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Full text of "Spherical astronomy"



- 



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 CO-ORDINATES. FORMULAE OF 
SPHERICAL TRIGONOMETRY. 

Page 

1. Formulae for the transformation of co-ordinates 1 

2. Their application to polar co-ordinates 2 

3. Fundamental formulae of spherical trigonometry 3 

4. Other formulae of spherical trigonometry 4 

5. Gauss s and Napier s formulae . 5 

6. Introduction of auxiliary angles into the formulae of spherical 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. Co-ordinate system of azimuths and altitudes 73 

3. Co-ordinate system of hour angles and declinations 74 

4. Co-ordinate system of right ascensions and declinations .... 75 

5. Co-ordinate system of longitudes and latitudes 77 

II. THE TRANSFORMATION OF THE DIFFERENT SYSTEMS OF 
CO-ORDINATES. 

6. Transformation of azimuths and altitudes into hour angles and 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 pole-star 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 circum-meridian altitudes . . 266 

8. The same problem, when the declination of the heavenly body is 
variable . 269 

9. Method of finding the latitude by the pole-star 271 

10. Method of finding the latitude, given by Gauss 275 

III METHODS OF FINDING BOTH THE TIME AND THE LATITUDE 

BY COMBINING SEVERAL ALTITUDES. 
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 spirit-level. 

1. Determination of the inclination of an axis by means of the spi 
rit-level 390 

2. Determination of the value of the unit of its scale 395 

3. Determination of the inequality of the pivots of an instrument . 398 

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. 



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XX 



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r stand 



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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 

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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 cos sin A 

read = 

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read between 
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read from 




INTRODUCTION. 



,1. TRANSFORMATION OF CO-ORDINATES. FORMULAE OF 
SPHERICAL TRIGONOMETRY. 

1. In Spherical Astronomy we treat of the positions 
of the heavenly bodies on the visible sphere of the heavens, 
referring them by spherical co-ordinates to certain great cir 
cles of the sphere and establishing the relations between the 
co-ordinates with respect to various great circles. Instead of 
using spherical co-ordinates we can give the positions of the 
heavenly bodies also by polar co-ordinates, viz. by the angles, 
which straight lines drawn from the bodies to the centre of 
the celestial sphere make with certain planes, and by the 
distance from this centre itself, which, being the radius of 
the celestial sphere, is always taken equal to unity. These 
polar co-ordinates can finally be expressed by rectangular 
co-ordinates. Hence the whole of Spherical Astronomy can 
be reduced to the transformation of rectangular co-ordinates, 
for which we shall now find the general formulae. 

If we imagine in a plane two axes perpendicular to each 
other and denote the abscissa and ordinate of a point by x 
and ?/, the distance of the point from the origin of the co-or 
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 co-ordinates 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 co-ordinates 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 co-ordinates, 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 co-ordinates: 

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 co-ordinates 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 co-or 
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 (7-4- 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 . 6-4-c 6 c 

sm-j^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 b-i-c b c 

cos T a cos ^ -- -.cos^asm -- - =sin^cos .cos^^lcos n 

Z Z 2 Z 

. , BC . . B C 6-f-c 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 : 

. 6-hc 
sin A sm 5 = a 

sin? J-cos <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 

6-hc 
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(6-f-c)~ 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): 



A-i-B 

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 right-angled 
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 right-angled 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 right-angled 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 </6-h-. : --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 -+- -j-r 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 

r-f- 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-{-asin2x-i- 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 2acosx-i-a 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:-t-a 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 An-i 

**) 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:r-4- }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) = = 

1-j- tang y tango: l-f-ntang* 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 



(n4-D (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: -t-4 ( . ) sin6r + ... (16) 
n-hl Vn-f-lx \n-j-l/ 

If we take here: 

n = cos a, 

we have: --- = tang 4 a 2 . 

n-f-1 

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 
ioI-a 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 

a-b 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 )H-Ttang - 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 : 

A-B sin ~2~ 180- (7 



sin - - 



a~b 

~2~ 180-C7 



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[lH-3cotang* 2 ] -,.... (19) 

sin* sin* 2 sin* 3 

In the same way we find from the equation: 

sin y = sin * -f- b 

y = x-\ Ktangs-^-r-H [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 
/"(a-hft-f-i), 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(tf-hf) the difference of 
f(a -l-20) and /"(a-f-w?). 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 (a-Hf) 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) 



/ (- 




o-|-3;/(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 /"(a-j-J), 
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 /""(aH-f) 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 = -y-g 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 (1-f-a?)*. 

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 H-2 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)4-7/-h21/"4-35/"" " 



22 

Hence we have to add to f(cf) 
-1-3 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 (aH-2) = 6". 3. Adding this to f" (4-f) 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 ^a-h^) - 

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)(n-4) 

1.2.3 1.2.3.4 1.2.3.4.5 

_ (n-f-2) (nH-1) 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 a-Hf 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) (n-f-2) 



1.2.3 1.2.3.4 1.2.3.4 

We find therefore: 

f" (a) + 1 



( n --2)( n -l)n(n+l)(nH-2) 

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+ln-l) 2) /lv 



(n4-2)(n-4-l)n(n-l)(n-2) 

~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 /" (a-hl). 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 4-1). 

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 (o-f-to) 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: 



- * [/"(a-H) - ^ [/ 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". H-2 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: 

/(oH-nuO =/(a) -f- n[f (a 4-^) /" (a 4-1) -+- 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/ " (a-f-i) - ...] 

^ = 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 : 

/(a4-nu;)=/(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 (a-f-n) -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"(a-t-ri) 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)(n-l) 
1 .2.3 

/ (aH-i) 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/(oH-j) 



etc. 
we find, as we have: 

(n-hl)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 -*^) / (a-H + 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 right-ascensions 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 right-ascension. 
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 -4-2 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?i-l) 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 -7-7-7 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) 

14-3? 
14-4?" 

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 5-2 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, ^o-4-y? 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-}- by-i- 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 4-logC 
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 -f-rV 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 H-d , 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? H-r 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 


4-0.12 


4 


43 


. 52 


-0.03 


19 


43 . 


32 


4-0.17 


5 


43 


. 31 


4-0.18 


20 


43 . 


12 


4-0.37 


6 


43 


. 67 


-0.18 


21 


43 . 


30 


4-0.19 


7 


43 


. 98 


-0.49 


22 


43 . 


72 


-0.23 


8 


43 


. 63 


-0.14 


23 


43 . 


25 


4-0.24 


9 


43 


. 83 


-0.34 


24 


43 . 


13 


4- 0.36 


10 


43 


. 79 


-0.30 


25 


43 . 


27 


-4-0.22 


11 


43 


. 54 


0.05 


26 


43 . 


34 


4-0.15 


12 


43 


. 18 


4-0.31 


27 


43 . 


15 


4- 0.34 


13 


43 


. 45 


4-0.04 


28 


43 . 


86 


-0.37 


14 


43 


. 68 


-0.19 


29 


43 . 


29 


4-0.20 


15 


43 


. 32 


4-0.17 


30 


43 . 


40 


4-0.09 










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~4-Va 4~ a" a" 4- . . = r V[aa\. 

Every term in \bn^\ is of the following form (A 1 -r-6)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 -f-A" 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 H-o *, 

0=w"-f-rt"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. 


= 


4-0 


.02 -+-x 4- 


0.2?, 





& 


.03 


ft 


Urs. min. 


= 


4-0 


.454- 


x 


4- 


8.23, 


4- 





.43 


ft 


Cephei 


= 


4-0 


. 104- 


X 


4- 


20.13, 


4-0 


.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. 


= 


4-0 


.594- 


X 


4- 


75.53, 


4-0 .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 


= 


4-0 


. 12 4- 


X 


4- 


142.13, 


4-0 


.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 


4-393.53, 


4- 





.51 


17 


Aurigae 


= 


- 


.804- 


X 


4-419.6^ 


4- 





.06 


ft 


Persei 


= 


2 


.164-* 4-481.23, 





.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] 

4-20.000 4-3014.80 -12.72 20.28 

1.301030 3.479259 1.104487, ^- 8.09 

Ian] =12.72 [66] [6n] 12.19 

[a 6]* = 4- 13.78 4-844586.1 3700.65 ^~ 8.15 

[*&|J 

4- 1.06 4-454452.0 -1917.41 [wn 2 ] = 4.04 

0.025306,, [66,] = 4-390134.1 [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 ar-f- 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 p-i-q = kn or p q kn, hence 
p = q-i-kn 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/j-fj 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 

-1-4- 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 sin604-X 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)A-l-a -\-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 co-ordinates to certain great circles of the sphere. 
Spherical astronomy teaches then the means, to determine the 
positions of the stars with respect to these great circles and 
the positions of these circles themselves with respect to each 
other. We must therefore first make ourselves acquainted 
with these great circles, whose planes are the fundamental 
planes of the several systems of co-ordinates and with the 
means , by which we may reduce the place of a heavenly 
body given for one of these fundamental planes to another 
system of co-ordinates. 

Some of these co-ordinates are independent of the diurnal 
motion of the sphere, but others are referred to planes which 
do not participate in this motion. The places of the stars 
therefore, when referred to one of the latter planes, must 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 co-ordinates, 
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 North-Pole of the earth, being there 
fore visible in the northern hemisphere of the earth is called 
the North-Pole of the celestial sphere, while the opposite is 
called the South-Pole. If we now imagine a line parallel to 
the equator of the earth, hence vertical to the former, it will 
on account of the diurnal motion describe a plane, whose 
intersection with the celestial sphere coincides with the great 
circle, whose poles are the Poles of the heavens and which 
is called the 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 Parallel-circles. 

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 co-ordinates. We 
imagine a great circle drawn through the star and the zenith 
and hence vertical to the horizon. If we now take the point 
of intersection of this great circle with the horizon and count 
the number of degrees from this point upwards to the star 
and also the number of degrees of the horizon from this point 
to the meridian, the position of the star is defined. The great 
circle passing through the star and the zenith is called the 
vertical -circle of the star; the arc of this circle between the 
horizon and the star is called the altitude, while the arc between 
the vertical -circle and the meridian is the azimuth of the star. 
The latter angle is reckoned from the point South through 
West, North etc. from to 360. Instead of the altitude 
of a star its zenith-distance is often used, which is the arc 
of the vertical circle between the star and the zenith, hence 
equal to the complement of the altitude. Small circles whose 
plane is parallel to the horizon are called almucantars. 

Instead of using spherical co-ordinates we may also de 
fine the position of a star by rectangular co-ordinates, 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 co-ordinates an instrument perfectly 
corresponding to them is used, the altitude- and azimuth -instrument. This 
consists in its essential parts of a horizontal divided circle, resting on three 
screws, by which it can be levelled with the aid of a spirit-level. This circle 
represents the plane of the horizon. In its centre stands a vertical column, 
which therefore points to the zenith, supporting another circle, which is 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 zenith-distance of any star towards which 
the instrument is directed, may be determined. 



74 

Besides this instrument there are others by which one can observe only 
altitudes. These are called altimeters, while instruments, by which azimuths 
alone are measured, are called theodolites. 

3. The azimuth and the altitude of a star change on 
account of the rotation of the earth and are also at the same 
instant different for different places on the earth. But as it 
is necessary for certain purposes to give the places of the 
stars by co-ordinates which are the same for different places 
and do not depend on the diurnal motion, we must refer the 
stars to some great circles, which remain fixed in the 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 polar-distance of the star. The great circle itself 
is called the declination -circle of the star. The declination 
is positive, when the star is north of the equator and 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 zenith-distance in 
the first system of co-ordinates. 

The arc of the equator between the declination-circle of 
the star and the meridian, or the angle at the pole measured 
by it, is called the hour-angle of the star. It is used as the 
second co-ordinate and is reckoned in the direction of the 
apparent motion of the sphere from east to west from 
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 co-ordinates, the de 
clination and the hour-angle, we may again introduce rectan 
gular co-ordinates 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 hour-angle 90. Denoting 
then the declination by d, the hour-angle by , we have: 

x = cos cos ?, y = cos sin t, z = sin S. 

Note. Corresponding to this system of co-ordinates we have a second 
class of instruments, which are called parallactic instruments or equatorials. 
Here the circle, which in the first class of instruments is parallel to the 
horizon, is parallel to the equator, so that the vertical column is parallel to 
the axis of the earth. The circle parallel to this column represents therefore 
a declination circle. If the points of the circles, corresponding to the 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 co-ordinates 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 co-ordinate, 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 co-ordinates, declination and 
right-ascension, we can again introduce rectangular co-ordi 
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 right-ascensions, the positive axis of y to the 
point, whose right-ascension is 90 . Denoting then the right- 
ascension by a , we have : 

x" = cos S cos , y" = cos sin , z" = sin d. 

The co-ordinates 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 hour-angle 
of the point of the equinox, which is called the sidereal time, 
while the time of the revolution of the celestial sphere is 
called a sidereal day and is divided into 24 sidereal hours. 
It is 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 hour-angle 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 right-ascension 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 right-ascension expressed in 
time. Hence when the right -ascension of a star which is 
culminating, is known, the sidereal time at that instant is 
also known by it*). 

*) The problem to convert an arc into time occurs very often. 

If we have to convert an arc into time, we must multiply by 15 and 
multiply the remainder of the degrees, minutes and seconds by 4, in order to 
convert them into minutes and seconds of time. 



77 

If the sidereal time at any place is 0, at the same 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 co-ordinates of the third system can be found by instruments 
of the second class, if the sidereal time is known. In one case these 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 
meridian-altitude of the star, if the latitude of the place is known. For such 
observations a meridian-circle is used. If such an instrument is not used for 
measuring altitudes but merely for observing the times of the meridian -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 right-ascensions are reckoned cannot be observed itself, 
it is more difficult, to find the absolute right-ascensions of the stars. 

5. Besides these systems of co-ordinates a fourth is 
used, whose fundamental plane is the ecliptic. Great circles 
which pass through the poles of the ecliptic and therefore 
are vertical to it, are called circles of latitude and the arc 
of such a circle between the star and the ecliptic is called 
the latitude of the star. It is positive or negative if the star 
is North or South of the ecliptic. The other co-ordinate, 
the longitude, is reckoned in the ecliptic and is the arc 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, 4x34-3 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 co-ordinates ft and A 
by rectangular co-ordinates, referred to three axes vertical 
to each other, of which the positive axis of z is vertical to 
the ecliptic and directed to the north -pole of it, while the 
axes of x and y are situated in the plane of the ecliptic, the 
positive axis of x being directed to the point of the vernal 
equinox, the positive axis of y to the 90 th degree of longitude, 
we have: 

x " = cos ft cos I , y " = cos /3 sin ^,, z" = sin ft. 

These co-ordinates are never found by direct observations, 
but are only deduced by computation from the other systems 
of co-ordinates. 

Note. As the motion of the sun is merely apparent and the earth really 
moving round the sun, it is expedient, to define the meaning of the circles 
introduced above also for this case. The centre of the earth moves round 
the sun in a plane, which passes through the centre of the sun and 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 66-5- 
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 
CO-ORDINATES. 

6. In order to find from the azimuth and altitude of 
a star its declination and hour -angle, we must revolve the 
axis of z in the first system of co-ordinates in the plane of 
x and z from the positive side of the axis of x to the positive 
side of the axis of z through the angle 90 (p (where cp 
designates the latitude), as the axes of y of both systems 
coincide. We have therefore according to formula (la) for 
the transformation of co-ordinates, 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 right-angled triangles or by the 
addition of a right-angled triangle is transformed into one. 
In the present case we must draw an arc of a great circle 
from the star perpendicular to the opposite side 90 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 co-ordinates: 

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 -f-cos 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 (S-i-cp") 
sin \ z . sin A- (/> -4) = cos \ t . sin | (8 cp) 
sin .] z . cos 15 (p A) = sin -5 t . cos A ($-4-9?). 

*) 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* ^^ ^ 

TT-K , 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 (e-h<?)] 
sin (45 4/?) cos^ (E X) = sin (45 +|J cos [45 I (s )] 
cos(45 $ ft) sin \ (JE-M) = sin (45 -!-) sin [45 $( )] 
cos (45 j/5) cos I (JF-|-4) = 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 co-ordinates 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) cosOE-H) = 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.dh-t-smad 
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 co-ordinates with respect to the plane 
of the horizon: 

x = cos A cos h, 

y = sin A cos h, 

z = sin h. 

If we revolve the axis of x in the plane of x and z through 
the angle 90 (f in the direction towards the positive side 
of the axis of 3, we find the new co-ordinates: 

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(t-T)=j r - 



n ,1 IT a (1 e 2 ) a cos y 2 , . 

As we have tor the ellipse r = - = , * tnis 

H-ficos-^ 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 = - - - , 

l-j-e 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 ecos-h 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 dE-i- 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 = Z-M244". 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 n-t-w the true daily motion on that certain day, 
24 hours of mean time are equal to 24 w hours of apparent 
time, hence we have: 

x : A L == 24 h : 24 h w, 

24 h 

or x = (A-L}~- 

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: 

H-14 m 31s, 3 m 53s, H-6 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 

3-6-042201 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". SG-rCaH-*)]- " 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- */(a-H) + ^"^ / (+*), 
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 -i-f(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 semi-upper diurnal arc 
of the star. If we know the sidereal time at which the star 
passes the meridian or its right ascension, we find the time 
of the rising or setting of the star, by subtracting the 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 semi-diurnal 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 4-O 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 4-6 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.Tl-r 

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 co-ordinates. 

In order to form a clear idea of the mutual motions of 
the equator and ecliptic, we must refer them to a fixed place, 
for which we take according to Laplace that great circle, 
with which the ecliptic coincided at the beginning of the year 
1750. Now Physical Astronomy teaches, that the attraction 
of the sun and moon on the excess of matter near the equator 



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 1750-f-: 

o = 23 28 18". 4- t* 0". 0000098423 

and the angle between the equator and the ecliptic at the time 
1750-M (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 1750-M; 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 
1750-M. 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 . 4-tJi . l t l *-f-*o 

tang 4 7t . sin j II-}- j = sin --- - - tang - , 

( I,-*- I \ l t l s 

tang ^ 7t . cos j/7-f- 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? 



_ 

2062bo-h .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 
co-ordinates of the star with respect to this plane, if we take 
the ascending node of the ecliptic on the fixed ecliptic of 
1750 as origin of the longitudes, are as follows: 

cos 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 co-ordi 
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 = 90-f-L 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: 

ZT-f- 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 1750-f- with the fixed ecliptic, is 
equal to L + /,, when /, is the lunisolar precession during 
the interval from 1750 to 1750 -f- 1. Hence the co-ordinates 
of the star with respect to the plane of the fixed ecliptic 
and the origin of the longitudes adopted last are: 
cos B cos (L -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 1750-f-, the right ascension reckoned 
from the origin adopted before, is equal to -+- a. We have 
therefore the co-ordinates 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 co-ordinates 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 1750-M , 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 = -f-33 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, <? = -j-20 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 4-sin 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 4-cos 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 ) 
cotangj-i/ , 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 north-pole and is called therefore the pole-star. 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 = H-43 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 pole-star. 

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(04-P), 

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(04-P) 

- [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" .11734-0". 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 (1-H), 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 = N-t- 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^-1-0.0096 cos 2 ([ 0.1294 cos 2Q 

0.0022 cos (0-hP)] dv 
and from this we find in the same way as in No. 5: 

^.~_ a )_ _i.7t56sinO [0.7445 sin } sin H-1 0000 cos O cos ] tang 
dv 

-+- 0.0206 sin 2^ + [0.0090 sin 2^ snuH-0.0097 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 
-4-0.2735 sin20-f-[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 (0-hP) 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 
H-0.1187sin20cos 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 (fi--y ) 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 0-f-P 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, 180-f-2,O (because these 
terms have the opposite sign) and 0-f-P, 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 co-ordinates 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 semi-major axis is C cos e = 9". 22, and 
whose semi-conjugate 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 semi-major axis 
as radius. Then it is obvious, that a radius of this circle 
must move through it in a time equal to the period of the 
revolution of the moon s nodes with uniform and retrograde 
motion*), so that it coincides with the side of the major axis 
nearest to the ecliptic, when the ascending node of the moon s 
orbit coincides with the vernal equinox. If we now let fall 
from the extremity of this radius a line perpendicular to the 
major axis, the point, in which this line intersects the 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 semi-major 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 semi-major 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 co-ordinates. As the 



place 
on the 



Fig. 3. 



first co-ordinate 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 co-ordinates 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 co-ordinates 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 yH-1". 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 4-0.0007271 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 
co-ordinates : 

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 co-ordinates at right angles 
with each other, of which the positive axis of z is directed 
towards the zenith of the place, whilst the axes of x and y 
are situated in the horizon, so that the positive axis of x 
is directed towards the south, the positive axis of y towards 
the west, the co-ordinates of the body with respect to these 
axes are : 

A sin z cos A , A sin z sin A and A cos z , 

where A denotes the distance of the object from the place 
and z and A are the zenith distance and azimuth seen from 
the place. 

The co-ordinates of the same object with respect to a 
system of axes parallel to the others but passing through the 
centre of the earth are: 

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 co-ordinates 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 co-ordinates 
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 co-ordinates 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 co-ordinates 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 co-ordinates 
referred to the plane of the ecliptic, it is necessary to know 
the co-ordinates of the place of observation with respect to 
the earth s centre referred to the same fundamental plane. 
But if we convert and 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 co-ordinates 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 

^(i-i) 

, 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 ( -t-a)] 

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 

= 4-15 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 w-f-l 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 co-ordinates 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 l-i-c^J 

or taking: 



co co a A P \ 
2, hence- -^=2a(l 5-1 
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>H-2ssins 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 4-2* 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 - = LT-r 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 1-f-mr, 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 )e--2/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 = 0-690613 
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 1-M (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 H-s(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 = -. -f-i- = so + rr- d ~- (-50) + ; -- d H (6-2 J.G), () 
H-(T oO; 29.6 1 d-r 1 d6 

but as the influence of the last two terms is small we may 
take for the sake of convenience: 

* ,_ U?*_ /_1V + " ( ^ 

~~ [l-f. 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 . 
dZi-jf. - - a + T ]/2/3. A [1/2 y, (2) - y, (I)] 

-I- - /2 ~ 4- (1-A- 



As we have: 

b 



rf /; 29.6 
we find: = - ^-g - - e (r - 50), 



172 

and likewise: 

p + dft = -2- -2-e(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.<T-50). 

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 1-f- ^, so that finally we get: 

180-f-s(* 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 
80H-7(* 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 4-1 2, s- ~~ 333728 804-5 .v 
_ 180 4- s(f 50) __ 804- 



180 4- q (/ 50) 80 4- q (r 8) 

1_ _1 

7 ~~ 1 4- B . (/- 50) 1 4-f 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 1-f-g 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= -Z-g 

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 "^ 12-27.5. 

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 = 90-hc, 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 
sin|r=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 object-glass and the eye-piece of 
a telescope (or the lense and the nerve 
of the eye). The positions of the object- 
glass and of the eye-piece 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 co-ordinates 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 co-ordinates of the eye-piece at the time , since during 
the interval t t we may consider the motion of the earth 
to be linear. If the relative co-ordinates of the object-glass 
with respect to the eye-piece are denoted by , i] and f , the 
co-ordinates of the object-glass 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 co-ordinate 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 object-glass 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 co-ordinates of the object-glass at the time t: 

I cos cos a -+ .r, 
I cos sin a -\-y, 
I sin <T -h z, 

and by the co-ordinates of the eye-piece 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 ) = -7-3 : 

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 co-ordinates 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 co-ordinates 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 semi-major 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 semi-major 
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 f-r J sec<? 2 [cos20sin2(H-cos 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 (&cose-Ha)sin2sinCQ-4-n) 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 -i-ft) > 

- 0".0003847 sin 2 sin (04-H) 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 (04-4 ) , 

$ <?= 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 0-M 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 = -f-18". 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 co-ordinates 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 co-ordinates 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 co-ordinates of the sun with respect to the earth are: 

X=RcosQ and r=/2sinQ 

where the semi-major 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 co-ordinates 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 semi-major 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 co-ordinates, the 
axes of x and y being the lines of intersection of the parallel 
circle and of the circle of latitude with the plane and if we 
refer the apparent place of the star affected with aberration 
to the mean place by the co-ordinates: 

x = (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 semi-minor 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 co-ordinates 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 semi-major 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 sem-imajor axis is 
0". 3113 cos (f and whose semi-minor 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 object-glass 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 object-glass at the time T, a and b the 
places of the object-glass and the eye-piece at the time t and 
finally a and b their places at the time , 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 
semi-major 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 co-ordinates 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 
co-ordinates , ?/, f, since the geocentric co-ordinates 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 co-ordinates 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 = S-t-T). 

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 [m-f-w 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 4-P) 

- 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 4-0".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, O-hO 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 

H-0".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 sin20-K 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 (0-f-P)] 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 sin20-Hi 3 sin(0 P) / 4 sin (0-f-P) 

,B = 9".223 1 cosO -I- 0".0896 cos 2^ 0".5509 cos 20 0".0093 cos(0H-P) 

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 0-f-P, 
the values given by him require the following corrections 
in order to make them correspond to the formulae given 
above : 

For 1750: 

dA 0.0090 sin ^ 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 cosiH-0.0019cos 2^ -H0.0291cos20 0.0093 cos(0-f-P). 



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 k-i-d 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 

Am-i- E=f, 
the terms for the right ascension become: 

f-t-gsm(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^-f-t cos^H-r//. 
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(G-j-a) 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 4-0). 

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 Y-pieces 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~i-b(t T), where a is the rate at the time T. Multiplying 
this by dt and integrating it between the limits t and 24-f-f, 
we find the rate between two successive culminations of a 
star, whose time of culmination is , equal to: 

24aH-24&(12-M 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 object-glass to the wire-cross is vertical to the axis 
of the pivots (the axis of revolution of the instrument), it 
describes when the telescope is turned a plane, which 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 spirit-level, whether the 
axis of the pivots is horizontal and we may also correct any 
error of this kind, since one of the Y-pieces can be raised or 
lowered by adjusting screws. The position of the line of 
collimation with respect to the axis can be examined by 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 wire-cross at the focus of 
this small telescope, it can be seen in the telescope of the 
meridian circle like any object at an infinitely great distance, 
since the rays coming from the focus of the collimator after 
their refraction by its object glass are parallel. Now if the 
angle, which the line of collimation makes with the axis of 
the meridian circle, differs by x from a right angle, the angles 
which the lines of collimation of the two telescopes make 
with each other in both positions of the meridian circle, will 
differ by 2x or the wire of the collimator as seen in the 

*) The complete methods for rectifying the meridian circle and for 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 pole-star, 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- &> 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 Y-pieces 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 -f-A 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 semi-diameter 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 clock-time 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 eye-piece towards the mercury, we see in 
the light field besides the wires also their reflected images. 
Therefore if we turn the telescope until the reflected image 
of the horizontal wire coincides with the wire itself, the line 
of collimation must be directed exactly to the nadir, hence 
we find by the reading of the circle the nadir point or by 
adding 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 
-r-|A<? 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 -4-0". 70, we have: 

Z> = -4-2326 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 = H-0".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 
= -r-e/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 -i-adD-+- 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 

= n-ha4- 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 -4-0". 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 -f-0 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 -J-0 S .01; furthermore the aberration is = 0\l 56, the 
nutation of the obliquity is -j-0".27, hence, as the mean 
obliquity was on that day 23" 27 34". 82, we find: 

Z> = -t-l 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 -f-2 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 circle-reading 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 zone-stars 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 4-f(r 50)]~ A , 
where A 1 H- q and /I = 1 -i-p. From this we get: 



dR A(r-50) 

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 (l-t- ) 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 pole-star 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 pole-star 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 -f-A and &H-A&. 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 -+- A-f- A& sin (0H-4) sec d + na cos (0-K4) 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 (O-h#). 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 semi-diameter 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 &: 

H-tfa 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 P-f- 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 -i-TiR 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 

co-ordinates is 7".0 and 0". 8) and 1830 Groombridge (which 
moves 5". 2 in right ascension and 5". 7 in declination). 

The elder Herschel first discovered a law in the direction 
of the proper motions of the stars, when comparing, a great 
many of them he observed, that in general the stars move 
from a point in the neighbourhood of the star 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 = -f-37 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^^ -f-dn,, 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 -f-l".16 
from the right ascensions and 4-0". 66 from the declinations 
and hence, taking into account their probable errors, he found 
the value of the constant of precession for 1790 equal to 
50". 23449 or greater than Bessel had found it by 0.01343. 
Further he found for the motion of the sun, as seen from a 
point at the distance of the stars of the first magnitude, 
0".321 from the right ascensions and 0".357 from the 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 -i-dn cos g cos (G D) adD -+- cos D sin$ sin ( A) ad A 

-hags m(G-D) 

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 co-ordinates 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 = -4-88 30 34". 680. 

But in Bessel s Tabulae Regiomontanae this place is: 

= 16 15 19". 530 8 = 4-88 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 4-0". 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 = 4-1". 501 189 A <? = 4-0". 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: 

,? = -f-0".56-hO".0202(* 1 800) -r- 1". 47 sin w 4-0". 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 co-latitude, 
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 co-ordinates 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 -r-Z: 

90-* = J (5 )H-I 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 
wire-cross, and only at the time of the greatest distance the 
vertical motion alone will be sufficient. If the reading of 
the azimuth circle is a in this position of the instrument and 
a , when the same observation is made on the other side of the 
meridian, ^~- is the reading of the circle corresponding to 
the zero of the azimuth. It is best to use the pole-star for 
these observations on account of its slow motion. 

A third method for determining the zero of the azimuth is that 
of taking corresponding altitudes. For as equal hour angles 
on both sides of the meridian belong to equal altitudes, it 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 wire-cross, then if the altitude 
of the telescope has not been changed but merely its azimuth, 
the arithmetical mean of the two readings of the circle is 
the zero of the azimuth. If the sun, whose declination changes 
in the time between the two observations, is observed, a 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 -f-A, 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 
pole-star 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) = T-TT ; =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 clock-time 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 semi-dia 
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 = -f-7 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 clock-time, 
in order to get the true time, is called the error of the clock* 
whilst the difference of the errors of the clock at two 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 
clock-times, 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 clock-times, 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 T-t- r T-f-r" T-j-.. .=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 4-3. 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 
circum-meridian 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 -+- -T-TV- 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 

- 180-4-,- -- 



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- , j-v 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 circum-meridian 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 circum-meridian 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 pole-star 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 z-t-y 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^cotgz-h^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 pole-star 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 pole-star. 

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 pole-star 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 -i-x) 

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 

iCd-hB)= 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 @ -f-7 

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 clock-times and 
subtract from it each clock-time 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 pole-star 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 

(zH-y) = 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+tfO-M (*-* ) 

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 clock-time 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 clock-time 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 
-4-90" 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 chronometer-time 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, <* = H-2 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 
circum-meridian altitudes. 

14. If the altitudes of two heavenly bodies have been 
observed as well as the interval of time between the two 
observations, we can find the time and the latitude at the 
same time. In this case we have the two equations: 
sin // = sin <f> sin -+- cos <p cos cos t, 
sin h sin cp sin -+ cos cp cos cos t . 

If then u and u are the clock-times 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 | (s-r-/?) 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 (s-f-) 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 = -rr-T -- 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(J-f-^) = 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 -4-y>) 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 log-book, 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 = H-3m 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 

<H-C" = 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 -i-A"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 U4-A) : sinp = cos h : cos 9?, 
hence : 

sin t : sin U-f-A) = 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 
,f-l-^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 (p-M) = * 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 (*H-A ) * 

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/V-hcT), 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 y-f- 

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) 

* A-4 
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 transit-instrument, before it was 
well adjusted: 

a Aurig-ae 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 4-45 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=5-H42".0 
and : 

A<?= 13". 2, 

and since we have: 

^ = 37 31 - 
and : 

^ = -t-2G 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 (Y-M) 
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) = -7-7,, ;>. [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 |(Y-f-) = 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: 



_ __ _sinj-j3in [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 clock-time when two stars are 
on the same vertical circle, if for instance we. observe the clock- 
time when two stars are bisected by a plumb-line, we can find 
the error of the clock at least approximately, when we know 
the latitude of the place and compute the time by means of 
the formulae given above. It is best to take as one of the 
stars always the pole-star, since it changes its place very 
slowly, a circumstance which makes the observation more 
easy. 



V. DETERMINATION OF THE ANGLE BETWEEN THE MERIDIANS OF 

TWO PLACES ON THE SURFACE OF THE EARTH, OR OF THEIR 

DIFFERENCE OF LONGITUDE. 

26. If the local times, which two different places on 
the surface of the earth have at the same absolute instant, 
are known, the hour angle of the vernal equinox for each 
place is known. But the difference of these hour angles, 
hence the difference of the local times at the same moment, 
is equal to the 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 semi-diameter of the earth is insignificant. The same is 
true for the eclipses of the satellites of Jupiter. 

These phenomena therefore would be very convenient 
for finding differences of longitude, since they are simply 
equal to the differences of the local times of observations, 
if they could be observed with greater accuracy. But 
since the shadow of the earth on the moon s disc is never 
well defined^ and thus the errors of observation may amount 
to one minute and even more, and since likewise the 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 electro-magnet so that, every time the current is broken 
for an instant and the armature is pulled away from the 
magnet by means of a spring attached to it, 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 
second-hand of the clock reaches the zero-second of a certain 
minute, the first second-mark on the paper will correspond 
to this certain second, and hence the second corresponding 
to any other mark is easily found. If then the current can 
also be broken at any time by a break-key in the hand of the 
observer, who gives a signal at the instant when a star is seen 
on the wire of the instrument, the time of this observation 
is also marked on the sheet, and hence it can be found with 
great accuracy by measuring the distance of this mark from 
the nearest second-mark. 

If the current goes to another observatory, whose 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 clock-times corresponding to the 
records on the chronographs are known, the difference of 
the two clocks can be obtained from every recorded second 
or better from the arithmetical mean of all. These differences, 
as obtained at both stations, differ again by twice the time, 
in which the current passes from one station to the other, 
and which in this way can be determined even with greater 
accuracy. A few such comparisons are already sufficient to 
give a very accurate result, since the accuracy of one 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 clock-timc=19 b 58 29s .56 A. A. clock-t. 
(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 diffei-ence 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 semi-diameters 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 co-ordinates). 
Then in the triangle between the pole of the ecliptic and 
the centres of the two bodies the three sides, namely the 
complements of the apparent latitudes and the sum or the 
difference of the apparent semi-diameters, are known; hence 
we can compute the angle at the pole, that is, the difference 
of the apparent longitudes of the two bodies at the time of 
observation and, applying the parallaxes in longitude, we find 
the difference of the true longitudes, as seen from the centre 
of the earth. From this, the relative velocity of the two 
bodies being known, we obtain the time of true conjunction, 
that is, the time, at which the two bodies have the same 



323 

geocentric longitude, and expressed in local time of the place 
of observation. If the beginning or end of the same eclipse 
or occultation has also been observed at another place, 
we find in the same way the time of true conjunction 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 co-ordinates, and f is the vertical angle of the 
generating triangle. 

We must now find the equation of the cone enveloping 
the two bodies and referred to a system of axes one of which 
passes through the centres of the two bodies. If then we 
substitute in place of the indeterminate co-ordinates ar, ?/, z 
the co-ordinates 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 
co-ordinates, whose origin is at the centre of the earth. Let 
the axis of y be directed towards the north pole of the equator, 
whilst the axes of z and x are situated in the plane of the 
equator and directed to points, whose right ascensions are 
a and 90 -+- a. Then the co - ordinates of the nearer body 
with respect to these axes are: 

z = & 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 co-ordinates 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 co-ordinates x and y will be the co-ordi 
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 co-ordinates 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 co-ordinates of this place with respect to a system 
of axes, whose axis of z is the line joining the two centres 
itself, are: 

| x, 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 semi-diameters of the two bodies and 
where the upper sign must be used for exterior contacts, the 
lower one for interior contacts. Now since the unit we 
use for G is the semi -diameter of the equator of the earth, 
we must refer r and r to the same unit. Therefore if k 
denotes the semi-diameter of the moon expressed in parts of 
the semi-diameter of the equator of the earth and h the ap- 



327 

parent semi-diameter of the sun seen at a distance equal to 
the semi-major axis of the earth s orbit, we. have, since: 



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-( < y-7 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 T-f- 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 -+- A-r] 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 T-f- 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 -f-t: 



~ 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 T-H 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 T-i-~ -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)-t-AsiniVA(<? d) -+-h COSJC^TT [>coszV y sin A ], 

or multiplying the upper equation by cos?/ , the lower one 
by sin \\) and adding them : 



i -f-i 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 </)cos-r(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 (> a-re 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 + AeH-A^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 .V-t- ^ 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 


4-2322 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 -1-0. 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) = scosN-l- 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 co-ordinates 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 co-ordinate 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 co-ordinates intersects this cylin 
der, is equal to the semi-diameter of the moon or equal to k. 
Hence we need not compute the co-ordinate z and we have 
simply : 

cos 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 co-ordinates 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 a-t-d () ) -, -- 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 t-ha?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 (iV-h 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 co-ordinates 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 co-ordinates of that 
point, which lies in the straight line drawn from the place 
on the earth to the point of contact of the two bodies, and 
whose distance from the vertex of the cone is equal to that 
of the latter point from the place on the surface of the earth. 

Hence - - 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: 

AT-hlSO" 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 H-d 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 H-T . 

The expressions (A) give then the particular points on 
the limb of the sun, where the contact takes place. 

For calculating an occultation the 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=r2V-M80 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 =4-0.58280, x = -f- 0.55718, / = 0.12133 
a = 106 55 . 8, d=-j-22 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 -f-l 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 semi-diameters of the two bodies, accordingly as the 
contact of the limbs nearest each other or that of the other 
limbs has been observed. If r be the horizontal semi-diameter 
of the moon, the semi-diameter affected with parallax will be : 



346 

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 semi-diame 
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|(A-hA") 
and put: 



we obtain: 

A=A"H-ar, (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. =H-22 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 semi-diameter 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 T-M-t-d?, 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- * tS-H- 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 T-M and T-\-t\ that is, the time 
j-l-i (_!_ ) which we will denote by T . Then we must 
wr ite T \(f and T -\-\(t f) in place of T-M and 
T-i-t\ 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 moon-cul 
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 clock-times of 
the transit of the moon and the moon-culminating 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 4-25 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 semi-diameter 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 semi-major axis of the earth s orbit being 
the unit of distance, are derived from the theory of their 
orbits, which they describe round the sun according to 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 co-ordinates of a point on the surface, 
referred to a system of axes in the plane of the meridian, 



359 

the origin of the co-ordinates being at the centre of the earth 
and the axis of x being parallel to the equator: 



a cos 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 semi-conjugate 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 -t-x 
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,] i-f-[a&,]fc 
*[*,] 4- [aft i] e-f-[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 4-11 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 = 4-1". 97 4- 1.1225^4- 5.6059 z 

2) x\ ^ 2 =4-0 . 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 serai-transverse 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 -V-Vi- 

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 
semi-diameter, 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-(y-rt*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;? H-rtsinp 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 semi-major axis of the moon s orbit. 



373 

Example. In 1752 February 23 Lalande observed at 
Berlin the declination of the lower limb of the moon: 

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: 

*<, =-t-21 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 T-f-r, 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: 

= 180H-2V > (^) 

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>H-cos^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 
-f-yfsin 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(M-N) 

Therefore in order to find the times of the ingress and 
egress for a place on the surface of the earth from the times 
of the ingress and egress for the centre of earth, we need 
only use R=^=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 = JV-f-i//, 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 
semi-diameters 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 


4-22 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 -j-c/, r) -j-c?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 D-i-(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^cosV-h/) 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(M--H) np d(R^r) 

ncos(M-N) 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 =H-10".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(B-r-r) 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 co-ordi 
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 co-ordi 
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 co-ordinates 
of any point, towards which the telescope is directed, could 
be read off directly on the circles. With every instrument, 
however, errors must be presupposed, arising partly from the 
manner, in which it is mounted, and partly from the imperfect 
execution of the same, and which cause, that the circles of 
the instrument do not coincide exactly with the planes of the 
co-ordinates, but make a small angle with them. The pro 
blem then is, to determine the deviations of the circles of 
the instrument from the true planes of co-ordinates, in order 
to derive from the co-ordinates observed on the circles the 
true values of these co-ordinates. 

Besides other errors occur with instruments, arising partly 
from the effect of gravity and temperature on the several 
parts of the instrument, partly from the imperfect execution 
of particular parts, such as the pivots, the graduation of the 
circles etc., and means must be had to determine these errors 
as far as possible, so as to find from the indications of the 



390 

instrument the true co-ordinates of the heavenly bodies with 
the greatest possible approximation." 

Besides these instruments, with which two co-ordinates 
of a body perpendicular to each other can be observed, there 
are still others, with which only a single co-ordinate or merely 
the relative position of two bodies can be observed. With 
regard to these instruments likewise the methods must be 
learned, by which the true values of the observed angles can 
be obtained from the readings. 



I. SOME OBJECTS PERTAINING IN GENERAL TO ALL INSTRUMENTS. 
A. Use of the spirit-level. 

1. The spirit-level serves to find the inclination of a 
line to the horizon. It consists of a closed glass tube so 
nearly filled with 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 air-bubble in every position of the level so 
places itself as to occupy the highest point in this curve. 
The highest point for the horizontal position of the level is 
denoted by zero, and on both sides of this point is arranged 
a graduated scale marked off in equal intervals and counting 
in both directions from the zero of the scale. If the level 
could be placed directly on the line, it would only be 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 zero-point 
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: 

B-A 

- ( 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 circle-end of the axis expressed in units of the scale. 
Finally if this number be multiplied by the value of the unit 
of the scale in seconds of arc, the result will be the eleva 
tion of the circle-end 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: 

^^(B-B ) 
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 T-shaped supporter, which 
rests on three screws and on which the level can be placed 
in two rectangular Y-pieces, in such a manner, that the 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 wire-cross 
of the collimator, and if this process be repeated after the 
inclination of the collimator has been somewhat changed by 
means of one of the foot -screws, then will the length of 
the unit of the scale be determined by comparing the 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 foot-screws, which is finely cut for this purpose and is 
provided with a graduated head. These instruments rest 
namely on three foot-screws which form a equilateral triangle. 
If now the level be set upon the horizontal axis of such an 
instrument and the axis be so placed, that the direction of 
the level shall pass through the screw a provided with the 
graduated head and therefore be perpendicular to the line 
joining the two other screws, then can the value of the 
unit of the scale be determined from the readings of the 
screw a and the corresponding motion of the bubble of the 
level, when the distance between the threads of the screw as 
well as the distance of the screw a from the line joining the 
two other screws are known. The value of the unit of the 
scale for the level attached to the supports of the 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 foot-screws of the 
instrument, the amount of the inclination in units of the scale 
can be read off on the level, whilst the same can be obtained 
in seconds by turning the telescope towards the object and 
reading off the circle in the new position. 



398 

3. The case hitherto considered, to determine by means 
of the level the inclination of a line upon which the level 
can be placed, never actually occurs with the instruments, 
but the inclination of an axis is always sought which is only 
given by a pair of cylindrical pivots on which the level must 
be placed. Even if the axis of the cylinders coincides with 
the mathematical axis of the instrument, nevertheless the 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. circle-end 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 circle-end 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 level-error 
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 
circle-end, and read off by different zenith distances of the telescope and then 
the mean of the readings in the horizontal position of the telescope is 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 

? -4-1 

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 m-f-1 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 division-line on the vernier. Then the arithmetical mean 
of these readings will be equal to the length of the vernier. 

5. If great accuracy is required for reading the circles, 
the instruments, for instance the meridian circles, are furnished 
with reading microscopes, which are firmly fastened either 
to the piers, or to the plates to which the Ys are attached, 
in such a manner, that they stand perpendicular over the 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 division-line 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 division-lines of the 
circle, for the object glass of the microscope can be moved 
farther from or nearer to the eye -piece, and thus the image 
of the space between two lines can be altered and can be 
made equal to the space through which the wire is moved 
by an entire number of revolutions of the screw. If the screw 
performs more than an entire number of revolutions, when the 
wire is moved from one division -line to the next, then the 
object glass of the microscopes must be brought nearer to 
the eye-piece; but since by this operation the image is thrown 
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 division-lines of the circle, 
and that a plane passing through the axis of the microscope 
and any radius of the circle is perpendicular to the latter. If 



*) It is better to use instead of a single wire two parallel wires and to 
bring the division lines of the circle exactly between these wires. 



405 

it is not rectified in this way, the image of a line moves a little 
sideways, when the circle is gently pressed with the hand, and 
thus errors would arise in reading off the circle, if it should 
not be an exact plane or should not be exactly perpendicular 
to the axis. If such a motion of the image arising from the 
gentle pressure of the hand be observed , the tube in which 
the object glass is fastened must be turned until a position 
is found in which such a pressure has no more effect upon 
the image. 

Since the distance of the microscope from the circle is 
subject to small changes, the error of run, that is the 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 zero-point we 
move the circle until the following one of the two lines is 
between the wires and then bring by the motion of the screw 
the preceding line between the wires *). If in this position 
the corrected reading of the screw is 120-j-p, then the read 
ing of the screw, if we had moved it from zero through 
exactly 120 seconds, would have been 120-f-p 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 120-f-ic. 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 120-f-w, the first interval is too 
large by x u = y or is equal to 120-h?/. 

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 =4-0". 3 X,, =4-1". 5 

v i 9 q v (\ n 

-*TA_ 3 Q """P" O O -^\- 210 ~~~~ U . D 

X 90 =4-3 .1 X a70 =H-0 .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 



4-15.1 
4-10.4 
-4-2.4 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 -f-5".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 -j-2".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 -f-l".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 " = -f-6". 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, H-2".l and 7".9. 
Now since the errors of the lines 90" and 45" have been found 
to be H-5".46 and -+-3".36, the corrected readings are 12".06, 
+ 2". 10 and --4". 54. The mean is 4". 83, and hence the 
error of the line 195 is 4- 7". 23, and that of 300" is 4-0".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 division-lines produced by the action of the force of gravity 
on the circle. The errors produced by the latter cause will 
change according to the position of a line with respect to 
the vertical line, so that the correction which must be applied 
to the reading for this reason will be expressed by a series 
of the form: 

a coss-h 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" -t-z from the 
zenith, all odd terms of the series are in those two cases 
equal but have opposite signs. Therefore if we measure the 
distance between two lines first in a position of the circle, in 
which the zenith distance of that line is z and afterwards in 
the opposite position, in which its zenith distance is 180-f-3, 
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, = 3-f-m; 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 = 3-r-w; 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 wire-cross, it is evident, that it 
has no influence at all upon the observations, since in that 
case the line joining those two centres (the line of 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 2z-f- a " cos 3z-f- .. b sin z (B 

- &"sin2z b "sm3z.. (lSQ+N)-i-a 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"-f-a ". 
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 -i-N ) -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 eye-piece 
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 180-f- 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 wire-crosses coincide. 
Then the telescope being directed first to the wire-cross of one 
collimator and then to that of the other, will describe exactly 
180. Hence if we read the circle in the two positions of 
the telescope, and denote the true zenith distance of the 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: 

180-t-=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 eye-piece 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 
wire-cross 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 wire-cross of the collimator, the 
angle, through which the telescope is turned, will be exactly 
180, and hence we can find, as before, those terms of the 
expression for the flexure, which depend upon 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 ~i-a cosz-i-a"cos2z-\-a" cos3z+... + a a"-f-a ", 

2i 

where Z= 180-1- 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 H-T(* 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 4-J/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 -& ^2-H&"-& "*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 
4-0.001085 
H- . 003435 
4- . 005435 
4- . 005535 



( /) sin ( 

4- . 002485 

- . 000865 
-0.001123 
4-0.001686 
4- . 004320 
4-0.002415 

- . 000882 

- . 001083 
4-0.001646 
4- . 004457 



sum 4-0.013056, 
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: 

, =4-0 . 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 -4-0 . 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?/) 
-t-6 1 (sin?/ sin M) will be in one case-f-ctj 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, -f-0 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} =H-0.0147H-/ 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 
co-ordinates of it can be obtained from the indications of 
the circles. We will denote by i the angle, which the line 
through both Ys makes with the horizontal circle, and by K 
the point, in which this line produced beyond that end on 
which the circle is, intersects the celestial sphere. The 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 co-ordi 
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 co-ordinates 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 co-ordinates 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 co-or 
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 co-or 
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 co-ordinates will then be: cos z and sin z cos (e A). 
This is true, when the circle is on the left side, whilst we 
must take A e instead of e A, when the circle is on the 
right side. If further we imagine the point to be referred 
to a system of axes, of which the axes x and y are in the 
plane of the instrument, the axis of y being directed to the 
point K, then the co-ordinate y of the point 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 co-ordinates: 

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 90-f-c, 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 level-error is found according to the rules given in 
No. 1 of this section by placing a spirit-level 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 
a-j-120" and a-f-140. 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 = 4-1 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 90-J-c 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 collimation-error, if d is known. 

If the telescope is attached to one extremity of the axis, 
its weight can produce a flexure of the axis, which renders 
the collimation-error variable with the zenith distance. When 
the telescope is horizontal, the flexure has no influence on 
the collimation-error, since it merely lowers the line of 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 collimation-error 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 
pole-star. For, if we observe the pole-star at the time t, 
read the circle and then reverse the instrument and observe 
the pole-star a second time at the time t\ we shall have the 
two equations : 

e = A -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 spirit-level 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 Z-f-|(/? w) an( l in the other Z-i-$(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 # -f-90 () 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 co-ordinates, that of the altitudes and 
azimuths, so the equatoreal corresponds to the second system, 
that of the hour angles and declinations. With this 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 co-ordinates of the 
point /f, referred to these axes, are: 

z == sin D, y = cos D cos T, x = cos D sin T. 

Further, the co-ordinates 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=t-t . 

The telescope is attached to the declination axis and we 
will assume, that the part of its line of collirnation towards 
the object-glass makes with the side of the axis, on which 
the circle is, the angle 90 -f- c, c being called the collima- 
tion-error. 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 co-ordinates 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 circle-end is 



443 

west of the telescope, the latter is directed towards a point, 
whose hour angle is less than that of the point K. 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 co-ordinates will be: 

z = sin , y = cos 8 cos (T T^, x = cos 8 sin ( T TJ). 
On the contrary, when the circle-end is east of the teles 
cope, these co-ordinates 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 
co-ordinates 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 J-M tang 8 -+- c sec (?, 
when the circle-end is east of the telescope, and: 

Tl =(t Z ) 90 -h A cos (< * ) tang <? i tang c sec S, 
when the circle-end 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: 

A-h* * -H90" 
and A-H 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 -f-A* Asin (T A) tang d== c sec 5 =1= i 1 tang #, 

where the upper sign is used, when the circle-end is west, the 
lower one, when it is east. 

We can also find these equations and the corresponding 
ones for the declination from the spherical triangle between 
the pole of the heavens P, the pole of the instrument // 
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 -j-A^ 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 pole-star, 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 : 

"Vj-f-r ~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 CZV-f- ) 
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 wire-cross. 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 circle-end is west 
and =r" 90 if the circle -end is east, 
we have to take instead of this hour angle: 

90 H-T"^ 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 tf-f- 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 co-ordinates of this point, referred to 
a system, whose axis of z is vertical, whilst the axes of x 
and y are situated in the plane of the horizon so that the 
positive sides of the axes of x and y are directed 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 co-ordinates 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 90-h-c, 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 co-ordinates 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 co-ordinates, so 
that the axis of x coincides with that of the former system, 
whilst the axis of y is not in the plane of the equator, but 
parallel to the axis of the instrument, then we have: 

y = sin c, 

and since the axes of z of these two systems make with each 
other the angle n, we have: 

sin c = sin n sin 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 
co-ordinates 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 H-rc, 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 T-j-r. If 
then A* is the error of the clock on sidereal time, then 
T-t-r-hA* 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 (op-hd) 
-h k -^ c sec 
cos 8 

Circle East +12h = T-f- 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 circle-end of the axis is directed, then 
the telescope will not move in the plane of the meridian, but 
it will describe the great circle A Z B Fig. 14 pag. 433. If 
now the star 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 H-c-4-/*), 

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 
wire-distances are found very accurately. For the pole-star, 
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 eye-piece is moved towards or 
from the object glass, until the star is seen distinctly. When 
this is the case, the wires are at the focus. In order to 
examine this more fully, we direct the telescope to an object 
at an infinite distance and bring it on the wire, and then 
slighty shifting the eye before the eye-piece we see, whether 
the object remains on the wire notwithstanding the motion. 
If this should not be the case, it shows, that the wires are 
not exactly at the focus, and they are too far from the 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 transit-instrument 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 pole-star. 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 wire-distances 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 wire-distances, 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 semi-diameter 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 wire-distances 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 
level-error can then be accurately determined by means of 
the spirit-level according to No. 1 of this section, when the 
inequality of the pivots is known from a large number of 
observations in both positions of the instrument. The 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 
circle-end in the two positions. But if we observe the image 



467 

of the star reflected from an artificial horizon, in which case 
the zenith distance is 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 level-error, 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 collimation-error 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 
micrometer-screw, 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 
wire-cross 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 |(a-f-6) 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 wire-crosses 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 wire-crosses. 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 eye-piece. 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 micrometer-screw 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 eye-piece so often for this collimat- 
ing eye-piece, Bessel proposed, to place simply outside upon 
the common eye -piece a plane glass in the right inclination 
or a small prism, and to reflect by means of it light into 
the telescope. It is true, a small part of the field is then 
only illuminated, but there is no difficulty in observing the 
reflected image^ provided that the glass or the prism is 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 b-i-u is known by means of the spirit-level, 
the error of collimation can be found from this equation, and 
conversely, if the error of collimation has been determined 
by other methods, the inclination of the axis of the pivots 
is found. Now if the instrument is reversed, and d denotes 
again the distance of the middle wire from its reflected image, 
taken again positive, when it is on the side towards the 
circle, we have: 

4 d = b -f- u c, 

and from both equations we obtain: 

c t* = J(rf-hrf ) 

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 spirit-level. For if one. end of the axis is raised or 
lowered by means of the adjusting screws, until the reflected 
image is made coincident with the middle wire, we have 
d = and hence c=b-\-u. Therefore if b-}-u is found by 
the spirit-level according to No. 3 of this section, this value 
is equal to the error of collimation. 

With the meridian circle at Ann Arbor the following 
observations were made in the two positions of the 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 = -f-0". 17, we have: 

c = -1-0". 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 \ (a-t-6) = 2-1 . 5655; the coincidence 
of the wires was 21^.5397, and since we must take (0-4-6) C, 
in order to find the error of collimation with the right sign, 

we obtain: 

c = -f-0".025S = -}-0".52. 



472 

Finally the two collimators were directed towards each 
other and the moveable wire was made coincident with the 
wire-crosses. Then the readings of the screw were: 

for the south collimator 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 level-error, the collimation-error 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 (c-f-0 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 H-A 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 pole-star, a meridian-mark 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 pole-star 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 meridian-mark will not be greater that those 
of the two Ys of the instrument, and since experience shows, 
that the azimuth of a well mounted instrument does not change 
more than a second during a day, the probable change of 
the line of collimation of the meridian- mark (that is, of the 
line from the centre of the lens to the centre of the small 
hole) will be less in the same ratio, as the length of the 
axis of the instrument is less than the focal length of the 
lens. Therefore if the length of the axis is 3 feet and the 
focal length of the lens is 550 feet, this change will not 
exceed T |. T of a second. The chief advantage of such a me 
ridian-mark 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 a-h(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 co-ordinates 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 co-ordinates, 
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 co-ordinates 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 semi-diameter *), 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== 9-0 )(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 pole-star must be observed at the upper 
and lower culmination. When the readings are freed from 
refraction, and from the errors of division and 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 wire-cross 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 
eye-piece 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 wire-cross is coincident with that of the collimator. 
In this position the circle is read. The same operation is 
repeated after the collimator has been turned 180 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 
object-end 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 co-ordinates, 
of which the axis of z is perpendicular to the horizon, whilst 
the axes of x and y are situated in the plane of the horizon 
so that the positive axis of x is directed to the north point 
and the positive axis of y to the east point, then the three 
co-ordinates of the point Q are: 

z = sin b , y = cos b sin k and x = cos b cos k. 

Further if we imagine another system of co-ordinates, 
whose axis of z is parallel to the axis of the heavens, and 
whose axis of y coincides with the corresponding axis of the 
first system so that the positive axis of x is directed to the 
point in which the equator intersects the meridian below the 
horizon, then the three co-ordinates of the point (), 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 90-j-G % , and that it is directed to an ob 
ject, whose declination is d and whose hour angle is , then 



486 

the three co-ordinates 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 
c-H/. 



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 spirit-level remains always on 
the axis, even when the instrument is being reversed, so 
that any disturbance of the level, which can be produced by 
its being placed on the axis, is avoided. When the level is 
reversed on the axis and observed in each position, b and b 
are obtained; but it is only necessary to leave it in the same 
position when the instrument is reversed, because the two 
readings of the level give then immediately b & , 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 (f-f-<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 : 

2sin-H<-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: 

l-cos; = 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 wire-distances 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 wire-distances, 
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/=-f-37 .24 

F//=H-73 .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 spirit-level. 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: 

= r-f- 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- = -r-rr-r-jK 

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 = -f-6".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-(r-hr) = 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 spirit-level 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 spirit-level. By observing a star in 
two positions of the horizontal axis which differ 180", double 
the zenith distance is determined in the same way as with 
the altitude and azimuth instrument, and everything that was 
said about the observation of zenith distances with that 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 -f-y-h/? = 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 
half-degree spaces, which are numbered as whole degrees, 
and thus the reading gives immediately the angle between 
the two objects. 



503 

When altitudes are observed with the sextant, an 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 90-f-c, 
we easily see, that c is the angle at the centre of the earth 
between the two radii , one passing through the plane of 
observation, the other drawn through a point of the small 
circle in which the surface of the cone is tangent to the earth. 
Hence if a denotes the radius of the earth, h the elevation 
of the eye above the surface of the water, we have : 



a 

cos c = - - , 
a 4- h 



and hence: 2 sin \ c~ = 



a-f- 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 horizon-glass) 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 horizon-glass 
has two adjusting screws. One is on the back -side of the 
reflector, which by means of it is turned round an axis 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 horizon-glass can be turned until 
the two images coincide and the two glasses are parallel. 
The reading in this position is the index error, which must 
be subtracted from all readings, in order to find the true 
angles between the observed objects. In order to correct 
this error, the alhidade is turned until the index is exactly 
at the zero of the scale and then the images of an object 
at an infinitely great distance are made coincident by turning 
the horizon-glass by means of the screw on its back. Usually 
however this error is not corrected, but its amount is 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 index-error, 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 index-error found by the sun is wrong. When afterwards 
altitudes of the sun are taken, this error has no influence, 
as long as the same colored glasses are employed which were 
used for finding the index error. But when other observa 
tions are made, for instance when lunar distances are taken, the 
index-error must be found by a star or by a terrestrial object. 

But when a terrestrial object is observed, whose distance 
is not infinitely great compared to the distance between the 
two reflectors, the index -error c as found by these 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 horizon-glass, /"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 index-error 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 index-error 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 wire-cross 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 horizon-glass in 
tersects the sphere in B and after its reflexion from the in 
dex-glass 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 horizon-glass 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 
index-glass which is turned, until the image of the piece is 
seen through the ^ole. Then the other piece is likewise placed 
before the index-glass so, that the wire is also seen through 
the hole. If then the wire passes exactly through the centre 
of the reflected 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 index-glass 
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 
plane-glas reflectors, which ought to be parallel, make a small 
angle with each other so that the reflectors have the form 
of prisms. Let then 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 -f-rt ( 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 
dex-glass. 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 index-glass out of its set 
ting, put it back in the reversed position and determine the 
index-error 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 index-glass strike the hori 
zon-glass always at the same angle, it follows, that the error 
arising from a prismatic form of this glass is the same for 
all positions of the index -glass and hence it has no effect 
upon the measured distances. 

Finally the sextant may have an excentricity, the centre 
on which the alhidade turns being different from that of the 



512 

graduation. This error must be determined by measuring 
known angles between two objects. If the angle is 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 horizon-glass have a glass-prism. 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 wire-distances 
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 
screw-head is equal to the distance between the two objects. 
If another observation is made by placing now the second 
object on the middle wire and bisecting the first object by 
the moveable wire, then it is not necessary to determine 
the coincidence of the wires, since one half of the difference 
of the two readings is equal to the distance between the two 
objects. If also the position-circle is read, first when the 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 H-2".382. 

If we take p = 207, A = 3". 5, we obtain the equations: 
= 0".011 - 0.306 rf A - 0.590 dp 
= 4-0".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 ring-micrometer, 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 <? <? = 4-4 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 -r-r 

^ = 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 



/^H-y 

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 JH-|tang|(<?4-<? )[cos<? 2 (15r) 2 cos S 2 (15-r ) 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 = H-7 36".9. 

With this we find: 



527 

log (<?-t-d) = 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 r-hr 
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 



r-r 



. 
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 
= sin-H# 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 (4-h 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 semi-diameter of 
the sun and r that of the ring. If we assume the centre of 
the sun to describe a straight line while passing through the 
field, we have a right angled triangle, whose hypothenuse 
is 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 R-t-r: 

(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)*[t-ht ][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 semi-diameter 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 semi-lenses 
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 semi-lenses are moved with respect 
to each other. When the semi-lenses are placed so that they 
form one entire lens, that is, when their centres coincide, 
we shall see in the telescope the image of any object, to 
which it is directed, in the direction from the focus of the 
lens to its centre. If then we move one of the semi -lenses 
through a certain number of revolutions of the screw , the 
image, made by that semi-lens wjiich is not moved, will 
remain in the same position, but near it we shall see another 



533 

image made by the other semi-lens in the direction from its 
focus to its centre. Therefore if there is another object 
in the direction from the centre of this semi-lens to the focus 
of the fixed lens, then the image of the first object made 
by this lens and that of the second object made by the 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 
semi-lenses was moved. 

In order that the plane of section may always pass 
through the two observed objects, the frame-work support 
ing the two slides with the semi-lenses is arranged so, that 
it can be turned around the optical axis of the telescope. 
Therefore if the heliometer has a position circle whose 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 eye-piece 
always over the images of the object made by the semi-lenses. 
For if one of them is moved so that its centre does not 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 
semi-lens. Therefore in order to see it distinctly, we must 
move the eye-piece just as far from the axis of the telescope 
and in the right direction, so that its focus and the image 
of the object coincide. 

Now the plane of section will not pass exactly through 
the centre of the position circle. We will call the reading 
of the moveable slide *) , when the distance of the optical 

*) We will assume here, that only one of the slides is moved and that 
the other always remains in a fixed position. 



534 

centre of the lens from the centre of the circle is a mini 
mum, the zero-point. It can easily be determined, if we find 
that position, in which the image of an object seen in the 
telescope does not change its place in the direction of the 
plane of section, when the object glass is turned 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 eye-piece, and we will assume, that for this position the 
readings of the three scales, namely those on the slides 
of the two semi -lenses and that on the slide of the 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 wire-cross must be corrected by means of 
its adjusting screws. 

We will assume, that when the image of an object made 
by one of the 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 eye-piece 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 
hour-circle and the declination-circle of the instrument ; these 
belong to that point of the heavens, towards which the axis 
of the telescope is directed. We will imagine then a 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 
co-ordinates 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 co-ordinates 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 
co-ordinates of the point s with respect to the focus. 

The co-ordinates must now be changed into such which 
are referred to the plane of the equator and the meridian, 
the positive axis of 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 co-ordinates 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$ H-a[sinp sin ?r] cos <?. 

From this we find the square of the distance r of the 
point s from the origin of the co-ordinates: 

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 co-ordinates to 
the point s makes then the following angles with the three 
axes of co-ordinates: 

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 declination-circle and 
the hour-circle, D and A by the readings of the slides of 
the object glass and the eye-piece, and p and 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^[(/) 4-Z>)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: 



-UK-A 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 

d-hd H-d"-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 

d-P _ 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*~ 

d-D_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 semi-lenses have 
been separated considerably, turn the frame of the object 
glass so that the index of the position circle is at the zero, 
and then make one image of an object coincident with the 
point of intersection of the wires *). If then also the other 
image can be brought to this point merely by turning the 
telescope round the declination-axis, the plane of section will 
be parallel to the plane in which the telescope is moving, 
and hence the collimation-error of the position circle will be 
zero. But if this should not be the case, then the object 
glass must be turned a little, until both images of an object 
pass over the point of intersection of the wires when the 
telescope is moved about the declination-axis. Then the 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 object-glass is turned about 
the axis of the telescope until the two images of an object 
move along this wire, then the reading of the position circle 
ought to be 90" or 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 object-glass 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 semi-lens is moved so that the index of its scale is 
at a considerable distance from the middle, we measure the 
distance between the two images of the wire by means of a 
theodolite. Comparing then with this angular distance the 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 screw-head. 
Let then S be the reading of the scale of the latter 
semi-lens 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 eye-piece, 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 semi-lens, s that of the scale of the eye-piece 
and a the observed angular distance between the two images, 
then we shall obtain a similar equation for R, in which S , s 
and a take the place of S, s and a. Now we can always 
arrange the observations in such a way that: 

S 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- - 

tuga-K( - 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 wire-cross, we 
can also place one of the wires parallel to the equator and then, 
after the two semi-lenses have been separated and turned so 
that the two images of a star move along this wire, ^observe 
the transits of the two images over the normal wires. 

The value in 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 (JV-f- <? ), 
or: 

tang sin 77 = cotang n . cosec (N-\- S ) 
tang cos 77 = cotang (2V-t- $ ). 

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 co-ordinates. 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 (7V-f-$ ) 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 (d-d } 

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 
-f-fc(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 -t-Dd , 

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 B-i-c > 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. 



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iOV 22 1944 



50m-8, 26 



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