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[Memoirs of the National Academy of Sciences, vol. iv. part ii. pp. 79–104, 1889.]

The determination of an orbit from three complete observations by the solution of the equations 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 simplicity and accuracy the relations between three positions in an orbit separated by small or moderate 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, but 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 unity at a unit's distance, and denote the vectors[1] drawn from the sun to the body in its three positions by and the lengths of these vectors (the heliocentric distances) by the accelerations corresponding to the three positions will be represented by 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 for the interval of time between the first and second positions, and for that between the second and third, and set for the second position,


We may therefore write with a high degree of approximation
From these six equations the five constants may be eliminated, leaving a single equation of the form
This we shall call our fundamental equation. In order to discuss its geometrical signification, let us set
so that the equation will read
This expresses that the vector is the diagonal of a parallelogram of which and are sides. If we multiply by and by in skew multiplication, we get
Our equation may therefore be regarded as signifying that the three vectors lie in one plane, and that the three triangles determined each by a pair of these vectors, and usually denoted by 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 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), for unit vectors drawn in the directions of the body, as observed, and for the three distances of the body from the places of observation. We have then
By substitution of these values our fundamental equation becomes
where (the geocentric and heliocentric distances) are the only unknown quantities. From equations (6) we also get, by squaring both members in each,
by which the values of may be derived from those of or vice versâ. Equations (7) and (8), which are equivalent to six ordinary equations, are sufficient to determine the six quantities or, if we suppose the values of in terms of to be substituted in equation (7), we have a single vector equation, from which we may determine the three geocentric distances

It remains to be shown, first, 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 of the 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 sun upon the line of sight. If we set

equations (8) become
Let us also set, for brevity,
Then may be regarded as functions respectively of therefore of and if we set
we shall have
To determine the value of we get by differentiation
But by (11)

Now if any values of (either assumed or obtained by a previous approximation) give a certain residual (which would be zero if the values of satisfied the fundamental equation), and we wish to find the corrections which must be added to 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,

This gives[2]
From the corrected values of we may calculate a new residual and from that determine another correction for each of the quantities

It will sometimes be worth while to use formulæ 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,


It is evident that is generally many times greater than or the factor in the case of equal intervals, being exactly ten times as great as or This shows, in the first place, that the accurate determination of is of the most importance for the subsequent approximations. It also shows that we may attain nearly the same accuracy in writing

We may, however, often do a little better than this without using a more complicated equation. For may be estimated very roughly as equal to Whenever, therefore, and are about as large as as is often the case, it may be a little better to use the coefficient instead of in the last term.

For then, we have the equation

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 must 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 term.[3]

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 is one value of to save the trifling labor of computing the value of For it is evident from the theory of equations that if and are the two roots,

Eliminating we have
Now is the value of which we obtain if we neglect the quadratic term in equation (24). If we call this value we have for the more exact value[4]
The quantities and might be calculated by the equations
But a little examination will show that the coefficients of 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
where denote values obtained from equations (20).

In making successive corrections of the distances it will not be necessary to recalculate the values of when these have been calculated from fairly good values of But when, as is generally the case, the first assumption is only a rude guess, the values of should be recalculated after one or two corrections of To get the best results when we do not recalculate we may proceed as follows: Let denote the values which have been calculated; respectively, the sum of the corrections of each of the quantities which have been made since the calculation of the residual after all the corrections of which have been made; and the remaining corrections which we are seeking. We have, then, very nearly


The same considerations which we applied to equation (21) enable us to simplify this equation also, and to write with a fair degree of accuracy


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 equation 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.[5]

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 identical 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 shortened 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 motion 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 certain 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 coefficients, etc.

In the process indicated there are certain economies of labor which should not be left unmentioned, and certain precautions to be observed in order that the neglected figures in our computations may not unduly infiuence 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 and the two heliocentric angles, which will be represented by and if we write for the true anomalies. These angles ( and ) may be determined from and and therefore from and the given intervals. For our fundamental equation, which may be written

indicates that we may form a triangle in which the lengths of the sides shall be and (let us say for brevity, ), and the directions of the sides parallel with the three heliocentric directions of the body. The angles opposite and will be respectively and We have, therefore, by a well-known formula,
. (34)

As soon, therefore, as the solution of our fundamental equation has given a sufficient approximation to the values of (say five- or six-figure values, if our final result is to be as exact as seven-figure logarithms can make it), we calculate 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 which occurs in the value of the semi-parameter, and the expression or which occurs both in the value of the semi-parameter and in the formulæ 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 quantities may, however, be determined accurately by the formula
With respect to the quantity a little consideration will show that if we are careful to use the same value wherever the expression occurs, both in the formulæ for the heliocentric angles and for the semi-parameter, the inaccuracy of the determination of this value from the cause mentioned will be of no consequence in the process of correcting the fundamental equation. For although the logarithm of as calculated by seven-figure logarithms from 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 ). We need not trouble ourselves 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 fulfilment of two conditions, viz:., that our values of the heliocentric distances and angles shall have the relations required by the funda- mental equation to the given intervals of time, and that they shall have the relations required by the exact laws of elliptic motion to the calculated intervals of time. The third condition, that none of these values shall difler too widely from the actual values, is of a looser character.

After the determination of the heliocentric angles and the semi-parameter, the eccentricity and the true anomalies of the three positions may next be determined, and from these the intervals of time. These processes require no especial notice. The appropriate formulæ will be given in the Summary of Formulæ.

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 paragraphs depend upon the quantity the numerical determination of which from and a critical to the second degree when the heliocentric angles are small. This was of no consequence in the process which we have called the correction of the fundamental equation. But for the actual determination of the orbit from the positions given by the corrected equation— or by the uncorrected 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 perhaps the most simple. Let us set

and for the length of the vector obtained by taking the square root of the sum of the squares of the components of the vector. It is evident that is the longer and the shorter diagonal of a parallelogram of which the sides are and The area of the triangle having the sides is therefore equal to that of the triangle having the sides each being one-half of the parallelogram. This gives
The numerical determination of this value of is critical only to the first degree.

The eccentricity and the true anomalies may be determined in the same way as in the correction of the formula. The position of the orbit in space may be derived from the following considerations. The vector is directed from the sun toward the second position of the body; the vector from the first to the third position. If we set

the vector will be in the plane of the orbit, perpendicular to and on the side toward which anomalies increase. If we write for the length of
will be unit vectors. Let and be unit vectors determining the position of the orbit, being drawn from the sun toward the perihelion, and at right angles to in the plane of the orbit, and on the side toward which anomalies increase. Then
The time of perihelion passage () may be determined from any one of the observations by the equation
the eccentric anomaly being calculated from the true anomaly The interval in this equation is to be measured in days. A better value of 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 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

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 likely to affect 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 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 formulæ in § XII of the summary may be used.



(For the case in which an approximate orbit is known in advance, see XII.)


Preliminary computations relating to the intervals of time.

times of the observations in days,

(after Gauss)

For control:


Preliminary computations relating to the first observation.

(components of ) the heliocentric coordinates of the earth, increased by the geocentric coordinates of the observatory.

(components of ) the direction-cosines of the observed position, corrected for the aberration of the fixed stars.

Preliminary computations relating to the second and third observations.

The formulæ are entirely analogous to those relating to the first observation, the quantities being distinguished by the proper suffixes.


Equations of the first hypothesis.

When the preceding quantities have been computed, their numerical values (or their logarithms, when more convenient for computation,) are to be substituted in the following equations:

Components of

For control:

Components of

Components of

For control:

Components of

Components of

For control:

Components of

The computer is now to assume any reasonable values either of the geocentric distances, or of the heliocentric distances, (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 By equations By equations


Calculations relating to differential coefficients.

Components of Components of Components of

These computations are controlled by the agreement of the three values of

The following are not necessary except when the corrections to be made are large:


Corrections of the geocentric distances.

Components of

(This equation will generally be most easily solved by repeated substitutions.)


Successive corrections.

are to be added as corrections to With the new values thus obtained the computation by equations are to be recommenced. Two courses are now open:

(a) The work may be carried on exactly as before to the determination of new corrections for

(b) The computations by equations and may be omitted, and the old values of etc., and may be used with the new residuals to get new corrections for by the equations

where denotes the former correction of (More generally, at any stage of the work, will represent the sum of all the corrections of which have been made since the last computation of etc.) So far as any general rule can be given, it is advised to recompute etc., and once, perhaps after the second corrections of unless the assumed values represent a fair approximation. Whether is also to be recomputed, depends on its magnitude, and on that of the correction of which remains to be made. In the later stages of the work, when the corrections are small, the terms containing may be neglected altogether.

The corrections of should be repeated until the equations

are nearly satisfied. Approximate values of may suffice for the following computations, which, however, must be made with the greatest exactness.


Test of the first hypothesis.

(approximate values from the preceding computations).

The value of 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 formulæ:

For adjustment of values: }}

For control:


For the second hypothesis.

(aberration-constant after Struve.)

These corrections are to be added to the logarithms of in equations and the corrected equations used to correct the values of until the residuals vanish. The new values of must satisfy the relation and the corrections must be adjusted, if necessary, for this end.

Third hypothesis.

A second correction of equations may be obtained in the same manner as the first, but this will rarely be necessary.


Determination of the ellipse.

It is supposed that the values of

have been computed by equations with the greatest exactness, so as to make the residuals vanish, and that the two formulæ for each of the quantities give sensibly the same value.

Components of Components of

For control only:

The computer should be careful to use the corrected values of (See VIII.) Trifling errors in the angles should be distributed.

For control:
Direction-cosines of semi-major axis.
Direction-cosines of semi-minor axis.
Components of the semi-axes.


Time of perihelion passage.

Corrections for aberration.

The threefold determination of affords a control of the exactness of the solution of the problem. If the discrepancies in the values of are such as to require another correction of the formulæ (a third hypothesis), this may be based on the equations

where denote respectively the values obtained from the first, second, and third observations, and the modulus of common logarithms.


For an ephemeris.

Heliocentric coordinates. (Components of .)

These equations are completely controlled by the agreement of the computed and observed positions and the following relations between the constants:


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 the excentric anomalies and the heliocentric distances which belong in the approximate orbit to the times of observation corrected for aberration.

Calculate as in § I, using these corrected times.

Determine by the equation

in connection with the relation

Determine so as to make

equal to either member of the last equation.

It is not necessary that the times for which are calculated should precisely agree with the times of observation corrected for aberration. Let the former be represented by and the latter by and let

We may find as above, using and then use to correct their values, as in § VIII.

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, § 169, viz.,

Times, 1805, September 5.51336 139.42711 265.39813
Longitudes of Ceres 95° 32' 18''.56 99° 49' 5''.87 118° 5' 28''.85
Latitudes of Ceres -0° 59' 34''.06 +7° 16' 36''.80 +7° 38' 49''.39
Longitudes of the Earth 342° 54' 56''.00 117° 12' 43''.25 241° 58' 50''.71
Logs of the Sun's distance 0.0031514 9.9929861 0.0056974
The positions of Ceres have been freed from the effects of parallax and aberration.


From the given times we obtain the following values:

Numbers. Logarithms.
133.91375 2.1268252
125.97102 2.1002706
259.88477 2.4147809
.4847187 9.6854897
.5152812 9.7120443



From the given positions we get: