1896
LIBRARY
UNIVERSITY OF CALIFORNIA.
01 FT OR
Received
Accession No.
Class No.
THE ELEMENTS
OF THE
FOUB INNER PLANETS
AND THE
FUNDAMENTAL CONSTANTS OF ASTRONOMY
BY
SIMON NEWCOMB
Supplement to the American Ephemeris and Nautical
Almanac for 1897
WASHINGTON
GOVERNMENT PRINTING OFFICE
1895
PREFACE.
THE diversity in the adopted values of the elements and
constants of astronomy is productive of inconvenience to all
who are engaged in investigations based upon these quanti-
ties, and injurious to the precision and symmetry of much of
our astronomical work. If any cases exist in which uniform
and consistent values of all these quantities are embodied in
an extended series of astronomical results, whether in the
form of ephemerides or results of observations, they are the
exception rather than the rule. The longer this diversity
continues the greater the difficulties which astronomers of
the future will meet in utilizing the work of our time.
On taking charge of the work of preparing the American
Ephemeris in 1877 the writer was so strongly impressed with
the inconvenience arising from this source that he deemed it
advisable to devote all the force which he could spare to the
work of deriving improved values of the fundamental elements
and embodying them in new tables of the celestial motions.
It was expected that the work could all be done in ten years.
But a number of circumstances, not necessary to describe at
present, prevented the fulfillment of this hope. Only now is
the work complete so far as regards the fundamental constants
and the elements of the planets from Mercury to Jupiter inclu-
sive. The construction of tables of the four inner planets is
now in progress, those of Jupiter and Saturn having already
been completed by Mr. HILL. All these tables will be pub-
lished as soon as possible, and the investigations on which
they are based are intended, so far as it is practicable to con-
dense them, to appear in subsequent volumes of the Astro-
nomical Papers of the American Ephemeris. As it will take
several years to bring out these volumes, it has been deemed
advisable to publish in advance the present brief summary of
the work.
HI
IV PREFACE.
The author feels that critical examination of this monograph
may show in many points a want of consistency and conti-
nuity. The ground covered is so extensive, the material so
diverse as well as voluminous, and the relations to be investi-
gated so numerous, that no conclusion could be reached on
one point which was not liable to be modified by subsequent
decisions upon other points. The author trusts that the diffi-
culties growing out of these features of the work, as well as
those incident to the administration of an office not especially
organized for the work, will afford a sufficient apology for any
defects that may be noticed.
NAUTICAL ALMANAC OFFICE,
U. 8. Naval Observatory, January 7, 1895.
I^IVWITT
CONTENTS.
CHAPTER I. GENERAL OUTLINE OF THE WORK OF COMPARING
THE OBSERVATIONS WITH THEORY.
Page.
1. Reduction to the standard system of Right Ascensions and
Declinations
2. Observations used
3. Semidiameters of Mercury and Venus. Table for defective
illumination of Mercury in Right Ascension 3
4. Tabular places from LEVERRIER'S tables. Reduction for
masses used by LEVERRIER 6
5. Comparisons of observations and tables 8
$ 6. Equations of condition. Method of formation 8
7. Method of determining the secular variations and the masses
of Venus and Mercury independently 10
8. Method of introducing the results of observations on transits
of Venus and Mercury ; separate solutions, A from meridian
observations without transits ; B, including both meridian
observations and transits 13
CHAPTER II. DISCUSSION AND RESULTS OF OBSERVATIONS OF
THE SUN.
$ 9. Method of treating observed Right Ascensions of the Sun.
Expression of errors of observed Right Ascension as error
of longitude 15
10. Treatment of observed Declinations of the Sun. Formation
of equations of condition for the corrections to the
obliquity and to the Sun's absolute longitude 16
$ 11. Formation of equations from observed Right Ascensions of
Sun 17
12. Solution of equations from Right Ascensions of the Sun.
Tabular exhibit of results of observations of the Sun's
Right Ascensions at various observatories during different
periods 20
13. Mass of Venus, derived from observations of the Sun's Right
Ascension 24
$ 14. Discussion of corrections to the Right Ascensions of the Sun
relative to that of the stars 25
v
VI CONTENTS.
Page.
15. Discussion of corrections to the eccentricity and perihelion
of the Earth's orbit 27
16. Results of observed Declinations of the Sun. Exhibit of
individual corrections to the absolute longitude and the
obliquity of the ecliptic at the different observatories
during different periods ^ 29
$ 17. Discussion of the observed corrections to the Sun's absolute
longitude 32
18. Discussion of the observed corrections to the obliquity of
the ecliptic 33
$ 19. Effect of refraction on the obliquity ; special investigation of
the secular change of obliquity as derived from observa-
tions of the Sun 35
$ 20. Concluded results for the obliquity, and its secular varia-
tion 39
21. Summary of results for the corrections to the elements of the
Earth's orbit and their secular variations as derived from
observations of the Sun alone . 41
CHAPTER III. RESULTS OF OBSERVATIONS OF THE PLANETS
MERCURY, VENUS, AND MARS.
22. Elements adopted for correction 43
$ 23. Introduction .of the corrections to the masses of Venus and
Mercury 45
24. Introduction of the errors of absolute Right Ascension and
Declinations of the standard stars 46
25. Introduction of the corrections to the secular variations.
Method of forming the normal equations by periods so as
to include the correction to the secular variation 49
$ 26. Dates and weights of the equations for the various periods. 52
27. Unknown quantities of the equations. Factors for changing
corrections of the unknown quantities into corrections of
the elements 55
28. Table of the values of the principal coefficients of the normal
equations '..'...' 56
29. Order of elimination 57
30. Treatment of meridian observations of Mercury. Effect of
want of approximation in the coefficients of the equations
of condition 58
31. Introduction of the equations derived from observed tran-
sits of Mercury 61
$ 32. Solution of the equations for Mercury 65
$ 33. Systematic discordances among the observed Right Ascen-
sions of Mercury in different points of its relative orbit.. 66
CONTENTS. VII
Page.
34. Comparison of the results derived from meridian observa-
tions of Mercury with those derived from transits over the
Sun'sdisk 69
35. Treatment of meridian observations of Venus 70
36. Results of observed transits of Venus 70
37. Equations derived from observed transits of Venus 75
38. Solutions of the equations from Venus 76
39. Comparison of the results of meridian observations of Venus
with those of transits 76
40. Solution of the equations for Mars. Inequality of long
period in the mean longitude and perihelion, indicated by.
observations 77
41. Reduction from the equator to the ecliptic 79
CHAPTER IV. COMBINATION OF THE PRECEDING RESULTS TO
OBTAIN THE MOST PROBABLE VALUES OF THE ELEMENTS
AND OF THEIR SECULAR VARIATIONS FROM OBSERVA-
TIONS ALONE.
42. Modifications of the canons of least squares 81
43. Relative precision of the two methods of determining the
elements of the Earth's orbit 86
$ 44. Concluded secular variations of the solar elements, as
derived from observations alone 87
$ 45. Common error of the standard declinations 89
$ 46. Definitive secular variations of all the elements from obser-
vations alone. Matrices of the normal equations for the
secular variations. Tabular statement of results 90
$ 47. Definitive corrections to the solar elements for 1850. . 95
CHAPTER V. MASSES OF THE PLANETS DERIVED BY METHODS
INDEPENDENT OF THE SECULAR VARIATIONS, WITH THE
RESULTING COMPUTED SECULAR VARIATIONS.
$ 48. Plan of discussion 97
49. Mass of Jupiter ; general combination of results 97
50. Mass of Mars. Prof. HALL'S value adopted 99
51. Mass of the Earth, derived from the preliminary value of the
solar parallax 99
52. Mass of Venus, derived from periodic perturbations 101
53. Mass of Mercury, from various sources 102
54. Theoretical values of the secular variations for 1850... 106
VIII CONTENTS.
Page.
CHAPTER VI. EXAMINATION OF HYPOTHESES AND DETERMINA-
TION OF THE MASSES BY WHICH THE DEVIATIONS OF THE
SECULAR VARIATIONS FROM THEIR THEORETICAL VALUES
MAY BE EXPLAINED.
55. Comparison of the observed and theoretical secular varia-
tions 109
$ 56. Hypothesis of nonsphericity of the equipotential surfaces
of the Sun Ill
57. Hypothesis of an intraniercurial ring 112
58. Hypothesis of an extended mass of diffused matter, like that
Which reflects the zodiacal light 115
$ 59. Hypothesis of a ring of planets outside the orbit of Mer-
cury. Elements of such a ring. This hypothesis the only
one which represents the observations, but too improbable
to be accepted 116
$ 60. Examination of the question whether the excess of motion
of the perihelion of Mars may be due to the action of the
zone of minor planets 116
61. Hypothesis that gravitation toward the Sun is not exactly
as the inverse square of the distance 118
62. Degree of precision with which the theory' of the inverse
square is established 119
63. Determination of the masses which will best represent the
observed secular variations of the eccentricities, nodes,
and inclinations 121
$ 64. Preliminary adjustment of the two sets of 1 masses. Result-
ing value of the solar parallax 122
CHAPTER VII. VALUES OF THE PRINCIPAL CONSTANTS WHICH
DEPEND UPON THE MOTION OF THE EARTH.
65. The processional constant 124
66. The constant of nutation, derived from observations 129
67. Relations between the constants of precession and nutation
and the quantities on which they depend 131
$ 68. The mass of the Moon from the observed constant of nuta-
tion --- 132
69. The constant of aberration 133
70. The values of this constant, derived from observations 135
71. The lunar inequality in the Earth's motion 139
72. The solar parallax derived from the lunar inequality 142
$ 73. Values of the solar parallax derived from measurements of
Venus on the face of the Sun during the transits of 1874
and 1882, with the heliometer and photoheliograph 143
$ 74. The solar parallax from observed contacts during transits of
Venus.. 145
CONTENTS. IX
75. Solar parallax from the observed constant of aberration and
measured velocity of light 147
76. Solar parallax from the parallactic inequality of the Moon. 148
i 77. Solar parallax from observations of the minor planets with
the heliometer 152
78. Remarks on determinations of the parallax which are not
used in the present discussion. Errors arising from dif-
ferences of color 154
CHAPTER VIII. DISCUSSION OF RESULTS FOR THE SOLAR PARAL-
LAX AND THE MASSES OF THE THREE INNER PLANETS.
79. Separate values of the solar parallax, and their general
mean 156
80. Rediscussion of the motion of the node of Venus 159
81. Possible systematic errors in determinations of the parallax. 164
82. Revised list of determinations 166
83. Definitive adjustment of the masses of the three inner
planets 168
84. Possible causes of the observed discordances 173
85. Adopted values of the doubtful quantities 173
86. Bearing of future determinations on the question. 175
CHAPTER IX. DERIVATION OF RESULTS.
87. Ulterior corrections to the motions of the perihelion and
mean longitude of Mercury 178
88. Definitive elements of the four inner planets for the epoch
1850, as inferred from all the data of observation 179
$ 89. Definitive values of the secular variations 182
90. Secular acceleration of the mean motions 186
91. The measure of time 188
92. The constant of aberration 188
93. The mass of the Moon 189
94. The parallactic inequality of the Moon 190
95. The centimeter-second system of astronomical units 190
\S 96. Masses of the Earth and Moon in centimeter-second units.. 191
^S 97. Parallax of the Moon 193
98. Mass and parallax of the Sun 194
99. Constant of nutation, and mechanical ellipticity of the
Earth 195
100. Precession 196
$ 101 . Obliquity of the ecliptic 196
102. Relative positions of the equator and the ecliptic at differ-
ent epochs for reduction of places of stars and planets . . 197
ELEMENTTlfFCONSTANTS.
OHAPTEK 1.
GENERAL OUTLINE OF THE WORK OF COMPARING THE
OBSERVATIONS WITH THEORY.
1. In logical order, the first step in the work consists in the
reduction of observed positions of the Sun and planets to a
uniform equinox and system of declinations.
The adopted standard of Eight Ascensions was that origi-
nally worked out in my paper on the Eight Ascensions of the
fundamental stars, found in an appendix to the Washington
Observations for 1870, and extended to a fundamental system
of time stars in the catalogue published in Yol. 1 of the Astro-
nomical Papers of the American Ephemeris. This system
coincides closely with that of the Astronomische Gesellschaft
and the Berliner Jahrbuch, about the epoch 1870, but the cen-
tennial proper motion is greater' by about 8 .08.
In Declinations, the adopted standard was that of Boss,
which has been used in the American Ephemeris since 1881,
and on which is based the catalogue of zodiacal stars just
referred to. But as Declinations generally are not immediately
referred to fundamental stars, the method of reducing obser-
vations to this system in Declination was not entirely uniform.
Observations used.
2. The following is a general statement of the observations
used, and the extent to which they were corrected, or re-re-
duced.
Greenwich. Dr. AUWERS courteously supplied me with the
results of his re-reduction of BRADLEY'S observations both of
the Sun and planets. From the beginning of MASKYLENE'S
work until 1835, the Greenwich observations were completely
re-reduced, utilizing, so far as possible, AIRY'S reductions. The
5690 N ALM 1 i
GENERAL OUTLINE. [2
data necessary for these observations were discussed in Prof.
SAFFORD'S paper, Vol. n, pt. n, which paper was prepared
for this purpose. In the case of the Greenwich observations
from 1835 onward, it was deemed sufficient to apply constant
corrections to the Eight Ascensions, determined from time to
time by comparisons of the adopted Eight Ascensions with
the standard ones. In the case of the Declinations, Boss's
special tables were used, but in the later years it was judged
sufficient to apply the constant correction necessary for reduc-
tion to Boss's standard.
Palermo. PIAZZT'S observations of the Sun and Planets were
completely re-reduced, the zero point of his instrument being
determined from the observed Declinations.
Paris. LEVERRIER'S reduction of the Paris observations
from 1801 onward was made use of, applying the correction
necessary to reduce the results to the adopted standard.
Konigsberg. BESSEL'S clock corrections were individually
corrected by the new positions of the fundamental stars, so
that practically the Eight Ascensions may be considered as
completely re-reduced.
In the case of the other observatories, it was deemed suffi-
cient to determine, by a comparison of the adopted or of the
concluded Eight Ascensions and Declinations of the funda-
mental stars with the standard catalogue, what common cor-
rections were necessary for reduction to the standard. When,
however, the period was covered by Boss's tables, the correc-
tion which he gives as varying with the Declination was ap-
plied. After more mature consideration, I am inclined to think
it would have been better to apply a constant correction to the
Declinations in every case, except those where the change
with the Declination was quite large.
Although these processes were somewhat heterogeneous, it
is believed that the main object of referring the Declinations
to a system of which the error would be a uniformly varying
quantity was fairly well attained. The subsequent determi-
nation of this error both in Eight Ascension and Declination
is a necessary part of the work.
3] OBSERVATIONS USED. 3
The following is a list of the observatories whose observa-
tions of the Sun and Planets were included in the work:
Greenwich 1750-1892
Palermo 1791-1813
Paris __-- 1801-1889
Konigsberg 1814-1845
Dorpat 1823-1838
Cambridge . 1828-1844
Berlin 1838-1842
Oxford, Radcliffe 1840-1887
Pulkowa 1842-1875
Washington _ 1846-1891
Leiden 1863-1871
Strassburg 1884-1887
Cape of Good Hope 1884-1890
The number of the meridian observations of the Sun, and
of the planets Mercury, Venus, and Mars, actually included in
the work is approximately as follows:
The Sun 40,176
Mercury _' 54 21
Venus 12, 319
Mars 4, 114
Total 62,030
Semidiameters of Mercury and Venus.
3. The reduction of the semidiarneter of the planets was a
point to which special attention was given. In the case of
Mercury, the adopted semidiameter at distance unity was 3".34.
The values adopted by the various observatories in reducing
their observations varied so little from this that in cases where
the original reductions were accepted no correction was applied
for the difference. So, also, when the observers applied a cor-
rection for reducing the observed center of light to the actual
center of the planet, no revision of this reduction was made.
Such was supposed to be the case with the Paris observations.
When the published Eight Ascension was that of the center
of light simply, a reduction to the true center was computed
by the empirical formula used in the Washington observations.
If we put i for the angle between the Earth and Sun as seen
from the planet, then 1 -f- cos i will represent the fraction of
4 GENERAL OUTLINE. [3
the apparent transverse diameter of the planet that is illu-
minated by the Sun. It was assumed that when the illumina-
tion was such that the thickness of the crescent approached
zero, the point observed would be two-thirds of the way from
the center of the planet to the limb, and that when the planet
was dichotomized the center of observation would be five-
twelfths of the way from the center to the limb. These con-
ditions, with the added one that when the planet was fully
illuminated the correction should vanish, suggested the em-
ployment of the formula
Correction = seinidiameter x (1-cos ^(5-f cos i)
This correction was to be multiplied by the sine or cosine of
the angle which the line of cusps made with the meridian to
reduce it to Right Ascension and Declination respectively.
The correction being practically the same whenever the
Earth and planet return to the same positions in anomaly, it
is possible to embody it in a table of two arguments, one
depending on the longitude of the Earth, the other on that of
the planet. Actually, however, the table was arranged in a
more convenient form, in which one argument is the date at
which Mercury last passed perihelion, and the other, its mean
anomaly. Owing to the importance which this correction may
assume, a partial transcript of the table actually employed for
the reduction in Right Ascension is given on the next page.
Read horizontally, the numbers show the corrections of the
argument through one revolution of the planet. Vertically,
they may be regarded as giving the successive corrections corre-
sponding to any one position of the planet, while the Earth
goes through a complete revolution. The table as actually
used extended to every 10, but the values for every 00 of
mean anomaly will suffice to show the general magnitude of
the correction.
The correction to the Declination was embodied in a similar
table, which it is not deemed necessary to print at present.
In the case of Venus, it seems scarcely possible to decide
upoiT a value of the semidiauieter, or a law of its apparent
change, which should apply to all parts of the orbit. After a
3]
SEMIDIAMETERS OF MERCURY AND VENUS.
careful examination of the data, it was decided to reduce all
the observations with the semidiameter
8 -^- 5 +0".20
when made with modern instruments, and to use a value 0".3
greater in earlier observations. The actual reductions of all
Correction for defective illumination of Mercury in R. A.
Arguments: Date of perihelion passage at side, and mean
anomaly "g" at top.
g=
*
60
120
1 80
240
300
360
s
s
s
s
s
J
s
Jan. o __
+.19
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03
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05
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13
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+ .01
Feb. 9__
. 10
17
.15
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. oo
19 ._
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. 18
. 10
.05
.01
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Mar. i ..
.06
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June 9 __
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Sept. 7 . .
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Oct. 7__
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+.03
6 GENERAL OUTLINE. [4
the principal series of observations were corrected to this value
of the element in question.
Observations of the estimated center of Venus, when made
more than one hundred days from superior conjunction, were
rejected altogether; when made within that limit, the point
observed was assumed to be the center of gravity of the illu-
minated portion of the disk, considered as a plane figure, and
the necessary reduction to the center was always applied.
A similar correction was applied to observations of the esti-
mated center of Mars. The Paris results, after 1830, and the
later Greenwich and Washington results, are published with
the reduction for center of light already applied, and in these
cases the published corrections were not changed.
Tabular places.
4. The tabular elements of the planets adopted for correc-
tion were those of LEVERRIER'S tables. These tables having
been continuously used in Astronomical Ephemerides since
1864, it was judged more convenient to adopt the theory on
which they were based as the provisional one to be corrected
than it was to construct a new provisional theory. As the tables
in their original form are extremely cumbrous to use, the
theory was partially reconstructed by making manuscript
tables of the principal perturbations, which were, however,
carried only to tenths of seconds. With these tables the
places of the planets were computed for dates previous to 1864.
As places of the Sun were necessary not only for direct com-
parison with observations of the Sun, but also for the geocen-
tric places of the planets, an ephemeris of the Sun's longitude
and radius vector was prepared for the entire period 1750-1864
to every fifth day, the lunar perturbation being omitted and
afterward applied for each date when required.
The method of deriving the final tabular places varied with
circumstances. When there was no accurate ephemeris avail-
able for comparison, which was the case before 1830, it was
necessary to compute a completely independent set of tabular
geocentric places. Sometimes these places were computed for
the moment of the individual observations, but more generally,
when the observations occurred in groups, an ephemeris was
4] TABULAR PLACES. 7
computed in order that the work might be checked by differ-
ences. After 1830 it was common to compute an ephemeris
for intervals of three, five, or ten days, thus deriving the cor-
rections necessary to reduce the published ephemerides of the
Berliner Jahrbuch or of the Nautical Almanac to those derived
from LEVERRIER'S tables.
Until this plan was mapped out, and work well in progress
upon it, it was not noticed that the planetary masses adopted in
LEVERRIER'S tables were so diverse that corrections to reduce
the geocentric places to a uniform system of masses would be
necessary. Although theoretically the necessary reductions
were very simple, I can not but feel that the application of
such corrections involves more or less doubt and uncertainty,
and that it would have been better to have constructed pro-
visional tables based on uniform masses quite independent of
those of LEVERRIER.
In Annales de V Observatoire de Paris, Vol. n, LEVERRIER
gives the following values of the masses used by him as the
basis of his provisional theory :
Mercury . . ^ = .000 000 333 . .
Venus 40T847 =- 0000024 885
Earth . QKA . ftQg =.000002
Mars 2 680 337 = - 000 00 373
The following table shows the factors by which these masses
were multiplied in the cases of the several planets in LEVER-
RIER'S final tables. They were controlled by induction from
the numbers of the tables themselves, the result of which was
found in all cases to agree with the statements in the introduc-
tion to the tables.
In the last line of the table is shown the factor used in the
present provisional theory.
GENERAL OUTLINE.
Mercury.
Venus.
Earth.
Mars.
In tables of
The Sun .
i 004.
o SCK
Mercury
j
Venus
i
j
j
Mars
o. 071;
i. 0026
Present work
j
i
o 86s?
As in the actual work the masses of Mercury and Venus
were to be determined from the observed periodic perturba-
tions which they produced, it was necessary that the perturba-
tions produced by them should all be carefully reduced to the
adopted standard. The reduction was less necessary in the
case of Mars, but was carried through all the work relating to
the Sun.
Comparison of observations and tables.
5. The result of each separate observation of each body was
compared with the tabular result thus derived. The residuals
were then taken and divided into groups. The interval
between the extreme dates of each group was always taken
so short that it could be presumed that the mean of all the
residuals would be the correction for the mean of all the dates.
The general rule was that the interval should not exceed four
or five days in the case of Mercury, or six or eight days in
that of Venus, and that not more than six or eight observa-
tions should be included in a single group. In taking these
means, weights were assigned to the results of each observa-
tory founded on the discordance of its residuals. Then to each
mean a weight was again assigned equal to the sum of the
weights of the individual residuals when these were few in
number, but not allowed to exceed a certain limit, how great
soever might be the sum of the individual weights.
Equations of condition.
6. Each mean result thus derived formed the absolute term
of an equation of condition for correcting the tabular elements.
The number of these equations was as follows:
Equations.
The Sun . 11,676
Mercury ___ 3, 929
Venus 4,849
Mars i, 597
6] EQUATIONS OF CONDITION. 9
In forming the equations of condition from observations of
the planets, I adopted the system suggested in the introduc-
tion to Vol. i of these publications, namely, the determination
of the solar elements not only from observations of the Sun
itself, but from observations of each of the planets. The reason
for this course is quite simple and obvious. An observation of
the position of a planet as seen from the Earth is the exact
equivalent of an observation of the Earth as seen from a
planet, and thus depends equally upon the elements of both
orbits. Hence, whatever elements of the Earth's orbit could
be determined by observations made from a planet can equally
be determined by observations made upon the planet. A
strong reason for proceeding upon this plan was found in the
very large errors, both accidental and systematic, to which
observations of the Sun are liable.
The advantages, however, have not proved relatively so
great as were anticipated. The eccentricity and perihelion of
the. Earth's orbit come out in the solution of the normal equa-
tions as functions of those of the planetary orbit to so great an
extent that their weight is much less than that which would
correspond to independent determinations from the same num-
ber of observations. On the other hand, the determination
of these elements from observations of the Sun proved to be
much more consistent than was expected, thus indicating a
high degree of precision.
The case is different with the Sun's mean longitude referred
to the Stars. Here systematic and personal errors enter so
largely that the results from Mercury and Venus appear to be
rather more reliable than those from the Sun itself. In the
case of these planets it fortunately happens that the weight of
the result derived for the Sun's mean longitude is not mate-
rially diminished by the uncertainty of the corresponding
element of the planet, the errors of the two mean longitudes
being nearly separated in a series of observations equally dis-
tributed around the orbit.
The systematic errors in observations of the Sun rendered
it unadvisable to determine the elements of the Earth's orbit
from observations of the Sun by a single system of equations.
The solar observations, therefore, were classified according to
10 GENERAL OUTLINE. [6
the observatory where made, and divided into periods rarely
exceeding eight years in length. The elements are separately
derived from the observations of each period. This system has
the advantage of eliminating to a large extent the injurious
effect of systematic and personal error upon the eccentricity
and perihelion of the Earth's orbit, and also enabling us to
judge of the precision of the corrections to those elements by
the discordance among separate results.
Meridian observations of the Sun and Planets are referred
to the fundamental stars, while the Eight Ascensions of the
latter are referred to the equinox, the position of which has
heretofore depended on observations of the Sun. The adopted
position of the fundamental stars therefore comes in, to a cer-
tain extent, as the basis of the work, and the constant parts
of their systematic corrections are among the results to be
derived.
Thus, in the case of the equations pertaining to the three
planets, the following corrections were introduced as unknown
quantities :
Correction of the mass of Mercury or of Venus.
Corrections to the elements of the orbit of the planet
observed.
Correction of the obliquity of the ecliptic.
Corrections to the Sun's mean longitude, eccentricity, and
longitude of perihelion.
Common corrections to the adopted Eight Ascensions and
Decimations of the fundamental stars.
In the case of Mercury an adopted hypothetical correction
of the ratio of the radius vector of the planet to that of the
Earth was also included in the equations, although little doubt
could be felt that the true value of such a quantity must be
zero. The reason for introducing it will be explained here-
after.
Determinations of the masses and secular variations.
7. The secular variation of all the preceding elements, the
mean distances excepted, was also introduced into the equa-
tions from observations of the planets. In addition to the
above elements, the mass of Venus appeared in the equations
7] MASSES AND SECULAR VARIATIONS. 11
derived from observations of the Sun, Mercury, and Mars, and
the mass of Mercury in the equations derived from obser-
vations of Venus. The coefficients of the masses, however,
depended wholly upon the periodic perturbations.
Were it quite certain that the secular variations arise
wholly from the masses of the known planets, the masses
could of course be derived from these variations, and the lat-
ter would appear in the equations of condition only through
the mass itself. On this hypothesis the secular variations
would not appear in the equations, but only the masses. But
it is well known that the perihelion of Mercury is subject to a
secular variation which can not be accounted for by any ad-
missible masses of the known disturbing planets. The same
thing may well be true of the secular variations of the other
elements. It is therefore necessary, in the absence of a known
cause for such deviations, to derive the masses of the planets
independently of the secular variations. In the case of Mars
the mass is obtained with all necessary precision from the sat-
ellites. It is, however, different in the case of Mercury and
Venus. Here no resource is left us but to determine them
from the periodic inequalities. As the inequality produced by
Venus in the Earth's longitude is rarely more than eight sec-
onds, it might seem that the coefficient would be too small to
obtain a sufficiently precise value of the mass. But in the
case of observations upon the Sun, Mercury, and Mars the
error of the determination of the mass in question may be
almost indefinitely reduced by multiplication and extension
of the observations without danger of systematic error.
To illustrate this, let us suppose the Sun's longitude to be
determined with a meridian instrument only once a year, say
at equal intervals of three hundred and s^xty-five days. Let
the longitudes thus observed be compared with an ephemeris
in which the elements are affected with only slight errors.
Leaving out of consideration the periodic perturbations pro-
duced by the planets, the comparison of the observed longi-
tudes with the tabular ones through an entire century should
be nearly constant. Any error affecting all the longitudes
alike would appear as a constant. The errors of mean motion
12 GENERAL OUTLINE. [7
would vary imiformly with the time. Thus the other elements
would be nearly constant, and could be still more approxi-
mately represented by a slight apparent secular variation.
Now let the disturbing action of a planet, say Venus, be in-
troduced. We should then have a series of deviations from the
law of uniform increase, which would enable us to evaluate
the mass of the planet. The value of this mass thus derived
would not be affected by any systematic error common to all
the observations, nor even by such an error which varied uni-
formly with the time. Nor would small errors in the adopted
elements of the Sun have any effect upon the result.
If this would be the case for observations made- only at a
certain point of the orbit, a fortiori would it be the case for
the observations made at various points of the orbit, since any
tendency to a systematic effect of the errors of observation
would thereby be ultimately eliminated.
Considerations almost identical apply to the case of observa-
tions upon either of the planets when we consider the action
of the other planet upon the planet observed and upon the
earth. But they do not apply to the case of the action of the
earth itself upon the observed planet, or vice versa. For ex-
ample, in the case of observations of Venus, we may suppose
that all observations made when Venus is at a certain point
of its relative orbit, near inferior conjunction, say one month
before inferior conjunction, are affected with a certain error
common to all observations made at that point of the orbit.
Since the perturbations produced by the third planet will in
the long run have all values, positive and negative, for these
several observations, the systematic error in question will not
affect the ultimate value of its mass. But the perturbations
of Venus produced by the Earth, as well as those of the Earth
produced by Venus, will not have all values in such a case, but
only special ones dependent on the relative position. Hence,
determinations of these masses might be affected by errors of
the kind in question. We conclude, therefore, that the mass
of the Earth can not be satisfactorily determined by the peri
odic perturbations which it produces in the motion of any
planet, nor that of Venus by observations on Venus through
its periodic perturbations of the Earth.
8] TRANSITS OF VENUS AND MERCURY. 13
In the solution of the equations of condition the method of
least squares has been used throughout, the arrangement of
the work, the choice of quantities to be corrected, and the
accuracy of the coefficients being so chosen as to minimize the
great mechanical labor of making the necessary multiplica-
tions. The adoption of this method was necessary in order to
separate, so far as possible, the various unknown quantities
and show to what extent their values were interdependent.
By no other method of combination could so large a number
of unknown quantities have been separately determined in a
way which would have been at all satisfactory. On the other
hand, in combining the final results and deciding upon the
values of the corrections to be adopted, the method has not
always been applied, for reasons which will be developed in
Chapter IY.
Introduction of results of observations on transits of Venus and
Mercury.
8. In the case of Mercury and Venus the observed transits
over the Sun give relations between the corrections to the
elements more accurate than those ordinarily derivable from
meridian observations. This is especially the case with Venus.
The value of these observations is greatly increased by the
fact that they are made when the planet is near inferior con-
junction, and therefore nearest to the Earth, and in a point of
the relative orbit where meridian observations are necessarily
most uncertain. In the case of Venus the error of the helio-
centric place will be more than doubled in the case of the geo-
centric place during a transit. As, however, the observation
of a transit gives no one element, but only an equation of con-
dition between the values of all the elements at the epoch, the
only way of treating it is to introduce the result as such an
equation, with its appropriate weight. The determination of
the proper weight is a difficult matter. The systematic errors
of meridian observations are such that the theoretical value
of the weights assignable to so great a mass as we have dis-
cussed would be entirely illusory. In fact so great is the
weight assignable to the observed transits of Venus that if
we should regard the results of each transit as a condition to
14 GENERAL OUTLINE. [8
be absolutely satisfied we should not be dangerously in error.
I conclude, therefore, that there is more danger of assigning
too small than too great a weight to these observations.
In order to determine what change was produced in the re-
sults by the use of the observed transits over the sun's disk,
two separate solutions of the equations of condition for Mer
cury and Venus were made. In the one, termed solution A,
the meridian observations alone were used; in the other,
termed solution B, the, combined equations formed by adding
the normal equations derived from the transits to those given
by the meridian observations were used.
In the case of solution A it was originally supposed that by
using the mean epoch of all the observing in the case of each
planet as that from which the time was to be reckoned, the
normal equations for the secular variations would be almost
completely separated from those for the corrections to the
elements themselves. The separation would be complete were
the observations at different epochs similarly distributed
around the orbit. But, as a matter of fact, it was found that
the accidental deviations from this symmetry were so consider
able that the separation could not be regarded as complete.
The solution was therefore made by successive approximations,
the terms depending on the secular variations being in the
first approximation dropped from the normal equations for the
corrections to the elements, and afterwards included when
approximately determined, and vice versa.
In the case of solution B, in which the transits were included,
such a separation did not occur, and the equations were solved
in the usual rigorous way for all the unknown quantities.
CHAPTER II.
DISCUSSION AND RESULTS OF OBSERVATIONS OF THE
SUN.
Treatment of the Eight Ascensions.
9. The meridian observations of the Sun have been treated
on a system different in some points from that adopted in the
case of the planets. It was possible to simplify the treatment
by supposing that the small latitude of the Sun was always a
definitely known quantity, so that when the observations were
corrected for it the apparent motion of the Sun could be sup-
posed to take place along the great circle of the ecliptic. This
allowed the correction of the elements to depend on but two
quantities the obliquity of the ecliptic and the Sun's true
longitude. Assuming the obliquity to be known, the longi-
tude of the Sun could always be determined from an observa-
tion of its Right Ascension. An observed Eight Ascension
being compared with a tabular one, the residual gives rise to
an equation of condition between the correction of the long-
itude, A, of the obliquity, , and of the Right Ascension of the
Sun, a:
da = cos s sec 2 ddX tan e sin 2ad?.
This equation may be used to express the error of the longi-
tude in terms of the error of the obliquity and of the Right
Ascension as follows :
#A = sec 8 cos 2 dda + J tan e sin 2Xds
= sec s cos 2 6$a + 0.21 sin 2Xds
The elements mainly to be determined from the observations
in Right Ascension being the eccentricity and perihelion of
the Earth's orbit, each of the coefficients of which go through
a period in a year, the effect of the small term 0.21 $s sin 2A
whose coefficient does not amount to 0".10 after 1800, and has
a period of half a year, will be practically without influence
15
16 OBSERVATIONS OF THE SUN. [10
on the result. The system was therefore adopted of deriving
the residual in longitude directly from the residual in Eight
Ascension by the formula
where
F = cos 2 3 sec e .
The residual #A in true longitude is then to be expressed in
terms of the residual 61" in mean longitude and of corrections
to the eccentricity and to the longitude of the perigee relative
to the Stars. In this expression the coefficient of the residual
in mean longitude was always taken as unity, the value of the
correction being so small in the case of LEVERRIER'S tables
that no appreciable error would result from this supposition.
Thus each residual in Eight Ascension would give rise to an
equation of condition of the form
61" + Pe"6n" + E6e" = tfA = ~F6a
We are here to regard 61" and 6n" as corrections to the
Eight Ascensions relative to the clock stars, and not to the
Sun's longitude or perigee simply. I shall therefore use the
symbol c instead of 61" to express the relative correction here-
after.
Treatment of the Declinations.
10. The declination of the Sun in the case supposed is a
function only of the longitude and obliquity. The equation
for expressing the observed correction in Declination in terms
of the corrections to these two quantities is
Ad = sin a6s -f cos a sin sSX
Thus each observation of the Sun's Declination gives rise to
an equation of condition of this form.
It is however to be supposed that the observations in Decli-
nation made at each observatory will be affected by a constant
error. If the observations are truly reduced to the standard
system of star places, this error will be that of the standard
system. As a matter of fact, however, observations made in
the daytime, especially on the Sun and at noon, are made
under circumstances so different from night observations on
11] FORMATION OF EQUATIONS IN RIGHT ASCENSIONS. 17
stars that we can not assume the error of the reduced declina-
tion to be necessarily the same as that of the star system.
We must, therefore, in each case, regard the constant error in
declination as something peculiar to the observatory and the
instrument, which may or may not be worthy of subsequent
discussion. Thus each residual in declination gives rise to
an equation of condition,
j# o 4. cos a sin (U -f sin ade = AS
Ad being the excess of observed over tabular declination,
and Ad Q the common error of all the measured declinations of
any one series.
Formation of the equations from Right Ascensions*
11. The method of treating the observed Eight Ascensions
of the Sun was suggested by the fact that they are peculiarly
liable to systematic and personal errors ; the former likely to
change with the seasons, and to be different for different in-
struments; and the latter to continue through the work of one
observer. It is now well understood that the observed Eight
Ascensions of the mean of the Sun's two limbs relative to the
fixed stars are affected by personal errors, no means of elimi-
nating which have yet been tried. In a series of observations
made by a. single observer, under uniform conditions, this error
would systematically affect only the relative mean of the Eight
Ascensions of the Sun and Stars, leaving the eccentricity and
perigee derived from the observations substantially correct.
On taking up the work it was also supposed that, owing to
the different effect of the Sun's rays upon the instrument at
different seasons, and the different circumstances under which
observations were made, the Eight Ascensions of the Sun
would be affected by errors varying in a regular way through
the year, but not wholly expressible as a term of single annual
period. It was therefore deemed best to consider the observa-
tions possibly affected by an error of double period, having the
form
x' cos 2g -f y' sin 2g
5690 N ALM 2
18 OBSERVATIONS OF THE SUN. [11
The introduction of the coefficients x 1 and y' added two more
terms to the equations of condition, which terms, however, did
not express any astronomical fact, but only the possible errors
of the observations.
An additional and very important element to be determined
from the observed Eight Ascensions was the mass of Venus.
The question now arose whether, by a uniform series of obser-
vations, extending through some definite period, the correc-
tions to the eccentricity and perigee and the coefficients x 1 and
y 1 could be completely separated from the coefficients of the
correction to the mass of Venus. Examination showed that
from such a series of observations, extending through eight
years, the mass of Venus could be determined irrespective of
all systematic errors repeating themselves with the season,
provided that the observations were equally distributed
throughout the year, or even that an equal number were made
at the same time through successive years. As neither of
these conditions are practically fulfilled it was judged best to
assume in the beginning that the systematic errors of an un-
known kind repeated themselves at each season during an
eight-year period, and that they could be expressed in the
form
c + x cos g H- y sin g + x 1 cos. 2g + y' sin 2g
x and y would appear as errors of eccentricity and perigee
which could not be eliminated.
The quantities actually introduced as the uuknown ones of
the equations of condition were as follows:
X, the factor of correction of the mass of Venus j
#, one-fifth the correction to the eccentricity;
y, one-fifth the correction e"dn" ;
x',y', one-tenth the coefficients expressing the .supposed
error of double period arising from all causes whatever ;
c, the constant correction to the Eight Ascension of the
Sun relative to the Stars.
The coefficient of c was supposed unity throughout. The
reduction of the residual in Eight Ascension to that in Longi-
tude and the other factors were taken from a table like the
following, of which the argument was the day of the year.
11] FORMATION OF EQUATIONS IN EIGHT ASCENSION. 19
Separate tables were constructed for 1802 and 1850, but they
were so nearly identical that no distinction need be made
between them. Furthermore, the error introduced by sup-
posing the mean anomaly to have the same value on the same
day of every year is entirely unimportant.
Table of coefficients for expressing errors of the Sun's Right
Ascension in terms of errors of the elements of the Earth's
orbit.
da
dl
dl
Coefficients of
da
,=0^
*-**
f
y'
Jan. I
1.09
o. 91
4- O.I
10.
-[- o. i
4-10.0
II
1.07
0-93
1.8
9.8
3-5
9-4
21
1.04
o. 96
3-4
9.4
6-5
7.6
31
I. OI
0.98
8.7
8.7
Feb. 10
0.98
I. OI
6.' 4
7.7
9.8
4- \\
20
o. 96
.04
+ 7.6
-6.5
+ 9-9
- 1.6
Mar. 2
0.94
.06
8.6
5.1
8.7
4.9
12
o. 92
.08
9.4
3-5
6.6
7.5
22
o. 92
.08
9-8
1.9
3-7
9-3
Apr. I
-93
.07
IO. O
O. I
+ 0.3
IO. O
II
0.94
.05
+ 9-9
4- 1.6
*> T
o* *
- 9-5
21
o. 96
. 03
9-5
3-2
6. i
7-9
May i
0.99
. OI
8.8
4.8
8.4
5-4
ii
I. 02
0.98
7-8
6.2
9-7
2. 2
21
1.05
0.95
6.6
7-5
9.9
I. 2
31
1.07
0.93
+ 5.3
+ 8.5
- 8.9
4-5
June 10
1.09
o. 91
3-7
9-3
6.9
7.2
20
I. 10
o. 91
2. I
9.8
4.1
9.1
30-
1.09
0.91
+ 0.4
IO. O
- 0.7
10.
July 10
1. 08
0.93
9.9
+ 2.7
9.6
20
1.05
0.95
3.0
+ 9-5
+ 5.8
- 8.2
30
1.03
0.07
4.6
8.9
8,2
5-7
Aug. 9
I. OO
. OO
6. i
8.0
9-6
+ 2.7
19
0.97
03
7-3
6.8
10.
0.8
29
-95
05
8.4
5-4
9.1
4. i
Sept. 8
0-93
.07
9.2
+ 39
+ 7.2
-6.9
18
0.92
.08
9-7
2-3
4.5
8.9
28....
o. 92
.08
10.
+ 0.6
4- 1.2
9-9
Oct. 8...
0-93
.07
9.9
1. 1
2. 2
9-7
18
o-9S
05
9.6
2.8
5-4
8.4
28....
0.97
I. 02
9.0
4.4
7.9
6.1
Nov. 7
I. 00
0.99
8.1
5.9
9-5
- 3-1
17
1.03
o. 96
7.o
7.2
IO..O
+ 0.3
Dec. 7~~"
.06
.08
0.94
o. 92
5.6
4- i
8.3
9.1
9-3
7-5
3.7
6.6
17
.09
o. 91
- 2. 5
9-7
4-9
+ 8.7
27....
.09
o. 91
-0.8
IO. O
1.6
+ 9-9
20 OBSERVATIONS OF THE SUN. [12
Finally, throughout the work the equations of condition
were expressed only in entire numbers, the decimals being
neglected. To lessen the number of equations of condition,
the residuals were divided into groups generally covering from
ten to fifteen days, the length of the group being determined
by the condition that the perturbations of Venus must not
change much during the period.
While the formation and solution of the equations of condi-
tion on this system were going on, it was found that the intro-
duction of the assumed coefficients x 1 and y' was a refinement
productive of little or no good result. In fact, the observa-
tions of the Sun proved to be much freer from annual sources
of error than I had supposed, as will be seen by the tables of
their results soon to be given. This is shown by the general
consistency of the corrections to the eccentricity and perigee
given by the work at the same or different observatories dur-
ing different periods.
In marked contrast to this is the discordance among values
of the correction c to the relative Eight Ascensions of the Sun
and Stars. This quantity it is that is affected by personal
error and possibly by the effect of the Sun on the instrument.
Under a perfect system of discussion it would be advisable to
determine it separately for each observer. This however was
practically impossible.
Solution of the equations.
12. For the purposes of forming and solving the normal
equations, the equations of condition were divided into groups
of generally from four to eight years, the exact lengths of
which will be seen from the following exhibit of results. The
equations for each period were solved on the supposition that
the corrections were constant during the period. Thus every
separate result is independent of every other, except so far as
they may depend on the same instrument or the same observer
at different times.
The first column shows the years through which the obser-
vations extend.
The second one shows to the nearest year the value of T
that is, the fraction of the century after 1850.
12] SOLUTION OF THE EQUATIONS. 21
The third column shows the value of //, or that factor which,
being multiplied by the adopted mass of Venus, is to be applied
as a correction to that mass, to obtain the value given by the
observations.
All systematic errors arising from the instrument and the
observer are so completely eliminated from the separate de-
terminations of X that they may be regarded as absolutely
independent of each other, that is as not affected by any
common systematic error.
We have next the relative weight assigned to each value
of //, which is determined in the usual way from the solu-
tion, and is, therefore, on a different scale for different ob-
servatories.
Next is given the value of c, or the apparent correction to
the Eight Ascension of the Sun, relative to the assumed Eight
Ascensions of the Stars, as given by observations during the
several periods and expressed in seconds of arc, followed by
the weights assigned to the separate results.
The next two columns, the corrections to the solar eccen-
tricity and to the longitude of the perigee, require no further
explanation.
Eespecting the weights ultimately assigned to these quanti-
ties, and to GJ it is to be remarked that they are the result of
judgment more than of computation. It is only possible to
enumerate in a general way with some examples the consider
ations on which they are based.
In assigning the weight of c the number of observers en-
gaged is an important factor in determining it. Other factors
are the steadiness of the atmosphere and the adaptation of the
instrument to this particular work. General consistency is
an important factor in the assignment. In this respect the
Cambridge observations are quite remarkable ; if their excel-
lence corresponds to their consistency they must be the best
ones made.
It will be seen that PIAZZI'S results are thrown out en-
tirely. The wide range of his values of c led to the inquiry
whether more consistent results would be obtained by taking
shorter periods, but it was found that the values of c varied
from time to time in such an irregular way that his instrument
22
OBSERVATIONS OF THE SUN.
[12
must have been affected by some extraordinary cause of error,
unless some mistake has been made in interpreting or treating
the observations.
The Oxford values of c are unusually discordant. The pre-
sumption that this discordance arises mainly from the special
personal equation in observations of the Sun, described on
page 17, derives additional weight from the greater relative
consistency of the values of 6e" and e"dn". I have therefore
allowed the values of these quantities to receive a fair weight.
The value of c for Paris, 1866-'70, has received a much re-
duced weight, solely on account of its excessive value. It
seems that the work of one observer who made many observa-
tions during this period was affected by an unusual system-
atic error.
Results of observations of the Sun's Right Ascension.
GREENWICH.
Years.
T
P'
w
c
W
6e"
*"<Jir"
za
!750-'62
-94
.027
20
+o'-33
i-S
a
+0.04
0.42
2
I765-'7I
.82
.041
10
+0-37
o-5
0.08
o. 64
I
1772-^8
-75
.022
10
+0.74
o-5
o. 16
0.49
I
'779-'85
-.68
-35
IJ
+2.89
0. 2
o. 1 8
o. 73
-5
ij86-'()2
-.61
037
8
+ i-5i
0. 2
0. 12
0.88
o
i793-'97
-55
. 114
5
+ 1.87
0. 2
O. 22
1.27
i798-'o2
So
+. 060
5
+ i. 02
O. 2
o. 42
i. 15
i8o3-'o6
45
. 002
5
+0.27
O. 2
o. 03
1.03
o
1807-' 10
.41
.068
5
0.34
O. 2
-0.32
I. 12
o
i8ii_'i4
37
095
3
-3-33
0. 2
+0.17
-I 08
o
i8i5-'i8
-33
-.052
6
1.99
o-5
0. 12
0-34
l8l9-'22
.29
+ .010
6
0.51
+0. 22
o. 19
I
1823-^6
25
054
6
i. 08
+0.05
o. 17
I
i827-'30
. 21
.045
6
0.42
o. 09
0-75
I
i83i-'34
17
+.016
7
+0.76
+0.04
o. 27
I
1835-38
!3
+ . O2O
8
4*1. 16
+0.26
+0.06
2
1839-42
.09
+ .061
8
+0.84
+0.32
+o. 10
2
1843-46
05
-.008
8
+0.15
2
+0.25
+0. 22
2
i847-'5o
.01
.045
8
0. 10
2
+0. 28
+0.02
3
i8si-'54
+.03
+.024
8
+0.40
3
+O. 22
-j-o. 02
3
1855-58
+.07
.032
9
+0.36
3
+0.15
+O. O2
3
i859-'62
+.11
.043
0.02
3
+ 0.25
+0. 22
4
i863-'66
+.15
.Ol6
8
+0-31
3
+0.23
0.05
4
i867-'7o
+.19
+ .031
8
+Q-35
3
+0-33
O. IO
4
i8 7 i-' 74
+.23
+ .021
8
+0. 12
3
+0.24
+q.os
4
1875-78
+.27
.008
8
. 0. 12
3
+0.26
+0.06
4
i879-'82
+31
+.017
8
o. 05
3
+0. 21
+o. 14
4
i883-'88
+.36
+ . OOI
13
0. 20
3
+0.18
+0.07
4
i889-'92
+.41
.025
8
0.44
2
+o. 24
+0. II
3
12J SOLUTION OF THE EQUATIONS. 23
Results of observations of the Sun's Right Ascension Continued.
PARIS.
Years.
T
/"'
w
c
w
6e"
e"fa"
W
iSoi-'oy
-.46
.025
H
_i'/ 7 8
0-5
+o'.'o8
n
0.23
1 808-' 1 5
-38
+.015
17
-0.65
o-5
O. OI
+O. 12
1816-22
-3i
. 050
H
+o. 18
0-5
o. 13
+0.32
i823-'29
.24
.050
10
-j-O. 01
0-5
0.31
O. O2
i837-'44
.09
-.034
19
+Q-33
i
o. 04
+O. IO
5
i8 4 5-'52
+ .01
-f-. 009
15
+0. 10
+0.04
+o. 10
5
i853-'59
+.06
+.014
15
+0.66
o. 04
+0.32
2
i86o-'65
+-I3
+.003
10
+0.38
+0.07
+o. 26
2
i866-'7o
+. 18
.000
7
+2.29
. 3
+- I 3
+0.40
2
i87i-'79
+ 25
+.048
ii
o. 26
o. 06
+O. 22
2
1 880-' 89
+ 35
-[-. 002
14
+ 0.44
+0.24
-i-o. 03
2
PALERMO.
i79i-'96
-.56
.079
o. 07
o
o. 06
o'.'ss
i797-'oi
-51
. 116
2.33
o
o. 29
o. 28
i8o2-'o5
-.46
. OOI
3-"
o
-0.05
o. 76
1 806-' 1 2
.41
+243
o
4-5-92
o
1.17
+ 1-55
o
CAMBRIDGE.
1 828-' 34
.21
+ .007
16
o'.'i3
2
+0.08
+0.12
4
i835-'40
. 12
-033
H
o. 1 8
2
+0.06
0. 06
4
i8|2-'47
-05
. 026
9
O. 21
2
+ 0.08
O. 12
4
i8so-'58
+ .0 4
. O24
20
0. II
2
+0.17
0.04
4
WASHINGTON.
1 846-' 5 2
. 01
-.038
5
-o."8 5
2
+0. 20
o. oo
3
i86i-'6s
+-I3
-.038
8
-0.53
4
+ O OI
o. oo
5
i866-' 73
+.20
.004
13
0. 22
4
+o. 18
o. 03
6
i8 74 -'8i
+ .28
-033
12
-0.45
4
+0.07
o. 16
5
i882-'9i
+ 37
. 002
17
0.79
4
+0.07
o. 07
5
KONIGSBERG.
II
i8i6-'23
3
+ .002
'3
+0.30
I
+0.07
0.28
3
1824-^0
23
.006
12
+O. O2
I
o. 1 6
+ O. II
3
i8 3 i-' 3 8
- 15
. O2I
15
+0.23
I
O. 12
+0.03
3
1 839-45
.08
. O2 1
12
+0.77
1
+0.08
+o. 20
3
24 OBSERVATIONS OF THE SUN. [13
Results of observations of the Sun's Right Ascension Continued.
OXFORD.
!
j
Years.
T S
w
e
w
fe"
e"W
w
i840-'49
05
.043
12
a
+2.49
-3
+0.24
0/17
2
i86o-'68
+.14
+.042
13
+ 1.96
o-3
4-0.08
o. 13
2
1 869-' 76
+.23
+ 054
IS
4-0. 92
0-3
4-0. 20
o. 04
2
i88o-'87
+.34
-.014
9
0.31
o-3
4-0.27
4-0.64
2
PULKOWA.
1 842-' 50
i86i-'7o
.04
4-.i6
4-. 047
4-. 002
II
10
+ i!'a
0.40
i
0. 12
4-0.05
4-0. 20
4-0.28
3
3
DORPAT.
I82 3 -' 3
i8 3 i-' 3 8
-23
-15
4-. 021
+ .008
I
4-0. 36
+0.45
!
0. 12
4-0.02
ii
O. 22
+0.03
2
2
CAPE OF GOOD HOPE.
1 884-' 90
+37
. 026
12
o. 36
3
-f-o! 02
+0. OI
4
STRASSBURG.
i883-'88
+.36
.014
12
-t!'s
2
4-0. 23
-fO/09
3
The mass of Venus.
13. The mean results for the mass of Venus given by the
work at the several observatories are shown as follows:
The probable error, where given at all, is that derived from
the discordance of the separate individual results at the par-
ticular observatory. In some cases there are only one or two
results; here no probable error could be assigned.
w' is the sum of the weights of the result at each separate
observatory, as given by the equations of condition. Were
all the observations of equal accuracy, these would be the
weights to be assigned to the separate results. Such not be-
14]
CORRECTIONS OF RELATIVE RIGHT ASCENSIONS.
25
ing the case, we choose for the actual weights certain numbers,
founded partly on a compromise between the mean errors fol-
lowing each result or upon the values of iv 1 , partly on a judg-
ment of the accuracy of the observations.
Values of //' for the mass of Venus.
/
w
W
Greenwich
. OI5-J-. 006
226
II
Paris ._. _ _ __ _
. OO7-4-. OOQ
146
c
Konigsberg ._
. OI2-J-. OIO
C2
7
Cambridge
. 018^. 009
Co
6
Dorpat _
+. 016
1C
i
Pulkowa
+- 02 5
21
i
Oxford _ . .
4-. oiA-L. 023
49
i
^^ashington
. oi84-. OOQ
cc
4
Cape
. 026
12
I
Strassburg _
. 014
12
I
Using the weights in the last column, we have for the mean
result
fi = - .0118 .0034.
The mean error i .0034 is that given by the discordance of
the separate results of the preceding table.
Corrections of relative Right Ascensions.
14. The true values of the remaining quantities c, 6e", and
e"Sn" are to be regarded as increasing uniformly with the
time and therefore of the form
x+Ty.
Here T is the time, and in the treatment of these particular
equations it is counted from 1850 in units of one century, so
that x is the value of the correction at this mean epoch.
The quantity designated by c is the same which, elsewhere
in this discussion, is represented by dl + a, so that
c = dl" -f a
I shall, however, for convenience, continue to use the designa-
tion c, or #+T y.
26 OBSERVATIONS OF THE SUN. [14
As the observations at Greenwich and Paris extend over
longer periods than at any other observatories, I shall first
solve them separately. The totality of the Greenwich obser-
vations give for c the following normal equations and solution :
43.4 x + 1.65 y= + 4".23
1.65 4- 4.24 = - I". 25
x = 4 0".ll
y = - 0".34
Those at Paris give the equations and solution
8.3 x + 0.04 y= + 1".22
0.04 4- 0.48 == 4- 0".77
x = 4- 0".14
y = 4 1". 59
If we combine all the other results into a single set of normal
equations, we have
40.2 # + 4.26 T/ = -10".84
4.26 4- 2.20 = - 3". 98
x=- 0".10
It will be seen that the results for t/, the secular motion, are
markedly discordant. Indeed, if we refer to the exhibit of
results, p. 23, we shall see that the values of c are much more
discordant than those of the other two quantities. To obtain
a definite value, founded on all the observations of the Sun's
Eight Ascension, I do not see that any better result can be
obtained than that found from a general solution of the com-
bined normal equations. The equations and their solution are
as follows :
91.9 #+ 5.95 # = -5".39
5.95 + 6.92 = - 4".46
x = - 0".02
y 0".63
or
61" 4- a = - 0".02 - 0".63T
15] CORK. TO THE SOLAR ECCENTRICITY AND PERIGEE. 27
Corrections to the solar eccentricity and perigee.
15. 1 have already mentioned the remarkable consistency
of the corrections to these elements given by the results at
different observatories and at different epochs. The eccen-
tricity is more consistent than the perigee. One cause for
this, the consideration of which will throw some light on the
relative merits of the observations, is that the error of Bight
Ascension depending on the Declination of the object observed
effects the eccentricity less than the perigee. It is well known,
from a comparison of the results, that the systematic differ-
ences in the Eight Ascensions of different star catalogues
vary somewhat with the Declination. Now, since the Sun's
Declination goes through an annual period, it follows that this
error will produce a systematic effect on both the eccentricity
and the perigee. But the effect will be much larger in the
case of the latter element than in the case of the former,
because of the nearness of the perigee to the winter solstice,
the difference being only some 10 or 12. Consequently the
extreme coefficients in the correction to the eccentricity have
nearly the same values, with opposite signs, for the same Decli-
nations in different seasons of the year. But it is different
with the perigee. The coefficient of this quantity is negative
from October until March, when the Sun is in south Declina-
tion, attaining its maximum value about January 1; while it
is positive during the remaining months when the Sun's Decli-
nation is north, attaining its maximum value about July 1.
A systematic difference in the errors of Eight Ascension will
therefore produce its full effect on the longitude of the perigee,
while its effect on the eccentricity will be but slight.
In this connection, the very large negative values of the cor-
rection to the perigee during the period when the old Green-
wich transit instrument was in use are quite remarkable.
The progressive change in the value of c is also remarkable in
this connection. It is to be remarked that the new transit was
mounted in 1816, but account was not taken of this fact in
grouping the equations. Hence it is only from the year 1819
that the results of the table are derived wholly from observa-
tions with the new instrument. The anomaly alluded to is
28 OBSERVATIONS OF THE SUN. [15
then seen to disappear. The fact that the abnormally large
corrections in c are positive before 1800 and negative after it,
while e" dn" is abnormally negative through the doubtful
period 1765-1815, complicates the theory of these errors. I
have not been able to consider them in detail, but have simply
rejected the results for de" and e" 6n" from 1786 to 1818, hav-
ing given them a gradually diminishing weight from BRAD-
LEY'S observations to the first epoch.
As in the case of c, I have made a solution for Greenwich
alone, Paris alone, the other observatories combined, and all
combined. The results are shown as follows :
1. From Greenwich observations :
8e" e"8Tt"
54.5# + 2.73s/ = + HM4; - 0".88
2.73 + 5.72 = -}- 1".82$ + 2".69
x= + 0".19; -0".04
y= + 0".22 ; + 0".49
2. From Paris observations :
de" e"dn"
17.0#+ 0.39 y= + 0".30; + 2".95 '
0.39 + 0.99 = + 0".29; + 0".33
x= + // .01; +OM7
y= + 0^.29 ; + 0".27
3. The equations and results from all the other modern
observations are
de" e"8n"
77.0#+ 4.99# = + 5". 58; + 0".35
4.99 + 3.68 = + 1".09; + 0".40
y= +0".22;
16] RESULTS OF OBSERVED DECLINATIONS OF THE SUN. 29
4. Finally, if we combine all the equations, we have
de" e"d7t"
US.Zx -f 8.11 y = + 17".02; + 2"A2
8.1 -4-10.39 =+ 3". 20-, + 3".42
a? =4. 0".10; 0".00
y=+ 0".23; + 0".33
In the case of the eccentricity the general accordance is
quite satisfactory, and for the perigee it is much better than
in the case c, the relative Eight Ascension.
Results of observed declinations of the Sun.
16. The Sun's absolute longitude can be found only from
observations of his declination, because this longitude is
referred to the equinox, which is defined only by the Sun's
crossing of the equator.
The corrections to the eccentricity and perigee, as just found,
are so slight that they may be neglected in determining the
correction of the absolute longitude from that of the declina-
tion. Thus, as already stated, the unknown quantities of the
equations given by the declinations are the corrections of the
mean longitude Z", and of the obliquity f, and a constant A6^
peculiar to each observatory, of which we take no further
account. The equation of condition given by each observa-
tion or group of observations is
Ad 4- A sin s 61" + Bfo = dd
where dd is the excess of the observed over the tabular decli
nation, and
A = cosec e = cos a
B
30
OBSERVATIONS OF THE SUN.
[16
The equations are grouped and solved for periods, as in the
case of the Eight Ascensions, with the results shown in the
following table:
Results of observations of the Sun's Declination.
GREENWICH.
Years.
T
<?/"
w
6.
IV
*
,.
w
1753-57
i758-'62
-95
.90
+0.78
+ I -5
0-34
1.81
I
I
2.43
1.94
0-34
1.81
I
I
1765-^0
.82
0.23
o. 95
0-5
+o. 20
0-95
o. 5
i77i-'78
. 75
+0.48
0-93
0-5
+ i. 25
0-93
o. 5
i779-'85
. 68
+ i. 23
1.09
0.99
1.09
o. 5
i786-'9i
.61
+0.48
o. 50
o. 3
f o. 15
o. 50
0-3
1792 '97
-55
+ 1. 12
o. 70
0. 2
0.35
o. 70
0. 2
1798-03
49
+ 0.41
1.02
0. I
0. 10
1.02
O. I
1 804-' i o
43
+O.I8
I.4I
O. I
0.84
I.4I
0. I
i8i2-'i6
36
-0.15
3
0-53
3
+0.48
0-53
3
l8l7-'22
30
o. 41
3
+o. 03
3
+0.40
+0.03
3
i823~'28
-.24
+0-43
3
0. 10
3
+0.08
0. 10
3
1 829- '34
.18
0.08
3
+0. 21
3
4-0.25
+0.21
3
1835 '40
. 12
0. 12
3
O. 2O
3
4-0.37
o. 13
3
1 84 1 -'46
. 6
+0. 21
3
4-0.13
3
4-0.47
+0. 12
4
1847-52
o
4-0.25
4
O. OO
4
o. 24
o. 15
4
1853-58
+. 6
-f'55
5
+o. 18
5
o. 26
0.05
5
1859-64
+ .12
+0.03
5
+0.28
5
0.46
+0. 12
5
i865-'7o
+.18
-0.23
5
0.15
5
4-0.05
0.36
5
i87i-'76
+.24
o. 15
5
+ 0.26
5
+o. 16
o. 1 6
5
i877-'82
+.30
o. 90
5
+0.22
5
4-0-34
+0.08
5
1 883-' 88
+.36
0.27
5
4-0-33
5
-0.14
-j-o. 02
5
i889-*92
+.4-1
0.05
3
4-0.19
3
4-o. 13
o. 07
3
1
PARIS.
i8oo-'o3
.48
-f-o. 01
I. 93
O. 45
i8o4-'o7
.44
4-0.7
+0.82
2.02
1 808-' 10
.41
+2.66
+ 1.60
. 95
i8ii-'i5
. 37
o. 92
I. 2O
i. 18
i8i6-'2i
3 1
4-0.58
+ 1.68
i. 42
l822-'28
. 25
+ 1.09
7
+o. 39
7
O. OI
i837-'42
. 10
+o. 79
7
o. 15
3
+0.40
1843 '48
4
+o. 43
2
o. 03
3
-f-o. iq
1849-^4
-f. 2
+ i. 19
2
O. OI
2
+ 1. 74
i855-'6o
-f. 8
+o. 35
o. 02
7
+ 1. 22
1 86 1 -'66
4-. H
+ 1.35
3
o. oo
T.
+O. 12
i867-'72
-f-.20
+o. 31
2
o. 67
2
+O. IO
i873~'77
-f. 25
O. Sq
2
+o. 04
2
4-1. 01
i878-'83
4--3 1
o. oq
2
o. 32
2
+o. q8
1 884-' 89
4-. 37
0.80
2
+o. 32
2
+o. 78
16] RESULTS OF OBSERVED DECLINATIONS OF THE SUN. 31
Results of observations of the Sun's Declination Continued.
PALERMO.
Years.
T
w
w
6e
W
J<?
Vt
W
r -j
//
I 46
o
//
O Qi
//
4-o. 78
//
o. o<>
o 4
i/y 1 U J
i SOA ' 1 1
- JJ
41
1 I 7O
o
O $2
4-O. 42
O. 52
O. A
1 0^4 i j
CAMBRIDGE.
1877 '78
T A
O "I
2
O. 77
4-o. So
O. 54
i
10 OJ O
1870 '44
.08
-[-o. 31
2
O. 2O
+o. 29
o. 41
i
1847 '57
oo
4-O. 21
2
4-o. 7,1
O. }2
-fo. 10
i
i854-'s8
+.06
o. 15
2
4-o. 74
o. 42
-4-o. 13
i
WASHINGTON.
i840-'49
. 02
0.28
4
o. 73
o. 47
o. 81
2
i86i-'66
+.14
O. II
4
o. 43
0.45
o. 25
2
i867-'72
+ . 20
4/-O. 74
4
o. 39
+0.28
o. 51
2
i8y3-'78
+ . 26
o. 58
4
o. 32
-(-o. 10
o. 45
2
1 879-' 84
+ 32
o. 31
4
o. 60
0.35
o. 72
2
i88s-'9i
-f-38
o. 02
4
0.05
o. 20
o. 18
2
KONIGSBERG.
i8i
I O7
1 820-' 23
28
o. 14
2
O. 22
- 59
O. 47
I
. 24
-(-0.65
2
4~- 49
o. 60
-f-o. 24
I
i828-'3i
. 20
41.08
2
+o. 09
o. 64
o. 1 6
I
1 832-' 34
. 17
o. 72
2
o. 15
I. 72
o. 40
I
1 837-' 44
. 09
0.66
2
o. 62
2. 24
o. 87
I
OXFORD.
1 840-' 45
1 846-' 5 1
i86i-'66
.07
. 01
+ H
+0-79
+0.35
-j-o. 36
2
2
2
+0.42
4-0-40
o. 81
+0.67
4-0.89
4-o. 10
4-0. 22
4-0. 20
I. OI
O. 2
0. 2
O 2
1 867-' 7 2
+.20
o. 1 6
2
o. 24
-j-O. 2Q
O. 44
O 2
1873-76
i88o-'83
+ .25
+ 32
0.38
-43
2
2
0-33
-f-O. 12
+0.29
o. 17
o 53
o 08
0. 2
O. 2
i88 4 -'87
-f-36
o. 24
2
-fo. 23
o. 19
4-o. 07
O. 2
OBSERVATIONS OF THE SUN. [16, 17
Results of observations of the Sun's Declination Continued.
PULKOWA.
Years.
61"
W 6 e
\\
W
1842^45 .06 -fO. 82 2 0.35 0.01 0.35 I
.02 o. 10 2 0.48 +0.07 0.48 .1
i86i-'65 +.13 0.53 2 0.48 0.30 0.48
i866-'7o -f. 18 +0.27 2 0.31 0.38 0.31
DORPAT.
i823-'28 .24 +0.99 2 1.26 +0.59 1.41 i
i829-'32 .19 -f- 99 2 o. 7 6 -f I -34 0.91 I
l8 33~'3 8 .14 4- 1 - 00 2 0.63 -fi-34 0.78 i
CAPE OF GOOD HOPE.
i884-'87 +.36 0.51 4 +0.05 +o. ii 0.07 2
i888-'9O -}- 39 0.84 4 -{-0.09 -f - 1 9 0.21 2
. STRASBURG.
i884-'88 +.36 0.57 4 0.05 0.77 4-0.12 2
LEIDEN.
i864-'69 -)-. 17 +0.14 4 o. 01 -(-0.27 0.24 2
i87o-'76 -(-.23 0.23 4 0.06 0.04 0.29 2
Correction to the Sun's absolute longitude
17. So far as mere instrumental measurement is concerned,
the correction d s should be determined with greater precision
than dl" in the ratio 5:2, because the errors in decimation
have to be divided by the factor sin s = 0.40, in order to form
dl". Allowing for this large increase in the source of error,
the values of 6 1" are more accordant than those of 6 8. This
is what we should expect. The values of the former quantity
depend mainly upon the comparison of observations made
17, 18] OBLIQUITY OF ECLIPTIC. 33
near the opposite equinoxes, when the snn has the same decli-
nation, and when the season is not greatly different. Indeed,
if the season changed exactly with the sun's declination, all
effects of annual change of temperature would be completely
eliminated from 61", as would also in any case any constant
error which is a function simply of the Sun's Declination. It
is therefore to be expected that the actual probable error of
this quantity will conform more nearly to that determined from
the residuals than in the case of the other.
For these reasons the value of dl" does not give rise to
much discussion. The general result from all the observa-
tories is, for dl", when developed in the form x -f- y T.
x = + 0".05
y = 0".97.
Obliquity of the ecliptic.
18. The determination of the obliquity rests upon an essen-
tially different basis from that of the absolute longitude, in
that it depends upon actual differences of measured Declina-
tions, which differences are still further complicated by the
fact that they are necessarily made at opposite seasons. A
more detailed discussion of them is therefore necessary, and
some modification may have to be made in the separate results
as adopted. The following special circumstances affecting the
observations are to be taken into consideration :
The BRADLEY Greenwich results for 1753-^2, are derived
from a manuscript communicated by Dr. AUWERS, containing
the results of his very careful reduction of BRADLEY'S ob-
served Declinations of the Sun, which were compared with
HANSEN'S tables. The corrections were reduced to those of
LEVERRIER'S tables by being computed at intervals suffi-
ciently short to permit of the reduction being interpolated with
all necessary precision. No reduction was applied either on
account of the constant error of the Declinations determined
by Dr. AUWERS himself, nor for reduction to the Boss system
of standard Declinations. Hence arises the large value of Ad
given by these Declinations. Consequently the value of df is
5690 N ALM 3
34 OBSERVATIONS OF THE SUN. [18
that given immediately by the instrument, on the system of
reduction adopted by Dr. AUWERS, in which I have supposed
that the Pulkowa refractions were used.
From 17G5 to 1816 the Greenwich observations were made
with the imperfect quadrant, the Declinations of which are
subjected to an error which is not constant. The neces-
sary corrections are derived by S AFFORD in Vol. n of the
Astronomical Papers. The corrections are those necessary to
reduce to Boss's system, and they vary with the Declination.
Hence the arc on which the obliquity depends is not that
measured with the instrument itself, but that so corrected as
to reproduce as nearly as may be the standard Declinations.
From 1812 onward the two mural circles were used. Up to
1830 no correction except the constant one derived by SAF-
FORD was applied to the Declinations as measured with these
instruments. Hence the arc of obliquity is that measured
with the instrument itself without being corrected by the
standard stars.
After 1830 the Declinations were corrected by the tables for
Greenwich given in Boss's paper. These corrections vary
somewhat with the Declination, and they are different also
for different periods. Hence we have here a period during
which the instrumental differences of Declination were cor-
rected to reduce them to the standard star- system.
If the standard system were subject to no farther error than
a constant one, common to all Declinations within the zodiac,
which common correction would be subject to a uniform change
with the time, this system would doubtless be the best one to
adopt in order to obtain the secular variation in the obliquity
of the ecliptic. But, as a matter of fact, the standard Decli-
nations are simply the mean results of Declinations measured
with different instruments. It is, therefore, a question whether
we shall get any better results by applying reductions to a
standard system than we should get by simply taking the
mean of the instrumental results, because the system is itself
only a mean of such results. It is true that the standard sys-
tem depends on more instruments than the obliquity, though
not on better ones; but it is also to be considered that the
reductions in the case of the Sun may be different from those
18, 19] OBLIQUITY OF ECLIPTIC. 35
in the case of the stars, owing to the very different conditions
in which the observations are made.
Another troublesome point arises from the refraction used
in the reductions. The effect of refraction is always to make
the measured obliquity less than the actual one; the correc-
tion to the obliquity on account of refraction is therefore a
positive quantity, which is a minimum for an observatory at
the equator and increase equally towards each pole. Some
values of the obliquity were derived from BESSEL'S refractions
of the Tabulae Regiomontance, and others from the Pulkowa
tables. Since the secular variation of the obliquity is more
important than the absolute value of the quantity, it is essen-
tial that the standard to which all determinations of the ob-
liquity are reduced should be as nearly as possible the same,
and therefore that the same refraction should be used. But in
reductions to standard star places we meet with the addi-
tional complication that the differences in the constant of
refraction might be wholly or partially eliminated by the
reductions to a standard system. It would therefore be a dif-
ficult question how far we should modify the values of 6s on
account of the use of different tables of refraction.
To avoid all these difficulties I have judged it best to make
the obliquity depend mainly upon absolute measures, the
reductions being made with the Pulkowa refractions.
Effect of refraction on the obliquity.
19. The determination of the average or most probable effect
on the obliquity produced by using the Pulkowa refractions,
instead of those of the Tabulce Regiomontanw, is easily deter-
mined. We divide the ecliptic into a number of equal arcs
throughout the year, and by equations of condition express
differences of refraction in terms of differences of Declination,
and hence differences of obliquity. We thus find that at
certain latitudes where observations were made, and where
BESSEL'S refractions were used in the reduction, the follow-
ing corrections are necessary to reduce the obliquity to the
ones given by the Pulkowa refractions:
Pulkowa; y = 59.S; Jf = 0".325
Greenwich; <p = 51.o; z/f = 0".20
Washington; q> = 3S.9; 4e = - 0".125
36 OBSERVATIONS OF THE SUN. [19
Hence I conclude that for .
Dorpat; As 0".29
Konigsberg; Js = /7 .26
Cambridge; Je = - 0".21
Cape Town; At = - 0".12
The corrections to the obliquity thus derived, depending
mainly on direct instrumental measurement, and reduced to the
Pulkowa refractions, are designated as 6'f . The results for this
quantity are given in the last column of the several tables.
In the case of BRADLEY'S Greenwich results, I have taken
as 6'e Dr. AUWERS'S results unchanged, assuming in the
absence of any specific statement that he has used the Pul-
towa refraction tables.
In the case of MASKYLENE'S observations, I have, by excep-
tion, used them as reduced to the standard star-system,
because we have no other results at these times, and the en or
of his instrument is so strongly shown that it would not do to
use the results unchanged. It will be seen, however, that
small weights are assigned, and that the weights diminish
towards the end of the 'series.
In the case of the Greenwich observations from 1812 to
about 1834, no change has to be made, as the results are gen-
erally or always purely instrumental, and Pulkowa refractions
are used in SAFFORD'S work.
From 1835 onward I have depended mainly on certain cor-
rected Greenwich reductions. First, for tf 7 , I have used the
results given by Mr. CHRISTIE in his very valuable paper on
the Greenwich Declinations, in M. E. A. S., Vol. XLV, where
the Declinations from 1836 to 1879 are reduced on a uniform
system. Later, I have adopted the corrected results given in
Appendix III to the Greenwich observations for 1887. In
each case the result has been reduced to the Pulkowa refrac-
tions.
The Paris results rest on a different basis from the others,
in that the zero point of the instrument depends wholly upon
LEVERRIER'S Declinations of the stars, and I fear it was not
always accurately determined. Observations near the winter
solstice are mostly referred to one set of stars; those near the
19J OBLIQUITY OF ECLIPTIC. 37
summer to another set, the error of which may be systemat-
ically different. Certain it is that the results during the early
years were very discordant. The weights as given in the table
are those assigned a priori, without sufficient reference to the
discordance of the older results. I have felt constrained to
evade a decision as to their treatment by entirely omitting
their results in the final discussion.
Iii the case of some other observatories it was difficult to
determine exactly what refractions had been used in each
special case and what reductions should be made. I have, how-
ever, determined the corrections in the best way I was able.
A precise determination of the secular change in the ob-
liquity is of more importance for our present object than a
precise determination of its amount. Hence a series of obser-
vations extending through a long period of time, and made on
a uniform system, has an advantage over a number of isolated
values, in that any constant error with which it may be
affected will be eliminated from the secular variation. Possi-
ble constant differences between the determinations of the
various observatories at different epochs will vitiate the sec-
ular variation, but the probable amount of this error may be
diminished by using a number of separate determinations,
such as are presented in the preceding table. In the Green-
wich transit circle we have a very uniform series, extending
over a period of forty years, but giving results systematically
different from other determinations. This series gives for the
correction to the obliquity :
Transit Circle, 1847->91 :
d'e = - 0".ll i 0".06 + (0".21 i 0".46) T . . . (a)
Here, in view of the uniformity of method and reduction,
we may regard the mean error of the centennial variation from
the discordance alone as a fair approximation to the probable
mean error. It will be seen that I have here included four
years (1847-'50) of the Mural Circle results.
Continuing the Greenwich series backward, the question
arises whether we can regard the results of the mural circle
from 1812 to 1850 as comparable with those of the transit circle.
38 OBSERVATIONS OF THE SUN. [19
There is certainly nothing in the table to indicate any system-
atic difference. From the combination of the two we have
M. C. andT. 0., 1812->50:
<J'6 = - 0".08 0".05 + (+ 0".14 i 0".23) T (1850) . . (b)
Here the mean error is naturally smaller than in the case of
the transit circle alone, but is now more subject to possible
systematic difference between the two instruments.
If we now go back to BRADLEY, we meet with the very diffi-
cult question, whether we should regard his results as best
comparable with the modern Greenwich observations, or with
modern observations in general. If we assume that the differ-
ence between the Greenwich and other modern results is due
to any cause which has remained unchanged since BRADLEY,
we should reach one conclusion; otherwise, we should reach
the other. The result of combining all Greenwich observa-
tions, with the weights as assigned, is
6'e = -0".ll + 0".50T (c)
In this combination I have used the weak results of MASKE-
LYNE, with the small weights assigned, although they depend
wholly upon the standard declinations of stars. In view of
the discordance between BRADLEY'S two results, this seems
the only admissible course.
Next in the length of time which they include come the Paris
observations, of which the results, with the. weights assigned,
are
6f= + 0".01 0".36T
I give this result in order that nothing may be omitted.
Undue weight has probably been assigned to the earlier
determinations; in any case the method of deriving it from
the original observations is so objectionable that no further
use is made of it. A satisfactory discussion of the observa-
tions would require a complete redetermination of the zero
points of the instrument from fundamental stars.
19, 20] DISCUSSION OF RESULTS OF OBLIQUITY. 39
If we omit the Greenwich, Paris, and Palermo results, and
combine all the others into a single set of equations of condi-
tion, we have the equations arid results :
36.9# + 0.26 y = - 14".37
0.26 + 1.88 = + 1".01
x = - 0".39
y=+ 0".59
Here x is the value of 6'e for 1860, and y its centennial varia-
tion. Transferring the epoch to 1850, as usual, the result is
d'e = - 0".45 + 0".59 T ..... (d)
No reliable mean error can be computed, owing to systematic
errors. In view of these, one mode of treatment would be to
form equations of condition in which a possible systematic
error at each observatory would appear as one of the unknown
quantities. By this . process we should get the same result
for the secular variation as if we made an independent determi-
nation from the work of each observatory. At most of the
observatories the period through which the observations are
made, with one instrument and on an unchanged plan, is too
short to render such a course advisable.
As a last combination, we shall combine the earlier Green-
wich results, up to 1810, with Palermo and with all the modern
results except Paris, first dividing the weights of the Green-
wich results by 2. We then have the equations
39.8 a? -1.82 y = - 17 ".12
- 1.8 + 3.47 = + 2".99
x = 0".40
y=+0".65 ....... ()
Concluded results for the obliquity.
20. The data on which these various results for the obliquity
rest show the following noteworthy features :
(1) That the correction given by the modern Greenwich
instruments, mural and transit circles, is markedly greater
40 OBSERVATIONS OF THE SUN. [20
than that given by other modern observations. This may be
most plausibly attributed to the atmospheric conditions
within the observing room.
(2) The minuteness of the change of the correction given
by these instruments during nearly eighty years. To this
circumstance is due the smallness of the centennial variation,
0".50, found from the totality of the Greenwich observations.
A comparison of BRADLEY with the mean of the T. C. results
only would have given a change of 0".97 in 117 years, or a
centennial change of about 0".80.
The long period, uniformity of plan, and systematic devia-
tion of the modern Greenwich observations lead me to consider
them as forming a series distinct from all others. We have
therefore the following two completely independent determi-
nations of the centennial variation :
(1) Modern Greenwich results: y = + 0".14 i 0".23
(2) All other results + 0".6o
To the latter no reliable mean error can be assigned. To
judge its reliability we may compare it with the results (), (c),
and (d)
Greenwich T. C., alone, + 0".21 0".46
Greenwich observations in general, -f- 0".50
Miscellaneous modern observations, + 0".59
We may, it would seem, fairly give double weight to the
result (2), thus obtaining, as the definite result from observa-
tions of the Sun alone:
Correction to LEVERRIER'S centennial variation of the obliq-
uity of the ecliptic (- 47".594)
+ 0".48 0".30
the mean error being an estimate from the general discordance
of the data.
For the constant part of the correction I take
tie (1850) = - 0".30
21] SUMMARY OF RESULTS. 41
Summary and comparison of results.
21. From what precedes we have the following as the values
of the unknown quantities, and of their secular variations, as
given by observations of the Sun alone.
de" =
Value for
1850.
+ 0".10 d
- 7/ .03
Cent,
var.
+ 0".23
0^.10
e"(dn"+a) =
0".00 J
r /7 .07
+ C^.33 i
;/ .12
dl"+a =
- 0".02
- G^.63
dl" =
+ 0^.05 J
- 0".12
- /7 .97 i
0^.23
ds =
- /7 .30 =
t 7/ .15
+ 77 .48 i
: 0".30
a =
-0".07
4- /7 .34
No estimate of the probable errors of these quantities would
be useful which did not take account of the systematic dif-
ferences between the results of different observatories. We
have therefore formed the mean outstanding residual correc-
tions given by the several observatories, as shown in the
tables which follow. Originally the scale of weights used for
the Greenwich observations did not correspond to that for the
other observatories; they were, therefore, divided by 2. As
used below, however, the change has been made in the case
of dl" by multiplying all the weights of the other observatories
by 2, and, in the case of 6s, by dividing the Greenwich weights
by 2.
The correction to the obliquity depends solely on 6'e ; but
the comparison has also been made with the values of <?,
which, it will be remarked, differ from the others in that
account is taken of the supposed variation of the systematic
correction with the declination. It is noteworthy that the
results are somewhat more accordant when this correction is
omitted and purely instrumental errors are used for the
obliquity.
The mean errors given in the preceding summary of results
are derived from the discordances in question, and may be
regarded as substantially real.
No use was made of the Paris results for 61" and ds for
the reason that they depend on decimations referred to star
42 OBSERVATIONS OF THE SUN. [21
places which may be affected by differences in different Eight
Ascensions. They are, however, retained in the table to show
the amounts of outstanding discordance.
Outstanding mean residual corrections to quantities depending
on the Sun's Right Ascension.
Greenwich +
0".09 - 0".03
54.5
Paris
0".09 + 0".17
17
Cambridge 4-
7/ .02 0".00
16
Washington
0".05 0".12
24
Konigsberg
/7 .08 4- /7 .08
12
Oxford . 4-
0".06 4- 0".02
8
Pulkowa
/7 .15 4- 77 .22
6
Dorpat
/7 .10 - /7 .03
4
Cape
7/ .16 7/ .ll
4
Strassburg 4-
77 .05 (V.03
3
Mean errors for
weight unity t \ =
/7 .34 /7 .39
Mean error of x i
/7 .03 i /7 .03
Mean error of y
/7 .10 /7 .12
Outstanding mean residual corrections to quantities
depending
on the Sunh
? Declination.
SI"
w de w
d'K
Greenwich - 7/ .06
64 4- 77 .31 29.6
4-0".! 7
Paris 4- /7 .45
4- 7/ .31
Palermo - 77 .39
- /7 .20 0.8
/7 .20
Cambridge - 0".05
8 4- 77 .35 4
4- /7 .14
Washington 4- /7 .07
24 - 7/ .22 12
- /7 .29
Konigsberg - 77 .20
10 + /7 .31 5.5
/7 .00
Oxford 4- /7 .14
14 + /7 .19 1.4
- 0".01
Pulkowa + /7 .12
8 - 77 .13 4
- /7 .13
Dorpat 4- 7/ .75
6 77 .49 3
- /7 .64
Cape - 7/ .35
8 4- /7 .10 4
- /7 .02
Leiden + /7 .10
8 4- 77 .17 2
- /7 .06 .
Strassburg - /7 .26
4 + /7 .08 4
4- /7 .25
for weight unity /7 .81
7/ .74
/7 .60
CHAPTEE III.
RESULTS OF OBSERVATIONS OF MERCURY, VENUS, AND
MARS.
Elements adopted for correction.
22. We first give an outline of the method of expressing the
observed corrections to the Eight Ascensions and Declinations
of each of the planets as linear functions of the corrections to
the tabular elements. This linear function forms the first
member of the equation of condition in its original form, and
the observed correction forms its second member.
Let us put
E, r, the radii vectores of the Earth and planet 5
L, the Sun's true longitude;
J, the inclination of the orbit of the planet to a plane
passing through the Sun's center parallel to the
plane of the Earth's equator;
N, the Eight Ascension of the ascending node of the
orbit on this plane;
U, the argument of heliocentric declination of the planet
or its angular heliocentric distance from the node
on the equator;
a, 6, the geocentric Eight Ascension and Declination of
the planet.
, the obliquity of the ecliptic;
We shall then have
a =/(r. E. L. J. X. U., *.) . . . . . . . .V (a)
For the correction to the tabular Eight Ascension arising
from symbolic corrections to these seven quantities, we have
the equation
Sa = A 63 + * tfN + %L SU + % Sr + * <fe
dJ dN du dr de
43
44 MERCURY, VENUS, AND MARS. [22
with a similar equation for the declination, formed from this by
writing <5 for a.
The relations by which these two equations are derived, as
well as the expressions for the differential coefficients they
contain, are given very fully in A. P., Yol. II, Part I, to which
reference may be made. The corrections tfN and tfU are not,
however, the most convenient ones to choose. It will be found
in the paper alluded to that they have been transformed by
measuring the longitude in orbit of the planet and that of the
perihelion from an arbitrary point in the orbit. As to this very
convenient device in celestial mechanics, it is to be remarked
that the "departure point" always disappears from the final
equations which determine the position of the planet. We
may, in fact, make abstraction of it by considering that its
introduction is equivalent to the following simple linear trans-
formations.
We put
w, the distance from the node to the perihelion ;
/, the true anomaly ;
g, the mean anomaly.
TT, the longitude of the perihelion ;
I, the mean longitude of the planet;
v, its true longitude;
these longitudes being counted from the departure point.
Then, we have the relations
#U = tfw 4- df --= 6v cos
6w = dn - cos JtfN (2)
61 = dn +dg
Hence,
dn = tfU -f cos JtfN df
(3)
The elements finally adopted for correction by the equations
of condition were
I. TT. e. J. N.
22, 23] ELEMENTS ADOPTED FOR CORRECTION. 45
The value of a, the mean distance, is known with such pre-
cision that its correction need not enter into the equations of
condition. The latter are formed by substituting in (1)
n+-e + - cos
dgj de dg (4)
dr x . dr ~, dr ^
dr = -=- de + -=- 61 =- o n
de dg dg
The coefficients of each equation of condition from the Eight
Ascension thus become
Coefficient of $J . . .
dS
XT da da
... ^-cosJ^-
de . . . *
(5>
u u rf i <* ^/ _. ^ ^ r
ro ^"^ ^ ^
da .d \dadr
u it se dtx d f 4- da dr
Wde + W fo
In the second members of the equations a is regarded as
a function of the seven quantities (a), as is also #, for which
a similar equation is to be formed.
The corrections of the solar eccentricity, perihelion, and
mean longitude were also introduced by putting in (1)
tfL = dl" + de" + dn"
de" dn" (6)
^R = *L de" + ^ dn"
de" dn"
Introduction of the masses of Venus and Mercury.
23. The correction to the mass of Venus was introduced
by taking the tabular perturbation produced by Venus on
the geocentric place of the planet at the mean date of each
equation as the coefficient of the unknown quantity to be
determined. In computing these perturbations regard was
46 MERCURY, VENUS, AND MARS. [23, 24
had to the action of Venus on the Earth as well as ou the
planet. On this system the unknown quantity finally found
would be the factor by which the adopted mass of the planet
must be multiplied in order to give the correction of that mass.
It has already been remarked that the mass of a planet can
not be determined free from systematic error by observations
made upon the planet itself. Hence, the mass of Venus can
be determined only from observations of Mercury and Mars,
and that of Mercury only from observations of Venus and
Mars. But the mass of Mercury is so minute that it would be
useless to attempt to determine it from observations either of
the Sun or Mars. It was therefore determined solely from the
periodic perturbations of Venus.
It has happened that the mass of Venus could not be deter-
mined in a reliable way from observations of Mars, owing to
a defect in the theory of the latter planet, which I shall men-
tion hereafter, and have not yet had time to correct. Practi-
cally, therefore, the mass of Venus is determined only from
observations of the Sun and of Mercury, and that of Mercury
from observations of Venus.
Correction of equinox and equator.
24. t Could all the observations be directly referred to a
visible equinox and equator, the corrections above enumerated
would have been the only ones which it was necessary to
include in the equations of condition. But, as a matter of
fact, the observations were all referred to an assumed system
of Right Ascensions and Decimations of standard stars my
own system in Eight Ascension and Boss's in Declination.
We must therefore introduce two additional unknowns into
the equations, which I have represented in the following way:
<*, the common error of the adopted Right Ascensions.
#, the common error of Boss's Declinations.
The first quantity will appear only in the equations derived
'from observed Right Ascensions and the second only in the
equations derived from Declinations, the coefficient being unity
in each case.
24] CORRECTION OF EQUINOX AND EQUATOR. 47
That the value of 6 found in this way should be regarded
as a correction to the Declinations of the equatorial stars will
appear by the following considerations. The mean heliocen-
tric orbit of a planet as projected on the celestial sphere is
undoubtedly a great circle. On the other hand, in view of the
systematic discordance always found to exist in measures of
absolute Declinations near the equator, and of the fact that
these absolute Declinations depend upon assumed constants
and laws of refraction, which are necessarily affected with
greater or less uncertainty, and are otherwise subject to
systematic errors, instrumental or personal, of an obscure
character, but strongly shown by a comparison ot.the Declina-
tions derived from the work of different observatories, it can
not be assumed that these Declinations are free from sys-
tematic error. JSow, m one circle ot Decimation, say the
equator, we may expect that the error will be nearly constant
around the sphere, since the causes of error will generally be
nearly constant for any one Declination. This conclusion is
confirmed by a comparison of the best star catalogues.
Moreover, between the zodiacal limits, the error in each par-
ticular case is not likely to differ very greatly from the error
at the equator. Even if the difference should be considerable
the various values of the error of the different Decimations
must have a certain mean value, so that in the case of each
particular star, or each region of the heavens, we may conceive
the actual error to be divided into two parts one the mean
value in question, and the other the deviation from this mean.
The latter is probably smaller than the former, and in any
case can not very well be determined from observations of the
planets. But the condition that the planet moves on a great
circle of the sphere admits of the mean value being deter-
mined with great precision. It should, therefore, be included
in the equations of condition.
The value of <*, the common error of all the Eight Ascen-
sions, can obviously not be determined from the equations in
.Eight Ascension alone, because the only result that such
observations can give us would be the values of the Eight
Ascensions referred to some assumed equinox. The coefficient
of a would therefore completely disappear from the equations
48 ' MERCURY, VENUS, AND MARS. [24
of condition in Eight Ascension. But since the same unknown
quantities are introduced into the equations of condition in
Eight Ascension and in Declination, the requirement that the
two sets of equations shall give common values of these
quantities does away with this indetermiriatiou and enables
determinate values to be found. In fact, this method does not
differ in principle from that usually adopted, in deriving the
Eight Ascensions of stars from observations of the Sun. The
latter consists in deriving the Sun's absolute longitude from
observations of its Declination and absolute Eight Ascensions
of the stars by comparing them with the Sun. In the same
way we may. consider that, in observations of the planet, the
Sun's absolute longitude is derived from observations of Decli-
nations of the planet, and then a comes out from the observa-
tions in Eight Ascension.
I have deemed it absolutely necessary that all the equations
of condition should be solved by the method of least squares.
By this method alone can the results of the observations as
regards separate values of the elements and constants be prop-
erly brought out. But the work of constructing and solving
a system of nine thousand equations of condition, each involv-
ing twenty unknown quantities, would be extremely laborious,
and might even require a century for its completion, if done in
the usual way. It was therefore necessary to adopt every
device by which the labor could be reduced to a minimum.
One device was the dropping of all superfluous decimals in the
coefficients of the equations. Since tbe errors thus produced
would be purely accidental, it follows that if the sum of the
products obtained by multiplying the value of each unknown
quantity by the error of its coefficient in the equation of con-
dition is but a small fraction of the necessary probable error
of the absolute term, no serious harm will result from the
errors of the coefficients.
Another device was the construction of tables for finding
the coefficients. Such tables relating to Mercury and Venus
are found in Vol. II, Part 1, of the Astronomical Papers.
These tables are, however, only given for one mean anomaly in
each case, and therefore require computations dependent on
the value of the other anomaly. They were therefore extended
24, 25] INTRODUCTION OF SECULAR VARIATIONS. 49
to tables of double entry, so that the value of the derivatives
of the geocentric Eight Ascension or Declination at any epoch
could be taken from the tables at sight. The arguments were
the mean anomaly of the planet and the day of the year at
which the planet last passed through its perihelion.
Introduction of the secular variations.
25. When the equations of condition are formed on the plan
just set forth, the unknown quantities will be the corrections
to the elements or to the mean longitude at the date of each
equation. But every one of the unknown quantities which
have been enumerated, the correction of the masses excepted^
is subject to a secular variation. Hence, instead of the
unknown quantities heretofore denned, we introduce two
others, the one the value of this unknown at some assumed
mean epoch, which, for reasons already set forth, must first
be determined from the observations; the other the secular
variation in a unit of time. The unknown quantities which
have been enumerated make twelve for each equation of con-
dition. Eleven of these are subject to a secular variation, so
that if the secular variations were introduced into the original
equations of condition they would each have twenty-three
unknown quantities.
The following device was employed to reduce to a minimum
the work of introducing and determining the secular variations
of the various elements :
Firstly, the whole time covered by the observations was
divided into periods, never exceeding ten years, except when
the observations were very few in number, or entitled to but
small weight. It was then assumed that no error would arise
from supposing the value of the unknown quantity to be the
same throughout the period as it was at the mid-epoch of the
period. The maximum absolute error thus arising would be
the secular variation during half the length of the period, and
the mean error the secular variation during one-fourth of the
period; but actually the effect of even this error would be
almost entirely nullified by the combination of positive and
negative coefficients throughout each period.
5690 N ALM 4
50 MERCURY, VENUS, AND MARS. [25
Let us now put
x,y,
the corrections to the elements at any epoch, T.
Let
a x+ b y + cz -f . . . ^=n
be an equation of condition between these quantities at this
epoch. From a system of such equations, extending through a
period numbered i, during which #, #, etc., may be considered
as constant, we derive normal equations of the form
[aa] t x+[db] t y + . . . = [an],
which I shall call partial normal equations, and which we
might solve so as to obtain the values of x, y, etc. This solu-
tion is not, however, necessary. The values of the unknown
quantities being really of the general form
we may imagine these values substituted in the normal equa-
tions (1), the value T, of t for the mean epoch of the period
being substituted for t.
Let us now suppose that we introduce the quantities # , 2/o, >
#', y', . . into the original equations of condition, using for t
the value r tj which pertains to the mean epoch of the period.
Our equation of condition will thus become
ax Q + fy/o 4- + ar t x f + br t y' + . . = n (3)
If from a system of conditional equations of this form we
form the normal equations for all the unknown quantities, the
results will be these :
Partial normal equation in x ;
[aa] f # -f- [db] t y + . . + rJaaJX + r^ab^y 1 + . . = [aw], (4)
25] INTRODUCTION OF SECULAR VARIATIONS. 51
Partial normal equation in x 1 j
T,[aa],a?o + T,[a&],y + . . + T 2 [aa],#' + rf [ab^y 1
+ . . = r t [an] t (5)
We conclude that the partial normal equations, when the full
number of unknown quantities is included, may be derived
from those of the form (1) by the following rules.
(1) Each partial normal equation in X Q , y , . . . is formed
from that in #, y, etc., by adjoining to the first member of the
equation the member itself multiplied by r and then changing
x, y, . . .to x j XQ'J and, in the products by r, changing
x, y, . . . into a?', y', . . .
(2) The partial normal equation in a?', y', . . . is formed
from the partial equation in x 0j y^ . . . by multiplying all
the terms throughout by the factor r.
The final or complete normal equations in all the unknown
quantities being formed by the addition of the partial normals,
the formulae for the coefficients are as follow :
For the final equation in X Q
[aa] = [aa]! + [aa], + . . . + [aa] n
[ab] = [a&]i+ [a&] 2 + . . . + [ab] n
[aa]' = n [aa]i + r 2 [aa] 2 + . . . + r n [aa],
[an] = [an]^ [an] 2 + . . . + [aw],
For the final equation in x'
[aa]" = r, a [aa]! + r, 2 [oa] 2 + . . . +T n [aa] n
. . . +r n *[ab] n
[an]" = n [an]i + T 2 [an] 2 + . . . + r n [an] n
The final equations for all the unknown quantities will then
be of the form
[oa] x + [aft] y + . . + [aa] 1 x' + . . . = [an]
... (8)
[aa]'xo+[ab] / y () + . . . +[aa]"0 / + . . . = [an]"
52 MERCURY, VENUS, AND MARS. [25,26
The epoch from which we count the time, r, is arbitrary.
An obvious advantage will be gained in counting it from the
mid- epoch of all the observations. Then we shall have, by
putting w^ w 2 , etc., for the sum of the weights for the different
periods :
MI r\ + w 2 r 2 + + w n r n = (9)
If the observations are then equally distributed around the
orbits of the planet and of the Earth it may be expected that
the coefficients
[.]', [aft]' .... (10)
will all nearly or quite vanish. Practically we may expect that
as observations are continued through successive revolutions
the ratios of these to the other coefficients will approach zero
as a limit. We may then divide the normal equations into two
sets, one containing the quantities x , y^ etc., and the other
#', y', etc. The coefficients (10) being small, the two sets of
normals will be nearly independent, and we may omit the
terms (10) in the first approximation, and introduce them in
one or two successive approximations so far as necessary.
The unit of time is also arbitrary. A certain advantage in
symmetry will be gained by so choosing it that the mean value
of T 3 shall not differ greatly from unity. It was found that
twenty-five years was a sufficiently near approximation to be
adopted for all three planets.
Dates and weights for epochs and periods.
26. As want of space makes impracticable the present publi-
cation of the great mass of material worked up, the following
particulars have been selected as those most likely to be use-
ful in judging and criticising the work. We give three tables,
showing the division of the dates of observation into periods,
and the weights for each period. The first column of each
table contains the number or designation of the period, as
found in the manuscript books. The second contains the
mean year of the period. The third column shows the time
26] DATES AND WEIGHTS FOR EPOCHS AND PERIODS. 53
of this mean period from the mid-epoch of the observations,
which is taken as follows :
For Mercury, 1865.0
Venus,
The next column contains the sum of the weights of the
equations in each period, as used 'in forming the normal equa-
tions. These were not, however, the weights actually used
in multiplying the coefficients of the equations of condition.
Owing to the diversity in the quality of the observations at
different times it was not found convenient to reduce the
equations at once to a uniform system of weights, and so dif-
ferent units of weight were selected for the older observations
and for the earlier observations. After the partial normal
equations were formed they were multiplied by the factor F,
necessary to reduce them to a standard in which the unit of
weight should correspond to the mean error
The sums of the weights reduced by these factors are shown
in the table.
In arranging the weights and selecting the factors it should
be remarked that a liberal allowance was made at each step
for probable constant errors, which results in the given
weights being much smaller than they would have been by
the theoretical treatment of the original observations. Not-
withstanding this allowance the final result seems to show
that it was still insufficient, and that the actual weights of
the results are less than would follow even from the final ones
as given. .
The partial normal equations for each period after being'
multiplied by the factors F, are added to form the final normal
equations as derived from meridian observations.
54 MERCURY, VENUS, AND MARS. [26
WeightSj epochs, and periods of partial normal equations.
MERCURY.
1
1
Right Ascension.
Declination.
Mean
year.
T Wt
(units of 257.)
F -
Mean
year.
T
(units of 25 y.
Wt.
F.
I
2
3
3i
3-2
4
5
5i
8-
6,
6,
8
9i
9-2
10,
I0 2
i
n 2
Ii 3
1766.60
1784. 22
1799.81
-3. 9360
-3.2312
2. 6076
26! I
*
1
1765-50
1782.99
-3. 9800
3. 2804
O. 2
4.9
1
f
1796.42
1802.37
1809. 1 8
1824. 83
2. 7432
-2.5052
2. 2328
I. 6068
5-0
39-9
52.8
74-1
I
I
1809. 53
2. 2188
18.9
i
1818. 79
1825. 80
1835-56
1.8484
1.5680
I. 1776
0.9
34.5
75-o
i
i
i
1833.84
1838. 26
1843. 97
1855-92
1862. 79
1867. 18
1872.64
1877.05
1882. 17
1886. 29
1889. 70
I . 2464
75-3
141.5
281.5
201. 5
189.5
294.5
214.0
204.5
171.5
338. o
176.0
1%
1
1
1
1
1
1
I 0606
1843. 74
1855.90
1863. 10
1867. 12
1872. 62
1877.12
1882. 24
1886.29
1889.82
o. 8504
o. 3640
o. 0760
+o. 0848
4-o. 3048
-j-o. 4848
4-o. 6896
-j-o. 8516
-j-o. 99*28
98.8
83-3
99-8
1 86. o
129.8
129.8
108.2
199.8
109.5
i
i
i
*
i
|
0.8412
o. 3632
o. 0884
-j-o. 0872
-l-o. 3056
-j-o. 4820
-fo. 6868
4-0.8516
-j-o. 9880
VENUS.
I
1755.83
4. 2868
"3
*
1759.69
4. 1324
7.0
i
2
1767.92
3. 8032
19.7
i
1770. 18
-3.7128
IO. O
i
3
1781.06
3- 2776
3-7
i
I793.25
2. 7900
13.5
i
4
1792.47
2.8212
12.3
i
1806. 73
2. 2508
65.5
*
5
1802. 64
2.4144
23-3
i
i8i5-59
1.8964
67.5
i
6
1810. 31
2. 1076
34-0
i
1823.75
1.5700
197.0
i
7
1816. 88
1.8448
42. 7
i
1836. 02
1.0792
762. o
8
1825.55
I.4 9 80
141.0
1844.08
o. 7568
650. o
9
1835.31
I. 1076
339-3
i
1854. 24
o. 354
333-o
10
1843.98
o. 7608
259.3
i
1861.43
o. 0628
749.0
li
1853-51
o. 3796
205.3
i
1868.06
-j-o. 2024
815.0
12
1861. 60
o. 0560
353-7
*
1875-32
-j-o. 4928
692.0
13
1868. 12
+o. 2048
466. o
1
1883. 15
4-o. 8060
819. o
i
H
1875- 38
+o. 4952
399-5
1
1888.56
4-1.0224
801.0
i
1C
1883. 09
-j-o. 8036
04. c
i
16
1888. 67
-j-I. 0268
D T^ J
520. 5
2
|
*/ *?
26,27] UNKNOWN QUANTITIES OF EQUATIONS. 55
Weights, epochs, and periods of partial normal equations.
MARS.
Right Ascension.
Declination.
T3
o
1
Mean
year.
T
(units of 25 y.}
Wt.
F.
Mean
year.
r
(units of 25 y.}
Wt.
F.
,
1757-43
3- 9428
25-3
i
1758.82
-3.8872
8.8
i
2
1770.55
3- 4i8o
II.
*
11773-79
-3- 2884
8.8
3
1787.82
-2.7272
10.
*
1794.48
2. 4608
13.0
i
4
1799.77
2. 2492
20.7
1
1804. 91
-.-2. 0436
47.0
i
5
1811.32
-1.7872
14.7
*
! 1813.00
I. 7200
30.5
1
6
1829. 17
1.0732
60. o
i
1828.04
I. Il84
93-o
7
1837- 39
o. 7444
121. O
i
1837. 18
o. 7528
371.0
8
1845- 39
o. 4244
76.3
t
1844.95
o. 4420
255-0
9
1853-36
o. 1056
90. o
*
1853. 02
o. 1192
245.0
10
1861. 07
-j-o. 2028
114. o
i
1860. 94
+0. 1976
306.0
ii
1869. 20
+o. 5280
124. o
*
1868. 80
-f o. 5120
197.0
12
1877.71
-j-o. 8684
132. o
i
1877. 38
+o. 8552
257.0
J 3
1883. 27
4-i. 0908
91. o
i
1883.26
+1.0904
1 60. o
14
1888. 85
+ 1.3140
115.5
i
1888. 48
+ 1.2992
167.0'
Unknown quantities of the equations.
27. For convenience in solving the equations of condition
the coefficients of the equations were multiplied by such
numerical factors as would reduce their general mean abso-
lute value to numbers of approximately the same order of
magnitude. Hence, the unknown quantities themselves are
not the corrections to the elements, but these corrections
divided by the adopted factors.
In the case of Mercury the absolute term was also multi-
plied by 10, so that effectively the factors in question were
reduced to one-tenth part of their value. The unknown
quantities of the equations are represented by the symbols
of the elements to which they relate inclosed in brackets.
For convenience of reference the following table is given,
showing the factors used in the case of each planet. In the
case of Mercury the column (a) shows the factors by which the
differential coefficients were actually multiplied; (b) the factor
by which the unknown quantity, as finally found, must be
56 MERCURY, VENUS, AND MARS. [27, 28
multiplied to obtain the correction as expressed in the last
column.. In the case of Venus and Mars these factors are the
same.
Factors by which the unknown quantities are to be multiplied to
obtain corrections of the elements.
Symbol of
unknown.
Mercury
Factor ior
Venus.
. Mars.
Corr. of
element.
w
(*)
[ ]
1
0.1
7
0.3
dm : m Q
( * ]
40
4
5
2
dl
1 JJ
30
3
6
2.5
dJ
[Nj
30
3
7
2.5
sin JdR
I ]
30
3
3
10^-7
de
f * ]
100
10
439
100-r7
d?r
1 * ]
100
2.056
3
1.3323
edn
1 * 1
10
1
4
4
de
["]
6
0.6
2.5
2
de"
["-]
6
0.6
2
2
e"dn"
I 1
10
1
1
5
Of
[ * ]
10
1
5
5
d
[I"]
10
1
4
3
dl"
The secular variation of each unknown in 25 years is
expressed sometimes by a suffixed 1, sometimes by an accent,
thus:
[1]' = [l]i = change of [I] in 25 years.
28. It may also be useful to give the values of the principal
coefficients in each of the normal equations. They are found
in the following table. Were the other coefficients all zero,
these numbers would indicate the weights of the different
unknown quantities as resulting from the solution. Several
of them were greatly diminished by the process of solution.
28, 29]
ORDER OF ELIMINATION.
57
Values of the principal diagonal coefficients in the normal
equations.
I
rtercury.
Venus.
Mars.
Symbol c
f
From
oefficien
:.
From mer.
observa-
From
transits.
Sum.
From mer.
observa-
From
transits.
Sum.
mer.
observa-
tions.
tions.
tions.
mm
5488
o
5488
^868
2929
8797
17887
11
I0 559
11308
21867
598i
3540
9521
20924
" JJ
15222
1296
16518
13232
7444
20676
28783
: NN ;
14176
2304
16480
I795I
1636
19587
32478
- ee '.
19015
5076
24091
5686
3350
9036
20119
7T 7T
8621
8352
16973
5290
1732
7022
20564
IIOOI
196
11197
11429
3598
15027
31460
'' e" e" ''
9757
508
10265
9586
665
10251
15909
'IT" TT"
9099
261
9360
5836
1895
7731
14911
' r // r //
^242
o
5242
' /"/" =
Or
13041
542
13583
11031
2349
I338o
15427
aa
13230
13230
335
o
335
25138
r 66 ''
24657
24657
15196
o
15196
53975
11 '
7014
67155
74169
6005
8983
14988
26689
JJ "
12366
9383
21749
9837
13014
22851
23440
; NN ;
"35
16682
27717
14724
2874
17598
29494
ee '
15437
29647
45084
5743
8610
'4353
24364
7T7T
6745
493 i 8
56063
4948
4483
943i
27131
ee
8488
1418
9906
8458
6306
14764
25675
: e " e ff "
8409
2937
11346
9805
1682
11487
22947
: TT" TT //=
8439
1513
9952
5242
4805
10047
17356
V'r'/ n
5432
o
5432
'. l " l " ".
11629
3126
I47S5
10677
5667
16344
20655
aa
11400
o
11400
297
o
297
33624
; 66 \
18716
o
18716
10772
o
10772
42405
NOTE. The coefficients for Mercury and Venus in this table are given as they
were used in the solution, after dropping the units from all the terms of the
equations, except those from transits of Mercury.
Order of elimination.
29. In dealing with so extensive a system of unknown
quantities it is impracticable to investigate the dependence of
each upon all the others. It is therefore essential to arrange
the unknowns in an order partly that of interdependence and
partly that of the liability of each to subsequent change by
discussion and adjustment. Hence, the mass of the planet.
Mercury or Venus, should be first eliminated, as being that
unknown which is least affected by changes in the final values
of the other unknowns. The secular variations, as derived
58 MERCURY, VENUS, AND MARS. [29, 30
from meridian observations, are nearly independent of the
corrections to the other elements. The solar elements are to
be subsequently determined by a combination of the results
of the observations of the Sun and of the three planets.
Guided by these considerations, the order of elimination
was, with some exceptions, as follows :
1. The mass of the disturbing planet.
2. The five elements of the observed planet.
3. The four elements of the Earth's orbit.
4. The corrections to the star-positions for the mid-epoch.
5. The secular variations of the eleven quantities (2), (3),
and (4), taken in the same order.
Treatment of meridian observations of Mercury.
30. In the case of Mercury the factors of the coefficients of
the equations were chosen large enough to admit of the deci-
mals being dropped from the products without prejudice to
the accuracy of the final result. This was done to facilitate
the formation of the normal equations. For the same reason
the factors were made so small that the absolute numerical
values of the coefficients should generally not exceed 13. As
this degree of precision is far short of that usually employed
for correcting the elements of a planet, it may be well to set
forth the considerations on which it is based.
Let any equation of condition as actually used be
ax-\- by + cz + . . . =n (a)
Let the coefficients a, &, etc., be affected by the mean errors
e, ', etc., so that the true equation should be
. . . = n
This true equation may be written in the form
ax + by + . . . = nx e'y . . . (b)
We may regard (b) as a rigorous equation, in which the error
of the second member is increased by the quantity
30] MERIDIAN OBSERVATIONS OF MERCURY. 5
and the only effect upon tlie precision of the results will be
that arising from this increased probable error. Let us esti-
mate its magnitude. From an examination of the tables used
in finding the coefficients I infer that the probable error of the
coefficient of n Avas 1, and that of all the other coefficients
0.6. The mean value of the unknown quantities was gener-
ally a small fraction of a second. We conclude, therefore,
that the probable or mean value of the error
ex fy i . . .
would in any case be only a small fraction of a second. More-
over, these errors would be purely accidental and not system-
atic, since the intervals of time between the equations were
generally so long that the coefficients for different equations
came from different tables, so that no error from omitted deci-
mals in any one equation would enter into the other equations.
Now, in view of the necessary systematic errors which affect
observations of the planets, there is no hope of approximating
to this degree of accuracy in the second members of the equa-
tions. Were the observations rigorously correct and the
values of the unknown quantities finally determined affected
by no error except that arising in this way, they would be
many times more accurate than we can hope to make them.
The errors might, in fact, be considered unimportant in the
present state of astronomy.
It has already been remarked that the scale of weights was
so taken that the unit of weight should correspond approx-
imately to a supposed mean error i 1".0 in the value of each
absolute term of an equation of condition, so far as the error
could be determined from the discordance of the original
observations. The corresponding probable error would be
dt 0".65. In the case of Mercury, however, modifications were
made which prevents this mean error from corresponding to
the unit of weight which would be found from the solutions in
the usual way. In the first place, the absolute members were
all multiplied by 10; in other words, the decimal point was
dropped from tenths of seconds, and no further account taken
of it. Secondly, in consequence of the probable error in the
coefficients of the normal equations arising from the irnperfec-
60 MERCURY, VENUS, AND MARS. [30
tions of tlie decimals, the final values of these coefficients
would be subject to probable errors ranging between 50 and
100 units. In consequence there would be no advantage in
retaining the last figure in the normal equations, and it was
dropped in all the subsequent solution and discussion of these
equations.
In dropping the last figure from the absolute term of the
normal equations we may consider that we are merely drop-
ping the tenths of seconds and that the units are once more
expressed in seconds. Thus, considering only the effect of
this operation, the unit of weight would correspond to a mean
error of 1.0 in units of the absolute term. But in dropping
off the last figure from the coefficients we practically reduce
the scale of weights, considered as multipliers of the equa-
tions, to one-tenth of their former value. On the other hand,
in expressing the unknown quantities in terms of the correc-
tions to the elements, we divide the multipliers by ten, so that
effectively we multiplied the coefficients in the equations of
condition, considering the unknown quantities to be defined
as on page 56, by 10. Since these coefficients are of the second
degree in the normal equations, it follows that the scale of
weights has in effect been increased ten fold. Hence the unit
of weight for the normal equations between the unknown
quantities as finally solved will correspond to the mean error
l = 1.0 X VI6 = 3.1
As the mean error is at best a rather indefinite quantity in a
case like the present, we may consider its value as 4 units and
even then as by no means rigorously determined.
Up to the time of writing no attempt has been made to
derive rigorously the weights of the unknown quantities from
the solution, because in the cases of most of the uukowns such
weights would be entirely illusory. The fact is that in solving
so immense a mass of equations, we must expect systematic
errors to vitiate many of the results. The observations of
Mercury, especially of its Eight Ascension, are not made on
.a uniform system 5 sometimes the limb is observed, sometimes
the apparent center or the center of light.
30, 31] TRANSITS OF MERCURY. 61
An ideally perfect system of reduction would require us to
reduce each separate observation with a semidiaineter corre-
sponding to the personal equation of the observer. This being
entirely impracticable, we must regard the reduction of the
observer's semidiameter to that used in the reductions as a
probable error. In fact, however, it will be of a systematic
character, varying at each point of the relative orbit of
Mercury, and going through a cycle of changes impossible to
determine in a synodic period of the planet. It is impracti-
cable to give even a full discussion of these errors; we shall,
however, meet with a proof of their magnitude.
Introduction of the equations derived from observed transits of
Mercury.
31. The relations between the elements of Mercury and the
Earth derived from this source are shown in my Discussion of
Transits of Mercury (A. P., Vol. I, Part VI.) On page 447 are
found expressions for those linear functions of the corrections
to the elements which are determined by the November and
May transits, respectively. With a slight change of notation
to correspond with that of the present paper, these functions
are as follows :
V = 1.487 61 - 0.487 dx - 1.137 tie - 1.01 dl" -f 1.19 e"dn"
+ 1.58 de"
W = 0.716 61 + 0.284 drt + 0.896 de - 0.97 61" - 1.11 e"dn"
- 1.62 de"
The values of V and W being derived from a series of transits
extending from 1677 to the present time, enable us to deter-
mine both these quantities at some epoch, and their secular
variations. The values derived from the transits, together
with their mean errors, are found on page 460 of the work in
question. Omitting the doubtful factor fc, introduced on
account of a possible variability of the Earth's axial rotation,
which was not proved by the transits, the values of V and W
were found to be as follows :
V = 0".90 i 0".31 + ( - 2 // .63 0';.59) (T - 1820)
W == + 0".84 0".25 -f (+ 1".84 i 0".60) (T - 1820)
62 MERCURY, VENUS, AND MARS. [31
The mean epoch for the transits is taken as 1820, to which
the zero values correspond. The values for 1865.0, the mid
epoch for the meridian observations, are, therefore, from the
transits alone
V = - 2".08 0".41
W = + 1".67 0".37
This, however, is only a first approximation to the quantities
which should be introduced. Since the meridian observations
help to determine the values of V and W, we should not
regard the reductions to 1865.0 as final, but retain the results
in the form (a).
Another element which is determined from the observed
transits of Mercury with greater precision than it can be from
meridian observations is the longitude of the node of the orbit
relatively to the Sun. In the paper quoted we have put
F = (30 -61"} sin i
and found from all the transits up to 1881,
N = - 0".16 i 0".27 + (0".28 dL 0".62) (T - 1820) (b)
The values of Y, W, and N, found from the discussion in
question, give rise to six conditional equations, which become
completely independent when we take as observed values the
secular motions and the absolute values at the mid-epoch of
observation. This mid-epoch is not the same for the May and
November transits. But I have assumed that no serious error
would be introduced by taking 1820.0 as the epoch for all three
of the quantities, Y, W, and X.
If we substitute for sin i 66 its value in terms of tfJ, etc.,
namely,
Sin idB=- 0.6018 J + 0.796 sin JtfN + 0.721 de (c)
and then for tf J, tfN, tff, their values in terms of the unknowns
of the equations of condition, we shall have
N = - 1.805 [J] + 2.394 [N] + 0.721 [*] - 0.122 [V] (d)
31] TRANSITS OF MERCURY. 63
Similar expressions will be found for the values of Y and W
by substituting for the corrections to the elements the unknown
quantities of the conditional equations, as already given.
Taking 1820.0 as the mid epoch, we may regard the inde-
pendent quantities given by the transits of Mercury to be the
six following ones :
Vo - 1.8 Y x ; W - 1.8 W,; N -
V, ; W, 5 N, -
Here Y , W , and N indicate values for 1865, the mid-epoch of
the meridian observations; and Y 1? W 1? and Nj. the variations
in 25 years. The six conditional equations thus found from
the transits may be written
Yo - 1.8 Yt = - 0".90 0".31
W - 1.8 W t = + 0".84 i 0".25
:N O - 1.8 N! = - 0".16 0".27
Y, = - 0".66 i 0".15
Wi = + 0".46 i 0".15
Ni = + /7 .07 /7 .15
Substituting for Y , YI, etc., their expressions as linear func-
tions of the unknowns of the conditional equations, we find
the following six equations, which are to be used as conditional
equations additional to those given by the meridian observa-
tions :
5.95 [1] - 4.87 [TT] - 3.41 [e] - 1.01 [l"\ + 0.71 [n"\ + 0.95 [e"]
-1.8)6.95[Z]! - 4.87 [ir]i 3.41 [e],- 1.01 [l"]i+ 0.71 [n"^
+ 0.95[e // ] 1 J = -O^O
Weight = 250
2.86 [1] + 2.84 [it] + 2.69 [e] - 0.97 [I 11 ] - 0.67 [n"\ - 0.97 [e"}
-1.8 {2.86 [I], + 2.84 [w-J! + 2.69 (e}, - 0.97 [l"^ - 0.67 [n"},
-C.97^ 7 '],} = + 7/ .84
Weight = 300
- 1.8 [J] + 2.4 [N] + 0.7 [f] - 0.12 [I"}
- 1.8 { - 1.8 [ J] x + 2.4 [NJi + 0.7 [f ]x - 0.12 [/ // ] l } = - 0".16
Weight = 400
64 MERCURY, VENUS, AND MARS. [31
5.95 [1], - 4.87 [TT]! - 3.41 [e], - 1.01 [l"^ + 0.71 [*"]i+ 0.95
= - 0".66
Weight = 700
2.86 [l]t + 2.84 [TT]! + 2.69 [e]i - 0.97 [Z"]i - 0.67 [TT"], - 0.97 [a' 7 ]!
= + 0".46
Weight = 700
-1.8 [J], + 2.4 [N] { + 0.7 [e], - 0.12 [I"}, = + 0".07
Weight = 1,600
The weights assigned to these several equations have been
determined by the following considerations:
We have already found that in the equations of condition
from the meridian observations as finally reduced, the scale of
weights has so come out as to show a practical mean error for
weight unity of about 4". Were this error purely accidental,
the weights of the conditional equations derived from the
transits would be determined in the same way, from the mean
errors assigned to them. But, as a matter of fact, the exist-
ence of systematic errors in the meridian observations is
shown, as will be subsequently explained, by the large value
found for the fictitious quantity 6r 2 . Since observations of
transits are made at the point of the relative orbits of Mercury
and the Earth, near which meridian observations are rarely
available, and are of a higher order of accuracy than meridian
observations, it follows from the theory of probabilities that
we should assign a larger relative weight to the observations
of the transits. How much larger does not admit of being
determined with numerical precision. Actually I have taken
the weights as if the mean error corresponding to weight
unity were between 5 and 6. In the case of the motion of the
node a still larger weight has been assigned to the secular
variation, from the belief that the accuracy of the determina-
tion from transits relative to meridian observations is in this
case of a yet higher order of magnitude than in the case of
31, 32] SOLUTION OF EQUATIONS FOR MERCURY. 65
the other elements. Whether this belief is justified or not
must be left to the decision of the future astronomer.
The first three of the preceding six conditional equations
may be treated in a way similar to that adopted for the
meridian observations. They express what is supposed to be
equivalent to observations of the three quantities V, W, and
N in 1820, when r 1.8. Hence, from the partial normals
in the six principal unknowns, [e], [>]... [>"], the com-
plete normals may be formed by multiplication by r and i*
(r = 1.8) in the way set forth in 25.
Solutions of the equations for Mercury.
32. In the case of Mercury and Venus, it is desirable to
know to what extent the results of the transits diverge from
those of the meridian observations. Hence, as already
remarked, two solutions of the equations were made, termed
A and B.
Solution A is that derived from the meridian observations
alone. Solution B is that of the normal equations formed
from both the meridian observations and the transits.
The results of the solutions in the case of Mercury are shown
in the following tables. The relation of the unknown quan-
tities given in the first columns, A and B, to the corrections
of the elements has been shown in a preceding section ( 27).
The upper half of the table shows the corrections to the
elements; the lower half those of the secular variations.
It will be seen that all the values, with a single exception,
come out less than a unit. In stating the corrections to the
elements, it must be remembered that, owing to the proximity
of Mercury to the Sun, the errors of geocentric place are much
less than those of the heliocentric elements, so that an error
in the latter indicates a proportionally smaller error in the
actual observations. For the same reason we must expect a
less degree of precision in the elements as finally derived than
in the case of the other planets.
5690 N ALM 5
66 MERCURY, VENUS, AND MARS
MERCURY.
Results of solutions of the normal equations.
[32, 33
Unknowns.
EM
Corrections of elements.
Symbol.
A.
B.
Symbol.
A.
B.
["]
o. 1478
o. 1207
O. I
6 m : m
o. 0148
O. OI2I
' /
o. 1342
0.0752
4-
ii
o. 537
0.301
: j
o. 2436
o. 2299
3-
6]
o. 731
o. 690
;N
o. 0227
0. 0201
3-
SinJdN
-o. 068
o. 061
t
-fo. 2074
+o. 2194
i.
6
4-o. 207
4-0.219
e
O. I2O2
4-0. 4094
3-
6e
o. 361
41.228
' TT
4-0. 5209
4-0. 2688
10.
6 7T
+5- 209
4-2. 689
' e"
-f o. 0669
4-0. 8397
0.6
(5 e f/
-f-o. 040
-f o. 504
V'
o. 2248
o. 7027
0.6
e" 6 TT //
o. 135
o. 422
>//
4-1. 1240
4-1.0566
2.
6r
4-2. 248
4-2. 113
6
o. 2310
o. 2556
I.
6
0.231
o. 256
/"
-0-0354
o. 0897
I.
61"
-o. 035
o. 090
a
4-0. 4803
4-0. 4930
I.
a
-fo. 480
+o- 493
/
J
o. 2060
o. 0114
o. 1209
-f-o. 0636
1 6.
12.
D t #
3. 296
o. 137
i. 935
-f o. 764
N;
4~o. looo
4~o. 0930
12.
SInjDt<m
-|-i. 200
4-1. 116
" e ~
4-o. 0681
4-o. 0966
4-
Dt 6 e
4-0. 272
4-0. 386
e
o. 1165
+o. 0987
12.
D t 6e
-i. 398
4-1.184
7T
o. 2385
o. 0252
40.
D t d7T
9- 540
1.008
e"
o. 1968
+- I 3 I 7
2.4
D t 6 e //
o. 472
4-0.316
7T // "
o. 1677
o. 1193
2.4
e/s D t 6 TT //
o. 402
o. 286
r"
4-o. 1108
4-o. 0806
8.
D t dr"
4-0. 886
4-0. 645
6
o. 1826
o. 1233
4-
D t <5
o. 730
o. 493
I"
o. 1442
0.3152
4-
D t d/ x/
-o. 577
i. 261
a
o. 3160
o. 1973
4-
D t a
i. 264
-o. 789
Mean epoch of corrections, 1865.0.
Discordance in the observed Right Ascensions of Mercury.
33. The most remarkable feature in the result is the value
of the quantity represented by [r"\. The unknown quantity
introduced with this symbol had as its coefficient the derivative
of the geocentric place as to the Earth's radius vector, and the
result would therefore be an apparent constant correction to
that radius vector. Since, however, the position of the planet
depends only on the ratio of the distances of the Earth and
Mercury, it follows that the actual correction may be regarded
as a correction to the ratio of the mean distances.
The determination of the mean distances by KEPLER'S
third law may be regarded as so unquestionable that the true
33] DISCORDANCE OF OBSERVATIONS. 67
value of this unknown quantity should be regarded as zero,
and the result as a purely fictitious one, arising from errone-
ous elements of reduction or systematic personal errors. It
was the possibility of the latter that led to its introduction.
When the planet is east of the Sun, observations are always
made on or near its west limb, or at least on some point west
of the true center, and vice versa. The value of dr" therefore
indicates that there is a remarkable systematic difference in
the observed Eight Ascension according as the planet is east
or west of the Sun, and therefore according to the illuminated
side. The sign of the result shows that the reduction to the
center of the planet was apparently too small. It is there-
fore of interest to learn according to what law this error
changed as the planet moved around its relative orbit.
It has up to the present time been impracticable to substi-
tute the unknown quantities in the original equations of con-
dition, and thus determine the separate residuals, and for the
purpose of investigating the present case such a substitution
is the less necessary, owing to the sniallness of the unknown
quantities. I have therefore simply determined the mean
correction to the Right Ascension given by all the observa-
tions during the various periods in six segments of the relative
orbit, near the elongations, and before and after the two
conjunctions. The results are shown in the following table.
Commencing with the moment of inferior conjunction, column
A contains the mean correction to the tabular Eight Ascension,
from observations made within about twenty days following.
Column B contains the observations made from twenty days
after the inferior conjunction until twenty days before superior
conjunction, a period during which the planet was generally
near its greatest west elongation. Column C contains the
observations made during the twenty days following and up
to superior conjunction. Then follow in regular order the
corresponding results when the planet was east of the Sun,
beginning with the twenty days following superior conjunc-
tion and going around to inferior conjunction.
68
MERCURY, VENUS,- AND MARS.
[33
Table showing the mean corrections to the tabular Right Ascen-
sion of Mercury in six segments of its relative orbit.
Epochs.
A
B
C
1765-1791.
" wt.
4-3. 24 4
" wt.
4-2. 61 5
" wt.
1793-1815.
-(-2.06 6
4-1.82 10
-f o. 97 4
1817-1839
+3- 06 6
4-1.79 24
-|-i. 13 24
1840-1849
+ 1.46 6
+ 1.48 18
o. 38 20
1850-1859
1860-1869.
4-3-72 4
4-1. 18 28
4-0. 77 20
+ 1.14 72
4-o. 08 1 6
4-O. 31 44
1870-1880
+ i. 18 25
4-o. 74 65
o. 20 6 1
1881-1892
4-1. 19 38
4-0. 98 63
o. 15 62
D
E
F
1765-1791
" wt.
4-0.92 . i. 5
" wt.
-{-1.30 10
" wt.
+o. 81 3
1793-1815.
+2. 82 5
4-i. 10 16
+ 1.85 5
1817-1830
4-O. 27 25
4-3. 76 24
i. 20 ;
1 84.0 1 840
j O 22 22
o S7 30
j_o 71; 3
i8co 18^0
J-O 60 14.
O 7Q 28
O 6< 4.
1860-1869
o. 44 5 ">
o. <u 69
V. \J^ ^
o. 35 1 6
1870-1880
o. 52 57
jj _?
i. 25 67
o. 30 24
1881-1892
o. 84 80
o. 73 102
o. 37 26
The remarkable feature of these results is the near approach
to constancy in the values of the numbers in each column,
after the secular variation is allowed for, and the large magni-
tude of the corrections. The most natural conclusion is that
the reduction from the limb of the planet or the observed
center of light to the true center was too small by an amount
which, at the mean distance of the Sun, must have been nearly
or quite a second of arc (cf. 3). The adopted semidiameter
3".4 seems so well established, both by micrometric measures
and by heliometer measures during transits of Mercury, that
such a correction to the diameter seems inadmissible.
I have not yet been able to enter upon the investigation of
the source of this anomaly. A very important question is that
of its influence on the results. Since a constant error in the
radius vector of a planet would have opposite effects on the
elements in different points of the relative orbit, it may be
inferred that the effect of the error would be nearly eliminated
33, 34] COMPARISON OF OBSERVATIONS OF MERCURY. 69
in an extensive series of observations distributed equally
between the two elongations. Actually, however, there seems
to have been an appreciable lack of symmetry in this respect,
as the influence of the unknown quantity upon the other
unknowns is not inconsiderable. Although the law of change,
as shown in the preceding table, does not correspond to the
magnitude of the coefficient of 6r", this coefficient being rela-
tively too small near inferior conjunction and too large near
superior conjunction, it is still probable that through the intro-
duction and elimination of dr" a large part of the injurious
effect is eliminated.
Comparison of transits and meridian observations of Mercury.
34. Another remarkable result which may be associated with
this is shown by the difference between the solutions A and B,
in the case of the eccentricity and perihelion not only of the
planet, but of the Sun. It will be seen that the meridian
observations alone give a negative correction to the eccen-
tricity of the planet, while, when the transits are included,
the correction becomes positive. That this is due to a system-
atic cause running through the observations is shown by the
fact that the same thing is true of the secular variation of
the eccentricity. This relation of the correction to its secular
variations holds true for three of the four relative elements,
and for the eccentricity and perihelion both of the planet and
of the Earth. In the case of the Earth's perihelion, however,
there is a nearer approach to conformity between the two
results.
There is yet another anomaly in this connection, which indi-
cates a very considerable systematic error in the older meridian
observations, which is not completely eliminated from the ele-
ments. If we take the values of the unknown quantities and
their secular variations, which result from the two solutions,
and substitute them in the linear functions of the corrections
to the elements derived from the transits alone, namely
V = 1.487 dl - 0.487 drr - 1.137 de - 1.01 dl" + 1.19 e"dn"
+ 1.58 de"
W = 0.716 dl + 0.284 dn + 0.896 de - 0.97 dl" - 1.11 e"d7t"
- 1.62 de"
70 MERCURY, VENUS, AND MARS. [34, 35, 36
we find the following results :
From meridian observations V = 2".99 + 0".69T
From November transits 1 .69 2 .63 T
From combined solution 2 .77 2 .30 T
From meridian observations W = -f- 0".89 4".55 T
From May transits alone +1 .39 + 1 .84 T
From combined solution +1 .39 + .42 T
We conclude that, had no transits ever been observed, the
errors of the elements and their secular variations, derived
from the great mass of meridian observations, would have
caused an error of some 5" per century in the heliocentric
place of the planet at the times of the May transits, and of
some 3" at the time of the November transits.
The fact that the combined solution B satisfies the transits
so much better than A, although the total weight of equations
A is so much greater than that of the transit equations, shows
that the meridian observations give only weak results for the
functions in question.
Meridian observations of Venus.
35. So far as the meridian observations are concerned, those
of Venus were treated on the same general plan as the observa-
tions of Mercury. The following are the principal points of
difference :
1. The hypothetical quantity dr" is omitted. Hence no
index to the consistency of the observations at different points
of the relative orbit can be derived from the solution.
2. Tenths of a unit were included in the coefficients of the
equations, and no modification was made in the units. The
units and tenths were, however, dropped in the final solution
of the normal equations.
Results of observed transits of Venus.
36. We put, at the time of a transit,
v, the longitude in orbit of Venus ;
Z, its mean longitude, or the mean vame of v;
fi, A, its ecliptic latitude and longitude;
L, the Sun's true longitude.
36 ] EQUATIONS OF CONDITION FROM TRANSITS OF VENUS. 71
Then
tf A = cos i 6v + sin 2 i 66
= 0.99S2 dv + 0.0592 sin i 36
We thus have, for the dates of the observed transits,
1761-'69 ; dp = 0.0592 6v + 0.9982 sin i S6
1874->82 ;/?= + 0.0592 Sv - 0.9982 sin * dd
I have discussed very fully the observations of the transits
of 1761 and 1769 in Astronomical Papers, Vol. n. The final
results which I shall use are found on page 404 of that volume.
Here I have put.
x, correction to A L ;
y, correction to /?,
the Sun's latitude being supposed to require no correction.
The values of x and y for 1769 are distinguished by an accent.
I have also represented by z 2 and 2 3 the corrections to the dif-
ference of the semidiameters of the Sun and planet, for the
respective internal contacts, to which may be added the. un-
known but probably nearly constant quantity due to personal
error in estimating the time of contact. From their very nature
these quantities do not admit of accurate determination, and
must therefore be eliminated from the equations. From the
observations of internal contact are derived the following four
equations :
1761 II; ,87a? + .50 y + z 2 = - 0".07
HI ; + .68 + .73 + 2 3 = - 0".06
1769 II; - .64 a? 7 - .11 y' + z 2 = - 0".27
III; + .84 - .55 + 2 3 = + 0".02
We have here more unknown quantities than equations, so
that it is not practicable to determine them all separately.
What I have done has been first to assume ^ = 2 3 . This pre-
supposes that the distance of centers at the estimated appa-
72 MERCURY, VENUS, AND MARS. [36
rent contact at egress is, in the general mean, the same as at
ingress. The result of any error in this hypothesis will be
almost completely eliminated from the mean latitude at the
two transits, but not from the longitude.
Still, the values of x and y can not be separately determined;
I have therefore so combined the equations as to obtain mean
values of x and y for the two contacts, assuming that this
would be the result of supposing these quantities to have the
same, values at both epochs. Calling these values x" and y",
we have by addition and subtraction, supposing z 2 = 23,
- 0.39 a?" + 2.55 y" = 0".12
3.03 #"4- 0.45 y" = 0".30
We thus have*
x" = + 0".09
y 1 ' = + 0".06
These corrections are not applicable to the coordinates from
LEVERRTER'S . tables as they stand, but to those quantities
as corrected by the following amounts :
A\ = 4- 0".25
Afi= 4-2".00
* In a second approximation to these quantities, which may be made
after the correction to the enteimial motion of the node is determined,
we should put, on account of this correction,
The solution would then give
I have carried through a more careful approximation in a subsequent
chapter.
36] EQUATIONS OF CONDITION FROM TRANSITS OF YENUS. 73
We thus find, for the corrections to LEVERRIER'S tables at
the epoch 1765.5,
d A - L = + 0".09 + 0".25 = + 0".34
3 = _ 0".06 2".00 .= + 1".94
and hence
dv = + 0".22 + 0".998 L
sin i 6 S = + 1".95 + // .059 6L
A still farther modification is required to the tabular longi-
tude on account of the correction to the mass of the Earth
used by LEVERRIER, and hence to the periodic perturbations
in longitude. This correction is + 0".20. We thus have for
the correction to the orbit longitude of Venus
dv = + 0".02 4- 0".998 6 L
For the results of the transits of 1874 and 1882 I have
depended entirely on the heliometer measures and photo-
graphs made by the German and American expeditions,
respectively. The definitive results of the German observa-
tions, as worked up by Dr. AUWERS, are found in Vol. V of
the German Keports on the Transits.* The American photo-
graphic measures of 1874 have not been officially worked up
and published, but a preliminary investigation from the data
contained in the published measures was made by D. P. TODD,
and published in the American Journal of Science, Vol. 21,
1881, page 491. The measures of 1882 have been definitively
worked up by HARKNESS, but only the results published.
They are found in the report of the Superintendent of the
TJ. S. Naval Observatory for the year 1890.
The corrections to the geocentric Eight Ascension and
Declination of Venus relative to the Sun thus derived are
* Die Venus-durchgiinge 1874 und 1882 Bericht uber die Deutsclien
Beobachtungen Fuufter Band, Berlin, 1893.
74 MERCURY, VENUS, AND MARS. [36
given in the following table. In taking the mean the weights
are not strictly those which would result from the probable
errors as assigned, but, in accordance with a general princi-
ple, independent results have received a weight more near to
equality than would be indicated by the mean errors.
1874: German, 6 E. A. = + 4.77 0.28
American, . . . + 4.14 0.30
Adopted, . . . +4.44
German, 6 Dec. = + 2.28 + 0.10
American, . . . -f 2.50 0.30
Adopted, . . . +2.34
1882: German, 6 E. A. = + 9.03 + 0.12
American, . . . + 9.10 + 0.08
Adopted, . . . +9.07
German, d Dec. = + 2.02 i 0.06
American, . . . +2.02 + 0.08
Adopted, . . . +2.02
We change these results successively to geocentric longi-
tude and latitude, heliocentric longitude and latitude, and
orbital longitude and latitude. The results of these several
changes are as follow:
Corr. in geoc. long.
Corr. in lat.
1874.
+ 3''.853
+ 2 .724
1882.
+ 8".077
+ 2. 971
Corr. in hel. long.
Corr. in hel. lat.
-1 .415
+ 1 .001
-2 .965
+ 1 .091
Corr. in orbital long.
Value of sin i 6
-1 .35
-1 .08
-2 .90
-1 .26
37] EQUATIONS FROM TRANSITS OF VENUS. 75
Equations from transits of Venus.
37. The corrections to the heliocentric positions of Yenus
and the Earth, as thus found, are now to be expressed in
terms of corrections to the elements. The results of this
expression are shown in the following equations:
Equations given by the corrections to the orbital longitude.
I. Epoch, 1765.5; T= 3.90 ; weight = 200
0.992 61 + 1.17 eSn + 1.62 de - 0.976 61" 1.81 e"dn"- 0.85 de"
= +0".02 O."15
II. Epoch, 1874.9 ; r = + 0.48; weight = 400
- 0".88/* + 1.009 61 - 1.223 edn - 1.596 6e - 1.030 61"
$*" + 0.817 6e" = - 1".35 0".08
III. Epoch, 1882.9; T = + 0.80; weight = 800
0".60,w+ 1.008 61 - 1.146 edn - 1.651 6e - 1.028 <M"+ 1.825 6"$w"
+ 0.900 $6" = 2".90 i 0.' / 027
Equations given by the corrections to the orbital latitude.
I. 1765.5; sin i66- 0.057 <"- 0.11 ^^^"-O.OS^^ + 1".95
i 0".10
II. 1874.9; sinid9-0.
/7 .04
III. 1882.9; sin^^-0.
i 0".019
The weights assigned to these three equations are, respec-
tively, 200, 600, and 1,600.
Before using these equations the corrections to the elements
were transformed into the unknown quantities denned in 27,
and their secular variations by multiplying the coefficients by
the factors given on page 56.
76
MERCURY, VENUS, AND MARS.
[38, 39
Solutions of the equations for Venus.
38. The parts of the normal equations formed from the
preceding conditional equations were added to the parts from
the meridian observations, and the resulting solution B
obtained. As in the case of Mercury, a solution A was made
of the normal equations derived from the meridian observa-
tions alone. The results are as follows :
VENUS.
Results of solutions of the normal equations.
Unknowns.
Factors.
Corrections of elements.
Symbol.
A.
B.
Symbol.
A.
B.
[>]
o. 0708
o. 0834
7-
<5 m : m
o. 496
0.^584
' / '
o. 1435
o. 1501
5-
61
-0.718
0-751
" J '
+o. 1156
+o. 1340
6.
6 T
+o. 694
+o. 804
N;
4-o. 0164
-j-o. 0106
7-
sinJrfN
+o. 115
+o. 074
e
+o. 0941
-j-o. 1003
3-
fit
+o. 282
-|-o. 301
7T ]
4-o. 0628
-j-o. 0764
3-
e6K
4-o. 1 88
+o. 229
e
4-o. 0246
+0.0271
4-
6e
-j-o. 098
4-o. 1 08
~ e "\
4-o. 0336
4-o. 0318
2-5
5e"
4-o. 084
+o. 080
~ir"'
o. 0274
0.0212
2.
e"tv"
o. 055
o. 042
' a
+o. 4742
+o. 4642
I.
a
+o. 474
+o. 464
i 6 ''
o. 0383
-o. 0375
5-
6
o. 192
o. 1 88
*///
o. 0768
o. 0743
4-
61"
o. 307
-o. 297
' / '
o. 1846
-o. 1983
20.
D t d/
3. 692
3. 966
:*:
+o. 0970
o. 0561
+o. 1088
o. 0594
2 4 .
28.
DtJ
sinJD t N
+2. 328
I-57I
+2.611
-1.663
e
+o. 1472
+o. 1644
12.
Dt e
+ 1.766
+ r -973
TT
+0.0555
4-o. 0698
12.
^D t 7r
+o. 666
+o. 838
-(-o. 0182
-j-O. O2O2
1 6.
Dte
+o. 291
+o. 323
"e"\
4-o. 0283
+0.0317
10.
Dt^ x/
+o. 283
7T //
+o. 0399
+o. 0506
8.
e" Dt TT //
+o. 3 J 9
4-o. 405
a
o. 0820
-o. 0347
4-
D t a
o. 328
o. 139
! <* s
0. 0020
O. OOO2
20.
Dt d
o. 040
o. 004
L/// -
--o. 0562
o. 0662
1 6.
EM/"
o. 899
i. 059
Mean epoch of correction, 1863.0
Comparison of transits of Venus with meridian observations.
39. To show to what extent the results of the meridian
observations differ from those of the observed transits over
the Sun, we form the values of the absolute terms of the
equations of condition, 37, first by substituting the values
A of the corrections, and then the values B. We thus have :
39, 40] EQUATIONS FROX TRANSITS OF VENUS. 77
Residuals in orbital longitude.
1765.5. 1874.9. 1882.0.
(a) From meridian obs. alone . - 0".07 - 1".36 - 2".54
( ft) From combined solution . + 0".04 - 1".43 - 2".78
(y) From transits alone . . . + 0".02 - I". 35 - 2".90
Discordance, (y)-(tf) . . +0".09 + 0".01 - 0".36
Discordance, (y)- (ft) . . - 0".04 + 0.08 -0".12
Residuals in orbital latitude.
1765.5. 1874.9. 1882.9.
(a) From meridian obs. alone . + 1".92 - 0".77 - 0".96
(ft) From combined solution . + 2".06 - 0".91 - 1".12
(y) From transits alone . . . + 1".95 - 1".08 - 1".26
Discordance, (y)- () + 0".03 - 0".31 - 0".30
Discordance, (y) (ft) . . - 0".ll - 0".17 - 0".14
It will be seen that the combined solution represents the
observations of the transits much better here than in the case
of Mercury.
Solution of the equations for Mars.
40. As the formation of the normal equations for Mars was
approaching its end, a singular discordance among the resid-
uals of the partial normal equations for different periods was
noticed. On tracing the matter out it appeared that while the
correction of the geocentric longitude of LEVERRIER'S tables
in 1845 and again in 1892 was quite small, the correction in
1862 was considerable. Now there is an inequality of long
period, about forty years, in the mean motion of Mars, depend-
ing on the action of the Earth, and having for its argument
15# 7 Sg. This coefficient is of the seventh order in the eccen-
tricities, and the terms of the ninth or even of the eleventh
order might be sensible in a development in powers of the
eccentricities and sines or cosines of multiples of the mean
longitudes. The conclusion which I reached was that the the-
oretical value of this coefficient was not determined with suffi-
cient precision. As the work of solving the equations could
not wait for a new determination and a new formation of the
absolute terms of the normal equations, it was decided to make
an approximate empirical correction to the theory. This was
used to correct the absolute terms of the partial normal equa-
78
MERCURY, VENUS, AND MARS.
tions for each period, and the solution was then proceeded
with. The chances seein to be that by this process the inju-
rious effect of the error upon the elements derived from the
equations would be inconsiderable; this is, however, a point
on which it is impossible to speak with certainty. It is the
intention of the writer to recompute the doubtful terms of the
perturbations, and, if possible, reconstruct the absolute terms
of the normal equations in accordance with the corrected
theory. Meanwhile, the present work necessarily rests on the
imperfect theory with the approximate empirical corrections,
which are as follow :
M = 0".30 cos (150' - 8-7 - 223)
edn = 0".15 cos (150' 80)
As the elements of Mars are derived wholly from meridian
observations, only one set of equations of condition was formed.
The results of the solution are shown in the following table :
MARS.
Unknowns.
Factors.
Corrections of elements.
Symbol.
Value.
Symbol.
Value.
[ tn/ ~\
. 02278
0-3
6 m : m
-o. 007
1 ]
. 44854
2.
61
-o. 897
N;
+ 05479
+.06724
2-5
2-5
SinJJN
+o. 137
+o. 1 68
e
+ .43803
V
6 e
+o. 626
TT
. 05056
1^-Q
6w
o. 722
e
+.07474
4-
6s
+o. 299
e"\
. 49898
2.
be"
o. 998
V /x "
.42409
2.
e' f 6 K"
o. 848
a
+ 18545
5-
a
+o. 927
; 6 '
. 04536
5-
6
o. 227
///'
+ .05786
3-
61"
+o. 174
" / "
+. 16605
8.
D t d/
+1.328
; JT
+ 13408
. 02263
10.
10.
D t J
SinJDtN
+1.341
o. 226
e
. 03180
V 1
D t <?
o. 182
TC
. 00928
A^Q
D t 7T
-o. 530
e
+. 06097
1 6.
Dt
+0.976
' //
. 12597
8.
l}^e //
1. 008
"TT //=
+.00853
8.
e" D t TT"
+o. 068
a
. 09670
20.
D t a
i. 934
" 6 ]
.01168
20.
Dt^
-o. 234
\l>'\
+. 13111
12.
D t rf/"
+1.573
41] REFERENCE TO THE ECLIPTIC. 79
Reference to the ecliptic.
41. In all the preceding determinations the planes of the
orbits are referred to the plane of the Earth's equator, or, to
speak more exactly, to a plane through the Sun parallel to the
Earth's equator. As in astronomical practice the ecliptic is
taken as the fundamental plane, it is necessary to investigate
the reduction of the elements from one plane to the other.
Let us consider the spherical triangle formed on the celestial
sphere by the plane of the orbit, the plane of the ecliptic, and
the plane of the Earth's equator. For the sides and opposite
angles of this triangle we have
Sides: N 6 y
Opposite angles : i 180 J e
When equatorial coordinates are used, the position of the
planet is considered as a function of the three quantities
N; J; e (a)
When ecliptic coordinates are used, the three corresponding
quantities are
0; I-, e (b)
Taking the set of quantities (a] as the fundamental parts of
the triangle, and expressing the corrections of the other parts
as functions of them, we have
6 i = + cos rp6J + sin ip sin JtfN cos Ode
sin i d 9 = sin fidJ + cos >/> sin J6N + cos * sin Ode
Taking (b) as the fundamental parts, we have for the correc-
tions to N and J
d J = cos fi6i sin ip sin i$6 + cos
sin J#N= sin 6i + cos sin idd cos J sin Ntfs
The numerical values assigned to the coefficients in these
equations are those corresponding to the mean epoch 1850.
The fact that they change somewhat in the course of a hundred
years has not been taken account of. The future astronomer
will meet with a real difficulty in that the corrections to a
80 MERCURY, VENUS, AND MARS. [41
set of elements at one epoch do not accurately correspond to
similar corrections at another epoch. It is impossible to do
away rigorously with the difficulty thus arising, except by
introducing a more general system of elements than elliptic
ones. The error is, happily, not important in the present state
of astronomy. The equations in question for the three planets
are as follow:
Mercury.
di = + .799 (5 J + .602 sin J fi X .688 fo
sin idO = - .602 6J + .799 sin J d N + .721 fa
Venus.
di = + .373 6 J + .928 sin J N - .255 fa
sin idS .928 d J + .373 sin J d N 4- .967 de
Mars.
di = .703 6 J + .712 sin J tf N - .664 6s
sin id 6 = - .712 3J + .703 sin J 3X + .747 fa
For the inverse relations we have
Mercury.
3J= .799 <H - .602 sin idti + .983 de
sin J SIS = .602 6i + .799 sin id 6 - .162 de
Venus.
6 J = .373 di .928 sin idO + .990 de
sin J d N = .928 di + .373 sin idB - .125 fa
Mars.
6 J = .703 <M - .712 sin id 6 + .998 fa
sin J N = .712 < + .703 sin id B - .052 fa
CHAPTER IY.
COMBINATION OF THE PRECEDING RESULTS TO OBTAIN
THE MOST PROBABLE VALUES OF THE ELEMENTS
AND OF THEIR SECULAR VARIATIONS FROM OBSER-
VATIONS ALONE.
In the two preceding chapters are derived four separate
values of the six corrections, <*, #, 6e, 61", de", and e"6n", and
of their secular variations, which pertain to the orbit and
motion of the Earth relative to the stars. We have now to
combine these four results so as to derive the most probable
values of the twelve unknown quantities in question.
Deviations from the method of least squares.
42. If we applied without modification the principles of the
method of least squares, we should first eliminate the elements
and secular variations for each planet from the normal equa-
tions given by observations of that planet, which would leave
us with three sets of normal equations, containing only the
twelve quantities depending on the motion of the Earth. We
should then reduce these normal equations to equality of
weight, by multiplying each of them by the appropriate
factors, and we should then consider the observed corrections
to the solar elements derived from observations of the Sun
alone as affording equations of condition to be reduced to the
adopted system of weights, and then multiplied by their coeffi-
cients and added to the normal equations. The solution of
the single set of normal equations thus formed would lead to
the definitive values of the solar elements and of their secular
variations, which, being substituted in the eliminating equa-
tions from each planet, would lead to the definitive elements
of the planet and of their secular variations.
This proceeding is not, however, advisable in the present
case, because, owing to the immense mass of material worked
5690 N ALM 6
81
82 PROBABILITY OF ERRORS. [42
up, the errors to be principally feared are not the accidental
ones, of which alone the method of least squares takes account,
but the systematic ones arising principally from personal
equation and imperfect reduction of the observations to the
actual center of the planet or of the Sun. These errors affect
different elements in very different ways and to different
amounts; from some they will be almost completely elim-
inated and from others they will not. We must therefore pro-
ceed by a tentative process, ascertaining at each step, so far
as possible, how each result .will come out before we accept it
as final, to be combined with other results. In doing this it
is necessary to deviate so widely from what are commonly
regarded as fundamental principles of the theory of the com-
bination of observations that a brief presentation of the prin-
ciples involved is appropriate.
It is frequently accepted as an axiom that when we have
several non-accordant determinations of the same quantity,
between which we have no reason for choosing, the most prob-
able value is the arithmetical mean. The operation of taking
the arithmetical mean is, in fact, the simplest application of
the method of least squares. The fundamental hypothesis on
which this method rests is that the probability of an error of
magnitude i x is given by the well-known exponential equa-
tion
h M**
<p (Ji, x)dx = e dx (a)
h, the modulus of precision, being a constant. It was shown by
GAUSS that this function for the probability follows rigorously
from the principle of the arithmetical mean. It therefore fol-
lows that the method of the arithmetical mean, and therefore
that of least squares, is rigorously correct only so far as the
law of error is expressed by the above exponential function.
It scarcely needs to be pointed out that, as a matter of fact,
the law of error in question is not true. Not only so, but in
astronomical experience it deviates from the truth in a way
admitting of precise statement. It presupposes that the mod-
ulus of precision is a determinate quantity. Were this the
case, then, to take a single instance, the probability of an
42] PROBABLE ERRORS AND WEIGHTS. 83
error five times as great as the probable error would be less
than 0.001, and the probability of an error six times as great
would be about 0.0001. This is not true, because, taking the
function q> (ft, x) as a basis, we may say that the modulus of
precision, ft, is nearly always in practice an uncertain quan-
tity. Let us then put
hi, lit, h 3 , . . .
for the possible values of ft, and
for the several probabilities that h has these respective values.
Then the probability function will become
<p(x)=Pi <P(h\, x) +Pz(p'(h*, oo) + ... (6)
Now this form can not be reduced to the form (a) with any
value whatever of the modulus h. If we make the closest rep-
resentation possible, we shall have a curve in which small
values and large values of x are relatively less probable as
compared with the facts than are intermediate values. To
show that this is the actual case, let us suppose that we have
three determinations of an unknown quantity. If we proceed
in the usual way, we should infer the value of ft, the measure
of precision, from the discordance of these three values. But
it is evident that this determination of ft w.ould be very uncer-
tain. Should the three values chance to be fortunately accord-
ant, then, proceeding in the usual way, our function would lead
to the conclusion that the probability of an error of a certain
magnitude in the mean was very small, when, as a matter of
fact, it might be very considerable.* The value of ft being
* To take a simple and quite possible instance, let three observations of
a star with a meridian circle give, for the seconds of declination, 0".4, 0".5,
and 0".6. By the canons of least squares the mean result would be
0".50 0".039
and the probability of an error as great as 0".l would come out about 0.08.
84 PROBABLE ERRORS AND WEIGHTS. [42
uncertain, the true form of the function is not (a} but (b). It
follows that we may lay down the following general rule :
The best value from a system of non- accordant determinations
is not the arithmetical mean, but a mean in which less iceiglit is
assigned to those results which deviate most widely from the
mean of the others.
I have considered the subject from this point of view in the
American Journal of Mathematics, Vol. VIII, p. 343, and given
tables for determining the weights to be assigned to the results
when the law of error is that derived from several hundred
observed contacts of the limb of Mercury with that of the Sun
during transits of the planet.
Another well-known defect in the method of least squares
is that it does not take any account of systematic errors. The
greater the number of observations that are combined, the
larger the proportion in which the errors of the results may
be due to the systematic errors in the observations or the
elements of reduction. Although such errors may elude inves-
tigation so far as their determination and elimination is con-
cerned, we may yet be able to point out their origin, and to
show to what extent they would influence each separate result.
Of some results we can say with entire confidence that they
are but slightly affected with systematic error 5 of others, that
they may be very largely so affected. In the latter case, the
weights of the results, as determined from the solution of the
normal equations, give no clue whatever to the probable mag-
nitude of the error.
The result of this is that in the following paper we are more
than once confronted with the following problem: Among
several determinations of a quantity one is known to be free
from systematic error and to be affected with a well determined
probable mean error, i e. There are also one or more other
determinations of which the probable error is unknown and
can not be determined, because we have no sufficient knowl-
edge of the probable effect of systematic errors upon the result.
What shall be the relative weight assigned to two such results
in order to obtain the mean? The decision of this question is
necessarily a matter of judgment, the grounds for which it
might be extremely prolix to state at length. An attempt has
42 1 PROBABLE ERRORS AND WEIGHTS. 85
been made in these cases to classify the results, so as to give
a general idea of what is likely to be their modulus of pre-
cision, and weight them accordingly.
Any attempt at numerical accuracy in such an estimate
would be labor thrown away. It has therefore been considered
sufficient in such cases to state what the conclusion of the
author is, leaving its revision and criticism to the future
investigator. Indeed, in some cases, as in that of the correc-
tion to the centennial motion of the Sun in longitude, a con-
venient round number has been chosen, very near to the result
of well-decermined weight.
We should be carrying the preceding conclusions too far if
they led us to a general distrust of the conclusions reached by
the method of least squares. The doctrines that there is a
necessary limit to the accuracy with which astronomical deter-
minations can be made; that systematic errors necessarily
affect every such determination 5 and that the canons of least
squares necessarily lead to illusory probable errors, are too
sweeping. We may lay down the general rule that if we have
a sufficient number of really independent determinations of an
unknown quantity, of which we individually know nothing
except that they are the results of actual measures, and not
mere guesses, then the arithmetical mean will be a definite
result, the probable deviation from which will actually follow
the law given by the canons in question with a closeness
which will continually increase with the number of independent
determinations.
If we have such knowledge of the relative values of the
various determinations as to assign greater weight to some
than to others, the result will be still better when those
weights are used, provided always that they are assigned
without undue bias in favor of those results which most nearly
approach the value supposed to be approximately correct.
These considerations lead me to a policy which I have
always adopted when it was easy to do so in the following
discussions, namely, that of so conducting the work as to
lead to as many independent determinations of a quantity
as possible, arid of always giving a less relative weight to such
sets of determinations as might from any cause whatever be
86 ELEMENTS OF EARTH'S ORBIT. [42,43
supposed affected by au important common source of error.
Where the independent determinations are few in number, the
computation of a definite probable error is impracticable, and
the probable mean error assigned is necessarily a result of a
judgment based on all the circumstances.
Relative precision of the tico methods of determining the elements
of the Earth's orbit.
43. When the system of determining the solar elements from
observations of the planets as well as of the Sun was originally
decided upon, it was supposed that the two methods would
give results not greatly differing in accuracy in the case of any
of the elements. This, however, is proved by the results not to
be the case. Attention has already been called to the extreme
consistency of the values found for the correction to the eccen-
tricity and perihelion of the Earth's orbit from observations of
the Sun. This consistency inspires us with confidence that
the probable errors of the corrections to the elements as given
do not exceed a few hundredths of a second. But the deter-
mination of these elements from observations of Mercury and
Venus may be seriously affected by the form of the visible
disks of those planets, which results in observations being
made only upon one limb when east of the Sun and the other
limb when west of it. Thus personal equation and the uncer-
tainty of the semidiameter to be applied in each case may have
an effect upon the result. But personal equation is likely to be
smaller in the case of Mercury than in that of Venus, owing
to the smalluess of its disk.
There is another circumstance which weakens the inde-
pendent determination of the Earth's eccentricity and perigee
from observations of the planets. If we define the orbit of a
planet, not as a curve, but as the totality of points which the
planet occupies at a great number of given equidistant moments
during its revolution, then it is easy to see that the general
mean effect of an increase of the eccentricity is to displace the
entire or.bit toward the point of the celestial sphere marked by
its aphelion, while the effect of a change of its perihelion is to
move the entire orbit in its own plane in a direction at right
angles to the line of apsides. The result is that in a series of
43, 44J SECULAR VARIATIONS OF THE SOLAR ELEMENTS. 87
observations of a planet from the Earth the corrections to the
eccentricity and perihelia of the two orbits can not be entirely
independent, and we can determine with entire precision only
two linear functions expressive of the relative displacements
just described. It may be admitted that, were observations
exactly similar in kind made around the entire relative orbit
in equal numbers, the effect of the principle systematic errors
would be nearly eliminated from the result. But we can not
rely upon this being the case, and even were it the case there
would probably be a residual effect which would be large in
proportion to the interdependence of the two sets of correc-
tions. But in this connection the important remark is to be
made that, so far as these systematic errors are invariable,
they would not affect the secular variations, but only the abso-
lute values of the elements. We may therefore assign greater
relative weights to the former than to the latter.
So far as we cau classify the results, I have concluded that
in the case of the secular variations of f, e"< and TI" , the weight
of the determination from Mercury and Venus might receive a
weight one-fifth that from the Sun. But in the case of the
absolute values of these quantities, it would seem from the
discordance of the results that the relative weight of the
planetary results should be much smaller.
In dealing with the common error, a, of the adopted Right
Ascensions of the stars, it is to be remarked that we may
regard the observations in Eight Ascension as fitted to give
the values of a + 61", while 61" necessarily depends solely
upon the observations of declination, in effect if not in form.
Hence, although the unknown quantities of the solution are
a and 61", I have deemed it best to derive the result by
regarding a + 61" as the quantity to be first found, instead of
a itself.
Secular variations of the solar elements.
44. The following table shows the corrections to the tabular
secular variations of the solar elements, as they have been
found from observations. In the cases of Mercury and Venus
the results of both solutions are given for the sake of compari
son, although only solution B is used. The relative weights
88
ELEMENTS OF EARTH'S ORBIT.
[44
have been determined by the considerations already set forth.
In the case of Mars, the final determinant of the solution for
the solar elements came out so- nearly evanescent as to show
that no reliable values could be obtained, a result which we
Corrections to the secular variations of the solar elements derived
from observations only.
Dt*
Dt<J/"
Dt#f*
From observations of
The Sun
" w.
4-0. 48 5
" w.
o. 97 i
" TC/.
+- 2 3 5
Mercury, solution A .
" " B
-4-0.27
-4-O 3Q I
0.58
i 26 i
0.47
1 O 32 I
Venus, solution A _ _
-f o. 29
o. 90
4-0.28
B
4-O. 32 I
i. 06 i
4-o. 72 i
Mars
+ 1.03 |
Mean
-1-0.48
I. IO
4-o. 26
Adopted _ . _ .
+0.48
. I.OO
4~O. 21
*"D t c57r"
D t (a + d/")
D t a
From observations of
The Sun
" IV.
4-0. 32 5
" w.
0. 63 2
//
4-0. 34
Mercury, solution A
0.40
-1.84
1.26
B
o. 29 i
2. 05 3
o. 79
Venus, solution A
4-o. 32
O. ??
" " B
4~o. 46 i
I. 2O 2
o. 14
Mars
Mean
-fo. 25
I. 4O
o. 30
Adopted
-l-o. 26
I. ^O
o. 30
might expect, because, in order to separate the principal ele-
ments of the Earth's orbit from those of the planet, observa-
tions should be continued all around the relative orbit, whereas,
as a matter of fact, they are generally made only near the time
of opposition. I have judged, however, that the correction to
the secular variation of the obliquity obtained by putting
D t dl" = 1".00 MI the equation for D t tf e might enter with
half the weight that it does in the cases of Mercury and
Venus. Before the final values and weights of the quanti-
ties in the table had been definitively revised, provisional
values were used in the subsequent part of the investigation.
44, 45] CORRECTION TO THE STANDARD DECLINATIONS. 89
These provisional values are given in the last line of the table.
Jt is also to be noted that the secular variations of e, e" TT, n"
and in the definitive theory and tables are those computed
from the adopted masses of the planets.
Correction to the standard of Declination.
45. The results for the secular variation of 3, the common
error of the standard Declinations within the zodiacal limits,
are not given in the table, as other data are available for its
determination. The following shows the separate values of
6 and its secular variation, derived from observations of the
planets to Saturn inclusive. For reasons already stated obser-
vations of the Sun are not used for this purpose.
" " w.
From observations of Mercury, 6 = 0.18 - 0.49 T ; 2
Venus, - 0.19- 0.00 T; 1
Mars, - 0.21- 0.23 T; 4
Jupiter, - 0.04 - 0.43 T ; 3
Saturn, + 0.04- 0.68 T; 4
Mean ; d= 0".09 0".42 T
Adopted ; <? = -0 .08-0 .50T
Not only observations of the planets but those of the fixed
stars are available for the determination of 6 and of its secular
variation. In the discussion of the Declinations derived from
observations with the Greenwich and Washington transit cir-
cles (Astronomical Papers, Vol. II), I have shown that the
Greenwich observations indicate, with some uncertainty, a
secular variation of the corrections to the standard declina-
tions which will give a value of about 0".55 for the secu-
lar variation of d. But BRADLEY'S Declinations, as reduced
by AUWERS, would give a still larger negative value, approxi-
mating to an entire second. As the value which we may
assume for d does not greatly influence the other elements,
I have adopted as a convenient probable result, the varia-
tion 0".50 T.
90 ELEMENTS OF EARTH'S ORBIT. [46
Definitive secular variations of the planetary elements from obser-
vations alone.
46. Having decided upon the adopted values of the six
quantities found in the last article, we regard them as known
quantities, and substitute them in the eliminating equations,
which give the values of the remaining secular variations.
As the unknown quantities in these equations are not the
corrections themselves, but certain functions of them, we pre
pare the following table, showing the formation of the quan-
tities which are to be substituted in the several equations.
The table scarcely seems to need any explanation, except that
the unknown quantities given in the three columns on the
right are formed by dividing the secular variations for twenty-
five years by the coefficients given in 27.
Adopted secular variations of the solar unknowns, to be substi-
tuted in the eliminating equations for the several planets.
D t <M" = -T'.OO;
D t tf =0 .50;
D t a = -0.30;
e"D t <y7r" = + .26;
D t 6e ff =+0 .21; [ e"
D t fo =+0.48;
To facilitate a judgment or rediscussion of this part of the
process, we give on the next three pages the normal equations
between all the secular variations which remain after the cor-
rections to the elements of the Sun and planets are eliminated
from the original normal equations. We give these rather
than the eliminating equations "which were actually used in
the substitution, because they show more fully the relations
between the unknown quantities, and can therefore be better
used iu any ulterior discussion. Regarding the preceding six
quantities as known, and substituting them in the normal
equations for the secular variations, we derive the definitive
values of the secular variations which relate to the planets.
They are shown in the next table. In the latter the values of
the solar elemen ts are repeated for the sake of completeness.
Mercury.
Venus.
Mars.
I"
1' =-0.250;
-0.0625;
-0.0833;
d
]' =-0.125;
-0.0250;
-0.0250;
a
]' =-0.075;
-0.0750;
-0.0150;
n"
]' = + 0.108;
+ 0.0325;
+ 0.0325;
e"
]' = + 0.087;
+ 0.0210;
+ 0.0262;
]/ = +0.120;
+ 0.0300;
+ 0.0300.
46] NORMAL EQUATIONS FOR SECULAR VARIATIONS. 91
I I
8
CO
3
+ 1 1 l
p
^
I
-T7 CO
CS O5
+
SI
^ o
r 3!
5. fl . .
co
<M
10
te oo
5 <
rH CO
i
10
s
8-
CO
1O
CO
I I
o
I
T*
rH
1
I
28
I
Oi
00
rH
rH
I
CO
i
+ + i
O GO CO
rH rH CO
CO rH C5
3 s
CM t-
I I I
^ 5 51
<M CO
vr
3 55
1 1
3
I I + I
3 3
CO <^ rH CO
rH rH IO
I + I +
i i 1
CO
+
OO
00
rH
+
CT *3
Z co
__ (N CO CO
rH <M
^$
^L 3
I I
00
o
00
-8
NORMAL EQUATIONS FOE SECULAR VARIATIONS. [46
CO
CO
CO
CO
TH <M O
O CO Tf
S *
i i
^ S
v .2
*
^ a
s I
CD
QO
GO
t^ rH 00
1 ^
I I
4- + I
rH CO CO'
rH <M
CO
i
CO T* rH
+ 4- + I +
I -I I
S
CO
QQ
t>
>
I I
CO t*-
TH CO
TH
O
rH IS
00 b-
CM rH
C5 t
CO CO rH
C2 -
^- ^ ^ r<
* I
S S
CO
^>
I I +
8 g 53
" S
I + +
I
z
9
1O t~
rH
I I +
^ CO
S ^
46] NORMAL EQUATIONS FOR SECULAR VARIATIONS.
CO
O O O CO
O b- JO CO O b-
H rH rH CM rH rH
I + I I + I
1 1 1 +
b'38
CO
+ +
cp CQ
CO
+ + i + + +
a ^
^^ co
rH O5
tr^" tr**
CO t-
Tfl b- CO O O
C5 SO rH IO O
lO rH b- rH C5
b- rH rH
+ + I I +
rH CO
CM Ci
8 a
^
1
o
3
I
tJO
ti
R rH
C^ CO
2L S
CO CM CO
Ci <M CO
b- rH <M
1
I I
(M CO
CM CO
CO iO
lO b-
b- rH O CO
05 rH CM CO
CM
a s
b- O CO
rH CO ^*
CO
cc
!
^
I I
^a
KJ T^
J5- ^
i
- 8
b-
i i
CM C5
1^- b-
O O
CM CO
^^ T"H
co o
tfHIVBESITT,
i
+
CO
CO
94
SECULAR VARIATIONS FROM OBSERVATIONS.
[46
Values of the secular variations as derived from observations
only.
Unknown. Corr.
Tables.
Result.
Mercury. D t e
D t i
-.0691 -0.83 + 4.19 + 3.36i0.50
+ .1577 +1.30 +116.94 +118.240.40
+ .0593J +0.83 + 6.31 + 7.14i0.80
+ .0815K +0.70 - 92.59 - 91.890.50
-.0967 -1.55
Yenus.
Earth.
Mars.
D t e +.1393 +1.67 - 11.13 - 9.46i0.20
eD t 7T +.0685 +0.82 - 0.53 + 0.290.20
D t -t +.1153 J -0.65 + 4.52 + 3.87i0.30
sintD t -.0592K -2.73 -102.67 -105.40+0.12
D t tfZ -.1919 -3.84
D t e +0.21 - 8.76 - 8.55i0.09
eV t 7t +0.26 + 19.22 + 19.480.12
+0.48 - 47.59 - 47.11 =fc 0.25
D t f
D t e
eD t ;r
D t i
siniD t
~D t 6l
.1190 -0.68 + 19.68 + 19.OOiO.27
+.0536 +0.29 +149.26 +149.550.35
+.1136J +0.17 - 2.43 - 2.260.20
-.0442N -0.76 - 71.84 - 72.600.20
-.0946 -0.76
The first column of numbers in this table gives the unknown
quantity as found immediately from the eliminating equations.
These quantities being multiplied by the factors given in
27, we have the corrections to the tabular secular varia-
tions, as given in the column "correction." The next column
gives the value of the tabular secular variations, which are
in all cases those actually adopted by LEVERRIER. In the
case of the Earth, as has been pointed out by STURMER and by
INNES, the secular variation of the radius vector does not cor-
respond to that of the longitude. But as that of the longitude
is the preponderating quantity in its effect on geocentric
46, 47] CORRECTIONS TO THE SOLAK ELEMENTS. 95
places, I have regarded the value of the eccentr city used in
the tables of the equation of the center as the tabular one to
be adopted.
The numbers in the column "Unknown," which are followed
by the letters J and N, are the respective values of [J] L and
[aST]i, which are changed to <5i and sin id 8 by the equations
of 41.
Finally, we have the results given in the last column for the
actual secular variations of the several elements as derived
from the preceding discussion of all the observations.
The result is followed by the probable mean error of each of
the quantities as estimated from the probable magnitude of
the sources of error to which they are liable. As in other
cases, these quantities are very largelv a matter of judgment,
because the probable errors as determined in the usual way
from the eliminating equations would be entirely unreliable.
Definitive corrections to the solar elements for 1850.
47. Leaving the above results to be subsequently discussed,
we go on with the solution of the equations. By a continuation
of the process just described, we regard the preceding secular
variations as known quantities, and substitute them in the
eliminating equations for the solar elements which are derived
from the normal equations for each planet. By this substitu-
tion, we reach three fresh sets of values of the corrections of
the solar elements themselves, one set from the observe tions
of each planet, which are to be reduced to 1850 and combined
with those already found from observations of the Sun, in
order to obtain the most probable result.
Here we meet with the same difficulty that confronted us in
the case of the secular variations. With the exception of the
Sun's mean longitude, we are to regard the results derived
from each of the planets as affected by obscure sources of
systematic error, the probable magnitude of which can only
be inferred from the general deviation of the quantities them-
selves. As in the former case, a is not regarded as a quantity
independently determined, but a-\- 61" has been taken instead.
The concluded value of a is "then found by subtracting 61' '/
from 61" -f- a. Since the corrections to the solar elements
pertain to each separate epoch, those derived from the obser-
96
ELEMENTS OF EARTH'S ORBIT.
[47
vations of the planets are severally reduced to 1850, and the
results are shown in the following table :
Separate values of the corrections to the solar elements for 1850,
after the above definitive values of the secular variations are
substituted in the eliminating equations from solution B,
reduced to 1850.
Je
*>>
6f
e "6K"
a + <J/"
a
From observations of
The Sun
//
. 30
//
+-5
(f
+ .10
//
. oo
//
. 02
//
.07
Mercury
+ I 3
H--7
+.48
47
-f .60
+ -53
Venus
-f. 13
.17
+.06
7
-f .34
+ -5
Mars
+ 2 5
+24
.83
.82
+ i. 18
+ 94
Adopted
. 20
.02
-f . 12
.04
+ -46
+ -48
These adopted values are employed in the subsequent stages
of the discussion, but are not in all cases regarded as definitive.
In the case of f the value 0".20 is that which I have actually
used in the subsequent determinations of the elements, but for
the final value of the obliquity it will be seen that I have
taken O'MS as more probable.
CHAPTER Y.
MASSES OF THE PLANETS DERIVED BY METHODS INDE-
PENDENT OF THE SECULAR VARIATIONS WITH THE
RESULTING COMPUTED SECULAR VARIATIONS.
48. The plan of discussion laid down in Chapter I contem-
plates the determination of the masses of each of the planets
from all data independent of the secular variations, in order
to determine how far the observed secular variations can be
reconciled with these masses. The following is a summary of
these determinations. The planets outside of Jupiter need no>
discussion, as the well-known determinations of their masses
are amply accurate for all our present purposes.
Mass of Jupiter.
49. One of the works connected with the present subject has
been the determination of the mass of Jupiter from the motions
of (33), Polyhymnia. My work on this subject has not yet been
printed in full, but I have given in Astronomische Nachrichten y
No. 3249 (Bd. 136, S. 130), a brief summary of the results. The
mass of Jupiter has been derived not only from the motions
of Polyhymnia, but from such other sources as seemed best
adapted to give a reliable -result. The following table, tran-
scribed from the publication in question, shows the separate
results and the conclusions finally reached :
Reciprocal of mass of Jupiter from wt
All observations of the satellites, 1047.82 1
Action on FATE'S comet (MOL.LER), 1047.79 1
Action on Themis (KRUEGKER), 1047.54 5
Action on Saturn (HiLL), 1047.38 7
Action on Polyhymnia, 1047.34 20
Action on WINNECKE'S comet (v. HAERDTL), 1047.17 10
1047.35
in. e. 0.065
5690 N ALM 7 97
98 MASS OF JUPITER. [49
It will be seen that the result from observations of the- satel-
lites has been assigned a very small weight. This course has
been indicated by the circumstances. Other conditions being
equal, the greater the mass of a planet the less the propor-
tionate precision with which that mass can be determined by
observations on the satellites. In any case, if the measures of
the distances between the satellites and the primary are' in
error by a small fraction, or, of their whole amount, then the
error of the mass will be in error by the fraction 3 a of its
amount. For reasons founded on the construction and use of
the heliometer, I doubt whether the absolute measures made
with those forms of that instrument which have been used in
determining the mass of Jupiter can be relied upon within
their three-thousandth part. If so, the determination of the
mass of the planet itself would be doubtful by its thousandth
part in each separate case. The chance of personal equation
between transits of the satellites and the planet vitiates in the
same way the results from observed transits of the planet and
satellites. Notwithstanding the great refinement of the dis-
cussion by KEMPF of observations made at Potsdam, and the
care with which he, SCHUR, and others have determined the
mass of Jupiter by a discussion of all the^observations of the
satellites, I can not conceive that the probable error of any
possible result they could derive would be less than 0.3 or 0.4
in the denominator.
In this connection the discordances between" the mass of
Saturn, found by Prof. HALL and by other observers from
observations of the satellites, are worthy of consideration.
They lead us to suspect that perhaps it is through good for-
tune rather than by virtue of their absolute reliability that
determinations of the mass of Jupiter from observations of the
satellites have agreed so well.
As to the weights assigned to the other results, only the last
needs especial mention. The probable error assigned by v.
HAERDTL to his result is very much smaller than that which
I find for the mean of all the results. But, as remarked in the
paper in question, it has received a smaller relative weight
than that corresponding to its assigned probable error, because
of distrust on my part whether observations on a comet can
49, 50, 51] MASSES OF THE EARTH, MARS, AND JUPITER. 99
be considered as having always been made 011 the center of
gravity of a well-defined mass, moving as if that center were
a material point subject to the gravitation of the Sun and
planets. This distrust seems to be amply justified by our
general experience of the failure of comets to move in exact
accordance with their ephemerides.
I propose to accept the value thus found,
Mass of Jupiter = 1 4- 1047.35
as the definitive one to be used in the planetary theories.
Mass of Mars.
50. In consequence of the minuteness of the mass of Mars,
measures of its satellites, especially the outer one, afford a
value of its mass much better than can be derived by its action
on the planets. When nearest the earth, the major axis of the
orbit of the outer satellite subtends an angle of 70". I can
not think that the systematic error to be feared in the best
measures, such as those made by Prof. HALL, can be as great
as half a second. It therefore Appears to me that the mean
error in adopting Prof. HALL'S value of the mass does not
exceed its fiftieth part. This is a degree of precision much
higher than that of any determination through the action of
Mars on another planet.
Prof. HALL'S measures of 1892 show a minute increase of
the mean distance given by his work of 1877. The result is
v>" = + 0.014
These observations, however, were made when the position of
the orbit of the satellite was unfavorable to an exact deter-
mination of the elements of motion. I have adhered to the
original value in the work of the present chapter.
Mass of the Earth.
51. I have already pointed out the difficulty in the way of
determining the mass of the Earth from its action on the
other planets. On tbe other hand, the solar parallax has, in
recent years, been determined in various ways with such
precision that the mass of the Earth to be used in the plan-
100 MASS OF THE EARTH. [51
etary theories can best be derived from it. The theory of the
relation between the mass of the Earth and its distance from
the Sun, as given by observations of the seconds pendulum
and the length of the sidereal year, is one of the best estab-
lished results of celestial mechanics. It is, in effect, the
principle on which the lunar theory is constructed. In this
theory the disturbing action of the Sun is necessarily a func-
tion of the ratio of the mass of the Sun to that of the Earth.
But in the accepted theory this ratio is eliminated through
the ratio of the lunar month to the sidereal year. From the
well-established ratio between the distance of the Moon and
the length of the seconds pendulum, the ratio of the masses
of the Sun and Earth come out of this theory with great
precision. It need not be developed here; it will suffice to
give the numerical result, which is that between the ratio M
of the mass of the Sun to that of the Earth and the mean
equatorial horizontal parallax of the Sun in seconds of arc
there exists the relation
7r 3 M = [8.35493]
I have derived seven values of the solar parallax by different
methods, of which the following are the preliminary results :
wt.
GILL'S observations of Mars, 1877, 8.780 .020 1
Contact observations, transits of Venus. 8.704 i .018 1
Aberration and velocity of light, 8.798 i .005 16
Parallactic equation of the Moon, 8.799 .007 5
Measures of small planets on GILL'S plan, 8.807 i .007 8
LEVERRIER'S method, 8.818 .030 0.5
Measures of Venus from Sun's center, 8.857 i .022 1
Mean result, n = 8".802 i 0".005
I have provisionally taken this mean as the most probable
value of the solar parallax derived from all sources except the
mass of the Earth. The above relation then gives
M = 332 040
51,52] MASS OF VENUS. 101
Taking for the mass of the Moon 1 4- 81.52, we have for the
ratio of the combined masses of the Earth and Moon to the
mass of the Sun
m"
328 016
a result of which the probable error may be regarded as some-
thing more than a thousandth part of its whole amount.
Mass of Venus.
52. The mass of Venus adopted in the provisional theory,
to which LEVERRIER'S tables were reduced, was .000 002 4885
= 1 +- 401847, which is that of LEVERRIER'S tables of Mer-
cury. In the preceding discussions the following three factors
of correction to this mass have been found :
From observations of the Sun . . .0118 =t .0034
From observations of Mercury . . .0121 =t .0050
From observations of Mars ... .0076 =t . ( ? )
Mean - .0119 .0028
The mean error assigned to the result from observations of
the Sun may be regarded as real, because the result is the
mean of a great number of completely independent determina-
tions, among which no common error is either a priori prob-
able or shown by the discordance of the results. In the
case of Mercury, however, as already remarked, the effect of
systematic errors is such that, although they are almost com-
pletely eliminated from the result, the mean error computed
in the usual way would be misleading. The weight assigned
is therefore largely a matter of judgment.
The fact that it was necessary to introduce an empirical
correction, with a period of about forty years, into the mean
longitude of Mars, vitiates the determination of the mass of
Venus from its action on that planet, because one of the prin-
cipal terms in the action of Venus on Mars has a period which
does not differ from forty years enough to make the determi-
nation of the mass independent of this empirical correction.
I have therefore assigned no weight to the result. We thus
102 MASS OF MERCURY. [52,53
have for the mass of Venus, as derived from the periodic per-
turbations of Mercury and the Earth produced by its action.
m' = 1 -^ 406 690 i 1140
Mass of Mercury.
53. The mass of Mercury which I have heretofore adopted,
1 -=- 7 500 000, was rather a result of general estimate than of
exact computation. The fact is that the determinations of
this mass have been so discordant, and varied so much with
the method of discussion adopted, that it is scarcely possible
to fix upon any definite number as expressive of the mass.
An examination of LEVERRIER'S tables of Venus shows that
with the mass of Mercury there adopted (1:3 000 000) Mercury
frequently produces a perturbation of more than one second
in the heliocentric longitude of Venus. When the latter is
near inferior conjunction, the actual perturbation will be more
than doubled in the geocentric place, so that the latter might
not infrequently be changed by 1", even if the mass of Mer-
cury be less than one-half LEVERRIER'S value. It was there-
fore to be expected that a fairly reliable value of the mass of
Mercury would be obtained from the periodic perturbations
of Venus.
Eeferriug to 27, it will be seen that the indeterminate mass
of Mercury appears in the equations in the form
1+7;,
3000000
From the solution B, 38, the value of /* comes out
^ = _ 0.0834
corresponding to a mass of Mercury of 1 : 7 210 000. But in
a subsequent solution of the equations, when the secular vari-
ations are determined from theory and substituted in the
normal equation for /v, we find
,u = - 0.0889
which gives
m = 1 -4- 7 943 000
The work of the present chapter is based on the former
value.
53] MASS OF MERCURY. 103
A consideration of the probable error of this result is impor-
tant. The fortuitous errors which mostly affect it are of the
class which I have termed semi- systematic. Under this term I
include that large class of errors which, extending through or
injuriously affecting a limited series of observations, cause the
probable error of a result to be larger than that given by the
solution of the equations, but which, nevertheless, like purely
accidental ones, would be eliminated from the mean result of
an infinite series of observations. To this class belong the
errors arising from personal equation in observing the limb of
Venus, or, what is the same thing, a difference between the
practical semidiameter corresponding to the observer and that
adopted in the reductions. We may suppose that, during a
period of several days, when Venus is not far from inferior
conjunction, its geocentric position is affected by a perturba-
tion produced by Mercury. Through the error alluded to, all
the observations made by any one observer, and in fact all
that are made anywhere, may be affected by a certain con-
stant error in Right Ascension. Near another inferior con-
junction the same state of things may be repeated, with the
perturbation in the opposite direction. If, now, tne observa-
tions were made by the same observer, and under the same
circumstances, the personal error would be eliminated from
the mean of these two results so far as the mass of Mercury is
concerned. But very frequently different observers will have
made the observations under the two circumstances, and dif-
ferent conditions will have prevailed. Thus, it is only through
the general law of averages that we can expect the effect of
these fortuitous but systematic errors to be completely elim-
inated. That they would be eliminated in the long run is
evident from the fact that there can be no permanent rela-
tion between the personal equations of the observers and the
changes in the action of Mercury upon Venus. Moreover,
Venus has been observed with a fair degree of accuracy
through more than half a century, and it seems reasonable
to suppose that during that time the errors in question would
nearly disappear.
It is clear from these considerations that the probable
error derived from the solution of the equations would be
104
MASS OF MERCURY.
[53
entirely misleading. But a probable error which ought to be
reliable can be obtained by a process similar to that which I
have adopted elsewhere in this paper, namely, dividing up the
materials into periods, and determining the probable error from
the discordances among the results of the several periods.
This probable error will be reliable, because there is no reason
why the same error should affect the mass of Mercury through
any two periods. I therefore take the partial normal equa-
tions in n derived from Eight Ascensions during the several
periods, substitute in them the values of the unknown quanti-
ties found from solution B, /* excepted, and thus form six-
teen partial normal equations in /i. These equations may be
changed into the corresponding equations of condition, of
weight unity, by dividing each by the square root of the
coefficient of the unknown quantity. The residuals then left
when the definitive value of the unknown quantity is substi-
tuted will be those from whose discordance the probable error
may be inferred.
The partial normal equations thus found from the Eight
Ascensions are as follow:
1750-'62.
44;
i= - 38
1830-'40.
5649;
i=- 831
1765-'74.
1265
-165
1840-'49.
2913
- 18
1775-'86.
15
- 5
1849->56.
2238
- 49
1787-'96.
209
4- 53
1857->64.
4506
- 129
1796->06.
345
4- 19
1865-'71.
7736
- 265
1806-'14.
439
4-135
1871-79.
7062
761
1814->19.
942
+ 2
1879-'86.
4958
- 407
1820-'30.
1786
330
1885-'92.
9561
-1306
Sum: 49668/<= -4095
^ = _ 0.0824 i .019
The difference between this value of yw, which is obtained
only to find the probable error, and that formerly found, arises
principally from the omission of the declination equations.
The mean error corresponding to weight unity comes out
ft = 4".2
53] MASS OF MERCURY. 105
which, as anticipated, is much larger than that which would
be given by the discordance of the original observations.
This does not mean that the original observations are affected
by any such mean error as 4".2, but that the discordances
between the 16 values of /* are as great as we should expect
them to be if the original observations were absolutely free
from systematic error, but affected by purely accidental errors
of this mean amount.
The results of the solution for the mass of Mercury may be
expressed in the form
. 10.32 d
~ '
7 210 000 ' 7 043 000
In all researches which have been made on the motion of
ENCKE'S comet by ENCKE, VON ASTEN, and BACKLUND, the
determination of this mass has been kept in view. The
results are, however, so discordant that, as already remarked,
scarcely any definitive result can be derived from them.
To this statement there is, however, one apparent exeeption.
In an appendix to his very careful and elaborate discussion of
WINNECKE'S comet, VON HAERDTL has derived the value of
the mass of Mercury from all the return of ENCKE'S comet as
worked up by VON ASTEN and BACKLTJND.* The only inter-
pretation which I can put upon his result is this : If we regard
the acceleration of the comet, which it is supposed results
from all the observations made upon it, as non-existent, the
following two masses of Mercury are derivable from the obser-
vations :
1819-1868, w = 1 4- 5 648 600 i 2000
1871-1885, m = 1 +- 5 669 700 600 000
He also finds, from the motion of WINNECKE'S comet,
m = 1 4- 5 012 842 697 863
* Denkschriften der Kaiserlichen Akadeinie der Wissenschaften, Vol.
56, p. 172-175. Vienna, 1889.
106 MASS OF MERCURY. (53,54
and from four equations of LEVERRIER
1 4- 5 514 700 100 000
The consistency of these results seems to me entirely beyond
what the observations are capable of giving, and I hesitate to
ascribe great weight to them. Moreover, the result implicitly
contained in these numbers, that the supposed secular accel-
eration of the comet disappears when we attribute the pre-
ceding mass to Mercury, merits farther inquiry.
The probable density of the planet may form a basis for at
least a rude estimate of its probable mass. The fact that the
Earth, Yenus, and Mars have densities not very different from
each other, while that of the Moon is 0.6 the density of the
Earth, leads us to suppose that Mercury, being nearest to the
Moon in mass, has probably a slightly greater density. Its
diameter at distance unity has been repeatedly measured and
found to be 6".6, or, roughly speaking, three-eighths that of the
Earth. Were its. density 0.7, its mass would therefore be
about 1 : 9,000,000. In view of the fact that the measured
diameter is probably somewhat too small, these consider-
ations lead us to conclude that the mass is probably between
1:6,000,000 and 1:9,000,000.
As the value of the mass to be used in investigating the
secular variations, I have adopted
v = 0.08
1 08
Mass of Mercury =
7 500 000
Secular variations resulting from theory.
54. In the Astronomical Papers, Vol. V, Part IV, were com-
puted the secular variations of the elements of the orbits in
question using, as the basis of the work, the values of the
THEORETICAL SECULAR VARIATIONS.
107
54]
masses whose reciprocals are found in the column A below.
In column B are cited the masses which I have decided upon.
A
B
Original
Adpofced
reciprocal
reciprocal
of mass.
of mass.
V
Mercury,
7 500 000
6 944 444
+ .080
Venus,
410 000
406 750
+ .0080
Earth + Moon,
327 000
328 000
-.00305
Mars,
3 093 500
3093500
Jupiter,
1047.88
1047.35
+ .00050
Saturn,
3501.6
Uranus,
22756
Neptune,
19 540
In the case of the Earth we have to add the secular varia-
tion of the perihelion produced by the non-sphericity of the
system Earth + Moon. For the principal term I have found,
D t e" d n" = + 0".129
The resulting values of the secular variations, expressed as
functions of v, v 1 , v"j v 1 ", are given in the following section:
Theoretical secular variations for 1850.
Mercury.
// // // // // //
D t e = + 4.22 +0.00i/+ 2.8 K' + l.lF // -0.1?/ / " = + 4.24
6D t 7Ti =+109.36+0.00 +56.8 +18.8 +0.5 = + 109.76
D t ?: =+ 6.76 -0.04 - 0.6 - 1.4 +0.0 =+ 6.76
sin i Dt ; = 92.12 0.33 -49.3 -12.2 1.2 =92.50
Venus.
D t e =- 9.58 1.30^+ 0.0^'- 4.9*/ // 0.2v /// =- 9.67
eDtTTt =+ 0.39-0.81+0.0 -3.9 +0.5 =+ 0.34
D t i =+ 3.43+0.76 + 0.0 + 0.0 -0.3 =+ 3.49
sin iT> t # =-105.92 +0.26 -29.2 -43.2 -1.2 =-106.00
108 THEORETICAL SECULAR VARIATIONS [54
Earth.
// // // // //
T> t e = - 8.57 -0.12F+ 1.3^' -1.6i'"' = - 8.57
eT> t 7t = + 19.36-0.18 + 5.8 +1.6 =+ 19.39
D t * =- 46.65-0.21 -28.3 -0.7 ==-46.89
Mars.
// // // // // //
D t e =+ 18.71 +0.03T/4- O.lv'4- 2.1v 7/ =4- 18.71
eDtTf! = + 148.824-0.06 + 4.6 +21.4 = + 148.80
D t i =- 2.34-0.04+12.0 +0.0 +O.OF x// =- 2.25
intD t <y=- 72.43-0.27 -25.1 - 7.4 -1.0 =- 72.63
CHAPTER VI.
EXAMINATION OF THE HYPOTHESES BY WHICH THE
DEVIATIONS OF THE SECULAR VARIATIONS FROM
THEIR THEORETICAL VALUES MAY BE EXPLAINED.
55. *The investigations of the present chapter are founded
on a comparison of the secular variations derived purely from
observations in Chapter IV, with those resulting from the
values of the masses obtained independently of the secular
variations in the last chapter. For the sake of clearness,
these two sets of secular variations and their differences are
collected in the following table. The mean errors assigned to
the theoretical values are those which result from the prob-
able mean errors of the respective masses. They are there-
fore not to be regarded as independent. The mean errors
given in the column of differences are those which result from
a combination of those of the other two columns. The errors
of the observed quantities must not, however, be judged from
those of the differences, because subsequent changes in the
masses of Mercury, Venus, and the Earth may produce a
general diminution in the discordances.
Mercury.
Observation. Theory. Diff. A \/w.
// // // // // // //
D t <? + 3.36+0.50 + 4.24 + .01 -0.88i.50 -0.86 2
6D t 7r + 118.24+0.40 +109.76+. 16 +8.4S+. 43 . .
D t i + 7.14+0.80 + 6.76+.01 +0.38^.80 +0.38 1J
siniD t # - 91.89+0.45 - 92.50+.16 +0.61+.52 +0.23 2.2
Venus.
D t e 9.46+0.20- 9.67+.24 +0.21+.31 +0.12 5
eD t 7T + 0.29 i 0.20 + 0.34+.15 -0.05+.25 . .
D t * + 3.87 0.30 + 3.49i.l4 +0.38i.33 +0.44 3J
siniD t # -105.400.12 -106.00i.12 +0.60i.l7 +0.52 8
109
110 COMPARISON OF SECULAR VARIATIONS. [55
JEartli.
Observation. Theory. Diff. A Vw.
// // // // // // //
D t e - 8.55 0.09 - 8.57 .04 +0.02 .10 +0.02 10
eD t 7T + 19.48 0.12 + 19.38+ .05 +0.10i. 13 . .
D t - 47.lliO.23 - 46.89+.09 0.22+.27 -0.46 4
Mars.
D t e + 19.OOiO.27 + 18.71+.01 -fO.29i.27 +0.29 3.7
0D t 7r 4-149.55iO.35 + 148.80+.04 +0.75+.35 ...
D t i 2.26+0.20- 2.25 +.04 -0.01 + .20 +0.08 5
siniD t # - 72.60 + 0.20 - *2.63+.09 +0.03+.22 -0.17 5
If we multiply the mean errors given by 0.6745, to reduce
them to probable errors, we shall see that only four of the
fifteen differences are less than their probable errors. The
deviations which call for especial consideration are the follow-
ing four :
1. The motion of the perihelion of Mercury. The discord-
ance in the secular motion of this element is well known.
2. The motion of the node of Venus. Here the discordance
is more than five times its probable error.
3. The perihelion of Mars. Here the discordance is three
times its probable error.
4. The eccentricity of Mercury. The discordance is more
than twice its probable error. It is to be remarked, however,
that the probable error of this quantity is very largely a
matter of judgment, and that its value may have been under-
estimated.
The deviations, if not due to erroneous masses, may be
explained on two hypotheses. One is that propounded by
Prof. HALL,* that the gravitation of the Sun is not exactly as
the inverse square, but that the exponent of the distance is a
fraction greater than 2 by a certain minute constant. This
hypothesis accounts only for the motions of the perihelia, and
not for any other discordances.
The other hypothesis is that of the action of unknown
masses or arrangements of matter. Since the latter hypothesis
* Astronomical Journal, Vol. XIV, p. 7.
55,56] NON-SPHERICITY OF THE SUN. Ill
would account for other motions than those of the perihelia, it
might seem that the existence -of the other discordances
tells very strongly in its favor. The hypotheses of possible dis-
tributions of unknown matter, therefore, have iirst to be con-
sidered.*
Hypothesis of non- sphericity of the Sun.
56. In a case where our ignorance is complete, all hypotheses
which do not violate known facts are admissible. Beginning
at the center and passing outward, the first question arises
whether the action may not be due to a non -spherical distri-
bution of matter within the body of the Sun, resulting in an
excess of its polar over its equatorial moment of inertia. The
theory of the Sun which has in recent times been most gener-
ally accepted is that its interior may be regarded as gaseous,
or rather as a form of matter which combines the elasticity
and mobility of a gas with the density of a liquid. Such
being the case, we may conceive that vortices of which the
axes coincide with that of rotation may exist in the interior
in such a way that the surfaces of equal density are non-
spherical. A very small inequality of this sort would suffice
to account for the motion of the perihelion of Mercury.
This hypothesis admits of an easy test. Whatever be the
nature or amount of the inequality, a simple computation
shows that to account for the observed phenomenon it is
necessary and sufficient that the equipotential surfaces at the
surface of the Sun should have an elliptic! ty of rather more
than half a second of arc. It can not, I conceive, be doubted
that the visible photosphere is an equipotential surface. We
have then to inquire whether there is any such ellipticity of
the photosphere as that required by the hypothesis. This
question seems completely set at rest by the great mass of
heliometer measures made by the German observers in con-
nection with the transits of Venus of 1874 and 1882, which
have been discussed by Dr. AUWERS. The general result is
* After carrying out the investigations of this chapter, I find that the
subject was studied on similar lines by Dr. P. HARZER nearly three years
ago, and that I made certain suggestions on the subject to Dr. BAUSCH-
INGER ten years ago. See Astrononiacliv Nachrichten, Vol. 109, p. 32, and
Vol. 127, p. 81.
112 INTRA-MERCURIAL GROUP. [56,57
that the mean of the equatorial measures are slightly less than
the mean of the polar measures, the difference, however, being
within the probable errors of the results. I conclude that
there can be no such n on- symmetrical distribution of matter
in the interior of the Sun as would produce the observed effect.
This same conclusion seems to apply to matter immediately
around the photosphere. An equatorial ring of planetoids, or
gaseous substances of the required mass, very near the photo-
sphere, would render the equipotential surfaces of the photo-
sphere elliptical to a degree which seems precluded by the
measures in question. At a very short distance from the sur-
face, however, the effect would be inappreciable.
Hypothesis of an intra-mercurial ring or group of planetoids.
57. Passing outward, we have next to consider the hypothe-
sis of an intra-mercurial ring adequate to produce the observed
phenomena. In a first approximation we may suppose the
ring circular. Its mass can not be determined, because it will
depend upon the distance ; we have to determine a certain
function of the mass and distance adequate to produce the
observed motion of the perihelion. Then we must inquire what
effect the ring will have on the secular variations of the other
elements, both of Mercury and of the other planets, and see if
these effects can be reconciled with observation. In the com-
putations I have assigned to the excess of motion the pro-
visional value 40 // .7. If the ring is not very distant from the
Sun the motion which it will produce in the perihelion of a
planet whose mean motion is n and whose mean distance is a
may be represented in the form
}JL being a function of the mass of the ring and of its radius,
which is nearly the same for all of the planets, so long as the
radius of the ring is only a small fraction of the distance of
Mercury. A first approximation to /* is
u = . m r 2
57 ) INTRA-MERCTJRIAL
m being tke ratio of its mass to that of the Sun alSSTFTEs radius.
Multiplying these motions in the case of the four planets by
their eccentricities, we find that the hypothetical ring will
produce the following secular variations :
Mercury, D t n 40.7; eD t n = 8.38
Venus, 4.6 0.031
Earth, 1.5 0.025
Mars, 0.34 0.031
Owing to the sinallness of the eccentricities the effect is
insensible, except in the case of Mercury, so that the ring will
not account for the observed excess of motion of the perihelion
of Mars.
Such a ring will necessarily produce a motion of the plane
of the orbit of Mercury or Venus, or of both, because it can
not lie in the plane of both orbits.
Let us put ii for its inclination to the ecliptic, and 61 for the
longitude of its node on the ecliptic; and let us put, also,
Pi = ii sin #!
q l = i, cos 6\
and let j?, p', ... , q, q', . . be the corresponding quan
tities for the planets. The theory of the secular variations
then shows that the ring will produce a motion of the plane of
the orbit of Mercury given by the equations
D t^i = *-ll (9i ~q} = 40".7 (q l - q)
Expressing the motions of p and q in terms of the motions of i
and 0, which is necessary, owing to the very different weights
of the determination of the motion of the planes of Mercury
and Venus in the direction of these two coordinates, we have
5690 N, ALM 8
114 INTRA-MERCURIAL GROUP. [57
the following expressions for these two motions, which we
equate to the observed excesses :*
- 4.96 4- 26.9 i 4- 28.4_p t = + 0.57 0.50
- 0.27 + 0.8 4-3.0 = 4- 0.63 0.12
0.00 4- 28.4 - 26.9 = + 0.50 0.80
0.00 4- 3.0 - 0.8 =4- 0.45 0.30
0.00 0.0 1.5 = - 0.25 0.25
Multiplying the conditional equations thus formed by such
factors as will make the mean error of each equation nearly
0".5, we have the following conditional equations for p {
and #1 :
27 q, 4- 28^! = + 5.53
3 + 12 = 4. 3.60
17 _ 16 =4- 0.30
5 - 1 = 4- 0.77
- 3 = - 0.50
The solution of these equations gives very nearly
^=4-0.12; 0i=4&
This great inclination seems in the highest degree improbable
if not mechanically impossible, since there would be a tend-
ency for the planes of the orbits of a ring of planets so
situated to scatter themselves around a plane somewhere
between that- of the orbit of Mercury and that of the invari-
able plane of the planetary system, which is nearly the same
as that of the orbit of Jupiter. Moreover, the motion of the
perihelion of Mars is still unaccounted for and that of the
node of Venus only partially accounted for, as shown by the
large residual of the second equation. In fact, the great incli-
nation assigned to the ring- comes from the necessity of repre-
senting as far as possible the latter motion.
* It will be noticed that iii forming these equations I have neither used
the final values of the absolute terms, nor taken account of the fact that
the observed motions of the planes are referred to the ecliptic. Changes
thus produced in the equations are too minute to affect the conclusion.
57, 58] ZODIACAL LIGHT. 115
There would of course be no dynamical impossibility in the
hypothesis of a single planet having as great an inclination as
that required. But I conceive that a planet of the adequate
mass could not have remained so long undiscovered. Whether
we regard the matter as a planet or a ring, a simple computa-
tion shows that its mass, if at the Sun's surface, would be
about ri that of the Sun itself, and one-fourth of this if at a
distance equal to the Sun's radius. We may conceive, if we
can not compute, how much light such a mass of matter would
reflect. Altogether, it seems to me that the hypothesis is
untenable.
Hypothesis of an extended mass of diffused matter like that which
reflects the zodiacal light.
58. The phenomenon of the zodiacal light seems to show
that our Sun is surrounded by a lens of diffused matter which
extends out to, or a little beyond, the orbit of the Earth, the
density of which diminishes very rapidly as we recede from
the Sun. The question arises whether the total mass of this,
matter may not be sufficient to cause the observed motion.
So far as the action of that portion of matter which is near
the Sun is concerned, the conclusions just reached respecting
a ring surrounding the Sun will apply unchanged, because we
may regard such a mass as made up of rings. Observation
seems to show that the lens in question is not much inclined
to the ecliptic, and if so it would produce a motion of the
nodes of Venus and Mercury the opposite of that indicated
by the observations.
There is another serious difficulty in the way of the hypoth-
esis. A direct motion of the perihelion of a planet may be
taken as indicating the fact that the increase of its gravitation
toward the Sun as it passes from aphelion to perihelion is
slightly greater than that given by the law of the inverse
square. This increase would be produced by a ring of matter
either wholly without or wholly within the orbit. But if we
suppose that the orbit actually lies in the matter composing
such a ring, the effect is the opposite; gravitation toward the
11 6 EXTRA-MERCURIAL GROUP. [58, 59, 60
Sun is relatively diminished as the planet passes from aphelion
to perihelion, and the motion of the perihelion would be retro-
grade.
It can not be supposed that that part of the zodiacal light
more distant from the Sun than the aphelion of Mercury is
even as dense as that portion contained between the aphelion
and the perihelion distances. The result in question must
therefore be due wholly to that part of the matter which lies
near to the Sun, and we thus have all the difficulties of the
intra-mercurial ring theory, with one more added.
Hypothesis of a ring of planetoids between, the orbits of Mercury
and Venus.
59. It appears that any ring or zone of matter adequate
to produce the observed effect must lie between the orbits of
Mercury and Venus. Its assignment to this position requires
a more careful determination of its possible eccentricity.
There will be six independent elements to be determined;
the mass, the mean distance, the eccentricity, the perihelion,
the inclination, and the node.
I find that the observed excesses of motion of the elements
of Mercury and Venus will be approximately represented by
elements not differing much from the following:
Total mass of group 37000000
Mean distance 0.48
Eccentricity of orbit 0.04
Longitude of perihelion . . , . 10
Longitude of node 35
Inclination to ecliptic* 7.5
Probable diameter at distance unity if
agglomerated into a single planet . 3".5
Considerations on the admissibility of the hypothesis Possible
mass of the minor planets.
60. Although the preceding hypothesis is that which best
represents the observations of Mercury and Venus, we can
not, in the present condition of knowledge, regard it as more
than a curiosity. True, it is plausible at first sight. Since,
60] POSSIBLE ACTION OF THE MINOR PLANETS. 117
as already remarked, any disturbing body of sufficient mass
to cause the observed excess of motion of the perihelion of
Mercury would change the position of the planes of the orbits,
and since observations give apparent indications of such a
change in the plane of the orbit of Venus, it might appear
that we have here a very good ground for the view that all
the motions are due to the attraction of unknown masses.
But the great difficulty is that the excess of motion of the
orbital planes is in the opposite direction from what we should
expect. A group of bodies revolving near the plane of the
ecliptic would produce a retrograde motion of the nodes. But
the observed excess is direct. A direct motion can be pro-
duced only in case the orbits are more inclined than those of
the disturbed planet. In admitting such orbits we encounter
difficulties which, if not absolutely insurmountable, yet tell
against the probability of the hypothesis.
The hypothesis carries with it the probable result that the
excess of motion of the perihelion of Mars is produced by the
action of the minor planets. I have considered the question
of this action in an unpublished investigation. From the prob-
able albedo and magnitude of the minor planets and the obser-
vations of BARNARD and others on their diameters, I have
determined the probable mass of each part of the group having
a given opposition magnitude. The result is that the number
of these bodies having such a magnitude appears to progress
in a fairly uniform manner through several magnitudes. The
ratio of progression may lie anywhere between the limits 2
and 3. Up to the limit 3 the total mass, if continued on to
infinity, could not produce any appreciable effect on the motion
of Mars. But if we suppose a larger ratio than 3 to prevail,
then the number of planets of smaller magnitude would be so
numerous us to form a zone of light across the heavens, as may
readily be seen by considering that the total amount of light
reflected from the planets of each order of magnitude would
form an increasing series, since the ratio between the brillian-
cies of two objects of unit difference in magnitude is only
about 2.5. We may therefore suppose that the faint band of
light which is said to be visible across the entire heavens as
a continuation of the zodiacal light, as well as the "gegen-
118 HALL'S HYPOTHESIS. [60,61
schein," is due to these minute bodies, and yet find their total
mass too small to produce any appreciable effect.
Whether we can assign to the components of such a group
any magnitude so small that they would be individually invis-
ible, and a number so small that they would not be seen
collectively as a band of light brighter than the zodiacal arch,
and yet having a total mass so large as to produce the observed
effects, is a very important question which can not be decided
without exact photometric investigations. It is, however, cer-
tain that if we could do so we should have to suppose a very
unlikely discontinuity in the law of progression between each
magnitude and the number of bodies having that magnitude.
It must therefore suffice for our present object that we regard
the hypothesis of such bodies as unsatisfactory.
Hypothesis that gravitation toward the sun is not exactly as the
inverse square of the distance.
61. Prof. HALL'S hypothesis seems to me provisionally not
inadmissible. It is, that in the expression for the gravitation
between two bodies of masses m and m' at distance r
Force =
the exponent n of r is not exactly 2, but 2 + 6, d being a very
small fraction. This hypothesis seems to me much more
simple and unobjectionable than those which suppose the
force to be a more or less complicated function of the relative
velocity of the bodies. On this hypothesis the perihelion of
each planet will have a direct motion found by multiplying its
mean motion by one-half the excess of the exponent of grav-
itation.
Putting
n = 2.000 000 1574
the excess of motion of each perihelion of the four inner
planets would be as follows. It will be seen that the evidence
in the case of Venus and the Earth is negative, owing to the
01] LAW OF GRAVITATION. 119
very small eccentricities of their orbits, while the observed
motion in the case of Mars is very closely represented.
Mercury,
42.34
8.70
Venus,
16.58
0.11
Earth,
10.20
0.17
Mars,
5.42
0.51
An independent test of this hypothesis in the case of other
bodies is very desirable. The only case in which there is any
hope of determining such an excess is that of the Moon, where
the excess would amount to about 140" per century. This is
very nearly the hundred- thousandth part of the total motion
of the perigee. The theoretical motion has not yet been com-
puted with quite this degree of precision. The only determi-
nation which aims at it is that made by HANSEN.* He finds
Theory. Obser. Diff.
// // //
Annual mot. of perigee, 146 434.04; 146 435.60; ^+1.56
Annual mot. of node, -69 676.76 69 679.62; -2.86
The observed excess of motion agrees well with the hypoth-
esis, but loses all sustaining force from the disagreement in
the case of the node. The differences HANSEN attributes
(wrongly, I think) to the deviation of the figure of the Moon
from mechanical sphericity.
Consistency of Hall's hypothesis with the general results of the
law of gravitation.
62. The law of the inverse square is proven to a high degree
of approximation through a wide range of distances. The close
agreement between the observed parallax of the Moon and
that derived from the force of gravitation on the Earth's sur-
face shows that between two distances, one the radius of the
Earth and the other the distance of the Moon, the deviation
from the law of the square can be only a small fraction of the
*Darlegung, etc.: Abhandhungen der Math.-Phys. Classe der Kon. Sdchsi-
Bchen Gesellscltaft der Wissenschaften, vi, p. 348.
120 HALL'S HYPOTHESIS. [62
thousandth part, or, we may say, a quantity of the order of
magnitude of the five- thousandth part.
Coming down to smaller distances, we find that the close
agreement between the density of the Earth as derived Iroin
the attraction of small masses, at distances of a fraction of a
meter, with the density which we might a priori suppose the
Earth to have, shows that within a range of distance extend-
ing from less than one meter to more than six million meters,
the accumulated deviation from the law can scarcely amount
to its third part. The coincidence of the disturbing force of
the Sun upon the Moon with that computed upon the theory
of gravitation, extends the coincidence from the distance of
the Moon to that of the Sun, while KEPLER'S third law
extends it to the outer planets of the system. Here, however,
the result of observations so far made is relatively less pre-
cise. We may therefore say, with entire confidence, as a
result of accurate measurement, that the law of the inverse
square holds true within its five- thousandth part from a dis-
tance equal to the Earth's radius to the distance of the Sun, a
range of twenty-four thousand times ; that it holds true within
a third of its whole amount through the range of six million
times from one meter to the Earth's radius; and within a
small but not yet well-defined quantity from the distance of
the Sun to that of Uranus, in which the multiplication is
twentyfold.
If HALL'S hypothesis contradicted these conclusions it would
be untenable. But a very simple computation will show that,
assuming the force to vary as r-(* + 8 \ d being a minute con-
stant sufficient to account for the motion of the perihelion of
Mercury, the effect would be entirely inappreciable in the ratio
of the gravitation of any two bodies at the widest range of
distance to which observation has yet extended. Although
the total action of a material point on a spherical surface sur-
rounding it would converge to zero when the radius became
infinite, instead of remaining constant, as in the case of the
inverse square, yet the diminution in the action upon a surface
no larger than would suffice to include the visible universe
would be very small.
63] CORRECTION OF MASSES. 121
Masses of the planets which represent the secular variations of
other elements than the perihelia.
63. On HALL'S hypothesis the secular variations of all the
elements other than the perihelia will remain unchanged.
Our next problem is to consider the possibility of represent-
ing the variations of the other elements by admissible masses
of the known planets. In 55 I have given a comparison of
the secular variations as they result from observations, with
their theoretical expressions in terms of corrections to a cer-
tain system of masses. When the equations thus formed are
multiplied by the factors Vw, which make the mean error of
each equation unity, we have the following system of equa-
tions, in which we put v = 10 #:
Ox
+ QY'
+ 2 v"
+ Ov"'
= - 1.7
r = - 1.8
- 1
- 2
= +0.5
+ 0.5
- 7
-108
- 27
- 3
= +0.5
+ 1.1
-65
- 24
- 1
= +0.6
+ 0.7
+25
- 1
= + 1.5
+ 1.3
+ 21
-234
-346
-10
= 4-4.2
0.0
-12
+ 13
-16
= + 0.2
+ 0.1
- 9
-123
- 3
= -2.0
-0.7
+ 1
+ 8
= +1.1
+ 1.3
2
+ 60
= + 0.4
-0.2
-14
-126
- 37
- 5
= -0.8
-0.2
The resulting normal equations are
5766 x 1563 v 1 4991 v" + 140 v 1 " = + 114
-1563 +101231 + 88556 +3455 670
- 4991 + 88556 + 122462 + 3750 = - 1446
+ 140 + 3455 + 3750 + 401 39
Along with the results of the solution of these equations I
place, for comparison, the values of Chapter Y, which have
been considered most probable.
From sec. var. From other sources.
Wx = v = + 0.070 + 0.08 0.20
v' = + 0.0100 .0056 + 0.0084 0.0028
v" = - 0.0183 .0052 - 0.00304 i 0.0015
v" 1 = - 0.0115 .067 + .0037 i 0.018
122 CORRECTION OF MASSES. [63, 64
By substitution in the conditional equations we find for the
mean error corresponding to weight unity
*i = i 1.14
In forming these equations they were reduced by multipli-
cation to a supposed mean error of 1. Speaking in a
general way we may therefore say that the representation. of
the secular variations, those of the perihelia being ignored,
by these corrections to the masses is satisfactory. Except for
the large discordance in the motion of the eccentricity of
Mercury the mean error would have been less than unity.
Comparing the two sets of values we find that the masses
of Mercury, Venus, and Mars agree well with those derived
from other sources. Very different is it with the mass of the
Earth. The discordance is here more than the hundredth
part of its whole amount, which involves a discordance of
more than the three-hundredth part in the value of the solar
parallax. Let us now proceed in the reverse order, and deter-
mine the value of the solar parallax from the mass of the Earth,
as derived from the preceding data.
Preliminary adjustment of the two sets of masses.
64. We make the best adjustment for this purpose by adding
to the equations of condition last given the additional ones
derived from the values of the masses discussed in Chapter V.
Multiplying each value of v by the factor necessary to reduce
the mean error of the second member of the equation to unity,
we have the following conditional equations :
50 x =4- 0.4
360 v' = + 2.9
50 v 1 " = 0.0
30 v"> = -f 0.42
Of the last two equations it may be remarked that the first is
that given by Prof. HALL'S original mass of 1877, while the
last is derived by Dr. HARSHMAN from HALL'S observations
of the outer satellite made during the opposition of 1892.
64] CORRECTION OF MASSES. 123
When we add to the normal equations already formed the
products of these last equations by the factors of the unknown
quantities, the system of normal equations is as follows :
8266 # - 1563 v 1 - 4991 r" + 140 r"' =+134
-1563 +230831 + 88556 +3455 = +374
-4991 + 88556 +122462 +3750 = -1446
+ 140 + 3455 + 3750 +3801 == -26
The solution of these equations gives the following values of
the unknown quantities :
x = + 0.0071 i .0120
v = + 0.071 i .120
v 1 = + 0.0084 i .0024
V n = _ 0.0177 i .0035
v'"= + 0.0027 i .016
Here again we note that, the Earth aside, the results for the
masses are quite satisfactory. The correction to Prof. HALL'S
original mass of Mars is so minute and so much less than its
probable error that we may consider this value of the mass to
be confirmed, and may adopt it as definitive without question.
The corrections to the masses of Mercury and Veuus are scarcely
changed. The mean residual is reduced to
8 = i 0.91
which is less than the supposed value.
We have, therefore, so far as these results go, no reason for
distrusting the following value of the solar parallax, which
results from that of the mass of the Earth thus derived:
7i = 8".759 ".010
The critical examination and comparison of this and other
values of the parallax is the" work of the next two chapters.
CHAPTER VII.
VALUES OF THE PRINCIPAL CONSTANTS WHICH DEFINE
THE MOTIONS OF THE EARTH.
The Precessional Constant.
65. The accurate determination of the annual or centennial
motion of precession is somewhat difficult, owing to its depend-
ence on several distinct elements, and to the probable system-
atic errors of the older observations in Right Ascension and
Declination. What is wanted is the annual motion of the
equinox, arising from the combined motions of the equator
and the ecliptic, relative to directions absolutely fixed in space.
As observations can not be referred to any line or plane which
we know to be absolutely fixed, we are obliged to assume that the
general mean direction of the fixed stars remains unchanged,
or, in other words, that the stellar system in general has no
motion of rotation. This is a safe assumption so far as the
great mass of stars of smaller magnitude is concerned. But it
is not on such stars that we have the earliest accurate obser-
vations. Moreover, observed Right Ascensions of these
fainter stars relative to the brighter ones are subject to possi-
ble systematic errors, arising from the personal equation being
different for brighter and fainter stars. In the case of the
stars observed by BRA.DLEY, there is frequently such commu-
nity of proper motion among neighboring stars that we can
noLbe quite sure that all rotation is eliminated in the general
mean. Under these circumstances we have only to make the
best use that we can of existing material.
We must also remember that observed Right Ascensions are
not directly referred to the equinox, but to the Sun, of which
the error of absolute mean Right Ascension must be deter-
mined. This again can be done only from observed declina-
tions, since by definition the equinox is the point at which
the Sun crosses the equator. It is also to be noted that the
clock stars which are directly compared with the Sun by no
124
65] THE PRECESSIONAL CONSTANT. 125
means include the whole list to be used as absolute points of
reference. We therefore have three separate steps in determin-
ing completely a correction to the adopted annual precession :
(1) The correction to tlie Sun's absolute mean Eight Ascen-
sion or longitude.
(2) The correction to the general mean Eight Ascension of
the clock stars relative to the Sun.
(3) The determination of the clock stars relative to the great
mass of stars.
It goes without saying that the determinations of these three
quantities are entirely independent of each other, and that the
precision of the result depends on the precision of each sepa-
rate determination.
The motion of the pole of the equator, on which the luni-
solar precession depends, may be determined by observed
Declinations quite independently of the Eight Ascensions. A
determination of the precession from the latter includes the
planetary precession, but as this has to be determined from
theory independently of observations, we have, in observed
Eight Ascensions and Declinations, two independent methods
of determining the motion of the equator.
It fortunately happens that the constant of precession is
not so closely connected with other constants that a small
error in its determination will seriously affect our general con-
clusions, or the reduction of places of the fixed stars, because
the effect of an error will be nearly eliminated through the
proper motions of the fixed stars, or the motions of the planets
in longitude. I have therefore satisfied myself with reviewing
and combining ,the four best determinations.
I pass over in silence the classic determinations of BESSEL
and OTTO STRUVE, because the material on which they depend
has been incorporated in more recent works. Of these the one
which seems entitled to most weight is that of Luowia STRUVE,
Bestimmung der Constante der Prcecession, und der eigenen
Bewegung des Sonnensy 'stems.* This work was suggested by
the completion of AUWERS' re-reduction of BRADLEY'S Obser-
vations, and of the Pulkowa standard catalogues for 1845,
*Me"moires de PAcademie Impe'riale des Sciences de St. Pe'tersbourg.
VII e SSrie. Tome xxxv, No. 3.
126 THE PRECESSIONAL CONSTANT. [65
1855, aiid 1865. It depends entirely on the BRADLEY stars,
and the result, when reduced to the most probable equinox,
may be regarded as the best now derivable from those stars,
or, at least, as not susceptible of any large correction.
He, of course, includes in his work the determination of the
motion of the solar system relative to the mass of the stars.
In addition to this, the possibility of a common rotation of
the BRADLEY stars around the axis of the Milky Way is con
sidered. This rotation I should be disposed to regard as zero
for the present.
In place of considering each of the 2,509 stars singly, he
divides the celestial sphere into 120 spherical trapezoids, each
covering 15 degrees in Declination, and an arc of Right
Ascension equal approximately to one hour of a great circle
at the equator. The question might be legitimately raised
whether a different system of weighting the trapezoids, founded
on a consideration and comparison of the proper motions in
Eight Ascension and Declination would not have been advis-
able. I am, however, fairly confident that no change in this
respect would have materially affected the result. With this
work of STRUVE I have combined those of BOLTE, DREYER,
and NYREN.
In the case of the Eight Ascensions it is necessary to reduce
all the results to the equinox determined in the last chapter.
From this chapter it appears that the standard Eight Ascen-
sions with which the reduction of the preceding investigations
have been made require a correction to the centennial motion
of 4- 0".30. Eeducing each determination to the equinox thus
defined, we have the following results for the general preces-
sion in Eight Ascension at the epoch 1800 :
L. STRUVE, from the comparison of
AUWERS-BRADLEY with the modern
Pulkowa Eight Ascensions . . . m = 46".050l ; w = 4
DREYER, from the comparison of
LALANDE'S Eight Ascensions with
those of SCHIELLERUP 46 .0611; w = 2
NYREN, by the comparison of BESSEL'S
Eight Ascensions with those of
S'CH JELLERUP 46 .0456 ; W = I
Mean 46 .0526
65] THE PRECESSIONAL CONSTANT. 127
The weights here assigned are of course a matter of judgment.
The general agreement of the results is as good as we could
expect.
From observed declinations we have
L. STRUVE, from the comparison of
AUWERS - BRADLEY with modern
Pulkowa catalogues w = 20".0495; iv = 2
BOLTE, from the comparison of LA-
LANDE'S Declinations with those of
SCILJELLERUP 20 .0537 ; w = 1
Mean 20 .0509
We have now to -combine these independent results. I pro-
pose to call Precessional Constant that function of the masses
of the SUE, Earth, and Moon, and of the elements of the orbits
of the Earth and Moon, which, being multiplied by half the
sine o twice the obliquity, will give the annual or centennial
motion of the pole on a great -circle, and being multiplied by
the cosine of tire obliquity will give the lunisolar precession
at any time. It is true that this quantity is not absolutely
constant, since it will change in the course of time, through
the diminution of the Earth's eccentricity. This change is,
however, so slight that it can become appreciable only after
several centuries. If, then, we put
p, the precessional constant, we have, for the annual general
precession in Eight Ascension and Declination
m = p cos 2 e H sin L cosec s
n Y sin cos
L being the longitude of the instantaneous axis of rotation
of the ecliptic, and H its annual or centennial motion. From
the definitive obliquity and masses of the planets adopted
hereafter, we find the following values of #, L, and , for 1800
and 1850:
lOg H =
1800.
1.67372;
1850.
1.67341
L =
173 2'.31;
173 29'.68
=
23 27.92;
23 27 .53
128 THE PRECESSIONAL CONSTANT. [65
We thus find the following values of p, the unit of time
being 100 solar years:
From Eight Ascensions, P = 5490.12; w = 2
From Declinations, p = 5489.44; w = 1
Mean, p = 5489 // .89
As the data used in STRUVE'S Investigation may be con-
sidered of a more certain kind than those used by the others,
we may compare these results with those which follow from
STRUVE'S work alone. They are
From Eight Ascensions, P = 5489.83
From Declinations, p = 5489.06
Giving double weight to the results from the Eight Ascen
sions, the results may be expressed as follows :
From STRUVE'S investigation, P = 5489.57
From the other two works, p = 5490.18
Before concluding this investigation, I had adopted as a pre-
liminary value
P = 5489".78
As this result does not differ from the one I consider most
probable, 5489".S9, by more than the probable error of the
latter, and diverges from it in the direction of the best deter-
mination, I have decided to adhere to it as the definitive
value.
The centennial value of p is subjected to a secular diminu-
tion of 0".00364 per century, owing to the secular diminution of
the eccentricity of the Earth's orbit. We therefore adopt
p = 5489.78 0.00364 T for a tropical century.
p = 5489.90 - 0.00364 T for a Julian century.
In the use of p I at first neglected the secular variation,
but have- added its effect to the results developed in powers
of the time.
66] THE CONSTANT OF NUTATION. 129
Constant of nutation derived from observations.
66. The determination of this constant from observations is
extremely satisfactory, owing to the completeness with which
systematic errors may be eliminated. If, with a meridian
instrument, regular observations are made through a draconitic
period, on a uniform plan, upon stars equally distributed
through the circle of Eight Ascension, the observations being
made daily through more than 12 hours of Eight Ascension,
all systematic errors in the determination of the nadir point
and all having a diurnal or annual period may be completely
eliminated from the constant in question. These conditions
are so nearly fulfilled in the observations with the Greenwich:
transit circle, and, to a less extent, in those with the Wash-
ington transit circle, that the results of the Wdrk with those
two instruments alone are entitled to greater weight than has
hitherto been supposed. I have, however, discussed quite
fully all previous determinations of which it seemed that the
probable mean error would be less than 0".10.
Eeferring to the volume on the subject to be hereafter pub-
lished, the results of the discussion are presented in the fol-
lowing table. The weights are assigned on the supposition
that weight unity should correspond to a mean error of about
0".07, or to a probable error of /7 .05, this probable value
being not entirely a matter of computation from the discord-
ance of. the separate results, but, to a certain extent, a matter
of judgment.
It.must be understood that the results below are not always
those given by the authors who are quoted, but that their dis-
cussion has,, wherever- possible, been subjected to a revision by
the introduction of modern data, or by what seemed to me
improved combinations. Thus, NYREN'S equations have been
reconstructed on a system slightly different from his, and have
been corrected for CHANDLER'S variation of latitude. PETERS'S
classical work has also been corrected by the introduction of
later data, and by a re-solution of his equations. The Green-
wich and Washington results have been derived from the dis-
cussion in Astronomical Papers, Vol. II, Part VI.
5690 N ALM 9
130 THE CONSTANT OF NUTATION. [66
Values of the constant of nutation derived from observations.
BUSCH, from BRADLEY'S observations with
the zenith sector 9.232 1
ROBINSON, from Greenwich mural circles . . 9.22 1
PETERS, from Eight Ascensions of Polaris . 9.214 4
LUND AHL, from Declinations of Polaris . . 9.236 1.5
NYREN, from v Urs. Maj 9.254 3
" " oDraconis 9.242 2.5
" " i Draconis 9.240 4
DE BALL, from WAGNER'S Eight Ascensions
of Polaris 9.162 3
DEBALL, from WAGNER'S Declinations of
Polaris . 9.213 3
DEBALL, from WAGNER'S Eight Ascensions
of51Cephei 9.252 3
DEBALL, from WAGNER'S Declinations of
51 Cephei 9.227 3
DEBALL, from WAGNER'S Eight Ascensions
of 6 Urs. Miu 9.208 3
DEBALL, from WAGNER'S Declinations of
d Urs. Min 9.263 3
Greenwich North-Polar Distances of South-
ern Stars, Series I 9.116 3
Greenwich North-Polar Distances of South-
ern Stars, Series II 9.201 3
Greenwich North-Polar Distances of North-
ern Stars, Series I 9.204 4
Greenwich North-Polar Distances of North-
ern Stars, Series II 9.223 4
Washington Transit Circle, southern stars . 9.217 6
" " u northern stars . 9.177 3
Greenwich, Eight Ascensions of Polaris . . 9.153 2
" Declinations of Polaris . . . . 9.242 2
" Eight Ascensions of 51 Cephei . 9.135 2
" Declinations of 51 Cephei . . . 9.162 2
" Eight Ascensions of 6 Urs. Min. 9.147 2
" Decimations of tfUrs. Min. . . 9.235 2
" Eight Ascensions of A Urs. Min. 9.161 1
" Declinations of A Urs. Min. 9.339 1
Mean . 9.210; wt. = 72
66,67] PRECESSION AND NUTATION. 131
The mean error corresponding to weight unity when derived
from the discordance of the results is 0".068, while the
estimate was i 0".070. We may therefore put, as the resulj
of observation
N = 9".210 0".008
Relations betiveen the constants of precession and nutation, and
the quantities on which they depend.
67. The formula of precession and nutation have been
developed by OPPOLZER with very great rigor and with
great numerical completeness as regards the elements of the
Moon's orbit, in the first volume of his Bahnbestimmung der
Kometen und Planeten, second edition, Leipzig, 1882. What
is remarkable about this Avork is that it constantly takes
account of the possible difference between the Earth's axis
of rotation and its axis of figure, a distinction which has
become emphasized by CHANDLER'S discovery since OPPOL-
ZER wrote. His theory however fails to take account of the
change in the period of the Eulerian nutation produced by
the mobility of the ocean and the elasticity of the Earth. But
this effect is of no importance in the present discussion.
From OPPOLZER'S developments, I have derived the follow-
ing expressions, in Avhich the numerical coefficients may be
regarded as absolute constants, so accurately determined that
no question of their errors need now be considered. These
results haA^e been derived quite independently of the similar
ones by Mr. HILL in the Astronomical Journal, Yol. XI, which
are themselves independent of OPPOLZER'S work. In these
formulae we have
K, the constant of lunar nutation of the obliquity of the
ecliptic, as defined by the equation As = ~N cos &, and
expressed in seconds of arc;
P, so much of the precession of the equinox on the fixed
ecliptic of the date, in seconds of arc and in a Julian
year, as is due to the action of the Moon ;
P 7 , so much of the same precession as is duejo^j^e^ action
of the Sun.
^ * ^l - ^-
Of
132 . MASS OF THE MOON. [67, 68
We thus have,
luni-solar precession = P + P 7
f, the obliquity of the ecliptic;
yu, the ratio of the mass of the Moon to that of the Earth ;
A, the mean moment of inertia of the Earth relative to axes
passing through its equator;
C, the same moment relative to its polar axis.
With these definitions we have,
General value. Special value for 1850.
X = [5.40289J cos s
P = [5.975052] cos f
P' = [3.72509] c
0-A
C
= [5.36542!
C-A
1 +
~ A = [5.937585] _JL_
C J 1 + /i C
; "^ = [3.68762]
C-^.
The special values for 1850 are found by putting for the
value of the obliquity of the ecliptic for 1850,
f = 22 27' 31". 1
The mass of the Moon from the observed constant of nutation.
68. From the two quantities given by observation, N and
P 4- P' = po, these equations enable us to determine the two
unknown quantities yu and
C A
As the easiest way of
showing the uncertainty of the Moon's mass, arising from
uncertainty of the precession and nutation, I give the value of
its reciprocal corresponding to different values of these quan-
tities in the following table :
Reciprocals of the mass of the Moon corresponding to different
values of the nutation-constant and luni-solar precession.
A
i
*="
50.35
50.36
50.37
81.81
81.86
81. 91
81-53
81.58
81.63
81. 25
81.30
81-35
68, 69J THE CONSTANT OF ABERRATION. 133
Taking for the constant of nutation the value just found,
N = 9 // .210 ".068
and for the luni-solar precession,
lh = 50" .36 ''.006
we have, for the reciprocal of the mass of 'the Moon and its
mean error :
- = 81.58 0.20
p
The Constant of Aberration.
69. In the determination of astronomical constants the inves-
tigation of the constant of aberration necessarily takes a very
important place, not only on its own account but on account of
its intimate connection with the solar parallax. A general
determination, founded on all the data available, was therefore
commenced by me as far back as 1890, before the fact of the
variation of terrestrial latitudes had been well established.
The successive discoveries of the law of this variation by
CHANDLER required such alterations in the work as it went
along that much of it is now of too little value for publication
in full. Happily the necessity for a new discussion of the best
determinations at Pulkowa has been done away with by the
papers of CHANDLER himself in the Astronomical Journal.
Quite apart from the disturbing influence of the revolution
of the terrestrial pole upon the determination of the constant
of aberration, this constant is itself the one of which the deter-
mination is most likely to be affected by systematic errors.
In this respect it is at the opposite extreme from the constant
of nutation. From the very nature of the case it requires a
comparison of observations at opposite seasons of the year,
when climatic conditions are different. In most cases the
determination must even be made at different times of day.
The effect of aberration on a star, for example, is generally at
one extreme when the star culminates in the morning, and at
the other extreme when it culminates in the evening. The
culminations at opposite seasons of the year are necessarily
134 THE CONSTANT OF ABERRATION. [69
associated with culminations at opposite times of the day.
Moreover, in observations to determine the constant of aber-
ration from Declination, the stars which give the largest coeffi-
cients are, for the northern hemisphere, tbose near 18 h of Eight
Ascension. Any error peculiar to the times or seasons at
which these stars are observed will therefore affect the result
systematically.
Eight Ascensions of close polar stars also lead to a value of
this constant. But the same difficulty still exists. In this
case the maxima and minima of aberration occur when the
star culminates at noon and midnight. Not only is the aspect
of the star different at the two culminations, but the effect of
any diurnal change in the instrument will be transferred to the
final result for the aberration.
The prismatic method of LOEWY is free from some of these
objections. But its application is extremely laborious, and we
have, up to the present time, only two determinations by it,
one by LOEWY himself, which is only regarded as preliminary,
and one by COMSTOCK, in which a large uncertain correction
for personal equation was applied.
Under these circumstances the seeking of results derived by
methods of the greatest possible diversity is yet more strongly
recommended than in the case of the other astronomical con-
stants. I have therefore used not only the PULKOWA deter-
minations, but all those made elsewhere which it seemed worth
while to consider. Notwithstanding the great amount of mate-
rial added to NYREN'S paper of 1883, it will be seen that the
probable error of the final result at which I have arrived is
greater than that which he assigns to his result. This is a
natural consequence of combining so many separate determi-
nations. The advantage is, however, that the assigned prob-
able error is more likely to be the real one. It is not to be
supposed that any of the systematic errors already indicated
would pertain to all observers and to all instruments. The
final outcome should be a result in which the discordances of
the separate determinations show the probable values of all
the actual errors, both accidental and systematic.
Determinations founded on the Eight Ascensions of circum-
polar stars are not affected by the motion of the terrestrial
69, 70] THE CONSTANT OF ABERRATION. 135
axis, uor are those founded on declinations of these stars, if
only the declinations are observed equally at both culmina-
tions. But determinations founded on declinations of stars
from upper culmination only are necessarily aifected by this
cause. If however the stars on which the determination is
based extend through the whole circle of Right Ascension the
effect of the cause in question may be wholly eliminated by a
suitable treatment of the equations of condition. To practically
eliminate the injurious effect it is not even necessary to deter-
mine the exact law of variation. In fact, if the stars observed
are equally scattered in Eight Ascension, the effect of the varia-
tion will be partially eliminated without taking account of it.
CHANDLER has shown that there are two periodic terms in
the variation of latitude, one having a period of one year, the
other of four hundred and twenty-seven days. I may remark
that this combination is in accord with my theory developed in
the Monthly Notices of the Royal Astronomical Society for March,
1S92. It was there shown that any minute annual change of
the position of the principal axis of inertia of the Earth a
change which might be produced by the motion of water, ice,
and air on its surface would appear as an annual term in the
latitude, six times as great as its actual amount,
Values of the constant of aberration derived from observations.
70. What I have done since this discovery by CHANDLER
has been to reexamine the determinations of the constant of
aberration made from time to time, to make such corrections
in their bases as seemed necessary, and more especially to
determine the correction to be applied to each separate result
on account of the periodic term in the latitude. No attempt
was made to rework completely the original material, except
in the case of the results of the Pulkowa and Washington
observations with the prime vertical transit. In the case of
the former, however, the preliminary results reached from time
to time were so accordant with those of CHANDLER that it is
a matter of indifference whether we regard them as belonging
to his work or to my own.
Owing to the very different estimates placed by the astro-
nomical world upon the Pulkowa determinations and 'those
136 THE CONSTANT OF ABERRATION. [70
made elsewhere, I have used the former quite apart from the
others. The complete discussion of each separate value is
too voluminous for the present publication, and is therefore
reserved for a more extended future publication. At pres-
ent it appears sufficient to judge the final result by the general
discordance of the material on which it rests, rather than by
a separate criticism of each particular case.
In the exhibit of results which follows it is to be remarked
that NYREN'S prime vertical observations do not receive a
weight as great, relative to the other Pulkowa determinations,
as would be given by their assigned probable errors. The
reason of this course is that one can not be entirely confident
that the results of any one observer with this instrument are
free from constant error arising from differences of personal
equation in observing a bright and a faint star. Many of the
Pulkowa observations are necessarily made in the morning or
evening twilight. In the case of an evening observation the
star will therefore be much fainter on account of daylight
when it transits over the east vertical than it will when it
transits over the west vertical one or two hours later. In the
case of morning observations the reverse will be true. It is
easy to see that if, in consequence of this difference of aspect,
the observer notes the passage of the faint image too late, the
effect will be to make the constant of aberration too large.
The existence of this form of personal equation, when transits
are recorded on the chronograph, is so well known that, had
NYREN'S observations been made in this way, I should not
have hesitated to ascribe the large values of his aberration
constant to this cause. Although it has never been shown
that any such personal equation exists when observations are
made by eye and ear, as KYREN'S were, yet when we consider
that we are dealing with quantities amounting only to one or
two huudredths of a second of arc, and that a personal equa-
tion of this kind, undiscoverable by ordinary investigation,
might affect the result by this minute amount, we can not but
have at least a suspicion that his values may be slightly too
large from this cause.
70! THE CONSTANT OF ABERRATION. 137
Separate results for the constant of aberration.
A. Standard Pulkowa determinations :
A b. wt.
Observations with Vertical Circle ; Polaris, by n
PETERS 20.51 2
Observations with Vertical Circle ; 7 miscellaneous
stars, by PETERS 20.47 2
Observations with Vertical Circle 5 1863-1870, Po-
laris, by GYLDEN 20.41 2
Observations with Vertical Circle; 1871-1875, Po-
laris, by XYREN 20.51 2
Observations with Prime Vertical; 1842-1844, by
STRUVE 20.48 4
Observations with Prime Vertical; 1879-1880 by
NYREN. . 20.52 6
Observations with Prime Vertical; 1875-1879, by
NYREN 20.53 1
Observations with Vertical Circle; 1863-1873, by
GYLDEN and NYREN . 20.52 2
WAGNER : Transits of three polar stars .... 20.48 5
From Eight Ascensions of Polaris; 1842-1844, by
LINDHAGEN and SCHWEIZER 20.50 2
Mean result: 20 // .493 0".011
This result may be regarded as identical with that found by
NYREN in 1882.
B. Other determinations:
Ab. e wt.
AUWERS, from observations with the n
zenith sector at Kew 20.53 .12 0.5
AUWERS, from WANSTED observations . 20.46 .12 0.5
PETERS, from BRADLEY'S observations
of y Draconis at Greenwich with zenith
sector, 1750-1754 20.67 0.5
BESSEL, from Eight Ascensions observed
by BRADLEY at Greenwich .... 20.71 i.071 0.5
LINDENAU, from Eight Ascensions of
Polaris observed at various observa-
tories between 1750 and 1816 . 20.45 .05 3
138 THE CONSTANT OF ABERRATION. | 70
Separate results for the constant of aberration Continued.
B. Other determinations Continued.
BRINKLEY, from observations of thirteen --/ft. /.
stars at Trinity College, Dublin, with 7/
the 8-foot circle 20.46 .10 1
PETERS, from STRUVE'S Dorpat observa-
tions of six pairs of circumpolar stars . 20.36 .07 2
RICHARDSON, from observations with the
Greenwich mural circles 20.50 .06 3
PETERS, from Right Ascensions of Polaris
at Dorpat . 20.41 6
LUNDAHL, from Declinations of Polaris
at Dorpat 20.55 5
HENDERSON and MCLEAR, from a 1 and
<* 2 Centauri . . . 20.52 .10 1
MAIN, from observations with the Green-
wich zenith tube . 20.20 .10 1
DOWNING, from observations of ADra-
conis with reflex zenith tube . ... . 20.52 .05 4
XEWCOMB, from observations of <*Lyra3
with the 'Washington prime vertical
transit, 1862-1867 20.46 0.4 6
NEWCOMB, from Right Ascensions of
Polaris observed with the Washington
transit circle, 1866-1867 . 20.55 .05 :J
KUSTNER, from observations of pairs of
stars by the TALCOTT method . . . 20.46 4
PRESTON, from observations with the-
TALCOTT method at Honolulu, 1891-
1892 20.43 .05 4
LOEWY, from his prismatic method . . 20.45 .04 5
COMSTOCK, using LOEWY'S method,
slightly modified 20.44 3
KiisTNER, from MARCUSE'S observations,
1889-1890 20.49 .018 4
WANACH, from Pulkowa prime vertical
observations . 20.40 .015 4
70, 71] THE LUNAR INEQUALITY. 139
Separate results for the constant of aberration Continued.
B. Other determinations Continued.
Ab. tvt.
From Greenwich Eight Ascensions of polar stars
with the transit circle 20.39 3
BECKER, from observations at Strasburg by the
TALCOTT method, 1890-1893 20.47 6
DAVIDSON, from similar observations at San
Francisco, 1892-1894 20.48 6
Mean result of B : Ab. const. = 20".463 0".013
The two results. A and B, differ by 0".030, a quantity so
much greater than their mean errors as to leave room for a
suspicion of constant error in one or both means.
The Lunar Inequality in the Ear til's motion.
71. The source of this inequality is the revolution of the
center of the Earth around the center of mass of the Earth
and Moon. The former center describes an orbit which is
similar to that of the Moon around the Earth. Since this
orbit is not a Keplerian eclipse, but is affected by all the per-
turbations of the Moon by the Sun, no such element as a semi-
major axis can be assigned to it. Instead of this I take as the
principal element of the orbit the coefficient of the sine of the
Moon's mean elongation from the sun in the expression for the
Sun's true longitude. This element is a function of the solar
parallax and of the mass of the Moon, which may be derived
from the folio wii>g expression. Let us put
yw ; the ratio of the mass of the Moon to that of the
Earth ;
r,A, /?; the radius vector, true longitude and latitude of
the Moon ;
r', A',/J'; the same coordinates of the Sun;
* ; the linear distance of the Earth's center from the
center of mass of the Earth and Moon.
140 THE LUNAR INEQUALITY. [71
We then have, for the perturbations of the Sun's geocentric
place due to the cause in question :
A log r 1 = - x cos ft cos (A-A 7 )
A\' = - t cos ft sin (A A 7 )
and
s
I have developed these expressions, putting
7T = 8".848
-.4
and taking for the Moon's coordinates the values found by
DELAUNAY. Putting
D; the mean value of A A 7
g, g 1 } the mean anomalies of the Moon and Sun, respectively,
u'-, the Sun's mean elongation from the Moon's ascending-
node;
the result for JA 7 is
//
A\' = 6.533 sin D
+ 0.013 sin 3 D
+ 0.179 sin (D + g)
0.429 sin (D g)
+ 0.174 sin (D g 1 )
- 0.064 sin (D + g')
-f 0.039 sin (3 D - g)
0.014 sin (D g g 1 }
0.013 sin 2 u 1
This value of the lunar inequality is substantially identical
with that computed from the tables and formula of LEVER-
71] THE LUNAR INEQUALITY. 141
RIER'S solar tables. The development of the numbers there
given lead to the value 6".534 of the principal coefficient.
We have now to find what value of the coefficient is given
by observations. The observations I make use of are (1) all
the observations of the Sun's Eight Ascension from early in
the century till 1864 ; (2) The heliometer observations of Vic-
toria made in 1889 on GILL'S plan and worked up by him.
I had intended to use all the observations of the Sun up
to the present time. 1 found however that those made after
1864 gave, by comparison with the published ephemerides,
inadmissible positive corrections to the coefficient. This cir-
cumstance gives rise to a strong suspicion that in the process
of interpolating the Right Ascensions of the Sun during at
least some years after 1864, the inequality in question was
rounded off to the amount of several hundredths of a second.
The results were therefore entirely omitted.
The results for previous years, when the inequality was
computed separately for every day of observation, are:
Greenwich,
1820-'64;
-.068
3.0
Paris,
1801-'64;
-.050
0.8
Konigsburg,
1820-'45;
-.054
1.2
Cambridge,
1828->58;
-.047
2.0
Dorpat,
1823->38;
+ .160
0.3
Pulkowa,
1842-'64;
-.058
0.5
Washington,
1846-'64:
.000
0.2
Mean, JP = 0".048 i 0".018
GILL'S result is given in the Monthly Notices, Royal Astro-
nomical Society, for April, 1894 (Vol. LIY, page 350.) It is
derived in the following way. In the solar ephemeris which
he used for comparison the lunar inequalities were computed
rigorously from the coordinates of the Moon, putting
n = 8".880
M = 1 -r- 83
To the coefficient P thus arising he found a correction,
= + 0".046
142 THE LUNAR INEQUALITY. [71, 72
The above values of n and // give, on the theory just devel-
oped,
P = 6".400
Thus GILL'S result is, in effect,
P-= 6".446
while mine, from observations of the Sun, is
6".533 0".048 = 6".485
I consider that these results are entitled to equal weight, and
that we may take, as the result of observation,
P = C>".465 i 0".015
Solar parallax from the lunar inequality.
72. With the mass of the Moon already found from the
observed constant of nutation,
// = 1 : 81.58 (1 i .0025)
we may now derive a value of the solar parallax quite inde
pendent of all other values. The relation between P, TT, and
the mass of the Moon is of the general form
// P = A' 7T
where fc is a numerical constant, and, for brevity,
We have found that the following values correspond to one
theory :
TT = 8".848; X = 82 5 P = 6". 533
Hence follows
log fc = 1.78207
so that we have
^P= [1.78207] n
The numerical values P = 6 /7 .465 and // = 82.58 now give
7r = 8 // .818 0".030
73J PARALLAX FROM TRANSITS OF VENUS. 143
Values of the solar parallax derived from measurements of Venus
on the face of the Sun during the transits of 1874 and 1882,
with the heliometer and photoheliograph.
73. I put these determinations into one class because they
rest essentially on the same principle. Both consist, in effect,
in measures of the distance between the center of Yenus and
the center of the Sun j the latter being denned through the
visible limb. , The method is therefore subject to this serjous
drawback : that the parallax depends upon the measured differ-
ence between arcs which may be from thirty to fifty times as
great as the parallax itself, the measures being made in
different parts of the earth.
The equations of condition given by the American photo
graphs of 1874 are found in Part I of Observations of the
Transit of Yenus, December 9, 1874 ; Washington, Government
Printing Office, 1880. A preliminary solution of these equa-
tions, the only one, however, to which they have yet been sub-
jected, was published by D. P. TODD, in the American Journal
of Science for June, 1881. (Yol. XXT, page 490.)
The photographs of 1882 have been completely worked up by
Professor HARKNESS, and the results are found in the Eeport
of the Superintendent of the Naval Observatory for 1889. The
equations derived from the German heliometer measures, with
a preliminary discussion of their results, are officially published
by Dr. AUWERS, in the Bericht iiber die deutschen Beobachtungen,)
Y, p. 710.
The separate results for the parallax, with the probable
errors assigned by the investigators, are as follows:
// // w. W
1874 : Photographic distances, n 8.888 0.040 6 1
Position angles, 8.873 0.060 3 3
Measures with heliometer, 8.876 i 0.042 5 5
1882: Photographic distances, 8.847 0.012 64 6
Position angles, 8.772 i 0.050 4 4
Measures with heliometer, 8.879 0.025 1C 10
Under w is given a system of weights proportionally deter-
mined from the probable errors as assigned. Using this sys-
tem, the mean result is
n =8".854 i ".016
144 PARALLAX FROM TRANSITS OF VENUS. [73
I conceive, however, that these relative weights do not cor-
respond to the actual precision of the measures. The very
small probable error assigned by Prof. HARKNESS to the result
of the photographic distances of 1882 does not include the
probable error of the angular value of the unit of distance on
the plate, which may arise from a number of sources, includ-
ing the possible deviation of the mirror of the instrument
from a perfect plane. From this error the position angles
are entirely free. I have, therefore, assigned another set of
weights, w', which seem to me to correspond more nearly to
the facts. The result of this system is
7t = 8".857 i ".016
This mean error is derived from the individual discordances,
and not from comparisons with the values of the parallax
otherwise determined. As there may be a fortuitous agree-
ment among the separate values, another estimate may be
made on the basis of the total mean error derived by AUWERS,
which includes all known sources of error. He finds 6 = ".032
for the combined heliometer results, to which I have assigned
weight 15. Hence, for the total weight 29, we have
<? = 0".023
The deviation of the above result from the mean of all the
other good ones is worthy of special attention. The deviation
is more than three times its mean error, and therefore between
four and five times its probable error. We must therefore
accept one of two conclusions, either the probable errors have
been considerably underestimated, or the method is aifected
with some undiscoverable soured of systematic error, which
makes it tend to give too large a result. The close accordance
of the six separate results, of which only a single one deviates
from the adopted mean by more than its probable error, and
that by only a little more, would give color to the view that
the error is a systematic one, and that through some unknown
cause Venus is always measured too low relatively from the
center of the Sun. I can not, however, think of any such cause.
If we determine the mean error from the deviations of the
separate results from what we know, in other ways, to be
74) PARALLAX FROM TRANSITS OF VENUS. 145
nearly the most probable value of the parallax, namely 8".80,
we have
Mean errror to weight 1 ; dz .148
Mean error of result dz.029
Solar parallax from observed contacts during transits of Venus.
74. The contact observations of 1761 and 1769 are discussed
in Astronomical Papers, Vol. III. I have also made a com-
plete discussion of those of 1874 and 1882, which, at the date
of writing, is unpublished. The separate results from each,
contact follow.
In the case of the second contacts of 1874 and 1882 it was
found necessary to divide the observations into two classes:
those of mean or true contact, and those of the formation of
the thread of light. In the case of the third contact no such
division was necessary, as the observations could generally
be referred to the same mean phase. The mean error which
follows each result is derived from the discordance of the
separate observations.
Values of the solar parallax from observed contacts of the limb
of Venus with that of the Sun.
1761, III; TT = 8. / 78dz / .12; w. = 8
IV; 8.75 dz. 20 3
1769, I;
9.04 d
z .17
4
II;
8.55 d
z .13
7
III;
8.72 d
- .09
14
IV;
9.01 d
z .12
8
1874, I;
8.95 d
z .24
2
II; M;
8.78d
z .061
30
II; L;
8.75 d
z .10
11
III;
8.76 d
z .045
57
IV;
8.74 d
- .09
14
1882, I;
8.93d
z .15
5
II; M;
8.76d
z .042
64
II; L;
8.72d
- .072
22
III;
8.88 d
- .042
64
IV;
9.07 d
- .12
8
5690 N ALM 10
146 PARALLAX FROM TRANSITS OF VENUS. [74
The weights assigned are determined by these mean errors,
taken on such a scale that unity is the weight for mean error
i ".336. The mean result of the whole series is
7t = 8".797 i ".023
This mean error is that resulting from the deviations of the
sixteen separate results from the general mean, which give for
the mean error corresponding to weight unity,
1 = ".42.
The excess of this mean error over that determined from the
equations themselves shows that the general discordance of the
several contacts is somewhat greater than would be inferred
from the individual discordances of the contacts inter se. This
is what we should expect from constant errors in the determi-
nations of parallax from each separate contact. I conceive,
however, that such constant errors are not likely to be large;
and we can not conceive that contact observations in general
are subject to any constant error tending to make the parallax
derived from them always too great or too small. I conclude,
therefore, that the mean error determined from the totality of
the results may be regarded as real.
It will be interesting to compare the separate results of
internal and external contacts. They are
// //
From internal contacts ; TT == 8.776 i .023
From external contacts; TT = 8.908 i .06
These mean errors are those derived from the concluded
results and they show that the external contacts are relatively
more discordant in proportion to the weights assigned than are
the internal ones. If we consider this discordance to indicate
a larger mean error, and therefore -assign a proportionally
smaller weight to the results of external contact, we have, for
the concluded result,
7t = 8".791 i ".022
As these two hypotheses seem about equally probable, I shall
adopt the mean result,
n = 8".794
75] PARALLAX FROM VELOCITY OF LIGHT. 147
Solar parallax from the observed constant of aberration and
measured velocity of light.
75. The question of the soundness of the proposition that
the aberration is equal to the quotient of the velocity of the
Earth in its orbit by the velocity of light is too broad a one to
be discussed here. I can only remark that its simplicity and
its general accord with all optical phenomena are such that it
seems to me it should be accepted, in the absence of evidence
against it.
In Astronomical Papers, Yol. II, page 202, I have given the
following determinations of the velocity of light in vacuo by
MICHELSON and myself, expressed in kilometers, per second :
4
MICHELSON at Naval Academy in 1879 299910
MICHELSON at Cleveland, 1882 299853
NEWCOMB at Washington, 1882, using only results
supposed to be nearly free from constant errors . 299860
NEWCOMB, including all determinations . . . , . 299810
I have concluded,
Velocity of light in vacuo, = 299860 i 30 k. m.
Taking as the equatorial radius of the Earth 6378.2 k. m.
(CLARK), the following table shows the values of the constant
of aberration corresponding to admissible values of the solar
parallax when this determination of the velocity of light is
accepted.
Ab. = 20.46 n = 8.8076
20.47 8.8033
20.48 8.7990
20.49 8.7946
20.50 8.7903
20.61 8.7859
20.52 8.7816
20.53 8.7773
20.54 8.7730
148 PAEALLACTIC INEQUALITY. [75, 76
We thus have for the values of the solar parallax resulting
from the two values of the constant of aberration already
derived :
// //
From Pulkowa determinations; Ab. = 20.493; n 8.793
From miscellaneous determinations; Ab. = 20.463; n = 8.806
Solar parallax from the parallactic inequality of the Moon.
76. I have derived a value of the parallactic inequality of the
Moon from the meridian observations made at Greenwich and
Washington since 1862. The determination of this inequality
is peculiarly liable to systematic error, owing to the fact that
observations have to be made on one limb of the Moon when
the inequality is positive, and on the other limb when it is
negative. Hence, if we determine the inequality by the com-
parison of its extreme observed effects on the Moon's longitude
or Eight Ascension, any error in the adopted semidiameter of
the Moon will affect the result by its full amount.
It does not seem practicable to make a reliable determina-
tion of the Moon's diameter, because it will necessarily be
made near the time of full Moon, when the illumination of the
extreme limb is less intense than near the quadratures, and
when some portions of the limb that might be visible if it were
illuminated by a perpendicular Sun will be thrown into shadow
by the horizontal one. For these reasons it may be expected
that the parallactic inequality determined by using observed
semidiameters of the Moon will be too large. I have therefore
adopted the plan of determining the inequality from each limb
separately. To show in regular progression the errors depend-
ing on the elongation from the Sun, I have classified the resid-
uals of observations according to the hour of mean time at which
the Moon passed the meridian ; and formed equations of con-
dition containing two unknown quantities, the one a constant
correction depending on the semidiameter, personal equation,
etc., and the other the parallactic inequality. The question is
further complicated by the fact that the majority of observa-
tions near are quadratures made during daylight, when it is
to be expected that the illumination of the atmosphere will
76] PAKALLACTIC INEQUALITY. 149
diminish the irradiation, and thus lead to a smaller apparent
sernidiameter. I have therefore sought to determine for the
two observatories, by a comparison of the observations, the
correction to be applied in order to reduce observations made
during daylight or twilight to what they would have been had
the sky not been illuminated. The reduction was smaller than
I had expected, and somewhat doubtful ; I have assigned pro-
portionally less weight to those observations where it was
necessary. The following are the equations of condition thus
formed. The unknown quantities are
#, a constant, depending on the semidiaineter, personal
equation, etc.;
2/, the correction to the parallactic inequality of the Moon
after reduction to the value 8".848 of the solar parallax.
GEEE^WICH.
Limb I.
4.6;
5.6
x + 0.93 y =
0.99
-0.53;
-0.72
wt. 0.2
0.6
6.5
0.99
-0.41
1
7.5
0.92
-0.59
1
8.5
0.79
-0.54
1
9.5
0.61
-0.13
1
10.5
0.38
-0.09
1
11.5
0.13
-0.06
1
Limb II.
//
12.5; #'-0.13 y =+0.20; wt. 1
13.5
-0.38
+ 0.16
1
14.5
-0.61
+ 0.28
1
15.5
-0.79
+ 0.54
1
16.5
-0.92
- 0.11
1
17.5
^0.99
-0.02
1
18.4
-0.99
+ 0.44
0.5
19.4
0.93
+ 1.21
0.2
150 PARALLACTIC INEQUALITY. [76
WASHINGTON.
. Limb I.
h
4.6 ; x + 0.93 y = 1.62 5 ict. = 0.2
5.6
0.99
- 1.26
0.4
6.5
0.99
-0.85
1
7.5
0.92
-0.64
1
8.5
0.79
-0.71
1
9.5
0.61
-0.71
1
10.5
0.38
- 0.48
1
11.5
0.13
-0.23
1
Limb II.
12.5;
x'- 0.13 y
= +0.41;
wt. = 1
13.5
-0,38
0.43
1
14.5
-0.61
0.52
1
15.5
-0.79
0.40
1
16.5
-0.92
0.72
1
17.5
-0.99
0.96
0.5
18.4
- 0.99
1.32
0.3
19.4
- 0.93
1.50
0.1
With these equations we have our choice to determine the
parallactic inequality by assigning a value to the semidiaineter,
or to eliminate the semidiameter from the normal equations.
In each case the equations give the following expressions for y:
Greenwich : Limb I; y = 0.55 1.23 x
" II; 0.28+1.230'
Washington : Limb I; y = 0.99 1.23.r
" II; -0.88 + 1.29 a?'
If we choose to utilize the observed diameters we have the fol-
lowing results:
From 66 transits of the Moon's diameter observed at Greenwich;
76] PARALLACTIC INEQUALITY.
From 33 transits observed at Washington :
We should thus have,
From Greenwich observations, y = 0.
From Washington observations, y = 0.23
If, on the other hand, we eliminate x from each pair of
normal equations, the final results for y will be
- x/ /x /f wf.
Greenwich : Limb I; 0.64 y = - 0.45; y = - 0.70 0.16 6
u II; 0.64 y = 0.00 ; y = 0.00 i 0.36 2
Washington : Limb I ; 0.64 y = - 0.52 ; y = - 0.81 i 0.16 6
II ; 0.53 y = - 0.32 ; y = - 0.60 0.27 3
The weighted mean of these results is
y= - 0".64 0".12
The resulting value of the solar parallax is
7t = 8".802 0".008
A very careful determination of the solar parallax was made
from the same theory by Dr. BATTERMAN, by means of occulta-
tions, and the result is discussed very fully in the publica-
tions of the Berlin Observatory. Dr. BATTERMAN'S definitive
result is
n = 8".794 ".016
I have slightly revised this result, by applying a correction
to the coefficient for the parallax adopted by Dr. BATTERMAN,
with the result
n = 8".7S9 i ".016
Accepting this result, and combining it with that already
found from meridian observations, the parallax from this
method will finally come out
n = 8".799 i ".007
This mean error may be regarded as belonging to the doubtful
class.
152 SOLAR PARALLAX FROM MINOR PLANETS. [76,77
While tin's work is passing through the press there appears
an important paper by FRANZ of Konigsberg,* giving the value
of the parallactic equation derived from observations on the
lunar crater Hosting A. The correction to HANSEN'S coeffi-
cient is found to be
- 2".10 i 0".30
The corresponding result for the solar parallax is
8".767 =t 0".021
We may combine the three results for the solar parallax
thus :
Greenwich and Washington meridian obser-
vations . . ........... n = 8.802; w = 5
BATTERMANN from occupations ..... 8.789; 2
FRANZ from crater Hosting A ..... 8.767; 1
Mean .......... 8.794 i ".008
Solar parallax from observations on minor planets with the
neliometer.
77. The fact that the determination of the parallaxes of the
small planets by comparison with neighboring stars is free
from the grave uncertainty attaching to similar observations
of Venus and Mars, owing to the absence of a sensible disk,
was long since pointed out by Dr. GALLE. In 1875 he pub-
lished a discussion of observations on Flora, made at nine
northern observatories, and at the Cape, Cordoba, and Mel-
bourne in the Southern hemisphere.t The result was
n = 8".873.
An examination of the residuals Of the several observatories
shows that in the case of at least one of the Southern observa-
tories there is a systematic difference of a considerable fraction
* Astronomische Nachrichten, Vol. 136, S. 354.
tUeber eine Bestimnmng der Sonnen-Parallaxe aus correspondirenden
Beobachtuugen des Planeten Flora, in October und November 1873.
Breslau, Maruscbke & Berendt, 1875.
77] SOLAR PARALLAX FROM MINOR PLANETS. 153
of a second. This fact seems to present our assigning any
appreciable weight to the final result.
In 1874, GILL, at Mauritius, made heliometer observations
of Juno, east and west of the meridian, with the same object.
The result was 8".765, or 8".815 when a discordant observation
was rejected. In this connection, only an allusion is necessary
to GILL'S expedition to Ascension in 1877, made for the pur-
pose of applying the method to Mars at the opposition of that
year.
Shortly afterwards GILL published in the first volume of The
Observatory a very exhaustive discussion of the methods of
determining the solar parallax, in which he showed that heli-
ometer observations of the minor planets, made either at a
single station not too far from the equator, or at two stations
in different hemispheres, afforded a method of measuring the
parallax more precise than any before applied.
Ten years elapsed before the plan was put into operation.
Then, in 1889 and 1890, a concerted system of observations was
made on the three minor planets, Victoria, Iris, and Sappho, at
a number of observatories in both hemispheres. The observa-
tions relating to Victoria were carried out most thoroughly,
in that a very careful triangulation of the stars of comparison
inter se was made at the observatories which took part in the
measures. The tabular data for the reductions were supplied
by the office of the Berliner Jahrbuch, and the reductions
and discussion were made by GILL himself for Victoria and
Sappho, and by Dr. ELKIN, on GILL'S plans, for Iris. The
three results, as communicated in advance of their complete
official publication, are
// //
From Victoria: n = 8.800 p. e. i 0.006
Iris: 8.825 p. e. 0.008
Sappho: 8.796 p. e. J= 0.012
I assign the respective weights 4, 2, and 1, thus obtaining,
as the final result of this method,
n = 8".807 i 0".006
I have included in a separate category GILL'S determina-
tion by Mars, at Ascension, in 1877, as published by the
154 UNCERTAINTY OF PARALLAX FROM MARS. [77, 78
Eoyal Astronomical Society (Memoirs Royal Astronomical So-
ciety, Vol. XLVI), for the reason that, owing to the disk of
Mars, and its reddish color, determinations made on it are
liable to errors peculiar to that planet, or at least different
from those which might come in in the case of the small
planets.
Remarks on determinations of the parallax which are not used
in the present discussion.
78. In the preceding discussion are given the results of
every modern method of determining the solar parallax with
which I am acquainted, except meridian and eqiiatorial obser-
vations on Mars. I have not used any of the results derived
from this source, owing to their large probable error, and
the suspicion of systematic error to which they are open.
One of these causes of error is to be found in the red color of
Mars. This cause will be pointed out and discussed very
fully in a subsequent section. Its effect would be to make the
observed parallax too large. Since, as a matter of fact, all
the determinations of Mars by meridian observations have
given a larger parallax than the generality of other methods,
color seems to be given to this suspicion. Apart from this,
the setting of the threads of a meridian circle upon the appar-
ent disk of Mars involves a visual estimate not comparable
with that of the bisection of the image of a star by the threads.
Hence, there is a chance of systematic personal error arising
from this source. The observations generally exhibit large
discordances, which may be attributed to one or the other of
these causes.
It may be objected to the inclusion of GILL'S Ascension
result that it should be rejected for the same reason, since the
color of the planet would affect heliometer observations and
meridian observations equally. I have, however, considered
it free from the objection in question, for two reasons. In the
first place, the result is not too large, but is, on the contrary,
the smallest of all the accurate measures. The principle that
when a result is open to a strong suspicion of being affected
by a cause which would cause it to deviate in one direction, it
is logical to conclude a posteriori that the cause has not acted
78] UNCERTAINTY OF PARALLAX FROM MARS. 155
if the deviation is found to be in the other direction, may not
be a perfectly sonnd one, bnt I have nevertheless acted upon
it. In the next place G-ILL himself, as a part of his discus-
sion, compared the observations when Mars was at different
altitudes, in order to determine whether the action of such a
cause was indicated, and found a negative result.
In 1890 an unsuccessful attempt was made, at the writer's
request, by Dr. W. L. ELKIN, to measure the effect in question,
by placing a refracting prism of very small angle over one of
the halves of a heliometer objective, and measuring the refrac-
tion thus produced. It was supposed that the dispersing
action of the prism would represent that of the atmosphere,
greatly magnified. The failure arose from the result that the
apparent mean refraction of the star produced by the prism
proved to be a function of the star's magnitude, ranging from
748".79 for a star of magnitude 2.55 to 751".61 for a star of mag-
nitude 6.95. The reason seemed to be that too powerful a prism
was used, so that the spectrum was quite sensible; then, in the
case of faint stars, the red portion of the spectrum was invis-
ible, so that the apparent mean refraction was greater than in
the case of the brighter stars. The mean of the observed
v displacements of Mars was 748".61, so that it was always less
for Mars than for the stars.*
An investigation of the question whether the same effect is
noticeable in meridian observations fails to show any relation
between the brightness of a star and its refraction. But this
does not disprove the relation between the refraction and the
color of a star.
On the whole it seems to me that, at least in the case of
Mars, we have here a cause so mixed up with personal error
in making the observations that the objective and subjective
effects can not be completely separated.
* Astronomical Journal, Vol. 10, page 97.
CHAPTER VIII.
DISCUSSION OF RESULTS FOR THE SOLAR PARALLAX
AND THE MASSES OF THE FOUR INNER PLANETS.
79. We have, in what precedes, found or collected nine
separate values of the parallax of the Sun, by methods of
which seven may be regarded as completely distinct, in the
sense that no one source of error is common to any two. Of
these seven the two most nearly associated are those which
utilize transits of Venus. These are similar only in the sense
of resting upon a determination of the relative parallax of
Venus and the Sun during the time of a transit. But the
only common elements which enter into the determination are
the ratio of the distances of the Sun and Venus, which is
determined with such certainty that we can not regard it as
subject to error. The methods of determining the parallax in
the two cases are . completely distinct from the beginning,
there being, I conceive, no common source of error affecting
an observation of contact of limbs and one of a distance
measured from the center of the Sun while Venus is in transit.
I have classified as if they were independent the values of
the parallax which follow from the Pulkowa determinations
of the constant of aberration, and those which follow from all
other determinations. Of course whatever doubts may affect
the theory of the assumed relation between the constant of
aberration and the velocity of light will equally affect both
determinations. I do not, however, conceive that there is
any source of error which can affect both the Pulkowa deter-
minations of the aberration and those made elsewhere. The
two could have been combined so as to give a single result
of the method ; but as the two values of the constant differ
by more than we should expect them to from their probable
errors, I have kept them separate, partly not to give a false
appearance of agreement of results, and partly to facilitate
the inception of any future investigation on the subject.
156
79] THE SOLAR PARALLAX. 157
I have also separated tbe result of GILL'S observations on
Mars, at Ascension, in 1877, from the determinations made by
the same method on the minor planets, because, owing to the
color and disk of Mars, the two results may be affected by
very different systematic errors. The only common systematic
error which seems likely to affect them is that arising from the
color of the object, which will be discussed hereafter.
Results of determinations of the solar parallax arranged in the
order of magnitude.
From the mass of the Earth resulting
from the secular variations of the wt.
orbits of the four inner planets . . . 8.759 i .010 9
From GILL'S observations of Mars at
Ascension . . . t 8.780 d= .020 2
From Pulkoica determinations of the
constant of aberration 8.793 .0046 40
From observations of contacts during
transits of Venus 8.794 i .018 3
From the parallactic inequality of the
Moon 8,794 .007 18
From determinations of the constant of
aberration made elsewhere than at
Pulkowa 8.806 .0056 28
From heliometer observations on the
minor planets 8.807 .007 20
From the lunar equation in the motion
of the Earth 8.825 i .030 1
From measurements of the distance of
Venus from t he Sun's center during
transits 8.857 i .023 2
The mean errors which follow each value are those which,
from a study of the determination, it seemed likely might
affect them, no allowance being made for mere possibility of
systematic error. The weights assigned are convenient small
integers, generally such as to make the weight unity corre-
spond to the mean error 0".30, allowance being made, how-
158 THE SOLAR, PARALLAX. [ 7j
ever, for doubt as to what value should be assigned to the
mean error and for the different liabilities to systematic error.
The mean result is
// //
From all determinations; n 8.797
Omitting the first result; n = 8.800 .0038
The last value differs from the preliminary value 8".802 of
Chapter V, from a change in the weights. It will be seen
that the different values are all as accordant as could be
expected, with the exception of the two extreme ones. In the
largest value we have a case the principles involved in which
have been discussed in Chapter IV.
We can not suppose the parallax to be materially greater
than 8".800, and may take it as probably less than this. Thus
the absolute error of the results of measures of Venus on the
face of the Sun may be considered as about 0".06 or 0".07,
which is four times the computed probable error. The prob-
ability against this, even in the case of one result out of eight
or nine, is so small that we must either regard the method as
being affected by some systematic error, or as affected by
an objective probable error larger than that assigned. It
seems to me the latter view is not untenable, in view of the
very wide range of the possibilities of error which might affect
a series of observations with a heliometer exposed to the Sun's
rays during a period limited to a few hours.
Again, in the photographic measures, the value of a second
of arc in length on the photographic plate enters as a some-
what uncertain element. In this connection it is to be
remarked that the measures of position angle on the photo-
graphic plates, which are not affected with this uncertainty,
although their probable error is quite considerable, give a
value of the solar parallax much smaller than the measures of
distance.
Much more embarrassing is the value which results from the
mass of the Earth. We here meet in another aspect the same
deviation which we encountered in determining the mass of
the Earth from the secular variations, and on which we post-
poned a conclusion ( 64). This determination rests very
79, 80] MOTION OF THE NODE OF VENUS. 159
largely on the motion of the node of Venus, as determined
from the transits of 1761 and 17G9. It is true that results of
meridian observations are combined with them ; but no expla-
nation is thus afforded of the difficulty, because the results of
these observations agree with those of the transits (v. 39).
What adds to the embarrassment and prevents us from wholly
discarding the suspicion that some disturbing cause has acted
on the motion of Venus, or that some theoretical error has
crept into the work, is that, of all the determinations of the
solar parallax this is the one which seems the most free from
doubt arising from, possible undiscovered sources of error. It
is, as we shall presently see, really entitled to twice the relative
weight assigned it. As, however, the determination rests
mainly on the motion of the node of Venus, and this again
mainly rests on the observations of the older transits, I have
made a reexamination of the results of these transits with a
view of reaching a more exact estimate of the sources of error
and the magnitude of the mean error. In this re-examination
I have regarded the Sun's parallax as a known quantity equal
to 8".798, and then obtained the results of the old observations
of the transits on the supposition that the only quantities to
be determined were the corrections to the relative heliocentric
positions of Venus and the Earth.
Rediscussion of the motion of the node of Venus.
80. In discussing the observations of 1761 and 1769 (Astro-
nomical Papers, Vol. II, Part V), I introduced a quantity
expressive of the error in the observed time of contact arising
from imperfections of the telescope and atmospheric absorp-
tion and dispersion. The constants on which these errors
depend are represented by symbols fc 2 and & 3 . As I have
worked up the observations, the ultimate result of each
observation of contact is the value of an unknown quantity,
dCj which, were there no imperfections of vision and were the
radii of the Sun and Venus accurately known, would represent
the correction to the tabular distance of centers. As a matter
of fact, however, we are to consider 6 c as equal to this correc-
tion increased by a rather complex combination of quantities
depending on the errors of the assumed semidiameters of
160 MOTION OF THE NODE OF VENUS. [80
Venus and the S.un, and the thickness of the thread of light
when it first became visible at second contact, or vanished at
third contact. The observations must be so combined as to
eliminate these quantities. What I have done is to represent
the undiscoverable minute correction to dc thus arising by
the symbol 2 for second contact, and 3 for third contact. In
the present re-examination the absolute terms are reduced to
the parallax 8".798 by putting 67r Q = - ".05 and n> = - ".025
in the final equations of the original paper. After each result
is given the mean .error with which it is affected, as deter-
mined by the investigation in question. When thus treated,
the equations which I have given on pages 391-398 of the
paper referred to give the following normal equations for tfc,
the indeterminates & 2 and & 3 being retained as such in order to
show their final effect on the result.
// //
1761. II 5 8.5 do = + 0.76 - 18.5 fc 2 0.78
III; 41.7 dc = - 2.81 - 19.2 k, 1.30
1769. II; 44.8 dc = - 8.00 - 104.1 fc 2 i 1.95
III; 12.1 do = + 0.31 - 16.0 fc 3 0.70
In order to vary the proceeding as much as possible from
that of the former investigation, I now express dc in terms of
dX and 6fi, which, for the time being, I take as the corrections
to the heliocentric longitude and latitude of Venus referred
to the Earth, and these again in terms of dv and sin 166,
which latter, for brevity, I call u. The first transformation is
made with the coefficients of p. 71, where we have put x and
y for 6\ and d/3, and the last by the equations
//
6X = 6v + 0.06 u
8p = u 0.06 v
Putting Ui for the value of u in 1765, we have, in consequence
of the known change in the motion of the node,
//
In 1761; u = Ui + 0.11
In 1769; u = ^ 0.11
80] MOTION OF THE NODE OF VENUS. 161
We thus have the four equations which follow for determining
6v and HI, the former being supposed the same at the times of
the two transits.
- .84 6v - .55 M! + * a = + 0.15 - 2.2 & 2 i 0.09
+ .73 - .69 + z 3 = + 0.01 - 0.5 fc 3 0.03
- .69 + .73 + 2 2 = - 0.10 - 2.3 A" 2 0.04
+ .81 + .60 + s 3 = + 0.10 - 1.3 & 3 0.06
Eliminating z. 2 and 3 by subtracting the first equation from
the third, and the second from the fourth, we have
.15 6v + 1.28 % = - o'.25 -~ o'.l A; 2 i 0.10
.08 <* + 1.29 ui = + 0.09 - 0.8 fc 3 0.07
We thus have for Ui the value
m = - 7/ .04 - 0.08 6v - 0.03 A: 2 - 0.36 fc 3 i O x/ .05
dv can not be determined independently of z 2 and 3 . Assum-
ing these quantities to be equal, we have already found it to
be only /7 .02, and may therefore, to determine its probable
effect upon the result by assigning to it the value
In the former paper I have found for k 2 and & 3 the values
fc 2 = + 0.040 i 0.040
Jc 3 = - 0.034 0.040
A preliminary correction of + 2 /7 .02 having been applied to
the tabular orbital latitude, we have, for the epoch 1765.5,
sin id 6 = + 1".99 i 0".06
Combining this result with that of the transits of 1874 and
1882, we have the following results, which are compared with
those of meridian observations :
//
Transits of Yenus alone ...... sin i D t $6 = 2.82
Meridian observations alone .... " 2.45
Combined solution ........ _ 2.71
Adjusted with other* results (46) . . . 2.73
Adopted ........... 2.77
5690 N ALM - 11
162 MOTION OF THE NODE OF VENUS. [80
The adopted result is the one which seems the most probable.
For the final probable error we are to include that of the pre-
cession and of the Sun's longitudes at the two epochs. We
may estimate the combined value of these at i 1", correspond-
ing to an error of 0".06 in sin i D t 66. Thus we have
sin i D t 66 = 2". 77 i // .084
I conceive this mean error to be as real as any that can be
determined in astronomy. This conviction rests upon the fact
(1) that the systematic errors affecting the four contacts are
shown to be small by the general minuteness of the four values
of dc; (2) that whatever systematic errors may affect the
formation or disappearance of the thread of light are almost
completely eliminated from the mean of the transits of 1761
and 1769 by the method in which the observations have been
combined. The accordance of the observations of external
contact made at the same transits strengthens this view.
The equation thus derived takes the place of the sixth
equation of 63 and should have twice the weight there
assigned. As the mass of the Earth determined by the secu-
lar variations rests mainly on this equation, I shall first con-
sider it alone. Expressing the theoretical secular variation of
sin i66 in terms of the above observed value, we find that the
observed motion of the node of Yenus gives the equation
0".26 v 29". 2 v 1 43".2 v" = + 0".48 i // .084 (a)
which gives for v" the value
v" = 0.0 ill 4- 0.006 v - 0.676 v 1 i .0019
The value of the solar parallax for v" is 8" .811. Hence,
for the value expressed in terms of the corrections to the
assumed masses of Yeuus and Mercury, this equation gives
n = 8".778 + 0".020 r 1".98V
We have found from the periodic perturbations
// //
v - _ 0.055 i .25
v 1 = + 0.0080 i .0025
80] SOLAR PARALLAX. 163
Whence,
// //
Y" = - 0.0168 i .0029
n = 8.762 i .0086
This result of observation, errors and unknown actions aside,
Fcan not suppose to be affected by any other mean error than
that here assigned.
We have now to consider how far this result may be recon-
ciled with the others by changes in the masses of Mercury
and Venus. No admissible change in the former could greatly
affect the result. The question then arises whether the dis-
crepancy may not be due to an error in the concluded mass
of Venus. In making so large a change in this element, we
meet with insuperable difficulties. The observed motion of
the ecliptic, which is a fairly well-determined quantity, indi-
cates a still further increase of this mass. We may put this
difficulty in another form. The observed motion of the node
of Venus is a relative one, consisting in the combined effect of
the motion of the ecliptic around an axis at right angles to the
node of Venus, and an absolute motion of the orbit of Venus
around nearly the same axis. This motion of the ecliptic
depends mainly on the mass of Venus ; the absolute motion
Of the orbit of Venus mainly on that of the Earth. If, now, we
determine the motion of the ecliptic from observation, we shall
find that the relative motion of the orbit of Venus still unac-
counted for is yet greater than we have supposed it to be, and
should therefore find a yet smaller mass of the Earth than that
heretofore concluded.
The determination of the mass of Venus already made from
observations of the Sun and Mercury seems to admit of no
doubt. We can not conceive that the mean of fifteen deter-
minations, made during one hundred and thirty years, at dif-
ferent observatories, which determinations are so separated as
to be entirely independent of each other, can be affected by
any considerable common error. The entire accordance of the
result thus reached from the periodic perturbations produced
by Venus with that from a combination of all the secular
variations, as shown in Chapter VI, strengthens the result
yet further. Unknown actions and possible defects of theory
164 SYSTEMATIC ERRORS OF PARALLAX. [tO, M
aside, it seems to me that the value of the solar parallax
derived from this discussion is less open to doubt from any
known cause than any determination that can be made.
Possible systematic errors in determinations of the parallax.
81. We have now to return to the other values, in order to
see to what extent they may be affected by systematic error.
I have already excused myself from discussing the validity of
the assumed relation between the constant of aberration and
the velocity of light, because there is nothing valuable to be
said on the subject, and have alluded to the possible sources
of systematic error in the Pulkowa determinations of aberra-
tion. It is worthy of attention here that the very best of these
determinations, that of NYR^N with the prime vertical transit,
in resp,ect to the care with which it was made, and the general
accordance of the entire work throughout, gives a result most
accordant with that under consideration. In fact, to the value
8". 77 of the solar parallax corresponds the value 20 // .55 of
the constant of aberration, which is larger by only // .02 than
the result of NYREN'S best determinations.
A.S for miscellaneous determinations of the constant, it is to
be remembered that the corrections applied to a part of the
separate values on account of the Chandlerian inequality of
latitude are somewhat doubtful, and the general mean mav
have been affected by a few huudredths of a second in conse-
quence. It is not, however, possible to determine the amount
of the correction, except by an exhaustive rediscussion of the
whole of the original observations, and even then the result
would still be doubtful.
Next in the order of weight we have the results of measures
on the minor planets with the heliometer, on GILL'S plan. I
have already remarked upon the possible error in such obser-
vations arising from the probable difference of color between
the planet and the star. A hypothetical estimate of the
amount of this error is worth attempting. Let us assume that
in the case of a minor planet the mean of the visible spec-
trum corresponds to the line D, and that in the case of a star
the same mean is halfway between the lines D and E.
81] SYSTEMATIC ERRORS OF PARALLAX. 165
The index of refraction of air has been determined inde-
pendently by KETTLER and LORENTZ for the different rays.
The mean of their results for the rays D and E is
For D, n = 1.000 2920
ForE, n = 1.000 2940
These results are accordant in giving a dispersion between
these two lines equal to about .0037 of the total refraction.
We have hypothetically taken the extreme possible difference .
between planet and star to be one-half of this. At an altitude
of 45, where the refraction is about 60", the error would be
0".ll. At an altitude of 30 the error would be 0".20. We
are thus led to the noteworthy conclusion :
If the difference between the spectra of a minor planet and a
comparison star is such that the means of their respective visible
spectra, or the apparent amounts of their respective refractions,
differ by one- tenth of the space between D and E, an error of
0" .02 or 0" .03 may be produced in the apparent parallax of the
planet.
The question thus arising maybe readily settled by measures
with the heliometer. The distances of pairs of stars differing
as widely as possible in color should be measured at different
altitudes, when one is nearly above or below the other, in
order to see what difference of refraction depending on the
color is indicated. A colored double star, such as ft Oygni,
might also be used for the same purpose.
The minor planets are of different colors. I am not aware
of any evidence that Victoria or Sappho differ in color from
the average of the stars, but 1 believe that Iris is somewhat
yellow, or reddish. Kow, in this connection, it is a significant
fact that the parallax found from observations of Iris, 8".82o,
is the largest by GILL'S method.
I have already remarked that the value of the solar parallax
derived from the parallactic equation of the Moon is one of
which the probable mean error is subject to uncertainty.
While it is true that the value may be smaller than that we
have assigned, we must also admit that it may be much larger.
The probable error of the determination by the lunar equa-
tion of the Earth is larger than that of any other method. At
166 RESULTS FOR THE SOLAR PARALLAX. [82
the same time I do not think that it is liable to systematic
error, and we must therefore regard the mean error assigned
as real.
Results for the solar parallax after making allowance for prob-
able systematic errors.
82. Let us now see whether we can reach a satisfactory
result by making a liberal allowance for the more or less
probable sources of systematic error just pointed out. The
modifications we make in the weights formerly assigned are
these: We reduce the weight of GILL'S Ascension result to
one-half, owing to the uncertainty arising from the color of the
planet Mars. We retain the Pulkowa determinations of the
constant of aberration with their full weight, but reduce the
weight of the miscellaneous determinations. In the case of
the parallactic inequality, we reduce the weight for the reasons
already given. We omit Iris from the determination from the
minor planets. We also reduce to one- half its former value
the relative weight assigned to measures of Venus on the Sun,
on the theory that the actual mean error must be larger than
that given by the discordance of results. Our combination
will then be as follows :
wt.
From the motion of the node of Venus .... n = 8.708 10
From GILL'S Ascension observations .... 8.780 1
From the Pulkowa constant of aberration . . . 8.793 40
From contacts of Venus with the Sun's limb . . 8.794 3
From heliometer observations on Victoria and
Sappho 8.799 5
From the parallactic inequality of the Moon . . 8.794 10
From miscellaneous determinations of the con-
stant of aberration 8.806 10
From the lunar inequality in the motion of the
Earth 8.818 1
From measures on Venus in transit 8.857 1
Mean result, ignoring the first ; 8".7965i .0045
This mean result still differs from that given by the motion
of the node of Venus by more than five times the probable
error of the latter, and is yet farther from the combined result
82] RESULTS FOR THE SOLAR PARALLAX. 167
of all the secular variations, so that no reconciliation is brought
about.
The embarrassing question which now meets us is whether
we have here some unknown cause of difference, or whether
the discrepancy arises from an accidental accumulation of
fortuitous errors in the separate determinations. We have
already discussed the former hypothesis, and have been unable
to find any reasonably probable cause of abnormal action.
The motion of the planes of the orbits is that which is least
likely to deviate from theory, because it is independent of
all forms of action depending upon distance from the Sun,
or upon the velocity of the planet.
An examination and comparison of all the results shows one
curious feature: the unanimity with which the secular varia-
tions speak against the large value of the solar parallax, or
of the mass of the Earth, as the one quantity at fault. The
adopted motion of the node of Venus is sustained not only by
the meridian observations, but by the external contacts at the
transits of 1761 and 1769, and, weakly, by a comparison of the
transits of 1874 and 1882.
If we determine the correction of the mass of the Earth from
other secular variations than that of the node of Venus, by
the equations of 63, we have, after eliminating the masses of
Mercury and Venus,
v" = -0.029; p. e. .018
If, instead of eliminating these values, we put
v = + .08; v 1 = + .0080;
we have
v" = -0.026; p. e. i .014
In each case the value of the parallax is yet smaller than that
found from the motion of the node of Venus. I have already
remarked that the observed motion of the ecliptic indicates
an increase of the mass of Venus.
The question thus takes the form, whether it is possible that
the mean of the eeven determinations of the solar parallax
TT = 8".797 i ".0035
168 DEFINITIVE ADJUSTMENT. [82, 83
can with reasonable possibility be in error by aii amount the
correction of which would bring it within the range of adjust-
ment of the other quantities.
From what has already been said of the systematic errors
to which every one of the determinations may be liable, it is
evident that we should have no difficulty in accepting the
necessary reduction of each of the separate values. The
improbability which meets us is not so much the amount of
the individual errors of the determinations as the fact that
seven of the eight independent determinations should all be
largely in error in the same direction.* Still, under the cir-
cumstances, we must admit this possibility, and make what
seems to be the best adjustment of all the results.
Definitive adjustment.
83. In making the definitive adjustment I shall proceed on
the supposition that no correction is necessary to the adopted
mass of Mars. I also go on the principle that no result is to
be rejected on account of doubt or discordance, except when
it is affected with a well-established cause of systematic error,
and shows a large deviation in the direction in which this
cause would act. At the same time it will be admissible to
diminish the weights in special cases, on account of causes of
systematic error which we know to exist, although we can not
determine the directions in which they would act ; and also on
account of deviations so wide as to show that the probable
error of the result must have been greatly underestimated.
Proceeding on this plan, we might reweight the last eight
results for the solar parallax, so as to get a result slightly
different from 8". 797. But 1 doubt whether such a reweight-
ing would not involve an objectionable bias.
We might diminish the weight of the result given by the
Pulkowa constant of aberration on the ground that no one
method should have so preponderating a weight as this has.
If we did so the result might be increased to 8".800. We
* For a very searching criticism of the systematic errors with which the
determinations of the solar parallax may be affected, reference may be
made to the first two articles by Dr. DAVID GILL, in Vol. I of The Observa-
tory.
83] DEFINITIVE ADJUSTMENT. 169
might very largely increase the relative weight assigned to
the heliometer observations on Victoria and Sappho, but no
admissible increase would appreciably change the result. We
might also diminish the relative weight of the largely dis-
cordant result derived from measures of Venus during transit.
But as, by throwing out this result altogether, we should only
diminish the mean by ".001, it is scarcely worth while to do
so. Altogether no rediscussion of the relative weights seems
necessary.
On the other hand, the weight which we assign to the mean
result will enter as a very important factor into the final
adjustment. This is a point on which it is impossible to reach
a positive numerical conclusion by any mathematical process.
If, as one extreme case, we consider that the mean error of
each separate result corresponds to i0 7/ .03 for weight unity,
we shall have a mean error of rt".0035 for the value 8". 797.
The result will not be very different if we determine the mean
error from the discordance of the eight separate results. On
the other hand, if we include the deviation of the result given
by the motion of the node of Venus, the mean error for weight
unity will be increased to i 0".0045. The latter is undoubt-
edly the most logical course, so long as we proceed on the
hypothesis that the deviations of the final adjustment can all
be explained as due to fortuitous errors. If we include a com-
parison with the results of all the secular variations we shall
have a yet larger mean error. To show the result of assigning
one weight or the other I shall make two solutions, A and B,
in one of which a less and in the other a greater weight will
be assigned.
To the value 8".797 i .005 or .007 of the solar parallax
corresponds
r" = - 0.049 i .0016 or .0025
According as we assign one weight or the other to this result,
we may take as the corresponding equation of condition of
weight unity
(A): 400^' =-2.0
r (BH 600," = -2.9 W
170 DEFINITIVE ADJUSTMENT. [83
The masses of Venus and Mercury, determined by methods
independently of the secular variations, also enter as conditions
into the adjustment. I have, however, made a revision of the
preliminary adjustment given in 64, the latter being based on
the results of 32-38; whereas it is better to use the defini-
tive results of the combination used in 46.
For the mass of Mercury the result found in 53 by the
last combination is
The values of the denominator corresponding to the mean
limits here assigned are
5 890 000 and 12 210 000
These limits are so wide as to include all admissible results for
the mass of Mercury. Moreover, we can not definitely say that
the value (6) of this mass is markedly greater or less than that
given by the weighted mean of all other results, since we
might so weight the latter as to give a result greater or less
without transcending the bounds of judicious judgment. I
conceive, therefore, that we are justified in reducing the mean
error to i 0.26, which will give as the equation of condition
r= - 0.055 i 0.25
and hence
40 x = - 0.22 i 1 (c)
When, in the normal equation for the mass of Venus, given
by the observations on Mercury, we substitute the values of
the secular variations found from the general combination of
46, the result is
v 1 = 0.0114
Combining this with the result from the Sun, we have
v 1 = - 0.0117
In view of the fact that the mass derived from observations of
Mercury may be affected by systematic errors of the kind
83] DEFINITIVE ADJUSTMENT. 171
shown and discussed in 53, the mean error formerly assigned
to this result should be somewhat diminished. The result is
406 600
From this we have
v' = + 0.0084 .0030
For the equation of condition of weight unity I take
330 v' = + 2.8 (d)
With these equations of condition we have to combine the
eleven equations of 63, which we use unchanged, except that
we double the weight assigned, to the sixth equation, that
derived from the motion of the node of Venus, on account of
the smaller probable error of the result of our preceding redis-
cussion, and use the value of the absolute term found in 80.
If we accept the view that all the perihelia move according
to the same law of gravitation toward the Sun, namely, that
expressed by HALL'S hypothesis, then the value of the quan-
tity 6 in the formula expressing the law of gravitation is so
well determined by the motions of Mercury that it becomes
legitimate to use the observed motions of the perihelia of the
other three planets as equations of condition. But since it is
not impossible that the minor planets between Mars and
Jupiter may have an appreciable influence on the motion of
the perihelion of Mars, it is a question whether we should not
exclude that motion from the equations.
The conditional equations given by the motions of the three
perihelia in question are found by comparing the results of
46, 54, and 61. They are
40 x + v 1 + 20 v" = + 1.0
-14+46 +0 = - 0.3 (e)
2 - 13 +61 = + 0.7
The conditional equations to be combined are the eleven
equations of 63, the sixth of which is to have double weight^
and the six equations (a), (c), (d), and (e).
172
DEFINITIVE ADJUSTMENT.
[83
The normal equations to which we are thus led are the
following, which show the results of the four combinations we
may make according as we use (A) or (B) for the equation
given by the mass of the Earth, and omit or include the third
equation (d), which is given by the motion of the perihelion
of Mars.
(a.) Including the motion of the perihelion of Mars.
9 607,r 7 147 7' ; 11 335j/" = + 220
= -587
= - 3388 (A)
= - 4328 (B)
7 147 + 267 174 + 168 727
11 335 + 168 727 + 406 300
1 1 335 , + 168 727 + 606 300
(/?.) Omitting the motion of the perihelion of Mars.
9 603# - 7 12lv' - 11 457 v" = + 219
- 7 121 + 267 003 + 169 520 = - 578
- 11 457 + 169 520 + 402 578 = - 3431 (A)
- 11 457 + 169 520 + 602 578 = - 4371 (B)
The results of the solutions in the four cases are:
Aa
A//
B
B/?
X -f
0.0147
+ 0.0142
+ 0.0161
+ 0.0158
V -j-
0.147
+ 0.142
-f 0.161
+ 0.158
V 1 +
0.004 34
+ 0.004 60
+ 0.003 10
+ 0.003 i>.")
v"
0.009 73
0.01005
- 0.007 70
0.007 87
1
-V- m
6 539 000
6 567 000
6 460 000
6 477 000
1
+ m'
408 230
408 120
408 730
408 670 ,
7T
8".783
8 // .782
8".789
8".788
I conceive that if the secular variations, especially the motion
of the node of Venus, are not affected by any unknown cause,
some mean between these should be regarded as the most
probable solution. The result does not, however, bring about
a satisfactory reconciliation. We still find ourselves confronted
by this embarrassing dilemma: Either there is something
abnormal in connection with the node of Venus, due to an
unknown cause acting on the planet, to some extraordinary
errors in the observations or their reduction, or to some error
in the theory on which the discussion is based, or the deter-
83, 84, 85] ADOPTED PARALLAX AND MASSES. 173
ruinations of the solar parallax are nearly all in error in one
direction by amounts which are, in more than one case, quite
surprising.
Possible causes of the observed discordances.
84. Two possible causes of discordance may be suggested,
one of which has not been touched upon at all in the preceding
chapters, and one perhaps inadequately. As to the hypothesis
of non-sphericity of the Sun, considered in 56, it may be
remarked that Dr. HARTZER shows that an ellipticity of the
Sun sufficient to produce the observed motion of the perihelion
of Mercury would cause a direct motion of 5".l in the motion
of the node of Venus. This would correspond to a change of
0".30 in the value siniD t # and would therefore go far toward
reconciling the discrepancy. But it is easy to see that this
cause would produce a secular motion of 2".6 in the inclina-
tion of Mercury. We have seen that the observed motion of
the inclination already exceeds the theoretical motion by 0".38;
so that introducing the hypothesis of ellipticity of the Sun we
should have a discrepancy of about S^.O between theory and
observation. This conclusion alone seems fatal to the theory,
which otherwise has been shown to be scarcely tenable.
The other possible cause is an inequality of long period ;
especially one depending on the argument \3l" 81' which
has a period of about two hundred and forty three years. A
very simple computation shows that the coefficient of this term
is only of the order of magnitude (V'.Ol.
It is a curious coincidence that if we had neglected to add
the mass of the Moon to that of the Earth, in computing the
secular variations, the discrepancy would not have existed.
Adopted values of the doubtful quantities.
85. The practical question which has been before the writer
in working out the preceding results is : What values of the
constants should be used in the tables of the celestial motions
of which the results of this discussion are to form the basis ?
Should we aim simply at getting the best agreement with obser-
vations by corrections more or less empirical to the theory ?
It seems to me very clear that this question should be answered
in the negative. No conclusions could be drawn from future
174 ADOPTED PARALLAX AND MASSES. [85
comparisons of such tables with observations, except after
reducing the tabular results to some consistent theory. The
imposition of such a labor upon the future investigator is not
to be thought of. Moreover, there is no certainty that the
tables which would best represent past observations would
also best represent future ones. Our tables must be founded
on some perfectly consistent theory, as simple as possible, the
elements of which shall be so chosen as best to represent the
observations.
In choosing the theory and its constants we have again a
certain range. If we accept the necessity of assuming the
secular variations of the orbits of Mercury and Venus to be
affected by the action of unknown masses of matter, then the
simplest course to adopt is to construct our theory on the sup-
position of a planet or group of planets between Mercury and
Venus.
It seems to me that the introduction of the action of such a
group into astronomical tables would not be justifiable. The
more I have reflected upon the subject the more strongly
seems to me the evidence that no such group can exist, and,
indeed, that whatever anomalies exist can not be due to the
action of unknown masses of matter.
Besides, the six elements of such a group would constitute
a complication in the tabular theory.
On the other hand, it did not seem to me best that we should
wholly reject the possibility of some abnormal action or some
defect between the assumed relations of the various quanti-
ties. What I finally decided on doing was to increase the theo-
retical motion of each perihelion by the same fraction of the
mean motion, a course which will represent the observations
without committing- us to any hypothesis as to the cause
of the excess of motion, though it accords with the result of
HALL'S hypothesis of the law of gravitation ; to reject entirely
the hypothesis of the action of unknown masses, and to adopt
for the elements what we might call compromise values between
those reached by the preceding adjustment and those which
would exist if there is abnormal action. The exigency of hav-
ing to prepare the tables required me to reach a conclusion on
this subject before the final revision of the preceding discus-
85,86] FUTURE DETERMINATIONS. 175
sion, so that the numbers used are not wholly based upon it.
The conclusions I have reached are these:
Since, if there is nothing abnormal in the theory, the solar
parallax is probably not much larger than 8".780, and if there
is anything abnormal it is probably as large as 8".795 or even
8' '.800, we may adopt the value 8". 790 as one which is almost
certainly too large on the one hypothesis and too small on the
other, and which is therefore best adapted to afford a decision
of the question.
For the mass of Venus I took, as an intermediate value,
m ' =1-1-408000
For the mass of Mercury I took
1 4- 6,000,000
Actually it seems that this mass is larger than the most prob-
able one on either hypothesis, though not without the range of
easy possibility.
With these values the outstanding difference between theory
and observation in the centennial motion of the node of
Venus is
A sin i D t = 0".25
If this difference arises wholly from the error of the theory,
then between the transits of 1874 and 2004 the accumulated
error would amount to 0".32 in the heliocentric latitude, and
about 0".8 in the geocentric latitude. Unless an improvement
is made in the method of determining the position of Venus
by observation, the twentieth century must approach its end
before this difference can be detected.
Bearing of future determinations on the question.
86. The following shows the influence which subsequent
determinations of the principal elements will have upon our
judgment as to the solution of the dilemma. The changes in
the second column will, by emphasizing the discordance
between the results, tend to confirm the hypothesis of an
abnormal defect in the theory, while the opposite ones, in the
last column, will tend to reconcile theory and observation :
USUVBRSITf
176
FUTURE DETERMINATIONS.
[86
Element or quantity.
Change tending to
confirm the dis-
cordance
between theory
and observation.
Change tending to
reconcile exist-
ing theory with
observation.
The solar parallax.
Increase.
Diminution.
Longitude of the node of Mercury.
Increase.
Diminution.
Longitude of the node of Venus.
Increase.
Diminution.
Constant of aberration.
Diminution.
Increase.
Mass of Venus.
Increase.
Diminution.
Mass of Mercury.
Diminution.
Increase.
Secular diminution of the obliquity.
Diminution.
Increase.
Among these constants are some the values of which can
scarcely be decisively obtained except by observations con-
tinued through half a century, or even through the whole
twentieth century, unless improvements are made in our pres-
ent methods of observing.
The improvement of others, however, is quite within the
reach of the astronomy of the present time. Among these
the constant of aberration and the solar parallax have the
first place. The more accurate determination of these quanti
ties thus assumes an importance which may justify some sug-
gestions on the subject.
The observations made on the European continent for the
detection and study of the variations of latitude have been
executed with such precision that we might look to them for a
marked improvement in the determination of the constant of
aberration, were it not for a single circumstance. In the gen-
eral average few are made after midnight, while the maxima
and minima of aberration occur in the morning and evening.
The extension of the system into the early morning therefore
seems desirable. Although these observations may scarcely
equal in accuracy those made by NYREN, with the prime
86] FUTURE DETERMINATIONS. 177
vertical transit, they have the advantage of not requiring so
long a period for a complete observation. The great disad-
vantage of the prime vertical instrument is that unless a star
culminates within a few minutes of the zenith, an hour, or
even several hours, will be required for the completion of a
determination, which may thus be made impossible by the
'advent of daylight. It may be remarked in this connection
that the northern latitudes of the European observatories are
favorable to the determination of the aberration-constant.
LOEWY'S method has over all others the great advantage of
being independent of the direction of the vertical. But its
application, and the reduction of the observations made with,
it, are laborious in a high degree.
So far as practical astronomy has yet developed, the best
method of directly measuring planetary parallax, and there-
fore the only one to be considered, is that of GILL. It there-
fore seems desirable that measures by this method should be
continued. At the same time it is very necessary that the
spectra of the small planets to be used should be carefully
studied photometrically, and that the probable influence of
coloration upon the measures should be investigated.
The necessity of completing the present work, and of pro-
ceeding immediately to the construction of tables founded
upon the adopted elements, prevent the author's awaiting the
mature judgment of astronomers upon the embarrassing ques-
tions thus raised. The regret with which he accepts this
necessity is weakened by the consideration that even if the
solar parallax which he has adopted requires the largest cor-
rection to which it can reasonably be supposed subject, namely,
one of 0".015, reducing the value of this constant to 8". 775,
the effect of the error will not be prejudicial to the astronomy
of the'immediate future.
More important will be the error /x .035 in the constant of
aberration. Yet a long-continued series of observations will
be necessary to establish even the existence of such an error,
and should it prove detrimental in any astronomical work the
evil will be easily remedied by a slight correction.
5690 N ALM 12
CHAPTER IX.
DERIVATION OF RESULTS.
Ulterior corrections to the motions of the perihelion and mean
. longitude of Mercury.
87. In 32 and 46 we have reached three values of the
correction to the tabular motion of the perihelion of Mercury.
Of these the first rests on meridian observations alone, the
second on the combination of meridian observations with trans-
its, and the third is derived by substituting in the eliminating
equations the corrections to the solar elements and their secular
variations which result from observations. The three values
thus reached are 9".54, 1".01, and + 6".34. The pro-
gressive divergence of these values, taken in connection with
the discrepancy pointed out in 33, leads us to distrust the
influence of the meridian observations upon the motion of the
perihelion. Under these circumstances I deem it advisable to
make such final corrections to the motions in mean longitude
and mean anomaly as will best satisfy all the observed transits
over the disk of the Sun. In doing this I am enabled to intro-
duce the results of a preliminary discussion of the transits of
1891 and 1894. By combining the observations of these two
transits with those of the older ones I derive the following
values of the functions Y and W defined in 31 :
// //
Y = -1.93- 3.03 T
W= + 1.50 + 2.04 T
The preliminary theory, so far as yet investigated, gives for
the values of this quantity,
// //
Y = - 2.44 3.40 T
W = + 1.38 + 1.3GT
178
87, 88] PERIHELION OF MERCURY. 179
Equating these values to the corresponding linear functions
of the corrections to /, TT, and their secular motions, we have
the equations,
// //
0.72 SI + 0.28 67f = + 0.12 + 0.68 T
+ 1 .49 - 0.49 = + 0.51 + 0.37 T
We find, from these equations,
// //
61 = +0.26 + 0.56 T
Sn = - 0.24 + 0.97 T
The preliminary values to which these corrections are appli-
cable are
// //
61 = +0.04- 1.33 T
6jr= + 5.83 + 6.34 T
The definitive values thus become
61 = + 0.30- 0.77 T
tf TT = + 5.59+ 7.31 T
Definitive elements of the f out inner planets for the epoch 1850, as
inferred from all the data of observation.
88. We have made a fourth solution of the normal equations
which give the corrections to the elements of each planet by
substituting in those equations the definitive values of all the
other quantities, including the values of the secular variations
derived from theory. In making this substitution for Mercury,
however, the ulterior corrections just found were not applied.
The values of the unknowns resulting from this solution are
shown in the first column of the next table. From these
numbers are derived the definitive elements for 1850, 'by the
following processes:
(a.) By multiplying the unknowns by the appropriate factor
given in 27, we have the corrections of the tabular elements
at the mid- epoch of observations for each planet. These cor-
rections are found in the second column.
(/?.) The preceding corrections are to be reduced from the
respective mid-epochs to 1850. This reduction is found by
180 DEFINITIVE QUANTITIES. [88
multiplying the definitive correction to the tabular secular
variation by the elapsed interval, and is shown in the third
column.
(y) We next have the value of the tabular elements for the
fundamental epoch 1850, January 0, Greenwich mean noon.
These numbers are those of LEVERRIER'S tables, with the
following modifications:
(d) The reduction from 1850, January 1, Paris noon, to
January 0, Greenwich noon
(f) The corrections to LEVERRIER'S values of the eccen-
tricity and perihelion which are necessary to represent those
terms in the perturbations of the mean longitude which depend
only upon the sine and cosine of the mean anomaly. The
theory is more symmetrical in form when all such terms are
included with those of the elliptic motion. In LEVERRIER'S
tables they have the following values:
Mercury 5 6v = 4- 0.030 sin I - 0.111 cos I
Venus; +0.010 +0.037
Earth; 0.067 -0.098
Mars; +1.061 +0.718
These terms of the longitude may be represented by the follow-
ing corrections to the elements:
Mercury; de = + 0.058 dn r= 0.0
Venus; -0.012 +2.3
Earth; +0.054 +1.4
Mars; +0.613 -1.0
Applying these corrections d and e to LEVERRIER'S tabular
quantities, we have the values of the tabular elements as given
in the fourth column. Then applying the preceding correc-
tions we have the definitive values given in the last column.
In some cases this derivation is modified. Instead of using
the correction to the perihelion, mean longitude and mean
motion of Mercury given by the unknown quantities of the
88] ELEMENTS FOR 1850. 181
equations, we have used the values for 1850 derived from the
discussion of the preceding section.
The quantities which give the position of the node and
inclination have been treated in the same way as their secular
variations. The symbols J and N indicate values of the
unknown quantities related to the corrections of the elements
J and N. These unknowns are then changed to corrections of
the elements by the factors of 27, and these again to correc-
tion of the inclination and node by the equations of 41.
In the case of the node of Venus two values are given. The
value (a) is that which follows immediately from the normal
equations. If we carry forward the position of the node just
derived to the mean epoch of the last two transits of Venus,
we find a discrepancy amounting to 2".04 in the longitude,
corresponding to a difference of 0".121 in the heliocentric lati-
tude. This is considerably larger than the probable error of
the results of the observations of the transits. It may, there-
fore, be questioned whether the latter are not entitled to a
greater relative weight than that assigned, owing to the prob-
able systematic errors of the meridian observations. A second
value (b) has therefore been derived from the observations of
the transits alone. In subsequent investigations we may
choose between these two values.
Formation of definitive elements of the four inner planets, for tlit\
epoch 1850 7 January 0, Oreemvich mean noon.
Mercury.
Unknown of Corr. of Red. to Tabular Concluded
equations. element. 1850. element. element.
// // // //
n -.0940 - 0.77 0.0 538106654.49 538106653.72
e - .0741 - 0.222 - 0.005 42 409.088 42 408.861
n + .6763 -1- 5.59 75 7 13.78 75 7 19'!s7
t .0402 + 0.30 323 11 23.53 323 11 23.83
i -.2762J-- 0.64 - 0.07 7 7.71 7 7.00
d -.0001N+ 3.88 - 0.27 46 33 8.63 46 33 12.24
182 DEFINITIVE QUANTITIES. [88,89
Formation of definitive elements, etc. Continued,
Venus.
Unknown of Corr. of Ked. to Tabular Concluded
equations, element. 1850. element. element.
n - .1783 - 3.57
210669165.04 210669161.47
e + .1463
0.439 - 0.165
1 411.522
1 411.796
129
7t + .0835 + 36.6 16.4
z _ .1330 - 0.67 + 0.46. 243
i + .0968 J + 0.31 + 0.12 3
0(a)+ .0126 N- 9.39 + 6.63 75
0(b) -20.36 +15.56
Earth.
27
57
23
19
14.3 129
44.34 243
34.83 3
52.21 75
75
27
57
23
19
19
34.5
44.13
35.26
49.45
47.41
1.10
0.12
2.4
0.15
0.02
129 602 767.84 129 602 766.74
3 459.334
100 21 43.4
23 27 31.83
99 48 18.72
Mars.
- .1094 - 0.88 68 910 105.38
- .1088 - 0.155 + 0.058 19 237.101
+ .1663 + 2.38 + 0.02 333
3 459.454
100 21 41.0
23 27 31.68
99 48 18.74
68 910 104.50
19 237.004
- .4029 - 0.81 + 0.05 83
- .0507 J + 0.18 - 0.01 1
+ . 1135 N+ 6.56 +1.34 48
17
9
51
23
52.47
16.92
2.28
53.02
333
83
1
48
17
9
51
24
54.87
16.16
2.45
0.92
Definitive values of the secular variations.
89. The definitive values of the secular variations, as inferred
from the adopted theories and the concluded values of the
masses, are shown in the following table, which gives in detail
the parts of which each quantity is made up.
The first four lines of the table give the values of the secular
variations as they result from the investigations found in Vol.
V, Part IV, of the Astronomical Papers, after correcting the
mass of each planet by its appropriate factor.
The motion of the perihelion first given, denoted by D t n\,
is measured along the plane of the orbit itself. The numbers
89 1 SECULAR VARIATIONS. 183
given being divided by the corresponding values of the eccen-
tricity we have the motion of the perihelion itself along the
plane. The symbols i and # represent the inclinations and
longitudes of the nodes referred at each epoch to the ecliptic
and equinox of 1850, regarded as fixed. The motions of these
elements are next to be referred to the fixed ecliptic of the
date. So referred, they are designated as D? i and D? 6. The
transformations to the latter quantities are made by comput-
ing an approximate value of the motion of the node due to
the motion of the ecliptic alone along the plane of the orbit
regarded as fixed.
If we put
,, the inclination of the fixed orbit of the planet at any epoch
TO to the moving ecliptic at any time;
61, the longitude of the corresponding node, h;
F, the distance from the node Q t to the instantaneous rota-
tion axis of the orbit at the epoch T ;
we shall have
D t v = H" cosec i\ sin (L" #1) (a]
If we compute v and H from the equations
H sin VQ = sin i D?
H cos r = D? i
and then find Av by integrating the value (a) of D t r from 1850
to the date we shall have
sin i D? # = H sin ( V Q + A v]
D i = H cos (v Q -f Av)
The change of D t ^ between 1850 and the extreme epochs has
been found so nearly uniform that it was sufficient to multiply
its value at the mid-epoch (1675 or 1975) by 2.5 to obtain Av.
Next, we have the changes in i and due to the motion of the
ecliptic, represented by T>]i and Df0, and computed by the
formula
D l t i= H f/ GOS(r /f -0)
sin i D[ 6 = H" cos i sin (v" 0)
184 DEFINITIVE QUANTITIES. [89
The planetary precession due to the motion of the ecliptic is
here omitted, to be afterwards included in the general preces-
sion. The sum of the two motions gives the actual variation
at each epoch, referred to a fixed equinox.
The motion of 6 itself thus found is increased by the general
precession, which gives the motion of 6 at each epoch.
The motion of the perihelion to be actually used in the tables
is equal to the motion of the node from the mean equinox, plus
the increase of the arc of the orbit between the node and
perihelion. The adopted value of this quantity is found by
increasing the motion of n\ by the following quantities:
1. The change due to the motion of the plane of the orbit.
2. The change due to the motion of the ecliptic.
The formulae for these two quantities are
(1) ; d] D t n = tan J i$m i D? d
(2) 5 <? 2 D t n = H" tan J i sin (L" - 0)
3. The excess of motion shown by observations in the case
of Mercury and Mars, and computed for all four planets as if
they gravitated toward the Sun with a force proportional to
r~ n where
n = 2.000 000 16120
The values of this correction are
//
Mercury; D t n 43.37
Venus; 16.98
Earth; 10.45
Mars; 5.55
4. The general precession.
5. In the case of the Earth, the motion arising from the
action of the Moon, of which the amount is
D t n" = 7".68
But the first two corrections drop out in this case.
The preceding transformations of the secular variations are
made with the original values of the elements e and i, as given
in Astronomical Papers, Vol. V, Part IV, pp. 337, 338.
89J
SECULAR VARIATIONS.
185
Secular variations of the elements of the four orbits at the three
epochs, 1600, 1850, and 3100, as inferred from the definitively
adopted masses.
Mercury.
1600.
1850.
2100.
D t e
+ 4.257
+ 4.227
+ 4.196
tfDtTTi
+ 109.524
+ 109.498
+ 109.475
DJtp
21.581
- 21.568
- 21.551
sinioD?0o
54.891
- 54.969
55.049
D?i
- 21.786
- 21.568
- 21.347
sintDfd
54.813
- 54.969
- 55.130
D{i
+ 28.884
+ 28.333
+ 27.785
sin t DJ B
- 37.196
- 37.397
37.595
D t *
+ 7.098
+ 6.765
+ 6.438
sin i D t
- 92.009
- 92.366
- 92.725
JD t 7T
1.06
1.06
- 1.06
D t 7T
5593.41
5598.70
5604.02
D t
4262.98
4266.12
4269.24
Venus.
D t e
- 9.959
- 9.866
- 9.772
eD t 7Ti
-f 0.384
+ 0.219
+ 0.060
Dfto
- 2.484
- 3.071
- 3.656
sin to DJ
- 59.005
- 59.112
- 59.229
D?t
- 3.049
- 3.071
- 3.091
sin*D?0
- 58.978
- 59.112
- 59.260
D<*
+ 6.690
+ 6.695
+ 6.697
sin i DJ
- 46.758
- 46.582
- 46.413
D t i
+ 3.641
+ 3.624
+ 3.606
sin i D t
- 105.736
- 105.694
- 105.673
JD t 7T
- 0.36
- 0.37
- 0.38
BtTT
5090.07
5072.44
5054.92
D t
3230.39
3237.98
3245.22
186 DEFINITIVE RESULTS. |89, 90
Secular variations of the elements of the four orbits, etc. Cont'd.
Earth.
1600. 1850. 2100.
D t e"
- 8.467
- 8.595
- 8.727
e"D t 7r"
D t 7T"
+ 19.293
6179.58
+ 19.210
6187.41
+ 19.139
6195.68
H" sin Li'
H" cos Li'
+ 4.370
47.113
+ 5.341
- 46.838
+ 6.305
. 46.550
log H"
L'o
L"
1.67500
1740 42'.04
171 12 / .83
1.67340
173 29'.68
1730 29 X .68
1.67187
1720 17M8
1750 46'.62
Po
p
5034.91
5018.28
- 46.761
5036.13
5023.82
- 46.838
5037.36
5029.38
- 46.847
Mars.
e D t 7T]
+ 18.775
+ 148.633
28.994
+ 18.706
+ 148.707
- 29.396
+ 18.623
+ 148.762
- 29.803
sin *o D?
34.023
34.012
34.017
D?*
- 29.482
- 29.396
- 29.309
sin t D?
- 33.605
34.012
34.445
DM
sin i DJ
+ 26.964
- 38.860
+ 27.104
38.551
+ 27.245
38.247
D t t
- 2.518
- 2.292
2.064
sin i D t
72.465
- 72.563
- 72.692
1>^
+ 0.08
6621.51
+ 0.07
6623.96
+ 0.06
6626.25
D t
2776.39
2776.87
2776.63
Secular acceleration of the mean motions.
90. The mean motions of the planets, like that of the Moon, are
subject to a secular acceleration arising from the secular vari-
ations of the elements of the orbits. The following formulae
for this acceleration are formed by differentiating the known
90] SECULAR ACCELERATIONS. 187
expressions for the variation of the longitude of the epoch in
the theory of the variation of elements. The notation is that
of Astronomical Papers, Vol. V, Part IV.
We compute for the action of an outer on an inner planet:
A = D <*\ }
B = - (D - D 2 2 D 3 ) c ( >
8
W- - (2 - 9 D + 3 D 2 + 4 D 3 ) c^
8
Then
D; = w' a n D t { A <J 2 + Be 2 - Ge' 2 + Wee' cos (n - n')\
For the action of an inner on an outer planet we compute
A' =-^(l + D)6^ )
B 7 = l (D + 2 D 2 + D 3 ) c ( ]
4
8
W = I (10 + 3 D - 9 D 2 - 4 D 3 ) <t\ }
8
D? 1 = m n' D t j A 7 a 2 + B'e 2 + We' 2 + Wee' cos (n - *') [
The symbol D t indicates the secular variation of the expres-
sion following it produced by the action of all the planets. The
unit of time must be the same one in which n is expressed.
The following table gives the results of this computation :
Secular change of the centennial mean motions.
Action of Mercury. Venus. Earth. Mars.
Venus, 0.0426
.
-0.0104
+ 0.0010
Earth, -0.0029
+ 0.0128
. . .
+ 0.0119
Mars, + 0.0003
-0.0001
- 0.0012
.
Jupiter, -0.0039
0.0046
-0.0308
+ 0.0004
Saturn, -0.0004
+ 0.0015
+ 0.0021
+ 0.0036
Total, -0.0495 +0.0096 -0.0403 +0.0169
188 DEFINITIVE QUANTITIES. [91, 92
The measure of time.
91. The fictitious mean Sun whose transit over any meridian
defines the moment of mean noon on that meridian is a point
on the celestial sphere having a uniform sidereal motion in the
plane of the Earth's equator, and a Eight Ascension as nearly
as may be equal to the Sun's mean longitude. If we put /* for
this uniform sidereal motion and add to JA the precession of the
equinox in Eight Ascension we have for the mean Eight Ascen-
sion of this fictitious mean Sun
T = TO + /i T + 4606 // .36 T + 1".394 T 2
From 88, 90, and 100 the expression for the Sun's mean
longitude, affected by aberration, is found to be
L = 2790 47' 58".2 + 129602766".74 T + 1 // .089 T 2
Equalizing the coefficients of T we find, for the mean Eight
Ascension of the fictitious mean Sun
r = 279 47' 58".2 + 1296027G6 // .74 T + l // .394 T 2
This differs from the mean longitude of the actual Sun by the
quantity
r - L = 0".305 T 2 = 8 .020 T 2
This difference is of no importance in the astronomy of our
time, but may result in an error of 2 s in the course of one thou-
sand years in the measurement of time by the actual mean
sun. We must leave to the astronomers of the future the
question how best to meet the question thus arising. Chang-
ing to time the expression for r, the difference or mean excess
of sidereal over mean time for the meridian of Greenwich
becomes
T = l& 39 ra 11 8 .880 + 24" O m K84449 t + 8 .0929 T 2
t being time in Julian years after 1850, January 0, Greenwich
mean noon.
Constant of aberration.
92. We first investigate certain fundamental constants con-
nected with the motion of the Sun, Earth, and Moon, on which
the precession and nutation depend.
92, 93] MASS OF THE MOON. 189"
From the adopted value of the solar parallax,
n = 8 // .790,
and the adopted velocity of light in kilometers per second,
Y = 299 860,
follows for the constant of aberration the value
A = 20 // .501
But if we accept the mean result of the solutions of 83 as
giving the most likely value of the solar parallax, we shall
have
n = 8".7854
Then 75 will give
A = 20".511
as the adjusted value of the constant of aberration.
Mass of the Moon.
93. By means of the equation of 71 between the lunar
inequality P in the motion of the Earth and the mass of the
Moon
/*'P = [1.78207] n
we may find a fresh value of the Moon's mass from the values
of 7t and P.
We have found from observation
P = 6".465 i .015
Thus follows, for the mass of the Moon, when 7r = 8".790,
yw = 1 : 81.32 i 0.20
Combining this with the value found from the constant of
nutation,
yw = 1 : 81.58 0.20
we have, as the definitive mass of the Moon,
* = !: 81.45 i 0.1
190 DEFINITIVE QUANTITIES. [94,95
Parallactic inequality of the Moon.
94. From the transformation of HANSEN'S lunar theory in
Astronomical Papers, Vol. I, it may be concluded that the solar
parallax and the parailactic inequality are connected by the
relation
P. I. = [1.16242] ln 7t
1 + V
= [1.15176] 7i
Hence we have, for the coefficient of the parailactic inequality
of the Moon, corresponding to n = 8 // .790,
124".66
Here the inequality is that in ecliptic longitude.
The centimeter -second system of units.
95. There are certain methods in physics by which the next
step in the course of our researches will be guided. The adop-
tion of a system of absolute units has simplified the methods
and conceptions of physics to such an extent that we may
find it advantageous to introduce a similar system into those
investigations of astronomy which are closely connected with
that science.
The fundamental units most widely adopted are the centi-
meter as the unit of length, the gram as the unit of mass,
and the second as the unit of time. There is, however, an
insuperable difficulty in the way of introducing the gram,
or any other arbitrary terrestrial unit of mass, into astronomy,
from the fact that the astronomical masses with which we are
concerned can not be determined with sufficient precision in
units of terrestrial mass. It is, therefore, quite common in
celestial mechanics to regard the unit of mass as arbitrary,
and to multiply this arbitrary unit by a factor which will
represent its attractive force upon a unit particle at unit dis-
tance. The introduction of this factor is, however, needless.
It is simpler to adopt the course of DELAUNAY and many other
writers, and regard the unit of mass as a derived one, based
on the units of time and length, by defining it as that mass
which will attract an equal mass at unit distance with force
95, 96] MASSES OF THE EARTH AND MOON. 191
unity. In this definition the unit of force retains its physical
meaning, as that force which, acting on unit mass, will pro-
duce a unit of acceleration in a unit of time.
The number of fundamental units is then reduced to two,
those of time and length, and the unit mass becomes a : derived
one of dimensions,
The centimeter as a unit of length wou
small for astronomical purposes, if we had to deal mainly r with
natural numbers, but it causes no inconvenience in logarith-
mic computations, and has the advantage of being assimilated
directly to the centimeter-gram-second system in physics.
We shall therefore adopt it, expressing our results, however.
in terms of other units whenever convenience will thereby be
gained.
I shall make clear this assimilation and the use of the unit
of mass as a derived one, by calling this the centimeter-
second system.
In the latter the definitions of units in the centimeter-
gram-second system will remain unchanged, except that the
derived unit of mass must be substituted for the gram. The
dimensions of units in the centimeter-second system will be
found by making the above substitution for M in the expres-
sions for those of the centimeter- gram -second system.
Masses of the Earth and Moon in centimeter -second units.
96. A fundamental quantity in the centimeter- second system
is the mass of the Earth. This mass will be by definition the
force of gravity of the Earth, if concentrated in a point at the
distance of one centimeter. Were the Earth a sphere of known
dimensions, it could be readily determined through the force
of gravity at any point on its surface. This being not the case,
we shall proceed on the accepted approximate theory that the
geoid is an ellipsoid of revolution, and that the force of gravity
at a point the sine of whose latitude is 1 : -v/3 is the same as
if the mass of the Earth were concentrated in its center.
The determination of this constant with astronomical preci-
sion is a difficult and we might say hitherto an insoluble prob-
192 DEFINITIVE Q UANTITIES. [96
lem, owing to the heterogeneity of the Earth and the absence
of determinations of the force of gravity over the surface of the
ocean. Although the limits of uncertainty thus arising can
not be set with any approach to precision, I do not think they
are such as to greatly impair the astronomical results which
are to be derived from them. Investigations in geodesy not
being practicable in the present work, I have, mainly from a
study of the work of G. W. HILL,* assumed for the length of
the seconds pendulum at the point the sine of whose latitude
is 1 : V3, which I shall call the mean latitude,
L! = 99.2715
With this we may compare HELMERT'S expression for the
length of the seconds pendulum in terms of the latitude
L = O m .990918 (1 + .005310 sin 2 tp)
which gives
L! = 99.2688
From these values of LI we have:
HILL. HELMERT.
Gravity at mean latitude, 979.770 979.745
Correction for centrifugal force, 2.260 2.260
Attraction of the Earth, 982.030 982.005
I also accept as the result of CLARKE'S investigation of 1880,
Equatorial radius of the Earth, 6378249 m
Reduction to mean latitude, 7245
Mean radius of the Earth, 6371004
From HILL'S and HELMERT'S numbers follows :
Logarithm mass of Earth expressed in centimeter- second units.
HILL. HELMERT.
20.600541. 20.600530.
* Astronomical Papers, Vol. Ill, p. 339.
96, 97 PARALLAX OF THE MOON. 193
From the adopted ratio of the mass of the Moon to that of the
Earth:
fji = 1 : 81.45
follows
Logarithm of the mass of the Moon in centimeter-second unite,
18.68965.
Parallax of the Moon.
97. From these results the distance of the Moon and the
relation between the mass and distance of the sun follow in a
very simple way. By the formulae of elliptic motion it follows
that when we put
w,w', the masses of any two bodies revolving around each
other in virtue of their mutual gravitation ;
a, the sernimajor axis of the relative orbit, which would
be the actual distance if the motion were circular;
n, their mean angular motion in unit of time;
we have the relation
a 3 ri l = m + m'
This relation is rigorous and independent of the adopted units
of length and time, provided we define the unit of mass in the
way already done. It follows that if the Moon in its revolu-
tion around the Earth were not subject to disturbance, its mean
motion in one second, and its distance expressed in centimeters,
would be connected by the relation
Log a 3 n 2 = log m" ( 1 + /*) = 20.605841
In the theories of DELAUNAY and ADAMS the quantity &, as
determined by this equation, is accepted as a fundamental
element, and it is shown that in consequence of the perturba-
tions produced by the Sun the constant 77 of the Moon's hor-
izontal parallax is connected with a by the relation
a sin 77 = 1. 000907 p
p being the radius of the Earth corresponding to 77
5690 N ALM 13
194 DEFINITIVE QUANTITIES. [97, 98
From the mean sidereal motion of the Moon in a Julian
century
1336 . 85136 revolutions
we find, for the co-logarithm of the motion in arc in one
second
log JL= 5.574841
and thus have for the undisturbed mean distance of the Moon
in centimeters
log a = 10.585174
and hence
log sin 77 = 8.219921
/ //
77 = 57 2.68
Red. to sine, .16
Constant of sin n in arc, 57 2.52
Using HELMERT'S length of the seconds pendulum we
should have found for this constant
3422".55
Mass and parallax of the Sun.
98. In the case of the motion of the center of gravity of the
Earth and Moon around the Sun the relation of 97 becomes
a' 3 n 12 = M] -f m" (1 + ^)
MI being the mass of the Sun. Replacing a' by TT, the parallax
of the Sun, and p the radius of the Earth, we find for the
ratio M of the mass of the Sun to the sum of the masses of the
Earth and Moon
M
m" (I 4- IJL) sin 3 n "
log M7r 3 = 8.349674
98, 99] SUN'S MASS AND PARALLAX. 195
The values of M corresponding to certain values of the mean
equatorial horizontal parallax of the Sun are as follows :
M
8.780
330514
8.785
329951
8.790
329388
8.795
328827
8.800
328266
Nutation and mechanical ellipticity of the Earth.
99. Begarding the mass of the Moon as known, we now
utilize the equations of 67 to obtain the constant of nutation
and the mechanical ellipticity of the Earth. The last two of
these equations give, for the absolute precessional constant,
when the Julian year is the unit of time,
p = [[5.975052] j-^ + 5310".o] ~ A
*
We have found, in 66, for a Julian year
p = 54".8990
We then have, for the mechanical ellipticity of the Earth,
~ A = 0.0032753
We also have, from the first equation of 66, for the constant
of nutation for 1850
N = 9".214
For the parts of the precessional constant which arise from
the action of the Sun and of the Moon, respectively, we have
//
Action of the Sun 17.3919
Action of the Moon . 37.5071
196
DEFINITIVE QUANTITIES.
Precession.
[100, 101
100. In order to develop the terms of the precession and
obliquity to higher powers of the time, I have extended their
computation one step backward and forward from the three
fundamental epochs, by extrapolation of H and L. The results
are as follows :
Year.
Motion of the ecliptic and equator.
log. K
1350
1.67666
168 56.13
- 46.613
2009.05
1600
1.67500
171 12.84
- 46.761
2006.92
1850
1.67340
173 29.68
- 46.838
2004.79
2100
1.67187
175 46.63
- 46.847
2002.66
2350
1.67039
178 3.50
- 46.789
2000.52
Centennial precessions for tropical centuries.
Year.
1350
1600
1850
2100
2350
In longitude
Lunisolar.
Planetary.
General.
5033.58 *
- 20.94
5012.64
5034.80
- 16.63
5018.17
5036.02
- 12.31
5023.71
5037.25
- 7.98
5029.27
5038.49
- 3.67
5034.82
In Right
Ascension.
4592.41
4599.38
4606.36
4613.35
4620.32
From these values we have the following general expres-
sions :
Annual precession in Eight Ascension;
Annual precession in longitude;
Centennial precession in longitude;
Total precession from 1850;
46.0636 + 0.0279 T
50.2371 + 0.0222 T
5023.71 +2.218 T
5023.71 T + 1.109 T 2
Mean obliquity of the ecliptic.
101. The expression for the mean obliquity when T is counted
from 1900 is
e = 23 27' 8".26 - 46".845T - 0".0059 T 2 + 0".00181 T 3
101, 102] PRECESSION. 197
Tables of the mean obliquity at different epochs.
Year.
Obliquity.
Year.
Obliquity.
/ //
o / //
1600
23 29 28.69
-2500
23 58 44.00
1650
29 5.31
-2000
55 38.99
1700
28 41.91
- J500
52 23.10
1750
28 18.51
-1000
48 57.70
1800
27 55.10
- 500
45 24.14
1850
27 31.68
41 43.78
1900
27 8.26
+ 500
37 57.97
1950
26 44.84
1000
34 8.07
2000
26 21.41
1500
30 15.43
2050
25 57.98
2000
26 21.41
2100
23 25 34.56
2500
23 22 27.37
Relative positions of the equator and ecliptic at different dates.
102. The motions expressed iii the preceding tables are, for
the most part, purely instantaneous ones, referred to the planes
of the ecliptic and equator of each separate epoch. For the
reduction of the places of the fixed stars from one epoch to
another, it is necessary to know the relative position of the
planes of the equator or ecliptic at the two epochs. We shall
therefore derive the fundamental quantities which express
the position of the equator and the ecliptic at any one epoch
relatively to their positions at a fundamental epoch taken at
pleasure. The latter we shall call zero position. Then, the
zero equator and ecliptic are those of the fundamental epoch;
the equator and ecliptic simply those of any other varying
epoch. So far as convenient, and as conducive to ease in
comparing our results with former ones, we shall use the nota-
tion of BESSEL.
To derive the equations for the motions, let us consider the
following four points of the celestial sphere:
E , the pole of the zero ecliptic.
E, the pole of the actual ecliptic.
P , the pole of the zero equator.
P, the pole of the actual equator.
198 DEFINITIVE QUANTITIES. |102
We put,
f T = PE , the obliquity of the equator to the zero ecliptic;
k = EE , the inclination of the two ecliptics;
770, the longitude of the node of the ecliptic on the zero
ecliptic, measured from the zero equinox of the date;
771, the longitude of the same node, measured from the actual
equinox;
A, the arc of the equator intercepted between the two eclip-
tics, or the planetary precession on the equator ;
if? a the total, lunisolar precession on the zero ecliptic from
the zero epoch to the actual epoch ;
w, the rate of motion of the pole of the equator;
T, the time, expressed in units of 250 years from the zero
epoch to any other epoch.
The position of the variable point E is denned by the quan-
tities k and J7 or 77i, which are themselves to be determined
through the values of x and L of 100.
The position of the variable point P is determined by the
condition that its motion is constantly at right angles to the
arc EP, and its velocity measured on the arc of a great circle
is given by the equation
ds
= n = P sin s cos s (a)
ci t
The positions of the equator and equinox relative to the
zero equator and ecliptic are then determined by the quanti-
ties 1, ij> and A. The spherical triangle P E E gives the follow-
ing equations:
sin A, sin 77! sin 7I
sin k " sin e i sin e
During a period of several centuries the quantities k and A are
so small that no distinction is necessary between them and
their sines. We may therefore put
A = k sin 77 t cosec 1 = k sin 77 cosec e (b)
We also have, from the law of motion of the pole of the
equator,
D t t = n sin A
D t fy = n cos A cosec e\
102]
MOTION OF THE EQUATOR.
199
As the value of 81 does not change by 0".6 from one epoch to
another, we may, without appreciable error, use f for ^ in the
formulae (b) and (c). To use these equations, we first obtain k
and 77! from the secular motion of the ecliptic, while n is com-
puted for any epoch from the formula (a). We then easily
develop the values of s-i and ip in powers of the time by the
equations (c). The values of n have no reference to any
special coordinates. From the table ot 100 it will be seen that
we may put
n = 2004".79 - 2".13 r'
r' being counted from 1850.
To find the value of III in each case, we remark that the
instantaneous values of L given in 100 show that the instan-
taneous node, or intersections of two consecutive ecliptics,
moves with so near an approach to uniformity that we may
take for the actual node between the ecliptics of any two
epochs TI and r 2 the mean of the instantaneous nodes for those
two epochs. For example, let it be required to find the value
of 77j for the node of the ecliptic of 2100 on that of 1850. We
have
For 2100
For 1850, referred to eq. of 2100
Concluded value of 77i ...
L = 175 46.63
L = 176 59.13
77! = 176 22.9
As the basis of our work we have computed the required
quantities for the zero ecliptics of 1600, 1850, and 2100,
respectively. The values of k and 77i for the ecliptics of two
hundred and fifty years before and after these epochs are as
follows :
Zero epoch.
1600
1850
2IOO
250 Y
+ 2 5 OY
k
n,
k
n,
ff
-118.48
118.07
-117.64
/
1 68 20.0
170 36.7
172 53-4
//
+ 118.07
+ 117.64
+ 117-23
/
174 5-9
176 22. 9
178 39.9
200 DEFINITIVE QUANTITIES. [102
Changing the. unit of time to two hundred and fifty years,
the equations (a) (b) and (c) give the following values of the
derivatives of fi and :
Zero-epoch. 250 Y +250Y 250 Y +250Y
1600 _ 1.4636 + 0.7400 12600.33 12573.65
1850 -1.1768 +0.4527 12603.44 12576.65
2100 -0.8898 +0.1665 12606.57 12579.71
At the respective epochs T> T \ vanishes, and Dr# has the
values of the luuisolar precession in longitude ( 100).
Developing in powers of r we have- the following results:
Zero-epoch. o / // // //
1600; e l = 23 29 28.69-+ 0.5509 r 2 - 0.1206 r 3
1850; ei = 23 27 31.68 + 0.4074 - 0.1207
2100; t = 23 25 34.56 + 0.2641 - 0.1206
1600; # = 12587.00 T - 6.67 r 2
1850; # = 12590.05 -6.70
2100; # = 12593.14 -6.72
// //
1600; A = 45.28 r - 14.83 T*
1850; A = 33.52 - 14.86
2100; A = 21.75 -14.88
These values of Si and # completely fix the position of the
equator at the time T relative to the zero ecliptic and equinox.
For the reduction of coordinates from one epoch to another
we must express the position of the equator at the time r. We
consider the triangle PE P , of which the sides and opposite
angles are designated
Sides, fi
Opposite angles, 90 - C 90 , #
If, in the Gaussian relations between the parts of this triangle,
we put
sin A (f, Q } = A (fi e) = A z/
102] MOTION OF THE EQUATOR. 201
and regard the cosine of this angle as unity, we have
tan J (C + Ci) = cos J (e t + e ) tan ip
If we develop the differences between the tangent and the
arc we find from these equations
r + c, = ^ cos A (fi + f ) (1 -
where we put z for the approximate value of C Ci
For the inclination 6 of the mean equator of the epoch r to
the zero equator, we have the equation
sin 9 =
cos
and then, by developing in powers of 6 and ^, we find
= ip sin f (1 + 4 C 2 ) (1 i ^ 2 cos 2 f )
We thus find
Zero-epoch. // // //
1600; C + Ci = 11543.79 T - 6.12 r 2 + 0.57 r 3
1850; 11549.44 - 6.L4 +0.57
2100; 11555.12 -6.16 +0.58
1600; C - Ci = 45.29 r - 9.92 r 2
1850; 33.53 -9.93
2100; 21.76 - 9.94
1600; e = 5017.30 T - 2.66 r 2 - 0.64 r 3
1850; 5011.97 - 2.67 - 0.64
21CO; 5006.64 -2.67 -0.65
To show the significance of the preceding quantities, con
sider once more the spherical quadrangle P E EP. Let these
202 DEFINITIVE QUANTITIES. [102
letters represent the positions of the poles on the celestial
sphere at any two epochs. In this quadrangle we shall have
Angle E P E = 90 - Ci
Angle E P P = 90 - C + A
SideP P = #
Let S be the position of a star on the celestial sphere. Its
polar distances at the two epochs will be P S and P S and its
Eight Ascensions will be determined by the angles P and P
of the triangle S P P.
Thus, if the Eight Ascension and Declination of S are given
for one epoch, we can find it for the other epoch by the solu-
tion of the triangle S P P when we have given the values of
the quantities 0, Ci, and -f A.
To find the values of these quantities from the preceding
formula, let T be the zero-epoch, expressed in calendar years,
and let -c be the interval between the two epochs, taken posi-
tively when the zero-epoch is the earlier one, and negatively
when it is the later one. We interpolate the coefficients of r
and its powers from the preceding formula to the epoch T.
Then by substituting the value of r in the formula we shall
have the values of the required quantities, and hence the data
for reducing the position of S from one epoch to the other.
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