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y.
\
SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE.
202
AN
INVESTIGATION
OP THE
ORBIT OF URANUS,
WITH GENERAL TABLES OF ITS MOTION.
t
BY
SIMON NEWCOMB,
FE0FES80B OF HATUEHATICS, DKITGD STATES NATT.
[AOOEFTED FOR PDBI.IC ATIOK, FKBRDART, 1S73.]
f
ADVERTISEMENT.
In tlic investigation of the Orbit of Uranus wliicli forms the subject of the accompanying memoir,
as well as in tliat of the Orbit of Neptune previously published in the Smithsonian Contributions,
a large amount of arithmetical computation has been required, especially in the reduction and com-
parison of observations. The cost of this, in accordance with the spirit of the Institution in
advancing science, has been defrayed from the income of the Smithson fund.
As required by the rules of the Institution, the accompanying memoir was referred to competent
authority for examination, and the persons selected for this purpose were Professor J. 11. C. CoQiu,
of the Nautical Almanac Office, and Professor Asaph Uall, of the Naval Observatory.
JOSEPH HENRY,
Secretary S- 1.
Wasuinoton, 1873.
PHn,ADRi.pnrA!
CULLI Na, l>R I KTBIt,
7U.'t Ja^ne Street.
^
PEEFACE.
The present work was undertaken as far back as the year 1859. But the labor
devoted to it at first amounted to little more than tentative efforts to obtain
numerical data of sufficient accuracy, and to decide upon a satisfactory method of
computing the gfcncral perturbations of the planet. The elements of Neptune
employed in the earlier computations were found to deviate too widely from the
truth to be used in computing the perturbations of Uranus with the first order of
accuracy, and it became necessary to correcit them. This was done during the years
18G-1 and 1865, and the investigation was printed by the Smithsonian Institution
in the latter year. It was then found that the adopted elements of Uranus also
differed too widely from the truth to serve is the basis of the Avork, and they were
provisionally corrected by a scries of heliocentric longitudes derived from observa-
tions extending from 1781 to 1861. FinalK' it was found that the adopted method
of computing the perturbations, that of the " variation of elements," though not
deserving of the disfavor into which it has fallen of late years, was practically
inapplicable to the computation of the most difficult terms, namely, those of the
second order with respccc to the disturbing forces. Indeed, it appeared to the
author that the only method of computing these terms which was at tiic same time
general, practicable, and fully developed, W! s that of Hansen. But, were this
method adopted, all that had previously been done would have been, useless, even
for the purpose of comparison and verification, owing to the expression of the co-
ordinates in terms of a disturbed mean anomaly. It appeared to the author that,
although this form of theory led to expressions having fewer terms than the other,
it was not without its relative disadvantages. Other considerations being equal, he
conceived that astronomers generally would gr:^atly prefer to see the perturbations
expressed directly in terms of the time, owin^' to the ease with which the results
of different investigators could then be compared, and with which corrections to
the Hieory may be introduced.
Under these circumstances the method deicribed in the first chapter of the
present paper was worked out. The question how much it contains that is essen-
tially new is one that the author has never clos.'ly examined: it is, however, certain
(iii)
iv
PREFACE.
that the mode of con8iilering the subject is well known, being that employed by
La Place, Ilerschcl, De rontecoulnnt, Encke, and perhaps others. The method
of forming the required derivations of the perturbative function from the analytical
development of that quantity, he has not seen elsewhere.
With these improved elements and methods the work was recommenced in 1868.
The curlier investigations being merely provisional, it has not been deemed neces-
sary to present them in the present work. Some of the results, corrected for
errors of the older elements, are, however, given for the purpose of comparison.
Although this investigation has absorbed the greater part of the author's leisure
for more than live years, it is only through the aid of the Smithsonian Institution
and Nautical Almanac that he has been enabled to bring it to a conclusion witliin
that time. At an early stage of the work Professor Henry responded favorably to
a request for aid by the employment of computers; it was, however, not found
practicable to use such aid until the perturbations had been completed, and the
provisional theory concluded. Then, the comparison of theory and observation,
and the construction of the tables, involved a large amount of mechanical compu-
tation, and on this part of the work a number of persons have been employed by
the Institution at various times, among whom may be mentioned Prof(;ssor F. W.
Uardwell, of the University of Kansas, and Dr. C. L. F. Kumpf, late of the Ob-
servatory of Leiden. Every part of the work has, however, been done under the
author's immediate direction, and, as nearly as possible, in the same way as if he
had done it himself, a result which, in one or two cases, has been attained only
by the expenditure of an amount of labor approximating that saved by the employ-
ment of the computer.
In presenting the steps of the investigation, the end has been kept constantly in
view to render as easy as possible the detection and correction of any error, or the
introduction of any alteration in the elements or other data. It is, of course,
impossible to present the steps of the computation with any approach to fulness
without far transcending the limits of the printed work : The results given are,
therefore, those which it was supposed would be most useful to the futme investi-
gator of the same subject. There is reason to believe that the original computa-
tions will ultimately become the property of the National Academy of Sciences, so
that they may always be referred to for the clearing up of any difficulty in the
printed text.
The author's acknowledgments arc due to Professor J. II. C. Coffin, Superin-
tendent of the Nautical Almanac, and Mr. E. J. Loomis, of the Nautical Almanac
Office, for reading the proof sheets of the last twelve tables during the absence of
the former abroad.
k
W'AsnisaTON, July 31, 1813.
TABLE OF CONTENTS.
Introduction .
PAOI
1
CHAPTER I.
METHOD OP DETERMININfUTnE rERTrRBATIONS OP THE T.ONOlTrDE, RADIUS VECTOR, AND
LATITUDE OF A 1'I.ANET BY DIRECT INTEORATIO.V.
Notatfon and general differential formulB) .....
Formation of the re'.iiired derivatives of the pcrturbntive ftmction .
Correction of these derivatives for terms of the second order .
Integration formula! for perturbations of radius vector
Development of functions of rectangular co-ordinates ....
Integration of perturbations of radius vector .....
Formnho for perturbations of longitude to terras of the second order
Motion of the orbital planes .......
Perturbations of the second order depending on the motion of the orbital planes
Reduction of the longitude to the ecliptic .....
E.\pressions for the latitude .......
R
10
12
14
n
23
24
25
27
29
CHAPTER II.
APPLICATION OP THE PRECEDINO METHOD TO THE COMPUTATION OP THE PERTURlLVTIONa
OF URANUS BY SATURN.
Data of computation .....
Numerical expressions for R and its derivatives
Perturbations of radius vector
Perturbations of longitude ....
Perturbations of latitude ....
81
34
44
49
61
I
CHAPTER III.
PERTURBATIONS OP URANUS PRODUCED BY NEPTUNE AND JUPITER.
Adopted elements of Xcptnno .....
Development of li and its derivatives for the action of Neptuno
The term of long period between Xcptune and Franus
Perturbations of the longitude produced by N'cptnno .
Perturbations of tlio radius vector prndneed by Neptune
Perturbations of the latitude produced by Neptune .
Perturbations produced by Jupiter
53
64
55
58
fiO
CI
63
(V)
yl TABLE OF CONTENTS.
CnAI'TEIl IV.
TEBMB OF THE SECOND OKDEIl DUE TU THE AOTIOM OF BATDRN.
rrrlimiiiary iiiTCHtipation of the orbit of So,tiirn ....
PcrturbatiniiH of Saturn mid Urnnus ......
Formation of llip oxprossioiiH for tlio terms of tlic second order
I'erlurbatioMH dopendins on the s(|iiarc of the mass of .Saturn
Perlurbatious depending on the product uf the muHava of Jupiter and Saturn
Cn AFTER. V.
COLLECTION AND TKAN8F0UMATI0N OF THE PRECEDINO PEKTt^IlBATIONB OF URANVB.
Terms Independent of the position of the di-sturbiiig planet
Secular Tariation.s .....
Au.xiliury expressions on which the perturbations depend
Kcduced expressions for the latitude of Uranus
Positions of Uranus resulting from the preceding theory
Elements III of Uranus ....
CHAPTER VI.
REDCCTIOS OF THE 0BSEBVATI0N8 OF URANl'9, AND THEIR COMPARISON WITH THE
PHECEDINO THEORY.
Rcdnction of the ancient observations .
Tlieir coniporison with the provisional theory .
Discussion of the modern observations
Reduction of the results to a uniform system .
Adopted positions of fundamental stars
Discussion of corrections to reduce the diiferent observations
Table of these corrections ....
Results of the observations from 1181 to 1830
Observations from 1830 to I8t2
Table to convert errors of right ascension and declination of
and latitude ....
Tabular summary of results of observations, 1830 to 1872
Corrections to be applied to (i -^ positioi.a of Uranus in the Berlin Jahrbuch and the Nautical
Almanac to reduce them to positions from the provisional theory
CHAPTER. VII.
FORMATION AND SOLUTION OF THE EQUATIONS OF CONDITION RESULTING FROM THE
PRECEDINO COMPARISONS.
Expressions of the observed corrections to the longitudes of the provisional theory in terms
of the corrections to the heliocentric co-ordinntes .....
Expressions of the same quantities in terms of the corrections to the elements of Uranus and
the mass of Neptune ......
Table to express errors of heliocentric co-ordinntes as errors of elements
Discussions and .solutions of the equations thns formed
Concluded corrections to the elements of longitude
Corrections to the inclination and node of Uranus
to a homogeneous system
Uranus into errors of longitude
that
V5
68
fi9
7G
17
79
80
81
93
98
99
ior>
110
111
111
113
115
120
122
126
127
131
151
158
101
162
105
173
173
TABLK OP CONTEiNTS.
CTIAPTEll Vlir.
COMPLETION AND ABBANOEMENT OF THE TIIEORV TO FIT IT FOB PERMANENT U8E.
Correction of llio covfllcients of tlio loiiff inequality bctwcon UnimiH uikI N'c|ituue for tlio
ternia of tiie second order .....
Concluded elements, or cloiuenta IV of Uranus
Lon((-pcriod ond secular pcrturbationfl of tlio ulcmoiits
Table of these perturbations from A.D. 1000 until A. I). 2200
Mean elements of Uranus .....
Expressions for tbo concluded theory of Uranus
CIlAl'TEll IX.
OENEBAL TABLES OF UBANU8.
Enumeration of the qnantitics contained in the several tables
Precepts for the use of the tables ....
K.xainplea of the use of the tables ....
Tables of Uranus ......
Subsidiary tables ......
vU
PAOI
178
181
183
184
IS4
186
190
195
198
20«
279
ERRATA.
Pages 100 to 105. In computing the latitude from the provisional theory the values of the
secular terms of iij and tk ou page 97 have been iatcruhaugcd. The provisioual latitude, therefore,
requires the correction
— 0"..')3 r Bin V + 0".63 T cos v
where
\—g+ 12° 45' -|-2<; sin g.
This correction is not applied in the subsequent investigation. Its circot would have boon to
change the value of b deduced on p. 170 l)y probably 0".2 or 0".3. Tlic effect on the other tlcuienta
of latitude would have been much smaller, and therefore unimportant.
Page 122, line 1,5. Add : the corrections in the si.xth column being omittoil.
Pago 151. Add foot-note: In forming these comparisons the corri-'ctions to the heliocentric
longitude in the si.xth column of the provisional cphemeris, pages 100 lo 105, are not applied.
Page 159. Equation 7. In this equation the coelTicienta of ix and ip have been multiplied by ^,
instead of f , the factor of il. The effect of this error enters into all the subsequent results, but in
the compari.sons of theory and observation it is corrected.
Page 184. The element hero represented by > (kappa) is the same which, in the preceding
chapters, has by mistake been represented by k, and which is defined on p. 24. The k of Chapter
VII 1 is, therefore, not the same with that of preceding chapters.
ON TIIK ORBIT OF URANUS.
INTRODUCTION.
The connection of (lie [jlunot Uranus witli the most brilliunt astronomicnl
acliicvenjcnt of tlic century lends u peculiar interest to its theory. The researches
of Adams and Le Vcrrier showed that the observed motions of that planet were
represented, at least approximately, by the action of i theoretical planet havinj^
the lonj^itude of Neptune. I'eirce showed that the ih tion of Neptune itself
aecoiuited for tiiese motions within the limits of possible error of the observations
nsi'd by Lc Verrier. It remains to be seen wlicth'-'- the n^'reement between theory
and observation still subsists wiien the comparati' '; few observations used by those
investi.; itors are reduced with tlie more relined data now at our disposal, and when
the great mass of additional observations made both befun; and since the date of
Lc \'errier's researches are included.
The circumstances connected with the discovery of Neptune have been so
exhaustively recounted by a number of authors that it would be difficult to add
anything not already familiar to astronomers without transcending our present
limits. I shall therefore confine myself to such nn account of previous researches
on the theory of Uranus as may give an idea of their nature and extent, and facili-
tate their comparison with the methods and results of the present investigation.
The perturbations used by Bouvard in his tables are those of the ^lecanique
Celeste. Although not affected with any striking error, the numerical methods
adopted in their computation are necessarily too rough to allow of much interest
attaching to their comparison with the results of the more recent researches.
It is essential to a clear understanding of subsequent researches that we classify
the methods which have been or may be adopted in the computation of the
general perturbations of the planets. This computation comprises two distinct
operations: (1) the development of the disturbing forces, or some quantities of
which these forces are fmictions ; (2) the integration of the equations of motion
inider the influence of these forces. In each of these operations three methods
have been employed.
In developing the perturbativc function, we have first the purely analytic method
used by the great geometers of the last century. In this metliod this function is
developed in powers of the eccentricities and mutual inclination of the orbits of
the two planets, and the numerical coefficients are found by substituting the values
of the elements in these expressions. A is only applicable when the eccentricities
1 Uaroh.lSTS. ( ^ )
:i\
THE ORBIT OF URANUS
and mutual inclination are small, and has for that reason fallen, of lute, into a
certain disrepute. The extended tables publislied Ly Le Verrier' liave, however,
added so much to its facility for use that it is not wholly unworthy of attention.
At the other extreme stands the purely mechanical method, in which spccifil
values of the disturbing force are computed for many combinations of the mean
anomalies of the two phniets, and tlic values of the coefficients in the general
expression for the force thence deduced.
Between these two stixnds what 1 conceive we may designate as the Cauchy-
Hansen method, in which the development is made mechanically with respect to
the one ])lanct, but the eccentric anomaly of the other is retained as an undeter-
mined quantity. The germ of this mctliod is found in several papers, by Cauchy,
in the earlier volumes of tlie Comiitca Rendiis of the French Academy, which have
since been combined into a liomogeneous memoir by Puiseux.^ Tlie object had in
view by these authors is only the computation of inequalities of long period. But
Hansen has taken up the essential principle of the metliod, first, in his piize memoir
on the perturbations of comets, crowned by the French Academy of Sciences, about
1848, and afterwards in his " Auseinaiiderscfzuiiff ei'ner zwcckmiimgcn ilethode znr
JJcrecIinung tier Storiiugcn (hr klcinen Planetcn"^ and applied it to the general
development of perturbations
Among the three metliods of integration, the first in point of analytical elegance
and generality, but the last in order of convenience in use, is that of the variation
of elements, a method with which the name of La Grange is inseparably associated.
In the second the direct integration of the differential equations wliich express
the perturbations of longitude, latitude, and radius vector is effected by special
devices.
In the first of these methods the problem is presented in this form : The equations
of motion being completely integrated for the action of the principal forces only,
how must the arbitrary constants of integration vary in order that the same expres-
sions may represent the motion of the planet under the influence of the disturbing
forces? In the second method, the same thing being presupposed, the question is,
what expressions must be added to the integrals of undisturbed motion in order
that the sum may represent the integrals of the disturbed motion X
The third is Hansen's method, in which the co-ordinates are partly expressed in
terms of a certain function of the time known as th? disturbed mean anomaly,
determined by the condition that the true longitude in the disturbed orbit shall be
the same function of the disturbed time that the longitude in tlie elliptic orbit is
of the simple time.
Although the last two metliods have a great advantage over the first in the com-
putation of the periodic perturbations, I conceive the first to be best adapted to
the computation of the secular variations, and perhaps, of terms of very long period
in the mean longitude and the elements of the orbit.
•*■
• Annales de V Ohservatoire Imperial de. Paris. Tome I.
' AiinnleK de roharrvaloire Jmpi'rinl de Paris. Tome A'll.
• AbhamUitnrjen der KOniglich, Sdchsischcn Oescllscha/t der Wisscnscha/lcn. Band V. VI, VII.
•I*
THE ORBIT OP URANUS.
3
In his researches on the motion of Uranus, the first thing done by Le Verricr
was to reconii»ute the perturbations by Jupiter and Saturn. It will sufficiently
describe his method of doing this to say that in the developments he used the
purely mechanical method for the action of Saturn, and the algebraic development
of tlie perturbative function for the action of Jupiter, while in the integration he
used the method of the variation of elements. After completing the perturbations
of the first order he made the earliest attempt at a complete determination of those
of the second order. Beginning with the terms of tliis order which arise from the
secular variations of the elements, he determines them by recomputing the terms
of the first order for the epoch 2300, and assuming that the general term will then
be given by interpolating between the two terms thus found, supposing them to
increase uniformly with the time. This proceeding has the sanction of such high
authority that it is worth while to call attention to its want of rigor. The dif-
ferential coefficient of each element being given in the form
(la J J,
, = k cos Ot.
(It
k being a function of the elements, the perturbation of the first order will be
(a = , sua lit.
b
When we take into account (he variation of /•, and suppose it of the form h^ -}- Ic't,
the process is equivalent to supposing that in this case
la = ' — sni bty
whereas it really contains the additional term,
J j cos bt,
which appears to be neglected in tlic process in question. It will be seen that the
neglected coefficient is equal to the secular variation of the term during the time
that its argument requires to increase by an amount equal to the unit radius. It
is tlierefore the more important the longer the period of the inequality.
To obtain the periodic terms of tlie second order Le Verrier begins by determin-
ing the ten principal terms of the perturbations of tiic elements of Saturn produced
by Jupiter. Next he takes up the terms in the mean longitude of Uranus which
depend on the square of the mass of Saturn. The only sensible terms he finds are
— T'.n sin {^— 30 — 0".35 cos (^' — S^)
+ 0".43 sin {i," — ^' + 40 — 0".2l cos (^' — 4^' + 40,
^, <^', and ^" being the mean anomalies of Uranus, Saturn, and Jupiter, respectively.
The terms depending on the product of the masses of Jtq)iter and Saturn are then
taken up. lifteen arguments are found the coefficients of wliich vary from a
small fraction of a second to one or two seconds, while a single one of long period
amounts to 32".
When the method of variation of elements is used, it is necessary not only to
determine these variations to quantities of tlie second order, but, in the tiansforma-
THE ORBIT OP URANUS.
tion of the perturbations of the elements into perturbations of the co-ortlinates, to
carry tliis transformation to terms of the second order also. This Le Verrier avoids
by showing that the terms of the lowest order with respect to the eccentricities
thus introduced are destroyed by certain terms in the perturbations of the elements,
so that it is only necessary to omit both classes of terms. These terms are of that
fictitious class which disappear of themselves by a simple change of elements.
When, instead of the eccentricity and longitude of the perihelion, we take h and /■•,
which represent the products of the eccentricity into the sine and cosine of this
longitude respectively, these terms disappear of themselves both from the perturba-
tions of the elements and of the co-ordinates. It is not likely that any of the
neglected terms of tliis class exceed 0".l.
As soon as the elements of Neptune were known, the nature of its general action
on Uranus became of interest. This subject was taken up by Prof. Peirce, whose
results are found in the Proceedings of the American Acatlemy of Arts and Sciences,
Vol. T, pp. 334-337. This paper is accompanied with a comparison of his theory
of Uranus with observations, to wliich simihir comparisons of the theories of Adams
and Le Verrier are added. Tliis comparative exhibit is of sufficient interest to be
given here. The numbers given arc probably excesses of computed over observed
longitudes.
Residual Differknceb between the Theoreticai, anp Observed Longitudes of Uranus, |
riio.M the Tueouies of 1'eirce, Le A'ekrieb, a.nd Adams.
From he Ver-
From Le Ver-
From Adams's
From I'eiroe's theory of Neptune
adopting for its
rier'a test or-
rier's original
original the-
mass
ory with his
from tlie mo-
hi!) best orbit
se(!0iid hypo-
That of Strnve
Tliat deduced
That deduced
dern ob^^erva-
of hypotheti-
tlietical planet
from his own
by Peirce from
by Peirce from
Year.
tions without
cal planet, of
of which mass
observations
Bond's & Las-
Bond's obser-
any external
which the
is
of the satellite
sel'a observa-
vations of Las-
planet.
mass IS
tions combined
sel's satellite
jjVi-
zi^-^-
Tr'tT^'
riHi-
•n\n-
ft
//
//
//
//
//
1000
+ 289.0
— 19.9
+ 50.0
— 124.7
+ 13.0
-f 0.8
1715
-1- 279
+ 5.5
— 0.6
— 99.6
4- lO.O
+ 8,7
1750
+ 230 9
— 40
— 4.0
— 102.4
— 12.7
+ 4,0
1709
-Y 123 3
+ 3.7
+ 1.8
— 67.0
— 10.0
— 0,0
1782
+ 20 5
+ 23
0.0
— 18.3
— 5.6
— 3.0
1787
+ 2,0
— 12
— 0,2
— 4.7
— 12
— 0,5
1792
— 7.8
+ 3
— 1.1
+ 1.0
-f 5
+ 0,3
1797
— 6.7
— 1.0
— 0,;')
-f 3 3
-f 0.8
+ 0,3
1803
— 34
+ 0.8
+ Ifi
+ 3.2
4- 1.2
+ 0,8
1808
+ 3.8
+ 0.8
0.0
— 1.3
— 0.6
— 0,4
1813
+ 4.5
— 09
— 1.0
— 2.3
+ M
— 0,3
1819
+ 3.8
+ 0.4
— 22
+ 0.9
+ 0.7
+ 1.0
1824
— 7.0
— 54
+ 1.1
— 1 6
— 1.9
— 2.0
1829
— 7.8
— 2.2
+ 2.0
+ 25
+ 1.3
+ 0.8
1835
— 45
— 0,8
— 1.2
-f 3.9
+ 24
+ 2.0
1840
+ 07
+ 22
+ 1.3
- 1.3
— 1,3
— 1.1
1845
+ 6.5
— 0,3
— 2.8
— 1 2
— 0.9
■4
i
In this paper Professor Peirce presents the results of a complete computation of
tlie general perturbations of Uranus by Neptune in longitude and radius vector,
THE ORBIT OF URANUS.
I r
but witliout any details v/hatcvcr of the investigation, or any statement of the
methods employed. Tlie minuteness of the residuals in the last column of the
preceding table shows that employing these perturbations by Neptune, and those
of Le Verrier by Jupiter and Saturn, we had a theory of Uranus from which quite
accurate tables might have been constructed. But this never seems to have been
done. The cphemeris of Uranus in the American Nautical Almanac was intended
to be founded on this theory, but the proper definitive elements do not seem to
have been adopted in the computations, as the cphemeris does not correspond with
the theory.
Although twenty-five years have elapsed since the epoch of these researches, I
am not aware of any published work of importance on the theory of Uranus during
the interval. Mr. T. H. Safford has, however, made a very extended investigation
of the subject, but has published nothing more than a brief general description of
his work, which may be ^ound in the Monthly Notices of the Royal Astronomical
Society, Vol. 22. Like Professor Teirce, he took I^e Vcrrier's perturbations by
Jupiter and Saturn, but, instead of using general perturbations by Neptune, he
computed the effect of the action of this planet by mechanical quadratures for the
whole period of the observations of Uranus, and thus corrected the elements and
the mass of Neptune from modern observations alone. The mass in question
deduced was
1
20039
Mr. Safford does not give the representation of the modern observations, but pre-
sents the following comparison of the ancient ones, alongside wliicli wo place for
comparison the corresponding numbers of Pcirce's theory and those of the present
investigation.
Excess op
Observe
Date.
No. of obs.
Safford.
ifino
1
+ 5".0
, 1-15
3
— 4.2
ltr>o
2
— 1 2
It 53
1
_ 0.2
1"5G
1
— 0.0
17G4
1
+ 0.4
nco
8
+ 4.5
Peirce.
Newcomb
— 0".8
-u"
— 8.T
— 8
— 4.0
+ CO
+ 2.0
— 1.4
6
THE OllBlT OF UKANUS.
CHAPTER I.
METHOD OP DETERMINING THE I'EUTURIJATIONS OF LONGITUDE, RADIUS
VECTOR, AND LATITUDE OF A I'LANET BY DIRECT INTEGRATION.
Let us conceive a plane determined by the condition tluit it shall pass through
the sun and contain the tangent to the' orbit of a planet at any moment. If the
planet were acted on by the sun alone, the position of this plane would be invariable,
but, under the influence of the disturbing forces of the other planets, it is subject,
at each instant, to a motion of rotation around the radius vector of tlie planet. We
may regard this as the instantaneous plane of tlic plaiu>t's orbit. Tlie disturbing
and the disturbed planet will each have its own instantaneous plane.
Let us now put : —
V, the longitiule of a planet counted from a determinate point in the instantaneous
plane of its orbit.
V, its distance from the node of intersection of its own orbit with that of another
planet,
the mutual inclination of the two orbits,
sin I y.
the radius vector of the planet,
its logarithm.
fi, the attractive force of the stui upon unit of matter at unit distance.
a, the mean distance corresponding to the observed mean motion of the planet,
determined by tlie condition
»r
m and n being as usual the mass and mean motion.
a„, the value of a corrected for the constants introduced by the perturbations, so
that, as in the elliptic motion, we have
p = logrt+/(/, p, ct),
we shall have in the disturbed motion
p = log CTo +/(^ e, ct) -|- periodic terms only.
«!, the mean distance of an outer planet, whetlier it be a disturbing or disturbed
planet.
», the logarithm of a.
a, the ratio of two mean distances, taken less than tmity.
li, the perturbative function.
o,
r,
P
%«
Mt«9
THE ORBIT OF UIl ANUS. J
h, the coefficient of any term of -i- R, so that we have
B = Z COS N
«1
m' being here the mass of the disturbing planet.
X, the mean distance of the planet from the node, or the mean vahic of v.
w, tlie distance of the perihelion from the node.
g, the mean anomaly.
I, the mean longitude, or the mean value of v.
4/, the angle of eccentricity so that c = sin t^^.
r„ the radius of the planet in the undisturbed ellipse,
r,, the quotient of r„ divided by the mean distance, wliich is a function of the
eccentricity and mean anomaly only.
T, the time after the epoch 1M50, Jan, 0, Greenwich mean noon, counted in Julian
centuries.
V, the integrating factors of the periodic terms, or the ratio , ^ being the change
of the angle in iniit of time.
«, the eccentric anomaly, and, in tlie tables, the argument of latitude.
We have for the value oi li
Mt»9
It
m
m r
(cos v cos v'-)-sin V sin v' cos y)
V 1^ — 2)-r'(cosvcosv'-|-sin v sin v' cos y)-\-r''' i
or, if we suppose r replaced by its value in p, namely
we shall have
With this value of R it is well known that the differential equations for the longi-
tude and radius vector of a planet arc
dh-
dr^ " (!+»')_ ^J^
de
d(' ^
r
fPv
de
+ 2,
dr dv
dt dt ~~
(^li
= F
cJp'
(1)
If we multiply the first of those equations by 2 ' P and the second by 2 J^ and
dt
dl
add them together, putting, for brevity,
f*^^ (/p 1^ r/? dv jy „
cJp f/<+ ov dt -^^''^-
(2)
and then integrate, we shall have
';:+y';:-'^'-^'+'"'='>,(c+fir.n,o
d(
8
THE ORBIT OF URANUS.
^~d^^ ~=t^{2c+2fiy,m+f) c3)
Lot us now represent by ,, that elliptic value of . which satfsfies the equation
Subtracting this equation from the last we have
^^t::zt^:js::^:::tz:'h'''': -' '-'^ -''- ^^^ ^^-^o-
the logarithm of the radh ^ ." he .t ,7 " "" 'T'f ''^'^""^'''^'^ "^ ^'"^'"S
for the latter its value '""''''' '"^'^'''^ °^ '^ "^'-'If' ^^c substitute
for the latter its value
and put
Then
^P = P — po.
^('•'-0 = »v^p-f,-„^y^etc.
Substituting these valiip<! in fl,« .,1,
tern, of ,|,e° second orfTr „d t"™™:? h ''"''l '""^'"K, ""' ''"^P-onl „„,y „
J* /2S\ t * ^ I /> ^ Hilt..
" dt^ + & ( V5p) = fc ( 2Sjyjidt + ^\_<^ (ro'V) , f^ ¥ rx.
• ^ ^ ^ dp) dC' I- -9,. ' (*)
an equation which gives the perturbations of radius vector
we siLi b4in by ^:::!:ir;^i:^^:::zz:t:7 "^ ^"'' '-'-' ''- '"--^
of 5v, K ^p, ap', ^y, and that from the e we wi h " ^r""''''? ^ '" *'" "•'"'''^
We must first carrv this innrnv,-.. . . . ^ ' *" "" '"''"""'^ approximation,
ber of (4). To effLuhi^rr; 7 '", ''V' '""^'""^ "^ ^ '" ^'^ ^^""^ "--
of the elements and , time w mtT "' '™'" '" ^--'"P--^* of ^ i" terms
to the quantities which or 'iJo ^^ ^Zu^'^'^t 't^'^'' ''''' ^^^"^^^
and r, is, in its developed form, a futtbn of^T ' '> '""?'"" "5 ^''Z ' P' P''
development being effected bv s nbsfih f / u L"' "' ^' "' "' "' ""'* ^ '^'^
values in terms of the second '"''''''*'"? ^"'^ ^''^ «-* set of quantities their
01 me second. Ihe substitution is as follows:
v = ;i + /},,
^-^'+^y, (5)
P = » + «/'5' ,
P' = »'+ <?'^',
THE ORBIT OF URANUS. 9
Fg being the equation of the centre, and ^g the part of p depending on the eccen-
tricity in the elliptic motion. It follows that if wc express the developed expression
for iJ as a function of X, X, g, ^, », x/, which we may do by putting
u = /I — g, (J = W — (f;
a = c', a! = c"';
we shall have by successive differentiation
~dX* ~ 5v* 'd-^ ~ 6\*
6R _ 6R «?p _ 6R
dTO~~ dp dv~ 6p
d-a" ~ dp"' d-o ~ 5p»
etc. etc. etc.
(6)
and in general
^m + n + m' + n-^ ^m+n+m'+n'^
5X" 5»" ^A,""' 5»"'' Sv™ df oiv'^'^p"
Thus, by expressing the developed R in the above form, we may find the derivative
of any order with respect to v, v', p and p', by taking the corresponding derivative
with respect to X, 7!, » and »'.
The developed R is usually expressed in the form
12 = 2 -"*'* cos {i'X + a +y'(j' +io))
a, being the mean distance of the outer planet, whether disturbing or disturbed,
and h a function of e, e", a, and y. Substituting for o its value in g, this equation
will become
72 = 2 ^^cos ((»'+/) 7:-^r(i+jn-jy-jg\
Putting for brevitv
the formulae (6) give
^v a,
|^ = -2^(^+yVcosiyr
0)
and in general
"dv^W"' dp" dp"
a Htroh, 1873.
^ " + •»'_
JV
hr
10
THE ORBIT OP URANUS.
The formation of the derivatives in the second member of this equation demands
attention. In the analytic development of the perturbativc function each value of
h is composed of a scries of terms each of the form
Ex A,
E being a function of ♦he eccentricities and mutual inclination, and A a function
of a of the form
(0)a'-.6^. + (l)a'^'^ + (2)a-^^^' + etc. + a^rjg', (8)
(0), (1), etc., being numerical coefficients connected with the coefficients F" tabu-
lated by Le Verrier, in Tome I of his Annalcs de I'Obscrvatoire, by the relation
(">=i.J".-.v
and J','> being, as usual, the coefficient of cos i^ in the development of
(1— 2acos<j>-fa=)-'
in multiples of cos ^, and » — 1 the sum of the exponents of the eccentricities in E.
It would have been much more convenient if in effecting this development the
derivatives of fe'j' had been taken with respect to » instead of a. In fact the
derivative ^ ^ when expressed in terms of the derivatives with respect to » is of
da '^
the form
«"--•=".--.:-+«. -^„;- + etc. + ».,.-^
<9a"
6to
Therefore, when expressed in terms of the derivatives with respect to », A will
be of the form
a'-' ( (0)' t<i' + (ly ^ + (2)' 1^ -f etc. ),
dA 6^A
from which the derivatives - - -, etc., may be foimd with great facility.
As in the actual developments of R which we possess, the values of A are given
in the form (8), we must find the expression for the first two derivatives of its
several terms with respect to », which we easily do by the application of the sym-
bolic formulae
D\ = a{Da-{-aLP,).
Beginning with the case of « = }, we have
'dr^'~'^'6a ^"- da?'
H^^
THE ORBIT OF URANUS.
11
V da"/
dn
= na"
^x>>
^a»
= 71
oo
^»
consequently wc have for the derivatives of A from formuloe (8)
6A_
= (0) a -^t + (1) ( a -^ + a» -^^,' ) + (2) ( 2a' -^^-,i + a' ^j)+ etc.
5a ~r - ^„, / -r v-y v-* "^oT ^ ""* ~5a»' + "' ^
-)+
etc.
The derivatives of A being formed in this way, those of h are immediately
deduced from the equations
6h 6 A
oVi
d^A
dri' ~ ^^ dra'
Wlien 8 is equal to |, ^ is of the form
a I (0') i'l' + (!)'« :^'^ + (2)' a' ^^''l + etc. I
I <9a ^a" 3
W
The quantity within parentheses is of the same form with A, in the case of s = |.
If we represent it by A' we sliall liave
6A / dA' A SA' , .
d^A
dri
,3 =^ + 2a-^-+a~^^
dA , oM'
A being the same form with A, tlie derivatives ^'"' and
5» (9ir
will be of the form
(9), substituting | for the index |, and (0)', (1)', etc., for (0), (1), etc.
In the case of s = | the derivatives are obtained in the same way, which is too
simple to need elucidation.
We have now to pass from the derivatives of h to those of * , the coefficients
of the perturbative function. The form of these derivatives will depend not on
whether the planet is disturbing or disturbed, but on whether it is an outer or
flq^
12
THE ORBIT OF URANUS.
inner one. Let us then suppose for the present, that a and « refer to the inner
planet, and put », for the logarithm of the mean distance of the outer cue. Wo
then have for the derivatives relatively to »
5- A
a, 1 d'h
5»»
a, dvf
n »
and for the first derivative relatively to », using the symbolic notation,
^A
^ = 1(Z)., -1)7».
5»j «i
The symbols in the second member being distributive, vire have by successive
differentiation
The quantity A is a function of o, the ratio of the mean distances or oi C'—'i,
being the neperian base. Hence
D.y* = — D,h,
which substituted in the last equation gives
h
6"
a, (-1)"
dK
(Z>, + 1)»A.
(10)
This formula gives for the first two derivatives
5»i
d«A
Substituting in the general formula; (7) these expressions for the derivatives
relatively to v and », we have expressions for the derivatives of E relatively to
V, V, p, p', it being understood, however, that all the quantities are expressed in
functions of the elements of elliptic motion.
In order to compute the perturbations of the second order we must carry E and
such of its derivatives as enter into the difierential equations (1) to quantities of
the first order with respect to the perturbations. Let us then represent by Vo, Vo',
Po7 p'o» yo. t^e elliptic values of v, V, p, p', and y, which we have assumed in the
Erst approximation to the perturbations, and by 5v, 6v', etc., the quantities to be
»
*
THE Oil BIT OF URANUS.
ta
added to v„, v,,', etc., to make the true vnluos of v, V, etc., whether perturbations or
corrections of the elements. \Vc shall then have
<9v„»
dm.
h\ +
dRo
d'R„
■ W +
JR^
d'R,
^ oiVg^v'o ^ C^VoOlpo
S'R,
d'Ro
^po^Vo ^ ^pu<5v'« ^ df\
.3p + -
ip +
5p +
5p'o
S'Ro
^V„c)p'o
^po^p'o
c'Voo'yo
Sfodyo
(11)
The value of DftR may be found either by cqiuition (2), or by differentiating with
respect to the time as introduced by the co-ordinates of the disturbed planet.
When quantities of the first order only are considered the latter operation is very
simple, but it is different when terms of the second order come in, because the true
longitude of the planet is then expressed in terms not only of its own mean longi-
tude, but also of the mean longitude of all the disturbing planets. The result can
still bo obtained in the same way by separating all the mean longitudes introduced
by the co-ordinates of the disturbed planet from those introduced by the co-ordinates
of the other until after the differentiation relatively to <'.
Let us now resume the equation (4), representing its second member by // Q, so
that it becomes
where
(It Vq
(12)
By the operations already given Q has become a known function of the time.
It is well known that the integration of (12) may be effected by finding two
values of rj^hp which satisfy this equation when the second member is neglected,
or, in other words, by finding two variables x and y which satisfy the equations
<r-x /^(l + vi) Q
<^>/ , /'(I +»»)„_
when the required integral is
'5p =
^
Ji/ dx
ill
yji
I yjxQdt — xjyQdtX.
The above differential equations are satisfied by the rectangular co-ordinates of the
planet in its assumed elliptic orbit. The position of the axes of co-ordinates being
arbitrary we shall take the line of apsides for the axis of X, the perihelion being
on the positive side. If we put
Co = sin 1^,
we have
dy dx , :r — -
■ yj/=A«(i + '
X
dt
dt
M (1 +770008 4-
|cos^= ^„
<l^
w
u
THE ORBIT OP URANUS.
Lot us, for convonionno, rpplacc x and y by twci otlier variables £ and yj connected
with them by the equutiuns
X = aC,
y = ari cos 4'.
f and Yi are then functions of the eccentricity and mean anomaly only, and may be
developed according to the multiples of the latter. Substituting the last three
expressions in the preceding value of r^bf it becomes
''»'^p=i+l{''/'^'^'-^^'^'^'M
If we put r,* for the value of r^ when the mean distance of the planet is put equal
to unity, so tluit r,, like ^ and yi contains only the eccentricity and mean anomaly,
we shall have
hf = "^"'^ I yif^a Qdt - ^fyja Qdl | (13)
We mnst now express ^ and y; in terms of the time, or of the mean anomaly.
Putting for the present n f )r the eccentric and o for the true anomaly, we have,
by the theory of the elliptic motion.
from which follow
X = >• cos V ^a (cos n — c),
y = r sin v = a cos ^ sin m,
^ = cos n — e,
ri ■=-. sin M,
As ^ and >? are to be expressed in the form
i=--\^Pi cos hj,
the finite integrals extending to all values of i from —oc to +00, we shall deduce
general expressions from 'p^ and q^ arranged according to the power of the eccen-
tricity. Since
« = jf -|- e sin tt,
•we have by Lagrange's theorem
cos V, = cos
or
g — esm'^fl ■'- ^ — etc.;
'' '' 2\ eg 3! fV
'sm
using the notation
We then have
C09 u^ — 2 -i — -.,
n + l.
n! = 1.2.3 n= r(H, + 1).
0! = 1! = 1.
THE ORIHT OF URANUS.
16
Siibstitutiiif,' ill tlie gciicnil term of the above scries for sin g its value in imaginary
cxpuuuutial functions
2 sin flf = yiTl' (o-»*^- » — cO^-\)
we find by the binomial theorem, using the notation of combinations,
«,_w(» — 1) (» — /,4-l) _ „!
l.'i.JiVr.a' "«!(»» — «)!
I n+l il4
'1+1 j
Differentiating n — \ times with respect to g, and putting together the first and
last terms, the one after the first, and that before the last, and so on, we find
_ 2"+' ^Tl'^^^'^^^'j} = (M -(- 1 )»-i(„('.+i)(^'-r ^ c-<"+"<' ' -0
— C(n — 1)"-' (c'— ""• -' + c-'"-')" '-"') + etc.
n+l 1^1
Substituting for the exponentials their values in circular functions, and dividing
by 2"+' we have
^"-' sin"+V
~ Og"-'
- L { (» + l;"-' cos (« + l)g- b{n - 1)"-' cos (n - \)g
* I n+l
+ C (« — 3)"-' cos (« — 3) j7 — etc. I
n+l )
the series terminating at the lust positive coeflficicnt of g. Substituting this last
value in the general term of the series which gives cos «, we have
cosM="F ~^^^ |(«+l)-'cos(n+l)flr-r'(„_l)''-icos(«-l)^+ctc. I
Let us now substitute for n another variable /, putting in the first term of the
last factor t = n -f 1, in the second / = ?i — 1, in the third t = »i — 3, etc. The
limits of finite integration with respect to t will then be
in the first term, -f-l to -\-<x.,
in the second term, — 1 to -f-oc,
in the third term, — 3 to +oc,
etc. etc.
But all the coefficients of g will then be i, and the formula supposes the factor of
cos ig to vanish whenever i is zero or negative; whence, those elements of the
finite integral in which i is negative must be omitted, and all the terms must be
taken between the limits + 1 and + oc. Making the proposed substitution we
have
X8 THE ORBIT OF URANUS.
-,rt9 ,, V J ^ o<-i * C'#>'+'_| : (7e'+* — etc. > cos lo
''°'"-raci-l)!2'-'' -(l+l)!2'+»];/ +(i+3)!2'+»„* j "^
- 1. (i-l) ! 2^» I 2^ i(t+l) <-.!. + 2«i(t+l) (t-l-2) (i+3) *+« i
We have, therefore, for all values of i different from zero
i'-«e'-» jl h ^^ _J_^- —J^^' -ctclrU)
'''=^-'=(i-l)!2--M <+^"2^i^+lT"^*4^2*(i+l)(i+2)(i4-^ •; '^
To obtain the value of p„ we remark that the only constant term in cos u arises
from the term — e sin^ jr ; its value is therefore — i e. The constant term in
^ = cos u — e is therefore | e, whence
l)o = -3e. (15)
The values of (ji may be obtained in a similar way by developing sin « by
La Grange's theorem. But the development is rather more complex, and it is
easier to derive them from 2>(. Let us take up the equations
g ^ cos tt — e
ri = sin u
u — e sin u = g
Considering m, like ^ and >?, as a function of the independent variables e and «/, we
have by differentiation
6u _ d (e sin w) _ _
6e Se ~
du _ d (cy;) _ sin M
' de '
6'e 1 — e cos u
1
Comparing («) and (I>)
du _ _______
dg 1 — c cos u
du . Su d^du 6^
de dg dudg dg
dg de
(a)
(i)
Putting in this equation for g and >? thcii developed values this equation becomes
2 ipt sin ig =2 - \f^ - sin ig
^ de
which gives by equating the coefficients of sin ig
(16)
*T
THE OliniT OF URANUS
n
The following are special values of i\ and g',, developed to the sixth power of
the eccentricities, as derived from the preceding formula; :
2)o = — 3e
11)2
<y — -..V. V, e
9216
P^
•2"
3
.<^^ +
IG'
.J 45 5o7 J
h'' ~128'^ +5120^
1,6,
■^* = 3^-ir/
Pi
^^ = m'
125 ,_4;375 g
27
>
Pt
2^
16807
46080'
, 1
8' ^192 9216
1
^2'
3
1
c» +
48
,_ 27 , 243 ,
^8*-" 128'' +5120*'
24-..' 1.-/
125
^^"=384'
27 .
3125
9216
. e"
2"
80
16807
'46080'
(16)'
Having the developed ^ and yi in terms of time, let us resume the equation (13).
As only purely linear operations arc performed on Q in this equation, it follows
that if we represent its several parts by Q^, (^., etc., and by (^p„ ^p^, etc., the values
^p obtained by putting Q=z Q^^ Q = Q^^ etc., we shall have
^p = ^p, + (^p2 + f tC.
We have, therefore, only to find the separate values of ro'5p corresponding to the
different terms of Q, and to take their sum. Let us then represent, as before, by
m
any one term of R.
3 April, 1873.
" h cos (iX + a +/u' +JL})
18
THE OUBIT OF URANUS.
We then have, considering only terms of the first order with respect to the dis-
tiirbing forces,
UtE = sm Jv,
(17)
fu,R =
m'ihv
cos N:
•where we put for brevity.
-n ^ —
cli , a, ,,
1'= -.;-,-,-—
Let us represent by Q^ the terms in Q which are of the first order with respect
to the disturbing forces, so that we have
The general term in B, will tlien give rise in Q^ to the term
m
d
a,
cosN.
a, • dx>
In the case of the action of an outer on an inner planet this expression becomes
ai V ' d-of
while in the contrary case it is
— ( 2tj7t — Ji — *^ M cos N,
both derivatives being taken with respect to the logarithm of the mear distance
of the inner planet.
In the integration it will be more convenient to substitute for ^' and X the mean
longitudes counted from the perihelion of the disturbed planet. If we put
the angle N will become,
Since corresponding to each set of values of i' and / there are several values of/' and
y, it will be convenient in the numerical computation to combine these different
terms into a single one, because after forming the derivatives of R there is no need
that u, <J and the other elements should appear in an analytical form. If we put
^'9
THE ORBIT OP URANUS.
19
Te for the coefBcicnt of — cos N in the preceding general term of Qa, this term will
become
g, = — h cos [/g>' + {i! + i +i) u] cos [tT + ig\
a,
_ — k sin [iV + (i' + i +j) w] sin [t7' + ig-\
«i
If we put
h = 2 i cos [ j'J + (»•' + i +y) o],
k.=Xlc sin [j'o' + (*' +i+j) u],
the sign 2 being extended so as to include all values of J and f which correspond
to tlie given values of i and i', we shall have for the general terms of Qa
^ I h cos (*Y + ii/) + /.-. sin (lY + if/) I ,
or, when we represent the angle i'l' -\- i<j by JV,
(2„ = ^''|/.-„c.siVi + A-;sinJ\^,|.
This we are to combine with the values of !^ and yi
y; =h^qj sin jg,
in the general integral formula (13). If we substitute them in this formula, and
represent by ft the coefficient of t in the value of N we shall have to integrate
differentials of the form
sm
cos
(iVi±*i/)
in which the coefficient of the time t in the angle is /t + in. Let us represent by
V, the integrating factor
[i-\-m'
The formula (13) will become by these substitutions, which, though a little comr
plex, offer no difficulty,
, 1 m'arr' +?
^<' = 16a.(i+,«-f|>-^^'^X
I i'+j-v+,\ 1/h.cos [N,-\-{i+j)g-\+k,sin [Ar,+(i+y)^]J
+ } r+t-r-j \ \ A-oCos [A\+{i-j)g]-\-k, si" [iV.+('-i)i/] \
+ 1 .'+,->'_, \ \ h cos [iV,-(i-y)«/]+A-.sin [iV;_(/-j)»/] \
-\-\v.,—v_j\ lAvcos[iVi-(i+y);7]+A-.sin[Ar,_(t+y)i/]|
The sign 2 of finite integration here includes the separate combination of every
value of i with every value of j, except those combinations which make the
♦ Tlio imlloes ? and j, in tlipsn eqnatidns, arc not to be ponfoundcd with the coefficients of x and u
In the general terms of R and <^. We need not use the latter at present.
ta!^
i
20
THE ORBIT OP URANUS.
coefficient of tfie time under the sign mn or cos vanish, and so render the corre-
sponding value of V infinite. These cases have to be treated separately.
To find, from the expression, the coefficient of the sine or cosine of cosine Ni-\-ug
in r,^Sp, we put, in the four lines of this equation, e : follows:
In the first, i-\-j=u.'.j= u — i;
" second, i — J=:u.'.j= i — u;
" third, — i-\-j=u.\j= u-\-i;
" fourth, — i —j = u .•. j= — u — i.
In the above expressions i and j being independent, and including all values
from — oc to -foe, »■ and u will also be independent, and include the same range
of values. Substituting for J its value in it the coefficient of
VI a r
becomes
16a,(l-f-m)
[k, cos ( A^i + itg) + k, sin {N, + ug)}
+ i>i ?-(«+() (''-( — »'(«+.■))•
Since q^ = — q^j this expression reduces immediately to
" \ + J'. '?.■+»)(»'("+.)— »'-i )'
or, substituting t — u for i in the second line
22 (j',?(,-„) + i',(-.)?() (''( — »'«-()•
Hence, writing N instead of Ni,
• 8ai(14-w»)
This expression fails for the particular case iV= vg, where the value of j'_„ will
be infinite. If we take each term of Q of the ibrm
m
(Ai"> COS vg + A;*"' sin vg).
and substitute in the general expression (13) it will be found that the terms in r^-fip
which have the infinite values of v as a factor are to be omitted, and replaced by
^ *a,(l + wi)"
(20)
The two parts of i-j'Sp thus found include all the terms of the first order with
respect to the disturbing forces. But when terms of the second order are taken
into account, we shall find terms in Q proceeding from secular variation in wliich
the time appears as a factor, outside the signs sin and cos. Let us represent such
of these terms as depend on any angle N by
Q = V}^± (k, cos N-\-k, sin N)
*
THE ORBIT OF URANUS.
31
;
and use the symbol j'j, as before, to represent the ratio of the mean motion of the
phinet to the coefficient of t in the angle ^-|- i<j, so that if fi represents the
coefficients of t in N we have
n
n
Vi =
^'+ in fi'
n
1
2'i',<7. -i
we find the expression
G Qdt=^ ] "''^^'- \ (r,V.-„ - vAnt) cos {N+ i,j) + (vi, h - V-Ant) cos (iV - tg)
(r.V.-. + vAnO sin (.V+ /r/) + (vi, t. + r_(7.-,.«0 si" C^"- «i/) J
r,?gf?< = ] '"-* I (r//.-. - vjhnt) sin (iV+yy) - (vljh - 1^-/.-.«0 sin (iVT-jV,)
- ('■;-/'•. + rjhnf) cos (iV+yy) + (vi/-, + v.^wO cos {N-jy) J .
If we now put for brevity
)•("/.•„ — !■(/.•.«< = s„
the general value of r^^l^p becomes
1 m'arT^
^^'-VG «,(! + "')
(<• - c, ) cos (.V+ {i +/) <7) + («< - «; ) sin(xV+ (t +y) </) "
+ ('■< - C-.) cos (.V+(t-i )(/) + («_,-«, )sin(.V+(;-y)gr)
+ (c,. — c_,) cos (.V— (i — y) fj) + (Af_, — s^ ) sin (.V— (i—j) (j)
, +(c_,-c_,)cos(.V-(i+y)i/) + (s_,-«_,)sin(i\r_(t+y)j/)
If, as before, we transform this expression by putting
in the first line j =^u — » ;
in the second " y = t — u;
in the third " j—i-\- u ;
in the fourth " j — l—u-
the value of r{'8^> reduces to
1 «»'«
is a,(l+J/0 ^
V 2 f i'.?(u-o (C..-0- c. ) cos (iV+ «^) -! i),7„_„ («<- «,.-„) sin (iV+ w^/) )
"'"' li'i'Z(a4.)(c,„,. )-"-<) cos (.V+«j/) -ri'*,.H)(-'~V+o)s»n(^V+«i/) i
or, putting t — u for i in the last line,
1 m'arT^
^ 2„^, 1 i'/Au-o -i'(,.-,,7, ! S (c>„-.)- '■.) cos ( A^+ vy) + («. - «„_.) sin (A^+ wjr) ( ;
HI
2'i
THE ORUIT OP URAXUS.
to which expression is to be added, in lieu of tlie terms which will have iufinite
values of p as a factor.
1 m'ar-hi-e
^aJi^^^^-^-^^^^^n
(22)
for' "o^ ^''' ^""^ *^'*' ^''''''''' °^ «. "' ''°' "^ *""* a] "' ''" "^ ^" *''^ expression
The formulae 19, 20, 21, and 22 give the complete expressions for the perturba-
tions of the logarithm of radius vector by successively substituting in it all tho
terms of Q.
Perturbations of Longilade,
We now pass to the perturbations of longitude. In the Mecanique Celeste
(Premiere Partie, Liv. ii. Chap, vi.), Laplace gives an equation (Y) by which the
perturbations of longitude, which are of the first order, may be derived from those
of the radius vector without the formation of any other derivatives of i? than those
which enter into Q. But tlie formula does not seem easily adapted to the case in
which the perturbations of the second order are taken into account, we shall
therefore derive all the perturbations of longitude from the second of equations (1)
liy integration this equation gives
do ft { rSE , )
dt=r"-\J6y<^(+(^\
IS
being the arbitrary constant of the integral, llepresenting, as before, by sub-
scnj.t zeros tlie values of the co-ordinates corresponding to the ellipse to which the
orbit IS supposed to reduce itself when the disturbing forces vanish, we have
fA'o n"-n cos i// fiO
because the constant to which the integral must reduce itself in the elliptic motion
a^>i cos T^ o w i-
- ~- Subtracting the last equation from the preceding, and putting
V — I'o = Sv, we find
dhv _ ,1 rdRj /I 1 V
1
Developing -^.^ to terms of the second order with respect to the disturbing force
-^ = ,\(l-2<V + 2V-etc.),
which, being substituted in the last equation by putting
/' =
gives
.d^v
r,' =
1+w'
-SR
' an ra/i
which is r.,; ous to quantities of the second order.
(23)
J
TUE ORBIT OF UUANUS.
33
1
The most convenient mode of making the numerical computation of the second
order terms by means of this equation will depend upon circumstances. If the
perturbations of longitude and radius vector of both planets are already known with
a sufficient degree of approximation for the computation of formula (11), it will be
more convenient to form at once the complete values of all the quantities which
enter into the equations (12), (13), (19) to 22), and (23), so that no steps of the
process shall have to be repeated. If such perturbations are not known, they
must first be computed, and it will then be necessary to begin with the perturba-
tions of the first order, and afterward add those of the second. There is, how-
ever, one class of terms of the second order which it will be most convenient to
take account of from the beginning, namely, those arising from the constant term
in fip and Sp. This is eff'ccted by correcting the mean distances for an approximate
value of these constants at the beginning of the computation, and then proceed-
ing in the usual way. This is in fact what we have supposed to be done in the
preceding investigation. The values of 6v, hV, hp, hf>' in formula (11) will then
contain only periodic terms.
In computing the terms of the first order we determine the value of 5p from the
equations (19) and (20), using the value of ^o in il»). '-I'l^en those of tv are
obtained by integrating the equation
lit
= l + m«^ ^v'^'-2"^°«^r.'-
(24)
Having found tlie values of hv and Sp for both planets, they are to be substituted
in (11), to obtain 5^, 5-^^and h ^!^. But, rigorously, Sv and ^v' arc not the
6v op
same with So and M, owing to the movement of the orbits of the planets, and the
corrections for Sy are also to be added. Considering, for the present, only the
perturbations of the second order, which depend on Iv, ^v', Sp, and 5p', we may
use the following equation for hE, and similar ones for its derivatives:
s« ='"*»+ '5 w+
vR
m
^p+'^-^-.^p',
6\' ^"'^ <?p ' ' dp'
Having thus found ?fR, and hence D'tf'B by differentiation, and then S
6R l<r(;-„V) , I'V
the quantity
6R
'6p
(2;3)
, wo form
'0
(2G)
/^ dii 1 '^"('•o"'V")
D',>RcU + 8 ^^-^^ </<^'
which is the difference between the value of <?« i» (1*^) and that of Q in (12).
The terms in ^p arising from bQ are then to be computed by tlie formulic (19),
(20), (21), and (22), when we shall have ^p accurate to quantities of the second
order. Let us represent these additional terms by d^p. Subtracting (24) multi-
plied by r,* from (23), recollecting that the ^p which appears in the second term
of the "former is really hp—hy we find, neglecting quantities of the third order,
idb
dt
'« = «n» |/S •^^ dt - 2^p/^^^» dt I - 2« cos ,j. (,^»p - 5p«)
24
THE ORDIT OP URANUS.
from whicli the terms of 5i' of the second order arc obtained by multiplying by
ri~'^ and integrating.
Motion of the Orbital Planca.
The general theory of the motion of the planes of reference, especially of the
motion of the instantaneous orbit, has been so often treated that I can scarcely
hope to add anything essentially new to it. 1 shall, however, endeavor to pre-
sent the differential equations of tlie motion in a simple and general form, and
one in which the geometrical conceptions of the problem shall be made as clear
as possible.
The orbital plane of each planet being at each moment osculatory to that part
of the orbit which the planet is actually describing, its only motion is one of rota-
tion around the radius vector of the planet as an instantaneous axis. This rota-
tion may be resolved into two others around any pair of rectangular axes fixed in
the moving plane. But the rotation produced by any one planet is most sim])ly
expressed when inferred to axes, one of which coincides with the common node of
the two orbit"*. The rotation produced by each separate planet must, therefore, be
first referred to its node on the moving orbit, and then the combined rotations
must be resolved into two aroinid axes assumed at pleasure. To effect this, let us
suppose positive rotation around an axis to be such that an observer looking from
the origin along the positive direction of the axis sees the right hand side of the
plane move downwards, and the left hand side upwards. Let us also denote the
first axis in the order of longitude the principal axis, or that of A' and that 90°
farther advanced the secondary axis, or that of Y. Let us now put
ilq, the instantaneous rotation around the axis of X;
dj), the instantaneous rotation around the axis of Y. Let us also put, relatively
to any disturbing planet,
d/;, the instantaneous rotation around the ascending node of the disturbing planet
on the orbit of the disturbed one.
die, that around the corresponding secondary axis.
Then, from the known equations for the perturbations of the inclination and
node of an orbit, we find, that, if any term of the perturbative function be repre-
sented, as before, by
"''* cos (i'A'-l-a-f/u'+y«),
the differential rotations y; and Jc will be given by the equations
dy; m'h
dt
die
dt
an
cos
T { (*' +J) coty -j- (t"+/) cosec y Um iV
711 an
cVt
cos N.
<f, cos il/dy
As E is actually developed, the mutual inclination y does not explicitly appear,
but is replaced by
a =^ sin 1 V.
*»* »
;. -tuv . -?" ■'aij^ff* *-aK
THE ORBIT OF URANUS.
90
Milking this substitution, and putting iilso
» + *+/+y= — '•
these equations become
at «, COS 4' cos ^y { 2a ' ' )
lU' m'lm COS \y Ch ,»
, =— , " , ' „ cosiV.
ut 2a, (08 -^ do
To pass to the general rotations </j> and dq, lot us represent by 0„ 0,j, etc, the lon-
gitudes of the ascending nodes of the several orbits of tlic disturbing planets on
that of the disturbed planet. We shall then have
dq_^
dl
dr,
dt
ScosO, "'" — 2 sinO,
dl
dp — „ dh\ , „ . ^ dr,
,,^=scoso<^;+vsmo,^y;.
(28)
These equations completely define tlic instantaneous motion of the orbital plane.
They cannot, however, be rigorously integrated in their present form because p
and q as integrals have no completely defined signification. To do this it is neces-
sary to express the difttrential rotations dp, dq, etc., in terms of the differentials
of any elements wo may select to define tlic position of tbe orbital plane, and then
to integrate the equations thus formed. But, for the purpose of constructing tables
of the planets we may consider ^), q, etc., to represent small rotations of the planes
of which the powers and products may be neglected, and the integration is then
quite simple.
Peiiitrhadons of the second order depending on the motion of the orhital planes.
R being a function of the five quantities of r, r\ v, v', and y, the motion of tlie
orbital planes introduces tcrnis of tiic second order by clianging the values of v, v',
and y. These terms we liavc hitherto neglected. To investigate them let us refer
the rotations of both planes as given by (28) to the node of the disturbing on the
disturbed planet as the principal axis. If wo represent by dy;,dK;d/;\ and (//.;' the
rotations corresponding to this axis, and designate by the subscript 1, the quantities
which refer to the disturbing planet whose action we are considering, and by 2, 3,
etc., the other planets, the equations (28) will be replaced by these
S' ---(«.-".) *■•
•».+v, !„(„,_„,)*.,
the summation commencing with / =: 2.
By formuhc of tbe same kind wo arc to find the difl"erential rotations dy;' and
dk' of the orbit of the disturbing planet, produced by the action of all the planets.
4 April, 1873.
Iw ^
96
THE ORBIT OF URANUS,
These rotations will be nroimd tlio snmo principal axis with the rotations dn and
*//.-, but around a secondary axis in the piano of the disturbing orbit, and therefore
making an angle y with the secondary axis of the disturbed orbit. A geometrical
construction will now sliow (p.ite simply that the infinitesimal rotations hn, hk, bn\
and hic' will produce the following changes in v, v', and y.
^v = cot ybk — cosec yhK
hV = cosec yU- — cot yhk ' (Op)
hy = hri — A»7
If wo substitute these values in the general fornndic (11) the terms of the second
order added to hR will be
(6R . dR
SD /o'^ . SR \
\6v ^°^ ^ + <5v' ^°^^^ ^'r^
-(^^cosccy+ ,^cotypk^
(30)
^k
The first two terms of this expression may be put into the form
{i(2f + ^:;)(— y+eoty)-j('^f__;^;?_)(._^_,,,^)|
- { K 7! + t^) (^"^^'^ y + cot y) + I ( f^ - I'JL) (eosec y - cot y) I hk.
But,
cosec y + cot y = cot i y = ^^^ ^ y,
a
cosec y — cot y = tan ^y= "
cos i y
and in the general term of R, by (7)
rR m'h
Ox ~ a,
0'+y)sin-Ar
Making these substitutions, and putting, as before,
the above value of SR reduces to
m'h
, w/cosjy clh . , .
(31)
,WJ
THE OllUIT OF URANUS.
9t
1!
Tlio currcsnoiiiliiig tLriim of h and h ', < uiul may ho ulitaiiicd in tliv satiie way
liy Miibntituting ^, uud '[ tur A' iu (:3()) uud coutiuuing the correspuuding sub-
stitutions of the general terms of tlie derivatives of It as given on page 9.
Tlic equation (31), besides being of tlie second order witli respect to the disturb-
ing forces, is also of tlie second order with respect to tlie mutual inclinations. For
Ms hkt hr„ and hy; arc of the first order with respect to both quantities, and, when-
ever ( is not zero, h is a quantity of the second order, containing a^ as a factor. It
is, therefore, only in exceptional cases that the terms of th(! second order depend-
ing on the motion of the orbital planes can become sensible.
JiediwIuM of (lie loiigUiiJe in the or/til to lo tjitude on (he ecl!j)lic.
The integration of (23) gives a value of ftv, which, added to the longitude iu
orbit corresponding to tlie pure elliptic motion gives the longitude in the disturbed
orbit, counted from a fixed point in tiie moving plane of that orbit. The position
of this fixed point is comi)letely determined by the condition that the instanta-
neous rotation of the plane in question around tiie axis perpendicular to itself is
always zero, so that the motion of-the point of reference is always perpendicular to
the direction of the plane. IJut, altliougli tliis instantaneous rotation is zero, the
integrated rotation is not rigorously zero when W(.> consider tin; terms of the second
order. It follows that the value of r, the longituoe in orbit, and the position of
the piano of the orbit do not rigorously determine the position of the planet: we
must also know how the fixed point of reference has changed its position iu con-
fiequence of the motions which the plane has undergone. Let us consider the
relative positions of this plane at two epochs. If the fixed point were equally
distant from the common node of the two planes, the integrated rotation of the
plane around its own axis would be zero. But, these distances not being equal,
tiieir difference is a correction to be ajiplied to the longitude of the planet in its
orbit. Suppose, now, that at the end of any time the inclination of the actual
orbit to tlie primitive orbit is <^, and the distance of its ascending node from the
present position of the moving axis of a; is 0. A rotaticm around tiu' line of nodes
will not change the quantity sought. But, if we represent the infinitesimal rota-
tion around an axis perpendicular to it by dr we shall have
cos dp — sin dif =; di\
dq and dk being the instantaniMins rotations around th(> resp(>ctive axes of a* and y.
By this rotation it is easy to see that the relative distance of any two fixed points,
one on each plane, from the node, will be altered by the quantity,
dr (cosec ^ — cot ^) = dr tan \ <^,
the relative longitude of the fixed jioint on the moving plane being increased by
this amount. Tiio correction to the longitude in orbit from tliis cause is, therefore,
(// — dr tan J </) = tan ^ ^ (cos 6 dp — sin dq).
'
28
THE OIIHIT OF U 11 AN US.
Counting the integrated values of p nnd q \\\ a. tlirection perpendicular to the
moving plane wo iiiivu
„ taiii)
sin = '
tun <p
cos = '
tun <p
which, heing substituted in the expression for dl, gives
(11:
COS(^
(tiui qdp — tun j>f/(/).
(32)
1 -f- cos tp
The approximate value of the integrated correction is therefore
5^ =2 /('/'(/'— My)-
For every pair of periodic terms in jt and </, such as
q = 8 sin fit, p=.8 cos fit,
U will contain the secular term — J /*' fit, which will be confounded with tlie mean
motion, and, if it were not so confounded, would in few or nono of tlie larger
planets amount to u second in a thousand years. If the secular terms in p and q be
q =z at ; 2) =^ n't
f.l will vanish. We hence conclude that these terms arc entirely unimportant in
the present state of astronomy, and that, if we consider the positions of the plane
of the orbit at two epochs, we may consider the points of departure in them to be
equally distant from their common node.
We have therefore only to consider the motion of the inclination and node duo
to the change of the position of the orbit and of the ecliptic. If we put
^, the inclination of the orbit of the planet to the cclipt'",
0, the longitude of its node counted on the ecliptic,
T, the longitude of the same node counted from tlie same - ' -'niut in the
moving plane of the orbit from which v is counted.
Then, the longitude of the planet on the ecliptic, or L, will be given by the
equation
tan ( L — 0) = cos ^ tan {v — t),
or, when developed in powers of <^,
L = v-\-e — r + D, (33)
where D is the reduction to the ecliptic, the value of which is
D = — tan^ J 4) sin 2 {v — t) + J tan' | <^ sin 4 (« — t) — etc.
Let us refer the instantaneous rotations of the orbit and of the ecliptic to the
fixed points of reference in the two planer ; q being the rotation around an axis
passing through the sun and the fixed point, and p that around an axis in 90°
greater longitude, and the accented quantities referring to the ecliptic. We then
have
t;-
V^J
m^waamiieitmm
'
»
Til K Oil HIT OF URANUS.
99
' — I*
'/o
ill
= co8T . -f-sinr
■cosO
lit
d<f
dt '
■ sin
dp
dt
dp'
dt
dL
dt
dt = '•°'"'*'''' (-'■" ^ 'dt + '°'^ '?<' ) ^®*^
+ cot <^( sinO'^^'f-cosO'J;)
dr ^ I • ih , ''/' \
4-cosec<Ji( sinO .' — cosO ' )
If wc dift'ercntiiite (;};3) ami substitute these values of and , we shall have
dt dt
dv , dD ^ . / dp . dn , „ </// . « </'/ \ /.i.-\
dt + ^^/^- tun i.^ (cost J^ -«mr-^^; +cos0 j^ -^mO--^; ) (J-O
If wc consider only quantities of the first order with respect to the disturbing
forces, wc may, in integrating, suppose t and equal and constant, and (p constiuit.
The integral will tlieu be
A = (' + i> + tan i <^ S cos {hh + W) — sin ((V + 'V) \ (3fi)
In the case of Uranus, tan <^ is so small that this equation will be sufficient for
a long tmie before and after our epoch.
In the application of the method to other planets the mode of operation must
depend on the circumstances of each particular case. The differential equations
(34) between 0, t, and <^ are rigorous, and their integrals may be approximated to
in various ways, out of which that best applicable to the particular case must be
selected.
Expressions for the latitude.
If the position of the orbital plane and of the ecliptic were each determined by
tlie i)receding formuUr, there would be no perturbations of the latitude, the lati-
tude itself being given rigorously by the equation
sin [i = sin ^ sin (« — t).
= sin (J) cos T sin v — sin ^ sin t cos v.
But the instantaneous values of ^ and t, or of sin ^ cos t and sin <^ sin t, are
troublesome to tabulate ; it will therefore, in practice, be found mor" convenient to
use only their mean values, and to consider their changes from this mean as per-
turbations of the latitude. Representing by the sign h the deviations from tiie
mean values, which are of course arbitrary, we have
cos /35/i = cos <^ sin {v — t) 5(^ — sin <^ cos (v — t) hr.
Let us substitute for S<^ and ^t their values given by the integration of (34) to
lJ
80
THE ORBIT OF URAXUS.
quantities of the first order, in which case and t may be assumed equal. Tliese
vahies are
h<p = sin T?p-\- cos r ^q
blii (pbr =^ cos <^ (cos T <^p — sin t ^q)
the terms dependent o;i ^j>' and f.q' being omitted because, I'cing purely secular,
they may be included in the mean values of <^ and t. Substituting in tlie expres-
sion for S^
cos ;3<V3 = cos ^ I sin vlq — cos vlp\. (37)
In the case of all the larger planets both cos (i and cos ^ may here be put equal
to unity, when the expression for h^i will become
hli = sin V hi — cos v }p. (38)
To develop this expression in purely periodic terms we must substiiute for v its
value in terms of the mean longitude or mean anomaly, namely,
V = Z + 2e sin jr -|- , e* sin Ig -j- etc. ;
suppose the terms of ;^j) and Iq depending on any argument, N to be
^^ = — a^siniV— «,.cos JV /jjqn
hq^= a', sin iV-|- «',.cos .V
and put 71 for the longitude of the perihelion, so tliat
then, to terms of the first order witli respect to the eccentricities, we have
5;i = — t; (ff, cos 7t + «', sin 7t) sin iV— e (a, cos 71 + «'„ sin 7i) cos iV
+ \ S((i, + «'e)cos7t + K — «e)sii»7i| sin(.V+ O)
+ \ \ ("c- — O i-os n + («'o + <'J sin n \ cos {N-\- y)
_|_ 1 J(a, — «',) cos 7t +(«', + ", 1 sin 7t I sin (.V— <j)
-i-i S(«„ + a',)cos7r-i-(«'e — 't.)sin7iicos(.V— g)
+ h''] («. + «'c) f OS 71 + («', — «,.) sin 71 \ sin" ( .V-f- 2r/)
+ ^e) (a, — a',) coSTt 4- («'-■ + ".I siuTil cos(.V+ 'ig)
+ ^ e j (a, — a',) cos 7t + («'« + «,.) sin 7r | sin (iV — 2.7)
+ i e| (rt„ + O cos 71 + K - fl.) sin 7t| cos (iV— 2*7)
The point of the orbit from which n and v are counted is entirely arbitrary,
and, in considering the action of but a single planet, it will he most convenient to
count them from the common node, in wliich case n must be replaced by o, and
fp and <^q by ^,k and fty;. Thus, 'educing the perturbations of the latitude imme-
diately from the formulje (27), we shall have
6/i = sin v 5)7 — cos V (^A.
(40)
I
THE ORBIT OF URANUS.
81
CIIAPTEIl II.
APPLICATION OF THE PRECKDING METHOD TO THE COMPUTATION OF THE
PERTURBATIONS OF URANUS BY SA.TURN.
Data of Conqnitaiion.
The elements of Uranus, adopted in this computation, were deduced from the
comparison of nine normal heliocentric longitudes at intervals of 3(597 days extend-
ing from 1781, December 26, to 1862, December 18, with corresponding provisional
places derived from the elements given in the " luvcsitigation of the Orbit of Nep-
tune," with perturbations produced by Jupiter, Saturn, and Neptune. As the
perturbations are to be entirely re-computed, and the elements to be re-corrected
from more extended scries of observations, all the details of this first approxima-
tion will be omitted. The resulting elements of Uranus are given in the follow-
ing table, togctlicr with the adopted elements of Saturn, which are nearly the same
as those employtd in tlie theory of Neptune, except that the inclination and lon-
gitude of the node have been corrected to agree witli observations: —
Elements II. of Urnnus.
Elrnionts I. of Saturn
m
168° 16' 31"
90° 4' 0"
$'-''
28 25 36.0
14 48 45.0
¥
73 11 58
112 20
f
4(5 20
2 29 39.2
c
.0469276
.0560050
e in seconds,
9679.5
11551.9
n
15426.10
43996.13
m
1
21000
1
3501.6
In romputing the perturbations cf tiio radius vector, one of the largest torms
will be a constant. To avoid the necessity of computing separately the perturba-
tions of the second order, which depend on this constant, we shall include an
approximate value of it in the mean distance. This approximate value is, in the
action of an outer or an inner planet, h\o^a = — \ m'Ma^D^ h'f. In the action of
an inner or an outer planet, Mog a' = + J mM(b"l^ -f a Dab'f). M being the
modulus of the system of logarithms.
Using the values of hf and a 7). h'l\ which ore found in different works relating
to Celestial Mechanics, wo find that the different planets produce the following
changes in 6 log a, the units being those of the seventh place of decimals: —
V
32
TUE ORBIT OF URANUS.
On Saturn.
On Uranns
Action of Venus,
+ 22
+ 22
Earth,
+ 24
+ 25
" Mars,
+ 3
+ 3
" Jupiter,
+10865
+8780
" Saturn,
+3081
" Uranus,
— 35
" Neptune,
— 9
— 119
" Sum,
+
.0010870
+
.0011792
S log a
+
.0001812
+
.0001965
The uncorrected mean distance is deduced from the mean motion by the rela-
tion
We thus have
Saturn.
Franns.
Uncorrected mean dist. (log)
0.979496
1.282901
Action of the planets
+ 181
+ 197
Corrected log a
0.979677
1.283098
The following functions of the elements are derived from the preceding ele-
ments by well known formuhc: —
y (mutual inclination) 1° 5 ' 24.4:"
log sin i y == (T 8.232373
log cos 1 y 9.9999307
T (long, of ascending node of Saturn on Uranus) 126° 44' 51"
o 41 31 40
w' 323 18 21
<y — 6) = (u) 281 46 41
log sin (u) —9.990759
log cos (u) +9.309888
fin2((,)) —0.39966
cos 2(g)) —0.91667
a 0.497249
1
a
4.04438
The following functions of a, necessary in computing the coefficients ?i, are
derived from llunkle's Tables, published by the Smithsonian Institution: —
«^A
THE ORBIT OF URANUS.
Values of a^Dlb'lK
i
6<«
aDab'»
a'Dlb'l'
a'/>V/J
a«i):6|
a'Z)'6'J
2.14447
0.33969
0.5878
1.081
3.44
13.6
1
0.55207
.68314
.4990
i.in
3.40
13.8
2
0.20836
.47198
.7396
1.152
3.59
13.9
3
0.08687
.28491
.7123
1.463
3.68
14.5
4
0.03793
.16270
.5632
1.596
4.30
15.1
5
0.01702
.09010
.3998
1.485
4.87
16.9
6
0.00777
.04896
.2653
1.231
4.98
19.1
7
0.00359
.02624
.1682
0.940
4.60
20.5
8
0.00168
.01392
.1022
0.675
3.91
20.4
9
0.00079
0.00733
0.0615
0.463
3.11
18.8
Dciivadrcs with re-fjtcct to (log
'({ = ») o/a
»/>:?/;>.
i n
= 0*
1
^2
3
4
DJ>*> IM
aDah^f) Dx
(oi^Z)^/,';') D.
WI^'J.'I') A,(a*Z);i'»
0.33969 0.9275
2.257
6.68
27.4
1
68314 1.1821
2.175
6.93
27.4
2
47198 1.
2116
2.631
7.05
28.3
3
28191 0.9972
2.888
8.07
29.2
*
16270 0.
7259
2.722
9.09
32.3
$
09010 0.4899
2.285
9.33
36.4
6
04896 0.3143
1.762
8.67
39.0
7
02624 0.1944
1.276
7.4i
38.9
1
01392 0.1161
0.879
5.93
36.0
ft 0.00733 0.0688
0.586
4.50
31.2
&
coiid (Jcrivativcs.
f
Dii'?
Z>;(a2)o6"?)
Diwi^f'V
2)-(a'^;6'|>)
©
0.9275
3.184
11.19
47.4
:t .
1.1821
3.357
11.28
48.2
t
1.2116
3.843
12.31
49.4
$
0.9972
3.885
13.85
53.4
,*
0.7259
3.448
14.53
59.6
t
0.4899
2.775
13.90
64.4
t
0.3143
2.076
12.19
65.0
Vo
lues of a"+^Dlb">
«
^
a'Djr^
.Wl'''^
«'/'i6|
$
1.865
2.674
8.104
30.8
^ 1,
1.267
2.844
7.77
30.8
f
0.761
2.412
7.63
29.9
. t'^
0.433
1.790
6.92
28.7
. t
0.240
1.224
5.73
26.8
ft
0.130
0.792
441
23.5
g^
0.070
0.493
3.2U
19.6
k
April. 1873.
8a
"f-
34
THE ORBIT OF URANUS.
Derivatives with rcs2>ect to {log a = ».)
1
V„{a'Dabp
=fi<»,7i<i>
4.539
13.452
65.1
1
4.111
13.46
54.1
2
3.173
12.45
52.8
3
2.223
10.50
49.5
4
1.464
8.18
44.0
5
0.922
5.99
36.7
6
0.563
4.19
29.2
Second derieatii'ca.
i
a^DvB^»
a,DvH<!
ia^DvBV
4.539
17.99
82.0
1
4.111
17.f.7
81.0
2
3.173
15.6:^
77.7
3
2.223
12.72
70.5
4
1.464
9.64
60.4
5
0.922
6.91
48.7
6
0.563
4.75
37.6
a,E^'>
a
,i5<?
ia,E<-i
Gfl,£<f
a,Dr,E<'> a,DvE'2
UJKE'V
1.267
4.111
13.46
54.1
4.111 17.57
81.0
1
1.313
3.856
12.95
54.0
3.856 16.80
79.8
2
0.850
3.167
11.98
51.8
3.167 15.15
75.8
3
0.500
2.
318
10.31
48.4
2.0I8 12.63
69.0
4
0.281
1.
573
8.24
43.1
1.573 9.82
59.6
5
0.155
1.014
6.18
36.6
1.014 7.20
49.0
The notation B'l^ and Ej; is that of liC Vcrvier in liis devolopment in the first
volume of '■^Anwdea de V Ohwrvnioirc Inq)erii.d dv Paris."
Numeric d exjircssion of R and its dciivatircs.
We next proceed to the computation of the coefficients h and their derivatives.
As an example of the most convenient form of computation we present in full that
of the coefficient of — cos (jX' — (t— 1)X — w) in the expression of R for the action
"1
of Saturn on Uranus. In this computation I use the tahles given by Le Verrier
in his "Annales de VOhsermtoire" tome i, ])ajTes 358-383, comparing the develop
ment with that of Professor Tcircc in the Astronomical Journal, vol, i, as a control.
(•aw
THE ORBIT OP URANUS.
86
y=o;
y=i-
•
-3
—2
—1
+1
2
3
4
6
6
i>
+4
3
2
1
—1
—2
—3
—4
6
(0)Xt<,'>
+0.5212+0.8334]+ 1-1041
—1. 10414
—0.83344
-0.6212
—0.3034 — 0.1702'-0.0929J
0)y.alJ^bf
—0.2849
—0.4720— 0.6831 —0.3397
—0.68314
—0.47198
-0.2849
—0.1627
—0.0901 -0.049(1
Ao, (50)(0
a, (50)(«)
-16.1775
+O.2303
+0.3614—15.7565
—0.3397
—0.78728
—1.30542
—0.8061
—0.4601
—0.2603
—0.1419
(0)6«j"
(i)x«/>-
—13.55
—11.25 — 5.52
0.00
0.00
+ 2.92
+ 5.73
+ 6.83
+ 6.46
+ 8.54
+ 7.7S
+ 4.78
+0.51
0.00
+ 1.18
+ 2.56
+ 3.25
+ 3.06
(2)X«««.
+ 1.43
+ 0.74
— 0.00
—0.59
— 1.00
— 2.22
— 2.65
— 2,82
— 2.40
(3)X«»fi'«
— 0.73
— 0.5S
— 0.59
—0.54
— 0.59
— 0.68
— 0.73
— 0.80
— 0.74
Aa,(51)
(1,(51)
+ 48.53
+ 4.71
— 4.31
— 3.31
+ 47.20
—0 62
— 1.69
+ 1.30
+ 6.46
+ 6.38
(0)X6'/'
mX'D,.
—18.70
—13.33
— 4.42
0.00
+ 4.42
+13.33
+18.76
+ 19.41 +17.01
+ 13.1o| +10.38; + 4.10' —0.68
— 1.37
+ 2.83
+ 6.27
+ 7.48 + 7.02
(2)X.'/>'-
+ 1.43
o.oo; — 1.00 —2.35
— 3.00
— 6.92
— 7.13
— 6.70
— 5.60
(3)X«'i^«
— 1.46
— 1.15 — 1.18 —1.08
— 1.18
— 1.15
— 1.46
— 1.60
— 1.48
ao,(D2)
1,(52)
+ 32.36
— 5.69 — 4.10 + 29.86' —4.11
1 1
— 1.13
+ 9.09
+16.44
+ 18.53
+16.96
(0)X"^,(')
—3.00
— 3.40 — 2.63 0.00
+ 2.63
+ 3 40
+ 3.00
+ 2.25
+ 1.55
(1)X<',£,(')
+2.32
+ 3.17 + 3.8(1 +4.11
+ 3.86
+ 3.17
+ 2.32
+ 1.57
+ 1.01
Aii,(tiO)
0,(60)
+ 16.18
+ 17.41 +4.11
+ 6.67
—0.68
— 0.23
+ 6.49
+ 6.32
+ 3.82
+ 2.60
1<X(50)
+6.611
+ 10.12, -441.22 -9.51
—50.049
1
-36.556
-22.57:1
—13.05
— 7.28 —3.97
J,'X(51)
—0.09
— 0.07 + 1.04 —0.01
— 0.035 + 0.028+ 0.103
+ 0.14
+ 0.14
Jee"x(52)
—0.08
— O.oo' + 0.45 —0.06
i — 017 + 0.138+ 0.24S
+ 0.28
+ 0.26
iec«X(tiO)
A
—0.05
+6.39
— 0.02
+ 9.97
+ 0.14 +0.03
+ 0.051 + 0.052+ 0.043
' i
+ 0.03
+ 0.02
' —439.59 —9 55
—50.050 —36.338-22 179
—12.00
— 6.8u| —3.60
The derivatives of h with respect to (log. a = r)) are computed in precisely the
same way by simply substituting for h[\ aDab'\\ etc., their derivatives with respect
to » as given in the above table of constants.
The quantities Ai,(50)"\ etc., which appear in the third series of terms above
express that part of the perturbations of Uranus caused by the action of Saturn
III units of tlic tliird pluro of dpoimiils.
36
THE ORBIT OF URANUS.
on the sun. They arc each of the form .Y X a~^ N being a numerical coefficient
given by Le Vorrier under tlio coefficient for each term. The derivative of tliis
expression with respect to » is — 2N X ct~S so that for the corresponding terms
in DJi and Dih we have
ADJi = — 2Ah
ADlh = + iSh
The values of h and its derivatives, corresponding to any one argument f and i,
are to be combined into two terms depending the one on the cosine, the other on
the sine of the argument. Let us represent by g the mean anomaly of Uranus,
and let us put T for the mean longitiide of Saturn counted from the perihelion of
Uranus, or, more exactly, for the arc X — w. Put also
Then, for each value of N there will be several values of P corresponding to dif-
ferent powers and products of the eccentricities and inclinations in h. Distin-
guishing these values and tlie corresponding values of h by subscript numerals, we
shall have a series of terms of B of the following form —
22 = ^
7i. cos (.V+ P,)
-{-h,cos{N-{-P'^)
+ 7»3Cos(iV+P'3)
-(- etc. etc.
and by putting
T'l-
h^ = 7<, COS P\ -(- hi cos P'„ -(- 7/3 cos P^ -\- etc.
h, = — 7*1 sin P\ — 7*2 sin P., — hj, sin P\ — uic.
iibove terms may bo condensed into
(41)
72 = 7j,, cos iV -f" '*» s^" ■^'
which are of the form supposed in the preceding theory.
In order that the derivative of R, with respect to the true longitude of Uranus,
may be expressed in the form
fiR m . „ , m ,-
„ = — r, sm iV + k.cosjY
&V M, f»,
we must, by (7), put
v. = — (i+j\) hi cos P\
'»c=— (i+j\) hi sin P\
■ (i -\-jt) hi cos P\ — etc.
(t -j- ji) '*2 sin P2 — etc.
(42)
j\> j-2i representing the several values of j in the different terms which correspond
to one and the same set of values of i and t".
jl
THE ORIJIT OF UK ANUS
87
To obtain the dcrivativo with respect to y wo notice that all tlie apprcciahlc
terms in the difleicnt values of /<, whicli depend upon the mutual inclination, are
of the form
li = ff- A,
where <t = sin J y. These equations give
v7i oh da , , 1 J ■
Aa cos ^y = ^ A sui y.
Consequently
6y da vy
OR
iJy
l^lsinycos(.V-|-/').
and the various terms depending on the same argument («', /) may be condensed
into two, exactly as in the case of It itself.
Tlic different co-efficients U and DJi, computed in the way already described, are
given in exlcnso in the following table. At the top of each individual column is given
the value of P, or of jij -\-j\'i\ corresponding to the values of h below, and imme-
diately under P is given its modified value, or P', to be used in condensing the
terms, putting for brevity
((.)) = (J — (j'.
P and P' are therefore regarded as constant angles the numerical values of the
sines and cosines of which may be obtained from the values of u and cj' already
given.
The condensed /t„ and h, are given in the two right hand columns.
All the numbers are given in units of the third place of decimals.
Values of h. 1
r
w U
p
=
(")
h.
h.
i i'
h
h
—!. + ■*
4- 0.02
—3, 3
+ 0.82
—2, 2
4- 0.88
—1, 1
4- 0.53
0,
4-10'?2.90
— 0.4975
4- 1072.80
+ 1—1
— 34Hl.3r.
— 0.83
— 34S1.42
— 1.33
2^ 2
+ 205.02
— 9.(13
4- 204.14
— 10.29
3,-3
+ 84.27
+ l.(!2
+ 84.77
+ 0.79
4,-4
■j- 35. SO
4- 1.54
+ 3(i.24
+ 0.90
5,-5
+ 15.48
+ 1.2
4- 15.81
+ 0.81
/'
10
/
U
u— 2u'
UJ - - 2w
1
u
i"
-(-)
— 2(u>)
+ (")
u-j-w
+2m>
K
h.
i i'
h
/,
h
/(
/.
—3, + 4
— 2.31
4- f..39
— 1.01
— 0.20
—2, 3
— 3.42
4- !).!t7
—0.95
+ .02
— 0.51
— 9.40
—1. 2
— 3.54
—439.59
—.03
+ .02
— 93.30
+ 430.35
0, 1
+287.30
— 9., 55
+0.13
— .02
—.04
4-. 09
+ 285.20
+ 9.19
+ 1-
+ 58.38
— 50.05
—0.02
— .05
—.03
+ 48.16
+ 48.90
2,-1
— 3!).7fiO
— 3(!.:W8
—0.074
— .071
—.016
— 47.138
+ 35.531
3—2
4- 35.13
— 2-2. 179
—0.105
— .313
—.007
+ 30.f,3
+ 21.44
4,-3
4- 20.3!)
— 12.(!0
—0, 1 1
+ -n
+ 17.90
+ 12.55
ft,— 4
4- 11.25
— CMC,
—0.10
4- .23
+ 9.98
+ 7.00
r.,— 5
4- ('..00
— 3.f.
+ .22
+ 5.38
+ 3.78
88
THE ORBIT OP URANUS.
Valuks of h.
P
— 2u
— W — U
— 2«'
P"
h
-2(«)
+ 2«
h.
/,.
i i'
h
h
h
—1, 3
+0.04
— 0.27
42.24
+0.11
+38.72
+ 17.03
0, 2
+0.00
+31.05
— 0.04
+0.18
+ 0.58
—31.15
+ 1. 1
—0.74
— 0.83
— 1.08
—0.01
— 0.03
+ 2.15
2,
+3.00
— 5.19
+ 1.89
+0.18
+ 0.32
+ 4.14
3,-1
+ 1.2(i8
— 0.200
+ 2.082
+0.111
— 1.912
+ 5.216
4,-2
+4.1(i
— 5.458
+ 1.707
+0.001
+ 1.44
+ 4.570
5,-3
+3.13
— 4.03
+ 1.20
+0.04
+ 1.13
+ 3.30
(1,-4
+2.11
— 2.74
+ 0.87
+0.02
+ 0.75
+ 2.31
7.-5
+ 1.35
— 1.70
+
0.55
+0.01
+ 0.49 1
+ 1.40
P
— 3«
— u — 2u
.Jui u
— 3„'
t
Id
P'
-(-)
— 2M
— :!(w)
t)U u
2«
h.
h.
i i
h
h
h
/(,
/)
h
+ 1.+2
—.12
—.11
—.14
+.02
+.17
+ .00
2, 1
+.01
—.07
+ .03
—.02
—.02
+ .01
+ .04
3,
+.18
-.40
+ .34
—.08
—.02 .
+ .04
— 15
+ .213
4,-1
+ .222
—.700
+.520
—.117
—.010
+ .020
—.325
+ .410
5,-2
+.424
-.847
+.,558
—.122
—.012
+ .021
—.170
+ .400
(!,-3
+.:i08
—.773
+ .408
—.107
—.008
+ .014
—.145
+ .400
:,— 4
+ .327
—.027
+ .305
—.080
—.005
+.009
—.107
+ .370
8,—;-.
_-t-Al __:
—.47
+.20
—.06
—.083
+ .280
A
— 4w 1 — u' Itio
— 2«' — 2u
ow u
— ij
P
i — («)
— 2(o)
— 3(..)
— 4(")
;..
/'.
i i'
h
h
h
h
h
4,
—.04
+.04
—.02
—.04
5,-1
+.02
—.08
+.08
—.04
+ .01
—.03
+ 003
<■),— 2
+ .030
—.105
+ .100
—.047
+ .008
—.045
+ .028
7—3
+ .043
—.114
+ .112
—.048
+ .008
—.049
+ .031
«,— 4
+ .042
—.100
+ .103
—.044
+ .007 1
—.044
+ .032
»,— 5
+.0;$.')
—.000
+ .080
—.030
+.000
—.037
+ .029
Vai.les of DJi.
P
lo b)
F
(«)
i i
D„h
/>../l
Dvho
D«h,
-4. +4
+ 3.18
—3, 3
+ 3.24
—2, 2
+ 2.39
—1, 1
+ 0.40
0,
+ 171.03
— 2.075
+ 171.51
+ 1—1
+8740.50
— 2.09
+ 8749.12
— 3.02
2, 2
+ 407.72
+ 20.83
+ 472.40
+ 18.05
3,-3
+ 277.00
+ 2.87
+ 278.25
— 0.36
4,-4
+ 153.87
+ 4.02
+ 155.46
+ 1.41
5,-5
+ 82.12
+ 4.00
+ 83,54
+ 1.80
THE OK HIT OP URANUS.
30
*-
Vai.i Ka (IF 7Vi. 1
F
— u
t
— W
u— 2u'
«' — 2-
+u'
u
F
-(")
— 2(«)
+ (")
Dvh
2w
Dvh,
Dvh.
i %•
Vvh
Dvh
Dvli
Dvh
Dvh
—3, +4
— 9.28
+ 19.4
— 5.32
— 19.00
-2, 3
— 9.74
+ 18.5
+ 2.05
— .06
—.12
+.08
— 7.97
— 17.43
-1, 3
— 4.fi3
+907.62
+ .05
— .08
—.13
+ .07
+ 1K0.44
—888.65
— ", 1
— 5.5(i.40
— 26.17
— .10
— .09
— 14
—.12
—561.82
+ 25.60
1,
+ 30.16
— 71.56
+ .04
— .19
—.11
+.02
+ 15..37
+ 69.80
2,-1
+ 2(1,-). 17
— 86.47
— .19
— .26
—.07
+247.58
+ 84., 30
3,-2
+ 83.10
— 74.60
— .42
+ .63
—.04
+ 68.34
+ 73.49
4,-3
+ 68.34
— 54.9
— .,55
+ .29
—.02
+ 5V.70
+ 53.9
6, 4
+ 49.04
— ;!6.9
— .58
+ .68
—.01
+ 42.19
+ 36.7
(>,— 5
+ 32.1
— 23.2
_ .53
+ .90
+ 28.1
+ 23.4
P
— 2u — w — w
— 2w'
F
-(")
Dvh
_2M
+ 2«
Dvh,
Dvh.
i {
i>o/l
Dvh
Dvh
-l,+3
+ 0.16
— 0.66
+ 86.02
+0.47
— 7S.76
—34.19
0, 2
+ 0.32
—64.55
+ 0.15
+0.60
— 12.92
+ (;2.54
+ 1. 1
+ 3.27
— 2.70
+ 4.69
+3.02
— 1.22
— 2.20
2,
+ 2.,')6
— 8.78
+ 5.00
+0.60
— 3.74
+ 6.00
3, 1
4-13.745
—15.870
+ 7.232
+0.463
+ 3.933
+ 12.153
4,-2
-flO.2,')
—IS. 93
+ 7.85
-1 0.32
— 0.77
+ 15.07
,5,_3
+ 10.76
— IS. 07
+ 6.99
+0.21
+ 0.69
+ 14.68
(1,-4
+ ',1.39
— 15.0
+ 5.59
+0.13
+ 1.22
+ 12.4
T,— 5
+ 7.3
—11.5
+ 4.21
+0.09
+ 1.2
+ 9.0
P
— 3«
_
_2« — u'
, — « — 2w'
•1 '
— u — u
P
i i
— (")
1 — 2(")
-3(„)
^u — u' +2u
Dvh',
Dvh.
Dvh
Doh
Dvh
Dvh
Dvh
1 Dvh
+ 1. + 2
+0.03
+0.18
+ 0.05
+0.22
+ .22
1 +.08
—0.31
—0.17
2, I
+ 0.12
—0.26
+o.:i3
— 0.09
—.08
+ .17
—0.08
—0.09
3,
+0.21
— 0.S9
+ 0.98
—0.31
—.09
+ .13
—0.59
+0.13
4,-1
+0.S9
—2, ((2
+ 1.S9
—0.54
—.09
+ .13
—0.84
+ 0.69
;'>,— 2
+ 1.0S
—3.00
+2.53
—0.67
-.08
+ .11
—1.38
+ 1.30
(•),— 3
+ 1.40
—3.47
+2.73
—0.69
—.07 +.09
— 1.33
+ 1.68
1,-4
+ 1.46
—3.44
+ 2.60
—0.(12
, —.05
+ .06
—1.20
+ 1.78
8,— >)
+ 1.32
—3.00
+ 2.20
—0.51
1
! +.04
—0.99
+ 1.62
P
— 4«
■ ,iui W
— 2a— 2«
— u — ow
-w 1
P
i i'
— (u.) — 2(u,)
_3(„)
Dvh
4(«)
Dvh,
Dvh.
Dvh
Dvh Dvh
1
■)vh
4,
+ .02
—.09 ' +.13
-.08
+ .02
—.06
—.01
fi,— 1
+ .0.5
—.22 , +.30
—.18
+ .04
—.14
—.02
(;,— 2
+ .10
—.38 ' +.48
—.26
+ .05
—.24
+ .01
■J,— 3
+ .15
—.52 +.62
—.32
+ .00
—.30
+ .04
«, -t
+ .18
—.59 +.66
—.33
+ .06
—.32
+ .09
n,— 5
+ .19
— .59 +.65
—.31
+ .05
—.31
+ .11
The values of I)vfi, needed in compntins; the perturbations of the second order
with respect to the masses being obtained in the same way, by the simple substitu-
tion of the second derivatives of the functions h,aDuh, etc., for those functions
themselves in the expressions for /*, it is not necessary to present the details of the
computation.
After obtaining h and its derivatives., ic will be found convenient to change the
arrangement of tlie terms. Ilitlurto we have kept in one series those in which
the sum of the indices are a constant. Now, we shall put together all those in which
40
THE OKBIT OP UIIANUS.
he index of the disturbing planet has the same vnhie, nrranfriug the individual
terms ot each scries according to the index of the disturbed phmet. Thus the
index of the product of any term, as h cos N, by any multiple of the mean anomaly
of tlic disturbed planet, as^y, "ill be found in the same series witli that of .V itself
and J lines above and below. '
The next process will be the formations of the required functions of the mean
anomaly of Uranus, |^J, '^J^, "', , bg r. Their values are as follows :-
rr= =
Po
= »
= 1
1.001 l();i +.0()().55()7 "''' "'^'~
+.()9;3!);J3 ros g -.04(i8SS«) cos g +.0468889 sin g +.0938294 cos g
Iniw. '"'!^ -.001(i494 cos -V +.0032988 sin 2y +.0055012 cos 2^
tZtt '" ? -;;;|"<)^^-^ -« 3y +.0002190 sin 3^ +.0003357 cos 3^
+.000020 cos 4y -.0000035 cos Ag +.0000142 sin 4g +.000020G cos Ag
Considering only those terms wuich are of the first order, the value of D'^R may
be found in two >yays, tlie agreement of whicli will aft'ord a check upon the entire
dcvvelopment of the perturbative function, and upon the computations of 7? and
These are (1) by direct differentiation, witli respect to the time as con.
ov
tamed in the mean anomaly of a single planet, whereby each term in R of the
will produce in D\R tlie term
R =■ /* cos N
a.
D\R = — "* inh siniV
and (2) by forming the expression
D\R = "'^ '•'■»
6'v (It
GR dp,
"^ Op lit '
As several "mechanical multiplications," like those indicated in tliis hst
expression, are to be performed, the following example of the fom o^ com-
putation IS presente.1. It exhibits the formation of the product of tliose terms
of -.: in wliich t' = _ 1 by
ih
- ' -^^ (sin)
m S"
X .04G915
X .002750
-'^ -10 +1 2 3 4 .5
+ .02 + l.fi.'i _2S7.42 +:!4,S1.14 + r,4.42 + C.988 +1.18 +0.10
I " -i- OH — l;!.48 +10,1.32 + 2 5r)3 4-0 ■?<? -Lft nr
U.08 -13.48 +.r.3...2 + ,,„ I 33 + ^J^^ +1'' +;o«
' ° " ^ 0«<' - 0.79 + 9.570 +0.15 + 02
+ 02
" — .048 + .58
i-_.79 + 9.58 + 0.15
X .0001G8 +.68 + 0.01
a, f)R di^ , . ^ T, '7 — ____
m-n-di dl ^''""^—^^ -^'^^ -^-•^■^'^ +3170.23 +217.28 +19.125 +2.24 +0.18
>f^i^t»ism''i"iimii&imm
THE ORBIT OF URANUS.
41
The Tinltiplicr^ on the left arc each one-half the cocfflcinnt cos j<j in the ex.
prcssion for ,." , and cacli product is phiced in the two columns corresponding
respectively to N-\-j<j and N — jy.
All the derivations of /?„ necessary in the computation of the perturbations of
the first order are given in the following tables. First we have the values of
D\R obtained by direct differentiation, as indicated in the preceding formidic.
, obtained by the formulir (7) and (42). The products
Next we have
dv
and
f)E
6.
"'" and of ^^' by ^^, being formed in the simple way just pointed out,
6R^
and with the values of'the component factors just given, their sum is next shown,
This sum should agree accurately with D',l{. The discrepancies are shown in the
next two columns. The only apparently large discrepancy is found in the argu-
ment Otf' — 5/. It probably arises from the incompleteness of the computation of R
and - , so far as they depend on this argument. As the entire term docs not
amount to 0".01, I have not sought to correct it.
The great value of this check arises from the fact that it gives a complete con-
trol of the correctness of the development of the perturbativc function, ab initio,
since the two valves of D',R are derived from diff'erent terms of that development.
dR
It also controls all the computations except that of . This quantity being
op
multiplied by quantities of the order of the eccentricities in the second value of
D'lR, an error in its value will produce a discrepancy of only ■^\f its own amount in
jy,R, and niny therefore be overlooked. The derivative in question must thertv
fore be checked by a complete duplicate computation.
In the column next following are given the integrating factors r, for which the
expression is
n 1
V =
i'u'-\- i)i
i + i'
For each value of /' the values of r arc therefore the reciprocals of a scries of num-
bers iu arithmetical progression, the common ditfcrcuce being unity.
April, 1873.
49
TIIK OIiniT OF URANUS.
'^'
/V« =
m'
- n X
<5v ~
- X
«1
(lU in'
'J V
Hi 1
I'oa
Kill
C(I9
cos
ttiri
0,
. . • •
4- 0.4S7
—1
244.31
....
'i
—
4S.15
4- 48. 9C
-f 10.20
4- 49.07
—
03.52
—118.82
3,
—
O.fU
-I- 8.28
-f- 4.48
4- 3.20
+
3.42
— 10.14
3,
+
0.45
+ 0.(!4
+ 0.50
— 0.08
+
0.74
4- 0.34
4,
+
O.IC
-f- 0.00
— 0.00
+
0.10
4- 0.01
— 2,-1
+ 0.08
4- 0.02
— 04
■+
0,08
— 0.05
— 1,
—
0.03
+ 2.15
+ 1.03
4- 1.34
L
1.25
— 0.11
0,
— 287.42
— 0.10
+
176.02
4- 34.70
1,
f3ISl.42
— 1.33
+ 3481.14
— 1.04
— .'J
2r7.70
4- 4.;i5
2,
+
94.28
f 71.00
+ 54.42
4- 71.23
200.44
— 119. .S3
•'!,
■V
.'■..74
+ 15.05
-1- 0.988
+ 9.490
2.02
— 17.37
4,
+
1.30
+ i.(;o
-f l.lH
4 0.42
4-
1.10
— 1.11
\
+
0.15
-f 0.15
-1- 0.10
+
0.17
— 0.01
-1,-2
-f
0.17
+ O.Ofi
4- 0.21
— 0.10
+
0.14
— o.u
0.
— ^.58
+ 30. 119
+
6.34
+ 31.39
1.
+
93.30
— 4;!0.35
4- 90.81
— 4;i0.3,•
—
87.14
—458.30
2,
—
40M.2S
_ 20.58
— 410.42
— 12.01
—
070.00
— 7.70
•■?.
—
91. HO
+ 04.32
— 57.01
— 04.98
—
98.97
— 94.93
4,
—
r..7G
-f IS. 30
4- 1.45
4- 12.90
—
0.07
— 19.05
5,
+
0.S8
+ 2.45
4- 1.30
4- 1.04
4-
1.50
— 1.79
«,
+
0.27
+ 0.17
4- 0.20
— 0.02
+
0.28
— 0.04
1,-3
38.72
— 17.03
— 38.75
— 17.29
4-
40.04
— 17.16
2,
+
1.02
+ IS. SO
i 5.31
4- 19.14
+
8.48
— 20.83
3,
—
254.31
+ 2.37
— 254.15
— 0.02
—
303.02
— 0.43
4,
—
71.84
+ 50.20
_ 61.49
+ 49.90
—
75.00
— 00.45
5,
—
5.()D
+ 10.95
— 0.23
4- 13.0
—
1.82
— 18.07
fi,
+
0.8V
-; 2.76
4- 1.29
4- 1.5
+
1.48
— 2.14
7,
+
0.34
-)- 0.24
4- 0.27
4-
0.35
— 0.07
3,-4
+
3.03
+ J8.78
4- 1.34
4- 18.78
4-
6.33
— 25.26
4,
—
144.90
4- 3.00
— 144.78
+ 1.5
—
191.70
— 2.31
s.
—
49.00
+ .35.00
— 38.04
+ 34.58
—
52.17
— 43.7
fi,
—
4.50
-f 13.80
— 0.9
4- 11.2
—
1.97
— 14.7
7,
+
0.75
+ 2.05
4- 1.1
4- 1.0
+
1.31
— 2.2
8,
+
0..35
+ 0.20
4- 0.3
+ 0.1
-f
0.30
— 0.1
5,-5
79.05
+ 4.05
— 78.9
4- 2.4
—
99.3
— 2.1
c,
—
32.3
+ 22.7
— 20.3
4- 22.3
—
33.5
— 27.2
7.
—
3.4
+ 10.4
— 1.1
+ 8.7
—
1.7
— 11.1
8.
+
0.7
+ 2.3
4- 0.9
4- 1.5
+
1.1
— 1.0
9,
+
0.3
+ 0.3
1
4- 0.3
4- 01
+
0.3
4- 0.1
w
T II K O II HI T O V U U A N U 8.
43
w
111'
olt </,.„
oV <"
DiMtTf
mncy.
V
<■.
<•.
a I'
= «,"X
Hill
t'()9
Hill
CU»
0,
— 0.01
—.01
. . . •
—1244.31
1,
— 4H.0')
-{■ 48.92
+ .10
—.04
+ 1.0
+ 32.78
— 20.90
2,
_ 0.03
+ 8.27
+ .01
—.01
+ 0.5
+ 4.00
— 1.80
.%
+ 0.4,')
+ 0.05
+ .01
+ i
+ 0.44
+ O.OH
4,
-1- 0.11
— 0.03
—.05
—.03
+ 0.25
_2,— I
— 0.03
-f 0.02
-.03
—.00
-0.200008
+ 0.08
— 0.08
— 1,
— 0.05
+ 2.10
—.02
+ .01
—0.259001
+ 1.23
— 1.23
0,
— 0.00
+ 0.09
—.00
+ .09
—0.350023
+ 270.02
+ 34.79
+ 1
-f34Sl.41
— 1.33
—.01
—0.539942
—1508.18
+ 5.78
2
+ 94.20
+ 71.00
—.08
—1.173030
+ 20.85
—280.59
ii.
+ .1.78
+ 15.03
+ .04
—.02
+0.75940
— 79.50
-}-194.17
4,
+ 1.27
+ 1.00
—.03
+0.871120
— 1.10
+ 1.79
T',
-1- 0.14
-(- 0.11
—.01
—.04
+ 0.4050
+ 0.03
+ 0.13
\ 2
+ 0.10
+ 0.10
—.01
+ .01
—0.149102
+ 0.19
— 0.13
0,
+ O.Ol
+ 0.07
+ .01
+ .07
—0.175312
+ 0.34
■1 31.39
+1.
+ 93.24
430.37
—.00
—.02
—0.212580
— 47.47
—275.33
— 40S.2S
— 20.00
— .(i-J
— 0,209971
— 897.05
+ 3.35
3,
— 91.90
+ 04.33
—.01
+ .01
—0.309800
— 100.93
—142.50
4,
— 5.T7
+ 18.17
—.01
—.13
—0.580813
— 7.44
— 41.13
5,
+ 0.91
+ 2.45
+ .03
— 1.42022
+ 4.05
— 8.75
c,
+ 0.29
+ 0.15
+ .02
—.02
+ 3.3797
— 1.53
+ 1.09
1,-3
— 38.78
— 17.03
—.00
— 0.1,32342
+ 29.73
— 12.05
2,
+ 1.02
+ 18.08
—12
—0.152528
+ 8.79
— 32.50
3,
— 254.29
-f 2.:i7
+.02
—0.179981
— 451.55
— 1.28
4,
— 71.88
-1- 50.13
—.04
—.07
—0.219482
— 107.20
— 88., 50
r',
— 5.09
-f 17.01
—.04
+ .00
—0.281202
— 5.00
— 27.01
G.
+ 0.91
+ 2.73
+ .04
—.03
—0.391210
+ 2.15
— 4.30
7,
+ 0.35
+ 0.19
+.01
—.05
—0.0426
+ 0.79
— 0.38
3,— 4
+ 3.10
+ 18.78
+ .07
+ 0.11893
+ 7.05
— 29.73
4,
— 145.0
+ 3.51
—.04
-.09
—0.13500
— 230.83
— 3.28
5,
— 49.9
+ 34.85
-.15
—0.15005
— 07.74
— 54.00
fi,
— 4.7
+ 13.95
—.20
+ .09
—0.18490
— 3.03
— 19.80
t,
-'r 0.8
+ 2.02
+ .05
—.03
—0.22085
+ 1.05
— 3.30
8,
+ 0.3
+ 0.2
—.05
—.00
—0.2934
+ 0.57
— 0.27
5,-5
— 79.3
+ 2.7
—.25
—1.35
—0.1080
— 110.4
— 3.5
fi,
— 32.4
+ 22.7
—.10
—0.1211
— 41.3
— 32.7
7,
— 3.5
+ 10.5
—.10
+ .10
—0.1377
— 2.6
— 13.9
8.
-(- 0.8
+ 2.3
+ .10
—0.1597
+ 1.3
— 2.0
fl.
+ 0.3
+ 0.2
—.10
—0.1901
+ 0.5
— 0.2
u
THE on BIT OF URANUS.
R'
The values of "' f D',Ii„ilt rire formed from ,' ITEhy simple multiplication by
V, and proper changes of sign. The vuUies of k^ and Ic, arc then formed by adding
the terras of 2 ' .' ( i>«iio''' to the corresponding terms of ',
111 "' "«
in' d^
Perturhalions of radiu6 vector.
Let us now resume equation (19), and put for brevity
(44)
■«.(l + »0
If we give to u the successive values 0, +1, —1, -f2, —2, -fS, —3, we have
+ i S< (iVA(-i) +i>u-i)V() ('—''(i-o) 1 ''".cos (.V+ g) -\-k. sin(iV+i/) |
r,=,^p=i/'X ] +|2.(i>,<z^,+„+i'(.+i)'7.X'*— »'-(M-i))l^"oCos(^— sr)+/.-,sin(JV^— «7);
-|- etc. etc. "tc. etc.
the finite integral being taken with respect to all values of i from — oc to + oc,
and the terms in which the angles A'iwf/ vanish being omitted. Piocceding far-,
thcr to expand with respect to /, if we collect similar terms we shall find the indi'
vidual terms in j-'.^p to be as follows:
j Pill (»'i — '-i)
•^ i + M'j ('•» — ''-0
Ml ("i — »'")
txnr) + (P'l^ + P'h) {»'a — ''-0
+ *^^ ^ +(M3 + M«)('V-*-.)
-f- etc. etc.
4-JJ^
»V/i ('o — I'-i)
-f (Mi+M^) (''2 — ''-:0
-|- etc. etc.
> I A'o cos iV -|- /i", sin iV (
. \hcQs{N+g)+k.smiN+g)\
■ ; Z-„ cos (iV - flf) + A-. sin (JNT - i/) (
•f i ^ • + (Pili + Pill) (''3 — v-i)
I 4- etc. etc.
1
{l;cos{N+2g)-\-l:s{n(N-\-2g)l
+ \M
+ i^r
{2Wi (»'j — J'-j) ■)
+ ( Piqs +i'3?.) (»^, - »'_3) \ J /.•„ cos ( A^- 2g) + Z-, sin (AT - 2g) \
4- etc. etc. )
+
p»q3 (»'» — n)
+ (pi74+r.?i)(»'« — »'-i)
+ (7'i72— i'//i)(»'2 — J-i)
-|- etc. etc.
j /.-. cos (iV+ 3sr) + 1; sin (JV + 3g) \
m
TUE ORBIT OP UBANUS.
i6
+ i^
+ (/"74+ivy.)(". -»■-)! I /,„ cos (iyr- 3^7; + /.-. sin (iyr_3j7) I
+ {Pi'h — iV/i) ("-1 — »'-2) I '
-|- etc, etc. I
A law of the factois of /.-, cos (iV+ «//) + A-. sin (^+ ««/) wliicli will be noticed
in the above expression, is this: lloprcsenting this i:tctor by A'„, wo have
the index i representing the coefficient of (j in iV, so that only half the valnes of
it' need be separately computed.
As the computation of r,«<^p from these formulie .an be arranged in such a way
as to be very simple, the computation of the termv i.i which the index i' is —1 is
Here presented quite fully. The logarithms only are omitted, being used only in
the cases in which they are more convenient than a table of products. In prac-
tice I find it convenient to write them in red ink immediately under the numbers
which tliey represent.
via
First, to find M, it will be noticed that in the expression .,"V' y the a in the
a,(l -j- VI)
numerator represents the moan <Hstance of the <//«/iuie7 planet, as deduced from the
observed mean motion by the equntion («'«' -:^u (1 -| "0 while «, represents the mean
motion of the onler planet. When the outer planet if the disturbed one, the ratio
"- would be unity, but that, to avoid r large class of second or.'-r terms, o, has
been corrected for perturbations in the beginning (p. 32). In the case of Uranus
disturbed by Saturn, we have in consequence
a
Whence
log = ;).<)91)8()3.
i/= 285.44
in iniits of the sixth place of decimals.
Computing the values of />( and q^ from (U?)' we find, for Uranus,
i M mi — —
•i -iVC/v/i— 7'//i) — I-
0.0784
10,044
3.343;>
0.0028
0.2;5.')8
O.llh
0.008
0.005
0.()();{
In units of the sixth
place of decinuds.
In the computation the first three lines are copied from previous pages.
46
THE OllBIT OF URAxNUS.
i = — 1
•
-4
—3
—2
—1
+1
2
3
4
5
V
-0.1460
-0.1700
-o.soeio
-0.35000
-0.85062
-0.68004
-1608.18
-1.17868
+6.76040
+0.87118
—1.10
1- 0.466a
40.03
k„
+0.08
+ 1.38
+276.62
+ 20.85
- 70.56
+142.56
•l— •-!
-0.08
-1.28
+ 84.70
— .28034
+ 6.7U
-286.60
+ 104.17
+2.04476
+ 1.70
—6.2938
+ 0.18
—0.6536
— .0887
— .14452
— .82301
+7.299:14
+ 0.0784
.,-._,
— .2047
— .3690
— .9675
+7.0190
+1.2218
+ 1.0O85
+ 1.4913
— 6.51b6
— 10.044
'i~'o
— .0.'),'?5
— .0910
— .1893
— .63361
+7.9330
—5.8883
—0.4056
+ 3.343:t
'i—'l
— .1797
— .3338
— .9140
+ 7.1100
+1.4110
+ 1.6392
—6.4417
-«- 0.0028
'..-».
— .39
—1.00
+6.96
+ 1.13
+0.82
+0.86
+ 1.41
— 0.23B8
»»— • V
.144
— .280
— .823
+7.299
+2.045
—6.29 J
+ 0.118
•a— '-1
— .3(;9
— .9i;8
+7.019
+ 1.222
+ 1.005
+ 1.-190
0.008
— .33
— .91
+ 1.41
+11.82
+7.93
+1.64
+O.Mi
—5.89
+ 0.005
+ 0.0003
1.00
+6.97
— .19
—20.60
•j-'-l
— .09
142.56
X{'|-'-,)
—12.35
— 39.9(;5
—117.327
+ 1040.59
+291.49
—897.2
—78.8
0.0784
X(',— -,)
— 0.02
— 0.03
— 0.075
+0.550
+0.10
+0.08
+0.1
— 05
— 10044
A'o
-12.87
-80.68
+0.914
-40.010
+ 1.901
-110.777
+ 1040.00
+201.57
+B0.I4
-807.1
-70.8
X (•,-•„)
+0..'i32
+ 6.364
—79.68
+ 4.1
+ 3.:tJ:l3
Xf»i-»-,)
— O.OCl
-1.12ii
—3.056
+23.770
+ 4.72
+ 6.48
-21.B
+ o.ot>a:
X(«j-»-j)
—0.001
-0,003
+ 0.020
+ 0.003
A',
— 0.20
—1.186
+80.187
- 1.135
—1.721
— 74.»«
+64.62
—17.4
— .2:i.')a
A-:.
Xfvj-O
—0.07
+o.oi;«
—0.205
+ 194
+80.14
—0.48
-74.1I6
+ 1.48
+64.6
+0.034
+ .118
X('j-»-,)
A'.
—0.04;;
—0.114
+0.!-2S
+ 1.022
-0.00!'
+0.144
— 1.577
+0.1-2
-0.86
+ 0.18
+ 1.66
+ 1.58
.008
A-*.
-0.048
+ 1.03
—.013
+ .003
.(H)5
X (•,-•„)
X {•,-»,)
+ .007
+ .035
-.011
+ .005
+ .004
-.007
-.003
+ 176117
+.004
^.000
+.(M2
+ 21698
h'„ h
— 1
— 2il
—11076
—23197
+ 991
2
K, h
—314
-45152
—15113
—51 •■•.
+ 19
K , h
— 5i!
+ 1712
-t-628
+ 5964
-71
Kj /■,
n
+ 2,-2
-1-2378
—8
-132
— 3
+ 72
+ 21
+ 126
K'k,
+ 8
A',/-„
+ 3
+ 1
r,i»,(.Oi.)
- 1
-
—0848
f 170557
-17508 -82458
-4151
-116
X.0-. ••
i
-43S
-l->2.-3 —-21
— lii.',;t
-1!)5
—438
+ 8283
—821
—1053 —195
— 5
X. 002751
—26
+:!(>
+485
-48
-62
—26 +485
— 11
— 4H
II
—62
X.OOOIfix
— 3
- 4
— 1
—2
-4-30
— 3
cos ^1^ (co.-»)
+ 8
+ 84
+ 26
-1112
-1393
+ 175235
—676
-10315 -22086
— 2982.'i4 +56614
-5227
— 1606
-875
— 10
+ 1
A, <.
— 39
+ 175 ! +21483
+ 12548
—31
A' ,t.
— 7
—6
— »(;37
—14556 +116
-2
A, i.
+ 36
—9
+ 104
+ 322
(1
n
—293
—306
— 1
A'f,i,
— 11
II
I)
- I
r,«», («in)
+ 1
+ 8
-1602
-0058
-812600
J 1*8204
+ 11044
4 281
X. 0409 15
{
— 79
—453
— 79
— Ilt.ilii
-453
+3i;(i'.i
-146116
+ 518
+ 3lil>!l
+ 13
+ 51H
II
— 5
-27
—860
+ 215
X. 002751
— 27
— HIIO
+ 215
+ 3M
x.oooiim
.^
— 53
4- 13
-1-2
'1
- 3
COs*l^(:.ill)
- »
- 151
- 21tlt3
341M6
-800351)
4 64020
f 18864
4!>«1
it"
• .
THE O K B 1 T OF URANUS.
47
'
In forming the next ten lines, it will be noticed that the value of r„ correspond-
ing to any vertical column is found u columns to the right. It is therefore
necessary to extend the line v two columns at cath end. The extension on the
right is, however, omitted for want of space. In performing the subtractions it
will be convenient to copy the j^'s again on the lower edge of a horizontal strip of
paper, and, in forming tlie differences r„ — r„' to lay the; strip above the line of r's,
and u — u' colunnis to the riglit.
On the left of each line of differences is written the factor by which that line is
to be multiplied.
The mode of formation of the A"s is evident from the formula.
It will be seen that the same computation which gives 7C gives also K_„, only
the latter belongs u columns to the right.
Each l\ and //, is multiplied in succession by all the A"s wliich lie below it in
the same cohuun, but the product by A', is to be written u columns to the right,
and that by A'_„ u columns to the left. The sum of the products in any one
colinnn gives the coefficient of ^?^ {Ig -\-i'l') in the development of r,*!^,
</i\ »«° cos 4/ , ,5 ... ,
This quantity being multiplied by ^n~ = ^.z we have cos 4!'p, which only
needs to be inultii)liod by sec v^ = l.OOllO:} to give 5p. The unito of ri\^p and
^p correspond to the ninth place of decimals.
All the periodic terms are to be treated in this manner, all the series of values
of l-^ and /•„ inchuling the constant term, being subjected to the same process.
Hut, when /' and / are both zero, v will be infinite. Here we sinii)ly omit the r,
treating it as if it were zero. We thus obtain the coni[)lete value of the terms
with constant coefficients in ;V-'\) and >^) wliich are given in the following table.
Tlie terms midtiplied by the time are still to be comptited. They are derived from
('20), wliich may be put in the form
Thisexp^ession is computed thus:
u
Pn
*<;"
«7..
*•;' /'.,/.•'"'
-qJ-V
— .t4l)-s;j
- 1 •.'44.31
+n->.i8
1
4-.!)!)!! '7
+ ;5-.Ms
4- .!»!»«)■
•i
— 'io.oo + ;}J.7:)
-f 2(1 >i)
2
+ .()t>:U
-t- 4.()G
+ smi
— 1.8G + ().(»!)
+ 0.04
We have now
^ /'„/.•':;' =
- _|_ OOS.Ct?;
\ }r:iii..i-":^ = + -.'nosn
-:iqj-r =
:.+ '.'().!»:} ;
iJ/'iv,./.- -+ -mn
r^\>
= — 'iU) lit
-\- '.'!)«() lit
cos If
+
'>{)(»S1 lit sin g
4- ■<<» III
COS '2l/
+
()!)7 /// sin 'i'/
+ -',7
COS %
+
'2i it sill ;{</,
in units of the ninth ])laee of decimals. The value of cos ^i.^p is obtained from
them by multiplying by /[', exactly as in the case of the constant terms.
48
THE ORBIT OF URAXUS.
n««p
cos
4«p
log My
V.
V.
9 V
cos 1
sin
COS
sill
0,
—210 ni
—70 nt
+ 139)i7
1,
-I-29HG H<
+29081 nl
+2H7«n<
+29033 nt
2,
-1-70 Hi
+097 nl
+ 209 »i<
+ 2084 >i<
3,
-f 2 »<
+ 24 H<
+ 14/1/
+ 139«<
0,
—355045
-354347
oc
1,
-I-14S75
—1503
—18411
—1496
2.45551
—2912
+14206
2,
— 2,S3
+ 49
—1530
—21
2.25448
—039
+457
3.
—20
— 1
—112
2
1.97839
—40
—8
—2,-1
— 1
+ 1
+ 3
— 9
1.7090
+ 1
+3
-1,
—10
+ 8
+ 34
—151
1.8098
+ 12
—99
0,
—9343
—1093
—1112
2!t92
2.00035
—2870
+ 10
+ 1,
-|-17(i557
—9058
+ 175?35
—21186
2.18780
+530512
+ 100
2,
—17508
—312600
—10315
—309359
2.52504
+ 1823
—23800
3,
—22453
+78204
-22980
+04029
3.28542
—13482
+ 18310
4,
—4151
+ 11044
—5227
+ 13804
2.39559
—293
+ 104
5,
—115
+281
—373
+ 901
2.1235
—13
-1-2
— 1
+ 1
+ 1
2
1.0292
+ 9
+1
0,
—5(5
— 27G
+32
— 100
1.0993
—329
—1.551
+1.
+710
+3723
+ 1075
+ 3728
1.78303
+.5874
+ 20114
2,
+20180
—13
+ 20583
+ 484
1.88083
—31000
+ 920
3,
+7797
+0512
+ 8824
-1-0796
2.02348
—0019
—6800
4,
+ 1534
+0202
+ 2054
+ 0230
2.22401
+243
—2172
5,
+ 2071
—5954
+ 2134
—5010
2.00780
+527
—421
0,
—053
+755
—550
+ 494
2.98438
—190
—20
1-3
—151
+ 04
—137
+ 75
1.5772
—1464
+ 053
2,
— i-)2
+ 221
+ 149
4 229
1.03880
+ 231
—833
3,
+4348
+ 15
+ 4420
+88
1.71074
—13050
4'
4,
+ 1504
+ 1280
+ 1774
+ 1314
1.79091
—3225
—3126
5,
+ 133
-{-085
+ 214
+756
1.9045
—18
— 1')43
«,
—111
+ 232
—109
+273
2.0479
+ 144
— 103
7.
—103
+ 87
—108
+ 100
2.2034
+ 50
—7
3,-4
—28
+ 121
+31
+ 122
1.5408
+ 181
—038
4,
+ 1223
+ n
+ 1245
S +41
1.5858
—.5578
— f.8
5,
+ 485
+ 389
+ 545
1 4 400
1.6488
—1721
—15(0
<■',
+38
! +201
+03
+ 221
1.7225
—50
—590
7,
—25
+53
—23
+ 03
1.8012
+70
—100
S.
—15
+ 7
—10
+ 9
1.9229
ft,—:)
+393
+ 12
+ 400
+ 18
1.4889
—2432
—74
fi.
+ 175
+ 138
4 194
+ 142
1.5385
—90S
' —700
7,
+ 14
+77
-+22
! 4 84
1.5940
—40
— .UO
8,
—10
+20
— 9
4 2»
1,05M<)
1 +40
—70
9,
— 5
+ 3
— 5
4 4
1.7:!45
^ims^
I ■itr-Bir r*— "*"•"" wnaniriiriinMiM
THE ORBIT OF URAXUS.
49
Pert 111 hations of Lomjitude.
The perturbations of the longitude are now to be computed by formulte (24).
To do this iu the most simple way we remark that the numbers given on page 42,
under the heading '^'^ are those represented in formula (42) by v, and v,. If
c*v
we put
nt = i
equation (24) may be put into the form
di
out we have from (42)
„ ill = S( V, sin N— V, cos N).
If now we represent the numerical values of cos i|-(^p, already found, by
2 (p, sin .V -f p„ cos N) ,
and if we substitute these expressions in the above value of . , , the latter will
become
d'v
'y^, = rr- S Kv. - 2l'.) si" ^V+ (V,. - 2p,) cos N\,
where we put for brevity
V, = Mn;.,
The nnmoriral expression for r,"" is given on page 40, and by multiplying the
quantities witliin brackets by tiiis expression, after tlie manner explained on pages
40 and 41, we form the terms of ' ,'f. Multiplying each of these terms by its
al
corresponding value of r, changing ros to s!n and sin to ros, we have the coefficients
iu the expressions for I'v given on page 50.
Al previously mentioned, before commencing the above computation, I had
conii.-''ed all the perturbations of Uranus by the method of "])erturbations of the
elements," using tlie formuhr developc^d in my Investigation of the Orliit of Ne|)-
tune. The two results are here placed side by sid(>, for the purpose of comparison.
The discrepancies in the various coefficients, expressed in thousandths of a second,
are shown iu the sixth and seunth cohnuns.
It will be seen that the largest discrepancies, and indeed the only ones (with a
single exception) exceeding one-tenth of a second, occur in the coefficients of the
terms 2/' — / aug %' — /. Here tiie errors are almost certainly in tlie computation
from perturbations of tlie elements. Owing to the limg pi-riod of the term 'i(/ — I
they would not become sensible iu the course of any one century.
7 April. 1873.
^
60
THE ORBIT OF URANUS.
Pebtl'bbatiu.ns or the Lonuitudk of Uranus produckd by
TUB ACTION OF SaTUBN, AND 1
UEI'ENUINU UN THE I'lUST I'UWEB OC TIIK DISTL'BBINU I'llUCKK.
From conip. prucuiliiig
From purt. of uluiiiviits
iu
Discrepancy.
0.434294 ip
9, I'
8in
coa
siu
cua
.{ill
cos
cos
8ia
ti
If
fi
If
It
II
0,
+ io.»(;90<
+ 10,9045/
.004.5/
1,
— 1.2:10(1/
+ 12.231 III
— 1.22H III
+ 12.271)1/
.002(1/
.040(1/
+ 13(1/
+ 128(1/
2.
— 0.01-2 lit
+ 0.717 H<
—0.072 nt
+ 0.720(1/
.003 ((/
+ !((/
+ 9(1/
3,
— O.OOiiil
+ 0043 «<
—0.004/1/
+ 0.043 ;i/
0,
'"'*'*
—1.541
1,
+ 8.;") 45
— 4.735
+ 2.844
+ 1.013
....
....
— 80
— 6
2.
+ 0.4(>l
— 0.1 ('.9
+ 0.133
+ 0.100
....
— 7
3,
-1- 0.028
— 0.005
+ 0.013
+ 0.014
—1,-1
-f o.o;;*;
+ 0.039
+ 0.032
+ 0.005
4
34
_ 1
0,
+ 1.282
+ 0.718
+ 1.280
+ 0.719
2
1
— 5
— 13
1,
—20.817
+ 8.522
— 20.873
+ 8.595
50
73
+ 701
— 100
2,
-11.890
-1-143.4(13
— 11.003
+ 143.405
797
2
— 45
—1.351
3,
+ 4!t.30
-|-115.8(i
+ 49.02
+ 110.08
320
220
—103
+ 280
4,
+ 2.1 13
+ S.CIO
+ 2.195
+ 5.021
02
5
— 24'+ 01
5,
+ 0.1 2(i
f 0.329
-j- 0.109
+ 331
17
2
- 2 + 4
0,-2
+ 0.017
— 0.017
+ 0,025
— 0.033
8
10
1.
+ 0.042
+ 0.814
+ 0.034
+ 0.818
8
4
+ 7
+ I"-.
2,
+ 4.110
— 0.009
+ 4,103
— 0.012
7
3
+ 89
+ 2
3,
+ 2.079
— l.(!07
+ 2.100
— 1 070
27
G9
+ 38
+ 30
4,
+ 0.048
— 1.830
+ 0.043
— 1.902
5
73
+ 9
+ 27
5,
+ 1.1C3
+ 2.950
+ 1.274
+ 2.991
111
35
+ 9
— 24
6,
+ 0.503
+ 0,378
4- 0,550
+ 0.445
53
07
2
+ 2
1,-3
+ 0.034
+ 0.012
+ 0,030
+ 0.015
2
3
2,
+ 0.037
— 0.041
+ 0,014
— 0.050
23
9
+ 1
+ 1
3,
+ 0.824
— 0,019
-1- 0,812
— 0.017
12
2
+ 19
4,
+ 0.355
— 0,207
+ 0,351
— 0.203
4
4
+ «
+ fi
5,
+ 0.047
— 0.105
-f 0,039
— 0,191
8
20
+ 1
+ 3
6,
— 0.02r.
— 0.003
— 0,028
— 0,000
2
3
+ 1
7,
— 0.053
— 0.032
— 0.053
— 0.018
14
_ 1
3, 4
+ o.oor,
— 0.022
— 0,005
— 0.023
11
1
+ 1
4,
+ 0,228
— 0,008
+ 0,221
+ 0.002
7
10
4- 5
5i
+ 0.103
— 0.077
+ 0.084
— 0,075
19
2
+ 2
f 2
fi,
4- 0,013
— 0,044
1- 0,013
— 0,057
13
+ 1
T,
_ 0.005
— 0,013
— 0.001
— 0.015
4
2
8,
— 0.003
— 0.002
+ 0.002
3
4
5,-5
+ 0.074
— 0.003
+ 0.071
3
3
+ 2
6,
+ 0.038
— 0.027
+ 0.023
— 0.020
15
1
f I'-f «
7.
-f 0.005
— 0.010
-f 0,005
— 0,025
ft
8,
— 0.002
— 0.004
— 0,003
2
1
"1 »
<4Ki*
TUE ORBIT OF URANUS.
61
cos N
Perturhationa of Latitude.
These are computed from the formuhc (27) and (40), no reductions being made
from hk and l/i to hp and hq, but the perturbations of the hititude being computed
directly from the former by (40). We have only to represent the expressions for
hk and hrihy
hk = — 2«„ cos N — 2a, sin N
hr, = Sa',, cos iV-|- 2a', sin N
and substitute u for n in the equations (40) from which f)3 is computed.
The principal stops of the computation are shown quite fully in the following
table. The values of
c7i 1 1 fJi
are first formed from those terms of Ii, on pages 37 and 38, which contain d as a
coeffici(!nt. Then, having for each original term of Ii
OR _ m' eh
dy a, dy
all the terms which have the same coefficients of?, and ?.' in N are combined into
two «lepoi!(ling on «/ and /' as shown in the case of Ii on page 3(5. The coefficients
of these terms, in units of the third place of decimals, are given in the columns
licadcd — .
Oy
The value of ' sin N being formed for each term of R, all the terms depending
on the same multiples of ?. and ?.' are combined into two, of which the coefficients
are given under tlie proper heading. Tiie terms of (t -\-j)ah sin N being formed
in like manner, we have, by adding the last two expressions, all the quantities
which enter into the formulic (27). To integrate these equations thus forming
the numerical values of (V.- and hv; we have only to multiply each term in the
second, third, eighth, and ninth columns of the table by the corresranding values
of '" '"' , for which we juay use the value of . '' „ already given.
<», cos 4/ sni 1
The quantities giv(>n in the four columns under hk and — hr, show the values
of — rt„ — «,., — («',, — rt'., corresponding to each argument. From these the
terms of 6;i arc formed by equation (40) with the modification mentioned above.
52
THE ORBIT OP URANUS.
PSRTIJllBATIONS OF TUB LATITUDE I'KODUCED BY SaTURN.
3 ''
dy
2-'*-«iUiV
'It
I ('+»'■'' sln^V
:;;=.,/,. X
ik
— i,
W
i i'
cos
sin
sin
009
sin
cos
sin
00s
sin
It
ens
tt
cos
sin
It
Bin
00s
0,
—10.82
+0.008
0.00
+ .008
+0.19S
1,
— 3.45
— 2.92
—1.94
— 0.11
+ 0.17
+0.84
- 1.77
+ 0.73
+0.203
-0.172
—0.104
—0.043
+0.246
+0.209
2.
4- 1.42
—11.10
+ 1.30
+10.73
+ 0.07
+0.05
+ 1.37
+ 10.78
-0.042
—0.326
+0.042
—0.317
—0.049
+0.0U
3,
4- 1.31
- 1.71
+1.31
+ 1.71
+ 0.01
0.00
+ 1.32
+ 1.71
—0.026
—0.033
+0.026 —0.033
—0.008
+0.003
-2,-1
+ 1.0
+ 0.1
—1.0
— 0.1
0.00
0.00
— 1.0
— 0.1
+0.012
—0.001
+0.012—0.001
0.000
0.000
-1,
— e.43
-52.«
+6.4
-82. 8
+ 0.03
+0.02
+ 6.4
—52.8
—0 098+0.804
—0.098—0.804
—0.004
+0.018
0,
— 0.97
+ 7.06
+1.53
+ 5.14
- 4.91
0.00
— 3.38
+ S.14
—0.144—0.145
+0.070+0.106
—0.661
+0.678
+1,
+46.67
0.00
0.00
+59.45
—0.02
+59.45
— 0.02
+1.482
—1.887—0.001
+0.081
—0.026
2,
— 1.84
— 2.81!
—0.99
— 0.20
+ 0.93+1.32
— 0.06
+ 1.02
-0.127+0.197
+0.004+0.070+2.218
+1.92C
3,
+ 0.90
- 6.98
+0.73
+ 0.44
+ 0.12+0.16
1
+ 0.90
+ 6.60
—0.358 —2 m
1
+0.358—2 630+0.091
-0.071
4.
+ 1.09
— 1.38
+1.09
+ 1.38
+ 0.02+0.01
+ 1.11
+ 1.39
—0.056
-0.071
+0.057—0.071
-0.048
+0.068
-1,-2
+ 7.6
- 1.5
—7.6
+ 1.5
O.UO 0.00
- 7.6
+ 1.5
+0 067
+"'"»
+0.067 +0 013
+0.(103
+0.004
0,
+ 13
+10.9
—1.3 +10.7
— 0.11+0.53
— 1.4
+ 11.2
+0.013'— 0.112
+0.014+0.116+0.008
+0.034
+1,
— 0.27
+ 9.03
+1.70 + 0.91
+ 1.85—7.35
+ 3.35
— 6.44
—0.003—0.112
—0.042—0.080
+0.040
-0.04S
2,
-14.51
0.00
o.do
— 7.01 —0.21
1
-7.01
— 0.21
—0.230
»
+0.112—0.003
1
—0.028
3,
— 2.85
— 2.29
—0.42 — 0.19
- 0.97+1.11
— 1.39
+ 0.92
— 0.0()2+0.050
+0.0,30+0.020
—0.063
-0.064
4.
+ 0.43
— 4.23
+0.44 + 3.67
+ 0.02+0.22
+ 0.46
+ 3.80
+0.015+0.146
—0.016+0.134
—0.080
—0 014
6,
4- 0.80
— 0.99
+0.80
+ 0.9!)
f 0.02+0.02
+ 0.82
+ 1.01
+0.067 +i> 083
-0.068+0.084
+0.008
—0.004
2,-3
— 1.17
+ 0.86
+1.22
+ 0.97
+ 0.09+0.33
+ 1.31
+ 1.30
—0.01(1 '0.*
—0.011 +0.012
+0.050
—0.046
3,
— 8.54
0.00
0.(10
— 4.34'
-..«
0.00
—0.090
+0.046
+0.004
-0.010
4,
— 2.08
— 1.65
—0.13
— 0.15
— O.fS +0.85
-1.1
+ 0.70
—0.026+0.022
+0.013
+0.009
—0.012
—0.015
6.
+ 0.14
— 2..'i0
+0.24
+ 2.00
0.00 +0.22
+ 0.24
+ 2.22
+0.002+0.041
—0.004
+0.037
—0.010
-o.ooa
«.
+ 0.55
— 0.66
+0.55
+ 0.6(!
+ 0.02+0.02
+ 0.57
+ 0.68
+0.013+0.015
—0.013+0.016
+0.004
—0.001
3,-4
— 0.72
+ 0.47
+0.82
+ 0.78
+ 0.09 +0.,32
+ 0.91
+ 1.10
_ 0.005 —0.003
1
—0 006 +0.008
+0.022
—0.020
4,
— 4.80
0.00
0.00
— 2.4.S —0.02
— 2.4S
— 0.02
— 0.0:!8.
1
+0.020
+0.002
—0.006
5.
-,..
- 1.09
-0.01
— 0.10
— 0.06 +0.59
— 0.6"
+ 0.49
—0.013 +0.010
+0.006
+0.004
—0 004
—0.003
6,
+ 0.12
— 1.3
+0.12
+ 1.0
— 0.02+0.19
+ o.in
+ 1.2
+0.001 +0.014
-0.001
+0.013
+0.0(>4
+0.002
5,-6
— 2.64
— 1.35 +0.04
— l.,35
+ 0.04
-0 017
+ 0.008
IXl L A" 77 T
S"^'"'"' ""•"•' {4 = -4". 77 7- CO. 7.
i
^■■■I
THE ORBIT OF URANUS.
53
C II AFTER III.
PERTURBATIONS PRODUCED BY NEPTUNE AND JUPITER.
The perturbations of Uranus by Neptune were originally computed with ele-
ments of both planets quite different from those finally adopted. But the last
computations, on which the concluded values of the perturbations depend, Averc
made with the concluded elements of Neptune found in my investigation of the
orbit of that planet.' They are as follows :
• >
7t,
4;j
17 .30
0,
l:j()
7 33
f.
3;J5
5 39
«?>'
1
47 1.6
«,
7864.935
e.
().00849(J2
log a.
l.-l
78141
Mass,
TtJiTU
rm»
(J
r
(J
Hence follow the following functions of the elements of Neptune and Uranus:
a = 0.G38195
12° 44' 58"
247 45 20
y = 1 30 29.6
c = sinjy= 0.013161
M= 37.522 (in units of 6th plito of decimals).
From these values of the elements are obtained llie following values of the
various terms in the development of the perturbacive function, and of v. As the
developments have been formed on the same principle as in the case of Saturn, it
is deemed unnecessary to give the details of the process. It is only necessary to
remark that the indices t' and i arc the coefficients of l' and g respectively, the
mean longitude of Neptune, or I', being counted from the perihelion of Uranus.
' Suiithsonian Cuntributions to Knowledgi', Vol. XV.
54
THK OR HIT OF UIIANUS.
A
CTIIIN or
"'x
Ni;i-n.Ni!.
1
/<-
Ml'
dlt
'!"='"' X
«-r'><
V
A.
"1
A.
0'"
"1
(?P "i
aj
•' i
*'»
I'O
00:1
sill
h
d.
0,
+n:ir..(n
4- 0.202
4-3 ;8.2(i
4-.3(18.2(>
+ 1
—
I».(i7
— I.2.'.
f 1.01
— 1.23
— (14.80
— 5.89
— 100 94
— 8.39 4- 1
+ 2
+
0.41
— 0.0.1
— 0.0
— 0.1
4- 2.98
4- 0.12
4- 3.82
4- O.Oli '4- 5
+ ■i
U
u
4- 0.33333
1,-3
+
0.21
— 001
4- 0.7
— 0.2
4- ].(!
4- 0.2
+ 2.1
4-0.1 — 0.40158
— U
—
fi.21
- O.ii.1
— 4.11 >
4- 1.27
— 211 5
-f- ((.(!
— 43.5
— 1.1 1 — 0.()7107
— 1
4-
i;u.i!»
+ (i.:ti»
4-i:i3iis
— 0.08
4-509. m
4- 0.77
4-10.'-,7.49
-f- 1.99 — 2.04023
(>
2.'). 4.')
-1041
— 18.0 1
4- 0.02
— 84 57
—11.47
— 84.57
— 11.47 4- 1 9(11:17
+ «
+
1.12
4- 1 19
4- 0.5
— 0.1
4- B.-.;i
4- 0.82
4- 7.31
4- 1.07 4 o.««2;i2
+ •■!
0.05
+ 0.01
4- 0.02
+ 0.02
— 0.24
— 0.32
4- 0.02 1 4- 0.39843
2,-4
+
O.S.I
— 0.03
4- 2.1
4- 2., 5
4- 0.1
4- 4.8
-- 59.7
— 0.335841
— 3
4-
13.5:1
— 1.01
4- 27.7
4- 3.1
4- 18.7
— 2.1
— 5.5
— 0.504971
t)
+
37.'..!I2
+ :m
4-751.59
— 0.7!
4-9 7.«2
4- 1.25
4-2491.(19
4- 2.72 — 1.02009
— 1
-
U0.090
— 3.1*1
— 117H03
4- 3.20:1
—171.79 -i:i.l:l
4- ,'(931.21
-f3oil.82 4-50.7820
+
3.31
4- 4U
4- 5 9
— 0.4
4- 12.24 4- l-'*4
4- 12.24
4- 1.84 4- 0.98009
+ 1
0.09
— 0.01
— 0.0!>
4- 0.02
— 0.51 — 0.08
— 0.(10
— 0.09 -j- 0.49512
3,- f.
4-
0.81
— 0.04
4- 27
4- 0.2
4- 3.0 — 1
4- S.5
— 0.2 — 0.28814
— 4
+■
12i>9
— 0.11 i
+ ;i(;.9
4- 3.8
4. 34.5 — 3.(1
4- 73.(1
— 0.7 — 0.40478
— 3
+
2lHi.:)7
— 0.04
+(i01.l:t
— 0.42
4-717.94 4- 081
-1-1535.81
4- 0.115 — (I8OOO
— 2
lij 3:t
— 9.8,S
-l.'i0.02
4-lll.M
— 190 :!3 —27.51
—(1:15. 2:1
—111.09 — 2 12559
— 1
+
■').ii.'i
+ 1.5ti
+ l:i.S
— :).l
4- 20.12 ; 4- 4.(15
4- 1.0
— 1.2 1 4- 1.88844
0.24
— 0.07
— 0.6S
4- 0.15
— 1 11 — 0.30
— 1.11
— 0.30 + U. 05378
4 - «
4-
0.7-.
— 0.07
+ 3.2
4- 0.3
-1- 3.4 — 0.2
4- 0.7
- 0.4 1 - 0.25249
— 5
4-
9.-17
— 0.79
-f 38.:»
4- 4.0
4- :is.5 — 3 9
4- 70.4
— 0.0 ' — 0:1:1777
— 4
4-
Ill.l.i
- 0.12
4-441.7
— O.H
4-5i2.2i 4- o.:i5
4-9(15.7
— (1.1 ; — O5I01I4
— 3
—
;j8.fi8
— 7.4t
— 149 10
4-22.:l7
— 17- 15 —27 95
—419.12
— 74.3(1 — 1.(410;
— a
4-
."i.2a-.
+ 1.^9J
4- 19.. 54
— 5.50
4- 25.:tO 4- 7.19
— 603.3
-185.4 4-25.:i910
— 1
0.3lJ
— 0.17
— 1.28
4- 1.49
— 1.90
— 0.70
— 1.21
— 0.37 + 0.90210
6,-7
4-
O.!!,"!
- 0.07
4- 3.5
4- 0.4
4- 3.5
— 0.3
4- 6.5
— 0.5 — 0.22468
—
4-
U.97
— 0.59
4- 3.5.1
4- 3.4
4- 3(1.7
— 3.4
4- liO.9
— 5.6 — 0.281184
— 5
4-
(13.03
— O.K!
4-315.3
4- 0.1
4-.3.54.S
— 0.1
4-012.0
— 0.7 — 40804
— •'.
—
27.43
— 5.32
— I:tt4
4-21.4
— 1 5:1.1!
1 -25.3
—304.8
_ 64. ti ; _ 0.08U29
— 3
4-
4.84
4- 1.81
4- 23.0
— 7.0
4- 27.82 4- 8.57
4- 92.2
■f 32.7 ' — 2 21043
-- 2
—
0.45
— 0.24
— 2.09
4- 0.90
— 2.77 — 1.14
+ 0.48
4- 0.57
4- 1.82073
— 1
■f
0.03
4- 0.02
4- 0.12
— 0.0(1
4- 0.18 4- 0.09
4- 0.15
4- 0.07
4- 0.0455
«,-7
4-
4.9:1
— 0.41
-f 29 ,8
4-2.7
4- 31.5 — 2.9
4- -19.0
— 4.4
— 0.253S
— (1
4-
3li lli
- 17
4-217.1
4- 0.4
4-2:111. IP — 0.4
4-387.4
- 1.1
— 0.3400
— 5
19.01
— 3 74
— Ul.ti
4-18.8
-125 7 -214
-22:1.8
— 40.0
— 0.5152
— 4
4-
4.11
+ 1..51I
-f- 2:1.9
— 7.8
4- 27 7(1 4- 9.01!
4- (13
4- 22.0
— 1.0(12-
— 3
0.4i»(»
- 0.271
- 2 79
4- 1 :io
— .3.42 — 1.55
4- 4i!.40
4- 20.01
4-10 9273
2
+
0.03(i
4- 0.02J
4- 0.20
— 0.13
-f- 0.28 4- 0.10
4- 0.14
-i- 0.00
4- 0.944:1
7,— 8
4-
3.41
- 0.28
4- 24.0
4- 2.1
4- 25.5 — 2.5
4- 37.8
— .3.5
— 2267
— 7
4-
2i>.!l(»
— M
4-14H.4
4- 0.4
-f-lilii — 0.(1
4-245.3
- 1.3
— 0.21114
— «
12.1III
— 2.. 54
— 811 2
4-15.4
— 98 11 —17 2
— 1(13.1
— 29.8
— 0.411:1
— R
4-
3 40
4- 1.31
4- 2:!.o
— 7.7
+ 2il.O : -f- 8.x
4- 49.8
4- 18.0
— 0.(i9o8
- 4
0.4x7
— 0.27ii
- 3.25
4- i-':i
— :i.79 — 1.87
— 12.83
— 0.99
— 2.311)8
— 3
4-
0.044
+ 0.o:i4
4- 0.28
— 0.19
4- o.:t7 4- 24
— 0.09
— 0.12
4- '-7577
8,- 9
4-
2.3.-.
— 0.20
4- 18.8
4. i.«
4- 19 n i — 2.1
4- 28.1
— 1.2
— 2032
— 8
4-
12.20
— 0.1:1
4- 9ti (1
4- o:i
4-105.0 — 0.7
4-1.54.8
— 1.3
— o,2.'prio
— 7
8 M
- 1.71
— (18. !t
4-12.1
- 75.5 -l:!.l
—117.7
— 21.3
— 0.3423
_ (i
4-
2>:9
+■ 1.05
4- 21.0
— 7.1
4- 2:t.2 4- 8.1
4- 40.0
4- 14.7
- 5205
— 5
45
-0.2i
— 3.5
4- 1.8
— 3.90 ! — 1 99
— 8.8
_ 4.H _ 1. 0855 1
— 4
4-
0.0411
+ 0.039
4- o.;t7
— 0.2«1
4- 0.45 4- 0.31
— 4..50
— 3.09
-f- 12. 0958
9.- 9
4-
7.0
— 0.1
4- «3.4
4- 0.1
4- (18.2 — 0.(1
4- 9(1.8
— 1.0
— 0.2207
— 8
BO
— 1.13
— 51 9
4. 9.1
— 5(1 3 1—9.9
— 84.0
— 15.2
— 0.29:11
— 7
4-
2.117
4- o.si
4- 18.1
1(1.4
4-19 8 1 4- 7.0
4- 31.8
4- 11.7
— 4147
— 6
liO
— 0.25
- 3.4
4- 1.9
— .1.78 — 2.02
- 7.1
— 4.1 — 0.7085 1
— 5
+
0.080
4- 0.042
4- 0.43
— 0.32
4- 0.51
4- 0.3G
4- 1.72
4- 1.38
— 2.4308
10,-10
4-
4.1
— O.l
4- 43.
4- 44.3
— 0.8
4- fil.
— 0.8
— 0.2040
— 9
—
3.9
— 0.8
- 40.
4- 6.
— 42.0
— 7.0
— BO.
— 11.
— 0.2503
— 8
4-
1.8
+ O.li
4- 14.7
— 5.(1
4- 1(1.1
4- (1.0
4- 25.1
4- 9.3
— 3446
— 7
—
0.3
— 0.2
— 3.1
4- 1.9
— 3.5
— 1.9
— 5.8
— 3.4
— 5259
— «
4-
0.0ft
+ 0.04
4- 0.4(1
— 0.37
4- 0.53
4- 0.40
4- 1.1(1 4- 0.9«
— 1.1092
THE on BIT OF URANUS
fiS
The term of hmg I\ri(Ml,
From the cx[)rossioiis for tlio perturbations of Uranus, sijbsoqucntly Riven, it will
be seen that several of the terms luive very large coeffieieuls, that of sin (-/'— J/)
being nearly an entire degree. Tiie magnitude of most of the terms in which I' is
even arises from the near approach to commensurability in the mean motions of
the two planets. Twice the mean annual motion of Neptune exceeds that of
Uranus by only 30:J".8. The elements ef the orbits of botli planets will there-
fore, in consequence of their mutual action, be affected with a slow oscillation,
having a period of about 4'2(>6 years. The employment of these large terms and
the great inconveniences to which they will give rise, especially in the corrections
of the elements of Uranus, may be avoided by the device employed in the theory
of Neptune, The following arc the essential featun^s of this method:
YJrat, all the perHnbations arising from that portion of the perturbntive func-
tion in which the coefficient of the time is 2m'— n or its multiples are considered
and developed as perturbations of the elements.
Secondly, tlic arbitrary constants to be added to the integrals of these perturba-
tions are so taken that the perturbations shall vanish at the epoch 1850.0.
In other words, the perturbations in question wiil be treated as producing secular
variations of the elements of the orbit, only, instead of being developed in powers
of the time, these variations will be retained in their rigorous form.
The formuliE for the conq)utations of the perturbations in question, arc as
follows :
Let
m
m
h cos («" I' -\-il -|-/(j' + j(j) = cos iV
be any term of the pcrturbativc function, 7i being a function of a, c, and a.
sin 4, = e
g = cos 4. tan J 4'
For each such term, compute
v =
n
i'u'-\- ill
A =
2 !h
L=^
cW =
ens 4/ ^
E =
— ''('.'/+ J ">t 4)
T =
1 =
l,'^ + (/+y)a/«.
M
THE OliniT OP U RAN US.
The corresponding perturbations of tlic demonts mny then bo put into the form
6 log a = MvA cos N-\- 5»o»
il = Mi'L »ii\ N -\- i(o,
ehn = Mve IVsin JV-f efi7i„
he = MvE cos N -\- Scq,
iy = Mi'f COS N-{- fyo,
tan y 5t = Mv T sin iV -\- tun y^To.
Here, 5»)o, 3?o» etc., arc arbitrary constants so taken that 5 log a, 3?, etc., shall
Tanish at the fundamental epoch.
All the terms depending on the same values of i' and / are to be combined into
a single one. And it will save labor to make this combination at as early u
stage as possible in the computation; that is, to nuiltiply the various values of /j,
S^Jl ^JL, _' , and 27 by the sines and cosines of I'u' -t-iw, and afterward proceed
Sv 6e Sa
with the sums of the products according to the proper modification of the formulir.
Thus arc obtained the following long period perturbations of the elements of
Uranus :
— 3474.3-2 sin (3/'
+ im.T* mi {At— '2i/) —
— 8.97 sin (()/'— Sij) +
4- 0.()4sin(W— 4J/) —
4-consfant=3320".lS.
r/)-j- 180.10 cos (2/'— fj)
54.10 cos (4/'— 2//)
5.03 cos (()/'— iiij)
0.53 cos (8/'— 4fj)
eSn = — 484.9() sin (2/'— //) +
+ 38.00 sin (4/'— 2J/) —
— 3.01 sin (r)/'—3</) -f
4- 0.33 sin (8/'— 4.7) —
-f constant = 405". 23.
0.73 cos (2/' — g)
7.0() cos (4/'— 2y)
1.38 cos (()/'— 3.7)
0. 15 cos (S/'— 4//)
= _ 484.21 cos (2?'— «/) —
+ 38.21 cos (4/'— 2j/) 4-
— 3.01 cos (0/'—3f/) —
4- 0.33cos(8/'— 4r/)4-
— constant = 158". 59.
3i)=4- 2277 cos (2r— .7) +
— 198 cos (4/'— 2)/) —
4- 18cos(G/'— 3(/) +
4" constant =: 630.
0.29 sin (2?'— .7)
7.10 sin (4/'— 2j/)
i.40sin(fi/'— 3.7)
(I 15 sin (8/'— 4(/)
120 sin(2r— g)
78 sin (4/'— 2.7)
7 sin (0/' — 3.y)
The variations of the elements wliich fix the position of the jdane of the orbit
arc here omitted, because their nature is such that it is indifferent in which form
they are developed.
These expressions are reduced to perturbations of the co-ordinates by the follow-
•-
'
THE OUDIT OP UTIANUS.
67
ing forniultr. Express the usunl dcvcloitmcnts of tlic longitude nnd logarithm of
radius vector in the form
« = ? -|- 2 V, sin i(j I
p = » -|- 2 ii, cos iff.
Put also
e .'
;„ OR,
oe
ir, = -' R,
Express any set of corresponding terms of the preceding perturbations in the form
}l z= L, sin N-\-L, con N
e}.l — chn = F. sin N-\-F, cos N.
le = Eo cos N-\- E, sin N\
hv = A, cos iV+ -4, sin N.
Wc shall then have
25r = S ( r\ F. + V\ E,.) sin (A' + »;/) + 2 ( V", F. - V, E,) sin (iS^- ij)
+ V ( V\ F, - V\ E.) cos (iV+ iy) + li ( V'\ /'„ + 1"« ^'.) cos (A'- ig)
+ 2ii
2:V = 2 (/?, E„ - /?", /^,) cos (A^+ ig) -\- 2 {K, E„ + iT, /-.) cos(A^- vj)
+ V (/^, E. + 7?", /',) sin (Ar+ tV) + 2 (/f'< E. - R\ /',) sin(A'- ij)
-\- 2)v
The numerical values of T", F, /?', and liT arc as follows :
r, = 1.098:35
F3= o.ini;}
F,= 0.00714
F4 = 0.00044
i?'.,= + 0.0'i;)48
jT,= — 0.90703
i?*, = — 0.070'20
71"!.= — 0.004GG
F', =
F'3 =
F'4 =
1.99945
0.11722
00714
0.00044
i?', = + 0.99917
Jl\ = + 0.07030
i^a = -j- 0.00467
The final results of the entire computations arc given in the following table:
In the columns ii;„ wc have the complete perturbations of the longitude computed
by the direct method from the values of M, ^j^ , (?, etc., already given. Next
wc have, under the caption t'r,, the perturbations of the true longitude deduced
from the long period perturbations of the elements, as set forth in the last para-
graph, omitting the constants added to the perturbations. Under hv^ we have the
8 April, 1873.
08
THE ORBIT OF URAXUS.
perturbations of the longitude deduced from all tlie remaining terms of the per-
turbutiuns of the elements. The sum of the columns ii'j and hvj shows the entire
perturbutionr, computed by the method of variation of elements. Thus, in })Vi
and hv-i -f- ^^3 we have two complete sets of perturbations computed by methods
entirely independent. The diftcrences of the results, expressed in thousandths of
a second, arc given in the last two columns of the table.
Tliis comparison gives rise to remarks similar to those suggested by the per-
turbations of Saturn computed by the same methods. The only terms in wliicli
the difference of results amounts to w much as one-tenth of a second are those of
very long period, and those very nearly the period of Ij anus, wliere a mon; accu-
rate vaiiK> is not at present of great importance, because the error will be com-
pensated by the corrections of the element during several centuries.
Pertuhbatioss or tii«
LoNOITUDE OF Uh.\NUH PIIODUCED BY NeI'TUNB.
poiH-y.
<
"i
ai'.
«i
J
Discri
f 'J
sin
cos
sin
cua
sill
cos
sin
COS
0,
— 0.4 202 <
—1
»
on.') lit
— 1
.isl m
—a
—
004 (1/
—0.009 nt
0,
_\
+
O.fittT
—
o.oss
—•2
+
o.o4(;
—
0.005
—i
+
o.ooa
1,— T
-f
0.147
0.001
....
+ 0.140
—0.002
1
1
2
+
2.r)0i»
—
0.01 !)
....
4- 2.509
—0.010
9
— 1
+
3!t.(i5S
—
o.oso
....
....
+39.073
—0.0.^1
15
1
+
4. 2;-) 7
—
o.r>ii
+ 4 249
— 0.47S
S
33
—1
+
0.2S0
—
o.o;i2
... *
+ 0.275
—0.032
6
-2
+
0.017
—
«.002
2,-4
+
2.ft7S
+
0.0 Jf
+ 2. so
-f 0.03
+ o.oos
—0.002
10
1
:I
+
41I.01.5
+
o.4l:i
+ 47.2:'
+ 0.4 4
+ 1.797
—0.021
2
—2
+
H40.!t;{
+
7.:1HS
-f HO5.04
+ 7.43
+ 35. .355
— .1.028
0.5
14
—I
—347-). 4
+
ISO., 'to
-3474. 32 +1S0.IO
- - 0.700
+ 0.095
3S0
ir>5
—
i(;2.o7
+
S.OI
— 101.90
-f S.OI
— 0.007
+0.015
43
15
—1
—
9.447
+
0.408
— 9,50
+ 0.47
3,— :>
__
0,070
0.000
...
— 0.077
-0.003
1
.T
—4
l.ir.j
0.000
■ . , ,
— 1.1,53
— O.dll
9
11
—a
17.2S(!
+
0.0-2S
....
— 17 2H5
+ 0.2-29
1
1
—2
—
22.0S5
+
4.o:i7
....
-2-2.077
+ 4.(t-J0
H
17
—1
o,(i7:{
+
0.0S2
• . . .
. t • .
— O.0S2
+0.079
9
3
—
0.0U7
+
0.000
4.-fi
.
0.0:50
+
0.00-2
— 0.01.)
-f O.OOv
— 0.027
—0.003
.1
—5
o.ftr.s
+
o.o:i7
— 0.25
+ 0.04
— 0.315
— 0.007
7
4
—4
— .
7.!tr,8
+
0.7')0
— 4. OS
4- 0.08
— 3. 90S
+ 0.059
20
11
—3
..—
7.'>.oo
+
12.s:i2
— 09.55
+ 11.07
— 5.7.13
-+I<I07
2S3
95
—2
+
14fi.7S
5I.21S
f 140.72
—54.10
+ 0.1-20
—0.079
00
39
—I
+
6.900
—
2.579
+ O.Sl
— 2.03
u
THE ORBIT OP URANUS.
69
I'EKTUKUATIONS of TilK
LuNuiTUDe — Voiilinueu
•
««,
««,
tv
1
tiiacrc
>ancj.
i' 9
sin
//
COS
Biti
cos
siu
COS
sin
COB
b.~ 7
—0.000
0.000
....
—0.015
—0.002
C
2
— (!
—0.103
+ 0.001!
, . , ,
—0.113
—0.004
10
10
— 5
— 0.!»«(!
—0.043
....
—0.004
—0.042
K
— 4
-f3.;jfifi
—0.(170
t . • •
+3.370
— o.(;(i2
4
8
— 3
+3.210
— l.lfiO
• . . •
+ 3.227
— l.lHfi
17
17
— 2
+0.077
—0 017
....
....
+ 0.075
—0.002
2
15
— 1
-fO.OOD
—0.001
«.— 7
—0 050
-0.004
0.00
0.00
—0,054
—0.002
4
2
— fi
— 0.3«7
—0.020
+0.02
—0.01
—0.423
—(,.013
in
3
— 5
-fl.2fil
—0.320
+ 0.40
—0.14
+0.S.55
— o.i(;o
«
11
— i
+ 7.7HI
— 2.S25
+o.«o
—2.54
+0.S57
— 0.3IH
124
33
— :i
—0.025
+5.073
— S.07
+5.03
+0.017
—0.022
72
C6
—0.437
40.240
—0.42
+ o.2i;
T,— «
—0.027
—0.002
— 7
— o.isn
—0.003
....
—0.180
—0.009
3
fi
— fi
+0.272
— o.om
+0.2(iO
—0.047
12
1
_ f>
—0.571
-< 0.217
— ().53S
+0.202
33
15
— 4
—0.450
+ 0.250
—0.410
f 0.255
40
5
— a
—0.010
-I 0.005
«,— i»
-(t.(ll3
0.000
— H
— 0.0!(4
—0.002
— 0.003
— <> 007
5
— 7
.) 125
—0,022
-0.115
- •24
10
2
— fi
—0.104
■\ 0,OM2
— 0.03S
4 0.017
—0. 1 45
+ 0.055
11
10
— 5
— O.H30
1 (»,4t'.3
— O.fifi
-t-0.30
—0.110
+ 0.007
(19
06
— 4
+ O.G47
—0,501
-fO.«4
—0.53
9,— It
—0,040
—0.001
— 0.04^^
—0.005
8
4
— H
+ 0.0fi3
— O.Oll
• • • ■
+ 0,002
—0,011
fil
— 7
— 0.055
+ 0.020
. . . •
• . • •
—0,050
-t 0.025
4
5
—
+ 0.OHO
— o.o4;t
• . • •
+0.0«3
—0,054
3
5
_^ r
-f 0.050
—0,050
10—10
—0.020
0.0(»0
....
—0.025
—0.005
5
&
—
4 0.035
—0. 005
— 8
—0.027
— OOIO
— 7
f0.02ti
—0,0 IT
— fi
+ 002
—0,070
Thoporturhatioiisof thol()p:!>''ith"isof the radin.s vrctor nro givon in nform similar
to thoso of the loDf^itiulc. lliwlcr ^p, we hiive Mio coinpUat' perturbution. Under
^p.j the I'ffoct of tlic perturbations of long period. But luuler ipa wc have only
the difference between ^p, and ^p„ it being deemed nnnece-ssary to present in full
the perturbations of the radius vector as computed by the other method. Ap, beinir
employed in computing ^», may, in fact, be regarded as completely checked ty its
affording a correct value of the latter.
In the last two columns f^p,. is reduced to common logarithsns by multiplying the
coefKcieuts by the modulus 0. i;}4'.'!)4.
60
THE OUHIT OF URANUS.
I'KKTL'KUATIU.NS Ut TIIK LodAlllTIIM of TIIK KAIHI.S VKCTKIl OK UlU.Nl 8 IMUH-I CKU
IIY KkI'TINK.
f
0,
1,-3
2
— 1
+ »
2,-4
— ;i
2
—I
3,— ."S
—4
_;{
-J
—1
4.-fi
—4
_4
_3
— r.
—r.
—4
_3
7,-7
— f.
— 5
—4
8,— s
—7
—0
—5
9,-9
_s
—7
— ••>
»P.
-f i:i8
+ 4
-f fi8
+.')'2:i
— (IS
— 8
+ «4
+i4i(;
+-.»(i02r)
-I IC.fiS
4-3!) 1 -J
— 3
— 3S
—■.•-> 7
— 1>S4
-I ir>
_ 1
— 10
—•.'.".4
— nr.tt
— 143
— 1(17
— 4
— 40
-1-103
-f 3'J
— 17
+ ■«•-'
-f ISO
+ i:5
— s
4 11
— 17
- 4
-t- ;i
— 7
— 19
2
4 ■■!
2
+ a
+ >
— c.
— 1
— 1
— \i
— 179
-I IK!
+ 194
— (J
—at
4 2
— 1
oo
— :ioo
— 01
— (■.-'
+ >
+ -.MI
+ 1&
4- 1
4I«»
+.ir.
+ «
4 i
— 7
— 3
4- 1
— 3
— II
— 1
+ a
*p«
-f 92
+ 1374
-1-19190
-f 1720
-f 39J7
1
tf.
Hill
cort
— 1
+ -'
— 12
4 •«•-'
— ISO
4-r.35
4 120
— 03
4 194
— ir.
s
— lis
— KiSi
— ir.4
4- 10
4-1C.4
4- H
— 20
'.>><2
— 70
— (U
4
01
— 11
— 13(i
— 7s
4- II
17
32
I(i
1
+ «
— 4
Mif
— I
2
— IS
4 '••
+ a
+ I
4- 6
4- *
-\- 3
1
COS 1
4-
00
4-
2
4
30
I--
27
30
—
3
+
I
4-
IS
L. •»■!.> 1
27
7
1
17
— ;
!29
—
24
+
7
r.
_
ri9
34
4-
1
17
•f
45
4-
17
— 1
— 7
+ U
4 7
— 3
(■ ■''
— 7
+ I
— 1
4- I
2 ■
3
« i
I
1
— 3
U
2
U
— 3
—23
4 1
— 1
— H
4- 4
4- I
4- «
-f (i
4 :i
4 a
4- I
(I
_ I
— 5
4- »
I (
u.
• -T^'iHifiiiflfrarltiiMf'W
1
i
1 1,
T
A
u
■)
THE OKBIT OF UUANUS.
61
reilTI.KIIATIUNH ur TIIK liATlTl'DK PUODL'CEI) BV NciTUNB. 1
dit >»'
J '* (liD .V
2f
I ('■+» r
'/"J
vr "i
XA "iii.V
J=«'.„X
ik
«T
tfl
1' 1
COS
liu
■Id
OM
Hill
ooa
ilu
001
•in
CM
CO.
■In
//
sin
r,
— 21.G5
0.00
-.012
1
+ l.ll^,^-2.7:l'— 4.0(1— l.tiUf .01— (Mil
— 3.!>ll— l.r.7i— 0.008
+0.021
+0.030
—0.012
+ .080 +.020
2
+ li».32— U.3!)+1<j.32, + 1).:J1»— .01
II 1 1
+ 111.31 +11.31)
— 0.073
— 0.03ti
—0.073
+0.(lJ)i
+ .01(i— .002
1,-3
+ 14..'.^+7.0 Ll4.7 1+7.11 1 !
-14 7 +7.0
+0.045
—0.022
—0 045
-0.022
+.0(17 —.(102
'J
— 1.1« --J.liO -f 4.li4—'J.40— 0.07+0.02
+ 3 1)7 2.4
— 0.( (1(1
+ 0.014
+0 (120
+ 0.012
+.201 —.04(1
—1
—34 2.1 1 1 +1.7tJ
+ 1.7(i ..
-0.540
+ (128
— (108 — (i:(7
+ :i.74 +'J.:i2— 2.H0 -- 1.48 -0.24
— 3.(14-1.48
- (1.057
+0.o3.'i
+0 04il
—0.022
+ .3(i4+.(if4
+»
+ 1S.77— 7.h7+ll).20+7.7i
+l(i.2i +7.75
1
—0.081
— (1.040
—0,083
+0.040
+.OU(i +.007
1
2,-4
+ 10.52+.1.0(!— lO.K '+6.0 !
-lO.tf +5.0
+0.027
—0.013
—0 027
—0013
+ .002— .001
—3
— 2.tiO— 2.38+ 3.(ji— 2.28 +ii.3ii+ 0.04
+ 4.01 -2.24
— 0.010
+ 0.(1(1!)
+ (1 Olil
+ 0.(8'!)
+ .0(j2-.(il4
—a
— 3:l.;j»' 1 > -f "J.^.s —ii.dl
-I- D.i-M — (i.dl _ (i.-i;4
+ ('78
+ .008 — (104
—1
+ &.M'+2.04— 2.12—1 04 — l..');i+0.04
— 3.115 - 1.(10 t— 2.283) [ + 0.K12] t + 1.431) [-0.31)3]
+ .32(1 +.075
u
+ 18.7U — a..V2 +l«.5li +i>.;i2+0.08
+ ll).(i4+l).32
1
-0.143
—0.072
— 0.04U +0.071
1
+ .002
3,-5
+ 7.4 '+3.ii ! - 7.4 +3..'. ;
- 7.4 +;i.5
+0.017
— O.OOH
1
— 0(117 —0(108
+ 002
—4
— 3.011 _2.0ii+ 3.12 2.0 +0.48+0.05
+ 3.(!() -2.0
— 0.010
4-0. oou
4 0(111
+ OI8l(j
+ .oi)_.lli4
—3
— 21.4 j 1 10 +7.87
+ 7.87
-0.128
40 041
-.(K^+.(l4l
•1
— •*
+ ii.5h+|.7.l— 1.20— 0..14— 1.08 +o.-J(!-_ 3.18—0.28
+0.108
— 0211
- (153
+ 0(104
+ 303 +.074
— 1
+ l:l.ii(; —7.28 +14.tih +7.(KI +0.1^ — 0.04 —14 8il +l!.!lii
—0.1 111)
— 0.1(1(1
—0 217
+ 0.101
— .{190 — .icu
— 1.61 +0.22 — l.Sl —0.22
1 1 i 1
— l.il -0.22
i
+ 0.008
+0.001
+ o(Ki»
— (1.001
— .013 u
1
4,-5
- 2I». -1.(13+ 2.53 ^1.7 fO..'.
+ 3.0 -1.7
— (I.1K>8
+0.(814
+ (08
+ 0(14
4-. 022 —.004
—4
— 17.4 1 +''-8 '
+ 5.8
— 0.070
+ 023
—Mill- +.(Ki(i
—3
+ (1.;U + I.34— O..'!!) —0.22 1.07 +0.:10 — 2. .'XI +0.08 -(0.(151
— O.OII
(1 (I'ji —0 001
4-.(l4^ -i-.olO
-i
+ 11.42- .'..37+10 57 f 5.04 +0.25 0.07 f 10.82 +4.117 I 1.8Vj) ( l (i5i;| l~2 1J71 l+o 1(77)
— (147
— I
— 1.72+0.32— 1.72 -0.32—0.112-1 '(.ua _ 1.74— o.iHi
1 1 1 1 1 ' 1 1
+ 0.013 +0.(KC
+0.013 1 — 0.C02
— 001
8,-e
- 2.4 — 1.4o'+ 1.8 - 1.4 +0.B + 2.3 - 1.4
—0.00.11
+0.0(13
+ 0(105 ' +0(103
+ O12-.003
—5
— 12.(1 (1 1 U (1 -f-4.0 1+ 4.0
— (i.(i:m
(I
+ 013 j
—.012
-4
+ 5.5h +1.(11- 0.22 -11.04 - 1.75+11.28- 1.117+0.24
+ (i.o3(i
- 0.(8(5
- (1 olll —0 (Nil
— 1(0 — (128
-3
+ (1.25— 3. hU+ 7.4.1 f3 5l+0.30— (i.('ll+ 7.73+3.45
+('.1('7
+ 0.0(17
+0 l:ii
—0 o51(
— (i;*4. i,(il
-2
— l.li(i+0.33— 1. (id —0.33— 0.(13 +0.01 — i.iiU— 0.32
III i
+0.024
+0.wi
+0.(iJ6
— (104
+ 018 (1
(J,-6
— 8.4 : n ' n +2.8 + 2.80
- 0.022
+0(Ml7
-did .(-.(812
— S
+ 4.(13 +0.7(1 — 0.03 4-0.04 - 1.4(1 +0.24— 1.40 +0.28
+0 011)
—(1.(8)3
— (Kill
- 0(01
— (128 — .(H18
—»
+ 4.(13 - •,;.71( + 5.15 +2.45 +0.31 —0. lo + 5.4il +2 35
+o.o3t
+ 0.(122 +0 045
—0 (111)
— Old
—3
— I.4J +0.31 — 1.45 —0.31 —0.04 -(-0.02 — 1.41) -0.21i l+O.llHi) [ + 0.(141] l + " IW] I— 0.038 J
+ .015^
7,-7
— S.(!
1
1 +1.1) '+ i.n
-0.012
+0.(^4
_.(Kl6+ dOl
— tf 4- .•;.ii:l -f (i.iii
—1.17 +0.2(1— 1.17+o.-'(i
40.('12
- (1.002
— O 001
—0 (ml
_ (108 — ,(KI4
-8
+ 2.411—2.(10+ 3.52 + 1.08 +(1.30 —(i.lo-(- 3.82+1. 5«
+ 0.013
+ (i.on
4'i.i2ii
— (Hih
+ .014
-4
- 1.111 +0.27— 1.1'J -0.27 -0.04 +0.02— 1.23.-0.25
-0.021
-(1.(05
—0.022
+ 0.004
+ .(8)2,
1
8,-8
— 4.0 ! +1.3 ' !f 13
-0.(818
+0.003
— (Ki3 +.002
—7
+ 2.711+0 50 —d.ltl +0.1(1
- O.lil +0.1(1
+ 0.007
—0.002
+ (1.1K12
_,(Ki.i— .ooa
— «
+ 1.411-1.41+ 2.3k +1.14 +0.27 -o.O!"
1- 2.(15 +1.05
+ (l.l«lli
+ (l.(Nl>i
+ 0.011
— 0(104
+ (814
—5
— 0.1(7 +0.22— (i.a7 — 0.22 +0.02— 0.U7— 0.20
1 , . 1 1
-0.(818
-0.(8)2
—0.(8(8
+0 (Ki2
+ (8)2
1
b,fular Utriii, M =4- I "25 7'
H = - 1 ■ -i:, r .-.HI T
Tlio terms iV.- niul fir;, wlii( h iirc inclosed in linieKets, are of very Ion;; period,
niul arc tliiTeCore omitted in I'urminy tiie valius of yj in tiie lii>t two (ohimns
«9
THE ORBIT OP URANUS.
Ihrlu)/ia!ioit8 praluccd by Jupiter.
The scries in which these pertmhations are expressed converge so rapidly that
I deem it unnecessary to present tiie details of the computation. Thoy have been
computed by both methods, and the separate and independent results are given in
the following table, where ht\ rcj.. scnts the perturbations computed by the method
developed in Chapter I, and it „ those computed by the method of varia.jn of
elements.
The apparently large discrepancy between the coefficients multiplied by the time
arises from the circumstances that in the form of development the mean motion,
and hence the mean anomalj ^j)pears affected by the perturbation fiV'.Ql. Accord-
ingly when we enter the table which gives the true longitude in terms of the mean
anonudy in the form
r = ? 4- 2c sin (/ — «)-{- etc.,
we may consider this quantity 31".2< as a secular variation of / — n produc'ar in v
the term
it)=^G2'.4c^cos(/ — 7t).
In rr, this term is left in its primitive form, while in h^^ t''f" value of/ is supposed
to include this term, and the secular terms are only those which arise from the
secular variation of the eccentricity and perihelion.
It is also to be remark«'d tliat the terms which are inde])endent of the mean
longitude of Jupiter, or thos(> in wliich /' = 0, are not comparable, as they corre-
spond to slightly diilurent < lliptic elements ia tltu tuu theories.
th
THE ORBIT OF URANUS.
63
I'EUTIRUATIONS or UllASLS BY
JlI'ITKR. 1
»y,
t
f.
Uiff.
C08 ^tf
3l»p 1
9 f
»iii
COS
sin
COS
sin
1
COS
coa
sia
cuu
sin
0,
1
2
a
— 0. IC.Od/
— 0.010/1/
+ai.2ii(;<
+a7.5S5n/
+ 2.207/1/
+ 0.135/1/
— 0.H;22/t/
— 0.00!t5/i/
— O.OOOti/i/
+31.1ftH2/
— 1.540(;/i/
— O.OSOO/i/
— 0.0054/1/
a
0,
2
a
+ 25.«57
+ I.a!l7
+ 0.073
— i.8.59 — l.aVr,
_ o.o«7 — o.oao
— 0.005
+ i.3(;i
+ 0.080
...
— lOOS!)
—401.5
— 10
—4387.
— 213.7
— 14.2
— 0.8
— 0.8
+ 0.5
j
— 32.8+ 1.1
— 1.8|
—1,-1
1
2
a
+ 0.027
+ 1.2(!9
— 5a.oc.'
— a.4;t'i
— 0.14^
+ 0.00!) + 0.017
+ 0.002 + 1.232
. O.OOJ — 53.0S4
.- 0.0112 — a.5tl5
— 0.047 — 0.104
1
+ 0.011
+ 0.001
— 0.002
— O.OHJ
— 0.050
10
37
20
70
in
2
1
2
10
a
0.0— 0.5
_ 5«.H_ 0.5
+ 25H5.7 + 0.1
+ l!)5.a+ 5.0
+ n.\+ 2.3
0.0
— 25.0
+ 1127.2
+ 82. s
+ 7.4
— 0.2
— 0.2
+ 2.2
+ 1.0
0.-2
1
SI
a
4
— 0.027
+ 1.1S2
+ 0.277
+ 0.074
+ 0.015
— 0.011 —0.031
+ 0.515 + 1.170
+ O.oao + o.2r,a
— 0.005 + 0.0X3
— 0.003 + 0.014
— 0.014
+ 0.515
+ 0.037
— O.OOS
— O.ooa
4
I'i
a
1
3
+ 1.4— 0.0
— 50.5 +24. !•
+ 8.5 + l.H
-j- 4.2 + 0.5
+ 0.0 + 0.2
+ O.f.
-24.0
+ 3.7
+ 1.8
+ 0.4
— 0.3
+ 10.7
+ l.C,
+ 0.8
+ 0.2
1,— a
2
.1
4
5
— ()a2
— 0.005
+ 0.025
+ 0.01 1
+ o.ooa
_ 0.034 — 0.025
_ 0.005
+ 0.015
— 0.001 + 0.037
— 0.017
— 0.025
— 0.001
— 0.010
+ 0.004
7
10
2ti
20
!)
1
n
*
I'KIITIRII.^TKINS OP TIIK LaTITI l)K. 1
U I'
ih
»1
tfl
t l'
2
a
—2—1
— 1
1
2
— ;v-2
2
Hill
'/
1 0.071
—0 012
—O.ooa
—O.OOfi
— 0.2!t7
—0.1 CI
\2.i\\\
4 0.070
4 0.055
+ 0.002
—0 no
— 0.014
COS
'/
—0.011
—0.075
— (».0I2
40,012
4 l.!»l!)
— 0. 101
+ tMI21
—0.088
— It.Oll
-U.O.'.O
COS
//
40.030
— 0.012
— (».003
40.000
4(».2!i7
— O.IS'.t
4 2. "128
-j o.oco
—0.055
—0.002
—0.10!)
—0.011
Kill
'/
4 o.oac.
) 0.075
40.012
4 0.012
-1 1 '.no
— 0.1 !I0
—0.002
_0.0><8
—0.011
1 O.OJfi
C(I8
//
-1 .oi;o
— .010
—.002
-( .005
4.001
— .4'.14
— .021
+ .002
4 .010
■\ .017
— .00!t
—.002
sill
//
-4 .or.5
4.047
—.007
—.004
—.003
4 .420
— .013
4.019
—.003
—.004
( .002
Sr.uhir l.riiis " *'■ \\\Z
««,) — 1.14 1 cos V
64
TUE OKUIT OF UUANUS.
CIIAPTKll IV.
TERMS OF THE SECOND OltDKll I'UOUUCKl) IJV THE ACTION OF SATURN.
PrcVr.uimiry Inrrstiijalhm nf the Orhil of Sulnrn,
Tou tlir" arourntr (letcnnination of the pcvtiuhatioiis of a planet it is essential
tlia'c tlie fiinctior.s of the time which are suhstitiitcil for the co-ordinates of each
]>la u't in tlie expression of the clisturhing forces shoultl approximately represent
the true places of the planet. The tlilferencc! between the true place an<l that
implicitly assumed in the investigation shoidd be so small and of such a character
that, when multiplied by tlie mass of the disturbing planet, and by the factors
introduced by the process of inti-gration, the result shall be insensible. If one of
these factors is so large as to make a perturbatiim of an order of magnitude api)roxi-
matiug tluit of the inetpiality which gives rise to it, it will represent an inecpialify
of very long period in the elements, which, though apparently sensible, may be
neglected for a great length of time.
The pertiirbations liitherto found have been computed on tin hypothesis that
the disturbing action of Saturn on Uranus is the same s if both planets moved in
the elliptic orbits corresponding to the adopted elements. "We luue given formulic
for the computation of the corrected perturbations when, to the co-ordinates of the
two planets corresponding to the adopted ellipse, wc add corrections represented
by h', 5r', fip, etc. These corrections are now to be taken of such magnitude that
when thus added they shall very nearly represent the actual motions of the planets.
Generally, it is considered sufficient to take for these corrections the pertiirba-
tions of the first order. But this presupposes that the elliptic elements are nearly
correct, which does not hold true in the case of the old elements of the outer
planets. Uouvard's Tables of Saturn, the elements of which have been adopted,
arc subject to recurring errors aint)unting to ;}()" or more. Moreover, when we
siibstitute the new and more accurate perturbations for the old and imperfect (mes
adopted in the tables, the chances are that the errors will be increased. Desirin^
that the theory shall be as far >is possible free from doubt, we begin with a j)rc-
liminary investigation of the orbit of Saturn, the design of which will be to give
the co-ordinates of that body in terms of the time with sufficient certainty and
accuracy to serve for compiiting the perturbations both of Jupiter and I'ranus,
As usual, the first ste]) in thi* investigation will be the determinations of the per-
turbations of the planet.
•>»
M 9
THE ORBIT OP URANUS.
w
Oeneral Peiinrhationa of Saturn.
TIio perturbations produced by Jupiter will be taken from the exhaustive prize
memoir of llunsen.* As the perturbations required are those of tlie co-ordinates,
it will be necessary to transform those of Hansen into tlie usual form. Hansen
gives the true anomaly v in the form
V = tj -\- nh -\- «?i sin (</ -|- mc^z) -f- "i si» 2 (</ -\- nh) -\- etc.,
f-nfj, etc., bcin<; the coefficients of the multiples of the mean anomaly in the usual
development of the elliptic true anomaly. Whence, neglecting the second power
of Mi'z,
hv = »t!;; (1 + Pi cos g -\- 2ej ros 2(/ -}- etc.).
To make the development sufficiently rigorous it is only necessary to increase <] by
I nhz in this expression. In the same way, we have for the perturbutions of log r,
hf = ^p„ -j- w'z («'" sin g -\- c'-' sin 2 «/ -{- etc.)
^Pj being Hansen's perturbation, and c'"' the negative coefficient of cos iij in the
development of the elliptic log r.
Hansen having adopted {otTr.ii "" t''*^ mass of Jupiter, it will be necessary to
multiply his perturbations by l.O'iUi to reduce them to Dessel's mass. Thus the
perturl)ations by Jupiter hereafter given have been obtained.
The perturbations by Uranus and Neptune have been computed by the preceding
general method, and are given in the following talde. In the table I' is the mean
longitude of iIm; disturbing planet, Uranus or Neptune, counted from the perihelion
of Saturn. f\^ is the ixuturbation of the Naperian logarithm, in units of the
seventh place oi'* decimals.
(iKN£ll.VI> I'EKTtaOATIU.NS Uf TUK LoNOITUne IN ORBIT AND TUK LuUAUlTIlM UlT THK RAIllLS
VECTdll or Satlr.n.
Ac
ion of Ursi
III*.
I
f <J
Av'tioii of Nept
Hue.
Hill
P
,«
_ - 1
'■ <J
I'OS
Hill
Hill
cua
cod
Hill
1,
—1
-1- o.ss
-Y 8.(iO
+ 0.92
+ 0.21
— 12.7
-f 146.(1
+ 7.9
— 3.5
1,-0
— 1
+0.23
+ 1.93
—0.05
0.00
— 2.2
+.39.9
—0.2
0.0
2.
— 1
2
— 0.42
— S.1',1
— l:i.:{it
— 0.17
— 2.5:{
— 0.25
1- 10.2 1 — 3.7
— IJti.S -1-21.4
— .•i!)2.5 4 '■'•♦•
2.
— 1
—2
-1-0.44
—1.11
-1.27
+0.02
+0.02
0.00
— 3.0
— 1S.5
—41.1
+ 0.1
—0.4
0.0
3,
~2
— ;i
f O.Ofi
— i.'^
— lo.ci
— 2.05
1- I. (12
i-2s.|
— 20, ss
— 1.47
■{ 0.5
f 10.. s
—247.
- (17.5
-1 :t9.1
+.!:i.O
+ 4'<7.
+ 42.8
3,-1
-2
—.1
-|0.02
1 0.1(1
—0.10
—0.03
—((.04
0.00
+ 0.2
+ 3.2
— 3.9
+ 0.4
+ 1.3
0.0
4,-1
-2
—4
+ O.Ofi
— o.r,2
f o.:i()
— 0.2ti
4- 0.05
4- O.MO
+ 0.7:1
f o.o;j
— 0.1
— 10.5
+ 10 5
— 10.fi
+ O.fi
— 12.5
—23.4
— 0.8
' UnfersiichiinifcM iilwr die gegi'iiseitigon StoruiiRPn di's JiipitcrH uiid Siitiini.s. Von P. A. Hon-
sen. Berlin, 1831.
9 April. I87S.
66
THE Oil BIT OF URANUS.
I have Hiibniitted tli(<s«; perturbations to such duplicate rompiitiitions niul otliri
checks us lead ine to believe that none of the terms can be in error by more than
a small fraction of n second, but, us they arc not intended to form the basis of a
definitive theory of Saturn, I do not vouch for their absolute precision.
In this provisional correction of the orbit of Sutuni only heliocentric longitudes
liave been employed. These were ilerived fttr a series of dates from Airy's rechic-
ti«m of the Ureenwich observations, the modern (Jrcenwich observations, and the
Washington observations.
For these dates the value of n^z for Saturn wns computed from the formulir
found on papes 189 and lJ)()of the work of Hansen, already quoted, omittin;; all
terms less than 1", and including only tenths of seconds in the results. The dates,
the resulting values of nh, of the factor <-, cos («/ -|- J iifz) -f- 'ie-i cos 2 (y -|- | uJz),
and of the concluded tv are as follows. The formuhe for Iv is
1 .()'.»1(> »»'3 j 1 -f e. cos (i/ + J ii'z) -f •ic, cos 2 (i/ + J nh) j .
,«r=1.021(>
n'z
Pfttn
Or. M
■uii Nuoii.
17 Jl
May
:n
n-)7
Aug.
7
17.VS
Aug.
27
17(51
Oct.
G
17();l
Nov.
1
17(!.>
Nov.
2:{
177:$
Feb.
2()
1780
May
24
17!)4
Nov.
k;
180'>
Feb.
2:{
is-.»;j
Nov.
l:{
I8;ii
F.'b.
18
18:}8
May
1!)
184.>
Aug.
17
l8.-,i
Nov.
l.»
1S()()
F.b.
14
18G7
May
l.j
nd
Fiu'(or.
m
— 1792.7
— 20:i*<.2
-215;j.2
— 2(»(54.5
-:no7.l
— ;{r)()4.()
— .'ir).)7.2
—2(540.7
-;i7:n.!)
— :M2').r>
— ;J0:J(5.8
-;5(571.7
-2777.9
—2220.9
— :{l."»9.:l
— ;j4(58.;i
— 2227.G
The perturbations by Uranus and Neptiuie were computetl from the values of
their terms just given. The principal terms, the sum of which make up tlie helio-
centric longitude resulting from the adopted elements, are shown in the first of the
following tables.
In the next table we have after the date the heliocentric longitude from Bouvard's
Tables, as deduced from the longitudes given in Airy's reductions of the tireeu-
wich Observations, from the Anhonomi'm/ioi Ju/nlnwh for 18;ll, and from the
Nautical Almanac. Then follow the corrections, roiighly deduced from ob.serva-
tions made near the opposition. Adding these columns, wc have the longitude
1947.7
—.0990
21;J4.2
— .()Go2
2212.')
— .0474
2"»4(5.->
+.0244
2880.;}
-f.0729
;j()9").l
-{-.1(>82
;j;J42.o
+ .0419
28.»S.2
— .09->G
:W21.I
+ .1017
:51S4.7
+.0529
27l(5.;J
+ .0944
;w78.-i
+ .0(5:59
297(5.7
— .08(5(5
2:J42.7
— .0721
2847.0
+ .08(5;$
:}l(il.4
+ .0740
2;n:5.i
— .081*i
,.
Tin: ouiJiT OK uiiAXua.
•7
from ubserviitiuu. To the ri^^'lit of thoHu uro tlie equations of condition for llic
correction of tiic dcnu'iits.
IWlurbttioim lijr
I.OIIK. of
I'lirilivUou.
M.'i>n
■ijiiiualjr.
Ei|nali<in of
uviitra.
R«il. to
Kcllptio.
Nuts-
tlou.
TruB
lougitudo.
Nwp-
tuu«.
II
Jupiter.
/ II
UrtDui.
II
O 1 II
O » II
/ 1'
/
II
O 1 II
8S 41 27.5
150 55 50.1
-1-2 3 5(1. fi
—20 52.7 ■
-45.7
—2.7
+ 1
»n.9
+ 15.8
250 12 25.8
SS 4<> 41.2
2311 13 7.1
—5 7 38. 1
—33 58,2
— 3(l.(i
+ 1.0
— 1
20.5
—11.8
310 Hi 4.1
H.S 47 :il.4
248 25 53.1
—ft 48 30.4
—35 53.2 1
-35.2
+ 17
— 1
35.8
—15.0
330 4(1 30.(1
8S 50 7.«
28(1 211 20.8
— ti Itl 1.3
—44 24.(1
- 3.3
+ 2.0
— 43.5
— 14.1
8 15 13.5
8H 51 51 «
311 44 13. ti
—5 5S.2
—52 37.1
f 11.0
+ 2.4
+ 43.3
— 4.9
34 43 22.0
Hs5:t;j.j.l :i;if, jV) aii.i _2 41 lo.ii
— i58 24.0 ,
f 10.2
+ 1.0
+ 1
3tl.(i + (1.7
tl2 12 1.0
8H 5fl :«'.».«
(15 40 2.3 -l-t! 35.0
-50 17.2 -
-32.0
—0.8
— 1
37.1
+ 4.8
150 38 54.0
8!> 5 4:i.4
154 8 4.8
+ 2 37 57.7
—44 0.7 i-
-53.8
—1.7
+ 1
37.5
—14.5
245 8 13.2
8!» 17 50. 7
331 11 10.4
—3 18 5.2
— tI2 17.0 + 0.0
+ 3.5
+ 1
30.0
— 14.1
50 5 7.9
89 2:» 55.7
50 5tl 17.0
+5 44 4tl.fi
_57 5.5 — l8,tl
+ 2.5
— 1
37.3
+ 1.4
154 2.7
8!) 42 7.0
325 22 2tl.O
—3 51 58.0
—50 3(1.8 -27.0
—3.0
1-1
21.(1
+ 14.7
50 23 3.0
8!» 4H 11. «
54 10 33.1
+5 25 2.(1
—til 11.7 +13.3
+2.5
— 1
33.2
— 7.4
148 21 11.1
8!»54 ir.i
142 44 37.0
-t-3 40 38 1
—4(1 17.0 — 3.4
+ 2.1
+ 1
20.7 — 4.3
235 34 37.4
90 20.4
231 18 40.0
—4 47 3ti.tl
—37 0.0 -
-35.5
+0.0
11.8 +14.8
315 52 52.2
no •> 24. ti
310 52 44.8 _4 21 53.0
—52 30.3
+ 4.7
—0.0
+ 1
8.8 —18.4
44 45 31.3
00 1 2 2H. 7
48 2ii 48.7 +5 1 4tl.4
—57 48,3
f30.3
—3.4
— 1
25.1 +14.1
142 42 31.4
•JO 18 32.8
137 52.« +4 8 31.2
I
—37 7.(1 -
1
— 8.2
+ 1.3
+ 1
21.(1 — 4.1
230 52 50.7
Dat«.
Tal.iilir
hiiigitU'la.
Otis.
cor.
I.OI1H. from
ubixtrvntiuu.
KquATioNs or Condition.
1
1751 Mav 31 250 13 38,0
II
— 8.5
250 13 20.5
(!3.7 = +0.fl()«,
—44*11 +0.nO««' +1.82»'«u
1757 AiiJt, 27 310 17 8.5
— is (1
310 It! 50.5
4(1.4 0.0 4
—30 —1.53 -fl.l4
1758 Aiitf. 27 3.10 47 37.0
— 11.3
330 47 23. ti
47.0 O.Oti
_40 — 1.7t! +0.80
17<11 Ocl. C.
8 15 13. S
+ 0.2
8 15 14.0
0.5 1.03
—30 —1.00 -0.50*
17t;3 No?. 1
34 4.! 42.3
-f-15.3
34 43 57.(1
35.0 l.tl7
—30 —1.(13 —1.31
17(15 X(»v. 2.1
tI2 12 2t».7
+ 211. 1
(12 12 41 1
40.1 1.11
— ;iH —0.88 — 1.88
1773 Ffl>. 211
15 It 40 3.8
+ IS.S
150 4(t 22.(1
HS.(( I.(l(
—28 +1.03 —0.70
1780 .M:iy 24
245 17.tl
+ T.8
245 25.4
72.2 OOtt
— I.s +0,7(1 +1,74
I704 Nov. Hi
5ll 5 38.8
fl4.7
5tl 5 53.5
45. tl 1.10
— ti —1.08 —1,78
lsM2 Ki'l). 23
154 7 2.7
i Kt.s
154 7 13.5
70.8 l,tl5
^, 2 f 1,85 —0.08
H-23 Xov. 13
50 23 31.5
+ 21.3
5tl 23 55. M
.52.2 l.tiO
+ 2ti —1.27 — l.titi
1S31 K.l). is
US 22 3S.5
f- 3.tl
MS 22 41.5
Otl.4 l.tl7
t 33 +1.75 — l.ltl
1S38 .MttV lit
235 WW 11.2
+ 3.4
235 .1(1 14.t;
O; a 0.01
} .35 1 1.08 -1 l,5tl
1845 Aii'tf. 17
315 53 42.2
fl5.7
315 53 57.0
(15.7 0.03
+ 42 —1,43 +1,28
1852 Nov. 15
44 411 15.4
+ 5.0
44 4(1 21.3
50. t) 1.00
1 57 —1,42 —1,53
IXCO Kcl). 14
142 H 111.2
— 13.7
1 12 44 2.5
01.1 i.tts
1 (;5 t I,tl3 —1,35
18)17 Miiy 15
230 54 17 s -1 r,.l
1
2.10 54 52.0
113.2 +0.02
+ (12 ^ 1,2,3 ( 1,45
A normal equotion for iV is ohtiiinod hy tnkiii}' tlic sum of all the rqnntions.
Tliiit for hi is formed by 8ul)triirtinj; the sum of the first seviMi from the sum of
the last seven, and those for <V' and e\i liy taking the sum of the equations in
whiih the coefficients of rV or e\i are jjreater than unity, after chanpin<; the signs
of tiie eciuations in whidi tliey are negative. The normals thus obtained arc
1'
(k)
TIIK OKU IT OF I- IIA N U S.
0.04 -i-5H7
■ 2.14 -f'207
- 2.11 — GO
These equations givo
— 'i.Ki.V
-f G.m;}
-1-21.58
-f a.Gi
- 3.irv.\j
— o.oy
+ 2.10
-flO.GG
-f lOGO.Jl
-f 230.;}
-H 208.1
-I- 63.8
if = -f 64.8 (Epoch, 1800.)
An = 4- 0.2G8
ie =-]- 12.6
c\j = 4- 8.2
— Substituting these viilues in tl)o seventeen equations of eondition we have the
following residuals, or e.\ce8ses of theoretical over observed longitudes:
1
-f 7.2
10
-1-10.7
3
— 4.3
11
— G.4
3
- 10.2
12
— 1.1
4
+ 21.7
13
— 0.4
6
— 9.0
14
0.0
6
— 7.1
15
+ 3.7
7
— 11.3
16
+ 4.1
8
-f 8.2
17
— 7.7
9
— 6.4
These residuals are much larger than they sliould be, and I scarcely know to
what cause to attribute their magnitude. The results are however amply reliable
for the purposes of the investigation, and lead to the following elements of Saturn:
n,
90 6 26
h
14 r,{) 3.2
e,
112 20
«.
2 29 39.2
fi,
4399().395
e.
.O'jGOGGO
log (« -f fta), 0.n7i)(i7G
Kpoch,
1850, Jan. 0, Clreenwich mean noon.
It will be seen that the adopted position of the plane of Saturn's orbit is retained.
It was corrected from observations before the perturbations were finally computed.
Of the above corrections, those of the epoch and mean motion need not be taken
account of in the corrections of the co-ordinates, since the mean longitude remains
in the formultc as an arbitrary quantity to the end. The effect of the correction
of the mean distance is insensible. The corrections of eccentricity and perihelion
are therefore alone to be retained. They are allowed for by adding to Iv and iif
the terms
I
u^
irisrwS5!U!a»ir«£v",«jaia*aiWlSS*aB«ftiKia^
T UK ORBIT >F U KAN US. W
fiv = 2^e sin </ — efiu cos g
= -f- 'i^'-'i si" J/ — 1(»'.4 tos(/;
ip = — ^e cos </ — c^u sin //
= — ll2".6 cos »/ — a". 2 cos </.
rbrturhaliona of Saturn and Uranus.
Tho following oxprcssioiis include, with tlu'st< corrections, all the perturbations of
Saturn and Urai us which can produce any apjircciublo perturbations of the second
ord( in their mutual action. In these expressions the initial letter of each planet
is put for its mean longitude counted from the perihelion of Uranus.
I
rEllTUBIIATIONS (ir SaTIB.N.
Argiiiueiit.
»o'
1
■in
fOH
IM>«
Hill
»
"
tt
II
S
+ 2.'). 2
f 21.4
— 2.7
+ 0.0
.S
+ l.ti4<
- 1.4H<
— 0.H0<
— 0.74/
S — J
— 2<l.l
+ fi.a
+ 17fi.7
_ 1.5.0
iS — J
— 42l.ti
— 8.8
— ll.'i.s
— 9.1
as — J
— .'.4.7
— 12.4
+ 12.3
— lfi.fi
2.S —2.;
+ :t2.«
— 1.0
+ 2H..5
— 1.1
:i.s —2,7
+ 27.7
— 21.3
— l.-i.l
_ 12.3
4S —2,/
f 2i(;.9
— 7(i«.a
+ 11.4
+ 403.0
6.S —2./
+2:iita.
— icita.
— .'■>8.8
— 34.4
CS — 2./
+ 1J3.0
— !l3.f.
— M.\
+ f)fi.!»
rr- s
-f H,«
f 0.3
+ 3.0
_ 0.1
2ir— s
+ 0.8
_ s.s
+ 0.1
+ 1.4
■2U — 2S
— I.J. 4
f 0.2
— 8.0
+ 0.1
:iU— S
— «.4
— 27.0
+ O.l
— 0.7
3U — 2S
-1- 1«.3
— 14.«
+ 8.8
+ 7.0
I'EIITL'RBATIONS OF I'llANllS. 1
Argument.
t
u
1
sin
coa
COS
(in
It
It
n
11
U
— n.iu
— 0.2f«
4 0.05<
— 0.14<
U— 8
— 2(I.S
+ s..--,
+3tl.O
— 4.0
217— 8
— ll.!»
+ 14:i.,5
— 2.0
— fi3.8
•.U7— 8
+ 411.3
+ 11.^.'.)
— 4.7
+ 13. a
2(7-28
+ 4.1
0.0
+ 4.2
+ 0,1
a (7 — 28
+ 2.1
— l.B
+ l.S
+ 1.4
4(1 — 28
f 0.6
— 1.8
-\- 0.4
+ 1.3
riU — 28
+ 1.2
-f ;t.o
f 0.4
— i.a
Let us now resume the equation
'^ ih H i ra
0^>
\,^
70
Til K OllHIT OF UllA NUS.
Brginiiinp witli thr last two trrins of tliin cxprissioii, it miiy be sliown at the
outset that thry are- (initc iimciiNilih', The vtivvt of llie constant terms in fip and
ip' iias alr.Muiy l)een inehuled by correeting the lofjaritbm of the mean distanee by
their amount; they are therefore omitted. Tlie Iar<,'e.st remaiiiinf» term is ti4', the
square of which k only ()".()•.'. la the proihict ri',Y tiic hirjjest terms urc
+ 0.014
— O.OlDsinf/
— 0.01 1 sin (% — 20
— 0.011 voH(4(j — 'H)
which may be entirely no<jlected.
We shall thereforo only consider in liQ the terms
rip
As already reninrked, It is rijjorously a function only of T', p, and p', T being
the angle niaile by the radii vectores of the two planets. Hut, in the analytical
development of /;, the qtuintity V is considered as a function of v, v', and y, so
that we have
^=/(p.p'.v, v',y).
In the previous computation of the perturi)ations of Uranus, wc have supposed li
to be a function of p,„ p'o, etc. The corrections to Jl and its derivatives with respect
to V and p are now giv«>n by the equations (11), with the modifications shown on
pages 24 to 27. The derivatives of Jf„ which enter into these e(iuations are formed
as follows: If, in the value of Ji produced by the action of Saturn on Uranus, wc
consider any term of the form
in'h --
cos iV
(I,
where
the accented (piantities always r(<ferring to Saturn, hut a, being the corrected mean
distance of L ranus, then we shall have the following terms in the derivatives of /f.
'
6R
f)It
6R
r9p
8^R
'6\'
■■— „ ('+.y)'*'i> -V
"1
-'f (/' + /) sin AT
"I
m' fh
„ cos ^V
</, tin
I'i'fl ,■,-.■, XT
— ., ('+^)-cosJV
"1
THK OUUIT OF UU ANUS.
dm m'h
iJUi
71
(•+;){«■'+;■) cos iv
■, = (A+ J3 )(* + J )»'"•"
6vd^
dvfif ''
fit
sm
6/ ''
an _ ivii
ov t'Vf'yj/
m' vh
(«-|-y)siii Y
h
(1, On
"''(,.+ ':;M(.'+y)Hi„.v
•9'«
c'fjcVj)'
It ?H / f'rt j^ f'VJ \
COS AT
All tlio numcrinil data lU'ccssary for the computation of these derivatives have
lieeii jjiveii in Chapter II. C'oml>iiiin^ the terms having the saiiie ar;{umeiit, we
fiiul tiio foUowiiiji values, omitting those given in Chapter II, and tliose which
arc derived from the others by mere addition. The terms of ' ., are also omit-
fVV
ir-n
ted, because they are sensibly the same with those of \ , ,, changing the algc-
' ' t/Vc/V
braic sign.
0,
dlt
0|
rru
a,
«9'« 1
m'
t*v'
wi' c^v^v'
«•' (5('(9v' 1
9 I'
sin
CIIH
sin
COB
Hill
COS
0,-1
—2
+0.2^*74
-fo.oonii
—0.0310
+ 0.0310
+ 0.2x72
+0.00(i7
+0.2(101
+0.00(10
—0.0322
1,
'2
— ;j
—0.0103
— :i.4Mii
— O.OiK'.M
+ 0.03S8
—0.. 11
—O.lit.lO
+0.4.i04
+0.0173
+ 0.0100
—0.0021
—0.4304
—0.0179
—0.0101
— 3.4S10
—0.1039
+0.03N9
+0.0249
_5.2(1S4
—0.0703
+0.0308
+0.1104
+ 0.45sn
+0.01(13
—3
—3
—0.001.')
—0.0.544
+0.4104
—0.0053
—0.0035
—0.0713
+ 0.0120
—0.01 111
+ 0.0021
+ 0. 142(;
—0.0171
-f 0.0300
—O.OOHO
—O.OCSO
+ 0.S233
— 0.022(»
+ 0.01,55
—0.17(10
— 1,(515
+0(I2HH
+ 0. 00f.fi
+0.2403
+0.0014
+0.0547
3,_I
—3
—3
—0.0070
+0.0670
+0.2542
—0.0003
— o.o(;5o
+0.0171
4 0.lo,'-.7
—0.00(15
—0.0210
-t 0.1013
+ 0.7(124
+0(1107
—0.1701
— 1.0S84
+ 0.0304
-I-0.2S40
—0.0071
4,-2
—3
—4
— O.OOH
+0.0515
+ 0.1448
—0.01.10
— O.Oi'.tO
—0.0015
+o.o;tfis
+ 0.1 0H7
— O.OOKi
—0.0103
+ 0.14.50
+O.57K0
+ 0.0213
—0.2144
—0.7(144
+0.0547
+0.2(147
—0.0005
78
TIIK OUWIT OF UK ANUS.
«i
oJi
n.
ffit
«.
irn 1
<J I
m'
d9
in i)\df'
"« ritdf 1
Bill
I'OH
«iii
con
Hill
Ct.i
0,
—1
—0.02(1
40.172
— 0.5(12
4'o.r.'5n
40.002
U
40.113
-0.04.')
—0.(110
1,
—a
-l-o.oto
— o.oo;{
-fO.HS'J
-} (lOlf)
48.741)
4 O.lHO
40.01-1
— 8.7.'>0
—0.17(1
40.070
— o.col
40.888
—0,227
40.011
-j-o.aia
—0.07 ft
-i B.I8»
40.100
2,-1
—a
40.084
40.018
-( 0.248
40.472
— 0.2:10
—0.04;
40.100
40.013
— o,:i22
—0.00(1
40.020
— l.OS.'i
•'1,-1
—a
40.013
4 o.oTil
40.oo<
4o.O(>8
40.278
40.01;'
—0.122
— o.n;i4
40.n'Jl
-1 0.21 ft
—0.007
-0.049
—0.344
4 0.010
40.028
—0.244
—1.2(10
4,-2 '
40 oir)
40.001
—0 001
-l-O.li.ft
40.020
- 111 'J
40.042
—0.071
40.001
4 0.01!)
— 0.8f)4
Till! iloiivations witli n'spcct to y iiixl tl)(> node \\\\\v hrcu omitted 'ircniiso tlicy
lire (jiiite iK^f'nsihle. Tin- terms of hii (lepeniliii;; on tlies<' deriviitives iire f,'iveii
liy equation (;J1>. In tlie ease of rramis diMturlu!! by Saturn the largest values
of the coelficient»
4,/Moti^; i/.tau Jy, J '^^^■
are only ahotit S)h, .vliile the larjjest roefficieiits in hh, hk\ and }y are less than 10".
l^''ne(> the lai'p>st terms in (31) \viil lie > tlu order of niiigiiitiide {)".h miiltiplied
•> ■ the mass of Saturn, and may therefore lie omitted entirely. Omitting them
the values of hit, h . »niJ ^ ., become
w tip
iltr^
6R
^r -f
rn
h' 4- . ho -4
PR . , f R , , , o'R
i'u , , f It . , . in J .
dv
'R
(Vt'V
trR
R
r-R
{R
,V'
= . ^W 4- , ,, hv' -\- . . hi) -\ . . ,ho'
»
All the separate faetors frmii which the second members of lhes( e(|natiniH
are formed have alreiuiy been given, lorming their |uodiu ts in tiie way described
in Chapter 11, we have the res ilt f: ven in tlie following tallies.
The expressions for hR are a;r;ii ged so that the vi.Iiie of P',hR can be olitained
(rom them by direct diflereiitiat'on. This is done by disfingiiisliing the time
inti'odiieed into R by the co-ord nates of I'ranus from that iiitroilticeil by the
co-orilinates of Saturn.
TUE OUBIT
OF URANUS.
78
2", (fV/
...•+^»vj
u s
COS
tf
U
S J
II
If
'.
— I.40<
— i.3a<
+ 1.
3—3
-f 37.5
-f7a.3«
2.
-1 0.0(!<
— 0.2W
+ 2.
-fa,
— '.»!I0
— 73
— .'>.^S
— 2<;i
0.— 1
— 0,24<
+ 0.20<
+<.
+ 23
— 42
1 '— '
-f (ta/
-f 0.04<
— l.O.V
- - iCrf
0,
ii
— 1;47
— aia
+ 0.02/
— 0.30<
'i
4 77a7
-\-:,i;\i;
«.
-f i.'.a
— loa
0,-3
+ o.n.M
— 0.8S»
».
+ 24
— 211
1,— J
— 12.<;'.«
f ll.tiO/
4,
2,— J
— 0.2»
-f 0.10/
a, -2
— » tilW
— 0.5'J/
-8.
-i.
6—2
f 00
— (',2.'i
+ o
1 7 ail
l.-'J
— I.(i0/
— 1.14/
,1
+ <<1
— 7f;
2,— :»
-f 0, •.':.<
— (t 211/
(
+ <ll
-f 7l!t
:«,— .1
-f- OS/
f » It-it
•A
— 20
+ «
4,— a
— o.aiK
— o.a«
-4,
r,— 2
+27..')
— a. It
:t.— 4
-t- O.I!l/
— 17/
-a.
+ 114.1
-1 2».4
4,-4
+ 0.1 j<
—2,
— 1,
4- lid
— «s2a
4 243
— t;ii4
(7 ir s
0,
-)-(;ii'j
-f f.22
1, u
4- 3H"
+ *"
1, 0—1
1- 2
f «
—4,
7-2
+ 70
— 2
1. 0-2
— 112
- i.-ia
—a.
+ 2!MI
fin
1, 0— J
f M
— IS
—2,
f".ia7
t "(Mi
l,fl-l
+ 5tl
— 2
-1.
— I07S
+ 1117
1,(2-1
— 4
f la
1,— 'fl
— 1
♦ I'.t
'■in, (,■)//.
I ^". \
l.f2-2
— ii:
f 2
*•
III ( tl V
* «?H *" I
1.-2 + 2
1 f:t-l
l,-.ifl
— 2a
— 2K
f ao
t 100
/•
K S
COH
ain
Vail. n(
l.fa-2
,140
I II.'
„
(/
tt
i,-;t+2
+ u
— 10
2,-
-1 — 1
— 120
— 3
— S,')2
r S J
a,
+ .'»«
-( s(M
4" I4H
0, I — i
f24fi
- 1
4,
— 139
+ a.'ia
\ I.44S
I 1
—2 Kill
— -2
'I
— .',0
t 1
1 l!
X-
4,
-1—8
— 122
— liO
+ 2(1
— 2ft
f 0.1 48
4 1 US
'l
■ 1
1 l>
•'«,
+ 24
— (iO
f 2.14S
—2, :— i
— 1.
f- 42
f iiias
— ia«
— M!t
f 2
— m
— MS
f 12
— SO
— n,_
t 1,
2,
-Ifl
— 2.'.0
_ 4 '
\ I!t7
n
-no
— 4.'.0
— 2.H.'.9
— iNsa
— o.s.'.a
2.'
a,
1
-1 as
I us
—a, a— 1
—i'.
-f n
f till
H7S
— «
— 1.-
0,
1,
-M2
1
1
— H
— 1'
— HO 1
f in
f 134
+ .13
— 3.H52
—2 sr.2
— 1 S.'.2
0,
1,-
-1 1
- no
-f 12
— I.K.'.2
—a, 4-1
•>
4- n
-221
( 2S
— h
a,'
1
1 « 1
-f 'i !
1
-t l.'lS
— 32
— O.S.'.iJ
< 148
1
+ 2'.t7
-2H«
1
111
10 MV.
IbTt.
74
TIIK ORBIT OF U KAN US.
U S
».
I, (I
■I (I
I,
1,
I. -a
i.
■'t,
4,
.■1,— 4
i,
0,
1,
-1,-1
0.
I
•i.
3.
it,
;«.
4,
f.,— <
.'l.— 3
4,
I
2 "; *//,«
+ 1.40/
— 0.1 2*
G
— 0.0.K
I '.'.10/
— 0.0(1«
n
4 12 nw
■f- 4H<
+ l.SiM
4 l.no(
— O.fiO*
-- tt'iU
) l.4t(
— 57/
— o.r,o<
— 45
I DiH
( a
j I4f)
t «
—I an
— Kl
— 4
- MH
t lOH
I UO
— 5a
_ I :i.l/
— 0..'.(1/
f 0.04/
_ i.ao/
— 0.510/
I II no/
-i O.tiO/
— 1 77/
— 1 u/
— 0.5 N/
f 0.011/
— I.:ii'>/
— 0.51/
+.119
f :iM»
-f 5
— (11
—.•ml
— 70
M
t ;w
— J.Vl
— 7
^ISI
4517
4 4
—S3
—Ian
8 -,,''«
in ^v
-I \.m
— 0.27/
— 0.04/
— 0.00/
} 2.10/
— O.I'M
— 0.91/
i l2.no/
♦ 21/
4 I.MSM
4 l.(,U
— 0.53/
— 0.10/
4- 1.43/
— O.'i.'./
0.52/
4 9
4ll>»
I TO
It!
— 4
4 •■«
4 I
H
— 14
t 145
— <;o
f HI
— IH
490«
-♦ M2
— 49
— 42
_ l.l.'i/
— o.4;i/
— 0.0/1/
4 O.OIM
— l.Xil/
— 7.«
— o.h:u
) 11.02/
4 0:17/
— 1 75/
— 1.11/
— n.!M
4 0.07/
— I.:i2/
— 5:1/
4 0.10/
-♦- 5
-( IllO
f41'.»
— 71
— •,'47
— :I2
— 12
— I
4 4
— no
— 13
) 7'.m
( a53
t H
f Ifl
— (-,7
-lao
— •,114
.-loa
4 0.14/
4 3.64/
— o.a4/
4 1 00/
I 07/
f a. 5 4/
— 0.19/
4 0.84/
— 13.52/
_ 42/
4 2.7W
— 1 «»4/
— m;u
— 0.2.5/
I 1.93/
— O.T4«
— 0.7 a/
—193
— 49
t 27S
— lit
I 103
I I
— 10
— I1»
- 10
I H4
( M
-202
.- 45
t 17
_ .15
— '.12
4 8.21/
-\ 0.110/
— 1.02/
— 12/
I 3.22/
I I Ix/
- 94/
12.34/
U. I 1/
2.52/
■ I 21/
7.V
- 10/
1.70/
( Ml
— I) 10/
110
12
■ 111
-709
5111
) l-.'5
( 347
•( <7
— 215
4 3M
— 9
4251
4 7
t Nlft
f«17
t 2
-» 31
4 41
4 H
4 .''2
4 Us
\-
J
T II K ORBIT 0¥ U K A N U S.
76
f
a "/
hl),lt
ij «.
a';^"
9«.
*'?^
m n
m
t;»
m
e/p
U S J
hill
It
CUM
Hill
COM
1*
Hill
n
0, 1 — 1
-ih:i
f 8
— H4
.\- 13
1,
1 •.'ir.'.i
— 60
fJ477
— 50
— 2'.I47
— ll'J
•.',
I •»
-f- H
— 7
+ la
— I7ii
— 17
•'«.
— y
h l-^
— 11
) ill
— 17
— 20
—2, a— 1
♦- H4
4 aj
f- ii
+ J!)
+ 9
+ 49
~i,
f mas
+ »H
-t li;4:l
+ HS
i I2ai
— ua
II,
— 114
__ i
t 41
-r II
1 1.
\- Hit
— H!»
f sa
— Hit
— I'til
— 112
-i'i,
— 4
-1- «
— 1
( »
— 11
— 18
—a, a— 1
1 H
( '.J»
1 -••■'
1 2S
— 42
} 41
—•J,
1 i.:j
- 4s
t 1 4a
_. ri4
— I'.M
— 104
—1,
J-47M
^2H0
f477
+ 'J7;>
+ i;.2;i
—2711
0,
(1
- .-.7
— «
— no
f 44
—a 4—1
f ^'I
_ H4
•f 1>
~ !l|
— 't')
— ia4
-J
— 4IJ
} 10
~-44'.»
1 7
i 7711
— «
—I!
f i'.iT
t -JHIl
f aiu
♦ 270
faiii
— 2.".8
1, .1-3
- airi
f 7 tail
— a!ia
+ 7a4a
f r,a!>
-+7722
•1,
( l'.tst»
— iiiii
-\ 1!W7
— 1271
4aao4
«2:>44
•t,
t ■iV\
7sa
f ''1
—1172
( i.-.l
f 1011
4,
— Hi
-HIS
— 101
-102
— laii
f 1J8
"•. \~i
(»
fliiil)
— »sr,
— r.ol
— 9i;rt
1,
— 77:n
^'•r.lii
-77aa
t .'Hiort
\ 121 an
-f HIl'.IH
a.
—aim
— -.'it'i
-aa:.
—iun
♦ 27a
{ !I10
a,
— 7i
— 7H
+ /)«
— iti
— (!0
-flO'J
—2, .')— a
l-lsi)
— iJii
♦ i:i',i
f f,
— aa7
— 37
—1,
_i;j.'i
— 7a;i
— 11 JO
—710
— 1222
— 1217
II,
II
{ Hi
— r.ii
) ln7
— no
1 1.
— r.l
t7l'.t
— iia
+ 72.-.
+ 72
f 1 1 ii'.i
)•-'.
t 4"
t IJ
— la
— tl
— 17
t !i7
—1, f,_j
f no .1
♦ 1.". 11
I mi; I
1 a2 ;»
— 127 .'•
) ail 4
—a,
tJx-.'a
— 7a i
f U4S,H
— I4.'>
—4 la (1
— 7:. 7
.»
(lay
— isii
♦ ia7
-,'..'.7
— 11 11
— 14111
~'\.
-K'Hja
1 1,111
— WNJ'.I
+ 11121
— I2'.>7a
— mi-.'ii
»,
II
1 7 HI
— »!ia
t77l
1 IIMI
--4, 7 --J
) •-•"(I
♦ *
t 'nn
+ 4-'.
— 2it;
1 4M
—a,
I S(|7
—441
t 7J7
— (ia2
— 112S
—977
-J,
1 IX7I
- 1 1 IJ
\ I'll a
-117 4
— a2art
— 2111.'.
—1,
— I'.as
— 1117
lll!l.'>
— |ltH7
- 207!
+'.101
0,
11
II
1 lait
1 (Ml
f H4
— 71
oAa
70
TIIK Oil HIT OV UllA NHS.
Fii the trrnis of hit intrtMliiccd liy ilu; pcrliirlmtions of Sahim, niimcly,
^ , it/ -^- , Aj)', tlic tlifri'rcutiatioii r<'|ircH«'nt(><l liy />*, Klumltl he |M>i'toriii(>(l liy con-
RidcriiiK At/ iiiul fif) as t-oiistant, altlit)U(r1i tlicy arc <-\|ir(>NS('(l art n fiiiu-tioii of tlio
iiu'un luii^ittiilc of l!raiiiiM, as wi-ll uh o( Saturn. 'I'lic mean loii;;itiiiii' of I'rantw
thus iutrodui'cii is tliinfori' rcprcMcutcd hy /", which ia regarded uh constant iu'
taking //,/?, and //^only supiMwd to vary.
Again, iu the teriiis " ' jf4- ^"•••''kc I'r and A/i reiireHent perturhations of
t^'v ' 5p ' '
Iratnis, their roinplete derivatives, with resjXM t to the time, i:r<> to he taiveu. Ihit
their exprexiiiuDs eontuin tlie mean lon^'itiuh- of Saturn us well as rrunus. 'llu?
mean hmgitudc! of Saturn thiiK introdiieeii is representiMl hy .S*. and iK to he eon-
fiidered varialde in ohtaining iy,hll, while S is eonsidi red eonstaat. The ratio of
the roeffieient of t to it in tln^ varioUH terms of thin part, of ftit is given to the
riglit of eaeli rorres|N)tiding ti rni.
'I'lie vahie of //,A/i' iM'ing onee ohtained, there is no hiUfjiT any disliiutinii
necessary hetween /', V\ or l>etween iS' and .S". 'Ihe himdar terms are liu-refore
eomltined hy jtiilfing .S" .V; /" r.
Ironi llie ul»<»ve vahies of 'J:!)J)) U and 'i,S ' , we hirni the following value of
'V
"' H'- '^" J' A/>'. /•'/'+ "' ^V~
and of the other fjuantities whiili enter th( |»er!urhatit»us «d" the ro-ordinates. We
liliall l>e;;in with thos<> lernis wliirli depetid mdy on th<- nnihial action of Saturn
and I'ranns, lieeansc tiie) are (i w and small, and tiu' only terms whit h are sl■n^illle
are those in whiih the rocHMi nt of tin- mean liinj,'itiiile of Saturn is I. \\e
shall therefore (online oursi'hes to these. And, instead of enntlii)inj,' the eoii-
iliiised furmnhe, Wf shall make the en;!i|)ntation in fnll liy (l-l).
".^Q
II
{AV
"',<!*«
in
in
CIIK
hIii
i.i
In
HI*
.iu
1
i tc.
t' n
„ „
n II
II II
II
It
*/ II
). 1
1 (7 ( 7S
III
t •*"<
1 T
1 i;i:<
it+o..'.;w i + ;i<it (i.M/
4- l:i4i) *>-.i'
f 4'.l-f (1 KVM
1 :i.i...
■O.dC'/
- S-f(Mll.'
~ L'l ) a.iei/ + (,-.'. -siic** 1- II f .' :;7'
I |:is , I K-H
S '-'It-
-1 s,">/
-If.l i •-' llr
•J
• - fid 1 i -ni — 2s 1 :t.su \— ■>',[) \*\M
t t.il :i (i.W
) -.'.'ill-
(1 i.!(
4- K.'.— -J 711/
■ :<•>:> ( ;il; ■ t ;.'|ti-_.i.v.i<
1 7 1 -i-WM
:;i t •.'..■!0<
— ;i(i (
•i 1 \t
1 14— 1 ti-J/
*
—•iwx 1 a l,><
, IJii -' IU
~'X\\'X )
II l.'i/
--l-'(» ) "J 741
i> '■
r>is 1 Hill
•»
am 1
foi
// n ' tt II
" '" 1
0—1
{ aoi- oiuHtii — o>i:i i imon-.v
1 a oii'j ) a (Midi)/ 1
\ (!'.> 1 ( iKIrtdf
1,
t nil , »(Mi:i:(# — (CK; t """i:i:t>
1 M lilt I ii.aaiii,/
1 (i.'is t a iiihMi/
o
♦ :i:;;i ( imuiti _ iw ( mi.iii
1 IMV.'a \ (>,(IO!f'l
( so:t_.(i (ian7i
•'l,
__ 1 ir>_ (itin(i-j< ♦ II71I Oi'ir."*
f >.'(',!) (1 (Mil W 1
4 II s;tu— (1 iii.Mu
4,
_. an i.iii'i
|ti l;t'j 11/
1, i> uta 1 II iMiiiii/
TlIK OUniT OF URANUS,
IT
The rompiitalitiM of tli<w tcrntH hciiifj oxtrrmcly roiindrx, t\ chcrk njKin tliiir
ncciirary is (Icsiru'ilc. In llic '. asc of tlic secular viiriiilit>iis ni' llic (■((••Hiciciiis, ilio
nil (Ui'ii'iits of the tiiiK" ar<' easily oMaiunl liy n-il).stinititi;^ in tlic inlef;mt«'ti per-
tiirltatiiMM tl»o vuriwtioui of tlie et;n iilrieity uml iKTiliclioa of Suturii. Thus I
htive fuuiitl
» n
fv := I 0.i)IO;J / sin ('i,j - t) - ().()(»•>* t «os (-2;/ — /)
4-0,0027 I sin (>ij — I) — O.Oiaw t io» {.iy — /)
Tlie jjfrefifi'Ht iliserepitury is AmiikI in i1m> coeffii ient of sin {i)ij — t), mnl it anionnts
to (I ((();|^/, or iiliont U" I in a lentiiry, IJutj owinj^ to the ;;re,it |Mr;o(i of this
term, nearly (iOO years, tliis dift'erenee, thirin^r imy one centurj, v ill lie nearly
t'liniinateil llH-on<^'h ihe mean !(>U|U'iliiil< ami nnan tiiotion.
It may aUo hi rei)iark<il tlial in tl>i'< ruse the terms derived ftom the pertiir-
bations of tlio elements are imdouhtidly thy correet ones, and will therefore ho
employed.
'I'he fi>rins whiel, tin' pveci-diii^ iiite^nition fail-i to give, owin;^ to the constant
t< mis iiUTodnc( d into ^ls(^ and yii\(^, an I'uund l>y {'22).
We tiiiiH have
w^; /),,;.;;' r- .x. (»■■.;«»
«:;■/,./.•;'=- -fo. '27
r,».^p 1= I ,)[„!' jtr.HC sin ,j - i)".-'! <os ,/{
.Sji = J J/,»/' ;(j".;i(;,,)s./ | ((■.•J7r,in//i
~- r- |i!'.()()tH((>;iS((»s;/ f (r.il(l()(K».".)sin</j.
'Ihe (i^rialest etreet of thi'se terms anioimls to less than tnu -twentieth of a
fieennd in a ceiitiny. Tin y may Iht rctore he inijic eied in tin present theory.
The other teiiiis lOiilaininf^ the MCjiiave o(' tile tiiiie are yet. smaller.
Apphiii;; the terms of the -.eeorld order tlilis t'lUnid to the terms of tile liis( order
di'pendini; on the iurre>p<)i'.d<ii}{ ar^nnieiits, the p( rturhatiou.sof I ramin h} Saturn
become
i
it
r
+«,
:; '■
KID
f«
CUK
u
Kit*
niii
-1, '
\ « ti trt
1 11 II^I'.I
f
_ a
ly 1
, 1 Jsj
, i ;••
-. II
— .11
1. -1
-^.HITM »
nmr
■J, K .,M\
t I7.'):t
—•-'11
J 1 II ,!7H 1
1 .»ir
\\M -.'i... —
11.04 7'
- S7
— .•im
:; 1 f.'.l !»!! t
j; T
\\\f< r.!i —
1 its T
—2.17
t.iii
II I 'i-Mt
( .'. f.r.c,
— .'..-i
( till
;i 1 -+ » i;:il
+ o.:ja»
— 4
-f 10
A
T here I' j'r"se,ii-t tli. liiii.- counted in crninries fvipin 1^."<<I.(I.
The utber teriM>i remain the same as f;iven on pi;,'e ■'")().
/V, 'iirhiiiiiiiiM i/ijM'iniini/ oil III' jirinliit I nf f/i, imti- s nf Jnplhr mnl St4iirH.
Ihe v;»iueiinl'<\/>^/i', <$ ' , ami A , il>'|N'ndiiiK <>h the proUtttt)* of the nuisM s
78
T UK O II U I T OF U U A X U S.
of Jupiter and Sutiirn, arc f»ivon on pago 74 The coniptitatioii from tlicso data
being coiulucU-<l in tlu! tamw way as in tlu> c asc of tli<> terms of tlie tirst order, it is
not necessary to give niueli more than the results. These are sii nvn in tlie fol-
lowing table. The indiees to the left represent tliie <-oefM(!ients of '.lie mean longi-
tudes of Uranus, Saturn, and Jupiter, all counted from the (n'rihelion of Uranus.
Column I' gives the ratio of the mean motion of Uranub to the coefficient of tho
time in each argument. The perturbations of the comnu)n logarithm of the radius
vector arc expresM-d in units of the seventh place of decimals.
V
lu
0.4343 «i>
U S J
Hin
VUH
tl
fi)»
Hill
0, l,~l
1,
3.
4,
— 0.23fi4
— 0.30i)f)
—0.4480
—0.8127
+ 0.002
—0.020
-f 004
+ 0.01(1
— 0.0(1.1
— 0.021
u
u
—2, 2,-1
o!
1.
— 0.2!tr,0
—0.4204
—0.7 2.') 4
— 2.(!420
+ I.I10!I0
—0.007
— 0. lo.s
— (111
+0.1 r,4
4-0 oo,>
—.0 (Mil
— .(I (107
—0,(112
—0,2117
-0.00,'.
— 2
1 1
1 >
u
—3, 3,-1
-2.
—I.
— 0.(!.5.')1
— I.81llt7
+2.1115
-1-0 ()7H«
fO.Ol2
1 17'>
(-0,07s
fo.007
— 0,002
— (i.dl.'l
fO.:-.12
) It. 021
1 1
— 2
t :i
—J. 4,-1
—2.
— 1,
-f0.7.'>»
-f-(),«.10
-1-0 301
.—0.0.10
f 0.013
1 o.ool
1 0.(I><1
1 0.00.')
— 0.003
— 1
f 1
1, 3,-2
•>
4,
—0.2170
—0.2771
— :{s:),i
—It t.215
_o,o()2
—O.O.'.I
—o.olo
+ 0.(t32
—0.0.12
1 (i,(i:;.'>
( O.'iO
t o.o.".2
-• 1
— 1
— 1
- 1
- 1
0. 4,-2
1,
~0,3(;27
— 0..5(i!i2
— 1 3210
1-4 ll.iO
—0 010
f o.(i7.">
— :;i'.i
— 0..M0
—0 001
— I.M
— 1 2'.i7
— l,>;t
— :>
— ,1
'2
1 II
— 2
— i!
0,
1.
2,
— 0.f)250
— I.IO.M
-)-10..'>ir.2
-f0.!1132
-1-0.4773
--0 2ri3
1.133
_o,.'>ii;
f II 201!
-f 0.012
—0 (131
— o,(;17
(Mi:(2
— ;i -j.ii
—(I.I '.12
— 3
— 43
♦ 2
— 2
■i 7
— :i(l
— 2
—4, 0,-2
—3,
—2,
— I.
0,
— 0.9»'t7
— lH.!l2rtO
-)-l.o.''(ris
4o..'>13(t
-j-0.33!»3
+ 1.824
-t 4O.C..'i0
-(-t;,2:i7
-(-o.4t.7
+(».0(i7
_. ;i 1 (1
-|O..Ml(l
— 7.HI".(i
_o ."i:!!!
—0.017
+ 10
^32
—(13
II
1 *••
1 l»
— 7'.l
— 2
-4. 7,-9
—2!
— 1.
—0,
+OMM
-|-0.3r)73
-f 2i;32
-f-0. 2(tH4
+0.1724
— tl O.'iO
— ii.oti;
— 0I'>
+0,000
— o.oo;i
) (t (132
1 (i:i2
1 (107
f 1
( 1
♦ 1
t 1
■1 1
^
TUK OHBIT OF UKANUS.
79
ClIAPTKIl V.
COLLKCTION AND TRANSI'OKMATION OF TIIK I'KECKDIXO PJIRTURIJATIOXS
OK U RAN US.
The tcrniH of tlu.' iicrtitrlmttonM whirh neither contain the clnmrnts of tlin
disturldn^ iiIaiictN, nor <li-|n'ii(l on the scrular variations of tlio eccentricity and
perilicliou, admit of being greatly ttiniplitied Ity u slight change in the arbitrary
elements. These terms are as follows:
(1) In the longitude of Utanus
Art inn of Jii|iii(>r, f .11 -illtK
Aflioii of Sutiirii, -|-l«) '.WMtl
Ai-liuii i>r N<'(ituiic, — M.4Ui'i'ii
T..I11I, H ''■''<"
4-j:>.(;ri7 sin 7 4- 1 .;i!)7 nin 2.7
• f «..'i4.'>Hin ^ -fit -Itll hin i!/
-f OCnxin ij -fO.IMf, din iij
-)-:i(.H'.)',)nin</ -\-\.'MiA\ii-2<j — (i.(i82coH(/
1 , HM con g —0. 0H7 rng 217
4.7:J'> O'lS ij — 0. Id'.t I OK 'iij
('2) In the value of cos i^p, units of 7th place of decimals.
A<-iii>n or Jiip'ti^r, — liKlso — I'.ii con // — :!:i I'n.f 2;/ — 2Kiii(; + I Kin 27
Ariii>n (>r .Siitnrii, — :t.'i4:i — H-l cos j; — l.'>c<is2i/ — l.^niny
Aciidu of X»'|»tiinc, -f- l'!S — Icosi/ — I ciiK iiy -I iNlajr
Tiiiiil, — l:i('.l| — C'TvDH'j — 4',»i'o«2(/ — ICiHJuj -f'liiin27
l,et ns first r(iii->id(>r tiio first or coiisfmit term in the perturlmtion of ea«h
ro-onlinatc. If we sM|i|tosc a change of hi in the mean nio'ion of a |ilauet, the
eorrespiindiiig change in i^p will Ix;
ip:
If, tlieti, we iiicreusi- the mean motion of Iranus Ity 41".1."»1, the cnrrespcmdiiig
rliange in Ap will he iHOt.j, imd in cos >^ip, — IHO'Ji'i. Subtracting these from
th«' above jK'turb'itioiis, the serular term in the mean motion will ilisaiipear, and
we shall hau- for the c(in>r:tMt tiTiu of cos i^Ap
j l.-):U
This same change in tlie mean motion will produce a sr^nlar term in the ('(pia-
tion of the centre of the samf nature witli that prodnceti by the s<'cular variation
of the perihelion. The differetices of the values of the secular terms, fouiul b)
!u' two inr^phrids emploviil in ('li;ipters H. :md TIT. (trocerd-. iVom thr fut tliaf in
the one case tin- i tb'it of tile above term in the nnan motion is included, and in
the other excluded.
£S
80
T II K U U li I T OF U U A N U S
If wo Hiiliiluct tlu! rffcct in qiu-Rtion when iicrcsBiiry, the ri'iniiiii«lrr will In- tlio
cflft'ct ut' the hcculur variation uC tin* longitude uC tlic ixriliclion of I'ranus, tu
whii'h we Hliall revert presently.
lA>t us next introduce Nueli ii ehnngp in the ereentrieity of rninu)* ns Himll |)ro-
duee the terni :)4".HiM) Hin «/, and uM-ertain its eti'vct ou the other turins, l-'ur tliiii
purpose we luuitt determine te by tite cunditiun
3
(2- 'c»)^c = 34-.Hl)!)
whii
ch gives
;c=n".404 = . 0000847.
A rlnmge of this nmonnt in h will introduee the foUowin;^ t«'rnjs in h nnd ^p
h = :14 ".H!)» sin (/ -f 2".()4M itin 'i,j
cos ^■^p = *0 — 844 eon <j — ftJ) eos 'iij.
Subtracting these terms from the expressions previously found we have
hvz= — 0". 1 44 sin •.>./ — (;".(>M-.> cos fj — 0".-J(i I cos Q;f.
COS 4,^p =i: -(- 4 51 1 -f- IGl cos ij -\~ 10 cos '2i/ — IG sin 'j -^ I »in 2</.
Again, let us put
c,^rt = 3".a42^ .0000102,
W(! shull have the elliptic terms
fiv = — (;".f)82 cos «7 — 0".:«)1 cos 'ij
con \J,^p = — 102 sin ij — 11 sin 2»/.
Subtracting these ex])ression8 tlie constant terms, independent of the mean hmgi-
tude of the disturbing planets, are reduced to
hr = - 0". 144 sin Qj -f- 0". 1 30 cos 2</.
cos <i,5p = 4 ') 1 1 -f 1 (17 (OS (T/ -j- 10 cos 'iij -\- 1 4fi sin y -|- 1 2 sin 'i>j.
0.43429 ^p = IDG!) -{- IS cosy -f 4 cos 2</ + G3 sin r/ -f- 5 sin 2y.
In the last equation we have introduced the constant -|-.0000(M>rt prtNluced in ^p
by the combined action of Venus, tlie Kartli, and Mars. Tlie effect of each planet
is computed by the approximate formula
^p=: J „,'(//;' +„/)aA7).
Sinihir ViiiladoiiD.
The following inequalities result from the si-cular variations of the ecceiitririty
nnd longitude of perihelion produced by each of the disturbing planets, 7' being
the time expressed in centuries.
From the variation of the eccentricity
ArUon of .lti))lt<-r,
Ai'timi of Siiliirii,
At'liiiti uf Neptune,
ti).
■ —I 'in;'/' Hill 7
— ',1 Hi>7'hiii ;/
— o.r)0'.'7'iiiii <j
— (Id"'.' 7' hill '.''7
— (l.Ti.'ls 7' hill •-'</
— o.oaoy'hiii 'iij
— 0. (Mi:. 7' Kill .'I;;
—II ti:i:!/>iii .lij
— U (iu-.!7'hiii -.Ig
* *
..
TUB ORBIT OF UllANUH.
81
Action nf .lupltiT,
Actiuii of Suiiirii,
Actiuo uf N('|ttum',
-[ li T fOK (/
-I- 1 T eon 8<;
-f Tif'cuii ity
The soculur vnriiition of thv lon|{itiid(> of the {icrihelion is
Artinn of .lii|iiti'r,
Artioii of Suluni,
Action of Ni'|ituno,
Total,
■\\t2.\r
4-llH,»y
+ r.i.ir
««== +2111.(17'
Thp rffort of this scnihir varintinii on tlic loiif^itudo niul radius vertor is
It II II
Action of JopiliT, «('-— ll.4(!rcoM j; — (I.C.TI T wh 2j — 0.0J7 Tcob 3<;
Anion of Siiliirii, —W.WTvmij — ll.(!.'il Tion 2</ — O.(i;{!t7'ro« 3«;
Action of Ne|»tuiiP, — 4.so7'ciiH»; — 2«<| 7'coii 2</ — O.ni«'i7'cob :i</
Total, — 27.:t:7'coH </ — I C03 7'co« 2j/ — 0. I02 7'co« 3jr
Anion rif Jupiter,
Action of Saturn,
Action of Ncptuno,
m.
-I'-'iirHin g
-in 7'Hin <j
- fiU 7' Hin g
— Nrnin Ig
— f 7' Bin 2^
— 3 7'«in2j/
For the piirpoHO of convonicntly t4(l)ii]atiii<; the prrttirliatinnH, wr tthull pxprrss
thrni in a form Hiniiliir to that adoptrd in the tlicory of Ncptinw*. Let uh hchTt,
from tiio tcnuH of tlic |H>riotlir pcrturhationH produced by any planet, nil thoM* in
wiiich tlic differcnco iK'twcm the indices •' and i" is tlie same, lor exaniph', in
the |H'rturbntiunK of tlie loiij^itiide produced by Jupiter, let us consiiler tlic terms
Ap=-}-l.V>(;i)sin( - ;)
— 8.495 Bin C^;—
-f l.lHisin( g—'il)
-f O.ori sin (3y - '2/)
— ().(»or) sin {'hj — ;}/)
-f ().onsin(4i/ — ;}/)
These terms niny be expressed in the form
-f 0.002 cos ( — /)
— 0.0!)'.>eos(27— /)
-fO..')l.'ieos( ij~'il)
— 0.00.') cos (% — '21)
— 0.001 cos (4«/ — ;j/)
>v=ih\\\ij X
-f cos;/ X ■
-1-0.094 sin (v —
-|- 0.5',»0 sin 'i{ij —
— 'i.'i'.Hi sin (./ - /)
-4- l.'.»')(isin%- /)
-f 0.(M»(i sin ;{(./ -
— 4.7r»4eo» ((/ — /)
— l.lOHeos'% — /)
-f-0.01(icos%— /)
— 0.090 cos (,'/ — /)
•} 0,."»lOeos'% — /)
II May. 1873.
89 T II K O Kit I T F U II A N U H.
In gciicrul, u scrius of tornis of the form
Soi lin {iA -f «:/) + 2 6, cos (I'A -f inj)
-f ia', sill {iA — ««/) -f S t', cos («\4 — «»/),
may be put in tliu form
\^ (»», — «',) COS iM — 2i (h, — li',) Hill iA ( ttin «(/
-f- I- ("( + "i) "i" '-1 + i (''( 4" ''i) "'" '■'^ { •'•"•"i/-
All the fioriodic terms rontaiiiing only if atul / in the nr^iimnits miiy he put into
tluH form by taking
A=:t/-/,
io that the roeffioicntx of Hiii /»■/ anil vm mj may all ho rxprcusnl ns n function of
tilt* HJn^lu variable ar^unHMit A.
Tlic ]K-rtnrbation8 of the eh'mentfl may be rnlnrcil to perturbations of the
civortlinates ex|»rc»sc<(l as the sum of several protiiirts of slowly varying functions
into the sines anil cosines of the multiples of <j. W'v have, in fact,
iv = fl
+ etc.
+ etc.
It appears, therefore, that all the perturbations in which the arguments contain
the mean longitudes of only two planets may be ])ut in tin- form
iv = (tt.c.O) -f (r.r. I ) cmq -\- (v.r.'i) cos 2if -|- •'tc.
-f (r./i.I ) sin^ -f- (r.M.'i) sin '2// -j- <'tc.
J/,^p = ( p.r.O) 4- ( ^}.l■. \ ) COS tj -\- ( p.c'i) cos 'it/ -j- etc.
-{- ( ('■"• I ) sin ij -\- { ^).l^,'i) sin 'ij -\- etc.
We have next to reduce to fbe same form those tenns which contain the mean
longitudes of both Jupiter and Saturn, and wbicli are given on |)age 78. We have
here twenty-four terms, each greater than O'.OI. As most of these terms depend
on three independ(-nt arguments, they cannot be included in a doidde entry table,
while, if we include them as perturbations of the hmgitude in tables of sin^jie
entry, we shall have to enter twenty-two tables with as many different arguments.
Hut, by taking, for the ar^nmetit .i, the miildle one in each series of argimients
wliicb depend on th(! same midtipl(>s of Jupiter and Saturn, and expressin;r the
terms above and below it in each series as coefficients of sin '/, cos ij, sin 2y, and
■ o
THE ORBIT OF URANUS, 18
COS 2j/, wo niuy ridiioc iho iiuni»M«r of urKununtB to liglit, niul ihr number of table*
to seventeen. Consider, for inutunce, the terniH of the second series,
-(U0Hsin(-i/4-'.».V-/) -0.(Mncos(-j/-t-2.S'-J)
— 0.0l4sin( 2S-J) -().orico8( '2S-J)
4-().l()4Min( ff-\-'iS-J) — O.'2(i7co8( g-\-'2S-J).
These terms may be ullowe«l for by ndding to (r.o.O), (i'.«.l), (t'.c.I), the termi
II "
(,, ,..0) = — 0.014 sin (2.V- J) - 0.0 1 2 cos {'2S — J)
(i'.«.l ) = + 0.2<»0 sin (2.V — J)-\- O.'iTi cos CiS—J)
(t'.f.l) = -f- 0.0:0 sin {2S-J) - 0.'i74 cos ('iS—J).
From the perturbations of longitude and radius vector already given, we readily
find the following values of (I'.c.O), («'.«. I), etc.
Adii.n 0/ JiijilfiT.
(»'.c.0)=-f53.0fi4 8in A^ —0.004 cos ^4,
— 0.277 sin 'ivl, -f 0.0;MJcos'.»/l,
— 0.02r>8in;J^l,
(».c.l)=-}- 2.226 sin yl , —0.000 cos Ay ((•.«.1)=— 0.001 sin yi, —4.7(54 cos yl,
_ 1.2r)(}sin2y1,-f0.510eos2i1, _0.&20sin2il, — 1.108eos2.4,
— 0.006 sin :L1, +0.016 cos 3.4,
-11.467' -I". 22 7'
((•.c.2)=-|- 0.121 sin i1,— O.OaScoH y1, (r.«.2)=— O-O-'jOsin yl,— O.l75cos yl,
-f- 0.012 sin 2.4, -0.01 4 cos 2.1, +0.00H sin 2.4, +0.042 cos 2.4,
4- 0.029 sin ;L4, —0.034 cos 3/1, -f0.034 sin 3.4, +0.035 cos 3.4,
-0".6r/' -0".07r
(«.c.3)=- 0.04 7'
(,..^.3)=^ _().()(),-, 7'
(().c.O)= + 1127cos yl,
+ 4 cos 2.1,
(p.f.l)=~ 2 sin .4, +.'.7 COM .4,
+ lO sin 2yl, —23 cos 2.1,
+ 137'
(p.c.2)=+ 7 cos yl, +17'
(().«.l)= + 10Ssin yl, + 2co8 yl,
+ 26 sin 2.1, +12cos2yl,
-120 7'
(p.«.2)==+ 7 sin yl, -87*
IMAGE EVALUATION
TEST TARGET (MT-3)
/.
1.0
I.I
liillll 112.5
|50
Ir-
.4^ IM
i.':
ii
2.0
1.8
1.25 1.4
1.6
:
<« 6" -
►
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WEBSTER, N.> 14580
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^
t.<s>
S'^.y
Qa
H
A
84
TUE ORBIT OP URANUS.
(«.o.0)=»j
(t).C.l) =
Action of Saturn,
//
It
'+20"MsinJ,
_ 0.062V
+8.580 cos A^
— 4.110 sin 2 A,
— 0.009 cos 2.4,
— 0.824 sin 3^,
—0.019 cos 3^,
— 0.228 sin 4A,
—0.008 cos 4yl,
— 0.074 sin bA^
—0.003 cos 5.4,
— 0.025 sin 6 A,
// "
[— 1.032*/ '^
— 2.121 sin 2^,
— 0.392 sin 3.4,
— 0.109 sin iA,
— 0.038 sin 5^1,
— U"
e»s;:fj».^.(-..)-(i'«:e!«-"-+(+?:»y-
—0.793 cos 2i4, —2.421 sin 2^, +2.037 cos 2^,
—0.308 cos 3^, —0.226 sin 3^, +0.318 cos 3i4,
—0.099 cos 4^, —0.055 sin 4^, +0.097 cos 44,
—0.027 cos 6i4, —0.027 sin bA^ +0.038 cos 5J,
.iir — 9".18r
<"»)H:"::~)"".+t'';:Sr)™^-<""-''=(l''::Sy-".+(+'::"T)-^
—1.813 sin 2.4, +0.631 cos %A^
—0.177 sin 3^, +0.013 cos 3^,
—0.044 sin 4^, +0.013 cos 4^4,
—0.016 sin bA^ +0.005 cos 54,
— 0".65 2' — 0".542'
— 0.665 sin 2^4, —1.847 cos 2.4,
— 0.081 sin 34, —0.153 cos 34,
— 0.013 sin 44, —0.044 cos 44,
— 0.005 sin 54, —0.016 cos 54,
(t).c.3) = — 2.265 sin 4,
— 1.163 sin 24,
+ 0.026 sin 34,
+ 0.005 sin 44,
+ 0.002 sin 54,
— 0".04 2'
— 0.126 sin 4,
— 0.503 sin 24,
+ 0.053 sin 34,
+5.656 ros 4,
+2.956 cos 24,
—0.063 cos 34,
— 0.013 cos 44,
—0.004 cos 54,
(U.C.4) =
+0.329 cos 4,
+0.378 cos 24,
—0.032 cos 34.
(p.r.O)
= + 106 sin 4,
— a sin 24,
(p.e.l):
: + 1364 sin 4,
46 sin 24,
sin 34,
sin 44,
+
+
+
+
+
(p.c.2) = -
(p.c.3) = -
+
+9sr
279 sin 4, —
27 sin 24, +
3 sin 34,
f7 2'
61 sin 4, —
24 sin 24, +
761 cos 4,
89 cos 24,
19 cos 34,
5 cos 44,
43 cos 4,
45 cos 24,
9 cos 34,
2 cos 44,
103 cos 4,
9 cos 24,
24 cos 4,
9 cos 24,
(u.s.3) = +5.656 sin 4, +2.205 cos 4,
+2.956 sin 24, +1.163 cos 24,
—0.063 sin 34, —0.026 cos 34,
—0.013 sin 44, —0.005 cos 44,
—0.004 sin 54, —0.002 cos 54,
— 0".03r
(u.8.4) = +0.329 sin 4, +0.126 cos 4,
-10.378 sin 24, .;-0.503 cos 24,
—0.032 sin 34, —0.053 cos 34,
(p.s.l) = _
(p..s.2) = -
(p.8.3)=-
^
33 sin 4,
1338 cos 4,
+
31 sin 24,
+
14 cos 24,
+
7 sin 34, .
+
5 cos 34,
+
2 sin 44,
+
1 cos 44,
—117 2*
—
103 sin 4,
+
281 cos 4,
+
9 sin 24,
+
27 cos 24,
+
3 cos 34,
—8 2*
._
24 sin 4,
+
61 cos 4,
+
9 sin 24,
—
24 cos 24,
THE ORBIT OF URANUS.
86
Action of Neptune.
A, = g-r
—0.08 cos ^3
—0.03 cos 2^3
+0.23 cos S.is
4-0.06 cos 4^3
—0.04 cos 5 At
—0.01 cos 6^,
,•
(w.c.0)=— 39.66 sin A^
—35.36 sin 2^,,
+17.29 sin Sylj
+ 3.91 sin 4yl3
+ 0.99 sin 5^3,
+ 0.42 sin 6^1^
+ 0.19 sin 7^3
+ 0.09 sin 8.43
+ 0.02 sin 9.I3
+11
{v.cA)^ — 6.77 sin A.^ — 0.53 cos A3
— 1.10 sin 2.I3 +0.07 cos 2-13
+23.25 sin 3.43 +4.04 cos 3.43
+ 6.05 sin 4.43 +1.06 cos 4^3
— 3.26 sin 5.43 — 0-66 cos 5^43
O.lj — 0.17 cos 6^
lA, —0.05 cos 7.43
8.43— 0.02 cos 8/I3
9/I3 —0.01 cos 9yl3
— 0.80 sin
— 0.24 sin
— 0.12 sin
— 0.06 sin
— 0.04 sin lO/l,
+1.99945e5i/
(tj.c.2)=— 0.43 sin A^ —0.03 cos A^
—0.03 sin 2.43 +0.01 cos 2.43
+0.75 sin 3^3 +0.08 cos 3^3
—0.10 sin 4^13— 0.08 cos 4.43
—3.20 sin 5.43— 1.17 cos 5.13
—0.83 sin 6.43—0.32 cos 6.4,
+0.57 sin 7.43 +0.22 cos 7^
+0.14 sin 8.13 +0.06 cos 8.43
+0.06 sin 9.43 +0-02 cos 9.43
+0.03 sin 10.4, —0.01 cos 10^3
+0.11 722c V
A,
3.43 +0.01 cos
4.43 — 0.05 cos
5,43 —0.02 cos
6.43 —0.02 cos
7.43 +0.25 cos 7 A,
+0.11 sin 8.4, +0.07 cos S.lj
—0.08 sin 9^13-0.05 cos 9/I3
—0.03 sin 10.43 —0.02 cos IO.43
+0.0071 4c V
((7.c.3)=— 0.02 sin
+0.04 sin
— 0.1 5 sin
—0.08 sin
—0.02 sin
+0.46 sin
3.43
5.43
6.43
(y.s.l)=— 0.49 sin ^3+ 1.75 cos A3
+0.12 sin 2/I3— 2.48 cos 2^
+4.04 sin 3^3 —20.92 cos 3.43
+1.07 sin 4^3— 5.42 cos 4^
—0.68 sin 5.43 + 3.47 cos 5^3
—0.17 sin 6.43+ 0.91 cos 6.4,
—0.04 sin 7^3+ 0.30 cos 7^,
—0.02 sin 8^13 + 0.12 cos 8^3
—0.01 sin 9.43 + 0.06 cos 9^
+ 0.04 cos 10^3
+1.99835Se
(«7.8.2)=— 0.03 sin ^3 +0.13 cos A
+0.02 sin 2.43— 0.16 cos 2^3
+0.08 sin 3.43— 0.60 cos 3^3
—0.08 sin 4^3 +0.15 cos 4J3
—1.17 sin 5.43 +3.22 cos 5^13
—0.32 sin 6^3 +0.86 cos QA^
+0.22 sin 7^-0.57 cos 7yl3
+0.06 sin 8.43— 0.14 cos 8^3
+0.02 sin 9.43— 0.06 cos 9^,
—0.01 sin 10.43 —0.03 cos IO.43
+0.117135e
(,;.«. 3)= +0.02 cos A^
+0.01 sin 3^1-0.04 cos 3^1,
—0.05 sin 4.43 +0.15 cos 4^1,
—0.02 sin 5.43 +0.08 cos 5.43
—0.02 sin 6.43 +0.02 cos 6.43
+0.25 sin 7/13—0.46 cos 7^3
+0.07 sin 8.43— 0.11 cos 8^
—0.05 sin 9^+0.08 cos 9^,
—0.02 sin \^A., +0.03 cos 10^,
+0.007 146e
86
TUB ORBIT OP UUiNUS.
Action of Neptune. — Continued.
A, = g — 1!
(«>.c.4)=— 0.06 sin 9^ —0.05 cos QA^ (/•.s.4)=
—0.09 sin 10^3 —0.08 cos 10^,
+0.00044e.'</
(p.c.O)=
+ 3 sin 3^3
+227 cos A^
+232 cos 2vl3
—229 cos 3^
— 59 cos 4.4g
— 17 cos 5^dj
— 7 cos 6i4s
— 3 cos lAf
+0.01018Je
(p.c.l)=+ 3 sin A,
+ 2 sin 2.13
+23 sin 3.43
+ 8 sin 4^3
— 9 sin 5.43
— 3 sin 6.I3
— 1 sin 7^i,
— 9 cos IA3
—141 cos 3^
— 39 cos 4^3
+ 43 cos 5 Jj
+ 13 cos 6^3
+ 5 cos 7^3
— 0.43322 :e
(p.c.2)=
— 1 sin 3^ ,
— 4 sin 4.^3
— 6 sin 5i43
— 2 sin 6.^3
+ 3 sin 7^3
— 0.03048:e
(p.c.3)=— 0.000202.'e
— 1 cos A3
— 6 cos 2 Jj
+ 6 cos 3^3
+ 5 cos 4.43
+ 17 cos 5^3
+ 7 cos 6^3
— 7 cos 7/I3
—0.05 sin 9^ +0.06 cos 9^,
—0.08 sin 10^3 +0.09 cos 10^
+0.00044Se
(p.«.l)=— 60 sin .4'
— 45 sin 2^3
—107 sin 3Jg
— 29 sin 4^
+ 47 sin 5 J3
+ 15 sin 6 J3
+ 5 sin 7.^3
— 3 cos A3
— 2cos2il3
—23 cos 3^3
— 8 cos 4.^3
+ 9 cos 5.^3
+ 3 cos 6^3
+ 1 cos 7.^3
+0.43394eS^
(p.2)=-
5 sin A3
8 sin 2vl3
8 sin 3 J3
5 sin 4^3
17 sin 5^3
7 sin 6^3
— 7 sin IA3
+0.03053e3^
(p.s.3)=+0.00203e5«7
+
+
+
+
+ 1 cos 3^3
+ 4 cos 4.^3
+ 6 cos 5.4,
+ 2COS6J3
— 3 cos 7 Jj
Perturbations of the Latitude.
(The secular terms being omitted.)
Action of Jupiter.
II II
(&.c.0)= 0.024 sin A^ —0.013 cos A^
(fe.s.l)=— 0.420 sin .^, +0.494 cos J,
(i.c.l)=+0.494 sin A^ +0.420 cos A^
(6.s.2)= +0.004 sin 2.41 —0.017 cos 2Ay,
(i.c.2)=— 0.017 sin 24i —0.004 cos 2 J,
t
THE ORBIT OP URANUS
87
Action of Saturn.
\
(b.c.oy.
=—0.08 sin A3
—0.03 cos 4
—0.03 cos 24
w
II
—0.01 cos 34
(6..
.1)=
=+1.34 sin A
+2.88 cos Ai
(i.c.l)=
- — 1.56 sin At
+2.50 cos 4
—0.01 sin 2Ja
—0.10 cos 2.4j
+0.02 sin 2At
—0.10 cos 24
+0.03 sin 3^
—0.06 cos 3^j
—0.04 sin 3A
—0.06 cos 34
-1-0.02 sin 4^2
—0.03 cos 4^j
—0.02 sin 4^
—0.02 cos 44
(b.s
•2) =
= —0.09 sin Ai
+0.10 cos A^
{b.c.2)=
=—0.09 sin At
— 0.05 cos 4
—0.05 sin 2^
—0.09 cos 2At
—0.01 cos 3^
+0.07 sin 2At
4-0.01 sin SAt
+0.02 cos 24
Action 0/
Neptune
(6.c.0)=
II
=+0.01 sin A3
—0.01 sin 2A3
—0.01 sin 34
+0.01 sin 4^
+0.01 sin 54
II
—0.04 cos 4
+0.00 cos 2^3
4-0.04 cos 34
+0.01 cos 44
//
//
-1-0.01 sin 64
{h.8
.1)=
= +0.13 sin A3
+0.16 cos A3
(l.c.\)=
= — 0.57 sin A3
+0.04 cos 4
+0.09 sin 2A3
+0.26 cos 2^3
—0.39 sin 2^3
-f0.06 cos 2^3
-f 0.08 sin 3^
4-0.28 cos 3^3
—0.33 sin 34
+0.07 cos 34
4-0.01 sin4^
4-0.03 cos 4il3
—0.07 sin 44
-j-0.01 cos 44
—0.02 sin 5^3
— 0.12cos5vl3
+0.10 sin 54
—0.03 cos 54
—0.01 sin 6 A
—0.03 cos 64
—0.01 cos 74
4-0.03 sin 64
+0.01 sin 74
(b.8
.2)=
=+0.01 sin A3
+0.05 cos A3
ib.c.2)=
=—0.07 sin A3
—0.03 sin 3^3
—0.09 cos 3^3
—0.05 cos 44
—0.04 cos 5^
+0.01 cos IA3
-J-0.09 sin 34
+0.05 sin 44
4-0.04 sin 5^3
+0.01 sin 64
—0.01 sin 74
—0.03 cos 34
■u
Action of Jupiter and Saturn.
(Terms multiplied by the product of their masses.)
N^= 2S— J
N.,=— U-^SS— J
K3=—2U-\-iS— J
Nt= SU-J-3S—2J
N^= 2U-\-4S—2J
JV„= 58— 'ir
Nt^—UT-\-(SS—2J
N^=—^U-\-lS-2J
M (
88
THE ORBIT OP URANUS.
Action of Jupiter and Saturn — Continued.
(Terms multiplied by the product of their masses.)
(t).c.O)=+ 0.08
+ 0.04
— 0.01
— 0.35
f — 0.55
t +40.65
— 0.05
(w.c.l)=+ 0.06
+ 0.18
— 0.03
— 0.02
— 0.44
( — 4.23
\ + 8.06
— 0.10
sin iVj
siniV^
sin JV,
siniVj
sin JV,
sin Nj
siniVj
sin iV,
sin Nj
sin J\r3
sin N^
sin JVj
siniVg
sin JV,
sin iVa
+ 0.51 cos N^
4- 0.01 cos JV3
4- 0.05 cos iVi
— 1.30 cos jv;
— 0.03 cos iVg)
—10.50 cos JVj i
+ 0.03 cos iVg
— 0.21 cos Ni {V.8
+ 0.01 cos iVj
-f 0.08 cos iV^3
-I- 0.09 cos iVi
— 0.61 cos iVj
— 3.87 cos ^;)
— 8.38 cos iV,)
+ 0.03 cos iV,
l)=+0.26 s
—0.04 s
+0.08 s
—0.02 s
+0.30 s
f +2.64 s
t +7.35 s
—0.04 s
njv;
niVe
niV,
+0.27 cos Nt
—O.ncosNt
+0.03 cos N^
+0.08 cos Nt
—0.58 cos N^
+4.64 cos JV„ )
+4.41 cos N, i
+0.00 cos Nt
(v.c.2)= I —0.24 sin JV; —0.22 cos iV, )
t +0.47 sin JVj —0.54 cos iV, )
(p.c.O)i=+l 1 sin Ni —3 cos JVj
(v.s,2)= ( +0.16 sin N^ +0.26 cos iV„ 1
I +0.54 sin N, +0.47 cos N, j
Two of these arguments, namely, 5S — 2J, and — Sg^GS — 2J, are of very
long period, that of the first being about 880, and that of the second about 1590
years. It will, therefore, be convenient to tabulate them both as functions of the
time for the time during which the theory is to be used. To make their effect as
small as possible during the period for which the provisional ephemeris is to be
computed, we shall suppose the longitude of epoch, mean motion, and longitude of
the perigee to be affected with the negative of the following corrections :
he= + 27.27,
^Tt = + 27.27,
5n = — 0.1172.
Reducing these corrections to corrections of the co-ordinates, and adding them to
the terms of long period in the true longitude and logarithm of radius vector, we
shall have for these terms.
(w.c.O) = — 0.546 sin JV;
+ 40.650 sin ^^
0.032 cos A^„
10.500 cos N, + 27.27 — 11.72 T
(«.s.l) = 2.63 sin JV„ + 4.64 cos N^ + 7.35 sin N, + 4.42 cos JV,
(D.c.l) = — 4.22 sin N, — 3.87 cos N^ + 8.06 sin iV, — 8.39 cos N, — 1.107*
THE ORBIT OF URANUd.
89
(».8.2) = + 0.16 sin N^ + 0.26 cos N, + 0.54 sin JV, + 0.47 cos N,
(».c.2) = — 0.24 sin N^ — 0.22 cos N^ + 0.47 sin N, — 0.54 cos N, — 0".07 T
(p.c.O) = 4- 2 cos iVe + 10 sin N., + 32 cos N^ + 22
(p.«.l) = — 41 sin A'; — 43 cos N^ + 82 sin N, — 85 cos N,— 12T
(p.c. 1) = — 29 sin N^ — 45 cos JV; — 73 sin iV, — 44 cos J\^,
The values of those and of the other socuhir terms and terms of long period for
the period during which Uranus has been observed, are given in the following
table :
(ac.O)
Nfpluiie
.TnpitHr and Saturn
Sum.
(long per.)
(long per.)
1700
ti
+85.54
//
+4.86
It
+90.40
17.50
+38.45
+1.90
+40.35
1760
31.25
1.48
32.73
1770
24.80
1.11
25.91
1780
19.09
0.80
19.89
1790
14.12
0.54
14.66
1800
9.89
+0.33
10.23
1810
6.42
0.17
6.59
1820
3.69
+0.07
3.76
1830
1.71
0.00
1.71
1840
+ 0,48
—0.02
+ 0.46
1850
0.00
0.00
0.00
1860
+ 0.27
+0.06
-f 0.33
1870
1.29
+0.16
1.45
1880
+ 3.06
+0..30
+ 3.36
Values op i
u.s.l)
(1)
(2)
(3)
(4) (r.)
Jupiter
Saturn
Neptune
Neptune Jupiter & Saturn
Sam.
(aec.)
(sec.)
(sec.)
(long per.) (long per.)
ff
//
ft
//
ft
rf
1700
+ 1.82
+13.77
+0.75
—212.73
—10.18
—200.57
1750
+1.22
+ 9.18
+0..50
—141.97
— 9.11
—140.18
1760
1.10
8.26
0.45
—127.76
— 8.83
—120.78
1770
0.97
7.34
0.40
—113.. 55
— 8.54
—113.38
1780
0.85
6.43
0.35
— 99.33
— 8.24
— 99.94
1790
0.73
5.51
0.30
— 85.11
— 7.93
— 80. .50
1800
0.61
4.59
0.25
— 70.90
— 7.61
— 73.06
1810
0.49
3.67
0.20
— 56.69
— 7.28
— 59.61
1820
0.36
2.76
0.15
— 42.49
— 6.94
— 46.16
1830
0.24
1.84
0.10
— 28.31
— 6.60
— 32.73
1840
+0.12
+ 0.92
+0.05
— 14.14
— 6.26
— 19.31
1850
0.00
0.00
0.00
0.00
— 5.91
— 5.91
1860
—0.12
— 0.93
—0.05
+ 14.12
— 5.57
+ 7.46
1870
—0.24
_ 1.84
—0.10
2S.23
— 5.23
20.82
1880
—0.36
— 2.76
—0.15
_
+ 42.32
— 4.89
+ 34.16
12 May, 1873.
90
THE ORBIT OP URANUS.
Valuks of (u.c. 1)
.<'^
(2)
(3)
(•*)
(5)
Jupiter
Saturn
N«ptnne
Nt-ptuna
Jupiter &rJAtarn
Sam.
(a«c.)
(auu.)
(»«o.)
(long p«r.)
(long per.)
//
//
n
/;
II
II
1700
+ 17.19
+10.07
+7.20
+37.63
—1.98
+76.71
1750
+ 11.4(5
+ 11.11
+4.80
+28.07
—2.28
+53.70
i7(;o
10.31
10.00
4.32
20.46
—2.30
48.73
1770
9.17
8.89
3.84
24.10
—2.45
43.55
1780
8.03
7.78
3.36
21.00
—2.53
38.23
nno
0.88
0.07
2.88
18.05
—2.01
32.77
1800
5.73
5.50
2.40
10.16
—2.09
27.16
1810
4.58
4.44
1.93
13.23
—2.76
21.41
1820
3.44
3.33
1.44
10.15
—2.83
15.53
18;{0
2.20
2.23
0.96
6.92
—2.90
9.49
1840
+ 1.15
+ 1.11
+0.48
+ 3.53
—2.97
+ 3.30
1850
0.00
0.00
0.00
0.00
—3.03
— 3.03
1800
— 1.15
— 1.11
--0.48
— 3.08
—3.08
— 9.50
1870
— 2.29
— 2.23
—0.96
— 7.51
—3.13
—10.11
1880
— 3.44
— 3.33
—1.44
—11.49
—3.16
—22.86
(i>.».2)
const. =
= — 0".14.
(1)
(2)
(3)
(4)
(5)
Sum.
//
//
II
n
//
II
1700
+0.11
+0.81
+0.04
—12.46
—0.72
—12.36
1750
+0.07
+0.54
+0.03
— S.31
—0.70
—8.51
1700
0.06
0.49
0.03
—7.48
—0.69
—7.73
1770
0.00
0.43
0.02
—6.64
—0.08
—6.95
1780
0.05
0.38
0.03
—5.81
—0.66
—6.10
1790
0.04
0.32
0.02
—4.98
—0.65
—5.39
1800
0.04
0.27
0,02
—4.15
—0.04
—4.60
1813
0.03
0.22
0.01
—3.32
—0.03
—3.83
1820
0.02
O.IC
0.01
—2.49
—0.0 1
—3.05
1830
0.01
0.11
+0.01
—1.00
—0.60
—2.27
1840
+0.01
+0.05
0.00
—0.83
—0.58
—1.49
1850
0.00
0.00
0.00
0.00
—0.57
—0.71
i8(;o
—0.01
—0.05
0.00
-1-0.82
—0.56
+0.06
1870
—0.01
—0.11
—0.01
1.65
—0.54
0.84
1880
—0.02
—0.16
—0.01
+2.48
—0.53
+ 1.62
(«.f.2)
const. =
+ 0".13.
(1)
(2)
(3)
(4)
(■^')
Snra.
//
n
It
It
tf
II
1700
+ 1.00
+o.ns
+0.43
+2.20
—0.11
+4.62
1750
+0.07
+0.05
+0.28
+ 1.08
—0.13
+ 3.29
17 CO
+0.00
0.59
0.25
1.55
—0.12
3.00
1770
+0.54
0.52
0.22
1.41
—0.12
2.70
1780
+0.47
0.40
0.20
1.26
—0.13
2.39
1790
+0.40
0..39
0.17
1.11
—0.13
2.07
1800
+0.34
0.33
0.14
0.94
—0.13
1.75
1810
+0.27
0.26
0.11
0.77
—0.14
1.40
1820
+0.20
0.20
0.08
0.59
—0.14
1.06
1830
+0.13
0.13
0.06
0.39
—0 14
0.70
1840
+0.07
+0.07
40.03
+0.20
—0.15
+0.35
1850
0.00
0.00
0.00
0.00
—0.15
—0.02
1800
—0.07
—0.07
—0.03
—0.21
—0.15
—0.40
1870
—0.13
—0.13
—0.00
—0.43
—0.16
—0.78
1880
—0.20
—0.20
—0.08
—0.08
—0.16
—1.19
w>
THE Oil BIT OP URANUS
91
(.•..S.3)
(1)
(2)
(3)
(4)
(S)
Jupiter
Saturn
Neptune
Neptune Jupiter & Saturn {
Bum.
(8W).)
(8«C.)
(««o.)
(long per.)
(long per.)
//
II
//
//
//
rr
noo
-f-0.01
+0.05
0.00
—0.76
—0.70
1750
0.00
+0.03
—0.51
—0.48
17 GO
0.03
—0.40
—0.43
1770
0.03
—0.40
— 0.3T
1780
0.02
—0.35
—0.83
1790
0.03
—0.30
—0.28
1800
0.02
—0.25
—0.23
1810
0.01
—0.20
—0.19
1820
0.01
—0.15
—0.14
1830
+0.01
—0.10
—0.09
1840
0.00
—0.05
—0.05
1850
0.00
0.00
0.00
18G0
0.00
+0.05
+0.05
1870
—0.01
+0.10
+0.09
1880
—0.01
+0.15
+ 0.14
(«.p.3)
(1)
(2)
O)
(4)
Sura.
//
//
II
//
II
1700
-fO.OC
+0.06
+0.02
+0.13
+0.27
1750
4-0.04
+0.04
+0.02
+0.10
+0.20
17C0
0.04
0.04
0.01
+ 0.09
+0.18
1770
0.03
0.03
0.01
+0.08
+0.i5
1780
0.03
0.03
0.01
+0.07
+0.14
1790
0.02
0.02
0.01
+0.00
+0.12
1800
0.02
0.02
0.01
+0.00
+0.11
1810
0.02
0.02
+0.01
+0.05
+0.09
1820
0.01
0.01
0.00
+0.04
+0.06
1830
+0.01
+ 0.01
6.00
+0.02
+0.04
1840
0.00
0.00
0.00
+0.01
+0.02
1850
0.00
0.00
0.00
0.00
0.00
ISfiO
0.00
0.00
0.00
—0.01
—0.02
1870
—0.01
—0.01
0.00
—0.02
—0.04
1880
— O.Ol
—0.01
0.00
—0.04
—0.00
For the uadi
t;.S VECTOR. 1
t'^ALUES OF (|).(
.0)
0)
(2)
<3)
(4)
(5)
Sum.
II
//
II
//
//
II
1700
+0.4
+3.5
—0.2
+ 104
+ 13
+ 181
1750
+0.3
+2.3
—0.1
+110
7
+120
1700
0.3
2.1
99
107
1770
0.2
1.9
88
5
95
1780
0.2
1.0
77
4
83
1790
0.2
1.4
07
3
72
1800
0.2
1.2
50
2
60
1810
O.I
0.9
45
2
48
1820
0.1
0.7
34
+ 1
36
1830
+0.1
0.5
22
23
1840
.0
+0.2
+ 11
+ 11
1850
.0
0.0
— 1
— 1
18fi0
.0
—0.2
— 11
— 2
— 13
1870
—0.1
—0.5
22
— 2
— 25
1880
—0.1
—0.7
— 33
— 3
— 37
«ll
92
THE ORBIT OF URANUS.
(p.«.l
1 count.
- + 63.
(1)
(2)
(3)
(4)
(.'•)
Jupiter
Saturn
Neptune '
NnptuiiB
Jupiter&i^atiirn
Snm.
(sec)
(aeo.)
(a«o.)
(long per.)
(long pur.)
//
//
n
«
It
/;
1700
+ 180
+ 170
+75
+390
—21
+809
1750
+ 120
+ 117
+50
+302
—25
+627
1760
108
105
45
280
20
675
1770
96
94
40
255
28
620
1780
84
83
35
228
29
403
1790
72
70
30
200
30
405
1800
CO
59
25
170
31
346
1810
48
47
20
139
33
285
1820
36
35
15
106
33
222
1830
24
23
10
73
34
158
1840
+ 12
+ 12
+ 5
+ 37
34
95
1850
35
+ 28
1860
— 12
— 12
— 5
— 39
36
— 41
1870
24
23
10
80
36
—110
1880
— 36
— 35
—15
—122
—37
-182
(p.r.l)
const. —
■ + 73.
(1)
(2)
(3)
(4)
(5)
Snm.
//
//
//
//
//
II
1700
—20
—147
—9
+2241
+104
+2242
1750
—13
— 98
—6
+ 1490
+ 94
+ 1546
1760
— 12
— 88
—5
1346
91
1405
1770
—10
— 78
—5
1197
88
1265
1780
— 9
— 69
—4
1047
85
1123
1790
— 8
— 59
4
897
82
981
1800
— 6
— 49
—3
747
78
840
1810
— 5
— 39
—2
598
74
699
1820
— 4
— 29
—2
448
11
557
1830
— 3
— 20
—1
299
68
416
1840
— 1
— 10
—1
+ 149
64
274
1850
61
+ 134
1860
+ 1
+ 10
+ 1
— 149
58
— 6
1870
3
20
1
298
54
— 147
1880
4
29
2
— 447
+ 51
— 288
(p.s.2
) const.
= +5.
(1)
(2)
(3)
(4)
tl
(5)
Sum.
n
//
II
;/
//
1700
+ 12
+ 12
+*
+ 28
+61
1750
+ «
+ 8
+fi
+21
+48
1760
7
7
5
19
43
1770
6
6
5
17
39
1780
6
6
4
15
36
1790
5
5
4
13
32
1800
4
4
3
11
27
1810
3
3
2
8
21
1820
2
2
2
6
17
1830
2
2
1
4
14
1840
+ 1
+ 1
+ 1
+ 2
10
1850
+ 5
1860
_ 1
— 1
—1
— 2
1870
2
— 2
—1
4
— 4
1880
— 2
— 2
—2
-6
— 7
"f
*
THE ORBIT OF URANUS.
93
(p.c.2) const.
-+4
(I)
(2)
W
(4)
(r>)
JupitHr
Saturn
Nnptuiia
Niiptuiie
JupitHr jiBatnrn
Som.
(seo. )
(8B0.)
(»O0.)
(long per.)
(long per.)
ti
'/
It
'/
//
//
noo
—1
—10
+157
+ 150
1750
—1
— 7
+ 105
+ 101
17(10
— 1
_ G
!)5
!I2
1770
—1
_ fl
84
81
17H0
— 1
_ 5
74
72
1790
—1
— 4
03
02
ISOO
_ 3
52
63
1810
_ 3
42
43
1820
— 2
31
33
1830
— 1
21
24
1840
_ 1
+ 10
13
IS.'iO
+ i
18(i0
+ 1
— 10
— 5
1870
1
21
_10
1880
+ 2
— 31
—25
ReiJuced E.ipres8iom for ihe Latitude of Uranus.
If we represent by F,, T'i, T^, the distances of Uranus from its descending nodes
on the respective orbits of Jupiter, Saturn, and Uranus, wc find tlie following
perturbations of the latitude, which arc independent of the mean longitude of the
disturbing planets.
5/3= — 0.0 11 4< cos F,
— 0.0477< cos Fj
— 0.0125/ cos Fs
-f 0.245
4- 0.386 sin «/ + 0.266 cos g
— 0.043 sin 2g -j- 0.006 cos 2ij.
To find how far the last five terms may be represented by simple corrections to
the elliptic elements, we first represent the cftect of minute corrections to the
inclination and node of Uranus as a function of its mean anomaly. Tutting « for
the argument of latitude of Uranus, we have to a sufficient degree of approxima-
tion
6/3 = sin m6<^ — sin <Ji cos iihO
u =<74-w+2esin(/
sin M = — e sin w
-\- cos (J sin g -\- sin o cos </
-\- e cos (J sin Ig -\-e sin o cos 2g
4
94
THE OIiniT OP URANUS.
COS u = — e cos (J
-j- cos (J cos 4) — sin (J sin g
-\- e cos (J cos '2(j — e sin o sin 2(/,
Substituting these values of sin u and cos u in the expression for 5/i, and putting
sin ^hO = 6'0, we liavo
5/3 = c cos uh'O — e sin o 3<^
+ ( ''OS u5(^ -(- sin u(VO) sin g -\- ( sin u^^ — cos ufi'O) cos q
-\- (e cos o5^ -\- e sin w6'0) sin 2^ -f" (»■ sin (j(^<^ — e cos urVO) cos 2tf.
To represent the numerical coefficients of sin g and cos if in Sjiif wc must put
cos uh<p + sin uh'O = ()".386
sin 0)5^ — cos u6'0 = .206,
Since u ^ 95° 3', this gives
h'O
0.231 ;
0.409:
5,iJ = — 0.013
//
+ 0.380 sin .7 + 0.200 cos g
-j- 0.018 sin 2,j -j- 0.013 cos '2g
Subtracting tins expression from the corresponding terms of (\i, wc have left
^ii=^ 0".258 — 0".()G1 sin 2g — 0".007 cos 2y.
The first term of this expression sliows that the mean orbit of Uranus at the
present time is a small circle of the sphere one-quarter of a second nortli of its
parallel great circle.
If we put
V = longitude of Uranus in its orbit, referred to the equinox and ecliptic of
1850, we have
F, = « — 127° 37
V, = v — 126 45
Fa = w — 155 32
Substituting these values in the first three terms of 8;3, and multiplying the last
term by the factor (1 + /u) by which the adopted mass of Neptune, jj^^^, must
be multiplied to obtain the true mass, we find
hiS = (4''.69 + l".14,u) Tcos v - (5".24 + 0".52/i)7'8in v.
To these terms must be added those which arise from the motion of the ecliptic.
In the absence of any exhaustive investigation of the obliquity and motion of
the ecliptic, I adopt the elements of Hansen, employed in his " Tables du Soleil,"
because they are a mean between the results of others, and are very accordant
with recent observations. The secular motion of the obliquity there employed is
— 46". 78.
*V
iS
THE ORHIT OF URANUS.
95
IIiiiiHon mentions — 5".39 ns the corrospomUng motion at the equinox of 1850,
found by Olufson, but I cunuot reproduce tliis result from the sccuhir diminution
witii nny musses of Mercury, Venus, and Murs, wliich seem to me probable. The
expressions in terms of the masses given by Lc Verrier ure {Aumles ile VObserva-
tvire Impcriul <le Paris, tome ii, p. 101),
Secular change = - 47.59 - 0.52i' - 28.90*-' - 0.83»'''
Mot. at eipiinox = + 5.89 + 0.021^ + 7.57^' + 0.73/'.
In this expression the masses of Mercury, Venus, and Mars are represented by
l+v 1 +
and
1-f
„.„. ' '! .«. respectively. The influence of admissible
3,000,000' 401,847' " ' 2,080,337' ^^ '"-"^ _ ^'
changes in the masses of the other plants is insensible.
From the researches of l-e Verrier on the motions of the four inner planets T
conclude that the following arc about the most probable distribution of the correc-
tions of the masses necessary to produce the motion of the obliquity given by
Hansen, namely,
V =- I
v' = — .018
v'"=- iV
These values give for the motion at the equinox of 1850
-f- 5". 43
Introducing the secular variation of these motions avo have, for the change in
the latitude of any celestial body near the ecliptic, arising from motion of the
ecliptic,
h(i = (5'.43 T-\- 0''.19 T-) cos v + (4G".78r— 0".0G T-) sin v.
Combining this with the change arising from the motion of the orbit of Uranus,
we find
hl3=\(lO".l2 + l".Uu)T-\-0".l9T^cosv
^ J(41".54 — 0".52,u) r— 0".06 ^ | sin v.
We may represent these expressions in the usual way by secular variations of
the inclination and node of Uranus. But, owing to the small inclination, and
consequent rapid motion of the node, it will be necessary to include the coefficients
of the second power of the time. On the other hand, no distinction between r
and 6 is necessary. Putting ^ for the inclination of the orbit, for the longitude
of the node referred to the equinox of 1850, and
2? = sin <J» sin 0,
q = sin^ cosO;
we have
sin /3 = — pcQ9V-\- q sin v
cos /35/3 = — hpco9v-\- 6q sin v.
i
96 THEORBITOPURANUS.
From the expressions for p and q wo obtain
cos <^ Di(p = sin D^p -j- cos D,q;
sin ^ i><0 = cos Dfp — sin Diq.
And, neglecting (A<?> ^ X sin <f , we have farther,
cos ^E^tip = sin ^ {Dfiy + sin eD";p + cos Oi)', j ;
sin ^D\0 = — 2 cos^DfiB,^ -\- cos OZ/jj' — sin OZ/V/-
Since <^ is only 46' wc may pnt cos ^ and cos (3 both equal to unity in these
expressions, while we have, for 1850,
sinO= 0.9573
cosO= 0.2890
D,p = — 10".12 — r.i4«
D,q =+41.54— 0.52;^
.b^,p = _ .38
L\q = — 0.12
logsin<^= 8.129G06.
The above formula; then cive
D,^ = + 2".3l — l".24,(t
Dfi = — 31(i7".5 4- 12'.%
D\^ = + 0".26
B'fi = + 5".6
^ = ^0 + (2".3l
= 0o — (31G7".p
1".24//) T'+O'.IS 7"
12".fi'</)7'+ 2.8 r.
or, adding Struve's precession, we have when is counted from the mean equinox
of date,
= Oo + (1857".7 + 12".6//) T-\- 3".9 T".
Using the values of <^ and given by these expressions, the latitude, secular
variation included, will be given by the expression
sin/3 = sin(f sin (t; — 0).
If we take from a. table, as the principal term of the latitude, the value of sin
<jio sin (y — 0), the secular term to be added will be
S(2'.31 — r.24«) T-\- 0".13 r j sin (« — 6).
If we represent, as before, by « the variable distance of the perihelion from the
node, this term will be allowed for by adding to (/>.s.l), (/>.c.l), etc., the terms
,
THE ORBIT OF U 11 AN US.
97
Jk
(b.c.0) = — e sin u5<^,
(h.K.l) = cosu^(J),
(i.f.l) = sinut^^),
(i.s.'i) = ecostaS<^,
(6.C.2) = csinuf^^;
Avlicrc
Tutting in the above expressions
l".->U)T-{-0''.ViT*
,,,3=95^3' + 3459 "7;
cos(j = — .0880 — .0167 r,
siu(o = + .99Gl — .00157;
we find
(!>.c.0)=:-(0".ll— 0".06/)7'
{h..U) = — (0 ••-'<> - -11")?'- 0".05r»
(/>.c.l) - (-i .30 - 1 .24-/) 7'+ .127'"
{I>.k2)^ -0.01 7-
(&.c.2)= (0.11-0.06.7)7.
We have, finallv, to consider the terms of long period in hy; nnd <'/.; which luvvc
been omitted froni the periodic perturbations produced by Neptune, in computing
the terms of <\i on page 61, and wliich arc as follows:
(^^1= l".43cos(2/'— J/)— 0''.39sin{2/'- g)
— 2 .12 cos (4/' — 2<j) + 1 .00 sin (4/'- 2*/)
-f .20 cos (6r — 3(7) — .04 sin (G/'— 3i/)
-f constant = C.OO
^•= 0".80cos(2?'— J/) — 2". 28 sin (2/'- j/)
— 1 .06 cos (4/' — 2i/) — 1 .85 sin (4/' — 2.'/)
+ .04 cos (6r — 3.7) + .19 sin (e/"— 3y)
-\- constant = 0".364.
For the period during which Uranus has been observed, these values of hri and tk
may be replaced by tlie following:
,^>. = _0"807'
^/• = + 0.277'
which arc to be nudtiplied by the factor 1 + ;,. The corresponding perturbation
of the latitude will be
}^S = sin v'v; — cos V k.
Tutting for v its approximate value
V =1 (/ -f- (J + 2t' sin g
and dtncloping to quantities of the first order with respect to the eccentricities,
we have . , . x
sin V = sin {g -f ■ c) + e sni (2,7 + 1.>) — f sm u
cos V = cos (j/ + u) -i- c cos (2r/ + (j) — c cos u.
13 M»y. 1873.
A
98
THE 11 B 1 T O 1' U U .\ N U S.
Substituting for o its value, 1'2"45', and for ly; and tJc their above vahics in the
expression for 6ti, we find that the terms of S3 in question will add the following
terms to (b.c.O), (ft.s.l), etc,
(bj;,0) = — .010 ^>7 + .046 Ik = + 0".027' (1 + fi)
(b.8.\) = + .975 A>7 + .221 hk = — .727' (1 + n)
(A.c.l) = 4- .221 hy, — .975 }k = — .44 7^(1 + ^)
(/m.2) = + .040 r^>7 + .011 Ik = — .047' (1 + //)
(/>.f.2) = - (b.c.O) = - .027^ (1 + fi)
These vahies will be employed in the construction of the provisional ephemeris,
but not in the tables.
Collecting all three classes of terms discussed in this section, we have the
fallowing constant and secular terms in (/>.f.O), [b.n.l), etc.
(ft.c.O) = + 0".26 + (-0".09 + O-OS:/)?"
(h.8.1) = (— 0".92 — 0".Glr/)7'— 0".05 7^
(h.c.l) = (+ 1 .86 — 1 .68,/) 7'+ .127"'
(6.S.2) = —0.06 — 0.057'
(6.C.2J = — .01 + (0 .09 — .08 /) T
PosUloiia of Uranus resulting from the preceding tlicory.
The next step in order is the preparation of an ephemeris of the planet for
comparison with observations. As this provisi(mal theory is, for future use, super-
seded by the tables appended to the present work, it seems unnecessary to enter
very fully into the details of the computation of the ephemeris. The perturba-
tions of the longitude, logarithm of radius vector, and latitude, were tirst com-
puted by the formulte already given.
hv = (tJ.c.O) -\- {v.c. 1 ) cos J/ -f (!'.c.2) cos 2// + etc. ,
4" {v.8-V) sin g -\- (v.8.2) sin 2g -\- etc.,
Jf5p = (p.c.O) + (p.c.l) cos .7 -f (p.c.2) cos 2.7 -j- etc.,
-f (p.s.l) sin flf + (p-^.S) sin 2^ + etc.,
S.3 = {b.c.O) -\- (J).c.l) cos^ + {b.s.l) sin g.
Each coefficient (v.c.O), {v.c.l), etc., is composed at most of the following quan-
tities:
1. The five classes of secular, long period, or constant terms, the separate values
of wliich, with the sum of all, are given on pages 89 to 93.
2. Periodic terms due to the action of Jupiter, Saturn, and Neptune, given on
pages 83 to 87.
3. Terms depending on the product of the masses of Jupiter and Saturn, given
on page 88, omitting those depending on iV„ and N^, because they arc given in
column 5 of the terms of the first class.
The sum of the perturbations thus computed is given in the third column of the
following ephemeris.
An approximate value of the perturbations produced by Neptune alone is inde-
pendently computed for every fourth date, and tlie result is given in the fourth
THE ORBIT OF URANUS.
99
column. The secular and long period terms are liere taken from columns (3) and
(4) of the tables on piigcs 89 to 93.
The elliptic co-ordinates were then derived from the following elements, which
are a little different from those employed in the computation of the perturbations.
Elements III.
of Uranus.
7t, 168°
15' 12".0
e, 28
25 29 .5
0, 73
11 58.0
4,,
46 20.0
e.
.0469436
e, (in sec.) 9682".81
n, 15426.196
log«, 1.2828989
Eed. to Ecliptic, — 9".37 sin 2 (y — 0)
The longitudes thus found arc corrected for lunar, but not for solar nutation,
and the results are given in the fifth column.
The column "correction" arises in this way: after the comparison of the eplie-
mcris with observations was nearly completed, it was found that some errors had
crept into the former, the most importmit of which was the employment of a mean
anomaly, q, corrected for secular variation of the perihelion in the computation
of the perturbations from the preceding formula;. As a large portion of the com-
putations on the provisional ephemeris had been made by assistants furnished by
tlie Smithsonian Institution and Nautical Almanac, I deemed it prudent to make
a careful rccomputation of the perturbations for every si.xtli date during the entire
period of the modern observations. The longitudes actually printed m the fifth
column are the results of the original incorrect computation, while the numbers
in the next column sliow the several corrections to be applied to obtain the results
of my final revised computation.
During the period of the modern observations the ephemeris is computed for
intervals'of 120 days, and the selected dates are all exact multiples of that interval
before or after the fundamental epoch, 1850, Jan. 0, Greenwich mean noon.
For convenience of reference the dates are numbered from an epoch earlier by 212
intervals, and the number is given in the second column.
Between 1796 and 1801 no observations wortli using were made on Uranus, the
ephemeris has. therefore, not been extended over this iiit(;rval.
.L
100
THE ORBIT OP URAxNUS.
IleUOCENTKlC
El'UEMEIUS OF UlLXNUS FllO.M THE I'BEOKIH.MJ TKOVIiSIONAl, TlIEORY.
[T
lu longitailea
are oorrwtfil fur lunar but not for solar natatioD.]
Dat».
flrftfiiwioh
No.
Sum of
lieituiba-
ApproxiniAlH
ptTtiirbutioiiH
producHd by
Neptune,
Longltado.
CorrfHition
Latitude.
Logarithm
K.iclins
lUtiUU UUOll.
tioiis.
Tector.
/ //
//
/ //
//
/ //
lf.90, Dec. 23
4-4 28.5
4-23C.1
59 40 35.2
—10 7.8
1.2870828
Dcf. 24
+ 4 28.5
59 41 16.5
—10 7.3
1.2876789
1712, April 2
+8 37.9
155 24 42.0
+45 58.7
1.2026650
April 3
+8 .ST.O
155 25 29.0
+45 58.8
1.2626640
1715, Mar. 4
-ffi 5«.3
4-136.2
109 10 21.4
+40 2.1
1.2623890
Mar. 5
-J-r> .58.2
169 11 7.7
+46 2.1
1.2623893
Mar. 10
-f (! 57.5
169 15 0.0
+46 1.7
1.2623911
April 2!)
-)-(i 51.1
4-134.8
109 53 42.2
+45 58.0
1.2624006
April 30
f(! 51.1
109 54 28.8
+45 57.9
1.2024102
1748, Oct. 21
—0 28.9
— 65.0
310 13 19.3
—41 27.0
1.3001855
Oct. 22
—0 28.8
316 13 58.8
—41 27.9
1.3001875
1750, Sept. 13
+0 6.3
— 51.3
323 44 14.9
—43 47.2
1.3013714
Sept. 14
-fO 6.3
323 44 54.0
—43 47.3
1.3013731
Oct. 14
fO G.6
— 50.8
324 4 22.1
—43 52.4
1.3014105
Oct. 15
+0 6.6
324 5 1.0
43 52.5
1.3014181
Dec. 3
-1-0 8.0
— 49.8
324 36 49.5
—44 0.0
1.3014850
Doc. 4
-1-0 8.0
324 37 28.5
—44 0.8
1.3014872
1753, Doc. 3
—0 5.6
— 25.1)
336 25 0.6
—40 1.1
1.3025789
Di'c. 4
— 5,6
330 25 39.3
—40 1.2
1.3025790
175f>, Sept. 25
—0 40.5
— 0.4
347 25 34.9
—40 9.2
1.3029303
Sept. 2(i
—0 49.5
347 20 13.6
—46 9.1
1.3029363
IICA, Jan. 15
-1-1 24.0
16 13 35.0
—38 36.9
1.3003508
Jan. Ifi
-1-1 24.0
16 14 14.2
—38 36.7
1.30034HC
17C8, Dec. 27
-f-1 33.1
30 3 35.2
—27 42.4
1.2955331
Dec. 2f)
-fl 33.1
+ 100.9
36 4 41.1
—27 41.5
1.2955274
Dec. 31
-f-1 33.1
30 6 5.0
—27 40.7
1.2955216
17fi9, Jan. 15
-1-1 33.5
36 16 4.6
—27 3,.2
1.2954753
Jan. 18
+ 1 33.5
36 18 4,5
—27 33.0
1.2954659
Jan. 21
-fl 33.5
+ 106.9
3(1 20 4,5
—27 31.6
1.2954566
Jan. 24
-f 1 33.5
3(1 22 4,4
—27 30.3
1.2954476
1781, Jan. 1
1
+:i 57.46
-1-178.9
8(1 36 13.23
+0.16
+ 11 3.03
1.2785972
May 1
2 .
4-3 53.41
88 2 34.70
+ 12 11.14
1.2780883
Aug. 2it
3 '
-f3 48.58
89 29 7.41
+ 13 18.33
1.2775834
Dec. 27
4 1
4-3 43.09
90 55 51.40
+ 14 25.14
1.2770827
1782, April 2(J
1
4-3 37.18
+180.4
92 22 4(1.89
-1-15 31.53
1.2765870
Anf;. 24
'1 I
43 30.97
93 49 53.89
+ 10 .37.47
1.2760968
Dec. 22
^ ,
-1-3 24.52
95 17 12.51
+0.18
+ 17 42.88
1.2756124
1783, April 21
8
-1-3 18.03
96 44 42.73
+ 18 47.73
1.2751345
Ang. 1!»
!)
-i-3 11.87
4 180.1
98 12 24.96
+ 19 51.93
1.2746633
Dec. 17
10
4-3 6.01
99 40 19.00
+ 20 55.07
1.2742012
1784, April 15
11
4-3 0.78
101 8 25.13
4-21 58.09
1.2737412
Aug. I.i
12
42 56.24
102 36 43.29
+23 0.92
1.2732907
Doc. 11
13
4-2 52.82
+178.6
104 5 13,79
+0.23
-1-24 2.46
1.2728469
1785, April 10
14
42 50.58
105 33 56.61
+25 3.14
1.2724105
Aug. H
15
+ 2 49.29
107 2 51,36
+20 3.05
1.2719806
Dec. (i
1(1
-42 49., 35
108 31 58,18
+27 1.98
1.2715564
1780, April 5
17
-42 50.53
+ 175.5
110 I 16.88
+ 27 59.97
1.2711405
Aug. 3
IH
-f2 53,29
111 30 47.68
+28 57.02
1.2707299
Dec. 1
in
4-2 57.31
113 30.11
+ 0.31
-f 29 53.03
1.2703265
1787, Mar. 31
20
43 2.38
114 30 23.81
+.30 47,92
1.2699294
July 2!)
21
+3 8.51
+ 171.0
116 28.62
+31 41,72
1.2095394
Nov. 2(1
22
4-3 15.48
117 30 44.11
+32 34.33
1.2091,551
\
+
T H K
11 B I T
OF UK AN US.
101
IlKLlOCENTllIO El'llE.MElllS OF UllANUS. — VunlillUed
Datrt.
Sam of lAPPf'f im»l« (
Logaritlim
Ore«iiwicli
i-.o.
. I p«rturl)iitioii»
Loiigitiiile. Correction.
Latitude.
Radius
vector.
tui-aii iiuoii.
-
tlOUB.
Neptune.
1
' " i
//
o / //
/ ft
1788, Mar. 25
23 '
+3 23.02 !
3 31.30 i
119 1 9.79
-1-33 25.72
1.2087775
July 2:!
Nov. 20
24
120 31 45.79
34 15.81
1.2084005
25
3 39.82 j
-1-105.1
122 2 31.13
4-0.28
35 4.58
1.2080423
1781), Mar. 20
2ri
3 38.07 j
123 33 25.90
35 51.99
1.2070852
July 18
Nov. 15
27
3 57.40 I
125 4 29.49
30 38.01
1.2073353
28
4 5.97
120 35 41.50
37 22.50
1.2009932
1790, Mar. 15
29
4 14.35
-fl57.8
128 7 1.70
38 5.50
1.2000592
July l;5
Nov. 10
30
4 22.48
129 38 29.88
38 47.00
1.2003330
31
4 30.00
131 10 5.37 -1-0.25
39 20.85
1.2000107
1791, Mar. 10
32
4 30.95
132 41 47.99
40 5.04
1.2057087
July 8
Nov. 5
33
4 42.95
-1-119.1
134 13 37.15
40 41.58
1.2054082
34
4 48.11
135 45 32. (;2
41 10.44
1.2051215
1792, Mar. 4
35
4 52.33
137 17 34.10
41 49.53
1.2(i48434
July 2
30
4 50.03
1,38 49 41.78
42 20.89
1.2045770
Oct. 30
37
4 58.32
-fl39.3
140 21 54.47
-1-0.40
42 50.35
1.2043210
1793, Fob. 27
38
4 59.50
141 54 12.11
43 18.00
1.2040700
Juno 27
39
4 59.85
143 20 34.79
43 43.81
1.20,38448
Oct. 25
40
4 58.84
144 59 1.04
44 7.75
1.2030207
1794, F(!b. 22
41
4 50.80
-fl2S.O
140 31 32.77
41 29.74
1.2034220
June 22
42
4 53.48
148 4 7.47
44 49.85
1.2032327
Oct. 20
43
4 49.13
+ 122.9
149 30 45. S4
-1-0.47
45 7.99
1.2030577
1795, I'el). 17
44
4 44.20
151 9 27.73
45 24.24
Juno 17
45
4 3S.00
152 42 12.55
45 3S.44
Oct. 15
1801, Jan. 17
4(i
4 31.04
154 15 0.15
45 50.04
02
+3 22.92
3 20.59
179 3 50.23
-1-44 34.51
1.2027599
Mar. 17
03
+ 57.3
180 30 54.114
44 12.H(i
1.2028920
Sept. 14
04
3 31.00
182 9 57.40
43 49.24
1.2030302
1802, Jan. 12
()5
3 30.20
183 42 57.23
43 23.00
1.2031922
Mar. 12
00
3 41.88
185 15 54.08
42 50.34
1.2033599
Sopt. 9
07
3 47.78
4- 43.7
1.80 48 40.54
-fO.35
42 20.97
1.2035387
1803, Jan. 7
08
3 54.01
188 21 35.11
•M 55.80
1.203'' 281
Mar. 7
01)
4 0.29
189 64 18.55
41 22.99
1.2039280
Sept. 4
1804, Jan. 2
70
4 0.43
191 20 50.80
40 48.28
1.2041380
71
4 12.95
-t- 30.7
192 59 30.15
40 11.79
1.2043591
Mar. 1
72
4 18.74
194 31 50.83
39 33.03
1.2045891
Aug. 21)
Dec. 27
73
4 24.38
190 4 17.13
+ 0.14
38 53.72
1.2048287
74
4 29.34
197 30 30.24
,
38 12.21
1.2050777
1805, April 20
Aug. 24
Deo. 22
75
4 33.' '
-f 18.0
199 8 35.97
37 29.07
1.2053300
70
4 37.^2
200 40 33.91
30 44.40
1.2050037
77
4 40.11
202 12 23.39
1
35 58.15
1.2058808
1800, April 21
Aug. 19
78
4 41.83
203 44 4.25
35 10.48
1.2001074
79
4 42.. '.5
-1- 5.9
205 15 30.11
' -1-0.21
34 21.31
1.2004031
Deo. 17
80
4 42.19
200 40 58.07
33 30.88
1.2007089
1807, April 111
Aug. 14
Dec. 12
81
4 40.52
208 IS 11.39
32 39.05
1.2070847
82
4 37.43
209 49 13.98
1
31 45.90
1.2074103
83
4 33.71
— 5.0
211 20 0.90
30 51.03
1.2077459
1808, April 10
Aug. 8
Deo. (!
84
4 28.30
212 50 49.02
29 50.12
1.2080922
85
4 21.03
214 21 20.40
+ 0.25
28 59.51
1.2084491
80
4 14.32
215 51 41.72
1
28 1.80
1.2088100
1801), April 5
87
4 5.84
— 15.4
217 21 51.97
27 3.03
1.2091945
Aug. 3
Doc. 1
8S
3 50.03
218 51 51.53
1
20 3.28
1.2095829
89
3 47.07
220 21 40.02
25 2.00
1.2099810
1810, Mar. 31
90
i 3 37.10
221 51 19.10
24 1.10
1.2703905
July 29
Nov. 20
91
; 3 27.15
— 24.3
223 20 47.07 -|-0.23
22 58.08
1.270X094
92
+3 17.52
224 50 4.83
1
4-21 55.38
1.2712377
102
THE ORBIT OF UUAXUS.
UkUOCENTHIC Fl'IIK.MElllS OF
UllANL'8. —
Cunlinufd.
Date.
Greenwich
No.
Sum of
pfitiiibii-
Ai>pioxiiiiatH
pfrMirhalioiis
pMMiuieil by
Loili^itUcle.
Corieutloii.
Liitltude.
Lu}.'HI'ltlllll
HailiiiH
uuaii iiuuu.
tiuUK.
veutiir.
Nfptuiie.
//
/ ft
//
1 II
/ //
1811,
Mnr. 20
93
+3 7.99
220
19 12.04
+20 51.52
1.2716752
July 24
94
2 59.12
227
48 9.00
19 40.95
1.2721215
Nov. 21
95
2 50.91
—32.1
229
10 55.87
18 41.09
1.2725700
1812,
Miir. 20
9(i
2 43.45
2.30
45 32.41
17 35.80
1.2730381
July 18
97
2 37.20
232
13 59.13
+0.11
10 29.34
1.2735072
Nov. 15
98
2 31.98
233
42 15.55
15 22.35
1.2739838
1813,
Mar. 15
99
2 27.80
—39.1
235
10 21.88
14 14.92
1.2744(i51
July 13
100
2 25.05
23(i
38 18.20
13 7.03
1.2749507
Nov. 10
101
2 22.98
238
3.95
11 58.79
1.2754409
1811,
Mar. 10
102
2 22.35
239
33 39.01
10 50.21
1.2759307
July 8
103
2 22.54
—44.0
241
1 4.03
+0.22
9 41.32
1.2704300
Nov. 5
104
2 24.24
242
28 19.59
8 32.19
1.2709372
1815,
Miir. 5
105
2 20.58
243
55 23.07
7 22.80
1.2774403
July 3
100
2 30.00
245
22 17.17
(1 13.39
1.2779459
Oct. 31
107
2 34.17
—49.3
240
48 59.79
5 3.80
1.^784533
181f.,
Feb. 28
108
2 39.00
248
15 31.47
3 54.09
1.2789(00
Juno 27
109
2 44.51
249
41 52.08
—0.01
2 44.40
1.27940S8
Oct. 25
110
2 50.43
251
8 1.42
1 34.82
1.2799725
1817,
Feb. 22
111
2 50.75
—54.4
252
33 59.49
+ 25.25
1.2804709
June 22
112
3 3.22
253
59 40.00
— 44.19
1.2809792
Oct. 20
113
3 10.03
255
25 21.18
1 53.44
1.28147«8
181S,
Feb. 17
114
3 10.03
250
50 44. 4S
3 2,48
1.2819759
Juno 17
115
3 22.90
—54.7
25 s
15 55.79
0.00
4 11.23
1.2824701
Oct. 15
110
3 28.90
259
40 55.14
5 19.73
1.2S29599
1810,
Feb. 12
117
3 34.28
2i:l
5 4 2.25
27.83
1.283440 7
Juno 12
lis
3 38.77
202
30 17.04
7 35.52
1.2839317
Oct. 1(1
119
3 42.48
—50.1
203
54 39.42
8 42.77
1.2844143
1820,
Feb. 7
120
3 45.03
205
18 49.19
9 49.52
1.284S927
June
121
3 40.37
200
42 40.29
—0.01
10 55. 7(;
1.2853070
Oet. 4
122
3 40.38
20 H
.■5il.07
12 1.40
1.2X5.'*3!I1
1821,
Feb. 1
123
3 45.07
—50.4
209
30 2.43
13 fi.44
1.2803087
Juno 1
124
3 42.35
270
53 21.. 55
14 10.88
1.2807757
Sept. 211
125
3 3S.08
272
10 27.90
15 14.57
1.2872402
1822,
Jiiii. 27
1 20
3 32.80
273
39 22.29
10 17.58
1.2,s77(i80
May 27
127
3 20.18
—50.2
275
2 4.25
—0.10
17 19.84
1.2^81032
Sept. 24
128
3 18.32
270
24 34.00
18 21.32
1.28,S(:214
182;?,
Jan. 22
129
3 9.08
277
40 52.38
19 22.01
1.2S90772
May 22
130
3 0.24
279
8 59.05
20 21.89
1.2895311
Sept. HI
131
2 50.30
—50.1
2so
30 54.82
21 20.91
1.2S99820
1824,
Jan. 17
132
2 40.10
■>K1
52 39.71
22 19.10
1.2904315
May Hi
133
2 29.74
2S3
14 14. k;
—0.07
23 10.40
1.2908770
Sept. 13
134
2 19.49
2S4
35 38.00
24 12.78
1.2913208
182'),
Jan. 11
1.35
2 9.25
—55.5
2S5
50 52.92
25 8.22
1.2917000
May 11
130
1 59.40
2S7
17 57.80
20 2.71
1.2921907
Sept. 3
137
1 50.21
2.SS
38 53.43
20 50.22
1.2920290
1820,
Jan.
138
1 41.44
2S9
59 40.04
27 48.71
1.2930570
May
139
1 33.58
—54.5
291
20 1H.03
—0.09
28 40.20
1.2934804
Sept. 3
140
1 20.44
292
40 47.50
29 30.70
1.2938987
1S2'7,
Jan. 1
141
1 20.21
294
1 H,.^3
30 20.10
1.2943110
May 1
142
1 14.89
295
21 22.15
31* 8.00
1.2947180
Aug. 29
143
1 10.52
—54.0
29(1
41 27.72
31 55.92
1.2951190
I»ee. 27
144
1 7.22
298
1 25.72
32 42,14
1.2955130
1828,
April 25
145
1 4.79
299
21 10.45
—0,07
;!3 27.24
1.2959002
Au)r. 23
140
1 3.51
300
41 0.14
34 11.25
1.2902790
Dee. 21
147
+ 1 3.11
—52.9
302
30.75
—34 54.11
1.29GG49C
TIIK OUIJIT OF UIIANUS.
103
m
IIkLIoCENTIIIC I-I'IIKMKUIS 01' Utt
ANUS.—
Continued.
Dftt«.
Sii
ni of
A|i|iroxiiiiat«
perlartmtiotirt
pniilucfil l,y
I/ocaritlim
Orefiiwicli
mtjnn noun
No.
|iBrturba-
tiuiiti.
LoDgl
uile.
Correction.
Latitude.
Itailiud
vector.
Ni-piuiia.
It
//
o /
"
/;
/ //
182U,
April
20
148
+ 1
3.(18
303 20
fi. 73
—35 35.77
1.2970111
Aug.
18
un
5.15
304 39
30.08
3(i 1(1.2(1
1.2973030
Dee.
Hi
150
7.37
305 58
4(1. 9fi
30 55.(10
1.2977052
1830,
April
15
151
10.14
—52.3
307 17
57.34
+0.08
37 33.08
1.2980308
Aug.
13
1.52
13.74
308 37
1.72
38 10.50
1.2983571
Dee.
11
153
17.(13
309 55
59.75
38 40.13
1.2980001
^831,
April 10
154
21.(19
311 14
51.(!8
39 20.49
1.2989038
Aug.
8
155
2(1. or,
—52.2
312 33
37.70
39 53.54
1.2992501
Dee.
(i
15(1
3(t.2(l
313 52
17.74
40 25.29
1.2995247
1832,
April
4
157
34.13
315 10
51.(13
+0.05
40 55.74
1.2997873
Aug.
2
158
37. 50
31(1 29
19.(17
41 24.90
1.3000379
Nov.
30
159
40.39
—52.2
317 47
41.94
41 52.05
1.3002708
1833,
.M.ir.
30
KiO
42.02
319 5
57.95
42 19.10
1.3005047
July
28
101
42.8(1
320 24
8.31
42 44.17
1.3007210
Nov.
25
lfi2
42.45
321 42
12.K9
43 7.88
1.3009280
1834,
•Mar.
25
1(13
40.92
—52.5
323
11.92
+0.15
43 30.24
1.3011231
.luly
23
1(14
38.01
324 18
5.47
43 51.18
1.3013390
Nov.
20
1(15
34.01
325 35
53.99
44 10.74
1.3014851
1835,
Mar.
20
l(i()
2s.7(i
320 53
37.59
44 28.92
1.3010515
July
IS
1(17
22.19
—52.0
328 11
10.27
44 45.09
1.3018084
Nov.
15
1(18
14.89
329 28
50.92
45 1.14
1.3019507
18311,
Mar.
14
l(i9
(1.50
330 4(1
21.41
+0.19
45 15.12
1.3020!)00
July
12
170
57.31
332 3
48.20
45 27.71
1., 3022205
Nov.
!»
171
47.51
—53.9
333 21
ll.7fi
45 38.98
1.302;!4S2
1837,
Mar.
!»
172
.37.27
334 38
32.35
45 48.84
1.3024012
July
7
173
2(1. (>9
3i5 55
50,40
45 57.28
1.3025050
Nov.
4
174
15.92
337 13
(1.1(1
~
40 4.45
1.. 3020022
183.S,
.Mar.
4
175
+ «
5.23
—55. 1
338 30
20.17
+0.17
40 10.04
1.3027503
July
•2
17(1
5.21
339 47
32.7(1
40 14.3(1
1.302S297
Oet.
30
177
15. 7 (i
341 4
43. si
40 17.29
1.3029008
183'J,
Feb.
27
178
25. 5 S
342 21
54.35
40 18. S3
1.3029034
Juno
27
179
34.82
— 50.9
243 39
4.45
4(1 19.00
1.3030170
Get.
25
180
43.(18
344 5(1
14.08
40 17.84
1.3030030
1840,
Feb.
22
isl
51.42
34(1 13
24.23
+0.17
40 15.28
1.3030990
June
21
182
..
58.3(1
347 30
34.70
40 11.29
1.3031209
0!t.
lit
1S3
4.34
—58.4
348 47
4(1.09
40 O.Ol
1.3031447
1841,
!• el).
1(>
184
9.47
350 4
58.23
45 59.35
1.3031528
.luiie
Hi
185
13.3(1
351 22
11.79
45 51.:>3
1.3031509
Oet.
14
18(i
1(1.11
352 39
2(1.97
45 41.91
1.3031392
1842,
Feb.
11
187
17.73
—(10.5
353 5(1
43.91
+0.15
45 31.07
1.3031104
Juuo
11
188
IS. i:t
355 14
2.89
45 18.90
1.3030817
Oet.
il
189
17.35
25(1 31
24.13
4: 5.38
1. .3030351
1843,
Feb.
(i
190
15.. 50
.357 48
47.74
44 50.50
1.3029707
Juuo
(i
191
12.83
— (U1.7
359 r,
1 3.(15
44 34.22
1. 302900(1
Oet.
4
192
9.27
23
42.19
44 10.04
1..302822S
1844,
Feb.
1
193
4.7(1
1 41
13.59
+0.07
43 57.59
1.3027207
May
31
194
59.91
2 58
47.55
43 37.23
1.3020173
Sept.
28
195
54.(11
—(14.9
4 1(1
24.34
43 15.53
1.3024947
1845,
Jail.
2li
19(1
49.19
5 34
3.83
42 52.43
1.3023588
May
2(i
197
43.90
(1 51
4(1.0(1
42 27.99
1.. 3022095
Sept.
23
198
38.92
a 9
31.09
42 2.22
1.3020472
1840,
Jan.
21
199
34.37
—(17.7
9 27
19.04
+0.15
41 .35.11
1.3018722
May
21
200
"
30.55
10 45
9.82
41 0.70
1.3010851
Sept.
18
201
—0
27.21
1 2 3
3.. 39
—40 37.01
1.3014801
104
THE OllBIT OP URANUS.
IIeliocentuiu Ei'Iikmeuis op UiiANUS. — Continued. 1
Date.
Hum of
Approjciiiinte
Logarithm
Qre«iiwioli
uieau noun.
No.
p«itiirba-
tiouH.
Heitiirballons
pi'odiicetl by
Neptniie.
Longitiule.
noireclloii.
LutltuJu.
UiKliUll
vector.
r tf
/;
o //
//
/ (/
184T, .Tan.
10
202
— 25.04
13 21 0.27
—40 0.03
1.3012758
M.iy
10
203
24.73
—70.2
14 39 0.21
39 33.75
1.3010544
Si'pt.
13
204
24.95
15 57 3.01
39 0.23
1.3008219
1848, .Inn.
11 '
205
20.43
17 15 10.40
+ 0.00
38 25.40
1.3005790
May
10 1
200
28.95
18 33 21.30
37 49.42
1.3003203
Sept.
7 '
207
32.75
—72.8
19 51 30.15
37 12.18
1.3000040
1849, .Ian.
5
208
37.59
21 9 65.34
30 33.73
1.2997923
May
5
209
43.47
22 28 19.02 1
35 54.10
1.2995115
Sept.
2
210
50.27
23 40 4S.13 1
35 13.30
1.2992224
Dfc.
31
211
57.74
Y') I>
25 5 22.50 1
—0.08
34 31.47
1.2989252
1850, April
30
212
1 0.18
20 24 2.38
33 48.40
1.2980203
Aug.
2S
213
1 15.11
27 42 48.39
33 4.37
1.2983078
Dec.
2(i
214
1 24.38
29 1 40.87
32 19.00
1.2979881
1851, April
25
215
1 34.33
—77.9
30 20 39.78
31 32.79
1.2970015
Aug.
23
210
1 44.28
31 39 45.91
30 45.40
1.2973287
Dfc.
21
217
1 54.21
32 58 59.55
—0.09
29 57.09
1.290989i
1852, April
Aug.
1!)
21.S
2 4.12
34 18 20.89
29 7.71
1.2900430
n
219
2 13.01
—80.5
35 37 50.43
28 17.33
1.2902910
Dec.
15
220
2 23.00
30 57 28.19
27 25.95
1.2959338
1853, April 14
221
2 31.54
38 17 15.00
20 33.02
1.2955701
Aug.
12
222
2 39.58
39 37 10.77
25 40.35
1.2952015
Dec.
10
223
2 40.02
—82.0
40 57 10.17
—0.07
24 40.17
1.2948270
1854, April
9
224
2 52.04
42 17 31.20
23 51.08
1.2944458
Aug.
7
225
2 57.77
43 37 50.24
22 55.08
1.2940583
Dec.
5
220
3 1.00
44 58 31.59
21 58.25
1.2930045
1855, April
4
227
3 4.03
—84.0
40 19 17.49
21 0.51
1.2932045
Aug.
2
228
3 5.35
47 40 13.93
1 20 1.97
1.2928582
Nov.
30
229
3 6.. 32
49 I 21.20
—0.11
19 2.00
1.2924449
185G, .Mar.
29
230
3 3.95
50 22 39.57
8 2., 55
1.2920239
July
Nov.
2" 231
3 1.24
— Sli.l
51 44 9.08
17 1.07
1.2915954
24
232
2 57.02
53 5 49.32
10 0.03
1.2911592
1857, Mar.
24
233
2 52.95
54 27 40.00
14 57.72
1.29071.55
■July
Nov.
22
234
2 47.58
55 49 42.98
13 54.71
1.2902043
19 235
2 41.05
—87.9
57 11 50.10
—0.10
12 51.08
1.2898057
1858, Mar.
19
230
2 35.18
58 34 20.10
11 40.89
1.2893395
July
Xov.
17
237
2 28.82
59 50 54.59
10 41.90
1.2888059
14
238
2 22.40
01 19 39.71
9 30.91
1,2883853
1859, Mar.
14
239
2 10.35
—88.0
02 42 35.14
8 30.87
1.2878981
July
Nov.
12 240
2 10.33
04 5 41.38
7 24.01
1.2874049
9 ' 241
2 5.17
05 28 57.59
—0.13
17.90
1.2809001
18G0, Mar.
8 242
2 0.07
00 52 24.15
5 10.84
1.2804023
July
Nov.
243
1 57.10
— S9.1
08 10 0.90
4 3.42
1.2858941
3
244
1 54.37
09 39 47.78
2 55.74
1.2853810
1801, Mar.
3
245
1 52.01
71 3 44.98
1 47.83
1.2848058
July
Oct.
1
240
1 51.92
72 27 52.38
_ 39.09
1.2843408
29
247
1 52.43
—89.3
73 52 10.31
—0.19
I-l- 28.00
1.2838257
1802, Feb.
20
' 248
1 53.51
75 10 38.24
1 37.00
1.2833029
June
20
1 249
1 55.99
70 41 10.05
2 45.40
1.2827788
Oct.
24
; 250
1 59.54
78 5.40
1
3 53.95
1.2822542
1803, Feb.
21
' 251
2 3.98
—88.5
79 31 4.97
5 2.40
1.2817290
June
21
i 252
2 9.43
80 50 15.07
10.80
1.2812047
Oct.
19
, 253
2 15.03
82 21 30.01
—0.04
7 19.09
1.2800819
1804, Fob.
10
1 254
2 22.52
83 47 7.95
8 27.25
1.2801009
.luiie
15
255
2 30.20
—^7.2
85 12 .'•)0.83
9 .35.22
1.2790421
Oct.
13
250
—2 38.39
80 38 44.88
+ 8 42.95
1.2791203
TUK OKU IT OF UUANUS.
105
IIi-.iiiucKNTUic Ki'iiKMKiiis OF UiiANi 1). — Voniltidfd.
Datfl.
Sum of
ApproxImRte
pertarlmtinna
prrdiiceil by
Nvptniie.
Logarithm
Oreeuwit'h
muftn iioou.
No.
perturl)*-
tiona.
Longitada.
Correction. Latitude.
Radius
vector.
/ tt
//
/ //
(/
/ If
18G5, Feb. 10
2.'-) 7
—2 47.06
«8 4 00.26
-f 11 52.20
1.2786142
Juno lU
2.')8
2 55.93
89 31 7.17
12 57.54
1.2781063
Oct. 8
2.') 9
3 5.01
—85.2
90 57 35.61
—0.09
14 4.34
1.2776029
1866, Feb. 5
260
3 13.91
92 24 16.00
15 10.74
1.2771044
Juno 5
261
3 22.61
93 51 8.39
16 16.70
1.2766112
Oct. 3
262
3 30.79
95 18 13.06
17 22.18
1.2761237
18CT, Jnn. 31
263
3 38.4i»
—82.5
96 45 29.91
18 27.15
1.2756419
Mny 31
264
3 45.25
98 12 59.44
19 31.54
1.2751656
Sept. 28
2(;5
3 50.96
99 40 41.71
—0.08
20 35.37
1.2746948
1868, Jim. 26
266
3 55.67
101 8 36.54
21 38.51
1.2742299
May 25
267
3 69.06
_79.1
102 36 44.31
23 40.96
1.2787709
Sept. 22
268
4 1.02
104 5 5.00
23 42.75
1.2733175
1869, Jnn. 20
269
4 1.71
105 33 38.41
24 43.69
1.2728693
Mny 20
270
4 0.81
107 2 24.67
25 43.81
1.2724264
Sept. 17
271
3 58.59
—75.1
108 31 23.. 55
—0.11
26 43.07
1.2719888
1810, Jan. 15
272
3 54.96
110 34.97
27 41.44
1.2715563
Mny 15
273
3 50.23
Ml 29 58.50
28 38.77
1.2711290
Sept. 12
274
3 44.34
112 59 33.91
29 35.16
1.2707070
1871, Jan. 10
275
3 37.90
— C9.8
114 29 20.79
30 30.41
1.2702903
May 10
276
3 30.56
115 69 19.11
31 24.57
1.2698791
Sept. 7
277
3 22.61
117 29 28.39
—0.1 8
32 17.57
1.2694735
1872, Jan. 5
278
3 14.50
lis 59 48.05
33 9.31
1.2690738
May 4
279
3 6.19
— C4.2
120 30 17.95
33 69.79
1.2686S04
Sept. 1
280
_2 57.05
122 57.87
+34 48.97
1.2682934
The next operation would bo to interpolate these co-ordinates to intervals of
time suitable for the computation of a geocentric ephemcris, to correct the longi-
tudes for solar nutation, and then to compute the geocentric right ascension and
declination. This operation has not, however, been completely carried out except
for most of the observations before ISIJO, and for three of the oppositions observed
since, the latter being computed only as a check upon the accuracy of the com-
parisons. As a general rule, it may be said that wherever a complete geocentric
ephemeris, with the heliocentric cphemeris from which it was computed, were
available, these ephemcrides were made use of in a manner which will be more
fully described hereafter, while, in all other cases, the geocentric places were
computed directly.
It may also be stated here that Hansen's Tahka tin Solc.il have been adopted as
giving the places of the sun to be used in computing the geocentric places.
H May, 1873.
100
TUK OIIHIT OF UIIANUS.
CIIAPTER VI.
REDUCTION OP TIIK OUSKUVATIOXS OK UUANUS, AND THEIR COMPARISON
WITH THE I'RECKDINU THEORY.
Tub observations of Uranus naturally tlivide thonisclvos into two distinct
classes. (1) The purely accidental ones, made jjrevioiis to the recognition of the
planet by Ilerseliel in 1781, and therefore witliout any suspicion on tlu; part of the
observers that the oltjeet was not u iixed star, and ('2) the systematic observations
made since.
The first class arc nearly all so uncertain in comparison with the second that I
have hesitated over the (piestion of employing them at all. If nothing but a
determination of the elements of Uranus were called for, they would certainly not
be worth using, since these elements miy be determined with entire certainty
from the observations which have been nuide during the entire revoluticm of the
planet since 1781. But the mass of Neptune is also to be determined, and it is
at least possible that these observations, uncertain though they are, may add
materially to the weight of this determination. I have, therefore, determined to
include them all, re-reducing them when there seemed to be good reason so to do.
The earliest observations are those of Flamstead, ])Td)lishcd in the Ilistoriic
Coelestis. The observations themselves, as printed, together with the principal
elements for reduction, are given in the following tables.
The first colunni of the table gives the name of the star. The second gives
the clock time of transit over the wire of the quadrant as given by I'lamstead,
The time, it will be seen, is only given to entire S(>conds. AVe must, therefore,
exj)ect to find a probable error, of which the mathematical minimum is ()\2o,
and of which the minimum we can reasonably expect is much greater.
Next we have the apparent right ascensions of the stars as comjjuted. For
these data I am indebted to I'rof. Coffin, Superintendent of the American
Epiiemcris. The mean ])laces are mostly derived from the " Star Tables of the
American Ephcmeris," and from the two Greenwich Seven Year Catalogues, while
the reduction to apparent place is made with the modern constants.
The fourth coliunn gives the a])parent clock correcticm for sidereal time, in which
is inchuled the eflect of deviation of the instrument from the meridian.
The clo( k keeping mean time, the errors are in the next column reduced to
those of sidereal time at the moment of the transit of Urainis.
The next two columns give the corrections for clock rate, and for deviation of
the instrument from the meridian, as inferred from the observations themselves,
both being referred to the time and position of the transit of Uranus.
'»
t
• ' W.mjH i .A. « R»U ' -lMfflWaj-
TIIK ORBIT OF URANUS.
107
'»
t
In the last column wo Imvo the seconds of conclnd.'d com-ction for clcx
instrument to be upplied to tlie observed time of transit of Uranus.
k and
IG'JO, Ihwmber 23.
Star.
a Arietis,
7t Arietis,
a Arietis,
f Arietis,
5 Arietis,
n Tauri,
Uranus,
A Tuuri,
Tiino of Tp.
Ii. III. n.
7 48 40
8 30 54
8 ;j:} 17
8 40 21
8 52 44
!) 27 47
9 41 4!)
y 45 3
U. A. of Htur.
II. III. H.
(.'lork
h. III. B
Iliijhl AscciiHion.
t".
m. «•
58 29.3
1 49 52.1 —5 58 47.9
2 32 8.4 —5 58 45.6 58 33.9
2 34 31.0 -5 58 4(5.0 58 34.7
2 41 38.0 —5 58 43.0 58 33.0
2 54 3.0 -5 58 41.0 58 33.0
3 29 13.0 -5 58 33.4 58 31.1
U.
«.
—1.1
—0.7
—0.7
— O.G
—0.5
—0.1
58 32.4
5() 49.3
58 39.0
3 4(J 31.1 -5 58 31.9
aViiginis,19 4 15 13 8 58.2 -5 55 1G.8
o Bootis, 19 58 31^ 14 1 34.0 -6 50 57.5
Hourly rate of clock,
Deviation of instrument for each degree of Z. D.,
Transit of Uranus,
Correction for clock and instrument (mean),
Observed 11. A. of the planet.
a Arietis,
7t Arietis,
a Arietis,
f Arietis,
h Arietis,
>7 Tauri,
Uranus,
A Tauri,
1690, Dcccmhcr 23.
Z. D. observed. Refraction.
2?)° 29' 10'
35
37
31
32
28
31
30
18 55
41
23 15
55 55
20 55
52 35
15 55
+0' 33"
4-0 41
+0
+0
+0
44
35
37
31
3()
84
16
13
20
18
23
21 12 3
Circle reading for Uranus, corrected for refraction,
Equatorial point on circle,
Docliiiation of Uranus, from observation,
Uranns,
6 Virginia,
e Virginis,
^ Virginis,
1712, Ajnil
h. m. a.
9 35 19
12 19
12 14 51
12 32 11
lil(jht Ascension.
li. m. B
12 47 52.4
13 2 31.5
13 20 5.7
Dev, C
B.
-1.2
+ 1.7
+2.9
—0.2
+0.5
—1.3
3i.6
32.9
32.5
33.8
33.0
32.5
0.0 —0.8 33.2
— 0-.6
—0.5
h.
9
-5
3
m.
41
49.0
58 32.8
43 16.2
DecUnafion.
Dcfliiuition.
21° 58' 53"
9 4
46 58
4 37
31 40
6 53
Eq. point.
51° 28' 36'
28 40
28
28
28
28
51
51
51
51
51
42
27
12
19
51 28 32
31° 53' 11"
51 28 30
+ 19 35 19
+47 33.4
-1-47 40.5
+47 54.7
The discordance of clock errors, and the time whic^h intervened between the
transit of the planet and tliat of the first star, seem to render an accurate reduction
impossible.
10b
TIIK Oil BIT OF URANUS.
nir), MarrJi 4. IHyht Aurrnnion
T.
II. III. K.
d Lconis, 1 1 50 lU
Urmius, 12 27 1
6 Virgiiiis, 1'2 49 41 11 45 23.G
II. A.
)l. III. K.
10 45 52.5
V.
)i. III. *.
1 4 2ti.5
-1 4 17.4
li. m.
0".
20.5
21.1
Clock time of transit of Uranus, 12 27 1
Correction for clock and instrument, — 1 4 20.8
Ilight ascension of Uranus from observation, 11 22 40.2
1715, March 4. Vcctinalion.
Z. I). II. Dec.
d Lconis, 40° 19' 40" +1' 0" +5° 8' C
Uranus, 46 33 10 +11
iVirginis, 40 13 20 +1 5 14 17
Circle reading for Uranus,
Equatorial point.
Observed declination of Uranus,
Eq. point.
51° 28' 46*
51 28 37
46° 34' 11"
51 28 42
+4 54 31
1715, Mirch 5. Right Ascension
T.
h. in. fl.
rfl-eonis, 11 46 24
Uranus, 12 22 59
b Virginis, 12 45 49
II. A.
h. m. 8.
10 45 52.5-
11 45 23.6
0.
h. in. R.
— 1 31.5
C
25.5
29.1
-1 25.4
h, m. 8.
12 22 59
—1 27.7
11 22 31.5
Transit of Uranus,
Correction for clock and instrument,
Observed right ascension of Uranus,
Tlie large apparent clock rate, and the colons after the time of transit, both
throw doubt on this observation.
Declination.
The circle readings for the stars are the same as on the day preceding, while
that for Uranus is 50" less. The declination is therefore 50" greater, or
+4° 55' 21".
1715, March 10. lti(jhl Ascension.
C.
h. m.
—0 40
-0 40
d Leonis,
p* Leonis,
Uranus,
b Virginis,
T.
h. m. a.
11 25 58
11 32 28
12 1 42
12 25 18
R. A.
h. m. n.
10 45 52.5
10 52 25.1
11 45 23.6
2.9
59.5
58.1
58.2
Clock time of transit of Uranus,
Correction for clock and instrument.
Observed right ascension of Uranus,
-0 39 54.4
h.
12
—0 39 58.6
11 21 43.4
m. B.
1 42
TIIK OIiniT OF URANUS.
109
UvcUiialio)!.
v]
tl Loon is,
^i' liconis,
Uranus,
b Virginis,
Z. n.
4(5° ID' JJ/J"
47 M M
40 27
4G 13 2Q
11.
+ 1' 1"
4-1 ;)
+1 1
-i-i
Dci'.
5° H' ()"
6 14 17
E.|. rt.
51° 'i< -i'r
51 21' 0!i
Circle reading for Uranus,
Eciuatorial point on circle.
Observed declinutiou of Uranus,
51
46
51
+5
a Lconis,
Uranus,
V Virginis,
17 Virginis,
X Virginis,
1715. Ajn-il 29
T.
ii. III. *.
8 42 11
8 50 44
9 G 55
9 43 38:
11 32 48
RitfJit Ascension.
n. A.
h.
11
ni. ••
G 28.3
11 31 14.1
12 8 G.5
13 57 47.G
C.
Ii. ni. •.
+2 24 17.3
4-2 24 19.1
^-2 24 28.5
+2 24 59.G
28 42
2ir 1
28 49
U 48
C
8.7
[6.4
19.8
33.1
Ti.e discordance of the clock corrections mak.-s a satisfactory determination ot
the right ascension very difficult. 1 deem it best to reject the doubt.l o bserva-
tiou of 17 Virginis, and the discordant one of . Virguus. 'ihc resuU wdl then
be
Observed transit of Uranus,
'^Jorrcction for clock and instrument,
)bscrved right ascension of Uranus,
h. in. «.
8 50 44
2 24 17,6
11 15 1.6
Z. D.
a Lconis, 43° 52' 40"
Uranus, 45 45 30
v Virginis, 43 20 20
17 Virginis, 44 34 10
X Virginis, GO 23 5
Circle reading for Uranus,
Mean equatorial point.
Observed declination of Uranus,
Declination
B.
-f 0' 55"
4-0 59
-fO 54
-fO 5G
4-1 41
Dpc.
7° 34' 51"
8
6
—8
7 11
53 33
55 42
Ef . Pt.
51° 28' 26"
51 28 25
51 28 39
51 28 G4
45= 46' 29"
b\ 28 38
4-3 42 9
The next observations in the order of time are two by Bradley, discovered by
Mr. Hugh Breen, but still unpublished. The following are tbe results as given
by Mr. Breer in the AstronomiscJie NachricJikn, No. 1463.
Mean Time. ^- ^■
1748, October 21, 7 6 18.4 21 4 37.93
1750, September 13, 10 8 57.8 21 40 0.23
N. P. D.
107 29
104 42 33.9
I
no
TIIK ORRIT OF URANUS.
Mr. Brccn remarks: "The right ascensions are very acciirute. It has been
assumed tiiat the N. P. D., on l"i5(), September 13, is identical with // ('apricorni,
with which it was compared. Tlie first observation was by the transit instrument,
and the second by tlic quadrant."
No ground is given for the above assumption respecting tile N. P. D. for tlie
second observation ; it may, therefore, be omitted as valueless.
In the year 1750 we have also two observatidns by Le Monnier at Paris. For
these, and all the other observations by the same observer, 1 shall adopt the results
given by Bouvard in the C'oimaissince dia Temps, for 1821, p. 341, with tlie cor-
rections indicated by Le Vcrrier, in Connahsance iJcs Temps, for 1849, pp. 125 and
126. The necessary uncertainty of the observations is sucli that, consid: ring that
Bouvard reduced them with the star positions of the " Fundumeula," scarcely
anything will be gained by a new reduction.
1753, December 3, we have another observation of right ascension by Bradley.
I adopt tlie result kindly communicated by my distinguished friend. Dr. Auwers.
1753, December 3,
Ii. m.
5 33.
R. A. = 22 23 21.59
1756, September 25. Observation by Mayer, at Gottiugen. I adopt the result
given by Bessel, in Fundamenta Astronomia:, p. 284.
li. m.
1756, September 25, 10 12.
R. A. = 348 54.5
Dec. = —6 1 49.4
The following is a tabular summary of the preceding results, with their com-
parison with the provisional theory. In the computation of the geocentric place
the places of the sun were derived from Hansen's Tables. I am indebted to Pro-
fessor Coffin for a duplicate computation of the geocentric places from the pro-
visional ephemeris, which was executed by Mr. Joseph A. Rogers.
- ate.
Right Asceuslon.
Deulinatiou.
Correction to theory. 1
Obserratlon.
i-
Observation.
£•
R. A.
Deo.
liOng.
dl
'ei
J
s
r)-A.
■ ep
H
II
h m s
s
1 II
II
It
II
IfiflO, Dec. 23
3 43 lfi.2
14.7
+ 19 35 19
7
+22
+ 12
+24
1.04
+.027
171.'), Mar. 4 11 22 40.2
38.7
4 54 31
48
+22
—17
+28
1.00
Mar. 5 11 22 31.5
29.1
4 55 21
49
+30
—28
+ 4*
1.00
Mar. 10 11 21 43.4
41.0
5 4i;
5(!
+30
_ 8
+30
1.00
Apr. 29 11 15 l.fi
1.5
-f 5 42 9
U
+ 1
— 2
+ 2
1.04
+.0.30
1748, Oct. 21 21 4 37.03
35.42
+37.0
+39.3
1.015
+ .0.50
1750, Sept. 13 21 40 0.23
o / "
57.90
II
+35.0
+35.7
1.05
+ .022
Oct. 14 324 15 24. f)
:47.r,
—15 1 40.4
47.0
+37.0
+ 0.0
+35.9
1.03
+ .045
THE OUBIT OF URANUS.
Ill
1
Date.
Right Asoenalon.
Declination.
Correction to theory.
• 1
ObserTatlon. o ,
Observation.
b
R.A.
Dec.
Long.
01
m
69
^ !
/
1 "
; o ' '1 1 //
O 1 It
tf
ti
ItoO, Doc. 3 324 34 53.5 15.4
_14 53 20.2
32.4
438.1
412.2
438.8
0.98
+.047
h 111 8 1 s
1753, DfC. 3 22 23 21.(10 19.34
1 o ' " "
n5f., Sept.25 348 .54.5,25.0
17()4 .Jiiii 15 12 37 39.0 50.4
433. S
435.0
1.00
+ .048
_ 1 49.4
+ 4 43 47.2
40.0
50.2
-i-9.5
(-17.4)
_3.4
(-V.0)
44,4 1
-0.5 \
-(-25.7
1.05
+ .013
17(!8, Dec. 27 31 2ti 52.0 32.0
Dfc. 30 31 20 45. S 38.0
12 15 35.0
12 14 65.4
30.0
55.9
419.4
4 7.8
413.0
1.02
+ .045
1709, .Inn. 15 31 22 7.7:55.8
Jiin 10 31 22 23.4 11.1
li 14 2(;.0:
12 14 30.3
29.7
37.0
411.9:
412.3
— 3.7:
—0.7
.Inn 20 31 24 0.0:43.7
12 15 19.0
18.«
422.9:
40.2 \
412.5
1.01
+ .049
.Inn. 21 31 24 33.8 14.1
Jan. 22 31 25 4.7 :47.7
Jan. 23 31 25 28.5 24.2
. ^
12 15 31.8
30.8
419.7
41.0
12 15 45.7
44.7
417. 0:
41.0
12 10 7.5
:59.(
, + ^■•^■-
47.9 J
AVlu-rc no declination has been observe! Ll.e observed corrections m right ascen-
sion have been changed to conretions in h.ngitnde on the hypothesis that the
theoretical latitude is correct. The approximate formula is
, . (Vx cos h ,
U-= . ,,, where
sin jL
cos /; = sin f cos a, f being the obliquity.
DISCUSSION OF TIIK MODKUN OnSKIlVATION'S.
TtciUctxon of the PiiblUhed Fesnlts of Olmrmilons lo a Uniforvi Sijstem.
We have now to discuss the great mass of observations made at the pvincipal
observatories of the world since the discovery of the planet by Herschel, m 1 .81
To make all the data of reduction rigorously homogeneous and uniform, it would
be necessarv to completely re-reduce the greater part of the observations maue
befop. 1850 using the modern values of the constants of reduction and to com-
pare each observation separately with the geocentric place deduced from the pro-
visional theorv. Such a reduction and comparison would be extremely desirable.
Their execution would, however, involve an amount of labor for greater than it is
now possible for the author to bestow upon the problem. A^ c must, therefore
adopt the reductions which have been already made, applying such systematic
corrections for reduction to a uniform system of star places as we have the means
readily to determine. No reduced places are employed unless we can find data
for some more or less accurate determination of these -^7'=^7';;' '^ "f^; "^^^J
necessitates the rejection of a great mass of observations made at the mil o o W -
vatories of the European contin.Mit, and published in the A.rononu.cJ.n ^a?•
riehten We still have the following rich collection of materials at our disposal .
112 THE OUBIT OF URANUS.
1. Observations at Greenwich, 1781 to 1872.
2 Paris. 1802 to 1827, and 1837 to 18G9.
3 Kcinigsberg, 1813 to 1835.
4 Vienna, 1822, and 1827 to 1839.
6 Speier, 1827-29.
G Cambridge, 1828 to 1842.
7 Edinburgh, 1836 to 1844.
8 Berlin, 1838 to 1842.
9 Vulkowa, 1841 and 1842.
10 Washington, 18G1 to 1872.
11 Leiden, 1863 to 1871.
12 Santiago, 1854 and 1855.
As to the general distribution of tliese observations in time, we may remark
that during the first three or four years the pUuiet was zealously observed at
Greenwich. Observations then began gradually to fall off until 1798, in which
year we find but one. From this time until 1814 only one or two observations
were made at each opposition. They become a little more numerous, until 1829,
when there is a sudden increase. Few interruptions have occurred since. With
regard to the other observatories it may be said that from 1802 until 1830 there is
a gradual increase in the number of observations, and that since the latter year
the number of observations is entirely satisfactory.
A great number of the observations were reduced with the star places of the
Tabula: liegioniontana', and the entire Paris series arc reduced with the star
positions of Le Verrier, given in his '■'■Anmilcti de V Obaervaloire Imperial de Paris"
Tome II. As a preliminary to the discussion of the systematic corrections to the
principal published reductions, I have prepared the following table, showing the
corrections which must be applied to the places of the equatorial fundamental
stars in the above catalogues to reduce them to the adopted standard, namely. Dr.
Gould's coast survey list in right ascension, and Auwcrs' standard in declination.
In the table of right ascensions tlie first column after name of the star gives the
annual variation of that co-ordinate for the epoch 1860.0, as derived from Le
Terrier's tables of right ascensions just cited. Next we have the corrcct'on to this
annual variation, expressed in units of the fourth place of decimals, ti reduce it to
that given in the " Star Tables of (he American EjJiemeris" the positions in which
are founded on Dr. Gould's Catalogue. The fourth column gives the correction to
the right ascensions of Lo Verrier for 18G0, in hundredths of a second of time.
Subtracting from this column sixth-tenths of the preceding, we have the corre-
sponding corrections for 1800. The last four columns give the corresponding num-
bers for the right ascensions of the Tabulai RegiomontaiivR.
The table of declinations shows, for different epofihs, the corrections necessary to
reduce the tabular positions to those given by Auwers in his paper on the declina-
tions of the fundamental stars
THE ORBIT OF URANUS.
118
•
I. RiaiiT Ascensions.
Corrections to
Ann. var. of _
Corrections to
Date. ann. var.
IStiU. A
nn. var.
R. A.
B. A.
Tah. Keg.
18B0. Ann. var.
B. A.
18(iO.
R. A.
1800.
18(iO.
1800.
_
a Andromeda),
8
+3.0844
+ 14
+ 2
_ 6
3.0840
+ 18
+ 8
— 3
y Pcgasi,
3.0801
+ 15
+ 3
_ G
3.0824
_- 8
— 9
— 4
a Arietis,
3.3C44
+ 12
_ 1
3.3036
+ 20
+ 6
— 6
o Coti,
3.1266
+ 10
+ 5
_ 1
3.1261
+ 9
+ 7
+ 2
a Tauri,
3.4346
_ 1
_ 1
3.4335
+ 10
+ T
+ 1
/I Orionis,
2.8191
+ G
+ 2
_ 2
2.8800
+ 3
+ 5
+ ^
3 Tauri,
3.1871
_ 4
+ 1
+ 3
3.7888
_21
— 4
+ 9
a Orionis,
3.2460
+ 5
+ ^
+ 1
3.2464
+ 1
+ 3
+ 2
a Gcminovani,
3.8409
— ',
+ 1
3.8386
+ 21
+ 18
+ 5
a Cauis Min.
3.1 i02
+ '^
+ 9
+ 6
3.1455
+ 12
+ T
fl Gcminorum,
3.6828
+ 5
+ 3
3.6807
+26
+ 13
— 3
a Hvdroj,
2.9485
+ 12
+ fi
— 1
2.9469
+28
+ 16
— 1
a Leoui.s,
3.2030
+ 15
+ 6
_ 3
3.2014
+ 31
+ 13
— 6
p Lcoiiis,
3.0054
+ 11
+ ^
— 3
3.0640
+25
+ 12
— 3
a Virginis,
3.1495
+ 20
+ 5
K
3.1497
+ 18
+ 2
— 9
a Bootis,
2.7325
+ 16
+ 3
— 1
2.7327
+ 14
+ 5
— 3
o' Libra;,
3.3044
+ 6
+ 3
_ I
3.3074
_24
— 6
+ 8
— 5
+ 3
— 3
— 5
— 5
— 4
+ 2
— 5
^- 3
— 2
_ 8
—12
— 6
—55
—.01
a CoroniB,
2.5318
+ 12
+ 1
— 6
2.5373
+ 17
+ 5
a Scrpcntis,
2.9488
+ 8
+ *
— I
2.9513
—17
— 7
— 4
a Seorpii,
3.6654
+ 16
+ -1
_ 6
3.6672
• Hcrculis,
2.1322
+ T
+ 2
_ 2
2.1319
+ 9
a Ophiuchi,
a, Lyra;,
y Aquilffi,
2.1808
2.0312
2.8520
+ 10
+ 2
+ 9
— 1
+ 2
_ fl
_ 2
_ 3
2.1183
2.0305
2.8546
+34
+ 9
—17
+ 15
+ 1
— 8
a A(inila!,
3 Aquilaj,
2.9281
2.9466
+ *
+ 5
+ 1
+ 3
_ 1
2.9281
2.9496
+ 4
—25
— 3
—12
o' Capriconii,
3.3338
+ *
+ 2
3.3349
■ 1
— 6
a Aquarii,
a I'iiscis Aust
3.0829
3.3311
+ 13
+ 8
+ ^
_ 4
_ 1
3.0822
3.3326
+-n
- 7
—16
a Pogasi,
Sum,
Mean,
2.9828
+ 11
+ 251
+8.5
+ 81
+ .02
— 1
—72
7 — 024
2.9830
+ 9
+210
+7.0
1
— 1
+71
+ .024
1
15 Uar. 1S73.
114
THE ORBIT OP URANUS.
■
II. DEC1>1NAT10((8.
Corrections to Taliu
» Regiomontani
Correellons to Le Verrier.
1780.
1800.
1820.
1840.
1820.
1840.
//
n
ir
If
//
//
a Andromcdfc,
-fO.3
+0.2
+0.1
0.0
+0.3
0.0
y Pegasi,
—0.2
+0.1
+0.4
+C.3
+0.1
+0.3
a Arictis,
—0.3
0.0
+ 0.3
+0.5
+0.3
+0.3
a Ccti,
—1.0
—0.1
+0.7
+ 1.G
+0.1
+0.7
a Taiiri,
0.0
+0.1
+0.1
+0.2
+0.1
0.0
|3 Orionis,
—0.5
+ 0.1
+0.C.
+ 1.3
+0.3
+0.7
3 Tauri,
—0.5
0.0
+0.f.
+ 1.1
+0.2
+0.5
a Orionis,
—1.3
— O.f.
0.0
+0.G
—0.2
+0.3
a aemiuornin,
—0.8
—0.5
—0 1
+0.3
+0.5
+ 1.0
a C<>!!:3 Min.
—0.3
0.0
+0.3
+0.G
+0.2
+0.5
3 Qcminorum,
—0.3
0.0
+0.3
+0.G
+0.2
+0.5
a HydroB,
—0.2
+0.3
+0.7
+ 1.3
+0.3
+0.7
a Lconis,
-{-0.3
+0.4
+0.4
+0.5
+0.4
+0.5
Lconis,
+0.2
+0.2
+0.2
+0.3
+0.2
+0.3
a A'irginis,
+0.3
+0.G
+ 1.1
+ 1.5
+0.5
+0.8
a Bootis,
+0.4
+0.4
+0.3
+0.3
+0.4
+0.4
a? LibriB,
+0.5
+0.5
+0.G
+0.G
+0.5
+0.fi
a CoroniB,
+0.7
+0.(5
+0.4
+0.3
+0.4
+0.4
a Serpentis,
+0.2
+0.G
+ 1.1
+ 1.5
+0.5
+0.9
a Scorpii,
+0.1
+o.r,
+ 1.0
+ 1.4
+0.4
+0.7
a llerculis,
+0.6
+0.8
+ 1.0
+ 1.3
+0.5
+0.7
a Opbiuchi,
+0.4
+0.5
+0.5
+ 0.G
+0.4
+0.G
a Lyns,
+0.7
+0.8
+ 0.0
+ 1.0
+0.4
+ 0.4
y AquiliB,
0.0
+0.3
+0.G
+ 1.0
+0.2
+0.4
a Aquiloe,
+0.1
+0.4
+0.7
+1.0
+0.1
+0.4
|3 AquiliB,
+0.1
+0.fi
+ 1.3
+ 1.7
+ 0.5
+0.8
»* Caprieonii,
+0.2
+0.0
+ 1.G
+2.3
+0.6
+ 1.1
a Aquarii,
—0.5
+0.1
+0.8
+ 1.5
+0,5
+ 1.0
a Piscia Aust.
+1.0
+ 3.4
+3.9
+5.3
+1.7
+2.3
a Pegasi,
+0.4
+0.3
+0.2
+0.1
+0.3
+0.2
Mean
+0.03
+0.3G
+0.C9
+ 1.03
+0.3G
+0.60
1
THE ORBIT OP URANUS.
115
The coxTection to the reductions to apparent place given in the Tabulae Rcgio-
montansc on account of the correction to the constant of Nutation is;—
In right ascension :
— 0".46 sin ft — 0".l8 sin Q sin a tan S — 0".24 sin Q cos a tan 8.
In declination:
— 0"18 sin a cos a + 0.24 cos Q sin a.
The terms which contain tan S as a factor may be entirely neglected, as they are
small, periodic, and contain tan h as a factor which is sometimes positive and
sometimes negative. I shall also neglect tlic corrections in declination, as their
sum is sensibly
0".21 sin (a - 8)
the effect of which will generally be confounded with the accidental errors of
observation.
The only correction we shall apply on account of nutation is, therefore,
^a = — 0\()30 sin Si.
The values of this expression at the dates when it is zero, a maximum, or a
minimum, are as follows : —
1778.5
1783.1
1787.7
1792.4
1797.0
1801.7
1806.3
1811.0
1815.6
1820.3
—.03
.00
+.03
.00
—.03
.00
+.03
.00
—.03
.00
1820.3
1825.0
1829.6
1834.3
183S.9
1843.6
1848.2
1852.9
1857.5
.00
+.03
.00
-.03
.00
+.03
.00
—.03
.00
Having adopted this system of standard positions, we may adopt two ways of
reducing the observations to it. One is to compare the positions of the stars
adopted in the published reductions with the standard, and apply the mean differ-
ence to the reduced place of the planet. Another is to make a similar com-
narison of the standard cata1.-ue with the positions of the fundamental stars
which have been deduced fr- m observations by a system of reduction uniform
with that employed in reducing Uie observations of the planet, and to regard the
mean difference as a correction appU :.ble to all the positions of the planet. If
the standard catalogue and the observations are both free fram systematic error,
the results obtained in these two ways should be substantially identical. 1 hese
are however, conditions which we cannot expect to find fulfilled. In the follow-
in-^ discussions I have sometimes used one, sometimes the other, and sometimes
116
THE ORBIT OP URANUS.
Mr
combined both, the choice being determined by circumstances. We shall con-
sider the different series of observations in succession.
Greeincicli Ohaeivatii/ns from 1781 to 1830,
These observations are completely reduced by Airy and comp;.red with Bou-
vard's Tables, in the work licduction of the Observations of Planets made at the
Royal Observatory., Greenwich, from 1750 to 1830, London, 1845, The con-
cluded positions given in this work depend mainly on the star places of the
TabuliE Regiomontanic, both in right ascension and declination. If we consider
the first four oppositions — 1781-1785 — as forming a single group of which the
mean epoch is 1783, wo find that the general correction to the TabultE Ilegio-
montanse for this epoch is
In right ascension, — 0'.030 ;
In declination, +0".08.
If, on the other hand, we consider only the particular stars compared with Uranus,
the result will be a little different. The number of times each of the fundamental
stars has been compared with Uranus, and the correction in right ascension cor-
responding to each star, are nearly as follows : —
a Arictis, JV =
= 2
Cor. =
-0.09
iVX c =
— .18
a Tauri,
2
—.01
— .02
y Pcgasi,
2
-.03
— .06
(i Tauri,
19
+.13
+2.47
a Orionis,
33
+.02
+0.66
a Canis Minoris,
33
—.02
—0.66
/3 Geminorum,
34
—.07
-2.38
a Leonis,
7
—.11
— .77
/i Leonis,
2
—.07
— .14
The mean correction from these data comes out — 0".008, differing by 0\022 from
the general mean correction. Our choice between the two (corrections depends
on whether we are to consider the relative positions of the Tabultc Regiomontamr,
or those of the standard catalogue, as nearest the truth at the epoch 1783, and
particularly upon whether we arc to consider the large correction to the proper
motion of (3 Tauri as real. In the absence of exact data for settling this ques-
tion, the mean of the two results, or — 0'.020, has been adopted.
A similar anomaly is exhibited by the declinations. It is probable that the
declinations of Uranus during this period mainly depend on stars in the first
twelve liours in right ascension, for which the mean correction is about — 0".,'50
instead of +0".08. I have adopted — ()".16. Changing these corrections to lon-
gitude and latitude, we have, during the period 1781-1786: —
Correction to observed longitude, = — 0".30 ;
Correction to observed latitude, — .19.
TllK ORBIT OP URANUS.
117
Dtuing the years 1788-1798 the above systematic iliffeience in right ascension
does not appear. The most probable correction seems to be
A8 = 0".00.
Mat. = —()". 10.
Aa=— 0'.02"j;
Wliencc A long. = — 0". 34 ;
Between the years 1800 and 1823 the stars used for comparison are so widely
scattered tliat I consider it sate to apply only the general mean correction for the
epoch 1813, which is
Aa = — '.005 ; Ac^ = +0".()6.
Whence A long. = 0".00 ; A hit. = -f .GG.
From 1825 to 1830 more than half the weight of the right ascf^nsion comes
upon the stars a, (3, and y AquiUv, the mean correction to wliich, during tliis
interval, is — 0'.035. The general mean correction at this epocli is -|-0'.002. I
think the right ascensions of these three stars in the Tabultc Kegiomontana; are
really too great at this epoch by the ehtire diff'ereiue of these results. We may,
in fact, hereafter regard the positions of the standard catalogue as sufficiently
accurate, 'llie mean corrections to be applied will llien be
A'( = — 0\017; A5 = +0".83.
Whence A long, =• — O'.O.J ; A hit. = -f<'"-^^-
From the year 1831 until the present time the Greeiuvich observations are regu-
larly reduced in the several annual volumes of observations. But a reduction of
the observations from 1831 to 1835, executed by Mr. Hugh Breen, is given in an
appendix to the volume for the year 18G4. The results here given differ from
those published by Pond in the several annual volumes for tlie same interval. The
right ascensions are altered only by a])plying the constant correction — 0".030,
which is found necessary to reduce Pond's right ascensions to those of the Tabula;
Regiomontana". This correction I have verified. The mean correction to reduce
the right ascensions of the Tabuhr Regiomontana; to oiir standard is at this time
-|-0\005. On the other liand, when we compare the concluded right ascensions
of stars within six hours of Uranus, as given by Pond in the (ireenwicli observa-
tions for 1834, with our standard, we find a mean correction of — ".034 to reduce
his positions to the standard, wliieh implies a correction — ".004 to Breen's reduc-
tion. The two residts being -|-'.005 and — ".004, I have applied no correction
whatever.
In the paper in question the declinations are comi>letely re-reduced, using im-
proved data of reduction, but, so far as I see, making no changes in Pond's
method. The results diff"er strikingly from those of Pond, and suggest the
desirableness of a complete re-examination of all Pond's determinations of decli-
nation. Having no catalogue of observed declinations of standard stars reduced
in this same way, we cannot directly determine the systematic correction to the
declinations. I therefore proceed as follows: A comparison of Pond's observed
declinations of standard stars with Aiiwers' normal catalogue show that the former
require the following corrections near the parallel of Uranus :
118
THE ORBIT OF URANUS.
In 1831 — 1".42;
1834 — 2.10.
Then comparing Airy's reduced declinations of Uranus with Pond's, we find tlie
following mean differences:
In 1831, Airy — I'ond = — 3". 18
1834, — 3 .50.
To reduce Airy to Auwers we must there apply to the declinations
In 1831 + 1".76
1834 + 1 .40.
I have regarded the correction -j- 1".60 as applicable throughout the period in
question.
1836-72.
During this interval the corrections in right ascension have been derived by the
following two sets of comparisons: (1) A comparison of the several collected six
and seven ycai catalogues with Gould's standard, from which it appears that they
require the following general corrections in right ascension :
Six year catalogue of 1840 + 0'.047
Six year catalogue of 1845 -f .002
Seven year catalogue of 1860 -f .003
Seven year catalogue of 1864 -\- .022
(2) A comparison of the corrections applied to the right ascensions of the indi-
vidual years to reduce them to the standard of the catalogue, as given in the
introduction to each catalogue. The sum of these two numbers gives the correc-
tions for each year.
A sliglitly different method is to regard the above correction for each catalogue
as applicable to all right ascensions which depend fundamentally upon that cata-
logue. I have sometimes combined both methods so as to derive what seemed to
be the most probable result, and sometimes used but one.
The corrections to the declinations during tlie interval in qnestion have been
derived from Auwers' " Tafeln zur Reduction der DccUnationen verxcJiicdener
Stentverzeichnime aiif ein Fundamentdxyfttem" Astronomische Nachncli/en, No.
1536. These tables include the Greenwich seven year catalogue for 1860, when
the correction corresponding to the declination of Uranus is about -|- 0".45. The
corrections for the previous catalogues vary between 0".35 and 0".68. The cor-
rection corresponding to the interval 1861-67 has been derived by a direct com-
parison with Auwers' declinations, and the result is -|- 0".44, agreeing with the
two preceding catalogues. But, on making a similar comparison with the annual
catalogue for 1S69, a considerable change was found, the correction being — 0".17,
a change of more than half a second. I shall use this correction for and after the
beginning of 1868, as the change is probably due to the introduction of a new
constant of refraction in the reduction of the observations for 1868 and subse-
quent years.
i
THE ORBIT OP URANUS.
119
Cambridge,
An extended scries of planetary observations was eommcnccd here by Professor
Airy, in 1827. Tlie series was continued by him and Professor C'hallis, his suc-
cessor, until 1842. During the first three or four years the combined right ascen-
sions depend on a few special stars, and mainly on a^ Capricorni. Taking the
mean correction to the adopted right ascensions of the stars actually compared as
they are given in the introduction to each annual volume, giving to each star a
weight proportional to the number of comparisons, the following corrections are
deduced :
1828 — OMO
1829-31 —0.10
1832-37 — O.in.
In the introduction to the volume for 1838 it is stated that the adopted right
ascensions are diminished by the average amount of ()'.()83, wliich would still
leave a correction of — ()M07. Actual comparisons in two subsequent years give
1840, A« = — 0'.087
1842, - .069.
Although the positions deduced from each year's work were adopted for clock
correction" the year following, without any change of equinox, it seems that there
was, eifcctivcly, a progressive change of about 0\0l annually in the equmox as
" No*" declinations were observed until 1830. On comparing the declinations
deduced from several years' work with Auwers, it was evident that the correction
increased with the polar distance of the star. The law of increase could be well
enough represented by supposing the correction proportional to N. 1 . D. i bus the
following corrections were deduced in three different years.
1".78 X N. P. D. jn^egrees
100
1834, h dec. = —
1840,
1842,
1".00 X N. P. D. in degrees
100
1".03 X N. P. D. in degrees
100
From which the correction for other years was deduced by interpolation. But,
on applying these corrections, the results were found systematically difterent from
those of other observatories, and on referring to Auwers' corrections to Airy s
Cambridge Catalogue, it appeared that the mural circle required a large correction
near the declination of Uranus during this period. The above results were there-
fore altered so as to conform as nearly as practicable to Auwers' law.
EiUnhtirgJi.
In reducing the observations of 1836 Henderson uses the right ««^">«;°"« °f
theTabulffi Ilegiomontana^, to which the general correction is at this epoch + .UU1.
199
THE OIIBIT OF URANUS.
But, if wc take only the stars near Uranus, with which the latter was necessarily
most frequently compared, the corrections will be negative. C'omparing the con-
cluded positions of the otars from a Serpentis through 0'' to fj Orionis, we Hud tho
following mean corrections :
In right ascension, — 0'.012; in declination, — 0".09.
In subsequent years it is stated that the adopted positions of clock stars used
each year ore derived from the right ascensions observed at Greenwich, Cam-
bridge, and Edinburgli, during the year or the two years preceding, without any
statement whether corrections were applied for ditfercnce of equinoxes. In some
subsequent years the following corrc-ctions are deduced, sometimes from the adopted
and sometimes from tho concluded positions :
1837, Aa = O'.OOO ;
1840, Aa = +0.015; A Dee. = O'.OO;
1844, Aa = +0 .070 ; A Dec. = +0 .45).
Far 18.
All the positions of planets given by Lc Verricr, in his '■'■Aniiahs tie VOhscrvn-
toire Impcmd de Paris: C>i/<6/-f«/t«M«"depcnd both in right ascension and N. P. D.
on his adopted positions of fundamental stars, the corrections to which iiuve
already been given. As the corrections to the in<lividual star places used by
Le Verrier are not generally of a systematic character, the general mean correc-
tion is employed, which is: —
In right ascension — 0'.024 + 0'.085 T,
In declination +0".12 + 1".207;
T being the fraction of a century after 1800.
In 1854 a new and larger catalogue was introduced, and for this and the follow-
ing years the correction in declination is derived from Auwers' tables.
A summary of the adopted corrections after 1830, as deduced from the pre-
ceding comparisons and discussions, is given in the following table: —
Table (if A
DOPTED SySTE.MATIO CcttUEOTIONS.
Year.
Greenwich.
Paris.
Kijiiigsberg.
Berlin.
Cambridge.
Eiiinbiirgh.
Aa
AS
Aa
A8
Aa
Aj
Aa
AS
Aa
AS
Aa
AS
s
n
8
n
B
//
8
II
8
8
1830
.00
....
+0.9
-.16
—1.7
.00
1831
.00
+ 1.8
.00
+0.5
—.02
+d.'9
....
+0.9
-.16
—1.6
1832
....
+ l.t
• • . •
—.02
+0.9
....
....
—.19
—1.5
....
1833
+ 1.6
—.04
....
—1.4
....
1834
+ 1.4
....
—.05
....
—1.3
....
1835
+ 1.2
....
. • • •
—.05
....
• at*
—1.2
....
1836
—.04
+ 1.0
+.01
+0.6
—.02
+ 1.0
—.04 +1.0
....
—1.0
—.01
1837
—.03
+0.9
....
> • • <
—.03 ....
—0.8
.00
1838
—.04
+0.8
....
....
—.02 1 . . . .
—.11
—0.5
+.01
1839
....
+0.T
....
....
.... 1+1.0
—.01
....
—.10
—0.3
+ .01
1840
....
+0.(!
....
....
1
.00
—.09
—0.2
+ .02
+0.1
1841
+0.5
....
1
.... j ....
+.01
—.08
—0.2
—.04
+0.2
'
TUE Oil BIT OF UUANVB.
191
Table or Adoi>tkd Systematic Cobbections.— Co«/tnu»(/.
Vuttr.
Oraonwluli.
I'irU.
Kilnlgslwrg.
U«rliil.
Caiulirlilgs.
K<lliil>urgh.
1842
1843
1844
1845
184(;
1847
1848
1849-53
1854-55
l85(J-(;o
1801
i802-f.5
l8(!li
18(17
1808
18(i0
1870
1871
1872
As
8
—.02
+ .07
+ .00
+ .08
+ .04
+ .05
+ .05
.00
.00
—.01
—.01
.00
.00
.00
.00
.00
+ .01
+ .01
+ .02
AS
II
+0.4
+ 0.4
+0.4
+ 0.4
+ 0.4
+ 0.4
+0.4
+ 0.4
+ 0.5
+0.5
+0.4
+0.4
+0.4
+0.4
—0.2
—0.2
—0.2
—0.2
U.2
A.
+ "01
+ .01
+ .01
+ .01
+ .02
+ .02
+ .02
+ .02
+ .02
+ .02
-1 .03
+ .03
+ .03
+ .03
+ .03
+.03
• . . .
AS
II
+0.0
+ 0.('.
+ 0.0
+0.0
+ 0.0
+
+0.0
+0.0
+0.2
+ 0.2
+ 0.2
+ 0.2
+ 0.2
+o.a
+0.2
+ 0.2
Ao
H
AS
Aa
8
+.03
+.04
+ .04
+ .04
+.03
+.02
+ .01
.00
—.01
At
n
+1.1
+ 1.1
+1.1
+ 1.1
+ 11
+1.1
+ 1.2
+ 1.2
+ 1.2
Aa
8
—.07
Sillll
.00
. . • •
....
AS
II
— o.:i
in(fo
-j 0.0
A» At
8
— 02 +0.8
+ .07 +0.4
+ .07 1 ^ 0.6
Wusliington
.00 —0.5
.00 +1.1
.00 j +1.1
.00 +1.2
.00 +0.0
.00 +0.4
.00 -4-0.4
.00 -i-0.4
Applyiiij? the prrroding corroctions to tho positions of tho planet as orijjinally
Tediux'd and publishi'd, we liavc a sciii-s of observed positions as nearly homo-
geneous as it is possible to make them with the means now at our command. Tlio
next step in order will be tiio computation of the geocentric place of the planet
from the provisional theory for the moment of every observation, to be compared
with the results of the latter. The complete execution of this labor, ab initio, is,
however, at i)rcsent impracticable, and it is proposed to diminish it by making use
of the published comparisons with the older tables. Tliis can be done without
danger of serious error, and with all the more case that owing to the great distance
of Uranus the errors of the solar tables are, for the most part, without appreciable
effect upon tho computed geocentric place of tho planet. Tlie method of making
the comparison is different with different series of observations, and each scries
must therefore be described and discussed separately. The general plan, how-
over, has been to replace observed and computed absolute positions by observed
and computed corrections to the geocentric positions deduced from Bouvard's
Tables. To carry out this plan it is necessary to have at our disposal an ephemeris
botli of the heliocentric and geocentric positions derived from these tables. Tlie
corrections to the latter given by the observations are then given by direct com-
])arison. To obtain the corrections given by the provisional theory, the heliocentric
longitudes, latitudes, and radii vectores given by tliat theory arc interpolated to
the dates of the heliocentric ephemeris from Botivard's Tables, and compared with
that ephemeris. The differences are tli(>n chang(Ml to differences of geocentric
place by the usual differential formuhr, and thus tlio cornn'tions given by theory
are derived. The difference between the two sets of corrections is the difference
10 May, 1873.
122
T II K 11 H IT OF U II A N U S.
between the provisional theory niul observation, A roti«b'ns(>(l ■nmmary of the
results for euch of the priiieipal tieries of observations is here presented.
Oirciivlch, nsl-lSHO.
In Airy's rediietions, ahcady refcrvi-d to, we liave given for the moment of each
individual obNervation u helioet^iitiie place computed from IJouvard's Tables, nnd
the geocentric longitudes and latitudes thenee deduced. The observed right
ftscensions and declinations are then changed to longitudes nnd latitudes, nnd thi;
apparent error of the tables thence deduced. The means of these errors nre taken
for groups of observations, and expressed in terms of tho errors of heliocentric
longitude, radius vector, and latitude. The mode in which these means have been
treated is fully shown in the following table. The first colunm gives tho mean
date of each individual group of observations. The ne.\t three give the mean
excesses of the co-ordinates interpolated from the heliocentric ephemeris, p. 100,
and corrected for solar nutation, over those printed in the "Compututions of tabular
place, etc.," in the Greenwich reductions. In tlie fifth column these corrections
nre changed to corrections of geocentric longitude. In the next two columns we
have tlie mean corrections to Bouvard's geocentric places given by observation. It
is the negative of the mean error of tabular place printed in the " Reductions,"
corrected by the numbers ulready given to reduce the star places to ii nniforiu
system. Then we have the difference between these two sets of corrections, or,
the mean correction to the geocentric place of the provisional theory as given by
observation. Lastly, we have the differential coefficients for expressing tlie errors
of geocentric in terms of the errors of heliocentric co-ordinates taken without
change from the Greenwich volume.
From ProTinional Theory.
From ObiiiTTationa.
~^
rorri-i'tlnn to
Mean Date.
Corroction to latmlar poaitlou in
UreHUwich Kuductiona.
Correction to
I'lOV.
'i'l»«ory.
dl
69
Long.
W.
Ut.
Geoo.
Oeoo.
H«l.
Noof
(1(«>0.
R. V.
long.
long.
lut.
Olw. 1 long.
1.
//
If
ff
If
II
1791, Oct.
10
— 7.8
• • • •
+1.1
— 7.5
— (5.4
+ 2.0
4
+ 1.1
+ 1.5
1.01
— 9
Nov.
13
— 7.5
+1.1
— 7.5
— 4.9
— 1.0
4
+2.0
— 2.7
1.04
— 6
Dec.
27
— 7.3
— i59
+ 1.0
— 7.7
— 5.(i
— 2.3
5
+2.1
— 3.3
1.05
+ 1
1782, J Ml.
31
— 7.4
+0.9
— 8.0
— 4.9
— 1.1
4
+3.1
— 2.0
1.04
+ fi
Mar.
4
— G.G
+0.9
- 7.1
— 4.(i
— 0.0
4
+2.5
— 1.5
1.01
+ 9
Oct.
10
— 5.0
+0.4
— 5.3
-3.C
— 4.0
4
+ 1.7
- 4,4 1.01
— 9
Nov
2(i
— 5.3
+ 0.3
— 5.3
— 0.0
— 5.0
3
—0.7
~ 5.3
1.04
— 5
1783, .Ian.
11
— 5.0
+0.3
— 5.4
— 3.9
— 3.7
3
+ 1.4
— 4
1.05
4-3
Feb.
Oct.
24
10
- 4.4
— 3.5
+0.2
—0.3
— 48
— 3.2
— 2.2
— 0.3
— 2.4
— 1.0
8
2
+2,0
— '..6
— 0.7
1.02
1. 00
+ 9
—10
+2.9
Nov.
1
— 3.2
—0.3
— 3.0
— 1.1
— 2.3
2
+ 1.9
— 2.0
1.02
— 9
Dec.
15
— 3.3
— i2i
—0.4
— 3.4
— 2.2
— 3.8
3
+ 1.2
— 3.4
1.05
— 3
1784, .Ian.
29
— 3.1
. • • •
—0.4
— 3.4
— 2 4
— 2.4
3
+ 1.0
— 2.0
1.05
+ 5
Mar.
Oct.
12
30
— 2.6
— 1.8
....
—0.5
—0.8
— 2.9
— 1.5
— 0.3
— 0.7
— 4.3
+ 1.4
3
2
+2.0
— 3.8
1.01
1.02
+ 9
— 9
+0.8
+ 2.2
Dec.
14
— 1.5
— i28
—0.9
— 1.5
— 2.4
+ 2.7
2
—0.9
+ 3.0
1.05
— 4
Til 13 OUUIT OF UUANUa.
198
Kiuiii rruvltluual Tliuury.
Kruiu OUarTtttiuua.
M««u Datii.
CurrHCtliin tn tnlmliir pnaltlou In
(li'ouiiwiuli iilmirvittioiiii.
Carrsatlon to
Correilloii to
Piov. riieoiy.
vl
6-/.
dl
Lo(i.
Lat.
n
Ohoo.
Oeoo.
llol.
No.of
n«oo.
H«l.
Long.
R. V.
long.
long.
lut.
UU.
long.
lat.
//
It
II
II
II
1785, Jhii. KI
— 1.2
— 1.0
— 1.3
— 1.4
— 1.4
2
—0.1
— 0.4
1.05
+ 8
KrI.. 1.1
— 1.0
— 1.1
— 1.3
— 1.2 i
— 1.1
2
+ 0.1
0.0
1.04
+ '
Mm. 20
Nov. H
— O.fJ
+ 0.2
—1.1
—1.5
— 0.9 ; — 2.1
+ 0.5 ' + 1.2
— 5.2
»
— 1.7
2
2
—1.2
— 4.1
1.01
1.02
+10
— 9
+0.7
— 0.2
1788, Mur. l.'l
Oct. 2«
+ a.l
+ 3.4
— 'i87
—3.2
—3.0
+ 1.8 + 4.4 1
— 4.9
— 2.2
3
2
+2^
— 1.7
1.03
1.00
+ 9
—10
+ 3.9
+ 3.0
—0.9
+ 1.4
178'J,Jan. 18
+ 4.2
-'224
—3.7
+ 4.5
+ 1U.4
—14.5
1
(+6.9) (—10.8)' 1.00
— 1
Apr. 8
Oct. 31
+ ^■1
+ 4.8
—3.8
—4.2
+ 4.2
+ 5.4
+ 9.4
+ 4.7
— 3.7
— 1.1
4
2
+ 5.2
+ 0.1
1.01
+ 10
—10
—0.7
+ 3.1
1.00
1790, Juu. 24
+ 4.9
4.4
+ 5.2
+ 3.1*
— 2.6
2
—1.4
+ 1.8
1.00
— 1
Nov. 5
+ 6.1
- 254
—4.8
+ 5.8
+ 5.0
— 2.3
3
—0.8
+ 2.5
1.00
— 10
1791, Jan. 29
+ 5.1
— 258
—6.0
+ 5.4 + 4.5
— 3.8
3
—0.9
+ 1.2 1.06
Apr. 14
+ 5.0
....
—5.2
+ 4.5 + 2.7
— 4.8
1
—1.8
+ 0.4 ! 1.01
+ 10
Nov. 10
+ 5.0
- 234
—5.4
+ 5.0 + 5.7
— 4.0
2
+0.1
~+ l74 1 1. 00
—10
1792, Feb. 5
Nov. 15
-f 5.2
+ 5.0
—5.0
—0.0
+ 5.5
+ fi.2
+ 4.2
+ 3.7
— 4.0
— 2.5
1
3
—1.3
+ 1.6
1.00
1.00
-10
—2.5
+ 3.5
1793, Feb. 8
t 5.4
—0.1
+ 5.7
+ 8.6
— 5.0
2
+ 2.8
+ 1.1 i 1.06
Nov. 14
+ 5.0
— 135
— G.4
+ 5.9
+ 10.4
— 6.3
1
+4.5
+ 0.1
0.99
—10
1794, Fob. 15
+ 5.7
— 100
—0.6
+ 0.0
+ 7.3
— 0.3
2
+ L3
+ 0.2
1.00
Nov. 19
+ 5.4
— 07
—0.7
+ 5.0
+ 4.1
— 5.1
4
—1.5
+ 1.6 f 1.00
—10
1795, Feb. 20
Nov. 29
+ 5.9
+ 5.4
— 40
— 13
—0.9
—7.0
+ 0.3
+ 5.5
+ 0.0
+ 3.9
— 8.0
— 3.4
3
2
—0.3
— 1.1
1.06
1.00
—10
—1.6
+ 3.6
179fi, Feb. 24
+ 5.3
- 8
—7.2
+ 6.6
+ 4.5
— 4.5
2
—1.1
+ 2.7
1.06
1797, Feb. 27
1800, Mur. 14
1814, Mfty 22
+ 5.1
+ 4.0
— 1.2
+ C2
+ 224
+ 217
—7.5
—8.4
—2.2
+ 5.4
+ 4.9
— 1.3
+ 4.7
+ 4.9
+ 1.1
— 4.2
— 8.7
+ 0.2
3
2
2
^0/7
+ 3.3
— 0.3
1.00
1.00
1.00
0.0
+2.4
~+ 2.4
1815, May 25
— 1.1
+ 214
—1.5
— 1.2
+ 1.4
— 1.0
4
+2.6
+ 0.5
1.00
1818, June 10
— 0.5
+ 413
—0.8
— 0.5
— 1.1
+ 4.8
2
—0.6
+ 5.6
1.00
1819, Juno 14
— 0.4
+ 483
+ 1.5
— 0.4
— 1.4
+ 3.7
4
—1.0
+ 2.2
1.00
1820, June Ifi
— 1
+ 498
+2.2
— 0.1
— 2.0
+ 4.0
2
—2.5
+ 1.8
1.06
1823, July 1
0.0
+ 538
+4.4
0.0
+ 0.2
+ 5.0
4
+ 0.2
+ 0.6
1.05
1825, July 11
— 2.4
+ 507
+5.8
— 2.5
— 4.8
+ 7.8
2
—2.3
+ 2.0
1.05
1820, July l(i
— 4.5
+ 582
+0.0
— 4.7
— 3.0
+ 8.3
4
+ 1.1
+ 1.7
1.05
1827, July 20
— 0.0
+ 02(
+7.2
— 0.9
— 7.2
+ 9.6
6
—0.3
+ 2.4
1.05
1828, July 23
— 9.8
+ 710
+7.9
— 10.3
— 0.9
+ 0.9
3
+3.4
— 1.0
1.05
1829, Aug. 7
—13.4
+ 83.'i
+8.5
—13.7
—16.3
+ 10.0
14
—2.6
+ 1.5
1.08
+ ?
Oct. 4
—14.2
+ SO,")
+8.0
—12.5
—13.0
+ 9.9
8
—0.5
+ 1.3
1.02
+ ^
1830, July 30
—17.8
+ 97C
+9.0
—18.8
—21.4
+11.1
3
—2.6
+ 2.1
Aug. 29
—18.2
+ 98!>
+9.1
—17.it
—20.0
+10.2
5
—2.1
-4- 1.1
Sept. 20
—18.5
+ 99i
+9.1
—17.2
—19.1
+ 9.1
12
—1.9
0.0
Oct. 14
— 18.9
+100f
+9.1
—10.8
—17.3
+ 9.9
11
—0.5
+ 0.8
Nov. 13
—19.3
+ 100J
+9.2
—16.8
—17.8
-\ a. 3
8
—1.0
+ 0.1
124
THE OUBIT OF URANUS.
Parh, 1801-1827.
A complete reduction of this series is found in Le V'errier's Anindvs lie I'Obser-
valoifc Imj}erud (Ic Purin, OhserontioHK, tome 1. No comparison with any ephemeris
is given here, nor i« there any complete ephemeris to compare them with. A
complete geocentric ephemeris was therefore computed from the provisional theory
for the principal groups of the Paris observations. Tlic individual observations
being compared with it, the resulting mean corrections arc given in the following
table :
Menn date.
Aa
AS
N.
Mcnn datfi.
Aa
AS
N.
1801, ^larch 24,
—".02
+ 1".2
2
1813, May 20,
+M9
+r.8
6
1802, April 1,
+ .08
+0.6
13
1814, May 27,
+ .21
+0.8
4
1805, April 22,
+ .10
+2 .2
13
1815, May 24,
— .02
+2 .2
5
1806, April 17,
— .01
-1.6
5
1816, June 1,
— .01
+0.8
7
1807, April 28,
+ .17
+0.4
16
1817, June 5,
— .08
+ 1.6
5
1808, April 28,
+ .02
+ 1.4
6
1818, June 7,
+ .12
+2 .2
9
1809, May 5,
+ .20
+0.1
9
1819, June 18,
— .07
+ 1.8
7
1810, April 30,
+ .22
+2 .6
16
1820, June 20,
-.20
—2.4
8.5
1811, Fcbr'y 18,
+ .21
+2 .2
3
1821, June 22,
+ .05
+ 1.0
5
1811, May 17,
+ .14
+2.6
8
1823, July 18,
+ .02
+ 1.8
5
1812, Febr'y 1(5,
+ .28
2
1824, July 13,
+ .04
0.0
7
1812, May 10,
+ .16
+3 .0
6
1827, July 25,
— .05
+0 .6
5
1813, Fcbr'y 25,
+ .44
3
Total number of observations in right ascension, 175.
The observations in this series exhibit numbers of discordances of that class
which leave the astronomer in doubt whether the observation should be retained
or rejected. This remark applios more especially to the declinations. If we de-
termine the probable error of an observation in declination by the condition that
it is that amount which the error fulls short of as often as it exceeds, it is found
to be about 2". Then, if the errors followed the commonly assumed law of proba-
bility, only about one in six of the errors should exceed 4", and one in twenty-
three 6". But errors of these magnitudes are much more numerous, the deviations
often amounting to six or eight seconds. I nave rejected only a few in which the
discordances approached 10".
BessiTs IConiifgherg Obxcrrntions, 1814-1835.
T have made a complete re-reduction of the right ascensions of this important
series, and of most of the declinations. In order to avoid the necessity of ap|)ly-
ing systematic corrections. Dr. Gould's right ascensions and Dr. Anwers' declina-
tions were used througho)it in these reductions. In this work a selection of
the fundamental stars observed by Bessel was made for each observation of the
planet, to be used for clock error. These were chosen so tliat the mean of their
right ascensions and declinations should be as near as practicable to those of
Uranus, a condition, however, which coidd not generally be fulfilled for the decli-
nations, owing to the southern position of the planet. Bessel's instrumental cor-
THE OUBIT OP URANUS.
126
roctions were applied to his observed times of transit over the mean wire, and the
resulting ti.ne was employed as that of transit. Each time, compared with the
computed right ascension of the star gave a value of the clock correction, which
was reduced to the time of transit of the planet by the known daily rate. It the
instrumental errors were always accurately determined, the mean of these clock
corrections would be used to obtain the right ascension of Lranus. But it was
frequently found that the clock error varied systematically with the declination of
the star, so that it was deemed advisable to add to the clock correction a term
varying as the simple declination, which was deduced from all the stars, and used
to reduce the correction to the parallel of Uranus.
It was intended to give the results of this reduction for each observation, but
on comparing the results with those of Fleming in the AstronomiscJie Xachnchten,
Band 30, it appeared that the results were not materially better than his. It
does not, therefore, seem necessary to give more than the mean results for each
opposition. .
From Bessel's declinations, with the old Cary circle, I was unable to obtain any
satisfactory results, owing, apparently, to a wai.t of knowh-dge of some pernl.ar.ty
„f tlie instnmunt. Fleming's reductions were therefore adopted, ihey are
designated by the letter F in the following list.
Mean Corrections to the Prormonal Eplicmeris ijiven ly Bo^scVs Oh^^crvalhns at
Koii l<jxher<), 1 8 U- 1 829.
AS K I Moan ilftto.
-]-2".i)F 9 1822, June 24,
_|_1 .8/.' 11 1823, July 4,
+ 1.2/'' 11 \ 1824, JulyC),
+2 .3/' 8 ! 1825, July K),
4-4 :.\F 13 i 1826, July 18,
+4.1 4 I 1828, July 25,
+ 1 .5 5 ! 1829, Aug. 1,
Total numbers of observations, 103.
Results of Ohscrvat!n»s at various Ohserratorlrs, from 1827 to 1829 lurlush-c.
During these three years we have, besides the observations already quoted, the
following: —
1 Observations bv Schwcrd, at Sp.-ier, of which the originals are given in
Astrouo,ulsvJ,r Ikohal.Um,jen amjrsicti; av/ <1rr Sler.irarte drs Kiiul.,1. J.>/z>ums lu
SjH;,rr vou F M. Mwerd, Speyer, 1S29-30, an.l of whi.h the reduced results
are found iu tlie Astronomiscbe Naehricliten, Ba-.id 8, S. 2(14. ^ , . ,
2. The series by Airy, at Cambridge, commenced iu 1828, and found in the
Cambridge Observations.
3. Littrow's Vienna Observations, found in the fiist series of Annahn drr A. A.
Stennvartc In Wlcn.
The mean corrections to the ].rovI. ional epbenieris given by these series are
shown iu the following table. The observations have been divided in the usual
Menu ilal?.
1814, May 2\
1815, May 25,
181(), May 27,
IS17, June 0,
1818, June 8,
1820, June 21,
1821, June 23,
Aa
+M1
+ .13
+ .()')
+ .13
+ .02
+ .02
+ .12
Aa
+M0
— .05
+ .01
+ .01
— .01
— .15
—.10
+r'.8
-1 .()/<'
— 1 .{)F
-2 .4/''
—3 .OF
-3 J)F
-1 .OF
N.
J
126
TUE ORBIT OP URAXUS.
way into groups of about a month each, and the mean date and mean correction
found for each group. The Paris ond Konigsberg results are repeated for tlie
sake of clearness. The small figuies show, as usual, the number of observations
employed in forming the mean.
Aa
Ad
Date.
Observatory.
Original. Corrected.
1827,
July 22,
Speier,
— 0M6, — 0M4
July 25,
Paris,
_() .0:3. —0 .05
+0".53
September 15,
Vienna,
— 0.11,,— 0.10
.0,.
October U,
Vienna,
—0.18, —0.17
-2.2,
1828,
July 25,
Kiinigsberg,
—0.15, —0.15
-3.5,
July 29,
Vienna,
—0.24, —0.20
-1.4,
August 14,
Vienna,
—0 .13,,, —0 .09
+ 1 .1,0
August 27,
Speier,
—0.10, —0.09
September 18,
Vienna,
—0.03, +0.01
+ 1.0,
September 25,
Cambridge,
_0.05, —O.IG
October 17,
Vienna,
—0 .13,, —0 .09
0.0„
October 17,
Cambridge,
—0 .02,,, —0.12
1829,
August 1,
Konigsbcrg,
_0.1()„ —0.10
— 1.0
August 6,
Cambridge,
+0.11, -0.08
-1.1.7
August 28,
Speier,
_0.04, —0.04
Soptembor 23,
Cambridge,
+0.21,0+0.05
November 6,
Cambridge,
+0.25, +0.09
Obnervations from 1830 io 1872.
Since the year 1830 heliocentric and geocentric ephcmerides of Uranus com-
puted from Bouvard's Tables are at our disposal. We make use of those in the
Berlin Astronomisches Jahrbuch for the years 1830 to 1833, and of those in the
Nautical Almanac from 1834 forward. The system of comparison is the same
as that already explained. That is to say, we deduce separately :
(1) Mean corrections to the geocentric longitude and latitude of Uranus hi the
ephereris as derived from observation.
(2) Mean corrections to the same, given by the provisional theory, as derived
from a comparison of the heliocentric positions of that tiieory with the heliocentric
positions in the ephemeris.
Then (1) — (2) is the correction to the provisional theory given by observation.
The process of forming (1) and (2) is shown quite fully in the following jiages.
Each individual printed observation was first compared with the printed ephemeris,
and a correction to the latter was thence deduced. AVhen this correction was
given with the observations themselves, it was of course not recomi)uted, uidess in
some doubtful cases. The observations were then divided into groups, usually of
about a month each, and coinciding in time with the grouping of the Greenwich
results. The mean of the dates and the mean of the corrections were then taken
separately fir each group and each observatory. The separate results arc shown
THE ORBIT OP URANUS.
127
I
J
in the proper columns of the following table, under the head "Mean dates,"
Mean cor. in R. A., and Mean cor. in Dec. These means are those given by the
observations as printed, without the application of the systematic corrections on
pages 120 and 121. In the columns " Corrected mean" these corrections are applied ;
this column would therefore exhibit no systematic differences between the results
of the different observatories, unless the observations of Uranus were affected by
errors different from those which affect the positions of the fundamental stars. A
careful comparison of the differences in various parts of the table shows that this
is unfortunately the case. A weight is next assigned to each individual result
depending on the number of observations, the general sufficiency of the data of
reduction, the mean discordance of the individual observations, and the quality of
the instruments. The critical reader will notice a lack of homogeneity among the
weights assigned, of which I shall speak presently. The mean of the separate
group-results is then taken with regard to these weights, and also the mean of tlie
mean dates, using for the latter the relative weights adopted for the several right
ascensions. Thus, we have a mean result derived from all the observations for
each month, or other group-period, which is written under the horizontal lines.
These corrections to right ascension and declination are next changed to correc-
tions of longitude and latitude, using for this purpose the following table, which
is computed from the formula; of Gauss :
cos E = sin E cos a sec b = sin e cos I sec 5
, cos /; . .
sin jP cos 5 ^
Al=: , - Aa
coso
cos b
Ab= — cos E cos 5 Aa + sin E Ab.
The differential coefficients in this table are expressed as a function of the right
ascension of Uranus only, which may be done because, owing to the small iuclina-
tiou and great distance of the planet, its geocentric position on the celestial sphere
is never more than about 2' from some point of the projection of its heliocentric
orbit. The coefficients of A« are multiplied by 15, that the right ascension may
be expressed in time.
To Convert Eiuiors of Riqht Asoe.nsio.v and Declination op Ukanus into errors op
LoNdlTUDE AND LATrfUIlE.
When the Ilight Asieiision exceeds 12', enter with R. A — 12^ aii.l oh-iiige the signs of Uie qnanlities
(' b , 61
6a od
■J. A.
0" 0""
10
20
30
40
60
1
10
20
30
40
60
Logniithms of
61
6a
+13.8+
+13.9+
6b
6a
-5 97+
6.96
6 95
5.92
6.87
6.81
-5.74+
5.66
6.56
5.45
5.33
5.20
61
6S
6b
6b
61
6a
1.1386
1.1387
1.1388
1.1389
1.1390
1.1391
1.1392
1.1393
1.1394
1.1394
1.1395
1.1396
6h 61 \ 6b
6a 66 ' 68
—0.7761 , +9.6000 . 9.9626
—0.7757 "" ,1 +9.5996 ~ „ 9.9626
—0.7743 "~ '... +9.5983 ~ ',, 9.9628
—0.7720 ~ ';; +9.5963 "~ f,' 9 9632
—0.7687 — ii +9.5934 ~ T^l 9.9637
—0.7643 ~ ll +9.5896 _ ;," 9 9645
-0.7588 ,,„ +9.5849 ., 9,9653
-0.7525 — ''■] +9 57114 ~ '{[ 9.9662
-0.7451 ~ I* +9.573(1 ~ 74 9 9673
-0.7 65 ~ ,'; +9.56.-.6 "" r*. 9.96^5
-0.7J70 -,; ;; +9.5573 - :;: 9.9698
-0.71B2 ~},g +9.5479 ^^^^ 9.9711
+0.40—
0.40
0.40
0.40
0.39
0.38
+0.38—
0.3K
0.37 ■
0.37
0.3(i
0.36
+0.92+
-4 0.92+
128
THE ORBIT OP URANUS.
To CoNVEiiT EiiRons or Uiqiit Asci
.\sioN AND Declinatki.n. — C'oiiliniicd.
B. A.
LogarithniB of
61
,5ft
61
O^ft
()l
eft
5Z
6b
da
6a
^5
dS
6a
6a
'66
+ 0.93+
21' 0"
1.1.395
-0.7<i44 „q
— 0.6915 \-.,
-(1.6773 -J;^-
-0.6618 \'.Z
-0.6450 -]f,
-0.6266 Zw
+ 9.5375 _ J.
+ 9.5260 }.^*
+ 9.5134 -|^^
+9.4995 X •'?
+0.4843 Jf.2
+9.4676 X}^I
9.9725
+13.8+
—5.06+
+0.34-
10
1.1, !95
9.9740
4,91
0.33
2(1
1.1395
9.9756
4,75
0.32
30
1.1395
9.9772
4,58
0.31
41)
1.1395
9.9788
4,41
30
BO
1.1394
9.9804
4,23
0.29
3
1.1394
— O.C069 niR
_0..'.854 -j'i:
—0.5368 —r;l^
-0.5094 -f'*
-0.4795 Z^^
+9.4495 ,.„
+9.4297 -}^,l
+9.4081 X^i?
+ 9.3844 -i'
+ n.358(i ^»°
+ 9.3302 _^J*
9.9821
+13.8+
—4,05+
+0.28—
+0.96+
10
1.1394
9.9837
3.86
0.27
20
1.1.393
9.9853
3.66
0.26
3U
1.1393
9.9869
8,45
0.24
40
1.1392
9.9883
3.24
0.23
60
1.1391
9.9898
3.U2
0.21
4
1.1390
-0.4468 „,„
-0.4109 _lll
—0.3710 _fji
-0.3267 -i^^
-0.2769 *i,
-0.2198 ZZl
+9.29S0 „.„
+9.2644 Zttt
+9.2259 ^.^\
+9.1828 "1
+9.1341 ,^»^
+9 0781X2^?
99912
+13.8+
—2.80+
+0.20—
+0.98+
10
1.1390
9.9925
2.58
0.18
20
1.1389
9.9938
2.36
0.17
30
1.1388
9.!'9.n
2.14
0.15
40
1.1387
9.:':.-U
1.91
0.13
60
1.1386
9.y' J
1.67
0.12
5
1.1385
—0.1539 -„,
-0.(1753 ~'l^.
-9979 -,?,^
-9.854 };l
—9.378 ■^^^
+9.0130 »-_
+8.9353 _^^J
+ 8.8397 _f^%
+8.7157 |:^"
+8.540 _}/■:'
+8.240 •^^J^
9.997'i
• +
—1.43+
+0.10—
+0.99+
10
1.1384
9.9984
1.19
0.08
20
1.1383
9.9990
0.95
0.07
30
1.1.382
9.9994
0.71
0.05
40
1.1381
9.9997
0.^8
0.03
50
1.1380
9.9999
0.24
0.02
6
1.1379
— oc
<X
+13.7+
0.00
0.00
+1,00+
10
1.1378
+9.378 jlo";
-(-9.«79 +fl
+ 9.854 +\l'
+9.978 +^;*
+0.0743 +^11
-87157 +1^"
«-^^" +1240
-89353 +95«
9.9999
—0.24+
—0.02+
20
1.1377
9.9997
0.48
0.03
30
1.1376
9.9994
0.71
0.05
40
1.1375
9.9990
0.95
0.07
50
1.1374
9.9984
1.18
0.08
7
1.1373
+02753 +6'»'
+S:.3260 +'^,
+0.3692 +t!^
+0.408S +^«J
—9.0130 ,..,
-9.0781 X*"
-9.1341 ±5^'*
-9.1828 Xl^i
-9.2259 X*2J
-9.2644 +33««,
9.9977
+1?.7 +
+1.42—
—0.10+
■ 1-0,99+
10
1.1373
9.9969
1.65
0.12
20
1.1372
9.9959
1.88
0.13
30
1.1371
9.9949
2.11
0.15
40
1.1371
9.9938
2.34
0.17
50
1.1370
9.9925
2.56
0.18
8
1.1370
+0.4448 , „„-,
+0,4773 X-^^
+0,.';071 J^l
+0,5344 +^l
+0.5597 +2:*,^
+0.6830 +2j'3
-9.2990 , ,,„
-9.3302 +111
-9.3586 Tf*
-9.3844 +^^8
-9.4(181 X^*J
-9.4297 +\ll
9.9912
+13.7+
+2.78—
—0.20+
+0.98+
10
1.1370
9.9898
3.00
0.21
20
1.1370
9.9883
3.22
0.23
30
1.1370
9.9869
3.43
0.24
40
1.1370
9.9853
3.03
0.26
50
1.1370
9 9837
3.83
0.27
9
1.1370
+0.6044 , ,„
+0.6241 + 11
+0.6425 +\f
+0,6594 + '^
+0.6749 +\'^,
+0.6892 +\jl
-9.4495 , ,„.
—9.4676 t\%
-9.4843 + 5^^
-9.4995 +j°2
-9.5134 t"^
-9.6260 +111
9.9821
+16.7+
+4.02-
—0.28+
+0.96+
10
1.1370
9.9804
4.20
0.29
20
1.1370
9.9788
4.38
0.30
30
1.1371
9.9772
4.55
0.31
40
1.1371
9.9756
4.72
0.32
50
1.1372
9.9740
4.38
0.33
10
10
1.1373
1.1374
+ 0,7023 , ,
+0.7434 X ^/j
+0.7510 + ^5^
. -9,5375 , J..
-9.5479 +^"*
— 9.M73 X 81
— 9.5(;66 X 74
—9 5730 X ,'I
-9.5794 + 5^*
9.9725
9.9711
+13.7+
+5.04—
6.18
-0.34+
0.35
+ 0,93+
20
1.1376
9 9698
5.31
0.36
30
1.1376
9.9685
5.43
0.37
40
1.1377
9.9673
6.54
0.37
50
1.1378
9.9602
6.64
0.38
11
1.1379
+0.7575 , .,
+0.7(i32 + "
+0.7678 + *°
+0.7713 + Z
+0.7737 X ^*
+0.7754 + ^
—9.5849 , .-
—9.5896 X QS
-9.5934 X on
-9.5963 + :^;
-9.69.K3 X i\
—9.6996 X 4
9.9i;53
+ 13.7+
+5.72—
-0.38+
+ 0.92+
10
1.1380
9.9645
5.79
0.38
20
1.1.381
9.9037
6. SO
(1.39
30
1.1382
9,9032
6.90
0.39
40
1.1. J83
9.9628
6.94
0.39
60
].13()5
9.9626
5.96
0.40
12
1.1386
+0.7761
—9.6000
9.9626
+13.8+
+6.97-
— «.40+
+0.92+
THE ORBIT OP URANUS.
129
Wc thus have, for the interval occupied by each group of observations, a mean
correction to the geocentric longitude and latitude of the planet given by obser-
vations, which are fountl in the ninth and tenth columns of the table, on the same
horizontal line with the mean corrections in right ascension and declination from
which they arc derived. The next step is to obtain the corresponding corrections
given by the provisional cphemeris.
This correction has been first obtained for every twentieth day of each of the
forty-two oppositions included in the table. The heliocentric longitude, latitude,
and radius vector were interpolated to the most convenient twenty-day' intervals',
and compared with the corresponding co-ordinates in the heliocentric eplicmcris.'
This cphemeris was of course the one corresponding to that with which the
observations were compared, namely, the Berliner Jahrb'uch for the years 1830-33,
and the Nautical Almanac for subsequent years. These comparisons arc fullj'
given at the end of this chapter, and the resulting corrections to the printed
cphemeris are given in the proper columns of the table.
These corrections to the heliocentric co-ordinates were then changed to corrections
of geocentric longitude and latitude by the following formula'. Put
»•', the projection of the planet's radius vector on the cclii»tic ;
p', the projection of the planet's distance from the earth on the same plane ;
p, this distance itself;
X,6\ the planet's heliocentric longitude and latitude ;
Z, the sun's geocentric longitude ;
Jl, its radius vector;
M, the modulus of the common logarithms;
il, if), the corrections to the geocentric longitude and latitude ;
5p, the correction to the common logarithm of tlie radius vector
Then
61
= ^|l+5cos(Z-;i)|
— ^ s:n (Z, — 7.) jf4-^,
p J/sm 1
hX
-\--r, sin(/. — ?.)tim,33^i
Ih = -'; I 1 -I- ilf tan V cos (Z - ?.) I h^
r'Ul
,„ tan ;i sui (Z — X) fU
PP
+ ''^1 { 1 + ;' cos (/. _ X) I sin i3 -,, ^P ,.-
The last tern? in hi and tl;e last two terms of fb have been omitted in the com-
l)utation, as they scarcely ever exceed a few hundredths of a second.
17 May, 1873.
130
THE ORBIT OP URANUS.
The values of H and ib are printed in the last two columns of the table,
formula for hb might have contained the additional term
ib = sin ISa
The
fiu being the correction to the obliquity of the ecliptic adopted in the ephcmeris
to reduce it to tliat employed in tiie provisional theory. This correction is, how-
ever, deferred until wc conio to form the equations of condition.
From the values of tl and hb thus obtained we are to find the mean values
during each group of observations. If these quantities varied uniformly, the
proper value would be that corresponding to the mean date of each group. But
the second ditferences arc so large that this value would generally be in error by
one- or two-tenths of a second. Owing to the minuteness of this difference, it has
been considered that when tiie mean date was near the middle of a twenty-day
interval, the correction fil interpolated to that date without regard to second
differences would furnish a sufficient approximation to the required mean value of
H during an interval of about 30 days. In other case the value of fl was inter-
polated to 5-day intervals through the period of each group of observations, and tho
mean value taken.
During the years '850-1863 the sun's longitude employed in the ephemeris
required a gradually .iicreasing correction, amounting at the latter date to about
3". A small correction of which the maximum value is about 0".15 was applied
to tl to reduce it to the value it would have had if Hansen's tables had been
employed.
The corrected mean values of II and !h thus obtained arc given in the last two
columns of the following table, being inclosed in brackets and printed innnediately
above the values of Al and Ab derived from observation.
1 deem it proper to mention that the mechanical labor of constructing these tables
of comparisons, in the manner just described, was in great part performed by Dr.
C. li. F. Kampf, who was enijdoyed by the Smithsonian Institution to assist me in
the work. Before using it I subjected the whole of the work to a careful revision,
altering especially the relative weights of the corrected means in many cases. As
the assigned weights now stand, each set of results which arc combined into a
i' ^-j- mean has its own unit of weight, which does not necessarily coincide with
that of any other set. The use of a uniform scale of weights through this scries
of observations, and the assignment to every final mean of a weight equal to the
sum of the weights of the quantities whose mean was taken, would have led to
weights in many cases quite fictitious, owing to the obvious presence of systematic
errors in the residts. For tliis reason I have made no further use of the weisrhts
found in this table, and their lack of homogcncousn(>ss therefore does no harm.
lesss
TUE 0U13IT OF URANUS.
131
T
Mean CoiuiiiCTioNS to tiik Kpiiemkhis ok Uranvs in tiik Bkiii-inku Jaiiuducu and the
Aaltical Almanac.
Obaervatory.
[It. A. of
Crauus. ]
Mean dates.
Obaerred oarrections in R.A.
Konip;sl)org,
Cambridge,
[20" 40'°]
KOnipKbcrg,
Cambridge,
[20' 3G'°]
Cnmbridgo,
[20" 37'"]
Cambridge,
[20" 3(1'"]
Cambridge,
[20» 3'i"']
Grcenwieh,
Cambridge,
[20" M'"}
Greenwicli,
Cambridge,
[20" l>-2'"]
Grcenwieh,
Cambridge,
[20" 50"']
Greenwich,
Konig.«berg,
Camliridge,
Vienna,
[21" U""]
Cambridge,
[21" 12"']
Cambridge,
Vienna,
[21" 0'"]
Cambridge,
A'ienna,
[21" lO"]
1830
July 29
July 29
July 29
Aug. 12
Aug. 25
Aug. 19
Sept. 19
Oct. 17
Xoir. 14
1831
Aug. 3
A ug. H
Mean.
Aug. G
Sept. 7
Sept. 15
Sept. 11
Xov.
Oct.
4
20
,s
— 1.5(i
—1.51
-l.f.5
-1.30
-1.4f.
— 1.34
—1.30
Oct. 31
1832
Aug. 9
Aug. 10
Aug. 15
Aug. 3
Aug. 12
Sept. 12
Get.
Oct.
12
Oct. 1
Nov.
Nov.
Ifi
9
-1.72
-1.70
-l.f.7
-1.00
-1.4S
-1.54
-2.02
-2.24
-1.99
-2.33
-1.97
-1.91
-i.;io
-1.S5
-1.89
Nov. 15
No. of
obd.
11
9
10
10
13
15
Corrected
Observed correotiona in Dec. Corr. to Qeocentrlo
Mean.
8
—1.50,
— 1.C7,
—1.00
-1.05,
— 1.52,
—1.58
-1.62,
—1.50,
-1.52,
—1.12.
—1.86,
—1.79"
—LOT,
-1.70,
—1.72
-1.4S.
—1.70.
— 1.C3
-2.02,
-2.24;
-2.18,
-2.33,
-2.19
-2.1c,.
-2.09,^
-1.90^
-2.01
-2.04,
-1.89,
-2.02
+4.0
-I-3.G
-f2.5
-f3.5
-f3.1
-fl.7
40.2
-fl.9
+ 1.1
No. of Corrected Lougitude.
obd. ueau.
n
-1-4.0
-f3.C
-fl.S
+5.3
+ 4.9
+3.4,
+ 1.1
4 2.5
15 +2.8
+ 2.0
Latitnde.
[-19.2] [+ 9.5]
—20.8 +10.3
[-18.6] [+ 9..5]
—20.8 + 9.1
[-17.2]
—21.2
[-16.7]
—19.5
[-10.0]
—19.8
[—23.8] [+10.0]
—23.4 -j-11.1
[—22.0] [+10.0]
—22.2 I +11.8
[— 20.7][+ 9.8]
—21.0 +10.8
[—28.4] [+10.3]
29.2
[—20.91
— 28.8
+ 11.9
[—25.8] [+10.2]
— 2G.G +11.1
[—25.2] [+10.1]
—27.4 . +10.5
132
THE 11 BIT OF U II A N U S.
Mean Coukeutions to the Kimiemeuis or Ukani-s.— C'o»<(Hwtrf
Obmirvatory.
[K. A. ur Mean dntei.
Drauui.]
ObBerr><d corrvutfons In B. A. Oliserrvd oorreotions In Deo.
Muan. No of Corrnclf il
ubi4. j uiuaii.
Oreciiwich,
Kdriip[.sl)i'r)f,
Cambridge,
[21" 32™]
Qroi'iiwich,
Cuiulji'i(l|ru,
Vieium,
[21'' 23"]
Oroeiiwicli,
Caiiiliriiijfe,
Viemin,
[21" 2G"]
Ciimbriilije,
Vieiiiiiv,
[21" 2(i™]
Cambridge,
Vie:iiia,
[21" 49"]
Grpenwicli,
Cambridge,
[21" 45'"]
Greenwich,
Cambridge,
Vienna,
[21" 41'"]
Cambridge,
Viunna,
[21" 41'"]
Ciinibiidge,
Vienna,
[21" 4:{'"]
Oreenwicli,
Kiinigsbcrg,
(Cambridge,
Vienna,
[22" 4'"]
1833
Aug.
22
Aug.
12
Aug.
15
Aug.
15
Sept.
18
Se|>t.
1!»
Sept.
11
Sept.
1.S
Oet.
11
Oct.
12
Oet.
14
Oet.
12
\ov.
\ov.
1«
14
Nov. in
1834
Aug. 1
Aug. 1;
Aug.
Sept.
Sept.
14
10
Hi
Sept. 13
Oct.
Oct.
Oct.
14
l(i
20
Oct. 17
Xov.
Nov.
Hi
12
Xov. 14
Dec.
Dec.
Dec.
1835
A ug.
Aug.
Aug.
Aug.
17
20
14
21
Aug. 17
R
-2.57
-2.57
-2.40
-2.3.3
-2.33
-2.25
-2.20
-2.37
-2.10
-2.24
-2.58
-3.07
-3.00
-2.00
-2.74
-2.(18
-2.75
-2.04
-2.53
-2.7!
-2.55
-3.25
-3.31
-3. 28
-3.30
M<'an.
7
5
11
5
10
4
12
10
11
4
5
18
I'l
4
11
20
1
H
-2.57,
-2.57.
— 2.5'.i.,
—2.58
-2.3.!,
-2.52.
-2.50,
—2.47
2. 25
— 2.48J
—2.43
-2.35,
o_28
—2.33
—3.11,
—2.83
—3.00,
-2^
—2.88
—2.74,
—2.87,
4 _
2.80,
—2.83
-2.83,
-2.58,
—2.77
1 _
2.00,
• 50'
—1.4
—1.5
—0.8
-f2.8
-f2.0
+ 0.1
-f2.2
+ 1.3
1.3
4.4
4.4
1.2
3.1
0.8
1.3
0.7
2.8
1.2
2.1
—2.88
—3.25, ' — fi.O
—3.31,
—3.47,:— 4.7
— 3.35„ _ 8.8
—3.39
No. of
oIm.
4
H
4
13
11
4
5
18
10
10
4
11
8
20
1
Cormotdd
uieuii.
+"•2,
— I .5.
—0.0
H-0 8.
+ 1.2,
+ 1.2
41-T.
-1-1.
—0.1
Corr. to Geocentriu
Lougltude. Latitude.
[—33.8]
— 35.0
[ + 10.8]
-f 10.9
[—32.2] [+10.8]
-33.4 -)-12.3
[-31.2] [+10.0]
—33.0 +11.0
[—30.3] [ + 10.3]
—0.1 I —32.0 , +10.4
—2.0
—3.5
—2.7
—3.0.
2 5
—2.0
-l-7,„
-2.1,„
—1.9
—2.0,,
i_[_40.0] [ + 11.1]
—39.7 +11.3
[-30,0] [ + 11.1]
—40.5 +11.2
[-37.5] [+10.9]
—39.5 i +11.0
■■' [_3f,.7] [+10.7]
2.0 —38.7 , +11.3
-2., 5.
— 1.2i
—2.4
-5.4,
-5.9.,
[—30.2] [ + 10.5]
—40.3 -f 11.5
:^Aj[_47.1] [+11.3]
-5.9 —48.0 +11.8
THE OKU IT OF URANUS.
133
Mean Corhkctions to tub Ki'iiKMKitm ok Uranus. — Continued.
Olifler»iitory.
[K. A. of
Uiauua.]
Canibridgo,
Vii'iiiia,
[22" 1'"]
Oi'ot'nwicli,
Cambridj^o,
[21'' 57'" I
Orcciiwicli,
C'umlji'iilgc,
[21" 57'"]
Orecnwicli,
Ciimbridge,
[22" 21'"]
Qi'ccnwirli.
Ciunbridifo,
I'Jiliidiui'^h,
V^ieniia,
[22" 20"]
Greoinvic'i,
Uaiiil)ridj?(',
Kdiiiliiii'^li,
Vii'iina,
[22" l(i">]
Qrcpnw it'll,
CamhridfTc,
Kdiribiii'>;li,
Vienna,
[22" l;)'"]
Green wii'li,
Camljrid^c,
E)diid)iii'^li,
V^ienna,
[22" 12"'l
Camhrid^o,
Edinljurgii,
[22" i:{"']
QreiMitt'ii'h,
[22" 40'°]
Ub8Hrr«(l uorraotions in R.A. , Observed correotiona in Deo. Corr. to Oeocebtrio
Mean dates.
1835
Sept. 15
Sept.
Sept.
Od.
Oct.
Oct.
II
15
10
17
Xnv. 27
Xov. 2()
Nov. 2(1
1836
July 22
.Inly 25
July 2;j
A lift.
Auf;.
A..-.
AiiK.
AngT
Sept.
Sept.
S.pt.
St'pt.
24
!J0
Hi
1!»
20
Mean.
— :M'?
— y.;50
—.3.27
—a. 11
-:!.2i
-i.oo
-.1.H0
-;{.<!
— .•i.7S
No. of
obii.
Corrected
luuuii.
Mean.
Sept. 15
Oct.
Oct.
Oct.
Oct.
12
1(1
15
11
Oct. K
Xov.
Xov.
Xov.
Xov.
i;i
l:i
17
«
Xov. 13
Dec.
Dec.
12
12
12
1837
July 22
-;i.(i7
-;i.5l
-4.05
-3.57
-3.59
-3.37
-3.7«
-3.50
-3.40
-3.29
—4.23
9
10
— 3.(;;i
—3. 78
—4.09
1 —3.77
-3.77
—3.70
—4.01
—3.59
7
10
8
9
7
12
8
10
11
8
8
3
s
— 3.3fi,
—3.35,
— 3.3(J
—3.27,
—3.30,
—3.29
—3.21,
—3.19;
-3.20
—3.84,
—3.89,
—3.8(1
--.82,
-3.(;3,
—3 97,
—4.10:,
— 3.8l„
— 3.S7
—3.81,
—3.89,
—4.02,
—3. ('.3;
—3.87
-3.71,
— i-70,
— 4.0(i,
—3.(11,
—3.77
—3.(13,
-—3.5(1,
— 3.79.J
—3.(10,
—3.03
—3.59,
—3.30,
—3.52
-4.20,
— 4.0
— (1.7
— 0.53
— 4.1
- . 5.0
— 4.5
-10.5
- 8.9
— 9.8
— 0.3
— 9.3
— 12.5
- 8.8
- 8.4
- 8.0
-10.4
- 8.0
- 8.5
- 7.9
-10.2
- 0.1
- 7.P
-10.2
8.0
8.8
-13.4
No. of I Corrected Loni;itudu.| Latitude,
obd.
12
7
1
C
10
8
9
8
II
4
10
11
7
—6.8
-5.. 3.
—5.3.
[-45.8]
—48.1
[+n.2]
4-11.0
—5.3
-4.4,
-5.7,
—5.1
-9.. 5,
-9.O.,
-9.0
- **-8,
-10.3,.,
- 9.3;
-11..5,,
—9.0
-T.8.
-9.4„
—8.(1,
-9.43
—8.8
_7.fi,
-9..5„
-7.0.
-9-2:,
—8.0
-S.l„
-8.9,
—9.2,
[_44.4] [+11.1]
40.9 -fll.4
[_4;i.O] [ + 10.7]
—45.0 +11.2
[— W.3] [ + 11.5]
50.5 +11.8
[-54.0] [+11.0]
50.0 +11.5
[-M.4] [+11.0]
—50.2 +12.1
[-51.8] [+11. .3]
54.8 +11.7
[-5fl.fi] [ + 11.1]
8.5 —52.8 -j-U.O
-0.0,
-8.8^
—8.9
—12.5
[—SO.O] [+10.9]
—51.5 I +10.1
[—(12.9] [+11.5]
—03.2 -j-12.0
184
T H K O U IJ I T O V U U A N U S.
Mkan Coukkutions
TU TIIK Kl'IIKMt
iiix or Uranus.
(>bii«rv«(l correct!
— Ciiiithiticil.
Obs«rv«(l
corrvvtiouH in It. A.
ma ill Uec.
Cnrr. to (l«oceiitrio
ObHerviitnry.
K. A. of
Mvau dates.
....
1
Urauua. )
Moan.
No. of
Ciirrecldd
No. of
CoiTfcl«d
Loiigltudv.
Latitude.
ulw.
Ult'ttU.
Muan.
"
obH.
ui«nu.
1887
H
H
H
'/
It
Oreonwicli,
Ang. IS
— 4.;!0
10
-4.33,
— 13.4
10
-12..5,,,
Cainhritljrt!,
Aug. IH
— 4.0!l
14
-4.2S,
— 12.(i
11
-13.4,,
Eiliiihurgli,
Aug. 22
—4.40
()
—4.40,
— 12.7
(i
-1-^.T.
Paris,
Aiij?. 14
—4.34
9
-4.3,3,
— 12.0
10
-11.4,„
Vii'iiiia,
Aug. 2'>
— 4.2'J
4
—4.33,,
— 12.(!
4
— 11.(1,
1
[_(i2.9] L + II.7JI
[22" Sfi-"]
Aug 18
—4.33^
— 12.5
—(14.1
+ 12.2
On'cnwifli,
Si'pt. 17
4.23
14
— 4.2(!,.i
— 13.4
14
-12.5,,
KoiiijfttlR'rj;,
Sept. 11
4.10
H
_4.13„
CuiMl)ri<l);(>,
Sept. 17
—4. or,
14
-4,25„
—11.9
15
— 1-' 7
Kdiiiljurgli,
Sept. 10
—4.311
4
—4.39,
— 11.5
3
— 11.5,
Piiris,
Sept. IS
—4.20
12
-4.19„
— 12.3
13
— 11 7;.,
Vii'niin,
Sept. l;{
—4.12
5
_4.ir..
— 13.9
6
-12.9,
[-(11,7]
[ 1 ll.fi]
[22" 32"']
Sejit. k;
....
4.21
....
—12.3
—(12.3
+ 11.5
Orccnwk'li,
Oet. k;
—4.13
11
_4.1(V,
— 12.9
11
-12.0,,
Ciuiibridgc,
Oel. 17
— 4.00
10
-4.19.
— ll.(i
11
— 12.4,,
Paris,
Oet. 17
— I.O;-)
4
—4.04,
— 11.2
4
-10.(1.
Vii'iino,
Oet. IS
—4.41
2
-4.47,
— 15.5
a
— 14.5i
[-59. S]
[H1.4]
[22" 2S'"1
Oet. 17
...
—4^ 1(1 "
—12.0
—(11.6
-1 11.3
Orcenwicli,
Nov. S
—4.10
5
—4.13,
— 12.4
5
— 11.5.
Caiiiliridgp,
Xov. 4
— 3.1M1
4
— 4.ir,;
— 12.3
a
-1.3.1:;
Paris,
Xcjv.
—4.07
3
— 4.0(1'
—11.3
3
-10.73
Vii^iiim,
Nov. 2
—4.0!)
2
-'■'i
—12.9
2
-11. 9j
[—59.0]
[-111.2]
[22" 27"']
Nov. (J
— 1.11
—11.7
—(10.7
-f 11.3
Grponwicli,
Dee. 2
— 4.().'i
2
— 4. OS,
— 14.0
2
-13.1,
Cniiil)ri)i};e,
Nov. 30
— 3.S7
(1
— 4.0(1..
— 11.(1
7
— 12.4.
I'orls,
Dee. H
—4.03
7
—4.02;
— 10.9
7
-10.3;
Vifiiiia,
Dec. S
—4.28
2
— 4.3l;
—13.0
2
— 12.0(
[_.<,7.7] L+"OJ|
[22" 28""]
Dec. 5
1838
—4.05
' ' ' ■
— ll.(!
—59.9
+ 11.1
Oropnwicl),
A ug. 20
—4.72
11
—4.70,
—15.9
n
-1.5.1,,
K(Mii(?sl)crjr,
Aug. 2.')
—4. (-.5
.5
_4.(1.\
— IS.C,
5
-17.(V,
Ciuubridjtc,
A ug. 1 '.)
— 4.07
11
—4.7 s.,
— 15.S
12
— 1(1.3,,,
Kdinburgli,
Aug. 2.T
— 4.!)(1
3
_4.1I.V,
— 15.0
4
-15.0.
Paris,
Aug. 20
4. SO
9
—4.79,
— 1(1.(1
9
— 1(1.0„
It
[-70. 9] r4-ll.7l 1
[22" 51"]
Aug. 21
...
—4.7(1
— 15.8
^-71.4-'
l_ 1 -J
+ 12.3
Qrwnwifh,
Sept. 12
4.(12
S
-4.(1(1.,
— 1(1.2
8
-15.4,
Koiii(»sl)iTg,
Sept. 1)>
— 4.tl.'-.
u
— 4.(17,,,
—IS. 5
14
—17.5,
raml)rid)?e,
Sept. k;
4.(11
11
—4.72,
— 14.9
12
-15.4,,
Edinburgh,
Sept. 15
—4.74
9
-4.7;!,
— 15.S
1
— I5.S.
Paris,
Sept. 12
—4.71
(i
—4.70.,
— IC.O
(!
-15.4,
Vii'iiiia,
Sept. 11
—4.71
12
-4.7.3,
— IS.l
12
-17.1,
Berlin,
Sept. t;
—4.72
7
—4.74,
—17.5
1
-1(1.5,
[_70.4]'[+11.7]
[22" 47°']
Sept. 1;')
....
— 1.70
....
—1(1.0
—70.7 -\-\'i.i)
III
.,.
THE ORBIT OP URANUS.
185
.,
Mean CoftRKoxioNs to the Ei'iikmerib or U
lANUS.
— Continued.
01)imrvnlnry,
(U. A. »r
Mean (lat«a.
Obaervi'd oorreotlona in R.A
OI>serTed correotlona In Deo
Corr. to Oeooentrio
1
1
UraiiuH. J
Mean.
No. of
CiirroeteJ
No. of
Corrpoled
Longitude
batltnde.
R
uIm.
liieau.
Mean.
obs.
lueau.
ff
!
1888
8
ff
If
,,
Orodiwicli,
Oct. 15
— 4.fi7
1
-4.71,
— 15.5
7
—14.7,
CaiiiliridfTc,
Oct. 17
—4.49
7
— 4.fiO,
— 15.3
G
— 15.M,
Kdhilnii'^li,
Oct. k;
— 4.5fi
12
—4.55:,
— 15.0
8
—15.0,
Piiri.s,
Oct. k;
—4.(11!
7
— 4.fi5,
— 15.9
7
—15.3,
Viciitia,
Oct. 14
— 4.00
9
— 4.fis',
—17.3
9
— 1(>.3,
[22" 44'"]
Ocl.^K!
— 4.fi3
— 10. 2
[— CH.f,] [f 11.5]
— C9.7 1 +10.8
Qreciiwicli,
\(iv. n
—4.52
fi
— 4.5fi,
— Ifi.O
fi
— 15 2,
Ciuul)ri(l(fo,
Nov. 1.')
— 4.;i7
10
-4.48,
— 15.1
9
— 15.fi,
K<liiiliiii'<;li,
Nciv. k;
—4.70
s
— 4.fi9,
— 17.4
2
— 17.4,
Vk'iina,
Nov. 7
4.93
4
—4.95,
— 18.3
4
-17.3
i
■ 1
r t\i\ 01
Till m
L — tiiL.ij H -t- 1 ■.•> [ 1
[22" 42-"]
Xov. 14
-4.^58
—15.8
— «8.8
+ 10.9
Oropiiwicli,
Dec.
4.fi0
2
— 4.fi4,
— 15.9
2
—15.1,
Ciiinliridp',
l>cc. 15
— 4.2fi
5
—4.37.,
— 15.1
7
-I5.fi
Eilinhtirgli,
iKc 17
—4. IS
3
—4.17;
—14.7
1
-14.7;
Paris,
Due. 5
—4.40
7
4.39,
—15.1
7
— 14.5,
[— fi5.8]
— «5.7
[+11.1]
+ 10.4
[22" 43"]
l>cu. 10
—4. 38
—15.0
1838
Qri'diwieh,
Aug. 22
— 5,2S
5
—5.32,
—21.3
5
— Sfi.fij
Cutnbridgi',
Aug. 24
—5. 1 1
S
-5.22,
—20.7
7
—21.0,
Paris,
A\ig. 23
■■. 2 »
3
-5.1 9:
—20.7
3
— 20.I3
fidinbiirgli,
Aug. 25
—5 . )
2
—5. 20,
[-79.7]
—79.7
[+11.7]
+ 11.0
[23" fi"']
Aug. 23
—5.22
....
. . .
—20.7
Grooiiwicli,
Sept. 10
—5.14
12
-5.18,
—21.0
12
—20.3,,
Kolliff.-iblTg,
Sept. 12
—5. 1 1
12
—5.13.
—21.7
12
— 20.7„
BlTJitl,
Sept. 10
—5.10
14
—5.11.
—21.1
14
—20.1,
Cainliridgc,
Sept. 17 '■
—5.12
11
-5.23:
—20.0
10
— 20.3,„
Paris,
Sept. 14
—5.23
10
—5.22.
—20.7
10
— 20.1,„
Ediiiliiirgli,
Sept. If, i
—5.10
17
-5.15;
—20.5
5
—20.55
ViiMiiia,
Sept. 17 1
—5.32
9
-5.33,
—20.8
9
-19.8,
[-V9.0]
—78.8
[+ ii.fi]
+ 11.0
[23" ;i"']
SeptrU
—5.17
—20.3
Gn'ciiwicli,
Oct. 12
— 5.1H
5
—5.22,
—20.2
5
—19.. 5.
Caiiiliridjfo,
Oct. 1.".
—5.07
S
—5.18,
—19.2
5
—19.5,
Eiliidxirgli,
Oct. 15
—5,14
10
-5.13,
— 19.9
8
—19.9,
ViiMiiia,
Oct. 10
—5.17
13
—5.18,
—20.5
13
—19.5,
[_77.4]Y+11.5l|
[22" 50'"]
Oct. 14
—5.18
—19.6
—78.7
+ 11.8
Grccnwii'li,
Nov. 12
—4. 98
fi
—5.02,
—20.2
fi
—19.5,
Cnrnhridgo,
Nov. Ill
— 4..S5
fi
— 4.9fi.,
— 19.1
5
—19.4,
Paris,
Nov. !»
—4.99
2
—4.98,
—20.4
2
—19.8,
Ediiiburgii,
Nov. 1 1
—4.90
3
—4.89,
—18.3
1
—18.3,
1
r-75.3] r+11.3]|
[22" 57'"]
Nov. 14
....
—4.97
—19 i j --75.7 1
+ 10.4
Grcenwieli,
Deo. fi
—4.84
2
—4.88,
—19.0
2
—18.3,
Cambridge,
Dec. 15
—4. S3
5
—4.94.,
—19.1
2
—19.4,
Paris,
Dec. 3
-4.90
3
—4.89,
-17.8
3
-17.2:
Ediiibursli,
Dec. 2H
— 4.9(!
4
—4.95;
— 18.5
3
— 18.5,
1
' [—73.9] [ + 11.0]
—18.2 j -74.(1 +11.2
[22" 57'"]
Due. 12
....
—4.92
l'.]Cy
TIIR OIiniT OF URANUS.
MkaN C'llUllKl'I'lilNS Til TIIK I'll'IIK.MKIllH 111' UllANIIH. — O'lilli II liril.
Ubattrfntory
[It. A of
Urnniin. ]
Orccnwic'li,
Ciiniltriil);)',
Eiliiiliiir);li,
[l>3'' 20'"]
OriMMiwich,
Kdiii^slii'r)r,
Cuiiiliriil)!0,
Kiliiiliiir(;li,
I'liris,
Berlin,
Viuiinii,
[2;!'' IH"]
(Ireonwicli,
Ciiinliridtfc,
Kiliiiliiii'gli,
I'lirU,
IK'rliii,
Vii'iiiiiv,
[23'' 14""]
Orcciiwicli,
("niiiliridgo,
Kdiiibui-jjcli,
I'liris,
ViiMino,
[23" 12"]
Ort'oii\vi4'li.
Ciiinln'idjji',
Kdiiiliurj;li,
Vii'ima,
[23" 12""]
Groonwicli,
I'liris,
[23'' ST"-]
Oi'oonwicli,
KoiHgsbi'i'g,
Berlin,
Edinbiirgli,
Paris,
ViiMinn,
[23" 33""]
OWwrvoil currHullniin In R. A. OIwitvhI ciiiTi'ilidiiH In I>i'i'. Ciirr tii (Iciii'i'iitrlo
No. of I Corri'uti'd LiingltuilH liStituile.
oIjm. iiioiiii.
-23,!),,
-2r).2,
-23.7,
[_S7.!.] [ + 11.4]
23. !l — S«.8 -j-l''"
-23.4.
-24.1,
-2..,\
-23.(1.
-•>:, 3,
-21.2,
—23.8
-23.0,
-^3.7,
-23.2,,,
-22.7,
— 2I..S
-21.5,
—23.3
00 n
--" «
-24.4,
-22. H,
-2.''). 5.
[_87.0] [+il.5]
—87.5 -f-11.4
—23.3
-22.7,
-22.5.,
—22.8
2.S.1,
2il.O,
28 3
2S.5,„
2-.U.,
2«.r,,
2S.(),
28.4,,
29.3,
[_85.8] [ + 11.4]
—8(1.3 +11.2
[_S3.7] [ + 11.3]
—84.4 -4-10.3
[—82.2] [ + 11.0]
—83.7 +10.4
[— !)fi.!l]
-2S..
[-nfi..5]
— i)(!.3
[ + 11.3]
+ 10.G
[ + 11.4]
+ l(».4
M *
1
TllK OUUIT Ol' UUANUS.
131
(i
1
MkaN ('<lUltECTU»N8 TO THE Kl'IIK.MKIllS Of U
ii\svH.—Conlinui:il. 1
OlmervBtory.
[11. A. of
Obaisrvtiil norrvctionn in R. A.
Ol)«ervcil cnrwctloiis lu l)«o.
Corr. to (iHoinntrlo
Meau
dnt«8.
1
""
UiaiiUH. ]
Mean.
No. of
CnrrHcrltiil
M«an.
No. nr rorrei:U'il
tiOii|;ltU(lH
t.utttiide.
8
(ibrt.
lUUIlll.
H
If
1 "
/»
1841
II
Orcciiwicll,
Oct.
17
—CM
t
— r..]:i,
— 2S.2
7
' —27.7,
Bi'i-liii,
Oct.
20
— .'■>.S7
2
—5,, ST,
— 2S.2
2
7
—27.2,
Kdiiihurgli,
Oft.
la
— (i.OO
4
-(1.04,
—27.3
4
1
—27.0.
Piiri.s,
Oct.
21
—11.10
1
— (l.O'.t,
—27.2
1
-2(1.(1,
V it'll mi,
Oct.
l(i
—(1.14
"J
—(1.14,
— 2S.S
!)
: —27.8.
[-94.0]
—91.1
[+11.2]
+ 10.5
[2a'' 2S'"]
Oct.
IS
—(1.0(1
... —27.4
Grconwicli,
Nov.
17
— .'■..n.->
5
— 5.09j
— 2S.0
5 —27.5
BlTlill,
Nov.
111
— (1.1;-)
2
-(1.15,
— 2s. ^
2 —27.4,'
Etiiiiliiirgli,
X..V.
2
' — f).8!»
a
— 5.!ia,
—2(1.5
4 — 2r,.:t,
Yiuiiim,
Nov.
U
— f>.!KS
4
— 5.!)S,
—30.0
4 —29.0
[2a'' 2(1'"]
Nov.
la
— (1.01
—27.2
[-92.0] [-1 11.0] 1
—93.3 -f 10.4 1
Grooiiwicli,
Dec.
14
—5.8(1
H
— 5.!»0,
—27.4
8
—2(1.9.
Bi'i'liii,
]).c.
l.i
— ."i.dO
a
— 5.(10|
—25.5
■ 3 —24.5,
K(liiiiiiii';.'li,
Dec.
20
— .^.(iH
4
—5.7:1,
■
Puri.s,
Dec.
1,^
—5.7s
4
— 5.77j
—2(1.7
4
—2(1.1.
[2a" 2(;"']
Dee.
k;
— 5.7s
^ 4
— 2U.¥~
[—90.5]
—89.9
[ + 10.7]
+ 9.8
1842
GrocnwU'li,
Aiijr.
lit
— (1.(12
5
— fi.r,4,
32. 1
(i
—31.7,
Ciiinliriilcc,
AUR.
24
— (i.4s
2
— fi.55,
-32.7
2
-33.0,
[2a" ol'"]
Ang.
21
—(1.(11
't
~32To'
[—105.(1]
— 103.0
[+11.1]
-t 10.0
Orponwicli,
Sept,
12
— fi.,W
4
—(1.57,
—32.2
4
-31.8,
KoniR.sbiT}?,
Sept.
is
—(1.(1!)
10
—(1.110, 1
-:!:;. 4
10
—32.4.
Uorliii,
Sept.
11
—(1.70
3
-(1.(1!.;
— ;!1.7
3
—30.7,
I'«ri.>!,
iScpt.
14
— fi.(15
5
—(1.(14.,
—32. 1
5
—31.5,
Caiiiliridpo,
Sept.
11
— (1.(10
1:5
— (l.(17„
—31.9
14
—32.2,,
K(iiiil)iir};li,
Sept.
2.!
— fi.57
4
—(1.5!),
—30.9
4
— 30.(lj
Tulkowii,
Sept.
17
-(1.(1 a
c.
— fi.(14,
Vienna,
Sept.
1',)
— c.sl
2
— (l.SI),
—33.5
2
— 32.5t
[2a" 47'"]
Sept.
iiV
— (l.dG
r._l 05.41 T-l-ll oil
—32.0
—104.3
L 1 J
+ 10.3
Orc'oiiwieli,
Oct.
20
— r,.(ir,
12
— fi.OS,
—31.8
12
—31.4,,
Borliii,
Oct.
2a
— (1.54
4
-(1.5.3-;
—30.7
4
—29.7,
J'liris,
Oct.
17
— (1.(14
12
-(i.(i:i,
—31.5
9
— 30.!!,,
Ciiinliridirp,
Oct.
17
-(1.(10
U
— (l.fi7.
—31.4
10
— 31.-,„
Kiliiilniri,'li,
Oct.
1.")
—(1.57
12
— fi..'-)!!,'
—31.1
8
—30. .s.
Vit'iiiiii,
Oct.
lU
—(1.74
4
— (I.7:!;
— 34.G
5
— ;):i.(i, 1
I_r_l 03.01 r-4 11.011
[2a'' 44'"]
Oct.
17
....
—(1.(14
...
—31.2
— 103.7
H 10.9
(irrcnwicli,
Nov.
2.'!
— (i.a(i
7
— fi.as,,
—31.0
— 30.r,,
Berlin,
Xov.
H
-(1.42
2
— (1.41)
— :!0.2
2
— 2!t.2,
Cniii1)ri(l!;e,
Nov.
17
—(1.40
7 '
—(1.47,
—30.7
7
—31.0,
iMliiilitirgli,
Nov.
111
— (>.a!)
=» ;
—(1.32,
Vienna,
Nov.
20
—(1.39
a !
— r,..ss.
—30.0
2
— 29.0j
'
[_100.S][-f lO.S]
— 100.2 -flO.O
i
[23'' 4.2'"]
Nov.
I'.l
—(1.41
....
1
— 30.0
18 May, 1873.
wmmmm
138
THE ORBIT OP URANUS.
MbAN CoUHrCTIONS to TUK Kl'IIEMKRlS »!»• I'llAMS. <\llllinl(n!.
[R. A. i.r
Urauii.'t.J
(llwfrvwl I'orrt'Oliviis in R. A.
H«an ilalini.
! Mi'iin.
No. of CorrHi'tml
(lbs. uiuaii.
Ort'fiiwifli,
Uorliii,
Piiris,
Onml)ri(!!?o,
K(linbui'(;li,
Pulkown,
1842
Dfc. K
Dec.
Doc
Dff.
Doc.
Dec.
[iJJ" 4r] Dec. K)
EtiinbiirKh,
Greenwich,
I'rtriH,
[0" C"]
Orcenwicli,
I'liri.s,
Kdinhiirfrli,
Piilkiiwtk,
! 1843
I Jiiri. <)
AiiR. 20
Au-. 20"
Sept. 17
i Sc|.t. l,-) 1
Sept, l.S !
I Sept. 22
»
-C,.,-!!
-G.Ki
-(;.2;j
-(',.22
-(;.2.')
-t;.2s
— ii.:!2
—7. II
— 7.20
-7.2.'J 1 ';)
-7.17 : II
-7.2(i 1 7
-7.1s 10
O.-t.
IS
Oct.
17
Oct.
17
Oct.
17
\.IV.
20
Nov.
11
Xov.
1.^
Nov.
1.-.
Nov.
15
[o" ;i'"] I Sept. ID
Green'.vich,
Paris,
L(ii!|i)lirir||,
[23" 5 It'"]
Ori'emvicli,
Koiii;.;slici'jf,
I'ari.s,
I'Mliiltiir};!),
Oreciiwicii,
I'Miiihurj!;,!,
I'liris,
r2:;'' ,-.(.'"]
I Jrcenwicl,,
I'arir!,
|;)ii 22"'l
Orcciiwicii,
Kiliiil)iir);li,
KOiii)?slicrj?,
I'liri.s,
[0" IH"-]
1841
•litii.
.Ian. I
.Ian.
Jan.
Ansj. I!»
A 11,1?. 17
Aug. 18
Sept. 2.")
Sept. li»
Sept. 17
Se;>t. 10
Sept. 17
-7.10
-7.07
M.OJ
-7.01
-(1.S4
-7.0;t
10
12
-fi.7o a
-(!.C.7 (1
-(i.72 2
-7.70
-7.7t
-7.(i7
-7.(!5
-7.U3
10
II
II
10
!0
8
-<i.i5,
— (i.22,
-(!.2',..
— (I.ISj
— (!.2S,
— (;.27
7 — -O.".')
I
7 ! —7.0.1,
a i —7. lit,
. .. —7.10
Olis»rv«d rorrpvtioiin in Deo.
Munn.
■.■iO.2
•.•il.4
■HO, It
.•!0. 1
-7.10,
-7,10.
—7.1!),
-7, IS,
—7,17
-7.0:!,
-7 0(1,
-(I. as,
—7.02
— fi.iU,,
-(!.s:{.
—7 . 02_,
— ti.Sd,]
— 0,',)0
— o.co,
-0.71;
— fi a
-7 (17,
-7. •;-.»,
—7. (IS
— 7.(i.s,
—7.(10,
-7.(14,
—7.02,
— 7.(ia
No. of
ob»
-.T1.4
-3-). 4
-;!(;. s
-3(1.8
-:!;">. 3
-37.1
■3,-i.l
■34.3
■34,4
■3(1.3
■3:.. 7
-34.1
-.3.-..,-|
-3!t,n
-40.0
-3<,).!»
-40.3
-40.7
-31). 4
14
5
10
10
3
II
10
10
15
Corrnittiil
iiii'iin.
-2!).S,
-30.4]
-3(1.3,
-30.4.
Corr. lo (ivovcnlrio
Iiongituile. Lulituil«,
—30.2
—31,0
—34.7-
-37.1,
—35.7
-35.0,
-.3(1.2,.
—3(1.4,
— 35.S
!4.i),
id. 5,
(4.7,
55.2
!3.')„
i3.4,
'•'■ ',1
!5.;{,.
31.5
-34.
-34.
-3!).
),
-31), 4,
-30,2
-30.
-31).
-30.
-38.;
-31).
. [— n8.0][-|-l0..5]
I — 08.2 I -f D.G
l~ !>'7.5]'[ f 1()..3J
:— 1)8.2 , -[- 8.(i
i
[-II4.,3][-FI0.7]
-111.1) : -f- !).G
[— II4.H[ f lO.d]
-1 12. ll' \ 1(1.0
[-1 12.(1] [f 10.. 5]
-110.(1 -f i).()
[_n().ij[ 1 10.4]
— 108.(1 -f 0.5
r— 1011,1 ir f !).!)]
— 105.0 )- 8.2
I
[-122.3] [h 10,2]
— 121,2 -|- 0.(1
■■• [—122.0] [-flO..(]
—120.(1 + D.a
THE ORBIT OF URAMJS.
13i)
Mean Couiikctions to tiik Kimikmkuis t)F Uh. nvs. — Co'ilinuvd.
OlxiorTatory.
[K. A. of
Uraiiua]
Mean datt'S
I
flrccnwii'li,
K(IIiiImii')j:Ii,
[0" If)'"]
Ori'i'invicli,
l'Miiiliur);li,
[0" 1-.""]
Kdiiiliiirgli,
I'uris,
[0'' 10"']
(Jrociiwirli,
Kiliiiliiir^li,
[O*- 11'"]
(Jivcir.vicli,
(Jri'iMiwirli,
Kinii>;sl)>i'}f,
I'aris,
[o* :j;r]
(}i nwii'li,
I'aris,
[0" iV]
(Jri'i'iivvii'li,
[0" :;T'"]
(In-ciiwicli,
[0^ l-.V"]
Ol'I'Cllwilll,
I'uris,
[tf .Ml"]
(Iri'i'iiwii'li,
Konigsberj^,
I'aris,
[0" 41'.'"]
(Jri'CMwirli,
I'uris,
1044
Oct. n
Oct. l:!
Oct. l.i
N(iv. 2(;
Nov. 1!»
Nov. 2.!
Doc. 1 S
Iter, i-i
Obii«rv«iloorr«ctioii8 in R.A. , Observed corrections in Deo.
Muau.
-■.7t)
, No. of TorrtH'tcd
Obl<. UIKUU.
l)c(
20
1845
.laii. I.'i
.liiii. 14
,laii. 14
.Vug. 2r.
Sept. IS
Sc|it. ;!o
Sept. 1 4
Sept.
Oct.
Oct.
j Oct.
! Nov.
I-
20
Dec. 1 1
1846
Sept. S
Si'pt. 12
Sept. II
Oct.
Oct.
Oct.
Oct.
Nov.
Nov.
[0'' 41'"] j Nov.
s
4
14
10
III
1 I
.M
-7.44
-7.41
-7.24
-7.10
^7. IS
-7.il!
-S.2.'>
-S.21
-S.lli
-S.24
-S.22
-S.Ol
-S.OC,
10
10
6
i lU
c
—S.SO
— S.7:!
-S.7I
-S.Cl
_H.(1!»
--s.r,2
— S.4S
H
1!)
7
II
8
—7. 1! 4,
^7.r.i,
— 7.r.!)
-7.:i.s,
-7.:(5,
— 7.;i7
—7. IS,
— 7.0'.>,
—7.14
.10,
.OS.
Iv.
No of ('orrec;? I
Mean. ' ub». I uioaii.
—SI). 2 !t — :!s.,s^
—40.,') 1 ; —40.0,
-a',).o
—7.
— s.
s,
— s.
— s,
(•S
IT.
1\
2:1,
.17
!•.
00,
.OS
.OS
—7.77
— s.
— s.
.-;(.. 7
—.'!!). n
— ;!s.4
-42.7
-42.(1
-4;!. 4
-4;i.',»
-42.2
-4:i.7
—4;!.:!
-40.2
_s.7.V, _4C..2
_s.72,„! — 4(i.O
— S.73 ; ....
_s.7o, — n;.s
_s (;i„ -_4( ■
— N.tis, _4(;.',
— S.ti;") I ...
— s.,^s.^
— S.47,
— H.M)
- (i.S
-4(1.4
9
Corr. to Ouuoentrio
Longitude.
Latitude.
L-I21.f.] [ + 10.2]
— 120.0 i -f 11.4
r-ns.2] [-1- '.).!)]
— llii.d + H.!<
'• [-I1.V7] [f 0.(1]
_;i,s 7 -11,1.7 f 7.1
10
a
10
(j
— as.o
— as.o
— 12. a
—42.1'
—42.4,
_4a.a;
— 42.r.
— ".s,„
— «a. 1,,
—42. a
—42.0
-114.
-112.
-i;ii.
-120.
n-
[-lai d] [
-lai
-12
■lao,
.I2s
■ 12S
■ 1211
o.r,]
7. a
0.(1]
0.0]
0.1
+
[+
« i
s
II
..•lO.S [-124. S] [ I
~4."..S
— 111.:!
— I22.S
ft..S]
S.O
0.(1]
7.0
-,).a]
0.(1
"'[-lao.oj [ j 0.2]
411.1 I —las. 2 ( s.a
7
II
-4(1.4,
_4a.s.
la — 4(1.1 „
[-l;!0.7] [ I 0.2]
4,-,.s I — i;!7.l I s.a
(1 ~M 4, I 1
— 4r> s ' I
— LJL[-ia(i.M] [f 0.0]
_4(i.o ] — ia.">.i I 7.(1
140
TlIK ORBIT OF UKANUS.
Mkan Correct.ons to tub Ei.,iKMLtt,8 o. Viij,svs.-Continued.
Ob8erT«torjr.
[U. A. of
Urauus.]
Grccnwifli,
Paris,
[()'■ SO""]
Green wicli,
I'liris,
[0'' 40'"]
Oreenwieli,
[■Jh ,;,„-|
Grcenwieli,
Pur-',
rjh 2'"]
Green Willi,
Puris,
[0'' ;.:'"]
Greeti\vi(rli,
Puris,
[0" a^-"!
Greenwich,
Pu-is,
[()" 54"']
Greenwieli,
Purls,
[1" !!»■"]
Greenwieli,
Paris,
Groenwieli,
Puris,
[!'' ir]
Greeiuvieli,
Pnris,
[I'' O'"!
Greenwieli,
Puris,
[1- <r'2
M«aii dates.
Observe,! oonectioim ii, R.A. Observe.Uorreotiong In D™.
Meau
1846
Dec. 15
Dec. k;
Dee. 1<;
1847
Jan. Hi
Jan. II
<>aii.
Sept.
J 2
3
o<-t.
Oct.
I
12
Oct.
10
\<n-.
Nov.
Nov.
«
Doe.
•Dee.
i!
12
Doc.
9
1848
Jan. 10
Jan. 11
Jan.
11 1
— 8.;}i
—8.25
— S.25
— S.I4
— tl.2I
— fl. 25
— !'. 2;j
— !M0
—II. OS
— s.no
—8.84
— S.8.1
— 8.5:j
Sept. 8 , —9.84
.Sept. 22 ■ —9.82
Sept. 17
Get.
Oct.
[P 15'"] ' Oet.
19 _n.83
li; —9.79
17
No,, i:) _9.C8
Nov. i;i I —9.55
Nov. l;i I
i
Dee. 14 ' —9.41
Dee. ',) _<).4;t
De.
12
1849
Jnn. n — !)..^7
Juii. is , — <). 11
Jan.
No. of Corrveleil
obx.
(i
18
9
10
T
II
11
luuun.
Mean.
-8.27., I — 4(!.0
-8.24, — 4(;.i
—8.20
-8.20,
-8^12,
—8. 15
— 9.ir,
—9.20,
—9.21
—9.0.-),
—9.0(1,
—9.01!
—8.91,
— 8.82J
—8.85
— 8.7.\
—9.79,
—9.80,,
—9.80
—9.77.
—45.4
—49.4
—49.0
—49.2
—49.2
— 49.:i
— (9.0
—48.5
—47.1
—47.fi
—5 1 . 1
—51.5
-!'.TT„: -52.2
—9.77
— O.^.-i,
— 9.5;t,
—9.59
— 52.:t
—52.2
— 9.:tfi, I _-51.3
—9.41, —51. a
— 9.;i8
— 9.:!T, —.''1.2
—9.09, —50.0
— 9.2:j
Nn.of
ob.i.
10
4
Corr. to Oeocentrio
Corrected
uieuii.
Longitude
—»■>■«,„
—45.5, ,.
* [— 1.34..5]
— i;J1.7
[— 1.S0.9]
lo
— 45.0
-42.7.
_-44-«.
—44.0 p— 129!g
—49.0 [—148.4]
—144.9
-48,fi„
-'•':''iC-i48.o]
—48.fi
— 48 S
-48.7,,
-145.6
[— 14fi.21
-48.8 : —142.(1
—4S.fi,
—47.9,
—48.2
-4r,.7.,
—47.0.
— 4(1.9
-50,7,
—50,',),.
[—143.0]
—140.fi
LatitDdf'
[+8.9]
+ 0.5
[+S.fi]
+ 7.4
[+8.7]
+ 0.8
[+8.7]
+V.8
[ + «..'■.]
+ 7.2
[+8.3]
-f (1.8
[—139.4]
— 13fi.2
—50.8
—51.8,
-5I.«,
—51.7
—51.9,
—51.0.
[—150,4]
[ + 8.1]
+ 0.0
[ + 8.0]
— 154.1 +7.5
[-15fi.4]
— 154.2
"•» , _151.8
— 50.8j
—50.7.1
-50.7 [-'•'•>'>■ 2] '[ + 7,7]
[-fS.O]
+ 0.9
[+8.0]
+0.3
-148.0'
— 50..S,
—49.4, '
^ [-147.7] [ + 7. .5]
—50.1 -HO.a', +0,(.
+0.2
THE ORBIT OP URANUS.
141
.1
Mean Coukections to the Ei>iiemeri8 ok Uranus. — CoiUinued.
Observatory.
LU. A. of
Uraiiua. ]
Oreciiwicli,
[!'' ai'"]
GiTomviili,
I'ari.s,
II'' yo'^j
Ort'L'invifli,
Paris,
[1" 2(rl
Qrcriuvicli,
I'uri.-i,
[1" ar]
Qrcrinvioli,
J'lliis,
[!'■ 24'"]
Qrccinvich,
[I" r.2'"]
Orcciuvioli,
Paris,
[I" 47'"]
Grcpriwifh,
Pari.s,
[!'■ 4:r]
Orccnwicli,
Koiiifr--l»'r(r,
[1" 41"-
Orccnwicli,
[I" 41'"]
Orccinvicli,
Koiiijtsltcrg,
Puris,
[2^ r-]
KoiiiKshorg,
Piiris,
roll 2"]
Mean dates.
Observed corrections in R. A. Observed correotions ill Deo.
1849
Scjit. 1(>
Oct.
Oct.
2.S
21
Oct.
Nov.
.Nov.
Mean.
s
—10.27
— 10.24
— 10.2(i
K; ' —10.12
17 —10.22
Nov. 17
Dec.
Dec.
l:)
lii
Dec. 14
1850
Jan. 7
•Ian. .')
Jan. G
Sept. C
Oct. 11
Oct. 17
Oct. 1(!
\(.v. r.
Nov. 10
Xov. 8
Dec. 7
Dec. l!i
Deo. 9
1851
Jan. Ill
Sept. II
Sept. 1!)
Sept. 14
Sept. 14
Oct. 22
Oct. 2:;
Oct 23
— o.n,')
- 0.71
— '.».77
— io.,s;i
— 10.02
— I9.S7
—10.70
— 10.«0
— 10..")0
— 10..'')0
—10.21
— 11.42
— 11.41
—11.5!)
— 11.7.^
— 11.4(i
No. of Correcteil
obi. mean.
Mean. No of Corrected
1 obs. uiuttii.
8
-10.27
— 10.24,
— 10.24^
—10.24
—10.12
— 10.20,,
— 10.17
— O.Oi"),
-- O.MC.
— 0.03
— 0.71,
— 0.75^
— 0.74
—10.81!
— 10.02.
— lO.xri.
— 10.87
— 10.70.
— 10..s7'j
— 10.82
— 10..'-.fi,
-10.48,
—10.54
—10.21
—11.42,
— 11.4li
— 11.40
—11.75,
— 11.44,
—11.50
— 5;i..'5
— 5;{.9
—54.8
—54.4
—54.1
-5.3.1
-54.4
-53.0
-53.0
-53.0
—55.3
—50.3
—.-.5.7
- -.(i.l
— .5.-.0
— 'vd.o
—54.8
-55.7
-58.4
3 —52.9
Corr. to Oeoceutric
Longitude. Latitude.
[-f 1(15.1] [-f 7.3]
-|-I(i0.8
-fO.2
[_ir,4.4] [+7.4]
— i.a.o 4-5.8
[-102.5]
—100.1
[— iro.4]
— 150.0
r+T.3]
-l-c.o
[+T.1]
-f5.7
"• i [_i:,o.8] [+0.0]
52.5 —153.8 ; +C.2
i i
9 -53.5 [-171.7] '[ + r,..5]
. ■ i —108.1 +5.9
—'•'•' [_173..3] [+0.4]
. . —55.3 —100.0 +5.3
fi i —55.3,
4 ' —.'•.5.5,
'[-172.4] [+0.4]
■55.4 -100.1 I +5.0
54.r.,
■ '_■__.'[— 100.0] [ ( 0.3]
—54.0 , —105.0 . 5.2
'
2
—54.4 '[_103..'>] [+0.0]
-10O.4 4-3.6
' [-1S0.0] [ + 5.7]
—170.9 I -f4 4
[_ismj [)r..8j
—57.4 . —170.1 1 4 4.3
142
TlIK Oil BIT OF URANUS.
MkaN ('0ltKEUTItlN8 TO TlIK Kl'IIKMKItIB or UUANUS. — Cimtiliuod.
Observatory.
IR. A. of
Uraiiu:' )
Orpcnwifli,
Piiiin,
Qreonwidi,
Piiri.s,
[1" .is™]
Orpciiwicli,
Paris,
[1" 54'"]
Orocnwicli,
I'liris,
[I- o-J"]
Orci'Mwii'li,
Qri'iMiwicli,
I'liris,
[•2" 18""]
Orpcinvicli,
I'liris,
Ort'c'iiwicli,
Koni)!sl)fr)^,
I'urix,
[2" 10"]
Qrccnwii'li,
I'uri.s,
Mean ilatei.
Observvdcorrtiutiniia ill li.A. <)lid«rvedaorractioD9 in Uecv
Mean.
1851
Nov. 3
Nov. a
s
Nuv.
\i)v. 22
.Nov. U
.Nov. 21
I>cc-.
22
•J."-.
Dfc. 24
1852
.lull. 11
.lull. 14
.lull. 12
.Si'pt. 1 2
Oct.
(M.
17
2.!
Oct. Ill
[2'' 9"]
Jreenwii'li,
'liris,
[2'' .lO""]
Jrconwic'li,
furis,
[2" J4'"]
Nov.
Nov.
i:i
1.^
Nov. 14
Dec.
Dec.
Dec.
lit
Hi
p.)
Dec lit
1853
.lull. 12
.lull
Juii.
Sept.
Sept.
1.-.
17
in
Sept. 17
Oct.
Oct.
27
Oct. 22
— ll.-.o
— II.4S
— II. 4.1
— li.;i;»
—II. 1.5
—11.00
— lo.sr.
— 1(».S!»
— 12. Of,
-12.2.!
-12.0(1
-11.72
-11.07
-11.77
-11 42
-11.40
No. of
obn.
-12.00 !,
-12. OS 5
-12,00 4
-12. .'is 1 2
-12.f.!l 7
-12.00 r,
CorrKuteil
luxau.
-IL.'iO
-ll.40!
-11.411
-11.41!
-ll.:!7,
-11.42 :
-ll.lo.l --.57.2
-lO.'.IS I —57.2
Ntan.
-.'■)7..'>
-iiS.I
-iis.;}
— 11.(10
-lo.s.v
. |(),S7,! —50.5
-10. so
—12.00.
— 12,2;!.
— 12.04
— 12. IS
— 12.00 :
— 12.00';
— l2.o;i !
-11.72,
— 11.75,
— 11.7:1
— 11.42,
— 11.44
— 11.4!
— I2.0tt.
— 12.50.
3l2,fi5
-12,00.
— 12,04'
—12,00
-5(1.5
-5-1.0
—50.0
-50.7
-00.2
— 5S,7
— 50,:l
—.50.0
—,'.0.7
—57.0
— 5S.S
—50.0
No. of Correi'tail
ub«. luuaii.
—57.1,
-57. .5,
—57.2
— 57.;k
-57.0,
— 50.Sj
— .50.0,
— 5(iT"
— 50.S,
—55.0,
—50.4
—50.1
— 5S.2,
—50.0,
— .5S.4
— 50.nj
—50.0,
—50.4
-5s.:i,
-5s.:t,
— 5S.4.,
— 5s.;!
—57.1
— 57.1
—50.3,
—50.4,
— 50.;j
-.58.4,
-5S.4,_
— 5S.4
C'i>rr. tu (ieooeiitrio
Luiigituile.
Lulituil«.
[-IS1.4]
-17S.1
[+•'■••7]
4-4.0
[_I70.0] [ + 5.0]
-177.0 -f;!.8
[-175.7]
-172.a
[+5.:!]
-i-3.8
[_l;:i.l] [,5 2]
— 100.4 ! -|-;!.I
[-1S7.4]
-1H4.5
[f4.0]
[_ISO.O] [ I 4.7]
-1S7.2 f;i.i
[-1SS.0] [4-4.7]
— 1S5.0 -|2.4
[_IM4.S] [f4.fi]
lsi.:i -1-2.8
[_1S1.0] [f4.4]
— 170.8
f2,0
[_104.fi] [fl.O]
— 101,0
-f '.!-
[-100,7] [-1-3.8]
-102,7 +1.7
THE ORBIT OP URANUS.
148
MKAN CullllKCTIONti TO THE Kl-lltlMKHIS OF UUANUS. Coutillllfd.
Olisfrvatory.
|K. A. of
Uiaiius. ]
Qrcenwicli,
1'uri.s,
[2" 31'"]
Paris,
r^ii St"'!
Grociiwicli,
I'liris,
[2" iir,-]
Paris,
[2'' 2r."']
Orccnwicli,
[2" r,:r]
(irccMiwich,
Piiris,
[2'' SO"]
flrccnwicli,
Paris,
Haiiliuj^i),
[2'' 4S'"]
(iri'cnwicli
I'aris,
Saiitiii;^(),
[2" -in
(jrccnwiili,
[2" 4-"J
(ircctiwicli,
Paris,
Orcciiwirli,
Paris,
Santiajro,
(Jrccnwicli,
Paris,
Santiago,
Mean dates.
1853
Nov. 12
Nov. l.")
Observed currectiona ill R. A. Observed corructloiiH iu Deo. : Curr. to (ieoceiitriu
Mean. | No. of
oIm.
Nov.
Dec.
14
12
1854
Jan. 1.1
.Ian. 20
Jan.
l''cl).
15
-12. CI
-12.G7
— 12.28
-12.02
-II. s7
-ll.H!)
.K
cpl. 21 _l.!.l-2
Oct.
Oct.
2(! ' _l:!,24
2y — 1;!,12
Oct. 27
Nov.
Nov.
Nov.
12
l(i
17
— 1:5.22
— I.!. 21
— l;l.20
Nov. li) j
!
Dec. n
l>cc. It
Dec. 1.-.
-12.!).'!
-l:i,H
-I2.s.-i
|)c<
14
1855
.Ian. is
O.'t.
Oct.
Oct.
Nov.
Nov.
Nov.
l:!
2."i
IT
II
17
It;
Nov. ir>
Dec.
Dec.
Dec.
Hi
IS
-l.i.Tl
-I. •!.!»:!
-I.'i.s.',
-|:!.I12
-l:!.s2
-i:i.4s
-l:!..'i2
-Hi .^.1
Dee. 1,S
Corrected ; Mean,
lui'aii.
14
10
7
I
10
10
3
-12.(l.'),„
-12.(1.5
-12.2C
-12.02,
-ll.sr.,
-ll.'Jti
-11.87
r> —13.12
-i:i.24,
-l.t.lO,
-i:i.22
-i;?.22,
-i:!.l!t,
-l:!.203
-l:!.20
-12.03,
-l:f.l2,
-!2.S5,
— 12.!iO
-I2.5.V
-i:!.7I„
— l:i.".tl,
— l:!.7.s
i
'— l.'t.s.^,
— l.f.c.o,
_I.!.S2.
— I.!. 7(1
— .'■iS.fi
— ."ifl.it
— r)it.7
~M.'.)
■;'i.').S
-.'■i7.C.
-.'.i;.(;
-ri7.2
-.'■.0.4
-50.2
50.0
-50.7
-5S.1
— 57.S
50.0
-5S.0
Noor
obn
14
8
— l:t.4S„, —57. .'I
'_l:!,50, — 5i;.7
i— i;!.5;i, I — 5S.1
— l:i,50 i ....
Correctvd Longituile. Latitude,
lueau.
-50.0,,
-58.(1.
-58.
-5S.4
-50.2,
-57.0,
-58.0
-57.G
5S.2
— 5S.7 ' I
-5S.7 I a
-57.. 5,
-57.0,
-5 7.. '5
-57.5,
-5S.5,
-.5H.I.
; —57.8
— 5S.I,
-5(;.7i
-50.1,
i — 5S.0
—5 (',.4
4 —
55
"t
55
"i
55
:i
57
1,
5(;
^.
5(;
.li.
—5(1.7
10
— 5(i.8,
:!
—5(1.5,
(>
-57.5,
—57.0
[_inr,.o] [-H!.t]
—I ',12.0 1 -I^I.U
[_i.,.T4] [+:t.(i]
187.0 -1-1.5
[-188.1] [ + 3..5]
— 183.0 . 4 0.1
[-184.5]' [4 3..5]
—182.1 I -[0.7
[-201.3] [ + 3.0]
—10(1.0 -i-1.7
[-203.7] [-f2.8]
lOJI.O -i 1.2
[—203.0] [-I2.T]
— 100.1 -i 1.2
[—200.1] [4 2.(1]
— 105.3 ; 40.7
[-104. n] [42-'']
—100.1 41.1
[—200.2] [4I.X]
—204.0 j 4 0.1
r-200.4] [41.7]
—205.3 —0.2
[-20(1.2] [41.'1]
, —202.1 i —0.1
144
TUE ORBIT OP URANUS.
Mean Coruections to the KpiiE.MGiiiii or Ubanus. — Con Hinted.
OlpucrTRtory.
[II. A. of
Uramid.]
Grcciiwitll,
Paris,
[2" OS""]
Grcciiwieli,
Paris,
[3" 27'"]
Qrci'iiwicli,
Paris,
[3" 22"°]
Orecnwifli,
Paris,
[3" K"-]
(Irconwicli,
Paris,
[3" la"]
Grecnwii'l),
[3" 40'"]
Qrconwifh,
Paris,
KoiiigsliiTg,
[3" 41'"]
Orci'iiwicli,
Paris,
KOiiijj.sln'rg,
[3" 3Cr]
Orccinvich,
I'uris,
[3" 32™. 2]
Orccnwirh,
Paris,
[3" 3r.O]
Qreeiiwich,
[4" 3"'.3]
flrooiiwldi,
Piiris,
Ub.-iervt'd uorreotiona <n R.A. Obserrvd corrections 111 Deo.
Mean dates.
Mean.
No. of Corrected
olix. iiicau.
1856 I
Jim. 22
Fob. 10
Jttii. 27
Oct.
Oft.
10
Id
Oct. k;
Nov. IS
.Nov. 17
Nov. l.S
Dec.
Due.
18
19
Deo. 19
1857
.Ian. 21
Jan. 12
Jan.
Oct.
19
8
Nov.
Nov.
Nov.
II)
19
Nov. 1 1
Pec. 1 1
Di'c. K!
I»oc. 8
Die. 13
1858
Jan. 17
Jan. Hi
17
Jan.
Feb.
Feb.
Feb.
Oct.
II
12
II
17
Nov.
Nov.
s
-12.97
-12.87
— 14.30
— 14.38
—14.23
—14.32
—13.92
—14.11
— 13..')9
—13.87
-14.08
-14.80
-14.83
-14.77
— 14..-..'i
—14.48
— 14.50
— 14.13
— 14.19
— 1 3. SI
— 13.91
1-12.97,
2 1—12.85,
10
3
13
8
10
1
'—12.94
I
-14.30.
— 14.30,
;— i4.3(r
I
— 14.23,
-14.30.
'—14.20
1
— 13.9,3,
—14.09,
—14.03
-13. CO,,
;_13.85,
'—13.00
—14.09
-14.81,
-14.81,
— 14.75,
—14.80
-14.. 50.
— 14.40.
— 14.54J
— 14.51
— 14.14,
— 14.17,
—14.15
Mean.
-50.1
-54.8
—53.8
—54.8
—54.0
—55,9
—55.0
—5,3.4
— 55.3
—49.1
—5 '.3
—51.5
—52.7
—53.2
—52.8
—53.7
— 52.8
—52.8
— 13.89,
—13.83
I
-15.00 5 —15.00
13.82,J —52.1
—52.7
-40.4
14 —15.10
20 : —15.19
10
9
I
[3" 58'».5] Nov. n
_15.I7,„' — 4S.2
— 15.17,
Noof Corrected
obs. I uicau.
—.55.0,
—54.0,
—55.4
—53.3
—53.3
I
—54.3,
\ —54.1
Corr. to Oeoceutrio
Longitude. , Latitude.
[-in9..5]:[+i.4]
—194.7
—0.5
5 j —55.4,
i —54.8.
[-213.8] [+0.7]
—211.0 i —1.4
[-214.7] [ + 0.7]
-210.2
—0.9
[-211.0] [ + 0.0]
•207.1
-1.2
—55.1
—52.9,,
^~""''' '[-200.4] [ + 0.5]
—53.5 1 —202.3 —0.4
I i
—48.0 [-210.8] [—0.3]
—213.0 . -1.9
—50., 8,
—51.3,
—51
—51.2
'[—218.9] [—0.4]
—215.5 —2.4
5 —52.7,
' — 52.0„
1 ! —52.5;
—52.0
—52.3,
13
9
10
1
[—217.1] [—0.5]
—212.2 : —3.0
[—211.8] [—0.0]
—52.0,
—52.4 I —207.5
-2.8
-51.0,„i
so r, I
"— 'i [_2n7.n] [—0.0]
—51.7 I —203.1 j —3.1
—45.9 [—220.0] [—1.4]
II
10
—47.7
—47.0
-210.2
-4.0
[—222.1] [—1.5]
47.7 i —218.0 I —3.8
TUB OIIBIT OF URANUS.
145
Mkan C'oiiRKCTioNs TO TiiK Ei'iiEMERis OF Uhanu8. — Continued.
Ob8«rTatory.
[it. A. «r
Uranus.]
Ciri'cnu'icli,
I'uris,
[3'' sa^.c]
Orconwioli,
Paris,
[3* 50"". 0]
(Jrccnwicb,
Paris,
[S*- 49"'. 3]
Grccnnifh,
[4" 20"". 8]
Orconwifh,
Paris,
[4" n-'.-i]
Groonwirli,
Paris,
[4*' 12"".:.]
Orccnwic'li,
Paris,
[4' S-.O]
Qreonwifli,
Paris,
[4" V'.O]
Ori'cnwii'li,
[4" 40'".!)]
Greenwich,
Paris,
[4*' 3.")'". 8]
Oreonwii'h,
Paris,
[4' 31-. C]
Groonwicli,
Paris,
Observed corrections in R. A.
Mean dates.
1858
I)<c Ifi
Dec. k;
Doc. 10
1859
Jan. 19
Jan. 10
Jan. lU
IVl). 18
Ftl). 1 1
Feb. 14
Oct. 2."(
Mean. No. of Corrwted
I ubs. uieau.
.\.>v. 17
Nov. IT
Dec. 1(1
l)<c. U
Dec. 14
18C0
Jan. If)
Jan. 15
Jan. IG
Pel), n
Pel). G
Feb7"l5
Oct. 13
Nov. li")
Nov. 22
Nov. IS
Dec. 12
Dec. 12
Dec. 12
1861
Jan. 13
Jan. 13
-l4.!ir,
-14. 7S
-14..^G
-14.72
-14. IS
-14.37
—15.43
-15.5fi
-l.-).i;l
-15.43
-15.4G
-15.08
-15.08
-14.fi4
-14.79
—15.52
-15.88
-15.79
-15.77
-15.79
-15.45
-15.52
[4'' 2fi"».8] I Jan. 13
-14.97,
—14.87
—14.57,
—14.70,
— 14.G1
—14.19,
-14.. 3,5:
G —
14.28
15.44
12
10
6
-15.57,
|-1_5.S9,
— 15.5S
—15.44,
—15.44,
— 15.44
-15.09,
|-15.0fi,
—15.08
—14.05,.
—14.77,;
—14.07
—15.52
-15.89„
j— 15.85
-15.78.
!— 15.77,
1-15.78
Observed corrections In Dec.
Mean. No.of Corrected
oits. mean.
It
-19.2
-49.5
-48.9
-4S.2
-48.1
-48.3
-42.0
-43.3
-4.!.0
-45.2
-44.5
-45.8
-44.3
—45.0
—35.2
-38 3
-37.3
15.49,,
—15.47
-41.5
-38.5
-40.5
-40.3
12
— 48.7.J
-49.3,
—48.9
-48. 4,
-48.0,
-48.3
-47.0,
-47.8
-42.1
[—220.7] [—1.0]
—215.1 —4.1
—42.8
—42.8,
—42.8
Corr. to Oeocentrio
Latitude. Longitude.
[—210.1]
—211.5
[—210.9]
—207.1
[—223.2]
-219.0
[-1..5]
—3.1
[-1.0]
—3.4
[-2.2]
—5.5
" [-224.7] [—2.2]
—44.7,
—44^3,
—44.5
■221.9
[—223.9]
20.7 -4.9
10
3
— 45..3,
—44.8
—44.5
[
—44.5
—34.7
[-
—4.0
[-2.3]
219.0] [-
210.1
-37.8.,
-37.1,
—37.0
-41.0,
—38.3.,
—40.0
-213.9] [-
-210.4
-223.0] ! [-
-218.3
[-
220.0]
223.5
-220.2]
-223.2
-40,1,
-■•"•*'■ [-222.4]
—40.1 1—219.3
-2.3]
-4.5
-2.3]
-5.0
-2.9]
-5.4
[-
3.1]
5.8
[-:
3.2]
; 8
[-
3.2]
5.8
19 May, 1873.
116
THE ORBIT OF UK AN US.
Mkan Cohhbctiuns to tub EPUEMERI8 or UiiANUB. — CoiUinuvd.
ObsprTRlory.
[R. A. of
Uraiiun. ]
flrcenwifh,
I'arig,
[4" 21-.0]
Orcenwicli,
[4" 6G"'.4]
Orrcnwii'h,
Paris,
Washington,
[4' 50".8]
Greenwich,
Paris,
[4" 44°'.0]
Greenwich,
Washington,
Paris,
[4" 43M]
Oreonwich,
[5" ic.a]
Greenwich,
Paris,
Leyden,
[5" 10". 7]
Ornonwich,
Paris,
Wasliington,
Lcydcii,
[5" 4"'.fi]
Greenwich,
Paris,
Wasliington,
Leyilcn,
[5* 2". 2]
Greenwich,
Paris,
[5'' 2"". 21
OliaerTwl correotions in R. A. Olwerved corrHctions in Drv.
Mvan (latea.
Uhru.
No. of CnrrMted
I ob». I nioan.
1861
Feb. 10
Feb. 5
Feb.
Nov.
8
10
■ 1.5.12 I 2
■15.23 1 1
— l.'i.!)!)
Dee. a
Dec. 10
Dec. 18
Dec. 12
1862
Jan. 19
Jan. 20
Jan. 19
Feb. 22
Feb. 19
Feb. 15
Feb.
XoT.
19
9
Dec. 12
Dec. 1 1
Dec. 8
Dec. 19
1863
Jan. 1 7
Jan. 20
Jan. 10
Jan. 16
Jan. 17
Feb. 15
Feb. 15
Fel). 9
Feb. 15
Feb. 15
-1(1.01
-Kl.04
-Hi. 10
-15.07
-15.72
-15. IS
-15. :U
-15.47
— lO.l.J
-10.27
-10.27
■10.10
-15.90
-15.94
-15.92
-10.03
Afar.
■Mar.
Mar."
4
11
fi
10
2
4
1-15.1:5,
— 15.20,
—15.15
!— 10.00,
1—10.02.
|_10.01„
.10.07.
10.03
—15.07,
-15.09,
—15.08
— 15.1.<«,
■—15.31,
1—15.44,
No.of Correotoii
M««n. oIm. I mean.
Corr. to OeocDDtrlu
-40.7
—15.30
15.55
15.47
15.04
15.57
JO
17
2
C
15.. 10
15.,30
2
4
—
—10.13
—10.27,
— I0.24j
-10.13,
-10.23
-15.90,,
-15.91,
-15.92.
-10.00,
-15.90
-15.55,.
-15.44,,
-15.04,
-15.54,
-15.50
-1.5.30,
-15.27,
-15.30
-35. 1
-34.1
-•J i . :s
-20.5
-•>■. '2
— 2S.3
—27.4
—27.9
-29.2
-2.S.0
-28.5
-29.0
-28.0
-40.3
Iiongitnile.
Latitude.
. ^ —40.3
—30.9 8 ' —30.5
—33.4 i 4 ' —33.0,
—32.8 13 ! —32.0,,
-34.3 7
-34.0 i 6
—24.5
—32.7
-34. 7„
—33.9,
—34.4
-33.9,
— 34.4,
[—217.9] [—3.1]
—215.0 I —0.0
[—220.2] [—4.0]
— ?23.4
—0.1
—34.1
—24.1
-20.8,
-20. .3,
25.2
—20.3
-27.9,,,
-27.2,
-27.9
[-227.3] [-4.0]
—224.4 —0.4
[—223.2] [—4.0]
—220.2 —0.3
[-217.5] [-4.0]
—215.1 —0.0
[-220.2] [—4.7]
—223.0
—7.1
[—227.0] [-4.7]
—225.4 —7.2
I'LL [—224.3]
—27.9
—28.4,
— 2.S.,5,
— 2S.0
' [—219.3]
— 21G.0
-28.0,
"^':*''_ [-215.9]
-28.2 —213.3
[-4.8]
-0.8
[-4.8]
—7.2
[-4.8]
—7.0
THE ORBIT OF URANUS.
141
Mean Cuiiuections tu tiik Kimiemkuih ur Uhanub. — Continued.
ObRerT«tory.
[K. A. Ill
Urauui..]
Paris,
Lcydcii,
[5" 34""]
Greenwich,
Leyden,
[5" 29~J
Washington,
Leyileii,
[5" as"]
\Vii<liiiif;ton,
[f)'' 21'°]
Orccnwifh,
[5" 21"]
Pnris,
Waxliiiigton,
L(ydeii,
(5" 4!)"]
Oreciiwiuh,
Parin,
Wasliiiigton,
Leydt'ii,
[5" 45™]
Wusliingtoii,
Lcydcn,
[5>' 41-]
Oreciiwlch,
Wusliingtoii,
[fl" 17"]
Orcenwicli,
I/cydcii,
Wttsliington,
[0" ll"]
Mean <lal«g.
1863
Nov. 2!)
Nov. 14
Nov. 21
Di-c.
Deo.
I!)
14
Dec. 18
1864
Juii. 15
Juii. 10
Jan. 14
I'eh. 15
May :{
Dec. 20
Dec. 10
Dec. 18
Dec. n
1865
Jan. 5
Jan.
15
Jan.
17
Jan.
10
Jan. 15
Feb. 14
Feb. 13
Feb. 14
Oct.
Oct.
10
18
Oct. 17
Dec.
Dee. 7
Dec. IS
Dec. 14
Obaervad corrautious In R.A. ObBervvd oorreotiODS in Duo.
Mean.
S
-1<!..'J3
-l(i.23
-10.33
-10.51
-10.01
-10.07
-15.74
— 15.57
-10.32
-1(;.22
-lO.l'.l
-10.2.S
-10.22
-It;. 11
-10.03
-15.77
-15.85
-15.50
-15.59
-10.13
-10.20
-10.17
No. of Corrected
obn. lueau.
B
—10.30,
—10.20,
—10.24
—10.32,
—10.48,
—10.30
—10.01,,
—10.04,
—10.02
—15.74
—15.57
—10.29,
—10.22,
—10.10,
—10.22
-10.28,
—10.19,
—10. II,
— 10.00,
—10.13
—15.77,
— 15.82,
—15.78
—15.59,
— 15.59.
—15.59
—10.13,
—10.23,
—10.17.
—10.17
M«an.
-18.4
-18.1
-20.0
-17.1
—20.0
-21.5
22.2
-13.2
-11.4
-12.4
-13.7
-15.1
-14.4
-14.7
-15.4
-15.0
— 2.4
4.8
3.8
No.of Corrected
obd. lueau.
II
4
-18.2.
-18.1,
Corr. to Oeoaenlric
Lougitade. Latitude.
-18.2
[—225.9]
—224.1
-19.0,
'^''•'' [—220.8]
-19.0 —225.9
-20.0
—21.8,
—12.8,
-12.4,
—12.0
-13.. 3,
-14.9,
[—224.0]
-221.4
[—219.0]
—217.8
[-210.1]
—215.0
[—225.1]
—223. 1
"'111, [—223.4]
-14.3 —220.0
-15.0
[—218.9]
—217.4
[—215.3]
—214.0
[-5.G]
—8.3
[-5.7]
—7.0
[-5.8]
—0.7
[-5.7]
[-5.0]
—7.3
[-0.5]
—8.4
[-0.5]
—8.8
[-0.4]
—8.0
—8.1
.- 2.0,
— 2.0
- 4.8,
- 4.0.
[—222.2]' [—7. .3]
— 4.7 —222.0 —8.8
148
THE ORBIT OP URANUS.
i
Mean CuKKKCTioNg to tug ICi'iikmkhis ur Uhani'b. — Cotiliiiucd.
OlmorTatorv.
(R. A. or
Unuaa.]
Qrccnwicli,
t'arix,
WaHliington,
Li-yilcn,
[fl" 5"']
Oreenwioh,
Washington,
Luyduii,
[C 0""]
Greenwich,
\Vti.shinglun,
Leyilt'ii,
£«" O"]
Washington,
[li" aT'"]
Qrccnwich,
VVusliington,
Qrccnwich,
Washington,
Leyilen,
[( " 31""]
Greenwich,
Washington,
Lcydon,
[-(jh 24m-]
Paris,
Wasliinglon,
Lcydcn,
U«an dates.
1866
Jan. 10
Jan. 1>2
Jan. 17
Jan. 17
Jan. 14
Feb. 14
Feb. 14
Feb. 14
Fub.~14
Mar. 12
.Mar. 9
Mar.
3
Mar.
Oct.
9
1.3
Nov.
Nov.
9
15
Nov. 14
Dec.
Dec.
Dec.
11
12
10
Dec. 11
1867
Jan. 21
Jan. 21
Jan. 9
Jan. 19
Feb. l.J
Feb. 15
Feb. 18
[C 20™] I Feb. 15
Greenwich, '
Wa.shington, ,
Mar.
Mar.
[C 19""] I Mar. 6
Greenwicii,
Washington,
[fl" 51"]
Dec.
Dec.
11
18
Dec. 13
Ubavrvvd vorreutloni in R.A
Mean
R
-1(1.20
-lfi.04
-IC.OO
-lfl.02
-15.81
-15.81
-15.75
-15.47
-15.49
-15.59
— 15.3fi
-15.fi5
-15.72
-10.01
-Ifi.OO
-15.87
-15.89
-15.93
-15.82
-15.72
-15. (i5
-15.54
-15.45
-15.32
-15.IJ1
-15.70
No. of
obi.
10
6
8
2
10
10
7
2
5
12
4
Correoted
u«au.
Ubucirved uorreotions in Dec.
Mean.
— ir..20,
— Ifl.OI,
— Iti.Ot),
— 15.99,
—16.09
-1.5.81,.
—15.81.
—15.72.
^5.79^
—15.47,
—15.49,
— 15.5(;,
—15.50
— 15.3«
7.2
n.3
7.«
7.1
9.1
8.4
8.fl
8.1
9.7
9.0
+ 4.4
—15.65, -j- 3.5
—15.72,, 4- 3.8
—15.71
— Ifi.Ol,, -t- 2.S
—Ifi.OO, -t- 2.0
— 15.84, -f l.«
—15.99
—15.89,
— 15.9;{.,
-15.79,
—15.90
— 15.fi9,
— 15.fi,5,
— 15.51,
-15.04
—15.4.5,
—15.32,
—15.40
— 15.fil,
—15.70.
— 15.C4
1.4
1.2
0.3
2.4
2.0
0.5
2.9
2.5
8.5
6.9
No. of Corrected
uba. mean.
11
8
8
10
9 — fi.S,
1 j — 6.1.
9 1 — 6. .5.
6 — 7.1,
— 0.6
Corr. to Oeocentrlo
Utitude.
Longitude.
8.7,
7.3,
8.6,
— 8.2
— T.7,
— 8.0,
— 9.0,
— 8.5
+ 5.5
+ 3.9,
+ 4.9,
-I- 4.0
10 -f 3.2.
7 -f 3.1,
i + l.fi,
-\- 2.9
5
12
4
10
3
1.0.
0.1.
0.3,
— 0.4
— 2.0,
— 0.9,
— 0..5,
1.1
2.5,
1.4.
— 2.1
+ 8.9.
+ 8.0,^
+ 8.5
[—221.6] [.
—221.0
7..1]
9.0
[— 217.3]'[.
—210.9
- 7-3]
— 8.2
[— 212.9]'[-
— 212.9
[-209.0]
-211.2
7.1]
8.5
[-
7..5]
7.9
[-21.5.1] [
—215.9
-218.2] [-
—219.7
7.8]
8.3
8.0]
9.7
[-217.8] [_ 8.0]
218.3 — 9.3
.214..5] [— 7.9]
—214.7 — 8.0
[-211.1]
—211.3
[—213.1]
—215.3
[- 7.8]
— 9.0
C- «■«]
— 10.4
THE ORBIT OP URANUS.
140
Mean Cobiikctions to the Epiiemkuis or Ukanub. — Continued.
ObiierTatory.
LK. A. of
Unuua.]
Qrccrnvicli,
[(;" 45'°]
Qrpcnwk'li,
Lvydcii,
VVusliiiigtoii,
Piii'is,
[•;'' 41'"]
LcydcM,
Wttsliirigton,
[(!'■ 3!)'"]
Wiisliiiipton,
[7" l(i"']
Wiisliiiijrton,
[7" Hi"']
flrccnwicli,
Wusliiiigtiin,
[7'' lO""]
Orpcnwirli,
Wusliiiigton,
[7'' T)"']
flrconwifli,
Wiisliiiigtuii,
Paris,
[7" l"-]
flri'ciiwich,
Wttsliiiijjfton,
[(i"" 59"°]
Oreenwicti,
[7'' Si™]
Qrconwicli,
[7" 25'"]
Oroeiiwic'h,
Washington,
[T" 21-]
Mean dstea.
Obiertttd oorreotloua in R.A.
Mean.
1868
Jan. 13
Fcl>. 15
I'VI). 5
Fub. 13
Ftb. 14
Mur. 1 1
Mill-. 19
Mar.
Oft.
15
18
Nov.
7
Doc.
Dec.
25
12
Dec. 18
1869
Jan. 19
Jan. 18
Jan. 18
Ffl).
Fi'l).
Fcl).
14
14
10
Feb. 13
Mar.
Mar.
3
7
Mar.
Deo.
5
8
1870
Jan. 19
Feb.
Feb.
Ifi
1(!
Feb. IG
— 15.C7
-15.43
-15.43
-15.50
-15.41
-15.02
-14.91
-14.57
-14.83
-15.24
-15.1(i
-15.28
-15.31
-14.81
-14.87
-14.50
—14.73
-14.58
-14.49
No. of
obit.
-15.04 • 10
-15.13 13
-15.09 10
10
Corrected
luuau.
8
-I5.r.74
—15.4.3.
— 15.40,
—15.50.
—15.38.
—15^44
— 14.99,
—14.91,
—14.95
—14.67
—14.83
—15.24.
— la.Ki,
—15.20
-15.28,
-15.31.
Ubaerred oorrootiona in Deo.
Mean.
-15.31
-f 0.9
+ 4.8
+ 5.9
-I- 4.9
-t- 4.0
+ 3.1
-fl2.3
-j-lfi.2
-fl5.1
+ 14.0
-1-12.0
-1-12.9
-15.04,, -1-11.1
-15.13„i -1-11.3
-15.0(1, -1-11.7
-Is. 08
-14.81,
-14.87,
-1-10.8
-fll.4
-14.84
-14.50, -1-21.1
—14.72.
-14.57.
-14.49,
-14.55
-1-19.0
-fl7.2
-flO.7
No.of
oba.
Corrected
uieau.
10
13
10
Corr. to Quocentria
Loiigltade. ' Latitude.
-f 6.7. [-213.4] [— 8.7]
-215.5 —10.1
+ 4.0.
+ CO,
[—210.3] [— 8.0]
-f 5.3
-f 4.0,
+ 4.5,
—212.3
-f- 4.0
-1-13.3
-f 17.3
+ 14.9.
+ 15.0.
+ 15.2
+ 12.4.
+ 13.4.
+ 13.3
+ 10.9,,
+ 11.7,.
+ 11. t
+ 10.6,
+ 11.9^
+ 11.3
+20.9
10 +18.8
+ 17.0
+ 17.1
— 9.7
-205.0] [— 8.6]
-205 5 1 — 9.2
198.3] [_ 8.7]
—201.5
—12.9
I- 0.1]
—205.0 — 9.3
[—202.0]
C-:
200.5] [_ 9.2]
210.3 —10.0
[— 207.1] '[— 9.3]
—211.4
-10.3
[—204.7] [— 9.2]
-208.1
-10.2
[—201.5] [_ 9.2]
—204.8 j _ 9.3
[— 197.1] i[— 9.8]
—202.1 —10.3
[—199.3] [_ 9.9]
—204.5 —10.8
[—197.2] [_ 9.9]
+ 17.0 —201.9 —10.9
150
THK ORBIT OF U II A N U S.
Mkan CoBBECTioNH TO TUB Ki>iiKMERi8 Of XJRASva.— Continued.
Oh»vtJtlotf.
[K A. vf
VtHUMi.]
Orm^nwieli,
Wa.sliiii)(ton,
[7" 18"]
Qropiiwicli,
[7" ■17"']
flreonwicli,
[7" 41'"]
Orpcnwit'h,
[7" 3S"']
Orponwicli,
[S" 11'"]
Greenwich,
Wusliiiijjtoii,
[8" (;■". 1]
flrcenwioli,
Washington,
[8" l-.O]
(JrocMwich,
VVusliington,
[7'' 57".9]
Washington,
Uruunwioh,
[7" 57'". 3]
M«SD >U.ea.
Obierruti oorractiona hi R.A.
M«au.
1870
Mar. I a
Alar. 1 1
Mar. 12
1871
Jan. 9
Feb. 15
Mar. 14
Dec. 21
1872
Jan. A
Jan. 21
Jan. 18
Fib. 15
Feb. 21
Feb. 17
Mar. 15
Mar. 15
Mar~15
April 8
April 8
April 8
8
— 14.-->!(
— 14. HO
— IH.fl!)
—13.92
—13.82
— 13.1G
-13.32
-13.31
-13.23
-13.24
-13.01
-13.07
-12.79
-12.77
No. of Correi'teil
uba. I luoan.
0I> irred correotioD* tn D«a
M«an.
14
3
14
10
— I4.2S,
— 14.3(»,
^14.29
—13.98
—13.91
—13.81
—13.15
—13.31
—13.31
—13.31
—13.22
—13.24
— 13.23
—13.03.
—13.07,
— 13.05^
— 12.79,
— I2.7li,
—12.78
-flfi.9
-1-15.9
-f24.8
-I-2I.8
-f20.7
-f 29.2
-f2.S.3
-f.7.8
-f2P..2
-l-2(;.4
-f-25.4
-1-24.5
-f24.9
-f25.0
No of Corrauted
ub<. lueitu.
14
+ lf..O
-|-l(i.3
Corr. to Oeocvutiio
Longitude.
Utitod*.
14
8
-f 1(1.2
-f24.fl
-f21.6
-1-20.5
-1-29.0
-f2S.l.
+ 28.2,
-f28.1
-l-2r..o,
+ 2«.H,
[-103.f.] [- 9.7]
—198.1
—10.1
[—190.0] [-10.2]
— 19(!.l j —10.7
[—188.8] [—10.1]
-194.4 I —11.5
[-184.9] [—10.0]
—192.8 I —11.5
[-177.9] [-10.4]
-18(5.5
-11.4
-f2(i.3
[—179.8] [—IO.fi]
188.3 —11.3
[_179.3] [—10.5]
-180.0 —11.3
-f2o.2.
_-\--t^i_ [—170.4] [-10.4]
-f25.1 —183.7 —11.1
-f25.3,
-f-24.8,
-1-25.2
[—172.4] [—10.2]
1 —180.2 I —10.8
1
TUB Oil HIT OF URANUS.
Iftl
1
COHIIECTIONM TO HE Al'I'MKI) TO TIIK PolilTKlNll (IF
UllANIH IN TriE
[iKIILIN jAIIHUlCri AND THE
NAuriuAk Almanac to hkuuce them to the I'uhitionh fhom the I'huvihiunal Theouy. 1
Dxte.
Htilloo«iitria,
Oenoentrlo. 1
n
Mf'f
'I'.
ib
It
1830,
July 24
— 18.0
■rinoi
+ 0.0
—10.3
+ 0.5
Aug. 13
18.3
1007
0.0
18.8
0,5
Sept. 2
18.4
1014
0.1
17.0
0.5
Bupt. 22
18.5
1020
O.l
J7.1
9.4
Oct. 12
18.7
1027
9.1
16.7
0.3
Nov. 1
10.0
1031
0.2
16.5
0.2
Nov. 21
—10.3
+ 1038
+ 0.2
—16.7
+ 0.1
1831,
July 19
_22.5
+ 1112
+ 0.5
—24.5
+ 10.0
Aug. H
22.0
II 1
0.5
23.7
10.0
Au(f. 2S
22.7
1123
0.6
22.7
10.1
8f'pt. n
22.8
1128
9.6
21.7
10.0
Oct. 7
22.0
1133
9.6
20.0
0,9
Oct. 27
23.2
1138
0.7
20.6
9.8
Nov. 16
23.5
1142
0.7
20.0
9.6
Dec. 6
—23.8
4-1140
+ 0.7
—21.0
9.5
1832,
Aug. 2
—26.0
+ 1108
+ 0.8
—28.7
+ 10.3
Aug. 22
27.2
1201
0.0
27.9
10.4
Sept. 11
27.4
1202
9.0
26.9
10.3
Oct. 1
27.6
1 205
0.0
26.1
10.2
Oct. 21
27.8
1208
10.0
25.4
10.2
Nov. 10
2H.0
1200
10.1
25.1
10.1
Nov. 30
—28.3
+ 1200
+ 10.2
—25.2
+ 10.0
1833,
July 28
—31.0
+ 1233
+ 10.2
—34.5
+ 10.7
Aug. 17
32.2
1236
10.3
33.7
10.8
Sept. 6
32.4
1240
10.3
32.7
10.8
Sept. 26
32.6
1242
10.3
31.8
10.7
Oct. 16
32.0
1245
10.3
31.0
10.5
Nov. 6
33.2
1247
10.4
30.4
10.4
Nov. 25
—33.5
+ 1250
+ 10.4
—30.2
+ 10.3
1834,
July 23
— 3S.0
+ 1264
+ 10.5
—41.2
+ 11.0
Aug. 12
38.4
1266
10.5
40.7
HI
Sept. 1
38.4
1268
10.6
30.6
11.1
Sept. 21
38.7
1270
10.6
38.6
11.1
Oct. 11
30. 1
1275
10.6
37.7
10.0
Oc.,. 31
30.4
1277
10.6
36.9
10.8
Nov. 20
30.7
1282
10.7
36.6
10.7
Dec. 10
—40.1
+ 1282
+ 10.7
—36.1
+ 10.5
1835,
July 18
— 13.0
+l.^oo
+ 10.8
—47.7
+ 11.3
Aug. 7
44.5
1311
10.7
47.6
11,3
Aug. 27
44.5
1313
10.7
46.6
11.3
Sept. 16
45.1
1316
10.7
45.8
11,2
Oct. 6
45.4
1318
10.8
44.8
11,2
Oct. 26
45.0
1321
10.7
44.0
10.9
Nov. 15
46.5
1324
lO.S
43.6
10.8
Dec. 5
—46.6
+ 1327
+ 10.7
— 42.5
+ 10.6
1836,
July 12
—51.0
+ 1361
+ 11.0
—.55.3
+ 11.4
Aug. 1
51.3
l:iC.4
n.o
55,2
11.5
Aug. 21
51.5
1365
11.1
54.6
11.7
Sept. 10
—51,0
+ 1368
+ 11.0
—53.7
+ 11.6
162
TUE ORBIT OF URANUS.
CoBRECTiONH TO BE APPLIED TO THE POSITIONS OF UiiANUs — Continued.
Dr.t«.
Heliocentric.
Geouentrio. 1
hJ^
Mfip
¥
u
hh
It
II
II
II
i83r>,
Sept. 30
—52.2
+ 1370
+ 11.0
—52.4
+ 11.5
Oli. 20
52.9
1372
11.0
51.0
11.3
Nov. 9
53. 1
1375
no
.50.5
11.1
Nov. 29
53.9
1377
11.0
.50.4
11.0
I>fC. 19
—03.8
+ 1380
+11.1
—49.8
+ 10.9
1831,
July 7
—58.3
+ 1387
+ 11.0
—02.9
+ 11.4
July 27
MA
i:!87
11.0
02.9
11.5
AuK- Hi
59.2
13.88
11.1
03. 1
11.7
Sept. 5
59.2
1389
11.1
02.0
11.7
S.-pt. 25
00.0
1391
11.0
01.3
11.5
Oct. 15
00.0
1.393
11.0
59.0
11.4
Nov. 4
fil.O
1394
11.0
59.1
11.2
Nov. 24
01.3
J 394
11.0
58.2
11.0
Deo. 14
—02.1
-} 1393
+ 11.0
—57.1
+ 10.9
1838,
Auff. 11
— fifi.4
+ 1395
+11.1
—71.1
+ 11.0
Aug. 31
07.2
1390
11. 1
71.0
11.7
Sept. 20
07.7
1395
11.1
70.2
11.7
Oc-t. !0
08.1
1392
11.1
08.9
no
Oct. 30
08.5
1390
11.1
07.0
11.4
Nov. 19
08.9
1388
11.1
00.4
11.2
Dec. 9
09.4
1388
11.1
05.8
11.1
Dec. 20
—09.7
+ 1389
+ 11.2
—05.4
+ 11.0
1839,
Aug.
—74.7
+ 1382
+11.1
—80.0
+ 11.0
Aug. 20
75.1
1381
11. 1
79.7
11.7
Sept. 15
75.5
1380
no
79.0
11.0
Oct. 5
70.0
1379
11.1
77.9
no
Oct. 25
70.4
1379
11.1
70.5
11.5
Nov. 14
70.8
1379
111
75.2
11.3
Dec. 4
77.2
1378
11.1
74.0
111
Dec. 24
77.0
1377
11.1
73.5
10.9
1810,
Jan. 13
—78.0
+ 1370
+ 11.0
—73.1
+ 10.7
June 24
—8'. 9
+ 1370
+ 11.0
— Sfi.O
+ 11.1
Aug. 20
^•z.i
1374
no
M7.;;
11.5
Sept. 9
83.4
1370
11.0
H7.S
11.0
Sept. 29
83.5
1377
n.o
80.4
11.5
Oct. 19
84.2
1370
no
85.5
11.4
Nov. 8
84.4
1370
no
83.5
11.3
Nov. 13
85.3
1377
no
82.7
11.1
Dec. 13
H5.5
1378
n.o
HI, 5
10.9
1841,
Jan. 7
—80. 1
+ lu-7
+ 10.9
—SI. 3
+ 10.7
June 10
-89.8
+ 1378
+ 10.8
—93.2
+ 10.8
Aug. 15
91.1
13S4
10.8
97.0
11.3
Sept. 4
91.5
13H5
10.8
90.9
11.4
Sept. 24
91.9
1388
10.7
90.1
11.3
Oct. 14
92.3
1 390
10 7
94.8
11.2
Nov. 3
92.9
1392
10.7
93 4
11.1
Nov. 23
93.4
1393
10.8
91. H
10.9
Dee. 13
93.8
I.;95
10.7
90.5
10,7
18-i'2,
J III). 2
94.3
1 .•19X
10 s
S9.7
IIJ.O
I'eli. 11
—95.0
-f U04
4 10.7
— S9.0
+ 10.3
\\
«
THE ORBIT OF URANtJS.
163
CouREtmoNs TO BE AppuKD TO TiiK POSITIONS OF Uranvs Coiilinued
Date. Hellootmtrlo.
OKOceiitrlc. 1
h?.
V^p
^i
11
ih
/'
' ft
ft
II
1842,
Tunc 11
— Of). 7
+ 1409
+ l'.'.(i
— 9S.5
+ lii.5
Aiiir. 10
!llt.l
14(tS
10. r,
105.3
10.0
jVii)f. ;i()
y.t.c
1409
10.7
105.8
10.2
Sept. lit
91t.9
1409
,M).(J
105.2
10.2
Oct. a
l(tO..T
1410
10.5
104.2
10.0
Got. 2!)
lltU.'J
1412
10.5
102.6
10.9
Nov. 18
.''11. a
14ia
10.5
100.8
10.8
Doc. 8
l(tl.8
1414
10.5
99.:'
10. (i
D.'c. 28
1(12.2
1415
10.5
9S.0
10.4
iH4a,
Jul). 17
-1(12.5
■tl417
-f 10.4
— 97.2
+ 10.2
All jr. ft
—107.3
+ 1422
+ 10.3
— M3.7
+ 10.7
.Auji. 2.'>
i!!-.-
1420
10.2
114.4
10.7
Sept. 14
Kts.O
1421)
'O.l
114.3
10. (i
Oct. 4
lOX.f)
1 420
10.1
113.4
10. fi
Oct. 24
lOil.O
1419
10.1
112.1
10.5
Nov. l:}
Kill. 4
1418
10.1
110.2
10.4
Dec. a
IIO.O
1118
10.1
lOsf,
10.3
Dec. 2;j
HO.4
1417
10. 1
lOfi.9
10.1
1844,
Jiiii. 12
— llit.S
-i 1418
+ 10.0
—105.9
+ 9.8
July ;iO
-11.-..1
-t 1 40C.
+ '•'(
— 121.2
~f 10.0
Aiijjr. lit
II.V4
IKia
'.1.8
122.3
10.2
i^i\>{. s
II. "..8
1 too
!l,S
122.7
10 3
Sept. 2s
in;, a
ia;i7
9.8
■ 2.4
10. a
Oil. ix
licit
l.'illC.
:i.7
i. ..3
10.2
Nin "
117.4
iaii2
o.c,
119.7 1 iO.O
Nov. 27
1 17.8
1 .iS9
o.c,
117. (! ! !I.S
Dec. 17
lis. a
ias4
9.(!
115.7 I 9.7
1845,
i1 II II . li
IIS.I!
lasi
'.1.5
111. 3 9.4
Juii. 2(<
— llll.l
+ ia78
+ 9.(1
— 1 13.3 + 9.4
A UK. 14
--12a.-)
+ 1340
+ 9.1
— 130.4 + 9.4
J^cpt. a
i2a.!»
laaa
9.4
l:!l.4 9.8
Sept. 2;t
124.4
la.ao
9.4
131.7
9.9
Oct. l;i
124.9
ia27
9.3
130.(i
9.8
Nov. 2
I2r),a
ia22
9.3
129.0
9.7
Nov. 22
12.-). 7
iai8
9 2 1
127.0
9 5
Dec. 12
12(1.2
laia
9.2
125.0
9.3
184P.,
.lull. 1
l2i;.7
iao9
9.2
123.2
9.2
Jan. 21
— 127.!
+ iaoa
+ 9.2
— 121.9
— 9.0
A up. 2'.i
— ia2.o
-1 125a
+ 8.8
—1 311.5
+ 9.2
.Sept. 18
ia2,a
1249
8.8
l;!9.9
9.2
■.
Oct. 8
ia2.8
124(i
8,8 lao.s
9.3
Oct. 2H
iaa,2
1242
8.7 1 ias.4
9.1
Nov. 17
\X).h
I2a8
8.7 13(1.4
9.0
Dec. 7
lai.i
I2a4
8.7 131.4
8.9
Dec. 27
lai.d
i2aa
H.C. 132.3
S.7
184T,
Jiin. ID
— lai.'j
4 i2ao
+ S.O 1 —130.5
+ 8.5
A up. 24
— lait.r)
+ 1197
+ 8.3 — ltd 7
-1 :«.fi
Sept. l:!
11(1.0
11114
S.3 14S.0
8.7
Oct. .!
'4(i.r>
mil
8.3 1 HS.3
8.7
Oct. 2;!
— ilO.9
+ 11^8
+ 8.2 -147.4
+ 8.6
ao May, 1
B73.
154
THE ORBIT OP URANUS.
Ill
Corrections to bb Applied to tub Positions op Uban
ua — Continued.
D«te.
Ileliocentrio.
Oeooentrlc.
67i
tt
Mh^
u
II
hb
It
184T,
Not. 12
—141.4
-fll85
+8.2
—145.8
+8.5
Dec. ■^i
141.0
11 83
8.1
143.7
8.3
Due. i2
142.4
1182
8.1
141.4
8.2
1843,
Jan. 11
_l42.8
-fll82
+8.1
—139.3
+8.1
Sept. 7
—147.0
+ 1105
+ 7.6
—150.0
+7.9
Sept. 27
14S.:(
1100
7.0
150.7
8.0
Oct. 17
14M.7
1104
7.0
150.4
8.0
Nov. 6
140.a
1104
7.0
155.3
8.0
Nov. 2(!
140.8
1103
7.0
1.53.2
7.9
Doe. U\
150.2
1102
7.0
150.7
7.8
1849,
.Full. 5
\■^{^^^
1101
7.5
148.3
7,5
Jan. 25
— 151.3
+ 1101
+7.5
—140.0
+7.4
Sept. 2
—155.8
+ 1149
+7.0
—104.0
+7.3
Sept. 22
150.2
1150
7.0
105.1
7.3
Oct. 12
15fi.5
1149
7.0
105.1
7.4
Nov. 1
150.8
1148
7.0
104.0
7.4
Nov. 21
157.4
1149
7.0
102,4
7.3
Deo. 1 1
157.9
1148
0.9
100.0
7,1
Dee. .tl
I5S.3
1U7
0.8
157.5
0.9
1850,
Jan. 20
— 158.0
+ 1147
+0.8
—155.0
+ 0.8
AiiR. 28
— lo;!.3
+ 1135
+0.2
—170.0
+0.4
Sept. 17
103.7
11. {3
0,2
172.0
0.5
On. 7
104.2
1131
0.2
173,3
0.5
Oet. 27
104.7
1129
0.1
173.1
0.4
Nov. 1((
105.2
1127
0,1
171.8
0,4
Dee. G
105.6
1120
0.1
109.4
0.3
Dec. 2fi
100.0
1127
0.0
100,7
0.2
1851,
Jan. 15
100.3
1127
0.1
103,8
0.1
l\\>. 4
—100.7
+ 1124
+0.0
—101.8
+ 5.9
Sept. 12
— 171.5
+ 1109
+5.5
—180.0
+5.7
Oet. 2
171.9
11 Of
5.5
181.4
5.8
Oet. 22
172.4
1103
5.5
181,0
6.8
Nov. 11
172.9
1103
6.4
180.8
5.7
Dee. I
173.3
1103
6.3
178,7
6 5
Dee. 21
173.8
1102
5.2
170.1
5.3
185-2,
Jan. 10
174.3
1101
6.2
173.4
6.3
Jan. 30
—174.7
+ 10U8
+5.1
— 170.2
+5.1
Sept.
— 179.0
+ 1074
+ 4.7
—180.8
+4.9
Sept. 2(;
179.4
1073
4.0
188.9
4.8
Oet. l(i
179.8
1071
4.5
180,7
4.7
Nov. 5
1H(».3
1008
4.5
180,0
4.7
>■• ■'. 25
180.7
1007
4.4
187,8
4.6
Ito. 15
181.2
1005
4.4
185,3
4.6
185.1,
Jan. 4
181.5
1004
4.4
182,3
4.5
Jiin. 24
181,8
1002
4.4
179.0
4.4
Fob. l;i
— 182.1
+ 1000
+4.3
—170.9
+ 4.2
Sept. 1
—185.8
+ 1055
+3.9
—192.4
■14.0
Sept. 21
180,2
1055
3.8
195.2
3,9
Oet. 1 1
— ISC. 5
+ 1055
+3,7
—190,7
+ 3,9
THE
ORBIT OF
URANUS.
165
CoRiiECTioNS TO BK Api'UED TO THE POSITIONS OP Ukanus — Continued.
Dat«.
Heliooentrio.
Oaooentrio. 1
ft
J/Sp
'i
hi
It
1853, Oct. 31
—isfi.s
+ 1054
+ 3.fi
— 19fi.8
+3.8
Nov. 20
1M7.1
1054
3.5
195.fi
3.7
Dec. 10
1H7.5
10f>»
3.5
193.6
3.B
Dec. 30
1«7.9
1053
3.5
190.7
3.6
1854, Jnn. 19
l«S.l
1053
3.5
187.4
3.5
Feb. 8
—188.5
-t-1052
+ 3.5
—184.5
+ 3.5
Sopt. Ifi
—192.2
+ 1034
+2.9
—200.5
+ 3.U
Oct. f,
i92.(;
1032
2.8
202.9
2.9
Oct. 2(5
193.0
1032
2.7
203.7
2.8
Nov'. IS
193.3
1032
2.r,
203.0
2.7
Dec. 5
193.7
1020
2.fi
201.3
2.7
Doc. 25
194.0
1025
2.5
198.fi
2.«
1855, Jan. 14
194.2
1021
2.5
195.2
2.5
Fclj, 3
—194.5
+ 1018
+2.4
—192.0
+ 2.4
Oct. 1
— 198.3
+ 1010
+ 1.8
—207.9
41.9
Oct. 21
19S.7
1009
1.7
209.fi
1.8
Nov. 10
199.0
1008
in
209.7
1.7
Nov. 30
199.3
1007
1.5
208.4
l.fi
Doc. 20
199.fi
loofi
1.5
20fi.O
l.fi
185i!, Jnn. !)
199.8
1005
1.5
202.fi
1.5
Jan. 2!)
2(10.1
1003
1.4
199.2
14
Feb. 18
— 200.3
+ 1001
+ 1.3
— 19«.l
+ 1,3
Oct. 1.^
—203.3
+ 9(12
+0.7
-213.9
+ 0.7
Nov. 4
203.7
9.-. 7
0.7
214.9
0.7
Nov. 24
203.9
953
0.7
214.5
0,7
Dec. 14
204.3
949
O.fi
212.4
0,fi
185T, Jan. 3
204.4
945
O.fi
209.3
O.fi
Jnn. 23
2(14.5
941
0.5
205. (;
0.5
Feb. 12
—204.7
+ 'J34
+0.4
—202.2
4 0.4
Sept. 20
— 20fi.9
+ 879
—0,2
—214
— 0.2
Oct. 10
207.2
87 4
0.3
217.1
0.3
Oc't. 30
207.5
808
0.3
2 is, 8
0.3
Nov. in
207.8
Kfil
0.4
219,0
4
Dec. fl
208.0
853
0.4
217,fi
0.4
Dec. 20
20S.1
840
0.5
215,0
0.5
1858, Jim. IS
20S.2
S39
O.fi
211,5
O.fi
Fell. 7
20S.3
830
O.fi
207,7
fi
Feb. 27
—208.4
+ 823
—O.fi
—204,2
—O.fi
Oct. f)
—210.4
4 741
— 13
—219,1
—1.3
Oct. 25
210.fi
732
1.4
221,4
1.5
Nov. 14
210.8
723
14
222.2
1.5
Dec. 4
211.0
713
1.5
221.9
l.fi
Dec. 24
211.2
704
1.5
219,7
l.fi
1859, Jan. 13
211.3
095
1.5
21fi,fi
1.5
Feb. 2
211.4
«8fi
l.fi
213,0
l.fi
Feb. 22
— 2I1.«
+ fi77
— 15
— 209,4
—1.5
Oct. 20
—213.1
4 57 fi
—2.1
222, s
—2.2
Nov. 9
213.2
509
2.1
224.5
2 2
Nov. 29
—213.4
1 + 5fil
—2. 1
—224.8
—2.2
156
THE ORBIT OF URANUS.
CuaBECTKlNH TO UK A
"i'LiED TO TUB TuMiTioNH OK Uranvd — Conlinuetl.
Date.
Uelioconlrie.
OttovcDlrle. 1
hX
Sifif
¥
hi
,M
11
It
it
If
185»,
nrc. 19
— 2I3..'>
-f554
—2.3
—223.5
—2.3
1»UU,
.lull. 8
2i3.r>
54fi
2.3
220. d
2.3
.Inn. 28
213.(5
539
2,3
217.4
2.3
Fob. 17
—213.7
+533
—2.3
—213.5
—2.3
Sept. 24
—214.7
-f 45fi
—2.8
—210.7
—2.9
Oft. 14
214.8
451
2.8
223. 1
2.9
Nov. 3
:
444
2.n
2-.?5.0
3.0
Nov. 2H
21;'.. 1
437
2.9
220.8
3.1
Dec. 1H
215.3
431
3.!)
220.3
3.3
1801,
.lull. 2
2ir..3
424
3.1
223.8
3.3
Jim. 22
2ir,.3
417
3.1
221.0
3.3
Fib. 12
-215.3
+ 411
—3.1
—217.3
—3.1
Oct. 29
—215.fi
+ ;!3!)
—3.7
—225.0 -3.9 1
Nov. 13
2l5.(i
333
3.8
220.9
4.0
Dec. 8
215.7
32!)
3.8
227.4
4.0
Dw. 2s
215.8
321
.1.8
220.3
4,0
1 80 2.
.Ian. 17
215,8
320
3.9
223,5
4.0
l'\-l). (i
215.8
315
4.0
220,0
4 1
Ffl). 2(!
—215.!)
4-310
—4.0
—210.0
—4.0
Oct. 24
— 21fl.O
424(5
-44
—224.2
— 4.fi
Nov. i:{
21t;.0
241
4.5
220.0
4 7
Dc, 3
21f..O
237
4 5
227.7
4.7
Dec. 23
215.0
2;!2
4.0
227.3
4.8
lues,
.litri. 12
215.8
22ti
4.7
225.0
4.9
F«-l>. 1
J 15. 8
21!)
4 7
221.8
4.8
Fob, 21
415.7
213
4.7
218,1
4.8
Mur. 13
-215.7
4-208
—4.8
—214,1
—4.8
Nov. 8
—215.3
4 13!»
—5,3
—224.fi
—5 5
Nt)V. 28
215.2
133
5,4
220.5
5.7
l>.M'. 18
215.1
n;fi
.'.,4
220.8
5.7
18G4,
.litii. 7
215.(1
120
."1.5
225,4
5.8
.lull. 27
214 !t
114
.'■.,5
222,7
5,7
Feb. ii;
214.8
108
5,fi
210,3
5,7
Mtticb 7
— 2U.8
4-103
—5,0
—215.3
— 5,fi
Oct. 13
—213.8
4 21
-fi,0
—217.8
—0,1
Nov. 2
213.7
Ifi
0,0
221.3
0,3
Nov. 22
21.3.7
!)
0,1
2 2 to
0,4
Dec. 12
213.fi
4- 1
0.2
225,1
0.5
1865,
.lull. 1
213.4
— (1
0,2
224 7
0.5
•Ian. 21
213.2
14
0,3
222,7
0,5
Feb. 10
213 1
20
0,2
2?0.fi
0,4
March 2
—213.0
— 27
—0,3
—215.7
—0.4
Oct. 8
—2ll.fi
—103
—0.8
-213,5
—0,9
Oct. 28
211.4
no
fi.S
217,2
7,0
Nov. 17
211.3
117
0,9
220,0
7,2
Pec. 7
211.1
124
0,9
222,0
7.3
Dee. 27
210.!)
132
r,!»
222 4
7.3
laort,
.lull. l*t
21(1 7
130
7,0
r,-M 5
7.4
Feb. 5
•'10.fi
i;5
(I.O
218,8
7.3
Feb. 2.''>
210.4
151
7.0
215.4
7 3
Mur. 17
—210.1
—158
—7 1
-211.2
—7 1
f ^
THE OUUIT OF URANUS.
167
CiiKRECTidNS TO iiE Ai'Pi.iEU TO THE I'oHiTiONS OF I'uAMs — Continued.
Date.
iI«lior«ntrio.
Oeocentrlo. 1
hx
II
J/^p
ft
hh
II
l8Cf.,
Oft.
3
—208.2
225
— 7.4
—207.3
— 7.4
Oct.
2.1
20S.1
2.30
7.4
211.5
7.0
Nov.
12
207.9
2.35
7.5
214.9
7.8
nee.
2
207.8
241
7.fi
21 7. «
8.0
Dee.
22
207.fi
24«
7. ft
218.8
8.0
ISf.T,
.lull.
11
207.4
251
7.fi
21S.5
8.0
Jan.
:tl
207.2
25fi
7.7
21(1.7
8.0
F.I,.
20
207.0
2(i2
7.7
213.8
7.9
Mar.
12
—20(1.7
— 208
— 7.7
—200.8
— 7.8
Nov.
27
—203.2
— 351
— 8.2
—211.4
— 8.0
Dec.
17
203.0
358
8.2
213.5
8.0
1808,
•la II.
(>
202.8
3(15
8.3
213.7
8.7
•Ian.
2(i
2(t2.4
371
8.3
212.7
8.7
Fell.
la
202.2
37(!
8.3
210.3
8
Miir.
t>
201.8
:',s-2
8.4
207.0
8.0
iMar.
2I-.
—201.4
— 387
— 8.4
—202
— 8.4
O.t.
12
— lOH.O
— 45S
— S.C,
— 107.2
— 8.0
XnV.
1
107.8
4(15
8.7
201.1
8.9
N..V,
21
107.5
473
8.8
203. 8
0.1
Kit.
11
107.2
4HI
8.8
20(1.2
0.2
I».r.
::i
ioi;.8
400
8.8
207.5
0.3
18(1 it,
.Ian.
20
1!h;.4
4!I8
8.8
207.1
0.3
Fell.
ii
loco
507
8.8
205.4
0.2
Mar.
1
lO.'i.C.
515
8.0
202.4
0.2
Mar.
21
-10,'). 3
— 524
— 8.;)
— l!is.4
— 0.0
Deo.
(i
—1 110.0
— 107.0
Dee.
2<i
IHO.5
los.o
1S70,
.Ian.
ir>
IH'.I.O
— (1(15
— 0.4
100.5
— 9.9
Fell.
14
ISH.IJ
<17(>
0.4
10H.7
0.0
Feb.
24
ISM
fiSCi
0.4
10(1.2
0.8
Mar.
ir,
1S7.5
ri07
0.4
103.0
9.0
April
;'»
— 1H7.2
— 708
— 0.4
-1x0. 4
— 0.4
De.v
1
— isi.r.
— S4(!
— O.fi
— 1.S(!.5
— 10.0
Dee,
21
isl 1
m5'.»
O.fi
IHO.O
10.1
1871,
•Ian.
It
l.SO.fl
K72
0.7
100,3
10.2
.Ian.
30
170.S
8.M3
0.8
100.1
10.3
Fc'l..
10
no 1
.X04
0.7
1X8.3
10.1
Mar
11
17X I
OOli
0.7
1M5.(!
10.0
Mar.
31
— 177.0
— '.110
— 0.8
— 181.2
— 0.9
Dee.
H!
— 171.C.
— los.l
— 10
—177.4
— 10.4
IH'i-i,
.Ian.
r.
171.0
1005
10. 1
170.3
10.0
Jan.
25
170. 5
1107
lO.U
HO. 2
10.0
Fell.
14
100.8
1 1 20
10.0
170.(1
10.5
Mar.
r.
100.3
1133
10.1
177.8
10.5
Mar.
2.*»
l(l.s.7
1115
10. 1
174.0
10.3
Apri
It
IC.M.I
11511
10.1
171.3
10 1
May
4
— 1(17.(1
— 11(18
— lo.l
— 1(1(1.3
— 10.0
■fli
V0
158
THE lUU T O r URANUS.
CIIAPTEll VII.
HI
FORMATION AND SOLUTION OF THE KQUATIOXS OF CONDITION RESULTING
FROM THK riUICKDlNU COMTARISONS.
In the preceding cliaptcr wo have ohtaiiicd from observations a series of cor-
rections to the {geocentric positions of Vrunns resulting from the provisional
theory. The furUier operations are as follows: —
1. To reduce all the corrections in right ascension and declination to correc-
tions in geocentric longitude and latitude. Most of the corrections arc already
so expressed, so that this reduction is necessary in only a few cases.
2. To find the mean value of the correction in geocentric longitude during each
opposition, and to express this mean value in terms of tiie correction to the helio-
centric cf)-ordinates.
;{. To express these corrections to the heliocentric co-ordinates in terms of cor-
rections to the elements of Uranus and the mass of Neptune.
4. To solve? the equations of condition thus formed.
The first of these jjroeesses is too sinii>le to mak(> it necessary to present any
details of it. With regard to the second 1 have sought, not the simple correction
to the geoeentri*: longitude, hut this correcticm multiplied 'jy smh a factor as it
was supposed would make the i)rol)ahle error of tin- correction 0".;"). The equations
for expressing the error of geoc(>ntric longitude in terms of trrors of heliocentric
longitude and radius vector have been given on page I'i!). The first observation
of I'lamstead, p. 107, gives the ('(piation
4- 2v>'=1.04.U + .n'27,^p
h?. being the correction to the h<lif)centric longitude, and f,p that to the Neperian
logaritlim of tin; radius vector. From the discordance of I'lamstead's clock errors
it nniy l»e estimated that the probable error of the first member of this ecpuition is
10". Therefore we divide the eepiation by 20, which gives
jV? = I'M = .0ry2,U + AWUp.
In the op])osition of 171") we have four observations. The best weve those of
March 4 and 10, of which wt' may estimate the ])robable error at 10", anu the
worst that of March 5, of which the i)r()babl(> error may be estimated at 20",
while that of April 2!) is intermediate in certainty. 1'he separate obscTvations
give the e<piations
\MftX; AVeight, 4
1.06^3,; "Weight, 1
l.OfxSX; "Weight, 4
1.04V;i-f .04V: Weiglit, 2.
March 4, ,V = -f-2.S"
March T), ,V = -f- •* ^
March 10, ,V^ + W
April 2!), <V=-f- -
Mean
hl—-{- 27.G = l.OiJtJAX -f.OOa V ; probal)le error = ± 6*.
.^
THE ORBIT OF URANUS.
169
Applying tlic correction — V.l for equinox, anil dividing by 12, the equation of
condition becomes
^\^fl = + 2".2 = ().0885;i.
In tliis way the following equations were obtained. It is deemed unnecessary
to give tlic details of the process, as it is one which every one can go over for himself
from the data already given, and can reproduce all the resjdts, except so far as
they depend on the relative weights assigned to the difl'cnint groups of observations
during one and the same opposition.
No.
Date.
Equations.
Number of
obHvrvatiuus
iu U. A.
1
1G91.0;
^o^i
= +
1.1 = .052r^J
.+
.001<^p
1
2
171 '1.2
vi
=-■ +
2.2 = .088
4
3
1748.8
k
= +
12.8 = .338
+
.017
1
4
1750.8
h
= +
11.8 = .345
+
.010
3
6
1153.9
h
= +
ll.G = .333
+
.016
1
6
n5G.7
I
= +
5.0 = .210
+
.003
1
7
17G9.0
!
= +
4.8 = .203
+
.010
8
8
1782.0
,}
= 4-
3.0 =1.370
21
9
1783.0
1
= +
1.25 = 1.030
—
.002
13
10
1784.0
1
= +
1.92=1.026
—
.008
13
11
1785.0
1
= —
0.2G = 1.034
+
.006
10
12
1788.0
i
.7
= +
1.23 = 0.084
+
.006
5
i:j
1789.0
J
= +
1.58 = 0.504
+
.008
6
14
1 "190.0
i
= —
0.52 = 0.514
—
,013
4
15
1791.0
.1
z=
O.'oG = 0.G84
—
.010
7
IG
1792.0
h
= —
0.12 = 0.340
—
.011
3
17
1193.0
J
= _
0.19=0.512
—
.015
5
18
1794.0
h
= +
0.79 = 0.344
—
.006
3
19
1795.0
-»
r= —
O.GG = 0.G83
—
.019
7
20
n9G.O
h
= —
O.G8 = 0.514
—
.015
4
21
1797.1
i
= —
0.35 = 0.528
3
22
1800.2
h
=
0.00 = 0.352
3
2;j
1801.2
h
= —
().2(> = 0.352
2
24
1S02.3
1
=^ +
O.S5=r^l.05
13
2)
l.S(t5.:5
1
= 4-
0.(i9 = 1.05
13
2G
1N()(;.3
i
= +
O.20 = 0..>2
5
27
1801.3
1
= -\-
2.34 = 1.045
16
28
1808.3
i
t= —
0.0 1 = 0.52
6
29
1809.3
1
-= +
1.80=0.70
9
30
1810.3
1
- +
2.39 = 1.05
16
31
1811.3
I
- +
1.49 = 1.01
.^_
).0l
11
32
1812.4
1
>
- +
O.S =0.53
8
33
IS 13. J
I
= +
1.2 =().5:{
9
34
1814.4
1
-=-h
1.1 = 1.05
15
160
1
r II K
OIUMT OF UllANUS.
No.
Pnle.
E<|unlioii»
.
Xninlier of
oltsiTvationa
ill It. A.
35
1815.4
3
V=+ 2.1
-=1.58,U
20
36
1818.4*
•J
= + 0.3
= 1.58
24
37
1819.4
i
= — 0.8
= 1.05
11
38
18-20.5
1
= — 1.3
= 1.05
14
3!)
18-21.5
i
it
= + 0.9
= 0.70
10
40
18:il>.5
i
= + 0.9
= 0.70
7
41
18-23.5
.1
= 0.0
= 0.70
11
4'i
18li4.5
1
= 4- 0.5
= 1.05
12
43
18-25.5
■»
= — 0.4
= 0.70
7
44
18-2(5.5
1
= + 0.3
= 1.05
11
45
18-27.7
2
= - '2.7
= *2.07
+ 0.04 ,^p
37
4G
18-28.7
-'i
= — *2.7
= *2.57
+ 0.07
67
47
18*29.7
n
= - 2.4
= '2.59
-1- 0.04
61
48
1830.7
n
= — 4.9
= 2.56
4- 0.07
73
49
1831.7
2
= 0.0
= *2.oa
-j- 0.05
54
60
183*2.7
2
^^ 2.2
= *2.07
-\- 0.05
65
51
1833.8
n
= — 4.1
= *2.58
4- 0.08
88
62
1834.8
n
= — 3.9
= *2.57
-j- 0.08
91
63
1835.8
n
= — 5.1
= *2.59
-f 0.05
82
64
183(5.8
3
= — 7.0
= 3.11
+ 0.08
157
65
1837.8
3
= — 3.r.
= 3.11
+ 0.01
162
6(i
1838.8
3
= — 1.6
= 3.11
H- 0.06
193
67
1839.8
3
= - 1.2
= 3.11
4- 0.06
170
68
1840.8
3
= — 1.5
= 3.11
-f- 0.06
1*24
69
1841.8
3
= + 0.8
= 3.10
-f- 0.06
108
60
184*2.8
3
= + 1.6
= 3.10
-f 0.06
169
61
1843.8
3
= + 4.7
= 3.12
+ 0.04
111
62
1844.9
3
= + 5.1
= 3.11
+ 0.03
106
63
1845.9
2
= + 4.2
= *2.08
+ 0.02
55
64
184G.9
3
= + (5.3
= 3.10
4- 0.04
98
65
1847.9
*2
= + 6.8
= *2.07
4- 0.03
74
66
1848.9
2
= + 4.4
= *2.()8
4- 0.02
59
67
1849.9
2
= + (5.6
= *2.08
4- <».04
33
68
1850.9
2
= + 7.2
= *2.08
4- 0.01
46
69
1851.9
2
= + «.3
= 2.07
4- 0.04
42
70
185*2.9
2
= + 6.6
= '2.07
+ 0.03
54
71
1853.9
•2
= -f 7.9
= *2.09
4-0.01
49
72
1854.9
2
= + 8.9
::= 2.09
4- 0.02
49
74
1855.9
2
= -l- 8.5
= *2.08
+ 0.04
48
76
1 85(5.9
2
= -f 7.9 .
= '2.08
+ 0,04
45
76
1858.0
n
= -f 10.3 :
^ *2.(;i
-|- 0.09
66
* The results for ISir, uikI IhU wore ipiiiillcil in this list tlirmigli oversight.
J
THE ORBIT OF URANUS.
161
Numlirr of
No.
Date.
EqilMinriR.
obHiTviitionii
ir
ill It A.
77
1859.0
2i''
/= +
10.6
= ) ,U + 0.03 -V
58
78
18G0.0
•4
= +
8. -2
— V'.CiO
+ 0.05
G4
79
IHGI.O
•_>
= +
(i.:j
= 2.0!)
+ o.o;j
41
«()
lS()'i.()
n
= +
7.0
= 2.(iO
4- 0.05
60
81
1H(5:J.0
n
= +
(>.<»
= 2.5!)
-f- 0.08
88
82
ls(i4.()
')
= +
4.:J
= 2.0!)
+ 0.0:{
35
8;j
i.s,;5.o
H
= +
y.i)
= 1.57
4- 0.04
37
84
18()().()
n
= +
1.1
= 2.(>0
4- 0.06
76
85
1S(H.0
n
=;
1.8
= 2.G0
+ 0.02
83
8(5
isr.s.o
2
=
:j.8
= 2.0!)
4- 0.04
40
87
i,s(;!).()
n
=: —
!).l
= 2.01
-f 0.02
r.s
8H
imo.o
•i
:^=
y.o
= 2.084
+ 0.054
31
8!)
1H71.0
u
=: —
10.(>
= 1.500
-f 0.040
21
!)()
187 2. 1
n
== —
ii).i
= 2.000
-f 0.070
50
inihor
of ohsiTX
atioi
IS ill 11
A.,
. . . .
. .
37()3.
1 -f 0.i»!):J!> CDS <j -\- 0.0055 cos 2y
'^ = 1.!)!)!) sin 7 4 0.1 17 sin 2'/ + 0.007 sin 3'/
' = _ 2.000 COS 7 _ 0. 1 1 7 co.s 2j — 0.007 cos 3y
AVc have next to express the vnlnes of A?, and .'j» in tcnn.s of tiic corn'ctions to
the elements. Dirt'erentiatinfi; the expressions
■A^l + (2/- — It') sin (/ — 7t) + Jfc" sill (2/ — 2.n) + ] ^e" sin »•» sin (3/ — 3:t) + etc.
,, = » -f !«' 4 ( — ' }- y) 'o^ (/ — 7t) — y- cos (2/ — ''n) — etc.,
with respect to /, ', and rr, and reducing tlie coc flicients to nuniliirs, we find
il
6v
d%
f)n
We liavc lierc put / for the mean longitude, or
/ =r «/ -|- £
whence
I'll vl
Also, from the expression for p
di
Si'
cJp _ cip _ 2^
rn (i on
The values of tliese coefficients wliich depend only on g arc. shown in the following
table :
::i May, 1873.
-.;=('
i'--(I-
t
Jl
— l^) sin U + ■?.'-" sin "i'J +
\t^) cos g — |c cos 2// — etc.
1fi
1U2
THE OUDIT OF UUAXU8.
d-K
dx
d?.
^l
s^>
c?p
9
6e
+ 1.099
Oe
«/.T
de
de
eiln
—2.124
—1.00
u
1.0!)!)
+0.0,'!!»
2,124
+ 0.001
1.00
—0.02
2
I.01t!»
0,070
2. 1 2:t
0.002
1,00
0.0:1
■A
l.Ollll
Oils
2.121
0,002
1.00
O.O.'i
4
l.t»!l!)
0,1. 'iS
2. IIS
o,oo:{
1.00
0.07
T)
+ 1.011')
+0. |;m;
—2.114
+ 0,004
— 1.00
— 0.00
I.OD!)
o,2:i:>
2.110
0,00:)
0.0!)
u, 10
7
1.00!)
0,271
2.ior)
oooc
0.0!)
0,12
«
l.O'.tS
0,.ll.l
2.00!)
0,001;
0.0!)
0,14
'.»
roos
0,:!.'i2
2.002
0,007
0.!l!)
oin
10
^ l.OilS
-|-o,;ioi
— 2.0xr>
+ o,oos
— O.OS
—0,17
11
1,007
4.iO
2.077
0,00!)
0. !IS
0.10
12
1.00'
ics
2.oi;o
0,010
0.!I8
0,21
l.'l
1.007
o,riO.".
2.o:i0
011
0.!I7
0.22
14
1.0!ll'>
.'.44
2,01!)
UOll
0.07
0.24
i:.
+ 1.0!)(!
40,, -.si
— 2,0,iS
40.012
—0.07
— 0.2»1
li;
I.OO.'i
0,1; IS
2,027
0,01:1
0.!)C
0.2s
n
1.0!»r>
o,cr,5
2,014
OOll
O.OC
0.20
IH
1.004
o,r,o:t
2,001
o,oi:)
O.!).")
o.;ti
1!)
1.004
0,720
1,0 S7
o.oir>
0.!)r>
0.:!:i
iiU
+ 1.00.J
+o.7cr>
— 1.07:1
40,01c
—0.04
_o.:i4
21
1.002
0,S01
1.057
0.017
0.0:1
o.;ic
22
l.ooi
o,n;ic
1,041
O.OIS
0.!):<
o.:i7
2:1
1.000
0,><72
i,!i2r.
o,ois
0.02
o,:!0
2.
1 000
0,!I07
1 007
0,010
0,01
0,41
2".
4 1 oso
40,012
— is'io
4 0,020
— 0,!ll
—0,4 2
2ii
LOSS
o.07r,
l,s72
0,0:>1
0,00
0,4 1
27
1.0H7
1, 010
i,.srp2
0,021
o.si)
0,4.-.
2S
l.oHi;
i,04;i
l.s:!2
0,022
0,HS
0.47
2'.>
i(i><r»
1,07c
I,M1
o,o-j;i
0.S7
4S
;ii)
+ I.0M4
+ l,los
— 1,700
40,02;!
— 0.S7
— O.j'iO
:!l
I,0S4
1.140
1,7C0
0.0:>4
o.sc
o.;'>l
32
l.o.s;i
1 172
1 717
o,o2r>
o.sri
o.r.:i
:;;t
1.0S2
1,203
1 721
0,02c
0.H4
o.:.4
HI
l.oso
l,2;!;j
1,700
0,02c
o..s:i
o.r.c
;!■.
-t 1,070
-1 1,2c 4
— l,(i7C
4 0,027
— 0.H2
— 0..-.7
lie.
l.o-H
1,204
l,r,.M
0.02H
0.x|
O.fiO
;i7
1.077
l,:i2:i
1.C2H
0,02s
O.XO
o.co
;iH
1.07C.
1 :i:.l
l,COl
0,020
0.70
o.<:2
:!!i
1.071
I,:i70
i,r.74
o.o:;o
0,7s
o,<i:i
Id
H l,07:i
-t 1,407
— i,r.is
4o.(i:;o
—0,77
— 0,C4
\\
1,072
1 4:: 1
i,r,2i
0,0:; 1
0,7C
o.cc
42
1,070
1.4 CO
1,404
o::i
0,71
0.C7
4:1
IOC!)
1 4s(!
1,4CC
(i:!2
o.7:t
O.CS
(4
IOCS
i.r.ii
1 , 4;!s
0,0:1:1
0,72
0.70
4:.
4 l,or,7
+ l.r.:ic
—1.400
40,0:;:!
—0.71
—0.71
4(;
1,0C5
i.r.ci
i,:!so
o.o:s4
0.70
0.71
47
1,0c 4
l.r)S4
l.:!.M
o,o;ti
O.CS
0.72
4x
l,OC.'<
l.COli
l.:!2o
o,!):):*
o,c7
0.74
4!l
l,ocl
l.r,20
1 . 200
o,o::c
o.cc
0.7'>
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-f l,oco
4 l,r,.-.i
— i.2(;o
40,0:1c
—0,04
— 0.7t;
r>i
l.o:.s
1,072
1 , 22!)
o,o;;7
o,c;i
0.77
fi2
l,or.7
1,002
1,107
00:17
0,C2
0.7 s
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1.712
iicr.
! 0,o;i7
0,C0
0.70
r>4
1.0,-.4
1,7:11
l,i:!:i
1 o.o:;s
o,r>!)
O.SO
.••5
+ l,0.-)2
-f 1 7r.o
' —1,100
4o,o;is
— o,r.7
— O.Sl
f>(l
l.o.M
1,7CS
I,(m;7
0,0:10
o,ric
0.S2
f.7
lOtO
1,7S5
l,o:i4
o,o:!0
o,r.4
o,x:i
f)H
: 1.017
1.H02
1.002
0,040
o,.-i:i
0,s4
n;>
M"i*i
H I,H17
—O.ocs
4 0,010
— 0,.M
— 0.s5
THK Oil HIT OF UUA XUS.
1«»
d>.
6'/.
('».
rV
y
r9p
9
+ 1.044
6e
edn
—o.nb
+0.041
lie
edn
00°
+l.«a:»
—0.50
—0.80
til
l.04:i
1.H48
o.ooi
0.041
0.40
0.87
(>i
1.041
l.s(12
0.S07
0.041
0.47
0.87
f.:i
1.040
1.H7.')
o.H;i2
0.042
0.4.'>
0.88
r,4
l.o:is
l.SSS
0.70,S
0.042
0.44
0.80
«5
+ l.o:iii
+ 1.001
_o.7(i;j
+ 0.04:1
—0.4 J
—0.00
«u-,
1.0 :»
1.012
0.72H
0.04:1
0.41
0.01
fiT
1 ().;:t
1.022
0.(103
0.04:1
0.30
0.01
«■<
l.O.il
l.o:i2
0.(150
0.041
0.37
0.02
(1»
l.o;to
1.012
0.(124
0.044
o.:to
0.0:1
70
+ I.02S
+ l.0".0
— 0.5SS
+ 0.041
—0.34
— (t.04
• 71
1.(127
l.OiV.t
0.55:1
0.041
o.:t3
0. 05
72
1 o2r>
i.0(;7
0.518
0.045
31
0.05
711
1.0 2; J
1.074
0.48:1
0.015
0.20
0.0(1
74
1 1121
I.OSO
0.447
0.0 »5
0.28
0.00
7'>
+ 1.010
+ i.osr>
—0.412
+0 1 15
— 0.20
—0.07
70
l.ois
1.000
o.:i7(i
U.(>4(i
0.24
0.07
77
l.oir,
1.001
o.:ui
0.04(1
0.22
0.07
7S
l.Olt
1.007
o.:j(t5
0.04(1
0.21
0.08
7'.t
l.oi:{
2.000
0.2(10
0.04(>
0. 1
0.08
MO
+ 1.011
+2.oo;t
_0.2:!:t
+0.()4C)
—0.17
—0.08
Hi
I.OIO
2.005
0.1' ,
040
0.10
0.00
H-i
l.oos
2.007
o.l(i:i
0.04(1
0.14
0.00
8;i
l.ooc,
2.007
0.128
0.((47
0.12
0.00
84
1.00.")
2 (»0(1
o.oo.t
0.047
0.11
0.00
«:>
+ l.oo:{
+ 2.005
— 0.057
+0.047
—0.00
— 1.00
Sli
1.002
2.(101
—((.022
0.047
((.07
1.00
H7
1 IMM)
2.(M(-j
+ii.oi:i
0.017
0.05
1.00
8S
O.It'.tS
2.000
0.048
0.047
o.o:>
1.00
m
0.007
1.007
082
0.047
— 0.02
1.00
Jto
+ 0.00.")
+ 1.00:1
+0.117
+0.((47
0.00
— 1.00
01
O.'.IO.!
l.oso
0.152
0.017
-f (1.02
1.00
!t-2
0.002
l.osi
0.1 8(!
0.017
0.03
1.00
',t:t
0.000
1.07S
0.220
0.047
0.05
1.00
04
o.oss
1.0-2
0.254
(( 01 ,
0.07
1.00
05
fO.OHT
+ 1.0(15
+ 0.287
+0.((47
+0.00
— 1.00
Oil
0. OS,-)
1.!I5S
o.:i2l
0.047
0.11
0.00
07
(t.OS»
1.050
0.;i54
0.047
0.12
0.00
OS
O.OSO
I.oi:!
o.:i87
0.04(;
0.14
0.00
00
o.oso
1 . o.i;{
0.421
0.04(1
0. HI
0.00
lIMI
+0.070
+ 1.021
+ 0.45:1
+ 0.040
+ 0.17 ■
— 0.08
101
0.077
1.014
0.487
0.040
O.IO
08
1112
(t.07«
l.OOli
0.510
0.040
0.21
O.08
lo:!
0.074
l.sit:{
0.551
0.040
0.22
0.07
lot
0.072
l.ss-2
0.582
0.040
0.24
0.07
10.")
+0.071
+ l.S(iO
+ 0.(114
+0.045
-fd 20
—0.07
10l>
o.oc.o
I.S57
0.(145
0.045
0.28
0.00
107
o.oi;s
is (4
0.(177
0.015
0.20
0.00
los
o.ot;7
>.S20
0.707
0.045
0.31
O.05
loo
O.Oli,")
1.S15
0.7:!7
0.014
0.33
0.05
no
+o.oi;4
+ 1.S(I0
+ 0.708
+0.014
+ 0.34
—0.04
111
0.0112
i.7sr,
0.708
0.044
0.30
0.03
112
0.0(11
1.770
0.827
0.044
0.37
0.02
U.i
0.0.")0
1.754
0.855
0.04:1
0.30
0.01
114
o.o:>s
l.7;is
0.SS4
0.043
0.41
0.01
lir.
0. o.")(;
+ 1.721
+ o.oi:{
+ 0.043
+0.42
— 0.00
11 c
0.0;")")
1.704
0.042
0.042
0.44
0,80
117
0.0.")4
l.lisii
0.070
0.042
0.45
0.S8
UH
o.o.w
l.ddH
0.007
(1.041
0.47
0>7
no
+0.0;") 1
+ 1.(150
+ 1.025
-\ 11.041
+ 0.40
_-0.h7
^:a?
m
li
T
Itil
T II K O U B I T OF UK A N US.
d>.
d-K
&/.
'■'('
''('
«5|'
a
0.
de
edn
d(
ve
e-/n
{■Mt"
-fO.lC.O
+ l.r.:il
+ l.o:.i
+ oil
+0.f)0
— o.sn
121
0.1»4!»
l.i'.ll
1 1.077
0.040
0.:-,l
o.sri
\.i-2
0.!»H
1..V.IJ
l.lol
0.040
:);l
0.M4
IJ3
!t47
l.ii71
1.1211
O.O.'I'.I
0..',4
O..H.J
124
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1.1. -.4
o.o:i<.)
u.r)i;
0.^*2
I3.i
-f0.'.t44
t l.'i.lO
+ 1 . 1 ><0
+o.o;iH
+».:)7
—O.Ml
I2*i
O.DI.'t
l..')0<.l
1.20.-)
o.o:is
\).M
{).m
127
0.1I42
I.4SS
I.22'.t
0.0.17
u.r.o
0.7'J
I2S
O.'.ltl
l.4i:ii
1 I.2...I
0.0.17
U Wi
0.78
1 1't)
O.'.MO
l.44:i
\ 1.277
o.o:i7
u.f-.a
0.77
l.'IU
-fl».'.t:il»
+ 1.421
+ 1..IU0
+ 0.0.111
+ 0.04
— 0.7li
III
0.'.t,l7
I.:i!i7
1 .122
o.o;in
O.lili
0.7.^
l:iJ
O.'.KKi
l.:i74
1 :IU
o:ir)
0.07
0.74
l;t;j
O.'.KIII
1 .l.-iO
l.:u;7
o.o;t4
O.II.S
0.72
III
0. !i;i.")
l.;i27
l..|M>(
O.O.'II
0.70
0.71
i.i;.
-|-o.!i:i4
-|-I.;i02
+ l.lll!»
+ o.o;!;i
-|0.71
—0.71
I;ir.
O.K.I.I
1 277
l.4:;it
o.o;i,'i
71
0.70
l:i7
0.'J.12
1 2.-.;i
I.4.-.I
U.ii:i2
72
o.os
l.is
(>.!t.ll
1.22S
1 470
O.'ll
0.74
0.07
111)
O.'.I.'IO
1.202
I.ISO
O.O.'il
0.7')
oil)
110
+lt.!l2!l
f 1.177
+ i.:)Os
+ 0.0.I0
+ 0.7ii
— 0.04
141
o.'.tis
l.l'il
l..'.27
u.(i:io
0.77
0.0:1
142
0.'.I27
1.124
l..^4.')
0.(120
7S
o.r>2
lt:i
0.!I27
l.lltM)
l..'>r>2
0.(I2S
0.7!»
0.110 1
111
O.llL'li
1 1172
l.'iXO
0.02S
O.NO
0..'.!)
11.)
)-il.'.l2.')
+ 1 1114
+ I..-.1H;
+ 11.1127
-t O.Ml
— o.;,7
Ill-,
O.'.t-.'l
1 Olli
1 i;i:i
0.ll2l-|
(I.X2
o.m;
117
0.!ij:|
O.'.N!)
1 i;2!»
o.ii2r>
o.s:i
0..'.4
US
o.:ij2
o.!ii;2
i.iiii
0.02.'')
O.Ml
o.r>;i
ll'.l
o.;iL'2
(l.'.l.ll
l.r..-.!i
0.II2I
0..S5
u..->i
ir>o
f(l.li2l
fo.ooc,
+ 1.1174
+0.02:1
+o.,s(;
—o.'ll)
i:.l
0.'.t21
0.S7S
l.i-iss
l»2:l
0.M7
0.4!l
ir>i
o.;i2o
0..s4'.t
1 702
0.022
0.M7
0.47
Ia3
0.!M!»
0.S20
1.714
0.021
0.,H.S
0.4:1
l.'.4
oil!'.)
0.7!»2
1.727
0.021
0.M9
0.44
l.i5
-|-0.'.ll<»
f0.7ii2
+ 1 710
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f O.'IO
—0.42
I'llt
0.!IH
o.7;i;j
1.7;-.l
O.OI'.I
O.'.H
0.41
1.^7
O.'JIS
0.704
i.7ii;i
o.ols
0.111
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l.->s
o.'.tn
o.i;7l
1 77.1
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0.<I2
o.:t7
i:i!»
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O.lll.')
l.7s.t
0.017
O.li.l
o.:n;
I no
+ii.:ilil
-f-ll.lil.')
+ l.7'.i:i
+ 0.011;
+ 0.'.i4
— o.:ri
1(11
II '.(!.■)
o..">s-i
1. HI I.I
dill.')
O.'.l;')
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lr,2
(I.'ll.j
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O.il.'i
o.::i
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O.'.lll
0.2!l
iiii
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O.tlll
0.2s
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f-O.'.Ill
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+ l.s:;i;
\ 0.012
+ 0.'.I7
— o.2ii
Dii;
(l.!)ll
(1. (:!.•(
l.Ni;i
O.OII
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0.2»
Iti7
II. '.Ij 1
II lll.'l
I.Mll
O.llll
0.07
0.22
ic.s
O.'.lj.l
o.:!7:i
I.S-,;-)
OHIO
O.'.IS
0.21
KiO
It. 11 I.I
:!»:t
I.HI'mI
O.OO'I
'.H
0.1 '.I
170
+o'.i|:i
+o.:ill
+ l.sr,-,
+ O.I)O.S
+ (I.!IS
—0.17
171
'112
0.2><0
1 H70
0.007
O.'I'.I
O.ill
172
o.;ii2
24!)
l.-<7.^
o.iior.
0.!l!l
14
I7.J
0.!II2
(l.2l!l
I.S70
o.ooi;
0.!l!)
12
174
0.!I12
0.1H7
I.HM2
0.00')
0.!I!1
on
175
+ 0.1III
+0.1. "ill
+ I.HKt
+ 0.004
+ 1.00
—0.0ft
170
O.'.H 1
0. 1211
l.sstl
0.00:1
1.00
0.07
177
o.'.tn
0.0!I4
I.HS.H
0.002
1.00
O.ll'i
17S
'.III
o.oi;:i
I .SS!»
0.002
1.00
o.o:t
I7!»
+ 0.'.l||
}o.o:il
+ I.HOO
+ 0.001
f 1.00
—0.02
Till-: OK HIT OF UK AN US.
160
In the oqiiutions of coiulitioii tea yenrs lias bivn ndoptrd for the unit of time,
ill onlcr to iiiuki- tlic gcMunil valiii.> of tlio cuctKciciitH uh iii'iiily iqinil as possihlr,
iiiul llic lime liiis hi'cii (duiitid from tlio cjiorli 1N;J0,(), in order to liave tlie jiosi-
tive and negative values of t in tlie efinutioiis mon; nearly i»aluiired. To distin--
giilHli theHu values of lie and iii they aru marked with an accent. 'I'iiis unit of
•)
time gives (),HJ)1 1 for the value of ' in arc, wliencu
('li
'I'lie e«|iiations of condition are now formed by putting in the preceding equa-
tions for heliocentric longitude and ratlins vector
6X
67.
f'?. , . {jT.
fu,
h-K = Th + : fit- + . " ,v + T e\n + : , V
vv
oe
t:a.1
O'i
• (Je I'H ve ' «<',T
1^
I'or the coefficients " have beiii taken one-hiiiidr<'dtli the iierturbatioiis of
da' '
loiif,'itu(l(! jModiucd by Neptune, as given in the heliocentric ephunuris at the end
of C'lia[it<r \'. 'I'he corrected muss of Neptune will tlitn be
Finally, 1 n-niark that all the preceding,' coniparistms are made with the helio-
centric epheiiicris as printed, without tiie correction indicated in the colnniu
adjoining it, but in the following eciuations this correction is for the first timo
introduced.
Kijualioiin I if fduilltioii yinii hi/ the ('ornrti:,)is in Loiiyitndr,
I
o.(».),v
— 0.70' «'
— O.IO.V
+0.0:U7t
+0.1 '2W
= + I'M
••>
0.10
— 1.11
-t-o.oi
—0.10
+0.1-2
= + '2 .2
a
o.;ji
- '>..>;»
-|-0.:{:{
-f ().")!
-0.-2-2
= +12.8
4
0.3-2
— 'i.'t'i
4-0. •27
-f0.">8
—0.18
= +11.8
T)
(».;«)
— 2.:Ji
+0.1-2
+o.r)0
— 0.00
= +ll.()
(1
().1!>
— 1.40
—0.01
-fO.:{0
0.00
= + 5.0
7
0.1!)
- 1.18
— O.'.'i)
+o.-,';j
+ 0.-2-2
= +4.8
8
1.41
— ().7rj
— '.'.(iO
—0.7")
+-2.47
= +28
y
l.OO
— ;■).() 1
— 1.08
—0.7-2
+ 1.8")
= + 1.1
i(»
1.07
— 4.01
-1.04
— o.so
+ 1.8.7
= +1.7
11
l.Oi)
— 4.S7
— l.S,")
— 1.04
+ 1.8.")
= — .")
li
o.-;j
— ;{.()()
— 1.04
—0.07
+ 1.10
= + 1 .0
V.i
o.:)4
— '.».•.'()
—0.71
—0.78
+0.82
= +1.4
11
().;m
— 2.10
— 0.08
—0.81
+ 0.81
= — .0
1.-)
0.7:J
— 'IS{\
— O.S-2
-l.Ki
+ l.o:J
= — .9
i(>
o.:n
- 1 ;{!)
— o.;u;
— o.(;o
+0.40
= - .2
n
0.."»()
— '.'.O")
—0.47
— 0.0"»
-f0.70
= — .4
IS
o.:n
— l.:5:J
— O.'JO
—0.0")
+0.44
= + .0
1!)
0.7:)
— '2.M
-0.:J8
-l.:]4
+ 0.8-2
= - 1 .0
!"1
166
T il E O K H 1 T O V U R A N U S.
20
0.56 y
— 1.89^,
-0.22,V
-1.04.S«
+(V58^m'
= — 0".9
21
0.5H
- 1.92
-0.13
-1.12
+0.55 '
= — (» .5
22
0.;J8
— 1.15
+0.09
—0.74
+ 0.-24
^ - .2
23
o.;}8
- 1.11
+0.15
0.73
+0.21
= - .3
24
1.15
- 3.18
+0.65
—2.13
+0..")0
= + .6
25
1.13
- 2.81
+ 1.!1
— 1.91
+0.19
= + .5
26
0.56
- 1.33
+0.62
—0.89
+0.05
= +0.1
27
1.12
- 2.54
+ 1.38
— 1.68
0.00
= + 2.1
2S
0.56
- 1.21
+0.75
—0.79
—0.04
=.- 0.1
29
0.70
- 1..V)
+ 1.09
—0.97
—0.10
= + 1.6
»0
1.11
— 2.19
+ 1.73
-1.32
-0.-23
=-+ 2.2
31
1.09
- 2.04
+ 1.81
-1 14
—0.-29
= + 1 .3
32
0.5(5
— 0.!>8
-H»-06
-0.51
—0.18
= + .7
33
0.55
— 0.!>1
-I i.OO
—0.43
—0.21
-+ 1.1
34
1.08
- 1 CO
+ 2.0'<
-0.71
—0.46
= + 1.5
33
l.(i:<
- 2.37
+ 3.11
—0.8-2
-0.74
= + 2 .(»
35'
1.07
- 1.45
+ 2.08
—0.38
—0.52
= + .4
35*
i.oe
— 1.33
+2.10
- 0.-22
—0.55
= + .8
30
1.58
- 1.S3
+ 3.18
- O.tiO
— 0.S5
= + 0.3
3-J
1.05
- 1.11
+ 2.10
+0.08
—0.59
= — .8
38
1.04
— 0.99
+ '>.07
+0.-2(}
— 0.59
= — 1 .3
39
0.69
— 0.58
+ l.3()
+0.27
—0.39
=z + .8
40
0.69
- 0.51
+ 1.33
+0.38
—0.39
= + .8
41
(».<)S
- 0.44
+ 1.29
-( 0.47
-0.39
.0
42
l.Ol
- ().,")6
+ J.88
+(t.si
—0.59
.4 0.3
4:)
0.67
- (».30
+ 1.20
+ 0.64
—0.38
—. — .5
44
1.00
— 0.35
41.72
) !.09
-0.57
=^4-0 .2
45
1.95
— 0..")0
+3.21
4 2,39
-1.12
— — 2 .5
46
2.42
- (>.:n
+3.77
+3.21
-1.3(i
=-- — 2 .()
47
2.42
- 12
+3 ,-,3
+ 3.52
-1.37
=^— 2.4
48
2.39
+ 0.12
+ 3 i
+3.(i9
— 1.33
= - 4 .9
49
1.91
-I- v».28
^2.;i7
4 3.14
-1.07
rrr _ 0.1
50
1.91
;- 0.48
+ 2.15
+3.30
— 1.08
= — 2 .4
51
2.37
+ 0.S3
+2.31
4 ! 32
-1.34
= — 4 .5
52
2.3(J
+ J. 06
+ 2.02
44 u
— 1.36
= - 4 .3
53
2.37
+ 1.32
+ 1.68
44.62
-l.:57
■=.— 5.5
54
2.85
+ l.S(>
+ 1.66
45.(11
-I. (IS
= — 7 .5
55
2,84
-f 2.17
+ 1.22
-] 5.7 1
-1.71
= - 4.1
5(J
2.84
f- 2.44
+ 81
45.84
-1.74
= — 2.1
57
2.8?
+ 2.71
+ 0.K»
45 8H
-1.78
= - 1 .7
5S
2.83
+ 3.0()
— 0.()(»
4 5..SS
-^ 1.S2
= - 2 .0
59
2.82
+ 3.28
-0.41
45.S4
— l.s(;
:-- f .3
60
2.83
+ 3.57
— (».N1
45„si
- l.iU
=- + 1 .2
61
2.85
+ 3.00
-1.27
4 5.78
— 1.98
- + 4 .4
62
2.85
+ 4.21
-1.74
-f 5.(il
-•2.02
== + 4 .M
63
1.91
+ 3.02
-1.42
43,(i9
— 1 .39
= + 4 .0
64
2.85
+ -l.-M
-2.50
45.:;2
-•2.11
= + 6.0
65
1.91
4- 3.3!»
— l.'tV'
4 3.J(i
^1.17
= + 5.6
66
1.92
-f 3.61
-2.18
43.3-2
-1.52
= + 4 .3
67
1.93
+ 3.8-2
-2.40
+ 3.17
l..-)6
=3 4 () .()
68
1.94
+ 4.05
— 2.65
4 2.0s
- 1.(10
.-:- f 7.3
69
1.94
+ 4.20
— 2. S3
+2.80
-1.61
== -( (i .5
k^ ■'«--:
THE Oil BIT OF URANUS.
1()7
70
1.}).-.,V
-t-
4.44^h'
-3.()4'e
-f2.59c,^n
— l.(58,'iH
= + (5".8
71
l.»8
-h
4.72
-3.23
+2.37
-1.73 '
= + 8.1
72
i.})y
+
4.93
-3.44
+ 2.15
-1.7(5
= +9.1
7:J
1.99
+
5.13
—3.58
+ 1.91
-1.78
= +8.7
74
•i.oo
+
5.35
—3.73
+ 1.(54
— 1.80
= + 8.1
7.)
'2.M
+
7.02
-4.N3
4 1.73
—2.30
= +10.(5
7(J
2.54
+
7.34
—4.9(5
+ 1.33
—2.29
= +10.9
77
2.5(}
4-
7.(54
-5.97
+0.97
—2.31
= + 8 .2
7s
2.»»7
+
(5.38
-4.14
+0.47
— 1.8(5
= + (5.3
7i»
2..">!)
+
S.25
—5.20
+0.21
—2.32
= + 7 .0
m
2.()(>
+
H.51
-5.21
-0.15
—2.29
= + (5 •<>
81
2.11
+
7.U
-4.19
-0.51
-1.N3
= + 4 .3
S'>
l.fiO
+
5.5(i
-3.13
— 0.5M
-1.3(5
= + 3 .9
Kl
'Hil
+
9.59
-5.12
— 1.37
—2.20
--+ 1.1
84
2.G8
+
9.!>2
— 5.00
— 1.80
-2.14
= - 1 .s
85
2.17
4-
8.22
—3.92
-1.75
-1.(57
= — 3 .8
m
2.73
+
h).()l
-4.97
— 2.59
-2.01
= — !).0
87
2.20
+
8.78
— 3.(i;j
—2.33
— 1.52
= — 9 .5
88
l.(>5
+
(5.74
— 2 58
- 1.9(5
— 1.08
= -10.5
8!)
2.77
+
11.(51
— 4.04
-3.(51
-1.(59
= — 19 .9
2S3.(54,V
+ 41l.3(i,^u'
— 151.()3^;
414.3(5
+ 1(519.14
— (5N9.ll
151.(53
— (SsD.ll
+ 557, S2
247.23
+ 2(!0.2(5
+ 3S.»5
ri(5.0:{
- 43(;.(t2
+ 122.S8
'I'lic lollowiiifj arc tlu' iipproximiifc normals to which these r()!iatioiis fii\e rise.
Inaccuracies lieiii;,' delected ill .^cMTal of the e()liatiolls ot" conditinil alter tiiese
normals were formed, they <lo not accurately corr<'sjioiid to those ((piatious as written.
+ 247.2:kA7i — 17(5.03,W =- ( 123'.5
+ 2(50.2(5 —43(5.02 ^. \ 103.2
-f 3S.45 +122.8S =— :t99.M
-I (lis. I.-, —19 (.(50 = -j 2(57 .3
- 191. (50 -|l(i{.13 :.:.— 12S.1
The values of tile u!i!>uov,-n ([iniitities deduced frc.n tlicse ufU'iiials were snlisti-
tnted ill tiie <i|natioiis of conditioti, and a fartiier approxi. nation was made iiy
sohini: tlie eipi ifioiis ;j;i\('n liy tin- residuals. The followiii:,' are the tirst approxi-
niatioas ^iveii liy tiie luMinals, and tlie tinaily roncluded ccnctions
I'rcliiiiiiiiiry. h'iiial.
— 15.00 ' — l:i.U
_ o.;V2 _ o.:U5
_ I.J.-, _ 4.04
10,Vn = f^/(',— I."i3 — 1.33
V, — 3, IS - 3.44
Mass of Neptune ,^l,^ ^,,,<,^.
The final \alues of the corrections Ix'in;^ snhstituted ir the i'(pi!itions Ieav(< the
following; s\st<'m of residuals, or outstanding escesses of tiie oliser\ed loiif^itudes
over theory. Colunni /'(V f^i\es the residual of the e((Uatioii itself; the prohahle
error of which lias always heeii jniliied to he 0."">, whih in c<'!uini'. ?! this residu.d
is dividetl hy/'to oltain the roidiial correction cd' the loii:,'itii(ie itself 'the \alues
of the factor* / are found with tin original (upuitious on pa),'es 159 and 1(50-
Cj?«»
168
THE OKIIIT OF URAXUS.
. of Eq. Year.
f)l
1 1691.0
-0.5
-10.
2 1T1"».'2
—O.S
— 10.
;{ IHS.H
I n'lO.H
T) 17 •»:{.})
(i 17,6.7
+1.5
-f O.M
— o.-i
+ ().()
- 1.0
7 1 7(;!).0
H 1 7S-.>.0
)) ns;t.()
10 ITSI.O
11 nsvo
12
i;j
u
i:.
Ki
17
IS
1!>
20
21
;^2
;j.v
1 7SS.0
1 7Si).0
175)0.0
nin.o
n!»2.o
17!):l.O
17!M.O
17!).").0
1706.0
17!»7.1
22 1S()0.2
2:{ I MO 1,2
24 I s( )•,'.:{
2.") lso.V;J
2(1 ls(Ki.:j
27 ls07.;{
2H 1M0H.:{
20 lson.:J
:J0 isio.:}
;Jl isll,;}
1 SI 2.4
isl:).4
IMI.I
IHI.VI
l^K! t
— 0.2
-fO.l
— O.j
+0.(i
— 1.0
+ 1.2
■fl.7
— (t.5
-0.7
0.0
-0.1
-0.6
— 0.")
~o.(;
-1.4
— 1.4
—0.6
-1.4
-1.0
O.l
1.0
-f-0.1
-0.1
—<».!)
-0.;)
-fO.l
—0.4
o.:»
- (».H
— 1.0
+ 0.2
— 0.;>
+ 0.6
— 1.0
+ l.S
+ U.4
— 1.0
- 1.0
0.0
0.2
-I I •■■»
- 0.i»
- 1.0
- 1.2
— 4.2
- 4.2
— 0.6
— 1.4
- 2.0
- O.l
- 2.0
+ 0.2
- 0.1
- Us*
— o.rt
•f 0.2
- 0.4
— 0.2
— «.H
So. of K(|. Ycnr.
;n
;js
;i!)
45
46
47
4S
4!)
50
51
52
1S17.4
ISIS.4
1SI!).4
1.S20.5
1N21.5
40 1S22.5
41 l,sv';j.5
42 1S'.'4.5
4:{ 1,S25.5
44 1S26.5
1S27.7
1S2S.7
is-,';). 7
1s:M»,7
|s:tl.7
ls:t2,7
lN:!:t,H
ls:n,M
5;{ ls:i.-,..s
54 l.s;{(i.S
55 ls:n.S
56 ls:{,s.H
57 ls:i!).S
5.S l,s|(),S
5!) ls||,S
60 1S4'.'.,S
61 lH4:J.s
(»2 1S44,!»
M IM.V!)
(it 1sk;,!»
65 lsl7,i)
M ls|slf
67 lH4i).i)
6s 1M.V),<>
6!) 1M51.0
70 Is.V.'.O
71 ls.-,;l.i)
'3?
-0.;}
—0.7
-1.4
— 1.4
+0.0
+ 1.4
+ 0.6
+0.7
+2.0
+ 1.0
+ 2.5
+ 2.S
-f 0.4
+ 41
+ 5
+0.1
— 1.4
-:1.4
—0.6
+0.5
-0.4
-1.7
— 0.:j
—0.5
+ 1 .6
+0.7
i o.(;
-0,4
+ 0.6
l.()
+0.2
|-o.:j
-1.2
- 1.2
-0.2
fit
n
— ().;)
— 0.5
— 1.4
— 1.4
+ 1.4
+ 2.1
+ 0.0
+ 1.2
+ 1.0
+ 2.0
+ 0.5
+ 1.0
+ 1.1
f 0,2
+ 2.0
+
0.!)
0.2
0.0
0.6
1.1
— 0.2
+ 0.2
— 0.1
— 0.6
— 0.1
— 0.2
+ 0.5
+ 0.2
+ ().:}
— o.l
+ o.:i
— o.s
+ 0.1
+ 0.2
— 0.6
— 0.6
— O.l
THE O U m T OF U K A N U S.
169
Ni). Ill' Ki]. Yi'iir.
NO i.sC);;.o
ni
H
Vi
IH.-)!.))
+ 0.5
+ 0.2
r.\
IH.V).!)
0.0
00
14
isr.G.'J
—()..■)
— o.-J
1")
is.ys.O
— O.'i
~ 0.1
It;
l.s.-,!).()
1 1).!)
-f o.i
^ •»
1 1
Ihdd.O
- 1.1
— o.i
"S
18(U.(>
— 0.(>
— (),:{
1!)
iw;-,>.()
— o.(;
- O.'J
40. v' -\- 0.1
No. of V.i\. Yi-'iir.
Pl
u
HI
is(;-i.o
+0.7
+
0.4
S-2
isd.j.o
+ 2.-i
+
i.r.
s:{
1,S(!().0
+ 1.0
-f-
0.4
Nt
1M)7.0
+ ()..->
-t-
o.'i
8.5
INIJS.O
+ 0.4
•1-
0.'v»
Hi
l.S(i!).0
-l.iJ
—
O.H
NT
IHIO.O
-o.l
0.0
HM
1.S71.0
-l.'i
—
o.s
W)
187 '2. 1
+0.4
+
o.'i
.\ simple •(limcc at tlic roiirsc of tlio rcsiilimls shows (1) lliat tlnir i>rol»iiliIc
visliic i.s fonsidcialiiy !j;i(at<i- tliaii the (inilmltlu error attril>nti'<l to tli • ciiuafioiis
ot' /.oiuiitinii, hciii},' more marly 0".7 than WS^y ami yet larger in tlic later y<-ars ;
('.') tliat diiriii',' rertaiii periods they are of a systematic eiiaracier. J).irin<,' llie
years I74H to \'>'t'-\ tlw oliservatioiis show a de<ided positive eorreeti'tii to tiie
theory of a ma;,'iiif iide },'reater than we can consider prol»al)le, anionntin;,' to aliont
one. third of a second of time in the mean of Ihadley's two ol)servation8 of 1748
and n.'»:). Ahont ISOO (he correction hecome.s ne<,'ative, and so continues for
'.'0 years with an averaj,'e v;dne <if ahont I". In l^.M it suddenly hecoim s
positive, and so continues until l8;j;l. From this year forward the residuals are
not systematic in character.
In order to sliow clearly the general coursr' of the out-itandin<,' corrections, they
hiiM' h' in <livided into groups, generally including aliout five years ciuh. The
mean outstanding concciion for each group, taken with respect to the vvi iglils
iiidirnted Ity the fa<tors y", is as follows. In the colunui i is shown what the
proliahle error of ihi- residual should he if the wi'ights iissigiicd to the several
equations were strictly correct, ami 110 Kystematio errors wcr- priMiit either in
theorv or ohservatioii.
$«
Y.ar,
m
f
n
ni.V'i
-^
Kt.
+ <!.
n.-)i.i
i
;i.7
+0.M
ntiK.o
--
1
i •-'. ■-. '
ns;!.:^
O.IM
+ <>.«':»
I".!»0.0
;
0.(i'.»
d <»■«<»
170.'i.O
—
O.."),")
t 0.11
ISO •.>.(>
—
l.'j:.
+ <t.l.")
1S( )!>,."»
1.00
+ (!.:il
isio..-.
o.:!7
+0.:j'2
IMI .'.
0.117
-\ 0.',':l
islii,.-)
o.:n
XS^AH
•Xi M..y
1873.
Ynir.
18'.>».8
-I
l.:,o
+0.1 ()
ISJO.7
+
0.01
+0.10
|8:r..'.>
—
0.'.>7
+ 0.00
I8;i!).s
—
0.17
+ <>.i>7
184 4.8
■1
0.1 1
+o.o:J
isio.o
—
o.-Jl
+0.11
is.-, 1.0
....
0.1 1
+0.11
Isdo.o
—
0.1 :J
+ 0.0!>
ls(i.',.o
■f
().:{()
+0.10
ls-,0.0
—
O.'Jl
ri <».n
M
i
'IF =
k
no
THE ORBIT OP URANUS.
|i^
A Miiiplc glance at the* rcsitluiils il hIiowji that tlicy nro much {greater Ihuii the
purely lu'cidi'iital residuals rehulting iVoni tlie theory of least sciuares. We may
divide th(> putisible cause!* of these sytiteuiatic enttrs iutu three classes.
1. iSi/nlt iiiallc h'rroiM of Olitnrnitwn. — These may result from deviation of the
line of ct>llimatiou of the iustruuuMit from u true great cireh-, or from any i»ecu-
liarify of th<? observer which leads to his registering the transit of I rauns ciirlier
or later than that of a hsed star. If we conntare the corrections lUrived fiom tin*
work of ditterent ohservafories as given in the last chapier, we shall hud fre<|nent
cases not only of systematic ditl'trences lietween <he results of diti'erent oitserva-
tories, hut li( tween those of tlie same ohservatDry in two siucessive years. An
instance which particularly attracted my attentiou on lirst prepaiing the c(uii-
ltaris(ms of theory and o!)servation is that of the (ireei'wich observations for ls;ii,
which, as compared with observations at the same ol>seivntory diiring the yem^
preceding and tnllowiiig, seem to be afieetid witii some constiint err.n in U. .V. of
about 'i". I tind that this disc ppaney can be attributed only to the original
observations.
'i. JilrmiM III ihi Tliriiin i-nitiintii il.- ■\'\\i'><v may arise tVotn errors in the jireceiling
theoretical computations, fiom the omission of the terms of th<' second otder \my-
(luccd by Neptune, from the adoption (d' an erroneous n\ass of Saturn. u\- from tb<>
attraction of an nnknuwn planet. NN'ith regard to the probability of tin >e ditl'i rent
siinrees of ernii' it may bi' rem iiKed that errors of C(im|iutation seem pnssibli' imly
in the terms of the seennd order, that the mass (d' Satnru is taken from the
exhaustive disenssiun of the Saturnian sjstem by IJessel, in which an error suffieient
to intluenee tin' theory of I r.uins si mss higldy improlmble, atid that a trans-
Neptunian planet large enuiigh to produce a sensible deviation of the orltit of
Iranus fntm an ellipse in the cotuNc (da eentnry vvimld be too large to have es<'a|ie(l
detection. The ehoii'e of tlie elliptic (liinenls(d' I raiins and N( jitniie is snrh
that the terms of the serond order, due to the a<tion of ^lcptune, can scare< ly
become sensi'tli' willun a eentnrv of tli<' i pm h.
U. I'li'i'iirH ill t/ir nirltiiiM I'l /iirllnim l>i/ ir/iir/i Tlnori/iinil O/tsrrriitiiiH iiri' idih-
jMiri'l.-~l\\ the methoil adopted for comparing tbeorv and olis<r\ation a nnmber of
small uncertainties incident to the iiii[ierfe:tions of the older data id' reduction
iieci'ssarily creep in. In the early observations tlie imperfections arise prim ip, illy
from the niiceriiiint} of the instrninenlal correefioiis, ,iiid tlie (ii>irs in the adopted
positions of llie Inmlanniilal stars, and indeed in nearly ai! the ilata of ritha (ion.
In tlie lale v .irs they arise principally from the great magtiitiwle (tf the correction
to Ho.ivanl's tables, and the cotiseqnent rapid ( liange of the corre<tion.H to the
geocentric epli>ii' rides, which make the dcti rminaJio!; of the corrections J^l sumI
N from theory and obsrr\ation somewhat niieertaiii. l'".rrors jVoni ftii-< source will
uecettsarilv Iw in put of a systematic < haracter. and. in view of tie : possibilitv,
I re!.,'ret not bating been able to completely re-r<(lue.- all tin- observations before
INIO, and to compare all since dircctlv with ephennridt s eoinpnti'd IVom llw
pr.nisional theory. In ordir, however, to test fjie ipiestion whetlnr tliej are
siMsilde, I leixi' prepired an e|iloMneris from tlie pro\isiftnal theorv for the three
relent oi'.positioiis of I'^ii! 2, \h{,} ;[. aiid In 7-.', and compind it dir.cily with
!
Til K 'I U MIT UK I' UA N US.
,171
,
till' ulisrnaUniiN. 'I'hc iMcfiu <i>r)M'«-ti(iMs ill <;('occtifric loii^^itiidc for •;i'(tiij>s of
(il»^<T\ati()iirt ail' ffi'.i'ii in <(iliiiiii»s ("i), riAuimi (I) sliuxviii}.'; ti;r < oriHTtioii givi'V
by till! work of tlie last i'haj)tii-.
<)|i|msiti<jii !S(.I -i
l^>:l .!
")
(••i)
+'r.«
K/'.-r
•i.!>
•i . K
;f ,0
:i.l,
2 »
•.',1 ,
Mcjm I -.Mi) f'i.()i
0)
i;;>
-1 '->".»>
-r •/'.«.
•J .;{
2 ,:»,
t.' 1
•,' 0,
;{ .;>
•> .*».
•J .)i
• » . « ,
-f -J .ti.'i
-l-i.5!
H7'i
(1) (•-')
__H.;i — H.l,
-7. a -7.5,
-11 -7.7,
—7 .H -7 .«,
— 7 .(ij —7 .82
A systciniitir ilHI'i r iicc of 0" IC) woiilii >i riii to lie indiciiti'd, iiiul on nrcomit of it
II con'cctiiid iif II", 10 was iiiipl'i'd, to !ii>' i»;iiij>aii,'^oii.'> of tlic la.-l frw jrais in
I'liniiid^ titr »'i|iiati!>iis I'f t'diiihtiiiii.
Ill uiu of tilt' po.ssiliility of >yst(!iialir i'rri)i< fmni tlii-. .soun-c it may he coii-
Kiili'icd that too f^rcat rclativi' wii-^iii !ia,s In t u ;!s,siirii(.(l to the rtsult.s of ilic later
oljscrvatioii.H. If tin- ii'siiliials aiisf froiu i rrois of coiiniarison and of (Ihoij, tliiir
jirohahli' Jiia;,'iiituili' is iwarly as j.'i-('at ai iuir ipiu Ii a,>. at aiintlii-r. It may llicrc-
ton- 111- niti'i'cstin;; to iiKin'.n' what r»'s<ik wi- vlnmld |,'rt it", iiistriul of asNi;,'iiiiii{
Muh ditt't'irnt wci;;!!^ to llir niniiiaiiMHis at didVniit ipoclis, «c M)n<,'li! nnh fur
llic l)i'st jjrncral a<.:iri".ii('i)l willi oliscrx.itiuns iliniu;; the jicviud ilir jilaiii't lias
lici'M ulisfTMil. Till' jKcri (liiij; >', >ti'iii of Diraii rrsidiiaU will riiahlc ns to liisciiss
tills ([ncstion i|iiiti' ra'-ily. In llir Hist solution wi' sliall rcji'i t the rrsnlts from
I'lamsti'ml's olwcrvations, Mwinf^ to tliiir assurni Mn< 'itaintj, and thosr fmiu
1,1' Monnicr's of \'l\'.). nwjn;;; to iIh' ]iii<..ilil<' nia!;!ilJ!i-!iii<iit ol' his ijiiadnuit, I'lic
rijuatioiis fioiii ihi' rciiiaiiiiiig n siilna'-, uill In- \\u (ollnwiiig:
l.(),W
— 7.()(V'm'
4().s,w
\\.le,Yn
-o.a,vy
■=r-.. 4.:r.-i w
i
I.I
-•).0
~ .'.0
- K.S
4 1 S
(►.IS
I
11
-4.1
l.»
- 1 (i
r i.<;
— f o.(i-,>
I
i.l
-:!.!»
-(>.()
"•.'.0
fi.'~'
r--^-0.iM
1
):i
-U
-f-O.I.)
-t>.l
fO.o
= -1 .'.>;■)
1
M
~'iA\
-Hl.;J
- I.M
o<»
=. — 1 (10
•I
1 1
-i:i
-1 1.7
1 ;i
.»
= - ;(7
I
1,1
1,7
* 'H)
o.l
tl.)
= - 0.:!:
l
1,0
11
ri\
(0.1
-tl.O
ss - 0,:i7
I
1,1)
,v
-) l.H
1 0,0
0.(t
ass -•{ 1 ,."iO
•»
10
0.0
1 1.4
f 1.4
- 0.:,
— iO.OI
:)
o;j
ho..-,
1 O.H
4»M
0.5
= -o.'j;
:)
(•.:»
f (>,!»
^01
,'<•
0(i
r^ -0 17
:i
!.(»
1 1 (
— «ii
-i I.IJ
-0.7
= 1 O.H
:»
10
! 1 '»
-ft?
f Mi
- JI.M
=:- -0 •.>!
:t
10
4*^.>
1.7
-fl.l
(».«>
= on
;j
10
fiU
- 'i.O
f l>. 1
0<l
•«-^o l;{
:t
11
, :1.7
-11
-0.4
0.11
= +0 Mi
:i
1.1
-; i.i
*>40
~l.i
-OM
= —0 .;»l
a
17-2
T II K O II It I T O F V K A N V S.
(living tlios«> uinofcon cqtififioiis «>i|iiiil wci^'lits, we liavr tliv second of flio
fi)lIowiii<,' soliilions, anil tiic second »»f the series (if residuals tiie Hrst eorres-
pDiidinj,' to liie |triinitive Nolntion. Solvin-; liieni a},'ain and assijrninf,' tlie wcijjlits
attaehed to the respective e.inations, wiiich 1 jndyo to he inose to wliich th<'y
are entitled when a liberal allowanec is made for systematic errors of observa-
tion an<l of comparison of the(iry with ohservations, nddinj? also the ecpiations
given l»y the observations of I'lanistead and he Moiniier, which are us follows;
^11
111
IC.iU) 0.0,W-O.Wn'-().U*e 4 O.OfS^Tt 4 ().UV= -0".-,; /, .,i,^ ; AVt, 1
I'M)
O.,' -\.i —(),;{ -|-().'.> -jo.-i ~ ^2 .'2 I
we have the second solution, and the third series of ri'siduals.
6*e, - (r.;!<)
i*ni — (» .;w
i*e —(>.:};)
efi^n I .'i.j
Ay _ 1 .()-2
m'
Vear.
1 a r il
^J
I 1) » 7 II
lU.SIIH M.S.
(-)
-0.21
-0.19
-0.1 -)
+0.1!)
—0.1!)
a;
1
ii;*)i.ii
-10.
^11.
-I-.'.
ni.vi
-10.
- !).
~ 7.
n:)i.i
+ :U
+ 0.0
+ 'i.O
IKiil.O
- 1.0
- '.>.()
— 1.!)
ns:{.:{
— O.IS
— o.:m
- 0.17
17!K).(»
-f o.d?
( <).!):{
+ O.H!)
n!).V(»
d.,').".
— 0.0!)
— 0.18
|s(i-,'.(»
- 1. •.'.'.
— 0.7S
- O.MT
iN)(i.:»
~ l.(M>
— ().(\!)
- 0.73
l^'KI..'!
- o.:J7
— 0.10
— 0.05
Is 11.:.
— ().:n
0."J(i
— O.'iH
isl!)..-,
— o.:t7
— o.;U
- o.;n
INJ4.S
+ l.:.()
+ 1.4(1
f 1.41
1.S-MI.7
-f o.!)l
-i- 0.!)0
"i- o.si
is:{:).-.»
— 0.'i7
- o.4;j
— ().4fi
1n:$!).s
- 0.17
— 0.5(i
— 0.48
1S44.M
-f O.U
— o.:Jl
— 0.18
|S|!).«t
- O.'il
— 0.70
- i)M
|s.")l.!>
- «.I4
— i)M
— ().;}«
IsiiO.l)
— M.l:{
— o.*i:{
- 0.14
|s(i,-,.(»
+ *.:*«
-f o.io
+ OM'i
lHl((M
— »:*1
-f O.su
-1 U.t4
ir
TIIK ORBIT OF URANUS.
173
It will 1)0 socn th;it the effect of these changes of weights is, that the older
observations are a little better, and the later a little worse represented. I concuivc
tliat our choice must lie between the first and third solutions, tiie first being the
more probable if we conceive the outstanding residuals to b.; due to errors of
observation only, and tlie third if we suppose; tlieni e(pially due to errors of com-
putation. On the whole, I consider the mean of tin; two to ho. about the most
probable, and this will giv(! the mass of Neptune very near the rouiul nu.aber
1
i;>ioo
which will be adopter! as the definitive value. T!ie definitive corrections to
Elements III (p. i>'J) will then be
be' {\X-W)
h (1N.>())
- '.V'At
- li.4.>
- 4.44
- 4.1i
- O.'i.j
- o.i:n
f'firi'trlioim la l/ii Iinl'niutiua and A'w/c.
These corrections have been derived entirely from the n\oderu observatiems, the
ancient oiu's lieing too micertain to add anytliing to tlie weight of the re-sidt. The
mode in which the correclif>n to tlie latitu<le el' the \in)viNi(inal epiiemeris has been
conchuleil from fhi- (ibservations ims been sufliciciitly explained : it is only necessary
to add tliat the immediate vesidts froiii tlie data of the preceding chapter reepiirc
two corrections, uanu'ly :
(1) A correction to the theoretieiil latitude' fi.r the change in the adopted mass
of Neptune. The value of tliis correction, as derived fioiu the data of ('liapt.r \',
is witii sulHcient aiiproximatiou
Jni = ()".'2.")7'cos(/.
(•Jl A ceunclioii to the observed latitude on 'ccouiit of the difference between
the oblitpiitv (if tlie ediptie adopted in tlii' various epliemerides compared, au<l that
(if UiDis.u's Tnhhxilii .Si,/,;/, which liaving been aelopled in the tlii'or) should Ix;
used tliiiiugl!"ut.
Applying tin- corree tiim {'J) (l)lo all tlie observed bititudes, we hiive the
fidlowin<i corrr, tious to the latitude of the provisional epheiiieris derived from all
tlie oliscrvafi.-iis of e.uh opiiositiou since HSl. The third (olumn gives tlur
number of ob-^i rvatioiis in declination, 'i'liesc nundters may, however, in somo
cases be iuiu luiat.' Tlie fourth and fifth < olinnns givu the sine ami cosine of the
ar-'umcnt nt latitude, to be used in forming the eepiations of condition.
174
THE O 11 B I T OF URANUS.
in^i
II ii
V^r.
¥
No. of oUi.
hill H
i'<i» II
Ytfiir.
V
Nu. Ol' ullH.
•lu u
vol i
17M2.0
—(1.4
21
+ 0.31
f O.!!.'.
1s:jo.7
+ 0.3
4S
— 0.M2
—0.57
n.s;i.o
— ;i.'>
13
0.3S
o.o:J
1 S3 1.7
(-1.0
23
0.«7
0.50
17M4.0
i7s.').0
—0.1
13
10
0.5d
0. so
O.Sj
is.ii 7
Is;t;t.7
+0.0
-to.n
20
f)4
—0.90
U.93
—0.44
0.37
I7SM.0
— 1.0
5
+0.71
+ 71
1.S34.7
+ 0.3
02
0.95
0.31
17H!t.O
H-(l.!l
1 ,1 i>
«
0.77
o.r,4
1S3.-I.7
+o.a
71
—9.97
—0.24
1 ilMI.O
17',ll.0
-1 •-'.(>
-1 l.U
4
7
0.s|
O.St;
0..'iS
0..-.I
ls.|(;.7
ls:l7.S
+ 0.1
0.0
13.'i
154
0.98
0.99
0.17
O.IO
li'.fi.O
-|1.S
:i
1 O.'.HI
+ u
Is.lS.S
— 0.-2
1S2
— 1.00
—0.03
I7!l,».0
ntu.o
-f.'.'J
3
0.03
0. '.I.I
o.:(7
30
is.m.s
1S40.S
—0 4
—0.3
142
lOli
1.00
0.99
+ 0.03
0.10
Kitr.o
17!t(;.0
4<l.7
-1 .•(.<•■
7
4
(0.07
O.'.III
4 <'.-•■-'
oil
1S41.S
IsIl'.S
—0 7
—0.7
lol
li.'i
— 0.9.S
0.97
+ 0.17
0.24
17'.I7.(»
4 .•i.i;
3
l.ou
fO.II-l
1SI3.S
—0.0
N.S
0.95
0.31
is(m.2
IMIM.-.'
0.0
+ I.-J
i
•-•
|0.!IS
0.07
— -.'l
o.'ji;
isll
ls|.-,.0
— 1.0
— (I.e.
H7
—0.93
0.90
+ 0.37
0.44
lso-.>.;i
1 1-^
13
0. !l i
0.31
ls4r..o
— 11
02
0.S7
0.50
lH(ir>.;t
iHO(i.:t
-l-l.l
— 1.4
3
r.
-t(>.S3
0.7S
— 11. "ii;
o.i;3
isn.o
|sls
—1.1
— 0.0
CO
.'>(;
— 0.H3
0.70
+ O.51;
0.112
lso7.;i
-fl.C,
d;
O.Ti
0.1,0
Is 10.0
— 1.0
■M
0.74
o.(;7
1X(IS.,|
+ I.S
n
1 <t.r>7
—0.74
Is.Ml.'.l
—0.7
4l>
— O.C.O
40.72
0.77
l.s(i'.».;t
+ l.;i
•1
O.l'.O
(ISO
1 s.-, 1 .
— 1 . •-•
3."»
O.IU
IMIO.,!
4 3.7
K!
o.:>3
O.S.)
1S.-.2.0
— 1.3
49
0.59
O.Hl
Hll .!
-|:i.4
11
t (.-,
0.s!(
Is.-,:!.o
— l.S
4S
— 0.53
+ 0.S5
o.ss
Mli'.4
4;i.(;
•;
!>.:;><
(1.03
1 s.i 1
— 1.0
47
0.47
lsl;i.4
4-J.»
t;
O.JO
0.01)
Is, -,.-1.0
— 1.3
40
0.41
0.91
|HU,4
-1 •J.S
i.t
f o.-.'i'
— OS
Is.Mi.O
— 1.1
41
—0.34
4 (1.04
0.97
1M|.>.4
■l:.'.!'
ii;
It
0.00
Is.iS.d
— 1.7
(;.')
0.20
isic.i
-|l.:j
H
o.im;
1.00
IS.iO.O
— !.7
fiii
0.19
0. 9»
1HI7..'.
-f.'.4
13
— III
— 1 no
isr.d.o
— 1.0
M
—0.12
4 0.09
1 110
isis.r>
+ I.J
5J2
0.110
1.00
lsi;i (1
—■'-.i
41
0.05
1 Hi '.!.;,
+ 1.7
7
0.17
0.00
lsr.:!.o
--1.0
.'^.2
40.02
1.00
lH;!l».r>
+ 1.0
— I'l
—0.07
lsc,:(.(i
— 1.7
H3
+ 0.10
+ 0.99
lx-.'l.'>
+ l.:i
10
::i
O.O.'i
Isr, 1.0
■>-2
39
O.IS
0.99
l>s-2-2.:>
+ 1.S
7
0.3S
O.OJ
lsi;.'i.(i
— \.l
37
0.25
0.97
\x-2:t.:>
+o.r,
7
—0. 1.^)
— O.Sft
l>-r,c,.i»
—0.9
72
+ 0.33
0.40
40.05
0.92
isoi.r.
— 0.-2
11
o.r.i
o.si;
1 si; 7. 11
— 0.(>
«3
lH->:,.:,
+ 1.5
5
0..-.7
O.si
ll'.CS.l
—0.0
32
0.47
O.SS
IH •.'(;. 5
+ 1.-J
7
— o.r.:{
—O.TS
lsr,!l 1
— o.s
(i.'i
+0.54
+ 0.S4
H:J7.(;
-1 1>.0
■I
I'lS
(1.73
IsTO.l
—0.4
32
o.i;o
o.so
Is2s.(i
— l.h
7
0.73
o.r.s
IS71.1
—0.7
21
4 0.(;(i
0.75
ls2".t.7
i
+(!.!»
9
— 0.7S
— O.f.3
Is7:J 1
—0.4
47
+ 0.72
+0.(19
It will be rcmomlicrcd tliut the oliscr^'orl (Iccliiiiitions have, ns for 11s imssililc,
liccn reduced to .Viiwers' .standard. \N'e li.ivc no |Misitive proof that this »<taiidard
is I'orrcct. If it Ik; affected hy a coiistaiit error, tlie residt will he that tlii' orhit
(if the planet 011 tlic celestial sphere, as didiiced from oliservatioii, instead of hein^
a i^nNit circle, as wa know the real orhit to he, will he 11 small one, and the com-
jiarison of a uniform series of observations evteiidin^' tliro!i;:h an entire revolntiim
of the planet, after niakin;; tlie be>t correction to tlie position of the orhit, will
leave a constant residual. Now, we can best ihtrriiiine lliis residual by includinjj
it us uu unknown ipiantity in oiir eipiations.
THE ORBIT OF URANUS.
1
i->
Aj^uin, the error of the stnndnril is not nrccssnrily coiistniit, Imt miiy contiiin a
term iiriiporliuiial tu tliu tiiiii', arising from erroneous proper motions of tlie
stundaril stars. 'I'lierefore, imstead of siipposing tlie residual eoiiMtaiit, we shall
suppose it of the form a -j- hi. Kaeli observed correction to the theoretical latitude
will then give the cqiuition,
bin «.^<^ — cos uiphO -\- a -(- h 7'= h^i.
To facilitate' tlie solution of these e(|natioiis they have been di\ided into groups,
each group usually comprehending three oppositions, and comliined into a single
etpiatiou multiplied hy such a factor as would make its proltalde error half a second.
'I'Ik! factor hy which the correction of the latitude is multiplied in the equation is
the same with tin; coeflieieiit of ii. The year 1H4().0 is taken as tin- epoch for A.
'I'hus we have the followin*':
KglATIciNS or I.ATITI UK.
t)iitf:< of o|i|iniiHlnnii.
No. of op|>.
KlIIMlllMI.
ITMS.O-S.l.d
4
i^f
— o.'.ifi'y 1 1 .On
— n.fifc
: —\.r,
I'SS.O-'.II.O
4
OS
— o.i; ^ 1.0
— O..W(
.. 4 1.1
I7'.>2.0 Itt.O
:i
o..">
—0.2 -|0.5
—0.2
^ -lO.!!
17!l.">.0 !I7.I
;i
a.:!
0.0 4 ».">
—0.2
. p».7
isno 2 o-.> .1
;i
(1.0
■fO.IJ -fl.O
—0.4
0. 4 1.0
lso.''i,;t 07. :l
:!
o.s
-1 0.(1 ^ 1.0
—0:1
= 4 "7
|HOM.;j 10. :i
;i
0.(1
-I 0..S 1 1.0
— o.;i
- 4 2.(1
isii.:t-i.: J
;i
0.4
-1 0.9 1 1.0
—0,1
- 4:1.1
isn.»-n;. J
.•t
0.2
• tl..'') -11.;^
—0.4
: f.TO
isi7.(-i!i.ri
:(
—0.1
• fl..'-) +1..'-.
— o.:i
1 4.S
1S20..'. •Jl'..''>
;:
— o..'>
4 1.4 4 !.•'■.
— o.:i
- 12.0
is-.';i. .'■>-•-'.■)..".
:!
-~0..'>
■1 O.ft 1 1.0
—0.2
>a 1 0.(1
ls-_'r,,.'i I's.c,
>*i
-0.7
-1 0.7 i 1.0
— 1
- I 0.(1
1 s I'll, r,;; 1.7
•■t
-1.2
.|0.S -1 l.Ti
—0.1
- 4 1.1
ls:!J.7 :;i 7
:i
-^2.S
-|l.l -|:i.o
—0.2
-^:-\\.^
Ih:-,;-, 7-::7.s
;{
— 2.;»
1 O..'. -) .'1.0
—0.1
-. -, o.:i
|s:js.,s-io.s
;i
-;! ()
—0. 1 +:i.o
0.0
m, !)
isil M-|;;.s
••1 — -J.'.i
-0.7 4:1.0
4(U
— — 2:1
isu.s.ic.s
.!
—•-'.7
-l.:i -1 ;i.o
4i».2
= -2.7
1S47 S-t!M>
•>
— 2. «
— 1.!) -1:1.0
4 o.;i
: — ;i.o
\<A).'.) :rl.'.)
•»
— l.'.l
— 2.:i -f:io
40.4
=. — :j.2
ls.-,:t.i( aU.W ' ;!
— 1.1
—2.1! 1 :i.o
4 0.4
= —4 1
|m,m;.!( ."I'.i j :i — o.s
--2.0 i :i.o
4 0..-.
= —4.5
isr.o i;-.'.o :; —n.-l
— :;.o 1 ;t.o
-• (I 11
; — r).s
isc.i.o i;.'..o :! 1 o..'!
— :i.o 4 :i.o
•io.T
= — .^.(!
iscc.o r,s .1
1 I.-'
— 2.S 4:1.0
4 o.s
— — 2 4
isc,'.).! 70 •_'
^ 1.1
— 1.(! 42.0
-1 0.(1
=r^— 1.2
1S7I.I 7i' 1 1 -•
-i i.«
- 1.4 I •-'.<>
4 o.(i
^:— l.l
Treating these eiiiiatioiis liy tlir iiictlnxl of least s([iiiiiis, we find the normal
e({uations
If It n 'I X
():5.C.0,^./) + {\.\ruf„\0— rri.li),,,— l.OSAr-. 4. t>s.;{:l
(;. l.>.^<^ -I (i!»..'i!>r/»V) — .V.MiO.e — 1 I. :.■)/< I lll.d'i
li
'it
1.O8A0 - 1 l.'v';"»</«N« -j- H.\K„t ■{■ i.l\)h
•i'.i.iW)
11
no
Til i: o It II I r o I'' II It \ N II s.
ir
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178.
THE ORBIT OF UUANUS.
CHAPTER VIII.
COMPLETION AND ARRANGEMKNT OF THE THEORY TO FIT IT FOR
PERMANENT USE.
In the preceding discussions the terms of the second order due to the action of
Neptune have been neglected, the elements of Uranus and Neptune being so chosen
that these terms can scarcely become sensible within a century of the epoch. But
this very choice will make them larger in the course of centuries tlian if mean
elements had been chosen. They will be most sensible in the case of the great
inequality of 4300 years between Uranus and Neptune, an inequality which will
make centuries of observation necessary to an accurate determination of the mean
elements of the two orbits. The uncertainty arising from the great inequality is
probably of the same order of magnitude with the omitted terms of the second
order, and, such being the case, the theory would really be made but little more
accurate by the addition of those terms. 1 conceive, however, that the theory will
be made much more satisfactory by the computation of at least the largest of the
terms in question, if only to arrive at a certain determination of their order of
magnitude, and of their effect on the planet during the period in which it has been
observed.
The term in que lion, being of very long period, may be most advantageously
treated by the method of variation of elements, more especially as it has in the
theory been already treated as sucii a perturbation. The largest of the pertur-
bations in question are those of the mean longitude which arc multiplied by the
square of the integrating factor r, which is nearly 51, but which also contain the
eccentricities as factors, and those of the eccentricity and perihelion which are
independent of the eccentricities, but are multiplied by only the first power of j;
Tliese terms will probably comprise nearly or quite nine-tenths of those arising
from the term of long period.
Let us begin with the perturbations of mean longitude. These are given by the
integration of the equation
-^^ = — ^m'ati' \ r/.-, sin (2Z'— ?— tt) + c% sin QlT—l—n') \
l\ and /i-j being functions of the ratio of the mean distances, or a. If we integrate
this equation, supposing all the quantities in the second member except /' and I to
be constant, and these two to be of the form nt -(- f, n and e being constants, wo
shall reproduce the principal term of long period already found. But in the
second approximation we must suppose all the elements variable. It is not, how-
ever, necessary to take into account the variations of «, n, and /r, bccau:,c these are
THE ORBIT OF URANUS.
179
fi within the brackets will thus become
of a lower order of magnitude. The perturbations to be added will be those of
I, I, e, e, 7t, and n.
The point from which the longitudes are counted being arbitrary, we shall take
the position of the perihelion of Uranus for 1850.0 as the origin, and put, as before,
g for the mean longitude '<f Uranus counted from tliis point, and let I' represent
the mean longitude of Neptune counted from the same point. The terms of
fl
de
eky sin i2l'—ff—bn) + eV.-, sin {2l'-g—{n'—7t))
or, if we put
<;' sin (ti' — 7t) = h'
e' cos (ti' — 7i) = //
and notice, that to terms of the first order we have, sin ^,n = hn., cos Sn = 1, we
shall have
^~ = — 3w*'a»r \{cl; + Jc/,;) sin X— {ckM + h'k,) cos iV^|
differentiating the quantities, of which the perturbations are to be considered with
respect to the sign b, we find for the terms of the second order.
S J, = — 3m'a»" I (/.\Se + /.-j-V.' + 7t7.-jSiV^) sin iV
d£
-|-((eZ;i + Jc'h) hN—eki^Ti — k.,^7i')cos Nl
We have now to substitute in this expression the numerical values of the quanti-
ties within parentheses. Those of the perturbations of Uranus have already been
given in Chapter III, but it is necessary to diminish them by the factor 0.145* for
the altered mass of Neptune. Those of Neptune are taken from my investigation
of the orbit of that planet (p. 38). The mass of Uranus there adopted is ^j^-^-^^
while the investigation of Dr. Von Asten,t from the observations of Struve and
others, shows it to be tttj Jjt^. The perturbations are therefore diminished by ^\.
In accordance with the system adopted throughout both investigations, constants
are added to all the perturbations to make them vanish at the epoch 1850.0. A
terra is also added to make ' also vanish at the epoch ; this corresponds to the
dt
constant which ought to be added to hi. The numerical values thus obtained, are :
* This factor was adopted before the niasa of Neptune to be employed had been finally decided
upon. Hence the difference between it and that in the preceding chapter,
f Mdmoires de rAcaddmie de St. Tdtcrsbourg, tome xviii, vii sdrie.
J|
180
Till ORBIT OP UUAJiUS.
8N= +
c57t = —
^C = -
Ih' = 4-
hfc = 4-
/.-, = — 1.234
k^ = -j- 0.452
h' = 4- 0.00695
Jc = — 0.0048G
72G0" sin N— 6658" + 4".26<
414 smiV+ 380
414 cosN— 165
120 sinJV— 110
120 COS.V+ 48
fin
{ft
Substituting these values in the expression for 5-,.2 and integrating twice, we
find, putting b for the coefficient of the time in iV, of which the value, taking the
century as the unit, is -f 0.1472, and putting Tfor the time in centuries.
il= „ ni'av^
|(411" +
1 no" 8 '»" 'X
- J • + 2".6 T) sin iV + (1837" + 'Y -51."4 ^ cos N
— 109".3 sin 2iV — 5".5 cos %N \
— 16".5»j'a»t¥4-c7'+c',
c and c' being the arbitrary constants of integration, which are to be chosen so that
both U and its first differential coefficient shall vanish at the epoch. Reducing to
numbers, we find
5? = (140".70 + 0'.327') sin iV
+ (232.60— 6. 377') cos iV
— 13 .60 sin 2JV
— .70 cos 2.V
— 0.037^^
+ 34 .27 T
— 46.76,
the last two terms being arbitrary.
When we carry the perturbations of the eccentricity and perihelion to quantities
of the second order, we are troubled by the introduction of large terms depending
on the square of the disturbing force, which disappear from the rigorous expres-
sions for the co-ordinates. These may be avoided by substituting for the eccen-
tricity and perihelion the quantities h and Ic determined by the condition
/t = e sin rt
k = c cos n
If, as before, we count the longitudes from the perihelion of Uranus at the epoch
1850, we should substitute ^n for n in these expressions. The values of h and /.;
will then be given by the integration of the equations
*''* ' 7 XT •
-J = matiiCi cos N
dk
lit
= — ni'ank, sin N.
THE OllBIT OF UUANUS.
181
Differentiating with respect to ^, we find for the terms of the second order
5 , = — ni'anhi sin Nh N
lit
.(Ik
— vi'anJii cos Nf) N.
Substituting for hNits numerical value just given and integrating, wc find
ih = m'uvJci \ —2895" sin .V— (G658" — -l^'iG/) cos iV+ 1815" sin 2.V|
— 3630"/«'a/i'i«< -j- constant ;
tk = m'avh\ 1-2895" cosiV+ (6658"— 4".2G0 sin iV^+ 1815" cos 2.V|
-|- constant;
the constants being so chosen that <Vt and hic shall vanish at the epoch.
Reducing the values of hh and hk to numbers, they become
tJi = 5".82 sin iV+ (13".40 — 0".867') cos N— 3".65 sin 2iV+ r'-OST— 2".61,
hk = 5 .82 cos N— (13 .40 — .86 T) sin N— 3 .65 cos 2.V + 12 .12,
the last two terms being arbitrary constants.
Computhig the values of these terms of fil, <Jt, and hk, for intervals of 50 years,
from 1600 to 2000, we find them to be as follows:
Year
n
,7t
hk
1600
—V.U
+0M0
— 0'.02
1650
—0.71
+0.05
0.02
ITOO
—0.31
+0.02
—0 .01
1750
—0.10
0.00
—0 .01
1800
—0 .01
.00
.00
1850
0.00
0.00
0.00
1900
0.00
.00
.00
1950
-j-0 .04
.00
—0.01
2000
+0.18
—0 .01
—0.02
We see that although the ultimate effect of these terms is very considerable,
their effect, during the period that Uranus has been observed, is insignificant.
Conchided Elements and Perturhntions of Uranus.
The corrections found in the last chapter being applied to the final provisional
elements (p. 99) give the following elements for 1850, affected by the great
inequality produced by Neptune:
Elements IV of Uranus.
Epoch, 1850, Jan. 0, Greenwich mean noon.
168° 15' 6".7
28 25 17.05
0,
■i|^
73 14 8.0
46 20 .54
.0469236
182 THE0RBIT0PURANU8.
c (in sec), 9678".69
n, 15425.752
log a, 1.2829072
log(/„ 1.2831044
Log «, includes, as before, tlie constant term in the perturbations of the logarithm
of the radius vector which, with tlic corrected mass of Neptune, is +.0001972,
To find the correspoi. ding elements at any other epoch, the following secular
and long-period perturbations are to be applied. Those produced by Neptune are
derived from the expressions in Cliapter III by correcting them for the new mass
of Neptune, and for the change in the value of tlie small divisor 2n'—u j/roduced
by the correction of the elements of Urauns. The logaritiims of the factors for
correction arc.
Correction of mass of Neptune 9.93598
Correction of divisor 0.00051
Log. factor for H 9.9349G
Log. factor for le, Sn, 5» 9.93547
Including the perturbations of the second order just found, we have, by putting
N=2r-!,,
= 1 13"^ 30' 4G".0 -f- 8° 2G' 51".9 T,
tl = (-2850".41 + 0".32r) sin N-\- (387".G7 - G".377') cos iV
+ 112 .72 sin 2.V — 47.28 cos 2N
— 7.72sin3iV + 4.33 cos 3.V
+ .55 sin 4iV — 0.4G cos 4Ar
— .037'== — 83".787'+ 2811 ".41.
M = — 412".18 sin iV+ (14".03 — 0".8G7') cos N
+ 29.20sin2.V— G .09 cos 2iV
— 3.11.sin3.V+ 1.19 cos 3.V
+ 0.28 sin 4.V— 0.13 cos 4.V
+ 14.7G7' +398.33
a- = — 41 1".53 cos N— (13".G5 — 0".8G 7') sin N
+ 29.33cos2iV+ G.17sin2iV
— 3.12cos3.V— 1.21 sin 3iV
+ 0.29 cos 4.V+ 0.13 sin 4iV
— 5.4537' —124.72
3n (in units of tlie 7th place of decimals).
= 1963 cos iV^+103sin iV
— 171cos2iV— G7sin2iV
+ 15 cos SAT + Gsin3iV
+ 511.0.
The perturbations of e&n and le are here replaced by those of h and k, defined
by the equations
7i =csin {7t'—n„)
k = e cos (ti — rr,)
.-1-^
THE ORBIT OF URANUS.
183
7to representing the perihelion of 1850 = 168° 15' 6".7. We then have Jor tlie
eccentricity and longitude of perihelion at any epoch
c sin (n; — 7t„) = f,h
e cos (7t — TTo) = Co -\- fil:
In the above terms multiplied by the time we have included the sectdar varia-
tions produced by Jupiter and Saturn. If the perturbations of the elements due
to cacli particular planet are required, we have
Action of Jupiter,
h7i = + 5". 737'; hk = — 0".G087'.
Action of Saturn,
,7i = + 5"MyT; th- = — 4.589 f.
Subtracting these from the above exjjrcssions all the remaining terms will be duo
to tlic action of Neptune. The values of U and bv are due entirely to the action
of Neptune.
For the sake of rigor, we may suppose the perturbations produced by each
planet to be multiplied by a factor representing the number by which tlie adopted
mass of the jdanet must be multiplied to obtain the true mass.
It will add to the homogeneousness of the tlieory to express the i)orturbations
of long period, wliich aw multiplied by the product of the masses of Jupiter and
Saturn, as perturbations of the elements. These terms, as found on page 88, are
{v.c.O) = — 0".55 sin N^ — 0".()3 cos iV,,
+ 40 .(iu sin iV, — 10 .50 cos iVj
(r.«.l) = -|- 2 .(54 sin iV,, + 4.()4cosjVo
-f 7 .35 sin iVj + 4. 41 cos iV^
(r.f.l) = — 4 .'.>3 sin iV„ — 3 .87 cos .V^
+ 8.06siniVj— 8. 38 cos .V,
These terms, together with the arbitrary corrections of the elements whi'ih have
been applied to make them very small at the epoch, may be replaced by tin follow-
ing corrections to the elements:
,y = — 0".55 sin iV,, — 0".03 cos N,
+ 40 .05 sin iV, — 10 .50 cos N.,
4--27".-27 — ll".7-.>7'.
,Vt =+ 2 .09 sin .V, + 1 .94 cos N^
' — -i.lSsiniV,-}- 3.71 cos iV^7
+ 1".'28.
a- = 4- 1 .3'2 sin iV„ -1- 2.32cosiVr„
4- 3 .08 sin iV, + 2 .21 cos JV,
fiv = 27 sin iV, + i04 cos iVj + 76 (in luiits of the 7th decimal).
The amount of the perturbations of the elements for every half century, from
the year 1000 to 2200, is given in the following table. Cohinm(l) gives the per-
turbations by Neptune, Saturn, and Jupiter, computed from the expressions
m
1'
m-
m
184
THE OR BIT OF URANUS.
on pago 182; column (2) those just given depending on tlic product of the masses
of Jupiter and Saturn.
Year.
hi
Ih
hk
hv
hx
h
^'\,
C?)
(•>,
(2)
^'l
(2)
(!)
(2)
II
1000
+2050.31
+100.09
+
42.00
+1.94
—389.51
+G.08
+ 1955
—140
—2.00
+ 1.20
1050
1841.17
149,08
27.79
3.00
375.05
0.08
1882
150
2.10
1.37
1100
l(i38.70
130.39
14.03
4.99
300.31
0.82
1802
169
2.12
1.64
1150
1444.87
122.87
+
1.45
0.01
343.49
6.47
1717
178
2.08
1.60
1200
1200.24
108.85
9.90
0.00
325.27
5.04
1026
183
2.02
1.77
1250
+ 1085.7(i
+ 94.07
—
19.84
+ 0.75
-305.09
+4.42
+ 1529
+ 183
—1.94
+ 1.80
1300
921. 7(i
80.07
28.34
0.49
284 90
2.80
1420
180
1.84
1.91
1350
7(!9.32
07.12
35.27
5.90
252.90
+ 1.10
1318
173
1.70
1.92
1400
fi29.00
54.35
40.50
5.08
239.78
—0.57
1205
101
1.64
1.91
1450
501.39
42.02
44.11
4.16
215.08
2.17
1080
147
1.37
1.85
1500
+ 387.18
^- 32.11
—
45.83
+3.24
—190.00
—3.52
+ 903
+ 129
—1.19
+1.74
1550
28G.05
22.78
45.03
2.42
104.84
4.52
835
110
0.99
1.60
1(100
200. fi5
15.38
43.40
1.80
138.31
6.12
704
90
0.80
1.42
1G50
129.43
9.34
39.20
1.41
111.22
5.30
608
08
0.01
1.20
1700
73.40
4.80
32.79
1.23
83.70
5.09
430
48
0.44
0.95
1750
+ 33.00
+ 1.90 —
24.10
+ 1.23
— 55.90
4.50
+ 288
+ 29
—0.27
+0.60
1800
8.51
0.33 _
13.20
1.30
— 27.93
3.81
+ 146
+ 11
—0.12
+0.34
1850
0.00
0.00
0.00
1.52
0.00
2.00
— 5
0.00
0.00
1000
7.04
0.71 +
15.05
1.03
+ 27.74
2.10
- 145
18
+0.10
-0.30
1950
31.47
2.23
33.70
1.59
55.09
1.34
290
26
0. 10
0.73
2000
+ 71.43
+ 4.25 +
54.20
+ 1.34
+ 81.81
0.73
— 433
— 32
4 0.20
—1.10
2050
127.33
0.52
77.12
85
107.77
0.34
574
30
0.20
1.47
2100
198.80
8.75
102.40
+0.13
132.07
0.14
711
27
0.17
1.83
2150
285.72
10.59
1.30.15
—0.77
150.33
0.11
843
20
0.11
2.17
2200
387.11
+ 11.77
+
100.15
—1.70
+ 178.48
—0.19
908
— 9
+0.02
—2.48
Mean Elements af Uranns.
If, instead of the elements of Uranus affected by the great inequalitj', we wish
the absolute mean elements, these are to be obtained by adding to the elements
already given the constants applied to the perturbations hi, hh, hk, and hn to make
the perturbations vanish at tiie epoch 1850.0, and also the corrections (p. 113)
which we have subtracted from the elements and added to the perturbations to
reduce the latter to a small quantity during the period for which the tables are
likely to be used. We thus find the following mean elements :
Elements V of Uranus. Epoch, 1850, Jan. 0, Greenwich mean noon.
l/ongitude of the jierihelion
Mean longitude at epoch,
Longitude of the node,
Inclination of the orbit,
Eccentricity,
Eccentricity in seconds,
Mean motion,
Log mean distance (uncorrected).
The same corrected.
True mass of Neptune,
170° 38' 48".7 4-8()i)8". fi
29 12 43.73 + 2811.4«
73 14 37.6 + 29 .6«
46 20 .92+ 0.38/t
.0463592 — 5236,a
9562".27— 108".b«
15424.797 — 0".838«
1.2829251 + 179h "
1.2831223 -j- 179;t
19700
*f4ai-
THE ORBIT OP URANIJS.
165
Supposing the ninss of Noptunc to be imccrtnin by oiu'-fiftietli of its entire
amount, wliieli is quite possible, it will be seen the longitude of the mean j)eii-
helion is from this cause uiicevtain by more than two minutes, the mean longitude
of Uranus itself by nearly a minute, and the mean motion by nearly two seconds
in a century.
It will be seen that the logarithm of the mean distance just given docs not
accurately correspond to that of elements I V' jjIus the constant term of iSv X 0.-i343,
as it should. This ditt'erenee arises from the rejection of the terms of the second
order in hr), which can not affect tlie geocentric longitude of the planet by a tenth
of a second for a number of centuries.
It is to be remarked that these mean elements arc those to be used in the
general theory of the secular varintion of the planetary orbits.
Cviirhuhd Theory a/ l^raiiKx.
The elliptic longitude and radius vector of Uranus, affected by the secular and
long period perturbations of the elements, will be given by the following equations.
Tut
I, = iij + f(„
(j = l — 7r,„
k = c, + tf>\
the zeros mdicating elements IV, and i7i, fil; and &l being the perturbations of
these three elements just given. Then
Elliptic longitude in orbit = Z
+ |2-]^e=-f ^e' I |/.sinj/-Acos^J
+ i 1 i! ~ 04 "' } I ^'' " ^'''^'^ ''" '^^ ~ ^^''' ~ ^'"^ '"' '^'A .
Nepcrian logarithm of >• = n -|- I*?- -|-^,\,e'
— I 1 — j^«'' I I /'Cosr/ + 7t sin^ J
- { 4 - 2I '"' ] 1 ^'' ~ ''"'^ "^"^ ^^ + ^'''' '"' ^^ }
24 May, 1873.
'V
T
1 I
I
I
M
'0
17
li
. 1 I
I 1
186 THE ORBIT OP URANUS,
~ 24 { ^'''' ~ "^'''''^ ""^ ^^ "^ ^'^''''' ~ '*'^ *'" ^^ }
— Ij J I (/.' — Gi7i^ 4- A') cos 4j + (.4/.-7t — 4M') sin iy I
In computliifif these oxprcssions it will bo sufficient for scvorul centuries before
or after 1850 to develop h, H; and <*/ to tlieir first dimensions: it will, liowever, l»e
more convenient to correct the mean anomaly ij for the perturbation hi before obtain-
infj the equation of the centre;. Dcvelopin<; the perturbations of 7i and k to terms
of the first order, we have for the effects of the perturbations of those elements:
fk
(('.c.l) = -(*2-^f',r)<7«
{v.8.2) = (Jo - y^o' ) '^*
(r.s.3)= ^^c,,-Uc
13 „„
(r.c.;J) = — "4 <'"' ««
103
(i-.c.4) = — .j^V ' ^«
(p.f.O) = hn -\- (\,()7c
(p...l) = _(l_|,„^),vA
(p.c'.l)-=-(l-yv)^/^
(p.c.2) = — 2^0 '7';
17
8
17
(p.«.3)
e.'^h
(p.c.3)=-- J^e,^7^
These coefficients for p must, of course, bo multiplied by the modulus 0.434394
to reduce the perturbations to those of the common logarithm of the radius vector.
*£i4."
THE ORBIT OF URANUS.
187
Among the elliiitir terms may l)c> iiuliulcil the effect of the following minute
constants introdjiced by tlie peitnibutions.
(ns. 2) =_()". 144
(w;.2) = + 0.l:{()
0.434;} (p.f.O) = -f l\)V2 in units of the 7th place
().4;34;j (().«. 1 ) = -j- 63 of decimals.
0.4343 (p.f.l) = + 73
0.4343 (p.«.2) = 4-5
0.4343 (p.r.2) = 4-4
This term (p.c.O) is that added as a eonection to the logarithm of the mean
distance.
To the coefficients (I'.sA), (r.cA), etc., arc still to be added the following periodic
terms : —
1. The periodic terms due to tin; action of Jupiter, given in Chapter V, omitting
the terms multiplied by T, which are inchuled in the perturbations of the
elements.
2. The periodic terms produced by Saturn, including those terms multiplied
J)oth by 7' and by sin A.^ or cos A.,, b»it omitting those multiplied by 7' only for
tlie same reason as in the case of Jui)iter.
3. The periodic terms produced by Neptune, multiplied by the factor 0.8(5294
on account of the correction to tlio mass of that planet, and omitting the terms
multiplied by hi, he, and elg.
4. The periodic terms multiplied by the product of the masses of Jupiter and
Saturn, given on page 88, omitting the terms nndtiplied by the sine and cosine of
N^ and N-, because tiiey are replaced by tiie terms of hi, hh, and hi; given on page
183, and tabulated in the columns headed (2) on page 184. The result will bo
the same whetlier we employ the terms of (r.c.O), (r.s.l), etc., given at the bottom
of page 88 and tlie top of page 89, omitting the nundwrs in the columns 2 on
page 184 from the expressions on page 18G, or whether we include the latter and
omit the former.
The true anomaly of Uranus will then be :
<jt)-\-hl-\- (equation of centre from elements IV, using for mean anomaly </o -\- hi)
-\- 1 (r.s.i) sin !ij -}- 2 {i:c.i) cos ty.
The logarithm of the radius vector will be:
log ;• in elliptic orbit from elements TV.
4" - (p-*-0 s'" i'J 4" '^ (p.c.i) cos iff
care being taken to multiidy the coefficients by the modulus where that has not
already been done. All the terms in Chapter V are so multiplied.
To pass from the true anomaly to the true longitude we must investigate tlic
secular motion of the jdanes of the orbit and of the ecliptic. The effi'ct of this
motion on <}>, 0, and t will be found by successive api)roximations from the formuke
188
THE OIUUT OF U II A NUS.
(;U), corrcctiiig tlic dutii for tlic new nuiss of Ncptiiiu'. \Vc shall also use the
saiuo niotiou of the (.•clii)tic adoptud on j). !).j. \\C liuvo thus:
til
— r.:>:\
^= + 5.4:}4-0".;J87'.
at
lit
lU
= — 5.17
:=_4(;.7S-f 0.1'27'.
Asa first approximation wc liave
= T = 7:J^ 14' 8" — 31G!)".'i7'
</) = 4G 20.54+ '2.48 7'
Substituting these values in (34) and integrating we find
<^ = <^,+ 2".477'4-(r.l37'^
= 0„ — 3168 .42 7' -j- 3 .00 7'=
T = Tu — 31(i8 .70 7* -i- 3 .00 7'-
For tabulating wo shall use, instead of and t, the distanro of the perihelion
from the ascending node, or 7t— t, and the ^aluc of corrected for Struve's pre-
cession. Since the mean motion has been derived without nniking any distinction
between t and 0, it will be necess;;iy to correct the motion of moan anonndy by
the difference of those quantities. We tims obtain for tiie values of the thvco
principal arguments: —
j/ = 220° 10' 10".35 + lo42;)74".86 7'+«
0)= 9;! 58.70+ 31(58 .7(57'— 3.00 r^
0= 73 14 8. 00 4- 1856 .82'/'-]- 4.127'^
Tf we represent all tlic inequalities of the true longitude by A/, so that we shall
have for the true anomaly
the argument of latitude will be
"=/+w.
The reduction to the ecliptic will tlien be
It = — (J)".37 + O'.OIGT') sin 2*^
the true longitude on the ecliptic referred to tlie mean equinox of date,
and the sine of the elliptic latitude,
sin /3o = sin <^„ sin 7*.
The perturbations of the latitude will be
(i.c.O) + (i.c.l) cos g -\- {h,a.\) sin g -\- etc.
I ■/*"■■* f
kJBJ
TIIK OUniT or U II ANUS.
189
Tl.o lu-riodic terms of (h.c.O), (h.xA), O'.c.l). etc., arc given in Chnpter \ , on
pa^cs 8G ami 87, und nro to be tuken witliout any farther niodiHeation than the
niultiplieution of those due to the uetion of Neptune by the factor U.8Ua, The
constant, seculur, and long period terms arc
/,.c.o =+o".2() — ir.i'ir — o.()ii(V + <'*'^6,^x
(6.*.1) = —0 .2'iT — O .057" + ().!)75rV -j- O.'i'21/ix
(h.c.l) = +'i .47 T 4- .127" + 0.221^); — 0.!)75ix
{h.H.2) = -0 M - .01 7' + 0.04fi,V + 0.01 Ihx
(i.f.'i) = _0 .01 -f- .127' + 0.011.V — 0.0465x
'rii(> vahies of hr^ and i^x to be used in tliose expressions are those the expressions
for whicli are given on page 97, and wliich are tabulated in the last two columns
of the table on page 184.
The following tables are bused on the elements and theory laid down in this
chapter.
Ul'
190
TUE Oil 15 IT OF UUANUS.
CHAPTER IX.
GENERAL TA15LES OP UUAXUS.
Emnmration of the Quantifies contained in the several Tahlca.
The first six tables are designed to give the values of tlic three arguments of
the elliptic motion, g, w, and 0, and of tlie nine arguments of the tables of pertur-
bations. The argument u is, however, diminislied by 3', the sums of the constants
added to the perturbations of (i-.c.O) to make these quantities positive, and by
10", the constant added to the reduction to ecliptic. The expressions for the argu-
ments of perturbations are as follows, the mean longitude of each planet, counted
from the perihelion of Uranus, being represented by the initial letter of the planet.
All these arguments are expressed in units, of wliich GOO make an entire circum-
ference, so that each unit is 3G'. The time t is counted in Julian years from the
fundanu iitul epoch,
1850, January 0, Greenwich mean n. on.
Ar;
1 =
J—
u
= 219. 1 0(
+ 43.44028/
2 =
S —
u
= 577.34!
1+13.22717/
3==
U-
X
= 88.88-i
t+ 3.50035<
4 =
J-
2S
= 497.(5
+ 9.8445/
5=:
3,^'-
U-
- ,7= 79.8
+ 3.3825/
() =
46' —
nr-
J= 57.1
+ Kl.OlO/
7 =
'2J -
3,S" -
-3r7=238.7
+ 18.()33/
8 =
'iJ —
\S -
2f/=2()1.3
+ 5.4058/
y^
IS -
'2 J -
3^=130.9
+ 19.992/
Table I gives the corrections which must be applied to the values of the argu-
ments at any time during the nineteenth century to reduce them to the corresponding
time in any preceding or following century between the Cliristian era and the year
2300 Since o and each contains a term proportional to the square of the time,
the correction for these quantities is not constant during each century, but is of
the form
ij + cj'T*
6) and J being constant during each century, and T being the fraction of the
century counted from its beginning.
Table II gives the value of 7, u — 3', — 10", and the above nine argiiments for
(ircenwich mean noon of Jan. of each leap year from 1752 to 1948, and for
J iniiary — 1 of the years 1800 and 1900, corresp'Muling to Decemb(>r 30 of the years
1799 and 1899. Tlu: corrections for the perturbations of long period are not
k3
THE ORBIT OP URANUS.
191
applied in this tabic. The numbers at the bottom of this table, in the line AfJi,
sliow the variation of the corresponding quantity in I'iO days, for the epoch 1850.0.
In tiie line "Factor 7'" is given the change of tliis variation in a century, while
Alii is the second difference for intervals of 120 days, liy means of tliese num-
bers, when the arguments are computed for any date, their values for other dates
at intervals of 120 days may be found by successive addition.
Table III gives the motion of tiie several arguments between the epochs of the
preceding table and the zero day of each montii in the course of a four-year cycle.
The variable motions, w and 0, correspond to the epoch 1850, and rigorously they
each require a correction for any other four-year cycle tlian that between 1848 and
1852. But, owing to tlie small inclination of the orbit of Uranus it is not neces-
sary that either « or should be exact, if only their sum is exact. Tlie column
0' of this table, therefore, gives the correction wiiicli nuist be applied to the motion
of at the end of a century (1050) in order that, being applied to alone, (j -)-
may be exact. Tliis correction is, in fact, tiiat for the secular variation of the
precession.
Tables IV and V give the motion of the arguments for days and hours. The
motion for hours is, liowever, not necessary in the case of any argument but (j, as
all the others can be readily enougli interpolated to fractions of a day.
Table VI gives the corrections to the arguments on account of the terms of long
period from 1000 to 22()(T. The terms in question are, in the case of Jupiter, tlic
great ineciuality produced by the action of Saturn, in the case of Neptune tlie
great inequality produced by Uranus, and, in the case of Uranus, the iiu'qualitics
in the mean longitude tabulated in the preceding chapter. The numerical expres-
sions are
U= 0.5:J5 sin (110° 21' + 40^ 45' 20" T)
hu=n
The corrections to the several arguments are
b arg. 1
= (7
U—H
b arg. 2
-rV
S ar?' 3 = (V
,").V=1.75,y
Xo correction to the mean longitude of Saturn is applied, all its inequalities being
taken account of in the terms of the second order.
The corrections, expressed in seconds, have been reduced to units of the argu-
ment by dividing them by 21(iO".
Outside the limits of the table these corrections must be computed from their
formuhe.
Table VII gives the equation of the centre, and the elliptic part of the logarithm
of the radius vector. No constant is ajiplied to the former, but the latter is dimi-
nished by .0()0:M00, the sum of the constants added to (ji.r.O) in Tables VIII, IX,
X and XVII.
192
THE ORBIT OF URANUS.
The formulae for the Tables are
Equation of centre = 193iV2".()() sin
-j- 5G7 .24 sin
4- 23 .05 sin 3j
+ 1 .07 sin 4<j
-\- .05 sin b<j
Elliptic log. r = 1.2833435
— .()()03400
— .0203618 cos g
— .0007 1G5 cos 2y
— .0000318 cos 3i/
— .0000016 cos 4</.
Table VIII gives the coefficients (y.c.O), (I'.c.l), etc., for tlio perturbations of the
longitude and logarithm of radius vector produced by the action of Jupiter. They
arc computed from the periodic terms of the fornuilic on page 83, with tiie addi-
tion of the following constants to make all the numbers of the table positive :
Constant of (e'.c.O) = 55".
(r.s.l)= 6.
(r.c.l)=7^ 4.
{v^ 2) = .20
(W.C.2) = .20
(p.c.O) = 1200
(p.s.l)= 150
(p.c.l)= 100
(p.,'(.2) = 10
(p.c.2)= 10
Table IX gives the periodic part of the coefficients due to the action of Saturn,
taken without change from the expressions on page 84, together with the secular
variations, the latter including only the terms of (t-.s.l), (t-.c.l), (i-.s.2), and (r.c.2),
which are multiplied by T and by sin Jj or cos A.^. The coefficients of T are given
hi the columns Sec. Var. and each number is increased by the constant 1".50 to
make it positive. The term — 0".06rsin A.^ in {r.c.O) is omitted entirely, as it will
not amount to a tenth of a second until after the year 2000. Tiie constant terms
added to the quantities of tiicse tables to make all tl-.e numbers positive, are :
'/ //
Constant of (r.c.O) = 30.
(r..9.1) = 150. + 1.50r
(I'.c.l) = 150. + i.5or
(M.2) = 130. -I- \MT
{r.c.2) = 130. + 1.507'
(V..«.3)= 8.
(t\c.3)= 6.
(r...4) =1.
((•.C.4) =1,
THE ORBIT OP URANUS.
193
Constant of (p.c.O) = 800
(p.8.1) = 1500
(p.c.l) = 1500
(p.s.2j = 400
(p.c.2) = 400
(p.s.3) = 100
(p.c.3)= 100
Tabic X gives the coefficients produced by the action of Neptune, computed
from the periodic tonus on pages 86 and 86 without any other change than the
multiplication of all the numbers by the factor 0.863 to reduce them to the new
mass of Neptune. The constants added to the several quantities, are
Constant of (I'.c.O) =
92.85
(v.s.l) =
20.00
(v.c.l) =
31.00
(v.s.2) =
5.00
{V.C.2) =
5.00
(r..s.3) =
1.00
(r.0.3) =
1.00
(r..s-.4) = ■
- 1.00
(r.6'.4) =
- 1.00
Constant of (p.c.O) =
400
(p.«.lj =
200
(p.c-.l) =
200
(p.s.2} =
40
(p.c.2) =
40
Tal)les XI to XVI give tlie terms of the second order and of short period which
contain tlie products of the masses of Jupiter and Saturn, which, with the constants
added to the niunbers of the several tables, are as follows:
{p.c.O) = + 0".m sin A, + 0".51 cos A, ; Table XII ; const = 0".6n
— 0.01 sin .1,-1- 0.05 cos ^;;
— 0.35 sin .1, — 1 .30 cos .4,;
— .05 sin .1,, -f- .03 cos .4,;
Sum of constants added to these tables
(r.«.l) = + 0-.2fi sin ^, -f- 0".27 cos A,;
— 0.04 sin /I, — 0.17 cos yl,;
-l-0.08sin y1„-|-0.03cos^1.,;
-0.02 sin A, -f- 0.08 cos .4;;
-I- .30 siiwl, — ,58 cos .4,;
— 0.04 sin A,;
25 Juno, 1373.
XUI;
= .05
XIV;
= .05
XV;
= 1 .35
XVI;
= 0.10
ibles
2.15
Table XI;
const = 0".40
XII;
= .20
XIII;
= 0.10
XIV;
= 0.10
XV;
.75
XVI;
0.10
li:
194
THE ORBIT OF URAXUS.
Table XI;
const
= 0".40
XII;
.20
XllI;
0.10
XIV;
0.10
XV;
0.75
XVI;
O.IO
in these ti
ibles
1.65
(V.C.I) = -I- 0".06 sin At - 0".27 cos A,;
+ 0.18sin^, -fO.Olcos^;
— .03 sin A^ + .08 cos A,r,
— 0. 0-2 sin it, -f 0.09 cos A;
— .44 sin A^ — .61 cos A^;
— 0.1 sin 4, + 0.03 cos J,;
Snm of constants added to (v.s.l) and (c.c.l) in these tables
The term of (p.c.O)
1 1 sin A^ — 3 cos A^
is omitted from the tables entirely.
Tables XVIIa and XVII i give the constant, secular, and long-period terms of
{r.fi.l) (v.c.l), computed from the fornuilic p. 186, with the following additions:
1. The constant terms introduced by the ])erturbations, given on p. 187.
2. Tlie negatives of the constants added to the tables VII to XVI inclusive to
make the numbers of those tables positive. The values of these terms are
Pert. Const.
Tobies Till t(
) XVI.
{l)-(2)
(v.s.l)
177".65
+ 1'
.50 7'
_177".65 — 1".507'
(V.C.I)
186 .65
+ 1
.50 r
— 186.65 — 1.50 7'
(r...2)
— 0".14
135 .20
+ 1
.50 7'
— 135.34 — 1 ..507'
(r.c.2)
+ 0.13
135 .20
+ 1
.507'
— 135.07 — 1.507'
(r..s.3)
9 .00
— 9 .00
(v.c.-i)
7 .00
— 7.00
(p.c.O)
[+1972]
— 1000
+ 1000
(p.«.l)
+ 63
1850
— 1787
(pel)
+ 73
1800
— 1727
(P....2.)
+ '">
450
— 445
(p...2)
+ i
450
— 446
The perturbation constant of (p.c.O), being added to log a in forming tiie elliptic
radius vector, is not included in this table.
Table XVIII gives the reduction to the ecliptic
— 9".37 sin 2>,.
The constant 10" is added to make the numbers always positive, wliich constant
has been already subtracted from ().
Table XIX gives the principal term of the latitude
4(i' 20".54 X sin ».
'fable XX gives the coefficients {h..s.l) and (h.c.]) for the jjerturbations of the
latitude produced by Jui)iter. They are given by the formuhc
(-/>.«.!)=: 0'.65 cos (,/— r+ 40^)
(h.c.l) = .65 sin (./— U -\- 40^)
The ronstnnt 0''.70 is added to make all the numbers of tlie tabl(> positive.
THE OUBIT OF URANUS.
195
Table XXI gives the conospouding coefficients for tlie action of Saturn, com-
puted from the expressions on p. 87 witli the addition of the following constants.
Const, of {h.c.O) = 0".lO
(i..v.l).-=;}..3()
(i.e. 1) = 3.10
(i.«.2) = .20
(6.C-.2) = .20
Tabic XXII gives the coefficients for the action of Neptune from the formuhe
on p. 87, all the numbers l)eing multiplied by the factor ().8();3 to reduce them to
the adopted mass of Neptune. The following constants are added :
To (i.f.O) .... 0".06
(i.s.l) 1 .00
{h.c.l) 1 .20
(/>.s.2) .20
(A.r.2) 0.20
Table XXIII gives the secular and long-period terms for various epochs com-
puted from the formula? of p. 189. The sums of the several constants added in
the three preceding tables are here subtracted again so that these expressions
become
h.rJ) — 2c= 0".10-0".12r— .01b^>7 + .046,^x
{/t.f.l) ~^c^ — o .00 — .227' — 0'.orj7'- + .975^>7 + .221Sx
(A.c.l) _ vc = _ 5 .00 + 2 .47T+ .12'/'^ + .22bV
(/>.«.2) — 2c = — .40 — .01 7' 4- .046,V + -Ollf^K
(/AC. 2) — 2c = — .41 + .127'+ .OIU*? — M68x
.9'iobx
Prcccjyfs for the nse of tli( Talks.
Express the date for which the position of Uranus is required in years, months,
days, and hours of Greenwich mean time, according to the Julian Calendar if the
date is earlier than 1500, according to the Gregorian Calendar if it is later than
IfiOO, and according to either calendar between these epochs.
Enter Tabj I with the beginning of the century, and take out the values of g,
«, J, 6, 0', and arguments 1 to 9. Multiply w' and (/ by the fraction of a century
corresponding to the date, and write the products with their proper algebraic signs
under o and 0. If the calendar is the Julian, the century marked ,7" must be takcMi,
and if the Gregorian, that marked G. Between the dates 1752 and 1951 it is not
necessary to enter Table I at all.
If Table I was not entered, enter Table II with the yenv, (U* the first preceding
year found therein. If Table I was entered, enter Table II between the year
1800 and 1896 as if the niimber of the c(>ntury wer<> changcul to 18. Take out
th(! values of <f, o, 0, and the arguments, and write them under the corresponding
quantities from Table I.
Enter Table III with the excess of the actual year over that with which Table II
was entered, and with the month. Write the corresponding values of //, w, 0,
and the arguments under the previous values. Multiply 0' by the fraction of a
196
THK Oil HIT OF URANUS.
\
ccn Mry after 1850, corresponding to the date with which Tahle II was entered,
and '.vrite the product under (), or add it to it in writing {). 11' Table 11 was
entered witii a date before 1850, this product is negative.
Enter Table IV with the day of tiie month and write down the corresponding
values of g, u, etc., inidcir the former values.
If the date does not corresjxmd to Greenwich mean noon, the motion of y for
the hours must be computed from Table V, and the other quantities must be
interpolated to the fraction of a day in entering Table IV.
Enter Table ^'I witli the year, find by interpolation tlie values of //, and argu-
ments 1, 2, and 3, corresponding to the date, and write them under the former
values.
Add up all the partial values of g, o, 0, and the arguments, attending to the
algebraic signs of tin; products. Snbtrac^t from tlie ar<j;uments as many times GOO
as ])o9sible, and the results will be the final values of those quantities.
Enter Table VII with g as the argument, the seconds being first reduced to frac-
tions of a minute, and interpolate the quantities £'and log r. When g exceeds 180°
the former (piantity is to receive the negative sign; the latter is always positive.
Enter Tables VIII to XVI inclusive with tbeir respective arguments, and take
out the values of the quantities (r.r.O), (r..s.l\ (v.c.l), etc., (p.c.O), (p.».l), etc., so
far as they are found in the tid)les, wvitiiig the qtiantities having the same desig-
nati(m under each other. In Table IX the quantities Sec. ^'ar. must be nndtiplied
by th(> centuries and fraction of a centiny of the actual date after 1850, and the
l)rodiHt must be included witli the corresponding quantities, (r.'i.l), (r.c.l), etc.
Before 1850 this product will always be negative; aftmward always positive. All
the quantities taken from these tables are positive except (f.«.4) and (r.c.4) in
Table IX, which are negative.
Add up all the partial values of (r.c.O), (r..'».l), etc., thtis obtained from Tables
VIII to XVI, and from their sum take the corresponding quantities obtained from
Tabl<> XVll by interpolathig to the date. The rc^quired quantities arc all given
in Table XVII A ; Table XVlIa being only an expansion of a i)art of XVIli for
the present century. The final values of (r.s.l), (r.c.l), (r.s.2), etc., (p.s.l), (p.*".!),
etc., thus obtained are to be multiplied by the sines and cosines of the correspond-
ing multiples of g, in doing which four place logaritlmis are sufficient if the coni-
jMitation is carefully mad(\ Tlie products are then all added togetlii^r, and to f/, w, 7s',
and (r.c.O); in the; case of r, and to log. r, (p.c.O) in the case of p. That is, we
are to form the expressions;
u==</-|-u-f-£+ (t'.c.O) --)- (r..f.l) sin r/ -f- (r.c.l) cos g
-f (r..s.2) sin 2/ + (<'.c.2) cos 2j
-f- etc. -\- etc.
log r — log r (from Table VII) + (p.c.O)
4- {p.«.l)sin g) +(p.c.l)cos g
-f (p.«.2) sin 2;/) -f (p.c.2) cos 1g
-|- (p.s.3) sin 3y) 4" (P'^'-'^)^'^'*%'
TlIK OUBIT OP URANUS.
197
u will then be the true argument of latitude, and log r the loguritlnn of the radius
vector with seven places of decimals.
Under n write 0; enter Table XVIII with the argument m and take out the
reduction to the ecliptic. Add it to u and 0, and the sum of the three quantities
will be the heliocentric longitude of Uranus referred to the mean equinox and
ecliptic of the date. Applying nutation the longitude will be reduced to the
true equinox.
Enter Table XIX with u as the argument, or, when u exceeds 180°, with
u — 180°, and take out the principal term of the latitude, which will be positive
when u is less than 180°, and negative when it is greater.
Enter Tables XX, XXI, XXII, and XXIII with their respective arguments,
the argument for the last being the date, and add up the various (piantities having
the same designation, noticing that in the first three tables all the (piantities are
positive, while in the last they are all negative except (i.c.O). Then form the
expression,
{h.c.O) + {h.s.1) sin g + (h.c.l) cos g + {h.M.2) sin 2j + a>.c.2) cos Qg,
and add it to the principal term of the latitude, with regard to the algebraic signs.
Tlie sum will be the heliocentric latitude of Uranus above the ecliptic of the date.
"SViien an ephemeris of Uranus is to be conq)uted for a series of years, some
modifications may be introduced, which will save the computer labor. In the first
place an ecpiidistant series of d.ites being selected for computation, it will be suffi-
cient to compute g, u, 0, and the arguments for every sixth, eighth, or tentli date,
and to fill in the arguments for the intermediate dates by adding the nearly con-
stant ditterences corresponding to the adopted intervals. The agreement of the
numbers thus obtained for the last date with those found by the original computa-
tion will prove the whole process. This interval may be as great as l'>() days
without detracting from the accuracy with which the places for the immediate
dates can be interpolated, and the diff'erences for this interval may be deduced
from the numbers at the bottom of Table II. If these numbers are used without
change the values of w and for the last date muy not always come out right.
But these errors, if less than a second, will be of no importance if the one quan-
tity comes out as much too great as the other is too small, and they may be avoided
entirely by making a small cliange in the constant difterence to be added.
Tables XI to XVI, inclusive, need be entered only for every third or fourth date,
and the sums of the quantities can be then interpolated to every date, and added
up with the corresponding quantities from the other tables.
Again, it will be found convenient to compute the sum of the small terms
(y.s.3) sin % + 0'<^--^) ^^^ '^.Z + ("•^''•*) ^'" "^'^ + {r.cA) cos 4r/, as well as the corre-
sponding terms of the radius vector, and all the terms of the latitude, not for the
dates adopted, but for every fourth entire degree of g. Having a series of values
computed in this way, the sum can be interpolated to the value of g corresponding
to the date. To facilitate the formation of the smaller products for entire degrees
of //, a table of products of numbers by the sine and cosine of every degre(> is
appended to these tables, by which the products in question can be formed at sight
ji
198
THE OUBIT OF URANUS.
wlioncvor the coefficient to bo nmltiplicd is less than 3'2". Tiie Viihies of these
coofficieiits, (f.«.;3), (r.r.'A), etc., corresjioiuling to the entire dej^ree.s of //, may ho
eitlier foiinod by interpohition at sight from tiiose corresponding to tiie dates of
computation, or the vahies of the arguments 2 and !i corresponding to tin;
required degrees of ij nniy be computed, and the values of {r.x.'.i), etc., correspond-
ing to these values of the arguments may be taken from Tables IX and X, while
Table XVII must bo entered with tlie corresponding dates.
If tlic lieliocentric ephomeris is coinput(;d for ten years at a time, tlio last of
those modifications in the mode of computation will greatly facilitate the computa-
tion of tiie smaUer terms. We first lind tlie date, and the values of arguments 1,
2, and 3, to one jdace of decimals, for some entire degree of y preceding that wiiicli
corresponds to the first date, and tlien find the dates and the values of the arguments
corresponding to successive values of (/, differing by 2"^ or 4°, until we pass the last
date of computation. We then take out the values of (?\«.3), (r.c.3), ((•.«.4), ((•.c.4),
(p.«.3), (p.c.3), (/(.c.O), (ft.fl.l), {h.c.l), {l)..i.2), and (/>.c.2), with these values of the
dates and iirguments, form their ])roduets by the sines and cosines of th(> corre-
sponding multij)les of ij by means of tiie supplementary tables, and add tiie proper
products together so as to fori:; three small tables witli tj as the argument. These
terms are then iuteipulated to the values of (j corresponding to tlie original dates
of computation.
As a iinit e.\ampl(< of the use of the Tables we will compute the heliocentric
co-ordinates of Uranus for Greenwich mean noon of tlie date 1753, Dec. 3. In
computing the argninents we sliall make use of Talde I, tiiongh it is not necessary
to do so. The computation of the arguments is as follows:
u
u
7
Arg. 1
Tabic 1, 1700
PriHliicl hv a.5:!02
Tablu Il,"ls52
111, Y. 1, Dec.
]V, 3 iliivs
VI, 175:! 02
o /
2-.ll 31
n
7.
0.73
43. 3S
(>.70
J2.30
30.48
:!50
04
/ //
7 5.33
+3.24
50 2.03
1 0.73
0.2(1
o t
350 2'J
73 14
//
11.4
4 4
450.002
300.010
83.252
0.357
0.507
22S 44
S 12
2
108 30
35. 1 1
35.50
0.15
1753, Dec. 3
94
7 11.50
72 44
17.87
240.218
Arg. 2
3
4
5
7
8
9
401
177
38
Tabic I, 1700
II, 18,52
HI, y. 1, Dec.
1 IV, 3 (lavs
' VI, 1753.0
477.310
3.785
25.340
0.100
—0.014
500.548
240.075
05.880
G.700
.020
+ .024
210
517
10
202
S7
130
00
32
537
270
30
59.4
272.1
10.3
0.0
175.3, Dec. 3
352. CI 7
1.52
355
201
240
341.8
10
THE OUBIT OF TJRAXUS.
li)9
(vr.O) (r.i.l) (I'.c.l) (i'.«.2) (v.c.2) (.'.s.S) (r.0.3) (r.xA) (r.r.i)
,,,,/, " '. " " " "
0.37 0.31
CO.'M i-M.H'i 1.42 11.07 0.32 1.04
0.04 5.3a 0.70 1.1'J — O.O;: —1.04
Table VIII
83.58
9.97
6.0.1
IX
21.32
22.24
223.49
X
71. OS
10.02
is. 30
XI
.13
.47
XII
.13
.3(1
.10
XIII
.or,
11
.01
XIV
.01
.04
.03
XV
2.38
1.40
1.12
XVI
.12
.09
.11
V
178. <;»
53.31!
250.24
Table XVII
— 2!)2.r),5
—140.91
—230.19
-j-109.35
(;7.!I3 244.58 2.12 12.2(1
—140.73 —131.03 —9.41 — (i.M5
— 72.80 -1-113.55 —7.29 -|-5.41
Tabic VIII
IX
X
XVII
(l>.''0)
251
1080
320
1003
1104
2707
(p.x.l)
188
741
259
118S
—1321
— 133
(P.''.l)
32
300
222
(p.s.2) (p.*-.2)
021
21
014
—500
+ 114'
042
—412
-f230
507
42
039
—300
+279
—0.03
{,,.... 3)
154
r I
—08
+50
+0.00
(,,.c.3)
111
HI
—04
l0(r(l'..S.l)— 2.37H7
lipjr sill +9.2004
luhr(p..s.l) -2.124
o /
30
108
91
1
7
2
— 1
1
9
U
K
(•!.<'.0)
(r.,s.l)siii <j
(i'.c.l) ens (J
(i'..s.2) sin 23
(y.e.2) cos 2^
(i).s.3) sin 3^
{v.c.'ii) cos 31/
(U..S.4) sin ig
(v.r.i) cos4</
u
e
li
Loiiir. mean Ki\. 330 25
Nntalion
203
72
40
44
lotr(r.c
loj; CIIS
log(|,.C.
II
30.18
11.50
40.21
58.08
-47.00
47.15
2H.43
4 4. 55
- 4.13
- 4.40
0.4T
0.41
57.42
17.87
7.95
23.24
+ 0.51
.1) + 2.0388
f/ _ 0.0012
1) +2.057
Tabic
l»g(i'.«.2)— 1.8021
loj? sin 2-/ —0.5010
log (p.«.2)+ 2.302
Ii>j?(ii.<'.2 + 2.
lojr cos 2,7 + 0.
lui;0,.r.2) + 2.
0,552
0041
440
XX ".00
X.\I 3.01
XXII _ .84
4.81 7 .80 6^59'
XXIII —4.28 — .88 —0.43
+ .53 + . 08 + (f. 1 0"
(h.r.\) (6.,s.2) (/».c.2)
0".02
5 .88 0.41
1 .30 0.18
0.17
_0.I7
0.34'
— 0.51_
""—0.17
Long, true Kq. 330 25 20.75
Tabic
(I'.f
(p..S
((>.(•
(p..s-
(p..
(p..s
(p.r
log
VII
<•)
1 ) **'" .'/
. 1 ) COS (I
.2) .sin 2;/
.2) cos 2.7
.3) sin 3f/
3) cos 3;/
I<ig )•
1.3023222
2707
— 2(!
—112
— 00
+257
+ ;<i
— 14
1.320035
l^^ Table XIX —0
(l).c.O) Table XXI
XXII
XXIII
(li.xA) sin g
{h.cA) I'OHg
(fix. 2) sill 2.7
(fc.f.2) cos23
Latitude —
40
40
3.
+ "
—
—
—
4
"73
.10
.09
.20
.11
.00
.00
.10
.35
As a second example wc will take the comptitation of an ophemoris for tiie years
187G and 1877. AVe take as the extreme dates 1875, December 15, and 1878, April
;}, between whii^h are seven intervals of I'iO days eneb, wbich wi- adopt as those
of compulation. We lirst form the uryiuueuts for the e.xtn'uie dates as follows :
■f'f
200
1 a i: O 11 D 1 T OF U U A N u s.
1. For 1875, Dec. 1,'> — 1875.90.
9
u
Arg. 1
Table 11, 1872
III, 3 Y. Doc.
IV, 15ihiy«
VI, 1875.90
a 1 It
314 25 6S.70
IG 4(! CC.7C
10 33.. 50
2.20
95
1
9 3
2
If
5.04
4.00
1.30
/ 1/
73 20 40.0
1 12.7
0.7
7
>
574.815
170.072
1.784
+ 0.420
147.097
For 1875, IH'o. 15
331 23 5.10
95 11 41.00
73 22 0.15
Arg. 2
3
4
5
C
7
8 9
Tubloll, 1872
III, 3 Y. Doc.
IV, 15 (liiv.s
VI, 1875.90
208.328
51.785
0.543
— 1
320.055
105.888
13.705
0.144
+ 2
114
39
154
13
423
05
1
49
73
1
380.3
21.1
0.2
577
78
1
For 1875, Doc. 15
179.739
l5;j
107
489
123
401.0
50
11. For 1878, April 3 = 1878.20.
9
M
Arg. 1
148.570
97.043
.357
.421
Table ir, 1870
III, 2 Y. April
IV, 3(1
VI, 1878.26
o 1 n
331 31 18.70
9 37 53.02
2 0.70
2.02
95
/ //
11 42.33
1 11.23
0.20
/ ;/
73 22 1.(1
41.7
0.1
5
5
For 1878, April 3
341 14 21.04
95 12 53.82
73 22 42.93
240.997
Arg. 2
3
4
5
G
7
8 9
Table II, 1870
III, 2 Y. April
IV, ;!(1
VI, 1878.20
321.237
29.732
.109
— .001
351.077
179.889
7.808
.029
4- .002
153
22
108
7
175
489
37
123
42
401.9 57
12.1 45
i
1
For 1878, April 3
187.788
175
520 105
414.0 102
We now fill in the values of (7, the arc;nments 1 — 9, and the times with which
Table XVII is to be entered, for the intermediate dates, by adding the nearly
constant differences dcdnced from the numbers at the bottom of Table II. The
seconds of <j arc first reduced to fractions of a minute, with which to enter Table
VII. In makinfT the subsequent computation we have used none of the devices
previously described except in the case of the small longitude terms, as follows :
THE OUBIT OF URANUS.
201
o
o
9
330
332
334
330
338
340
342
8?
270
270
2H2
288
204
300
son
iy
240
248
250
204
272
280
288
1)
n
>>
/'
II
II
/'
/,,
.(..3)
— 1.14
— 1.13
— 1.13
— l.ir,
— 1.23
— 1.33
— 1.48
(•'
.c'.:t)
— 2.4fi
— a-.-io
— 2. CO
— 2.7(1
— 2.88
— 3.01
— 3.10
(l
.^.4)
-f o.r.o
-f O.fiO
+ 0.,'')0
+ 0.55
+ 0.50
+ 0.43
+ 0.30
(l
.f.4)
— 0.10
— 0.20
— 0.30
— 0.40
— 0.47
— 0.57
— 0.04
(l
.X.3) Bill 3;;
+ 1.14
+ 1.12
+ 1.11
+ 1.11
+ 1.12
+ 1.15
+ 1.20
(t
.(•.;i) COS It;/
.00
— 0.20
— o.,^5
— 0.85
— 1.17
— 1.50
— 1.80
(l
.0.4) sill 4;/
— .,11
_ .M
— ..17
— .55
— ..50
— .42
— .34
(l
.r.4) ecis 4f7
4- .0,'j
+ .07
+ .07
+ .04
— .02
— .10
_ .19
Siiiu
+ O.fiS
+ 0.37
+ 0.00
— 0.25
— 0.57
— 0.87
— 1.19
It will bo s(>en tliat wo hiivo licro romputod twicu as many numbers as arc
nocos.sary to intcrpoliito with all attainable aixuiacy.
Tiio rest of tiio computatiim is fully jjivcn on the four followin«,' paj^os. First
wo have the values of ij and the nine arf,'umonts for the intermediate dates, filled
in by suecessivo additicm of the nearly constant diflerence. The ar<,'uinents thus
obtained for the last date may be compared with those just cimiputed on the pre-
ceding page.
The numerals in tin; first columns of the sceticms of computation following
indicate the argum(-nts with which tables arc entered to obtain tlie separate values
of the quantities (r.c.O), (r.«.l), (r.c.l), etc. Tlio negative terms in Table XVII
being taken from the sum of all the periodic terms from Tables VIII to XVI with
argument 1 to 9, we have the final values of {r.c.O), {v.s.l), etc.
The final computation of the products («.«.(') sin i;/, etc., and the addition of the
separate terms which make up the three co-ordinates, are shown on page 1205. The
expressions c.O, .s.l, etc., are employed for brevity, instead of (r.c.O), {v.s.l) sin </, etc.
The lon"itude finally given by the tables is referred to the mean equinox, and
must therefore be corrected for nutation before bouig used to compute the geocentric
place.
June, 1873
IPs
909
TI! !•: OIIBIT OF URANUS.
D.l«, 1
l»7.'>, l>H«. 16 I87'l, Apr i:i
Auk M
Di'c. »
Xm, Apr. H
Aiii.' H
!>..•, 4
1>7K. Apr 3
l»75.tf5D.
/
l»7(i.'>3.
lb7ii lili.
lb7i).U4(l.
11)77. ilijtf.
/
In77,r)li7,
f
lh77. !•-.'>.
/
/
/
o /
/
Hal 23. 081!
332 47.554
331 12.021
335 30.480
337 0,!»57
338 25,425
339 49.803341 14.3(ll|
ArK. 1
U-.(t!l7
ini.ac.s
175.(13!)
I80.!(l(t
204.182
218,453
232.724
240.0!t5
2
32ll.H,'ii'>
32.'). (Mil
aj',i.34(l
333.(102
3as.(i;is
342.383
34(1.720'
351,(174
a
iT.i.-ai)
1H0.«.S!(
1^2.038
l83. 1M8
ls.»,y;i8
1.S5.488
18(1.038
187.788
4
\:>:i
150.
150.
1(13.
ico.
109.
172.
175.
5
lea.
• • • . •
• . . . •
175.
ts
4S1).
4!H.
500.
505.
511,
5io.
521.
527,
1
i-.'a.
12!(.
135.
141.
117,
153.
ICO,
100,
8
4U1.C
403.4
405.2
407.0
408,7
410.5
412.3
414.1
•1
(D.O.O) 1
6(1.
(13.
(10.
10(1.2!)
75.
103.08
82.
88.
95.25
95.
89,02
101.
83.21
los.ol
107.74
0!l,il8
2
ir,..so
14.71
13.(1(1
12.(15
11,08
1((,70
0,00
0,00
S
72.4a
73.(10
74.77
75.04
77.10
78.20
70.42
80.50
.,■.;>
.58
.58
.57
.50
.50
.55
,54
l>
.(12
.02
.02
.03
.03
.03
.03
.03
7
.07
.07
.0(1
.00
.00
,00
.05
.05
8
l.CS
1.(15
i.(ia
1.00
1.58
1.55
1.53
1.50
(.'.<-.0)
.011
I'jH.di)
.011
li)8.4(i
.00
107.10
.00
lft4.02~
.08
.08
.08
.07
175.05
101.07
180.55
181.18
(r.«.l)l
fi.84
7.(18
8.40
8.00
9.44
9.74
9.01
0.08
:!
132.20
12.').48
118 82
112.21
105.07
99.2»
92.89
80.09
SVC. i
.IS
. 20
•)•)
.23
. 25
.27
.29
.31
:i
l.(l!l
1.07
1.08
1.12
].10
1.20
1.41
1.50
4
.13
.12
.11
.10
.10
.00
.09
.OX
f)
.!'.»
.10
.10
.10
.lit
.20
,20
.20
ii
.04
.04
.04
.05
.05
.00
.00
.07
7
.14
.14
.13
.13
.12
.12
.11
.10
H
1.28
1.28
1.27
1.2(1
1,20
1.25
1,24
1,23
y
z
.OS
142.17
.08
13r>.28
.08
130.34
.08
,07
118.34
.07
112.33
.07
100.27
.07
124.30
100.20
Tab. XVII
(r.,..l)
(u.e.l)l
— ir)4.21
—153.84
_lf,;j,47_
— 23.13
0.38
— 153.10
— 28.74
—152.73
—152.30
—151 99
— 45.72
— 151.(12
— 51.33
— 12.04
— 17.5(1
fi.03
— 34.30
— 40.03
5.fi4
0.07
0.85
0.01
0.S2
0.59
•2
5.01
fi 00
7-31
8.00
10.80
13.00
15.49
18.25
soc. •!
.CI)
.71
.72
.74
,75
.77
.79
.so
••i
23.20
24.00
24.05
25.82
20,08
27.53
28.37
20. 1
4
.47
.48
.40
.50
.50
.51
.frtJ
.52
f)
.37
.37
.37
.:;7
.37
.37
.37
.37
(i
.1(1
.17
.17
.18
.18
.18
.18
.18
7
.1.-.
.14
.13
.13
.12
.12
.11
.10
8
0. (iC.
• 0.(14
0.(13
0.01
0.00
0.50
0,57
0.50
.07
3(!.42
.07
38.70
.0(1
41.21
.00
.((5
.04
.04
.03
50,50
43.08
40.00
50,02
53.20
TuIlXVII
— 20.'). hi;
— 20(1.08
—20(1.20
—200.51
—200.73
—200.04
—207,15
—207,37
(f.r.l)
— 1(',!).44
— 1(17.38
—105.08
—102.53
— 150.83
.20
— 150.92
— 153.89
— 150.78
(W.S.2) 1
.or.
.10
.15
.21
.31
.34
.37
2
54.12
40.37
44.78
40.37
30,15
32.11
28.30
24.70
sec. 2
.3!)
.42
.44
.47
.40
.51
.53
.50
3
2
2.05
2.19
2.31
2.45
2.00
2.70
2.91
32,08
3.07
28.70
5r..fi2
52.08
47.08
43.50
30.50
35.00
Tuh.XVII
—1.34.34
—134.32
—134.31
— 80.03
—134.20
—134.27
— 04.77
— 134.20
— 08.57
— 134.24
— 102.i0
—134.22
— 105.52
|(.-..s.2)
— 77.72
— 82,24
_ 00.79
T H E II IJ IT OF U 11 A N U S
W3
l)M»,
iHT.'i, D.'O. ID
Ih7li,l»,'j6.
Ih7i). Apr. l:(, Aux. 11 I Of,'. »
Itilii.M.i. \ lb7il UU. I lf<U7.»4ll.
9
(u.c.2) 1
2
IOC. 2
3
S
Tall. XVI 1
(«,o.2)
(u.«.3) 2
3
Tub. XV 11
(o.tr.3)
331 23.0S(>
.31
2"> 34
.711
0.23
32.(14
-131). .'>7
-IU3.03
(u.0.3) 2
3
Tab. XVII
(o.f.3)
(u.hA) 2
3
(v.sA)
0.02
0.H7
7.70
— «.02
—1.13
8.19
1.34
4.. '^3
-7.0(1
O > O /
132 47.. '154 334 12.02133:) 3(1. 4S0
.30
2H.1)3
."(1
(1.40
3(1.30
— 13(i.fi0
—100.20
o.so
0,01
7. HO
— H.02
— 1.12
3.12
1.34
4.4(1
—".(((1
— 2.r)3
1.4S
— OSS
-|-o.(;(»
—2.00
1.4S
— O.SH
-f 0.(10
(u.c.4)2
3
(■U.C.4)
(p.<.'.0) 1
2
3
Tab.XVll
(ff^)
(p.».l) 1
.J
3
S
Tab.XVll
(p.S.l)
0.S3
— 1.00
—0.17^
1230
OS
11
0(18
(p.c.l) 1
2
3
Tab.XVll
(p.c.l)
2307
248
2S.J-)
173
32,')(1
— 10S4
+ 1272
123
12(1(1
CO
14,'>5
-1077
0.77
—1.02
-0.2')
10(13
00
10
0(18
2140
240
2S22
KIH
3230
-10S7
1243
.20
32.71
,77
(1,5(1
40,33
— 13(1,(10
— 0(1.27
112
1202
66
13H0
-lOSl
f..H4
_ 0.04_
7.78
—8.01
— 1.13
3.05
l.>14_
4,30
—7.0(1
—2.(17
.20
36.73
.78
6.70
44.50
—13(1.62
— 02.12
6,78
0,08
'^^ 7.76
—8.01
~^1.15"
2.08
1.34
4.32
—7,06
—2.74
1.47
— 0.H8
-+-0.50
0.72
—1.03
—0.31
800
lol
9(18
"l077
522 — 601
230
2S07
1(15
3202
-1080
1213
07
1138
67
1.302
-1086
- 684
1.45
-0.80
-f 0.5(1
0.66
—1.04
-0.38
741
104
967
~i82^r
221
2780
161
lh77, Apr. H I
l»77.l!tl!». I
AiiK. II
lh77 f.l>7.
IIhii. 4
Ml MM.
o / I o > o /
U37 0.057 338 25.425 339 49.893
.20
40.03
.70
6.85^
48,86
—136.64
— 87.78
6,70
1,01
7.71
— H.Ol
— 1.20
.30
45.35
.79
(1.99
63.43
—136.06
30
40 03
.HO
7.10
68.13
—136.67
— 83.28
— 78.64
6.61
1.04
6.51
1.08
2,00
1,33
4,23
-7.0(1
-2.83
1,43
— o.si)
4(1.54
0.61
—1.05
—0.44
504
108
s
967
1677
211
2760
158
3171
-1001
1180
3138
-1 003
1145
83
1074
67
1224
—1000
— 766
69
1000
68
1146
—1094
— 848
7.65
—8.91
2.82
_l,33
4,15
—7.0(1
—2.01
7.50
—8.01
"—1.32
2.75
1.31
4.06
-7.0(1
—3.00
1.40
—0.01
+0.40
0.57
— 1,0(1
—0.40
460
113
8
967
1548
203
2748
155
3106
— 1006
~^ 1110
55
047
70
1072
-1098
— 926
3.36
—0.03
+ 0.43
0.51
— 1.07
—0.56
314
118
8
06(1
I»<78, Apr 3
IH-M 2!i4.
341 14.361
.31
54.60
.NO
7,20
63.00
— 136.69
— 73.69
6.37
1,11
7.48
—8.01
1436
104
2723
151
3068
-1008
1070
-1.43
2.67
1.29
3.06
-7.06
-3.10
1.33
_ .04
+0.30
0.47
— 1.08
—0.61
246
126
9
966
1347
187
2606
148
3031
2000
1031
41
886
72
31
827
74
909
-2002
-1003
932
2006
—1074
ilftl
i-'<i
204
THE ORBIT OF URANUS.
T)ate, 1
1875, Dot. 16
1 875.1.(65.
187fi, Apr. 13
1870.283.
o /
332 47.654
181
25
Aug. 11
l87ti.Uia.
o /
334 12.021
189
25
lief.
18(i7.»40.
o /
335 .36.489
196
24
4l>77. Apr. 8
lt>77.2ii».
o '
337 0.!)57
204
24
228
—460
—232
Ann. 6
1877.597.
Dfo. 4
1877.925.
1878, Apr. 3
1878.264.
9
(p.8.2) 2
3
S
Tab. XVI I
(p.8.2)
o /
331 23.0S(i
174
2fi
o r
.338 25.425
213
24
o /
339 49.!- 93
223
23
o /
.341 14.361
232
23
200
—459
20C
—459
2i4
—459
220
—459
—239
237
—460
~— 223~
246
—460
255
—460
—259
—253
—245
576
34
—214
—205
(p.c.2) 2
3
S
Tab. XVI I
(p.c.2)
559
33
5fiS
33
584
35
6!)2
36
600
37
607
39
614
40
592
-404
+ 128
601
—464
610
—464
619
—465
628
—465
n;3
637
—465
646
—465
654
—466
137
146
154
39
—101
172
181
188
(p.s.3) 2
Tab.XVII
(p.8.3)
(-^.^.3) 2
Tab.XVII
(p.c.3)
28
—101
31
—101
34
—101
43
— 101
47
— 101
52
—101
57
—101
— 73
— 70
— 67
— 62
— 58
— 54
— 49
175
—102
73
— 44
155
— lOJ
+ 53
0.12
0.11
0.07
0.30
159
—102
57
162
—102
166
—102
169
—102
67
172
—102
177
—102
75
60
64
70
(6.C.0) 2
.'t
Tab.XXIIl
(6.C.0)
0.12
0.11
0.07
0.13
0.11
0.07
0.13
0.11
0.07
0.31
0.14
0.11
0.07
0.32
0.14
0.11
0.07
0.32
0.15
0.11
0.07
0.15
0.11
0.07
0.30
0.31
0.33
0.33
(6.S.1) 1
•2
3
Tab.XXlIl
(6.S.1)
0.30
0.12
l.ll
1.53
—5.23
—3.70
1.21
O.'.X!
1. 01
0.22
0.09
1.11
1.42
—5.24
—3.82
0.16
0.06
1.12
0.12
0.05
1.12
0.08
0.04
1.13
1.25
—5.24
—3.99
0.06
0.04
1.13
1.23
—5.25
4.02
0.05
0.05
1.14
1.24
—5.25
0.06
0.06
1.14
1.26
—5.25
— 3.99
1.34
—5.24
3.90'
1.06
1.17
1.00
3.23
—4.43
—1.20
1.2!)
—5.24
—3.95
—4.01
(6.^.1)1
2
3
S
Tab XXIII
(6.P.1)
1.14
1.06
1.00
0.!)8
1.28
0.99
3.25
—4.43
— 1.18
0.04
0.15
—0.48
—0.29
0.34
0.10
—0.38
+0.06
0.!10
1.39
0.99
3.28
—4.42
— 114
0.80
1.51
0.99
3.. 'id
4.41
—1.11
0.70
1.63
0,98
3.31
—4.4 1
—1.10
O.fil
1.75
0.98
3.1:i
—4.45
-1.27
3.20
—4.44
—1.24
3.34
—4.40
— 1.06
(6.8.2) 2
3
Tab.XXIIl
(6.8.2)
02
O.K!
— 0.4S
-0.30
0.31
0.10
— 0.3>i
4-0.03
0.02
0.16
—0.48
—0.30
0.03
0.16
-0.48
—0.29
0.04
0.15
—0.48
—0.29
0.05
0.15
—0.48
—0.28
0.06
0.15
—((.48
~— 0.27
30
0.11
—0.38
+0.09
0.07
0.15
—0.48
—0.26
(6.C.2) 2
3
Tab.XXIIl
(6.„.2)
0.32
0.10
—0.38
0.33
0.10
—0.38
f0.05
0.34
0.11
—0.38
0.35
0.11
—0.38
0.36
0.11
—0.38
40.09
+0.04
+0.07
+ 0.08
THE ORBIT OF URANUS,
205
Date, 1
187.'), D™. 15
; Ih". , April 13
1 AnK- 11
Dtto. 9
1877, April 8
A UK- •!
Deo. 4
1878, April 3
IbTS.aOO.
1 lbTti.2»3.
1 lH7ti.(il2.
lb7li.94U.
lS77.2lii).
1877.697.
1877.925.
1878.2S4.
r
/
1 ° '
/
o /
/
o /
O 1
9
331 23.080
333 47.554
334 12.021
335 30.489
337 0.957
338 25.425
339 49.893
341 14.301
Iog(u.8.1)
— i.osor.
—1.2445
—1.3042
—1.4585
—1.5305
—1.0024
-1.6001
—1.7104
8111 g
_9.(;ho3
— 9.(;(;oi
—9.0380
—9.0159
—9.5910
—9.50.55
—9.5375
—9.5073
l0g(p.8.1)
+3.1045
3.0944
3.0838
3.0719
3.0588
3.0453
3.0294
3.0132
log(u.e.l)
—2.2290
—2.2237
—2.2177
.^2.210
—2.20.30
—2.1957
—2.1872
—2.1784
cos 7
+ 9.9434
9.9491
9.954
9.9594
9.9041
9.9085
9.9725
9.9703
log(p.f.l)
—2.718
—2.779
—2.835
—2.884
—2.928
—2.907
—3.0013
—8.0310
l0)r(«.,S.2)
— 1.890.')
—1.9151
— 1. 9370
—1.9580
-1.9700
—1.9937
—2.0093
—2.0233
sill iff
—9.9247
— 9...102
—9.8941
—9.8703
— 9.85(;0
—9.8350
—9.8111
—9.7840
log(p.s.2)
—2.413
—2.403
—2.389
—2.378
—2.305
—2.348
—2.330
—2.312
log(u.,..2)
— 2.01(;7
—2.0009
—1.98.35
—1.9044
—1.9434
—1.9202
—1.8951
—1.8074
CDS 2(/
+9.7335
9.7049
y.7932
9.8189
9.8421
9.8030
9.8821
9.8994
K>fr(p.<-.2)
+2.107
2.137
2.104
2.187
2.212
2.230
2.258
2.274
o ' n
■ '/
O f
/ '.•
t ft
/ n
O / II
1 II
A!7
1 24 28.0(15
1 24 2S.O00
1 24 ':>..000
1 24 28.007
1 24 28.008
1 24 28.008
1 24 28.009
a
331 23 fj.lO
332 47 33.225
334 12 1.291
3,35 30 29.357
337 57.424 338 25 25.492
339 49 53.500
341 14 21.63
u
95 11 41.00
95 11 61.40
95 J2 1.81
95 12 12.21
95 12 22.01
95 12 33.01
95 12 43.42
95 12 53.82
E
—2 42 49.15
—2 35 33.18
—2 28 10.i;3
—2 20 41.70
—2 13 0.93
—2 5 20.43
— 1 57 40.55
— 1 49 49.00
c.O
3 18.i;9
3 18.40
3 17.10
3 14.02
3 11.07
3 6.55
3 1.18
2 55.05
8.1
6.77
8.03
lO.Oi
11.87
13.43
14.72
15.70
10.51
e.l
—2 28.72
—2 28.80
—2 28.02
—2 28.00
—2 27.13
—2 25.93
—2 24.43
—2 22.80
«.2
1 5.34
1 0.88
1 7.87
1 8.28
1 8.12
1 7.40
1 0.13
1 4.20
r.2
— 50.27
— 68.31
— 59.. '^O
—1 0.71
—1 1.03
—1 0.70
— 69.87
— 58.45
(3 + 4)
0.40
0.25
0.03
— 0.19
— 0.41
— 0.03
— 0.83
— 1.00
11
03 .'■)3 2.28
05 24 57.89
00 50 59.12
08 29 5.08
70 1 17.15
71 33 33.48
73 5 54.37
74 38 19.30
e
73 22 0.15
73 22 0.20
73 22 12.37
73 22 IS. 48
73 22 24. CO
73 22 30.71
73 22 30.. s2
73 22 42.93
R
2.5,-i
2.91
138 47 7.00
1.2(!44735
3.25
3.01
3.98
4.37
4.79
6.22
143 1 7.61
Ldiigitudc
l.n 15 5.01
140 19 14.74
141 51 27.77
143 23 45.73
144 50 8.50
140 28 35,98
l"f? '•o
1.2i;4-392
1.2042190
1.2(i39778
1.2037480
1.2035319
1.2033280
1.2031370
c.O
2307
2140
1977
1821
1077
1548
1436
1347
s.l
—009
—508
—528
—487
—447
—408
—309
— 332
e.l
—458
—535
—015
—097
—780
—801
—941
—1017
S.2
+ 218
200
192
179
107
152
138
12:
C.2
+ 09
80
91
101
113
120
138
149
8.3
+ 73
09
05
CO
64
49
43
37
C.3
+ •*
1.2048990
8
13
19
U
29
80
41
log r
1.2040135
1.2043391
1.2G40774
1.2038293
1.2035954
1.2033701
1.2031720
1 H
/ II
' //
> It
' //
3«
+ 41 30.04
42 8.4S
42 .'IS. 54
43 0.78
43 33.14
43 57.02
44 20.23
44 41.19
c.O
4().:!0
0.30
0..I1
0.31
0.32
0.32
0.33
0.33
.'.■l
+ 1.77
1.75
1.70
1.03
1.57
1.48
1.38
1.29
e.l
—1.11
—1.1(1
—1.08
—1.07
— 1.05
—1.03
—1.03
— 1.00
8.2
+0.25
0.21
0.23
0.22
0.21
0.19
0.17
0.10
0.2
Latitudo
+ 0.02
0.03
42 9.70
0.03
42 39.73
0.04
43 7.91
0.05
0.06
0.07
0.07
44 42.04
+ 41 37.87
43 34.24
43 68.04
44 21.15
li:
206
THE ORBIT OF URANUS.
TABIiE
I. CoUEKCTIONS OF AKOUMENTS TOR PAST AND FUTURE t'ENTURl
E8.
Arg. 1
Century.
9
u
u
6
0'
O
/
tt
t
//
//
o
1 It
tt
OJ
207
15
59.32
343
52
17.36
+ 108.00
351
C 26.89
—148.32
408.924
100
275
45
34.18
344
40
54.12
102.00
351
34 55.39
—140,08
552.952
200
344
15
9.04
345
41
24.88
96.00
352
3 32.13
—131.84
96.980
300
52
44
43.90
346
35
49.64
90.00
352
32 17.11
—123.00
241.008
400
121
14
18.76
347
30
8.40
84.00
353
1 10.33
—115.36
385.036
500
180
43
53.02
348
24
21.10
-1- 78.00
353
30 11.79
—107.12
529,004
COO
258
13
28.48
349
18
27.92
72.00
£j3
59 21.49
— 98.88
73.092
700
326
43
3.34
350
12
28.08
60.00
354
28 39.43
— 90.64
217.120
800
35
12
38.20
351
6
23.44
00.00
354
58 5.61
— 82.40
301.148
900
103
42
13.06
352
12.20
54.00
355
27 40.03
— 74.16
505.170
1000
172
11
47.92
352
53
54.96
+ 48.00
355
57 22.69
— 05.92
49.204
1100
240
41
22.78
353
47
31.72
42.00
356
27 13,59
— 57.08
193.232
1200
309
10 57.64
354
41
2.48
30.00
356
57 12.73
— 49.44
337.200
1300
17
40
32.50
355
34
27.24
30.00
357
27 20.11
— 41.20
481.288
1400
80
10
7.36
356
27
40.00
. 24.00
357
57 35.73
— 32.96
25.310
1500,1
154
39
42,22
357
20
58.70
+ 18.00
358
27 59.59
— 24.72
169.344
15000
154
32
39.89
357
20
57.89
18,00
358
27 59.08
— 24.72
168.1.55
1(100
22.1
2
14.75
358
14
4.05
12.00
358
58 31.18
— 16.48
312,1.H3
1700
291
31
7.37
359
7
5.33
0.00
359
29 11.47
— 8.24
450,092
1800
0.00
0.00
0.00
0.00
0.00
0.000
1000
08
28
52.63
52
4S.67
— 6.00
30 56.7V
+ 8.24
143,908
2000
130
58
27.49
1
45
31.43
— 12,00
1
2 1.83
16.48
287,936
2100
205
27
20.11
2
38
8.11
— 18.00
1
33 15,08
24.72
431.845
2200
273
56
12,74
3
30
38.78
— 21.00
2
4 30.57
+ .32.96
575.755
TABLK
11— A
HaUMKNTS FOR THE BEQINNINQ OK EACH FoURTH
Year 17
52—1948.
Year.
9
U
Arg. 1
o
1
ti
o
t
ft
O
t
tt
1 , 1752
160
15
8,10
94
6
10,48
7r
43
42.30
102.101
1756
177
23
31.10
94
8
17.46
1 *d
44
56.25
335.8(12
1760
194
31
54.09
94
10
24.43
72
40
10.22
509,023
1704
211
40
17.09
94
12
31,39
72
47
24.21
83,384
1708
228
48
40.08
94
14
38.34
7i
"H
38,20
257.115
1772
245
57
3.08
94
10
45.28
72
49
.52.21
430.907
1776
■ 1780
263
5
20.07
94
18
52.22
72
51
0.23
4.068
280
13
49.00
94
20
59,14
72
52
20,27
178.429
1784
297
22
12.00
94
23
6.05
72
53
34.32
352,190
1788
314
30
35.05
94
25
12.96
72
54
48,38
525,951
1792
331
38
58.05
94
27
19,85
72
56
2,40
99,712
1796
348
47
21,04
94
29
26,74
72
57
16 54
273.473
1800
6
55
1,.>S0
94
31
33,53
72
5H
3(1,59
447,115
■
TlIK ORBIT OF URANUS.
207
TA15LE I
. — Continued.
Century.
2
3
4
5
6
7
8
9
OJ
191.528
299.485
280
512
102
61
470.0
15
100
314.245
49.520
04
250
503
124
410.6
214
200
430.902
399.555
449
589
424
188
351.2
413
300
559.079
149.590
233
327
285
251
291.7
13
400
82.396
499.025
18
05
146
314
232.3
212
500
205.113
249.600
402
403
1
378
172.9
411
fiOO
327.830
599.095
187
142
468
441
113.5
10
700
450.547
349.730
571
480
329
604
54.1
209
800
573.204
99.705
356
218
100
507
594.6
409
900
95.981
449.800
140
550
51
31
536.2
8
1000
218.098
199.835
624
295
512
94
475.8
207
1100
341.415
649.870
309
33
373
157
410.4
400
1200
404. 1?2
299.905
93
.•^il
234
221
357.0
5
13110
680.849
49.940
478
109
95
284
297.6
205
1400
109.506
399.975
202
448
660
347
238.1
404
1500 J
232.283
150.010
47
186
417
411
178.6
3
1500CJ
231.921
149.914
47
186
417
410
178.3
2
1000
354.038
499.949
431
524
278
473
118.8
201
1700
477.319
249.975
216
202
139
537
59.4
401
1800
0.000
0.000
1
11100
122.081
350.025
385
338
461
63
540.6
199
2000
245.398
j 100.000
109
76
322
127
, 481.2
398
2100
308.079
460.080
554
415
183
IflO
421.8
697
2200
490.700
200.111
338
153
44
253
' 362.3
196
TABLE U. — Continued.
Year.
2
3
4
5
6
229
7
8
9
1762
481.104
346.855
133
348
213
331.6
678
1766
534.013
369.856
172
362
296
287
353.2
58
1760
586.921
373.858
212
375
362
302
374.8
138
1704
39.830
387.869
251
389
429
436
396.4
218
1768
92.739
401.860
200
402
495
511
418.1
298
1772
145.047
415.862
330
416
562
585
439.7
377
1776
198.666
429.803
369
429
28
60
461.3
467
1780
251.465
443.865
408
443
95
134
482.9
537
1784
304.373
467.866
448
467
161
209
504.6
17
1788
357.282
471.868
487
470
227
283
526.2
97
1792
410.191
485.860
527
484
204
358
547. S
177
1790
403.100
499.870
566
497
360
433
569.4
257
1800
515.972
513. S62
6
611
427
607
691.1
337
208
THE ORBIT OF URANUS.
TAHLE
II. — Continued.
Year.
V
(J
Arg. 1.
/
/'
t i>
/
tt
1800
5
55
1.8m
94
31 33.53
72
58
30.59
447.115
1801
23
3
24.80
94
33 40.39
72
59
44.71
20.876
1808
40
11
47.79
94
35 47.25
73
58.84
194.638
1812
57
20
10.79
94
37 54.09
73
2
12.98
368.399
18IG
74
28
33.78
94
40 0.93
73
3
27.13
542.100
1820
91
36
56.77
94
42 7.76
73
4
41.30
115.921
1824
108
45
19.77
94
44 14.58
73
5
65.48
289.683
1828
125
53
42.76
94
46 21.38
73
7
9.67
463.444
1833
143
2
5.76
94
48 28.18
73
8
23.88
37.205
1830
160
10
28.75
94
50 34.97
73
9
38.10
210.966
1840
177
18
51.75
94
52 41.75
73
10
52.33
384.727
1844
194
27
14.74
94
54 48.52
73
12
6.58
558.488
1848
211
35
37.74
94
56 55.28
73
13
20.84
132.249
1852
228
il
0.73
94
59 2.03
73
14
35.11
306.010
185G
245
6a
23.7b
95
1 8.77
73
15
49.40
479.771
1800
203
46.72
95
3 15.50
73
17
3.70
53.532
1804
280
9
9.71
95
5 22.22
73
18
18.01
227.293
1868
297
17
32.71
95
7 28.94
73
19
32.34
401.054
1872
314
25
55.70
95
9 35.64
73
20
46.67
674.815
1876
331
34
18.70
95
11 42.33
73
22
1.03
148.576
1880
348
42
41.69
95
13 49.01
73
23
15.39
322.337
1884
5
51
4.69
95
15 55.69
73
24
29.77
496.098
1888
22
59
27.68
95
18 2.35
73
25
44.16
69.860
1892
40
7
50.68
95
20 9.01
73
26
58.57
243.621
1896
57
16
13.67
95
22 15.05
73
28
12.98
417.382
1900
74
23
54.43
95
24 22.20
73
29
27.36
691.024
1904
91
32
17.43
95
26 28.83
73
30
41.81
164.785
1908
108
40
40.42
95
28 35.44
73
31
56.26
338.546
1912
125
49
3.41
95
30 42.05
73
33
10.74
512.307
1916
143
57
26.41
95
32 48.64
73
34
25.22
86.068
1920
160
5
49.40
95
34 55.23
73
35
39.72
259.830
1924
177
14
12.40
95
37 1.81
73
36
54.23
433.591
1928
194
23
35.39
95
39 8.38
73
38
8.75
7.. 35 2
1932
211
30
58.39
95
41 14.94
73
39
23.29
181.113
1930
228
39
21.38
95
43 21.48
73
40
37.84
354.874
1940
245
47
44.38
95
45 28.02
73
41
52.40
628.635
1944
262
56
7.37
95
47 34.55
73
43
6.97
102.396
1948
280
4
30.36
95
49 41.07
73
44
21.56
276.158
A '1'
1
24
28.007
10.411
6.100
14.2715
Factor T
4-0.222
—.020
+ .027
—.0012
A.'S
+ .0007
—.0001
+.0001
THE Oil BIT OF URANUS.
209
TABLE U.— Continued.
2
3
4
5
6
427
7
8
9
337
1800
r)ir).972
513.802
5
511
507
591.1
1804
r)08.881
527.803
45
524
493
582
12.7
417
1808
21.790
641.805
84
538
500
50
34.3
497
1812
74.098
555.806
124
.551
20
131
.55.9
577
181(1
127.007
609.808
103
605
93
205
77.5
67
1820
180.510
583.809
202
578
159
280
99.2
137
1824
233.424
597.870
242
592
225
354
120.8
217
1828
2.S0.333
11.872
281
292
429
142.4
297
1832
339.242
25.874
320
19
358
503
104.0
377
1830
392.150
39.875
300
32
425
678
185.7
457
1840
445.059
53.870
399
40
4 91
52
207.3
537
1844
497.908
07.878
439
59
557
127
228.9
17
1848
550.870
81.879
478
73
24
201
250.5
97
1802
3.785
95.880
517
87
90
270
272.1
177
1850
50.094
109.882
.557
100
157
350
293.8
257
iHfiO
109.002
123.883
591!
114
223
425
315.4
337
18(!4
li;2.51I
137. S85
35
127
290
499
337.0
417
1808
215.420
151.880
75
Ml
350
574
358.0
497
1812
208.328
105.888
114
154
( 23
49
380.3
577
isto
321.237
179,889
153
108
489
123
401.9
57
1880
374.140
193.890
193
ISl
555
198
423.5
137
1884 r
427.054
207.892
232
195
22
272
445.1
217
1888
479.903
221.893
271
208
88
347
400.7
297
1892
532.872
235.895
311
2:>2
155
421
4X8.4
37C
1800
585.780
249.890
350
235
221
490
510.0
450
11)00
38.053
203.889
390
249
287
570
531.0
530
11)04
91.502
277.890
429
2i;2
354
45
553.2
16
11)08
144.470
291.891
408
270
420
119
574.9
96
1912
197.379
305.892
508
290
487
194
590.5
176
1910
2.50. 2S8
319.894
547
303
553
208
18.1
256
1920
303.197
333.895
5S7
317
20
343
39.7
336
1924
350.105
347.897
20
3:!0
80
417
01.4
410
1928
409. OH
301.898
05
341
152
492
83.0
496
1932
401.9.!
-;..900
105
357
219
500
104.0
576
1930
514.831
389.901
144
371
2S5
41
i 120.2
56
1940
507.740
403.1^2
184
384
352
110
147.8
130
1944
20.049
417.994
223
398
41S
1 90
109.5
216
1948
73.557
431.905
1.1500
202
411
485
205
0.1
, 191.1
1.78
296
6.6
4.3457
3.2
1.1
5.4
— .0001
-f .0002
S
7 June. 187
3.
210
THE ORBIT OF UIIANUS.
TAI5LK III. — IlEDiiCTioN OF THE EpociiH AND Akqumentb to tue beuinmnu or EACH Month
IN A Cycle of fouu ykaus.
YearO
January
Ft'bruary
March
April
May
June
July
Au^cust
St'ptemljer
October
Xovcinber
Ueccuiber
Year!
January
Fi'liruary
March
April
May
Juno
July
August
Scptoniber
October (»
Nuvembor
Ueceiuber
Year 2
January
February
March
April
May
June
July
August
September
October
November
Ueceiuber
Years
January
February
March
April
Jfay
June
July
Auirust
S('|)teniber
October
November
Ducembur
9
0.00
21 411.24
42 14.00
1 4 3.24
. 9
10
10
II
11
11
12
12
14
14
14
15
15
10
Hi
IG
2o
4(>
8
2!)
51
12
;54
55
10.24
60. 4S
0.48
55.71
44.95
51.95
41.19
48.19
4 17 37.42
4 39 2(1.1)6
4 59 9.19
5 20 58.43
42
3
25
40
8
29
51
12
59
20
41
3
25
40
8
29
15
37
58
20
42
3
25
4U
5.43
54.07
1.07
50.90
40.14
47.14
30.38
43.38
8 34 32.01
8 50 21.85
9 10 4.38
9 37 53.02
0.02
49.80
60.80
40.09
35.33
42.33
31.57
38.57
12 51 27.80
13 13 17.04
13 32 59.57
13 54 48.81
55.81
45.05
62.05
41.28
30.52
37.52
20.70
33. 7C
0.00
2.09
5.21
7.89
10.60
13.19
15.79
18.48
21.17
23.77
20.40
29.06
31.75
34.44
30.87
39.56
42.10
44.85
47.46
50.14
62.83
55.44
68.13
1 0.73
1 3.43
I 6.11
1 8.64
1 11.23
1 13.83
1 l(i.52
1 19.12
1 21.81
1 24.50
1 27.10
1 29.79
1 32.40
1 35.08
1 37.77
1 40.20
1 42.89
1 46.50
1 48.18
1 60.79
1 63.48
1 66.17
I 68.77
2 1.46
2 4.06
//
0.00
1.68
3.05
4.63
6.15
7.73
9.25
10.83
12.40
13.93
16.61
17.03
18.01
20.18
21.61
23.18
24.71
20.28
27.81
29.38
30.96
32.48
34.00
35.59
37.16
38.74
40.10
41.74
43.20
44.84
40.30
47.94
49.52
51.04
52.62
54.14
1
1
1
1
55,72
57.29
68.72
1 0.29
1.82
3.40
4.92
6.50
1 8.07
1 9.60
1 11.17
1 12.70
&
0.00
0.00
0.00
O.Ol
0.01
0.01
0.01
0.01
0.01
0.02
0.02
0.02
0.02
0.02
0.03
0.03
0.03
0.03
0.03
0.04
0.04
0.04
0.04
0.04
0.04
0.05
0.05
0.05
0.05
0.(]^
0.06
0.06
0.00
0.00
0.00
0.07
0.07
0.07
0.07
0.07
0.08
0.08
0.08
0.08
0.08
0.08
0.09
0.09
Arg. 1
0.000
3.087
7.136
10.823
14.391
18.078
21.647
25.333
29.019
32.587
30.274
39.842
43.529
47.210
50.640
64.233
57.801
01.488
65.056
68.742
72.429
76.997
79.0S4
83.252
86.939
90.020
93.956
97.643
101.211
104.898
108.400
112.153
115.840
119.4(17
123.094
120.662
130.349
l:i4.o;i(!
137.366
141.053
144.021
148.308
151.8(6
155.563
169.249
102.S17
100.504
170.072
TlIK ORBIT OP URANUS.
211
TAULK III
— Cuntinued.
2
3
4
5
6
7
8
9
YearO
January
0.000
0.000
0.0
February
1.123
0.297
1
1
1
0.5
2
March
2.209
0.575
2
1
3
3
0.9
3
April _
3.25a
0.872
2
1
4
4
1.3
5
May
4.382
1.159
3
1
5
1.8
7
June U
6.505
1.457
4
1
7
7
2.2
8
July
C.5!)l
1.745
5
2
8
9
2.7
10
August
7.714
2.042
2
10
10
3.1
12
Scptombcr
8.837
2.339
7
o
11
12
3.0
13
October
«J.y23
2.02G
7
3
12
14
4.0
15
November
11.040
2.923
8
3
14
15
4.5
17
December
12.132
3.211
9
3
15
17
4.9
18
Yearl
January
13.254
3.508
10
3
17
19
5.4
20
February
14.377
3.805
11
4
18
20
6.9
22
March
15.301
4.073
11
4
19
22
0.3
23
April
1C.514
4.370
12
4
21
23
0.7
25
May
17.000
4.058
13
4
22
25
7.2
27
June
18.723
4.955
14
5
24
27
7.0
28
July
10.809
5.242
15
25
28
8.1
30
August
20.932
6.539
15
5
20
30
8.5
32
September
22.054
5.830
10
28
31
9.0
33
October
23.140
0.124
17
29
33
9.4
35
November
24.203
0.421
18
30
34
9.9
37
Uecomber
25.340
0.709
19
32
30
10.3
38
Year 2
January
21). 472
7.000
i^O
1
33
37
10.8
40
February
27.505
7.303
20
7
34
39
11.3
42
March
28.009
7.571
21
7
30
40
11.7
43
April
29.732
7.808
22
■y
37
42
12.1
45
May
30.818
8.150
23
8
3S
44
12.0
47
Juuo
31.941
8.453
24
8
40
45
13.0
48
July
33.027
8.741
25
8
41
47
13.5
60
August
34.150
9.038
25
9
43
48
13.9
52
Sei)tembcr
35.272
9.335
20
44
50
14.4
53
October
3(1.358
9.022
27
9
45
51
14.8
56
November
37.481
9.919
28
<t
47
53
15.3
57
December
38.507
10.207
29
10
1
48
55
15.7
58
Year 3
1
January
39,090
10.504
30
1 10
50
50
10.2
00
February
40.813
10.801
30
10
\ 51
58
10.7
02
March
41.827
11.070
31
11
1 52
59
17.1
03
April
42.950
11.307
32
' 11
54
01
17.5
. 05
1
May
44.030
11.0.54
33
1 11
55
02
18.0
i 07
Juno
45.159
11.951
34
11
57
04
18,4
08
July
40.245
12.239
35
12
58
05
18.9
70
A ugust
47.308
12.530
35
12
69
07
19.3
72
September
48.490
12.833
30
12
01
08
19.8
73
October
49.570
13.121
37
12
02
70
20.2
75
Noveml)er
50.099
13.418
38
13
04
72
20.7
77
December
51.785
13.705
39
13
Oj
73
21.1
78
a
}i
^m
212
THK OllUlT OF UKANLS.
!r
T.UILK IV.—
.Motion oir Auoiments for D.wa.
Day».
9
u
1
3
3
4
5
G
7
»
9
1 II
II
II
1
42.23
0.09
0.05
0.119
0.03(1
0.010
u
u
2
1 24.47
0.17
0.10
238
0.072
0.019
3
2 (■..70
0.2(i
0.15
0.357
0.109
0.029
u
4
2 48.93
0.35
0.20
0.470
0.145
0.039
u
0.1
5
3 31.17
0.43
0.2ft
0.595
0.181
0.048
0.1
f.
4 13.40
0.52
0.30
0.714
0.217
0.058
u
0.1
7
4 55. (;3
0.01
0.3(!
0.833
0.253
0.007
0.1
8
5 37.87
0.09
0.41
0.951
0.290
0.077
0.1
!)
(! 20.10
0.78
40
1.070
0.320
0.08(5
u
0.1
10
7 2.33
0.87
0.51
1.189
0.302
0.090
U.l
11
7 44.57
0.95
0.50
1.308
0.398
0.105
1
0.2
12
8 2(i.80
1.04
O.fil
1.427
0.434
0.115
]
O.-J
i;i
9 9.03
1.13
O.fifi
1.54G
0.471
0.125
0..
14
9 51.27
1.21
0.71
1.C05
0.507
0.134
*
0.2
15
iO 33.50
1.30
0.70
1.784
0.543
0.144
1 : 1
0.2
Id
11 15.73
1.39
0.81
1.903
0.579
0.153
1 1
0.2
17
11 57.97
1.47
0.80
2.022
0.010
0.163
1
0.2
18
12 40.20
1.50
0.91
2. Ml
0.052
0.173
0.3
19
13 22.4.!
1.(15
0.97
2.200
0.C88
0.182
1
U.3
20
14 4.f.7
1.74
1.02
2.378
0.724
0.193
^
0.3
21
14 4r,.90
1.82
1.07
2.497
0.700
0.201
0.3
22
15 29.14
1.91
1.12
2.filfi
0.797
0.211
0.3
23
1(1 11.37
2.00
1.17
2.735
0.833
0.220
1
0.3
24
K; 53. CO
2.08
1.22
2.854
0.8G9
0.230
0.3
25
17 35.84
2.17
1.27
2.973
0.905
0.240
0.4
2(;
18 18.07
2.2(i
1.32
3.092
0.941
0.249
0.4
27
19 0.30
2.34
1.37
3.211
0.978
0.259
1
0.4
2«
19 42.54
2.43
1.42
3.330
1.014
0.208
0.4
29
20 24.77
2.52
1.47
3.449
1.050
0.278
0.4
'■
30
21 7.00
2.(10
1.52
3.508
1.080
0.28S
0.4
31
21 49.24
2.G9
1.57
3.087
1.123
0.297
0.5
TAIU-I'] v.— MoTio.N OF <j K(ia Ilouits.
Hours.
.7
lloura.
.'/
Hours.
V
Hours.
//
II
II
II
o.no
fi
10.50
12
21.12
18
31.07
I
1.70
7
12.32
13
22.88
19
33.43
2
3.52
8
14.08
14
24.04
20
35.19
3
5.28
9
15.84
15
20.40
21
3ii.95
4
7.04
10
17.(10
IG
28.10
22
38. 71
5
8, so
11
19,30
17
29.92
23
40.47
fi
10.50
12
21.12
18
31.07
24
42,23
The period of nr{i;nments 1 to 9 is 600.
Ill January and Fi-liruiiry of thoso years which, though divisible by 4, are not
leap years, namely, 1700, ISOO, 15)00, 2100, etc., Table IV must be entered with
a number 1 greater than the real day of the month.
TUE OllBIT OF URANUS.
213
TAHLE VI.— CullllKUTlONS OK AllUUMENTS KOtt TKIIM.S (IF LtiNll PKlllnn.
Year.
u
1000
1010
1U20
1030
1040
lOJO
lOUO
1070
lOSO
lU'JO
1100
1110
11^0
1130
1110
1150
IKiO
1170
IKSO
11 'JO
1200
1210
1220
1230
1210
1250
12U0
1270
12S0
1200
1300
1310
1320
1330
1340
1350
I3(i0
1370
13S0
13'J0
1400
1410
1420
1430
1440
1450
14(10
1470
14«0
14!»0
1500
1510
1520
1530
1540
/ II
+30 51.00
3I> (•..49
35 22.14
84 37. '.)7
33 54.00
+33 10.95
32 2(;.71
31 43.41
31 0.37
30 17.01
+20 35.15
2S 53.01
2'< 11.20
27 20.71
2(i 48.5U
+2ii 7.71
25 27.27
2t 47.1(1
21 7.42
23 28.00
+22 40.0!)
22 10.51
21 32.31
20 54.57
20 17.23
-10 40.31
10 3. S3
is 27. so
17 52.21
+
17 17.0'J
+ lfi 42. n
IC. S.25
15 34.55
r- 1.31
14 2S.(;3
+ 13 5r,.44
13 24. 7(!
12 53. (;i
12 22.00
11 52.00
+ 11 23,35
10 54.35
10 25.02
5.S.04
9 30.74
4.01
37. SM
12.34
47.30
23.01
+ r. 59.20
(; 311.14
n 13.(11
5 51.00
+ 5 30.40
-0.014
0.021
0.020
0.03S
0.04S
-0.0.50 —
0.(172
O.IISO
0.702
0.71S
-0.735
0.752
0.770
0.700
0.800
-0.820
0.848
0.800
0.883
0.899
-0.914
f 0.927
0.039 ,
0.950
I 0.000
'—0.907
' 0.972
! 0.975
! 0,977
I 0.977 j
'—0.974
! 0.907
I 0.059
0.948
j 0.030
—0.021
0,003
! 0.8S3
O.SOl
0.837
— O.SIO
0.780
(.748
0.714
0.078
—0.041
0.(101
0.500
0.518
0.475
—0.431
().3S0
0.340
0.293
—0.247
-1.024
1.003
0.083
0.902
0.942
■0.022
0.902
0.882
0.802 1
0.842 :
-0.822
0.803
0.783
0.704 !
0.745 I
-0.720
0.707 i
0.089 j
0,070 i
0.052 I
-0.034
0.010
0.598
0,5S1
0.5U4
—0.547
0.530
0.513
0.407
0.4lil
—0,405
0.140
0,433
0,417
0.402
— 0.3S7
0.372
0.358
0.344
0.330
—0.317
0.303
0.200
0.277
j 0.204
—0.252
0.240
0.228
0.210
0.205
—0.104
j 0.1S3
! 0.172
I 0.102
— 0. 153
Year.
+ 1.702
1.750
1.720
1.0S4
1.040
+ 1.014
1.570
1.544
1.500
1.471
+ 1,430
1 . 405
1.370
1.337
1.304
+ 1.270
1.237
1.200
1.172
1.141
+ 1.110
1.07 ■>
1.040
1.017
0.980
+0.957
0.028
0. 898
O.S70
0.842
+0.814
0.7 SO
0.758
0.730
0.704
+0.(;77
0.051
0.020
0.002
0.578
+0.555
0.530
0.508
0.485
0.402
+0.441
0.420
0.399
0.378
0.359
1550
1500
1570
1580
1500
1000
1010
1020
1030
1040
1050
1000
1070
lOsO
1090
1700
1710
1720
1730
1710
1750
171.0
1770
1780
1700
1800
181)1
1802
1803
1801
1805
1800
1807
180S
1809
1810
1811
1812
1813
1814
1815
1810
1817
1818
1819
1820
1821
1822
1823
1824
+5
4
4
4
3
II
9.73
49.70
30.:!2
11.57
53.48
+3 30,03
3 19.20
3 3.14
2 47.09
2 32.89
+ 2 18.77
1
1
1
-fl
1
52.55
40.40
29.05
18.32
8.28
58.92
50.24
42.2(1
+
+0
+8!'84
8.49
8.15
7,82
7,50
+7.18
0.87
0.57
0.27
6.
31.90
28.30
22,44
17.22
12.08
— jj
•34
•33
•32
— 31
•30
•30
.2<)
• 2y
+5.00 _
5.41 '^'^
5.14 --7
4.88 -^^
4.02 -'^
.98
+0,340
1825
0,322
1820
0,303
1827
0,284
1828
+0.208
1829
+4.37
4,13'
3,s9
3,00
3.44
+ 3.23_
3.02 •
2. 82 ■
2.03 ■
2.44 •
+2.20_
2.09
1,92
1,70
j + 1.01
.24
.24
•23
.22
.21
—0.200
0,153
0,1 (h;
0.000
—0,013
+ 0.034
0.078
0,120
0.102
0.203
+0,242
0,278
0.312
0.345
0,375
fO,403
0,429
0,452
0,472
0,489
+0,503
0,514 I
0,522
0,527
0,520
+0,528
0,528
0,528
0,527
0.527
+0,527
0.520
0.520
0.525
0,525
+ 0,524
0,523
0,523
0.522
0.522
+ 521
0.520
0.520
0.510
0,518
17
■»7
.16
•■5
— li
+ 0.517
0.51(1
0.515
0.514
0.513
+0.512
I 0.511
I 0.510
i 0.509
—0.508
-0.144
0.134
0.125
0.117
0. 108
-0.100
0.092
0.085
0.078
0,071
-0,004
0.058
0.052
0,040
0,041
-0,030
0,031
0,027
0.023
0,019
-0,015
0.012
0.009
0.007
0.005
-0,004
0,004
0.004
0.004
0,004
—0,003
0,003
0.003
0.003
0.003
—0.003
0.003
0.003
0,003
0.002
-0.002
0,002
0,002
0,002
0,002
-0,002
0.002
0.002
0.002
0.002
-0.001
0.001
0.001
0.001
-0.001
+ 0.252
0.230
0.210
0.205
0.189
+ 0.175
0.101
0.148
0.135
0.123
+0.112
O.lOl
0.091
0.081
0.072
+ 0.003
0.054
0.040
0.030
0.032
+ 0.020
0.021
0,017
0.013
0,010
+0,007
0,007
0,007
0,007
0,007
+ 0,000
0,000
0,000
0.000
O.OOG
+0.005
0.005
005
0.005
0.005
+ 0.004
0.004
0.004
0.004
0.004
+ 0.003
0.003
0.003
0.003
0.003
+0.002
0.002
0.002
0.00 a
+0.002
214
TUE ORBIT OF URANUS.
TAULIO y\.—(!i>iilii,ti,il.
Toar.
1830
1831
1S32
1833
1834
1835
1830
1837
1838
1839
1840
1841
184-2
1843
1844
1845
184(!
1847
184 8
1841)
1850
1851
1852
1><53
1854
1855
185fi
1857
1858
1859
18(10
18(il
I8fi2
18()3
I8(i4
1805
1800
1807
1808
1809
1770
1871
1872
1873
1874
1875
1870
1877
1878
1879
1880
1881
1882
18.S3
1884
.oo
.00
0.00
0.00
0.00 ,
+ 0.01+-°'
0.03 ■°'
•03
+.04
04
+ 0.00
0.10
0.14
0.19 -"S
0.24
+0.30
•05
.06
0..'i7+-°7
((.45
0.53
0.02
+0.71
0.81"
0.92
1.04
I.IG
oS
.08
.09
.09
.10
.11
.12
.12
•13
+ 1.2!) ,
1.43 +-'4
1.57 -li
1.72 -5
1.88 -'f
.16
+ -''^*+ 17
2.21 I '
2.39
2.57
+2.70
1.32 — "*
1.19 '-^
1.07 •'•'
0.95 •'"
.11
+0.84
0.74
0.04
0.55
0.47
+ 0.39
0.32"
0.20
0.20
0.15
+0.11
0.07
0.04
0.02
0.00
—.10
. 10
,09
.08
.08
-.07
.oO
.06
■05
.04
—.04
•03
.01
.02
.00
.18
.18
.19
.20
+2.90
3. 1
3.37
3.59
+3.82 , •^■'
^ +.24
+ .20
.21
.22
+0.507
0.500
0.505
0.503
0.502
+0.501
0.500
0.498
0.497
0.495
+0.494
0.493
0.491
0.490
0.488
+0.487
0.485
0.483
0.482
0.481
+0.479
0.477
0.470
0.474
0.472
+0.471
0.409
0.407 I
0.405
0.403
+0.401
0.459
0.457
0.455
0.453
+0.451
0.449
0.440
0.444
0.442
+0.440
0.438
0.430
0.433
0.431 I
+0.429 I
0.420
0.424 i
0.422 I
+0.419 ■
+0.417 !
0.414 I
0.412 '
0,409
+0.407 I
—0.001
0.001
0.001
0.001
O.OUi
(I
-0.001
O.ool
-0.001
0.001
0.001
0.001
0.(»01
-0.001
o.ool
0.001
o.ool
-0.001
-0.002
0.002
0.002
0.002
-0.002
I
+0.002
0.002
0.002
0.002
0.002
+ 0.001
0.001
0.001
o.ool
+0.001
0.001
0.001
o.ool
o.ool
+0.001
0.001
o.ool
0.001
0.002
+0.002
0.002
0.002
0.002
+0.002
+0.003
0.003
0.003
0.003
+0.003
Yuur.
1 885
1880
1887
18S8
1889
1890
18111
1892
I81t3
1894
1895
1890
I8II7
1898
1899
1900
1901
1902
1903
1904
1905
1 900
191(7
19((8
1909
1910
11(20
1930
1940
11(50
1 90((
11(70
198((
199((
2000
2010
202((
203((
204((
2050
2000
2((70
2((s((
2((1I0
2100
2110
2120
213((
2140
2150
2100
2170
2180 !
2190 I
2200
4.79
5.05
+ 4.00 ,
^4.30+'^*
26
27
+5.33 ,
5.59+ ',
5.87 •-•'^
(1.10
0.45
.29
.29
•3°
7.00+ '»'
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335
424
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40
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THE OlllJIT OF UllANUS.
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20.57
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34.61
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42.76
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51'
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10'
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3
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51 27 52
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10'
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40
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88°
10'
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10'
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40
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80'
20
20
20
20
20
21
21
21
21
21
21
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201IOI
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21702
22:!SI1
22its0
2:157 I
211(17
217(11
25:!5I
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123
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5'/'
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50.S
595
595
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505
595
595
595
595
59^'
595
595
59^'
595
595
595
595
595
595
59^'
595
595
595
595
59 1
595
595
591
59 1
59 1
591
59 1
594
59.5
59 1
59.1
591
59.1
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59.!
59.:
59.1
59-!
592
592
592
59-'
59"
592
59'
59"
590
5''<9
590
280
50'
40
.•10
20
lU
279'
50'
40
:iO
20
10
278
50'
40
30
20
10
277'
50'
III
30
20
10
276
50'
40
:!0
20
10
275
50'
40
30
20
10
274
50'
40
:to
20
10
273
5ii'
4(1
:io
20
10
272
50'
40
30
20
(0
27 V'
50'
40
:;o
20
10
270
K
90"
10'
20
30
40
50
9r
10'
20
30
40
50
02"
10'
20
:to
40
50
93'
10'
20
30
40
50
94'
10'
20
30
40
50
95"
10'
20
;io
to
50
96°
10'
20
30
40
50
07°
10'
20
30
40
5(1
08°
111'
20
30
40
50
99'
10'
20
30
40
50
100
i 22
I 22
> 22
1 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 21
5 20
5 20
5 20
5 20
5 20
5 20
5 20
5 20
5 20
5 111
5 10
5 111
5 10
5 10
5 10
5 10
5 10
5 IS
5 IS
5 l.S
5 IS
5 IS
5 IS
5 17
5 17
5 17
5 17
5 17
5 17
5 111
5 1(1
5 1(1
5 111
5 15
5 15
5 15
5 15
5 15
5 II
5 I I
5 II
5 II
5 It
0.05
5.(1H
2.15
5S.45
51.50
50. 5H
4(1.40
42.0(1
37.57
32.01
2S.01I
23.11
17.07
12.(17
7.20
I.5S
55.70
4 0.S5
43.74
37.4 s
;!I.(I5
24.17
17.72
10. H2
3.7(1
5(1.51
40.1(1
41.(12
3:!.o:t
211.07
IS. 0(1
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1.5(1
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41.42
35.(11
211.115
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4.01
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4. 54
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5.62
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8.81
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71
90
10.06
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10. -,7
10.52
10.68
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10. ()9
11.15
• ' .50
I l..|6
11.61
11.77
I 1.92
12.08
12 2.1
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12.5.1
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1.28
37187
:!777(l
3S3(!5
38051
311:43
40131
40710
41307
41H04
424S2
4:10(10
43(1.55
44242
44S28
45114
4(1000
4I15S.^.
47170
4 7755
4s:i;{0
4sll2:i
40507
500110
501173
51250
5Is:js
52420
5:tO(l2
5:f5s:i
541(13
54743
55:12:1
55II02
5(14 Si
570(10
571138
5821(1
5M71I4
50371
5004S
(10525
(IIIOI
(11(177
(12252
H2S27
ii:t4(il
(i:i075
(14548
(15120
(1511112
(1112(14
(ir,s:tr,
(17 107
(17077
(1S547
110117
(loiisii
711255
7IIS23
71301
71058
i?.t;
270"
5«9
50
5«9
40
5«9
30
5«9
20
588
10
5. 88
268'
58S
bU
5«7
40
5,88
30
5«7
20
586
10
5«7
2«H'
5 86
50'
586
40
586
30
S«5
2U
5«5
10
S«5
267^
5'^4
5(1'
S«4
41»
5«4
30
5«.5
20
SH.5
10
S«1
266"
S«2
50'
5«2
40
S''^-'
30
5«'
20
580
10
5,80
5 So
579
579
579
57«
57«
573
577
577
577
57'-'
S7'->
575
575
571
574
57.5
57->
572
572
572
57'
570
570
570
5'>9
$(>')
568
568
5 ''7
, 1 -i
265°
50'
40
30
20
10
264°
50'
40
30
20
10
263"
50'
40
30
20
10
262"
5(1'
40
30
20
10
261"
.50'
40
30
20
Id
200
220
THE ORBIT OF URANUS.
TABLE \\l. -Continued.
E
100°
10'
20
30
40
50
lor
10'
20
;to
40
50
102'
10'
20
30
40
f)0
103°
10'
20
30
40
SO
104°
lu'
20
30
40
50
105°
10'
20
30
40
50
106°
10'
20
30
40
50
107°
10'
20
30
40
50.
108°
10'
20
80
40
50
109°
10'
20
30
40
50
UO^
14 4.80
13 51.96
13 38.96
13 25.81
13 12.51
12 59.06
12 45.45
12 31.(19
12 17.78
12 3.72
11 49.50
11 35.14
5 11 20.62
5 11 5.95
5 10 51.13
5 10 36.16
5 10 21.04
5 10 5.77
5 9 50.35
9 34.78
9 19.06
9 3.19
8 47.17
8 31.01
14.69
58.22
41.61
24.85
7.94
50.88
6 33.67
6 16.32
5 58.82
5 41.17
5 23.37
5 5.42
5
6
5
4 .'■)9
4 59
4 59
4 .^S
4 5S
4 58
4 57
4 57
4 57
47.33
29.10
10. -:i
!)2.19
33.52
14.70
55.74
3(>.63
17.38
57.99
38.45
18.77
58.94
38.97
18.H6
5 s. 60
38.21
17.67
56.98
36.15
15.18
54.07
32.81
11.42
4 56 49.88
2.84
3.00
3»5
,?-45
3.61
,V7<5
4.06
4.22
4- 36
4-52
4.67
4.S2
4-97
5->-
5-27
5-42
5-57
5-72
5-87
6.02
6.16
6.32
6.47
0.61
6,76
6.91
7.06
7.21
7-3S
7-5°
7-6.S
7. So
7-95
S.09
S23
8.38
8.53
8.67
8.82
8.96
9.11
9-25
9-39
9-54
9.68
9-83
19.97
20. 1 1
20.26
2°-,l9
20.,S4
20.69
20.83
20.97
21. 1 1
21.26
2' -39
21 54
Li>;f. r
1.23
71958
72525
73091
73656
74221
74785
75348
75911
76474
77036
77598
78159
78720
792SO
79839
80398
80956
81514
82071
82628
83184
83739
84294
84849
85403
!S5'.I56
86508
87059
87610
88160
§8709
89258
89806
90354
90902
91449
91995
9-.'540
9;i(),s5
931:29
94172
94715
95257
9579S
96338
96S78
97417
97955
98493
99030
99566
*00101
*n(l636
*01170
*01703
♦02236
*(»2768
*(»3299
♦03829
♦04359
♦ (tJssH
I •1.29
S^'7
566
565
565
564
563
563
Sf'3
5f'2
562
561
561
560
559
559
55S
558
557
557
55^'
555
555
555
554
553
552
55'
55»
55°
549
549
548
548
54S
547
546
545
545
544
543
543
542
541
540
540
539
5.?8
538
537
5,^6
535
535
534
533
533
532
S\^
530
530 I
529
1/
260'
110°
50'
10'
40
20
30
30
20
40
10
60
259°
111°
50'
10'
40
20
30
30 j
20
40
10
50
258'
112°
50'
10'
40
20
30
30
1 20
40
10
60
25T
113°
50
10'
> 40
20
30
30
20
40
10
50
l256°
114°
50'
10'
40
20
30
30
20
40
10
50
255°
115°
50'
10'
40
20
30
30
20
40
10
5u
1254^
116°
50'
10'
40
20 1
30
30
20
40
10
50
253°
in°
60'
10'
40
20
30
30
20
40
10
50
252°
118°
, 50'
10'
i 40
20
1 30
30
20
40
10
50
251°
119'
50'
10'
1 40
20
1 30
30
1 20
40
1 10
50
2.'J0'
120
! a
E
49.88
28.20
fi.39
44.44
22.34
0.10
54 37.72
54 15.2!
53 52.55
53 29.76
6.83
4 53
4 52 43.76
52 20.55
51 57.20
51 33.72
51 10.10
50 46.34
50 22.44
49 58.41
49 34.24
49 9.94
48 45.50
48 20.92
47 66.21
47 31.36
47 6.38
46 41.26
46 16.00
4 5 50.61
45 25.09
44 59.44
44 33.66
44 7.74
43 41.69
43 15.50
42 49.19
42 22.74
41 56.16
41 29,45
41 2.ro
40 35.62
40 8.52
39 41.28
39 13.91
38 46.41
38 1S.7S
37 51.02
37 23 13
55.12
26.98
58.71
35 30.. il
35 1.78
34 33.13
34 4.35
3:! 35. M
33 6.41
32 37.25
32 7.96
31 38.55
21.68
21.81
21.95
22. 10
22.24
22.38
22.51
22.66
22.79
2--.93
23-°7
:J3.2i
23-35
23.48
23.62
23. 7O
23.90
24.03
24.17
26.05
4 31 9.61
L(i(r. r
129
04888
05416 \
05944 5
06471 5
06997 \
07522 5
08047
0857 1 ^
09094 ^
09616 \
10137 ^
10658 ^
11178
11(;97 5
12215 5
12732 5
13248 5
13764 5
14279
14793 5
15306 5
1581H 5
16329 5
16>i40 5
5
17350
17859 \
18367 \
I8f<74 \
193.><0 \
19886 ^
20391
20895 \
21398 ^
21900 ^
224(11 \
229(11 ^
23401 .
23S<.|<) ^
24397 Z
24.s-.,4 ^
25390 ^
258.S5 ^
263S0 ,
26S74 ^
27366 ]
27S57 \
28348 ]
28,><37 'I
29325
29812 ]
30299 ';
307.X5 ]
31270 Z
31754 ]
— ■" 4
327 1 9 J
33200 \
336S1 ]
341(;l J
34640 ;;
3.1II7
1.29
\- i
!i
THE OllBIT OF UK AN US.
221
TAULK Vn.—Conlinued.
1520°
10'
20
30
40
50
121"
10'
20
30
40
50
122'
10'
20
30
40
50
123'
10'
20
30
40
50
124°
10'
20
30
40
50
125°
10'
20
30
40
50
E
31 !).01
30 39.35
30 9.56
29 3fl.fi5
29 9.(il
28 39.45
28 O.lfi
27 38.75
27 8.22
2(i 37.50
21! (1.78
25 35.88
25 4. 85
24 33.71
24 2.44
23 31.05
22 59.54
22 27.91
4 21 50.10
4 21 24.29
4 20 52.29
4 20 20.18
4 19 47.94
4 19 15.59
126°
12
4.35
10'
11
30.37
20
10
5(1.27
30
4
10
22.00
40
9
47.73
50
9
13.29
127^
8
38.74
10'
4
8
4.07
20
4
7
29,29
30
fi
54.40
40
ft
19.40
60
5
44.29
128'
10'
20
30
40
60
129'
10'
20
30
40
50
,')
4
3
3
2
2
1
59
59
Log. r
43.12
10.53
37. S3
5.01
32.07
50.01
4 15 25.84
4 14 52.55
4 14 19.14
4 13 45.02
4 13 11.98
4 12 3S.22
3 58
9.00
33.72
58,27
22,72
47,05
11.27
35.38
59.38
23,27
47.05
10,73
34.29
130' i 3 57 57.75
29.66
29.79
29.91
30.04
30.16
30.29
30.41
.so- 53
30.66
30.78
30.90
3103
3114
3«-27
31-39
3i-5»
31-75
31-87
32.00
32.11
32-24
32-35
32-47
32-59
32-70
32.82
32.94
33-06
33-17
33- 29
33-41
33-52
33-<>4
33-76
33-87
33-98
34- to
34-21
34-33
34-44
34-55
34-67
34- 7«
34-89
35-00
35-"
35-23
35-34
35-45
35-55
35-67
35-78
35-89
36,00
36. 1 1
36.22
36-32
36-44
36-54
1.29
35117
35593
3(;009
30544
37018
37491
37903
38434
3(S904
39372
39840
40300
40771
41230
41700
42103
42025
43080
43540
44005
44403
44919
45375
45829
40282
407.35
47180
47030
4S080
48534
48981
4!I428
49x73
.•.0317
50701
51204
51045
520S5
52524
52902
53399
53834
542l!8
54702
55134
j 555(;5
i 55995
1 50424
50^52
57279
57705
5sl.i0
5S554
58977
59399
59820
00239
00057
01074
01489
01903
1.29
476
476
475
474
473
472
47«
470
468
468
466
465
465
464
463
462
461
460
459
458
456
456
454
453
453
45 «
450
450
44S
447
447
445
444
444
443
441
440
439
43^
437
435
434
434
432
43'
430
429
42S
427
426
425
424
423
422
421
419
418
4'7
415
414
240'
50'
40
30
20
10
239'
50'
40
30
20
10
238°
50'
40
30
20
10
23T
50'
40
30
20
10
23U'
50
40
30
20
10
235°
50'
40
30
20
10
234'
50'
40
30
20
10
233'
50'
40
30
20
10
232'
50'
40
30
20
10
231^
"'0'
40
.30
20
10
230"
n
130'
10'
20
30
40
50
131°
10'
20
30
40
50
132^
10'
20
30
40
50
133'
10'
20
30
40
50
134'
10'
20
30
40
50
135'
10'
20
30
4J
fO
B
3 57 57.75
3 57 21.10
3 60 44.35
3 50 7.48
3 55 30.61
3 54 53.43
3 64 10.25
3 53 38.97
3 53
3 52
3 51
3 51
1.58
24.08
40.48
8.78
3
3
3 49
50 39.97
49 53.00
15.04
3 48 30.92
3 47 68.70
3 47 20.38
3 40 41.95
3
40 3,42
3 45 24,79
3 44 40.00
3 44 7.23
3 43 2S.30
3 42 49.27
3 42 10.14
3 41 30.91
3 40 51.58
3 40 12.15
3 39 32.02
3 3S
3 38
3 37
3 30
3 30
53.00
13.28
33.40
53,54
13,52
35 33.41
136'
3
34 53.21
10'
3
34 12.90
20
3
33 32,50
30
3
32 52.01
40
3
32 11.42
50
3
31 30.73
137'
3
30 49.95
10'
3
;!0 9.08
20
3
29 2S.11
30
3
28 47.05
40
.3
28 5.90
50
3
27 24.00
138'
10'
20
30
40
50
139°
10'
20
30
40
50
140°
3 20 4.3,32
3 20 1.S9
3 25 20,37
3 21 3S,70
3 23 57.05
3 2.! 15.20
3 22 33.38
3 21 51.41
3 21 9, .35
3 20 27.20
3 19 4 4.90
3 19 2.03
3 18 20.22
36-65
36.75
36.87
36-97
37.08
37-«8
37-28
37-39
37-50
37.60
37-70
37-81
37-9"
38-02
38.12
38.22
38.32
38.43
38-53
38-63
38-73
38-83
38.93
39-03
39- "3
39-23
39-33
39-43
39-53
39-62
39.72
39.82
39.92
40,02
40.11
40.20
40.31
40.40
40.49
40.59
4^.69
40.78
40.87
40.97
41.06
41.15
41.24
41.34
41.43
41.52
41,61
41.71
41.79
41.88
Log. r
1.29
61903
62317
62729
63140
63550
63959
64300
04773
05179
65583
65987
06389
60790
07190
67588
07985
08382
68777
69171
69504
69955
70345
70734
71122
71509
71895
72270
72602
73044
73425
73805
74184
74501
74937
75312
75080
70059
70430
7 0800
77109
77537
77903
78208
7S032
7S995
79350
79710
SOU75
80433
80789
81144
81498
81S51
82202
82552
414
;12
411
410
409
407
407
406
404
404
402
401
400
398
397
397
395
394
393
391
390
389
388
387
3S6
384
383
382
381
380
379
377
376
375
374
373
37»
370
369
368
366
365
364
363
361
360
359
358
356
355
354
353
35 >
350
41.07
8''90I
349
42,06
83249
348
42.15
83595
346
42.24
83910
345
42.33
84284
344
42.41
343
84027
1.29
230'
50'
40
30
20
10
229'
50'
40
30
20
10
228°
50'
40
30
20
10
227"
50'
40
30
20
10
226'
60'
40
30
20
10
225'
50'
40
30
20
10
224°
60'
40
30
20
10
223°
50'
40
30
20
10
222°
50
40
30
20
10
221°
50'
40
30
20
10
220'
I i
! \
IIP
222
THK ORBIT OF URANUS.
TAUl
r
K VII
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iiiiK'd.
9
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Loff.
9
E
I'|>K-
r
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1.29
n t
tr
1.30
140°
3 IS 20.22
84027
220'
150°
2 33 20.93
02798
262
260
210°
10'
3 17 37.72
42-5°
84908
341
50'
10'
2 32 39.79
47.14
030(;0
50'
20
3 l(i 55.13
42-59
85308
340
40
20
2 31 52.59
47-20
03320
40
30
3 l(i 12.45
42.68
85647
339
30
30
2 31 5.32
47-27
03579
259
258
256
30
40
3 15 2!l.(18
42-77
85985
336
20
40
2 30 17.98
47 34
03837
20
50
3 14 4U.83
42-85
80321
10
60
2 29 30.58
47-40
04093
10
42-94
335
47-47
25s
141°
3 14 3.89
80050
219°
151°
2 28 43. n
04348
209°
10'
3 13 20.87
43.02
80990
334
50'
10-
2 27 55.58
47-53
47.60
47.66
04002
«S4
50'
20
3 12 37.78
43- « I
87322
332
40
20
2 27 7.98
04854
252
40
30
3 11 54. 5G
•43-20
87053
33^
30
30
2 20 20,32
05105
251
30
40
3 11 11.28
43.28
43-36
87983
330
20
40
2 25 32.59
47-73
05355
250 1
20
50
3 10 27.92
88312
329
10
50
2 24 44.80
47-79
47-85
05004
249
10
43-45
327
*47
142°
3 9 44.47
88039
326
218°
152°
2 23 50.95
05851
208°
10'
3 9 0.94
43-53
88905
50'
10'
2 23 9.04
47.91
00090
245
50'
20
3 8 17.32
43.62
89290
325
40
20
2 22 21.00
47-98
00340
244
40
30
3 7 33.f)2
43-70
89014
324
30
30
2 21 33.02
48.04
00583
243
60
40
3 (1 49.83
43-79
43-86
89930
322
20
40
2 20 44.92
48.10
48.16
00824
241
20
50
3 G 5.97
902 '
321
10
50
2 19 60. 7G
07004
240
238
10
43-95
1
48.22
143°
3 5 22.02
9057'.
n°
153°
2 19 8.54
48.28
07302
237
236
234
233
207'
10'
3 4 ;i7.99
44-03
9()S95
-■^o'
10'
2 18 20.20
07539
50'
20
3 3 53. .H8
44- 1 1
91212
3'-
40
20
2 17 31.92
48.34
48.41
48.46
07775
40
30
3 3 9.09
44.19
91528
3'6
30
30
2 10 43,51
08009
30
40
3 2 2.^). 42
44.27
9IS42
3'4
20
40
2 15 55,05
OS242
20
50
3 1 41.08
44-34
44-43
92155
313
311
10
50
2 15 0.53
48. 5 2
48.58
08473
231
230
10
144°
3 5n.r,5
92400
216°
154°
2 14 17.95
48. 6 J
08703
22S
206°
10'
3 12.14
44.51
92770
310
,Ml'
10'
2 13 29.31
08931
50'
20
2 59 27.55
44-59
9;iOS5
309
40
20
2 12 40,02
48. 69
09158
227
226
40
30
2 TiS 4 2.89
44-66
9;!393
308
30
30
2 11 51, SO
48. 76
09384
30
40
2 .-.7 5S.14
44-75
9:!700
307
20
40
2 11 3,05
48.81
4R.87
09008
224
20
50
2 57 13.32
44.82
94005
305
10
50
2 10 14.18
09831
223
10
44.90
304
48.92
222
145°
2 50 28.42
44-98
94.109
215'
155°
2 9 25,20
10053
220
219
205°
10
2 55 43.44
91012
3,°3
50'
10'
2 8 30.28
48.98
10273
50'
20
2 54 5S.39
45-05
94 913
301
40
20
2 7 47.24
49-04
10192
40
30
2 54 13.2(!
45- '3
95213
300
30
30
2 G 5.-<.15
49.09
10709
217
216
214
30
40
2 53 2S.05
45-21
45.28
9,"i511
298
20
40
2 9,01
49.14
10925
20
50
2 52 42.77
li.">S08
297
10
50
2 5 19.81
49.20
11139
10
4;,- 36
296
49.26
213
146°
2 51 57.41
90104
214°
156°
2 4 30.55
11352
212
204°
10'
2 51 11.97
45-44
90398
294
50'
10'
2 3 41.24
49-3'
11504
50'
20
2 50 2<i.4()
45-5'
90091
293
40
20
2 2 51. S8
49-36
11774
210
209
208
206
40
30
2 49 40,87
45-59
45.66
90983
292
30
30
2 2 2,47
49-41
119H3
30
40
2 48 55.21
97273
290
20
40
2 1 13.00
49-47
12191
20
50
2 48 9.48
45-73
97502
289
10
50
2 23.48
49-52
12397
10
45.81
2 88
49-57
205
147°
10'
20
2 47 23.07
2 4(1 37.79
2 45 51.83
45-88
45-96
46.03
46. 10
46. 1 7
46.24
97850
98130
98421
,86
2«5
213°
50'
40
157°
10'
20
1 59 33.91
1 58 44.29
1 57 54. (;l
49.62
49.68
12002
12s 05
13007
203
202
200
198
»97
196
203°
50'
40
30
40
2 45 5.80
2 44 19.70
98705
9.S9SS
284
2*^3
30
20
30
40
1 57 4.S8
1 50 15.10
49-73
49- 78
132(17
13405
30
20
50
2 43 33.53
99209
281
2S0
10
50
1 55 25.28
49.^2
49.88
13002
10
148°
2 42 47.29
46.31
46.39
46-45
46.53
46.60
46.66
99549
212'
158°
1 54 35.40
13798
202°
10'
2 42 0.98
99.*<27
2 78
50'
10'
1 53 45.4 7
49-93
13992
194
60'
20
2 41 14.59
*(I01()4
277
40
20
1 52 65.50
49-97
14185
193
40
30
2 40 28. U
*(in3S0
276
1 30
30
1 52 5.48
50-02
14377
192
30
40
2 39 41.01
*(l(m54
274
20
40
1 51 15.41
50.07
14507
190
189
188
20
50
2 38 55.01
♦00920
272
271
10
50
1 50 25.29
50.12
50- > 7
14750
10
149'
2 38 8.85
46. 73
♦01197
i 211'
159-=
1 49 .35.12
14944
1S6
185
•83
182
180
201°
10'
1 2 37 21.02
♦014 07
270
; 50'
10'
1 48 44,91
50-21
15130
50'
20
2 30 34.81
46. Si
46.87
♦O1730
26()
40
20
1 47 54,05
50.26
15315
40
80
2 35 47.94
♦02004
26S
30
30
1 47 4,34
50-31
15498
30
40
2 35 I. 01
46.93
♦02270
266
! 20
40
1 40 13.98
50.36
15080
20
50
{ 2 34 14.00
47.01
♦02535
265
! 10
50
1 45 23.58
50.40
15800
10
47.07
263
5... 44
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160'
2 33 20.93
♦0279^^
210'
160
1 44 33.14
10039
200°
1 „
1
♦1.30
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1
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__
THE Oil 11 IT OF URANUS.
223
TABLK VII
.— Cont
inited.
9
E
Log. r
K
Log.
r 1
Q 1 II
>t
l.SO
1 n
II
1.80
IQO^
1 44 33.14
10039
200°
170°
52 57.33
24087
89
88
86
84
83
82
190°
10'
1 43 42.05
SO'49
10210 '
77
76
50'
10'
52 4.81
52-5'
24176
50'
20
1 42 52.11
50-54
S°-5S
50.62
50.O7
50-71
10392 '
40
20
51 12.28
52-53
52-56
52.58
52.60
52.62
24204
40
40
1 43 1.53
1 41 lO.'Jl
10500 '
10739 '
74
73
30
20
30
40
50 19.72
49 27.14
24350
24434
30
20
50
1 40 20.24
1C310 ;
71
70
10
50
48 34.54
24517
10
161°
1 39 29.53
17080
68
67
66
64
63
62
199°
mr
47 41.92
52.64
52.66
52.6.;j
52-70
52.72
52-74
24599
80
189°
10'
20
30
1 3,S 38.7S
1 37 47.98
1 3G 57.14
5°-75
50.80
50.84
50.89
17248 '
17415
17581
50'
40
30
10'
20
30
40 49.28
45 50.02
45 3.93
24079
24758
24835
79
77
75
74
73
50'
40
30
40
1 3G 0.25
17745
20
in
44 11.23
24910
20
50
1 35 15.33
50.92
50.96
17908 J
10
50
43 18.51
24984
10
162°
1 34 24.37
18070
60
198°
172°
42 25.77
52.76
26057
188°
10'
1 33 33.37
51.00
18230 '
50'
10'
41 33.01
25128
71
50'
20
1 32 42.33
5' 04
l.')388 '
S^
40
20
40 40.24
52-77
26198
70
69
67
65
64
40
ao
40
I 31 61.24
1 31 0.12
510')
51.12
51.16
51.20
18545 '
18700
57
55
30
20
30
40
39 47.45
38 64.04
52-79
52.81
_ _ y _
25207
25334
30
20
50
1 30 8.90
18854
54
53
10
50
38 1.82
52.82
52.84
25399
10
163°
10'
1 29 17.70
1 28 20.52
51-24
19007
19158
51
197°
50'
173°
10'
37 8.98
30 10.12
52.86
52-87
52.89
26403
25525
62
61
187°
50'
20
1 27 35.25
51-27
19308
50
48
40
20
35 23.25
25580
40
30
1 20 43.93
51-32
19450
30
30
34 30.30
25045
59
58
56
30
40
1 25 52.58
5«-35
19003
47
46
20
40
33 37.45
52.91
25703
20
60
1 25 1.19
5'-39
19749
10
50
32 44.53
52.92
25759
10
S»-42
44
52-93
55
164°
1 21 9.77
51-46
19893
193°
174°
31 51.00
25814
53
52
50
48
47
46
186'
10'
1 23 1S.31
20035
42
50'
10'
30 58.00
52-94
52.96
25807
50'
20
; 22 20.81
5'-5o
20170
41
40
20
30 5.70
25919
40
30
40
1 21 35.28
1 20 43.71
51-53
5>-57
51.60
51-64
20310
20454
40
138
■•^6
'35
30
20
30
40
29 12.73
28 10.75
52.97
52.98
25909
20017
30
20
50
1 19 52.11
20590
10
50
27 20.70
52-99
53-00
2(!004
10
165°
10'
20
1 19 0.17
1 18 8.80
1 17 17.10
5'-67
5'-7°
20725
20858
20090
32
195'
50'
40
175°
10'
20
20 33.70
25 40.75
24 47.72
53-01
53-03
53-04
53-04
53-o6
53-07
20110
20154
20197
44
43
42
40
39
37
185°
50'
40
30
1 10 25.30
5'-74
21121
31
30
30
23 54.08
20239
30
40
1 15 33.59
5^-77
51. So
5'-83
21250
29
20
40
23 1.04
20279
20
50
1 14 41.79
21377
27
26
10
50
22 8.58
20318
10
166°
1 13 40.90
5 1. 86
21503
194°
176°
21 15.51
S3-08
53.08
20355
35
184°
10'
1 12 58.10
21028
'25
50'
10'
20 22.13
20300
50'
20
1 12 0.20
51.90
21751
'23
40
20
10 29.35
20424
34
40
30
1 11 14.27
51-93
51.96
51.98
52.02
21873
22
30
30
18 30.25
53-'o
53-'o
53-12
53-12
20450
32
30
40
1 10 22.31
21993
20
20
40
17 43.15
20480
30
20
50
1 9 30.33
22112
'9
118
10
50
10 50.03
20515
29
28
10
167°
10'
1 8 38.31
1 7 40.27
52-04
52.08
22230
22;!40
ri6
193°
50'
177°
10'
15 50.01
15 3.78
53-13
53 13
53.14
53- '5
53-15
53-15
20543
20509
26
2 z
183°
50'
20
1 54.19
22400
114
40
20
14 10.05
20504
23
23
40
30
1 2.08
52.11
22573
'■3
30
30
13 17.51
20017
30
40
1 5 9.94
5-'-' 4
52.16
52-19
22084
( 1 1
20
40
12 24.30
20039
20
20
50
1 4 17.78
22793
109
108
10
50
11 31.21
2G059
19
10
163°
1 3 25.59
22901
192'
178'
10 38.00
53- J 6
53.16
5316
20078
1 7
182°
10'
1 2 33.37
52.22
23008
107
50'
10'
44.00
20005
16
60'
20
1 1 41.13
5 =-24
23113
■05
40
20
8 51.74
20711
■5
13
1 J
40
30
1 48.80
52.27
2.3217
104
30
30
7 68.58
20720
30
40
59 50.57
52-29
23320
'O,? _
20
40
7 5.41
53-17
20739
20
50
59 4.25
52-32
52.35
23421
lol
100
10
50
12.24
53-17
53-17
20750
10
10
169°
58 ii.no
23521
191°
173°
5 10.07
53-18
20700
8
181°
10'
57 19.53
52-37
23019
98
50'
10'
4 25.89
20708
50'
20
50 27.13
52-40
23710
97
40
20
3 32.72
53-17
53-18
53-18
53-18
20775
7
40
30
, 55 34.71
52-42
23811
95
30
30
2 39.54
20780
5
30
40
' 54 42.27
5244
52-46
23904
93
20
40
1 40.30
20784
4
20
50
53 49.81
23990
92
10
60
53.18
20780
2
10
52.49
91
53-18
1
170'
52 57.32
1.30
190°
180°
0.00
20787
1.30
180°
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J
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1
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4't
i|: :ii
i
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l.cHi
m
11
224
TllK OllBlT OF UUANUS.
TAISLK VIII, Ana.
1. — Action of Juimter.
Arjt.
(c.f.O) Dili'.
(,-.,.. 1)
It
0.14
(u.c.l)
(U.S. 2)
{v.cM)
(p.c.O)
2331
(p.«.l)
1 (p.f.l)
55.03 ,'
55.58+°- 55
It
4.42
0.10
tl
0.11
104
1
134
1
0.13
4.42
0.10
0.12
2.-!31
100
134
2
5(i.l3 °-5S
5(;.(I8 °-55
.,... Ill
59.97 °55
0-54
«i.o(i+°-ss
Cl.dO °-S4
C2.15 °-5S
(;2.(!9 °-54
0-54
0.12
4.41
0.10
0.12
2330
107
134
3
0.11
4.41
0.10
0.12
2330
109
134
4
O.ll
4.41
0.10
0.12
2330
171
135
5
0.10
4.40
0.10
0.13
2329
172
135
C.
0.09
4.40
o.ll
0.13
2328
174
135
■:
0.09
4.39
0.11
0.13
2328
170
135
8
0.08
4.39
0.11
0.13
2327
177
135
9
0.08
4.38
0.11
0.14
2320
179
130
10
0.08
4.38
0.11
0.14
2324
180
130
11
0.07
4,37
0.11
0.14
2323
182
130
12
0.07
4.37
0.11
0.15
2321
184
137
13
0.07
4.30
0.11
0.15
2320
185
137
U
0.08
4.30
0.11
0.15
2318
187
137
15
03.23
0.08
4.35
0.11
0.15
2310
1S8
137
l(i
03.78 +°-SS
0.08
4.35
0.11
0.10
2311
190
137
17
04.32 °-54
0.09
4.34
11
0.10
2312
191
138
IS
04.80 °-54
0.09
4.34
0.11
0.10
2310
193
138
■J
05.40 °S4
0-53
05.93
0.10
4.33
0.11
0.17
2308
194
138
20
0.11
4.33
o.ll
0.17
2305
190
138
21
00.47 +°S4
0.12
4.32
o.ll
0.17
2303
198
139
22
07.00 °-53
0.13
4.32
0.11
0.17
2300
199
139
23
07.54 °-54
0.14
4.31
0.11
0.18
2297
200
139
24
08.07 °-53
0-53
08.00 ,
0. 15
4.30
0.11
0.18
2294
202
140
25
0.10
4.30
0.10
0.18
2291
203
140
2(!
09.14 +°S4
0.18
4.29
0.10
0.19
2288
205
140
27
09.06 °-52
0.20
4.29
0.10
0,19
2284
200
140
2S
70.19 °-53
0.21
4.28
0.10
0,19
2281
208
140
29
70.72 -53
71.24 ,
71.70 +°-5^
0.23
4.28
0.10
0.20
2277
209
141
30
0.25
4.27
0.10
0.20
2274
210
141
31
0.27
4.27
0.10
0.20
2:i70
212
141
32
7-> 28 °'5^
0.29
4.20
0.10
21
2200
213
141
33
72^80 °-5^
0.31
4.20
0.10
0.21
2202
214
142
34
73.32 III
74.34 +°-
0.33
4.25
0.10
0.21
2258
210
142
35
0.35
4.25
0.10
0.22
2254
217
142
30
0.38
4.24
0.10
0.22
2249
218
142
37
74.85 °-
0.40
4.24
0.10
0.22
2245
219
143
38
75.30 °-5
0.43
4.23
0.10
0.23
2240
221
143
39
0.30
70.87 +°--2°
0.40
4.23
0.09
0.23
2230
222
143
40
0.49
4.22
0.09
0.23
2231
223
143
41
0.52
4.22
0,09
0,23
2220
224
144
42
77.37 °-i:°
0.55
4.22
0,09
0,24
2221
225
144
43
78.80
0.58
4.21
0.09
0,24
2210
227
144
44
0.01
4.21
0.09
0,24
2210
228
144
45
0.04
4.21
0.09
0,25
2205
229
144
4(i
79.35+°- 49
0.08
4.20
0.09
0.25
2200
230
145
47
79.84 °-49
80.32 °-4f
80.80 °-48
0.48
81.28 ,
81.70 +°-48
0.71
4.20
0.09
0.25
2194
231
145
48
0.75
4.20
0.09
0.20
2188
232
145
49
0.79
4.20
(.09
0.20
2182
233
145
50
0.82
4.20
0.08
0.20
2170
234
140
51
0,80
4.19
0.08 .
0.20
2170
235
140
52
82.23 °-47
0.90
4.19
0.08
0.20
2104
230
140
53
82,70 °-*7
0.94
4.19
0.08
0.27
2158
237
140
54
83.17 °-+J
0.46
0.98
4.19
0.08
0.27
2151
238
140
55
83.03
1.02
4.19
0.07
0.27
2145
239
140
5li
84.10 +°-*7
1,07
4.19
0.07
0.28
2138
240
140
57
84.55 °-45
85.01 °;f
1.11
4,19
0.07
0.28
2132
241
140
5S
l.lfi
4,19
0.07
0.28
2125
242
147
59
85.40 °-*5
0-4.1
85,91
1.20
4.19
0.07
0.28
2118
243
147
r.o
1.25
4.19
0.07
0.29
2111
241
147
TUE OllBIT
OF URANUS
225
TA15LE
VllI, Ano. \.—C
inlinii.d.
Arg.
no
(ii.f.O) Diff.
(o.s.l)
(v.c.X)
(».«.2)
It
0.07
(i'.f.2)
(p.r.O)
(,,.«. 1) 1
(p.O'.l)
II II
85.01 .
1.25
4.10
II
0.29 j
2111
244
147
(ii
80.30 +°-'*5
1.30
4.19
0.07 1
0.20
2104
244
147
(12
80.80 °-*^
1 34
4.20 1
0.00
0.29
2007
245 I
148
03
87.24 °-*-*
1.39
4.20
0.00
0.29
2000
240
148
04
8 .08 °-*-*
1.44
4.20
0.00
0.29
2082
247
148
05
0.43
8S.11 .
1.49
4.20
con
0.30 I
2075
248
148
Oli
88.53+°-*'
1.54
4.21
0.00
0.30
2007
248
MS
07
88.00 °-«
1.00
4.21
0.00
0,30
2059
249
148
OS
80.38 °--*'
1.05
4.21
0.05
0.30 '
2052
250
148
0!)
80.80 °;]J
90.21 ,
1.70
4.22
0.05
0.31
2044
250 1
148
70
1.75
4.22
0.05
0,31
2030
251
149
71
00.02+°- 4'
1.81
4.23
0.05
0.31 i
2028
253
149
7i
91.03 "-4'
1.80
4.23
0.05
0.31 ;
2020
252
149
7:i
91.43 °»°
1.92
4.24
0.05
0.31 I
2012
253
149
74
91.83 °-+^
1.98
4.24
0.04
0.32
2003
253
149
7.1
0-39
92.22 ,
2.03
4.25
0.04
0.32
1995
254
149
7.">
02. 02+°- 4^
2.09
4.20
0.04
0.32
1980
254
149
77
ot.oo °-^i
2.15
4.20
0.04
0.32
1978
255
149
7'f
9.!.3S °--^f
2.21
4.27
0.04
0.32
1000
255
149
7'.»
80
93.7li °--]^
0.38
94.14 ,
2.27
2.33
4.2s
4.29
0.04
0.04
0.32
0,32
1000
10.--)2
25 G
250
149
149
81
94.51 +°-.>7
2.30
4.30
0.03
0,32
r.M3
257
149
82
04.87 °--5^
2.45
4.31
0,03
0.33
1934
257
149
83
95.23 °-^^
2.51
4.32
0.03
0,33
1925
257
149
84
95.50 °-^^
0.36
05.05 ,
2.57
4.33
0.03
0.33
1010
258
149
85
2.04
4.34
0,03
0.33
1000
258
149
80
9!. 30 *-°--^5
2.70
4.35
0.03
0,33
1807
258
149
87
9;.04 °'5-*
2.70
4.30
0.03
0.33
1SS8
259
149
8S
01!. OS °--' +
2. S3
4.37
0.03
0.S3
1878
250
149
8!>
07.32 -35
97.05 ,
2.80
4.38
0.02
0.33
1809
259
149
00
2.90
4.30
0.02
0..33
1859
200
148
01
07.07 +°--'^
3,02
4,41
0.02
0.33
1850
200
148
02
9^.20 °-3-'
3.00
4,42
0.02
0.33
1840
ac'o
148
03
os.oi °-3^
3.15
4,43
0.02
0.33
1830
200
148
94
OS. 02 °-3'
0.31
90.23 ,
3.22
4.45
0,02
0.33
1820
200
148
05
3.29
4.40
0.02
0.33
ISIO
201
148
01")
90 53+°- 3°
3,35
4.4S
0.02
0,34
l.'^OO
201
148
07
00. S3 °-3°
3.42
4.40
0.02
0.34
1790
201
148
OS
100.12 °"}
3,49
4.51
0,01
0.34
1780
201
m
00
100.70 ,
3.50
4.52
0.01
0.34
1770
201
147
100
3.03
4.54
0.01
0.34
1700
201
147
101
100.97+°-;
101.25 ° :,
3.00
4.50
0.01
0.34
1750
201
147
102
3,70
4,57
0.01
0.34
1739
201
147
103
•"'.'V2 :
3.83
4,50
0.01
0.34
1729
201
140
104
3.00
4. .01
0.01
0.34
1718
201
140
1 05
102.04 ,
3,97
4,03
0,01
0,34
: 1 708
201
140
100
102.29+°- S
4,04
4,05
0.01
0,34
i 1097
201
145
107
102.54 °-',^
4,11
4.07
0.01
0.34
lOSO
201
145
lOS
102. 7S °- 1
4.18
4.09
0.0 1
0,34
1070
201
145
109
103.02 °;;J
103.25
4.25
4.71
0.01
0,34
1005
201
145
110
4.32
4.73
1 0.01
0.34
1054
201
144
111
103.48+°- '5
4.30
4.75
0,01
0.34
1043
201
144
112
103.70 °-"
4.40
4.77
0.01
0.34
1032
201
144
113
103.02 °-"
4,53
4,70
0,01
0.34
1022
200
143
114
10.13 -;
101.34 ,
4.00
1 4.81
0,01
0.34
1011
200
143
115
4.07
4,83
0.01
0.33
1000
200
142
lift
104.54+°-^°
4.74
4.S5
0,01
0,33
15SS
200
142
117
104.74 °-"°
4.81
4.87
0,01
0.33
1577
200
142
118
104.03 °-'^
l«^ll 0.8
4.88
4.90
O.Ol
0.33
1500
259
141
119
4.05
4.92
0.01
0.33
1555
250
141
120
105.29
5.01
4.94
0,02
0.33
1544
250
140
1
19 July, 1878.
„! \
if!
!«i I
■ I
226
THE ORBIT OF UIIANUS.
TAIILF
VIII, A
no. 1. — Continued.
Ai'n.
(iJ.c.O) Diir.
(i>.».l)
{v.cA)
(V.H.i)
(0.C.2)
(p.f.O)
(p.».l)
(P.f.l)
t, n
n
tt
tt
tt
120
•"■'■'•2« ,„ .,
5.01
4.94
0.02
0.33
1544
259
140
121
105.40+°-
5.08
4.97
0.02
0.33
1533
259
140
122
105.03 °-'/
105.79 °'^
105.95 °"^
o> 1 ^
6.15
4.99
0.02
0.33
1521
258
139
12;j
5.22
5.01
0.02
0.33
1510
258
1 39
124
6.29
6.04
0.02
0.33
1499
258
138
135
100.10 ,
100.25+°- ;S
5.30
6.00
0.02
0.33
1487
257
138
12(!
6.43
6.09
0.02
0.33
1470
257
137
127
100.39 °-^^
6.50
6.11
0.02
0.33
1404
257
137
12S
100.53 °- j
6.57
5.14
0.02
0.33
1453
250
130
129
100.05 °;;;
100.78
6.04
6.16
0.02
0.33
1441
250
135
IM
5.71
6.19
0.02
0.32
1430
250
135
l.Jl
100.«9+°-"
6.78
5.21
0.03
0.32
1418
255
134
l;!2
107.01 °-'*
6.85
6.24
0.03
0.32
1407
255
134
l:!.l
107.11 °-'°
5.91
6.20
0.03
0.32
1395
254
133
131
107.22 °-"
0.09
6.98
5.29
0.03
0.32
1384
254
132
135
107.31 ,
fi.05
6.32
0.03
0.32
1373
254
132
13(!
107.40+°-°'>
fi.lt
6.34
0.04
0.32
I.'IOO
253
131
137
107.49 °°'l
107.57 °-°'^
fi.18
6.37
0.04
0.32
1349
253
131
13S
0.25
6.40
0.04
0.32
1337
252
130
139
107.04 °-°/
0.06
C.31
6.42
0.04
0.32
1325
252
129
140
107.70
107.70+°°^
107. S2 °-°'^
0.38
6.45
0.04
0.32
1314
251
129
141
fi.44
6.48
0.05
0.31
1 302
251
128
142
G.51
5.51
0.05
0.31
1290
250
127
143
107..S7 °-°5
0.57
6.53
0.05
0.31
1278
250
120
144
107.91 °-°-t
0.04
107.95
fi.G4
6.50
0.05
0.31
1207
249
120
1!5
0.70
5.59
( 5
0.31
1 255
249
125
lie
107.98 +°-°3
0.70
6.01
0.06
0.31
1243
248
124
147
los.oi °-°-<
0.83
5.04
0.00
0.31
1231
248
123
US
10X.03 °-°^
fi.H9
5 07
0,00
0.31
1220
247
123
149
108.05 °-°^
0.00
0.95
5.09
0.00
0.31
1208
247
122
150
108.05 ,
7.01
5.72
0.07
0.31
1190
240
121
ir.i
108.00+°°'
7.08
6.75
0.(»7
0.31
11S4
240
120
152
108.05-°°'
7.14
5.78
0.07
0.31
1172
245
119
153
108.04 °-°'
7.20
5. SO
0.07
0.30
1101
244
118
154
108.03 °-°'
0.02
7.25
6.83
O.OS
0.30
1149
244
118
155
i-^.Ol
7.31
6.80
0.08
0.30
1137
243
117
15r,
, 'T.98-°-°-'
7.37
6.S8
0.08
0.:i0
1 1 20
243
110
157
107.95 °-°-]
7.43
5.91
0.09
0.30
III4
242
115
15S
'"'•"» TA
7.49
6.94
0.09
0.30
1102
241
114
159
107.87 °-°!
0.05
107.77 °-°S
7.54
6.97
0.09
0.30
1090
241
113
IfiO
7.00
6.99
0.10
0.30
1079
240
112
1(11
7.00
6.02
0.10
0,30
1007
9 to
112
1(;2
107.70 °-°/
107.04 °-°^
107.56 °-°^^
0.07
107.49
7.71
0.04
0.10
0.30
1055
239
111
103
7.77
0.07
0.11
0,30
1044
23S
110
1G4
7.82
6.10
O.ll
0.29
1032
238
109
ir,5
7.87
0.12
0.11
0.29
1020
237
108
ii'.a
7.93
6.15
0.12
0.29
1009
230
107
107
107.31 °-°^
107.22 °-°?
7.98
6.17
0.12
0.29
997
230
100
lOS
8.03
6.20
0.12
0.29
!)S0
235
105
109
8.08
6.22
0.13
0.29
974
234
104
170
107.01
8.13
6.25
0.13
0.29
903
234
103
171
100.89-°-'^
8.18
6.27
0.14
0. 29
951
2.-13
102
172
100.77 °-'-'
8.23
6.29
0.14
0.29
940
232
101
173
100.05 °-"
8.28
6.32
0.14
0.29
928
232
100
174
100.52 ° '•'
0.14
100.38
8.33
6.34
0.15
0.29
917
231
99
175
8.37
6.30
0.15
0,29
900
230
98
170
100.24-°-'-*
8.42
6.39
0.15
0.29
894
230
97
177
100.10 ° ';}
105.94 °-'^
105.78 °-'^
0. 16
8.40
6.41
0.10
0,29
8s:i
229
90
17S
8.51
6.43
0.10
0. 29
872
228
95
179
8.55
6.45
0.17
0.29
801
228
94
180
105.02
8,00
6.47
0.17
0.29
850
227
93
THE OIllJIT OP URANUS.
227
TAlUiK
VIII, Ann. \.—C
onlinued.
Arg.
(u.c.O) Diir
(1..8.I)
{>y.v.\)
(.'.«.2)
(y.f.2)
(f.c.O)
(p.O.l)
(p.C.l)
180
105.f.2
8.00
n
6.47
0.17
0.29
850
227
93
181
105.45-°- 7
105.27 °' ^
105.09 °- '^
8.04
6.50
0.17
0.29
839
220
92
182
8.08
0.52
0.18
0.29
828
220
91
183
8.72
0.54
0.18
0.29
810
225
90
184
104.90 °;;j
8.7(1
C.50
0.19
0.29
805
224
89
185
8.80
fi.57
0.19
0.29
794
224
88
18(>
104.51-°'^°
8.84
C.59
0.19
0,29
784
223
87
ls7
104.31 °-\°
8.88
6.01
0.20
0.29
773
222
80
188
104.10 °-
8.92
0.03
0.20
0.29
702
222
85
18«
103.89 °:^;
8.9(i
0,05
0.21
0.29
751
221
84
190
103.07 „ ,,
8.99
6.07
0.21
0.29
740
221
83
191
103.44-°-^
9.03
0.08
..21
0.29
730
220
82
192
103.21 °-\\
9.07
0.70
0.22
0.29
719
219
81
193
102.98 °':';
9.10
0.71
0.22
0.29
708
219
80
194
102.49 „ ,.
9.13
0.73
0.22
0.29
698
218
79
195
9.1 7
0.74
0.23
0.29
688
217
78
I9f>
102.24 -°-^S
101.72 °,,
9.20
0.70
0.23
0.29
677
210
77
197
9.23
0.77
0.24
0.29
607
210
70
198
9.20
0.78
0.24
0.29
057
215
75
199
101.18 ,
9.29
6,79
0.24
0.29
640
215
74
200
9.32
6.81
0.25
0.29
030
214
73
201
100.90-°-^„
9.35
0.82
0.25
0.29
620
213
72
202
100.(12 °-^'*
9.38
6.83
0.25
0.29
610
213
71
ms
100.33 °'9
9.41
0.84
0.20
0.29
600
212
70
204
100.04 °-"9
O.JO
9.43
6.85
0.20
0.29
590
211
69
205
99.74
9.40
0.80
0.26
0.29
680
211
08
20li
99.44 -°-3°
9.48
0.80
0.27
0.29
577
210
07
207
99.13 °-5'
9.51
0.87
0.27
0.29
507
210
00
208
98.82 °--5'
9.53
6.88
0.27
0.29
557
209
05
209
98.5(- °-^-
0.32
98.18
9.55
0.88
0.28
0.29
548
208
64
210
9.58
6.89
0.28
0.29
538
208
63
211
97. 85-°' 3^
9.00
0.89
0.28
0.29
629
207
02
212
97.52 °-3-
9.02
0.90
0.29
0.29
520
200
01
2l;5
97.18 °\]
9.04
0.90
0.29
0.29
510
200
00
214
9(1.84 "-^1
911.49
9.00
0.90
0.29
0.29
601
205
59
215
9.0s
0.91
0.30
0.29
492
205
58
21(1
9(1.1 4 -°-3S
9.70
0.91
0.30
0.29
483
204
57
217
95.79 °-35
95.43 °-t
9.71
0.91
0.30
0.29
474
204
50
218
9.73
0.91
0.31
0.29
404
203
55
219
95.0G -3;
9.75
6.91
0.31
0.30
450
202
54
220
9.70
6.91
0.31
0.30
448
202
53
221
94.32-°-"
93.94 °-3-^
93.5(! °-3«
9.78
6.90
0,31
0.30
439
201
52
222
9.79
6.90
0.32
0.30
430
201
51
22;]
9.81
0.90
0.32
0.30
422
200
50
224
«^-^^ c:5
9.82
6.90
0.32
0.30
413
200
49
225
92.78
9.83
6.89
0.33
0.30
405
199
48
22«
92.39-°- 3^
9.85
6.88
0.33
0.30
398
198
47
227
91.99 °-^°
9.80
0.88
0.33
0.30
390
198
40
228
»!-'"•» o\i
9.87
0.87
0.33
0.30
382
197
40
229
90.77
9.88
6.86
0.34
0.30
374
197
45
230
9.89
0.85
0.34
0.30
305
196
44
231
90.30-°-^'
9.90
0.84
0.34
0.30
357
190
43
232
89.94 °-*^
9.91
0.83
0.34
0.30
849
195
42
233
89.51 ° •♦3
9.91
0.82
0.34
0.30
342
194
41
234
89.09 °--*^
0.44
88.r,5
9.92
0.81
0.35
0.30
334
194
41
235
9.93
0.80
0.35
0.30
327
194
40
23()
88.22 -°-*3
9.94
0.79
0.35
0.30
320
1 93
89
237
87.78 °-4;^
9.94
0.78
0.35
0.30
312
192
88
238
87.34 °-44
9.95
0.70
0.35
0.30
305
192
38
239
80.89 °-'*5
0-45
8(1.44
9.95
0.74
0.30
0.30
298
191
37
240
9.90
0.73
0.30
0.31
291
191
36
! 1;
j'l
^.1
11
228
THE ORBIT OP URANUS.
TAHLE VIII, Alio. l.—CoiUinm-<l. |
Arg.
(i'.<'.0) Diff.
(.-.«.!)
(v.r.l)
C.73
(r.«.2)
(u.c.2)
It
0.31
(,,.r.O)
(,,.«.!)
O'.'-.l)
240
II II
9.96
0.36
291
191
36
241
85. 99-°- 45
85.63 Ijl
9,96
6.71
0.30
0.31
284
190
35
342
9.96
0.09
0.36
31
278
190
35
24.'i
85.08 °-'»5
9.97
6.67
0.36
0.31
271
189
34
244
«^"» lil
9.97
6.66
0.37
0.31
265
189
33
245
9.97
6.64
0.37
0.31
268
188
33
24fi
83.08-°- 47
9.97
0.61
0.37
0.31
252
188
32
247
83.21 y,i
82.73 If.
9.98
0.59
0.37
0.31
246
1S7
31
24S
9.98
6.57
0.37
0.31
240
187
31
249
9.98
6.55
0.37
0.31
234
180
30
250
81.77
9.98
6.53
0.37
0.31
228
186
29
251
81.28-°-49
9.98
0.51
0.38
0.31
222
185
29
252
80.80 °-4»
9.98
6.48
0.38
0.31
210
185
28
253
80.30 °-5°
9.97
6.46
0.38
0.31
211
184
28
254
79.81 °-49
0.50
9.97
6.43
0.38
0.31
205
184
27
255
9.97
6.40
0.38
0.31
200
183
27
25 (i
78. 82-°- 49
9.97
6.37
0.38
0.30
195
183
26
257
78.31 °-5i
9.90
6.34
0.38
0.30
190
182
26
25S
77.81 °-5°
9.90
6.31
0.38
0.30
184
182
25
250
77.30 °Si
0.51
70.79
9.90
0.28
0.38
0.30
179
181
25
2f.O
9.95
6.25
0.38
0.30
175
181
24
20 1
70. 28-°- 5'
9.95
6.22
0.38
0.30
170
180
24
202
75.77 °-5'
9.95
6.19
0.38
0.30
105
180
24
2fi;i
75.25 °-S»
9.94
6.16
0.33
0.30
101
179
23
204
74.73 °-S*
0.52
74.21
9.94
6.13
0.38
0.30
150
179
23
205
9.93
6.10
0.38
0.30
152
178
22
200
73.09-°-S2
9.93
6.06
0.38
0.30
148
178
22
207
73.10 °-53
9.92
6.03
0.38
0.30
144
177
22
208
72.04 °'52
9.92
5.99
0.38
0.30
140
177
21
209
72.11 °-53
0.54
9.91
5.96
0.38
0.30
137
no
21
270
9.90
5.92
0.39
0.30
133
170
21
271
71.04 -°-53
9.90
5.88
0.39
0.30
129
175
20
272
70.51 °-53
9.89
5.85
0.39
0.30
120
175
20
273
09.97 °S4
9.88
6.81
0.39
0.30
122
174
20
274
09.43 °S4
■•■54
03.89
9.88
6.77
0.39
0.30
119
174
20
275
9.87
5.73
0.39
0.30
no
174
20
270
08.35-^-54
9.80
6.69
0.39
0.29
113
173
19
277
07.81 °-S4
9.85
6.65
0.39
0.29
110
172
19
27.S
07.20 °-5S
9.84
5.61
0.39
0.29
107
172
19
279
Cfi.72 If
00.17
9.84
5.57
0.39
0.29
105
171
19
280
9.83
6.52
0.39
0.29
102
171
19
281
C5.02-°-5S
9.82
5.48
0.39
0.29
100
170
19
282
05.07 °S5
9.81
5.44
0.39
0.29
97
170
18
283
04.52 °-S5
9.80
5.40
0.39
0.29
95
169
18
284
63.97 °-S5
0.56
03.41 ^
9.79
5.36
0.39
0.29
93
109
18
285
9.78
6.31
0.39
0.28
91
108
18
286
02.80— °- 55
9.77
6.26
0.39
0.28
90
108
18
287
02.31 °-55
9.77
5.23
0.39
0.28
88
167
18
2SS
01.75 °-56
9.70
5.17
0.39
0.28
80
167
18
289
01.19 °S6
°-5S
00.04 ^
9.75
6.13
0.38
0.28
85
100
18
290
9.74
5.08
0.38
0.28
84
100
18
291
60.08— °- 5''
9.73
6.03
0.38
0.27
82
165
19
292
59.52 °-5^
9.72
4.99
0.38
0.27
81
164
19
293
58.90 °-5^
9.71
4.94
0.38
0.27
80
164
19
294
58.40 °-56
0,56
9.70
4.89
0.38
0.27
80
163
19
295
57.84 .
57.28-^°-5°
50.72 °-5»
9,69
4.84
0.38
0.27
79
163
19
290
9.68
4.80
0.38
0.27
78
102
19
297
9.07
4.75
0.38
■ 0.26
78
102
19
298
50.16 °-5°
55.00 °'5o
0.56
5.V04
9.00
4.70
0.38
0.26
77
101
20
299
9.65
4.05
0.38
0.26
77
101
20
300
9.64
4.60
0.38
0.2')
77
100
20
THE ORBIT OP URANUS.
229
TAIJLI'
VIII, A
mi. \.—t
Continued.
Arg.
(i).r.O) Diir.
(...,.1)
(v.c.l)
(u.(..2)
(u.e.2)
(p.r.O)
(p.".!)
(,.<■.!)
300
n II
65.04_„ cfi
5t.48-°-S^
53.92 °-S^
49.44
9.04
ft
4.00
0.38
0.2c
77
160
20
3Ul
9.03
4.55
0.38
0.20
77
159
20
302
9.02
4.50
0.38
0.25
77
159
20
303
9.01
4.45
0.38
0.25
77
158
21
304
9.00
4.40
0.38
0.25
78
158
21
305
9.59
4.35
0.38
0.25
78
157
?1
300
9.58
4.30
0.38
0.25
79
150
22
307
9.57
4.25
0.38
0.24
80
150
22
30S
9.50
4.20
0.38
0.24
81
155
22
309
9.55
4.15
0.38
0.24
82
154
23
310
9.54
4.10
0.38
0.24
83
154
23
311
48.88 -°-S6
9.53
4.05
0.38
0.23
84
153
24
312
48.33 °S5
9.52
4.01
0.38
0.23
85
152
24
313
47.77 °'5^
9.51
3.94
0.38
0.23
87
152
24
3U
47.22 "-55
0.56
9.50
3.89
0.38
0.23
88
151
25
315
40.00
9.49
3.84
0.38
0.23
90
150
25
31G
40.11 -°-5S
9.48
3.79
0.38
0.22
92
150
20
317
45.50 °-5S
9.47
3.74
0.38
0.22
94
149
20
318
45.01 °SS
9.40
3.09
0.38
0.22
90
14S
27
319
44.40 °-5S
°-S5
43.91
9.45
3.03
0.38
0.22
98
148
27
320
9.44
3.. 58
0.38
0.21
100
147
28
321
43.30 -°-5S
9.43
3.53
0.38
0.21
103
140
28
322
42.81 °-5S
9.42
3.48
0.38
0.21
105
140
29
323
42.27 °-5-»
9.41
3.43
0.38
0.21
108
145
29
324
41.72 °-55
0.54
41.18
9.40
3.38
0.38
0.20
III
144
30
32.-)
9.40
3.33
0.38
0.20
114
144
30
32(!
40.04-=-S-»
9.39
3.28
0.38
0.20
117
143
31
327
40.10 °-S4
9.38
3.23
0.38
0.20
120
142
32
32S
39.50 °-S4
9.37
3.18
0.38
0.19
123
141
32
329
39.03 °S3
0.54
9.30
3.14
0.37
0.19
127
140
33
330
38.49
9.35
3.07
0.37
0.19
130
140
34
331
37.90-°- 5 3
9.34
3,02
0.37
0.19
1.34
139
34
332
37.43 °-S3
9.33
2.08
0.37
0.18
138
138
35
333
30.90 °S3
9.32
2.93
0.37
0.18
141
137
30
334
30.37 °'53
0.52
9.31
2.88
0.37
0.18
145
130
30
335
9.30
2.83
0.37
0.18
150
130
37
33B
35.33-°- 5^
9.29
2.78
0.37
0.17
154
135
38
337
34.81 °-S^
9.29
2.73
0.37
0.17
158
134
38
338
34.29 °-52
9.28
2.09
0.37
0.17
102
133
39
339
33.78 -5;
33.20
9.27
2.04
0.37
0.17
107
132
40
340
9.20
2.59
0.37
0.17
172
132
41
341
32.75 -°-5;
9.25
2.54
0.37
0.10
170
131
42
342
32.24 °- '
9.24
2.50
0.37
o.in
181
130
42
313
31.74 °-°
9.23
2.45
0.37
0.10
186
129
43
344
31.23 °-5'
0.50
30.73 „ ,„
9.22
2.41
0.37
O.IG
191
128
44
345
9.21
2.30
0.37
0.15
197
127
45
340
30.24-°- ^2
9.20
2.31
0.37
0.15
202
126
40
347
2»-T* li°
9.20
2.27
0.37
0.15
207
120
40
348
2»-25 III
9.19
2.23
;7
0.15
213
125
47
349
2«-T6 °:S
27.79 -"--^S ■
9.18
2.18
0',7
0.15
218
124
48
350
9.17
2.14
0.37
0.14
224
123
49
351
9.10
2.09
0.37
0.14
230
122
50
352
27.31 °-''^
20.83 °-+^
9.15
2.05
0.37
0.14
230
121
51
353
9.14
2.01
0.37
0.14
242
120
51
354
20.30 °--*7
0.47
9.13
1.97
0.37
0.14
248
119
52
355
25.89
9.12
1.93
0.37
0.14
255
118
53
350
25.42 -°-'*7
24.90 °-t^
24.50 °-'t^
24.04 °-t^
0.46
23.58
9.12
1.89
0.37
0.13
201
117
64
357
9.11
1.85
0.37
0.13
208
116
55
358
9.10
1.81
0.37
0.13
274
115
50
359
9.09
1.77
0.37
0.13
281
114
57
300
9.08
1.73
0.38
0.13
288
113
58
w-
1
•A
Clii
I HI
ma
li
:i f:
230
T 11 K O U n I T () V II U A N IT S.
TAIU-K VI ir, A
lUl, \. — (
'iinlinitiil.
Art?.
(tM'.O) Diir.
(V.H.I)
(''■<•. 1)
ft
(•'.«. 2)
(r.,-.2)
It
(p.r.O)
(,...1)
(,...•.1)
II It
n
tt
ano
^'''-'"^ „ ,c
9.08
1.73
0.38
0.13
288
113
68
8(11
23.13-°^S
0.07
1.70
0.38
0.13
29 1
112
58
8ii-i
22(19 °^^
9.00
1.0»
0.38
0.13
301
111
59
8)13
22.24 °-5
9.05
1.02
0.38
0.13
308
110
60
8(U
21.80 °;^}
9.04
1.59
0.38
0.12
310
109
61
snr)
9.03
1.55
0.37
0.12
323
108
62
son
20. 94-°-^^
9.02
1.52
0.37
0.13
330
107
63
3(17
20.51 °|
9.01
1.48
0.37
0.13
338
106
64
ac.s
20.0H °^-
9.00
1.45
0.37
0.12
345
105
(15
SCO
19.(10 °;]j
19.21
8.99
1.42
0.37
0.11
353
104
60
370
8.9S
1.39
0.37
0.11
8(11
103
66
.ni
18.83 ''°-*'
8.97
1.30
0.37
0.1 1
3(19
102
67
373
18.42 °-»'
8.9i;
1.32
0.37
0.11
877
101
68
37;J
18.03 °-^°
8.95
1.29
0.37
0.11
385
100
(19
874
375
n.oa °-*''
0.40
17.23
8.94
8.93
;.27
"i.24
0.37
0.37
0.11
O.II
393
401
99
98
70
71
37(>
l(1.83-°-3'>
8.92
1.21
0.37
0.11
410
97
72
377
10.44 °-."
Ifi.Ofi °-^'J
15 -i °-^'^
0.38
15.30
8.91
1.18
0.37
0.11
418
9(1
73
871
8..S9
1.10
0.37
0.11
420
95
71
371)
3S0
8.88
8.87
1.13
1.11
0,37
o.;i7
0.11
0.10
435
414
91
9:1
75
76
8S1
14.93 -°"
14..',7 °-'^
13.85 °»^
13.50
8.80
1.08
0.37
0.10
4,^2
92
77
8S2
8.85
1.00
0.37
0.10
401
91
7S
3S3
8.83
1.04
0.37
0.10
470
90
78
3S4
3M5
f:.82
8. SI
1.02
1.00
0.37
0.37
0.10
0.10
479
488
89
88
79
80
3S(I
13.15-°'"
8.79
0.98
0.37
0.10
497
87
81
3s;
12.80 °--^3
8.78
0.90
0.37
0.10
50(1
8(1
8:'
3HS
12.47 °-]l
8.77
0.94
0.37
0.10
514
85
8'
88 y
12.13 III
11. Si
8.75
0.92
0.37
0.10
525
84
3!)()
8.74
0.90
0.37
0.10
534
82
391
11.48-^-"
8.73
0.88
0.37
0.10
544
81
8(.
392
11.10 °-
8.71
0.87
0..37
0.10
554
80
86
393
10.85 °-
8.70
0.85
0.3(1
0.10
5(14
79
87
394
10.54 °;];
^0 -^ 70
8.(18
0.84
0.3(i
0.10
5f3
78
88
395
8.07
0.83
0.3(i
0.10
5s;i
77
89
39(i
9.93-°-^°
8.05
0.82
0.3(i
10
593
76
90
397
9.(14 III
8.(!.<;
SO
3(1
0.10
(103
75
91
39 S
9.. 35 °- ^
8.03
0.79
0.3(5
0.10
013
74
92
399
'^.00
0.7.8
0.30
0.10
C23
73
93
400
«-T!^ 0,,
8.5S
0.7T
0.30
0.10
033
72
9.3
401
8..52-°-
8.. 57
0.70
0.3(1
0.10
043
71
94
403
«-2'^ olJ
8.55
0.7('.
0.30
0.10
653
70
95
4113
v.fls °- !
8.53
0.75
0.30
0.10
604
69
96
4U4
^■7=' L'^
8.51
0.74
0.35
O.IO
674
C8
97
405
7.47
8.49
0.74
0.35
0.10
684
67
98
40(!
7 .)2-°='S
8.47
0.74
0.35
0.10
695
66
98
407
(iMs °-^-*
8.40
0.73
0.35
0.10
700
65
99
40 S
fl.75 °13
8.44
0.73
0.35
0.10
71(5
64
100
409
G.51 °-^-*
0. 22
8.42
0.73
0.34
0.10
727
63
101
410
0.29
8.:!9
0.73
0.34
0.10
737
62
101
411
G.()7-°"
8.37
0.73
0.34
0.10
. 748
61
102
412
5.85 °"
8.35
0.73
0.34
0.10
759
60
10.^
413
5.04 °"
8.33
0.73
0.34
0.10
770
59
104
414
6.14 °-'^
0. 20
8.31
73
0.34
0.10
781
58
104
415
5.24
8.28
0.73
0.34
0.10
792
57
105
410
5.04-°-^°
4.80 °''^
8.2(1
0.74
0.33
0.10
803
56
106
417
8.24
0.74
0.,33
0.10
814
55
107
41S
4.C.7 °"'
8.21
0.74
0.33
0.10
825
54
107
419
4.33
8.19
0.75
0.33
0.10
836
53
108
420
8.17
0.76
0..33
0.10
847
52
109
TIIK OniJIT OF URANUS.
231
T.VHLK
VIII, Ami. 1 —f
iiiUniird.
Arg.
(h.<!.o) Diir.
(.....1)
(v.v.l)
(u.».2)
(>K>:2)
(,.r.0)
(P.-.I)
(p.C.l)
n tf
ft
n
tf
420
<-''3 „,,
8.17
0.70
0.33
0.10
847
53
109
421
4.1(1-° 'J
4.00 °'^
8. It
0.70
0.33
0.10
858
51
110
4'2!)
8.11
0.77
o.;;3
0.11
870
50
110
4-i:t
8.85 °-\l
8.00
0.78
0.33
0.11
881
50
111
424
8.0(1
0.79
0.33
11
893
49
111
4'2.'>
8.04
0.80
o.;;3
0.11
1)04
48
112
421!
^■*'^ oA
8.01
0.81
0.31
0.11
010
47
113
427
3.29 °^
7.98
0.83
0.31
0.11
037
40
113
428
3.1(1 °\l
7.95
0.83
0.31
0.11
938
45
114
429
3»^ o.W
7.!I2
0.85
0.31
0.11
950
44
114
430
2.n3
7.89
0.80
0.31
0.11
901
44
116
431
2.S2-°'"
7.80
0.87
o.;io
0.11
073
43
110
4;t2
2.72 °''°
7.83
0.89
0.30
0.11
984
43
110
4:t;{
2.(12 °'°
7.80
0.!(0
0.30
0.11
990
41
117
434
2.53 °°2
o.o8
2.45
7.77
0.93
0.29
0.11
1008
40
118
435
7.74
0.94
0.29
0.11
1019
40
118
4;t(J
2.37 -°°']
7.71
0. !(5
0.29
0.12
1031
39
119
4;{7
2 i)() °o8
7.07
0.97
0.29
0.12
1043
38
119
4:i.s
2:2.3 °°'^
7.04
0. 119
0.28
0.12
1054
38
120
43!)
2.1(1 °°7
0.0s
7.(10
1.01
0.28
0.12
1000
37
120
440
2.11 :
7.57
1.03
0.28
0.12
1078
30
121
441
2.00 -°°S
7.54
1.05
0.28
0.13
1089
35
121
442
2.03 °°^
7.50
1.07
0.27
0.12
1100
35
123
443
1.98 °°-*
7.47
1.09
0.27
0.12
1113
34
123
444
l.!)5 °°^
0.03
1.92
7.43
1.11
0.27
0.12
1124
33
123
445
7.39
1.14
0.20
0.12
1137
33
123
4411
, ,,,,-0.02
7.3(1
1.10
0.20
0.12
1149
32
123
447
1.88 °°'
7.32
1.18
0.20
0.12
1100
32
124
44S
1.87 °°'
7.28
1. 21
0.20
0.12
1172
31
124
44!)
1.87 °'°°
0.00
7.24
1.23
0.25
0.12
1184
30
125
4:.o
1.87 ,
7.20
1.20
0.25
0.12
1190
no
125
451
1.88+°'°'
7.10
1.28
0.25
0.12
1208
30
125
453
l.!l() °°-
7.12
1.31
0.24
0.12
1220
29
120
453
1.92 "■°-'
7.08
1.33
0.24
0.12
1 232
28
120
454
0.03
1.98 ,
7.01
1.30
0.24
0.13
1243
28
127
455
7.00
1.39
0.23
0.12
1255
28
127
45(;
2.01 +°°.^
0.90
1.42
0.23
0.12
1207
27
127
457
2 0(1 °°-^
0.91
1.45
0.23
0.12
1279
27
128
45H
2.11 °°3
0.87
1.47
0.22
0.12
1291
20
128
45!)
2.1(1 °°^
0.07
2.23
0.83
1.50
0.23
0.12
1302
20
128
4(->0
0.78
1.53
0.22
0.12
1314
20
128
4(i!
0.74
1..50
0.21
0.12
1320
25
129
402
0.(19
1.59
0.21
0.12
1338
25
129
4(13
2.44 °'°7
0.0. >
1.02
0.21
0.12
1350
25
129
4(14
2.53 °°'^
0.09
2-''2 ,„„„
0.00
1.05
0.20
0.12
1301
25
130
4f,5
0.50
l.fiS
0.20
0.12
1373
24
130
4r>a
2^1+0.00
0.51
1.72
0.20
0.12
1385
24
130
4fi7
2«2 °- !
0.40
1.75
0.19
0.13
1390
24
130
4fiS
2'92 :
0.41
1.78
0.19
0.13
1408
24
130
4(19
304 ::;-;
0.37
1.81
0.19
0.13
1420
23
131
■170
3.15
0.32
1.84
0.18
0.13
1431
23
131
471
•J .)s+°'.'5
0.27
1.88
0.18
0.13
1 1 13
23
131
472
3.41 °'-^
0.22
1.91
0.18
0.12
1 455
23
131
473
3.. 55 °'4
0.17
1.94
0.17
0.12
1400
23
131
474
3.(19 °'-*
0.14
3.83 ,
0.12
1.98
0.17
0.12
1478
23
132
475
0.07
2.01
0.17
0,12
1489
23
132
47G
3. 98+°- '5
4.14 °'^
0.02
2.04
0.1(1
0.12
1501
23
132
477
5.90
2.07
0.10
0.12
1512
23
132
478
4.31 °;7
5.91
2.11
0.10
0.12
1524
23
132
479
4.48 °'7
0.17
4.05
5.80
2.14
0.15
0.12
1535
23
132
480
5.81
2.18
0.15
0.12
1540
23
132
iMii:
.i *
)■ H
i IP
f
i i!
ii ;!
233
THE ORBIT OF URANUS.
TABLK
VIII, A
KO. 1. — ContinvriK
A.V'.
(f.c.O) Diff.
(v.x.l)
(•'•'•. 1)
(U.S.2)
(o.r.2)
Cp.c.o;
(f)."!)
(P.e.l)
If II
tt
$t
ft
ft
480
-;Uo..8
5.81
2.18
0.15
0.12
1540
23
133
4Sl
5.75
2.-1
0. 15
12
1558
23
132
4,S2
5«=2 nl^
5.70
2.24
0.15
0.12
1509
23
133
483
5.21 ""^
5.05
2.28
0.14
0.12
1 5S0
23
132
484
^•^" .r^
5.59
2.31
0.14
0.12
1591
23
133
485
5. CI ,
5.54
2.35
0.14
0.12
! 002
24
133
4Si;
5.81+°"
5.48
2 38
0.13
0.12
1013
24
133
487
fi.03 °-
6.43
2.42
0.13
0.12
1 024
24
133
48S
f'-'-^l o.
5.37
2.45
-.1.13
0.11
10,35
24
134
481)
6.70 ,
5.32
2.48
0.12
O.ll
1040
25
134
490
5.20
2.52
0.12
0.11
1057
25
133
491
f;.!l3+°-'-5
5.20
*2.55
0.12
0.11
1008
25
133
492
7.17 °-^-t
5.1;-)
2.59
0.12
0.11
1079
25
133
493
7.41 "•'■>
5.119
2.02
0.11
0.11
1090
25
133
494
7.(!(! °%5
O. 20
6.U3
2.00
0.11
0.11
1700
20
133
495
7.92 , ' .
8.1,S+°-^^
8.-I4 °-^^
4.98
2.09
O.ll
0.11
1711
27
133
49G
4.92
2.72
0.11
O.ll
1721
27
133
49T
4.80
2.70
0.10
0.11
1732
28
133
498
8.71 °H
4.80
2.79
O.IO
0.11
1742
28
l.i3
499
8.99 ""-'^
0.27
9 20 ^ ,
4.75
2.83
0.10
0.11
1753
28
133
500
4.09
2.80
0.10
on
1703
29
133
501
9.55+°-^'^
4.03
2.89
0.10
o.ll
17-4
30
133
502
9.84 °-''
4.57
2.93
0.09
0.10
'784
30
133
503
10.13 °'^'^
4.51
2. 91'.
0.09
0.1 1»
1794
31
133
604
10.43 °'^
10.74 ,
4.44
2.i'9
0.09
0.10
1804
31
133
505
4.39
3.02
0.09
0.10
1S14
32
133
&0I>
ll.0;-.+°'^'
4.33
3.0;!
0.09
0.10
1824
32
133
507
11. 3(, °''
4.28
3 09
0.08
0.10
1834
33
133
508
11. (!8 °y-
4 22
3.12
0.08
(1. 10
1844
34
133
509
12.00 '"■•'^
°J3
4.10
3.15
0.08
0,10
1S53
35
133
510
12.33 ,
12.r,.!+°-|3
4.10
3.18
0.08
0.09
1 803
!>;"»
133
511
4.04
3.21
0.08
0.09
1873
30
133
5i:
13.00 °34
8.93
3.25
0.07
0.09
lss2
37
133
;-.l:i
13.34 l\\
3.92
3.28
0.07
0. 19
1892
38
133
614
13.G8 °;j^^
'4.03 ,
14.39+°'^"
14.75 °'^^'
15.11 °-|^
8.80
3.31
0.07
0.09
1901
39
133
515
3.S0
3.34
0.07
0.09
1910
40
133
510
3.74
3.37
0.07
0.09
1920
40
133
517
3.08
3.10
0.07
0.09
1 929
41
133
518
3.02
3.13
0.97
0.09
1938
42
133
519
3.50
3..<i6
0.07
0.08
l'.M7
43
133
520
3. 50
3.49
0.00
08
1 9,'^„i
44
132
521
3.41
3.51
0.00
O.CM
1904
45
132
522
3.38
3.54
0.00
0.08
1973
40
132
523
1C..99 °'5
3.32
3.57
O.Oi!
0.08
1982
47
132
524
17.77 ,
3.27
3.00
0.06
0.08
l'J;»0
48
132
525
3.21
3.02
0.06
0.08
1999
49
132
52fi
18.17+°'°
3.15
3.0."i
0.06
0.08
2007
50
l:;2
527
18.07 °-t°
3.09
3.07
0.00
0.08
2010
51
132
528
18.97 °-'°
3.03
3.70
0.06
0.07
2024
52
1 32
529
19.38 °-»'
0.41
19.79 ,
2.97
3.73
0.05
0.07
2032
53
132
530
2.91
3.7.'>
0.05
, 0.07
2040
55
132
531
20.21 +°-''
2. MO
3.77
1«.05
"0.07
2048
50
132
532
20.(12 °-"
2 80
3. so
0.05
0.07
2050
67
132
533
21.05 °«
2.75
3.82
0.(i5
0.07
2(103
58
132
534
21.47 °-»'
0.43
21.90 ,
2.09
3.84
0.05
0.07
2071
69
131
535
2.03
3.87
0.05
O.OT
2079
01
131
531)
.,J ;,4+o-14
2.57
3.8i)
0.05
0.07
2080
62
131
537
"VT °-!^
2.52
3.91
0.05
0.07
2094
03
131
538
23.21 °^*
2.40
3.93
0.05
0.07
2101
04
131
539
23.. IC, "■'•S
0.44
24.10
3.40
3.90
0.05
0.00
2108
66
; 131
1
540
2.35
3 OS
0.05
0.00
2115
67
131
THE ORBIT OF URANUS.
233
TAHLK
VIII, A
10. 1. — Concluded.
Aig.
(o.e.O) Diff.
(u.s.l)
(...c.l)
(0.8.2) ]
0>.6'.2)
(p.c.O)
{p.»l)
(p.o.l)
ft ff
fr
ft
//
640
2*10. ,A
2.35
3.98
0.05
0.00
2115
07
131
f)41
2.2'J
4.00
C.')5
0.00
2122
08
131
542
2.24
4.02
0.05
0.00
2129
70
131
643
2.18
4.04
0.05
0.00
2135
71
130
644
2.13
4.C.'
0.05
O.OG
2142
73
130
545
2.08
4.07
0,05
0.06
2149
74
130
51()
2.02
4.0!)
0.05
0.00
2155
75
130
547
27.;i2 °-^^
l.i)7
4.11
0.05
0.00
2102
77
130
54S
2«-2T o:£
1,'J2
4.13
0.05
0.00
2108
78
130
l>i'J
1.87
4 14
0.05
0.06
2174
80
130
550
2S.75 o
1.H2
4.',;
0.05
0.00
2180
81
130
551
2!t.2.i+°-*°
1.77
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0.05
0.00
2180
83
130
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1.72
4,19
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1.07
4.20
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8".
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1.48
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4,31
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0.00
2238
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1.25
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2250
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2201
100
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o.;if;
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113
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0.09
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132
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50, 10+° 5'
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2320
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l:!t
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on
23:tl
104
i;u
it
30 July. 1873
23 i
THE ORBIT OF URANUS-
t ;li
(.,
I!!i
TAHLR IX, Ami. 2—
VCI'ION
oi- Saturn.
1
Arg.
(c.r.O) Dill.
1
(i.'.s.l) Dili. Sco.var.
{r.c.D Dili, g
L'c.var.
It
(r.*i.2) Diff. S
t'c.var.
(v.u.2) Dill". Sec.var.
II II
// //
/'
// //
/'
// /'
"
38.54 ,
139.94 ,
2.53
293.78 ,
0.50
182.02 , „
1.77
244.07
0.12
1
S8.63+°-°9
141.38 + ;-'|
2.52
293.81 +°°3
0.55
183.80 + '-'"
1.70
244.10-°-57
243.52 °-f^
0.12
2
38.72 °-°5
142.82 -^l
2.50
293.83+°-°^
0.54
184.90 '-'^
) 74
0.12
a
38.81 °°9
144.26 '-^^
2.49
293.83 °°°
0.53
180.12 '■'"
1.73
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4
38.90 °°l
145.70 '•'»■*
2.48
293.82-°°'
0.52
187.27 '-'5
1.71
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0.08
1.44
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5
38.98 ,
Kf.14 ...
2.47
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0.51
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1.70
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148.58+ -y
2.40
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0.50
189.50 +''-^
1.09
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T
39. 1(! °-°l
150.01 1 •♦•?
2.45
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0.49
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1.67
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8
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154.32
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1.01
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157.19 '•^■5
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39.75 °°^
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1.57
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2.30
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104.32 '■•♦^
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IS
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105.74 '•'^
2.33
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0.39
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1.50
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1.49
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108.58
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0.37
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174.22 '•^'
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290:53 °-.'°
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27
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1.29
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34
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2 17. 43 -°-'-'-'
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193.55 •••'?
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214.59 °-''^
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194.90 1:
'33
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•>■>, ci 4-0.85
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0.13
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15
47
205.54 '5'
1.90
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0.19
229.51 °-''>
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231.07 °■7^^
0.76
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1 14
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1 93
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••°5
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50
43.20 ,
209.44
1.92
277.91 ,
0.17
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1.04
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43.32 +°' =
210.73 +'■*:>
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212.01 '•-'„
1.89
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((.10
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201,41 '•°'^'
0.17
53
43.54 °- '
213.29 ' ^;^
1.88
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0.10
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1.(10
200.34 '■°l
0.18
54
43.05 °"
214.57 '
1.87
275.17 °-7°
O.lfi
234.77 °l'
0.99
1911.20 '"'^
0.18
0.12
1.27
0.72
0.71
1.08
55
43.77 ,
215.^4
1.H5
271.45
0. 15
235.48 , ,
0.97
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0.19
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217.10 + '-^'^
1.84
273.72-°-"
0.15
2;i0.l8+°J°
230.H7 °p
237.51 °f7
23<.20 ° '"
0.(15
0.90
197.09-'°'^
0.20
57
44.00 °-
2H..I5 ''^S
1.82
27' 98 °'''*
0.15
0.95
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0.2((
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44 12 °"
21 9. CO ' '^
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0.15
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59
44.21 °"
220. '<4 '"^
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27i:4o °;:;
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44.35
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1.7S
27((.C9
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((.91
192 >;7
(i..'2
THE ORBIT OF URANUS.
235
TARLE IX,
Aro. 2
. — Con
billed.
Arg.
(u.«.3)
(i).c.3)
(U.8.4)
(U.C.4
(p.c.O)
(p.s.l)
(p.C.l)
(p.s.2) (p.c.2
(p.K.3)
(p.c.Sj
11.40
ft
14.53
tr
1.58
//
1078
182
1513
711
306
137
85
1
11.52
14.48
1.59
1.66
1079
182
1526
711
303
137
85
9
U.fil
14.43
1.00
1.05
1080
183
1539
710
293
137
85
3
11.75
14.38
1.01
1.04
1080
184
1551
709
295
137
85
4
11.87
14.33
1.62
1.03
1681
185
1504
70 J
291
137
85
5
11.98
It. 28
1.03
1.62
1682
187
1.577
707
288
137
85
6
12.10
It. 22
1.04
1.01
1083
188
1590
700
284
137
85
7
12.21
14.10
1.04
1.00
1083
190
1003
704
281
138
84
8
12.32
14.10
1.05
1.50
1083
191
1015
703
278
138
84
9
12.43
14.04
1.66
1.58
1682
193
1028
701
274
138
84
10
12.54
13.97
1.07
1.50
1082
195
1641
700
271
138
84
11
12.05
13.91
1.68
1.55
1081
197
1054
099
267
138
84
12
12. 7G
13.84
1.09
1.54
1080
199
1006
09 (
264
138
84
l:i
12.87
13.77
1.09
1.53
1079
201
1079
090
260
138
84
U
12.97
13.70
1.70
1.52
1078
203
1091
095
257
137
83
15
13.07
13.03
1.71
1.50
1077
200
1704
691
253
137
83
|{>
13.18
13.50
1.71
1.49
1075
20 <
1710
692
250
137
83
IT
13.28
13.49
1.72
1.48
1074
211
1729
091
240
137
83
IS
13.38
13.41
1.73
1.47
1072
21 1
1741
6-19
243
137
82
19
13.48
13.33
1.74
1.45
1070
217
1754
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239
137
82
20
13.58
13.25
1.74
1.44
1008
220
1700
085
230
137
82
:>1
13.(18
13.17
1.75
1.43
1000
223
1778
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233
137
82
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13.78
13.09
1.75
1.42
1004
227
17V0
681
230
137
81
2:j
13.S7
13.01
1.70
1.40
1002
230
n02
07.1
220
137
81
24
13.97
12.92
1.7G
1.39
1000
23 4
1S14
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14. on
12.83
1.77
1.37
1058
2:!7
1 820
075
220
137
81
2fi
14.15
12.74
1.78
1.30
1050
241
ls:!s
073
217
137
81
27
U,2t
12.05
1.78
1.35
1053
245
Is5(t
071
214
137
80
2S
14.33
12.50
1.79
1.34
1051
219
ls(;2
00 i
211
137
80
29
14.41
12.47
1.79
1.32
1048
253
1874
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208
137
80
no
11.50
12.37
1.79
1.31
1045
258
issi;
004
205
137
79
;ti
14.5S
12.28
1.S0
1.30
1042
202
189s
002
202
137
78
3J
H.c.r,
12.19
1.80
1.28
10.38
2i;7
1910
051)
190
137
78
3:1
14.74
12.09
1..S0
1.27
1035
272
1922
057
190
137
78
34
14.82
11.99
1.8.;.
1.25
1031
277
lOiU
054
193
137
78
3.")
11.90
11.80
l.sl
1 24
1027
2S2
1940
052
no
137
77
,'<i;
H.97
11.79
1.81
i "'2
ir,23
2-^7
1'.l5-t
050
187
137
77
37
15. ((5
11.09
1.81
1 21
1 1; 1 9
^'.>-2
1970
647
184
137
77
3S
15.12
11.59
1.82
1.20
1015
297
19«2
045
isl
137
70
39
15.19
11.49
1.82
1.18
1011
3tt2
1993
042
179
137
76
40
15. 2n
11.38
1.82
1.17
lOOS
30S
201 15
Olii
170
137
70
41
15.33
11.28
1.S2
1.15
1002
313
20 1<;
037
17 !
137
70
42
15.39
11.17
1.82
1.11
1597
319
2028
035
171
137
70
43
15.40
11. oo
1.82
1. 13
1592
325
2039
032
los
137
75
44
15.52
10.95
1.82
1.11
1587
3.31
2051
030
160
137
74
4.'>
15.58
10. St
1.S2
1.10
1 582
337
2003
027
103
1.37
74
4(i
15.(;4
10.,.;
1.82
1.0S
1577
313
2073
2 4
100
137
74
47
15.70
10.02
1.82
1.07
1572
349
20S4
021
158
137
73
4S
15.7C
10.51
1.82
1.00
1500
355
2095
0, 1
155
137
73
49
15. Ml
10.40
1.82
1.04
1501
302
2100
1
153
137
73
r>o
15.86
10.28
1.S2
1.03
1550
308
2117
1 ■
150
137
72
61
15.91
10.17
l.x-i
1.01
1550
375
2128
r,i(i
148
137
! 72
5ii
15.90
10.05
1.82
1.00
1545
381
213S
007
145
137
72
r)3
10.01
9.94
l.sl
0.08
1539
388
21 19
00 4
H3
137
72
f)4
10.05
9.82
1.81
0.97
1533
394
2159
001
140
137
72
55
10.09
9.70
1.81
0. 90
1527
401
2170
598
13S
137
■ 71
5n
10.13
9.59
1.81
0.94
1521
408
2180
595
135
137
71
57
10.17
9.47
1.80
0.93
1515
415
2191
592
133
137
71
58
10.20
9.35
1.80
0.92
150!)
422
2201
589
1.11
137
71
59
10.23
9.23
1.80
0.90
1502
430
2212
5, SO
129
137
71
1
fiO
10.20
9.11
1.80
0.89
1490
437
2222
683
127
137
1 ''
236
THE ORBIT OP URANUS.
TAIJLE IX, Ami. 2.— Continued.
\rK. '(w.c.O) Diir. (r.».l) Diir. Sccvur. (I'.f.l) DilT. Stcvur'! u.«.2) Diff. Sec.vnr,
(il
«2
f,3
C4
C5
Gl>
(!7
68
69
70
71
7-2
73
74
75
7t!
77
78
7 it
80
81
82
83
84
85
8<;
87
8S
89
90
91
92
93
94
95
9i>
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
IIS
119
44.35 .
44.47+°'^
44.59
44.71
44.83
44.95
45.07
45.19
45.31
45.44
45.56
45. CS
45.80
45.92
40.04
46.16
46.29
46.41
40.53
4U.65
46.77
46.,S9'
47.01
47.13
47.24
0.12
0. 12
0.12
0. 12
-1-0.12
O. 12
O. 12
O.I3
0. 12
-t-0.12
0.12
O. 12
O. 12
O.I2
+ °-M
O. 12
O. 12
O. 12
O. 12
,+O.I2
O. 12
O. 12
O. II
O.I2
4 7.. 16 ,
47.48+°'^
47.5!)
47.71
47.82
o. I I
O. 12
O. II
O.I 2
47.94 ,
48.05+°-"
48.16
48.27
48.38
o. I I
o. I I
O.I I
O.ll
48.49 ,
48. ,59+°'°
4,S.70 °"
48.50 °'°
48.91 °"
o.io
49.01 ,
49.11+°'°
49.21 °'°
49.31 °'°
O. ID
49.50 J „
49.59+°-°''
49.77 °?'^
*'J«5 .In
0.09
49.94 ,„„o
•'•»" c"oS
f-'-'-'fi col
fiO.34 ,
50.41 +°-°7
f.0.48 °-°7
50.,55 °°7
50.62
222.07 ,
223.30 +
224.52
225.73
2a,:.. 94
228.14
229.33 +
230.51
231.69
232.86
234.02 ,
235.17"^
236.31
237.45
238.53
2.39.70 ,
240.81''"
241.91
243.00
244.09
245.17 ,
246.24"^
247.. 30
248. ;{5
249.38
250.41
251.43
252.44
253.44
254.43
+
.14
•14
•'3
.11
. 10
.09
.09
.08
.07
.06
■°5
•03
■°3
.02
.01
.00
0.99
0.98
S:^i+°-''r
257.34 °-'^^
2.^8.29 °-'^5
259.23 °-'^-'
260.16
261.08
261.99
262.88
263.76
4.0.92
0.91
0.N9
0..S8
0.87
^65l9+°-«'^
S'l"-"'* o.«4
268.00
0.81
268. SI
269.61
270.41
271.19
271.96
120 50.69
0.07
0.07
+0.S0
o.Xo
0.78
0-77
°-7S
272.71 ,
273.45+°-7-»
274.18
274.90
275,60 ,
0.09
276,29 , .„
276.97+°'^'^
277.63
278.28
278.!»2
0-73
0.72
0,70
279,55
o.M)
0,^15
o.().\
0.63
1.78
1.76
1.75
1.73
1.72
1.70
1.69
1.67
1.66
1.64
1.63
1.62
1.60
1.59
1.57
1.56
1.55
1.53
1.52
1.50
1.49
1.18
1.46
1.45
1.43
1.42
1.40
1.39
1.38
1.36
1.35
1.33
1.32
1.30
1.29
1.28
1.26
1.25
1.23
1.22
1.20
1.19
1.17
1.16
1.14
1.13
1,11
1,10
1,09
1.07
1.06
1,05
1.03
1.02
1.00
0.99
0,98
0,97
0.96
0,94
270.69
269.89"
269.09
268.28
267.45
266.61
265.76"
264.89
264.02
263.13
262.23
261.32"
260.39
259.46
258.52
257. .56
256.59-
255.61
254.62
253.62
252.61
251.59-
250.56
249.51
248.45
247.38
246.30"
245.21
2.44.12
243.01
241.89
210.76-
239.62
238,48
237.33
236.16
234.98"
233.79
232.60
231.39
2.30.18
228.96-
227.73
226.50
225.25
223.90
222.73"
221.46
220.18
218.88
217.58
216.28"
214.97
213.65
212.33
210.99
209.65"
208.31
206.96
205.60
-0.80
0.80
0.81
0.83
0.S4
-0.85
0.87
0.87
0.89
0.90
-o 91
0-93
0-93
0.94
0.96
-0.97
0.98
0.99
00
0,93 204,23
0.14
0.13
0.13
0.13
0.13
0.12
0.13
0.12
12
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.11
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.13
0.13
0.13
0,13
0.13
0.13
0.14
0.14
0,1 t
0.15
0.15
0.16
0.16
0.16
0,17
0,17
0,18
0,18
0,1s
(,19
0,19
0.20
0.20
0,21
0,21
0,22
0.22
238
239
240
240
241
241,
242,
243,
243.
244.
244.
245.
245.
246.
246.
247.
247.
248.
248.
249.
249.
249.
250.
250.
251.
251.
251.
252.
252.
252.
252.
253.
253.
1253.
253.
254.
254,
254.
254.
254,
255,
255,
255.
25.'-),
^g+0.64
12 °-^5
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0.60
53+°-58
10 °S7
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.y.2 °-55
0.54
79 °-5'
0.50
0.49
0.48
2fi ,
7.) +0.47
18 "--tS
(;., 9-44
05 °-»-5
0.42
JJ7+0.40
26
64
29
78
0.38
01
°-37
37
7[+o-34
04 °-'53
36 °-^'
67 °--^'
0.29
50
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0.23
98
0.26
0.2;
21
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63
82
99
o, 19
0.17
o. 16
255,
255,
255.
255.
256,
256.
256,
256.
256.
255,
255,
255,
255,
255,
0.13
O. 12
O. I!
C.09
^.,-l-o.oS
90 °°7
95 °-°S
<)<) 004
0.03
02 ,
04+°°'
04 °°°
03-°-°'
00 °°|
0.04
96
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76
67
0,23 1255,56
I
0,07
o.oS
0.09
0.1 1
0.91
0.90
0.98
0.87
0.86
0.84
0.83
0.82
0.81
0.79
0.73
0.77
0.75
0.74
0.73
0.71
0.70
0.69
0.68
0.66
0.65
0.64
0.63
0.62
0.61
0.60
0.58
0.57
0.56
0,55
0,54
0,53
0,52
0.51
0.50
0,49
0,48
0,47
0,46
0.45
0,44
0,43
0.42
0.41
0,40
0,39
0,39
0,38
0.37
0.36
0,35
0,34
0,33
0.33
0.32
0.31
0.311
0,29
0,28
' r.e.2) DilT. Scc.var,
192.67
191.55'
190.42
189.29
188.14
186.99
185.83'
184.67
183.51
182.34
181.16
r....97"
178.78
177.59
176.40
175.20
173.99-
172.78
171.. 56
170.34
169.11
167.88"
166.65
165.41
164.17
162.92
ici.rj-
160.42
159.16
157.90
156.63_
155..'!6~
154.09
152.82
151.55
150,27
I4s,!('.r
147,71
146,42
145,13
143.84
142.55"
141.26
139.97
13><.n7
1 37.. 37.
136,07"
134.77
133.47
132.17
130.87
129,57"
128,2(;
1 21;. 96
125,66
121 36
123,05"
121,75
120.45
119,15
12
'3
13
»S
'5
16
16
16
'7
18
»9
«9
•9
>9
20
27 I17,s5
-1. 21
1. 21
1.22
1.22
>-23
-1.23
1.24
1.24
'•-'5
1.2O
1.26
1.27
-1.27
1.27
1.27
1.27
1.28
-1.28
1.2.S
1.29
1.29
1.29
-1.29
1.29
1.29
1.30
1.30
-1,30
1.30
1.30
1.50
1.30
-1.30
'•3>
1,30
1.30
1.30
-•■3'
1.30
1.30
1.30
'•30
0.22
0.23
0.23
0.24
0.25
0.25
0.26
0.27
0.28
0.28
0.29
0.30
0.31
0.32
0.33
0.34
0.34
0.35
0.36
0.37
0..38
0.39
0.40
0.41
0,42
0,43
0,43
0,44
0.45
0.46
0.47
0.48
0.49
0..50
0.51
0.52
0.54
0.55
0.56
0.57
0.58
59
0.60
0.61
0. 1!2
0.63
0.'!5
0.66
0.67
0.68
0,69
0,70
0,71
0.73
0.74
0.75
0.76
0.77
0.78
0.80
81
TUB ORBIT OP URANUS.
287
T.\I5LE IX,
Abo. 2
. — Continued.
1
Avg.
(r.,s.3)
(..C.3)
(V.sA)
((!.C.4>
(p.e.O)
(p.«.l)
(p.C.l)
(p.«.2)
(p.c.2) (p.8.3) (h.c.3)|
r,o
If
10.20
It
9.11
n
1.80
It
0.89
1490
437
2222
583
127
137
71
fii
lo.i;9
9.00
1.79
0.88
1489
445
2233
580
125
137
70
(12
10.32
8.88
1.79
0.87
1483
453
2243
577
123
137
70
(i:i
10.35
8.70
1.78
0.85
1477
400
2254
574
121
137
70
G4
10.38
8.04
1.78
0.84
1470
40S
2204
570
119
137
69
(15
10.41
8.52
1.7S
0.83
1404
470
2275
507
117
130
69
or,
1G.43
8.40
1.77
0.81
1457
484
2285
504
115
130
68
f.7
10.45
8.28
1.77
0.80
1451
492
2295
500
113
130
68
(IS
10.47
8.10
1.70
0.79
1444
501
2300
557
112
130
67
(19
10.49
8.04
1.75
0.78
1438
509
2310
553
110
130
67
70
10.50
7.91
1.75
0.70
1431
518
2320
550
108
130
66
71
10.51
7.79
1.74
0.75
1424
527
2330
640
107
130
60
72
10.52
7.07
1.74
0.74
1417
530
2340
543
105
130
60
73
10.53
7.55
1.7!
0.73
1410
545
2350
640
103
135
65
74
10.54
7.43
1,72
0.71
1403
554
2305
530
102
135
65
75
10.54
7.31
1.72
0.70
1390
603
2375
533
100
135
04
7(1
10.54
7.19
1.71
0.09
1389
572
23S4
630
99
135
64
77
10.54
7.07
1.70
0.08
1382
581
2394
520
VJS
135
64
78
10.54
0.95
1.70
0.07
1374
590
2403
523
90
135
63
79
10.53
0.83
1.09
0.05
1307
599
2413
519
95
135
63
80
10.53
0.71
1.08
0.04
1.300
008
24 2f!
510
94
135
02
81
10.52
0.00
1.07
0.03
13.53
018
2431
512
93
135
02
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10.52
0.48
1.00
0.02
1340
027
2440
608
92
134
01
83
10.51
0.30
1.05
0.01
1339
037
2449
505
91
134
01
84
10.50
0.24
1.04
0.00
1332
640
2457
501
89
134
00
85
10. 4S
0.13
1.03
0.59
1325
050
2400
498
88
134
CO
80
10.40
0.01
1.02
0,5S
1318
004
2475
494
87
134
59
87
10.44
5.89
1,01
0.57
1311
070
24 S3
490
80
134
59
88
10.42
5.78
1,00
0.50
1304
084
2492
4X7
85
134
58
89
10.40
5.00
1.59
0.55
1290
090
2500
483
84
133
58
00
10.38
5..55
1.5S
0.54
1289
700
2509
480
84
133
57
ill
10.35
5.44
1.57
0.53
12SI
717
2517
470
83
133
57
92
10.33
5.32
1.50
0,52
1273
727
2525
472
82
133
50
93
10.30
5.21
1,55
0,51
1 200
73S
25.34
409
82
132
50
94
10.27
5.09
1.51
0.50
1258
749
2542
405
81
132
55
!I5
10.21
4. 98
1.5:t
0,50
12.50
700
2550
401
81
132
55
ilCi
10.20
4.87
1.52
0,49
1242
771
255 S
457
80
131
64
!I7
10.17
4.70
1,51
0,4S
1234
782
2500
454
80
131
54
9S
10.13
4.04
1,50
0.48
1220
793
2574
45(1
80
131
63
99
10.09
4.54
1.49
0.47
121S
804
25M2
440
79
130
53
100
10.05
4.44
1.48
0.40
1210
815
2590
442
79
130
52
KiL
10.01
4.33
1,40
0.40
1202
S20
259X
43S
79
130
51
102
15.90
4.23
1.45
0.45
1194
83S
2005
435
79
129
51
103
15.92
4.13
1,41
0.44
1180
H49
2013
431
78
129
5((
104
15.87
4.02
1.43
0.44
1178
801
2020
42S
78
129
60
105
15.83
3,92
1.42
0.43
1170
872
2027
424
78
128
49
100
15.78
3.82
1.41
0,43
1102
8><3
2034
420
7s
128
49
107
15.73
3,72
l..i9
0.42
1154
895
2041
410
78
128
48
108
15.08
3.02
1.3S
0,42
'140
907
20 47
413
78
127
4S
109
15.02
3.52
1.37
0,41
1138
919
2054
409
78
127
47
110
15,50
3.43
1.30
0,41
1130
931
2001
405
78
120
40
111
15.51
3,34
1,.!4
0, 10
1122
943
2(;os
402
78
120
40
112
15.45
3,24
1,33
0.10
lilt
955
2074
39S
79
125
45
113
15.39
3.15
1,32
0. !(l
11(10
907
20><1
394
125
45
114
15.33
3,(((>
1,31
0.39
109S
979
20S7
391
79
124
44
115
15.27
2.97
1.29
0,39
1090
991
2093
3S7
80
124
44
no
15.21
2.88
l.-.'S
0,3S
10H2
1003
2099
383
SO
123
44
117
15.) 4
2. HO
1,20
0,:!8
1074
1010
2705
380
80
1 22
43
lis
15.08
2.71
1,25
0.3S
1000
1028
2711
370
SI
122
43
119
15.01
2.02
1.24
0.3S
1059
1(141
2710
373
81
121
42
120
14.94
2.54
1.23
0.37
1051
1053
2722
370
82
120
42
238
THE ORBIT OP URANUS.
i
I
"II
TAIJLK IX, .
Vro. 2. — Continued.
Ai'ff.
(v.c.O) Diff. ,11.8.1) Diff. Soc.var.
KV.C.l)
Diff. Sco.var.
(V.g.2) Dill'. Scc'.var.
'' 'I II
(i'.c*.2) Dilf. Sccvar.
11 II
tf ft
tt
tf
It
/'
// tt
II
120
60.09
279.55 , ,
280.10+°-^'
0.93
204.23
0.23
255,50
C.27
117,85
0.81
121
50.70 +°-°J
50.82 °°^
50.88 °°^
0,92
202. 8(!'
-> 37
1.38
1.38
0.24
255.44-°"
0.20
110.55-'-3°
0.82
122
0.90
201.48
0.24
2,55.31 °'^
0.20
115,25 •-■'°
0.84
12:j
281.35 °59
0.89
200.10
0.25
255.10 °- S
254.00 °'^'
0.25
113,95 '••5°
0.85
124
50.93 °°5
281.92 °-5J
0.88
198.71
« ■ M)
0.25
0,25
112,05 '--5°
0,87
0.05
0.56
1..10
0.17
1.29
125
50.98^^ -
282.48 ,
0.87
197.31
0.20
254.83
0,24
111.30
0.88
121!
51.03 +°°3
28,3. 02+°- 5*
0.85
195.91'
-1.40
0.27
254.04-°- ^
0,23
110.07-! -^5
0.89
127
51.08 °-°-^
283 55 °-5.?
0.84
194.50
1. 41
0.27
254,44 °- °
0,23
108.77 •^°
0,91
12H
51.13 °°-^
2,84.07 °-5;
0.83
193.09
1.41
0.28
2.54,23 "•
0,22
107,48 ]:l
0.92
129
61.17 °°-*
0.05
284.57 °:^;
0.81
191.07
1.42
1.4J
0.28
254,00 l:ll
0.22
100,19 1:;^
0.94
130
51.22
51.20 +°°-*
285.00
0.80
190.25
0.29
25.3.70
0.21
104.91
0.95
i;ii
285. 53+°- -'J
0.79
IS8.82'
-'•43
0.30
253,51-°- -5
0.21
103.03-'-=^
0.96
1.12
51.30 °°-*
285.99 °:''
0.7 <
187.39
«.4,^
0.31
253.24 °-7
0,20
102.35 '-^^
0,98
133
51.34 °°-*
280.44 °-'5
0.70
1S5.95
1.44
0.32
252,90 °-'^
0.20
101.07 '-^^
0,99
134
51.37 °°'
0.03
280.87 °-''
0.42
0.75
184.51
1.44
1.44
o.;i3
252,07 °'^
0,31
0.19
99.79 '-^^
1.27
1.01
135
51.40
287.29
0.74
183.07
0.33
252,30
0.19
98.52
1.02
130
51. 43+°°^
287.09 +°-*°
0.73
181.02-
-1-43
0.34
252,04 -°''^
0.18
97.25 -'-7
1.03
137
51.45 °°'
2 •(8! 08 °-,59
0.72
180.17
1-45
0.35
251.71 °-"
0. 1 8
95.99 '-^
1.05
138
61.47 °°-'
288.45 °--'7
0.70
178.72
1.45
0.30
251.30 °-35
0.17
94.73 '•-^'
l.OG
139
51.49 °°-
0.02
288.81 °--5''
°j5
0.09
177.26
1.46
1.46
37
251.00 °i^
0-37
0.17
93.47 '--^
1.26
1.08
140
61.51
289.10
0.08
175.80
0.38
250.03
O.lfi
92.21
1.09
141
61.52+°°'
289.49+°.'-'
0.07
174.33-
-1.47
0.39
250.24-°- 39
0.10
90.90— '--5
1.10
142
51.53 '^•°'
289.80 °->'
0.00
172.80
1.47
0.10
249.84 °-»°
0.15
89.71 '-^5
1.12
143
51.54 °°'
290.10 °-5°
0.05
171.39
1.47
0.41
249.43 °-»'
0.15
88.40 '--5
1,13
144
61.65 °°'
0.00
290.39 °-'>
0.27
0.03
109.92
1.47
1.48
0.42
249.01 °-^'
0.44
0.14
87.22 '-'•*
1.24
1.15
145
51.55
200.00
290.92 +°--''
0.02
108.44
0.42
2-««-'''' „ ,r
0.14
85.98
1.16
140
61.55 °°°
0.01
100.90-
-!..(,'!
0.43
248.12-°-'»5
J,. ,,,, 0.46
247.18 °-f
0.14
84.75-'-^3
1.17
147
51.55 °°°
291.10 °-^
0.00
105.48
1. 48
0.44
0.13
83.52 '--'3
1.19
14<
51.54-°°'
2'.ll,38 °-"
0.58
104.00
1.48
0.45
0.13
K.. ...) "-23
1.20
149
51.53 °°"
0.01
291.59 °-'
0.20
0.57
';;2.52
1.48
1.49
0.40
240.09 °-^''
0.50
0.12
81,07 '■''
1,22
1.22
150
61.52
201.79
0.50
101.03
0.47
-+'l'" 0..
0.12
79.85
1.23
151
61.50-°°^
291.97+°'°
0.55
159.54-
-I.. 19
0.!8
245.08-°-5'
0.12
7.S.(i4-'--'
1.24
152
51.49 °°'
292.13 °"'
0.51
158.05
1.49
0.49
21.5.15 °-5-3
0. 1 2
77,44 ••^°
1.20
153
51.47 °°-
292.28 °'-i
0.53
150.50
1,49
0.50
244.01 °-5j
0.11
70.24 \l°
1.27
154
51.45 °°-
292.42 °'-*
0.52
155.07
1.49
0.51
244.00 °-5-!
0,50
0.11
75.04 '•^°
1.29
0.02
0. 12
I 49
1. 19
155
51.43
292.54
0.51
153.58
0.52
243,50
O.U
73.85
72.(;7-- '^
71.49 • ';
1.30
150
51.40-°°'
292.04+°'°
0.50
152.09"
-1.49
0.54
242,92 — °-S"
0.11
1.31
157
51.37 °°'
292.73 °°'^
0.49
150.00
1.49
0..-.5
242 .■(3 ° ■-■''>
0.11
1.33
158
51.34 °°'>
292, KG °°7
O.H
149.11
1.49
0.50
241,73 °f°
0.10
70.;!2 -
1.34
159
61.30 °■°^
292.80 °°''
0.47
147.02
1.49
0.57
241,12 °'"
0.10
09.15 '-'J
1.16
1.30
0.04
0.04
'5°
0.63
100
51.20
292.90
0.40
140.12
0.58
240.49
0.10
07.99
l.:i7
101
61.22-°°»
292.9.3+°°,'
0.45
144.03-
-1.49
0.59
239.,S)-°-^'4
0.10
0(1. S4 — '-'5
1.39
102
51.17 °°'
292.91 °°'
0.4t
143.13
1.50
0.00
239.20 °''S
0.10
05.09 '-'5
1.40
103
61.12 °°'
292.94 °°°
0. (3
141.03
1.50
0.01
238.54 °-^''^
0.10
04.55 '-'4
1.42
104
61.07 °°'
0.06
292.92"°°"
0.42
140.14
1.49
0.02
237.87 °-^'7
0,10
0-.t.41 '•'■*
1.43
0.04
'5°
0.69
'•'3
105
61.01
5o.y5-°-°^'
50.89 °°'^
292.88
0.41
138.04
0.04
237,18
0.09
02.28
1.45
loo
292 m;)— °°.^
0.40
137.15-
-1.49
0.05
230. 4 8-°- 7°
0.(19
01. 10-''^
1.47
107
292^70 °■°^
0.40
135.00
1.49
0.00
2,!5.77 °-''
0.<I0
(lO.O.'i '•"
1.48
108
50.82 °°J
50.70 "°^'
292.08 °°^
o.;i9
134.17
1.49
0.07
235.05 °"-
0.09 ;
5S.04 '•"
1.50
109
292 58 °' '°
0.38
132.08
1.49
0.08
231,32 °-7,'>
0.09
57.84 '•'°
1.51
0.07
0. 1 1
1.48
0,74
1.09
170
50.09
292,47
0,37
131.20
0.09
233,58
232..S2— °7''
0.09
50,75
1.53
171
60.02-"°7
292., 'U ~°-'-'
0.30
129.71-
-1.49
0.70
0.09 ;
.5.5.(;7-'°«
1.54
172
.'.0.54 °'°^
2i|.) 10 °- '5
0.35
IJS.23
1.48
0.71
232.05 "■~l
(1.(19
54.59 '•"«
1.50
173
50.40 °°''
2!»2!o3 °'^'
0.:!5
1 ■-'•;. 75
1..18
0.73 !
i>;j| .>1 O.7.S
0.10
.53. -.2 '"^
1.57
174
50.37 °°'J
0.09
291,80 °'7
0,19
0.,i4
125.27
I. ;«
1.4S
■ '■• i
23(».48 °i"
0,80
0,10 ;
52.45 '•°7
1,05
1.59
175
50.28
291.07
0..33
123.79
0.7.^
229,08
228.s7-°-^'
228.05 °Z'
227,22 °-^'
220.37 °-'^P
0.8O
0.10 '
51.40
1.00
170
.'.0.19-°°''
291.40-°-'
0.32
122.32-
-1.47
0.7<>
0. 10
50.,!5-'°-'5
I.Ol
177
•-"••O 000
291. 2» ° ■'■
0.,!1
120.85
1.47
0.77
o.lo
49.31 '"^
1.03
17S
•'•"•»! n?o
291,01 °^'
0.,'!l
119.38
I. 17
0.79
0.11
4.S28 '•°'
1.04
179
^■•••«l 0°;::
290,70 °"'^
0,27
0.,!0
U.S. 91
...17
..46
0.80
0.11
1
47,20 '°-'
1. 01
1.00
ISO
4'.I.,S1
290 49
0,29
110.45
O.Sl
225.51
0.11 '
40.25
1.07
,T^£tf ^aiwN*'**— ~ -^ -. . -T -
'
THE ORBIT OP URANUS.
239
TABLE IX
Alio. 2. — Continued.
Arg.
{V.S.3)
(u.f.3)
((>.«. 4)
{u.cA) (p.c.O)
y^f.S.\){f.C.\)
(p.«.2)
(f.c.'l)\ (|,.».3)
(p.c.3j
120
It
14.04
n
2.54
1.23
0.37
1
1051
1053
2722
370
82
120
42
121
14. «7
2.4(!
1.22
0.37
1043
1 0(;5
2727
300
82
120
42
122
14.H0
2.38
1.20
0.37
1035
1078
2732
302
83
120
41
12;$
14.73
2.30
1.19
0.37
1028
1090
2737
359
83
119
41
124
14. (iO
2.22
1.18
0.37
102U
1103
2742
855
84
119
40
125
14.59
2.15
1.17
0.37
1012
1110
2747
352
85
119
40
12(5
14.52
2.07
1.15
0.37
1004
1129
2751
348
85
118
39
127
14.44
2.00
1.14
0.37
99(i
1141
2750
344
80
118
39
12S
14.37
1.93
1.13
0.37
989
1154
2700
341
87
117
38
12 'J
14.29
l.SC
1.11
0.37
9;1
1107
2705
337
88
117 •
38
l.!0
14.21
1.79
1.10
0,37
973
IHO
2709
334
89
110
37
l.il
14.13
1.72
1.09
0,37
9(;5
1193
2773
330
90
115
37
i;i2
U.OC)
l.C.O
1.07
0,37
958
1 200
2777
327
91
115
30
i;j3
13.08
1.59
l.OCi
0,37
950
1219
2781
324
92
114
30
134
13.90
1.53
1.05
0.38
942
1232
2785
321
93
113
35
l;i-)
13. H2
1.47
1.04
0.38
035
1245
27S9
318
04
113
35
l:i(i
13.74
1.41
1.02
0.38
927
1258
2793
314
95
112
35
l.J7
13.(;(>
l.:!5
1.01
0.38
920
1271
2790
311
90
111
34
l.JS
13.57
1.30
1.00
0.39
012
12S4
2^i00
308
97
110
34
131)
13.49
1.24
0.99
0.39
905
1298
2803
305
99
109
33
no
13.40
1.19
0.97
0.39
897
1311
2800
302
100
108
33
Ml
13.32
1.14
O.liC,
0.40
889
1325
2S(I9
298
)(il
108
33
142
13.24
1.09
0.!15
0.40
882
1338
2.sn
295
103
107
32
14:{
13.15
1.04
0.94
0. II
875
13.-2
2X14
292
104
100
32
144
13.07
0.99
0.93
0.41
807
1300
2810
289
100
105
32
145
12.08
0.94
0.91
0.41
850
1379
2819
280
107
105
31
1411
12.90
0.90
0.90
0.42
852
1393
2821
282
109
104
31
147
12.81
0.8f,
0.X9
0.42
844
1407
2823
279
no
103
31
14S
12.73
0.82
O..SS
0.43
837
1420
2825
270
112
102
30
149
12.04
0.78
0.87
0.44
829
1133
2S20
273
113
101
30
150
12.55
0.74
O.SC,
0.44
822
144 7
2><28
270
115
100
30
151
12.40
0.70
0.S5
0.45
815
1401
2829
207
117
99
SO
152
12.38
o.(;7
0.84
0.40
807
1474
2831
204
118
98
29
15.i
12.20
0.04
0.83
0.40
800
14SS
2S32
201
120
97
29
154
12.20
0.01
0.82
0.47
793
1502
2X34
258
122
90
29
155
12,11
0.58
0.81
0.48
780
1515
2835
250
124
95
28
15(1
12.03
0.55
0.80
0.49
778
1529
28,3(1
253
120
04
28
157
11.94
0.52
0.79
0.49
771
1543
2X37
250
128
93
28
15(S
11. 8(;
0.50
0.78
0.50
704
155(;
2S37
247
129
92
28
151)
11.77
0.47
0.77
0.51
750
1509
2838
244
131
91
27
ICO
11.08
0.45
0.7(1
0.52
749
r^s.'J
2838
242
133
90
27
ICl
11.59
0.43
0.75
0.52
742
1597
2s;!H
2.39
135
90
27
102
11.51
0.41
0.74
0.53
735
1011
2838
2.30
137
89
27
KI.J
11.42
0.40
0.73
0.54
728
1024
2S37
233
139
88
27
KU
11.33
0.38
0.72
0.55
7:>1
10.38
2837
230
141
87
27
1(!5
11.24
0.37
0.7!
0.5(5
714
1052
2X30
228
143
80
27
Kid
11.10
0.3(>
0,71
0.57
707
1005
2x:!.-,
225
145
85
27
1(!7
11.07
0.35
0.70
0.58
700
1079
2x34
222
147
84
27
i(;,s
10.09
0.34
O.CiO
0.50
093
1093
2x.!3
219
149
83
27
Kil)
10.91
0.3il
0.C8
0.00
C80
1700
2833
210
JM
82
27
170
10.82
0.32
0.08
o.(;i
f.79
1720
2X31
214
153
82
27
171
10.74
0.32
0.07
o.(;2
072
17.!4
2829
211
150
81
27
172
lO.O'i
0.3)
O.CO
0.03
005
1717
2X28
208
158
80
27
173
10.57
r ■
O.C.C,
0.04
059
1701
2820
20(1 '
1(10
79
27
174
10. 4!'
..31
0,05
o.cs
G52
1775
2S24
204
102
78
28
175
10.41
0.31
0.05
0.00
045
1788
2822
202
105
77
28
I7(i
10.3:»
0.3!
0.04
0.07
038
1802
2820
199
107
70
28
177
10 2-)
0.31
(..03
0.08
032
1815
::i7
197
170
75
28
17S
10. li
0.32
0.03
0.09
025
1 829
2X15
1 95
173
74
28
170
10.00
0.32
0.03
0.70
018
18J2
2812
192
175
73
28
180
10.01
0.33
0.02
0.71
Oil
1S55
2810
190
178
72
28
--CiJ
d
940
THE ORBIT OF U 11 ANUS.
I i •■
i I
TAHLK IX,
Alio. 2. — Continued.
\Tg.
(y.c.O) Diir.
(w.s.l) Diir. Sfc.viir
O'.C.I)
Diir. Sfc'.var.
(0.H.2) Dill'. Sir.var
(,i'.c,2) Dill'. Sce.var.
// II
// ft
/r
It
It
II
II It
II
It It
ft
18U
49.81
'^""•*!' ^ ,„
0.29
Ilfi.45
-1.46
1.46
0.81
225.51
224.C4-°-^7
0.11
4C.25
1.C7
181
49.71-°'°
290.20-°-"'
0.28
114.99
0.82
0.11
45.24-'°"
1 . (18
182
49. CO °]\
289.90 °-^°
0.28
113.53
0.84
223 77 °-**'
222.89 °-"**
0.11
**-'^ ".Z
1.70
188
49.49 °"
289.59 °-^'
0.27
112.08
>-4S
0.M5
0.12
43.25 °'>'>
1.71
184
49.37 III
289. 2G ^^
0'35
0.20
' 1I0.C3
>-45
«4S
0.8C
221.99 °-9°
0.91
0.12
^^-^« o::j;
1.73
185
*^'^^ n ,,
2*^8.91 „ ,
288.55-°- 3^
o.2r,
109.18
0.88
^-'-^"* on.
0.12
i;;!!--?
1.71
18(1
49.13-°''^
0.25
107.74
—'•44
0.M9
220.IC-°-''^
0.12
1.75
187
49.00 °'^
0.24
101;. 30
1.44
0.90
219.23 °-"^3
0.12
39.40 °-'^5
1.77
188
48.87 l\]
287.78 111
0.24
104.87
'■43
0.91
21s. 29 °-'^-*
0.13
3S.4C °;l\
1.78
189
^«-^^ l:\l
=^^^-" ix.
0.23
103.44
'•43
1.42
0.93
217.34 °'^^
0.96
0.13
■'^■^■^ V,\
l.so
I'JO
^'^••'■l „,.
28C.95 „ .,
0.22
102.02
0.94
2'«-;^'^ „„,
0.13
•'"■'•'■'1 n n.
1.81
191
48.47-°- j
28C.05 °-;^^
21
100.00 ■
—1.42
0.95
215.41 -°-9
0.13
•'!-'-7''~!oo
1.82
192
48.33 °' ;^
0.21
99. 1 9
1. 41
1. 41
1.40
1.40
0.97
214.44 °'97
213.4C °-9«
0.14
•^^-^O o! 9
33.91 'Z
1.S4
19a
^^•>» o 1
285.58 °-]l
0.20
97.78
0.9S
0.14
1.S5
194
*«"^ lit
0.20
9C.38
1.00
212.4C ;:°°
0.14
l.,S7
195
47.00
2S4.no
0.19
94.98
1.01
211.45
0.15
■ji;;;--f
l.ss
19a
47. 75-°- '5
284.09—°--'''
0.19
93..59-
-1-3')
1.02
210.43-'-°*
0.15
1.90
197
47. CO °'5
47.44 °'^
283..5fi °-53
0.18
92.20
'•.?9
..38
1.04
209.40 '-"3
0.15
S;S =t
1.91
198
283.01 °-S5
282.45 °-5'^
0.18
90. .S2
1.05
20.S.37 '-°3
15
1 '13
199
47.28 °'f
0.17
89.44
1.38
1.07
207.33 •-°-*
O.IC
28.77 °''^
1.94
o. i6
0-57
'•37
'•OS
0.S2
200
47.12
281.88
0.17
88.07
-1.36
1.0s
2(»C.28
O.IC
27.95 „
1.9c
201
4(!.95-°'7
2«tl.29-°-S9
2S().C8 °-^'
280.0C °-^'
279.43 °p
0.17
8C.7r
1.09
205.22-'-°'^
0.17
27.14-°-^'
1.97
202
4C.7S °-'7
0.1c
85. ;m
' .55
1.11
204.15 ^■°l
0.17
2C.34 °'^°
1.99
20:i
4i;.Cl °'7
0. 1 C
84.01
' 3,S
1.12
203.07 '-°'^
0.1s
2-V55 °'7'>
2.00
204
4C.44 °'7
0.1c
82. CC
'•.?5
1.14
201.98 '■°'>
0.18
24.78 °-77
2.U1
O.I 8
0.65
« 34
1.09
0.77
205
4C. 2fi
27'*. 78 ..
'>7s 12-°-''''
^77:44 °f
0.15
81.32
1.15
200. S9
0.19
24.01
2.03
20(i
4C.08-°;«
0.15
79.99"
-'•.33
I.IC
199.79-'-'°
O.-JO
2.'t.2C-°'"
2.04
207
45.90 °'^^
0.15
7H.C7
'3^
LIS
19S.C8 '-"
0.20
22.52 ''■*
2.05
201
4.^.71 °"^
27r,.75 °-'^9
0.14
77.3C
'3'
1.19
197. 5C '-'^
0.21
21.79 °'!-'
2. OC
209
45.52 °"^
0.19
27C04 °-7'
0.72
0.14
70.05
'3'
1.30
1.21
19C.43 ■-'•'
'•'3
0.21
^'-0^ 0:;:
2.08
210
45..'53 ^ ,
275.32
O.M
74.75
1.22
195.30
0.22
20 3C „ ,
2.09
211
4.V14-°"'
274..58-°-^^
0.13
73.40"
-1.29
1.28
1.23
194. IC-'-'-*
0.23
19.CC-°'/°
2.10
212
4i.94 °"
273.83 °-7j
0.13
72,18
1.25
193.01 '-'5
191.85 '-'^
190. CO '-'"
0.23
IS 97 °' "^
17. C4 . ,
2.12
2I:(
4474 III
273.0C °!I
0.13
70.91
1.27
1.2c
0.24
2.13
2U
44.54 °" 272.28 "•'"
0.13
r.'i i;4
1.27
1.28
0.25
2.14
0.20 0.79
1.27
1. 17
' 0.64
21.-)
44,34 271 49 „
0.12
n^ ,;
1.29
189.52 „
0.2c
17.00 f
2.1c
21 li
44.13-°-^' 270.CS-°-^'
4.3.92 °'! 2'19.SC °-^^
4!. 71 °-" 2-.9.03 °'^''
0.12
C7.12
-1.25
1.31
18S.34 -'-'^
0.27
15 13 ,
2.17
217
0.12
fi5.S8
1.24
1..-S2
1S7.15 \\l
0.28
2.1s
21 S
0.12
C4.C5
'•23
1.34
1S5.9C '-'^
0.28
2.19
219
43.50 III 2C8.18 °;«|
0.12
C3.42
'-23
1.22
1.35
184. 7C '•'°
1.20
0.29
^•r.3 -°
2.21
320
43.28 , 2fi7..32
0.11 '
fi2.20
1.37
I83.5C . _
0.29
1^05 „ ,,
2 2"
821
43.0c -°" 2CC.44-°'^'''
0.11
CO. 99
-1. 21
1.38
1S2.35--
0.30
IV'r°:
1 v> xO -'
2.23
222
42.84 °ll 2C5.55 °>^'>
0.11
59. XO
1. 19
1.40 '
isi.u -;
0.31
2 25
223
42.C2 °" 2'U.C5 °''°
on
5^.(11
1.19
i.>8
1.11 :
179.92 -
0.32
12:27 °-5-l
2.2c
224
42.40 °" 2C3.73 ° ^^
0.11
57.43
1.43 1
17.S.C9 '■'•'
0.33
11.73 °-:|
2.27
0.23 0.93
1
1. 17
'•23
0.52
325
42.17 2C2.S0
0.11 ;
5C.2C
-1.16
1.14
177.4C
0.34
11.21
2.29
32(!
41.94-°-*' 2'!1.85-°''.f
0.11
.55.10-
1.45
17 c. 22 •"'•*-♦
0.34
10.70-°-^'
2..-i0
327
41.71 °-^' 2C0.M9 °'>^
Oil
53. 9C
1.14
K47
174.97 '--5
0.35
10.20 °-?°
2.31
328
41.48 °-^.?
259 92 °^'
Oil
52.82
1. 14
1.48
173.72 '-*5
172.4C '-"'5
3C
9.72 °-»'^
2.32
329
41.24 °'-*
2.58:91 "-'^'^
0.11
51.C9
'•'3
1.50
0.37
9.25 °-»J
2.34
1
0.24
1. 00
1.12
'•27
0.46
230
41.00
257.94
0.11
.50.57
1.51
171.19
0.3s
8.79
2.35
2:tl
40.7C-°-^-+ 25'i.03-'°'
0.11
49.4C-
-I.I I
1.52
1r,9.92-'--7
0.39 1
S.35-°-*-»
2.3I-,
•2:ii
40.52 °='-* 25.5.92 ' °'
0.11
4S.37
1.09
1.54
1CS.C5 '--•
0.40
1 ^2 °--»3
2.37
2;i:l
40.28 °-t 251.89 ■■°'
0.11
47.28
1.09
1.08
I.5.'i
IC7.37 '•*'„
0.41
7:50 °"»^
2.3S
2.U
40.03 °^-' 2.53.85 '"^
0.11 !
4C.20
1.57 ICC. 09 '--'"
0.42
7.00 °-»'
2. 39
0.25 1.06
1
1.07
'29
0-39
2:55 ,
39 78 252.79
0.11 '
45.13
1.58 Ifil.SO
0.12
fi.Tf
2.41
23fi ;
39.53-°- '5
251.72^ '"^
0.11
44.08"
-1.05
l.co ir,3 51 -'•''>
0.43
c..-t2-°--^;'
2.42
2:i7 1
39.28 °-^^
250. C4 '"'^
0.12
43.05
1.03
L'!l 1C2.22 '^^
0.44
5.9C °--^'^
2.4.!
2H 1
39.03 °l^
249.55 '-""^
0.12
42.02
1.03
1.03 ICO. 92 '■•''^
0.45
5. CI °--«
2.14
2.9 1
3S.77 °-^ 248.45 j-
0.12 ;
41.00
1 .02
1^'54 ir.9.C2 '-■'°
0.4C
5.27 °--^-*
2.45
1.02
'3'
I
°32
240
38.51 247.33
0.12
1
39.98
l^Ofi 158.31
0.47
4.95
2.4C
THE ORBIT OF URANUS.
241
TABLE IX
Aua. i.— Continued.
Art,'.
(i>.«.3)
(C.C.3)
(0.8.4)
(w.f.4;
(p.c.OJ
(p.8.1,
(p.f.l,
(p.8.2
> (p.c.2
(p.«..3;
(p.c.3)
180
10.01
tt
0.33
//
0.02
tt
0.71
Oil
1855
2810
190
178
72
28
181
9.93
0.34
0.03
0.72
004
1869
2807
188
180
71
28
183
9.85
0.35
0.61
0.73
598
1882
2804
186
183
70
28
183
9.78
0.36
0.01
0.75
591
1895
2801
184
185
69
28
184
9.70
0.37
0.61
0.76
584
1908
2798
182
188
68
28
185
9.03
0.38
0.60
0.77
577
1921
2794
180
190
07
29
186
9.55
0.39
0.00
0.78
571
1935
2791
178
192
60
29
187
9.47
0.41
0.00
0.79
564
1948
2787
176
195
05
29
188
9.40
0.43
0.60
0.80
557
1901
2783
174
197
04
29
189
9.33
0,44
0.59
0.81
551
1974
2779
172
200
03
30
190
9.26
0.45
0.59
0.83
544
1987
2775
171
202
62
30
191
9.19
0.47
0.59
0.84
538
2000
2771
109
205
01
30
193
9.12
0.50
0.59
0.85
531
2013
2700
107
207
(,0
30
193
9.05
0.,'-.2
0.59
0.86
525
2025
2702
105
210
59
31
194
8.D8
0.64
0.59
0.87
519
2038
2757
103
212
58
31
195
8.91
0.56
0.59
0.88
512
2051
2752
161
215
57
32
196
8.84
0.59
0.59
0.90
500
2004
2747
159
218
50
32
19T
8.^8
0.01
0.59
0.91
600
2076
2742
157
220
65
32
198
8.71
0.04
0.59
0.92
494
20>^9
2737
1 50
223
54
33
199
8.65
0.06
0.59
0.93
488
2101
2731
154
225
63
33
200
8.59
0.69
0.59
0.94
482
2113
2726
153
228
52
34
201
8.53
0.72
0.59
0,95
470
2120
2720
151
231
51
34
202
8.47
0.75
0.59
0.97
470
2138
2714
150
233
50
35
20:j
8.41
0.78
0.59
0.98
404
2160
2708
148
23(!
49
35
204
8.36
0.81
0.00
0.99
459
2102
2702
147
239
4S
30
205
8.30
0..S4
0.60
1.00
453
2174
2695
140
242
47
30
206
S.25
0.«T
0.00
l.OI
447
2186
2089
H4
244
47
37
207
8.19
0. 90
0.01
1.02
441
2198
2082
142
247
46
37
20S
8.14
0.93
0.01
1.03
430
2210
2675
III
250
45
38
209
8.08
0.90
0.01
1.04
430
2222
20(;h
140
253
44
39
210
8.03
1.00
0.03
1.0.-)
425
2234
2061
138
250
43
39
211
7.98
1.03
0.02
1. 00
419
2215
2051
137
259
43
40
212
7.93
1.06
0.62
1.07
413
2257
2040
135
202
42
41
2l;i
7.88
1.10
0.03
1.08
408
2268
2039
134
205
41
42
214
7.84
1.13
0.03
1.09
402
2280
2031
133
208
40
43
215
7.79
1.17
o.ot
1.10
397
2291
2024
132
271
39
43
2ir.
7.74
1.21
0.04
1.11
392
2302
2616
130
274
39
44
217
7.70
1.24
0.05
l.U>
386
2313
2008
129
277
38
44
218
7.05
1.28
0.00
1.13
380
2324
2000
128
2S0
37
45
219
7.01
1.32
0.66
1.14
375
2335
2592
127
283
30
46
220
7.57
1.35
0.07
1.15
370
2340
2584
120
286
35
40
221
7.53
1.39
0.08
1.10
304
2357
2575
125
2S9
35
47
222
7.49
1.43
0.09
1.17
359
2307
2567
125
292
34
47
22{
7.45
1.46
0.09
1.18
354
2378
2558
124
295
33
48
224
7.41
1.50
0.70
1.19
S49
2389
2550
123
298
32
49
225
7.38
1.54 !
0.71
1.20
344
2399
2541
122
301
32
50
22(1
7.34
1.58
0.71
1.20
339
2409 1
2532
121
304
31
50
227
7.31
1.02
0.72
1.21
334
2420 ;
2523
121
307
30
51
228
7.28
1.06
0.73
1.22
329
2430
2513
120
310
30
52
229
7.25
1.70
0.74
1.23
324 1
2440
2504
119
313
29
53
230
7.22
1.74
0.75
1.24
819
2450
2494
lis
316
28 1
54
231
7.19
1.78
0.75
1.24
314
21i>0 ■
2484
lis
318
28 I
54
2:12
7.17
1.81
0.76
1.25
sn
2470 1
2474
117
?21
27
55
2:!:j ;
7.14
1.85
0.77
1.25
305
2480 i
2464
110
321
27
50
234
7.11
1.89
0.78
1.26
300 :
2489 !
2454
110
327
26 ]
57
235
7.09
1.93
0.79
1.27
296
2499 j
2444
115
330
25 ''
58
23fi
7.07
1.97
80
1.27
291
2508 ;
2433
114
333
25
59
237
7.05
2.00
0.81
1.28
2«7
2518 1
2423
114
330
24
60
238
7.03
2.04
0.82
1.28
283
2527 ■
2412
113
339
23
01
239
7.01
2.08
0.83
1.29
278
2536
2401
113
342
22
62
240
fi.99
2.12
0.84
1.30
274
2545 ■
2390
113
345
21
53
31 Jul7.
1873
^^^.m^t^
r
2ii
THE OUBIT OF U RAN US.
TABLE IX, Aho. 2
. — Continued.
Arg.
(u.c.O) Uiir. j(u.s.l) Diir. S
or.vor.
(u.c.l) Diir. Soc.vor.
(.U.S. 2) 1)
IT. 8
cc.vur.
(V.C.I) Diir. S
L'e.vor.
tl H It
/
ti
ft H
It
It
It
It
/' /;
//
240
38.51 f 247.33
38.25-°-^^ 24f..20-'
37.99 °- 7 245.00 '
0.12
39.98
1.66
158.31
0.47
4.95
2.46
241
■13
0.12
38.98-'-°°
3S.0O °'^l
1.67
157.00-'
•3'
0.48
*■'•■* "o'lo
2.47
242
'4
0.12'
1.09
155.69 '
3'
0.49
4.34 °-°
2.48
243
37.73 °- ^
243.91
'S
.16
18
0.13
■J7 ni °97
1.70
154.37 '
•3*
0.50
4.00 °- '^
2.49
244
3^-*« III
242.75 1
0.13
1.71
153.05 ■
■i-
•3»
0.51
3.79 °-
0.25
2.50
245
37.20 „ „
241.57
.18
0.13
•''•''•12 „„,
1.73
151.73 ,
0.53
3 •'•* „,,
2.51
24G
3r..93-°-^7
240.39-
0.13
34.l9-°-W
1.74
150.40"
33
0.54
3.;!0-°- 1
2.52
247
SC.Ofl °-
239.20
.19
0.13
33.27 °:i\
1.75
149.07
•33
0.55
30T °- f
2.53
243
30.39 °-
237.99 '
. 21
0.14
"2.36 °'5'
1.76
147.74
■ii
0.50
2-^« o'
2.54
249
3«-ia l\]
236.77 j
. 23
.23
0.14
3l-*« °oH
1.78
140.41 J
■a
•34
0.57
2"« l\l
2.55
250
35.85 . ,,
235.54 ,
0.14
30.57 „ ,,
29.70-°-^?,
^;-?? °s
^'•'^ 0.S2
1.79
145.07 ,
0.58
2^8 „,,
2.56
251
3.-..58-°-'J
35.31 l\l
^^ oM
3^-7^ 0.28
234.30"
24
■ n
26
.28
0.14
1.80
143.73
•34
.34
•34
•34
•35
0.59
2'l~o ?
2.57
252
233.05 1
0.15
1.S2
142.39
0.00
2"'' i
2.58
253
231.80
0.15
1.83
141.05
0.01
2 02 °- 1
2.59
254
2.50.54 1
0.16
1.85
139.71 ,
0.02
'■«» TA
2.60
255
.34.19-°*?
229.20
.28
0.10
2.5.52-°-^'
1.80
138.. 30
0.64
1.78
2.61
256
227.98-'
0.10
1.88
137.01-'
•35
0.05
1.08-°-'°
2.61
257
33 91 °-^l
3.3.03 °-^^
226.08 '
30
0.17
24.73 °-"
2.3.95 °ll
1.89
135.00 '
•35
0.00
1.00 °°^
2.62
258
225.37 '
3'
0.17
1.91
134.31 '
•35
0.07
l.,53 °-°/
1.47 °-°^
2.03
259
33.34 l^l
224.06 '
•3«
0.18
23.18 °-"
1.92
132.90 \
•35
0.68
2.04
0.28
•3»
0-75
•35
0.04
200
33.00
32.78-°'^
222.74
0.18
22.'»3_„,^
1.94
131.01
0.09
'■^3 „„,
2.05
2ni
221.41-'
■M
0.18
21.69-°-74
1.95
130.20"'
•35
•36
0.70
l-^O^oof
2. 00
2C2
32.49 °\l
220.07 '
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0.19
20.97 °-^
1.97
128.90 '
0.71
1.397°-°'
2.07
2fi3
32.21 °'^
218.72 '
•.?S
0.19
20.26 °- '
1.98
127.. 55 '
•35
0.73
'-^0+°°
2.07
2U4
31.92 °-'9
0.29
217.36 '
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•37
0.20
»«-S« 0^8
1.99
120.20 :
•35
35
0.74
1 -"a °°
0.03
2.08
2r,5
SI"''
215.99
0.20
18.88 „ ,
18.21-°-^'
n.56 °-^.
16.92 °;l\
2.01
124.8.'>
0.75
'■4!^ ,„„-
2.69
2fir,
3!..34-°-'''
214.02—'
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0.21
202
123.50—'
•35
0.-0
1..5u+°-°5
, -,. 0.06
2.70
2(;7
31.05 °-"'
213.24 '
0.21
2.03
122.14 '
0.77
2.71
2R8
30.70 °-\l
211.85 '
•39
0.22
2.04
120.79
•35
0.T9
'•"3 °'°
2.71
2U9
30.47 °:\i
210.45 '
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40
0.22
2.0G
119.44 \
■35
•35
O.HO
1.72 ^-°^
'■° O.IO
2.72
270
^^ 1" « ,„
209.05
0.23
'^■50 ol6
2.07
118.09
O.SI
1.82 .
2.73
271
29.88-°-^°
207.04-'
41
0.24
2. OH
110.74-'
•35
0..X2
'•'•'^"^o";!
2.74
272
29.59 °-;2
200.22 '
4^
0.24
2.10
115.39 '
•35
0.S4
2-07 i
2.74
273
29.29 °-°
204.79 '
43
0.25
'•'•"* o'»
2.11
114.04 '
•35
0..S5
2 21 ,•*
23: °-"'
2.75
274
29.00 °:;;j
203.36 '
43
44
0.25
•3.30 :\\
2.13
112.70 1
•34
•35
0.87
2.75
275
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201.92
0.20
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2.14
111. .35 ,
O.ss
2. 55 ,„ ,
2.70
270
2^-41 ~^'°
2IM).4S— '
44
0.27
'233 -°-^^
2.15
110.01-'
•34
O.Sil
2-7*'*'oM
2.77
277
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199.03 '
46
47
0.27
10.86 Ti
2.17
108.07
•34
•34
■34
■ii
O.'.ll
2.95 °-
2.77
27S
-'^•«3 0"
197.57 '
0,28
2.18
107.33 '
0.92
"■'T o"t
2.78
279
i^7-5* ::2^
190.11 J
0.28
2.20
105.99
0.94
3^*0 l\\
2.78
280
2SI
27 25
194.64
193.17—'
47
48
0.29
0.30
":«S-:S
2.21
2.22
104.00
103.33
■ii
(t.95
o.!m;
^tv^,
2.79
2.79
2S2
2'l"fi o'.o
191.09 '
0.31
"■•■^a T\
2.23
102.00 J
■ii
0.9.S
2.S0
2S3
20.30 „•^-
100.20 '
49
0.32
"•'0 040
2.24
100.67 I
■ii
0.99
^•^^ 00
2.80
284
2G.07 l\l
188.71 J
49
49
0.33
«-^» o'.S
2.25
99.34 ,
■ii
•32
1.01
4 78 ■^
*■"' 0.32
2.81
285
S'i.fS „ ,„
1S7.23
0.34
831
2.27
98.03
1.02
5.10 ,
2.81
2sr.
25. 48-°- 3°
185.72—'
50
0.34
7. 94-°- 37
2.28
96.70-'
•3»
1.(13
.5.43 +°-«
2.82
287
25.19 °-^,^
184.21 '
5«
0.35
7.59 y,\
2.29
95.39 '
•3«
1.05
5.78 °-3S
2.82
288
24.89 °-3°
182.70 "
5'
0.36
7.25 °-3^
2.30
94.08 '
3«
1.00
6.14 "--J"
2.83
289
24.00 °-2
0.29
181.19 '
5'
5»
0.37
6-93 IM
0.30
2.31
92.77 '
•3'
•3°
1.08
6.51 °--57
0.39
2.83
290
2*3' „,n
179.67
0.38
"•^ „ ,„
2.32
91.47
1.09
6.90 ,„ .
2.84
291
24.02-°- ^9
178.15-'
52
0.39
6.34-°- 9
2.33
90. 1 7-'
•30
1.10
7.30+°-'t°
2.84
292
23.73 °;\l
176.03 '
5»
0.40
6.07 °-%7
5.81 °-f
2.34
88.S7 '
■30
1.12
7-^2 TA
2.S5
293
23.44 °- 9
175.10 '
53
0.41
2.36
87.58 '
.29
1.13
8.15 °-^-5
2. 85
294
'■^^■'5 °1^
173.57 '
53
53
0.42
•^•^^ :;:]
2.37
80.29 1
.29
.28
1.15
u r.n 3-44
**-•'" 0.46
2.86
295
22.80 172.04
0.43
•'■••3< „ „
2.38
85. ni
1.10
9.05 , ,,
2.86
29<i
22..57-°-^^
2-'- 29 l\l
17J.50-'
54
0.43
•'•'3-:o
2.39
83.74—'
•27
1.17
^■•''-^o\l
2.86
297
108.96 '
54
0.44
4.93 °-f°
-^" CIS
2.40
«2.47 '
■-7
1.19
10.01 °-^'^
2.«7
298
22-00 °i^
107.42 '
54
0.45
2.42
81.20 '
■-7
1.20 !
10.51 °--°
2.S7
899
2M1 l\l
105.88 J
54
55
0.40
2.43
79.93 '
.26
1.22
i
11.02 °-5'
°53
2.88
300
21.42
104.33
0.47
4.44
2.44
78.07
1.23 1
11.55
2.88
THE ORBIT OF URANUS.
243
TAIJI
.K IX,
(».c.4)
Ahu. 2
. — Con
inucd.
Arg.
(».«.3)
(...0.3)
(U.8.4)
(p.c.O)
(M.l)
(p.C.l)
(p.«.2)
113
(p.c.2)
845
(P.«.3)
21
\l..v.\\)
240
n
C.99
2.12
n
0.84
ft
1.30
274
2545
2390
03
241
C.97
2.16
0.85
1.31
270
2554
2379
113
S48
21
04
242
C.96
2.20
0.86
1.31
205
2502
2308
113
350
20
65
243
6.94
2.24
0.87
1.31
201
2571
2357
112
353
19
00
244
C.92
2.28
0.88
1.31
257
2579
2340
112
356
10
07
245
6.90
2.31
0.90
1.32
253
2587
2334
112
359
18
68
246
fi.88
2.3?
0.91
1.32
249
2595
2322
112
S62
18
09
247
0.87
2.38
0.93
1.32
245
2003
2311
112
365
17
70
248
6.85
2.42
0.93
1.32
242
2611
2299
112
308
16
72
249
6.84
2.45
0.94
1.33
238
2018
2287
111
371
16
73
250
6.83
2.48
0.95
1.33
234
2020
2275
111
374
15
74
251
0.82
2.52
0.90
1.33
230
2033
2203
111
377
15
75
252
6.82
2.55
0.97
1.33
227
2041
2251
111
380
i4
70
253
6.81
2.53
0.98
1.33
223
2048
2239
111
383
14
77
254
6.80
2.61
0.99
1.34
219
2050
2220
111
386
13
78
255
6.79
2.64
1.00
1.34
215
2603
2214
111
389
13
79
25(i
6.79
2.07
1.02
1.34
212
2070
2201
111
392
13
80
257
0.78
2.70
1.03
1 34
208
2077
2189
112
395
12
82
25.S
6.78
2.73
1.04
1.34
204
2084
2170
112
398
12
83
259
0.77
2.76
1.05
1.34
201
2091
2104
112
401
12
84
2C0
0.77
2.79
1.00
1.34
193
2698
2151
112
404
11
85
201
0.77
2.82
1.07
1.34
195
270)
2138
112
407
11
80
202
0.77
2.85
1.09
1.33
192
2711
2125
113
410
11
87
2(13
0.77
2.88
1.10
1.33
189
2717
2112
113
413
11
89
2U4
0.77
2.90
l.U
1.33
180
2723
2098
113
410
11
90
20.5
0.77
2.93
1.12
1.33
183
2729
2085
114
419
11
91
20 r.
0.77
2.95
1.13
1.32
180
2735
2072
114
422
11
92
207
0.77
2.98
1.14
1.32
177
2740
2058
115
42.".
11
93
20S
6.77
3.00
1.15
1.32
174
2745
2045
115
427
11
94
209
0.77
3.03
1.10
1.32
172
2751
2031
110
430
n
90
270
0.77
3.05
1.17
1.31
109
2756
2018
116
433
11
07
271
0.77
3.07
1.18
1.31
106
2761
2004
117
430
11
98
272
0.77
3.09
1.20
1.31
164
2760
1990
117
439
10
99
273
0.73
3.11
1.21
1.30
101
2771
1977
IIS
442
10
100
274
0.78
3.13
1.22
1.30
159
2770
1903
119
444
10
102
275
0.78
3.15
1.23
1.29
156
2781
1949
119
447
10
103
270
0.79
3.17
1.24
1.28
154
2785
1935
120
450
10
104
277
0.79
3.19
1.25
1.28
151
2790
1921
121
452
10
105
27«
0.80
3.20
1.26
1.27
149
2794
1907
121
455
10
100
279
0.81
3.22
1.27
1.27
140
2798
1892
122
457
10
107
280
6.81
3.23
1.28
1.20
144
2802
1878
123
400
10
108
281
6.82
3.25
1.2S
1.25
142
2800
1804
124
403
10
no
2S2
0.83
3.20
1.29
1.25
139
2809
1849
124
405
10
III
283
0.83
3.27
1.30
1 24
137
2813
1835
12,-|
4 OS
10
112
284
0.84
3.28
1.31
1.23
135
2810
1820
120
471
11
113
285
6.84
3.29
1.32
1.22
133
2819
1805
127
474
11
115
280
6.85
3.30
1.33
1.22
131
2822
1791
127
470
11
no
287
0.85
3.31
1.34
1.21
129
2825
1770
I2S
479
II
117
288
6.86
3.32
1.34
1.20
127
2828
1701
12!l
482
11
118
289
6.86
3.32
1.35
1.19
120
2830
1740
130
484
12
120
290
0.87
3.33
1.36
1.18
124
2833
1731
131
487
12
121
2'.»1
6.88
3.33
1.37
1.17
123
2835
1710
132
49(1
12
122
293
0.88
3.34
1.38
1.16
121
2837
1702
133
492
12
123
293
0.89
3.34
1.38
1.15
119
2838
1080
131
495
13
125
294
6.89
3.35
1.39
1.14
118
2840
1071
130
497
13
120
295
6.90
3.35
1.40
1.13
110
2841
1656
137
500
13
127
290
6.90
3.35
1.40
1.12
115
2843
1641
138
503
13
128
297
6.90
3.35
1.41
1.11
114
2844
1620
139
505
14
130
29 S
6.91
3.35
1.42
1.10
112
2845
1011
141
508
14
131
299
6.91
3.35
1.12
1.09
111
2847
1590
142
510
14
132
300
0,92
3.35
1.4i
l.O-i
no
2848
15sl
143
512
15
133
:; li
244
T II U O It in T or URANUS.
TAIU.K l\, Aii.i, 2
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(K.O.O) Dlff.
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(U.K. 8) Diir. Soc.var.
(U.C.3) Diir. Mvc.var.
ft
tl
tt
II
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tt tt
11
300
81. 4J
104. »:>
0,47
4.44
2.44
78.07
1.23
11.55 ,
2.88
itiil
21.13-°'^
102.78"
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0,4 8
4.31-° ■]
2.45
77.42-"S
1.24
12.09 +°S-«
2.88
■M-i
20.85 I'Jt
80.57 °''l
Hi 1.23
•55
0,49
^'» T.'
8.40
70.17 '-'5
1,20
12.04 °-5S
2.88
:ioa
159.08
•■55
1,56
0,50
8.47
74.93 '•■'■♦
1.27
13.81 °57
8.89
au4
80.29 "••'^
158.12
0,51
8.48
73.70 '•'•»
1.29
13.79 °S»
8.89
O.JS
'•55
0.07
'•23
059
soft
80.01
I5fi,57
-1,56
0,53
•'•"* ««,
8.49
72.47
1.30
14.38 , ,
2.80
:>()i>
10.73 -°-^«
15ft. or
0.54
3.80-°°5
8.50
71. 25-'"
1.31
14.99+°-^'
8.80
307
30 S
10.45 °' «
10.17 °'1
153.40
151.00
'■55
"S6
1,56
0.55
0.50
3.86 °°^
8.51
2.52
70.03 \\\
08.88
1.33
1.34
15.01 °^'
10.24 °-^i
8. MO
8.90
30'J
18.00 °'^
150.34
0.57
a-«4l°°
8.53
07.02 •■^°
1.30
10.88 "ij
8.90
0.J7
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1.20
0.66
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1,56
'•55
1.56
1.56
0.58
l^'-"
2.54
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1.37
l''.54 , ,.
2.90
311
18.3,i-°'
U7.23'
0.50
2. 55
05.23- -^^
114.05 \\^^
02.87 ■ •;
1.39
18.-21+°-?7
18.90 °^'^
2.90
312
1,8.09 °'
145.07
0,00
2! 5*0
1.40
8.90
313
17,82 °
144.12
0.01
2.57
1.42
19.00 °- °
8.90
314
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142.50
0.02
8.58
01.70 \\l
1.43
^"••" ::;;
8.90
315
17,28
111,00
0,04
■»'".«.
2.5ft
00.54
1.45
21.03
2.91
31(!
17.02-° -''^
130.44"
1.56
0,05
4,27+°"
2.59
50.39 -'-'S
1.47
21. 70+°- 73
8.91
317
10.75 °'J
137.88
O.CiO
4.40 °''
2.i;o
58.24 '-'S
1.48
22.51 °V5
2.91
31.S
1G.49 °'^'
10.23 °-^'^
136.33
= •55
1,56
0.07
4.55 °';
4.71 °'^
2.01
57.10 ' '4
1.50
23.27 °-7''
8.91
319
134.77
0.08
2.02
55.97 ' '»
1.51
24.04 ° 77
8.91
0.26
'55
0. 18
I. 12
0.78
320
15,07 ,
15.7l-°-^<^
133.22
0.00
4.89 ,
2.03
54.85
1.53
24.82
2.91
321
131.07"
-'•55
0.70
6.08 -♦-°'">
2.04
53,74 -'•"
1.54
25.02 -r°'*°
2.91
322
15.46 "•'5
130.12
'■55
0.71
6.29 °"
8.05
52.04 ''o
1.56
20.42 °'^°
8.91
323
15.21 ° 'S
128.57
'•55
0.7:.
5.51 °'-»
8.05
61.54 ' '°
1.57
27.24 ° "^
2.90
324
14.90 °-^5
127.02
'•55
'54
0.74
5.75 °-'-»
2.00
50.45 '•°'J
1.08
1.59
28.07 °",^
0.8?
i: 90
325
14.71
125.48
0.75
"■"O . „
2.07
49.37
1.(10
28.92
2.90
32(1
14.47-° ^^
123.04"
-'54
0.70
„.0T+o- 7
2.08
48.30 — '°7
1.01
29.77+0-85
2.90
327
14.23 °'-^
122.41
'•5.?
0,77
fi.50 °=«9
2.09
47.23 ' °7
1.03
30.63 °-^^
2.90
328
13.00 °-^-«
120.88
'•53
0.79
fi.,87 °'-^'
8.09
40.17 ' °''
1.04
31.51 o-^S
8.89
32»
13.75 °^-t
0.24
119.35
• 53
'53
0.80
'■'» :i;
2.70
45.13 '"-^
1.03
1.00
32.40 °-89
0.90
8.89
330
13.51
117.82
O.xl
!:--+o..„
2.71
44.10
1.07
33.30
2.89
331
13.2,S-°*-^
110.30"
-1.52
0. 82
2.72
43.(»7— '■°-'
1.08
34.21 +°-9'
2.89
332
1J.04 °'*
114.78
• 5^
0.84
8.24 °-"
2.72
42.00 ' °'
1.70
35.13 °-9»
2.89
333
12.81 °-"-'
113.20
'5^
0,85
2.73
41.0fi * °'^
1.71
30.00 °-93
2.88
334
12.5S °-»
0.23
111.74
'52
' 5«
0.80
9.02 °;;;°
2.74
40.00 '•°°
0.99
1.73
37.01 °'''.5
0.96
2.88
33:)
12.35
110.23
0.8S
2.75
39.07
1.74
37.97
2.88
331!
12.13-°"
108.72"
-1,51
0.80
2.75
38.10— "•'>7
1.75
38.94 +-°-97
2.88
337
11.01 °"
107,22
1.50
0.90
10.33 °-'5
11.20 °-',^
2.70
37.14 °-'/'
1.77
39.91 °-''7
2.88
33S
11,110 °-
105.7 I
'■49
0.0 1
2.7 7
30.19 °-9S
1.78
40.89 °'i^
2.87
33'J
'in n :
1111,24
1.49
0.93
2.77
35.24 °-9S
1.80
41.89 ^■°°
2.87
0. 2 1
'49
0.49
0.94
1. 01
340
•'•-« ^ ,.
102,75
-.,4s
0.01
ii:ilii
2.78
34.30
1.81
42.90
2.87
341
11,05-°
101.27"
0.05
2.78
33.38 -«-9»
1.82
43 <,i2-t-'-°*
2.87
842
'"■^^ o
00.80
' -t?
0.07
2.7!t
32.47 °'"
1.84
44.95 ••°3
2.80
343
10.04 °- °
98.33
'47
0,08
lis ^
2.-9
31.57 °''°
1.85
45.99 ^°*
2.80
344
0.20
90.80
'■47
1,46
1.00
2.80
ao.«8 °.«0
1.87
47.03 '"i
1.06
2.80
345
10.24
95. 4 r
1 01
14.40 ,
2,80
20.80
1.88
48.09
2.85
3411
10.05-°"'
03.05"
-'■45
1.02
15.05-t-°-59
15.05 °^°
10.20 °-^"
10.89 °-^'»
0.65
8,81
28.93— °-^7
1.90
4y.i5+'-°^
2.85
347
9.85 °'^°
92.50
'45
1.04
-Lx.\
28.08 "•p
1.91
.'>0.22 " °7
2.85
34S
y.fiii °">
91.00
'44
1.05
8.8-i
27.2.1 °«S
20.39 °-^-»
1.93
61.30 '"^
2.85
34'J
U.47 °"'
89.03
'43
1.07
8.82
1.94
52.40 ' '°
2.84
0.19
1.42
0.82
1. 10
350
9.28
88.21
1.08
17.54 , .,
l8.20+°-^'^
18.88 °-?'*
10.57 °-^"^
2.83
25.57, _
1.00
53.50
2.84
35 1
9,10-°'t
80.70"
-1,42
1.00
2.83
i,^.,.:-o.8i
1.97
64.01 -t-'"
2.83
352
8,02 °'^
85.38
1,41
1 11
8.84
83.97 °'^'>
1.!I9
55.73 ' '^
2.83
353
8.74 °'**
83.98
1,40
1.12
2.84
83.18 °-79
2.00
50.80 '-'3
2.82
354
8.57 °'7
0.17
82.58
1.40
'•39
1.14
20,28 °;;;
8.84
22.40 °78
0.70
2.01
57.99 '■'•'
« '5
2.88
355
8.40
81.10
1.15
21 00j^„ ,.
2.85
21-«* „„
2.03
59.14
2.81
35(i
8.23-°- 'J
8.07 °'^'
7.01 °'^
7.75 °-^
70.81"
-1,3s
1.10
21. .4+0. 4
2.85
20.8!|— °"S
2.04
00.29 +'-'5
2.80
357
78.44
'37
1.18
22.49 °"
23,2.5 °ll
84.03 °r
0.79
2,85
20.10 °"
2.05
f.1.45 '"5
2.80
35S
35 !l
77.07
75.71
'•37
1.35
'35
1.19
1.21
8.. so
19.43 °73
18.71 °'^
0. 70
2.00
2.08
02.02 ''7
03.80 J J;^
2.79
2.79
8(;o
7.fiO
74.30
1.22
34.82
a. 80
18.01
2.09
64. 9d
2.78
THE Oil HIT OP URANUS.
24S
TAULK IX,
A HO, S
.^Cmtdnued.
Arg.
(».«,3)
it
8.92
(U.C.8)
3.35
(«.f.4)
n
1.43
{v.oA)
{(..c.O
110
(p.".!)
1581
(h.«.2
143
(^..•.2
(,,.».. 3)
( p.t'.3
133
1.08
2848
512
15
;iui
(1.92
8.35
1.43
1.07
1 09
2S4!I
1566
144
515
15
134
!1(I2
<!.93
3.85
1.44
1.06
108
2849
1.551
146
517
15
185
:to;<
(1.93
8.34
1.44
1.05
107
2850
15:i5
147
519
16
137
ao'
(1 93
8.34
1.45
1.04
106
2850
1620
149
622
16
138
an.".
(1.94
8.84
1.45
1.03
105
2850
l.M)5
150
624
17
139
:ioii
(1.94
8.33
1.46
1.U1
104
28,50
1490
152
626
17
140
SUIT
(1.94
3.33
1.46
1.00
103
2850
1474
153
628
18
141
itds
(! '.!(
8.32
1.47
0.99
103
28,50
1459
154
631
'.8
142
auo
(J. 95
8.31
1.47
0.98
102
2819
1444
166
633
19
143
a 10
6.95
3.31
1.47
0.96
102
2819
1429
157
635
20
144
an
(i.95
8.30
1.47
0.95
101
28(8
1413
158
538
20
145
ai-j
(1.95
3.29
1.48
0.94
101
2847
1398
160
640
21
146
aia
6.05
3.28
1.4S
0.93
KM)
2847
1383
162
642
22
147
au
6.94
8.27
1.48
0.92
100
281(1
1368
163
645
23
148
ai5
6.94
8.26
1.48
0.90
9!)
2815
1352
165
547
24
149
aifl
6.94
3.25
1.48
0.89
99
2843
1337
167
649
24
1.50
an
6.94
8.23
1.48
0.88
99
2842
1322
168
651
25
151
ais
6.93
3.22
1.48
0. 86
98
2810
1307
170
6.54
26
152
ai<j
6.03
3.21
1.48
0.85
98
2838
1291
171
666
27
168
320
6.92
3,20
1.48
0.84
98
2836
1276
173
668
27
154
a-.'!
6.92
3.18
1.48
0.83
98
2834
1261
175
660
28
1.56
ai'3
6.91
3.17
1.48
0.81
98
2831
124(1
176
562
29
156
a-ja
6.91
3.15
1.48
0.80
98
2828
1231
178
564
29
157
324
6.90
3.14
1.48
0.79
99
2825
1216
180
666
80
168
325
6.89
3.12
1.48
0.77
99
2822
1202
181
668
31
159
:i2(i
6.HS
8.11
1.48
0.76
99
2819
1187
183
570
32
159
327
6.H7
3.10
1.47
0.75
100
28 15
1172
185
672
33
160
328
6.«6
8.08
1.47
0.74
100
2812
1157
I8(i
673
33
161
32!l
6.85
3.06
1.47
0.72
101
2808
1143
188
575
34
162
330
6.83
3.04
1.46
0.71
101
2804
1128
100
577
35
163
331
6.82
8.03
1.46
0.70
102
2800
1113
102
679
86
164
332
6.81
8.01
1.46
0.69
103
2796
1098
193
581
87
165
333
6.79
2.99
1.45
0.67
103
2792
108 4
195
583
38
165
334
6.77
2.97
1.45
0.66
104
2788
1069
197
585
39
166
335
6.75
2.96
1.44
0.65
105
2784
1054
199
587
40
167
33fi
6.74
2.94
1.44
0.64
106
2779
1039
201
589
41
168
337
6.72
2.92
1.43
0.63
107
2775
1025
203
591
42
169
33S
6.70
2.90
1.43
0,61
108
2770
1010
204
592
43
169
339
6. OS
2.88
1.42
0.(10
109
2765
906
206
594
44
170
340
6.66
2.87
1.42
0.59
110
2760
081
208
696
45
171
341
6.64
2.85
1.41
0.58
HI
2755
967
210
698
46
171
342
6.62
2.8a
1.40
0.57
112
2750
953
212
.599
47
172
343
6.59
2.81
1.40
0.56
114
2744
938
214
601
48
173
344
6.57
2.79
1.39
0.55
115
2739
924
217
608
49
173
345
6., 55
2.78
1.38
0.53
116
2733
910
219
605
60
174
340
6.52
2.76
1.37
0. 52
117
2727
896
221
606
61
174
347
6.50
2.74
1.36
0.51
119
2721
882
223
608
62
175
34H
6.47
2.72
1.36
0.50
121
2715
868
226
610
53
176
34!)
6.44
2.70
1.35
0.49
122
2709
865
228
611
54
176
350
6.41
2.69
1.34
0.48
124
2703
841
230
613
66
177
35 1
6.37
2.67
1.33
0.47
126
2696
828
232
614
67
177
352
6.34
2.65
1.32
0.46
128
2689
814
235
616
68
178
353
6.31
2.64
1.31
0.45
130
2682
801
237
617
69
178
354
6.28
2.62
1.30
0.44
132
2675
788
230
619
60
178
355
6.24
2.60
1.29
0.43
134
2668
775
241
620
61
179
35r.
6.21
2.59
1.28
0.42
136
2660
762
244
621
62
179
357
6.18
2.57
1.26
0.41
139
2653
749
246
623
63
180
35.S
6.14
2.56
1.25
0.40
141
2645
736
248
624
64
180
359
6.10
2.51
1.24
0.39
144
2637
723
250
626
65
180
3(10
6.07
2.53
1.23
0.38
146
2629
710
252
627
67
181
I'l
346
THE ORIJIT OP UilANUS.
m
! .i
■■:
TAHL
U IX, Aim. 8
(c.f.i) iiiir. s
— Co,
'c.vur.
tinued.
Arg.
(i-.o.O) Diir.
ft t»
(iM,.l) IHir. Soo.vur.
(ox. 2) Ditr. S
•c. var.
(o.c.2) Diff. Scf.viir.
// n II
ff If
fl
If n
It
t' tl
/*
afio
''■''■O r. ,r
■J4.3n
1.22
25.(13+°-^'
27.29 °;\\
2.8r,
17.33-°^*
2,00
*"'*"8j.. ■„
2.78
3til
7.45 -°-'5
7:t.02-'J-*
1.23
2.8(1
2,10
(1(1.17+ ■"
2.77
S(i2
7.30 °- 5
7I.«!» ■•«3
1.25
2.87
1(1.(1(1 °f7
1(1.00 °"^
15.35 °"5
2,12
(17.37 • °
2.77
3fi3
^••■^ o ^.
70.37 11^
1.21!
2.87
2,13
118.58 •
2.7(1
3U4
7.01 °'^
(10.05 '^'
1.28
2.87
2.15
00.79
2.75
O.I4
> 3°
0.64
1.22
3«!5
3(iri
1.20
1.31
20.00 ,^„„
2.88
2.88
u:ol-°t
13.48 °-^'
12.88 °'°
2 111
2.17
72.23^' ;,
2.74
2.74
3tiT
«■«» o ■
1.32
30.78 °''°
2.88
2.18
73.4(1 :•
2.73
3C8
(!.4,S °'-^
1.34
31.(11 y'
2.88
2. 1
74.70 \\\
2.72
36 9
C.3(! °''
(;2.(;,i ''7
1.35
32.(11 °'>^
2.80
12.20 °-5'^
2.21
''••'•'^ .-.2^
2.71
O. 12
1.25
0.93
0-57
310
r> 24
f.l.I'.S . ,,
1.37
:<•■'•''•» 1 „ r
2.80
11.72
2.22
2.71
3-1
fi.l2-°''
«"H--^1
1.38
34.40+°'^5
35.45 °'>^
2.8<.)
"■'^~o 5
2! 23
2.70
372
«.00 °''
5S.01 •
1.40
2.80
10.(13 °-^
2.25
V.KU !
2.(10
373
5.8» °"
57.f.H ^
1.41
3(1.42 °'^7
37.40 °:^«
2. 8!(
10.10 °- ■
2.2(1
^"" :;
2.(18
374
^•" :;;;
5(1.4(1 ,:,„
1.43
2.80
0.50 °-5'
0.50
2.27
2.(17
37.")
S.fiH
55.2fi
1.44
38 40
2.80
0.00
2.20
83.5B ,
2. fid
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5.57-°"
54.07-''9
52 .SO -'^
1.45
30.41+'°"
2.80
8.(10-°-l''
2.30
84.85+'-^'>
2.(1(1
377
5.47 °'°
1.47
40.43 '°^
2.8il
8.12 °-»;^
7(,(l °-»''
2,31
8(1.14 '■''>
2.(15
37S
5.:i,s """J
51.72 "7
1.4S
41.47 ■•°*
2,8!)
2.32
87.43 '•-"
2.(14
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6.30 °°'''
50.57 ''5
1.50
42.52 '"S
2,80
7.21 °-«S
2.34
88.73 •■•'°
2.(13
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1.15
I.Ob
0.43
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3S0
5.22
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1.51
43.68 ,
2.80
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1.52
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2.80
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47.17 ' ;;
1.51
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2.8<l
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2.37
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2,80
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2.38
03.00 •'"
2.50
384
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44.0(i ' '°
1.57
47.05 ' "
2.80
2.39
05.31 '■•'-
2.58
0.07
1.09
1.12
0.36
t i.i
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4.H4
4;?.H7
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2.80
4.83
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2.57
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41.74 '"^
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50.20 + ' '^
2.80
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2.42
07.07 +'■•'■'
2.57
3S7
4.71 ^■°':
1.(11
51.31 ' ';»
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2.43
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2.5(1
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4.(;5 °°^'
4o.(;o "S
1.(13
52.50 ' "^
2.SS
3,8:i °
2.44
100.(14 '•■'*
2.55
38U
4. CO °°S
30. (!5 '°-»
1 (14
53.(17 '-'7
2.88
3,52 °-5'
2.45
101.08 '••'•*
2.54
0.04
1.02
1. 18
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300
4.-1(5
3s.(i:i
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54.85
2. 88
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2.4(1
103,33 j^, ,.
2,53
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4.52-° °-»
:i7.(;2-'°'
1.(17
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2.8,S
2 0(1-° -J
2 70 °;^
2.47
101.(18 +'•■'■■'
2,52
3!l-2
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:t5.(!.-, °''l
1.(10
57.25 ' -'
2.8K
2.48
10(1.03 '-'5
2.51
3'.);!
4 45 °°-'
1.70
5s.4(l ' -'
2,87
2 15 °''
2,40
107.:i8 'll
2,50
3i)4
4.41 °°-»
0.0 ;
••^■^•' o..;6
1.71
50.(18 J '1*
2.87
2,22 °-'
0.21
2,50
2.40
305
4..'tS
3:! 71
1.73
(1(1.01
2,87
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2.51
110.10 , , ,
2.47
vm;
4.:..-.-^^°-'
32.77 -°''^
1.71
(12.15 + ' •'■♦
2.S7
'■'*'"o ^
2.52
111.4.1+ ••
2.4(1
3!I7
4.:t:J °-^-
31. S5 '^•'^
1.75
(13,40 ' -5
2,S7
'•'•■=2 °^
2.53
w>.^-> •'
2.45
3!iS
4.31 °°-
30.04 °'i
1.7(1
.14.(17 •
.15.05 ;-«
2.8C,
l:^» li
2.54
III.IS [f
2.44
asiy
4.:i0 °°;
0.01
="•■"3 :i;
1.78
2.8(1
2.55
1.5.55 \-H
2.43
4on
4.20
S;;i--i
1.70
(17.23 , . ,
2.8r,
l'« n..
2.5(1
'"■•"-.L. „
2.42
401
4.2S-°°'
1.80
(ls..-,2+ -">
2.8(1
l'"~o •
2.57
lls,-20"^ •''
2.41
40i>
4.2s °°°
.-.42 °;''
1.S2
(10,82 ••°
2.85
"'•"• 00
2.58
lio,.;(i [ll
2.40
4ii;t
4U4
4.2,S °°°
4 2S °-°°
^5:7^ °:?
1.83
1.85
71.13
72.45 '•''
2.85
2.84
2.50
2.(10
121,03 •■"
122,11 •',
2.30
2.38
'•-4-0.01
' 0.82
1 .53
0.07
'■37
4 Of)
4 ''0
2».01
1.80
73,78 .
2.8»
1 "■.■'■''
2.(10
123.78 , , ^
2.311
4 (It;
4(17
40S
40!l
:.^o+°°:
2:!.3:t ° !,
1.88
I.SO
1.01
1.02
2,81
2.8;|
2.83
2.82
0.(10 -:;°5
0.5.1 °°|
0.00
2. Ill
2.(12
2.(13
2.(14
.25,l.l+-
12(1,53 ■•"
120,28 •"
'■37
2.35
2.34
2.33
2.32
410
4.:iO
21. or,
1.04
8(1,50.
2.82
"•'••2l„„,
2.(15
1:10.(15
2.31
411
4 4.) 4-00,^
20.;t3-°7.?
1.05
81,04 + 'V^
2,8->
0,51+°°
2. '1(1
l:i2.02 + ''7
2.30
412
4:4(1 °-°-'
10.C.2 °J'
1.07
83.33 ' '■'
2,81
"•■'^ on-
2.(17
1:1:1,10 [■^>l
2.20
4l:l
4..'-.0 °°-»
IS.ilH ° ">
lS-<5 °'''*
0.O7
l.os
84.73 ' '°
2.81
0.(12 °°S
0.(18 °'°^'
o.oH
2.(17
1:11.77 '••"
2.27
414
4.54 °°'
o.oi;
1.00
8(1.14 ' '"
1.41
2.80
2.(l.s
1:11;, It '-'^
'37
2.20
415
4.50
1 17. 5S
2.01
87.55
2.80
"f'''j_o„>
2.(10
1:17 51 ,
2.25
41(1
4.(14+°°.^
lC,.03-°'^
2.02
88 .17+''"
'.'.70
0,85+°°"
2,;o
l.!S,8s + ''37
2.21
417
4.70 °°^'
1C..30 °M
15.(;7 °','
2.03
00.30 ' '■'
2 70
0,0(1 °"
2,VI
110 25 '••"
2.22
4H
4.7(! °°f
2.01
01.8-2 '••■'
2.78
1,08 °''
2 71
111,,.. '37
2.21
41'.)
4.,S2 °°'
0.06
, i:..o(: °'^"
2.011
i 0.!.2(1 ' ■"
I 45
2.78
1 1.22 °''
o'5
2.72
11 -.',00 '■■'/
'■3'>
2 20
420
4.S8
14 47
2.0,
04.7!
2.77
1.37
2,7:1
lll.:!5
2.19
.J^
^
THE OR BIT OP U 11 AN US.
isn
TAULK IX,
Abo. 2. — Continued.
AfR.
(U.8.3)
(U.C.3)
2.53
(i'.».4)
(...r.4)
(p.r.O) (p.«.l)
710
fp.«.2)
253
(p.c.2)
027
(,..H.3)
67
(P.c.3)
181
SCO
It
f.07
If
1.23
0.38
140
2029
aiii
0.02
2.51
1.22
0.38
148
2021
097
254
028
08
I8l
.S(>2
5.98
2.50
1.21
0.37
151
2013
085
250
629
(.9
182
•MV.l
5.94
2.49
1.20
0.30
153
2005
072
259
031
70
183
■Mi
5.90
2.48
1.19
0.35
156
2590
000
201
032
71
183
3fi5
5.85
2.46
1.17
0.35
!59
2588
047
203
033
72
183
aiiti
5. Hi
2.45
1 10
0.34
102 2579
035
205
034
74
183
3 (17
5.7r,
2.44
1.15
0.34
104 2570
023
207
035
75
183
3r)8
5.72
2.44
1.14
0.33
107
2502
010
270
630
70
183
3(i9
5.C7
2 43
1.12
0.32
170
2553
698
273
038
77
184
370
5.fi3
2.43
1.11
0.32
173
2544
580
274
639
78
184
371
5.58
2.41
1 10
0.31
170
25;i5
.•>74
270
040
79
184
37.
5.53
2.41
1.08
0.31
180
2520
502
279
041
80
184
373
5.48
2.40
1.07
0.30
183
2510
550
281
042
81
184
374
5.43
2.39
1.05
0.30
180 , 2507
539
283
043
h3
184
375
6.38
2.39
1.04
0.29
190 2498
528
285
645
83
181
370
5.33
2.39
1.03
0.2!.
194 2488
517
288
040
84
184
377
5.28
2.38
I.OI
0.29
197 2479
500
290
047
85
|8(
3:»
5 22
2.38
1.00
0.28
201
2409
495
292
048
80
184
379
5.17
2.38
0.98
0.28
205
2400
485
294
649
87
184
380
5.12
2.38
0.97
0.28
208
24.50
474
297
650
88
184
3S1
5.07
2.38
0.90
0.28
212
2440
403
299
051
89
184
3H2
5.01
2.38
0.94
0.27
210
2429
453
302
052
90
184
3S3
4.911
2.39
0.93
0.27
220
2419
443
304
053
91
184
3S4
4. '.to
2.39
0.91
0.27
224
2408
4.33
300
054
92
184
3S-,
4.84
2.39
0.90
0.27
228
2397
423
309
6.55
94
184
3sr,
4.79
2. 40
0.88
.i..27
232
o-- ., ,
413
312
650
95
184
3S7
4.73
2.40
0.80
0.27
230
2374
40i
: 1 4
0.57
90
184
3.SS
4.07
2.41
0.8'.
0.27
210
230:!
394
317
057
97
183
3.S.»
4. Ill
2.42
0.81
0.27
245
2:!5!
384
319
058
98
183
3!t0
4.55
2.43
0.83
0.27
249
2310
375
322
059
99
183
3!ll
4 49
2.44
0.81
0.27
253
2328
300
321
OOO
100
183
■M-i
4.43
2.40
0.79
0. 27
258
2'ilO
357
327
001
102
183
3it:t
4.37
2.47
0.78
0.27
202
2.!05
318
329
002
103
18a
391
4.31
2.48
0.76
0.27
200
2293
339
332
662
104
183
39')
4.21
2.49
0.75
0.27
271
2281
330
334
603
105
182
:!9<i
4.18
2.51
0.73
0.28
270
2209
322
330
661
100
183
397
4.12
i.r.i
0.72
0.28
280 2257
313
339
665
107
182
398
4.00
2.55
0.70
0.28
«<5 ! 2245
304
341
060
109
181
399
4.00
2.57
0.09
0.28
29U
2232
296
341
007
no
181
400
3.93
2.59
0.07
0.29
294
2220
288
340
667
111
181
401
3.87
2.112
0.00
0.29
299
2207
280
318
608
112
IHI
402
3.. 80
2.04
0.05
0.30
304
2195
273
351
008
113
180
403
3.74
2.00
O.f.3
0.30
309
2182
205
353
609
114
!nO
404
3.08
2.09
0.0-i
0.31
314
2170
258
355
609
115
180
405
3.02
2.72
o.oo
9.31
320
2157
251
358
009
110
179
400
3. ^.5
2 75
0.59
0.31
325
2144
241
301
009
no
179
407
3.49
2.78
0.57
0.32
331
2131
237
303
670
117
179
40S
3.43
2.81
0.50
0.33
337
2119
230
305
670
118
178
409
3.31!
2.84
0.55
0.33
312
2100
221
308
670
119
178
410
3.30
2.88
0.53
0.34
34S
2093
217
371
670
1 20
178
411
3.21
2. 92
0.53
0.35
35 I
2080
211
37."
070
121
177
412
3.17
2.95
0.50
O.30
359
2000
205
370
670
122
177
413
3. 1 1
2.99
0. 19
0.30
305 2053
199
378
671
123
170
414
3.04
3.03
0.48
0.37
•171 2030
193
381
071
123
170
415
2.98
3.07
0,40
0.38
3. '7 2020
187
382
671
124
175
41f.
2.92
3. II
0.45
0.38
,183 2012
isl
38(1
671
125
171
417
2.85
3.15
0.44
0.39
;.89
I!I98
17"!
388
671
120
174
418
2.79
3.20
0.42
0.40
39)
1 9S5
no
391
071
120
173
419
2. 73
3 24
(1
0.41
401
1971
1(,5
393
672
127
173
420
2.67
B.29
0.40
0.42
407
195Y
160
390
672
128
172
248
THE ORBIT OF URANUS.
i •'
TABLE IX, Alio. 2
. — Continued.
Arg.
(u.c.O) Diff.
(i).8.1) Diir. Sccvor.
(tj.c.l) DiOf. Secvar.
(U.S. 2) Diff. Secvar.
Cu.c.2) Diff. Scfvar.
ff ft
ft ft
/f
n
/
II
ti n
»
>l n
If
430
<-88 ,„„,
14.47
2.07
94.71 ,
2.77
1.37 .
2.73
144.35 ,
2.19
421
*-"5+oo
13.90-°"
2.08
90.10 +
45
46
2.70
l-'^"*"o'8
2.74
14.5.72+ -37
2.18
422
so'-J III
0.03
13.35 °-53
2.10
97.02 '
2.70
2.74
147,09 -37
148.45 -36
2.10
423
12.81 °-54
2.11
99.09
47
2.75
l.!12 °'°
2.75
2.15
424
12.28 °S3
2.13
100.50 1
47
2.75
ai3 °''
2.75
149.80 '••'S
2.13
0.51
48
0.22
>-35
425
5-2fi ,„„„
11.77
2.14
102.04
48
2.74
2. •'?•"> , „ ,
2.70
151.15 , ,
2.12
4M
5-a'^"^ooo
ii.28--°-;^9
10.80 °-»?
10.34 If^
2.15
103.52 + ;
2.73
2.M+°- ?
2.77
I52,50 + '-3S
2.11
427
5.44 °°'^
2.17
105.01 J
49
2,73
2«^
2.77
1.53.84 \\\
2.09
428
5-W °?^
2.18
100.50 J
49
2.72
-'» o's
3.39 °:
2.78
155.10 35
2.08
429
sea °- °
9.89 °-»S
2.20
108.00 :
5°
2.72
2.78
150.54 '-'5
2.06
o.io
0-43
5«
0.30
1-34
430
5. 73 ,„ ,„
9.40
2.21
109.51 ,
2.71
3.09 ,
2.79
157.88 ,
2.06
431
6.83 +°-°
9.05 -°-»'
2.22
111.02+;
5«
2.70
4.00+°-^
2.79
150.22 + j^;
2.04
432
lit -"
8.00 °-]1
'■'' oil
i.h
112..53 .
5'
2.09
^■■''^ ot
2.80
100., 55 , •"
2.02
433
2.24
114.04 :
5«
2. 0-l
4 07 °-'^-'
2.80
101.88 ••»
103.20 i:^;
2.01
434
2.25
115.56 \
5^
53
2.07
2.81
1.99
435
0.29
IM
2.27
117.09 ,
2.00
5.39 , .
2.81
104.52
1.98
43i;
fi.41+O.I2
T,-2:i—°-3i
2.28
118.62 + '
53
2.00
.5.77 +°-^'"
2.82
105.84 + ''3'
1.91
437
fi.54 °'3
c.h °'J'
2.29
120.16 '
54
2.05
0.17 °-l°
2.82
107.15 '-31
1.95
438
0.07 °^'
0.02 °-3°
2.30
121.70 '
54
2.04
0.58 °->'
2.83
108.40 '-3'
1.94
439
6.80 °'-5
0.14
6.34 °-^8
0.27
2.31
123.24 '
54
54
2.03
7.00 °-''
0.44
2.83
109.76 '-3°
1.30
1.92
440
0.94
0.07
2.32
124.78 ,,
2.02
■J^* . ..
2.84
171.00
1.91
441
7.08 +°'4
5.H2-°='S
2.33
120.33 + '
55
2.01
7 H!.+°-»5
2.84
172.35+'-''
1.00
442
7.22 °"l
6.. 59 °'i
2.34
127.88
55
2,00
8.84 °-*"
2.85
173.04 '•-''
1.88
443
7.37 °'5
5.38 "-'
2.30
129.43 "
55
2.50
2.85
174.113 '-'2
1.87
444
7.52 °'5
5.18 °-°
2.37
130.98 ■
55
56
2.58
9.33 °-»''
2.80
170.21 '•^'*
1.85
0.15
0.18
o-5«
1.27
445
8.31 °"^
5.00
2.38
132.54 ,
2.57
0.84 , ^
2.80
177.48
1.84
44r.
4. 83-°- '7
2.39
I34.10 + '
56
56
56
56
2.57
10. .30+°- 5'
2.80
17H.75 + '-7
180.01 '•^''
1.83
447
4.08 °'5
2.40
135.00 '
2.50
10.8!) °-5.i
2.87
1.81
448
4.55 °'3
2.42
137.22 '
2.55
11.44 °-55
12.00 °S'^
2.87
181.20 '-'5
1.80
449
4.43 °'^
2.43
138.78 1
2.54
2.88
182.51 '-^5
1.78
0.17
0.10
57
0-57
1.24
450
8.48 ,
4.M -
2.44
140.35
2.53
12.57 , ,
2.88
183.75
1.77
451
8.05+°- "7
4.25-°°?
2,45
lll.!)2 +
57
2,52
i;uo+°"
2.88
184.1I0 + ''-''
1.70
452
8.82 °\l
9.18 °'»
4.19 °°'^
2,40
14.3.49 '
57
2.51
13.70 ° '°
14.37 °-''
2.88
180,:!2 '-3
1.74
453
4.14 °°S
2,47
115,00 '
57
2.50
2.8{. 7.44 '•"
1 73
454
4.10 °°-^
2.48
140.03 1
57
2.49
2.89 ..((.00 "
1.71
0.18
O.Ol
57
1. 21
455
9.54+°'"
4.09
2.49
148.20
2.47
I?;!!; o:^J
^^ oS
0. 70
19.02 , ,
2.89
189.87
1.70
45(i
4.09 °°,
2.50
14'.».77+;
57
2.40
2.80
101.07+'"
1.09
457
!»" tin
4.11+°°-
2.51
1.M.34 '
57
2.45
2.80
192.27 If
1.07
458
«3^ 00
<••* ooi
2.52
152.91 ;
57
2.44
2.90
.93.40 \W
194.04 ;:;■;
105.81
1.06
459
10.11 °^
0. 19
10.30 ,
<18 toi
0.07
4.25 ,
2.53
154.48 J
57
57
2.43
2.90
1.04
4 no
2.54
150.05 ,
2 42
2.00
1.03
4fil
10.50+°-^°
4.33 + °°«
2,.'->5
157.02 + 1
57
57
2.41
19.73+°-
2.!i0
100.08 + ''J
108.14 \\'l
1.01
41'. 2
10.70 °' °
<•*=« 0!
2,50
159.19
2,10
20,45 °'^
2.90
1.00
403
"'•'•'0 o'.o
^•^'•■^
2,57
100.70
57
2,311
21.19 °-lt
2.!I0
100.20 -^
1,58
4G4
'I'" 0:;:
*•"« ::;j
4.83 , ,
2,58
102.33 ,
57
56
2.38
^'••'^ o.?-,
2.90
200.43 \\\
1.57
405
11.31
2.5!)
103.89 ,
2.30
22,71 . .
2.91
201.50
1.55
40(1
II r,o+0-2'
4 q,,+o.ir,
2.59
105.45+'
^6
56
2,35
23,4S+°"
2.01
202.08-t-'''
1,53
407
11.73 °-'
b.V-J °'»
2.00
ir,7 01 '
2,34
24,27 °i'^
25,07 °-^°
25.88 °-^'
2.91
203,70 ' "
1,52
4 OS
11.94 °='
5.37 °'°
2.01
108.57
2.33
2.91
204,00 '■"
l.;.0
409
12.15 °^'
5.59 °"
2.02
170.13 1
2.32
2.91
200.00 '■'°
1,49
0.}2
0.2J
55
0.82
1.09
470
12.37 ,
5. 82
2.03
171,08
2.31
20,70 ^ „
2,01
207.00
1.47
47!
|o r,<)+o.zi
0.07+°'.'!
2.04
173,23 + '
55
2.30
27.53 +°«-»
28.37 °";^-»
2,01
208.17+'°"
1,40
472
12:81 °"
0.34 °»7
2.05
174.78 '
55
2,29
2,01
200.21 ^■°l
1.44
473
13,03 °-''
0.02 "-''^
2.05
170.33 '
55
2 27
29 22 °-2
2,00
210..iO '•°^
1.43
474
13.20 °-^-'
0.91 °'''
2.00 17 7. 87 '
54
2,20
30.0!) °-^7
2,90
211.30 •■°<*
1,41
0.2.5
°-3>
I
54
0.88
1.04
475
13.49 ,
7.22 ,
2.07
179.41
Ho.ns-*"'
2.25
31.80+°'*''
2,00
212,40 .
1 40
470
l.!.72+°-'-]
7.55 +°-^.^
2. OH
54
2.21
2,!M) 213, 43 + '°'
1,30
477
13.95 °'
7.90 °-."
8.20 °-3'i
2.0!)
182.48 '
53
2.23
32,70 °''°
2,00
214.45 '•°'
1.37
47H
14.18 °'^'
2.0!) 184.01 '
53
2.21
33,07 °'^'
2,80
215.47 '•°'
1.30
479
14.41 °'-
8.03 °-"
2.70 185.54 '
53
2.20
34,69 °'"
2,80
210.48 '■°'
1.34
480
o.a4
°J9
I
5»
0-93
0.99
14.05
o.oa
2.71 187.00
2.19
35.52
2.80
217.47
1.13
"
TUB
ORBIT OP UUANUS.
241)
TABLE IX
Alio. 2. — Continued.
Arjj.
(r.y.:i)
(r.c.3)
3.29
(o.i..4)
(v.c.i)
(p.e.O)
407
((..«. 1]
(p.C.lj
(f-x.2]
(,.c.->
{,>..s.3
(p.f.3)
4 '20
ft
2.fi7
0.40
It
0.42
1957
100
390
072
128
172
421
2.<!1
;t.;!4
0.39
0.43
413
1943
155
399
072
129
172
422
2.55
3.39
0.38
0.44
419
1930
150
402
073
129
171
42.J
2.49
1 3.44
0.30
0.45
425
1910
140
405
073
130
171
424
2.43
i 3.50
0.35
0.40
431
1903
141
407
673
131
171
425
2.. 37
3.55
0.34
0.47
437
1889
130
410
073
132
170
42(i
2..il
3.(!l
0.33 ■
0.49
443
1875
132
413
673
133
170
427
2.25
3.(!(!
0.32
0..50
450
I8(i2
128
410
673
133
109
42S
2.20
3.72
0.31
0.51
4.50
1848
124
418
073
134
169
429
2.14
3.78
0.30
0.52
403
1834
120
421
673
135
108
4:10
2.0s
3.S4
0.29
0.53
409
1820
117
424
073
130
167
4.tl
2.o:j
3.90
0.2s
0.51
470
1.800
lit
420
673
130
167
4.J2
1.97
3.9(i
0.27
0.55
482
1791
111
429
073
137
lOG
4:t;J
1.92
4.03
0.20
O..57
489
1777
108
432
073
138
105
4:i4
1.8(')
4.10
0.25
0.58
496
1702
105
434
073
139
105
435
I. SI
4.10
0.24
0.59
503
1748
102
437
072
139
164
4:!(5
1.7G
4.23
0.23
0.00
510
1733
1(»0
440
072
140
103
4:17
1.70
4.30
0.22
0.02
517
1719
9S
442
072
141
103
4.(8
l.(i5
4.37
0.22
0.03
524
1704
90
445
071
142
102
4.19
l.C.O
4.45
0.21
0.04
531
1 0!IO
94
447
071
142
161
4»0
1..55
4.52
0.20
0.00
539
1075
92
450
071
143
ICO
441
1.50
4.00
0.19
0.07
540
KiOO
91
45.1
070
144
159
442
1.48
4.07
0.18
0.08
553
1045
90
455
070
144
159
4(:i
1.41
4.75
0.18
0.70
500
1031
8'.1
458
009
145
158
444
l..t7
4.83
0.17
0.71
507
l(il(i
8S
4(10
009
145
157
445
1..12
4.91
0.17
0.73
.574
1001
S7
403
009
140
150
44(i
1.2s
4.99
0.10
0.74
582
li,Hi\
87
400
608
140
150
447
1.24
5.07
0.10
0.75
589
1571
NO
408
008
147
155
444
1.20
5.15
0.15
0.77
590
1557
80
471
007
147
154
449
l.ir,
5.21
0.14
0.79
003
1542
80
473
007
147
153
450
1.12
5.32
0.14
0.80
010
1527
SO
470
007
148
152
451
1.0s
5.40
0.13
0. 82
018
1512
S7
479
000
148
152
452
1.U4
5.49
0.13
0. 83
025
1497
87
4S1
600
149
151
45:j
1.01
5.57
0.12
0..85
0.!3
1483
8S
484
0(i5
149
150
454
0.97
5.*il>
0.12
0.80
040
14 OS
88
480
005
149
149
455
0.94
5.75
0.12
O.SS
048
1453
89
480
004
150
148
45(!
0.91
5. SI
0.11
0.89
05(!
143S
90 i
492
0(i 1
150
148
4". 7
O..S.S
5.91
0.11
0.9(1
0(i t
1 423
92 ;
49t
00.3
15(»
147
4:.s
0.S5
0.03
0.11
0.92
071
1 409
93 '
497
0(i2
151
140
459
D.H-i
0.13
0.10
0.94 ,
079
1394
95
499
001
151
145
4(iO
o.so
0.23
0.10
0.95 '
087
1379
90
502
001
151
144
4 lit
0.77
0.32 i
0.10
0.97
(i95
1304
98 1
505
(ifiO
151
144
41; 2
0.75
O.tJ
0.10
0.99
703
1319
100
.507
(i(iO
152
143
4c;;j
O.7.!
r,..-,i
0.10
1.00
711
i:i:!5 j
102 1
51(1
(i59
152
142
4U4 1
0.71 1
C.lil
0.10
1.02 !
719
1320
105
512
058
152
142
405
o.r,9
0.71 '
0.(i;i
1.03
727 1305 1
lOS
515
057 '
1.52
141
4lili
O.tlS
O.sl
0.(19
1.0.-.
735 1290
llo
517
(i57
l.V<
140
4(i7 1
o.tin
0.91
O.O!)
1.0(i
713 1275
113
52((
(i5ii
153
139
4(i8 j
0.(!4
7.01
0.09
1.08
751
1201
IKi
52!
055
153
139
4(!» 1
0.(!:l
7.11
0.09
1.10
759
1210
119
525 j
05 »
154
13S
470
O.C'^
7.21
0.09
1.11
707
1232
123 '
52S !
653
154
137
471
0.fl2
7.31
0.10
1.13 !
775
1217
127 ■
5.!l
Ii52
151
130
17 2
(Mil
7.42
0.10
1.15
784
1203
1.11
533
(i5l
155
l:t(i
47:1
ll.lill
7.52
0.10
1.10
792
II 89
1.35
.53ii
Ot9
155
1. 15
47 «
O.till
7.ii2 1
0.10 ,
1.18
800
1175
139
639 j
048
155
134
475
O.)>0
7.72
0.10
1.19
809
IIOI
143
542
047
150
13.3
47(!
O.liO
7. S3
0.11
1. 21
817
1117
I4S
545 i
010
150
1 3.!
477
O.tiO
7.9»
o.lt
1.23
82(i
113.! ,
15.t
548 t
044
1.50 ,
132
47><
O.iiO
N.dl
0.11
1.21
834
1119
15S
.551 i
013
15'i
III
479
O.til
8.15
12
1 . 25
843
1105 1
103
55 1
1
012
15ii
130
4St)
O.lil
8.20
0.12
1.27
851
1091
108
,55(1
040
150
130
2 July.
lavs.
250
TIIK ORBIT OF
URANUS.
A
TAULE IX, Aro. 2
. — Cunliinied.
Arff.
' (u.c.O) IUff.
yv.g.l) l»ilT. Sec.vnr
iu.e.l) Diir. S
cf.vur.
(t'.».2) Diff. Scf.vur.
(('.f.2) Diff. Si'c.vnr
»/ //
tf II
ft
II II
'»
//
II
//
II If
II
4«0
14.f.5 ,
9.02 ,
2.71
187.00
2.19
35.52 ,
2.89
217.47 ,
1.33
4K|
14.H9+°-^-»
9.4:1 +°-»
2.72
188.57 '*"''5'
2.18
30.4(1 +
3-94
2 89
2l8.40+°-W
1.32
4H3
15.13 11^,
9.S5 °-''
2.72
190.08 ''5'
2.10
37.41
3.y6
3.y.8
2.89
219.44 °'^?
220.40 °'-'l'
l.:io
4.S3
'''-*T n'^
'0.29 °-»^
2.73 191.59 '-S'
2.15
38.37
2.88
1.29
4M
15.r,i °-^-t
10.74 °-»5
2.74
193.09 50
2.14
39.35
2.88
221. :i5 °'^5
1.27
o.-M
0.47
1.50
- y.S
0.94
4Hr.
15.S5 ,
11.21 ,
2.74
194.59 ,
2.12
40.33 ,
2. 88
000 01)
1.20
4MI!
i(;.o',» +"•-'•♦
ii..;9+°f
2.75
19.1.08 + '-'''
2.11
41.32 +
3.99
2.88
.'.r( oJ+°W
1.25
4«7
ir,..i3 °;-»
12.19 °5°
2.7.1
197..57 '•■♦^
2.1.1
42.32
1. 00
2.88
'"•"4 1.3 °'^'
1.2:1
4HS
l(i.5S °-*3
12.71 °='
2.77
I99.(»5 '■•♦•^
2.09
43,:!3
I.OI
2.87
225;04 °'-"
1,22
489
l.i..3 °]l
'"< 1.11
2.77
200.53 ;;i;
2.07
44.35
1.02
1.0 J
2.87
225.94 y°
0.09
1.20
4'.>'>
17.08
13.79 , ^
2.7s
2.12,00
2.00
45.38
2.87
220.8.3 , „„
1.19
4!li
17.;i3+o-^i
14. 35+°- 56
2.79
203.4.1 + '-«^^
2.05
10.42 +
1.04
2.87
227.71 +°'!'^
lis
4:'2
1-.5S 0-25
n.93 "s**
2.79
204.92 "■-♦''
2.03
47.40
1.04
2.8r,
228.58 °'^7
1.1(1
4!».t
17.H3 °-5
15.52 °;'^
2 80
20(1.37 '•+''
2.02
48.51
1.05
2.8(1
229.43 °'^5
1.15
4'.t4
is. 09 o^**
0-5
lfi.l3 °f'
0.62
2.80
•J07.81 '••♦•♦
>-4J
2.00
49.58
1.07
1.07
2.8(;
230.27 °Z^
o.«3
1.13
4!t5
IS.. '{4
lfi.75 , ^
2.81
209,24
1.99
50.f,5
2.85
231.10, „
1.12
4'.M!
ls.t;o+°-^''
I7.39+°-^'*
2.81
210,07 + '•■♦■5
1.98
51.73'
.oS
2.85
"31 <i->+°''^
l.lo
4'.t7
l.s.«5 °--^
1,S.04 °^^
2.82
212.09 '• '-
1.90
52. SO
,09
2. 85
232:7:! °'*'
1.09
4 IIS
lit. 11 °-^
1S.71 °''l
2.82
213.50 '■■♦'
1,95
5:1.92
[. lo
2.85
2;i3.52 °"
1.07
4!)!)
19.3ti °^''
0.26
19.39 °^"^
0.70
2.83
214.90 '•■*°
1.40
1.93
.55.03
. 1 1
. 1 1
2.84
234. :iO °''^
0.77
1.00
5(10
in.(i2
20.09
2.83 2If..30
1.92
.50.14 ,
2.84
2.35,07 , ,
1.04
501
l!t.8S+°-^"
20.,S0'^°''
2.S3 217..;9 + '-''2
1.91
57.2(1 +
.12
2.83
2.15.83 '--•■^'^
1.03
■M\-2
20.14 °-'^'
21.53 °"'
2.84 21 9.. 17 '■■'
1.89
58.38
. I 2
2.83
230.58 "■'■■^
1 01
f.o:t
20.40 ° -'■'
00 on °- "4
2.84 220.44 '•''
1.8S
59.52
.14
2.82
237. :'l °'-'
I.(>0
.'•.04
20. CO °'^('
iiiioi °';
2.84 221,81 '•■'/
1.80
00.(17
•'.S
2.82
238.(13 °'^"
0.9;;
0.26
0.76
1.36
'.S
0.71
r.or.
20. (12
23.77 , u
2.85 223.17,
1.85
01.82
2.81
238.74 ,
0.97
.')OI'i
2!.lS+°-''
2t.55+°J''^
2.85 224.52 + '''5
1 84
02.98 +
r. 16
2.80
2:19.44 ^°?°
0,9.1
r.(»7
21.4; °-^'
•25.:i4 °- ';
2.85 :'•'"). s.'-, ■•3.?
1.82
01. U
. 16
2.80
240,12 °;;'
240,79 °-f/
9 95
^(tS
2 I.e.!) °'^-'
2.;. 15 °i'
2.85 227.18 '■■'•'
1.81
05.31
•'7
2.79
0,94
i')0!t
21, It.) °-^'
2.1.97 °''
2.8.1 228.50 '••'"
1.79
00.49
. 18
2.79
241.45 °'''
0.9''
C.J 6
0.S4
'•3'
1.19
0.65
510
>)■) ■))
30.40 \,„
2.80 229.81
1.78
07.08
2.7s
242.10, ,
0.91
r.i2
.')i,{
2147 +°-'''
22.72 °%^
.>•> C(.^ 0.26
2.8(1 |23l.ll+'-'°
2.87 i232.:l9 '-'^
2.87 2:i3..;7 '■"
1,77
1,75
1,74
fis 87 +
70,(i7
71,28
r.19
.20
.21
2.77
2,77
2,70
242.73+°- ;•'
243.30 ° '■
213.98 °^'
0.90
0.S8
0.87
514
2^:21 °-'''
3 1.2s °'^
2.87 : 234.94 ' •;/
1.72
72 49
.21
2,75
241.58 °-'':^
0.8(i
0-5
o.yo
1 , JO
, 22
o.5,^
515
2:1.49
32. H ,
2.8S !2:iC..20 ,
! 71
73 71 ,
271
?»5.1fi ,
0.84
5ii;
2:1, 75+°'^^
■•'•""•'"^00
2.88 2:!7.45"'" ■;^
1,(19
71,9:! +
.22
2,71
21.5,71 -^°-
0.83
517
24 00 °-j
34.01 °'>'
2.88 2;t8.(19 "■*
1,(18
7'1, K!
-,1
2,73 940,20 °-^-1
0.82
51 H
2».2r, °-°
«4.95 I'lJ
2.88 239. '12 '■"'■'
1,00
77 :J9
■-,?
2,72 2i'0..8O ''■♦
0.8]
51'J
24.:-.! "''^
=«-"" l.U
2.89 1241.13 '-■'
1.05
78,(1:1
.2.\
2,72
2(7 .33 ° ■^■'
0.79
0.25
1 I.JO
.24
05'
5-.>0
24 715
3r..sfi ,
2,89 212,33 , .
1.(13
79.S7
2,71
2(7,84 ,
0,7s
5J1
2.-..02+°-'*
:i7.H3+°''7
2,89 '2I3,52 + '- ',J
2.89 1 214. 7.1 '' ■
1.02
81,12 +
•25
2.7.>
2(8.3 +°--';°
(..77
5-.>-i
25 27 °-'-^
3SS2 °"'>
1.00
82,:i7
•-'5
2.09
24 8,83 °-'''
0.75
52:!
521
2.-...-.2 °-^
1 .00
40.83 ,.„,
2.89 ; 2(5.87 ; ;;,
2.89 247,03 .
'■ '.1
1,59
1,57
8:1,(12
84.88
• Ji
j6
.j6
2.08
2.07
219:10 ° 'J
0,71
0.73
525
2fi.0-i
41.85 .
2.89 248 18
1.50
80. 1 1
87.41 +
2,00
250,21 ,
0.71
52ii
211.27 +°--?>
42.88 + '°'
2.89 219.31 +"'3
1 55
•-'7
.28
.j.S
2, 0(1 ■
250,C,4+° •»3
0,70
527
2ii.52 ° -S
4:1.92 '°'
2,89 [250,43 '•'■'
1 53
88. CK
2,05
251, Of, °-^^
0,09
52S
2i;.7ii °--*
44.97 '"^
2.89 1251.53 ' '■'
1 52
S9.'.I7 '
2.01
251,47 °-"
o,ris
521>
27.00 ° ■'^
40.04 '°7
2.89 252.. 13 '*°
1.50
91.25
.28
2.03
25l,8(i °-39
0.0(1
0.24
1.08
1.09
.29
0. 58
5:10
27.24
47.12 ,
2.89 05:1.72 ,
1 49
92.51
93.83"*"
2.02
252.24 , '
0.05
5111
27. 4v: + '-'■»
48 21+' "^
2.8-. ,254 79 + '-°;
".89 25 -..85 '■°''
1,4S
.29
2.01
252.(10+°3<5
0.04
5:t2
27.72 °''-«
49.31 ' ■°
1 4ft
95.12
29
2,00
252 95 °'''''
o.i;:i
5:!:t
27. '.m °-'»
50.42 '
2.89 25.1.90 '"S
I 15
9.1.42
-.la
2,59
25.T29 °-'*
0. 02
5.14
2S.19 °-.'
51.53 ' ■'
2.89 ,257.93 ' "■'
1.43
97.72
30
2.58
2.53.(11 °-'-'
0.01
0. 2,i
' '.5
1 1.02
30
0-3'
5;r.
2S.42
52 ;,f, .
2.88 ,258,95
1 42
99.. t2
2,58
253. !)2 ,
0.(10
MC,
2s (;5+°-^'
;,380 + ''^
2.88 259. 9.; + '"°'
1.10
loo.:!2 +
•30
2,57
o-,4 oo+°-30
2.54;.5('. °-''^
0,5 s
5:!7
OS ss °-'
54.95 ' '5
5.1.11 '■'^'
57.27 "^'
2.88 ; 2.10.95 °''1
l.:i9
1.11.(13
3"
2.5t!
0,57
5:is
2'.U1 °-'
2.88 '2.11.93 °'^
i.:i7
102 91
3'
2.55
2.54.77 l\l
0, 5(1
r,;i9
29.:t:i °-''
2.88 2.12.90 °'''''
l.:!(l
101.25
3'
2.5 J
255.02 °^5
0,:-.5
0.22
1 . 1 .8
o<)5
'3'
0,24
540
•»;..55
58. »5
2,88 203,85
1.34
105.5(5
2. 53
2.55.20
0,54
THE ORBIT OF URANUS.
251
*
TABLE IX,
Am. S
. — Continued.
Arj?
(r.«.3)
(u.f.3)
(W.8.4)
(0.C.4)
1
(p. 8.1)
(p.C.1)
(p.».2) (p.c.2)
1
(p.«.3)
(P.O. 3)
480
O.C.l
8.2fi
ft
O.ll
ff
1.27
851
1001
108
550
640
156
130
481
0.02
8, 31)
0.12
1.28
8(10
1070
173
559
C39
156
129
482
0.(!4
8.47
0.13
1.30
808
1003
179
501
037
156
128
483
o.i;5
8.58
0.13
1.31
877
1040
184
504
030
156
128
484
0.«t!
8. GO
0.14
1.33
885
1035
190
500
634
156
127
4s:>
0.(!7
8.79
0.14
1.35
894
1021
190
509
033
156
120
4Hr.
0.1.0
8.00
0.15
1.3fi
902
1007
202
571
632
156
125
4S7
0.71
O.Ol
0.15
1.37
911
094
208
574
630
156
124
4t»S
073
0.11
0.1(1
1.39
919
980
214
570
629
166
124
480
0.75
9.22
0.17
1.40
927
900
220
579
627
156
123
400
0.77
0.33
0.17
1.42
930
953
227
581
620
156
122
401
0..80
0.43
0.18
'..43
945
040
234
584
024
156
121
402
0.82
0.54
0.10
1.44
954
920
241
580
023
150
120
'0:t
0.85
0.(15
0.10
1.4(1
903
913
249
580
021
150
120
;i>4
88
9.75
0.20
1.47
972
000
250
501
619
150
119
4 or.
0.01
9.8fi
0.21
1.40
081
887
204
504
018
150
118
40(;
0.04
9.07
0.22
1 . 50
000
874
272
500
1 (1
150
n7
407
0.08
10,07
0.23
1.51
090
801
280
509
015
150
117
4 OS
1.02
10.18
0.23
1.53
1008
848
2HH
001
013
155
no
400
lot!
10.29
0.24
1.54
1017
835
290
0O4
on
155
115
500
1.10
10.39
0.25
1.55
1020
822
304
000
010
155
115
501
1.14
10.50
0.2(1
1.57
1035
810
312
008
008
1.55
114
502
1.10
10. CO
0.27
1.58
1(144
707
321
on
000
155
114
503
1.24
10.70
0.28
1..59
105:)
785
320
013
(104
154
113
504
1.20
10,81
0.29
1.(10
1003
772
338
615
002
154
113
50;-.
1.34
10,01
0,30
l.r>2
1072
700
340
f.l7
000
154
112
501',
1.30
11, (tl
0.31
1.(13
1081
748
355
020
598
154
112
607
1.44
11.12
0.32
1.(14
1 000
7.30
304
022
500
154
111
50S
1.50
11.22
0.33
1.(15
1009
724
37.{
021
504
154
11(1
500
l.afi
11.32
0.34
1.(1(1
1108
712
382
020
502
153
llo
510
1.(12
11.42
0.35
1.07
1117
700
391
028
500
153
109
511
l.<;8
11.52
0.37
i.r.8
1120
088
401
031
588
153
109
512
1.74
ll.ti2
0.38
1.(19
1135
077
410
033
585
153
108
51.t
1.80
11.71
0.30
1,70
1114
000
420
035
583
153
108
514
1.87
11.81
0.40
1.71
11.53
054
430
037
581
153
107
515
1.04
11 01
0.41
1,72
Il(!2
043
440
030
578
1,53
100
5111
2.01
12,00
0.42
1,73
1171
032
450
042
570
153
100
517
2.08
12.00
0,43
1 74
IHO
021
40(1
014
573
152
100
518
2. Hi
12.10
0.45
1,75
1189
010
471
040
571
152
105
619
2.23
12.28
0.4«
I.7fi
1197
599
481
048
500
152
105
520
2.31
12.37
0.47
1.77
1 20(1
588
403
050
.500
152
104
■,il
2. 30
12.111
0.40
1.78
1215
577
503
052
504
152
104
522
'i■^^
12.5.-.
0.50
1.70
1221
507
514
051
501
151
103
52;j
2.55
12.(13
0.51
1.70
1 2:!3
557
525
050
558
151
103
524
2.(!3
12.72
0.52
1.80
1242
540
530
058
550
151
103
525
271
12.81
0.5 «
1,81
1250
530
548
000
553
151
102
52(i
2.80
12,80
0.55
1,82
1 250
r,..,!
550
002
550
151
102
527
2.88
12.08
0.57
1,83
12117
..17
571
004
518
150
101
528
2.07
1 1.0(1
0.58
1,83
1 270
5(17
583
000
515
150
lot
520
3.0ti
l.t.14
0.(10
1.84
1281
408
595
008
5(2
ho
100
5110
3.15
13.22
O.CI
1.85
1203
488
fi07
07O
540
15(1
100
5,{1
3.24
13 :10
0.(12
1 85
1 .
479
018
(;-2
537
150
100
5,12
3.;' 4
13.37
n,(it
1,8(1
1310
470
O.'iO
073
535
U'.l
00
5;i:i
3.43
13.45
0.(15
1.8(1
1318
4111
042
075
532
140
00
634
3.53
13.52
0.(17
1.87
1327
452
064
077
520
149
08
535
3.03
13.59
0.fi8
1.87
1335
443
000
078
527
140
98
53r.
3.73
13.(1(1
0.(10
1.88
1313
43 4
078
080
521
148
98
537
3.83
13,73
0.71
1.88
1352
420
(100
0S-.<
521
14s
08
538
3.03
13,8(1
0.72
1.80
I.'IOO
417
7(12
083
5 1 !»
lis
07
530
4.03
13,87
0.71
I.SO
13HS
400
711
osr.
510
I4S
07
510
4.13
13.o;i
0.75
1.00
1370
401
72'1
osci
513
117
07
35:;
THE OUBIT OF URANUS.
11
TAHI.K IX, Ami i
. — (Unwluili'd.
Arj5.
(D.c.O) Diff.
(u.D.l) Diir. Sfc.var
(c.p.l) Diir. !•
M'c.viir.
I i'.(i.2) Diff, Sec.vnr
'(i^.c.2) Diff. Sic.vnr.
540
29.55
58.45 , , „
2.88
203.85
1.34
105.50
If
2.53
2.55.20 ,
0.54
541
L>9.77+°"
59. ('.3 + '''^
2.S8
204.79 +°-9^
1.33
100.88 +
•3»
2.52
2.-.5.49+°-'
0.53
543
29,99 °'^'
00. X2 '•"'
2..S8
205.71 °'^'
1.32
108.19 '
■,i'
2.r,l
:2.-.5.70 °-'
0,52
543
30.-.>l in
02.02 ■•-°
2.87
200.02 °-'^'
1.30
109.51 ;
•,»J
2.50
2.M.90 °;°
,250.08 °''^
0.17
0,51
544
30.43 °-
03.24 ;•;-
2.87
207.52 l;^
1.29
110.83 J
■32
•3^
2.49
0.50
545
54fi
547
••M.O.; ll\
04.40 ,
05.09 + ' -'3
00.93 ••-'♦
2.87
2.87
2.87
208.40 , „
.,,;,, 07+O.S7
270.' ?2 °-':;5
1.27
1.20
1.25
112.15 ,,
113.47+1
114.80 ,
3-'
•33
2.47
2.40
2.45
250.25 , ,
2.Ml,41+°"^
250..55 °'-*
0.49
0.48
0.47
548
549
08.18 ;•-'
09.43 \]l
2.80
2.80
270.90 °-'^-*
271.79 °'^3
1.21
1.2 J
110.12
117,45 J
■3-s
32
2.44
2.43
2,-.0.08 °'3
250.79 I''
0. 10
0.40
0.45
550
551
0. 19
71.90+ -'
73.23 \:i
'-«o ,:3o
2.80
2.80
272.00 , „ „
273.40+°-^°
•--T4.I8 °!
271.95 °t^
1.21
1.20
118,77 ,,
120.09'^,
32
2.42
2.41
2.J0.S9 ,
2.V198+°°9
0.03
0.44
0.43
652
5.".3
554
2.85
2.85
2.84
1,18
1,17
1.15
121.42 ,
122.74
124.07 ,
32
32
2,40
2.39
2.38
0.42
0.41
0.40
555
32.f.3 , -
32.Sl+°''^
33:17 °;«
77.10 ,
2.S4
270.41
1.14
125,39 ,
2. .30
257,18 ,
0.40
550
7«.40 + '--'°
2.84
277.10+°-7^
1 12
120.71+'
32
2.35
257,20+°°*
0.39
55T
79.71 •••'■
2.83
277.80 °^^°
1.11
128.03 '
3-'
2,31
2.-.7.2> °°°
0,38
558
81.03 '••5^
2.83 27 s.. 55 °'"}
1.09
129.35 '
3-'
2.33
257, 1 9 -'^■°'
0,37
559
33.35 °'^
82.35 '-3-
2.82
279.23 °"';
1.08
130.07 '
3-'
2.32
257.10 °°3
0.30
0.17
>•.?,?
0.66
34
0.04
5f;o
33.r.-.> ,
83.08 ,
2.82
279. «9
1. 00
132.01
2.31
257,12
0.35
5i;i
33.(;9+°-'7
85.01+'-"
2.82
2s().54+°-''5
1.(15
1:13.33 + '
32
2.30
2.--7,o7-°-°-;
0.34
5(!2
33.Hfi °-'7
80.-35 '•■'•*
2.81
2si.n °'''
1.03
131.05
32
2 29
257,00 °°l
0.33
6(;3
34.93 °';
34.19 °'^'
0. :6
87.09 '■•'■*
2.81
2S1.78 °-^'
1.02
135.97
3-
2.27
0.33
5U4
89.04 '-3;
i,,;6
2.80
2S2.38 °-^°
°-59
1.01
137.28 \
3*
32
2.20
2.-,o',s2 ° '°
0, 1 1
0.32
5r.5
34 .n5 , ,
31.51 +°'f'
90.40 , ,
91.70+'-"^
2.80
282.97
0.99
13^,00
2.25
-•^'■•'' 0,,
0.31
5i;(i
2.79
2s3.54+°-5'
0,98
139.91 +,
.1 (
2 21
2.-,0 59-°"
30
5<i7
9.1.13 ;••"
2.79
? 4.09 °"
0,97
141.2?
3(
2,23
2.-,o45 °;^
0.29
5(18
34. S2 °'''
91.50 -7
2.78
2^4.03 °-'^-*
0,90
142,54
32
2.21
250, .30 °'5
250.14 °;^
0. la
0.29
5(19
34.97 °'5
o.is
2.78
2M5.10 °-^-^
°-5'
0.94
143,s5 J
3'
3'
2.20
0.28
5T0
35.12 ,
3.'..27"^°-'5
9><.04+ 3
2.77
2S5.07 ,
9.93
145.10 ,
2.19
255,90
0.27
!)71
2.70
2S0.10+°I2
■1.92
140.40 + !
30
2.18
255,77-°:^
0.20
572
3.V41 l[-\
1(111.03 '••''^
2.70
280.04 ^-l:'
0.90
147.70 J
30
2.10
2.V..57 °- °
0.20
573
3:V5:, °- -^
101.42 '••"
2.75
2S7.10 °-''^'
0.89
14!1.05
29
2.15
255.30 °-
0.25
574
O.I.^
I()2.S2 '■•'°
1.40
2.75
2S7.55 "•■'■^
-I,?
0.87
1^.0,35 ,
30
29
2.13
^^^•'^ :.]i
0.25
575
57(1
35.95+° '•
101.22 ,
l(l.-,.r,2 + ''°
2.74
2.73
287.98 ,
28^.40+°'-
0.8fi
0.85
151.01 ,,
l-.2.9::"^'
20
2.12
2.1]
25 1 , S9 r
254,03-°:;
0.24
0.23
577
3n.08 °'-^
107.03 '•;''
2.73
2'.!^ so °-^°
2S9.18 °-'''^
0.S3
154,22
2')
2.09
2.-i4.30 ^-^l
0.23
578
3(;.2i I'l
10S.44 [^'
2.72
0.W2
155.50
2.08
0. 22
579
3(i.33
0.12
109.85 J-^J
2.72
289.55 °-'J
o.j6
0,80
150,77 ,
-'7
28
2.00
^^^'^" :.^
0.22
5>*0
5S1
3(;.(;9 °'-'
3.;.s, °;-:
'"■'-"4-1 12
112.08+ ■■'
lU.lO ["[l
11.-..52 •;^
110.95 ;;]j
2.71
2.70
289.91 ,
290.25 +°-'^
0.79
0,78
158.05 ,,
159.32''"
27
2.05
2.01
-•'"■••«_o V
0.21
21
5^2
5^3
2.09
2.08
290.57 °-;'
290. S8 °'''
0,77
0,70
100. .-o
101,S5
27
2,02
2.01
0.20
0.20
684
3.1.93 °;;
2.07
291.18 :-^
0,75
103,11 ,
26
1.99
0.19
585
37.04
118.38 ,
2,07
291.40
0,73
104,37
1.98
251,75
0.19
58(5
37.1:. +°"
119.81 +'•'»•'
2.00
291.72+°-'^
72
105,02 + '
2,i
1.97
251, 30 -°-'''
0.18
5.S7
37 2fi °- ' '
121.24 '•■'■?
2.05
291.90 °-*
0,71
100,v7 '
25
1.95
250,90 °-'°
0.18
5S8
37.;;7 °"
122.07 ' '■'
2.04
292.19 ° -■'
0,70
KWII '
■4
1.94
2.-.0.55 °-t'
0.17
589
87.48 °"
0. 10
121.11 '^-^
>-43
2.03
292.41 °-'
0. 20
0.09
109,35 '
-M
1.92
2.-10,13 °-"
0.44
0.17
5'10
37.59 ,
125.54 ,
2.02
292.01
O.OS
170.58
1.91
219.09
0.10
591
37.(;>s+°'°
120. 98 + '-'-*
2.(11
292.79 +°''^
0,07
171, H1+'
23
1.90
249.24-°-;^
248,78 °-t'^
0.10
592
37.78 ^'^
128.41 '•«
2.00
292.90 ° 'J
293.12 °'
0,00
173,03 '
, 22
1.88
0.15
593
37.88 °'
129.85 ' •♦^
2.59
0.04
171,25 '
,22
1.87
248,31 °-*l
217.83 °-»**
0.15
594
37.98 °'°
131.29 '••♦■♦
2.58
293.20 °''*
0.03
175.40 '
.21
1.85
0.14
0. 10
1.44
0.12
.21
o.";o
595
3S.08 ,
132.73, ,,
2.58
293.38
0,02
170.07 ,
1.84
217,33
0.14
59A
3S.18+°''"
,,.,.,7 + 1.44
2.57
293.49+°"
0,01
177. X7+'
.20
l.«3 240 S2- 5
0.14
697
3>< 27 ° °''
135.01 '•••♦
2.50 I293.5X °°'>
0.00
17".t.(t7
. ,;o
l.Sl 210,30 °-'''
0.13
598
38.30 °"''
1. •17.05 ' '*
2.55 293 00 °°'^
0,5 s
1HO,20 '
II)
1,S0 21577 ° •'^•'
0.13
699
3S.45 "■'"' i;is..-,o ' •'^
2 51 293.73 °"'
0.57
1><I,U '
, iS
1,7s 21,-,. 2:1 °%»
0.12
o.fv i.K
1 oi
. iS
(too
i 38.54 ,l:!9.n4
2.53 ,2;i3.78
0.50
1H2,0J
1,77 214,07
0.12
:
THE
ORBIT OF
URANUS
•
253
TAB
.K IX, Alio. 2.—ComUided.
Arg.
.')40
(y.«.3)
{v.c.H)
(0.8.4)
ft
0.75
i(u.c.4
(^.r.O
(p.-S.lJ
(p.f.lj
720
(p.<..2
(p.c.2
(,).«.3,
147
[ (p.e.3
97
4.13
13.93
1.90
1370
401
080
513
f>4l
4.23
13.99
0.70
! 1.90
I3H1
393
738
088
510
147
97
542
4.34
14.05
0.78
1.90
1392
380
751
089
507
147
90
54:{
4.45
14.11
0.79
1 . 90
1400
a78
703
091
503
140
90
544
4.50
14.17
0.81
1.91
1408
371
770
092
500
140
95
545
4.07
14.23
0.83
1.91
1410
303
7S8
093
497
140
95
54ti
4.79
14.28
0.84
1.91
1124
350
801
095
491
140
95
547
4.90
14.34
0.85
1 1.91
1431
319
813
090
491
140
94
54S
5.01
14.39
0.87
1.92
1439
343
820
098
4S7
145
94
549
5.13
14.44
0.89
1.92
144(;
330
t'38
099
484
145
94
550
5.24
14.49
0.90
1.92
1454
329
851
700
4SI
145
93
551
5.35
14.54
0.92
1.92
1401
323
804
702
47s
145
93
552
5.47
14.58
0.93
1.92
1408
310
877
703
475
145
93
55;{
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TIIK OK II IT OF U KAN US.
205
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23 WEST MAIN bTRflE'
WEBSTER, N.Y. )45t:0
(716) 873-4503
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181
1«2
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184
185
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187
188
9 189
20
21
22
23
24
25
2(>
27
28
29
32
33
34
35
3(i
37
38
39
40
41
42
43
44
45
190
191
192
193
194
195
19(>
197
198
19 199
200
201
202
203
204
205
20()
207
208
209
30 210
31 21 1
212
213
214
215
2 III
217
218
219
220
221
222
223
224
225
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0.30
0.29
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0.26
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0.23
0.22
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0.18
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50
51
52
53
55
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57
58
59
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01
02
03
04
05
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07
08
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70
71
72
73
74
75
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77
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40 I 220
47 ; 227
48 I 228
49 229
54 234 1.09
230
231
232
233
235
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239
240
241
242
243
244
245
240
247
248
249
250
251
252
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254
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204
85
205
80
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87
207
88
208
89
209
90
270
0,80
7.10+°-
7,42 °-
7.73 °-
8.05 °-
o.
8.37 ,
8.70+°-
9.02 °-
9.35 °-
9.07 °'
90
91
92
93
94
95
90
97
98
99
270
271
272
273
274 j
275'
270 \
2771
278
279 I
110
111
112
113
114
115
110
117
118
119
120
121
122
123
124
125
3°
120
32
127
31
128
3-!
129
32
130
33
131
3^
132
,>,.•,
133
3-!
134
33
R
10.00 ,
10.33+°-"
10.05 °-3^
10.98 °'"
11.30 °-3^
0-33
11.03 ,
11.95+°- 3^
12.27
12,58
12.90
0.32
0.31
0.32
0.30
1001 280 :
101 i 281
102: 282
103 1 283
104 I 284,
105 285:
100! 280 !
107^ 2S7
lOS 2S8
109 289
13.20
13.51 +°-3'
13.81
14.11
14,40
0.30
0.30
o, 29
0,28
290
291 I
292
293
294:
295
2911
297 i
298
299
300
301
302
303
304
I
305
3(10
307
30S
309
310
311
312
313
314
14.08 , „
14,90+°-=^
15,24 °-^'^
15.51 °%7
15.77 °'^
0.25
10.02 ,
10.27 +°-'5
10.51 °-^^
10.74 °-'3
10.90 °"
0.22
17.18
n,38
17.58
17.77
17.95
4-0.20
0.20
o. 19
0.18
0.17
18.12 ,
18 27+°-'5
19.42 °'5
1>^<'« o.M
18,80
18.91+°"
19,01 °-'°
19.09 °-°'^
19.10 °°7
0.07
19.23
]<l,.2,s+°°5
19,32
19.35
19.30
0,04
0,03
O.OI
O.OI
10.00
135.315 19.37
135 315
130 310
137, 317
1381 318
1391 319
140' 320
141 ! 321
142 322
143' 323
144 1 324
145 ; 325
140 j 320
147 327
148 328
149 329
1501 330
151 331
152 332
153 333
154 , 334
155 335
150
157
158
330
337
338
170
171
172
173
174
175
170
177
17s
179
ISO
159 339
100 [ 340
101 j 311
102] 342
103, 343
104' 344
105 345
100, 340
107: 347
108 348
109 349
350
351
352
353
354
355
350
357
35H
359
300
R
19.37
19.30"
19.35
19.32
19.28
19.23
19.10
19.09
19.01
18.91
18.80
18.09'
18.50
18.42
18.27
18.12
17,95"
17.77
17.58
17.38
17.18
10.90'
10.74
10.51
10.27
10.02
15.77"
15.51
15.24
14.90
-O.OI
O.OI
0.03
0.04
0.0s
—0.07
0.07
0.08
O.IO
O.II
-O.II
0.13
0.14
0.15
-0.17
0.18
0.19
0.20
0.20
-0.22
0.22
0.23
0.24
0.25
-0.25
0.26
0.27
0.28
0.28
14,40-°f
0.30
0.30
0-3I
13.81
13.51
13.20
12.90"
12,58
12,27
11.95
-0,30
0.32
0.31
0.32
0,32
11,03
11.30"
-0'33
10,98
0.32
10.05
°-33
10.33
0.32
0-33
10.00
i I
f.1 I
|,l
THE ORBIT OF URANUS
213
TABLE XIX.—
I'lUNCIl
AL Tku.m of the Latitude. A
lUiLMENT M.
u
(3
U
3
u
(3
180°
/ /'
360°
190°
t n
350°
200'
/ "
340°
0°
0.00
8.09
180°
10'
8 2.82
7.96
7.96
170°
20°
15 50.98 ,
15 58.58 I
60
160°
10'
8.09
50'
10'
8 10.78
50'
10'
e t\
50'
20
10.18
8 09
40
20
8 18.74
40
20
16 6.17 III
16 13.75 I'l^.
16 21.32 l\l
16 28.88 '-^S:
16 36.43 '
16 43,98 l'l\
16 51.52 l\-l
40
30
24.27
8.09
30
30
8 20.09
7-95
30
80
30
40
32.35
8.08
20
40
8 34.04
7-95
20
40 i
20
50
1°
40.44
48.53
8.09
8.09
8.09
8.08
10
179°
50
11°
8 42.59
8 50.54
7-95
7'95
7-94
7-93
10
169°
60
21°
10
159°
10'
20
5(5.02
1 4.70
50'
40
10'
20
8 58.48
9 0.41
50'
40
10'
20 ;
50'
40
30
1 12.79
8.09
30
30
9 14.34
7'93
7.92
7.92
7.91
30
30
16 59.05 '
•SI
•5°
30
40
1 20.87
8.08
20
40
9 22.26
20
40 ;
17 6.57 ;
20
50
1 28.9G
8.09
8.0S
10
50
9 30.18
10
50 .
17 14.08 '
10
2°
10'
1 37.04
1 45.12
8.08
8.08
8.08
8.08
8.08
8.08
178°
50'
12°
10'
9 38.09
9 45.99
7.90
7.90
7.90
7.89
7.89
7.89
168°
50'
22' ;
10'
17 21.58
17 29.07 ;
•49
.48
.48
•47
.46
1,58°
50'
20
30
40
1 53.20
2 1.28
2 9. 30
40
30
20
20
30
40
9 53. ao
10 1.79
10 9.68
40
30
20
20
30 i
40 ;
17 36.55 I
17 44.03 ;
17 51.50 '
40
30
20
50 ,
2 17.44
10
50
10 17.57
10
50 '
17 58.90 ;
•45
10
3"
10'
2 25.52
2 33.00
8.08
177=
50'
13°
10'
10 25.40
10 33.34
7.88
7.87
167=
50'
23° '
10'
18 fi.41
18 13.85 I
•44
•43
.42
.41
.41
.40
•39
•37
•37
■ii
•35
•33
•33
•31
• ^0
157°
50'
20
2 41.07
8.07
40
20
10 41.21
40
20
13 21.28 I
40
30
2 49.75
8.08
30
30
10 49.08
7.87
7.S6
7. SO
7-85
7.8.=;
7,84
7-8.3
7-8,3
30
30
18 28.70 '
30
40
2 57.82
8.07
20
411
10 56.94
20
40 1
18 36.11 '
20
60
4°
3 5.89
3 13.90
8.07
8.07
10
176°
50
14°
11 4.80
11 12.65
10
166°
50 1
24' 1
18 43.52 ;
18 50.92 ,
10
156°
10'
3 22.03
8.07
50'
10'
11 20.50
50'
10'
18 58.31 i
50'
20
3 30.09
8.c0
•10
20
11 2S.34
40
20
19 5.68 i
40
30
3 3S.10
8.07
30
30
11 36.17
30
30
19 13.05 -i
30
40
3 40.22
S.06
20
40
11 44.00
20
40
19 20.40 i
20
50
5'
3 54.28
4 2.34
8.06
8.0O
8.05
8.06
10
175°
50
15'
11 51.82
U 59.63
,.h2
7. Si
7. So
7.S1
7.80
7-79
7. 78
7-79
7.7;!
7-77
7-75
7-75
7-74
7-74
7-7.3
10
165°
50
25'
19 27.75 ;
19 35.08 !
10
155°
10'
4 10.39
50'
10'
12 7.43
50'
lo'
19 42.41 i
50'
20
4 18.45
40
20
12 15.24
40
20
19 49,72 i
40
30
4 211.50
8.05
30
30
12 23.04
30
30
19 57.02 i
•30
.29
• 27
.20
30
40
4 34.50
8.o(j
20
40
12 30. S3
20
40
20 4.32 i
20
50
6"
4 42.61
4 50.05
8. 05
8.04
8.04
10
174°
50
16°
12 38.61
12 40.40
10
164°
50
26°
20 11.61 ;
20 18.88 .
10
154°
10'
4 5s.(;'.i
50'
10'
12 54.18
50'
10'
20 26.14 .
.20
50'
20
5 (;.73
8. 04
8.03
8.0J
40
20
13 1.95
40
20
20 33.40 .
f.24
40
30
5 14.70
30
30
13 9.70
30
31)
20 40.64 .
J. 24
30
40
5 22.79
20
40
13 17.45
20
40
20 47. 8S
h 22
20
50
5 30. S2
8.0J
8.0J
8- 0,1
8.02
10
50
13 25.19
10
50
20 65.10
?-2I
10
T
5 3.-i.s5
173°
17°
13 32.93
163°
27°
21 2.31
7.20
153°
10'
5 41). US
50'
10'
13 40.06
."lO'
10'
21 9.51
7.19
7.IS
50'
20
5 54.90
40
20
13 48.39
7-7.1
7-7^
7-/1
7.70
7.O9
7-^19
7. 08
7-O7
7.O7
7.00
7.(>5
7-65
7,04
7-f',3
7.O2
7.61
7.O0
40
20
21 16.70
40
30
2.92
8.C2
S.o_'
30
30
13 56.11
30
30
£1 23. S8
7-17
7.1O
30
40
(i 10.94
20
40
114 3.S2
20
40
21 31.05
20
50
IS. 95
8. 01
8.01
10
5')
'\ 14 11.52
10
50
21 38.21
1
7.14
10
8°
10'
2l!.9(!
31.97
S.oi
172°
50'
18'
10'
14 19.21
It 26.90
162'
50'
28'
10'
:;l 45.35
21 52.49
7.14
■? I 2
162°
50'
20
(i 42.97
8.00
40
20
14 34.58
40
20
21 59.61
■7 13
40
30
1) 50.97
8.00
30
30
' 14 42.25
30
30
22 0.73
/• '-^
30
40
(i 5S.97
8.00
20
40
14 49.92
20
40
22 13.83
7.10
7,09
7,08
20
50
7 (•,.90
7-9';
8.00
10
50
14 57.58
10
50
22 20.92-
10
9°
7 14.90
171°
19'
15 5.23
161°
29°
22 28.00
7,07
"7 0^
151°
10'
7 22.95
7-99
50'
10'
15 12.S8
50'
10'
22 35.07
5(i'
20
7 .'i(l.9;i
7.9S
40
20
15 20.52
40
20
22 42.12
7-05
7-°3
701
40
30
7 3S.91
7.98
30
30
15 28.15
30
30
22 49.17
30
40
7 4(1. S8
7-97
20
40
15 35.77
20
40
22 56.20
20
50
7 54.85
7.97
10
50
15 43.38
10
50
23 3.23
/ -^.1
7.01
! 10
10°
8 2.S2
7-97
170°
20°
15 50.98
160'
30°
23 10.24
150°
190°
!i
350°
■u
200°
/5
340'
u
210°
a
830°
u
35 July, 1873.
271
THE ORBIT OF TTRaNUS,
TABLE
XIX, Alio. M. — Cuiilin
ued.
u
' 3
u
P
u
3
210^
/ 1/
330"
220"
/ »/
320"
230°
/ V
310°
30"
2;3 10.24
150"
40"
29 47.27
6 t9
6., 7
6.16
6.14
6.13
140
50"
35 29.99
130°
10'
23 17.24
7.00
6.99
6.1,7
6.97
(>')5
50'
10'
29 53.40
50'
10'
35 35.18
5- 19
50'
20
23 24.23
23 31.20
40
30
20
30
29 59.03
30 5.79
40
30
20
3('
35 40.35
35 45.50
5'7
5-'5
40
30
40
23 3S.17
20
40
30 11.93
20
4j
35 50.03
5- '3
20
r)0
23 45 12
10
50
30 13.06
10
60
35 55.75
512
10
6.94
6. II
5.10
31°
23 52.00
6-93
6.91
0.90
6.89
6.88
0.86
149"
41"
30 24.11
6.10
6.08
6.06
6.05
6.03
6.02
139°
51°
30 0.85
5.08
129°
10'
23 5S.!)i)
50'
10'
30 30.27
50'
10'
3(! 5.93
50'
20
ao
40
50
24 5.!»0
24 12.80
24 ll).(i!)
24 20.57
40
30
20
10
20
30
40
50
30 3f;.35
30 42.41
30 48.40
30 54.49
40
30
20
10
20
30
40
60
30 11.00
30 10.05
30 21.07
30 20.08
5-07
5-°5
5.02
5.01
4-99
40
30
20
10
32'
24 33.43
6.85
6.84
6.83
148°
42°
31 0.51
138°
52"
30 31.07
128°
10'
21 40.28
50'
10'
31 0.51
6.00
50'
10'
30 30.04
4-97
50'
20
24 47.12
40
20
31 12.50
5-99
40
20
30 40.99
4-95
40
30
24 53.05
30
80
31 18.47
5-97
30
30
30 45.93
4.94
30
40
2.-I 0.77
6.82
20
40
31 24.43
5-96
20
40
30 50.84
4.91
20
M
25 7.57
6.80
10
50
31 30.37
5-94
10
50
30 55.73
4. 89
10
33=
25 14.3r,
0.79
6.7s
6,76
6-75
6.74
6.7.'
147"
43"
31 30.29
5-92
137°
53"
37 0.01
4.88
127°
10'
25 21.14
50'
10'
31 42.20
5-91
50'
10'
37 6.47
4.86
50'
20
25 27.90
40
20
31 48.09
S-«9
40
20
37 10.31
4-84
40
;{0
25 34.05
30
30
31 53.90
S-«7
30
30
37 15.13
4. 82
30
40
25 41.39
20
40
31 59.82
5. 86
20
40
37 19.93
4.80
20
CO
25 48.11
10
50
32 6.00
5-84
10
60
37 24.72
4-79
10
34"
25 54.82
6.71
6.70
6.68
6.68
6.66
6.65
6.6,5
6.62
146°
44"
32 11.49
5.81
136"
54"
37 29.48
4.76
126°
10'
■2C, 1.52
50'
10'
32 17.30
50'
10'
37 34.22
4-74
50'
20
;io
2() 8.20
.'<i 14.88
40
30
20
30
32 23.09
32 28.87
5-79
5-78
576
40
30
20
30
37 38.95
37 43.00
4-73
4-7'
40
30
40
2r, 21.54
20
40
32 34.03
20
40
37 48.34
4. 68
20
50
35"
20 28.19
20 34.82
10
145"
60
45"
32 40.38
32 40.11
5-75
5-73
10
135"
60
55°
37 53.01
37 57.00
4.67
4-('5
4-63
10
125"
10'
20 41.44
50'
10'
.■12 51.82
5-7'
50'
10'
38 2.29
50'
20
20 4S.04
6.60
6.59
6.58
6.56
6.55
6-54
6.52
6.5.
6.50
6.,i8
6.47
40
20
32 57.51
5 •''9
5.68
40
20
3S fi.OO
4.61
40
:iO
20 54.03
30
30
33 3.19
30
30
38 11.49
4-59
30
40
27 1.21
21)
40
33 8.S5
5.66
564
563
5.61
20
40
3S 10.07
4-58
20
50
27 7.77
10
50
33 14.49
10
50
38 20.02
4-55
10
36"
27 14.32
144"
46"
33 20.12
134"
56°
38 25.15
4-53
124"
10'
27 2(1. 8(i
50'
10'
3;i 25.73
50'
10'
38 29.00
4-51
50'
20
27 27.38
40
20
33 31.32
5-.=;9
5-58
40
20
3S 34.10
4-5°
40
;iO
27 33.89
30
30
33 30.90
30
30
38 38.03
4-47
30
40
27 40.39
20
40
33 42.40
20
40
38 43.08
4-45
20
50
27 40.87
10
50
33 48.00
5-54
5o3
10
50
38 47.52
4-44
4.41
10
37"
10'
27 53.34
27 59.79
6-45
6.44
0.4J
6.4,
6.40
6.58
6.37
6-35
6.3 1
6. J9
6.28
6.26
6.25
6.24
6.22
6.20
143"
50'
47^
10'
33 53.53
33 59.04
5-51
133'
50'
57"
10'
38 51.93
38 6(!.33
4.40
123"
5o'
20
28 0.23
40
20
3t 4.53
5-49
40
20
39 0.70
4-37
40
;!o
28 12.05
31)
30
34 10.00
5-47
30
30
39 5.00
4-3'^'
30
40
28 10. 01)
20
40
34 15.40
5-46
20
40
39 9.40
4.34
20
50
28 25.40
10
50
34 20.!)0
5-44
10
60
39 13.71
4-31
10
38"
28 31.84
142'
480
34 20.32
5-42
132'
58"
39 18.01
4'.5o
122"
10'
28 38.21
50'
10'
3t 31.72
5-40
511'
10'
39 22.29
4,2s
50'
20
28 44.50
40
20
34 37.11
5-39
40
20
39 20.51
4- -'5
4
.'iO
28 50.89
30
30
34 42.48
5-37
30
30
39 30.78
4.24
30
40
28 57.22
20
40
34 47.83
5-35
20
40
39 34.99
4.21
20
50
29 3.53
10
50
34 53.10
5-33
10
50
39 39.19
4.20
10
39"
29 9.82
141"
49'
34 58.47
5-31
131°
59"
39 43.30
417
121°
10'
29 10.10
50'
10'
35 3.77
5-30
5.28
5.26
50'
10'
39 47.52
4. 16
50'
20
29 22.30
40
20
35 9.05
40
20
39 51.05
4J,'
40
MO
29 28.01
30
30 :
35 14.31
30
30
39 55.77
4.12
30
41)
29 34.85
20
40 ,
35 19.55
5*4
20
40
39 59. s 7
4. 10 1
20
50
29 41.07
10
50 ;
35 24.78
5-23
5.21
10
50
40 3.91
4.07
4.06
10
40'
29 47.27
140
50'
35 29.09
130
60"
40 8.00
120"
9.90'
?
320"
u
230"
310'
u
240
300"
' 1
r
THE ORBIT OF URANUS.
215
TA15LE XIX, Alia. u.— CoiUinued.
u
240
60°
10'
20
30
10
50
61°
10'
20
30
40
50
62°
10'
20
30
40
50
63°
10'
20
30
40
50
64°
10'
20
30
40
60
65°
10'
20
30
40
50
66°
10'
20
30
40
60
67°
10'
20
30
40
50
68°
10'
20
30
40
50
69"
10'
20
30
40
50
70°
260°
8.00
12.03
10.04
20.04
24.01
27.90
31.89
3'>..S0
40 39.09
40 43.50
40 47.41
40 51.24
40 55.05
40 5.><.84
2.00
0.35
10.07
13.78
17.40
21.13
24.77
2K.39
31.99
35. 5G
39.12
42.05
40.17
49.00
5.!. 13
50. 5S
O.Ol
3.42
o.so
10.17
13.51
10.83
20.13
23.41
20.07
29.91
42 33.12
42 30.32
42 39.49
42 42.04
42 4."). 77
42 4S.S7
4 2 51.90
42 55.02
42 58.00
43 1.08
4.08
7.05
10.01
12.94
15.85
18.74
21.00
24.45
27.27
30.00
32.84
3
4- 03
4.01
4.00
3-97
3-95
3-93
391
3«9
3-«7
3-85
383
3.S1
3-79
3. 76
3-75
3-7^
371
3.6S
367
3(>4
3.62
3.60
3-57
356
3-53
3-5^
3-49
3-47
3'4.=;
3-43
3- 4 1
3-3«
3-37
3-34
332
3i°
3,28
3.26
3-^4
3-2I
3.20
317
43
43
43
43
43
43
43
43
43
43
43
31.=;
3- '.I
3.10
3°9
3.06
3°4
3.02
3.00
2.97
2.96
2-93
2.91
2.89
2.86
2.85
2.82
2.79
2.78
300°
120"
50'
40
30
20
10
U9°
50'
40
30
20
10
118°
50'
40
30
20
10
U7°
50'
40
30
20
10
116°
50'
40
30
20
10
115°
50'
40
30
20
10
114"
50'
40
30
20
10
113°
50'
40
30
20
10
112°
50'
40
30
20
10
111°
50'
40
30
20
10
110°
290°
u
250°
70°
10'
20
30
40
50
•71°
10'
20
30
40
50
72°
10'
20
30
40
50
73°
10'
20
30
40
50
74°
10'
20
30
40
50
75°
10'
20
30
40
50
76°
10'
20
30
40
50
77°
10'
20
30
40
50
78°
10'
20
30
40
50
79°
10'
20
30
40
50
80^
260°
43 32.84
43 35.00
43 38.33
43 41.04
43 43.73
43 40.40
43 49.04
43 51.00
43 r4.20
50.84
59.40
1.93
4.44
0.93
!>.39
11.83
14.25
10.05
19.03
21.38
23.71
44 20.02
44 28.30
44 30.57
44 32.82
44 35.04
44 37.23
44 39.40
44 41.55
43.08
45.79
47.87
49.93
51.97
53.98
55.97
57.94
44 59.89
45 1.81
43
43
44
41
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
45
l45
45
45
45
45
45
3.71
5.59
7.44
9.27
11.08
12.80
14.02
10.30
18.08
19.77
21.44
23.09
24.71
20.31
27.89
29.45
3it.98
32.49
33.97
35.43
30.87
38.29
2.76
2-73
2.71
2.69 i
2.67
2.64
2.O2
2.60
2.58
2.55
2-53
2.51
2.49
2.46
2.44
2.42
2.40
2.38
2-35
2-33
2-3'
2.28
2.27
2.25
2.22
2.19
2.17
215
2-13
2. t 1
2.0S
2.06
2.04
2.01
1.99
1.97
1-95
1.92
1.90
1. 88
1,85
1.S3
i.Ri
1.78
1.7O
1.74
1.72
1.69
1,67
,.65
1.62
1.60
1.58
1.56
1-53
1. 48
1.46
J. 44
1.42
290°
110°
50'
40
30
20
10
109°
50'
40
30
20
10
108°
50'
40
30
20
10
107°
50'
40
30
20
10
106°
50'
40
30
20
10
105°
50'
40
30
20
10
104
5(1'
40
30
20
10
103°
50'
40
30
20
10
102°
50'
40
30
20
10
101'
50'
40
30
20
10
100'
280°
n
u
260°
80°
10'
20
30
40
50
81°
10'
20
30
40
50
82°
10'
20
30
40
50
83°
10'
20
30
40
50
84°
10'
20
30
40
50
85°
10'
20
30
40
50
8G°
10'
20
30
40
50
87'
111'
20
30
40
50
88°
10'
20
45 38.29
45 39.08
45 41.05
45 42.40
45 43.73
45 45.03
45 40.30
45 47.50
45 48.79
45 50.00
45 51.18
45 52.34
45 53.48
54.59
55.08
50.75
57.79
58.81
59.80
0.78
1.73
2.00
3.50
4.44
30
40
50
89°
10'
20
30
40
50
90°
270°
45
45
45
45
45
45
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
41;
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
40
5.30
0.14
0.95
7.74
8.50
9.24
9.90
10.05
11.32
11.97
12.59
13.19
13.70
14.31
1 1 . 84
15.35
15.83
10.29
10.72
17.13
17.52
17. S!)
18.23
18.55
18.84
19.11
19.30
19.59
19.79
19.97
40 20.12
40 20.25
20.30
20.44
20.50
20.53
20.54
3
1-39
1-37
1-35
1-33
1.30
1.27
1.2O
1-23
1. 21
i.iS
1. 16
1.14
I. II
1.09
1.07
1.04
1.02
0.99
0.98
0-95
0-93
0.90
0.8.S
0.86
0.S4
0.81
0.79
0.76
0.2:
0.09
o.r,7i
0.05
0.62
0.60
°-S7
°o5
0-53
0.51
0.48
0.46
0-43
0.4T
°-39
°-37
0-34
0.32
0.29
280°
100°
50'
40
30
20
10
99°
50'
40
30
20
10
98°
50'
40
30
20
10
97°
50'
40
30
20
10
96°
50'
40
30
20
10
95'
50'
40
30
20
10
94°
511'
40
30
20
10
93°
50'
40
30
20
10
92°
40
40
4li
40
40
0.27
50'
0.25
40
0.23
30
0.20
20
o.iS
10
0.15
91°
013
50'
0. 1 1
40
0.08
30
0.06
20
0.03
10
O.OI
90°
270°
u
f!
276
THE ORBIT OF URANUS.
Tablk
XX.
XXI.
XXII.
Aim.
1
2
3
(6.S.1)
(6.C..1)
(6.C.0)
(6.«.1)
(6.C.1) (//.S.2)
(6.e.2)
(6.C.0)
(b.S.l)
(h.c.V
(6.M.2I
(6.C.2)
//
//
tt
"
tt
//
If
II
U
n
1.20
1.12
0.04
5.99
5.42
0.21
0.17
0.0"
1.58
1.34
0.07
o.n
10
1.15
1.17
0.03
6.13
5.23
0.19
0.18
0. 06
1.63
1.14
0.08
0.23
20
1. 10
1.21
0.03
6.25
6.04
r.18
0.18
0.(15
1.67
0.93
0.11
0.29
30
1.04
1.25
0.(12
6.34
4.84
(,.17
0.18
0.04
1.68
0.72
0.15
0.32
40
0.98
1.28
0.02
6.41
4.62
0.16
0.18
0.03
1.67
0.51
0.21
0.33
50
0.92
1.31
0.02
6.45
4.40
0.15
0.18
0.03
1.60
0.33
0.27
0.32
GO
0.8(i
1.33
0.(12
6.45
4.18
0.16
0.18
0.(12
1.48
0.18
0.32
0.28
70
0.79
1.34
0.02
6.40
3.94
O.lfi
0.18
O.ol
1.32
0.09
0.35
0.23
80
0.72
1.35
0.02
6.32
3.69
0.16
0.17
0.(!0
1.13
0.07
0.36
0.18
90
O.GIl
1.35
0.(13
6.19
3.43
0.17
0.16
0.00
0.94
0.13
0.35
0.13
100
0.59
1.34
0.03
G.02
3.15
0.17
0.15
O.OI
0.77
0.25
0.32
0.09
110
0.52
1.32
0.04
5.81
2.87
0.18
0.14
0.02
0.66
0.41
0.29
0.07
120
0.4f.
!.30
0.04
5.56
2.57
0.19
0.12
0.04
0.62
•.1.60
0.25
0,(1(1
130
0.40
1.27
0.04
6.29
2.26
0.20
0.11
0.06
0.64
0.7 V
0.22
O.Ofi
140
0.34
1.24
0.05
4.99
1.96
0.20
0.10
0.08
0.72
0.91
0.20
0.07
mo
0.28
1.20
0.05
4 (,'i
1.66
0.20
0.09
0.09
0.83
1.00
0.18
0.08
1(10
0.23
1.15
0. 05
4.36
1..S8
0.20
0.08
0.10
0.95
1.04
0.17
0.09
no
0.19
1.10
O.OG
4.03
1.11
0.20
0.08
O.U
1.04
1.04
0.17
0.10
ISO
0.15
1.04
O.OG
3.69
0.86
0.19
0.07
0,11
1.11
l.OI
0.16
0.10
190
0.12
0.98
0.06
3.36
0.65
0.17
0.07
0.11
1.14
0.97
0.14
0.11
200
0.09
0.92
O.OG
3.03
0.56
0.16
0.08
O.ll
1.14
0.93
0,13
0.12
210
0.07
0.8*'"
O.Ofi
2.71
0.32
0.14
0.08
0.11
1.11
0.91
0,11
0.13
220
o.or,
0.79
O.Ofi
2.30
0.20
0.12
0.09
0.10
1.07
0.91
0.10
0.15
230
0.05
0.72
O.OG
2.08
0.12
0.1 1
O.U
0.10
1.02
0.92
0.10
0.18
240
0.05
O.GG
0.07
1.77
0.08
0.09
0.13
0.09
0.98
0.9C
0.10
0.21
250
o.on
0.59
0.07
1.48
0.08
0.07
0.15
0.09
0.96
1.00
0.12
0.23
2fiO
0.08
0.52
0.07
1.21
0.10
0.05
0.17
0.08
0. 95
1.05
0.14
0.25
270
0.10
0.4G
0.08
1.05
0.16
0.03
0.20
0.08
0.94
1.09
0.16
26
2S0
0.13
0.40
0.08
0.72
0.26
0.02
0.22
0.07
0.95
1.13
0.19
0.26
290
0.1 fi
0.34
0.09
0.52
0.38
0.01
0.25
O.Ofi
0.95
1.16
0.22
0.25
300
0.20
0.28
0.10
0.35
0.54
0.01
0.27
O.Ofi
0.94
1.18
0.23
0.23
310
0.25
0.23
0.11
0.22
(1.73
0.01
0.29
O.Ofi
0.93
1.20
0.24
0.21
320
0.30
0.19
0.12
0.12
0.94
0.02
0.31
0.07
0.91
1.22
0,23
0.19
330
0.3rt
0. 15
0.13
0.0(1
1.18
0.03
0,33
0.08
0.88
1.25
0,22
0.18
340
0.42
0.12
0.14
0.04
1.44
0.05
0.35
0.09
0.80
1.28
0.20
0.17
350
0.48
o.no
0.15
O.Ofi
1.72
0.07
0.36
0.09
0.84
1.34
0.18
0.17
3r,o
0.54
0.(17
O.lfi
0.12
2.01
0.10
0.37
0.10
0.85
1.40
0.16
0.18
370
O.fil
O.Ul)
0.17
0.21
2.31
0.13
0.37
0.11
0.88
1.46
0.15
0.19
380
0.G8
0.05
0.18
0.34
2.62
0.16
0.37
0.12
0.92
1.51
0.14
0.20
390
0.74
0.05
0.19
0.50
2.92
0.19
0.36
0.12
0.98
1.54
0.13
0.21
400
0.81
O.Ofi
0.20
0.70
3.21
0.23
0.35
0.11
1.03
1.54
0.13
0.22
410
0.88
0.08
0.20
0.92
3.54
0.26
0.31
0.10
1.07
1.50
0.12
0.23
420
0.94
0.10
0.21
1.17
3.84
0.80
0.33
0.08
l.Ofi
1.44
0.12
0.25
430
1.00
0.13
0.21
1.45
4.14
0.33
0.31
0.06
1.01
1.37
0.12
0.27
440
1.06
O.IG
0.21
1.75
4.42
0.36
0.29
0.04
0.91
1.30
0.12
0.29
450
1.12
0.20
0.21
2.06
4.70
0.38
0.27
0.03
0.77
1.28
0.12
0.32
4 CO
1.17
0.25
0.21
2 39
4.96
0.40
0.25
0.02
O.fil
1.31
0.14
0.35
470
1.21
0.30
0.20
2.72
5.20
0.41
0.23
0,01
0.47
1.40
0.17
0.38
4 SO
1.25
0.3G
0.19
3.06
5.42
0.42
0.21
0.(11
0.36
1.56
0.21
0.39
490
1.2H
0.42
0.18
3.39
5.62
0.42
0.20
0.01
0.32
1.76
0.27
0.39
500
1.31
0.48
0.17
3.71
5.79
n.42
0.18
0.01
0.36
1.96
0.32
0.37
510
1.33
0.54
O.lfi
4.02
5.92
0.41
0.17
0.01
0.46
2.15
0.37
0.33
520
1.34
O.Gl
0.14
4.31
6.01
0.39
0.16
0.02
0.62
2.21
0.40
0.27
530
1.35
0.G8
0.13
4.58
6.06
0.38
0.16
0,02
0.81
2.35
0.40
5.20
540
1.33
0.74
O.U
4.84
6.08
0.35
0.15
0.03
1.00
2.34
0.38
0.14
550
1.34
0.81
0.10
5.07
G.Ot
0.33
0.15
0.03
1.18
2.2fi
0.33
0.08
5i;o
1.32
0.88
0.09
5.28
5.98
0.30
0.15
0.04
1.31
2.11
0.27
0.05
570
1.30
0.94
0.07
5.48
5.88
0.28
0.16
0.(14
1.41
1.94
0.20
0.04
5 so
1.27
1.(10
O.Ofi
5.67
5.75
0.2fi
0.16
0.(15
1.48
1.74
0.14
0.07
590
1.24
l.dCi
05
6.81
5.59
0.23
0.17
0.06
1.53
1.51
0,1(1
0.11
000
1.20
1,12
0.04
5.99
5.42
0.21
0.17
0.(1(1
l..^S
1.34
0.07
0,17
THE ORBIT OF URANUS.
8TT
T.vm.KXXlH.
Year. (6.C.0) (fc.ti.l), {b.c.l)
1300 +0
fifil— 3.S5'_12.74
llilO ; O.flSl 3.81
13-20
0.04,
.78
1330 ' 0.03 3.74
1340 I 0.01 3.70
12.0,0
12.:)0
1-2.40
12.37
(t.».2) (6.C.2)
.0,341—0.07
0.34 i 0.00
0.341 0.05
Year. (6.c,0) (ft.s.l) (b.c.l) V>-i-'^)\ (l'-'-'-'^)
1350 +0.00—3.00—12.27
1300
0.59; 3.03
1370 1 0.58
1380 0.57
1390 0.50
1400 +0.55
0.54 I
0.53,
0.52 i
3.00
3.57
3.54
12.17
12.07
11.07
11.80
—3.51 —11.70
1410
1420
1430
1440
3.48
3.40
3.44
0.51 1 3.43
1450 +0.5" —3.42
11.05
11.54
11.43
11.32
0.34
0.34
-0.34
0.34
0.34
0.34
0.34
0.04
0.03
'—0.02
0.00
0.80
0.88
0.87
1800 +0.15
1810 , 0.14
1820 ! 0.13
1830
0.12
1840 0.11
1850 +0.10
1800 0.09
-4.00 —0.00
1870
0.08
_0.35'— 0.80
0.35 j 0.85
0.35 0.84
0.35 1 0.83
0.35; 0.8-
1880 : 0.07
1890
0.00
4.08
4.75
4.83
4.92
-5.00
5.09
5.18
5.27
5.30
5.80
5.01
5.41
5.21
ff
ft
—0.44
—0.40
0.45
0.45
0.45
0.44
0.45
0.43
0.46
n .it\
0.42
n ii
—5.00 —0.40 —I
4.79! 0.47 1
4.58 1 0.47
1900 '+0.05! —5.45
1010 I 0.04
1020 0.03
1030
0.02
1400
1470
1480
1490
1500
1510
0.49
0.48
0.40
0.45
+0.44
0.43
-11.21 —0.35
3.41! 11.10
3.40 1 10.08
3.40 1 10.80
3.40 10.74
_:?,40 —10.02
.10 10.50
1520 0.42
1530 I 0.41
1540 i 0.40
3.41
3.42
10.38
10.20
3.43 10.14
1000 '+0.34
1010 ' 0.33
1C20 i 0.32
1030 0.32
1040
1050
1 000
1070
1080
1090
1700
1710
1550 +0.39—3.45 —10.01
1.500 ' 0.38: 3.47 0.88
1570 0.371 3.40 0.75
1580 0.30! 3.51
1590 0.351 3.r.3
0.35
0.30
0.30
0.30
-0.30
0.30
0.30
0.3-
0.37
-0.80
0.70
0.78
0.77
0.70
— 0.
0.7t
0.73
(1.72
0.71
1040 t + 0.01
1950
1 000
1070
1080
1000
2000
0.00
—0.01
0.02
0.04
0.05
5.55
5.04
5.74
5.84
—5.94
0.04
6.14
6.25
6.30
0.40
0.,39
4.30; 0.48 i 0.38
4.141 0.48i 0.37
I
_3.02 — 0.49 —0.30
3.09
3.40
3.22
2.97
0.40
0.35
0.50 0.34
0.50 j 0.33
0.50 1 0.32
_') 73—0.51 —0.30
2.48' 0.51; 0.29
2.24 0.52! 0.28
1.99 0.52 1 0.27
1.73 1 0.521 0.20
_0.00 —0.47 —1.471— 0.531-0.25
_n.37 — O.TO
0.70
9.01
0.47
—3.50 — 0..33
0.31
+0.30
0.20
0.28
0.27
0.20
+0.25
0.24
1720 ' 0.23
3.50
3.02
3.05
3.09
0.19
0.05
8.00
0.37'
0.37;
0.37 I
0.38
0.00
0.08
0.07
2010 I 0.07, 0.58
2020 ! 0.0S|
2030
2040
0.00
0.00 0.80
0.10 i 0.92
1.211 0.5:i
0.04
0.00
0.54
0.54 1
0.24
0.23
0.22
0.38, 0.541 0.20
-0.11' —7.03 L-0.10'-0..55'— 0.19
2050
•2000 1 0.12! 7.14 :+o.
IS 0.55; 0.18
_0.3S — 0.00
0.381 0.05
0.30! 0.04
0.30
0.03
0.22
0.21
1730
1740
1750
1700
1770
1780
1790
1800 +0.15 —
3.77
S.liU -
8.44
3.81
8.20
3.85
8 13
3.00
1.00
—3.05
— 7.79
' 4.00
7.03
4.00
7 40
4.12
7.20
4.18
t.l2
i o.\
— 0.05
0.10
4.32
0.18
4.38
0.17
4.45
0.10
1 n 1 ^
4.53
I r.n'
0.30, 0.02
_0.39'— 0.01
0.40 1 0.00
0.40] 0.50
0.40 0.58
0.41
-0.41
0.41
0.42
0.42
0.42
0.43
2070
2080
2090
0.13;
7.20
0.15 1 7.37
0.10
2100 '—0.17
2110 ! 0.18
2120 • 0.19
2130 i 0.21
2140 O.22I
7.48
0.47 0.50
0.70 0.50
1.05 0.50 j
0.17
0.10
0.15
_7 00 +1.35 —0.57' —0.14
7.72 1 1.05 0.571 "■''*
7.84
7.07
8.09
2150 —0.23 —8.21
2100 i 0.241
2170 ' 0.251
2180 0.27!
8.33
8.44
1.00 0.57;
0.12
2.27 0.58 0.10
2.58 1 0.58
+2.90—0.58
3.22i 0.50
3.541 0.50
3.87 i 0.50
0.09
—0.08
0.07
0.00
0.05
2100
0.281
2200 —0.20 — S
210
2220
2230
0.301
0.311
0.33!
0.77
0.43
0.50
0.58
0.43
0.49
0.39
0.44
0.48
0.20
0.44
0.47
- 0.00
—0.44
—0.40
2240 , 0.34
2250 —0.35 —9
2200
2270
2280
2290
2300 —0.42 —10
0.30
0.38
0.30
0.40
1
08
4.20
0.00
0.03
SO
+ 1.54
—0.00
—0.02
92
4.88
0.00
—0.01
04
5.23
0.01
0.00
10
5 58
0.01
+0.01
.28
5.94
0.01
0.03
40
+ 0.29
—0.02
+0.04
52
0.05
0.02
0.05
04
7.01
0.03
0.07
70
7.37
0. 03
0.08
.88
t.74
0.03
0.10
.00
+8.10
-0.04
+ 0.11
( (
! ii
in 'i
TABLE FOR FORMING THE PRODUCTS OF GIVEN NUMBERS BY THE
SINE OR COSINE OF A GIVEN ANGLE.
This tab. 3 is formed for thn especial purpose of facilitating the formation of the
products («.«.3) siu 3r,, (r.c.3) cos JJ./, etc., (,.«.!) sin g, (p.cl) cos j, for entire degree^
tf q It is so arranged that the required products can be taken out at siglit.
Supposing the number to be given in seconds and decimal fractions ot a second,
we first seek the given angle at the top or bottom of the page, and then enter one
of the first nine lines of the table with the fraction part of the second, niterpolatu.g
for the hundredths. We then add the result mentally to the number corrcspondnig
to the entire seconds. The algebraic signs at the sides of the angles are those of
the sines or cosines corresponding to the angle and to the column above or below.
If the number does not exceed 3" we can enter the table as if it were ten times
greater, and remove the decimal pouit one place to the left ni the result.
For exanii>le, to find the value of
2l".67 sin 280' + 2".25 cos 280°
we find the angle 280° at the bottom of a pair of columns, the right hand one being
the sine column. Entering this column with 0.67 as the argument, we find O.CUi.
Entering with 2.1, we find 20.68, to which adding 0.66, we have 21".:U as the
sine product. Entering the other column with 22.5, and moving the decimal point,
we find 0".39 for the cosine product. Noticing the algebraic signs on each side
of 280°, we find the result to be - 21" M + 0'.39 = - 20".<J5.
( 279 )
naa
280
TAULE OF rilODUCTS OF SINKS ALD COSINES.
+
v+
4-
2" +
+
3^ +
+
4°+
+
6°+
+ 179 —
+ 178 —
+ 177 —
+ 178 —
+ 176 —
— 181 —
— 182 —
—183 —
— 184 —
— 185 —
—359 J-
—358 +
-3£
gin
>7 +
cos
— 356 +
—355 +
sin
cos
sin
cos
xin
cos
sin
cos
0.1
0.00
0.10
0.00
0.10
0.01
0.10
0.01
0.10
0.01
0.10
0.1
0.2
0.00
0.20
0.01
0.20
0.01
0.20
0.01
0,20
0.02
0.20
0.2
0.3
0.01
0.30
0.01
0.30
0.02
0.30
0.02
0,30
0.03
0.30
0.3
0.4
0.01
0.40
0.01
0.40
0.02
0.40
0.03
0.40
0.03
0.40
0.4
0.5
0.01
0.50
0.02
0.50
0,03
0,50
0.03
0.50
0.04
0.50
0.5
O.fi
0.01
o.(;o
0.02
0.(10
0.03
0,(10
0.04
O.CO
0.05
0.(10
0.6
0.7
0.01
0.70
0.02
0.70
0,04
0.70
0.05
0.70
0.0(1
0.70
0.7
0.8
0.01
o.so
0.03
0.80
0.04
0.80
O.Ofi
0.80
0.07
0.80
0.8
0.9
0.02
0.90
0.03
0.00
0.05
0.90
O.Ofi
0.90
O.OS
0.00
0.9
1.0
0.02
1 00
0.03
1.00
0.05
1.00
0.07
1.00
0.00
1.00
1.0
2.0
0.03
2.00
0.07
2.00
0. 10
2.00
0.14
2.00
0.17
1 . 09
2.0
3.0
0.0,)
3.00
0.10
3.00
O.lfi
3.00
0.21
2.00
0.2(1
2.09
3.0
4.0
0.07
4.00
0.14
4.00
0.21
3.99
0.28
3.99
0.35
3.08
4.0
6.0
0.09
5.00
0.17
5.00
0.20
4.99
0..35
4,09
0.44
4.08
5.0
6.0
0.10
fi.OO
0.21
0.00
0.31
5,99
0.42
5,09
0.52
5.08
6.0
7.0
0.12
7.00
0.24
7.00
0.37
fi,09
0.49
fi.08
0.61
6.07
7.0
8.0
0.14
8.00
0.28
8.00
0.42
7 09
0..'>fi
7.08
0.70
7.07
8.0
9.0
O.K!
9.00
0.31
8.09
0.47
8.09
0.03
8.98
0.78
8.07
9.0
10.0
0.17
10.00
0,35
0.90
0.52
0.99
0.70
0.98
0.87
O.Ofi
10.0
11.0
0.1!)
11.00
0.38
10.90
0.58
10.08
0.77
10.07
0.96
1 O.Ofi
11.0
12.0
0.21
12.00
0.42
11.00
O.fi 3
11.98
0.S4
11.07
1,05
11.05
12.0
1.3.0
0.23
13.00
0.45
12.00
O.fiS
12.98
0.01
12.07
1,13
12.05
13.0
14.0
0.24
14.00
0.49
13.99
0.73
13.98
0.98
13.97
1.22
13.05
14.0
15.0
0.2fi
15.00
0.52
14.99
0.70
14.98
1.05
14.06
1.31
14.94
150
IG.O
0.28
Ifi.OO
0.5fi
15.99
0..S4
15.98
1.12
15.96
1.30
15.04
16.0
17.0
0.30
17.00
0.50
Ifi.OO
O.SO
lfi.98
1.10
16.96
1.4S
lfi.04
17.0
18.0
0.31
IS. 00
O.C.S
17.09
0.04
17.98
1.2(;
17.06
1.57
17.03
18.0
19.0
0.33
10.00
O.fi (5
IS. 99
0.00
18.97
1.33
18.05
l.Gfi
18.93
19.0
20.0
0.3:)
20.00
0.70
10.09
1.05
19.07
1.40
10.05
1.74
19.92
20.0
21.0
0.37
21.00
0,73
20.09
1.10
20.07
1.4fi
20.05
1.S3
20.02
21.0
22.0
38
22, Ou
0.77
21.99
1.15
21.07
1.53
21.95
1.02
2 1 . 02
22.0
23.0
0.40
23.00
0,sO
22.09
1.20
22.97
1 . (10
22,04
2.00
22.01
23.0
24.0
0.42
24.00
0.84
23.09
1.2fi
23.97
1.(17
23,94
2.00
23.01
24.0
25.0
0.44
25.00
0..^7
24.98
1.31
24.97
1.74
24 94
2.18
24.00
25.0
2ti.0
0.4n
2(!.00
», '
25.98
1.3fi
25.90
l.sl
25,94
2.27
25.00
2(1.0
27.0
0.47
27.00
0.04
2fi.98
1.41
2fi.96
1.S8
2(1.03
2.35
2fi.90
27.0
28.0
0.40
28.00
0.08
27.98
1.47
27. 00
1,05
27.93
2.44
27. SO
2S.0
29.0
0.51
20.00
1.01
2S.98
1.52
28.90
2.02
28.93
2.53
28.80
20.0
30.0
0.52
30.00
1.05
29.98
1.57
29.96
2.00
20.03
2.fil
20.80
20.0
cos
Bin
CO.S
sin
cos
sin
cos
.sill
cos
sin
+ 2-
71°—
+ 27
2 —
+27
3 —
+ 2-
14 —
+ 2T
5 —
— 2e
9 —
—26
8 —
— 2£
n —
— 2(
56 —
—26
5 —
— J
n +
— £
(2 +
— £
3 +
— S
4 +
— £
•5 +
+ i
J9 +
+ £
18 +
+ £
17 +
+ i
36 +
+ £
15 +
'
J
TABLE OP PRODUCTS OF SINES AND COSINES.
281
J
+
6 +
+ -7^ +
+
8"+
+ r+
+ 10^
+ 174 —
+ 173 —
+ 172-
+ 171 —
^ 170 —
— 186 —
— 187 —
— 188 —
— 189 —
— 190 —
— 354 +
— 358 +
—352 +
— 351 +
— 350 +
Kill
CD.i
ttin
C(JH
sin
cos
sill
tus
Hin
cos
0,1
0.01
0,10
0.01
0,10
0.01
0.10
0.02
0.10
0.02
O.IO
0,1
0.2
0.02
0,20
0,02
0.20
0.03
0.20
0.03
0.20
0.03
0.20
0.2
0,3
0.03
0..30
0.01
0.30
0.04
0.30
0.05
0.30
0,05
0.30
(1.3
0,4
0.04
0.40
0.05
0.40
0.00
0.40
0.06
0.40
0.07
0.39
0.4
0.5
0.05
0.50
0.00
0,50
0,07
0,50
0.08
0.49
0.09
0.49
0,5
0. li
0.00
0.00
0.07
0,00
0.08
0.59
0.09
0.59
0,10
0.59
0.0
0.7
0,07
0.70
0.09
0,09
0.10
0.09
0.11
0.09
0,12
0.09
0.7
O.S
O.OS
0.80
0,10
0.79
0.11
0.79
0.13
0.79
0,14
0,79
0.8
0.9
O.OD
0.90
O.lt
0.89
0.13
0.80
O.U
0.«9
0.10
0,89
0.9
1.0
O.lO
0,99
0.12
0.99
0.14
0.99
0,10
0.99
0.17
0,98
1.0
2.0
0.21
1.99
0.21
1.99
0.28
1.98
0.31
1.98
35
1,97
2.0
3.0
0,31
2.98
0.37
2.98
0.42
2.97
0.47
2.90
0.52
2,95
3.0
4.0
42
3.98
0,49
3.97
O.uO
3,90
0.03
3.95
0.09
3,94
4.0
5,0
0.52
4.97
0.01
4.90
0.70
4.95
0.78
4.94
0.87
4,92
5.0
fi,0
0.03
5.97
0.73
5.90
0.84
5,94
0.94
5.93
1.04
5.91
0,0
7.0
0.73
0.90
0.85
0.95
0.97
0.93
1,10
0.91
1.22
0.89
7.0
8,0
0.H4
7.90
0.97
7.94
1,11
7.92
1,25
7,90
l..i9
7.88
8.0
9,0
0.94
8.95
l.IO
8,93
1.25
8.91
1,41
8.89
1.50
8.80
9.0
10.0
1,05
9.95
1.22
9,93
1.39
9.90
1.50
9.88
1.74
9.85
10.0
11,0
1.15
10,94
1.34
10.92
1.53
10.89
1,72
10.80
1.91
10.83
11.0
12.0
1.25
11.93
1,40
11.91
1.07
11,88
1,88
11.85
2.08
11.82
12,0
13.0
1.30
12,93
1,58
12.90
1.81
12,87
2,03
12.84
2.20
12,80
13.0
14.0
1.40
13.92
1.71
13.90
1.95
13,80
2.19
13.83
2.43
13,79
14.0
15.0
1.57
14.92
1.83
14.89
2.09
14.85
2,35
14.82
2.00
14.77
15,0
10.0
1.07
15.91
1.95
15.88
2.23
15.84
2.50
15.80
2.78
15.70
10.
17. D
1.7S
10.91
2.07 10.87 1
2.37
10.83
2.00
10.79
2.95
10.74
17.0
1S,0
l.ss
17.90
2.19
17.87
2,51
17.82
2,82
17.78
3.13
17.73
18,0
19.0
1.99
18.90
2.32
18.80
2.04
18.82
2.97
18.7T
3.30
18.71
19,0
20.0
2.09
19.89
2.44
10.85
2.78
19,81
3.13
19.75
3.47
19.70
20.0
21.0
2.20
20.88
2.50 20.84
2.92
20.80
3.29
20.74
3.05
20.08
21.0
22,0
2.30
21.88
2.08 21.84
3.00
21.79
3,44
21.73
3.82
21.07
22.0
23.0
2.40
22.87
2.80
22.83
3.20
22,78
3.00
22.72
3.99
22.05
23.0
24.0
2.51
23.87
2.92
23.82
3.34
23,77
3.75
23.70
4.17
23.04
24.0
25.0
2.01
24. SO
3.05
24,81
3.48
24,70
3.91
24.09
4.31
24.02
25.0
20,0
2,72
25.80
3.17
25,81
3,02
25.75
4.07
25.08
4.51
25.01
20.0
27.0
2,82
20.85
3.29
20,80
3.70
20.74
4.22
20.07
4.09
20.59
27,0
2S.0
2,93
27.85
3,41
27.79
3,90
27.73
4.38
27.00
4.80
27.57
28,0
29.0
3,03
28.84
3.53
28.78
4.04
28.72
4.54
28.04
5.04
28.56
29.0
30.0
3.14
29.84
3.00
29.78
4.18
29.71
4.09
29.03
5.21
29.54
30.0
cos
sin
COS
siti
cos
sin
cos
sin
cos
sin
+ 2'
76 —
+ 277 —
+ 2'
IB —
+ 279 —
+ 2-
—
— 2(
84 —
— 263 —
— 2(
32 —
— 281 —
— 2(
30 —
— {
:6 +
- 97 +
— i
)3 +
— 89 +
— 1(
)0 +
+ i
J4 +
+ 83 +
+ J
32 +
+ 81 +
+ i
JO +
3
a Aug
ust. 1873
.
282
TABLE OF PRODUCTS OF SINKS AM) COS INKS,
n
|i! ;
+ 11° +
+ 12° +
+ 13°+
+ 14°+
+ 1B° +
+ 169 —
+ 168 -
+ 167 -
+ 166 —
+ 16B —
— 191 —
— 192 —
— 193 —
— 194 —
— 105 —
— 349 +
— 348 +
— 847 +
— 346 +
— 846 +
sin
t'08
«in
(•(IH
xln
cos
RJn
cos
Rin
cog
0.1
0.02
0.10
0.02
0,10
0.02
0,10
0,(12
0,10
0.03
0.10
0.1
0.2
O.lit
0.20
O.dt
0,20
cot
0.19
0,05
0,19
0.05
0.19
0.2
0.3
O.OC,
0.2'J
(l,0(>
0,29
0.(17
0.29
0.07
0,29
0.08
0.29
0.3
0.4
O.OH
0.39
0.08
0,39
0.09
0.39
0,10
0,39
0.10
0.39
0.4
O.ft
O.IO
0.49
0.10
0,(9
0.11
0.49
0.12
0,49
0.13
0.48
0.5
O.C)
0. II
0.59
0.12
0.59
0.13
0,58
0.15
0,58
0,10
0.58
O.fi
0.7
0. l:!
O.CO
0,15
0.08
0. 1
0,08
0,17
0.08
0.18
0,08
0.7
0.8
o.i:.
0.79
0.17
0.78
O.IS
0.78
0,19
0,-8
0,21
0,77
0,8
0.9
0.17
0.88
0.19
0.88
0.20
0.88
0.22
0,87
0,23
0.87
0.9
1,0
fl.IO
0.9S
0,21
0.98
0.22
0.97
0.24
0.97
0.20
0.97
1,0
2.0
o.;!s
i.9(;
0.42
l.9(;
0,45
1,95
0,48
1.94
0.52
1,93
2.0
3.0
0..M
2.91
o.(;2
2.93
0,07
2.92
0.73
2,91
0,78
2.90
3.0
4.0
0.7(1
3.93
0.83
3.91
0.90
3.90
0.97
3.88
1.04
3.80
4.0
f).0
O.Of)
4.91
1,04
4.89
1.12
4,87
1,21
4,85
1.29
4,83
5.0
fi.O
1.14
5.89
1,25
5.87
1.35
5.85
1,45
5,s2
1.55
5,80
6.(1
7.0
l.:!4
n.87
1,4(1
fi.85
1.57
0.82
1.09
0.79
1.81
0,70
7,0
H.O
i.r>.{
7.85
i,(;r,
7 83
1.80
7,79
1.94
7.70
2.07
7,73
8.0
0.0
1.72
8.83
1.87
8.80
2.02
8.77
2.18
8.73
2.33
8.09
9.0
10.0
i.ni
9.82
2,08
9.78
2,25
9.74
2,42
9,70
2,59
9.06
10.0
11.0
2.10
l((.80
2.29
10. 7c.
2^47
10.72
2,00
10.07
2,s5
1(1,03
11.0
12.0
') .)()
11.78
2,49
11.74
2,70
11,09
2,90
11.04
3,11
11,59
12,0
l.i.O
2'.4S
12.7(!
2,70
12,72
2,92
12,07
3,14
12.01
3.;io
12.50
13,0
14.0
2.(17
13.74
2.91
13.09
3.15
13.G4
3.39
13.58
3.02
13.52
14.0
in.o
2.Sfi
14.72
3,12
14.07
3.37
14,02
3.03
14,55
s.y-H
14.49
15.0
lil.O
.•J.O,')
15.71
3.33
15.05
3,00
15.59
3.87
15.52
4.14
15.45
10.0
17.0
:i.24
l(i.(!9
3.5:i
10.03
3,82
10.50
4,11
10.50
4.10
10.42
17.0
18.0
.'i.4;!
17.07
3,7t
17.01
4,(15
17.54
4,35
17.47
■<,00
17.39
18.0
ly.o
•J. an
18. C5
3.95
18.58
4,27
18.51
4.00
18.44
4,92
18.35
19.0
20.0
3.S2
19.03
4.10
19.50
4,50
19.49
4.84
19.41
5,18
19.32
20.0
21.0
4.01
20.01
4,37
20.54
4,72
20.40
5.08
20.38
5.44
20.28
21.0
22.0
4.2(1
2I.(;0
4,57
21.52
4.95
21.44
5.32
21.35
5.09
21.25
22.0
2:t.0
4.:i9
22.58
4,78
22.50
5.17
22.41
5,50
22.32
5.95
22.22
23.0
24.0
4.58
23.50
4.99
23.48
5.40
23.38
5.81
23.20
0.21
23.18
24.0
25.0
4.77
24.54
5.20
24.45
5.02
24.30
0.05
24.20
0.47
24.15
25.0
2(i.O
4.!Mi
25,52
5.41
25.43
5.85
25, .",3
0.29
25.23
0.73
25.11
20.0
27.0
5.15
2(i.50
5,(;i
20.41
0.07
20,31
0.53
20.20
0.99
20,08
27.0
2.S.0
5.:!4
27.49
5,82
27.39
0.3()
27.28
0,77
27.17
7.25
27.05
28.0
29.0
5.53
28.47
6.03
28.37
0.52
28.20
7.02
28.14
7.51
28.01
29.0
30.0
5.72
29.45
0,24
29.34
G.75
29.23
7.20
29.11
7.70
2S.98
30.0
cos
sin
cos
sill
cos
sin
OIKS
sin
t'O.S
sin
+ 281 —
+ 2£
J2 —
+ 2f
}3 —
+ 2e
14 —
+ 2f
J5 —
— 259 —
— 2E
>8 —
— 257 —
— 2E
•8 —
— 2£
>5 —
— 101 +
— IC
>2 +
— 103 +
— 10
4 +
— IC
)5 +
+ "79 +
+ -
(8 +
+ 77 +
+ '■
16 +
+ ^
'5 +
TAin
,K OF PRODUCTS OF 8
N !•: S
A Nl)
C08IT
s'BS.
283
+ 10° +
+ ir+
-t 18°+
4 19°+
+ 20°+
+ 164 —
+ 163 —
+ 162 —
+ 161 —
1 160 —
— 196 —
— 197 —
._ 198 —
— 199 —
— 200 —
— 344 +
— 343 +
— 342 +
— 341 +
— 340 +
Bill
CHS
Hin
cos
hIii
C'OH
KIM 1
j
fos
sill
cog
O.l
0.03
0.10
0.03
0.10
0.03
0,10
0.03 ;
0.09
0.03
0.09
0.1
o.a
0.li)>
0.1!)
O.OO
0.19
O.OO
(I 1!)
0.07
0. 1 9
0.07
0,19
0.2
0.3
o.os
0.2!)
0.09 ,
0.29
0.09
0.2!)
0.10 !
28
0.10
0,28
0.3
0.4
0.1 1
0.38
0.12
0.38
0.12
0.38
0.13 ,
0.38
0.14
0,38
4
0.5
0.14
0.48
0.15
0.48
0.15
0.48
0.10
0.47
0,17
0,47
0.5
o.c,
0.17
0.58
0.18
0.57
0.19
057
0.20
0.57
0,21
0,50
O.li
0.7
0.1!)
0.07
0.20
0.07
0.22
0,07
0,23
0.06
0,24
0,06
0.7
0.8
0.22
0.77
0.23
0.77
0.25
0.76
0,20
0.76
0,27
0,75
0.8
0.9
0.25
0.87
0.26
0.86
0.28
0.80
0.29
0.85
0.31
0,^)0
0.9
1.0
0.28
0.96
0.29
O.OO
0.31
0.95
0.33
0.95
0.34
0,94
1.0
L'.O
0.55
1.92
0.58
1.01
0.02
l.!)0
0,r,5
1.89
0,08
1,88
2.0
;i.o
0.H3
2.88
0.S8
2.87
0.93
2.S5
0.'.'
2.S4
1,03
2.S2
3.0
4.0
1.10
3.85
1.17
3.83
1.24
3.80
1.30
:;.78
1,37
3'70
4.0
5.0
1.38
4.81
1.40
4.78
1.55
4.76
1.03
4.73
1.71
4.70
5.0
r,,o
l.(;5
5.77
1.75
6.74
1.85
5.71
1 95
5.67
2.05
5.04
6.0
V.o
1.113
0.73
2.05
6.09
2.10
0.06
,.28
f..02
2.3 9
0.58
7.0
8.0
2.21
7.09
2.34
7.05
2.47
7.01
2.00
7.56
2.74
7.52
8.0
n.,;
2.48
8.65
2.03
8.01
2.78
8.56
2.93
8.51
3.08
8.40
9.0
10.0
2.7(5
0.01
2.92
9.56
3.09
0.51
3.0')
9.40
3.42
9.40
lO.O
11.0
3.03
10.57
3.22
10.52
3.40
10.46
3,5X
10.40
3.76
10.34
n.o
12.0
3.31
11.54
3.51
11.48
3,71
11.41
3,91
11.35
4.10
11.28
12.0
13.0
3.58
12.50
3.80
12.43
4.02
12.36
4,23
1::.29
4.45
12.22
13.0
14.0
3.80
13.46
4.09
13.39
4.33
13.31
4.50
13.24
4.7!)
13.10
14.0
15.0
4.13
14.42
4.39
14.34
4.04
14.27
4.88
14. IS
5.13
14.10
15.0
ICO
4.41
15.38
4.08
15.30
4.!I4
15.22
5.21
15.13
5.47
1 J. 04
10.0
n.o
4. CO
10.34
4.97
10.26
5.25
10.17
5.53
16.07
5.81
i5.!»7
17.0
is.O
4.',)li
17.30
5.26
17.21
5.56
17.12
5.80
17.02
0.10
10.91
18,0
10.0
5.24
18.26
5.56
18.17
1
5.87
18.07
0.19
17.96
6.50
17.85
19.0
20.0
5.51
19.23
5. 85
19.13
0.18
19.00
0.51
IS. 91
O.fvl
18.79
20.0
21.0
5. 79
1 20.19
0.14
20.08
0.4 9
1 1!1.'.I7
6.84
19. MO
T.18
; 19.73
21.0
22.0
fi.Oll
21.15
0.43
21.04
0.80
2().!t2
7.10
20.80
7.52
20.07
22.0
23.0
0.34
22.11
0.72
! 22.00
7.11 i 'Jl.87
7.49
21.75
7.87
21.01
23.0
24.0
G.02
23.07
7.02
22.05
1
7.42
22.83
7.81
22.60
8.21
22.55
1
24.0
25.0
O.SO
24.03
7.31
]
23.91
7.73
23.78
8.14
2 ".04
8.55
23.49
25
20.
7.17
24.99
7.00
24.80
S.(K!
2t.73
8.40
'J4.58
8.8!)
24.43
20.0
27.0
7.44
25.05
7.89
25.82
8.34
25.08
S.79
25.53
9.23
25.37
27.0
2S.0
7.72
' 20.92
8,19 20.78
8.05
20.03
9.12
i 20.47
9.58
20.31
28,0
2'J.O
T.it'J
27.88
8.48
27.73
8.90
27.58
j
9.44
27.42
9.92
27.25
29.0
30.0
8.27
28.84
8.77
28.69
9.27
, 28.53
9 7T
28.37
10.26
28.19
30.0
con
sin
cos
sin
co.s
sia
cus
BlU
C08
I sin
+ 286 —
+ 287 —
+ 288 —
+ 2
89 —
+ 290 —
— 254 —
— 253 —
— 252 - -
— 2
51 —
— 250 —
— 106 +
— 107 f
— 108 +
— 1
09 +
— 110 +
+ "74 +
+ 73 +
+ ^2 +
+
71 +
+ 70 +
■dB
284
TABLE OF PKODUCTS OF SINES AND COSINES,
+
21 +
+ 22° +
+ 23' +
+ 24°+
+ 25°+
■i 159 —
+ 158 —
+ 157 —
+ 156 —
+ 155 —
— 201 —
—202 —
— 203 —
— 204 —
— 205 —
— 839 +
— 338 +
— 337 +
— 336 +
— 335 +
sin
cos
Hin
cos
sin
cos
sin
COS
sin
COS
0.1
0.04
0.09
0.04
0.09
0.04
0.09
0.04
0.09
0.04
0.09
0.1
0.2
0.07
0.19
0.07
0.19
0.08
0.18
0.08
0.18
0.08
0.18
0.2
0.3
O.ll
0.28
0.11
0.28
0.12
0.28
0.12
0.27
0.13
0.27
0.3
0.4
0.14
0.31
0.15
0.37
0.10
0.37
0.16
0.37
0.17
0.36
0.4
0.5
O.IS
0.4t
0.19
0.46
0.20
0.46
0.20
0.46
0.21
n.45
0.5
O.G
0.22
0.56
0.22
0.56
0.23
0.55
0.24
0.55
0.25
0.54
0.6
0.7
0.25
0.05
0.20
0.05
0.27
0.04
0.28
0.04
0.30
0.63
0.7
O.S
0.2!)
0.75
0.30
0.74
0.31
0.74
0.33
0.73
0.34
0.73
0.8
0.9
0.32
0.84
0.34
0.83
0.35
0.83
0.37
0.82
0.38
0.82
0.9
1.0
o..3r.
0.93
0..37
0.93
0..S9
0.92
0.41
0.91
0.42
0.91
1.0
2.0
0.72
1.87
0.75
1.85
0.78
1.S4
0.81
1.83
0.85
1.81
2.0
3.0
l.OS
2.80
1.12
2.78
1.17
2.76
1.22
2.74
1.27
2.72
3.0
4.0
1.43
3.73
1.50
3.71
1.56
3.08
1.03
3.65
1.69
3.63
4.0
5.0
1.70
4.67
1.S7
4.04
1.95
4.60
2.03
4.57
2.11
4.63
5.0
().0
2.15
5.60
2.25
5.56
2.34
5.52
2.44
5.48
2.54
5.44
0.0
7.0
2.51
0.54
2.1 '
0.49
2.74
6.44
2.85
6.39
2.96
6.34
7.0
8.0
2.H7
7.47
3.00
7.42
3.13
7.36
3.25
7.31
3.38
7.25
8.0
U.O
3.23
8.40
3. 37
8.34
3.52
8.28
3.06
8.22
3.80
8.16
9.0
10.0
3.58
9.34
3.75
9.27
3.91
9.21
4.07
9.14
4.23
9.00
10.0
U.O
3.94
10.27
4.12
10.20
4.30
10.13
4.47
10.05
4.05
9.97
11.0
12.0
4.30
11.20
4.50
11.13
4.09 11.05
4.88
10.96
6.07
10.88
12.0
13.0
4.(ii>
12.14
4.87
12.05
5.08
11.97
5.29
11.88
5.49
11.78
13.0
14.0
5.02
13.07
5.24
12.98
5.47
12.89
5.69
12.79
6.92
12.69
14.0
15.0
5.3S
14.00
5.02
13.91
5.80
13.81
6.16
13.70
6.34
13.59
15.0
Ki.O
5.73
14.94
5.99
14.83
0.2;.
14.73
6.51
14.02
6.76
14.50
10.0
n.o
(l.O!)
15.S7
0.37
15.70
6.04
15.65
0.91
15.53
7.18
15.41
17.0
IH.O
fi.45
10.80
0.74
10.09
7.03
10.57
7.32
10.44
7.61
10.31
18.0
I'J.O
6.x I
17.74
7.12
17.02
7.42
17.49
7.73
17.36
8.03
17.22
19.0
20.0
7.17
18.07
7.19
18.54
7.81
18.41
8.13
18.27
8.45
18.13
20.0
21.0
7.53
19.01
7.S7
19.47
8.21
19.33
8.54
19.18
8.S7
19.03
21.0
22.0
7.S.S
20.54
8.24
20.40
8. CO
20.25
8.95
20.10
9.30
19.94
22.0
23.0
8.24
21.47
8.li2
2 '..33
8.99
21.17
9.35
21.(»1
9.72
20.85
23.0
24.0
8.00
22.41
8.99
22.25
9.38
22.09
9.70
21.93
10.14
21.75
24.0
25.0
8. or,
23.34
9.37
23.18
9.77
23.01
10.!7
22.84
10.57
22.66
25.0
2(1.0
9.32
24.27
9.74
24.11
10.10
23.93
10.58
23 75
10.99
23.58
20.0
27.0
O.C.S
25.21
10.11
25.03
10.55
24.85
10.98
24.07
11.41
24.47
27.0
2S.0
10.03
20.14
10.49
25.96
10.94
25.77
11.39
25.58
11.83
25.38
28.0
2i».0
10.39
27.07
10.8(1
26.89
11.33
20.69
11.80
20.49
12.26
26.28
29.0
30.0
10.75
) 28.01
11.24
27.82
11.73
27.62
12.20
27.41
12.68
27.19
30.0
cos
.sin
co.s
sin
cos
sin
cos
sin
COS
sin
6 —
+ 2
91 —
+ 2!
?2 —
+ 293 —
+ 29
4 —
+ 2£
— 2-
49 —
-2'
18 —
— OAl —
— 24
[B —
— 24
t5 -
— 1
11 +
— 1
12 +
— 113 +
— 11
4 +
— 11
6 +
+
89 +
+ i
J8 +
+ 67 +
+ e
8 +
-r €
15 +
TABLE OP PRODUCTS (
OF SINES
AND
COSINES.
285
+ 26"+
+ 27" +
+ 28°+
+ 29" +
+ 30'+
+ 154 —
+ 153 —
+ 152 —
+ 151 —
+ 150 —
— 206 —
— 207 —
— 208 —
— 209 —
— 210 —
— 334 +
— 333 +
— 332 +
— 331 +
— 330 +
sin
cos
sin
cos
sin
cos
sin
cos
sin
cos
0.1
0.04
0.00
0.05
0,00
0.05
0.00
0.05
0.09
0.05
0.09
0.1
0.2
0.09 :
0.18
0.00
0.18
0.00
0.18
0.10
0.17
0.10
0.17
0.2
0.:^
0.13 :
0.27
0.14
0.27
0.14
0.26
0.15
0.26
0.15
0.26
0.3
0.4
O.IS
0.36
0.18
0.36
0.10
0.35
0.10
0.35
0.20
0.35
0.4
0.5
0.22
0.45
0.23
0,45
0.23
0.44
i
0.24 !
0.44
0.25
0.43
0.5
O.fi
0.26 1
0.54
0,27
0,53
0.28
0.53
0.20
0.52
0.30
0.52
0.6
0.7
0.!J1
0.63
0.32
0.62
0.33
0.62
0.34
0.61
0.35
O.Cl
0.7
0.8
0.35
0.72
0.36
0.7i
0.38
0.71
0.30
0.70
0.40
0.69
0.8
0.9
0.39
O.sl
0.41
0.80
0.42
0.79
0.44
0.70
0.45
0.78
0.0
1.0
0.44
0.00
0.45
0.80
0.47
0.88
0.48
0.87
0.50
0.87
1.0
2.0
O.SH
1.80
0.01
1.78
o.ot
1.77
0.07
1.75
1.00
1.73
2.0
y.o
1.32
2.70
1.36
2.67
1.41
2.65
1.45
2.62
1.50
2.60
3.0
4.0
1.75
3.60
1.82
3.56
1.88
3.53
1.04
3.50
2.00
3.46
4.0
5.0
2.10
4.40
2.27
4.46
2.35
4.41
2.42
4.37
2.50
4.. 33
50
c.o
2.(53
5.30
2.72
5.35
2..S2
5.. 30
2.01
5.25
3.00
5.20
6.0
t.o
3.07
6.20
3.18
6.24
3.20
6.18
3.30
6.12
3.50
6.06
7.0
s.u
3.51
7.10
3 63
7.13
3.711
7.06
3.88
7.00
4.00
6.03
8.0
9.0
3.0,".
8.09
4.00
8.02
4.23
7.05
4.36
7.87
4.50
7.70
0.0
10.0
4.3^^
8.00
4.51
8.01
4.60
8.83
4.85
8.75
5.00
8.66
10.0
11.0
4.S2
0.80
4.00
0.80
5.16
0.71
5.33
9.62
5.50
0.53
11.0
12.0
5.2(1
10.79
5.45
10.69
5.63
10.60
5.82
10.50
6.00
10.30
12.0
13.0
5.70
•
11.68
5.00
11. 5S
6.10
11.48
6.30
11.37
6.50
11.26
13.0
14.0
6.14
12.58
6.36
12.47
6.57
12.36
C.79
12.24
7.00
12.12
14.0
l.'i.O
fi.58
13.48
fi.Sl
13.37
7.04
13.24
7.27
13.12
7.50
12.09
15.0
)(>.0
7.01
14.38
7.26
14.26
7.51
14.13
7.76
13.00
8.00
13.86
16.0
n.o
7.45
15.28
7.72
15.15
7.08
15.01
8.24
14.87
8.50
14.72
17.0
IH.O
7. SO
16.18
8.17
16.04
8.45
15.89
8.73
1.5.74
9.00
15.50
18.0
19.0
8.33
17.08
8.63
16.03
8.92
16.78
9.21
16.62
9.50
16.45
10.0
20.0
8.77
17.08
O.OS
17.82
0.30
17.66
0.70
17.40
10.00
17.32
20.0
21.0
0.21
: 18.87
0,53
18.71
0..'S(1
is.,'^4
10.18
18.37
10.50
18.10
21.0
22.0
0.(14
10.77
0.00
10.60
10.33
10.42
10.67
10.24
11.00
10.05
22.0
2:1.0
10. OS
20.67
10.44 ■ 20.40
10.80
20.31
11.15
20.12
11.50
10.02
23.0
24.0
10.52
21.57
10.00
21.38
11.27
; 21.10
11.64
20.09
12.00
20.78
24.0
25.0
10.06
22.47
11.35
22. 2S
11.74
22.07
12.12
21.87
12.50
21.65
25
2li.0
11.40
23.37
11.80 23,17
12.21
22.06
12.61
\ 22.74
13.00
22.52
26.0
27.0
11. S4
24.27
12.26 24.06
12.68
23.84
13.00
23.61
13.50
23.38
27,0
2H.0
12.27
25,17
12.71 24.05
13.15
24,72
13.57
24.40
14.00
24.25
2S.0
2'.).0
12.71
26.07
13.17
25.84
13.61
25,61
1
14.06
25.36
14.50
25.11
20.0
30.0
13.15
26.06
13.62
i
' 26.73
14.08
! 26.49
14. .M
26.24
15.00
25.08
30.0
CO..!
.sin
96 —
cos
sin
cos
sin
cos
sin
cos
sin
+ 2
+ 297 —
+ 298 —
+ 299 —
+ 300 —
—3
44 —
— 243 —
— 242 —
— 241 —
— 240 —
— ]
Lie +
— 117 +
— 118 +
-119 +
— 120 +
+
64 +
+ 63 +
+ 62 +
+ 61 +
+ 60 +
286
TABLE OF niODUCTS OF SIXES AND COSINES.
t
1 1 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2.0
8.0
4.0
5.0
(i.O
7.0
«.0
!».0
10.0
11.0
12.0
l.f.O
14.0
15.0
in.o
17.0
IH.O
19.0
20.0
21.0
22.0
2:t.O
24.0
25.0
2tl.O
27
2S.0
29.0
30.0
+ 31° +
+i4r —
— 211 —
—329 +
sm
0.05
0.10
0.15
0.21
0.20
0.31
0.30
0.41
0.40
0.52
1.03
1.55
2.01!
2.5S
3.09
3.01
4.12
4. 04
5.15
5.(;7
fi.lS
0.70
7.21
7.73
8.24
8.70
9.27
9.79
10.30 I
li).X2
11.33
il.85
12.30
12.88
13.39
13.91
14.42
14.94
15.45
(■C).S
+ 32°+
+148 —
—212 —
—328 +
cos
0.09
0.17
0.26
0.34
0.43
0.51
0.00
0.09
0.77
O.Sfi
1.71
2.57
3.43
4.29
5.14
6.00
0.86
7.71
17.14
18.00
18.86
19.71
20.57
21.43
22.29
23.14
24.00
24.86
25.71
sin
+ 301°—
—239 —
— 121 +
+ 59 +
sm
0.05
0.11
0.16
0.21
0.26
0.32
0.37
0.42
0.48
0.53
1.06
1.59
2.12
2.65
3.18
3.71
4.24
4.77
5.30
5. S3
6.36
89
42
95
18
01
I
0.08
0.17
0.25
0.34
0.42
0.51
0.59
0.68
0.76
0.85
1.70
2.54
3.39
4.24
5.09
5.94
6.78
7.63
8.48
9.33
10.18
11.02
11.87
12.72
13.57
14.42
15.26
9.54
10.07 16.11
10.60
11.13
11.66
12.19
12.72
13.25
13.78
I4.:!i
14.84
15.37
16.96
17.81
18 lu;
19.51
20.35
21.20
22.05
22.90
23.75
24.59
15.90 25.44
co.s sin
+ 302 —
—238 —
— 122 +
+ 58 +
+ 33°+
+147 —
—213 —
—327 +
sill
0.05
0.11
0.16
0.22
0.27
0.33
0.38
0.44
0.49
CU9
I 0.08
0.17
0.25
0.34
0.42
0.50
0.59
0.67
0.75
0.84
1.68
2.52
3.35
4.19
5.03
5.87
6.71
7.55
H.39
9.23
10.06
10.90
11.74
12.58
13.42
14.26
15.10
10.35 : 15.93
5.45
5.99
6.54
7.08
7.62
8.17
H.71
9 26
9.80
10.89
11.44
11.98
12.53
13.07
13.62
14.16
14.71
15.25
15.79
16.77
17.61
1.S.45
19.29
20.13
20.97
21.81
22.64
23.48
24.32
16.31 25.16
C(is sill
+303 —
—237 —
— 123 +
+ 57 +
+ 34°+
+ 148 —
—214 —
— 326 +
0.06
0.11
0.17
0.22
0.28
0.34
0.39
0.45
0.50
0.56
1.12
1.68
2.24
2.80
3.36
3.91
4.47
5 03
5.
6.
6
7.
7.
8.
8.
9.
10.
10
59
15
71
,27
83
39
95
51
07
62
11.18
11.74
12.30
12.86
13.42 ' 19.90
cos
0.08
0.17
25
0.33
0.41
0.50
0.58
0.66
0.75
0.83
1 . 66
2.49
3.32
4.15
4.97
5.. 80
6.63
7.46
8,29
9. 1 2
9.95
10.78
11.61
12.44
13.26
1(.09
14.92
15.75
16.58
17.41
18.24
19.07
13.98
14.54
15.10
15. (',6
16.22
20.73
21.55
22.38
23.21
24.04
16.78 1 24.87
COS sin
+ 304 —
— 236 —
— 124 +
+ 58 +
+ 35°+
+ 145 —
— 215 —
— 325 +
0.06
0.11
0.17
0.23
0.29
0.34
0.40
0.46
0.52
0.57
1.15
1.72
2.29
cos
0.08
0. 1 6
0.25
0.33
0.41
0.49
0..57
0.66
0.74
0.82
1.64
2.46
3.28
2.87 '■ 4.10
3.44 i 1.91
4.02 i 5.73
4.59 ! 6.55
6.16 ' 7.37
5.74
6.31
6.S8
7.46
8.03
8.60
9.18
9.75
10.32
10.90
11.47
12.05
12.62
13.19
13.77
8.19
9.01
9. S3
10.65
11.47
12.29
13.11
13.93
14.74
15.56
16.38
17.20
is. 02
1S.,S4
19.66
14.34 I 20.48
14.91 i 21.30
15.49 22.12
16.06 22.94
16.63 23.76
17.21
24.57
sin
+ 305 —
—235 —
— 125 +
+ 55 +
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
IS.O
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.
29.0
30.0
TABLE OF PK0DUCT8 OF SINES AND COSINES.
287
+ 36°+
+ 37°+
+ 38°+
+ 39°+
+ 40°+
+ 144 —
+ 143 —
+ 142 —
+ 141 —
+ 140 —
— 216 —
— 217 -
— 218 —
— 219 —
— 220 —
— 324 +
— 323 +
— 322 +
— 321 +
— 320 +
sin j
cos
sin
cos
sin
cos
sin
cos
sin
co.s
0.1
0.00
0.08
0.00
0,08
0.00
0.08
0.00
0,08
0.00
0.08
0.1
0.2
0.12
0.10
0.12 1
0,10
12
0.10
0.13 !
0.10
0.13
0.15
0.2
0.3
0.18
0.24
0.18 !
0,24
0.18
0.24
0.19
0.23
0.19
0.23
0.3
0.4
0.24
0.32
0.24
0.32
0.25
0.32
0.25
0.31
0.20
0.31
0.4
0.5
0.20
0.40
0.30
0.40
0,31
0.39
0.31
0.30
0.32
0.38
0.5
0.0
o.^;'^ ;
0.49
0.30
0.48
0.37
0.47
0.38
0.47
0.39
0.40
0.0
0.7
0.41 >
0,57
.0.42
0.50
0.43 i
55
0.44 ■
0.54
0.45
0.54
0.7
0.8
0.47
0.05
0.48 i
0,04
0.40
0.03
0.50
0,02
0,51
0.01
0.8
O.U
o.5:j .
0.73
0.54
0.72
0.55
0.71
0.57
0.70
0.58
0.09
0.9
1.0
0.59 i
0.81
0.00
0.80
0.02
0.79
0.03
0.78
0.04
0.77
1.0
2.0
1.18 :
1.02
1.20
1.00
1.23
1.58
1.20 ;
1.55
1.29
1.53
2.0
;{.o
1.70
2.43
1.81
2.40
1.85
2.30
1,80
2,33
1.93
2.30
3.0
4.0
2.^5
3.24
2.41
3.19
2.40
3.15
2,52
3.11
2.57
3.06
4.0
5.0
2.04
4.05
3.01
3.09
3.08
3.04
3.15
3,80
3.21
3.83
5.0
0.0
;{.:):!
4.85
3.01
4.79
3.(;o
4,73
3.78
4.00
3.86
4.00
6.0
7.0
4.11
5.00
4.21
5.59
4.31
5,52
4.41
5.44
4.50
5.30
7.0
8.0
4.70
0.47
4.81
0,39
4.93
0,30
5.03
0.22
5.14
6.13
8.0
9.0
5.29
7.28
5.42
7.10
5.54
7,09
5.00
0.90
5.79
0.89
9.0
10.0
5. 88
8.00
0.02
7.99
0.10
7.88
6.29
7.77
6.43
7.00
10.0
11.0
0.47
8.90
0.02
8.78
0.77
8.07
0,92
8.55
7.07
8.43
11.0
li'.O
7.05
9.71
7.22
9.58
7.39
9.40
7.55
9.33
7.71
0.10
12.0
l.i.O
7.04
10.52
7.82
1(1.38
8.00
10.24
8.18
10.10
8.30
9,90
13.0
14.0
8.23
11.33
8.43
11.18
8.02
11.03
8.81
10.88
9.00
10,72
14.0
l.'i.O
8.82
12.14
9.03
11,98
0.23
11.82
0.44
11.00
9.04
11.49
15.0
10.(1
9.40
12.94
9.03
12.78
9.85
12.01
10.07
12.43
10.28
12.20
10.0
n.o
9.09
13.75
10,23
13.58
10,47
13.40
10.70
13.21
10.93
13.02
17.0
IH.O
10. 5S
14.50
10.83
14.38
11.08
14.18
11.33
13.99
11.57
13.79
18.0
19.0
11.17
15.37
11.43
15.17
11.70
14.97
11.90
14.77
12.21
14.55
19.0
20.0
11.70
10.18
12.04
15.07
12,31
15.70
12.50
15.54
12.80
15.32
20.0
21.0
12.34
10.99
12.04
10.77
12.93
1(;.55
13.22
10.32
13.50
10.09
21.0
22.0
12.0.{
17.80
13,24
17.57
13.54
17.34
13.85
17.10
14.14
10.85
22.0
2.'!.0
1.!.52
18.(il
1.3.84
' 18.37
14.10
18.12
14.47
17.87
14,78
17.02
23.0
24.0
14.11
19.42
14.44
10.17
14.78
; 18.91
15.10
18.(i5
15.43
18.39
24.0
2:").0
14.09
20.23
15.05
10,97
15.30
19.70
15.73
19.43
10.07
19.15
25.0
21).
15.28
21.03
15.05
2(1.70
IC.dl
20.49
10.30
20.21
10.11
19.92
20.0
27.0
15.87
21.84
10.25
21.50
10.02
21. 2«
10.09
i 20.08
17.30
20.08
27.0
2^.0
1(').40
22.05
10.85
22,30
17.24
22,(10
17.02
21.70
18.00
21.45
28.0
29.0
17.05
23.40
17.45
23.10
17.85
22,85
\
18.25
22.54
18.04
22.22
20.0
30.0
17.03
24.27
18.05
23.90
18.47
23.04
18.88
23.31
19.28
22.98
30.0
cog
sin
cos
sin
(,'os
sin
cos
sin
cos
1 sin
+ 308 —
+ 307 —
+ 308 —
+ 309 —
+ 310 —
— 234 —
— -233 —
— 232 —
— 231 —
— 230 —
— 126 +
— 127 +
— 128 +
— 129 +
— 130 +
+ 54 +
+ 53 +
+ 52 +
+ 51 +
+ 50 +
388
fir :!
■ ^ :'
TABLE OF PRODUCTS OF SINES AND COSINES.
+ 41°+
+ 42°+
+ 43°+
+ 44°+
+ 4B°+
+ 139 —
+ 138 —
+ 137 —
+ 136 —
+ 135 —
221 —
—222 —
— 223 —
—224 —
—225 —
— 819 +
— 318 +
— 817 +
— 318 +
— 315 +
sin
C03
sill
cos
sill
cos
sin
cos
sill
cos
0.1
0.07
0.08
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.1
0.2
0.13
0.15
0.13
0.15
0.14
0.15
0.14
0.14
0.14
0.14
0.2
0.3
0.20
0.23
0.20
0.22
0.20
0.22
0.21
0.22
0.21
0.21
0.3
0.4
0.20
0.30
0.27
0.30
0.27
0.29
0.28
0.29
0.28
0.28
0.4
0.5
0:33
0.38
0.33
0.37
0.34
0.37
0.35
0.30
0.35
0.35
0.5
O.fi
0.39
0.45
0.40
0.45
0.41
0.44
0.42
0.43
0.42
0.42
0.6
0.7
0.46
0.53
0.47
0.52
0.48
0.51
0.49
0.50
0.49
0.49
0.7
0.8
0.52
0.00
0.54
0.59
0.55
0.59
0.50
0.58
0.57
0.57
0.8
0.9
0.59
0.68
00
0.67
0.01
0.06
0.63
0.65
0.04
0.64
0.9
1.0
O.fifi
0.75
0.67
0.74
0.08
0.73
0.69
0.72
0.71
0.71
1.0
2.0
1.31
1.51
1.34
1.49
1.30
1.40
1..39
1.44
1.41
1.41
2.0
3.0
1.97
2.26
2.01
2.23
2.05
2.19
2.08
2.16
2.12
2.12
3.0
4.0
2.C2
3.02
2.08
2.97
2.73
2.93
2.78
2.88
2.83
2.83
4.0
5.0
3.28
3.77
3.35
3.72
3.41
3.00
3.47
3.60
3.54
3.54
5.0
fi.O
3.94
4.53
4.01
4.46
4.09
4.39
4.17
4.32
4.24
4.24
6.0
7.0
4.5'.)
5.28
4.08
5.20
4.77
5.12
4.80
5.04
4.95
4.95
7.0
8.0
5.25
6.04
5.35
5.95
5.40
5.85
5.56
6.75
5.60
5.66
8.0
9.0
5.90
0.79
0.02
0.69
6.14
6.58
6.25
(;.47
0.30
0.36
9.0
10.0
6.50
7.55
6.09
7.43
6.82
7.31
6.95
7.19
7.07
7.07
10.0
11.0
7.22
8.30
7.36
8.17
7.50
8.04
7.64
7.91
7.78
7.78
11.0
12.0
7.87
9.06
8.03
8.92
8.18
8.78
8.34
8.63
8.49
8.49
12.0
l:i.0
8.53
9.81
8.70
9.60
8.87
9.51
9.03
9.35
9.19
9.19
13.0
14.0
9.18
10.57
9.37
10.40
9.55
10.24
9.73
10.07
9.90
9.90
14.0
l.i.O
9.84
11.32
10.04
11.15
10.23
10.97
10.42
10.79
10.61
10.61
15.0
1(!.0
10.50
12.08
10.71
11.89
10.91
11.70
11.11
11.51
11.31
11.31
16.0
17.0
11.15
12.83
11. 3S
12.03
11.59
12.43
11.81
12.23
12.02
12.02
17.0
18.0
11.81
13.58
12.04
13.38
12.28
13.16
12.50
12.95
12.73
12.73
18.0
19.0
12.47
14.34
12.71
14.12
12.90
13.90
13.20
13.67
13.44
13.44
19.0
20.0
13.12
15.09
13.38
14.86
13.04
14.03
13.89
14.39
14.14
14.14
20.0
21.0
13 78
15.85
14.05
15.01
14.32
15.30
14.59
15.11
14.85
14.85
21.0
22.0
14.43
10.60
14.72
10.35
15.00
16.09
15.28
15., S3
15.50
15.56
22.0
23.0
15.09
17.36
15.39
17.09
15.09
10.82
15.98
16.54
10.20
16.26
23.0
24.0
15 75
18.11
16.06
17.84
16.37
17.55
16.67
17.20
10.97
16.97
24.0
25.0
16.40
18.87
16.73
18.58
17.05
18.28
17.37
17.98
17.08
17.68
25.0
2fi.0
17.06
19.62
17.40
19.32
17.73
19.02
18.00
18.70
18.38
IS. 38
20.0
27.0
17.71
20.38
18.07
20.06
18.41
19.75
18.70
19.42
19.00
19.09
27.0
28.
18.37
21.13
18.74
20.81
19.10
20.48
19.45
20.14
19. SO
19.80
28.0
29.0
19.03
21.89
19.40
21.55
19.78
21.21
20.15
20.80
20.51
20.51
29.0
30.0
19.08
22.64
20.07
22.29
20.40
21.94
20.84
21.58
21.21
21.21
80.0
cos
sin
cos
sin
cos
sin
cos
Kin
cos
sin
+ 3]
LI —
+ 31
2 —
+ 313 —
4 31
4 —
+ 3]
5 —
— 22
3 —
— 22
8 —
— 227 —
— 22
6 —
— 2S
5 —
— 15
J1 +
-13
2 +
— 133 +
— 13
4 +
— 13
5 +
+ 4
Q +
+ 4
8 +
+ 47 +
+ 4
6 +
+ 4
5 +