GL:
UC-NRLF
/ On the deter-N
mination of
elliptic or-
bite from 3
complete ob-
servations.
N.A.S. 4. 8th
Mem.
Students'
. Observatori
NATIONAL ACADEMY OF SCIKNCES.
V 01,. TV.
EIGHTH MEMOIR.
ON THE DETERMINATION -OF ELLIPTIC ORBITS FROM THREE
COMPLETE OBSERVATIONS.
NATIONAL ACADEMY OF SCIENCES.
VOL. IV.
EIGHTH MEMOIR.
ON THE DETERMINATION OF ELLIPTIC ORBITS FROM THREE
COMPLETE OBSERVATIONS.
79
ON THE DETERMINATION OF ELLIPTIC ORBITS FROM THREE COMPLETE
OBSERVATIONS.
By J. WILLARD GIBBS.
The determination of an orbit from three complete observations by the solution of the equa-
tions which represent elliptic motion presents so great difficulties in the general case, that in the
first solution of the problem we must generally limit ourselves to the case in which the intervals
between the observations are not very long. In this case we substitute some comparatively simple
relations between the unknown quantities of the problem, which have an approximate validity for
short intervals, for the less manageable relations which rigorously subsist between these quantities.
A comparison of the approximate solution thus obtained with the exact laws of elliptic motion
will always afford the means of a closer approximation, and by a repetition of this process we may
arrive at any required degree of accuracy.
It is therefore a problem not without interest it is, in fact, the natural point of departure in
the study of the determination of orbits to express in a manner combining as far as possible sim-
plicity and accuracy the relations between three positions in an orbit separated by small or mod-
erate intervals. The problem is not entirely determinate, for we may lay the greater stress upon
simplicity or upon accuracy ; we may seek the most simple relations which are sufficiently accurate
to give us any approximation to an orbit, or we may seek the most exact expression of the real
relations, which shall not be too complex to be serviceable.
DERIVATION OF THE FUNDAMENTAL EQUATION.
The following very simple considerations afford a vector equation, not very complex and quite
amenable to analytical transformation, which expresses the relations between three positions in
an orbit separated by small or moderate intervals, with an accuracy far exceeding that of the
approximate relations generally used in the determination of orbits.
If we adopt such a unit of time that the acceleration due to the sun's action is nnity at a
unit's distance, and denote L he vectors* drawn from the SUB to the body in its three positions by
* Vectors, or directed quantities, will be represented iu this paper by German capitals. The following notations will
be used in connection with them.
The sign = denotes identity iu direction as well as length.
The sign -f- denotes geometrical addition, or what is called composition iu mechanics.
The sign denotes reversal of direction, or composition after reversal.
The notation $l-$$ denotes the product of the lengths of the vectors and the cosine of the angle which they
include. It will be called the direct product of $[ and 3. If x, y, z are the rectangular components of $(, and x 1 , y',
i' those of S3,
Sl-Vl may be written W and called the square of fl.
The notation SlXg will be used to denote a vector of which the length is the product of the lengths of Jl and 93
and the sine of the angle which they include. Its direction is perpendicular to Jl and 2J, and on that side on which
H. Mis. 597 - 6 81
M789809
82 MEMOlltS OF THE NATIONAL ACADEMY OF SCIENCES.
tt], Wf, 9t], and the lengths of these vectors (the hHiix-i'ntric distances) by ri, r,, r 3 , the acceler-
OJ kU 01
ations corresponding to the three positions will be represented by j . _ Z , _ -. Now the
motion between the positions considered may be expressed with a high degree of accuracy by an
equation of the form
having five vector constants. The actual motion rigorously satisfies six conditions, viz., if we write
TJ for the interval of time between the first and second positions, and TI for that between the second
and third, and set t=0 for the second position,
for t= r,,
a,_ s u d8t JR,.
dt* ~ -j?'
for f=0,
aj 01 w H y\2 .
a? = ~T?
for tssTi,
m _ g\ nl lK 3
// J " "??
We may therefore write with a high degree of approximation :
Ki=8
,=
rotation from ft to SJ appears connter-clock-wise. It will be called the ikew product of Jl and SJ. If the rectan-
gular components of ft and <B are x, y, t, and x 1 , y 1 , t 1 , those of fix 8) will be
yz'zy', zx'xz', ryyr'.
The notation (JISC) denotes the volume of the parallelepiped of which three edges are obtained by laying off
the vectors JI, 8, and G from any same point, which volume is to be taken positively or negatively, according as
the vector G falls on the side of the plane containing ?l and ^, on which :i rotation from *l to 5) appears counter-
clock-wise, or on the other side. If the rectangular components of ft, *l, and G are x, y, z ; x 1 , y 7 , z 1 ; and x", y", t",
(*<&) =
x y z
x 1 y' .-'
x" y" t"
It follows, from the above definitions, that for any vectors ft, 8, and G
=, *x=-x, ($G)=(^G)=(G*^)=
and
(8G)=-( x G) = -(G X )=<?(* X *);
also that ft-8, <lx8, re distrihutive functions of ft and 3), "'! (fl'HG) a dintributive function of ft, 8, and G,
for example, that if ft=
and to for $ and G.
The notation (ft\G) ' identical with that of Lagrange in tho M/caniqtie A*alyti<ptt, exci-|it that tin-re its use is
limited to unit vectors. The signification of ft X 4^ ' dourly r.-lat.-d to, hut n,.t id.-ntu-al with, that .if Ih.- nota-
tion [r,r.] commonly used to denote the double area of a tuan^l. il- t. i n.iu. .1 l>v two |HiHitiuini in un orbit.
MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 83
From these six equations the five constants SI, S, (i, 25, may be eliminated, leaving a single
equation of the form
where
f. _
A,=-
This we shall call our fundamental equation. In order to discuss its geometrical signification,
let us set
so that the equation will read
0. (3)
This expresses that the vector 2 S 2 is the diagonal of a parallelogram of which niSRi and n 3 9R 3 are
sides. If we multiply by S 3 and by 9l| , in sfcetc multiplication, we get
=0, (4)
whence
ii\ . . vi ,n . . ,n ill . . iii
(5)
Our equation may therefore be regarded as signifying that the three vectors Sfti, SR 2 , $3 lie in one
plane, and that the three triangles determined each by a pair of these vectors, and usually de-
noted by [*V 3 ], [rir 3 ], [n^], are proportional to
Since this vector equation is equivalent to three ordinary equations, it is evidently sufficient
to determine the three positions of the body in connection with the conditions that these positions
must lie upon the lines of sight of three observations. To give analytical expression to these
conditions, we may write d, & 2 , (S 3 for the vectors drawn from the sun to the three positions of
the earth (or, more exactly, of the observatories where the observations have been made), gi , g 2 , g,
for unit vectors drawn in the directions of the body, as observed, and pi, pi, p 3 for the three
distances of the body from the places of observation. We have then
(6)
By substitution of these values our fundamental equation becomes
where PI, PI, P3,ri,r t ,r 3 (the geocentric and heliocentric distances) are the only unknown quanti-
ties. From equations (6) we also get, by squaring both members in each,
(8)
by which the values of r t , r 2 , r 3 may be derived from those of pi, p z , p-j, or vice versa. Equations
(7) ami (8), which are equivalent to six ordinary equations, are sufficient to determine the six
84 MIIMOIKS OF TI1K NATIONAL AUADK.MY OK SCIENCES.
quantities r lt r f , r 5 , pi, /a,, /3j; or, if we suppose the \ dues of n, r a , r 3 in terms of pi, p,, p^ to be
substituted in equation (7), we have a single vector equation, from which we may determine the
three geocentric distances pi, pi, p 3 .
It remains to be shown, fust, how the numerical solution of the equation may be performed,
and, secondly, how such an approximate solution of the actual problem may furnish the basis of
a closer approximation.
SOLUTION OP THB FUNDAMENTAL EQUATION.
The relations with which we have to do will be rendered a little more simple if instead of each
geocentric distance we introduce the distance of the body from the foot of the perpendicular from
the Min upon the line of sight. If we set
equations (8) become
ri^qf+tf, rf=qf+pf, r 3 *=q 3 *+p 3 *. (11)
Let us also set, for brevity,
=-(i-^)(e 2 +/o s &), *-*(i+)(fc+/>*). (12)
Then S,, 2,, g, may be regarded as functions respectively of PI, p, pa, therefore of ji, 3 , q 3 ,
and if we set
g'=g-', "=?% &"=f\ (13)
aqi dq, dq 3 '
and
s=e,+ 2 -i-S3, (U)
we shall have
<i2='dq l + "eiq 1 +"'dq3. (15)
To determine the value of g', we get by differentiation
But by (11)
*"!_ ?1 . _.
^-r,"
Therefore
(IS)
Now if any value* of ?,, q,, q t (either assumed orolitain.-d hy a previous a|)pn>xiinatioii) ivi-
certain reM.lual : (.vl.irh woul.l I,,, /.,-n, if tin- raldefl of ,,,, ,,., , h sati.slie.l the fuu.lameutal
equation), and we wish to tin.l the eornvtions J, h , J,, : . J ft( w |,i,.|, must b,- a.l.l.-d to ,,, . ,/.. ,,
"^rjO^+r, )
t "3
MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 85
to reduce the residual to zero, we may apply equation (15) to these finite differences, and will have
approximately, when these differences are not very large,
^. (19)
This gives*
(23''"') (e<s'"') (g'e")
J ' (S'S'^} ^=- ( @/"g'") J fc=- ( Wl^y
From the corrected values of q\,qz, q 3 we may calculate a new residual 3, and from that determine
another correction for each of the quantities q\, q%, q 3 .
It will sometimes be worth while to use formulas a little less simple for the sake of a more
rapid approximation. Instead of equation (19) we may write, with a higher degree of accuracy,
, (21)
where
(22)
S'" =
It is evident that 1" is generally many times greater than S' or 2'", the factor .B 2 , in the case of
equal intervals, being exactly ten times as great as A^B, or A 3 B 3 . This shows, in the first place,
that the accurate determination of Aq^ is of the most importance for the subsequent approxi-
mations. It also shows that we may attain nearly the same accuracy in writing
2 2 2 (23)
We may, however, often do a little better than this without using a more complicated equation.
For J'+I'" may be estimated very roughly as equal to \\". Whenever, therefore, Aq v and Aq,
are about as large as Jg 2 , as is often the case, it may be a little better to use the coefficient &-
instead of J in the last term.
For Jg 2 , then, we have the equation
. (24)
is easily computed from the formula
which may be derived from equations (18) and (22).
The quadratic equation (24) gives two values of .the correction to be applied to the position of
the body. When they are not too large, they will belong to two different solutions of the problem,
generally to the two least removed from the values assumed. But a very large value of Aq% muse
not be regarded as affording any trustworthy indication of a solution of the problem. In the
majority of cases, we only care for one of the roots of the equation, which is distinguished by
being very small, and which will be most easily calculated by a small correction to the value which
we get by neglecting the quadratic terni.t
* These equations are obtained by taking the direct products of both members of the preceding equation with
" X '", '" X 2', and 2' x 2", respectively. See foot-note on page 81.
tin the case of Swift's comet (V, 1880), the writer found by the quadratic equation .247 and .116 for cor-
rections of the assumed geocentric distance .250. The first of these numbers gives au approximation to the position '
of the earth; the second to that of the comet, viz., the geocentric distance .134 instead of the true value .1333. The
coefficient -f a was used in the quadratic equation; with the coefficient \ the approximations would not be quite so
good. The value of the correction obtained by neglecting the quadratic term was .070, which indicates that the
approximations (in this very critical ease) would l>e quite tedious without the use of the quadratic term.
H; MKMOIKS oi- THE NATIONAL ACADEMY OF SCIENCES.
When a comet is somewhat near the earth we may make use of the fact that the earth's orbit
is one solution of the problem, i. e., that f>t is one value of Jg 2 , to save the trifling labor of com-
puting the value of (2"2'"2')- For it i* i-vident from the theory of equations that if p 1 and z
are the two roots,
_(2'2 [ (22'"2')
Pt ~ z ~l(l"2" / 2 l ) ~f(3"2'""2'j
Eliminating (1"2'"'), we have
(p,-
whence
Now - m] is the va l e f ^9> wnicl1 we obtain if we neglect the quadratic term in equa-
tion (24). If we call this value [dq,], we have for the more exact value*
Jft -._M!jL
9 H.L*] < 26 >
p>
The quantities dq\ and Jg 3 might be calculated by the equations
2
But a little examination will show that the coefficients of Aqf in these equations will not generally
have very different values from the coefficient of the same quantity in equation (24). We may
therefore write with sufficient accuracy
J$i=[Jgi]+J,-MJ Aq 3 =[Aq 3 ] + Jq t -[Aq 2 ], (28)
where [dq\], [^<7z], l-^ft] denote the values obtained from equations (20).
In making successive corrections of the distances q\, <ft, q 3 it will not be necessary to recalcu-
late the values of 2', 2", 2'", when these have been calculated from fairly good values of q\,q2,q 3 .
But when, as is generally the case, the first assumption is only a rude guess, the values of 2', 2",
2'" should be recalculated after one or two corrections of q } , q 2 , q 3 . To get the best results when
we do not recalculate 2', 2", 2'", we may proceed as follows: Let 2', 2", 2'" denote the values
which have been calculated; Dq t , Dqi, Dq 3 , respectively, the sum of the corrections of each of
the quautities q } , q t , <fr, which have been made since the calculation of 2', 2", 2'" ; 2 the residual
after all the corrections of q t , q t , q 3 , which have been made; and Jft, J<fe, dq 3 the remaining
corrections which we are seeking. We have, .then, very nearly
The same considerations which we applied to equation (l'l) enable us to simplify this equation
also, and to write with a fair degree of accuracy
(30)
(31)
where
In the eaae mentioned in the preceding font-in<ti-. fnm [^g a ]= .079 and pt.'ift, we get Jg,= .ll.V.,
in ennilily the ume value M that obtained lir ralrulntinu tin- quadratic t<-nn.
MEMOIKS OF THE NATIONAL ACADEMY OF SCIENCES. 87
CORRECTION OF THE FUNDAMENTAL EQUATION.
When we have thus determined, by the numerical solution of our fundamental equation,
approximate values of the three positions of the body, it will always be possible to apply a small
numerical correction to the equation, so as to make it agree exactly with the laws of elliptic
motion in a fictitious case differing but little from the actual. After such a correction, the equa-
tion will evidently apply to the actual case with a much higher degree of approximation.
There is room for great diversity in the application of this principle. The method which
appears to the writer the most simple and direct is the following, in which the correction of the
intervals for aberration is combined with the correction required by the approximate nature of
the equation.*
The solution of the fundamental equation gives us three points, which must necessarily lie in
one plane with the sun, and in the lines of sight of the several observations. Through these points
we may pass an ellipse, and calculate the intervals of time required by the exact laws of elliptic
motion for the passage of the body between them. If these calculated intervals should be iden-
tical with the given intervals, corrected for aberration, we would evidently have the true solution
of the problem. But suppose, to fix our ideas, that the calculated intervals are a little too long.
It is evident that if we repeat our calculations, using in our fundamental equation intervals short-
ened in the same ratio as the calculated intervals have come out too long, the intervals calculated
from the second solution of the fundamental equation must agree almost exactly with the desired
values. If necessary, this process may be repeated, and thus any required degree of accuracy
may be obtained, whenever the solution of the uncorrected equation gives an approximation to
the true positions. For this it is necessary that the intervals should not be too great. It appears,
however, from the results of the example of Ceres, given hereafter, in which the heliocentric mo-
tion exceeds 62, but the calculated values of the intervals of time differ from the given values by
little more than one part in two thousand, that we have here not approached the limit of the
application of our formula.
In the usual terminology of the subject, the fundamental equation with intervals uncorrected
for aberration represents the first hypothesis, the same equation with the intervals affected by cer-
tain numerical coefficients (differing little from unity) represents the second hypothesis, the third
hypothesis, should such be necessary, is represented by a similar equation, with corrected coeffi-
cients, etc.
In the process indicated there are certain economies of labor which should not be left un-
mentioned, and certain precautions to be observed in order that the neglected figures in our com-
putations may not unduly influence the result.
It is evident, in the first place, that for the correction of our fundamental equation we need
not trouble ourselves with the position of the orbit in the solar system. The intervals of time,
which determine this correction, depend only on the three heliocentric distances r 1? r 2 , r 3 and the
two heliocentric angles, which will be represented by VtVi and v 3 v 2 , if we write Vi, v 2 , v 3 for
the true anomalies. These angles (v 2 Vi and v 3 v t ) may be determined from r t , r 2 , r 3 and n } , 2 , 713,
and therefore from r^ r 2 , r 3 and the given intervals. For our fundamental equation, which may be
written
0, (33)
indicates that we may form a triangle in which the lengths of the sides shall be w,ri, w 2 r 2 , and n 3 r 3 ,
(let us say for brevity, s b s 2 , s } ,) and the directions of the sides parallel with the three heliocentric
directions of the body. The angles opposite s } and s 3 will be respectively v 3 v t and 2 v\. We
have therefore, by a well-known formula,
tan
2
(34)
'2 '~V"(si
* 3
* When an approximate orbit is known in advance, we may correct the fundamental equation at once. The
formula) will be given in the Summary, $ XII.
SH MK.M01RS OF T1IK NATIONAL A( AKK.MV OK SCIKNCES.
As soon, therefore, as the solution of our fundamental equation has given a sutlicient approx-
imation to tlie values of r,, '.., c, (say five- or six-figure values, if our final result is to he as exact
as seven-figure logarithms ran make it), we calculate n,, ,, -., with seven figure logarithms by
equations (2). and the heliocentric angles by equations (34).
The semi-parameter corresponding to these values of the heliocentric distances and angles is
given by the equation
The expression i nj+Ha, which occurs iu the value of the semi-parameter, and the expres-
sion nir, ;t,r,+ 3 r 3 , or *, j+ 3 , which occurs both in the value of the semi-parameter and in the
formula) for determining the heliocentric angles, represent small quantities of the second order (if
we call the heliocentric angles small quantities of the first order), and cannot be very accurately
determined from approximate numerical values of their separate terms. The first of these quanti-
ties may, however, be determined accurately by the formula
^*2 f*3
With respect to the quantity i *j+ 3 , a little consideration will show that if we are careful to
use the same value wherever the expression occurs, both iu the formula? for the heliocentric angles
and for the semi-parameter, the inaccuracy of the determination of this value from the cause men-
tioned will be of no consequence in the process of correcting the fundamental equation. For,
although the logarithm of *i * 2 +* 3 as calculated by seven figure logarithms from r t , r 2 , r 3 may be
accurate only to four or five figures, we may regard it as absolutely correct if we make a very
small change in the value of one of the heliocentric distances (say r 2 ). We need not trouble our-
selves farther about this change, for it will be of a magnitude which we neglect in computations
with seven-figure tables. That the heliocentric angles thus determined may not agree as closely
as they might with the positions on the lines of sight determined by the first solution of the
fundamental equation is of no especial consequence in the correction of the fundamental equation.
which only requires the exact fulfillment of two conditions, viz., that our values of the heliocen-
tric distances and angles shall have the relations required by the fundamental equation to the
given intervals of time, aud that they shall have the relations required by the exact laws of
elliptic motion to the calculated intervals of time. The third condition, tlfat Sone'of these values
shall differ too widely from the actual values, is of a looser character.
After the determination of the heliocentric angles aud the semi-parameter, the eccentricity ami
the true anomalies of the three positions may next be determined, aud from these the intervals of
time. These processes require no especial notice. The appropriate formula will be given in the
Summary of Formula-.
DETERMINATION OF THE ORBIT FROM THE THREE POSITIONS AND THE INTERVALS OF TIME.
The values of the semi-parameter and the heliocentric angles as given in the preceding para-
graphs depend upon the quantity , * 2 +* 3 , the numerical determination of which from ,, 2 , and * 3
is critical to the second degree when the heliocentric angles are small. This was of no conse-
quence iu the process which we have called the cm-rcctiou of the fundamental fi/Hation. But for
the actual determination of the orbit from the positions given by the corrected equation or by
the uncorrecti-d equation, when we judge that to be sufficient a more accurate determination of
this quantity will generally be necessary. This may be obtained in different ways, of which the
following is pei haps the most simple. Let us set
5 4 = 5 :i -5,. (37)
and 4 for the length of the vector 2, obtained by taking the square root of the sum of the squares
of the components of the vector. It is evident that x t in the longer aud 4 the shorter diagoual of
MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 89
a parallelogram of which the sides are , and 3 . The area of the triangle having the sides 2 , * 3
is therefore equal to that of the triangle having the sides *,, 3 , 4 , each being one-half of the
parallelogram. This gives
4-3), (38)
and
S.-S +8 =
The numerical determination of this value of *! 2 +s 3 is critical only to the first degree.
The eccentricity ami the true anomalies may be determined in the same way as in the correc-
tion of the formula. The position of the orbit in space may be derived from the following consid-
erations. The vector S 2 is directed from the sun toward the second position of the body ; the
vector (4 from the first to the third position. If we set
the vector < 5 will be in the plane of the orbit, perpendicular to < 2 and on the side toward
which anomalies increase. If we write 5 for the length of 5 ,
-?? and ?
2 5
will be unit vectors. Let 3 and 3' be unit vectors determining the position of the orbit, 3 being
drawn from the sun toward the perihelion, and 3' at right angles to 3, in the plane of the orbit,
and on the side toward which anomalies increase. Then
3= -cos v 2 ??-sin t> 2 ^ (41)
*2 *5
3' = sin r 2 2 +cos e^? (42)
* 2 * 5
The time of perihelion passage (T) may be determined from any one of the observations b y
the equation
-.(tT)=E-e &in E, (43)
a*
the eccentric anomaly E being calculated from the true anomaly v. The interval t T in this
equation is to be measured in days. A better value of T may be found by averaging the three
values given by the separate observations, with such weights as the circumstances may suggest.
But any considerable differences in the three values of T would indicate the necessity of a second
correction of the formula, and furnish the basis for it.
For the calculation of an ephemeris we have
R = -ae3+cos E 3+sin E by (44)
in connection with the preceding equation.
Sometimes it may be worth while to make the calculations for the correction of the formula
in the slightly longer form indicated for the determination of the orbit. This will be the case
when we wish simultaneously to correct the formula for its theoretical imperfection, and to correct
the observations by comparison with others not too remote. The rough approximation to the orbit
given by the uncorrected formula may be sufficient for this purpose. In fact, for observations
separated by very small intervals, the imperfection of the uncorrected formula will be likaly to af-
fect the orbit less than the errors of the observations.
The computer may prefer to determine the orbit from the first and third heliocentric positions
with their times. This process, which has certain advantages, is perhaps a little longer than
90 Mi:.Miil;s ()1 THE NATIONAL ACADEMY OF SCIENCES.
that here given, and does not lend itself quite so readily to successive improvements of the
hypothesis. When it is desired to derive an improved hypothesis from an orbit thus determined,
the formula) in XII of the summary may be used.
SUMMARY OF FORMULA
WITH DIRECTIONS FOB USB.
[For the case in which an approximate orbit is known in advance, see XII.]
I.
Preliminary computations relating to the intervals of time.
( t t , ? 3 =times of the observations in days.
log fc=8.2355814 (after Gauss)
T 1 =k(t 3 -t t ) r 3 =fc(<z-*i)
A _<3-* A _t*-t t
'- '-
p_-riTi-> R ..nT 1 r 3 T 3 D
12 ~T2~
Forcoutrol: Ai
II.
Preliminary computations relating to the first observation.
Xi, FI, Z\ (components of Gi)=the heliocentric coordinates of the earth, increased by the geocen-
tric coordinates of the observatory.
fi, 7 C: (components of &)=the direction-cosines of the observed position, corrected for tin-
aberration of the fixed stars.
Preliminary computations relating to the second and third observations.
The fin-mill. e are entirely analogous to those relating to the first observation, the quantities
being distinguished by the proper suffixes.
III.
Equations of the first hypothesis.
When the preceding quantities have been computed, their numerical values (or their loga-
rithms, when more convenient for computation,) are to be substituted in the following equations:
rmnponents of 5
For control:
MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 91
Components of 2'
Components of (
For control :
- *r --* i f~ f, ift, \
Components of
P"=- "
Components of
3
For control :
Components of (g'"
3// , 3R 3 q 3
The computer is now to assume any reasonable values either of the geocentric distances, p,,
p 2 , p 3 , or of the heliocentric distances, r,, r 2 , r 3 , (the former in the case of a comet, the latter in the
case of an asteroid,) and from these assumed values to compute the rest of the following quantities:
By equations III,, III'. By equations III.,, III'. By equations III 3 , III'".
9i <fe q a
'g r \ log r 2 log r 3
l S #1 log JRz log R 3
log (1+^,) log (!-#,) log (i+R 3 )
log P' log P" log P">
& A A
a! a" a 1 "
y' y" y>
92 MK.MOII.'S OK Till-: NATIONAL ACADKMY OF S( 'I KNCKS.
IV.
Calculations relating to differential coefficients.
Components of 2" X 2'" Components of 2'" X 2' Components of 2' X 2"
a t =ft"y'"y"ft>" at=fl'"y'-y l "fl' a 3 =/3'y"-y'ft"
b t =y"a"'-a"y"' b t =y'"i*'-a"'y' b 3 =y'a"a'y"
C l =a"ft l "-P"a'" Ct=a"'/3'-/3'"a l c 3
These computations are controlled by the agreement of the three values of 0.
The following are not necessary except when the corrections to be made are large :
V.
Corrections of the geocentric dixtmices.
Component* of 2.
i -
y=y }+ y i+ y 3 ft
(This equation will generally be most easily solved by repeated substitutions.)
VI.
Successive corrections.
J^u Jg z , J^ 3 are to be added as corrections to </,, 172, / 3 . With the new values thus obtained
the computation by t-cjuations III,, 111^, III., are to be recoin ..... need. Two courses are now open:
(a) The work may be carried on exactly as before to the determination of new corrections for
0i > ft.ft.
(b) Tin- commutations by equations III', III", III'", and IV may be omitted, and the old valm s
of a, , ft,, ci, flj, etc., O, and L may be used with the new residuals v /?, y to get new corrections
for 91, 9>, q 3 by the equations
0,
where Dq t denotes the fornirr correction of </ 2 . (More generally, at any stage of the work, Dq t will
n-prcwnt the sum of all tin- cm-icctions of </., which have been made since the last computation of
<I| ,*!,<'
So far aa any general rule can In- ^-ivcn, it is advised to recompute a,, 6,, etc., and G once,
after the second <>! ivi-timi- of ./i '/ '/ unless tli' assumed values represent a fair
MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 93
approximation. Whether L is also to be recomputed, depends on its magnitude and on that of
the correction of g 2 , which remains to be made. In the later stages of the work, when the cor-
rections are small, the terms containing L may be neglected altogether.
The corrections of q lt q 2j q 3 should be repeated until the equations
=0 /J=0 y=Q
are nearly satisfied. Approximate values of r t , r 2 , r 3 may suffice for the following computations,
which, however, must be made with the greatest exactness.
VII.
Test of the first hypothesis.
log r t , log r 2 , log r 3 , (approximate values from the preceding computations.)
The value of s s 2 may be very small, and its logarithm in consequence ill determined. This
will do no harm if the computer is careful to use the same value computed, of course, as carefully
as possible wherever the expression occurs in the following formulae.
tan (v 2 t>i )=^-
tan (tf 3 _t> 2 =
N
t
tan %(v 3 Vi)=-
For adjustment of values: ( 3 -
. e sin
U sin *() 1
e cos
2, COS J(t> a 1
tan
For control : e cos <v 2 =- 1
*2
a- *
tan J.Z7i= tan Jv, tan JjE7 2 =e tan 2 tan ^E 3 f tan
sin _E 2 ea$ sin -& 3
! sin E,ea$ sin R
94 MEMOIltS OF T11E NATIONAL ACADEMY OF SCIENCES.
VIII.
For the second hypothesis.
<yn=.0057613il-(/3 z -p 3 ) (aberration-constant after Struve.)
yr,=.0057613A-(A-Pz) lo g (.0057613fc) =5.99610
A log r,=log r,-log (r, ..i^ rfr,)
% A log r 3 =log T 3 log (r 3c ic. tfr 3 )
A log (r,r 3 )=J log TI+ A log T 3
A log '= J log r, A log r s
J lOg ,=J
J log ,= J log
J log B 3 = J log
These corrections are to be added to the logarithms of A t , A 3 , BI, B 2 , B 3 , in equations IIIi,
III,, IIIj, and the corrected equations used to correct the values of q it g 3 , g 3 , until the residuals a,
ft, y vanish. The new values of A t , A 3 must satisfy the relation Ai+A 3 =;l., and the corrections
J log A tt J log A 3 must be adjusted, if necessary, tor this end.
Third hypothesis.
A second correction of equations IIIi, III 2 , III 3 may be obtained in the same manner as the
first, but this will rarely be necessary.
IX.
Determination of the ellipse.
It is supposed that the values of
1, fil, Pi, 2, ftt, Y*l 3> /*3 K3>
r\, r 2 , r 3 , Ti",, Kj, B 3 , ,, 2 , 3 ,
have been computed by equations III,, III Z , III 3 with the greatest exactness, so as to make the
residuals a, ft, y vanish, and that the two formulae for each of the quantities *,, z , 3 give sensibly
the same value.
Components of 2 4 Components of 2 5
MEM01ES OF THE NATIONAL ACADEMY OF SCIENCES. 95
For control only : ?
*(*-*,) (*- 3 )
tan (1?,-*,)=
tan i r ,-tr 2 =
K
The computer should be careful to use the corrected values of AI, A 3 . (See VUL) Trifling
errors in the angles should be distributed.
PP
e sin
.
. ,
sin t? 3 -i
e cos = r '
,
2 cos ^(t>3 t
tan ^r 3 +ri) e
p
For control: ecosr z = 1
Direction-cosines of semi-major-axis.
,__cosr 2 _ sin r 2
cos r 2 sin r 2/?
m= fh PS
5
Direction-cosines of semi-minor-axis.
5
Components of the aemi-axes.
MI;.MOII;S 01 Tin-: NATIONAL ACADEMY OK SCIENCES.
x.
Time of perihelion passage.
Corrections for aberration.
tan AA'i f tan .4r, 6t, = .00576 13pi
tail Jf?,=f tan J, 6t. f = . 0057613 pt
tan J-E 3 =f tan 4r, <tt 3 =-.0057613/>,
log .0057613=7.76052
,+<,- T=*- 1 a?(.E 1 -e sia 1?,)
t t +6t t -T=k- l a*(E 1 -e sin J? 2 )
3-e sin E 3 )
The threefold determination of T affords a control of the exactness of the solution of the
problem. If the discrepancies in the values of T are such as to require another correction of the
formula- (a third hypothesis), this may be based on the equations
A log n= A log T 3 =
where T,,,, T tll , T (i) denote, respectively, the values obtained from the first, second, and third
observations, and M the modulus of common logarithms.
XI.
For an ephemeris.
ft
T)=E esin E
Heliocentric co-ordinates. (Components of
x= ea,+a, cos E+b f sin E
y=ea r +a r cos E+b, sin E
z=ea,+a, cos E+b, sin E
These equations are completely controlled by the agreement of the computed and observed
positions and the following relations between the constants :
a,b.+ajb,+a.b,=0 o,*+a,+a.=a z b.'+bf+bf^l-*)*
XII.
When an approximate orbit is known in advance, we may use it to improve our fundamental
equation. The following appears to be the most simple method :
Find theexcentric anomalies E t , E } , #,, and the heliocentric distances r,, r*, r 3 , which belong
in the approximate orbit to the times of observation corrected for aberration.
Calculate B tt &,, as in I, using these corrected times.
Determine A,, A 3 by the equation
. . .
sm (Et E,) e sin E 3 +e aiu E t sin (E t E t )e sin E^+e sin E,
in connection with the relation Ai+A 3 =l.
MEMOIRS OP THE NATIONAL ACADEMY OP SCIENCES.
Determine B 2 so as to make
B,
97
4 siii kEz
sin E 3
equal to either member of the last equation.
It is not necessary that the times for which H,, E 2 , E 3 , r,, r 2 , r 3 , are calculated should pre-
cisely agree with the times of observation corrected for aberration. Let the former be represented
by ti', t 2 ', t 3 ', and the latter by *,", 2 ", t 3 " ; and let
A log r,=log (t 3 "-tt")-\og (t 3 '-t 2 '),
A log r 3 =log (^''_^')_log (tz'-li 1 ).
We may find BI, B 3 , AI, A 3 , B 2 , as above, using f/, V, t 3 ', and then use
correct their values, as in VIII.
Jlogr z to
NUMERICAL EXAMPLE.
To illustrate the numerical computations we have chosen the following example, both on
account of the large heliocentric motion, and because Gauss and Oppolzer have treated the same
data by their different methods.
The data are taken from the Theoria Motus, 159, viz:
Times, 1805, September
Longitudes of Ceres
5. 51336
95 32' 18". 56
139.42711
99 49' 5". 87
265. 39813
118 5' 28". 85
Latitudes of Ceres . ...
59' 34". 06
+7 16' 36". 80
+7 38' 49". 39
Longitudes of the Earth
Logs of the Sun's distance. ..
342 54' 56". 00
0.0031514
117 12' 43". 25
9. 9929861
241 58' 50". 71
0. 0056974
The positions of Ceres have been freed from the effects of parallax and aberration.
I.
From the given times we obtain the following values:
Control :
Numbers.
Logarithms.
t>t,
133. 91375
2. 1268252
(j *
125.97102
2. 1002706
< 3 -t t
259. 88477
2. 4147809
At
.4847187
9. 6854897
At
. 5152812
9. 7120443
ri
. 3358520
r.
. 3624066
Hi
9. C692113
-B,
. 3183722
ft
9. 562391G
H. Mis. 597-
r,r 3 r=2.4959081
MKM01US OK TIIK NATIONAL ACADKMY OF SCIENCES.
II.
From the given positions we get :
lo K -Y,
i,.,)-,
9.983.V.1:,
J.I711748
.log-Y,
11796
9. 94
+
1" -V,
V
9. c.7:
9.951
l-K fi
'"I! Ji
8.9845270
-
'"K
9.22H2738
7
log,
log
9. OV.I0294
9.94K
7.
;i. 10-J)549
+
log C>
9. 124d.-'l:t
+
IVrt,
.9314993
^_
G 3 '^ 3
. r,:,;K304
.8645336
+
PS
.1006681
+
J'
. 7i:itW4
+
III.
The preceding computations furnish the numerical values for the equations IIIi, III', 111,,
III", III a , III'", which follow. Brackets indicate that logarithms have been substituted for
nuuilu-rs.
\\'o have now to assume some values for the heliocentric distances r t , r t , r 3 . A mean propor-
tional between the mean distances of Mars and Jupiter from tin- Hun suggests itself as a reasonable
assumption. In order, however, to test the convergence of the computations, when the assump-
tions are not happy, we will make the much less probable assumption (actually much farther
from the truth) that the heliocentric distances are an arithmetieal mean between the distant >
<if Mars and Jupiter. This gives .526 for the logarithm of each of the distances r,, r 2 , r 3 . From
these assumed values we compute the first columns of numbers in the three following tables.
qi=pi .3874081
a,= -[8.C700167]('/,-!t.5901555)(l + ^,
/?,= [9.G833924](</!+ .0900552)( !+/.',) > III,
7/, = -[7.9242047](g,+ .3874081)(l+jR,).
a! =-.04C775- [8.67002 J ,-!", \
/3'= .482383+[Q.683M\Ri-P'/ii V III'
r ':=-.008399-[7.924L > 0]R I -P' ri )
4q,
.66731
.04
.00104:54
+ .0000006
9i
+
3.22606
S75
9.61317
2.5142134
2.514--M40
log r,
+
. :.,'(iOO
. I::i960
-i791
. 4'.-
. 42&.':!77
log It,
8.091V-1
8.364XU
i'.i74(l
, i;i-:,
log (! + /.',)
+
. oor,:):!
. 0091)34
.0104199
.0104010
log 1"
+
8. 01967
MM
H. :v.i:.7 10
8.3951457
i
+
.30136
BG06
. :!:!'.-
.:w. is
Pi
+
L6I
1.307304
1.88
i. :-.
r\
.03072
.OHUfl
.034
.0949011
a 1
.O.'>0505
ft'
+
.47K
10949
r'
~~
.00818
.007960
rS= qj >+. 1006681
^,=[0.3183722]r,- J
a,= + [!.22S27;{SJ( (/ ,+ 1.7-Si;>
/S 2 =-[9.9008l)0](.7,- .03(;i:l9)(l /t' 2 ) ^ HI,
y,= -[9.ioi. i i;.-ii!i|!v.- J
a"- .101H51-[0.2L > S27]/i > 2 +P"a 2 \
/S"=-.!)7717-f['.).'.)(MIS]/, > ,+ /"Vy 2 (ill
y"=-A-CMl + \\\.W2C.:,\K. i +l > "y t )
MEMOIES OP THE NATIONAL ACADEMY OP SCIENCES.
99
Aqu
- .77826
+ .005042
+ .0013222
+ .0000021
?2
+
3. 34235
2. 56409
2. 569132
2. 5704542
2. 5704563
log r 2
+
. 52600
.412233
. 4130733
.4132934
. 4132937
log K,
+
8. 74037
9. 081673
9. 0791524
9. 0784920
log(l-B,)
+
9. 97543
9. 944142
9. 9444866
9. 94457G1!
log P
+
8.71411
9. 199120
9. 1954270
9. 1944:,98
a.
+
.81059
. 638489
. 6397466
.640(1760
A
3. 05379
2. 172660
2. 1787230
2. 1803116
r*
.28858
. 181843
. 1825486
. 1827338
a"
+
.20182
.2491854
ft"
1. 08177
1.2018221
r"
. 13464
. 1400944
q 3== p 3 -.5599304
r 3 2 =g 3 +.7130624
jR 3 =[9.5G23916]r 3 - 3
.
-
3 = -[9.3810737 J (,+ 1.5798163) (l+R 3 ) j
/? 3 = [9.6537308] (q,- .4630521) (1+JJ,) ( III,
y 3 = [8.83612561( 2 3-f .5599304) (l + R 3 ) )
a"'=-.240477-i9.38107] J R 3 -P'"o 3 ^
p"= +.450537+ [9.65373] Z? 3 -P'"y9 3 V III'"
y"'=+.068569+[S.83G13]R 3 -P'" ;r3 )
4qa
.80780
.04055
+ .0025316
+ .0000031
93
+
3.24945
2.44165
2.40110
2.4036316
2. 4036347
log r : ,
+
0. 52600
. 412217
.4057319
. 4061394
. 4061399
log JR,
+
7. 98439
8. 325742
8. 3451948
8. 3439733
log (1 + B 3 )
+
.00417
. 009099
. 0095108
. 0094843
log 1""
+
7. 91715
8.357016
8.3817516
8. 38019SI3
ay
1. 17253
.987590
. 9785152
.9790776
ft
+
1. 26749
. 910305
. f 924956
. 8936069
73
+
. 26373
. 210171
. 2075292
.2076940
a'"
. 22847
. 2222335
ft'"
+
.44441
. 4390163
ylll
.06690
. 0650888
IV.
The values of a', ft', etc., furnish the basis for the computation of the following quantities :
i=-. 01254
, = + .01726
Cj= .15746
02= -.03517
b 2 =-. 00525
Ct=-. 08526
a 3 = .07232
b 3 =-. 00845
c 3 = -.04050
For_(? we get three values sensibly identical. Adopting the mean, we set
#=.01006.
We also get
#=-.00998, i=.02322.*
V.
Taking the values of i, 2 , etc., from the columns under IIIi, II I 2 , III 3 , we form the residuals
=-.06058, /5=-.16692, j/=-- 05557 -
From these, with the numbers last computed, we get
d= -.65888, C 2 =-.76983, (7 3 =-.79939,
* It would have been better to omit altogether the calculation of H and L, if the small value of the latter could
have been foreseen. In fact, it will be found that the terms containing L hardly improve the convergence, being
smaller than quantities which have been neglected. Nevertheless, the use of these terms in this example will illus-
trate a process which in other cases may be beneficial.
]00 MM MOIL'S OF T1IK NATIONAL ACADKMY OK SCIKNCKS.
which might be used as corrroctions lor our values of </,, </.., </,. To get more accurate values for
t hese corrections we set
or J<f 2 =-.
which gives
J$ .77826.
The quadratic term diminishes tin- value of Jq t by .00843. Subtracting the same quantity
from C| and C t we get
Jgi= .66731, Jg 3 = -.80780.
VI.
Applying these corrections to the values of g,, 3,, </ 3 we compute the second numerical columns
under equations III,, HI 2 ,and III 3 . We do not go on to the computations by equations Ill',etc.,
but content ourselves with the old values of a,, 61, etc., G>, and L, which with the new residuals,
= .012595, 6=. 044949, ;/=.003012,
give
Ci= .04567, C,=.004952, C,= .04064.
This gives
^(? 2
As the term containing L lias increased the value of dq t by .00009, we add this quantity to C\
ami (' , and get
Aqi = .04558, 4q 3 = - .04055.
With these corrections we compute the third numerical columns under equations III], etc.
This time we recompute the quantities a 7 , etc., with which we repeat the principal computations <>t
IV, and get the new values:
o, = -.0167215 o,=-.0335815 a 3 = -.0743299
&,= + . 0149145 bt= .0054413 b 3 = -.0098825
c,= .1576886 c,= .0779570 c*= -.0474318
O=.0090929
The quantities // and L we neglect as of no consequence at this stage of the approximation.
With these values the new residuals,
or= + .0002919, /S=-.0000044, ^=+.0000288,
give
^, = 0, =+. 0010434, J-(%=+.OOI3L>22, 4q 3 =C 3 = + .0025316.
These correct i(n> furnish the basis for the fourth columns of numbers under equations III,,
etc., which give the residuals
=+. 0000002, /*= + . 0000009, >/= + .0000001,
and the new corrcrtioiiM
3= + . 0000031.
MEMOIRS OF THE NATIONAL ACADEMY OP SCIENCES.
101
The corrected values of q l1 g 2 , q 3 give
log r,=0.4282377, log r 2 =0.4132937,
log r 3 =0.4061399.
We have carried the approximation farther than is necessary for the following correction of
the formula, in order to see exactly where the uncorrected formula would lead us, and for the
control afforded by the fourth residuals.
VII.
The computations for the test of the uncorrected formula (the tirst hypothesis) are as follows :
Number or arc.
Logarithm.
Number or arc.
Logarithm.
n
0. 42S2377
e
+
8.9025438
r*
0. 4132937
+
9. 9652259
r 3
0. 4061399
a
0. 4419546
AiBirr*
+
.01174865
8. 0699879
tan 1 i'i
35 41' 39". 75
9. 8563809
B&i~ 3
-(-
. 11980944
9. 0784911
tan ij !
19 53' 28". 93
9. 5584981
AsB-irz" 3
_^_
. 01137670
8. 0560162
tan Jc:i
4 13' 52". 55
8. 8691380
N
-j.
. 14293479
9. 1551380
tan $Ei
33 33' 0". 17
9. 8216068
81
-f-
1. 3308476
0. 1241283
tan lE, a
IB 28' 6". 35
9. 5237240
83
-j-
2. 2796616
0. 3578704
tan i_E 3
3 54' 24". 21
8. 8343639
83
8
881
+
1.3417404
2. 4761248
1. 1452772
0. 1276685
0. 3937725
0. 0589106
sin E\
sin 3
sin E 3
67 6' 0". 34
36 56' 12". 70
7 48' 48' . 42
9. 9643473
9.7788272
9. 1333734
8 8a
-f
0. 1964632
9. 2932812
883
1. 1343844
0. 0547602
e<& sin EI
.3387061
9.5298230
R
_|_
9. 5065898
ea% sin E?
. 2209545
9. 3443029
P
+
0. 4391732
ea$ sin E 3
. 0499861
8. 6988491
tan i( 3 vi)
tan i(i> 3 3 )
+
15 48' 10". 82
15 39' 36". 38
9. 4518296
9. 4476792
a* (Ei-EJ
+
2. 4226307
0. 3842872
tan i(s i)
31 27' 47". 20
9. 7866915
a 3 (EzEv)
+
2.3391145
0. 3690515
e sin i(3+"i)
8. 7099387
^3 calc.
_J.
2. 3048791
0. 3626482
e cos i(3-j-i>i)
-f-
8. 7872701
Z"l C.1C.
+
2. 1681461
0. 3360885
tan i(t>3-j-i)
39 55' 32". 31
9. 9226686
VIII.
The logarithms of the calculated values of the intervals of time exceed those of the given
values by .0002416 for the first interval (r 3 ) and .0002365 for the second (TI). Therefore, since the
corrections for aberration have been incorporated in the data, we set for the correction of the
formula (for the second hypothesis)
A log r, = .0002365 A log r 3 =-.0002416
This gives
A log A !=. 0000026 A log A 3 = .0000025
A log #!= .0004872 A log 5 2 = -.0004782 A log B 3 = .0004665
The new values of the logarithms of A\ , A 3 are
log ^,=9.6854923 log 4 3 =9.7120418
104
MEMOIRS OP THE NATIONAL ACADEMY Ol SC1KNCKS.
The equations for an ephemeris will then be :
T=180G, June 23.96378, Paris mean time
[2.8863140](*-T)=-Einxma.-[4.2216530] sin E
Heliocentric coordinates relating to the ecliptic.
*= + .1820765-[0.35302Gl] cos -[0.1827783] sin E
y= -.1244853+ [0.1878904] cos -[0.3603153] aiu E
*=-.0373987+[9.6656285] cos +[9.3320758] sin E
The agreement of the calculated geocentric positions with the data is shown in the following
table:
Times, 180f>, September
6.51336
139. 42711
265. 39813
Second hypothesis:
9532'18". 88
9949' 5". 87
118 5'28".52
0".32
0".00
-0". 3:)
059'34". 01
716'36". 82
738'49". 34
errors ...... .... ..
0".05
0". 02
0". 05
Third hypothesis:
9532'18". 65
9949 / 5". 82
118 5'28". 79
0".09
-0".05
0". 06
latitudes
059'34". 04
716'36". 78
738'49". 38
errors
0". 02
-0".02
-0".01
The immediate result of each hypothesis is to give three positions of the planet, from which,
with the times, the orbit may be calculated in various ways, and with different results, so far as
the positions deviate from the truth on account of the approximate nature of the hypothesis. In
some respects, therefore, the correctness of an hypothesis is best shown by the values of the goo-
centric or heliocentric distances which are derived directly from it. The logarithms of the helio-
centric distances are brought together in the following table, and corresponding values fiom
Gauss* and Oppolzerf are added for comparison. It is worthy of notice that the positions given
by our second hypothesis are substantially correct, and if the orbit had been calculated 1'roni the
first and third of these positions with the interval of time, it would have left little to be desired.
log r, .
logr,.
log r, .
First hypothesis ..........
.4282377
4i:fc*i:!7
40613' *l
.42H-,'7-->
.413
1061996
Third hypothesis
.42K
.4132808
. 4IHJ2003
Gauss:
First hypothesis ...
4323934
.4114720
4094712
St-ciuid hypothesis....
Third hypothesis
Fourth hypothesis
. I-.-.H773
. rj-i-ii
. 42827'.
.4129371
.4132107
.41:12317
. 407197. r >
.4lHJ4ii'.l7
.4062033
OppolziT:
Firnt hypothesis
4281340
413330
I ur, u ','.;
Second h.vpnth-
Third hypothesis
. 4282794
. 42-
.4132801
.40til'.l7li
. 4/H52009
fn comparing the different methods, it should be observed that the determination of the posi-
tions in any hypothesis by (lauss's method requires successive cot red ions of a single independent
variable, a corresponding determination by ( >ppolzer'a method requires the successive corrections
of two independent variables, while the eonvsp<inding determination by the method of the pres-
ent paper requires the successive corrections of three independent variables.
Theoria mot. ,liiil)rHtiiiinnni <|.T Kiin'tru niul Ham-lm, 2d ed., vol. J, p. 394.