COMET

Comet. In the article Astronomy we have given a general description of the orbits and physical appearances of comets. In the present article it is our purpose to give a solution of the problem of determining the approximate elements of a comet's orbit from three geocentric observations,—a problem of very great importance in astronomy, since it is only from the circumstance of moving in the same orbit, that the identity of a comet, in its successive revolutions, can be inferred with certainty.

The astronomy of comets may be said to have originated with Tycho Brahe, who showed that, by reason of the smallness of their parallaxes, their distances from the earth must be greater than that of the moon. Before this era they had been generally regarded as meteors casually engendered in the atmosphere, and having no permanent duration or connection with the solar system. The conclusions of the Danish astronomer were confirmed by the observations of Galileo, Snell, and Kepler, on the comet which appeared in the beginning of the seventeenth century; and as it was now certain that the comets do not belong to the earth, but move in the region of the planets, the determination of their orbits, and their relations to the other known bodies of the universe, became a problem of the highest interest. Kepler attempted to represent their apparent paths by supposing them to move in straight lines, with a variable velocity; and on this hypothesis the orbits of several comets were computed by Dominic Cassini, with a degree of success that might appear remarkable, were it not known that the apparent path of a comet differs little from a straight line during a considerable portion of the time that it is visible from the earth. But it could not fail soon to be perceived that the hypothesis of rectilinear motion is not only improbable in itself, but inconsistent with accurate observation. Hevelius demonstrated, in his Cometographia, published in 1668, that the path of the comet of 1655 was curvilinear, and concave towards the sun; and he even supposed the orbit might be a parabola, without hinting, however, that the sun is placed in the focus. But the individual who, before the grand discoveries of Newton, approximated most nearly to a correct knowledge of the true nature of the cometary orbits, was Dörfel, protestant minister of Wieda, a village near Plauen, in Upper Saxony. Dörfel observed the comet of 1680-1, from the 22d of December till the end of the following January, and recognised its identity on its emerging from the sun's rays, after passing through the perihelion of its orbit; and he proved the orbit to be a parabola, having the sun in the focus. The observations of Dörfel were published in 1681, consequently before the appearance of the Principia, in a small work, which has become so extremely rare, that neither Montucla nor Delambre had been able to procure a copy of it. To Dörfel, therefore, belongs the honour of having been the first to point out the true orbit of a comet, though his merit has passed almost unnoticed in consequence of the more important discoveries which followed so soon after. It is not known whether Newton was aware of the observations of Dörfel; but it is certain that the latter had no idea of the great principle of attraction, in virtue of which the comets as well as the planets perform stated revolutions about the sun. In his hands the parabolic orbit was simply an astronomical hypothesis, connected with no system, and framed only to satisfy a few observations. Newton, on the other hand, demonstrated that the comets form a part of the solar system, that they describe orbits about the sun in vir-

tue of his attractive force, obeying the same laws of motion as the planets; and he showed, moreover, in what manner the elements of their orbits may be computed, on the parabolic hypothesis, from three geocentric observations of their longitudes and latitudes.

Newton has given, in the Principia, two different solutions of the problem of determining a parabolic orbit from three observations. In the first he supposes that a small portion of the trajectory may be regarded as a straight line, described with a uniform motion, so that its segments, intercepted by straight lines drawn from the earth to the comet, at the times of the respective observations, are proportional to the intervals of time between the observations. According to this supposition, the ratio of these segments is known. It follows, also, since the portion of the trajectory included between the extreme observations is a straight line, that its projection on the plane of the ecliptic is also a straight line, and divided similarly to the trajectory itself, by the projections of the straight lines joining the places of the earth and the comet. In order, therefore, to determine the projected orbit, it might seem to be only necessary to draw a straight line in such a manner that its segments, intercepted by straight lines whose positions are given (being determined by the observed longitudes), shall have to each other given ratios. But this problem, in the case in which there are only three observations, or three given straight lines, is altogether indeterminate; for having assumed any point whatever in one of the three given straight lines, it is always possible to draw through that point a straight line, which shall be divided into segments having a given ratio, by the other two given straight lines. But it is obvious that the position of the transversal line will be different for every different position of the assumed point; consequently the number of solutions is infinite. When the number of observations is four, the problem is to draw a straight line, so that it may be divided into parts having given ratios to one another, by four straight lines given in position. In this state of the data, it becomes, generally speaking, determinate, and admits of only a single solution. This problem had been already resolved by Dr Wallis and Sir Christopher Wren, and to their solutions Newton added two others of his own. There is still a case, however, in which, even when four straight lines are given, the problem is indeterminate; and it happens, by a remarkable coincidence, that it is this case precisely which occurs in the application to the cometary motions. If the position of the four given lines is such that two transversal lines can be drawn so that the segments of both of them are respectively proportional to given lines, then innumerable other straight lines may be drawn which shall be all cut in the same ratio. Now this is the case with regard to the four straight lines which join the places of the earth and comet at the instants of the respective observations. During the short interval that elapses between the extreme observations, the earth, as well as the comet, may be regarded as moving in a straight line with a uniform velocity, so that the orbit of the earth, and the trajectory of the comet, are both divided into segments proportional to the intervals between the observations, and consequently proportional to each other. This circumstance, which was first pointed out by Boscovich, renders the problem inapplicable to astronomy; for although the earth's motion should not be regarded as entirely rectilinear and uniform, yet the data necessarily approach so

near to the indeterminate case, that no satisfactory conclusion can be deduced from them. Hence all attempts to compute the orbit of a comet in this manner have failed. (See Playfair's Memoir on the Origin and Investigation of Porsims, in vol. iii. of the Edinburgh Transactions, or in the third volume of his Works.)

The second method proposed by Newton forms Proposition 41 of the third book of the Principia. Though unexceptionable in theory, it leads, when actually applied to the determination of an orbit, to computations excessively complicated and laborious, and is consequently now entirely disused. It is probable that it was this method which was followed by Dr Halley, who computed the orbits of a great number of comets, though he has nowhere explained the process of computation he adopted. It is illustrated by Dr David Gregory in his Astronomia Physica, and by Lemonnier in his Théorie des Comètes.

The problem of the comets remained for many years in the state in which it was left by Newton, and astronomers were still in want of a practical method by which a parabolic orbit could be determined with tolerable ease and accuracy. In the absence of a direct solution, it was usual to have recourse to the methods of trial and error, and to approximate, by successive suppositions, to an orbit that would nearly represent the observations. This method was practised by Bradley, and by Lemonnier in his Institutions Astronomiques. Lacaille also published a method of the same kind in 1746, which was long followed by the continental astronomers, and is recommended by Delambre as possessing some advantages over those which are now generally preferred. In fact, all the methods which have yet been devised involve more or less of hypothesis, and require to be corrected by successive trials. Various other solutions of the problem were published about the same time, among which, those of Sejour, Hennert, Templehoff, and Bouguer, may be mentioned, though they are now entirely abandoned. The celebrated Boscovich, also, in the third volume of his Opera Pertinentia ad Optimam et Astronomiam, undertook to consider this subject; and the solution which he gave is remarkable, as being the first in which the velocity of the comet in its orbit was taken into account as one of the essential conditions of the problem. It is, however, excessively complicated, and contains a mixture of algebraic formulae and graphical operations susceptible of very little accuracy. Delambre, in speaking of this method, doubts if it was ever employed in the actual computation of an orbit by any astronomer, excepting the author himself, or perhaps some of his friends, though it is destitute neither of science nor address. (Astr. du XVIII^e Siècle, p. 653.) But the most important work which appeared on the subject of the cometary motions, previous to the investigations of Lagrange and Laplace, was that of Lambert, entitled Insigniores Orbitæ Cometarum Proprietates. Among the formulae which it contains relative to the motion of a body in a parabolic orbit, there is one, extremely remarkable on account of its elegance, which gives the time of describing an arc in terms of its chord, and the two radii vectores drawn through its extremities. This formula, which retains the name of Lambert's Theorem, though Gauss has shown that it really belongs to Euler, is of great use in the cometary theory, as it affords a very easy means of correcting the hypothetical values which it is necessary to substitute in the equations of the problem. Lambert also pointed out rules for determining, without solving the equations, whether the distance of the comet from the sun is greater or less than the distance of the sun from the earth; a knowledge of which circumstance affords considerable aid in forming a first hypothesis, and serves to abridge the preliminary computations.

Notwithstanding the repeated attempts of so many celebrated astronomers and mathematicians, the problem of determining a cometary orbit had not received a direct or satisfactory solution, when it was taken up by Lagrange in the Berlin Memoirs for 1778. After examining and pointing out the defects of the different methods then known, Lagrange gave a new method, which has formed the basis of the greater part of the analytical solutions which have been proposed since. The principle of his analysis is at once simple and direct. It will be recollected that the data of the problem are three observed positions of the comet, or three geocentric longitudes and latitudes. Lagrange first gives formulae to express the co-ordinates of the comet at the epochs of the first and third observations, in terms of its co-ordinates at the second observation; he then proceeds to substitute these values in the differential equations of motion; and in this manner, without making any hypothesis respecting the nature of the orbit, he obtains the necessary number of independent equations for eliminating the three co-ordinates, together with their differential co-efficients, leaving three final equations, in which the unknown quantities are the three distances of the comet from the earth at the epoch of the respective observations. On eliminating two of these distances, the resulting equation is found to rise to the eighth degree. It is easily reducible to the seventh; but unfortunately there are no other means of determining its roots than by repeated trials according to the methods of false position. Assuming, however, that a sufficiently approximate value of the comet's distances from the earth has been found in this way, it is thence easy to deduce its corresponding radii vectores, or distances from the sun, together with the angles they include, from which data all the elements of the orbit can be determined without difficulty.

Theoretically considered, the solution of Lagrange may be considered as perfect; but when it comes to be actually applied in the computation of an orbit, difficulties arise which can only be overcome, or eluded, by certain modifications of the general method, which the illustrious author has not explained. He did not himself attempt the applications, or he could not have failed to perceive, and would doubtless have provided for, the circumstances which, in following his method, occasion so much embarrassment to the computer. Lagrange made some important additions to his original analysis in the Berlin Memoirs for 1783, and it is also explained in the second volume of the Mécanique Analytique.

Another method of obtaining a solution of the problem, entirely different from the above, was proposed by Laplace in the Memoirs of the Academy of Sciences of Paris for 1780, and is developed with all the necessary details, in the second book of the Mécanique Céleste. The peculiarity of Laplace's method consists in its being supposed that a small portion of the orbit is accurately known from observation; that is to say, it is assumed that the longitude and latitude of the comet at a given epoch, together with the first and second differentials of those quantities, have been precisely determined. From these data the elements of the orbit can be deduced more simply than by any other method yet proposed; but it is found in practice that the determination of the first and second differentials of the comet's longitude and latitude is generally a matter of difficulty or uncertainty, and in some cases altogether impracticable. Laplace had at first imagined, that when a considerable number of observations have been obtained, as is most frequently the case, the differential values of the longitude and latitude would be found with greater accuracy by combining the whole of the observations; but it was subsequently shown by Legendre, that when more than a few observations are employed, the errors unavoid-

Comet. ably introduced in the arithmetical operations necessary for interpolating the observed places of the comet, are such as to increase, instead of diminishing, the errors of observation. It is therefore advantageous to confine ourselves to the number of observations strictly necessary, namely three; and to employ the remaining observations in correcting the elements deduced from these three. With three observations only, it cannot be expected that the results will have a high degree of accuracy; but when approximate values of the elements of an orbit have been found, it is in general easy to introduce such corrections as will give an orbit representing the observations within the limits of unavoidable errors. Laplace's method affords the simplest means of obtaining such approximations of any hitherto proposed, and is applicable perhaps in as great a number of cases as any other; it is consequently very frequently adopted by computers. Cases indeed occur in which it fails altogether, as happened in regard to the comet of 1814; but the same thing takes place with every other method that has yet been devised; nor is it always possible to discover, till actual trial has been made, the particular circumstances connected with the position or motion of the comet, which render one method more applicable than another.

Of the numerous methods of determining the approximate elements of an orbit from three observations, that of Olbers is one of the simplest and most direct, though the approximations which it gives are not very exact. Olbers supposes the orbit to be a parabola; a supposition which gives one equation more than the number of unknown quantities. He supposes, moreover, that the chord of the arc which joins the places of the comet at the epoch of the first and third observations, is divided, by the radius vector corresponding to the middle observation, into segments proportional to the intervals of time between the observations. The last supposition can only be admitted when the intervals between the observations are nearly equal; if a considerable inequality exists in the intervals, and more especially, if, at the same time, the radius vector of the comet is less than that of the earth, it leads to very inaccurate results. The author, indeed, shows how the errors of the hypothesis may be in some degree corrected; but, after all, his approximation includes only quantities of the second order in respect of the time. This may be sufficient in some cases; but in general it will be necessary to include quantities of the third order. The method of Olbers is explained at considerable length in the third volume of Delambre's Astronomie.

In 1806 Legendre published his Nouvelles Méthodes pour la détermination des Orbites des Comètes; a work very remarkable, not only by reason of the important light it threw on the question under consideration, but also on account of its containing the first exposition of the method of combining the results of a great number of observations, known by the name of the method of least squares, and which is so extensively useful in many questions of natural philosophy and astronomy. The method of determining an orbit given in this memoir is in some respects the same as that of Lagrange; but the equations of the problem are put under a form which admits of an easier application, and gives surer results; and the circumstances are fully developed and explained, which had so frequently caused Lagrange's method to fail in actual trial. Legendre also adopted a different method of treating the fundamental equations of the problem. Instead of deducing the elements of the orbit from two radii vectores and the contained angle, he eliminates those radii, and deduces expressions for the three rectangular co-ordinates of the comet at the epoch of the mean observation, and their three differential coefficients regarded as functions of the time. When the co-

ordinates and their differential co-efficients have been determined, the elements of the orbit can be easily deduced from them by known methods applicable to orbits of all descriptions. In the case in which the intervals between the observations are equal (and this condition can in general be satisfied by interpolating the observed places of the comet), the equations take a remarkably simple form, and the analytical solution is obtained with great facility. In the general case, that is to say, when the intervals between the observations are unequal, the equations are somewhat more complicated; but they have still an analogous form, and the additional labour required for their computation is perhaps compensated by the absence of preliminary interpolation, and by the greater certainty that arises from employing directly the data furnished by observation. This method of determining an orbit is exhibited in a very clear and elegant manner by Pontécoulant in his Théorie Analytique du Système du Monde, tome ii.

In the Transactions of the Royal Society for 1814, Mr Ivory has given a method of obtaining a first approximation to the orbit of a comet, explained with all the perspicuity and analytical elegance which so peculiarly characterize the writings of this eminent and justly celebrated mathematician. Mr Ivory's solution is founded on the same principles as that of Olbers, but it gives a nearer approximation, and is also adapted to a case in which the ordinary methods fail, namely, when the comet and the earth are nearly at the same distance from the sun. Like Olbers, and indeed Newton, Mr Ivory supposes the chord of the parabolic arc, described in the interval between the extreme observations, to be divided by the radius vector corresponding to the middle observation in the ratio of the times between the observations; a supposition which, as we have already remarked, though it simplifies the analysis and computations, cannot always be admitted, and which, in fact, leads, in some cases, to very sensible errors. It has been objected to this solution, that it is needlessly encumbered with auxiliary formulae.

The facility with which an orbit can be computed according to this method, induced Legendre to resume the investigation of the subject, in order to obtain, if possible, from the same method, more approximate elements. In 1820 a second supplement was published to the Nouvelles Méthodes, containing a solution grounded on the same principles as those of Olbers and Ivory. Instead of assuming, however, like those two mathematicians, the chord of the parabolic arc to be divided by the radius vector of the middle observation in the ratio of the elapsed times, Legendre finds an expression for the value of each of its segments in terms of the time; and the degree of approximation ultimately obtained depends on the number of terms of the series expressing those values that are included in the computation. This method possesses several advantages. In the first place, it requires no preliminary interpolations; in the second place, it is simple, and can be understood without a very high degree of algebraic skill; and lastly, which is of most importance, it will be found to give, we believe, in a great majority of cases, especially when the intervals between the observations are considerably unequal, a nearer approximation than any of the other methods, with the same degree of labour. For these reasons we have adopted it in the present article. In particular cases other methods may be found more applicable; but the same objection may be made to those of Laplace, Olbers, and indeed every other which has been proposed. No method will answer equally well in every case.

The last work on this subject to which we shall refer is a paper by Mr Lubbock, in the Memoirs of the Royal Astronomical Society, vol. iv. part i., 1830. We have already remarked, that the final equation which gives the radius

Comet vector of the comet, or its distance from the earth, rises to the seventh degree, and that consequently the solution of the problem can only be obtained by the method of trial and error. Mr Lubbock proposes the following method of finding a value of the radius vector by means of a quadratic equation. The accelerating forces acting on the comet, in the direction of the three rectangular co-ordinates of its orbit, the sun's mass being unit, are,

from the integration of which the following equation is obtained, which gives the square of the velocity, namely,

But since $r^2\; =\; x^2\; +\; y^2\; +\; z^2$, we have also, in consequence of the above three equations,

therefore,

Assuming the intervals between the observations to be equal, and rejecting powers of the time above the square, the author first expresses the differential co-efficients in equations (1) and (2) in terms of finite differences. He next substitutes for the variables their values in terms of $\backslash rho$, the comet's distance from the earth at the second observation; then, making $a$ infinite, and neglecting quantities divided by $r^3$, he eliminates $\backslash frac\{1\}\{r\}$, and obtains a quadratic equation

which gives an approximate value of $\backslash rho$. By this means, the necessity of having recourse to repeated trials is obviated; but the accuracy of the equations is impaired by the approximate values introduced, and as the intervals are supposed equal, the method is not always applicable.

The reduction of the problem to the solution of a quadratic equation is not new; it was already accomplished by Sejour, and also by Olbers. (See Delambre, Astr. du XVIII^e Siècle, p. 733.)

In all the methods which we have here alluded to, it is assumed that the comet moves in a parabolic orbit, and their success is owing entirely to this assumption. But it is certain that the true orbit of some comets, and most probably of all comets, is an ellipse, and not a parabola. The question will therefore naturally suggest itself, whether it would not be better to investigate an ellipse at once, than to concern ourselves about a parabolic orbit, which, with whatever care it may be computed, can never be perfectly exact, nor even be expected to satisfy observations made at considerably distant intervals. To this it may be replied, that on the first appearance of a comet it can seldom be of any use to seek an ellipse, the computation of which is greatly more tedious. For every practical purpose it is sufficient to have a parabola which represents the observations within a few seconds; first, because in the case of a comet the observations are scarcely ever more accurate than the computations founded on this assumption; and, secondly, because the approximation obtained by supposing the orbit to be a parabola, will enable us to determine the position of the node, the inclination of the orbit, and the perihelion distance, more exactly than we can expect to find them on the return of the comet, the perturbations of the planets giving rise to variations in the elements of the orbit probably exceeding the errors of the first computation. It is enough that we are able to recognise its identity. When this has been done we have the periodic time, and consequently the greater axis

of the orbit. With these data we are enabled to compute the other elements by the methods employed for determining the elliptic orbits of the planets. Comet.

The problem of determining a cometary orbit resolves itself into two parts. In the first it is required to assign an orbit which will represent all the observations made during a single appearance of the comet; and in the second, to compute the effects of the disturbing forces of the planets, so as to predict the changes which will take place in the form and position of the orbit during an indefinite number of revolutions. The perturbations, however, which are computed in the same manner whether the disturbed body be a comet or a planet, form a problem of PHYSICAL ASTRONOMY by far too important to be undertaken in this place. We will therefore confine ourselves to the approximate determination of a parabolic orbit from three observations. The method which we have followed, as we have already intimated, is that which was proposed by Legendre in the second Supplement to his Nouvelles Méthodes.

1. Let $C^0,\; C,\; C\text{'}$ be the places of the comet at the instants of the first, second, and third observations respectively. Let the interval between the first and second observations be denoted by $\backslash delta$, and that between the second and third by $\backslash delta\text{'}$; then, if the epoch, or instant from which the time is reckoned, be made to fall on the second observation, and be denoted by $t$, the epochs of the first and third observations will be respectively $t\; -\; \backslash delta$, and $t\; +\; \backslash delta\text{'}$. Next, let $x,\; y,\; z$, be the rectangular co-ordinates of $C$ referred to the plane of the ecliptic and a straight line drawn through $S$, the centre of the sun, to the first point of Aries, the origin of the co-ordinates being at $S$. Similarly, let $x^0,\; y^0,\; z^0$, be the co-ordinates of $C^0$, and $x\text{'},\; y\text{'},\; z\text{'}$, those of $C\text{'}$. The first step of the process is to find expressions for $x^0,\; y^0,\; z^0$, and $x\text{'},\; y\text{'},\; z\text{'}$, in terms of $x,\; y,\; z$, the latter being regarded as functions of the time $t$.

By Taylor's theorem we have

in which series the intervals $\backslash delta$ and $\backslash delta\text{'}$ may be regarded as so small, that the values of $x^0$ and $x\text{'}$ will be determined with sufficient exactness when the terms affected with the fifth, and all the higher powers of $\backslash delta$ and $\backslash delta\text{'}$, are omitted.

Let $r^0,\; r,\; r\text{'}$, be the three radii vectors of the comet corresponding to the three observations, or the distances $SC^0,\; SC,\; SC\text{'}$. The accelerating forces acting on the comet at $C$ in the directions of the co-ordinates are expressed by the equations

the mass of the sun being assumed to be unit, and the mass of the comet being neglected in comparison of that of the sun.

Comet. By differentiating the first of the above equations, we get

or, by making $\backslash frac\{r\; dr\}\{dt\}\; =\; k$,

Differentiating again, and making the necessary reductions, we obtain

If we now substitute these values of $\backslash frac\{d^2x\}\{dt^2\}$, $\backslash frac\{d^2x\}\{dt^2\}$, and $\backslash frac\{d^2x\}\{dt^2\}$, in the foregoing series for $x^2$ and $x\text{'}$, we shall have, on rejecting the superfluous terms,

For the sake of abridging, let us assume

and we shall have

It is evident that $y^2$, $y\text{'}$ and $z^2$, $z\text{'}$ may be expressed exactly in the same manner; hence the co-ordinates of the comet at the first and third observations are given in terms of its co-ordinates at the middle observation and their differential co-efficients, by the following system of equations, due to Lagrange:

If from these equations we eliminate $\backslash frac\{dx\}\{dt\}$, $\backslash frac\{dy\}\{dt\}$, $\backslash frac\{dz\}\{dt\}$, we shall find, on making $w\; =\; v\text{'}u\text{'}\; -\; v\text{'}u$,

$u\text{'}x^2\; -\; ux\text{'}\; =\; vx$, $u\text{'}y^2\; -\; uy\text{'}\; =\; vy$, $u\text{'}z^2\; -\; uz\text{'}\; =\; vz$; (2)
but from the values of $v$, $u$, $v\text{'}$, $u\text{'}$, given above, we have evidently

therefore, by eliminating $v$, $u$, and $w$, from equations (2), there results

$x(y^2\; z\text{'}\; -\; y\; z\text{'}^2)\; +\; y(z^2\; x\text{'}\; -\; z\; x\text{'}^2)\; +\; z(x^2\; y\text{'}\; -\; x\; y\text{'}^2)\; =\; 0$,
an equation to a plane, which denotes that the three points $C^0$, $C$, $C\text{'}$ are situated in the same plane with $S$, the origin of the co-ordinates.

2. Let the chord of the arc $C^0\; C\text{'}$ intersect the radius

SC in D, and let the ratio of $C^0\; D$ to $CD$ be that of $\backslash lambda$ to $\backslash lambda\text{'}$. Make $\backslash lambda\; +\; \backslash lambda\text{'}\; =\; 1$, and the chord $C^0\; C\text{'}\; =\; c$; we have then $C^0\; D\; =\; \backslash lambda\; c$ and $CD\; =\; \backslash lambda\text{'}\; c$. Further, let $DC\; =\; e\; r$, so that $SD\; =\; r(1-e)$, $e$ being an infinitely small quantity of the second order in respect of $\backslash theta$ and $\backslash theta\text{'}$, which are supposed infinitesimals of the first order.

In order to find the ratio of $\backslash lambda$ to $\backslash lambda\text{'}$, suppose the three points $C^0$, $D$, $C\text{'}$, to be projected on the plane of the ecliptic; then the projections of the lines $C^0\; D$ and $CD$ will necessarily be to one another in the same ratio as these lines themselves, that is, as $\backslash lambda$ to $\backslash lambda\text{'}$. But the co-ordinates of the projections of $C^0$, $D$, and $C\text{'}$, in the direction of the axis of $x$, are respectively $x^2$, $(1-e)x$, and $x\text{'}$; therefore also $(1-e)x\; -\; x^2\; :\; x\text{'}\; -\; (1-e)x\; ::\; \backslash lambda\; :\; \backslash lambda\text{'}$, whence

and, since $\backslash lambda\; +\; \backslash lambda\text{'}\; =\; 1$,

Substituting in this last equation the values of $x\text{'}$ and $x^2$ given by equations (1), we get

and by proceeding in the same manner, we shall find relatively to $y$ and $z$,

In these last equations the quantities $x\; y\; z$, $\backslash frac\{dx\}\{dt\}$, $\backslash frac\{dy\}\{dt\}$, $\backslash frac\{dz\}\{dt\}$ are independent of any particular values of $v$, $u$, $v\text{'}$, $u\text{'}$; they cannot therefore hold true, unless we have also the two equations of condition $\backslash lambda\; v\text{'}\; +\; \backslash lambda\text{'}\; v\; =\; 1\; -\; e$, and $\backslash lambda\; u\text{'}\; +\; \backslash lambda\text{'}\; u\; =\; 0$. Joining to these the equation $\backslash lambda\; +\; \backslash lambda\text{'}\; =\; 1$, we shall have, in order to determine $\backslash lambda$, $\backslash lambda\text{'}$, and $e$, the following system of equations:

From the second and third of these we get $\backslash lambda\; =\; \backslash frac\{u\}\{u\text{'}\; -\; u\}$,

$\backslash lambda\text{'}\; =\; \backslash frac\{u\text{'}\}\{u\text{'}\; -\; u\}$; and consequently from the first, $1\; -\; e$

The accuracy of the values now found of $\backslash lambda$, $\backslash lambda\text{'}$, and $e$, will depend on the number of terms of the series represented by $u$, $u\text{'}$, and $w$, that are taken into account. We shall distinguish three different degrees of approximation.

1. Retaining only the first terms of the series, or those which depend on the simple powers of $\backslash theta$ and $\backslash theta\text{'}$, we have $u\; =\; -\backslash theta$, $u\text{'}\; =\; \backslash theta$, $w\; =\; \backslash theta\; +\; \backslash delta$, and $u\text{'}\; -\; u\; =\; \backslash theta\; +\; \backslash delta$, whence

consequently $\backslash frac\{\backslash lambda\}\{\backslash lambda\text{'}\}\; =\; \backslash frac\{\backslash theta\}\{\backslash theta\text{'}\}$ that is to say, the chord which joins

the places of the comet at the extreme observations is divided, by the radius vector belonging to the middle observation, into segments proportional to the intervals between the observations. This is the approximation of Newton

Comet. and Olbers. In the method we are about to follow, the results thus obtained serve only to direct the subsequent calculations.

II. By including a second term of the series represented by $u$, $u\text{'}$, and $w$, we have

whence

and therefore by division

In like manner, from the equation $\backslash lambda\text{'}\; =\; \backslash frac\{u\text{'}\}\{u\text{'}-u\}$ we find

Whence we have in this case the following values of $\backslash lambda$, $\backslash lambda\text{'}$, and $e$, namely,

The values of $\backslash lambda$, $\backslash lambda\text{'}$, and $e$, thus found, will be sufficiently exact in ordinary cases; but it will sometimes be necessary to carry the approximation farther, and to include the third terms of the series represented by $u$, $u\text{'}$, and $w$.

III. In order to obtain convenient expressions when a third term of the series is taken into account, it will be necessary to introduce the extreme radii vectors $r^0$ and $r\text{'}$. For this purpose we may suppose $r^0\; =\; r\; -\; dr$, and $r\text{'}\; =\; r\; +\; dr$; but we have already assumed $k\; =\; \backslash frac\{dr\}\{dt\}$,

therefore $dr\; =\; \backslash frac\{kdt\}\{r\}\; =\; \backslash frac\{k\backslash theta\}\{r\}$. Hence

whence $(r^0\; r)^\{\backslash frac\{3\}\{2\}\}\; =\; r^2\; -\; \backslash frac\{3\}\{2\}r\; k\backslash theta$, and consequently

In the same manner we find

Substituting these expressions in the series denoted by $u$ and $u\text{'}$, we get

consequently,

whence by division,

By proceeding in a similar manner with respect to $\backslash lambda\text{'}$, we find

To find $e$, we have in this case

hence, on dividing and neglecting the terms which involve the fourth and higher powers of $\backslash theta$ and $\backslash theta$,

But $r^0\; =\; r\; -\; \backslash frac\{k\backslash theta\}\{r\}$, $r\text{'}\; =\; r\; +\; \backslash frac\{k\backslash theta\}\{r\}$, therefore $r^0\; r\text{'}\; =\; r^2\; +\; k$

$(\backslash theta\; -\; \backslash theta)r\; -\; \backslash frac\{k^2\backslash theta\backslash theta\}\{r\}$. The last term of this expression is a

quantity of the order of those that may be neglected; therefore simply

Substituting this in the above expression for $1\; -\; e$, it becomes

It follows, therefore, that this third degree of approximation gives

3. We now proceed to apply the above results to the solution of the problem.

Let $\backslash alpha^0,\; \backslash alpha,\; \backslash alpha\text{'}$, be the three geocentric longitudes of the comet at the instants of the first, second, and third observations respectively; $\backslash beta^0,\; \backslash beta,\; \backslash beta\text{'}$, its three geocentric latitudes; $\backslash rho^0,\; \backslash rho,\; \backslash rho\text{'}$, the three curvate distances, that is to say, the three distances of the comet from the earth projected on the plane of the ecliptic; $A^0,\; A,\; A\text{'}$, the three corresponding heliocentric longitudes of the earth; and $R^0,\; R,\; R\text{'}$, the three radii vectors of the earth. Finally, let $X^0\; Y^0,\; XY,\; X\text{'}Y\text{'}$, be the co-ordinates of the earth, referred to the centre of the sun, so that

For the sake of abridging, let us assume

then, on transferring the origin of the co-ordinates from the centre of the sun to the earth, we shall have relatively to the points C', C, C,

It has already been shown (2) that

and it is obvious that the same condition must be fulfilled in regard to $y$ and $z$, or the projection of the chord C'C' on the other two co-ordinate planes. Hence we have the following system of equations:—

Substituting in these last equations (11) the values of $x,\; y,\; z,\; x^0,\; y^0,\; z^0,\; x\text{'},\; y\text{'},\; z\text{'}$, given in equations (10), and making at the same time

on which equations the solution of the problem essentially depends. Before we proceed to their discussion, however, it will be necessary to find more convenient expressions for $M$ and $N$.

4. In order to express $X^0,\; Y^0,\; X\text{'},\; Y\text{'}$, the co-ordinates of the earth, which belong to the extreme observations, in a function of the time, and of the co-ordinates $X,\; Y$ relative to the time $\backslash theta\; =\; 0$, let us assume, as in § 1,

and we shall have

the symbols $V,\; U,\; R,\; K$, having now the same signification relatively to the orbit of the earth which $v,\; u,\; r,\; k$, had before relatively to the orbit of the comet.

In consequence of these last equations, and of the equations (4), we have

and substituting these values, in the first of equations (12),

Substituting now for $u\text{'},\; u,\; V,\; V\text{'},\; U,\; U\text{'}$, their values in terms of $\backslash theta$ and $\backslash theta\text{'}$, we obtain $u\text{'}V\; -\; uV\text{'}\; =\; \backslash theta\; +\; \backslash theta\text{'}\; -\; \backslash frac\{\backslash theta\backslash theta\text{'}\}\{2R^3\}(\backslash theta\; +\; \backslash theta\text{'})$

but

therefore, by dividing, and retaining only the terms which do not exceed the third order in respect of $\backslash theta^2$ and $\backslash theta\text{'}$,

whence

In like manner we have

therefore, by division,

The value of $M$ therefore becomes

It will be remarked that the quantity $\backslash frac\{\backslash theta\backslash theta\text{'}\}\{2R^3\}\; -\; \backslash frac\{K\backslash theta\backslash theta\text{'}\}\{2R^5\}(\backslash theta\; -\; \backslash theta\text{'})$

is exactly analogous to that which has already been denoted by $e$. Let us therefore suppose it to be represented by $E$, and the last equation will become

but on rejecting quantities involving higher powers than the third of $\backslash theta$ and $\backslash theta\text{'}$, we have obviously

therefore ultimately

Since the epoch or instant from which the time is reckoned is supposed to fall on the second observation, it is evident that $\backslash frac\{\backslash theta\; -\; \backslash theta\text{'}\}\{3\}$ is the instant corresponding to the mean time of the three observations. Let the value of $X$ at this instant be denoted by $X\_1$, and the corresponding values of $R$ and $A$ by $R\_1$ and $A\_1$, then

By proceeding exactly in the same manner with regard to $N$, we shall find

therefore, on substituting for $X$ and $Y$ their values in terms of $R$ and $A$, we have at the mean time of the three observations

5. We now resume the equations (13). The unknown quantities which they contain are the three curvate distances $\backslash ell^0,\; \backslash ell,\; \backslash ell\text{'}$, together with $r$, the radius vector of the comet, which is involved in the values of $\backslash lambda,\; \backslash lambda\text{'},$ and $e$. They may be combined in various ways, and the different combinations will afford so many different expressions for $\backslash ell^0,\; \backslash ell,$ and $\backslash ell\text{'}$, which will be more or less easily computed according to the circumstances of the case. The most obvious solu-

Comet. tion will be obtained by eliminating $\backslash xi$, $\backslash zeta$, and $\backslash xi\text{'}$. For this purpose the third of equations (13) gives

and on substituting this value of $\backslash xi\text{'}$ in the first and second of these equations, they become respectively

On eliminating $\backslash lambda\; \backslash xi\text{'}$ from the two equations now found, we get

an equation which becomes by reduction

and on restoring the values of $M$, $N$, $m$, $n$, $p$, &c. given by equations (14) and (9), and recollecting that $\backslash sin.\; (x-y)\; =\; \backslash sin.\; x\; \backslash cos.\; y\; -\; \backslash cos.\; x\; \backslash sin.\; y$, we get

Let us assume
$P\; =\; \backslash sin.\; (A\_1\; -\; a\text{'})\; \backslash tan.\; b\text{'}\; -\; \backslash sin.\; (A\_1\; -\; a)\; \backslash tan.\; b\text{'}$
$Q\; =\; \backslash sin.\; (a\text{'}\; -\; a)\; \backslash tan.\; b\; +\; \backslash sin.\; (a\; -\; a\text{'})\; \backslash tan.\; b\text{'}\; +\; \backslash sin.\; (a\; -\; a\text{'})\; \backslash tan.\; b\text{'}$;
this assumption gives

but on carrying the approximation only to quantities of the third order in respect of the time, $\backslash frac\{e-E\}\{1-e\}\; =\; \backslash frac\{\backslash theta\; \backslash theta\}\{2\}\; \backslash left(\; \backslash frac\{1\}\{r^3\}\; -\; \backslash frac\{1\}\{R^3\}\; \backslash right)$;

therefore if we make $\backslash frac\{\backslash theta\; \backslash theta\; P\; R\_1\}\{2Q\}\; =\; h$, the last equation will become

6. The equation which has now been obtained contains two unknown quantities $\backslash xi$ and $r$; but another equation between the same variables may be readily found from the equations (10). These give, on restoring the values of $m$, $n$, $p$, $X$, and $Y\_1$

whence, by squaring and adding
$x^2\; +\; y^2\; +\; z^2\; =\; \backslash xi^2\; (1\; +\; \backslash tan^2\; b)\; +\; 2\backslash xi\; R\; \backslash cos.\; (A-a)\; +\; R^2$;
but $x^2\; +\; y^2\; +\; z^2\; =\; r^2$; and $1\; +\; \backslash tan^2\; b\; =\; \backslash frac\{1\}\{\backslash cos^2\; b\}$;
therefore

We have now arrived at two equations between $r$ and $\backslash xi$, by means of which the values of those quantities become known. In order to eliminate $r$, let equation (16) be put under the form

the square of which is

then, by substituting in this the value of $r^6$ given by equation (17) raised to the cube, there results

an equation of the eighth degree in respect of the unknown quantity $\backslash xi$. It is evident, however, that on being developed, $h^2\; R^6$ will disappear from both sides of it, after which the whole will be divisible by $\backslash xi$; it belongs, therefore, in fact, only to the seventh degree. On being resolved by approximation it will give a value of $\backslash xi$, but in practice it is more commodious to determine $r$ and $\backslash xi$ simultaneously from the equations (16) and (17), by the ordinary methods of false position.

7. Although the equations (16) and (17) thus appear to lead to a complete solution of the problem, it seldom happens in practice that an orbit can be determined in this manner. The reason of this is, that the quantity which we have represented by $Q$ is always very small (it is not difficult to prove that it is of the third order in respect of $\backslash theta$), and liable to be greatly affected by the errors of observation. If $Q\; =\; 0$, we have the equation

$\backslash sin.\; (a\text{'}\; -\; a)\; \backslash tan.\; b\; +\; \backslash sin.\; (a\; -\; a\text{'})\; \backslash tan.\; b\text{'}\; +\; \backslash sin.\; (a\; -\; a\text{'})\; \backslash tan.\; b\text{'}\; =\; 0$, which denotes that the three apparent places of the comet are situated in the same great circle of the sphere. The value of $Q$ therefore depends solely on the deviation of the comet's apparent path from a great circle of the heavens, during the short interval between the extreme observations; and this deviation is in general so small, that a slight error of observation, which it is impossible to guard against, may cause it to become evanescent, or even to change its sign. On this account the equation (16), which is called by Legendre the doubtful equation, can only be employed in very favourable circumstances; and it becomes necessary, in the greater number of cases, to have recourse to other methods of solution. Some conclusions may, however, be deduced from it, which will materially assist in guiding the subsequent calculations.

In the first place, as the curtate distance $\backslash xi$ is necessarily a finite quantity, if the denominator of the fraction expressed by $h$ vanishes, that is, if $Q\; =\; 0$, the numerator must vanish at the same time. Now this can happen only

in two ways: either by supposing $P\; =\; 0$, or $\backslash frac\{1\}\{r^3\}\; =\; \backslash frac\{1\}\{R^3\}$.

If we suppose $P\; =\; 0$, we have the equation

$\backslash sin.\; (A\_1\; -\; a\text{'})\; \backslash tan.\; b\text{'}\; -\; \backslash sin.\; (A\_1\; -\; a)\; \backslash tan.\; b\text{'}\; =\; 0$, which indicates that the place of the sun at the instant $\backslash frac\{\backslash theta\; -\; \backslash theta\}\{3\}$, corresponding to the mean time of the three observations,

is situated in the same great circle of the sphere with the three apparent places of the comet. Now it is easy to see, that although the sun may not be exactly in the same great circle with the three apparent places of the comet, yet if it is near one of the intersections of that circle with the ecliptic, the quantity $P$ will be very small, and the equation (16) in consequence will not lead to any satisfactory result.

In the second place, if $Q$ vanishes while the value of $P$ continues so great that its sign would not be changed by supposing its component parts to vary to the extent of the

errors of observation, we must then have $\backslash frac\{1\}\{r^3\}\; =\; \backslash frac\{1\}\{R^3\}$ or $r\; =$

$R$; that is to say, the comet and the earth are at the same distance from the sun. This circumstance, therefore, also renders equation (16) indeterminate. It may be remarked, however, that in this case we obtain an easy approximation to the value of $\backslash xi$ from equation (17), which, when $r$ is equal to $R$, becomes

Comet. where $180\; +\; A$ is the longitude of the sun at the instant of the middle observation.

In the third place, when the quantities $P$ and $Q$ are both so great that their signs can be determined without ambiguity, equation (16) will show whether the distance of the comet from the sun is greater or less than the radius vector of the earth. For since $\backslash xi$ is necessarily positive, the sign of $\backslash frac\{P\}\{Q\}$ must be the same as that of $e\; -\; E$ or

$\backslash frac\{1\}\{r^2\}\; -\; \backslash frac\{1\}\{R^2\}$; consequently, the sign of $\backslash frac\{P\}\{Q\}$ being known, we also know whether $r$ is greater or less than $R$.

8. Since the state of the observations will seldom permit equation (16) to be employed with advantage, it becomes necessary to attempt some other combination of the three fundamental equations (13). Instead of eliminating the three curtate distances $\backslash xi$, $\backslash xi^0$, $\backslash xi\text{'}$ from these equations, we may determine two of them, for example $\backslash xi^0$ and $\backslash xi\text{'}$, in terms of the third, $\backslash xi$; by which means we shall obtain at least the ratios of these distances to one another. This may be effected in various ways; but we shall confine ourselves to the two following combinations, the one or other of which will be applicable in every case.

Let the first of the equations (13) be multiplied by $N$, and the second by $M$; the difference of the products is $\backslash lambda\; \backslash xi\; (Nm\text{'}\; -\; Mn\text{'})\; +\; \backslash lambda\text{'}\; \backslash xi^0\; (Nm^0\; -\; Mn^0)\; -\; (1-e)\; \backslash xi\text{'}\; (Nm\; -\; Mn)\; =\; 0$. Combining this with the third equation, so as to eliminate first $\backslash xi\text{'}$ and then $\backslash xi^0$, we get

and, on substituting the values of $M$, $N$, $m$, &c. before found,

These two equations will give the ratios $\backslash frac\{\backslash xi^0\}\{\backslash xi\}$, $\backslash frac\{\backslash xi\text{'}\}\{\backslash xi\}$, in all

cases when the common denominator (the quantity we have already denoted by $P$) is not so small as to render the expressions indeterminate. In general they will be most applicable when the motion of the comet in latitude is greater than its motion in longitude.

The other combination to which we have alluded, is formed by employing only the two first of the equations (13). Let the first be multiplied by $n\text{'}$, and the second by $m\text{'}$; the difference of the products gives

In like manner, if we multiply the first by $n^0$, and the second by $m^0$, in order to eliminate $\backslash xi^0$, we find

On substituting the values of $M$, $N$, $m$, $n$, &c. in these last equations, they become

The equations now found will be most applicable when

the motion of the comet in longitude is greater than its motion in latitude.

It may be proper to remark, that on employing the equations (18) or (19) to determine the ratio of the curtate distances, we may suppose in the first trials $e\; =\; 0$,

$E\; =\; 0$, $\backslash lambda\; =\; \backslash frac\{\backslash theta\}\{\backslash theta\; +\; \backslash theta\text{'}\}$, $\backslash lambda\text{'}\; =\; \backslash frac\{\backslash theta\text{'}\}\{\backslash theta\; +\; \backslash theta\text{'}\}$. On this supposition the equations (19) become simply

11. From what has now been shown it may be inferred that the equations (13) can in general be used only in determining the ratios of $\backslash xi^0$ and $\backslash xi\text{'}$ to $\backslash xi$; on account of the smallness and consequent uncertainty of the denominator $Q$, they leave the absolute value of $\backslash xi$ indeterminate. It is therefore necessary to seek some other equation among the variables employed, before we can obtain a complete solution of the problem. For this purpose we must have recourse to some of the properties connected with the comet's motion in its orbit. No hypothesis has as yet been made respecting the nature of the orbit. If we suppose it to be a parabola, the laws of parabolic motion will give a new equation, without introducing any other variable than those which have already been employed, which may either be combined with equation (16), for the purpose of determining $r$ and $\backslash xi$ directly, or which may be employed to correct the results obtained by assigning a hypothetical value to $\backslash xi$ in the equations (18) or (19). There are various methods of introducing the condition derived from the nature of the parabolic motion; but, in the method of solution which we here follow, it is most convenient to employ the theorem of Lambert, which gives the time of describing a parabolic arc in terms of its chord, and the two radii vectors drawn to its extremities. Before demonstrating this theorem, however, it will be proper to show how the chord may be computed.

Let $e$ be the chord which joins $C^0\; C\text{'}$. The co-ordinates of its extremities being $x\text{'}\; y\text{'}\; z\text{'}$, $x^0\; y^0\; z^0$, we have

that is,

But by reason of the equations (8) and (10), $x\text{'}x^0\; =\; \backslash xi\; \backslash xi^0\; n\text{'}\; m^0\; +\; m^0\; \backslash xi^0\; R\text{'}\; \backslash cos.\; A\text{'}\; +\; n\text{'}\; \backslash xi^0\; R\text{'}\; \backslash cos.\; A^0\; +\; R\text{'}\; R^0\; \backslash cos.\; A\text{'}\; \backslash cos.\; A^0$, $y\text{'}y^0\; =\; \backslash xi\; \backslash xi^0\; n\text{'}\; n^0\; +\; n^0\; \backslash xi^0\; R\text{'}\; \backslash sin.\; A\text{'}\; +\; n\text{'}\; \backslash xi^0\; R\text{'}\; \backslash sin.\; A^0\; +\; R\text{'}\; R^0\; \backslash sin.\; A\text{'}\; \backslash sin.\; A^0$, and $z\text{'}z^0\; =\; \backslash xi\; \backslash xi^0\; p\text{'}\; p^0$, therefore, by adding and substituting for $m$, $n$, $p$, &c. their values given in equations (9)

hence, denoting the right hand side of this equation by $F$,

Comparing this last equation with

which is given by the triangle $C^0\; S\; C\text{'}$, we have $F\; =\; r\text{'}r\text{'}\; \backslash cos.\; \backslash phi$, or $\backslash cos.\; \backslash phi\; =\; \backslash frac\{F\}\{r\text{'}r\text{'}\}$, $\backslash phi$ being the angle included between the two radii vectors $r\text{'}$ and $r^0$.

12. We now proceed with the demonstration of Lambert's theorem.

According to the doctrine of central forces, when a

Comet. body describes an orbit in virtue of a force varying inversely as the square of the distance, the square of the velocity at the distance $r$ is given by the equation

in which $a$ is the major axis of the orbit, and the sum of the masses is supposed equal to unit. In the case of the parabola, the major axis is infinite, and the equation becomes simply

Let $D$ be the perihelion distance $SP$, and $\backslash psi$ the true anomaly, or the angle $PSC$. We have then $x\; =\; r\; \backslash cos\; \backslash psi$, $y\; =\; r\; \backslash sin\; \backslash psi$, whence $\backslash frac\{dx^2\}\{dt^2\}\; +\; \backslash frac\{dy^2\}\{dt^2\}\; =\; \backslash frac\{dr^2\; +\; r^2\; d\backslash psi^2\}\{dt^2\}$, and consequently

But from the property of the parabola $r\; =\; 2D\; -\; r\; \backslash cos\; \backslash psi$; whence $r(1\; +\; \backslash cos\; \backslash psi)\; =\; 2D$, or $r\; \backslash cos^2\; \backslash frac\{1\}\{2\}\backslash psi\; =\; D$, therefore

Let $\backslash tan\; \backslash frac\{1\}\{2\}\backslash psi\; =\; u$; then $dr\; =\; 2D\; du$; and $du\; =\; \backslash frac\{\backslash frac\{1\}\{2\}d\backslash psi\}\{\backslash cos^2\; \backslash frac\{1\}\{2\}\backslash psi\}\; =\; \backslash frac\{1\}\{2\}d\backslash psi(1\; +\; \backslash tan^2\; \backslash frac\{1\}\{2\}\backslash psi)$, whence $d\backslash psi\; =\; \backslash frac\{2du\}\{1+u^2\}$. Substituting these expressions in the equation $\backslash frac\{2\}\{r\}\; =\; \backslash frac\{dr^2\; +\; r^2\; d\backslash psi^2\}\{dt^2\}$, it

becomes $dt^2\; =\; 2D^3(1+u^2)^3\; du^2$, whence $dt\; =\; \backslash sqrt\{2D^3\}(1+u^2)\; du$; which, on being integrated, becomes

Let $u\text{'}$, $u^\backslash circ$ be the values of $u$ in this equation at the times $t\; +\; \backslash theta$, $t\; -\; \backslash theta$ respectively; we have then

and, by subtracting,

In order to transform this expression into another containing only $r^\backslash circ$, $r\text{'}$ and $c$, let $\backslash psi^\backslash circ$ be the anomaly corresponding to $C^\backslash circ$, $\backslash psi$ that corresponding to $C$, $\backslash varphi\; =\; \backslash psi\; -\; \backslash psi^\backslash circ\; =$ angle $C^\backslash circ\; SC$, and $\backslash tan\; \backslash frac\{1\}\{2\}\backslash varphi\; =\; v$. We have then $\backslash psi\; =\; \backslash psi^\backslash circ\; +\; \backslash varphi$, and consequently $\backslash tan\; \backslash frac\{1\}\{2\}\backslash psi\; =\; \backslash tan\; \backslash frac\{1\}\{2\}(\backslash psi^\backslash circ\; +\; \backslash varphi)\; =\; \backslash frac\{\backslash tan\; \backslash frac\{1\}\{2\}\backslash psi^\backslash circ\; +\; \backslash tan\; \backslash frac\{1\}\{2\}\backslash varphi\}\{1\; -\; \backslash tan\; \backslash frac\{1\}\{2\}\backslash psi^\backslash circ\; \backslash tan\; \backslash frac\{1\}\{2\}\backslash varphi\}$, that is, $u\text{'}\; =\; \backslash frac\{u^\backslash circ\; +\; v\}\{1\; -\; u^\backslash circ\; v\}$. Now, if we put

we shall have evidently

therefore, substituting $\backslash frac\{u^\backslash circ\; +\; v\}\{1\; -\; u^\backslash circ\; v\}$ for $u\text{'}$,

that is,

hence the formula for the time becomes

Again, the triangle $C^\backslash circ\; SC$ gives the equation

but $\backslash tan\; \backslash frac\{1\}\{2\}\backslash varphi\; =\; v$; therefore, from the trigonometrical formulae, $\backslash cos\; \backslash varphi\; =\; \backslash frac\{1-v^2\}\{1+v^2\}$, whence, substituting and reducing,

For the sake of abridging, we shall put

whence $r\text{'}\; +\; r^\backslash circ\; =\; s\; +\; d$, $c\; =\; s\; -\; d$, and $(r\text{'}\; +\; r^\backslash circ)^2\; -\; c^2\; =\; 4sd$. Comparing these with the above equation, we

Now the polar equation to the parabola gives

therefore $sd\; =\; \backslash frac\{r\text{'}r^\backslash circ\}\{1+v^2\}\; =\; D^2\; \backslash frac\{(1+u^\{\backslash circ\; 2\})^2\}\{(1-u^\backslash circ\; v)^2\}$, and consequently,

From the same values of $r^\backslash circ$ and $r\text{'}$ we get

but $r\text{'}\; +\; r^\backslash circ\; =\; s\; +\; d$, and $\backslash frac\{1+u^\{\backslash circ\; 2\}\}\{1-u^\backslash circ\; v\}\; =\; \backslash frac\{\backslash sqrt\{sd\}\}\{D\}$, therefore

Substituting these values of $v$ and $\backslash frac\{1+u^\{\backslash circ\; 2\}\}\{1-u^\backslash circ\; v\}$ in the equation for the time, it becomes

that is

or

and, finally, on restoring the values of $s$ and $d$,

This is the very elegant theorem found by Euler, but commonly attributed to Lambert, by means of which we are enabled to correct the hypothetical values successively given to $\backslash varphi$, till a value of that quantity is found which will satisfy all the equations as nearly as the observations will admit.

13. We shall now collect into one view the different processes indicated in the preceding investigations; and in order to throw every necessary light on the subject, we shall give at the same time an example of the actual computation of an orbit from Legendre. The comet is that of 1805, and the data from which the computation is made are contained in the following table:

Comet. I. The first step is to compute the arc described by the mean motion of the earth in the time $\backslash theta\; +\; \backslash theta\text{'}$. Let this be represented by $\backslash Theta$; then since $\backslash theta$ and $\backslash theta\text{'}$ are expressed in mean solar days, we have $\backslash Theta\; =\; \backslash frac\{2\backslash pi(\backslash theta\; +\; \backslash theta\text{'})\}\{365.25638\}$, $2\backslash pi$ being the circumference of a circle whose radius is unit, and 365.25638 the mean solar days in a sidereal year. The logarithm of $\backslash Theta$ will therefore be found by adding the constant logarithm 8.2355821 (which is the logarithm of $\backslash frac\{2\backslash pi\}\{365.25638\}$) to the logarithm of $\backslash theta\; +\; \backslash theta\text{'}$.

From the above table we have $\backslash theta\; =\; 8.00182$ days, $\backslash theta\text{'}\; =\; 4.03418$ days, $\backslash theta\; +\; \backslash theta\text{'}\; =\; 12.03600$ days; hence taking the logarithms of these numbers, and adding 8.2355821, we have the following results expressed in logarithms:

It may be here remarked, that although the computations are considerably shortened when $\backslash theta\; =\; \backslash theta\text{'}$, a condition which may in general be satisfied by interpolation, yet it is preferable to avoid any preliminary calculations, and to employ the direct results of observation. The intervals, however, should not be greatly different; and as the solution is founded on the supposition that quantities of the fourth order in respect of the time may be neglected, the longest interval cannot safely exceed ten days. In the present example, $\backslash theta$ is considerably greater than $\backslash theta\text{'}$; it therefore affords no facilities for computation, and the approximations must be carried as far as the method admits.

II. The places of the sun at the time of the respective observations are next to be taken from the solar tables. These places, increased by $180^\backslash circ$, give the three longitudes of the earth $A^\backslash circ$, $A\text{'}$, $A\text{'}\text{'}$. The same tables also give the corresponding distances $R^\backslash circ$, $R\text{'}$, $R\text{'}\text{'}$; and the quantities $A\_1$ and $R\_1$, which denote the longitude and radius vector of the earth at the time $t\; +\; \backslash frac\{1\}\{2\}(\backslash theta\; +\; \backslash theta\text{'})$, corresponding to the mean epoch of the three observations, will be obtained with sufficient accuracy by assuming $A\_1\; =\; \backslash frac\{1\}\{2\}(A^\backslash circ\; +\; A\text{'}\; +\; A\text{'}\text{'})$, and $\backslash log.\; R\_1\; =\; \backslash frac\{1\}\{2\}\; \backslash log.\; (R^\backslash circ\; R\text{'}\; R\text{'}\text{'})$.

At the epoch given in the above table, we find from the solar tables,

By taking a mean among these numbers we shall have

III. With the results now found, we may proceed to compute the quantities $P$, $Q$ from the expressions given in § 5, namely,

For this purpose the following logarithms are required:

On computing $P$ and $Q$ by means of these numbers, we find $P\; =\; 0.0046585$, and $Q\; =\; 0.0002617$. The signs of both are positive, and though $Q$ is thus a very small number, yet its sign could not become negative without sup-

posing errors in the observations amounting to four or five minutes. By reason of the smallness of $P$ and $Q$, no use can be made of the equation (16); we can only infer from it, since $P$ and $Q$ have the same sign, that ; that is to say, the distance of the comet from the sun is less than the sun's distance from the earth.

IV. It is now necessary to make a supposition with regard to the value $e$. In general a few trials will suffice to give an approximate value, by means of which $e^\backslash circ$ and $e\text{'}$ may be computed from the formulæ (18) or (19), according to the different cases. In the first trials we may suppose $E\; =\; 0$,

$e\; =\; 0$, $\backslash lambda\; =\; \backslash frac\{\backslash theta\}\{\backslash theta\; +\; \backslash theta\text{'}\}$, $\backslash lambda\text{'}\; =\; \backslash frac\{\backslash theta\text{'}\}\{\backslash theta\; +\; \backslash theta\text{'}\}$; but when approximations to the values of $e^\backslash circ$ and $e\text{'}$ have been obtained on this supposition, we must employ the more exact expressions given in the formulæ (6.) In the present case, on account of the great inequality between $\backslash theta$ and $\backslash theta\text{'}$, the values of $e^\backslash circ$ and $e\text{'}$ deduced from the supposition of $E\; =\; 0$, $e\; =\; 0$, &c. would not have that degree of approximation which would render them of any use in directing the subsequent hypotheses; we may therefore have recourse at once to the second degree of approximation. The formulæ (6) give

from which we deduce

In order to convert these formulæ into numbers, we must recollect that if $b$ is a small number in comparison of $a$, we have $\backslash log.\; (a\; +\; b)\; =\; \backslash log.\; a\; +\; \backslash frac\{b\}\{a\}\; M$, where $M$ denotes the modulus of the tables, or the number 4342945, the logarithm of which is 6.6377843; and reciprocally, if we have $\backslash log.\; x\; =\; \backslash log.\; a\; +\; b$, then $x\; =\; a\; +\; \backslash frac\{ab\}\{M\}$. According to these formulæ

Now $\backslash log.\; \backslash frac\{\backslash theta\; +\; \backslash theta\text{'}\}\{\backslash theta\}\; =\; 0.1772935$; and to find the number $\backslash frac\{\backslash theta(2\backslash theta\; +\; \backslash theta\text{'})\}\{6r^3\}\; M$, we have $2\backslash theta\; +\; \backslash theta\text{'}\; =\; 20.03782$ days, therefore

If therefore we denote by $(L)$ the number whose logarithm is $L$, we shall have

and on substituting this value of $r$ in the above two equations, we shall obtain the values of $\backslash ell^0$ and $\backslash ell\text{'}$.

Comet. the intervals $\backslash delta$ and $\backslash delta\text{'}$, it is necessary to recommence the calculations, and to determine $\backslash lambda$, $\backslash lambda\text{'}$, and $e$ from the more accurate expressions which form the third degree of approximation. Let

from which, employing the approximate values of $r^3$, $r\text{'}$, and $r\text{'}$, already obtained, there results

If we now proceed to substitute these results in the equations (19), we shall find the following expressions for $\backslash xi^0$ and $\backslash xi\text{'}$:

in which the only indeterminate quantity is $\backslash xi$. We must now make another hypothesis with respect to $\backslash xi$. If we assume the value given by the last approximation, namely $\backslash xi\; =\; (9.7650651)$, we shall find that it is considerably too small, and that the first approximation is consequently insufficient, owing to the imperfect values it gave of the quantities $\backslash lambda$ and $\backslash lambda\text{'}$, and $e$.

Let us suppose $\backslash xi\; =\; 0.6045\; =\; (9.7813963)$. By proceeding as before with the corrected values now found, we obtain

Suppose, therefore, $\backslash xi\; =\; 0.6055\; =\; (9.7821141)$. Computing as above, we shall find

Let $n\; =\; \backslash frac\{2501\}\{8973\; +\; 2501\}\; =\; 0.218$; the mean between the two values of the same quantity in the two hypotheses is $B\; +\; (A\; -\; B)(0.218)$, by taking which, there results $r^3\; =\; (9.8880427)$, $r\text{'}\; =\; (9.7886828)$, $r\text{'}\; =\; (9.7314028)$. These values being considerably different from those found by the first approximation, it becomes necessary to have recourse to a third operation.

The values of $\backslash lambda$, $\backslash lambda\text{'}$, and $1\; -\; e$, deduced from the values of $r^3$, $r\text{'}$, $r\text{'}$ now found, are

In order to find a more exact value of $A\_1$ than results

from taking the mean of the three longitudes of the earth, we may make use of the equations (13), namely,

in which $M\; =\; (e\; -\; E)\; R\_1\; \backslash cos.\; A\_1\; =\; \backslash pi\; \backslash cos.\; A\_1$, $N\; =\; \backslash pi\; \backslash sin.\; A\_1$, and we shall have in the present example $\backslash pi\; \backslash cos.\; A\_1\; =\; \backslash lambda(9.8713530)\; +\; \backslash lambda\text{'}(9.9378419)\; -\; (1\; -\; e)(9.8969484)$

$\backslash pi\; \backslash sin.\; A\_1\; =\; \backslash lambda(9.8162347)\; +\; \backslash lambda\text{'}(9.6874819)\; -\; (1\; -\; e)(9.7792587)$

whence, substituting for $\backslash lambda$, $\backslash lambda\text{'}$, and $(1\; -\; e)$, the above values,

The value of $A\_1$, thus found, differs considerably from that which was employed in the previous approximation, namely, $A\_1\; =\; 36^\backslash circ\; 0\text{'}\; 41\text{'}\text{'}$.

On substituting $\backslash pi$ for $(e\; -\; E)\; R\_1$, the equations (19) become

whence, making $A\_1\; =\; 36^\backslash circ\; 39\text{'}\; 20\text{'}\text{'}$,

from which two equations we shall obtain the nearest approximation of which this method of solving the problem is susceptible.

Suppose $\backslash xi\; =\; (9.7790000)$. On going through the same computation as before, we shall find

Suppose, secondly, $\backslash xi\; =\; (9.7791000)$. This supposition will give

From these errors, the mean to take between A and B, the two values of the same quantity in the two hypotheses, is $B\; +\; (A\; -\; B)(0.41)$, whence the following solution:

VI. It now remains only to indicate the method of computing, from the foregoing solution, the elements of the parabolic orbit.

It was shown in § 12 that $v\; =\; \backslash tan.\; \backslash frac\{1\}\{2\}\; \backslash phi\; =\; \backslash frac\{\backslash sqrt\{r^3\; -\; sd\}\}\{\backslash sqrt\{sd\}\}$, and also that $v\; =\; \backslash sqrt\{D\}\; \backslash cdot\; \backslash frac\{\backslash sqrt\{s\}\; -\; \backslash sqrt\{d\}\}\{\backslash sqrt\{sd\}\}$. Comparing these

Comet. values, $\backslash sqrt\{r^2\; -\; sd\}\; =\; \backslash sqrt\{D\}(\backslash sqrt\{s\}\; -\; \backslash sqrt\{d\})$, and, consequently,

Having obtained the perihelion distance $D$, the true anomaly $\backslash psi^0$ of the point $C^0$ will be given by the equation of the parabola, namely, $r^0\; \backslash cos^2\; \backslash frac\{1\}\{2\}\; \backslash psi^0\; =\; D$, from which

The calculation of $D$ and $\backslash psi^0$ is as follows:

Having thus obtained the perihelion distance and the true anomaly, we must now find the time required for describing this anomaly. For this purpose it is usual to employ a table constructed from the following property of bodies moving in a parabolic orbit, namely, if $T$ and $T\text{'}$ be the times of describing the same anomaly in two parabolas whose perihelion distances are respectively $D$ and $D\text{'}$,

then $T\; :\; T\text{'}\; ::\; D^\{\backslash frac\{3\}\{2\}\}\; :\; D\text{'}^\{\backslash frac\{3\}\{2\}\}$. Hence, if a table is formed showing the time $T$ in which every degree, &c. is described in a parabola whose perihelion distance is $D\text{'}\; =\; 1$, it will give the time $T$ in which the same anomaly is described in the parabola whose perihelion distance is $D$, by

simply multiplying the tabular time by $D^\{\backslash frac\{3\}\{2\}\}$; for when

$D\text{'}\; =\; 1$, we have $T\; =\; D^\{\backslash frac\{3\}\{2\}\}\; T\text{'}$. The time $T$ must be added to the epoch of the first observation, if the comet is approaching the perihelion of its orbit, but subtracted from that epoch if it has already passed the perihelion.

But instead of employing a table of parabolic motion, the time may be computed directly from the formula

observing to add to the logarithm of $T$ the constant logarithm 1.7644179 (the compliment of 8.2355821), in order that $T$ may be expressed in mean solar days.

Computing from this formula, there results in the present case,

Epoch of first observation, October....22.68488

Passage through perihelion, November 18.04383.

To determine the position of the orbit, we must first

compute the heliocentric latitudes and longitudes. Let $\backslash alpha^0$, $\backslash alpha\text{'}$ be the heliocentric longitudes of the points $C^0$ and $C\text{'}$; $\backslash beta^0$, $\backslash beta\text{'}$ their heliocentric latitudes; then, denoting for an instant by $s$ the comet's distance from the earth, we shall have $r\; \backslash sin\; \backslash beta\; =\; s\; \backslash sin\; b$; but $s\; =\; \backslash xi\; \backslash sec\; b$, therefore

These two formulae, which are very easily computed, give the heliocentric latitudes at the extreme observations. In the present case they become

Now, to compute the heliocentric longitudes, let us suppose $K$ to be the projection of $C$ on the plane of the ecliptic. In the rectilinear triangle $ESK$, formed by joining the positions of the earth, the sun, and the projected place of the comet, we have $EK\; =\; \backslash xi$; $SK\; =\; r\; \backslash cos\; \backslash beta$; the angle $ESK$, opposite to $EK$, is $\backslash alpha\; -\; A$, the difference between the heliocentric longitude of the comet and the longitude of the earth; and the angle $KES$, opposite to $SK$, is $\backslash alpha\; -\; A$, the difference between the geocentric longitude of the comet and the longitude of the sun; therefore

from which, since $\backslash frac\{\backslash xi\}\{r\}\; =\; \backslash frac\{\backslash sin\; \backslash beta\}\{\backslash tan\; b\}$ we deduce for the first and third observations,

In the present case these formulae become

Let $N$ be the ascending node, $\backslash gamma$ its longitude, and $I$ the inclination of the orbit. In the spherical triangle $NCK$, the side $NK\; =\; \backslash alpha\; -\; \backslash gamma$, $CK\; =\; \backslash beta$, and the angle $CNK\; =\; I$. Therefore, by Napier's analogies, $\backslash cot\; I\; =\; \backslash cot\; \backslash beta\; \backslash sin(\backslash alpha\; -\; \backslash gamma)$, which, on substituting the longitudes $\backslash alpha^0$ and $\backslash alpha\text{'}$, gives the double equation

from which we have

whence

But if $A$ and $B$ represent two arcs, then, by the trigonometrical formulae,

and

whence the above equation becomes

consequently