THEORIA XMOTVS CORPORVM C OELE STIV M IN SECTIONIBVS CONICIS SOLEM AMBIENTIVM A V C T ORE CAROLO FRIDERICO GAVSS. THEORY OF THE MOTION OF THE HEAVENLY BODIES MOVING ABOUT THE SUN IN CONIC SECTIONS: A TRANSLATION OF GAUSS'S "~THEORIA MOTUS." WITH AN APPENDIX. BY CHARLES HENRY DAVIS, COMMANDER UNITED STATES NAVY, SUPERINTENDENT OF THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAO. BOS TO N: LITTLE, BR O W N' AND COMPANY. 1857. Published under the Authority of the Navy Department by the Nautical Almanac and Smithsonian Institution. TRANSLATOR'S PREFACE. IN 1852, a pamphlet, entitled The Computation of an Orbit from Three Complete Observations, was published, under the authority of the Navy Department, for the use of the American Ephemeris and Nantical Almanac, the object of which was to excerpt from various parts of GAUSS's Thleoria Motus, and to arrange in proper order the numerous details which combine to form this complicated problem. To these were added an Appendix containing the results of Professor ENOKE'S investigations, Ueber den Ausnahmefall einer doppelten Bahnbestimmung aus denselben drei geocentrischen Oertern (Abhandlungen der Akademie der Wissenschaften zu Berlin, 1848), and also Professor PEIRCE'S Graphic Delineations of the Curves showing geometrically the roots of GAuss's Equation IV. Article 141. After this pamphlet was completed, the opinion was expressed by scientific friends that a complete translation of the Thleoria Motus should be undertaken, not only to meet the wants of the American Ephemeris, but those also of Astronomers generally, to whom this work (now become very rare and costly) is a standard and permanent authority. This undertaking has been particularly encouraged by the Smithsonian Institution, which has signified its high estimate of the importance of the work, by contributing to its publication. And by the authority of Hon. J. C. DOBBIN, Secretary of the Navy, this Translation is printed by the joint contributions of the Nautical Almanac and the Smithsonian Institution. The notation of GAUSs has been strictly adhered to throughout, and the translation has been made as nearly literal as possible. No pains have been spared to secure typographical accuracy. All the errata that have been noticed in ZACH's Monatliche Correspondenz, the Berliner Astronomisches Jahrbuch, and the Astronomische Nachrichten, have (v) vi TRANSLATOR'S PREFACE. been corrected, and in addition to these a considerable number, a list of which will be found in GOULD'S Astronomical Journal, that were discovered by Professor CHAUVENET of the United States Naval Academy, who has examined the formulas of the body of the work with great care, not only by comparison with the original, but by independent verification. The proof-sheets have also been carefully read by Professor PHILLIPS, of Chapel Hill, North Carolina, and by M/r. RUNKLE and Professor WINLOCK of the Nautical Almanac office. The Appendix contains the results of the investigations of Professor ENCIE and Professor PEIRCE, from the Appendix of the pamphlet above referred to, and other matters which, it is hoped, will be found interesting and useful to the practical computer, among which are several valuable tables: A Table for the Motion in a Parabola from LEVERRIER'S Annales de L'Observatoire Imperial de Paris, BESSEL'S and POSSELT's Tables for Ellipses and Hyperbolas closely resembling the Parabola, and a convenient Table by Professor HUBBARD for facilitating the use of GAUSS's formulas for Ellipses and Hyperbolas of which the eccentricities are nearly equal to unity. And in the form of notes on their appropriate articles, useful formulas by BESSEL, NICOLAI, ENCKE, GAUSS, and PEIRCE, and a summary of the formulas for computing the orbit of a Comet, with the accompanying Table, from OLMERS'S Abhandlung ueber die leichteste und be. quemste Methode die Bahn eines Cometen zu berechnen. Weimar, 1847. CONTENTS. PAGE PREFACE. e s o o. ix F I R S T BOOK. GENERAL RELATIONS BETWEEN THE QUANTITIES BY WHICH THE MOTIONS OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED. FIRST SECTION. - Relations pertaining simply to position in the Orbit... 1 SECOND SECTION.- Relations pertaining simply to Position in Space.... 54 THIRD SECTION. - Relations between Several Places in Orbit... e.. 100 FOURTH SECTION. - Relations between Several Places in Space.. ~ 153 SECOND BOOK. INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC OBSERVATIONS. FIRST SECTION.- Determination of an Orbit from Three Complete Observations... 161 SECOND SECTION. - Determination of an Orbit from Four Observations, of which Two only are Complete....... 234 THIRD SECTION. - Determination of an Orbit satisfying as nearly as possible any number of Observations whatever...... 2 9 FOURTH SECTION. — On the Determination of Orbits, taking into account the Perturbations. 274 APPENDIX......279 TABLES....... 329 (vii) CAMBRIDGE: PRINTED BY ALLEN AND FARNHAM. PREFACE. AFTER the laws of planetary motion were discovered, the genius of KEPLER was not without resources for deriving from observations the elements of motion of individual planets. TYcHo BRAHE, by whom practical astronomy had been carried to a degree of perfection before unknown, had observed all the planets through a long series of years with the greatest care, and with so much perseverance, that there remained to KEPLER, the most worthy inheritor of such a repository, the trouble only of selecting what might seem suited to any special purpose. The mean motions of the planets already determined with great precision by means of very ancient observations diminished not a little this labor. Astronomers who, subsequently to KEPLER, endeavored to determine still more accurately the orbits of the planets with the aid of more recent or better observations, enjoyed the same or even greater facilities. For the problem was no longer to deduce elements wholly unknown, but only slightly to correct those already known, and to define them within narrower limits. The principle of universal gravitation discovered by the illustrious NEWTON b (ix) X PREFACE. opened a field entirely new, and showed that all the heavenly bodies, at least those the motions of which are regulated by the attraction of the sun, must necessarily, conform to the same laws, with a slight modification only, by which KEPLER had found the five planets to be governed. KEPLER, relying upon the evidence of observations, had announced that the orbit of every planet is an ellipse, in which the areas are described uniformly about the sun occupying one focus of the ellipse, and in such a manner that in different ellipses the times of revolution are in the sesquialteral ratio of the semiaxes-major. On the other hand, NEWTON, starting from the principle of universal gravitation, demonstrated a' priori that all bodies controlled by the attractive force of the sun must move in conic sections, of which the planets present one form to us, namely, ellipses, while the remaining forms, parabolas and hyperbolas, must be regarded as being equally possible, provided there may be bodies encountering the force of the sun with the requisite velocity; that the sun must always occupy one focus of the conic section; that the areas which the same body describes in different times about the sun are proportional to those times; and finally, that the areas described about the sun by different bodies, in equal times, are in the subduplicate ratio of the semiparameters of the orbits: the latter of these laws, identical in elliptic motion with the last law of KEPLER, extends to the parabolic and hyperbolic motion, to which KEPLER'S law cannot be applied, because the revolutions are wanting. The clue was now discovered by following which it became possible to enter the hitherto inaccessible labyrinth of the motions of the comets. And this was so successful that the single hypothesis, that their orbits were parabolas, sufficed to explain the motions of all the comets which had been accurately observed. Thus the system of universal gravitation had PREFACE. xi paved the way to new and most brilliant triumphs in analysis; and the comets, up to that time wholly unmanageable, or soon breaking from the restraints to which they seemed to be subjected, having now submitted to control, and being transformed from enemies to guests, moved on in the paths marked out by the calculus, scrupulously conforming to the same eternal laws that govern the planets. In determining the parabolic orbits of comets from observation, difficulties arose far greater than in determining the elliptic orbits of planets, and principally from this source, that comets, seen for a brief interval, did not afford a choice of observations particularly suited to a given object: but the geometer was compelled to employ those which happened to be furnished him, so that it became necessary to make use of special methods seldom applied in planetary calculations. The great NEWTON himself, the first geometer of his age, did not disguise the difficulty of the problem: as might have been expected, he came out of this contest also the victor. Since the time of NEWTON, many geometers have labored zealously on the same problem, with- various success, of course, but still in such a manner as to leave but little to be desired at the present time. The truth, however, is not to be overlooked that in this problem the difficulty is very fortunately lessened by the knowledge of one element of the conic section, since the major-axis is put equal to infinity by the very assumption of the parabolic orbit. For, all parabolas, if position is neglected, differ among themselves only by the greater or less distance of the vertex from the focus; while conic sections, generally considered, admit of infinitely greater variety. There existed, in point of fact, no sufficient reason why it should be taken for granted that the paths of comets are exactly Xii PREFACE. parabolic: on the contrary, it must be regarded as in the highest degree improbable that nature should ever have favored such an hypothesis. Since, nevertheless, it was known, that the phenomena of a heavenly body moving in an ellipse or hyperbola, the major-axis of which is very great relatively to the parameter, differs very little near the perihelion from the motion in a parabola of which the vertex is at the same distance from the focus; and that this difference becomes the more inconsiderable the greater the ratio of'the axis to the parameter: and since, moreover, experience had shown that between the observed motion and the motion computed in the parabolic orbit, there remained differences scarcely ever greater than those which might safely be attributed to errors of observation (errors quite considerable in mrost cases): astronomers have thought proper to retain the parabola, and very properly, because there are no means whatever of ascertaining satisfactorily what, if any, are the differences from a parabola. We must except the celebrated cormet of HALLEY, which, describing a very elongated ellipse and frequently observed at its return to the perihelion, revealed to us its periodic time; but then the major-axis being thus known, the computation of the remaining elements is to be considered as hardly more difficult than the determination of the parabolic orbit. And we must not omit to mention that astronomers, in the case of some other comets observed for a somewhat longer time, have attempted to determine the deviation from a parabola. However, all the methods either proposed or used for this object, rest upon the assumption that the variation from a parabola is inconsiderable, and hence in the trials referred to, the parabola itself, previously computed, furnished an approximate idea of the several elements (except the major-axis, or the time of revolution depending on it), to be corrected by only slight changes. Besides, it PREFACE. Xiii must be acknowledged, that the whole of these trials hardly served in any case to settle any thing with certainty, if, perhaps, the comet of the year 1770 is excepted. As soon as it was ascertained that the motion of the new planet, discovered in 1781, could not be reconciled with the parabolic hypothesis, astronomers'undertook to adapt a circular orbit to it, which is a matter of simple and very easy calculation. By a happy accident the orbit of this planet had but a small eccentricity, in consequence of which the elements resulting from the circular hypothesis sufficed at least for an approximation on which could be based the determination of the elliptic elements. There was a concurrence of several other very favorable circumstances. For, the slow motion of the planet, and the very small inclination of the orbit to the plane of the ecliptic, not only rendered the calculations much more simple, and allowed the use of special methods not suited to other cases; but they removed the apprehension lest the planet, lost in the rays of the sun, should subsequently elude the search of observers, (an apprehension which some astronomers might have felt, especially if its light had been less brilliant); so that the more accurate determination of the orbit might be safely deferred, until a selection could be made from observations more frequent and more remote, such as seemed best fitted for the end in view. Thus, in every case in which it was necessary to deduce the orbits of heavenly bodies from observations, there existed advantages not to be despised, suggesting, or at any rate permitting, the application of special methods; of which advantages the chief one was, that by means of hypothetical assumptions an approximate knowledge of some elements could be xiv PREFACE. obtained before the computation of the elliptic elements was commenced. Notwithstanding this, it seems somewhat strange that the general problem,To determine the orbit of a heavenly body, without any hypothetieal assunption, from observations not embracing a great period of timbe, and not allowzin a selection,vith a view to the application of special methods, was almost wholly neglected up to the beginning of the present century; or, at least, not treated by any one in a manner worthy of its importance; since it assuredly commended itself to mathematicians by its difficulty and elegance, even if its great utility in practice were not apparent. An opinion had universally prevailed that a complete determination from observations embracing a short interval of time was impossible, —an ill-founded opinion, —for it is now clearly shown that the orbit of a heavenly body may be determined quite nearly from good observations embracing only a few days; and this without any hypothetical assumption. Some ideas occurred to me in the month of September of the year 1801, engaged at the time on a very different subject, which seemed to point to the solution of the great problem of which I have spoken. IUnder such circumstances we not unfrequently, for fear of being too much led away by an attractive investigation, suffer the associations of ideas, which, more attentively considered, might have proved most fruitful in results, to be lost from neglect. And the same fate might have befallen these conceptions, had they not happily occurred at the most propitious moment for their preservation and encouragement that could have been selected. For just about this time the report of the new planet, discovered on the first day of January of that year with the telescope at Palermo, was the subject of universal conversation; PREFACE. XV and soon afterwards the observations made by that distinguished astronomer PIAZZI from the above date to the eleventh of February were published. Nowhere in the annals of astronomy do we meet with so great an opportunity, and a greater one could hardly be imagined, for showing most strikingly, the value of this problem, than in this crisis and urgent necessity, when all hope of discovering in the heavens this planetary atom, among innumerable small stars after the lapse of nearly a year, rested solely upon a sufficiently approximate knowledge of its orbit to be based upon these very few observations. Could I ever have found a more seasonable opportunity to test the practical value of my conceptions, than now in employing them for the determination of the orbit of the planet Ceres, which during these forty-one days had described a geocentric arce of only three degrees, and after the lapse of a year must be looked for in a region of the heavens very remote from that in which it was last seen? This first application of the method was made in the month of October, 1801, and the first clear night, when the planet was sought for* as directed by the numbers deduced from it, restored the fugitive to observation. Three other new planets, subsequently discovered, furnished new opportunities for examining and verifying the efficiency and generality of the method. Several astronomers wished me to publish the methods employed in these calculations immediately after the second discovery of Ceres; but many things-other occupations, the desire of treating the subject more fully at some subsequent period, and, especially, the hope that a further prosecution of this investigation would raise various parts of the solution to a greater *By de ZACH, December 7, 1801. 2 xvi PREFACE. degree of generality, simplicity, and elegance, - prevented my complying at the time with these friendly solicitations. I was not disappointed in this expectation, and have no cause to regret the delay. For, the methods first employed have undergone so many and such great changes, that scarcely any trace of resemblance remains between the method in which the orbit of Ceres was first computed, and the form given in this work. Although it would be foreign to my purpose, to narrate in detail all the steps by which these investigations have been gradually perfected, still, in several instances, particularly when the problem was one of more importance than usual, I have thought that the earlier methods ought not to be wholly suppressed. But in this work, besides the solutions of the principal problems, I have given many things which, during the long time I have been engaged upon the motions of the heavenly bodies in conic sections, struck me as worthy of attention, either on account of their analytical elegance, or more especially on account of their practical utility. But in every case I have devoted greater care both to the subjects and methods which are peculiar to myself touching lightly and so far only as the connection seemed to require, on those previously known. The whole work is divided into two parts. In the First Book are developed the relations between the quantities on which the motion of the heavenly bodies about the sun, according to the laws of KEPLER, depends; the two first sections comprise those relations in which one place only is considered, and the third and fourth sections those in which the relations between several places are considered. The two latter contain an explanation of the common methods, and also, and more particularly, of other methods, greatly preferable to them in practice if I am not mistaken, by means of PREFACE. xvii which we pass from the known elements to the phenomena; the former treat of many most important problems which prepare the way to inverse processes. Since these very phenomena result from a certain artificial and intricate complication of the elements, the nature of this texture must be thoroughly examined before we can undertake with hope of success to disentangle the threads and to resolve the fabric into its constituent parts. Accordingly, in the First Book, the means and appliances are provided, by means of which, in the second, this difficult task is accomplished; the chief part of the labor, therefore, consists in this, that these means should be properly collected together, should be suitably arranged, and directed to the proposed end. The more important problems are, for the most part, illustrated by appropriate examples, taken, wherever it was possible, from actual observations. In this way not only is the efficacy of the methods more fully established and their use more clearly shown, but also, care, I hope, has been taken that inexperienced computers should not be deterred from the study of these subjects, which undoubtedly constitute the richest and most attractive part of theoretical astronomy. GOTTINGEN, March 28, 1809. C FIRST BOOK, GENERAL RELATIONS BETWEEN THOSE QUANTITIES BY WHICH THE MOTIONS OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED. FIRST SECTIONo RELATIONS PERTAJNING SIMPLY TO POSITION IN THE ORBIT. 1o IN this work we shall consider the motions of the heavenly bodies so far only as they are controlled by the attractive force of the sun. All the secondary planets are therefore excluded from our plan, the perturbations which the primary planets exert upon each other are excluded, as is also all motion of rotation. We regard the moving bodies themselves as mathematical points, and we assume that all motions are performed in obedience to the following laws, which are to be received as the basis of all discussion in this work. I. The motion of every heavenly body takes place in the same fixed plane in which tile centre of the sun is situated. II. The path described by a body is a conic section having its focus in the centre of the sun. III. The motion in this path is such that the areas of the spaces described about the sun in different intervals of time' are proportional to those intervals. Accordingly, if the times and spaces are expressed in numbers, any space whatk ever divided by the time in which it is described gives a constant quotient. 1 2 RELATIONS PERTAINING SIMPLY [BooK 1. IV. For different bodies moving about the sun, the squares of these quotients are in the compound ratio of the parameters of their orbits, and of the sum of the masses of the sun and the moving bodies. Denoting, therefore, the parameter of the orbit in which the body moves by 2p, the mass of this body by u, (the mass of the sun being put = 1), the area it describes about the sun in the time I by Fg, then tvp(1+t) will be a constant for all heavenly bodies. Since then it is of no importance which body we use for determining this number, we will derive it from the motion of the earth, the mean distance of which from thle sun we shall adopt for the unit of distance; the mean solar day will always be our unit of time. Denoting, moreover, by n the ratio of the circumference of the circle to the diameter, the area of the entire ellipse described by the earth will evidently be y'p, which must therefore be put -1-g, if by I is understood the sidereal year; whence, our constant becomes _ 2 __ In order to ascertain the numerical value of this constant, hereafter to be denoted by k, we will put, according to the latest determination, the sidereal year or t- 365.2563835, the mass of the earth, or t =354710 0.0000028192, whence results log 2i........ 0.7981798684 Compl. log t.. 7.4374021852 Compl. log. (1+) o o 9.9999993878 log k..... 8.2355814414 k- J 0.01720209895. 2. The laws above stated differ from those discovered by our own KEPLER in no other respect than this, that they are given in a formn applicable to all kinds of conic sections, and that the action of the moving body on the sun, on which depends the factor / (1 — + ), is taken into account. If we regard these laws as phenomena derived from innumerable and indubitable observations, geometry shows what action ought in consequence to be exerted upon bodies moving about SECT. 1.] TO POSITION IN THE ORBIT. 3 the sun, in order that these phenomena may be continually produced. In this way it is found that the action of the sun upon the bodies moving about it is exerted just as if an attractive force, the intensity of which is reciprocally proportional to the square of the distance, should urge the bodies towards the centre of the sun. If now, on the other hand, we set out with the assumption of such an attractive force, the phenomena are deduced from it as necessary consequences. It is sufficient here merely to have recited these laws, the connection of which with the principle of gravitation it will be the less necessary to dwell upon in this place, since several authors subsequently to the eminent NEWTON have treated this subject, and among them the illustrious LA PLACE, in that most perfect work the Mecanique Celeste, in such a manner as to leave nothing further to be desired. 3, Inquiries into the motions of the heavenly bodiesso far as they take place in conic sections, by no means demand a complete theory of this class of curves but a single general equation rather, on which all others can be based, will answer our purpose. And it appears to be particularly advantageous to select that one to which, while investigating the curve described according to the law of attraction, we are conducted as a characteristic equation. If we determine any place of a body in its orbit by the distances x, y, from two right lines drawn in the plane of the orbit intersecting each other at right angles in the centre of the sun, that is, in one of the foci of the curve, and further, if we denote the distance of the body from the sun by r (always positive), we shall have between r, x,,y the linear equation r + ax -x-. (y y-, in which a, (P, y represent constant quantities, y being from the nature of the case always positive. By changing the position of the right lines to which x,y, are referred, this position being essentially arbitrary, provided only the lines continue to intersect each other at right angles, the form of the equation and also the value of y will not be changed, but the values of a and:3 will vary, and it is plain that the position may be so determined that [I shall become = 07 and a, at least, not negative. In this way by putting for a, 7, respectively e, p, our equation takes the form r 4- ex =- p. The right line to 4 RELATIONS PERTAINING SIMPLY BooK I. which the distances y are referred in this case, is called the line of apsides, p is the serni-parameler, e the eceenlricity; finally the conic section is distinguished by the name of ellipse, parabolc, or hfyperbola, according as e is less than unity, equal to unity, or greater than unity. It is readily perceived that the position of the line of apsides would be fully determined by the conditions mentioned with the exception of the single case where both a and (3 were O O; in which case r is always _p, whatever the right lines to which X,,y are referred. Accordingly, since we have e - 0 the curve (which will be a circle) is according to our definition to be assigned to the class of ellipses, but it has this peculiarity, that the position of the apsides remains wholly arbitrary, if indeed we choose to extend that idea to such a case. 4. Instead of the distance x let us introduce the angle v, contained between the line of apsides and a straight line drawn from the sun to the place of the body (the radius vector), and this angle may commence at that part of the line of apsides at which the distances x are positive, and may be supposed to increase in the direction of the motion of the body. In this way we have x - r cos v, and thus our formula becomes r - -os from which immediately result the following 1 + e cosv conclusionsI. For v - 0, the value of the radius vector r becomes a minimum, that is, 1+ e this point is called the perihelion. II. For opposite values of v, there are corresponding equal values of r; consequently the line of apsides divides the conic section into two equal parts. III. In the ellipse, v increases continuously from v =0, until it attains its maximum value, I Pe, in aphelion, corresponding to v =180~; after aphelion, it decreases in the same manner as it had increased, until it reaches the perihelion, corresponding to v- 3600. That portion of the line of apsides terminated at one extremity by the perihelion and at the other by the aphelion is called the mc1/or SECT, 1.] TO POSITION IN THE ORBIT. 5 axis; hence the semi-axis major, called also the nmean distanee, =; the dis- p tance of the middle point of the axis (tle centre of thie ellipse) from the focus will be 1 - ea, denoting by a the semi-axis major. IV. On the other hand, the aphelion in its proper sense is wanting in the parabola, but r is increased indefinitely as v approaches + 180~, or - 180~. For v -+ 180O the value of r becomes infinite, which shows that the curve is not cut by the line of apsides at a point opposite the perihelion. Wherefore, we cannot, with strict propriety of language, speak of the major axis or of the centre of the curve; but by an extension of the formulas found in the ellipse, according to the established usage of analysis, an infinite value is assigned to the major axis, and the centre of the curve is placed at an infinite distance from the focus. V. In the hyperbola, lastly, v is confined within still narrower limits, in fact between v = — (lS0- W), and v = + (180- V), denoting by p the angle of which the cosine _-. For whilst v approaches these limits, r increases to infinity; if, in fact, one of these two limits should be taken for v, the value of r would result infinite, which shows that the hyperbola is not cut at all by a right line inclined to the line of apsides above or below by an angle 180~-, y. For the values thus excluded, that is to say, from 180 — y to 180~ + H-% our formula assigns to r a negative value. The right line inclined by such an angle to the line of apsidles does not indeed cut the hyperbola, but if produced reversely, meets the other branch of the hyperbola, which, as is known, is wholly separated from the first branch and is convex towards that focus, in which the sun is situated. But in our investigation, which, as we have already said, rests upon the assumption that r is taken positive, we shall pay no regard to that other branch of the hyperbola in which no heavenly body could move, except one on which the sun should, according to the same laws, exert not an attractive but a repulsive force. Accordingly, the aphelion does not exist, properly speaking, in the hyperbola also; that point of the reverse branch which lies in the line of apsides, and which corresponds to the values v 1800, r P- might be considered as analogous to the aphelion. If now, we choose after the manner of the 6 RELATIONS PERTAINING SIMPLY [BOOK I. ellipse to call the value of the expression l ee even here where it becomes negative, the semi-axis major of the hyperbola, then this quantity indicates the distance of the point just mentioned from the perihelion, and at the same time the position opposite to that which occurs in the ellipse. In the same way I ep, that is, the distance from the focus to the middle point between - e epieen these two points (the centre of the hyperbola), here obtains a negative value on account of its opposite direction. We call the angle v the true anomnalby of the moving body, which, in the parabola is confined within the limits — 1800 and + 1800, in the hyperbola between - (180~ — tp) and + (180~ J- ), but which in the ellipse runs through the whole circle in periods constantly renewed. Hitherto, the greater number of astronomers have been accustomed to count the true anomaly in the ellipse not from the perihelion but from the aphelion, contrary to the analogy of the parabola and hyperbola, where, as the aphelion is wanting, it is necessary to begin from the perihelion: we have the less hesitation in restoring the analogy among all classes of conic sections, that the most recent French astronomers have by their example led the way. It is frequently expedient to change a little the form of the expression 1 + e cos v }e- P P is - P (1 + e) e2-v + (1-e n V Accordingly, we have in the parabola r = 2 2 v 9 in the hyperbola the following expression is particularly convenient, p cos 9 r 2 cos a (v + ip) c _os I ( ) iE~;CT. l.] TO POSITION IN THE ORBIT. 7 6. Let us proceed now to the comparison of the motion with the time. Putting, as in Art. 1, the space described about the sun in the time It g, the mass of the moving body — /t, that of the sun being taken -1, we have g -kt/I V/(1- +). The differential of the space r- I rr d, from which there results kt1/p (1 + - -) =frrdv, this integral being so taken that it will vanish for I- 0. This integration must be treated differently for different kinds of conic sections, on which account, we shall now consider each kind separately, beginning with the ELLIPSE. Since r is determined from v by means of a fraction, the denominator of which consists of two terms, we will remove this inconvenience by the introduction of a new quantity in the place of v. For this purpose we will put tan I v.v/ 1 e tan 1 E, by whiph the last formula for r in the preceding article gives "rOS 21 (Cos E2 sin21 E) = (1-EoS). (+) CtS21V St l+e e -ee Moreover we have and conseqvuently dv E acos2AE Cos21vV 1 +e n hence rrddv rpdE PP -,, (1 -e cos E) d E, d- /(1 —-e -- e e) " and integrating, kl V V'p (1- 4+ ) = 1 PP (E-e sin E) + Constant. (e — e e) Accordingly, if we place the beginning of the time at the perihelion passage, where v — O, E- 0, and thus constant O0, we shall have, by reason of 1 e a, BE- e sin _ E t- +( )e In this equation the auxiliary angle E, which is called the eccentric anomcdy, must be expressed in parts of the radius. This angle, however, may be retained in degrees, etc., if e sin E and ktV (1 ~-) are also expressed in the same manner; these quantities will be expressed in seconds of arc if they are multiplied by the 8 RELATIONS PERTAINING SIMPLY [BOOK I. number 206264.81. We can dispense with the multiplication by the last quantity, if we employ directly the quantity k expressed in seconds, and thus put, instead of the value before given, k = 3548".18761, of which the logarithm 3.5500065746. The quantity kt V ~i) expressed in this manner is called the a mnean anmonzaly, which therefore increases in the ratio of the time, and indeed every day by the increment IC/ (1 +) called the ewan daily mnotion. We shall denote a the mean anomaly by tL. 7. Thus, then, at the perihelion, the true anomaly, the eccentric anomaly, and the mean anomaly are 0; after that, the true anomaly increasing, the eccentric and mean are augmented also, but in such a way that the eccentric continues to be less than the true, and the mean less than the eccentric up to the aphelion, where all three become at the same time - 180~; but from this point to the perihelion, the eccentric is always greater than the true, and the mean greater than the eccentric, until in the perihelion all three become - 360'~ or, which amounts to the same thing, all are again 0. And, in general, it is evident that if the eccentric Ei and the mean HA answer to the true anomaly v, then the eccentric 360~- -E and the mean 360~ - 1 correspond to the true 3600 -v. The difference between the true and mean anomalies, v - 11 is called the eeqtalion of the centre, which, consequently, is positive from the perihelion to the aphelion, is negative from the aphelion to the perihelion, and at the perihelion and aphelion vanishes. Since, therefore, v and X run through an entire circle from 0 to 360~ in the same time, the time of a single revolution, which is also called the periodic lime, is obtained, expressed in days, by dividing 360~ by the mean daily motion ki/(1 V+t) from which it is apparent, that for difa ferent bodies revolving about.the sun, the squares of the periodic times are proportional to the cubes of the mean distances so far as the masses of the bodies, or rather the inequality of their masses, can be neglected. SECT. 1.] TO POSITION IN THE ORBIT. 9 8. Let us now collect together those relations between the anomalies and the radius vector which deserve particular attention, the derivation of which will present no difficulties to any one moderately skilled in trigonometrical analysis. Greater elegance is attained in most of these formulas by introducing in the place of e the angle the sine of which = e. This angle being denoted by %, we have \V(1-ee)=cos), \/(1+e)=cos(45 -- 2)V2, V(1 -e) = cos (45~ + p) /2, 1 + etan (45 ) (1 + e) + V(l} - e)c- 2 os, ( + e) -- (1 - e) = 2 sin The following are the principal relations between a, p, r, e, (p, v, Ep 2 I. p2= acos2 q I. r = 1 e cos v III. r =a(1 - e cos E) IV co os v e cos E-e 1 +ecosv 1-ecosE IV. cosin E = V ~- ~s;~' or cos v 1 -- e v. sin ~E= - l/ (1 -cos E) sin i v I + e cosv - 1v ( ) sin V I sin v 1(1 - e) VI. cos1E=-/1(1+-cosE)=cosivV14edos -Cos 1 V ( ) _-Cos V VII. tan lE tan vtan(45~ - ) VIII. sin Ez r sin v cos c r sin v p a cos q IX. r cos - a (cos E e)- 2 - oos (2 a + o +c 45o) cos (e E — 45+) X. sin &(V-B)=sin (pE) in V ssin v sin sin XI. sin ( + E) os sinv _ Cos sinE XII M1- = E —- e sin b. 2 10 RELATIONS PERTAINING SIMPLY [BooK I. 9. If a perpendicular let fall from any point whatever of the ellipse upon the line of apsides is extended in the opposite direction until it meets the circle described with the radius a about the centre of the ellipse, then the inclination to the line of apsides of that radius which corresponds to the point of intersection (understood in the same way as above, in the case of the true anomaly), will be equal to the eccentric anomaly, as is inferred without difficulty from equation IX. of the preceding article. Further, it is evident that r sin v is the distance of any point of the ellipse from the line of apsides, which, since by equation VIII. it - a cos sp sin E1, will be greatest for ~- 90~, that is in the centre of the ellipse. This greatest distance, which - a cos yc -p raap, is called the se~mi-axis inzor. In the focus of the ellipse, that is for v - 90Q~ this distance is evidently _-p, or equal the semi-parameter. 10. The equations of article 8 comprise all that is requisite for the computation of the eccentric and mean anomalies from the true, or of the eccentric and true from the mean. Formula VII. is commonly employed for deriving the eccentric from the true; nevertheless it is for the most part preferable to make use of equation X. for this purpose, especially when the eccentricity is not too great, in which case BE can be computed with greater accuracy by means of X. than of VII. Moreover, if X. is employed, the logarithm of sine E required in XII. is had immediately by means of VIII.: if VII. were used, it would be necessary to take it out from the tables; if, therefore, this logarithm is also taken from the tables in the latter mnethod, a proof is at once obtained that the calculation has been correctly made. Tests and proofs of this sort are always to be highly valued, and therefore it will be an object of constant attention with us to provide for them in all the methods delivered in this work, where indeed it can be conveniently done. We annex an example completely calculated as a more perfect illustration. SECT. 1.] TO POSITION IN THE ORBIT. 1]. Given v = 3100 55' 29".64, cp 14 12' 1".87, log r = 0.3307640; p, ap,.EY, are required. log sin e... 9.3897262 log cos v.... 9.8162877 9.2060139 whence e cos v = 0.1606993 log ( + e cosv)., 0.0647197 log r... 0.3307640 logp e.. 0.3954837 log cos2.. 9.9730448 log a. 0.4224389 log sin v.. 9.8782740 nz log.... 0.0323598.5 9.8459141.5n log sin I... 9.0920395 log sin i (v — E). 8.9379536.5n, hence I (v E —) 40 58' 22".94; v —E= - 90 56' 45".88; E= — 320~ 52' 15".52. Further, we have Calculation of log sin E by formula VIII. log206264.8. 5.3144251 logsinv.... 9.8135543n log- 206264.8. 5.3144251 r log ein seconds 4.7041513 log cosyp... 9.9865224 log sinE... 9.8000767n log sinE.... 9.8000767T 4.5042280 n, hence e sin E in seconds = 31932'.14- 80 52' 12'.14; and M — 329~ 44' 27".66. The computation of E by formula VII. would be as follows: - Iv= 155~27'44".82 log tan Iv. 9.6594579n 450~- ~ T-= 37~ 53'59'.065 log tan (45~- ]?). 9.8912427 log tan EB.... 9.5507006n whence IE=- 160~26'7".76, and E- 3200 52'15/.52, as above. * The letter n afixed to a logoarithfm signifies t.at the nlumber correspondingf to it is negaie. 12 RELATIONS PERTAINING SIMPLY [BoooK I 11. The inverse problem, celebrated under the title of Kepler's problemn, that of finding the true anomaly and the radius vector from the mean anomaly, is much more frequently used. Astronomers are in the habit of putting the equation of the centre in the form of an infinite series proceeding according to the sines of the angles X, 2 X, 3, etc., each one of the coefficients of these sines being a series extending to infinity according to the powers of the eccentricity. We have considered it the less necessary to dwell upon this formula for the equation of the entre, which several authors have developed, because, in our opinion, it is by no means so well suited to practical use, especially should the eccentricity not be very small, as the indirect method, which, therefore, we will explain somewhat more at length in that form which appears to us most convenient. Equation XII., 1E i-+ e sin E, which is to be referred to the class of transcendental equations, and admits of no solution by means of direct and complete methods, must be solved by trial, beginning with any approximate value of E, which is corrected by suitable methods repeated often enough to satisfy the preceding equation, that is, either with all the accuracy the tables of sines admit, or at least with sufficient accuracy for the end in view. If now, these corrections are introduced, not at random, but according to a safe and established rule, there is scarcely any essential distinction between- such an indirect method and the solution by series, except that in the former the first value of the unknown quantity is in a measure arbitrary, which is rather to be considered an advantage since a value suitably chosen allows the corrections to be made with remarkable rapidity. Let us suppose E to be an approximate value of BL and x expressed in seconds the correction to be added to it, of such a value as will satisfy our equation - E + x. Let e sin e, in seconds, be computed by logarithms, and when this is done, let the change of the log sin E for the change of 1" in E itself be taken from the tables; and also the variation of log e sin E for the change of a unit in the number e sin e; let these changes, without regard to signs, be respectively, u,, in which it is hardly necessary to remark that both logarithms are presumed to contain an equal number of decimals. Now, if E approaches so near the correct value of E SECT. 1.] TO POSITION IN TIHE ORBIT. 13 that the changes of the logarithm of the sine from a to E + x, and the changes of the logarithm of the number from e sin 8 to e sin (e + x), can be regarded as uniform, we may evidently put e sin (e + x) -e sine - t the upper sign belonging to the first and fourth quadrants, and the lower to the second and third. Whence, since e+ ( -= i+ e sin (e + x), we have x (M+ e sin - e), and the correct value of E, or E -+ x -- + M+ e sin ~ + ((Mx+ esine — ), the signs being determined by the above-mentioned condition. Finally, it is readily perceived that we have, without regard to the signs,: I; - 1: e cos e, and therefore always p >;, whence we infer that in the first and last quadrant MJI+ e sin e lies between e and E + x, and in the second and third, E + x between E and -.-+ e sin e, which rule dispenses with paying attention to the signs. If the assumed value E differs too much from the truth to render the foregoing considerations admissible, at least a much more suitable value will be found by this method, with which the same operation can be repeated, once, or several times if it should appear necessary. It is very apparent, that if the difference of the first value E from the truth is regarded as a quantity of the first order, the error of the new value would be referred to the second order, and if the operation were further repeated, it would be reduced to the fourth order, the eighth order, etc. Moreover, the less the eccentricity, the more rapidly will the successive corrections converge. 12. The approximate value of B, with which to begin the calculation, will, in mo?,-, i,- z't wou be necessary to repeat the calculation anew until it underwent no further ehdage. When the quantity x shall be found, g will be got by the formula sin2 Ig 2. These precepts refer to the first case, in which cosf is positive; in the other case, where it is negative, we put and [14*] LH- - whence equation 12* properly reduced passes into this, [15*] H=f(Y+1) YY0 Y-I~ Y and H can be determined, accordingly, by this cubic equation, whence again x will be derived from the equation [16*] -L mmP MM In the first approximiation will be taken for H; will be taken from table ITI. with the value of x deiived from H by means of the equations 15*, 16*; hence, by formula 14*, will be had the corrected value of 11 with which the calculation, will be repeated in the same, manner. Finally, the angle g will be determined from x in the sacme way as ink the first case. 92. Although the equations 15, 15*, can have three real roots in certain cases, it will, notwithstanding, never be doubtful which should be selected in our problem.. Since hi is evidently a positive quantity, it is readily inferred from the theory of equations, that equation 15 has one positive root with two imaginary or two negative. Now since m 161 122 RELATIONS BETWEEN SEVERAL [BOOK I. miust necessarily be a positive quantity, it is evident that no uncertainty'remains here. So far as relates to equation 15*, we observe, in the first place, that L is necessarily greater than 1; which is easily proved, if the equation given in article 89 is put under the form L=_ Moreover, by substituting, in equation 12*, YV (L. - w) in the place of 31 we have Y+ I- = (L; - x) X,, and so yF 1>(1_x)X>4 +4 xF 5 4.+ 6 3 ~4.6.8X~ etc. > 4 3 3. 5 3.5 7 3.57. 39 and therefore Y>. Putting, therefore, Y= - + Y, Y' will necessarily be a positive quantity; hence also equation 15* passes into this, Y'3~+2YY'+ (I1-H)Y +A -4 II-0~ O which, it is easily proved from the theory of equations, cannot have several positive roots. Hence it is concluded that equation 15* would have only one root greater than -i,t which, the remaining ones being neglected, it will be necessary to adopt in our problem. 93. In order to render the solution of equation 15 the most convenient possible in cases the most frequent in practice, we append to this work a special table (Table I.), which gives for values of h from 0 to 0.6 the corresponding logarithms computed with great care to seven places of decimals. The argument Ii, from 0 to 0.04, proceeds by single ten thousandths, by which means the second differences vanish, so that simple interpolation suffices in this part of the table. But since the table, if it were, equally extended throughout, would be very voluminous, from h - 0.04 to the end it was necessary to proceed by single thousandths only; on which account, it will be necessary in this latter part to have regard to second differences,if we wish to avoid errors of some units t If ina fact we suppose that our problem admits of solutionl. SECT. 3.] PLACES IN ORBIT. 123 in the seventh figure. The smaller values, however, of I are much the more frequent in practice. The solution of equation 15, when I exceeds the limit of the table, as also the solution of 15', can be performed without difficulty by the indirect method, or by other methods sufficiently known. But it will not be foreign to the purpose to remark, that a small value of g cannot coexist with a negative value of cosf, except in an orbit considerably eccentric, as will readily appear from equation 20 given below in article 95.-t 94. The treatment of equations 12, 12', explained in articles 91, 92, 93, rests upon the supposition that the angle g is not very large, certainly within the limit 66~ 25', beyond which we do not extend table III. When this supposition is not correct, these equations do not require so many artifices; they can be most securely and conveniently solved by trial wzithout a change of form. Securely, since the value of the expression 2 g - sin 2 g sin3 g in which it is evident that 2g is to be expressed in parts of the radius, can, for greater values of g,be computed with perfect accuracy by means of the trigonometrical tables, which certainly cannot be done as long as g is a small angle: conveniently, because heliocentric places distant from each other by so great an interval will scarcely ever be used for the determination of an orbit wholly unknown, while by means of equation 1 or 3 of article 88, an approximate value of g follows with almost no labor, from any knowledge whatever of the orbit:' lastly, from an approximate value of g, a corrected value will always be derived with few trials, satisfying with sufficient precision equation 12 or 12'. For the rest, when two given heliocentric places embrace more than one entire revolution, it is necessary to remember that just as many revolutions will have been completed by the eccentric anomaly, so that the angles E'-E, v'-v, either both lie between 0 and 360~, t That equation shows, that if cosf is negative, q must, at least, be greater than 900 - g. 124 RELATIONS BETWEEN SEVERAL [BooK I. or both between similar multiples of the whole circumference, and also f and g together, either between 0 and 1800, or between similar multiples of the seinicircum-ference. If, finally, the orbit should be wholly unknown, and it should not appear whether the heavenly body, in passing from the first radius vector to the second, had described a part only of a revolution or, in addition, one entire revolution, or several, our problem would sometimes admit several different solutions however, we do not dwell here on this case, which can rarely occur in practice. 95, We pass to the second matter, that is, the determination of the elements from the angle g when found. The major semiaxis is had here immediately by the formulas 10, 10O, instead of which the following can also be used:2 mnz cosf V r r' kIltt yy sin2 g 4yy rr cos2fsin2g l[7a ] ac - 2 AftfcosfV r r' k tt YY sin2 4 YYrr' cos2fsin2g' The minor semiaxis b V/ap is got by means of equation 1, which being combined with the preceding there results [8 (y r r' sin 2 f)2 C[1-8] (:Yrrsin2f)2 Now the elliptic sector contained between two radii vectores and the elliptic arc is i k /2 p also the triangle between the same radii vectores and the- chord IT rrt sin 2f: wherefore, the ratio of the sector to the triangle is asy: 1 or Y: 1. This remark is of the greatest importance, and elucidates in a beautiful manner both the equations 12, 12": for it is apparent from this, that in equation 12 the parts ma, (l+X)2, X(l+- )-, and in equation 12' the parts X, (L-x)2, X(LZ-_x)2, are respectively proportional to the area of the sector (between the radii vectores and the elliptic arc), the area of the triangle (between the radii vectores and the chord),? the area of the segment (between the arc and the chord), because the first area is evidently equal to the suml or difference of the other two, accordIng (a"s — v lies between 0 and 1800, or between 180~ and 360~. In the case SECT. 3.] PLACES IN ORBIT. 125 where v'- v is greater than 3600 we must conceive the area of the whole ellipse adced to the area of the sector and the area of the segment just as malny times as the motion comprises entire revolutions. Mforeover, since b = a cos p, from the combination of equations 1, 10, 10*, follow [19] cos o sing tanf [1siLn g tan f [19'] COS~o T — 2 (L- sina-g)I whence, by substituting for 1, Ltheir values from article 89, we have [20] cos sin fsin g 20] - os cos g + 2 tari2 2 co This formula is not adapted to the exact computation of the eccentricity when the latter is not great: but from it is easily deduced the more suitable formula sin2 (f+ g) +- tan2 2 co to which the following form can likewise be given (by multiplying the numerator and denominator by cos2 2 co) [22] ~tan2 AD? ln -sin (f g) os2 (f g)sin22 sin2I (f y +) -F co2 I2 (- g) sin22 co' The angle q can always be determined with all accuracy by either formula, using, if thought proper, the auxiliary angles of which the tangents are tan 2 co tan 2 co sin I (f-g) sin 2 (f+ g) for the former, or sin 2 o sin 2 to tan 2 (f ~7)I tan (f/+ g) for the latter. The following formula can be used for the determination of the angle G, which readily results from the combination of equations 5, 7, and the following one not numbered, [23] tanG~ G =(r' -- r) sin g (r'+ ) cos g- 2 cosflrri' from which, by introducing wis easily derived 126 RELATIONS BETWEEN SEVERAL [BooK I. sin g sin 2 co L2i 2ae C os"2 c sn I (f-g) sin (f+ g) + in22 W cosg The ambiguity here remaining is easily decided by means of equation 7, which shows, that G must be taken between 0 and 180~, or between 180~ and 3600, as the numerator in these two formulas is positive or negative. By combining equation 3 with these, which flow at once from equation II. article 8, 1 2e. - sm f sin r r p 1 1 2 2e +,~+ - cosf cos F the following will be derived without trouble, [25/1 tanF=2 oX;~/~- ~+ (r' - r) sinf 2 cos g rr'-(r + r) cosf; from which, the angle cw being introduced, results [261 tanP F cos2 2 sin sinf sin 2 o, os2 c sin (f-g) sin (f- g) -s n2 osf The uncertainty here is removed in the same manner as before. —As soon as the angles F and a shall have been found, we shall have v F-f, v' Pf -f, whence the position of the perihelion will be known; also E- G-g, E' CG +yg Finally the mean motion in the time I will be -2 g-2 e cos G sing, the agreement of which expressions will serve to confirm the calculation; also, the epoch of the mean anomaly, corresponding to the middle time between the two given times, will be - -e sin G cosg, which can be transferred at pleasure to any other time. It is somewhat more convenient to compute the mean anomalies for the two given times by the formulas - e sin E, E' -e sin' and to make use of their difference for a proof of the calculation, by comparing it with a2 SECT. 3.] PLACES IN ORBITo 127 96. The equations in the preceding article possess so much neatness, that there may seem nothing more to be desired. Nevertheless, we can obtain certain other formulas, by which the elements of the orbit are determined much more elegantly and conveniently; but the development of these formulas is a little more abstruse. We resume the following equations from article 8, which, for convenience, we distinguish by new numbers: — Ie sin -s/a sinI E V(1+e) IL cos cos -cos (1-e) II. sin v'V/ =sin + E' (1 e).... COS av' -= Cos I E'(1 -- e). We multiply I. by sin i (F + g) II by cos i (F +g), whence, the products being added, we obtain Cos (f+ ) / sin E sin i (+g) V (1 + e) + cos 1Ecos (F+g) - (1-e) or, because / (I + e) - cos - P +- sin i %, /(1 — e) = coS 1 -- Sin 21 %p cos I (f]g) a = cos i Cos (IF-A- ) -+sin I -- cos 1 (F + G). In exactly the same way, by multiplying III. by sin 1 (F-g), IV. by cos It (F-g1), the products being added, appears cos (f+)/ a = cos 1 T cos ( F-1 CG -— ) -sin I q cos A (-+G- C). The subtraction of the preceding from this equation gives cos (f4-g)( ( /L)=2 cos l p singsin I (F- C ), or, by introducing the auxiliary angle w, [27] cos t (f -+g) tan 2 w -sin ( -G) cos p sing/7 128 RELATIONS BETWEEN SEVERAL [BooiK L By transformations precisely similar, the development of which we leave to the skilful reader, are found [28] sco 2 -(f =cos (P-G ) cos p sing / 4/_/ [29] cos E (f-y) tan 2 w = sin - (E+ C) sin A 9) sing aa [30] sin (f-g)=cos( C(2) sin 2ip sing v cos 2 to When the first members of these four equations are known, - (F- G) and will be determined from 27 and 29; and also, from 29 and 30,in the same manner, 2(P+ G) and S 4aa sin r rp sin g V - = the doubt in the determination of the angles 1 (P-CG), a (P — G), is to be so decided that P and Q may have the same sign as sin g. Then I 9p and 4 aa sing V = R will be derived from P and Q., From R can be deduced sin' ga and also sin2f/ r / Hi RR n unless we prefer to use the former quantity, which must be + ( (2(1+ in2 i g) cosf) + (-2(L - sin2 g) cosf), for a proof of the computation chiefly, in which case a and p are most conveniently determined by the formulas b sinf rrt r sin g cos 0 p } O Several of the equations of articles 88 and 95 can be employed for proving the calculation, to which we further add the following: - 2.tan2. e sin C sin cos 2 co a a SECT. 3.] PLACES IN ORBIT. 129 2 tan 2 o P e sin p sinf cos 2 w \/ rr' 2 tan 2 co 2-t 2 -- tan s sin G sin f — tan p sin F sin g. cos 2 co Lastly, the mean motion and the epoch of the mean anomaly will be found in the same manner as in the preceding article. 97. We will resume the two examples of article 87 for the illustration of the method explained in the 88th, and subsequent articles: it is hardly necessary to say that the meaning of the auxiliary angle co thus far adhered to is not to be confounded with that with which the same symbol was taken in articles 86, 87. I. In the first example we have f - 3~ 47' 26'/865, also log. = 9.9914599, log tan (45~ + wi) - 9.997864975, o = - 8' 27.006. Hence, by article 89, log sin2iAf.. 7.0389972 log tan22 2 w. 5,3832428 logcosf.... 9.9990488 log cosf... 9.9990488 7.0399484 5.3841940 log 0.0010963480 = log 0.0000242211 and thus l= — 0.0011205691, -5- -- 0.8344539. Further we have logk i... 9.5766974 2 logk.. 9.1533948 C. log r r'.. 9.0205181 C. log8cossf.. 9.0997636 log mn m.. 7.2736765 log ( + — 1)... 9.9214023 7.3522742 The approximate value, therefore, of h is 0.00225047, to which in our table II. corresponds logyu -= 0.0021633. We have, accordingly, log mm 7.2715132, or - 0001868587 17 130 RELATIONS BETWrEEN SEVERAL [BOOK I. whence, by formula 16 x-= 0.0007480179: wherefore, since 5 is, by table III., wholly insensible, the values found for h, y, x, do not need correction. Now, the determination of the elements is as follows: - logx..... 6.8739120 log sin ig... 8.4369560, g 5 1~ 34' 2".0286, i (f +y) 3~27'45".4611, i (f —g) 19'41'.4039. Wherefore, by the formulas 27, 28, 29, 30, is had log tan 2 o 7.6916214 n C. log cos 2 C.. 0.0000052 log cos 2 (f-+g). 9.9992065 log sin (f-+g).. 8.7810188 log cos (f- -g). 9.9999929 logsin l (f —g).. 7.7579709 log P sin 1i (F — ) 7.6908279 n log Q sin I (F-+ G). 7.6916143 n logPcos 2 (F — G) 8.7810240 log Q cos (F+C G). 7.7579761 (F — G) - 40 38' 41".54 log P - log R cos i1 p 8.7824527 1 (F+ G)-C 319 21 38.05 log Q - log R sin p 7.8778355.F_ 314 42 56.51 Hence I q- 7~ 6' 0~".935 v- 310 55 29.64 9_ 14 12 1.87 V - 318 30 23.37 logR... 8.7857960 GC _ 324 0 19.59 For proving the calculation. 320 52 15.53 log 2cosf... 0.1500394 Wt -327 823.65 2log(l+ x) = log 8.6357566 8.7857960 Ilogrr'.... 0.3264939 log sin..... 9.3897262 logsinf.... 8.8202909 log 206265... 5.3144251 C.log sing 1.2621765 loge inseconds.. 4.7041513 logh.. 0.4089613 log sinE.....'9.8000767n log cos.. 9. 9865224 log sin E'.... 9.7344714 n logp. 0.3954837 logesinE o,.. 4.5042280n log a 0.4224389 log e sin'.... 4.4386227 n SECT. 3.] PLACES IN ORBIT. 131 logk.. 3.5500066 e sin E -- 31932".14 -=- 8~ 52'12".14 loga a 0.6336584 e sin E' — 27455.08 - - 7 37 35.08 2.9163482 Hence the mean anomaly for the logt o. 1.3411160 first place = 329044'27/".67 4.2574642 for the second 334 45 58.73 Difference = 5 1 31.06 Therefore, the.mean daily motion is 824".7989. The mean motion in the time t is 18091".07- 50 1'31".07, II. In the other example we have f =31 27'38".32, o -- 21'50".565, 1 - 0.08635659, log m 9.3530651, mm mm-l or the approximate value of Ah 0.2451454; to this, in table II., corresponds loggyy 0.1722663, whence is deduced 0.15163477 x - 0.06527818, hence from table III. is taken 0.0002531. Which value being used, the corrected values become h = 0.2450779, logyy = 0.1722303, m = 0.15164737, x = 0.06529078, - 0.0002532. If the calculation should be repeated with this value of e, differing, by a single unit only, in the seventh place, from the first; hA, logyy, and x would not suffer sensible change, wherefore the value of x already found is the true one, and we may proceed from it at once to the determination of the elements. We shall not dwell upon this here, as it differs in nothing from the preceding example. III. It will not be out of place, to elucidate by an example the other case also in which cosf is negative. Let v'- v =224~ 0' 0", or f= 112~ 0' 0", log r- 0.1394892, log r' — 0.3978794, t- 206.80919 days. Here we find co - + 40 14'43" 78, L 1.8942298, log Ml1M= 0.6724333, the first approximate value of log H= 0.6467603, whence by the solution of equation 15* is obtained Y= 1.591432, and afterwards x -0.037037, to which, in table III., corresponds - 0.0000801. Hence are derived the corrected values log H- 0.6467931, Y= 1.5915107, x=- 0.0372195, 5 — 0.0000809. The calculation being repeated 132 RELATIONS BETWEEN SEVERAL [BooK I. with this value of 5, we have x -0.0372213, which value requires no further correction, since a is not thereby changed. Afterwards is found 1g = 110 7'25".409 and hence in the same manner as in example I. ( (F-) = 3~ 33' 53".59 log P - log R cos I? 9.9700507 ~ (P-+ G) =- 8 26 6.38 log Q — logRsin~ o. 9.8580552 F-= 11 59 59.97 = 370 41'34".27 v= — 100 0 0.03 A= 75 23 8.54 v'= + 123 59 59.97 logR...... 0.0717096 G -- 4 52 12.79 For proving the calculation. -- x17 22 38.01 M E 2-1 7 2238 019 log y -2 cosf.. 0.0717097 E'=. ~27 7 3.59 The angle qs in such eccentric orbits is computed a little more exactly by formula 19*, which gives in our example 5 = 750 23' 8".57; likewise the eccentricity e is determined with greater precision by the formula e = 1-2 sin2 (45~- I 0), than by e sin p; according to the former, e = 0.96764630. By formula 1, moreover, is found log b-6 0.6576611, whence logp = 0.0595967, log a = 1.2557255, and the logarithm of the perihelion distance log =loga(1-e)= -logbtan(45~- p)= —.7656496. It is usual to give the time of passage through the perihelion in place of the epoch of the mean anomaly in orbits approaching so nearly the form of the parabola; the intervals between this time and the times corresponding to the two given places can be determined from the known elements by the method given in article 41, of which intervals the difference or sum (according as the perihelion lies without or between the two given places), since it must agree with the time 1, will serve to prove the computation. The numbers of this third example were based upon the assumed elements in the example of articles 38, 43, as indeed that very example had furnished our first place: the trifling differences of the elements obtained here owe their origin to the limited accuracy of the logarithmic and trigonometrical tables. SECT. 3.] PLACES IN ORBIT. 133 98. The solution of our problem for the ellipse in the preceding article, might be rendered applicable also to the parabola and hyperbola, by considering the parabola as an ellipse, in which a and b would be infinite quantities, p - 90~, finally E, E'' g, and G = 0; and in a like manner, the hyperbola as an ellipse, in which a would be negative, and, B,' g, G, q, imaginary: we prefer, however, not to employ these hypotheses, and to treat the problem for each of the conic sections separately. In this way a remarkable analogy will readily show itself between all three kinds. Retaining in the PARABOLA the symbols p, v, v', Ff, r, r', I with the same sign nification with which they had been taken above, we have from the theory of the parabolic motion: - [1] V = cos (t-f) [2] -Ccos 2 ( +f) 2kt= t tan (+f)-tan (-) - tanl 2 (E+-f)- + tan3 ( ta3 ( —f) p2 = (tan t (F+f)- tan (F- )) ( + tn tan (P4-f tau I (F-f) + X (tan 1 (E +f )-tan X ( f —))2) -_2 sinfVrr' (2 cos frr' 4 sin2fr r p P 3pp whence [3] k -I 2sinfcosf. rr' 4sinsf (rr')2 + 3pA Further, by the multiplication of the equations 1, 2, is derived [4] _,=cos + cosf and by the addition of the squares, [5] P (Id+-r') =1 + cos -F os 134 RELATIONS BETWEEN SEVERAL [BooK I. Hence, cos F being eliminated, 2 rr' sin2f [6] - r+' + i, 2 cosfV rr If accordingly, we adopt here also the equations 9, 9*, article 88, the first for cosf positive, the second for cosf negative, we shall have, r1ri P ~sin2f Vr r' L[7] -2 1cosf [ 7*] sin2fV r r [7] -- _ 2L cosf which values being substituted in equation 3, preserving the symbols mn, M, with the meaning established by the equations 11, 11*, article 88, there result 1 2 [8] m-=l +l ~, [8*] X =- 1~ + A1oL These equations agree with 12, 12', article 88, if we there put g = 0. Hence it is concluded that, if two heliocentric places which are satisfied by the parabola, are treated as if the orbit were elliptic, it must follow directly from the application of the rules of article 19, that x 0; and vice versa, it is readily seen that, if by these rules we have x 0, the orbit must come out a parabola instead of an ellipse, since by equations 1, 16, 17, 19, 20 we should have b _- m, a= s0o - 90. After this, the determination of the elements is easily effected. Instead of p, either equation 7 of the present article, or equation 18 of article 95 t might be employed: but for F we have from equations 1, 2, of this article tan cotan f sin 2 uo cotan f, if the auxiliary angle is taken with the same meaning as in article 89. We further observe just here, that if in equation 3 we substitute instead of p its value from 6, we obtain the well-known equation -kt (r + r' - cosf. Vrr') (r + r' - 2 cosf.r r' )~ / 2. t Whence it is at once evident that y and r express the same ratios in the parabola as in the ellipse. See article 95. SECT. 3.j PLACES LN ORBIT. 135 99. We retain, in the HYPERBOLA also, the symbols p, v, v', r, F r, r' t with the same meaning, but instead of the major semiaxis a, which is here negative, we shall write — a; we shall put the eccentricity e = in the same manner as above, article 21, etc. The auxiliary quantity there represented by u, we shall put for the first place —, for the second = Cc. whence it is readily inferred that e is always greater than 1, but that it differs less from one, other things being equal, in proportion as the two given places are less distant from each other. Of the equations developed in article 21, we transfer here the sixth and seventh slightly changed in form, C[1] cos t v ( e +. /)(e-1) a [2] sin -=(v/ (e+)c G3 cOS9 d (e 1) a [3] cos l2 2 +V /Ce+ ) Cc rl )[4] sin 12v = a v 1 (e + 1) a From these result directly the following:[5] sinF=- C- a ebeV'; [6] sionf=- a (c I)V i e e l, [7] cosF=(e(e+!) (C+))2 [8] cosf=(e(a+, c a+e) 2\r re ~ Again, by equation X. article 21, we have r - = e + - 136 RELATIONS BETWEEN SEVERAL [BooK I. and hence, [9] r = a e(G(7,)( >-), [I [10] r'+ e (C(C+ )(+ )-2. This equation 10 combined with 8 gives /+r- (c+-) cosf. rr' [ll] J= A (, — )2 Putting, therefore, in the same manner as in the ellipse T - r cosf = 1 +21, or — 1-2 L, according as cosf is positive or negative, we have 8(l-4 (8c-V/cV)2)cosf. Vrr' 1:*] 1 = -8 (L+ - (Vc-V )2) cosf irr' [~12]t~) 1 The computation of the quantity I or L is here made with the help of the auxiliary angle o in the same way as in the ellipse. Finally, we have from equation XI. article 22, (using the hyperbolic logarithms), t, a: e ( e~a I a_ V)_ log 0 —c+ — log Ic _~ 1 U2 _8 e(CL+ - c )-2) Cosf.V r or, C being eliminated by means of equation 8, kt (C —) cosf. rr' 1 kt a + 1 - - l(c-og-Cc)-2 logC In this equation we substitute for a its value from 12, 12*; we then introduce SECT. 3.] PLACES IN ORBIT. 13~ the symbol m or X, with the same meaning that formulas 11, 1*, article 88 give it; and finally, for the sake of brevity, we write c c -- 4 log c 4c1)2 - Z; C)4 (~a_ 1)8 _ PX_ from which result the equations [13] 24= (I )2+( )3 Z -[13*] J= -(L d+ 7)2 + (L + 2)' Z, which involve only one unknown quantity, z, since Z is evidently a function of z expressed by the following formula, Z= (l + 2 z) (z zz) —log ( (1 (+z)+ /z) 3 2 (Z +Z )2 100. In solving the equation 13 or 13*, we will first consider, by itself, that case in which the value of z is not great, so that Z can be expressed by a series proceeding according to the powers of z and converging rapidly. Now we have (I +2 z) 2 ( + z) = 2 + 2 2 + 87 s log -( z) (I () -6- Z) -- V - and so the numerator of Z is 3 zt- + Z -...; and the denominator, 2 z 3 z...2 whence, Z A In order to discover the law of progression, we differentiate the equation 2 (o + Z (I 2 z) 1(z+ z z) - log (\/(I ) + +X)n whence results, all the reductions being properly made, adZ 2 ( + ZZ) 2(Id + 3 Z (1 + 2 ) \/( + )- 4 ( + ), 18 138 RELATIONS BETWEEN SEVERAL [BooK 1 or (2 2 + 2 Z 4 - (3 + 6 z) Z, whence, in the same manner as in article 90, is deduced 4.4.6 4..8 4.6.8.10 3 4 6 8 10.12 3 — Z 527zz —- 9 e3.5.7c. 33.5.5577. 9 3. 57.9. 1 It is evident, therefore, that Z depends upon - z in axactly the same manner as X does upon x above in the ellipse; wherefore, if we put 1 4 + F (Z +) 4 also will be determined in the same manner by z as d, above, by x, so that we have [14] = 1 + etc., or, 1 1- g~ x + +P1 + 2% - zzi 1 -4 - _o. 1 + etc. In thlis way the values of t are computed for z to single thousandths, from z O up to z = 0.3, which values are given in the third column of table I1l By introducing the quantity 5 and putting Z/or~ (L + Z) _ also [15] ++ h, or [LS~j — f SECT. 3.] PLACES LN ORBIT. 139 equations 13, 13* assume the form, (Y+ 1) YY[16'] ( -F l) fY and so, are wholly identical with those at which we arrived in the ellipse (15, 15*, article 91). Hence, therefore, so far as h or H can be considered as known, y or Y can be deduced, and afterwards we shall have [17] z=I — [17*] - rij LeL. From these we gather, that all the operations directed above for the ellipse serve equally for the hyperbola, up to the period when y or Y shall have been deduced from h or H; but after that, the quantity ni rn.2n ilL -1, or L- yi which, in the ellipse, should become positive, and in the parabola, 0, must in the hyperbola become negative: the nature of the conic section will be defined by this criterion. Our table will give 4 from z thus found, hence will arise the corrected value of h or H, with which the calculation is to be repeated until all parts exactly agree. After the true value of z is found, e might be derived from it by means of the formula a = I + 2 z + 2 + (a + ZZ)7 but it is preferable, for subsequent uses, to introduce also the auxiliary angle n, to be determined by the equation tan 2 n -- 2 V (z- zz) hence we have e =tan 2 - 1 (- an" 2 n) = tan (45 +)- n) 140 RELATIONS BETWEEN SEVERAL [BOOK I, 102. Since y must necessarily be positive, as well in the hyperbola as in the ellipse, the solution of equation 16 is, here also, free from ambiguity:t but with respect to equation 16*, we must adopt a method of reasoning somewhat different from that employed in the case of the ellipse. It is easily demonstrated, from the theory of equations, that, for a positive value of Ht, this equation (if indeed it has any positive real root) has, with one negative, two positive roots, which will either both be equal, that is, equal to 6 /5 — - = 0.20601, or one will be greater, and the other less, than this limit. We demonstrate in the following manner, that, in our problem (assuming that v is not a large quantity, at least not greater than 0.3, that we may not abandon the use of the third table) the greater root is always, of necessity, to be taken. If in equation 13' in place of M is substituted Y / (L + z),we have Y+ = (L +z)Z> (1+ -)Z, or 4 4. 6 4. 6.8 s _ etc. 3> ~ 3.-5 - 3.5.7 -3.5.7. etc whence it is readily inferred that, for such small values of z as we here suppose, Y must always be > 0.20601. In fact, we find, on making the calculation, that z must be equal to 0.79858 in order that (1 + z) Z may become equal to this limit: but we are far from wishing to extend our method to such great values of z. 103. When z acquires a greater value, exceeding the limits of table III., the equations 13, 13* are always safely and conveniently solved by trial in their unchanged form; and, in fact, for reasons similar to those which we have explained t It will hardly be necessary to remark, that our table II. can be used, in the hyperbola, as well as in the ellipse, for the solution of this equation, as long as h does not exceed its limit. + The quantity Hf evidently cannot become negative, unless > 6; but to such a value of C would correspond a value of z greater than 2.684, thusn far exceeding the limits of this method. SECT. 3.] PLACES IN ORBIT. 141 in article 94 for the ellipse. In such a case, it is admissible to suppose the elements of the orbit, roughly at least, known: and then an approximate value of n is immediately had by the formula tan 2 n- Si-fV 1) c~(e e-1') which readily follows from equation 6, article 99. z also will be had from n by the formula 1 -cos 2 n sin2 n 2 cos 2 n cos 2 n and from the approximate value of a, that value will be deduced with a few trials which exactly satisfies the equation 13, 13'. These equations can also be exhibited in this form, sin2 n *)(- sin2n os ta2n 2 hyp loggtan (45oj+ ) cos2n rn= (l cos 2n) +2 ( 1 cos 2 j ltan3 2 n ta 2 tan32 2 22 n hyp.logtan(45~q-n) -- -- (L + cos / - 2 (L -+ cos 2 n) and thusg z being neglected, the true value of n can be deduced. 104o It remains to determine the elements themselves from z, n, or c. Putting a / (e e X 1) = (, we shall have from equation 6, article 99, [18] /? = sinf ~rg tan 2 n combining this formula with 12, 12%, article 99, we derive, [191] V(ee —1)_tann t"af tan 2 n 9 (I - -) [19*] tan A, V _ tanf tan 2 n [19'] tan - (- L+z)' whence the eccentricity is conveniently and accurately computed; a will result from (3 and / (e e - 1) by division, and p by multiplication, so that we have, 142 RELATIONS BETWEEN SEVERAL [BooK I. a 2(1-z) cosf. Vrr' 2rnmcosf.V/rr' kktt tan2 2 n yy tan2 2 n 4 y y rr' cos2ftan2 2 n -2 (L + z) cosf. r r' - 2.MIcosf. f/rr' k tt tan2 2 n Y Ytan2 2 n 4 YYrr' cos2ftan2 2 n9' sinf.tanf.Vrr' _yysinf.tanf.Vrr' rr'sin2f\2 sinf. tanf. / r r' _ y y sinf tsinf. tan r r r r'Yrr' sin 2 f 2 (L + z) 2mm M kt J - 22(L+z) 2MM 2f ft The third and sixth expressions for p, which are wholly identical with the formulas 18, 18, article 95, showv that what is there said concerning the meaning of the quantities y, Y, holds good also for the hyperbola. From the combination of the equations 6, 9, article 99, is derived - e sin f. ( C- -); by introducing therefore yV and o, and by putting C= tan (450 -+ N), we have 2 sin V tan 2 co [20] tan 2 N — sin fcos 2 sin f cos 2 o C being hence found, the values of the quantity expressed by u in article 21, will be had for both places; after that, we have by equation II., article 21, t;an - v C-c tan 2 -(C+ c) tan I i _Cc-i ta(Cc + 1) tlan tp or, by introducing for C c, the angles N, n, [21] tan' v sin (N-n) cos (ZV-+ n) tan I P sin (N+ n) [22] tan 2 v'= -cos (N- n) tan - Lp' Hence will be determined the true anomalies v, v', the difference of which compared with 2f will serve at once for proving the calculation. Finally, the interval of time from the perihelion to the time corresponding to the first place, is readily determined by formula XI., article 22, to be 8 a Y(2 e cos (NV- n) sin (- n) log tan (450 + ) \' cos 2 Ncos 2 n — tanhp. l og ( -ia n) SECT. 3.] PLACES IN ORBIT. 143 and, in the same manner, the interval of time from the perihelion to the time core responding to the second place, all 2 e cos (N-n) sin (N+ n)t. ke cos (2 N)osi2(N+ n) 2 -hyp.log tan (45~ + -r) tan (46~ + n)) If, therefore, the first time is put T — t, and, therefore, the second T- + I t, we have [23] a (eanN _ lo tan (45~ + N)) cos2ng whence the time of perihelion passage will be known; finally, [24] =a{ ( e tan 2. log tan (450 +n )) [24] t —.~ \cos2N which equation, if it is thought proper, can be applied to the final proof of the calculation. 105. To illustrate these precepts, we will make an example from the two places in articles 23, 24, 25, 46, computed for the same hyperbolic elements. Let, accordingly, -v v = 48"12' 0", or 24 6' 0", log r = 0.0333585, logr' = 0.2008541, t = 51.49788 days. lHence is found C - 2 45' 28".47, 1 0.05796039, mm or the approximate value of A = 0.0644371; hence, by table II., logy/y = 0.0560848, m m 0.05047454, z = 0.00748585, to which in table III. corresponds- 0.0000032. Hence the corrected value of h is-0.06443691, log/y = 0.0560846, m 0.05047456 z- =0.00748583, which values require no further correction, because' is not changed by them. The computation of the elements is as follows: 144 RELATIONS BETWEEN SEVERAL [BooK Ie log z.. 7.8742399 log tanf.... 9.6506199 log(1 +-z). e 0.0032389 log tan 2 o. o 8.9387394 0log \/ (. +.z) ~ e8.9387394 C. log (I-z).. 1.2969275 log 2.. 0.3010300 log tan..... 9.8862868 log tan2n. 9.2397694 370 341 59 *77 2 n 90 51'11".816 (it should be 370 35' 0") n' =4 55 35.908 logsinf.... 9.6110118 C. log J sinf o 0.6900182 log/r r'.. 0.1171063 log tan 2.... 8.9848318 C. log tan 2 n. 0.7602306 C. log cos 2 C. 0.0020156 log3..eoo 0.4883487 log sin yl e**.... 9.7852685 log tan r.. 9.8862868 log tan 2 N. 9.4621341 log a.0.....6020619 21V = 16 946.253 logp e 0.0@3746355 N =8 4 53.127 (they should be 0.6020600 and 0.3746356) N — n 3 9 17.219 N2V-qz- n 13 0 29.035 log sin (NV-n) a 8.7406274 log sin (N+ n).. 9.3523527 C. log cos (N+ n). 0.0112902 C. log cos (N-n). 0.0006587 log cot i p. o 0.4681829 log cot 21 p.. 0.4681829 log tan l. o 9.2201005 log tan v'.... 9.8211943 v u= 9~ 25' 29".97 v' 33~ 31' 29".93 v 18 50 59.94 v'- 67 2 59.86 (it should be 18~ 51' 0") (it should be 67~ 3' 0") log e..... 0.1010184 log e. 0.1010184 log tan 2N. o 9.4621341 log tan 2 n... 9.2397694 C. log cos2 n 0.0064539 C. logcos2a o 0.0175142 9.5696064 9.3583020 number 0.37119863 number 0.22819284 hyp log tan (45 +-N)- 0.28591251 hyp log tan (45~ +n)- 0.17282621 Diffierence 0.08528612 Difference -- 0.05536663 SECT. 3.] PLACES IN ORBIT. 145 log...... 8.9308783 log...... 8.7432480 loga. 0.90309 loga.. 0.9030928 C. log k..... 1.7644186 C. log k..... 1.7644186 log T..... 1.5983897 log 2..... 0.3010300 AdT 39.66338 log.... 1.7117894 t1 51.49788 Therefore, the perihelion passage is 13.91444 days distant from the time corresponding to the first place, and 65.41232 days from the time corresponding to the second place. Finally, we must attribute to the limited accuracy of the tables, the small differences of the elements here obtained, from those, according to which, the given places had been computed. 106. In a treatise upon the most remarkable relations pertaining to the motion of heavenly bodies in conic sections, we cannot pass over in silence the elegant expression of the time by means of the major semiaxis, the sum r + r', and the chord joining the two places. This formula appears to have been first discovered, for the parabola, by the illustrious EULER, (Miscell. Berolin, T. VII. p. 20,) who nevertheless subsequently neglected it, and did not extend it to the ellipse and hyperbola: they are mistaken, therefore, who attribute the formula to the illus trious LAMBERT, although the merit cannot be denied this geometer, of having independently obtained this expression when buried in oblivion and of having extended it to the remaining conic sections. Although this subject is treated by several geometers, still the careful reader will acknowledge that the following explanation is not superfluous. We begin with the elliptic motion. We observe, in the first place, that the angle 2f described about the sun (article 88, from which we take also the other symbols) may be assumed to be less than 3600; for it is evident that if this angle is increased by 360~, the time is increased by one revolution, or 3 a2.360o 3 a- -_2 X 3 6 5.25 days. 19 14:6 RELATIONS BETWEEN SEVERAL [BooK 1. NNoW, if we denote the chord by Q, we shall evidently have Q = (r' cos v'-r cos v)2 + (r'sin v'-r sin v)2, and, therefore, by equations VIII., IX., article 8, - - ab (cos E' — cos E)2 + a a co, (sin _' - sin E)2 -4 a a sin2 g (sin2 G + cos2 s cos2 G) = 4 aa sin2 g (1 -e e cos2 G). We introduce tlhe auxiliary angle h such, that cos h = e cos G; at the same time, that all ambiguity may be removed, we suppose Ih to be taken between 0~and 180~, whence sin h will be a positive quantity. Therefore, as g lies between the same limits (for if 2 g should amount to 3600 or more, the motion would attain to, or would surpass an entire revolution about the sun), it readily follows from the preceding equation that Q _ 2 a sing sin h, if the chord is considered a positive quantity. Since, moreover, we have r + r' - 2 a (1 - e cos g cos G) 2 a (1.- cos g cos ), it is evident that, if we put h-g - Y, h +-g, we have, [1] r+r'- 2 (I -cosY)- 4 sin2 -p J, [2] r + r'+Q —2 a (I - cosE) — 4 a si2 2. Finally, we have k g a2 (2 g -2 e sin cos G) - a2 (2 g - 2 sin g cos h), or [3] k = cad (e-sin e- (8-sin ))o Therefore, the angles Y and E can be' determined by equations 1, 2, from s- + r', o, and a; wherefore, the time t will be determined, from the same equations, by equation 3. If it is preferred, this formula can be expressed thus: k = a are COS 2 a — (r +- ) —Q _ sin are cos 2-(r+r)-Q 2a-(r+r')+e * 2a-(r+rl)+Qg -arc cos 2 + sin are cos a But an uncertainty remains in the determination of the angles 8, e, by their cosines, which must be examined more closely. It appears at once, that d must lie between - 1800 and + 180~, and e between 0~ and 360~: but thus SECT. 3.] PLACES IN ORBIT. 147 both angles seem to admit of a double, and the resulting time, of a quadruple, determination. We have, however, from equation 5, article 88, cosf. V/rr'- a (cosg - cos h) - 2 a sin l Y sin Ie: now, sin i e is of necessity a positive quantity, whence we conclude, that cosf and sin 1 6 are necessarily affected by the same sign; and, for this reason, that a is to be taken between 00 and 1800, or between- 180~ and 0~ according as cosf happens to be positive or negative, that is, according as the heliocentric motion happens to be less or more than 180~. Moreover, it is evident that a must necessarily be 0~, for 2f- 180~. In this manner 6 is completely determined. But the determination of the angle E continues, of necessity, doubtful, so that two values are obtained for the time, of which it is impossible to determine the true one, unless it is known from some other source. Finally, the reason of this phenomenon is readily seen: for it is known that, through two given points, it is possible to describe Iwo different ellipses, both of which can have their focus in the same given point and, at the same time, the same major semniaxis; but the motion from the first place to the second in these ellipses is manifestly performed in unequal times. 107. Denoting by X any arc whatever between - 180~ and -+ 180~, and by s the sine of the arc A x, it is known that, A S+ 8+ 1.35 1. 3.5 as7 + etc. Moreover, we have 1.15 1.1.3 s7 - etc. )2.4 2.4.6 and thus, 1.3 7 1'.3.59- sin x -4 ( s8 + - ~ S. + e 2.4 +9- e 24 6s-+ etc. * A circle being described from the first place, as a centre, with the radius 2 a - r, and another, from the second place, with the radius 2 a - r', it is manifest that the other focus of the ellipse lies in the intersection of these circles. Wherefore, since, generally speaking, two intersections are given, two different ellipses will be produced. 148 RELATIONS BETWEEN SEVERAL [BOOK I. We substitute in this series for s, successively Vr+rrfQ and X r+fQa and we multiply the results by a2; and thus obtain respectively, the series, 1 (r - ) + (r + - +, (r + r' ) + etc. + Q + etc. (r81 (- + r' + e) a a (r + r' 18432 s(rr'+Q)1+t etc. the sums of which we will denote by T, U: Now it is easily seen, since 2 sin i _ /r — a the upper or lower sign having effect according as 2f is less or more than 18.00, that a (6-sin 6) = T, the sign being similarly determined. In the same manner, if for e is taken the smaller value, inferior to 180~, we have a(e - sin 8)- U; but the other value, which is the complement of the former to 360, being taken, we evidently have a~ (- sin ) = a1 360~- U Hence, therefore, are obtained two values for the time t, U+T and ao 3600 U+iT Pk ~ k k 108. If the parabola is regarded as an ellipse, of which the major axis is infinitely great, the expression for the time, found in the preceding article, passes into SECT. 3.] PLACES IN ORBIT. 149 but since this derivation of the formula might perhaps seem open to some doubts, we will give another not depending upon the ellipse. Putting, for the sake of brevity, tan v-, tan v 6'&,we have r- p(l +8), r' — 2p(1+'8/'), 1 00 2 09 I vi 4 2 0' cos v _- Cos:' sin v si-m -' ~ -te1+0e7 0 -1 8+01' 1sU +0 01- 1$ +e0' Hence follow r' cos' — r cos s v p (8 6 --''), r' sin v- r sin v - (6'-O ), and thus Q Q } = pp (68 — )2 (4 + (68 + )2). Now it is readily seen that' -- =- sinf is a positive quantity: putting, therefore, (1 + I (8/ + 8)2) - - we have Q-p ('61-)' Moreover, - r r'- p (2 + 88 +') 22-p (n X + ( — 8)2 wherefore, we have r-Fr- t-= ( _ (+ 8))2 From the former equation is readily deduced, =v+ ( —o) as X and 6'- 6 are positive quantities; but since I (6'- 6) is smaller or greater than A, according as i (6 6)2= + 6- 61 cosf is positive or negative, we must evidently, conclude from the latter equation that in which the upper or lower sign is to be adop-ed according as the angle d(' scribed about the sun is less than 180, or more than 180.o 150 RELATIONS BETWEEN SEVERAL [BOOK I. From the equation, which in article 98 follows the second equation, we have, moreover, 2 ( t kt) A ( p2 whence readily follows, A-~- ((~ f/+ 2r)2F( r ) )+ the upper or lower sign taking effect, as 2f is less or more than 1800~ 109. If, in the hyperbola, we take the symbols a, c c, with the same meaning as in article 99, we hayve from equations VIII., IX., article 21,' os,- r cos — ~ (e- ) ( a-c C r's v' r sin v - (c C ) V \(ee- 1) and consequently, Q= _ ja CL -8/e 7 1 ) (C+4). Let us suppose that' is a quantity determined by the equation since this is evidently satisfied by two values, the reciprocals of each other, we may adopt the one which is greater than 1. In this manner Moreover, r~ r _ +(ec+ ) (C+ -4) a ((c + 4) (rV ) ), and thus, r~ir'~Q e_ a (\ v /VI)2 SECT. 3.] PLACES IN ORBIT. 151 Putting, therefore, r+ / +r+ r'4a -, 4a we necessarily have \/y-2CY in, but in order to decide the question whether - - C is equal to + 2n or 2 n, it is necessary to inquire whether y is greater or less than c: but it follows readily from equation 8, article 99, that the former case occurs when 2f is less than 180~, and the latter, when 2f is more than 180~o Lastly, we have, from the same article, =2m/(1+ ( 2-2loge= e jy7~- ( —( —)- logc7 +logg47 3 +'J.... _2m(I +n-m) +2nV(1 +- nn) -2 log ((1 -+ mm) + nm) + 2 log (/ (I + izn) + n) the lower signs belonging to the case of 2f> 1800. Now, log (V/(1 +mqm)+-m) is easily developed into the following series — 1.3m5 1.3.5 n -- -.4. I 7 4- etc. m-~ 3-'4 2. 4. 6 7 + e This is readily obtained from d log (V (I + min) + m) d m There follows, therefore, the formula 2m i/ (I + vim) - 2 log (8/ (1 + mm) + mn)-= 4 ( I me. 2 m5 lr e 2+ - etc.), and, likewise, another precisely similar, if in is changed to n. Hence, finally, if we put 1 1 T _- 1 (- + r _)l ( (- + etc. — 32 (rr -Q) + etc. -18 2 ( +a-'+4)% + eveo 152 RELATIONS BETWEEN SEVERAL. PLACES IN ORBIT. [Booi 1. we obtain which expressions entirely coincide with those given in article 107, if a is there changed into -- a. Finally, these series, as well for the ellipse as the hyperbola, are eminently suited to practical use, when a or a possesses a very great value, that is, where the conic section resembles very nearly the parabola. In snch a case, the methods previously discussed (articles 85-105) might be employed for the solution of the problem: but as, in our judgment, they do not furnish the brevity of the solution given above, we do not dwell upon the further explanation of this method. FOURTH SECTION. RELATIONS BETWEEN SEVERAL PLACES IN SPACE. 110. THE relations to be considered in this section are independent of the nature of the orbit, and will rest upon the single assumption, that all points of the orbit lie in the same plane with the sun. But we have thought proper to touch here upon some of the most simple only, and to reserve others more complicated and special for another book. The position of the plane of the orbit is fully determined by two places of the heavenly body in space, provided these places do not lie in the same straight line with the sun. Wherefore, since the place of a point in space can be assigned in two ways, especially, two problems present themselves for solution. We will, in the first place, suppose the two places to be given by means of heliocentric longitudes and latitudes, to be denoted respectively by A, X',': the distances from the sun will not enter into the calculation. Then if the longitude of the ascending node is denoted by 2, the inclination of the orbit to the ecliptic by i, we shall have, tan tan i sin (I- ), tan /'_ tan i sin (Y — ). The determination of the unknown quantities 2, tan i, in this place, is referred to the problem examined in article 78, II. We have, therefore, according to the first solution, tan i sin ( -- 2) -- tan p, tain COS ( ) _ tan'-tan 3 cos (3!- ) tanicos(X-) ( 2sinq (;3- x) 20 (153) 154 RELATIONS BETWEEN SEVERAL [BOOK I. likewise, according to the third solution, we find g by equation tan ( h X + A' __ g ) -- sin (3 -i a) tn (X' — ) not by the logarithms of their tangents: but, for determining i, recourse rmust be had to one of the formulas tani- tan B_ tan g t tan a s in ( -- sin ( - ) Finally, the uncertainty in the determination of the angle by its tangent will be decided so that tani may become positive or negative, according as the motion projected on the ecliptic is direct or retrograde: this uncertainty, therefore, can be removed only in the case where it may be apparent in what direction the heavenly body has moved in passing from the first to the second place; if this should be unknown, it would certainly be impos;ible to distinguish the ascending from the descending node. After the angles 2, i, are found, the arguments of the latitude u, u', will be obtained by the formulas, tan -u _ tan tuo = tan (A'- ) which are to be taken in the first or second semicircle, according as the corresponding latitudes are north or south. To these formulas we add the following, one or the other of which can, at pleasure, be used for proving the calculation:cos -- cos cos (s - ), cos' -- cos' cos (- ), SlCsin ti =__ I sin sin. sinn - se (a t- u) - sin (- + l' 2 2) cos B cos p sin (iA'-A ) cos COS p slr i)(c + =os s_ n (-'u) =sin (' s cos Cosi s Cos It SECT. 4.] PLACES IN SPACE.. 55 111. Let us suppose, in the second place, the two places to be given by means of their distances from three planes, cutting each other at right angles in the sun; let us denote these distances, for the first place, by x, y, -, for the second, by x', y/', Z, and let us suppose the third plane to be the ecliptic itself; also the positive poles of the first and second planes to be situated in N, and 90~ -- N. We shall thus have by article 53, the two radii vectores being denoted by r, r/ x = r cos U cos (N- 2) + r sin u sin (N- 2) cos i, y r sin u cos (N — 2 ) cos — r cos u sin (N —' ), z = r sin tu sin i = r'cos cos (N- ) -r' sin' sin (N — s ) cos i, - r?' sin u' cos (N- ) cos i — cos i'e sin (N- 8), p —' sr smn U sin i, Hence it follows that xi -yx' = rr' sin (t' -tC) cosin (N- S ) sin i, xY' -x' -Y rr' sin (Wt - u) cos i. From the combination of the first formula with the second will be obtained N- g and r r' sin (/' - u) sin i, hence and from the third formula, i and r-r' sin (' - t) will be obtained. Since the place to which the coo3rdinates x', y, z, correspond, is supposed pos terior in time, iU' must be greater than ui: if, moreover, it is known whether the angle between the first and second place described about the sun is less or greater than two right angles, rr'sin (' - u)sin i and rr'sin (zit'- i) must be positive quantities in the first case, negative in the second: then, accordingly, N- q is determined without doubt, and at the same time it is settled by the sign of the quantity xy' -yx' whether the motion is direct or retrograde. On the other hand, if the direction of the motion is known, it will be possible to decide from the sign of the quantity xy' -y x' whether it' - - is to be taken less or greater than 180~. But if the direction of the motion, and the nature of the angle 156 RELATIONS BETWEEN SEVERAL [BOOK I. described about the sun are altogether unknown, it is evident that we cannot distinguish between the ascending and descending node. It is readily perceived that, just as cos i is the cosine of the inclination of the plane of the orbit to the third plane, so sin (N — ) sin i, cos (N — ) sin i, are the cosines of the inclinations of the plane of the orbit to the first and second planes respectively; also that r r' sin (u'- u) expresses the double area of the triangle contained between the two radii vectores, and zy'-yz z''- zx', X y'-TY-Yx',T the double area of the projections of this triangle upon each of the planes. Lastly, it is evident, that any other plane can be the third plane, provided, only, that all the dimensions defined by their relations to the ecliptic, are referred to the third plane, whatever it may be. 112. Let x", "', z'" be the coordinates of any third place, and u" its argument of the latitude, r" its radius vector. We will denote the quantities r' r" sin (tu" -, t') r r" sin (u" -), r r' sin (u' - -), which are the double areas of the triangles between the second and third radii vectores, the first and third, the first and second, respectively, by n, i', n". Accordingly, we shall have for x", ye", ", expressions similar to those which we have given in the preceding article for x, zy Z, and f, y', z'; whence, withthe assistance of lemma I., article 78, are easily derived the following equations - 0 x -- n'w'. - "x' 0 = - n'y/ + f"ye, o nz -- n' +- n"I Let now the geocentric longitudes of the celestial body corresponding to these three places be a, a', a"; the geocentric latitudes, (n, i',7 a"; the distances from the earth projected on the ecliptic, y, Y', Y"; the corresponding heliocentric longitudes of the earth, L L', L"; the latitudes, B, B', B", which we do not put equal to 0, in order to take account of the parallax, and, if thought proper, to choose any other plane, instead of the ecliptic; lastly, let D, f,' D", be the distances of the earth from the sun projected upon the ecliptic. If, then,,,, are expressed SECT. 4.] PLACES IN SPACE. 157 by means of L, B, D, a, (Y, and the coordinates relating to the second and third places in a similar manner, the preceding equations will assume the following form: - [1] 0 n (Y cosa + D cosL ) -' (I' cos a' + D' cos') + -]" (Y" coS a" + D" COS L"), [2] 0 - n (6 sin a + D sin L) - n' (d' sin a' + iD' sin L') + 1" (d" sin a" + Y' sin L"), [3] 0 -n ( tan + DtanB) -n z' (d' tan (3' +t-D'tanB') + 1Z" (Y" tan P" + D" tan B"). If ac, (3, D,, B, and the analogous quantities for the two remaining places, are here regarded as known, and the equations are divided by nf, or by n", five unknown quantities remain, of which therefore, it is possible to eliminate two, or to determine, in terms of any two, the remaining three. In this manner these three equations pave the way to several most important conclusions, of which we will proceed to develop those that are especially important. 113. That we may not be too much oppressed with the length of the formulas, we will use the following abbreviations. In the first place we denote the quantity tan (3 sin (a" a') + tan (3' sin (a - a") + tan (" sin (a'- a) by (0. 1. 2): if in this expression, the longitude and latitude corresponding to any one of the three heliocentric places of the earth are substituted for the longitude and latitude corresponding to any geocentric place, we change the number answering to the latter in the symbol (0. 1. 2.) for the Roman numeral which corresponds to the former. Thus, for example, the symbol (0. 1. I.) expresses the quantity tan (3 sin (1' - a') + tan i' sin (a -- ) + tan B' sin (a' -- a), also the symbol (0. 0. 2), the following, tan (3 sin (a" - 1) + tan B sin (a -a") +- tan " sin (1 --- a). We change the symbol in the same way, if in the first expression any two helio 158 RELATIONS BETWEEN SEVERAL [BooK IO centric longitudes and latitudes of the earth whatever, are substituted for two geocentric. If two longitudes and latitudes entering into the same expression are only interchanged with each other, the corresponding numbers should also be interchanged; but the value is not changed from this cause, but it only becomes negative from being positive, or positive from negative. Thus, for example, we have (0. 1. 2) -- (0. 2. 1) = (1. 2. 0) -— (1. 0. 2) - (2. 0. ) -(2. 1. 0). All the quantities, therefore, originating in this way are reduced to the nineteen following - (0.1. 2) (0. 1. 0), (0. 1. I.), (0. 1.11), (0. 0. 2), (0. 1. 2), (0. 11.2), (0. 1. 2), (I. 1. 2), (II. 1. 2), (2.1. II.), to which is to be added the twentieth (0. I. II.). Moreover, it is easily shown, that each of these expressions multiplied by the product of the three cosines of the latitudes entering into them, becomes equal to the sextuple volume of a pyramid, the vertex of which is in the sun, and the base of which is the triangle formed between the three points of the celestial sphere which correspond to the places entering into that expression, the radius of the sphere being put equal to unity. When, therefore, these three places lie in the same great circle, the value of the expression should become equal to 0; and as tlhis always occurs in three heliocentric places of the earth, when we do not take account of the parallaxes and the latitudes arising from the perturbations of the earth, that is, when we suppose the earth to be exactly in the plane of the ecliptic, so we shall always have, on this assumption, (0. I. II.) 0, which is, in fact, an identical equation if the ecliptic is taken for the third plane. And further, when B, B', B", each, - 0, all those expressions, except the first, become much more simple; every one from the second to the tenth will be made up of two parts, but from the eleventh to the twentieth they will consist of only one ternm. SECT. 4.] PLACES IN SPACE. 159 114. By multiplying equation [1] by sin a" tan B" - sin " tan (3", equation [2] by cos 1" tan (3" - cos a" tan B", equation [3] by sin (9" - a"), and adding the.products, we get, [4] 0= ((0. 2.I.)' + (0. 2.II.)D) —n' ((12.I.)' -(I. 2. II.)D'); and in the same manner, or more conveniently by an interchange of the places, simply [5] 0 - n ((0. 1. 1.) + (0. 1. I.) D) D) + n" ((2. 1. I.) d" + (II. 1. I.) D") [6] 0 _ n' ((1. 0. 0.)6' + (I. 0. 0.)D') -- " ((2. 0. 0.)"' + (II. 0. 0.) D"). If, therefore, the ratio of the quantities z, n', is given, with the aid of equation 4, we can determine 6' from 6, or 6 from 6"; and so likewise of the equations 5, 6. From the combination of the equations 4, 5, 6, arises the following, 7 (. 2. I.) + (0. 2. 11.) D (1. 0. O.)' + (I. 0. 0.)D' (2. 1. 1.) + (II. 1. I.) ZY' [7](0. 1. I.) (0. 1. I1.) D (1 2. 1.)'+ (I. 2. I.) D' (2. 0. 0.) " + (II. 0 O.)D,- 1 by means of which, from two distances of a heavenly body from the earth, the third can be determined. But it can be shown that this equation, 7, becomes identical, and therefore unfit for the determination of one distance from the other two, when B=B'=B"- O0 and tan (3'tan O' sin (L1- a) sin (L" -L') + tan (3 tan ( sin (19- a') sin (L - 9) + tan ( tan (' sin (L" - a") sin (Z' — 1) - 0. The following formula, obtained easily from equations 1, 2, 3, is free from this mnconvenience - [8] (0. 1. 2.) 66'8" -+ (O. 1. 2) D'"- + (O. I. 2) D'6d' +- (0O L. II,) D"6d' 4- (0.1. II.h) D'D"6' + (0. 1 II.) DD"'- + (0. 1. 2) DD'6'- + (0. I. I1) DD'D" = 0. By multiplying equation 1 by sin' tan p"- sin a" tan (', equation 2 by cos a" tan ('- cos a' tan (3", equation 3 by sin (a" - a'), and adding the products, we get 9] o =n ((. 1. 2) 6 -+ (0o.. 2).) D- a' (L. 1. 2) D' -+ i" (ll. 1. 2)f' 160 RELATIONS BETWEEN SEVERAL PLACES IN SPACE. [BooK I. and in the same manner, [10] 0 = n (0o.. 2.)D —n' ((O.. 2)+(0. I. 2) ) +n"(0.l. 2)YD [1L] 0 = n (O. L. O) D — n' (O. 1. J.) D'+ n" ((o..2) o"+( (o.1 I.) y'). By means of these equations the distances 8, 6', Y", can be derived from the ratio between the quantities n, n', n", when it is known. But this conclusion only holds in general, and suffers an exception when (0. 1. 2) 0. For it can be shown, that in this case nothing follows from the equations 8, 9, 10, except a necessary relation between the quantities nz, n, n, and indeed the same relation from each of the three. Analogous restrictions concerning the equations 4, 5, 6, will readily suggest themselves to the reader. Finally, all the results here developed, are of no utility when the plane of the orbit coincides with the ecliptic. For if [(, (f', f"' B, B B" are all equal to 0, equation 3 is identical, and also, therefore, all those which follow. S E C OND B OOK. INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC OBSERVATIONS. FIRST SECTION. DETERMINATION OF AN ORBIT FROM THREE COMPLETE OBSERVATIONS. 115. SEVEN elements are required for the complete determination of the motion of a heavenly body in its orbit, the number of which, however, may be diminished by one, if the mass of the heavenly body is either known or neglected; neglecting the mass can scarcely be avoided in the determination of an orbit wholly unknown, where all the quantities of the order of the perturbations must be omitted, until the masses on which they depend become otherwise known. Wherefore, in the present inquiry, the mass of the body being neglected, we reduce the number of the elements to six, and, therefore, it is evident, that as many quantities depending on the elements, but independent of each other, are required for the determination of the unknown orbit. These quantities are necessarily the places of the heavenly body observed from the earth; since each one of which furnishes two data, that is, the longitude and latitude, or the right ascension and declination, it will certainly be the most simple to adopt ithree geocerzric places which will, in general, be sufficient for determining the six unknown elements. This problem is to be regarded as the most important in this work, and, for this reason, will be treated with the greatest care in this section. 21 (161) 162 DETERMINATION OF AN ORBIT FROM [BOOK II. But in the special case, in which the plane of the orbit coincides with the ecliptic, and thus both the heliocentric and geocentric latitudes, from their nature, vanish, the three vanishing geocentric latitudes cannot any longer be considered as three data independent of each other: then, therefore, this problem would remain indeterminate, and the three geocentric places might be satisfied by an infinite number of orbits. Accordingly, in such a case, four geocentric longitudes must, necessarily, be given, in order that the four remaining unknown elements (the inclination of the orbit and the longitude of the node being omitted) may be determined. But although, from an indiscernible principle, it is not to be expected that such a case would ever actually present itself in nature, nevertheless, it is easily imagined that the problem, which, in an orbit exactly coinciding with the plane of the ecliptic, is absolutely indeterminate, must, on account of the limited accuracy of the observations, remain nearly indeterminate in orbits very little inclined to the ecliptic, where the very slightest errors of the observations are sufficient altogether to confound the determination of the unknown quantities. Wherefore, in order to examine this case, it will be necessary to select six data: for which purpose we will show in section second, how to determine an unknown orbit from four observations, of which two are complete, but the other two incomplete, the latitudes or declinations being deficient. Finally, as all our observations, on account of the imperfection of the instruments and of the senses, are only approximations to the truth, an orbit based only on the six absolutely necessary data may be still liable to considerable errors. In order to diminish these as m-ach as possible, and thus to reach the greatest precision attainable, no other method will be given except to accumnulate the greatest number of the most perfect observations, and to adjust the elements, not so as to satisfy this or that set of observations with absolute exactness, but so as to agree with all in the best possible manner. For which purpose, we will show in the third section how, according to the principles of the calculus of probabilities, such an agreement may be obtained, as will be, if in no one place perfect, yet in all the places the strictest possible. The determination of orbits in this manner, therefore, so far as the heavenly bodies move in them according to the laws of KEPLET, will be carried to the SECT. 1.] THREE COMPLETE OBSERVATIONS. 163 highest degree of perfection that is desired. Then it will be proper to undertake the final correction, in which the perturbations that the other planets cause in the motion, will be taken account of: we will indicate briefly in the fourth section, hlow these may be taken account of; so far at least, as it shall appear consistent with our plan. 116. Before the determination of any orbit from geocentric observations, if the greatest accuracy is desired, certain reductions must be applied to the latter on account of nutation, precession, parallax, and aberration: these small quantities may be neglected in the rougher calculation. Observations of planets and comets are commonly given in apparent (that is, referred to the apparent position of the equator) right ascensions and declinations. Now as this position is variable on account of nutation and precession, and, therefore, different for different observations it will be expedient, first of all, to introduce some fixed plane instead of the variable plane, for which purpose, either the equator in its mean position for some epoch, or the ecliptic might be selected: it is customary for the most part to use the latter plane, but the former is recommended -by some peculiar advantages which are not to be despised. When, therefore, the plane of the equator is selected, the observations are in the first place to be freed from nutation, and after that, the precession being applied, they are to be reduced to some arbitrary epoch: this operation agrees entirely with that by which, from the observed place of a fixed star, its mean place is derived for a given epoch, and consequently does not need explanation here. But if it is decided to adopt the plane of the ecliptic, there are two courses which may be pursued: namely, either the longitudes and latitudes, by means of the mean obliquity, can be deduced from the right ascensions and declinations corrected for nutation and precession, whence the longitudes referred to the mean equinox will be obtained; or, the latitudes and longitudes will be computed more conveniently from the apparent right ascensions and declinations, using the apparent obliquity, and will afterwards be freed from nutation and precession. The places of the earth, corresponding to each of the observations, are coin 164 DETERMINATION OF AN ORBIT FROM [BOOK IL puted from the solar tables, but they are evidently to be referred to the samle plane, to which the observations of the heavenly body are referred. For which reason the nutation will be neglected in the computation of the longitude of the sun; but afterwards this longitude, the precession being applied, will be reduced to the fixed epoch, and increased by 180 degrees; the opposite sign will be given to the latitude of the sun, if, indeed, it seems worth while to take account of it: thus will be obtained the heliocentric place of the earth, which, if the equator is chosen for the fundamental plane, may be changed into right ascension and declination by making use of the mean obliquity. 117. The position of the earth, computed in this manner from the tables, is the place of the centre of the earth, but the observed place of the heavenly body is referred to a point on the surface of the earth: there are three methods of remedying this discrepancy. Either the observation can be reduced to the centre of the earth, that is,freed from parallax; or the heliocentric place of the earth may be reduced to the place of observation, which is done by applying the parallax properly to the place of the sun computed from the tables; or, finally, both positions can be transferred to some third point, which is most conveniently taken in the intersection of the visual ray with the plane of the ecliptic; the observation itself then remains unchanged, and we have explained, in article 72, the reduction of the place of the earth to this point. The first method cannot be applied, except the distance of the heavenly body from the earth be approximately, at least, known: but then it is very convenient, especially when the observation has been made in the meridian, in which case the declination only is affected by parallax. Moreover, it will be better to apply this reduction irmaediately to the observed place, before the transformations of the preceding article are undertaken. But if the distance from the earth is still wholly unknown, recourse must be had to the second or third method, and the former will be employed when the equator is taken for the fundamental plane, but the third will have the preference when all the positions are referred to the ecliptic. SECT. 1.] THREE COMPLETE OBSERVATIONS. 165 118. If the distance of a heavenly body from the earth answering to any observation is already approximately known, it may be freed from the effect of aberration in several ways, depending on the different methods given in article 71. Let t be the true time of observation; 6 the interval of time in which light passes from the heavenly body to the earth, which results from multiplying 493s into the distance; I the observed place, i' the same place reduced to the time I + 6 by means of the diurnal geocentric motion; l" the place 1 freed from that part of the aberration which is common to the planets and fixed stars; L the true place of the earth corresponding to the time I (that is, the tabular place increased by 20".25); lastly,'L the true place of the earth corresponding to the time I- 6. These things being premised, we shall have I. l the true place of the heavenly body seen from'L at the time t -6. II. 1' the true place of the heavenly body seen from L at the time t. III. I" the true place of the heavenly body seen from L at the time t — 6. By method I., therefore, the observed place is preserved unchanged, but the ficttiious time -6 is substituted for the true, the place of the earth being coInputed for the former; method II., applies the change to the observation alone, but it requires, together with the distance, the diurnal motion; in method III., the observation undergoes a correction, not depending on the distance; the fictitious time - 8 is substituted for the true, but the place of the earth corresponding to the true time is retained. Of these methods, the first is much the most convenient, whenever the distance is known well enough to enable us to compute the reduction of the time with sufficient accuracy. But if the distance is wholly unknown, neither of these methods can be immediately applied: in the first, to be sure, the geocentric place of the heavenly body is known, but the time and the position of the earth are wanting, both depending on the unknown distance; in the second, on the other hand the latter are given, and the former is wanting; finally, in the third, the geocentric place of the heavenly body and the position of the earth are given, but the time to be used with these is wanting. 166 DETERMINATION OF AN ORBIT FROM [BOOK II. Whiat, therefore, is to be clone with our problem, if; in such a case, a solution exact with respect to aberration is required? The simplest course undoubtedly is, to determine the orbit neglecting at first the aberration, the effect of which can never be important; the distances will thence be obtained with at least such precision that the observations can be freed from aberration by some one of the methods just explained, and the determination of the orbit can be repeated with greater accuracy. Now, in this case the third method will be far preferable to the others: for, in the first method all the computations depending on the position of the earth must be commenced again from the very beginning; in the second (which in fact is never applicable, unless the number of observations is sufficient to obtain from them the diurnal motion), it is necessary to begin anew all the computations depending upon the geocentric place of the heavenly body; in the third, on the contrary, (if the first calculation had been already based on geocentric places freed from the aberration of the fixed stars) all the preliminary computations depending upon the position of the earth and the geocentric place of the heavenly body, can be retained unchanged in the new computation. But in this way it will even be possible to include the aberration directly in the first calculation, if the method used for the determination of the orbit has been so arranged, that the values of the distances are obtained before it shall have been necessary to introduce into the computation the corrected times. Then the double computation on account of the aberration will not be necessary, as will appear more clearly in the further treatment of our problem. 119. It would not be difficult, from the connection between the data and unknown quantities of our problem, to reduce its statement to six equations, or even to less, since one or another of the unknown quantities might, conveniently enough, be eliminated: but since this connection is most complicated, these equations would become very intractable; such a separation of the unknown quantities as finally to produce an equation containing only one, can, generally speaking, be regarded SECT..] THREE COMPLETE OBSERVATIONS. 167 as impossible,* and, therefore, still less will it be possible to obtain a complete solution of the problem by direct processes alone. But our problem may at least be reduced, and that too in various ways, to the solution of two equations X -, Y- O0 in which only two unknown quantities x, y, remain. It is by no means necessary that x, y, should be two of the elements: they may be quantities connected with the elements in any manner whatever, if only, the elements can be conveniently deduced from them when found. Moreover, it is evidently not requisite that X, Y, be expressed in explicit fnunctions of X, y: it is sufficient if they are connected with them by a system of equations in such manner that we can proceed from given values of x, y, to the corresponding values of X, I. 120. Since, therefore, the nature of the problem does not allow of a further reduction than to two equations, embracing indiscriminately two unknown quantities, the principal point will consist, first, in the suitable selection of these unknown quantities and arsrawncgnet of the equations, so that both X and Y may depend in the simplest manner upon x, y, and that the elements themselves may follow most conveniently from the values of the former when known: and tlhen, it will be a subject for careful consideration, how values of the unknown quantities satisfying the equations may be obtained by processes not too laborious. If this should be practicable only by blind trials, as it were, very great and indeed almost intolerable labor would be required, such as astronomers who have determined the orbits of comets by what is called the indirect method have, nevertheless, often undertaken: at any rate, the labor in such a case is very greatly lessened, if, in the first trials, rougher calculations suffice until approximate values of the unknown quantities are found. But as soon as an approximate determination is made, the solution of the problem can be completed by safe and easy methods, which, before we proceed further, it will be well to explain in this place. X When the observations are so near to each other, that the intervals of the times may be treated as infinitely small quantities, a separation of this kind is obtained, and the whole problem is reduced to the solution of an algebraic equation of the seventh or eighth degree. 168 DETERMINATION OF AN ORBIT FROM [BOOK II. The equations X - 0 Y- 0 will be exactly satisfied if for x and y their true values are taken; if, on the contrary, values different from the true ones are substituted for x and y, then X and Y will acquire values differing from 0. The more nearly x and y approach their true values, the smaller should be the resulting values of X and Y, and when their differences from the true values are very small, it will be admissible to assume that the variations in the values of X and Y are nearly proportional to the variation of a if y is not changed, or to the variation of y, if x, is not changecl. Accordingly, if the true values of x and y are denoted by, 7, the values of X and Y corresponding to the assumption that w - + A, y + p will be expressed in the form in which the coefficients a, p, y, can be regarded as constant, as long as X and p remain very small. Hence we conclude that, if for three systems of values of x, y, differing but little from the true values corresponding values of X, Y have been determined, it will be possible to obtain from them correct values of x, y so far, at least, as the above assumption is admissible. Let us suppose that, for x a, 1 -b we have X- A Y- B, x I a' y b X~- XA' Y B', X a", y - b" X- A" Y-= B", and we shall have A —a (a —)+: (I —t), ~ y(d —)+~ (b -), A = a (a" - ) +l (b' —al) B'- (d — ) +- 0 (bi — V) A" —a (c, —) +/~ (a" —~), B"'y (" —g) - (B" —, ). From these we obtain, by eliminating a, p, y, y, a_ (A'BO"- A"B') + a' (A"B- A B") + a" (A B' -A'B) A'B" -- A"B' +- A"B -A" - A B' -- A'-B 9 b (A'B" - A"B') + b' (A"B — A B") + b" (A B' - A'B) A'B" — A"B'- + A"B -A B" + A B' — A'B or, in a form more convenient for computation, (a' - a) (A"B-A B") + (a- a) (A B'-A'B) -- -'B A"B-l + B —A B" +'- A B' -- A'B 2 H, -- _6) (A"B -- A B) +- (g — a) (A' — A'B) t7~J _'B --- 86 -- ~81 e- B --- 73 - ni SECT. 1.] THREE COMPLETE OBSERVATIONS. 169 It is evidently admissible, also, to interchange in these formulas the quantities 1, 1, A, AB, with c', 1', A', B' or with a", 13" A", B". The common denominator of all these expressions, which may be put under the form (A' - A) (B" - B) - (A' - A) (B' - B), becomes (cc-gy)((a'- a) (b" - b) - (a"'- a) (b'- )): whence it appears that a, a', a", 3, 1' 1" must be so taken as not to make aft - a' ca Yl b - b6'- 6' otherwise, this method would not be applicable, but would furnish, for the values of 5 and i;, fractions of which the numerators and denominators would vanish at the saime time. It is evident also that, if it should happen that a 4- (3 y 0, the same defect wholly destroys the use of the method, in whatever way a, a', a",, 1', 13", may be taken. In such a case it would be necessary to assume for the values of X the form ax + (p"t +-F- + + + o and a similar one for the values of J7 which being done, analysis would supply methods, analogous to the preceding, of obtaining from values of X, Y, computed for four systems of values of x, y, true values of the latter. But the computation in this way would be very troublesome, and, moreover, it can be shown that, in such a case, the determination of the orbit does not, from the nature of the question, admit of the requisite precision: as this disadvantage can only be avoided by the introduction of new and more suitable observations, we do not here dwell upon the subject. 121. When, therefore, the approximate values of the unknown quantities are obtained, the true values can be derived from them, in the manner just now explained, with all the accuracy that is needed. First, that is, the values of X, Y, corresponding to the approximate values (a, 1) will be computed: if they do not vanish for these, the calculation will be repeated with two other values (a',') differing but little from the former, and afterwards with a third system (a", 1") i22 170 DETERMINATION OF AN ORBIT FROM [BooK RI. unless X: Y have vanished for the second. Then, the true values will be dedlucedl by means of the formulas of the preceding article, so far as the assumption on which these formulas are based, does not differ sensibly from the truth. In order that we may be better able to judge of which, the calculation of the values of X, Yr will be repeated with those corrected values; if this calculation shows that the equations X 0, Y- 0, are, still, not satisfied, at least much smaller values of X7 Y, will result therefrom, than from the three former hypotheses, and therefore, the elements of the orbit resulting from them, will be much more exact than those which correspond to the first hypotheses. If we are not satisfied with these, it will be best, omitting that hypothesis which produced the greatest differences, to combine the other two with a fourth, and thus, by the process of the preceding article, to obtain a fifth system of the values of x, y; in the same manner, if it shall appear worth while, we may proceed to a sixth hypothesis, and so on, until the equations X - 0 -O 0, shall be satisfied as exactly as the logarithmic and trigonometrical tables permit. But it will very rarely be necessary to proceed beyond the fourth system, unless the first hypotheses were very far from the truth. 122. As the values of the unknown quantities to be assumed in'the second and third hypotheses are, to a certain extent, arbitrary, provided, only, they do not differ too much from the first hypothesis; and, moreover, as care is to be taken that the ratio (a" - a): (b" - b) does not tend to an equality with (a'- a): (' —b), it is customary to put a'- a, /'- b. A double advantage is derived from this; for, not only do the formulas for n, sn, become a little more simple, but, also, a part of the first calculation will remain the same in the second hypothesis, and another part in the third. Nevertheless, there is a case in which other reasons suggest a departure from this custom: for let us suppose X to have the form X'- x, and Y the form Y'-y, and the functions X', Y', to become such, by the nature of the problem, that they are very little affected by small errors in the values of x, y, or that d/) \_dy/' \dY/' \(dY kEcT. 1.1 THREE COMPLETE OBSERVATIONS. 171 may be very small quantities, and it is evident that the differences between the values of those functions corresponding to the system x 5, y - and those which result from x -a, y 6, can be referred to a somewhat higher order than the differences 5 - a, rj - 6; but the former values are X', Y' -,i, and the latter X' - a + A, I' - 6 + B, whence it follows, that a + A, 6 + B, are much more exact values of x, ~y, than a, 6. If the second hypothesis is based upol these, the equations X- 0, Y- 0, are very frequently so exactly satisfied, that it is not necessary to proceed any further; but if not so, the third hypothesis will be formed in the same manner from the second, by making 0 X - a + A +A'A 6"= b'+B' b-VB + B', whence finally, if it is still not found sufficiently accurate, the fourth will be obtained according to the precept of article 120. 123. We have supposed in what goes before, that the approximate values of the unknown quantities x, y, are already had in some way. Where, indeed, the approximate dimensions of the whole orbit are known (deduced perhaps from other observations by means of previous calculations, and now to be corrected by new ones), that condition can be satisfied without difficulty, whatever meaning we may assign to the unknown quantities. On the other hand, it is by no means a matter of indifference, in the determination of an orbit still wholly unknown, (which is by far the most difficult problem,) what unknown quantities we may use; but they should be judiciously selected in such a way, that the approximate values may be derived from the nature of the problem itself.' Which can be done most satisfactorily, when the three observations applied to the investigation of an orbit do not embrace too great a heliocentric motion of the heavenly body. Observations of this kind, therefore, are always to be used for the first determination, which may be corrected afterwards, at pleasure, by means of observations more remote from each other. For it is readily perceived that the nearer the observations employed are to each other, the more is the calculation affected by their unavoidable errors. Hence it is inferred, that the observations for the first de 172 DETERMINATION OF AN ORBIT FROM [BOOK II. termination are not to be picked out at random, but care is to be taken, frst, that they be not too near each other, but then, also, that they be not too distant from each other; for in the first case, the calculation of elements satisfying the observations would certainly be most expeditiously performed, but the elements themselves would be entitled to little confidence, and might be so erroneous that they could not even be used as an approximation: in the other case, we should abandon the artifices which are to be made use of for an approximate determination of the unknown quantities, nor could we thence obtain any other determination, except one of the rudest kind, or wholly insufficient, without many more hypotheses, or the most tedious trials. But how to form a correct judgment concerning these limits of the method is better learned by frequent practice than by rules: the examples to be given below will show, that elements possessing great accuracy can be derived from observations of Juno, separated from each other only 22 days, and embracing a heliocentric motion of 7~0 35'; and again, that our method can also be applied, with the most perfect success, to observations of Ceres, which are 260 days apart, and include a heliocentric motion of 62~ 55'; and can give, with the use of four hypotheses or, rather, successive approximations, elements agreeing excellently well with the observations. 124. We proceed now to the enumeration of the most suitable methods based upon the preceding principles, the chief parts of which have, indeed, already been explained in the first book, and require here only to be adapted to our purpose. The most simple method appears to be, to take for X, y, the distances of the heavenly body from the earth in the two observations, or rather the logarithms of these distances,or the logarithms of the distances projected upon the ecliptic or equator. Hence, by article 64, V., will be derived the heliocentric places and the distances from the sun pertaining to those places; hence, again, by article 110, the position of the plane of the orbit and the heliocentric longitudes in it; and from these, the radii vectores, and the corresponding times, according to the problem treated at length in articles 85-105, all the remaining elements, by which, it is evident, these observations will be exactly. represented, whatever values may SECT. 1.] THREE COMPLETE OBSERVATIONS. 173 have been assigned to x, y. If, accordingly, the geocentric place for the time of the third observation is computed by means of these elements, its agreement or disagreement with the observed place will determine whether the assumed values are the true ones, or whether they differ from them; whence, as a double comparison will be obtained, one difference (in longitude or right ascension) can be taken for X, and the other (in latitude or declination) for Y. Unless, therefore, the values of these differences come out at once _ 0, the true values of X, y, may be got by the method given in 120 and the following articles. For the rest, it is in itself arbitrary from which of the three observations we set out: still, it is better, in general to choose the first and last, the special case of which we shall speak directly, being excepted. T'his method is preferable to most of those to be explained hereafter, on this account, that it admits of the most general application. The case must be excepted, in which the two extreme observations embrace a heliocentric motion of 180, or 360, or 540, etc., degrees; for then the position of the plane of the orbit cannot be determined, (article 110). It will be equally inconvenient to apply the method, when the heliocentric motion between the two extreme observations differs very little from 180~ or 3600, etc., because an accurate determination of the position of the orbit cannot be obtained in this ease, or rather, because the slightest changes in the assumed values of the unknown quantities would caase such great variations in the position of the orbit, and, therefore, in the values of X, Y that the variations of the latter could no longer be regarded as proportional to those of the former. But the proper remedy is at hand; which is, that we should not, in such an event, start from the two extreme observations, but from the first and middle, or from the middle and last, and, therefore, should take for X, Y, the differences between calculation and observation in the third or first place. But, if both the second place should be distant from the first, and the third from the second nearly 180 degrees, the disadvantage could not be removed in this way; but it is better not to make use, in the computation of the elements, of observations of this sort from which, by the nature of the case, it is wholly impossible to obtain an accurate determination of the position of the orbit. Moreover, this method derives value from the fact, that by it the amount of 174 DETERMINATION OF AN ORBIT FROM [Boo II. the variations which the elements experience, if the middle place changes while the extreme places remain fixed, can be estimated without difficulty: in this way, therefore, some judgment may be formed as to the degree of precision to be attributed to the elements found. 125. We shall derive the second from the preceding method by applying a slight change. Starting from the distances in two observations, we shall determine all the elements in the same manner as before; we shall not, however, compute from these the geocentric place for the third observation, but will only proceed as far as the heliocentric place in the orbit; on the other hand we will obtain the same heliocentric place, by means of the problem treated in articles 74, 75, from the observed geocentric place and the position of the plane of the orbit; these two determinations, different from each other (unless, perchance, the true values of X, y, should be the assumed ones), will furnish us X and Y, the difference between the two values of the longitude in orbit being taken for X, and the difference between the two values of the radius vector, or rather its logarithm, for Y. This method is subject to the same cautions we have touched upon in the pree ceding article: another is to be added, namely, that the heliocentric place in orbit cannot be deduced from the geocentric place, when the place of the earth happens to be in either of the nodes of the orbit; when that is the case, accordingly, this method cannot be applied. But it will also be proper to avoid the use of this method in the case where the place of the earth is very near either of the nodes, since the assumption that, to small variations of x,y, correspond proportional variations of X, Y7 would be too much in error, for a reason similar to that which we have mentioned in the preceding article. But here, also,'may be a remedy sought in the interchange of the mean place with one of the extremes, to which may correspond a place of the earth more remote from the nodes, except, perchance, the earth, in all three of the observations, should be in the vicinity of the nodes. SECT. 1.] THREE COMIPLETE OBSERVATIONS. 175 126. The preceding method prepares the way directly for the tMird. In the same manner as before, by means of the distances of the heavenly body from the earth in the extreme observations, the corresponding longitudes in orbit together with the radii vectores may be determined. With the position of the plane of the orbit, which this calculation will have furnished, the longitude in orbit and the radius vector will be got from the middle observation. The remaining elements may be computed from these three heliocentric places, by the problem treated in articles 82, 83, which process will be independent of the times of the observations. In this way, three mean anomalies and the diurnal motion will be known, whence may be computed the intervals of the times between the first and second, and between the second and third observations. The differences between these and the true intervals will be taken for X and Y. This method is less advantageous when the heliocentric motion includes a small arc only. For in such a case this determination of the orbit (as we have already shown in article 82) depends on quantities of the third order, and does not, therefore, admit of sufficient exactness. The slightest changes in the values of x, y, might cause very great changes in the elements and, therefore, in the values of X, Y, also, nor would it be allowable to suppose the latter proportional to the former. But when the three places embrace a considerable heliocentric motion, the use of the method will undoubtedly succeed best, unless, indeed, it is thrown into confusion by the exceptions explained in the preceding articles, which are evidently in this method too, to be taken into consideration. 127. After the three heliocentric places have been obtained in the way we have described in the preceding article, we can go forward in the following manner. The remaining elements may be determined by the problem treated in articles 85-105, first, from the first and second places with the corresponding interval of time, and, afterwards, in the same manner, fromn the second and third places and 176 DETERMINATION OF AN ORBIT FROM [lBOOK II. the corresponding interval of time: thus two values will result for each of the elements, and from their differences any two may be taken at pleasure for X and Y. One advantage, not to be rejected, gives great value to this method; it is, that in the first hypotheses the remaining elements, besides the two which are chosen for fixing X and Y, can be entirely neglected, and will finally be determined in the last calculation based on the corrected values of x, y, either from the first combination alone, or from the second, or, which is generally preferable, from the combination of the first place with the third. The choice of those two elements which is, commonly speaking, arbitrary, furnishes a great variety of solutions; the logarithm of the semi-parameter, together with the logarithm of the semi-axis major, may be adopted, for example, or the former with the eccentricity or the latter with the same, or the longitude of the perihelion with any one of these elements: any one of these four elements might also be combined with the eccentric anomaly corresponding to the middle place in either calculation, if an elliptical orbit should result, when the formulas 27-30 of article 96, will supply the most expeditious computation. But in special cases this choice demands some consideration; thus, for example, in orbits resembling the parabola, the semi-axis ma-:or or its logarithm would be less suitable, inasmuch as excessive variations of these quantities could not be regarded as proportional to changes of z, y: in such a case it would be more advantageous to select!. But we give less a time to these precautions, because the fifth method, to be explained in the following article, is to be preferred, in almost all cases, to the four thus far explained. 128. Let us denote three radii vectores,obtained in the same manner as in articles 125, 126, by r, r/, r"; the angular heliocentric motion in orbit from the second to the third place by 2f, from the first to the third by 2f, from the first to the second by 2f", so that we have f'=f -[-f"; next, let r' r" sin 2 f _ n, r r sin 2f' n', r r' sin 2f"-' n"; SECT. 1.] THREE COMPLETE OBSERVATIONS. 177 lastly, let the product of the constant quantity / (article 2) into the intervals of the time from the second observation to the third, from the first to the third, and from the first to the second be respectively, 8, 6' 6". The double computation of the elements is begun, just as in the preceding article, both from rr'f" and 6", and from r' r",f, 6: but neither computation will be continued to the determination of the elements, but will stop as soon as that quantity has been obtained which expresses the ratio of the elliptical sector to the triangle, and which is denoted above (article 91) by y or - Y. Let the value of this quantity be, in the first calculation, i", in the second, i7. Accordingly, by means of formula 18, article 95, we shall have for the semi-parameter p the two values:/p',,, and p -o But we have, besides, by article 82, a third value, 4 rr'r"' sinfsinf' sin/fI P n n' + nI 9 which three values would evidently be identical if true values could have been taken in the beginning for x and y. For which reason we should have 0 i/n/,+ —,' I 4 0" r r'r" sin f sin f' sin f"_ n' 0 0" lf nen" - 2 g "rr'r" cosfcosf' cosf"' Unless, therefore, these equations are fully satisfied in the first calculation, we can put X log n nO" - - + 9!' 0 0" 2 2/rr r'r cos fcosf' cosf' This method admits of an application equally general with the second explained in article 125, but it is a great advantage, that in this fifth method the first hypotheses do not require the determination of the elements themselves, but stop, as it were, half way. It appears, also, that in this process we find that, as it can be foreseen that the new hypothesis will not differ sensibly from the truth, it will be sufficient to determine the elements either from r, r',f", 6", alone, or from r', r",f, 68 or, which is better, from r, r"f', 6'0 23 178 DETERMINATION OF AN ORBIT FROM [BOOK II. 129. The five methods thus far explained lead, at once, to as many others which differ from the former only in this, that the inclination of the orbit and the longitude of the ascending node, instead of the distances from the earth, are taken for x and y. The new methods are, then, as follows:I. Fromn x and y, and the two extreme geocentric places, according to articles 74, 75, the heliocentric longitudes in orbit and the radii vectores are determined, and, from these and the corresponding times, all the remaining elements; from these, finally, the geocentric place for the time of the middle observation, the differences of which from the observed place in longitude and latitude will furnish X and Y. The four remaining methods agree in this, that all three of the heliocentric longitudes in orbit and the corresponding radii vectores are computed from the position of the plane of the orbit and the geocentric places. But afterwards:II. The remaining elements are determined from the two extreme places only and the corresponding times; with these elements the longitude in orbit and radius vector are computed for the time of the middle observation, the differences of which quantities from the values before found, that is, deduced from the geocentric place, will produce X and Y: III. Or, the remaining dimensions of the orbit are derived from all three heliocentric places (articles 82, 83,) into which calculation the times do not enter: then the intervals of the times are deduced, which, in an orbit thus found, should have elapsed between the first and second observation, and between this last and the third, and their differences from the true intervals will furnish us with X and Y: IV. The remaining elements are computed in two ways, that is, both by the combination of the first place with the second, and by the combination of the second with the third, the corresponding intervals of the times being used. These two systems of elements being compared with each other, any two of the differences may be taken for X and Y: V. Or lastly, the same double calculation is onl only continued to the values of SECT. 1.] - THREE COMPLETE OBSERVATIONS. 179 the quantity denoted by y, in article 91, and then the expressions given in the preceding article for'X and Y, are adopted. In order that the last four methods may be safely used, the places of the earth for all three of the observations must not be very near the node of the orbit: on the other hand, the use of the first method only requires, that this condition may exist in the two extreme observations, or rather, (since the middle place may be substituted for either of the extremes,) that, of the three places of the earth, not more than one shall lie in the vicinity of the nodes. 130. The ten methods explained from article 124 forwards, rest upon the assumption that approximate values of the distances of the heavenly body from the earth, or of the position of the plane of the orbit, are already known. W hen the problem is, to correct, by means of observations more remote from each other, the dimensions of an orbit, the approximate values of which are already, by some means, known, as, for instance, by a previous calculation based on other observations, this assumption will evidently be liable to no difficulty. But it does not as yet appear from this, how the first calculation is to be entered upon when all the dimensions of the orbit are still wholly unknown: this case of our problem is by far the most important and the most difficult as may be imagined from the analogous problem in the theory of comets, which, as is well known, has perplexed geometers for a long time, and has given rise to many fruitless attempts. In order that our problem may be considered as correctly solved, that is, if the solution be given in accordance with what has been explained in the 119th and subsequent articles, it is evidently requisite to satisfy the following conditions:- First. the quantities x, y, are to be chosen in such a manner, that we can find approximate values of them from the very nature of the problem, at all events, as long as the heliocentric motion of the heavenly body between the observations is not too great. Secondly, it is necessary that, for small changes in the quantities x, y, there be not too great corresponding changes in the quantities to be derived from them, lest the errors accidentally introduced in the assumed values of the former, prevent the latter from being considered as approximate. 180 DETERMINATION OF AN ORBIT FROM [BooK II. Thirdly and lastly, we require that the processes by which we pass from the quantities x, y, to X, Y, successively, be not too complicated. These conditions will furnish the criterion by which to judge of the excellence of any method: this will show itself more plainly by frequent applications. The method which we are now prepared to explain, and which, in a measure, is to be regarded as the most important part of this work, satisfies these conditions so that it seems to leave nothing further to be desired. Before entering upon the explanation of this in the form most suited to practice, we will premise certain preliminary considerations, and we will illustrate and open, as it were, the way to it, which might, perhaps, otherwise, seem more obscure and less obvious. 131. It is shown in article 114, that if the ratio between the quantities denoted there, and in article 128 by n,, n', were known, the distances of the heavenly body from the earth could be determined by means of very simple formulas. Now, therefore, if n n' should be taken for x, y, 0 0" (the symbols 8, 6', 8", being taken in the same signification as in article 128) immediately present themselves as approximate values of these quantities in that case where the heliocentric motion between the observations is not very great: hence, accordingly, seems to flow an obvious solution of our problem, if two distances from the earth are obtained from x, y, and after that we proceed agreeably to some one of the five methods of articles 124-128. In fact, the symbols i7,;i" being also taken with the meaning of article 128, and, analogously, the quotient arising from the division of the sector contained between the two radii vectores by the area of the triangle between the same being denoted by', we shall have, n 0_ a/ n"F 0" or/' t_7en n ~ n' O' r/ 1 n' O' 9" SECT. 1.] THREE COMPLETE OBSERVATIONS. 181 and it readily appears, that if in, no if, are regarded as small quantities of the first order, - 1, n' - 1, i" - 1 are, generally speaking, quantities of the second order, and, therefore, 0 0" the approximate values of x,y, differ from the true ones only by quantities of the second order. Nevertheless, upon a nearer examination of the subject, this method is found to be wholly unsuitable; the reason of this we will explain in a few words. It is readily perceived that the quantity (0. 1. 2), by which the distances in the formulas 9, 10, 11, of article 114 have been multiplied, is at least of the third order, while, for example, in equation 9 the quantities (0. 1. 2), (I. 1. 2), (II. 1. 2), are, on the contrary, of the first order; hence, it readily follows, that an error of the second order in the values of the quantinfl ties n, -n produces an error of the order zero in the values of the distances. Wherefore, according to the common mode of speaking, the distances would be affected by a finite error even when the intervals of the times were infinitely small and consequently it would not be admissible to consider either these distances or the remaining quantities to be derived from them even as approximate; and the method would be opposed to the second condition of the preceding article, 132. Putting, for the sake of brevity, (0. 1. 2) = a, (O. I. 2)Y = -b, (0. 0. 2) D =+ 1, (0. l. 2) DI= + d, so that the equation 10, article 114, may become ad'=b+c, n+J, the coefficients c and d will, indeed, be of the first order, but it can be easily shown that the difference c - d is to be referred to the second order. Then it follows, that the value of the quantity cn+dn" n -r- n" 182 DETERMINATION OF AN ORBIT FROM [BOOK IL1 resulting from the approximate assumption that n: n"- 6:6" is affected by an error of. the fourth order only, and even of'the fifth only when the middle is distant from the extreme observations by equal intervals. For this error is c 0+d0" cn+dn" 00"(d-c) (e -,7) 0 +0Of" n+n" - (O +ol) (/'O+,I0") where the denomi nator is of the second order, and one factor of the numerator 6 6" (d- c) of the fonrth, the other i" - rj of the second, or, in that special case, of the third order. The former equation, therefore, being exhibited in this form, ad" =bfcn+dn" nz +n" n+W+, n', it is evident that the defect of the method explained in the preceding article does not arise from the fact that the quantities n, n" have been assumed proportional to 6,P6", but that, in addition to tih, n' was put proportional to 6'. For, indeed, in this way, insteadl ofthefactor + / the less exact value 0+ 1 is introduced, from which the true value 2 yr/rr'r" cosfcosf' cosf" differs by a quantity of the second order, (article 128). 133. Since the cosines of the anglesff'f", as also the quantities i), ni" differ from unity by a difference of the second order, it is evident, that if instead of the approximate value 0 0"1 is introduced, an error of the fourth order is committed. If, accordingly, in place of the equation, article 114, the following is introduced, Ic b -r-~dlfI + 00", mr 0ote+o2ielfn t anB error of the secondr ordter wipll show itself in the vpalue of -the dilstanrce d"~ w67hen% SECT. 1.] THREE COMPLETE OBSERVATIONS. 183 the extreme observations are equidistant from the middle; or, of the first order in other cases. But this new form of that equation is not suited to the determination of Y', because it involves the quantities r, r', r"', still unknown. Now, generally speaking, the quantities r, r,, differ from unity by a quantity r,,, r'ferfro uitybya qt of the first order, and in the same manner also the product "r: it is readily perceived that in the special case frequently mentioned, this product differs from unity by a quantity of the second order only. And even when the orbit of the ellipse is slightly eccentric, so that the eccentricity may be regarded as a quantity of the first order, the difference of rr?/ can be referred to an order one degree higher. It is manifest, therefore, that this error remains of the same order as before if, in our equation, 20, is substituted for s whence is obtained the equation, 2rrwhen2re' following form, ad' = b +e 0 +, dO" + 0" =6 + c'0 (1+ 2 r'8) In fact, this equation still contains the unknown quantity r', which, it is evident nevertheless, can be eliminated, since it depends only on d' and known quantities. If now the equation should be afterwards properly arranged, it would ascend to the eighth degree. 134. From the preceding it will be understood why, in our method, we are about to take for x, y, respectively, the quantities n-= P, and 2 (t-n" 1)r Q. For, in ithe frst place, it is evident that if P and Q are regarded as known quantities, 6' can be determined from them by means of the equation a = b + c+dP (1+ Q) and afterwards a, A"' by equations 4, 6, article 114, since we have n ( P n — l+p r n2-= 1- + ( I r' ~ In the second place, it is manifest that, 8 6" are, in the first hypothesis, the 184 DETERMINATION OF AN ORBIT FROM [BOOK II. obvious approximate values of the quantities P, Q, of which the true values are precisely Of/ rl rrS Of/ 0 i" r'/i' cosf cosf cosf"' from which hypothesis will result errors of the first order in the determination of d', and therefore of Y, Y' or of the second order in the special case several times mentioned. Although we may rely with safety upon these conclusions, generally speaking, yet in a particular case they can lose their force, as when the quantity (0. 1. 2), which in general is of the third order, happens to be equal to zero, or so small that it must be referred to a higher order. This occurs when the geocentric path in the celestial sphere has a point of contrary flexure near the middle place. Lastly, it appears to be required, for the use of our method, that the heliocentric motion between the three observations be not too great: but this restriction, by the nature of the very complicated problem, cannot be avoided in any way; neither is it to be regarded as a disadvantage, since it will always be desired to begin at the earliest possible moment the first determination of the unknown orbit of a new heavenly body. Besides, the restriction itself can be taken in a sufficiently broad sense, as the example to be given below will show. 135. The preceding discussions have been introduced, in order that the principles on which our method rests, and its true force, as it were, may be more clearly seen: the practical treatment, however, will present the method in an entirely different form which, after very numerous applications, we can recommend as the most convenient of many tried by us. Since in determining an unknown orbit from three observations the whole subject may always be reduced to certain hypotheses, or rather successive approximations it will be regarded as a great advantage to have succeeded in so arranging the calculation, as, at the beginning, to separate from these hypotheses as many as possible of the computations which depend, not on P and Q, but only on a combination of the known quantities. Then, evidently, these preliminary processes, common to each hypothesis, can be gone through once for all, and the hypotheses themselves are reduced SECT. 1.] THREE COMPLETE OBSERVATIONS. 185 to the fewest possible details. It will be of equally great importance, if it should not be necessary to proceed in every hypothesis as far as the elements, but if their computation might be reserved for the last hypothesis. In both these respects, our method, which we are now about to explain, seems to leave nothing to be desired. 136. We are, in the first place, to connect by great circles three heliocentric places of the earth in the celestial sphere, A, A' A" (figure 4), with three geocentric places of the heavenly body, B, B', B", and then to compute the positions of these great circles with respect to the ecliptic (if we adopt the ecliptic as the fundamental plane), and the places of the points B, B', B", in these circles. Let a, a', a" be three geocentric longitudes of the heavenly body, t,(' P(" latitudes; 1,1', 1", heliocentric longitudes of the earth, the latitudes of which we put equal to zero, (articles 117, 72). Let, moreover, y, 7'/ y be the inclinations to the ecliptic of the great circles drawn from A, A, A", to B, B' B", respectively; and, in order to follow a fixed rule in the determination of these inclinations, we shall always measure them from that part of the ecliptic which lies in the direction of the order of the signs from the points A, A', A", so that their magnitudes will be counted from 0 to 360~, or, which amounts to the same thing, from 0 to 1800 north, and from 0 to - 180~ south. We denote the arcs AB, A'B', A"B," which may always be taken between 0 and 180~, by A, I', Y". Thus we have for the determination of y and a the formulas, tan / [1] tan sin ( tan ta - 1) [2] tan d t (Cl) To which, if desirable for confirming the calculation, can be added the following, siny= ain cos -=cos/ cos (e — ). We have, evidently, entirely analogous formulas for determining y7 4I' 7', 4". Now, if at the same time ( = 0, ca-I= O or 1800, that is, if the heavenly body should 24 186 DETERMINATION OF AN ORBIT FROM [BOOK I. be in opposition or conjunction and in the ecliptic at the same time, y would be indeterminate. But we assume that this is not the case in either of the three observations. If the equator is adopted as the fundamental plane, instead of the ecliptic, then, for determining the positions of the three great circles with respect to the equator, will be required the right ascensions of their intersections with the equator, besides the inclinations; and it will be necessary to compute, in addition to tine distances of the points B, B' B", from these intersections, the distances of the points A, A', A" also from the same intersections. Since these depend on the problem discussed in article 110, we do not stop here to obtain the formulas. 137. The second step will be the determination of the positions of these three great circles relatively to each other, which depend on their inclinations and the places of' their mutual intersections. If we wish to bring these to depend upon clear and general conceptions, without ambiguity, so as not to be obliged to use special figures for different individual cases, it will be necessary to premise some preliminary explanations. Firslly, in every great circle two opposite directions are to be distinguished in some way, which will be done if we regard one of them as direct or positive, and the other as retrograde or negative. This being wholly arbitrary in itself, we shall always, for the sake of establishing a uniform rule, consider the directions from A, A', A" towards B B' B" as positive; thus, for example, if the intersection of the first circle with the second is represented by a positive distance from the point A, it will be understood that it is to be taken from A towards B (as D" in our figure); but if it should be negative, then the distance is to be taken on the other side of A. And secondly, the two hemispheres, into which every great circle divides the whole sphere, are to be distinguished by suitable denominations; accordingly, we shall call that the stperior hemisphere, which, to one walking on the inner surface of the sphere, in the positive direction along the great circle, is on the right hand; the other, the izferior. The superior hemisphere will be analogous to the northern hemisphere in regard to the ecliptic or equator, the inferior to the southern. SECT. 1.] THREE COMPLETE OBSERVATIONS. 187 These definitions being correctly understood, it will be possible conveniently to distinguish both intersections of the two great circles from each other. In fact, in one the first circle, tends from the inferior to the superior hemisphere of the second, or, which is the same thing, the second from the superior to the inferior hemisphere of the first; in the other intersection the opposite takes place. It is, indeed, wholly arbitrary in itself which intersections we shall select for our problem; but, that we may proceed here also according to an invariable rule, we shall always adopt these (D,i DI D" figure 4) where the third circle A"B" passes into the superior hemisphere of the second A'B', the third into that of the first, AB, and the second into that of the first, respectively. The places of these intersections will be determined by their distances from the points A' and A", A and A", A and AX which we shall simply denote by A'D, A"D, AD', AD', AD", A'D". Which being premised, the mutual inclinations of the circles will be the angles which are contained, at the points of intersection D, D', D", between those parts of the circles cutting each other that lie in the positive direction; we shall denote these inclinations, taken always between 0 and 180~, by e, -', s". The determination of these nine unknown quantities from those that are known, evidently rests upon the problem discussed by us in article 55.'We have, consequently, the following equations:[3] sin t E sin 1 (A'D + A"D) sin 1 ( - 1')sin 1 (r" +- y'), [4] sin I e cos 1 (A'D + A"D) cos I (l"-I') sin (y" -'), [5] cos 2 E sn (AD - AD) - sin A' AD) sin ( ) cos 1 (" +') [6] cos 1 e cos (A'D -A"D)- cos I (l"-l') cos"- y'(). 1 (A'D -A"D) and sin 2I E are made known by equations 3 and 4, I (A'D A"D) and cos -1 by the remaining two; hence A'D, A"D and e. The ambiguity in the determination of the arcs I (A'D + A"D) I (A'D - A"D), by means of the tangents, is removed by the condition that sin I a, cos I F, must be positive, and the agreement between sin e, cos -2- E, will serve to verify the whole calculation. The determination of the quantities AD'I, l"ID', E, A D" A'Dl" e" is effected in precisely the same manner, and it will not be worth while to transcribe here the eight equations used in this calculation, since, in fact, they readily appear if we change 188 DETERMINATION OF AN ORBIT FROM [BooK II. A'D A"D e F — I'"7' for AD' A"Da'' I"- 7" or for AD' A'D" "' - I7' r respectively. A new verification of the whole calculation thus far can be obtained from the mutual relation between the sides and angles of the spherical triangle formed by joining the three points D, D', D", from which result the equations, true in general, whatever may be the positions of these points, sin (A D'- AD) _ sin (A'D - -A'D) sin (A"D -- A'') sin Sine' - sin e"i Finally, if the equator is selected for the fundamental plane instead of the ecliptic, the computation undergoes no change, except that it is necessary to substitute for the heliocentric places of the earth A, A', A" those points of the equator where it is cut by the circles AB, A'B', A"B"; consequently, the right ascensions of these intersections are to be taken instead of 1, 1', 1", and also instead of A'D, the distance of the point D from the second intersection, etc. 138. The {lird step consists in this, that the two extreme geocentric places of the heavenly body, that is, the points B, B", are to be joined by a great circle, and the intersection of this with the great circle A'B' is to be determined. Let B* be this intersection and d'- a its distance from the point A'; let a*c be its longitulde, and A,3 its latitude. We have, consequently, for the reason that B, B*, B" lie in the same great circle, the well-known equation, 0 - tan p sin (a" - a*) - tan (* sin (a" - a) + tan (i" sin (a*t - a), which, by the substitution of tan y' sin (aC* -') for tan (3, takes the following form: - 0 - cos (-') (tan sin (a" -')- tan g" sin (a -')) - sin (cC* -') (tan (3 cos (a - I') + tan y' sin (a" - oa) - tan f" cos (a -1)), Wherefore, since tan (a* - )- cos y' tan (' - a) we shall have, tan (Y' - -) tan 8 sin (c' —') tan' sin (o -') v:~ie.- cos y' (tan: cos (.-I') -tan g' cos ( l -')) -- sin 7' sin (, - a) SECT. 1.] THREE COMPLETE OBSERVATIONS. 189 Thence are derived the following formulas, better suited to numerical calculations. Putting, [7] tan ( sin (a" --') - tan P" sin (a - 1') = -, [8] tan ( cos (a" - 1') - tan (3" cos (a -') = T7sin t1, [9] sin (a" -a) - Tcost, we shall have (article 14, II.) [10] tan(8-a)-T sin (t~ +I) The uncertainty in the determination of the arc (A'- a) by means of the tangent arises from the fact that the great circles A'B", BB", cut each other in two points; we shall always adopt for B* the intersection nearest the point B', so that a may always fall between the limits of - 90~ and + 900, by which means the uncertainty is removed. For the most part, then, the value of the arc i (which depends upon the cztrvatucre of the geocentric motion) will be quite a small quantity, and even, generally speaking, of the second order, if the intervals of the times are regarded as of the first order. It will readily appear, from the remark in the preceding article, what are the modifications to be applied to the computation, if the equator should be chosen as the fundamental plane instead of the ecliptic. It is, moreover, manifest that -the place of the point B* will remain indeterminate, if the circles BB", Al'B" should be wholly coincident; this case, in which the four points A', B, B', " lie in the same great circle, we exclude from our investigation. It is proper in the selection of observations to avoid that case, also, where the locus of these four points differs but little from a great circle; for then the place of the point B*, which is of great importance in the subsequent operations, would be too much affected by the slightest errors of observation, and could not be determined with the requisite precision. In the same manner the point B*, evidently, remains indeterminate when the points B, B" coincident in which case the position of the t Or when they are opposite to each other; but we do not speak of this case, because our method is not extended to observations embracing so great an interval. 190 DETERMINATION OF AN ORBIT FROM [BOOK II. circle BB" itself would become indeterminate. Wherefore we exclude this case. also, just as, for reasons similar to the preceding, those observations will be avoided in which the first and last geocentric places fall in points of the sphere near to each other. 139. Let C, C', C" be three heliocentric places of the heavenly body in the celestial sphere, which will be (article 64, III.) in the great circles AB, A'B', A"B", respectively, and, indeed, between A and B, A' and B', A" and B"; moreover, the points a CTc C" will lie in the same great circle, that is, in the circle which the plane of the orbit projects on the celestial sphere. We will denote by r, r', r", three distances of the heavenly body from the sun; by i, Q', p"t its distances from the earth; by R R R' -", the distances of the earth from the sun. Moreover, we put the arcs C' C/', CC, CC' equal to 2 f, 2f', 2f", respectively, and r'r" sin 2f = n, rr" sin 2f' - n, rr' sin 2f" - n". Consequently we have f' =f-+f", AC-+ CB _, A'' "+ C'B' —cg A"C"- + C"B"= "'; also, sin 8 sin A C sin GB r e R sin d' sin A' C' sin C'B' r' - QR' sin'' sin A" C(" sin C"B",-'- RB" Hence it is evident, that, as soon as the positions of the points 6 C' C" are known, the quantities r, r,' r", d',) -" can be determined. We shall now show how the former may be derived from the quantities from which, as we have before said, our method started. SECT. 1.] THREE COMPLETE OBSERVATIONS. 191 140. We first remark, that if 1\ were any point whatever of the great circle CC6' C") and the distances of the points C, C' C"A from the point N were counted in the direction from C to CN so that in general NC" — NC'= 2f NC" - NC- 2f', NC' - C 2f", we shall have I. 0 -sin 2f sin NC — sin 2 f' sin NCA + sin 2f" sin NCN. We will now suppose N to be taken in the intersection of the great circles BB*B", CC'C'", as in the ascending node of the former on the latter. Let us denote by C, G', "C, T, Z', i", respectively, the distances of the points C, C', C", D, D', D" from the great circle BB*B", taken positively on one side, and negatively on the other. Then sin (, sin', sin i", will evidently be proportional to sin VC, sin NrC', sin 1NC", whence equation I. is expressed in the following form:0 = sin 2f sin X - sin 2f' sin G' + sin 2f" sin r7"; or multiplying by rrr% II. 0 n r sin G - n'r' sin W' + "'r" sin G". It is evident, moreover, that sin ( is to sin gy, as the sine of the distance of the point C from B is to that of D' from B, both distances being measured in the same direction. We have, therefore, sin T' sin GB -sn Gi sin (A D' — ) 4 in precisely the same way, are obtained, sin $" sin CB sin (A Y" — ) f sin! sin C'B* sin $TI sin C'B* sin G' (sin A'D]- 8' + a) sin (A'D" —'- +)' -n sin - s sin C"B' sin T' sin C'GB'" - sin (A"D - "') sin (A"2D' - f")' Dividing, therefore, equation II. by r" sin (", there results, r sin CB sin (A"D' - ") r'sin'B* sin(A"D-6") -A r', sin CG"B"' sin (A D' - ) n' r" sin CG"B" sin (A'D —'+6) _+ _ 192 DETERMINATION OF AN ORBIT FROM [BOOK II. If now we designate the arc C'B' by z, substitute for r, r' r" their values in the preceding article, and, for the sake of brevity, put R sind sin (A"Df' — ") R'l" sin'" sin (AD' -) 6) R' sin d' sin (A"D - ) b R[12] sin 6' sin (A'D -' - a) our equation will become II. 0 = an - -bn sin (z —a) +__n" sin z The coefficient b may be computed by the following formula, which is easily derived from the equations just introduced: — R' sin n' sin (A D" - -) R sin 6 sin (A''"- -,Y +) - For verifying the computation, it will be expedient to use both the formulas 12 and 13. When sin(A'D" —-'+- ) is greater than sin (A'D —'+ a), the latter formula is less affected by the unavoidable errors of the tables than the former, and so will be preferred to it, if some small discrepancy to be explained in this way should result in the values of b; on the other hand, the former formula is most to be relied upon, when sin (A'D" - d'+ a) is less than sin (A'D - 6'+ 4); a suitable mean between both values will be adopted, if preferred. The following formulas can be made to answer for examining the calculation; their not very difficult derivation we suppress for the sake of brevity. a sin ("- 1) b sin (I"- ) sin (' - ) +sin ( - ) R RI sin 6' + El' R' sin d' /Ucos / cos ~I" R" sin A" sin (A D'- 8) sin F' in which (article 138, equation 10,) U expresses the quotient _S Tsin (t- + y') sin(' - ) cos (' —)' 141. From P = —, and equation III. of the preceding article, we have ( --'"P) P+P - a bn sin (z- ). P+-I sinz SECT. 1.] THREE COMPLETE OBSERVATIONS. 193 thence, and from Q 2 +n' 1)/3 and ff si is obtained, Q sin z4 P+ 1 sin z - 2R1sin ~); P+a 8in (z —o), or, 2'3S in = ( +- cos a sin (z —) - sin a cos (z ). Putting, therefore, for the sake of brevity, 1 [14] 2 R'3sin3'sin- 7 and introducing the auxiliary angle aw such that sin a P+a- COS, we have the equation IV. e Q sin w sin4 z = sin (z - -- ) from which we must get the unknown quantity z. That the angle o may be computed more conveniently, it will be expedient to present the preceding for mula for tan co thus: —(P + a) tan a p(_)+ (L) Whence, putting, b [15] os6 d, cos a COS a cos a we shall have for the determination of co the very simple formula, tan o =- " (P+ a) P+d We consider as the fourth step the computation of the quantities a, b, c, d, e, 25 194 DETERMINATION OF AN ORBIT FROM [BoOK II. by means of the formulas 11-16, depending on given quantities alone. The quantities b, c, e, will not themselves be required, only their logarithms. There is a special case in which these precepts require some change. That is, when the great circle BB" coincides with A"B", and thus the points B, B* with D', D respectively, the quantities a, b would acquire infinite values. Put, ting, in this case, ]R sin e sin (A'DY' —' - a) _R sin d' sin (A Uf- ) = in place of equation III. we shall have n' sin (z- a) sin z whence, making,tsin a tan tos P+- (I — t cos a)' the same equation IV. is obtained. In the same manner, in the special case when a - 0, c becomes infinite, and =- 0, on account of which the factor e sin ow, in equation IV., seems to be indeterminate; nevertheless, it is in reality determinate, and its value is P+a 2 R'3 sin3' (b - 1) (P4+ d)' as a little attention will show. In this case, therefore, sin z becomes _R' sin At i1 2 () (P + a) 142. Equation IV., which being developed rises to the eighth degree, is solved by trial very expeditiously in its unchanged form. But, from the theory of equations, it can be easily shown, (which, for the sake of brevity we shall dispense with explaining more fully) that this equation admits of two or four solutions by means of real values. In the former case, one value of sin z will be positive; and the other negative value must be rejected, because, by the nature of the problem, it is impossible for r' to become negative. In the latter case, among the values of sin one will be positive, and the remaining three negative, -when, SECT. 1.] THREE COMPLETE OBSERVATIONS. 195 accordingly, it will not be doubtful which must be adopted, - or three. positive with one negative; in this case, from among the positive values those, if there are any, are to be rejected which give z greater than 6' since, by another essential condition of the problem, )' and, therefore sin (d'-Z), must be a positive quantity When the observations are distant from each other by moderate intervals of time, the last case will most frequently occur, in which three positive values of sin z satisfy the equation. Among these solutions, besides that which is true, some one will be found making z differ but little from Y,' either in excess or in defect; this is to be accounted for as follows. The analytical treatment of our problem is based upon the condition, simply, that the three places of the heavenly body in space must fall in right lines, the positions of which are determined by the absolute places of the earth, and the observed places of the body. Now, from the very nature of the case, these places must, in fact, fall in those parts of the right lines whence the light descends to the earth. But the analytical equations do not recognize this restriction, and every system of places, harmonizing of course with the laws of KEPLER, is embraced, whether they lie in these right lines on this side of the earth, or on that, or, in fine, whether they coincide with the earth itself. Now, this last case will undoubtedly satisfy our problem, since the earth moves in accordance with these laws. Thence it is manifest, that the equations must include the solution in which the points C. C', C" coincide with A, A' A" (so long as we neglect the very small variations in the elliptical places of the earth produced by the perturbations and the parallax). Equation IV., therefore, must always admit the solution z Y', if true values answering to the places of the earth are adopted for P and Q. So long as values not differing much from these are assigned to those quantities (which is always an admissible supposition, when the intervals of the times are moderate), among the solutions of equation IV., some one will necessarily be found which approaches very nearly to the value For the most part, indeed, in that case where equation IV. admits of three solutions by means of positive values of sin z, the third of these (besides the true one, and that of which we have just spoken) makes the value of z greater than Y', and thus is only analytically possible, but physically impossible; so that it can 196 DETERMINATION OF AN ORBIT FROM [BOOK II. not then be doubtful which is to be adopted. But yet it certainly can happen, that the equation may admit of two distinct and proper solutions, and thus that our problem may be satisfied by two wholly different orbits. But in such an event, the true orbit is easily distinguished from the false as soon as it is possible to bring to the test other and more remote observations. 143. As soon as the angle z is got, r' is immediately had by means of the equation, R'sind' sin z Further, from the equations P _- and III. we obtain, n'r' (P+ a) R' sin r' n b sin (z-)' nfrl 1 nlr' " — P n Now, in order that we may treat the formulas, according to which the positions of the points C, a", are determined from the position of the point C', in such a manner that their general truth in those cases not shown in figure 4 may immediately be apparent, we remark, that the sine of the distance of the point C' from the great circle CB (taken positively in the superior hemisphere, negatively in the inferior) is equal to the product of sin e" into the sine of the distance of the point C' from D", measured in the positive direction, and therefore to - sin e" sin C'D" - - sin e" sin (z + AX'D"-'); in the same manner, the sine of the distance of the point C" from the same great circle is- sin 8' sin C"D'. But, evidently, those sines are as sin CC' to sin CC", or as - to, or as r" to n'r'. Putting, therefore, C"D' =i', we have r r' rr V. r" sin ("f- fl1n'r'1 e sin e" A'8" V. /' sill 5/' 12~ ~ sin'sin 0 s i. n (z -]Precisely in the same way, putting CD' = 4, is obtained nrlo sin e VI. r sin/' = —-. sin (z - A'D —-Y) VII. rsin ( -+ AD"-AD') =r"P sn,sin(4.,+ A"D- A"D'). SECT. 1.] THREE COMPLETE OBSERVATIONS. 197 By combining equations V. and VI. with the following taken from article 139, VIII. r" sin (4"- A"D' + Y")) R" sin a", IX. r sin ( AD' + A) - - X sin 6P the quantities rt, ", r, r",will be thence derived by the method of article 78. That this calculation may be more conveniently effected, it will not be unacceptable to produce here the formulas themselves. Let us put R sin C [17] sin (A D' —) )' R" sin _" [18] sin (A"D' —- d' ) 9] cos (A D'- 1 R sin a 9 cos (A"D' —V) - ['~] R sin d'" The computation of these, or rather of their logarithms, yet independent of P and Q, is to be regarded as the flZh and last step in the, as it were, preliminary operations, and is conveniently performed at the same time with the computation of a, b, themselves, or with the fourth step, where a becomes equal to Making, then,.n%' sins -~.sin P sin (o + S'D - 0') =p, n'r sin d n.sin e' sin (z + ASY'-[) 2p " (X"p -- 1) - q", we derive &' and r from r sin -p, r cos i- q; also, ~" and r" from r" sin i" —py", and r" cos i;'- ". No ambiguity can occur in determining C and C", because r and r" must, necessarily, be positive quantities. The complete computation can, if desired, be verified by equation VII. There are two cases, nevertheless, where another course must be pursued. That is, when the point D' coincides with B, or is opposite to it in the sphere, or when AD'-A_ 0 or 1800, equations VI. and IX. must necessarily be iden 198 DETERMINATION OF AN ORBIT FROM [BooK II. tical, and we should have x -, 7p - 1= 0, and q, therefore, indeterminate. In this case, " and r" will be determined, in the manner we have shown[, but then. and r must be obtained by the combination of equation VII. with VI. or IX. We dispense with transcribing here the formulas themselves, to be found in article 78; we observe, merely, that in the case where ADL'- aY is in fact neither - O nor 180, but is, nevertheless, a very small arc, it is preferable to follow the same method, since the former method does not then admit of the requisite precision. And, in fact, the combination of equation VII. withVI. or IX. will be chosen according as sin (A D"- AdD') is greater or less than sin (AD'- A). In the same manner, in the case in which the point D', or the one opposite to it, either coincides with B" or is little removed from it, the determination of' and r" by the preceding method would be either impossible or unsafe. In this case, accordingly, and r will be determined by that method, but C" and r" by the combination of equation VII. either with V. or with VIII., according as sin (A"D - A"') is greater or less than sin (A"D' - J/). There is no reason to fear that D' will coincide at the satne tizme with the points B, B", or with the opposite points, or be very near them; for the case in which B coincides with B", or is but little remote from it, we excluded above, in article 138, from our discussion. 144. The arcs 5 and C" being found, the positions of the points G C", will be given, and it will be possible to determine the distance CC"' 2f' from G,," and I'. Let u, W'"be the inclinations of the great circles AB, A"B" to the great circle CC" (which in figure 4 will be the angles C" CD' and 180' - CC"D', respectively), and we shall have the following equations, entirely analogous to the equations 3-6, article 137:sinf' sin A (u-f+ it) = sin - A' sin 1 ( + - "), sinf' cos 1 (" -+ u) -- cos 1 E sin 1 (- -"), cosf' sin I ("` - i) = sin X' cos i (o -t- C"), cosf' cos 1 (it" - -) - cos 1' E oS (- - ) SECT. 1.] THREE COMPLETE OBSERVATIONS. 199 The two former will give 2 (" H+ ut) and sinf' the two latter - (it" - t) and cosf'; from sinf' and cosf' we shall havef'. It will be proper to neglect in the first hypotheses the angles (it" + ui) and 1 (it" - t), which will be used in the last hypothesis only for determining the position of the plane of the orbit. In the same way, exactly, f can be derived from e, C'D and C"D; alsof" from a", CD" and C'D"; but the following formulas are used much more conveniently for this purpose:sin 2f= r sin 2f' -, n! sin 2 f" r" sin 2 f' in which the logarithms of the quantities,2 n, — are already given by the preceding calculations. Finally, the whole calculation finds a new verification in this, that we must have 2f + 2f"- 2f'; if by chance eany difference shows itself, it will not certainly be of any importance, if a11 the processes have been performed as accurately as possible. Nevertheless, occasionally, the calculation being conducted throughout with seven places of decimals, it may amount to some tenths of a second? which if it appear worth while, we may with the utmost facility so distribute between 2f and 2f" that the logarithms of the sines may be equally either increased or diminished, by which means the equation _p _ r sin 2af" n1 r'/ sin 2f n will be satisfied with all the precision that the tables admit. When f and f" differ a little, it will be sufficient to distribute that difference equally between 2f and 2f"/ 145. After the positions of the heavenly body in the orbit have been determined in this manner, the double calculation of the elements will be commenced, both by the combination of the second place with the third, and the combination of the first with the second, together with the corresponding intervals of the times. 200 DETERMINATION OF AN ORBIT FROM [Booli II. Before this is undertaken, of course, the intervals of the times themselves require some correction, if it is decided to take account of the aberration agreeably to the third method of article 118. In this case, evidently, for the true times are to be substituted fictitious ones anterior to the former, respectively, by 493,o, 493', 493~." seconds. For computing the distances (), c/, we have the formulas: - R sin (A D'- r) r sin (A D' —) P sin ( — AD'+ _d) sin_' R' sill (d' - z) r' sin (Y- z) sin z sin,',_ R" sin (A"D' — ") / r"sin (A"D'- ") sin (" t A"D'+ - ) sin 6' But, if the observations should at the beginning have been freed from aberration by the first or second method of article 118, this calculation may be omitted; so that it will not be necessary to deduce the values of the distances Q, (', Q", unless, perhaps, for the sake of proving that those values, upon which the computation of the aberration was based, were sufficiently exact. Finally, it is apparent that all this calculation is also to be omitted whenever it is thought preferable to neglect the aberration altogether. 146. The calculation of the elements — on the one hand from r' r", 2f and the corrected interval of the time between the second and third observations, the product of which multiplied by the quantity ic, (article 1,) we denote by 6, and on the other hand from r, r' 2f" and the interval of time between the first and second observations, the product. of which by k will be equal to 6" -- is to be carried, agreeably to the method explained in articles 88-105, only as far as the quantity there denoted by y, the value of which in the first of these combinations we shall call n, in the latter n". Let then f0"q_- t r'r'6" 0 0' Q 0 m - r' r r',"r cosf cosf' cosf" and it is evident, that if the values of the quantities P, Q, upon which the whole calculation hitherto is based, were true, we should have in the result P'- P, SECT. 1.] THREE COMPLETE OBSERVATIONS. 201 Q' — Q. And conversely it is readily perceived, that if in the result PP'-P, Q,'_ the double calculation of thle elements from both combinations w-ould, if completed, furnish numbers entirely equal, by which, therefore, all three observations will be exactly represented, and thus the problem Wvholly satisfied. But when the result is not P' P' 0-' Q, let P'-I P Q Q be taken for X and Y, if, indeed, P and Q were taken for x and y; it will be still more convenient to put log P - x2 log O _ yog og _ogP'- log P X, log Q' - log Q = Y. Then the calculation must be repeated with other values of x, y. 147. Properly, indeed, here also, as in the ten methods before given, it would be arbitrary what new values we assume for x and y in the second hypothesis, if only they are not inconsistent with the general conditions developed above; but yet, since it manifestly is to be considered a great advantage to be able to set out from more accurate values, in this method we should act with but little prudence if we were to adopt the second values rashly as it were, since it may easily be perceived from the very nature of the subject, that if the first values of P and Q were affected with slight errors, P' and Q' themselves would represent inmuch more exact values, supposing the heliocentric motion to be moderate. Wherefore, we shall always adopt P' and Q' themselves for the second values of P and Q, or log P' log Q' for the second values of x and y, if log P, log Q are supposed to denote the first values. Now, in this second hypothesis, where all the preliminary work exhibited in the formulas 1-20 is to be retained without alteration, the calculation will be undertaken anew in precisely the same manner. That is, first, the angle w will be determined; after that z, r', -, -,/, r,, r", f' f, f,. From the difference, more or less considerable, between the new values of these quantities and the first, a judgment will easily be formed whether or not it is worth while to compute anew the correction of the times on account of aberration; in the latter case, the intervals of the times, and therefore the quantities d and d", will remain the same as before. Finally, hi, ~)" are derived from f, ", r",f", r, r' and 26 202 DETERMINATION OF AN ORBIT FROM [BooK IH. the intervals of the times; and hence new values of P' and Q', which comnmonly differ much less from those furnished by the first hypothesis, than the latter from the original values themselves of P and Q. The second values of X and Y will, therefore, be much smaller than the first, and the second values of P', Q', will be adopted as the third values of P, Q, and with these the computation will be resumned anew. In this manner, then, as from the second hypothesis more exact nunmbers had resulted than from the first, so from the third more exact numbers will again result than from the second, and the third values of P', Q' can be taken as the fourth of P, Q, and thus the calculation be repeated until an hypothesis is arrived at in which X and Y may be regarded as vanishing; but when the third hypothesis appears to be insufficient, it will be preferable to deduce the values of P, Q, assumed in the fourth hypothesis froms the first three, in accordance with the method explained in articles 120, 121, by which means a more rapid approximation will be obtained, and it will rarely be requisite to go forward to the fifth hypothesis. 148. When the elements to be derived from the three observations are as yet wholly unknown (to which case our method is especially adapted), in the first hypothesis, as we have already observed,, 0 d ", are to be taken for appyroxinate values of P and Q, where d and 0" are derived for the present from the interxwls of the tines not corrected. If the ratio of these to the corrected intervals is expressed by t': I and u": 1, respectively, we shall have in the first hypothesis, X- log -o - log iYf + log n - log y", Y: log t, -+ log,y't - log A2 - log a;" + Comp. log cos f + Comp. log cosf' + Comp. log cos f" + 2 log r' - log r - log r'. The logarithms of the quantities pA, p," are of no importance in respect to the remalining terms; log.it and log j", which are both positive, in X cancel each other in rome measure, whence X possesses a small value, sometimes positive, sometimes negative; on the other hand, in Y some compensation of the positive terms Comnz. bC lo sf, Comlp. log cosef' Comp. log cosf" arises also from the negative SECT. 1.] THREE COMPLETE OBSERVATIONS. 203 terms log A, log <', but less complete, for the former greatly exceed the latter. In general, it is not possible to determine any thing concerning the sign of log G-r Now, as often as the heliocentric motion between the observations is small, it will rarely be necessary to proceed to the fourth hypothesis; most frequently the third, often the second, will afford sufficient precision, and we may sometimes be satisfied with the numbers resulting from even the first hypothesis. It will be advantageous always to have a regard to the greater or less degree of precision belonging to the observations; it would be an ungrateful task to aim at a precision in the calculation a hundred or a thousand times greater than that wbhicl the observations themselves allow. In these matters, however, the judgment is sharpened more by frequent practical exercise than by rules, and the skilful readily acquire a certain faculty of deciding where it is expedient to stop. 149. Lastly, the elements themselves will be computed in the final hypothesis, either fromf, r', r", or fromf", r, r', carrying one or the other of the calculations through to the end, which in the previous hypotheses it had only been requisituK to continue as far as i9'/"; if it should be thought proper to finish both, the agreement of the resulting numbers will furnish a new verification of the whole work. It is best, nevertheless, as soon as ff',f", are got, to obtain the elements fromr the single combination of the first place with the third, that is, from f'r, r" and the interval of the time, and finally, for the better confirmation of the comnputation, to determine the middle place in the orbit by means of the elements found. In this way, therefore, the dimensions of the conic section are made known, that is, the eccentricity, the semi-axis major or the semi-parameter, the place of the perihelion with respect to the heliocentric places ~ C', C", the mean motion, and the mean anomaly for the arbitrary epoch if the orbit is elliptical, or the time of perihelion passage if the orbit is hyperbolic or parabolic. It only remains, therefore, to determine the positions of the heliocentric places in the orbit with respect to the ascending node, the position of this node with reference to the equinoctial point, and the inclination of the orbit to the ecliptic (or the 204 DETERMINATION OF AN ORBIT FROM [Boox II. equator). All this may be effected by the solution of a single spherical triangle. Let 2 be the longitude of the ascending node; i the inclination of the orbit; g and g" the arguments of the latitude in the first and third observations; lastly, let 1 — 2 h ", I' - A". Calling, in figure 4, 2 the ascending node, the sides of the triangle 2 AC will be AlD'-, g, A and the angles opposite to them, respectively, 4i 180~ — 7, e-. We shall have, then, sin 1- i sin - ) sin I (AD' — 0) sin (A + u) sin i cos 1 (g + h) -cos 1 (AD'- s-) sin 1 (y - u) cos i sin (g-sh) -sin i (AD'-) cos ~ (r + t) os icos ( - ) cos - (A' - ) cos (y - ). The two first equations will give 1 (g+h) and sin - i4 the remaining two B (g-h) and cos 2 i; from g will be known the place of the perihelion with regard to the ascending node, from A the place of the node in the ecliptic; finally, I will become known, the sine and the cosine mutually verifying each other. We can arrive at the same object by the help of the triangle g A" C', in which it is only necessary to change in the preceding formulas the symbols g, 7i, A, n, y, il into g", /h", A"l, ", y", Z'. That still another verification may be provided for the whole work, it will not be unserviceable to perform the calculation in both w-ays; when, if any very slight discrepancies should show themselves between the values of 4 9, and the longitude of the perihelion in the orbit, it will be proper to take mean values. These differences rarely amount to Osl or 08.2, provided all the computations have been carefully made with seven places of decimals. When the equator is taken as the fundamental plane instead of the ecliptic, it will make no difference in the computation, except that in place of the points A, A" the intersections of the equator with the great circles AB, A'B" are to be adopted. THREE COMPLETE OBSERVATIONS. 206 150. We proceed now to the illustration of this method by some examples fully explained, which will show, in the plainest manner, how generally it applies, and how conveniently and expeditiously it leads to the desired result.* The new planet Juno will furnish us the first example, for which purpose we select the following observations made at Greenwich and communicated to us by the distinguished MASCKELYNE. Mean Time, Greenwich. App. Right Ascension. App. Declination S. 1804, Oct. 5 10h 51m 6s 3570 10' 22".35 60 40' 8" 17 9 58 10 355 43 45.30 8 47 25 27 9 16 41 355 11 10.95 10 2 28 From the solar tables for the same times is found Longitude of the Sun utation Distance from Latitude of Appar. Obliquity of from App. Equin. the Earth. the Sun. the Ecliptic.... Oct. 5 1920 28' 53".72 - 15".43 0.9988839 -0".49 23~ 27 59".48 17 204 20 21.54 15.51 0.9953968 + 0.79 59.26 27 214 16 52.21 + 15.60 0.9928340 0.15 59.06 We will conduct the calculation as if the orbit were wholly unknown: for which reason, it will not be permitted to free the places of Juno from parallax, but it will be necessary to transfer the latter to the places of the earth. Accordingly we first reduce the observed places from the equator to the ecliptic, the apparent obliquity being employed, whence results, * It is incorrect to call one method more or less exact than another, That method alone can be considered to have solved the problem, by which any degree of precision whatever is, at least, attainable. %Wherefore, one method excels another in this respect only, that the same degree of precision may be reached by one more quickly, and with less labor, than by the other. 206 DETERMINATION OF AN ORBIT FROM [Boost IL App. Longitude of Juno. App. Latitude of Juno. Oct. 5 3540 44'54".27 40 59'31".59 17 352 34 44.51 -6 21 56.25 27 351 34 51.57 -7 17 52.70 We join directly to this calculation the determination of the longitude and latitude of the zenith of the place of observation in the three observations: the right ascension, in fact, agrees with the right ascension of Juno (because the observations have been made in the meridian) but the declination is equal to the altitude of the pole, 51~ 28' 39". Thus we get Long. of the Zenith. Lat. of the Zenith. Oct. 5 24~0 29' 46 53' 17 23 25 47 24 27 23 1 47 36 Now the fictitious places of the earth in the plane of the ecliptic, from which the heavenly body would appear in the same manner as from the true places of the observations) will be determined according to the precepts given in article 72. In this way, putting the mean parallax of the sun equal to 8".6, there results, Reduction of Longitude. Reduction of Distance. Reduction of Time. Oct. 5 - 22'".39 1 0.0003856 _ 0s.19 17 -27 21 0.0002329 - 0.12 27 -35.82 + 0o.o000o2085 -0.12 The reduction of the time is added only that it may be seen that it is wholly insensible. After this, all the longitudes, both of the planet and of the earth, are to be reduced to the mean vernal equinox for some epoch, for which we shall adopt the beginning of the year 1805; the nutation being subtracted the precession is to be added, which, for the three observations is respectively 11".87, 10".23, 8".86, SECT. 1.] THREE COMPLETE OBSERVATIONS. 207 so that - 3".56 is to be added for the first observation, 5".28 for the secondl - 6".74 for the third. Lastly the longitudes and latitudes of Juno are to be freed from the aberration of tlhe fixed stars; thus it is found by well-known rules, that we must subtract from the longitudes respectively 19".12, 17".11, 14".82, but add to the latitudes 0".53, 1l.18, i".75, by which addition the absolute values are diminished since south latitudes are considered as negative. All these reductions being properly applied, we have the correct data of the problem as follows:Times of the observations reduced to the meridian of Paris o Oct. 5.458644 17.421885 27.393077 Longitudes of Juno, a,. a, a" 3540 441 31".60 352~ 34'22".12 351~ 34'30".01 Latitudes, (, (i', (f ".... — 4 59 31.06 -6 21 55.07 -7 17 50.95 Longitudes of the earth,l 1 I', 1" 12 28 27.76 24 19 49.05 34 16 9.65 Logs. of the distances, R1, R", 9.9996826 9.9980979 9.9969678 Hence the calculations of articles 136, 137, produce the following numbers, 7, Y7" 7".... 1960 0O 8".36 1910 58 0".33 1900 411 401/.17 Y,,',' c".... o... 18 23 59.20 32 19 24.93 43 11 42.05 ]ogarithms of the sines.. o 9.4991995 9.7281105 9.8353631 A'D, AD' AD".. 0. 232 6 26.44 213 12 29.82 209 43 7.47 A"D, A"D'. A'D"..... 241 51 15.22 234 27 0.90 221 13 57.87 E El, E, e 0..... 0 2 19 34.00 7 13 37.70 4 55 46.19 logarithms of the sines... 8.6083885 9.0996915 8.9341440 log sin I e'.. 0.. 8.7995259 log cos e...... 9.9991357 Moreover, according to article 138, we have log tan.3.... 8.9412494 n log tan f".... 9.1074080 n log sin (a"-l'). 9.7332391 n logsin (a —') o. 9.6935181n log cos ("- l'). 9.9247904 logcos(a- l').o 9.9393180 208 DETERMINATION OF AN ORBIT FROM [BOOK 11. Hence log (tan ( cos (a'- l') - tan 3" cos (a -')) = log Tsin 1 8.5786513 log sin (a - a) log T cos.... e... 8.7423191 n Hence t- 145~ 32' 57".78 log T.. 8.8260683 t - + Y — 337 30 58.11 log sin (1 + y')... 9.5825441 nz Lastly log (tan P sin (a" —') - tan s" sin (a --')) = log.o 8.2033319 it log T sin (t - Y')....... 8.4086124 n whence log tan (' - a).. 9.7947195'T - a - 31~ 56' 11".81, and therefore -- 0~ 23' 13".12. According to article 140 we have A"D' — Y = 191~ 15' 18".85 log sin 9.2904352 n log cos 9.9915661 i ADI -( =_ 194 48 30.62 cc "C 9.4075427 n C " 9.9853301G A"D - - = 198 39 33.17 "" cc 9.5050667 n A'D -6' a = 200 10 14.63 " 9.5375909 n AD" -6 = 191 19 8.27 " " 9.2928554n A'D" — 6' - a = 189 17 46.06 " " 9.2082723n Hence follow, loga.. o 9.5494437, a- +- 0.3543592 log b o o 9.8613533. Formula 13 would give log b- 9.8613531, but we have preferred the formei1 value, because sin (A'D -' — + o) is greater than sin (A'D" - -' + 0). Again, by article 141 we have, 3 logR'sin' o. e 9.1786252 log 2...... 0.3010300 log sina... 7.8295601 7.3092153 and therefore log c - 2.6907847 log b...... 9.8613533 log cos..o 9.9999901 9.8613632 SECT. 1.] THREE COMPLETE OBSERVATIONS. 209 whence b 0.7267135. Hence are derived COS a d - - 1.3625052, log e _ 8.3929518 n Finally, by means of formulas, article 143, are obtained, log.... 0.0913394 n log x".... 0.5418957 n log X.... 0.4864480 n log ".... 0.1592352 n 152. The preliminary calculations being despatched in this way, we pass to the first hypothesis. The interval of time (not corrected) between the second and third observations is 9.971192 days, between the first and second is 11.963241. The logarithms of these numbers are 0.9987471, and 1.0778489, whence log 3- 9.2343285, log C"- 9.3134303. We will put, therefore, for the first hypothesis, x- log P - 0.0791018 y= log Q- 8.5477588 Hence we have P - 1.1997804, P + a = 1.5541396, P + d - - 0.1627248; loge. o 8.3929518 n log (P + ). 0.1914900 C. log (P + d) 0.7885463 n log tan o.. 9.3729881, whence w - + 13~16'51".89, + o -- + 13~40' 5"o01.o log Q... 8.5477588 logc... 2.6907847 log sin o 9.3612147 log Qe sin o. 0.5997582 The equation Q c sin w( sin4 z= sin (z — 13o40' 5".01) is found after a few trials to be satisfied by the value z = 14~ 35' 4".90, whence we have log sin z - 9.4010744, log r'= 0.3251340. That equation admits of three other solutions besides this, namely, 27 210 DETERMINATION OF AN ORBIT FROM [BooK II. z — 320 2' 28" z- 137 27 59 z — 193 4 18 The third must be rejected because sin z is negative; the second because z is greater than d'; the first answers to an approximation to the orbit of the earth of which we have spoken in article 142. Further, we have, according to article 143, R' sin Y' log b.. 9.8648551 log (P + a)... 0.1914900 C. log sin (z- -).... 0.6103578 n' r' log.... 0.6667029 logP. 0.0791018 log,,.. 0.5876011 z+ A'D -d' =z- 1990 47' 1".51 214~ 22' 6".41; log sin 9.7516736 n z +A'D" —'=z — + 188 54 32.94 203 29 37.84; log sin 9.6005923n Hence we have log p = 9.9270735 n, log p" = 0.0226459 n, and then log q = 0.2930977 n, log q'" 0.2580086 n, whence result 5 = 203~ 17' 31".22 log r - 0.3300178." — 110 10 58.88 logr"- 0.3212819 Lastly, by means of article 144, we obtain i (U" + u) - 205~ 18' 10".53 (u"- u) - - 3 14 2.02 f'= 3 48 14.66 logsin,2f'... 9.1218791 log sin2f'... 9.1218791 log r..... 0.3300178 log r"..... 0.3212819 C.log... 9.3332971 C.log n.... 9.4123989 log sin2f... 8.7851940 logsin 2f". 8.8555599 2 f= 3~ 29" 46'.03 2 f" 40 6' 43".28 The sum 2f - 2f" differs in this case from 2f' only by 0".01. SECT. 1.] THREE COMPLETE OBSERVATIONS. 211 Now, in order that the times may be corrected for aberration, it is necessary to compute the distances (, ~',?" by the formulas of article 145, and afterwards to multiply them by the time 493s, or 0d.005706. The following is the calculation, logr.... 0.33002 logr' o.. 0.32513 log r".... 0.32128 log sin (AD' -) 9.23606 log sin (G'- z) 9.48384 logsin(A"9D' — ") 9.61384 C.logsin. 0.50080 C.logsinY'. 0.27189 C. log sin 6". 0.16464 logQ... 0.06688 log'. o 0.08086 logg... 0.09976 log const... 7.75633 7.75633 7.75633 log of reduction 7.82321 7.83719 7.85609 reduction- 0.006656 0.006874 0.007179 Observations. Corrected times. Intervals. Logarithms. I, Oct. 5.451988 I1d1.963023 1.0778409 17.415011 9L 27.970887 0.9987339!IL 27.385898 The corrected logarithms of the quantities C, 6", are consequently 9.2343153 and 9.3134223. By commencing now the determination of the elements from f, r', r", 8 we obtain log n= 0.0002285, and in the same manner from f" r, r', C"we get log <"- 0.0003191. We need not add here this calculation explained at length in section III. of the first book. Finally we have, by article 146, log 8.... 9.3134223 2 logr'.... 0.6502680 C. log.... 0.7656847 C. logrr".. o 9.3487003 log..... 0.0002285 log 8 8".... 8.5477376 C. log ". ~ ~ 9.9996809 C. log n".. 9.9994524 log1P'.... 0.0790164 C. log cosf.. 0.0002022 C. log cosf'. o 0.0009579 C. log cos f" O. 0.0002797 log Q'..... 8.5475981 The first hypothesis, therefore, results in X =- 0.0000854, Y= - 0.0001607. 212 DETERMINATION OF AN ORBIT FROM [BOOK ILo 153. In the second Iihypothesis we shall assign to P, Q, the very values, which in the first we have found for Pt Q'. We shall put, therefore, x = log P - 0.0790164 y = log Q - 8.5475981 Since the calculation is to be- conducted in precisely the same manner as in the first hypothesis, it will be sufficient to set down here its principal results: -..... o oe130 15 38".13 ee 2100 8'24".98 13 38 51.25 logre.... 0.3307676 log Qcsin o. 0.5989389 logr".... 0.3222280.....O e. c e14 33 19.00 2([ + U). 205 22 15.58 log r'... 0.3259918 9(u — u).o. — 3 14 4.79 log.... 0.6675193 2f 7 34 53.32 1g ~ef ~. 3 29 0.18 log,.... 0.5885029 2f. 4 553.12 203 16 38.16 It would hardly be worth while to compute anew the reductions of the times on account of aberration for they scarcely differ Is from those which we have got in the first hypothesis. The further calculations furnish log — 0.0002270, logff'" 0.0003173, whence are derived log P' 0.0790167 X + 0.0000003 log Q' 8.5476110 Y= +- 0.0000129 From this it appears how much more exact the second hypothesis is than the first. 154. In order to leave nothing to be desired, we will still construct the thllird zypobte sis, in which we shall again choose the values of P', Q', obtained in the second SECT. 1.] THREE COMPLETE OBSERVATIONS. 213 hypothesis as the values of P, Q. Putting, therefore, x = log P 0.0790167 y - log Q -- 8.5476110 the following are found to be the principal results of the calculation:o.... 13~ 15'38".39 ~"... 2100 8'25".65 Co +a o.... 13 38 51.51 log r.... 0.3307640 log Qc sin co. 0.5989542 logr"... 0.3222239 14 33 19.50 (2 u"+ ).. 205 22 14.57 logr... 0.3259878 A (T"-u)... -3 14 4.78 log-. e 0.6675154. 7 34 53.73 af 2 329 0.39 logn..... 0.5884987 2/".... 4 5 53.34 All these numbers203 di 16er 38.41 nishedll these numbers differ so little frthatom those which the second hypothesi further correction.* We may, therefore, proceed to the determination of the elements from 2f',, r" 6', which we dispense with transcribing here, since it has already been given in detail in the example of article 97. Nothing, therefore, remains but to compute the position of the plane of the orbit by the method of article 149, and to transfer the epoch to the beginning of the year 1805. This computation is to be based upon the following numbers:AD'- - 90~55'51".41 (/ +t)= 202 18 13.855 (Y —tt) — 6 18 5.495 whence we obtain 2 (g + -) -- 196o 43'14".62 (g- h) - -4 37 24.41 i = 6 33 22.05 * If the calculation should be carried through in the same manner as in the preceding hypotheses, we should obtain X 0O, and Y —- 0.0000003, which value must be regarded as vanishing, and, in fact, it hardly exceeds the uncertainty always remaining in the last decimal place. 214 DETERMINATION OF AN ORBIT FROM [BooK 11. We have, therefore, h -201~ 20' 39".03, and so g- I - - 171~ 7'48".73; furthery -192~ 5' 50".21, and hence, since the true anomaly for the first observation is found, in article 97, to be 310055'29".64, the distance of perihelion from the ascending node in the orbit, 241' 10'20".57, the longitude of the perihelion 520 18' 9".30; lastly, the inclination of the orbit, 13~ 6'44".10. If we prefer to proceed to the same calculation from the third place, we have, A"'D - " 24~ 18' 35".25 (y"+ 1")= 196 24 54.98 ~ (Z" — ") — =-5 43 14.81 Thence are derived -(g" + ")- 211~ 24' 32".45 (g"'- ") - 11 43 48.48 -- 6 33 22.05 and hence the longitude of the ascending node, I" - h" 1710 7' 48".72, the lon gitude of the perihelion 52~ 18' 9".30, the inclination of the orbit 13~ 6'44".10, just the same as before. The interval of time from the last observation to the beginning of the year 1805 is 64.614102 days; the mean heliocentric motion corresponding to which is 53293".66 =14 48' 13".66; hence the epoch of the mean anomaly at the beginning of the year 1805 for the meridian of Paris is 3490 34' 12".38, and the epoch of the mean longitude, 410 52' 21".68. 155. That it may be more clearly manifest what is the accuracy of the elements just found, we will compute from them the middle place. For October 17.415011 the mean anomaly is found to be 332~ 28' 54".77, hence the true is 315~ 1' 23".02 andlogr", 0.3259877, (see the examples of articles 13, 14); this true anomaly ought to be equal to the true anomaly in the first observation increased by the angle 2 f, or to the true anomaly in the third observation diminished by the angle 2f, that is, equal to 3150 1' 22".98; and the logarithm of the radius vector should be 0.3259878: the differences are of no consequence. If the calculation SECT. 1.] THREE COMPLETE OBSERVATIONS. 215 for the middle observation is continued to the geocentric place, the results differ from observation only by a few hundredths of a second, (article 63;) these differences are absorbed, as it were, in the unavoidable errors arising from the want of strict accuracy in the tables. We have worked out the preceding example with the utmost precision, to show how easily the most exact solution possible can be obtained by our method. In actual practice it will rarely be necessary to adhere scrupulously to this type. It will generally be sufficient to use six places of decimals throughout; and in our example the second hypothesis would have given results not less accurate than the third, and even the first would have been entirely satisfactory. We imagine that it will not be unacceptable to our readers to have a comparison of the elements derived from the third hypothesis with those which would result from the use of the second or first hypothesis for the same object. We exhibit the three systems of elements in the following table:From hypothesis III. From hypothesis II. From hypothesis I. Epoch of mean long. 1805 410 52' 21".68 410 52' 18".40 420 12' 37'.83 Mean daily motion.. 824'.7989 824t.7983 823".5025 Perihelion.. 52 18 9.30 52 18 6.66 52 41 9.81 CP.... 1..4 14 12 1.87 14 11 59.94 14 24 27.49 Log of semi-axis major. 0.4224389 0.4224392 0.4228944 Ascending node... 171 7 48.73 171 7 49.15 171 5 48.86 Inclination of the orbit. 13 6 44.10 13 6 45.12 13 2 37.50 By computing the heliocentric place in orbit for the middle observation from the second system of elements) the error of the logarithm of the radius vector is found equal to zero, the error of the longitude in orbit, 0'.03; and in computing the same place by the system derived from the first hypothesis, the error of the logarithm of the radius vector is 0.0000002, the error of the longitude in orbit, 1.31.E And by continuing the calculation to the geocentric place we have, 216 D)ETERMINATION OF AN ORBiT FROM FBooi t From hypothesis II. From hypothesis I. Geocentric longitude 352~ 34' 22".26 352~ 34' 19'.97 Error 0.14 2.15 Geocentric latitude 6 21 55.06 6 21 54.47 Error... 0.01 0.60 156. We shall take the second example from Pallas, the following observations of which, made at Milan, we take from von ZACH's 3fonalliche Correspondenz, Vol. XIV., p. 90. Mean Time, Milan. App. Right Ascension. App. Declination S. 1805, Nov. 5d14/N 14m 4s 780 20' 37".8 27~ 16' 56".7 Dec. 6 11 51 27 73 8 48.8 32 52 44.3 1806, Jan. 15 8 50 36 67 14 11.1 28 38 8.1 We will. here take the equator as the fundamental plane instead of the ecliptic, and we will make the computation as if the orbit were still wholly unknown. In the first place we take from the tables of the sun the following data for the given dates: — Longitude of the Sun Distance from Latitude of from mean Equinox. the Earth. the Sun. Nov. 5 2230 14' 7".61 0.9804311 +0".59 Dec. 6 254 28 42.59 0.9846753 0.12 Jan. 15 295 5 47.62 0.9838153 -0.19 We reduce the longitudes of the sun, the precessions A- 7".59, +- 3".36, -2".11, being added, to the beginning of the year 1806, and thence we afterwards derive the right ascensions and declinations, using the mean obliquity 230 27' 53".53 and taking account of the latitudes. In this way we find SECT. 1.] THREE COMPLETE OBSERVATIONS. 217 Right ascension of the Sun. Deci. of the Sun S. Nov. 5 220~ 46' 44.65 150 49'431'.94 Dec. 6 253 9 23.26 22 33 39.45 Jan. 15 297 2 51.11 21 8 12.98 These places are referred to the centre of the earth, and are, therefore, to be reduced by applying the parallax to the place of observation, since the places of the planet cannot be freed from parallax. The right ascensions of the zenith to be used in this calculation agree with the right ascensions of the planet (because the observations have been made in the meridian), and the declination will be throughout the altitude of the pole, 45~ 28'. Hence are derived the following numbers: - Right asc. of the Earth. Decl. of the Earth N. Log of dist. from the Sun. Nov. 5 40~ 46' 48".51 15~ 49' 48".59 9.9958575 Dec. 6 73 9 23.26 22 33 42.83 9.9933099 Jan. 15 117 2 46.09 21 8 17.29 9.9929259 The observed places of Pallas are to be freed from nutation and the aberration of the fixed stars, and afterwards to be reduced, by applying the precession, to the beginning of the year 1806. On these accounts it will be necessary to apply the following corrections to the observed places: — Observation I. Observation II. Observation III. Right asc. Declination. Right asc. Declination. Right asc. Declination. Nutation - 12".86 - 3".08 - 13f.68 - 3".42 - 13f.06 - 3".75 Aberration - 18.13 -- 9.89 — 21.51 - 1.63 — 15.60 + 9.76 Precession + 5.43 + 0.62 + 2.55 + 0.39 -- 1.51 - 0.33 Sum -25.56 -12.35 -32.64 - 4.66 — 30.17 + 5.68 28 218 DETERMINATION OF AN ORBIT FROM [BooeK Il Hence we have the following places of Pallas, for the basis of the compu. tation;Mean Time, Paris. Right Ascension. Declination. Nov. 5.574074 78~ 20', 12".24 -27~ 17' 9".05 36.475035 73 8 16.16 -32 52 48.96 76.349444 67 13 40.93 -28 38 2.42 157. Now in the first place we will determine the positions of the great circles drawn from the heliocentric places of the earth to the geocentric places of the planet. We take the symbols W,'2, Wf, for the intersections of these circles with the equator, or, if you please, for their ascending nodes, and we denote the distances of the points B, B'N B' from the former points by A, A', zi. In the greater part of the work it will be necessary to substitute the symbols X, I', I", for A, A', A", and also z, z, X" for 4, 4', 4"; but the careful reader will readily understand when it is necessary to retain A, A' A"' 4, 6', 6, even if we fail to advise him. The calculation being made, we find Right ascensions of the points Wf,', a.". 233~ 54' 57".10 253~ 8' 57".01 276~ 40' 25".87 y, y',7y"..... 1 51 17 15.74 90 1 3.19 131 59 58.03 z,z',z".... 1 215 58 49.27 212 52 48.96 220 9 12.96,84 8, "....... 56 26 34.19 55 26 31.79 69 10 57.84 7'D, D' WD"... 23 54 52.13 30 18 3.25 29 8 43.32 "'D, ~"D', T'D"..T 33 3 26.35 31 59 21.14 22 20 6.91 47 1 54.69 89 34 57.17 42 33 41.17 logarithms of the sines 9.8643525 9.99998.85 9.8301910 log sin e..... I 9.8478971 log Cos i E... 9.8510614 SECT. 1.] THREE COMPLETE OBSERVATIONS. 219 The right ascension of the point 2f' is used in the calculation of article 138 instead of 1'. In this manner are found log T sin I..... 8.4868236 n log Tcost..... 9.2848162 n Hence t = 189~ 2'48".83, log T= 9.2902527; moreover, t + y' = 2790 3' 52".02, logS....... 9.0110566 n log Tsin (t-+').. 9.2847950n whence A'- a = 208~ 1' 55".647 and o r 4~ 50' 53".32. In the formulas of article 140 sin Y, sin Y', sin Y" must be retained instead of a, b and b, and also in the formulas of article 142. For these calculations we have t("D' z- " = 171~ 50' 8".18 log sin 9.1523306 log cos 9.9955759 n ifDI' — = 174 19 13.98 " " 8.9954722 " " 9.9978629n C"D -A" _ 172 54 13.39 " 9.0917972 WI'D -A'+ a = 175 52 56.49 CC 8.8561520 W D"-A = 173 9 54.05 C 9.0755844 D'-,d4 +- a = 174 18 11.27 " G 8.9967978 Hence we deduce log x = 0.9211850, log X = 0.0812057 n log z" = 0.8112762, log a" - 0.0319691 n log a = 0.1099088, a + 1.2879790 log b - 0.1810404, log - 0.0711314, whence we have log 6 = 0.1810402. We shall adopt log b -0.1810403 the mean between these two nearly equal values. Lastly we have log e 1.0450295 d -- -+ 0.4489906 log e = 9.2102894 with which the preliminary calculations are completed. 220 DETERMINATION OF AN ORBIT FROM [BOOK II. The interval of time between the second and tlhird observations is 39.874409 days, between the first and second 30.900961: hence we have log 6 - 9.8362757, log 6"- 9.7255533. We put, therefore, for the first Iypothesis, x -log Pz 9.8892776 y log Q- 9.5618290 The chief results of the calculation are as follows: - ot + o _ 20~ 8' 46".72 log Q sin w 0.0282028 Thence the true value of z is 21~ 11'24".30, and of log r', 0.3509379. The three remaining values of z satisfying equation IV., article 141, are, in this instance, z ~ 63~ 41' 12" -- 101 12 58 - 199 24 7 the first of wlhich is to be regarded as an approximation to the orbit of the earth, the deviation of which, however, is here much greater than in the preceding example, on account of the too great interval of time. The following numbers result from the subsequent calculation:-,.... 195 12' 2"'.48 "...e e.e. 196 57 50.78 logr..... 0.3647022 log r".... ~ 0.3355758 [(u"t-+t-)... 266 47 50.47 (u' —u)... — 43 39 5.33 2f'... 22 32 40.86 2f..... 13 5 41.17 2f"..... 9 27 0.05 We shall distribute the difference between 2f' ancl 2 f + 2 fP which in this case is 0".36, between 2f and 2f" in such a manner as to make 2f 130 5' 40".96, and 2f" 9~ 26' 59".90. The times are now to be corrected for aberration, for which purpose we are to SECT. 1.] THREE COMPLETE OBSERVATIONS. 221 put in the formulas of article 145, AD' -- _ -- D'- + d - - i'Az)' -- S" - A "D' D-" +,,_, i;. We have, therefore, logr.. 0.36470 logr'.. 0.35094 logr"... 0.33557 log sin (AD'' —) 9.76462 log sin ('- z) 9.75038 logsin (A"D' — ") 9.84220 C.logsin.. 0.07918 C. logsin'. 0.08431 C. logsinY".. 0.02932 log const... 7.75633 log const.. 7.75633 log const... 7.75633 7.96483 7.94196 7.96342 reduction of 0.009222 0.008749 0.009192 the time Hence follow, Observations. Corrected times. Intervals. Logarithms. I. Nov. 5.564852 IIe 36.466286 30d901434 1.4899785 39.873966 1.6006894 IHI. 76.340252 whence are derived the corrected logarithms of the quantities 8, 8" respectively 9.8362708 and 9.7255599. Beginning, then, the calculation of the elements from r/, r" 2f, 0, we get log X = 0.0031921, just as from r, r', 2f", 0" we obtain log " = 0.0017300. Hence is obtained log P'= 9.8907512 log Q'-= 9.5712864, and, therefore, X== + 0.0014736 Y= + 0.0094574 The chief results of the second hypolhesis, in which we put =- log P = 9.8907512 y=log Q = 9.5712864 are the following:to + a.... 200 8' 0".87...., 1950 16' 59".90 log Qc sin co.. 0.0373071 ".e.o e. 196 52 40.63 z... a o. 21 12 6.09 logr... 0.3630642 log r'..... 0.3507110 log.... 0.3369708 222 DETERMINATION OF AN ORBIT FROM [BOOK II. i (u" + u) 2670 6'10".75 2f'..... 220 32' 8".69 2 (t-u).. -43 39 4.00 2f..... 13 154.65 2f"... 9 30 14.38 The difference 0."34, between 2f' and 2f-+ 2f" is to be so distributed, as to make 2f- 130 1' 54".45, 2f" - 9~ 30' 14".24. If it is thought worth while to recompute here the corrections of the times, there will be found for the first observation, 0.009169, for the second, 0.008742, for the third, 0.009236, and thus the corrected times, November 5.564905, November 36.466293, November 76.340280. Hence we have log...... 9.8362703 log"...... 0.0017413 log6"6 @0. a 9.7255594 log P'..... 9.8907268 logiy.... 0.0031790 log Q'..... 9.5710593 Accordingly, the results from the second hypothesis are X _ — 0.0000244 Y- - 0.0002271. Finally, in the third h1ypoltesis, in which we put x- log P - 9.8907268 y- log Q 9.5710593 the chief results of the calculation are as follows:o +..+.. 20~ 8' 1".62 logr".... 0.3369536 log QCsin e.. 0.0370857 i (u"+u)... 267 5 53.09 z...... 21 12 4.60 (t" -u)...-43 39 4.19 logr'.... 0.3507191 2f'.... 22 32 7.67......195 16 54.08 2f.... 13 1 57.42'*...... 196 52 44.45 2f".... 9 30 10.63 logr..... 0.3630960 The difference 0".38 will be here distributed in such a manner as to make 2f = 13~ 1' 57".20, 2f" - 9 30' 10".47. * This somewhat increased difference, nearly equal in all the hypotheses, has arisen chiefly from this, that a had been got too little by almost two hundredths of a second, and the logarithm of b too great by several units. SECT. 1.] THREE COMPLETE OBSERVATIONS. 223 Since the differences of all these numbers from those which the second hypothesis furnished are very small, it may be safely concluded that the third hypothesis requires no further correction, and, therefore, that a new hypothesis would be superfluous. Wherefore, it will now be proper to proceed to the calculation of the elements from 2f', g', r, r": and since the processes comprised in this calculation have been most fully explained above, it will be sufficient to add here the resulting elements for the benefit of those who may wish to perform the coniputation themselves: - Right ascension of the ascending node on the equator... 158~ 40' 38".93 Inclination of the orbit to the equator..... 11 42 49.13 Distance of the perihelion from the ascending node... 323 14 56.92 Mean anomaly for the epoch 1806....... 335 4 13.05 Mean daily (sidereal) motion.......... 770".2662 Angle of eccentricity,... e e o. 14 9 3.91 Logarithm of the semi-axis major........ 0.4422438 158. The two preceding examples have not yet furnished occasion for using the method of article 120: for the successive hypotheses converged so rapidly that we might have stopped at the second, andl the third scarcely differed by a sensible amount from the truth. We shall always enjoy this advantage, and be able to do without the fourth hypothesis, when the heliocentric motion is not great and the three radii vectores are not too unequal, particularly if in addition to this, the intervals of the times differ from each other but little. But the further the conditions of the problem depart from these, the more will the first assumed values of P and Q differ from the correct ones, and the less rapidly will the subsequent values converge to the truth. In such a case the first three hypotheses are to be completed in the manner shown in the two preceding examples, (with this difference only, that the elements themselves are not to be computed in the third hypothesis, but, exactly as in the first and seconcldhypotheses, the quantities a), ii", P' Q', X, Y); butL then, the last values of P', Q' are no longer to be taken as 224 DETERMINATION OF AN ORBIT FROM [BOOK II. the new values of the quantities P, Q in the new hypothesis, but these are to be derived from the combination of the first three hypotheses, agreeably to the method of article 120. It will then very rarely be requisite to proceed to the fifth hypothesis, according to the precepts of article 121. We will now explain these calculations further by an example, from which it will appear how far our method extends. 159. For the dtird example we select the following observations of Ceres, the first of which has been made by OLBERS, at Brernen, the second by HARDING, at Got tingen, and the third by BESSEL, at Lilienthal. Mean time of place of observation. Right Ascension. North declination. 1805, Sept. 513h 8 54' 950 59' 25" 220 21' 25" 1806, Jan. 17 10 58 51 101 18 40.6 30 21 22.3 1806, MSay 23 10 23 53 121 56 7 28 2 45 As the methods by which the parallax and aberration are taken account of, when the distances from the earth are regarded as wholly unknown, have already been sufficiently explained in the two preceding examples, we shall dispense with this unnecessary increase of labor in this third example, and with that object will take the approximate distances from voN ZACl's oneatdliche Correspondcen, Vol. XI., p. 284, in order to free the observations from the effects of parallax and aberration. The following table shows these distances, together with the reductions derived from them:Distance of Ceres from the earth.. 2.899 1.638 2.964 Time in which the light reaches the earth 23m49s 13m28s 245n21s Reduced time of observation... 12h 45m 5s 10" 45m 235 9h 59m 32s Sidereal time in degrees... 3550 55' 970 59' 2100 41' Parallax in right ascension..... + 1".90 + 0".22 - 1".97 Parallax in declination.. 0 -- 2.08 - 1.90 - 2.04 SECT. 1.] THREE COMPLETE OBSERVATIONS. 225 Accordingly: the data of the problem, after being freed from parallax and aberration, and after the times have been reduced to the meridian of Paris, are as follows: - Times of the observations. Right Ascension. Declination. 1805, Sept. 5, 12h 19m 14s 950 59' 23".10 22~ 21' 27'.08 1806, Jan. 17, 10 15 2 101 18 40.38 30 21 24.20 1806, May 23, 9 33 18 121 56 8.97 28 2 47.04 From these right ascensions and declinations have been deduced the longitudes and latitudes, using for the obliquity of the ecliptic 230 27' 55".90, 230 27' 5411".59, 230 27' 53".27; the longitudes have been afterwards freed from nutation, which was for the respective times + 17".31, + 17".88, - 18".00, and next reduced to the beginning of the year 1806, by applying the precession + 15".98, - 2".39, - 19".68. Lastly, the places of the sun for the reduced times have been taken from the tables, in which the nutation has been omitted in the longitudes, but the precession has been added in the same way as to the longitudes of Ceres. The latitude of the sun has been wholly neglected. In this manner have resulted the following numbers to be used in the calculation:Times, 1805, September 5.51336 139.42711 265.39813 a, a, a... X 950 32' 18".56 990 49' 5".87 1180 5 28".85,t /', ~e"...... o -0 59 34.06 + 7 16 36.80 + 7 38 49.39 1, 1', 1"...... 342 54 56.00 117 12 43.25 241 58 50.71 log R, log R', log R" 0.0031514 9.9929861 0.0056974 The preliminary computations explained in articles 136-140 furnish the following:Y, 7', 7"/. ~ 1 3580 55' 28".09 1560 52' 11"..49 1700 48' 44".79 Y, o', (Y".... 112 37 9.66 18 48 39.81 123 32 52.13 A'D, AfD', A.D". 15 32 41.40 252 42 19.14 136 2 22.38 A"D, A"D'l A'D".. 138 45 4.60 6 26 41.10 358 5 57.00,7 e',...l 29 1 8 8.21 170 32 59.08 156 6 25.25 29 226 DETERMINATION OF AN ORBIT FROM [BOOK II. a = 8~ 52' 4".05 log e - 0.8568244 log cc 0.1840193 n, a = —1.5276340 log x - 0.1611012 log b - 0.0040987 logzx" 9.9770819 n log c - 2.0066735 log X - 9.9164090 n d - 117.50873 log X" = 9.7320127? The interval of time between the first and second observations is 133.91375 days, between the second and third, 125.97102: hence log 8 - 0.3358520, log 6"- 0.3624066, log -= 0.0265546, log t6"-= 0.6982586. We now exhibit in the following table the principal results of the first three hypotheses: — I. II, III. logP = X 0.0265546 0.0256968 0.0256275 log Q = y 0.6982586 0.7390190 0.7481055 o+ - 7 15' 13".523 7 14' 47".139 70 14' 45"e.071 log QC sin o 1.1546650 n 1.1973925 n 1.2066327 n 7 3 59.018 7 2 32.870 7 2 16.900 log r' 0.4114726 0.4129371 0.4132107 160 10 46.74 160 20 7.82 160 22 9.42 ~.4" 262 6 1.03 262 12 18.26 262 14 19.49 log r 0.4323934 0.4291773 0.4284841 log r" 0.4094712 0.4071975 0.4064697 21 (2u"-qtu) 262 55 23.22 262 57 6.83 262 57 31.17 (uc" —_,: 273 28 50.95 273 29 15.06 273 29 19.56 2f' 62 34 28.40 62 49 56.50 62 53 57.06 2f 31 8 30.03 31 15 59.09 31 18 13.83 2f" 31 25 58.43 31 33 57.32 31 35 43.32 log 17 0.0202496 0.0203158 0.0203494 log' 0.0211074 0.0212429 0.0212751 log P' 0.0256968 0.0256275 0.0256289 log Q' 0.7390190 0.7481055 0.7502337 X 0.0008578 - 0.0000693 -0.0000014 Y'I — + 000407604 +- 0.0090865 +- 0.0021282 SECT..]'THREE COMPLETE OBSERVATIONS. 227 If we designate the three values of X by A, A"; the three values of Y by B, B', B"; the quotients arising from the division of the quantities A'B" —A"B', A"B -AB"l AB'- A'B, by. the sum of these quantities, by ic, 1, k', respectively, so that we have Ik+ c'+ "- 1; and, finally, the values of log P' and log Q' in the third hypothesis, by 3 and N, (which would become new values of x and y if it should be expedient to derive the fourth hypothesis from the third, as the third had been derived from the second): it is easily ascertained from the formulas of article 120, that the corrected value of x is 3- Ic (A' +- A") -'A", and the corrected value of y, N- kc (B' + B") -'B". The calculation being made, the former becomes 0.0256331, the latter, 0.7509143. Upon these corrected values we construct the fourth Ipotlhesis, the chief results of which are the following:t -C- o +.... 7~ 14'45".247 log. r"... 0.4062033 log Qcsinw. 1.2094284n (zu' +u)... 2620 57138".78 z...... 7 212.736 (t- -u)... 273 29 20.73 logr'.. e. 0.4132817 2f'..... 62 55 16.64;.......160 22 45.38 2f... 31 19 1.49 262 15 3.90 2f".. 31 36 15.20 logr..... 0.4282792 The difference between 2f' and 2 f + 2f" proves to be 0".05, which we shall distribute in such a manner as to make 2f 31~ 19' 1".47, 2f"- 31 36' 15".17. If now the elements are determined from the two extreme places, the following values result: - True anomaly for the first place....... e 289~ 7 391".75 True anomaly for the third place........ 352 2 56.39 Mean anomaly for the first place........ 297 41 35.65 Mean anomaly for the third place........ 353 15 22.49 Mean daily sidereal motion... 769".6755 Mean anomaly for the beginning of the year 1806. 322 35 52.51 Angle of eccentricity, p... 4 37 57.78 Logarithm of the semi-axis major.... 0.4424661 Dy computing from these elements the heliocentric place for the time of the 228 DETERMINATION OF AN ORBIT FROM [BooK II. middle observation, the mean anomaly is found to be 3260 19'25".72, the logarithm of the radius vector, 0.4132825, the true anomaly, 320~ 43' 54".87: this last should differ from the true anomaly for the first place by the quantity 2f", or friom the true anomaly for the third place by the quantity 2f, and shoulcld therefore, be 320~ 43'54".92, as also the logarithm of the radius vector, 0.4132817: the difference 0".05 in the true anomaly, and of eight units in the logarithm, is to be considered as of no consequence. If the fourth hypothesis should be conducted to the end in the same way as the three preceding, we would have X — 0 Y 0.0000168, whence the following corrected values of x and y would be obtained, x log P- 0.0256331, (the same as'in the fourth hypothesis,) y log Q- 0.7508917. If the fifth hypothesis should be constructed on these values, the solution would reach the utmost precision the tables allow: but the resulting elements would not differ sensibly from those which the fourth hypothesis has furnished. Nothing remains now, to obtain the complete elements, except that the position of the plane of the orbit should be computed. By the precepts of article 149 we have From the first place. From the third place. f9 ee*e.........Oi 3540 9 444".22 "l * e 570 5Y 0".91 h... 261 56 6.94 h".. 161 0 1.61 ~i....* *o10 37 33.02 10 37 33.00 g........ ee**80 58 49.06 80 58 49.10 Distance of the perihelion 65 2 4.47 65 2 4.52 from the ascending node Longitude of the perihelion 146 0 53.53 146 0 53.62 The mean being taken, we shall put i = 100 37' 33".01, Q - 80~ 58' 49".08, the longitude of the perihelion - 1460 0' 53".57. Lastly, the mean longitude for the beginning of the year 1806 will be 108~ 36' 46".08. SECT. 1.] THREE COMPLETE OBSERVATIONS. 229 160. In the exposition of the method to which the preceding investigations have been devoted, we have come upon certain special cases to which it did not apply, at least not in the form in which it has been exhibited by us. We have seen that this defect occurs firs, when any one of the three geocentric places coincides either with -the corresponding heliocentric place of the earth, or with the opposite point (the last case can evidently only happen when the heavenly body passes between the sun and earth): second, when the first geocentric place of the heavenly bodly coincides with the third; 1/ird, when all three of the geocentric places together with the second heliocentric place of the earth are situated in the same great circle. In the first case the position of one of the great circles AB, A'B' A"B" and in the second and third the place of the point B*, will remain indeterminate. In these cases, therefore, the methods before explained, by means of which we have shown, how to determine the heliocentric from the geocentric places, if the quantities P, Q, are regarded as known, lose their efficacy: but an essential distinction is here to be noted, which is, that in the first case the defect will be attributable to the method alone, but in the second and third cases to the nature of the problem; in the first case, accordingly, that determination can undoubtedly be effectecl if the method is suitably altered, but in the second and third it will be absolutely impossible, and the heliocentric places will remain indeterminate. It will not be uninteresting to develop these relations in a few words: but it would be out of place to go through all that belongs to this subject, the more so, because in all these special cases the exact determination of the orbit is impossible where it would be greatly affected by the smallest errors of observation. The same defect will also exist when the observations resemble, not exactly indeed, but nearly, any one of these cases; for which reason, in selecting observations this is to be recollected, and properly guarded against, that no place be chosen where the heavenly body is at the same time in the vicinity of the node and of opposition or conjunction, nor such observations as where the heavenly body has nearly returned in the last to the geocentric place of the first observation, nor, finally, such 230 I)ETERMINATION OF AN ORBIT FROM [BOOK II. as where the great circle drawn from the middle heliocentric place of the earth to the middle geocentric place of the heavenly body makes a very acute angle with the direction of the geocentric motion, and nearly passes through the first and third places. 161. We will make three subdivisions of the frsl case. I. If the point B coincides with A or with the opposite point, Y will be equal to zero, or to 1800; y,,' e" and the points D', D", will be indeterminate; on the other hand, y, y7" e and the points D, B*, will be determinate; the point C will necessarily coincide with A. By a course of reasoning similar to that pursued in article 140, the following equation will be easily obtained: — 0 _, sin (z- a) R' sin c'' sin (A"D-'") gN sin z t' sin o" sin ( /A'D- }- + 6) It will be proper, therefore, to apply in this place all which has been explained in articles 141, 142, if, only, we put a- 0, and b is determined by equation 12, article 140, and the quantities z, r', -, n will be computed in the same manner as before. Now as soon as z and the position of the point C' have become known, it will be possible to assign the position of the great circle CC', its intersection with the great circle AB1B, that is the point' C, and hence the arcs CC', CC", C'C", or 2f", 2f' 2g. Lastly, from these will be had ntr'sin 2f r,, n'r'sin 2 f n sin 2f' I r n sin 2-f'. II. Every thing we have just said can be applied to that case in which B" coincides with A" or with the opposite point, if, only, all that refers to the first place is exchanged with what relates to the third place. III. But it is necessary to treat a little differently the case in which B' coincides with A' or with the opposite point. There the point C' will coincide with A'; y', e, e and the points D, D" B*, will be indeterminate: on the other hand, the intersection of the great circle BB" with the ecliptic,t the longitude of which t More generally, with the great circle AA"': but for the sake of brevity we are now considering that ease only where the ecliptic is taken as the fundamental plane. SECT. 1.] THREE COMPLETE OBSERVATIONS. 231 may be put equal to i'+ m-, may be determined. By reasonings analogous to those which have been developed in article 140, will be obtained the equation R sin a sin (A'D'-8") sin (+' R" sin A" sin (AD' -) - Rr sin (-r' 2 )-. Let us designate the coefficient of n, which agrees with a, article 140, by the same symbol a, and the coefficient of z'r' by (3: a may be here also determined by the formula Rsin (' + t- l) We have, therefore, 0 an +'r''+ n", which equation combined with these, n n! pl 7 Q 2 Q+2 - 8 produces 3(P+1) r4+r'3+ t Q-O0 ~' (p+ l) r,~ dr,~ + ~ Q = 0, Pda whence we shall be able to get r/, unless, indeed, we should have 0, in which case nothing else would follow from it except P - - a. Further, although we might not have (3 - 0 (when we should have the third case to be considered in the following article) still ( will always be a very small quantity, and therefore P will necessarily differ but little from -a: hence it is evident that the determination of the coefficient (P- I) P+a is very uncertain, and that r', therefore, is not determinable with any accuracy. Moreover, we shall have n'r' P-+ a n'r' PJ a n A 7 =- -~P after this the following equations will be easily developed in the same manner as in article 143, n'r' sin /snI rs~~ ~~~in (7: 7sn{ —', 232 DETERMINATION OF AN ORBIT FROM [BooK II. r" sing" ='- Vsin sin (l' — 1) r sin ( A - AiD') r"P s sin ("- A"D'), from the combination of which with equations VIII. and IX. of article 143, the quantities r, 4, r", i'" can be determined. The remaining processes of the calculation will agree with those previously described. 162. In the second case, where B" coincides with B, D' will also coincide with them or with the opposite point. Accordingly, we shall have AD' - a and A"D'- " either equal to 0 or 1800: whence, from the equations of article 143, we obtain n' r' sin e'R sin a n - sin E sin (z + A'D - d') n r' sin' Af't sin A" n" _- &sin e" sin (z + A'Z" - l')' R sin d sin e" sin (z + A'D" - Y') = PR" sin Y' sin e sin (z + AD —'). Hence it is evident that z is determinable by P alone, independently of Q, (unless it should happen that A'D" - A'D, or _ A'D + 180~, when we should have the third case): z being found, r' will also be known, and hence, by means of the values of the quantities nWr na n d n9 also and - and, lastly, from this also Q= 2 (,+ r 3. Evidently, therefore, P and Q cannot be considered as data independent of each other, but they will either supply a single datum only, or inconsistent data. The positions of the points C, C" will in this case remain arbitrary, if they are only taken in the same great circle as C'. In the third case, where A', B, Bf' B", lie in the same great circle, D and D" will coincide with the points B", B, respectively, or with the opposite points: hence is SECT. l.] THREE COMPLETE OBSERVATIONS. 233 obtained from the combination of equations VII., VIII., IX., article 143, R sin 6 sin e" X sil (If-I) Rf" sin " sin n E R" sin ( - I') ~ In this case, therefore, the value of P is had from the data of the problem, and, therefore, the positions of the points C, C', C", will remain indeterminate. 163. The method which we have fully explained from article 136 forwards, is principally suited to the first determination of a wholly unknown orbit: still it is employed with equally great success, where the object is the correction of an orbit already approximately known by means of three observations however distant from each other. But in such a case it will be convenient to change some things. When, for example, the observations embrace a very great heliocentric motion, it will no longer be admissible to consider Of, and 0d" as approximate values of the quantities P, Q: but much more exact values will be obtained from the very nearly known elements. Accordingly, the heliocentric places in orbit for the three times of observation will be computed roughly by means of these elements, whence, denoting the true anomalies by v, v', v", the radii vectores by r', r", the semi-parameter by p, the following approximate values will result -- p sin (v — _ v) 4r'4 sin I (v'-v) sin - (v"-v') 7r'sin (vr vl) ~ Q p cos - (V' — V) With these, therefore, the first hypothesis will be constructed, and with them, a little changed at pleasure, the second and third: it would be of no advantage to adopt P' and Q' for the new values since we are no longer at liberty to suppose that these values come out more exact. For this reason all three of the hypotheses can be most conveniently despatched at the same time: the fourth will then be formed according to the precepts of article 120. Finally, we shall not object, if any person thinks that some one of the ten methods explained in articles 124-129 is, if not more, at least almost equally expeditious, and prefers to use it. 30 SECOND SECTION. DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, OF WHICH TWO ONLY ARE COMPLETE. 164. WE have already, in the beginning of the second book (article 115), stated that the use of the problem treated at length in the preceding section is limited to those orbits of which the inclination is neither nothing, nor very small, and that the determination of orbits slightly inclined must necessarily be based on four observations. But four complete observations, since they are equivalent to eight equations, and the number of the unknown quantities amounts only to six, would render the problem more than determinate: on which account it will be necessary to set aside from two observations the latitudes (or declinations), that the remaining data may be exactly satisfied. Thus a problem arises to which this section will be devoted: but the solution we shall here give will extend not only to orbits slightly inclined, but can be applied also with equal success to orbits, of any inclination however great. Here also, as in the problem of the preceding section, it is necessary to separate the case, in which the approximate dimensions of the orbit are already knownn from the first determination of a wholly unknown orbit: we will begin with the former. 165. The simplest method of adjusting a known orbit to satisfy four observations appears to be this. Let x, y, be the approximate distances of the heavenly body from the earth in two complete observations: by means of these the corresponding heliocentric places may be computed, and hence the elements; after this, (234) SECT. 2.] DETERMINATION OF AN ORBIT. 235 from these elements the geocentric longitudes or right ascensions for the two remaining observations may be computed. If these happen to agree with the observations, the elements will require no further correction: but if not, the differences X, Y, will be noted, and the same calculation will be repeated twice, the values of x, y being a little changed. Thus will be obtained three systems of values of the quantities, y, and of the differences X, Y, whence, according to the precepts of article 120, will be obtained the corrected values of the quantities z, y, to which will correspond the values X - 0 Y - 0. From a similar calculation based on this fourth system elements will be found, by which all four observations will be correctly represented. If it is in your power to choose, it will be best to retain those observations complete from which the situation of the orbit can be determined with the greatest precision, therefore the two extreme observations when they embrace a heliocentric motion of 90~ or less. But if they do not possess equal accuracy, you will set aside the latitudes or declinations of those you may suspect to be the less accurate. 166. Such places will necessarily be used for the first determination of an entirely unknown orbit from four observations, as include a heliocentric motion not too great; for otherwise we should be without the aids for forming conveniently the first approximation. The method which we shall give directly admits of such extensive application, that observations comprehending a heliocentric motion of 30~ or 40~ may be used without hesitation, provided, only, the distances from the sun are not too unequal: where there is a choice, it will be best to take the intervals of the times between the first and second, the second and third, the third and fourth but little removed from equality. But it will not be necessary to be very particular in regard to this, as the annexed example will show, in which the intervals of the times are 48, 55, and 59 days, and the heliocentric motion more than 50~. Moreover, our solution requires that the second and third observations be complete, and, therefore, the latitudes or declinations in the extreme observations 236 DETERMINATION OF AN OBIT FROM FOUR OBSERVATIONS, [BOOK II. are neglected. We have, indeed, shown above that, for the sake of accuracy, it is generally better that the elements be adapted to two extreme complete observations, and to the longitudes or right ascensions of the intermediate ones; nevertheless, we shall not regret having lost this advantage in the first detenrmination of the orbit, because the most rapid approximation is by far the most important, and the loss, which affects chiefly the longitude of the node and the inclination of the orbit, and hardly, in a sensible degree, the other elements, can afterwards easily be remedied. We will, for the sake of brevity, so arrange the explanation of the method, as to refer all the places to the ecliptic, and, therefore, we will suppose four longitudes and two latitudes to be given: but yet, as we take into account the latitude of the earth in our formulas, they can easily be transferred to the case in which the equator is taken as the fundamental plane, provided that right ascensions and declinations are substituted in the place of longitudes and latitudes. Finally, all that we have stated in the preceding section with respect to nutat tion, precession, and parallax, and also aberration, applies as well here: unless, therefore, the approximate distances from the earth are otherwise known, so that method I., article 118, can be employed, the observed places will in the beginning be freed from the aberration of the fixed stars only, and the times will be corrected as soon as the approximate determination of the distances is obtained in the course of the calculation, as will appear more clearly in the sequel. 167. We preface the explanation of the solution with a list of the principal symbols. We will make t, 1', f1, t'1, the times of the four observations, a, a ac', a"', the geocentric longitudes of the heavenly body,, (3', /(3", /"',/ their latitudes, r, r', r", r% the distances from the sun, ( /; (/ a"/, the distances from the earth, 1, 1', 1", 1"', the heliocentric longitudes of the earth, SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 237 B, B', B", B"' the heliocentric latitudes of the earth, B7 R', R R"' the distances of the earth from the sun, (201), (n 12), (n23), (n 02), (n 13), the duplicate areas of the triangles which are contained between the sun and the first and second places of the heavenly body, the second and third, the third and fourth, the first and third, the second and fourth respectively; (aq 01), (a 12), (a 23) the quotients arising from the division of the areas 1 (n 01), 1 (n 12), 1 (n 23), by the areas of the corresponiding sectors; p, (n 12) pi, ( 12) (n 0)' (n 23), Q - ((n 01) + (n 12) )r ((12) + (n 23) )/3 (n 02) (n 13) v, v', v", v'", the longitudes of the heavenly body in orbit reckoned from an arbitrary point. Lastly, for the second and third observations, we will denote the heliocentric places of the earth in the celestial sphere by A, A", the geocentric places of the heavenly body by B' B", and its heliocentric places by C', C'". These things being understood, the first step will consist, exactly as in the problem of the preceding section (article 136), in the determination of the positions of the great circles A'C'B', A"''CB", the inclinations of which to the ecliptic we denote by y', y": the determination of the arcs A'B' —', A"TB":= A" will be connected at the same time with this calculation. Hence we shall evidently have A/ -- - (e'q' + 2 Q'R' cos d' + R'R') /'r= ~ ("Q+ " co 2 + cos 6" R+"R""), or by putting I' + R' cos' - x', "- + " cos A" = x, B' sin Y' = a', R" sin A" = c", V ("x" + a''). 168. By combining equations 1 and 2, article 112, the following equations in syms bols of the present discussion are produced:0 (n 12) cos B sin (1- a) - (n 02) (' cos B sin (-a'- a) + B' cos B'sin (1'- a)) -- (ns 01) (W" cos (I" sin (a" - a.) — + R" cos B" sin (I"- a)), 238 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BOOK 1]. 0 = (n 23) (Q' cos P' sin (a"'- a') + -' cos B' sin (a"' —I')) ( 13) (Q"cos ""sin (a"'- a") + R" cos B" sin (a"' — I)) + (n 12) R"' cos B"' sin (a" -- I"). These equations, by putting R' cos B' sin ('- a) RI cos 6' COS' Sin (a- a).R cos B" sin ("-'- "1) - RB COS 6"t coS j/ Sin (a"' -- c') b' cos Bo sin ("' -') B'cos 6= = cos,B sin (a"' - a') R" cos B' sin (1"' - a) BCOS cs X' cos'" sin ('" - ) R cos B sin (I - a) cos f3 sin (a"-a) -?"' cos B"' sin (a"l -') ) cos 3' sin (a"' -' a') - cos sin (a' — a) - cos/3' sin (a'" —- a) - and all the reductions being properly made, are transformed into the following: —-' ( + P') (x + b') = x + U" + tp', 1+ (x'f' + d- a'a) fe (1 t. P) (_ -- "'"or, by putting besides, - r' - "'P"- c",' (1 +.P ) =d", ~..se~~,a/p1',,llp jU" (I + PI ) = 6r, into these, I.(d' d (x +') (x'x' J aa'') SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 239 II.,"t +-ct'+ d " (i'.~b) + + 3) With the aid of these two equations x' and x' can be determined from a, b' c', d, Q'. a", b", ct" d", Q". If, indeed, x' or x" should be eliminated from them, we should obtain an equation of a very high order: but still the values of the unknown quantities x', x", will be deduced quickly enough from these equations by indirect methods without any change of form. Generally approximate values of the unknown quantities result if at first, Q' and Q" are neglected; thus -- d 0,+ d" (,'+.c) + d'd'b' 1 -d'd", c' d' (b' c")+ d'd"b'l 1- d'dd" But as soon as the approximate value of either unknown quantity is obtained, values exactly satisfying the equations will be very easily found. Let, for example, d' be an approximate value of i' which being substituted in equation I., there'results x =/"; in the same manner from i" - " being substituted in equation II., we may have x'= X'; the same processes may be repeated by substituting for x' in I., another value d' +v', which may give a"' - "+ v"; this value being substituted in II., may give a! x'X' + Ni'. Thereupon the corrected value of x' will be'+ lr' -X —'' and the corrected value of i' a,, (a — x,) X, If it is thought worth while, the same processes will be repeated with -the corrected value of x' and another one slightly changed, until values of Xa a! satisfying the equations I., II. exactly, shall have been found. Besides, means will not be wanting even to the moderately versed analyst of abridging the calculation. In these operations the irrational quantities ('x' + ca'a'), (x"x"' + ca"a") are conveniently calculated by introducing the arcs oZv a!', of which the tangents are 240 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BOOK II. at aU respectively a, ~, whence come V (x' + da'a) = " — cosi/ (xax +N ha") — sin z - cos z' I These auxiliary arcs, which must be taken between 0~ and 1800, in order that r', r", may come out positive will, manifestly, be identical with the arcs C'B' C"B" whence it is evident that in this way not only r' and r", but also the situation of the points C', C", are known. This determination of the quantities x', x" requires a, a", b;/ b" c, d d, Q', Q" to be known, the first four of which quantities are, in fact, had from the data of the problem, but the four following depend on P, P". Now the quantities P', P", Q', Q", cannot yet be exactly determined; but yet, since iII p't — Auet' (Xl01) It-t (y12)' IV. ~p, t —t' (ij23) — t.,- e(l12)~ v q'-q Wk (t' —2 t) (t"- t') )r'r (r1 01) ( 12) cos - (v'- v) cos 2 (v"- v) cos I (to-v"I) ) vI. 2" —I k (#,? —#') (-_1), V.Q- k(to t- 7r"' (V 12) (a 23) cos (v"- /) cos 1 (v"- v') cos - (v"'"-v', ) the approximate values are immediately at hand, thel t - t-l t - t e t"t('' --, A (t'- k t) (t" —' ), Q" -- k (t" — t') (t' " -- t"), on which the first calculation will be based. 169. The calculation of the preceding article being completed, it will be necessary first to determine the are C' C". Which may be most conveniently done, if, as in article 137, the intersection D of the great circles A''B' A"C"B", and their mutual inclination e shall have been previously determined: after this,will be found from, C'D a-' +B'D7 and CD = --- RB"D, by the same formulas SECT. 2.] OF'WHICH TWO ONLY ARE COMPLETE. 241 whiCh we have given in article 144, not only C'C" v" - v', but also the angles (a', U",) at which the great circles A'B', A"B" cut the great circle C'C". After the arc v"- v' has been found, v'- v and r will be obtained from a combination of the equations r sin (v' v) Pt-v rsin (v' --- v"') 1 + P r'sin (v- v') and in the same manner, r"' and v"' - v" from a combination of these:i'" sin (v' - - v") r' sin (v"P-rv') r"" sin (v"' - v" + vt- vi') - ~p,1 " si ( v"- v') All the numbers found in this manner would be accurate if we could set out in the beginning from true values of P', Q', P", Q": and then the position of the plane of the orbit might be determined in the same manner as in article 149, either from A'C zi' and 7', or from A"C', u" and y"; and the dimensions of the orbit either from r r"f t, t", and v"- v' or, which is more exact, from r, r"' 4' 1t", v"' - v. But in the first calculation we will pass by all these things, and will direct our attention chiefly to obtaining the most approximate values of P', P" Q', Q". We shall reach this end, if by the method explained in 88 and the following articles, from r, r', v' —v, V'-t we obtain (I 01)' cc r, " v"/ — v'l - ift ( 12) vc Pr P"' —V"'_ Ptn~t~gt c (V 23). We shall substitute these quantities and also the values of r, r' r", rCos 1 ('-), etc., in formulas III.-VI., whence the values of P', Q', P", Q" will result much more exact than those on which the first hypothesis had been constructed. With these, accordingly, the second hypothesis will be formed, which, if it is carried to a conclusion exactly in the same manner as the first, will furnish much more exact values of P', Q', P"' Q", and thus lead to the third hypothesis. These processes will continue to be repeated, until the values of P'T Q' P", Q" seem to 31 242 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BOOK II. require no further correction, how to judge correctly of which, frequent practice will in time show. When the heliocentric motion is small, the first hypothesis generally supplies those -values with sufficient accuracy: but if the motion includes a greater arc, if, moreover, the intervals of tile times are very unequal, hypotheses several times repeated will be wanted; but in such a case the first hypotheses do not demand great preciseness of calculation. Finally, in the last lhypothesis, the elements themselves will be determined as we have just indicated. 170. It will be necessary in the first hypothesis to make use of the times 1, 1', ", 1"t' uncorrected, because the distances from the earth cannot yet be computed: as soon, however, as the approximate values of the quantities x', x" have become known, we shall be able to determine also those distances approximately. But yet, since the formulas for Q and c"' come out here a little more complicated, it will be well to put off the computation of the correction of the times until the values of the distances have become correct enough to render a repetition of the work unnecessary. On which account it will be expedient to base this operation on those values of the quantities x', x", to which the last hypothesis but one leads, so that the last hypothesis may start with corrected values of the times and of the quantities P', PI", Q', Q". The following are the formulas to be employed for this purpose — VII.?' =' — A' cos d', VIII.," — R —" cos A", IX. Q cos - -R cos B cos (a —l) + 1~ PI: (a' cos g' cos (a' - a) + R' cos B' cos (I'- a)), (Q"cos "cos (a'"-a) -t" — cos B" cos ("-)) X. esind-_ R sin B I —+ P' ('ine+Bf' sin B') p, (" sin I" + B" sin Bs") SECT. 2.] OF WIHCH TWO ONLY ARE COMPLETE. 243 XI. c"' cos 3 "'- - B"' cos B"' cos (a"' -1"') + I+. + P r/(/ cos (" cos (a' - a")+R cos B cos (a's - 1")) 77 ()' cos (3' cos (a" - a') -+ R' cos B' cos (a"' - 1')), XII. "'" sisi" — sin B"' + + ( sin "' sin B") P" (1I + ) p,, ( sin (3' + sin B'). The formulas IX.-XII. are derived without difficulty from eqluations 1, 2, 3, article 112, if, merely, the symbols there used are properly converted into those we here employ. The formulas will evidently come out much more simple if B, B', B" vanish. Not only (, but also (3 will follow from the combination of the formulas IX. and X., and, in the same lmanner, besides r', also (3" from XI. andl XII.: the values of these, compared with the observed latitudes (not entering into the calculation), if they have been given, will show with what degree of accuracy the extreme latitudes may be represented by elements adapted to the six remaining data. 171. A suitable example for the illustration of this investigation is taken from Vesta, which, of all the most recently discovered planets, has the least inclination to the ecliptic.* We select the following observations made at Bremen, Paris, Lilienthal, and Milan, by the illustrious astronomers OLBERS, BOUVARD, BESSEL, and ORIANI: — X Nevertheless this inclination is still great enough to admit of a sufficiently safe and accurate determination of the orbit based upon three observations: in fact the first elements which had been derived in this way from observations only 19 days distant from each other (see vON ZACH'S Jlonatliche Correspondenz, Vol. XV. p. 595), approach nearly to those which were here deduced from four observations, removed from each other 162 days. 244 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BOOK II. Mean time of place of observation. Right Ascension. Declination. 1807, March 30, 12h 33m 178 *183~ 52' 40'.8 110 54' 27".0 N. May 17, 8 16 5 178 36 42.3 11 39 46.8 July 11,10 30 19 189 49 7.7 3 9 10.1N. Sept. 8, 7 22 16 212 50 3.4 8 38 17.OS. We find for the same times from the tables of the sun, Longitude of the Sun Nutatio Distance from Latitude of Apparent obliquity from app. Equinox. n. the Earth. the Sun. of the Ecliptic. March 30 90 21' 59".5 + 16.8 0.9996448 +- 0".23 230 27' 50".82 May 17 55 56 20.0 + 16.2 1.0119789 — 0.63 49.83 July 11 108 34 53.3 + 17.3 1.0165795 - 0.46 49.19 Sept. 8 165 8 57.1 +-16.7 1.0067421 -+0.29 23 27 49.26 The observed places of the planets have, the apparent obliquity of the ecliptic being used, been converted into longitudes and latitudes, been freed from nutation and aberration of the fixed stars, and, lastly, reduced, the precession being subtracted, to the beginning of the year 1807; the fictitious places of the earth have then been derived from the places of the sun by the precepts of article 72 (in order to take account of the parallax), and the longitudes transferred to the same epoch by subtracting the nutation and precession; finally, the times have been counted from the beginning of the year and reduced to the meridian of Paris. In this manner have been obtained the following numbers:1, t t, 1t'", o. 89.505162 137.344502 192.419502 251.288102 a, a', a", a"'. 1780 43' 38".87 1740 1' 30".08 187 45' 42".23 213 34' 15".63 (d, (, (3", 3"'".. 12 27 6.16 10 8 7.80 6 47 25.51 4 20 21.63,t', l",1"'.. 189 21 33.71 235 56 0.63 288 35 20.32 345 9 18.69 log R, R', R",R"' 9.9997990 0.0051376 0.0071739 0.0030625 Hence we deduce 7' = 1680 32' 41".34, Y' 62 23 4".88 log a' 9.9526104, "= — ]173 5 15.68, d" 100 45 1.40, loga"- 9.9994839, SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 245 b' - 11.009449, x' = — 1.083306 log X - 0.0728800, logt' = 9.7139702n b"=- 2.082036, x"=+ 6.322006, log"'- 0.0798512n log/"=9.8387061 A'D - 37~ 17' 51"150, A"D 89~ 24' 11".84, e - 90 5' 5".48 B'D -- 25 5 13.38, B"D - - 11 20 49.56. These preliminary calculations completed, we enter upon the frsG hypolkesis. From the intervals of the times we obtain log k (t' - 1) - 9.9153666 log k (1" - -') 9.9765359 log k (1' -1") = 0.0054651, and hence the first approximate values log P' 0.06117, log (1 + P') = 0.33269, log Q' 9.59087 logP"= 9.97107, log (1 + P") = 0.28681, log Q"- 9.67997, hence, fuirther, c' - 7.68361, log d' = 0.04666 n c"= + 2.20771, log d"= 0.12552. With these values the following solution of equations I., II., is obtained, after a few trials: — X 2.04856, z' - 23~ 38' 17", log r'= 0.34951 x"- 1.95745, z"_ 27 2 0, logr"= 0.34194. From z', z" and e, we get C' C" v" -v' = 170 7' 5": hence v' - v, r, v" - v", /r" will be determinable by the following equations: - log r sin (v' - v) 9.74942, log r sin (v' - v + 17~ 7' 5") - 0.07500 logr"'sin(v"'-v")= 9.84729, logr"'sin(v"'-v"+ 17 7 5") 0.10733 whence we derive v'- v - 14~ 14' 32", log r = 0.35865 v"' —v"= 18 48 33, logr"'= 0.33887. Lastly, is found log (n 01) = 0.00426, log (n 12) =0.00599, log (n 23) 0.00711, and hence the corrected values of P', P", Q'', Q" 246 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [IBOOI I.. log P' - 0.05944, log Q' 9.60374, log P"_= 9.97219, log Q"= 9.69581, upon which the second hypoltesis will be constructed. The principal results of this are as follows: - c' =-7-.67820, log d' - 0.045736 n c" - 2.21061, logd"- 0.126054 x' = 2.03308, z' - 23~ 47' 54", log r' - 0.346747, x"-=1.94290, z" 27 12 25, logr'= 0.339373 C' C"=v"- v'= 170 8' 0" v' - = 14~ 21' 36", log r - 0.354687 v"'-v"- 18 50 43, logr"'= 0.334564 log (n 01) = 0.004359, log (n 12) - 0.006102, log (n 23) - 0.007280. Hence result newly corrected values of P', P", Q', Q", log P' = 0.059426, log Q'=- 9.604749 log P" = 9.972249, log Q" = 9.697564, from which, if we proceed to tlhe third hypothesis, the following numbers result:-' =- 7.67815, log c' = 0.045729 n'" =- + 2.21076, log d"- 0.126082 =' - 2.032557 z' - 23~ 48' 14", log r' 0.346653 " —1.94235, z"= 27 12 49, logr"= 0.339276 C' C"= v' —= 17~ 8' 4" v'-v - 14~ 21' 49", log r - 0.354522 v'-v"= 18 51 7, logr"'=- 0.334290 log (n 01)= 0.004363, log (n 12) = 0.006106, log (n 23) = 0.007290. If now the distances from the earth are computed according to the precepts of the preceding article, there appears:Q'= 1.5635, P"- 2.1319 log Q cos =_ 0.09876 log "' cos (3"' = 0.42842 log Q sin ( -- 9.44252 log Q"' sin'"'- 9.30905 (= 120 26' 40'" ("= 4~ 20' 39" log = -0.10909 log ("'- 0.42967. SECT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 247 Hence are found Corrections of the Times. Corrected Times. I. 0.007335 89.497827 II. 0.008921 135.335581 III. 0.012165 192.407337 IV. 0.015346 251.272756 whence will result newly corrected values of the quantities P', P", Q', Q", log P' 0.059415, log Q' - 9.604782, log P"i= 9.972253, log Q" - 9.697687. Finally, if the fourith hypol/esis is formed with these new values, the following numbers are obtained:c' - 7.678116, log d' = 0.045723 "- + -2.210773, log d"= 0.126084 xi =2.032473 z'- 23~ 48' 16".7, log r' 0.346638 N"-=1.942281, a"=-27 12 51.7, logr" =0.339263 v"- v' =170 8' 5".1, i (u"- +) -= 176 7' 50".5, 1 (t —u"u)= 4 33 23".6 v' -v - 14 21 51.9, log r = 0.354503 v -v"' 18 51 9.5, logr"=- 0.334263 These numbers differ so little from those which the third hypothesis furnished, that we may now safely proceed to the determination of the elements. In the first place we get out the position of the plane of the orbit. The inclination of the orbit 7~ 8' 14".8 is found by the precepts of article 149 from y', %, and A'C' =' -', also the longitude of the ascending node 1030 16'37".2, the argument of the latitude in the second observation 94~ 36' 4".9, and, therefore, the longitude in orbit 197~ 52' 42".1; in the same manner, from y', i", and A" C" d"'- a" are derived the inclination of the orbit = 70 8' 14". 8, the longitude of the ascending node 1030 16'37".5, the argument of the latitude in the third observation 1110 44' 9".7, and therefore the longitude in orbit 215' 0' 47".2. Hence the longitude in orbit for the first observation will be 1830 30' 50".2, for the fourth 233~ 51' 56".7. If now the dimensions of the orbit are determined from "'- t, r, r"', and v"'- v = 50~ 21' 6".5, we shall have, 248 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS. [BOOK II. True anomaly for the first place... 293~ 33r 43'.7 True anomaly for the fourth place... 343 54 50.2 Hence the longitude of the perihelion.. 249 57 6.5 Mean anomaly for the first place..... 302 33 32.6 Mean anomaly for the fourth place..... 346 32 25.2 Mean daily sidereal motion........ 978".7216 Mean anomaly for the beginning of the year 1807. 278 13 39.1 Mean longitude for the same epoch... 168 10 45.6 Angle of eccentricity q. e... 5 2 58.1 Logarithm of the semi-axis major.... 0.372898 If the geocentric places of the planet are computed from these elements for the corrected times 1 1', 1" 1"', the four longitudes agree with a, ac', " a"CC', and the two intermediate latitudes with (3', (3", to the tenth of a second; but the extreme latitudes come out 12~ 26' 43".7 and 4~ 20' 40".1. The former in error 22".4 in defect, the latter 18".5 in excess. But yet, if the inclination of the orbit is only increased 6", and the longitude of the node is diminished 4' 40", the other elements remaining the same, the errors distributed among all the latitudes will be reduced to a few seconds, and the longitudes will only be affected by the smallest errors, which will themselves be almost reduced to nothing, if, in addition, 2" is taken from the epoch of the longitude. TIHIRD SECTION. THE DETERMINATION OF AN ORBIT SATISFYING AS NEARLY AS POSSIBLE ANY NUMBER OF OBSERVATIONS WHATEVER. 172. IF the astronomical observations and other quantities, on which the computation of orbits is based, were absolutely correct, the elements also, whether deduced from three or four observations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of KEPLER), and, therefore, if other observations were used, they might be confirmed, but not corrected. But since all our measurements and observations are nothing more than approximations to the truth, the same must be true of all calculations resting upon them, and the highest aim of all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unknown quantities. This problem can only be properly undertaken when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most accurate manner possible. It then can only be worth while to aim at the highest accuracy, when the final correction is to be given to the orbit to be determined. But as long as it appears probable that new observations will give rise to new corrections, it will be convenient to relax more or less, as the case may be, from extreme precision, if in this way the length of the computations can be considerably diminished. We will endeavor to meet both cases. 32 (249) 250 DETERMINATION OF AN ORBIT FROM [BOOK IT. 173. In the first place, it is of the greatest importance, that the several positions of the heavenly body on which it is proposed to base the orbit, should not be taken from single observations, but, if possible, from several so combined that the accidental errors might, as far as may be, mutually destroy each other. Observations, for example, such as are distant fromn each other by an interval of a few days, -or by so much, in some cases, as an interval of fifteen or twenty days, - are not to be used in the calculation as so many different positions, but it would be better to derive from them a single place, which would be, as it were, a mean among all, admllitting, therefore, much greater accuracy than single observations considered separately. This process is based on the following principles. The geocentric places of a heavenly body computed from approximate elements ought to differ very little from the true places, and the differences between the former and latter should change very slowly, so that for an interval of a few days they can be regarded as nearly constant, or, at least, the changes may be regarded as proportional to the times. If, accordingly, the observations should be regarded as free from all error, the differences between the observed places corresponding to the times 1, t', 1", ", and those which have been computed from the elements, that is, the differences between the observed and the computed longitudes and latitudes or right ascensions and declinations, would be quanti-e ties either sensibly equal, or, at least, uniformly and very slowly increasing or decreasing. Let, for example, the observed'right ascensions a, a', a", cc"', etc., correspond to those times, and let a + 6, a' -', a"+ (l' ca"' +cY", etc., be the computed ones; then the differences Y, 6', (Y', r'", etc. will differ from the true deviations of the elements so far only as the observations tbheinselves are erroneous: if, therefore, these deviations can be regarded as constant for all these observations, the quantities A, d',', A", etc. will furnish as many different dleterminations of the same quantity, for the correct value of which it will be proper to take the arithmetical mean between those determinations, so far, of course, as there is no reason for preferring one to the other. But if it seems that the same degree of accuracy cannot be attributed to the several observations, let us assume SECT. 3.] ANY NUMBER OF OBSERVATIONS. 251 that the degree of accuracy in each may be considered proportional to the numbers e e', e", e"' etc. respectively, that is, that errors reciprocally proportional to these numbers could have been made in the observations with equal facility; thlen, according to the principles to be propounded below, the most probable mean value will no longer be the simple arithmetical mean, but e e - + e'e'" -F- e+e" - + e"'e"'"' -f-+ etc. ee + e'e' + e" + ee"'e"' +- et.c Putting now this mean value equal to Jd we can assume for the true right ascens-i: a - - j, a' -~- t' -- J, a' A- a" —- /, "Jra+ A A, respectively, and then it will be arbitrary, which we use in the calculation. But if either the observations are distant fron each other by too great an interval of time, or if sufficiently approximate elements of the orbit are not yet known, so that it would not be admnissible to regard their deviations as constant for all the observations, it will readily be perceived, that no other difference arises from this except that the mean deviation thus found cannot be regarded as common to all the observations, but is to be referred to some intermediate time, which must be derived from the inclividual times in the same manner as X from the corresponding deviations, and therefore generally to the time e e t +'e't' A- e. "t' - effefft/1 - etc. e e + e'e' +- e"e" t+ e"'e"' + -etc. Consequently, if we desire the greatest accuracy, it will be necessary to compute the geocentric place from the elements for the same time, and afterwards to free it from "the mean error J, in ordler that the most accurate position may be obtained. But it will in general be abundantly sufficient if the mean error is referred to the observation nearest to the mean time. What we have said here of right ascensions applies equally to declinations, or, if it is desired, to longitudes and latitudes: however, it will always be better to compare the right ascensions and declinations computed from the elements ilmmediately with those observed; for thus we not only gain a much mnore expeditious calculation, especially if we make use of the methods explained in articles 53-60, but this method has the additional advantage, that the incomplete observations can also be made use of; and besides if every thing should be referred to longitudes and latitudes, there 252 DETERMINATION OF AN ORBIT FROM [BooK II. would be cause to fear lest an observation made correctly in right ascension, but badly in declination (or the opposite), should be vitiated in respect to both longitude and latitude, and thus become wholly useless. The degree of precision to be assigneld to the mean found as above will be, according to the principles to be explained hereafter, / (ee + e'e' +'- e" + e'e" + etc.); so that four or nine equally exact observations are required, if the mean is to possess a double or triple accuracy. 174. If the orbit of a heavenly body has been determined according to the methods given in the preceding sections from three or four geocentric positions, each one of- which has been derived, according to the precepts of the preceding article, from a great many observations, that orbit will hold a mean, as it were, among all these observations; and in the differences between the observed and computed places there will remain no trace of any law, which it would be possible to remove or sensibly diminish by a correction of the elements. Nowwhen the whole number of observations does not embrace too great an interval of time, the best agreement of the elements with all the observations can be obtained, if only three or four normal positions are judiciously selected. How much advantage we shall derive from this method in determining the orbits of new planets or comets, the observations of which do not yet embrace a period of more than one year, will depend on the nature of the case. When, accordingly, the orbit to be determined is inclined at a considerable angle to the ecliptic, it will be in general based upon three observations, which we shall take as remote from each other as possikble: but if in this way we should meet with any one of the cases excluded above (articles 160-162), or if the inclination of the orbit should seem too small, we shall prefer the determination from four positions, which, also, we shall take as remote as possible from each other. But when we have a longer series of observations, embracing several years, more normal positions can be derived from them; on which account, we should SEaCT 3.] ANY NUMBER OF OBSERVATIONS. 253 not insure the greatest accuracy, if we were to select three or four positions only for the determination of the orbit, and neglect all the rest. But in such a case, if it is proposed to aim at the greatest precision, we shall take care to collect and employ the greatest possible number of accurate places. Then, of courser more data will exist than are required for the determination of the unknown quantities: but all these data will be liable to errors, however small, so that it will generally be impossible to satisfy all perfectly. Now as no reason exists, why, from among those data, we should consider any six as absolutely exact, but since we must assume, rather, upon the principles of probability, that greater or less errors are equally possible in all, promiscuously; since, moreover, generally speaking, small errors oftener occur than large ones; it is evident, that an orbit which while it satisfies precisely the six data, deviates more or less from the others, must be regarded as less consistent with the principles of the calculus of probabilities, than one which, at the same time that it differs a little from those six data, presents so much the better an agreement with the rest. The investigation of an orbit having, strictly speaking, the maxiazzmtn probability, will depend upon a knowledge of the law according to which the probability of errors decreases as the errors increase in magnitude: but that depends upon so many vague and doubtful considerations - physiological included — which cannot be subjected to calculation, that it is scarcely, and indeed less than scarcely, possible to assign properly a law of this kind in any case of practical astronomy. Nevertheless, an investigation of the connection between this law and the most probable orbit, which we will undertake in its utmost generality, is not to be regarded as by any means a barren speculation. 175. To this end let us leave our special problem, and enter upon a very general discussion and one of the most fruitful in every application of the calculus to natural philosophy. Let TV T, V'", etc. be functions of the unknown quantities p, q, r. S, etc., t the number of those functions, v the number of the unknown quantities; and let us suppose that the values of the functions found by direct observation are VFM, 3' - 31', V" - 31" etc. Generally speaking the 254 DETERMINATION OF AN ORBIT FROM [B30O HII. determination of the unknown quantities will constitute a problelm, indetermlninate, determinate, or more than determinate, according as Ft < v, t =v, or Lt >v.* We shall confine ourselves here to the last case, in which, evidently, an exact representation of all the observations would only be possible when they were all absolutely free from error. And since this cannot, in the nature of things, happen, every system of values of the unknown quantities p, q, r, s, etc., must be regarded as possible, which gives the values of the functions V- -;i 1v'- -' V'" - v", etc., within the limits of the possible errors of observation; this, however, is not to be understood to imply that each one of these systems would possess an equal degree of probability. Let us suppose, in the first place, the state of things in all the observations to have been such, that there is no reason why we should suspect one to be less exact than another, or that we are bound to regard errors of the same magnitude as equally probable in all. Accordingly, the probability to be assigned to each error i will be expressed by a function of z which we shall denote by T Az. Now although we cannot precisely assign the form of this function, we can at least affirmn that its value should be a maximum for J- 0, equal, generally, for equal opposite values of, anld, should vanish, if, for A is taken the greatest error, or a value greater than the greatest error: c AJ, therefore, would appropriately be referredl to the class of discontinuous functions, and if we undertake to subs-titute any analytical function in the place of it for practical purposes, this must be of such a form that it may converge to zero on both sides, asymptotically, as it were, from -- 0, so that beyond this limit it can b.e regarded as actually vanishing. Moreover, the probability that an error lies between the limits A and X c- d z differing from each other by the infinitely smnall difference d A, will be expressed by cp4 dA; hence the probability generally, that the error lies between D and e If, in the third case, the functions V, F', V' should be of such a nature that / + 1 -- v of them, or more, might be regarded as functions of the remainder, the problem would still be more than determinate with respect to these functions, but indeterminate with respect to the quantities p, q, r, s, etc.; that is to say, it would be impossible to determine the values of the latter, even if the values of the fiunctions V F, VF, etc. should be given with absolute exactness: but we shall exclude this case firom our discussion. SECT. 3.] ANY INUMBER OF OBSERVATIONS. 255 D', will be given by the integral fp. d.d extended from z =D to a- D'. This integral taken from the greatest negative value of z to the greatest positive value, or more generally from Jz- - oo to d- + co must necessarily be equal to unity. Supposing, therefore, any determinate system of the values of the quantities p, q, r, s, etc., the probability that observation would give for 1 the value 3i will be expressed by q (f- V),7 substituting in V for p, q, r, s, etc., their values; in the same manner sp (31'- V), p (1"- V1"), etc. will express the probabilities that observation would give the values 1', 3n1, etc. of the functions V7, V", etc. Wherefore, since we are authorized to regard all the observations as events independent of each other, the product p (p —v) p ( (Xv" —v") etc., = Q will express the expectation or probability that all those values will result together from observation. 176. Now in the same manner as, when any determinate values whatever of the unknown quantities being taken, a determlinate probability corresponds, previous to observation, to any system of values of the functions V, 1', V7 etc.; so, inversely, after determinate values of the functions have resulted from observation, a determinate probability will belong to every system of values of the unknown quantities, from which tlhe values of the functions could possibly have resulted: for, evidently, those systems will be regarded as the more probable in which the greater expectation had existed of the event which actually occurred. The estimation of this probability rests upon the following theorem:I nf, cly hypotesi's H being mztade, the probability of anyl cdieerlinate event E is h, clzd it calzoter ypothesis H' being madae exclilding {Ae formner and equally probcable in. itself, tie probability of the samnze event is h': then I sacy, wuen k he event E has ccltclcy occ2urred7, tMat the probability th1at Hd waCGS the true hlypothesis, is to the pprobabili4y that H' wvas the true hlpothesis, as h to h'. For demonstrating which let us suppose that, by a classification of all the circumistances on which it depends whether, with H or H' or some other hypothesis, 256 DETERMINATION OF AN ORBIT FROM [BOOK II. the event E or some other event, should occur, a system of the different cases is formed, each one of which cases is to be considered as equally probable in itself (that is, as long as it is uncertain whether the event En or some other, will occur), and that these cases be so distributed, that among them in which should be assumed in such a mode as would give may be found the hypothesis occasion to the event. m /H E n 11 different from E m' H' B nr H' different from E in" different from H and H' E ntl different from H and H' different from B Then we shall have m+n,, +, m'+ n' moreover, before the event was known the probability of the hypothesis H was.. + n m + -- m' +i m' + m"- + na' but after the event is known, when the cases n, n', n" disappear from the number of the possible cases, the probability of the same hypothesis will be + nz' + m in the same way the probability of the hypothesis H' before and after the event, respectively, will be expressed by m' -q- n' m' and m + n _+- mn' j n'-+ i_" + n" and -- m'+ +'n" since, therefore, the same probability is assumed for the hypotheses Hf and H' before the event is known, we shall have VI + n - ns' - n', whence the truth of the theorem is readily inferred. Now, so far as we suppose that no other data exist for the determination of the unknown quantities besides the observations V7= V[' - T', V J"1",', SECT. 3.] ANY NUMBER OF OBSERVATIONS. 25'7 etc., ancld, therefore, that all systems of values of these unknown quantities were equally probable previous to the observations, the probability, evidently, of any determninate system subsequent to the observations will be proportional to!2 This is to be understood to mean that the probability that the values of the unknown quantities lie between the infinitely near limits p and p + dp, q and Gq+ dq, r and r + d r, s and s + ds, etc. respectively, is expressed by 2 d2pdqdrds........, etc., where the quantity X will be a constant quantity independent of p,, r, s, etc.: and, indeed, will, evidently, be the value of the integral of the order v, fv2dpdqdrds........, etc., for each of the variables p, qi r, s, etc., extended from the value - o to the value + oo 177. Now it readily follows from this, that the most probable system of values of the quantities p, q, r, s, etc. is that in which ~2 acquires the maximum value, and, therefore, is to be derived from the v equations d52 dS~ d.Q dS~ 0,0 — 0 0,- O -- 0 etc. dp dq dr ds These equations, by putting V — v Jf V-1' v, F'- 1- = v", etc., and d A t 4 assume the following form — d v d d v' d of dV + etc.-0 dp V + - ~p + d-p'p -+ eta -O dv dv' dv" d V r Vt @ ++- + etc.- = o, dv, d' d,, dv, Tr t'v -[- ~ ~''nu 3 - etc.Tr — O, dv dv' d:', V v - 9 -v- +T d(v"+- etc.=- 0. Hence, accordingly, a completely determinate solution of the problem can be obtained by elimination, as soon as the nature of the function fp' is known. Since 33 258 DETERMINATION OF AN ORBIT FROM [BOOK II. this cannot be defined a priori, we will, approaching the subject from another point of view, inquire upon what function, tacitly, as it were, assumed as a base, the common principle, the excellence of which is generally acknowledged, depends. It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several clirect observations, made under the samle circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it. By putting, therefore, V= V' 1" etc. =-p, we ought to have in general, T' (L-p) + p' (' — ) +' (' --— p) -+ etc. = 0, if instead of p is substituted the value I (M+ M' + M" + etc.), wnatever positive integer M expresses. By supposing therefore, f!'- -;1"- etc. = M-,m/, -we shall have in general, that is, for any positive integral value of nt, so' (M — 1) x- (1 -- j) T' (- x)5 whence it is readily inferred that BA must be a constant quantity, which we will denote by k. Hence we have log X d = ] k ldd + Constant, 9)L4 z ekAAo denoting the base of the hyperbolic logarithms by e and assuming Constant = log x. Moreover, it is readily perceived that ic must be negative, in order that S2 may really become a maximum, for which reason we shall put and since, by the elegant theorem first discovered by LAPLACE, the integral fe-hhAA dz SECT. 3.] ANY NUMBER OF OBSERVATIONS. 259 fromJ A - to J - zI - is -, (denoting by i the semicircumference of the circle the radius of which is unity), our function becomes p=d e-hhAA 178. The function just found cannot, it is true, express rigorously the probabilities of the errors: for since the possible errors are in all cases confined within certain limits, the probability of errors exceeding those limits ought always to be zero. while our formula always gives some value. However, this defect, which every3 analytical function must, from its nature, labor under, is of no importance in practice, because the value of our function decreases so rapidly, when h A has acquired a considerable magnitude, that it can safely be considered as vanishing. Besides, the nature of the subject never admits of assigning with absolute rigor the limits of error. Finally, the constant h can be considered as the measure of precision of the observations. For if the probability of the error A is supposed to be expressed.in any one system of observations by h -hhAA e and in another system of observations more or less exact by h''A e- hh' AA the expectation, that the error of any observation in the former system is contained between the limits - 6 and + Y will be expressed by the integral f - ehhAAdA taken from A - 6J to A =+ Y; and in the same manner the expectation, that the error of any observation in the latter system does not exceed the limits - 6' and + Y' will be expressed by the integral e-f:/h8AA d A extendedl from A = - J' to A = + Y': but both integrals manifestly become 260 DETERMINATION OF AN ORBIT FROM [BOOiK II. ecjual when we have h - h'61 Now, therefore, if for example' = 2 i, a double error can be coimmitted in the former system with the same facility as a single error in the latter in which case, according to the common way of speaking, a dlouble degree of precision is attributed to the latter observations. 179. We will now develop the conclusions which follow from this law. It is evident, in order that the product S2 - jhy 7C fbe h h (zv + VW e o a... ) may become a maximum, that the sum v v + v'v' + v"v"- + etc., must become a minimum. Therefore, that will be the most probable system of values of the unkn~own gcunztities p, q, r, s, etc., in which the suvm of the squares of the dierenes between the observed and comptuled values of the fncetions V, V', V", etc. is a minintoln, if -the same degree of accuracy is to be presumed in all the observations. This principle, which promises to be of most frequent use in all applications of the mathematics to natural philosophy, must, everywhere, be considered an axiom with the same propriety as the arithmetical mean of several observed values of the same quantity is adopted as the most probable value. This principle can be extended without difficulty to observations of twequal accuracy. If, for example, the measures of precision of the observations by mneans of which V - X V' = Yt V' - A", etc. have been found, are expressed, respectively, by A, h', h", etc., that is, if it is assumed that errors reciprocally proportional to these quantities might have been made with equal facility in those observations, this, evidently, will be the same as if, by means of observations of equal precision (thle measure of which is equal to unity), the values of the functions hV K' V', h'" V", etc., had been directly found to be h, a'I', h"lf", etc.: wherefore, the most probable system of values of the quantities p, q, r, s, etc., will be that in which the sum of hA v v - A'h'v'v' + hh//"v"v" + etc., that is, inz zwhich hie sulm of the squares of the c ijerences between the actzally observed and coimpted valhes mulitplied by 7-tmbers tha7t nmcasu-re the degree of precision, is a mn-imum'.n. In this way it SECT. 3.] ANY NUMBER OF OBSERVATIONS. 261 is not even necessary that the functions V, T' F" etc. relate to homogeneous quantities, but they may represent heterogeneous quantities also, (for example, seconds of are and time), provided only that the ratio of the errors, which might have been committed with equal facility in each, can be estimated. 180. The principle explained in the preceding article derives value also from this, that the numerical determination of the unknown quantities is reduced to a very expeditious algorithm, when the functions V, V', V", etc. are linear. Let us suppose 7'- M'- v' - m' + p -+- 6'q + c'r +- ds + etc. T _ — = v" - Ain" + aLlp + bq + c"r + d"s + etc, etc., and let us put av + au' -+ d'v" + etc. P by + b'v' + b"v" + etc._ Q ev + c'v' -+ "v" + etc. R dv + d'v'-+ d"v"+ etc.- S etc. Then the v equations of article 177, from which the values of the unknown quantities must be determined, will, evidently, be the following:P - 0, Q — 0, R 0, A - 0, etc., provided we suppose the observations equally good; to which case we have shown in the preceding article how to reduce the others. We have, therefore, as many linear equations as there are unknown quantities to be deternmined, from which the values of the latter will be obtained by common elimination. Let us see now, whether this elimination is always possible, or whether the solution can become indeterminate, or even impossible. It is known, from the theory of elimination, that the second or third case will occur when one of the equations P O, Q-= 0 R -0, 8= 0, etc., beingl omitted, an equation can be formed from the rest, either identical with the 262 DETERMINATION OF AN ORBIT FROM [Boox 11. onmitted one or inconsistent with it, or, which amounts to the same thing, when it is possible to assign a linear function aP + [Q + YR + cH-+ etc., which is identically either equal to zero, or, at least, free from all the unknown quantities p, gq r s, etc. Let us assume, therefore, aP+ rJ Q + yrR+ S+ etc.=. We at once have the identical equation (V[ + -n) v + (v' + nI') v' + (v" + mc") v" + etc. pP + Q + rR + sS+ etc. If, accordingly, by the substitutions P = ax, q - x, r = yx, s - (x, etc. we suppose the functions v, v', v', to become respectively, -- + x, X- m' +- )'x, - m" -j- /+ x, etc., we shall evidently have the identical equation (h + X'f' + aX"Y + etc.) xx- (Xm + Xrn' + X/in" etc.) x = x, that is, XX + lX'X + AX" + etc. 0, X m + X + X'F"rn" + Fetc. - 0: hence it must follow that X - 0 i = 0, X" — 0, etc. and also x -. Then it is evident, that all the functions V,,' F", are such that their values are not changed, even if the quantities p, q, r, s, etc. receive any increments or decrements whatever, proportional to the numbers a, r7, 7, 6, etc.: but we have alrefady mentioned before, that cases of this kind, in which evidently the determination of the unknown quantities would not be possible, even if the true values of the functions V7, 7', 7" etc., should be given, do not belong to this subject. Finally, we can easily reduce to the case here considered, all the others in which the functions V, 7, V", etc. are not linear. Letting, for instance, mx,;, g, a, etc., denote approximate values of the unknown quantities p, q r7 S9 etc., (which we shall easily obtain if at first we only use v of the it equations V773 7' -', VT"= M", etc.), we will introduce in place of the unknown quantities the others, p2,,, r', S', etc., putting r - a +-', = X + q', r = ( + rt, S -a + s', etc.: the values of these new unknown quantities will evidently be so small that their SECT. 3.] ANY NUMBER OF OBSERVATIONS. 263 squares andl products may be neglected, by which means the equations become linear. If, after the calculation is completed, the values of the unknown quantities Ap', r', Sq' etc., prove, contrary to expectation, to be so great, as to make it appear unsafe to neglect the squares and products, a repetition of the same process (the corrected values of p, q, r, s, etc. being taken instead of a, p, ), c, etc.), will furnish an easy remedy. 181. When we have only one unknown quantity p, for the determination of which the values of the functions ap +i-, p +> e', a'> + 2n", etc. have been found, respectively, equal to 31f i1', V'", etc., and that, also, by means of observations equally exact, the most probable value of p will be A a m +- a'' -- a"m'z - etc. a a + a'a'+ a -'a+ etc. 7 putting m, in', in", respectively, for 1M —n, t1 - i-', i" - in", etc. In order to estimate the degree of accuracy to be attributed to this value, let us suppose that the probability of an error J in the observations is expressed by h ehhAA Hence the probability that the true value of p is equal to A +2p' will be proportional to the function eh h ((ap —m)2+ (ap-m')2(a"p- m")2+ etc.) if A +p' is substituted for p. The exponent of this function can be reduced to the form, -- h (aa +' + d' + etc.) (pp - 2pA + B), in which B is independent of p: therefore the function itself will be proportional to e-h h(a a +aa'+ a"a" etc.) p'p It is evident, accordingly, that the same degree of accuracy is to be assigned to the value A as if it had been found by a direct observation, the accuracy of which would be to the accuracy of the original observations as h! (aca + a'a'c aa"-+ etc.) to it or as 2/ (an A a'a' + a/a" + etc.) to unity. 264 1)ETERMNINATION OF AN ORBIT FROM [BOOK.I 182. It will be necessary to preface the discussion concerning the degree of accuracy to be'assigned to the values of the unknown quantities, when there are several, with a more careful consideration of the function v v + v'v'+ v"v" + etc., which we will denote by W. I. Let us put d — 2= + aP — + q-+yr-+ as -+ etc., dp also Wf PfP1= WI and it is evident that we have p' - P, and, since d W'__ d W 2p' dp' dp dp a dp that the function W' is independent of p. The coefficient\ a aa + a'a' + d"a"detc. will evidently always be a positive quantity. II. In the same manner we will put 2 d7 =' _' -'-S + y'r + ad's + etc., also W'. and we shall have dW pdp, d -~ p'p' Q p — and -, dq a a adq whence it is evident that the function W"' is independent both of p and q. This would not be so if (3' could become equal to zero. But it is evident that 1W' is derived from v v + v'v' + v"v" *- etc., the quantity p being eliminated from v, vf v", etc., by means of the equation p' - 0; hence, (' will be the sum of the coefficients of gq in vv, v'v' v"vY" etc., after the elimination; each of these coefficients, in fact, is a square, nor can all vanish at once, except in the case excluded above, in which the unknown quantities remain indeterminate. Thus it is evident that 1' must be a positive quantity. SECT. 3.] ANY NUMBER OF OBSERVATIONS. 265 II. By putting again, + _ r" _ + d"s + etc., and TV" - - drp I we shall have R' — also W" independent of p, and q, as well as r. Finally, that the coefficient of r" must be positive is proved in the same manner as in IL In fact, it is readily perceived, that y" is the sum of the coefficients of rr in vv, v'v', v"v" etc., after the quantities p and q have been eliminated from v, v', v", etc., by means of the equations p' = 0, q' -0. IV. In the same way, by putting d W" = s "=' + S' + etc., W = — we shall have at- g pt-a'f 6" a WIV independent of p, q, r, s, and A"' a positive quantity. V. In this manner, if besides p, q, r, s, there are still other unknown quantities, we can proceed further, so that at length we may have W=-'pp' + q' + r r'r +, s's' + etc. + Constant, in which all the coefficients will be positive quantities. VI. Now the probability of any system of determinate values for the quanm tities p, q, r, s, etc. is proportional to the function e-hhW; wherefore, the value of the quantity p remaining indeterminate, the probability of a system of determinate values for the rest, will be proportional to the integral f e-a"w dp extended from p —- oo to p — + -o which, by the theorem of LAPLACE, becomes h-oa e-h ( q'-r' dr'- s's'-+et.) therefore, this probability will be proportional to the function e-hhW'. In the, same manner, if, in addition, a is treated as indeterminate, the probability of a 34 266 DETERMINATION OF AN ORBIT FROM [BOOK II. system of determinate values for r, s, etc. will be proportional to the integral fe-hW' d q, extended from = —oo up to g= -+ o, which is h, 3- I [ e-hh (z r'r'+ Y s's'+ etc.). or proportional to the function e-hhW"o Precisely in the same way, if r also is considered as indeterminate, the probability of the determinate values for the rest, s, etc. will be proportional to the function ehhw"', and so on. Let us suppose the number of the unknown quantities to amount to four, for the same conclusion will hold good, whether it is greater or less. The most probable value of s will be -b,, and the probability that this will differ from the truth by the quantity a, will be proportional to the function e-4h"'f;7 whence we conclude that the measure of the relative precision to be attributed to that determination is expressed by VYV, provided the measure of precision to be assigned to the original observations is put equal to unity. 183. By the method of the preceding article the measure of precision is conveniently expressed for that unknown quantity only, to which the last place has been assigned in the work of elimination; in order to avoid which disadvantage, it will be desirable to express the coefficient d'" in another manner. From the equations P=f R =r'+ p +7,s'-S, + r +'+;p it follows, that p',', s, can be thus expressed by means of P, Q, R, 8, p'_ P g'Q ~I) -L SO P SECT..] ANY NUMBER OF OBSERVATIONS. 267 -r'=R~t' Q +'2, = X + e"R, + O" Q + w"PN, so that g, I'T 3', rt", 3", S" may be determinate quantities. We shall have, therefore (by restricting the number of unknown quantities to four), +,, +,,,, P + A,,, Q + aR+,,, S. Hence we deduce the following conclusion. The most probable values of the unknown quantities p, q, r, s, etc., to be derived by elimination from the equations P- 0, Q- 0, R- 0, S- 0, etc., will, if P, Q, R, 5 etc., are regarded for the time as indeterminate, be expressed in a linear form by the same process of elimination by means of P, Q, R, 7, etc. so that we may have p= L + AP +BQ + CR + DS+ etc. q — L' + A'P + B' + CR' + aD'S+ etc. r - L'+ A"P + B" Q + C"R + D"S + etc. - L"'+ A"P + B"' Q + C"'R + D'"'S + etc. etc. This being done, the most probable values of p, q, r, s, etc., will evidently be LP, L, L"p, L"' etc., respectively, and the measure of precision to be assigned to these determinations respectively will be expressed by 1 1 1 1 2' B" V "' v "" etc., the precision of the original observations being put equal to unity. That which we have before demonstrated concerning the determination of the unknown quantity s (for which a1v answers to D"') can be applied to all the others by the simple interchange of the unknown quantities. 184. In order to illustrate the preceding investigations by an example, let us suppose that, by means of observations in which equal accuracy may be assumed, we have found 268 DETERMINATION OF AN ORBIT FROM [BooK II. p —- g+2r= 3 3p+2q-5r - 5 4p+ +4r-21, but from a fourth observation, to which is to be assigned one half the same accuracy only, there results - 2p + 6 + 6 r -- 28. We will substitute in place of the last equation the following:-— 2-+ 3 g-+ 3 r — 14, and we will suppose this to have resulted from an observation possessing equal accuracy with the former. Hence we have P=27p+- 6Gq 88 Q — 6p+S15q-r 70 R2?= + E-54r- 107, and hence by elimination, 19899p = 49154 -+- 809 P — 324 Q + 6 R 737qg- 2617- 12P+ 54 Q —R 6633 r =12707 +- 2P- 9 Q + 123 R. The most probable values of the unknown quantities, therefore, will be p - 2.4-70 g = 3.551 r -- 1.916 and the relative precision to be assigned to these determinations, the precision of the original observations being put equal to unity, will be r 19899 for p... 809..4.96 809 for.... V/54 7 3.69 5- 3. for r.....3. /22P1 = 7.34. SECT. 3.] ANY NUMBER OF OBSERVATIONS. 269 185. The subject we have just treated might give rise to several elegant analytical investigations, upon which, however, we will not dwell, that we may not be too much diverted from our object. For the same reason we must reserve for another occasion the explanation of the devices by means of which the numerical calculation can be rendered more expeditious. I will add only a single remark. When the number of the proposed functions or equations is considerable, the computation becomes a little more troublesome, on this account chiefly, that the coefficients, by which the original equations are to be multiplied in order to obtain P, Q, B, 8 etc., often involve inconvenient decimal fractions. If in such a case it does not seem worth while to perform these multiplications in the most accurate manner by means of logarithmic tables, it will generally be suffi-cient to employ in place of these multipliers others more convenient for calculation, and differing but little from them. This change can produce sensible errors in that case only in which the measure of precision in the determination of the unknown quantities proves to be much less than the precision of the original observations. 186. In conclusion, the principle that the sum of the squares of the differences between the observed and computed quantities must be a Ininimunm may, in the following manner, be considered independently of the calculus of probabilities. When the number of unknown quantities is equal to the nuiber of the observed quantities depending on them, the former may be so determined as exactly to satisfy the latter. But when the number of the former is less than that of the latter, an absolutely exact agreement cannot be obtained, unless the observations possess absolute accuracy. In this case care must be taken to establish the best possible agreement, or to diminish as far as practicable the differences. This idea, however, from its nature, involves something vague. For, although a system of values for the unknown quantities which makes cdl the differences respectively 270 DETERMINATION OF AN ORBIT FROM [BooK II. less than another system, is without doubt to be preferred to the latter, still the choice between two systems, one of which presents a better agreement in some observations, the other in others, is left in a measure to our judgment, and innumerable different principles can be proposed by which the former condition is satisfied. Denoting the differences between observation and calculation by J, Jl'// etc., the first condition will be satisfied not only if X d + J'J'+ zJ"J" + etc., is a minimum (which is our principle), but also if J4 + J'4 + a4 + etc., or J6+1'6 +i "6 + etc., or in general, if the sum of any of the powers with an even exponent becomes a minimuni1. But of all these principles ours is the most sinmple; by the others we should be led into the most complicated calculations. Our principle, which we have made use of since the year 1795, has lately been published by LEGENDRE in the work Nouvelles netlhodes Pour la delerminaion~z des orbiles ldes conaeles, Paris, 1806, where several other properties of this principle have been explained, which, for the sake of brevity, we here omit. If we were to adopt a power with an infinite even exponent, we should be led to that system in which the greatest differences become less than in any other system. LAPLACE made use of another principle for the solution of linear equations the nunmber of which is greater than the number of the unknown quantities, which had been previously proposed by BoscovIii, namely, that the sum of the errors themselves taken positively, be made a minimum. It can be easily shown, that a system of values of unknown quantities, derived from this principle alone, must necessarilyt exactly satisfy as many equations out of the number proposed, as there are unknown quantities, so that the remaining equations come under consideration only so far as they help to determinie the choice: if, therefore, the equation V - M, for example, is of the number of those which are not satisfied, the systeml of values found according to this principle would in no respect be chlanged, even if any other value N had been observed instead of X, provided that, denoting the computed value by n, the differences!- nA, N- n, were affected by the same signs. Besides, LAPLACE qualifies in some measure this principle by adding * Except the special cases in which the problem remains, to some extent, indeterminate. SECT. 3.] ANY NUMBER OF OBSERVATIONS. 271 a new condition: he requires, namely, that the sum of the differences, the signs remaining unchanged, be equal to zero. Hence it follows, that the number of equations exactly represented may be less by unity than the number of unknown quantities; but what we have before said will still hold good if there are only two unknown quantities. 187. From these general discussions we return to our special subject for the sake of which they were undertaken. Before the most accurate determination of the orbit from more observations than are absolutely requisite can be commnenced, there should be an approximate determination which will nearly satisfy all the given observations. The corrections to be applied to these approximate elements, in order to obtain the most exact agreement, will be regarded as the objects of the problem. And when it can be assumed that these are so small that their squares and products may be neglected, the corresponding changes, produced in the computed geocentric places of a heavenly body, can be obtained by means of the differential formulas given in the Second Section of the First Book. The computed places, therefore, which we obtain from the corrected elements, will be expressed by linear functions of the corrections of the elements, and their comparison with the observed places according to the principles before explained, will lead to the determination of the most probable values. These processes are so simple that they require no further illustration, and it appears at once that any number of observations, however rem6te from each other, can be employed. The same method may also be used in the correction of the parabolic orbits of comets, should we have a long series of observations and the best agreement be required. 188. The preceding method is adapted principally to those cases in which the greatest accuracy is desired: but cases very frequently occur where we may, without hesitation, depart from it a little, provided that by so doing the calcula 272 DETERMI:NATION OF AN ORBIT FRO.M [BOOK 11. tion is considerably abridgecld, especially when the observations do not embrace a great interval of time; here the final determination of the orbit is not yet proposed. In such cases the following method may be employed with great advantage. Let complete places L and L' be selected from the whole number of observations, and let the distances of the heavenly body from the earth be computed from the approximate elements for the corresponding times. Let three hypotheses then be framed with respect to these distances, the computed values being retained in the first, the first distance being changecd in the second hypothesis, and the second in the third hypothesis; these changes can be made in proportion to the uncertainty presumed to remain in the distances. According to these three hypotheses, which we present in the following table, yp. I. Lyp. II. Hyp. HI. Distance * corresponding to the first place, D D + 8 D Distance corresponding to the second place, D) D' -DI i) let three sets of elements be computed from the two places L, L', by the methods explained in the first book, and afterwards from each one of these sets the geocentric places of the heavenly body corresponding to the times of all the remaining observations. Let these be (the several longitudes and latitudes, or right ascensions and declinations, being denoted separately), in the first set.. M 3 M'1 ", etc. in the second set. M+ i + a', 1"+ a", etc. in the third set.... Me — ( s M'- -:', M"+p, etc. Let, moreover, the observed places be respectively.. N', AN" etc. Now, so far as proportional variations of the individual elements correspond: It will be still more convenient to use, instead of the distances themselves, the logarithms of the curtate distances. SECT. 3.] ANY NUMBER OF OBSERVATIONS. 273 to small variations of the distances D, D', as well as of the geocentric places computed from them, we can assume, that the geocentric places computed from the fourth system of elements, based on the distances fromn the earth D + -r'/ + -y 6, are respectively M+ a x + py,' +- a'x + -', M" + a"x + p"Jy etc. Hence,, y, will be determined, according to the preceding discussions, in such a manner (the relative accuracy of the observations being taken into account), that these quantities may as far as possible agree with N, N', N", etc., respectively. The corrected system of elements can be derived either from L, L' and the distances D + x6, D' + xw', or, according to well-known rules, from the three first systems of elements by simple interpolation. 189. This method differs from the preceding in this respect only, that it satisfies two geocentric places exactly, and then the remaining places as nearly as possible; while according to the other method no one observation has the preference over the rest, but the errors, as far as it can be done, are distributed among all. The method of the preceding article, therefore, is only not to be preferred to the former when, allowing some part of the errors to the places L, Le', it is possible to diminish considerably the errors in the remaining places: but yet it is generally easy, by a suitable choice of the observations L, L', to provide that this difference cannot become very important. It will be necessary, of course, to take care that such observations are selected for L, L', as not only possess the greatest accuracy, but also such that the elements derived from them and the distances are not too much affected by small variations in the geocentric places. It will not, therefore, be judicious to select observations distant from each other by a small interval of time, or those to which correspond nearly opposite or coincident heliocentric places. 35 FOURTH SECTION. ON THE DETERMINATION OF ORBITS, TAKING INTO ACCOUNT THE PERTURBATIONS. 190. THE perturbations which the motions of planets suffer from the influence of other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several entire revolutions, according to circumstances, the motion would differ so little from the motion exactly described, according to the laws of KEPLER, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it would not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been accurately observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the ellitic motion. Since the determination of the elliptic elements with which, in order that the observations may be exactly represented, the perturbations are to be combined, supposes a knowledge of the latter; so, inversely, the theory of the perturbations cannot be accurately settled unless the elements are already very nearly known: the nature of the case does not admit of this dclifficult task being accomplished with complete success at the first trial: but the perturbations and the elements can be brought to the highest degree of perfection only by alternate corrections (274) SECT. 4.] ON THE DETERMINATION OF ORBITS. 275 often repeated. Accordingly, the first theory of perturbations will be constructed upon those purely elliptical elements which have been approximately adjusted to the observations; a new orbit will afterwards be investigated, which, with the addition of these perturbations, may satisfy, as far as practicable, the observations. If this orbit differs considerably from the former, a second determination of the perturbations will be based upon it, and the corrections will be repeated alternately, until observations, elements, and perturbations agree as nearly as possible. 191. Since the development of the theory of perturbations from given elements is foreign to our purpose, we will only point out here how an approximate orbit can be so corrected, that, joined with given perturbations, it may satisfy, in the best manner, the observations. This is accomplished in the most simple way by a method analogous to those which we have explained in articles 124, 165, 188. The numerical values of the perturbations will be computed from the equations,for the longitudes in orbit, for the radii vectores, and also for the heliocentric latitudes, for the times of all the observations which it is proposed to use, and which can either be three, or four, or more, according to circumstances: for this calculation the materials will be taken from the approximate elliptic elements upon which the theory of perturbations has been constructed. Then two will be selected from all the observations, for which the distances from the earth will be computed from the same approximate elements: these will constitute the first hypothesis, the second and third will be formed by changing these distances a little. After this, in each of the hypotheses, the heliocentric places and the distances from the sun will be determined from two geocentric places; from those, after the latitudes have been freed from the perturbations, will be deduced the longitude of the ascending node, the inclination of the orbit, and the longitudes in orbit. The method of article 110 with some modification is useful in this calculation, if it is thought worth while to take account of the secular variation of the longitude of the node and of the inclination. If f(, Y',denote the heliocentric l:liitudes freed from the periodical perturbations; X, X', the heliocen 276 ON THE DETERMINATION OF ORBITS, [BooK IIJ tric longitudes;, 2 -- +, the longitudes of the ascending node; i, i + -Y, the inclinations of the orbit; the equations can be conveniently given in the following form':tan 3 - tan i sin ( — ), tan i tan+ ) tan (' = tan i sin ('m- A- - Q ). tan (i -q- 6) tan i This value of -tan (i n 8) acquires all the requisite accuracy by substituting an approximate value for i: i and 2 can afterwards be deduced by the common methods. Moreover, the sum of the perturbations will be subtracted from the longitudes in orbit, and also from the two radii vectores, in order to produce purely elliptical values. But here also the effect, which the secular variations of the place of the perihelion and of the eccentricity exert upon the longitude in orbit and radius vector, and which is to be determined by the differential formulas of Section I. of the First Book, is to be combined directly with the periodical perturbations, provided the observations are sufficiently distant from each other to make it appear worth while to take account of it. The remaining elements will be determined from these longitudes in orbit and corrected radii vectores together with the corresponding times. Finally, from these elements will be computed the geocentric places for all the other observations. These being compared with the observed places, in the manner we have explained in article 188, that set of distances will be deduced, from which will follow the elements satisfying in the best possible manner all the remaining observations. 192. The method explained in the preceding article has been principally adapted to the determination of the first orbit, including the perturbations: but as soon as the mean elliptic elements, and the equations of the perturbations have both become very nearly known, the most accurate determination will be very conveniently mnade with the aid of as many observations as possible by the method of article 187, which will not require particular explanation in this place. Now if the number of the best observations is sufficiently great, and a great interval SECT. 4.] TAKING INTO ACCOUNT THE PERTURBATIONS. 277 of time is embraced, this method can also be made to answer in several cases for the more precise determination of the masses of the disturbing planets, at least of the larger planets. Indeed, if the mass of any disturbing planet assumed in the calculation of the perturbations does not seem sufficiently determined, besides the six unknown quantities depending on the corrections of the elements, yet another, j, will be introduced, putting the ratio of the correct mass to the assumed one as 1 + t to 1; it will then be admissible to suppose the perturbations themselves to be changed in the same ratio, whence, evidently, in each one of the computed places a new linear terlm, containing A, will be produced, the development of which will be subject to no difficulty. The comparison of the computed places with the observed according to the principles above explained, will furnish, at the same time with the corrections of the elements, also the correction A,. The masses of several planets even, which exert very considerable perturbations, can be more exactly determined in this manner. There is no doubt but that the motions of the new planets, especially Pallas and Juno, which suffer such great perturbations from Jupiter, may furnish in this manner after somine decades of years, a most accurate determination of the mass of Jupiter; it may even be possible. perhaps, hereafter, to ascertain, from the perturbations which it exerts upon the others, the mass of some one of these new planets. APPENDIX@ T'IE value of t adopted in the Solar Tables of HANSEN and OLUFSEN, (Copenhagen, 1853,) is 365.2563582. Using this and the value of /, = 354936' from tlhe last edition of LAPLACE'S Sysl2me du ilionde, the computation of k is log 2 - o.... 0.7981798684 Compl. log t.... 7.4374022154 Compl. log (1l).. 9.9999993882 log k..... 8.2355814720'k —. o. 0.01720210016. 11. The following method of solving the equation 1 — E- e sin E, is recommended by ENCKE, Berliner Aslronomisches Jahrblc2h, 1838. Take any approximate value of BE, as e, and compute 1' E - e" sin E * The numbering of the Notes of the Appendix designates the articles of the original work to which they pertain.'(279) 280 APPENDIX. e" being used to denote e expressed in seconds, then we have dMl- dE (1 - e cos E), or X — M'i - E-E -e e" (sin E sin e) (E -- )(1 - ecosE), if E- e is regarded as a small quantity of the first order, and quantities of the second order are neglected for the present:- so that the correction of E is -- 1 -e cos and a new approximate value of e is 11- i +- -e cos en with which we may proceed in the same manner until the true value is obtained. It is almost always unnecessary to repeat the calculation of 1 - e cos e. Generally, if the first E is not too far from the truth, the first computed value of I - e cos E may be retained in all the trials. This process is identical with that of article 11, for X is nothing more than d log sin E cos E d -] sin E' if we neglect the modulus of BRIGGS'S system of logarithms, which would subsequently disappear of itself, and d log (e" sin E) _ 1 d (e sinE) e sin E' therefore, f. 1 -Xv 1 -- e cosE' E and and the double sign is to be used in such a way that X shall always have the same sign as cos E. In the first approximations when the value of E differs so much from E that the differences of the logarithms are uncertain, the method of this note will be found most convenient. But when it is desired to insure perfect agreement to the last decimal place, that of article 11 may be used with advantage. APPENDIX. 281 As an illustration, take the data of the example in article 13. Assume E - 326~ and we find log sin E 9.74756n log cos - 9.91857 log e" 4.70415 log e 9.38973 log e" sin e 4.45171n log e cos e 9.30830 e" sin E - -- 28295f -= 7~ 51' 35" 1 - e cos e =.79662'_ ~ e - e" sin e = 3330 51' 35" log (1 - e cos e) 9.90125 M- M'= -- 4960" log M W' 3.69548n iJ -M' -2 M' M — m _ — 6226" log os 3.79423n 1- e cos s -- 1~ 43' 46"~ And for a second approximation, e = 326~- 10 43' 46"'= 3240 16' 14" log sin e 9.7663820n log e" 4.7041513 log e" sin E 4.4705333 n e" sin E - 29548/",36 - 80 12' 28".36 M' ~ 332~ 28' 42".36 log (1 - e cos e) 9.90356 M-1 M' = + 12".41 log (M — M') 1.09377 -M- -M _ + 1510.50 - log MK —W 1.19021 1- e cos e - -cos 0 which gives E= 3240 16' 14" + 15".50 = 3240 16'29"o50. 18. Putting q -- p= perihelion distance, log x 8.0850664436, we have =a 1(3 tan' v + tan3 v); 36 282 APPENDIX. a table may be computed from this formula, giving v for values of, as the argument, which will readily furnish the true anomaly corresponding to any time from the perihelion passage. Table IIa is such a table. It is taken from the first volume of Anzcles de I'Observaloire inpcriacle de Pcais, (Paris, 1855,) and differs from that given in DELAMBRE'S Astronomy, (Paris, 1814,) Vol. III., only in the intervals of the argument, the coefficients for interpolation, and the value of k with which it was computed. The true anomaly corresponding to any value of the argument is found by the formula v = v-0 + A1 (, -,0o) + A2 (r -,0o)2 + (, - Zo)3 A3 + A4 (, — _0)4. The signs of A1, A2, A3, are placed before the logarithms of these quantities in -the table. iBURtCKHARDT S table, BowDrTcH's Appendix to the third volume of the Jfiecanigue Celesle, is similar, except that log X is the argument instead of;. Table IIa contains the true anomaly corresponding to the time from perihelion passage in a parabola, the perihelion distance of which is equal to: the earth's mean distance from the sun, and the mass / equal to zero. For if we put -- 1,, 0, we have =t. By substituting the value of x in the equation - 3 (3 tan I v + tan3 2' ) it becomes: 27.40389544 (3 tan I v + tan3 1 v) 1.096155816 (75 tan 1 v + 25 tan3 v; and therefore, if we put t' - 0.912279061, 75 tan I v + 25 tan3 I v = x' c log x' 9.9601277069 BARKER's Table, explained in article 19, contains x', for the argument v. The Jieaz dcailIy motion or the quantity 3, therefore, of BARKER'S Table may be obtained from table IIa, for any value of v, by multiplying the corresponding value of c by I'. The following examples will serve to illustrate the use of the table. Given, the perihelion distance g=- 0.1; the time after perihelion passage t= 6.590997, to find the true anomaly. APPENDIX. 283 Assuming u = 0O, we find,r- 208.42561 T- 200. v — =o -- 8.42561 vO - 110~ 24' 46".69 A1 ('- no) -+ 1~ 14' 42".42 A2 ( -_ To)2 _ 2' 20".19 A3 ( — ) -)3 _=+ 4". 76 A4 (T — )o) -- 0".16 v -- 1110 37' 13".52 or = — 208.42561 o- = 210. — %= -- 1.57439 v = 111~ 50' 16".87 Al ( - 0o) - 12' 58".96 A2 (c - )-0)2 4".35 A3 (- -co)' - 0".03 A4 (r-0)4 = — 0".00 v 111~ 37' 13".53 The latter form of calculation is to be preferred because the value of i —o is smaller, and therefore the terms depending on (T - o), (f - o)2, (f - o)3 are smaller, and that depending on (r - )4 is insensible;, and it is the only form of which all the appreciable terms are to be found in the table. Beyond r = 40000, the limit of the table, we can use the formula, v = 180~- [6.0947259] ()- [6.87718] (!)-[7.313] () etc., in which the coefficients expressed in arc are given by their logarithms. For - = 40000, for example, we have v - 180~- 10~ 6' 6".87 - 3' 8".41 - 0".44 - 1690 50' 44".28. If v is given, and it is required to find c, we have v-s - AV A2( T _q)2 _A(i-T 0)8. 284 APPENDIX. For a first approximation the terms depending on the square and third power of c - cto may be neglected, and the value of r - %c thus found can be corrected so as to exactly satisfy the equation. If v exceeds 169~, the formula --- [1.9149336] tan i v + [1.4378123] tans3 v may be used instead of the table. Thus, for v = 169~ 50' 44".28, log tan i v. 1.0513610 1.9149336 925.33 2.9662946 log tan' I v..3.1540830 1.4378123 39074.67 4.5918953 _ 40000.00 This method will often be found more convenient than the table, even where v is less than 169~. 35. Table Va contains BESSEL'S table here referred to, in a slightly modified form; and also a similar table by POSSELT, for the coefficients v' and v" in the formula of article 34, to = v + i V' + VA V + 31,, + etc., it is taken from ENCKE'S edition of OLBERS Abhacndluny iiber die leichleste utnd bequemste Meitode die Balzn eines Conmeen zu ber.echnen (Weimar, 1847). The following explanation of its construction and use is taken from the same work, with such changes as are needed to adapt it to the notation of the preceding articles - If we put % —tan 1w c tan i v APPENDIX. 285 the formulas of article 34 become I - 1 C - 2 5 W —21 t-I 2 2 ( 1 +- 22)2 V W T to + ~ ( 1+ -t2- 2) (1(1 +2)4v Ile $3 ~.~5+ 5g3,037 + 3 7' —— " (I 1 q +2), 35 The second equation, in which v is expressed in terms of w, is that given by BESSEL, lMonatliche Corres~pondeinz Vol. XII., p. 197. He also gives the third coefficient of the series, but has computed a table of only the first two. POSSELT, in the Ze[ts1cIrjft fiir Astlronomnie nd verwuandte Wissenlschaflen, Vol. V., p. 161, has given the first equation; he has also given three coefficients of the series, but a table of the second only, since BESSEL'S table will give the first coefficient simply by changing the sign. POSSELT has changed the sign of the second ooefficient also. Instead of the logarithms as given in the tables of BESSEL and POSSELT, the corresponding numbers are given in table Va, and to avoid large numbers, 0.01 is taken as the unit of 6. Putting tan x -= the table contains A3 - 2 5 206265 JaB A- - ot - -9 + 3 7206265 pi 7- i76TI o0 i(6 q- 31)~ 10000 (Ia+ w2)406265 So that when x - tw we have v — w t+ A (1006) + B(100 6)2 And when x - w = v-1 - (100 6)-B' (100 6)2 It seems unnecessarvy to recompute the table in order to be certain of the accuracy of the last place, or to extend it further, as its use is limited. For 286 APPENDIX. absolute values of a greater than 0.03, and for values of x considerably greater than 90~, the terms here given would not be sufficient. In such cases the method of 37 and the following articles should be used. Example. - For IHALLEY'S comet, logY- 8.5099324, and t- 63d.43592, we have by table IIa, w 990 36' 55".91 and by table Va, A= - 417.45 1st cor. + 22' 30".63 B — + 3.111 2d cor. +- 32".57 v = 99 59 59".11 which, rigorously, should be 1000; so that d is in this case too great. Inversely, we find, for v - 100~, v — 100~ 0' 00".00 A =- 426.78 st cor. - 23' 0".83 B = ~- 0.297 2d cor. - 3".11 w-= 990 36' 56".06 which agrees nearly with the preceding value. The change of the table to the present form has been made under the supervision of D'ARREST. 39. When table IIa is used instead of BARKER'S table, w is the value of v, which corresponds to the argument at 40. If we put E,,-v (_1- A+a) - (1 t A + C 1 - A+ C the formulas for computing the true anomaly and radius vector are tan E v = E~, y tan A w r = Eq sec2 Ive APPENDIX. 287 Table la for the Ellipse contains log EY, and log Er for the argument A, together with the logarithms of their differences corresponding to a change of a unit in the seventh decimal place of the argument. It was computed by Prof. J. S. HUBBARD, and has been used by him for several years. Since it was in type, a similar table, computed by Mr. A. MARTII, has appeared in the Aslronomische Nachzrichien, Vol. XLIII., p. 122. The example of article 43 will furnish an illustration of its use. Formulas expressing the differentials of the true anomaly and radius vector in a very eccentric ellipse, in terms of the differentials of the time of perihelion passage, the perihelion distance and the eccentricity may be obtained from the equations of this article. If we put B - 1, C- 0, we have, article 39, tan I w -}- + tan3 -1 ccwhich, by article 20, gives dw 3 at t - Sd-I Idq 5 + 5 da. 2 -s4 W 75 2q7 We also have, article 40, log tan -2 v = log tan I w - - log (I- Atan2 - w) + log7 and, therefore, dv Cos2 1wdw dy 7 A do 2 sin I v cosv w (11pf~ ~1 -A B dv a cos2 j w dF. 3atCos2 dq - dt dgq sin v 75 tanIw(1I -tA) 2 q 75 tan (1 - t A) tCos2IW -t A dB + z da d7 +75'tan Iw (-A) i7L1 Ag which, by putting 2 1 Kz75 tan acsw(I-fA 3 N (1+e) (1+9e) 288 APPENDIX 01-A 10 (1 - e) (1 + 9 e) is reduced to dv si v _-_ KdT- RL t dg q - [KMt - N — O P] d e, observing that d tI — dT, if T denotes the time of perihelion passage. If we differentiate the equation q (1+ e) 1 - e cos v we find r q - 2 q2 sin2 2 V r2 e sin v q q (1+e) q( e) dv These formulas are given by NICOLAI, (JJIonatliche Correspondenz, Vol. XXVl,. p. 212). The labor of using them is greatly abridged by the fact that K, L, L, etc., are computed once for all, and that the quantities needed for this purpose are those required for computing the true anomaly and radius vector. If the ellipse so nearly approaches the parabola that, in the coefficients, we may assume tan i v y tan i w k V 2cos2 1 v 2 ql tan I v the values of d v and dr assume a much more simple form. In this case we should have K5sin v kV 2 cos' v sin - v k V 2 COS4 2v _ k 2 q 2 qa tan v qI _s 20 tan!IF,- sin v (N+- O0 P) sinv =v [(1 +- e) (1 + 9 e) 7 (1 + 9 e)2 [+r 44tan2v 21 e 8tan v = (l - e) ( 1 +9 e) (+) (1 +9 and consequently, du.s d -- 2 q T -. ra 2 (1 + 9 e) (1 + e) (1 9 e) deo APPENDIX. 289 This form is given by ENCIKE (Berliner Astronoimisches Jahlrbuch, 1822, page 184.) If we put e - 1 in the coefficient of de it becomes dv_ kt/2q 2 tan 1 v. If we substitute the value of dv in the expression for dr given above, it may be reduced to the form d r =z-I sinvdTf cosvdgq + ( i + - r talvma) de 41. The time I may be found from table IIa, by multiplying the value of z corresponding to wu by 45. Table la for the hyperbola is similar to that for the ellipse, and contains log ]Ev and log Et for the formulas tan I v E -E y tan A w r-E sec2 i v. The differential formulas of article 40, of the Appendix, can be applied to the hyperbola also, by changing the sign of A and of 1 - e in the coefficients. 56. As the solution here referred to may sometimes be found more convenient than the one given in articles 53-57, the formulas sufficient for the use of practical computers are given below. Using the notation of 50 and the following articles, the expressions for the rectangular coordinates referred to the equator are, x - r cos u os r sin ut sin g cos i (1) y - r cos u sin Q cos e -+ rsin u cos ~ cos icos --- r sin u sin i sin e z = r cos u sin 2 sin e + r sin u cos 2 cos i sin -- r sin zu sin i cos e 37 290 APPENDIX. which can be put in the form x - r sin a sin (A + u) (2) y r sin b sin (B + u) z = r sin c sin ( C+ u) or x r sin a sin A cos u + r sin a cos A sin u (3) y r sin b sinB cos u + r sin b cos B sin u z - r sin c sin Ccos u + r sin c cos ( sin u equations (3), compared with (1) give sin a sin A = cos 2 sin a cos A -sin g cos i (4) sin b sin B n s in b cos e sin b cos B = cos 2 cos i cos E — sin i sin sin csin C= sin sine sin c cos = cos cos isin +sin i cos e. By introducing the auxiliary angle E tan E-= tan i cos g we shall find cotan A - tan g cos i cotan B os i cs (+ ) tan g cos E cos e Cotan C _cos i sin (E+ e) tan g cos E sin e sin a Ccos _ sin g cos t sin A cos A sin b sin 2 cos e cos g cos i cos e - sin i sin ~ sin B cos B *in c sin g sin cos 09 cos i sin e + sin i cos e sin C cos C sin a, sin b, sin ce are always positive, and the quadrants in which A, B, C are to be taken, can be decided by means of equations (4). The following relations between these constants, easily deducible from the foregoing, are added, and may be used as checks: tan 1 sin 6 sin c sin ( — B) sin a sin 1 APPENDIX. 291 cos a — in 9 sin si cos b - cos 9 sin i cos E -- cos / sin e cos C - cos 9 sin i sin E + cos Z cos C sin2 a + sin2 b + sin2 c = 2 cos2 cos2 s cos2 I- 1 cos (A - B) -- cotan a cotan b cos (B — C) -=- cotan b cotan c cos (A- C) - cotan a cotan c. 58. If in the formulas of article 56 of the Appendix, the ecliptic is adopted as the fundamental plane, in which case - 0; and if we put =7C long. of the perihelion sina-k_ A=-K —(7 —;2) sin c=kz CE-X-(2v- ) we shall have k, sin (K. —( — g )) - -cos g k cos (K — ( - g )) -- - sin g cos i k, sin K _- cos g. cos ( -- ) - sin 2 sin ( - ) cos i kcos K= -[cos; sin ( -- ) + sin ga cos ( -- ) cos which can easily be reduced to the form, k sin K - cos2 { i cos v -- sin2 I i cos (Ti - 2 ga) k1 cos., -- [cos2 ^ i sin 2 -+- sin2 / isin (n -2 a )] and in like manner we should find k, sin KE- cos2' i sin - sin2 i i sin (7 - 2 ) kcos cos'Ky - cos2 1 i cos - sin2 l icos ( -- 2 g ) k sin K- sin i sin ( - ) kz cos X -- sin i sin (~ - g~ ) 292 APPEXDIX. If these values are substituted in the general expression for coordinates, a k cos yq cos Ksin E -+ a k sin K(cos E- e) and if we put a cos qp = b cos i a co2 1 i si a I + tan[s2 1 (ito -2 _ - a Cos2 i2insLl - l Sinl it- 2)si a Ccos2 i sin- [1 — tan,2 1.sin (-2 g) A' L sin i Cos (r I-COS J a sin i sin (7- )-A" b sin icos ( - ) - B" the coordinates will be xA (cosE-e)+B sinE=A (1-esecE)+B sinE yzA' (cosE-e) +B'sin E=_A' (1-esecE)+B' sinE z = A" (cos E - e) + B" sin E A" (1 - e sec E) + B" sin E. If the equator is adopted as the fundamental plane instead of the ecliptic, the same formulas may be used, if 7, m, and i are referred to the equator by the method of article 55. Thus, if 96 denote the right ascension of the node on the equator, for 9, g, and i, we must use 26, 2 —(nm- )-J4, and z respectively. This form has been given to the computation of coordinates by Prof. PEIRCE, and is designed to be used with ZECH'S Tables of Addition and Subtraction Logarithms. Exanmple. -The data of the example of articles 56 and 58, furnish S = 158~ 30' 50".43, 7c- 122~ 12' 23".55, i- 11~ 43' 52".89 when the equator is adopted as the fundamental plane; and also log b - 0.4288533. Whence we find log cos (c - 2 s ) 9.9853041 n log sin (i - 2 ) 9.4079143 log sec mx 0.2732948 n log cosec -r 0.0725618 log tan2 i 8.0234332 log tan2 I1 8.0234332 logG - 8.2820321 log c' 7.5039093 APPENDIX. 293 1 1 add. log c 0.0082354 C. sub. log - 9.9916052 log cos a 9.7267052 n log cos i 9.7267052 log cos2 1 i 9.9954404 log cos2 I i 9.9954404 log a 0.4423790 log b 0.4288533 log A 0.1727600 n log B' 0.1426041 n 1 1 add. log o 0.0013836 C. sub. log 2 9.9986120 log sin fA 9.9274382 log sin u 9.9274382 log cos2 ~ i 9.9954404 log cos2 i i 9.9954404 log b 0.4288533 log a 0.4423790 log B 0.3531155 n log A' 0.3638696 This method may also be used to compute k and K for the general formula of article 57. Thus: — add. log - 0.0082354 C. sub. log 1 9.9916052 log cos m 9.7267052 n log cos F 9.7267052 n log cos2 1 i 9.9954404 log cos2 4 i 9.9954404 log k/ sin K- 9.7303810 n log ky cos y, 9.7137508 n 1 1 adcd. log 1 0.0013836 C. sub. log 9.9986120 C C log sin m-. 9.9274382 log sin m 9.9274382 log cos2 1 i 9.9954404 log cos2 i. 9.9954404 log k, cos K, 9.9242622 n log ky sin K] 9.9214906 log tan K, 9.8061188 log tan Ky 0.2077398 n log cos K' 9.9254698 n log sin K, 9.9294058 log k =- 9.9987924 log ki, 9.9920848 K =- 212~ 36' 56".1 Ky - 121~ 47' 28".1 It will not be necessary to extend the example to the final expressions for Xzy, j, a's illustrations of similar applications of the Addition and Subtraction Logarithms are given in the directions accompanying ZECH's Tables. 294 APPENDIX. 59. If r, b6, and I denote the radius vector, the heliocentric latitude and longitude of any planet, the rectangular coordinates referred to three axes,- of which that of x is directed towards the vernal equinox, that of 0, parallel to the earth's axis, and that of y, 90~ of right ascension in advance of x,-will be as in case II. x r cos b cos I y -r cos b sin 1 cos e - r sin 6b sin i z r cos 6 sin e sin l- r sin 6b cos and by putting cos u = Cos b cos I sin b sin I cos b sin u = sin 0 cos 0 tan - tan b sin I they assume the following forms convenient for computation: - = r Cos U y - r sin u cos (6 + E) z = r sin u sin (8 + e)> 74. The following are the solutions and examples from the Monalliche Correspondenz referred to in this article, adopting the notation of article 74, and using L' to denote the longitude of the Sun. Given, 9, L', 1, b, i, n, to find u, r, /, and the auxiliary angles A, B, C, etc. 1. cos (L'-D ) tan b tan A sin A tan (L' -V ) tan zu sin (L'- 1) sin (A +-J i) 2. sin (L'-l) tan i tan B cos B sin b tan (L' — ) - tan cos (L'- g ) sin (B +-b) cos i sin (L' - ) tan b an sin Csin (L' - ) - t sin (L - l) tan i sin (C+ L' -- ) cos i 4 cos (L' — ) tanb t an D sin 1) tan (L' — g ) cos (L' — I) _ tan u cos (L'- l) tan i sin (D + L'- 1) cos i APPENDIX. 295 The angle u is to be taken between 0~ and 180~ when b is positive, and be. tween 180~ and 360~ when b is negative. When b 0~, the body is in one of the nodes of its orbit, in the ascending node when sin (L'- 1) and sin (1 — ) have the same sign; and in the descending node when they have opposite signs. It is immaterial in which of the two quadrants that give the same tangent, the auxiliary angles A, B, C, etc.,-are taken. In the following examples they are always taken between + 90~ and — 900~ IIL tan b sin sin (L - *) r 5. tail -E sin (1- ) sin (i- E) sin u R 6. tan i sin (1l- d )= tanF X cos Fsin (L' - ) sin b r sin (F — b) sin u cos 1i R 7. cosi tan u- tan cos Gin(L'I) sin (I- -G) cos u R 8. tan (l- ) tan H/ sin Hsin (L' -1) r cos i sin (H —u) sin (I- 2f) - R 9. tan b tan I sin Icos (L' - ) r sin i cos (I-S l ) sin (u - 1) R 10. sin i cos (1- g ) tan tu tan K cos Ksin b cos (L' - ) r sin (K- b) cos u R sin C sin (L - I) sin L r cos (+ L'- l) tan (L' - ) cosan sin (u —L) cos (L'-)- ) R 12. sin Dcos (L' -- ) tan l nM sin Mco r cos (D+' — 2)CoS i- sin (t - ) Cos (' -- ) - III. 13. r sin sn sin b 14. R sin E sin (L'- g ) sin i R cos E sin (L' - ) sini z sin (i- E) sin b sin (i - E) sin (I - 2 ) cos b R cos Fsin (L' — ) tan i R sin F sin (Lt- g) sin (I — ) sin (F- b) sin () sin (F-b) Other expressions for A4 may be obtained by combining 13 with all the formulas II. Examples: - Given, g= — 806 59'12".07,'- 281~ 1'34".99, -= 53~ 23'2".46, i= 10~37'9".55, z) — 3~ 6' 33".561, log R= 9.992615 8. 296 APPENDIX. 10, log tan b 8.7349698 n log sin A 8.8381955 n log cos (L' - ) 9.9728762 n log tan (L' -- ) 9.5620014 Clog sin (L' - b) 0.1313827 n Clog sin (A + i) 0.9350608 log tan A 8.8392287 n log tan u 9.3352577 1 A - 30 57' 2".136 u -- 120 12' 37".942 A+ -- 6~ 40' 7".414 20, log sin (L' - 1) 9.8686173 n log cos B 9.9953277 log tan i 9.2729872 log sin b 8.7343300 n C. log cos (L'-2) 0.0271238n log tan (L'- ) 9.5620014 log tan B 9.1687283 C. log sin (B +- b) 1.0360961 B - 8~ 23' 21".888 C. log cos Z 0.0075025 B + b - 50 16' 48".327 log tan u 9.3352577 n 30, log sin (L' - ) 9.5348776 n log sin C 9.1243583 n log tan b 8.7349698 n log sin (L'- ) 9.5348776 i a. log sin (L' - 1) 0.1313827 n Clogsin(C+-L'- ) 0.6685194 n C. log tan i 0.7270128 C. log cos i 0.0075025 log tan C 9.1282429 n log tan u 9.3352578 n C- -7039' 7".058 C+ i-' -- 192~ 23' 15".864 40, log cos (L' -- ) 9.9728762 n log sin D 9.5735295 n log tan b 8.7349698 n log tan (L' — 2 ) 9.5620014 C. log cos (L'- 1) 0.1714973 n log cos (' - 1) 9.8285027 n C. log tani 0.7270128 C. log sin(D+-L' —) 0.3637217 i log tan D 9.6063561 n C. log cos i 0.0075025 D = - 210 59' 51"'.182 log tan u 9.3352578 n D L' —I=- 205~ 38' 41".348 APPENDIX. 297 50~ log tan 6 8.7349698 n log sin E 9.0661081 log sin (I — ) 9.6658973 n log sin (L' —') 9.5348776 n log tan BE 9.0690725 C. log sin (i-E) 1.1637907 E — 6~ 41' 12".412. log sin u 0.6746802 n i-E= 30 55' 577".138 log R 0.4394566 logr log R + log R 0.4320724 60, log tan i 9.2729872 logcos 9.9983674 log sin (l — ) 9.6658973 n log sin b 8.7343300 n log tan F 8.9388845n log sin (L' - ) 9.5348776 n F — _ 4 57' 53".955 C. log sin (F — b) 1.4896990 n F-b -- 1~ 51' 20".394 C. log sin u 0.6746802 n C. log cos i 0.0075025 n log17 0.4394567 7,0 log cos i 9.9924975 log cos G 9.9903922 log tan u 9.3352577 n log sin (L'- 1) 9.8686173 n log tan C 9.3277552 n C. log sin (/ — -C) 0.5705092 n G -- 12 0' 27".118 C. log cos u 0.0099379 1 — -- -- 15~ 35' 42".492 log 0.4394566 8~e log tan (I-/ ) 9.7183744 n log sin H 9.6717672 n log cos i 9.9924975 log sin (L' — 1) 9.8686173 n log tan IT 9.7258769 n C. log sin (H- u) 0.5649695 n H- -280~' 39".879 UC. log sin (l —) 0.3341027 n II- --- 15~ 48' 1".937 logX 0.4394567 38 298 APPENDIX. 90. log tan b 8.7349698 n log sin l 9.4991749 n C. log sin i 0.7345153 log sin (L' - ) 9.9728762 nz C. log cos (1 —g) 0.0542771 C. log sin (t - I) 0.9674054 log tan I 9.5237622 n log _ 0.4394565 I — - 18~ 23' 55".334 u — _ 6~ 11' 17".392 10O. log sin i 9.2654847 log cos K 9.9997290 log cos (l —) 9.9475229 log sin b 8.7343300 n log tan u 9.3352577 n log cos (L'- -Q) 9.9728762 n log tan K 8.5482653 n C. log sin (K- b) 1.7225836.K -- 2 1' 26".344 C. log cos u 0.0099379 K- b- 1~ 5' 7".217 loge'0.4394567 11~. -+ EL' —I — 219~ 59'25".474 log sin C 9.1243583 n log sin L 9.5279439n log sin (L' — i) 9.8686173 n G. log sin (t- L) 0.8843888 C. log cos( (C —L' -- ) 0.1156850 n C. log cos (L' —) 0.0271238 in C.log tan (L' — ) 0.4379986 logo 0.4394565 C. log cos i 0.0075025 log tan L 9.5541617 n L= — 19~ 42' 32".533 u-L - L_ 7~ 29' 54".591 12~. 130. D+L' —-= 1780 2' 31".738 logr 0.4320724 log sin D 9.5735295 n log sin ut 9.3253198 n log cos (L'- ) 9.9728762 n log sill i 9.2654847 C.logcos(D+L' — 2) 0.00092536 n C.log sin b 1.2656700n C.logcosi 0.0075025 log d 0.2885469 logtan f( — L) 9.5541618 n APPENDIX. 299 76. If in the equations of article 60, x-X — cos 8 cos a y Y- i cos y sin a z Z — = sin d a denoting the right ascension, and Y the declination, we suppose X, Y, Z known, we have d x -- cos d sin a ossd a -- os a sin cos d dy - sin a cos Y dJ + d cos a cos d da - zd sin a sin Y d Y d z - sins d d - + cos r dd. Multiply the first of these by sin a, and subtract from it the second multiplied by cos a, and we find cos d cca - dx sin a + dy sin a. Multiply the first by cos a and add to it the second multiplied by sin a, and we find d x cos a + dy sin a = cos 6d,z- -zsin 6 d. Multiply this equation by - sin a and add it to the third of the differential equations above multiplied by cos Y and we find - dx cos a sin a - dy sin a sin 6T - dz cos =- zd d8 and, therefore, sin a COS a cos d da ll- dx+ dy cos ax sin 6 x-sinc asin 8 cos a d - dx- a S — do Fromn the formulas of article 56 of the Appendix are obtained dx x dy y dz z;-r-' d r-r dr rd- xcotan (A ),cotan (B + t) d = cotan C+ -u) dx dy. dz - x U r sin ucos, dos br sin U cos C, and the partial differentials dx dy dz --- 6-}/,1 E-y Cln e |-x f.ng t~ - C0 e, si 300 APPENDIX. whence dx=r dr + xcotan (A +-u) d v xcotan (A + u) dc r - [x cotan (A + t) + y cose + A- sin ] d + r sin u Cos a di dy ==dr + y cotan (B+-tu) dv+ycotan (B+-u) d a - [y cotan (B + u) - x cos E] d Q + r sin u cos b di dz= - d r -zcotan (C + u) d v+ z cotan (C + t) d i -[z cotan (C-t+u) - -xsine] d 2 +rsinu coscdi. These formulas, as well as those of 56 may be found in a small treatise tfe6er die D/iferentialfornelnz fiir Comeelen-Ba/lnen, etc., by G. D. E. WEYER, (Berlin, 1852). They are from BESSEL'S AbhaiZdlungy iber den Olers'schen Cometen. 90. GAuss, in the Berliner Astrono2misches Jefarbuceh for 1814, p. 256, has given another method of computing 5, and also 5 of article 100. It is as follows:We have — = x —6 -=x 9x ='o This fraction, by substituting for X the series of article 90, is readily transformed into -- 2 2.8 3.8.10 4.8.10.12 3 5 8. 10.12.14 4 105 -9- 9 -.11 -x + 9.11.13 - 9.I.13.15 x - etc.) Therefore, if we put 2.8 3.8.1o A = — l+- x+ 8 x2 + etc., we shall have x -15 X+ 80 - L-5 AX2 175 y means of wich n always be by mneans of which $ can always be found easily and. accurately. APPENDIX. 301 For ~, article 100, it is oniy necessary to write o in place of x in the preceding formulas. A may be computed more conveniently by the following formula:1..3.5.7 1.3..5.5.7.9 A=g (I 2 1( + X +, X2 1~ —2.,.91.3 s~etc.) 142. PROF. ENCKE, on the 13th of January, 1848, read a paper before the Royal Academy of Sciences at Berlin, entitled Ueher dend Ausnialhefall elner doppelen Bcdthizestimnnwyg cans denselbenz drei geocentriscklen Oer/en, in which he entered into a full discussion of the origin of the ambiguous case here mentioned, and the manner in which it is to be explained. The following paragraphs, containing useful instructions to the practical computer, embody the results of his investigation:By putting m c Q sin w q== (W + cr Equation IV., 141, becomes, for r'> B' m sin z =sin (z - q) and for r' < B' m sin4 z =sin (z + q) m)z is always positive. The number and the limits of the roots of this equation may be found by examining both forms. Take the first form, and consider the curves, the equations of which are y rsin4 z, MI = sin (z - q) y and y' being ordinates, and v abscissas. The first differential coefficients are 302 APPENDIX. There will, therefore, be a contact of the curves when we have m sin4 z- sin (z - a) and 4 m sin3 Z cos Z = cos (z - ) or when 4 sin ( - ) cos z cos (z - q) sin z which may be more simply written sin (2 z - q) - sin q. When the value of z deduced from this equation satisfies nz sin" z - sin (z - f) then there is a contact of the curves, or the equation has two equal roots. These equal roots constitute the limits of possibility of intersection of the curves, or the limits of the real roots of the equation. For the delineation of both curves it is only necessary to regard values of -- q between 00 and 180~, since for values between 1800 and 360~ the solution is impossible; and beyond 3600 these periods are repeated. The curve y' - sin (z - q) is the simple sine-curve, always on the positive side of y', and concave to the axis of abscissas, and has a maximum for z -- q 90~. The curve y _- sin4 z is of the fourth order, and since it gives d 4 zm sin3 z cos z -= m sin 2 z- sin 4 z dz d2y - 12 in sin2 z cos2 z-4 n sin4 - dZ2 =4 mn sin2 z (1 + 2 cos 2 z) 2 nz (cos 2 z - cos 4 z) d Y - 4 m (sin 2 z - 2 sin 4 z) d - -8 m (cos 2 z - 4 cos 4 z) it has a maximum for = 900 APPENDIX. 303 and a point of contrary flexure for z = 600, and z = 120'. From z 00 to z = 600 it is convex to the axis of abscissas, from 600 to 1200 it is concave, and convex from 120' to 1800. For osculation, the three equations, m11 sin4 z - sin ( -q) 4 m sin8 z cos z = cos (z - q) 4m sin2Z (I + 2 cos 2 z)=- sin(z-q) miust coexist, or rn sin4Z - sin (z - q) sin (2 z -q) 2 -- 5 sin q cos 2 o - 8 In this case we should have sin (2 ) - Aco) = Aos q -a sinq consequently, tan q = and sin q= or z=450 - -2 1sin'*. From these considerations we infer that for the equation M sin4 Z = sin (z - q) or even when it is in the form 2 sin' - 2 m ncos q sin5 jsin- sin-= 0 of the eighth degree, there can only be four real roots; because, in the whole period from z - q = 0' to z - - 360" only four intersections of the two curves are possible on the positive side of the axis of ordinates. Of these, three are between z = 0 and z = 1800, and one between 1800 and 1800 + q; or, inversely, one between 00 and 1800, and three between 1800 and 1800 +q; consequently, there are three positive and one negative roots, or th~ree negative and one positive roots for sin z. 304 APPENDIX. Contact of the curves can exist only when for a given value of q,'- -1- + 1 sin1- 3 sin gl and sin (- - q) sin4 z' If the contact of the curve of the fourth order with the sine-curve is without the latter, then will mn' constitute the upper limit, —for mn greater than this values of the roots will be impossible. There would then remain only one positive and one negative root. If the contact is within the sine-curve, then will the corresponding in" constitute the lower limit, and for mn less than this, the roots again would be reduced to two, one positive and one negative. If q is taken negative, or if we adopt the form nz sin4 2 = sin (z -+ q) 180 - z must be substituted for z. The equation n2 sin8 z - 2 m cos g sin5 z +- sin2,- sin2 g = 0 shows, moreover, according to the rule of DESCARTES,. that, of the four real roots three can be positive only when q, without regard to sign is less than 900, because in is always regarded as positive. For y greater than 90~, there is always only one real positive root. Now since one real root must always correspond to the orbit of the Earth, that is, to r'- R'; and since sin 6', in the equation, article 141, — _R' sin 6' sin z is always positive, so that it can be satisfied by none but positive values of z; an orbit can correspond to the observations only when three real roots are positive, or when g without regard to its sign is less than 90~. These limits are still more narrowly confined, because, also, there can be four real roots only when mn lies between in' and mn" and when we have siny<1, or sin qz can never be fulfilled. Or the middle root coincides with Y', then will the problem be solved only by the smallest root. Or, finally, the greatest of the, three roots differs least from 4'. in which case the choice must lie between the two smaller 39 3 0 6 APPENDIX. roots. Each of these will give a planetary orbit, because each one fulfils all the conditions, and it will remain to be determined, from observations other than the three given ones, which is the true solution. As the value of rn must lie between the two limits m' and in", so also nmust all four of the roots lie between those roots as limits which correspond to;n' and m". In Table IVa. are found, therefore, for the argument q from degree to degree, the roots corresponding to the limits, arranged according to their magnitude, and distinguished by the symbols z', z"x z', z. For every value of m which gives a possible solution, these roots will lie within the quantities given both for 9n' and n", and we shall be enabled in this manner, if d is found, to discern at the first glance, whether or not, for a given m and q, the paradoxical case of a double orbit can occur. It must, to be sure, be considered that, strictly speaking, Q' would only agree exactly with one of the z's, when the corrections of P and Q belonging to the earth's orbit had been employed, and, therefore, a certain difference even beyond the extremest limit might be allowed, if the intervals of time should be very great. The root z'%v for which sin z is negative, always falls out, and is only introduced here for the sake of completeness. Both parts of this table might have been blended in one with the proviso of putting in the place of z its supplement; for the sake of more rapid inspection, however, the two forms sin (x - q) and sin (z + q) have been separated, so that q is always regarded as positive in the table. To explain the use of Table IVa. two cases, are added; one, the example of Ceres in this Appendix, and the other, the exceptional case that occurred to Dr. GOULD, in his computation of the orbit of the fifth comet of the year 1847, an account of which is given in his Astronoinical Journal, Vol. I., No. 19. I. In our example of Ceres, the final equation in the first hypothesis is [0.9112987] sin4 z sin (z - 7 49' 2".0) and r' -- 24~ 19' 53".34 the factor in brackets being the logarithm. By the table, the numerical factor lies between n' and na and this' answers to concerning which there can be no hesitation, since z" must lie between 10~ 27' and 87~ 34'. Accordingly, we APPENDIX. 307 have only to choose for the z' which occurs in this case, and which, as we perceive, is to be sought between 7~ 50' and 10~ 27'. The root is in fact z= 7~ 59' 30".3, and the remaining roots, z -- 26 24 3 z" — 148 2 35 z -- 187 40 9 are all found within the limits of the table. 2. In the case of the fifth comet of 1847, Dr. GOULD derived from his first hypothesis the equation [9.7021264] sin4 sin (- + 32~ 53' 28".5). He had also f -- 133~ 0' 31". Then we have sin g < -, and the inspection of the table shows that the factor in the parenthesis lies between m' and nm"; therefore, there will be four real roots, of which three will be positive. The given d' approximates here most nearly to Z"'I about which, at any rate, there can be no doubt. Consequently, the paradoxical case of the determination of a double orbit occurs here, and the two possible values of z will lie between 880 29' - 1050 59' and 105 59 — 131 7 In fact, the four roots are, 950 31' 43".5 z 1 —17 31 13.1 I"' —:137 38 16.7 zI ~ 329 58 35.5. By a small decrease of In without changing q, or by a small decrease of g without changing m, a point of osculation will be obtained corresponding to nearly a mean between the second and third roots; and on the contrary, by a small increase of n without changing q, or a small increase of q without changing in, a point of osculation is obtained corresponding to nearly a mean between the first and second roots. 308 APPENDIX. We have, therefore, the choice between the two orbits. The root used by Dr. GOULD was z", which gave him an ellipse of very short period. The other observations showed him that this was not the real orbit. M. D'ARREsT was involved in a similar difficulty with the same cornet, and arrived also at an ellipse. An ellipse of eighty-one years resulted from the use of the other root. " Finally, both forms of the table show that the exceptional case can never occur when 6' < 63~ 26'. "It will also seldom occur when Y' < 90~. For then it can only take place with the first form sin (z - ), and since here for all values of q either the limits are very narrow or one of the limits approximates very nearly to 900, so it will be perceived that the case where there are two possible roots for 6' < 90~ will very seldom happen. For the smaller planets, therefore, which for the most part are discovered near opposition, there is rarely occasion to look at the table. For the comets we shall have more frequently' > 90~; still, even here, on account of the proximity to the sun, 6' > 150~ can, for the most part, be excluded. Consequently, it will be necessary, in order that the exceptional case should occur, th-at we should have in general, the combination of the conditions d' > 900 and g between 0~ and 32~ in the form sin (z - q), or between 22~ and 36~ 52' in the form sin (. + q)." Professor PEIRCE has communicated to the American Academy several methods of exhibiting the geomnetrical construction of this celebrated equation, andl of others which, like this, involve two parameters, some of which are novel and curious. In order to explain them, let us resume the fundamental equation, mn sin4 z - sin (z.- q). 1. The first method of representation is by logarithmic -curves; the logarithm of the given equation is log n + 4 log sin z - log sin (z - ). If wae construct the curve y = 4 log sin z, APPENDIX. 309 and also the same curve on another scale, in which y is reduced to one fourth of its value, so that y = log sin z, it is plain that if the second curve is removed parallel to itself by a distance equal to q in the direction of the axis of z, and by a distance equal to - log m in the direction of the axis of y, the value of z on the first curve where the two curves intersect each other will be a root of the given equation; for, since the point of intersection is on the first curve, its coordinates satisfy the equation, y - 4 log sin z, and because it is on the second curve its coordinates satisfy the equation, y+ -- logm- log sin (z - ); and by eliminating y from these two equations we return to the original equation, m sin4z -- sin (z 0 - ). A diagram constructed on this principle is illustrated by figure 5, and it will be readily seen how, by moving one curve upon the other, according to the changeable values of g and m, the points of intersection will be exhibited, and also the limits at which they become points of osculation. On this and all the succeeding diagrams, we may remark, once for all, that two cases are shown, one of which is the preceding example of the planet Ceres, in which the four roots of the equation will correspond in all the figures to the four points of intersection D, DI', D" D"', and the other of which is the very remarkable case that occurred to Dr. GOULD, approaching the two limits of the osculation of the second order, the details of which are given in No. 19 of his Asiromomical Jouirncal, and the points of which are marked on all our diagrams C, C', CG", C"' 2. The second method of representation is by a fixed curve and straight line, as follows. (a.) The fundamental equation, developed in its second member, and divided by am cos z, assumes the form sin4 z cos q ( - -(tan z -tan ~) cos z m By putting x = tan b, -= tang, a _ cos 310 APPENDIX. the roots of the equation will correspond to the points of intersection of tlle curve sin4 z X4 C (1 + X2) with the straight line y - a (x — b). [Figs. 6 and 6'.] It will be perceived that the curve line, in this as in all the following cases under this form, is not affected by any change in the values of mz and q, and that the position of the straight line is determined by its cutting the axis of x at the distance tan q from the origin, and the axis of y at the distance - sin q from the origin. The tangent of its inclination to the axis is obviously equal to cosy which may in some cases answer more conveniently for determining its position than its intersection with the axis of y. (b.) The development of the fundamental equation divided by in sin z, is sin3 =z in- (cotanq - cotanz); and by putting z -- cotan z b -- cotan q sin q a m the roots of the equation correspond to the intersection of the curve y= sin8 z(1 x2) with the straight line y a -(b x). [Fig. 7.] The position of the straight line is determined by its cutting the axis of x at a distance equal to cotan q from the origin, and the axis of y at a distance equal to cos q from the origin. This form of construction is identical with that given by M. Binet in the Jourzal de l'~cole Polytec/miique, 20 Cahier, Tome XIII. p. 285. His method of fixing the position of the straight line is not strictly accurate. This mode of representation is not surpassed by either of the others under this form. (c.) The fourth root of the fundamental equation developed, and divided by cos (z - g), assumes the form /m cos g (tan (z - ) - tan g) - / (i (Z — q) ) cos (z - q) APPENDIX. 311 By putting - =tan (a - ) b - tanq a - /mn cos q the roots of the equation correspond to the intersection of the curve ~/(sin (z -- )):1 Y ~Cos ( (z- q) + with the straight line y =- a (x + b). [Fig. 8.] The straight line cuts the axis of x at a distance equal to - tan and the axis of y at a distance equal to / nm sin q, from the origin. (d.) The development of the fourth root of the fundamental equation divided by sin (z - q) is, 3 ~m sin q (cotan (z - q) + cotan q) - cosec (- q). By putting x = cotan (z - M) b = cotan g a - m sin q the roots of the equation correspond to the intersection of the curve with the straight line y = a (x + b). [Figs. 9 and 9'.] The straight line cuts the axis of x at a distance equal to - cotan q, and the axis of y at a distance equal to / nm cos q, from the origin. (e.) From the reciprocal of the fundamental equation multiplied by rk its roots may be seen to correspond to the intersection of the curve r - cosec4 z with the straight line r -= cosec (z - q). [Figs. 10 and 10'.] Both these equations are referred to polar cooirdinates, of which r is the radius vector, z the angle which the radius vector makes with the polar axis,.n the distance of the straight line from the origin, and q the inclination of the line to the polar axis. 312 APPENDIX. (f). Fromi the reciprocal of the fourth root of the funtlamnlental equation, its roots may be seen to correspond to the intersection of the curve r -cose4 C with the straight line r 1- cosec (T + ) in which (p - q. [Fig. 11.] Both these equations are referred to polar coordinates, of which p is the angle which the radius vector r makes with the polar axis, / - the distance of the straight line from the origin, and q the inclination of the line to the polar axis. 3. The third mnethod of representation is by a curve and a circle. (a.) The roots of the fundamental equation correspond to the intersection of the curve r -- sin4 z with the circle 1 r =- sin (z - z). [Fig. 12.]. Both these equations are referred to polar coordinates, of which r is the radius vector, z the angle which the radius vector makes with the polar axis, - the radius of the circle which passes through the origin, and 90~q + is the angle which the diameter drawn to the origin makes with the polar axis. (b.) From the fourth root of the fundamental equation it appears that its roots correspond to the intersection of the equation r= / sin T with the circle r n/ sin (p + q) [Fig. 13], in which p (z- q) is the inclination of the radius vector to the polar axis, mn is the diameter of the circle which passes through the origin, and 90~ —q is the inclination of the diameter drawn through the origin of the polar axis. In these last two delineations the curve I' K_''I" incloses a space, within which the oentre of the circle must be contained, in order that there should be four real roots, and therefore that there should be a possible orbit. The curve APPENDIX. 313 itself corresponds to the limiting points of osculation denoted by Professor ENCKE'S M' and n", and the points K and K' correspond to the extreme points of osculation of the second order, for which ENCKE has given the values gq 36~ 52' and di' 4.2976, and gn" = 9.9999. On the delineations, X is the centre of the circle for our example of Ceres, and S' the same for Dr. GOULD'S exceptional case. A careful examination of the singular position of the point S' will illustrate the peculiar difficulties attending the solution of this rare example. 159. We add another example, which was prepared with great care to illustrate the Method of Computing an Orbit from three observations published in pamlphlet form for the use of the American Ephemeris and Nautical Almanac iia 1852. It furnishes an illustration of the case of the determination of two orbits from the same three geocentric places, referred to in article 142. We take the following observations, made at the Greenwich Observatory, from the volume for the year 1845, p. 36. Mean Time, Greenwich. Apparent Right Ascension. Apparent Declination. m. h. s. O / / 1845. July 30, 14 5 10.8 339 51 15.15 S. 23 31 34.60 Sept. 6, 11 5 56.8 332 22 39.30 27 10 23.13 Oct. 14, 8 19 35.9 328 7 51.45 26 49 57.23 From the Ncautical Almanac for the same year, we obtain Longitude of the Sun Distance from the Latitude of the Apparent Obliquity Date. n Nutation. from App. Equinox. Nutation.Earth. Sun. of the Ecliptic. 0 i/ I 1 i July 30. 127 40 11.32 +14.99 0.0064168 -0.17 23 27 28.13 Sept. 6. 164 9 40.85 +14.06 0.0031096 +0.21 28.41 Oct. 14. 201 21 12.49 +12.16 9.9984688 +0.53 28.05 The computation is arranged as if the orbit were wholly unknown, on which account we are not at liberty to free the places of Ceres from parallax, but must transfer it to the places of the earth. 40 14 APPENDIX. Reducing the observed places of the planet from the equator to the ecliptic, we find Date. App. Longitude of Ceres. App. Latitude of Ceres. July 30. 332 28 28.02 S. 13 54 52.47 Sept. 6. 324 35 58.87 14 45 30.00 Oct. 14. 321 4 54.55 13 5 35.33 And also, Date. Longitude of Zenith. Latitude of Zenith. July 30. 11 6 N. 53 26 Sept. 6. 4 49 56 22 Oct. 14. 1 4 58 4 The method of article 72 gives Date. Reduction of Longitude. Reduction of Distance. Reduction of Time. July 30. +16.32 +0.0001368 -0.070 Sept. 6. - 7.10 1421 -0.065 Oct. 14. -26.95 0907 -0.071 The reduction of time is merely added to show that it is wholly insensible. All the longitudes, both of the planet and of the earth, are to be reduced to the mlean vernal equinox for the beginning of the year 1845, which is taken as the epoch; the nutation, therefore, being applied, we are still to subtract the precession, which for the three observations is 28".99, 34".20, and 39".41, respectively; so that for the first observation it is necessary to add - 43".98, for the second, - 48'.26, and for the third,- 51".57. Finally, the latitudes and longitudes of Ceres are to be freed from the aberration of the fixed stars, by subtracting fromn the longitudes 18".76, 19".69, and 10".40, respectively, and adding to the latitudes - 2.02, + 1.72, and + 4.02, numbers which are obtained from the following formulas of Prof. PEIRCE - a rn cos (o - a) see a: nr sin (O - a) sin a; where 0 sun's longitude, and mn - aberration of o. APPENDIX. 315 The longitudes of the sun were corrected for aberration by adding 20".06, 20".21, and 20".43, respectively, to the numbers given in the Tcauical Almanaiac. These reductions having been made, the correct data of the problem are as follows - Times of observation. For Washington Meridian, I July 30. 372903. Sept. 6. 248435. Oct. 14. 132915. Ceres'slong. a, a', a" 330~ 27' 25".28 324 34 50.92 321 3 52.58 latitudes (3, (:' -3" 13 54 54.49 - 14 45 28.28 -- 13 5 31.31 Earth'slong. 1, 1/' 1" 307 39 43.66 344 8 45.49 21 19 53.97 logs. of dist. R, R', R" 0.0064753 0.0031709 9.9985083 By the formulas of Arts. 136 and 137, we find 7, 7,'. 7i... 3290 25' 34".81 2180 11' 22".38 194 59 35.15,', " Y..... 28 12 56.84 24 19 53.34 61 6 50.78 log Y', Y', C" sines 9.6746717 9.6149131 9.9422976 A'D, AD', A D" 199~ 45' 41".00 204~ 8' 25".14 203~ 56' 46".56 A" D, A" D A' D", 233 54 11.72 233 31 23.54 199 30 24.04 ~, d', e",Z 27 32 45.72 142 37 25.44 115 4 41.1]( log e, E, e" sines, 9.6650753 9.7832221 9.956992 log sin e' 9.9764767 log cos 2I d 9.5057153 And by article 138, log Tsint.... 6.2654993n log T cos t 9.2956278 n wherefore t =180~ 3' 12".63,log... 9.2956280 t+7-' - 380 14' 35".01, log sin (t t- y') 9.7916898 log S...... 8.6990834 log Tsin (t —).. 9.0873178 Whence log tan (d' )... 9.6117656' - a - 22~ 14' 47".47 and o = 2~ 5' 5"o87O 316 APPENDIX. By articles 140-143, we find XI DI - j"' -. 172' 24' 32".76 log sin 9.1208995 log cos 9.9961773 n A DI (Y 175 55 28.30 8.8516890 9.9989004 n A"D-Y" 172 47 20.94 9.0987168 AD - 6't-o =177 30 53.53 8.6370904 A=C175 43 49.72 8.8718546 A'D"-J'+a=177 15 36.57 8.6794373 loga....0.0095516P a- 1.0222370 log b... 0.1389045. Formula 13, which serves as a check, would give log b == 0.1389059. We prefer the latter value, because sin (A' D J-' + a) is less than sin (A' D" (Y/ + a). The interval of the time (not corrected) between the second and third obserm vations is 37.884480 days, and between the first and second 37.875532 days. The logarithms of these numbers are 1.5784613, and 1.5783587; the logarithm of k is 8.2355814; whence log 6 = 9.8140427, log 3" -- 9.8139401. We shall put, therefore, for the first hypothesis xr=logP= = 9.998974 y ~'=log Q= 0 0 6" 9.6269828 and we find C o 50 43- 56".13 wt-co=7 49 2.00 log Qcsin C 0.9112987 It is found, by a few trials, that the equation Q c sin w sin' = sin (z + 70 49' 2"'.00) is satisfied by the value = 70591 30".30, whence log sin o 9.1431101, and =~ - c= 0.4742939. APPENDIX. 317 Besides this solution, the equation admits of three others, z-= 260 24' 3" z=148 2 35 z= 187 40 9 The third must be rejected, because sin z is negative; the second, because z is greater than d'; the first answers to the approximation to the orbit of the earth, of which we have spoken in article 142.* The manner of making these trials is as follows. On looking at the table of sines we are led to take for a first approximation for one of the values, z = 8 nearly, or 80 + x. Then we have logsinz....... 9.14356+89x log sin4 z....... 6.57424 + 356 log Q e sino..... 0.91130 logsin(z-o-o)... 7.48554-+-356 z- o - =- a 00 10' 52"+ -5 X o —=-7 49 3 z =7 59 55 + -12 x, nearly -- 8 --. For the second approximation, we make z 70 59 30" O+ x; and have log sinz..... 9.1431056 + 150 x log sin z..... 6.5724224 + 600 x Q c sin..... 0.9112987 log sin (z — o —a). 7.4837211 + 600 xt z - a -= 0~ 10' 28".27 + -- x' nearly. to)+-o=7 49 2. 00 z- 7 59 30. 27 + - x'=- 7~ 59' 30".30. The process is the same for the other roots. * See article 142 of the Appendix, 3]18 APPENDIX. Again, by art. 143 we obtain — s185~ 10' 31"'.78 ~" - 189 25 30.25 log r 0.4749722 log r" - 0.4744748 A (u" + A-) = 264~ 21' 48".61 (u" — u) =288 49 5.19 2f = 6 57 7.46 2f" 6 56 32.68 The sum 2f -r-2f" which is a check, only differs by 0".20 from 2f', and the equation r' sin 2f n is sufficiently satisfied by distributing this 0".2 equally between 2f and 2f", so that 2f —= 6~597".36, and 2f"- 6~56'32".58. Now, in order that the times may be corrected for aberration, the distances Q, Q', Q" must be computed by the formulas of Art. 145, and then multiplied into the time 493B or Od.005706, as follows: — log r....... 0.47497 logsin(AD-)... 9.51187 comp. log sin.. 0 0.32533 log Q 0.31217 log const 7.76054 log of reduction 8.07271 Reduction - 0.011823 log r/ 0.47497 log sin (d — z) 9.44921 comp log sin I', 0.38509 log of reduction 0.30927 Reduction, 0.011744. * The constant of aberration is that of M. Struve, APPENDIX. 319 log r"........ 0.47447 log sin (A" D'-C")... 9.84253 log sin A"....... 0.05770 log of reduction.... 0.37470 Reduction = 0.013653 Observations. Corrected Times. Intervals. Logarithms. I. July 30. 361080 II. Sept. 6. 236691 37.875611 1.5783596 III. Oct. 14. 119260 37.882569 1.5784395 Hence the corrected logarithms of the quantities 8, 0" become 9.8140209, and 9.8139410. We are now, according to the precept of Art. 146, to commence the determination of the elements from the quantitiesf, r, r", 8, and to continue the calculation so far as to obtain in, and again from the quantities f", r, r, 8" so as to obtain'I". log..... 0.0011576 log "..... 0.0011552 log.... 9.9999225 log Q.... 9.6-309476 From the first hypothesis, therefore, there results X- 0.0000251, and Y= 0.0029648. In the second hypothesis, we assign to P and Q the values which we find in the first hypothesis for P' and Q'. We put, therefore, x-log P- 9.9999225, y=log Q- 9.6309476. Since the computation is to be performed in precisely the same manner as in tile first hypothesis, it is sufficient to set down here its principal results:-.. e...... 5043 561110 7 591 34" 98 (7+o-A... e. 1.97 logr'....... 0.4749037 log Qc sinw... 0.9142633 log -.... 0.7724177 no.147 320 APPENDIX. nf log, 0.7724952 ~(u u).. 264' 21' 50".64 1850 10'39" 64 1 (u"-u).... 288 49 5.57.189 2542.36 2f'..13 53 58 82 log r...... 0.4748696 2f.. 6 57 15 58 logr"...... 0.4743915 2f". 6 56 43 41 In this case we distribute the difference 0".1 7 so as to make 2f = 60 51' 15".49 and 2f"/ 60 56' 43".33. It would not be worth while to compute anew the reductions of the time on account of the aberration, for they'scarcely differ 1" from those which we derived from the first hypothesis. Further computations furnish log i7 = 0.0011582, log ii" = 0.0011558, whence are deduced logP'= 9.9999225, X-0.0000000 log Q' - 9.6309955, Y- 0.0000479. From which it is apparent how much more exact the second hypothesis is than the first. For the sake of completing the example, we will still construct the third hypothesis, in which we shall adopt the values of P' and Q' derived from the second hypothesis for the values of P and Q... Putting, therefore, r = logP== 9.9999225 y = log Q=9.6309955 the following are obtained for the most important parts of the computation: — (r). 50 43' 56".1I0 4.. 1890 25' 42".45 7 49 1.97 logr.... 0.4748690 log Qc sinw u.... 0.9143111 logr-"...... 0.4'743909.7 59'35"/.02 I (u+ u).... 2640 21' 50".64 logUr...... 0.4749031 1(U"-u).... 288 49 5.57 n r' log n r 0.7724168 2f'...f.. 13 53 58.94 nr lognv I. 0.7724943 2f.A. 6 57 15.65 5.. ~~1850 10' 39".69 2f"...... 6 56 43.419 APPENDIX. 321 The difference 0".2 between 2f' and 2f + 2f" is divided as in the first hypothesis, making 2f 6~ 57' 15".55, and 2f"- 60 56' 43".39. All these numbers differ so little from those given by the second hypothesis that it may safely be concluded that the third hypothesis requires no further correction; if the computation should be continued as in the preceding hypotheses, the result would be X= 0.0000000, Y= — 0.0000001, which last value must be regarded as of no consequence, and not exceeding the unavoidable uncertainty belonging to the last decimal figure. We are, therefore, at liberty to proceed to the determination of the elements from 2f', r, r", 8' according to the methods contained in articles 88-97. The elements are found to be as follows:Epoch of the mean longitude, 1845,.... 278~ 47' 13".79 Mean daily motion,..... 771".5855 Longitude of the perihelion,.... 1480 27' 49".70 Angle of eccentricity,........ 4 33 28.35 Logarithm of the major semi-axis.... 0.4417481 Longitude of the ascending node,.... 800 46' 36".94 Inclination of the orbit,........ 10 37 7.98 The computation of the middle place from these elements gives a' — 3240 34' 51".05, 9'= - 140 45' 28".31 which differ but little from the observed values a'-= 324~ 34' 50".92, d'-=- 140 45' 28".28. 41 322 APPENDIX. FORMULAS FOR COMPUTING THE ORBIT OF A COMET. Given Mean times of the observations in days,', F, /"' Observed longitudes of the comet, a', a", a"' Observed latitudes of the comet, g', /"', g"' Longitudes of the sun, A', A", A"' Distances of the sun from the earth R', RB" R"' Required The curtate distances from the earth9, Q' q", I"' Compute L tan " t"'- e' m sin (a'-A") - tanll sin (d'-iA") e/ t" —' tan a"'-m sin (" — A') and by means of this, approximately, e' =H Q'. R"' cos (A"'- A')- R' g cos (G - A') R'" sin (A'" — A') = g sin ( G -- A') g is the chord of the earth's orbit between the first and third places of the earth. G the longitude of the first place of the earth as seen from the third place. III. -11- cos (a"'- a') = h cos 5 cos (H- ad') sin (a"'- a') = h cos X sin (H- a'") Htan _"'- tan': — h sin CO h is always positive. If AN is a point, the coordinates of which, referred to the third place of the earth, are'cos a', Q' sin a', Q tan (, then are he', H, APPENDIX. 323 the polar coordinates of the third place of the comet, (that is, the distance, longitude and latitude,) referred to the point N as the origin. IV. cos t cos ( G- I).- cos q g sin 9 - A cos (I cos (a' - A') cos p' R' sin vp' — B' cos 3"' cos (a"' — A"') - cos V"' R"' sin /J'" - B"' By means of,,' ip"', A, B' B"', Olbers's formulas, become:2 -- (h' — COS p)2 + A2 r2 (ei' sec'- R' cos y')2 + B'2 r/2=- (Me' see l"' _ R"'" cos p"'")2 + B"'2 The computation would be somewhat easier by V. h cos 3' —f', g cos S -f' R' cos p'=- c' h Cos 3"''" 4 cos f g cos /p-f"'" R"' cost"' o'.~. ~t + -A2,42 _ + 2 A 2 r'2z — +c/ 2'f/122 =( +U C/'2 "' 2 in which u - he'-g cos cp VI. A value of u is to be found by trial which will satisfy the equation 3 2 t(r'+ "'+ -) — -(-'+ r" — k)j M t, in which log m'- 0.9862673 If no approximate value for e' or for r' or r"' is otherwise known, by means of which an approximate value of u can be found, we may begin with 4,,l Ba 324 APPENDIX. This trial will be facilitated by Table IJIa, which gives u corresponding to by means of which is found k, which corresponds rigorously to r, r" and t" —t': ( (e"-C- > E)' in which log x -- 8.5366114. The process may be as follows: For any value of u compute k, r' r", by V, and with r', r" compute ], with which Ft is to be taken from Table'Ia, and a value of k is to be computed which corresponds to the r' r, 1"-1 used. And u is to be changed until the second value of k shall agree exactly with that computed by V. Then we have u —+g cos (q 0'"~ Mq7~'. VII. os~ (a' - A') -- R'- r' cos 13' cos (a' — A') g/sina' - A') = r'cos b' sin (( - A')' tan (' r' sin b' c" Cos (a"'- At") - "'1 - r." cos 1"' cos (' - A"') " sin (a"' - A"') = /" cos 1"' sin ("'- A"') e"' tan ("' -- r' sin b"'. FIRST CONTROL, The values of r', r", obtained from these formulas, must agree exactly with those before computed. b'; 1", 1", are heliocentric longitudes and latitudes of the comet. The motion is direct when "'- 1' is positive, and retrograde when "'-1' is negative. APPENDIX. 325 VIIlo =- tan' tan i sin (1' —- ) tan b"' - tan b' cos (I"' —') tan i cos (1' ) sin (g1"' - tan ) cos i the inclination is always positive, and less than 900~ The upper signs are to be used when the motion is direct; the lower when it is retrograde. IXo tan( tn (L g ) tan ("' -- ) tan (- )o cos i cos i L' and L"' are the longitudes in orbit. SECOND CONTROL. The value of k before computed must be exactly k - /I [r2' + r"2 - 2' /" cos ( -I' EI)]. X. 41 _cos I (L' - ) Vq cos (L"'"-I') cosec- (L"' -- L') sin (L' — v'r' r'/ V" q a, the longitude of the perihelion, is counted from a point in the orbit from which the distance, in the direction of the order of the signs, to the ascending node, is equal to the longitude of the ascending node. XL The true anomalies are AV = L' tt- aI., With these the corresponding M' and 31"' are to be taken from BARKEnas Table, and we have then the time of perihelion passage 326 APPENDIX. in which M' and M'" have the sign of v' and v"'; the constant log n is log n -_ 0.0398723. The upper signs serve for direct, the lower for retrograde motion. For the use of Table IIa instead of BARKER'S Table, see Article 18 of the Appendix. THIRD CONTROL. The two values of T, from i, and t", must agree exactly. XII. With T q, g, i2 1", A", R", compute a" and i", and compare them with the observed values. And also compute with these values the formula tan ff sin (a - A"l) If this value agrees with that of m of formulas I., the orbit is exactly determined according to the principles of Olbers's Method. That is, while it satisfies exactly the two extreme places of the comet, it agrees with the observations in the great circle which connects the middle place of the Comet with the middle place of the Sun. If a difference is found, 1ZM can be changed until the agreement is complete. TABLES. T A B L E I. (See articles 42, 45.) ELLIPSE. HYPERBOLA. A Log B C T Log B C T 0.000 O O 0.00000 0 O 0.00000.001 0 0.00100 0 0.00100.002 0 2.00200 0 2.00200.003 1 4.00301 1 4.00299.004 1 7.00401 1 7.00399 0.005 2 11 0.00502 2 11 0.00498.006 3 16.00603 3 16.00597.007 4 22.00704 4 22.00696.008 5 29.00805 5 29.00795.009 6 37.00907 6 37.00894 0.010 7 46 0.01008 7 46 0.00992.011 9 56.01110 9 55.01090.012 11 66.01212 11 66.01189.013 13 78.01314 13 77.01287.014 15 90.01416 15 89.01384 0.015 11 7 1()3 0.01518 17 102 0.01482.016 C 19 118.01621 19 116.01080.017 i 22 133.01723 21 131.01677.018 24 149.01826 24 147.01774.019 27 166.01929 27 164.01872 0.020 30 184 0.02032 30 182 0.01968.021 33 203.02136 33 200.02065.022 36 223.02239 36 220.02162.023 40 244.02343 39 240.02258.024 43 265.02447 43 261.02355 0.025 47 288 0.02551 46 283 0.02451.026 51 312.02655 50 306.02547.027 55 336.02760 54 330.02643.028 59 362.02864 58 355.02739.029 63 388.02969 62 381.02834 0.030 67 416 0.03074 67 407 0.02930.031 72 444.03179 71 435.03025.032 77 473.03284 76 463.03120.033 82 503.03389 80 492.03215.034 1 87 535.03495 85 523.03310 0.035 0 92 567 0.03601 91 554 0.03404.036 1 97 600.03707 96 585.03499.037 103 634.03813 101 618.03593.038 108 669.03919 107 652.03688 *039 114 704.04025 112 686.03782 040 I 120 741.04132 118 722.03876 L 7 __...........1 _._ 2 TAB LE I. ELLIPS E. HYPERBOLA. A | Log B C T Log B C T 0.040 120 741 0.041319 118 722 0.038757.041 126 779.042387 124 758.039695.042 133 818.043457 130 795.040632.043 139 858.044528 136 833.041567.044 146 898.045601 143 872.042500 0.045 152 940 0.046676 149 912 0.043432.046 159 982.047753 156 953.044363.047 166 1026.048831 163 994.045292.048 173 1070.049911 170 1037.046220.049 181 1116.050993 177 1080.047147 0.050 188 1162 0.052077 184 1124 0.048072.051 196 1210.053163 191 1169.048995.052 204 1258.054250 199 1215.049917.053 212 1307.055339 207 1262.050838.054 220 1358.056430 215 1310.051757 0.055 228 1409 0.057523 223 1358 0.052675.056 236 1461.058618 231 1407.053592.057 245 1514.059714 239 1458.054507.058 254 1568.060812 247 1509.055420.059 263 1623.061912 256 1561.056332 0.060 272 1679 0.063014 265 1614 0.057243.061 281 1736.064118 273 1667.058152.062 290 1794.065223 282 1722.059060.063 300 1853.066331 291 1777.059967.064 309 1913.067440 301 1833.060872 0.065 319 1974 0.068551 310 1891 0.061776.066 329 2036.069664 320 1949.062678.067 339 2099.070779 3' 29 2007.063579.068 350 2163.071896 339 2067.064479.069 360 2228.073014 349 2128.065377 0.070 371 2294 0.074135 359 2189 0.066274.071 381 2360.075257 370 2251.067170.072 392 2428.076381 380 2314.068064.073 403 2497.077507 390 2378.068957.074 415 2567.078635 401 2443.069848 0.075 426 2638 0.079765 412 2509 0.070738.076 437 2709.080897 423 2575.071627.077 449 2782.082030 434 2643.072514.078 461 2856.083166 445 2711.073400.079 473 2930.084303 457 2780.074285.080 485 3006.085443 468 2850.075168 TABLE I. 3 ELLIPSE. HYPERBOLA. A Log B C T Log B C T 0.080 485 3006 0.085443 468 2850 0.075168.081 498 3083.086584 480 2921.076050.082 510 3160.087727 492 2992.076930.083 523 3239.088872 504 3065.077810.084 535 3319.090019 516 3138.078688 0.085 548 3399 0.091168 528 3212 0.079564.086 561 3481.092319 540 3287.080439.087 575 3564.093472 553 3363.081313.088 588 3647.094627 566 3440.082186.089 602 3732.095784 578 3517.083057 0.090 615 3818 0.096943 591 3595 0.083927.091 629 3904.098104 604 3674.084796.092 643 3992.099266 618 3754.085663.093 658 4081.100431 631 3835.086529.094 672 4170.101598 645 3917.087394 0.095 687 4261 0.102766 658 3999 0.088257.096 701 4353.103937 672 4083.089119.097 716 4446.105110 686 4167.089980.098 731 4539.106284 700 4252.090840.099 746 4634.107461 714 4338.091698 0.100 762 4730 0.108640 728 4424 0.092555.101 777 4826.109820 743 4512.093410.102 793 4924.111003 758 4600.094265.103 809 5023.112188 772 4689.095118.104 825 5123.113375 787 4779.095969 0.105 841 5224 0.114563 802 4870 0.096820.106 857 5325.115754 817 4962.097669.107 873 5428.116947 833 5054.098517.108 890 5532.118142 848 5148.099364.109 907 5637.119339 864 5242.100209 0.110 924 5743 0.120538 880 5337 0.101053.111 941 5850.121739 895 5432.101896.112 958 5958.122942 911 5529.102738.113 975 6067.124148 928' 5626.103578.1.14 993 6177.125355 944 5724.104417 0.115 1011 6288 0.126564 960 5823 0.105255.116 1029 6400.127776 977 5923.106092.117 1047 6513.128989 994 6024.106927.118 1065 6627.130205 1010 6125.107761.119 1083 6742.131423 1027 6228.108594.120 1102 6858.132643 1045 6331.109426 4 TAB LE 1. ELLIPSE. HYPERBOLA. A Log B C T Log B C T 0.120 1102 6858 0.132643 1045 6331 0.109426.121 1121 6976.133865 1062 6435.110256.122 1139 7094.135089 1079 6539.111085.123 1158 7213.136315 1097 6645.111913.124 1178 7334.137543 1114 6751.112740 0.125 1197 7455 0.138774 1132 6858 0.113566.126 1217 7577.140007 1150 6966.114390.127 1236 7701.141241 1168 7075.115213.128 1256 7825.142478 1186 7185.116035.129 1276 7951.143717 1205 7295.116855 0.130 1296 8077 0.144959 1223 7406 0.117675.131 1317 8205.146202 1242 7518.118493.132 1337 8334.147448 1261 7631.119310.133 1358 8463.148695 1280 7745.120126.134 1378 8594.149945 1299 7859.120940 0.135 1399 8726 0.151197 1318 7974 0.121754.136 1421 8859.152452 1337 8090.122566.137 1442 8993.153708 1357 8207.123377.138 1463 9128.154967 1376 8325.124186.139 1485 9264.156228 1396 8443.124995 0.140 1507 9401 0.157491 1416 8562 0.125802.141 1529 9539.158756 1436 8682.126609.142 1551 9678.160024 1456 8803.127414.143 1573 9819.161294 1476 8925.128217.144 1596 9960.162566 1497 9047.129020 0.145 1618 10102 0.163840 1517 9170 0.129822.146 1641 10246.165116 r 1538 9294.130622.147 1664 10390.166395 1559 9419.131421.148 1687 10536.167676 1580 9545.132219.149 1710 10683.168959 1601 9671.133016 0.150 1734 10830 0.170245 1622 9798 0.133812.151 1757 10979.171533 1643 9926.134606.152 1781 11129.172823 1665 10055.135399. 153 1805 11280.174115 1.686 10185.136191.154 1829 11432.175410 1708 10315.136982 0.155 1854 11585 0.176707 1730 10446 0.137772.156 1878 11739.178006 1752 10578.138561.157 1903 11894.179308 1774 10711.139349. 158 1927 12051.180612 1797 10844.140135.159 19052 12208.181918 181.9 10978.140920.160 1977 12366.183226 1842 11113.141704 TABLE I. 5 ELLIPSE. HYPERBOLA. A Log B C T Log B C T 0.160 1977 12366 0.183226 1842 11113 0.141704.161 2003 12526.184537 1864 1.1249.142487.162 2028 12686.185850 1887 11386.143269.163 2054 12848.187166 1 910 11523.144050.164 2080 13011.188484 1933 11661.144829 0.165 2106 13175 0.189804 1956 11800 0.1 45608.166 2132 13340.191127 1980 11940.146385.167 2158 13506.19245 2 2003 12081.147161.168 2184 13673.193779 2027 12222.147937.169 2211 13841.195109 2051 12364.148710 0.170 2238 14010 0.196441 2075 12507 0.149483.171 2265 14181.197775 2099 12651.1502'55.172 2292 14352.199112 2123 12795.151026.173 2319 14525.200451 2147 12940.151795.174 2347 14699.201793 2172 13086.152564 0.175 2374 14873 0.203137 2196 13233 0.153331.176 2402 15049.204484 2221 13380.154097.177 2430 15226.205832 2246 13529.154862.178 2458 15404.207184 2271 13678.155626.179 2486 15583.208538 2296 13827.156389 0.180 2515 15764 0.209894 2321 13978 0.157151.181 2543 15945.211253 2346 14129.157911.182 2572 16128.212614 2372 14281.158671.183 2601 16311.213977 2398 14434.159429.184 2630 16496.215343 2423 14588.160187 0.185 2660 16682 0.216712 2449 14742 0.160943.186 2689 16868.218083 2475 14898.161698.187 2719 17057.219456 2502 15054.162453,188 2749 17246.220832 2528 15210.163206.189 2779 \ 17436.222211 2554 15368.163958 0.190 2809 17627 0.223592 2581 15526 0.164709.191 2839 17820.224975 2608 15685.1 65458.192 2870 18013.226361 2634 15845.166207.193 2900 18208.227750 2661 16005.166955.194 2931 18404.229141 2688 16167.167702 0.195 2962 18601 0.230535 2716 16329 0.168447.196 2993 18799.231931 2743 16491.169192.197 3025 18998.233329 2771 16655.169935.198 3056 19198.234731 2798 16819.170678.199 3088 19400.236135 2826 16984.171419.200 3120 19602.237541 2854 17150.172159 ~~~6 ~TABLE I. ELLIPSE. HYPERBOLA. A | Log B C T Log B C T 0.200 3120 19602 0.237541 2854 17150 0.172159.201 3152 19806.238950 2882 17317.1728993.202 3184 20011.240361 2910 17484.173637.203 3216 20217.241776 2938 17652.174374.204 3249 20424.243192 2967 17821.175110 0.205 3282 20632 0.244612 2995 17991 0.175845.206 3315 20842.246034 3024 18161.176579.207 3348 21052.247458 3053 18332.177312.208 3381 21264.248885 3082 18504.178044.209 3414 21477.250315 311.1 18677.178775 0.210 3448 21690 0.251748 3140 18850 0.179505.211 3482 21905.253183 3169 19024.180234.212 351 6 22122.254620 3199 19199.180962.213 3550 22339,256061 3228 19375.181688.214 3584 22557.257504 3258 19551.182414 0.215 3618 22777 0.258950 3288 19728 0.183139.216 3653 22998.260398 3318 19906.183863.217 3688 23220.261849 3348 20084.184585.218 3723 23443.263303 3378 20264.185307.219 3758 23667.264759 3409 20444.186028 0.220 3793 23892 0.266218 3439 20625 0.186747.221 3829 24119.267680 3470 20806.187466.222 3865 24347.269145 3500 20988.188184.223 3900 24576.270612 3531 21172.188900.224 3936 24806,272082 3562 21355.189616 0.225:3973 25037 0.273555 3594 21540 0.190331.226 4009 25269.275031 3625 21725.191044.227 -4046 25502.276509 3656 21911.191757.228 4082 25737.277990 3688 22098.192468.229 4119 25973.279474 3719 22285.193179 0.230 -4156 26210 0.280960 3751 22473 0.193889.231 4194 26448.282450 3783 22662.194597.232 4231 26687.283942 3815 22852.195305.233 4269 26928.285437 3847 23042.196012.234 4306 27169.286935 3880 23234.196717 0.235 4344 27412 0.288435 3912 23425 0.197422.236.4382 27656.289939 3945 23618.198126.237 4'421 27901.291445 3977 23811.198829.238 4459 28148.292954 4010 24005.199530.239 4498 28395.294466 4043 24200.200231.2410 4537 28644.295980 4076 24396.200931 __-i;. llluiS _ TABLE I. ELLIPSE. HY PE R B OLA. A Log B C T Log B C T 0.240 4537 28644 0.295980 4076 24396 0.200931.241- 4576 28894.297498 4110 24592.201630.242 4615 29145.299018 4143 24789.202328.243 4654 29397.300542 4176 24987.203025.244 4694 29651.302068 421.0 25185.203721 0.245 4734 29905 0.303597 4244 25384 0.204416.246 4774 30161.305129 4277 20584.205110.247 4814 30418.306664 4311 25785.205803.248 4854 30676.308202 4346 25986.206495.249 4894 30935.309743 4380 26188.207186 0.250 4935 31196 0.311286 4414 26391 0.207876.251 4976' 31458.312833 4449 26594.208565.252 5017 31721.314382 4483 26799.209254.253 5058 31985.315935 4518 27004.209941.254 5099 32250.317490 4553 27209.210627 0.255 5141 32517 0.319048 4588 27416 0.211313.256 5182 32784.320610 4623 27623.211997.257 5224 33053.322174 4658 27830.212681.258 5266 33323.323741 4694 28039.213364.259 5309 33595.325312 4729 28248.214045 0.260 5351 33867 0.326885 4765 28458 0.214726.261 5394 34141.328461 4801 28669.215406.262 5436 34416.330041 4838 28880.216085.263 5479 34692.331623 4873 29092.216763.264 5522 34970.333208 4909 29305.217440 0.265 5566 35248 0.334797 4945 29519 0.218116.266 5609 35528.336388 4981 29733.218791.267 5653 35809.337983 5018 29948.219465.268 5697 36091.339580 5055 30164.220138.269 5741 36375.341181 5091 30380.220811 0.270 5785 36659 0.342785 5128 30597 0.221482.271 5829 36945.344392 5165 30815.222153.272 5874 37232.346002 5202 31033.222822.273 5919 37521.347615 5240 31253.223491.274 5964 37810.349231 5277 31473.224159 0.275 6009 38101 0.350850 5315 31693 0.224826.276 6054 38393.352473 5352 31915.225492.277 6100 38686.354098 5390 32137.226157.278 6145 38981.355727 5428 32359.226821.279 6191 39277.357359 5466 32583.227484.280 6237 39573.358994 5504 32807.228147 $ TABLE 1. ELLIPSE. HYPERB 0 LA. Log, B T Log B C T 0.280 6237 39573 0.358994 5504 32807 0.228147.281 6283 39872.360632 5542 33032.228808.282 6330 40171.362274 5581 33257.229469.283 6376 40472.363918 5619 33484.230128.284 6423 40774.3650566 5658 33711.230787 0.285.6470 41077 0.367217 5697 33938 0.231445.286 6517 41381.368871 5736 34167.232102.287 60564 41687.3700529 5775 34396.232758.288 6612 41994.372189 5814 34626.233"413.289 6660 42302.37385,3 5853" 348 5 6.234068 0.29() 6708 42611 0.3755521 58930 350O8 7 0.234721.291 6756 4292~2.377191 59032 3053 19.235374.292 6804 43233.378865 5972 3555'2.236025.293 6852 4030547.380542 6012 3 5785a.236676.294 6901 43861.382222 6052 36019.237326 0.295 ~6950 44177 0.3083906 6092 36253 0.287975.296 6999 44493.385593 6132 36489.2388023.297 7048 44812.387283 6172 36725'.239271.298 7097 45131.388977 6213 36961.239917.299 7147 45452.390673 6253 37199.240563.300 7196 45774.392374 6294 37437.241207 T A B L E II o (See Article 93.) 9 ~~~~~h log y yh log y h logy 0.0000 0.0000000 0.0040 0.0038332 0.0080 0.0076133.0001.0000965.0041.0039284.0081.0077071.0002.0001930.0042.0040235.0082.0078009.0003.0002894.0043.0041186.0083.0078947.0004.0003858.0044.0042136.0084.0079884 0.0005 0.0004821 0.0045 0.0043086 0.0085 0.0080821.0006.0005784.0046.0044036.0086.0081758.0007.0006747.0047.0044985.0087.0082694.0008.0007710.0048.0045934.0088.0083630.0009.0008672.0049.0046883.0089.0084566 0.0010 0.0009634 0.0050 0.0047832 0.0090 0.0085502.0011.001059 e.0051.0048780.0091.0086437.0012.0011556.0052.0049728.0092.0087372.0013.0012517.0053.0050675.0093.0088306.0014.0013478.0054.0051622.0094.0089240 0.0015 0.0014438 0.0055 0.0052569 0.0095 0.0090174.0016.0015398.0056.0053515.0096.0091108 ~.0017.0016357.0057 0054462.0097.0092041.0018.0017316.0058.0055407.0098.0092974 e.001827 0059.0056353.0099.0093906 0.0020 0.0019234 0.0060 0.0057298 0.0100 0.0094838.0021.0020192.0061.0058243.0101.0095770.0022.0021150.0062.0059187.0102.0096702.0023.0022107.0063.0060131.0103.0097633.0024.0023064,0064.0061075.0104.0098564 0.0025 0.0024021 0.0065 0.0062019 0.0105 0.0099495.0026.0024977.0066.0062962.0106.0100425.0027.002593 0067.0063905. 0107.0101355.0028.0026889.0068.0064847.0108.0102285.0029.0027845.0069.0065790.0109.0103215 0.0030 0.0028800 0.0070 0.0066732 0.0110 0.0104144.0031.0029755.0071.0067673.(0111.0105073.0032.0030709.0072'.0068614.0112.0106001.0033.0031663.0073.0069555.0113.0106929.0034.0032617.0074.0070496.0114.0107857 0.0035 0.0033570 0.0075 0.0071436 0.0115 0.0108785.0036.0034523.0076.0072376.0116.0109712.0037.0035476.0077.0073316.0117.0110639.0038.0036428.0078.0074255.0118.0111565.0039.0037380.0079.0075194.0119.0112491.0040.0038332.0080.0076133.0120.0113417 2 10 TABLE lI. h log yy h log yy h log y y 0.0120 0.0113417 0.0160 0.0150202 0.0200 0.0186501.0121.0114343.0161.0151115.0201.0187403.0122.0115268.0162.0152028.0202.0188304.0123.0116193.0163.0152941.0203.0189205.0124.0117118.0164.0153854.0204.0190105 0.0125 0.0118043 0.0165 0.0154766 0.0205 0.0191005.0126.0118967.0166.0155678.0206.01 91905.0127.0119890.0167.0156589.0207,0192805.0128.0120814.0168.0157500.0208.0193704.0129.0121737.0169 <.0158411.0209.0194603 0.0130 0.0122660 0.0170 0.0159322 0.0210 0.0195502.0131.0123582.0171.0160232.0211.0196401,0132.0124505.0172.0161142.0212.0197299.0133.0125427.0173.0162052-.0213.0198197.0134.0126348.0174.0162961.0214.0199094 0.0135 0.0127269 0.0175 0.0163870 0.0215 0.0199992.0136.0128190.0176.0164779.0216.0200889.0137.0129111.0177.0165688.0217.0201785.0138.0130032.0178.0166596.0218.0202682.0139.0130952.0179.0167504.0219.0203578; 0.0140 0.0131871 0.0180 0.0168412 0.0220 0.0204474.0141.0132791.0181.0169319.0221.0205369.0142.0133710.0182.0170226.0222.0206264.0143.0134629.0183.0171133.0223.0207159.0144.0135547.0184.0172039.0224.0208054 0.0145 0.0136465 0.0185 0.0172945 0.0)225 0.0208948.0146.0137383.0186.0173851.0226.0209842.0147.0138301.0187.0174757.0227.0210736.0148.0139218.0188.0175662.0228.0211630.0149.0140135.0189.0176567.0229.0212523 0.0150 0.0141052 0.0190 0.0177471 0.0230 0.0213 416.0151.0141968.0191.0178376.0231.0214309.0152.0142884.0192.0179280.0232.0215201.0153.0143800.0193.0180183.0233.0216093.0154.0144716.0194.0181087.0234.0216985 0.0155 0.0145631 0.0195 0.0181990 0.0235 0.0217876.0156.0146546.0196 -.0182893.0236.0218768.0157.0147460.0197.0183796.0237.0219659.0158.0148374.0198.0184698.0238.0220549,0159.0149288.0199.0185600.0239.0221440.0160.0150202.0200.0186501.0240.0222330 TABLE II. 11 h log y y logyy h log y y 0.0240 0.0222330 0.0280 0.0257700 0.0320 0.0292626.0241.0223220.0281.0258579.0321.0293494.0242.0224109.0282.0259457.0322.0294361.0243.0224998.0283.0260335.0323.0295228.0244.0225887.0284.0261213.0324.02960935 0.0245 0.0226776 0.0285 0.0262090 0.0325 0.0296961.0246.0227664.0286.0262967.0326.0297827.0247.0228552.0287.0263844.0327.0298693.0248.0229440.0288.0264721.0328.0299559.0249.0230328.0289.0265597.0329.0300424 0.0250 0.0231215 0.0290 0.0266473 0.0330 0.0301290.0251.0232102.0291.0267349.0331.0302154.0252.0232988.0292.0268224.0332.0303019.0253.0233875.0293.0269099,.0333.0303883.0254.0234761.0294.0269974.0334.0304747 0.0255, 0.0235647 0.0295 0.0270849 0.0335 0.0305611.0256.0236532.0296.0271723.0336.0306475.0257.0237417.0297.0272597.0337.0307338.0258.0238302.0298.0273471.0338.0308201.0259.0239187.0299.0274345.0339.0309064 0.0260 0.0240071 0.0300 0.0275218 0.0340 0.0309926.0261.0240956.0301.0276091.0341.0310788.0262.0241839.0302.0276964.0342.0311650.0263.0242723.0303.0277836.0343.0312512.0264.0243606.0304.0278708.0344.0313373 0.0260 0.0244489 0.0305 0.0279580 0.03453 0).0314234.0266.0245372.0306.0280452.0346.0315095.0267.0246254.0307.0281323.0347.0315956.0268.0247136.0308.0282194.0348.0316816.0269.0248018.0309.0283065.0349.0317676 0.0270 0.0248900 0.0310 0.0283936 0.0350 0.0318536.0271.0249781.()311.0284806.0351.0319396.0272.0250662.0312.0285676.0352.0320255.0273.0251543.0313.0286546.0353!.0321114.0274.0252423.0314.0287415.0354.0321973 0.0275 0.0253303 0.0315 0.0288284 0.0355 0.0322831.0276.0254183.0316.0289153.0356.0323689.0277.0255063.0317.0290022.0357.0324547.0278.0255942.0318.0290890.0358.03254035,.0279.0256821.0319.0291758.0359.0326262.0280.0257700.0320.0292626.0360.0327120 l; I _ I --!^1~~... 12 TAI3LE II. h logyy h log yy h 0log yy 0.0360 0.0327120 0.040 0.0361192 0.080 0.0681057.0361.0327976.041.0369646.081.0688612.0362.0328833.042.0378075.082.0696146.0363.0329689.043.0386478.083.0703661.0364.0330546.044.03948056.084.0711157 0.0365 0.0331401 0.045 0.0403209 0.085 0.0718633.0366.0332257.046.0411537.086.0726090.0367.0333112.047.0419841.087.0733527.0368.0333967.048.0428121.088.0740945.0369.0334822.049.0436376.089.0748345 0.0370 0.0335677 0.050 0.0444607 0.090 0.0755725.0371.0336531.051.0452814.091.0763087.0372.0337385.052.0460997.092.0770430.0373.0338239.053.0469157.093.0777754.0374.0339092.054.0477294.094.0785060 0.0375 0.0339946 0.055 0.0485407 0.095 0.0792348.0376.0340799.056.0493496.096.0799617.0377.0341651.057.0501563.097.0806868.0378.0342504.058.0509607.098.0814101.0379.0343356.059.0517628.099.0821316 0.0380 0.0344208 0.060 0.0525626. 0.100 0.0828513.0381.0345059.061.0533602.101.0835693.0382.0345911.062.0a'4155'6.102.0842854.0383.0346762.063.0549488.103.0849999.0384.0347613.064-.0557397.104.0857125 0.0385 0.0348464 0.065 0.05a' 65285 0.105 0.0864235.0386.0349314.066.0573150.106.0871327.0387.0350164.067 1.0580994.107.0878401.0388.0351014.068.0588817.108.0885459.0389.0351864.069.0596618.109.0892500 0.0390 0.0352713 0.070 0.0604398 0.110 0.0899523.0391.0353562.071.0612157.111.0906530.0392.0354411.072.0619895.112.0913520.0393.0355259.073.0627612.113.0920494.0394.0356108.074.0635308.114.0927451 0.0395 0.0356956 00.075 0.0642984 0.115 0.0934391.0396.0357804.076.0650639.116.0941315.0397.0358651.077.0658274.117.0948223.0398.0359499.078.0665888.118.0955114.0399-.0360346.079.0673483.119.0961990.0400.0361192.080.0681057.120.0968849 TABLE II. 13 h log yy h log y y h log y y 0.120 0.0968849 0.160 0.1230927 0.200 0.1471869.121.0975692.161.1237192.201.1477653.122.0982520.162.1243444.202.1483427.123.0989331.163.1249682.203.1489189.124.0996127.164.1255908.204.1494940 0.125 0.1002907 0.165 0.1262121 0.205 0.1500681.126.1009672.166.1268321.206.1506411.127.1016421.167.1274508.207.1512130.128.1023154.168 s.1280683.208.1517838.129.1029873.169.1286845.209.1523535 0.130 0.1036576 0.170 0.1292994 0.210 0.1529222.131.1043264.171.1299131.211.1534899.132.1049936.172.1305255.212.1540565.133.1056594.173.1311367.213.1546220.134.1063237.174.1317466.214.1551865 0.135 0.1069865 0.175 0.1323553 0.215 0.1557499.136.1076478.176.1329628.216.1563123.137.1083076.177.1335690.217.1568737.138.1089660.178.1341740.218.1574340.139.1096229.179.1347778.219.1579933 0.140 0.1102783 0.180 0.1353804 0.220 0.1585516. 141.1109323.181.1359818.221.1591089.142.1115849.182.1365821.222.1596652.143.1122360.183.1371811.223.1602204.144.1128857.184.1377789.224.1607747 0.145 0.1135340 0.185 0.1383755 0.225 0.1613279.146.1141809.186.1389710.226.1618802. 147.1148264.187.1395653.227.1624315.148.1154704.188.1401585.228.1629817.149.1161131.189.1407504.229.1635310 0.150 0.1167544 0.190 0.1413412 0.230 0.1640793.151.1173943.191.1419309.231.1646267.152.1180329.192.1425194.232.1651730.153.1186701.193.1431068.233.1657184.154.1193059.194.1436931.234.1662628 0.155 0.1199404 0.195 0.1442782 0.235 0.1668063.156.1205735.196.1448622.236.1673488.157.1212053.197.1454450.237.1678903.158.1218357.198.1460268.238.1684309.159.1224649.199.1466074.239.1689 705.160.1230927.200.1471869.240.1695092 j i 14 TABLE II. h logyy h log yy y h log yy 0.240 0.1695092 0.280 0.1903220 0.320 0.2098315.241.1700470.281.1908249.321.2103040.242.1705838.282.1913269.322.2107759.243.1711197.283.1918281.323.2112470.244.1716547.284.1923286.324.2117174 0.245 0.1721887 0.285 0.1928282 0.325 0.2121871.246.1727218.286.1933271.326.2126562.247.1732540.287.1938251.327.2131245.248.1737853.288.1943224.328.2135921.249.1743156.289.1948188.329.2140591 0.250 0.1748451 0.290 0.1953145 0.330 0.2145253.251.1753736.291.1958094.331.2149909.252.1759013.292.1963035.332.2154558.253.1764280.293.1967968.333.2159200.254.1769538.294.1972894.334.2163835 0.255 0.1774788 0.295 0.1977811 0.335 0.2168464.256.1780029.296.1982721.336.2173085.257.1785261.297.1987624.337.2177700.258.1790484.298.1992518.338.2182308.259.1795698.299.1997406.339.2186910 0.260 0.1800903 0.300 0.2002285 0.340 0.2191505.261.1806100.301.2007157.341.2196093.262.1811288.302.2012021.342.2200675.263.1816467.303.2016878.343.2205250.264.1821638.304.2021727.344.2209818 0.265 0.1826800 0.305 0.2026569 0.345 0.2214380.266.1831953.306.2031403.346.2218935.267.1837098.307.2036230.347.2223483.268.1842235.308.2041050.348.2228025.269.1847363.309.2045862.349.2232561 0.270 0.1852483 0.310 0.2050667 0.350 0.2237090.271.1857594.311.2055464.351.2241613.272.1862696.312.2060254.352.2246130.273.1867791.313.2065037.353.2250640.274.1872877.314.2069813.354.2255143 0.275 0.1877955 0.315 0.2074581 0.355 0.2259640.276.1883024.316.2079342.356.2264131.277.1888085.317.2084096.357.2268615.278.1893138.318.2088843.358.2273093.279.1898183.319.2093582.359.2277565.280.1903220.320.2098315.360.2282031............ TABLE II. 15 h log y h log yy h log yy 0.360 0.2282031 0.400 0.2455716 0.440 0.2620486.361.2286490.401.2459940.441.2624499.362.2290943.402.2464158.442.2628507.363.2295390.403.2468371.443.2632511.364.2299831.404.2472578.444.2636509 0.365 0.2304265 0.405 0.2476779 0.445 0.2640503.366.2308694.406.2480975.446.2644492.367.2313116.407.2485166.447.2648475.368.2317532.408.2489351.448.2652454.369.2321942.409.2493531.449.2656428 0.370 0.2326346 0.410 0.2497705 0.450 0.2660397.371.2330743.411.2501874.451.2664362.372.2335135.412.2506038.452.2668321.373.2339521.413.2510196.453.2672276.374.2343900.414.2514349.454.2676226 0.375 0.2348274 0.415 0.2518496 0.455 0.2680171.376.2352642.416.2522638.456.2684111.377.2357003.417.2526775.457.2688046.378.2361359.418.2530906.458.2691977.379.2365709.419.2535032.459.2695903 0.380 0.2370053 0.420 0.2539153 0.460 0.2699824.381.2374391.421.2543269.461.2703741.382.2378723.422.2547379.462.2707652.383.2383050.423.2551485.463.2711559.384.2387370.424.2555584.464.2715462 0.385 0.2391685 0.425 0.2559679 0.465 0.2719360.386.2395993.426.2563769.466.2723253.387.2400296.427.2567853.467.2727141.388.2404594.428.2571932.468.2731025.389.2408885.429.2576006.469.2734904 0.390 0.2413171 0.430 0.2580075 0.470 0.2738778.391.2417451.431.2584139.471.2742648.392.2421725.432.2588198.472.2746513.393.2425994.433.2592252.473.2750374.394.2430257.434.2596300.474.2754230 0.395 0.2434514 0.435 0.2600344 0.475 0.2758082.396.2438766.436.2604382.476.2761929.397.2443012.437.2608415.477.2765771.398.2447252.438.2612444.478.2769609.399.2451487.439.2616467.479.2773443.400.2455716.440.2620486.480.2777272 16 TABLE II. h log y y h log y y h log yy 0.480 0.2777272 0.520 0.2926864 0.560 0.3069938.481.2781096.521.2930518.561.3073437.482.2784916.522.2934168.562.3076931.483.2788732.523.2937813.563.3080422.484.2792543.524.2941455.564.3083910 0.485 0.2796349 0.525 0.2945092 0.565 0.3087394.486.2800151.526.2948726.566.3090874.487.2803949.527.2952355.567.3094350.488.2807743.528.2955981.568.3097823.489.2811532.529.2959602.569.3101292 0.490 1 0.2815316 0.530 0.2963220 0.570 0.3'104758.491.2819096.531.2966833.571.3108220.492.2822872.532.2970443.572.3111678.493.2826644.533.2974049.573.3115133.494.2830411.534.2977650.574.3118584 0.495 0.2834173 0.535 0.2981248 0.575 0.3122031.496.2837932.536.2984842.576.3125475.497.2841686.537.2988432.577.3128915.498.2845436.538.2992018.578.3132352.499.2849181.539.2995600.579.3135785 0.500 0.2852923 0.540 0.2999178 0.580 0.3139215.501.2856660.541.3002752.581.3112641.502.2860392.542.3006323.582.3146064.503.2864121.543.3009890.583.3149483.504.2867845.544.3013452.584.3152898 0.505 0.2871565 0.545 0.3017011 0.585 0.3156310.506.2875281.546.3020566.586.3159719.507.2878992.547.3024117.587.3163124.508.2882700.548.3027664.588.3166525.509.2886403.549.3031208.589.3169923 0.510 0.2890102 0.550 0.3034748 0.590 0.3173318.511.2893797.551.3038284.591.3176709.512.2897487.552.3041816.592.3180096.513.2901174.553.3045344.593.3183481.514.2904856.554.3048869.594.3186861 0.515 0.2908535 0.555 0.3052390 0.595 0.3190239.516.2912209.556.3055907.596.3193612.517.2915879.557.3059420.597.3196983.518.2919545.558.3062930.598.3200350.519.2923207.559.3066436.599.3203714.520.2926864.560.3069938.600.3207074 TABLE III. (See Articles 90, 100.) 17 xor z xorz 0.000 0.0000000 0.0000000 0.040 0.0000936 0.0000894.001.0000001.0000001.041.0000984.0000938.002.0000002.0000002.042.0001033.0000984.003.0000005.0000005.043.0001084.0001031.004.0000009.0000009.044.0001135.0001079 0.005 0.0000014 0.000001 4 0.045 0.0001188 0.0001128.006.0000021.0000020.046.0001242.0001178.007.0000028.0000028.047.0001298.0001229.008.0000037.0000036.048.0001354.0001281.009.0000047.0000046.049.0001412.0001334 0.010 0.0000058 0.0000057 0.050 0.0001471 0.0001389.011.0000070.0000069.051.0001532.0001444.012.0000083.0000082.052.0001593.0001500.013.0000097.0000096.053.0001656.0001558.014.0000113.0000111.054.0001720.0001616 0.015 0.0000130 0.0000127 0.055 0.0001785 0.0001675.016.0000148.0000145.056.0001852.0001736.017.0000167.0000164.057.0001920.0001798.018.0000187.0000183.058.0001989.0001860.019.0000209.0000204.059.0002060.0001924 0.020 0.0000231 0.0000226 0.060 0.0002131 0.0001988.021.0000255.0000249.061.0002204.0002054.022.0000273.062.0002278.0002121.023.0000306.0000298.063.0002354.0002189.024.0000334.0000325.064.0002431.0002257 0.025 0.0000362 0.0000352 0.065 0.0002509 0.0002327.026.0000392.0000381.066.0002588.0002398.027.0000423.0000410.067.0002669.0002470.028.0000455.0000441.068.0002751.0002543.029.0000489.0000473.069.0002834.0002617 0.030 0.0000523 0.0000506 0.070 0.0002918 0.0002691.031.0000559.0000539.071.0003004.0002767.032.0000596.0000575.072.0003091.0002844.033.0000634.0000611.073.0003180.0002922.034.0000674.0000648.074.0003269.0003001 0.035 0.0000714 0.0000686 0.075 0.0003360 0.0003081.036.0000756.0000726.076.0003453.0003162.037.0000799.0000766.077.0003546.0003244.038.0000844.0000807.078.0003641.0003327.039.0000889.0000850.079.0003738.0003411.040.0000936.0000894.080.0003835.0003496 18; TABLE III. X or z x or z 0.080 0.0003835 0.0003496 0.120 0.0008845 0.0007698.081.0003934.0003582.121.0008999.0007822.082.0004034.0003669.122.0009154.0007948.083.0004136.0003757.123.0009311.0008074.084.0004239.0003846.124.0009469.0008202 0.085 0.0004343 0.0003936 0.125 0.0009628 0.0008330.086.0004448.0004027.126.0009789.0008459.087.0004555.0004119.127.0009951.0008590.088.0004663.0004212.128.0010115.0008721.089.0004773.0004306.129.0010280.0008853 0.090 0.0004884 0.0004401 0.130 0.0010447 0.0008986.091.0004996.0004496.131.0010615.000912().092.0005109.0004593.132.0010784.0009255.093.0005224.0004691.133.0010955.0009390.094.0005341.0004790.134.0011128.0009527 0.095 0.0005458 0.0004890 0.135 0.0011301 0.0009665.096.0005577.0004991.136.0011477.0009803.097.0005697.0005092.137.0011654.0009943.098.0005819.0005195.138.0011832.0010083.099.0005942.0005299.139.0012012.0010224 0.100 0.0006066 0.0005403 0.140 0.0012193 0.0010366.101.0006192.0005509.141.0012376.0010509.102.0006319.0005616.142.0012560.0010653.103.0006448.0005723.143.0012745.0010798.104.0006578.0005832.144.0012933.0010944 0.105 0.0006709 0.0005941 0.145 0.0013121 0.0011091.106.0006842.0006052.146.0013311.0011238.107.0006976.0006163.147.0013503.0011387.108.0007111.0006275.148.0013696.0011536.109.0007248.0006389.149.0013891.0011686 0.110 0.0007386 0.0006503 0.150 0.0014087 0.0011838.111.0007526.0006618.151.0014285.0011990.112.0007667.0006734.152.0014484.00121.43.113.0007809.0006851.153.0014684.0012296.114.0007953.0006969.154.0014886.0012451 0.115 0.0008098 0.0007088 0.155 0.0015090 0.0012607.116.0008245.0007208.156.00152!).0012763.117.0008393.0007329.157.001.5502.0012921.118.0008542.0007451.158.0015710.0013079.119.0008693.0007574.159.0015920.0013238.120.0008845.0007698.160.0016131.0013398 _ _ J TABLE III. 19 x or z x or z 0.160 0.0016131 0.0013398 0.200 0.0025877 0.0020507.161.0016344.0013559.201.00261554.0020702.162.0016509.0013721.202.0026433.0020897.163.0016775.0013883.203.0026713.0021094.164.0016992.0014047.204.0026995.0021292 0.1 65 0.0017211 0.0014211 0.205 0.0027278 0.0021490.166.0017432.0014377.206.0027564.0021689.167.0017654.0014543.207.0027851.0021889.168.0017878.0014710.208.0028139.0022090.169.0018103.0014878.209.0028429.0022291 0.170 0.0018330 0.0015047 0.210 0.0028722 0.0022494.171.0018558.0015216.211.0029015.0022697.172.0018788.0015387.212.0029311.0022901.173.0019020.0015558.213.0029608.0023106.174.0019253.0015730.214.0029907.0023311 0.175 0.0019487 0.0015903 0.215 0.0030207 0.0023518.176.0019724.0016077.216.0030509.0023725.177.0019961.0016252.217.0030814.0023932.178.0020201.0016428.218.0031119.0024142.179.0020442.0016604.219.0031427.0024352 0.180 0.0020685 0.0016782 0.220 0.0031736 0.0024562.181.0020929.0016960.221.0032047.0024774.182.0021175.0017139.222.0032359.0024986.183.0021422.0017319.223.0032674.0025199.184.0021671.0017500.224.0032990.0025412 0.185 0.0021922 0.0017681 0.225 0.0033308 0.0025627.186.0022174.0017864.226.0033627.0025842.187.0022428.0018047.227'.0033949.0026058.188.0022683.0018231.228.0034272.0026275.189.0022941.0018416.229.0034597.0026493 0.190 0.0023199 0.0018602 0.230 ( 0.0034924* 0.0026711.191.0023460.0018789.231.0035252.0026931.192.0023722.0018976.232.0035582.0027151.193.0023985.0019165.233.0035914.0027371.194.0024251.0019354.234.0036248.0027593 0.195 0.0024518 0.0019544 0.235 0.0036584 0.0027816.196.0024786.0019735.236.0036921.0028039.197.0025056.0019926.237.0037260.0028263.198.0025328.0020119.238.0037601.0028487.199.0025602.0020312.239.0037944.0028713.200.0025877.0020507.240.0038289.0028939 20 TABLE III. x or C x or z 0.240 0.0038289 0.0028939 0.270 0.0049485 0.0036087.241.0038635.0029166.271.0049888.0036337.242.0038983.0029394.272.0050292.0036587.243.0039333.0029623.273.0050699.0036839.244.0039685.0029852.274.0051107.0037091 0.245 0.0040039 0.0030083 0.275 0.0051517 0.0037344.246.0040394.0030314.276.0051930.0037598.247.0040752.0030545.277.0052344.0037852.248.0041111.0030778.278.0052760.0038107.249.0041472.0031011.279.0053118.0038363 0.250 0.0041835 0.0031245 0.280 0.0053598 0.0038620.251.0042199.0031480.281.0054020.0038877.252.0042566.0031716.282.0054444.0039135.253.0042934.0031952.283.0054870.0039394.254.0043305.0032189.284.0055298.0039654 0.255 0.0043677 0.0032427 0.285 0.0055728 0.0039914.256.0044051.0032666.286.0056160.0040175.257.0044427.0032905.287.0056594.0040437.258.0044804.0033146.288.0057030.0040700.259.0045184.0033387.289.0057468.0040963 0.260 0.0045566 0.0033628 0.290 0.0057908 0.0041227.261.0045949.0033871.291.0058350.0041491.262.0046334.0034114.292.0058795.0041757.263.0046721.0034358.293.0059241.0042023.264.0047111.0034603.294.0059689.0042290 0.265 0.0047502 0.0034848 0.295 0.0060139 0.0042557.266.0047894.0035094.296.0060591.0042826.267.0048289.0035341..297.0061045.0043095.268.0048686.0035589.298.0061502.0043364.269.0049085.0035838.299.0061960.0043635.270.0049485.0036087.300.0062421.0043906 I~ _ TABLE Ia. 21 ELLIPSE. HYPERBOLA. A Log E, Log duff. Log Er Log diff. Log E, Log diff. Log E,.. Log duff. 0.000 0.0000000 9.2401 0.0000000 9.6378 0.0000000 9.2398 0.0000000 9.6378.001.0001738.2403 9.9995656.6381 9.9998263.2395.0004341.6375.002.0003477.2406.9991309.6384.9996528.2392.0008680.6372.003.0005217.2408.9986959.6386.9994794.2389.0013017.6370.004.0006958.2413.9982607.6389.9993061.2386.0017350.6367 0.005 0.0008701 9.2416 9.9978252 9.6391 9.9991329 9.2383 0.0021682 9.6365.006.0010445.2418.9973895.6394.9989598.2381.0026010.6362..007.0012190.2420.9969535.6396.9987869.2378.0030337.6360.008.0013936.2423.9965173.6399.9986141.2375.0034660.6357.009.0015683.2428.9960807.6402.9984414.2372.0038981.63054 0.010 0.0017432 9.2430 9.9956439 9.6405 9.9982688 9.2369 0.0043299 9.6352.011.0019182.2433.9952068.6407.9980963.2366.0047615.6349.012.0020933.2435.9947695.6410.9979240.2363.0051928.6347.013.0022685.2438.9943319.6412.9977517.2360.0056239.6344.014.0024438.2443.9938941.6414.9975796.2357.0060547.6342 0.015 0.0026193 9.2445 9.9934560 9.6417 9.9974076 9.2354 0.0064853 9.6339.016.0027949.2448.9930176.6420.9972357.2351.0069156.6336.017.0029706.24,53.9925789.6423.9970639.2348.0073456.6334.018.0031465.2455.9921400.6425.9968923.2345.0077754.6331.019.0033225.2458.9917008.6428.9967207.2342.0082049.6329 0.020 0.0034986 9.2460 9.9912614 9.6430 9.9965493 9.2339 0.0086342 9.6326.021.0036748.2460.9908217.6433.9963780.2336.0090632.6323.022.0038510.2465.9903817.6436.9962068.2333.0094920.6321.023.0040274.2470.9899415.6438.9960357.2330.0099205.6318.024.0042040.2472.9895010.6441.9958648.2328.0103487.6316 0.025 0.0043807 9.2475 9.9890602 9.6444 9.9956939 9.2325 0.0107767 9.6313.026.0045575.2477.9886192.6446.9955232.232)2.0112045.6311.027.0047344.2480.9881779.6449.9953526,.2319.0116320.6308.028.0049114.2485.9877363.6452.9951821.2316.0120592.6306.029.0050886.2487.9872945.6454.9950117.2313.0124862.6303 0.030 0.0052659 9.2490 9.9868524 9.6457 9.9948414 9.2310 0.0129130 9.6301.031.0054433.2494.9864100.6459.9946712.2307.0133395.6298.032.0056209.2497.9859674.6462.9945012.2304.0137657.6295.033.0057986.2499.9855245.6465.9943313.2301.0141917.6293 V.034.0059764.2502.9850813.6468.9941615.2298.0146175.6290 0.035 0.0061543 9.2504 9.9846378 9.6471 9.99939918 9.2295 0.0150430 9.6288.036.0063323.2509.9841940.6474.9938222.2292.0154683.6285.037.0065105.2512.9837499.6476.9936 528.2290.0158933.6283.038.0066888.2514.9833056.6478.9934834.2287.0163180.6280 I.039.0068672.2516.9828610.6481.9933142.2284.0167426.6278.040.0070457.2519.9824161.6484 Al.9931450.2281.0171668.6275 22 TABLE Ia. ELLIPSE. HYPERBOLA. A Log E, Log diff. Log Er Log diff. Log E, Log diff. Log E,.. Log Diff. 0.040 0.0070457 9.2519 9.9824161 9.6484 9.9931450 9.2281 0.0171668 9.6275.041.0072243.2524.9819709.6487.9929760.2278.0175908.6273.042.0074031.2526.9815255.6489.9928071.2275.0180146.6270.043.0075820.2531.9810798.6492.9926383.2272.0184381.6267.044.0077611.2533.9806339.6494.9924696.2269.0188614.6265 0.045 0.0079403 9.2536 9.9801877 9.6497 9.9923010 9.2266 0.0192844 9.6262.046.0081196.2538.9797412.6500.9921325.2263.0197072.6260.047.0082990.2543.9792944.6502.9919642.2260.0201297.6257.048.0084786.2546.9788474.6505.9917960.2258.0205520.6255.049.0086583.2548.9784001.6508.9916279.2255.0209740.6252 0.050 0.0088381 9.2550 9.9779525 9.6511 9.9914599 9.2252 0.0213958 9.6250.051.0090180.2555.9775046.6514.9912920.2249.0218174.6247.052.0091981.2558.9770564.6516.9911242.2246.0222387.6245.053.0093783.2560.9766079.6519.9909565.2243.0226597.6242.054.0095586.2565.9761592.6521.9907890.2240.02308050.6240 0.055 0.0097391 9.2567 9.9757102 9.6524 9.9906215 9.2237 0.0235011 9.6237.056.0099197.2570.9752609.6527.9904542.2235.0239214.6235.057.0101004.2072.9748113.6529.9902869.2232.0243415.6232.058.0102812.2577.9743615.6532.9901198.2229.0247614.6230.059.0104622.2579.9739114.6535.9899528.2226.0251810.6227 0.060 0.0106433 9.2582 9.9734611 9.6538 9.9897859 9.2223 0.0256003 9.6225.061.0108245.2584.9730103.6541.9896191.2220.0260194.6222.062.0110058.2589.9725593.6543.98945a25.2217.0264383.6220.063.0111873.2591.9721080.6546.9892859.2214.0268570.6217 o064.0113689.2094.9716565.6548.9891195.2211.0272753.6215 0.065 0.0115506 9.2598 9.9712047 9.6551 9.9889531 9.2208 0.0276935 9.6212.066.0117325.2601.9707526.6554.988 869.2206.0281114.6210.067.0119145.2603.9703002.6557.9886208.2203.0285291.6207.068.0120966.2606.9698475.6560.9884548.2200.0289465.6205.069.0122788.2610.9693945.6562.9882889.2197.0293637.6202 0.070 0.0124612 9.2613 9.9689413 9.6565 9.9881231 9.2194 0.0297807 9.6200.071.0126437.2617.9684878.6567.9879574.2191.0301974.6197.072.0128264.2620.9680340.6570.9877918.2189.0306139.6195.073.0130092.2622.9675799.6573.9876263.2186.0310301.6192.074.0131921.2625.9671255.6576.9874610.2183.0314461.6190 0.075 0.0133751 9.2629 9.9666708 9.6578 9.9872957 9.2180 0.0318618 9.6187.076.0135583.2632.9662159.6581.9871306.2177.0322773.6185.077.0137416.2634.9657606.6584.9869655.2174.0326926.6182.078.0139250.2638.9653051.6587.9868006.2172.0331076.6180.079.0141086.2641.9648492.6590.9866358.2169.0335224.6177.080 0142923.2643.9643931.6592 1 9864711.2166.0339370.6175 ~~~~~~~~~~~~~~~~~~~~~~~~~~~... TABLE Ia. 23 ELLIPSE. HYPERBOLA. A Log Ev Log diff. Log Er Log diff. Log Ev Log diff. Log Er. Log Diff. 0.080 0.0142923 9.2643 9.9643931 9.6592 9.9864711 9.2166 0.0339370 9.6175.081.0144761.2646.9639367.6595.9863065.2163.0343513.6172.082.0146601.2649.9634800.6598.9861420.2160.0347654.6170.083.0148442.2652.9630230.6600.9859776.2157.0351793.6167.084.0150284.2655.9625657.6603.9858133.2155.0355930.6165 0.085 0.0152128 9.2659 9.9621081 9.6606 9.9856491 9.2152 0.0360064 9.6163.086.0153973.2662.9616503.6609.9854850.2149.0364196.6160.087.0155819.2665.9611922.6611.9853210.2146.0368325.6158.088.0157667.2668.9607337.6614.9851572.2143.0372452.6155.089.0159516.2671.9602749.6617.9849934 o2140.0376577.6153 0.090 0.0161367 9.2674 9.9598159 9.6620 9.9848298 9.2138 0.0380699 9.6150.091.0163218.2677.9593566.6623.9846663.2135.0384819.6148.092.0165071.2680.9588970.6625.9845028.2132.0388937.6145.093.0166925.2684.9584371.6628.9843395.2129.0393052.6143.094.0168781.2687.9579769.6631.9841763.2126.0397165.6141 0.095 0.0170638 9.2690 9.9575164 9.6634 9.9840132 9.2123 0.0401276 9.6138.096.0172497.2693.9570556.6636.9838502.2121.0405385.6136.097.0174357.2696.9565945.6639.9836873.2118.0409491.6133.098.0176218.2700.9561331.6642.9835245.2115.0413595.6131.099.0178081.2703.9556714.6645.9833618.2112.0417696.6128 0.100 0.0179945 9.2706 9.9552095 9.6648 9.9831992 9.2109 0.0421796 9.6126.101.0181810.2708.9547472.6650.9830367.2107.0425893.6123.102.0183677.2712.9542847.6653.9828743.2104.0429988.6121.103.0185545.2715.9538218.6656.9827121.2101.0434080.6118.104.0187414.2718.9533586.6659.9825499.2098.0438170.6116 0.105 0.0189285 9.2722 9.9528951 9.6662 9.9823879 9.2095 0.0442258 9.6114.106.0191157.2725.9524314.6664.9822259.2093.0446343.6111.107.0193030.2728.9519673.6666.9820641'.2090.0450426.6109.108.0194905.2731.9515030.6670.9819023.2087.0454507.6106.109.0196781.2734.9510383.6673.9817407.2084.0458585.6104 0.110 0.0198659 9.2738 9.9505734 9.6676 9.9815791 9.2081 0.0462661 9.,6101.111.0200538.2741.9501081.6678.9814177.2079.0466735.6099.112.0202418.2744.9496425.6681.9812563.2076.0470807.6096.113.0204300.2747.9491766.6684.9810951.2073.0474876.6094.114.0206183.2750.9487105.6687.9809340.2070.0478943.6092 0.115 0.0208067 9.2754 9.9482440 9.6690 9.9807730 9.2067 0.0483008 9.6089.116.0209953.2757.9477772.6692.9806121.2065.0487071.6087.117.0211840.2760.9473101.6695.9804513.2062.0491131.6084.118.0213729.2763.9468428.6698.9802905.2059.0495189.6082.11-9.0215619.2767.9463751.6701.9801299.2056.0499245.6080.120.)217511.2770.9459071.6704.9799694.20541.0503298.6077 _ r_ i 24 TABLE Ia. ELLIPS E. HYPERBOLA., A Log Ev Log diff. Log Er Lo diff. Log E2, Log diff. Log E,.. Log Diff. 0.120 0.0217511 9.2770 9.9459071 9.6704 9.9799694 9.2054 0.0503298 9.6077.121.0219404.2773.9454388.6707.9798090.2051.0507349.6075.122.0221298.2776.9449702.6709.9796487.2048.0511399.6072.123.0223193.2779.9445013.6712.9794885.2045.0515446.6070.124.0225091.2783.9440321.6715.9793284.2043.0519490.6068 0.125 0.0226990 9.2786 9.9435626 9.6718 9.9791684 9.2040 0.0523533 9.6065.126.0228889.2789.9430927.6721.9790085.2037.0527573.6063.127.0230791.2792.9426226.6724.9788487.2034.0531611.6061.128.0232693.2795.9421521.6727.9786890.2032.0535647.6058.129.0234597.2799.9416813.6729.9785294.2029.0539681.6056 0.130 0.0236503 9.2802 9.9412103 9.6732 9.9783699 9.2026 0.0543712 9.6053.131.0238410.2805.9407389.6735.9782105.2023.0547741.6051.132.0240318.2808.9402672.6738.9780512.2021.0551768.6049.133.0242228.2812.9397952.6741.9778920.2018.0555793.6046.134.0244139.2815.9393229.6744.9777329.2015.0559816.6044 0.135 0.0246052 9.2818 9.9388503 9.6747 9.9775739 9.2012 0.0563836 9.6041.136.0247966.2822.9383773.6749.9774150.2010.0567854.6039.137.0249882.2825.9379041.6752.9772562.2007.0571870.6037.138.0251799.2828.9374305.6755.9770975.2004.0575884.6034.139.0253717.2831.9369567.6758.9769390.2001.0579895.6032 0.140 0.0255637 9.2834 9.9364824 9.6761 9.9767805 9.1998 0.0583904 9.6029.141.0257558.2838.9360079.6764.9766221.1996.0587911.6027.142.0259481.2841.9355331.6767.9764638.1993.0591916.6025.143.0261405.2844.9350580.6770.9763057.1990.0595919.6022.144.0263331.2848.9345825.6773.9761476.1988.0599919.6020 0.145 0.0265258 9.2851 9.9341067 9.6775 9.9759896 9.1985 0.0603917 9.6018.146.0267187.2854.9336307.6778.9758317.1982.0607913.6015.147.0269117.2857.9331543.6781.9756739.1979.0611907.6013.148.0271048.2861.9326775.6784.9755162.1977.0615899.6010.149.0272981.2864.9322005.6787.9753586.1974.0619888.6008 0.150 0.0274915 9.2867 9.9317231 9.6790 9.9752011 9.1971 0.0623876 9.6006.151.0276851.2871.9312455.6793.9750437.1969.0627861.6003.152.0278789.2874.9307675.6796.9748864.1966.0631844.6001.153.0280728.2877.9302892.6798.9747292.1963.0635825.5999.154.0282668.2880.9298106.6801.9745721.1960.0639804.5996 0.155 0.0284610 9.2884 9.9293317 9.6804 9.9744151 9.1958 0.0643780 9.5994.156.0286553.2887.9288524.6807.9742582.1955.0647755.5992.157.0288498.2890.9283728.6810.9741014.1952.0651727.5989.158.0290444.2893.9278929.6813.9739447.1949.0655697.5987.159.0292392.2897.9274127.6816.9737881.1946.0659665.5985.160.0294341.2900.9269321.6819.9736316.1944.0663631.5982 TABLE Ia. 25 ELLIPSE. HYPERBOLA. A Log E- Log diff. Log Er Log diff. Log Ev Log diff. Log E.. Log diff. 0.160 0.0294341 9.2900 9.9269321 9.6819 9.9736316 9.1944 0.0663631 9.5982.161.0296292.2903.9264512.6822.9734752.1941.0667595.5980.162.0298243.2906.9259700.6825.9733189.1938.0671556.5978.163.0300197.2910.9254885.6828.9731627.1936.0675516.5975.164.0302152.2913.9250067.6831.9730066.1933.0679473.5973 0.165' 0.0304109 9.2916 9.9245245 9.6833 9.9728506 9.1930 0.0683428 9.5971.166.0306067.2920.9240421.6836.9726947.1928.0687381.5968.167.0308026.2923.9235592.6839.9725389.1925.0691332.5966.168.0309987.2926.9230761..6842.9723831.1922.0695281.5963.169.0311949.2930.9225926.6845.9722275.1920.0699228.5961 0.170 0.0313913 9.2933 9.9221089 9.6848 9.9720719 9.1917 0.0703172 9.5959.171.0315879.2936.9216247.6851.9719165.1914.0707114.5956.172.0317846.2940.9211403.6854.9717611.1912.0711055.5954.173.0319815.2943.9206555.6857.9716059.1909.0714993.5952.174.0321784.2946.9201704.6860.9714507.1906.0718929.5949 0.175 0.0323756 9.2950 9.9196850 9.6863 9.9712957 9.1904 0.0722863 9.5947.176.0325729.2953.9191992.6866.9711407.1901.0726795.5945.177.0327704.2956.9187131.6869.9709859.1898.0730724.5942.178.0329680.2960.9182266.6872.9708311.1895.0734652.5940.179.0331657.2963.9177399.6875.9706764.1893.0738578.5938 0.180 0.0333636 9.2966 9.9172528 9.6878 9.9705218 9.1890 0.0742501 9.5935.181.0335617.2970.9167654.6881.9703673.1887.0746422.5933.182.0337599.2973.9162776.6884.9702129.1885.0750341.5931.183.0339582.2977.9157895.6886.9700587.1882.0754259.5928.184.0341568.2980.90153011.6889.9699045.1879.0758173.5926 0.185 0.0343555 9.2983 9.9148123 9.6892 9.9697504 9.1877 0.0762086 9.5924.186.0345543.2987.9143232.6895.9695964.1874.0765997.5922.187.0347533.2990.9138338.6898.9694425.1871.0769906.5919.188.0349524.2993.9133441.6901.9692887.1869.0773812.5917.189.0351517.2997.9128540.6904.9691350.1866.0777717.5915 0.190 0.0353511 9.3000 9.9123635 9.6907 9.9689813 9.1863 0.0781619 9.5912.191.0355507.3003.9118727.6910.9688278.1861.0785520.5910.192.0357505.3007.9113816.6913.9686743.1858.0789418.5908.193.0359504.3010.91089016.6916.9685210.1855.0793315.5906.194.0361505.3014.9103983.691.9.9683678.1853.0797209.5903 0.195 0.0363507 9.3017 9.9099062 9.6922 9.9682146 9.1850 0.0801102 9.5901.196.0365511.3020.9094138.6925.9680615.1847.0804992.5899.197.0367516.302..9089210.6928.9679086.1845.0808881.5896.198.0369523.3027.9084278.6931.9677557.1842.0812767.5894.199.0371532.3031.9079343.6934.9676029.1839.0816651.5892.200.0373542.3034.9074405.6937.9674502.1837.0820533).5889 26 TAB LE Ia. ELLIP SE. HYPERBOLA. __ -. r _ A Logg E I Log diff. Log E Log diff. Log Ev Log diff. Log Er. Log Diff. 0.200 0.0373542 9.3034 9.9074405 9.6937 9.9674502 9.1837 0.0820533 9.5889.201.0375554.3037.9069463.6940.9672976.1834.0824413.5887.202.0377567.3041.9064518.6943.9671451.1831.0828291.5885.203.0379582.3044.9059569.6946.9669927.1829.0832166.5882.204.0381598.3047.9054617.6949.9668404.1826.0836040.5880 0.205 0.0383616 9.3051 9.9049662 9.6952 9.9666882 9.1823 0.0839911 9.5878.206.0385635.3054.9044703.6955.9665361.1821.0843781.5876.207.0387656.3058.9039741.6958.9663841.1818.0847649.5873.208.0389679.3061.9034775.6961.9662321.1815.0851514.5871.209.0391703.3065.9029806.6964.9660803.1813.0855377.5869 0.210 0.0393729 9.3068 9.9024833 9.6967 9.9659285 9.1810 0.0859239 9.5867.211.0395757.3071.9019857.6970.9657768.1808.0863099.5864.212.0397786.3075.9014877.6974.9656253.1805.0866956.5862.213.0399817.3078.9009894.6977.9654738.1802.0870812.5860.214.0401849.3081.9004907.6980.9653224.1800.0874665.5858 0.215 0.0403883 9.3085 9.8999917 9.6983 9.9651711 9.1797 0.0878517 9.5855.216.0405918.3088.8994924.6986.9650199.1795.0882367.5853.217.0407955.3092.8989927.6989.9648687.1792.0886214.5851.218.0409994.3095.8984927.6992.9647177.1789.0890060.5849.219.0412034.3099.8979923.6995.9645667.1787.0893903.5846 0.220 0.0414076 9.3102 9.8974915 9.6998 9.9644159 9.1784 0.0897745 9.5844.221.0416120.3106.8969904.7001.9642651.1782.0901585.5842.222.0418165.3109.8964889.7004.9641145-.1779.0905422.5839.223.0420211.3112.8959881.7007.9639639.1776.0909258,5837.224.0422260.3116.8954849.7010.9638134.1774.0913091.5835 0.220 0.0424310 9.3119 9.8949824 9.7013 9.9636630 9.1771 0.0916923 9.5833.226.0426362.3123.8944795.7016.9635127.1768.0920753.5830.227.0428415.3127.8939762.7019.9633625.1766.0924580.5828.228.0430470.3130.8934726.7022.9632123.1763.0928405.5826.229.0432527.3133.8929687.7025.9630623.1760.0932229.5823 0.230 0.0434580 9.3137 9.8924644 9.7028 9.9629124 9.1758 0.0936050 9.5821.231.0436645.3140.8919597.7031.9627625.1755.0939870.5819.232.0438707.3144.8914547.7035.9626128.1752.0943687.5817.233.0440770.3147.8909493.7038.9624631.1750.0947503.5814.234.0442835.3151.8904436.7041.9623136.1747.0951317.5812 0.23,5 0.0444902 9.3154 9.8899375 9.7044 9.9621641 9.1745 0.0955128 9.5810.236.0446970.3158.8894310.7047.9620147.1742.0958938.5808.237.0449040.3161.8889242.7050.9618654.1740.0962745.5806.238.04511 11.3165.8884170.7053.9617162.1737.0966551.5803.239.0453184.3168.8879094.7056.9615670.1734.0970355.5801.240.0455259.3171.8874015.7059.9614180.1732.0974157.5799 TABLE Ia. 27 E LLIPSE. HYPERBOLA. A Log E | Log diff. Log Er Log diff. Log E, Log diff. Log Er, Log Diff. 0.240 0.0455259 9.3171 9.8874015 9.7059 9.9614180 9.1732 0.0974157 9.5799.241.0457335.3175.8868932.7063.9612690.1729.0977957.5797.242.0459413.3179.8863846.7066.9611202.1727.0981755.5794.243..0461493.3182.8858756.7069.9609714.1724.0985551.5792.244.0463575.3186.8853663.7072.9608227.1722.0989345.5790 0.245 0.0465658 9.3189 9.8848566 9.7075 9.9606741 9.1719 0.0993137 9.5788.246.0467743.3193.8843465.7078.9605256.1716.0996927.5786.247.0469830.3196.8838360.7081.9603771.1714.1000716.5783.248.0471918.3200.8833252.7084.9602288.1711.1004502.5781.249.0474008.3203.8828140.7087.9600805.1709.1008287.5779 0.250 0.0476099 9.3207 9.8823025 9.7090 9.9599324 9.1706 0.1012069 9.5777.251.0478193.3210.8817906.7094.9597843.1704.1015850.5775.252.0480288.3214.8812783.7097.9596363.1701.1019628.5772.253.0482385.3217.8807657.7100.9594884.1698.1023405.5770.254.0484483.3221.8802526.7103.9593406.1696.1027180.5768 0.255' 0.0486583 9.3224 9.8797392 9.7106 9.9591929 9.1693 0.1030953 9.5766.256.0488685.3226.8792254.7109.9590453.1691.1034724.5763.257.0490788.3231.8787113.7112.9588977.1688.1038493.5761.258.0492893.3235.8781968.7116.9587502.1685.1042259.5759.259.0495000.3238.8776819.7119.9586029.1683.1046024.5756 0.260 0.0497109 9.3242 9.8771666 9.7122 9.9584556 9.1680 0.1049787 9.5754.261.0499219.3245.8766510.7125.9583084.1678.1053548.5752.262.0501331.3249.8761350.7128.9581613.1675 1057308.5750.263.0503445.3252.8756186.7131.9580143.1673.1061065.5748.264 0505560.3256.8751019.7134.9578673.1670.1064821.5746 0.265 0.0507677 9.3260 9.8745848 9.7137 9.9577205 9.1668 0.1068574 9.5743.266.0509796.3263.8740673.7141.9575737.1665.1072326.5741.267.0511917.3267.8735495.7144.9574270.1662.1076076.5739.268.0514040.3270.8730312.7147.9572804.1660.1079824.5737.269.0516164.3274.8725126.7150.9571339.1657.1083570.5735 0.270 0.0518290 9.3277 9.8719936 9.7153 9.9569875 9.1655 0.1087314 9.5733.271.0520418.3281.8714742.7157.9568412.1652.1091056.5730.272.0522547.3284.8709544.7160.9566949.1650.1094797.5728.273.0524678.3288.8704343.7163.9565487.1647.1098536.5726.274.0526811.3292.8699137.7166.9564027.1644.1102272.5724 0.275 0.0528946 9.3295 9.8693928 9.7169 9.9562567 9.1642 0.1106007 9.5722.276.0531082.3299.8688715.7173.95 61108.1639.1109740.5719.277.0533220.3303.8683498.7176.9559650.1637.1113471.5717.278.0535360.3306.8678278.7179.9558193.1634.1117200.5715.279.0537502.3310.8673053.7182.9556736.1632.1120927.5713.28).05039646 3313.8667825.7185.9555281.1629.1124652.5710 28 TABLE Ia. ELLIPSE. HYPERBOLA. A _| Log Ev, | Log Er Log diff. Log EE Log diff. Log E. Lo Log Diff. 0.280 0.0539646 9.3313 9.8667825 9.7185 9.9555281 9.1629 0.1124652 9.5710.281.0541791.3317.8662593.7188.9553826.1627.1128375.5708.282.0543939.3320.8657357.7192.9552372.1624.1132097.5707.283.0546087.3324.8652117.7195.9550919.1622.1135817.5704.284.0548238.3327.8646873.7198.9549467.1619.1139534.5701 0.285 0.0550390 9.3331 9.8641625 9.7201 9.9548015 9.1617 0.1143250 6,9699.286.0552546.3335.8636374.7204.9546564.16i.1146964 i (98.287.0554700.3338.8631118.7208.9545115.1612.1150677 (5t9J.288.0556858.3342.8625859.7211.9543666.1609.1154387 05693.289.0559018.3345.8620596.7214.9542218.1606.1158096.5691 0.290 0.0561179 9.3349 9.8615329 9.7217 9.9540771 9.1604 0.1161803 9.5689.291.0563342.3353.8610058.7221.9539325.1601.1165508.5687.292.0565507.3356.8604783.7224.9537879.1599.1169211.5685.293.0567674.3360.8599504.7227.9536435.1596.1172913.5683.294.0569842.3364.8594221.7230.9534991.1594.1176612.5680 0.295 0.0572013 9.3367 9.8588935 9.7233 9.9533548 9.1591 0.1180310 9.5678.296.0574185.3371.8583644.7236.9532106.1589.1184006.5675.297.0576359.3375.8578349.7240.9530665.1586.1187699.5673.298.0578535.3379.8573051.7243.9529224.1584.1191391.5671.299.0580713.3383.8567748.7246.9527785.1581.1195081.5668.300.0582893.3387.8562442.7249.9526346.1578 0.1198768 9.5666 e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TABLE IIa. 29 0 o. v0. Log Al Log A2. Log Aa. 0 / ().00 -+3.7005216 -0.00000 — 9.695 2 2 47 11.83 3.7000079 0.47160 9.691 4 5 34 0.00 3.6984710 0.76930 9.681 6 8 20 1.19 3.6959236 0.93987 9.664 8 11 4 52.82 3.6923863 1.05702 9.641 10 13 48 13.31 -+3.6878872 -1.14430 -9.610 12 16 29 42.39 3.6824613 1.21171 9.571 14 19 9 1.36 3.6761493 1.26497 9.525 16 21 45 53.23 3.6689972 1.30744 9.470 18 24 20 2.89 3.6610547 1.34135 9.405 20 26 51 17.15.+3.6523748 -1.36825.-9.329 22 29 19 24.78 3.6430121 1.38929 9.239 24 31 44 16.52 3.6330224 1.40535 9.130 26 34 5 44.97 3.6224621 1.41714 8.994 28 36 23 44.51 3.6113863 1.42520 8.814 30 38 38 11.23 -+3.5998496 -1.43003 — 8.538 32 40 49 2.74 3.5879044 1.43201 — 7.847 34 42 56 18.02 3.5756011 1.43149'+8.237 36 44 59 57.33 3.5629877 1.42877 8.585 38 47 0 2.00 3.5501091 1.42410 8.753 40 48 56 34.33 -+3.5370077 -1.41772 +8.857 42 50 49 37.39 3.5237227 1,40983 8.928 44 52 39 14.95 3.5102905 1.40060 8.978 46 54 25 31.32 3.4967444 1.39o20 9.013 48 56 8 31.24 3.4831149 1.37878 9.038 50 57 48 19.82 -+3.4694297 -1.36645 +9.056 52 59 25 2.41 3.4557140 1.35333 9.067 54 60 58 44.53 3.4419903 1.33952 9.073 56 62 29 31.82 3.4282790 1.32512 9.076 58 63 57 29.99 3.4145981 1.31021 9.075 60 65 22 44.74 A+3.4009637 -1.29486 +9.071 64 68 5 26.60 3.3738900 1.26308 9.056 68 70 38 21.86 3.3.471520 1.23025 9.035 72 73 2 13.17 3.3208214 1.19672 9.008 76 75 17 40.91 3.2949510 1.16277 8.978 80 77 25 22.94 +3.2695785 -1.12863' +8.945 84 79 25 54.44 3.2447291 1.09447 8.910 88 81 19 47.97 3.2204185 1.06044 8.874 92 83 7 33.52 3.1966546 1.02665 8.837 96 84 49 38.62 3.1734393 0.99319 8.798 100 86 26 28.52 +3.1507694 -0.96012 +8.760 104 87 58 26.32 3.1286388 0.92749 8.721 108 89 25 53.18 3.1070382 0.89534 8.682 112 90 49 8.43 3.0859565 0.86370 8.643 116 92 8 29.76 3.0653811 a 0.83257 8.605 30 TABLE IIa. o -o vo. Log Ai. Log A2. Log As. 116 92 8 29.76 +3.0653811 -0.83257 +8.605 120 93 24 13.33 3.0452984 0.80199 8.567 124 94 36 33.98 3.0256943 0.77194 8.529 128 95 45 45.25 3.0065544 0.74244 8.491 132 96 51 59.60 2.9878638 0.71347 8.454 136 97 55 28.43 +2.9696079 -0.68505 +8.418 140 98 56 22.24 2.9517723 0.65716 8.382 144 99 54 50.68 2.9343427 0.62979 8.346 148 100 51 2.62 2.9178052 0.60293 8.311 152 101 45 6.25 2.9006462 0.57658 8.276 156 102 37 9.12 +2.8843526 -0.55071 +8.242 160 103 27 18.23 2.8684116 0.52534 8.209 164 104 15 40.03 2.8528110 0.50043 8.176 168 105 2 20.49 2.8375388 0.47598 8.143 172 105 47 25.18 2.8225838 0.45198 8.111 176 106 30 59.23 +2.8079349 -0.42841 +8.080 180 107 13 7.45 2.7935817 0.40526 8.049 184 107 53 54.28 2.7795141 0.38253 8.018 188 108 33 23.87 2.7657223 0.36020 7.988 192 109 11 40.10 2.7521971 0.33826 7.959 196 109 48 46.58 +2.7389297 -0.31670 +7.930 200 110 24 46.69 2.7259114 0.29551 7.901 210 111 50 16.87 2.6944032 0.24407 7.831 220 113 9 55.67 2.664c2838 0.19472 7.764 230 114 24 20.89 2.6354467 0.14732 7.700 240 115 34 4.97 +2.6077961 -0.10174 - +7.637 250 116 39 35.94 2.5812455 0.05786 7.577 260 117 41 18.16 2.5557170 0.015;56 7.519 270 118 39 32.86 2.5311401 9.97476 7.463 280 119 34 38.67 2.5074507 9.93535 7.409 290 120 26 51.98 +2.4845910 -9.89725 +7.356 300 121 16 27.30 2.4625078 9.86038 7.305 310 122 3 37.49 2.4411532 9.82467 7.256 320 122 48 34.01 2.4204831 9.79006 7.208 330 123 31 27.11 2.4004569 9.75648 7.161 340 124 12 25.97 +2.3810379 -9.72387 +7.116 350 124 51 38.87 2.3621918 9.69219 7.072 360 125 29 13.25 2.3438873 9.66139 7.029 370 126 5 15.87 2.3260956 9.63142 6.987 380 126 39 52.85 2.3087898 9.60224 6.947 390 127 13 9.75 +2.2919450 -9.57381 +6.907 400 127 45 11.66 2.2755384 9.54610 6.868 420 128 45 48.63 2.2439555 9.49269 6.794 440 129 42 16.43 2.2138871 9.44176 6.72.3 460 130 35 2.66 2.1851991 9.39310 6.655 TABLE IIa. 31 To. VO. Log Al. Log A2. Log A3. 460 13~0 35 2.66 +2.1851991 -9.39310 +96.655 480 131 24 30.82 2.1577741 9.34654 6.589 500 132 11 1.09 2.1315086 9.30188 6.527 520 132 54 50.84 2.1063114 9.25901 6.467 540 133 36 15.19 2.0821011 9.21777 6.409 560 134 15 27.33 +2.0588051 -9.17805 +96.353 580 134 52 38.80 2.0363588 9.13976 6.299 600 135 27 59.81 2.0147037 9.10278 6.247 640 136 33 45.52 1.9735615 9.03246 6.148 680 137 33 45.39 1.9350140 8.96649 6.055 720 138 28 48.27 +1.8987593 — 8.90438 +95.968 760 139 19 33.81 1.8645446 8.84571 5.885 800 140 6 34.57 1.8321564 8.79012 5.807 850 141 0 45.22 1.7939648 8.72451 5.714 900 141 50 30.05 1.7580440 8.66275 5.627 950 142 36 24.37 +1.7241428 — 8.60441 +95.544 1000 143 18 57.20 1.6920492 8.54915 5.466 1050 143 58 32.66 1.6615826 8.49665 5.392 1100 144 35 30.95 1.6325881 8.44666 5.321 1150 14a5 10 9.20 1.6049315 8.39896 5.254 1200 145 42 41.98 +1.5784963 — 8.35333 +95.189 1250 146 13 21.82 1.5531804 8.30962 5.127 1300 146 42 19.55 1.5288937 8.26767 5.068 1350 147 9 44.57 1.5055568 8.22735 5.011 1400 147 35 45.11 1.4830989 8.18853 4.956 1450 148 0 28.40 +1.4614567 -8.15110 +94.903 1500 148 24 0.83 1.4405738 8.11498 4.851 1600 149 7 55.10 1.4008865 8.04631 4.754 1700 149 48 6.25. 1.3636849 7.98190 4.663 1800 150 25 5.10 1.3286785 7.92126 4.576 1900 150 59 16.75 +1.2956243 -— 7.86398 +94.495 2000 151 31 1.89 1.2643177 7.80971 4.418 2100 152 0 37.76 1.2345845 7.75814 4.345 2200 152 28 18.85 1.2062750 7.70903 4.275 2300 152 54 17.45 1.1792601 7.66216 4.208 2400 153 18 44.05 +1.1534272 -7.61732 +94.145 2500 153 41 47.70 1.1286779 7.57435 4.084 2600 154 3 36.21 1.1049254 7.53310 4.025 2700 154 24 16.39 1.0820930 7.49344 3.969 2800 154 43 54.21 1.0601125 7.45526 3.914 2900 155 2 34.93 +1.03898230 -7.41844 +93.862 3000' 155 20 23.19 1.0184698 7.38289 3.811 3200 155 53 38.39 0.9795803 7.31529 3.715 3400 156 24 7.80 0.9431040 7.25186 3.625 3600 156 52 14.00 0.9087603 7.19213 3.540 32 TABLE IIa. To. VO. Log A1. Log A2. Log As. 3600 15~6 52 14.00 +0.9087603 — 97.19213 +93.540 3800 157 18 15.42 0.8763145 7.13568 3.459 4000 157 42 27.29 0.8455688 7.08218 3,383 4200 158 5 2.33 0.8163545 7.03133 3.311 4400 158 26 11.25 0.7885269 6.98289 3.242 4600 158 46 3.15 +0.7619607 — 96.93664 +93.176 4800 159 4 45.83 0.7365469 6.89238 3..113 5000 159 22 25.99 0.7121902 6.84996 3.053 5200 159 39 9.45 0.6888063 6.80923 2.995 5600 160 10 6.00 0.6446674 6.73234 2.885 6000 160 38 9.17 +0.6036264 — 96.66082 +92.783 6400 161 3 45.36 0.5652780 6.59398 2.688 6800 161 27 15.57 0.5292915 6.53125 2.599 7200 161 48 56.78 0.4953934 6.47215 2.514 7600 162 9 2.89 0.4633554 6.41629 2.435 8000 162 27 45.39 +0.4329843 — 96.36332 — 92.359 8400 162 45 13.90 0.4041157 6.31297 2.287 8800 163 1 36.52 0.3766081 6.26499 2.219 9200 163 17 0.16 0.3503393 6.21916 2.154 9600 163 31 30.72 0.3252029 6.17531 2.091 10000 163 45 13.32 +0.3011054 -96.13326 +92.031 10500 164 1 20.80 0.2723199 6.08303 1.959 11000 164 16 27.66 0.2448894 6.03516 1.891 11500 164 30 40.23 0.2186921 5.98944 - 1.826 12000 164 44 3.94 0.1936223 5.94568 1.764 13000 165 8 42.90 +0.1465042 -95.86343 +91.646 14000 165 30 55.26 0.1029147 5.78733 1.538 15000 165 51 4.63 0.0623627 5.71652 1.437 16000 166 9 29.58 0.0244528 5.65032 1.342 17000 166 26 24.88 9.9888624 5.58817 1.254 18000 166 42 2.53 +9.9553241 — 95.52959 +91.170 19200 166 59 18.90 9.9174751 5.46348 1.076 20400 167 15 11.32 9.8819393 5.40141 90.987 21600 167 29 51.00 9.8484507 5.34290 90.904 22800 167 43 27.11 9.8167866 5.28758 90.825 24000 167 56 7.28 +9.7867585 — 95.23512 +90.750 26000 168 15 26.77 9.7399215 5.15328 90.633 28000 168 32 51.95 9.6965794 5.07755 90.525 30000 168 48 41.17 9.6562474 5.00706 90.424 32000 169 3 8.84 9.6185347 4.94116 90.330 34000 169 16 26.46 +9.5831221 -94.87926 +90.242 36000 169 28 43.36 9.5497452 4.82093 90.159 38000 169 40 7.19 9.5181828 4.76576 90.080 40000 169 50 44.28 9.4882481 4.71343 90.00t5..;._ TABLE JIla. 33 77 Log ju. Log Diff. 7 Log u. Log Duff.? Log pi. Log Duff. 0.00 0.00000 00 0.30 0.00167 33 3.0594 0.60 0.00735 26 3.4468.01.00000 18 1.556.31.00179 01.0754.61.00763 61.4585.02.00000 72 1.857.32.00191 12.0910.62.00792 74.4703 0.03 0.00001 62 2.0354 0.33 0.00203 67 3.1062 0.63 0.00822 68 3.4822.04.00002 89.1614.34.00216 66.1211.64.00853 45.4941.05.00004 52.2589.35.00230 10.1356.65.00885 08.5061 0.06.00006 52 2.3385 0.36 0.00243 99 3.1498 0.66 0.00917 59 3.5182.07.00008 88.4057.37.00258 34.1638.67.00951 03.5304.08.00011 61.4639.38.00273 15.1774.68.00985 42 5427 0.09 0.00014 70 2.5152 0.39 0.00288 43 3.1911 0.69 0.01020 81 3.5551.10.00018 16.5617.40.00304 20.2044.70.01057 23.5677.11.00021 99.6031.41.00320 45.2175.71.01094 73.5805 0.12 0.00026 18 2.6410 0.42 0.00337 20 3.2304 0.72 0.01133 35 3.5934.13.00030 74.6767.43.00354 45.2433.73.01173 15.6066.14.00035 68.7097.44.00372 22.2557.74.01214 19.6200 0.15 0.00040 99 2.7404 0.45 0.00390 50 3.2681 0.75 0.01256 52 3.6336.16.00046 68.7694.46.00409 31.2807.76.01300 22.6476.17.00052 75.7966.47.00428 67.2930.77.01345 36.6618 0.18 0.00059 20 2.8222 0.48 0.00448 58 3.3053 0.78 0.01392 02 3.6765.19.00066 03.8466.49.00469 06 -3173.79.01440 31.6915.20.00073 25.8701.50.00490 11.3293.80.01490 32.7070 0.21 0.00080 86 2.8924 0.51 0.00511 75 3.3411 0.81 0.01542 18 3.7231.22.00088 86.9135.52.00533 98.3529.82.01596 03.7397.23.00097 25.9340.53.00556 83.3647.83.01652 02.7570 0.24 0.00106 04 2.9538 0.54 0.00580 30 3.3764 0.84 0.01710 33 3.7751.25.00115 23.9729.55.00604 41.3882.85.01771 19.7942.26.00124 83.9914.56.00629 19.4000.86.01834 86.8144 0.27.00134 84 3.0090 0.57 0.00654 65 3.4117 0.87 0.01901 65 3.8360.28.00145 25.0261.58.00680 80.4233.88.01971 95.8593.29.00156 08.0430.59.00707 66.4350.89.02046 29.8846 0.30 0.00167 33 3.0594 0.60 0.00735 26 8.4468 0.90 0.02125 29 3.9128.31.00179 01.0754.61.00763 61.4585.91.02209 92.9452.32.00191 12.0910 I.62.00792 74.4703 n.92 a.02301 60 34 TABLE IVa. m sin z4 sin (z - q). m and q positive. f _F~ s.S I I ZT II mr! I IV q1 q | ~ | I m' m I I mI m 1 4.2976 9.9999 1 0 1 20 1 20 89 4'0 89 4'0 177 37 180 55 181 0 2 3.3950 9.9996 2 0 2 40 2 40 89 20 89 20 175 14 181 51 182 0 3 2.8675 9.9992 3 0 4 0 4 0 89 0 89 0 172 52 182 46 183 0 4 2.4938 9.9986 4 0 5 20 5 20 88 40 88 40 170 28 183 42 184 0 5 2.2044 9.9978 5 0 6 41 6 41 88 19 88 19 168 5 184 37 185 0 6 1.9686 9.9968 6 0 8 1 8 1- 87 59 87 59 165 41 185 32 186 0 7 1.7698 9.9957 7 1 9 22 9 22 87 38 87 38 163 18 186 28 186 59 8 1.5981 9.9943 8 1 10 42 10 42 87 18 87 18 160 52 187 23 187 59 9 1.4473 9.9928 9 2 12 3 12 3 86 57 86 a57 158 28 188 18 188 58 10 1.3130 9.9911 10 3 13 25 13 25 86 35 86 35 156 3 189 13 189 57 1 1 1.1922 9.9892 11 5 14 46 14 46 86 14 86 1 4 153 37 190 9 190 56 12 1.0824 9.9871 12 7 16 8 16 8 85 52 85 52 151 10 191 4 191 54 13 0.9821 9.9848 13 9 17 31 17 31 85 29 85 29 148 43 191 59 192 52 14 0.8898 9.9823 14 12 18 53 18 53 85 7 85 7 146 14 192 54 193 49 15 0.8045 9.9796 15 16 20 17 20 17 84 43 84 43 143 45 [ 193 49 194 46 1 6 0.7254 9.9767 1 6 20 21 40 21 40 84 20 84 20 141 14 194 44 195 42 17 0.6518 9.9736 17 26 23 a 23 5 83 55 83 55 138 42 195 39 196 38 18 0.5830 9.9702 18 33 24 30 24 30 83 30 83 30 136 9 196 33 197 33 19 0.5185 9.9667 19 41 25 56 25 56 83 4 83 4 133 34 197 28 198 28 20 0.4581 9.9629 20 51 27 23 27 23 82 37 82 37 130 58 198 23 199 22 21 0.4013 9.9588 22' 2 28 50 28 50 82 10 82 10 128 19 199 17 200 15 22 0.3479 9.9545 23 15 30 19 30 19 81 41 81 41 125 38 200 11 201 8 23 0.2976 9.9499 24 31 31 49 31 49 81 11 81 11 122 55 201 6 202 0 24 0.2501 9.9451 25 49 33 20 33 20 80 40 80 40 120 9 202 0 202 51 25 0.2053 9.9400 27- 10 34 53 34 53 80 7 80 7 117 20 202 54 203 42 26 0.1631 9.9345 28 35 3 6 28 36 28 79 32 79 32 114 27 203 47 204 32 27 0.1232 9.9287 30 4 38 5 38 5 78 55 78 55 111 30 204 41 205 22 28 0.0857 9.9226 31 38 39 45 39 45 78 15 78 15 108 27 205 35 206 11 29 0.0503 9.9161 33 1841 41 27 41 27 77 33 77 33 105 19 206 28 207 0 30 0.0170 9.9092 35 5 43 13 43 13 76 47 76 47 102 3 207 21 207 48 31 9.9857 9.9019 37 1 45 4 45 4 75 56 75 56 98 37 208 14 208 36 32 9.9565 9.8940 39 9 47 1 47 1 74 59 74 59 95 0 209 6 209 24 33 9.9292 9.8856 41 33 49 6 49 6 73 54 73 54 91 6 209 58 210 11 4 9.9040 9.90409.8765 44 21 51 22 51 22 72 38 72 38 86 49 210 50 210 58 35 9.8808 9.8665 47 47 53 58 53 58 71 2 71 2 81 8t 53211 41 211 46 36 9.8600 9.8555a 52 31 57 13'57 13 68 47 68 47 75 40 212 32 212 33 d7 9.8443 9.8443 I 63 26 63 26 63 26 63 26 63 26 63 26 213 15 213 15 q' 360 52' 11.64' sin q' = 0.6 TABLE IVa. 35 m sin z4 = sin (z + q). m and q positive. _. I...'V q o m mP m r' m m 1 4.2976 9.9999 2 23 90 20 90 20 178 40 178 40 179 0 359 0 359 5 2 3.3950 9.9996 4 46 90 40 90 40 177 20 177 20 178 0 358 0 358 9 3 2.8675 9.9992 7 8 91 0 91 0 175 0 175 0 177 0 357 0 357 14 4 2.4938 9.9986 9 32 91 20 91 20 174 40 174 40 176 0 3 56 0 356 18 5 2.2044 9.9978 11 55 91 41 91 41 173 19 173 19 175 0 355 0 355 23 6 1.9686 9.9968 14 19 92 1 92 1 171 59 171 59 174 0 354 0 354 28 7 1.7698 9.9957 16 42 92 22 92 22 170 38 170 38 172 59 353 1 353 32 8 1.5981 9.9943 19 7 92 42 92 42 169 18 169 18 171 59 352 1 352 37 9 1.4473 9.9928 21 32 93 38 93 3 167 57 167 57 170 58 351 2 351 42 10 1.3130 9.9911 23 57 93 25 93 25 166 35 166 35 169 57 350 3 350 47 11 1.1922 9.9892 26 23 93 46 93 46 165 14 165 14 168 55 349 4 349 51 12 1.0824 9.9871 28 50 94 8 94 8 163 52 163 52 167 54 348 6 348 56 13 0.9821 9.9848 31 17 94 31 94 31 162 29 162 29 166 51 347 8 348 1 14 0.8898 9.9823 33 46 94 53 94 53 161 7 161 7 165 48 346 11 347, 6 15 0.8045 9.9796 36 15 95 17 95 17 159 43 159 43 164 44 345 14 346 11 16 0.7254 9.9767 38 46 95 40 95 40 158 20 158 20 163 40 344 18 345 16 17 0.6518 9.9736 41 18 96 5] 96 5 156 55 156 55 162 34 343 22 344 21 18 0.5830 9.9702 43 51 96 30 96 30 155 30 155 30 161 27 342 27 343 27 19 0.5185 9.9667 46 26 96 56 96 56 154 4 154 4 160 19 341 32 342 32 20 0.4581 9.9629 49 2 97 23 97 23 152 37 152 37 159 9 340 38 341 37 21 0.4013 9.9588 51 41 97 50 97 50 151 10 151 10 157 58 339 45 340 43 22 0.3479 9.9545 54 22 98 19 98.19 149 41 149 41 156 45 338 52 339 49 23 0.2976 9.9499 57 5 98 49 98 49 148 11 148 11 155 29 338 0 338 54 24 0.2501 9.9451 59 51 99 20 99 20 146 40 146 40 154 11 337 9 338 0 25[ 0.2053 9.9400 62 40 99 53 99 53 145 7 145 7 152 50 336 18 337 6 26 0.1631 9.9345 65 33 100 28 100 28 143 32 143 32 151 25 335 28 336 13 27 0.1232 9.9287 68 30 101 5 101 5 141 55 141 55 149 56 334 38 ]335 19 28 0.0857 9.9226 71 33 101' 45 101 45 140 15 140 15 148 22 333 49 334 25 29 0.0503 9.9161 74 41 102 27 102 27 138 33 138 33 146 42 333 0 333 32 30 0.0170 9.9092 77 57 103 13 103 13 136 46 136 46 144 55a 332 12 332 39 31 9.9857 9.9019 81 23 104 4 104 4 134 56 134 56 142 59 331 24 331 46 32 9.9565 9.8940 85 0 105 1 105 1 132 59 132 59 140 51 330 36 330 54 33 9.9292 9.8856 88 54 106 6 106 6 130 54 130 544 138 27 329 49 330 2 34 9.9040 9.8765 93 11 107 22 107 22 128 38 128 38 135 38 329 2 329 10 35 9.8808 9.8665 98 7 108 58 108 58 126 2 126 2 132 13 328 14 328 19 36 9.8600 9.8555a 104 20 111 13 111 13 122 47 122 47 127 29 327 27 327 28 q' 9.8443 9.8443 116 34 116 34 116 34 116 34 116 34 116 34 326 45 326 45 q'=- 36~ 52' 11.64" sin q' = 0.6 36 TAB LE Va. x. A. Diff. B. Diff. B'. Diff. 0 - 0.o00 — 9.60 — 0.000 — 11 — 0.000 — 34 1 9.00 9.00 0.011 11 0.034 34 2 17.99 8.98 0.023 12 0.067 33 3 26.95 8.95 0.034 11 0.101 34 4 35.88 8.91 0.045 11 0.134 33 5 - 44.77 - 8.87 -0.057 12 -0.167 -33 6 53.61 8.80 0.068 11 0.200 33 7 62.37 8.73 0.080 12 0.232 32 8 71.07 8.65 0.092 12 0.263 31 9 79.67 8.56 0.104 12 0.294 31 10 - 88.18 -8.46 -0.117 — 13 -0.324 -30 11 96.58 8.34 0.129 12 0.353 29 12 104.86 8.22 0.142 13 0.382 29 13 113.01 8.08 0.156 14 0.409 27 14 121.02 7.94 0.169 13 0.436 27 15 -128.88 - 7.79 - 0.183 -14 - 0.461 -25 16 136.59 7.62 0.197 14 0.486 25 17 144.12 7.43 0.211 14 0.509 23 18 151.47 7.27 0.226 15 0.531 22 19 158.63 7.08 0.241 15 0.552 21 20 -165.60 -6.86 — 0.256 — 15 — 0.571 - 19 21 172.35 6.65 0.27.1 15 0.590 19 22 178.89 6.43 0.287 16 0.606 16 23 185.20 6.20 0.303 16 0.622 16 24 191.28 5.96 0.319 16 0.636 14 25 — 197.11 -5.71 — 0.336 - — 17 — 0.648 -12 26 202.69 5.45 0.352 16 0.659 10 27 208.00 5.18 0.369 17 0.668 9 28 213.05 4.91 0.386 17 0.676 7 29 217.81 4.63 0.403 17 0.682 6 30 — 222.30 — 4.34 -0.419 — 16 — 0.687 - 4 31 226.48 4.04 0.436 17 0.690 3 32 230.37 3.74 0.453 17 0.692 1 33 233.95 3.42 0.470 17 0.692 0 34 237.21 3.10 0.486 16 0.691 -4 2 35 -240.15 — 2.78 -0.502 -— 16 -0.688 + 4 36 242.76 2.45 0.518 16 0.683 5 37 245.04 2.11 0.534 16 0.677 6 38 246.98 1.77 0.549 15 0.670 8 39 248.57 1.41 0.564 15 0.661 9 40 -249.80 -1.06 -0.578 — 14 -0.651 +11 41 250.68 0.70 0.591 13 0.639 12 42 251.20 0.33 0.604 12 0.627 13 | ___~~~~~..... TABLE Va. 37 X. A. I'Duff. I B. Diff. B'. DifF. 42 -251.20 - 0.33 -0.604 - 12 -0.627 +13 43 251.34 + 0.04 0.615 11 0.613 15 44 251.11 0.42 0.626 11 0.597 16 45 250.50 0.80 0.636 10 0.580 17 46 249.51 1.18 0.645 8 0.563 18 47 -248.13 ~ 1.57 -0.652 - 7 -0.544 ~19 48 246.36 1.96 0.659 6 0.524 20 49 244.20 2.36 0.664 4 0.503 21 50 241.64 2.76 0.667 3 0.482 22 51 238.68 3.16 0.669 1 0.459 23 52 -235.31 + 3.57 -0.669 + 1 -0.436 +23 53 231.54 3.98 0.667 2 0.412 24 54 227.35 4.39 0.664 4 0.387 25 55 222.76 4.80 0.659 6 0.361 26 56 217.75 5.22 0.651 9 0.335 26 57 -212.32 + 5.64 -0.641 + 11 -0.309 +26 58 206.47 6.06 0.629 13 0.282 27 59 200.20 6.47 0.615 15 0.255 27 60 193.52 6.90 0.598 18 0.227 28 61 186.40 - 7.32 0.579 20 0.200 27 62 -178.87 + 7.74 -0.557 ~ 23 -0.172 +28 63 170.91 8.17 0.532 26 0.144 28 64 162.52 8.60 0.504 29 0.116 28 65 153.70 9.03 0.474 32 0.088 28 66 144.46 9.45 0.440 35 0.061 27 67 -134.79 + 9.88 -0.403 + 38 -0.033 +28 68 124.69 10.31 0.363 41 -0.006 27 69 114.16 10.74 0.320 45 +0.021 27 70 103.20 11.17 0.273 49 0.048 27 71 91.81 11.60 0.222 52 0.074 26 72 - 80.00 ~12.03 -0.168 + 56 +0.099 +25 73 67.75 12.46 0.110 59 0.124 25 74 55.07 12.89 0.049 63 0.148 24 75 41.97 13.32 +0.016 67 0.172 24 76 28.43 13.72 0.086 71 0.195 22 77 - 14.47 +14.18 +0.159 + 75 +0.216 +21 78 0.07 14.61 0.237 80 0.237 21 79 + 14.76 15.04 0.319 84 0.257 20 80 30.02 15.47 0.405 88 0.276 1 9 81 45.70 15.89 0.496 93 0.294 18 82 + 61.80 -[ —16.32 +0.591 + 97 +0.311 +16 83 78.34 16.76 0.691 102 0.326 15 84 95.32 17.19 0.795 106 0.340 13 38 TABLE Va. x. A. Diff. B. Diff. 3'. Diff. 84 _+ 95.32 +17.19 - 0.795 +106 +0.340 + 13 85 112.72 17.62 0.904 111 0.352 12 86 130.56 18.06 1.018 116 0.363 10 87 148.84 18.49 1.137 121 0.373 9 88 167.54 18.92 1.261 126 0.381 7 89 + 186.69 +19.36 + 1.390 +132 +0.386 + 5 90 206.27 19.80 1.525 137 0.390 3 91 226.29 20.24 1.665 142 0.392 1 92 246.75 20.68 1.810 148 0.392 - 1 93 267.65 21.13 1.961 154 0.390 3 94 + 289.01 -t21.58 + 2.118 +159 +0.385 - 6 95 310.82 22.03 2.280 165 0.378 8 96 333.08 22.49 2.449 171 0.368 11 97 355.80 22.95 2.623 178 0.355 14 98 378.99 23.42 2.805 184 0.339 17 99 -+ 402.65 +23.89 + 2.992 +191 +0.320 - 21 100 426.78 24.37 3.187 198 0.297 25 101 451.40 24.86 3.388 204 0.270 28 102 476.51 25.36 3.596 212 0.240 32 103 502.12 25.86 3.812 220 0.205 37 104 + 528.24 +26.38 + 4.036 +227 +0.165 - 42 105 554.88 26.90 4.267 235 0.121 47 106 582.04 27.43 4.506 240 0.071 53 107 609.75 27.99 4.755, 250 +0.015 59 108 638.02 28.55 5.012 261 -0.048 66 109 + 666.85 +29.11 + 5.278 +271 — 0.117 - 72 110 696.27 29.72 5.554 281 0.193 80 111 726.29 30.33 5.841 292 0.278 89 112 756.93 30.96 6.138 302 0.371 98 113 788.21 31.61 6.446 314 0.474 108 114 + 820.15 +32.28 + 6.766 +326 — 0.587 — 119 115 852.77 32.98 7.099 339 0.712 131 116 886.11 33.70 7.445 353 0.849 144 117 920.18 34.45 8.806 368 1.000 158 118 955.02 35.22 8.181 383 1.166 174 119 + 990.65 +-36.05 + 8.572 +399 -1.348 -191 120 1027.13 36.91 8.980 417 1.548 209 121 1064.47 37.79 9.407 436 1.767 230 122 1102.71 38.73 9.853 456 2.009 253 123 1141.93 39.71 10.320 478 2.274 278 124 +1182.14 +40.74 +10.809 +501 -2.566 -306 125 1223.41 41.82 11.323 527 2.886 336 126 1265.78 42.96 11.863 554 3.239 370 TABLE Va. 39 x. A. Diff. B. Diff. B,. Diff. 126 +1265.78 -t- 42.96 + 11.863 + 0.554 - 3.239 - 0.370 127 1309.33 44.16 12.431 0.584 3.627 0.408 128 1354.11 45.43 13.031 0.616 4.055 0.449 129 1400.20 46.78 13.663 0.651 4.526 0.496 130 1447.67 48.20 14.333 0.690 5.047 0.547 131 +1496.61 + 49.72 + 15.043 + 0.731 - 5.621 - 0.605 132 1547.11 51.33 15.796 0.777 6.257 0.669 133 1599.28 53.04 16.597 0.827 6.960 0.741 134 1653.20 54.87 17.451 0.883 7.739 0.821 135 1709.02 56.82 18.363 0.945 8.603 0.912 136 +1766.84 + 58.91 + 19.341 + 1.013 9.563 - 1.014 137 1826.84 61.15 20.389 1.088 10.631 1.128 138 1889.15 63.55 21.517 1.171 11.820 1.258 139 1953.95 66.14 22.732 1.265 13.148 1.406 140 202.1.43 68.92 24.047 1.371 14.633 1.573 141 +2091.79 + 71.90 + 25.475 -+ 1.490 - 16.295 - 1.765 142 2165.28 75.15 27.027 1.623 18.163 1.984 143 2242.15 78.65 28.722 1.774 20.263 2.234 144 2322.68 82.47 30.576 1.946 22.631 2.523 145 2407.20 86.58 32.615 2.143 25.309 2.856 146 +2496.06 + 91.16 + 34.862 + 2.368 - 28.344 - 3.242 147 2589.66 96.11 37.351 2.626 31.794 3.713 148 2688.45 101.56 40.115 2.924 35.730 4.224 149 2792.96 107.54 43.199 3.272 40.233 4.836 150 2903.74 114.13 46.659 3.677 45.403 5.566 151 +3021.46 +121.43 + 50.553 + 4.153 - 51.366 - 6.437 152 3146.88 129.53 54.966 4.717 58.267 7.469 153 3280.84 138.56 59.987 5.385 66.295 8.705 154 3424.37 148.67 65.737 6.185 75.677 10.202 155 3578.59 160.01 72.357 7.155 86.700 12.024 156 +3744.88 +172.81 + 80.042 + 8.328 99.726 - 14.260 157 3924.79 187.33 89.014 9.767 115.221 17.023 158 4120.22 203.89 99.577 11.548 133.773 20.471 159 4333.38 222.87 112.111 13.777 156.174 24.815 160 4566.94 244.78 127.132 16.603 183.404 30.348 161 +4824.14 +244.78 +145.317 +20.209 -216.860 - 37.483 162 5108.93 270.26 167.550 24.869 258.371 46.802 163 5426.19 300.11 195.056 31.062 310.464 59.156 164 5782.01 335.39 229.674 39.353 376.683 75.318 165 6184.14 377.50 273.762 50.636 462.100 98.618 166 +6642.49 +428.33 +330.946 +66.405 -574.089 — 130.816 167 7170.07 490.43 406.573 88.993 723.733 177.025 168 7784.18 567.43 508.933 122.256 928.140 246.403 16'3 8508.45 i 651.086 1214.530 C O N S T A N T S. Log. Attractive force of the Sun, k in terms of radius, 0.0172021 8.2355814 k in seconds, 3548".18761 3.5500066 Length of the Sidereal Year (HANsEN and OLurSEN), 365d.2563582 2.5625978 Length of the Tropical Year, 1850, 365d.2422008 2.5625809 Horizontal equatorial parallax of the Sun (ENCKE),* 8".5776 0.9333658 Constant of Aberration (STRUVE), 20".4451 1.3105892 Time required for light to pass from the Sain to the Earth, 497$.827 2.6970785 Radius of Circle in Seconds of arc, 206264".806 5.3144251 in Seconds of time, 13750s.987 4.1383339 Sin 1" 0.000004848137 4.6855749 Circumference of Circle in Seconds of arc, 1296000"' 6.1126050 in Seconds of time, 864008 4.9365137 in terms of diameter, a 3.14159265 0.4971499 General Precession (STRUVe) 50".2411 + 0".00022681t Obliquity of the ecliptic (STRUVE and PETERS), 230 27' 54"'.22 - 0.4645 t-.0000014t2 in which I is the number of years after 1800 Daily precession, 1850, 0".1375837 9.1385669 Modulus of Common Logarithms, Jl 0.4342945 9.6377843 * The Constants of Parallax, Aberration, etc., are those used in the American -Ephemeris, and the authority for them may be found by reference to the volume for 1855. (40) :D jZ9. ~ ~ ~ ~ -i. 3. E /E XG B ID Ai' AL7 ,,.i ~ ~ ~ ~ ~ i D D' (1.,. \ Y I -]ig 7. GI I P': x 0 or' I~ ~ ~ ~ ~~ G 8. ID" / 01' & ID" IxT 0 F7%. 9.' Io ~XZ . "a \ A// \ \,,, a (1 oS -.(1, -~~~~~~~~~~~~~~~~~,"] l 11, ma~~~~~~~~~~~~~~~~~~~~] l Slt \ l~~~~~~~~~~,