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SMITHSONIAN CONTRIBUTIONS TO KNOWLEDGE.

. 199

AN

INVESTIGATION

OF THE

ORBIT OF NEPTUNE,

WITH GENERAL TABLES OF ITS MOTION.

BY

SIMON NEWCOMB,

PROFESSOR OF MATHEMATICS, UNITED STATES NAVY.

[ACCEPTED FOB PUBLICATION, MAY, 1865.]

COMMISSION

TO WHICH THIS PAPER HAS BEEN REFERRED.

Admiral C. H. DAVIS, U. S. N.
Prof. STEPHEN ALEXANDER.

JOSEPH HENRY,

Secretary S. I.

COLLINS, PRINTKB,
PHILADELPHIA.

TABLE OF CONTENTS.

CHAPTER I.

INTRODUCTION.

SECT. PAGB

1. Introductory remarks . . 1

2. Account of Walker's theory ....... 1

3. Account of Kowalski's theory ....... 3

4. Form of Kowalski's equations of condition, and origin of the difficulties arising

from it .......... 4

5. Objects of the present investigation ...... 6

CHAPTER II.

PROVISIONAL THEORY OF NEPTUNE.

6. Formulas for the perturbations of longitude and radius vector ... 8

7. Formulas for the perturbations of latitude ..... 10

8. Secular variations ......... 13

9. Theory of the action of an inner on an outer planet through the Sun . 13

10. Development of the preceding theory according to the powers of the ratio of the

mean motions ......... 16

11. Method of treating the long-period perturbations of the elements produced by

Uranus .......... 19

12. Adopted elements, masses, and constants of theory, for perturbations . 20

13. Computation of the perturbations by Uranus, Saturn, and Jupiter . . 22
Action of Venus, the Earth, and Mars ...... 31

14. Indirect perturbations by Saturn .... 31

15. Collection of the long-period and secular perturbations of the elements . 32

16. Collection of the perturbations of the co-ordinates — Comparison with Peirce and

Kowalski ...... 33

17. Formulas for computing an ephemeris . . 36

18. Elimination of the elliptic terms ...... 36

19. Elements and formulas of the provisional theory . . . . .38

20. Heliocentric and geocentric positions resulting from the provisional theory . 41

CHAPTER III.

DISCUSSIONS OF OBSERVATIONS OF NEPTUNE.

21. Choice of observations, and method of discussing them . . 44

22. Reduction of Lalande's observations, May 8-10, 1795 . 45

23. Probable error and value of the Lalando positions . • . .49

24. Method of treating the modern observations . . . 49

25. Mean corrections of ephemerides of Neptune, given by different observatories . 51

26. Investigation of the systematic differences between the results of different obser-

vatories . ..... 53

i TABLE OF CONTENTS.

SEPT. PAGE

27. Concluded Normal Eight Ascensions and Declinations . . . 5(J

Systematic discrepancies still remaining between different authorities . 61

. 28. Longitudes and latitudes compared with the theory . . . .62

CHAPTER IV.

RESULTS OP THE COMPARISON OF THE THEORETICAL WITH THE OBSERVED POSITIONS OP

NEPTUNE.

29. Formulas for corrections of the elements ..... 65

30. Equations of condition — Method of treating them . ... . .67

31. Solution of the equations — Residual errors ..... 70

32. Impossibility of correcting the mass of Uranus . . . . .72

33. Impossibility of yet detecting an extra-Neptunian planet — Almost perfect accord-

ance of theory with observation during the nineteen years of observations . 72

34. Position of the plane of the orbit ...... 73

35. Concluded corrections and final values of the elements of Neptune . . 74

CHAPTER V.

TABLES OP NEPTUNE.

36. Fundamental theory on which the tables are founded ... 76

37. Data given in the several tables ...... 77

38. Elementary precepts for the use of the tables ..... 82

39. Examples of the use of the tables ....... 84

Tables 88-110

ON THE ORBIT OF NEPTUNE.

CHAPTER I.

INTRODUCTION.

THE errors of the published ephemerides of Neptune are now increasing very
rapidly. In 1SG3, Walker's ephemeris was in error by 33", and Kowalski's by
22". Both ephemerides may be 5' in error before the end of the present century.
The orbit of this planet is, therefore, more uncertain than that of any other of
the larger members of our system. The uncertainty arises from the insufficiency
of the data at the command of those astronomers who have hitherto investigated
the motions of this planet. These motions are so slow that it is impossible to
determine the elements of the orbit with accuracy from observations extending
through only a few years. In Walker's investigations the errors of observation
are multiplied more than a hundred times in the elements deduced from them, on
account of the smallness of the arc through which the planet had moved.

The time has now come when the orbit can be determined with some approach
to accuracy. The planet has moved through an arc of nearly 40° since its dis-
covery, and the errors of observation will be multiplied only ten or twelve times
in the errors of the elements. In commencing the work of a revision of the theory
of Neptune, it will be well to glance at the past and present state of our know-
ledge on this subject.

Approximate elements of this planet, neglecting the effect of perturbations,
were computed by several astronomers within a year or two after its discovery.
But the work of preparing a theory which should include the perturbations
produced by all the other planets seems to have been left entirely in the hands
of Professor Peirce and Mr. Sears C. Walker.

§ 2. All the first approximations to the elements showed that the mean motion
was very nearly half that of Uranus. It was, therefore, for some time doubtful
whether the mutual action of the two planets might not be such as to render the
period of Neptune exactly double that of Uranus, and thus present us, on a much
grander scale, with a phenomenon similar to that exhibited by the satellites of
Jupiter. Professor Peirce's first perturbations of Neptune were computed on this
hypothesis, and published in the Monthly Notices of the Royal Astronomical
Society, Vol. VIII, p. 40. The eccentricity of Neptune was neglected, but that
of the disturbing planets was included in the perturbations.

With these perturbations, the ancient observations of Lalande, and the vast
number of modern observations made in nearly every active observatory in the
world during 184G and 1847, Mr. Walker computed his "Elliptic Elements I." of

1 May, 1865. 1

2 THE ORBIT OF NEPTUNE.

Neptune. The longitude of perihelion referred to the mean Equinox of Jan. 1,
1847, eccentricity, and mean daily motion were as follows :

n = 48° 21' 2".93
e — .00857741.
n - 21".55448.

This mean motion rendered it certain that the supposed relation between the
mean motions of the planets Uranus and Neptune had no foundation in fact.
Professor Peirce thereupon revised his theory, and published the new perturb-
ations in the Proceedings of the American Academy, Vol. I, p. 286.

The near approach to commensurability of the mean motions renders the
general theory of the mutual action of Uranus and Neptune extremely complex.
Twice the mean motion of the latter exceeds that of the former by only 320"
according to Walker, or 304" according to my first revision of his elements. The
terms in the perturbations which contain this very small quantity as a divisor
will, therefore, be very large. Considered as perturbations of the elements, their
period will be more than 4000 years. We have an analogous instance in the 900
year equation of Jupiter and Saturn. But in the latter case the perturbations
of the mean motion are of the third order with respect to the eccentricities and
inclinations, while in Uranus and Neptune they are of the first order. From this
circumstance it happens that, notwithstanding the smaller masses of the dis-
turbing planets, the perturbation of the mean motion is as great in the case of
the planets in question as in that of Jupiter and Saturn, and that of the other
elements enormously greater. In fact, the perihelion of Neptune oscillates through
a space of eight degrees in consequence of the terms in question. Such a perturb-
ation as this, four degrees on each side of the mean, is, I think, found nowhere
else in our system. Moreover, a change of 1" in the mean motion of the planet
will produce a change of nearly 2' in the coefficient of this perturbation. Any
attempt to determine its magnitude with accuracy will, therefore, be hopeless.

But the difficulties connected with these terms can be avoided in the case of a
theory which is designed to be exact for a period of only a few centuries. Not-
withstanding the great magnitude of the general integrals of the perturbations,
if we take these integrals between limits not exceeding a couple of centuries, we
shall find them so small as not to involve serious difficulty. Their effect on the
co-ordinates can then be developed in powers of the time, and the values thus
obtained will not be subject to any uncertainty of moment. This is substantially
the course adopted by Professor Peirce. He says of the terms in question :

" These coefficients will vary very sensibly by a change in the value of the
mean motion of Neptune, arising from a more accurate determination of its orbit.
But the principal effect of these terms can for a limited period, such as a century,
for instance, be included in the ordinary forms of elliptic motion, and the residual
portion will assume a secular form which is no more liable to change from a new
correction of the mean motion of Neptune than the other small coefficients of
the equations of perturbations."

Accordingly, subducting from the terms in question a series of expressions

THE OKBIT OF NEPTUNE. 3

which would result from arbitrary changes in the elliptic orbit, there is left a
small residual, mostly developed in powers of the time, and only amounting to a
few seconds in a century, which alone is retained.

With the new perturbations, and revised normal places of Neptune, Mr. Walker
obtained the following final set of elements, which he denominated Elliptic Ele-
ments II.:

7i = 47° 12' 6".50
£1= 130 4 20 .81
= 328 32 44 .20
1 46 58 .97.
e =.00871946.
It — 21".55448.
Epoch, Jan.l, 1847.

From these elements and perturbations we have a continuous ephemeris of
Neptune since the time of its optical discovery. From 1846 till 1851 inclusive,
this ephemeris is found in the Appendix to Vol. II of the Smithsonian Contri-
butions to Knowledge ; for 1852, in Vol. Ill of the same series, and also in the
Astronomical Journal; and for subsequent years, in the American Ephemeris and
Nautical Almanac.

All the modern observations on which these elements were founded were made
in the yetars 1846-47, while the planet was moving over an arc of only two and
a half degrees. Considering that the complete determination of the elements
requires, effectively, four observed longitudes, all in different parts of the orbit,
and that three of these positions are included in a space of less than three degrees,
it must be admitted that an accurate determination of the elements was, under
the circumstances, impossible, owing to the imperfections of the observations. As
already remarked, the errors of observation would be multiplied several hundred
times in the elements. Hence, with the best possible observations, the elements
would be uncertain by one or more minutes. But the observations themselves
were mainly differential ones; and it is very doubtful whether the positions of the
stars of comparison were as well determined as the position of the planet itself
could be determined by a series of good meridian observations.

§ 3. The theory of Neptune was next taken up by Professor Kowalski, of the
University of Kasan. His work was published under the title of " Eeclierc1t.es
sur les mouvements de Neptune, suivSes des tables de cette planete, Kasan, 1855."
The long-period perturbations of the elements are here developed, in their general
form, as perturbations of the co-ordinates. There are, therefore, a much larger
number of terms having large coefficients in this theory than in that of Professor
Peirce.

Owing to this change in the form of the perturbations, the two theories cannot
be directly compared. But the ephemerides resulting from each theory can be
compared directly with observation, and corrections of the elements thence ob-
tained. It is thus found that the elements in question require, approximately,
the following corrections in order that the ephemerides may agree with obser-
vations to 1863 :

4 THE ORBIT OF NEPTUNE.

Theory of Walker. Theory of Kowalski.

%n _ 4° 11' • _ 4° 12' "

Se — 0 0 52 — 0 0 51

& — 0 3 6 2 53

Sn 8.4 8.5

Thus, it seems that the theory of Kowalski is, on the whole, no nearer the truth
than that of Walker, although it was founded on observations up te 1853, when
the planet had moved through an arc of sixteen degrees since its optical discovery.*
The cause of this failure to derive a more accurate result is an accidental mistake
in the computation of the perturbations of the radius vector by Jupiter, as I have
more fully pointed out in the Monthly Notices of the Royal Astronomical Society
for December, 1864.

§ 4. The form which Professor Kowalski finds his equations of condition to
assume is illustrative of an interesting and important principle of the method of
least squares. By the comparison of his provisional theory with observations,
forty-four equations of condition are obtained for the corrections of the four
elements 7t, e, e, and n. It is then inquired whether it is possible to determine
the orbit of Neptune from the modern observations alone, omitting that of La-
lande, the planet having moved through an arc of sixteen degrees. Treating
the equations derived from the modern observations alone by the method of least
squares, four normal equations are obtained. Two of these equations are, omitting
the terms involving the correction of the mass of Uranus, which we do not need,

— 10.4994 jn — 21.2GG1 8s + 13.0QS8 e%n + 40.2211 «e - — 324".65,
26.9661 3n — 73.2702 & + 40.2211 e&n + 139.9907 fe = — 886 .63,

and the other two can be transformed into the following :

— 10.4994 in — 21.2661 be + 13.0073 efa + 40.2219 & = — 324.50,

— 26.9661 hi — 73.2702 fc + 40.2219 efa + 140.0009 be = — 886.77.

It will be seen that the last two equations are very nearly identical with the
first two. Hence it is concluded that the modern observations alone give only
two independent relations between the four unknown quantities sought, and do
not suffice, therefore, to determine the elements of Neptune.

Now, the identity in question does not prove that the modern observations are
insufficient to determine the elements, because it is the necessary result of tltc ///<>i/r.
of treating equations of the kind in question by tlie meffuod of least M/tuircx. This
can be most easily shown by a theorem in determinants. By the elementary
principle of determinants, if we have a number of linear equations between the
same number of unknown quantities, of the form

* The differences of the two values of An and Se, which are so small, do not correctly represent the absolute
differences of the two theories, owing to the great difference of longitude of perihelion in the two theories
proceeding from the different forms given to the perturbations. The real difference Kowalski — Walker is

given Ijy the equations

<).e8in7r=+ I",

fi.e cos it = — 13.

THE ORBIT OF NETTUNE.

i» + % + CjZ + etc ..... = «!,
«2x + % + c.2z + etc ..... rr n.,,
etc* etc. etc. etc. etc.;

each unknown quantity is given in the form

etc.,

in which R represents the determinant formed from all the coefficients a, b, etc.
in the given equations, and A1} A.2, etc. the partial determinants, obtained by
omitting column a, row 1, column a, row 2, etc.

If, now, the number of equations is greater than that of the unknown quantities,
and they are solved by the method of least squares, the form of the solution will
be the same as the above, except that for R will be substituted the sum of the
squares of all the determinants R, formed by solving separately every combination
of such number of the given equations as is equal to the number of unknown
quantities, and for Al} A.2, etc., certain powers and products of the partial deter-
minants which enter into the separate solutions. Hence, if these determinants
are very small, the corresponding determinants in the solution by least squares
will be very small quantities of the second order. But the determinants will all
be very small if the equations are nearly equivalent to a number less than that
of the unknown quantities ; that is; if they can be put into the form

rp

aX+ (3Y+yZ+ etc. + p = n4,
a'X+ @'Y+ yZ+ztc. + p' = «6,
etc. etc. etc. etc. etc. etc.;

the quantities X, Y, Z, etc. being less in number than the unknown quantities,
and p being a very small linear function of the unknown quantities. If the p's
vanish, all the determinants will vanish with it ; whence, if they are very small,
the determinants will be very small likewise. Calling a system of equations
identical when they really give fewer independent relations than there are un-
known quantities, the theorem sought may be expressed as follows :

If a system of equations differ from identity by a very small quantity, the normal
equations derived from them ivill be identical to small quantities of tlie second order.

Hence, if such a system of equations is to be solved by least squares, it will be
necessary to carry the solution to nearly twice as many decimals as are necessary
in the original coefficients. Thus, in the case under consideration, as Professor
Kowalski considered it necessary to retain four places of decimals in the coefficients
of the unknown quantities, it would have been necessary to include at least six
or seven decimals in the normal equations, instead of only four.

But the necessity for so long a numerical calculation can be avoided by a suitable
transformation of the equations of condition. If the equations are identical,
they really give certain linear functions of the unknown quantities less in number

6 THE ORBIT OF NEPTUNE.

thcin the unknown quantities. We may then substitute these linear functions
themselves in place of an equal number of the unknown quantities. If the
equations are not absolutely identical, the coefficients of the other unknown quan-
tities will not entirely vanish by the substitution, and thus we shall still have
the whole number of unknown quantities, only the coefficients of certain of them
will be very small. The solution by least squares can then be performed without
trouble, because the extra decimals will be necessary only in multiplying by the
very small coefficients, when they can be introduced with ease. Afterward the
values of the original unknown quantities can be deduced from those of the linear
functions, and the unknown quantities which have been retained.
Suppose, for example, that the equations of condition are

aft -\- It^j -\- CjZ — TO!
a.2x -f-

cnz = nn

A simple inspection, or, at least, an attempt to solve three of the most diverse
of the equations, will show if the given n equations are really equivalent to only
one or two. Then we should put

X— ax + $y + yz

the coefficients a, /?, y, being entirely arbitrary, and so taken that when X and Y
were substituted for x and y the coefficients of z should be as small as possible.
It would conduce to simplicity if a and /?', or a' and /?, could each be made zero,
which could always be done.

If we attempt to correct the elements of a planet's orbit by observations extending
over only a few degrees, the equations of condition will necessarily be of the kind
referred to. Hence a transformation of this kind will be advisable. An example
will be given in the correction of the orbit of Neptune from observation.

§ 5. Ten years have elapsed since the publication of Kowalski's theory, and
no general revision of the orbit has been published by any astronomer, so far as
the writer is aware. The observations which have accumulated in the mean time
would seem sufficient to fix the elements exactly enough to give the place of the
planet within 5" during the remainder of the present century. It is, therefore,
proposed,

1. To determine the elements of the orbit of Neptune with as much exactness
as a series of observations extending through an arc of forty degrees will admit of.

2. To inquire whether the mass of Uranus can be concluded from the motions
of Neptune.

THE ORBIT OF NEPTUNE. 7

3. To inquire whether those motions indicate the action of an extra- Neptunian
planet, or throw any light on the question of the existence of such a planet.

4. To construct general tables and formulas by which the theoretical place of
Neptune may be found at any time, and, more particularly, at any time between
the years 1COO and 2000.

In giving the steps of an investigation like this, the true end should be to furnish
the means whereby every step can be corrected, or verified if already correct, and
to start only from admitted data. Sometimes a result will necessarily depend, to
a certain extent, on an act of judgment, as in assigning relative values to different
determinations of the same element. In this case data should be given for a
revision of the judgment, as far as this may be thought desirable.

Such, with very few exceptions, is the rule adhered to in the present paper.
The data are the published volumes of astronomical observations, and the funda-
jnental formulae of celestial mechanics. The steps will nearly or quite always
be so short that any one may be verified from the preceding one without much
labor.

The author is indebted to the courtesy of the Astronomer Royal, of the late
Captain James M. Gilliss, and of Professor G. W. Hough, for the observations made
at Greenwich, Washington, and Albany in the years 1863 and 1864, which have
added greatly to the reliableness of the results of his investigation.

WASHINGTON, April, 1805.

THE OIII3IT OF NEPTUNE.

CHAPTER II.
PROVISIONAL THEORY OF NEPTUNE.

§ 6. ALL the perturbations have been computed by formulae founded on the
method of La Grange ; the development of the perturbative function in series,
and the variation of arbitrary constants.

The following notation is used :

I rz mean longitude.

/I — mean longitude, counted from ascending node of inner planet on outer one.

<£> rz inclination of orbit to the ecliptic.

<y — mutual inclination of two orbits to each other.

a zz ratio of the mean distances.

u =1 sin \ y.

f zz mean anomaly.

o zz distance of the perihelion from the ascending node of the inner planet on
the outer one.

For the other elements the almost universal notation of astronomers is adopted.
The elements which pertain to the outer planet (Neptune) are distinguished by
an accent.

The potential of the disturbing force exerted by one planet upon another, usually
called the perturbative function, may be developed into an infinite series of terms,
each of which shall be of the form

m -, cos (t'T -f- fa +/u' +ja>)

U

in which i, i', j, and/ are numerical coefficients. 7i is a function of the ratio of the
mean distances, the eccentricities, and the mutual inclination of the orbits.

Then, by the theory of the variation of arbitrary constants, any term of the
perturbative function in the action of an inner on an outer planet will cause the
following differential variations of the four elements which determine the form
of the orbit, and the position of the planet in it. Putting

e — sin 4> (j rz cos ^ tan | ^ ;
we have ^- — — 2 mi'ha'ri sin N.

dt

( <11> dh )

- >• cos N, ( 1 )

da f

dt' YJd^^ "c

-j— zz mn'h •] /' cot A//' + *y [ sin ^
it t

V /

f/Tt' ,/77i

-77- zz mn' cot -f -r- cos N.

dt dd

THE ORBIT OF NEPTUNE. 9

From the first equation and the relation between the mean distance and mean
motion, we obtain

3£zz9irf&ffim*y>

dt

These equations are entirely rigorous, provided that we regard the elements in
the second member as variable. But they can be integrated only by successive
approximations. In a first approximation the elements are regarded as constant
Equations similar to (1) for the elements of all the planets whose action is taken
into account being integrated in this way, the resulting values may be substituted
in the second members of (1), and a new integration be performed.

In the case of Neptune, however, the variations of the elements are so slow
that a single approximation will be amply sufficient for a period of several cen-
turies, provided that we adopt suitable values of the elements in the second
members ; that is, if we add such constants to the integrals that the latter shall

n'
be very small for the present time. Putting v = - —

J i'n'-{-'in

we shall have, on the supposition that the elements as they enter into the second
member are constant,

log a' = mvA cos N-\- a'0, d = mvE cos jV+ e?0, (2)

I' = mvL sin N-\- n'0t + e'0, ri zz mvWsin N-{- rf0)

A, L, E, and W being given by the equations
A = 2 i'h,

E= — h(fcoi<V + W, (3)

*=«* + *

a'0, n'0 e', &0, and rt0 are arbitrary constants, dependent on the position and velo-
city of the planet at a given epoch. a0 and n0 are, however, dependent on each
other.

For the perturbations of the true longitude in orbit, and the logarithm of the
radius vector, we shall have, omitting accents,

E}sm(N-- f] — le*mv{eL

mv{eL — eW— E}sin(N + /) -\e>mv { eL — eW+
emv{eL — eW— E}$m.(N— 2/) —etc.
+lemt>{eL—eW+E}an(N+2f) (4)

+ V <?mv {eL — e W— E } sin (N— 3/)
+ ¥ ezmv { eL — eW + E } sin (N+ 3/)
+ W e*mv { eL — eW— E } sin (N— 4/)
+ etc.

2 May, 1865.

10 THE ORBIT OF NEPTUNE.

5 log r •=. & log «

+ mv { 2 i'h + J ?#— i e3^ } cos JV — A rm v { c£ — c TF— 3# } cos (N—/)
+ lmv{eL — eW—E}c,os(N- /) + A^mvj eZ — eTF+ 3# } cos (V-|-/)
— \mv{eL — eTF+ J£}cos(JV + /) — etc.

+ i em* { eL — e W— E } cos (N— 2/) (5)

— \emv\eL — eW + E] cos(N + 2/)
+ H e2mi/ { eL — eW— E } cos (N— 3/)
— etc.

By these formulae all the perturbations of the longitude and radius vector have
been computed, except that the computation was so conducted as to reject all
terms above a certain order with respect to the eccentricities. The sum of all
the factors (functions of the ratio of the mean distance) of any power of the
eccentricity in any coefficient in the perturbations of the co-ordinates will generally
be much smaller than each individual factor, as we shall presently show. If, for
example, we have

to = e* (/+/+/") sin N

the sum /+/+/" will, in general, nearly destroy itself, being much smaller than
the individual components,/,/', and/". Hence, if the computation is arranged so
as to include any one of the/'s, it should include all. This end may be attained
by omitting from /*, its differential coefficients, and 7<coti]/, all terms of a higher
order with respect to the eccentricities than the assigned limit. Thus, h being of
the form

if we limit ourselves to the power « + 1> we should put

7 dh (7x,

n — d 'x^; a -7 - = da •=-=• ',
da da

s7t cot ^ = set*-1 Xl + se's + ' (— J Xl + xa) .

*

§ 7. Perturbations of latitude.

The equations which determine the change in the plane of a planet's orbit are

dtt a'ri dR

_

dt ' ~ sin q>' cos fy dq>'

d<p' _ a'ri dR (6)

dt ' sin <p' cos fy ' dO1

R being a function of /I, ^,', w, o', and y, each of which depends on the position
of the plane of the orbit, we have

d R _ dR d^ dR <k>_ , dR dX dR d^ dR
d$ ~ <fa d$ ~*~ fa d$ + dX d<j>' ~T~ dti d$ "^ dy
dRdR fa dR rfw dR dX dR <k>'dR

THE ORBIT OF NEPTUNE. 11

The values of the second of each pair of differential coefficients can easily be
determined geometrically. /I, w, /I', etc., it will be remembered, represent the dis-
tance of certain points on each orbit from the ascending node of the disturbing
planet on the disturbed one : the infinitesimal changes in those quantities, produced
by infinitesimal changes in the position of the plane of either orbit, will be due
entirely to the changes in the position of that node. Let us put

x1 = distance of common node from ascending node of disturbed planet on the
ecliptic.

x zz same quantity for disturbing planet.

By drawing the diagram, it will readily be seen that by a change in <£>' the
common node will be moved forward on the disturbed planet by the amount

+ sin x1 cot yd<p',
and on the disturbing planet by the amount

+ sin x' cosec yd<p',
while Y will be varied by the amount

— cos x'd<p'.
In like manner, by a change in 0', the corresponding changes will be

— cos x' sin <£>' cot ydQ',

— cos x" sin <£>' cosec ydff,

— sin x' sin
We therefore have

cZ/l da

1 do'

zz —. — -M zz cos x cot y,

O I 1-1 ,*' fl(V I '

I do

-jfT, — cos x cosec y,

dy 1 dy

•f-. = — cos x'; -. — , jfe = — sin x'.

' ' ''

Also, by the differentiation of the representative term of R,

dR mi'h . ,T dR mj'li . ,,

-7 - - zz -- r sin N, -T- = -- -- sin N,

d% a' drf a!

dR mill . ,., dR mjJi . ,r

-^- = --- Bin JV, -j— = -- r sin N,

dh a' f?u a'

dR dR du m dh

-j- zz -, -- r~ zz \ —, -T-

dy du dy a' du

12 THE ORBIT OF NEPTUNE.

Substituting these expressions for the differential coefficients in the values of

.
^ and rfr we have

- =. — h sin y> sin N { (i' +/') cot y + (i -\-j) cosec y } - -j- cos \y cos it cos N.

(id) tt Q> du

-
(id)

1 dR

m
—
(t

sin <£> dQ

Let us now put

.. , ,.
i' +/)co

m dli
cosecvj — i ~, -r-

It may be remarked that i will then be the coefficient of the longitude of the
common node of the orbits in the usual development of the perturbative function.
The above equations may then be put into the form

dR m -I • , • »T m , 1 . f. I 1 • I • AT 1 m 'fa I »7

-r = — —rih cosec y sm / sin N — -7 (V + f) ft tan A y sm x' sin N — 1 — ? -5- cos -1 y cos x cos N.
dip a! a ^ a du

1 dR m , . . ,r wi , ,, , . . ,,. m dh

-~j- — —th cosec y cos x' sin IV -(- — (tT +/) A tan J, y cos x' sin /v — £ — y cos

sin <f dtf

Substituting these expressions in (6), and integrating, we shall have the values
of && and &$', the perturbations of the inclination and node.

For the perturbations of the latitude, counted in the direction perpendicular to
the plane of the orbit, we shall have

Where

Putting

$(? = %$ sin (^ — 0') — sin $W cos (tf — 9)
= mv sec ty{T+I} sin (N+ V)
+ mv sec 4/ { T— 1} sin f^— F)

T— \ ~ cos } y ; /= ^ 7i { t cosec y + (tv +/) tan J y }
V= true distance of planet from common node.

i — T

i — *

(7)

and developing Fin terms of ^, and/ to terms of the second order with respect
to the eccentricity, we shall have

+ <f sin (#;+*,+ /)
— d sin (N'+ %— /)

-f-

sn

T— »,— /)

(8)

THE ORBIT OF NEPTUNE. 13

For the perturbations of the constants which determine the position, of the

orbit, we put

p = sin <£ sin 6 ; q = sm<pcosQ;

t — longitude of common node of the two orbits.

We then have

$// =z 2 mv {/sin t cos N — ^cos t sin N] ;

&c( = 2 mv { I cos T cos N-}- Tsin t sin N } . (9)

Or, «f/= mv{(I— T)sm(N+r) — (I+T)sm(N— *)}>,

V = mv { (I— T) cos (N+ T) + (7+ T) cos (N—r) } ;

§ 8. The equations (2) and (9) determine the periodic perturbations of the
elements. For the secular variations, which proceed from those terms of the
perturbative in which both i' and •*' are zero, the same expressions apply, only

changing

v sin JVinto n't Cos N ;
v cos JVinto — n't sin N.

We therefore have, for the secular variations,

dl'

-jj- — mri L0 cos N ;

de! .

-^— — — mn A) sin Jy ;
dt

= mn'W0cosN; (10)

-~- •=. — 2 mn' {I0 sin t sin N-\- T0 cos t cos N} ;

U-t

^r — — 2 mn' { IQ cos r sin N — Tu sin r cos N} .
ctt

Owing to the smallness of the eccentricity of Neptune, it will l>e advisable to
substitute the rectangular co-ordinates of the centre of its orbit for the eccentri-
city and longitude of perihelion. The perihelion itself is subject to changes so
great that it would otherwise be necessary to develop the perturbations to
quantities of a higher order than the first. We shall, therefore, put

II-=.G sin 7i ; lt — e cos n.

For the secular variations of h and k, we then have, to a sufficient degree of
approximation,

dk

sn

§ 9. Development of the action of an inner on outer planet through the Sun.
The perturbations which one planet produces on another may be divided into
two distinct parts.

14 THE ORBIT OF NEPTUNE.

1. Those produced by their direct attraction on each other.

2. Those produced by the displacement of the Sun by the attraction of the
disturbing planet. The co-ordinates of the disturbed planet being counted from
the centre of the Sun, the displacement of the Sun not only changes the value
of the co-ordinates by changing their origin, but also by modifying the attraction
of the Sun itself.

The perturbations of both classes may be included in the same formulas, and
the total perturbations computed in the same way that those of the first class
are, by a very simple modification of those functions of the ratio of the mean
distances which enter into the different values of h. But in the case of the action
of an inner on an outer planet more than twice as far from the Sun, this method
will be subject to this serious inconvenience; that the perturbations of the
elements are many times greater than those of the co-ordinates. Referring to
formulas (4) and (5), it will be remembered that L, E, and W really express per-
turbations of the mean longitude, perihelion, and eccentricit}r, and it will be seen
that the perturbations of the true longitude fe are expressed as a function of the
perturbations of those elements. Now, having in this way computed the perturb-
ations of any co-ordinate which depend upon the different terms of the perturbative
function, when we collect those coefficients which are multiplied by the sine or
cosine of identical angles, we shall frequently find that their sum will nearly vanish,
as has been already remarked. As this circumstance depends on a theorem of
some importance, which will furnish a valuable check on the developments we
shall presently give, it is worth while to trace it to its origin.

The elements of a planet depend on its position and its velocity at a given epoch ;
each element is a function of the co-ordinates, their differential coefficients, and
the time, or, representing an element by a, and putting, for shortness,

t dx _ dy „ dz

t-dt^-di^-dt'

we have six equations of the form

an=f(x,y, z, £, >?, £, f) (12)

When we express the co-ordinates as a function of the elements and the time, we
have

x, y, or z —f (al} a.2, a3, a4, a6, ae, f) (13)

Substituting for the elements the values just given, £, y, and £ must vanish iden-
tically in the value of each co-ordinate. If, now, the changes in £, >y, and ^ are
of a higher order of magnitude than those in x, y, and z, the co-ordinates will he
subject to smaller variations than the elements.

Suppose, now, that one of the co-ordinates is affected with an inequality of
which the period is very short compared with that of the revolution of the planet.
Represent it by

c sin (pnt -f- e ) .
Its differential coefficient will be

pnccos

THE 0 KBIT OF NEPTUNE. 15

Since the elements contain this coefficient, and therefore include terms in which
the large number p multiplies the coefficient of the angle, their perturbations will
be much larger than that of the co-ordinate. But, in passing from the perturb-
ations of the elements to those of the co-ordinates, these large terms must destroy
each other.

Let us apply this principle to the case under consideration. That portion of
the perturbative function which arises from the action of an inner planet on the
Sun may be developed in a series of terms of the form

* t (X

c representing a number, not a line.

It therefore becomes infinite when a is infinitely small.

The second differential coefficient of the perturbation of any rectilineal co-
ordinate of the outer planet will be of the order of magnitude

. dR me

-j-f — -r cos ^>
da' a-

putting

N= iT + & + G.

Q

If we integrate this differential, and develop the quantity ., . according to

i n ~\~ in

7? o

the powers of — =. ~, the largest terms in the first differential coefficient of the

(I,',

n a'*
co-ordinates will be of the form

me .
— sm 2V.

This also will become infinite when a is infinitely small; and since the perturb-
ations of the elements contain these terms, it follows that they also will be infinite
in this case. Finally, by another integration, we shall have for the largest per-
turbations of the co-ordinate itself

men

-^j- cos N,

which will vanish when a is infinitely small. Hence, in the case under consider-
ation, although the perturbations of tlie elements become infinite, those of the co-ordi-
nates vanish.

The co-ordinates referred to arc linear. The order of magnitude of the angular
co-ordinates, or the logarithms of any linear co-ordinate, will be given by dividing
by a'. We shall, therefore, have for largest term in the perturbations

8v, $3, or & log r =. mca N.

cos

Hence, when we collect the perturbations of the co-ordinates due to the cause in
question, all terms of a higher order of magnitude than this ought to destroy each
other identically.

16 THE ORBIT OF NEPTUNE

§ 10. That portion of the perturbative function which is due to the action of

the inner planet on the sun is

r'

-5 cos V

r

V being the angular distance between the planets. Developing it in a series of
terms of the form

• cos (m +

\JL
f*

h will be of the form -# c being a numerical coefficient, multiplied by powers of

GC-

the eccentricities and mutual inclinations.

From this development, and the equations (3), (4), (5), (7), and (8), I have
computed the following analytical values of the coefficients for the perturbations
of the longitude, latitude, and logarithm of radius vector.

a

Vft

:^ 25

a

6 ^32—

(17)

2 Vu — 3

=e

= e2 (— \l v, + V ^9 —

THE OR13IT OF NEPTUNE.

17

= net (—
= ue( 2

The values of JVW are as follows :

N<» = X — K

= - a,' + 2 A — o

— 2 a/ — a — <y

— a *f- o' - 2 o
= _^'+3^ — 2o
= 2A— a -co'
= 3 A' — ^ — 2<y

_a,'_2o)

— 3o>'
—a'

— vf

(19)

^ — 2/1' — 2/1 —

— 2

—2

6>

zz 3

(20)

A'+ 2^— 2 o' —
4/1' — /L — 3o>'

From these values of N the corresponding values of v are derived, remembering
that

ri

~ '

i' and i being the coefficients of /I' and 2. respectively in the value of N.

The check on the correctness of the preceding values of V, R, and B may now

n

be applied by developing v in powers of — , and retaining only the first term ;

/&

that is, by putting v z= ^, v2 =. 0. Making these substitutions, all the values of

%

V, R, and B will be found to vanish. In other words, [j? will be the lowest power
of /j. which will enter into the values of V, R, or B, as we have already shown from
a priori considerations.

For convenience, we shall give the values of V, R, and B developed according to
the powers of n, the ratio of the mean motion's, a form similar to that in which
the lunar inequalities are developed in the theory of the moon. Putting

we have

sy—

May, 1865.

18 THE ORBIT OF NEPTUNE.

We shall also put

V rV
V — — —
yi-a*-ff

_R _cR

^-a3-^'

B cB

c being a constant, equal to unity if we neglect the change of mean distance pro-
duced by the action of other planets. We then have

! sin N,

8 log r = mac^R^ cos N, (21)

sin N.

Substituting the above developments for the vs in V, R, and B, we have

Vf» = (1 — uz — i-e2) (— 1 —i? — 6 fj.s — 19 ft4)
+ ^(1 — 2(j.2 — 30p8)

VV =€€>(- |- f;U2- \lt?)

i^2+ fl«3+ f|M4)
1«2+ V3+ 25^)
3^2— 27 ^3 — 135 ^4) (22)

_ ^ (_ y _ y ^ _ 2|

25

_l_42 — 24

=^ ( I— 6^2— 3(V — 116 ^4) (23)

zre2 ( fTV^2- i1^3)

THE ORBIT OF NEPTUNE. 19

Such arc the formula} by which we shall proceed to compute the perturbations
of Neptune by Jupiter, Saturn, and Uranus.

It will be noticed that the coefficient of ft vanishes identically in the last de-
velopments. I have not completely investigated this law, but it seems to arise
from the circumstance that that portion of the perturbation in question which
proceeds from the change in the origin of co-ordinates is independent of (i, while
that portion which is caused by the modified attraction of the Sun is of the order
of magnitude ^. It furnishes a yet more valuable check than the last on the
developments.

§ 11. Allusion has already been made to the complications introduced into the
theory of Neptune by the near approach of its mean motion to double that of
Uranus, and the consequent oscillation of all the elements of its orbit in a cycle
of 4300 years of duration. In order to construct a dynamical theory which should
be correct within a tenth of a second through the whole of one of these cycles, it
would be necessary to include many terms dependent on the second, and perhaps
some dependent on the third power of the masses of the disturbing planets.

If this task were accomplished, the necessary uncertainty in the mass of Uranus
and the elements of Neptune would destroy all the value of the theory. A change
of one-tenth in the mass of Uranus would produce a change of 200" in the co-
efficient of the perturbation of the mean longitude. The mean motions of Walker
and Kowalski being each about 8" in error, the place of the planet from this cause
alone would be in error by nearly 10° at the end of a cycle.

After much careful consideration of different ways of relieving the theories of
Uranus and Neptune from the complexities introduced by the large perturbations
referred to, I finally determined to develop them not as perturbations of the co-
ordinates, but of the elements. It will readily be seen that if the eccentricity or
perihelion is greater than the mean during several revolutions of the planet, there
will be a perturbation in the radius vector and longitude having nearly the same
period with the revolution of the planet, although the latter may really scarcely
wander from a true elliptic orbit during an entire revolution. In such a case it
is clearly best, in constructing a theory designed to remain of the highest degree
of exactness for only a few centuries, to take not the mean values of the elements,
but their values at a particular epoch during the time the theory is expected to
be used.

In doing this, we shall be treating the change in the elements in the same way
that the secular variations are usually treated. These variations are really
periodic, and in a perfect theory would have to be treated as such. But the
elliptic elements on which all our planetary theories are founded are not mean
elements, but elements brought up by secular variation to the epoch 1800 or 1850.

Thus, our perturbations of the elements will be of the form

01 IT

Sa zz c -f aj + 2a2 {kt-^-s},
cos

iji which a' is the secular variation proper, k a small coefficient equal to 2 ri — n
or its multiples, and c a constant added to the integral, of such value as to make
&a vanish at the epoch 1850.

20

THE ORBIT OP NEPTUNE.

§ 12. Adopted dements and masses.

The elements of Neptune adopted in the computation of the perturbations are
obtained by correcting those of Walker so as to agree with the Lalande obser-
vations, and as nearly as possible with seven normal places derived from the
modern observations from 1846 to 18G3. The latter series is thus represented
within a second of arc. As these elements are merely provisional, it is not worth
while to give any details of the corrections, except their amounts, which are as
follows :

&« = — 4° 11' 18".6 ; n = 43° 3' 18".G
be = — .00025451; e =.00846495
&n = — 8".40G ; n = 7864".3G8

& - _ 3' 5".92 ; e = 335 5 31.10
log a = 1.4 780405

i 1° 47' 1"

n 130 7 20
Epoch, Jan. 0, 1850, Greenwich, M. noon.

To obtain the value of log a, the mean motion was diminished by the secular
variation of the longitude of the epoch = 21".354. A more exact value of this
quantity will appear, in the course of our computations, to be 21".4426.

The provisional inclination and longitude have been taken from Walker without
change, as the small corrections which his values of these elements may require
will not affect the perturbations.

The adopted elements of Uranus, Saturn, and Jupiter, with their functions
used in the theory for the same epoch, are as follows :

Uranus.

Saturn.

Jupiter.

71

167° 34' 21"

90° 4' 0"

11° 54' 51"

£

28 27 14

14 48 40

159 50 20

i

0 46 30

2 29 28.8

1 18 41.1

0

73 14 14

112 22 14

98 56 10

n

15425.030

43996.127

109256.72

e —

.0466972

.0560050

.0482273

log a

1.2837047

0.9802225

0.7162201

t

335°38'

77°56'

355°52'

u

.0131517

.0083880

.0082735

a

0.638195

0.317301

0.1727703

m

t

i

i

These elements of Uranus have been obtained by applying to Feircc's values
of the mean elements (Appendix to American Ephemeris and Nautical Almanac,
1860-64, p. 4) approximate long-period perturbations of the elements produced
by Neptune at the epoch 1850. The elements of Jupiter and Saturn are from
Hanson's prize memoir on the mutual perturbations of those planets, and are, sub-
stantially, the same as Bouvard's.

THE ORBIT OF NEPTUNE.

21

The values of those constants which depend on the ratio of the mean distances
are as follows, using the notation of the Mecanique Celeste :

L— URANUS AND NEPTUNE.

c/i(;

rf^T

«s i

d«K

d'V?

I

Bi

Jo,

' da*

1 da*

da*

0

2.26969

0.72903

1.8326

6.4384

35.17

1

-1.68379

6.05279

—13.0023

65.5556

—259.42

2

0.37751

0.95867

2.1283

6.7135

35.99

3

0.20310

0.72530

2.2389

7.4924

36.95

4

0.11422

0.52446

2.1024

8.2270

39.52

5

0.06593

0.36954

1.8319

8.5192

43.00

G

0.03870

0.25606

1.5157

8.302

46.01

7

0.02299

0.17533

1.2085

7.679

47.57

8

0.01379

0.11900

0.9365

6.804

47.27

9

0.00832

0.0802

0.7100

5.818

45.18

10

0.0051

0.054

0.533

j<0

"**l

"^

da"

0.8966

26.5493

2.80

3.2907

11.9366

60.92

2.4710

11.0760

59.76

1.7806

9.6115

57.07

1.2524

7.9427

52.59

0.8668

6.3301

46.80

0.5931

4.9065

40.34

0.4023

3.7215

33.83

0.2711

2.7738

27.69

0.1817

2.0381

22.20

0

1

2
, 3

4
5
6

7
8
9

It will be observed that in 5|, atff, and their differential coefficients, we have
included those multiples of — which are introduced by the action of Uranus on
the Sun. It seemed less laborious to do this than to make, a separate computation
of the terms produced by this cause. But for Saturn and Jupiter —^ is so large
that it will be better to use the developments previously given.

oc

22

THE ORBIT OF NEPTUNE.

II.— SATURN AND NEPTUNE.

i

•

;; (i)

•3

rffS

3'raT

</a2

(/a3

0

2.05341

0.11342

0.14186

0.0986

1

0.33010

.35745

.08964

.1313

2

.07890

.16509

.19632

.1075

3

.02091

.06476

.14027

.1878

4

.00581

.02383

.07517

.1701

5

.00166

.00847

.03514

(0

tiff

0

0.8045

0.4003

1

0.3686

.5242

2

.1443

.3456

3

.0531

.1794

4

.0189

5

0066

III.— JUPITER AND NEPTUNE.

,«

*J

<raT

d%?

I

61

a — s

a3 i-

2

Ja

da*

da3

0

2.01518

0.03088

0.0330

0.0067

1

0.17474

.17876

.0124

.0139

2

.02267

.04592

.0483

.0074

3

.00327

.00989

.0202

.0221

4

.00049

.00199

.0061

.0125

0

0.3699

0.4209

1

.0948

.2005

2

.0204

.0634

3

•.0041

.0168

4

0.0008

§ 13. From these data the coefficients 7t of the different terms of the perturb-
ative function, their differential coefficients, and the perturbations of the co-
ordinates, are found to be as in the following table. The N's, it will be seen, are
grouped according to the values of their constant parts, /of -j-jb.

h, its differential coefficients, L, W, and E, are given in units of the third J>/<K;>
of decimals, to avoid writing zeros. The logarithms are reduced to the common
base, 10, .and are expressed in units of the seventh place.

THE ORBIT OF NEPTUNE.

23

ACTION OP URANUS.

I. — / iz: 0 ; /' ~ 0
tt 7 »/

I

= 0.

^,

0

+ 1

2

3

4

5

6

7

8

9

»

0

— 1

— 2

— 3

— 4

-6

— 6

— 7

— 8

— 9

A

+ 1135.50

840.31

+ 187.98

+ 100.3

7 + 55.83

+ 31.75

+ 18.29

+ 10.63

+ 6

2 +3.7

a.Dai

l

+ 367.47

+ 3026.10

+ 478.7

+ 359.3

+ 256.9

+ 178.5

+ 121.4

+ 81.2

+ 53

6 +36

Mh

+ 6

Hi;

4

r.

.52

— 4.1

— 7.6

— 8.7

— 8.4

— 7

4

-6.2

— 5

0 -4

IDuh.

— 10.84

— 2.66

— 8.36

— 6.1

— 4.3

— 3.0

— 2.1

— 1.4

— 1

0

• L

+ 3005.97

+ 1747.45

+ 1919.99

+ 1232.4

+ 799.6

+ 519.5

+ 336.3

+ 216.9

+ 138

9 +91

tf W

6

;«;

-t

1S

53

— 4.1

— 7.6

— 8.7

-8.4

— 7

t

-6.2

— 6

i

E

0

+ 3.56

— 1.0

— 1.3

— 0.9

— 0.7

— 0.5

— 0.3

— 0

2

it

//

tt

it

n

tt

tt

tt

„

J!-i-sin

JV

—

17.853

— 9.808

— 4.19

7 —2.043

-1.061

— 0.672

—

0.317

— 0.17

8 —0.102

6u-t-sin (N

—

f}

4

0.(

ia

— 0.112

— 0.06

6 — O.OS2

-0.028

— 0.0

IK

—

0.012

— 0.00

8

N

h

jf

O.I

(HI

— 0.096

— 0.05

J —0.037

-0.025

-0.0

17

—

0.011

— 0.00

8

N

+

0.(

KK>

— 0.001

— 0.00

1

N+if

—0.001

— 0.001

— 0.00

1

6 log r -f- cos N

+ 361

— 81

— 43

:-24

14

— 8

— 5

— 3

-r2

N-j

e

0

— 1

— 1

0

0

0

0

0

0

N+f

0

+ 1

+ 1

0

0

0

0

0

0

S/3 -s- sin (.2V— F)

— 0.030

+ 0.055

+ 0.02

3 + 0.014

+ 0.008

+ 0.005

+ 0.003

+ 0.00

1

(Ar+ r)

+ 0.083

+ 0.030

+ 0.01

1 + 0.007

+ 0.004

+ 0.002

+ 0.001

+ 0.00

n.-y=i;/ = -

L;

, = o.

i'

— 7

6

6

— 4 —3

— 2

—1

0

+ 1

2

3

4

6

6

7

t

+ 7

6

6

4 3

2

1

0

—2

— 3

1

— 5

— 6

— 7

/i

+ 0.21

+ 0.26

+ 0.30

+ 0.33 +0.3

2 +0.24

+ 0.05

— 0.24

— 0.32

0.91

+ 0.35

+ 0.'

18 +0.50

+ 0.4€

+ 0.40

a.Dah

+ 1.8

+ 1.9

+ 1.9

+ 1.65 +1.1

6 +0.38

— 0.66

— 1.32

-1.50

+ 1.05

+ 0.20

+ 1.:

4 +2.00

+ 2.37

+ 2.5

De'll

+ 24.8

: .".(

5

+ 35

.1

+ 38.5 +37.4

+ 27.93

+ 5.69

— 28

.77 —

-38.29

— 1

J7.13

+ 40.77

+ 66.1

+ 58.0

+ 53.0

+ 44.6

L
e'W

+ 4.8
+ 24.8

+5.2
+ 30.5

+ 5.4
+ 35.1

+ 5.1 + 4.1
+ 38.5 +37.4

+ 2.10
+ 27.93

— 0.82
+ 5.69

— 2.24

— 28.77 -

-4.80
-38.29

— 3.01

— 107.13

+ 2.4
+ 40.77

+ 5.1
+ 56.1

+ 6.8
+ 68.0

+ 7.3
+ 53.0

+ 7.2
+ 44.5

E

+ 24.6

+ 30.3

+ 34.9

+ 38.3 + 37.2

+ 27.81

+ 6.65

— 28.95-

-38.47

— 107.20

+ 40.61

+ 55.E

+ 67.6

+ 52.4

+ 43.7

n

/

r

,

t

tt n

tt

tf

„

it

it

„

n

n

it

&l -f- sin N

+ 0.007

+ 0.009

+ 0.011

+ 0.013 + 0.01

4 + 0.011

— 0.008

4

•0.049

+ 0.015

— 0.008

— 0.01

3—0.014

— 0.012

— 0.010

«D-^sin(JV— /)

— 0.072

— 0.103

— 0.143

— 0.196—0.25

5—0.284

— 0.115

-0.783

— 1.095

+ 0.277

6 + 0.236

+ 0.179

+ 0.130

N — 2J

— 0.001

— 0.001

— 0.002

— 0.002—0.00

3—0.003

— 0.001

-0.008

— 0.012

+ 0.003

+ O.OC

8 + 0.002

+ 0.002

+ 0.001

& + f

— 0.002

-1

0.002

J log r ^ cos N— f

— 1

— 1

— 1

— 2 —3

— 3

— 1

— 8

ll

+ 3

+ 3

+ 2

+ 2

+ 1

Ill.-j=-l;j' =

0;

, = 0

i'

— 5

_4

— 3

— 2

— 1

0

+ 1

2

3

.

4

5

6

7

8

i

+ 6

5

4

3

2

1

0

— 1

2

— 3

— 4

— 6

— 6

— 7

h

+ 6.5

+ 9.0

+ 11.4

+ 12.

r —219.7

— 17.00

— 62.71

57.57

3 —46.20

— 33.2-

23.66

— 16.41

— 11.20

— 7.6

af)ah

-t

34.1

4

36.0

+ 32.1

H

17.

t +444.1

— 59.5 -

- 120.65

— 1

r,i.f>7

— 170.23

— IS

7.H

— 1

35.8 -

-110.3

— 86.4

— 66.2

De'h

— 2.0

— 1.7

— 1.3

— 1.

) + 1.01

— 1.50

— 1.29

+ 0.61

+ 2.96

+ 4.81

) +6.0

+ 6.4

+ 6.2

+ 5.7

IDuh

0

0

0

+ 0.

i + 2.75

+ 2.31

+ 3.23

• -

3.61

+ 3.48

+

3.01

) -

-2.6

+ 2.2

+ 1.4

+ 1.2

L

-f

95.5

+

10R.6

+ 108.7

4

7;>.

S +223.3

— 153.0 -

- 178.40

+ 85

IIC..7-

— 871.70

— 6S

4.2

— <

43.5 -

-331.0 -

-244.5

— 180

e'W

— 2.0

— 1.7

— 1.3

— 1.

) + 1.01

— 1.50

— 1.29

+ 0.61

+ 2.96

+ 4.8

> +6.0

+ 6.4

+ 6.2

+ 6.7

E

+ 0.1

+ 0.2

+ 0.2

+ 0.

L —0.95

0.00

+ 0.27

+ 0.48

+ 0.67

' + 0.5(

i +0.5

+ 0.4

+ 0.3

+ O.S

n

n

„

„

„

„

n

„

tt

„

it

if

SI -t- sin 2V

+ 0.138

+ 0.184

+ 0.220

+ 0.201

i +0.750

— 0.766

— 1.752

+ 2163.60

5 +9.279

+ 3.09!

+ 1.631

+ 0.854

+ 0.504

+ 0.308

n

ff

v — sin fJV f)

+ 0.004
+ 0.004

+ 0.004
+ 0.005

+ 0.004
+ 0.005

+ 0.00.
+ 0.00.

1 +0.006
> 0

+ 0.001
+ 0.001

— 0.005
0

+ 0.116
+ 0.104

+ 0.054
+ 0.04S

+ 0.036
+ 0.032

+ 0.025
+ 0.022

+ 0.018
+ 0.0161

+ o.oi n

+ 0.012

i log r -f- cos If

— 2

— 2

— 3

— 3

+ 31

0

— 26

+ 61

+ 29

+ 17

+ 11

+ 6

+ 4

(^V — /")

+ 1

1

<^+y)

— 1

—

1

^.^yr-r)

0

0

0

+ 0.00.

+ 0.004

+ 0.012

+ 0.036

— 0.046

— 0.021

— 0.011

— 0.007 •

-0.004

- o.oor

sin(JV+ F)

+ 0.001

+ 0.014

+ 0.012

+ 0.028

— 0.028

— 0.011

— 0.006

— 0.004 -

-0.002

-0.001

_

24

THE ORBIT OF NEPTUNE.

IV.- -j = 0;j' = — 1; i = 0.

*' —5
• +6

— 4
S

— 3
4

— 2
3

— 1

2

0

1

1
0

2
— 1

3

1

4
— 3

5
— 4

6
— 5

7
— 6

8 9

-7 —8

ft —0.68
aDaJl —4.17
Vf'h —79.9
I H'ik 0
i —11.5
J5 —80.1

—0.91

— 4.56
— 107.5
0
— 13.2
— 107.5

— 1.1.'

— 4.2;:

— 133.8
0
— 13.5
— 133.9

— 1.21
— 2.71
— 142.5
— 0.1
— 10.3
— 142.6

— 0.72

--- 11.97
-85.17 +
— 0.16
—0.62
-85.24 +

+ 32.76
— 54.83
38fi5.es
— 0.52
— 27.80
3865.40

+ 12.73
+ 14.12
+ 1503.24
— 0.51
+ 21.87
+ 1502.86

+ 4.267
+ 47.61
+ 504.23
— 0.611
-557.07
+ 504.02

+ 12.016
+ 33.38
+ 1418.28
— 0.72
+ 214.18
+ 1418.32

+ 9.00

:. :\:;.su
r 1061.5
— 0.68
+ 147.4
+ 1061.7

+ 6.43
+ 30.55
+ 758.9 -
— 0.6
+ 111.0
+ 759.4 -

+ 4.47
+ 25.64
t- 527.1
— 0.5
+ 83.6
h 527.7

+ 3.04 +
+ 20.5 +
+ 358.7 + 2
— 0.4 —
+ 62.0 +
f 359.3 + 2-

2.05 + 1.36
15.8 + 11.8
40.1 + 158.8
0.3 0
15.3 + 32.1
10.7 +159.4

//

/

n

tt

it

n

n

n

•

5( -r- sin ,2V —0.017
jo^siu(JV— /) +0.232
Jf—2f +0.002
N— 3/ 0
A> / 0

— 0.022
+ 0.364
+ 0.004
0
0

— 0.027
+ 0.543
+ 0.006
0
0

— 0.026
+ 0.721
+ 0.008
0
0

-0.002

l n..-,::] -
+ 0.006
0
0

— 0.139
-38.716
— 0.411
— 0.006
— 0.002

+ 0.215
— 29.524
— 0.313
— O.IXB
— 0.001

- 141.69

— 2.279
+ 30.174
+ 0.321
+ 0.004
— 0.011

— 0.770
+ 11.061
+ 0.117
+ 0.001
— 0.008

— 0.383

+ 5.239
+ 0.055
+ 0.001
— 0.005

— 0.216
+ 2.720
+ 0.029
0
— 0.003

— 0.128 — (

+ 1.478 +(
+ 0.016 +(
0 I
— 0.002 — (

(.077 —0.040
).823 +0.467
1.009 +0.005
) 0
1.002 —0.001

03 jr— 2/ o

0

0

0

0

+ 8
0

0
+ «
0

+ 2
— 408
— 5

+ 6
— 311
— 4

— 17
+ 318
+ 4

— 8
+ 116

+ 1

— 5

+ 55
+ 1

— *\

+ 29
0

— 2 —

+ 15 +
0

1 0

9 +6
0 0

tt

ii

tt

„

„

,i

5|3-^siiifjV + T) 0

0
0

0
0

0
0

0
0

— 0.003
— 0.003

— 0.005
— 0.005

— 0.005
— 0.010

— 0.002
— 0.005

— 0.001
— 0.003

— 0.001
— 0.002

0
— 0.001

0 0
0 0

y._/--2j/=lit=:0.

i'
i

— 4
5

— 3
4

-2
3

—1

2

0

1

1

0

2
— 1

3

— 2

4
— 3

5
— 4

6
— 5

7
— 6

8
— 7

A

a Pa/i
2ZVA

+ 0.03 +0.02
+ 0.1 + 0.0
+ 6.5 + 4.6

— 0.0
+ 0.2
— 15.9

J +0.01
0.0
+ 1.4

+ 0.02
+ 0.08
+ 6.1

+ 0.006
+ 0.08
+ 1.3

— 0.0145
+ 0.014
— 3.87

— 0.033
— 0.11
— 7.9

— 0.0
— 0.2,
— 11.0

6 —0.05
— 0.34
— 13.0

5 —0.06
— 0.40
— 13

— 0.05
— 0.44
— 12

— 0.05
— 0.44
— 11

„

n

a

H

it

n

•

n

„

it

{I -f- sin JV + 0.001 0
«r-i-8in(Ar+/)— 0.011 —0.009

0

+ 0.04

0

J— 0.005

+ 0.001
— 0.026

+ 0.002
— 0.013

+ 0.56

+ 0.007
— 0.084

+ O.W
— 0.0,

5 +0.00
7 —0.04

4 +0.003
4 —0.032

+ 0.00'
— 0.02-

>. +0.002
— 0.019

vi._ y

sliis-iMSst

t'

t

— 4

+ 5

— 3

4

— 2 — 1
3 2

0

1

1

0

2
— 1

3456
_2 —3 —4 —5

7
— 6

8
— 7

&
2ZVA

— .002
— 1.0

— .002
— 0.9

— .001 — M
-0.7 —0.6!

1 — .002
— 1.04

— .004
— 2.10

— .0058
— 2.80

— .007 +.008 +.012 +.01
— 3.47 +2.86 +6.9 +7.8

i +.018
+ 8.6

+ .018
+ 8.8

tt tt

II

0

ti

0

n

0 0

0

0

|

+ 0.21

— 0.001 000
// n n n

0^

0

it

S«-i-Bin(JV— /)

+ 0.002

+ 0.002

+ 0.002 +o.a

12 +0.005

+ 0.021

— 0.037 +0.015 +0.020 +0.02)

) + 0.017

+ 0.015

VII.— y=l;y = 0; t =

-2

i

— 4

+ 5

-

3
4

— 2 —
3

1 0

2 1

1

0

2
1

3

2

456
-3 —4 —5

7
— 6

8
— 7

h
IDu

A

— 1.3

-1.7

— 1.9 —2.

2 —23

— 1.85

— 0.031
— 1.17

— 0.66 —0.3 —0.1 0

0

0

«( +- sin JV

•

+ 1.11

n

„

tt

it it

„

/

,,

ti

83 -5- sin 0

vr-F)

— 0.004-

-o.oo-

— 0.010 — 0

015 —0.02!

— 0.03

6—0.58

+ 0.007 +0.002 0 0

0

0

.

THK ORBIT OF NEPTUNE.

25

VIIL— y=0;/ = l; 6 =

— 2

t

i

+ 4
+ 5

3

4

— 2
3

— 1

2

0
1

1

0

2
— 1

3

— 2

4

— C

5
— 4

6
— 5

7
— 6

IDe'h

+ 1.6

+ 1.8

+ 1.8

+ 1.6

+ 2.3

+ 0.4 -

t-0.10

— 0.09

— 0.1

B —0.21

— 0.17

— 0.14

IDu/t

+ 0.5

+ 0.6

+ 0.6

+ 0.5

+ 0.75

+ 0.15 -

-0.033

— 0.03

— 0.0

3 —0.06

— 0.04

0

„

n

it

it

it

it

n

tt

it

ar-i-si

l(AT+/

— O.OC

13 —0.004

— 0.005

— 0.005

— 0.012

— 0.004

— 0.001

— 0.0

01 —0.00

1 0

0

„

„

n

//

it

tt

20-!- sin (A*— I")

+ 0.0(

1 +0.001

+ 0.002

+ 0.002

+ 0.004

+ 0.001

0

0

0

0

0

ix.— y

=_*;/=o;t=b.

i'

— 3

_.,

— 1

0

i

2

3

4

5

0

7

8

9

t

— 5

4

3

2

i

0

— 1

— 2

— 3

— 4

— 5

— 6

— 1

h
Dt'lt

+ 0.66
+ 2.07
— 0.06

+ 0.64
+ 1.36
— 0.01

— 17.57
+ 30.S
+ 0.19

+ 0.10
+ 1.36
: 0.07

+ 0.214
+ 5.03
+ 0.12

+ 2.766
+9.55
+ 0.06

+ 3.750
+ 15.48
— 0.17

+ 3.944
+ 19.71
— 0.51

+ 3.660
+ 21.67
— 0.86

+ 3.161
+ 21.6
— 1.15

+ 2.58
+ 20.2
— 1.35

+ 2.04
4 18.0 -
— 1.4

+ 1.57
H5.4
— 1.4

L

+ 6.4

+ 4.7

+ 33.-

+ 2.9

+ 10.3

+ 16.3

+ 5.32

— 565.6

+ 112.77

+ 80.22

+ 64.9

453.1 -

H41.9

„

n

t.

n

n

n

„

ff

*

•

//

it

//

SI -r- sin JV

+ 0.009

+ 0.008

+ 0.068

+ 0.007

+ 0.034

+ 0.080

+ 0.050

— 71.93

—

1.253

—0.425

— 0.227 -

-0.138 -

-0.087

2i>-rin(V— /•

0

0

0

0

0

0

+ 0.002

0

0.021

— 0.010

— 0.007 -

-0.005 -

-0.003

(A- + /

0

0

0

0

0

0

+ 0.001

0

0.018

— 0.009

— 0.007 •

-0.005 -

-0.003

•TT1 • 1 . ^ — "I

{,

-I

-2

— 1

0

1

2

3

4

5

6

7

8

9

i

h5

4

3

2

1

0

— 1

—

2

— 3

^— 4

— 5

— 6

— 7

h

0.113 —

0.107 -

-0.0!K)

+ 2.814

— 0.325

— 1.305

— 1.90

3 —2.06

— 1.954

— 1.70

0 —1.4

0 — 1.

2 —0.86

a Hah

0.50 —

0

42 -

-0.40

- 6.48

— 1 .50

— 3.38

— 6.03

—

i.30

-9.63

— 10.01

— 9.1

2 —8.

•4 — 7.5*

He'lt —

3.3 —

•i

5 —

10.5

H

;

.".1.8

-38J)

— 154.2

— 224.07

— 24.

1.19

— 2

28.57

— 197.7

— 161 .t

— 127.

! —96.9

L

1.4 —

i

2

-1.1

-3.5

— 6.11

— 0.38

42!

—

57 .2!t

— 40.86

— 33.2

— 27.

I —22.3

K —

3.3 —12.5 —

10.5

+ 331.8

-38.7

— 154.2

— 224.05

— 243.20

— 228.68

— 197.8

— 161.S

— 127.

) —97.0

„

H

„

„

n

it

„

„

„

*

//

//

n

SI -f- sin A" —

0.002 —0.002 -

-0.0112

— 0.015

— 0.011

— 0.03C

— 0.00

! +38.09

+ 0.63C

+ 0.21

7 +0.11

6 +0.01

55 + 0.04i

ai'-s-ninr.V— /) +
(-V-2/I

0.038 + 0.0 12 4
0 0

-0.012
0

— 1 .lift!
— 0.017

+ 0.257
+ 0.003

+ 1.514

+ o.oie

+ 4.25
+ 0.04

0
5

— 5.074
— 0.054

— 2.10
— 0.02

2 —l.K
2 —0.01

1 —0.61
2 —0.01

•A— 0.402
17 — 0.004

5 log r -^ cos A7

0

0

0

0

0

0

— 2

+ 5

+ 2

+ 1

+ 1

0

(A'-/)

0

0

0

— 17

4-3

+ 16

+ 44

— 53

— 22

— 12

— 7

— 4

XI

-/=0;/=-i

i'

— 3

— 2

0

l

2

3

4

5

6

7

8

9

t

45

4

3

2

i

0

— 1

— 2

—

3

—4

— 6

— 6

— 7

A
a.Da.h

0
+ .03

0
+ .03

0

+ .03

+ .01

t-o.oc

+ 0.01

+ 0.18

+ 0.14
+ 0.25

+ 0.169
+ 0.655

+ 0.2677
+ 0.810

+ 0.259
+ 1.018

+ 0.228
+ 1.12

+ 0.206
+ 1.11

+ 0.152
+ 1.04

+ 0.118
+ 0.92

De'lt
L
E

+ 1.1

+ 1.1

+ 1.1
+ 1.1

+ 1.1
+ 1.1

+ 2.08
+ 2.1

+ 1.76
+ 0.4
+ 1.8

+ 32.51
+ 0.50
+ 32.50

+ 40.15
+ 0.36
+ 40.15

+ 63.13
— 39.2
+ 63.13

+ 60 52 4
+ 7.24
+ 60.54 4

52.90 4
+ 5.15
- 52.92 4

- 42.98
+ 4.4
- 43.01

1- 34.1
+ 3.5
t-34.1

+ 25.8
+ 2.2
+ 25.8

„

ti

tt

/

„

n

it

n

n

S! -i- sill N

0

0

0

0

+ 0.001

+ 0.002

+ 0.003

— 4<

19

— 0.080

-0.027

-0.015

— 0.009

— 0.005

„

n

„

it

n

„

n

a

n

n

n

So-f-sinOV— f
UV-2/

) —0.003
) 0

— 0.00
0

4 —0.004
0

— 0.010
0

— 0.012
0

— 0.319
— 0.003

— 0.757
— 0.008

+ 1.344
+ 0.014

+ 0.564
+ 0.006

+ 0.301
+ 0.003

+ 0.178
+ 0.002

+ 0.110
+ 0.001

2 logr -j- cos (A' —

f) 0

0

0

0

0

— 3

— 8

+ 15

+ 6

+ 4

+ 2

+ 1

Hay, 1865.

26

THE ORBIT OF NEPTUNE.

xn.-,-=-3;y=o.

\

6
— 3

L

+ 51.5

« + -»*

+ 4.37

XIII.-/ = — 2; $- — 1.

5

0

+ 3

1
2

2
1

3
0

4

— 1

5

— 2

6
— 3

7
— 4

8
— 5

9
— 6

10

II1.1 7

£

2iVA

+ 64.5

+ 1.4

+ 6.0

+ 23.4

+ 41.7

+ 55.17

— 38.5
— 3.26

+ 62.63

+ 68.87

+ 53.4

+ 44.8

rr

rr

„

rr

rr

„

rr

rr

rr

/,

° N—Zf

— 0.091

— 0.003

— 0.015

— 0.076
— 0.001

— 0.201
— 0.002

— 0.503
— 0.005

+ 0.727
+ 0.008

+ 0.320
+ 0.003

+ 0.189
+ 0.002

+ 0.116

+ 0.001

XIV.-/ = -l;/ = -2.

i'
1

0

+ 3

1

2

2
1

3
0

4
— 1

5
— 2

6
— 3

7
— 4

8
— 5

9
— 6

10
— 7

£

2/Vft

— 03

— 1.8

— 2.8

— 10.9

— 20.7

— 28.4

+ 10.0
rr

— 3330

— 31.7

— 28.5

— 24.8

rr

rr

rr

rr

rr

+ 0.85

rr

rr

„

rr

8r -h sin (N—f)

0

+ 0.004

+ 0.007

+ 0.036

+ 0.100

+ 0.259

+ 0.387

+ 0.172

+ 0.101

— 0.065

XV.^=0;/ = _3.

5

0

+ 3

1
2

2
1

3
0

4
— 1

5

— 2

6
— 3

7

— 4

8
— 5

9
— 6

10

— 7

2fle'ft

0

+ 0.1

+ 03

+ 1.2

+ 2.3

+ 3.6

+ 4.25

+ 4.47

+ 4.24

+ 3.8

+ 3.3

rr

rr

„

„

rr

rr

rr

rr

•. + -«(*-/>

0

0

— 0.001

— 0.004

— 0.011

— 0.033

+ 0.052

+ 0.023

+ 0.013

+ 0.009

THE OKBIT OP NEPTUNE.

27

ACTION OF SATURN.

if

0

1

2

3

4

i

0

— 1

— 2

— 3

4

h
aDah
De'h
\Duh
L
E

+ 1026.84
+ 57.04
+ 0.780
— 0.773
+ 2167.76
0

+ 164.82
+ 178.60
— 1.092
— 0.99
+ 794.46
— 0.70

+ 39.15
+ 82.47
— 1.56
— 0.44
+ 268.80
— 0.33

+ 10.3
+ 32.3
— 1.02
— 0.17
+ 91.9
— 0.1

+ 2.8
+ 11.9
— 0.5
— 0.06
+ 31.2
— 0.1

SI -H sin N-

— 10.186

— 1.723

— 0.394

— 0.100

<ff -H- sin (N — /)
(*+/}

— 0.109
— 0.091

— 0.027
— 0.022

— 0.008
— 0.007

— 0.003
— 0.003

<51ogr-~ cos./V

— 89

— 21

— 6

— 2

(5/3-=-sm(JV+ V)
N— V

+ 0.004
+ 0.022

+ 0.001
+ 0.005

+ 0.001

i'

— 3

— 2

— 1

0

1

2

3

i

+ 3

2

1

0

— 1

— 2

— 3

h
aDah
De'h
L

+ 0.02
+ 0.09

+ 2.8
+ 0.24

+ 0.04
+ 0.13
+ 5.0
+ 0.39

+ 0.05
+ 0.09
+ 5.89
+ 0.33

— 0.0171
— 0.058
— 2.020
— 0.159

— 0.044
— 0.11
— 5.17
— 0.36

+ 0.139

+ 0.12
+ 16.5
+ 0.68

+ 0.124

+ 0.25
+ 14.7
+ 0.89

//

//

,,

,,

n

//

61 ~ sin N

+ 0.001

+ 0.002

+ 0.004

+ 0.005

— 0.004

— 0.004

11 N-Zf

- 0.024
0

— 0.064
— 0.001

— 0.151
— 0.002

— 0.133
— 0.001

+ 0.211
+ 0.002

+ 0.124
+ 0.001

i log r -=- cos (N — /)

0

-1

— 2

+ 1

0

0

V

— 2

— 1

0

1

2

3

4

i

+ 3

2

1

0

— 1

— 2

— 3

h

+ 4.23

+ 8.48

— 3.175

— 28.49

— 13.47

— 5.32

— 1.96

aDah

+ 8.4

+ 7.5

— 7.14

— 32.54

— 28.59

— 16.5

— 8.1

De'h

— 0.2

— 0.1

— 0.11

+ 0.15

+ 0.52

+ 0.5

+ 0.4

L

+ 27.0

+ 34.5

— 20.63

— 36.65

— 106.60

— 49.4

— 21.9

E

0

0

0

+ 0.12

+ 0.11

+ 0.1

0

SI -H sin N

+ 0.108

+ 0.200

— 0.217

— 2.158

+ 1.747

+ 0.355

+ 0.101

(Su-a-sin (N /)

+ 0.002

+ 0.002

— 0.001

— 0.034

+ 0.025

+ 0.007

+ 0.003

(tf+f)

+ 0.002

+ 0.002

— 0.001

— 0.020

+ 0.022

+ 0.006

+ 0.003

<? log r H- cos N

— 1

— 2

0

— 71

+ 18

- +5

+ 1

28

THE ORBIT OF NEPTUNE.

IV.-/=0; /' = -!.

h

aDah
De'h

L

E

JV— 3/

i log r -=- cos N
(W- /)

— 3

+ 4

— 0.07

-8.1

-8.1

+ 0.049
+ 0.001
0

0

— 2
3

— 0.17

— 0.50
-19.5

— 1.5
-19.5

— 0.006

— 1
2

— 0.30

— 0.57
-35.5

— 1.9
-35.5

— 0.011

_|_0.156 +0.410
-j- 0.002 + 0.004
0 0

0

0

1

+ 0.116

+ 0.38
+ 13.8

+ 1-1
+ 13.8

+ 0.012

— 0.291

— 0.003

0

0

— 3
0

1
0

+ 9.171
+ 1.560

+ 1083.89
— 1.46

+ 1083.86

— 0.086

(—127.63)

— 1.348

— 0.015

+ 29
-1344

— 17

2
— 1

+ 5.704
+ 6.43

+ 673.05
+ 36.64

+ 673.10

— 0.600

+ 22.056
+ 0.233

+ 0:003

— 9

+ 232
+ 3

+ 2.37
+ 5.02

+ 277.7
+ 18.6

+ 277.8

//
. — 0.134

+ 3.995
+ 0.042

2

+ 42
+ 1

4
— 3

+ 0.89
+ 2.8

+ 103.6
+ 8.6

+ 103.6

//
— 0.040

+ 0.955
+ 0.010

-j

+ 10

+ 0.32
+ 36.7
+ 36.7

+ 0.249
+ 0.003

De'h

— 2

+ 3

+ 0.7

— 0.006

2
— 0.04

//
0.000

0

1

+ 0.07

— 0.002

1
0

— 0.26

+ 0.031

2
— 1

— 0.44

— 0.014

3
— 2

— 0.39

//
— 0.006

VI.- j=l; j' = — 2.

i
De'h

rfw-s-sin (N — /)

— 2
+ 3

0.000

— 1

0.000

0.000

1
0

— 0.11

//

+ 0.013

2
— 1

— 0.14

//
-0.005

— 2
+ 0.46

//

+ 0.007

VIL— =

V
i

h

L

SI -=- sin N

\

+ 3

+ 0.3
+ 0.4
+ 1.5

+ 0.006

0
2

— 0.03

— 0.05

— 0.2

— 0.001

1
1

+ 0.18
+ 0.34
+ 0.95

2
0

+ 0.66
+ 1.44

+ 2.22

+ 0.008 +0.065

+ 0.48
+ 1.51

+ 5.7

— 0.129

4
— 2

+ 0.27
+ 1.08
+ 3.2

— 0.026

5
— 8

THE ORBIT OF NEPTUNE.

29

vm._y - — !;/-

i'

_,

0

1

2

3

4

5

i

+ 3

2

1

0

— !

2

— 3

h

— 0.02

0.0

— 0.04

— 0.50

— 0.41

— 0.23

— 0.11

aDah

— 0.04

— 0.02

— 0.11

— 0.61

— 0.88

— 0.72

— 0.45

De'h

— 2.1

— 0.49

— 5.17

— 69.02

— 47.95

— 26.96

— 12.92

L

— 0.1

0.0

— 0.30

— 0.96

— 4.20

— 2.41

— 1.40

tt

//

tt

tt

tt

<W-r- sin N

0

0

— 0.003

— 0.028

+ 0.096

+ 0.020

+ 0.007

tt

tt

<5u-:-sin(Ar— /•)

+ 0.016

+ 0.005

+ 0.092

+ 3.477

— 2.178

— 0.442

— 0.129

N— If

0

0

+ 0.001

+ 0.037

— 0.023

— 0.005

— 0.001

fi log r -r- cos (2V — /)

0

0

+ 1

+ 37

— 23

— 6

-1

=0 = — 2.

V
i

+ 3

0
2

1
1

2
0

3
— 1

4

— 2

5
— 3

h
aDah
De'h
L

0
0

+ 0.08
0

0
0

+ 0.05
0

+ 0.004
+ 0.008
+ 1.00
+ 0.02

+ 0.081
+ 0.022
+ 19.12
+ 0.05

+ 0.083
+ 0.098
+ 19.63
+ 0.72

+ 0.05
+ 0.11
+ 11.64
+ 0.45

+ 0.02
+ 0.08
+ 5.71
+ 0.24

«H-Bin(JV— /)

0

0

0

+ 0.001

— 0.016

— 0.004

tt
— 0.001

D A--2/

— 0.001

— 0.001

- 0.018

— 1.126

— 0.012

+ 0.891
+ 0.009

+ 0.191
+ 0.002

+ 0.057
+ 0.001

6 log r ~- cos (TV — /)

— 12

+ 9

+ 2

+ 1

V
i

— 1

+ 3

0
2

1

1

2
0

3
— 1

4
— 2

\Duh

+ 0.30

.+ 0.77

+ 1.68

+ 0.77

+ 0.30

+ 0.11

//

//

//

//

it

tt

(5/3-=- sin (JV— Vj

+ 0.002

+ 0.008

+ 0.030

+ 0.045

— 0.013

— 0.002

«•'

+ 2

3

4

5

6

t

+ 1

0

— 1

— 2

— 3

De'h

+ 043

+ 2.34

+ 2.44

+ 1.72

+ 0.7

//

//

tt

tt

//

<?!>-=- sin (# — /)

— 0.007

— 0092

+ 0.180

+ 0033

+ 0.008

30

THE ORBIT OF NEPTUNE.

XII.-^ = — !;/ = — 2.

¥

i

+ 2
+ 1

3
0

4
— 1

5
— 2

6
— 3

De'k

— 0.14

— 1.75

— 2.00

— 1.5

— 1.0

//

ir

it

i>

if

«v^a\n(N—f)

+ 0.001

+ 0.068

— 0.148

— 0.028

— 0.010

XIIL— y=0;/ = — 3.

i'
i

+ 2
+ 1

3
0

4
— 1

6

— 2

. 6
— 3

De'h

+ 0.02

+ 0.28

+ 0.40

+ 0.30

+ 0.3

ft

rf

//

//

ti

iv-~ sin (AT—/)

0

— 0.011

+ 0.030

+ 0.006

+ 0.003

THE ORBIT OP NEPTUNE. 31

ACTION OF JUPITER.

The direct action of Jupiter is so nearly insignificant that the details of the
computation are omitted. The only terms in the longitude exceeding one hun-
dredth of a second, and not sensibly confounded with the elliptic elements of
Neptune, are

0".278sin (% — X)
+ 0 .019 sin 2 (tf — A

ACTION. OF VENUS, EARTH, AND MARS.

The only appreciable effect of the attraction of these planets is found in the
relation between the radius vector and the mean motion. The coefficients of the
perturbative function which correspond to the case when both i' and i are zero
introduce changes as below into the secular variation of the longitude of the
epoch. Those which correspond to the term in which N=X — o' introduce con-
stants as below into the logarithm of the radius vector. For the sake of com-
pleteness we include the similar perturbations produced by Jupiter, Saturn, and
Uranus, as already computed :

de

It

Action of Venus, + 0".0403 — 11

Earth, + 0 .0444 — 12

Mars, + 0 .0059 — 2

Jupiter, +15.3571 —4240

Saturn, +4.8687 —1344

Uranus, + 1 .1261 —311

Total, 21.4425 — 5920

The principal term of y , and, indeed, the entire portion not multiplied by the

at

second power of the eccentricity, is

while the principal term in 8 log r is

8 lo r = —

The effect of these terms might, therefore, have been included in the mean
distance as a single term, without appreciable error.

§ 14. Perturbations of Neptune l>y Saturn throuyh the Sun.

These perturbations, it will be remembered, have been omitted in the preceding
computations, from reasons already set forth. They have been computed by
formula) (16) — (19), and are as follows:

32

THE ORBIT OF NEPTUNE.

- 20".536 sin (X — X)

0.007 sin (2 a/— 2 ;i — o' +
+ 0.530 sin (— a' + 2 a — o)

— 0 .059 sin (A, -co')

— 0 .340 sin (2 X — a — o')

+ 0 .022 sin (— 51' + 3 a — 2 o
. 0 .007 sin (A/ + a — 2 u)

— 0 .002 sin (2 a, — o — o')

ACTION OF SATURN.

8 log r =
+ 345 cos (a/ —

+ 10 cos (— 3,'+ 2 A — w)

— 2 cos (* — o')

+ 3 cos (2 a' — a, — o')

§ 15. Perturbations of the elements. — Collecting and adding up the coefficients
of all sines or cosines of the same angle in the perturbations, \ve find them as
below. For 2, and a, their values, I — t and -n — t, are substituted. We shall
first collect those terms which are developed as perturbations of the elements,
namely, the secular variations, and all terms in the action of Uranus in which
i' — 2 i. We find them to be as follows :

+ 125".67 sin (2 7' — 7)

— 0.42sin (27' — I — 2n)
-0.36 sin (2? — l+n — if)

+ 0.14sin (2 1 — I + Tt'— TI)

— 30".93 sin (4 Z' — 2 7 — TI)
+ 8 .03 sin (4? — 27 — Tt7)

• 0.03sin(47' — 27+7^ — 27t)

+ 2".62 sin (6 7' — 3 7 — 2 7t)

— 1. 37 sin (62'— 37 — rt — n)
+ 0.17sin(67'— 37 — 2 71')

+
+
+

2163".60sin(27' — 7 — TI)
14 1.69 sin (2 7' — 7— TI')

0.56 sin (27' — 7+ TT' — 2 ?t)
0.21 sin (27' — 7+7t — 27t/)
1 .08 sin (27' — 7 +71 —2 T)
O.OSsin (27' — 7+Tt' — 2r)

= + 125".G7 cos (2 ?' — 7)

+ 0 .42 cos (21' — I — 2 TT)
—0.36 cos (27' — I — it + 71}
+ 0.14cos (27' — 7+ rf — n)

— 30".93 cos (4 7' — 2 7 — n)
+ 8 .03 cos (4 7' — 2 7 — 7t')

— 0 .03 cos (4 7' — 2 7+ it— 2*)

+ 2".62 cos (6 7' — 3 7 — 2 n)
—1.37 cos (6 7' — 3 7 — Tt' — 7t)
+ 0 .17 cos (6 I — 37 — 2 TI')
+0".003U

a = — 1232 cos (2 7' — 7 — n)
+ 92 cos (2 7' — 7 — rf)

+ 85 cos (4 I' — 2 7 — 2 n)
— 44 cos (4 7'— 27 — T^ — n)
+ 6 cos (4 7' — 2 7— 27^)

— 71".93 sin (4 7' — 2 7 — 2 n)

+ 38 .09 sin (4 7' — 27 — TI' — 7t)

— 4 .99 sin (4 7' — 27 — 2 7t')

+ 4".36sin (67'— 37— 3 TI)
-3.27 sin (6 7' — 3 7 — Tt7 — 27t)
+ 0 .85 sin (G ¥ — 3 7 — 2 rt — n}

— 0.08 sin (07'— 37— 3^)

+ 2V.4425 i

THE ORBIT OF NEPTUNE.

33

= — rMlsin (2 f — I — 71+ T)

— 0 .72 sin (2 P — I — « — T)
+ 0 .16 sin (2 I' — I — 71!+ r)
+ 0 .15 sin (2 ? — Z — TI'— T)

— 2".98sin (4P — 2Z — T)

0".0110 «

— l".ll cos (2 Z' — Z — n + T)
+ 0 .72 cos (2 P — I — n — T)
+ O.lGcos (2Z' — Z — 7t+ T)

— 0 .15 cos (2 1' — I — rf— T)

— 2".98cos(4Z' — 2Z — T)

+ 0".0001<

§ 16. Perturbations of the co-ordinates — Comparison with PEIRCE and KOWALSKI.
— The first column of the following tables gives the coefficients according to
Peirce (Proceedings of the American Academy, Vol. 1, pp. 287-291) ; and the
second, the values according to Kowalski (Recherches sur les mouvements de
Neptune, pp. 14-16). In the case of Uranus, Peirce's coefficients have been
increased by £ + 3*5-* ^° reduce his mass of Uranus to the adopted one. The
coefficients enclosed in parentheses are not comparable, as they include the effect
of terms now developed as perturbations of the elements, and therefore omitted
from the perturbations of the co-ordinates. The perturbations of the radiua
vector have been reduced to logarithms by multiplying by

I.— ACTION OF URANUS.

c5 log r =

-206".91)

(_244".40)

+ 3'

'.002 sin (I' — 1) (—2284)

(— 2289)

+ 314 cos (I' — I)

f 10 .24

+ 10 .02

+ 9

.994 sin 2 (f — I) +

167

+ 163

+ 162 cos 2 (I — I)

f 2 .01

+ 2 .02

+ 1

.960 sin 3 (t — I) +

40

+ 69

+ 38 cos 3 (I — I)

f 0 .64

+ 0 .62

+ 0

.610 sin 4 (V — I) +

14

+ 38

+ 18 cos 4 (P — I)

f 0.25

+ 0 .27

+ 0

.237 sin 5 (V — I) +

5

+ 23

+ 5 cos 5 (V — /)

f 0 .11

+ 0 35

+ 0

.104 sin 6 (J' — I) +

2

+ 11

+ 1 cos 6 (f — I)

f 0 .05

+ 0 .27

+ 0

.041 sin 7 (I' — I)

f 0 .02

+ o

.017 sin 8 (f — I)

f 0 .01

+ o

.007 sin 9 (P — I)

+ 0".002 sin (— 4 V + 4 I — it" + it)

+ 0

.016 sin (— 3 1' + 3 I — it' + IT)

(- 0.11)

(- 0.73)

— 0

.103 sin (— 2 f + 2 1 — it' + IT)

(— 16.29)

(— 16.79)

— 0

.048 sin (— 1' + I — V + it)

(+ 0.66)

(+ 0.71)

+ 0

.045 sin ( 1'— l — ir' + it)

Q

.011 sin ( 2 1' — 2 I — it' + it)

+ 0

.003 sin ( 3 1' — 8 I — it1 + it)

+ 0

.003 sin ( 41' + 41 — IT" + it)

+ o

.002 sin ( 5 1' + 5 I — it' + it)

— 0.01

— 0".009 sin (— 5 1' + 6 1 — it)

— 0.01

— 0

.014 sin (— 41' + 51 — ?r)

— 0.02

— 0

.024 sin (— 3 f + 4 i — ir)

— 0.04

— 0.08

— 0

.033 sin (— 2 /' + 3 1 — IT)

+ 0.19

+ 0.19

+ 0

.183 sin (— 1' + 2 I — it) +

2

+ 2

+ 3 cos (— V + 2 1 — it)

+ 0.27

— 1.31

+ 0

.274 sin ( / — IT)

5

+ 11

— 5 cos (

l-it)

— 0

.238 sin ( /' — TT)

— 10 cos ( V

— it)

(1979.72)

(1955.50)

+ 4

.365 sin ( 21'— 1 — it) (—

1141)

(-1127)

+ 43 cos ( 21' —

I — it)

(+69.86)

(+68.73)

+ 9

.563 sin ( 31' — 21 — it) (+

693)

(+ 663)

+ 58 cos ( 3 1 —

21 — it)

— 1.78

-1.78

— 1

.721 sin ( 41' — SI — it)

28

— 27

— 27 cos ( 41' —

31 — it)

— 0.33

— 0.59

— 0

.375 sin ( 51' — 41 — it)

7

— 5

— 7 cos ( 61' —

41-it)

— 0.12

— 0.29

— 0

.134 sin ( 61' — 51 — it)

3

— 1

— 2 cos ( 61' —

51 — it)

— 0.06

— 0

.057 sin ( 7 V — 6 1 — ir)

2

— 2 cos ( 71' —

61— it)

— 0.04

— 0

.022 sin ( 81'— 71 — a)

_ 2cos( 81' —

71 — it)

— 0-01

-0

.009 sin ( 9 1' — 8 / — ir)

5 May, 1865.

34

THE ORBIT OF NEPTUNE.

ACTION OF URANUS (Continued).

p.

(-0.01)

tv —
K. N.
+ 0".001 sin (— 5 V + 6 1 — it1)
+ 0 .002 sin (— 4 V + 5 1 — it')
— 0 .002 sin (— 3P + 4J — Tr1)
— 0 .015sin(— 21' + 31 — it1)

P.

K.

6 log T =

jr.

+ lcoa(— SI' + 4 1 — IT)
— lcos(— 21' + 31 — it)

(-0.11)

— 0

.109 sin (— I' + 2 I — it1)

+

2

+

2cos(— I' + 21 — it)

(+ 2.33)

(+ 2.65)

— 0

.177 sin ( I — it1)

/

21)

(-

24)

+

2cos(

l-it)

+ o

.209 sin ( /' — it1)

+

5 cos (

I

-it)

(— 124.83)

(—132.51)

— 0

.466sin( 21'— I — it1)

(+

95)

(+

97)

—

13 cos (

2V

— l — it)

(- 17.45)

(— 18.37)

— 2

.477 sin ( SI' — 21 — it')

(-174)

(-

184)

—

15 cos (

SI' — 21 — it)

-+ 0.46

+ 0.53

+ 0

.452sin( 41' — SI — it1)

+

7

+

6

+

9 cos (

it

— 31 — it)

+ 0.11

+ 0.07

+ 0

.101sin( 51' — 41— it1)

+

2

+

1

+

1 cos (

6f

— 41 — it)

+ 0.04

— 0.23

+ o

.027 sin ( 6 V — 5 1 — it1)

3

+

1 cos (

9*

— 52 — it)

+ 0.01

+ o

.014 sin ( 7 1' — 6 I — it1)

+ o

.010 sin ( SI' — 11 — it')

+ o

.006 sin ( 9 V — 8 1 — it1)

+ 0".002 sin (3 V — 2 1 + <J — 2 it)

— 0

.016 sin (4 V — 3 I + a' — 2 it)

+ o

.002 sin (5 V — 4 1 + u' — 2 r)

(The terms in which the constant of the argument is n- — 2 it, it — 2 r, and it — 2 T* are yet smaller, and arc

neglected.)

+ 0".017 sin (— e + 3 I — 2 it)

— 0 .001 sin ( 2 I — 2 it)

~"

— 0 .007 sin ( V + I — 2 it)

— 0 .009 sin ( 2V —2 it)

(+ 0.75)

— 0.151 sin ( 91'— 1 — 2 it)

(—65.45) (—64.61)

— 0 .587 sin ( 4 V — 2 I — 2 it)

(— 5.10) (— 5.62)

— 1 .310 sin ( 51' — 31 — 2 it)

+ 0 .258 sin ( 6 V — 4 I — 2 it)

+ 0.061sin( 71' — 51— 2it)

+ 0 .027 sin ( 8 V — 6 1 — 2 it)

+ 0 .010 sin ( 9 V — 7 I — 2 it)

+ 0 .004 sin ( 10 V — 8 1 — 2 it)

+ 0".005 sin ( V + I — it> — it)

+ 0 .006 sin ( 2 V —it1 — it)

(+16.08) (+17.01)

+ 0 .098 sin (81'— I — it' — it)

(+ 33.73) (+ 36.67)

+ 0 .366sin( \l' — 21 — it' — it)

(+ 2.56) (+ 3.35)

+ 0.688sin( 5 V — 3 1 — it1 — it)

— 0 .136sin( 62' — 4Z — rf — it)

— 0 .032 sin ( 7 1' — 5 1 — it1 — it)

— 0 .019sin( 81' — 61 — it1 — it)

— 0 .010sin( 9f — 71 — it' — it)

— 0 .005sin(10f — SI — it' — IT)

— 0".003sin( 22 — 2^)

— 0 .011 sin ( V + 1 — 2 it')

(-1-04) (-1.15)

+ 0 .005 sin (3 V — 1 — 2 it')

(-4.29) (—4.79)

— 0 .046 sin (4 V — 2 1 — 2 it')

(—0.33) (—0.21)

— 0 .090sin(5/' — Zl — 2it")

+ 0 .020 sin (6 V — 4 I — 2 it1)

+ 0 .005 sin (7 «' — 61 — 2rt')

+ 0 .002 sin (8T — 6Z — 2it')

— 0".003sin( l'+ l — 2r)

— 0 .016 sin (3 i'— 1 — 2r)

— 0 .022 sin (4 I' - 2 I — 2 r)

— 0 .041 sin (5 1' — 3 1 — 2 r)

+ 0 .006 sin (6 V — 4 1 — 2 T)

Latitude.

it)

N.

it)

— 0"

.004 sin

(-41-

+ 5 / — r)

it)

— 0

,008 sin

(-81-

+ 41-T)

it)

— 0

023 sin

(_2J' + 3i — T)

it)

— 0

,056 sin

(- '

+ 21-T)

it)

+ o

.040 sin

(

l-r)

it)

+ 0,

106 sin

( f

-')

it)

+ o

.320 sin

( 2V

- l-r)

it)

+ o

.060 sin

( 31'

— 21 — r)

— 0

,063 sin

( 41'

— 3l — r)

— 0,

01 6 sin

( 5F

— 41 — T)

— 0.

005 sin

( Gl'

— 5J — T)

— 0

.004 sin

( TV

— 6 / — T)

THE ORBIT OF NEPTUNE.

35

II.— ACTION OF SATURN.

P. K. N.

— 18".60 — 18".12 — 18".552 sin (V — I)

+ 0.15 + 0.15 + 0 .141 sin 2 (V — I)

+ 0 .02 + 0 .03 +0 .012 sin 3 (V — I)

+ 0 .06 +0 .000 sin 4 (V — I)

+ 0".002 sin (— V + I — * + «•)
- 0 .006 sin ( 2 V — 2 I — ir1 + »r)

<! log r =

P. K. N.

+ 398 + 393 + 397 cos (f .
+ 4 0+4

+ 0 .54 + 0 53 + 0".524 sin (—
+ 0 .01 — 0 .10 +0 .008 sin (
+ 1 .319 sin (

- 0 .28 + 1 .09 - - 0 .280 sin (

- 0 .02 - - 0 .17 - 0 .023 sin (

— 0 .004 sin (

I' + 21 -ir) + 12 + 11 + 9cos(— I' + 21 — it)

*—•«•) — 2 + 2 cos ( I — K)

— ff) — 34 cos ( V — TT)

2 V — I — n) — 6 - - 20 - - 7 cos ( 2 V — I — TT)

3J'_ 21 — TT) — lcos( 31' — 21 — *)

4l' — 8l — K)

- 0 .08 - - 0 .09 - - 0".080 sin ( I — TT')

+ 0 .136 sin ( I' — u')

- 0 .22 + 3 .85 - - 0 .228 sin (2 I' — I — TT')
+ 0 .01 + 0 .04 +0 .008 sin (3 I' — 2 I — TT')

+ 0 .001 sin (4 I' — 3 I — K')

+ 0".022 sin (— I' + 3 I — 2 jr)
+ 0 .10 - - 0 .008 sin ( I' + I — 2 TT)

+ 0 .004 sin ( 21' — 2 w)

+ 0 .13 +0 .037 sin ( 3 I' — 1 — 2*)

- 0".002 sin ( 2 I — n' — ?r)

- 0 .002 sin ( V + I — TT' — TT)
+ 0 .020 sin (2 I' — V — jr)

+ 0 .10 — 0 .029 sin (31'— I — *' — ir)

+ 0".005 sin (21' — 2 *•')
- 0 .75 +0 .006 sin (3 I' — I — 2 TT')

,5/3 =

+ 0".309 sin ( I — T)

+ 0 .045 sin ( I' — T)
— 0 .005 sin (2 P — I — T)

— 3 — 3 — 1 cos ( I — ir')

+ 3 + 47 + 5 cos (21' — I — *•')

III.— ACTION OF JUPITER.

Sv =

.

_ 34".09 — 32".C7 — 34".121 sin (/' — I)
+ 0.02 + 0.03 + 0.019sin2(J' — I)

— 0 .14 +0 .003 sin 3 (f — I)

+ 0.11 - 0 .009 sin (2V— 21 — TT' +

+ 0. 82 +0 .84 + 0-.801 sin (— P + 2 I — TT)

— 0 .07 + 0 .003 sin ( I — TT)

+ 2 .358 sin ( I' — TT)

- 0 .01 + 0 .19 — 0 .010 sin ( 21'— I — n)

- 0.14 - - 0 .15 -- 0".143sin( I — w')

+ 0.117 sin ( I' —ir')
— 0 .42 - - 0 .48 — 0 .432 sin (2 I' — I — *)

S log r =

P. K. N.

+ 719 + 683 + 701 cos (V — I)

0 + 1 cos 2 (I' — I)

+ 17 + 17 + 18 cos (— /' + 2 / + »r)
+ 51 cos (— I' — «•)

— 6 — 27 — 2 cos ( / — «•')

— 2cos( /' — n-')

+ 6 + 135 + 7 cos (2 V — I — V)

36 THE ORBIT OF NEPTUNE.

ACTION OF JUPITER (Continued).

6v = (! log r =

P. K. N. P. K. N.

+ 0".030 sin (— /' + 3 I — 2 TT)
— 0 .011 sin ( I' + J — 2 TT)
+ 0 .004 sin ( 2 I' — 2 w)

+ 0.10 — 0".005 sin (2 I — ir1 — TT)
+ 0 .028 sin (2 V — ir1 — «)

<?/? =

+ 0".5G4 sin (I — T)
+ 0 .039 sin (I' — r)

By comparing the different authorities for the coefficients, it will be seen that
while our present results agree very well with those of Professor Peirce, the
agreement with Professor Kowalski is in many cases very far from being satis-
factory. It will be observed that the latter differ most in the case of those terms
whose coefficients depend on the action of the disturbing planets on the Sun, and
we have also seen that these terms are ordinarily developed as small differences
of very large quantities. They are, therefore, the terms, into which errors would
most easily creep.

The terms enclosed in parentheses are not of great importance, because they
are for a long period sensibly confounded with the elliptic elements. Notwith-
standing that one of these terms amounts to more than half a degree, and others
to several minutes, the effect of the whole of them could scarcely be discovered
from all the observations hitherto made on Neptune.

§ 17. For the purpose of tabulating and computing an ephemeris, it is expedient
to change the form of the perturbations by Uranus. Consider any two terms in
which the coefficients of 7 are equal, but of opposite signs :

&v =PI sin { si' — i A — u } -f- Pi sin { si' -\-iA — o> }
where

The terms may then be put in the form

{ (p.2 — p^) sin o sin iA + (PI + p\) cos a cos i A } sin si'
{ (pz — pj) cos 0 sin i A — (p2 -f p^) sin &> cos i A } cos si'

So that we may put

$v = to. + P,.t sin I' + P,t cos Z' + Ps.2 sin 2 I + Pc.2 cos 2 1'
8 log r — 5 log r0 -f- Rtl sin Z' -f- P^ cos Z'

where &v, P, and R are functions only of A, and may be tabulated as such.

§ 18. For Jupiter and Saturn, if we neglect those terms of which the coefficients
are less than 0".03, it will be more convenient to tabulate the perturbations
directly. This course we shall adopt, except with reference to those perturb-
ations which depend on the mean longitude of Neptune alone, and do not contain
the mean longitude of the disturbing planets. These have been omitted by both

THE ORBIT OF NEPTUNE. 37

Peirce and Kowalski, as may be seen by reference to the preceding values of their
coefficients. They are, in fact, very nearly confounded with the elliptic motion
of the planet, but not exactly. We shall, at present, retain only the small resi-
duals, after subducting those portions which are sensibly elliptic. The entire
terms are as follows :

1. In the longitude.

Action of Uranus, + 0".385 sinZ — 0".092 cosZ — 0".014 sin 2 1 — 0".002 cos 2 1
Saturn, + 0.099sinZ — 1.412cosZ — 0.018sin2Z — 0.020cos2Z
Jupiter, +2.393 sinl — 0.567 cosZ + 0.018 sin 2? — 0.029 cos 2Z

Total, + 2.877sinZ —2.071 cos? — 0.014sin2Z — 0.051 cos2Z (a)

2. In the logarithm of radius vector.

Action of Uranus, -f- 1 sin Z +14 cos I
Saturn, — 34 sin I
Jupiter, — 11 sin I — 51 cos I

Total, —44 sin Z —37 cos Z (Z>)

«

Changes in the functions e sin 7t and e cos n, represented by 5/t and 8k, will pro-
duce the following changes in the longitude and log r,

to = 2 Sfc sin Z — 2 Mi cos Z + f (We — 7tM) sin 2 Z — | (7^7i + Jfik) cos 2 Z
<5 log r — — M §/i sin Z — M &k cos Z.

Taking the elliptic terms to be subducted so that the coefficients of sin Z and cos Z
shall vanish, we must put

M = + 1".03G ; &• = + 1".438,
which will produce the inequalities

to = + 2".877 sin Z — 2".071 cos Z + 0".007 sin 2 Z— 0".037 cos 2 Z
6 log r — — 21 sin Z — 30 cos Z.

Subtracting these elliptic inequalities from (a) and (I), we have for the residuals

to = — 0".021 sin 2 I — 0".014 cos 2 Z
<51ogr — — 23 sin Z — 7 cos Z.

So that the constants of /*„ etc. are

Constant of Psl — 0

P*= 0

Ps2-_0".021
Pc,= — 0.014
#,! = — 23
3a = - 7

38

THE ORBIT OF NEPTUNE.

The constant terms in the coefficients Bsl and Bcl, which give the perturbations
of the latitude, may be omitted without any error amounting to one hundredth
of a second.

§ 19. The form of the preceding perturbations being different from that of the
perturbations computed by Professor Peirce, the elliptic elements are next pro-
visionally altered, so that the provisional theory shall be substantially identical
with that already adopted. Small corrections have also been applied to the
constants which determine the plane of the orbit.

The provisional elements finally adopted for correction are as follows :

e = 335° 5' 25".97

n=
7i =
fc =
p=
q=

7864.421
+ 1192 .93
+ 1279 .36
+4910.17
—4137.46

Epoch, 1850, Jan. 0, Greenwich mean noon. Unit of time, 365.25 days.

e= 0.00848055

e (in seconds) 1749".24
i 1°47' 1".95

n 42 59 52 .0

n 1-30 7 6.7

log a 1.4787523

The perturbations of the preceding elements are expressed in the following
form :
Put M = 21' — I

Then,
and

T — Number j)f centuries after 1850, Jan. 0.
M — 281° 43' 48" + 8° 26' 10".7 T;

$h = 125".42 sin ( M— 0° 16'.3)
+ 36 .08 sin (2 M+ 1° 50')
+ 3. 58 sin (3 M+ 3° 42')

+ 1".32 T + constant.

3Jfe = 126".17 cos ( M— 0° 6'.2)
+ 36 .08 cos (2 M + 1° 50')
+ 3 .58 cos (3 M + 3° 42')

+ 0".31 T -\- constant.

— 2247".52 sin ( M — 170° 32' 23") 5 log a = 1286 cos ( Af + 9° 8')

+ 98 .57 gin (2M+ 183° 24'.1) + 115 cos (2 M+ 4° 0')

+ 6 .81 sin (3 M + 186° 14') + constant.

+ 2144".26 T + const. + const. X T.

+ 2 .98 sin (2 M — 155° 38')
+ 1".10 T + constant.

M— 61° 0')
+ 2 .98 cos (2 M — 155° 38')

+ 0".01 T + constant.

THE ORBIT OF NEPTUNE.

39

The constants being so taken that the perturbations, and also the differential
coefficient of 81, shall all vanish at the epoch 1850.0. These perturbations are
given for the beginning of every tenth year, from 1GOO to 2000, in the following
table :

SECULAR AND LONG-PERIOD PERTURBATIONS OF THE ELEMENTS OF NEPTUNE FROM

1GOO TO 2000.

Date

a

d logrt

dh

dk

Sp

4

¥

¥

tt

tr

It

n

tt

n

n

1600

— 149.21

— 473

+ 22.33

— 45.28

— 4.68

+ 0.79

+ 8.58

— 116.52

10

137.63

455

21.18

43.85

4.48

0.79

8.29

111.82

20

126.50

437

20.05

42.39

4.29

0.78

7.99

107.12

30

115.83

418

18.94

40.90

. 4.10

0.77

7.69

102.42

40

105.62

400

17.85

39.37

3.90

0.76

7.39

97.73

50

— 95.88

— 381

+ 16.77

— 37.81

— 3.71

+ 0.75

+ 7.09

— 93.04

60

86.60

362

15.71

36.22

3.52

0.73

6.78

88.36

70

77.78

344

14.67

34.60

3.32

0.71

6.47

83.68

80

69.42

325

13.65

32.94

3.13

0.69

6.15

79.00

90

61.52

307

12.66

31.25

2.94

0.67

5.83

74.33

1700

— 54.09

— 288

+ 11.69

— 29.53

— 2.75

+ 0.64

+ 5.50

— 69.66

10

47.13

269

10.74

27.77

2.56

0.61

5.17

64.99

20

40.65

250

9.81

25.99

2.37

0.58

4.84

60.32

30

34.65

231

8.90

24.17

2.18

0.55

4.49

55.66

40

29.13

212

8.01

22.33

2.00

0.52

4.14

51.01

50

— 24.09

— 193

+ 7.14

— 20.45

— 1.81

+ 0.48

+ 3.79

— 46.36

60

19.52

174

'6.30

18.54

1.62

0.44

3.43

41.71

70

15.43

154

5.48

16.60

1.44

0.40

3.08

37.06

80

11.81

135

4.69

14.63

1.25

0.35

2.71

32.42

90

8.68

115

3.93

12.62

1.07

0.30

2.34

27.79

1800

— 6.03

— 96

+ 3.20

— 10.59

— 0.89

+ 0.25

+ 1.96

— 23.15

10

3.86

77

2.50

8.53

0.71

0.20

1.58

18.52

20

2.17

57

1.83

6.44

0.53

0.15

1.20

13.88

30

0.96

38

1.19

4.32

0.35

0.10

0.81

9.25

40

— 0.24

— 19

+ 0.58

— 2.17

--0.17

+ 0.05

+ 0.41

— 4.63

50

0.00

0

0.00

0.00

0.00

0.00

0.00

0.00

60

— 0.24

+ 19

— 0.54

+ 2.20

+ 0.17

0.06

— 0.41

+ 4.62

70

0.96

38

1.04

4.42

0.34

0.11

0.82

9.25

80

2.17

57

1.51

6.68

0.51

0.17

1.24

13.87

90

3.86

77

1.93

8.96

0.68

0.23

1.67

18.48

1900

— 6.03

+ 96

— 2.32

+ 11.26

+ 0.85

— 0.29

— 2.10

+ 23.09

10

8.68

115

2.67

13.59

1.01

0.35

2.54

27.70

20

11.81

133

2.98

15.94

1.17

0.42

2.98

32.30

30

15.44

152

3.25

18.31

1.33

0.49

3.43

36.90

40

19.53

171

3.47

20.70

1.48

0.56

3.89

41.49

50

— 24.10

+ 190

— 3.65

+ 23.11

-L1.64

— 0.63

— 4.35

+ 46.09

60

29.14

209

3.79

25.54

1.80

0.70

4.81

50.69

70

34.66

227

3.89

27.99

1.95

0.77

5.28

55.28

80

40.66

246

3.94

30.46

2.11

0.84

5.75

59.88

90

47.13

265

3.94

32.95

2.26

0.91

6.23

64.47

2000

— 54.09

+ 284

— 389

+ 35.46

+ 2.42

— 0.98

— 6.71

+ 69.06

ftp and $q refer to the fixed ecliptic of 1850.0, 5p'and V*° tne movable ecliptic
of the date, the motion being that adopted in Hanson's " Tables du Soleil," and
concluded from the secular diminution of the obliquity there given.

The corrections to the true longitude, latitude, and radius vector derived from
the pure elliptic elements require corrections for these perturbations as follows :

40

THE ORBIT OF NEPTUNE.

dv „ . dv „ . dv

For the period during which Neptune has been observed, we have, to a sufficient

degree of approximation,

dv

dl

,

-jj- — 2 COS /,

an,

d log r

-
dh

-f- =. COS V,

dp

= 1,

dv 0-7

-jj- = 2 sm Z:
dk

d log r _
~~dk~~~

d{3

-j— = sin v.

dq

Theyalues of Ps_i, Pc,t, etc., derived from the perturbations by Uranus, are, putting
A =: mean longitude of Uranus, minus that of Neptune,

0"

.68.1

sin

A

—

5"

.000 cos

A

0

.400

sin

24

— ,

11

.410 cos

2A

+

0

.044

sin

3.4

4-

2

.031

cos

SA

+

0

.006 sin

4.4

+

0

.462

COS

44

+

0

.009 sin

54

4-

0

.165

COS

-'..I

+

0

.001

sin

64

+

0

.076

COS

64

—

0

.003

sin

7 A

+

0

.035

COS

74

—

0

.002

sin

84

4-

0

.017

COS

84

—

0

.002

sin

9 A

+

0

.008

COS

94

—

0

.001

sin

10 A

4-

0

.004

cos KM

r.3 = — o".02i

— 0 .254 cos A

— 0 .867 cos 2 A

— I .821 cos 3 A
+ 0 .355 cos 4 A
4- 0 .083 cos 5 A
4- 0 .039 cos 6 A
4- 0 .018 cos 7 A
+ 0 .008 cos 8 A
+ 0 .004 cos 9 A

R^ = — 23

4- 4 cos 2 A

— 0".038 sin A

— 0 .035 sin 2 A

— 0 .147 sin 3 A
4- 0 .023 sin 4 4
4- 0 .006 sin 5 A

,i = 4- 0".328 cos A
4- 0 .005 cos 2 4

— 0 .078 cos 3 A

— 0 .022 cos 4 X

— 0 .009 cos 5 A

— 0 .006 cos 6 A

— 58 sin A

— 66 sin 2 A
4- 34 sin 3 A
4- 7 sin 4.4
4- 3 sin 5 A
4- 2 sin 6 A

4- 0".116 sin .4
4- 0 .048 sin 2 A

— 0 .017 sin 3 A

— 0 .003 sin 4 A

+

4-

.208

sin

A

—

0"

559

cos A

+

10

.892

sin

2A

—

0

.330 cos 2 A

—

l

.989

sin

34

+

0

.078

cos 3 A

—

0

.418

sin

4A

+

0

.018

cos 4 A

—

0

.135

sin

5 A

+

0

.013

cos 54

—

0

.056

sin

SA

+

0

.003

cos 6 A

—

0

.023

sin

1A

—

0

.009

sin

8^4

—

0

.004

sin

94

— 0 .002 sin 10 A

— 0".014

— 0 .022 cos A

— 0 .027 cos 2 A

— 0 .135 cos 34
4- 0 .025 cos 4 .4
4- 0 .008 cos 5 A
4- 0 .002 cos G .4

— 46 cos A

— 70 cos 2 A
4- 32 cos 3 A
4- 9 cos 4 A
4- 3 cos 5 A
4- 2 cos 6 A

4- 0".148 cos A
4- 0 .002 cos 2 ^

— 0 .035 cos 3^4

— 0 .009 cos 4 A

— 0 .004 cos 5 .4

4- 0".228 sin A
4- 0 .803 sin 2 4
4- 1 .849 sin 3 A

— 0 .355 sin 4 A

— 0 085 sin 5 A

— 0 .041 sin 6 A

— 0 .018 sin 1 A

— 0 .008 sin 8 A

— 0 .004 sin 9 A

— 1 sin 3 A

4- 2 sin 4 4

— 0".256 sin A

— 0 .105 sin 2 A
4- 0 .036 sin 3 A
4- 0 .008 sin 4 A

THE ORBIT OF NEPTUNE. 41

The other terms in the longitude, logarithm of r, and latitude, representing the
mean longitude of the planet by the initial letter of its name, are :

Sv, = — 2". 949 sin A — 0".002 cos A 6r,= 314 cos A

— 9.942sin2J. — 0.094cos24 + 162cos2X

— 1.967 sin 3 A + 0 .016 cos 3 A + 38 cos 3 A

— 0.610sin4.4 +0.004cos44 + 13cos4^1

— 0.237 sin 5 A -f 6 cos 5 A

— 0.104sin64 + 2cos64

— 0 .041 sin 7 A

— 0 .017 sin 8 A

— 0 .007 sin 9 A

+ 18".552sin (S — N) + 397 cos (S — W)

- 0 .137 sin 2 (S—;/) + 4 cos2 (S — JV)

— 0 .012 sin 3 (S — #)

— 0".524cos(2S— N) + 10 sin (2 S — N) + lcos(2S — JV)

- 0.058sin5 + 0.047cosS + 4sin(S — 2N) + 4 cos (S — 2 JV)
+ O.lGGsin (S— 2N) — 0 .436 cos (S — 2 N) + 70lcos (J—N)
+ 34.121sin (J — N) + 4sm(2J—N') + 18cos(2^— N)

- 0.011 sin 2 (J— N) — 5nin(J—2N) + 4 cos (J— 2N)
+ 0 .783 sin (2 J— N) — 0 .104 cos (2J— JV)

— 0 .101 sin J" + 0.007 cos J

+ 0 .326sin(^ — 2N) + 0 .297 cos (J — 2 If)

<5/3. = — 0".302 sin S + 0".OG5 cos S + 0".041 sin J + 0".5G3 cos J.

It will be observed that in the perturbations of the longitude by Jupiter and
Saturn we have neglected a number of small terms, the coefficients of the four
largest of which are each about 0".03. The probable error in the theory pro-
duced by this neglect is 0".04, and it was judged best, therefore, not to encumber
it with them. But, should any one wish to include their effect, it can readily be
calculated. Then, we have

Provisional longitude of Neptune, referred to the mean equinox
rr Precession, + Longitude in pure elliptic orbit, from elements page 39
+ M + CP..I + 2 3&) sin Z + (Ptl — 2 3A) cos I + P,.2 sin 2 1 + Pc,2 cos 2 1 + Sv,

-f- Reduction to ecliptic.

Common logarithm of the radius vector

= Log. radius vector in elliptic orbit
- .0005920 + 8a + (fltl — MSK) sin I + (R^ — MM) cos I + 8r0.

Latitude —

Latitude in elliptic orbit (the longitude being increased by the perturbations),

sin v + £,! — 8 cos v

I is the mean longitude of Neptune, and v its true longitude in orbit, referred
to the mean equinox of 1850.0.

§ 20. These formulae give the following heliocentric positions of Neptune :

6 May, 1865.

42

THE ORBIT OF NEPTUNE.

Heliocentric co-ordinates of Neptune, referred to the mean equinox of date,

for each ISOtft day, Greenwich mean noon.

Date.

Longitude.

Latitude.

log T.

0 / It

0 / H

1795, May 9,

215 5 20.12

+ 1 47 59.80

1.4817427

1846, Jan. 21,

325 28 41.54

— 0 28 26.90

1.4774075

July 20,

326 33 58.15

0 30 23.48

3215

1847, Jan. 16,

327 39 13.82

0 32 19.44

2356

July 15,

328 44 28.74

0 34 14.63

1510

1848, Jan. 11,

329 49 43.21

0 36 9.15

1.4770685

July 9,

330 54 57.50

— 0 38 2.92

1.4769892

1849, Jan. 5,

332 0 11.98

0 39 55.91

9135

July 4,

333 5 27.14

0 41 48.06

8420

Dec. 31,

334 10 43.37

0 43 39.37

7742

1850, June 29,

335 16 1.07

0 45 29.78

7104

Dec. 26,

336 21 20.50

— 0 47 19.25

6503

1851, June 24,

337 26 42.10

0 49 7.76

5935

Dec. 21,

338 32 6.10

0 50 55.27

5396

1852, June 18,

339 37 32.77

0 52 41.71

4880

Dec. 15,

340 43 2.18

0 54 27.04

4383

1853, June 13,

341 48 34.62

— 0 56 11.23

3895

Dec. 10,

342 54 10.02

0 57 54.26

3405

1854, June 8,

343 59 48.28

0 59 36.04

2907

Dec. 5,

345 5 29.19

1 1 16.58

2395

1855, June 3,

346 11 12.34

1 2 55.79

1856

Nov. 30,

347 16 57.47

— 1 4 33.67

1281

1856, May 28,

348 22 43.95

1 6 10.17

0671

Nov. 24,

349 28 31.22

1 7 45.23

1.4760028

1857, May 23,

350 34 18.63

1 9 18.83

1.4759357

Nov. 19,

351 40 5.78

1 10 50.93

8652

1858, May 18,

352 45 52.17

— 1 12 21.48

7918

Nov. 14,

353 51 37.46

1 13 50.50

7185

1859, May 13,

354 57 21.65

1 15 17.89

6454

Nov. 9,

356 3 4.60

1 16 43.65

5739

1860, May 7,

357 8 46.65

1 18 7.79

5049

Nov. 3,

358 14 27.84

— 1 19 30.24

4394

1861, May 2,

359 20 8.66

1 20 50.95

3779

Oct. 29,

0 25 49.45

1 22 9.96

3210

1862, April 27,

1 31 30.52

1 23 27.19

2691

Oct. 24,

2 37 12.30

1 24 42.64

2220

1863, April 22,

3 42 55.16

— 1 25 56.24

1797

Oct. 19,

4 48 39.57

1 27 7.97

1423

1864, April 16,

5 54 25.74

1 28 17.84

1093

Oct. 13,

7 0 14.10

1 29 25.77

0801

1865, April 11,

8 6 4.93

— 1 30 31.79

1.4750547

THE OllBIT OF NEPTUNE.

43

From these heliocentric positions are concluded the following apparent geocentric
positions, corrected for aberration, for the dates of the normal places to be given
in the next chapter.

Date.

Geocentric
Longitude.

Geocentric
Latitude.

Date.

Geocentric
Longitude.

Geocentric
Latitude.

O / /'

O / "

0 //

O / It

1795, May 9,

214 37 19.1

+ 1 50 34.4

1856, Aug. 8,

349 54 3.3

— 1 8 44.8

1846, Oct. 14,

325 31 34.9

— 0 31 56.0

Sept. 13,

348 58 37.4

1 9 27.2

Nov. 14,

325 23 23.2

0 31 44.0

Oct. 26,

347 56 48.8

1 9 8.2

1847, July 20,

329 41 22.0

— 0 35 25.9

Nov. 17,

347 40 53.1

1 8 35.3

Aug. 17,

329 7 18.3

0 35 47.7

1857, Aug. 13,

352 5 16.5

— 1 12 6.2

Oct. 8,

827 52 10.4

0 35 58.8

Sept. 21,

351 4 2.0

1 12 46.1

Nov. 18,

327 36 56.3

0 35 38.6

Oct. 24,

350 16 9.6

1 12 28.3

1848, July 25,

331 58 5.0

— 0 39 22.8

Dec. 8,

349 54 1.4

1 11 11.8

Aug. 29,

331 3 15.8

0 39 55.2

1858, Aug. 18,

354 23 52.6

— 1 15 21.0

Oct. 6,

330 8 55.3

0 39 57.8

Sept. 23,

353 19 17.2

1 15 56.6

Nov. 17,

329 49 22.4

0 39 32.8

Oct. 28,

352 28 49.2

1 15 34.8

1849, Sept. 1,

333 15 38.6

^0 43 52.7

Dec. 12,

352 8 46.7

1 14 11.8

Oct. 15,

332 15 32.4

0 43 49.2

1859, Aug. 21,

356 30 2.9

— 1 18 25.8

Nov. 25,

332 4 16.7

0 43 17.1

Sept. 23,

355 37 44.6

1 19 0.0

1850, Aug. 28,

335 39 38.5

— 0 47 42.7

Nov. 8,

354 34 29.4

1 18 23.0

Oct. 15,

334 31 9.5

0 47 41.1

Dec. 14,

354 22 52.5

1 17 9.3

Nov. 20,

334 15 23.8

0 47 9.5

1860, Aug. 20,

358 47 48.1

— 1 21 17.7

1851, Sept. 2,

337 48 58.1

— 0 51 33.7

Sept. 21,

357 54 28.1

1 21 56.4

Oct. 14,

336 48 10.9

0 51 30.0

Oct. 31,

356 58 22.4

1 21 32.2

Nov. 20,

336 28 31.0

0 50 54.1

Dec. 13,

356 36 24.7

1 20 4.6

1852, Aug. 7,

340 46 11.0

— 0 54 51.6

1861, Aug. 22,

1 2 42.4

— 1 24 6.0

Sept. 5,

340 0 10.3

0 55 19.5

Sept. 18,

0 21 7.6

1 24 42.2

Oct. 12,

339 5 43.0

0 55 14.8

Oct. 30,

359 16 38.4

1 24 24.6

Nov. 28,

338 43 23.4

0 54 23.3

Dec. 7,

358 50 3.3

1 23 7.9

1853, Sept. 1,

342 24 47.7

— 0 58 55.9

1862, Aug. 24,

3 17 34.7

— 1 26 46.7

Oct. 15,

341 19 0.2

0 58 52.4

Sept. 23,

2 31 7.7

1 27 25.7

Nov. 24,

340 56 3.0

0 58 4.7

Nov. 6,

1 25 26.1

1 26 57.3

1854, Aug. 30,

344 46 17.8

— 1 2 27.5

Dec. 15,

1 4 10.0

1 25 30.0

Sept. 24,

344 5 33.2

1 2 37.0

1863, Aug. 28,

5 29 46.2

— 1 29 23.7

Oct. 27,

343 23 17.3

1 2 15.3

Sept. 27,

4 42 49.6

1 30 0.3

Dec. 5,

343 12 2.8

1 1 19.1

Nov. 17,

3 30 58.0

1 29 12.5

1855, Aug. 10,

347 34 54.2

— 1 6 28.0

Dec. 12,

3 18 18.2

1 28 12.4

Sept. 8,

346 49 57.0

1 6 1.8

1864, Aug. 7,

8 9 20.6

— 1 30 53.7

Oct. 22,

345 45 6.2

1 5 50.1

Oct. 1,

6 52 56.7

1 32 27.0

Nov. 29,

345 24 25.3

— 1 4 54.8

Nov. 12,

5 51 37.8

1 31 49.1

Dec. 17,

5 32 27.7

— 1 30 23.1

The next step is to deduce positions of Neptune from observations, in order to
compare them with the above theoretical positions.

CHAPTER III.
DISCUSSION OF THE OBSERVATIONS OF NEPTUNE.

§ 21. DURING the four years following the discovery of Neptune, observations
of this planet, both meridian and extra-meridian, were very numerous. If the
results of all these observations were free from constant errors, and, therefore,
strictly comparable both with themselves and with subsequent observations, their
combination would give very accurate positions of the planet. Unfortunately,
however, we cannot assume that observations of different kinds, made at different
observatories, are strictly comparable, nor have we, in many cases, the data for
reducing them to a common standard.

Let us consider, for instance, the meridian observations. Under the title of
" Meridian Observations of Neptune," we find in astronomical periodicals series of
observed Right Ascensions and Declinations. But right ascensions and declinations
can never be really observed with any instrument. Only times of transit, and the
readings of micrometers and other instruments, are really observed. The right
ascensions and declinations of the planet are concluded from the observations, by
the aid of a great number of subsidiary data, some relating to the stars, others
to the instrument. Respecting these data we have, in most cases, absolutely no
information whatever. But a knowledge of some of them, at least, is indispen-
sable. Even if we grant that the instrumental errors are in all cases perfectly
known for every observation, we still do not know either the names or the assumed
right ascensions of the stars used in determining clock errors. Hence we cannot
use the results, because the right ascensions given in standard catalogues not
unfrequently differ by a second of space.

The declinations of the planet are sometimes determined by comparison with
standard stars, sometimes by measures of nadir distance, combined with the lati-
tude of the observatory. The Paris observations are reduced by the former
method ; those of most other observatories, by the latter. Using the latter method,
it would naturally be supposed that the declinations from the observations of all
observatories of which the latitudes are well determined ought to agree. But
such is far from being the case. Compare, for instance, the declinations of funda-
mental stars concluded from observations with the great transit circle at Green-
wich with those in the Tabulse Reductionum of Wolfers, and we shall find that
for stars more than 45° from the pole, the Greenwich positions are systematically
nearly a second south of Wolfers', an amount greater than the probable error of
a single isolated observation. We cannot impeach either authority. Wolfers'
positions depend on such authorities as Pond, Struve, Argelandcr, Henderson,
Airy, and Bessel. The conscientious care bestowed on the reduction of the
Greenwich observations would seem to render their results unimpeachable.
Besides, from a comparison of Winnecke's observations of his " Mars Stars" in

44

THE ORBIT OF NEPTUNE. 45

1862 with those of Greenwich, it would seem that the meridian circle of Pulkowa
gives declinations an entire second farther south than those of the great transit
circle ; so that had the Pulkowa instrument been employed on fundamental stars,
their declinations would have been 2" less than Wolfers'. On the other hand, the
Cambridge (Eng.) mural circle places the fundamental stars even farther north
than Wolfers, and the Washington mural nearly as far north.

It is foreign to our present purpose to speculate upon the causes of these dis-
crepancies ; we are concerned only with their existence and amount. Their
existence renders it absolutely necessary to correct the declinations as well as the
right ascensions in order to reduce them to a common standard ; and no obser-
vations have been used unless data for these corrections could be obtained.

This rule necessitates the entire rejection of nearly all the vast mass of obser-
vations on which Walker's theory was founded. In the case of the micrometric
comparisons, no sufficient data seem to exist for determining the positions of the
comparison stars ; the results are, therefore, heterogeneous in their character.
However valuable they might have been when made, it will not be admissible,
to combine them with the fifteen years of meridian observations made since.
Micrometric observations were almost given up after 1850, and the planet was left
to be followed by the meridian instruments of the larger observatories. The
superior accuracy of this class of observations may be inferred from the fact that
the comparatively small error in Walker's radius vector is made evident by them
even during the period of construction of Walker's theory.

A similar remark applies to the meridian observations. Four years of obser-
vations made at a great number of observatories may be indiscriminately combined
on the supposition that the systematic as well as the accidental errors will destroy
each other, particularly if each series extends through the entire period. But, as
few or none of these series made at observatories able to publish any thing but
their results are continued later than 1849, it will not do to assume that the mean
of their systematic errors, as fixed by the standard we have assumed, would vanish.

The observations which fulfil the conditions we have indicated are made at
observatories, as follows :

Ancient observations.
Paris, by Lalande, May 8 and 10 1795.

Modern observations.

Greenwich, 1846 to 1864.

Cambridge, 1846 to 1857.

Paris, 1856 to 1861.

Washington, 1846 to 1850.

Washington, 1861 to 1864.

Hamburg, 1846 to 1849.

Albany, 1861 to 1864.

§ 22. Reduction of Lalande s two observations of Neptune, May 8-10, 1795.
The first of these observations is found in the Comptes Rendus, tome 24, p. 667.
The second is in the Histoire Celeste, p. 158, and is the eighth star of the firsf

46

THE OKBIT OF NEPTUNE.

column. They were made with the large mural quadrant of the observatory
attached to the Military School. The Histoire Celeste does not seem to contain
any definite information as to the observer or observers by whom the observa-
tions were made.

The stars of comparison which I shall select for the determination of the errors
of the instrument and clock are the following :

May 10.

a Virginis,

1 Virginis,
/I Virginis,

2 Librae,
f.i Libra1,
£' Librae.

May8.

/? Virginis,
$ Corvi,
q Virginis,
•$/ Virginis,
a Virginis,
h Virginis,
x Virginis,
X Virginis,
2 Librae,
s Libra?.

These lists, I believe, include all of Bradley 's stars observed by Lalande on
the dates in question within the zone of the planet, for which reliable modern
positions can readily be obtained. Their positions for the year 1795 were obtained
as follows. The positions given by Bessel in the Fundamenta Astronomic were
reduced by the precessions there given to the mean equinox and equator of ITU"). (I.
The modern positions were obtained from the Greenwich Twelve Year Catalogue,
the Greenwich observations, or Eumker's Catalogue, and were also reduced to
1795.0 with Bessel's precessions. The difference of the results, being supposed
due to proper motion, was divided proportionally to the time, and the concluded
true position for 1795 obtained. As Lalande's observations arc subject to errors
of several seconds, any farther refinement in investigating the positions of the
stars would be a waste of labor. In the following table is exhibited the position
of the star at the two epochs, referred to the mean equinox and equator of 1795.0,
with the modern authorities, and the concluded mean positions for 1795.0 :

For 1795.0. Concluded.

Seconds of

Year of

> r l ,..,

Secondsof

Star.

R. A., 1755.

R. A., modern
epoch.

modern
epoch.

Modern
authority.

Dec. 1755.

modern
Dec.

R. A.

Dec.

6 Corvi,

h. m. s.
12 19 16.36

s.
15.87

1850

12 T. C.

0 / //

— 1522 13.2

//

28.7

h. m. s.

12 I'.l 16.1.1

/ //
— 15 22 19.7

q Virginis,

12 23 12.94

12.50

1840

12 Y. C.

— 819 9.8

9.7

12 23 12.7:!

— Sl'.i '.!.)•

Y< Virginis,

12 43 42.28

42.40

1840

12 Y. C.

— 825 17.5

21.9

12 43 42.34

— s 25 in.r>

t Virginis,

13 15 54.88

54.32

1859

Or. Obs. 1859

— 11 38 6.2

10.0

13 15 54.116

-1138 7.7

h Virginis,

1322 11.38

11.30

1845

12 Y. C.

— 9 611.5

15.1

13 22 11.34

— 96 13.1

K Virginis,

14 1 58.50

58.85

1840

12 Y. C.

— 91839.3

37.3

14 1 58.67

— 9 18 :;s.-1

\ Virginis,

14 8 2.29

2.41

1840

12 Y. C.

— 12 25 11.2

9.7

14 8 2.35

— 1225 10.5

2 Libraj,

14 12 25.07

25.17

1842

Rumker.

— 1046 5.9

11.7

14 12 25.12

— 1040 8.'i

fi Libra),

1438 C.33

6.19

1845

12 Y. C.

— 1317 6.0

8.4

14 38 6.27

— 1317 7.1

f Librae,

14 43 16.60

' 16.11

1842

Rumker.

— 11 3 6.2

6.3

14 43 16.37

— 11 3 6.2

t Librae,

1513. 6.60

6.14

1842

Rumker.

— 93422.7

57.6

1513 0.89

— 9 84 2!).r,

THE ORBIT OF, NEPTUNE. 47

The above places were reduced to the dates of observation with the constants
of the Tabulae Regiomontange.

The apparent positions of/? Virginia and a Virginis are derived from the same
work, correcting the Declination of the latter by + 0".60. The former is not used
for index error, owing to its distance from the zone of Neptune.

Intervals of wires.

0

On attempting to test the wire intervals of Lalande, H. C., p. 576, the interval
of the third wire was found to exhibit well-marked systematic discrepancies. The
observations of May 10 concur very well in indicating a diminution of OMO; and
this correction has been applied to Lalande's intervals. The interval for wire 1 has
not been changed.

Deviation of- instrument.

The next quantity required is the deviation of the instrument from the circle
of Eight ascension of the planet. On using Lalande's value of this correction,
stars of different altitudes, even in the zone of observation, gave inadmissible dis-
crepancies. It is found necessary to reduce the value to less than half. This
will be readily seen from the table below.

Clock error, &c.

The following tables give, for each star and each date —

The number of wires observed, ^ meaning a doubtful observation.

The concluded time of transit over the middle wire.

Lalande's correction to this time for deviation of the middle wire, this deviation
being supposed to vanish at the circle reading for Neptune, viz.: 60° 7'.

The correction for deviation actually applied, derived from the comparison of
clock corrections given by {$ Virginis and 8 Corvi.

Seconds of apparent R. A. of star.

The clock correction, using Lalande's deviation.

The clock correction, using the concluded deviation.

The weight assigned to the result for clock correction, depending on the number
of wires, and the proximity of the star to the planet.

For the second observation the deviation is of less importance than for the
first, the planet being near the middle of the zone, and the mean of the cor-
rections, therefore, very small.

48

THE ORBIT OF NEPTUXK.

1795, May 8.

Name of
star.

N.

T.

D'.

D.

R. A.

C'.

C.

W.

h. m. s.

a.

a.

S.

a.

s.

/? Virginia,

1}

11 39 42.67

— 8.80

— 1.90

1.86

22 99

21.09

0

S Corvi,

2

12 18 55.85

+ 0.81

+ 0.40

17.31

2o!ec

21.06

1

q Virginia,

2

12 22 52.10

—0.60

— 0.30

13.84

22.34

22.04

2

4 Virginia,

1

12 43 22.00

—0.58

— 0.12!)

43.53

22.11

21.82

1

a Virginia,

3

13 14 4.23

—0.24

— 0.12

26.09

22.10

21.98

4

A Virginia,

2

13 21 50.55

—0.44

— 0.22

12.67

22.66

22.34

3

n Virginia,

3

14 1 38.57

—0.40

— 0.20

0.09

21.92

21.72

6

A Virginia,

3

14 7 41.80

+ 0.20

+ 0.10

3.81

21.81

21.91

6

2 Libras,

2

14 12 4.45

— 0.11

— 0.05

26.53

22.19

22.13

5

E Librae,

1J

15 12 46.07

— 0.34

— 0.17

7.89

22.16

21.99

'>

May 10.

a Virginia,

2

13 14 3.55

— 0.24

— 0.12

26.09

» 22.78

22.C.6

1

i Virginia,

2

13 15 32.90

+ 0.06

+ 0.03

65.99

28.08

23.06

1

?. Virginia,

2

14 7 41.20

+ 0.22

+ 0.11

3.82

22.40

22.51

2

2 Libras,

2

14 12 3.40

— 0.11

— 0.06

26.53

23.24

23.19

2

fi Librie,

1

1437 45.10

+ 0.39

+ 0.20

7.79

22.80

22.49

1

f Librae,

2

14 42 64.95

— 0.06

— 0.03

17.88

^>-> i)ij

22.98

1

We have then

Clock time of transit of planet,
Correction for clock and instrument,
Concluded apparent Right Ascension,

or,

May 8.

14 11 36.50
+ 21 .94
14 11 58.44
212° 59' 3G".G

May 10.

14 11 23.50
+ 22 .82
14 11 4G .32
212° 56' 34".8

We use Bessel's refractions,
perature of the air, we have :

May8,

May 10.

Beginning of observations,
End "

Declinations.
For the height of the Barometer, and the tem-

in.

Bar. = 28 pou. 61. = 30.37 Eng. ; T= 13 Reau. = 61.2 Fah.
Bar. = 28 pou. 3.1 1. = 30.12 Eng. ; T= 13.7 Reau. = 02.8 Fah.
Bar. = 28 pou. 1.5 1. = 30.07 Eng. ; T= 13 Reau. = 61.2 Fah.

The equatorial points on the circle are concluded as follows :

May 8.

May 10

t

Name of

Mar.

Observed
Z. Dist.

Refrac-
tion.

Declination.

Equato-
rial point.

Name of
star.

Observed
Z. Dist.

Refrac-
tion.

Declination.

Equato-
rial point.

48° 49'

48° 49'

O t If

/ //

n

0 //

/ n

O 1 It

tt

S Corvi,

64 952

1 57.3

— 1522 29.0

20.3

rt Virginia,

5853 2

1 34.2

— 10 517.3

18.9

q Virginia,

57 7 11

28.0

— 8 19 17.6

21.5

t Virginia,

I;D 2.-> r.-t

1 40.2

— 11 38 14.2

20.0

\ji Virginia,

57 13 17

28.4

— 82526.6

18.8

A Virginia,

61 12 50

1 43.5

— 1225 15.4

18.1

a Virginia,

5853 0

34.3

— 10 5 17.3

17.0

2 Librae,

59 33 59

1 36.8

— 1046 13.3

22.6

A Virginia,

5754 5

30.6

— 9 619.0

16.6

ft Librae,

62 443

1 47.3

— 13 17 10.7

18.6

K Virginia,

58 637

31.4

— 9 18 43.0

25.4

f Librae,

59 50 50

1 37.9

— 11 3 9.6

18.4

A Virginia,

61 12 43

43.5

— 12 2515.2

11.3

2 Librae,

59 33 57

36.9

— 104(1 13.1

20.8

' Librae,

58 22 13

32.4

— 9 3431.7

13.7

THE ORBIT OF NEPTUNE. 49

Taking the means of the separate results for equatorial point, we have, for the
apparent declinations of Neptune —

May 8. May 10.

•

o ' // o i ft

Observed circle reading, 60 8 17 60 7 19

Refraction, 1 39.0 1 39.0

Corrected circle reading, 60 9 56.0 60 8 58.0

Equatorial point, 48 49 18.4 48 49 19.6

' Apparent declination, — 11 20 37.6 - 11 19 38.4

§ 23. Probable errors of these positions.

So far as we can judge from the discordance of the clock errors, and equatorial
points derived from the several stars, the probable error of a single observation
over a single wire in right ascension would appear to be about Os.27, and the pro-
bable error of a single observed zenith distance about 2". 2. The agreement of
the difference of the two observations with the computed motion of the planet
shows that neither observation is affected with any abnormal error. We conclude,
therefore, that the probable error of the normal place derived from the two obser-
vations is about 2". 8 in R. A. and 1".5 in declination.

Notwithstanding the magnitude of these probable errors, the observations will
be very valuable during the remainder of the present century, owing to the weight
with which they enter into the expressions of the elements. But in the twentieth
century the observations made after 1846 will enable astronomers to compute the
position of the planet in 1795 with a much higher degree of accuracy than La-
lande could observe it.

A similar remark applies to Lament's accidental zone observations in 1845.
Valuable during the first two or three years, they afterward ceased to be so,
because the theory soon became more accurate than the observation for an epoch
so near the time of optical discovery. Had they been made in 1820, they would
still have been valuable.

Reduction of the modern observations.

§ 24. The modern observations will be treated in the following manner. The
observations of each year will be divided into four groups, according to the time
of culmination of the planet. The first group will include all observations made
after

h. m. h. m.

13 30 m. t.

Second, between 10 30 and 13 30.

Third, « 7 30 and 10 30.

Fourth, all made before 7 30.

The mean correction derived from each group will at first be regarded as the
true correction applicable to the mean of the times of observation. This involves
the supposition that the error of the ephemeris is changing uniformly during each
series of observations. If we could compare with an ephemeris of the heliocentric

7 May, 1865.

50 THE ORBIT OF NEPTUNE.

place of the planet, this hypothesis would be sufficiently near the truth for an
entire year or more. But the error of geocentric place would be subject to an
annual period though the errors of the heliocentric place should be invariable.
Let us estimate the error of the hypothesis in question. Put

r — radius vector of Neptune.

D •=. difference of longitude of Sun and Neptune.

fe, 5r, errors of heliocentric longitude and radius vector.

Then the errors of geocentric longitude will be, appi'oximately,

. /.. cosZ>\ $r .
fr^l + — j+yi«n.ft

Of this expression the part

8v Sr . „

- cos D + -j sin D
r r2

will not be regularly progressive, but will change with the sine and cosine of D,
the period of which is about 368 days.

The integral of this expression gives 'for the mean value of the error, while/)
is increasing from D0 to D^

$v sin Dl — sin D0 fir cos />, — cos /},

T A — A " ? ~A^A

By putting

t l- -u,

and developing according to powers of 5, this expression becomes

This, plus the error of heliocentric longitude, is the mean error which will be
given by a series of observations equally scattered through a period + 8 on each
side of the mean epoch D. But what we really want is the error at the mean
epoch itself; that is,

«• i OV n t -Of • 7-,

dv + — cos D + -7, sm D;
T r

so that we must correct the mean error actually found by the quantity

or, since 8 is generally about 1J, and r about 30,

.027(1 eo-i + ^ri

THE CEBIT OF NEPTUNE.

51

The maximum value of &v being less than 30", the first term will be entirely
neglected. The value of &r sometimes amounts to .018, so that the correction
arising from the second term may sometimes amount to 0".ll. We shall, there-
fore, take account of it in a few cases.

The ephemeris which will be compared with observation in order to deduce
normal places of the planet will be the same with which the Greenwich obser-
vations are compared, namely, "Walker's ephemeris until the year 1854, and
Kowalski's ephemeris in subsequent years. It will be remembered, however,
that these ephemerides are used only for the purpose of obtaining normal places,
and in order to save the trouble of comparing every individual observation with
the provisional theory.

§ 25. Mean corrections of the Ephemeris of Neptune given by observations at the
different observatories, without correction for systematic differences.

GREENWICH.

CAMBRIDGE.

Dale.

K. A.

Dec.

No.

Date.

R.A.

No.

Dec.

No.

»

If

s

It

18-16, Oct. 14,

-0.050

+ 0.48

12

1846, Oct. 13,

— 0.014

10

+ 1.51

8

Nov. 16,

— .070

+ 0.55

7

Nov. 7,

.000

14

+ 1.43

15

1847, July iv,.

— .160

+ 2.05

4

1847, July 27,

— .062

6

+ 2.30

4

Aug. 20,

— .097

+ 2.23

10

Aug. 22,

— .090

18

+ 2.25

17

Oct. 3,

— .145

+ 1.43

10

Oct. 8,

— .100

14

+ 2.18

13

Nov. 24,

— .056

+ 1.70

8

Nov. 20,

— .022

13

+ 0.66

14

1848, July 28,

— .062

+ 1.16

4

1848, July 22,

— .056

8

+ 0.88

8

Aug. 81,

+ .002

— 0.20

8

Aug. 27,

— .048

19

+ 1.85

19

Oct. 7,

— .104

+ 0.01

14

Oct. 9,

-.018

16

+ 0.75

17

Nov. 16,

+ .022

+ 0.11

4

Nov. 19,

+ .009

10

+ 1.05

11

1840, Sept. 3,

— .027

+ 0.65

6

1849, Aug. 21,

— .087

16

+ 0.72

18

Oct. 17,

+ .080

+ 1.70

8

Oct. 15,

— .038

16

— 0.17

16

Nov. 28,

— .060

4-1.96

5

Nov. 22,

+ .090

11

+ 0.21

2

1850, Aug. 27,

— .079

— 1.00

13

1850, Aug. 29,

— .089

10

+ 0.48

11

Oct. 16,

+ .040

+ 0.20

13

Oct. 16,

+ .011

14

— 0.13

15

Nov. 24,

+ .020

— 0.62

16

Nov. 23,

+ .039

12

+ 0.14

13

1851, Sept. 1,

— .162

— 1.07

16

1851, Sept. 4,

— .054

18

— 1.62

19

Oct. 12,

+ .060

— 0.97

4

Oct. 17,

+ .028

11

— 1.61

10

Nov. 9,

— .040

— 1.94

5

Nov. 28,

+ .014

9

— 1.91

10

1852, Aug. 7,

— .260

— 2.34

5

1852, Aug. 29,

— .037

15

— 1.53

15

Sept, 11,

— .160

— 2.44

10

Oct. 11,

— .048

10

— 2.52

11

Oct. 12,

— .140

— 3.36

10

Dec. 4,

+ .038

7

— 2.99

7

Nov. 22,

— .080

— 2.27

6

1853, Sept. 2,

— .018

4

— 2.48

5

1853, Sept. 1,

— .250

— 2.59

14

Oct. 24,

— .134

13

— 3.04

13

Oct. 11,

— .177

— 2.93

16

Nov. 27,

+ .031

11

— 2.53

12

Nov. 19,

— .160

— 2.71

8

1854, Sept. 4,

1.314

11

— 3.59

12

1854, Aug. 30,

— .420

— 3.60

13

Oct. 11,

— .273

15

— 4.38

3

Sept. 21,

— .370

— 3.94

11

Nov. 24,

— .165

'4

— 6.17

8

Oct. 27,

— .310

— 3.68

7

Dec. 5,

— .300

— 4.36

4

f

If

1855, Aug. 10,

— .189

— 0.84

7

1855, Sept. 8,

— 0.046

12

+ 0.48

9

Sept. 8,

— .046

— 0.06

16

Oct. 12,

+ 0.50

6

Oct. 22,

+ .183

+ 0.80

6

Dec. 10,

+ 0.206

9

+ 3.07

7

Nov. 29,

+ .177

+ 1.51

6

1856, Sept, 12,

— 0.099

9

+ 0.05

8

1856, Aug. 8,

— .220

— 1.06

10

Oct. 29,

+ 0.120

8

+ 1.73

7

Sept. 13,

— .080

— 1.06

7

Nov. 28,

+ 0.164

5

+ 2.50

6

Oct. 26,

+ .076

+ 1.41

9

1857, Sept. 14,

— 0.030

9

— 0.83

12

Nov. 17,

+ .123

+ 1.67

6

Oct. 25,

+ 0.104

5

— 0.34

5

1857, Aug. 14,

— .356

— 2.43

6

Dec. 11,

+ 0.175

8

+ 0.16

9

Sept. 22,

— .130

— 0.50

12

Oct. 24,

+ .oso

+ 0.29

5

Dec. 5,

+ .130

+ 0.16

10

1858, Aug. 18,

— .301

— 1.74

14

Sept. 24,

— .260

— 1.81

13

Oct. 25,

— .200

-1.00

16

Dec. 10,

— .058

— 0.76

11

1850, Aug. 19,

— .500

— 3.27

9

Sept. 28,

— .416

— 8.11

17

Nov. 3,

— .315

— 2.56

15

Dec. 16,

— .328

— 1.46

10

52

THE OllBIT OF NEPTUNE.

GREENWICH (Cont.).

PARIS (Walker).

Date.

H. A. Dec.

No.

Date.

R. A. No. Dec.

No.

a

l

1860, Aug. 20,

_ 0.760 —5.02

4

1856, Sept. 14,

— 0.669 ' 12 —3.96

- 8

Sept. 20,

— 0.685 —3.86

13

Oct. 2"),

— 0.606 14 —8.91

15

Oct. 31,

— O.G62 —3.38

15

Dec. l!l.

— 0.470 2

Dec. 13,

_ 0.630 —4.73

2

1857, Sept. 19,

— 0.768 10 —4.98

10

1861, Aug. 22,

_ 0.040 —5.41

4

Oct. 25,

— 0.825 13 —5.95'

12

Sept. 18,

— 0.861 —5.62

16

Dec. 14,

— 0.729 7 —6.13

7

Oct. 30,

— 0.861 —5.18

12

Dec. 7,

— 0.993 —5.14

7

1862, Aug. 24,

— .12 —7.20

6

PARIS (Kowalski).

Sept. 25,

— .162 —6.76

13

Date.

R. A. No. Dec.

No.

Nov. 4,

... .139 —7.00

11

g

Dec. 17,
1863, Aug. 28,

— .18 —7.06
_ .585 —9.91

4
2

1858, Sept. 21,
Oct. 27,

— 0.291 18 —1.14
— 0.235 19 —0.76

17
19

Sept. 23,
Nov. 12,
Dec. 15,

_ .461 —8.62
— .388 —8.33
— .375 —8.41

9
4

2

1859, Aug. 23,
Sept. 23,
Nov. 17,

— 0.630 5 —2.64
— 0.474 9 —2.50
— 0.337 11 —1.92

5
8
14

1864, Oct. 3,

— .680 »— 11.03

8

Dec. 7,

— 0.430 2 —2.20

2

Nov. 8,

— .639 —10.52

7

1860, Sept. 29,

— 0.608 6 —3.75

6

Oct. 31,

— 0.618 12 —3.51

10

1861, Sept. 28,

— 0.960 7 —5.56

7

WASHINGTON (Walker's Eph.).

Nov. 4,

— 0.948 17 —5.52

17

Date.

R. A. No. Dec.

No.

Dec. 9,

— 0.890 5 —5.60

4

1

1846, Nov. 9,
1847, Aug. 23,

4-0.096 10 -)- 2. 1C
— 0.205 15 +2.27

7
6

HAMBURG.

Oct. 14,

— 0.023 9 4-1-80

7

Date.

R.A. Dec. No.

Nov. 8,

+ 0.052 5 +1.96

3

s "

1848, Aug. 30,

— 0.076 10 +0.80

14

1846, Oct. 7,

_ 0.098 —1.43 9

Oct. 2,

— 0.121 12 +1.71

10

Nov. 21,

— 0.131 +0.10 16

1849, Sept. 11,

— 0.154 5

1847, Aug. 22,

_ 0.061 —1.43 17

Oct. 12,

— 0.014 8 +1.06

6

Oct. 10,

— 0.102 +0.10 22

1850, Oct. 13,

— 0.032 9 + 0.94

25

Nov. 26,

— 0.051 +0.32 12

Nov. 11,

— 0.068 5 +1.25

13

1848, July 20,

0.000 —0.20 7

Aug. 27,

— 0.118 —2.17 15

1861, Oct. 29,

— 1.695 6 —8.70

5

Oct. 4,

_ 0.043 —2.15 12

Dec. 16,

— 1.593 11 —8.64

10

1849, Sept. 3,

_0.014 —0.92 9

1862, Sept. 23,

— 2.000 2

Oct. 10,

_ 0.008 —0.50 16

Nov. 14,

— 1.860 3 — 9.85

2

Nov. 25,

+ 0.071 —0.71 10

Dec. 12,

— 1.827 C —10.9

1

1863, Oct. 13,

— 2.22 3 —13.82

5

Nov. 12,

— 2.19 3 —13.52

6

ALBANY (Walker).

Dec. 8,

— 2.054 5 —12.76

7

Date.

R. A. No. Dec.

No.

1864, Aug. 7,

— 2.69 3 —15.5

4

Nov. 17,
Dec. 20,

— 2.52 5 —14.6
— 2.38 5 —14.0

12

2

1862, Aug. 25,
Sept, 21,

— 1.927 3 —13.05
— 1.905 14 —13.67

4
17

Oct. 81,

— 1.815 9 —13.46

10

Dec. 17,

— 1.732 6 -12.73

6

ALBANY (Kowalski).

1863, Sept. 27,

— 2.228 12 — 14.96

1

Date.

R. A. No. Dec.

No.

Nov. 6,

— 2.145 10 —15.38

9

g '

Dec. 14,

— 2.053 6 —14.80

6

1861, Sept. 1,
Nov. 11,

— 0.778 5 —6.02
— 0.825 8 —5.42

5

8

1864, Sept. 29,
Nov. 9,

— 2.490 6 —17.70
— 2.437 9 -16.29

6
9

Dec. 14,

— 0.847 9 —5.44

9

Dec. 14,

— 2.360 3 —16.27

3

THE ORBIT OF NEPTUNE.

53

§ 20. Corrections to the observed 2>ositions in order to render them strictly com-
parable with each other.

These corrections have been derived from a comparison of the positions of the
ten fundamental clock stars, from y Aquilre to a Ceti inclusive, given by obser-
vations at the different observatories, with the adopted standard positions. The
standard right ascensions are those of Dr. Gould, prepared for the United States
Coast Survey. The declinations are those of Wolfers in the " Tabulte Reduc-
tionum," diminished by 0".50. Both are given in the following table :

R. A. 1850.0.

Annual
var. 1850.

Dec. 1850.0.

Annual
var. 1850.

Cor. to Am. Eph.

R. A.

Deo.

h. m. s.

s.

O r If

ft

7 Aquiloo,

19 39 7.08

+ 2.853

+ 1015 5.02

+ 8.41

+ 2

— 2

a Aquils!,

19 43 27.82

2.928

+ 8 28 33.45

9.13

— 1

+ 5

ft Aquilie,

1947 50.f,7

2.948

+ 62 8.08

8.62

+ 3

+ 5

a2 Cupricorni,

20 9 43.69

3.335

— 13 020.90

10.77

+ 2

+ 13

a Aquarii,

21 58 4.fi8

3.084

— 1 2 47.41

17.28

+ 4

+ 8

a Pegnsi,

22 57 17.50

2.983

+ 14 23 57.36

19.30

0

+ 1

a Andromeda;,

0 038.57

3.085

+ 28 1543.72

19.91

+ 2

0

7 Peg.-isi,

0 5 30.99

3.081

+ 14 20 57.67

20.04

+ 2

+ 1

a Arietis,

1 58 43. G4

3.304

+ 2244 1.97

17.29

+ 1

0

a Ceti,

2 54 2(5.58

+ 3.130

+ 3 29 52.45

+ 14.42

+ 6

+ 5

In reducing the Albany observations, it was found advisable to add o Piscium to
the number of standard stars for determining these corrections. Its assumed
position is

R. A. 18GO.O.

23 51 7.42

Declination 1800.0.

+ 6

0 5' 17". 9

The observed mean right ascensions and declinations of these stars, reduced to
the beginnings of the several years, have been compared with those derived from
the above table, giving the result from each star a weight proportional to the
number of observations when the observations were few in number, but giving
each result equal weight when they were numerous. Thus the following sys-
tematic corrections have been derived:

OllKKXWICII.

CAMBRIDGE.

WASHINGTON.

R. A.

Dec.

m

R. A.

Dec.

R. A.

Dec.

s

"

"

»

"

1810

+ 0.044

— 0.04

1846

-1.21

1840

+ 0.034

— 0.34

47

+ 0.059

— 0.19

47

-0.99

47

+ 0.057

— 0.41

48

+ 0.08

48

— 0.25

48

+ 0.036

— 0.57

49

:-,(>

— 1.51

— 0.52

49

50

s
— O.OS8

— 0.05
— 0.80

49 )

50 j

+ 0.0:19

— 1.04

51

— 0.020

— 0.22

51

— 0.052

— 1.17

01

+ 0.058

— 0.89

52

+ 0.21

52

— 0.20

63

— 0.09

54

+ 0.18

53

— 0.75

04

— 1.37

r,.-,

+ 0.05

54

— 0.044

-0.73

50

— 0.038

+ 0.30

57

— 0.052

00

— 0.005

01

0.003

+ 0.24

02
63

+ 0.63
+ 0.27

PABIS.

ALBANY.

OY. Cat.

»

n

8

„

of 1854

— 0.020

1850

+ 0.020

— 0.57

1801

+ 0.006

+ 0.17

7V. Cot.

58

+ 0.021

— 0.23

02

000

0.00

of 1800

+ 0.002

00

+ 0.023

— 0.53

03

000

+ 0.78

04

000

+ 0.97

54 THE ORBIT OF NEPTUNE.

REMARKS ON THE PRECEDING CORRECTIONS.

GREENWICH.

The corrections actually applied to the right ascensions from 1848 to 1853 have
been derived by comparing the corrections on p. IV. of the introduction to the Green-
wich six-year catalogue for 1854 with the corrections given by that catalogue,
namely, — 0'.020. From 1857 to 1864 the corrections have been derived in the
same way from the seven-year catalogue for 18GO. The entire list of corrections
is as follows :

1846,

+ 0'.044

47,

+ 0 .059

48,

+ 0 .052

49-55,

— 0.010

56,

— 0.025

57-61,

— 0.008

62-64,

+ 0 .002

The corrections to the declination have been concluded from year to year from
the table.

CAMBRIDGE.

One consistent set of adopted right ascensions having been used in the re-
ductions of the Cambridge observations, the constant correction

— Os.046

has been applied to the right ascensions throughout. The declinations have been
corrected as follows :

1846-47, — 1".12

1848-57, — 0 .58

WASHINGTON.

The corrections to the Washington right ascensions from 1S46 to 1850 have
]>ecn derived from a general comparison of twenty-five fundamental stars near
the equator with the results of the Greenwich observations. The mean -f ()".(J42
has been adopted as the constant correction for those years. After 1861, no
correction is needed, Dr. Gould's Right ascensions having been adopted in the
reductions.

The corrections to the declinations for 18G1 have been derived from those for
1862. The latter were diminished by 0".20 for error of nadir point, while no such
correction was applied to the former.

HAMBURG.

Having applied to Charles Rumker, Esq., M.A., of the observatory at Hamburg,
fur information respecting the data used in the reduction of the Hamburg obser-
vations of Neptune, I was informed that both right ascensions and declinations

THE ORBIT OF NEPTUNE. 55

depended on the positions of the Nautical Almanac stars. For the years 1846-47,
the Nautical Almanac right ascensions require the constant correction — 0'.003,
and in 1848-49 the correction +0*.049, to reduce them to those adopted.

The declinations do not seem so easily reducible to our adopted standard.
They are, therefore, not included.

All the Washington, and some of the Paris and Albany, observations having
been compared with Walker's Ephemeris in years subsequent to 1855, the fol-
lowing corrections have been applied for differences of Ephemerides :

To Paris Corrections.

Date. R. A. Dec.

s

1856, Sept. 14, +0.54 +4.68
Oct. 25, +0.65 +5.75

1857, Sept. 19, +0.676 +5.02
Oct. 25, +0.80 +5.82
Dec. 14, +0.80 +5.92

To Washington and Albany Corrections.

Date. R. A. Dec.

1861,

Oct.

29,

+

0

s
.76

+

5

.5

Dec.

16,

+

0

.70

+

4

.9

1862,

Aug.

25,

+

0

.90

+

6

.3

Sept.

21,

+

0.85

+

6.2

23,

+

0.85

+

6

.-2

Oct.

31,

+

0.76

+

5

A

Nov.

14,

+

0.75

+

5

.0

Dec.

12,

+

0.70

+

5.1

17,

+

0.70

+

5

.1

1863,

Sept.

27,

+

0.845

+

5

.9

Oct.

13,

+

0.81

+

5.6

Nov.

6,

+

0.78

+

5.3

12,

+

0.77

+

5.5

Dec.

8,

+

0.

73

+

5.

2

14,

+

0.73

+

5.2

1864,

Aug.

7,

+

0.91

+

6.2

Sept.

29,

+

0.

87

+

6.

0

Nov.

9,

+

0.

90

+

5.

9

17,

+

0.

88

+

5.

5

Dec.

14,

+

0.

82

+

5.

8

20,

+

0.

82

+

5.

5

56 THE ORBIT OF NEPTUNE.

§ 27. The concluded corrections of the ephemeris for normal dates generally
near the mean of the means have been concluded by applying to the corrections
of pp. 51, 52 the following corrections :

1. Correction for systematic error given by fundamental stars.

2. Reduction, when the change of error was rapid, from the dates of the means
to the dates of the normals.

3. 0.027 ^-jf sin D for second differences of error, when 8r > .01.

4. Correction just given for difference of epheme rides.

The results are given in the following table. The small figures show the
relative weights assigned to the separate results, which are, to a certain extent, a
matter of judgment, but which are assigned without any reference to the magni-
tude of the correction itself.

THE ORBIT OF NEPTUNE.

57

CORRECTIONS TO THE TABULAR RIGHT ASCENSIONS GIVEN BY THE DIFFERENT

OBSERVATORIES, WITH THE CONCLUDED CORRECTIONS AND CONCLUDED NORMAL

RIGHT ASCENSIONS.

(The units are hundredths of seconds of time.)

Gr.

Cam.

Par.

Wash.

Ham.

Con-
cluded.

Tab.
R.A.

R. A. from
Observation.

s.

h. m. s.

1846, Oct. 14,

-15

-6.

-10,

— 4

55.02

21 51 54.98

Nov. 14,

-*,

-53

+ 145

-13,

+ 1

22.99

21 51 23.00

1847, July 26,

-9,

-11.

9

1.94

22 8 1.85

Aug. 17,

— 4.

-14,

-166

-9,

— 11

51.90

22 551.79

Oct. 8,

-95

-142

+ 25

-103

— 6

3.42

22 1 3.36

Nov. 18,

0,

—7,

+ 93

-5,

0

4.28

22 0 4.28

1848, July 25,

-la

-10,

+ 5,

- 3

49.85

22 16 49.82

Aug. 29,

•fS.

— 9,

-8,

-7i

— 2

21.55

22 13 21.53

Oct. 6,

-6.

-6,

-85

+ 1.

— 5

53.89

22 953.84

Nov. 17,

+ 73

—4,

+ 3

38.40

22 838.43

1849, Sept. 1,

-4.,

-13,

-12,

+ 8,

-7

51.43

22 21 51.36

Oct. 15,

+ 74

-8,

+ 36

+ •*,

+ 2

2.74

22 18 2.76

Nov. 25,

— 7.

+4,

-2,

+ 12.

+ 1

19.06

22 17 19.07

1850, Aug. 28,

-»,

-14,

— 10

62.59

2231 2.49

Oct. 15,

+ 35

-4,

+ 1.

+ 1

43.45

22 26 43.46

Nov. 20,

+ !•

-1,

-23

+ o

42.94

22 25 42.94

1851, Sept. 2,

-17,

—Mb

— 15

15.94

22 39 15.79

Oct. 14,

+ 53

-2,

+ 2

2G.94

22 35 26.96

Nov. 20,

— 5,

-8,

— 4

11.92

22 34 11.88

1852, Aug. 7,

-273

— 27

25.60

22 50 25.33

Sept. 5,

— 17,

— S2

— 15

34.10

22 47 33.95

Oct. 12,

-16,

-9,

— 13

9.83

2244 9.70

Nov. 28,

- 93

-1,

— 7

44.70

22 42 44.63

1853, Sept. 1,

-266

-9,

— 24

40.14

22 56 39.90

Oct. 15,

— 18,

-18,

— 18

34.47

22 52 34.29

Nov. 24,

-17,

-2,

— 8

7.41

22 51 7.33

1854, Aug. 30,

-48,

-373

— 41

32.47

23 5 32.06

Sept, 24,

-385

— 38

1.25

23 3 0.87

Oct. 27,

-324

— 83,

— 33

23.38

23 0 23.05

Dec. 5,

-31,

-21,

— 27

40.04

22 59 39.77

1855, Aug. 10,

-20U

— 20

2.20

23 16 2.00

Sept. 8,

- 6a

-106

— 7

16.50

23 13 16.43

Oct. 22,

+ 17,,

+ 17

15.60

23 9 15.77

Nov. 29,

+ 17,,

+ 15,

+ 16

57.33

23 7 57.49

8 May, 1865.

58

THE ORBIT OF NEPTUNE.

CORRECTIONS TO THE TABULAR RIGHT ASCENSIONS GIVEN BY THE DIFFERENT

OBSERVATORIES, WITH THE CONCLUDED CORRECTIONS AND CONCLUDED NORMAL

RIGHT ASCENSIONS (Cont.).

(The units are hundredths of seconds of time.)

Gr.

Cam.

Par.

Wash.

Albany.

Con-
cluded.

Tab.
E.A.

R. A. from
Observation.

s.

h. m. s.

1856, Aug. 8,

-25M

— 25

41.68

23 24 41.43

Sept. 13,

llu

-145

-153

— 12

17.95

23 21 17.83

Oct. 26,

+ 5,8

+ 6,

+ 6

28.76

23 17 28.82

Nov. 17,

+ 11,,

+ lll

+ 11

28.90

23 16 29.01

1857, Aug. 13,

-37,,

— 37

50.90

23 32 50.53

Sept. 21,

-142,

-5.

-62

— 12

6.08

23 29 5.96

Oct. 24,

+ 2

8.91

23 26 8.93

Dec. 8,

+ H«

+ 13S

+ 92

+ 11

45.10

23 24 45.21

1858, Aug. 18,

-40M

— 40

58.46

23 40 58.06

Sept. 23,

-27*

-275

— 27

29.38

23 27 29.11

Oct. 28,'

-2032

-21,

— 20

22.93

23 34 22.73

Dec. 12,

— 6

6.89

2333 6.83

1859, Aug. 21,

-51*.

-61,

— 52

14.82

23 49 14.30

Sept. 23,

— 46,0

— 452

— 46

3.51

2346 3.05

Nov. 8,

-32a

-323

— 32

9.94

2342 9.62

Dec. 14,

— 33B

— 41,

— 33

25.27

23 41 24.94

1860, Aug. 20,

-77,,

— 77

45.29

23 57 44.52

Sept. 23,

-69ffi

-592

— 68

30.53

23 54 29.85

Oct. 31,

— 67ffi

-60,

— 66

3.89

23 51 3.23

Dec. 13,

-636

— 63

40.91

23 49 40.28

1861, Aug. 22,

-95,.

— 95

4.99

0 6 4.04

Sept. 18,

-8730

-942

-778

— 86

33.38

0 3 32.52

Oct. 30,

— 87,,,

-935

-88,8

-828

— 88

36.27

23 59 35.39

Dec. 7,

- 10015

-87,

— 84,,

— 84,

— 89

56.66

23 57 55.77

1862, Aug. 24,

— 112,6

— 103,

— 109

24.31

0 14 23.22

Sept. 23,

— 11625

- 115.

- 10515

— 113

34.92

0 11 33.79

Nov. 6,

114M

-111,8

-105,o

— 112

33.12

0 7 32.00

Dec. 15,

- US,,

— 113*

— 1038

— 112

12.76

0 6 11.64

1863, Aug. 28,

-1595

— 159

34.01

0 22 32.42

Sept. 27,

— 146M

-1418

— 138,o

— 144

42.64

0 19 41.20

Nov. 17,

- 138,o

-1418

— 1368

— 140

17.57

0 25 16.17

Dec. 12,

— 1376

- 13212

-1328

— 133

29.48

0 14 28.15

1834, Aug. 7,

-178,

— 178

22.99

0 32 21.21

Oct. 1,

- 1688

-1628

— 167

44.63

0 27 42.96

Nov. 12,

— 16315

-164,5

— 154,

— 162

58.31

0 23 56.69

Dec. 17,

- 15615

— 154,

— 157

45.62

0 22 44.05

THE ORBIT OF NEPTUNE.

50

CORRECTION TO THE DECLINATIONS, WITH THE CONCLUDED DECLINATIONS.

Gr.

Cam.

Par.

Wash.

Albany.

Con-
cluded.

Tab.
Dec.

Concluded Dec.
from Obs.

1846, Oct. 14,

n

+ 0.44

+ 0.42

+ 0.4

ft

20.6

of n

— 13 31 20.2

Nov. 14,

+ 0.53

+ 0.33

+ 1.8,

+ 0.9

54.7

13 33 53.8

1847, July 26,

+ 1.92

+ 1.2,

+ 1.7

31.0

— 12 8 29.3

Aug. 17,

+ 2.0,

+ 1.1,

+ 1.83

+ 1.7

48.0

12 20 46.3

Oct. 8,

+ 1.2,

+ 1.1,

+ 1.4,

+ 1.2

6.2

1247 5.0

Nov. 18,

+ 1.63

-0.53

+ 1-5:

+ 0.7

2.6

12 52 1.9

1848, July 25,

+ 1.2,

+ 0.32

+ 0.8

39.6

— 112338.8

Aug. 29,

+ OJ,

+ 1.3,

+ 0.2,

+ 0.5

44.9

11 43 44.4

Oct. 6,

+ o.i5

+ 0.23

+ 1.1,

+ 0.5

3.1

12 3 2.6

Nov. 17,

+ 0.2,

+ 0.52

+ 0.4

32.9

12 932.5

1849, Sept. 1,

— 0.93

+ 0.1,

— 0.4

55.1

— 10 59 55.5

Oct. 15,

+ 0.23

-0.83

+ 0.0,

— 0.2

33.0

11 21 33.2

Nov. 25,

+ 1.4,

-0.43

+ 0.3

5.2

11 25 4.9

1850, Aug. 28,

-1.55

-0.12

— 1.1

52.6

— 10 10 53.7

Oct. 15,

— 0.35

-0.7,

-0.15

— 0.3

59.4

10 35 59.7

Nov. 20,

-1.0,

-0.43

+ 0.24

— 0.5

15.3

10 41 15.8

1851, Sept. 2,

-1.8,

-2.2,

— 1.6

27.6

— 9 26 29.2

Oct. 14,

- 1.2,

-2,2,

— 1.7

3.1

949 4.8

Nov. 20,

— 2.22

— 2.52

— 2.4

45.9

9 55 48.3

1852, Aug. 7,

-2.1,

— 2.1

44.2

— 8 22 46.3

Sept. 5,

-2.2,

-2.13'

— 2.2

37.0

8 40 39.2

Oct. 12,

-8.1,

-3.12

— 3.1

5.7

9 1 8.8

Nov. 28,

-2.13

— 3.6,

-2.7

42.2

9 844.9

1853, Sept. 1,

-2.45

-3.1,

— 2.5

53.0

— 7 48 55.5

Oct. 15,

-2.7,

— 3.G3

— 3.1

58.4

8 14 1.5

Nov. 24,

— 2.5,

-3.13

— 2.9

58.5

822 1.4

1854, Aug. 30,

-3.44

-4.23

-3.7

39.1

— 6 57 42.8

Sept. 24,

— 3.84

— 3.8

32.9

7 13 36.7

Oct. 27,

—8.5,

-5.1,

— 3.9

30.6

7 29 34.5

Dec. 5,

-4.12

— 5.72

— 4.9

58.1

733 3.0

1855, Aug. 10,

— 0.83

— 0.8

54.9

— 55455.7

Sept. 8,

0.05

-0.12

0.0

59.8

6 12 59.8

Oct. 22,

+ 0.93

+ 0.2,

+ 0.7

4.1

638 3.4

Nov. 29,

+ l.G3

+ 2.52

+ 2.0

15.2

— 6 45 13.2

60

THE ORBIT OF NEPTUNE.

CORRECTION TO TUB DECLINATIONS, WITH THE CONCLUDED DECLINATIONS (Cont.).

Gr.

Cam.

Par.

Wash.

Albany.

Con-
cluded.

Tab.
Dec.

Concluded Dec.
from Obs.

1856, Aug. 8,

n
— 0.84

— 0.8

n
21.4

0 1 II

— 5 3 22.2

Sept. 13,

-0.7,

— 0.5,

+ 0.1,

— 0.4

48.3

5 25 48.7

Oct. 26,

+ 1-7,

+ 1-1,

+ 1.3,

+ 1.4

45.4

5 49 44.0

Nov. 17,

+ 2.0,

+ 1.7,

+ 1.9

29.2

5 55 27.3

1857, Aug. 13,

— 2.0,

— 2.0

40.8

— 4 14 42.8

Sept. 21,

—0.1.

-1.33

-0.4,

— 0.5

29.9

4 39 30.4

Oct. 24,

+ 0.7,

— 0.9,

— 0.5,

— 0.2

6.4

4 58 6.6

Dec. 8,

+ 0.6,

-0.4,

— 0.6,

0.0

39.0

5 5 39.0

1858, Aug. 18,

-1.8,

— 1.3

44.6

— 32545.9

Sept. 23,

-1.34

-1-4,

— 1.3

55.2

3 48 56.5

Oct. 28,

— 0.65

— 1.0,

— 0.8

34.7

4 8 35.5

Dec. 12,

— 0.35

-0.3

14.2

4 15 14.5

1859, Aug. 21,

-2.9,

— 3.0,

— 2.9

26.2

— 2 35 29.1

Sept. 23,

-2.7,

- 2.9,

— 2.8

45.6

2 56 48.4

Nov. 8,

-2.15

-2.33

— 2.2

19.5

3 21 21.7

Dec. 14,

-1.0,

— 2.6,

— 1.5

48.4

3 24 49.9

1860, Aug. 20,

-4.7,

— 4.7

13.8

— 143 18.5

Sept. 23,

— 3.65

-4.2,

— 3.8

3.1

2 5 6.9

Oct. 31,

•-3.15

-4.0,

— 3.4

0.3

2 27 3.7

Dec. 13,

-4.4,

— 4.4

24.1

2 34 28.5

1861, Aug. 22,

-5.2,

— 5.2

4.6

— 052 9.8

Sept. 18,

-5.4,

-6.1,

— 5.8,

— 5.6

10.8

1 9 16.4

Oct. 30,

-4.9,

-6.1,

-4.1,

— 5.3,

— 5.1

35.1

1 34 40.2

Dec. 7,

-4.9,

-6.1,

-4.6,

— 5.3,

— 5.2

59.8

— 1 44 5.0

1862, Aug. 24,

— 6.6,

— 6.6

53.0

— 0 0 59.6

Sept. 23,

-6.15

— 7.50

— 6.1

56.3

0 20 2.4

Nov. 6,

— 6.3<

-5.7,

-8.4.

— 6.1

38.0

0 45 44.1

Dec. 15,

-6.4,

-6.7,

-7.60

— 6.5

45.8

— 05252.3

1863, Aug. 28,

— 9.6,

— 9.6

11.6

+ 049 2.0

Sept. 27,

-8.4,

— 8.92

-8.4,

— 8.5

61.5

0 29 53.0

Nov. 17,

-8.1,

-8.7,

— 9.34

— 8.8

13.6

+ 0 2 4.8

Dec. 12,

-8.1,

— 8.2,

-8.8,

— 8.6

53.1

-0 2 1.7

1864, Aug. 7,

- 10.7,

— 10.7

53.8

+ 1 50 43.1

Oct. 1,

— 10.8,

- 10.7,

— 10.7

19.0

1 19 8.3

Nov. 12,

- 10.2,

— 10.5,

- 9.4,

— 10.1

3S.7

0 55 28.6

Dec. 17,

- 9.9,

— 9.5,

- 9.7

21.3

+ 049 11.6 .

THE ORBIT OF NEPTUNE. 61

REMARKS ON THE PRECEDING TABLE.

The processes to which we have subjected the observations ought, it would
seem, to eliminate every source of constant differences between those made at
different observatories. But there are still two well-marked cases of systematic
differences in the right ascensions, namely, in the Cambridge observations of the
first five years, and the Albany observations of the last four. The differences
between the corrections finally concluded from all the observations, and those
concluded from Cambridge and Albany, are, it will be seen, as follows :

Date. Cone. — Camb. Date. Cone. — Albany.

S S

1846, Oct. + 0.02 1861, Sept. — 0.09
Nov. +0.06 Oct. —0.06

1847, July, +0.02 Dec. —0.05
Aug. + 0.03 1862, Aug. — 0.06
Oct. +0.08 Sept. -0.08
Nov. +0.07 Nov. —0.07

1848, July, + 0.07 Dec. — 0.09
Aug. + 0.07 1863, Sept. — 0.06
Oct. +0.01 Nov. —0.04
Nov. +0.07 Dec. —0.01

1849, Sept. + 0.06 1864, Oct. - 0.05
Oct. +0.10 Nov. -0.08
Nov. —0.03 Dec. —0.03

1850, Aug. + 0.04
Oct. + 0.05
Nov. + 0.01

The constancy of signs here exhibited can hardly be attributed to chance in
the case of Cambridge, and not at all in the case of Albany. The only cause
to which I can attribute it is a habit of registering the transit of Neptune earlier
or later than that of a bright star. Such a habit would seem to pertain to the
observer rather than the instrument, and, therefore, less to be feared as the number
of observers is increased. On account of its possible existence, the weights of the
results of any one observatory have not been supposed proportional to the number
of observations, but each has been subject to a constant probable error of at least
0'.02 when observations were made by eye and ear, and Os.01 when made with
chronograph, however great the number of observations.

Albany exhibits the anomaly that the real systematic error seems greater than
the probable accidental error. The latter is of the smallest class, as might be
anticipated from the facts that the observations are made with a first-class in-
strument, in a good atmosphere, and are recorded with the electro-chronograph.
They have, therefore, been treated in such a way that, while they should enter
the absolute longitudes with a very small weight, they should enter the relative
longitudes at different times of the year, in other words, the radius vector, with

62 THE ORBIT OF NEPTUNE.

as much weight as those of any other observatory. This has been effected by
applying the constant correction — Os.04 to all the results before combining
them.

Anomalies somewhat similar are exhibited by the Paris declinations from I860
to 1861, and by the Washington declinations of 1861. In the case of Wash-
ington, they may be accounted for by the circumstance that the systematic cor-
rections for 1861 depend mainly on observations made in 1863, very few declina-
tions of fundamental stars being observed in 1861-62. But it does not seem so
easy to account for the discrepancy between the Paris and Greenwich results.
A comparison of them shows that while the Paris observations systematically
place the ten fundamental stars adopted as our standard about 0".8 farther north
than Greenwich, their positions of Neptune, and of some small stars near the
equator, substantially agree.

§ 28. The preceding normal right ascensions and declinations are next con-
verted into apparent ecliptic longitudes and latitudes, for the purpose of com-
parison with the provisional theory. For this purpose Hansen's obliquity of the
ecliptic has been adopted, so as to agree with the motion of the ecliptic adopted
in the preceding chapter. In the following table we give for each date — 1. The
longitude from observation, obtained as just stated. 2. The seconds of longitude
from provisional theory, as given on p. 43. 3. The excess of the theoretical
over the observed longitude. 4, 5, 6. The corresponding quantities relative to
the latitude.

THE OK 15 IT OF NEPTUNE.

GEOCENTRIC APPARENT LONGITUDES AND LATITUDES OF NEPTUNE DERIVED

FROM OBSERVATION.

Longitude.

Latitude.

TJ, _ ~

T^WOT* rif

Date.

jji roi 01
Theory.

! . 1 I ' ' 1 ' < \

Theory.

Observation.

Theory.

Observation.

Theory.

1795, May 9,

O 1 tf

214 37 20.4

//
19.1

ff

-1.3

+ 1 50 33.3

34.4

+ 1.1

1846, Oct. 14,

325 31 35.0

34.9

— 0.1

— 0 31 55.8

56.0

— 0.2

Nov. 14,

325 23 24.2

23.2

— 1.0

0 31 43.5

44.0

— 0.5

1847, July 26,

329 41 22.9

22.0

— 0.9

0 35 24.1

25.9

— 1.8

Aug. 17,

329 7 18.9

18.3

— 0.6

0 35 45.7

47.7

— 2.0

Oct. 8,

327 52 10.4

10.4

0.0

0 35 57.6

58.8

— 1.2

Nov. 18,

327 36 56.9

56.3

— 0.6

0 35 38.3

38.6

— 0.3

1848, July 25,

331 59 6.7

5.0

— 1.7

0 39 21.8

22.8

— 1.0

Aug. 29,

331 3 16.6

15.8

— 0.8

0 39 54.0

55.2

— 1.2

Oct. 6,

330 8 55.3

55.3

0.0

0 39 56.8

57.8

— 1.0

Nov. 17,

329 59 22.8

22.4

— 0.4

0 39 32.3

32.8

— 0.5

1849, Sept. 1,

333 15 38.7

38.6

— 0.1

0 43 51.8

52.7

— 0.9

Oct. 15,

332 15 32.6

32.4

— 0.2

0 43 48.6

49.2

— 0.6

Nov. 25,

332 417.1

16.7

— 0.4

0 43 15.7

17.1

— 1.4

1850, Aug. 28,

335 39 38.5

38.5

0.0

04741.9

42.7

— 0.8

Oct. 15,

334 31 10.3

9.5

— 0.8

0 47 39.9

41.1

— 1.2

Nov. 20,

334 15 24.4

23.8

— 0.6

047 8.6

9.5

— 0.9

1851, Sept. 2,

337 48 58.5

58.1

— 0.4

0 51 32.6

33.7

— 1.1

Oct. 14,

336 48 12.7

10.9

— 1.8

0 51 30.0

30.0

0.0

Nov. 20,

336 28 32.5

. 31.9

— 0.6

0 50 53.3

54.1

— 0.8

1852, Aug. 7,

340 46 10.7

11.0

+ 0.3

0 54 49.8

51.6

— 1.8

Sept. 5,

340 0 11.1

10.3

— 0.8

0 55 18.8

19.5

— 0.7

Oct. 12,

339 543.2

43.0

— 0.2

0 55 14.6

14.8

0 2

Nov. 28,

338 43 24.0

23.4

— 0.6

0 54 23.4

23.3

+ 01

1853, Sept. 1,

342 24 48.0

47.7

— 0.3

0 58 54.1

55.9

— 1.8

Oct. 15,

341 19 0.8

0.2

— 0.6

0 58 52.0

52.4

— 0.4

Nov. 24,

340 56 4.3

3.0

— 1.3

058 4.5

4.7

— 0.2

1854, Aug. 30,

344 46 18.1

17.8

— 0.3

1 2 25.8

27.5

— 1.7

Kept. 24,

344 533.4

33.2

— 0.2

1 2 35.8

37.0

— 1.2

Oct. 27,

343 23 17.5

17.3

— 0.2

1 2 14.4

15.3

— 0.9

Dec. 5,

343 12 3.1

2.8

— 0.3

1 1 19.4

19.1

+ 0.3

1855, Aug. 10,

347 34 55.1

54.2

— 0.9

1 5 26.8

28.0

— 1.2

Sept. 8,

346 49 57.9

57.0

— 0.9

1 6 1.4

1.8

— 0.4

Oct. 22,

34545 7.4

6.2

— 1.2

1 5 50.2

50.1

+ 0.1

Nov. 29,

345 24 25.6

25.3

— 0.3

1 4 53.7

54.8

— 1.1

G4

THE ORBIT OF NEPTUNE.

GEOCENTRIC APPARENT LONGITUDES AND LATITUDES OF NEPTUNE DERIVED

FROM OBSERVATION (Cont.).

Longitude.

Latitude.

I-'i-]', ,1- r\f

T?

Date.

Observation.

Theory.

-Lji rui ui

Theory.

Observation.

Theory.

-ftri'or oi
Theory.

1856, Aug. 8,

O / tf

349 54 4.2

n
3.3

ff

— 0.9

— 1 843.7

44.8

— 1.1

Sept. 13,

348 58 37.9

37.4

— 0.5

1 9 26.5

27.2

— 0.7

Oct. 26,

347 56 49.5

48.8

— 0.7

1 9 7.1

8.2

— 1.1

Nov. 17,

347 40 53.7

53.1

— 0.6

1 834.0

35.3

— 1.3

1857, Aug. 13,

352 5 16.8

16.5

— 0.3

1 12 5.6

6.2

— 0.6

Sept. 21,

351 4 3.1

2.0

— 1.1

1 12 45.6

46.1

— 0.5

Oct. 24,

350 16 10.7

9.6

— 1.1

1 12 28.3

28.3

0.0

Dec. 8,

349 54 2.4

1.4

— 1.0

- 1 11 11.8

11.8

0.0

1858, Aug. 18,

354 16 21.7

20.9

— 0.8

1 15 19.7

21.0

— 1.3

Sept. 23,

353 19 17.3

17.2

— 0.1

1 15 56.1

56.6

— 0.5

Oct. 28,

352 28 49.3

49.2

— 0.1

1 15 34.2

34.8

— 0.6

Doc. 12,

352 848.3

46.7

— 1.6

1 14 11.6

11.8

— 0.2

1859, Aug. 21,

356 30 3.7

2.9

— 0.8

1 18 25.2

25.8

— 0.6

Sept. 23,

355 37 45.0

44.6

— 0.4

1 18 59.5

60.0

— 0.5

Nov. 8,

354 34 30.3

29.4

— 0.9

1 18 22.8

23.0

— 0.2

Dec. 14,

354 22 53.5

52.5

— 1.0

1 17 8.5

9.3

— 0.8

1860, Aug. 20,

358 47 47.9

48.1

+ 0.2

1 21 17.0

17.7

— 0.7

Sept. 23,

357 54 28.9

28.1

— 0.8

1 21 55.4

5(i.4

— 1.0

Oct. 31,

356 58 23.5

22.4

— 1.1

1 21 31.1

32.2

— 1.1

Dec. 13,

356 36 25.2

24.7

— 0.5

1 20 4.5

4.6

— 0.1

1861, Aug. 22,

1 243.6

42.4

— 1.2

1 24 4.8

6.0

— 1.2

Sept. 18,

0 21 9.6

7.6

— 2.0

1 24 41.8

42.2

— 0.4

Oct. 30,

359 16 40.6

38.4

— 2.2

1 24 24.0

24.6

— 0.6

Dec. 7,

358 50 4.3

3.3

— 1.0

1 23 7.0

7.9

— 0.9

1862, Aug. 24,

3 17 37.0

34.7

— 2.3

1 26 46.0

46.7

— 0.7

Sept. 23,

2 31 9.8

7.7

— 2.1

1 27 24.2

25.7

— 1.5

Nov. 6,

1 25 28.0

26.1

— 1.9

1 26 56.1

57.3

— 1.2

Dec. 15,

1 4 11.5

10.0

— 1.5

1 25 29.0

30.0

— 1.0

1863, Aug. 28,

5 29 46.2

46.2

0.0

1 29 22.8

23.7

— 0.9

Sept. 27,

4 42 51.8

49.6

— 2.2

1 29 59.4

60.3

— 0.9

Nov. 17,

3 30 59.6

58.0

— 1.6

1 29 12.1

12.5

— 0.4

Dec. 12,

3 18 20.4

18.2

— 2.2

1 28 12.1

12.4

— 0.3

1864, Aug. 7,

8 9 22.8

20.6

— 2.2

1 30 52.9

53.7

— o.s

Oct. 1,

6 52 59.8

56.7

— 3.1

1 32 26.8

27.0

— 0.2

Nov. 12,

5 51 40.4

37.8

— 2.6

1 31 48.0

49.1

— 1.1

Dec. 17,

5 32 30.3

27.7

— 2.6

— 13022.6

23.1

— 0.5

CHAPTER IV.

RESULTS OF THE COMPARISON OF THE THEORETICAL WITH
THE OBSERVED POSITIONS OF NEPTUNE.

§ 29. THE first question of the present chapter will be whether the observations
of Neptune can be satisfied within the limits of their probable errors by suitable
changes in the elements of the orbit of Neptune and the masses of the disturbing
planets.

No admissible change in the mass either of Jupiter or Saturn will sensibly
affect the perturbations of Neptune. The mass of Uranus will, therefore, be the
only one the correction of which need be taken into account.

The errors of the provisional latitude of Neptune are so small that the errors
of the longitude in orbit may be taken as sensibly the same with the errors of
ecliptic longitude. The latter give equations of condition between the following
unknown quantities.

Correction of the mean longitude of Neptune,
mean motion of Neptune,
eccentricity X sm- perihelion of Neptune. "
eccentricity X cos- perihelion of Neptune.
" " mass of Uranus.

But if we attempt to solve by least squares the equations between these cor-
rections, we shall be met with the difficulty set forth in the introduction, and our
normal equations will be equivalent to only three, unless we include a great
number of decimals in the computation. We shall, therefore, make a linear
transformation of the unknown quantities, on the principles already referred to,
and suggested by the following considerations.

The true longitude of Neptune has been less than its mean longitude, and its
true motion has been greater than its mean motion, ever since its optical discovery.
From these circumstances the difficulty in question arises. We may obviate it
by substituting for the mean longitude and mean motion of Neptune during an
entire revolution its average longitude and heliocentric motion during the period
of the modern observations. Suppose an imaginary planet to move uniformly in
the orbit of Neptune in such a way that its average longitude and motion have
been the same as the average longitude and motion of Neptune during the last
nineteen years, and let x be its longitude, 1850, Jan. 0, and xf its annual motion.
We may then make the eccentricity and perihelion of Neptune to depend ana-
lytically upon the deviation of its motion from that of the hypothetical planet,
as it must depend really, because this deviation is the only real datum which we
to reason from, the Lalande observations excepted. It is to be remarked

9 May, 1865. 65

66 THE ORBIT OF NEPTUNE.

that both the longitude and motion of the hypothetical planet are entirely
arbitrary.

For the differential coefficients of the elements with respect to the heliocentric
co-ordinates, we have

- — 1 + 2 k cos I + 2 h sin I.
(.IE

dv dv

dn ~ de '

dv
dh

-JT — — 2 cos I — f h sin 2 I — | k cos 2 I.

t3Q

-TT=. 2 sin I + I k sin 2 7 — | h cos 2 I.
ct/c

\ dr .

- -j— — /•: sin I — li cos t.

« ae

1 d> _ J^ t_ dr_
a dn ~ 3 an a de

7r — — sin I + h — k sin 2 I + A cos 2 I.
a dh

— z= — cos 7 + Z; — 7i sin 2 Z — k cos 2 Z.
a dk

In accordance with what has been proposed, we shall substitute for s and n the
quantities x and a', connected with them by the relations

x •=. e + ah -\- (3k ... .

af = n + a'h

a and ft being approximately the average values of — 2 cos Z and + 2 sin Z during
the last nineteen years, and a' and ft' the average values of 2 n sin Z and 2 n cos Z
during the same time. We shall take

a = — 1.77 a' = — 0.018

ft = — 0.85 P' = + 0.073.

Then, considering v as a function of x, y, It, and /•;, and enclosing the new dif-
ferential coefficients in parentheses, we have, by suitable transformations,

dv\ dv {dv\_dv (c]^\ — dr_. i1^]— —
di ' Vcfcey dn ' \dx / de ' \dstf / dn

dv

/dv \ dv . dv

( -71- ) — 77 (« + « 0 -j-

\dh / dh ' de

THE OKJ3IT OF NEPTUNE. 67

( ,

Putting 2, for the geocentric longitude, .and A for the distance from the earth,
the differential coefficients of the geocentric with respect to the heliocentric co-
ordinates will be

<fa r

-7- — — cos (v — /I), ...

dv A (4)

dh a

a -=— — — sm (v — /I) ;
dr A

and the coefficients of the equations of conditions will be

(ft dh dv dh I dr

— — .___ ,_ I fi ._..

dx dv de dr a de

dh _ d?i dv eft I dr
dxf dv dn dr a dn

dv\ d/l 1 /dr

/(ft \ _ (ft /dv_\ (ft 1 /dr\ (5)

\dh ) — dv~ \dh) ~* a fo a \dh)

d/l 1 /dr_\

adr~ a\dk)

The perturbations in the geocentric longitude of Neptune produced by Uranus
will be —

Jn

1. Perturbations of the true heliocentric longitude multiplied by y- ;

* /7^

2. Perturbations of radius vector multiplied by -y-, for which has been taken

dr

„ a dh

n°°r^Md^

Of course the effect of the long-period and secular perturbations of the elements
produced by the action of Uranus must be included in the perturbations of
Neptune.

Representing by p the factor by which the assumed mass of Uranus must be
multiplied, so that the true mass shall be

21000'

the computed perturbations produced by Uranus will be the coefficients of fi in
the equations of condition.

§ 30. The residuals in longitude thus give the following equations between the
unknown quantities, which are numbered in the order of time, but grouped
somewhat differently.

68

THE Oil BIT OF NEPTUNE.

No.

Date.

Equation.

P.

M.

1

1795, May 9,

0 = 1.02<5z — 55.7<5z' + 2.45WA + 3.742<tt + 34.C,u —1.3

J

i

4

1847, July 26,

1.02 —2.2 +0.009 —0.001 +1.17 —0.9

4

2

5

1847, Aug. 17,

1.04 —2.4 +0.010 +0.003 +1.34 — O.G

10

5

2

1846, Oct. 14,

1.02 —3.8 +0.035 +0.0:M +1.79 —0.1

6

2

6

47, Oct. 8,

1.03 —2.7 +0.012 +0.014 +1.66 0.0

8

3

3

1846, NOT. 14,

1.01 —3.7 +0.034 +0.025 +1.82 —1.0

7

2

7

47, Nov. 18,

1.01 —2.7 +0.011 +0.017 +1.73 —0.0

7

2

8

1848, July 25,

1.04 —1.2 —0.011 —0.008 +1.16 —1.7

5

o

9

1848, Aug. 29,

1.04 —1.4 —0.010 0.000 +1.39 —0.8

8

3

12

40, Sept. 1,

1.04 _o.4 —0.027 —0.006 +1.47 —0.1

0

2

15

50, Aug. 28,

1.04- +0.7 —0.042 —0.011 +1.55 0.0

5

2

10

1848, Oct. 6,

1.03 —1.7 —0.008 +0.007 +1.61 0.0

8

3

13

49, Oct. 15,

1.03 —0.7 — 0.02G +0.002 + 1.69 —0.2

8

?,

16

60, Oct. 15,

1.03 +0.4 —0.041 —0.002 +1.79 —0.8

7

2

11

1848, NOT. 17,

1.01 —1.7 —0.009 +0.010 +1.08 —0.4

4

1

14

49, NOT. 25,

1.01 —0.7 —0.026 +0.005 +1.74 —0.4

6

2

17

60, NOT. 20,

1.01 +0.4 —0.041 +0.001 +1.85 —O.G

7

2

21

1852, Aug. 7,

1.04 +3.0 —0.002 —0.020 +1.85 +0.3

3

1

18

1851, Sept. 2,

1.04 +1.7 —0.054 —0.013 +1.70 —0.4

6

2

22

52, Sept. 5,

1.04 +2.8 —0.003 —0.014 +2.00 —0.8

5

2

25

63, Sept. 1,

1.04 +3.9 —0.069 —0.016 +2.31 —0.3

5

2

19

1851, Oct. 14,

1.03 +1.4 —0.054 —0.005 +1.95 —1.8

4

1

23

62, Oct. 12,

1.03 +2.5 —0.003 —0.008 +2.16 —0.2

5

2

26

53, Oct. 15,

1.04 +3.5 —0.070 —0.008 +2.44 —0.6

6

2

20

1851, NOT. 20,

1.01 +1.3 —0.053 —0.003 +2.00 —0.0

4

1

24

62, NOT. 28,

1.01 +2.4 —0.002 —0.004 +2.22 —0.6

4

1

27

53, NOT. 24,

1.01 +3.4 —0.069 —0.004 +2.53 . —1.3

5

2

28

1854, Aug. 30,

1.04 +5.0 —0.072 —0.016 +2.02 —0.3

6

2

32

55, Aug. 10,

1.04 +6.1 —0.071 —0.018 +2.89 —0.9

8

3

36

66, Aug. 8,

1.04 +7.2 —0.067 —0.017 +3.31 —0.9

8

3

29

1854, Sept. 24,

1.04 +4.8 —0.073 —0.012 +2.75 —0.2

4

1

33

65, Sept. 8,

1.04 +6.0 —0.072 —0.014 +3.11 —0.9

9

3

37

56, Sept. 13,

1.05 +7.0 —0.070 —0.011 +3.44 —0.5

9

3

30

1854, Oct. 27,

1.03 +4.5 —0.073 —0.006 +2.84 —0.2

6

2

34

55, Oct. 22,

1.04 +5.6 —0.074 —0.006 +8.14 -1.2

7

2

38

56, Oct. 26,

1.03 +6.6 —0.072 —0.004 +3.57 —0.7

9

3

31

1854, Dec. 5,

1.01 +4.4 —0.072 —0.004 +2.85 —0.3

4

1

35

65, NOT. 29,

1.02 +5.4 —0.074 —0.003 +3.17 —0.3

8

3

39

56, NOT. 17,

1.02 +6.5 —0.072 —0.002 +3.59 —O.G

8

3

40

1857, Aug. 13,

1.04 +8.2 —0.061 —0.014 +3.83 —0.3

7

2

44

68, Aug. 18,

1.04 +9.3 —0.052 —0.011 +4.39 —0.8

. 9

3

48

59, Aug. 21,

1.04 +10.3 —0.040 —0.007 +4.94 —0.8

9

3

41

1857, Sept. 21,

1.04 +8.0 —0.005 —0.007 +3.97 —1.1

9

3

45

58, Sept. 23,

1.05 +9.1 —0.056 —0.005 +4.51 —0.1

9

3

49

69, Sept. 23,

1.05 +10.1 —0.044 —0.002 +5.05 —0.4

10

3

42

1857, Oct. 24,

1.04 +7.7 —0.067 —0.002 +4.05 —1.1

6

2

46

58, Oct. 28,

1.04 +8.8 —0.059 +0.001 +4.57 —0.1

10

3

50

69, Nov. 8,

1.03 +9.7 —0.047 +0.005 +5.11 —0.9

9

3

43

1857, Dec. 8,

1.01 +7.5 —0.066 +0.001 +4.09 —1.0

9

3

47

58, Dec. 12,

1.01 +8.5 —0.058 +0.004 +4.55 —1.6

8

3

61

69, Dec. 14,

1.02 +9.5 —0.047 +0.007 +5.09 —1.0

8

3

62

1860, Aug. 20,

1.04 +11.4 —0.024 —0.004 +5.56 +0.2

6

2

66

61, Aug. 22,

1.04 +12.5 —0.005 —0.001 +6.10 —1.2

6

2

GO

62, Aug. 24,

1.04 +13.5 +0.016 +0.003 +6.79 —2.3

9

3

THE ORBIT OF NEPTUNE.

69

No.

Date.

Equation.

P.

M.

53

1860, Sept. 23,

0 = 1.05<Jz +11.2Jz' — 0.028<!A . + 0.002<!/t +5.67^ —0.8

10

3

57

61, Sept. 18,

1.05 +12.3 —0.009 +0.004 +0.20 —2.0

11

3

61

62, Sept. 23,

1.05 +13.4 +0.012 +0.008 +6.88 —2.1

11

3

54

1860, Oct. 31,

1.04 +10.8 —0.033 +0.008 +5.71 —1.1

9

3

58

01, Oct. 30,

1.04 +11.9 —0.015 +0.011 +6.31 —2.2

10

3

62

62, Nov. 6,

1.U4 +12.9 +0.005 +0.015 +6.89 —1.9

11

3

55

1860, Dec. 13,

1.02 +10.5 —0.033 +0.011 +5.68 —0.5

4

1

59

61, Dec. 7,

1.02 +11.0 —0.010 +0.014 +6.25 —1.0

11

3

63

02, Dec. I'),

1.02 +12.6 +0.004 +0.018 +6.81 —1.5

10

3

64

1803, Aug. 28,

1.04 +14.0 +0.040 +0.006 +7.42 0.0

4

1

68

04, Aug. 7,

1.04 +15.6 +0.070 +0.007 +7.94 —2.2

5

2

05

1863, Sept. 27,

1.05 +14.4 +0.036 +0.012 +7.50 —2.2

10

4

69

04, Oct. 1,

1.05 +15.4 +0.063 +0.015 -+8.16 —3.1

8

3

66

1803, Nov. 17,

1.04 +13.9 +0.028 +0.019 +7.49 —1.6

9

3

70

64, Nov. 12,

1.04 +15.0 +0.054 +0.022 +8.18 —2.6

10

4

67

1863, Dec. 12,

1.02 +13.7 +0.027 +0.021 +7.45 —2.2

9

3

71

64, Dee. 17,

1.02 +14.7 +0.052 +0.024 +8.13 —2.6

8

3

In order to lessen the labor of solving these equations, they have been divided
into groups, with respect to the years of observation, and the difference of helio-
centric longitude of the earth and planet. The nineteen years of modern obser-
vations have been divided into seven groups, of which the first and last each
include two years, and each of the intermediate ones three years. Then, in each
group of years, the equations which pe'rtain to corresponding times of the year
are grouped together, and will be combined into one.

The numbers in column P. are assumed as the " measure of precision" of the
residuals of each equation. These numbers were inferred from the numbers and
excellence of the observations on which each normal was founded, the unit of
precision was assumed to correspond to the probable error 1".5, and no equation
was allowed to have a precision exceeding 11. Hence the assumed probable error

of each equation is

But the residuals left after the final solution show that

the measures of precision attached to the modern positions are too great, and

2" 4
that their probable errors are really about -^-.

Column M. gives the number by which the individual equations must be mul-
tiplied in order that when those of each group are added together, the precision

P

of their sum may be 2. It is approximately o /-» n being the number of indi-

vidual equations in the group.

To make the solution more convenient with respect to decimals, the coefficients
of <5.r' will all be multiplied by 10, and those of <§7t and 8k divided by 10, after
condensing the equations in the manner proposed.

Thus the following twenty-nine homogeneous equations are obtained :

70

THE ORBIT OF NEPTUNE.

0 = 0.25&e

— 1.39X10*

+ 0.14!

+ 9.36^

+ 8.60

— 0.3

2.04

— 0.44

%+ 0.18

— 0.02

+ 2.3

1.8

5.20

— 1.20

+ 0.50

+ 0.15

+ 6.7

— 3.0

5.13

— 1.57

+ 1.06

+ 0.90

+ 8.6

— 0.2

4.04

1.28

+ 0.90

+ 0.84

+ 7.1

— 3.2

2.08

— 0.24

— 0.22

— 0.16

+ 2.3

3.4

7.28

— 0.36

— 1.68

— 0.34

+ 10.2

— 2.6

8.24

— 0.64

— 1.84

+ 0.23

+ 13.5

— 2.2

5.05

— 0.23

- 1.43

+ 0.22

+ 8.9

— 2.4

1.04

+ 0.30

— 0.62

— 0.20

+ 1.8

+ 0.3

6.24

+ 1.68

— 3.72

— 0.86

+ 12.1

— 3.0

5.16

+ 1.34

— 3.20

— 0.37

+ 11.2

— 3.4

4.04

+ 1.05

— 2.53

— 0.15

+ 9.3

— 3.8

8.33

+ 4.99

— 5.58

— 1.37

+ 23.8

— 6.0

7.31

+ 4.38

— 4.99

— 0.87

+ 22.1

— 4.4

7,23

+ 4.00

— 5.10

— 0.36

+ 22.7

— 4.9

7.13

+ 4.01

— 5.10

— 0.19

+ 23.1

— 3.0

8.32

+ 7.52

— 3.98

— 0.82

+ 35.7

5.4

9.42

+ 8.16

— 4.95

— 0.42

+ 40.6

— 4.8

8.29

+ 7.09

— 4.52

+ 0.14

+ 37.1

— 5.2

9.10

+ 7.65

— 5.13

+ 0.36

+ 41.2

-10.8

7.28

+ 8.83

-0.10

— 0.01

+ 43.8

— 8.9

9.44

+ 11.07

— 0.75

+ 0.42

+ 56.4

-14.7

9.36

+ 10.68

— 1.29

+ 1.02

+ 56.7

— 15.6

7.14

+ 8.31

— 0.69

+ 1.07

+ 44.9

— 8.0

3.11

+ 4.58

+ 1.80

+ 0.20

+ 23.3

— 4.4

7.35

+ 10.38

+ 3.33

+ 0.93

+ 54.5

— 18.1

7.27

+ 10.17

+ 3.07

+ 1.45

+ 55.2

— 15.2

6.12

+ 8.52

+ 2.37

+ 1.35

+ 46.7

— 14.4

§ 31. Treating these equations by the method of least squares, but leaving 0
indeterminate for the present, we have the fdur normals

1277.7LC + 935.29(10^) — 350.59. s + 22.46*! + 5431.7/t -1263.39 =

935.29 +1010.58 -190.60 + 24.58 +5178.1 -1240.38

- 350.59 - 190.60 + 305.88 + 90.30 - 935.5 + 146.81

22.46 + 24.58 + 90.30 +101.33 + 317.9 74.89

(S)

The solution of these equations gives the following values of the unknown

quantities in terras of

8x=+ 0.650 — 2.0670
& = + 0.0800 — 0.3420
M=+8.76 —12.180
&fc= — 3.79 — 7.640

(9)

THE ORBIT OF NEPTUNE.

71

Substituting these values of the corrections in equations (7), we have the fol-
lowing residuals, which are grouped, as before, according to the time of year of the
normals on which the equations were founded.^ Thus, the first residual of each
series of modern observations corresponds to positions of Neptune observed when
the planet culminated after 13'' 30'" during the years to which the series belongs.

h. m. h. m.

The second, to observations between 10 30 and 13 30
The third, to observations between 7 30 and 10 30
The fourth, before 7 30

We first give the residuals from the equations (7), each of which is supposed
to be of equal precision ; then the numbers by which the errors of observation
are multiplied to reduce them to the assumed standard of precision derived from
(G), column M. ; and, finally, the apparent errors of the theory derived from ob-
servations themselves, formed by dividing the residuals of the equations by the
measures of precision.

Residuals of equations. Acfual ™cfln resi<1«^ or ap-

parent errors ot theory.

1st series, \ ''
179-3, { 1

-1.8ft

i

+ 2"3

— 73(1

/-0.7

— 0.60

2

— 0.35

— 0.300

2d scries, j — 0.2

— 0.80

5

-0.04

— 0.160

1846-1847, + 2.4

+ 1.50

5

+ 0.48

+ 0.300

(-1.1

+ i.4p

4

-0.28

+ 0.350

— 2.3

— 0.70

2

— 1.15

— 0.350

3d series,

+ 0.5

— 1.50

7

+ 0.07

— 0.21^

1848-1850,

+ 1.0

+ 0.60

8

+ 0.12

+ 0.080

-0.7

+ 0.8//

5

-0.14

+ 0.160

+ 0.8

— 0.40

1

+ 0.80

— 0.400

4th series,

— 0.6

— 0.50

6

— 0.10

— 0.080

1851-1853,

— 1.6

5

— 0.32

— 2.5

+ O.G0

4

— 0.62

+ 0.150

— 1.0

— 2.7ft

8

— 0.12

— 0.340

5th series,

— 0.2

— 1.20

7

— 0.03

— 0.170

1854-1856,

-1.4

+ 0.40

7

— 0.20

+ 0.060

+ 0.4

+ 1%

7

+ 0.06

+ 0.140

+ 2.8

— 1.7ft

8

+ 0.35

— 0.210

6th series,

+ 3.7

— 0.40

9

+ 0.41

— 0.050

1857-1859,

+ 1.8

+ 0.90

8

+ 0,22

+ 0.110

— 3.4

4-2,20

9

— 0.38

+ 0.240

+ 2.8

— 1.40

7

+ 0.40

— 0.200

7th series,

— 0.6

— 0.50

9

— 0.07

— 0.060

1860-1862,

— 2.5

+ 1.6p

9

— 0.28

+ 0.180

+ 2.2

+ l.fy

7

+ 0.31

+ 0.240

72 THE ORBIT OF NEPTUNE.

Residuals of equations. Actual """"' """'""Is or »1>-

parent, errors ot theory.

8th series,
1863-1864,

+ 2.8% — l.2(i 3 + o.93 —

—2.4 —0.9^ 7 —0.34 — 0.13p

—0.3 +0.7^ 7 -0.04 + 0.10^

— 2.0 + 0.9,u 6 —0.33 +0.15^

§ 32. The coefficients of (i, taken negatively, represent the changes which would
be produced in the residuals if we suppose the mass of Uranus to be nothing. It
will be seen that these coefficients are generally smaller than the residuals them-
selves, and that their actual effect on the modern residuals never amounts to
more than four-tenths of a second. Supposing that the modern observations
cannot be relied on within this limit of error, we should arrive at this remarkable
result, — that if the planet Uranus were unknown, its existence could scarcely be
inferred from all the observations hitherto made on Neptune, unless these were
combined in such -a way as to show the systematic error of the theoretical radius
vector. In fact, the orbit of Neptune, computed without regard to the perturb-
ations of Uranus, would only exhibit an error of 9" when compared with Lalande's
position ; and a discussion of the modern observations would exhibit no sensible
error in the heliocentric longitudes. This circumstance furnishes a very good
illustration of the propriety of developing the long-period perturbations, the co-
efficients of which amount to whole minutes, as perturbations of the elements
which shall vanish at the epoch 1850.

Under these circumstances, no reliable correction of the mass of Uranus can be
concluded from the motions of Neptune. The solution of the preceding residuals
does, indeed, indicate an increase of this mass by one-third, which seems altogether
inadmissible, and is certainly very unreliable. Of the twenty-nine residuals,
fifteen indicate an increase of the mass, thirteen a diminution, and for one the
coefficient of ^ vanishes : so that the increase of the mass of Uranus is indicated
only by the fact that the residuals which favor it are generally a little larger than
those which do not.

§ 33. If Uranus could scarcely be detected from the motions of Neptune, much
less can an extra-Neptunian planet, unless it happened to be nearly in conjunction
with Neptune at the present time, and to have a much greater mass than Uranus,
— a highly improbable combination of circumstances. That there is no present
indication of any such action is shown by the smallness of the apparent mean
errors of theory in heliocentric longitude and radius vector during the whole
period from 1846 to 1864. The following table shows the mean value of these
errors during each of the seven series of modern observations, and the error of
the geocentric longitude of the Lalande observations, putting fj. zz 0. The error
of radius vector is expressed as error of annual parallax. It will be remembered
that the first of the four equations of each series arise from observations made
about half-way between the first quadrature and the opposition, the second at
opposition, the third between opposition and last quadrature, and the fourth near
the last quadrature. Each series, therefore, gives four equations of the first
degree between the errors of heliocentric longitude &>•, and annual parallax fy>.

THE OKBIT OF NEPTUNE. 73

The coefficient of 8v will be sensibly unity, and that of fy will vary from about
• — 0.5 to + 1.0 in each series.

Error of theory by tJie Lalande observations.

+ 2".3

(It will be remembered that the probable error of the Lalande position was
estimated at 2".8 ; but, owing to the over-estimate of the comparative precision of
the modern observations, the weight assigned to this position in the equations of
condition corresponded to a probable error of rather more than 4".)

By modern observations.

Limiting dates. Error of longitude. Error of parallax.

1846-47, — O."o5 — o"l8

1848-50, -0.08 —0.03

1851-53, —0.07 +0.55

1854-56, —0.08 0.00

1857-59, + 0.22 + 0.23

1860-62, + 0.11 + 0.18

1863-64, + 0.02 + 0.28

These errors are as small as could be expected if the theory were perfect.
There is, therefore, no indication of the action of an extra-Neptunian planet.
But this fact does not militate against the existence of such a planet. The per-
turbations of a planet, and its elliptic elements, develop themselves, not in pro-
portion to the time, but in proportion to the square of the arc described. .In
order, therefore, to determine the errors of a slow-moving planet with as much
accuracy as those of a quick-moving one, we must observe it through a period pro-
portioned to its time of revolution. And we cannot detect a deviation of long period
from an elliptic orbit until we have accumulated data much more than sufficient
for the exact determination of the elliptic elements. For example, when the
position of Neptune was determined from the perturbations of Uranus, the latter
planet had been regularly observed through an arc of some 270°. Moreover, the
two planets had been in conjunction in 1824. They are also remarkably near
each other when in conjunction. Yet, with all these circumstances so favorable
to the development of large perturbations, Uranus only wandered about 5" from
an elliptic orbit during the entire period of the modern observations.

Perturbations will, at first, be developed in proportion to the square of the arc
passed over. Therefore, had Uranus been observed through an arc of only 120°,
the perturbations by Neptune would have been indicated only by deviations in
heliocentric longitude of less than I". It is, therefore, almost vain to hope for the
detection of an extra-Neptunian planet from the motions of Neptune before the
close of the present century.

§ 34. Determination of the position of the plane of the orbit of Neptune.

To determine the corrections of the constants p and q, which determine the

10 Hay, 1865.

74

THE ORBIT OF NEPTUNE.

position of the plane of the orbit, '\ve shall divide the residuals of latitude into
five groups, the last one including three years, and each of the others four years.
To find the heliocentric angular distance of the planet above the plane of its
assumed orbit, we shall take an indiscriminate mean of the errors of geocentric
latitude of each group, multiply it by 0.98 to reduce it to heliocentric error, and
correct it for the mean error in longitude.

The mean errors of geocentric latitude, with the equations to which they give
rise, are as follows. The probable errors of each modern mean is estimated at
0".15 : so that the Lalaude position is entitled to a precision of ^

Limiting Dates.

tp

Equation of Condition.

n

n

1795,

+ 1.1

0=+0.08% —0.0585?

+ 0.11

1846-49,

— 0.97

— 0.866 —0.500

— 0.96

1850-53,

— 0.75

— 0.934 —0.358

— 0.75

1854-57,

— 0.71

— 0.978 —0.208

— 0.71

1858-61,

— 0.67

_0.999 —0.052

— 0.68

1862-64,

— 0.79

— 0.996 +0.084

— 0.80

The solution of which by least squares gives

lip = — 0".73 ; Sq = — 0".41.

The residuals, multiplying the first by 10 to reduce it to actual observed error,
are

1795, + o"7

1846-49, —0.13

1850-53, + 0.07

1854-57, + 0.09

1858-61, + 0.07

1862-64, —0.10

So that the Lalande observation is represented within 0".7, notwithstanding
the small weight with which it enters the equations. In fact, if p and q were
determined from the modern observations alone, the Lalande position would still
be represented within about 0".7.

§ 35. Concluded elements of Neptune.

From equations (1) and (2) of this chapter, we have

& =te + 1.77 M +0.85 87,:;
0.0185A - 0.073&fe;

the concluded corrections to the

So that, making the mass of Uranus ^T
provisional elements of § 19 are

THE ORBIT OF NEPTUNE.

& = + 12M
«»= + 0.5144
37* = + 8.76
•&fc = — 3.79
$p = — 0.73
fy = _ 0.41

Applying these corrections to the provisional elements of § 19, they become

e = 335 5 38.91
n= 7864.9354

h= +1201.69
k= +1275.57
2)= +4909.44
q= —4137.87

75

CHAPTER V.

TABLES OF NEPTUNE.

§ 30. F/II/I/IIHH i/ttt/ theory.

The fundamental theory on which these tables are founded is as follows :

1. Undisturbed elements of Neptime, referred to the m«ut <t-/!j>/!i- <ni<! Kji/i/id.r <>/
the epocli.

h zz eccentricity X *mc perihelion zz -f- 1201.69
k — eccentricity X cos perihelion zz -f- 1275.57
p •=. sine inclination X gine node zz -j- 4909.44
q zz sine inclination X cos node zz — 4137.87
n zzmean motion in 365^ days zz 7864.935
E zz mean longitude at epoch zz 335° 5' 38".91
Epoch 1850, Jan. 0, Greemvich mean noon.

From these expressions we deduce

•n zz43°17'30".3

e - 0.0084962

log a zz 1.4781414

Period zz 164.782 Julian years.

In log a we have included the constants of log ?• introduced by the action of
the planets, and also the effect of the secular variation of the longitude of the
epoch, both of which are computed on p. 31.

2. Secular and long-period, perturbations of the above elements.
These are taken without change from the table p. 39.

The elements being corrected by the addition of these perturbations for the
epoch of computation, we thence deduce the elliptic place of the planet.

3. Perturbations of the co-ordinates.

To the elliptic place of the planet we apply corrections for- periodic perturb-
ations of the co-ordinates, as follows :
To the longitude in orbit,

Ptl sin I + Pc.j cos I + Ps,2 sin 2 / -f Pc.2 cos 2 1 + &v
To the logarithm of the radius vector,

Rsl sin I + Rcj cos I + 5r0.

To the north latitude, computed with the true longitude in orbit,

B, , sin v -\- J3cl cos v -f fy?o-

70

TABLES OF NEPTUNE. 77

All these quantities have the same values as in § 19, pp. 40 and 41.

The elliptic values of the co-ordinates being thus corrected, we have the helio-
centric co-ordinates resulting from the concluded theory.

To facilitate this computation, the following tables are constructed. They are
designed to give the means of determining, for any date between the years 1600
and 2000, the principal auxiliary quantities which will be needed in computing
the place of the planet from the above theory. Many of these quantities are
modified so that the computer shall be troubled as little as possible with difference
of signs. Thus, to all the quantities Ps, Pa Rs, etc. constants are added so that
they shall always be positive, and so that the signs of the products which form
the perturbations shall be the same as those of sin /, cos I, etc. Again, constants
are added to all the perturbations of the longitude and radius vector, to make
them positive.

§ 37. Data given in the several tables.

TABLE I. gives the values of the " epochs and arguments" for the beginning of
each fourth year from 1800 to 1952 inclusive, the years 1800 and 1900 beginning
with Greenwich mean noon of Jan. 0, and all the other years with that of Jan. 1.

Pis simply the number of the four-year cycle before 1900, by which I' and &

1900 Y

of the next table must be multiplied, or - — 7 , adding a unit for fractions.

I is the mean longitude in orbit of Neptune, affected with the long-period per-
turbations of that clement, p. 39, and referred to the mean equinox of 1850.0.

y is the negative of the longitude of the node affected by perturbations, counted
on the orbit of the planet from that point which is equally distant from the node
of 1850 with the equinox of 1850, and diminished by 1°, the sum of the constants
added to the equations of longitude.

6 is the longitude of the node, referred to the mean equinox of the epoch, and
diminished by 1', the constant added to the reduction to the ecliptic.

In the arguments 1 to 9 inclusive, the circle is divided into 400 parts. Repre-
senting the mean longitude of a planet, referred to the equinox of 1850.0 by its
initial letter, the values of the different arguments are as follows :

Arg. 1 = U — N,

" 2 = S —N,

" 3 = J — N,

« 4 = 2S— N,

" 5 = S,

« G = S —2N,

" 7 = 2J— N,

« S = J,

" Q=J —2N.

•>>

Thus, Arg, 1 gives the difference of the mean longitudes of Uranus and Neptune,
expressed in parts 100 of which make a quadrant ; and so of the other arguments.

At the bottom of the table the expression A[}^)is the change in the longitude
or the argument during that 180 days which commences with 1850, Jan. 0.

78 TABLES OF NEPTUNE.

Fact. T gives the change in A[{g0) during a century : so that the change in any
180-day period within one or two centuries of the epoch may be found by mul-
tiplying Fact. T by the fraction of a century after 1850.0 at which the 180-day
period commences, and applying it to Aj}^).

A$so) gives the second difference for any series of 180-day periods within one or
two centuries of 1850 : so that, knowing the first value of Aj}^,, we can find a series
of values by successive addition.

The period of 180 days has been selected as a convenient one for computing a
heliocentric ephemeris. If any other period, represented by JVdays, be preferred,
the corresponding values of A(1) and A(2) are found by multiplying

and

A$o> by j.

TABLE II. gives the change of each longitude and argument for the first day
of each month during a four-year cycle. The change in I is given for that cycle
which begins with 1900 and ends with 1904. Column ?' gives, in units of the
second decimal of seconds, the change in column I during one cycle. Hence,
multiplying I' by the whole number P of the preceding table, and adding the
units of the product to the hundredths of seconds of /, we have the change of
mean longitude during the cycle numbered P in Table I. The correction is
positive for years before 1900, because the mean motion is diminishing.

0 must be corrected in precisely the same way ; but here the correction is nega-
tive before 1900.

Rigorously, both y and 0 require correction similar to I. But it is not requisite
that either of these quantities should be accurate within a second, so long as their
sum is exactly equal to the precession diminished by 1° 1'. The four-year changes
of both y and 6, which destroy each other, are, therefore, neglected ; but the change
in 0 due to the secular variation of the constant of precession (0".0227) is allowed
for by the correction P6'.

TABLE III. gives the reduction from the first to the subsequent days of any
month, or the motion of the epochs and arguments during a number of days one
less than those on the left of the table.

TABLE IV. gives the corrections to be applied to the longitudes and arguments
for the epochs 1800 -H to reduce them to the epochs 1GOO + t, 1700 + 1, and
1900 -f- 1, respectively. They are expressed in the form

a0 + T X Fact. T + T- X Fact. T\

in which Tis the fraction of a century.

TABLE V. gives the expressions for the perturbations of the longitude produced
by Uranus. To each of the expressions 7^, and Pct 14" has l>een added, and to
Ps2 and Pcx 3" has been added. Hence, when these quantities, as given in the

TABLES OP NEPTUNE. 79

tables, are multiplied by sin I, cos I, sin 2 Z, and cos 2 Z, the sum will be too great by
the quantity

14" sin I + 14" cos I + 3" sin 2 I + 3" cos 2 Z,

which expression has been subtracted from the equation of the centre. The con-
stant 14" has been added to 5^.

TABLE VI. gives the principal perturbations of the longitude produced by
Saturn, namely,

18".552sin (S— N)

— 0 .141 sin 2 (S—N)

— 0.012 sin 3^—^)
+ (const. = 19".000)

TABLE VII. gives the principal perturbations of the longitude produced by
Jupiter, namely,

34".121sin (J—N)
• 0.011sin2(t7— N)
+ (const. = 35".000)

TABLE VIII. gives the term

— 0".524cos(2S — N)
+ (const. = 0".GOO)

TABLE IX. gives the terms

- 0".058 sin S + 0".047 costf
+ (const. = 0".100)

•

TABLE X. gives the terms

+ 0".1GG sin (S — 2 N) + 0".43G cos (S — 2 N)
+ (const. = 0".500)

TABLE XI. gives the terms

+ 0".783 sin (2 / — N) — 0".1G4 cos (2J — N)
+ (const. =: 1'MOO)

TABLE XII. gives the terms

— 0".101 sin J + 0".097 cos J

+ (const. = 0".200)

TABLE XIII. gives the terms

+ 0".32G sin (J—2N) + 0".297 cos (J — 2 N)
+ (const. = 0".500)

TABLE XIV. will be more easily understood after we have explained the table
of equation of the centre.

TABLE XV. is composed of the four following parts :

1. The equation of the centre in the undisturbed ellipse of 1850.0, or,

80 TABLES OF NEPTUNE.

+ 2551-.117 sin I — 2403".35Scos
+ 1.163 sin 2 I — 18 .580 cos 2 I
0.088 sin 3 1 — 0.104 cos 3 1

2. The change in the equation of the centre produced by the perturbations of
the elements h and k during that revolution of the planet which commenced
1779, Jan. 4, and ends 1943, Oct. 15. This change is represented by

2 Me sin I — 2 M cos I,

M and 8k being taken from the table on p. 39 for the times corresponding to the
various values of I during the period in question.

3. The terms

— 14" sin 7 -14" cos I

— 3 sin 2 I — 3 cos 2 I

introduced to destroy the effect of the constants added to the values of PsA, Pcl,
Ps,2, and Pc.2 to render them positive.

4. The constant

3529",

?

added to render all the numbers of the table positive.

During the revolution to which Table XV. corresponds, the planet passed from
180° mean longitude, and returned to the same point in the heavens ; whence the
table begins and ends with this value of I. But since the commencement of the
table corresponds to the values of 7* and k in 1779, and the end to these values
in 1943, they do not correspond with each other. The sum of the constants
added to Tables V. to XV. inclusive is 1°, which has been subtracted from y in
Table I.

Table XIV. is formed by subtracting the values of M- and M during the revo-
lution of Table XV. from the values of the same elements 164.78 years earlier
or later. Or, we have

APU= 2(&' — M-o)
APC., = — 2(8# — Mo)

M' and 8fr representing the values of M and Sk at any epoch, and &7i0 and <$/,•„ their
values at that date of the period 1779-1943 when the planet had the same mean
longitude as at the epoch in question.

The sum of the sixteen quantities P^ sin I, Pcj cos 7, P,2 sin 2 7, P^ cos 2 7, 8v d to 9) ,
7, y, and the equation of Table XV. will give the true distance of the planet
from its ascending node, which we represent by u.

TABLE XVI. gives the reduction to the ecliptic for the years 1800, 1900, and
2000, together with the change of the reduction for a century. The constant

60"
has been added to render all the numbers of the table positive.

TABLES OF NEPTUNE. gj

The sum of u, 6, and the reduction to the ecliptic gives the true ecliptic longi-
tude of the planet, referred to the mean equinox of the date.

Tables of the radius vector.
TABLE XVII. gives the values of

-B.J + 150, and R^ + 100.

The expressions for E^ and #cl are given on p. 40, § 19, and the units are those
of the seventh place of decimals. Rsl + 150 must be multiplied by sinZ, and
HC.I + 100 by cos I, and the products included in the perturbations of log r.

TABLE XVIII. gives the principal terms of the perturbations of the logarithm
of the radius vector produced by Uranus, as given on p. 41. The constant added
is 209.

TABLE XIX. gives the perturbations of the same element by Saturn, namely,

39 7 cos (S — N)
+ 4 cos 2 (S — N)
+ (const. = 400)

TABLE XX. gives the perturbations of the same element by Jupiter, namely,

701 cos (J—N)
+ (const. = 700)

The units of these tables are those of the seventh place of decimals.
TABLE XXI. is formed of the four following quantities.

1. A constant formed by applying the necessary corrections to the logarithm
of the mean distance. We have

Mean motion, including its perturbations, 7864.935

Secular var. long, epoch, _j_ 21.443

Elliptic mean motion, 7843.492

To which corresponds log a = 1.4787334

Constants of perturbations of log r (p. 31), 5920

Negative of constants added to Tables XVIII.-XX., — 1309
Constant to be substituted for log a in expression for log radius vector, 1.4780105

2. The elliptic log r — log a, namely,

+ .0000078

-.0026857 cos I —.0025301 sin I
— .0000014 cos 2 I — .0000235 sin 2 I

3. The effects of the perturbations of h, k and a during the same revolution to
which Table XV. corresponds, represented by

MSh MSk

r-^r- Sin I — :r- COS Z+ 6 log «,

sin 1" sin 1"

M being the modulus of the common system of logarithms.

11 May, 1865.

82 TABLES OF NEPTUNE.

4. The terms

— 150 sin I — 100 cos I

introduced to destroy the effects of the constants added to Rsl and 7?cl.

TABLE XXII. gives the values of B^ and Bcl (p. 40). The constant 0".30 has
been added to each of these quantities to render them positive.

TABLES XXIII. and XXIV. give the perturbations of the latitude produced by
Saturn and Jupiter respectively, no constants being added.

TABLE XXV. gives the values of log sin i, to be added to log sin u in order to
obtain the elliptic latitude. They, as well as 6, have been obtained from the
formulae

sin i sin 0 —p + 8p + 0".30
sin i cos 6 =. q -f- &q — 0 .30

The values of ftp and 8q being taken from the table p. 39, and the corrections
+ 0".30 being applied to destroy the effect of the constants added to Bsl and Bcl.

§ 38. Elementary precepts for the iise of the tables.

Express the date for which the position of Neptune is required, in years, months,
and days of Greenwich mean time, according to the Gregorian Calendar.

If the date is between 1800 and 1955 inclusive, enter Table I. with the year,
or the first preceding year found therein, and take out the values of /, y, 6, and
Arguments 1—9 inclusive. Note also the value of P. If the date is not between
the above limits, enter as if the number of the century were 18.

Enter Table II. with the excess of the actual year above that with which
Table I. was entered, and with the month. Write the values of 7, y, 6, and the
arguments under those from Table I. Multiply I' and 0', the former interpolated
to the day of the month, by P of Table I., and write the units of the product under
the hundredths of seconds of I and 0, paying attention to the algebraic signs.

Enter Table III. with the day of the mouth, and write down I, &c., under the
former values.

If the date is without the limits 1800-1955, enter Table IV. with the century,
write the principal quantities under their proper heads, as before ; multiply column
"Fact. T" by the entire fraction of the century represented by the date, and
column "Fact. T2" by the square of this fraction, and write the products under
their proper heads.

Add up all the partial values of I, y, 6, and the arguments thus obtained,
attending to the algebraic signs of the products, subtracting from the arguments
as many times 400 as possible, and we have the final values of those quantities.

Enter Table V. with the final value of Arg. 1, and take from it the five quan-
tities there found. Multiply the first four of them as follows, using logarithms
or natural numbers as may be most convenient :

PiA by sine of Z,
P,..! by cosine of f,
P,z by sine of 2 I,
Pc.2 by cosine of 2 /.

TABLES OF NEPTUNE. 83

But if the* date is earlier than 1779 or later than 1943, PsA and P^ must first
be corrected from Table X1Y.

Write these four products under each other, remembering that their algebraic
signs will be the same as those of the sine and cosine of I and 2 I, unless the cor-
rections make Psl or Pc.i negative. Write under them the fifth quantity, &V

Enter Tables VI. to XIII. inclusive, with the arguments at the top of each.
Take out the eight remaining values of &v.

Enter Table XV. with Z, first reducing the minutes and seconds to decimals
of a degree, and take out the corresponding equation by interpolation to second
differences.

Under these fourteen quantities write Z and y, add up the sixteen lines, and call
the sum u.

Under u write 0 ; enter Table XVI. with u (reduced to hundredths of a degree)
as the side argument, and the year as the top argument, and take out the reduction
to the ecliptic. Add it to u and 0, and the sum will be the heliocentric longitude
of Neptune referred to the mean equinox and ecliptic of the date.

Enter Table XVII. with argument 1, and take out the values of R^ and R^.
If the date is previous to 1779 or subsequent to 1943, multiply the values of
APsi and AP^ from Table XIV. by 10.53, and correct R^ and B^ as follows :

Rsl by 10.53 APcA,
Rc.i by — 10.53 AP,i,

adding the units of these products to the last figures of R,.i and R^. Then multiply

Rsl by sine of Z,
Rcl by cosine of I,

and write down the products with the algebraic sign of sine I and cos I respectively.

Enter Tables XVIII. to XX. with their proper arguments, and write the results
under the products thus found.

Enter Table XXI. with the argument Z, and take out the corresponding number,
the first two figures of which are at the top of each column. Write it so that
the last figure (the seventh place of decimals) shall be under the last figures of
the former numbers.

The sum of the six numbers thus found will be the common logarithm of the
radius vector of Neptune.

Enter Table XXII. with argument 1, and take out S^ and B^. Multiply the
former by sin Z and the latter by cos Z.

Enter Tables XXIII. and XXIV. with their proper arguments, and take out
the corresponding numbers, applying the proper algebraic signs.

Take the sine of i from Table XXV., and multiply it by the sine of u (u having
already been found).

The sum of the five quantities thus found, each taken with its proper algebraic
sign, will be the north latitude of Neptune above the plane of the ecliptic of the date.

Thus we shall have the heliocentric co-ordinates of the planet. The computer
can then pass to the geocentric place by the method which he prefers.

If an ephemcris is wanted during a series of years, it will not be necessary to

84

TABLES OF NEPTUNE.

take the arguments from Tables I.-IV. more than once in three or four, or even
five, years. The intervals of computation are first to be chosen, and need not he
less than 180 days for the heliocentric place. Then compute the values of I, y, 0,
and the arguments for the first date of the series, and again for a date an integral
number of intervals (not generally exceeding ten) later. The longitudes and
arguments for the intermediate dates may then be found by continual addition
of the differences for 180 days (if this is the interval) from the bottom of Table I.

§ 39. Examples of (lie use of the tables.

As a first example, we will compute an ephemeris of the heliocentric positions
of Neptune for the years 1865 to 18G8 inclusive. The intervals of computation
will be 180 days, and we commence with the date 1864, Oct. 13, and end with
1869, March 21, between which are nine of the assumed intervals. We first
compute the epochs and arguments for the extreme dates as follows :

1. FOR 1864, OCTOBER 13.

I

y

0

Arg. 1

2

Table II., 1864,
Table III., Year 0, Oct.,
Fact. X 9,
Table IV., Day 13,

5 40 58.30
1 38 19.86
.14
4 18.39

228 54 54.37

7.88

0.35

130 15 49.25

29.82
— .01
1.31

91.96
1.75

0.08

199.49
8.87

0.37

Epochs & Args. 1864, Oct. 13,

7 23 36.69

228 55 2.60

130 16 20.37

93.79

208.23

Arg.

3

4

5

6 7

8

9

Table II., 1864,
Table III., Year 0, Oct.,
Table IV., Day 13,

243.86
23.47
1.03

5
19
1

206
10
0

193 93.9

7 48.8
0 2.2

250
25
1

287
1

For 1864, Oct. 13.

268.36

25

216

200 144.9

276

200

2. FOR 1869, MARCH 21.

I

y

0

Arg. 1

2

Table II., 1868,
Table III., Year 1, March,
Fact. X 8,
Table IV., Day 21,

14 25 17.73
2 82 81.25
.19
7 10.64

228 55 30.no
12.23

0.57

130 18 28.26
46.25

— .01
2.18

101.30
2.72

0.13

244.11

12.!>s

0.61

For 1869, March 21,

17 4 59.81

228 55 49.19

130 19 16.68

104.15

257.70

Arg.

3

4

5

6 7

8

9

Table II., 1868,
Table III., Year 1, March,
Table IV., Day 21,

369.03
36.41
1.71

104
29

1

260
16

1

990 Qf\^ O

«l_n •)•>•>..'

10 75.7
0 3.6

385
39
2

858

34
2

For 1869, March 21,

7.15

134

277

-):)s 33 2

26

389

TABLES OF NEPTUNE.

85

The epochs and arguments for the intermediate dates are now formed by suc-
cessive additions of the change in 180 days, deduced from Table I. T, the fraction
of a century after 1850, being 0.148, the first differences for 180 days, with the
arguments, are found to be as follows :

Also

1864, Oct. 13

1865, Apr. 11

1866, Apr. 6

1868, Sept. 22

1869, Mar. 21

o / //

I 1 4 35.908

7 23 36.69

8 28 12.598

9 32 48.505

16 023.920

17 459.818

AZ --0012

1 435.908

1 435.907

1 435.905

1 435.898

y 5.177

22855 2.60

22855 7.777

228 55 18.131

2285544.013

228 55 49.189

0 19.590

130 16 20.37

1301639.960

130 16 59.550

130 18 57.093

130 19 16.684

Arg. 1 1.150

93.79

94.940

96.090

102.990

104.140

2 5.497

208.23

213.727

219.224

252.206

257.703

3 15.421

268.36

283.781

299.202

391.729

7.151

4 12.20

25.

37.2

49.4

122.6

134.8

5 6.7

216.

222.7

229.4

269.6

276.3

6 4.3

200.

204.3

208.6

234.4

238.7

7 32.03

144.9

176.93

208.96

1.14

33.17

8 16.6

276.

292.6

309.2

8.8

25.4

9 14.2

260.

274.2

288.4

373.6

387.8

° /

0 /

0 1

0 /

0 /

21

1447

1656

19.6

32 1

3410

o

O

0

0

0

I (in Dec. of dog.)

7.3935

8.4702

9.5468

16.0066

17.0833

LONGITUDE.

//

n

n

ff

ff

P.J

23.76

24.04

24.30

25.57

25.72

Pai

22.50

22.14

21.77

19.35

18.93

P.,

4.75

4.67

4.59

4.02

3.92

A,

1.76

1.67

1.59

1.21

1.17

P,, sin I

3.06

3.54

4.03

7.05

7.55

J\ , cos /

22.32

21.90

21.47

18.60

18.09

I\,*m 21

1.21

1.36

1.50

2.13

2.20

PC* cos 2 I

1.70

1.60

1.51

1.03

0.96

SVi

11.15

11.49

11.83

13.79

14.12

Sv,

16.57

15.01

13.41

5.30

4.26

9®t

5.03

1.98

0.88

30.58

38.83

8vt

0.12

0.17

0.23

0.78

0.87

dVg,

0.07

0.07

0.08

0.13

0.14

3v6

0.06

0.05

0.04

0.04

0.05

dv1

1.80

1.53

1.15

0.95

1.35

3vs

0.26

0.29

0.31

0.28

0.25

dl\

0.06

0.08

0.13

0.64

0.73

Tab. XV.

02359.17

0 24 53.01

0 25 47.59

0 31 29.74

0 32 29.03

I

7 23 36.69

8 28 12.60

9 32 48.50

16 023.92

17 459.81

y

22855 2.60

22855 7.78

228 55 12.96

228 55 44.01

228 55 49.19

u

236 43 41.87

237 49 12.46

238 54 45.62

245 28 58.97

246 34 47.43

0

130 16 20.37

130163996

130 16 59.55

130 18 57.09

130 19 16.68

Red. Eel.

14.23

15.01

1588

22.33

23.62

Longitude

7 016.47

8 6 7.43

912 1.05

154818.39

16 54 27.73

86

TABLES OF NEPTUNE.

RADIUS VECTOR.

-B..1
*u

22
155

24

158

26
163

41
176

44

178

R,A sin I
R,.i cos I
S log rt
5 log r,
8 log r,
Prin. term

3
154

82
10
306
1.4750064

156
77
17
524
1.4749650

4

158
73
23
691
1.4749250

11
169
48
129
1894
1.4747074

13
170
44
154
1896
1.4746754

logr

1.4750679

1.4750427

1.4750199

1.4748825

1.4748581

LATITUDE.

log sin M
log sin i
log sin /90

5..:

-BC.I

7?,! sin Z
.fic , cos Z

*&

»A
A

9.922246

8.492852
8.415098

//
0.47
0.01

+ 0.05
+ 0.01
+ 0.28
— 0.54
— 1 29 25.02

9.927565
8.492842
8.420407

0.46
0.00

+ 0.06
0.00
+ 0.26
— 0.56
— 13031.03

9.932667
8.492831
8.425498

0.45
0.00

+ oV
0.00
+ 0.24
— 0.55
— 13135.09

9.958964
8.492764
8.451728

0.38
0.00

+ 0.10
0.00
+ 0.08
+ 0.12
— 13717.28

9.962660

8.492753
8.455413

n

0.37
0.00

+ 0.11
0.00
+ 0.05
+ 0.25
— 138 7.04

Latitude

- 1 29 25.22

-13031.27

— 13135.33

— 13716.98

— 138 6.63

Inserting the results for the five middle dates, the computations of which have
been omitted in printing, for want of space, we have the following heliocentric
ephemeris of Neptune :

Dale.

Longitude (mean
equinox of date).

Logarithm of radius
vector.

Latitude.

01 II

O ' It

18C4, Oct. 13,

7 0 16.47

1.4750679

— 1 29 25.22

1865, Apr. 11,

8 6 7.43

1.4750427

— 1 30 31. '-'7

Oct. 8,

9 12 1.05

1.4750199

— 1 31 35.:;::

1866, Apr. Q,

10 17 57.51

1.4749986

— 1 3237.:;<»

Oct. 3,

11 23 56.84

1.4749778

— 13337.41

1867, Apr. 1,

12 29 58.92

1.4749567

— 1 3435.41

Sept. 28,

13 36 3.52

1. 4749:542

-1 3531.38

1868, Mar. 26,

14 42 10.14

1.4749097

— 1 8625.26

Sept. 22,

15 48 18.39

1.4748825

— 1 37 16.98

1869, Mar. 21,

16 54 27.73

1.4748531

- 1 38 6.63

These co-ordinates being interpolated to every ten days, and corrected for
nutation, the geocentric co-ordinates may then be computed and corrected for
aberration in the usual way.

TABLES OF NEPTUNE.

87

As another example, let us compute the heliocentric position of Neptune for
Greenwich mean noon of 1795, May 9, the epoch of the normal place derived
from Lalande's two observations.

I

/ 9

Arg. 1

Arg. 2

Of ff

Table I., 1892,

66 51 12.69 228 5C

48.26 130 34 22.60

157.30

111.85

Table II., 3' May,

71623.32 0 C

35.01 0 212.33

7.77

37.13

, 2 XV

.13

— .01

Table III., Day 9,

0 252.26

0.23 0.87

0.05

0.24

Table IV., 1700,

141 31 19.97 359 42

21.53 358 53 55.63

166.69

85.00

Fact. T X -9536,

+ 45.60

f6.58 —8.75

— .05

— .36

Fact. T' X -91,

+ 0.22

0 0

+ .01

-.02

1795, May 9,

215 42 34.19 228 42

51.61 129 30 22.67

331.77

233.84

21 =

= 71.25

I = 215.7095

Arg. 3

4 5

6

7

8

9

Table I., 1892,

320.05

298 1?

56 38

314.2

394

246

Table II., 3" May,

104.18

82 L

15 29

216.6

112

96

Table III., Day 9.

0.68

1

0 0

1.4

1

1

Table IV., 1700,

70.66

327 2^

[2 328

298.6

228

313

Fact. T X .9536,

+ 0.12

\

0 —1

+ 0.2

0

0

1795, May 9,

95.69

307 '

'3 394

31.0

335

256

Longitude.

Radius vector.

Latitude.

//

u

P.,- 16.69

R,i

242

Bal 0.39

PC, 0.38

Bcl

77

^c.i 0.73

P., 5.16

PC., 1.97

P.., sin / — 9.74

K,A sin I

— 141

log sin u 9.998700

Pclcos 1 — 0.31

BcA cos I

— 63

log sin i 8.494395

P.'2 sin 2 I + 4.90

drl

234

log sin fa 8.493095

Pc2cos2/ +0.63

Sr,

60

Svl 24.08
8v, 9.48
dv, 69 05

Sr3
Prin. term

747
1.4816441

ff

7?,, sin 1 —0.22
B'cl cos 1 — 0.59

Ut/3 Ui/.VtJ

<5u4 0.54
Sv,, 0.07

logr

1.4817278

'*,3, —0.06
<5/32 — 0.46

5v6 0.92

/?„ +147 0.82

Svi 1.32

<5u8 0.34

Latitude + 1 46 59.49

to, 0.06

Eq. Cent. 1 7 1.63

•

I 2154234.19

y 228 42 51.61

u 8534 8.77

0 129 30 22.67

Ecd. Ecliptic 52.26

Long. (Mean Eq.) 215 5 23.70

Nutation - 15.90

Long. (True Eq.) 215 5 7.80

*~

88

TABLES OF NEPTUNE.

TABLE I.

EPOCHS AND ARGUMENTS FOR THE BEGINNING OF EACH FOURTH YEAR FROM
1800 to 1952.

Year.

P

I

y

e

1

1800

25

225 51 36.90

228 43 40.52

129 33 27.21

342.62

1804

24

234 35 57.56

228 44 22.73

129 36 5.97

35f.95

1808

23

243 20 18.15

228 45 04.92

129 38 44.75

361.28

1812

22

252 4 38.66

228 45 47.10

129 41 23.54

370.62

1816

21

260 48 59.10

228 46 29.27

129 44 2.35

379.95

1820

20

269 33 19.46

228 47 11.42

129 46 41.18

389.28

1824

19

278 17 39.74

228 47 53.56

129 49 20.02

39S.62

1828

18

287 1 59.94

228 48 35.69

129 51 58.88

7.95

1832

17

295 46 20.07

228 49 17.81

129 54 37.75

17.29

1836

16

304 30 40.12

228 49 59.92

129 57 16.64

26.62

1840

15

313 15 00.09

228 50 42.02

129 59 55.54

35.95

1844

14

321 59 19.98

228 51 24.11

130 2 34.45

45.29

1848

13

330 43 39.80

228 52 6.18

130 5 13.38

54.02

1852

12

339 27 59.54

228 52 48.24

130 7 52.33

63.96

1856

11

348 12 19.20

228 53 30.30

130 10 31.29

73.29

1860

10

356 56 38.79

228 54 12.34

130 13 10.26

82.03

1864

9

5 40 58.30

228 54 54.37

1-30 15 49.25

91.96

1868

8

14 25 17.73

228 55 36.39

130 18 28.26

101.30

1872

7

23 9 37.08

228 56 18.40

130 21 7.28

110.63

1876

6

31 53 56.36

228 57 0.40

130 23 46.31

119.96

1880

5

40 38 15.56

228 57 42.39

130 26 25.35

• 129.30

1884

4

49 22 34.68

228 58 24.36

130 29 4.42

138.63

1888

3

58 6 53.72

228 59 6.32

130 31 43.50

147.97

1892

2

66 51 12.69

228 59 48.26

130 34 22.60

157.30

1896

1

75 35 31.58

229 0 30.20

130 37 1.71

166.64

1900

0

84 19 28.87

229 1 12.09

130 39 40.75

175.97

1904

— 1

93 3 47.60

229 1 54.01

130 42 19.89

185.30

1908

— 2

101 48 6.26

229 2 35.92

130 44 59.04

194.64

1912

— 3

110 32 24.84

229 3 17.82

130 47 38.20

203.97

1916

— 4

119 16 43.35

229 3 59.71

130 50 17.38

213.31

1920

— 5

128 1 1.78

229 4 41.58

130 52 56.58

222.64

1924

— 6

136 45 20.13

229 5 23.44

130 55 35.80

231.98

1928

— 7

145 29 38.40

229 6 5.29

130 58 15.02

241.31

1932

— 8

154 13 56.60

229 6 47.14

131 0 54.26

250.65

1936

9

162 58 14.72

229 7 28.99

131 3 33.51

259.98

1940

— 10

171 42 32.77

229 8 10.80

131 6 12.77

269.32

1944

— 11

180 26 50.73

229 8 52.61

131 8 52.06

L'TS.65

1948

— 12

189 11 8.62

229 9 34.41

131 11 31.35

287.99

1952

— 13

197 55 26.43

229 10 16.20

131 14 10.66

291.0-2,

O / "

H

ft

ZA(1*»

1 4 35.943

5.182

19.583

1.150

Fact. T

— 0.237

— 0.033

+ 0.044

0.0

A (180)

— 0.0012

— 0.0002

+ 0.0002

0

TABLES OF NEPTUNE.

89

TABLE I.

EPOCHS AND ARGUMENTS FOR THE BEGINNING OF EACH FOURTH YEAR FROM
1800 TO 1952 (Continued).

Year.

2

3

4

5

6

7

8

9

1800

285.66

241.09

22

137

35

333.1

92

390

1804

330.27

366.26

121

191

70

193.2

227

105

1808

374.88

91.44

220

245

104

53.2

362

221

1812

19.49

216.61

319

300

139

313.3

97

336

1816

64.10

341.78

18

354

174

173.3

232

52

1820

108.71

66.96

117

8

209

33.4

367

167

1824

153.32

192.13

216

63

244

293.4

101

283

1828

197.94

317.31

315

117

279

153.5

236

398

1832

242.55

42.48

14

171

314

13.5

371

114

1836

287.17

167.65

113

226

349

273.5

106

229

1840

331.78

292.82

212

280

384

133.6

241

345

1844

376.40

18.00

310

334

19

393.6

376

60

1848

21.02

143.17

9

388

54

253.7

111

176

1852

65.64

268.34

108

43

88

113.7

245

291

1856

110.25

393.51

207

97

123

373.8

380

6

1860

154.87

118.68

306

151

158

233.8

115

122

1864

199.49

243.86

5

206

193

93.9

250

237

1868

244.16

369.03

104

260

228

353.9

385

353

1872

288.73

94.20

203

314

263

214.0

120

68

1876

333.35

219.37

302

369

298

74.0

255

184

1880

377.98

344.54

1

23

333

334.0

390

299

1884

22.60

69.71

100

77

368

194.1

125

15

1888

67.22

194.88

199

132

3

54.1

259

130

1892

111.85

320.05

298

186

38

314.2

394

246

1896

156.47

45.22

397

240

72

174.2

129

361

1900

201.06

170.29

96

295

107

34.3

264

77

1904

245.69

295.46

195

349

142

294.3

399

192

1908

290.32

20.63

294

3

177

154.4

134

308

1912

334.94

145.80

393

58

212

14.4

269

23

1916

379.57

270.97

92

112

247

274.5

4

138

1920

24.20

396.14

191

166

282

134.5

138

254

1924

68.82

121.30

290

221

317

394.6

273

369

1928

113.45

246.47

389

275

352

254.6

8

85

1932

158.08

371.64

88

330

387

114.7

143

200

1936

202.72

96.81

187

384

22

374.7

278

316

1940

247.35

221.97

286

38

57

234.8

13

31

1944

291.98

347.14

385

92

92

94.8

148

147

1948

336.61

72.31

84

147

126

354.9

283

262

1952

381.25

197.47

182

201

162

214.9

17

378

A<"
i_M180l

5.497

15.421

12.20

6.7

4.3

32.03

16.6

14.2

Fact. T

+ .001

0

0

0

0

0

0

0

A(180)

0

0

0

0

0

0

0

0

12 Hay, 1863.

90

TABLES OP NEPTUNE.

TABLE II.

REDUCTION OF THE EPOCHS AND ARGUMENTS TO THE FIRST DAT OF EACH MONTH
IN A CYCLE OF FOUR YEARS.

I

V

y

0

ff

1

Year 0,

O I II

ft

t ft

Jan. 1,

0 0 0.00

0.00

0.00

0 00.00

0.00

0.00

Feb. 1,

0 11 7.50

0.16

0.89

0 3.37

— 0.01

0.20

Mar. 1,

0 21 31.94

0.32

1.72

0 6.53

— 0.01

0.38

Apr. 1,
May 1,

0 32 39.44
0 43 25.41

0.48
0.64

2.61
3.48

0 9.90
0 13.17

'—0.02
— 0.03

0.58
0.77

June 1,

0 54 32.92

0.80

4.37

0 16.54

— 0.04

0.97

July 1,

1 5 18.89

0.96

5.23

0 19.81

— 0.05

1.16

Aug. 1,

1 16 26.39

1.12

6.12

0 23.18

— 0.05

1.36

Sept. 1,

1 27 33,89

1.29

7.01

0 26.55

— 0.06

1.56

Oct. 1,

1 38 19.86

1.45

7.88

0 29.82

— 0.07

1.75

Nov. 1,

1 49 27.37

1.61

8.77

0 33.19

— 0.08

1.95

Dec. 1,

2 0 13.34

1.77

9.64

0 36.46

— 0.08

2.14

Yearl,

Jan. 1,

2 11 20.84

1.93

10.53

0 39.83

— 0.09

2.34

Feb. 1,

2 22 28.34

2.09

11.42

0 43.20

— 0.10

2.54

Mar. 1,

2 32 31.25

2.25

12.23

0 46.25

— 0.11

2.72

Apr. 1,

2 43 38.75

2.41

13.12

0 49.62

— 0.11

2.U1

May 1,

2 54 24.72

2.57

13.98

0 52.89

— 0.12

3.10

June 1,

3 5 32.22

2.73

14.87

0 56.26

— 0.13

3.30

July 1,

3 16 18.19

2.89

15.74

0 59.53

— 0.14

3.49

Aug. 1,

3 27 25.70

3.05

16.63

1 2.90

— 0.14

3.09

Sept. 1,

3 38 33.20

3.21

17.52

1 6.28

— 0.15

8.89

Oct. 1,

3 49 19.17

3.37

18.39

1 9.54

— 0.16

4.08

Nov. 1,

4 0 26.67

3.53

19.28

1 12.92

— 0.17

4.28

Dec. 1,

4 11 12.64

3.69

20.15

1 16.18

— 0.17

4.47

Year 2,

Jan. 1,

4 22 20.15

3.85

21.04

1 19.56

— 0.18

4.67

Feb. 1,

4 33 27.65

4.01

21.93

1 22.93

— 0.19

4.87

Mar. 1,

4 43 30.55

4.17

22.74

1 25.98

— 0.20

5.05

Apr. 1,

4 54 38.06

4.33

23.63

1 29.35

— 0.20

5.24

May 1,

5 5 24.03

4.49

24.49

1 32.62

— 0.21

5.44

June 1,

5 16 31.53

4.65

25.38

1 35.99

— 0.22

5.63

July 1,

5 27 17.50

4.81

26.25

1 39.26

— 0.23

5.83

Aug. 1,

5 38 25.00

4.97

27.14

1 42.63

— 0.23

6.02

Sept. 1,

5 49 32.50

5.13

28.03

1 46.00

— 0.24

6.22

Oct. 1,

6 0 18.47

5.29

28.90

1 49.26

— 0.25

6.41

Nov. 1,

6 11 25.97

5.45

29.79

1 52.64

— 0.26

6.G1

Dec. 1,

6 22 11.94

5.61

30.66

1 55.90

— 0.26

6.80

Year 3,

Jan. 1,

6 33 19.44

5.77

31.55

1 59.27

— 0.27

7.00

Feb. 1,

6 44 26.95

5.93

32.44

2 2.64

— 0.28

7.20

Mar. 1,

6 54 29.85

6.09

33.25

2 5.69

— 0.29

7.38

Apr. 1,

7 5 37.35

6.25

34.14

2 9.06

— 0.29

7.58

May 1,

7 16 23.32

6.41

35.00

2 12.33

— 0.30

7.77

June 1,

7 27 30.82

6.57

35.80

2 15.70

— 0.31

7.97

July 1,

7 38 16.79

6.73

36.77

2 18.97

— 0.32

8.16

Aug. 1,

7 49 24.29

6.89

37.66

2 22.34

— 0.32

8.36

Sept. 1,

8 0 31.80

7.05

38.55

2 25.72

— 0.83

8.55

Oct. 1,

8 11 17.76

7.21

39.40

2 28.98

— 0.34

8.75

Nov. 1,

8 22 25.27

7.37

40.30

2 32.36

— 0.34

8.94

Dec. 1,

8 33 11.24

7.53

41.17

2 35.62

— 0.35

9.14

Columns V and tf interpolated to the day of the month must be multiplied by the integer,
P, of Table I. (not interpolated), and the units of the product added to the hundred ths of
seconds of I.

TABLES OF NEPTUNE.

91

TABLE II.

REDUCTION OF THE EPOCHS AND ARGUMENTS TO THE FIRST DAT OF EACH MONTH
IN A CYCLE OF FOUR YEARS (Continued).

2

3

4

5

6

7

8

9

Year 0,

Jan. 1,

0.00

0.00

0

0

0

0.0

0

0

Feb. 1,

0.95

2.66

2

1

1

5.5

3

2

Mar. 1,

1.83

5.14

4-

2

1

10.7

6

5

Apr. 1,

2.78

7.80

6

3

2

16.2

8

7

May 1,

3.70

10.37

8

4

3

21.6

11

10

Juno 1,

4.64

13.02

10

6

4

27.1

14

12

July 1,

5.56

15.59

12

7

4

32.4

17

14

Aug. 1,

6.50

18.25

14

8

5

37.9

20

17

Sept. 1,

7.45

20.90

16

9

6

43.5

22

19

Oct. 1,

8.37

23.47

19

10

7

48.8

25

22

Nov. 1,

9.31

26.13

21

11

7

54.3

28

24

Dec. 1,

10.23

28.70

23

12

8

59.7

31

26

Year 1,

Jan. 1,

11.18

31.36

25

14

9

65.2

34

29

Feb. 1,

12.12

34.01

27

15

9

70.7

37

31

Mar. 1,

12.98

36.41

29

16

10

75.7

39

34

Apr. 1,

13.92

39.07

31

17

11

81.3

42

36

May 1,

14.84

41.64

33

18

12

86.6

45

38

June 1,

15.79

44.29

35

19

12

92.2

48

41

July 1,

16.70

46.86

37

20

13

97.5

50

43

Aug. 1,

17.65

49.52

39

22

14

103.0

53

46

Sept. 1,

18.51

52.18

41

23

15

108.5

56

48

Oct. 1,

19.51

54.75

43

24

15

113.9

59

51

Nov. 1,

20.46

57.40

45

25

16

119.4

62

53

Dec. 1,

21.38

59.97

47

26

17

124.7

65

55

Year 2,

Jan. 1,

22.32

62.63

50

27

17

130.2

68

58

Feb. 1,

23.27

65.29

52

28

18

135.7

70

60

Mar. 1,

24.12

67.68

54

29

19

140.7

73

62

Apr. 1,

25.07

70.34

56

30

20

146.2

76

65

May 1,

25.99

72.91

58

32

20

151.6

79

67

Juno 1,

26.93

75.57

60

33

21

157.1

81

70

July 1,

27.85

78.14

62

34

22

162.4

84

72

Aug. 1,

28.80

8079

64

35

23

167.9

87

74

Sept. 1,

29.74

83.45

66

36

23

173.4

90

77

Oct. 1,

30.66

86.02

68

37

24

178.8

93

79

Nov. 1,

31.60

88.67

70

38

25

184.3

96

82

Dec. 1,

32.52

91.24

72

40

25

189.6

98

84

Year 3,

Jan. 1,

33.47

93.90

74

41

26

195.1

101

87

Feb. 1,

34.42

96.56

76

42

27

200.7

104

89

Mar. 1,

35.27

98.96

78

43

28

205.7

107

91

Apr. 1,

36.22

101.61

80

44

28

211.2

110

94

May 1,

37.13

104.18

82

45

29

216.6

112

96

Juno 1,

38.08

106.84

84

46

30

222.1

115

98

July 1,

39.00

109.41

87

47

30

227.4

118

01

Aug. 1,

39.94

112.06

89

49

31

232.9

121

03

Sept. 1,

40.89

114.72

91

50

32

238.4

124

106

Oct. 1,

41.81

117.29

93

51

33

243.8

126

108

Nov. 1,

42.75

119.95

95

52

34

249.3

129

111

Dec. 1,

43.67

122.52

97

53

34

254.6

132

113

92

TABLES OF NEPTUNE.

TABLE III.

REDUCTION FROM

THE FIRST TO SUBSEQUENT DAYS OF ANY MONTH.

Days.

I

y

e

i

2

3

4

5

6

7

8

9

f H

H

H

1

0 0.00

0.00

0.00

0.00

0.00

0.00

0

0

0

0

0

0

0

2

0 21.53

0.03

0.11

0.01

0.03

0.09

0

0

0

0

o

0

0

3

0 43.06

0.06

0.22

0.01

0.06

0.17

0

0

0

0

4

0

0

4

1 4.60

0.09

0.33

0.02

0.09

_ 0.26

0

0

0

0

5

0

0

5

1 26.13

0.11

0.44

0.03

0.12

0.34

0

0

0

0.7

0

0

6

1 47.C6

0.14

0.54

0.03

0.15

0.43

0

0

0'

0.9

0

0

7

2 9.19

0.17

0.65

0.04

0.18

0.51

0

0

0

1

1

0

0

8

2 30.73

0.20

0.76

0.04

0.21

0.00

1

0

0

1

3

1

1

9

2 62.26

0.23

0.87

0.05

0.24

0.08

1

0

0

1.

4

1

1

10

3 13.79

0.26

0

.98

0.06

0.27

0.77

1

0

0

1.

0

1

1

11

3 35.32

0.29

1

09

0

.06

0.30

0.86

1

0

0

1

8

1

1

12

3 66.86

0.32

1

.20

0.07

0.34

0.94

1

0

0

2.0

1

1

13

4 18.39

0.35

1

.31

0

.08

0.37

1.03

1

0

0

2

2

1

1

14

4 39.92

0.37

1

.41

0

.08

0.40

1.11

1

0

0

0

3

1

1

15

6 1.45

0.40

1

.52

0.09

0.43

1.20

1

1

0

2_

5

1

1

16

6 22.99

0.43

1.63

0

.10

0.46

1.28

1

1

0

27

1

1

17

644.52

0.46

1

.74

0

.10

0.49

1.37

1

1

0

2.

9

2

1

18

6 6.05

0.49

1.85

0

.11

0.52

1.46

1

1

0

3.

1

2

1

19

6 27.58

0.52

1

.96

0.12

0.56

1.54

1

1

0

3.

2

2

1

20

6 49.11

0.54

2

.07

0

.12

0.58

1.63

1

1

0

3.

4

2

1

21

7 10.65

0.57

2.18

0

.13

0.01

1.71

1

1

0

3.

6

2

2

22

7 32.18

0.60

2

.29

0

.13

0.64

1.80

1

1

1

3.

8

2

2

23

7 53.71

0.63

2

39

0

.14

0.07

1.88

2

1

1

3.

0

2

2

24

8 15.24

0.66

2

.50

0.15

0.70

1.97

2

1

1

4.

1

2

2

25

8 36.78

0.69

2

.61

0

.15

0.73

2.06

2

1

1

4.

3

2

2

26

8 58.31

0.72

2

.72

0

.16

0.76

2.14

2

1

1

4.

4

2

2

27

9 19.84

0.75

2

83

0

.17

0.79

2.23

2

1

1

4.

6

2

2

28

9 41.37

0.78

2

.94

0

.17

0.83

2.31

2

1

1

4.

8

3

2

29

10 2.91

0.80

3

.05

0

.18

0.86

2.40

2

1

1

4.

9

3

2

30

10 24.44

0.83

3

.16

0

.18

0.89

2.48

2

1

1

5.

1

3

2

31

10 45.97

0.86

3.26

0.19

0.92

2.57

2

1

1

5.

3

3

2

In January and February of 1700, 1800, and 1900, Table III. must be entered

with a number of days

1 greater than the real day of the month.

TABLE IV.

CORRECTIONS FOR PAST AND FUTURE CENTURIES.

1600 Fact. T

Fact. 7s

1700

Fact. T

Fact. 7" 1900

Fact. T Fact. T2

O f ft f

n

0 1 I'

tr

" a i >i

It n

i

283 1 52.88 + 94.09

+ 1.03

141 31 19.97

+ 47.82

+ 0.24 2182751.97

— 48.17 +0.17

.V

359 24 35.91 -j- 14.05

0

359 42 21.53

+ 6.90

0 01731.67

— 6.92 0

e

35748 0.68 —18.59

0

358 53 65.63

— 9.17

0 16 13.54

+ 9.19 0

Arg. 1

333.41 —0.08

+ 0.01

100.69

— 0.05

+ 0.01 233.35

+ 0.02 —0.10

2

170.36 —0.69

— 0.08

85.00

— 0.38

— 0.02 315.40

+ 0.46 -+(1.111;

8

141.19 +0.22

+ 0.04

70.66

+ 0.13

0.00 329.20

— 0.17 0

4

255. — 2

0

327.

— 1

0 74.

0 0

5

85. —1

0

242.

0

0 158.

0 0

C

256. _1.

0

328.

— 1

0 72.

+ 1. 0

7

196.9 -1-0.5

0

208.0

+ 0.2

0 101.2

— 0.4 0

8

55. 0

0

228.

0

0 172.

0 0

9

227. 0

0

313.

0

0 87.

0

TABLES OF NEPTUNE.

93

TABLE.

V.

VI.

VII.

Arg.

1

2

3

P..I

Diff.

Pa

Diff.

/YJ

PC.*

d,,

Diff.

<Ji>j

Diff.

*b

Diff.

II

„

It

n

;/

it

//

„

«

n

„

//

0

• 0.38

13.22

0.54

2.84

13.85

19.00

35.00

1

0.36

O.O2

13.49

0.27

0.54

2.92

13.33

0.52

19.29

0.29

35.54

o-54

2

0.35

0.01

13.76

0.27

0.53

3.00

12.80

0.53

19.57

0.28

36.07

0.53

3

0.34

O.OI

14.03

0.27

0.53

3.08

12.28

0.52

19.86

0.29

36.61

0.54

4

0.33

O.OI

14.30

0.27

0.53

3.16

11.76

0.52

20.14

0.28

37.14

0.53

0.00

0.27

0.51

0.29

0.54

5

0.33

14.57

0.54

3.25

11.25

20.43

37.68

6

0.33

o.oo

14.84

0.27

0.54

3.33

10.74

0.51

20.72

0.29

38.21

0.53

7

0.34

O.OI

15.11

0.27

0.55

3.41

10.24

0.50

21.00

0.28

38.75

0.54

8

0.34

o.oo

15.38

0.27

0.56

3.50

9.74

0.50

21.29

0.29

39.28

0.53

9

0.36

0.02

15.66

0.28

0.58

3.58

9.25

0.49"

21.57

0.28

39.81

0.53

0.02

0.27

0.49

0.28

0.53

10

0.38

15.93

0.59

3.66

8.76

21.85

40.34

11

0.40

O.O2

16.21

0.28

0.61

3.74

8.29

0.47

22.14

0.29

40.87

°'53

12

0.43

0.03

16.49

0.28

0.04

3.83

7.83

0.46

22.42

0.28

41.39

0.52

13

0.46

0.03

10.76

0.27

0.66

3.91

7.38

0.45

22.70

0.28

41.92

0.53

14

0.50

0.04

17.04

0.28

0.69

3.99

6.94

0.44

22.98

0.28

42.44

0.52

0.05

0.28

0.43

0.28

°-53

15

0.55

17.32

0.72

4.07

6.51

23.26

42.97

16

0.60

0.05

17.60

0.28

0.76

4.15

6.09

0.42

23.54

0.28

43.49

0.52

17

0.06

0.06

17.88

0.28

0.80

4.23

5.69

o 40

23.81

0.27

44.01

0.52

18

0.73

0.07

18.17

0.29

0.84

4.31

5.30

0.39

24.09

0.28

44.53

0.52

19

0.81

0.08

18.45

0.28

0.88

4.39

4.93

0.37

24.37

0.28

45.04

0.51

0.08

0.29

0.36

0.27

0.51

20

0.89

18.74

0.93

4.46

4.57

24.64

45.55

21

0.98

0.09

19.02

0.28

0.97

4.54

4.22

0.35

24.91

0.27

40.06

0.51

22

1.08

0. 10

19.31

0.29

1.03

4.61

3.89

o-33

25.18

0.27

•46.57

0.51

23

1.19

0. II

19.59

0.28

1.08

4.68

3.58

0.31

25.45

0.27

47.07

0.50

24

1.30

0. II

19.88

0.29

1.14

4.76

3.28

0.30

25.72

0.27

47.67

0.50

0.13

0.28

0.28

0.27

0.50

25

1.43

20.16

1.20

4.83

3.00

25.99

48.07

2fi

1.56

0.13

20.44-

0.28

1.27

4.89

2.73

0.27

26.25

0.26

48.56

0.49

27

1.70

0.14

20.73

0.29

1.34

4.96

2.48

0.25

26.52

0.27

49.05

0.49

28

1.88

o.i 6

21.01

0.28

1.41

5.02

2.25

0.23

26.78

0.26

49.54

0.49

29

2.02

o. 1 6

21.29

0.28

1.49

5.08

2.03

O.22

27.04

0.26

50.02

0.48

0.17

0.28

O.2I

0.26

0.48

30

2.19

21.57

1.56

5.14

1.82

27.30

50.50

31

2.38

0.19

21.84

0.27

1.64

5.20

1.63

O.I9

27.55

0.25

50.97

0.47

32

2.57

0.19

22.11

0.27

1.73

5.25

1.46

0.17

27.81

0.26

51.44

0.47

33

2.77

O.20

22.38

0.27

1.81

5.30

1.30

o.i 6

28.06

0.25

51.91

0.47

34

2.98

O. 21

22.66

0.27

1.90

5.34

1.16

0.14

28.31

0.25

52.38

0.47

0.23

0.26

0. 12

0.25

0.46

35

3.21

22.91

1.99

5.38

1.04

28.56

62.84

86

3.44

0.23

23.17

0.26

2.09

5.42

0.93

O. II

28.80

0.24

63.29

0.45

37

3.08

0.24

23.42

0.25

2.18

5.45

0.84

0.09

29.04

0.24

53.74

0.45

38

3.93

0.25

23.07

0.25

2.28

5.48

0.70

0.08

29.28

0.24

54.18

0.44

39

4.19

0.26

23.92

0.25

2.38

5.51

0.69

0.07

29.52

0.24

64.63

0.45

0.27

0.24

0.05

0.24

0.44

40

4.46

24.16

2.48

5.53

0.64

29.76

65.07

41

4.74

0.28

24.39

0.23

2.58

5.54

0.60

0.04

29.99

0.23

55.50

0.43

42

5.03

0.29

24.62

0.23

2.69

5.56

0.58

O.O2

80.22

0.23

66.92

0.42

43

6.88

0.30

24.84

O.22

2.79

5.56

0.57

O.OI

30.45

0.23

50.34

0.42

44

5.64

0.31

25.05

0.21

2.90

5.56

0.57

O.OO

30.68

0.23

60.70

0.42

0.31

O.2O

O.O2

O.22

0.41

45

5.95

25.25

3.00

5.56

0.59

30.90

67.17

40

6.28

0.33

25.45

O.2O

3 11

5.55

0.61

O.O2

31.12

O.22

57.57

0.40

47

G.G1

0.33

25.64

0.19

321

5.54

0.65

0.04

31.34

0.22

57.07

0.40

48

6.94

0.33

25.82

o.i 8

3 32

5.53

0.70

0.05

31.55

0.21

68.30

0.39

49

7.29

0.35

25.99

0.17

!!.42

5.51

0.77

O.07

31.76

O.2I

58.75

0.39

0.35

0.17

0.08

0.21

0.39

GO

7.64

20.16

3.52

5.48

0.85

31.97

59.14

94

TABLES OF NEPTUNE.

TABLK.

V.

VI.

VII.

Arg.

1

2

3

P..I

Diff.

-Pe.1

Diff.

P+

*••!

<5t>j

Diff.

*,

Diff.

*•

Diff.

n

n

n

«

tr

H

//

ft

ft

n

//

n

50

7.64

26.16

3.52

5.48

0.85

31.97

59.14

51

8.00

0.36

26.31

0.15

3.63

5.45

0.94

0.09

12.17

O.2O

59.52

0.38

62

8.36

0.36

26.45

0.14

3.73

5.41

1.03

0.09

32.38

O.2I

59. 89

0.37

53
54

8.73
9.11

0.37
0.38

26.59
26.71

0.14

0.12

3.83
3.92

5.37
5.33

1.11
1.26

O. II
0.12

82.68

32.77

O.20

o. 19

60.25

00.01

0.36
0.36

0.38

O. II

0.13

0.19

0.35

55

9.49

26.82

4.02

5.28

1.39

32.96

60.90

56

9.87

0.38

26.93

0. II

4.11

5.23

1.53

O.I4

33.15

0.19

61.30

0.34

57
58

10.26
10.65

0.39
0.39

27.02
27.10

0.09
0.08

4.20
4.29

5.17
5.11

1.67
1.83

O.I4

o.i 6

33.34
88.62

019

0.18

61.68

61.96

°-33
0.33

59

11.05

0.40
0.40

27.17

0.07
O.O6

4.38

5.04

2.00

0.17
o.i 8

33.70

o.i 8

0.17

62.29

°-33
0.32

60

11.45

27.23

4.40

4.97

2.18

33.87

62.01

61

11.85

0.40

27.28

0.05

4.55

4.90

2.37

0.19

34.04

0.17

62.91

0.30

62

12.25

0.40

27.32

0.04

4.62

4.82

2.56

0.19

34.21

0.17

68.22

0.31

63

12.05

0.40

27.34

O.O2

4.70

4.74

2.76

0.2O

34.37

o.i 6

68.62

0.30

64

13.06

0.41

27.36

O.02

4.76

4.65

2.97

O.2I

34.53

o. 1 6

63.81

0.29

0.41

0.00

O.2I

o. 1 6

0.29

65

13.47

27.36

4.83

4.57

3.18

34.69

04.10

66

13.88

0.41

27.35

O.OI

4.89

4.48

3.40

0.22

34.84

0.15

64.38

0.28

67

14.29

0.41

27.33

0.01

4.94

4.39

3.63

0.23

34.99

0.15

64.66

0.27

68

14.70

0.41

27.29

0.04

5.00

4.29

3.87

0.24

35.14

0.15

04.91

0.26

69

15.10

0.40

27.25

0.04

5.04

4.20

4.11

O.24

35.28

o. 14

05.17

0.26

0.41

0.06

0.25

0.14

0.24

70

15.51

27.19

5.09

4.10

4.36

35.42

65.41

71

15.92

0.41

27.12

0.07

5.13

3.99

4.61

0.25

35.55

0.13

65.05

0.24

72

16.32

0.40

27.04

0.08

5.16

3.89

4.86

0.25

35.68

0.13

65.88

0.23

73

16.72

0.40

26.95

0.09

5.19

3.79

5.13

O.27

35.81

0.13

00.10

O.22

74

17.12

0.40

26.84

O. II

5.22

3.69

5.39

O.26

35.93

0. 12

66.32

O.22

0.39

0.12

0.27

O. II

O.2I

75

17.51

26.72

5.24

3.58

5.66

36.04

66.53

76

17.90

. o-39

26.59

0.13

5.25

3.48

5.94

0.28

36.16

0. 12

66.72

O.I9

77

18.29

0.39

26.45

O.I4

5.20

3.37

6.21

0.27

30.27

0. II

60.92

O.2O

78

18.67

0.38

26.30

0.15

5.27

3.27

6.50

0.29

36.37

0. 10

67.11

o. 19

79

19.05

0.38

26.14

o.i 6

5.27

3.16

6.78

0.28

30.47

0. IO

67.28

0.17

0.37

o.ig

0.29

O. IO

0.17

80

19.42

25.96

5.27

3.05

7.07

30.57

67.45

81

19.78

0.36

25.78

0.18

5.26

2.95

7.36

0.29

86.68

0.09

67.01

o.i 6

82

20.14

0.36

25.58

0.10

5.25

2.85

7.65

0.29

30.75

0.09

07.70

0.15

83

20.49

0.35

25.37

0.21

5.23

2.74

7.94

0.29

30.83

0.08

07.91

0.15

84

20.84

o-3S

25.15

O.22

5.21

2.64

8.24

0.30

30.91

0.08

68.05

o. 14

0.34

O.22

0.30

0.07

0.13

85

21.18

24.93

5.18

2.54

8.54

36.98

68.18

86

21.51

0.33

24.69

O.24

5.15

2.45

8.83

0.29

37.05

0.07

68.80

0. 12

87

21.8:5

0.32

24.44

0.25

5.11

2.35

9.13

0.30

37.12

0.07

68.41

O. II

88

22.14

0.31

24.18

0.26

5.07

2.25

9.43

0.30

37.18

0.06

68.62

O.II

89

22.45

0.31

23.U1

0.27

5.02

2.16

9.73

o. 30

37.24

0.06

68.61

0.09

0.29

0.27

0.29

0.05

0.09

90

22.74

23.64

4.97

2.07

10.02

37.29

68.70

91

23.02

0.28

23.35

0.29

4.92

1.98

10.82

0.30

37.34

0.05

68.78

0.08

92

23.30

0.28

23.06

0.29

4.86

1.96

10.82

0.30

87.88

0.04

58.86

O.O7

93

23.50

0.26

22.75

0.31

4.80

1.82

10.92

0.30

37.42

0.04

08.91

O.o6

94

23.81

0.25

22.44

0.31

4.74

1.74

11.21

0.29

37.45

0.03

68.97

0.06

0.24

0.32

0.30

0.03

O.OJ

95

24.05

22.12

4.67

1.66

11.51

37.48

09.02

90

24.28

0.23

21.80

0.32

4.60

1.59

11.80

0.29

37.51

0.03

oo.no

O.O4

97

24.50

O.22

21.47

0.33

4.53

1.53

12.10

0.30

37.53

O.O2

69.09

0.03

98

24.71

O.2I

21.13

0.34

4.45

1.46

12.38

0.28

87.66

O.O2

69.1 1

O.O2

99

21.91

O.2O

20.78

0-35

4.37

1.40

12.07

O.29

37.50

O.OI

09.12

O.OI

0.19

0.35

O.29

0.00

O.OO

100

86.10

20.43

4.28

1.35

12.96

37.56

69.12

TABLES OP NEPTUNE.

95

TABLE.

V.

VI.

VII.

Arg.

1

2

3

P.1

Diff.

Pc.l

Diff.

PS.l

-Pc.2

fe,

Diff.

rfi)2

Diff.

*•.

Diff.

n

//

tl

It

n

//

It

n

It

n

II

n

100

25.10

20.43

4.28

1.35

12.96

37.56

69.12

101

25.26

o. 1 6

20.08

°-35

4.20

1.30

13.24

0.28

37.57

O.OI

69.12

o.oo

102

25.42

o.i 6

19.72

0.36

4.11

1.25

13.52

0.28

37.56

O.OI

69.11

O.OI

108

25.57

0.15

19.35

0.37

4.02

1.21

13.79

0.27

37.56

o.oo

69.09

O.O2

104

25.70

0.13

18.98

0.37

3.93

1.17

14.08

0.29

87.54

O.O2

69.06

0.03

0.12

0.37

0.26

O.OI

0.04

105

25.82

18.61

3.84

1.14

14.34

37.63

69.02

106

25.92

O. IO

18.23

0.38

3.75

1.11

14.00

0.26

37.51

O.O2

68.97

0.05

107

20.02

O. IO

17.86

o-37

3.65

1.09

14.87

0.27

37.48

0.03

68.91

0.06

108

26.10

0.08

17.47

0.39

3.56

1.07

15.13

0.26

37.45

0.03

68.85

0.06

109

26.16

0.06
O.O6

17.09

0.38
0.38

3.46

1.05

15.38

0.25
0.25

37.42

0.03
O.O4

68.78

0.07

0.08

110

26.22

16.71

3.36

1.04

15.03

37.38

68.70

111

26.26

O.O4

16.32

0.39

3.27

1.04

15.87

0.24

37.33

O.O5

68.62

0.08

112

26.28

0.02

15.94

0.28

3.17

1.04

10.11

0.24

37.28

0.05

68.52

O. IO

113

26.30

0.02

15.56

0.38

3.07

1.04

16.34

0.23

37.23

0.05

68.41

O. II

114

20.29

O.OI

15.17

0.39

2.98

1.06

16.57

0.23

37.17

O.O6

68.30

O. II

O.OI

0.38

0.23

O.O6

0. 12

115

26.28

14.79

2.88

1.07

16.80

37.11

68.18

110

20.25

0.03

14.41

0.38

2.78

1.09

17.02

0.22

37.05

O.O6

68.05

0.13

117

26.20

0.05

14.03

0.38

2.69

1.11

17.24

O.22

36.98

O.07

67.91

O.I4

118

26.15

0.05

13.65

0.38

2.60

1.14

17.44

O.2O

36.90

0.08

67.76

O.I5

119

20.08

0.07

13.27

0.38

2.51

1.18

17.65

O.2I

36.82

0.08

67.61

0.15

0.08

0.37

0.19

0.08

o.i 6

120

20.00

12.90

2.42

1.21

17.84

36.74

67.45

121

25.90

0. 10

12.53

0.37

2.34

1.26

18.03

O.I9

36.65

O.Og

67.28

0.17

122

25.79

0. II

12.17

0.36

2.26

1.30

18.22

o. 19

36.55

O. IO

67.11

0.17

128

25.07

0.12

11.81

0.36

2.18

1.35

18.40

0.18

36.46

0.09

66.92

0.19

124

25.54

0.13

11.45

o. 36

2.10

1.41

18.57

0.17

36.35

O. II

60.73

0.19

O.I4

0.35

0.17

O.I I

O.2O

125

25.40

11.10

2.03

1.46

18.74

36.24

66.53

12(3

25.24

o.i 6

10.70

0.34

1.95

1.52

18.89

0.15

36.13

O. II

66.32

O.2I

127

25.07

0.17

10.42

0.34

1.89

1.59

19.05

o.i 6

36.02

o.i i

66.10

O.22

128

24.89

o.i 8

10.09

°-33

1.82

1.66

19.19

0.14

35.90

0.12

65.88

0.22

129

24.70

0.19

9.77

0.32

1.76

1.73

19.33

0.14

35.77

0.13

65.65

0.23

0.21

0.32

0.13

0. 12

O.24

130

24.49

9.45

1.70

1.80

19.46

35.65

65.41

131

24.28

O.2I

9.14

0.31

1.65

1.88

19.59

0.13

35.51

o. 14

65.17

O.24

132

24.05

0.23

8.84

0.30

1.60

1.96

19.70

O. II

35.38

0.13

64.91

O.26

133

23.82

0.23

8.55

0.29

1.55

2.04

19.82

0.12

35.24

O.I4

64.65

O.26

134

23.57

0.25

8.27

0.28

1.51-

2.12

19.92

C. 10

35.09

O.I5

64.38

0.27

O.26

0.27

0. IO

0.15

0.28

135

23.31

8.00

1.47

2.21

20.02

34.94

64.10

136

23.04

0.27

7.73

0.27

1.44

2.29

20.11

0.09

34.79

O.I5

63.81

O.29

137

22.77

0.27

7.48

0.25

1.41

2.38

20.19

0.08

34.63

o.i 6

63.52

0.29

138

22.49

0.28

7.23

0.25

1.39

2.47

20.27

o 08

34.47

o.i 6

63.22

0.30

139

22.19

0.30

7.00

0.23

1.37

2.56

20.34

O.O7

34.31

o.i 6

62.91

0.31

0.30

0.23

O.O6

0.17

0.30

140

21.89

6.77

1.36

2.65

20.40

34.14

62.61

141

21.58

O.JI

6.50

O.2I

1.35

2.74

20.46

O.o6

33.97

0.17

62.29

0.32

142

21.27

0.3!

6.36

O.2O

1.34

2.83

20.50

O.O4

33.79

0.18

61.96

0.33

143

20.95

0.32

6.17

0.19

1.34

2.92

20.54

O.O4

33.61

0.18

61.63

0.33

144

20.63

0.32

5.99

o.i 8

1.35

3.01

20.58

0.04

33.43

o. 18

61.30

0.33

0.34

0.17

O.O2

0.19

0-34

145

20.29

5.82

1.35

3.10

20.60

33.24

60.96

146

19.96

o-33

5.67

0.15

1.37

3.18

20.63

O.O3

33.05

0.19

60.61

0.35

147

19.01

0.35

5.53

0.14

1.39

3.27

20.64

O.OI

32.85

O.20

60.25

0.36

148

19.26

0.35

5.40

0.13

1.41

3.36

20.65

O.OI

32.66

O.I9

59.89

0.36

149

18.91

0.35

5.28

0.12

1.44

3.44

20.65

o.oo

32.45

O.2I

69.52

0.37

0.35

O. II

o.oo

O.2O

0.38

150

18.56

5.17

1.47

3.52

20.65

32.25

59.14

96

TABLES OF NEPTUNE.

TABLE.

V.

VI.

VII.

Arg.

1

2

3

P..I

Diff.

Pc.1

Diff.

p^

P^

<5»,

Diff.

fo,

Diff.

*•

Diff.

//

it

n

n

If

II

//

//

//

M

It

//

150

18.56

5.17

1.47

3.52

20.62

32.25

59.14

151

18.20

0.36

5.08

0.09

1.50

3.60

20.64

O.OI

32.04

O.2I

68.75

0.39

152

17.84

0.36

6.00

0.08

1.54

3.68

20.62

O.O2

31.83

0.21

58.36

0.39

153

17.47

0.37

4.93

0.07

1.59

3.76

20.59

0.03

31.02

O.ZI

57.97

0.39

154

17.10

o-37

4.87

0.06

1.63

3.83

20.56

0.03

31.40

0.22

67.57

0.40

0.36

0.04

O.O4

0.22

0.40

155

16.74

4.83

1.69

3.90

20.52

31.18

67.17

156

16.37

0.37

4.80

0.03

1.74

3.97

20.48

O.O4

30.95

0.23

56.76

0.41

157

16.00

°-37

4.79

O.OI

1.80

4.03

20.43

0.05

30.73

0.22

66.34

0.42

158

15.64

0.36

4.78

O.OI

1.85

4.09

20.37

O.o6

30.50

0.23

56.92

0.42

159

15.27

0.37

4.79

O.OI

1.92

4.15

20.31

O.O6

30.26

O.24

55.50

0.42

0.37

0.03

O.O7

0.23

0-44

160

14.90

4.82

1.98

4.21

20.24

30.03

55.06

161

14.54

0.36

4.85

0.03

2.05

4.2G

20.16

0.08

29.79

O.24

54.03

0.43

162

14.18

0.36

4.90

0.05

2.12

4.30

20.08

0.08

29.55

O.24

54.18

0.45

103

13.82

0.36

4.97

0.07

2.20

4.35

19.99

0.09

29.31

O.24

53.74

0.44

164

13.46

0.36

5.04

0.07

2.27

4.39

19.90

0.09

29.00

0.25

53.29

0.45

0.35

0.09

O. IO

0.25

0.45

165

13.11

5.13

2.35

4.42

19.80

28.81

62.84

166

12.76

°-35

5.23

0. 10

2.43

4.45

19.70

O. IO

28.56

0.25

52.38

0.46

107

12.42

0.34

5.34

0. II

2.51

4.48

19.59

O.I I

28.30

O.26

51.1U

0.47

168

12.08

0.34

5.46

0. 12

2.59

4.50

19.48

0. II

28.04

O.26

51.44

0.47

109

11.74

0.34

6.60

0.14

2.67

4.52

19.36

0.12

27.79

0.25

50.97

°-47

o-33

0.14

0. 12

0.27

0.47

170

11.41

5.74

2.75

4.53

19.24

27.62

50.50

171

11.09

0.32

5.90

o.i 6

2.83

4.54

19.11

0.13

27.26

O.26

50.02

0.48

172

10.77

0.32

6.07

0.17

2.92

4.54

18.98

0.13

27.00

O.26

49.64

0.48

173

10.46

0.3:

6.25

0.18

3.00

4.54

18.84

O.I4

26.73

0.27

49.05

0.49

174

10.16

o. 30

6.44

0.19

3.08

4.53

18.70

O.I4

26.46

0.27

48.56

0.49

0.29

0.21

0.1S

0.27

0.49

175

9.87

6.65

3.16

4.68

18.55

20.19

48.07

176

9.58

0.29

6.86

O.2I

3.25

4.51

18.40

O.I5

25.92

0.27

47.57

0.50

177

9.31

0.27

7.09

0.23

3.33

4.49

18.24

o.i 6

26.64

0.28

47.07

0.50

178

9.04

0.27

7.32

0.23

3.41

4.47

18.09

0.15

26.87

0.27

46.67

0.50

179

8.78

O.26

7.56

0.24

3.49

4.44

17.92

0.17

25.09

0.28

40.00

0.51

0.25

0.26

o.i 6

0.28

0.52

180

8.53

7.82

3.57

4.41

17.70

24.81

45.54

181

8.29

0.24

8.08

0.26

3.04

4.38

17.59

0.17

24.52

0.29

45.03

0.51

182

8.06

0.23

8.35

0.27

3.71

4.34

17.41

o.i 8

24.24

0.28

4-1.52

0.51

183

7.84

O. 22

8.62

0.27

3.79

4.30,

17.24

0.17

23.96

0.28

44.01

0.51

184

7.63

0.21

8.91

0.29

3.85

4.25

17.06

(5. 1 8

23.67

0.29

43.49

0.52

0.20

0.29

0.18

0.28

0.52

185

7.43

9.20

3.92

4.20

16.88

23.39

42.97

186

7.24

0.19

9.49

0.29

3.98

4.14

16.70

0.18

23.10

0.29

42.44

0.53

187

7.06

0.18

9.80

0.31

4.04

4.09

16.51

0.19

22.81

0.29

41.92

0.52

188

6.90

0.16

10.11

0.31

4.10

4.03

16.32

0.19

22.52

0.29

41.39

0.53

189

6.76

0.15

10.42

0.31

4.15

3.96

10.13

0.19

22.23

0.29

40.87

0.52

0.14

0.33

0.19

0.29

o-53

190

6.61

10.76

4.20

8.90

15.94

21.94

40.34

191

6.48

0.13

11.07

0.32

4.25

3.83

15.75 .

O. IO

f

21.65

0.29

39.81

o-53

192

6.36

O. II

11.40

0.33

4.29

3.76

15.55

0.20

21.36

0.29

39. 28

0.53

193

6.26

O. IO

11.74

0.34

4.33

3.69

15.36

0.19

21.06

0.30

38.75

0.53

194

6.17

0.09

12.08

0.34

4.36

3.61

15.16

O.2O

20.77

O.29

38.21

0.54

0.07

0.34

O.2O

0.30

°-53

195

6.10

12.42

4.39

3.64

14.96

20.47

37.68

196

6.03

0.07

12.76

0.34

4.42

3.46

14.76

O.2O

20.18

O.29

37.14

0.54

197

5.98

0.05

13.11

0.35

4.44

3.38

14.56

O.2O

19.89

O.29

36.61

0.53

198

5.94

0.04

13.46

0.35

4.46

3.30

14.36

O.2O

10.59

0.30

30.07

0.54

199

5.92

O.O1

13.80

0.34

4.48

3.22

14.16

O.2O

19.30

0.29

35.54

o-53

O.OI

0.36

O. 2O

0.30

0.54

200

5.91

14.16

4.49

3.13

13.96

19.00

35.00

TABLES OF NEPTUNE.

97

TABLE.

V.

VI.

VII.

Arg.

1

2

3

P..1

Diff.

JVi

Diff.

f..2

Prt

fv,

Diff.

iv,

Diff.

S*3

Diff.

//

n

//

If

n

II

n

/*

„

n

tl

n

200

5.91

14.16

4.49

3.13

13.96

19.00

35.00

201

5.91

O.OO

14.51

0.35

4.49

3.05

13.76

O.2O

18.70

0.30

34.46

0.54

•2i>'2

5.92

O.OI

14.86

0.35

4.49

2.97

13.56

O.2O

18.41

0.29

33.113

0.53

203

5.95

0.03

15.21

0.35

4.49

2.88

13.36

O.2O

18.11

0.30

33.39

0.54

204

6.00

0.05

15.55

0.34

4.48

2.80

13.16

O.2O

17.82

0.29

32.86

0.53

0.06

0.35

0.19

0.29

0.54

205

6.06

15.90

4.47

2.72

12.97

17.53

32.32

206

6.12

0.06

16.25

0.35

4.45

2.64

12.77

O.2O

17.23

0.30

31.79

0.53

207

6.21

0.09

10.59

0.34

4.43

2.56

12.57

O.2O

16.94

0.29

31.25

0.54

208

6.30

0.09

16.93

0.34

4.41

2.48

12.38

O.I9

16.64

0.30

30.72

0.53

209

6.41

0. II

17.27

0.34

4.38

2.41

12.19

0.19

16.35

0.29

30.19

0.53

O.I2

0.33

0.19

0.29

°-53

210

G.'53

17.60

4.34

2.33

12.00

16.06

29.66

211

6.65

0.12

17.93

°-33

4.31

2.26

11.81

O.I9

15.77

0.29

29.13

0.53

212

6.80

O.I5

18.26

0.33

4.27

2.19

11.62

0.19

15.48

0.29

28.61

0.52

213

6.96

o.i 6

18.57

0.31

4.22

2.13

11.43

0.19

15.19

0.29

28.08

0.53

214

7.14

o.i 8

18.89

0.32

4.17

2.06

11.25

0.18

14.90

0.29

27.56

0.52

0.18

0.31

o.i 8

0.29

0.53

215

7.32

19.20

4.12

2.00

11.07

14.61

27.03

216

7.51

0.19

19.50

0.30

4.07

1.94

10.89

o.i 8

14.33

0.28

26.51

0.52

217

7.72

0.21

19.79

0.29

4.01

1.88

10.72

0.17

14.04

0.29

25.99

0.52

218

7.94

O.22

20.08

0.29

3.95

1.83

10.55

0.17

13.76

0.28

25.48

0.51

219

8.16

O.22

20.36

0.28

3.88

1.78

10.38

0.17

13.48

0.28

24.96

0.52

0.24

0.27

o.i 6

0.29

0.51

220

8.40

20.63

3.82

1.73

10.22

13.19

24.45

221

8.65

0.25

20.90

0.27

3.75

1.69

10.06

o.i 6

12.91

0.28

23.94

0.51

222

8.91

O.26

21.15

0.25

3.68

1.65

9.90

o.i 6

12.63

0.28

23.43

0.51

223

9.18

0.27

21.40

0.25

3.60

1.62

9.74

o.i 6

12.36

0.27

22.93

0.50

224

9.45

0.27

21.64

0.24

3.53

1.59

9.59

0.15

12.08

0.28

22.43

0.50

0.29

0.23

0.15

0.27

0.50

225

9.74

21.87

3.45

1.56

9.44

11.81

21.93

226

10.04

O.JO

22.08

0.21

3.37

1.54

9.30

0.14

11.54

0.27

21.44

0.49

227

10.34

o. 30

22.29

O.2I

3.29

1.52

0.16

0.14

11.27

0.27

20.95

0.49

228

10.65

0.3I

22.49

O.2O

3.22

1.50

9.03

0.13

11.00

0.27

20.46

0.49

229

10.97

0.32

22.67

o.i 8

3.14

1.49

8.90

0.13

10.74

0.26

19.98

0.48

0.33

0.18

0. 12

0.26

0.48

230

11.30

22.85

3.06

1.49

8.78

10.48

19.50

231

11.63

0.33

23.01

o.i 6

2.97

1.49

8.66

0.12

10.21

0.27

19.03

0.47

232

11.97

0.34

23.K!

0.15

2.89

1.49

8.54

0. 12

9.96

0.25

18.56

0.47

233

12.32

0.35

23.30

0.14

2.81

1.50

8.44

O.IO

9.70

0.26

18.09

0.47

234

12.67

0.35

23.42

O.I2

2.73

1.51

8.33

0. II

9.44

0.26

17.62

0.47

0.36

O.I2

0.09

0.25

0.46

235

13.03

23.54

2.65

1.53

8.24

9.19

17.16

230

13.39

0.36

23.64

O. IO

2.57

1.55

8.14

O.IO

8.94

0.25

16.71

0.45

237

13.76

0.37

23.73

0.09

2.49

1.58

8.06

0.08

8.69

0.25

16.26

0.45

238

14.12

0.36

23.81

0.08

2.42

1.60

7.97

0.09

8.45

6.24

15.82

0.44

239

14.50

0.38

23.87

O.O6

2.34

1.64

7.90

O.O7

8.21

0.24

15.37

0.45

0.37

O.O6

O.O7

0.24

0.43

240

14.87

23.93

2.27

1.68

7.83

7.97

14.94

241

15.25

0.38

23.96

0.03

2.20

1.72

7.77

O.O6

7.74

0.23

14.50

0.44

242

15.63

0.38

23.99

0.03

2.13

1.76

7.71

O.O6

7.50

0.24

14.08

0.42

243

16.01

0.38

24.00

O.OI

2.06

1.81

7.66

O.O5

7.27

0.23

13.66

0.42

244

16.39

0.38

24.00

O.OO

2.00

1.87

7.61

0.05

7.05

O.22

13.24

0.42

0.38

O.OI

O.O4

0.23

0.41

245

16.77

23.99

1.93

1.92

7.57

6.82

12.83

246

17.16

0.39

23.96

0.03

1.88

1.98

7.54

0.03

6.60

O.22

12.43

0.40

247,

17.54

0.38

23.92

0.04

1.82

2.04

7.62

O.O2

6.38

0.22

12.03

0.40

248

17.92

0.38

23.86

0.06

1.77

2.11

7.50

0.02

6.17

O.2I

11.64

0.39

249

18.30

0.38

23.79

0.07

1.72

2.18

7.48

O.O2

5.96

O.2I

11.25

0.39

0.37

0.08

O.OO

O.2I

0.39

250

18.67

23.71

1.67

2.25

7.48

5.75

10.86

13

May, 1865.

98

TABLES OF NEPTUNE.

TABLE.

V.

VI.

VII.

Arg.

1

2

3

P..I

Diff.

Pc.1

Diff.

P*

PC.,

6vl

Diff.

*t

Diff.

*ll

Diff.

n

n

ft

n

H

H

tt

ff

//

tt

//

tt

250

18.67

23.71

1.67

2.25

7.48

5.75

10.80

251

19.05

°-38

23.62

0.09

1.63

2.32

7.47

O.OI

5.55

O.2O

10.48

0.38

252

19.42

0.37

23.51

O. II

1.59

2.40

7.48

O.OI

5.34

O.2I

10.11

0.37

253

19.78

0.36

23.39

0. 12

1.55

2.48

7.50

O.O2

5.15

O.I9

9.75

0.36

254

20.15

0.37

23.26

O.I3

1.52

2.56

7.52

O.O2

4.95

O.2O

9.39

0.36

0.36

O.IS

O.O2

O.I9

o-35

255

20.51

23.11

1.60

2.64

7.54

4.76

9.04

256

20.86

°-35

22.96

0.15

1.48

2.72

7.58

O.O4

4.57

O.lg

8.70

o-34

257

21.21

0.35

22.78

o.i 8

1.40

2.81

7.61

O.O3

4.39

0.18

8.37

0.33

258

21.55

0.34

22.00

o.i 8

1.44

2.90

7.66

0.05

4.21

0.18

8.04

0.33

259

21.89

0.34

22.41

0.19

1.43

2.98

7.71

0.05

4.03

o.i 8

7.71

°-33

0.33

O.2O

0.06

0.17

0.32

260

22 22

22.21

1.43

3.07

7.77

3.86

7.39

201

22.55

0.33

21.99

O.22

1.43

3.16

7.84

O.O7

3.69

0.17

7.09

0.30

262

22.8(5

0.31

21.76

0.23

1.43

3.25

7.91

O.O7

3.53

o.i 6

6.78

0.31

263

23.17

0.31

21.52

0.24

1.44

3.33

7.99

0.08

3.37

• o. 1 6

6.48

0.30

264

23.47

0.30

21.27

0.2S

1.46'

3.42

8.08

O.Og

3.21

o.i 6

6.19

0.29

0.29

O.26

O.O^

0.15

0.29

265

23.76

21.01

1.47

3.51

8.17

3.06

5.90

266

24.04

0.18

20.74

0.27

1.50

3.59

8.28

O. II

2.91

0.15

5.62

0.28

267

24.32

0.28

20.46

0.28

1.52

3.68

8.38

0. 10

2.76

0.15

5.35

0.27

238

24.58

0.26

20.17

0.29

1.56

3.76

8.50

0. 12

2. 02

0.14

5.09

0.26

269

24.83

0.25

19.87

0.30

1.59

3.85

8.62

0.12

2.49

0.13

4.83

0.26

0.25

0.30

O.I 2

0.14

0.24

270

25.08

19.57

1.63

3.93

8.74

2.35

4.59

271

25.31

0.23

19.25

0.32

1.68

4.01

8.88

0.14

2.23

0.12

4.35

0.24

272

25.53

O.22

18.93

0.32

1.72

4.08

9.02

0.14

2.10

0.13

4.12

0.23

273

25.74

O.2I

18.59

0.34

1.78

4.16

0.17

O.IS

1.98

0. 12

3.90

O.22

274

25.93

0.19

18.26

0.33

1.83

4.23

9.32

O.IS

1.87

0. II

3.08

0.22

O.I9

0.35

o.i 6

O. II

O.2I

275

26.12

17.91

1.89

4.30

9.48

1.76

3.47

276

26.29

0.17

17.56

0-35-

1.96

4.36

9.65

0.17

1.65

0. II

3.27

O.2O

277

20.46

0.17

17.20

0.36

2.02

4.43

9.82

0.17

1.54

O. II

3.08

0.19

278

26.60

O.I4

16.84

0.36

2.09

4.49

10.00

o.i 8

1.45

0.09

2.89

0.19

279

26.74

O.I4

16.47

0.37

2.16

4.55

10.19

0.19

1.35

0. 10

2.72

0.17

0.13

0.38

0.19

0.09

O.I7

280

26.87

16.09

2.24

4.60

10.38

1.26

2.55

281

26.98

0.1 1

15.71

0.38

2.32

4.65

10.57

0.19

1.18

0.08

2.39

0.16

282

27.07

0.09

15.33

0.38

2.40

4.69

10.78

O.2I

1.10

0.08

2.24

0.15

283

27.16

0.09

14.94

0.39

2 48

4.73

10.98

O.2O

1.02

0.08

2.09

0.15

284 '

27.23

0.07

14.55

0.39

2.57

4.77

11.20

O.22

0.95

0.07

1.95

0.14

O.O6

0.39

O.22

0.06

0.13

285

27.29

14.16

2.66

4.80

11.42

0.89

1.82

286

27.34

0.0S

13.77

o-39

2.75

4.83

11.64

O.22

0.83

0.06

1.70

0.12

287

27.37

0.03

13.37

0.40

2.84

4.85

11.87

0.23

0.77

0.06

1.59

O. II

288

27.38

0.01

12.97

0.40

2.93

4.87

12.10

0.23

0.71

0.06

1.48

O.I I

289

27.39

0.01

12.58

0.39

3.03

4.88

12.34

0.24

0.67

0.04

1.38

O. IO

O.OI

0.40

0.25

0.05

0.08

290

27.38

12.18

3.12

4.89

12.59

0.62

1.30

291

27.35

0.03

11.78

0.40

3.21

4.90

12.84

0.25

0.58

0.04

1.22

O.oS

292

27.31

0.04

11.38

0.40

3.31

4.90

13.09

0.25

0.55

0.03

1.15

0.07

293

27.26

0.05

10.98

0.40

3.40

4.90

13.34

0.25

0.52

0.03

1.09

O.06

294

27.20

0.06

10.58

0.40

3.50

4.89

13.60

O.26

0.49

0.03

1.03

0.06

0.08

0.39

0.27

O.O2

0.05

295

27.12

10.19

3.59

4.87

13.87

0.47

0.98

296

27.03

0.09

9.80

0.39

3.69

4.85

14.13

0.26

0.46

O.OI

0.94

O.O4

297

20.92

O.I I

9.41

0.39

3.78

4.83

14.40

0.27

0.44

O.O2

0.91

0.03

298

26.80

0. 12

9.02

0.39

3.87

4.80

14.68

0.28

0.44

O.OO

0.89

O.O2

299

26.67

0.13

8.64

0.38

3.96

4.76

14.95

0.27

0.43

O.OI

0.88

O.OI

0.14

0.38

0.2g

O.OI

O.OO

300

26.53

8.26

4.06

4.73

15.23

0.44

0.88

TABLES OF NEPTUNE.

99

TABLE.

V.

VI.

VII.

Arg.

1

2

3

-P..!

Diff.

Pc.l

Diff.

P.3

P*

6vl

Diff.

Svt

Diff.

ivt

Diff.

It

a

II

n

tl

n

„

//

n

n

»

//

300

26.53

8.26

4.06

4.73

15.23

0.44

0.88

301

26.37

o.i 6

7.89

0.37

4.14

4.68

15.51

0.28

0.44

o.oo

0.88

o.oo

302

20.20

0.17

7.52

0.37

4.23

4.64

15.80

0.29

0.45

O.OI

0.89

O.OI

303

26.02

o.i 8

7.16

0.36

4.31

4.59

16.08

0.28

0.47

O.O2

0.91

O.O2

304

25.82

O.2O

6.80

0.36

4.40

4.43

16.37

0.29

0.49

O.O2

0.94

0.03

O. 20

0.35

0.28

0.03

O.O4

305

25.62

6.45

4.47

4.47

16.65

0.52

0.98

306

25.40

O.22

6.10

°-35

4.55

4.41

16.95

0.30

0.55

0.03

1.03

0.05

307

25.17

O.2J

5.76

0.34

4.62

4.34

17.24

0.29

0.58

0.03

1.09

O.o6

308

24.92

0.25

5.43

0.33

4.69

4.27

17.53

0.29

0.62

0.04

1.15

O.O6

309

24.67

0.25

5.11

0.32

4.76

4.20

17.82

0.29

0.66

O.O4

1.22

0.07

O.26

0.31

0.30

0.05

0.08

310

24.41

4.80

4.83

4.12

18.12

0.71

1.30

311

24.14

0.27

4.49

0.31

4.89

4.04

18.41

0.29

0.76

0.05

1.38

0.08

312

23.85

0.29

4.19

0.30

4.94

3.96

18.71

0.30

0.82

0.06

1.48

O. IO

313

23.56

0.29

3.90

0.29

5.00

3.87

19.00

0.29

0.88

O.O6

1.59

O. II

814

23.26

O.JO

3.62

0.28

5.05

3.78

19.29

0.29

0.95

O.O7

1.70

0. II

0.32

0.27

0.29

O.O7

0. 12

315

22.94

3.35

5.09

3.69

19.58

1.02

1.82

316

22.62

0.32

3.08

0.27

5.13

3.59

19.87

0.29

1.09

O.O7

1.95

0.13

317

22.29

0.33

2.83

0.25

5.17

3.50

20.16

0.29

1.17

0.08

2.09

0.14

318

21.95

0.34

2.59

0.24

5.20

3.40

20.45

0.29

1.25

0.08

2.24

O.I5

319

21.61

0.34

2.36

0.23

5.23

3.30

20.73

0.28

1.34

0.09

2.39

O.I5

0.36

O.22

0.29

0.09

0.16

320

21.25

2.14

5.25.

3.20

21.02

1.43

2.55

321

20.89

0.36

1.93

O.2I

5.27

3.10

21.29

0.27

1.53

O. IO

2.72

0.17

322

20.53

0.36

1.73

O.2O

5.28

2.99

21.57

0.28

1.63

O. IO

2.90

0.18

323

20.15

0.38

1.54

0.19

5.29

2.89

21.84

0.27

1.73

O. IO

3.09

0.19

321

19.78

0.37

1.36

0.18

5.30

2.78

22.12

0.28

1.84

O. II

3.28

0.19

0.39

o.i 6

0.26

0. 12

O.2O

325

19.39

1.20

5.29

2.68

22.38

1.96

3.48

326

19.00

0.39

1.04

o.i 6

5.29

2.57

22.65

0.27

2.07

O. II

3.70

O.22

327

18.61

0.39

0.90

0.14

5.28

2.46

22.91

0.26

2.19

0.12

3.92

0.22

328

18.22

0.39

0.77

°-'3

5.26

2.36

23.16

0.15

2.32

O.I3

4.14

O.22

«29

17.82

0.40

0.65

0. 12

5.24

2.25

23.41

0.25

2.45

0.13

4.37

0.23

0.41

O. II

0.25

0.13

O.24

330

17.41

0.54

5.22

2.15

23.66

2.58

4.61

331

17.01

0.40

0.45

0.09

5.19

2.05

23.90

0.24

2.72

0.14

4.85

O.24

332

16.60

0.41

0.36

0.09

6.15

1.95

24.13

0.23

2.86

O.I4

5.11

O.26

333

16.19

0.41

0.29

O.O7

5.12

1.85

24.36

0.23

3.01

O.I5

5.37

O.26

334

15.78

0.41

0.24

O.O5

5.07

1.75

24.58

O.22

3.16

O.I5

5.64

0.27

0.41

O.O5

O.22

O.IS

0.28

335

15.37

0.19

5.02

1.65

24.80

3.31

5.92

336

14.95

0.42

0.16

O.O3

4.97

1.56

25.00

0.20

3.47

0.16

6.21

0.29

337

14.54

0.41

0.13

0.03

4.92

1.47

25.21

O.2I

3.63

o.i 6

6.50

0.29

338

14.13

0.41

0.12

O.OI

4.85

1.38

25.40

O.I9

3.79

o.i 6

6.80

0.30

339

13.72

0.41

0.12

0 00

4.79

1.29

25.59

0.19

3.96

0.17

7.11

0.31

0.41

O.OI

O.I?

0.17

0.30

340

13.31

0.13

4.72

1.21

25.76

4.13

7.41

341

12.90

0.41

0.1 6

0.03

4.65

1.13

25.93

0.17

4.30

0.17

7.73

0.32

342

12.49

0.41

0.19

0.03

4.57

1.05

26.09

o.i 6

4.48

0.18

8.06

0.33

343

12.09

0.40

0.24

0.05

4.49

0.98

26.24

0.15

4.66

o.i 8

8.39

0.33

344

11.69

0.40

0.30

0.06

4.41

0.91

26.39

0.15

4.85

0.19

8.72

0.33

0.40

0.07

0.13

0.19

0.34

345

11.29

0.37

4.32

0.85

26.52

6.04

9.06

346

10.89

0.40

0.46

0.09

4.24

0.79

26.65

0.13

5.23

0.19

9.41

°-35

347

10.50

0.39

0.55

0.09

4.15

0.73

26.76

0. II

5.42

0.19

9.77

o. 36

348

10.12

0.38

0.65

O. IO

4.05

0.68

26.86

O. IO

6.62

0.2O

10.13

0.36

349

9.73

0.39

0.77

O.I2

3.96

0.63

26.95

0.09

5.83

0.21

10.50

0.37

0.38

0. 12

0.08

O.2O

0.38

350

9.35

0.89

3.86

0.59

27.03

6.03

10.88

100

TABLES OF NEPTUNE.

TABLE.

V.

VI.

VII.

Arg.

1

2

3

/>,,

Diff.

-P..1

Diff.

P.,

P«

*,

Diff.

*,

Diff.

*.

Diff.

n

n

n

It

It

It

n

n

,i

n

n

n

350

9.35

0.89

3.86

0.59

27.03

6.03

10.88

351

8.98

0.37

1.02

0.13

3.77

0.55

27.09

O.O6

6.24

O.2I

11.20

0.38

352

8.62

0.36

1.17

0.15

3.66

0.52

27.15

O.O6

O.2I

11.66

0.40

353

8.26

0.36

1.32

0.15

3.56

0.49

27.20

0.05

6.6fl

O.2I

12.05

0.39

354

7.90

0.36

1.48

o.i 6

8.46

0.47

27.23

0.03

6.88

O.22

12.45

0.40

0.34

0.18

0.03

O.22

0.40

355

7.56

1.66

3.36

0.45

27.26

7.10

12.85

356

7.22

0.34

1.83

0.17

3.26

0.43

27.27

O.OI

7.32

O.22

13.26

0.41

357

6.89

0.33

2.02

0.19

3.15

0.42

27.27

o.oo

7.55

0.23

13.I18

0.42

868

6.56

0.33

2.22

O.2O

3.05

0.42

27.25

O.02

7.78

0.23

14.10

0.42

359

6.25

0.31

2.42

O.2O

2.95

0.42

27.22

0.03

8.01

0.23

14.52

0.42

0.31

0.21

0.04

0.23

0.43

360

5.94

2.63

2.85

0.42

27.18

8.24

14.95

361

5.64

0.30

2.85

0.22

2.75

0.43

27.12

O.06

8.48

O.24

15.39

0.44

362

5.35

0.29

3.07

O.22

2.64

0.44

27.05

O.O7

8.72

0.24

15.84

0.45

363

5.06

0.29

8.30

0.23

2.54

0.46

26.97

O.O8

8.96

0.24

16.28

0.44

364

4.79

0.27

3.54

0.24

2.45

0.48

26.86

O. II

9.20

O.24

16.73

0.45

0.26

O.24

0. II

O.24

0.45

365

4.53

3.78

2.35

0.50

26.75

9.44

17.18

366

4.27

0.26

4.03

0.25

2.26

0.53

26.62

0.13

9.69

0.25

17.64

0.46

3ii7

4.03

0.24

4.28

0.25

2.16

0.57

26.47

0.15

9.94

0.25

18.11

0.47

368

3.79

0.24

4.53

0.2J

2.07

0.60

26.31

0.16

10.19

0.25

18.58

0.47

369

3.56

0.23

4.79

O.26

1.98

0.65

26.13

o.i 8

10.45

O.26

19.05

0.47

0.21

O.26

0.19

0.25

0.47

370

3.35

5.05

1.89

0.69

25.94

10.70

19.52

371

3.14

O.2I

5.31

0.26

1.81

0.74

25.74

O.2O

10.96

O.26

20.00

0.48

372

2.94

O.20

5.58

0.27

1.72

0.79

25.51

O.23

11.22

0.26

20.48

0.48

373

2.76

0.19

5.85

0.27

1.65

0.84

25.27

O.24

11.48

O.26

20.07

0.49

374

2.58

0.17

6.12

0.27

1.57

0.90

25.02

0.25

11.75

O.27

21.46

0.49

0.17

0.27

0.27

O.26

0.49

375

2.41

6.39

1.50

0.96

24.75

12.01

21.95

376

2.24

0.17

6.66

0.27

1.42

1.02

24.46

0.29

12.28

0.27

22.44

0.49

377

2.09

O.I5

6.94

0.28

1.36

1.08

24.16

0.30

12.65

0.27

22.04

o. 50

378

1.95

0.14

7.22

0.28

1.29

1.15

23.84

0.32

12.82

0.27

23.44

0.50

379

1.81

0.14

7.49

0.27

1.23

1.21

23.51

0.33

13.09

0.27

23.95

0.51

0.13

0.28

0.35

0.27

0.51

380

1.68

7.77

1.17

1.28

23.16

13.36

24.46

381

1.56

0.12

8.04

0.27

1.11

1.35

22.80

0.36

13.63

0.27

24.97

0.51

382

1.45

O. II

8.32

0.28

1.06

1.42

22.42

0.38

13.91

O.28

26.48

0.51

383

1.34

0. II

8.60

0.28

1.01

1.49

22.03

0.39

14.19

O.28

25.09

0.51

384

1.24

O.IO

8.88

0.28

0.96

1.57

21.62

0.41

14.46

0.27

26.51

0.52

0.09

0.27

0.41

O.28

0.52

385

1.15

9.15

0.92

1.64

21.21

14.74

27.03

386

1.06

0.09

9.43

o.i 8

0.87

1.72

20.77

0.44

15.02

0.28

27.50

0.53

387

0.98

0.08

9.70

0.27

0.83

1.79

20.33

0.44

15.30

O.28

28.08

0.52

388

0.91

0.07

9.98

0.28

0.79

1.87

19.88

0.45

15.68

O.28

28.61

0.53

389

0.84

0.07
O.o6

10.25

0.27
0.27

0.76

1.95

19.42

0.46
0.48

15.86

O.28
0.29

29.13

0.52
0.53

390

0.78

10.52

0.72

2.03

18.94

16.15

20.66

391

0.72

O.O6

10.79

0.27

0.70

2.10

18.46

0.48

16.43

0.28

30.19

0.53

392

0.67

0.05

11.06

0.27

0.67

2.18

17.97

0.49

16.71

O.28

30.72

0.53

393

0.62

0.05

11.34

0.28

0.64

2.26

17.47

0.50

17.00

0.29
n

31.25

°-53

394

0.57

0.05

11.60

0.26

0.62

2.35

16.97

0.50

17.28

O.28

31.79

0.54

0.04

0.28

0.52

O.29

°-S3

395

0.53

11.88

0.60

2.43

10.45

17.57

32.32

396

0.50

0.03

12.14

o.*6

0.59

2.51

15.94

0.51

17.86

O.29

32.86

0.54

397

0.46

0.04

12.42

0.28

0.57

2.59

15.42

0.52

18.14

O.28

33.39

°-53

398

0.43

0.03

12.68

0.26

0.56

2.67

14.90

0.52

18.43

O.29

33.93

0.54

399

0.41

0.02

12.95

0.17

0.55

2.75

J4.37

0.53

18.71

O.28

34.46

°-53

0.03

0.27

0.51

O.29

0.54

400

0.38

13.22

0.54

2.83

13.85

19.00

35.00

TABLES OP NEPTUNE.

101

TABLE.

VIII.

IX.

X.

XI.

XII.

XIII.

TABLE.

VIII.

IX.

X.

XI.

XII.

XIII.

Arg.

4

5

6

7

8

0

Arg.

4

5

6

7

8

9

*,

*,

*h

*,

*,

6va

*<

*>,

,

•6

*

*.

to.

0

0.08

0.15

0.94

0.94

0.30

0.80

200

1.12

0.05

0.06

1.26

0.10

0.20

10

0.08

0.14

0.96

1.06

0.28

0.84

210

1.12

0.06

0.04

1.14

0.12

0.16

20

0.10

0.13

0.97

1.19

0.26

0.88

220

1.10

0.07

0.03

1.01

0.14

0.12

30

0.13

0.12

0.96

1.31

0.24

0.91

230

1.07

0.08

0.04

0.89

0.16

0.09

40

0.18

0.10

0.95

1.43

0.22

0.93

240

1.02

0.10

0.05

0.77

0.18

0.07

50

0.23

0.09

0.93

1.54

0.20

0.94

250

•0.97

0.11

0.07

0.66

0.20

0.06

60

0.29

0.08

0.89

1.64

0.18

0.94

260

0.91

0.12

0.

11

0.56

0.22

0.06

70

0.36

0.07

0.85

1.72

0.15

0.93

270

0.84

0.13

0.

15

0.48

0.25

0.07

80

0.44

0.06

0.79

1.79

0.13

0.90

280

0.76

0.14

0.21

0.41

0.27

0.10

90

0.52

0.05

0.73

1.85

0.12

0.87

290

0.68

0.15

0.27

0.35

0.28

0.13

100

0.60

0.04

0.67

1.88

0.10

0.83

300

0.60

0.16

0.33

0.32

0.30

0.17

110

0.68

0.04

0.60

1.90

0.09

0.78

310

0.52

0.16

0.40

0.30

0.31

0.22

120

0.76

0.03

0.52

1.90

0.08

0.72

320

0.44

0.17

0.48

0.30

0.32

0.28

130

0.84

0.03

0.45

1.87

0.07

0.66

330

0.36

0.17

0.

55

0.33

0.33

0.34

140

0.91

0.03

0.38

1.83

0.06

0.59

340

0.29

0.17

0.

62

0.37

0.34

0.41

150

0.97

0.03

0.31

1.77

0.06

0.52

350

0.23

0.17

0.69

0.43

0.34

0.48

160

1.02

0.03

0.25

1.69

0.06

0.45

360

0.18

0.17

0.

75

0.51

0.34

0.55

170

1.07

0.03

0.19

1.60

0.07

0.38

370

0.13

0.17

0.81

0.60

0.33

0.62

180

1.10

0.04

0.14

1.50

0.07

0.32

380

0.10

0.16

0.86

0.70

0.33

0.68

190

1.12

0.05

0.09

1.40

0.09

0.26

390

0.08

0.15

0.91

0.80

0.31

0.74

200

1.12

0.05

0.06

1.26

0.10

0.20

400

0.08

0.15

0.94

0.94

0.30

0.80

TABLE XIV.

If the date is earlier than 1779, Jan.

4, or later than 1943, Oct. 15, the

values of Psl and Pcl must be corrected as follows, the argument being the

year :

Year.

AP..

APci Year.

AP..i

APc.i

Year.

AP..!

APC., *

II

„

1 61 4.2

— 56.83

— 31.88 1700.0

— 65.57

— 24.94

1943.8

+ 72.87

+ 16.62

1620.0

— 57.44

— 31.46 1710.0

— 66.53

— 24.02

1950.0

+ 73.39

+ 15.88

1 630.0

— 58.49

— 30.73 1720.0

— 67.50

— 23.04

1960.0

+ 74.21

+ 14.68

1640.0

— 59.52

— 29.98 1730.0

— 68.46

— 22.04

1970.0

+ 75.02

+ 13.44

1650.0

— 60.56

— 29.18 1740.0

— 69.40

— 21.00

1980.0

+ 75.81

+ 12.18

1660.0

— 61.58

— 28.36 1750.0

— 70.32

— 19.92

1990.0

+ 76.58

+ 10.88

1670.0

— 62.60

— 27.54 1760.0

71.22

— 18.82

2000.0

+ 77.32

+ 9.52

1680.0

— 63.62

— 26.70 1770.0

— 72.10

— 17.68

1690.0

— 64.60

— 25.84 1779.0

— 72.87

— 16.62

. Between 1779 and 1943, Ps and Pc require no correction.

For dates earlier

than 1614 or later than 2000, the corrections must be computed from the

formulas.

•

102

TABLES OF NEPTUNE.

TABLE XV.

EQUATION OF THE CENTRE.

i

Equation.

Diff.

I

Equation.

Diff.

I

Equation.

Diff.

0

0 / II

n

o

O 1 II

n

0

O 1 II

//

180

1 38 54.42

225

057 42.12

270

0 17 6.17

181

1 38 10.17

44-25

22G

05641.74

60.38

271

0 1624.17

42.00

182

1 37 25.21

44.96

1-11

055 41.40

60.34

272

015 42.'.i2

41.25

183

1 3G 39.56

45-65

228

05441.12

60.28

273

0 15 2.45

40.47

184

1 35 53.23

46.33
46.99

229

0 53 40.90

60.22
60.12

274

0 14 22.76

39 69
38.90

185

1 35 6.24

230

0 52 40.78

275

0 13 43.86

186

1 34 18.60

47.64

231

0 51 40.76

60.02

270

013 5.77

38.09

187

1 33 30.33

48.27

232

0 50 40.87

59.89

277

0 1228.50

37-^7

188

1 32 41.43

48.90

233

04941.12

59-75

278

0 11 52.00

36-44

189

1 31 51.93

49.50

234

0 48 41.53

59-59

279

0 11 1G.40

35.60

50.09

59-4"

34-75

190

1 31 1.84

235

0 47 42.12

280

0 1041.71

191

1 30 11.17

50.67

236

0 46 42.90

59.22

281

010 7.82

33-89

192

1 29 19.93

51.14

237

0 45 43.90

59.00

282

0 9 34.81

33.01

193

1 28 28.15

51.78

238

044 45.13

5^-77

283

0 9 2.0'.)

32.12

194

1 27 35.84

52.31

239

0 43 4G.61

58.52

284

0 831.4G

3'-i3

52.81

58-*5

30-33

195

1 26 43.02

240

0 42 48.36

285

0 8 1.13

196

1 25 49.70

53.32

241

0 41 50.39

57-97

286

0 7 31.72

29.41

197

1 24 55.90

53.80

242

0 40 52.72

57.67

287

0 7 3.23

28.49

198

1 24 1.64

54.26

243

0 39 55.36

57.36

288

0 G 35.07

27.56

199

1 23 6.93

54- 7 >

244

0 38 58.34

57.02

289

0 G 9.0G

26.61

SS-'S

56.67

25.67

200

1 22 11.78

245

038 1.67

290

0 5 43.39

201

1 21 16.21

55-57

246

037 5.37

56.30

291

0 5 18.08

24.71

202

1 20 20.25

55.96

247

0 36 9.45

55-92

392

0 4 54.94

23-74

203
204

1 19 23.90
1 1827.19

56-35
56.71
57.06

248
249

035 13.94
0 34 18.84

55-5»
55.10
54.66

293
294

0 432.18
0 4 10.40

22.76
21.78
20.80

205

1 17 30.13

250

0 33 24.18

295

0 3 49.00

206

1 16 32.73

57-40

251

0 32 29.97

54.21

296

0 3 29.80

19.80

207

1 15 35.02

57-71

252

0 31 36.22

53-75

297

0311 .00

18.80

208

1 14 37.01

58.01

253

0 30 42.96

53.26

298

0 2 .ri3.21

17.79

209

1 13 38.72

58.29

254

0 29 50.19

52.77

299

0 2 :;o.-ll

16.77

58.56

52-25

15.76

210

1 12 40.16

265

0 28 57.94

300

0 2 20.08

211

1 11 41.35

58 81

256

028 6.22

51.71

301

0 2 5.05

>4-73

212

1 10 42.32

59-oj

257

0 27 15.05

51.17

302

0 1 52.26

13.69

213

1 943.08

59- *4

258

0 26 24.43

50.62

308

0 1 39.00

12.66

214

1 8 43.64

59-44

259

0 25 34.40

50.03

304

0 1 27.97

1 1.63

59.61

49-45

10.58

215

1 744.03

260

0 24 44.95

-805

0 1 17.39

216

1 644.26

59-77

261

02356.11

48.84

306

0 1 7.85

9-54

217

1 5 44.35

599'

2G2

023 7.89

48.22

307

0 0 59.30

8.49

218

1 444.32

60.03

2G3

0 22 20.31

47-58

308

0 051.93

7-43

219

1 3 44.18

60.14

264

0 21 33.37

46.94

309

0 045.55

6.38

60.23

46.27

53'

220

1 243.95

265

0 20 47.10

310

0 040.24

221

1 1 43.65

60.30

289

020 1.50

45.60

311

0 0 30.00

4 24

222

1 043.30

60.35

207

0 19 16.60

44-90

312

0 032.82

3.18

223

0 59 42.92

60.38

268

0 18 32.40

44.20

313

0 0 30.71

2.1 I

224

0 58 42.62

60.40

269

0 17 48.92

43.48

314

0 0 29.07

I.O4

60.40

4^-75

0.03

225

0 57 42.12

270

017 6.17

315

0 0 29.70

TABLES OP NEPTUNE.

103

TABLE XV.

EQUATION OF THE CENTRE (Continued).

i

Equation.

Diff.

I

Equation.

Diff.

I

Equation.

Diff.

o

0 1 II

//

O

O 1 II

If

O

O 1 II

H

315

0 029.70

0

018 11.18

45

1 024.20

316
317

0 0 30.80
0 0 32.00

I.IO

2.16

1

2

0 18 55.91
0 19 41.40

44-73
45-49

46

47

1 1 26.40
1 2 28.66

62.2O
62.16

318

0 03G.20

3.24

3

0 20 27.63

46.23

48

1 3 30.65

62.09

319

0 040.50

4.30

4

0 21 14.59

46.96

49

1 4 32.67

62.02

5.38

47.67

61.91

320

0 045.88

5

0 22 2.26

50

1 5 34.58

321

0 052.33

6.45

6

022 50.63

48-37

51

1 6 36.37

61.79

322

0 0 59.84

7-5'

7

0 23 39.68

49.05

52

1 7 38.02

61.65

323

0 1 8.42

8.58

8

0 24 29.40

49-72

53

1 8 39.51

61.49

324

0 1 18.06

9.64

9

0 25 19.78

50.38

54

1 940.82

61.31

10.70

51.01

61.11

325

0 1 28.76

10

0 26 10.79

55

1 10 41.93

326
327
328

0 1 40.52
0 1 53.34
0 2 7.21

11.76
12.82
13.87

11
12
13

0 27 2.43
0 27 54.67
0 28 47.50

51.64

52.24
52.83

56
57
58

1 11 42.82
1 12 43.46
1 13 43.85

60.89
60.64
60.39

329

0 2 22.13

14.92

14

0 29 40.91

53-41

59

1 14 43.96

60. ii

'5-97

53.96

59.82

330

0 2 38.10

15

0 30 34.87

60

1 15 43.78

331

0 2 55.11

17.01

16

0 31 29.38

54-5'

61

1 16 43.28

59.50

332

0 3 13.1ii

18.05

17

0 32 24.41

55-°3

62

1 17 42.45

59-'7

333

0 3 32.24

19.08

18

0 33 19.95

55-54

63

1 18 41.27

58.82

334

0 3 52.35

20.11
21.13

19

0 34 15.97

56.02
56.50

64

1 19 39.71

58.44
58.05

335

0 4 13.48

20

0 35 12.47

65

1 20 37.76

886

0 4 35. (52

22.14

21

036 9.42

56.95

66

1 21 35.41

57-65

337
338

0 458.78
0 5 22.94

23.16
24.16

22
23

0 37 6.81
038 4.61

57-39
57.80

67
68

1 22 32.62
1 23 29.39

57.21
56.77

339

0 5 48.09

*5-i5

24

0 39 2.81

58.20

69

1 24 25.70

56.31

26.15

58.59

55-83

340

0 6 14.24

25

040 1.40

70

1 25 21.53

341

0 G41.37

27.13

26

041 0.35

58-95

71

1 26 16.87

55-34

342

0 7 9.47

28.10

27

0 41 59.64

59.29

72

1 27 11.69

54.82

343

0 7 38.54

29.07

28

0 42 59.26

59.62

73

1 28 5.97

5428

344

0 8 8.57

30.03

29

04359.19

59-93

74

1 28 59.71

53-74

30.97

60.22

53-'7

345

0 839.54

30

0 44 59.41

75

1 29 52.88

346

0 911.46

31.92

31

0 45 59.90

60.49

76

1 30 45.46

52.58

347

0 9 44.30

32.84

32

0 47 0.63

60.73

77

1 31 37.45

51.99

348

0 10 18.07

33-77

33

048 1.60

60.97

78

1 32 28.82

5'-37

349

0 10 52.74

34-67

34

049 2.78

ftl.lS

79

1 33 19.55

5°-73

3S-58

61.38

50.09

350

0 11 28.32

35

050 4.16

80

1 34 9.64

351

012 4.79

36.47

36

051 5.71

61.55

81

1 34 59.06

49.42

352
353

0 1242.14
0 13 20.35

37-35
38.21

37
38

052 7.41
0 53 9.25

61.70

61.84

82
83

1 35 47.80
1 36 35.85

48.74
48.05

354

0 13 59.42

39-°7

39

05411.20

61.95

84

1 37 23.19

47-34

39.92

62.05

46.61

355

0 14 39.34

40

0 55 13.25

85

1 38 9.80

356

0 1520.09

40.75

41

0 56 15.37

6.2. 12

86

1 38 55.67

45-87

357

0 16 1.66

4'-57

42

0 57 17.54

62.17

87

1 39 40.80

45-'3

358

0 16 44.04

42.38

43

0 58 19.75

62.21

88

1 4025.15

44-35

359

0 17 27.22

43.18

44

05921.98

62.23

89

1 41 8.73

43-58

43.96

62.22

42.78

300

01811.18

45

1 024.20

90

1 41 51.51

104

TABLES OF NEPTUNE.

TABLE XV.

EQUATION OF THE CENTRE (Concluded).

i

Equation.

Diflf.

I

Equation.

Diff.

I

Equation.

Diff.

0

Of ft

//

o

O / /'

ft

o

O ' H

//

90

I 41 51.51

120

1 66 10.41

150

1 54 54.44

91

1 42 33.48

41-97

121

1 56 23.47

13.06

151

1 54 35.48

18.96

92

1 43 14.63

41.15

122

1 56 35.47

12. OO

152

1 54 15.50

19.98

93

1 43 54.95

40.31

123

1 56 46.39

10.92

153

1 53 54.51

20.99

94

1 44 34.42

39-47
38.61

124

1 60 5G.24

9.85
8.78

154

1 63 32.52

21.99
22.99

95

1 45 13.04

125

1 67 5.02

155

1 53 9.53

96
97

1 45 50.78
1 4<! 27.65

37-74
36.87

126
127

1 67 12.72
1 57 19.34

7.70
6.62

166
157

1 62 45.55
1 52 20.59

23.98
24.96

98

1 47 3.62

35-97

128

1 57 24.87

5-53

158

1 51 54.06

25.93

99

1 47 38.69

35-0?
34.16

129

1 57 29.33

4.46
3-37

159

1 51 27.70

26.90
27.85

100

1 48 12.85

130

1 57 32.70

1GO

1 50 59.91

101

1 48 46.08

33.23

131

1 57 34.99

2.29

101

1 5031.11

28.80

102

1 4!) 18.38

32.30

132

1 57 30.20

1. 21

102

1 50 1.38

29.73

103

1 49 49.73

3'-35

133

1 57 30.32

O. 12

103

1 49 30.71

30.67

104

1 50 20.13

30.40

134

1 67 35.36

0.96

104

1 48 59.13

31.58

29.44

2.05

32.49

105

1 50 49.57

135

1 57 33.31

105

1 48 20.64

106

1 51 18.03

28.46

136

1 57 30.19

3.12

10P,

1 47 53.25

33 39

107
108

1 51 45.52
1 52 12.01

27.49
26.49

137
138

1 57 25.98
1 57 20.70

4.21
5.28

107
108

1 47 18.97
1 46 43.82

34.28
35-'5

109

1 52 37.51

25.50

139

1 67 14.34

6.36

109

1 46 7.80

36.02

24.50

7-43

36. 88

110

1 53 2.01

140

1 57 6.91

170

1 45 30.92

111

1 53 25.50

23.49

141

1 56 58.41

8.50

171

1 44 53.20

37.72

112

1 53 47.07

22.47

142

1 5(5 48.84

9-57

172

1 44 14.65

38-55

113

1 54 9.42

21.45

143

1 56 38.21

10.63

173

1 43 35.28

39-37

114

1 54 29.84

20.42

144

1 56 26.52

11.69

174

1 42 55.10

40. 1 8

19.38

12.74

40.97

115

1 54 49.22

145

1 66 13.78

175

1 42 14.13

116

1 55 7.56

18.34

146

1 55 59.99

'3-79

176

1 41 32.38

4' 75

117
118

1 55 24.8.')
1 55 41.10

17.29
16.25

147

148

1 55 45.16
1 65 29.28

14.83
15.88

177
178

1 404'.t.85
1 4(1 6.57

4^ 53
43.28

119

1 55 56.28

15.18

149

1 55 12.38

16.90

179

1 39 22.55

44.02

14.13

17-94

44-75

120

1 56 10.41

150

1 64 54.44

180

1 38 37.80

TABLES OF NEPTUNE.

105

TABLE XVI.

REDUCTION TO TIIK ECLIPTIC.

Argument u.

1800

1900

2000

Diff.
100 Y.

Argument «.

1800

1900

2000

Diff.
100 Y.

0

O

O

O

If

II

II

„

o

O

O

0

„

//

„

rr

0

90

180

270

60.00

60.00

60.00

o.oo

135

315

135

315

110.23

109.72

100.21

o.5i

1

89

181

269

58.25

58.26

58.28

0.01

134

316

136

314

110.20

100.70

109.18

0.51

2

88

182

268

56.50

56.53

56.57

0.03

133

317

137

313

110.11

100.61

109.10

0.51

3

87 .

183

267

54.75

54.80

54.84

0.05

132

318

138

312

109.06

100.45

108.05

0.50

4

86

184

266

53.01

53.07

53.14

0.07

131

319

139

311

109.74

109.24

108.74

0.50

5

85

185

265

51.27

51.36

51.45

0.09

130

320

140

310

109.47

108.97

108.47

0.50

6

84

186

264

49.55

49.66

49.76

O.I I

129

321

141

309

109.14

108.64

108.14

0.50

7

83

187

203

47.85

47.97

48.09

0. 12

128

322

142

308

108.74

108.25

107.76

0.49

8

82

188

262

46.15

46.29

46.43

O.I4

127

323

143

307

108.29

107.80

107.32

0.49

9

81

189

261

44.48

44.63

44.79

O.I5

126

324

144

306

107.78

107.30

106.81

0.48

10

80

190

260

42.82

42.99

43.16

O.I7

125

325

145

305

107.21

106.73

106.25

0.48

11

79

191

259

41.18

41.37

41.56

o. 19

124

326

146

304

106.58

106.11

105.64

0.47

12

78

192

258

39.57

39.78

39.98

O.2I

123

327

147

303

105.90

105.44

104.07

0.46

13

77

193

257

37.98

38.20

38.43

0.12

122

328

148

302

105.15

104.70

104.24

0.45

14

76

194

256

36.42

36.66

36.90

0.24

121

329

149

301

104.35

103.91

103.46

0.44

15

75

195

255

34.88

35.14

35.40

O.26

120

330

150

300

103.50

103.06

102.62

0.44

16

74

196

254

33.38

33.65

33.93

O.27

119

331

151

299

102.60

102.17

101.74

0.43

17

73

197

253

31.91

32.20

32.49

O.29

118

332

152

298

101.64

101.22

100.80

0.42

18

72

198

252

30.47

30.78

31.08

0.31

117

333

153

297

100.64

100.23

99.82

0.41

19

71

199

251

29.07

29.39

29.71

0.32

116

334

164

296

99.58

99.18

98.78

0.40

20

70

200

250

27.71

28.03

28.37

0.33

115

335

155

295

98.48

98.09

97.69

0.39

21

69

201

249

26.39

26.73

27.07

0-34

114

336

156

294

07.33

96.95

96.57

0.38

22

68

202

248

25.11

25.46

25.82

0.35

118

337

167

293

90.13

95.77

95.40

0.36

23

67

203

247

23.87

24.23

24.60

0.36

112

338

158

202

94.89

94.54

04.18

°-35

24

66

204

246

22.67

23.05

23.43

0.38

111

339

169

291

93.61

93.27

92.93

0.34

25

65

205

245

21.52

21.91

22.31

0.39

110

340

160

290

92.29

91.96

91.63

0.33

26

64

206

244

20.42

20.82

21.22

0.40

109

341

161

289

90.93

90.61

90.29

0.32

27

63

207

243

19.36

19.77

20.18

0.41

108

342

162

288

89.53

80.22

88.02

0.31

28

62

208

242

18.36

18.78

19.20

0.42

107

343

163

287

88.09

87.80

87.51

0.29

29

61

209

241

17.40

17.83

18.26

0.43

106

344

164

286

86.62

86.35

86.07

0.27

30

60

210

" 240

16.50

16.94

17.38

0.44

105

345

165

285

85.12

84.86

84.60

0.26

31

59

211

239

15.65

16.09

16.54

0-44

104

346

166

284

83.58

83.34

83.10

0.24

32

58

212

238

14.85

15.30

15.76

0.45

103

347

167

283

82.02

81.80

81.57

O.22

33

57

213

237

14.10

14.56

15.03

0.46

102

348

168

282

80.43

80.22

80.02

0.21

34

56

214

236

13.42

13.89

14.36

0.47

101

349

169

281

78.82

78.63

78.44

O.I9

35

55

215

235

12.79

13.27

13.75

0.48

100

350

170

280

77.18

77.01

76.84

0.17

36

54

216

234

12.22

12.70

13.19

0.48

99

351

171

279

75.52

75.37

75.21

0.15

37

53

217

233

11.71

12.20

12.68

0.49

98

352

172

278

73.85

73.71

73.57

0.14

38

52

218

232

11.26

11.75

12.24

0.49

97

353

173

277

72.15

72.03

71.91

0.12

39

51

219

231

10.86

11.36

11.86

0.50

96

354

174

276

70.45

70.34

70.24

O. IO

40

50

220

230

10.53

11.03

11.53

0.50

95

355

175

275

68.73

68.64

68.55

0.09

41

40

221

229

10.26

10.76

11.26

0.50

94

356

176

274

66.99

06.93

66.86

O.O7

42

48

222

228

10.04

10.55

11.05

0.50

93

357

177

273

65.25

65.20

65.16

0.05

43

47

223

227

9.89

10.39

10.90

0.51

92

358

178

272

63.50

63.47

63.43

0.03

44

46

224

226

9.80

10.30

10.82

0.51

91

359

179

271

61.75

61.74

61.72

O.OI

45

45

225

225

9.77

10.28

10.79

0.51

90

0

180

270

60.00

60.00

60.00

O.OO

14 May, 1865.

106

TABLES OF NEPTUNE.

TABLE XVII.

COEFFICIENTS FOB PERTURBATIONS OF LOG. RADIUS VECTOR.

Argument 1.

0

50

100

150

200

250

300

350

*u

Rc.1

/?...

^c.1

Ba

•KC.I

B..I

K*

A.I

flc.1

tf..i

Rc.l

£u

flc.l

*..i

«n

0

131

23

40

26

34

170

176

142

131

45

78

142

212

170

214

26

i

130

23

37

28

36

172

178

139

129

45

79

145

214

168

212

24

2

130

23

35

30

39

174

180

136

126

45

81

147

216

166

209

°2

3

129

23

82

32

41

176

181

133

124

45

82

150

218

163

207

20

4

129

23

30

35

44

178

183

130

122

46

84

162

220

160

204

18

5

128

23

28

37

46

180

184

127

120

46

86

155

222

158

202

17

6

127

23

26

39

49

181

185

124

118

47

88

157

224

156

199

16

7

127

22

24

42

51

183

186

121

116

47

91

160

226

163

197

15

8

126

21

22

45

54

185

186

119

113

48

93

162

227

151

194

15

9

126

21

20

48

57

185

187

116

111

49

96

165

229

148

191

14

10

125

20

18

51

60

186

187

113

109

50

98

167

230

146

189

13

11

124

19

16

54

63

186

187

110

107

51

100

169

232

143

187

12

12

123

18

14

57

66

187

187

108

104

52

103

171

233

140

184

11

13

123

18

13

60

70

187

187

105

102

63

105

173

235

138

182

10

14

122

17

11

64

73

188

187

102

100

54

108

175

237

135

179

9

15

121

16

10

67

76

188

187

99

97

56

110

177

238

132

177

9

16

120

15

9

70

79

189

187

96

95

57

113

178

239

129

175

9

17

118

15

8

73

82

190

186

93

93

59

116

180

240

126

172

9

18

117

14

8

76

86

191

185

90

92

60

119

181

241

122

170

9

19

115

13

7

79

89

192

184

88

90

61

122

183

241

119

167

9

20

113

13

7

82

92

192

183

85

89

63

125

184

242

116

165

9

21

111

13

6

85

95

192

183

83

87

65

128

185

242

113

163

9

22

110

13

6

88

98

190

182

80

85

67

131

186

243

109

161

10

23

108

12

5

92

102

189

182

78

83

70

134

187

244

105

160

10

24

106

12

5

95

105

189

181

76-

81

72

137

187

244

102

158

11

25

104

11

4

98

108

188

180

74

80

74

140

188

244

98

156

11

26

102

11

4

102

111

187

178

72

79

76

143

189

244

95

154

12

27

100

10

4

105

115

187

177

70

78

78

146

190

244

92

152

12

28

98

10

5

109

118

186

175

67

77

80

150

191

244

88

150

12

29

95

9

5

113

121

185

173

65

76

83

153

192

243

85

148

13

30

93

9

G

116

125

184

171

63

75

85

156

192

243

82

147

13

31

91

9

6

119

128

183

170

61

74

88

158

192

243

70

146

14

32

88

9

7

122

131

181

168

60

73

90

161

191

242

76

144

14

33

86

9

7

126

134

180

167

59

72

93

164

190

242

73

143

15

34

83

9

8

129

137

178

166

57

72

96

107

189

241

70

142

15

35

81

9

8

132

140

177

165

56

71

99

170

188

240

67

141

16

36

78

9

9

135

143

175

163

54

70

102

173

188

239

64

140

17

37

76

10

10

138

146

173

1110

53

70

105

176

187

238

00

139

18

38

73

11

12

140

149

171

158

61

69

108

180

187

237

57

138

18

39

70

12

-13

143

-151

169

155

50

69

110

183

186

235

54

138

19

40

67

13

16

146

154

167

153

49

69

113

186

186

234

51

137

20

41

64

14

17

148

156

165

151

48

69

116

189

185

232

48

136

21

42

62

15

19

151

159

162

148

48

70

119

191

184

230

45

136

21

43

59

16

21

153

162

160

146

48

70

121

194

183

228

42

135

22

44

57

16

22

156

164

157

144

47

71

124

197

181

226

39

134

22

45

54

17

24

158

166

155

142

47

72

127

200

180

224

37

134

22

46

51

18

26

160

168

152

140

46

73

130

202

178

222

35

133

23

47

49

20

28

163

170

150

138

45

74

133

205

176

220

32

133

23

48

46

22

30

166

172

147

136

45

75 .

136

207

174

218

30

132

23

49

43

24

32

168

174

145

133

45

77

139

210

172

216

28

131

23

50

40

26

84

170

176

142

131

45

78

142

212

170

214

26

131

23

NOTE.— Before 1779 and after 1943, we have

.i = 10.53 A ^c, 1614 — 1778 Slogr = -

.i = — 10.53 Ap.. 1943 — 2108 s l°S r = + 314-

TABLES OF NEPTUNE.

107

PERTURBATIONS OF LOGARITHM OF RADIUS VECTOR.

TABLE XVIII.

TABLE XIX.

TABLE XX.

Argument 1.

Argument 2.

Argument 3.

Arg.

0

50

100

150

0

50

I
100

150

0

50

100

150

0

743

387

58

5

801

681

396

119

1401

1196

700

204

50

1

743

378

54

5

801

676

390

115

1401

1188

689

196

49

2

743

3fi9

51

6

801

671

383

111

1400

1180

678

188

48

3

742

361

48

6

801

666

377

107

1400

1172

667

181

47

4

740

352

44

7

800

662

371

103

1399

1103

656

174

46

5

738

343

41

7

800

657

365

99

1398

1155

645

167

45

6

736

334

38

7

800

652

859

95

1397

1147

634

160

44

7

733

325

35

8

799

647

353

91

1396

1138

623

163

43

8

730

316

33

9

798

642

346

88

1396

1130

613

146

42

9

726

307

30

9

797

637

340

84

1393

1121

602

139

41

10

722

298

28

10

796

632

334

80

1392

1112

691

133

40

11

718

290

26

11

795

627

• 328

77

1390

1103

680

127

39

12

713

282

24

11

794

621

322

73

1388

1094

570

121

38

13

708

274

23

12

793

616

316

70

1386

1085

659

115

37

14

702

267

21

13

791

610

310

67

1384

1075

648

109

36

15

690

259

19

14

790

605

304

64

1382

1066

537

103

35

18

690

251

17

14

788

600

298

61

1379

1057

526

97

34

17

684

244

' 16

15

787

604

292

67

1377

1047

515

91

33

18

678

236

14

15

785

589

286

54

1374

1038

504

85

32

19

672

229

13

16

783

583

280

51

1370

1028 ,

493

80

31

20

665

222

12

16

781

578

274

48

1367

1018

483

75

30

21

658

215

10

16

779

573

268

45

1364

1008

473

70

29

22

651

207

9

17

777

567

262

43

1300

998

463

65

28

23

643

200

8

18

775

661

257

40

1356

988

452

60

27

24

635

192

6

19

772

655

251

38

1352

978

442

66

26

25

626

185

5

20

770

549

245

36

1348

968

432

52

25

20

617

179

4

20

707

643

239

34

1344

958

422

48

24

27

608

172

3

20

705

537

234

31

1339

948

412

44

23

28

699

166

3

21

702

532

228

29

1335

937

402

40

22

29

590

160

2

21

759

526

223

27

1330

927

392

36

21

30

580

154

2

21

756

520

218

25

1325

917

382

33

20

31

570

148

2

22

753

513

213

23

1320

907

372

30

19

32

501

143

2

22

750

507

207

22

1315

896

362

26

18

33

561

137

2

23

747

500

202

20

1309

885

353

23

17

a4

542

131

3

23

743

494

196

19

1303

874

343

21

16

35

532

125

3

24

740

488

191

18

1297

863

334

18

15

36

522

120

3

24

730

482

186

17

1291

852

325

16

14

37

513

115

3

25

733

476

181

16

1285

841

315

14

. 13

38

503

110

4

25

729

470

176

14

1279

830

306

12

12

39

494

105

4

20

726

464

171

13

1273

820"

297

10

11

40

484

100

4

26

722

458

166

12

1267

809

288

8

10

41

474

95

4

26

718

452

161

11

1261

798

279

7

9

42

404

91

4

27

714

446

156

10

1254

787

270

6

8

43

455

86

4

27

711

439

151

10

1247

777

2C,2

4

7

44

445

81

4

27

707

432

146

9

1240

766

253

3

6

45

435

77

4

28

703

427

141

8

1233

755

246

2

5

40

42C

73

4

28

699

421

137

8

1226

744

237

1

4

47

416

69

4

28

(idl

415

132

8

1219

733

228

0

3

. 48

406

65

5

29

690

408

128

7

1212

722

220

0

2

49

397

61

5

29

685

402

123

7

1204

711

212

0

1

50

387

58

5

29

081

396

119

7

1196

700

204

0

0

350

300

250

200

350

300

250

200

350

300

250

200

Arg.

108

TABLES OF NEPTUNE.

TABLE XXI.

PRINCIPAL TERM OF THE LOGARITHM OF THE RADIUS VECTOR.
Argument Z.

i

1.4

I

1.4

I

1.4

o

o

O

180

806G76

240

815373

300

788978

181

7112

436

241

5192

181

301

8352

626

182

7540

428

242

5001

191

302

7722

630

183

70GO

420

243

4799

202

303

7089

633

184

8371

411

244

4587

212

304

G455

634

404

223

637

185

808775

245

814364

305

785818

186

9169

39+

246

4130

234

306

6179

639

187

9554

385

247

3885

*45

307

4538

641

188
189

9931
810299

377
368

248
249

3031
3366

254
265

308
309

8896

3252

642
644

358

274

645

190

810657

250

813002

310

782007

191

1005

348

251

2807

285

311

1961

646

192

1344

339

252

2513

294

312

1814

647

193

1674

33°

253

2208

3°5

313

0667

647

194

1994

310

254

1894

3'4

314

0020

647

311

324

647

195

812305

255

811570

315

779373

196

2606

301

256

1237

333

316

8726

647

107

2897

291

257

0895

342

317

8079

647

198

3178

281

258

0543

352

318

7432

647

199

3449

271

259

0182

361

319

6786

646

261

37°

646

200

813710

260

809812

320

776140

201

3961

251

261

9433

379

321

540.5

645

202

4202

241

?62

9045

388

322

4Kol

644

203

4432

. 230

263

8648

397

323

42(18

643

204

4653

221

2C4

8242

406

324

3507

641

2IO

415

640

205

814864

265

807827

325

772927

206

5064

20O

266

7404

4*3

326

2289

638

207

5252

188

2(17

6072

432

327

1054

635

208

5430

178

2G8

6532

440

328

1022

632

209

5597

167

269

6084

448

329

0303

629

156

455

626

210

815753

270

805629

330

709707

211

5899

146

271

5166

463

331

014:?

624

212

6034

"35

272

4696

470

332

8622

621

213

6159

115

273

4217

479

333

7804

618

214

6273

114

274

3731

486

334

7200

614

103

495

61 1

215

816376

275

803236

335

760679

216

6468

92

276

2735

501

336

0(172

607

217

6549

81

277

2227

508

337

5470

602

218

6619

70

278

1712

5'5

338

4S72

598

219

6G78

59

279

1191

521

339

4279

593

49

528

588

220

816727

280

800663

840

763691

221

6763

36

281

0128

535

341

3108

5«3

222

6789

26

282

799588

540

342

2530

57«

223

6804

IS

283

9041

547

343

1057

573

224

6807

3

284

8488

553

344

1389

568

7

558

563

225

816800

285

797930

345

7G082G

226

6782

18

286

7366

564

340

0270

556

227

6752

3°

287

6796

570

347

759719

55'

228

6711

4'

288

6221

575

348

9174

545

229

6659

5*

289

5041

580

349

8G3G

S38

230

816596

63

290

795056

585

350

758104

S32

231

6523

73

291

4467

589

851

7579

5*5

232

6439

«4

292

8873

594

352

7001

518

233

6344

95

293

3274

599

853

6549

512

234

6238

1 06

294

2671

603

354

6045

504

117

608

497

235

816121

295

792063

355

755548

236

5993

128

296

1452

fill

356

5059

489

237

6854

'39

297

0838

614

857

4578

481

238
239

5704
5544

150
1 60

298
299

0221
789601

617
620

868

359

4104
3G38

474
466

171

623

457

240

815373

300

788978

• J

300

753181

TABLES OF NEPTUNE.

109

TABLE XXI.

PRINCIPAL TERM OF THE LOGARITHM OF THE RADIUS VECTOR (Continued).
Argument I.

i

1.4

I

1.4

I

1.4

0

0

o

0

753181

GO

744487

120

772170

i

2731

450

61

4685

198

121

2811

641

2

2290

441

62

4894

209

122

8454

643

3

1858

432

63

5114 '

220

123

4098

644

4

1434

424

64

6346

232

124

4744

646

4'5

242

648

5

751020

65

745588

125

775392

6

7

0015
0218

405
397

66
67

6841
6105

253
264

126

127

6041
6691

649
650

8
9

749830
9462

388
378
369

68
69

6380
6065

275
285
296

128
129

7343
7996

652
653
654

10

749083

70

746961

130

778650

11

8724

359

71

7207

306

131

9303

653

12

8375

349

72

7584

3'7

132

9956

653

13

8030

339

73

7910

326

133

780009

653

14

7700

33°

74

8246

336

134

1262

653

320

346

653

15

747386

75

743592

135

781915

16

7076

310

76

8948

356

136

2567

652

17

6777

299

77

9314

366

137

3219

652

18

6488

289

78

9089

375

138

3869

650

19

6210

278

79

750074

385

139

4518

649

268

395

648 .

20

745942

80

750469

140

785166

21

5685

257

81

0873

404

141

5812

646

22

5439

246

82

1286

4'3

142

6456

644

23

5204

Z35

83

1707

421

143

7098

642

24

4980

224

84

2137

430

144

7738

640

214

439

638

25

744766

85

752576

145

788376

26

4563

203

86

3024

448

146

9011

635

27

28

4372
4191

191
181

87
88

3480
3944

456
464

147
148

9642
790271

631

629

29

4021

170

89

4417

473

149

0897

626

159

480

622

30

743862

00

754897

150

791519

31

3715

'47

91

5385

488

151

2138

619

82

3580

'35

92

5881

496

152

2753

615

SB

8 156

124

93

6384

5°3

153

3304

fill

34

3344

112

94

6895

5"

154

3971

607

102

519

604

35

748213

95

757414

165

794575

36

3152

90

96

7940

526

166

5174

599

37

3073

79

97

8472

53*

157

5708

594

38

3007

66

98

9010

538

158

6357

589

39

2968

54

99

9553

543

159

6941

584

40

742910

43

100

760103

55°

160

797519

578

41

2879

3'

101

0660

557

161

8093

574

42

2800

'9

102

1223

563

162

8661

568

43

2852

8

1(13

1792

569

163

9223

562

44

2856

4

104

2367

575

164

799779

556

45

742872

if

105

762948

58i

165

800330

55'

46

2899

27

106

3533

585

166

0875

545

47

2937

38

107

4123

59°

167

1413

538

48

2987

5°

108

4718

595

168

1944

53'

49

3048

61

109

6317

599

169

2468

5*4

50

743120

7*

110

765922

605

170

802985

5'7

51

3204

84

111

6531

609

171

3496

5"

52

8300

96

112

7143

612

172

4000

504

53

8408

1 08

113

7759

616

173

4497

497

54

8529

121

114

8379

620

174

4986

489

132

624

4.82

55

743661

115

769003

175

805468

.f Ud>

56

3804

'43 "

116

9631

628

176

6942

474

57

3958

'54

117

770262

631

177

6408

466

58

4123

165

118

0895

633

178

6866

458

59

4300

'77

119

1531

636

179

7317

45'

CO

744487

187

120

772170

639

180

807759

442

110

TABLES OF NEPTUNE.

TABLE XXII.

COEFFICIENTS FOR PERTURBATIONS OF LATITUDE.

Argument 1.

Arg.

0

100

200

800

-B..1

-Bc.1

A,

Ba

B..I

B*

B*l

Ba

//

//

It

n

n

n

n

n

0

0.52

0.40

0.41

0.00

0.04

0.18

0.14

0.58

10

0.56

0.35

0.31

0.00

0.04

0.18

0.21

0.05

20

0.61

0.31

0.23

0.03

0.04

0.17

0.30

0.70

30

0.67

0.27

0.15

0.06

0.03

0.18

0.38

0.73

40

0.72

0.23

0.09

0.10

0.02

0.20

0.45

0.73

50

0.73

0.18

0.05

0.13

0.01

0.23

0.50

0.70

60
70

0.72
0.67

0.13
0.08

0.03

0.02

0.16

0.17

0.01
0.02

0.28
0.34

0.52
0.52

0.65

0.59

80

0.59

0.04

0.03

0.18

0.04

0.42

0.51

0.62

90

0.50

0.01

0.04

0.18

0.08

0.50

0.51

0.40

100

0.41

0.00

0.04

0.18

0.14

0.58

0.52

0.40

PERTURBATIONS OF LATITUDE.

TABLE XXIII.

TABLE XXIV.

Arg.

Arg. 5.

Arg. 8.

0

100

200

300

0

100

200

300

It

//

ff

H

„

„

II

„

0

— 0.30

+ 0.06

+ 0.30

— 0.06

+ 0.04

+ 0.56

— 0.04

— 0.56

10
20
30
40

— 0.29
— 0.27
— 0.24
— 0.21

+ 0.11
+ 0.16
+ 0.19
+ 0.23

+ 0.29
+ 0.27
+ 0.24
+ 0.21

— 0.11
— 0.16
— 0.19
— 0.23

+ 0.13
+ 0.21
+ 0.29
+ 0.36

+ 0.55
+ 0.52
+ 0.48
+ 0.43

— 0.13
— 0.21
— 0.29

— 0.36

— 0.55
— 0.52
— 0.48
— 0.43

50

— 0.17

+ 0.26

+ 0.17

— 0.26

+ 0.43

+ 0.37

— 0.43

— 0.37

GO
70
80
90

— 0.12
— 0.08
— 0.03
+ 0.02

+ 0.28
+ 0.30
+ 0.31
+ 0.81

+ 0.12
+ 0.08
+ 0.03
— 0.02

— 0.28
— 0.30
— 0.31
— 0.31

+ 0.48
+ 0.52
+ 0.55
+ 0.56

+ 0.30
+ 0.22
+ 0.14
+ 0.05

— 0.48
— 0.52
— 0.55
— 0.50

— 6.30
—0.23
— 0.14
—0.08

100

+ 0.06

+ 0.30

— 0.06

— 0.30

+ 0.56

— 0.04

— 0.56

+ 0.04

TABLE XXV.

VALUES OF SIN i FOR EVERT TEN YEARS.

Year.

1600

1700

1800

1900

0

8.498705

8.496503

8.494292

8.402066

10
20
30
40

8485
8265
8045
7825

220

6282
6061
5840
6619

221
221
221
221
221

4071
3849
3627

8404 III

1X12 "4
ir.i-.i "3
1896

im »4

50

8.497605

8.495398

8.493182

8.490947

60
70
80

'JO

7385 *10
7165
6944
C724 -

5177
4956
4735
4513

221

221
221
22£

221

2'.i;V.) "3
2736
2513
2289 "4
"i

0723 "4
0-108
0274

8.490(149 "••

100

8.496503

8.494292

8.492006

8.489824

PUBLISHED BY THE SMITHSONIAN INSTITUTION,
WASHINGTON, D. C.

3 A N D ARY, 1866.

•..
ail •'-.

-?;•;%'-»*::;•*-•

yMms^m^

U.C. BERKELEY LIBRARIES

, : 9R8S

J^.H

UNIVERSITY OF CALIFORNIA LIBRARY