OF
R. Tracy Crawford
A8TRUHUMI
THE SECULAR VARIATIONS
OF THE
ELEMENTS OF THE CEBITS
OF THE
FOUR INNER PLANETS
COMPUTED FOR THE EPOCH 1850.0, G. M. T
BY
ERIC DOOLITTLE
\
UNIVERSITY OF PENNSYLVANIA
Extracted from THE TRANSACTIONS OF THE AMERICAN PHILOSOPHICAL SOCIETY,
N.S., Vol. XXII, Part 2
PHILADELPHIA
1912
To MY FATHER,
PROFESSOR CHARLES L. DOOLITTLE,
THIS WORK
is INSCRIBED.
ASTROUOVY
[Extracted from the TRANSACTIONS OP THE AMERICAN PHILOSOPHICAL SOCIETY, N. S., Vol. XXII., Part 2.]
THE SECULAR VARIATIONS OF THE ELEMENTS OF THE ORBITS OF THE FOUR
INNER PLANETS COMPUTED FOR THE EPOCH 1850.Q G. M. T.
BY ERIC DOOLITTLE.
(Read March 1, 1912.)
TABLE OF CONTENTS.
THE THEORY.
1. Introduction 39
2. The method of GAUSS 40
3. HILL'S first modification of GAUSS'S method 42
4. HILL'S second modification. The work of CALLANDREAU and INNES 47
5. The method of HALPHEN and its modifications by ARNDT and INNES 49
THE COMPUTATION.
6. The elements of the orbits and the adopted masses 52
7. The formulas employed in the computation 53
8. The values of the preliminary constants 56
9. The radii vectores and the true anomalies 59
10. The separate results:
Mercury by Venus 61 Earth by Mercury 123
" Earth 65 " " Venus 127
" Mars r." . . 70 " " Mars 132
" Jupiter 77 " " Jupiter 138
" Saturn 82 " " Saturn 142
" Uranus 86 " " Uranus 146
" " Neptune 89 " " Neptune 149
Venus by Mercury 93 Mars by Mercury 152
" Earth .100 " " Venus : 156
" " Mars 104 " " Earth 160
" " Jupiter 108 . " " Jupiter 164
" Saturn 112 " " Saturn 168
" Uranus 116 " " Uranus 173
" Neptune 120 " " Neptune 176
11. The final values of the perturbations 179
12. Comparison with the results of observation
13. Comparison with SEELIGER'S hypothesis on the constitution of the Zodiacal Light 185
. 37
1. INTRODUCTION.
The usual method of determining the secular variations of the elements of any
planet is the well-known one based upon the development of the perturbing function
into an infinite series whose successive terms involve continually higher powers of
the eccentricities and the mutual inclination. This method possesses two advantages.
The first is that when an extreme degree of accuracy is not required, so that higher
terms of the development may be disregarded, it is the simplest method available;
and, in the second place, since the coefficients of all terms are general literal expres-
sions, the change produced in the value of any variation by a change in the assumed
values of one or more of the elements can readily be ascertained by a simple substi-
tution of the more accurate values. On the other hand, this method possesses the
disadvantage that the complexity of the expansion grows rapidly greater as the order
of the included terms is increased, so that a slight increase in the desired accuracy
greatly increases the labor of the computation.
The integral methods, founded upon the celebrated theorem of GAUSS (I) ,* are
wholly free from this latter disadvantage, for if it is desired to include all terms to
the twenty fourth order this can be done by a computation which is less than twice
as long as that required when the approximation is stopped at terms of the eleventh
order. But the integral method, though thus extremely accurate, leads only to the
numerical values of the variations dependent upon the values of the elements assumed ;
if they are desired for some other epoch at which the various elements possess different
values from those adopted, or if an improved value of any of the elements becomes
known, they can only be found by an entire repetition of the computation.
The only determinations of the secular perturbations of the four inner planets
which are in any sense modern ones are the classic investigation of LE VERRIER (T)
and the computation of NEWCOMB (15> . The latter furnishes the most accurate values
of these variations so far determined ; the series were extended to terms of the eighth
order, only those terms of this order being included, however, which seemed likely
to be most important, and in some cases terms of the tenth order were included,
though usually by induction merely.
In both of the above computations the usual expansion into an infinite series
was employed. As the GAUSSIAN method is so extremely accurate, and as its formulas
throughout are wholly different from those hitherto employed, it seemed that an
* These symbols wherever they occur refer to the list of titles at the end of the present paper.
39
40 THE SECULAR VARIATIONS OF THE ELEMENTS
application of it to a re-determination of these variations based upon the most ac-
curate values of the several elements now obtainable would be of value. The results
of this work will be found in the following pages; the final comparison with the
earlier results is given in Article 11, and the comparison with the results of observa-
tion in Articles 12 and 13. The epoch throughout is 1850.0, G. M. T.
In the four following articles an attempt is made to state briefly the essential
features of the various methods of computing secular variations which are founded
on GAUSS'S theorem, but for a detailed account of the long and often complex trans-
formations which are involved, the original papers must be consulted.
2. THE METHOD OF GAUSS.
The equations which express the complete variations of the elements of the
orbit of any body revolving about the sun when it is disturbed in its motion by the
presence of a third body, may, as is well known, be put in a variety of different
forms; the form selected as the basis for all developments founded on GAUSS'S method '
is that in which three rectangular components of the disturbing force enter into
the expressions for the differential coefficients. Thus, if R denote the component
lying in the direction of the radius vector of the disturbed body, positive outward
from the sun; S, the component lying in the plane of the orbit of the disturbed body
and perpendicular to the radius vector, positive in the direction of motion; and W,
the component perpendicular to this plane and positive northward, we will have
for the variation of the eccentricity of the orbit of the disturbed body,
de tfn cos<p
dt = fe'(l + m) ^ sm + ( - cos " + cos ^ J'
with similar expressions for the variations of the six remaining elements.*
In the original memoir of GAUSS the determination of the secular terms of
these expressions was given a geometrical aspect. Thus, since each variation may
obviously be expressed in terms of the two single variables M and M' , the secular
term in question will be that given by the equation,
[de-\ 1 r 2 ' ("'de ,..,..,
[dt 1=4^1 1 dt dMdM '
* The usual notation is adopted throughout. Thus a, e = sin ip, i, 12, ir, n, and L are respectively the half major
axis, the eccentricity, the inclination, the longitude of the ascending node, the longitude of perihelion, the mean motion
and the longitude at the epoch of the disturbed body; M, E, v and r are respectively its mean, eccentric and true anomalies
and its radius vector, m is its mass, k* is the mass of the sun, and m = >nk 2 . The same letters with accents refer to
the disturbing body.
Watson, Theoretical Astronomy, pp. 516-523; Oppolzer, Lehrbuch zur Bahnbestimmung, Vol. II, p. 213; Tisserand,
Mecanique Celeste, Vol. I, pp. 431-433, etc. The final forms of the equations expressing the other variations may be
inferred from those stated at the end of Article 7.
OF THE ORBITS OF THE FOUR INNER PLANETS. 41
and this is the same as,
[de~\ a 2 ncos<f f 2 " f 1 f 2ir _ 1 f 2 ' , ,, , 1 , ,,
= o 70/1 sm t>- 5- /MM' + (cos v + cosE)- ^- SdM' dM,
\_dt !joo 27r 2 (l + m) J L 2irJ 2irJ J
since the variable of the first integration enters the expression only through R and S.
In the equation as thus written R and S are supposed to contain the mass, ra ', as a
factor so that if Ri and Si are the corresponding values produced by a unit mass,
R = m 'Ri and S = m 'Si.
If we now imagine an infinitely thin elliptic ring which coincides with the orbit
of m', whose total mass is equal to the mass ra</, and the density of any portion of
which is proportional to the time occupied by m' in describing that portion of its
orbit, we will have for the three components of the attraction exerted by any portion
dm f ,
R.dmo', Sidm'o, and
and integrating about the entire ring, we find for the complete components,
flit /*2n- /2jr
Rdmo, I Sidmo', and I W4m<>'-
Jo Jo
But by the conditions,
dt dM'
OTo' " T 2;r '
and hence the components are,
J rtn 1 r>2 1 /*2ir
m^'RidM', 5- m Q 'S t dM' and ~- mo'T^id/lf,
<i7T J ^TT J /7T J
which are identical with
^ f '" fldM', ^ (^ SdM' and g. if" W^dM'.
Thus the expressions giving the secular variations are seen to be the same whether
these are derived from the moving planet or from the elliptic ring.*
The work of GAUSS contains no application to the determination of secular vari-
ations nor are all the formulas necessary for this purpose there developed; the first
integration alone is effected, and it is shown that by changing first to the variable
E' and afterward introducing a new variable, T, each of the complicated integrals
may be made to depend upon elliptic integrals whose values GAUSS obtained by the
introduction of a new algorithm called by him the Arithmetico-geometrical mean.
* Other interesting geometrical aspects of the problem are treated by Bour < 5) , Hill < s "' (38) , and Halphen < 28) , but
for brevity a detailed account of these is here omitted.
42 THE SECULAR VARIATIONS OF THE ELEMENTS
The first application of GAUSS'S method was made by NicoLAi (2) , who determined
by it the secular variations of the Earth's orbit, but the results only were published.*
The first development of the method is by CLAUSEN (3) who also applied it to a determi-
nation of the perturbations of Tuttle's Comet produced by the action of Jupiter (4) ,
dividing the disturbed orbit into 120 parts with reference to the true anomaly. It
was next, in 1867, applied by ADAMS ((!) to the orbit of the November meteors with a
special view to ascertaining the cause of the steady progression of the node of the
orbit, but in this investigation certain small terms were neglected by ADAMS and the
solution of a fundamental cubic equation which occurs in the original method was in
this manner avoided.
No further applications of GAUSS'S method seem to have been made until after
the publication of HILL'S extensive development (8) and modifications of it in 1882.
3. HILL'S FIRST MODIFICATION OF GAUSS'S METHOD.
Although the first of the above integrations may be rigorously effected, the
value of the second must be approximated to by a mechanical quadrature about the
orbit of m, a greater or less number of terms being employed in the quadrature
according as the disturbed orbit is more or less eccentric. Since either the true,
eccentric, or mean anomalies may be selected as the variables, it becomes of im-
portance to decide which of these must be chosen in order to render the quadrature
most accurate. It is readily proved! that the inequalities of distribution of a series
of points on an elliptic orbit corresponding to a series of equidistant values of the
eccentric anomaly are of the order of the square of the eccentricity while for the
other two anomalies they are of the order of the first power of this quantity, and
therefore HILL has employed the eccentric anomalies throughout his development,
although SEELiGER (9) showed that a still higher accuracy will be obtained if the true
anomalies are chosen.
If, therefore, we decide to make the integrations with reference to the eccentric
anomalies, we will obtain, since
dM = ^dE, AM' =-- r -,dE', and r' = a' (I - e' cos E'),
[de~\ ncos<p I f 2 " f 2ir .
j, = ,,,, r -r \ sin v Kar(I e cos E)
L^JOO & 2 (1 + TO) 47T 2 J J
+ (cos v + cos E) Sar(l - e' cos E')]dEdE'dt,
*See Article 11.
t See Tisserand's Mecanique Celeste, Vol. I, page 442,
OF THE ORBITS OF THE FOUR INNER PLANETS. 43
and writing,
1 r 2 " nr 1 C 2 " nr
R = ^ I ,R(l-e' cosE')dE', S = ^ ,S(l-e' cos E')dE',
JIT JQ TOO zir JQ mo
the expression for the secular variation will become,
[del m'n 1 T 2 "
'- cos <p JT- [sin v-Ro + (cos v + cos E)S u ]dE.
at Joo 1 -f- TO ZTT J
In order to find the values of R , S , and W , it is first necessary to express
R, S and W in terms of E' '. For this purpose that part of the disturbing force arising
from the action of the disturbing planet upon the sun need not be included, for it is
known that this has no secular term.| Considering therefore only the action of m'
upon m, it is evident from a figure that R, S and W will have the values,
_ m o' I r> cos # r 1
I /
_ TOO' r' sin & cos y
TT7 '
: -^r sm y>
and also that
A-' = r 2 - 2rr' cos + r' 2 ,
in which & is the angle included between the radii vectores, A is the distance between
the two bodies, and y is the inclination of the plane which includes r and r' to the
plane of the orbit of the disturbed body.
If n and n' denote the angular distances respectively of the perihelia of the two
orbits from the ascending node of the orbit of m' upon the orbit of m, and if / be their
mutual inclination, we will have,
cos = cos (v + n) cos (v r + II') + sin (v + U) sin (v' + n') cos /,
sin cos 7 = sin (v + n) cos (v' + II') + cos (v + II) sin (v' + II') cos I,
sin sin 7 = sin / sin (v' + n')
The values of n, n', and / are obtained from the original elements by a direct
solution of the spherical triangle whose sides are n and n', and in which the angle
included between these sides is I. (See Article 7.)
tSee Hill's "On Gauss's Method <8 >, , . ," page 321.
44 THE SECULAR VARIATIONS OF THE ELEMENTS
If we now eliminate v' from the above expressions by the equations,
r' cos v' = a' (cos E' - e'), r' sin v' = a' cos <p' sin E', r' = a' (1 e' cos E'),
the resulting equations giving R, S, W, and A will be expressed wholly in terms of
the variable E' . In order to simplify these results, we assume certain new auxiliaries
defined by the equations,
k cos (.K-n) = cos IT, k sin (K-U) =- cos / sin IT, k' cos (K'-Il) = cos 7 cos II',
k' sin (K' - II) = - sin II',
A = r z + 2ka'e'r cos (v + K) + a'\
B cos e = ka'r cos (v + K) + o'V,
B sin e = /r'a' cos <p' r sin (v + K'),
A c = ka' cos (v + K), A s = k'a' cos <?' sin (v + K')
B c = - ka' sin (v + K), B s = k'a' cos v ' cos (v + A'')
C c = a' sin n' sin /, C s = a' cos <p' cos II' sin /.
C = a'V 3 ,
when the desired expressions become,
-*, R = 4, (cos E' - e') + A s sin E' - r
W?o
^U = 5 c (cos ' - e') + 5 8 sin '
W?0
~ W = C c (cos ' - e') + C, sin #'
2
A 2 = A-2B cos (' - e) + C cos
In order to effect the integrations, GAUSS here introduced a new variable, T,
connected with E' by the relations,
N sin E' = a + a' sin T + a" cos T
N cos E' = /3 + 0' sin 7 1 + 0" cos 7 T
N = 7 + 7' sin T + j" cos 7 7 ,
the quantities , a', a", 0, 0' . . . being subject to the conditions that
(N sin #') 2 + (N cos #') 2 - JV 2 and sin 2 T + cos 2 T - 1
shall be identically zero, and also being so chosen that the coefficients of sin T, cos T,
and sin T cos T shall vanish in the expression 2V 2 A 2 which therefore must take the
form,
G - G' sin 2 T + G" cos 2 T.
From these conditions it is derived that the coefficients G, G' and G" in the trans-
OF THE ORBITS OF THE FOUR INNER PLANETS. 45
formed expression for JV 2 A 2 must severally satisfy the cubic equation,
x(x - A)(x + C) + B 2 x + B 2 C sin 2 e = 0,
and hence that they must be the roots of this equation. By substituting for x the
successive values, -- (7, 0, a' 2 cos 2 <p' and + A, the first member is seen to take in
succession the corresponding values,
- B 2 C cos 2 e
+ B 2 C sin 2 e
- a' 4 cos 2 <p' r 2 sin 2 1 sin 2 (v + II)
+ B*-(A + C sin 2 e).
Since, even when cos (v + K) has its maximum negative value, the value of A
exceeds that of (r a') 2 , it is evident that A is always positive, and therefore that
the above equation has one negative root which lies between C and 0, one positive
root lying between and a' 2 cos 2 <p', and that the third root lies between this value
and + A. The roots are represented by G", G', and G, respectively, and thus
G", G' and G are always positive quantities, the last being the largest and the first
the smallest except when <p' exceeds 45, a case not met with in any of the planetary
orbits.
Since a, j3, y, a', 0' . . . must retain the same values whatever the values of
E' and T, we may, by writing the equations arising from the three conditions above
stated and equating the coefficients of the like terms in the two members, obtain
a series of equations which are sufficient for the determination of these quantities
in terms of G, G', G" and the other known auxiliaries. Upon substituting the resulting
expressions for sin E' and cos E' in the equations defining R , S , and T^o, and noticing
that JV 2 A 2 may be written,
G - G' sin 2 T + G" cos 2 T = (G' + G"} 1 1 - ^-^'sin 2 T },
I tr + tr J
we obtain each of the components in the form,
m s sin 2 T + m c cos 2 T
or
1 I'""
e = -
If we now write,
G
/I I fin - 7,
and consider that from LANDEN'S well-known transformation,
46 THE SECULAR VARIATIONS OF THE ELEMENTS
c ~
and also notice that
r' 2 dr
Jo (l-c 2 sin*
* sin 2 TdT
cos* TdT
![/ TT\
= ?L v c>2 /
it is evident that each of the above three integrals becomes expressible wholly in
terms of the rapidly convergent series of LANDEN.
For the purposes of the present computation HILL (S) has computed to ten places
the logarithms of the quantities
K = sec 2 KL, L' = L ~ B , and N u = sec 2 (1 + '),
and these correct to eight places are tabulated at intervals of one tenth of a degree
for all values of from = to = 50.
From a direct substitution it is now seen that the final resulting values of Ro,
S and W are as follows, in which the symbols N, P, Q, etc., are written for abbrevi-
ation and have the meanings stated in Article 7 :
7? = - N - QG' + VJS,
S = PF* + VJ,
W = PF, + VJ 3
The integration with respect to E' having been thus entirely completed, that
in regard to E is effected by mechanical quadratures. Since each variation is a
function of E alone, it follows by the principles of quadratures that if any one of them
be expanded into a periodic series involving the sines and cosines of E and its multiples,
the secular term of the series, which is rigorously equal to |ir I f(E)dE, may be
I/O
also obtained by forming the values of f(E) for 2j equidistant values of E, from
E = to E = 360, and dividing the sum by 2j. The expression thus obtained,
will be subject only to the error involved in dropping those terms which contain a
multiple of E not lower than 2j. An inspection of the known forms of the series
which express the variations renders it evident that the error thus committed is of
OF THE ORBITS OF THE FOUR INNER PLANETS. 47
the order 2j in terms of the eccentricities and mutual inclinations of the orbits except
in the one case of the variation of the Mean Longitude, in which, as this variation
depends wholly upon the expansion of 2(r/a)JRo> it is of the order 2j + 1.
The resulting equations giving the values of all the secular variations are those
stated in Article 7.
4. HILL'S SECOND MODIFICATION OF GAUSS'S METHOD. THE WORK
OF CALLANDREAU AND INNES.
In HILL'S second modification of GAUSS'S method (8> , the well-known expressions
for the roots of a cubic equation when this is solved by the trigonometric method are
introduced, and thus, throughout the integrals, the quantities p, q and 0' occur instead
of the roots G, G' and G", the equations connecting these quantities being,
G = 2 9 sin60 - + P, G' = 2gsin
G" = 2g sin (60+ I') -p.
It was shown in GAUSS'S original memoir (1) that
dT
f
(m 2 cos 2 T + n 2 sin 2 T) * J ( m ' 2 COS 2
if m' = \(m + w) and w' = V mn, and that by repeating this transformation by the
employment of the equations,
m " = i( m > + n '), n" = Jrnfri,
m'" = \(m" + n"), n'" = JriW,
etc. etc.,
m ( *> and n (k) very rapidly approach a single limit, p., which GAUSS named the Arith-
metico-geometrical Mean. It thus follows that our first integral is equal to 7r/2^,
and that integrals of the form
p (sin 2 T - cos 2 T)dT
J (m 2 cos 2 T + n 2 sin 2 T)*
become equal to ir/2 w/ju in which w is a very rapidly converging series involving
m, n, m', n', etc., in its successive terms.
The integral expressions which actually enter into the equations for .R , S , and
W o are
(f\f \ -/o
60 - Q ) - r -r^
3 / 4 (m 2
n 2 )
48 THE SECULAR VARIATIONS OF THE ELEMENTS
V3 w sin 6'
in which
tf
A
o
and
the values of # , , and T^ being connected by comparatively simple relations with
these quantities and with known auxiliaries.
HILL accordingly suggested that tables of these functions should be computed,
and this was first done by MONS. 0. CALLANDREAU (I:!> who however adopted as an
argument the quantity a defined by the relation
1
1
cos
1 + a 6'
V COS Q
* <5
and tabulated the logarithms of the functions r 4 n 4 x(0') and <K0') *- x(0') at intervals
of 0.001 from = 0.000 to a = 0.400; of 0.002 from = 0.400 to a = 0.600 and of
0.005 from this point to the extreme value, = 1.000. This paper repeats the
derivation of all formulas necessary when the second method alone is employed,
essentially as this was given by HILL, and also contains a direct proof that R , S ,
and W can be expressed wholly in terms of the complete elliptic integrals, F and E.
Similar tables were also computed by MR. R. T. A. lNNES (22) , the functions
here tabulated being (1 -- )/(! + 4 ) \l/(6') and \l/(6') + *(0') to the argument 0'
at intervals of one degree, from 0' = 90 to 0' = + 90.
Whether the first or second methods be employed, the values of the integrals
involved may also, as was pointed out by HiLL (38) , be approximated to with great
rapidity by the use of JACOBI'S Nome, q (American Journal of Mathematics, Vol. 23,
page 321. In the Astronomical Journal, No. 511, a brief application is given to a case
in the action of Venus on the Earth). This function is defined by the equation,
q = e~'x,
in which K' is the complete elliptic integral of the first kind complementary to K,
from which there may be derived,
KE=(1+ 3V + 5V ) - U + <? 2 + <? 6 ' )
OF THE ORBITS OF THE FOUR INNER PLANETS. 49
The values of log [(//tan 2 0] computed to ten decimal places for each degree of 8
from 6 = to 6 = 45 are given by lNNES (39) . When exceeds 45, the values of K
and E are readily obtained from their expressions in terms of the complementary
complete integrals whose moduli are sin (ir/2 0), and to which the table is therefore
directly applicable.
Lastly, in the second method, HILL recommends that the quadratures be per-
formed upon the quantities a/r R , a/r S and a/r W directly, all constant and
evanescent factors which appear in the expressions for the variations being removed
from under the integral signs and reserved until the integration has been completed.
5. THE METHOD OF HALPHEN AND ITS MODIFICATIONS BY ARNDT
AND INNES.
It was first pointed out by BRUNS (29) that the periods of the elliptic functions
of the first and second integrals can be evaluated without a knowledge of the three
roots, but it was HALPHEN (28) who first applied this remarkably elegant method of
analysis to the present problem. It was shown by him that if o> and 17 are the two
periods in question, then R , So, and W may be obtained in the form aw + br\, in
which a and b are rational functions of the coefficients of the cubic equation and w
and TJ are expressible in terms of certain hyper-geometric series in which the common
variable is an absolute invariant of the elliptic functions.
The three integrals entering into the problem have the form,
/"
Jo
IdT
7 1 ) 3 '
in which / has the values 1, sin 2 T and cos 2 T, respectively, in the three cases; by
introducing the new variable, s, defined by the relation,
G + G" s-G'
G' + G" ' s - G '
these become,
G> + G " r-o" te G + G " r-o ds
~Jl ~^s (s ' G}> ~ 2 ~^~i v! (s '
and
n
respectively, in which
f/~< i r<u\(r< r\ir<t i r>\
n = ((j -\- (j ; ((JT Cr ) (u -f- Cr ;
and
S = - 4( - G)(s - G')(s + (?")
50 THE SECULAR VARIATIONS OF THE ELEMENTS
Introducing the WEIERSTRASSIAN r function through the relation
C ds
U = %
u being the elliptic integral of the first kind and 61, e 2 , and e 3 the roots of the cubic
equation increased by one third of the coefficient of x 2 , and considering that from the
theory of these functions,
s - G = r(u) - ei, T(w) = d, r(w + w') = e 2 , and T(w') = e 3 ,
the first integral will become,
2 ^~ f" " f r () - e 'l d = 2 \ e ^ + ~ ( + ') - - ' w 'l
J u L " f J
ff and o-' being the second WEIERSTRASSIAN functions, which are connected with
the periods, w and rj, by the equations,
(<> + ') = 77 + ?;'; -co' = )?'.
The three integrals consequently take the final forms,
_G' + G" n G + G", n G'-G,
2 ~ - (eico + 77) ; 2 - (^co + ?;), and 2 - ~- (630) + 77).
fv ra 71
A direct substitution of these expressions for the integrals in the equations which
define .R > So and Wo leads, after some reduction, to forms which are seen to contain
only these integrals themselves, the coefficients of the cubic equation with other
known auxiliaries, and the quantity n. But if, for brevity, we write the original
cubic equation in the form,
x 3 - P,x 2 + P*x - P, = 0,
and let
X = P, 1 - 3P 2 and p = PjP 2 - 9P 3 ,
then the invariants, g 2 and g 3 , and the absolute invariant, g, will have the values,
to = |X; 0i = A(2PiX - 3p), and g = 2 3 * 270 3 2 ,
and w will be given by,
n ~ = leC^ 3 - 27^ 3 2 ),
in which the last factor is the discriminant. Thus, except for <o and TJ, our final
expressions are obtained wholly in terms of the coefficients of the cubic equation,
and a knowledge of the roots becomes unnecessary.
In the paper by BRUNS, before referred to, it is shown that w and i\ are directly
OF THE ORBITS OF THE FOUR INNER PLANETS. 51
expressible in terms of a hyper-geometric series whose variable is the absolute invari-
ant, g. By a simple transformation the relations may be placed in the following
forms, which are more convenient in practical application.
A. , o-
' 12' l > g
DR. Louis ARNDT i30) has fully developed this method, deriving all the formu-
las necessary for its application and stating tables for F(w) and F(T)) for values of
(g - l)/0 from (g - l)/g = 0.000 to (g - !),/</ = 0.980, the interval being 0.001.
In a recent paper by INNES (SI) the complete formulas for this method are derived
when the quadrature is applied directly to the expressions (a/r)R 0) (a/r)S Q and
(a}r)Wo, as suggested in the second method of HILL. The development is nearly
identical with that of ARNDT except that the forms of the hyper-geometric series
are slightly changed, the variable,
, * Vg - 1
sin 2 2 = *?=-
being preferred. The values of the logarithms of
F () = F(!, I 2, sin 2 -} and F (77) = F (- g, | 2, sin 2 * Y
were published by MR. FRANK RoBBiNS (32) for all values of i, at intervals of one
degree from i = 1 to i = 90, the computation having been made to ten places and
published to seven, and these tables, computed with seven place logarithms, have
been extended from i = 90 to i = 180 by MR. C. J. MERFiELD (33) .
Although the preceding methods are of great mathematical elegance, it is doubt-
ful whether their formulas lead to so accurate results as those of HILL'S first method
when seven place logarithms are employed. (See the computations of Jupiter on
Mars (24) and of Saturn on Mars (25> , Article 10.) Moreover, when the method is
applied which is explained in the computation of Jupiter on Mercury (Article 10),
the roots of the cubic equation are so readily obtained that the avoidance of its
solution becomes a matter of no practical importance. Accordingly HILL'S first
modification of GAUSS'S method has been employed throughout all of the following
computation.
THE COMPUTATION.
6. THE ELEMENTS OF THE ORBITS AND THE ADOPTED MASSES.
The values adopted for the elements of the several orbits, to serve as the basis
for this computation, were taken in each case from HILL'S "New Theory of Jupiter
and Saturn." (16) Those of the four inner planets will be found on page 192; those
of Jupiter and Saturn on page 558; of Uranus on page 109, and of Neptune on page
161. The epoch throughout is 1850.0 G. M. T.
The values of the masses finally selected by HILL, and here adopted, will be
found on page 554 for Mercury, Venus and the Earth; on page 192 for Mars; on page
19 for Jupiter and Saturn, and on page 161 for Neptune. The mass of Uranus as
stated in the "New Theory " is 1 -r- 22640, but at DR. HILL'S suggestion this is here
diminished to 1 -4- 22800, (A. J., No. 316). The value assumed for the mass of
Mercury when the first of these computations were made was 1 -r- 5000000, but all
of 'the results are here changed to agree with the value 1 -r- 7500000 stated below. It
seems not improbable that even this latter fraction is too large, but the true value of
this element is still very uncertain.
X
i
S2 I e
n
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
o / //
75 7 13.62
129 27 42.83
100 21 39.73
333 17 51.74
11 54 31.67
90 6 41.37
168 15 6.70
43 17 30.30
o / //
7 7.71
3 23 35.01
0.00
1 51 2.24
1 18 42.10
2 29 40.19
46 20.54
1 47 1.68
46 33' 8^63
75 19 53.08
48 23 54.59
98 56 19.79
112 20 49.05
73 14 8.00
130 7 31.83
0.20560476
0.00684311
0.01677114
0.09326803
0.04825511
0.05606025
0.0469236
0.0084962
5381016^260
2106641.357
1295977.416
689050.784
109256.626
43996.21506
.15425.752
7864.935
logo
1-j-m
Mercury
9.5878217
7 500 000
Venus
9.8593378
408 134
Earth
0.0000000
327 000
Mars
0.1828971
3 093 500
Jupiter
0.7162374
1 047.879
Saturn
0.9794956
3 501.6
Uranus
1.2831044
22 800
Neptune
1.4781414
19 700
52
OF THE ORBITS OF THE FOUR INNER PLANETS. 53
7. THE FORMULAS EMPLOYED IN THE COMPUTATION.
The following formulas are written in the order in which they were applied.
When the right hand member appears in two different forms, one of these was used
in the first computation and the other in the duplication, though sometimes other
obvious modifications were made use of in the several cases differing from those which
are here written.
The values of /, n, and n' were obtained from the general equations:
sin / sin (H w) = sin i' sin (ft' ft) ,
sin I cos (II w) = sin i cos i' + cos i sin i' cos (ft' ft)
= cos i cos i' [ tan i + tan i' cos (ft' ft)],
sin / sin (n' w') = sin i sin (ft' ft),
sin/ cos (II' a/) = cos i sin i' sin i cos i' cos (ft' ft),
= cos i cos i' [tan it tan i cos (ft' ft)].
When the Earth is the disturbing body, these become,
/ = i; n = 180 + co; n' = 180 + *' - ft;
and when the Earth is the disturbed body,
/ = i'- n = TT - ft'; II' = *-' - ft'.
As i, i' and / are always small, eight place logarithms were generally here used to
insure the accuracy of n and n'.
The auxiliaries k, k', K, K' and C were then found from the relations:
k sin (A' - n) = - cos I sin n'; k cos (K - n) = cos n'; k' sin (K' - n) = - sin n';
A;' cos (K'-U) = cos / cos n'; C = o'V 2 ,
and their values were tested by the equations,
tan .7 = p ; tan \(K - K' + 90) cot |(A' + K' - 90 - 211) = "!" ( ^, ~ "\ ,
v sin (^11 -p G)
sin ( K- K') = sin I tan / sin (K' - n) sin (K - n) cot n'.
The orbit of the disturbed planet being then divided into 2j parts in regard to
the eccentric anomaly, the following equations were applied to each point of division,
of which those marked with an asterisk are test equations upon the sums of the
functions corresponding respectively to the odd and even points of division of the
orbit. The sums corresponding to the odd points are designated by Si, those to the
even points by S 2 , and
2 = S, + S 2 .
54 THE SECULAR VARIATIONS OP THE ELEMENTS
r sin v = a cos ip sin E,
r cos v = a (cos E e),
r 2 = a 2 (1 - 2 e cos # + e 2 cos 2 .E),
(the last equation giving the value of r 2 for use in A, N, and J 3 . Since
i log r 2 = log T-,
this affords also an independent test of r).
*S,t; + 180 = S 2 *>; *Z^ = Z 2 r = ja.
A = r 2 + 2&aYr cos (w + X) + a' 2 = [r + ka'e' cos (t> + X)] 2 + a' 2 [l - fcV 2 cos 2 (v +
(the second form used with ZECH'S tables in the duplication).
*SiA = ^A = ja 2 + |jaV + j[a' 2 - 2kaa'ee' cos X]
J3 sin e = A;'o' cos ^>'r sin (v + K'}
B cos e = ka'r cos (t> + K) + a' 2 e'
*2iB sin e = S 2 B sin e = jk'aa' cos ^>' e sin ^'
*2iB cos = 2 2 B cos e = j[a'~e' kaa'e cos A']
g = B 2 C sin 2 e
To effect the solution of the cubic equation, h and I were found from the equations,
the very convenient test equation, hi = B* -AC, being applied to each pair of values.
The first approximation to G was then obtained from
G = h ~ h(h -I)'
and further approximations by successive applications of
G(G - iy
(The number of trials required never exceeded three.) G' and G" then follow from
the equations,
G' = (A - C - G) + (A -C- GY + ; G" =
and we have for verification,
ft i rt/ _ rtii _ A ri . fir _ L _i_ _ a _ . fin _
^'-'' = "
.
G")'
OF THE ORBITS OF THE FOUR INNER PLANETS. 55
(In some cases the first approximations to G were found by,
sin e' = - 3 ; G = 2q sin (60 - &') + p,
the solution being then finished as before).
The modulus, (c = sin 6), of the elliptic integrals employed in the computation
was separately found by the two equations,
C* T I /""' /"" I firr
sin 2 6 = Q , Q,, ; tan 2 6 = Q _ Q> ,
and with 6 as an argument the values of log K , log L ', and log N were taken from
the tables of HILL'S memoir (8) , the interpolation being effected in both directions to
second differences by the well-known formulas,
in which n + n' = 1.
The logarithms of Af, P, Q, and F were then obtained from,
o .
"(G + G") 3 ' ~(G + (?") 2 ' y "(? + "' W
the first three being verified by similar operations performed upon the values of 2 t
and 2 2 formed from the respective logarithms, and the last by the use of ZECH'S
tables and also by the equation,
V = ar-(G + G'T l [GN + G"(N - L ')]K .
The following auxiliaries were next obtained :
Ji' = a' 2 cos 2 <p'[l - sin 2 / sin 2 (v + n)] + G"
= [a' cos >p' + a' cos <p' sin 7 sin (v + n)][a' cos <p' a' cos <?' sin / sin (v + II)] 4- G",
J 2 = ka'e'r sin (v + K) - |a' 2 cos 2 <p' sin 2 1 sin 2(i> + n)
= ka'e'r sin (t; + K) - a' 2 cos 2 ^' sin (v + n) cos (v + n) sin 2 /,
the second form being employed with ZECH'S tables in the duplication
Jz = cos 2 <p' sin I cos 7 r sin (t; + H) -- ' sin / sin II' r 2 ,
tZ CL
a'e'
*Si/ 3 = SaJa = - ja- cos 2 ^' sin I cos / e sin n --- sin / sin n' S^ 2 .
(Z
= a' 2 sin <p' cos <p' cos / fi sin e,
56 THE SECULAR VARIATIONS OF THE ELEMENTS
*2iF 2 = S 2 F 2 = jk'aa' 3 ee' cos 2 <p' cos 7 sin K',
F 3 = -- sin 49' cos y>' sin I r cos (v + II) .B sin e.
There were next obtained,
B = - N -QG' + 7JY; S = PF* + VJ,; W = PF, + VJ a ;
fl<"> = -R sin E; 5<-> = -S ; W = .Sfl<">; (c) = .
r r z? zj
and the very accurate test equation,
sin <p !-Ai (>) + cos 9? B (c) = 0,
was applied.
These values were then substituted in the following series of equations, and
the final values of the differential coefficients obtained:
[dc ~\ TH'TI 1
~dt Joo = 1 + m ' COS *" ' 2?- 2 f sin " ' Ro + ^ cos " + cos
m'n cos ? 1
di 1
sJ o =
mn
[dftl m'n
,.
a< Joo 1 +
sec <p 1
^-s-.S
TO sin i 2j
TO r n If O r- P 1. , . 9 v[d x ~\ , .,iTd1
= ^r- - s-.ZI 2-J2p | + 2sin 1 7 +2sin 2 ^, -,. .
oo 1 + TO 2j a 2 L rf/ Joo 2 L rf/ Joo
<lt
When the Earth is the disturbed body, the third and fourth equations are re-
placed by,
" dp ~| m'n 1
- sec ip-- n -.2 sin ( + ) (r
L + TO 2j
m'n I _
[da ~\
ZJ l ,-
In this case
rdxl =
L dt Joo
and the last term of the expression for [dL/dt] 00 disappears, but the first two equations
remain unaltered.
8. THE VALUES OF THE PRELIMINARY CONSTANTS.
The values obtained for those constants which are direct functions of the ele-
ments of the orbits in the several cases are shown in the following tables. The last
OF THE ORBITS OF THE FOUR INNER PLANETS.
57
columns of these tables contain the differences between the values of K K' formed
directly and the same angles obtained from the test formula of the preceding article.
The other test equations were also exactly satisfied.
An examination of the formulas of the preceding article renders it evident that
with any two planets / will have the same value whether the inner or the outer
planet is the disturbing one, while the value of II in the first case will differ 180 from
that of n' in the second, and that of n' in the first case will similarly differ 180
from that of II in the second. These conditions will be seen to be here satisfied
very exactly, the minute discrepancies which occur being due to the fact that in some
places eight place logarithms were employed, in others seven, and in still others the
attainment of a higher accuracy throughout the entire computation was sought by
the use of the dash, ( ), which was placed above the last figure of each logarithm for
which the interpolation led to a value coinciding more nearly with the mean of the
two adjacent figures than with either one of them. In combining such logarithms
the effect of the dash was taken into consideration by methods which are obvious.
Mercury by
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Mercury by
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Venus by
Mercury
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
II
II'
K
4
7
5
6
6
6
7
20
9
17
23
19
1
42.982
7.710
10.165
15.310
44.130
17.399
42.654
230
208
209
218
229
211
223
39
34
13
5
26
43
12
31.39
4.99
54.31
54.72
43.69
10.39
39.15
284
233
107
154
244
304
191
54
48
24
49
17
49
16
1.27
31.10
19.31
24.01
50.53
47.06
25.42
305
334
101
63
345
266
324
43
57
45
8
17
43
34
2.40
50.59
36.16
32.52
17.36
32.95
3.91
K'
log k
logfc'
log<7
resid.
305
47
57.49
9.9988328
9.9999176
5.3891826
0.007
334
33
18.85
9.9978879
9.9988719
6.4491252
0.002
101
53
33.05
9.9983990
9.9998432
8.3052599
0.004
63
24
30.76
9.9995281
9.9978563
8.7995614
0.002
345
30.90
9.9978013
9.9994926
9.4563012
0.013
267
3
12.48
9.9982188
9.9991396
9.9089914
0.000
324
24
7.31
9.9968502
9.9998757
8.8147322
0.001
II
II'
K
4
20
42.980
104
54
1.27
50
39
31.37
54
19
21.08
3
23
35.010
234
7
49.75
205
1
46.65
29
8
21.75
1
56
2.460
208
26
43.81
52
18
22.07
156
9
18.63
2
15
11.352
247
36
52.56
130
2
45.43
117
32
48.56
2
3
12.046
281
7
33.71
241
43
52.16
39
24
36.81
2
37
16.883
233
30
46.37
272
18
13.25
321
12
24.44
2
46
38.369
265
47
34.23
179
34
46.34
86
12
46.11
58
Venus by
Mercury
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Earth by
THE SECULAR VARIATIONS OF
K' log k
54 9 38.85 9.9992531
29 3 44.28 9.9998637
156 7 24.89 9.9998450
117 35 25.70 9.9998033
39 22 46.29 9.9997836
321 12 41.79 9.9995460
86 12 49.67 9.9999999
/ II
THE ELEMENTS
log k' log C
9.9994984 7.8017097
9.9993746 6.4491252
9.9999075 8.3052599
9.9998610 8.7995614
9.9999375 9.4563012
9.9999992 9.9089914
9.9994896 8.8147322
II'
resid.
0.000
0.003
0.000
0.001
0.003
0.001
0.001
K
Mercury
Venus
o
7
3
/
23
7.710
35.010
53
25
48
1
31.10
46.65
O
28
54
34
7
n
4.99
49.75
25
330
i
25
56
13.33
48.79
Mars
1
51
2.240
51
57
45.14
284
53
57.15
127
3
21.25
Jupiter
Saturn
1
2
18
29
42.100
40.190
1
348
25
19.94
50.68
272
337
58
45
11.88
52.32
88
10
27
13
5.264
49.89
Uranus
46
20.540
27
7
31.73
95
58.70
292
6
31.40
Neptune
1
47
1.680
330
14
7.90
273
9
58.47
57
4
3.92
Earth by
K'
log A;
logfc'
logC
resid.
Mercury
Venus
25
330
3
51
36.28
5.12
9.9992608
9.9994999
9.9974965
9.9997387
7.8017097
5.3891826
0.002
0.008
Mars
127
4
14.72
9.9997885
9.9999850
8.3052599
0.009
Jupiter
Saturn
88
10
27
16
10.859
6.88
9.9998865
9.9999411
9.9999998
9.9996473
8.7995614
9.4563012
0.010
0.003
Uranus
292
6
34.66
9.9999609
9.9999997
9.9089914
0.005
Neptune
57
4
14.94
9.9997902
9.9999994
8.8147322
0.001
Mars by
n
n'
K
Mercury
5
9
10.165
287
24
19.31
29
13
54.31
258
16
20.56
Venus
1
56
2.460
232
18
22.07
28
26
43.81
203
52
27.49
Earth
1
51
2.240
104
53
57.15
231
57
45.14
232
57
4.23
Jupiter
1
26
6.381
149
47
4.35
188
22
45.31
321
24
28.37
Saturn
2
21
52.110
176
17
59.42
293
4
38.76
243
12
17.28
Uranus
1
11
40.460
120
39
30.31
315
36
26.40
165
2
41.49
Neptune
2
22
41.388
152
49
56.12
222
47
52.02
290
3
32.71
Mars by
K'
log k
log k'
logt'
resid.
Mercury
Venus
258
203
4
50
28.68
49.03
9.9995819
9.9999439
9.9986621
9.9998087
7.8017097
5.3891826
0.017
0.012
Earth
232
55
19.79
9.9998596
9.9999141
6.4491252
0.009
Jupiter
Saturn
321
243
24
14
9.72
23.99
9.9999971
9.9996870
9.9998667
9.9999432
8.7995614
9.4563012
0.007
0.007
Uranus
165
3
26.31
9.9999538
9.9999519
9.9089914
0.003
Neptune
290
35.50
9.9998274
9.9997986
8.8147322
0.006
OF THE ORBITS OF THE FOUR INNER PLANETS.
59
9. THE RADII VECTORES AND THE TRUE ANOMALIES.
The values of log r and v for the points of division employed in the four different
cases are given in the following tables. In each case the equations,
2 ir = 2 2 r = ja, and 2 : y + 180 = S 2 t>
were exactly satisfied, and the values of r were also obtained from the equation stated
in Article 7 for obtaining the value of r 2 .
E
MEKCUBY.
log r
V
E
VENUS.
logr
V
O
i
//
O
O
1
//
9.4878584
0.00
9.8563557
0.00
15
9.4916716
18
25
28.96
15
9.8564576
15
6
6.54
22.5
9.4963313
27
32
14.93
30
9.8567564
30
11
47.87
30
9.5026623
36
32
7.50
45
9.8572313
45
16
40.52
45
9.5195925
54
4
7.02
60
9.8578493
60
20
24.50
60
9.5407098
70
50
41.41
75
9.8585680
75
22
44.64
67.5
9.5522314
78
55
7.36
90
9.8593378
90
23
31.50
75
9.5640735
86
46
40.73
105
9.8601064
105
22
42.20
90
9.5878217
101
51
53.65
120
9.8608213
120
20
20.31
105
9.6103385
116
9
54.15
135
9.8614342
135
16
35.65
112.5
9.6207149
123
3
1.59
150
9.8619040
150
11
43.65
120
9.6303194
129
46
44.60
165
9.8621990
165
6
4.12
135
9.6467730
142
49
52.77
180
9.8622996
180
0.00
150
9.6589887
155
27
29.02
195
9.8621990
194
53
55.88
157.5
9.6633518
161
39
20.97
210
9.8619040
209
48
16.35
165
9.6664956
167
48
0.75
225
9.8614342
224
43
24.35
180
9.6690267
180
0.00
240
9.8608213
239
39
39.69
195
9.6664956
192
11
59.25
255
9.8601064
254
37
17.80
202.5
9.6633518
198
20
39.03
270
9.8593378
269
36
28.50
210
9.6589887
204
32
30.98
285
9.8585680
284
37
15.36
225
9.6467730
217
10
7.23
300
9.8578493
299
39
35.50
240
9.6303194
230
13
15.40
315
9.8572313
314
43
19.48
247.5
9.6207149
236
56
58.41
330
9.8567564
329
48
12.13
255
9.6103385
243
50
5.85
345
9.8564576
344
53
53.46
270
9.5878217
258
8
6.35
285
9.5640735
273
13
19.27
292.5
9.5522314
281
4
52.64
300
9.5407098
289
9
18.59
315
9.5195925
305
55
52.98
330
9.5026623
323
27
52.50
337.5
9.4963313
332
27
45.08
345
9.4916716
341
34
31.04
60
THE SECULAR VARIATIONS OF THE ELEMENTS
E
THE EARTH.
logr
V
E
MARS.
log r
V
O
O
1
//
O
O
i
II
9.9926546
0.00
.1403760
0.00
22.
5
9.9932181
22
52
14.25
30
.1463201
32
47
24.62
30
9.9936460
30
29
2.39
45
.1532670
48
54
53.41
45
9.9948189
45
41
0.84
60
.1621567
64
44
46.64
60
9.9963428
60
50
8.59
90
.1828971
95
21
5.913
67.
5
9.9972036
68
23
26.41
120
.2026920
124
31
47.15
90
0.0000000
90
57
39.46
135
.2106341
138
39
52.35
112.
5
0.0027784
113
23
5.92
150
.2166313
152
34
23.40
120
0.0036266
120
49
43.50
180
.2216237
180
0.00
135
0.0051200
135
40
31.82
210
.2166313
207
25
36.60
150
0.0062624
150
28
37.29
225
.2106341
221
20
7.65
157.5
0.0066776
157
51
53.72
240
.2026920
235
28
12.85
180
0.0072232
180
0.00
270
.1828971
264
38
54.087
202.
5
0.0066776
202
8
6.29
300
.1621567
295
15
13.36
210
0.0062624
209
31
22.71
315
0.1532670
311
5
6.59
225
0.0051200
224
19
28.18
330
.1463201
327
12
35.38
240
0.0036266
239
10
16.50
247.
5
0.0027784
246
36
54.08
270
0.0000000
269
2
20.54
292.
5
9.9972036
291
36
33.59
300
9.9963428
299
9
51.41
315
9.9948189
314
18
59.16
330
9.9936460
329
30
57.61
337.
5
9.9932181
337
7
45.75
10. THE SEPARATE RESULTS.
The values found for the intermediate auxiliary functions which depend upon ",
as well as the final perturbations of the four inner planets in each case are now stated
in the following tables. The results of the application of the more important test
equations are also shown, but all of the test equations of Article 7 were also applied,
and each computation (except the first), was, after its completion, duplicated from the
beginning, the forms of the equations being changed in the duplication when this
was possible.
OF THE ORBITS OF THE FOUR INNER PLANETS.
61
MERCURY.
ACTION OF VKNUS ON MERCURY.
E
A
B cos t
B sin e
1000000 Xff
h
0.619543952
+ 0.13308441
- 0.18036925
0.7970904
0.52358614
30
0.627434998
+ 0.22218381
- 0.06982371
0.1194506
0.52390836
60
0.647116316
+ 0.24372756
+ 0.07193966
0.1268000
0.52384406
90
0.675632886
+ 0.19194286
+ 0.20693555
0.0491867
0.52344851
120
0.706503003
+ 0.08070542
+ 0.29899200
2.1902889
0.52319742
150
0.730295757
- 0.06017874
+ 0.32344233
1.5631633
0.52358280
180
0.738317327
- 0.19295989
+ 0.27373528
1.8358797
0.52446104
210
0.727259050
- 0.28205939
+ 0.16318979
0.6524819
0.52500778
240
0.701243272
- 0.30360314
+ 0.02142638
0.0112481
0.52470755
270
0.669559472
- 0.25181838
- 0.11356958
0.3160138
0.52391075
300
0.641856586
- 0.14058090
- 0.20562585
1.0359483
0.52329644
330
0.624398293
+ 0.00030325
- 0.23007624
1.2969588
0.52323374
z,
4.054580456*
- 0.17962654f
+ 0.28009822J
5.9972554
3.14309264
2 2
4.054580456
- 0.17962659
+ 0.28009814
5.9972551
3.14309193
E
G
G'
G"
o
0.09593332
0.52358258
0.09595274
0.000015866
O
25
/
20
n
53.90
30
0.10350215
0.52390782
0.10350489
0.000002203
26
23
25.33
60
0.12324776
0.52384346
0.12325032
0.000001964
29
59.15
90
0.15215988
0.52344311
0.15217844
0.000013171
32
37
46.70
120
0.18328109
0.52318510
0.18331625
0.000022837
36
17
45.71
150
0.20668846
0.52356735
0.20672760
0.000023681
38
55
52.70
180
0.21383179
0.52444977
0.21385942
0.000016369
39
41
12.31
210
0.20222678
0.52500393
0.20223677
0.000006145
38
21
51.31
240
0.17651123
0.52470749
0.17651140
0.000000121
35
27
1.91
270
0.14562423
0.52390915
0.14562996
0.000004142
31
49
7.06
300
0.11853565
0.52329155
0.11855723
0.000016698
28
25
30.42
330
0.10114005
0.52322787
0.10117042
0.000024501
26
5
20.70
Si
0.91134083
3.14305994
0.91144736
0.000073855
194
13
23.40
S 2
0.91134154
3.14305922
0.91144808
0.000073843
194
13
23.80
* 6a 2 + 3aV + 6[o' 2 - 2kaa'ee' cos K] = + 4.054580460.
t 6[a'V - kaa'e cos K] = - 0.17962650.
t - Qk'aa' cos <p'-e sin K' = + 0.28009816.
62
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION
OF VENUS ON MERCUBY.
E
log*.
log Lo'
log N log N log P
logQ
0.06678154
0.36107029
0.27485672 9.0518226 9.9748963
9.6076810
30
0.07267844
0.36875602
0.28344481 9.0869397 0.0171823
9.6511278
60
0.08883727
0.38974368
0.30686976 9.1792738 0.1306112
9.7669404
90
.0.11429390
0.42259487
0.34345542 9.2994384 0.2842725
9.9240134
120
0.14429575
0.46098687
0.38608356 9.4147448 0.4383831
0.0821541
150
0.16872258
0.49199359
0.42040790 9.4960335 0.5500428
0.1974492
180
0.17620114
0.50144271
0.43084933 9.5225000 0.5845077
0.2336318
210
0.16325170
0.48506821
0.41274966 9.4887994 0.5335323
0.1813814
240
0.13698082
0.45165804
0.37573831 9.4055654 0.4173889
0.0613864
270
0.10823963
0.41480523
0.33478930 9.2928156 0.2691020
9.9083455
300
0.08503270
0.38481172
0.30136861 9.1761377 0.1234343
9.7587487
330
0.07094409
0.36649704
0.28092115 9.0860236 0.0150983
9.6482336
Si
0.69812922
2.54971331
2.07576629 5.7500443 1.6692215
9.5105424
2 2
0.69813034
2.54971496
2.07576824 5.7500501 1.6692301
9.5105508
E
logV
J/
1000 X Ji J 3
1000 X Fj
9.6076650
0.521404654
- 2.7049984 - 0.024195167
+ 0.6439191
30
9.6511256
0.520191194
- 0.6256166 - 0.032354328
+ 0.2492710
60
9.7669384
0.521003862
+ 1.8268277 - 0.030122813
- 0.2568249
90
9.9240003
0.522559008
+ 2.6401451 - 0.018096342
- 0.7387609
120
0.0821316
0.523207843
+ 2.0172777 + 0.000503667
- 1.0674025
150
0.1974260
0.522626872
+ 1.0213969 + 0.020692271
- 1.1546906
180
0.2336159
0.521405157
+ 0.3980040 + 0.037057786
- 0.9772364
210
0.1813754
0.520383901
+ 0.3750436 + 0.045213990
- 0.5825883
240
0.0613863
0.520288911
+ 0.7059590 + 0.042976535
- 0.0764923
270
9.9083414
0.521364846
+ 0.6938449 + 0.030947096
+ 0.4054437
300
9.7587319
0.522844254
- 0.4295092 + 0.012350057
+ 0.7340853
330
9.6482088
0.523030858
- 2.2812182 - 0.007832614
+ 0.8213734
Si
9.5104690
3.130154681*
+ 1.8135608 + 0.038570065
- 0.9999517
S2
9.5104775
3.130156679
+ 1.8235957 + 0.038570073
- 0.9999517
* 2i(J,' - G") = + 3.130080826.
2t(Ji - G") = + 3.130082836.
OF THE ORBITS OF THE FOUR INNER PLANETS.
63
E
30
60
90
120
150
180
210
240
270
300
330
2!
2 2
ACTION OF VENUS ON MERCUKY.
10000 X F 3 #o 1000 X -So W, R w 1000 X S<">
- 0.24640136 0.05971623 - 0.4883004 - 0.009827036 0.00000000 - 1.5879204
- 0.00762098 0.06444673 - 0.0208471 - 0.014490447 + 0.10127648 - 0.0655216
- 0.09149188 0.08146579 + 0.7212240 - 0.017625381 + 0.20314033 + 2.0766362
- 0.49802851 0.11164771 + 0.7947010 - 0.015287000 + 0.28842180 + 2.0529668
- 0.89441990 0.15077545 - 0.4917115 + 0.000363093 + 0.30587345 - 1.1518366
- 0.92808606 0.18433784 - 2.4881611 + 0.032271915 + 0.20211372 - 5.4561938
- 0.56751805 0.19359808 - 3.0725539 + 0.063241297 0.00000000 - 6.5837414
- 0.13320998 0.17488253 - 1.4207354 + 0.068605698 - 0.19174661 - 3.1154764
+ 0.01209827 0.14154301 + 0.6131517 + 0.049504580 - 0.28714394 + 1.4363109
- 0.19301611 0.10799082 + 1.3152406 + 0.025022994 - 0.27897487 + 3.3976883
- 0.46971306 0.08194590 + 0.7289596 + 0.007023584 - 0.20433749 + 2.0989094
- 0.49748432 0.06575439 - 0.1643583 - 0.003535808 - 0.10333144 - 0.5165700
- 2.25744598 0.70904446 - 1.9892305 + 0.092680137 + 0.01753235 - 3.7116419
- 2.25744596 0.70906002 - 1.9841603 + 0.092587352 + 0.01775908 - 3.7031067
Ro COS V
E
RO sin v
+ (cos v + cos E)So
+ ( -sec 2 <f + l ) sin v
\a )
W cos u
Wo sin u
-2 -Bo
a
- 0.00097660
- 0.05971623
- 0.008630594
- 0.004699311
- 0.09487655
30
+ 0.03833160
- 0.05180530
- 0.006100201
- 0.013143840
- 0.10594280
60
+ 0.07755258
- 0.02541165
+ 0.002882594
- 0.017388064
- 0.14618182
90
+ 0.10909895
+ 0.02454507
+ 0.009914505
- 0.011635936
- 0.22329541
120
+ 0.11643405
+ 0.09565744
- 0.000337472
+ 0.000133975
- 0.33255092
150
+ 0.08098459
+ 0.16537954
- 0.032192272
- 0.002265848
- 0.43432170
180
+ 0.00614511
+ 0.19359808
- 0.055541660
- 0.030242129
- 0.46680548
210
- 0.07011629
+ 0.16039915
- 0.041182679
- 0.054870101
- 0.41204381
240
- 0.10947718
+ 0.08954949
- 0.009624818
- 0.048559933
- 0.31218788
270
- 0.10595403
+ 0.01957233
+ 0.007191939
- 0.023967193
- 0.21598164
300
- 0.07680517
- 0.02822239
+ 0.005196776
- 0.004724857
- 0.14704334
330
- 0.03941900
- 0.05265111
- 0.003501678
+ 0.000490095
- 0.10809245
2!
+ 0.01287279
+ 0.26545474
- 0.066055174
- 0.105480319
- 1.49964599
2 2
+ 0.01292582
+ 0.26543968
- 0.065870386
- 0.105392823
- 1.49967781
sin (/> |4i (<) + cos <p-BiJ
c) = - 0.0000000083.
64
THE SECULAR VARIATIONS OF THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
[de/dt] w
[dx/dt] w
[di/dt]oQ
= + 11321.398 TO'
= +1133127.6 TO'
60449.278 m'
- 792605.00 TO'
= +1127216.0 TO'
= -1326653.0 TO'
log eoeff.
p 4.0539001
p 6.0542788
n 4.7813911
n 5.8990568
p 6.0520072
n 6. 1227573
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] m = +0.027739414
[dxAft]oo = +2.7763615
[di/dtlw = -0.14811133
[dfi/dfloo = -1.9420214
[dr/dt] m = +2.7618772
= -3.2505323
[dL/di]
w
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Hill. Method of Gauss.
[de/dt] m
e[dw/dt]oo
[di/dt] w
GITI i r/vo '/"//i
bill I [Ctlfcy CttJOO
[dx/dt] w
+0.02780
+0.56811
-0.14812
-0.23648
-3.2769
+0.02774
+0.57086
-0.14806
-0.23665
+0.0277391
+0.567852
-0.1481112
-0.2367447
+2.776347
-3.250522
+0.0277394
+0.567855
-0.1481113
-0.2367449
+2.776361
-3.250532
NOTES.
This is the only one of the twenty eight computations that was not duplicated,
but the values of 6 were computed by two different formulas and all known test
equations were applied. As an illustration of his first modification of GAUSS'S
method HILL published this complete computation from exactly the same elements
as here employed, and DR. Louis ARNDT states that he has verified the results and
found them correct. (Bulletin de la Societe des Sciences Naturelles de Neuchatel,
Vol. XXIV). INNES states however that the test arising from the constancy of the
major axis is not satisfied, the residual being 0.00075 (M. N., Vol. LII, page 87),
but this statement is an error, for the residual obtained from HILL'S figures is
0.0000000088, a practically exact agreement with that here obtained.
OF THE ORBITS OF THE FOUR INNER PLANETS.
65
Upon comparing the present computation with that of HILL, the following
slight discrepancies may be noticed :
n', K and K' differ by less than 0".l from HILL'S values, a difference doubtless
due to the fact that the preliminary computation was here effected with eight place
logarithms while HILL employed but seven. The value of I for 330 should be
0.10114009 instead of 0.11014009, and G" for 180 should be 0.00001637 instead of
0.00001617. These are misprints merely. The values of the logarithms of K ,
LQ, and N Q in HILL seem to be slightly in error throughout, a double interpolation
to second differences from HILL'S values of 6 giving with the three functions most
in error,
Hill's Values.
For E = 60, log N 0.30686978 0.3068691
For E = 150, log L ' 0.49199342 0.4919942
For E = 180, log L ' 0.50144261 0.5014421
The effect of these differences upon the final coefficients is, however, almost in-
appreciable.
It is evident from an inspection of the final sums that a division into twelve parts
is necessary in this case, the terms from the sixth to the eleventh orders, inclusive,
amounting to l/600th of the whole for [di/dt] 00 and to l/1200th of the whole for
ACTION OF THE EARTH ON MERCURY.
Bcos
+ 0.29403604
+ 0.32704393
+ 0.25768623
+ 0.10454706
- 0.09134000
- 0.27748723
- 0.40401673
- 0.43702459
- 0.36766690
- 0.21452774
- 0.01864062
+ 0.16750659
- 0.32994198f
- 0.32994198
* 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos A] = + 6.90363352.
t 6[a'V - kaa'e cos A] = - 0.3299419S.
t - Qk'aa' cos <p'-e sin A' = + 0.20460788.
E
A
1.10386215
30
1.11164085
60
1.12870093
90
1.15278960
120
1.17861164
150
1.19808875
180
1.20368356
210
1.19273750
240
1.16934301
270
1.14208711
300
1.11943230
330
1.10628960
s,
6.90363359*
2 2
6.90363341
B sin
KXXXTxff
h
- 0.13175730
0.04882864
1.00008277
+ 0.06103685
0.01047875
1.00111148
+ 0.24661358
0.17106421
1.00173668
+ 0.37524793
0.39606073
1.00128597
+ 0.41247210
0.47853578
1.00022588
+ 0.34831240
0.34124242
0.99978030
+ 0.19995995
0.11246339
1.00066133
+ 0.00716576
0.00014443
1.00216106
- 0.17841099
0.08952996
1.00285589
- 0.30704518
0.26517354
1.00212493
- 0.34426952
0.33336685
1.00067710
- 0.28010968
0.22068934
0.99977580
+ 0.20460782t
1.23378883
6.00623965
+ 0.20460808
1.23378921
6.00623954
66
THE SECULAR VARIATIONS OF THE ELEMENTS
E
ACTION OF THE EARTH ON MERCURY.
G G' G"
0.10349811
1.00007732
0.10355071 0.00004715
O
18
46' 28.61
30
0.11024810
1.00111031
0.11025877 0.00000949
19
22 58.72
60
0.12668299
1.00171716
0.12683714 0.00013464
20
51 17.40
90
0.15122236
1.00123943
0.15152995 0.00026105
22
54 42.04
120
0.17810449
1.00016768
0.17843084 0.00026815
25
4.02
150
0.19802718
0.99973772
0.19824193 0.00017219
26
27 9.31
180
0.20274096
1.00064732
0.20281046 0.00005541
26
45 34.58
210
0.19029517
1.00216104
0.19029526 0.00000008
25
50 0.71
240
0.16620585
1.00284522
0.16627021 0.00005369
24
1 53.32
270
0.13968091
1.00209425
0.13990074 0.00018915
21
57 13.77
300
0.11847394
1.00063933
0.11879216 0.00028045
20
10 34.78
330
0.10623253
0.99975109
0.10646458 0.00020734
19
3 48.08
2,
0.89570634
6.00609403
0.89669152 0.00083949
135
35 52.71
2 2
0.89570625
6.00609384
0.89669123 0.00083930
135
35 52.63
E
logtfo
0.03586144
log Lo'
0.32053269
log Wo log N log P
0.22947519 8.5993188 8.9197434
logQ
8.8287400
30
0.03828768
0.32372814
0.23305754 8.6307049 8.9534609
8.8632763
60
0.04451494
0.33191845
0.24223558 8.7125510 9.0428624
8.9539831
90
0.05408514
0.34447345
0.25629369 8.8165734 9.1597444
9.0722159
120
0.06487989
0.35858872
0.27208270 8.9130565 9.2712667
9.1849500
150
0.07303960
0.36922627
0.28397010 8.9788974 9.3482020
9.2629066
180
0.07483693
0.37156571
0.28658308 8.0002544 9.3712099
9.2865324
210
0.06949181
0.36460451
0.27880654 8.9738846 9.3366140
9.2517535
240
0.05973313
0.35186484
0.26456369 8.9063079 9.2556584
9.1696145
270
0.04950056
0.33846383
0.24956627 8.8114799 9.1479627
9.0600557
300
0.04158092
0.32806159
0.23791429 8.7102233 9.0374863
8.9477383
330
0.03700237
0.32203558
0.23116016 8.6301758 8.9522476
8.8613541
2i
0.32140725
2.06253200
1.53285453 2.8417119 4.8982271
4.3715583
2 2
0.32140716
2.06253178
1.53285430 2.8417160 4.8982316
4.3715621
OF THE ORBITS OF THE FOUR INNER PLANETS.
67
ACTION OF THE EARTH ON MERCURY.
E
30
60
90
120
150
180
210
240
270
300
330
Si
io g y
8.8287147
8.8632712
8.9539113
9.0720772
9.1848079
9.2628156
9.2865032
9.2517534
9.1695861
9.0599551
8.9475885
8.8612431
4.3711117
4.3711156
1000 X F,
J,' J 2 J,
0.9963685 - 0.0084115617 - 0.04554861
0.9875038 - 0.0046146363 - 0.08975347
0.9853935 + 0.0065532595 - 0.10655243
0.9913722 + 0.0136249408 - 0.09143424
0.9979641 + 0.0119853300 - 0.04844492
0.9998176 + 0.0047535770 + 0.01089164
0.9963768 - 0.0029434883 + 0.07066640
0.9902153 - 0.0070671940 + 0.11485779
0.9854773 - 0.0058653024 + 0.13162975
0.9862780 - 0.0010764519 + 0.11649805
0.9932756 + 0.0016290397 + 0.07352222
0.9996406 - 0.0026304325 + 0.01421265
5.9548558* + 0.0029472768 + 0.08527241
5.9548275 + 0.0029898031 + 0.08527242
Ko 1000 X So 1000 X TF <">
1000 XPj
+ 2.1929308
- 1.0158793
- 4.1045657
- 6.2455186
- 6.8650672
- 5.7972107
- 3.3280756
- 0.1192649
+ 2.9694212
+ 5.1103718
+ 5.7299237
+ 4.6620645
- 3.4054328
- 3.4054372
1000 X <S ( ">
- 0.18791333
0.020434768
- 0.3847186
- 3.085993
0.000000000
- 1.2510793
30
+ 0.04317409
0.021303927
- 0.4280932
6.547345
+ 0.033478638
- 1.3454762
60
- 0.07397385
0.025621038
+ 0.1363110
9.590571
+ 0.063887706
+ 0.3925293
90
- 0.49750230
0.033590134
+ 0.7062450
- 10.865960
+ 0.086774000
+ 1.8244563
120
- 0.86425140
0.043556201
+ 0.5521780
7.575463
+ 0.088361000
+ 1.2934800
150
- 0.83674077
0.051546760
- 0.4218487
+ 1.808294
+ 0.056517506
- 0.9250562
180
- 0.43280690
0.053431000
- 1.3516860
+ 13.566615
0.000000000
- 2.8963373
210
- 0.01035886
0.048659649
- 1.2877184
+ 20.505307
- 0.053351926
- 2.8237889
240
- 0.07819780
0.040457621
- 0.3317444
+ 19.465045
- 0.082075038
- 0.7771130
270
+ 0.18040025
0.032377496
+ 0.5949004
+ 13.349022
- 0.083641596
+ 1.5368184
300
- 0.46718624
0.026190705
+ 0.7690320
+ 6.465459
- 0.065308303
+ 2.2142904
330
- 0.46610559
0.022213151
+ 0.2265587
+ 0.990810
- 0.034907444
+ 0.7120630
2i
- 1.94793392
0.209691333
- 0.6106280
+ 19.245092
+ 0.004865365
- 1.0242299
2 2
- 1.94793368
0.209691117
- 0.6099562
+ 19.240128
+ 0.004869178
- 1.0209836
* 2,(J,' - G") =
5.9540163.
2, (j ' G") =
5.9539882.
68
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF THE EARTH ON MERCURY.
COS V Ro
E
BUI v m>
+ (cos v + cos E)S<>
. (r ,.\j 100 X TFo cos 100 X Wo sin u
+ I - sec* if + 1 1 sin t'So
- 0.000769437
- 0.020434774 - 0.2710273 - 0.1475729
30
+ 0.011967951
- 0.017591057 - 0.2756307 - 0.5938896
60
+ 0.024315363
- 0.008157573 - 0.1568518 - 0.9461439
90
+ 0.032727249
+ 0.008319102 + 0.7047203 - 0.8270793
120
+ 0.032844271
+ 0.028781472 + 0.7040911 - 0.2795215
150
+ 0.022159505
+ 0.046499174 - 0.1803831 - 0.0126962
180
+ 0.002703372
+ 0.053431012 - 1.1914874 - 0.6487582
210
- 0.017924683
+ 0.045456375 - 1.2308941 - 1.6399928
240
- 0.030714227
+ 0.026434445 - 0.3784440 - 1.9093573
270
- 0.031808132
+ 0.005466897 + 0.3836684 - 1.2785779
300
- 0.024103783
- 0.010000896 + 0.4783818 - 0.4349396
330
- 0.012845686
- 0.018098650 + 0.0981246 - 0.0137335
Si
+ 0.004275559
+ 0.070053686 - 0.5016340 - 4.3662934
2 2
+ 0.004276204
+ 0.070051841 - 0.5003946 - 4.3659693
i
in ,.M."> + COB *.'
> = - 0.00000000016.
DIFFERENTIAL COEFFICIENTS.
log coeff.
//
[de/dt}^ = +3752.8345 TO' p 3.5743594
[dx/dt] M = +299037.72 m' p 5.4757260
[dildt] M = -4591.3713 m' n 3.6619424
[dB/<ft] M = -328217.95 TO' n 5.5161623
[d7r/d*]oo = +296589.74 TO' p 5.4721561
[dL/dt] m = -390282.17 TO' n 5.5913787
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'
//
[d/dt] w = +0.011476557
[dx/dt] m = +0.91448833
[dildt]<n = -0.014040890
[dQ/dt]oo = -1.0037245
[d7r/di]oo = +0.90700208
[dLldtln = -1.1935233
0.03246657
0.03502118
0.04597425
0.06718020
0.09606764
0.12145034
0.12883338
0.11464781
0.08923343
0.06475510
0.04699652
0.03651581
0.43957179
0.43957044
OF THE ORBITS OF THE FOUR INNER PLANETS.
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dtlw +0.01153 +0.01147 +0.0114766
e[dr/dt] M +0.18668 +0.18799 +0.186484
[di/dtlao -0.01414 -0.01404 -0.0140409
sin i [dfl/di]oo -0.12219 -0.12233 -0.122360
-1.1942 -1.19352
NOTES.
As a' and e' are both small in this case, the sums, up to and including .R , are in
very exact agreement. But as / and e are unusually large, the final sums differ
considerably, the greatest discrepancy being in W cos u, which shows that a neglect
of the terms from the 6th to the llth orders would produce an error in [di/dt] QO of
slightly more than 1 /1000th of the whole value of this coefficient.
70
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON MERCURY.
E
A
B cos
B sin e
<7
h
2.3984504
+ 0.12138918
+ 0.45632916
0.0042055
2.3024514
15
2.3737047
- 0.02025060
+ 0.40611421
0.0033309
2.3025214
30
2.3556032
- 0.14407564
+ 0.32017434
0.0020703
2.3025321
45
2.3456111
- 0.24164776
+ 0.20436670
0.0008435
2.3024889
60
2.3447258
- 0.30631712
+ 0.06658267
0.0000896
2.3024184
75
2.3533241
- 0.33367707
- 0.08378742
0.0001418
2.3023625
90
2.3710514
- 0.32186246
- 0.23649650
0.0011296
2.3023658
105
2.3967846
- 0.27167903
- 0.38113729
0.0029337
2.3024640
120
2.4286852
- 0.18654648
- 0.50785318
0.0052088
2.3026710
135
2.4643476
- 0.07226635
- 0.60800831
0.0074658
2.3029751
150
2.5010251
+ 0.06337314
- 0.67477750
0.0091956
2.3033343
165
2.5359018
+ 0.21112855
- 0.70361065
0.0099983
2.3036852
180
2.5663693
+ 0.36093056
- 0.69254258
0.0096862
2.3039570
195
2.5902662
+ 0.50257031
- 0.64232765
0.0083325
2.3040870
210
2.6060491
+ 0.62639549
- 0.55638795
0.0062519
2.3040409
225
2.6128739
+ 0.72396747
- 0.44058010
0.0039202
2.3038234
240
2.6105920
+ 0.78863682
- 0.30279611
0.0018517
2.3034773
255
2.5996753
+ 0.81599664
- 0.15242602
0.0004692
2.3030742
270
2.5810993
+ 0.80418226
+ 0.00028298
0.0000000
2.3026229
285
2.5562149
+ 0.76399902
+ 0.14492387
0.0004242
2.3023962
300
2.5266327
+ 0.66886612
+ 0.27163969
0.0014902
2.3022243
315
2.4941374
+ 0.55458616
+ 0.37179487
0.0027917
2.3021798
330
2.4606272
+ 0.41894655
+ 0.43856414
0.0038844
2.3022350
345
2.4280691
+ 0.27119119
+ 0.46739720
0.0044120
2.3023425
2,
29.7509107*
+ 2.89391842f
- 1.41728084J
0.0450638
27.6343997
2 2
29.7509107
+ 2.89391853
- 1.41728059
0.0450638
27.6344002
* 12a 2 + 6aV + 12[a' 2 -
t 12[a'V - kaa'e COB K\
J 12fc'aa' cos <p' e sin
- Zkaa'ee' cos A') = 29.7509106.
= + 2.89391844.
A"' = - 1.41728062.
OF THE ORBITS OF THE FOUR INNER PLANETS.
71
E
ACTION OF MARS ON MERCURY.
G G'
0.0758032
2.3016305
0.0957141
0.0190900
12
51
3.65
15
0.0509875
2.3018785
0.0717874
0.0201570
11
28
40.15
30
0.0328754
2.3021359
0.0509293
0.0176577
9
54
3.88
45
0.0229265
2.3023282
0.0338958
0.0108086
7
59
27.93
60
0.0221116
2.3024014
0.0237650
0.0016363
6
1
37.49
75
0.0307659
2.3023354
0.0326775
0.0018845
7
2
5.28
90
0.0484898
2.3021481
0.0572743
0.0085668
9
43
5.24
105
0.0741248
2.3018919
0.0890147
0.0143178
12
11
37.36
120
0.1058185
2.3016404
0.1249596
0.0181104
14
22
46.33
135
0.1411768
2.3014735
0.1626257
0.0199473
16
17
11.47
150
0.1774951
2.3014531
0.1994129
0.0200367
17
54
21.70
165
0.2120208
2.3016063
0.2327627
0.0186630
19
13
8.83
180
0.2422166
2.3019140
0.2604177
0.0161582
20
12
26.11
195
0.2659835
2.3023097
0.2806562
0.0128954
20
51
34.72
210
0.2818125
2.3026974
0.2924401
0.0092842
21
10
38.08
225
0.2888548
2.3029783
0.2954612
0.0057613
21
10
27.64
240
0.2869190
2.3030785
0.2900893
0.0027715
20
52
41.30
255
0.2764054
2.3029737
0.2772408
0.0007349
20
19
35.05
270
0.2582114
2.3026922
0.2582114
0.0000000
19
33
51.59
285
0.2336230
2.3023071
0.2344977
0.0007857
18
38
24.76
300
0.2042127
2.3019157
0.2076391
0.0031178
17
36
2.08
315
0.1717618
2.3016103
0.1791035
0.0067722
16
29
5.95
330
0.1381965
2.3014548
0.1502129
0.0112362
15
19
13.32
345
0.1055309
2.3014695
0.1221038
0.0156999
14
6
53.42
2,
1.8741623
27.6251620
2.0110657
0.1276658
185
31
50.77
2 2
1.8741617
27.6251624
2.0118270
0.1284276
185
48
12.56
72
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON
MERCURY.
E
logtfo
log LJ
logtfo
log JV
logP
logQ
0.01657774
0.29504379
0.20087027
8.0316820
7.5954802
7.8669295
15
0.01319292
0.29055307
0.19582511
8.0355546
7.5943701
7.8655109
30-
0.00979383
0.28603832
0.19075139
8.0547661
7.6099058
7.8800682
45
0.00636436
0.28147805
0.18562486
8.0870691
7.6401445
7.9084927
60
0.00361341
0.27781623
0.18150720
8.1291203
7.6819573
7.9481379
75
0.00492729
0.27956556
0.18347444
8.1771100
7.7316276
7.9980604
90
0.00943284
0.28555854
0.19021211
8.2272783
7.7853440
8.0537440
105
0.01490785
0.29282894
0.19838217
8.2762397
7.8395128
8.1098440
120
0.02081742
0.30066160
0.20717937
8.3211159
7.8908949
8.1628540
135
0.02681707
0.30859802
0.21608807
8.3595540
7.9366442
8.2098882
150
0.03254501
0.31616047
0.22457224
8.3896939
7.9743208
8.2484993
165
0.03762579
0.32285663
0.23208058
8.4101312
8.0019112
8.2766735
180
0.04171228
0.32823434
0.23810787
8.4198969
8.0178772
8.2928778
195
0.04453613
0.33194630
0.24226678
8.4184648
8.0212321
8.2961421
210
0.04594748
0.33380024
0.24434354
8.4057702
8.0116018
8.2861294
225
0.04593448
0.33378318
0.24432441
8.3822399
7.9892733
8.2631894
240
0.04461766
0.33205341
0.24238678
8.3488317
7.9552231
8.228375
255
0.04222050
0.32890264
0.23885673
8.3070780
7.9111256
8.1835072
270
0.03902753
0.32470208
0.23414923
8.2591387
7.8593690
8.1310520
285
0.03533705
0.31984157
0.22870029
8.2078386
7.8030574
8.0742275
300
0.03141955
0.31467566
0.22290683
8.15CG450
7.7459663
8.0168746
315
0.02748746
0.30948385
0.21708212
8.1095325
7.6924008
7.9633068
330
0.02367787
0.30444739
0.21142959
8.0706478
7.6468600
7.9179598
345
0.02004728
0.29964170
0.20603415
8.0437755
7.6135016
7.8848519
Si
0.31918262
3.69919207
2.58841642
8.8145868
3.7748004
7.0335140
s s
0.31939818
3.69947951
2.58873971
8.8145879
3.7748012
7.0338946
OF THE ORBITS OF THE FOUR INNER PLANETS.
73
ACTION OF MARS ON MERCURY.
E
logf
/'
Ji
7.8624692
2.3161036
+ 0.034714816
15
7.8607965
2.3114562
+ 0.028725995
30
7.8759318
2.3036617
+ 0.023022893
45
7.9059548
2.2939379
+ 0.017028648
60
7.9477527
2.2850812
+ 0.009527241
75
7.9976170
2.2883310
- 0.000387926
90
8.0517340
2.2994643
- 0.012768514
105
8.1064943
2.3097716
- 0.026802369
120
8.1586286
2.3171693
- 0.041120811
135
8.2052458
2.3210340
- 0.054165965
150
8.2438461
2.3213532
- 0.064480743
165
8.2723467
2.3185110
- 0.070890916
180
8.2891363
2.3131717
- 0.072601961
195
8.2931581
2.3062073
- 0.069243148
210
8.2839812
2.2986444
- 0.060886015
225
8.2618557
2.2916136
- 0.048054879
240
8.2277453
2.2862701
- 0.031734816
255
8.1833367
2.2836646
- 0.013368882
270
8.1310520
2.2845306
+ 0.005188260
285
8.0740447
2.2890080
+ 0.021821056
300
8.0161484
2.2963713
+ 0.034481515
315
7.9617277
2.3049376
+ 0.041716329
330
7.9153379
2.3123762
+ 0.043244020
345
7.8811861
2.3164879
+ 0.040207969
S,
7.0037635
27.6341976*
- 0.133414115
2 2
7.0037641
27.6349607
- 0.133414088
*Si(Ji' -G") =27,
5065318.
2 2 (-/i' - G") = 27
,5065331.
- 0.08283714
- 0.12496689
- 0.15748255
- 0.17817560
- 0.18564570
- 0.17939386
- 0.15985316
- 0.12835823
- 0.08705254
- 0.03874380
+ 0.01328579
+ 0.06550049
+ 0.11434922
+ 0.15650567
+ 0.18909426
+ 0.20988694
+ 0.21745674
+ 0.21127772
+ 0.19176389
+ 0.16024221
+ 0.11886354
+ 0.07045517
+ 0.01832594
- 0.03396170
+ 0.19026829
+ 0.19026812
- 0.09798270
- 0.08720058
- 0.06874762
- 0.04388148
- 0.01429659
+ 0.01799079
+ 0.05078038
+ 0.08183755
+ 0.10904591
+ 0.13055114
+ 0.14488783
+ 0.15107882
+ 0.14870229
+ 0.13792019
+ 0.11946725
+ 0.09460109
+ 0.06501619
+ 0.03272882
- 0.00006076
- 0.03111796
- 0.05832630
- 0.07983156
- 0.09416821
- 0.10035921
+ 0.30431767
+ 0.30431761
74
THE SECULAE VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON MERCURY.
E
ft
#0
1000 X So
100 X Wv
fl<>
+ 0.006125234
0.005413043
- 0.1331172
- 0.05793911
0.000000000
15
+ 0.004244733
0.005395946
- 0.1341971
- 0.08902862
+ 0.004501863
30
+ 0.002091460
0.005581735
- 0.1069853
- 0.11749705
4- 0.008771561
45
+ 0.000394535
0.005978460
- 0.0544826
- 0.14331159
+ 0.012778556
60
- 0.000202379
0.006587462
+ 0.0157366
- 0.16470025
4- 0.016426268
75
+ 0.000673582
0.007397547
+ 0.0931210
- 0.17804909
4- 0.019496561
90
+ 0.003010065
0.008379102
+ 0.1659322
- 0.17823953
4- 0.021645905
105
4- 0.006397644
0.009479745
+ 0.2230373
- 0.15960687
+ 0.022459598
120
+ 0.010124788
0.010622853
+ 0.2557091
- 0.11755697
4- 0.021550254
135
+ 0.013355105
0.011710967
+ 0.2593958
- 0.05060872
4- 0.018676901
150
+ 0.015340435
0.012635664
+ 0.2351775
+ 0.03775311
4- 0.013854131
165
+ 0.015613320
0.013293529
+ 0.1902485
+ 0.13831063
4- 0.007415508
180
+ 0.014107795
0.013605711
4- 0.1366988
+ 0.23722085
0.000000000
195
+ 0.011176655
0.013535474
+ 0.0883172
+ 0.31912559
- 0.007550472
210
+ 0.007500348
0.013096767
+ 0.0561741
+ 0.37133333
- 0.014359702
225
+ 0.003911780
0.012350565
+ 0.0447317
4- 0.38738325
- 0.019696946
240
+ 0.001183526
0.011390146
+ 0.0503246
+ 0.36844987
- 0.023106835
255
- 0.000166297
0.010320547
+ 0.0628133
+ 0.32211263
- 0.024451640
270
+ 0.000001635
0.009239635
+ 0.0697179
+ 0.25931088
- 0.023868989
285
+ 0.001425688
0.008225494
+ 0.0610457
+ 0.19093564
- 0.021678655
300
+ 0.003528129
0.007331827
+ 0.0329172
4- 0.12533212
- 0.018282391
315
+ 0.005583303
0.006590510
- 0.0111915
+ 0.06726176
- 0.014086776
330
+ 0.006923128
0.006018069
- 0.0617561
4- 0.01815027
- 0.009457250
345
+ 0.007124120
0.005623237
- 0.1063107
- 0.02290733
- 0.004691492
Si
+ 0.069734164
0.109902014
+ 0.7165294
4- 0.78161752
- 0.006827048
s 2
4- 0.069734168
0.109902021
+ 0.7165286
+ 0.78161728
- 0.006826894
i"> + cos <f B <c) = + 0.000000000104.
OF THE DEBITS OF THE FOUR INNER PLANETS.
75
ACTION OF MARS ON MERCURY.
1000x-.RoCosB+
E
1000 X S<"
+ B ' " - 10 X W " c s u 10 X W 8in
-2-#o
C '
( - secV + 1 1 sin vSf,
a
-0.43288822
-0.2662344
- 5.4130432
-0.5088501 - 0.2770661
-0.008600192
15
-0.43258505
+ 1.4484906
- 5.1972533
-0.6072560 - 0.6510373
-0.008648634
30
-0.33624961
+3.1443027
- 4.6032128
-0.4946402 - 1.0657799
-0.009175713
45
-0.16468928
+4.7703837
- 3.5917373
-0.1836694 - 1.4212974
-0.010218569
60
+0.04531053
+6.2357682
- 2.1327368
+0.2693639 - 1.6248266
-0.011820509
75
+0.25408211
+7.4151862
- 0.2308954
+0.7622000 - 1.6090993
-0.014007787
90
+0.42865608
+8.1659686
+ 2.0547246
+ 1.1559864 - 1.3566975
-0.016758205
105
+0.54706506
+ 8.3522568
+ 4.6004959
+ 1.3031443 - 0.9215481
-0.019968410
120
+ 0.59900000
+7.8723845
+ 7.2196098
+ 1.0926174 - 0.4337649
-0.023429816
135
+0.58504581
+6.6852166
+ 9.6761487
+0.5003962 - 0.0756834
-0.026827124
150
+0.51571179
+4.8307371
+ 11.7119607
-0.3765995 - 0.0265069
-0.029771109
165
+0.41003953
+2.4394871
+ 13.0838349
-1.3270492 - 0.3897732
-0.031867221
180
+0.29291257
-0.2733976
+ 13.6057094
-2.0833919 - 1.1343956
-0.032806220
195
+ 0.19034846
-3.0319641
+ 13.1877783
-2.4169344 - 2.0838761
-0.032447216
210
+0.12318210
-5.5396099
+ 11.8615372
-2.2290420 - 2.9698842
-0.030857525
225
+0.10088867
-7.5290371
+ 9.7823314
-1.5918740 - 3.5316447
-0.028292299
240
+ 0.11788553
-8.8108894
+ 7.2045350
-0.7163503 - 3.6141908
-0.025122156
255
+0.15406833
-9.3069299
+ 4.4325573
+0.1350559 - 3.2182941
-0.021739497
270
+0.18010328
-9.0565656
+ 1.7602448
+0.7452938 - 2.4836977
-0.018479268
285
+0.16656410
-8.1932578
- 0.5835177
+ 1.0058653 - 1.6229225
-0.015575563
300
+0.09477911
-6.8986293
- 2.4659985
+0.9273370 - 0.8431253
-0.013156194
315
-0.03382955
-5.3509513
- 3.8502686
+0.6070918 - 0.2895753
-0.011264707
330
-0.19409650
-3.6857779
- 4.7671354
+0.1797507 - 0.0251579
-0.009892996
345
-0.34269706
-1.9808199
- 5.2732726
-0.2254930 - 0.0403424
-0.009012936
2!
+ 1.43430766
-4.2819430
+36.0361948
-2.0385248 -15.8550934
-0.229869903
2 2
+ 1.43430113
-4.2819391
+36.0362016
- 2.0385225 - 15.8550938
-0.229869963
DIFFERENTIAL COEFFICIENTS.
w
log coeff.
1 fif> //r/1
I U-C' / Ltt-lQQ
1879.077
TO' n 3.2739445
[dxMJoo
= +76914.75
TO' p 4.8860096
[dildt] m
934.0667
TO' n 2.9703779
[dn/dt] m
= -59594.26
TO' n 4.7752044
[dTr/dt} m
= +76470.27
TO' p 4.8834926
[dL/dt] w
= -101879.0
TO' n 5.0080846
76 THE SECULAR VARIATIONS OF THE ELEMENTS
FINAL VALUES CORRESPONDING TO THE ABOVE VALUES OF m'.
= -0.00060742746
[dx/d4 = +0.024863343
[dt'/dfloo = -0.00030194497
[dQ/dfloo = -0.019264347
[drfdt] m = +0.024719659
= -0.032933242
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dtlw
-0.00060
-0.00061
-0.000607
<
>,[dir/dt]oo
+0.00508
+0.00511
+0.005082
[dt'/dflw
-0.00030
-0.00030
-0.000302
sin i
[dQ/d*]oo
-0.00234
-0.00235
-0.002348
[dL/d/]oo
-0.0331
-0.032933
NOTES.
On account of the very large values of the eccentricities of both orbits and their
high mutual inclination, the approximate test is here wholly inapplicable if but
twelve points of division are employed. Thus the two sums differ by 1 38' 46". 90
for 6 and by 40' 42". 47 for , while the sums of the functions immediately dependent
upon these quantities differ by proportionate amounts. When the number of points
of division is increased to twenty-four, the final sums are in almost exact agreement,
showing that the combined effect of all terms from the llth to the 23rd orders is
wholly inappreciable. The greatest difference which arises in the variations from
the employment of twenty-four points of division, instead of twelve, occurs in the
case of [di/dt] 00 and here produces a decrease of but three units in the seventh decimal
of the logarithm of the coefficient,
OF THE ORBITS OF THE FOUK INNER PLANETS.
77
E
30
60
90
120
150
180
210
240
270
300
330
30
60
90
120
150
180
210
240
270
300
330
ACTION OF JUPITER ON MERCURY.
A
B cos e
Bam f
p
27.23340536
+2.0282403
+ 1.4219711
9.05679111
27.14356714
+ 1.0282450
+ 1.6206004
9.02684503
27.06879996
+0.0526625
+ 1.2863778
9.00192264
27.03145602
-0.6371012
+0.5088565
8.98947466
27.04270097
-0.8562240
-0.5036265
8.99322298
27.09836241
-0.5459922
-1.4797775
9.01177679
27.18120744
+0.2104677
-2.1580381
9.03939180
27.26787830
+ 1.2104630
-2.3566674
9.06828209
27.33631108
+2.1860457
-2.0224444
9.09109301
27.37048776
+2.8758091
-1.2449232
9.10248524
27.36240999
+3.0949315
-0.2324396
9.09979265
27.31308297
+2.7846997
+0.7437107
9.08335031
163.22483480*
+6.7161237f
-2.2081997J
54.28221419
163.22483460
+6.7161234
-2.2082005
54.28221412
- 8'
9
h
I
0111
88 49 5
0.12745094
27.006742
+0.163630
89 2 45
0.16554389
27.007491
+0.073044
89 19
0.10430339
27.007569
-0.001801
89 39 30
0.01632120
27.006899
-0.038475
89 41 40
0.01598743
27.006246
-0.026577
89 11 43
0.13801114
27.006455
+0.028875
88 41 46
0.29354831
27.007536
+0.110638
88 20 11
0.35007250
27.008599
+0.196247
87 48 4
0.25781881
27.008698
+0.264581
88 19 32
0.09768920
27.007728
+0.299728
88 22 22
0.00340551
27.006562
+0.292816
88 35 32
0.03486337
27.006176
+0.243874
532 41' 57"
0.80250439
162.043353
+0.803287
533 9 13
0.80250130
162.043348
+0.803293
f
80.552426
80.826371
81.050843
81.156986
81.117340
80.952183
80.714558
80.466952
80.265977
80.156946
80.170254
80.311910
483.871398
483.871348
G
27.006566
27.007263
27.007426
27.006877
27.006224
27.006265
27.007132
27.008116
27.008341
27.007593
27.006557
27.006128
162.042246
162.042242
* 6o 2 + 3o% 2 + 6[a' 2 - 2Jtaa'ee' cos K] = + 163.22483477.
t 6[o' 2 e' - koa'e cos A'] = + 6.7161238.
J - Sk'aa' cos >' e sin K' = - 2.2082004.
78
THE SECULAR VARIATIONS OF THE ELEMENTS
G'
ACTION OF JUPITER ON MERCURY.
G"
log ZV
0.188801
0.024995
5
6
8.07
0.0025877
0.2764500
30
0.123075
0.049804
4
35
5.15
0.0020887
0.2757852
60
0.061321
0.062980
3
53
7.75
0.0014995
0.2750001
90
0.011983
0.050435
2
45
10.71
0.0007524
0.2740043
120
0.014441
0.040995
2
35
41.32
0.0006684
0.2738923
150
0.087484
0.058419
4
12
38.17
0.0017613
0.2753489
180
0.173639
0.062597
5
21
37.00
0.0028566
0.2768082
210
0.248823
0.052092
6
3
11.63
0.0036449
0.2778582
240
0.297071
0.032133
6
20
5.30
0.0039929
0.2783216
270
0.311476
0.011613
6
16
40.56
0.0039213
0.2782263
300
0.293250
0.000430
5
59
8.38
0.0035638
0.2777502
330
0.249105
0.005182
5
34
4.63
0.0030827
0.2771094
o
/
//
Si
1.028523
0.224130
29
15
47.82
0.0151689
1.6582224
2 2
1.031946
0.227545
29
26
50.85
0.0152513
1.6583323
E
logtfo
log AT
logP
logQ
logV
0.1799707
6.4183196
3.8310274
5.1664192
5.1659174
30
0.1792229
6.4468140
3.8580378
5.1937562
5.1927566
60
0.1783398
6.5219986
3.9320091
5.2678436
5.2665795
90
0.1772196
6.6157906
4.0252257
5.3607256
5.3597128
120
0.1770937
6.7009450
4.1105921
5.4459161
5.4450927
150
0.1787322
6.7589559
4.1694988
5.5052851
5.5041125
180
0.1803735
6.7800058
4.1918462
5.5278954
5.5266396
210
0.1815544
6.7609474
4.1741434
5.5101707
5.5091259
240
0.1820755
6.7044320
4.1187250
5.4544932
5.4538485
270
0.1819683
6.6198775
4.0347585
5.3701732
5.3699402
300
0.1814329
6.5255910
3.9403890
5.2755478
5.2755392
330
0.1807123
6.4489104
3.8629284
5.1980770
5.1979730
Si
1.0792861
9.6512920
4.1245888
2.1381153
2.1336169
S 2
1.0794097
9.6512958
4.1245926
2.1381878
2.1336210
OF THE ORBITS OF THE FOUR INNER PLANETS.
79
ACTION OF JUPITER ON MERCUKY.
E
JV
J.
J 3
F t
Vi
26.907806
-0.08848287
-1.4438584
-1.8440911
+0.12702123
30
26.754757
-0.00410757
-2.3330959
-2.1016844
+0.05043578
60
26.779325
+0.16212886
-2.4984534
-1.6682462
-0.05353664
90
26.922525
+0.18465069
-1.8956909
-0.6599133
-0.05566955
120
27.032841
+0.04256694
-0.6863554
+0.6531309
+0.07759141
150
27.046752
-0.14516517
+0.8055488
+ 1.9190575
+0.24215469
180
26.945408
-0.26169947
+2.1803373
+2.7986639
+0.29255865
210
26.802715
-0.24648650
+3.0696706
+3.0562500
+0.18231424
240
26.714620
-0.10777763
+3.2352193
+ 2.6228182
+0.00934620
270
26.757114
+0.06770763
+2.6325527
+ 1.6144859
-0.07863175
300
26.911783
+0.13569580
+ 1.4231210
+0.3014406
-0.02506303
330
27.011072
+0.02690450
-0.0689744
-0.9644853
+0.08731406
Si
161.291783*
-0.11756837
+2.2100104
+2.8637163
+0.42791782
2 2
161.294935
-0.11649642
+2.2100109
+2.8637104
+0.42791747
E
1000 X Ro
1000,000 X So
100,000 X W,
1000 X R (n >
100,000 X S<">
0.12949124
-0.25462237
- 2.1070330
0.00000000
-0.82801500
30
0.13531974
-0.15796961
- 3.6329106
+0.21265173
-0.49649079
60
0.16094783
+0.15688051
- 4.6204172
+0.40133398
+0.45170948
90
0.20322837
+0.35279259
- 4.3458069
+0.52500398
+0.91137646
120
0.25064630
+0.20287634
- 1.9026678
+0.50847837
+0.47523891
150
0.28657259
-0.17989808
+ 2.6073819
+0.31420703
-0.39449173
180
0.29756923
-0.44461029
+ 7.3765067
0.00000000
-0.95269261
210
0.28082350
-0.33962404
+ 9.9405132
-0.30790355
-0.74474862
240
0.24483424
+0.03827880
+ 9.2004732
-0.49668759
+0.08966829
270
0.20310516
+0.33360150
+ 6.1619379
-0.52468566
+0.86179960
300
0.16660105
+0.28219872
+ 2.6818065
-0.41543058
+0.81254093
330
0.14104055
-0.02790130
- 0.1024400
-0.22164184
-0.08769243
Si
1.25008989
-0.01899829
+ 10.6286684
-0.00230582
+0.04845000
S 2
1.25008991
-0.01899894
+ 10.6286755
-0.00236831
+0.04975249
sin <f \ 4i ( "'+ cos if
Bo (c) = + 0.00000000000073.
*2,(Ji' -G") = 161.067653.
S 2 (Ji' - G") = 161.067390.
80
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION or JUPITER ON MERCURY.
E
1000 X [ft. sin v +
(cos v + cos E)S(,]
1000 X Wi, cos u 1000 X W a sin u
1000 X
DIFFERENTIAL COEFFICIENTS.
log cqeff.
[dx/dt] w
[di/dt]oo
+ 3.3470577 m' p 0.5246632
+ 1613.8089 m' p 3.2078521
51.404941 m'
- 1550.4039 TO'
+ 1602.2454 TO'
n 1.7110049
w 3.1904449
p 3.2047290
[dL/dt] w = -2312.2863 m' n 3.3640416
(-';*)
-0.00509245
-0.12949124
-0.018505013
-0.010075877
-0.20573445
30
+0.07792118
-0.11047546
-0.015293859
-0.032953015
-0.22244969
60
+0.15333585
-0.04994118
+0.007556597
-0.045582053
-0.28880400
90
+0.19816106
+0.04884221
+0.028185071
-0.033078773
-0.40645673
120
+0.19031365
+0.16372520
+0.017684089
-0.007020516
-0.55282671
150
+0.12222497
+0.25901646
-0.026009473
-0.001830673
-0.67519875
180
+0.00889221
+0.29756923
-0.064784149
-0.035274618
-0.71750180
210
-0.11061196
+0.25859897
-0.059670973
-0.079503145
-0.66165318
240
-0.18859570
+0.15601907
-0.017887815
-0.090249100
-0.54000750
270
-0.19945173
+0.03508579
+0.017710220
-0.059019459
-0.40621028
300
-0.15503995
-0.05982940
+0.019842790
-0.018040859
-0.29894808
330
-0.08443003
-0.11301587
-0.001014512
+0.000141991
-0.23185401
Zi
+0.00381361
+0.37805168
-0.056093501
-0,206243023
-2.60382254
22
+0.00381349
+0.37805210
-0.056093526
-0.206243074
-2.60382263
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[de/dfloo = +0.00319413
[d x /dt] w = +1.540072
[di/d/]oo = -0.049056191
[dtt/dtlw = -1.4795642
[djr/dflw = +1.5290366
= -2.2066350
OF THE ORBITS OF THE FOUR INNER PLANETS. 81
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] m
+0^00320
+0.00320
+0.003194
e[dw/dt]M
+0.31437
+0.31664
+0.314377
[di/dt]oo
-0.04907
-0.04905
-0.049056
sin i [dfl/dt]oo
-0.18042
-0.18037
-0.180368
[dL/dt] M
-2.2078
-2.20663
NOTES.
The above results were published in 1896 in A. J., No. 386. In 1911, upon
applying to the various computations all of the test equations devised or learned of
by that time, a slight error was detected in the value of F z for 240. This rendered
the values of W , W cos u, W sin u, [di/dt] 00 and [dQ/dt} 00 incorrect.
In this computation the device was for the first time applied of finding the root
G by approximations and then depressing the cubic equation and solving the resulting
quadratic equation directly. When a' and hence g is large, some such device becomes
necessary as the solution by HILL'S formulas involves a great amount of labor.
Thus, while but three approximations were necessary with the Earth on Mercury, no
less than eleven were required in some cases with Mars on Mercury, and in the latter,
as well as in the present case, if the formulas of HILL'S second method are employed
the angle 6' will be found so nearly equal to 90 as to render the values of the roots
obtained from it but little better than first approximations. Accordingly all the
remaining computations have been effected by the method here outlined, a method
which, since the approximation to the value of G is always very rapid, leads so quickly
to the values of the roots that special devices for avoiding the solution of the cubic
seem unnecessary.
The final sums are here practically in exact agreement, showing that the effect
of all terms from the sixth to the eleventh orders inclusive is inappreciable.
82
THE SECULAH VARIATIONS OF THE ELEMENTS
ACTION OF SATURN ON MERCURY.
E
A
B cos t
/.' sin .
9
h
91.40055452
7.9236831
-0.7566872
0.1637322
90.704247
22.5
91.41316673
8.0026121
+0.6479546
0.1200578
90.704730
45
91.37113281
7.5285556
+ 1.9837667
1.1253368
90.704973
67.5
91.28182268
6.5736833
+3.0473832
2.6555561
90.704844
90
91.15980423
5.2833659
+3.6768805
3.8659866
90.704445
112.5
91.02405557
3.8540430
+3.7764200
4.0781377
90.704075
135
90.89484135
2.5033169
+3.3308508
3.1725737
90.704038
157.5
90.79086206
1.4368221
+2.4080044
1.6581198
90.704442
180
90.72697664
0.8169216
+ 1.1483775
0.3771123
90.705145
202.5
90.71250894
0.7379931
-0.2562644
0.0187792
90.705802
225
90.75006384
1.2120495
-1.5920765
0.7248182
90.706059
247.5
90.83489465
2.1669211
-2.6556933
2.0167735
90.705781
270
90.95505790
3.4572383
-3.2851902
3.0861872
90.705088
292.5
91.09266180
4.8865615
-3.3847297
3.2760409
90.704321
315
91.22635509
6.2372881
-2.9391612
2.4702909
90.703855
337.5
91.33481374
7.3037834
-2.0163140
1.1625662
90.703855
2,
728.48478638*
34.9624190}
+ 1.5667604t
14.9860379
725.637848
2 2
728.48478617
34.9624206
+ 1.5667608
14.9860312
725.637850
*8a i + 4a s e* + 8[a' J - 2kaa'ee' cos A'] = + 728.48478640.
1 8(a'V - kaa'e cos K] = + 34.9624198.
t - Kk'aa' cos >' e sin K' = + 1.5667610.
OF THE ORBITS OF THE FOUR INNER PLANETS.
83
E
ACTION OF SATURN ON MERCURY.
G G'
G"
+0.410351
90.704227
0.4147226
0.0043526
3
53' 5078
22.5
+0.422480
90.704715
0.4256040
0.0031100
3
56 31.53
45
+0.380202
90.704836
0.4105578
0.0302188
3
59 47.96
67.5
+0.291021
90.704520
0.3703889
0.0790439
4
2 4.89
90
+0.169403
90.703974
0.3081768
0.1383038
4
1 12.34
112.5
+0.034023
90.703579
0.2300012
0.1954823
3
55 22.98
135
-0.095154
90.703652
0.1455476
0.2403156
3
44 5.05
157.5
-0.199537
90.704241
0.0683029
0.2676388
3
29 2.13
180
-0.264126
90.705099
0.0149026
0.2789825
3
15 29.12
202
.5
-0.279250
90.705800
0.0007394
0.2799874
3
11 3.15
225
-0.241953
90.705971
0.0294521
0.2713169
3
17 46.29
247.5
-0.156844
90.705536
0.0901199
0.2467190
3
29 20.24
270
-0.035987
90.704713
0.1675089
0.2031209
3
39 39.24
292.5
+0.102383
90.703922
0.2482641
0.1454820
3
46 28.97
315
+0.236543
90.703554
0.3215440
0.0847000
3
50 7.92
337.5
+0.345001
90.703713
0.3789646
0.0338216
3
52 2.72
O
/ //
v
+ 0.559279
725.636026
1.8124124
1.2513111
29
41 58.70
Zo
+0.559277
725.636026
1.8123850
1.2512850
29
41 56.61
*
ACTION OF
SATURN ON MERCURY.
E
log A'o
log LO'
log #
log N log P
logQ
0.00150875
0.27501245
0.
17835367 5.6285747 1.9882904
3.8492800
22.5
.00154356
0.
27505883
0.
17840585 5.
6455608 2.0053300
3.8663219
45
.00158664
0.27511623
0.17847042 5.6919307 2.0514967
3.912626(1
67.5
0.00161702
0.
27515671
0.
17851595 5.
7568907 2.1160330
3.9773995
90
0.00160533
0.27514113
0.
17849842 5.
8276384 2.1862035
4.0478488
112
.5
0.00152867
0.27503898
0.
17838353 5.
8929411 2.2508613
4.1127652
135
.00138529
0.27484791
0.
17816860 5.
9445922 2.3018923
4.1639869
157.5
.00120533
0.27460806
0.
17789879 5.
9773700 2.3341639
4.1963617
180
0.00105402
0.27440638
0.17767193 5.
9884810 2.3449566
4.2071875
202.5
0.00100673
0.27434335
0.
17760102 5.9770716 2.3334678
4.1956990
225
.00107884
0.
27443945
0.
17770914 5.
9440470 2.3006205
4.1628231
247.5
.00120882
0.
27461270
0.17790402 5.
8922401 2.2492258
4.1113306
270
.00133101
0.
27477556
0.
17808722 5.
8268942 2.1844672
4.0463801
292.5
.00141515
0.
27488768
0.
17821333 5.
7562165 2.1144600
3.9761077
315
0.00146116
0.
27494901
0.
17828232 5.6914235 2.0503133
3.9116762
337.5
0.00148559
0.27498156
0.
17831893 5.
6452895 2.0046971
3.8658214
2!
.01101104
1.
19868812
1.
42524172 6.5435816 7.4082403
2.3018085
2 2 0.01101087
1.
19868787
1.
42524142 6.
5435802 7.4082388
2.3018068
84
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF SATURN ON MERCURY.
E
logy
Ji' J 2
J,
p t
3.8492539
90.058195 -0.59752575
- 6.047237
+ 3.829727
22.5
3.8663033
89.638288 -0.20995430
- 7.910876
- 3.279413
45
3.9124452
89.669758 +0.36734212
- 8.328272
-10.040194
67.5
3.9769269
90.090673 +0.71855783
- 7.235732
-15.423345
90
4.0470220
90.582478 +0.67967125
- 4.799462
-18.609342
112.5
4.1115969
90.879839 +0.35660262
- 1.390299
-19.113128
135
4.1625507
90.892982 -0.04825504
+ 2.472681
-16.858027
157.5
4.1947623
90.670885 -0.36411899
+ 6.201244
-12.187337
180
4.2055204
90.332824 -0.49291773
+ 9.227617
- 5.812141
202.5
4.1940258
90.018885 -0.40939877
+ 11.090997
+ 1.296998
225
4.1612018
89.853321 -0.15646464
+ 11.507770
+ 8.057780
247.5
4.1098562
89.914463 +0.15541932
+10.414614
+ 13.440932
270
4.0451660
90.199819 +0.35977661
+ 7.978084
+ 16.626927
292.5
3.9752380
90.576579 +0.29271946
+ 4.569179
+ 17.130715
315
3.9111697
90.780941 -0.07365934
+ 0.706820
+ 14.875613
337.5
3.8656191
90.580690 -0.50178023
- 3.021123
+ 10.204923
2,
2.2943297
722.370318* +0.03796748
+ 12.718001
- 7.929657
22
2.2943284
722.370302 +0.03804694
+ 12.718004
- 7.929655
ACTION OF SATURN ON MERCURY.
E
F t
10000 X fto 1000000 X So 1000000 X W,
1000 X ft'"'
1000000 XS'>
-0.2217173
0.20835581 -0.38501122 - 4.2759258
0.0000000
-1.2520309
22.5
+0.0670688
0.21359630 -0.18752118 - 5.8140290
+2.6067657
-0.5980245
45
-0.2247552
0.23766012 +0.18723103 - 6.8102258
+5.0798256
+0.5659595
67.5
-0.9885705
0.27944181 +0.47990507 - 6.8742610
+7.2389467
+ 1.3456245
90
-1.8298399
0.33354940 +0.47167629 - 5.3763783
+ 8.6166490
+ 1.2184913
112.5
-2.2911663
0.39056810 +0.12052582 1.8384740
+8.6416600
-0.2886458
135
-2.1148434
0.43919690 -0.40799123 + 3.5527836
+7.0044048
-0.9201906
157.5
-1.3919042
0.46950930 -0.83324243 + 9.6804438
+3.9006063
-1.8089208
180
-0.5106652
0.47592790 -0.91983080 +14.8005740
0.0000000
-1.9709759
202.5
+0.0653940
0.45862525 -0.61203702 +17.3393050
-3.8101033
-1.3286969
225
+0.0610980
0.42282850 -0.06578321 +16.6811060
-6.7433594
-0.1483686
247.5
-0.4587079
0.37651300 +0.43874423 +13.4040390
-8.3306788
+ 1.0507430
270
-1.1366346
0.32772690 +0.65346841 + 8.8350920
-8.4662365
+ 1.6881183
292.5
-1.5401525
0.28276850 +0.49946088 + 4.2959022
-7.3251220
+ 1.4004581
315
-1.4203565
0.24587472 +0.10699355 + 0.5601261
-5.2554073
+0.3234187
337.5
-0.8596732
0.22009701 -0.26508243 - 2.2258014
-2.6861019
-0.8453756
2,
-7.3977141
2.69112025 -0.35924718 +27.9671518
+0.2358762
-0.4955782
2 2
-7.3977118
2.69111927 -0.3592470? +27.9671246
+0.2365727
-0.4955464
sin
<f- j^li'" + COS <f-l
V c) = + 0.000000000034.
*2j
(Ji' - G") = 721.
119007. 2 2 (J/ - G") = 721.119017.
OF THE ORBITS OF THE FOUR INNER PLANETS.
85
E
1000 X [ft) sin v
+ (COS V + COS E)S ]
1000X
ACTION OF SATURN ON MERCURY.
[
"COS"
o
1000000 XWt cos u 1000000 X Wo sin u
1000 X -2- Bo
a
-0.000770023
-0.0208355810
- 3.7553310
- 2.0447571
-0.033103367
22.5
+0.009535644
-0.0190997850
- 3.2422806
- 4.8260289
-0.034604548
45
+0.019486084
-0.0136593860
- 0.8728046
- 6.7540641
-0.040621607
67.5
+0.027699067
-0.0044469025
+ 2.0656152
- 6.5565758
-0.051490988
90
+0.032545329
+0.0078015052
+ 3.4868920
4.0923123
-0.066709877
112.5
+0.032625198
+0.0215155020
+ 1.6174933
- 0.8739008
-0.084259712
135
+0.027148276
+ 0.0344565700
- 3.5128323
+ 0.5313050
-0.100609845
157.5
+0.016337313
+0.0439769600
- 9.5267356
- 1.7182294
-0.111738865
180
+0.001839662
+0.0475927900
-12.9986018
- 7.0776661
-0.114756176
202.5
-0.013287669
+0.0439638930
-11.8447875
-12.6630324
-0.109148595
225
-0.025446823
+0.0337808020
- 6.8547683
-15.2076154
-0.096860250
247.5
-0.031966152
+0.0197522140
1.0475614
-13.3630400
-0.081227519
270
-0.032206863
+0.0054309792
+ 2.5393222
- 8.4623118
-0.065545379
292.5
-0.027462505
-0.0063965152
+ 2.7411608
- 3.3076901
-0.052103952
315
-0.019770536
-0.0145922640
+ 0.5055592
- 0.2411456
-0.042025673
337.5
-0.010655672
-'0.0192899750
- 2.2254410
- 0.0400332
-0.035657730
2i
+0.002825107
+0.0799753913
-21.4625646
-43.3484220
-0.560232174
22
+0.002825224
+0.0799754154
-21.4625365
-43.3485702
-0.560231909
DIFFERENTIAL COEFFICIENTS.
//
log coeff.
[de/dJfe = + 1
.8596825 m' p
0.2694389
[d x /dt] M = +256
.04618 m' p
2.4083183
[di/dt] w = 14
.751452 m' n
1.1688348
[da/dfl M = -244
.39983 m' n
2.3881009
[dir/dt] m = +254
.22335 m' p
2.4052154
[dL/dfloo = -373
.17967 m' n
2.5719180
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
u
[de/df]) =
+0.00053109524
[dx/d*] M =
+0.073122627
[di/dt] M =
-0.0042127757
[dtt/dt] M =
-0.069796619
[dir/dt]o =
+0.072602050
[dL/dtlw =
-0.10657405
86
THE SECULAR VARIATIONS OF THE ELEMENTS
[de/dt],
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss,
oo
sin i [dtt/dt] m
+o'.00053
+0.00053
+0.0005311
+0.01494
+0.01503
+0.0149273
-0.00421
-0.00421
-0.0042128
-0.00853
-0.00850
-0.0085087
-0.1070
-0.106574
NOTES.
The considerable disagreement of the first sums is caused by the rather large
value of e'. The very exact agreement toward the close of the computation shows,
however, that all terms above the 15th order are wholly inappreciable, the total
effect of all terms from the 8th to the 15th orders inclusive occurring with [de/dt] o
and amounting to but 1 /30000th of the value of this coefficient.
ACTION OF URANUS ON MERCURY.
E
A
B cos e
B sin .
a
h
368.36907643
16.94656315
-5.8755608
27.995613
367.49553
45
368.87526360
22.1817430
-3.9726882
12.798540
367.49652
90
369.14240318
24.5977597
+ 1.1481399
1.069008
367.49606
135
369.01848670
22.7793354
+6.4872119
34.127748
367.49535
180
368.57162431
17.7916836
+8.9169755
64.480235
367.49625
225
368.05910248
12.5565017
+7.0141016
39.896574
367.49742
270
367.78562896
10.1404905
+ 1.8932751
2.906824
367.49671
315
367.91587927
11.9589113
-3.4457976
9.628773
367.49535
2,
1473.86873228*
69.4764969t
+6.0828297J
96.451680
1469.98455
2 2
1473.86873205
69.4764914
+6.0828277
96.451635
1469.98464
* 4a 2 + 2aV + 4[a' 2 - Zkoa'ee' cos K] = + 1473.86873246.
t 4[a'V - kaa'e cos A'] = + 69.4764933.
t - 4k'aa' cos J e sin A" = + 6.0828300.
OF THE ORBITS OF THE FOUR INNER PLANETS.
87
ACTION OF URANUS ON MERCURY.
E
I
G G' G"
8
O / U
+0.06261
367.49532 0.30919 0.24638
2 13 39.23
45
+0.56779
367.49643 0.62373 0.05584
2 27 51.84
90
+0.83539
367.49605 0.83888 0.00347
2 44 38.85
135
+0.71220
367.49510 0.82501 0.11257
2 53 41.24
180
+0.26443
367.49577 0.57178 0.30687
2 48 5.47
225
-0.24927
367.49712 0.22774 0.47670
2 30 27.70
270
-0.52202
367.49669 0.01474 0.53674
2 13 6.46
315
-0.39041
367.49528 0.05839 0.44873
2 7 39.31
2,
+0.64041
1469.98383 1.73458 1.09346
9 59 30.01
2 2
+0.64031
1469.98393 1.73486 1.09383
9 59 40.09
E
log jr.
log L ' log No log N log P
logQ
0.00049251
0.27365789 0.17682994 4.7157168 9.8582891
2.3270039
45
0.00060285
0.27380499 0.17699541 4.7796309 9.9227977
2.3913072
90
0.00074754
0.27399787 0.17721240 4.9163276 0.0598121
2.5282833
135
0.00083193
0.27411036 0.17733893 5.0341227 0.1774639
2.6460770
180
0.00077916
0.27404002 0.17725980 5.0782321 0.2210425
2.6898771
225
0.00062422
0.27383348 0.17702747 5.0332669 0.1754670
2.6444777
270
0.00048849
0.27365254 0.17682391 4.9151227 0.0570006
2.5260593
315
0.00044928
0.27360025 0.17676511 4.7787835 9.9208203
2.3897669
Zj
0.00250770
1.09534833 0.70812605 9.6253991 0.1961442
0.0712235
22
0.00250828
1.09534908 0.70812692 9.6258039 0.1965489
0.0716217
E
logF
Ji' Ji J 3
Fi
2.3266400
366.509888 -2.2679092 -1.6777615
+ 100.81429
45
2.3912247
363.119586 -0.5138777 -3.4254958
+ 68.16435
90
2.5282782
365.160838 +2.2768225 -2.9100648
19.70006
135
2.6459108
367.567473 +0.7238013 -0.4333073
-111.30915
180
2.6894241
366.570373 -1.5747120 +2.5538323
-152.99965
225
2.6437740
364.094438 -1.1669936 +4.3014333
-120.34967
270
2.5252669
364.090394 +1.5139175 +3.7858697
- 32.48527
315
2.3891044
367.299619 +1.4067899 +1.3092445
+ 59.12384
Si
0.0696092
1462.331493* -0.0518812 +1.7518757
-104.37069
2,
0.0700139
1462.082117 -0.4497199 +1.7518747
-104.37063
*Si(J,' - G") = 1461.238038.
Z 2 (./i' - G") = 1460.987287.
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF UKANUS ON MERCURY.
E
F 3 1000000 X Ro 1000000000 X <S 1000000 X TFo 100000 X ft""
1000000 X S'">
- 7.546840 2.572319 - 40.838728 -0.3564792 0.00000000
-0.13280483
45
- 0.473969 2.902866 6.943520 -0.8432739 +0.62046843
-0.02098878
90
- 1.504590 4.048391 + 74.582644 -0.9823294 +1.04582909
+0.19267091
135
-14.059822 5.410853 + 15.278801 -0.1938530 +0.86293520
+0.03446006
180
-17.382080 5.928252 -102.476485 +1.2462639 0.00000000
-0.21958237
225
- 5.500356 5.225873 - 69.412255 +1.8932075 -0.83343288
-0.15655363
270
+ 1.222261 3.978026 + 47.038051 +1.2690546 -1.02765154
+0.12151438
315
- 5.177088 2.987389 + 39.388463 +0.3202880 -0.63853471
+0.11906293
,
-25.211249 16.526988 - 21.694518 +1.1765099 +0.01817755
-0.03820191
Ii
-25.211235 16.526981 - 21.688511 +1.1763686 +0.01143604
-0.02401942
JS
lOOOOOOXlffi sin4- 1000000 > < -#ocosv+
OM. + WJW /' \. -, lOOOOOOXTF.coau 1000000 XTF sin
( -sec 2 v> + ll sin vS \
1000000 X- 2- ft
a
-0.0816775 -2.5723195 -0.31307783 -0.17046914
4.0868774
45
+2.3415248 -1.7140875 -0.10807474 -0.83631981
- 4.9616693
90
+3.9465628 +0.9815683 +0.63709735 -0.74771500
- 8.0967815
135
+3.2460626 +4.3319665 +0.19167313 -0.02898997
-12.3950199
180
+0.2049530 +5.9282527 -1.09453081 -0.59596603
- 14.2942. r )M>
225
-3.0528880 +4.2563802 -0.77797582 -1.72597460
-11.9712672
270
-3.9027069 +0.7238029 +0.36474303 -1.21550896
- 7.9560519
315
-2.3679844 -1.8134015 +0.28908589 -0.13789041
- 5.1061400
2!
+0.1671314 +5.0613044 -0.40576826 -2.72965913
-34.4339707
2 2
+0.1667150 +5.0608577 -0.40529154 -2.72917479
-34.4340964
sin p 1^,0 + cos v Bo (c > = - 0.00000000000024.
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] m = + 0.21975650 TO' p 9.3419417
[dxldt]m = +32.406731 TO' p 1.5106352
[dildt] M =-- 0.55745051 TO' 9.7462063
[rfn/<ft] M = -30.777028 TO' n 1.4882267
[dw/dt]^ = +32.177180 TO' p 1.5075480
[dLldt] M = -45.859693 TO' n 1.6614312
OF THE ORBITS OF THE FOUR INNER PLANETS. 89
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
= +0.0000096384435
[d x /dt] m = +0.0014213479
[dtYdflw = -0.000024449584
[dQ/dt] m = -0.0013498699
[dT/dfloo = +0.0014112801
[dL/dt] m = -0.0020113907
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
// // //
{de/dt] m +0.00000 +0.00001 +0.0000096
e[dw/dt] w +0.00029 +0.00029 +0.0002902
[dt'/dfloo -0.00001 -0.00002 -0.0000244
sin i [dtt/dt] w -0.00016 -0.00016 -0.0001646
NOTES.
In the results of this computation, published in A. J., No. 398, the residual from
the test equation which arises from the constancy of the major axis was stated very
much too large. Its true value is as here given.
A comparison of the above figures with the corresponding tabulation for Saturn
on Mercury and Uranus on Venus shows 'that a division into eight parts is fully
sufficient. The effect of all terms of the 4th and higher orders may, however, in
some cases amount to l/1000th of the whole.
ACTION OF NEPTUNE ON MERCURY.
E
A
B COS e
Bsin e
g
h
904.45979027
+ 15.1625927
- 5.380877
1.889897
904.17356
45
904.50658592
+ 17.0408109
+ 3.150684
0.647955
904.17446
90
904.46648308
+ 12.3030460
+ 10.652425
7.406776
904.17363
135
904.36745259
+ 3.7246157
+ 12.729927
10.577523
904.17288
180
904.26302599
- 3.6693551
+ 8.166225
4.352861
904.17435
225
904.20989585
- 5.5475736
- 0.365336
0.008712
904.17570
270
904.24366422
- 0.8098086
- 7.867076
4.039717
904.17446
315
904.34902898
+ 7.7686225
- 9.944584
6.455136
904.17292
Si
3617.43296356*
+22.9864685f
+ 5.5706981
17.689251
3616.69599
2 2
3617.43296334
+22.9864755
+ 5.570691
17.689326
3616.69596
* 4o 2 + 2aV + 4[a' 2 - 2kaa'ce' cos K] = + 3617.4329635,
t 4[a'V - kaa'e cos A'] = + 22.986469.
J 4fc'oo' cos <f e sin A'' = + 5.570695,
90
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF NEPTUNE ON MERCURY.
E
I
G
G'
G"
e
1 II
0.22096
904.17356
0.2300460
0.0090860
55 54.56
45
0.26686
904.17446
0.2695190
0.0026589
59 38.88
90
0.22759
904.17362
0.2591945
0.0316047
1 1 39.23
135
0.12931
904.17286
0.1906740
0.0613537
57 23.73
180
0.02341
904.17434
0.0820755
0.0586556
42 53.31
225
0.03107
904.17570
0.0003070
0.0313770
20 20.99
270
0.00394
904.17445
0.0688415
0.0649018
41 48.59
315
0.11084
904.17291
0.1564755
0.0456255
51 23.82
s,
0.57589
3616.69597
0.6401575
0.1642481
3 22 15.69
22
0.57594
3616.69593
0.6169755
0.1410151
3 8 47.42
E
log A'o
log /,'
logtfo
log N log P
logQ
0.00008616
0.27311614
0.17622049 4
.1292404 8.4898441
1.3492047
45
0.00009807
0.27313202
0.17623836 4
.1927246 8.5533496
1.4127095
90
0.00010478
0.27314097
0.17624841 4
.3291694 8.6897764
1.5491508
135
0.00009080
0.27312233
0.17622746 4.4470372 8.8075977
1.6669838
180
0.00005070
0.27306887
0.17616731 4
.4915054 8.8520137
1.7113924
225
0.00001141
0.27301649
0.17610837 4
.4469773 8.8074580
1.6668178
270
0.00004818
0.27306552
0.17616353 4
.3290882 8.6895869
1.5489683
315
0.00007281
0.27309835
0.17620048 4,
.1926694 8.5532210
1.4125965
2,
0.00028982
1.09239150
0.70479974 7.2790034 4.7212211
6.1587162
2 2
0.00027309
1.09236919
0.70477467 7
.2794084 4.7216261
6.1591075
E
logF
J,'
J.
J3
F,
1.3491992
897.83542
-6.8010114
- 59.717292
+41.026915
45
1.4127079
890.85490
+ 1.7283795
- 93.077013
-24.022644
90
1.5491318
899.76750
+6.4446186
- 62.851349
-81.220264
135
1.6669469
904.08417
-1.3097917
+ 13.254268
-97.060333
180
1.7113571
897.88500
-6.6871696
+ 90.657875
-62.264029
225
1.6667989
891.04449
-2.2340656
+ 124.017084
+ 2.785531
270
1.5489293
894.36354
+5.9480179
+ 93.790933
+59.983153
315
1.4125691
903.73833
+2.4206062
+ 17.685807
+75.823263
2!
6.1586174
3589.85146*
-1.0955445
+ 61.880167
-42.474225
2 2
6.1590228
3589.72189
+0.6051284
+ 61.880146
-42.474183
-G") = 3589.68721.
- G") = 3589.58087.
OF THE ORBITS OF THE FOUR INNER PLANETS.
91
ACTION OF NEPTUNE ON MERCURY.
E
F j 1000000 X Ko 1000000000 X So 1000000000 X W 1000000 X fi"
1000000 X <S (n)
- 2.928627
0.6591808 -15.070783 -133.45316 0.000000
-0.04900917
45
- 0.320722
0.7449111 + 4.384511 -240.74232 +1.592198
+0.01325344
90
- 8.201274
1.0513250 +22.423107 -222.59983 +2.715911
+0.05792607
135
-13.630154
1.3989891 - 6.706655 + 61.47306 +2.231138
-0.01512625
180
- 6.745208
1.5178690 -34.846090 +466.35639 0.000000
-0.07466679
225
+ 0.065738
1.3382857 -10.354891 +575.81202 -2.134326
-0.02335460
270
- 3.847073
1.0317860 +21.345904 +331.94457 -2.665434
+0.05514330
315
- 7.846125
0.7779772 + 6.529864 + 45.70113 -1.662874
+0.01973839
r,
-21.722182
4.2601608 - 6.147862 +442.24897 +0.050477
-0.01060659
22
-21.731263
4.2601631 - 6.147171 +442.24389 +0.026136
-0.00548902
E
1000000000 Xtffo sm v lwooooooo ><L-tf<> c s <' 1000000000 1000000000
1000000000
7*
+ (cosw+cos)<S
o] fr 2 \ . "! XWoCOsw XWosmu
x-afa
30.1416
- 659.1808 -117.20523 - 63.81760
- 1047.3000
45
+ 608.8428
- 430.4075 - 30.85375 -238.75703
-1273.2255
90
+ 1024.2533
+ 261.0141 +144.36884 -169.43527
-2102.6500
135
+ 855.3050
+ 1105.9013 - 60.78178 + 9.19306
-3204.7615
180
+ 69.6922
+ 1517.8690 -409.57745 -223.01267
-3659.9008
225
- 792.9698
+ 1080.1647 -236.61842 -524.94878
-3065.7028
270
-1014.1308
+ 169.4383 + 95.40526 -317.93876
-2063.5719
315
- 621.4949
- 466.5348 + 41.24897 - 19.67525
-1329.7429
Si
+ 49.6731
+ 1289.1406 -287.00858 -774.20430
-8873.4227
S 2
+ 49.6831
+ 1289.1237 -287.00498 -774.18800
-8873.4327
sin <f \A i w + cos if L
t (c) = + 0.000000000000013.
DIFFERENTIAL COEFFICIENTS.
log coeff.
[deldt]<n = + o'.065401848 TO' p 8.8155900
[dx/dt] m = + 8.2544736 TO' p 0.9166894
[di/dt] w = - 0.39452600 TO' n 9.5960756
[drfdt]
w
= - 8.7298700 TO'
= + 8.1893623 TO'
n 0.9410078
p 0.9132501
[dL/d*]oo = -11.826130 TO' n 1.0728427
92 THE SECULAR VARIATIONS OF THE ELEMENTS
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[de/dt} 00 = +0.00000331989
[dx/dtlw, +0.00041900885
[di/dt]oo = -0.00002002670
[dV/dt] w = -0.00044314061
[dw!dt] m = +0.00041570371
[dL/dt] w = -0.00060031125
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] w +o'.()0000
+0.00000
+o'.0000033
c[d7r/d<]oo +0.00009
+0.00009
+0.0000855
[di/dt] m -0.00001
-0.00002
-0.0000200
sinz [dtt/dt] m -0.00005
-0.00005
-0.0000508
NOTES.
In the final results of this computation, published in A. J., No. 398, the value of
the residual arising from the equation [da/dt] 00 = is greatly overstated. Its true
value is that given above.
The very large disagreement in G', G", 6, etc., arises from the large values of e'
and I but the gradual lessening of the discrepancies as the end of the computation is
approached shows that terms of the 8th and higher orders are wholly inappreciable.
The greatest effect produced by all terms of the 4th and higher orders here occurs
with [de/dt] 00 and amounts to but 1 /10000th of the value of this coefficient.
OF THE ORBITS OF THE FOUR INNER PLANETS.
93
VENUS.
ACTION OF MEKCURY ON VENUS.
E
A
B cos
B gin e
1000 x g
0.73249627
+0.19271542
+0.22036223
0.30759837
15
0.70628935
+0.12839518
+0.25427819
0.40956972
30
0.67778527
+0.05734848
+0.27076206
0.46439247
45
0.64892716
-0.01558282
+0.26869056
0.45731389
60
0.62168257
-0.08542863
+0.24820463
0.39023784
75
0.59791027
-0.14742905
+0.21070058
0.28121662
90
0.57923027
-0.19735886
+0.15873412
0.15960636
105
0.56691636
-0.23181556
+0.09584678
0.05819218
120
0.56180731
-0.24845074
+0.02632410
0.00438952
135
0.56425026
-0.24613080
-0.04509594
0.01288203
150
0.57407787
-0.22501408
-0.11354628
0.08166866
165
0.59061890
-0.18653936
-0.17436209
0.19258116
180
0.61274521
-0.13332873
-0.22339892
0.31613445
195
0.63894853
-0.06900848
-0.25731488
0.41941058
210
0.66744400
+0.00203821
-0.27379880
0.47486769
225
0.69628968
+0.07496950
-0.27172719
0.46770903
240
0.72352217
+0.14481528
-0.25124139
0.39984532
255
0.74728586
+0.20681571
-0.21373734
0.28938127
270
0.76596211
+0.25674559
-0.16177082
0.16577156
285
0.77827946
+0.29120222
-0.09888350
0.06193800
300
0.78339732
+0.30783739
-0.02936080
0.00546066
315
0.78096661
+0.30551754
+0.04205926
0.01120554
330
0.77115125
+0.28440072
+0.11050957
0.07735875
345
0.75461912
+0.24592608
+0.17132542
0.18593159
2i
8.07130162*
0.356320051
-0.018220301
2.84733165
St
8.07130156
0.35632016
-0.01822015
2.84733161
* 12a 2 + 6aV + 12[a' 2 -
- 2kaa'ee' cos K] = 8.07130158.
1 12[a'V - kaa'e cos K]
= + 0.35632010.
t 12fc'aa' cos <p' e sin
K' = - 0.01822024.
0.58840054
0.56402284
0.53673295
0.50812387
0.48000449
0.45436878
0.43333042
0.41895978
0.41292353
0.41598961
0.42773114
0.44670373
0.47092914
0.49830819
0.52683118
0.55465379
0.58013161
0.60184004
0.61860745
0.62954599
0.63407808
0.63196045
0.62329194
0.60851491
6.33299247
6.33299197
94
THE SECULAR VARIATIONS OF THE ELEMENTS
E
ACTION OF MERCUKY ON VENUS.
G G'
G"
0.13776127
0.58723527
0.14259981
0.00367327
29
50
12.97
15
0.13593205
0.56231468
0.14274284
0.00510263
30
41
37.56
30
0.13471786
0.53456025
0.14296704
0.00607649
31
40
18.70
45
0.13446883
0.50568767
0.14321940
0.00631437
32
42
44.77
60
0.13534362
0.47761736
0.14342737
0.00569662
33
44
34.65
75
0.13720703
0.45239646
0.14351084
0.00433149
34
40
36.18
90
0.13956539
0.43206750
0.14340426
0.00257595
35
25
4.49
105
0.14162212
0.41845750
0.14309623
0.00097183
35
52
44.73
120
0.14254931
0.41288415
0.14266320
0.00007451
36
33.62
135
0.14192619
0.41587654
0.14225701
0.00021775
35
48
50.91
150
0.14001227
0.42706500
0.14202490
0.00134648
35
20
41.47
165
0.13758072
0.44529837
0.14203102
0.00304495
34
40
10.49
180
0.13548161
0.46890720
0.14224327
0.00473972
33
51
11.13
195
0.13430588
0.49597000
0.14257523
0.00593116
32
57
11.97
210
0.13427836
0.52451120
0.14293246
0.00633413
32
1
25.39
225
0.13530143
0.55262582
0.14323803
0.00590863
31
6
52.56
240
0.13705610
0.57856627
0.14343948
0.00481804
30
16
21.83
255
0.13911137
0.60079677
0.14351100
0.00335627
29
32
27.98
270
0.14102020
0.61804514
0.14345226
0.00186975
28
57
29.65
285
0.14239902
0.62934391
0.14328793
0.00068684
28
33
26.40
300
0.14298478
0.63406059
0.14306246
0.00006020
28
21
52.79
315
0.14267170
0.63192420
0.14283210
0.00012415
28
23
50.37
330
0.14152485
0.62303406
0.14265313
0.00087040
28
39
39.46
345
0.13976975
0.60786141
0.14256873
0.00214548
29
8
52.62
Si
1.66229562
6.31855399
1.71486964
0.03813556
384
9
26.15
S 2
1.66229608
6.31855323
1.71487036
0.03813555
384
9
26.54
OF THE ORBITS OF THE FOUR INNER PLANETS.
95
ACTION OF MEKCURY
ON VENUS.
E
log K,
log Lo'
logtfo
logtf
logP
logQ
0.09428087
0.39679023
0.31472571
0.0090498
0.8627996
0.5522553
15
0.10017341
0.40440450
0.32320948
0.0415726
0.9381720
0.6108796
30
0.10716555
0.41342180
0.33324961
0.0806578
1.0282684
0.6810018
45
0.11492344
0.42340404
0.34435531
0.1248163
1.1296769
0.7598999
60
0.12293987
0.43369427
0.35579389
0.1716324
1.2368681
0.8431970
75
0.13049856
0.44337414
0.36654485
0.2174860
1.3415449
0.9243732
90
0.13670353
0.45130400
0.37534554
0.2575171
1.4325547
0.9947294
105
0.14065788
0.45634998
0.38094249
0.2862203
1.4972527
1.0445040
120
0.14178785
0.45779080
0.38254019
0.2989083
1.5248859
1.0655419
135
0.14009658
0.45563407
0.38014861
0.2935151
1.5107656
1.0544719
150
0.13608370
0.45051253
0.37446742
0.2714379
1.4582282
1.0140442
165
0.13043971
0.44329885
0.36646127
0.2367595
1.3768369
0.9516100
180
0.12381666
0.43481823
0.35704266
0.1945715
1.2784801
0.8761594
195
0.11676724
0.42577304
0.34698954
0.1495758
1.1741125
0.7959472
210
0.10975153
0.41675185
0.33695547
0.1054452
1.0722608
0.7174326
225
0.10314673
0.40824135
0.32748247
0.0647779
0.9789193
0.6452104
240
0.09725105
0.40063000
0.31900464
0.0292994
0.8980200
0.5823493
255
0.09229597
0.39422222
0.31186325
0.0001263
0.8320547
0.5308426
270
0.08845847
0.38925289
0.30632243
9.9779738
0.7825625
0.4919641
285
0.08587486
0.38590395
0.30258708
9.9633062
0.7504867
0.4665316
300
0.08464926
0.38431433
0.30081370
9.9564277
0.7363980
0.4550694
315
0.08485629
0.38458290
0.30111334
9.9575313
0.7406137
0.4578944
330
0.08653835
0.38676424
0.30354673
9.9667119
0.7632399
0.475140o
345
0.08969689
0.39085723
0.30811150
9.9839477
0.8041353
0.5067244
Zi
1.32942669
5.01604517
4.05980799
1.3196325
13.0745661
8.7488847
Z 2
1.32942756
5.01604627
4.05980919
1.3196348
13.0745711
8.7488890
96
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MERCURY ON VENUS.
E
log V
/'
Ji.
J 3
1000 X F,
0.5489818
0.14641488
+0.046569366
+0.007078444
-6.6249758
15
0.6061455
0.14799570
+0.053805593
+0.005998321
-7.6446250
30
0.6750909
0.14917691
+0.057280924
+0.004274648
-8.1401963
45
0.7534266
0.14962155
+0.056746961
+ 0.002024896
-8.0779189
60
0.8370288
0.14915409
+0.052252409
-0.000597633
-7.4620305
75
0.9194294
0.14784239
+0.044134510
-0.003414224
-6.3345072
90
0.9916527
0.14602955
+0.032988273
-0.006232935
-4.7721868
105
1.0433053
0.14427332
+0.019614339
-0.008861684
-2.8815397
120
1.0654487
0.14317010
+0.004953440
-0.011121317
-0.7914084
135
1.0542014
0.14310862
-0.009985226
-0.012857840
+ 1.3557653
150
1.0124149
0.14408811
-0.024195183
-0.013952904
+3.4136578
165
0.9480753
0.14573205
-0.036737898
-0.014331875
+ 5.2420265
180
0.8709298
0.14748132
-0.046798849
-0.013968920
+6.7162708
195
0.7897501
0.14882171
-0.053732494
-0.012888777
+7.7359211
210
0.7111602
0.14942892
-0.057095855
-0.011165046
+ 8.2314924
225
0.6396416
0.14920890
-0.056671026
-0.008915214
+8.1692132
240
0.5779992
0.14827058
-0.052477157
-0.006292610
+7.5533276
255
0.5279163
0.14686716
-0.044770588
-0.003475962
+ 6.4258044
270
0.4903757
0.14532896
-0.034036602
-0.000657227
+4.8634820
285
0.4659576
0.14399775
-0.020965237
+0.001971497
+2.9728356
300
0.4550194
0.14316554
-0.006416142
+0.004231074
+0.8827039
315
0.4577910
0.14302191
+0.008631431
+0.005967519
-1.2644706
330
0.4744060
0.14361483
+0.023141851
+0.007062502
-3.3223626
345
0.5048724
0.14483260
+0.036096092
+0.007441419
-5.1507310
2,
8.7105081
1.75532379*
-0.003833505
-0.041341924
+0.5477741
2*
8.7105125
1.75532376
-0.003833543
-0.041341924
+0.5477737
* 2,(J,' - G") = 1
.71718823.
S 2 (j 2 ' _ G") = 1
.71718811.
OF THE ORBITS OF THE FOUR INNER PLANETS.
97
ACTION OF MERCURY ON VENUS.
E
1000 X F 3
R
So
W
BW
<">
+0.12855762
-1.0113711
+0.11654330
+0.02599380
0.0000000
+0.16223015
15
+0.28853773
-1.0855630
+0.15095431.
+0.02672256
-0.3910151
+0.21008155
30
+0.43551240
-1.1839703
+0.18420453
+0.02487781
-0.8232930
+0.25617923
45
+0.52994293
-1.3088599
+0.21275230
+0.01862044
-1.2857216
+0.29555816
60
+0.54641763
-1.4594255
+0.23029016
+0.00532100
-1.7533282
+0.31946706
75
+0.48046967
-1.6276742
+0.22753508
-0.01781201
-2.1774255
+0.31512321
90
+0.34977540
-1.7935752
+0.19439994
-0.05167293
-2.4796011
+0.26875615
105
+0.18941983
-1.9243133
+0.12616235
-0.09195653
-2.5651527
+0.17410988
120
+0.04248891
-1.9847155
+0.03108865
-0.12787906
-2.3681413
+0.04283321
135
-0.05148113
-1.9570464
-0.06917608
-0.14733858
-1.9039342
-0.09517474
150
-0.06711060
-1.8525243
-0.15091802
-0.14550282
-1.2730044
-0.20741354
165
+0.00001044
-1.7023416
-0.20114407
-0.12716787
-0.6051233
-0.27625376
180
+0.13212520
-1.5390947
-0.22013939
-0.10126618
0.0000000
-0.30227222
195
+ 0.29405223
-1.3852923
-0.21561000
-0.07503510
+0.4924233
-0.29612150
210
+0.44259806
-1.2520945
-0.19638955
-0.05218717
+0.8604054
-0.26990714
225
+0.53811675
-1.1428753
-0.16935246
-0.03375802
+ 1.1118587
-0.23300075
240
+0.55512295
-1.0569737
-0.13887026
-0.01942443
+ 1.2611694
-0.19133211
255
+0.48911303
-0.9922445
-0.10732663
-0.00839923
+ 1.3226839
-0.14811570
270
+0.35776744
-0.9463685
-0.07579537
+0.00013574
+ 1.3083458
-0.10478640
285
+0.19621617
-0.9174587
-0.04456334
+ 0.00686904
+ 1.2273328
-0.06171770
300
+0.04762640
-0.9042927
-0.01348260
+0.01232297
+ 1.0864013
-0.01870356
315
-0.04835263
-0.9063942
+0.01780841
+0.01685710
+0.8903706
+0.02473967
330
-0.06620421
-0.9240677
+0.04973149
+0.02067163
+0.6425655
+0.06916321
345
-0.00136711
-0.9584216
+0.08262385
+0.02378861
+0.3452194
+0.11498675
V
*-l
+ 2.90467772
-5.9084737
+0.01066288
-0.40860954
-3.5384806
+0.02421404
2*
+2.90467759
-5.9084840
+0.01066372
-0.40860959
-3.5384837
+0.02421607
sin <p
! ( "> + cos <p B (c) = + 0.000000033.
98
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MERCURY ON VENCJS.
Ro COS V
E
[o sin v
L
( \
, Wo cos u
Wo sin u
-2-flo
+ (cos v+cos E)S(
a 860 *" /
vSo
a
0*
+0.2330866
+ 1.0113711
+0.015230839
+0.021064164
2.0089005
15
+0.0087244
+ 1.1264720
+0.009475283
+0.024986282
2.1567748
30
-0.2767659
+ 1.2080644
+ 0.002459142
+0.024755972
2.3539076
45
-0.6298334
+ 1.2226121
-0.003043897
+0.018369962
2.6050539
60
-1.0391050
+ 1.1217558
-0.002204101
+0.004843036
2.9088638
75
-1.4586399
+0.8508163
+0.011332120
-0.013742297
3.2495836
90
-1.7948636
+0.3765263
+0.042079553
-0.029990049
3.5871504
105
-1.9215248
-0.2668018
+0.086138100
-0.032190511
3.8554443
120
-1.7441606
-0.9487539
+0.127283754
-0.012324508
3.9830138
135
-1.2790794
-1.4880990
+0.145357191
+0.024082111
3.9330324
150
-0.6591294
-1.7579567
+0.132584634
+0.059936528
3.7270060
165
-0.0490234
-1.7488888
+0.098503636
+0.080428204
3.4271881
180
+0.4402788
-1.5390947
+0.059336027
+0.082061396
3.0992536
195
+0.7728021
-1.2274791
+0.026854296
+0.070065081
2.7888981
210
+0.9628351
-0.8906639
+0.005513895
+0.051895060
2.5190296
225
+ 1.0443017
-0.5731024
-0.005195888
+0.033355748
2.2968106
240
+ 1.0518039
-0.2937729
-0.007836004
+0.017773731
2.1211810
255
+ 1.0129579
-0.0559799
-0.005257515
-0.006550236
1.9880041
270
+0.9468650
+0.1451147
+0.000109448
-0.000080285
1.8927371
285
+0.8649660
+0.3177537
+0.006402057
-0.002489441
1.8316679
300
+0.7723975
+0.4708818
+0.012250672
-0.001332937
1.8023971
315
+0.6691423
+0.6125550
+0.016656295
+0.002594179
1.8040170
330
+0.5508294
+0.7487960
-f 0.018894170
+0.008386104
1.8371831
345
+0.4092814
+0.8824106
+0.018479728
+0.014979907
1.9041728
2,
-0.5559282
-0.3477320
+0.405702029
+0.226988212
31.8406236
2 2
-0.5559251
-0.3477313
+0.405701406
+0.226989461
31.8406376
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt]oo = -
97592.111m' n4
.9894147
[dxldtiw = -8920493.9 TO' n 6
.9503889
[di/dt] M = +
71223.820 TO' p4
.8526253
[dQ/(ft]oo = +
673299.06 TO' p5
.8282080
[dirldt} w = -8919313.6 TO' 6
.9503315
[dL/dt]oo = +5590689.3 m' p 6
.7474654
OF THE ORBITS OF THE FOUR INNER PLANETS.
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[de/d/Joo = -0.013012279
= -1.1893992
= +0.0094965089
= +0.089773204
= -1.1892420
[dL/dt] w = +0.74542525
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt]^ -0.01304
-0.01301
-0.013012
e[dir/dt]oo -0.00810
-0.00814
-0.008138
[di/dt}^ +0.00950
+0.00949
+0.009497
sin i [dfl/d<]oo +0-00529
+0.00531
+0.005301
[dL/dfloo +0.747
+0.745425
NOTES.
This computation was originally made with but twelve points of division, but
it was found that, notwithstanding the small eccentricity of the orbit of Venus, the
values of e' and I are here so large that the tests which arise by comparing the sums
of the functions were, toward the close of the computation, entirely inapplicable.
The sums for [de/dt\ 00 agreed to but a single significant figure, while those for [dx/dt] o,
[di/dt] 00 , and [dQ/di\ 00 agreed to but two. It will be noticed that the increase of the
number of points of division almost wholly removes the discrepancy.
Notwithstanding the entire disagreement of the test equations when but twelve
points of division were employed, it is evident that this number would have been
sufficient. The greatest error would have occurred with [dLjdt] OQ , its amount being
0".00000016, showing that with this coefficient the sum of all terms of an order
higher than the 12th amounts to but 1 /4000000th of the remaining terms.
100
THE SECULAR VARIATIONS OF THE ELEMENTS
.E
30
60
90
120
150
180
210
240
270
300
330
A
1.53711333
1.52932317
1.51985488
1.51125435
1.50583081
1.50503256
1.50906482
1.51684286
1.52628659
1.53487476
1.54031063
1.54113325
9.13846106*
9.13846095
ACTION OF THE
EARTH ON VENUS.
B cos t
B am t
1000 Xff
+0.64403672
+0.34841048
0.03414346
+0.38337105
+0.61703357
0.10708849
+0.02331740
+0.71967967
0.14568124
-0.33964844
+0.62884478
0.11122751
-0.60827014
+0.36886855
0.03827088
-0.71057033
+0.00941083
0.00002491
-0.61913831
-0.35321171
0.03509098
-0.35847281
-0.62183471
0.10876150
+0.00158081
-0.72448100
0.14763156
+0.36454672
-0.63364618
0.11293247
+0.63316838
-0.37366983
0.03927365
+0.73546881
-0.01421211
0.00005681
+0.07469486f
-0.01440384t
0.44009177
+0.07469500
-0.01440382
0.44009169
1.00234463
1.00372554
1.00314070
1.00120600
0.99983131
1.0003553T
1.00226402
1.00370935
1.00326747
1.00133527
0.99985008
1.00036791
6.01069819
6.01069936
E
G
G'
G"
0.53448743
1.00227181
0.53462398
0.00006372
46
55
3.02
30
0.52531637
1.00350237
0.52574251
0.00020298
46
22
33.07
60
0.51643292
1.00284204
0.51701255
0.00028098
45
53
55.40
90
0.50976708
1.00097979
0.51021108
0.00021779
45
33
45.30
120
0.50571824
0.99975382
0.50587139
0.00007567
45
20
44.51
150
0.50439599
1.00035525
0.50439608
0.00000005
45
14
29.83
180
0.50651954
1.00219338
0.50665928
0.00006911
45
19
11.90
210
0.51285225
1.00348844
0.51328431
0.00021116
45
39
53.44
240
0.52273786
1.00296095
0.52332564
0.00028127
46
15
21.69
270
0.53325823
1.00109414
0.53371072
0.00021137
46
54
18.41
300
0.54017928
0.99976461
0.54033745
0.00007270
47
19
22.49
330
0.54048407
1.00036779
0.54048430
0.00000011
47
18
38.53
2,
3.12607525
6.00978661
3.12783029
0.00084345
277
3
39.01
2 2
3.12607397
6.00978778
3.12782900
0.00084346
277
3
38.58
* 6a s + 3aV + 6[o' ! - 2fcaa'ee' cos A'] = 9.13846101.
t 6[a'V - koa'e cos A'] = + 0.07469471.
I - 6k'aa' cos <p' e sin A' = - 0.04440383.
OF THE ORBITS OF THE FOUR INNER PLANETS.
101
ACTION
OF
THE EARTH ON
VENUS.
E
logtfo
log Lo'
logJVo
\ogN
logP
logQ
q
.26000336
0.60598257
0.54577154
9
.8305330
0.4344894
0.3752914
30
o
.25288171
.59719127
0.53614961
9.8233230
0.4173018
0.3578664
60
.24673133
0.58958546
0.
52781896
9
.8197364
0.4066135
0.3462012
90
.24246622
0.58430371
0.
52203035
9.8196999
0.4029640
0.3412105
120
.23974361
0.58092896
0.51833025
9.8208350
0.4019120
0.3392393
150
.23844519
0.57931865
0.
51656434
9
.8213596
0.4003696
0.3377696
180
.23942219
0.58053038
0.
51789317
9.8218870
0.4004544
0.3387987
210
.24375790
0.58590391
0.
52378440
9.8244982
0.4071947
0.3466789
240
.25132635
0.59526901
0.
53404476
9.8301980
0.4226554
0.3628370
270
.25983876
0.60577958
0.
54554439
9.8370023
0.4416487
0.3819851
300
.26543283
0.61267386
0.
55308969
9
.8405753
0.4533906
0.3937357
330
.26526804
0.61247091
0.
55286781
!)
.8378789
0.4500302
0.3905869
2, .
1
.50265967
3.56497024
3.
19694837
8.9637646
2.5195152
2.1561032
2 2
1
.50265782
3.56496803
3.
19694594
8.9637618
2.5195089
2.1560973
E
logF
j.
J,
J
i
F,
0.3752596
0.9974825
+0.004202020
-0.047236424
-0.005832177
30
0.3577653
0.9964541
+0.010025286
-0.058127293
-0
.010328762
60
0.3460609
0.9970986
+0.013405599
-0.053273840
-0.012046994
90
0.3411015
0.9987570
+0.012206904
-0.033976547
-0.010526476
120
0.3392013
0.9997616
+0.006516023
-0.005406098
-0
.006174632
150
0.3377695
0.9991244
-0.001172845
+0.024782063
-0
.000157532
180
0.3387641
0.9974877
-0.007608410
+0.048499056
+0
.005912547
210
0.3465734
0.9964670
-0.010818504
+0.059389945
+0.010409132
240
0.3626968
0.9970681
-0.010873469
+0.054536468
+0.012127366
270
0.3818798
0.9987050
-0.008962465
+0.035239146
+0.010606846
300
0.3936994
0.9997505
-0.005884342
+0.006668715
+0.006255003
330
0.3905868
0.9991427
-0.001520954
-0.023519435
+0.000237902
2,
2.1556821
5.9886490*
-0.000242579
+0.003787877
+0
.000241113
2 2
2.1556763
5.9886502
-0.000242578
+0.003787879
+0
.000241110
*
- 1 (t)
r/ _ G") =
5.9878055.
- ^ (J
Y _ G") .=
5.9878067.
102
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF THE EARTH ON VENUS.
E
1000 X F,
So
1000 X So
TF
B<"
S<">
+0.20122394
0.42126107
-5.890097
-0.11153496
0.0000000
-0.008199111
30
+0.06017487
0.40675651
-4.150418
-0.13232119
+0.2828447
-0.005772122
60
-0.29485277
0.40441131
-0.984602
-0.11894044
+0.4858527
-0.001365876
90
-0.50823894
0.41101802
+0.151227
-0.07580674
+0.5682286
+0.000209069
120
-0.36563277
0.41646740
-1.349097
-0.01272800
+0.4969244
-0.001858754
150
-0.00856116
0.41403810
-2.948800
+0.05391799
+0.2845157
-0.004052670
180
+0.20680805
0.40710579
-1.730893
+0.10632289
0.0000000
-0.002376680
210
+0.06559223
0.40536093
+2.554045
+0.13208012
-0.2785530
+0.003510140
240
-0.29105383
0.41526071
+7.029019
+0.12494379
-0.4954847
+0.009684416
270
-0.50707628
0.43289840
+7.732079
+0.08349760
-0.5984781
+0.010689527
300
-0.36741788
0.44451245
+3.199250
+0.01546615
-0.5340294
+0.004438118
330
-0.01281582
0.43893384
-3.068000
-0.05784756
-0.3052198
-0.004266768
s,
-0.91092526
2.50901873
+0.273580
+0.00352943
-0.0467370
+0.000322113
S 2
-0.91092510
2.50900580
+0.270133
+0.00352022
-0.0466619
+0.000317176
_ [fl sin v
+ (coav+coaE)S a ]
-0.01178019
30 +0.19740436
60 +0.35044516
90 +0.41100745
120 +0.36078900
150 +0.21090709
180 +0.00346179
210 -0.20590981
240 -0.36545661
270 -0.43294230
300 -0.38308894
330 -0.22607885
S, -0.04562979
2 2 -0.04561206
sin <f\Ai ( '' + coa <fB
-
Ro cos v
.('-sec? +\\smvsl
\o see *" / S " J
-0.42126107
-0.35572418
-0.20183130
+0.00311509
+0.20803111
+0.35633148
+0.40710579
+0.34919573
+0.19760050
-0.01250179
-0.22551772
-0.37629530
-0.03587269
-0.03587897
= + 0.00000000814.
TF.COSM
Wt, sin u
-2-flo
a
-0.065352939
-0.09038273
-0.83675673
-0.013079792
-0.13167315
-0.80869188
+0.049268295
-0.10825646
-0.80605518
+0.061732775
-0.04399687
-0.82203004
+0.012668751
-0.00122668
-0.83578481
-0.049130989
-0.02221027
-0.83298365
-0.062298971
-0.08615914
-0.81978340
-0.013955080
-0.13134085
-0.81552660
+0.050403535
-0.11432600
-0.83336328
+0.067326200
-0.04938656
-0.86579680
+0.015375410
-0.00167292
-0.88598306
-0.052873488
-0.02346769
-0.87266531
+0.000064081
-0.40202393
-5.01772646
+0.000019626
-0.40207539
-5.01770028
OF THE ORBITS OF THE FOUR INNER PLANETS. 103
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] w = - 16017!410w' n 4.2045923
[dx/dtlw = -1840673.3 m' n 6.2649767
[di/dt] 00 = + 14.695 m' p 1.1671802
[dQ/<ft] M = -2385136.3 TO' n 6.3775132
[ArAftJoo = -1844854.1 TO' n 6.2659621
[dL/dt] m = -1765973.3 m' n 6.2469841
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] w = -0^04898290
[d x /dt}w = -5.6289701
[di/dt] m = +0.000044940
[dn/dt}^ = -7.293993
[dw/dt] m = -5.6417558
[dL/dt] m = -5.4005288
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt]oo
-0.04875
-0.04896
-0.048982
e[dTr/dt]oo
-0.03873
-0.03852
-0.038607
[dildt]oo
+0.00006
+0.00004
+0.0000449
sin i [dtt/dt]w
-0.43154
-0.43169
-0.431698
[dL/dt] m
-5.397
-5.4005
NOTES.
The close agreement of the sums of the functions toward the beginning of the
computation is here caused by the smallness of the term a'V; the ratio of the major
axes is, however, so large that the expansion of the perturbing function is not very
rapidly convergent. The greatest error arising from a division of the orbit into but
six parts would here occur with the coefficient [dttjdiloa, its amount being 0".0004,
which is l/16000th of the whole.
104
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF
E
A
B cos e
2.6510232
-0.78427108
30
2.6356206
-0.87195528
60
2.6766222
-0.66613155
90
2.7630522
-0.22195070
120
2.8717556
+0.34156979
150
2.9736020
+0.87343460
180
3.0412895
+ 1.23113141
210
3.0566810
+ 1.31881567
240
3.0156549
+ 1.11299237
270
2.9292114
+0.66881112
300
2.8205208
+0.10529076
330
2.7187000
-0.42657423
2,
17.0768662*
+ 1.34058170f
2
17.0768672
+ 1.34058118
E
I
G
O
0.32836475
2.30159781
30
0.31341820
2.30194311
60
0.35238060
2.30214618
90
0.43608420
2.30201186
120
0.54393270
2.30165423
150
0.64776765
2.30144287
180
0.71834405
2.30164190
210
0.73432570
2.30208690
240
0.69089570
2.30230785
270
0.60177725
2.30207736
300
0.49293070
2..30165418
330
0.39347570
2.30144156
2,
3.12684850
13.81100215
2,
3.12684870
13.81100366
* 6a 2 + 3aV + 6[a' 2 -
Zkaa'ee' cos K] = 17.076!
t 6[a'V - kaa'e cos K]
= + 1.34058156.
t 6fc'aa' cos <p' e sin
K' = - 0.0182328.
MARS ON VENUS.
/.' sin .
+0.4410282
-0.1200582
-0.6497891
-1.0062238
-1.0938564
-0.8892048
-0.4471057
+0.1139805
+0.6437115
+ 1.0001462
+ 1.0877783
+0.8831271
-0.0182332J
-0.0182329
G'
0.33433455
0.31388465
0.36444375
0.46014840
0.56837725
0.66243750
0.72188175
0.73455360
0.69835610
0.62106785
0.51868770
0.41360955
3.20608110
3.20570155
0.003928189
0.000291101
0.008527164
0.020447915
0.024164644
0.015968474
0.004037199
0.000262374
0.008368400
0.020201647
0.023896846
0.015750931
0.072922442
0.072922442
G"
0.00510484
0.00040288
0.01016345
0.01930383
0.01847156
0.01047414
0.00242983
0.00015516
0.00520478
0.01412953
0.02001680
0.01654686
0.06139126
0.06101240
2.30246275
2.30200670
2.30404590
2.30677230
2.30762720
2.30563865
2.30274975
2.30215960
2.30456350
2.30723845
2.30739440
2.30502860
13.82884350
13.82884430
o /
22 33
21 40
23 41
27 1
30 11
32 38
34 6
34 23
33 30
31 34
28 47
25 31
172 54
172 51
26.00
58.40
4.25
51.11
38.49
59.04
10.10
46.69
59.53
45.83
46.93
1.93
5.30
23.00
OF THE ORBITS OF THE FOUR INNER PLANETS.
105
ACTION OF MARS ON VENUS.
E
log A!.
log ZV
log JVo log N log /'
logQ
0.05236235
0.34221622
0.25376720 9.0799241 8.6961571
8.9706997
30
0.04824440
0.36381569
0.24772073 9.0778391 8.6903136
8.9633892
60
0.05820469
0.34986592
0.26232763 9.0871721 8.7089460
8.9854537
90
0.07644597
0.37365893
0.28892064 9.1058581 8.7480486
9.0290445
120
0.09671054
0.39993150
0.31822634 9.1294237 8.7983322
9.0821385
150
0.11444649
0.42279102
0.34867356 9.1516340 9.8464806
9.1313354
180
0.12581914
0.43738414
0.35989311 9.1660121 8.8784042
9.1634092
210
0.12819869
0.44043118
0.36327720 9.1681182 8.8842476
9.1692445
240
0.12114959
0.43139835
0.35324262 9.1574139 8.8625240
9.1475124
270
0.10649224
0.41255431
0.33228403 9.1373395 8.8203390
9.1048461
300
0.08740987
0.38789402
0.30480686 9.1137453 8.7700379
9.0527515
330
0.06771795
0.36229172
0.27622192 9.0929018 8.7249711
9.0040125
2,
0.54165618
2.34869015
1.85226376 4.7336910 2.7144014
4.4019649
2 a
0.54154574
2.34854285
1.85209808 4.7336907 2.7144005
4.4018722
E
logF
Ji'
J 2 J,
F t
S.9695199
2.3059508
+0.040158678 -0.039429682
-0.09502735
30
8.9632959
2.2999322
-0.012465787 -0.068607310
+0.02586866
60
8.9831123
2.3089841
-0.060962061 -0.080069400
+0.14000859
90
9.0246325
2.3187331
-0.093160848 -0.070744754
+0.21680880
120
9.0779450
2.3192082
-0.101285499 -0.043131936
+0.23569076
150
9.1289682
2.3119133
-0.083189540 -0.004629729
+0.19159498
180
9.1628614
2.3032758
-0.042922865 +0.034445271
+0.09633687
210
9.1692095
2.2997005
+0.009559512 +0.063622976
-0.02455913
240
9.1463381
2.3040271
+0.060251438 +0.075085208
-0.13869908
270
9.1016472
2.3135287
+0.094787774 +0.065760616
-0.21549926
300
9.0481938
2.3207258
+0.103054621 +0.038147729
-0.23438118
330
9.0002162
2.3179852
+0.082763214 -0.000354612
-0.19028541
Si
4.3879705
13.8621718*
-0.001705675 -0.014952810
+0.00392861
2 2
4.3879695
13.8617930
-0.001705688 -0.014952813
+0.00392864
*2,(J,'-G") = 13
.800780G.
SzG/i' - G") = 13
.8007806.
106
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MAKS ON VENUS.
E
1000 X F s
Ro 1000 X -So
1000 X Wo
fiw
1000 X <S<">
+ 2.8021516
0.06350844 -0.9770071
- 3.5365233
0.00000000
-1.3600097
30
- 0.4518739
0.06287361 +0.1223535
- 6.3268753
+0.04372018
+0.1701610
60
- 0.0998524
0.06461988 +1.2993923
- 7.7066718
+0.07763320
+ 1.8025651
90
+ 3.5312016
0.06860530 +2.2776450
- 7.2896438
+0.09484620
+3.1488232
120
+ 6.8298095
0.07412522 +2.6943730
4.7318401
+0.08844541
+3.7122436
150
+ 6.5062493
0.07970667 +2.2590820
- 0.1661617
+0.05477227
+3.1047600
180
+ 2.8799133
0.08340064 +1.0358414
+ 5.2294311
0.00000000
+ 1.4223079
210
- 0.4389788
0.08380114 -0.4699266
+ 9.3597858
-0.05758587
-0.6458416
240
- 0.1552801
0.08095204 -1.6671420
+ 10.5057004
-0.09659109
-2.2969487
270
+ 3.4223024
0.07610223 -2.2705920
+ 8.5365166
-0.10521063
-3.1390719
300
+ 6.6966141
0.07080065 -2.2876620
+ 4.6568417
-0.08505867
-3.1735299
330
+ 6.3844559
0.06631737 -1.8208410
+ 0.3034394
-0.04611486
-2.5323023
Zi
+ 18.9533560
0.43740687 -0.0977956
+ 4.4169380
-0.01557115
+0.1066283
S 2
+ 18.9533565
0.43740632 -0.0977209
+ 4.4170612
-0.01557271
+0.1065284
\Ro sin v
1 Ro cos v
E
1 (T \ n 1000
X W cos u 1000 X W sin u
-2-flo
+ ( COS v+ COS E)S (
+ 1 sec 2 <f + 1 1 sin vSo \
\i / J
a
-0.001954014
-0.06350844
2.072195
- 2.865834
-0.12614768
30
+0.031835191
-0.05421922
0.625404
- 6.295890
-0.12500201
60
+0.057445987
-0.02972268 +
3.192309
- 7.014411
-0.12879747
90
+0.068588125
+0.00502476 +
5.936279
4.230780
-0.13721060
120
+0.061265775
+0.04210058 +
4.709813
- 0.456037
-0.14875771
150
+0.035700934
+0.07141598 +
0.151409
+ 0.068446
-0.16035811
180
-0.002071682
+0.08340064
3.064140
- 4.237688
-0.16794274
210
-0.040838000
+ 0.07318497
0.988919
- 9.307396
-0.16859554
240
-0.068190156
+0.04377273 +
4.238101
- 9.612920
-0.16245809
270
-0.076084930
+0.00506196 +
6.883205
- 5.049116
-0.15220446
300
-0.063800118
-0.03106656 +
4.629519
- 0.503716
-0.14111682
330
-0.036506252
-0.05549212 +
0.277348
+ 0.123100
-0.13184872
Si
-0.017304208
+0.04497627 +11.633407
-24.690606
-0.87522051
2 2
-0.017304932
+0.04497633 +11.633918
-24.691636
-0.87521944
sin if J.AI'" + cos <f
B (c) = + 0.0000000018.
OF THE ORBITS OF THE FOUR INNER PLANETS. 107
DIFFERENTIAL COEFFICIENTS.
log coeff.
= - 6075.5972 m' n 3.7835890
= +2307588.8 m' p 6.3631584
[difdt] m = + 4084.7434m' p 3.6111648
[dQ/dfloo = - 146478.61 m' n 5.1657742
[dw/dt] w = +2307332.0 m' p 6.3631101
= - 307497.75 m' n 5.4878419
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[de/dt] w = -0.0019639882
[d x /dt} 00 = +0.74594759
[dt'/dfloo = +0.0013204280
[dfl/dflw = -0.047350446
[d7r/d<]oo = +0.74586465
[dL/dt] w = -0.099401232
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
sn
[de/dt]w
-0^00195
-0.00196
-0.001964
[dw/dilw,
+0.00510
+0.00510
+0.005104
[di/dtlw
+0.00131
+0.00132
+0.001320
[dtt/dt] w
-0.00280
-0.00281
-0.002802
[dLfdt] w
-0.099
-0.099401
NOTES.
The close agreement of the final sums shows that, notwithstanding the high
eccentricity of the orbit of Mars, the expansion of the perturbing function is quite
rapidly convergent for this case. The greatest error arising from a division into but
six parts would here occur with the coefficient [dQJdt]^, and would amount to
1 /50000th of the whole value of this coefficient.
108
THE SECULAR VARIATIONS OF THE ELEMENTS
E A
27.41848845
30 27.28099847
60 27.22724715
90 27.27164617
120 27.40230326
150 27.58420446
180 27.76860046
210 27.90607834
240 27.95980514
270 27.91539376
300 27.78474891
330 27.60287233
Zi 165.56119337*
2 2 165.56119353
E I
0.347678
30 0.210701
60 0.157583
90 0.202099
120 0.332232
150 0.513472
180 0.697718
210 0.835716
240 0.890108
270 0.845842
300 0.714692
330 0.532182
Zi 3.140010
2 2 3.140011
t 6[a'V - kaa'e cos K] = + 7.9087800.
| - Gfc'aa' cos <p' e sin K' = 0.1367433,
ACTION OF JUPITER ON VENUS.
B COS e
B sin t
o
h
-0.4215334
+3.3076466
0.68960365
27.007779
-1.8560261
+ 1.9912702
0.24993195
27.007266
-2.4400062
+0.1352278
0.00115264
27.006632
-2.0169963
-1.7631551
0.19594869
27.006516
-0.7003432
-3.1952096
0.64351702
27.007039
+ 1.1571588
-3.7772161
0.89930104
27.007701
+3.0577930
-3.3532271
0.70874081
27.007851
+4.4922856
-2.0368516
0.26150512
27.007330
+5.0762663
-0.1808092
0.00206064
27.006665
+4.6532564
+ 1.7175743
0.18594838
27.006520
+3.3366031
+3.1496275
0.62528757
27.007025
+ 1.4791014
+3.7316353
0.87772776
27.007658
+7.9087796f
-0.13674401
2.67036233
162.042990
+7.9087799
-0.1367430
2.67036294
162.042990
G'
G"
e
1 II
27.0068207
0.4107944
0.06215867
7
35 44.57
27.0069201
0.2483150
0.03726867
5
53 53.53
27.0066304
0.1578551
0.00027037
4
23 18.41
27.0062448
0.2334496
0.03108029
5
40 35.79
27.0061457
0.3936568
0.06053126
7
26 34.62
27.0064436
0.5728576
0.05812876
8
46 58.16
27.0068530
0.7344471
0.03573164
9
42 56.80
27.0069600
0.8475114
0.01142507
10
16 14.68
27.0066621
0.8901968
0.00008571
10
27 38.97
27.0062568
0.8541659
0.00806094
10
17 29.01
27.0061444
0.7465850
0.03101258
9
45 50.06
27.0064305
0.5886244
0.05521465
8
52 22.88
162.0392563
3.3335352
0.18979023
49
22 3.43
162.0392558
3.3449238
0.20117838
49
47 34.05
a'ee'cosK] = 165.5611934,
OF THE ORBITS OF THE FOUR INNER PLANETS.
109
ACTION
OF JUPITER ON VENUS.
E
log/fo
log Lo'
log No log N
logP
logQ
0.00574768
0.28065745
0.18470221 7.4290891
4.8448027
6.1813194
30
0.00346017
0.27761215
0.18127767 7.4282000
4.8416643
6.1774038
60
0.00191340
0.27555166
0.17896019 7.4297378
4.8423401
6.1772233
90
0.00320448
0.27727161
0.18089469 7.4332724
4.8466166
6.1822034
120
0.00551791
0.28035168
0.18435840 7.4378460
4.8533279
6.1897695
150
0.00769513
0.28324824
0.18761506 7.4422392
4.8606851
6.1974532
180
0.00942826
0.28555246
0.19020526 7.4452930
4.8667491
6.2034501
210
0.01054435
0.28703562
0.19187230 7.4462011
4.8699179
6.2064140
240
0.01094130
0.28756298
0.19246500 7.4447133
4.8693318
6.205706T
270
0.01058711
0.28709242
0.19193614 7.4412095
4.8651139
6.2015516
300
0.00952253
0.28567775
0.19034610 7.4366174
4.8583732
6.1950025
330
0.00785489
0.28346069
0.18785389 7.4321742
4.8509265
6.1876739
2i
0.04307108
1.69535398
1.12103716 4.6232965
9.1349247
7.1524707
Zi
0.04334613
1.69572073
1.12144975 4.6232963
9.1349242
7.1526998
E
log V
Ji'
J*
J 3
Ft
6.1800737
27.032601604
+0.14514043
-0.9796507
-4.3121624
30
6.1766560
27.002428557
+0.10192507
-1.0502482
-2.5960090
60
6.1772178
26.980449138
+0.02691302
-0.8390983
-0.1762960
90
6.1815797
27.031356993
-0.07055224
-0.4027785
+2.2986164
120
6.1885563
27.065863046
-0.15999110
+0.1417995
+4.1655781
150
6.1962891
27.048572013
-0.20267009
+0.6487166
+4.9243381
180
6.2027348
27.006174576
-0.17674090
+0.9821448
+4.3715859
210
6.2061853
26.976509919
-0.09316228
+ 1.0527424
+ 2.6554331
240
6.2057043
26.979786076
+0.01124440
+0.8415929
+0.2357201
270
6.2013902
27.007935305
+0.09774464
+0.4052732
-2.2391928
300
6.1943816
27.036474318
+0.11682513
-0.1393049
-4.1061538
330
6.1865682
27.045934111
+0.16010585
-0.6462222
-4.8649146
Si
7.1486685
162.101348758* -0.00660902
+0.0074833
+0.1782719
2j
7.1486684
162.112736898
-0.00660905
+0.0074833
+0.1782712
* z,(J,' - G") =
161.911558528.
2 2 (Ji' - C") =
161.911558518.
110
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF JUPITEK ON VKNUS.
E
Ft
1000 X flo 100000 X So 1000 X W 1000 X <"> 100000 X <"'
+0.06417143
1.3439836 -0.81929943 -0.14785238 0.0000000 -1.1404782
30
-0.01379964
1.3378723 -0.27202988 -0.15783804 +0.9303113 -0.3783208
60
-0.00425156
1.3439282 +0.28211779 -0.12622122 +1.6145721 +0.3913643
90
+0.08385809
1.3588647 +0.54292475 -0.06059607 +1.8786178 +0.7505883
120
+0.16287446
1.3765538 +0.50194299 +0.02305118 +1.6424891 +0.6915652
150
+0.15397825
1.3917093 +0.38822446 +0.10305711 +0.9563448 +0.5335546
180
+0.06595223
1.4018961 +0.39768404 +0.15712801 0.0000000 +0.5460578
210
-0.01357095
1.4066592 +0.47042587 +0.16914113 -0.9666180 +0.6465278
240
-0.00563629
1.4053074 +0.35504098 +0.13510528 -1.6767973 +0.4891671
270
+0.08123098
1.3964225 -0.08725072 +0.06503287 -1.9305409 -0.1206233
300
+0.15970886
1.3800824 -0.66641223 -0.02064190 -1.6580071 -0.9244718
330
+0.15112247
1.3602249 -0.99122256 -0.09822805 -0.9458544 -1.3785254
2i
+0.44281913
8.2517515 +0.05107414 +0.02056897 -0.0777432 +0.0532044
2 2
+0.44281920
8.2517529 +0.05107192 +0.02056895 -0.0777394 +0.0532012
E
1000 X[Ro sin v
+ (cosv+cosE)S<>]
1000 x| -Ro cost)
, 1000 X Wo cos u 1000 X W sin u - 2 -
/; \ . 1 (j
~\- 1 - SGC^ (f -\~\ isin v *oo I
-0.0163860
-1.3439836 -0.086632804 -0.11981268 -0.0026695736
30
+0.6682014
-1.1590573 -0.015602101 -0.15706504 -0.0026598877
60
+ 1.1706512
-0.6601482 +0.052284193 -0.11488325 -0.0026786592
90
+ 1.3587959
+0.0201574 +0.049346057 -0.03516889 -0.0027177294
120
+ 1.1829926
+0.7039968 -0.022943878 +0.00222159 -0.0027625280
150
+0.6850081
+ 1.2114930 -0.093907380 -0.04245200 -0.0027999142
180
-0.0079537
+ 1.4018961 -0.092067787 -0.12732924 -0.0028229786
210
-0.7073256
+ 1.2159042 -0.017870798 -0.16819438 -0.0028299915
240
-1.2164221
+0.7037025 +0.054502775 -0.12362395 -0.0028202318
270
-1.3963839
+0.0113009 +0.052437627 -0.03846518 -0.0027928449
300
-1.2058917
-0.6713715 -0.020520787 +0.00223277 -0.0027507212
330
-0.7013026
-1.1657067 -0.089781837 -0.03984932 -0.0027043278
2i
-0.0930097
+0.1340921 -0.115378288 -0.48119476 -0.0165046924
2 2
-0.0930067
+0.1340915 -0.115378432 -0.48119481 -0.0165046855
sin (p \A i (<) + cos (f
Bo (c) = + 0.0000000000028.
OF THE ORBITS OF THE FOUR INNER PLANETS. Ill
DIFFERENTIAL COEFFICIENTS.
u log coeff.
[de/dt] M = - 32.654970 m' n 1.5139493
[d x /dt] m = +6879.8159 TO' p 3.8375768
[di/dt] M = - 40.510972 TO' n 1.6075727
[<&/#], = -2854.6599 TO' n 3.4555544
[dir/eft]oo = +6874.8117 TO' p 3.8372608
[dL/dt] m = -5799.7390 TO' n 3.7634084
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
= -0.031162921
[dxldt]<n = +6.5654682
[di/dt] M = -0.038659982
[dQ/dt] w = -2.7242270
= +6.5606924
= -5.5347410
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
-0.03117 -0.0311629
+0.04482 +0.04491 +0.0448955
[di/dt] M -0.03865 -0.03865 -0.0386600
sin i [daAft] M -0.16114 -0.16122 -0.1612345
[dL/dt] w -5.535 -5.5347410
NOTES.
The term aV is here so large that the sums of the functions B, e, G', G", and 0,
as well as those of the functions immediately dependent upon these quantities are
in great disagreement; but, as the expansion of the perturbing function is here rapidly
convergent, the final sums agree almost exactly. The greatest effect of all terms from
the 6th to the llth orders is here produced with the coefficient [di/dt]^ and amounts
to 0".00000002.
112
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF SATURN ON VENUS.
E
A
B cos t
B sin e
g
I
92.09886822
+ 10.392665
+4.3401600
5.386574
90.704833
30
91.77434432
+ 7.489741
+6.4166597
11.773870
90.705340
60
91.37861335
+ 3.936959
+6.7658078
13.090030
90.705178
90
91.01772022
+ 0.686282
+5.2940524
8.014522
90.704527
120
90.78837022
1.391274
+2.3957451
1.641279
90.704015
150
90.75201312
- 1.739027
-1.1525126
0.379833
90.704174
180
90.91838168
- 0.263800
-4.3999687
5.536055
90.704848
210
91.24289345
+ 2.639122
-6.4764687
11.994380
90.705369
240
91.63859982
+ 6.191904
-6.8256188
13.322485
90.705222
270
91.99948069
+ 9.442580
-5.3538604
8.196630
90.704548
300
92.22884291
+ 11.520135
-2.4555545
1.724250
90.704026
330
92.26522467
+ 11.867891
+ 1.0927030
0.341433
90.704165
S,
549.05167620*
+30.386589f
-0.179429U
40.700673
544.228122
*}
549.05167647
+30.386589
-0.1794266
40.700668
544.228123
E
I
G
G'
G"
e
i a
+ 1.108078
90.7041702
1.1599384
0.0511977
6
38 1.200
30
+0.783048
90.7038960
0.9248448
0.1403539
6
12 59.177
60
+0.387477
90.7035801
0.6213419
0.2322660
5
33 35.601
90
+0.027235
90.7035526
0.3116929
0.2834826
4
38 20.618
120
-0.201603
90.7038159
0.0673334
0.2687363
3
29 4.472
150
-0.238118
90.7041279
0.0164527
0.2545245
3
7 43.816
180
-0.072423
90.7041756
0.2137666
0.2855179
4
14 53.293
210
+0.251567
90.7039071
0.5115373
0.2585083
5
16 44.963
240
+0.647420
90.7035910
0.8267173
0.1776657
6
2 4.051
270
+ 1.008976
90.7035405
1.0926850
0.0827021
6
32 0.533
300
+ 1.238859
90.7038135
1.2542286
0.0151565
6
47 36.301
330
+ 1.275102
90.7041229
1.2780897
0.0029452
6
49 30.415
s,
+3.107808
. 544.2231463
4.1433262
1.0305401
32
45 14.918
2 2
+3.107809
544.2231470
4.1353023
1.0225166
32
37 19.522
* 6o 2 + 3aV + 6[a' 2 -
Zkaa'ee' cos K\ = 549.05167622.
t 6[a'V - kaa'e cos K\
= + 30.386587.
J 6fc'aa' cos if' e sin
K' = - 0.1794290.
OF THE ORBITS OF THE FOUR INNER PLANETS.
113
ACTION OF SATURN ON VENUS.
E
logtfo
log /V log No log N log P
logQ
0.00437971
0.27883659 0.18265468 6.6396204 3.0027123
4.8644028
30
0.00384466
0.27812417 0.18185352 6.6392492 3.0007786
4.8628053
60
0.00307377
0.27709752 0.18069890 6.6400076 2.9996349
4.8619714
90
0.00213852
0.27585161 0.17929758 6.6416827 2.9995754
4.8620009
120
0.00120578
0.27460866 0.17789947 6.6438207 3.0006087
4.862S099
150
0.00097200
0.27429705 0.17754895 6.6458518 3.0024609
4.8645568
180
0.00179285
0.27539102 0.17877951 6.6472415 3.0046480
4.8670287
210
0.00277061
0.27669370 0.18024473 6.6476233 3.0065929
4.8690060
240
0.00362228
0.27782805 0.18152050 6.6468912 3.0077706
4.8699374
270
0.00424801
0.27866126 0.18245753 6.6452313 3.0078525
4.8696688
300
0.00459387
0.27912172 0.18297533 6.6430830 3.0068083
4.8683601
330
0.00463699
0.27917912 0.18303987 6.6410259 3.0049227
4.8664247
Si
0.01866826
1.66288356 1.08452839 9.8606643 8.0221827
9.1945101
Y o
0.01861079
1.66280691 1.08444218 9.8606642 8.0221829
9.1944623
E
logF
J,' J 2 J 3
F 2
4.8640969
90.64262161 +0.26581387 -3.1529304
-22.089684
30
4.8619670
90.77821079 +0.41797337 -2.4126056
-32.658241
60
4.8605842
90.92403274 +0.41481456 -1.0167115
-34.435270
90
4.8603076
90.98237184 +0.27423234 +0.6607234
-26.944615
120
4.8612040
90.92121232 +0.07649557 +2.1702330
-12.193388
150
4.8630356
90.85355551 -0.10022240 +3.1073439
+ 5.865829
180
4.8653231
90.87694181 -0.22507782 +3.2209585
+22.394088
210
4.8674621
90.89557561 -0.30595380 +2.4806343
+ 32.962649
240
4.8688763
90.86858755 -0.34689500 +1.0847396
+34.739680
270
4.8691748
90.78219477 -0.32168268 -0.5926962
+27.249019
300
4.8682696
90.66900008 -0.19522740 -2.1022048
+ 12.497792
330
4.8664071
90.60246403 +0.02557719 -3.0393154
- 5.561423
2j
9.1883540
544.90239611* -0.01007622 +0.2040844
+ 0.913218
2 2
9.1883541
544.89437255 -0.01007598 +0.2040844
+ 0.913218
* 2,(J,' - G") =
543.87185601.
2 2 (J,' - G") =
543.87185600.
114
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF SATURN ON VENUS.
E
F 3 1000 X Ro 100000 X S 1000 X W 1000 X B (n)
100000 X <S<>
-0.15178108 0.21824849 -0.02789044 -0.023072725 0.00000000
-0.03882394
30
-0.76855930 0.21811010 -0.02299898 -0.017634117 +0.15166640
-0.03198543
60
-1.16657105 0.21852784 -0.04315220 -0.007491896 +0.26253554
-0.05986233
90
-0.94659087 0.21909707 -0.07037836 +0.004695343 +0.30289965
-0.09729742
120
-0.32871848 0.21963056 -0.06653491 +0.015732681 +0.26206084
-0.09167023
150
+0.06775179 0.22023472 -0.01412236 +0.022675427 +0.15133936
-0.01940900
180
-0.15599309 0.22103492 +0.06128516 +0.023605169 0.00000000
+0.08415031
210
-0.77884429 0.22186276 +0.10918347 +0.018202966 -0.15245807
+0.15005585
240
-1.18017310 0.22225231 +0.09717617 +0.007900358 -0.26518905
+0.13388702
270
-0.95986467 0.22180036 +0.03944793 -0.004483114 -0.30663694
+0.05453639
300
-0.33810816 0.22057741 -0.01719598 -0.015556236 -0.26499777
-0.02385490
330
+0.06476239 0.21916657 -0.03744378 -0.022338577 -0.15240100
-0.05207429
2,
-3.32134496 1.32027153 +0.00368780 +0.001117351 -0.00559044
+0.00382593
-"2
-3.32134495 1.32027158 +0.00368792 +0.001117928 -0.00559070
+0.00382610
E
m v [ff ,' 100 Xf-Socost;
J p T 1000 X TFo cos u 1000 X TFo sin u
1000X-2-Ro
1 + ^sec z Y,+ lJsin!%Sol
a
-0.00055781 -0.21824849 -0.013519262 -0.018697064
-0.43351010
30
+0.10930468 -0.18874417 -0.001743111 -0.017547753
-0.43363510
60
+0.18946670 -0.10888722 +0.003103342 -0.006818927
-0.43556030
90
+0.21909678 +0.00009174 -0.003823625 +0.002725091
-0.43819414
120
+0.19022138 +0.10978821 -0.015659442 +0.001516257
-0.44076418
150
+0.10971092 +0.19096268 -0.020662231 -0.009340619
-0.44307990
180
-0.00122570 +0.22103492 -0.013831594 -0.019129016
-0.44509495
210
-0.11216824 +0.19142744 -0.001923256 -0.018101079
-0.44635520
240
-0.19279204 +0.11058265 +0.003187081 -0.007228982
-0.44602556
270
-0.22179790 +0.00072885 -0.003614847 +0.002651640
-0.44360072
300
-0.19184805 -0.10885437 -0.015464964 +0.001682669
-0.43964535
330
-0.11088191 -0.18905102 -0.020417779 -0.009062352
-0.43573540
Si
-0.00673552 +0.00541570 -0.052184839 -0.048675063
-2.64060044
2 2
-0.00673567 +0.00541552 -0.052184849 -0.048675069
-2.64060046
sin <f Mi w + cos <p Bo w = + 0.00000000000029.
OF THE ORBITS OF THE FOUR INNER PLANETS. 115
DIFFERENTIAL COEFFICIENTS.
n
log coeff.
[de/dt]w
= - 2.3648522 TO'
n 0.3738040
[dx/dt] w
= +277.85744 TO'
p 2.4438220
[di/dt] w
18.322835 TO'
n 1.2629927
[dQ/dfloo
= -288.76199 TO'
n 2.4605400
[dir/dt]oo
= +277.35124 TO'
p 2.4430301
[dL/dt] w
= -927.63054 TO'
n 2.9673751
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] w = -0.00067536338
[dxldt] M = +0.079351564
= -0.0052327048
= -0.082465731
= +0.079207000
[dL/dt] w = -0.26491624
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
-0.00067
-0.00067
-0.00067536
+0.00055
+0.00054
+0.00054202
-0.00523
-0.00523
-0.00523270
-0.00489
-0.00488
-0.00488077
-0.265
-0.26491624
[di/dt] m
sin i [dQ/dt}oo
[dL/dt]
NOTES.
As in the previous case, the considerable disagreement of the sums of the functions
near the beginning of the computation nearly disappears as the work progresses, show-
ing that the convergence of the expansion of the perturbing function is here very
rapid. The greatest error which would have arisen from the neglect of all terms from
the 6th to the llth orders would have here occurred with the coefficient [dx/dt] 00 and
would have amounted to 1 /70000th part of the remaining terms.
116
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF URANUS ON VENUS.
E
A
B cos (
/ >' sin i
9
i
369.8294733
28.016529
- 8.627090
60.35594
367.496220
30
370.1021929
30.912376
- 2.059265
3.43885
367.496155
60
370.0319613
30.136215
+ 5.076263
20.89679
367.496140
90
369.6376057
25.896024
+ 10.867539
95.77536
367.496165
120
369.0247981
19.327964
+ 13.762797
153.60480
367.496245
150
368.3577351
12.191933
+ 12.986248
136.75991
367.496230
180
367.8151465
6.400029
+ 8.745974
62.03086
367.496275
210
367.5424142
3.504185
+ 2.178150
3.84740
367.496245
240
367.6126214
4.280346
- 4.957378
19.92946
367.496135
270
368.0069649
8.520535
- 10.748653
93.69136
367.496165
300
368.6197847
15.088597
-13.643909
150.96247
367.496240
330
369.2868724
22.224622
-12.867361
134.26735
367.496240
^
2212.9337853*
103.2496801
+ 0.356657}
467.78032
2204.977255
^
2212.9337852
103.249675
+ 0.356658
467.78025
2204.977200
E
G
G'
G"
+ 1.522310
367.495771
1.6238940
0.1011370
O
3
55
40^93
30
+ 1.795095
367.496129
1.8003166
0.0051977
4
1
9.46
60
+ 1.724880
367.495985
1.7573874
0.0323564
4
5.50
90
+ 1.330495
367.495453
1.5044398
0.1732316
3
52
23.80
120
+0.717605
367.495105
1.0990547
0.3803068
3
38
8.91
150
+0.050560
367.495217
0.6363646
0.5847917
3
18
7.21
180
-0.492075
367.495816
0.2329563
0.7245710
2
55
22.89
210
-0.764775
367.496217
0.0134531
0.7782012
2
39
26.72
240
-0.694455
367.495988
0.0708724
0.7651843
2
43
51.72
270
-0.300145
367.495472
0.3769271
0.6763791
3
3
57.85
300
+0.312600
367.495121
0.8167025
0.5029833
3
25
59.41
330
+0.979685
367.495243
1.2686692
0.2879853
3
43
48.64
2,
+3.090865
2204.973786
5.6008673
2.5065388
20
39
9.36
2 2
+3.090915
2204.973731
5.6001704
2.5057866
20
38
53.68
* 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos K] = 2212.9337852.
t 6[a'V - kaa'e cos K] = + 103.249685.
t 6fc'aa' cos <p' e sin A" = + 0.356657.
OF THE ORBITS OF THE FOUR INNER PLANETS.
117
ACTION OF UKANUS ON VENUS.
E
logtfo
log!/ logJV log AT logP
logQ
0.00153256
0.27504417 0.17838936 5.7255240 0.8698244
3.3385415
30
0.00160469
0.27514027 0.17849746 5.7265670 0.8711895
3.3398056
60
0.00159052
0.27512139 0.17847622 5.7286906 0.8732303
3.3418760
90
0.00149009
0.27498757 0.17832569 5.7313185 0.8753928
3.3441876
120
0.00131281
0.27475131 0.17805993 5.7337420 0.8770917
3.3461011
150
0.00108264
0.27444453 0.17771485 5.7353151 0.8778752
3.3470878
180
0.00084825
0.27413211 0.17736340 5.7356234 0.8775397
3.3468789
210
0.00070103
0.27393587 0.17714264 5.7345896 0.8761825
3.3455607
240
0.00074042
0.27398839 0.17720173 5.7324870 0.8741636
3.3435328
270
0.00093336
0.27424555 0.17749102 5.7298710 0.8720155
3.3413115
300
0.00117044
0.27456155 0.17784648 5.7274385 0.8703090
3.3394395
330
0.00138191
0.27484340 0.17816353 5.7258446 0.8695041
3.3384162
Si
0.00719500
1.64759892 1.06733712 4.3835054 5.2421585
0.0563696
2 2
0.00719372
1.64759719 1.06733519 4.3835056 5.2421594
0.0563691
E
logF
JY J 2 J,
ft
3.3383922
367.09942052 -0.77238411 -13.377206
+ 148.77538
30
3.3397979
366.74089875 -0.18040606 -16.560521
+ 35.51232
60
3.3418282
366.88453611 +0.52246333 -15.273725
- 87.54088
90
3.3439319
367.40160988 +0.87586900 - 9.861620
-187.41225
120
3.3455398
367.86665960 +0.72768333 - 1.774376
-237.34142
150
3.3462248
367.95568703 +0.32642368 . + 6.821041
-223.94969
180
3.3458097
367.72285452 +0.04295564 +13.621489
-150.82559
210
3.3444123
367.51508313 +0.01342241 +16.804806
- 37.56251
240
3.3424036
367.61069788 +0.04551264 +15.518012
+ 85.49071
270
3.3403134
367.89469814 -0.13535069 +10.105905
+ 185.36209
300
3.3386972
367.98729038 -0.54951786 + 2.018660
+235.29114
330
3.3379911
367.66272932 -0.88324571 - 6.576756
+221.89949
2i
0.0526706
2205.17145901* +0.01671297 + 0.732854
6.15066
S 2
0.0526714
2205.17070625 +0.01671263 + 0.732855
6.15055
*2,(J,'-G") =
2202.66492021.
Zi(Ji' - G") =
2202.66491965.
118
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF URANUS ON VENUS.
E
F 3 1000 X Ro 1000000 X So 1000000 X W
1000 X fi <n)
1000000 X S ( ">
- 4.0226412 0.02650922 -0.05811066 -2.918780
0.00000000
-0.08089097
30
- 0.1770896 0.02652281 -0.01305234 -3.621493
+0.01844306
-0.01815231
60
1.6152186 0.02667654 +0.04940526 -3.356828
+ 0.03204873
+0.06853681
90
- 6.9332675 0.02691093 +0.05269544 -2.182313
+0.03720412
+0.07285096
120
-10.8407275 0.02710189 -0.01759673 -0.401343
+0.03233767
-0.02424436
150
- 9.4434565 0.02715609 -0.09660941 +1.506695
+0.01866093
-0.13277471
180
- 4.1342719 0.02707758 -0.10424233 +3.017072
0.00000000
-0.14313458
210
- 0.2013093 0.02694790 -0.02527822 +3.713882
-0.01851786
-0.03474102
240
1.5455370 0.02684358 +0.07399797 +3.412656
-0.03202946
+0.10195266
270
- 6.7883563 0.02677475 +0.10841711 +2.207472
-0.03701585
+0.14988562
300
-10.6594103 0.02669990 +0.05468656 +0.432404
-0.03207678
+0.07586323
330
- 9.2743255 0.02659629 -0.02803268 -1.439065
-0.01849416
-0.03898596
2!
-32.8178065 0.16090871 -0.00185993 +0.185181
+0.00028016
-0.00191721
2,
-32.8178047 0.16090877 -0.00186010 +0.185178
+0.00028024
-0.00191742
1000 X f Bo sin D + 1000 X [- So cos v +
r
E
, \ 1 /r \ 1 100000 X ^o C08 " 1000000 X We sin u
(cos v + cos E] So 1 (~ secV + 1 1 sin vSo \
1000 X -2- Ro
-0.00011622 -0.02650922 -1.710234
- 2.365243
-0.05265561
30
+0.01331757 -0.02293687 -0.357980
- 3.603757
-0.05273125
60
+0.02323048 -0.01311517 +1.390488
- 3.055297
-0.05317053
90
+0.02690995 +0.00028955 +1.777154
- 1.266575
-0.05382187
120
+0.02340803 +0.01365914 +0.399475
- 0.038680
-0.05438924
150
+0.01366523 +0.02346772 -1.372927
- 0.620648
-0.05463405
180
+0.00020848 +0.02707758 -1.767867
- 2.444894
-0.05452574
210
-0.01335043 +0.02340859 -0.392395
- 3.693094
-0.05421519
240
-0.02324178 +0.01343114 +1.376698
- 3.122646
-0.05387086
270
-0.02677488 -0.00003361 +1.779939
- 1.305659
-0.05354950
300
-0.02314722 -0.01330732 +0.429867
- 0.046772
-0.05321706
330
-0.01342562 -0.02295918 -1.315326
- 0.583802
-0.05287735
Zi
+0.00034177 +0.00123615 +0.118427
-11.073532
-0.32182904
2 2
+0.00034182 +0.00123620 +0.118465
-11.073535
-0.32182921
sin v Mi (t) + cos v B <c) = + 0.000000000000020.
OF THE ORBITS OF THE FOUR INNER PLANETS. 119
DIFFERENTIAL COEFFICIENTS.
n log coeff.
[defdl] M = + 0.12000343m' p 9.0791936
[dx/dt] M = + 63.424159 TO' p 1.8022547
[di/<ft]oo = + 0.04158807 TO' p 8.6189687
= -- 65.693091 TO' n 1.8175197
= + 63.308999 TO' p 1.8014655
[dL/dt] m = -113.109825 TO' n 2.0535003
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'
[dejdt] m = +o'.0000052633084
[dx/dt}^ = +0.0027817616
[di/dtloo = +0.000001824038
= -0.0028812762
= +0.0027767109
= -0.0049609570
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] w +0^00000 +o!o0001 +o!o00005263
e[dirldt] M +0.00002 +0.00002 +0.000019001
[dildt] m +0.00000 +0.00000 +0.000001824
sin i \daldi\m -0.000165 -0.00017 -0.000170530
NOTES.
That a division into eight parts is here fully sufficient is shown by the agreement
of the final sums. Thus the greatest effect produced by all terms from the 4th to the
7th order is seen to occur with the coefficient [dx/dt] o and to amount to but
0".00000004.
120
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF NEPTUNE ON VENUS.
E
A B cos f B sin
h
904.77843877 + 9.109428 +21.528911 30.253556
904.17419
45
904.51260261 - 6.657698 +16.194152 17.117826
904.17356
90
904.39237664 -14.030230 + 1.286186 0.107979
904.17298
135
904.48820486 - 8.689452 -14.462107 13.651972
904.17345
180
904.74393521 + 6.236084 -21.825589 31.093114
904.17407
225
905.00974680 +22.003203 -16.490829 17.750770
904.17359
270
905.12994825 +29.375736 - 1.582864 0.163538
904.17281
315
905.03414463 +24.034972 +14.165428 13.097596
904.17334
2i
3619.04469887* +30.691018f - 0.593356f 61.618187
3616.69405
2,
3619.04469890 +30.691025 - 0.593356 61.618164
3616.69394
E
1 G G' G"
e
O / //
0.53898 904.17415 0.5952295 0.0562134
1 32 17.018
45
0.27377 904.17354 0.3309885 0.0571984
1 11 14.024
90
0.15413 904.17298 0.1548950 0.0007710
45 6.502
135
0.24949 904.17343 0.2998555 0.0503537
1 7 39.561
180
0.50459 904.17403 0.5654480 0.0608162
1 30 28.921
225
0.77088 904.17357 0.7955805 0.0246764
1 43 33.466
270
0.89187 904.17281 0.8920680 0.0002028
1 48 0.658
315
0.79553 904.17332 0.8133615 0.0178097
1 44 14.706
2,
2.08957 3616.69397 2.2076405 0.1180034
5 35 53.099
2 2
2.08967 3616.69386 2.2397860 0.1500382
5 46 41.757
ACTION OF NEPTUNE ON VENUS.
E
log K log La' log No log N log P
logQ
0.00023476 0.27331427 0.17644338 5.1378654 9.4986215
2.3580297
45
0.00013986 0.27318776 0.17630105 5.1395214 9.5001506
2.3595431
90
0.00005608 0.27307605 0.17617538 5.1436917 9.5042639
2.3636152
135
0.00012618 0.27316951 0.17628052 5.1479186 9.5085363
2.3679232
180
0.00022568 0.27330217 0.17642976 5.1497409 9.5104807
2.3698895
225
0.00029563 0.27339543 0.17653468 5.1481063 9.5089743
2.3683773
270
0.00032162 0.27343007 0.17657365 5.1439578 9.5048849
2.3642800
315
0.00029957 0.27340068 0.17654059 5.1397097 9.5005900
2.3599901
Si
0.00083814 1.09312256 0.70562217 0.5752556 8.0182509
9.4558143
2 2
0.00086124 1.09315338 0.70565684 0.5752558 8.0182511
9.4558336
* 4o 2 + 2aV + 4[a' J - Zkaa'ee' cos K] = 3619.04469884.
f 4[a'V - kaa'e cos A'] = + 30.691024.
t - 4k'aa' cos *' e sin A'' = - 0.593359.
OF THE ORBITS OF THE FOUR INNER PLANETS.
121
ACTION OF NEPTUNE ON VENUS.
E
logF
Jl' J2
J>
Ft
2.3579959
902.1174996 +0.0278169
-43.343250
-165.19752
45
2.3595087
903.0233587 +1.1892143
-32.830864
-124.26237
90
2.3636147
904.1640829 +0.1518779
- 2.911684
9.86927
135
2.3678929
903.3068137 -1.1744691
+28.888013
+ 110.97189
180
2.3698530
902.1221024 -0.3409776
+43.940408
+ 167.47400
225
2.3683624
902.9705135 +0.9081699
+33.428015
+ 126.53886
270
2.3642798
904.1592634 +0.1562063
+ 3.508843
+ 12.14577
315
2.3599794
903.2945923 -0.9279474
-28.290864
-108.69539
Si
9.4557434
3612.5629483* -0.0050765
+ 1.194317
+ 4.55298
2s
9.4557434
3612.5952782 -0.0050323
+ 1.194300
+ 4.55299
E
Pi
100000 X Bo 100000000 X So 1000000 X Wo
100000 X B<>
1000000 X S<>
+0.5839042
0.6821434 -0.4573134 -0.9883466
0.0000000
-0.006365877
45
-3.9412736
0.6867470 +2.3281642 -0.7513860
+0.6746065
+0.032343148
90
-0.4777116
0.6961065 +0.3193229 -0.0672756
+0.9623607
+0.004414608
135
+4.0782664
0.7008255 -2.3820088 +0.6740551
+0.6818058
-0.032772474
180
+0.6001079
0.7010447 -0.2565237 +1.0297346
0.0000000
-0.003522313
225
-4.0073139
0.7005531 +2.5294517 +0.7805505
-0.6815408
+0.034801040
270
-0.5873116
0.6967370 +0.4002346 +0.0811602
-0.9632323
+0.005533205
315
+3.9893101
0.6879052 -2.4698966 -0.6479491
-0.6757442
-0.034312118
Si
+0.1189889
2.7760316 +0.0057204 +0.0552726
-0.0008716
+0.000059623
y
+0.1189890
2.7760308 +0.0057105 +0.0552705
-0.0008727
+0.000059596
E
inonm v r ; , 100000 X -Bo cos w
ItAAJlH) X l/VO Sin V innnnn ^, lir
, , . _, 100000 X Wo cos u
+ (cos v+cos)S ] , IT 2 N . 1
\a sec *" r l " J
100000 X W sin u
1000 X - 2 - Bo
a
-0.00091463
-0.68214344 -0.057911307
-0.080091019
-0.013549511
45
+ 0.49123773
-0.47994176 +0.012282960
-0.074127847
-0.013668483
90
+0.69608806
+0.00540218 +0.005478547
-0.003904555
-0.013922131
135
+0.49653747
+0.49458425 -0.066499046
-0.011017258
-0.014084333
180
+ 0.00051305
+0.70104468 -0.060336403
-0.083444904
-0.014116841
225
-0.49655471
+0.49418261 +0.012013897
-0.077124930
-0.014078857
270
-0.69672347
+0.00396738 +0.006544151
-0.004800407
-0.013934740
315
-0.49226067
-0.48055578 -0.064023059
-0.009971443
-0.013691533
Si
-0.00103699
+0.02827080 -0.106225012
-0.172240885
-0.055523223
S 2
-0.00104018
+0.02826932 -0.106225248
-0.172241478
-0.055523206
sin <f ^A\ M + cos
v . BO M = - 0.0000000000000085.
*2,W,'-G"1 =3i
312.4449449.
i' - G") = 3612.4452400.
122 THE SECULAR VARIATIONS OF THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] w = - - 0.0054696734 m' n 7.7379614
[dx!dt] m = +21.756678 TO' p 1.3375926
[di/dt] m = -- 0.55945727 TO' n 9.7477669
[dB/(ft]oo =-15.327159 w' n 1.1854617
[drldt] m = +21.729810 TO' p 1.3370559
[dL/dt] M = -29.268164 TO' n 1.4663954
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'
[defdt] m = -0 / .00000027764841
[dx/dt]^ = +0.0011044000
[difdt] M -0.000028398849
[dQ/dt] w = -0.00077802855
[dT/ft]oo = +0.0011030360
[dL/dt] w = -0.0014856935
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
// // //
oo -0.00000 -0.00000 -0.00000028
e[drldt] m +0.00001 +0.00001 +0.00000755
[di/dt] w -0.00004 -0.00003 -0.00002840
sin ?: [dQ/dt] M -0.00006 -0.00005 -0.00004605
NOTES.
The large disagreement of the sums of the functions near the beginning of the
computation is caused, as in previous cases, by the presence of the term a'V. The
greatest disagreement in the final sums occurs in the second column and shows that
the effect of all terms from the 4th to the 7th orders is to produce a change of
0".00000003 in the value of [d x /dt] 00 .
OF THE ORBITS OF THE FOUR INNER PLANETS.
123
EARTH.
ACTION OF MERCUKY ON THE EARTH.
E
A
B COS e
It sin e
1000 xg
h
1.25770017
+0.37398164
+0.15686177
0.15586327
1.11042819
30
1.20879374
+0.24429166
+0.30606106
0.59337027
1.06575342
60
1.14343370
+0.05583076
+0.37253457
0.87910918
1.00282212
90
1.07923573
-0.14090326
+0.33847094
0.72569217
0.93669980
120
1.03345314
-0.29319554
+0.21299723
0.28738066
0.88633014
150
1.01830180
-0.36023958
+0.02973421
0.00560044
0.86904450
180
1.03773900
-0.32407097
-0.16221302
0.16667904
0.89145124
210
1.08650443
-0.19438096
-0.31141230
0.61430099
0.94482277
240
1.15158355
-0.00592005
-0.37788591
0.90454667
1.01121409
270
1.21564099
+0.19081399
-0.34382220
0.74882000
1.07228810
300
1.26156411
+0.34310630
-0.21834847
0.30200208
1.11392105
330
1.27699637
+0.41015028
-0.03508545
0.00779765
1.12755062
2,
6.88547367*
+0.14973214f
-0.01605383J
2.69558090
6.01616683
2 2
6.88547306
+0.14973213
-0.01605374
2.69558152
6.01615921
E
G
G'
G"
0.14093752
1.11028337
0.14207046
0.00098811
21
1
34.45
30
0.13670586
1.06515341
0.14124977
0.00394390
21
37
28.51
60
0.13427712
1.00181061
0.14149060
0.00620197
22
30
21.00
90
0.13620147
0.93572981
0.14260963
0.00543817
23
21
59.90
120
0.14078854
0.88589477
0.14348474
0.00226084
23
53
48.72
150
0.14292284
0.86903562
0.14297679
0.00004507
23
55
58.88
180
0.13995330
0.89120229
0.14152378
0.00132152
23
34
54.52
210
0.13534720
0.94401808
0.14077439
0.00462250
23
2
51.29
240
0.13403500
1.01019211
0.14138996
0.00633298
22
24
31.27
270
0.13701843
1.07154031
0.14266461
0.00489838
21
43
51.55
300
0.14130860
1.11364215
0.14347758
0.00189008
21
9
39.20
330
0.14311129
1.12754360
0.14316663
0.00004830
20
52
41.84
Si
0.83130008
6.01302530
0.85343712
0.01899550
134
34
49.16
2 2
0.83130709
6.01302083
0.85344182
0.01899632
134
34
51.97
* 6a 2 + 3aV + 6[a' 2 - 2kaa'ee' cos K] = 6.8854738.
t 6[a'V - kaa'e cos K] = + 0.14973211.
| - Gfc'aa' cos <p'-e sin K' = - 0.01605375,
124
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MEKCUKY ON THE
EARTH.
E
logtfo
log LQ
logtfo
log N log P
logQ
.04527343
0.33291492
.24335185
9.
9618524 0.2031270
0.1593841
30
0.04797642
0.33646399
.24732687
9.
9917424 0.2701720
0.2100521
60
.05211516
0.34189221
.25340453
0.
0396020 0.3745624
0.2895406
90
0.05634203
0.34742859
0.25960067
0.0958412 0.4959354
0.3817747
120
.05903754
0.35095528
.26354624
0.
1435571 0.5975343
0.4586143
150
.05922392
0.35119901
.26381890
0.1631586 0.6362374
0.4879174
180
.05742739
0.34884899
.26118992
0.
1459441 0.5935535
0.4565142
210
0.05475392
0.34534936
.25727397
0.
1016262 0.4927722
0.3817985
240
.05164961
0.34128199
0.25272142
0.0482257 0.3752715
0.2938290
270
.04846610
0.33710662
0.24804652
0.
0004822 0.2736102
0.2165394
300
0.0-1587421
0.33370403
.24423576
9.9673367 0.2060765
0.1640904
330
.04461832
0.33205428
0.24238775
9.
9536821 0.1814324
0.1439179
2l
.31137734
2.04959742
1
.51844973
0.3065179 2.3501251
1.8219725
Sj
.31138070
2.04960185
1
.51845468
0.
3065326 2.3501595
1.8219999
E
logF
Ji'
J,
J,
F,
0.1589092
0.14310992
+0.032517759
+0.009293505
-0.004694216
30
0.2080805
0.14534318
+0.064629382
+0.012521451
-0.009159128
60
0.2862522
0.14795099
+0.079427870
+0.011088166
-0.011148401
90
0.3786919
0.14823938
+0.072184328
+0.005377209
-0.010129019
120
0.4572602
0.14575314
+0.044711897
+0.003081410
-0.006374115
150
0.4878898
0.14319523
+0.004967377
-0.012020967
-0.000889820
180
0.4557266
0.14344333
-0.035694279
-0.019045640
+0.004854357
210
0.3791987
0.14602945
-0.066230415
-0.022272938
+0.009319268
240
0.2904985
0.14803624
-0.078997000
-0.020838344
+0.011308544
270
0.2141065
0.14763142
-0.071296738
-0.015126732
+0.010289159
300
0.1631852
0.14536909
-0.045398479
-0.006668768
+0.006534256
330
0.1438950
0.14322488
-0.007686344
+0.002269483
+0.001049961
Si
1.8118319
0.87366271*
-0.003432232
-0.029252491
+0.000480425
2i
1.8118624
0.87366354
-0.003432410
-0.029252494
+0.000480421
*s,(J
V - G") =
0.85466721.
St(Ji - G") =
0.85466722.
OF THE ORBITS OF THE FOUR INNER PLANETS.
125
ACTION OF MERCURY ON THE EARTH.
E
1000 X F 3 R a So Wo <">
gto)
-0.3347370 -0.9146346 +0.03939096 +0.012865148 0.0000000
+0.04006286
30
-0.1102483 -0.9755942 +0.08729250 +0.020012484 -0.4949864
+0.08857904
60
+0.5661785 -1.0850638 +0.12713088 +0.022775691 -0.9476313
+0.12820596
90
+ 1.0162060 -1.2358930 +0.14090497 +0.016043819 -1.2358930
+0.14090497
120
+0.7859945 -1.3865198 +0.10290744 -0.005719611 -1.1907761
+0.10205179
150
+0.1010653 -1.4553467 +0.01142556 -0.036530916 -0.7172557
+0.01126198
180
-0.3579653 -1.3946588 -0.08289463 -0.055794474 0.0000000
-0.08152732
210
-0.1348464 -1.2530893 -0.12959913 -0.053749951 +0.6175749
-0.12774376
240
+0.5468018 -1.1065943 -0.12737525 -0.039380718 +0.9503695
-0.12631602
270
+ 1.0072426 -0.9942912 -0.09740907 -0.022874526 +0.9942912
-0.09740907
300
+0.7898462 -0.9252302 -0.05560180 -0.008440800 +0.8080489
-0.05607200
330
+0.1166998 -0.8987688 -0.00911126 +0.003338197 +0.4560076
-0.00924555
Zi
+ 1.9961187 -6.8127015 +0.00355760 -0.073694764 -0.3799890
+0.00640527
*
+ 1.9961190 -6.8129832 +0.00350357 -0.073760893 -0.3802614
+0.00634761
E
. Rocosv
Ho sm v ,,, , . , . . .
m o , l r i \ i\ o " o cos (v + JT) TV o sm (v + IT)
+ (cosv+cosE)S +(-sec 2 ip + llsmv<So
-2-fl
o
+0.0787819 +0.9146346 -0.002313800 +0.012655370
1.7985901
30
-0.3440930 +0.9286742 -0.013088473 +0.015139066
1.9228487
60
-0.8219877 +0.7498962 -0.021560174 +0.007341046
2.1519297
90
-1.2380823 +0.2610826 -0.015731601 -0.003149764
2.4717860
120
-1.2947997 -0.5330552 +0.004304192 +0.003766681
2.7962929
150
-0.7369917 -1.2550382 +0.011990880 +0.034506913
2.9529690
180
+0.1657893 -1.3946588 -0.010034649 +0.054884684
2.8360980
210
+0.8424959 -0.9617143 -0.034466377 +0.041244714
2.5425788
240
+ 1.0791999 -0.3473975 -0.036894585 +0.013770630
2.2317474
270
+0.9957850 +0.1781428 -0.022567368 -0.003736033
1.9885826
300
+ 0.7530386 +0.5475911 -0.006510755 -0.005371886
1.8349430
330
+0.4402010 +0.7837111 +0.001148460 +0.003134422
1.7714297
Z,
-0.0399777 -0.0629896 -0.073009771 +0.087046525
13.6496011
2j
-0.0406851 -0.0651418 -0.072714479 +0.087139318
13.6501948
sin <p J.Ai ( '> + cos ? Bo M = + 0.00000006.
126 THE SECULAR VARIATIONS OP THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de!dt]w = - 8710J780TO' n 3.9400270
[d x /dt] M = [dTT/dt] m =--824986.23 TO' n 5.9164467
[dp/dt]*, = + 18814.333 TO' p 4.2744888
[dqfdt] m = - 15740.112 TO' n 4.1970078
[dL/dt] w = +2948201.7 w' p 6.4695572
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
= -o(X)l 1613570
[d x /dt] m = [drfdt] m = -0.10999815
[dpfdt] M = +0.0025085775
[dq/dt] w = -0.0020986812
[dL/dt] M = +0.39309355
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] m
-0.00116
-0.00116
-0.001 16136
e[dir/dt]w
-0.00184
-0.00184
-0.00184479
[dp/dt] M
+0.00250
+0.00251
+0.00250858
[dq/dt]w
-0.00209
-0.00210
-0.00209868
[dL/dt] w
+0.3931
+0.39309355
NOTES.
Although / and e' are here very large, the error in the approximate test with
e, G, G', G" and 6 is small in consequence of the smallness of the factor a'. As we
approach the end of the computation, however, the difference of the sums steadily
increases, indicating the rather slow convergence of the perturbing function. The
greatest difference is in the coefficient [dTr/dt] OQ where terms from the fifth to the
eleventh orders inclusive amount to one sixtieth part of the remaining terms and
produce an effect of 0".0018 in the value of [dir/dt] 00 . A division into twelve parts
is thus necessary in this case, but a comparison with the computation of the action of
Mars on Mercury, and especially with the similar case of Mercury on Venus, where
twenty-four points of division are employed, renders it evident that more than twelve
points are in the present case unnecessary.
OF THE ORBITS OF THE FOUR INNER PLANETS.
127
The agreement with previous values is exact. The results obtained by HILL
in the "New Theory, 11 pages 511 and 512, are,
These are, however,
Venus on the Earth.
= +0.0025049 [dq/dt] m = -0.0020956
but provisional -values. (See the note to the computation of
E
A
ACTION
logB
OF VENUS ON
i
THE EARTH.
e'
o /
,,
O
1
H
1.49844749
9.8537612
331
3.89 0.07950559
7
55.62
30
1.50411382
9.8546442
1 19
46.170 0.08070333
5
48
28.79
60
1.51484369
9.8567645
31 34
37.20 0.08262793
4
19
13.62
90
1.52786583
9.8597314
61 38
57.20 0.08465948
3
12
16.55
120
1.53974201
9.8626537
91 29
35.875 0.08634022
2
33
43.11
150
1.54723847
9.8644870
121 8
43.24 0.08741009
2
9
29.44
180
1.54824407
9.8645724
150 43
29.14 0.08768573
1
52
5.43
210
1.54243719
9.8629683
180 22
35.176 0.08700647
1
54
16.46
240
1.53142595
9.8603568
210 12
5.09 0.08536569
2
37
52.10
270 '
1.51826327
9.8576205
240 13
17.66 0.08309965
4
8
28.42
300
1.50652764
9.8554195
270 23
50.222 0.08090038
5
56
51.93
330
1.49931255
9.8540767
300 40
21.95 0.07954648
7
7
42.75
Z,
9.13923084*
9.1535280
1085 23
41.42 0.50242554
24
20
41.81
2 2
9.13923113
9.1535281
905 23
41.40 0.50242550
24
20
42.41
E
1000 X r'
lOOOXs
G
G' 1000000 X G"
e
O
O
,
,,
2.7380635
0.002936418
0.97594422
0.522484505 5.79
47
1
40.84
30
2.3200483
0.000006753
0.98353171
0.520557558 0.04
46
40
41.61
60
1.7893068
0.003473631
0.99543723
0.519388564 6.67
46
14
52.33
90
1.3770128
0.009946420
1.00773136
0.520128988 18.98
45
55
31.27
120
1.1340363
0.013007536
1.01774388
0.521998108 24.54
45
44
23.02
150
0.9732049
0.009615238
1.02406979
0.523162060 17.95
45
37
23.13
180
0.8464697
0.003140184
1.02571566
0.522509836 5.92
45
32
21.65
210
0.8529468
0.000000563
1.02173849
0.520674148 0.00
45
32
59.47
240
1.1449685
0.003259143
1.01199593
0.519411649 6.09
45
45
34.89
270
1.7299249
0.009581104
0.99828868
0.519968638 18.53
46
11
45.77
300
2.3843853
0.012589278
0.98467983
0.521847851 24.49
46
43
7.27
330
2.7841227
0.009256115
0.97615654
0.523149586 18.07
47
3
38.83
Zi
10.0372301
0.038406190
6.01151675
3.127640513 73.50
277
2
0.00
2 2
10.0372604
0.038406192
6.01151657
3.127640978 73.57
277
2
0.08
* 6o 2 + 3a 2 e 2 + 6[o' 2 -
2kaa'ce' cos K] = 9
13923110.
128
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF VENUS ON THE EARTH.
E
logtfo
log Lo'
log AT log N log P
logO
0.26147483
0.60779696
0.54775634 0.2626427 0.8915846
0.8209715
30
0.25683878
0.60207812
0.54149917 0.2549482 0.8714495
0.8036590
60
0.25122074
0.59513848
0.53390180 0.2468811 0.8459860
0.7827661
90
0.24707162
0.59000660
0.52828039 0.2420422 0.8253429
0.7669697
120
0.24470700
0.58707937
0.52507272 0.2404872 0.8122693
0.7579113
150
0.24322990
0.58524987
0.52306751 0.2402489 0.8048244
0.7529792
180
0.24217345
0.58394092
0.52163264 0.2400756 0.8019574
0.7506787
210
0.24230579
0.58410492
0.52181242 0.2408209 0.8062462
0.7532935
240
0.24496050
0.58739327
0.52541673 0.2444416 0.8214721
0.7646769
270
0.25055055
0.59430993
0.53299440 0.2516543 0.8474359
0.7853846
300
0.25737182
0.60273603
0.54221917 0.2600985 0.8762226
0.8090118
330
0.26191246
0.60833646
0.54834645 0.2649133 0.8941948
0.8237322
2i
1.50190834
3.56408503
3.19599940 1.4946267 5.0494920
4.6860161
S 2
1.50190910
3.56408590
3.19600034 1.4946278 5.0494937
4.6860181
E
log V
Ji'
1000 X Ji 1000 X J 3
1000 X Ft
0.8209685
0.522862677
- 3.0632731 + 1 2.629050
+ 1.2372987
30
0.8036589
0.521939851
-0.7336504 +24.877829
-0.0593369
60
0.7827628
0.521368465
+2.4504401 +30.338229
-1.3457284
90
0.7669603
0.521723193
+5.0835797 +27.547130
-2.2771885
120
0.7578992
0.522632205
+5.8345100 +17.252379
-2.6041325
150
0.7529704
0.523191642
+4.4234178 + 2.212471
-2.2389562
180
0.7506758
0.522862807
+ 1.7386808 -13.542647
-1.2795084
210
0.7532935
0.521968683
-0.9069972 -25.791387
+0.017127o
240
0.7646739
0.521377068
-2.6859269 -31.251723
+ 1.3035189
270
0.7853753
0.521675619
-3.5987100 -28.460585
+2.2349789
300
0.8089994
0.522582150
-4.0331250 -18.165873
+ 2.5619225
330
0.8237230
0.523186452
-4.0263352 - 3.126030
+ 2.1967465
2,
4.6859794
3.133685372*
+0.2413060 - 2.740585
-0.1266292
2 2
4.6859813
3.133685439
+0.2413047 - 2.740573
-0.1266288
* S,(J,' - G") = 3.
133611872.
2,(J,' - G") = 3.
133611869.
OF THE ORBITS OF THE FOUR INNER PLANETS.
129
ACTION OF VENUS ON THE EARTH.
E
1000 XF 3
RO 100 x So w,, --RO
<>
a
+0.06535471
- 1.8283274 -1.0644424 +0.08413477 1.7976643
0.0000000
30
-0.00196302
- 1.7898617 -0.5109526 +0.15828193 1.7638656
-0.9081206
60
-0.00570449
- 1.7535729 +0.5420211 +0.18393250 1.7388680
-1.5314810
90
+0.05916590
- 1.7367356 +1.4494077 +0.16147390 1.7367356
-1.7367356
120
+0.12885901
- 1.7362036 +1.6510300 +0.09963433 1.7507629
-1.4910928
150
+0.13426042
- 1.7386892 +1.0760758 +0.01338361 1.7639423
-0.8568990
180
+0.06988985
- 1.7361351 +0.1682926 -0.07583153 1.7652520
0.0000000
210
-0.00059748
- 1.7337538 -0.5029615 -0.14614352 1.7589352
+0.8544665
240
-0.00787443
- 1.7442502 -0.6981624 -0.18183297 1.7588768
+ 1.4980034
270
+0.05404190
- 1.7747204 -0.6225129 -0.17324740 1.7747204
+ 1.7747204
300
+0.12215392
- 1.8154792 -0.6714245 -0.11610019 1.8002552
+ 1.5855471
330
+0.12777090
- 1.8402326 -0.9613070 -0.01982985 1.8135046
+0.9336770
?i
+0.37267857
-10.6139684 -0.0726856 -0.00606309 10.6116792
+0.0609767
2 2
+0.37267862
-10.6139933 -0.0722505 -0.00608133 10.6117037
+0.0611087
,-, . Ra'cOS V
M
S
+ (cost)+cosE)So + ( -sec ! ^+lJsint>/So
WL, sin ( + IT)
-0.010825990
-0.0212889 +1.8283274 -0.01513166
+0.08276286
30
-0.005184831
-0.9168211 +1.5373039 -0.10351881
+0.11973727
60
+0.005466048
-1.5259137 +0.8639706 -0.17411620
+0.05928500
90
+0.014494077
-1.7367344 -0.0001389 -0.15833153
-0.03170097
120
+0.016373004
-1.5075995 -0.8612810 -0.07497804
-0.06561473
150
+0.010606704
-0.8754611 -1.5022509 -0.00439303
-0.01264209
180
+0.001655167
-0.0033659 -1.7361351 -0.01363832
+0.07459502
210
-0.004957609
+0.8630786 -1.5036464 -0.09371241
+0.11214236
240
-0.006923567
+ 1.5048615 -0.8818403 -0.17035373
+0.06358327
270
-0.006225129
+ 1.7745751 -0.0173138 -0.17092102
-0.02829602
300
-0.006771025
+ 1.5786949 +0.8963895 -0.08955312
-0.07388837
330
-0.009754749
+0.9169364 +1.5955426 -0.00682218
-0.01861936
2i
-0.001026363
+0.0253885 +0.1094311 -0.53777107
+0.14072305
2i
-0.001021537
+0.0255735 +0.1094965 -0.53769898
+0.14062119
sin <p \A\ W + cos (f
Bo (c> = - 0.0000000083.
130 THE SECULAR VARIATIONS OF THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
u log coeff.
[de/dt] w = + 5503.0089 m' p 3.7406002
[dx/dt\ = [dirfdt] = +1409586.4 TO' p 6.1490917
[dp/dt}^ = + 30388.832 TO' p 4.4827140
[dq/dt] m = - 116164.73 TO' n 5.0650743
= +4584354.6 TO' p 6.6612782
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF
[de/(ft] M = + 0.013483339
[dx/dt] = [dTr/dt] w = + 3.4537341
[dp/dt] w = + 0.074457966
[dq/dt]oo = - 0.28462399
[dL/dt] w = +11.232473
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
TO'
[de/dt] m
+ 0.01344
+o'.01348
+ 0.0134833
e[dw/dt]
+ 0.05796
+0.05792
+ 0.0579231
[dpldt] M
+ 0.07450
+0.07446
+ 0.0744580
[dq/dt]oo
- 0.28454
-0.28462
- 0.2846240
[dL/dt] w
+ 11.2298
+ 11.232473
NOTES.
This computation is of special interest because, notwithstanding the low eccen-
tricities of both the Earth and Venus, the perturbing function is but slowly con-
vergent for this case. In 1893, the computation was effected by MR. R. T. A. INNES
who employed HILL'S second modification of GAUSS'S method, using in the work
manuscript tables prepared by himself. (See M. N., Vol. LIII, No. 6. The tables
were afterward published in M. N., Vol. LIV, No. 5.) The values of [dp/dt] 00 and
[dq/dt]oo were also obtained by HILL in the "New Theory," pages 511 and 512.
As the results of INNES differed considerably in some cases from those hitherto
obtained, particularly in the case of [de/dt] o, which agreed to the first two figures
only with the values of LEVERRIER and NEWCOMB, and in the case of [dq/dt] QO , which
OF THE ORBITS OF THE FOUR INNER PLANETS. 131
differed in the fourth figure from the value given by HILL, and in order to make the
comparison more exact, the roots in the present paper were computed by the formulas
of the second method, their values being afterward verified by those of the first. It
was found that the functions tabulated by MR. INNES are substantially correct,
though the last two significant figures of all functions from R to the end usually
differ, doubtless owing to the inaccuracy of the tables employed by MR. INNES.
Using the values as given by him, all of this part of his computation was duplicated,
with the result that an error was found in his value of [de/dt] 00 , while for [dq/dt] a
and the other coefficients his values were found to be substantially correct. The
various values here referred to are as follows:
Innes. Hill.
[de/dt]oo
e[dw/dt]m
[dp/dflw,
[dq/dtloo
+ 0.013476*
+ 0.057915
+ 0.074459
- 0.284623
+ 11.232490
a
+0.0744329
-0.2845280
It will be noticed that the results of INNES are in almost exact accordance with
those here given. The disagreement of the value of [dqldt] 00 as derived by GAUSS'S
method with that found by HILL is, however, a more serious matter, and is almost
the sole cause of the considerable disagreement of the values of this variation in the
complete perturbations of the Earth's orbit, the values of [dqldt] OQ from the action
of all of the other planets except Venus agreeing with those obtained by HILL very
exactly. Using the values tabulated on page 510 of the "New Theory " and the
formulas of page 511, I have duplicated the computation by HILL'S methods and
find the same results as he obtained. It is to be noticed that the theory of the motion
of the ecliptic here given by HILL was to serve a temporary purpose only, the numerical
values of the coefficients stated by LEVERRIER in the Annales, Vol. II, pages 94 to 96,
being employed without a re-computation of them.
* The uncorrected value was + 0".013156.
132
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON THE EARTH.
E
A
B COS e
/; -in
g
h
3.12005845
-0.6857946
+ 1.1901000
0.028604007
2.3106194
22.5
3.04885416
-1.0809801
+0.7480691
0.011301685
2.3059358
45
3.01959381
-1.2762880
+0.1890610
0.000721878
2.3030821
67.5
3.03677529
-1.2419833
-0.4018201
0.003260792
2.3034537
90
3.09782583
-0.9832901
-0.9346186
0.017641228
2.3070230
112.5
3.19346899
-0.5395909
-1.3282200
0.035628705
2.3122967
135
3.30912609
+0.0215645
-1.5227032
0.046826348
2.3167796
157.5
3.42714623'
+0.6147456
-1.4884581
0.044743825
2.3179851
180
3.52951891
+ 1.1496460
-1.2306997
0.030588915
2.3148501
202.5
3.60064084
+ 1.5448315
-0.7886687
0.012561720
2.3089820
225
3.62970226
+ 1.7401394
-0.2296607
0.001065206
2.3043625
247.5
3.61232195
+ 1.7058351
+0.3612206
0.002635147
2.3044350
270
3.55118902
+ 1.4471417
+0.8940191
0.016141855
2.3088270
292.5
3.45562818
+ 1.0034423
+ 1.2876203
0.033483868
2.3139342
315
3.34017005
+0.4422870
+ 1.4821032
0.044362582
2.3163153
337.5
3.22234876
-0.1508942
+ 1.4478582
0.042336214
2.3148421
S,
26.59718442*
+ 1.8554059f
-0.16239891
0.185952019
18.4818589
2 2
26.59718440
+ 1.8554060
-0.1623987
0.185951956
18.4818645
* 8a s + 4a 8 e' + 8[a' ! - 2kaa'ee' cos A'] = 26.59718442.
t 8[a'V - kaa'e cos K] = + 1.8554056.
t - Sk'aa' cos v ' e sin A" = - 0.1623983.
E
OF THE ORBITS OF THE FOUR INNER PLANETS.
ACTION OF MAES ON THK EARTH.
G G' G"
133
22.5
45
67.5
90
112.5
135
157.5
180
202.5
225
247.5
270
292.5
315
337.5
2,
?2
0.7892434
0.7227227
0.6963160
0.7131259
0.7706071
0.8609765
0.9721508
. 1.0889654
1.1944730
1.2714631
1.3051440
1.2876912
1.2221663
1.1214982
1.0036591
0.8873110
7.9537596
7.9537539
2.3024091 0.8127396 0.0152860 36 42 22.96
2.3028298 0.7325284 0.0066997 34 27 16.75
2.3028870 0.6969608 0.0004498 33 23 4.72
2.3025627 0.7159948 0.0019779 33 55 44.11
2.3020189 0.7853689 0.0097577 35 54 24.65
2.3015508 0.8891330 0.0174106 38 41 59.61
2.3014739 1.0076482 0.0201918 41 42 38.47
2.3019609 1.1223087 0.0173190 44 30 20.12
2.3028661 1.2173683 0.0109112 46 46 8.30
2.3036994 1.2810025 0.0042567 48 15 58.20
2.3038996 1.3059610 0.0003540 48 50 43.66
2.3033085 1.2897047 0.0008871 48 27 7.65
2.3023363 1.2343370 0.0056800 47 8 13.64
2.3016065 1.1465149 0.0126889 45 3 3.25
2.3014626 1.0370980 0.0185863 42 25 11.06
2.3018396 0.9202987 0.0199852 39 31 20.00
18.4193535 8.0974817 0.0812168 332 52 47.46
18.4193582 8.0974855 0.0812251 332 52 49.69
ACTION OF MAKS ON THE EARTH.
E
log A'
log L '
log N a
log AT
logP
logQ
0.14793515
0.46562072
0.39121913
9.5856600
9.3211682
9.6118228
22.5
0.12867513
0.44104101
0.36395436
9.5698260
9.2838199
9.5702568
45
0.12011431
0.43007010
0.35176641
9.5662159
9.2715711
9.5556249
67.5
0.12442259
0.43559482
0.35790544
9.5749532
9.2853792
9.5702743
90
0.14089818
0.45665643
0.38128232
9.5949794
9.3237440
9.6123158
112.5
0.16647531
0.48915019
0.41726416
9.6240919
9.3826551
9.6760625
135
0.19735020
0.52805532
0.46020914
9.6588907
9.4553466
9.7533001
157.5
0.22941061
0.56809820
0.50425204
9.6947360
9.5321279
9.8336349
180
0.25803549
0.60355500
0.54311538
9.7259997
9.6009117
9.9047936
202.5
0.27840311
0.62861983
0.57051120
9.7469169
9.6490816
9.9542006
225
0.28660968
0.63868139
0.58149048
9.7530543
9.6666751
9.9720145
247.5
0.28101715
0.63182707
0.57401211
9.7427949
9.6495832
9.9542876
270
0.26293394
0.60959543
0.54972342
9.7180756
9.6011932
9.9045601
292.5
0.23607945
0.57638319
0.51334444
9.683858o
9.5314041
9.8327841
315
0.20515940
0.53784176
0.47098807
9.6465515
9.4533990
9.7520425
337.5
0.17455362
0.49936282
0.42855178
9.6122456
9.3799495
9.6749680
V
1
1.61903635
4.27007615
3.72979435
7.2494269
5.6940086
8.0664741
2 2
1.61903696
4.27007713
3.72979553
7.2494229
5.6940004
8.0664686
134
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON THE EARTH.
E
logF
Ji'
J 2
J,
Ft
9.6084099
2.3152378
+0.11029083
+0.061812671
-0.25644059
22.5
9.5687497
2.3059045
+0.06946998
+0.074883922
-0.16119255
45
9.5555233
2.2995328
+0.01805514
+0.077080482
-0.04073852
67.5
9.5698283
2.3016313
-0.03654213
+0.068068172
+0.08658346
90
9.6101298
2.3103260
-0.08632858
+0.049219187
+0.20138993
112.5
9.6721978
2.3186977
-0.12376087
+0.023403207
+0.28620250
135
9.7488617
2.3215901
-0.14288822
-0.005449606
+0.32810925
157.5
9.8298619
2.3181659
-0.14041555
-0.032946848
+0.32073022
180
9.9024333
2.3108630
-0.11642269
-0.054902532
+0.26518896
202.5
9.9532839
2.3034774
-0.07451078
-0.067974156
+0.16994090
225
9.9719383
2.2994233
-0.02127963
-0.070171605
+0.04948689
247.5
9.9540965
2.3004768
+0.03479475
-0.061160169
-0.07783516
270
9.9033316
2.3061702
+0.08485465
-0.042311544
-0.19264158
292.5
9.8300222
2.3139376
+0.12119708
-0.016495194
-0.27745411
315
9.7479652
2.3199983
+0.13850882
+0.012358484
-0.31936096
337.5
9.6705453
2.3208585
+0.13455789
+0.039856619
-0.31198183
s,
8.0485930
18.4831415*
-0.01520968
+0.027635537
+0.03499338
2 2
8.0485855
18.4831497
-0.01520963
+0.027635553
+0.03499343
*S,(y,' -G") = 18.4019247.
Zt(Ji' - G") = 18.4019246.
OF THE ORBITS OF THE FOUR INNER PLANETS.
135
ACTION OF MARS ON THE EARTH.
E
1000 X Ft
flo
-So
w.
RW
gw
- 5.019874
0.22207195
-0.00895625
+0.02403761
0.00000000
-0.00910901
22.5
- 1.341522
0.21055435
-0.00524961
+0.02748415
+0.08184381
-0.00533223
45
+ 0.173061
0.20752103
-0.00112516
+0.02773157
+0.14850060
-0.00113867
67.5
- 1.404619
0.21281662
+0.00313244
+0.02500875
+0.19788699
+0.00315267
90
- 5.191525
0.22627677
+0.00726155
+0.01896286
+0.22627677
+0.00726155
112.5
- 9.004074
0.24749832
+0.01089521
+0.00882887
+0.22720045
+0.01082573
135
-10.631982
0.27522212
+0.01347761
-0.00609013
+ 0.19233060
+0.01331965
157.5
- 9.129495
0.30647387
+0.01430898
-0.02537639
+0.11549298
+0.01409065
180
- 5.368216
0.33606721
+0.01279803
-0.04599728
0.00000000
+0.01258694
202.5
- 1.527816
0.35740451
+0.00883760
-0.06172287
-0.13468590
+0.00870276
225
+ 0.177177
0.36477534
+0.00302208
-0.06569865
-0.25491209
+0.00298666
247.5
1.210722
0.35578971
-0.00342959
-0.05556595
-0.32661056
-0.00340772
270
- 4.837361
0.33265813
-0.00898150
-0.03579916
-0.33265813
-0.00898150
292.5
- 8.543560
0.30146830
-0.01237466
-0.01405700
-0.28031952
-0.01245460
315
-10.135232
0.26942963
-0.01319021
+0.00403826
-0.19280199
-0.01334851
337.5
- 8.672134
0.24201044
-0.01181400
+0.01658570
-0.09407098
-0.01199994
2,
-40.833952
2.23402218
+0.00430615
-0.07881492
-0.21326424
+0.00357711
2s
-40.833942
2.23401612
+0.00430637
-0.07881474
-0.21326273
+0.00357732
sin
<p l-Ai (a) + cos
V Bo> = +
0.0000000050.
136
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MARS ON THE EARTH.
E
Ro sin v +
(cos v + cos E) So
( - sec' if + ij sin vS a
Wo COS (!)+)
Wo sin (v + T )
-2 a /J
-0.01791249
-0.22207195
-0.00432317
+0.02364565
-0.43669510
22.5
+0.07214538
-0.19805102
-0.01506202
+0.02298943
-0.41458381
45
+0.14689806
-0.14657925
-0.02300258
+0.01548939
-0.41012014
67.5
+0.20021147
-0.07256855
-0.02452836
+0.00487823
-0.42290136
90
+0.22612318
+0.01831801
-0.01859384
-0.00372284
-0.45255354
112.5
+0.21867465
+0.11830141
-0.00734127
-0.00490454
-0.49817351
135
+0.17313163
+0.21584084
+0.00340234
+0.00505112
-0.55697199
157.5
+0.08900258
+0.29475425
+0.00517812
+0.02484247
-0.62244500
180
-0.02559607
+0.33606721
-0.00827262
+0.04524724
-0.68340683
202.5
-0.15101812
+0.32435071
-0.03316015
+0.05205879
-0.72588458
225
-0.25917515
+0.25670929
-0.05360955
+0.03797800
-0.73820246
247.5
-0.32389097
+0.14753236
-0.05413656
+0.01252224
-0.71614631
270
-0.33246073
+0.02354205
-0.03531845
-0.00584698
-0.66531623
292.5
-0.28957300
-0.08808428
-0.01192486
-0.00744291
-0.59906678
315
-0.21131668
-0.16946344
+0.00233483
+0.00329486
-0.53246901
337.5
-0.11585773
-0.21387153
+0.00359251
+0.01619196
-0.47652121
s,
-0.30030825
+0.31236276
-0.13738304
+0.12113644
-4.47573530
2 2
-0.30030574
+0.31236335
-0.13738259
+0.12113567
-4.47572256
DIFFERENTIAL COEFFICIENTS.
[dxldt] m =
= - 48641.893m'
= +3016769.1 m'
log coeff.
n 4.6870105
p 6.4795421
= +
[dq/dt] w =
= - 724628.93
19626.398 m' p 4.2928406
22258.695 m' n 4.3474997
m' n 5.8601 157
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
= -0.015723904
[d x /dt] m = [d7r/d<]oo = +0.97519611
[dpfdt] M = +0.0063443986
[dq/dt] m = -0.0071953108
[dL/dt] m = -0.23424243
OF THE ORBITS OF THE FOUR INNER PLANETS.
137
Leverrier.
[<fe/cft]oo -0.01573
[dw/dt] m +0.9754
[dp/dt] w +0.00635
[dq/dtlw -0.00721
[dL/dt] w -0.2337
COMPARISON WITH OTHER RESULTS.
Innes. Hall. Newcomb. Method of Gauss.
-0.015722 -0.0157232 -0.01572 -0.0157239
+0.975224 +0.9751387 +0.9755 +0.9751961
+0.0063401 +0.0063444 +0.00634 +0.0063444
-0.0071898 -0.0071952 -0.00719 -0.0071953
-0.23469 -0.2342416 -0.2342424
NOTES.
In the "New Theory," Page 511, HILL points out that the convergence of the
expansion of the perturbing function is slow in this case, the terms of the fifth order
in the inclinations and eccentricities amounting to one per cent, of those of the first
order. He stated that a computation by GAUSS'S method would be very desirable
and consequently this was effected by DR. ASAPH HALL, JR., in July, 1891 (A. J.
No. 244), and by INNES in November, 1891 (M. N., Vol. LII, Nos. 2 and 7). HALL'S
computation is the first application of GAUSS'S method made after the publication
of HILL'S memoir.
Both HALL and INNES employed the values of the elements stated by LEVERRIER;
Hall divided the orbit of the Earth into twelve parts and INNES into sixteen. The
values of [dpldt] 00 and [dq/dt] Q given by the latter were however in error owing to a
misprint that occurred in HILL'S original paper in the value of J 3 ] in M. N., Vol. LII,
No. 7, INNES pointed out this error but did not re-compute the variations.
The final results of the present paper were printed in A. J., No. 518, but the
values there given are all slightly incorrect owing to errors in some of the preliminary
constants, which remained undetected even in the duplication. Upon devising new
test equations these were always applied to all computations previously made and in
this way the errors affecting practically every figure of the present computation were
discovered. The work was then both repeated and duplicated so that it is hardly
possible that any errors can yet remain in it.
The latter part of INNES' computation was also duplicated, the values of J 3 , W Q ,
[dp/dt] and [dq/dt] being freed from the errors referred to by him. It is these corrected
values which are given above.
It will be noticed that the agreement of the results here given with those of
HALL is very exact notwithstanding the difference of the original elements used in
the computation. The divergences from those of INNES are more considerable,
probably because the latter computer did not employ the accurate tables of HILL.
138
THE SECULAR VARIATIONS OF THE ELEMENTS
The values obtained in the "New Theory " for the motion of the plane of the
ecliptic are,
[dp/dt] m = +0.0063362
[dq/dt]^ = -0.0072112
ACTION OF JUPITER ON THE EARTH.
E
A
B cos t
B sin t
g
28.04923872
+ 1.4444306
+5.1077212
1.6444314
30
27.80097551
-1.1738070
+4.4818820
1.2661417
60
27.62449781
-3.1281533
+2.6317812
0.4365772
90
27.56719541
-3.8949449
+0.0531514
0.0001781
120
27.64447424
-3.2687173
-2.5630644
0.4140764
150
27.83557589
-1.4172691
-4.5158555
1.2854086
180
28.08919180
+ 1.1633052
-5.2819683
1.7585429
210
28.33731437
+3.7815432
-4.6561285
1.3665053
240
28.51351085
+5.7358901
-2.8060283
0.4963014
270
28.57067253
+6.5026807
-0.2273984
0.0032594
300
28.49353432
+5.8764527
+2.3888176
0.3596892
330
28.30271399
+4.0250047
+4.3416085
1.1881266
2:
168.41444773*
+7.82320801
-0.52274091
5.1096185
Si
168.41444770
+7.8232076
-0.5227406
5.1096197
E
I
G
G'
G"
O
0.977739
27.0061276
1.0387013
0.0586223
30
0.729882
27.0062768
0.7909419
0.0592752
60
0.554312
27.0065429
0.5826671
0.0277442
90
0.497498
27.0066658
0.4975109
0.0000133
120
0.574342
27.0065199
0.6004570
0.0255347
150
0.764478
27.0062522
0.8240511
0.0577594
180
1.017523
27.0061314
1.0803044
0.0602760
210
1.266039
27.0062772
1.3067275
0.0387224
240
1.443204
27.0065561
1.4565395
0.0126169
270
1.500952
27.0066843
1.5010367
0.0000804
300
1.423446
27.0065354
1.4332595
0.0092925
330
1.231720
27.0062551
1.2681196
0.0346927
Si
5.990566
162.0384133
6.1919288
0.1940866
2 2
5.990569
162.0384114
6.1883877
0.1905434
* 6o 2 + 3aV + 6[a' 2 -
Zkaa'ee' cos K] = 168.41444773,
f 6[a' e' kaa'e r.os K]
= + 7.8232074.
-
| Gk'aa' cos <f>' e sin
K' - - 0.5227409.
h
27.008467
27.008061
27.007154
27.006666
27.007100
27.008066
27.008637
27.008243
27.007275
27.006689
27.007056
27.007962
162.045689
162.045687
O
e
/ //
11
36 58.64
10
12 32.08
8
38 31.93
7
48 2.72
8
45 10.87
10
23 57.24
11
50 45.30
12
53 16.35
13
29 4.47
13
38 11.24
13
21 37.74
12
40 46.32
67
42 8.95
67
36 45.94
OF THE ORBITS OF THE FOUR INNER PLANETS.
139
ACTION OF JUPITER ON THE EARTH.
E
logtfo
log Lo' log N, log N log P
logQ
0.01351621
0.29098220 0.19630730 7.8502195 5.2763937
6.6141228
30
0.01041684
0.28686620 0.19168189 7.8490834 5.2711158
6.6083484
60
0.00744942
0.28292144 0.18724767 7.8522627 5.2713543
6.6075955
90
0.00606351
0.28107775 0.18517479 7.8588567 5.276992T
6.6125603
120
0.00764272
0.28317853 0.18753669 7.8670773 5.2864976
6.6227349
150
0.01081185
0.28739101 0.19227173 7.8747484 5.2973550
6.6346279
180
0.01406123
0.29170555 0.19712004 7.8798617 5.3067059
6.6445510
210
0.01667410
0.29517157 0.20101379 7.8810685 5.3120661
6.6499953
240
0.01827428
0.29729273 0.20339625 7.8780194 5.3119679
6.6497435
270
0.01869373
0.29784858 0.20402051 7.8714850 5.3063886
6.6440330
300
0.01793523
0.29684340 0.20289160 7.8631935 5.2968004
6.6344668
330
0.01613325
0.29445437 0.20020816 7.8553923 5.2858029
6.6235786
Si
0.07887909
1.74292385 1.17449955 7.1906339 1.7497196
9.7732144
2 2
0.07879328
1.74280948 1.17437087 7.1906342 1.7497203
9.7731434
E
logF
Ji' Ji J,
F 2
6.6129514
27.064741582 +0.24634548 +0.02063093
-6.6623169
30
6.6071626
27.061450021 +0.21012324 +0.32748544
-5.8459973
60
6.6070396
27.022786983 +0.12128740 +0.54805936
-3.4327952
90
6.6125600
26.992014082 +0.00316191 +0.62325071
-0.0693287
120
6.6222233
27.021540830 -0.11740027 +0.53291210
+3.3431631
150
6.6334726
27.060747670 -0.21222668 +0.30124976
+ 5.8903093
180
6.6433468
27.066395282 -0.25546365 -0.00966336
+6.8895984
210
6.6492222
27.041108427 -0.23112984 -0.31651859
+6.0732768
240
6.6494917
27.008005002 -0.14158677 -0.53709407
+3.6600761
270
6.6440314
26.992057684 -0.01087596 -0.61228634
+0.2966099
300
6.6342813
27.004932957 +0.12157000 -0.52194700
-3.1158827
330
6.6228858
27.037481542 +0.21569955 -0.29028297
-5.6630276
Zi
9.7693340
162.188402634* -0.02524781 +0.03289796
+0.6818428
S 2
9.7693345
162.184859426 -0.02524778 +0.03289801
+0.6818424
* s,(J,' - G") = 161
.994316084.
zl/i' - G") = 161
,994316076.
140
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF JUPITKK ON THE EARTH.
E
F,
1000 X fio 1000 X So 1000 X W 1000 X B (n)
1000 X S (n)
-0.14994500
3.5906058 -0.02485742 +0.00562843 0.0000000
-0.02528143
30
-0.11198359
3.5669877 -0.02409495 +0.13045170 +1.8097792
-0.02445007
60
-0.03628181
3.5812719 -0.01504685 +0.22107406 +3.1277000
-0.01517409
90
+ 0.00006601
3.6318358 -0.00001620 +0.25540271 +3.6318358
-0.00001620
120
-0.04119166
3.7069905 +0.01547114 +0.22249821 +3.1836522
+0.01534248
150
-0.12070192
3.7863426 +0.02555514 +0.12714491 +1.8660682
+0.02518930
180
-0.16035007
3.8464128 +0.02722801 -0.00750003 0.0000000
+0.02677889
210
-0.12100025
3.8690539 +0.02153453 -0.14361296 -1.9068325
+0.02122624
240
-0.04149417
3.8484336 +0.01189802 -0.24048156 -3.3051260
+0.01179908
270
+0.00005467
3.7925083 +0.00121404 -0.26976404 -3.7925083
+0.00121404
300
-0.03599894
3.7183323 -0.00934064 -0.22557067 -3.2473996
-0.00941964
330
-0.11169659
3.6453213 -0.01884026 -0.12397381 -1.8495240
-0.01911793
s,
-0.46526165
22.2920469 +0.00535226 -0.02435156 -0.2411734
+0.00404529
S 2
-0.46526167
22.2920496 +0.00535230 -0.02435149 -0.2411816
+0.00404538
E
SSSL.
1000 X|-#i> cos v
.. 1000 XW a cos (+) 1000XW sin(y+ir)
OJ / f \ ' r 1
+ ( sec- if + 1 1 sin r<So
\o / J
1000X-2-Ro
a
-0.0497148
-3.5906058 -0.00101227 +0.00553665
- 7.0607738
30
+ 1.7678935
-3.0981989 -0.08531744 +0.09868428
- 7.0303590
60
+3.1124039
-1.7713812 -0.20927556 +0.07125641
- 7.1024820
90
+3.6313250
+0.0608776 -0.25043240 -0.05014132
- 7.2636717
120
+3.1675401
+ 1.9264211 -0.16743708 -0.14652740
- 7.4761534
150
+ 1.8214372
+3.3200889 -0.04173395 -0.12010043
- 7.6826719
180
-0.0544560
+3.8464128 -0.00134888 +0.00737774
- 7.8218420
210
-1.9439518
+3.3453092 -0.09208972 +0.11020057
- 7.8505000
240
-3.3167069
+ 1.9516989 -0.22529979 +0.08409148
- 7.7614089
270
-3.7919952
+0.0611766 -0.26614166 -0.04405982
- 7.5850167
300
-3.2561646
-1.7957525 -0.17399247 -0.14355750
- 7.3742992
330
-1.8818157
-3.1224540 -0.04265147 -0.11640599
- 7.1847533
Zi
-0.3970983
+0.5667933 -0.77836605 -0.12182259
-44.5969593
2 2
-0.3971070
+0.5667994 -0.77836664 -0.12182271
-44.5969726
sin <f %A i ( *> + cos if
Bo w = - 0.0000000000093.
OF THE ORBITS OF THE FOUR INNER PLANETS. 141
DIFFERENTIAL COEFFICIENTS.
n log coeff.
[de/dilw = - 85.760340 m' n 1.9332865
[dx/dt]ao = [dv/dilw = +7298.7450 TO' p 3.8632482
[dp/dt] m = - 26.316855 TO' n 1.4202340
[dq/dt] m = - 168.14734 TO' n 2.2256900
[dL/dt] w = -9631.7202 TO' n 3.9837038
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] m = -0^081841849
[d x /dt] w = [dTr!dt] w = +6.9652565
[dp/dt]w = -0.025114405
[dq/dt] m = -0.16046446
[dLfdt] w = -9.1916336
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcoml). Method of Gauss.
[de/dt]oo
-0.08182
-0.08182
-0.0818418
e[dir/dl] 00
+0.11679
+0.11677
+0.1168153
[dp/dtlw
-0.02501
-0.02511
-0.0251144
[dq/dt] m
-0.16041
-0.16047
-0.1604644
[dL/dt] m
-9.1916
-9.1916336
NOTES.
The very close agreement of the sums toward the end of this computation is
owing to the circularity of the two orbits and to their small mutual inclination. It is
evident that a division into eight parts would have been sufficient, while the errors
arising from a division into only six parts would have been almost inappreciable.
In this, as in several other cases, the divergence from the last figure of NEW-
COMB'S results is rather larger than was to have been expected. The values stated
by NEWCOMB were computed to one more significant figure than was published to
insure the accuracy of the final figure given. The uncertainty of this figure is evi-
dently due to neglected terms in the series employed by LEVERRIER and NEWCOMB.
In the present case we obtain for [deldt] 00 ,
Computed from the six even points of division 0".0818428
Computed from the six odd points of division .0818409,
142
THE SECULAR VARIATIONS OF THE ELEMENTS
and the difference between any two corresponding values for any other coefficient is
even less than this.
The values of the coefficients which define the motion of the plane of the ecliptic
are stated by HILL as follows:
[dp/dt]^ = -0.0251149
[dq/dt] w = -0.1604628
E A
92.9909218
30 92.7594069
60 92.3168471
90 91.7819295
120 91.2980364
150 90.9947748
180 90.9533006
210 91.1846750
240 91.6269534
270 92.1617302
300 92.6457640
330 92.9493070
Zj 551.8318232*
2 2 551.8318234
E I
+2.001263
30 +1.768419
60 +1.323475
90 +0.787549
120 +0.304998
150 +0.004086
180 -0.036365
210 +0.193697
240 +0.633602
270 +1.167320
300 +1.652663
330 +1.958598
21 +5.879636
2 2 +5.879669
t 6[a'V - fcao'e cos K] = + 29.6608842.
t - 6k'aa' cos <p' e sin K' = - 0.1707000.
ACTION OF SATURN ON THE EARTH.
B cos .
B sin t
d
+14.3293908
+ 1.6679163
0.795517
+ 12.2250618
+6.1218324
10.716773
+ 8.1696352
+8.9277816
22.792314
+ 3.2497638
+9.3339213
24.913194
1.2162762
+7.2314183
14.953684
- 4.0318145
+3.1836393
2.898337
- 4.4424293
-1.7248166
0.850721
- 2.3380978
-6.1787321
10.916915
+ 1.7173265
-8.9846857
23.083782
+ 6.6371970
-9.3908217
25.217872
+ 11.1032373
-7.2883175
15.189927
+ 13.9187769
-3.2405388
3.002863
+29.6608843f
-0.1707036t
77.665945
+29.6608873
-0.1706996
77.665954
G
G'
G"
90.703603
2.0057342
0.0043727
90.703703
1.8341642
0.0644170
90.704603
1.4944314
0.1681447
90.705370
1.0517493
0.2611465
90.705257
0.5874560
0.2806341
90.704380
0.1809884
0.1765509
90.703605
0.0803976
0.1166593
90.703690
0.4578844
0.2628566
90.704569
0.9146652
0.2782376
90.705348
1.3729264
0.2025015
90.705263
1.7502260
0.0956818
90.704378
1.9757281
0.0167564
544.226899
6.8329103
0.9437302
544.226868
6.8734407
0.9842289
'ee' cos K] = 551.8318229.
h
90.703702
90.705031
90.707414
90.708424
90.707081
90.704732
90.703708
90.705020
90.707394
90.708453
90.707143
90.704751
544.236442
544.236411
e
O
i n
8
33 39.757
8
18 56.347
7
46 25.450
6
53 59.891
5
36 19.643
3
35 46.006
2
40 11.401
5
6 24.385
6
34 30.181
7
33 52.413
8
11 49.887
8
31 21.103
39
22 56.319
40
20.145
OF THE ORBITS OF THE FOUR INNER PLANETS.
143
ACTION OF
SATUKN ON
THE
EARTH.
E
logKo
log La
logA^o
log AT
logP
logQ
0.00730944
.28273527
.18703836
7.
0561506
3.4235951
5.2855436
30
0.00689441
.28218320
.18641769
7.0572862
3.4236026
5.2857705
60
0.00602140
.28102171
.18511178
7.0610565
3.4252108
5.2877346
90
0.00473960
.27931572
.18319348
7.0664172
3.4279697
5.2907291
120
0.00312447
.27716505
.18077485
7.0719166
3.4311334
5.2937173
150
0.00128428
.27471328
.17801715
7.
0761000
3.4338677
5.2956443
180
0.00070759
.27394462
.17715249
7.0778799
3.4354590
5.2968496
210
0.00259230
0.27645617
.17997757
7.
0767946
3.4354874
5.2978905
240
0.00430241
.27873368
0.18253896
7.0731165
3.4339314
5.2966961
270
0.00570043
.28059458
.18463152
7.
0677983
3.4311901
5.2938284
300
0.
00669846
0.28192252
0.18612461
7.0622487
3.4279906
5.2902830
330
0.00724349
0.28264754
.18693973
7.0579727
3.4252033
5.2872040
Zi
0.02816377
1
.67552285
1.09874105
2.4023686
6.1773201
1.7508241
2,
0.02845451
1
.67591049
1
.09917714
2.4023690
6.1773208
1.7510668
E
log V
J,'
/,
/i
Ft
5.2855175
90.7004946
+0.12827478
-0.7968027
- 8.486434
30
5.2853861
90.7506534
+0.29200243
+ 1.2416991
-31.148158
60
5.2867312
90.7742572
+0.41637229
+2.9535293
-45.424968
90
5.2891707
90.7991434
+0.49225830
+3
.8800058
-47.491413
120
5.2920417
90.8302700
+0.45906255
+3.7728791
-36.793788
150
5.2945889
90.8046167
+0.26449000
+2
.6608537
-16.198502
180
5.2961519
90.8127812
-0.06165231
+0.8418949
+ 8.775944
210
5.2963206
90.9507883
-0.39625124
-1
.1966086
+31.437674
240
5.2950355
90.8893132
-0.59035199
-2
.9084413
+45.714495
270
5.2926201
90.7428396
-0.56521208
-3
.8349184
+47.780934
300
5.2897120
90.6423828
-0.36124653
-3
.7277897
+37.083286
330
5.2871040
90.6419564
-0.09682848
-2
.6157627
+ 16.488008
Si
1.7451898
544.6494990*
-0.00954121
+0.1352696
+ 0.868535
2 2
1.7451904
544.6899978
-0.00954107
+0.1352689
+ 0.868543
*2,(J
i' - G") =
543.7057688.
z s (J
i' - G") =
543.7057689.
144
THE SECULAR VARIATIONS OF THE ELEMENTS
E
30
60
90
120
150
180
210
240
270
300
330
Si
E
ACTION OF SATURN ON THE EARTH.
F, 1000 X Ro 100000 X So 100000 X Wo 10000 X R M 100000 X S<>
-0.3555837 0.57362783 +0.02247623 -1.5471185 0.0000000 +0.0228596
-1.2681621 0.57437868 -0.26276188 +2.3619011 +2.9142195 -0.2666346
-1.2912829 0.57674453 -0.40344654 +5.6791610 +5.0370180 -0.4068583
-0.3956591 0.58127800 -0.31429014 +7.5403759 +5.8127800 -0.3142901
+0.5220384 0.58774791 -0.09358683 +7.4052797 +5.0477180 -0.0928086
+0.5361282 0.59426527 +0.08130415 +5.2579217 +2.9287882 +0.0801402
-0.3802586 0.59796705 +0.11726800 +1.6546186 0.0000000 +0.1153337
-1.3248861 0.59688219 +0.07295990 -2.4035193 -2.9416851 +0.0719154
-1.3648579 0.59139747 +0.07709136 -5.7742147 -5.0790610 +0.0764503
-0.4663692 0.58408733 +0.18082475 -7.5353293 -5.8408733 +0.1808248
+0.4731390 0.57795107 +0.28959440 -7.2511167 -5.0475302 +0.2920434
+0.5221422 0.57454813 +0.25136029 -5.0525337 -2.9150792 +0.2550648
-2.3968057 3.50543586 +0.00939762 +0.1666094 -0.0418552 +0.0070201
-2.3968061 3.50543960 +0.00939707 +0.1688164 -0.0418499 +0.0070205
lOOOXtft sinv 1000 xT-flo cos w
+ (coav+cosE)X } _
if \
h ( ~ sec 2 ^+ 1 I sin t'o'c
J
a
+0.00044952
-0.57362783
+0.002782496
-0.015218912
-0.0011280149
30
+0.28684099
-0.49762977
-0.015447198
+0.017867337
-0.0011320724
60
+0.49964759
-0.28807540
-0.053760700
+0.018305034
-0.0011438222
90
+0.58124893
+0.00316289
-0.073936353
-0.014803463
-0.0011625560
120
+0.50564826
+0.29959114
-0.055727122
-0.048767876
-0.0011853529
150
+0.29142597
+0.51791202
-0.017258560
-0.049666046
-0.0012057929
180
-0.00234536
+0.59796705
+0.002975836
-0.016276382
-0.0012159913
210
-0.29539381
+0.51865750
-0.015412218
+0.018443264
-0.0012111028
240
-0.50861524
+0.30174597
-0.054096837
+ 0.020191247
-0.0011927133
270
-0.58403554
+0.00617932
-0.074341448
-0.012307244
-0.0011681747
300
-0.50182267
-0.28668159
-0.055931013
-0.046147489
-0.0011462092
330
-0.28712395
-0.49766091
-0.017382540
-0.047441087
-0.0011324063
S,
-0.00703790
+0.05091934
-0.213757340
-0.087914378
-0.0070121038
S,
-0.00703741
+0.05092105
-0.213778317
-0.087907239
-0.0070121051
sin <p yAi M + cos <f
Bo (e > = + 0.00000000000028.
OF THE ORBITS OF THE FOUR INNER PLANETS. 145
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dtlw = - 1.5163927 TO' n 0.1808117
[dx/dt] m = [dT/dfloo = + 655.70924 TO' p 2.8167113
[dp/dt] m = - 18.991017 TO' n 1.2785482
[dq/dt] m = - 46.179399 TO' n 1.6644483
= -1514.4911 TO' n 3.1802667
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF
[de/dt] M = -0.00043305713
[dx/df], = [dir/d4 = +0.18725991
[dp/dfloo = -0.0054235259
[dq/dt] M = -0.013188086
= -0.43251400
m
COMPARISON WITH OTHER RESULTS.
Lcverrier. Newcomb. Method of Gauss.
[de/dt] w -0.00044 -0.00043 -0.00043306
e[dirldt} w +0.00315 +0.00314 +0.00314056
[dp/dt] m -0.00542 -0.00542 -0.00542353
[dq/dt] M -0.01317 -0.01318 -0.01318809
[dL/dt] M -0.4325 -0.43251400
NOTES.
Here, as in the previous case, the approximate tests completely fail with the
angle e, the roots G, G', G" , and with the functions which immediately depend upon
these quantities. The close agreement of the final sums shows, however, that the
expansion of the perturbing function is quite rapidly convergent for this case.
The values obtained by HILL in the "New Theory " are:
[dp/dt] w = -oo054237 [dq/dt]^ = -0.0131883
The agreement of the final results here obtained with all other values is satisfactory.
146
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF URANUS ON THE EARTH.
E
A
B cos t B sin e g
h
369.9391833
+24.383407 -17,162615 247.29194
367.49698
45
370.9299673
+34.837638 - 7.159449 41.56718
367.49556
90
370.9628887
+34.937206 + 7.512108 45.76306
367.49557
135
370.0188613
+24.623780 +17.957661 261.51163
367.49708
180
368.6506847
+ 9.938828 +18.058344 264.45220
367.49706
225
367.6596194
- 0.515405 + 7.755178 48.77249
367.49561
270
367.6264169
- 0.614971 - 6.916382 38.79263
367.49553
315
368.5707253
+ 9.698453 -17.361932 244.44864
367.49688
Si
1477.1791736*
+68.644470f + 1.191455t 596.29983
1469.98514
Z 2
1477.1791733
+68.644466 + 1.191458 596.29994
1469.98513
E
/
G G' G"
e
O
O
i a
+ 1.63126
367.495141 1.9739873 0.3408899 4
33 0.174
45
+2.62346
367.495250 2.6661957 0.0424235 4
55 28.953
90
+2.65637
367.495229 2.7027882 0.0460735 4
57 40.407
135
+ 1.71085
367.495130 2.0584810 0.3456945 4
38 13.695
180
+0.34269
367.495095 1.0379451 0.6933002 3
55 55.015
225
-0.64694
367.495249 0.163776T 0.8103506 2
56 52.565
270
-0.68006
367.495243 0.1303078 0.8100787 2
53 46.998
315
+0.26290
367.495069 0.9586081 0.6938970 3
50 28.866
S,
+3.95026
1469.980708 5.8450284 1.8903423 16
20 22.594
2 2
+3.95027
1469.980698 5.8470609 1.8923656 16
21 4.079
ACTION OF URANUS ON THE EARTH.
E
log A'
log LO' log No log N log P
logQ
0.00205713
0.27574316 0.17917560 6.1388849 1.2833195
3.7524062
45
0.00241015
0.27621390 0.17970508 6.1440956 1.2897058
3.7584989
90
0.00244638
0.27626178 0.17975893 6.1514873 1.3001369
3.7689401
135
0.00213675
0.27584925 0.17929492 6.1638868 1.3084161
3.7775217
180
0.00153562
0.27504825 0.17839395 6.1668770 1.3097851
3.7792009
225
0.00086277
0.27415148 0.17738519 6.1617903 1.3035252
3.7729672
270
0.00083285
0.27411159 0.17734031 6.1515209 1.2932165
3.7626532
315
0.00146560
0.27195493 0.17828898 6.1419973 1.2848106
3.7542155
S,
0.00687198
1.10116478 0.71466879 4.611770(1 5.1864579
3.0632003
2 2
0.00687557
1.10116956 0.71467417 4.6117700 5.1864577
5.0632033
* 4a 2 + 2aV + 4[o' 2
2kaa'ee' cos A:] = 1477.1791732.
t 4[a'V - kaa'e cos K] = + 68.644468.
t + 4fc'aa' cos v' e sin K' = + 1.191454.
OF THE ORBITS OF THE FOUR INNER PLANETS.
147
ACTION OF URANUS ON THE EARTH
.
E
logF
Ji' J*
Ji
Fl
3.7519032
367.8221780 -0.84734235
+2.2088876
+301.43340
45
3.7584363
367.4766488 -0.35515557
+4.6641774
+ 123.58384
90
3.7688721
367.4892672 +0.38057614
+4.3579828
-129.67133
135
3.7770116
367.8350259 +0.86234922
+ 1.4696662
-309.97871
180
3.7781780
368.1745883 +0.82112849
-2.3088335
-311.71664
225
3.7717715
368.2455011 +0.34414218
-4.7641198
-133.86711
270
3.7614579
368.2514548 -0.29842856
-4.4579225
+ 119.38809
315
3.7531918
368.1823028 -0.79535630
-1.5696087
+ 299.69549
Si
5.0604111
1471.7374883* +0.05593372
-0.1998856
- 20.56648
2 2
5.0604112
1471.7394786 +0.05597953
-0.1998849
- 20.56649
E
r,
1000000 X Ra 1000000 X So 1000000 X W,
1000000 X -R<">
1000000 X S<">
+3.5560766
68.949325 +0.10018820 +1.2544323
0.000000
+0.10189713
45
+0.4865778
69.825413 +0.03716997 +2.6752556
+49.966575
+0.03761606
90
+0.8230258
71.523975 -0.03529126 +2.5611571
+71.523975
-0.03529126
135
+4.0393764
73.045932 -0.11453773 +0.8877061
+51.045918
-0.11319535
180
+3.8028425
73.443068 -0.14342145 -1.3776247
0.000000
-0.14105578
225
+0.5809413
72.487200 -0.06580265 -2.8156195
-50.655465
-0.06503154
270
+0.709709S
70.794418 +0.06221357 -2.5724979
-70.794418
+0.06221357
315
+3.7847601
69.352325 +0.12685775 -0.8818747
-49.628040
+0.12838021
Si
+8.8916547
284.710786 -0.01631094 -0.1345332
+ 0.729557
-0.01223634
2*
+8.8916556
284.710870 -0.01631266 -0.1345325
+ 0.728988
-0.01223053
1 0ftOOOn V 1 7?n rct<* u
E
1 000000 X[R sin v
X L 1000000 1000000
1000x-2 r -K
+ (cosv+cosE)S l
l] +^Bec'>+l)<rfaS.] X^ cos ("+")
KWosm(v-\-ir)
a
+ 0.200376
-68.949325 -0.2256099
+ 1.2339773
-0.13558593
45
+50.011805
-48.728590 -2.2190515
+ 1.4942570
-0.13799470
90
+71.514500
+ 1.128956 -2.5113156
-0.5028130
-0.14304794
135
+ 51.201671
+52.095595 -0.4959295
-0.7362581
-0.14782435
180
+ 0.286843
+73.443068 -0.2477661
+ 1.3551611
-0.14934959
225
-50.554733
+51.949429 -2.2975222
+ 1.6276075
-0.14669365
270
-70.785500
+ 1.062876 -2.5379547
-0.4201590
-0.14158884
315
-49.442739
-48.631449 -0.5098809
-0.7195307
-0.13705976
ft
+ 1.216219
+ 6.685575 -5.5226463
+ 1.6661664
-0.56957230
2 2
+ 1.216004
+ 6.684985 -5.5223841
+ 1.6660757
-0.56957246
sin <p \Ai ( *> + cos .
, p . J3 <c> = - 0.00000000000025.
* 2,(J,' - G") = 1469.8471460.
2(Ji' - G") = 1469.8471130.
148 THE SECULAR VARIATIONS OF THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
n log coeff.
[<fe/tt]oo = + 0.39395664 TO' p 9.5954484
[dxfdtlw, = [drldt] M = +129.13143 TO' p 2.1110320
[dp/dt] w = + 0.53988815 TO' p 9.7323038
[dq/dt] w = - 1.7895101 TO' n 0.2527342
[dL/(ft]oo =-184.51950 TO' n 2.2660422
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF
TO
= +0.000017278801
[d x /dt] M = [Ar/dfloo = +0.0056636605
= +0.000023679306
= -0.000078487295
= -0.0080929604
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[(fe/tft]oo +0^00002 +0^00002 +o!oOOO 172788
e[d7r/d<]oo +0.00009 +0.00010 +0.0000949860
[dpldt] m +0.00002 +0.00002 +0.0000236793
[dq/dt] m -0.00008 -0.00008 -0.0000784873
-0.0081 -0.0080929604
NOTES.
It will be noticed that, owing to the very small mutual inclination, the approxi-
mate tests are here more exactly satisfied than even in the case of Saturn, where
twelve points of division were employed. It is therefore evident that eight points
are fully sufficient and that the greatest error arising from a division into only four
points (which occurs with the coefficient [dir/dt] 00 ), could not amount to more than
1 /20,000th of the whole.
The results obtained by HILL are :
[dpldt] M = +0.0000237 [dqldt] m = -0.0000785
exactly agreeing with those here given.
OP THE ORBITS OF THE FOUR INNER PLANETS.
149
ACTION OF NEPTUNE ON THE EARTH.
E
A
905.47785591
45
905.10315254
90
904.80486595
135
904.75792710
180
904.98963355
225
905.36405558
270
905.66206098
315
905.70928102
Zi
3620.93441639*
2 2
3620.93441624
E
I
1.23952
45
0.86483
90
0.56594
135
0.51893
180
0.75119
225
1.12575
270
1.42319
315
1.47027
2,
3.97984
2 2
3.97977
E
log A'
0.00047205
45
0.00035874
90
0.00022606
135
0.00019146
180
0.00031307
225
0.00044509
270
0.00052334
315
0.00053136
2i
0.00153452 1
y
0.00152666 1
B cos e B sin g
h
+23.748411 +24.815277 40.194861
904.17306
+ 1.127297 +28.979673 54.817494
904.17306
-17.814345 +15.920217 16.543607
904.17365
-21.980762 - 6.713038 2.941510
904.17372
- 8.931313 -25.661837 42.984089
904.17317
+ 13.689802 -29.826236 58.066960
904.17304
+32.631450 -1.6.766778 18.349806
904.17361
+36.797860 + 5.866476 2.246398
904.17374
+29.634203f - 1.693121f 118.072363
3616.69349
+29.634196 - 1.693125 118.072362
3616.69355
G G' G"
e
O 1 it
904.17301 1.274452 0.034882
2 10 50.926
904.17299 0.930076 0.065186
1 54 4.344
904.17363 0.596627 0.030667
1 30 33.476
904.17372 0.525125 0.006195
1 23 20.558
904.17312 0.809936 0.058696
1 46 33.999
904.17297 1.180229 0.054414
2 7 3.526
904.17359 1.437325 0.014120
2 17 46.322
904.17374 1.471958 0.001688
2 18 49.394
3616.69335 4.118340 0.138365
7 45 44.723
3616.69342 4.107388 0.127483
7 43 17.822
ACTION OF NEPTUNE ON THE EARTH.
og/V log Wo logtf logP
logQ
'363061 0.17679925 5.5513789 9.9124729
2.7719098
347955 0.17662931 5.5555725 9.9164864
2.7759190
330267 0.17643033 5.5658262 9.9265957
2.7859899
'325655 0.17637844 5.5760491 9.9367960
2.7961727
'341867 0.17656083 5.5803398 9.9411989
2.8006208
'359467 0.17675882 5.5762686 9.9373080
2.7967497
'369899 0.17687617 5.5661355 9.9273173
2.786753T
370968 0.17688820 5.5557902 9.9169945
2.7764257
405094 0.70666658 2.2636803 9.7075847
1.1452735
404045 0.70665477 2.2636803 9.7075848
1.145267T
* 4a" + 2aV + 4[a' 2 - 2kaa'ee' cos K\ = 3620.93441628.
t4[a'V-fraa'ecos A] = +29.634198.
| - 4k'aa' cos *.'. e sin A"' = - 1.693118.
150
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF NEPTUNE ON THE EARTH.
E
logF
Ji' Ji Ja
F 2
2.7718889
903.9916758 +0.58828709 -13.723633
-190.54625
45
2.7758798
904.1719863 +0.01502688 + 7.632175
-222.52292
90
2.7859714
903.5306465 -0.23467770 +24.659009
-122.24477
135
2.7961690
903.3686400 +0.17398234 +27.382788
+ 51.54664
180
2.8005856
904.0154903 +0.15962945 +14.207956
+ 197.04666
225
2.7967170
904.1717548 -0.46647484 - 7.147853
+229.02334
270
2.7867446
903.5394237 -0.52719220 -24.174694
+ 128.74518
315
2.7764247
903.3535982 +0.26287497 -26.898466
- 45.04625
2!
1.145190o
3615.0772363* -0.01395336 + 0.968638
+ 13.00082
2 2
1.1451905
3615.0659793 -0.01459065 + 0.968644
+ 13.00081
E
F,
100000 X Ro 10000000 X So 100000 X We 100000 X #<">
10000000 X -S ( ">
- 5.0649535
1.7793453 +0.19215285 -0.08120439 0.0000000
+ 0.19543047
45
- 6.5852856
1.7972262 -0.17462602 +0.04549983 +1.2860826
-0.17672179
90
- 1.8342882
1.8362237 -0.24659977 +0.15062685 +1.8362237
-0.24659977
135
- 0.4453113
1.8790643 +0.15337661 +0.17125249 +1.3131267
+0.15157902
180
- 5.4164244
1.9016607 +0.27295016 +0.08971979 0.0000000
+0.26844799
225
- 6.9853355
1.8852158 -0.09387083 -0.04482073 -1.3174254
-0.09277065
270
- 2.0485736
1.8383392 -0.21373143 -0.14796412 -1.8383392
-0.21373143
315
- 0.3483062
1.7940628 +0.11988971 -0.16075330 -1.2838186
+0.12132856
Si
- 14.3642397
7.3555689 +0.00477181 +0.01117813 -0.0021155
+ 0.00354726
2 2
- 14.3642386
7.3555691 +0.00476947 +0.01117829 -0.0020347
+ 0.00341514
E
100000X[osin, I00000x[-Rocos, imm l(mQQ
+ (eosv+cosE)St,\ . (r , ,\ r, 1 XWoCos (W+JT) XlFosin (W+JT)
I- I ^i<l<- ,n -I- 1 1 Q1T1 JlX., 1
100000x-2-flo
a
v / .1
+0.0038431
-1.7793453 +0.01460463 -0.07988026
- 3.4990076
45
+ 1.2834469
-1.2580636 -0.03774087 +0.02541381
- 3.5518258
90
+ 1.8360068
+0.0258636 -0.14769558 -0.02957146
- 3.6724475
135
+ 1.3107603
+ 1.3464279 -0.09567261 -0.14203579
- 3.8026901
180
-0.0054590
+ 1.9016607 +0.01613612 -0.08825681
- 3.8671076
225
-1.3159048
+ 1.3499919 -0.03657334 +0.02590924
- 3.8151452
270
-1.8380449
+0.0351057 -0.14597725 -0.02416657
- 3.6766784
315
-1.2819528
-1.2550750 -0.09294409 -0.13116027
- 3.5455738
2i
-0.0036540
+0.1832847 -0.26293208 -0.22187510
-14.7152411
2 2
-0.0036504
+0.1832812 -0.26293091 -0.22187301
-14.7152409
sin if \A i (> > + cos <f
. B = + 0.0000000000000014.
* 2,(J,' - G") = 3614.9388718.
2,(J,' - G") = 3614.9384968.
OF THE ORBITS OF THE FOUR INNER PLANETS. 151
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] m = -0.011831221m' n 8.0730296
[dx/dt] m = [dTr/dt]^ = +35.402545 TO' p 1.5490345
[dp/e&]oo = -- 0.71895833 TO' n 9.8567037
[dq/dt] m = -- 0.85200049 TO' n 9.9304399
[dL/dt] w = -47.671428 TO' n 1.6782582
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'
[de/dt] w = -0.00000060056972
[d x !dt] w = [dw/dt] w = +0.0017970838
[dp/dtlao = -0.000036495344
[dg/dt] m = -0.000043248757
[dL/dt] m = -0.0024198698
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt} 00 0.00000 0.00000 -0.00000060057
e[drldt]oo +0.00003 +0.00003 +0.00003013915
[dpldt] m -0.00004 -0.00004 -0.00003649534
[dg/dt] m -0.00004 -0.00004 -0.00004324876
NOTES.
The mutual inclination is here nearly twice as great as in the case of Uranus, and
yet the convergence of the perturbing function is more rapid because the eccentricity
of Neptune is so much smaller than that of Uranus. Hence, although the sums of
e, G, G', G", etc., are in great disagreement, the final sums from which the differential
coefficients are obtained are almost identical. The greatest error arising from a divi-
sion into only four parts occurs with the coefficient [dp/dt] 00 and amounts to but
0". 000000000002
The results of HILL are :
= -00000366 [dqldt] n = -0.0000435
152
THE SECULAR VARIATIONS OF THE ELEMENTS
E
30
60
90
120
150
180
210
240
270
300
330
MARS.
ACTION OF MERCURY ON MARS.
A
B sin (
B COS e
1000 x g
2.01395126
-0.51051059
-0.07779H4
1.6508943
2.19162773
-0.49427471
+0.22547307
1.5475600
2.44453125
-0.33152771
+0.47957033
0.6962245
2.71228886
+0.06587754
+0.61641571
0.0274906
2.92685114
+0.23149508
+0.59934132
0.3394636
3.02703005
+0.48090911
+0.43292228
1.4649936
2.97859089
+0.61553465
+0.16175058
2.4000193
2.79081655
+0.59929870
-0.14151362
2.2750785
2.51771727
+0.43655177
-0.39561101
1.2072061
2.23986183
+0.17090155
-0.53245622
0.1850128
2.03539750
-0.12647099
-0.51538178
0.1013191
1.95541421
-0.37588517
-0.34896288
0.8949940
14.91703931*
+0.31507221f
+0.25187830J
6.9351269
14.91703923
+0.31507194
+0.25187834
6.9351295
1.87197670
2.04795479
2.29696077
2.56269605
2.77862654
2.88207295
2.83609064
2.64795925
2.37176681
2.09074126
1.88662892
1.81063180
14.04205038
14.04205609
G'
G"
0.13564010
1.87146851
0.14234545
0.00619717
16
20
8.69
30
0.13733849
2.04755912
0.14301881
0.00528466
15
35
30.16
60
0.14123602
2.29682015
0.14348918
0.00211253
14
34
33.48
90
0.14325835
2.56269162
0.14333762
0.00007484
13
41
0.67
120
0.14189014
2.77858020
0.14279207
0.00085559
13
8
25.31
150
0.13862265
2.88188764
0.14237834
0.00357038
12
59
49.56
180
0.13616579
2.83577714
0.14242175
0.00594246
13
12
30.22
210
0.13652285
2.64761705
0.14287917
0.00601413
13
42
7.60
240
0.13961601
2.37153873
0.14339402
0.00354993
14
24
9.49
270
0.14278611
2.09069583
0.14344844
0.00061690
15
12
58.81
300
0.14243412
1.88659813
0.14284089
0.0003759S
15
59
29.51
330
0.13844795
1.81033610
0.14221981
0.00347617
16
27
51.54
2!
0.83698218
14.04078286
0.85728335
0.01903366
87
39
16.70
2 a
0.83697639
14.04078736
0.85728218
0.01903708
87
39
18.34
* 6a 2 + 3a 2 e ! + 6[a' s - Zkaa'ee' cos K] = 14.91703924.
t - Cfc'oo' cos >' e sin K' = + 0.31507212.
I 6[a'V - A-oaV cos A'] = + 0.25187831.
OF THE ORBITS OF THE FOUK INNER PLANETS.
153
ACTION
OF MERCURY ON
MARS.
E
log /Co
log V
logtfo
log N log /'
logQ
0.02698253
0.30881667
0.21633345
0.0802043 9.8417843
0.0229194
30
0.02453876
0.30558607
0.21270775
0.0315423 9.7124165
9.9318941
60
0.02139874
0.30143126
0.20804354
9.9863199 9.5646986
9.8328372
90
0.01882471
0.29802213
0.20421541
9.9544525 9.4350567
9.7499589
120
0.01734193
0.29605696
0.20200830
9.9396879 9.3478315
9.6977395
150
0.01696133
0.29555239
0.20144156
9.9427992 9.3179221
9.6840260
180
0.01752419
0.29629857
0.20227967
9.9632970 9.3524331
9.7119954
210
0.01887658
0.29809086
0.20429260
9.9992756 9.4496853
9.7797276
240
0.02088535
0.30075154
0.20728036
0.0456467 9.5950386
9.8772472
270
0.02335208
0.30401640
0.21094579
0.0914150 9.7545935
9.9819419
300
0.02583676
0.30730232
0.21463400
0.1193985 9.8751689
0.0582665
330
0.02741722
0.30939106
0.21697800
fl.1150661 9.9072726
0.0734518
Si
0.12996950
1.81065732
1.25057932
0.1345541 7.5769549
9.2010052
2 2
0.12997068
1.81065891
1.25058111
0.1345506 7.5769465
9.2010002
E
log V
/i'
J 2
J,
Ft
0.0211422
0.148654076
-0.10722959
-0.015479687
+0.015330048
30
9.9305073
0.148321182
-0.10335718
-0.012048361
+0.014842503
60
9.8323421
0.145601872
-0.06932482
-0.006505466
+0.009955397
90
9.7499432
0.143412552
-0.01386724
-0.000353153
+0.001978227
120
9.6975735
0.143648964
+0.04858051
+0.004751607
-0.006951532
150
9.6833581
0.145958760
+0.10133761
+0.007449454
-0.014441148
180
9.7108663
0.148399366
+0.13001805
+0.007034457
-0.018483798
210
9.7785043
0.148942861
+0.12663289
+0.003626261
-0.017996253
240
9.8764415
0.146960494
+0.09193479
-0.001870362
-rO.013109151
270
9.9817831
0.144077345
-0.03534552
-0.007999542
-0.005131979
300
0.0581594
0.143355345
-0.02759166
-0.013127432
+0.003797779
330
0.0724209
0.145910972
0.07970449
-0.015871553
+0.011287401
2,
9.1965248
0.876620117*
+0.06638728
-0.025196883
-0.009461257
2 2
9.1965168
0.876623672
+0.06638711
-0.025196894
-0.009461249
* 2,(J,' - G") =
0.857586462.
S,(J,' - G") =
0.857586592.
154
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF MERCURY ON MARS.
E
1000 X F, Ro So W,, fl<>
8*
+0.3749543 -1.1968194 -0.10192940 -0.015991438 0.00000000
-0.07377742
30
+0.9451901 -1.0712026 -0.08041942 -0.009779348 -0.38240323
-0.05741705
60
+0.8478648 -0.9676664 -0.04346887 -0.004110828 -0.57689880
-0.02992414
90
+0.1645035 -0.9003960 -0.00725842 -0.000153771 -0.59093068
-0.00476371
120
-0.4045215 -0.8699380 +0.02266377 +0.002278066 -0.47241848
+0.01421151
150
-0.2449197 -0.8749740 +0.04587694 +0.003542283 -0.26566469
+0.02785885
180
+0.5450969 -0.9160775 +0.06265312 +0.003737622 0.00000000
+0.03761136
210
+ 1.2365773 -0.9949389 +0.07097316 +0.002525788 +0.30208776
+0.04309857
210
+ 1.1824210 -1.1083446 +0.06401101 -0.000941853 +0.60188486
+0.04013864
270
+0.4525829 -1.2337308 +0.03097697 -0.007413721 +0.80969859
+0.02033021
300
-0.2401087 -1.3158841 -0.02869644 -0.015188691 +0.78449755
-0.01975474
330
-0.2482279 -1.2994033 -0.08505077 -0.018952193 +0.46386745
-0.06072369
Si
+2.3057068 -6.3747320 -0.02476681 -0.030217122 +0.33706513
-0.03149479
Z 2
+2.3057062 6.3746456 -0.02490154 -0.030230962 +0.33665520
-0.03161682
D . Ro cos v
E
RO sin v
+ (cos ,+cos E)S a + g BCC' ,+l) sin S.
-2 -Bo
a
-0.2038588 +1.1968194 -0.004111709 +0.015453801
2.1703888
30
-0.7173754 +0.8165787 -0.007231890 +0.006582964
1.9693580
60
-0.9154636 +0.3357086 -0.004043881 +0.000738872
1.8450804
90
-0.8957943 -0.0984951 -0.000144266 -0.000053225
1.8007921
120
-0.7408612 -0.4547261 +0.001481632 +0.001730419
1.8210138
150
-0.4834776 -0.7324563 +0.000768358 +0.003457946
1.8912954
180
-0.1253062 -0.9160795 -0.000961015 +0.003611962
2.0030408
210
+0.3338218 -0.9514367 -0.001700736 +0.001867379
2.1505951
240
+0.8448003 -0.7366597 +0.000887111 -0.000316419
2.3200626
270
+ 1.2254640 -0.1770222 +0.007311027 +0.001229687
2.4674617
300
+ 1.1635310 +0.6123066 +0.011609114 +0.009794123
2.5090379
330
+0.5585553 +1.1811278 +0.005822238 +0.018035720
2.3888945
Si
+0.0228415 +0.0373693 +0.004861252 +0.031012758
12.6686243
2 2
+0.0211938 +0.0382962 +0.004824731 +0.031120471
12.6683967
sin p JA,<'> + cos v Bo (e) = + 0".000000008.
OF THE ORBITS OF THE FOUR INNER PLANETS. 155
DEFERENTIAL COEFFICIENTS.
u log coeff.
[de/dtlw = + 2517.5250 m' p 3.4009738
[d x /dt] w = + 46380.761 TO' p 4.6663379
[di/dt] w =+ 558.61256 m' p 2.7471107
[dn/di]oo =+110961.28 TO' p 5.0451714
[dir/dt] M = + 46438.628 TO' p 4.6668794
=+1455134.1 TO' p 6.1629030
TO
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF
[de/dt]^ = +0.00033567000
[dx/dt] 00 = +0.0061841007
[di/dt] w = +0.000074481672
[cKl/dt] w = +0.014794833
[dw/dt] m = +0.0061918174
[dL/dt] w = +0.19401785
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt} 00 +0.00036 +0.00033 +0.0003357
e[dTr/dt] 00 +0.00058 +0.00057 +0.0005775
[di/dt]w +0.00008 +0.00007 +0.0000745
sin i [dtt/dt] w +0.00047 +0.00048 +0.0004778
NOTES.
On account of the large eccentricities of both orbits and the high mutual incli-
nation, the coefficients of the expansion diminish but slowly. Thus the combined
effect of all terms from the 6th to the llth orders is l/30th of the whole with [de/dt]o ,
l/90th with [dw/dt] o, and 1 /200th with [di/dt] <>. Yet all of the variations are very
small on account of the smallness of the mass of Mercury. A comparison with the
computation of Mars on Mercury renders it evident that a division into twelve parts
is sufficient and that terms of orders above the eleventh are wholly inappreciable.
156
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF VENUS ON MARS.
E
A
B COS e
B sin e
1000 xg
h
2.41946745
-0.9101348
-0.4038413
0.003995803
1.8967820
30
2.47732532
-0.5530947
-0.8457642
0.017525935
1.9549495
60
2.63301427
-0.0217107
-1.0499345
0.027008907
2.1104659
90
2.85220984
+0.5416339
-0.9616453
0.022657521
2.3292431
120
3.07987480
+0.9859909
-0.6045536
0.008954722
2.5566759
150
3.25131035
+ 1.1922952
-0.0743414
0.000135408
2.7282304
180
3.31318850
+ 1.1052676
+0.4869209
0.005808973
2.7904410
210
3.24523275
+0.7482272
+0.9288436
0.021138190
2.7227529
240
3.06934808
+0.2168441
+ 1.1330145
0.031452377
2.5468467
270
2.84005464
-0.3465004
+ 1.0447250
0.026741552
2.3172324
300
2.62248756
-0.7908574
+0.6876330
0.011585000
2.0993294
330
2.47124772
-0.9971620
+0.1574211
0.000607167
1.9481304
2,
17.13738066*
+0.5853997f
+0.2492390t
0.088805782
14.0005408
2 2
17.13738062
+0.5854002
+0.2492388
0.088805773
14.0005387
G
G'
10000 X G"
0.5226610
1.8967805
0.5226665
0.0403053
31
39
49.77
30
0.5223513
1.9549433
0.5223748
0.1716188
31
7
34.86
60
0.5225239
2.1104579
0.5225564
0.2449047
29
50
30.35
90
0.5229423
2.3292377
0.5229663
0.1860051
28
17
2.93
120
0.523174o
2.5566741
0.5231829
0.0669458
26
53
44.43
150
0.5230555
2.7282304
0.5230555
0.0009489
25
58
2.26
180
0.5227230
2.7904401
0.5227279
0.0398246
25
38
46.94
210
0.5224554
2.7227494
0.5224737
0.1485921
25
58
48.56
240
0.5224769
2.5468406
0.5225067
0.2363523
26
55
59.83
270
0.5227977
2.3172260
0.5228262
0.2207297
28
21
35.72
300
0.5231337
2.0993259
0.5231478
0.1054853
29
56
49.75
330
0.5230928
1.9481302
0.5230936
0.0059582
31
12
36.97
2i
3.1366929
14.0005190
3.1367881
0.7338180
170
55
41.07
Si
3.1366950
14.0005170
3.1367900
0.7338528
170
55
41.30
* 6o 2 + 3aV + 6[o' 2 - Zkaa'ee' cos K\ = 17.13738065.
t 6[a'V - kaa'c cos A'] = + 0.5854002.
\ - 6fc'aa' cos <p' e sin K' = + 0.2492389.
OF THE ORBITS OF THE FOUR INNER PLANETS.
157
ACTION OF VENUS ON MAKS.
I
logtfo
log LO
log # log N log P log Q
0.10710693
0.41334627
0.33316554 0.1537292 0.0110396 0.2088768
30
0.10323050
0.40834940
0.32760279 0.1420610 9.9681345 0.1785258
60
0.09431348
0.39683240
0.31477271 0.1149515 9.8630204 0.1053425
90
0.08414014
0.38365383
0.30007678 0.0820056 9.7312249 0.0148651
120
0.07564158
0.37216263
0.28775224 0.0524079 9.609667o 9.9324836
150
0.07024983
0.36559243
0.27991042 0.0325881 9.5264184 9.8766175
180
0.06843912
0.36323215
0.27727293 0.0260737 9.4979593 9.8576733
210
0.07032296
0.36568773
0.28001691 0.0339679 9.5296356 9.8789748
240
0.07586487
0.37290310
0.28807661 0.0551373 9.6180289 9.9372082
270
0.08461922
0.38427537
0.30077024 0.0858518 9.7401822 0.0216495
300
0.09502686
0.39775494
0.31580088 0.1191144 9.8727052 0.1128332
330
0.10383003
0.40912260
0.32846370 0.1449403 9.9748268 0.1837860
Zi
0.51639284
2.31668149
1.81684091 0.5214139 8.4704208 0.154417o
S 2
0.51639268
2.31668136
1.81684084 0.5214145 8.4704222 0.1544186
E
log V
Ji'
J 2 J 3 f\
0.2088756
0.52281610
-0.0030557479 -0.012760938 +0.0014450479
30
0.1785212
0.52261080
-0.0058421642 -0.016264386 +0.0030263615
60
0.1053364
0.52273704
-0.0069459413 -0.015093265 +0.0037569353
90
0.0148609
0.52303324
-0.0063117580 -0.009561780 +0.0034410135
120
9.9324822
0.52319006
-0.0041023613 -0.001152260 +0.0021632475
150
9.8766174
0.52307981
-0.0007324438 +0.007882145 +0.0002660128
180
9.8576725
0.52281605
+0.0030484592 +0.015121070 -0.0017423280
210
9.8789719
0.52262308
+0.0062562314 +0.018625039 -0.0033236408
240
9.9372033
0.52266847
+0.0079298727 +0.017454980 -0.0040542164
270
0.0216445
0.52292962
+0.0074476695 +0.011924014 -0.0037382940
300
0.1128306
0.52316808
+0.0048298806 +0.003513972 -0.0024605273
330
0.1837858
0.52311928
+0.0008867096 -0.005521489 -0.0005632929
Si
0.1544005
3.13739580*
+0.0017041620 +0.007083559 -0.0008918410
2i
0.1544017
3.13739582
+0.0017042445 +0.007083543 -0.0008918399
* 2,(J/ - G") = 3,
13732242.
2,(J,' - G") = 3.
13732243.
158
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF VENUS ON MARS.
E
100000 X F,
Ro 1000 X So 1000 X W<>
<'
S<">
- 2.705324
-1.4244790 -3.460778 -20.67006
0.0000000
-0.002504942
30
- 0.803003
-1.3866027 -6.000157 -24.54092
-0.4949963
-0.004283931
60
+ 5.500746
-1.3028006 -6.111878 -19.19609
-0.7766973
-0.004207440
90
+ 9.816975
-1.2077652 -4.678320 - 9.84176
-0.7926572
-0.003070385
120
+ 7.633806
-1.1282518 -2.631099 - 0.95528
-0.6126954
-0.001649853
150
+ 0.880725
-1.0779056 -0.461908 + 5.93579
-0.3272800
-0.000280495
180
- 3.932916
-1.0618130 +1.648220 +10.88332
0.0000000
+0.000989445
210
- 2.161955
-1.0812437 +3.609363 +14.08781
+0.3282934
+0.002191792
240
+ 4.374565
-1.1352348 +5.187600 +15.12313
+0.6164876
+0.003252928
270
+ 9.225323
-1.2184712 +5.773047 +12.58406
+0.7996835
+0.003788855
300
+ 7.735212
-1.3155488 +4.427365 + 4.61417
+0.7842976
+0.003047815
330
+ 1.648024
-1.3961376 +0.822269 - 8.41473
+0.4984001
+0.000587075
Si
+ 18.606089
-7.3681280 -0.940570 -10.20081
+0.0113925
-0.001072047
2 2
+ 18.606089
-7.3681260 -0.935706 -10.18975
+0.0114435
-0.001067089
E
. Rt, cos v
Rosmv
+ (cos v + cos )S +T- sec 2 v +lj sin vS,
Wo sin u
-2^0
a
-0.0069216
+ 1.4244790 -0.005314674
+0.019975127
2.5832417
30
-0.7611745
+ 1.1593980 -0.018148167
+0.016519714
2.5492068
60
-1.1839523
+0.5449664 -0.018883467
+0.003450266
2.4840909
90
-1.2020643
-0.1220026 -0.009233404
-0.003406539
2.4155306
120
-0.9266826
-0.6439879 -0.000621304
-0.000725630
2.3617340
150
-0.4956901
-0.9571936 +0.001287534
+0.005794466
2.3299411
180
-0.0032964
-1.0618130 -0.002798312
+0.010517416
2.3216920
210
+0.4917082
-0.9631868 -0.009486007
+0.010415471
2.3371562
240
+0.9297081
-0.6522758 -0.014244152
+0.005080676
2.3763511
270
+ 1.2126214
-0.1251906 -0.012409750
-0.002087273
2.4369424
300
+ 1.1939212
+0.5533939 -0.003526734
-0.002975358
2.5083988
330
+0.7575012
+ 1.1728178 +0.002585060
+0.008007817
2.5667361
2i
+0.0027764
+0.1647626 -0.045388643
+0.035322497
14.6355085
2 2
+0.0029019
+0.1646422 -0.045404734
+0.035243656
14.6355132
sin<f> j4i (<) + coaip
Bo (c) = O."0000000073.
OF THE ORBITS OF THE FOUR INNER PLANETS. 159
DIFFERENTIAL COEFFICIENTS.
[de/dt] w = + 324.6318 m' p 2.5113911
[dx/dfloo = + 201915.56 TO' p 5.3051698
[di/dt]oo 5236.2608 m' n 3.7190213
[dtt/dt] m + 126021.28 TO' p 5.1004439
[dTT/dtlw = + 201981.28 m' p 5.3053112
[dL/dt] m = +1681713.6 m' p 6.2257520
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[dg/dfloo = +0.0007954049
[dx/dfloo = +0.49472856
[di/di]oo = -0.012829757
[dO/d<]oo = +0.30877426
[dr/dfloo = +0.49488961
[dL/dt] m = +4.1204933
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
loo +0.00080 +0.00079 +0.000795
e[drldt]ao +0.04618 +0.04614 +0.0461574
[difdt] m -0.01280 -0.01284 -0.012830
sin i [dfl/d<]oo +0.00993 +0.00998 +0.009972
[dL/dt] M +4.117 +4.120493
NOTES.
The very close agreement of the sums of the functions near the beginning of the
computation is caused by the great circularity of the orbit of Venus. The discrepan-
cies increase however as the work proceeds because of the high eccentricity of Mars and
the rather large mutual inclination. All terms from the 6th to the llth orders, in-
clusive, produce an effect equal to l/30th of the whole in the very small coefficient
[de/dt]oo, and 1 /1000th of the whole in [dttfdt} 00 . Yet it is evident that terms of the
twelfth and higher orders are wholly inappreciable.
160
THE SECULAR VARIATIONS OF THE ELEMENTS
E
A
2.88085183
30
2.95824702
60
3.13279096
90
3.36510700
120
3.59664230
150
3.76166099
180
3.80855449
210
3.72106141
240
3.52632175
270
3.28390784
300
3.06247042
330
2.91764743
Si
20.00763175*
S 2
20.00763169
ACTION OF THE
EARTH ON MARS.
B cos f
B sin
1000 X g
-0.8153552
-1.1018750
0.34149937
-0.0872040
-1.3962265
0.54832354
+0.6917421
-1.2860904
0.46523043
+ 1.3127653
-0.8009781
0.18045398
+ 1.6094626
-0.0708746
0.00141289
+ 1.5023341
+0.7085891
0.14122581
+ 1.0200854
+ 1.3285566
0.49646115
+0.2919341
+ 1.6229077
0.74082034
-0.4870119
+ 1.5127725
0.64368353
-1.1080347
+ 1.0276597
0.29704612
-1.4047322
+0.2975561
0.02490366
-1.2976042
-0.4819076
0.06532098
+0.6141908J
+0.6800452f
1.97319103
+0.6141906
+0.6800443
1.97319077
1.8833414
1.9597848
2.1332651
2.3654511
2.5975998
2.7632686
2.8103218
2.7223893
2.5269386
2.2841910
2.0633533
1.9197321
14.0148199
14.0148169
E
I'
G
G'
G'
o
0.9972292
1.8831367
0.9976157
0.00018178
46
42
32.02
30
0.9981810
1.9594937
0.9987522
0.00028018
45
33
35.15
60
0.9992447
2.1330727
0.9996552
0.00021818
43
12
19.69
90
0.9993747
2.3653953
0.9995068
0.00007633
40
32
44.90
120
0.9987612
2.5975995
0.9987622
0.00000054
38
19
18.09
150
0.9981111
2.7632397
0.9981913
0.00005120
36
56
40.43
180
0.9979514
2.8102243
0.9982259
0.00017698
36
35
10.16
210
0.9983909
2.7222314
0.9988211
0.00027246
37
17
7.37
240
0.9991020
2.5267718
0.9995236
0.00025487
38
58
33.58
270
0.9994356
2.2840897
0.9996671
0.00013009
41
25
15.69
300
0.9988359
2.0633420
0.9988593
0.00001208
44
5
20.00
330
0.9976341
1.9196952
0.9977050
0.00003411
46
7
50.45
2,
5.9911243
14.0141470
5.9926419
0.00081443
247
53
13.54
2 2
5.9911274
14.0141449
5.9926435
0.00084437
247
53
13.98
* Go 2 + 3o 2 e 2 + 6[a' 2 - Zkaa'ee' cos K] = 20.00763172.
t 6[a'V - fcaa'e cos A'] = + 0.6141907.
t - 6fcW cos <p' e sin K' = + 0.6800448.
OF THE ORBITS OF THE FOUR INNER PLANETS.
161
ACTION
OF THE EARTH ON MABS.
E
log A'
log Lo'
lOgtfo
log N log P
logQ
0.25724275
0.60257672
0.54204483
0.3085062 0.3612353
0.5756272
30
0.24243067
0.58425966
0.52198206
0.2796590 0.2795065
0.5094350
60
0.21406633
0.54897788
0.48324184
0.2277018 0.1185796
0.3818936
90
0.18497596
0.51250382
0.44306081
0.1727907 9.9374590
0.2419337
120
0.16284593
0.48455413
0.41218099
0.1292685 9.7846780
0.1268772
150
0.15007482
0.46834273
0.39423491
0.1040949 9.6895845
0.0569033
180
0.14686278
0.16425584
0.38970669
0.0998549 9.6665742
0.0407933
210
0.15317152
0.47227918
0.39859494
0.1168787 9.7192209
0.0805051
240
0.16915889
0.49254542
0.42101792
0.1535252 9.8408510
0.1719333
270
0.19422097
0.52412777
0.45588060
0.2048055 0.0114576
0.3019483
300
0.22440852
0.56187400
0.49741669
0.2597585 0.1924850
0.4426015
330
0.24970710
0.59326698
0.53185208
0.3003845 0.3271715
0.5489966
2,
1.17458520
3.15478399
2.74560896
1.1786151 9.9644030
1.7397260
2 2
1.17458104
3.15478014
2.74560540
1.1786131 9.9643999
1.7397219
E
logF
JV
J 2
J,
F,
0.5755790
0.99892684
-0.018227992
+0.028809050
+0.018467466
30
0.5093633
0.99952648
-0.022898545
+0.020515873
+0.023400805
60
0.3818419
0.99990323
-0.021379458
+0.006120062
+0.021554920
90
0.2419173
0.99967015
-0.013761084
-0.010518960
+0.013424123
120
0.1268771
0.99911770
-0.001690224
-0.024941756
+0.001187862
150
0.0568937
0.99877639
+0.011675876
-0.033284760
-0.011875978
180
0.0407608
0.99892204
+0.022549333
-0.033314577
-0.022266656
210
0.0804536
0.99942130
+0.027739796
-0.025024224
-0.027199994
240
0.1718817
0.99985593
+0.025696938
-0.010634071
-0.025354120
270
0.3019193
0.99982014
+0.017055202
+0.006002126
-0.017223615
300
0.4425985
0.99929729
+0.004464507
+0.020427743
-0.004987052
330
0.5489877
0.99880864
-0.008398230
+0.028776417
+0.008076789
2,
1.7395389
5.99602303*
+ 0.011413104
-0.013533549
-0.011397580
Zo
1.7395349
5.99602310
+0.011413015
-0.013533528
-0.011397570
*Z,(Ji' - G") = 5.99517860.
2 2 (J/ - G") = 5.99517873.
162
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF THE EARTH ON MARS.
E
1000 XF,
n c* "nr
ft/Q Ofl If
R<>
fl
-0.13911285
-2
.0302096 -0.026171976 +0.10810004
0.0000000
-0.018943511
30
-0.51397488
-1
.9019956 -0.029451014 +0.06531256
-0.6789837
-0.021027135
60
-0.65316258
-1
.6889696 -0.023181583 +0.01388518
-1.0069217
-0.015958290
90
-0.40694037
-1
.4884239 -0.012395858 -0.01871310
-0.9768538
-0.008135411
120
-0.02612649
-1
.3462165 -0.001540204 -0.03342035
-0.7310608
-0.000965798
150
+0.08995598
-1
.2702099 +0.007499119 -0.03789964
-0.3856685
+0.004553852
180
-0.20223795
-1
.2578222 +0.014435140 -0.03668660
0.0000000
+0.008665574
210
-0.63957154
-1
.3082364 +0.019136251 -0.03045222
+0.3972143
+0.011620521
240
-0.80757787
-1
.4237326 +0.020598-119 -0.01635704
+0.7731558
+0.012916412
270
-0.54879816
-1
.6023546 +0.016496265 +0.01146539
+ 1.0516264
+0.010826511
300
-0.11741630
-1
.8174946 +0.004601712 +0.05641742
+ 1.0835451
+0.003167836
330
+0.07369507
-1
.9931947 -0.012572900 +0.10202141
+0.7115406
-0.008976672
2,
-1.94563404
-9
.5644451 -0.011258492 +0.09193865
+0.1187184
-0.011117777
2 2
-1.94563390
-9.5644151 -0.011288137 +0.09173440
+0.1188753
-0.011138334
Ro COS V
E
Ro sin v
+ (c,osv + cosE)S
. (r \ . Wa cos u
+ ( - seo 2 <f + 1 j sin i'.S
W sin u
2 RQ
a
-0.0523440
+2.0302096 +0.027794619
-0.10446568
3.6817114
30
-1.0803151
+ 1.5681913 +0.048299056
-0.04396512
3.4967331
60
-1.5490313
+0.6794308 +0.013659050
-0.00249569
3.2204119
90
-1.4807799
-0.1636143 -0.017556368
-0.00647718
2.9768478
120
-1.1074128
-0.7656902 -0.021736270
-0.02538610
2.8179922
150
-0.5982290
- 1 . 1 202 1 78 - 0.008220822
-0.03699731
2.7456152
180
-0.0288703
- 1 .2578222 +0.009432839
-0.03545319
2.7502731
210
+0.5690362
-1.1796147 +0.020504958
-0.02251409
2.8278111
240
+ 1.1509410
-0.8419068 +0.015406346
-0.00549521
2.9802545
270
+ 1.5938315
-0.1824413 -0.011306572
-0.00190172
3.2047092
300
+ 1.6480563
+0.7072282 -0.043121307
-0.03637964
3.4654744
330
+ 1.0579845
+ 1.6887208 -0.031341647
-0.09708793
3.6643987
2,
+0.0613389
+0.6114494 +0.001435277
-0.20967551
18.9161075
2 2
+0.0615274
+0.6110240 +0.000378605
-0.20894335
18.9161151
sin ip |Ai (>) + cos
f
B <> = + 0.000000102.
OF THE ORBITS OF THE FOUR INNER PLANETS. 163
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] w = + 7024.3393 TO' p 3.8466055
[d x /dt] w = + 749340.69 TO' p 5.8746793
[di/dt]<n = + 104.61082 TO' p 2.0195766
[dtt/dt] w = - - 747594.66 TO' n 5.8736662
[dvldt] m = + 748950.76 TO' p 5.8744532
[dL/dt] m = +2175235.9 TO' p 6.3375064
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[<fe/cft]oo = +0.021481158
[d x /dt] w = +2.2915614
[dildt] m = +0.00031991074
[dQ/<ft]oo = -2.2862242
[dr/dt] m = +2.2903688
*dLldt] m = +6.6520970
COMPARISON WITH OTHER RESULTS.
Leverrier. Neweomb. Method of Gauss.
[de/dt]oo
+0.02151
+o!o2148
+o!o2148116
e[dirldt] M
+0.21276
+0.21374
+0.21361818
[difdt] m
+0.00030
+0.00032
+0.00031991
sin i [dtt/dt] m
-0.07391
-0.07379
-0.07383093
[dL/diloo
+6.638
+6.6520970
NOTES.
As in all cases in which Mars is the disturbed body, the gradual increase in the
discrepancies in the sums of the functions as the computation proceeds is caused prin-
cipally by the large value of e. The greatest effect which is here produced by the
inclusion of all terms from the fifth to the eleventh orders occurs with the coefficient
[dirldt] QQ and amounts to 0".0007. It is evident that a division into twelve parts is
fully sufficient.
164
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF JUPITER ON MARS.
E
A
B cos f
B sin f
g
h
29.52014024
+6.924444
-4.477639
1.2637451
27.008366
30
29.73057269
+ 8.555992
-0.736229
0.0341655
27.006182
60
29.83402815
+8.090126
+3.325863
0.6972206
27.007669
90
29.81017790
+5.651672
+6.620206
2.7625134
27.011679
120
29.66910846
+ 1.894010
+8.264081
4.3047762
27.014415
150
29.44492347
-2.175995
+7.817013
3.8516159
27.013253
180
29.19030067
-5.467791
+5.398794
1.8371936
27.009343
210
28.96977017
-7.099343
+ 1.657384
0.1731141
27.006389
2-10
28.84611897
-6.633477
-2.404708
0.3644905
27.006977
270
28.85987156
-4.195019
-5.699050
2.0472288
27.010168
300
29.01103882
-0.437358
-7.342925
3.3985953
27.012567
330
29.25541959
+3.632647
-6.895857
2.9973538
27.011662
2,
176.07073531*
+4.369955t
+2.7634661
11.8660213
162.059336
2 2
176.07073538
+4.369955
+2.763467
11.8660215
162.059333
E
o
30
60
90
120
150
180
210
240
270
300
330
2,
I
G
G'
G"
27.0064605
27.0061300
27.0066036
27.0074648
27.0078870
27.0074649
27.0066097
27.0061336
27.0064416
27.0071624
27.0075479
27.0071916
162.0415503
162.0415473
* 6a 2 + SaV + 6[o' ! - 2fcaa'ee' cos A'] = 176.07073528.
t 6[a'V - kaa'e cos K] = + 4.369954.
t - Qk'aa' cos *>' e sin K' = + 2.763466.
2.448742
2.661358
2.763328
2.735467
2.591662
2.368639
2.117926
1.900349
1.77611T
1.786671
1.935440
2.180725
13.633208
13.633209
2.4695960
0.0189481
17
39'
53.68
2.6618860
0.0004753
18
17
56.75
2.7737001
0.0093077
18
43
15.01
2.7765210
0.0368400
18
48
58.35
2.6581522
0.0599626
18
28
29.57
2.4330416
0.0586151
17
39
45.34
2.1522663
0.0316074
16
30
39.82
1.9039718
0.0033673
15
24
39.18
1.7842098
0.0075644
14
55
27.23
1.8310752
0.0413982
15
15
16.64
2.0032755
0.0628165
16
2
14.95
2.2348561
0.0496603
16
53
32.61
13.841199!)
0.1902067
102
20
0.26
13.8413516
0.1903562
102
20
8.87
OF THE ORBITS OF THE FOUR INNER PLANETS.
165
ACTION OF JUPITEU ON MARS.
E
logtfo
log L '
log No log N log P
logQ
0.03165483
0.31498612
0.22325507 8.3476456 5.7990872
7.1391284
30
0.03402451
0.31811150
0.22676030 8.3623570 5.8175286
7.1576473
60
0.03565117
0.32025551
0.22916442 8.3954324 5.8524487
7.1929772
90
0.03602468
0.32074766
0.22971622 8.4366023 5.8931984
7.2342427
120
0.0346975G
0.31899875
0.22775524 8.4742980 5.9283892
7.2695995
150
0.03164635
0.31497492
0.22324250 8.4991683 5.9492925
7.2899854
180
0.02757619
0.30960108
0.21721367 8.5057537 5.9513987
7.2909894
210
0.02396325
0.30482489
0.21185335 8.4928484 5.9346402
7.2731852
240
0.02244999
0.30282273
0.20960576 8.4633475 5.9029922
7.2413643
270
0.02347170
0.30417466
0.21112344 8.4239469 5.8638333
7.2029262
300
0.02598821
0.30750252
0.21485868 8.3844577 5.8269721
7.1668224
330
0.02889111
0.31133796
0.21916251 8.3560126 5.8027960
7.1428978
?!
0.17801795
1.87416671
1.32185284 0.5709348 5.2612881
3.3008810
2-2
0.17802160
1.87417159
1.32185832 0.5709353 5.2612888
3.3008846
E
logF
li'
Ji J,
F 2
7.1387522
27.0207890
-0.20899481 +0.30968500
+ 5.840160
30
7.1576379
27.0065721
-0.03630747 -0.02674093
+ 0.960260
60
7.1927926
27.0099958
+0.15284461 -0.36413185
- 4.337906
90
7.2335125
27.0290264
+0.31348417 -0.61207993
- 8.634698
120
7.2684108
27.0492502
+ 0.40064241 -0.70414548
-10.778797
150
7.2888221
27.0526590
+0.38540712 -0.61566134
-10.195688
180
7.2903609
27.0334483
+0.26823852 -0.37034175
- 7.041617
210
7.2731181
27.0094582
+0.08088102 -0.03392190
- 2.161717
240
7.2112135
27.0106120
-0.12278062 +0.30345694
+ 3.136448
270
7.2021019
27.0363209
-0.28345023 +0.55139880
+ 7.433241
300
7.1655732
27.0521350
-0.35632057 +0.64347029
+ 9.577337
330
7.1419111
27.0423420
-0.32638726 +0.55499853
+ 8.991231
2i
3.2971031
162.1762303*
+0.13362954 -0.18200685
- 3.604375
V
3.2971036
162.1763786
+0.13362735 -0.18200677
- 3.604371
* S,(J,' - G") = 161
.9860236.
2(Ji' - G") = 161
.9860224.
166
THE SECULAR VARIATIONS OF THE ELEMENTS
E
30
60
90
120
150
ISO
210
240
270
300
330
Si
E
F,
-0.11464084
-0.02209150
+0.08535418
+0.09095508
-0.02126155
-0.14774803
-0.16666131
-0.05846157
+0.07437981
+0.10831696
+0.01978452
-0.09401607
-0.12304519
-0.12304513
fto sin v +
(cos v + oos E)f><,
ACTION OF JUPITER ON MARS.
# 1000 X -So 1000 X W 1000 X R M 1000 X <S ( ">
0.011523729 +0.08005004 +0.4190397 0.000000 +0.05794093
0.011964482 +0.01088811 -0.0398940 +4.271140 +0.00777378
0.012921829 -0.07058125 -0.5615348 +7.703673 -0.04858841
0.014185469 -0.13852471 -1.0407895 +9.309934 -0.09091387
0.015433331 -0.17072189 -1.3081943 +8.381048 -0.10705259
0.016299442 -0.15776374 -1.2103390 +4.948932 -0.09580229
0.016504164 -0.10615120 -0.7376107 0.000000 -0.06372374
0.015978601 -0.03427562 -0.0686500 -4.851517 -0.02081393
0.014896574 +0.03689409 +0.5347724 -8.089562 +0.02313475
0.013593080 +0.09184683 +0.8860645 -8.921148 +0.06027915
0.012429971 +0.12131684 +0.9434385 -7.410440 +0.08351498
0.011688531 +0.11863615 +0.7635196 -4.172629 +0.08470265
0.083709598 -0.10919337 -0.7100892 +0.584719 -0.05477408
0.083709605 -0.10919298 -0.7100884 +0.584712 -0.05477451
i
flo COS V +
Ir \ . 1000 X Wo cosu 1000 X JTo sin u -2-R,,
+0.000160100
-0.011523729
+0.10774325
-0.4049515
-0.020897865
30
+0.006498115
-0.010046690
-0.02950189
+0.0268547
-0.021996162
60
+0.011621455
-0.005638033
-0.55239000
+0.1009292
-0.024638457
90
+0.014136555
+0.001045999
-0.97645422
-0.3602495
-0.028370938
120
+0.012896599
+0.008458998
-0.85083667
-0.9937043
-0.032306104
150
+0.007784436
+0.014315488
-0.26253500
-1.1815227
-0.035231972
180
+0.000212302
+0.016504164
+0.18965406
-0.7128120
-0.036086942
210
-0.007299886
+0.014215603
+0.04622538
-0.0507547
-0.034538456
240
-0.012311629
+0.008381401
-0.50369057
+0.1796589
-0.031182525
270
-0.013542396
+0.001084104
-0.87379071
-0.1469683
-0.027186159
300
-0.011129593
-0.005518211
-0.72109467
-0.6083574
-0.023700620
330
-0.006127613
-0.009949901
-0.23455823
- 0.7265980
-0.021488839
Si
+0.001449234
+0.010664590
-2.33061460
-2.4392371
-0.168812513
2,.
+0.001449211
+0.010664603
-2.33061467
-2.4392385
-0.168812526
sin v> Mi w + cos <p
B (c) = - 0.00000000019.
OF THE ORBITS OF THE FOUR INNER PLANETS. 167
DIFFERENTIAL COEFFICIENTS.
log coeff.
[<fe/cB]oo = + 165.70584 m' p 2.2193378
[dx/dtln =+13074.175 m' p 4.1164143
[di/dt] m = - 268.82366m' n 2.4294675
[dQ/dt] w = -- 8712.2760 m' n 3.9401316
[dT/dfla, = +13069.631 TO' p 4.1162634
[dL/dt] 00 = -19334.282 TO' n 4.2863281
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] m = + 0.15813453
[dxldt] m +12.476799
[di/dt] m = -- 0.25654077
[dQ/dflflo = - 8.3142000
[dirfdt] w = +12.472464
[dL/dt] w = -18.450874
COMPARISON WITH OTHER RESULTS.
LeveiTier. Newcomb. Method of Gauss.
[de/dt}oo
+ 0.15810
+0.15818
+ 0.1581345
e[dirfdt] m
+ 1.16323
+ 1.16372
+ 1.1632822
[di/dfloo
- 0.25648
-0.25655
- 0.2565408
sin i [dttjdt] w
- 0.26864
-0.26850
- 0.2684974
[dL/dt]oo
-18.450
-18.450874
NOTES.
The very exact agreement of the final sums shows that for this case the expansion
of the perturbing function is highly convergent. The greatest effect of all terms
from the sixth to the eleventh orders inclusive occurs with [de/dt] 00 , and amounts to
but l/100000th of the whole variation.
This computation has been twice effected by DR. ARTHUR B. TURNER from the
same elements as are here employed. His first computation was made by HILL'S
first method, exactly as here, and was presented as a Thesis to the Faculty of the
168
THE SECULAR VARIATIONS OF THE ELEMENTS
Graduate School of the University of Pennsylvania, 1902. The values of the functions
in this computation agree practically throughout with those here given, but the last
two figures usually differ because eight place logarithms are here employed in certain
parts of the work.
DR. TURNER'S second computation, (A. N., 3065), was made according to the
method developed by DR. Louis ARNDT (SO) . The two papers taken together are of
high value since they afford a means of comparing the labor and accuracy appertaining
to the two very different methods. It is DR. TURNER'S opinion that while the form-
ulas of ARNDT'S method are presented in a more symmetric form yet they are less
accurate in application than those of DR. HILL. This is confirmed by the circum-
stance that the residual from the equation [da/dt}^ = is 300 times larger with the
former method than with the latter.
DR. TURNER'S results from ARNDT'S method, which agree almost exactly with
his earlier values and with those here obtained, are as follows:
[de/diloo = + 0.1581330
[dx/<ttjoo = +12.47677
[dildt] M = - 0.2565480
[dti/dt]* = - 8.314194
ldr/dt]m = +12.47244
[dL/dtloo = -18.45083
ACTION OF SATURN ON MARS.
E
A
B sin (
B cos <
9
h
92.2326164
-11.747198
- 0.835808
39.461155
90.708401 T
30
93.1075498
- 13.263728
+ 6.495252
50.307460
90.7105920
60
94.0543836
-10.902482
+ 13.616320
33.990063
90.7106532
90
94.8268069
- 5.296150
+ 18.619311
8.020876
90.7086748
120
95.2215463
+ 2.053054
+20.163679
1.205319
90.7070348
150
95.1291338
+ 9.175915
+ 17.835613
24.076867
90.7078833
180
94.5669406
+ 14.163873
+ 12.258914
57.367408
90.7108082
210
93.6819094
+ 15.680401
+ 4.927852
70.309758
90.7131088
240
92.7148797
+ 13.319160
- 2.193216
50.728826
90.7125327
270
91.9323586
+ 7.712824
- 7.196207
17.010933
90.7095237
300
91.5477178
+ 0.363621
- 8.740573
0.037809
90.7066608
330
91.6603255
- 6.759241
- 6.412507
13.064630
90.7063166
^
560.3380844*
+ 7.250028f
+ 34.2693 16}
182.790580
544.256090(1
2 2
560.3380840
+ 7.250021
+34.269314
182.790524
544.2560991
* 6d 2 + SaV + 6[a' 2 -
t Gfc'aa' cos if' e sin
\ 6[a'V - kaa'e cos A']
Ikaa'ee' cos A'] = + 500.3380843.
A' = + 7.250024.
= + 34.269314.
OF THE ORBITS OF THE FOUR INNER PLANETS.
169
E
ACTION OF SATUKN ON MARS.
G G'
G"
30
60
90
120
150
180
210
240
270
300
330
Zi
z,
E
1.2382581
2.1110005
3.0577732
3.8321748
4.2285543
4.1352933
3.5701752
2.6828133
1.7163897
0.9368777
0.5550998
0.6680516
14.3662501
14.3662411
log Ko
90.70353S2 1.5278682 0.2847474 8
90.7043316 2.3529760 0.2357148 9
90.7063778 3.1798913 0.1178425 10
90.7076570 3.8561238 0.0229312 11
90.7068811 4.2318481 0.0031400 12
90.7048170 4.2015371 0.0631774 12
90.7035495 3.7462608 0.1688274 11
90.7043024 2.9540530 0.2624034 10
90.7062481 2.0020241 0.2793497 9
90.7074346 1.1081936 0.1692270 6
90.7066561 0.5558543 0.0007499 4
90.7047169 0.8409314 0.1712800 6
544.2332508 15.2437468 0.8546569 57
544.2332595 15.3138149 0.9247338 57
log LO' log No log N log I'
6 50.425
42 47.930
59 5.400
55 59.135
28 42.340
31 7.643
58 46.967
50 18.088
6 39.626
48 32.560
29 34.176
3 29.499
9 38.934
52 14.855
logQ
0.00656257
0.28174175
0.18592136
7.5317335
3.8955043
5.7586694
30
0.00942345
0.28554606
0.19019807
7.5468281
3.9148639
5.7782711
60
0.01207449
0.28906810
0.19415617
7.5819826
3.9546474
5.8179375
90
0.01427105
0.29198398
0.19743287
7.6263318
4.0028084
5.8660110
120
0.01561994
0.29377356
0.19944338
7.6674180
4.0458810
5.9092061
150
0.01572227
0.29390930
0.19959586
7.6949830
4.0730267
5.9366461
180
0.01438391
0.29213375
0.19760114
7.7028806
4.0781506
5.9420499
210
0.01175197
0.28863978
0.19367515
7.6895882
4.0604630
5.9243808
240
0.00828439
0.28403181
0.18849593
7.6581064
4.0241928
5.8876296
270
0.00461510
0.27914998
0.18300711
7.6156278
3.9778730
5.8401825
300
0.00200564
0.27567456
0.17909843
7.5727522
3.9331412
5.7942078
330
0.00365091
0.27786617
0.18156337
7.5415145
3.9024823
5.7646287
V
^-1
0.05893094
1.71642353
1.14471671
5.7148732
3.9315172
5.1097001
2 2
0.05943475
1.71709527
1.14547243
5.7148733
3.9315172
5.1101201
170
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF SATUKN
ON MARS.
E
logF
/'
J'Z
J 3
F 2
F,
5.7569714
90.9876364
-0.64905242
+0.2442109
+ 59.776028
-2.2333582
30
5.7768667
90.9027545
-0.80993896
-1.6448282
+ 67.492953
-2.2386206
60
5.8172358
90.7031717
-0.67748734
-3.0908929
+55.477667
-1.0572468
90
5.8658745
90.5722076
-0.29321392
-3.7064154
+26.949646
+0.0320703
120
5.9091874
90.5928360
+0.18262899
-3.3264146
-10.447038
-0.2313819
150
5.9362703
90.7254544
+0.58278840
-2.0527628
-46.691968
-1.7836963
180
5.9410456
90.8717164
+0.80450796
-0.2268316
-72.073367
-3.2467872
210
5.9228190
90.9409399
+0.82304924
+ 1.6620728
-79.790278
-3.2597888
240
5.8859649
90.8876171
+0.67261531
+3.1078694
-67.775016
-1.8125358
270
5.8391722
90.7221965
+0.40924575
+3.7232565
-39.246977
-0.2549584
300
5.7942033
90.5707339
+0.07363159
+3.3433908
- 1.850296
+0.0267586
330
5.7636057
90.8202137
-0.30508990
+2.0700081
+34.394639
-1.0495556
Si
5.1046084
544.6137115*
+0.40684409
+0.0513320
-36.892022
-8.5545513
2 2
5.1046084
544.6837666
+0.40684061
+0.0513310
-36.891985
-8.5545494
E
R*
1000000 X So
1000 X W a
1000 X B ( ">
100000 X S ( ">
1000 X[flo sin v+
(cos v-\- cos E) So]
0.0017097618
+ 9.903284
+0.01219946
0.0000000
+0.7168085
+0.0198066
30
0.0017745243
+ 7.025377
-.0.10023833
+0.6334785
+0.5015908
+0.9730086
60
0.0019262900
+ 5.499323
-0.20387007
+ 1.1484062
+0.3785755
+ 1.7472855
90
0.0021375714
+ 5.593738
-0.27213044
+ 1.4028896
+0.3671175
+2.1277319
120
0.0023569201
+ 3.205770
-0.27013289
+ 1.2799221
+0.2010205
+ 1.9382857
150
0.0025168546
- 4.917236
-0.17936963
+0.7641820
-0.2986000
+ 1.1679255
180
0.0025606191
- 16.044422
-0.02369074
0.0000000
-0.9631644
+0.0320888
210
0.0024720520
-22.805150
+0.13539869
-0.7505788
-1.3848465
-1.0986739
240
0.0022843230
-19.928906
+0.23709997
-1.2404980
-1.2496588
-1.8606368
270
0.0020608533
- 9.038484
+0.25685358
-1.3525394
-0.5931965
-2.0510270
300
0.0018652798
+ 2.997952
+0.20817962
-1.1120336
+0.2063802
-1.6842325
330
0.0017413378
+ 9.774776
+0.11927113
-0.6216313
+0.6978895
-0.9263634
2,
0.0127031938
- 14.366999
-0.04021465
+0.0757967
-0.7100385
+0.1925973
2
0.0127031934
-14.366879
-0.04021500
+0.0758006
-0.7100452
+0.1926017
* 2,(J,' - G") =
543.7590546.
2 Z (J,' - G") =
543.7590328.
OF THE ORBITS OF THE FOUR INNER PLANETS.
171
1000 X |~- flocosv
L
f
E
( \ -i
1000 X W cos u
1000 X W sin u
1000X-2-flo
- sec 2 ip + 1 1 sin vSo
a
-1.7097618
+0.00313672
-0.01178931
- 3.1005914
30
-1.4844381
-0.07412689
+0.06747540
- 3.2623831
60
-0.8120508
-0.20054993
+0.03664321
- 3.6729182
90
+0.2105546
-0.25530900
-0.09419278
- 4.2751426
120
+ 1.3414125
-0.17569179
-0.20519295
- 4.9336670
150
+ 2.2292256
-0.03890712
-0.17509913
- 5.4402943
180
+2.5606191
+0.00609135
-0.02289425
- 5.5988851
210
+ 2.2161495
-0.09117052
+0.10010366
- 5.3434506
240
+ 1.3285850
-0.22331936
+0.07965468
- 4.7817011
270
+0.2102889
-0.25329570
-0.04260337
- 4.1217062
300
-0.8010985
-0.15911711
-0.13424046
- 3.5565885
330
-1.4740745
-0.03664087
-0.11350351
- 3.2013711
38,
+ 1.9077055
-0.74945012
-0.25781908
-25.6443513
2 2
+ 1.9077060
-0.74945010
-0.25781973
-25.6443479
sin
+ cos <p (c) = + 0.000000000031.
DIFFERENTIAL COEFFICIENTS.
[deldt] m
[d*/dt] w
[di/dt] m
= + 22.022051 m'
= +2338.7360 TO'
86.444970 TO'
- 920.85894 TO'
[dw/dt] w = +2338.2557 m'
-2935.3283 TO'
log coeff.
p 1.3428578
p 3.3689812
n 1.9367398
n 2.9641931
p 3.3688920
n 3.4676567
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
= +0.0062891406
= +0.66790508
[di/dt] M = -0.024687281
[dtt/dt] M = -0.26298236
[d7r/d<]oo = +0.66776785
= -0.83828212
172 THE SECULAR VARIATIONS OF THE ELEMENTS
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] w +0.00627 +0.00629 +0.0062891
e[dw/dt] w +0.06226 +0.06226 +0.0622814
[di/dt] m -0.02467 -0.02468 -0.0246873
sin i [dtt/dt] m -0.00852 -0.00849 -0.0084927
[dL/dt} w -0.838 -0.8382821
NOTES.
As in other similar cases, the great disagreement of the sums of the functions
near the beginning of the computation arises principally from the term a' 2 e', but the
remarkably close agreement of the final sums shows that the expansion of the per-
turbing function for this case is very convergent. The greatest effect produced by all
terms from the sixth to the eleventh orders inclusive here occurs with [de/dt] 00 and
amounts to but l/100000th of the value of this coefficient.
DR. SAMUEL G. BARTON has effected this computation from the same elements
as are here employed, making use of the formulas developed by DR. ARNDT (SO) . (A
Thesis presented to the Faculty of the Graduate School of the University of Pennsylvania,
1906}. The results* obtained by him are as follows:
[de/dt] m = +0.0062897
e[dw/dt] w = +0.0622817
[di/dl] w = -0.0246873
sin i [dtt/dt] M = -0.0084927
[dL/dt] w = -0.8382857
The agreement is thus practically exact.
It is the conclusion of DR. BARTON that in spite of the greater symmetry of the
equations employed in the method of ARNDT, computations effected by them are
somewhat less accurate than when the methods of HILL are employed. His residual
arising from the equation derived from the constancy of the major axis is eight times
greater than that here obtained. (See the notes to the computation of the action of
Jupiter on Mars, where it is shown that DR. TURNER came to the same conclusion.)
OF THE ORBITS OF THE FOUR INNER PLANETS.
173
ACTION OF URANUS ON MARS.
E
A
B cos t B sin e g
h
367.8110283
- 8.331256 + 6.828502 37.81310
367.49598
45
368.2057075
- 5.369694 -15.243795 188.44191
367.49649
90
370.1698856
+ 12.404411 -28.797937 672.53375
367.50044
135
372.5672534
+34.579222 -25.894092 543.74175
367.50030
180
373.9791848
+ 48.165033 - 8.233286 54.97149
367.49639
225
373.5643097
+45.203469 +13.839014 155.31082
367.49635
270
371.5799361
+27.429370 +27.393154 608.52085
367.49979
315
369.2027636
+ 5.254555 +24.489302 486.34483
367.49949
Zi
1483.5400347*
+79.667558f - 2.809567J: 1373.83919
1469.99259
2 2
1483.5400342
+79.667552 - 2.809571 1373.83931
1469.99263
* 4a 2 + 2a 2 c + 4[a' 2 -
2kaa'ee' cos K] = 1483.5400348.
t 4[a' e' kaa'e cos A']
= + 79.667564.
| - 4fcW cos ?' e sin A" = - 2.809567.
E
I
G G' G"
8
O
1 II
- 0.49589
367.49570 0.157536 0.653148 2
41 22.746
45
- 0.10173
367.49510 0.667671 0.768004 3
34 47.145
90
+ 1.85851
367.49543 2.57438(1 0.710869 5
25 12.441
135
+ 4.25601
367.49622 4.582935 0.322847 6
37 54.392
180
+ 5.67185
367.49598 5.698515 0.026250 7
10 10.482
225
+ 5.25702
367.49519 5.337362 0.079182 6
58 20.716
270
+ 3.26920
367.49525 3.718991 0.445245 6
6 25.010
315
+ 0.89233
367.49588 1.682505 0.786567 4
41 47.833
Si
+ 10.30367
1469.98230 12.149421 1.835512 21
23 10.679
2 2
+ 10.30363
1469.98238 12.270473 1.956599 21
*
52 50.086
ACTION OF URANUS ON MARS.
E
log A,
logLo' logATo log AT logP
logQ
0.00071814
0.27395868 0.17716832 6.6153321 1.7572439
4.2264770
45
0.00127262
0.27469773 0.17799967 6.6411665 1.7838479
4.2533081
90
0.00292086
0.2768938-1 0.18046982 6.7024753 1.8471866
4.3168538
135
0.00437720
0.27883326 0.18265093 6.7600912 1.9076561
4.3771079
180
0.00511859
0.27982020 0.18376077 6.7833377 1.9325907
4.4018149
225
0.00484000
0.27944936 0.18334377 6.7609874 1.9097463
4.3789860
270
0.00371007
0.27794494 0.18165196 6.7037350 1.8501248
4.3196094
315
0.00219200
0.27592287 0.17937772 6.6423516 1.7859125
4.2555483
2,
0.01246766
1.10861766 0.72305087 6.804880T 7.3871460
7.2647550
2 2
0.01268182
1.10890322 0.72337209 6.8048967 7.3871627
7.2649502
174
THE SECULAR VARIATIONS OP THE ELEMENTS
ACTION OF UBANUS ON MARS.
E
logF
Jl' J,
J,
F,
4.2255130
368.030126 +0.3910957
+5.9910041
-117.85624
45
4.2521750
368.257943 -0.6874254
+ 1.3122864
+263.09961
90
4.3158056
368.150829 -1.4287110
-4.4833753
+497.03672
135
4.3766319
367.663776 -1.2456453
-8.0008361
+446.91789
180
4.4017762
367.403228 -0.3170169
-7.1797383
+ 142.10202
225
4.3788692
367.559090 +0.6967658
-2.5011942
-238.85387
270
4,3189529
367.911230 +1.2249811
+3.2942931
-472.79092
315
4.2543883
368.137688 +1.1030033
+6.8119278
-422.67209
Zi
7.2620477
1471.495412* -0.1296511
-2.3778169
+ 48,19158
2,
7.2620644
1471.618496 -0.1333016
-2.3778161
+ 48.49154
E
F 3
1000 X fto 1000000 X So 1000000 X W
t 1000 X -K (n)
1000000 X S (B >
+ 1.1362797
0.2059026 -0.0165525 +10.07610
0.0000000
-0.0119808
45
- 5.0397791
0.2189646 +0.3708684 + 2.31469
+0.1087907
+0.2605870
90
- 8.3838519
0.2523915 +0.5397194 - 9.33604
+0.1656447
+0.3542182
135
- 1.8404722
0.2886690 +0.6481209 -19.05928
+0.1256756
+0.3990450
180
+ 1.6518880
0.3050752 +0,4171470 -18.09445
0.0000000
+0.2504179
225
- 5.0490913
0.2898966 -0.2732646 - 6.02536
-0.1262101
-0.1682478
270
- 8.9126276
0.2535451 -0.7948804 + 6.80306
-0.1664018
-0.5216806
315
- 2.5789640
0.2193839 -0.6003932 +12.22078
-0.1089990
-0.4218605
2,
-14.5083118
1.0169144 +0.1454335 -10.55133
-0.0007571
+0.0709747
S,
-14.5083066
1.0169141 +0.1453315 -10.54917
-0.0007428
+0.0695237
E
1000 X [flo sin v +
(cos v + cos E)So]
1000 xl-flocosv +
/ r x -, 1000000 X W cos u 1000000 X T^o sin w
( sec 8 <p + 1 J sin vSo 1
1000 X -2 ^flo
-0.00003310
-0.20590256 +2.590763
- 9.737343
-0.37339681
45
+0.16554695
-0.14335624 +2.077126
1.021440
-0.40904755
90
+0.25124093
+0.02461951 -8.758941
- 3.231492
-0.50478291
135
+0.18971131
+0.21763709 -8.485224
-17.006255
-0.61541380
180
-0.00083429
+0.30507521 +4.652436
-17.486113
-0.66705785
225
-0.19106859
+ 0.21804505 +5.009009
- 3.348852
-0.61803079
270
-0.25236576
+0.02523742 -6.708828
1.128399
-0.50709023
315
-0.16617611
-0.14329578 -6.836536
-10.129621
-0.40983085
Si
-0.00199222
+0.14902958 -8.224570
-31.583347
-2.05232780
2 2
-0.00198644
+0.14903012 -8.235625
-31.566168
-2.05232299
sin if \A i '*' + cos ip
BO M = - 0.0000000000013.
* 2,(Ji' - G") = 1469.659900.
2i(Ji - G") = 1469.661897.
OF THE ORBITS OF THE FOUR INNER PLANETS. 175
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] w = - o'.341 19354m' n 9.5330008
[dx/dt]w = +274.05283 TO' p 2.4378343
[di/dt] w = - 1.4239452 m' n 0.1534933
[dn/dfe]oo = -169.16430 TO' n 2.2283087
[dTT/dt] M = +273.96460 TO' p 2.4376945
[dL/dt] m = -352.43262 TO' n 2.5470761
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF TO'.
[de/dt] w = -o!o00014964631
[dx/dt]w = +0.012019862
[di/dtlw = -0.000062453743
= -0.0074194879
= +0.012015994
[dL/dt] w = -0.015457573
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt]m -0.00001 -0.00001 -0.000014964631
e[dirfdt] m +0.00112 +0.00112 +0.0011207080
[di/dt]m -0.00007 -0.00006 -0.000062453743
sin f [da/dtfo -0.00023 -0.00024 -0.00023960370
[dL/dt] w -0.015 -0.015457573
NOTES.
The greatest error produced in this case by a division into only four parts occurs
with the coefficient [dx/dt] 00 and amounts to but 0". 0000001. It is evident that,
notwithstanding the disagreement of the sums of the functions in the first part of
the computation, a division into eight parts is fully sufficient.
176
THE SECULAR VARIATIONS OF THE ELEMENTS
ACTION OF NEPTUNE ON MARS.
E
A B cos e B sin e g
ft
906.38891911 +21.92636 -39.01745 99.36880
904.17365
45
906.94215394 +47.61371 -15.38170 15.44333
904.17339
90
907.26271705 +49.05172 +19.61543 25.11471
904.17633
135
907.17710690 +25.39796 +45.47311 134.97137
904.17696
180
906.72119273 - 9.49146 +47.04426 144.45927
904.17395
225
906.14776209 -35.17881 +23.40851 35.76678
904.17317
270
905.80700342 -36.61681 -11.58863 8.76590
904. 17576
315
905.91280912 -12.96307 -37.44631 91.52721
904.17627
2i
3626.17983231* +24.86981t +16.05361J 277.70868
3616.69969
S 2
3626.17983205 +24.86979 +16.05361 277.70869
3616.69979
E
1 G G' G"
e
i tt
2.15000 904.17353 2.20007 0.049953 2
51 33.458
45
2.70349 904.17337 2.70981 0.006303 3
8 30.703
90
3.02112 904.17630 3.03031 0.009166 3
19 25.738
135
2.93488 904.17680 2.98505 0.050008 3
19 16.746
180
2.48197 904.17377 2.54493 0.062779 3
4 42.112
225
1.90932 904.17313 1.92986 0.020498 2
39 43.146
270
1.56597 904.17575 1.57215 0.006167 2
23 40.258
315
1.67127 904.17616 1.72990 0.058516 2
32 56.175
2,
9.21906 3616.69935 9.34746 0.128065 11
39 21.566
S 2
9.21896 3616.69945 9.35462 0.135325 11
40 26.770
ACTION OF NEPTUNE ON MARS.
E
log Kn log LO' log Na log N log P
logQ
0.00081164 0.27408332 0.17730851 6.0300471 0.3915788
3.2510798
45
0.00098011 0.27430787 0.17756111 6.0560291 0.4178275
3.2773355
90
0.00109700 0.27446368 0.17773639 6.1154021 0.4773508
3.3368810
135
0.00109536 0.27446148 0.17773391 6.1708446 0.5327514
3.3923012
180
0.00094087 0.27425556 0.17750228 6.1926623 0.5543538
3.4138826
225
0.00070344 0.27393908 0.17714626 6.1704766 0.5318928
3.3913614
270
0.00056913 0.27376004 0.17694484 6.1148768 0.4761252
3.3355658
315
0.00064494 0.27386109 0.17705853 6.0556543 0.4169531
3.2764317
s,
0.00341864 1.09656260 0.70949202 4.4529882 1.8994085
3.3374091
2 2
0.00342385 1.09656952 0.70949981 4.4530046 1.8994248
3.3374297
* 4a 2 + 2aV + 4[a' 2 - Ikaa'ee' cos K] = 3626.17983218.
t 4[a'V - kaa'e cos K] = + 24.869793.
t - 4fc'aa' cos <p' e sin K' = + 16.05361.
OF THE ORBITS OF THE FOUR INNER PLANETS.
177
ACTION OF NEPTUNE ON MAES.
E
log V JY J,
J,
Ft
3.2510498 903.898094 +0.3009875
+ 15.528734
+299.48600
45
3.2773317 903.965295 -0.6661748
- 12.963466
+ 118.06523
90
3.3368755 902.840060 -0.3701497
-34.790528
-150.56206
135
3.3922712 902.874950 +0.9173304
-37.166389
-349.03766
180
3.4138449 903.910920 +1.0320306
-18.699362
-361.09725
225
3.3913491 904.099942 -0.1708692
+ 9.792741
-179.67656
270
3.3355621 903.071970 -0.8044972
+31.619710
+ 88.95077
315
3.2763966 902.764465 +0.0451253
+33.995664
+287.42636
2,
3.3373322 3613.721044* +0.1583712
- 6.341446
-123.22254
2 2
3.3373486 3613.704652 +0.1254117
- 6.341450
- 123.22263
E
F, 1000 X Ro 1000000 X S 1000000 X TFo
1000 X fl ( ">
1000000 X S<">
-10.033551 0.05357158 +0.12743633 +2.7656535
0.00000000
+0.09223957
45
- 4.253985 0.05690845 -0.09526013 -2.4561209
+0.02827449
-0.06693360
90
+ 2.323772 0.06500834 -0.12559190 -7.5560767
+0.04266503
-0.08242607
135
- 5.662284 0.07385617 +0.10733754 -9.1730340
+0.03215420
+0.06608723
180
-14.586465 0.07791030 +0.13821863 -4.8544449
0.00000000
+0.08297417
225
- 7.712269 0.07407227 -0.10322243 +2.4086872
-0.03224828
-0.06355357
270
+ 1.985936 0.06494182 -0.14759132 +6.8479047
-0.04262137
-0.09686429
315
- 2.681773 0.05659771 +0.08359968 +6.4235154
-0.02812010
+0.05874051
2,
-20.310308 0.26143204 -0.00752826 -2.7969634
+0.00004366
-0.00407662
2 2
-20.310311 0.26143460 -0.00754534 -2.7969523
+0.00006031
-0.00565943
E
1 000 V 1 /? rrm v
lOOOXtflosinv ^L 1000000
1000000
1000 X 2 T -R
+ (oos i)+cos E)So] fr 2 \ . , cl XWaCosu
X Wo sin u
a
+0.000254873 -0.053571580 + 0.7111033
- 2.6726716
-0.09715013
45
+0.042763861 -0.037538560 - 2.2040406
+ 1.0838516
-0.10631064
90
+0.064736687 +0.005812013 - 7.0890057
- 2.6153925
-0.13001668
135
+0.048623034 +0.055602449 - 4.0838509
- 8.2138132
-0.15745404
180
-0.000276437 +0.077910304 + 1.2481721
- 4.6912376
-0.17035369
225
-0.048771753 +0.055759058 - 2.0023922
+ 1.3387309
-0.15791475
270
-0.064644985 +0.006352181 - 6.7530485
- 1.1358370
-0.12988365
315
-0.042545549 -0.037317274 - 3.5934362
- 5.3243555
-0.10573015
2l
+0.000070138 +0.036502918 -11.8827788
-11.1151387
-0.52740415
s.
+0.000069593 +0.036505673 -11.8837299
-11.1155862
-0.52740958
sin v iAi (s > + cos <p Bo M = + 0.0000000000012.
* 2,(J/ - G") = 3613.592979.
2t(Ji - G") = 3613.569327.
178 THE SECULAR VARIATIONS OF THE ELEMENTS
DIFFERENTIAL COEFFICIENTS.
log coeff.
[de/dt] w = + 0.011982 m' p 8.0785350
[dx/dt] M = +67.128215 TO' p 1.8269051
[di/dt] m = -- 2.0560028m' n 0.3130237
[dl2/cfc]oo = -59.551438 m' n 1.7748923
[dT/dt] w = +67.097154 TO' p 1.8267041
= -90.590942 TO' n 1.9570848
FINAL VALUES CORRESPONDING TO THE ABOVE VALUE OF m'.
[de/dtlw = +o!()0000060823
[dx/dt] w = +0.0034075236
[dt/<ft]oo = -0.00010436562
[dtt/dt] w = -0.0030229161
[dir/dtlw, = +0.0034059472
= -0.0045985255
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] w +0^00000 +0^00000 +o!o0000060823
+0.00032 +0.00032 +0.00031766599
-0.00011 -0.00011 -0.00010436562
sin i [dtt/dt] m -0.00009 -0.00010 -0.000097621545
[dL/dt] w -0.004 -0.0045985255
NOTES.
The agreement of the sums of the functions is much more exact throughout than
in the preceding computation because e' is here so much smaller. The greatest effect
produced by all terms from the fourth to the seventh orders inclusive is but 0". 000001,
and it is evident that the terms of the eighth and higher orders are wholly inappreci-
able.
OF THE ORBITS OF THE FOUR INNER PLANETS.
179
11. THE FINAL VALUES OF THE PERTURBATIONS.
Combining the results of the preceding pages, we now obtain the values of the
perturbations stated in the following tables. For comparison with these, the results
obtained by LEVERRIER (V) and NEWCOMB (IS) are added, all of the results being reduced
to the values of the masses here adopted and stated in Article 6.
SECULAR PERTURBATIONS OF MERCURY.
(Epoch 1850.0, G. M. T.)
Action of
r*i
LdtJoo
r&i
ldt],
r*-.
Udoo
Venus
+0^027739414
+2.7763615
-0.14811133
Earth
+0.011476557
+0.91448833
-0.014040890
Mars
-0.000607428
+0.02486334
-0.000301945
Jupiter
+0.00319413
+ 1.5400720
-0.049056191
Saturn
+0.000531095
+0.07312263
-0.004212776
Uranus
+0.000009638
+0.00142135
-0.000024450
Neptune
+0.000003320
+0.00041901
-0.000020027
-0.04234673
+5.3307482
-0.21576761
Action of
rda-i
LdUoo
f-1
LdUoo
1-dL-l
LJ
Venus
-1.9420214
+2.7618772
-3.2505323
Earth
-1.0037245
+0.90700208
-1.1935233
Mars
-0.01926435
+0.02471966
-0.03293324
Jupiter
-1.4795642
+ 1.5290366
-2.2066350
Saturn
-0.06979662
+0.07260205
-0.10657405
Uranus . .
-0.00134987
+0.00141128
-0.00201139
Neptune
-0.00044314
+0.00041570
-0.00060031
-4.5161641
+5.2970646
-6.7928096
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt]oo
e[dir/dt]oo
[di/dt]oo
sin i [dtt/dt] m
[dL,'dt] w
* Exclusive of the action of Uranus and Neptune.
f This unexpectedly large difference is a gradual accumulation from all of the computations Thus, the residuals,
Newcomb-Gauss, are, in the several cases: 0".00300, 0".00151, 0".0000.3, 0".00227, and 0".00010, the sum of which is
the difference as found above.
+0.04246
+0.04234
+010423467
+ 1.08946
+ 1.09601
+ 1.0891018t
-0.21586
-0.21570
-0.2157676
-0.55017
-0.55041
-0.5505495
-6.8190*
-6.79281
180
THE SECULAR VARIATIONS OF THE ELEMENTS
SECULAR PERTURBATIONS OF VENUS.
(Epoch 1850.0, G. M. T.)
Action of
(-del
LdUoo
r*n
L<ftJoo
r-A-i
UJoo
Mercury
-01)13012279
l'l893992
+o'6o94965089
Earth
0.04898290
56289701
+0 000044940
Mars
0.001963988
+0 74594759
+0 001304280
Jupiter
0031162921
+6 5654682
038659982
Saturn
0000675363
+0 07935156
0052327048
Uranus
+0 000005263
+0 00278176
-1-0 0000018240
Neptune
-0.000000278
+0.00110440
-0.0000283988
-0.09579247
+0.5762842
-0.033057385
Action of
ran
1 dUoo
r d *~i
Udoo
nun
LdUoo
Mercury
+ o'()897732
1 1892420
+ / 7454252 t i
Earth
- 7.293993
56417558
5 4005288
Mars
- 0.0473504
+0 74586465
09940123
Jupiter
- 2.7242270
+6 5606924
5 5347410
Saturn
- 0.0824657
+0 07920700
26491624
Uranus
- 0028813
+0 00277671
00496096
Neptune
0.0007780
+000110304
00148569
-10.061922
+0.5586460
-10.5606087
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
[de/dt] M - 0.09558
e[dTr/dt] M + 0.00366
[di/diloo - 0.03318
sin i [dO/di]oo - 0.59530
[dL/dt] M -10.549
-0.09576
+0.00392
-0.03306
-0.59551
- 0.0957925
+ 0.0038229
- 0.0330574
- 0.5955192
-10.5606087
SECULAR PERTURBATIONS OF THE EARTH.
(Epoch 1850.0, G. M. T.)
Action of
I"*]
L<Joo
f-1 =F^1
LdUoo LrfUoo
[dp-]
I dt Joo
Mercury
-o'6oi 1613570
- OJ0999815
+o'o025085775
Venus
+0.013483339
+ 3.4537341
+0 074457966
Mars
-0.015723904
+ 097519611
+0 0063443986
Jupiter
-0.081841849
+ 6 9652565
-0 025114405
Saturn
-0.0004330571
+ 0.18725991
-0.0054235259
Uranus
+0.0000172788
+ 0.00566366
+0.0000236793
Neptune
-0.0000006006
+ 0.00179708
-0.0000364953
-0.085660150
+ 11.4789092
+0.052760195
OF THE ORBITS OF THE FOUR INNER PLANETS.
181
Action of
l-dj-j
Ldt Joo
rdL-l
1 dt Joo
Mercury
-(X002098681
+ 0^3930935
Venus
-0.28462399
+ 11.232473
Mars
-0.007195311
- 0.2342424
Jupiter
-0.16046446
- 9.1916336
Saturn
-0.013188086
- 0.4325140
Uranus
-0.0000784873
- 0.0080930
Neptune
-0.0000432488
- 0.0024199
-0.46769226
+ 1.756664
COMPARISON WITH OTHER RESULTS.
Leverrier. Newcomb. Method of Gauss.
-0.08569
e[dir/dt] w +0.19254
[dpldt] M +0.05290
[dq/dt] m -0.46754
[dL/dt]w +1.7570*
-0.08563
+0.19248
+0.05276
-0.46768
-0.085660
+0.192514
+0.052760
-0.467692
+ 1.756664
The values of [dp/dt}oo and [dq/dt} QO obtained by HILL in the "New Theory"
are given below. These were regarded as provisional results only, and were derived
from the numerical values of the coefficients in the expansion of the perturbing
function stated by LEVERRIER in the Annales, Vol. II.
It may also be of interest to add the results obtained by the first application
ever made of the method of GAUSS. This was a computation by NICOLAI of the
secular perturbations of the Earth, the final values only being published, in BODE'S
Berliner Jahrbuch, 1820, pages 224-226 (Aug. 30, 1817). These results are here
reduced to the values of the masses stated in Article 6.
[de/dt] w
e[dirjdt]w
[dp/dt]oo
[dq/dt] w
Hill.
+0.0527225
-0.4676079
Nicolai.
-o!()8606
+0.19283
+0.05182
-0.46738
* Exclusive of the action of Neptune. If the value of this found above is included, we have [dL/<i(]oo = 1".7546 ;
-a less exact agreement.
182
THE SECULAR VARIATIONS OF THE ELEMENTS
SECULAR PERTURBATIONS OF MARS.
(Epoch 1850.0, G. M. T.)
Action of
f-1
Lddoo
[41
UtJoo
r*-]
UUoo
Mercury
+o!6o0335670
+ 0061841
-l-o'bo0074482
Venus
+0.000795405
+ 4947286
0128 < >Q7 1 V7
Earth
+0.021481158
+ 2 2915614
-un 000*31 QQ11
Jupiter
+0.15813453
+ 12 476799
2 s )fi l i4077
Saturn
+0.006289141
+ 6679051
024687281
Uranus
-0.000014965
+ 00120199
000062454
Neptune
+0.000000608
+ 00034075
000104366
+0.18702155
+ 15.952606
-0.29383023
Action of
rdo-l
LdiJoo
Fl
UUoo
f-1
ldt] w
Mercury
+ o'6l479483
+ o'6o61918
+ o"l940178
Venus
+ 0.30877426
+ 0.4948896
+ 4 1 9 04933
Earth
- 2.2862242
+ 2.2903688
+ 6 6520970
Jupiter
- 8.3142000
+ 12.472464
18 450874
Saturn . ...
- 0.26298236
+ 6677678
8382821
Uranus
- 0.00741949
+ 0120160
0154576
Neptune
- 0.00302292
+ 0.0034059
00045985
-10.5502799
+ 15.947104
- 8.342604
COMPARISON WITH OTHER RESULTS.
Leverripr. Neweomb. Method of Gauss.
+0.18703
+ 1.48645
[K/ctt]oo -0.29375
sin i [dtt/dfloo -0.34099
-8.358*
+0.18706
+ 1.48787
-0.29385
-0.34066
+0.187022
+ 1.487355
-0.293830
-0.340709
-8.34260
12. COMPARISON WITH THE RESULTS OF OBSERVATION.
From a discussion of all the available observations of the planets and of the
Sun, NEWCOMB has derived the most probable values of the preceding coefficients
based upon observations alone. These will be found summarized in a convenient
form on pages 107 and 108 of The Elements of the Four Inner Planets and the Funda-
mental Constants of Astronomy (Supplement to the American Ephemeris and Nautical
Almanac, 1897).
* The value of [dLldt]oo arising from the action of Mercury was not stated by Leverrier. The value as found above
has been added to his series of values in order to obtain this sum.
OF THE ORBITS OF THE FOUR INNER PLANETS. 183
In order to compare the values here obtained with those given by NEWCOMB
it is necessary to notice that the values of i and fl stated by him are measured from
the movable equator and equinox and that it is therefore necessary to free the values
of [di/dt] 00 and [dQ/dt] 00 here given from the changes caused by the motion of the
ecliptic itself. For this purpose we first compute p and L from the equations,
[dp-] \dq1
p sin L = and p cos L = Mr ,
L dt Joo L dt Joo
the secular variations being those which belong to the Earth's orbit, and then add
the quantities p cos (L fl) to the several determinations of [di/dt] Q o and p X
cos i sin (L Q) to those above given for sin i [dQ/dt]o . In this manner the values
stated in the following tabulation are obtained.
In a similar way it might appear necessary to add the correction,
e tan \i ( sin i ^7 + P sin (L ft) J
to the values obtained for e [dwjdt] o, the first part arising from the change due to the
plane of the orbit and the second from that produced by the motion of the ecliptic.
And in the case of the Earth's perihelion, there is a secular motion due to the lack of
sphericity of the Earth-moon system which is expressed by the equation,
dir^ mm' (a '\ 2 .
TT = I*** ' 7 7\} 'I ) > '
dt Joo 2 (m + m') 2 \a/'
the accented letters applying to the moon (Annales de I'Observatoire de Paris, Vol. IV,
pages 42-46). Employing the values of a' and m' given in the Astronomical Papers
of the American Ephemeris, Vol. IV, page 11, this correction is found to be
+0".0157884. But these last two corrections need not here be applied because the
values of the variations obtained by NEWCOMB from observation have already been
freed from their effects.
MERCURY.
Newcomb. Method of Gauss. Observ. 5i Si t
[de/dt] M
e [dv/dt]m
[dildt}oo
+0.0423
+ 1.0960
+0.0676
+0.0423
+ 1.0891
+0.0674
+o!t)336
+ 1.1824
+0.0714
-o!()087
+0.0864
+0.0038
-0.0087
+0.0933
+0.0040
0.0050
0040
0.0080
sin i [dn/dfloo -0.9250 -0.9234 -0.9189 +0.0061 +0.0045 0.0045
184
THE SECULAR VARIATIONS OP THE ELEMENTS
VENUS.
Newcomb. Method of Gauss. Observ. Si
[de/dt} w
-0.0958
-0.0958
-0.0946
+0.0012
+0.0012
0.0020
e [dv/dt]oo
+0.0039
+0.0038
+0.0029
-0.0010
-0.0009
0.0020
[dildt]oo
+0.0034
+0.0034
+0.0029
-0.0005
-0.0005
0.0030
sin i [dttldt]m
-1.0600
-1.0603
-1.0540
+0.0060
+0.0063
0.0012
EARTH.
Newcomb. Method of Gauss. Observ.
i
[de/dt]^
-0.0856
-0.0857
-0.0855
+0.0001
+0.0002
0.0009
e. [dir/dt}oo
+0.1925
+0.1925
+0.1948
+0.0023
+ 0.0023
0.0012
[d*!dt} M
-0.4677
-0.4677
-0.4711
-0.0034
-0.0034
0.0023
Newcomb.
MARS.
Method of Gauss. Observ.
[de/dt] M
+0.1871
+0.1870
+0.1900
+0.0029
+0.0030
0.0027
e [dw/dt] M
+ 1.4879
+ 1.4874
+ 1.4955
+0.0076
+0.0081
0.0035
[di/dt] w
-0.0225
-0.0229
-0.0226
-0.0001
+0.0003
0.0020
sin i [dn/dt]oo
-0.7263
-0.7251
-0.7260
+0.0003
-0.0009
0.0020
In the above tabulation the column headed 61 expresses the residuals from the
computation of NEWCOMB and that headed 5 2 states the residuals from the results
here obtained. It will be noticed that the differences are very minute throughout,
the only appreciable improvement arising from the more accurate computation
occurring in the case of the node of Mercury, where the residual is reduced by its
fourth part.
The last column contains the mean errors of the observational results. If we
multiply these by 0.6745 to reduce them to probable errors, we observe that in seven
cases the residuals are less than the probable errors; in five cases they vary from one
to three times as great but that in each of these cases where the divergence is greatest
a slight change in the value of the masses will correct the disagreement, and that
in the remaining three cases the difference is very much greater than can be ascribed
to errors either in the adopted masses, the computation, or to errors in the obser-
vations themselves. These three cases are:
1. The motion of the perihelion of Mercury.
2. The motion in the node of Venus.
3. The motion of the perihelion of Mars.
OF THE ORBITS OF THE FOUR INNER PLANETS. 185
The first of these is the well-known discordance. The second is well established,
the discordance between observation and theory being nearly eight times the probable
error, nor can the uncertainty remaining in the values of the masses account for more
than a small part of the discrepancy. NEWCOMB estimates the mean error of the
computed value arising from this uncertainty as not more than 0".0012, so that with
this included the residual is nearly six times the probable error. The third dis-
cordance is the least of the three, but as the masses of Jupiter and Saturn, the principal
disturbing planets for this case, are accurately known, the uncertainty of the com-
puted results is almost negligible. NEWCOMB estimates the mean error of the result
of computation arising from the uncertainties in the masses of all the planets as here
but 0".0004, so that the residual remains between three and four times as large as
the probable error.
13. COMPARISON WITH SEELIGER'S HYPOTHESIS ON THE CON-
STITUTION OF THE ZODIACAL LIGHT.
Many hypotheses have been made for the purpose of explaining the discrepancies
shown in the preceding article. In general, either the assumption is made that
NEWTON'S Law of Gravitation is not strictly accurate* or else that certain additional
matter in the solar system must be considered whose attraction has not hitherto
been allowed for.| The most recent and the most plausible investigation of the
second kind is that effected by SEELIGER (IO)I (11)i (12> who seeks to account for all of
the appreciable discrepancies by the perturbing effect of the cloud of particles known
as the zodiacal light.
What the true form of this cloud is, and still more, what the law of the dis-
tribution of its density is, is very uncertain. J SEELIGER assumes that it can be
roughly conceived as made up of two homogeneous ellipsoids of revolution whose
semi axes have the values 0.24 and 1.20, respectively. Both the eccentricities of
these ellipsoids and the position of the equator of the outer one can vary within wide
limits without greatly altering the values of the perturbations which they produce;
the distance from the focus to the center in each of them is arbitrarily chosen as
equal in length to ten times the semi minor axis, and the equator of the outer one is
assumed to be coincident with the plane of the equator of the sun. The respective
densities and also the two constants which define the equatorial plane of the first
ellipsoid remain as unknown quantities whose values are to be determined.
* See Tisserand's Mecanique Celeste, Vol. IV, Pages 494-542.
fSee Newcomb's " Astronomical Constants. . . ." (1 ", Pages 110-120.
t See the article, "The Zodiacal Light" by Newcomb, in the Encyclopaedia Britannica, Vol. XXVIII.
186
THE SECULAR VARIATIONS OF THE ELEMENTS
From the known formulas which express the attraction exerted by an ellipsoid
upon a point either wholly within or without its surface, the expression for the per-
turbing force in any case can readily be written, and from this the equations for the
variations of the various elements are derived, each equation containing five unknown
quantities whose values are to be so determined as to best account for the excess of
the variations observed over those heretofore obtained from the .theory. As the
ellipsoids are assumed to be symmetrical with respect to their axes of rotation,
however, they will cause no appreciable perturbation of any eccentricity. The
variation of the obliquity of the Earth's orbit was also not considered by SEELIGER.
There remain therefore but ten discrepancies to be represented; namely, those of
the four perihelia, those of the three nodes and those of the three inclinations. These
ten discrepancies form the absolute terms of ten corresponding equations which con-
tain five unknown quantities. It is to be noticed that in the "Astronomical Constants
..." two tables of the theoretical variations are stated by NEWCOMB; the first,
on page 109, are those computed from the values of the various masses assumed in
Chapter V; the second, on page 185, are those computed from the definitively adopted
masses. The latter values of the masses are in closer accordance with those assumed
in the present paper than the former; the first values are, however, the ones adopted
by SEELIGER in the computation.
The final results are as in the following table:
Newcomb.
Method of
Gauss.
Per. caused by
Zod.L't.
Final Residuals.
Prob. Errors.
Newcomb.
Meth. of Gauss.
MERCURY.
//
//
//
//
//
//
edit
+8.64
+9.33
+8.49
+0.15
+0.84
0.29
sin i<Kl
+0.61
+0.45
+0.62
+0.01
-0.17
0.54
di
+0.38
+0.40
+0.49
-0.11
-0.09
0.35
VENUS.
edw
-0.10
-0.09
+0.05
-0.15
-0.14
0.17
sin idU
+0.60
+0.63
+0.60
0.00
+0.03
0.22
di
-0.05
-0.05
+0.20
-0.25
-0.25
0.11
EARTH.
edit
+0.23
+0.23
+0.09
+0.14
+0.14
0.09
MAKS.
edw
+0.76
+0.81
+ 0.56
+0.20
+0.25
0.24
sin idSl
+0.03
-0.09
+0.21
-0.18
-0.30
0.14
di
-0.01
+0.03
-0.01
0.00
+0.04
0.15
The first two columns of the table contain the residuals from the masses employed
in the present paper; the third column states the perturbations caused by the zodiacal
light when its elements are derived from the residuals of NEWCOMB 's first tabulation.
As the five elements were so determined as to represent NEWCOMB'S first residuals as
OF THE ORBITS OF THE FOUR INNER PLANETS. 187
accurately as possible, their agreement with these is naturally more exact than with
the values here stated. Thus the first agreement for the motion of Mercury's peri-
helion is exact while here the discrepancy is considerable. On the other hand, the
greatest discrepancy when the results are compared with the first tabulation, and
which occurs in the motion of the node of Mars, is slightly lessened when the new
masses are employed.
As the five elements were determined to represent the ten residuals of NEWCOMB'S
computation as accurately as possible, the numbers of the fourth column are, as
might have been expected, generally smaller than those of the fifth. It may justly
be inferred, however, that SEELIGER'S hypothesis is capable of greatly reducing those
discrepancies whose values are sufficiently large to establish their reality, without
at the same time unduly increasing any of the smaller ones.
The last column contains NEWCOMB'S estimate of the total probable errors
arising both from the errors of observation and from the uncertainties in the values
of the adopted masses.
The elements of the zodiacal light derived by SEELIGER are as follows:
Density of inner ellipsoid = 2.52 X 10~ n times the Sun's density.
Density of outer ellipsoid = 0.0026 X 10~ n times the Sun's density.
Total mass = 35000 X 10" 11 times the Sun's mass.
Inclination of equator of I = 6. 95
Longitude of node of I = 40.03.
The unit of time throughout this article is the Julian Century.
BIBLIOGRAPHY.
LIST OF WORKS ON GAUSS'S METHOD AND RELATED SUBJECTS WHICH ARE REFERRED
TO IN THE PRECEDING PAGES.
1. GAUSS. Determinatio Attractionis quam in punctum quodvis positionis datae exerceret planeta si
ejus massa per tot am orbitam ratione temporis quo singulae partes describuntur uniformiter
esset dispertita. Werke, Vol. Ill, pages 333-357.
2. NICOLAI. Neue Berechnung der Secular Anderungen der Erdbahn. Bode's Astronomische Jahrbuch,
1820, pages 224-226.
3. CLAUSEN. Alia solutio problematis a celeberrimo Gauss in opera " Determinatio attractionis. . . "
tractati. Crelle's Journal, Vol. VI, 1830, page 290.
4. - Bestimmung der Bahn und der Umlaufszeit des Tuttle'schen Cometen. Beobachtungen der
Kaiserlichen Univcrsitats Sternwarte Dorpat, Vol. XVI.
5. BOUK, EDMOND. Thesis presentees a la faculte des sciences de Paris, 1855.
188 THE SECULAR VARIATIONS OF THE ELEMENTS
6. ADAMS. On the November meteors. Monthly Notices, Vol. XXVII; Collected Works, Vol. II, pages
194-200.
7. LEVERHIER. The secular perturbations of the elements of the orbits of the planets. Annales de 1'Ob-
servatoire de Paris. Mercury, Vol. V, pages 6 and 7; Venus, Vol. VI, page 6; The Earth, Vol. IV,
pages 11 and 12; Mars, Vol. VI, page 189.
8. HILL. On Gauss's method of computing secular perturbations. Astronomical Papers of the American
Ephemeris, Vol. I, pages 317-361.
9. SEELIGER. Ueber das von Gauss herruhrende Theorem die Sacularstorungen betreffend. Astrono-
mische Nachrichten, Vol. XCIV, 1879.
10. - Ueber die sogenannte absolute Bewegung. Sitzungsberichte der konigliche Akademie der Wis-
senschaften zu Munchen. Vol. XXXVI, pages 85-137.
11. - Ueber die empirischen Gleider in der Theorie der Bewegung der Planeten Merkur, Venus, Erde
und Mars. Vierteljahrschrift der Astronomischen Gesellschaft, Vol. XLI, pages 234-240.
12. - Das Zodiakallicht und die empirischeii Gleider in der Bewegung der inneren Planeten. Sitzungs-
berichte der koniglichliche Akademie der Wissenschaften zu Munchen. Vol. XXXVI, pages
595-622.
13. CALLANDREAU. Calcul des variations seculaires des elements des orbites. Annales de 1'Observatoire
de Paris, 1885, Vol. XVIII.
14. TISSERAND. Traite de Mecanique Celeste. Vol. I, pages 431-442; Vol. IV, pages 494-542.
15. NEWCOMB. Secular variations of the orbits of the four inner planets. Astronomical Papers of the
American Ephemeris, Vol. V, pages 301-378.
16. HILL. A new theory of Jupiter and Saturn. Astronomical papers of the American Ephemeris, Vol. IV
17. NEWCOMB. The elements of the four inner planets and the fundamental constants of astronomy.
1895. Supplement to the American Ephemeris and Nautical Almanac, 1897.
18. HALL, ASAPH, JR. Secular perturbations of the Earth from the action of Mars. Astronomical Journal,
No 244.
19. SEE. Secular perturbations of Uranus from the action of Neptune. Astronomical Journal, No. 316.
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