
so that t\&t is half the interval of time which would have
been observed, if A had been equal to zero, is:
A< = t.^a.
524
But we have:
15 2 t cos S
hence: M= 15 . ** cos * Aa
c?
or since we have 15 cos d = /LI:
Further the tangent of the angle rc, which the chord
described by the body makes with the parallel, is:
= (15
where A^ 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 d tang n = ^   r  s ,
(la A) cos d
and since we must add to the time  the correction
X
s or:
cos o
15 cos  A cos ^ 2
we have, neglecting the product of A<? and
In the example given above the change of the right as
cension in 24 h was 1 15 , and that of the declination was
1 17 , hence we have:
log A = 8.71551 n
and
log A J= 8.72694 j*;
further we have:
log d = 2.71538 , log ft = 2.52468,
and with this we find:
Z>) = 0". 75 and A TT ) = ~ 7 " 10.
The change of the right ascension is also taken into
account, if we multiply the chord by ~ , where A
ouuU
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 = _M.tia
g 3GOO "3600"
where M is the modulus of the common logarithms, that is,
0.4343. Now since this number is nearly equal 48 times
15 multiplied by 60 and divided by 100000, we have ap
proximately :
___
3600 ~~ eoTlOOOOO
therefore we must subtract from the constant logarithm of
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 con
1 ^ W
stant logarithm of = c s was 7.48667, we must now take
instead of it 7.48817, and we obtain:
2 . 24304
1 . 72428
cos^l 9.92563
cosJS 9 .99415
s> z)==8W 75a
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:
or:
526
cos r = sin D sin S + cos D cos S cos 15 T,
(15 \ 2
T I ,
hence :
(S Z>) 2 =r 2 cos<? 2 (15r) 2 [cos/) cos S] cos 5(15 r) 2
= r 2 cos $ 2 (lor) 2 (S Z>) sin S cos ^(15r) 2 .
If we take the square root of both members and neglect the
higher powers of d D, we have :
S  D = [r > _ cos 8 * (15 T )>]4  (JZLg)
2[r 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 jthe correction sought. We have
therefore :
S D = d \ sin S cos 8 (15 r) 2 ,
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:
S D = d \ sin S cos S (15 r ) 2 ,
and hence:
8 S = d d+ [tang 8 cos 2 (lor) 2 tang S cos <? 2 (lor ) 2 ],
instead of which we can write without any appreciable error:
3 S = d JHtang(<?4<? )[cos<? 2 (15r) 2 cos S 2 (15r ) 2 ],
or since:
cos<? 2 15 a T 2 =r 2 d
and
cosd 2 15 2 T 2 =r 2 rf 2 ,
also
S S^d d + t tang  (8 t 5) (d f d) (d d) .
Hence the correction which is to be applied to the dif
ference of declination computed according to. the formulae
of No. 34, is:
In 1850 May 30 Petersen s comet, whose declination was
74 9 was compared with a star, whose declination was
73 52 . 5. The computation of the formulae of No. 34 gave:
(/= 8 56". 7, rf = H7 36".9.
With this we find:
527
log (<?td) = 1.90200,,
log (d d) = 2 . 99721
Compl log 206265 = 4 . 68557
Compl log 2 = 9 . 69897
tang 1 (<T + 8) = 0^54286
"9 . 82661"
Correct. = 0". 67.
Hence the corrected difference of declination was:
h 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:
ft f f.i = r [sin y + sin cp ] = 2 r sin j (<p + y ) cos \(cp 90 )
jit, // = r [sin y sin y> ] = 2r cos  (y> h 95 ) sin (99 y )
Further we have:
S 3 8
cos <f ( cos cp 2 cos j (90 f 9 s ) cos T (9 P y )
and hence:
* = tang i ((f> h gp ) JF^fl == tan g T fa ~~
Therefore if we put:
; ;
 tang 4 and ^; ^ ~~~ tang B.
we obtain:
2 cos A cos B
2 sin
2 cos J. sin B
sin (4 f 5)
^ ;
sin ( J. E)
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 du 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 Aste
rope and Merope of the Pleiades, whose declinations are :
= 24 4 24". 26
and
<? =23 28 6". 85
and half the observed intervals of time were *) :
18s. 5 and 5G*.2.
With this we find:
log (ft fi ) = 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 ob
serving 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 inter
val of time between the two moments converted into arc,
for one star by T and for the other by T :
cos r = sin sin D f cos S cos D cos i
cos r = sin sin D + cos cos D cos T .
If we write:
+ > . 
  1   instead of o and ^  ~  instead of u
and then subtract the two equations, we obtain:
S r r rhr
tang D = cotang sin sm
T T T h T
tang cos  cos g
Therefore if we put:
*) The stars of the Pleiades are especially convenient for these obser
vations 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
cotang   sin   = a cos A
rr
.
tang ^ cos = a sin A,
we find D from the equation:
. fr+r
  ft C1Y1 I
tang D = a sin .  + A
(B)
When thus D has been found, we can compute r by
means of one of the following equations:
sin ^ r 2 =sin  (8 Z)) 2 4 cos S cos D sin ^ r 2 ,
or
sin i r 2 = sin  ( Z)) 2 + cos 5 cos Z) sin A r 2 .
If we put here:
sin i T
(C)
sm \ r
we obtain :
sin i r 2 = sin i (8 D} 2 sec y
= sinH# Z)) 2 sec/,
and
 . (Z))
cos/
The solution of the problem is therefore contained in
the formulae (4), (B), (C) and (Z>).
When the radius of the ring is determined by one of
these methods, the declinations of the stars must be the ap
parent 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:
and
8 +57" cotang (#+# ),
where
tang J ZV= cotg gp cos ,
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:
*, s 57"sin(? 8)_
34
530
instead of which we may write:
57" sin (5 e?)
The difference of declination thus corrected must be
employed for computing the value of the radius of the ring.
These methods of determining the radius of the ring are
p o
entirely dependent on the declinations of the stars. There
fore stars of the brighter class, whose places are very accu
rately known, 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 may be possible
that there is a constant difference between the observations
of bright and faint objects; therefore Peters of Clinton has
proposed another method, by which the radius is found by
observing 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 accu
rately known.
We find namely from the equation // = r sin y :
r = t u f 2 r sin (45 : 4 9")  .
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:
V> "f M
<p A} 13.
Hence we have:
r = (JL h 2r sin [45 { (A f 75)],
or because the last term is very small:
r = ^ [1 4 2 sin (45 4 (4h B))] 5
= f*[2 sin (A + 13)}.
Since suitable stars for this method can be found any
where, it is best, to select stars near the meridian and high
above the horizon so that the refraction has no influence
upon the result. In case that a chronograph is used for the
observations, this method is especially re commend able,
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 mi
crometer 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 f r, if R denotes the semidiameter 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 72 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 ex
terior contacts, expressed in arc and multiplied by the co
sine of the declination. Therefore, denoting half this inter
val of time by f, we have the equation:
(R + r ) 2 = (S DY H (15 t cos (?).
For interior contacts we find a similar equation in which
/ , i. e. half the interval of time between the interior contacts
occurs instead of , and R r instead of Rtr:
(R _ r y* = ( z>) 2 + (15 1 cos <?) 2 .
In these two equations the times t and t must be ex
pressed in apparent solar time in order to account for the
proper motion of the sun. If we eliminate now (S D) 2 , we
obtain :
(R H r) 2 (R rY = (15 cos <?) 2 [t 2 t *},
and
_ (15 cos S)*[tht ][t t ! ]
4R
The sun was observed with one of the ring micrometers
at the Bilk observatory, w]jen its declination was + 23 14 50"
and its semidiameter 15 45". 07, as follows:
34*
532
Exterior contact: Interior contact:
Immersion 10 h 31 m 8 . 2 Sidereal time 10 h 32 IU 30 s . 8
Emersion 34 m 47* .5 33 25 . 3.
From this we find half the intervals of time expressed
in sidereal time equal to I 1 " 49 s . 65 and O m 27 8 .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 4 m 8 s .7. We have therefore:
,= 109*. 33 and t = 27* . 17,
and we find:
r = y 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.
On the ring micrometer compare the papers by Bessel in Zach s Monat
liche 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 par
allel to the dividing plane or plane of section and perpen
dicularly to the optical axis. The entire number of revolu
tions which the screws make in moving the two semilenses
can be read on the scales attached to the slides which hold
the lenses, and the parts of one revolution are obtained by
the readings of the graduated heads of the screws. There
fore if the equivalent of one revolution of the screw in sec
onds of arc is known, we can find the distance through
which the centres of the semilenses are moved with respect
to each other. When the semilenses 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 semilens wjiich is not moved, will
remain in the same position, but near it we shall see another
533
image made by the other semilens in the direction from its
focus to its centre. Therefore if there is another object
in the direction from the centre of this semilens 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 semi
lens which was moved, will coincide, and the angular distance
between these two objects can be obtained from the num
ber of revolutions of the screw, through which one of the
semilenses was moved.
In order that the plane of section may always pass
through the two observed objects, the framework support
ing the two slides with the semilenses is arranged so, that
it can be turned around the optical axis of the telescope.
Therefore if the heliometer has a position circle whose read
ings 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 pos
ition 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 eyepiece
always over the images of the object made by the semilenses.
For if one of them is moved so that its centre does not co
incide 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
semilens. Therefore in order to see it distinctly, we must
move the eyepiece 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 mini
mum, the zeropoint. 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 180. 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 eyepiece, 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 eye
piece, 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 accom
plished by directing the telescope to a very distant object
and turning both position circles 180. 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 chan
ges its place, the wirecross must be corrected by means of
its adjusting screws.
We will assume, that when the image of an object made
by one of the semi lenses is on the wire cross, the reading
of the scale is s and that of the position circle, corrected for
the index error, /?; at the same time let the reading of the
scale of the eyepiece be rr, and that of its position circle n.
Let a be the distance of the zero point from the centre of
the position circle, and t and S the corrected readings of the
hourcircle and the declinationcircle 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 rect
angular system of axes, the axis of and ?/ being in the
plane of the wire cross so that the positive axis of is di
rected to 0, and the positive axis of ;/ directed to 90 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 wo put then:
s h = e and cr h E ,
and denote by / 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 i] is positive, and if the angle of posi
tion is either in the first or the fourth quadrant, then the
coordinates of the point s are:
e cos p a sin p , e sin p cos p , /
and those of the point 6 :
e cos n a sin n , a sin TC a cos it , 0.
Hence the relative coordinates of s with respect to 6
will be:
= e cos p e cos 7f a [sin p sin n]
r, = e sin p sin 71 + a [cos p cos n] (a)
and if celestial objects are observed, whose distance from
the focus of the telescope is infinitely great compared to ,
we can assume, that these expressions are also those of the
coordinates of the point s with respect to the focus.
The coordinates must now be changed into such which
are referred to the plane of the equator and the meridian,
the positive axis of a? 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 par
allel to the axis of the heavens and directed to the north pole.
For this purpose we first imagine the axis of g to be
turned in the plane of  towards the axis of through the
angle 90 <); then the new coordinates will be in the plane
of the equator, and we shall have :
=  sin 8 + cos 8
= sin S I cos S.
Then we turn the new axis of g in the plane of g ?/
forwards through the angle 270 M, in order that it may
become the positive axis of #, and we obtain:
x = cos t + ?/ sin t
y = sin t 77 cos t
If we eliminate now g , ?/, we find:
x == cos S cos t H  sin S cos t t rj sin t
y = cos S sin t + 1 sin S sin t rj cos t
z = sin S  cos 8,
or substituting the values of g, >/, taken from the equa
tions (a) : *
536
x = I cos 8 cos t ( [e cos p s cos n] sin <? cos * + [e sin ;> e sin TT] sin *
a [sinp sin TT] sin $ cos Z  a[cos/> cos n\ sin
y = / cos $ sin t f [e cos p s cos TT] sin 8 sin [e sin /> e sin n\ cos
a [sin/> sin TT] sin sin 2 a [cos/? cos TT] cos t
z = lsmd [ecosp ecos7r]cos$ Ha[sinp sin ?r] cos <?.
From this we find the square of the distance r of the
point s from the origin of the coordinates:
r 2 = l~ h [e cos p e cos n]  f [e sin p e sin TT] 2 + 4 a 2 sin 7(7? TT) 2 .
The line drawn from the origin of the coordinates to
the point s makes then the following angles with the three
axes of coordinates:
cos a = , cos ft = and cos y =
r r 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 S cos t , cos /? = cos S sin t\ cos y = sin ,
therefore if we put:
= Z>, = A and = d,
and also for the sake of brevity:
1 + [D cos /> A cos n] 2 h [D sin /? A sin TT] 2 h 4 rf 2 sin (/ TT) 2 = ^4
we obtain:
. cos 8 cos t f [Z) cos A cos TT] sin 8 cos <
cos ff cos F =
V A
[D sin p A sin 7t] sin <
^/T~
d [sin p sin TT] sin $ cos Z d [cos /> cos n] sin
,,. . cos 8 sin t\\D cos A cos TT] sin ^ sin
S sin = 
[Z) sin p A sin n] cos t

VA
d[sinp sin 71] sin ^sin t\ d[cosp cos 7t] cos t
VA
sinS [D cosp Acos7r]cosJ
VT
d [sin p sin TT] cos 8
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
, t, A, TT, d and p
remain the same, while instead of Z>, d 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;
arid all quantities occurring in the second members of these
equations will be obtained by the readings of the instrument,
namely # and t by the readings of the declinationcircle and
the hourcircle, D and A by the readings of the slides of
the object glass and the eyepiece, and p and n by the read
ings of the two position circles. Hence we can find by means
of these equations cT, , r>" and t". It is true, the instru
ment does not give the quantities r), , & and n 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 S"  fi and t"  t perfectly
accurate.
In case that the observed stars are near the pole, we
must find t)", d , t" and t by means of the rigorous formulae
(6), but in most cases we can use formulae, which give im
mediately d" d and " , although they are only approxima
tely true. First we may take d equal to zero. If then we de
velop the divisor in the equation for sine) in a series, and
retain only the first terms, we find:
sin S sin S = [D cos p A cos n] cos 8 + $ [D cos p A cos ?r] 2 sin S
H j [D sin p A sin n] 2 sin $,
or according to the formula (20) of the introduction, retain
ing only the squares of the quantities put in parenthesis :
S S = [D cos p A cos n] y [D sin p A sin n] tang S.
For the other star we find in the same way:
S" S= [D cosp ACOSTT] 4 [D 1 sin p AsinTrJ tang S,
and hence we obtain:
8" =[D Z> ] cos />+ tang [( 4 /> )sin/j 2Asin7r][Z> Z> jsin/>, (c)
an equation, by means of which the difference of the decli
538
nations of the two stars is found from the readings of the
instrument.
In order to find also the difference of the riorht ascen
O
sions we multiply the first of the equations (6) by sin , the
second by cos t and add them. Then we get:
cos 8 sin (t  =
. 4 [D cos p A cos n] 2 4 [D sin p
and in a similar way:
*n , ;/ N D sin p AsinTr
cos o sin (t t ) = .
I/I 4 f //cos/) AcosTr] 2 +[> sinp AsinTr] 2 "
If we neglect the squares of D, D and /\, and introduce
the right ascensions instead of the hour angles, these equa
tions are changed into:
cos (a a) = D sin p A sin TT
cos 8" (a" ) = D sin p A sin ?r,
and if we write here instead of 6 and d" :
and write $ <)" instead of sin (5 ()"), and 1 instead of
cos ($ #"), we obtain :
( a) cos  (S 1 + ") = [D sin p A sin 71] [I h f tang 5 (tf" # )]
(" ) cos .V (5 4 5") = [D 1 sin /; A sin TT j [ 14  tang 8 (S" 5 )],
and hence:
(a" a ) cos  ((? 4 5") = (/> />) sin p 4 i tang ^ [5" <T) [/> 4 D] sin ;>
tang ^A sin ?r [^" $ ],
and if we substitute instead of d" d the value found before
(D D ^cosp
we find:
(" ) cos  (<? 4<T) = (D D) sin/j
tang^[(/) 4Z>)sin;?~2Asin7r][Z) Z>] cos/7, (rf)
If now we put:
M = tang 5 [(/) 4 Z>) sin 7? 2 A sin TT], (^4)
we can write in the equations (c) and (d) sin ?/ instead of
the small quantity ?/, and add in the first terms of the equa
tions the factor cos u. Then we obtain :
y> _S = (D  Z)) cos (p 4 n)
a" = 4 (7V 7)) sin (/> 4 t/.) sec .V (^ 4 5").
We have assumed 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 & , 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 semi 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
must put \ (I) D) instead of D D in the above formulae.
Instead of the angle p + u, we obtain from the two obser
vations now p f u and p + ?/", and hence we shall have :
*t.y 2A, = a h
and
u = .j tang [(s f s 2 /*) sin p 2 (a /?) sin n\
S"8 = (// If) cos (p f M)
a" o= h  (// Z>) sin (/? h M) sec * ( + 5").
If we wish to find t)" <V and " expressed in sec
i y _ jj
onds and u expressed in minutes, we must multiply  
by the equivalent of one unit of the scale in seconds of arc
and the expression for u by QTTJ Now we can always
arrange the observations so, that we can neglect the term
dependent on p ;r, because we have
u = 0, when a = and n = 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 for, 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 mo
tion in right ascension and declination to be proportional to
the time. However it is more convenient to calculate the dif
ference 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 f, t\ t" etc. be the times of observation, and T their
arithmetical mean, and put:
tTr, t T=r , t"T=r",etc.
Further let p, /? , p" etc. be the angles of position corres
ponding to those times, P that corresponding to the time T,
and A and /\() the change of the right ascension and de
clination in one second of time, assuming that r, T etc. are
likewise expressed in seconds of time. Then we have:
We shall have as many equations as angles of position
have been observed, and if n is the number of observations,
we obtain:
UKA 7 H; ,i*a&8 + "A9*  ,
/ da 2 dado do n
where we can take:
2.22 sin I r 2 . f 2^
 instead of
n n
if we have tables for these quantities.
Likewise we obtain from the observed distances the dis
tance D corresponding to the arithmetical mean of the times:
541
dhd Hd"K.,
We must now find the expressions for the differential
coefficients. But we have:
D sin P = (a a ) cos
c,
or: tangP= s s; cos
Z> 2 = ( a ) 2 cos d 2 + (0" 8 ) 2 ,
and we easily find:
dP cos S cos P dP sinP dZ) d/)
=  ^ = ri  = cos o sin P. r = cos P
da D do D da do
dP _ 2 cos ? 2 sin P cos P d 2 P = 2 sin P cosP
d 2 Z) 2 do 2 Z) "
d 2 P 2 cos 0" sin P 2 cos 8
d~a~d~ ~~D*~~ ~~D*~
dD_cosS cosP 2 d 2 Z)_sinP 2 d 2 > cos S sin P cos P
do 2 " D d~ D"* da.dS~ D
If we put:
A cos S = c sin /
A 0^ == c cos 7,
we obtain :
_ /? 4 p h ^ H . . . _ sin_(Pri.?0_cos_(Pz: jO 2 ^ 2
"n D 2 n
__ ... _ , sin(P
D
or denoting by M the modulus of the common logarithms:
^_^_ d
log D = log
n JLJ u
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<* and j\d 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
86400 2 R*
and the term in the equation for D by:
100000 . 60 2
86400 ^TR^
542
But if we make use of the tables for 2 sin \ r 2 , so that
we take:
_  +... _ sin (P ^
and
we must multiply these terms respectively by
60. 206265 2
86400*. .15*.
and
__
86400 2 .^Tlo 2
41. It is still to be shown, how the zero of the posi
tion 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 semilenses 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 *). If then also the other
image can be brought to this point merely by turning the
telescope round the declinationaxis, the plane of section will
be parallel to the plane in which the telescope is moving,
and hence the collimationerror 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 declinationaxis. Then the read
ing of the position circle in this position is its error of colli
mation.
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
*) For this purpose it is convenient to have double pantile! wires, so
that the middle of the field is indicated by a small square.
543
wires, this must be parallel to the equator. If then the semi
lenses are separated, and the objectglass 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 270. 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 ob
served 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 objectglass of a heliorneter,
we see distinctly the double image of the wire at its focus.
Therefore if one of the semi lenses is in that position, in
which the index is exactly at the middle of the scale, while
the other semilens 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 dif
ference 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 screwhead.
Let then S be the reading of the scale of the latter
semilens and S the reading of the scale of the other semi
lens which remains always in the same position, finally s
that of the scale of the eyepiece, then we have, if b and c
are the angles, which straight lines drawn from the points
S and S to the focus make with the axis of the telescope:
(.s S ) R = 206265" tang b
(S .s) R = 206265" tang c,
where R is the value in arc 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 h c.
If we eliminate b and c by means of the last equation,
we find the following equation of the second degree:
(.  S.) (S  .) tang a . 2 + ( S.) = ** ,
from which we obtain:
R _ (S  )  tf(S  Sp) 2 + 4 (s ^SQ j QS
206265 2 S ) (S s) tang a
Let then S be the reading . of the scale in the second
position of the semilens, s that of the scale of the eyepiece
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 S = S<> S and s S = S s
and then we find from the difference of the two equations :
_R_ _ (S S) V(S Sr~ +16 (^^oX^
~
206265 4 (s S ) (S s) tang f (o h a )
When 5 S y and S s have the same sign , and if
we put:
we find for #:
206265 
tugaK(  4$,} OS )
= 206265
^ 5
But when 5 8 and S s have opposite signs, and if
we put:
we find for /?:
^ = 206265
sin /S
= 206265
W (.>
When = S and s = S , we obtain for /? instead of
the equations of the second degree the following:
545
f ) 2ol65 = tang "
R
hence :
R = 20G265  .^A.y_L.^
for which we can also write:
These formulae can be used also in case, that the dia
meter 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 wirecross, we
can also place one of the wires parallel to the equator and then,
after the two semilenses 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 arc of one revolution of the screw is va
riable with the temperature and hence it must be assumed
to be of the form:
R = a b(t * ).
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 mil dem Fraunhoferschen Heliometer Beobachtungen
anzustellen.
and
Bessel, Theorie eines mit einem Heliometer versehenen Aequatoreals.
Astronomische Untersuchungen, Bd. I. Konigsberger Beobachtungen
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 de
clinations 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 correc
tion, 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 pu
blished 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:
n tang z,
where z denotes the apparent zenith distance and a is a fac
tor variable with the zenith distance, which for
. 2 = 45 is equal to 57". 682
and decreases when the zenith distance is increasing so that
for 2 = 85 it is equal to 51". 310.
By means of these tables others can be calculated, whose
argument is the true zenith distance and by means of which
the refraction is found by the formula:
s o = ft tang ,
where /? is again a function of . We have therefore:
tang
hence :
= z z 4 ft 1 tang ft tang g,
or denoting:
(* *) by AC* *)
also :
A (z z} = (? tang  ft tang g. (a)
This is the expression for the correction, which must be
applied to the observed difference of the apparent zenith dis
tances in order to find the difference of the true zenith dis
tances.
*) This remark is not true for micrometers with which distances and
angles of position arc measured.
547
If we denote by ft that value of /?, which corresponds to :
2 "
and which is derived from the equation:
o = ft Q tang ,
we have:
(f tang = /? tang g + 1 ^ tang (g  g) }...
"bo
/? tang g = j3 tang g  4 j tang g (g ;  g) 4 . . .
ago
If we write in all terms of the second member, except
the first, tang ^ instead of tang and tang , the terms con
taining the second differential coefficients will be the same,
and we have with a considerable degree of accuracy:
ft tang g ft tang g = /9 [tang g tang g]
a&o sec g
Therefore if we put:
rf^o sec ^ n 
we obtain by means of (a):
A (z 1 2) A: [tang g tang g]
where & must be computed with the value:
2
and since we can take, neglecting the second power of :
tang g tang g== ~=v
we have :
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 c . and we find:
dz
A (s 1 z) = k ~ ., ,
az cos g "
or if we put now:
35*
548
* sec
ir H^r^sin 2 Co 206265 , (/I)
t/z ( d
we finally obtain:
_ ^_ _z_
cos C 2
The following example will serve to show how accura
tely 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
87 20 87 5 27". 4 14 32". 6
30 14 54 . 8 155.2
40 24 20 . 7 39 . 3
50 33 44 .5 16 15 .5
88 43 6 . 4 53 . 6.
From this we obtain the following values of ft:
87 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 c ? , that is, the variations
of ft Q corresponding to a change of c equal to one second:
87 30 0". 0019750
40 .0021767
50 .0023967.
If we compute now the values of A;, we find, since the
logarithms of ~ are :
87 30 0.0271
40 . 0287
50 . 0307,
the following values for the logarithms of k:
Jc
87 30 6.0505
40 6.0155
50 5.9771
where k is expressed in parts of the radius.
549
If we take now:
2 = S7 10 and z = S750 ,
and hence:
 _ 2 = 40 ,
we have by means of the common tables of refraction:
= 87 24 47". 8
=88 7 23 .0,
hence :
= + 42 35". 2
= S746 5".4.
If we suppose now that z z and are given, and
compute A (X *) by means of the formulae {A) and (#),
we find, since the value of log k corresponding to is
5.9925:
A (2 2) = + 2 35". 4,
hence :
= h42 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 argu
ment is the zenith distance. Such tables have been publi
shed in the third volume of the Astronomische Nachrichten
in Bessel s paper ^Ueber die Correction wegen der Strahlen
brechung 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 baro
meter.
For computing the difference of the true zenith distan
ces to 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 declina
tions. For this purpose the following formulae are the most
convenient, since it is also necessary, to know the parallactic
angle :
sin sin ij = cos cp sin t
sin cos r] = cos8 sin cp sin S cos cp cos t a
cos = sin $o sin cp + cos S cos cp cos t .
550
Putting:
cos n = cos tp sin t ( ,
sin n sin N= cos tp cos t
sin n cos N= sin 90,
we have:
sin sin 77 = cos n
sin g cos 77 = sin n cos (.AT") <? n )
cos = sin n sin (JVf <? ),
or:
tang sin 77 = cotang n . cosec (N\ S )
tang cos 77 = cotang (2Vt $ ).
The quantities cotang n and iV 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 coordinates. For if ft tang is the refrac
tion for the zenith distance f,
$ tang t sin ri . , , ,, , . . i ,
p.    is the refraction in right ascension
and
ft tang cos i] the refraction in declination.
But we have:
. sin 77 sin rj sin 77 . sin 77
^y ~ ft tang ^ cos 1= k tang ^ cos y ~ k tang e oosi
tang sin 17 o
. _ cos < ,
(d d) f fc .  (a a),
_
,  . 
ad,, d
and likewise we find:
/3 tang g cos ,  ft tang cos rj = k . _
c?a
rf. tang g,, cos 770 ,
h k . ( a),
aa
where (5 and denote the differences of the appa
rent right ascensions and declinations.
551
Differentiating the formulae for:
tang sin 17
~ and tang cos rj
cos o
we obtain:
jj _
cos S tang 2 sin TJ cos y tang sin rj_ tang o"
dS cos o^
. tang sin 77
a ^
= 1 tang cos 17 tang S + tang g 2 sin vj 2
 [tang 2 cos ?7 2 + 1]
= tang 2 cos 77 sin 77 cos $ f tang sin 77 sin J,
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 ascen
sion 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:
*9~fi tang cos 17,
for the second:
tang
and hence we obtain by means of the formulae in No. 43:
ta
A ( y  S) =  k .
552
or substituting the values of the differential coefficients:
A / ; v __ , /*; *s tang 2 sin /; cos ?? tang sin/; tang$
cos 8
A (8 <?) = ( 8) [tang 2 cos 77 2 f 1].
These formulae receive a more convenient form if we
introduce the auxiliary quantities cotang n and N. For, sub
stituting the values given in No. 42 for:
tang g sin ij and tang cos 17
we obtain:
A / ; N__/^ N Ct 20 )
sin (7Vf$ ) 2 cos 8
and
45. The ring micrometer. If the refraction 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 dif
ferences 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 micro
meter :
A ( a > a ) = k ( 8) tang 2 sin /o cos 77 tang g sin 77 tang <?
cos<?
and (a)
A ( 8) = k (S 8) [tang 2 cos y * + I}.
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 de
clination. Now if h and h 1 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 decli
nation computed from the observations:
8D_.
553
where D is the declination of the centre of the ring and p
is half the chord. Since:
tang sin 77
d .
cos o
dt
and
, d . tang cos 77
~~~dt
we have:
f TY tan & 2 cos ^ sin? ? ~+~ tan S S sin ^tang ^
and likewise for the other star:
^ > / /s n . tang " 2 cos 77 sin r/ 4 tang sin 77 tang <?
=*(*/>) ~cos>~
or if we write in both equations 0? 7 A> an( ^ f ^o instead of
u, 77, c) and , ?/, c> , that is, if we neglect terms of the order
of k(d D) 2 , we obtain:
A ( a > _ a ) yt (^ _ $) tan ? ?. 2 COS ^ si M "*" tan ^So sin _^o _tan_g ^
If we unite this with the first part of the correction,
which is given by the first of the equations (a), we find:
if i \ in ^ tang g 3 sin 2 77
A ( ) = K (d d) (A)
cos d
Further we have:
If we put rV D = d and denote by h the value of h
for the centre of the field, we have:
d o
r 2 (^^ ) 7 dd (dd }
dd> k ^ ~"dd r ~ ^
hence :
7 /^; _ V\ 2
t 1 ~~ tan gSo cos r; tang^ 4 tang
A; (5 5) [1 tang cos 77 tang ^ f tang 2 sin r; 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 ( 8) = k (8 8) [tang  cos 2i? + tang cos 77 tang <? ]
X [1 h tang 2 sin 7? 2 tang cos 77 tang <? ]
for the expression of the complete correction of the difference
of declination. Here we can in most cases neglect the terms
multiplied by tang and thus we obtain simply :
A (8 1 S) = k (3 9) tang 2 cos 2^ (Z?)
r 2
 k (S S) _ [tang 3 sin i? a H I].
Example. In 1849 Sept. 9 the planet Metis was ob
served at Bilk and compared with a star, whose apparent
place was:
a = 22h I" 1 59 s . 63 , $ = 21 43 27". 08.
The observations corresponding to 23 i! 23 " 19 s . 3 sidereal
time, were:
=+ 1 m 9s. 65 =4 17 24". 75
8 D = 5 17". 5, 8 D = + 6 34". 2
(? 5 = 11 51". 7 and we have r = 9 26". 29.
Now if we compute and /; with
* = lh20M5s=20 11 , (? = 21 49 . 4 and <p = 5l 12 . 5
we obtain:
, cotangn = 9. 34516 N=31l . 9
j? = 1255 .3 g = 759 . 6.
From the tables for ^ we find for this zenith distance:
log A = 6. 42 14,
and then the computation of the corrections by means of the
formulae (#) is as follows:
log k = 6 . 4214  sin 2 ^ 9 . 6394 . 0667
log (8 8) = 2 . 8523,, . 4273 cos (? 9 . 9677
tang 2 = 1 1 1536 cos 2 rj 9 . 9542 A( ) = 1".25
"6 . 4273,, 1 term of A (8 rV) = 2".41
sin TJ 2 8. 6990
log (tang 2 sin 77 2 H 1) = . 2335
log/ 2 5.5061
^
5.0133.
D)(S D) 5.0975,,
II term of A (8 8) h 0". 82
A ( ) = 1". 25
A(<? ^) = 3". 23.
555
Hence the corrected differences of right ascension and
declination are:
== + 17 23". 50
> _ $=ii 54". 93.
4(5. The micrometer with which angles of position and dis
tances are measured. If a and tf ft denote the dif
ferences of right ascension and declination affected with re
fraction, and a a and d d the same differences freed
from it, we have:
, tang sin rj
a d = a k ( 8} ~~~d~S
tang g sin?y
where the values of the differential coefficients ought to be
computed with the arithmetical means b 9  , r/ ^ and  ^
We have therefore:
tangg sinj/
d (a 1  ) =  k (3 ^
^ tang g sin 77
ffc(a ; a)
and likewise:
 Substituting the values of the differential coefficients found
in No. 43, we get:
_ tang 2 sin rj cos ?? tang g sin ?y tang ^
d (a  ) = A: (5  5)  ~^sT~
I A; ( ) [tang g 2 sin ?/ 2 tang cos ?? tang 5+1]
rf (5 5) = k ( S~) [tang g 2 cos T? 2 + 1]
+ k(a a) [tangt 2 COST; sin?; cos 5+ tang sin r? sin 5].
But, if A and ;r denote the apparent distance and the
apparent angle of position, we have:
cos 8 ( ) = A sin TC
and
8 8 = A cos TT,
hence:
cos 5 ( n)
and A = cos 5 ( ) sin ?r + ( 8) cos TT.
556
If then A and n denote the true distance and the true
angle of position, we have:
, cos 71 cos 8d(a ) sin nd(8 8)
TC = 7T + 
A
A = A ~f sin TT cos Sd (a a) f cos n d ( ).
If now we substitute here the values of d(a ) and
rf(<5 t)) which were found before, and introduce in them
A and n instead of a a and <)" d, we obtain:
Jt = it + fc tang  [sin ?r cos 77 cos n cos ?r f sin 77 sin 77 sin 7t cos cnr
cos rj cos ?y cos ?r sin n sin 77 cos 77 sin ?r sin n\
fc tang ^ [cos n cos ?r sin r, tang $ H sin 7t cos TT cos 77 tang 8
4 sin TT sin n sin 77 tang 8]
h ^ sin TT cos TT A: sin n cos TT,
or if we neglect the terms multiplied by tangC:
n = 7t k tang 2 sin (TT 77) cos (TT 77).
Further we get:
A = i\ + k A tang 2 [sin TT cos TT sin 77 cos 77 f sin n  sin 77 f cos n~ cos 77 2
h sin n cos TT sin 77 cos 77]
A: A tang [cos ?r sin ?r sin 77 tang ^ f sin TT sin ?t cos 77 tang 8
sin n cos TT sin 77 tang 8]
+ A; A [sin 7t 2  cos ?r 2 ],
or if we neglect the terms multiplied by tangc:
A = A f k A [tang  cos (n //) + 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. 1 1 of the first section
and the third of the differential equations (11) in No. 9 of
the introduction the variation of the angle ?/, which the de
clination circle makes with the circle of latitude:
cos 8 drj = sin e . sin a dk + cos a c?c,
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 t] 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 pre
cession and nutation, it follows that the change of p is equal
to that of r t taken with the opposite sign, and that therefore:
cos 8 dp sin e sin a d k cos a ds. (a)
Since the lunisolar precession does not change the obli
quity of the ecliptic, we find the annual change of the angle
of position by .precession from the equation
s dp dl
cos o  = sin a sin e >
dt dt
or:
dp *
L = n sm sec o
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, and rT for the arithmetical mean of the ti
mes, and to multiply the value of ~ found from them by
the interval of time.
In order to find the changes produced by nutation, we
must substitute in (a) instead of dl and de 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 == I 20" . 0644 sin sec S f [ 6" . 8650 sin O H 0". 0825 sin 2 1
0". 5054 sin 2 Q] sin sec S
 [9" . 2231 cos O  0" . 0897 cos 2 O
f 0". 5509 cos 2 Q] uos a sec <?,
or if we make use of the notation adopted in No. 1 of the
fourth section:
dp = A . n sin a sec S f B cos a sec #,
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 dis
tance 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 tDd ,
where C= 20". 445 cos s cos 0, D 20". 445 sin
c = sec 8 cos a, c = tang s cos 8 sin 8 sin
d = sec 8 sin , d = sin 8 cos n.
Now if ). and v denote the differences of the right as
censions 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 :
where : A c = sec S sin a . I + sec S tang 8 cos . v
Ac/= sec 8 cos a . k f sec 8 tang S sin . v
A c = sin S cos a . I [tang s sin 8 + cos 8 sin a] v
Ac/ = sin 8 sin a . k f cos S cos . v.
Hence, substituting these expressions we have :
cos Al = {?[ sin n . I + tang 8 cos a . / ] h D [cos . k + tang <? sin , r]
hv (7 [sin $ cos a . A f (tang s sin 8 f cos $ sin a) v\
D [sin 8 sin . k cos 8 cos a . v\
But, if we denote the distance and the angle of position
by s and P, we have:
* . sin P = 1 cos 8
* . cos P = *>,
hence:
A cos #
s =/ J cos d + v, tangP= ,
and therefore :
s . As = cos <? 2 k . A A h v kv cos <? sin S P (6V H Z) c/ ).
If we substitute herein the values of A^ and A^ found
before as well as the values of c and d , we find after an
easy reduction :
.s . A s = [I cos 8 f // ] [ C (tang sin 8 h cos c? sin a) \ D cos $ cos a]
or : A* = C v . s [tang c sin $ f cos $ sin a] }/). .s cos $ cos .
Further we have:
s 2 dP v cos $ . A^ & cos $ A* ^ sin (^ [Cc h />c/ J,
and if we substitute the values of A>t? A^ c and c? , we find
again after a simple reduction:
dP= 6 tang 8 cos a f D tang 8 sin a.
559
Therefore if we introduce the following notation:
, n ,. .
== sec o sm
bO
. sec cos
J)== 60
60
== _ ^_ f
tang o" sin a s
rf = d== cos o cos ,
where the factors   and  or , have been added in
bO w 206265
order to find the corrections of the distance and of the angle
of position expressed respectively in seconds of arc and mi
mites of arc, then we have:
Observed distance = True distance \cC\ dD
Observed angle of position = True angle of position for the beginning of the year
+ a A+b Bic > C+<?D.
Since c, rf, 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 circum
ference 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, printed by A. W. SCHADE, Stallsclireiberstr. 47.
UNIVERSITY OF CALIFORNIA LIBRARY,
BERKELEY
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iOV 22 1944
50m8, 26
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