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T R A N S A C T I
OF THE
AMERICAN PHILOSOPHICAL SOCIETY,
1L ..ajj HELD AT PHILADELPHIA,
FOR PROMOTING USEFUL KNOWLEDGE, ,
VOLUME XIX.-NEW SERIES.
PART I.
*
1 . A New Method of Determining the General Perturbations of the Minor Planets. By
William McKniyht Bitter, M.A.
ARTICLE IT. An Essay on the Development of the Mouth Parts of Certain Insects. By John B,
Smith, Xc.D.
PUBLISHED BY THE SOCIETY,
AND FOR SALE BY
THE AMERICAN PHILOSOPHICAL SOCIETY, PHILADELPHIA
N. TRUBNER & CO., 57 and 59 LUDGATE HILL, LONDON.
ASTRONOMY LIBRARY
TRANSACTIONS
OP THE
AMERICAN PHILOSOPHICAL SOCIETY
AETICLE I.
A NEW METHOD OF DETERMINING THE GENERAL PERTURBATIONS OF
THE MINOR PLANETS.
BY WILLIAM MCKNIGHT RITTER, M.A.
Read before the American Philosophical Society, February 28, 1896.
PREFACE.
In determining the general perturbations of the minor planets the principal diffi-
culty arises from the large eccentricities and inclinations of these bodies. Methods
that are applicable to the major planets fail when applied to the minor planets on
account of want of convergence of the series. For a long time astronomers had to be
content with finding what are called the special perturbations of these bodies. And
it was not until the brilliant researches of HANSEIST on this subject that serious hopes
were entertained of being able to find also the general perturbations of the minor
planets. HANSEN'S mode of treatment differs entirely from those that had been pre-
viously employed. Instead of determining the perturbations of the rectangular or
polar coordinates, or determining the variations of the elements of the orbit, ho regards
these elements as constant and finds what may be termed the perturbation of the
time. The publication of his work, in which this new mode of treatment is given,
entitled A.useinandersetzung einer zweckmassigen Methods zur Berechnung der absoluten
A. P. s. VOL. xrx. A
6 A NEW METHOD OF DETERMINING
Storungen der Ideinen Planeten, undoubtedly marks a great advance in the determina-
tion of the general perturbations of the heavenly bodies.
The value of the work is greatly enhanced by an application of the method to a
numerical example in which are given the perturbations of Egeria produced by the
action of Jupiter, Mars, and Saturn. And yet, notwithstanding the many exceptional
features of the work commending it to attention, astronomers seem to have been de-
terred by the refined analysis and laborious computations from anything like a general
use of the method ; and they still adhere to the method of special perturbations devel-
oped by LAGRANGE. HANSEN himself seems to have felt the force of the objections
to his method, since in a posthumous memoir published in 1875, entitled Ueber die
Stdrungen der grossen Planeten, insbesondere des Jupiter s, his former positive views
relative to the convergence of series, and the proper angles to be used in the argu-
ments, are greatly modified.
HILL, in his work, A New Theory of Jupiter and Saturn, forming Vol. IV of
the Astronomical Papers of the American Ephemeris, has employed HANSEN'S
method in a modified form. In this work the author has given formulae and devel-
opments of great utility when applied to calculations relating to the minor planets, and
free use has been made of them in the present treatise. With respect to modifica-
tions in HANSEN'S original method made by that author himself, by HILL and others,
it is to be noted that they have been made mainly, if not entirely, with reference to
their employment in finding the general perturbations of the major planets.
The first use made of the method here given was for the purpose of comparing the
values of the reciprocal of the distance and its odd powers as determined by the pro-
cess of this paper, with the same quantities as derived according to HANSEN'S
method. Upon comparison of the results it was found that the agreement was prac-
tically complete. To illustrate the application of his formula, HANSEN used Egeria
whose eccentricity is comparatively small, being about -%. The planet first chosen
to test the method of this paper has an eccentricity of nearly ^. And although
the eccentricity in the latter planet was considerably larger, the convergence of the
series in both methods was practically the same. It was then decided to test the
adaptability of the method to the remaining steps of the problem, and the result of the
work has been the preparation of the present paper.
HANSEN first expresses the odd powers of the reciprocal of the distance between
the planets in series in which the angles employed are both eccentric anomalies. He
then transforms the series into others in which one of the angles is the mean anomaly
of the disturbing body. He makes still another transformation of his series so as to
be able to integrate them.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 7
In the method of this paper we at first employ the mean anomaly of the dis-
turbed and the eccentric anomaly of the disturbing body, and as soon as we have the
expressions for the odd powers of the .reciprocal of the distance between the bodies,
we make one transformation so as to have the mean anomalies of both planets in the
arguments. These angles are retained unchanged throughout the subsequent work,
enabling us to perform integration at any stage of the work.
In the expressions for the odd powers of the reciprocal of the distance we have,
in the present method, the La Place coefficients entering as factors in the coefficients
of the various arguments. These coefficients have been tabulated by RUNKLE in a
work published by the SMITHSONIAN INSTITUTION entitled New Tables for Determin-
ing the Values of the Coefficients m the Perturbative Function of Planetary Motion j
and hence the work relating to the determination of the expressions for the odd powers
of the reciprocal of the distance is rendered comparatively short and simple.
In the expression for A 2 , the square of the distance, the true anomaly is involved
In the analysis we use the equivalent functions of the eccentric anomaly for those of
the true anomaly, and when making the numerical computations we cause the eccentric
anomaly of the disturbed body to disappear. This is accomplished by dividing the
circumference into a certain number of equal parts relative to the mean anomaly and
employing for the eccentric anomaly its numerical values corresponding to the various
values of the mean anomaly.
Having the expressions for the odd powers of the reciprocal of the distance in
series in which the angles are the mean anomaly of the disturbed body and the
eccentric anomaly of the disturbing body, we derive, in Chapter II, expressions for
the J or Besselian functions needed in transforming the series found into others in
which both the angles will be mean anomalies.
In Chapter III expressions for the determination of the perturbing function and
the perturbing forces are given. Instead of using the force involving the true anom-
aly we employ the one involving the mean anomaly. The disturbing forces employed
are those in the direction of the disturbed radius- vector, in the direction perpendicular
to this radius-vector, and in the direction perpendicular to the plane of the orbit.
Having the forces we then find the function W by integrating the expression
dW A dQ . D
f - = A.a r - - B . ar
n . dt dg
in which A, and B are factors easily determined.
8 A NEW METHOD OF DETERMINING
From the value of W we derive that of W by simple mechanical processes, and
then the perturbations of the mean anomaly and of the radius-vector are found from
n
. te = nfw.dt
C<tf
I
J dy
being a particular form for g.
The perturbation of the latitude is given by integrating the equation
u
~~
COS*
C being a factor found in the same manner that A and B were.
It will be noticed that in finding the value of n . &z two integrations are needed ;
in finding the perturbation of the latitude only one is required.
The arbitrary constants introduced by these integrations are so determined that
the perturbations become zero for the epoch of the elements.
In all the applications of the method of this paper to different planets the circum-
ference has been divided into sixteen parts, and the convergence of the different series
is all that can be desired. In computing the perturbations of those of the minor
planets whose eccentricities and inclinations are quite large, it may be necessary to
divide the circumference into a larger number of parts. In exceptional cases, such as
for Pallas, it may be necessary to divide the circumference into thirty-two parts.
In the different chapters of this paper the writer has given all that he conceives
necessary for a full understanding of all the processe s as they are in turn applied
And he thinks there is nothing in the method here presented to deter any one with
fair mathematical equipment from obtaining a clear idea of the means by which astron-
omers have been enabled to attain to their present knowledge of the motions of the
heavenly bodies. The object always kept in mind has been to have at hand, in conve-
nient form for reference and for application, the whole subject as it has been treated by
HANSEN and others. Thus in connection with HAN SEN'S derivation of the function
TF, to obtain clearer conceptions of some matters presented, the method of BBUNNOW
for obtaining the same function has also been given. In some stages of the work
where the experience of the writer has shown the need of particular care the work is
THE GENERAL PERTURBATIONS OF THE MINOR PLACETS. 9
given with some detail. And while the writer is fully aware that here he may have
exposed himself to criticism, it will suffice to state that he has not had in mind those
competent of doing better, but rather the large class of persons that seems to have
been deterred thus far, by imposing and formidable-looking formulae, from becoming
acquainted with the means and methods of theoretical astronomy. In the present
state of the science there is greatly needed a large body of computers and investiga-
tors, so as to secure a fair degree of mastery over the constantly growing material.
The numerical example presented with the theory for the purpose of illustrating
the new method will be found to cover a large part of the treatise. The example is
designed to make evident the main steps and stages of the work, especially where
these are left in any obscurity by the formula themselves. As a rule, the formula are
given immediately in connection with their application and not merely by reference.
It has been the wish to make this part of the treatise helpful to all who desire to
exercise themselves in this field, and especially to those who desire to equip themselves
for performing similar work.
The time required to determine the perturbations of a planet according to the
method here given is believed to be very much less than that required by the unmodi-
fied method of HANSEL. Nearly all the time consumed in making the transforma-
tions by his mode of proceeding is here saved. The coefficients b (i) are much more
quickly and readily found by making use of the tables prepared by RUNKLE, giving
the values of these quantities. Doubtless experience will suggest still shorter pro-
cesses than some of those here given and thus bring the subject within narrower limits
in respect to the time required. If we compare the time demanded for the computa-
tion of the perturbations of the first order, with respect to the mass, produced by
Jupiter, with the time needed to correct the elements after a dozen or more oppositions
of the planet, computing three theoretical positions for each opposition, it is believed
there will not be much difference, if any, in favor of the latter.
Again, when we wish to find only the perturbations of the first order, experience
will show where many abridgments may safely be made. And whenever the positions
of these bodies are made to depend upon those of comparison stars whose places are
often not well determined, it will be found that the quality of the observed data
does not justify refinements of calculation.
One of the things most needed in the theory of the motions of the minor planets
is a general analytical expression for the perturbing function which may be applicable
to all these small bodies. Thus if we had given the value of all in terms of a periodic
series, with literal coefficients and with the mean anomalies of the planets as the argu-
A. P. S. VOL. XIX. B.
10 A NEW METHOD OF DETERMINING
merits, we would at once have a , f by differentiation. And since
only two multiplications would be needed in finding the value of , whose expres-
sion has been given above.
In the present paper we have dealt only with the perturbations of the first order
with respect to the mass. The method has been employed in determining those of the
second order also for two of the minor planets ; but as those of Althaea, the planet em-
ployed in our example, have not yet been found, it was thought best not to give any-
thing on the subject of the perturbations of the second order, until the perturbations of
this order, in case of this body, are known.
The writer desires here to record his obligations to Prof. Edgar Frisby, of the
U. S. Naval Observatory, Washington, D. C., and to Prof. George C. Comstock,
Director of the Washburne Observatory, Madison, Wis., for kindly furnishing him
with observations of planets that had not recently been observed; to Mr. Cleveland
Keith, Assistant in the office of the American Ephemeris, for most valuable assistance
in securing copies of observed places. And to Prof. Monroe B. Snyder, Director of
the Central High School Observatory, Philadelphia, he is under special obligations for
the interest manifested in the publication of this work, and for continued aid and most
valuable suggestions in getting the work through the press.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 11
CHAPTER I.
Development of the Reciprocal of the Distance Between the Planets and its Odd
Powers in Periodic Series.
The action of one body on another under the influence of the law of gravitation
is measured by the mass divided by the square of the distance. If then A be the dis-
tance between any two bodies, this distance varying from one instant to another, it
(1 \ 2
J in terms of the time. If
r and r' be the radii-vectores of the two bodies, the accented letter always referring
to the disturbing body, we -have
A 2 = r 2 + r' 2 2rr H.
If we introduce the semi-major axes a, ', which are constants, and their relation
/
a a , we obtain
a '
// being the cosine of the angle formed by the radii-vectores.
Let the origin of angles be taken at the ascending node of the plane of the dis-
turbed, on the plane of the disturbing, body. Let II, II', be the longitudes of the peri-
helia measured from this point; also let/,/', be the true anomalies. The angle
formed by the radii-vectores is (f + IT) (/ + H) ; and the angles / + IT,/ + IT,
being in different planes, we have
H - cos (/ -f II) cos (/ + IT) + cos /sin (/ + II) sin (/' + IT), (2)
I being the mutual inclination of the two planes.
To find the values of n, IT, 7", let <I> be the angular distance from the ascending
node of the plane of the disturbed body on the fundamental plane to its ascending
12
A ]STEW METHOD OF DETERMINING
node on the plane of the disturbing body. Let ^ be the angular distance from ascend-
ing node of the plane of the disturbing body on the fundamental plane to the same
point.
If TT, TI', are the longitudes of the perihelia,
8, &', the longitudes of the ascending nodes on the fundamental plane adopted,
which is generally that of the ecliptic,, we have
IT = n - - Q' - - .
(3)
The angles 4>, 4 1 ? & &', are the sides of a spherical triangle, lying opposite the
angles t\ 180 , 7,
^ 'i', being the inclination of disturbed and disturbing body on the fundamental
plane.
The angles /, 4>, ^, are found from the equations
sin | /sin \ (^ + <) ~ sin | (& -- 8') sin | (i + *)
sin | /cos | (^ + 4>) = cos J (8 -- Q>') sin | (z -- i')
cos | /sin | (^ 4>) = sin \ (Q Q') cos \ (i + i)
cos J /cos I (^ - - $) =. cos | (8 -- Q>') cos | (^ -- z 7 )
In using these equations when Q is less than Q f we must take \ (360 + Q, &')
instead of \ (Q - - Q').
We have a check on the values of /, <J>, ^ by using the equations given in HAX-
SEN'S posthumous memoir, p. 276.
Thus we have
cos p . sin q
cos p . cos q
cos p . sin r
cos p . cos r
sin jp
sin /sin 4>
sin / cos 4>
sin / sin (^
sin /cos (^
cos 1
sin ^ . cos (Q, Q,')
COS l'
cos i . sin (8 Q')
cos (8 &')
sin ^' sin (8 Q')
sin ;j
cos p . sin (^ - - q)
sin .;; . cos (i -- q}
sin (^ q)
cos jj . cos (i - - q}
(5)
THE GENERAL PERTURBATIONS OP THE MINOR PLANETS. 33
To develop the expression for (-V we put
cos /. sin IT = k sin K, sin II' k L sin K^ )
cos n' = k cos Kj cos /cos II' = k L cos .STj, j
and hence
II = cos/". cos/" . & cos (n - - K) -\- cos /'. sin f . k L sin (II K^
- sin/*. cos/' . k sin (II - - ^T) -}- sin/, sin/' . A?i cos (II JSQ.
Introducing the eccentric anomaly F, we have
cos/" zz - (cos e), sin/ 1 = - . cos <p . sine,
e being the eccentricity, and $ the angle of eccentricity ; and find
T T 1
- ... //= cos c . cos E' . k cos (n K] cos F' . ek cos (n K]
a a
- cos . t'k cos (n K) + ee'k cos (II K)
+ cos F . sin F' . cos <?>' . fa sin (n K^ sin e' . e . cos <p' . fa sin (II
- sin e . cos F' . cos <?> . fc sin (II .5T) + sin .e' . cos <|> . k sin (n J5T)
+ sin . sin t' . cos $ . cos $' . ^ cos (II jffi).
Substituting the value of r . 7 , . // in the expression for f-J we have
/ j\ -
( J = 1 + a 2 2e . cos F + e 1 cos 2 e 2 aee'k cos (II /iT)
+ 2ae'k cos (II K) cos e 2ae' cos <p . ^ sin (n K) sin F
- [2aV -- 2a<?& cos (n K) + 2flfc cos (II 7T) cos e
- 2a cos ^ . k sin (II JT) sin F] . cos t'
- 2ae cos ^)' . fa sin (II JQ + 2a cos ^) cos q> . k^ cos (II JSTj) sin F
+ 2a cos <^' . A;, sin (II Id) cos e] . sin f'
a 2 6 /2 .COS V.
Putting yj, /^ , y 2 , for the coefficients of cos f', sin t', cos V, respectively, and y for
the term not affected b}^ cos t' or sin e', we have the abbreviated form
y y] . cos F' ft . sin F -f- 7^ cos V. (7)
14 A NEW METHOD OF DETERMINING
/ A \ -
In this expression for ( J , y c , y 1? and (3 are functions of the eccentric anomaly
of the disturbed body ; y-> is a constant and of the order of the square of the eccen-
tricity of the disturbing body.
In the method here followed the circumference in case of the disturbed body will
be divided into a certain number of equal parts with respect to the mean anomaly, g.
mu f -11 4.1 u no 360 360 o 360 -, 360
The various values 01 q will then be (Jr. , z. , d. , . . . . n 1.
n 7 'it ' n ' n
For each numerical value of #, the corresponding value of F is found from
g e sin F.
Before substituting the numerical values of cos F, sin F, for the n divisions of the cir-
cumference, the expressions for y^ y l7 /3 , will be put in a form most convenient for
computation.
Let
p. sin P - 2a 2 e - 2aJc cos (n K) \
e f ($)
p. cos P =: 2a cos <>' Jc L sin (II J5Q, J
and
fr=/'cosp- 1 (9)
we find
($Q r=/sin F =. 2a . cos <p . cos <>'. & x cos (II JKi). sin F + p cos P. cos F ep . cos P
yi =fcosF= (Z'j? * p sin PJ. cos F 2a . cos <p . k sin (n -- K) . sin -\- ep . sin P.
And from these equations we find, since
/. sin (F P) f. sin Fcos P /cos F. sin P
/. cos (F~ P) r=/cos F. cos P + /sin P. sin P,
/. sin (P P) rr [2a, . cos $ . cos <p' . ^ cos (II JT,). cos P
+2a.cos (?>.& sin(Il JT). sin PJ . sin F + p 2ot 2 - sin P ] . cos F ep
f. cos (F P) = [2a . cos ^ . cos <' . ^ cos (11 - - K { ) . sin P
2a . cos <p . ^ sin (FT K] . cos P] . sin F -f 2a 2 . - . cos P . cos F
e
THE GENERAL PERTURBATIONS OF THE MIISTOR PLANETS.
If we now put
v sin V rz 2a . cos <p . Jc sin (II K]
v cos V zz 2a . cos ^) . cos fy' . ~k cos (II /iQ
iv sin TF = p 2a 2 . - . sin P
e
10 cos W v . cos ( "F" P)
?! sin JFi=: v . sin ( V P}
^u l cos Wi= 2a 2 . * . cos P,
(10)
we get
Further, if we put
/. sin (F P) w. sin (s
f. COS (jF P) = M! . COS (e + TTi).
= 1 + a 2 2a 2 . e' 2 ,
we have
= j . e . cos
e'y i
(11)
(12)
or,
R 2e . cos F + e 2 . cos 2 e + e' ./cos
(13)
AVe find the value of y. 2 from
The constants, ^, JT, & b 1^, ;^, P, n^, IF", w,, TT^, ^, are found, once for all, from
the equations given above. For every value of e we have the corresponding value of
/and Pfrom equations (11) ; hence, also the values of _/sin ^,/cos F, which are the
values of /# and y lt Equation (13) furnishes the value of y by substituting in it the
various numerical values of e, as was done for j3 Q and y^ The value of the coefficient
y 2 being constant, we thus have given the values of (-J for as many points along
the circumference as there are divisions.
16 A NEW METHOD OF DETERMINING
We can put
f- ] y yi c s ?' ft sin e' -{- / 2 . cos . V
\ u> /
in the form
= [(7- g. cos (*'-#)] [1-^. cos (E' -,)],
in which the factor 1 q l . cos (V - - Qi) differs little from unity. For this purpose, if
we perform the operations indicated in the second expression, and then compare the
coefficients of like terms, we find
y = C+q.q.s'mQ. sin Q l
YL zz q . cos Q -f- q^ . C COS Q {
ft = . sin Q + #1 <? sin Qi
0=rin-(C+ft)
The last of these equations is satisfied by putting
Q, = --Q.
The remaining equations then take the form
yi = to 4-^-0). cos Q
y-2 = 2 . ffi
ft = (q qi.C). sin Q
The expressions
q . sin Q ft + ]
g.coBQzry,-, [
5 l . C. sin Q =
g L . (7. cos Q V J
satisfy the relations expressed by the second and fourth of equations (15), where
c= yo + <;.
"We have now to find expressions for the small quantities , q, f found in these
equations,
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 17
Equations (16) give
q.q,. Csm 2 Q= (ft + )..
The equation
7o= O q.q^sm^Q
then becomes
(yo + ?K=(ft + )f (a)
From (16) we have, also,
from which, since y. 2 = q . q^ and (7= y Q + f, we obtain
and hence
Equations (16) give again
(y,->0f = (& + )? (c)
When is known, is found from (a) ; and the difference between (a) and
gives >7 when f is known.
The equations (a) and (c) give
A. P. 8. VOL. XIX. C.
18 A NEW METHOD OF DETERMINING
Deduce the values of (3 Q + , ^ Y\ from (a) and (d), substitute them in (c), we find
The last equation then takes the form
This equation furnishes the value off; and with f known, we find , >?, from equations
already given. The three equations giving the values of the quantities sought are
=OJ
Finding the values of f, , >7, from these equations, and arranging with respect to
preserving only the first power, we have
/2
(9)
Substituting these values in equations (16), they become
h C sin =.
n
(17)
noting that C = y + f .
If more accurate values of f, , >?, are needed than those given by equations (#),
we proceed as follows :
Substitute the value of f given by (</) in the second term of the first of equa-
tions (/'), we find, up to terms including y 2 2 ,
4.
(18)
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 19
The last two of (/) give also
c ' * *
_
Introducing the values of/, F, given by (11), putting
(19)
we have
f = % .sin^
so that
<7= yo + .sin 2 .F. (20)
Moreover, since
72 ? = 2'. cos IF,
we find from the expressions for f , 77, given above,
& + =/.'. sin JP, .
7i~>7 =/>?'. COS ^
if
Substituting these in the expressions for q sin , q cos Q, they become
2 sin Q=f. .sin^ 7
(22)
20 A NEW METHOD OF DETERMINING
The value of q^ is found from
?L = r * (23)
The quantities q, q^ Q can be expressed in another manner. The equations (22)
give
<f f 2 . f 2 . sin ' 2 F+f 2 . vf 2 . cos -F;
from which we derive
Q=F + r= . sin 2F + | - - sin 4.P+ etc.
* ~t~ v v ^n~V''
log. gr = log./+ 1 log. (f- . sin 2 .F+ ^ cos 2 ^).
Since % 2 and ^^ agree up to terms of the third order, the equations for ' and >/
give
^-V - g (y + /) .
or
Further
' 2 sin *F-\- >?' 2 cos 2 F =. 1 + 2 -^ (% . sin 2 F ^' cos ~F) -
and
J log. (f 2 sin 2 ^+ >/ 2 cos 2 ^) = ^ (% sin 2 ^ ^' cos
TT (z s^ 11 2 ^ %' cos
Substituting the values of , ^', (7, given before, we find
( K sin ^- ^ cos *F) = JL + - - ? + cos
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 21
The equation y., nz q . q { gives
lg- 72 = log. q + log. ^
Putting:
log. ff = log.
we have for 3^
log. g 1 = k>g. y.
Writing s for the number of seconds in the radius, and ^ for the modulus of
the common system of logarithms, we find
(24)
in which
log. q =\og.f+y
log. 0i =
x = s . sn s - sn
(25)
cos 2^-X - -- 2 cos
And for C we have from the first of (15)
C^yo + ^.sin 2 ^. (26)
By means of the last three equations we are enabled to find the values of
ft <?? <?i> > w ^h the greatest accuracy. The equations (17), where not sufficiently
approximate, will, nevertheless, furnish a good check on the values of these quantities.
(A\ 2
J are thus known; and substituting their
values corresponding to the various values of g, we have the values of (~\ for the
different points of the circumference.
22
A NEW METHOD OF DETERMINING
Using the values of (7, </, </ b Q, just found, HILL, in his New Theory of Jupiter
and Saturn, has given another expression for (-} which we shall employ.
To transform
= (<7 q . cos (V - Q) )(!-&. cos (.' + Q))
into the required form we put
(27)
= seel . seel fr
V (7 ~
Then
( V z= C [l sin x - cos ( e ' )] [l s i n 1 cos ( e +
C [sec 2 1% (1 sin % . cos (e' - - (2))] [sec 2 1;^ (1 sin ^ . cos (e' +
sec 2 |ri
sec
C
cos
sec
sec
Substituting the values of a, &, JV^ we get
- N n [1 + a 2 2a cos (V )]" 2 [l + 6 2 2& cos (V +
(28)
We compute the values of a, &, JVJ corresponding to the different values of g, and
check by finding the sums of the odd and the even orders, which should be nearly the
same. If we put
[1 + a 2 2a cos (E' Q)]~ s - [J J> + & (1) . cos + 6 (2) . cos 29 + & ( " . cos 3,9 + etc.]
[1 + Z> 2 26 cos (e' + Q)]~ s - [l ^ (0) + BV . cos (e'+ Q) + B-\ cos 2 (/ + ) + etc.]
where s = ^, =: e' , we are enabled to make use of coefficients already known.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 23
For 2 . cos 0, write x + - , and then we have
7
+ a 2 2a cos 0]~ s = [l + a 2 a (x + ^
Expanding we have
, I r+l S _+J S + 3 S + 4 55
1*2 3 ~T~ ~5~ * C>
_ a~|~ s _ -j, s a . s s + 1 a 2 , s^ s + 1 s -f- 2 a 3 , s 8 + 1 s+2 g + 3 a*
^J 1 * * ~" 1 * 2 " ^ 2 " t " I ' 2 ' 3 ' ^ 3 ' 1 '
__
2 * 3 '
,s s+ 1 s_+2 s + 3 s + 4 a 8 . ,
1" 2 3 4~ ^5~ '^r 5 " 1
And hence, for their product, we have
/ s a + i s + 2 v 2 8 + 3 . a * + etc.! / l
" u ~2 3~; ^^ J r " *
-4- f s S -^ 2 , /s \2 s + l s + 2 4 / s + l\2 s + 2 s + 3 6
" Ll * 2 " VU ' 2 3 U'~2 ;'~3"~'~~4~- tt
, (s s + 1 , + 2y 8 + 3 + 4 8 -l / o, 1
u ~2 3 ; ~4~ "T^ a tc> J r
-1- P 1_1 S +J /.S.L/'M 2 ?-l !+-? S + 3 5
Ll ' 2 3 VI / ' 2 3 4
fs s+l\ 2 s + 2s + 3s + 4 7|
+ (r T-) 4 -5 a + etc -
+ etc.
But aj + 1 - = 2 cos 0, or + ^ = 2 . cos 20, a? 3 + \ = 2 . cos 30, etc.,
1 / ' /y* ' /T 1 * * '
24 A NEW METHOD OF DETERMINING
and hence
^
= 2,a[l + 1^ .* + 1, (- +3* * + ? .Ct 1 ^ 2 )
s _)_ 2 o i s s+ls + 2s + 3 4
, a + l /s + 2\2 s + 3 S + 4 . , t "I
I ' ~2" V 3" V ' ~T~ "5"
s + 3 j , . s+1 S + 3 S+ 4 , (29)
_0 81 , ,
-^T-~2~'~T~- a L - f . 4 -.a r*- 4 5
s s + 1 s + 2 s + 3 s + 4 s + 5 6 ,
"r-"^"-~3^'~"4"' :, 6 C ''
and generally
>= 2 . - . '
.... . . . . .._
Since s = ^, we find from these expressions the values of the 6 {0 coefficients for
different values of n.
RUNKLB has tabulated the values of 5 (?) in a paper published by the SMITHSONIAN
INSTITUTION. Thus the value of
[1 + 2 2a cos (s' )] ~
is obtained with great facility.
_ n
The value of [1 + 6 2 26 cos (e + Q)~]~ 2 is found in the same way.
We now let
c"> = i.^r.^.cos2^|
s (i) =i.JSr. (i) .sm2.iQ)
And hence have
c f0) = i . N. .Z? (0)
etc.= etc.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
25
Multiplying the series [| 6 (0) + & (1) . cos + Z> (2) . cos 20 + 6 (3) . cos 30 + etc.]
by [J (0} + B (Y > cos (e' + Q) + ^ (2) . cos 2(e' + Q) + etc.] ,
noting that (& e ', and arranging the terms with respect to cos *0, sin iQ,
we find
= . c
(0)
. c
(2)
. c
(0)
etc.
cos
gin
sin 20
cos 30
sin 30
etc.
(31)
c L cos / = & (i) . c (0)
sn
^ Cr' .
C * '
and we find
(2)
( a j = A: t [cos J . cos iO + sin ^ . sin .
= Tci cos (*0 JSTJ =, ki . cos (i Q
K t ).
Subtracting and adding the angle ig t this becomes
-} i- &,cos \i(0 Q} KI + (iq i
A I \ 4^ 7 / \ /
^z ^ cos * ( Q g] KI cos . i (g e') lc>i . sin i
If we put
(32)
(33)
i sin . t (g e') (34)
^ K = - ^ cos p ( Q K g K )
Ki, J
(35)
h i
A. P. S. VOL. XIX. D.
26 A NEW METHOD OF DETERMINING
n being the number of divisions, we find
fa\ l c] / A (S) / , \ (Qa\
( } A L K . cos i (g K E J AI K . sin ^ (g K e K ) (ob )
\ A)
If now, for the purpose of multiplying the series together, we put
(0 (0 J& ^
A^-^C^.
we have
. i>v ^
() (o ()
. t . = 2 $, . cos y# + 2 fi/,; . sin v</ J
= [2 [, cos j># + 2 Cl> smvg] cosi(ge')[2 S Lv cos r#-f-2 $, sin w/] sh (#'
(38)
Performing the operations indicated we get
(c) (c) (c)
22 cos (gr is'}. Cj-, cos vg 22 J C<, cos \_(i-\- v) 9 *V ] +22 1 (7 (> cos [(* i^) ^ te'J
(s) () (s)
22 cos (ig ie'). C i>v sin vg =. 22 J C^> sin [( + v) ^ w'] 22 J C^> sin [(* v) g tV]
(c) (c) (c)
22 sin (V? &') /S',,, cos v^r =. 22 J ^> sin [(t-f v) ^ *V] 22 J ^> sin [(* v) ^r ie'~]
(s) (s) (s)
22 sin (ig ie f ) S i>v sin vg 22 J S liV cos [(*+ v) g tV] 22 1 /S t> cos [(*' v) g u'~\
Summing the terms we find
(c) () () (c)
= 22 i ( C,,, =F , J cos [( =F v) ^ &'] =F i 22 ( a> , J sin [(i =F v) g u'] (39)
(c) (c)
From the formula of mechanical quadrature just given, we have C it , 8 it , when
(c) (c)
v n ; but we know that they are J . CJ, , J ^ o? as shown by their derivation.
Thus
(c) (c) (c) (c)
AI = J Q o + O { i cos (/ + C/ 2 cos 2g + etc. I (c) ()
w ( S ) > = 2 C it v cos ^ -f 2 C-, sin vg
+ <7 U sin gr + C it 2 . sin 2^r + etc. J
(s) (c) (c) (c)
At = \ BW + Ski cos ^ + Si ,2 cos 2^ + etc. ^ (c) w
w ( J = 2/S z> cos r^ + 2/S' /> sin vg.
+ /S/ ; i sin # + /S i; 2 sin 2g + etc. J
Hence where v = 0, each series is reduced to its first term.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 27
In the application of the very general formulae care must be taken to note the
signification of the various terms employed.
In case of
(c) 2
= ~ \ - cos P (ftflO KI, J
n
(s) 2
= &/, siu [ (&-#*) KI, J >
>i
n shows the number of divisions of the circumference ; and we divide by ^ m form-
ing Jc t K to save division when forming the coefficients c,,, s v .
The index and multiple i shows the term in the series
Ji (0) + Z> (1) cos (V Q) -\- & (2) cos 2(Y Q) -f Z> (3) . cos 3(e' - - Q) -(- etc.
The double index i, x shows the term of the series of La Place's coefficients and
the particular point in the circumference.
The index v shows the general term of the series expressing the values of
(c) (s)
AI :IK , -Af,,, when we give to v values from v r= 0, to the highest value of v needed in
the approximation.
2
In .&,,, ^*(* g K ) -5^ for each value of i 9 there are n values of each
it/
quantity.
(c) (c) () (c) ()
The next step is to express the w values of A , ^ , A l , u4 2 ? A > etc., respec-
tively in terms of a periodic series. And since these quantities are functions of the
mean anomaly #, if we designate them generally by Y, of which the special values are
~y ~y ~y ~V~
we have
Y Jc + c, cos g + c 2 cos 2g + etc. )
+ i sin ^ + s 2 sin 2g -\- etc. )
The values of c,, , in this series are found from the n special values of Y.
28 A NEW METHOD OF DETERMINING
From
(C) (8)
I , or AI | c + ^! cos g + c 2 cos 2</ -f- etc.
+ ! sin g + s 2 sin 2# + etc.,
and similarly, for every other value of x in A it K , A it K , we have a check on the values of
, s,, in each series. Thus if in case of sixteen divisions of the circumference we
take g =r 22 , 5 and find the value of the series, the sum of the terms must equal the
(c) (s)
value of At, , A it K , corresponding to g = 22 . 5. And this check should be employed
on each series, using that value of g that gives the most values of c, and s v . If i
nds to i 9, we
In the equation
(e) ()
extends to i 9, we have ten separate checks for the values of A lt A it KJ respectively.
Y= \C Q + <?! . cos g -{- c. 2 . cos 2g + c 3 . cos 3f/ + etc.
+ i sin g + s 2 . sin 2^r + s 3 . sin 3</ -f etc.,
if the circumference is divided into twelve parts, each division is 30. Then for the
special values of Y we have
YQ = Jc + c t + c 2 + c 3 + etc.
F! = \c Q + G! . cos 30 + c, . cos GO 6 + c 3 cos 90 + etc.
+ s, sin 30 + s. 2 sin 60 + s s sin 90 + etc.
F 2 = Jc + G! . cos 60 + c. 2 . cos 120 + c s cos 180 + etc.
+ s x sin 60 + s., . sin 120 + s a sin 180 + etc.
r n = Jcb + C A . 330 + c 2 . cos 300 + c 3 cos 270 + etc.
+ s 1 . 330 + s. 2 . sin 300 + s 3 sin 270 + etc.
In the same way we proceed for any other number of divisions of the circum-
ference.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 29
Now let
(0.6 )=r +:r 6 ($) =r -r 6
(1.7 )=Y 1 +Y, (I) nri-F,
(2.8)=:F 2 +:r 8 . (|) = r 2 -F 8
= F 5 + F u (A)r= F 6 - Fu
Then
3(c + 2c 6 )=: (0.6)+ (2. 8) + (4. 10)
3(c -2c c ) = (1.7)+ (3. 9) + (5. 11)
3(c 2 + c 4 ) = (0.6) [(2 . 8) + (4 . 10)] sin 30
3(c 2 - c 4 ) = [(1.7)+ (5.11)] sin 30 (3.9)
3(s. 2 + s 4 ) = [(1.7)-- (5 . 11)] cos 30
3(,9 2 - s 4 ) = [(2 . 8) - - (4 . 10)] cos 30
3( Cl + (%)= (f) + [(I) -(^)] sin 30
6.c 3 = (#)-( + (A)
3( Sl + , 6 ) = [(I) + ( T y ] sin 30 +
3( Sl - 8 5 ) = [(f ) + (A)] cos 30
The values of these coefficients can be easily verified by finding the values of
each one from the sum for all the different values of Y as given in the series for
V V V V
- 1 - o? * i> -* 2> J- 11
When we divide the circumference into sixteen parts, each division is 22. 5. We
find the values of Y 0) Y^ Y 2 , . . . . Y^ as in the case of twelve divisions. To find
the values of c v and s,,, in the case of sixteen divisions, we put
(0.8 )=Y a +Y s (f) =T,-T,
(i.9)=r,+ r, (i) =r,-r,
(2.10)= T,+ r 10 (A)= r 2 - r IO
(7 . 15) = F, + F u ( A) = F 7 - F B
30 A NEW METHOD OF DETERMINING
(0.4) = (0.8) +(4.12) (0.2) = (0.4) + (2.6)
(1.5) = (1.9) +(5.13) (1.3) = (1.5) + (3. 7)
(2. 6) ='(2. 10) + (6. 14)
(3. 7) = (3. 11) + (7. 15).
Then
4(c 2 + O = (0.8) -(4.12) '
4fo- c ) = { [(1 . 9) - (5 . 13)] - [(3 . 11) - (7 . 15)] j cos 45
4(s 2 + s 6 ) - \ [(1 . 9) - (5 . 13)] + [(3 . 11) - (7. 15)] } cos 45
4(s 2 -s,) =(2. 10) -(6. 14)
8.d=(0.4) (2.6)
8. s 4 = (1.5) (3.7)
4(c, + c,) = (f) + [(A) - ( T 6 4)] cos 45
4( Cl - e,) = [(i) - (A)] cos 22 5+ [(&) - (A)] cos 67 5
4(c, + c 3 ) = (-1) - [(A) - (A)] cos 45
4(c, - c 5 ) = [(I) - (A)] sin 22 . 5 - [(-ft-) - ( T %)] sin 67 . 5
4( Sl + S7 ) = [(i) + (^) ] sin 22 . 5 + [(&) + (A)] sin 67 5
4(s, - ,) = [(T%) + (A)] cos 45 + (A)
4(s 3 + *) = [ft) + (A)] cos 22 ." 5 - [( T T ) + (A)] cos 67 . 5
cos 45- (A).
When the circumference is divided into twenty-four parts, each part is 15.
Let
(0.12)= r o + F J2 (0.6) = (0.12) + (6. 18) (4) = (0.12) -(6. IS)
(1.13)= F x + F 13 (1.7) = (1.13) + (7. 19) (*) = (1.13) -(7. 19)
(2.14) = F 2 + r i4 (2.8) = (2.14) + (8.20) (|) = (2 . 14) - (8 . 20)
(11.23)= F u + F 23 (5. 11) = (5. 17) + (11. 23) (^-) = (6 . 17) - (11 . 23)
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 31
Then
6(c + 2 . c 12 ) = (0 . 6) + (2 . 8) + (4 . 10)
6(c 2 + c jo ) = () + [(I) - (^)] sin 30
60, - Clo ) = [(*)-- (A) ] cos 30
6(c 4 + c. ) = (0 . 6) [(2 . 8) + (4 . 10)] sin 30
6(c t c 8 ) = [(1 . 7) + (6 . 11)] sin 30 (3.9)
G)s, + 10 ) = [(I) + ( T S T)] sin 30> + (f)
6(s 2 -s 10 ) = [(*) + (A)] cos 30
6(s 4 + s s ) = [(I) (^)] cos 30
Q(s,-s s )
Further, let
23
Then
6(0, + c,, ) = (A) + [(A) (M)] cos 30 + [(A) (A)] cos 60
6(c, c,,) = [(A) (H)] cos 15+ [(A) (A)] cos45 + [( 1 'V) (A)] c s75 c
6(c 3 + c, ) = (A) - (A) + (A)
6(c 3 - c, ) = j (A) - (M) -[(A) - (A)] - [(A) - (A)] I cos 45
6(cj + c,) = (A) [(A) (H)] cos 30 + [(A) (A)] cos 60
6(c 5 -c 7 ) = [(A)-(tt)] sin 15 - [(A) - (A)] sin 46 + [(A)-(A)]W
6(., + ) = [(A) + ()] sin 15 + [(A) + (A)] sin 45 + [(A) + (A)] *in 75
6( 8l ,,) = [(A) + (if)] sin 30 + [( A) + (A)] sin 60 + (A)
6(b + s, ) = { (A) + (tt) + (A) + (A) - [(A) + (A)] } cos 45
6( Si + , ) = [(A) + (H)] cos 15 - [(A) + (A)] cos 45 + [(^) + ( T V)] cos 75
6(s 5 -s 7 ) = [(A) + (if)] sin 30= - [(A) + (A)] sin 60 + (A).
32 A NEW METHOD OF DETERMINING
When the circumference is divided into thirty-two parts, each part is 11. 25
Let
( 0.16)= r + Y u (0.8 ) = (0.16) + ( 8.24) (0.4) = (0.8 ) + (4
( 1.17)= F,+ r 17 (1.9 ) = (1.17) + ( 9.25) (1.5) = (1.9 ) + (5
( 2.18)= F 2 +r M (2. 10) = (2. 18) + (10. 26) (2.6) = (2.10) + (6.14)
(15.31)= F 15 +F 31 (7. 15) = (7. 23) + (15. 31) (0.2) = (0.4 ) + (2.6 )
(1.3) = (1.5 ) + (3.7 )
( 8.24) (f) = (0.8 ) (4.
( 9 . 25) Q) = (1 . 9 ) (5 .
: (|) = (2. 10) -(6. 14)
- (7 . 23) (15 . 31) (f) = (3 . 11) (7 . 15)
Then
8(c +2.c 16 ) = (0.2) + (1.3)
8 (c 2 . c 16 ) = (0 . 2) (1 . 3)
8 (c 2 - Cl4 ) = [(i) - (A)J cos 22 .5 + [(A) - (A) J cos 67 . 5
-' \* / 4 I 12/ \^"/
8 (c 4 c 12 ) = [(I) (f )] cos 45
+ c 10 ) =(|)-[( T 2 o)-(T 6 4)]cos45
- Clo ) = [(i) -(A)] sin 22 .<> 5 - [(A) ~ (A)] s^ 67 . 5
16. c 8 =(0.4) -(2. 6)
8 (s 2 + s u ) = [(i) + (A)] sin 22 . 5 + [(A) + (A)] sin 67 . 5
8(s 2 -s 14 ) = [(A) -(A)] cos 45 + (A)
R fa -- o ^
O ^o 4 6i 2 y
8 (s 6 s 10 )
/2\
VS/
[tt) + (A)] cos 22.5-[(A) + (A)] cos 67 .' 5
[(A) -(A)] cos 45 -(A)-
TUB GENERAL PERTURBATIONS OF THE MINOR PLANETS. 33
Further, let
\ 5 \ V
3TV * M
And besides, let
[(A) - (tf )] cos 11-.25 + [(A)
[( T V) - (tf )] sin 11.25 - [(^)
cos 78.75
sin 78.75
[(A) - (**)] cos 22.5 + [(JL) _ (i|)] cos 67.5
A) - tt!)] sin 22.5 - [( A ) - (if)] sin 67.5
( A) - (tf)] cos 33.75 + [( A) - (if )] cos 56.25
(T S ) - ttf)] sin 33.75 - [(,fr) - (i|)] n 56.25
'" = (A) + [(A) - (it)] cos 45
'" = ( A) - [(A) -ttf)] cos 45 .
' = [( iV) + (if)] sin 11.25 + [(A) + (A)] ^n 78.75
= [( iV) + ()] cos 11.25 - [(A) + (A)] cos 78.75
? = [(A) + ()] ^ 22.5 + [(^) + (fi)] sin67.5
' = [(AM tt4)]cos22.5 -[(^) + (io)] cos 67.5
" = [(A) + (tt)] sin 33.75 + [(^) + (J|)] sin 50.25
>" = [(A) + ()] cos33.75- [(Jj-) + (W-)] cos56.25
"' = [(A) + (41)] cos 45
'" = [(A) + ()] cos 45
A. P. S. VOL. XIX. E.
34 A NEW METHOD OF DETERMINING
Then
S( Cl + Cu) = A" + A'
8( Cl C*) = A + A"
8 (* + ) = &" + 3'
8 fa - c 13 ) = [ A A" + B + JB"] cos 45
8 (c 3 - c u ) = [ J. - -4" - (-5 + J?"
8(c 7 + c 9 ) = A"' A
S( Si 8^=C"'+C'
8 ( 3 + *!) = [D + D" ~ ( C - C")] cos 45
8 (8 5 + ) = ID + Z>" + (7 C"] cos 45
8(8 6 ) = />' --D'"
8(s 7 + s 9 ) = D i>"
8(5 7 s 9 ) = C""+ C".
The expressions for the determination of the values of c v and s v , just given, are
found in HANSEN'S Ausewandersetzung, Band I, Seite 159-164.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 35
CHAPTER II.
Derivation of the Expressions for BESSEL'S Functions for the Transformation of
Trigonometric Series.
The value of Qj given thus far is found expressed in a series of terms the argu-
nents of which have the eccentric anomaly of the disturbing body as one constituent.
3ut as the mean anomaly of both bodies is to be employed, it will be necessary to make
me transformation ; and the next step will be to develop the necessary formulae for this
mrpose. HANSEN, in his work entitled Eatwickelung des Products einer Potenz des
Radius Vectors et cet., has treated the subject of transforming from one anomaly into
mother very fully ; what is here given is based mainly on this work.
Calling c the Naperian base, and putting
ni f*y 1- fti' /iY 1
y c > y >
ve have
'= (cos e + V 1 sin <0 (cos e'+' V 1 sin e')>
ilso
, ,__ _ I gin *,
=. cos (i e i' e') + V 1 sin (i e' i' e').
Denoting the cosine and sine coefficients of the angles (is i' e') by (, i' 9 c)
d (i, i', s) respectively, the series
F^^ (i,i, c) cos (ie i f e' ) 22 V T(^, i', s) sin (ie i'e 1 ) (1)
;an be put in the form
F = i 2 2 5 (*; i', c) - V^ (*, t% s) | y 1 y' 1 ". (2)
36 A NEW METHOD OF DETERMINING
In a similar manner we get
F = J . 2 2 { ((i, h', c)) V T(te A', 8)) y< . a'-*', (3)
where
z' = c-*^.
We have now to find the relation between y and z.
Let
g n the mean anomaly,
and s = the eccentric anomaly.
Then from
g zz 8 e sin e,
introducing V 1? we get
g V l = eV 1 esine V
Since
2 V l.sinerry 2T 1 ,
we find
from
7/ r yi
y
we obtain
V 1 =log.,
V l = log.y,
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 37
and
(4)
Thus
g V 1 = log. z log.
and hence
z = y.c "2" . (5)
From
z = y . c~ ^(y-y~'\
we have
and
Let I be denoted by X ; then
&* - y~\ (8)
and
c ^(y~y ) c l y . c~ il ^ . (9)
But
C- 7lA -V . C^ A ^rz Tl ^ . y + T-O 2/ 2 - jVs ' ^ "^" 1234 ' &* ^ e ^ Ct )
38 A NEW METHOD OF DETERMINING
and
+ & . y + 2 . f + J . ys
Performing the operations indicated, we have
2 liars
etc
\ / _ 2 .
O ^ +
, ; ' 3p h ^ s \ f
+ 1^3 - I^2^4 db etc.] (y < -
F Ctc
, ^"^ /-i _ W _, _ A^ 4
" 1.2..m V " l.w-fl 1.2.ml.
LC
__ ^2 , JW . ^ 6 , ^ 8
1 2 .2 2 p 2 2 3 3 I 2 2 2 3 2 .4 2 '
(l'3)3 ,'6)6 ,757 \ /
+ * - i4 + TO - K^ki etc -) ( - y
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
39
As we may write li in place of i, we have, thus, also given the value of c l i(y V l )
put
+ 00 ( TO)
+ GO (m)
Then, from the preceding developments, we see that
( ni)
(m)
m (m)
m (m)
(m)
Again
(-3)
(1) (2)
+ 00 (m) (0) (1) (2)
-oo e/AA . y m = Jti + '/M . 2/ + ^A . f
(-1) (-2)
(-3)
'~ 3 + etc.
/ 3 -f etc.
3 + etc.
'~ 3 + etc.
+ co (TO)
Comparing the values of 2_ oo Jl^ . y~ m and
have
7~ 7 7,-i ^ 3 W KX' f -1
^tt :- J hx -Kk-- ^j + p-^ i^;3^r etc., tor y ,
(1) (1)
" "
(-2) (2)
(2) (2)
etc. r= etc.
7,3)3 1655 L7)7
ft AT | ft A /tA .. ^ -i
-^ - 2 - + p- 23 j jTg^g^ dz etc., for y\
+ 11 A - . ft O
,20204 ^F etc., for y \
i.i 1 . .'J Ji ,i .O.4
etc., for 2/ 2 ,
1.2 P.2.3 1 2 .2 2 .3.4
etc.
(10)
(ii)
(12)
(13)
40 A NEW METHOD OF DETERMINING
Comparing the values of 2_* J, lX - y m and c 7 ^ 2T 1 )'
we get the same expressions for y m and y~ m .
(1) (2)
We see from the values of 7 AA , J hK , etc., found above, that the general
term is
J,WI)TO Awi+2 j^wi+2 ^jWi+4 ^[wi+4
T _ __ _ I I pf"O
" A ~ 1.2...m r.2..m.m+l " I 2 .2 2 ...m.m-f l.m+2
1 1.2.m+l.m+2
Further, we have
and, by putting m = 7i i t
this becomes
Ji J n.i /-i r\
* "h\ y (^j
Let
+ 00 (h) ^
+ " (*>
Multiplying the second of these equations by z~ h . dg,
we obtain
+ co (i)
\z- h .dg = 2_ P h .dg.
Integrating between the limits + n and 7t,
we have
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 41
From
z = c gv l = cos g + V 1 sii
we have
dz == ( sin g -f V 1 cos g} dg ;
also
z V 1 V 1 cos g sin g.
Therefore
dz z V 1.<<7,
and (17) becomes
In like manner we find
'(A) i /
n. 1 . j
" 27r 1 A : rTJ
Integrating by parts we have
h
h -C y^.z^.dz. (18)
O / 1~ * ' \J /
x-, .
Comparing this value of Q t with that of P h , we obtain
or
(0 ,- (A)
s-fc ( 19 >
A. P. 8. VOL. XIX. F.
42 A NEW METHOD OF DETEEMINING
Thus we have, between the mean and the eccentric anomaly, the relations
In the application of these relations, since
MO
the expression for F is changed from
F = J 2 2 { (;, f^, r c) - V- 1 ft *\ *) j y*. 2/' -
into
F-^^ jfttVc) V lft*V) tf.
The other value of JP is
((t, ^', c)) - V- 1 ((*, ^,
A comparison of these two values gives
(-*') $
((, 7^', c)J = 2 P_, t , (, t', c) z 2 . , . J"//v (*, ^', c)
In transforming from the series indicated by (i, i', c) into that of ((?, A', <?)), it is
evident that h' is constant in each individual case, and *' is the variable.
Thus we find, beginning with i' = ^',
7/ (h'h r ) -,, _ -, O-(v-i))
((i, h', c)) = | . 7 y v (t, A', c) + ^ . ^ v 0; *' 1, c) + etc.
^'+ 1 c + etc -
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 43
To transform from f(t, ', e)J into (i 9 i' 9 c)
we have
(i, ', c ) = 2 _,* } ((*, A', c)) = 2 Ji,* * } ((*; h', c)).
Here, i' is the constant, and h' the variable ; and for the different values of 7i', begin
ning with h' i' 9
we find
(0) ((i'-l)-t'))
(V, i' 9 c) = J f v (, *' e)j + /(i'-i) v ((, *'' 1, c)) + etc.
-t-e^+Dv (0> v + l, c)) +etc.
The expression
^ - l^rn ( l ' " 1^+1 + 1.2.m+l.m+2 ~" 1.2.S.m+l,m+2.i+3 ^ GtC<
enables us to find the value of J hx for all values of m.
A simpler method can be obtained in the following manner :
,._i\ (0) (1) (-1) (2) (-2)
-* )
Putting c i~ y ~ in the form
(0)
= J + J . y-J y+ +J
fl-^ Fl-% fl~2 -
we have, for the differential coefficient relative to y,
(1) (2) (-1) (2)
,+2.e7. e .y
""2" """2"
If we multiply the second member of the first equation by ^|(1 + y 2 )> we have
an expression equal to the second member of the second expression, and by comparing
the two we find
(m+l) (m-1) ^. (m)
(22)
44 A NEW METHOD OF DETERMINING
Let
(m)
h-iT
-f^> - (23)
then
(m) (m-1)
J e J e ' 2
From this general expression we find
(1) (0)
(2) (1) (0)
etc. zz etc. z= etc.
(m)
From the values here given, since jjj-f- is put equal to p m , we have, by increas-
ing m by unity,
(m+l)
"
Putting -7 zz r w , equation (22)
h-^
takes the form
^m Pm+l I J- t
From this we find
1
I'm -
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 45
We also have
i
~ -
(25)
a form more convenient in the applications.
(m)
The general expression for J is
2
(m) (0)
e=J,e'Pl-p2.p3.. -Pm (26)
- -
where
,() I* , I
1? etc "
if we put l h^.
From the expression
( i') i (h'-V)
((*, h', c)) = 2P_ A / (*, tf, c) = 2 ^ J h w (i, i, c)
it is evident that when &' z= 0, or when both f and h' are zero, this expression cannot
be employed.
To find the values for these exceptional cases let us resume the equation
/I
When h = we have
(i)
46 A NEW METHOD OF DETERMINING
The equation
z = y.c- ( y~y~^
gives
dz dy e /- , _ 2\ j.. /9Q\
~ ^ (.* T~ y ) dy.
/
Hence
d) /.i-T^-i
--v 1
,
1
jo is a whole number
,-"V
except when jt? = 1, when this integral is 27tV 1-
Hence it follows that
When i = 0, we have
(i) (-i)
PQ =. PQ ZZ 5
(0)
Po = 1.
Using the expression
(-i') (-i') (-t'-l)
^ /,/ ,,\\ v p /,' ' ^,\ p / 7 * /' r "i _i_ p , / ? " /'_i_ 1 r \
z, li , cjj 2, . x'y ^, * > c ; -^ -A' I,*? * > 6 ^ n^ -L-W \^ - T^ J-j y
we have
((o,o,c))_=(o,o,c)-ai'(o,i,c)
for the constant term, the double value of this term being employed.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
For Ji' 0, we have
((1, 0, c)) = (1, 0, c) - X (1, 1, C ) - x (1, - 1, c)
((1, 0, *)) = (1, 0, 8) - V (1, 1, 8 ) - X (1, - 1, 8)
. ((2, 0, o)) = (2, 0, c) - X' (2, 1, c) - X' (2, - 1, c)
. ((2, 0, 6-)) = (2, 0, 8 ) - X (2, 1, 8 ) - X (2, - 1, 8 )
etc. etc.
In what precedes we have put
g r= the mean anomaly,
e i= the eccentric anomaly,
c the Naperian base,
z cf*-\
y = tf v ~\
and obtain
where c~ T& V ^ is expressed in a series, the general term of which is
.m
1.2.m+l.m+2 1.2.3.m+l.m4-2.m+3
Thus
l.m-j-I 1.2.m-f-l.m-f-2 1.2.3.m+l.m+2.m+3
etc. )y"
We have also put
and since
(m) (m)
48
have found
A NEW METHOD OF DETERMINING
(m)
(ft-O
if
Again supposing
we have found
m =. h i.
(i)
p -fc
* *
(i)
(A-t)
Thus we have
(h-i)
y
(h-i) r - "i
Jh\ |_cos * + sin is V 1 J 5
(A i)
Equating real and imaginary terms, we have
i A=00 _i (h ~ i}
COS *V=:-. 2 A= _ Jh\ COS
t - A=oo J (h-i)
siuie=-.^ h ^J h , .sin
(29)
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
We notice that
(i) (-i)
For all other values of i
(0)
Po = 1.
(0
o = 0.
49
If a large number of the J functions are needed they are computed by means of
equations (24) to (27), as shown in the example given in Chapter V.
If we wish to determine any of them independently we have from
()__ hm)m r- ^ fcp
** Z 1.2...m L " Lm+^ 1.2.
fr 4 .
m+l.m+2 i.2.3.m+l.w+2.m+3
d '
(0)
(i)
(2)
(3)
1 - '- -\ - . - -J- etc.
14 4 16 36 64 '
7,2 p i 7,4 p* 7,6 p e ~]
I ' I gf Q
" 2 '4 " 12 '16 " 144 '64 "
1.2 3 ' 4 ~^~ 24 ' 16
T.2.3
. .-
44 40 16
w (j, e-Y r / 2 ^ ~\
^ = ra3 i 4[ 1 --j-f etc J
(30)
(m)
In these expressions we have written for X its value Je.
Since h has all values from &=+co to --co we find any value of J" AA by at-
tributing proper values to h.
From equations (29) we find the values of the functions cos is, sin ie, in terms of
cos kg, sin hg, and the <7 functions just given ; always noting that when h 0, we
have only for i 1, Je as the value of the function.
We can employ equation (22) when only a few functions are needed, or as a
check.
A. P. s. VOL. xix. G.
50 A NEW METHOD OF DETERMINING
It may be of value to have y* in terms of z h and the J functions. Prom the sec
ond of equations (20) we Jiave
(0) (1) (2)
z 2 + -J 3 A.* 3 +etc.
J K . z- 1 - - i<7 2A . z~ 2 iJg, . z~* etc.
(0) (1) (2)
- Z" 1 - Z~ 2 *~ 3 + etc.
(2) (3) (4)
J.z J.*. cfe. etc.
(1) (0) (1)
2/ +2 = - ye/A . Z + | / 2 A 2 2 + |^3A - & + et C.
(8) (4) (5)
-ty.z- 1 -%J*.z-* -KsA-^" 8 -etc.
(i) (0) _ 2 (i)
(3) (4) (5)
Then from
y~ l = 2 cos
y* y~ l = 2 V 1 sin ie
we find the values of cos e, sin 8, cos 2e, sin 2e, etc.
In case of the sine, as for example when i =. 1, we have
y y~ l 2 V 1 sin e ; but in z z~ l = 2 V 1 sin g,
we have the same factor, 2 V 1> in the second member of the equation.
From
r =. a (1 e cos e)
we find
f- J zr 1 2e cos e + e 2 cos 2 e
^ V = 1 + 2e cos e + 3e ? cos 2 f + ^e 3 cos 3 e + etc.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 51
For (- ) we have
\a/
(r \ 2
- J = 1 -f 2 C " 2e cos + Je 2 cos 2e
/
But
d :_/rj\ _ 2e gin e - Q _ e cos e j ^1 - 2e sin e,
(Zgr Vav 7 rfy
and
p (0) (2)-, p (1) (3)-. p (2) (4)-.
sin = U7 A + J,, sin ^ + \ \ J^ + J 2A sin fy + i I Ju + ^ A sin 3^ + etc.
L _l L_ _l l_ J
Multiplying by 2e . d^ we have for the integral of -
(0) (2) ~i 9^> r (1) (3) n
+J* J cosg--^\_J 2 , + J 2 , Jcos2#-- 3A + 3 A cosS etc.
where c = 1 + f e 2 .
By means of (22) this becomes
/r\2 (i) (2) (3)
( a ) 1 -f f e~ f eA cos g J 2J , cos 2g ^ cos 3^ etc.
( \ 2
-J ", we have
Se 2 . cos 2 e == |e 2 (1 + cos 2e), 4e 3 cos 3 a = e 3 (3 cos e -f cos 3e),
5e 4 . cos % = |e 4 (3 + 4 cos 2e + cos 4e), 6e 5 . cos 5 e z= T %e 5 (10 cos e + 5 cos 3e + cos 5e),
7e 6 cos 6 e =z ^e 6 (10 + 15 cos 2e + 6 cos 4* + etc.)
and hence
~ 2 = 1 + + * 4 + ^ + etc.
+ [2e + 3e 3 + f|e 5 + etc.] cos e
+ [f ^ + ^-e 4 + -y^e 6 + etc.] cos 2e
+ [e 3 + f|e 5 + etc.] cos 3e
M e + etc -] cos 4e
52 A NEW METHOD OF DETERMINING
Attributing to i proper values in equation (29) we find the expressions for cos e,
cos 2?, cos 3e, etc. We then multiply these expressions by their appropriate factors and
thus have the value of ( r \~'
\aj
(2)
, r . / r \~ c
(a j ~ 2 & COS & (a) ~ 2 - Ei
(2) (-2)
The following are the values of R t and R L to terms of the seventh order of e.
(2)
J.TI - 2e -}- -%e - - -
(2)
-fl/ - " -& ~~
/2
(2)
& ~ ~ 256 e
(2)
7? _ . _2 5 fP _|_ 625 .o?
^^S 192 e l4608 e
(2)
-"6 -^O 60
(2)
7? _ 2401 P 1
**! 2"3040 6 '
^o" 2 = = 1 + e 2 + f e 4 + -W + etc.
(-2)
r> 1
J.4/4 -
24 240^
(-2)
J __ l Q 9 7 ff> 1 6 6 2 1 g7
(-2)
r> 1223.06
** 1 6 6
See HANSEN'S Fundamenta nova, pp. 172, 173.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
(2) ( 2)
We add also the differential coefficients of R, , R, relative to e.
l 7 Is
(2)
dJR a o .
de
53
(2)
de
= - - 1* + tt* ^Wo 6 etc
(2i
w
(2)
4875^
e
de
etc.
(-2)
dR<)
(-2)
de
(-2)
re -4- etc.
etc.
e + Se 3 +
2 + f e 2 +
5e + le 3 4
+
? 4 +
_
64
(-2)
de
(-2)
4608 ^
54 A NEW METHOD OF DETERMINING
The value of -, found by integrating d( r \ 2e . sin e . dg, is
(Jj \Cu /
, r i (1) (2) (3)
- 2 = I + fc 2 f J K cos g ft/a* cos % t f J" 3A cos 3g etc.
(2)
In terms of the R t functions,
r 2 (2) (2) (2)
- ZT: 1 + -le 2 jRi cos ^o cos 2q R^ cos og etc.
a 2 2 * y
Again, since
Let
then
dg r*
we have
a 2 T> ( ~ 2) -
2 = ^- cos ^^
2
and hence
^!~ 2) =
The coefficients represented by C t designate the coefficients of the equation of
the centre.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 55
Using the values of the d coefficients given by LE YERRIER in the Annales de
I'Observatoire Imperial de Paris, Tome Premier, p. 203, we have
f-g = 4: (I) - 2 (|) 3 + | (iy + JftL (|) 7 + W (|)o] sin g
I) 2 ~ V (I) 4 + V (l) c + If (I) 8 + etc. ] sin 2g
(I) 4 - W (I) 6 +* ^iP (I) 8 - etc. ] si
(I) 6 ^W^ (I) 8 + etc. ] sin Qg
_l_ [10661993 /_ \9 "1 Q
|_10080V2/ J sm y y
Converting the coefficients into seconds of arc, and writing the logarithms of the
numbers, we have for the equation of the centre,
+ [5.9164851 (I) 5.6154551 (|) 3 + 5.5362739 (|) 5 + 5.787506(|) 7 + 6. 25067 (|) 9 ] sin g
+ [6.0133951 (|) 2 6.1797266 (f) 4 + 6.067753 (|) 6 + 5.59571 (|) 8 ]sin2^
+ [6.2522772 (|) 3 6.6468636 (|) 5 + 6.690089 (|) 7 6.22336 (|) 9 ] sin 3g
+ [6 5491111 (|) 4 7.093540 (|) 6 + 7.27643 (|) 8 ] sin 4#
+ [6.8775105 (|) 5 7.533150 (|) 7 + 7.82927 (|) 9 ] sin 5g
+ [7.225760 (|) 6 7.96973
+ [7.587638 (|) 7 8.40484
+ [7.95944 (|) 8 ] sin 8#
+ [8.33880 (|) 9 ] sin 9g
56
CHAPTER III.
Development of the Perturbing Function and the Disturbing Forces.
By means of the formulas given in the preceding chapter, the functions ^(^),
ft . a2 (;!) 3 j etc., can be put in the desired form. The next step is to determine the com-
plete expression for the perturbing function, and also the expressions for the disturb-
ing forces.
If k 2 is taken as the measure of the mass of the Sun, and m the relation between
the mass of the Sun and that of a planet, the mass of the planet is represented
by mk 2 .
If x, y, z, be the rectangular coordinate of a body, those of the disturbing body
being expressed by the same letters with accents, the perturbing function is given in
the form
i xx'+yy'-\-zz f
~l
J
" 1+m J r' 3
Now
A 2 = (af x)* + (y' - 1/) 2 + O'-z) 2 ,
hence
n =*[*_'*
1-f-m L^J r' z
If a fl is regarded as expressed in seconds of arc, and if we put
s = 206264". 8, p = -=-., =-,
1-J-m a "
we have
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
Finding the expression for (H] first by the method of HANSEN, we let
h ! ' u , . k . cos (n K), h' = ** .cos<p. cos ^ . ^ . cos (n
I '* .cos<p.Jc. sin (n K ), I ! \_ . cos $' . ^ . sin (n K } ),
and have, if we make use of the eccentric anomaly,
(77) = h . cos e^) 2 . cos/' ehffi . cos/' I. sin e . () 2 . cos/'
, 7/ /a'\ 2 sin/"' 7,/a'\ 2 sin/' 7 , . /a'\ 2 sin/'
- I .cos e( J . - , el'( , . , -\-h f . sm e ( ,) . ^
\r/ cos ?> \r/ cos ^ \r / cos ^
Putting
cos/' = y\ . cos g' + y' 2 . cos 2g' + y' 3 . cos 3g' -f etc.
= ^ . sin </' + 3' 2 . sin 2gr' + V . sin 3</ + etc.
we find
cos
'2(hy'. 2 A'5' 2 ) cos
2(A/ 2 4
etc.
'x cos ( </ ) +
e)
'* cos
''i I'b'i) sin ( g' e)
eU\ sin ( g' )
2 Z'^'.J sin ( 2g' F)
sin ( 2^r' )
Z'^' 2 ) sin ( ( 2g' e )
etc.,
where
(0) (2)
(0) (2)
']-
etc.
etc.
57
A. P. s. VOL. xix. H.
58 A NEW METHOD OP DETERMINING
When the numerical value of (H) has been found from this equation we trans-
form it into another in which both the angles involved are mean anomalies. For this
purpose we compute the values of the J functions depending on the eccentricity, e, of
the disturbed body just as has been done for the disturbing body. The values of the
(0) (1)
J functions can be checked by means of the values of JJ, A , </ /iA , given in ENG EL-
MAN'S edition of the AWiandlungen von Friedrich Wilhelm Besxel, Erster Band, seite
103-109, or by equations (30) 2 .
Thus by means of the equation
(m+l) (m-l)_ m (.,)
VIA ~r ^/>\ T-T A
fl.A
(m) (0) (1)
we are enabled to find J hK if J hK , J hK are known.
It must be noted that the argument of BESSEL'S table is 2 . h%, or 2 . fcl, or he.
a)
Thus if it is sought to find the value of J 2X , we enter the table with 2 . 2A, or 2e as the
argument.
When we need the functions for k from h =. 1 to It 4, we must find the
(3) (2) (1) (0) e (-2)
values of $J ' , J \J , \J e , -, and - - \J e .
4 2 <i 2- i 22 * 2"
(1) (0) (3)
The values of ^ . J and J e we take from the table. To find J c we have
2 ' 2" 2" 4 2"
(3) (1) 2 (2)
** e ^ ~^~ ~ c" v
JD 2 I- (0) 1 (I)-,
v e "T r~Tl e/ e + ~T J . e
4-% 4. T L 4 T 4 4^J
For / e we have
(2) (0) JL_ (1)
o J ~e ~T o e-t/ C .
-
(2)
And for J e we have
(2) (0) 1 (1)
e_ Je^ + ~<rJe_
2 2 2 2
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 59
The expression for (H) can be put in a form in which both the angles are mean
anomalies. Thus, resuming the expression for (H),
sin/ 7
COS CP'
(H) = h . cos e f a ,J cos/' e/i ( a , J cos /' I . sin e f a , J . cos /'
+ 1 .cos , (Y. sin/ ; -rf'( a ;) 2 . ?!/; + A-. 8 i n E . CY .
VrV cos / \r'/ cos <p' \r' J
in which
A = ~.^.cofl (n JfT)
a 2 v
A' ZZ ^ . COS < . COS ^)' . &! . COS (II - - K^) ZZ Jff . -
7 At 7 /r-r TT" \ 1 V Sin
t - , . cos <h . A; . sin ( II 7t ) zz sw . -
a" a
) . A^j
we find the expressions for (~,Ycosf 9 f^J -^, as follows. "We put as before
(p) " cos/ = /! cos gr' + 7' 2 cos 2#' + y' s cos 3g' + etc.
(r'T cos V = 5/1 Shl ^ + ^ 8ln ^' + ^ Sln ^' + etC "
r
If we differentiate , cos f relative to q' we have
a J y
_ cos/' <tf __ r' gin f df _ __ sin
a' ' dg' a! ' ' ' dg' ~ cos
dr' aVsin/' df a' 2
since 7 - f-, / zz -., . cos of) ;
d cos <' d' r n
and hence
^ .cos/.
' 2 ^
60 A NEW METHOD OP DETERMINING
Similarly, in the case of - _ we have
J ? a' cos <p'
_# /V sin f\ _ _ _ of* sin ./"
dg' 2 \af cos <p'J " r' 2 ' cos ?>''
T> * r ^ j r sn /
out - cos / rr cos e e, and -. sin r
a' a' cos <p'
Hence
_^ 2 Ccos/') _ a^
" r' 2 >S *
a' 2 sin _
cos
() (2)-| r (1) (3) -I
cos e' = ^ + / A , J cos <;' + J LS^ 4, J cos 2gr' + etc.
r (o) (2)~| r (i) (3)-i
sm f' \jf v + J K , J sin^' + j[J, A , + ^ J sin "2g f + etc.
From the values of cos e' and sin e' we have
a' 2 r T (Q} I 2 n , r (1) (3) ~i r ( <2) (4 n
,T, COS / : |_e7 v J,, J COS #' + 2 |y 2A -- e^ J COS 20' + 3 [^ -- e/^ J COS 3(/' + etc
a' 2 sin /' r (0) (2) ~i r w ( :5 n r ( 2 >
rCsi = L^' +^'J sin ^ + 2 L^'+ ^ J sin %' + 3 L^'+
We now assume
*
(i ~ i) (ui) i
A + ^
.>*_ f^* ^*
(i'-i) (t'+i)-i p (i'-n (i'+i)-i
- -^ J ^=r[^-.+ ^].
Comparing these expressions for y'?, S' i>9 with those found in the expression for
'2 * -/*'
^2 S ~ / given above, we see that the relation between them is i' 2 .
r 2 cos ( *
9
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 61
The expressions for cos F, sin F, are the same as those of cos F', sin F', if we omit
the accents.
Hence if we perform the operations indicated in the expression for (H), we have
- i-
cos' ( ig V) ^ WtfvS&r#*'} sin ( *7-*V) (2)
y* and i' having all positive values.
Attributing to i and i' particular values, we find, noting that $ = 0, and 8' = 0',
-^ \]i . w\ + Wx ] cos ( g - - g') - - \ [U l7 \ + Z'y^'J sin ( g - - g>)
1P-71/1 *'W'i ]cos( flr-- 3r')-j-4[>Vi /VA] sin ( flf-- 5r')
P-7o-/i cos ( ^') - - i Z'^'i sin( ^')
2 [A . y,/ 2 + A'. W J cos ( 5r 2gr') - - 2 PV*+ Z>^ 2 ] sin ( ^ 2g')
2 [A. yi / 2 A'. WJ cos( g 2g') 2 [Z.V 2 ^'J sin ( </ 2<?')
2/i. W ' 2 cos( 2^)--2Z'.^' 2 sin( %r')
I [* yi/3 + *'. ^^' 3 ] cos ( ^r - 3^/') - - f P^^+Z'.y^] sin ( g- 3g')
etc. etc.
'.W'J cos ( 20 0') i[Z.Vi f-y^'J sin (-2^ ^
The numerical value of (If) given by (1) must first be transformed into a series
in which both the angles involved are mean anomalies before it can be compared with
the value given by the equation just found.
If we find the value of (Z7) from the preceding equation, it can be checked by
means of the tables in BESSEL'S Werke.
The expression for {.i f a .} is known; and with the expression for (If) just given,
we obtain the value of
The next step is to obtain expressions for the disturbing forces.
62 A NEW METHOD OF DETERMINING
Let v the angle between the positive axis of IK and the radius- vector measured in
the plane of the disturbed body, here called the plane of X. Y. The differential coeffi-
cient of the perturbing function H relative to the ordinate Z perpendicular to this
plane is found by differentiating LI relative to z and afterwards 'putting z = 0.
Thus from
O _ ' P . _ rr' -]
' l"+m LA r*'
_ m' Fl xx'-\- yy'~\- zz'~\
~ ~~^ J'
A* = (x-i*)* + (y-y>? +
= r* + r' 2 --2rr' H,
we find
dQ _ m' I" _!_ dJ _ _ ' r dH~}
dv ' ' l-\-m L A* ' dv r" dv J
dQ m' T 1 (r r'm _ H~\
dr^ '' : r+^i L " T 2 V J j " 7 2 J'
j/-v w*' r 1 j A > dz~l
dl - -- To-^A z'. /3 L
l-fm L ^ r 3 J 7
_
A dA ,dH A dJ , Tr dJ 2'
A rr rr , A zr r r //. - rz --
dv dv dr dz A
Hence
dtl _ m'
dv ~ 1-fri
dti _
dr ' l-f
' f 1 1 ~1 , r -T ni' r 2
ff H _
m L A 3 r'*_\ 1-fm ' A 3
<Zi2 7H' Fl 1 ~1 T i '
, = T , -?5 3 sin * r s
d^ l-fm L A 3 r /J J
where
H' - sin (/ + n) cos (/' + H') cos /cos (/+ II) sin (/' -f IT)
2' = r' . sin Jsin (/' -f "HO-
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 63
As before the origin of angles here is at the ascending node of the plane of the dis-
turbed body on the plane of the disturbing body, and the plane of reference is that
of the disturbed body.
If we differentiate the expressions for r ", ( , ', we find
dr
d, 2 & dQ m' 3
i 2 + T ~i - , - V-n=
dr* dr 1 +m J 5
+ ' (-IA-,) rr'H-2^--.^
l+m \/J 3 r 3 / l+m /J <i
l+m Vd 3 r' 3 J i+1
l+m '
r ^ = ,^~ . 4 (r 2 rr'H) sin Jr' sin (/' + IT)
- 6 v w /
m' 3 . o 7- ,o .;>/ /., . T-T.X
= r -. ? sm ~Ir' z sin -( /' + II')
l?/i J 5 v>/
i = rr- ( Va A-,) sin 1. r sin (/+ n)
l+m V J 3 rv Vt/
sm - r sm
m' 3 . 97- , / a' i TT\ / ^-/ i TT/\ m' / 1 1
sm " J - rr sm ( f + n) sm (/ + n } + cos
To eliminate ^T from some of these expressions we find from
that
_ i*+r j^
2J " 2l
The expression for r then becomes
dr
-> y-^ / iT~~ f*>. 9 T ^^
di2 m i r 2 r 2 1 r TT\
" -.-- ti
dr 14-m I 24 s 2J r' 2 J
dr
From the value of A 2 we have, further,
r^rr'H r' 2 r 2
2J 5
64 A NEW METHOD OF DETERMINING
and hence
*- _ in Bin j ^ Bin
J 5 J :1 J
_
2
the latter of which, by means of the expression for , becomes
' |~ r' 2 r 2 1 ~| r / /. . T^N m' . r r . / ~ ,
-- sin Jr sin (f+ H) sin I -- sm (f II)
m L J 5 3J :! J 1m r' 3
The expression for A 2 also gives
(r^rr'HJ _ (r' 2 r 2 ) 2 _^^_ i J
4J 6
by means of which we find
- m ' l"3(r /2 r 2 ) 2 _ / 2 i J_~|_ ^_ J^ rr
'~ Im" L 4J^~ " J 3 " 4 J J 1+m ' r 72 '
.
T dr'~ +
If we put, for brevity,
(J) = ,. sin J; 2 sin
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 65
the expressions which have been given for the forces, together with the perturbing
function, are
r" d * S
+ ('!") = ^(".Y
\<tr / \ J/ L a a,
\ drdZ/ \ A) L a' 2 a 2 a 2 J a a'
a\ r> sin 2 / r' 2 . 0/ /., . nA /o\
J -it a - sin -u + n ) - K J
>
a
f 9 sin r / * ,
aa = - " sin n -
drdZ' \J/ a' 2 a 2 a 2 a a
\3 sin /
r sin (/+ n)
Vt/
i 2 /\3 sin / r . /
^ U ~^~ a Sm
sin 2 / r' / /, , r-r.x r . / / , o/a\ 3 cos /
'
The form given to these expressions is the one best adapted to numerical compu-
tations ; and the equations are readily derived from the preceding in which the magni-
tudes occur in linear form. ,
Thus from
r __ __ _ 1 _r TJ "1
~ " 2J " " r'*
'9 9
* - . ^ y**
rfr " 1+m L 2J~
A. P. S. VOL. XIX. I.
66 A NEW METHOD OF DETERMINING
we have
fji ra 3 r' 2 __ a 3 rH /z a j*_ ^a" 2 r rr
Z 2 La** J* a''~J s J~ 2*J "a" *r"* ( 'a*
where, as before,
/^ /'\ 2 r TT a '
rz - . ( - t ) . . H. a=-
a \rj a a
.
1-fm
In a similar manner all the other expressions for the forces have been derived.
When we compute only perturbations of the first order with respect to the mass
we need the perturbing function
= p C
and the forces
1 J ~1 I fa\
" ;,- a ,] " If ( J - -
sin (/ + n') + (/)
The other forces are only needed when we take into the account terms of the sec
ond order also with respect to the mass.
An inspection of the expressions for the forces shows that besides the functions
we need expressions for the magnitudes
'\ 2 1 r 2 sin / r f . / /. T-r,\ sin I r
(r'\ 1 r 2 sin / r f
a') ' *'*>-*
a
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 67
When these are known we multiply the function ^a 2 (^j by
F/r'V 2 1 r 2 ~| sin/ r' / /> , T-T,\ sin/r . / / , T-T\
( /) - 2 -i U -/sinff + n'), sm (/ + n),
L Va / a a 2 J a a a a
the function fj.a ( a J by
r n i r 2-]2 Q cin r r ' . rV'2 1 s
F-?3- I > in (/+ n/ )&-i:=
3 ^ . ^ . sirf (/ + no, | *=-' r sin (/ + n) g - i, H
t* ^ * t* L t/> 6t Ci J
3 ^ r sin (/ + n) ^sin (/ + H').
i /T. \/ / fj' \<J /
We will now find the expressions for (/), (/)', (^)", and for the various factors
just given, that are the most convenient for numerical computation.
We have
(J) = ? sin /. Bin (/ + !!').
Putting, for brevity,
- cos d>' sin / cos II'
a 2
I' - l ~ sin I sin IT,
and noting that
/'\2<,iTi / 7 r <> ( <2 n r (l) (3) ~i
(1) ^ = [^ + J v J sin flf + 2[y 2V + ^ J sin 2g' + etc.
/a'\ 2 r (o) (2) ~i r (1) (3) ~i
(-,) cos f = [Jv Jx J cos^' + 2|_/ 2V J^ J cos 2g' + etc.
68
we have
A NEW METHOD OF DETERMINING
r (0) (2) -I [- (0) ('2)-j
(1) = I |y A , + ,/ A , J sin (- - g') + I' [y* J A , J cos (- 0')
r o) ( 3 )~i r (l) (3 n
+ 26 [_<7 2A , + ,7 2A , J sin ( 20') + 26' |y w '^' J cos ( 2 ^')
r (2) (4)~1 [- (2) (4)~1
+ 36 [_J W + e/ 3V J sin ( 3g') + 36' [_ J 3A ^ J cos ( 80')
+ etc. + etc.
The value of (/)' is found from
From
r'
- 1
cos e
we find
Expanding,
+ (fe' 2 +|e' 4 + etc.) cos 2^'
+ ^e' 3 cos 30' + ^J-e' 4 cos 40' + etc. ;
which, for brevity, we write,
(~>) = PO + 2 pi cos 0' + 2 p 2 cos 20' + 2 p 3 cos 30' + etc.
But
r sin f (0) (2)
(2) ~1 f (1) (3) ~1
+ ^ J sin ^ + i L^A + J<* J sin % + etc.
r f (0) (2)-i r (i) (3)-i
~ .cos/ = fe+ [_j; J x J cos0 + I |y 2A J ^ J cos20 + etc.
(3)
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
69
Putting
I = ~ f . cos <p sin / cos IT,
r w ( 3
'., I ,/ 2A e/
etc.
w =
sin </
etc.
For (/)" we have the expression
Putting
we have
Zi - : a s sin / sin IT,
2 2 |_f 2A T- ' 2A J
etc.,
-f^e.po ^
I 7
+ ?i . p! ^ COS ( g g'
- 2 hep! cos ( g'
H~ ^i p2 y\ cos ( ^ 2g'
~\~ ^l Ps 7i COS ( <7 2of'
\ / y
-22^. p 2 cos ( - 2g'
etc.
2 . - cos J, and using the p f coefficients as for (/)',
" z : ** '^ + i a . P! cos ( g') +
(4)
To obtain an expression for the factor
have that for f-V.
\a/
. p 2 cos ( %r') + etc. (5)
^>) 2 - * 2 -H it is only necessary to
70 A NEW METHOD OF DETERMINING
In terms of the eccentric anomaly we have, at once,
: : 1 2e cos 6 er cos 2
a
= 1 + |e 2 2e cos e + Je 2 cos
Substituting the values of cos e, and cos 2e, we have
r \2 (1) (2)
-; = 1 4- fe 2 f 7 * cos | J" 2A cos 2#
To find an expression for the factor ^L . ^ s i n (/ + H'), for brevity, we let
O. CL
, _ sin7
(1) (2) (3)
" etc.
_ _ cog ,
?* ^i n / i /4
and from the known expressions for -, -A, -> cos/ , we get
a a
r (o) (2)~i r w (3) ~l
I J + J* \ * sin^r' + J [^ + J, v J Cl sin 2gr' + etc.
'
r (0) (2)-i r (i) (3)n
|e'c 2 + L^A' J* J G, cos g' + i L^ix' J-* J C 2 cos 2^r' + etc.
In the same way, if
sin I sin / . ^
ft, = - . cos ct> cos n, c 4 = - - . sin IL
a a
we find
(0) (2)-i r (i) (3)
(6)
By means of the expressions for the factors
(.-)'
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 71
just given, we can form those for
3 -' 2 1 r 2 ~ 2
3 p-' 2 _ 1 r 2 ~|
4 La 72 a 2 a*J
3 sin
2 ~~~a
3 sin 2 7 r' 2 . 2 / /, ,
-r- . -.sm- (/> + n'
+ n) . sin (f +
72 A NEW METHOD OF DETERMINING
CHAPTER IY.
Derivation of the Equations for Determining the Perturbations of the Mean Anomaly,
the Radius Vector, and the Latitude, together with Equations for Finding
the Values of the Arbitrary Constants of Integration.
HANSEN'S expressions for the general perturbations are
cos*
where
/ o) 1 + 2- h * p - [cos (/ )-
In this chapter we will show how these expressions are derived from the equations
of motion, and from quantities already known.
The equations for the undisturbed motion of m around the Sun are
dt*
(1 + m} y ~ -
-? + V (1 + m) *- =
dt* } r*
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 73
The effect of the disturbing action of a body m' on the motion of m around the
Sun is given by the expressions
,79/V - X x'\ ,79/V - V y' \ ,T>( ' Z' - Z Z' \
mKr( - ;), wifcv y -"7-), m,'Tc( ,).
\ A 3 r/ J \ J 3 r' 3 /' V J 3 r' 3 J
Introducing these into the equations given above we have in the case of dis
turbed motion
dt*
(1)
dP
dt 2 ' r 3
The second members of equations (1) show the difference between the action of
the body m' on m and on the Sun. The action of any member of bodies m', m" ', m"',
etc., can be included in the second members of these equations, since the action of all
will be similar to that of m'.
The second members can be put in more convenient form if we make use of the
function
m' /I xa/-\-yy'-\-zz' \
+m
Differentiating relative to x
dtt _ m! (_ 1 d^
J x - - i+m \ j" ' d. x
But since
A 2 = (x' - xy + (y -
we have
<LA
dx
A. P. S. VOL. XIX. J.
74
and hence
A NEW METHOD OF DETERMINING
a\ dtt , (x' x x' \
+ m) = ra' f ).
/ dx \ A 3 r'V
In the same way we derive the partial differential coefficients with respect to
y and z.
The equations (1) then become
+ Jc 2 (1 + m) - 3 == tf (1 + m)
dy
dQ
dz
(2)
Let X, Y, Z, be the disturbing forces represented by the second members of
equations (2),
It,, the disturbing force in the direction of the disturbed radius-vector,
S 9 the disturbing force, in the plane of the orbit, perpendicular to the disturbed
radius-vector, and positive in the direction of the motion.
If f be the angle between the line of apsides and theradius- vector, the angle be-
tween this line and the direction of 8 will be 90 + /./We then have
In case of J?, we have
and for S,
From these we find
r r '
y i o x
- -r- O .
r r
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 75
If we wish to use polar coordinates we have
da
dx
=r R cos f 8 sin /
- = It, sin/ + JS cos /.
From
x r cos /, y = r sin/",
we find
dx = dr cos / rdf sin /
dy zz dr sin / + rdf cos/'
d 2 x = d 2 r cos/ rcPfswf 2dr dfsm f rdf 2 cos /
d?y = d 2 r sin/ + rd 2 fcosf-\- 2drdfcoaf rdf 2 sin/
From the expressions for dx and dy we find
dy cos f. dx sin/= r df
dxcosf. -f (Z?/sm/
and hence
dtt I dQ . , dQ j.
- = .-r^sm/- -^ cos/
rfa; r ^/ rfr
d& I dQ . dti . r
- - - -j* cos / + - - sin/:
(Zy r df dr
from which we see that
If we multiply the expression for d?x by cos /, that of d 2 y by sin / and add,
we obtain
d?x cos / -f- ^ 2 2/ s i n / d 2 r r df 2 .
76 A NEW METHOD OF DETERMINING
In a similar manner we find
d 2 y cos f - - d 2 x sin /' =. r d 2 f -\- 2dr df.
Operating on equations (2) in the same way, we have
d*x / , d' 2 it /. , & 2 (l-j-m) -*-r / -T7- / 73
-55. cos/ + -=| . sm/ + ' X. cos / + Y. sm/ = R
dt* '' dt r 1 J
- .sin/ = T.cosf-X duf=S
Comparing the two sets of equations, we have
2*f =f(l + i)i"
dt dt ; r df
(3)
The second members of equations (1) and (2) are small, and in a first approxi-
mation to the motion of m relative to the Sun, we can neglect them. The integration
of equations (2) introduces six arbitrary constants ; and the integration of equations
(3) introduces four. These constants are the elements which determine the undis-
turbed motion of m around the Sun. Having these elements, let
the semi-major axis,
n the mean motion,
g the mean anomaly for the instant t =. 0,
e the eccentricity,
<2> the angle of eccentricity,
7t the angle between the axis of x and the % perihelion,
v the angle between the axis of x and the radius- vector,
/o the true anomaly,
e the eccentric anomaly.
These elements are constants, and give the position of the body for the epoch, or
for t = 0. Let us now take a system of variable elements, functions of the time, and
let them be designated as before, omitting the subscript zero, and writing # in place
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 77
of 7t . The former system may be regarded as the particular values which these
elements have at the instant t = 0.
In Elliptic- motion we have
e e sn e
r cos /' = a cos e ae
r sin f =. a cos <> sin e
v - f + X
a?n 2 - F (1 -f- m)
ISTow let n& be the mean anomaly which by means of the constant elements gives
the same value for the true longitude that is given by the system of variable elements.
Further, let the quantities depending on n^z be designated by a superposed dash, and
let the true disturbed value of r be given by the relation r = r (1 + v).
We have then
~ e e sm i
r cos f rr a cos e -
r sin f = cos <> sin e
V = f + 7t
a 77 ~~ #*"' I I 1 171 1
Q /CQ .^ A/ I JL j II v I
We will now first give BRLTNNOW'S method of finding expressions for the pertur-
bation of the time, and of the radius vector.
Neglecting the mass ra, multiplying the first of equations (1) by y, the second
by x, we have
dy dx
C being the constant of integration.
Introducing
cos /' x -* and sin f '=. -.
/ /i 7 * r'
78 A NEW METHOD OF DETERMINING
into equations (2), neglecting the mass w, we find
d*x _|_ k*. cos f -^~
dt* 7* 2
(4)
(i If t L ' . Sin f -rr
j ,n \ n ^^~"
We have also
dx v dr /. df
/~>nQ T ^ i* si n r __
i, -^ l_'*JO / . ~~z~. / Olll / - -'
dt * dt *. dt
dy / dr /. df
jr = * m f-jr + rc f-^r
dt J 'dt "^ 'dt
and hence
or
and
In the undisturbed motion we have
being the semi-parameter.
Hence
THE GE^EKAL PERTURBATIONS OF THE MESTOK PLACETS. 79
From these relations we derive
(5)
and also
' "
i/P I _ - 1 f V'fr .Q r dt
^=r - X I =^. A}/ . Ul> //\
If we eliminate from equations (4), noting that
1 -^ = l--
p dt k p
we have
dz fcsin/
p
neglecting the constants of integration.
Since r rz r (1 + v), we have also
= x
The equations (7) then become
.
y dt dt ^/p~ J \ p
From the equations
= a cos F OQ^O = 0o cos > sn
(7)
dt dt ~/p J-\ p
(8)
80
we have
dx =: - a sin e ds
dy = cos <p . cos
Then since
tl dg = r - de, df= cos <p . gdg, d J- = h a =
a o r Tir 2 j/>o
using the values of sin e, cos e, in terms of sin/, cos/^ we find
dx k sin/ dy cos/-|-e e
^ i/S~' ^ "~t i/^T
ft
And these give
k sin/' dx i/p<>
l/p dz i/p
k cos/ _ dy VP ke
!/_p~~ rfz i/p
VP*
v. . i
ft
The equations (8) then become
_
, dy T (1 , x d5 _ I/Pol = f(
dt dz L V ; d< j/p j j
n f
the constant being included in the integral.
SLU ^ # r ) ^
" dt dz L^ ' dt Vp l ~J p
-to , ft r n , ^d L . . T/J.-I
We will now transform equations (9), and for this purpose we multiply the first
-~
by , the second by -~ , and noting that
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
81
we have
dv_
dt
_ **f . Sr \ dt + si" / r (
) j \
Now multiply the first of (9) by y, the second by x, putting for J its value
VP
given by (6), noting that
we have
* ~ <fy - 7,
v) ^ . = I - l f. Sr dt - y C(X- Sin/ . Sr)dt
dt fc,/V Cp K/nV 7>
(11)
We can write -- in the form
dt
dz
dz
__.
We have
r ' dt dz dt
2
Wn . - . . COS n , Of 3
cos
2
Making use of these relations we find
dz 1
and for ,- given above we have
^ dz yp
v ' 'dT
A. P. S. VOL. XIX. K.
82 A NEW METHOD OF DETERMINING
The equation (11) is thus changed into
dz
= f (l + 2^) Srdt-
J v )
The equations (10) and (12) can be put in briefer form.
Let
X s = X- sin / 8r, Y c = Y+ cos/+ ^ 8r.
P P
Then
- M 4_ 8in / f
** + i*:J
jo. _
~ (13)
The values of a?, /, found in these equations we get from
_ _
From the expressions for , -^ , we have also
dz 7 dz
sin/ __ . j ( _
(14)
. ,_< + etc.
(15)
The quantities given by equations (14) and (15) are found in equations (13)
without the integral sign. They can be put under the sign of integration and regarded
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 83
as constant if we designate all magnitudes in these factors dependent on t by a Greek
letter.
We thus obtain
d(z l C(-\ O I Vo\ o J* 2 T/-\r TT r\ J4
^ r= 1(1 -f 2 ' to } Sr at | ( X, . v Y e .t)dt
at k |/P( J \ \''T> ' k vPo J
(16)
These equations include terms of the second order with respect to the mass. If
we put
W=
we get
/i/v
(17)
J
In equations (17) </ is the mean anomaly for t = ; N is the constant of inte-
gration in the value of v.
From the value of W given above, we have
dW
Now since
> 1 d~\
m/. . T -
J r djj
- -- sm. .
dr J r dj
-T^ . , rffl . 7- 1 rf^l
Y = )sm/.- hcos/.-.-
^ dr J r dfj
84
A NEW METHOD OF DETERMINING
neglecting the common factor Jc~ (1 + m),
we have
dt
19
VP*\
- cos
A/i2 . >T 1 c?i2 7-\ r , fsin/" f/fi , (cos /+,) dfl r ~l
4- ~ ~ I sin / 4- cos / I r + . f 4- - . . r
^T/^o \*r f r d/ ' 7 / ? AT/P. L p d/ 1 jj d/ 1 ? J
And as
this becomes
v p sin 0, ^ = p cos
dt
-.- d
df
4- 2p cos Q . sin /'. - v 4- 2p . - cos o cos /' -)- 2p .
r r dj p aj
2p
oos.co/dfi,A
P
<v
]* l <
xdi2 , xdfl
2psm (/ Q)~ 4- 2" cos (/ o) T
f '
+ 2 P- cos (/- Q) 4- 2e cos M
But
2e p-cos "- 2po = 2 1 (e p cos o po) = p . 2
also
,
V/Po
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 85
Hence since & 2 (1 + m) is included in X, Y 9 72, S, we have
dw 2 (18)
dt
+ 2/ io p.sin(/ G>)^-
If we write h*. a Q cos 2 <> in place of Jc 2 in equation (18), we have the same ex-
pression for - - as that given by HANSEN.
Equations (17) and (18) are fundamental in HANSEN'S method of computing the
perturbations. We will now give HANSEN'S method of deriving them.
Using the same notation as before, we have, since
a 1-f-ecos/
r cos V
also
r cos 2 <f>q
a ~ ~ 1 -f e c cos / '
hence
r.a l+ecos/ cosVo
r.a t cos V * l-j-P cos/' '
Using f + 7t % in place of f, and developing, we get
r - a . _ r-{-rcosf.ecos(x 7r )-f r sin /. e sin (/ 7r )
r.a c ~ o,cos 2 ^^ QA
Let us put
e sin (% 7t ) z= >7 cos 2 <^ ,
ecos (^ 7t ) =:
since e = sin <|>, we have
cos 2 <?> cos Vo (1 2e ^ cos 2 ^> P - cos 2 ^> 7 2 )-
86 A NEW METHOD OF DETERMINING
With this value of cos 2 <2>> and r cos >2 ^ e o r cos/",
we find
r.a _ a cosVo e .r cos f -\-rcosf (^ cos 2 y -\-e )-{- r sin f.r] cos
cos
a t cos Vo~H r cosy. I 1 cos Vo~l~ ? ' si 11 /- ^ cos Vo
a cos Vo ( 1 2e cos Vo 2 cos Vo 1 ?' 2 j
and hence
1-f-^. .CO8j^-f-^. . sin/'
r.a 1 2e cos'Vo* 2 (
From
d v _ ^/ _ df dz
dl dt dz dt 9
and
we have
. COS
111 like manner we find
df o 2
- =. UQ. =7 - COS d) -
az r
We have therefore
n.a?,r 2 . cos
(Z< ^fl.a 2 - **' 2 cos
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 87
If we put 1 + 6, substitute the values of a . and cos 2 <2>, we get
"n * f n '
(20)
(1 2e cos Vo^ 2 cos :
Further, in the case of v, we have
Then since
and
we have
If we let
we find
1 2e? ? cos Vo- 2 cos 'Vo- 1 ? 2
cos/. + - . sin/.
= 1 e cos - - cos
h . _ (1+6)*
= (l + 5)ir,
B
88 A NEW METHOD OF DETERMINING
From the latter we have
Hence
__ > ^ I
/i /) dt
If we put
f
we have
We have yet to express r in terms of the elements.
From
and from
COS
we have
_ / \ ^ cos y>
V / ' CfOS <> '
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 89
or
h _ an cos ?>
A cos (p ' a c
If we put
cos
we have
7 an
cos <p
These values of h and k being substituted in the expressions for W, is found
Civ
expressed in terms of the elements and of v, in a very simple form. To find the rela-
tion between --- and v, we use the equation
dt
7~i9!
(1 + *) 2 =
and as this is also equal to
h
dt
we find
dz h
(22}
dt ' ' h '* -^*
For the purpose of keeping the formulae simple and compact, HANSEN makes use
of the device of designating the time, and the functions of the time other than the
elements, by different letters.
Thus for t, r, e, /, , v, x, y, we write,
r, p, >7, o, f,'/?, , v, respectively.
Whenever we integrate, these new symbols are to be treated as constants, noting
that the original symbols are used after integration.
A. P. s. VOL. xix. L.
90 A NEW METHOD OF DETERMINING
If in equation (21) we introduce <r instead of t we shall have
where
.._ coso .7._. sn a
o
ft, />, /> "
We have also
The coordinates of a body vary not only with the time but also with the variable
elements. In computations where the elements are assumed constant, that part of the
velocity of change in the coordinates arising from variable elements must, evidently,
be put equal to zero. Coordinates which have the property of retaining for them-
selves and for their first differential coefficients the same form in disturbed as in undis-
turbed motion, HANSEN calls ideal coordinates.
If L be a function of ideal coordinates, it can be expressed as a function of the
time and of the constant elements. Thus let the time, as it enters into quantities
other than the elements, be itself variable and, as before, designated by t.
The function dependent on , r, and the elements we designate by A. Then
dL ~dA
dt ' dr
or
where the superposed dash shows that after differentiation r is to be changed into
Let us write the equation (24) in the form
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
Differentiating relative to r, we have
dPf
The differentiation of (23) also relative to t gives
_ , C
2 ~ " ' ' '
dr />, ' 1/? 3 ' dr
Eliminating - by means of (24), we have
e h
dr
dTF 2/3
"" r
JO
Substituting in the expression for J- we have
dW
Since ^ is an ideal coordinate, we get from this
^T being the constant of integration, and the dash having the same signification as
before.
This expression for v is a transformation of that given in the equation
1 2e, cos Vo- 2 cos Vo- 1 ?*
. sn
Since 2; is also an ideal coordinate, we have from (23)
2 (26)
being the constant of integration and being the mean anomaly for t = 0.
92 A NEW METHOD OF DETERMINING
When we consider only terms of the first order with respect to the disturbing
force, f changes into r, and we have
n<p = n t + c + n ( WQ dt
(27)
_._. ., / /dW \ _
r := _Zv s | (- - ) a
J v czr y j
where
~\7f7~ f) *^ *^o "1 I O /* /^ I ^i P * /*OQ\
and p and o are functions of T, being found from
n r + c = YI e Q sin Y}
p cos G) a cos >7 a Q e
p sin G) zz a cosji) sin >?.
Also in the last two terms of WQ, - is put equal to unity.
"o
When terms of the order of the square and higher powers of the disturbing force
are considered, f cannot be changed into r. In this case let
n t
Likewise let
where
w^J is a function of r and
According to Taylor's theorem we have
W= TTo
.
ar ' dr
the value of W Q being given by (28).
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 93
We then have
dW_ _ dW, ^ ^ L . ^ ,
dS dr dr* <0 * 2 ' d^'^
Retaining only terras of the second order, the equations (25) and (26), replacing
by $z, give
v z dt
dr
(29)
The equation (26) has been put in simpler form by HILL. For this purpose from (21)
and (22) we have
~
dt ~ dt
Hence
Developing the second member and adding W, we have
= n, t + #o + o 2 dt. (30)
dh .
Vb OLCJJ IB IAJ CA.JJ1COO _ ^ttQ
dt
we find
The next step is to express " an d in terms of the disturbing force. From (19)
dt dt
COS
94 A NEW METHOD OF DETERMINING
Using these values of and vj 9 and e p cos o = cos 2 <> p, in equation (28), we
find
. . ,
ft, a cos Vo V*o cos Vo
Since
T _ an _ |/l-f-m
" cos? "
we have from the expression of h already given,
dt
By means of
- 1 =1 e cos /.
r
cos.?
we may transform the expressions
dv a 2
= . n cos
df r 2
an
. /.
e sm /,
cos
into
r - 37 ^ cos (f ") he cos (# 7t co) + sin (/ o) . Ae sin (% n
t
dr / j? \ T / \ / /^~ \ i /
= sm (j co) . he cos (% 7t o) cos (/ o) . fte sm (^ 7t
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 95
Multiplying the first of these equations by cos (f G>), the second by sin (/ o>),
and adding the results, we have
he cos (jc 7t G>) = (r~ h) cos (/ o>) + ~ sin (/).
at ctt
Substituting this value of h . e . cos (# 7i o) in the expression for TF" > noting
that
we have
-nr^ 2.fc../ ^ cQg ,-j._ x r dv . 2h 9 .p ^ gin / -_ s dr_
Differentiating relative to the time t alone, r remaining constant, and having care
that all the terms of the expressions be homogeneous, we have
c^r /' civ
and
dh _ F(l+m) d 2 u . fcV d 2 v
Substituting
\ p d*r
+ ) r
96 A NEW METHOD OF DETERMINING
we have
= *, 3f- cos ( / - o) - 1 + - - [<=os ( / - ) -1]
~ 1 J
-
dt ( r h~a cos 1 $P O J * \ dv
sn -
< 30 V
d W
This expression for -^ is the one used by HANSEN in his Auseinandersetzung.
It is given in a much simpler form in his posthumous memoir, and as the latter is the
form in which we will employ it, we will now give the process employed by HANSEN
to effect the transformation.
Substituting first the value of h, omitting the dash placed over certain quantities,
noting that in the posthumous memoir <?> takes the place of o, and remembering that
we are here concerned only with terms of the first order with respect to the mass, we
have
/? \ ( dQ \
o) r( }
\dr )
From the relation
p z= a(l e 2 ) ep cos G)
we have
v
_ -i ep cos (u
dW
An inspection of the value of -^- shows that its expression consists of three
parts, one independent of r, the other two multiplied by p cos o, and p sin G>, re-
spectively.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 97
Put
= dB [ + *Y,p _
dt dt dt \a l ) dt a
V
t e j
and we have
dS q a f [~ae cos / , e cos / , 1 ~| / cZi2 \ , ae sin /
dF 9 (Faces/' , (cos/-f-e) ~| ( d& \ , a sin/ ^ <?^ \ )
** ~ T 2 \ TV? J ~T~ ~ * V ~J / I '
ndt ,/l _P 2 ( L r 1 e z J \ df / r \ dr / )
But
hence
o j F a sin / , sin/ ~| /^_\_ a cos/
l L r TU^e^ J\d/ r
df _ o? 5 - "2 ae cos / , e cos / ,
: ^r^ " " TT=^)^" ' ' a-*) 1 :
dr ae sin /
r e
= a cos /;
^ }
ds = 3a
mZ^
^F
ndt
- (~ V
wd< " " y 7 ! e 2 \de )
Again from
fd&\ . . (d\ (d\ , (dtt\ fdr\
\dg) ' ' \df) \dg) \dr ) \dg )
A. P. S. VOL. XIX. M.
t/
9b A NEW METHOD OF DETERMINING
we have
dQ\ _ (dtt\ ?' _ (dQ\ resin/
J ' \dcjJ ayie* \drJ a(le>) '
Eliminating f J from the expression for - , we have
dY 2 (a 2 (l e 2 ) r 2 fd&\ r sin f /dQ\
- - J - - - d ( ] _l- - ' n. f [ _ ]
ndt~~le 2 \ a*e \dg ) r a T /l e 2 \dr J
A (// - fi^ 1 ^
In the same way we find z - '
dQ\ r~a cos/ ^ - e sin 2 /
r* sin/ /rfw\ pa cos/ ^
__ 2^
But if we employ the relation
x,/1 ^>2\ I
r . re cos/
fl2\
C y
, , | ':
-H"^ A (T
a cos/
in the term, - - Vl e2 'A^ ^ ne preceding expression, the whole term becomes
Crcos/ e re ~i /d&\
a (1 e 2 )? v 71 " e" a (1 e 2 )d P Vdr /
Using the equation
zr re cos/ r + (1 6 2 )?
multiplying by
e
CL T I -
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
adding^ to the preceding, it becomes
[-
~~ La
rcosf 2e
Further, we have
dg
rr
U sm/ + ?
cos f
sn
"Reducing this expression in the same manner as employed before, it becomes
/' -i 2 r cc
s)J ~ a
d r~r , r 3 sin/ ~i 2 r cos/-{- 3 a e
8 ^
a 2 (l-e s )
Multiply this by cZf/, the last expression for - becomes
ttdv
o
;j~
the integral to be so taken that it vanishes at the same time with g.
d3 dY d>F
Substituting these values 01 - , , , m
ndt ' ndt ' nrf^ '
dS dT
f p \ ?..
=- 4- ( - cos 6) + f e ) -f - sm
ndf nrf^ \a / ndta^
this expression can be made to take the simple form
dW
in which
-
a 2 (l e 2 ) r 2 2 /> sin w /-/2
99
J
1 e 2
a 1
e a
100
A NEW METHOD OF DETERMINING
Since
d . r 2 _ r sin /
a e .
tfig.de
^
cos />
we have
_^L r r^_
a 2 edJ L a 8 . rfe
2(1
These expressions for A and ^ can be much simplified.
Thus from
r 2 e 4
- = 1 + f e 2 (2e J e 3 ) cos # ( J e 2 J- e 4 ) cos 2# J e 3 cos 3g - cos 4g etc.,
^JL^J- *^ tft^VVV** ^L^'tt^ ^ ^^r-t. x{!^6^ ^.
p*
and a similar expression for -, we get
cZ 7*^ / f > ^\ / ^^\
a , e ' d = (2 - J sin ^ + (e -J
e sn
e 3 sin 4^ + etc.,
/LoT
/ La.
"
lain
r--- sin 4flr etc.,
a 2 . de
4e = e (2 f e 2 ) cos # ( e f e 3 ] cos 2g f e 2 cos 3^ | e 3
cos
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
From which we obtain
101
2e 2 )cos(y g) B
+ (e + ^)cos(y 20)
-J cos y
-30)
- (2
-(e + e 4 )sin(y 20)
cos
sin (7 Sg)
+ 3 cos (y 40)
(32)
Jtjv.
These are the expressions of A and B whose values are used in the numerical compu-
tations.
When we have the coefficients of the arguments in which y is + 1> and 1, we
obtain the coefficients of the arguments in which y is i * with very little labor.
d W
Let us resume the expression for =- , that is,
ndt
dW
ndt
A and B having the values given before.
Since can be put in the form
a 2
we have
a-j/1 e 2 e.dg
d.-
7 o r '? *\d BW 7
sm fc</. 2 - cos / = - = -/ - cos Jcq.
a de Z. de
A NEW METHOD OF DETERMINING
C I f ' a "\
J | (*)"
But since
dR(k) '
* 1
w " w - \ e T Ts e ; cos
a 2
(^e 3 -^e 5 ) cos 3^ etc.
we have
^7 (0)
= 3e.
Hence the integral just given is simply -- sin kg.
A and B can then be written
A = - 3 + -L, f( 2 ^ coso + 3e } &=f~.
2
sn
Putting
- 2 =: 2 jR (K) cos A; y,
a
we have likewise
2 ^cos co =
a de
Introducing these values of 2 -cos w, and 2^ sin w into the expressions for ^1 and
, after integration relative toj^we can write Win the form
~i . *^>\,
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
103
where
de
li
i'g'
CTand V being two functions depending alone on JL.
Putting x + 1, and 1, we have
and hence
Thus we find
2
or putting
we have
de
de
,0) _ (-
7T Y
U j r>(i) ) r ^(i)
2^- 2-
T
,d)
de
- de
de
de
de
'.JL M&"V
,(-i)
_ W a (D + ^W a (-D
(33)
104 A NEW METHOD OF DETERMINING
The values of YI (K] and O (K) are readily found from
") "K;
/O n 1 t$ I 1 ^5\ ^rvc A/ /I y^2 1 y>4 _j_ J_ ^o\ f*r*si *A f\/ 1
~1^/C"""^|,C/ '" Q o (5 I Ov& Y
cos
We have
etc., = etc.
^^ 3e
de
de
(3 # 45 ff
de ~(* e ' 64 e >
(1 p* 4
V3 6 ~5
de
etc. = etc.
For >7 (2) we have
^(2)
= a*- A ^ - - T k e 5 ) + a - A ^ 3 + 3k ^);
or
^< 2 > = 46-1^-^^^ . . _ (34)
For (2) we get at once
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 105
In a similar way we have
In case of the third coordinate we also compute the coefficients of the arguments
having no angle y from those having i y. For this purpose, putting K = in the
expression for OL (K} we have
5?rr-
j. ty
de
v de
where
For >7 (0) we then have
Perturbation of the Third Coordinate.
Let 5 the angle between the radius- vector and the fundamental plane,
i the inclination of the plane of the orbit to the fundamental plane,
v <r the angular distance from the ascending node to the radius-vector.
"We have then
sin & rr sin i sin (v <r).
If we use for i and cr their values for the epoch and call them i and & , o being
the longitude of the ascending node, we have
sin & = sin i sin (v & ) -f s ;
s is the perturbation.
Thus we find
s rr sin ? sin (v a) sm i Q sin (v Go),
A. P. S. YOL. N. XIX.
106 A NEW METHOD OP DETERMINING
Putting
p =. sin i sin (cr & ) , # = sin i cos (cr & ) sin i ,
we find
s = q sin (v &) p cos (v &<>)
Instead of s, let us use
r
U = S
and we have
u= ~ q sin ( & ) - 1 p cos
Introducing r and calling R the new function taking the place of u, we have,
putting 6) + 7t for v, 7t being the longitude of the perihelion, <U- ~it~jl '
dE dq p . , dp p , .
IF = df ^ Sm ( u + ^o So) ^ "^ COS (co + 7t So).
To find and we will employ the method given by WATSON in the eighth
fit Uv
chapter of his Theoretical Astronomy.
Thus a and (3 being direction cosines we have
z l = a x + p y;
also
-,
Zi = r sin i sin (v ff).
But
fa \ _ >($ =. r cos v, and y = r sin v.
Hence
3 L =: x sin sin G -}- y sin i cos cr ?
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 107
and
a = sin i sin cr, (3 zz sin i cos cr.
The values of p and q then are given by the equations
p - - a cos /3 sin &,
2 zz - - a sin &o + /? cos & sin ^ ;
from which we have
dp da dp
- = _co S ffi --sma -,
dq da dp
- = - 8 mS -+oo 8 ffi -.
From the equation z l zz a a? + /2 y we have, first regarding a and /3 as constant,
then regarding x and ^/ as constant,
dzA dx ~ dy
dt) ~~ a W"rP ^ -
rf^-i da dp
Differentiating the first of these, regarding all the quantities variable, we have
d 2 *! _ da dx dp dy d?x d*y
~d?~ ' ' dt ~di ~~ ~dt ~dt ~ ' a dt* ' " ^ df '
^i being the component of the disturbing force parallel to the axis z l9 and X and
!Fthe other two components, we have
Z a X + j3 Y+ Zcos i.
"Writing for X and Y their values
tfx .x d v , 70/1 \y
_ + ^ 2 (l + m) ;3 , - + Ar(l + m) - ,
108 A NEW METHOD OF DETERMINING
and reducing by means of
we have
* = ;+ + * a + ) !i +
or
Comparing this with the other expression for --* , given above,
d V
we have
da dx dp dy
From this equation, and the value of [ _- j , since
L ' / / J
dy dx
we find
da
dt
dp
dt
= h r cos i sin v Z ,
= krcos icosv Z .
Substituting these values in the expressions for --- and --- ,
we have
dp
dt
= hr cos i sin (v
dq
- = hr cos i cos (v Q> ) Z .
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 109
Introducing these values into the expression for
dt
we have
= 7i r cos * cos (v & ) p - sin (o> + 7t & ) Z
at @
- h r cos i sin (v & ) -- cos (w + ^o So)
a Q
h r cos i -- |sin G> cos (0 So (^o So)) I Z
a L
- h r cos * - cos o sin (v So (^o So)) -
li r cos i p - sin (o> f) .
Introducing n = - - , and h zz
we have
==-- sin (o / ) a 2 - cos i . (37)
Let
1 ' P / s>\
~i/r=?'o o' sm ^~~f>>
then
cos *.nc?^
To find an expression for (7 similar to those for J. and J5 we have, first,
l rp r P r ^~1
(7 = - sin G).- cos/ coso). sin/ .
/l e 2 La a a, a
110 A NEW METHOD OF DETERMINING
Substituting the values of r cos/, ? sin /J given before, and similar ones for
- cos a, - sin a, we find
fe V afde ) \a?edg J \a<?edb/ \a 2 de J '
Substituting the values of these factors we obtain for C the expression
O =. (1 - - -J- e 2 ) .sin (y - g)
- * 6?) Bin (y-
+ f e 2 sin (y
-fe 8 sin (
_ d W du
Having found the expressions for and
ndt ndt . cos i
we have, finally, for determining the perturbations, the following expressions
n&z = n f W dt,
u /
cos i J
dQ
dW
(38)
Two integrations are needed to find nbz. We first find W from ; then, form-
ndt
ing TFand J from W we have n^z and v by integrating these quantities. In
dW
the integration of - we give to the constants of integration the form
ndt
& + &! cos y + & 2 sin y + ? (2) ^i cos 2 y -f >y (2) fc, sin 2 y +
r ^
I *
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
Ill
Then in case of A - we have
J &i sin y - - i & 2 cos
i -i)
>? (2) ^ sin 2 7 - - V 2) fc 2 cos 2 y + ete. ^ "J
In the second integration we call the two new constants C and JV^ and the con-
stants of the results are in the forms
C + & nt + &! sin # & 2 cos g + % ? (2) &i sin 2 # - - | >y (2) & 2
JV~
>y 2 cos 2
^ >7 (2) A?! cos 2 ^ - - J >? (2) ^ 2 sin 2
, 5.
In case of the latitude the constants are iven in the form
sin g
sn
cos
7
The constants are so determined that the perturbations become zero for the epoch
of the elements. Hence also the first differential coefficients of the perturbations
relative to the time are zero. We substitute the values of g and g' at the epoch in
ni fl
the expressions for nz. v, - - , - - (nbz), etc., including in g' the long period term.
cos % nat
Putting the constants equal to zero, and designating the values of nfiz, v, etc., at
the epoch by a subscript zero, we have the following equations for determining the
values of the constants of integration: . 13;
C + ^ sin gr -- 7c 2 cos </ + i >? (
cos g
sn
cos
cos
cos 2r/ -fetc. + (nbz) Q g*
=
sn
sin g + I cos gr + >? (2) Zi sin 2^r + >y (2) ^2 cos 2^4- etc. + (~-) =
l> cos 2,7 -
cos j, - , sn g
112 A NEW METHOD OF DETERMINING
To find ki and k>. we derive from the preceding
, [cos<y c + >? (2) cos 2 </ + 57 (3) cos 3 # H-jetc.J + k> [sin (j + >y (2) sin 2
! [sin gr + 2 7< 2 > sin 2 g + 3 >y (3) sin 3 g +)^etc.] - - & 2 .[cos g + 2 >7 (2) cos 2
The value of N is found further on.
Having Jd we find ^ from
- fc e ^ - 3 Z + 3 ~- (nte)o + 6 ( v ) = 0.
We have
' 2 e
6 () e L , jY 3- & A^i 2 A) ?
Av^ where Z Q is the eonstant- of W.
Let us find the expressions for the constants N and K, K being the constant of
integration in the expression for 5 - .
"0
The equation (22) we can put in the form
The differentiation of nz relative to the time gives
dz
- 1 + &o + &o + &i + periodic terms,
Ct6
where Z Q =. - - 32".7162, in the case of Althaea, and Z^ the part to be added when
terms of the second order of the disturbing force are taken into account.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 113
The expression for v is
v =. N -\- periodic terms.
The approximate value of being 1, the complete expression for the integral of d ~
is given by
=. 1 + & 3 + periodic terms,
& a being the constant of integration.
Putting (3^ 2 4z^ 3 + etc.) - - 2v (- - l) = FI + periodic terms, and substi-
it/ \tL /
tuting this expression, together with those of v and - , in the expression for , we
/I dt
have, preserving only the constant terms,
It is necessary now to find the value of k & in terms of the constants. If in the
expression for given by equation (18) we write for p , its equivalent cos 2 <p
Ctt
- e p cos o , we will have
T 1. 1.4 f \ 7.2 si~ ... ^ .Jf)*
:}dt
We also have
Selecting from the expression for dW Q the terms not containing p cos w and
p sin to, we have
A. P. S. VOL. XIX. O.
114 A ]STE\V METHOD OF DETERMINING
If the eccentric anomaly is taken as the independent variable we have for the
complete integral
C f 7i 2 \ / dQ \
WQ =. & 4- &! cos YI -- k sin YI 7i I [ 1 - - 2rr ) ( >/ ) dt.
J V *o 2 ; \VJ ..&
Introducing the true anomaly instead of the eccentric, we have,
cos <a -f- e sin a> cos H
since cos >7 = , sin YI =. ,
^ 1 -4- e cos (a \- e cos
iA J ^ /;2 ^
- . r '}
^
1< /" C V * t C^a W^
Neglecting the terms having p cos o and p sin G> we have in TFo the constants
and e h.
h,
The integral of d T is
\ '\_ ^
Y^
From the expression for d 7 we find
7i
h
Integrating this, making use of the value of 7i , and adding the constants, we have
And since the quantities under the sign of integration do not have any constant terms
we can write
2 - r = l + & + ek v + periodic terms
I,
*
7i
= 1 + & 3 + periodic terms
THE GEISTEKAL PERTURBATIONS OF THE MINO^B-PLANETS. 115
Since ( - - Ij is a quantity of the order of the disturbing force we have
from which we get
Now putting
7 O / *-!
(JT- -1) - ~ ( '- -l) rb etc. =: HI + periodic terras,
substituting this expression and those for
~ h 7) /io
/) h ' ft '
the preceding expression for
2
gives, preserving only constant terms,
h
Introducing this value of ^ 3 into the expression for JV it becomes
^= - K 4 A; 4 e^ + 3^ ) + i (3 ^ + 2/^-
Preserving only the terms of the first order we have
To find the value of A", the constant of integration in case of . - , we have
''o
h =2 1 + K + periodic terms,
116 A NEW METHOD OF DETERMINING
also
' 1 4- &3 + periodic terms.
h
From these we get
j-i +!'-!= A-
/? h
Hence
K = \ +H,~l (fc -f ek,,)
or, neglecting the term of the second order,
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 117
CHAPTER Y.
Numerical Example Giving the Principal Formula Needed in the Computation
Together with Directions for their Application.
ALTH^A 119. JDPITER.
g - 332 48' 53".2 g' = 63 5 48.6
7i = 11 54 21.1 1 7i' = 12 36 59.4 ]
I I
= 203 51 51.5 } 1894.0 & = 99 22 59.9 } 1894.0
i= 5 44 4.6 J i' = 1 18 36.9 J
$ = 4 36 24.9 $' = 2 45 57.2
n = 855".76428 n' - 299".12834
log n - 2.9323542 log n' - 2.4758576
log a = 0.4117683 log a' = 0.7162374
The epoch is 1894 Aug. 23.0.
The elements of Jupiter are those given by HILL in his New Theory of Jupiter
and Saturn, in which the epoch is 1850.0. Applying the annual motion of 57". 9032
in 7t', of 36". 36617 in &', to HILL'S value of 7t', and of &', we have the values given
above. The mass of Jupiter is TOTT-ST^' ^he elements of Althsea are those given
in the Berliner Astronomisches Jalirbuch for 1896. The ecliptic and mean equinox
are for 1890. To reduce from 1890 to 1894 we employ the formula of WATSON in
his Theoretical Astronomy, pp. 100-102.
i' = i + YI cos (Q, 6)
Q' = Q + (t' t) -- YI sin (Q6) cot . i'
dl
t t) + YI sin (& 0)
A NEW METHOD OF DETERMINING
B = 351 36' 10" + 39".79 (t 1750) 5".21 (f t)
YI - 0".46S (f
4* = 50".246.
dt
These expressions for ?'', &' and 7t', can be used for the disturbed body as well as
for the disturbing body by considering the unaccented quantities to be those given,
and the accented quantities those whose values are to be found for the time, f.
HARKNESS, in his work, The Solar Parallax and Its Related Constants, using the
most recent data, gives the following expressions for 0, >?, and , when referred to
(' u
1850.0:
6 = 353 34' 55" + 32".655 (t - - 1850) -- 8".79 (t t),
vi = 0".46654
dl
dt
- [50".23622 + 0".000220 (t-- 1850)]
Let an:-,
n
we have then
(i = 0.34955
2^ = 0.69910
3^ = 1.04865
4^ - 1.39820
5^ = 1.74775
6^ = 2.09730
etc. =. etc.
Hence
1_ S(i = .04865,
2_ Qu = .09730.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 119
This shows that the arguments (g 3#'), and (2g 6*7'), have coefficients in the
final expressions for the perturbations greatly affected by the factors of integration.
In case of the argument (g 3g'), \ve should compute the coefficients with more deci-
mals ; also those of (0 3g') and (2g 3g'), since in the developments the coefficients
of these affect those of (g 3</').
From
sin i /. sin J (*P + 4>) = sin J (& &') sin J (i-ht)
sin i- /. cos \ (V + <I>) = cos (& &') sin J (i i')
cos i /. sin i (*P 4>) = sin J (
cos i 7. cos J (V 4>) = cos (&
where, if &' > ^, we take | (360 + Q &'), instead of i (^ ^') 5 we find
//
* = 116 15' 36.7
3>= 11 50 33.9
I- 6 11 35.3
An independent determination of these quantities is found from the equations
cos 2) sin q sin i' cos (& Q')
cos j9 cos g cos i'
cos jp sin r =. cos i' sin (& ^')
cosjacosf cos (S S')
sin jt? sin i' sin (S Q')
sin /sin O sinp
sin /cos $ zr cos p sin (i q)
sin /sin ( 1 P r) sinj? cos (* g)
sin /cos ( 1 P r) =. sin (^ g)
cos / = cos p cos ( g).
120 A NEW METHOD OF DETERMINING
From
n =TI --& --$>
IT = n' & ip
we have
n = 156 11' 55".7 , H' = 156 58' 22". S.
Then from
k sin K r= cos 7 sin II'
~k cos .BT = cos n'
& t sin TTj =. sin II'
&i cos ^ = cos / cos II'
p sin P = 2a 2 - 2a^ cos (H
^ cos P = 2a cos (|)' &i sin (n - - 7
u sin V = 2a cos $ ^ sin (II - - K)
v cos >"= 2a cos <|) cos <|>' A^ cos (n
e'
tw sin TF= ;^ 2a 2 - sin P
e
t.
w cos TF"= v cos ( F" P)
Wi sin TFi = v sin ( T r - - P)
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 121
we find
K = 157 5' 36".6 log k = 9.999614
X l =156 51 7.4 log ki = 9.997849
P = 93 3 27.0 \ogp = 9.932748
V 359 6 2.4 log v = 0.601463
TF=266 4 39.5 log w = 0.605196
15 380 log w, = 0.601352
Then from
R- 1 +a 2 2aV% y 2 = aV 2 ,
we have
log 72 = 702855 , log y, = 7.976024.
The values of the quantities from II to ^ 2 should be found by a duplicate compu-
tation without reference to the former computation, since any error in these quantities
will affect all that follows.
We now divide the circumference into sixteen parts relative to the mean anomaly,
and find the corresponding values of the eccentric anomaly E from
g zz E esin E ,
where e is regarded as expressed in seconds of arc. Substituting the sixteen values
of in the equations
/sin (F P) = w sin (E W ) --
we obtain the corresponding values of /' and F.
A. P. s. VOL. xix. P.
122 A NEW METHOD OF DETERMINING
Then in a similar manner from
\ogq-\ogf-\-y
where s = 206264".8, logX = 9.63778,
we find the values of Q, (7, log q , x , and y.
Thus we have found all the quantities entering into the expression
Instead of this, we use the transformed expression
Q" = N n (1 + a 2 2acos (E'Q)}-~> (1 + V 2&cos (E' + Q))~ " ,
and have, for finding the values of JVj a, &-, the equations
=. sm
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 123
NN _2i r (0) (1) (2) n
' O)) ' 2 := 16,, + &.. cos(J' O)+6 cos2(^' O)+etc.
i^y/ i j* .**-** v t' / ' "' v \j / l
L_ 7T IT -n _1
(1) (2)
n cos (.#' + Q) + ^ cos2 (^' + Q)
2 2
+ etc.]
For finding the values of the coefficients in these expressions we use RUNKLE'S
Tables for Determining the Values of the Coefficients in the Perturbatiee Function of
Planetary Motion, published by the Smithsonian Institution. With the sixteen values
of a as arguments we enter these tables and find at once the corresponding values of
(1) (2) (3)
(o) fe, 6. 6-. o* (0) a 3 (i) a 2 (2)
& ! , then those of - , - 2 , -" , etc., etc. ; - . 1% ,&<>* > 5 *J ' etc '' etc> > where P 1S found
Y a a a p 2 P 2 > 2
a 2
from the sixteen values of /5 2 = - - 2 .
A. ~~ Cv
Since & in (1 2& cos (j^ r + )) is very small it will suffice to put
i^' = l ,!!"=:>
2 2
(1) (1)
B, =36, ^ 5 =56.
2" 2"
Then from
() (i)
\ -\-r T> r /~\
r, -i- N rS
^n 2 M
2" 2"
(i) (0
, = t.JV B
2 2
we have, in case of ^ ( J,
124 A NEW METHOD OF DETERMINING
and, for ,ua 2 f?) ,
(0) 3 (I) 3 (1) 3
6-3 =JVr C = .#3&cos2 * = JT 36 sin
We divide by 8 to save division after quadrature.
With these values
values of k t , A, , from
(0 (0 (*)
With these values of c n9 s,,, and the values of the coefficients & M , we find the
2" 2" 2
>'-l) J'+lK (1)
C M ~T~ I ^ n
2" 2" X 2" 2 ' 2
(&,
\ 7,
V't- 1 ; \ (.1
TT / TT
_(i-l) -, (i+1) \ (1)
"% ' \
For i 0, we find A; from
(0) (0) (1) (1)
2 n w - n n
2" 2" 2" 2"
Then in case of / f from
(c)
where m' is the mass of the disturbing body and s = 206264/ / 8
and from
[i(Q g)
(s)
f> = i m' s a? Jc ( sin \i(Q g) A*] ,
(a\ 3 ( c ) *)
^J , we find the values of 4 t and 4 f) for the 16 different points of the
circumference, and the various terms of the series.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 125
(C) (8)
Again, since A lt K , A L K are given in the forms
(c) (c) (s)
A it K zO^v cos v g + 2 C, sin v g
() (c) (s)
J.^ = 2$> cos y# + 2$, sin >gr,
(c) () (c)
we have the following equations to find the values of the coefficients Q>, CJ; r , o<,^
w
(0.8 )= F + Y s () = F -F 8
/9 1 fh V-4- V ( % \ V - . V
\A.\J) _t 2 -j- _i IQ VTTT/ 2 -* 10-
^
/7ir;\ T^_j_y" ( i \ V V
( i.LO) J-T ~ ' J-15 \TS) --7 -"-M
(0.4) = (0.8 ) + (4.12)
(1.5) = (1.9 ) + (5.13)
(2.6) = (2.10) + (6.14) ' (0.2) = (0.4) + (2.6)
(3.7) = (3.11 1 ) + (7.15) (1.3) = (1.5) + (3.7)
4 (c + 2c 8 ) = (0.2)
4(c -2c 8 ) = (1.3)
4 (c 2 + <%) = (0.8) (4.12)
4 (c 2 - c 6 ) = { [(1.9) - (5.13)] [(3.11) (7.15)] } cos 45 C
4(s 2 + * 6 ) = | [(1.9) (5.13)] + [(3.11) (7.15)]| cos 45 C
4(s 2 - s 6 ) = (2.10) -(6.14)
8c 4 = (0.4) (2.6)
8s,- (1.5) (3.7)
126 A NEW METHOD OF DETERMINING
4 ( Cl + c 7 ) = (f ) + [(-&) (A)] cos 45
4 ( Cl c 7 ) = [(i) (yV)] cos 22.5 + [( TT ) (A)] cos 67.5
4 (c 3 + c 3 ) = (f) [(A) (&)] cos 45
4 (c 3 c 5 ) = [( | ) (yV )] sin 22.5 [(fV) (fV)] sin 67.5
4 Oi + s 7 ) = [( | ) + (yV)] sin 22.5 + [(^-) + ( 1 %)] sin 67.5
4 ( A 7 ) = [(A) + (A)] cos 45 + ( T %)
. 4 ( S3 + 6 ) - [( i ) + ( T 7 5 )] cos 22.5 - [(yV) + (fV)] cos 67.5
4 (8 3 -. 5 ) = [(A) + (A)] cos 45 -(A)
The values of c,, s,, must satisfy the equation
(c) (s)
A it K , or A it K = \ c + C L cos g + c 2 cos 2g + etc.
+ S A sin g -\- s> sin 2# -f etc.
i answering to i in 6 H , and x being any one of the numbers, from to 15 inclusive,
"2
into which the circumference is divided. We use c t , s v as abbreviated forms of C^
(s) (c) (c) (s)
C ijV , etc. Having found the values of c,, from the 16 different values of JL , A ly A^
(C) (S) (C) (8) / a \ / a \
^4 2 , A 27 . . . A 9 , Ac,, both for [i \ ^) and ^a 2 ^/i we have the values of these func-
tions given by the equation
(c)
The values of the most important quantities from the eccentric anomaly E to' c t ,
s needed in the expansion of ^ f j and pa 2 l-J , are given in the following tables,
first for ^ ( a -\ , and then for ^cr I a j , when not common to both.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
127
Values of Quantities in the Development of /*(-) and^a' 2 (-) .
I
9
^
E + W
E + W,
F P
F
( 0)
1 If
0.0
1 II
266 4 39.5
O I II
266 15 38.0
1 II
266 21 17.2
O 1 II
359 24 44.2
( i)
24 24 4.2
i> ( .() 28 43.7
290 39 42.2
290 8 7.8
23 11 34.8
( *)
48 26 37.2
314 31 16.7
314 42 15.2
313 40 58.4
46 44 25.4
( 3)
71 52 24.9
337 57 4.4
338 8 2.9
336 53 39.3
69 57 6.3
( 4)
94 35 14.0
39 53.5
50 52.0
359 41 1.3
92 44 28.3
( 5)
116 36 51.7
22 41 31.2
22 52 29.7
21 59 7.8
115 2 34.8
( < ; )
138 4 29.4
44 9 8.9
44 20 7.4
43 47 3.8
136 50 30.8
( T)
159 8 !!>.<;
65 12 59.1
65 23 57.6
65 8 48.4
158 12 15.4
( )
180 0.0
86 4 39.5
86 15 38.0
86 13 41.4
179 17 8.4
( ! >)
200 51 40.4
106 56 19.9
107 7 18.4
107 15 14.8
200 18 41.8
(10)
221 55 30.6
128 10.1
128 11 8.6
128 28 47.5
221 32 14.5
(11)
243 23 8.3
149 27 47.8
149 38 46.3
150 8 27.6
243 11 54.6
(12)
265 24 46.0
171 29 25.5
171 40 24.0
172 23 51.4
265 27 18.4
(13)
288 7 35.1
194 12 14.6
194 23 13.1
195 17 19.4
288 20 46.4
(14)
311 33 22.8
217 38 2.3
217 49 0.8
218 43 0.9
311 46 27.9
(15J_
335 35 55.8
241 40 35.3
241 51 33.8
242 28 57.5
335 32 24.5
V
1613 47 17.9
w
1433 47 18.6
9
L g-/-
y
X
Q
Log. q.
Log. a
( )
0.612427
.001251
ii
- 12.2
1 II
359 24 32.0
0.611176
01706582
( i)
0.612078
.000860
'+431.5
23 18 46.3
0.611218
0.706349
( 2)
0.609315
.000081
+598.0
46 54 23.4
0.609234
0.705534
( 3)
0.605242
+.000981
+390.0
70 3 36.3
0.606233
0.704403
( 4)
0.601312
+.001292
- 58.6
92 43 29.7
0.602604
0.703241
( 5)
0.598569
+ .000846
476.9
114 54 37.9
0.599415
0.702241
( 6)
0.597310
+.000091
626.7
136 40 4.1
0.597401
0.701493
( ?)
0.597194
.000956
435.1
158 5 0.3
0.596238
0.701011
( )
0.597621
.001322
- 15.7
179 16 52.7
0.596299
0.700788
( '')
0.598109
.000997
+408.7
200 25 30.5
0.597112
0.700494
(10)
0.598532
.000152
+618.1
221 42 32.6
0.598380
0.700021
(11)
0.599177
+.000777
+496.6
243 20 11.2
0.599954
0.699872
(12)
0.600584
+.001278
+ 96.7
265 28 55.1
0.601862
0.700504
(13)
0.603163
+.001032
363.1
288 14 43.3
0.604195
0.702020
(14)
0.606734
+.000148
600.1
311 36 27.8
0.606882
0.704038
(15)
0.610302
.000825
452.4
335 24 52.1
0.609477
0.705810
2 1
4.823835
+ 3
0.5
1613 47 17.4
4.823838
5.622201
i?
4.823834
2
0.7
1433 47 17.9
4.823842
5.622200
128
A NEW METHOD OF DETERMINING
Values of Quantities in the Development of
and
9
X
Zi
Log. b.
Log. a.
a.
Log. IT.
/ //
i n
( )
53 23 45.3
7 57.83
7.063818
9.701484
0.502902
9.695669
( i)
53 26 41.3
7 57.78
7.063792
9.701945
0.503437
9.695880
( 2 )
53 14 15.6
7 59.97
7.065778
9.699988
0.501173
9.695892
( 3)
52 54 33.7
8 3.30
7.068781
9.696876
0.497594
9.695837
( 4)
52 28 55.6
8 7.35
7.072405
9.692804
0.492951
9.695616
( 5)
52 6 31.2
8 10.95
7.075601
9.689226
0.488907
9.695421
( 6)
51 53 41.2
8 13.23
7.077613
9.687169
0.486597
9.695400
( 7)
51 46 50.0
8 14.55
7.078774
9.686068
0.485364
9.695430
( 8 )
51 49 41.2
8 14.49
7.078721
9.686526
0.485877
9.695629
( 9)
52 52.3
8 13.57
7.077913
9.688321
0.487889
9.696120
(10)
52 18 36.9
8 12.12
7.076635
9.691160
0.491089
9.696905
(11)
52 36 21.2
8 10.34
7.075061
9.693986
0.494294
9.697532
(12)
52 49 37.5
8 8.19
7.073153
9.696093
0.496699
9.697631
(13)
52 58 10.6
8 5.58
7.070825
9.697448
0.498251
9.697141
(14)
53 5 12.5
8 2.58
7.068133
9.698559
0.499527
9.696354
(15)
53 13 54.4
7 59.70
7.065534
9.699932
0.501109
9.695743
r
77.553783
3.956815
77.569096
r
77.553803
3.956845
77.569088
(0)
(i)
(i)
(0)
(i)
(2)
9
Log. !<?!
Log. icj
Log. fa.
Log. l l
Log. & t
Log. 6 t
"2"
"2
"2
2
2
f
( o)
8.792579
6.16064
4.47527w
0.332110
9.748094
9.329969
( i)
8.792790
5.98934
6.02920
0.332186
9.748669
9.331018
( 2)
8.792802
4.98551n
6.16173
0.331867
9.746235
9.326571
( 3)
8.792731
6.05070n
5.97267
0.331369
9.742375
9.319511
( 4)
8.792526
6.16734n
5.14693n
0.330730
9.737346
9.310298
( 5)
8.792331
5.9821 9n
6.05562n
0.330182
9.732946
9.302224
( 6)
8.792310
4.93934
6.17378n
0.329872
9.730425
9.297590
( 1)
8.792340
6.03383
6.01614w
0.329707
9.729076
9.295111
( 8)
8.792539
6.17549
4.57507^
0.329776
9.729636
9.296143
( 9)
8.793030
6.05359
5.99045
0.320045
9.731836
9.300183
no)
8.793815
5.23282
6.17067
0.33047>
9.735322
9.306586
(n)
8.794442
5.94812n
6.07618
0.330914
9.738805
9.312970
(12)
8.794541
6.16466^
5.36611
0.331246
9.741407 i
9.317738
(13)
8.794051
6.07296/1
5.94202n
0.331460
9.743073
9.320808
(14)
8.793264
5.23742n
6.16200n
0.331637
9.744461
9.323327
(15)
8.792653
5.97789
6.04134??
0.331858
9.746165
9.326443
2
2.647715
77.912926
74.508222
I'
2.647721
77.912945
74.508268
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
129
Values of Quantities in the Development of ^ and
(3)
(4)
(5)
(6)
(7)
(8)
(9)
9
Log. &j
Log. Jj
Log. &!
Log. l l
Log. &J
Log. b l
Log. &j
2
"2"
1
2"
2"
2
2
( )
8.954999
8.60017
8.2570
7.9215
7.5915
7.2654
6.9426
( i)
8.956515
8.60214
8.2594
7.9244
7.5947'
7.2691
6.9468
( 2)
8.950082
8.59373
8.2490
7.9120
7.5804
7.2528
6.9286
( 3)
8.939865
8.58036
8.2326
7.8926
7.5578
7.2271
6.8997
( 4)
8.926521
8.56292
8.2110
7.8668
7.5280
7.1932
6.8617
( 5)
8.914818
8.54760
8.1921
7.8444
7.5020
7.1636
6.8285
( 6)
8.908100
8.53882
8.1812
7.8314
7.4870
7.1466
6.8094
( 7)
8.904506
8.53411
8.1754
7.8244
7.4789
7.1373
6.7991
( 8)
8.906000
8.53606
8.1778
7.8273
7.4822
7.1411
6.8033
( 9)
8.911861
8.54373
8.1872
7.8386
7.4953
7.1561
6.8201
(10)
8.921142
8.55588
8.2024
7.8565
7.5160
7.1796
6.8464
(11)
8.930392
8.56797
8.2172
7.8742
7.5367
7.2031
6.8728
(12)
8.937298
8.57701
8.2285
7.8875
7.5520
7.2205
6.8923
(13)
8.941742
8.58283
8.2355
7.8960
7.5618
7.2317
6.9048
(14)
8.945388
8.58760
8.2415
7.9030
7.5700
7.2410
6.9152
(15)
8.949898
8.59349
8.2488
7.9117
7.5800
7.2524
6.9280
2
71.449530
68.55219
65.7484
63.0060
60.3071
57.6402
54.9995
I'
71.449597
68.55223
65.7482
63.0063
60.3072
57.6404
54.9998
3
(1) (i)
(0)
(1)
(2)
(3)
9
Log.i^
Log. -i c 3
Log. | s 3
Log. % 6 3
Log. 6 3
Log. 6 8
Log. 6 3
"2"
2
"2".
2
2
2
( o)
8.183917
5.42374
3.73837n
0.280319
0.417421
0.200612
9.961097
( i)
8.184550
5.25307
5.29293
0.281000
0.418474
0.202090
9.963016
( 2)
8.184586
4.24928ft
5.42550
0.278120
0.414013
0.195824
9.954877
( 3)
8.184421
5.31430ft
5.23627
0.273612
0.406981
0.185917
9.941987
( 4 )
8.183758
5.43028n
4.40987ft
0.267827
0.397890
0.173060
9.925223
( 5)
8.183173
5.24454n
5.31797??
0.262860
0.390004
0.161858
9.910585
( 6)
8.183110
4.20163
5.43607w
0.260054
0.385513
0.155458
9.902210
( T)
8.183200
5.29621
5.27852n
0.258559
0.383116
0.152039
9.897732
( 8)
8.183797
5.43847
3.83805w
0.259184
0.384116
0.153464
9.899598
( 9)
8.185270
5.31804
5.25490
0.261621
0.388024
0.159038
9.906900
(10)
8.187625
4.49962
5.43747
0.265530
0.394254
0.167901
9.918485
(11)
8.189506
5.21681ft
5.34487
0.269488
0.400515
0.176758
9.930076
(12)
8.189803
5.43364n
4.63509
0.272484
0.405223
0.183435
9.938754
(13)
8.188333
5.34047/1
5.20953ft
0.274429
0.408267
0.187732
9.944350
(14)
8.185972
4.50257n
5.42714ft
0.276036
0.410773
0.191265
9.948948
(16)
8.184139
5.24121
5.30466ft
0.278037
0.413885
0.195644
9.954643
S
65.482568
2.159554
3.209203
1.421019
79.449192
Z'
65.482592
2.159606
3.209266
1.421076
79.449289
A. P. 8. VOL. XIX. Q.
130
A NEW METHOD OF DETERMINING
Values of Quantities in the Development of l*(~) and [*a 2 ( J
(4)
.(5) 1 (6)
(7)
(8)
(9)
9
Log-. &o
^D O T
Log. 6 3
Log. 6 3
Log. 6 3
Log. 6 3
Log. 6 3
2
f
2
2
2
2
( o)
9.70884
9.4484
. 9.1822
8.9118
8.6383
8.3621
( i)
9.71121
9.4512
9.1854
8.9155
8.6423
8.3665
( 2)
9.70116
9.4393
9.1716
8.8998
.8.6247
8.3471
( 3)
9.68524
9.4203
9.1496
8.8747
8.5965
8.3158
( 4)
9.66450
9.3955
9.1207
8.8418
8.5595
8.2747
( 5)
9.64638
9.3739
9.0956
8.8131 8.5273
8.2389
( 6)
9.63600
9.3614
9.0813
8.7968
8.5089
8.2184
(I)
9.63043
9.3549
9.0735
8.7880
8.4991
8.2077
( 8)
9.63276
9.3576
9.0766
8.7914
8.5030
8.2119
( 9)
9.64181
9.3684
9.0893
8.8058
8.5191
8.2298
(10)
9.65617
9.3856
9.1093
8.8287
8.5449
8.2585
(11)
9.67052
9.4028
9.1292
8.8515
8.5705
8.2868
(12)
9.68125
9.4156
9.1440
8.8684
8.5893
8.3078
(13)
9.68816
9.4237
9.1537
8.8791
8.6015
8.3213
(14)
9.69382
9.4305
9.1614
8.8882
8.6118
8.3329
(15)
9.70087
9.4389
9.1711
8.8992
8.6240
8.3464
2
77.37450
75.2339
73.0471
70.8269
68.5804
66.3134
I'
77.37462
75.2341
73.0474
70.8269
68.5803
66.3132
9
Log. k
Log. lc l
Log. & 2
Log. & 3
Log. Jc 4
Log. 7c 5
Log. k 6
Log. k 7
( o)
8.824187
8.54492
8.12562
7.750420
7.39550
7.0523
6.7168
6.4105
( i)
8.824302
8.54433
8.12588
7.751220
7.39678
7.0540
6.7190
6.4054
( 2)
8.823605
8.53875
8.11916
7.742693
7.38634
7.0416
6.7046
6.3714
( 3)
8.822665
8.53172
8.10982
7.730361
7.37091
7.0232
6.6832
6.3298
( 4)
8.821701
8.52543
8.09963
7.716100
7.35261
7.0007
6.6565
6.2932
( 5)
8.821143
8.52236
8.09246
7.705215
7.33807
6.9826
6.6349
6.2764
( 6)
8.821183
8.52300
8.09009
7.700585
7.33130
6.9737
6.6239
6.2809
( ?)
8.821397
8.52470
8.08981
7.699023
7.32855
6.9698
6.6187
6.2913
( 8)
8.821810
8.52671
8.09164
7.701551
7.33151
6.9732
6.6226
6.3027
C 9)
_8.822444
8.52829
8.09567
7.707159
7.33895
6.9824
6.6337
6.3093
(10)
8.823323
8.52965
8.10077
7.715298
7.35002.
6.9965
6.6506
6.3129
(11)
8.824009
8.53059
8.10550
7.723069
7.3607GI
7.0100
6.6669
6.3147
(12)
8.824233
8.53159
8.10915
7.728940
7.36874
7.0202
6.6793
6.3196
(13)
8.824055
8.53359
8.11233
7.733450
7.37462
7.0274
6.6879
6.3342
(14)
8.823809
8.53721
8.11622
7.738311
7.38053
7.0345
6.6960
6.3608
(15)
8.823826
8.54164
8.12113
7.744423
7.38795
7.0433
6.7062
6.3901
S
70.583851
68.25726
64.85258
61.793910
58.89655
56.0927
53.3503
50.6520
I'
70.583841
68.25722
64.85260
61.793920
58.89653
56.0926
53.3505
50.6512
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
131
Values of Quantities in the Development of fi(^\ and f.ia 2 ( a \'.
<7
Log. fc 8
Log, & 9
JT, K> JT S K t K. JZi K, -X.
( o)
6.0606
5.7378
- 0.6
- 0.4
/
- 0.3
- 0.3
- 0.3
- 0.3
0.3
- 0.3
( ] )
6.0636
5.7413
. +20.3
+ 12.9
+11.4
+ 11.1
+10.6
+ 10.1
+ 9.5
+ 8.3
( 2)
6.0454
5.7212
4-27.9
+ 17.8
+15.6
+ 15.2
+ 14.6
+ 14.0
+13.5
+ 12.5
( '>)
6.0178
5.6904
+ 18.4
+ 11.7
+10.2
+ 10.0
+ 9.7
- 9.4
+ 9.2
+ 8.8
( 4)
5.9830
5.6515
- 2.8
- 1.8
- 1.6
- 1.5
- 1.5
- 1.5
- 1.5
- 1.5
( 5)
5.9541
5.6191
22.7
14.5
-12.7
12.0
-11.8
-11.6
11.4
-11.0
( 6)
5.9391
5.6019
29.8
19.0
-16.7
-15.7
-15.3
14.9
-14.5
13.7
( *)
5.9316
5.5934
20.7
-13.2
11.6
10.9
-10.5
10.1
- 9.7
- 8.9
( 8)
5.9364
5.5985
- 0.7
- 0.5
- 0.4
- 0.4
- 0.4
- 0.3
- 0.3
0.3
( )
5.9512
5.6151
+ 19.3
+ 12.3
+ 10.9
+ 10.2
+ 9.8
+ 9.4
+ 9.0
+ 8.2
(10)
5.9737
5.6405
4-29.1
+ 18.6
+ 16.4
+15.3
+ 14.9
+ 14.5
+ 14.1
+13.3
(11)
5.9959
5.6656
+23.4
+ 14.9
+13.1
+ 12.3
+ 12.1
+11.9
+ 11.7
+11.3
(12)
6.0124
5.6842
- 4.5
+ 2.8
+ 2.5
- 2.4
+ 2.4
+ 2.3
+ 2.3
+ 2.2
(13)
6.0251
5.6968
-17.0
-10.8
9.5
8.9
- 8.8
8.7
- 8.6
- 8.4
(14)
6.0341
5.7083
28.1
17.8
-15.7
-14.7
-14.3
13.9
13.6
13.0
(15)
6.0468
5.7224
21.0
-13.4
-11.8
-11.0
-10.6
-10.2
- 9.8
- 9.0
JT
45.3439
- .5
.3
.2
+ .3
.3
w
45.3441
.1
+ .8
.1
9
Log. Jc
Log. \
Log. k 2
Log. & 3
Log. & 4
Log. & 5
Log. Jc 6
Log. 7
( )
8.465272
8.60289
8.38621
8.14674
7.89481
7.6341
7.3679
7.0975
( i)
8.466247
8.60407
8.38777
8.14874
7.89694
7.6369
7.3712
7.1013
( 2)
8.462637
8.59849
8.38030
8.13935
7.88563
7.6238
7.3561
7.0843
( 3)
8.457236
8.59018
8.36903
8.12505
7.86829
7.6033
7.3326
7.0577
( 4)
8.450550
8.58006
8.35509
8.10719
7.84645
7.5774
7.3026
7.0237
( 5)
8.445362
8.57214
8.34391
8.09259
7.82837
7.5559
7.2776
6.9950
( 6)
8.443224
8.56872
8.33868
8.08543
7.81922
7.5446
7.2645
6.9800
( "0
8.442508
8.56750
8.33651
8.08224
7.81495
7.5395
7.2581
6.9726
( 8)
8.444020
8.56954
8.33902
8.08521
7.81840
7.5433
7.2623
6.9771
8.444679
8.57452
8.34564
8.09354
7.82847
7.5551
7.2760
6.9925
(10)
8.453274
8.58206
8.35573
8.10632
7.84401
7.5734
7.2971
7.0165
(11)
8.458368
8.58906
8.36522
8.11851
7.85895
7.5912
7.3176
7.0400
(12)
8.461465
8.59345
8.37153
8.12680
7.86927
7.6036
7.3320
7.0564
(13)
8.461922
8.59532
8.37468
8.13126
7.87506
7.6105
7.3405
7.0660
(14)
8.461886
8.59651
8.37704
8.13471
7.87957
7.6163
7.3472
7.0739
(15)
8.462852
8.59905
8.38088
8.13992
7.88616
7.6242
7.3564
7.0845
_y
68.69172
66.90360
64.93175
62.85706
60.7165
58.5297
56.3095
I'
68.69184
66.90364
64.93185
62.85719
60.7166
58.5300
56.3096
132
A NEW METHOD OF DETERMINING
Values of Quantities in the Development of
and
9
Log. & 8 Log. & y
J5T, K, K, (Q-g)-^ m-9)-K* 3(Q-g)-K 3
1
/ / I o / a i it
( )
6.8240 6.5478
0.1
0.1 0.1 359 25.1 358 49.5 358 13 55.0
( i)
6.8280 6.5522
+4.4
+4.4 +4.4 28.5 1 24.6 2 14 57.0
( 2)
6.8092 6.5317
+ 6.0
+ 6.0 +(>.0 l 26.5 3 31.0 5 27 34.4
( 3)
6.7795 6.4988
+3.9
+3.9 +3.9 2 15.2 4 55.5 7 30 33.9
( 4)
6.7414 6.4566
0.6
0.6 0.6 2 46.3 5 28.8 8 12 2.4
( 5)
6.7093 6.4209
47
4.7 -4.7 2 47.3 5 3.8 7 26 37.11
( 6)
6.6921 6.4016
6.2
6.2 6.2 2 9.9 3 39.1 5 16 57.0
( 7)
6.6837 6.3923
4.3
4.3 4.3 55.7 1 23.2 1 56 :5!i.u
( 8)
6.6887 6.3976
0.2
0.2 0.2 359 17.6 358 34.3 357 51 3.3
( 9)
6.7058 6.4165
+4.0
_|_4.0 +4.0 357 36.2 355 38.7 353 35 39.5
(10)
6.7327 6.4463
+ 6.1
+ 6.1 +6.1 356 13.4 353 6.5 349 51 14.9
(11)
6.758!) 6.4752
+5.0
+5.0 +5.0 355 26.8 351 25.5 347 17 2<>.4
(12)
6.7773 6.4958
+ 1.0
+ 1.0 -1-1.0 355 24.4 350 55.0 346 24 12.8
(13)
6.7883 6.5081
3.5
3.5 3.5 356 1.7 351 40.2 347 23 40.2
(14)
6.7976 6.5187
6.0
6.0 6.0 357 4.6 353 30.7 350 5 3.8
(15)
6.8093 6.5317
4.5
_4.5 _4.5 358 15.9 356 3.1 353 56 22.5
V
54.0630 51.7961
.0
.0 .0 1793 47.8 1781 22 3.6
I'
54.0628 51.7957
+ .3
+ .3 + .3 1433 47.3 1421 21 59.1
9
42-<
l) %
^(Q-9
) -^
; 6(e-0
)-^
'( Q 9\
K
im-9\
) A's 9(
:-</)-
-K,
( 0)
o
357
/
38.5
o
357
/
3.0
o
356
/
27.5
o
355
i
52.1
355
o /
16.7
354
41.2
( 1)
3
3.9
3
53.2
4
42.5
5
31.8
6
21.1
7
10.5
( 2)
7
22.4
9
17.4
11
12.4
13
7.3
15
2.2
16
57.1
( 3)
10
4.4
12
38.3
15
12.2
17
46.0
20
19.8
22
53.6
( 4)
10
55.5
13
39.0
16
22.5
19
6.0
21
49.5
24
33.0
( 5)
9
50.6
12
15.0
14
39.4
17
3.9
19
28.4
21
52.8
( 6)
6
55.9
8
35.6
10
15.3
11
54.9
13
34.5
15
14.2
( 7)
2
30.9
8
5.5
3
40.1
4
14.7
4
49.3
5
23.9
( 8)
357
8.0
356
24.9
355
41.7
354
58.6
354
15.5
353
32.4
( 9)
351
31.8
349
27.7
347
23.6
345
19.5
343
15.4
341
11.3
(10)
346
34.9
343
17.8
340
0.7
336
43"7
333
26.7
330
9.6
(11)
343
8.5
338
58.9
334
49.3
330
39.7
326
30.1
322
20.5
(12)
341
53.2
337
22.1
332
51.1
328
20.0
323
48.9
319
17.9
(13)
343
7.7
338
52.3
334
36.9
330
21.5
326
6.1
321
50.7
(14)
346
40.5
342
16.6
339
52.7
336
28.8
333
4.9
329
41.1
(15)
351
50.4
349
44.9
347
39.4
345
33.8
343
28.2
341
22.7
2-
1744
6.5
I'
1384
6.0
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
133
In the expansion of
()
(c) (s)
(c) ()
(c) (s)
(C) . (8)
9
A
A, A,
-4-2 -a-2
4 A
M.% -#.3
-^4 - ( *4
//
it n
n rr
// //
// //
( 0)
13.13109
6.9027 .0701
+2.6281 .0539
+ 1.10745 .03418
+.4889 .0201
( i)
13.13458
6.8933 -f.0571
2.6294 +.0647
1.10917 +.04356
.4901 +.0262
( 2)
13.11352
6.8033 +.1712
2.5849 +.1588
1.08348 +.10356
.4751 +.0615
( 3)
13.08513
6.6912 -f .2633
2.5254 +.2176
1.04890 +.13827
.4553 + -0809
( 4)
13.05615
6.5922 +-3192
2.4646 +.2364
1.01333 +.14604
.4353 +-0840
( 5)
13.03939
6.5457 +.3187
2.4259 +.2150
0.99004 +.12935
.4224 +-(H33
( 6)
13.04058
6.5584 +-2479
2.4172 +.1543
0.98367 +.09095
.4190 +.0509
( <)
13.04700
6.5880 +.1067
2.4198 +.0585
0.98375 +.03339
.4190 +.0184
( )
13.05942
6.6190 .0816
2.4317 .0606
0.98937 .03712
.4218 .0211
( 9)
13.07850
6.6377 .2779
2.4464 .1863
0.99667 .11189
.4249 .0633
(10)
13.10500
6.6498 .4389
2.4645 .2979
1.00593 .18002
.4287 .1023
(11)
13.12573
6.6578 -.5301
2.4816 .3742
1.01487 .22886
.4322 .1310
(12)
13.13248
6.6727 .5359
2.4991 .3995
1.02497 .24789
.4373 .1431
(13)
13.12612
6.7090 .4658
2.5224 .3693
1.03984 .23254
.4463 .1354
(14)
13.11967
6.7727 .3458
2.5559 .2907
1.06142 .18555
.4600 .1090
(15)
13.12018
6.8478 .2074
2.5954 .1791
1.08668 .11537
.4760 .0683
y
104.75791
53.5708 .7340
+ 20.0460 .5531
8.26962 .34421
+3.5661 .1992
\v
104.75663
53.5705 .7354
+20.0463 .5531
8.26992 .34409
+3.5662 .1992
(C) (8)
(c) (s)
( c ) ( s )
(c) (s)
(c> (s)
9
A Q A Q
Aj Arj
// //
II II
II II
// //
// //
( )
+.2217 .0114
+.1023 .0063
+.0505 .0036
+ .0226 .0019
+.0107 .0010
( i)
.2223 +.0151
.1027 +.0085
.0498 +.0048
.0226 +.0025
.0108 +.0014
( 2)
.2138 +-0350
.0978 +.0194
.0451 +.0105
.0211 +.0057
.0099 +.0030
( 3)
.2028 +.0454
.0916 +.0249
.0401 +.0128
.0192 +.0071
.0089 +-0038
( 4)
.1916 +.0465
.0856 +.0252
.0365 +.0126
.0176 +.0070
.0080 +.0037
( 5)
.1848 +.0401
.0821 +.0215
.0356 +.0109
.0167 +.0059
.0076 +.0030
( 6)
.1832 +.0277
.0815 +.0147
.0368 +.0078
.0166 +-0040
.0076 +.0021
( 7)
.1833 +.0099
.0816 +.0052
.0384 +.0028
.0168 +.0014
.0077 +.0007
( 8 )
.1847 .0116
.0823 .0062
.0394 .0035
.0169 .0017
.0078 .0009
( 9)
.I860 .0346
.0826 .0185
.0388 .0102
.0168 .0051
.0077 .0026
(10)
.1870 .0561
.0827 .0301
.0372 .0160
.0166 .0083
.0075 .0043
(11)
.1880 .0722
.0827 .0389
.0354 .0199
.0163 .0108
.0072- .0056
(12)
.1904 .0793
.0837 .0429
.0350 .0216
.0163 .0120
.0072 .0062
(13)
.1956 .0756
.0867 .0411
.0369 .0210
.0173 .0116
.0077 .0060
.2041 .0613
.0918 .0336
.0414 .0180
.0190 .0096
.0087 .0051
(15)
.2140 .0387
.0978 .0214
.0468 .0120
.0210 .0062
.0098 .0033
S
+ 1.5765 .1105
+.7077 .0598
+.3219 .0318
+.1467 .0168
+.0674 .0087
i'
+ 1.5768 .1106
+.7078 .0598
+.3218 .0318
+.1467 .0168
+.0674 .0086
134
A NEW METHOD OF DETEKMINING
In the expansion of jj. a 2 1 ^
(e)
(C) (8)
(c) (8)
(<0 ()
(c) ()
9
A
A, A,
A.^ -4 2
A.% A.%
A A,
it
n n
// //
n n
n n
( o)
23.3520
+32.0569 0.3301
+ 19.4613 0.4009
+ 11.2092 0.3464
+ 6.269 0.258
( i)
23.4045
32.1423 -j-0.41 99
19.5273 +0.5272
11.2569 +0.4603
6.300 +0.347
( 2)
23.2107
31.7192 +1.0033
19.1618 +1.2486
10.9731 +1.0737
6.096 +0.802
( 3)
22.9239
31.1043 +1.3503
18.6375 +1.6470
10.5748 +1.4097
5.813 +1.041
( 4)
22.5737
30.3821 +1.4503
18.0367 +1.7240
10.1342 +1.4580
5.516 +1.063
( 5)
22.3056
29.8387 +1.2952
17.5937 +1.5122
9.8190 +1.2644
5.310 +0.912
( 6)
22.1960
29.6180 +0.9110
17.4156 +1.0505
9.6988 +0.8734
5.239 +0.626
( *)
22.1595
29.5473 +0.3342
17.3564 +0.3782
9.6618 +0.3118
5.219 +0.222
( 8)
22.2368
29.6867 0.3713
17.4552 -0.4367
9.7264 0.3654
5.259 0.204
( 9)
22.4249
30.0100 1.1187
17.6808 1.3068
9.8617 1.0915
5.331 0.786
(10)
22.7157
30.5036 1.8033
18.0224 2.1155
10.0630 1.7762
5.436 1.285
(11)
22.9837
30.9679 2.3042
18.3471 2.7150
10.2558 2.2962
5.536 1.667
(12)
23.1482
31.2707 2.4810
18.5835 2.9616
10.4121 2.5144
5.627 1.839
(13)
23.1725
31.4193 2.3026
18.7500 2.7837
10.5580 2.3763
5.739 1.748
(14)
23.1706
31.5386 1.8212
18.9291 2.2155
10.7412 1.9027
5.895 1.409
(15)
23.2222
31.7564 1.1097
19.1791 1.3716
10.9764 1.1843
6.091 .882
r
182.6038
246.7758 3.4423
147.0656 4.1071
82.9580 3.5000
+45.337 2.564
w
182.5968
246.7862 3.4356
147.0719 4.1125
' 82.9644 3.4985
+ 45.339 2.563
(C) (8)
(C) (8)
(c) ()
(C) (8)
(c) ()
9
4 ^-5
A A
A A
( )
+3.440 0.177
+ 1.863 .115
+1.000 .072
+.532 .044
+ .282 .027
( i)
3.458 +0.240
1.874 +.157
1.005 +.098
.535 +.060
.283 +.036
( 2)
3.318 +0.550
1.781 +.356
.944 +.221
.497 +.134
.260 +.076
( 3)
3.130 +0.706
1.660 +.453
.868 +.279
.450 +.167
.231 +.098
( 4)
2.937 +0.713
1.540 +.453
.797 +.276
.409 +.164
.208 +.095
( 5)
2.812 +0.606
1.467 +.381
.756 +.232
.377 +.133
.196 +.078
( 6)
2.772 +0.413
1.448 +.260
.748 +.157
.383 +.092
.195 +.053
( ?)
2.766 +0.146
1.446 +.091
.750 +.055
.386 +.032
.197 +.019
( 8)
2.789 0.175
1.459 .110
.757 .053
.389 .039
.199 .023
( 9)
2.824 0.522
1.474 .329
.760 .199
.389 .117
.197 .067
(10)
2.870 0.855
1.491 .540
.759 .326
^.385 .192
.193 .111
(11)
2.915 1.115
1.505 .705
.757 .425
.379 .251
.187 .144
(12)
2.963 1.235
1.528 .783
.767 .473
.382 .280
.188 .162
(13)
3.042 1.179
1.582 .753
.803 .457
.404 .272
.201 .158
(14)
3.164 0.957
1.670 .615
.867 .378
.446 .227
.227 .133
(15)
3.312 0.604
1.775 .391
.942 .243
.495 .147
.259 .087
I
24.253 1.723
12.780 1.094
+ 6.639 .648
+3.423 .392
+ 1.752 .232
v/
24.259 1.722
12.783 1.095
6.641 .660
+3.415 .395
1.751 .225
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
135
. (0 (s) (c) (s)
The Quantities %C ttV , JC1 >: , $> , J$ (> , arranged for Quadrature in the Expansion of
u
G)-
<=0
i-l
,-,=2
< = 8
< = 4
; = 5
, = 6
,=,
(c)
(c)
L
+[209.51454]
+53.571
.735"
II
+20.046
.553
+8.26978
.34414
//
+3.566
.199
+ 1.576
.110
+.707
.060
(7 tC>
^1,1
+.25653
+.548
+.382
+.22949
+.129
+.071
+.038
v=l
(8)
a!i 8)
.25027
+ 1.706
.122
+ 1.273
.046
+.78997
.01129
+.456
+.002
+.253
+.005
+.138
+ .006
(c)
+.022
+.017
+.00807
+.003
+.001
.000
(c)
+.00463
+.257
+.096
+.05847
+.038
+.024
+.013
,2 }
.170
.003
+.01835
+.017
+.007
+ .004
*_2
(8)
+.12279
+.128
+.080
+.04667
+ .026
+.015
+ .001
(c)
.
+.065
+ .048
+ .03063
+.018
+.010
+.006
(c)
+ .03070
+.020
+.007
+.00662
+.005
+.002
+.001
(s)
.003
+.002
+.00216
+.002
+.001
+.001
v=3-
(8)
+ .05945
+.041
+.023
+.01319
+ .006
+ .003
+.002
(c)
.
000
.001
.00217
.002
.001
.001
'<
+.00037
+.001
+.00030
<&
000
+.00052
a!f
+.00055
000
+.00076
gS
.001
.00103
136
A NEW METHOD OP DETERMINING
(c) (s) (c)
The Quantities |Q>, , \G^ V , J$>
(s)
r t> , arranged for Quadrature, in the Expansion of
,/a\8
(i)
fcO
i=i
(-=2
i-3
i=4,
1=6
4 = 6
4*zz 7
Q
i=9
f (0
n
V<,0
n
+[364.6002]
+246.7810
+ 147.068
+ 82.9613
n
+45.338
+24.256
n
+ 12.781
+ 6.640
II
+3.419
+ 1.751
(c)
t,0
3.4388
-4.110
3.4992
2.562
-1.722
-1.095
.654
.392
.228
-
(c)
a,i
+4.3500
+4.6277
+3.873
+2.8862
+ 1.956
+ 1.253
+.771
+.461
+ .270
+.154
(s)
+ 7.8438
+ 9.373
+ 7.9505
+5.816
+ 3.910
+2.488
+ 1.514
+ .898
+.521
a!r
-1.8014
-1.1511
.801
.3643
.106
+.017
+.062
+ .078
+ .058
+ .049
(c)
;
+ .1015
+.104
+.0731
+.043
+.024
+ .011
.008
+ .003
+.001
(c)
.2566
+.0899
+.294
+.3888
+.384
+.327
+.252
+.193
+ .134
+.086
(8)
+.1010
+.296
+.3297
+.302
+.239
+.173
+ .116
+.078
+.047
<
+ 1.1803
+ 1.1209
+.883
+.6281
+.418
+ .266
+.162
+ .093
+ .058
+.031
(c)
$.2
L
+.3367
+.400
+ .3459
+.255
+.170
+.106
+ .065
+.034
.018
0,8
+ .1113
+.1140
+.099
+.0809
+ .066
+.049
+.035
+.024
+ .013
+ .012
Si'*
.0170
.000
+.0059
+ .012
+.015
+ .015
+.015
+ .013
+.008
~
(s)
+.5132
+ .6602
+.317
+ .2097
+.130
+.076
+.043
+ .020
+.012
+ .002
(c)
.0138
.030
.0344
.032
^.027
.020
.005
.010
.005
<c
+ .0177
+.0085
+.003
+ .0028
+ .002
+.002
.000
+.001
.000
+ .001
-4
(s)
+.0117
+.005
+.0061
+.005
+.006
+.004
+ .004
+.001
+ .001
a!r
+.0182
--
+.0172
+.016
+.0134
+.010
+ .006
+.005
+.003
.002
+.001
(c)
.0109
.022
.0182
.016
.012
.008
+ .002
.003
.001
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
137
The quantities Q,,, Q,,, etc., of the preceding tables have been divided by 2 to
save division after quadrature. To check the values of these coefficients we will take
the point corresponding to g 22.5, using the equation
(c) ()
A! , or A! I CQ + Ci cos g + G z cos 2# + etc.
+ #1 sin # -f- $> sin 2y + etc.,
noting that the tables give one-half of the values of these quantities.
Thus we have
i=l i 2 i\ i 2
c
i n
2^1,0
+53.571
n
+20.046
(c)
If!
2*^1,0 -
- 0.735
- 0.553
(c)
CM
+ 1.013
+ .707
#u =
+ 1.306
+ .974
Cu
.01)4
.032
#u =
+ .040
+ .031
(c)
+ .363
+ .135
# =
.240
.004
-1,2
+ .181
+ .114
(c)
#1,2 =
+ .092
+ .070
(c)
(s)
Q, 8
+ .015
+ .005
#J,3 -
.005
+ .004
d!s
+ .077
+ .043
# =
.001
(c)
(s)
Ci,4
. .
#1,4 -
. .
rf'
V/^4
(c)
#1,4 =
2
II
+55.126
//
+21.018
V
+ 0.458
+ 0.521
P
+ 6.891
+ 2.627
1.^
+ 0.057
+ 0.065
^f
+ 6.893
+ 2.629
=
+ 0.057
+ 0.065
In this way we check the values of these quantities for all values of **, in case of
both p(5), and po().
Applying to the coefficients of the two preceding tables the formula
(a\ n () () (*) ( c ) i-
2 ) = J22( a> =F #> ) cos [( =F v)g iE'] T JS2( 0,, dh #^ ) sin [(* T v)^
2 3
noting that J has been applied, we have the values of ^ f ^ J , ^a Q J that follow :
A. P. s. VOL. xix. R.
138
A NEW METHOD OF DETERMINING
[l
G)
9 E'
COS
sin
1
COS
sin
//
+[209.51455]
//
//
+[364.6002]
//
1
+0.25653
0.25027
+4.3500
1.8014
2
+0.00463
+0.12279
0.2566
+ 1.1803
3
+0.03070
+0.05945
+0.1113
+0.5132
4
+0.00037
+0.00055
+0.0177
+0.0182
2 1
+0.023
0.041
+ 0.1310
0.6464
1 1
+0.427
0.193
0.0112
-1.4577
1
-1.158
+ 0.101
3.2161
+ 1.0496
1 1
+53.571
+0.735
+ 246.7810
+3.4388
2 1
+ 2.254
0.144
+ 12.4716
-1.2526
3 1
+0.087
+ 0.063
+0.1909
+0.7842
4 1
+0.016
+ 0.041
+ 0.0970
+0.6740
1 2
+0.099
0.287
2
+0.098
0.129
0.001
-1.283
1 2
0.891
+0.029
5.500
+ 0.697
2 2
+20.046
+0.553
+ 147.068
+ 4.110
3 2
+ 1.656
0.063
+ 13.246
0.905
4 2
+0.093
+0.032
+0.590
+0.483
3
+0.00446
0.01101
' +.0750
0.1753
1 3
+0.04011
0.07730
+.0591
0.9741
2 3
0.56048
+0.00322
5.0643
+0.2912
3 3
+8.26978
+0.34414
+ 82.9613
+3.4992
43
+ 1.01947
0.01936
+ 10.8367
0.4375
5 3
+0.07682
+0.01603
+0.7185
+0.2822
6 3
+ 0.00879
+0.01536
+0.0868
+0.2441
1 4
+0.003
0.004
+0.053
0.098
2 4
+0.020
0.044
+ 0.082
0.674
3 4
0.326
0.005
3.859
+ 0.062
4 4
+ 3.566
+0.199
+45.338
+2.562
5 4
+0.585
0.001
+ 7.772
0.149
6 4
+0.055
+0.008
+0.687
+0.163
7 4
+0.078
+0.162
2 5
+0.005
+0.045
H- 0.033
0.049
3 5
+0.016
0.025
+0.088
0.095
4 5
0.182
0.007
2.657
0.041
5 5
+ 1.576
+0.110
+ 24.256
+ 1.722
6 5
+ 0.325
+0.004
+ 5.163
0.006
7 5
+0.031
+0.004
+ 0.567
+ 0.436
4 6
+0.009
0.008
+0.079
0.269
5 6
0.100
0.006
-1.717
0.073
6 6
+0.707
+0.060
+ 12.781
+ 1.095
7 6
+0.176
+ 0.005
+ 3.260
+0.050
8 6
+0.018
0.005
+0.426
+0.057
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 139
We have next to transform the expressions for ^ f^J and fj.a 2 (*~\ just given
into others in which both the angles involved are mean anomalies.
From
beginning with m =. 5, we find the values of r 5 for values of e f from f to e' 4 .
Then we find
Putting m =. 4, we find the values of r 4 as in the case of r 5 . Then we get p from
(0)
We proceed in this way until we finally have the values of p { . Then we find J h , ^ or
from
2" J
(0) 7<
j i 72 i i_
/' & ^~~ I j
n 2 4
where I
(m)
and 7^/^ from
(m) (0)
T , f, = 7 ,
/l -2 .1
The details of the computation are as follows :
140
A NEW METHOD OF DETERMINING
Computation of the J functions.
* =
y
e'
1*
2e'
2 G
3e'
n f
^ e
4,6'
log.?
8.38251
8.68354
8
.85963
8.98457
9
.08148
9.16066
9.22761
9.28560
log. r b
2
.31646
2.01543
1
.83934
1.71440
1
.61749
1.53831
1.47136
1.41337
tog. #3
7
.68354
7.98457
8
.16066
8.28560
8
.38251
8.46169
8.52864
8.58663
log. r 4
2
.21955
1.91852
1
.74243
1.61749
1
.52058
1.44140
1.37445
1.31646
log.n Iog.jp 5
4
.53601
3.93395
8
.58177
3.33189
3
.13807
2.97971
2.84581
2.72983
Zech
- 1
5
-12
-20
-31
-45
-62
-81
2
.21954
1.91847
1
.74231
1.61729
1
.52027
1.44095
1.37383
1.31585
log. p 4
7
.78046
8.08153
8
.25769
8.38271
8.47973
8.55905
8.62617
8.68415
log. r s
2
.09461
1.79358
1
.61749
1.49255
1.39564
1.31646
1.24951
1.19152
Diff.
4
.31415
3.71205
3
.35980
3.10984
2.91591
2.75741
2.62334
2.50737
Zech
2
-9
-19
34
-52
-76
-103
- 135
2
.09459
1.79349
1
.61730
1.49221
1.39512
1.31570
1.24848
1.19017
tog. 3
^j _M. o
7.90541
8.20651
8.38270
8.50779
8.60488
8.68430
8.75152
8.80983
log. r 2
1.91852
1.61749
1.44140
1.31646
1.21955
1.14037
1.07342
1.01543
Diff.
4.01311
3.41098
3.05870
2.80867
2.61467
2.45607
2.32190
2.20560
Zech
4
-17
38
-67
105
-152
-206
269
1.91848
1.61732
1.44102
'1.31579
1.21850
1.13885
1.07136
1.01274
log. p 2
8.08152
8.38268
8.55898
8.68421
8.78150
8.86115
8.92864
8.98726
log. r t
1.61749
1.31646
1.14037
1.01543
0.91852
0.83934
0.77239
0.71440
Diff
3.53597
2.93378
2.58139
2.33122
2.13702
1.97819
1.84375
1.72714
Zech
13
-51
-114
-202
315
454
-618
807
1.61736
1.31595
1
13923
1.01341
91537
0.83480
0.76621
0.70633
log. pi
8.38264
8.68405
8.
86077
8.98659
9.08463
9.16520
9.23379
9.29367
log. Z|
3.53004
4.73716
5:
43852
5.93828
6.
32592
6.64264
6.1)1044
7.14240
tog- 7
2.92798
4.13210
4.
83646
5.33622
5.
72386
6.04058
6.30838
6.54034
- tog. I 2
6.7650271
7.36708n
7.
7l926n
7.9691471
8.
16296n
8.3213271
8.45522n
8.5712071
Diff
3.
83704
3.23498
2.
88280
2.6321)2
2.
43910
2.28084
2.14684
2.03086
Zech
-7
-25
-57
-101
-157
-227
-308
402
log.( P + -)
6.
76495n
7.3669371
.?.
7186971
7.96813n
8.
1613971
8.3190571
8.45214n
8.5671871
4 /
3.
23505
2.63307
2.
28131
2.03187
1.
83861
1.68095
1.54786
1.43282
Zech
26
101
227
-401
625
896
1213
1575
log. J" (0)
9.
99974
9.99899
9.
99773
9.99599
9.
99376
9.99104
9.98787
9.98425
log. jpx
8.
38264
8.68405
8.
86077
8.98659
9.
08463
9.16520
9.23379
9.29367
log. J w
8.
38238
8.68304
8.
85850
8.98258
9.
07838
9.15624
9.22166
9.27792
tog. ^
8.
08152
8.38268
8.
55898
8.68421
8.
78150
8.86115
8.92864
8.98726
log. <7 (2)
6.
46390
7.06572
7.
41748
7.66679
7.
85988
8.01739
8.15030
8.26518
log. p s
7.
90541
8.20651
8.
38270
8.50779
8.
60488
8.68430
8.75152
8.80983
log.JV
4.
36931
5.27223
5.
84)018
6.17458
6.
46476
6.70169
6.90182
7.07501
lOO". 7?,
7.
78046
.8.08153
8.
25769
8.38271
8.
47973
8.55905
8.62617
8.68415
teg. J (4)
2.
14977
3.35376
4.
05787
4.55729
4.
94449
5.26074
5.52799
5.75916
THE GENERAL PERTURBATIONS OP THE MINOR PLANETS.
141
Noting that log. (e/ (0) 1) =: log. ( I 2 + ~), K ?', and I h'X, we form
the following: tables :
h'
Log.j[>(C
x 1 (1) 1 (2)
^
J3) ! (4)
A /V Log.^/A/v
1
6.7649n
8.38238
6.4639
4.3693
2.1498
2
7.0658/1
8.38201
6.7647
4.9712
3.0527
3
7.2415n
8.38138
6.9404
5.3231
3.5807
4
7.3661/1
8.38052
. 7.0647
5.5725
3.9551
5
7.4624/1
8.37941
7.1610
5.7658
4.2456
6
7.5409n
8.37809
7.2392
5.9235
4.4826
7
7.6070/1
8.37656
7.3052
6.0567
4.6828
8
7.6641n
8.37483
7.3621
6.1719
4.8562
i' h'= 2 7i' 1 ~ti -J-:
Value of
,-/ (h'-V)
T
h' h> *
h' -6
A'=7 ^'=8
l* : = 2 .'=3
*=* *=5
1 4.97l2n 6.4639/t 6.76495/z
8.38201 6.9404
5.5725 4.2455
....
.... ....
2 3.3537/1 4.6703ra 8.68341n
7.36693n 8.68241
7.3657 6.0668
4.7835
.... ....
3 6.9410
8.85913/1 7.71869/r
8.85764 7.6381
6.4006
5.1598
4 4.9714n
7.36675 8.98344/1
7.96813n 8.98147
7.8413
6.6588 5.4583
5
5.6702/1 7.6393
9.07949n 8.1614/1
9.07706
8.0042 6.8709
6
6.1012n
7.8432 9.15756/1
8.3190n
9.15471 8.1402
7 For h' =0,
6.4176/1 8.0061
9.22320n
8.4521w 9.21993
8 we have
6.6689n
8.1423
9.27965/1 8.5672/1
9 8.38251n
6.8777/1
8.2594 9.32905n
In computing the values of the J functions, the lines headed Zech show that
addition or subtraction tables have been used. For convenience, (J" (0 - 1) is em-
ployed instead of J (0 \ its values being found in the line headed log. ( V + -J.
142 A NEW METHOD OF DETERMINING
From the expression
h' being the multiple of g', and being constant, and i' being variable, we have
(A'+l) 9 (V+2)
' etc.
Now for h' = +1? we have, if we write the angle in place of the coefficient,
((ig - g')} = 1 J SS fa - &) + | </ A , . ( ^ _ 2 ^') + etc.
(2) (3)
, S? ^ + JS7') ~ f ^' SS (^ + 2^0 - etc. ;
and for 7i' 1, we have
(-2) Qg m ^ (-3)
(0) (1)
4- 1 ,7 *, c - os (in 4- ~K'\ 4- -2- ,7" , c - os (in 4- 2 TV'} 4- oto
I i e/ A' sm \ fc i/ I * / I l t/ A' sin V, " i *iJu* / n^ CLO.
Since
(TO) (TO) (wt) (m) ( m) (TO)
the last two expressions give
(0) (1)
((ig g')) = J K , Z (ig E') 2^ SS (ig W) etc.
(2) (3)
e/ A , s T n S (^ + ^) 2e7 v SS (f> + ^') - etc.,
(2) (3)
((ig + g')) = Jy SS (^ - -#') 2e7 v SS (ig - %E') etc.
(0) (1)
+ /v SS (^ + JS? ; ) - 2^ ^ (ig + 2^') etc.
THE GENERAL PERTURBATIONS OF TFIE MINOR PLANETS. 143
And for the particular case of i = 1, we have
(0) (1) (2)
((ff - 00) = ' SS ( g - E') 2J S ( g - 2JS7') + 3J % ( g - 3JB") =F etc.
<2) (3) (4)
- Jv SS($r + E')-2J Sg? (# + 2^') - 3</ A , SS (<7 + 3^') -etc.
(2) (3)
((g +
(0) (1) (2)
SS ( flr + 2E') + 37 V s c & s ( g + 3^') =F etc.
(0) (0)
Instead of J^, , we use (J K > 1), as has been noted.
If we put h f = + 2, we have
(1) (0) (-1)
((ig 2g')) - I J w SS (ig .#') + f ,7 2V ^ (^ 2^') + f e/ w SS (^ - 3-ET') + etc.
(3) (4)
' ^ + 2J0') - etc.
- '
-
In the table giving the values of - J K> > , we have, under h' = 2, which applies to
//'
the equation just given,
(1) (3)
for i' = 1, log. i<7 2A , = 8.38201 log. ( i J^,) = 4.9712^;
(0) (4)
for i - 2, log. ( %J w l) = 7.36693w log. ( f <7 2A ) = 3.3537^ ;
(i)
for tf = 3, log. ( f J w ) = 8.85913/i etc. = etc.
etc., etc. = etc.
(3) (4) ^
We find the values of J J^'? f Sw in the table under 7^' = 2. We see that
i' (h'-i')
these are the forms of the function J k ,^ when h 2, and i' =: 1 and i' = 2.
In the expansion of the coefficient of (ig h'g') indicated above by ((tg &'ff'))t
we have coefficients of angles of the form (ig + i'E'). These can readily be put into
the form ( ig i' E'\ but the form employed is convenient in the transformation.
144
Arranging the functions fi ( a \ ^0? ( a -\ in this form, we have
Log. n a - Log.
9 E'
COS
sin
COS
sin
1
0.0637ft
9.0043
0.5074ft
0.0210
2
8.9912
9.1106ft
7.0000ft
0.1082ft
3
7.6493
8.0418ft
8.8751
9.2437ft
1 + 1
9.6304
9.2856
8.0493
0.1637
1 1
1.72893
9.8663
2,3923
0.5364
1 2
9.9499ft
8.4624
0.7404ft
9.8432
1 3
8.6032
8.8882ft
8.7716
9.9886ft
1 4
7.4771
7.6021ft
8.7243
8.9912*
2+ 1
8.3617
8.6128
9.1173
9.8105
2 - 1
0.3530
9.1584ft
1.0959
0.0978ft
2 2
1.30203
9.7427
2.1675
0.6138
2 3
9.7486ft
7.5079
0.7045ft
9.4642
2 4
8.3010
8.6435ft
8.9138
9.8287ft
2 5
6.6990
7.6532
3-1
8.9395
8.7993
9.2808
9.8944
3-2
0.2191
8.7993ft
1.1221
9.9566n
3 3
0.91750
9.5368
1.9189
0.5440
3 4
9.5132ft
7.6990ft
0.5865ft
8.7924
3 5
8.2041
8.3979ft
8.9445
8.9777ft
4 1
8.2041
8.6128
8.9868
9.8287
4 2
8.9685
8.5051
9.7709
9.6839
4 3
0.0082
8.2869ft
1.0348
9.6410ft
4 4
0.5522
9.2989
1.6565
0.4085
4 5
9.2601ft
7.8451ft
0.4244ft
8.6128ft
4 6
7.9542
7.9093ft
8.8976
9.4298ft
5-3
8.8855
8.2049
9.8564 '
9.4506
5 4
9.7672
7.0000ft
0.8905
0.1732ft
5 5
0.1976
9.0414
1.3848
0.2360
5 6
9.0000n
7.7782ft
<j 0.2347ft
8.8633ft
6 3
7.9440
8.1864
8.9385
9.3876
6 4
8.7404
7.9031
9.8370
9.2122
6 5
9.5119
7.6021
0.7129
7.7782ft
6 6
9.8494
8.7782
1.1066
0.0394
6 7
0.0224ft
8.8451ft
7 6
*
0.5132
8.6990
7 7
0.8222
9.8156
7 8
9.7973ft
8.7924ft
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
145
We will now give examples to illustrate the application of the tables for trans-
forming from eccentric to mean anomaly, in case of the function
For the angle 3g 3g'.
/ (h'-i')
cos
sn
(K = 3)
Log. Product.
Product.
3
1
8.9395
8.7993
6.9404
5.8799
5.7397
+ .00008
+ .00005
3
2
0.2191
8.7993ft
8.68241
8.9015
7.6817ri
+ .07970
- .00303
3
3
0.91750
7.5368
7.71869n
8.6362ft
5.2555ft
- .04327
- .00180
3
4
9.5132ft
7.6990ft
8.98344ft
8.4966
6.6824
+ .03139
+ .00048
3 -
5
8.2041
8.3979ft
7.6393
5.8434
6.0372ft
+ .00007
- .00011
+8.26978
+0.34414
+8.33775
+0.33973
1 1
1.72893
9.6304
9.8663
9.2856
For the angle g og'.
(h' = 0)
8.38251w 0.11144w 8.2488n
8.38251 8.0129^ 7.6681ft
1.29259
- .01030
+0.25653
.01773
- .00466
0.25027
1.04636 0.27266
For the angle g -f- g'.
1 1
1.7289
9.8663
' = - 1)
6.4639ft
8.1928ft 6.3302ft
.016
+0.427
.000
+0.193
+0.411 +0.193
A. P. 8. VOL. XIX. S.
146 A NEW METHOD OF DETERMINING
For the angle og og'.
1 0.0637n 8.3825w 8.4462 ... -f .02794
+ 104.75727
+ 104.78521
For the angles represented by (ig #')> there may be cases when there are sensi-
ble terms arising from g + 12', g + 2.E7', etc. ; if so, we use the column for li' = - - 1,
and apply the proper numbers of this column to the coefficients of the angles named.
Likewise in the case of (ig -f #'), there may be terms arising from the product of the
numbers in the column li' = 1 and the coefficients of the angles g -f E', etc. This
will be made clear by an inspection of the two expressions
(0) (1)
((ig - g'}} - J (ig - - E') - 2<7 A (ig - - 2E') etc.
(2) (3)
ig -W)-- etc.,
(2) (3)
((ig + flO) = - J S (* E') 2,7 A , SS (ig - 2E') -- etc.
(0) (1)
+ J SS (ig + -#') - 2e/v SS (^ + 2J7') etc.
where ((ig #')), ((ig + ^')) represent not the angles but their coefficients.
In retaining the form (ig + i'E') instead of the form ( ig -- i'E') we can per-
form the operations indicated without any change of sign in case of the sine terms.
Making the transformations as indicated above, we obtain the following expres-
sions for the functions - and
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
147
9 9'
COS
sin
COS
sin
//
+ 104.78521
//
//
+182.3777
//
1 1.04636
0.27266
1.6046
-1.9194
2 0.05031
+0.12527
0.5606
+ 1.1949
3
-f 0.02860
+0.05793
+ 0.1067
+0.4943
2 1
0.1274
0.6468
1 1
+ 0.411
0.193
0.0830
-1.4558
1
1.162
+0.107
3.2141
+ 1.1107
1 1
+ 53.583
+0.734
+246.9027
+3.4023
2 1
+ 1.286
0.171
+ 5.3656
-1.4496
3 1
+ 0.014
+0.066
0.3758
+0.8304
2
+ 0.070
0.127
0.085
-1.242
1 2
+ 0.399
+0.053
+ 0.456
+0.848
2 2
+ 20.093
+0.551
4-147.392
+4.049
8 2
+ 1.056
0.086
+ 7.214
1.137
4 2
+ 0.027
+0.033
0.086
+0.537
0-3
+ 0.00815
0.01707
+ 0.0718
0.2352
1 3
+ 0.04342
0.07447
+ 0.0041
0.9231
2 3
0.40733
+0.03392
+ 2.0442
+0.5514
3 3
-f 8.338
+0.340
+ 83.537
+3.432
4 3
+ 0.675
0.036
6.432
0.659
5 3
+ 0.028
+0.010
+ 0.079
+0.449
2 4
+ 0.027
0.043
+ 0.050
0.637
3 4
0.275
+0.023
+ 2.174
+2.592
4 4
+ 3.628
+0.197
+ 46.016
+2.512
5 4
+ 0.397
0.013
4.828
0.323
6 4
+ 0.021
+0.008
+ 0.156
+0.188
3 5
+ 0.020
0.023
+ 0.080
0.074
4 5
+ 0.167
+0.012
+ 1.762
+ 0.241
5 5
+ 1.623
+0.109
- 24.829
+ 1.565
6 5
+ 0.224
0.004
+ 3.306
0.148
4 6
+ 0.012
0.008
+ 0.077
0.250
5 6
+ 0.092
+0.007
4.535
+0.150
6 6
+ 0.731
+0.059
+ 13.312
+ 1.085
148
A NEW METHOD OF DETERMINING
The transformation should be carefully checked by being done in duplicate, or
better by putting the angle ig = 0, in all the divisions of the two functions, having
thus only the angles (0 E'\ ( 2E'), (0 3E'), etc., etc. ; also (0 g'\ (0
20r'), etc. Adding the coefficients in each division of the functions before and after
transformation, and operating on the sums before transformation as on single members
of the sums, the results should agree with the sums of the divisions of the transfor-
mations given above.
The transformations of these functions were checked by being done in duplicate,
but we will give the check in case of another planet. We have for the logarithms of
the sums before transformation, and for the sums after transformation the following :
9
y
COS
sin
g
g'
o -
- 1 i
.85407
1.62090n
0-
- 1
-
- 2 1
.25778
1.51473n
-
2
-
- 3 9
.7024,*
1.26993n
-
-3
-
-4
.TlOln
0.9147n
0-
- 4
-
-5
.6632n
0.3899n
-
-5
-
- 6
.4387n
9.0934
-
- 6
-
- 7
.1222w
9.8069
-
- 7
-
- 8 9
.5965n
9.8865
-
- 8
For the angle
(0-1),
(0
-2),
a
0.041 -\
0.024
+ 1.722
1
.007
0.
873 H
1.578
.042
+
.076
000
0016
4- .037
+ 1
.346
4-71.
462
-41.774
.012
4- 18.104
32
.019
.714
+ 70.
4- 70.
548
573 -
- 40.188
-40.196
4- 19.809
4- 19.811
-32
-32
.318
.319
COS
II
4- 70.548
4- 19.809
4- 0.906
4.540
4.707
3.059
0.623
0.071
sm
- 40.188
- 32.318
- 19.352
9.263
3.313
0.330
4- 0.739
4- 0.615
3.
+ .062 .037
-j- .871 1.574
4- .003 4- .097
. -f- .494 4- .791
.020 .011
.504 - 18.618
4- 0.906 + 19.352
_|_ 0.902 19.355
The numbers in the last line of each case are the sums of the divisions after con-
version when ig is put = 0.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 149
To have close agreement it is necessary that all sensible terms in the expansion of
and ^a 2 Qj be retained. In the expressions for these functions given a large
number of terms and some groups of terms have been omitted as they produce no
terms in the final results of sufficient magnitude to be retained.
In transforming a series it will be convenient to have the values of the (/functions
on a separate slip of paper, so that by folding the slip vertically we can form the pro-
ducts at once without writing the separate factors.
o
The numerical expressions for ^(^) and ^a- 2 HJ being known, we need next to
have those designated by (H) and (Z), which represent the action of the disturbing
body on the Sun.
To find (H) we use two methods to serve as checks. We have first
(//) = i[A 7iyi ' + fc'SA'] cos (g - 00 - J[%/ + ZyA'] sin (g - cf)
<
+ i [ftyiyi' - $1 cos (- g - 00 i [ZV/ - ZVA'] sin (- g - g')
+ i fyoyl cos ( Sf) ~ i p M' sin ( 00
2 ' + ft'SAI cos (g 2g') - 2[M 1 y 2 ' + /'yAl sin (0 20')
' -- ^A'] cos (^ 2^) -h 2[Z^ 2 ' 7yA'] sin ( cj
+ 2 7i W ./ cos ( 2</') 2?yA' sin (
f [^iy 8 ' + ' cos - 3< - *' sin
+ etc.
@
where
(0) (2) (0) (2)
(1) (3) (1) (3).
(2) (4) (2) (4)
3A - ^3A ] ^3 = IK* + /3A Jj
and similar expressions for y/, ^', y. 2 ', 5 2 ', etc. ; noting that y = 3e.
150 A NEW METHOD OF DETERMINING
The other expression for (H ) is
(H) = #h yi ' - - h'K] cos ( E g') + i[Z yi ' - - Z'V] sin ( E g')
+ it V + W] cos (J0 - g) - jpy/ + I'W sin (^ ~ </')
- ehyi cos ( </') + eZ'V sin ( #')
4. 2[fty 2 ' /*'&>'] cos ( E 2g') + 2[fy 2 ' - - Z'^'] sin ( E 2g
cos r - 2 sn
cos ( 2g') + ieZ'^' sin ( 2gr')
+ etc. + etc.
In both expressions for (H) we have
=&cos(n K)
h' = t cos <}> cos <?>' ^ cos (n JKi) zz
JT)
^ sin (n K) - I
where as before
v
ft = . 206264."8 and a = -.
I -f-m a
In the second expression the eccentric angle of the disturbed body appears and we
must transform the expression into one in which both angles are mean anomalies.
With the eccentricity, e, of the disturbed body we compute the J functions just as
we did in case of e' of the disturbing body.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
151
We have in case of Althaea
(0)
Log.G/-!)
Log. J
Log. J
Log. J
Log. J
\
Log. J
(4)
e
f
2e
7.20740w
7.80894n
8.16025n
8.40890n
9.99930
9.99719
9.99368
9.98872
8.60344
8.90341
9.07774
9.20016
6.90632
7.5077
7.8587
8.1068
5.0329
5.9356
6.4630
6.8365
3.0347
4.2384
4.9418
5.4403
(h-i)
From these values we may form a table of -- J^ as was done for the disturbing
body. The values of these quantities can be checked by means of the tables found
in ENGELMANN'S edition of BESSEL'S Werke, Band I, pp. 103-109.
Finding the numerical value of (H) first by the second expression, we get
E
9'
COS
a
sin
1 -
i
+48.154
+0.651
-i -
i
+ 0.188
0.102
o -
i
- 3.884
0.044
i-
2
+ 4.644
+0.062
-i -
2
+ 0.018
0.010
-
2
- 0.374
0.004
1 -
3
+ 0.37800
+0.00510
1 -
3
+ 0.00141
0.00081
3
- 0.03048
0.00036
To transform we change from (hE i'g'} into (i'g 1 TiN). Making the transfor-
mation, writing also the values found from the first expression for the sake of compari-
son, and the value of (I) which will next be determined, we have
152
A NEW METHOD OF DETERMINING
(I)
9 9'
1
COS
II
- 5.826
sin
a
0.066
COS
//
- 5.824
sin
a
0.066
sin cos
// //
+4.799 +2.043
2
- 0.560
0.006
- 0.562
0.006
+0.463 +0.197
3
- 0.04566
0.00057
- 0.04575
+0.038 +0.016
-1 1
+ 0.149
0.103
+ 0.180
0.103
1 1
+48.076
+0.650
+48.079
+ 0.650
1 2
+ 4.631
+0.062
+ 4.605
+0.062
1 -3
+ 0.37740
+0.00502
+ 0.37738
+0.00510
2 1
+ 1.927
+0.026
+ 1.927
+0.030
2 2
+ 0.186
+0.002
+ 0.186
+0.002
2 3
+ 0.011
0.000
+ 0.015
0.000
To find the numerical value of (/) needed in case of the function a 2 ( -), we have
\(LZ/
(I) = &'! sin (- tf) + 6y x cos (- g')
+ 4 65' 2 sin ( 20r') + 4 6y a cos ( 2g f )
+ 9 M's sin ( 3gr') + 9 6y s cos (
+ etc.
+ etc.
where
b = ~ cos d>' sin J cos II', &' = ^ sin I sin II'.
Having the values of ^ Q), ^a 2 (**-} , (^), and (/), we next find those of
r\
all,
j <>
-r-, and a 2 ,
dr dz 7
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 153
from
where
(1) (2) (3)
e 2 - - ^t7 A cos # JJk cos 2g | J" 3A cos 3g etc.
o? n T T' (0 > (2 > Q ) < 3 )
-, sin (/' + IT) = [<7 A , + 7 V ] G! sin^' - - \\J W + ,7 2X ,] Cl sin 2^' etc.
(0) (2) (1) (3)
-f |e'c 2 [J^/ 7 A ' ] c 2 cos (/' J [t^ A / J^' ] c 2 cos 2^ etc.
and C 2 being given by the equations
sin/ , . T-T,
= -- COS d) COS II
a
sin/ . T-J.
2 = smll'.
a
We find
_I 2 ^] - [9.5769400] 2 [8.38238] cos g' 2 [6.46366]cos 2g' etc.
+ 2 [7.99450] cos0 + 2 [6.29667] cos2^-f- etc.
_ si ^" ^ sin ( / ' + n') = [7.18046] + 2 [8.39074] sin #' + 2 [6.77809] sin 2#'
CC Of
2 [8.01941] cos#' 2 [6.40668] cos 20'
A. P. 8. VOL. XIX. T.
154 A NEW METHOD OF DETERMINING
In multiplying two trigonometric series together, called by HANSEN mechanical
multiplication,
let GC A the coefficients of the angles /\.x in case of the sine,
fin those of the angles fix in case of the cosine,
/ those of the angles vy in case of the sine,
and S p those of the angles py in case of the cosine.
The following cases then occur :
a A sin /la? . $ p cos py rz | a^ p sin (%x -\- py) + J a A<^> sin tyx py)
ft. cos fix . y v sin vy \ (3^, sin (fix + vy) i (3^ sin (px vy)
^ cos fix . $ p cos py = J j3^ p cos (fix + p?/) + i ftA cos (^x py)
a A sin %x . y v sin vy = J a A / v cos (/(x + vy) 4~ 2 < K y v cos (/la? vy).
In every term of the second members the factor \ occurs. Hence before multiplying
we resolve the coefficients of one of the factors into two terms, one of which is 2,
Performing the operations indicated, we have the values of n, ar , a 2 that
follow :
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
155
ar\
dr
g
g' cos
sin cos
sin
COS
sin
o
+ 104.78521
"
+ 16.5202
"
+0.2828
"
1
1.04636
.27266
- 2.4398
- .6940
2.6311
+ 6.0177
2
-
.05031
+.12527
.3040
+ .3928
- .059
+ .239
3
-
+ .02860
+.05793
+ .0274
+ .1494
- .017
.017
1
1
.231
.090
.431
.355
.000
.129
- 1
+ 4.662
+.173
- 1.166
+ .481
-1.743
4.157
1
- 1
5.504
+.084
+ 18.839
+ .190
+ .318
+ .068
2
- 1 .641
.201
- 1.652
- .577
-k596
+3.580
3
- 1
+ .014
+.066
.240
+ .288
- .059
+ .232
2
+ .632
.121
+ .497
- .414
- .020
.149
1
2
4.206
.009
- 9.136
+ .200
2.474
6.095
2
2
+ 19.907
+.549
+45.566
+ 1.270
+ .095
.067
3
2
+ 1.056
.086
+ 1.642
- .441
- .922
+2.011
4
2
+ .027
+.033
.115
+ .180
- .064
+ .194
-3
+ .05390
.01764
+ .0718
.0602
.030
+ .017
1
O
.33396
.07957
.4443
- .3306
.045
.166
2
- 3
+ .39221
+.03380
- 2.1788
+ .1339
1.424
3.658
3
-3
+ 8.338
+.340
+27.227
+ 1.087
.064
.134
4
3
.675
.036
+ 1.796
- .269
.519
+ 1.099
5
o
+ .028
+.016
+ .043
+ .157
.042
+ .123
2
4
+ .027
.043
.054
.210
.046
.146
3
- 4
.275
+.023
.880
+ .908
.784
2.078
4
4
+ 3.628
+.197
+ 15.430
+ .882
.038
.106
5
4
.397
.013
+ .883
- .137
.282
+ .586
6
- 4
+ .021
+.008
.013
+ .063
.031
+ .083
3
-5
+ .020
.023
.034
.078
+ .020
.130
4
5
.167
+.012
.281
+ .044
.411
1.150
5
-5
1.623
+.109
+ 8.605
- .543
+ .024
.227
6
- 5
+ .224
.004
+ 1.061
+ .064
.158
+ .311
4
6
+ 0.012
.008
0.075
0.095
5
6
.092
+.007
2.225
+ .026
6
- 6
+ .731
+.059
+ 4.559
+ .386
156 A NEW METHOD OF DETERMINING
Having all we differentiate relative to ft. and obtain a c "*.
dg
We then form the three products, A.a "\ B . ar ( ' -- ), C . ar( - j . To this end
ilg \dr / \dz /
we find A, B, C y from
^ = 3 + 2[2 + e a ]cos(y- g) B = -2 [1+- f] sin (y - g)
+ 2 [f + f] cos (y 2g) -2 [| + f] sin (y-20)
(y Sg) - 2 |e 2 sin (y -
^-2f cos (/
-j- etc. - etc.
+ etc.
The numerical values of Ay B, G in case of Althsea are
A=-3 } )
+ 2 [0.302429] cos (7 g) B = -- 2 [0.001399] sin (7 - 0)
+ 2 [8.604489] cos (7 - - 2g) - 2 [8.604489] sin (y- - 20)
- 2 [9.304508] cos 7 - 2 [8.606234] sin r
+ 2 [7.2076] cos (7 30) - 2 [7.3836] sin (7 - - 30)
O = + 2 [9.697567] sin (7 - 0)
+ 2 [8.30066] sin (7 - - 20)
- 2 [8.77953] sin 7
+ 2 [7.08265] sin (7
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
For the three products we then have
157
A (d&
A - a (j;
o. '-
7
9
9'
sin
COS
sin
COS
sin
COS
a
,,
II
1
-
+ 2.1035
0.5371
+ 1.1341
0.6804
1.3464
3.0038
1
i
-
.012
+ .565
.4021
+ .3723
+ .1287
+ .2411
-1
i
-
.2530
+ .0439
32.9502
+ .0549
.3877
- .4802
1
2
-
.192
+ .299
.0153
- .1657
- .0049
+ .0228
-1
2
-
+ 2.079
- .597
- 1.1310
+ .6821
+ 1.2995
+2.9772
1
3
-
+ .261
+ .457
.1263
- .3720
+ .083
+ .2404
1
. 2
I
+ .462
+ .181
+ .432
.348
.076
+ .243
1
1
I
.266
.015
+ .453
+ .461
1.881
f4.454
1
I
10.992
+ .153
18.335
+ .187
+ .354
- .642
1
I
+ .462
+ .181
.477
- .349
.228
+ .572
1
1
1
- 3.680
- .815
+ .929
- .559
- .815
-1.785
1
1
- 1
+ 1.119
.013
.449
.476
+ 1.906
4.470
1
2
- 1
.342
+ .477
+ .306
+ .276
+ .067
+ .098
l
2
I
11.301
+ .249
+ 18.336
- .188
- .178
.359
1
3
1
+ 2.360
.843
.929
+ .559
+ .785
+ 1.760
1
4
- 1
.033
+ .381
.264
- .276
1
1
2
+ .232
.000
.232
.060
+ .194
1
2
+ 6.837
+ .026
+ 7.300
+ .235
-1.230
+3.029
-1
2
....
....
.001
+ .009
1
1
2
80.684
+2.195
45.412
+ 1.264
+ .178
.371
-1
1
2
.848
+ .002
+ .132
- .406
.139
+ .290
1
2
- 2
+ 1.633
- .735
- 3.470
- .384
.467
1.010
l
2
- 2
+ 16.433
.240
- 7.317
- .235
+ 1.239
3.036
1
3
2
.422
+ .316
'+ .048
+ .168
+ .024
+ .023
l
3
2
79.078
+ 2.254
+ 45.412
-1.264
.053
.273
1
4
2
- 7.937
.500
.213
+ .384
+ .454
+ .981
1
5
2
.408
+ .255
+ .198
- .163
1
-3
+ .5985
.1553
+ .4644
.3261
.0482
+ .157
1
1
-3
- 2.6517
+ .1927
- 1.1042
+ .1641
- .7083
+ 1.8160
l
1
-3
.0661
+ .0161
+ .0541
+ .0737
+ .0123
+ .0180
1
2
- 3
50.140
-[- 1.905
27.2994
+ 1.0854
+ .043
- .174
1
2
- 3
+ .828
.1733
.5308
+ .3287
.062
+ .136
1
3
3
.380
.492
- 2.8964
- .2201
.256
.558
-1
3
3
+ 3.482
.073
1.1112
.1645
+ .707
1.818
1
4
- 3
+ .263
+ .190
.115
+ .147
+ .010
+ .005
1
4
- 3
49.676
+2.079
+27.299
1.083
+ .029
- .206
- 1
5
3
6.395
.264
+ 3.899
+ .217
+ .257
+ .534
158
dQ
dg
c. <t.
7
9 9'
sin
COS
sin
COS
sin
COS
//
//
n
//
//
II
i
1 4
.165
.170
....
.038
+ .115
i
2 4
2.229
+ .187
+ .264
+ .939 .389
+ 1.029
1
2 4
-f .011
+ .017
....
+ .008
- .014
1
3 4
29.032
+ 1.564
15.481
+ .915 + .022
.083
1
3 4
+ .058
- .187
.089
+ .175 - .024
+ .051
i
4 4
- 1.063
- .287
- 1.504
- .098 - .140
.300
1
4 4
+ 1.268
- .024
.022
- .938 + .390
1.033
1
5 4
28.751
+ 1.597
+ 15.479
- .915 + .033
- .129
i
6 4
4.543
- .108
+ 1.506
+ .098
1
2 5
.160
.136
+ .002
+ .088
1
3 5
- 1.654
+ .132
.063
+ .063
.206
+ .570
1
3 5
+ .012
- .014
.001
.003
+ .001
+ .008
i
4 5
16.185
4-1.082
- 8.661
+ .544
- .034
- .038
-1
4 5
+ .015
- .148
.045
+ .076
.035
+ .004
1
5 5
1.061
- .158
- 1.412
.036
.080
- .168
i
5 5
+ .294
- .017
+ .062
.063
+ .206
- .563
1
6 5
16.038
+1.100
+ 8.661
- .544
1
3 6
.121
.063
i
4 6
1.088
+ .086
+ 2.052
+ .038
1
5 6
- 8.707
+ .703
- 4.516
+ .387
-i
7 6
8.818
+ .711
+ 4.516
.387
Next from
dW
ndt
we find the value of . Then we find W and from
ndt cosi
dW
ndt
COS*
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 159
We first form a table giving the integrating factors. From log. n' zz 2.4758576,
log. n = 2.9323542, we have - = 0.34954524.
i i'
*+*'*'
n
Log. (<+#)
/ 1 M
i -/ft'
Loaf, \i-\-i' )
\ ft /
/ 1 \
g '(i+i~)
i i'
ft
-Log-la+i'g)
2 1
2.34954
0.37098ft
9.62902ft
3 3
+ 1.95136
0.29034
9.70966
l i
1.34954
0.13018ft
9.86982ft
4 3
+2.95136
0.47002
9.52998
1
.34954
9.54350ra
0.45650ft
5 3
+3.95136
0.5968
9.4032
i l
+ .65045
9.813217
0.186783
1 4
.398181
9.60008ft
0.39992n
2 1
+ 1.65045
0.21760
9.78240
2 4
+ .601819
9.77946
0.22054
3 1
+2.65045
0.4233
9.5767
3 4
+1.601819
0.20461
9.79539
4 1
+3.65045
0.5624
9.4376
4 4
+2.601819
0.41528
9.58472
1 2
1.69909
0.23021ft
9.76979ft
5 4
+3.601819
0.5565
9.4435
2
.69909
9.8446ft
0.1554ft
6 4
+4.601819
0.6630
9.3370
1 2
+ .30091
9.478423
0.521577
2 5
- .252274
9.40187
0.59813
2 2
+ 1.30091
0.11425
9.88575
3 5
+ 1.252274
0.09770
9.90230
32
+ 2.30091
0.36190
9.63810
4 5
+2.252274
0.35263
9.64737
4 2
+3.30091
0.5186
9.4814
5 5
+3.252274
0.5122
9.4878
5 2
f4.30091
0.6336
9.3664
6 5
+4.252274
0.6286
9.3714
3
1.04864
0.02062ft
9.97938ft
3 6
- .902729
9.9556
0.0444
1 _3! .04863572
8.6869553ft
1.3130447ft
4 6
+ 1.902729
0.2794
9.7206
2 3
+ .95136
9.97835
0.02165
5 6
+2.902729
0.4628
9.5372
In regard to this table we may add that the form of the angles is (ig + i'g')
+ i> &\ g = (i +_ i' n ^\ nt. The differential relative to the time is ($ _j_ i' 7 -\ ndt.
The preceding table is applied by subtracting the logarithms of the column headed
log. (i + i' n -) 9 or by adding the logarithms of the column headed log. ( .^).
\ft/ *~T~*n
We will now give the values of ^, IF, and -^-., remarking that in the inte-
ndt 7 cosi 7
grations the angle 7 is constant ; after the integrations it changes into g.
160
A NEW METHOD OF DETERMINING
dW
w
ccm
7
9 9'
sin
COS
COS
sin
COS
sin
1
//
-f- 3.2376
//
1.2175
//
1.2175n
n
+ 3.2376n
//
3.0038 id
//
1.3464 nt
1
1
.3901
+ .9373
.3901
.9373
- .1287
+ .2411
i
1
+ 32.6972
-f .0988
- 32.6972
+ .0988
+ .3877
.4802
1
2
.2073
- .4647
+ .1036
+ .2323
- .0024
+ .0114
1
2
+ .9480
- .0851
.4740
.0425
- .6497
- 1.4886
j
3
+ .1350
+ .0850
.0450
+ .0283
- .028
+ .0801
1
2 1
+ .894
.167
+ .383
+ .07
.033
.10
1
-1 1
.187
+ .446
.115
.330
0.62
1.60
1
1
- 29.327
- .340
- 83.900
.973
+ 1.013
+ 1.84
1
1
.015
+ .530
.045
- 1.516
.652
- 1.64
1
1 1
+ 4.609
1.374
7.087
- 2.112
+ 1.264
- 2.74
-1
\ I
.670
- .489
1.030
.752
1.370
3.21
1
2 1
.036
+ .753
+ .022
+ .456
- .040
+ .06
1
2 1
+ 7.035
+ .061
4.263
+ .038
+ .107
.21
1
3 1
.019
- .254
+ .007
+ .096
l
3 1
+ 1.431
- .284
.540
.107
- .296
+ .670
1
4 1
.297
+ .105
+ .081
+ .029
1
1 2
.03
.11
1
2
+ 14.145
+ .261
+ 20.207
.373
1.76
- 4.33
1
1 2
126.276
4-3.459
+419.660
+ 11.503
- .59
- 1.23
-1
1 2
.716
- .408
2.380
- 1.356
+ .46
+ .96
1
2 2
1.837
-1.119
1.410
.860
+ .36
.78
1
2 2
+ 9.116
.475
7.008
.365
- .95
- 2.34
1
3 2
.470
+ .484
.204
+ .210
- .01
+ .01
-1
3^-2
- 33.666
-f .990
+ 14.632
.430
+ .02
.12
i
4 2
.017
-f .125
.005
+ .038
I
4 2
8.150
- .116
2.469
.035
- .14
+ .30
l
5 2
.210
-f .092
+ .050
+ .021
1
3
+ 1.0629
.4814
+ 1.0136
+ .4591
.05
.15
1
1 3
1.5475
-f- .3568
- 31.8180
7.335
-14.56
37.33
l
1 3
.0120
+ .0898
.2452
- 1.847
+ .25
.37
1
2 3
77.4394
+2.9904
+ 81.400
+ 3.139
.04
.18
-1
2 3
+ .2972
+ .1554
.3124
+ .1631
+ .06
+ .14
1
3 3
3.' ,64
.7121
+ 1.679
.365
+ .13
.28
1
3 3
-f 2.3706
.2375
1.216
.122
.36
.91
1
4 3
.148
-f .337
.050
+ .115.
.00
.00
1
4 3
22.377
+ .996
+ 7.413
+ .338-.
.01
.07
1
5 3
2.496
.047
+ .627
.012
.06
+ .13
1
1 4
.165
.414
.096
.29
1
2 4
1.965 ',;
+1.126
+ 3.265
+ 1.871
+ .647
+ 1.71
1
3 4
44.513
+ 2.479
+ 27.790
+ 1.548
.014
.05
1
3 4
.031
.012
.019
.007
+ .015
+ .03
1
4 4
2.567
.385
.986
.148
+ .054
.12
1
4 4
+ 1.002
.963
.385
.370
.150
.40
1
5 4
.022
+ .057
+ .006
+ .016
1
5 4
- 13.272
+ .682
+ 3.686
.190
.009
.04
1
6 4
3.037
- .010
+ .660
.002
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
161
sin
sin
// //
It II H "
1
3 5
1.717 + .195
+ I-
374
+ .156
.!<:-; .4(5
1
3 5
+ .011 - .017
009
+ .014
.'>0! .01
1
4 5
- 24.846 -j-1.626
+ 11.
030
+ .722
.015 + .02
1
4 5
.030 - .072
+
013
.032
tie .00
1
5 5
2.473 - .194
+
760
.060
.05
1
5 5
+ .356 - .080
110
.024
.18
1
6 5
.089 + .160
4 '.
021
+ .038
1
6 5
7.377 + .556
+ 1.735
.130
_ 1
7 5
+ 1.413 + .036
270
+ .007
1
4 6
.964 + .124
.507
+ .07
1
5 6
- 13.223 +1.090
-|- 4.555
+ .38
1
5 6
.167 + .023
+ .057
+ .06
1
6 6
.946 - .002
.242
.00
1
6 6
2.098 - .040
.538
.01
1
7 6
3.302 + .324
+ .674
+ .09
i
The part of W independent of y arising
from the
factor,
3, in the value of
-4,
has not yet been given. Its integral, or J 3a \^-j
, is the
following:
X.
5 /
dQ
V
J
3a
'
V
dg
9 9' cos
sin
9
9'
cos
, '
sin
10 + 3.1392
+ .8181
4 -
- 3
2.74
+ .14
2
+ .1509
- .3757
5 -
-3
- .11
_..;
3-0
.0858
- .1738
2 -
- 4
.VI
+ .4:5
I 1
.51
+ .20
3 -
- 4
- 1.54
.13
j i
25.39
- .39
4 -
- 4
-1(1.74
.tu
2 1
+ 2.33
+ .73
5 -
- 4
- 1.6*
r .u.3
3- 1
.04
.22
6 -
- 4
' s
1 2
+41.934
+ .090
3 -
-5
.14
4.16
2 2
91.80
2.53
4
- 5
- .89
.OB
3 2
- 4.13
+ .34
5 -
-5
- 7.49
.50
4
.10
- .V2
6 -
-5
- .96
+ .02
1 - 3
20
4.9099
4 -
- 6
- .07
+.05
2 3
- 2,1
5 -
- 6
.48
.04
3 3
B.4S
6 -
* r,
8.85
.2-7
A. P. S. VOL. XIX. U.
A NEW METHOD OF DETERMINING
Having the values of the coefficients of ( y + ig + i'g'\ both for IK and - ~
^3 OOS t/'
we have next to find those of (vy + ig + i'g'\ and of (Oy + if) + i'g) in the case
c U
of ..
COS i
The expressions for this purpose are
(2) _ _ _ 1,$ _ 1 f
384
__
For Althaia we find
log. >? (2) = 8.60309 log. >7 (8) =z 7.38308 log. >7 (0) = 9.08196/*
We multiply the coefficients of (dh 7 + # + *'^') by >7 (2) , and >y (3) , respectively,
to find those of ( 2/ -f ig + V), (=b 3/ + ^^ + *'y ).
In case of (O/ + /(/ + '^') in the expression for -^-. we add the coefficients of
OOS t
(+ y + ig + i'g') to those of ( y + ig + i'g) and multiply the sum by y (0) .
We will give a few examples to show the formation of W, and A -
2 d r
With these two we give at once also their integrals, which are n$z and v respec-
tively.
W
' d Y
(0 - 0)
u cos sin sin cos
- 1 !-<> 32.6972 +.0988
- 2 2 .0190 +.0017
32.7162
32. 7162 nt
+ 16.3486 +.0494
+ .0190 +.0017
+.0511
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
!r _id^v
163
(1-
// //
- 1 2 - .474 + .042
1 +3.139 + .818
2_1_0 -1.314 - .004
1 1.2175/3* +3.2376/1*
-0)
+ .2.37 + .021
-1.314 + .004
+ 1.351 1.2l75ni + .856 +3.2376n -1.077 ,6087n + .025 1.6188n
+ 4.59 - 1.2175n 2.07 3.2376w 0.54 +.6087w 0.58 1.6188n
(-1
12 1 + .383 + .070
_1 01 - .045 -1.516
- 2 1 1 - .041 - .030
1 1 .513 + .200
-i)
+.191 .035
+ .022 .758
+.041 .030
0.216 -1.246
+ .16 - .92
+.254 .823
+.19 +.61
(i-
- 2 3 1 .022 .004
- 1 2 1 4.263 + .038
01 1 - 25.390 - .390
1 1 - 83.900 - .973
-1)
+ .022 .004
+ 2.131 +.019
-41.950 +.486
-113.574 1.329
174.61 +2.04
39.798 +.501
+ 61.19 +0.77
. j TIT
In the integration we apply the proper factor to each term of W, J -p-, and
obtain the values of n&z, v, except in case of the terms (ig + og').
Let us take the term (g og') or (1 0), and let ^ the integrating factor to
be applied.
Let c, a, d, 5, represent the cos, sin, nt cos, nt sin terms respectively.
164
A NEW METHOD OF DETERMINING
Thus we have
c d
a a
+ 1.351 1.2175n<
a
+.856 +3.2376n;
and hence
+1.351 +3.2376 1.2175w
or, since ^ is unity,
// //
+4.59 -1.2175n<
[id
a
.856
2.07
' I' s
1.2175
3.2376.
-3.2376/iJ
In case of the term (2 0), ^ is g.
In the way indicated we derive the values of nbz, and v. In the case of --
COS I
we have the values at once without another integration as was necessary for nbz and v.
In the value of W given above the arbitrary constants of integration have not
been applied.
We give these constants in the form
cos y
Then in case of A
-
sin y
we have
cos 2y -f
sn
etc.
yc L sin y 5& 2 cos y + >7 (2) ^ sin 2/ - - >? (2) ^2 cos 2/ i etc.
Having TF from the integration of --^-, we form W from the value of W and
converting y into g.
We thus have from the equation
dz
dt
+ (1".351 + Jd) cos g -|- (0".856 + Jc 2 ) sin ^
1".2175^ cos ^ + 3".2376nt sin ^
+ ( ".284 + >7 (2) ^0 cos 2g + (0".589 + >7 (2) & 2 ) sin
- /7 .0488w?5 cos 2g + ".1298^^ sin 2g
i etc. etc.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 165
In the second integration the constants of nbz and v are designated by C and N
respectively, and the complete forms are
C + Ictfit + Jci sin g & 2 cos g + i>? (2) &! sin 2g J>? (2) Jc 2 cos 2g i etc.
_ZV - pi cos g - - p 2 sin g - - J>/ 2) Jc L cos 2</ J>? (2) k 2 sin 2# etc.
In case of the latitude the constants of integration have the form
Z + h sin g + L cos g.
We thus find
n* = C +[1 + Aso 32".7162]/i<
+ [4".59 + A;,] sin g + [ 2".07 AJ cos g
- l".2175nt sin g 3".2376w^ cos g
+ [ 0".ll + J>7 (2) k^\ sin 2^r + [ 0".31 i>7 (2) ^] cos 2g
0''.0244^ sin 2cr Q".QQ4Qnt cos 2</
,/ 7
i etc. rb etc.
v = + 0".0511frf + -2V
4 [ 0".54 -- PJ cos ^ + [ 0".58 p z ] sin ^
+ 0".6087ri^ cos g l".6188w< sin #
+ [0".05 -- i>7' 2) y cos 2g + [ ".24 iV 2) & 2 ] sin 2/
etc. =b etc.
u , = 1 + 0".3616 + 0".3623n*
+ [1".52 + y sin ^ + [ 0".68 + ZJ cos gr
1".3464^ sin g - 3".OQ38n* cos g
+ 0".32 sin 2g - 0"16 cos 2g
0".0539nt sin 2g 0'M204^ cos 2g
-4- etc. =b etc.
COS I
166
A NEW METHOD OF DETERMINING
9'
sin
-
-0
-f- 4.59 +
a
-
0.11 +
- 1
+ 3.10
- 2
3.00
- 3
+ 0.23
- 1
-174.61
-2
+263.97
- 3
+ 25.15
-4
+ 5.71
- 5
+ 1.64
- 6
+ .49
2
+ 185.18
- 4
1.10
-3
+410.16
- 1
5.25
-3
- 37.24
- 2
+ 6.77
-4
+ .90
-3
+ .92
-5
+ .17
-4
+ .34
- 1
+ .16
expressions for i
V u ' j. 1
m v in ta hi
ular form are the following :
*>** * J.J 1 vCv M
COS*
%
u
V
cos i
cos
COS
sin
sin
COS
+& nt
n
+ ^
+Z + 0.36
//
//
32.71 62w*
+ .0511n<
+ .3623n
_ 2.07 & 2
U.o4 -^K l
_ .58 _ i& 2
+ L52 + Z,
.68 + 7
It
"
//
//
3.2376n
+ 0.6087?i
- 1.6188rtf
1.3464n
3.0038n<
//
//
n
If
g| L-fiWfe
+ .05 ^ w kj
9 J. 1 -yi(^) l/^ 1
24 ' 77 ?y tv
+ .32
.16
"
ff
H
//
.0649w
+ .0244n
.0649n
.0539w*
.1204w
- 3.09
+ 2.12
- 1.54
- 4.83
- 2.03
+ 1.92
1.30
.95
+ 1.30
+ .61
- 1.76
+ .12
+ .89
.37
+ .25
+ 2.04
+ 61.19
+ .77
+ 2.69
+ 1.26
- 7.21
156.21
- 4.24
- 1.15
.57
- 0.81
- 18.30
.56
- 1.60
.60
- 0.35
4.68
.29
+ .03
+ .02
- 0.11
1.45
.09
.05
.50
.04
+ 2.10
- 43.27
+ .07
- 6.64
2.70
.71
+ .36
.01
.47
.17
87.44
+ 14.64
+ 3.15
+ 4.43
+ 1.73
+ .87
+ 4.02
+ .62
1.98
+ -9
+ 8.03
+ 16.07
+ 3.78
38.24
14.92
+ .04
7.08
.01
.52
+ .20
.86
1.05
.70
+ 1.31
+ .50
+ .04
.69
+ .05
.24
+ .03
.03
.33'
.04
+ .28
+ .10
+ .01
.38
.00
.92
+ .19
+ .61 ,
1.62
.63
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS.
167
The constants of integration are now to be so determined as to make the pertur-
bations zero for the Epoch. The following equations fulfill this condition :
C
sin g - & 2 cos g -f J>? (2) ^ sin 2g J>y (2) Jc 2 cos
etc.
cos y + c 2 sn g +
cos# - - J& 2 sin ^ - -
sin g - - po cos g +
cos 2# + >? (2) & 2 sin 2# -f etc.
cos 2y - - ^ (2) & 2 sin 2g etc.
=
sn
cos
etc. + (^) =
+ Zi sin ^ + Za cos g + >y (2) ^ sin 2g -{- >? (2) Zs cos 2g + etc. + (- A =
VCOS 7-y Q
- i 2 sin g
cos 2 ^ I sin 2r -- etc.
To find ^ and k 2 we have
[cos g e + V 2) cos 2# + >/ (3) cos 3g + etc.] + k 2 [sin ^ + ^(^ sin 2g + etc.]
- 3Z -f 6 ( v )o + 4- (^)o =
[sin ^ + 2>?( 2 ) sin 2g + 3>? (3) sin 3^ + etc.] Jc 2 [cos g + 2^ cos 2^ + etc.]
where
= 32".7162,
being found from
li.
ndt
6W.
SJS
\
We have also
/ "V
VQ ~~ 662.
i
The symbols (n^) , (v) , etc., represent the values of n$z, v, etc., at the Epoch.
168 A NEW METHOD OF DETERMINING
To find the values of the angles (ig + i'g) at the Epoch we have
y = 332 48' 53".2
g' - 63 5 48 .6
The long period inequality, 5 Saturn 2 Jupiter, is included in -the value of y'.
From these values of g and g' we find the various arguments of the perturbations.
Then forming the sine and cosine for each argument, we multiply the sine and cosine
coefficients of the perturbations by their appropriate sines and cosines.
In forming ~ (nbz\ etc., we can make use of the integrating factors, multiply-
itdv
ing by the numbers in the column (i -f *"' - ). Having their differential coefficients we
proceed as in the case of (?2&z), etc.
We thus find
) = + 401".7, Wo = + 180".6, (-JL) = 22". G
t V _,S = 391".6, -4r Wo = + 70".5,
nd v x wrf^ v " n^
And from these we have
k = + 412".8, Jc. 2 = 82" .9, 7c - --26".21, 1 = 0".0
I, = - 45".2, 1 2 = + 0".4, JV= + 28".3,
(7 = 332 44' 12".6.
The new mean motion is found from (1 32".7162 26". 21) nt, which gives
n 855".5196. "With this value of n we find the only change is in the coefficients
of the argument (1 3), having + 405".29 instead of 410".16, and 86".30 instead
of 87".44.
The constant G now has the value
C ~ 332 44' 16".3.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 1G9
Introducing the values of the constants of integration into the expressions for
nz 9 r, and , we have
nz - 332 44' 16".3 + 855".5196 1
+ 417" A sing + 80".8 cos#
3".2376/cos#
+ 16".4 sin 20 + 3".0 cos 2^
0".0244 nt sin ( 2g - 0".0649 nt cos 2g
db etc. etc.
= + 28".3 +
- 206". 9 cos cj + 40".9sin#
8' .2 cos 2g +
cosi
etc. i etc.
0".4 -i 0".3623n<
1".3464 TI^ sin ^ 3".0038 nt cos
I".5sin20 0".2cos2<7
0".0539 nt sin 2r/ - 0".l 204 nt cos 2g
From the expressions of the perturbations that have been given, and the elements
used in computing the perturbations, except that we use C in place of g and the new
value of the mean motion, we will compute a position of the body for the date 1894,
Sept. 19, 10 h 48 m 52 s , for which we have an observed position. From a provisional
ephemeris we have an approximate value of the distance; its logarithm is 0.14878.
A. P. s. YOL. XTX. v.
170 A NEW METHOD OF DETERMINING
Reducing the above date to Berlin Mean Time, and applying the aberration
time, we have, for the observed date, 1894, Sept. 19, 72800,
g - 339 19' 38".l, g' - 65 24'.1.
Forming the arguments of the perturbations with these, we find
n&z - + 4' 43".2, v - + 3".6, tt . = 2".8.
cost
To convert v into radius as unity and in parts of the logarithm of the radius
vector we multiply by the modulus whose logarithm is 9.G3778, and divide by 206264".8.
Thus we have from v + 3".6, the correction, + .000008, to be applied to the loga-
rithm of the radius vector.
In case of z= 2 '.8, we have
cost
i-- _7".19.
Converting into radius as unity, we have z' = - - .000035. The coordinate z' is per-
pendicular to the plane of the orbit. As we will use coordinates referred to the
equator we have, to find the changes in a?, y, z, due to a variation of 0', which we have
designated by &?', the following expressions :
fix = (sin i sin &) &?'
by ( sin i cos Q, cos e cos i sin e) $z'
bz =. ( sin i cos & sin e -\- cos i cos e) &&'
where e is the obliquity of the ecliptic.
For 1894 we find
fa = ( .0404) ^', 5y = ( .3123) &', & = (+.9491) &'
And for the date we have
&B = + .000001 % = + .000011 * = .000033
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 171
With i - 5 44' 4".6, Q zz 203 51' 51".5, e = 23 27' 10".8,
we compute the auxiliary constants for the equator from the formulae
cotg A = tg cos , ty E = ^ * ,
cos g3
COto- B C Si COS ( -I- e)
fy Q COS " COS
COto- (7 ____ cos * sin (E 9 + e)
'
cos
sin a = cos ^, sin I = sin ^ cos ^ sin (7 =
'
sn
sin B sin (7
The values of sin a, sin 6, sin c are always positive, and the angle E is always
less than 180.
As a check we have
, . __ sin b sin c sin ((7 B)
y ^ ~"
sin a cos A
We find
A = 293 45' 29".3, B = 202 59' 46".9, C - 210 45' 55".0
log sin a 9.999645, log sin I 9.977735, log sin c = 9.498012
Applying n$z + 4' 43". 2 to the value of g, we have
Tie = 339 24' 21".5
By means of g or nz = E -- e sin E we find
E = 337 39' 23".4
Then from
I sin J v \/a(l + e) sin J E
* cos $ v = ttl e cos E
172 A NEW METHOD OF DETERMINING
we find
v - 335 50' 12".2, log i\ = 0.378246
where v is the true anomaly.
Calling u the argument of the latitude we have
u v + TI-- Q = 143 52' 41".8.
Hence
A + u = 77 38' ll".l, 23+u = 346 52 28".7, C+u = 354 38' 36".8.
And from
x =. r sin a sin (A + u)
y = r sin & sin (13 -j- u)
z =. r sin c sin ( C + w),
where
log r log ?-! + 5 log r = log r, + .000008,
we have
x + 2.331894, y = .515433, *==.-- .070208.
The equatorial coordinates of the Sun for the date of the observation are
X - 1.002563 Y - + .045198 Z - + .019611.
Applying the corrections &c, ^//, ^, we have
x -t-fa+X= + 1.329332, y + fy + F= .470224, + & + 2T = .050630.
THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 173
Then from
4 y + fy + y t \ z+ dz + Z . z -f 3z + Z
tfl a =r J t g d rz - sin a = - COS a,
x -f- 8x -f- J' i/ -f to -f y a; -f- dx + ^
sin 8
we have, giving also the observed place for the purpose of comparison,
*
tt c = 340 31' II" A & c = - - 2 3' 23". I log A rz 0.149514.
a - 340 33 49.1 & = - - 2 2 25.4
where the subscript c designates the computed, and the subscript o the observed place.
Both observed and computed places are already referred to the mean equinox of
1894.0. If the observed position were the apparent place we should have to reduce
the computed also to apparent place by means of the formula)
Aa = / + y sin (G + a) ty &
AS = y cos (G + a),
the quantities/", y, and G being taken from the ephemeris for the year and date.
If the observed position has not been corrected for parallax we refer it to the cen-
tre of the Earth by means of the formulae
A _ TT p cos <f>' sin (a 0)
_i (X ' .
J cos 3
tg a>'
tciy =:
COS (a 0)
_ TT p *sin <p r sin (y S)
A v _ TT p
sin
where
a is the right ascension, $ the declination, A the distance of the planet from the
Earth, $' the geocentric latitude of the place of observation, the siderial time of
174 A NEW METHOD OP DETERMINING^ THE GENERAL PERTURBATIONS, ETC.
observation, p the radius of the Earth, and n the equatorial horizontal parallax of the
Sun.
For the difference between computed and observed place we have
C r= 2 37". 7 in right ascension, and C =. - - 57". 7 in declination.
By the method just given we have found the positions of the planet for several
dates and have compared with the observed places. The comparison shows outstand-
ing differences too large to be accounted for by the effects of the perturbations yet to
be determined, which are the perturbations of the second order, with respect to the
mass, produced by Jupiter, and the perturbations produced by the other planets that
have a sensible influence. We have therefore corrected the elements that have been
used in the computations thus far made, by means of differential equations formed for
this purpose, employing as the absolute terms in these equations the differences be-
tween computation and observation for the several dates. A solution of the equations
has given corrections to the elements that produce quite large effects on the computed
place. Thus recomputing the position of the planet for the date given above with the
corrected elements we find
a c = 340 33' 44".5 , 5 C = 2 2' 15".6.
And since
a = 340 33' 40". I , 8 = 2 2' 25".4
we have, for the difference between computed and observed place,
C O = 4" .6 in right ascension, and C O + 9".8 in declination.
STATISTICS LIBRARY
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!
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-~, -- v- RwHra