a town of Bengal, district of Purnia, on the bank of the Gorace. This stream, according to Bishop Heber, has here the width of the Thames at Vauxhall; and being navigable at all seasons of the year, the town has become a great emporium of commerce. Long. 89.11. E., Lat. 23.52. N. Is 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 1665 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 Dörfl, protestant minister of Wieda, a village near Plauen, in Upper Saxony. Dörfl 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 Dörfl 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 Dörfl, 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 Dörfl; 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 virtue 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 Parisons, 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 Opticam 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 XVIIIe 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 Orbitae 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- 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 Orbes 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 vectors 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 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,

$$\frac{d^2x}{dt^2} + \frac{x}{r^3} = 0, \quad \frac{d^2y}{dt^2} + \frac{y}{r^3} = 0, \quad \frac{d^2z}{dt^2} + \frac{z}{r^3} = 0;$$

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

$$\frac{dx^2}{dt^2} + \frac{dy^2}{dt^2} + \frac{dz^2}{dt^2} - \frac{2}{r} + \frac{1}{a} = 0.$$

(1)

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

$$\frac{d^2r^2}{dt^2} = \frac{dx^2}{dt^2} + \frac{dy^2}{dt^2} + \frac{dz^2}{dt^2} - \frac{1}{r};$$

therefore,

$$\frac{d^2r^2}{2dt^2} = \frac{2}{r} - \frac{1}{a} - \frac{1}{r}, \text{ or } \frac{d^2r^2}{2dt^2} - \frac{1}{r} + \frac{1}{a} = 0.$$

(2)

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 $\varrho$, the comet's distance from the earth at the second observation; then, making $a$ infinite, and neglecting quantities divided by $r^2$, he eliminates $\frac{1}{r}$, and obtains a quadratic equation which gives an approximate value of $\varrho$. 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 XVIIIe 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.

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', C, C' 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 $\delta$, and that between the second and third by $\delta'$; 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 $t - \delta$, and $t + \delta'$. 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 $x^\circ, y^\circ, z^\circ$, be the co-ordinates of C', and $x', y', z'$, those of C'. The first step of the process is to find expressions for $x^\circ, y^\circ, z^\circ$, and $x', y', z'$, in terms of $x, y, z$, the latter being regarded as functions of the time $t$.

By Taylor's theorem we have

$$x^\circ = x - \frac{dx}{dt} \cdot \delta + \frac{d^2x}{dt^2} \cdot \frac{\delta^2}{2!} - \frac{d^3x}{dt^3} \cdot \frac{\delta^3}{3!} + \ldots,$$

$$x' = x + \frac{dx}{dt} \cdot \delta' + \frac{d^2x}{dt^2} \cdot \frac{\delta'^2}{2!} + \frac{d^3x}{dt^3} \cdot \frac{\delta'^3}{3!} + \ldots,$$

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

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

$$\frac{d^2x}{dt^2} = -\frac{x}{r^3}, \quad \frac{d^2y}{dt^2} = -\frac{y}{r^3}, \quad \frac{d^2z}{dt^2} = -\frac{z}{r^3},$$

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

\[ \frac{dx}{dt} = -\frac{dx}{r^2 dt} + \frac{3xdr}{r^2 dt} = \frac{3xdr}{r^2 dt} - \frac{dx}{r^2 dt} \]

or, by making \( r = k \),

\[ \frac{dx}{dt} = \frac{3k}{r^2} x = \frac{1}{r^2} \cdot \frac{dx}{dt}. \]

Differentiating again, and making the necessary reductions, we obtain

\[ \frac{d^2x}{dt^2} = \left( \frac{3}{r^2} \frac{dk}{dt} - \frac{3k^2}{r^3} + \frac{1}{r^5} \right) x + \frac{2k}{r^3} \cdot \frac{dx}{dt}. \]

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

\[ x^o = x \left( 1 - \frac{1}{r^2} \cdot \frac{6}{1 \cdot 2} - \frac{3k}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} \right) \]

\[ + \frac{dx}{dt} \left( -\frac{1}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} + \frac{2k}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} \right) \]

\[ x' = x \left( 1 - \frac{1}{r^2} \cdot \frac{6}{1 \cdot 2} + \frac{3k}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} \right) \]

\[ + \frac{dx}{dt} \left( \frac{1}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} + \frac{2k}{r^2} \cdot \frac{6}{1 \cdot 2 \cdot 3} \right). \]

For the sake of abridging, let us assume

\[ v = 1 - \frac{6}{r^2} - \frac{3k}{r^2}, \quad u = -\frac{6}{r^2} + \frac{3k}{r^2}, \]

\[ v' = 1 - \frac{6}{r^2} + \frac{3k}{r^2}, \quad u' = \frac{6}{r^2} - \frac{3k}{r^2}, \]

and we shall have

\[ x^o = vx + u \frac{dx}{dt}, \quad x' = vx + u' \frac{dx}{dt}. \]

It is evident that \( y^o y' \) and \( z^o z' \) 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:

\[ \begin{align*} x^o &= vx + u \frac{dx}{dt}, \\ y^o &= vy + u \frac{dy}{dt}, \\ z^o &= vz + u \frac{dz}{dt}. \end{align*} \]

If from these equations we eliminate \( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \), we shall find, on making \( w = vu' - v'u \),

\[ w^o = ux - uz, \quad w^o = uy - uz, \quad w^o = uz - uz; \]

but from the values of \( v, u, v', u' \), given above, we have evidently

\[ w = (v + u) - \frac{(v + u)^2}{6r^2} + \frac{k}{4r^2} (v + u)^2 (v - u); \]

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

\[ x(y'z' - yz') + y(z'x' - zx') + z(x'y' - xy') = 0, \]

an equation to a plane, which denotes that the three points \( C^o, C, C' \) are situated in the same plane with \( S, \) the origin of the co-ordinates.

2. Let the chord of the arc \( C^oC' \) intersect the radius \( SC \) in \( D, \) and let the ratio of \( C^oD \) to \( CD \) be that of \( \lambda \) to \( \lambda'. \) Make \( \lambda + \lambda' = 1, \) and the chord \( C^oC' = e; \) we have then \( C^oD = \lambda e \) and \( CD = \lambda' e. \) Further, let \( DC = er, \) so that \( SD = r(1-e), \) \( e \) being an infinitely small quantity of the second order in respect of \( \delta \) and \( \delta', \) which are supposed infinitesimals of the first order.

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

\[ (1-e)x - x^o; x' - (1-e)x :: \lambda : \lambda', \]

whence

\[ \lambda \{ x - (1-e)x \} = \lambda' \{ (1-e)x - x^o \}; \]

and, since \( \lambda + \lambda' = 1, \)

\[ \lambda x' + \lambda x^o = (1-e)x. \]

Substituting in this last equation the values of \( x' \) and \( x^o \) given by equations (1), we get

\[ (\lambda x' + \lambda x^o) x + (\lambda x' + \lambda x^o) \frac{dx}{dt} = (1-e)x; \]

and by proceeding in the same manner, we shall find relatively to \( y \) and \( z, \)

\[ (\lambda x' + \lambda x^o) y + (\lambda x' + \lambda x^o) \frac{dy}{dt} = (1-e)y; \]

\[ (\lambda x' + \lambda x^o) z + (\lambda x' + \lambda x^o) \frac{dz}{dt} = (1-e)z. \]

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

\[ \lambda x' + \lambda x^o = 1-e, \quad \lambda x' + \lambda x^o = 0, \quad \lambda + \lambda' = 1. \tag{3} \]

From the second and third of these we get \( \lambda = \frac{-u}{u'-u}; \)

\[ \lambda = \frac{u'}{u'-u}; \quad \text{and consequently from the first, } 1-e = \frac{u'v - uv}{u'-u} = \frac{w}{u'-u}. \quad \text{Hence} \]

\[ \lambda = \frac{-u}{u'-u}, \quad \lambda' = \frac{u'}{u'-u}, \quad 1-e = \frac{w}{u'-u}. \tag{4} \]

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

I. Retaining only the first terms of the series, or those which depend on the simple powers of \( \delta \) and \( \delta', \) we have \( u = -\delta, \) \( u' = \delta, \) \( w = \delta + \delta, \) and \( u' - u = \delta + \delta, \) whence

\[ \lambda = \frac{\delta}{\delta + \delta}, \quad \lambda' = \frac{\delta'}{\delta + \delta'}, \quad e = 0; \tag{5} \]

consequently \( \lambda = \frac{\delta}{\delta'}, \) 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. 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 \( u \), \( u' \), and \( w \), we have

\[ u = \frac{\delta}{6r^3}, \quad u' = \frac{\delta}{6r^3}, \quad w = \frac{\delta}{6r^3} + \frac{(\delta + \delta')^2}{6r^3}, \]

whence

\[ \lambda = \frac{u - u'}{u - u} = \frac{\delta (1 - \frac{\delta^2}{6r^3})}{\delta + \delta' - \frac{\delta^2 + \delta'^2}{6r^3}}, \]

and therefore by division

\[ \lambda = \frac{\delta}{\delta + \delta'} + \frac{\delta}{\delta + \delta'} \cdot \frac{\delta - \delta'}{6r^3}. \]

In like manner, from the equation \( \lambda' = \frac{u'}{u - u} \) we find

\[ \lambda' = \frac{\delta}{\delta + \delta'} - \frac{\delta}{\delta + \delta'} \cdot \frac{\delta - \delta'}{6r^3}. \]

But \( 1 - e = \frac{\delta + \delta'}{6r^3} = 1 - \frac{3\delta^2}{6r^3} \)

Whence we have in this case the following values of \( \lambda \), \( \lambda' \), and \( e \), namely,

\[ \lambda = \frac{\delta}{\delta + \delta'} + \frac{\delta}{\delta + \delta'} \left( \frac{\delta - \delta'}{6r^3} \right), \]

\[ \lambda' = \frac{\delta}{\delta + \delta'} - \frac{\delta}{\delta + \delta'} \left( \frac{\delta - \delta'}{6r^3} \right), \]

\[ e = \frac{\delta}{2r^3}. \]

The values of \( \lambda \), \( \lambda' \), 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 \( u \), \( u' \), 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 \( r^o \) and \( r' \). For this purpose we may suppose \( r^o = r - dr \), and \( r' = r + dr \); but we have already assumed \( k = \frac{rd}{dt} \),

therefore \( dr = \frac{kd}{r} \). Hence

\[ r^o = r - \frac{kd}{r}, \quad r' = r + \frac{kd}{r}, \]

whence \( (r^o r')^3 = r^3 - \frac{3}{2}r^2k \), and consequently

\[ \frac{1}{6(r^o r')^3} = \frac{1}{6r^3 - 9r^2k} = \frac{1}{6r^3} + \frac{k^2}{4r^5}. \]

In the same manner we find

\[ \frac{1}{6(r^o r')^3} = \frac{1}{6r^3 + 9r^2k} = \frac{1}{6r^3} - \frac{k^2}{4r^5}. \]

Substituting these expressions in the series denoted by \( u \) and \( u' \), we get

\[ u - u = \frac{\delta}{6(r^o r')^3}, \quad u' = \frac{\delta}{6(r^o r')^3}, \]

consequently,

\[ \lambda = \frac{u - u'}{u - u} = \frac{\delta}{\delta + \delta'} \left( \frac{1 - \frac{\delta^2}{6(r^o r')^3}}{6(r^o r')^3} \right), \]

whence by division,

\[ \lambda = \frac{\delta}{\delta + \delta'} + \frac{\delta}{\delta + \delta'} \left( \frac{\delta^2}{6(r^o r')^3} - \frac{\delta^2}{6(r^o r')^3} \right). \]

By proceeding in a similar manner with respect to \( \lambda' \), we find

\[ \lambda' = \frac{\delta}{\delta + \delta'} - \frac{\delta}{\delta + \delta'} \left( \frac{\delta^2}{6(r^o r')^3} - \frac{\delta^2}{6(r^o r')^3} \right). \]

To find \( e \), we have in this case

\[ 1 - e = \frac{w}{u - u} = \frac{\delta + \delta'}{6r^3} + \frac{k}{4r^3} (\delta + \delta') (\delta - \delta'), \]

\[ = \frac{(\delta + \delta')}{6r^3} + \frac{k}{4r^3} (\delta + \delta') (\delta - \delta'), \]

hence, on dividing and neglecting the terms which involve the fourth and higher powers of \( \delta \) and \( \delta' \),

\[ 1 - e = 1 - \frac{\delta^2}{2r^3} + \frac{k\delta (\delta - \delta')}{2r^3}. \]

But \( r^o = r - \frac{kd}{r} \), \( r' = r + \frac{kd}{r} \), therefore \( r^o r' = r^2 + k (\delta - \delta') - \frac{k^2d^2}{r} \). The last term of this expression is a quantity of the order of those that may be neglected; therefore simply

\[ \frac{1}{r^o r'} = \frac{1}{r^2 + k (\delta - \delta')r} = \frac{1}{r^2} - \frac{k (\delta - \delta')}{r}. \]

Substituting this in the above expression for \( 1 - e \), it becomes

\[ 1 - e = 1 - \frac{\delta^2}{2r^3}, \quad \text{whence} \quad e = \frac{\delta^2}{2r^3}. \]

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

\[ \lambda = \frac{\delta}{\delta + \delta'} + \frac{\delta}{\delta + \delta'} \left( \frac{\delta^2}{6(r^o r')^3} - \frac{\delta^2}{6(r^o r')^3} \right), \]

\[ \lambda' = \frac{\delta}{\delta + \delta'} - \frac{\delta}{\delta + \delta'} \left( \frac{\delta^2}{6(r^o r')^3} - \frac{\delta^2}{6(r^o r')^3} \right), \]

\[ e = \frac{\delta^2}{2r^3}. \]

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

Let \( \alpha^o, \alpha', \alpha'' \) be the three geocentric longitudes of the comet at the instants of the first, second, and third observations respectively; \( b^o, b', b'' \) its three geocentric latitudes; \( g^o, g', g'' \), the three curvate distances, that is to say, the three distances of the comet from the earth projected on the plane of the ecliptic; \( A^o, A', A'' \), the three corresponding heliocentric longitudes of the earth; and \( R^o, R', R'' \), the three radii vectors of the earth. Finally, let \( X^oY^o, XY, X'Y' \), be the co-ordinates of the earth, referred to the centre of the sun, so that