THEORETICAL ASTRONOMY RELATING TO THE MOTIONS OF THE HEAVENLY BODIES REVOLVING AROUND THE SUN IN ACCORDANCE WITH THE LAW OF UNIVERSAL GRAVITATION EMBRACING A SYSTEMATIC DERIVATION OF THE FORMULAE FOR THE CALCULATION OF TIlE GEOCENTRIC AND HELIOCENTRIC PLACES, FOR THE DETERIINATION OF THE ORBITS OF PLANETS AND COMETS, FOR THE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COM'PUTATION OF SPECIAL PERTURBATIONS; TOGETHER WITH THE THEORY OF THE COMBINATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. ZltZ1 lumtmervral'taampl)es anlt 3uxmliary FTal el BY JAMES C. WATSON DIRECTOR OF THE. OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF MICHIGAN - PHILADELPHIA J. B. LIPPINCOTT & CO. LONDON: TRUBNER & CO. 1868 Entered, according to Act of Congress, in the year 1868, by J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. PREFACE. THE discovery of the great law of nature, the law of gravitation, by NEWTN, prepared the way for the brilliant achievements which have distinguished the history of astronomical science. A first essential, however, to the solution of those recondite problems which were to exhibit the effect of the mutual attraction of the bodies of our system, was the development of the infinitesimal calculus; and the labors of those who devoted themselves to pure analysis have contributed a most important part in the attainment of the high degree of perfection which characterizes the results of astronomical investigations. Of the earlier efforts to develop the great results following from the law of gravitation, those of EULER stand pre-eminent, and the memoirs which he published have, in reality, furnished the germ of all subsequent investigations in celestial mechanics. In this connection also the names of BERNOUILLI, CLAIRAUT, and D'ALEMBERT deserve the most honorable mention as having contributed also, in a high degree, to give direction to the investigations which were to unfold so many mysteries of nature. By means of the researches thus inaugurated, the great problems of mechanics were successfully solved, many beautiful theorems relating to the planetary motions demonstrated, and many useful formulae developed. It is true, however, that in the early stage of the science methods were developed which have since been found to be impracticable, even if not erroneous; still, enough was effected to direct attention in the proper channel, and to prepare the way for the more complete labors of LAGRANGE and LAPLACE. The genius and the analytical skill of these extraordinary men gave to the progress of Theoretical Astronomy the most rapid strides; and the intricate investigations which they successfully performed, served constantly to educe new discoveries, so that of all the problems relating to the mutual attraction of the several planets 3 4 PREFACE. but little more remained to be accomplished by their successors than to develop and simplify the methods which they made known, and to introduce such modifications as should be indicated by experience or rendered possible by the latest discoveries in the domain of pure analysis. The problem of determining the elements of the orbit of a comet moving in a parabola, by means of observed places, which had been considered by NEWTON, EULER, BOSCOVICH, LAMBERT, and others, received from LAGRANGE and LAPLACE the most careful consideration in the light of all that had been previously done. The solution given by the former is analytically complete, but far from being practically complete; that given by the latter is especially simple and practical so far as regards the labor of computation; but the results obtained by it are so affected by the unavoidable errors of observation as to be often little more than rude approximations. The method which was found to answer best in actual practice, was that proposed by OLBERS in his work entitled Leichteste und bequemste Methode die Bahn eines Cometen zu berechnen, in which, by making use of a beautiful theorem of parabolic motion demonstrated by EULER and also by LAMBERT, and by adopting a method of trial and error in the numerical solution of certain equations, he was enabled to effect a solution which could be performed with remarkable ease. The accuracy of the results obtained by OLBERS'S method, and the facility of its application, directed the attention of LEGENDRE, IVORY, GAUSS, and ENCKE to this subject, and by them the method was extended and generalized, and rendered applicable in the exceptional cases in which the other methods failed. It should be observed, however, that the knowledge of one element, the eccentricity, greatly facilitated the solution; and, although elliptic elements had been computed for some of the comets, the first hypothesis was that of parabolic motion, so that the subsequent process required simply the determination of the corrections to be applied to these elements in order to satisfy the observations. The more difficult problem of determining all the elements of planetary motion directly from three observed places, remained unsolved until the discovery of Ceres by PIAZZI in 1801, by which the attention of GAUSS was directed to this subject, the result of which was the subsequent publication of his Theoria Motus Corporum Coelestium,-_a most able work, in which he gave to the world, in a finished form, the results of many years of attention PREFACE. 5 to the subject of which it treats. His method for determining all the elements directly from given observed places, as given in the Theoric Motus, and as subsequently given in a revised form by ENCKE, leaves scarcely any thing to be desired on this topic. In the same work he gave the first explanation of the method of least squares, a method which has been of inestimable service in investigations depending on observed data. The discovery of the minor planets directed attention also to the methods of determining their perturbations, since those applied in the case of the major planets were found to be inapplicable. For a long time astronomers were content simply to compute the special perturbations of these bodies from epoch to epoch, and it was not until the commencement of the brilliant researches by HANSEN that serious hopes were entertained of being able to compute successfully the general perturbations of these bodies. By devising an entirely new mode of considering the perturbations, namely, by determining what may be called the perturbations of the time, and thus passing from the undisturbed place to the disturbed place, and by other ingenious analytical and mechanical devices, he succeeded in effecting a solution of this most difficult problem, and his latest works contain all the formulae which are required for the cases actually occurring. The refined and difficult analysis and the laborious calculations involved were such that, even after HANSEN'S methods were made known, astronomers still adhered to the method of special perturbations by the variation of constants as developed by LAGRANGE. The discovery of Astrcea by HENCKE was speedily followed by the discovery of other planets, and fortunately indeed it so happened that the subject of special perturbations was to receive a new improvement. The discovery by BOND and ENCKE of a method by which we determine at once the variations of the rectangular co-ordinates of the disturbed body by integrating the fundamental equations of motion by means of mechanical quadrature, directed the attention of HANSEN to this phase of the problem, and soon after he gave formule for the determination of the perturbations of the latitude, the mean anomaly, and the logarithm of the radius-vector, which are exceedingly convenient in the process of integration, and which have been found to give the most satisfactory results. The formulae for the perturbations of the latitude, 6 PREFACE. true longitude, and radius-vector, to be integrated in the same manner, were afterwards given by BRjNNOW. Having thus stated briefly a few historical facts relating to the problems of theoretical astronomy, I proceed to a statement of the object of this work. The discovery of so many planets and comets has furnished a wide field for exercise in the calculations relating to their motions, and it has occurred to me that a work which should contain a development of all the formule required in determining the orbits of the heavenly bodies directly from given observed places, and in correcting these orbits by means of more extended discussions of series of observations, including also the determination of the perturbations, together with a complete collection of auxiliary tables, and also such practical directions as might guide the inexperienced computer, might add very materially to the progress of the science by attracting the attention of a greater number of competent computers. Having carefully read the works of the great masters, my plan was to prepare a complete work on this subject, commencing with the fundamental principles of dynamics, and systematically treating, from one point of view, all the problems presented. The scope and the arrangement of the work will be best understood after al examination of its contents; and let it suffice to add that I have endeavored to keep constantly in view the wants of the computer, providing for the exceptional cases as they occur, and giving all the formule which appeared to me to be best adapted to the problems under consideration. I have not thought it worth while to trace out the geometrical signification of many of the auxiliary quantities introduced. Those who are curious in such matters may readily derive many beautiful theorems from a consideration of the relations of some of these auxiliaries. For convenience, the formula are numbered consecutively through each chapter, and the references to those of a preceding chapter are defined by adding a subscript figure denoting the number of the chapter. Besides having read the works of those who have given special attention to these problems, I have consulted the Astronomische Nachrichtein, the Astronomical Journal, and other astronomical periodicals, in which is to be found much valuable information resulting from the experience of those who have been or are now actively engaged in astronomical pursuits. I must also express my obligations to the publishers, PREFACE. 7 Messrs. J. B. LIPPINCOTT & CO., for the generous interest which they have manifested in the publication of the work, and also to Dr. B. A. GOULD, of Cambridge, Mass., and to Dr. OPPOLZER, ofVienna, for valuable suggestions. For the determination of the time from the perihelion and of the true anomaly in very eccentric orbits I have given the method proposed by BESSEL in the Monatliche Correspondenz, vol. xii.,-the tables for which were subsequently given by BRtNNOW in his Astronomical Notices,-and also the method proposed by GAUSS, but in a more convenient form. For obvious reasons, I have given the solution for the special case of parabolic motion before completing the solution of the general problem of finding all of the elements of the orbit by means of three observed places. The differential formula and the other formule for correcting approximate elements are given in a form convenient for application, and the formule for finding the chord or the time of describing the subtended arc of the orbit, in the case of very eccentric orbits, will be found very convenient in practice. I have given a pretty full development of the application of the theory of probabilities to the combination of observations, endeavoring to direct the attention of the reader, as far as possible, to the sources of error to be apprehended and to the most advantageous method of treating the problem so as to eliminate the effects of these errors. For the rejection of doubtful observations, according to theoretical considerations, I have given the simple formula, suggested by CHAUVENET, which follows directly from the fundamental equations for the probability of errors, and which will answer for the purposes here required as well as the more complete criterion proposed by PEIRCE. In the chapter devoted to the theory of special perturbations I have taken particular pains to develop the whole subject in a complete and practical form, keeping constantly in view the requirements for accurate and convenient numerical application. The time is adopted as the independent variable in the determination of the perturbations of the elements directly, since experience has established the convenience of this form; and should it be desired to change the independent variable and to use the differential coefficients with respect to the eccentric anomaly, the equations between this function and the mean motion will enable us to effect readily the required transformation. 8 PREFACE. The numerical examples involve data derived from actual observations, and care has been taken to make them complete in every respect, so as to serve as a guide to the efforts of those not familiar with these calculations; and when different fundamental planes are spoken of, it is presumed that the reader is familiar with the elements of spherical astronomy, so that it is unnecessary to state, in all cases, whether the centre of the sphere is taken at the centre of the earth, or at any other point in space. The preparation of the Tables has cost me a great amount of labor, logarithms of ten decimals being employed in order to be sure of the last decimal given. Several of those in previous use have been recomputed and extended, and others here given for the first time have been prepared with special care. The adopted value of the constant of the solar attraction is that given by GAuss, which, as will appear, is not accurately in accordance with the adoption of the mean distance of the earth from the sun as the unit of space; but until the absolute value of the earth's mean motion is known, it is best, for the sake of uniformity and accuracy, to retain GAUss's constant. The preparation of this work has been effected amid many interruptions, and with other labors constantly pressing me, by which the progress of its publication has been somewhat delayed, even since the stereotyping was commenced, so that in some cases I have been anticipated in the publication of formulae which would have here appeared for the first time. I have, however, endeavored to perform conscientiously the self-imposed task, seeking always to secure a logical sequence in the development of the formulTe, to preserve uniformity and elegance in the notation, and to elucidate the successive steps in the analysis, so that the work may be read by those who, possessing a respectable mathematical education, desire to be informed of the means by which astronomers are enabled to arrive at so many grand results connected with the motions of the heavenly bodies, and by which the grandeur and sublimity of creation are unveiled. The labor of the preparation of the work will have been fully repaid if it shall be the means of directing a more general attention to the study of the wonderful mechanism of the heavens, the contemplation of which must ever serve to impress upon the mind the reality of the perfection of the OMNIPOTENT, the LIVING GOD! OBSERVATORY, ANN ARBOR, June, 1867. CONTENTS. THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FORMULAE FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. PAGE Fundamental Principles......................................................................... 15 Attraction of Spheres....................................................................... 19 Motions of a System of Bodies................................................................. 23 Invariable Plane of the System................................................................ 29 Motion of a Solid Body........................................................................... 31 The Units of Space, Time, and Mass........................................................ 36 Motion of a Body relative to the Sun............................................... 38 Equations for Undisturbed Motion............................................................ 42 Determination of the Attractive Force of the Sun....................................... 49 Determination of the Place in an Elliptic Orbit.......................................... 53 Determination of the Place in a Parabolic Orbit........................................ 59 Determination of the Place in a Hyperbolic Orbit....................................... 65 Methods for finding the True Anomaly and the Time from the Perihelion in the case of Orbits of Great Eccentricity.................................................... 70 Determination of the Position in Space...................................................... 81 Heliocentric Longitude and Latitude......................................................... 83 Reduction to the Ecliptic........................................................................ 85 Geocentric Longitude and Latitude..................................................... 86 Transformation of Spherical Co-ordinates.................................................. 87 Direct Determination of the Geocentric Right Ascension and Declination........ 90 Reduction of the Elements from one Epoch to another................................ 99 Numerical Examples............................................................................ 103 Interpolation........................................................................................ 112 Time of Opposition............................................................................... 114 9 10 CONTENTS. CHAPTER II. INVESTIGATION OF THE DIFFERENTIAL FORMULAE WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATIONS OF THE ELEMENTS OF ITS ORBIT. PAGE Variation of the Right Ascension and Declination....................................... 118 Case of Parabolic Motion....................................................................... 125 Case of Hyperbolic Motion.................................................................... 128 Case of Orbits differing but little from the Parabola...................................... 130 Numerical Examples.............................................................................. 135 Variation of the Longitude and Latitude.................................................... 143 The Elements referred to the same Fundamental Plane as the Geocentric Places 149 Numerical Example...................................................................... 150 Plane of the Orbit taken as the Fundamental Plane to which the Geocentric Places are referred................................................................ 153 Numerical Example................................................................................ 159 Variation of the Auxiliaries for the Equator............................................... 163 CHAPTER III. INVESTIGATION OF FORMULAE FOR COMPUTING THE ORBIT OF A COMET MOVING IN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION OF THE GEOCENTRIC DISTANCE. Correction of the Observations for Parallax................................................ 167 Fundamental Equations................................................................. 169 Particular Cases........................................................................... 172 Ratio of Two Curtate Distances............................................................. 178 Determination of the Curtate Distances...................................................... 181 Relation between Two Radii-Vectores, the Chord joining their Extremities, and the Time of describing the Parabolic Arc..............1...8....................... 184 Determination of the Node and Inclination................................................ 192 Perihelion Distance and Longitude of the Perihelion................................... 194 Time of Perihelion Passage..................................................................... 195 Numerical Example......................................................................... 199 Correction of Approximate Elements by varying the Geocentric Distance........ 208 Numerical Example............................................................................ 213 CHAPTER IV. DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF THE CONIC SECTION. Reduction of the Data............................................................................. 220 Corrections for Parallax................................................................. 223 CONTENTS. 11 PAGE Fundamental Equations............................................................................ 225 Formule for the Curtate Distances............................................................ 228 Modification of the Formulae in Particular Cases.......................................... 231 Determination of the Curtate Distance for the Middle Observation.................. 236 Case of a Double Solution........................................................................ 239 Position indicated by the Curvature of the Observed Path of the Body............ 242 Formule for a Second Approximation.................................................. 243 Formule for finding the Ratio of the Sector to the Triangle........................... 247 Final Correction for Aberration......................................................... 257 Determination of the Elements of the Orbit................................................ 259 Numerical Example................................................................................ 264 Correction of the First Hypothesis........................................................... 278 Approximate Method of finding the Ratio of the Sector to the Triangle........... 279 CHAPTER V. DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS, OF WHICH THE SECOND AND THIRD MUST BE COMIPLETE. Fundamental Equations.2........................................................................ 282 Determination of the Curtate Distances...................................................... 289 Successive Approxim ations...................................................................... 293 Determination of the Elements of the Orbit............................................... 294 Numerical Example................................................................................ 294 Method for the Final Approximation...................................................... 307 CHAPTER VI. INVESTIGATION OF VARIOUS FORMULA FOR THE CORRECTION OF THE APPROXIMATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY. Determination of the Elements of a Circular Orbit...................................... 311 Variation of Two Geocentric Distances...................................................... 313 Differential Formul.............................................................................. 318 Plane of the Orbit taken as the Fundamental Plane...................................... 320 Variation of the Node and Inclination...................................................... 324 Variation of One Geocentric Distance...................................................... 328 Determination of the Elements of the Orbit by means of the Co-ordinates and V elocities....................................................................................... 332 Correction of the Ephemeris.................................................................... 335 Final Correction of the Elements........................................................ 338 Relation between Two Places in the Orbit................................................. 339 Modification when the Semi-Transverse Axis is very large............................ 341 Modification for Hyperbolic Motion..................................................... 346 Variation of the Semi-Transverse Axis and Ratio of Two Curtate Distances..... 349 12 CONTENTS. PAGE Variation of the Geocentric Distance and of the Reciprocal of the Semi-Transverse Axis....................................................................................... 352 Equations of Condition............................................................ 353 Orbit of a Comet.................................................................................... 355 Variation of Two Radii-Vectores............................................ 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OF THE COMBINATION OF OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OF OBSERVATIONS. Statement of the Problem....................................................................... 360 Fundamental Equations for the Probability of Errors................................... 362 Determination of, the Form of the Function which expresses the Probability... 363 The Measure of Precision, and the Probable Error...................................... 366 Distribution of the Errors........................................................................ 367 The Mean Error, and the Mean of the Errors.............................................. 368 The Probable Error of the Arithmetical Mean............................................ 370 Determination of the Mean and Probable Errors of Observations................... 371 Weights of Observed Values........................................................................ 372 Equations of Condition............................................................................ 376 Normal Equations.................................................................................. 378 Method of Elimination.......................................................................... 380 Determination of the Weights of the Resulting Values of the Unknown Quantities................................................................................................ 386 Separate Determination of the Unknown Quantities and of their Weights........ 392 Relation between the Weights and the Determinants..................................... 396 Case in which the Problem is nearly Indeterminate...................................... 398 Mean and Probable Errors of the Results................................................... 399 Combination of Observations................................................................... 401 Errors peculiar to certain Observations...................................................... 408 Rejection of Doubtful Observations............................................................ 410 Correction of the Elements....................................................................... 412 Arrangement of the Numerical Operations.................................................. 415 Numerical Example.....................................4.............................. 418 Case of very Eccentric Orbits................................................................... 423 CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. Fundamental Equations......................................................................... 426 Statement of the Problem......................................................................... 428 Variation of Co-ordinates.............................................................. 429 CONTENTS. 13 PAGE Mechanical Quadrature...........................4................................ 433 The Interval for Quadrature.................................................................. 443 Mode of effecting the Integration.............................................................. 445 Perturbations depending on the Squares and Higher Powers of the Masses...... 446 Numerical Example................................................................................ 448 Change of the Equinox and Ecliptic......................................................... 455 Determination of New Osculating Elements................................................ 459 Variation of Polar Co-ordinates........................................................ 462 Determination of the Components of the Disturbing Force........................... 467 Determination of the Heliocentric or Geocentric Place............................... 471 Numerical Example.............................................................................. 474 Change of the Osculating Elements............................................................ 477 Variation of the Mean Anomaly, the Radius-Vector, and the Co-ordinate...... 480 Fundamental Equations.................................................................... 483 Determination of the Components of the Disturbing Force............................ 489 Case of very Eccentric Orbits........................................................ 493 Determination of the Place of the Disturbed Body.................................. 495 Variation of the Node and Inclination..................................................... 502 Numerical Example............................................................................... 505 Change of the Osculating Elements........................................................... 510 Variation of Constants............................................................................ 516 Case of very Eccentric Orbits.................................................................. 523 Variation of the Periodic Time................................................................ 526 Numerical Example............................................................................. 529 Formulne to be used when the Eccentricity or the Inclination is small.............. 533 Correction of the Assumed Value of the Disturbing Mass.............................. 535 Perturbations of Comets................................. 536 Motion about the Common Centre of Gravity of the Sun and Planet............... 537 Reduction of the Elements to the Common Centre of Gravity of the Sun and Planet............................................................................................ 538 Reduction by means of Differential Formulae............................................. 540 Near Approach of a Comet to a Planet...................................................... 546 The Sun may be regarded as the Disturbing Body........................................ 548 Determination of the Elements of the Orbit about the Planet........................ 550 Subsequent Motion of the Comet............................................... 551 Effect of a Resisting Medium in Space...................................................... 552 Variation of the Elements on account of the Resisting Medium..................... 554 Method to be applied when no Assumption is made in regard to the Density of e the Ether......................................... 556 f14 CONTENTS. TABLES. PAGE I. Angle of the Vertical and Logarithm of the Earth's Radius.............. 561 II. For converting Intervals of Mean Solar Time into Equivalent Intervals of Sidereal Time.................................................................. 563 III. For converting Intervals of Sidereal Time into Equivalent Intervals of Mean Solar Time............................................................. 564 IV. For converting Hours, Minutes, and Seconds into Decimals of a Day... 565 V. For finding the Number of Days from the Beginning of the Year...... 565 VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit............................................................. 566 VII. For finding the True Anomaly in a Parabolic Orbit when v is nearly 180~ 611 VIII. For finding the Time from the Perihelion in a Parabolic Orbit........... 612 TX. For finding the True Anomaly or the Time from the Perihelion in Orbits of Great Eccentricity............................................................ 614 X. For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits.................................................. 618 XI. For the Motion in a Parabolic Orbit................................. 619 XII. For the Limits of the Roots of the Equation sin (z'- ) mvo sin4 a'.. 622 XIII. For finding the Ratio of the Sector to the Triangle........................... 624 XIV. For finding the Ratio of the Sector to the Triangle........................... 629 XV. For Elliptic Orbits of Great Eccentricity............................. 632 XTVI. ForHyperbolic Orbits................................................................ 632 XVI. For Hyperbolic Orbits. 632 XVII. For Special Perturbations........................................................... 633 XVIII. Elements of the Orbits of the Comets which have been observed......... 638 XIX. Elements of the Orbits of the Minor Planets................................... 646 XX. Elements of the Orbits of the Major Planets................................... 648 XXI. Constants, &c............................................................................. 649 EXPLANATION OF THE TABLES............................................................... 651 APPENDIX. —Precession............................................................................ 657 Nutation........................................................................... 658 A berration.......................................................................... 659 Intensity of Light............................................................... 660 Numerical Calculations......................................................... 662 THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FORMULE FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMIPUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. 1. THE study of the motions of the heavenly bodies does not require that we should know the ultimate limit of divisibility of the matter of which they are composed,-whether it may be subdivided indefinitely, or whether the limit is an indivisible, impenetrable atom. Nor are we concerned with the relations which exist between the separate atoms or molecules, except so far as they form, in the aggregate, a definite body whose relation to other bodies of the system it is required to investigate. On the contrary, in considering the operation of the laws in obedience to which matter is aggregated into single bodies and systems of bodies, it is sufficient to conceive simply of its divisibility to a limit which may be regarded as infinitesimal compared with the finite volume of the body, and to regard the magnitude of the element of matter thus arrived at as a mathematical point. An element of matter, or a material body, cannot give itself motion; neither can it alter, in any manner whatever, any motion which may have been communicated to it. This tendency of matter to resist all changes of its existing state of rest or motion is known as inertia, and is the fundamental law of the motion of bodies. Experience invariably confirms it as a law of nature; the continuance of motion as resistances are removed, as well as the sensibly unchanged motion of the heavenly bodies during many centuries, affording the 15 16 THEORETICAL ASTRONOMY. most convincing proof of its universality. Whenever, therefore, a material point experiences any change of its state as respects rest or motion, the cause must be attributed to the operation of something external to the element itself, and which we designate by the word force. The nature of forces is generally unknown, and we estimate them by the effects which they produce. They are thus rendered comparable with some unit, and may be expressed by abstract numbers. 2. If a material point, free to move, receives an impulse by virtue of the action of any force, or if, at any instant, the force by which motion is communicated shall cease to act, the subsequent motion of the point, according to the law of inertia, must be rectilinear and uniform', equal spaces being described in equal times. Thus, if s, v, and t represent, respectively, the space, the velocity, and the tisme, the measure of v being the space described in a unit of time, we shall have, in this case, s vt. It is evident, however, that the space described in a unit of time will vary with the intensity of the force to which the motion is due, and, the nature of the force being unknown, we must necessarily compare the velocities communicated to the point by different forces, in order to arrive at the relation of their effects. We are thus led to regard the force as proportional to the velocity; and this also has received the most indubitable proof as being a law of nature. Hence, the principles of the composition and resolution of forces may be applied also to the composition and resolution of velocities. If the force acts incessantly, the velocity will be accelerated, and the force which produces this motion is called an accelerating force. In regard to the mode of operation of the force, however, we may consider it as acting absolutely without cessation, or we may regard it as acting instantaneously at successive infinitesimal intervals represented by dt, and hence the motion as uniform during each of these intervals. The latter supposition is that which is best adapted to the requirements of the infinitesimal calculus; and, according to the fundamental principles of this calculus, the finite result will be the same as in the case of a force whose action is absolutely incessant. Therefore, if we represent the element of space by ds, and the element of time by dt, the instantaneous velocity will be ds V dt' which will vary from one instant to another. FUNDAMENTAL PRINCIPLES. 17 3. Since the force is proportional to the velocity, its measure at any instant will be determined by the corresponding velocity. If the accelerating force is constant, the motion will be uniformly accelerated; and if we designate the acceleration due to the force byf, the unit off being the velocity generated in a unit of time, we shall have v =ft. If, however, the force be variable, we shall have, at any instant, the relation =dv fdt' the force being regarded as constant in its action during the element of time dt. The instantaneous value of v gives, by differentiation, dv d2s dt dt2' and hence we derive d2s =dt; (1) - d/;2 so that, in varied motion, the acceleration due to the force is measured by the second differential of the space divided by the square of the element of time. 4. By the mass of the body we mean its absolute quantity of matter. The density is the mass of a unit of volume, and hence the entire mass is equal to the volume multiplied by the density. If it is required to compare the forces which act upon different bodies, it is evident that the masses must be considered. If equal masses receive impulses by the actioii of instantaneous forces, the forces acting on each will be to each other as the velocities imparted; and if we consider as the unit of force that which gives to a unit of mass the unit of velocity, we have for the measure of a force F, denoting the mass by M, F =-Mv. This is called the quantity of motion of the body, and expresses its capacity to overcome inertia. By virtue of the inert state of matter, there can be no action of a force without an equal and contrary reaction; for, if the body to which the force is applied is fixed, the equilibrium between the resistance and the force necessarily implies the development of an equal and contrary force; and, if the body be free to move, in the change of state, its inertia will oppose equal and 2 1 8'THEORETICAL ASTRONOMY. contrary resistance. Hence, as a necessary consequence of inertia, it follows that action and reaction are simultaneous, equal, and contrary. If the body is acted upon by a force such that the motion is varied, the accelerating force upon each element of its mass is represented by dt, and the entire motive force F is expressed by dv F-MA dt' 211 being the sum of all the elements, or the mass of the body. Since ds — dt' this gives d2'S which is the expression for the intensity of the motive force, or of the force of inertia developed. For the unit of mass, the measure of the force is d2s dt2' and this, therefore, expresses that part of the intensity of the motive force which is impressed upon the unit of mass, and is what is usually called the accelerating force. 5. The force in obedience to which the heavenly bodies perform their journey through space, is known as the attraction of gravitation; and the law of the operation of this force, in itself simple and unique, has been confirmed and generalized by the accumulated researches of modern science. Not only do we find that it controls the motions of the bodies of our own solar system, but that the revolutions of binary systems of stars in the remotest regions of space proclaim the universality of its operation. It unfailingly explains all the phenomena observed, and, outstripping observation, it has furnished the means of predicting many phenomena subsequently observed. The law of this force is that every particle of matter is attracted by every other particle by a force which varies directly as the mass and inversely as the square of the distance of the attracting particle. This reciprocal action is instantaneous, and is not modified, in any degree, by the interposition of other particles or bodies of matter. It is also absolutely independent of the nature of the molecules themselves, and of their aggregation. ATTRACTION OF SPHERES. 19 If we consider two bodies the masses of which are n and m', and whose magnitudes are so small, relatively to their mutual distance p, that we may regard them as material points, according to the law of gravitation, the action of m on each molecule or unit of mI will be - and the total force on n'l will be P2P mt~ The action of tm' on each molecule of m will be expressed by 2-, and its total action by m' -7The absolute or moving force with which the masses m and m' tend toward each other is, therefore, the same on each body, which result is a necessary consequence of the equality of action and reaction. The velocities, however, with which these bodies would approach each other must be different, the velocity of the smaller mass exceeding that of the greater, and in the ratio of the masses moved. The expression for the velocity of n', which would be generated in a unit of time if the force remained constant, is obtained by dividing the absolute force exerted by m by the mass moved, which gives -7 and this is, therefore, the measure of the acceleration due to the action of in at the distance o. For the acceleration due to the action of w' we derive, in a similar manner, p2 6. Observation shows that the heavenly bodies are nearly spherical in form, and we shall therefore, preparatory to finding the equations which express the relative motions of the bodies of the system, determine the attraction of a spherical mass of uniform density, or varying from the centre to the surface according to any law, for a point exterior to it. If we suppose a straight line to be drawn through the centre of the sphere and the point attracted, the total action of the sphere on the point will be a force acting along this line, since the mass of the sphere is symmetrical with respect to it. Let din denote an element 20 THEORETICAL ASTRONOMY. of the mass of the sphere, and p its distance from the point attracted; then will dm express the action of this element on the point attracted. If we suppose the density of the sphere to be constant, and equal to unity, the element dm becomes an element of volume, and will be expressed by dm = dx dy dz; x, y, and z being the co-ordinates of the element referred to a system of rectangular co-ordinates. If we take the origin of co-ordinates at the centre of the sphere, and introduce polar co-ordinates, so that x r cos sP cos 0, y = r cos p sin 0, z - r sin.p, the expression for dmn becomes dm r- 2 cos p dr dy dO; and its action on the point attracted is r2 cos p dr drp dO df= If we suppose the axis of z to be directed to the point attracted, the co-ordinates of this point will be' = O, y' = O, z' - a, a being the distance of the point from the centre of the sphere, and, since p (x -x')2 + (y _ y') + (Z )2, we shall have p2 = a2 2ar sin p +- r. The component of the force df in the direction of the line a, joining the point attracted and the centre of the sphere, is df cos y, where r is the angle at the point attracted between the element dmn and the centre of the sphere. It is evident that the sum of all the components which act in the direction of the line a will express the total action of the sphere, since the sum of those which act perpen ATTRACTION OF SPHERES. 21 dicular to this line, taken so as to include the entire mass of the sphere, is zero. But we have a -- z + p cos r, and hence a-rSilm o a - r sin (p cos = P The differentiation of the expression for p2, with respect to a, gives dp a — r sin p - cos r. da p - Therefore, if we denote the attraction of the sphere by A, we shall have, by means of the values of df and cos r, r cos ( dr d d -O dp dA - 2' a P da or d dA = -r2 cos p dr d dO d -P. da The polar co-ordinates r, (o, and 0 are independent of a, and hence d 2 cos 0 dr do dO dA — P da Let us now put dV=- cos c dr d dO (2) p and we shall have dV A = da' Consequently, to find the total action of the sphere on the given point, we have only to find V by means of equation (2), the limits of the integration being taken so as to include the entire mass of the sphere, and then find its differential coefficient with respect to a. If we integrate equation (2) first with'reference to 0, for which p is constant, between the limits 0 = 0 and 0 = 2r, we get V= 22ffr~ cos dr This must be integrated between the limits 5= + - 7r and o = - ~ir; 22 THEORETICAL ASTRONOMY. but since p is a function of (p, if we differentiate the expression for p2 with respect to op, we have r cos dp =- - P dp, a and hence V-~ffr dr dp. Corresponding to the limits of jp we have p a- r, and p a + r; and taking the integral with respect to p between these limits, we obtain -g__ 4~ f:r2 dr. Integrating, finally, between the limits r = 0 and r r,, we get irr V- 4 I';, being the radius of the sphere, and, if we denote its entire mass by m, this becomes a, Therefore, dV m da aG from which it appears that the action of a homogeneous spherical mass on a point exterior to it, is the same as if the entire mass were concentrated at its centre. If, in the integration with respect to r, we take the limits r' and r", we obtain A 4 = (v"3 - r'3) 3 2a and, denoting by m0 the mass of a spherical shell whose radii are r" and r', this becomes A 2a"' Consequently, the attraction of a homogeneous spherical shell on a point exterior to it, is the same as if the entire mass were concentrated at its centre. The supposition that the point attracted is situated within a spherical shell of uniform density, does not change the form of the FUNDAMENTAL PRINCIPLES. 23 general equation; but, in the integration with reference to p, the limits will be p =r + a, and,o r - a, which give V= - 44xr dr; and this being independent of a, we have dV A - -0. da Whence it follows that a point placed in the interior of a spherical shell is equally attracted in all directions, and that, if not subject to the action of any extraneous force, it will be in equilibrium in every position. 7. Whatever may be the law of the change of the density of the heavenly bodies from the surface to the centre, we may regard them as composed of homogeneous, concentric layers, the density varying only from one layer to another, and the number of the layers may be indefinite. The action of each of these will be the same as if its mass were united at the centre of the shell; and hence the total action of the body will be the same as if the entire mass were concentrated at its centre of gravity. The planets are indeed not exactly spheres, but oblate spheroids differing but little from spheres; and the error of the assumption of an exact spherical form, so far as relates to their action upon each other, is extremely small, and is in fact compensated by the magnitude of their distances from each other; for, whatever may be the form of the body, if its dimensions are small in comparison with its distance from the body which it attracts, it is evident that its action will be sensibly the same as if its entire mass were concentrated at its centre of gravity. If we suppose a system of bodies to be composed of spherical masses, each unattended with any satellite, and if we suppose that the dimensions of the bodies are small in comparison with their mutual distances, the formation of the equations for the motion of the bodies of the system will be reduced to the consideration of the motions of simple points endowed with forces of attraction corresponding to the respective masses. Our solar system is, in reality, a compound system, the several systems df primary and satellites corresponding nearly to the case supposed; and, before proceeding with the formation of the equations which are applicable to the general case, we will consider, at first, those for a simple system of bodies, considered as points and subject to their mutual actions and the action of the forces which correspond to the 24 THEORETICAL ASTRONOMY. actual velocities of the different parts of the system for any instant. It is evident that we cannot consider the motion of any single body as free, and subject only to the action of the primitive impulsion which it has received and the accelerating forces which act upon it; but, on the contrary, the motion of each body will depend on the force which acts upon it directly, and also on the reaction due to the other bodies of the system. The consideration, however, of the variations of the motion of the several bodies of the system is reduced to the simple case of equilibrium by means of the general principle that, if we assign to the different bodies of the system motions which are modified by their mutual action, we may regard these motions as composed of those which the bodies actually have and of other motions which are destroyed, and which must therefore necessarily be such that, if they alone existed, the system would be in equilibrium.?We are thus enabled to form at once the equations for the motion of a system of bodies. Let in, in', m"( &c. be the masses of the several bodies of the system, and x, y, z, x', y', z', &c. their coordinates referred to any system of rectangular axes. Further, let the components of the total force acting upon a unit of the mass of n, or of the accelerating force, resolved in directions parallel to the co-ordinate axes, be denoted by X, Y, and Z, respectively, then will'nX, mi Y, mZ, be the forces which act upon the body in the same directions. The velocities of the body mi at any instant, in directions parallel to the co-ordinate axes, will be dx dy dz dt' dt' dt' and the corresponding forces are dx dy dz cdt adt dt By virtue of the action of the accelerating force, these forces for the next instant become dx dy dz nm -- + mXdt, m +- m Ydt, m dt + nmZdt, which may be written respectively: MOTION OF A SYSTEM OF BODIES. 25 d dx md + rXdt, dy dy dy c_ 4 mdd - __d 7 dt dt dt m dt +n md- dt —md — dt- + mZdt. The actual velocities for this instant are dx dx dy dy dz dz E + drd d dt, d t + d dt, dt dt dt' dt..dt' and the corresponding forces are dx dx dy dy dz dz m -- m+ d - + nd md d-' dt dt dt dt dt? dt Comparing these with the preceding expressions for the forces, it appears that the forces which are destroyed, in directions parallel to the co-ordinate axes, are dx -md d- +- mvXdt, - mda d- m Ydt, (3) dt - md-~ -- mZdt. dt In the same manner we find for the forces which will be destroyed in the case of the body m': dx' - mid d -+ i'X'dt, -- m'd - + n' Y'dt, dzt m'd d- + m'Z'dt; and similarly for the other bodies of the system. According to the general principle above enunciated, the system under the action of these forces alone, will be in equilibrium. The conditions of equilibrium for a system of points of invariable but arbitrary form, and subject to the action of forces directed in any manner whatever, are 2X, = O Y, O O, 2Z, - O, xz( Y - Xy) --, z (Xt - Z,) = O, (ZY - Y,-=l 0; in which X, Y,, Z,, denote the components, resolved parallel to the 26 THEORETICAL ASTRONOMY. co-ordinate axes, of the forces acting on any point, and x, y, z, the co-ordinates of the point. These equations are equally applicable to the case of the equilibrium at any instant of a system of variable form; and substituting in them the expressions (3) for the forces destroyed in the case of a system of bodies, we shall have d2x zm~ - taX =- 0, dt' dt'y d2z dt2 Zndt -- L -Z_0, (4) (dt" Y dt- ) n (Y- Xy).0, t( d Y d2y __ m( Y dt2 dzt)- m (Xz - Yz) (y- -- (z) y ~ 0; which are the general equations for the motions of a system of bodies. 8. Let x,, y,, z,, be the co-ordinates of the centre of gravity of the system, and, by differentiation of the equations for the co-ordinates of the centre of gravity, which are i Zmx _my imz 2S~ m' y~,- 2m' we get'E d2x vm d2y vm d2z dx rndt-2 d2y, dt'z, __d__ cdt m i dt2 2mr dt2 2m Introducing these values into the first three of equations (4), they become d2x, 2mX d2y, 2m Y dz, ZnmZ dt2 min dt2 2mn dt2 n' ( from which it appears that the centre of gravity of the system moves in space as if the masses of the different bodies of which it is composed, were united in that point, and the forces directly applied to it. If we suppose that the only accelerating forces which act on the bodies of the system, are those which result from their mutual action, we have the obvious relation: mX - rm'X', m Y- - mn' Y', nZ - m'Z', MOTION OF A SYSTEM OF BODIES. 27 and similarly for any two bodies; and, consequently, 2mX = O, m Y- O, 2mZ = 0; so that equations (5) become 0, d dt,' t o d 0 t2 -0. dt2 dt~ dd2 Integrating these once, and denoting the constants of integration by c, c', c", we find, by combining the results, dx+ dy,2 dz + _ 2 2 + - c2 +.c2 dt2 and hence the absolute motion of the centre of gravity of the system, when subject only to the mutual action of the bodies which compose it, must be uniform and rectilinear. Whatever, therefore, may be the relative motions of the different bodies of the system, the motion of its centre of gravity is not thereby affected. 9. Let us now consider the last three of equations (4), and suppose the system to be submitted only to the mutual action of the bodies which compose it, and to a force directed toward the origin of coordinates. The action of m' on m, according to the law of gravitation, is expressed by ~2, in which p denotes the distance of m from mn'. To resolve this force in directions parallel to the three rectangular axes, we must multiply it by the cosine of the angle which the line joining the two bodies makes with the co-ordinate axes respectively, which gives XmW' (~ x) n__' (y' — y) mn' (' - z) f t P Po Further, for the components of the accelerating force of m on m/, we have X' - m (x - x') y - m (y - Y) Z - m (z - z') XI~?'?~' ~ ~' m ( Y - Xy) + ml' ( 7VX - Xy ) = 0, and generally m (Y. - Xy) = 0. (6) 28 THEORETICAL ASTRONOMY. In a similar manner, we find Zm (Xz - Zx) 0, (7) Zmn (Zy - Yz) = 0. These relations l11 not be altered if, in addition to their reciprocal action, the bodies of the system are acted upon by forces directed to the origin of co-ordinates. Thus, in the case of a force acting upon m, and directed to the origin of co-ordinates, we have, for its action alone, Yx =Xy, Xz= - Zx, Z = YZ, and similarly for the other bodies. Hence these forces disappear from the equations, and, therefore, when the several bodies of the system are subject only to their reciprocal action and to forces directed to the origin of co-ordinates, the last three of equations (4) become m ( d Y dt2 ), Z dt2.dt" ) I dz d2y\ C, the integration of which gives 2m (xdy - ydx) = cdt, mn (zdx - xdz) - ddt, (8) mn (ydz - zdy) - c"dt, c, c', and c" being the constants of integration. Now, xdy -ydx is double the area described about the origin of co-ordinates by the projection of the radius-vector, or line joining n with the origin of co-ordinates, on the plane of xy during the element of time dt; and, further, zdx - xdz and ydz - zdy are respectively double the areas described, during the same time, by the projection of the radius-vector on the planes of xz and yz. The constant c, therefore, expresses the sum of the products formed by multiplying the areal velocity of each body, in the direction of the co-ordinate plane xy, by its mass; and c', c", express the same sum with reference to the co-ordinate planes xz and yz respectively. Hence the sum of the areal velocities of the several bodies of the system about the origin of co-ordinates, each multiplied by the corresponding mass, is constant; and the sum of the areas traced, each multiplied by the corresponding mass, is proportional to the time. If the only forces which operate, are those INVARIABLE PLANE. 29 resulting from the mutual action of the bodies which compose the system, this result is correct whatever may be the point in space taken as the origin of co-ordinates. The areas described by the projection of the radius-vector of each body on the co-ordinate planes, are the projections, on these planes, of the areas actually described in space. We may, therefore,conceive of a resultant, or principal plane of projection, such that'he,sum of the areas traced by the projection of each radius-vector.on this plane, when projected on the three co-ordinate planes, each being multiplied by the corresponding mass, will be respectively equal to the first members of the equations (8). Let a, 9, and 7 be the angles which this principal plane makes with tye co-ordinate planes xy, xz, and yz, respectively; and let S denote the, sum of the areas traced on this plane, in a unit of time, by the projection of the radius-vector of each of the bodies of the system, each area being multiplied by the corresponding mass. The sum S will be found to be a maximum, and its projections on the co-ordinate planes, corresponding to the element of time dt, are S cosa dt, S cos dt, S cos dt. Therefore, by means of equations (8), we have c -= cos a, C, - cos, " cos r, and, since cos2a + cos2 + cos2r = 1, S2_ C2 + c'2 + Ct"2 Hence we derive c c cos a os0" cosr = — C2 + c02 + cr/2 These angles, being therefore constant and independent of the time, show that this principal plane of projection remains constantly parallel to itself during the motion of the system in space, whatever may be the relative positions of the several bodies; and for this reason it is called the invariable plane of the system. Its position with reference to any known plane is easily determined when the velocities, in directions parallel to the co-ordinate axes, and the masses and co-ordinates of the several bodies of the system, are known. The values of c, c', c" are given by equations (8), and 30 THEORETICAL ASTRONOMY. hence the values of a,?, and r, which determine the position of the invariable plane. Since the positions of the co-ordinate planes are arbitrary, we may suppose that of xy to coincide with the invariable plane, which gives cos 8 = 0 and cos r 0, and, therefore, c' = 0 and c" = 0. Further, since the positions of the axes of x and y in this plane are arbitrary, it follows that for every plane perpendicular to the invariable plane, the sum of the areas traced by the projections of the radii-vectores of the several bodies of the system, each multiplied by the corresponding mass, is zero. It may also be observed that the value of S is constant whatever may be the position of the co-ordinate planes, and that its value is necessarily greater than that of either of the quantities in the second member of the equatity. 2 - C + e'2 + c"]2, except when two of them are each equal to zero. It is, therefore, a maximum, and the invariable plane is also the plane of maximum areas. 10. If we suppose the origin of co-ordinates itself to move with uniform and rectilinear motion in space, the relations expressed by equations (8) will remain unchanged. Thus, let x,, y,, z, be the coordinates of the movable origin of co-ordinates, referred to a fixed point in space taken as the origin; and let xw, y0, -, x0', y,', z'o, &c. be the co-ordinates of the several bodies referred to the movable origin. Then, since the co-ordinate planes in one system remain always parallel to those of the other system of co-ordinates, we shall have X= xi + -, Y - Y + Yo, Z = Z,.+ and similarly for the other bodies of the system. Introducing these values of x, y, and z into the first three of equations (4), they become ( d2x, d2x) ) ( d, d2y, d)d The condition of uniform rectilinear motion of the movable origin gives - __ - __ ___ 0 d\_ __ ~ dtY —, d~= O, dt- dt. dt2 MOTION OF A SOLID BODY. 31 and the preceding equations become d2x 2 0-~27nX==0, -2 - 2xY=- O, d~ — mY-O, (9) 8d2z Im dt- mZ - 0. Substituting the same values in the last three of equations (4), observing that the co-ordinates x,, y,, z, are the same for all the bodies of the system, and reducing the resulting equations by means of equations (9), we get n O ( Yo ~ dt ) ~ ( Y. - Xy) = O0, ( ) d2X0 d22zo fn Yo d - - zo ~dt2 - zm (Zy0 - Yzo) = O. Hence it appears that the form of the equations for the motion of the system of bodies, remains unchanged when we suppose the origin of co-ordinates to move in space with a uniform and rectilinear motion. 11. The equations already derived for the motions of a system of bodies, considered as reduced to material points, enable us to form at once those for the motion of a solid body. The mutual distances of the parts of the system are, in this case, invariable, and the masses of the several bodies become the elements of the mass of the solid body. If we denote an element of the mass by dA, the equations (5) for the motion of the centre of gravity of the body become dt2 dty dtz d dm% =f= J'Xdmy, d=' fYdmn, mnd —2 rZdm, (11) the summation, or integration with reference to dm, being taken so as to include the entire mass of the body, from which it appears that the centre of gravity of the body moves in space as if the entire mass were concentrated in that point, and the forces applied to it directly. If we take the origin of co-ordinates at the centre of gravity of the body, and suppose it to have a rectilinear, uniform motion in space, and denote the co-ordinates of the element dm, in reference to this origin, by x0, yo, z, we have, by means of the equations (10), 32 THEORETICAL ASTRONOMY. d( Y — -- d dim - (Yxo -Xyo) dm - 0, fz0 dtr — Yo dt2 / f( d t2 o d2" ) dm -X - Zx0) dm = 0 (12) the integrao dtion wih r(Zye o YZo) t i d t the integration with respect to dcm being taken so as to include the entire mass of the body. These equations, therefore, determine the motion of rotation of the body around its centre of gravity regarded as fixed, or as having a uniform rectilinear motion in space. Equations (11) determine the position of the centre of gravity for any instant, and hence for the successive instants at intervals equal to dt; and we may consider the motion of the body during the element of time dt as rectilinear and uniform, whatever may be the form of its trajectory. Hence, equations (11) and (12) completely determine the position of the body in space,-the former relating to the motion of translation of the centre of gravity, and the latter to the motion of rotation about this point. It follows, therefore, that for any forces which act upon a body we can always decompose the actual motion into those of the translation of the centre of gravity in space, and of the motion of rotation around this point; and these two motions may be considered independently of each other, the motion of the centre of gravity being independent of the form and position of the body about this point. If the only forces which act upon the body are the reciprocal action of the elements of its mass and forces directed to the origin of coordinates, the second terms of equations (12) become each equal to zero, and the results indicated by equations (8) apply in this case also. The parts of the system being invariably connected, the plane of maximum areas, or invariable plane, is evidently that which is perpendicular to the axis of rotation passing through the centre of gravity, and therefore, in the motion of translation of the centre of gravity in space, the axis of rotation remains constantly parallel to itself. Any extraneous force which tends to disturb this relation will necessarily develop a contrary reaction, and hence a rotating body resists any change of its plane of rotation not parallel to itself. We may observe, also, that on account of the invariability of the mutual distances of the elements of the mass, according to equations (8), the motion of rotation must be uniform. 12. We shall now consider the action of a system of bodies on a MOTION OF A SOLID BODY. 33 distant mass, which we will denote by M. Let x,, y, z0, x o,', z0', &c. be the co-ordinates of the several bodies of the system referred to its centre of gravity as the origin of co-ordinates; x,, y,, and z, the co-ordinates of the centre of gravity of the system referred to the centre of gravity of the body M. The co-ordinates of the body m, of the system, referred to this origin, will therefore be X = x, -- X0, y= y, +-yo, -, +- Z, and similarly for the other bodies of the system. If we denote by r the distance of the centre of gravity of m from that of M1, the accelerating force of the former on an element of mass at the centre of gravity of the latter, resolved parallel to the axis of x, will be mx r3 and, therefore, that of the entire system on the element of M, resolved in the same direction, will be mx r We have also'' (X- + xo) 2 + (y, + yo) 2 + (, +- Z) 2, and, if we denote by r, the distance of the centre of gravity of the system from M, r,2 x,2 + y,2 + Z-2. Therefore _3 r= (X + ox) (r,2 + 2 (, Xo + y, yo + z,o) + r2) We shall now suppose the mutual distances of the bodies of the system to be so small in comparison with the distance r, of its centre of gravity from that of AI, that terms of the order r02 may be neglected; a condition which is actually satisfied in the case of the secondary systems belonging to the solar system. Hence, developing the second factor of the second member of the last equation, and neglecting terms of the order r2, we shall have x +x x 3x, (x,0 + y, Yo + Z, o) r3 r3 3 t5 and mxIV Tv?,m 2mxo 3x, }=, + -m - 3~- (x,2mxo + y,2myO + z,omzO). r,3 r 3 r,3, 3 34 THEORETICAL ASTRONOMY. But, since x, Yo, 0, are the co-ordinates in reference to the centre of gravity of the system as origin, we have ZmXo -= 0, -Zmy O, mzo = 0, and the preceding equation reduces to IrnX ICm 2 -mx r3 r 3 In a similar manner, we find ny TZin rmZ TZin r3 r,3t r3 r,3 The second members of these equations are the expressions for the total accelerating force due to the action of the bodies of the system on M1 resolved parallel to the co-ordinate axes respectively, when we consider the several masses to be collected at the centre of gravity of the system. Hence we conclude that when an element of mass is attracted by a system of bodies so remote from it that terms of the order of the squares of the co-ordinates of the several bodies, referred to the centre of gravity of the system as the origin of co-ordinates, may be neglected in comparison with the distance of the system from the point attracted, the action of the system will be the same as if the masses were all united at its centre of gravity. If we suppose the masses n, n', m", &c. to be the elements of the mass of a single body, the form of the equations remains unchanged; and hence it follows that the mass 1M is acted upon by another mass, or by a system of bodies, as if the entire mass of the body, or of the system, were collected at its centre of gravity. It is evident, also, that reciprocally in the case of two systems of bodies, in which the mutual distances of the bodies are small in comparison with the distance between the centres of gravity of the two systems, their mutual action is the same as if all the several masses in each system were collected at the common centre of gravity of that system; and the two centres of gravity will move as if the masses were thus united. 13. The results already obtained are sufficient to enable us to form the equations for the motions of the several bodies which compose the solar system. If these bodies were exact spheres, which could be considered as composed of homogeneous concentric spherical shells, the density varying only from one layer to another, the action of MOTION OF A SYSTEM OF BODIES. 35 each on an element of the mass of another would be the same as if the entire mass of the attracting body were concentrated at its centre of gravity. The slight deviation from this law, arising from the ellipsoidal form of the heavenly bodies, is compensated by the magnitude of their mutual distances; and, besides, these mutual distances are so great that the action of the attracting body on the entire mass of the body attracted, is the same as if the latter were concentrated at its centre of gravity. Hence the consideration of the reciprocal action of the single bodies of the system, is reduced to that of material points corresponding to their respective centres of gravity, the masses of which, however, are equivalent to those of the corresponding bodies. The mutual distances of the bodies composing the secondary systems of planets attended with satellites are so small, in comparison with the distances of the different systems from each other and from the other planets, that they act upon these, and are reciprocally acted upon, in nearly the same manner as if the masses of the secondary systems were united at their common centres of gravity, respectively. The motion of the centre of gravity of a system consisting of a planet and its satellites is not affected by the reciprocal action of the bodies of that system, and hence it may be considered independently of this action. The difference of the action of the other planets on a planet and its satellites will simply produce inequalities in the relative motions of the latter bodies as determined by their mutual action alone, a.nd will not affect the motion of their common centre of gravity. Hence, in the formation of the equations for the motion of translation of the centres of gravity of the several planets or secondary systems which compose the solar system, we have simply to consider them as points endowed with attractive forces corresponding to the several single or aggregated masses. The investigation of the motion of the satellites of each of the planets thus attended, forms a problem entirely distinct from that of the motion of the common centre of gravity of such a system. The consideration of the motion of rotation of the several bodies of the solar system about their respective centres of gravity, is also independent of the motion of translation. If the resultant of all the forces which act upon a planet passed through the centre of gravity, the motion of rotation would be undisturbed; and, since this resultant in all cases very nearly satisfies this condition, the disturbance of the motion of rotation is very slight. The inequalities thus produced in the motion of rotation are, in fact, sensible, and capable of being indicated by observation, only in the case of the earth and moon. It has, indeed, 36 THEORETICAL ASTRONOMY. been rigidly demonstrated that the axis of rotation of the earth relative to the body itself is fixed, so that the poles of rotation and the terrestrial equator preserve constantly the same position in reference to the surface; and that also the velocity of rotation is constant. This assures us of the permanency of geographical positions, and, in connection with the fact that the change of the length of the mean solar day arising from the variation of the obliquity of the ecliptic and in the length of the tropical year, due to the action of the sun, moon, and planets upon the earth, is absolutely insensible, -amounting to only a small fraction of a second in a million of years,-assures us also of the permanence of the interval which we adopt as the unit of time in astronomical investigations. 14. Placed, as we are, on one of the bodies of the system, it is only possible to deduce from observation the relative motions of the different heavenly bodies. These relative motions in the case of the comets and primary planets are referred to the centre of the sun, since the centre of gravity of this body is near the centre of gravity of the system, and its preponderant mass facilitates the integration of the equations thus obtained. In the case, however, of the secondary systems, the motions of the satellites are considered in reference to the centre of gravity of their primaries. We shall, therefore, form the equations for the motion of the planets relative to the centre of gravity of the sun; for which it becomes necessary to consider more particularly the relation between the heterogeneous quantities, space, time, and mass, which are involved in them. Each denomination, being divided by the unit of its kind, is expressed by an abstract number; and hence it offers no difficulty by its presence in an equation. For the unit of space we may arbitrarily take the mean distance of the earth from the sun, and the mean solar day may be taken as the unit of time. But, in order that when the space is expressed by 1, and the time by 1, the force or velocity may also be expressed by 1, if the unit of space is first adopted, the relation of the time and the mass-which determines the measure of the forcewill be such that the units of both cannot be arbitrarily chosen. Thus, if we denote by f the acceleration cue to the action of the mass m on a material point at the distance a, and by f the acceleration corresponding to another mass am' acting at the same distance, we have the relation 91~ f _ nm f',-'X MOTION RELATIVE TO THE SUN. 37 and hence, since the acceleration is proportional to the mass, it may be taken as the measure of the latter. But we have, for the measure off, d2s f~ dt2 Integrating this, regarding f as constant, and the point to move from a state of rest, we get - s _.ft2. (13) The acceleration in the case of a variable force is, at any instant, measured by the velocity which the force acting at that instant would generate, if supposed to remain constant in its action, during a unit of time. The last equation gives, when t 1, f= 2s; and hence the acceleration is also measured by double the space which would be described by a material point, from a state of rest, during a unit of time, the force being supposed constant in its action during this time. In each case the duration of the unit of time is involved in the measure of the acceleration, and hence in that of the mass on which the acceleration depends; and the unit of mass, or of the force, will depend on the duration which is chosen for the unit of time. In general, therefore, we regard as the unit of mass that which, acting constantly at a distance equal to unity on a material point free to move, will give to this point, in a unit of time, a velocity which, if the force ceased to act, would cause it to describe the unit of distance in the unit of time. Let the unit of time be a mean solar day; k2 the acceleration due to the force exerted by the mass of the sun at the unit of distance; andf the acceleration corresponding to the distance r; then will k2 r2 and k2 becomes the measure of the mass of the sun. The unit of mass is, therefore, equal to the mass of the sun taken as many times as k2 is contained in unity. Hence, when we take the' mean solar day as the unit of time, the mass of the sun is measured by k2; by which we are to understand that if the sun acted during a mean solar day, on a material point free to move, at a distance constantly equal to the mean distance of the earth from the sun, it would, at the end of that time, have communicated to the point a velocity which, if 38 THEORETICAL ASTRONOMY. the force did not thereafter act, would cause it to describe, in a unit of time, the space expressed by k2. The acceleration due to the action of the sun at the unit of distance is designated by k2, since the square root of this quantity appears frequently in the fornulhe which will be derived. If we take arbitrarily the mass of the sun as the unit of mass, the unit of time must be determined. Let t denote the number of mean solar days which must be taken for the unit of time when the unit of mass is the mass of the sun. The space which the force due to this mass, acting constantly on a material point at a distance equal to the mean distance of the earth from the sun, would cause the point to describe in the time t, is, according to equation (13), s i- kt. But, since t expresses the number of mean solar days in the unit of time, the measure of the acceleration corresponding to this unit is 2s, and this being the unit of force, we have k2t2 1; and hence k Therefore, if the mass of the sun is regarded as the unit of mass, the number of mean solar days in the unit of time will be equal to unity divided by the square root of the acceleration due to the force exerted by this mass at the unit of distance. The numerical value of k will be subsequently found to be 0.0172021, which gives 58.13244 mean solar days for the unit of time, when the mass of the sun is taken as the unit of mass. 15. Let x, y, z be the co-ordinates of a heavenly body referred to the centre of gravity of the sun as the origin of co-ordinates; r its radius-vector, or distance from this origin; and let mn denote the quotient obtained by dividing its mass by that of the sun; then, taking the mean solar day as the unit of time, the mass of the sun is expressed by k, and that of the planet or comet by mk2. For a second body let the co-ordinates be x', yi, z'; the distance from the sun, r'; and the mass, m'k2; and similarly for the other bodies of the system. Let the co-ordinates of the centre of gravity of the sun referred to any fixed point in space be C, V, t, the co-ordinate planes being parallel to those of x, y, and z, respectiVely; then will the MOTION RELATIVE TO THE SUN. 39 acceleration due to the action of n on the sun be expressed by ~2r and the three components of this force in directions parallel to the co-ordinate axes, respectively, will be 2X 2 Y 2 mk ~ mk~ Y mkw_ r3 r3 7 r3 The action of m' on the sun will be expressed by m'lc2 m'2 k2 m', r3 r" and hence the acceleration due to the combined and simultaneous action of the several bodies of the system on the sun, resolved parallel to the co-ordinate axes, will be,2,v, _ 2 Illy _2_mz r3 r7 r3' The motion of the centre of gravity of the sun, relative to the fixed origin, will, therefore, be determined by the equations d$ MX d 2, Zmy rn p Z n dt2 ~- — 3-, dt2.d2 3 (14 Let p denote the distance of ra from m'; p' its distance from mi", adding an accent for each successive body considered; then will the action of the bodies mn', n", &c. on m be k2.~ - p2 of which the three components parallel to the co-ordinate axes, respectively,' are k2m2xt kyt zt k -- 2 p^__3,p k^Wn -X'p3 3 The action of the sun on m, resolved in the same manner, is expressed by Px k2y k2z r 3' r3. r3' which are negative, since the force tends to diminish the co-ordinates x, y, and z. The three components of the total action of the other bodies of the system on m are, therefore, 40 THEORETICAL ASTRONOMY. kx - -A~ + k2 lnd (zt X) ky m'(y-y) r-T- + ~.zY' — y)P3 kz + k2fn/ (z' z) r3 P0 and, since the co-ordinates of m referred to the fixed origin are + x, +4 Y + Z, the equations which determine the absolute motion are d2 d4v k2x 2m' (X' - x) dlrl d J ~ _ ~y),^ dt + d2 +,-r 3k' k ) +t + + k2yy- (15) d2C d2Z k~z m' (Z' -- z) dt2 + + --- z ( a the symbol of summation in the second members relating simply to the masses and co-ordinates of the several bodies which act on m, d2~ d2'7 d2. exclusive of the sun. Substituting for dt2 d-' and their values (it dt2 d(t given by equations (14), we get d2I X 1 X X X d + k2 ( m) = - ~ \) = l2y + k2(11 + W -y~ k'2f'( y — ~ \(16) dt2 + -(l + -) r3 3 r3 dz+ k2 (1 + _n) 3 k2m ( 2 z -z ) dt2 r \p3r I Since x, y, z are the co-ordinates of m relative to the centre of gravity of the sun, these equations determine the motion of m relative to that point. The second members may be put in another form, which greatly facilitates the solution of some of the problems relating to the motion of rn. Thus, let us put i-' 1 x'+yy. +- zz l -' X-"+ +yy"+ \ (&, l + in \ p r13 /+ r3 + 1+ --- ~ +T pi- r —+' (17) and we shall have for the partial differential coefficient of this with respect to x, d\, _ m' 1 dp x'\ " / I dp' x" \ \ dx- l- - n\ p2 d r'3 + - + — bl pt2 d- r.3. + MOTION RELATIVE TO THE SUN. 41 But, since 2 - (x - X)2 + (y - y)2 + (Z - )2, p12 (XI' - )2 + (y y)2 + (z - (Z)2, we have dp x' x- dp' x" x dx p' dx'' and hence we derive dQg' I xx x' \ " { x" x x"\ \ dx) 1 +m\ rI 3 1 +m)m\ rp )+...&c. or ( _7 \ I( _ _ - _ (1+ ) d fo p 3- +, dx! p) We find, also, in the same manner, for the partial differential coefficients with respect to y and z, (+n)( dQ ) ( Y, — Y ) -- Z,' (1 +'i) ( d{ - ) 2m' Z — z- ~ 3 The equations (16), therefore, become d2X 22 \ dV+~k(l +k m)y h k'(l +m ddy dt 2 ) ( 1 )(r3 =k(1 +)(dS P), d-r 2(1 +)4 (1+)( (18) dt2 + k2 + ) 3 k(1 + )( d ) It will be observed that the second members of equationu (16) express the difference between the action of the bodies mn', mn", &c. on m and on the sun, resolved parallel to the co-ordinate axes respectively. The mutual distances of:the planets are such that these quantities are generally very small, and we may, therefore, in a first approximation to the motion of m relative to the sun, neglect the second members of these equations; and the integrals which may then be derived, express what is called the undisturbed motion of m. By means of the results thus obtained for the several bodies successively, the approximate values of the second members of equations (16) may be found, and hence a still closer approximation to the actual motion of m. The force whose components are expressed by the second members of these equations is called the disturbing force; 42 THEORETICAL ASTRONOMY. and, using the second form of the equations, the function l2, which determines these components, is called the perturbing function. The complete solution of the problem is facilitated by an artifice of the infinitesimal calculus, known as the variation of parameters, or of constants, according to which the complete integrals of equations (16) are of the same form as those obtained by putting the second members equal to zero, the arbitrary constants, however, of the latter integration being regarded as variables. These constants of integration are the elements which determine the motion of m relative to the sun, and when the disturbing force is neglected the elements are pure constants. The variations of these, or of the co-ordinates, arising from the action of the disturbing force are, in almost all cases, very small, and are called the perturbations. The problem which first presents itself is, therefore, the determination of all the circumstances of the undisturbed motion of the heavenly bodies, after which the action of the disturbing forces may be considered. It may be further remarked that, in the formation of the preceding equations, we have supposed the different bodies to be free to move, and, therefore, subject only to their mutual action. There are, indeed, facts derived from the study of the motion of the comets which seem to indicate that there exists in space a resisting medium which opposes the free motion of all the bodies of the system. If such a medium actually exists, its effect is very small, so that it can be sensible only in the case of rare and attenuated bodies like the comets, since the accumulated observations of the different planets do not exhibit any effect of such resistance. But, if we assume its existence, it is evidently necessary only to add to the second members of equations (16) a force which shall represent the effect of this resistance,which, therefore, becomes a part of the disturbing force,-and the motion of m will be completely determined. i16. When we consider the undisturbed motion of a planet or comet relative to the sun, or simply the motion of the body relative to the sun as subject only to the reciprocal action of the two bodies, the equations (16) become d2 k2(l+ -0, dt2 r3 dt + k2(1: + ) 0, (o, d+ 2- i1 + a) = 0. dt2z +2( +m) - 3O. MOTION RELATIVE TO THE SUN. 43 The equations for the undisturbed motion of a satellite relative to its primary are of the same form, the value of k2, however, being in this case the acceleration due to the force exerted by the mass of the primary at the unit of distance, and m the ratio of the mass of the satellite to that of the primary. The integrals of these equations introduce six arbitrary constants of integration, which, when known, will completely determine the undisturbed motion of m relative to the sun. If we multiply the first of these equations by y, and the second by x, and subtract the last product from the first, we shall find, by integrating the result, xdy - ydx dt c being an arbitrary constant. In a similar manner, we obtain xdz - zdx ydz - zdy dt dt ~ If we multiply these three equations respectively by z, y, and x, and add the products, we obtain cz -c y + C"x - 0. This, being the equation of a plane passing through the origin of co-ordinates, shows that the path of the body relative to the sun is a plane curve, and that the plane of the orbit passes through the centre of the sun. Again, if we multiply the first of equations (19) by 2dx, the second by 2dy, and the third by 2dz, take the sum and integrate, we shall find +dx'+d +d'+2k(+ ( + J xd + ydy + zdz 0. dt2 d But, since r2 = x - y2 +'2, we shall have, by differentiation, rdr xdx + ydy - zdz. Therefore, introducing this value into the preceding equation, we obtain dX2 + dy2 + d 2(1 + m) + h (20) A be-i +a r = o, (to dtb r h being an arbitrary constant. 44 THEORETICAL ASTRONOMY. If we add together the squares of the expressions for c, c', and cl", and put c2 + e2 + Gc2 = 4f2, we shall have (x' + y2 + Z ) (dx2 + dy2 + -dz') (xdx + ydy + zdz)2 42 dt' dt or dt dt~ - 4f' 2 dx2- + dy + dz2 _r2dr2 4f 21) If we represent by dv the infinitely small angle contained between two consecutive radii-vectores r and r + dr, since dx2 + cdy + dz2 is the square of the element of path described by the body, we shall have dx2 + dy + dz2 - dr2 + r'2dV2 Substituting this value in the preceding equation, it becomes r2dv 2fdt. (22) The quantity r2dv is double the area included by the element of path described in the element of time dt, and by the radii-vectores r and r + dr; and f, therefore, represents the areal velocity, which, being a constant, shows that the radius-vector of a planet or comet describes equal areas in equal intervals of time. From the equations (20) and (21) we find, by elimination, dt -rdr (23) l/2rk2 (1 +- m) -- hr2 — 4f( Substituting this value of dt in equation (22), we get dv- - 2fdr (24) rl/2rk2 (1 +- ) - hr2 — 4f2 which gives, in order to find the maximum and minimum values of r, dr rV/2rk(1 + m) -hr-24f2 dv- 2f or 2rk2(1 + n) - hr2- 4f2 = 0. Therefore k2(1 +M) 4f2,4(1 + m)2 h + h h2 and 1 k2l(1+m)) 4f2 k+(1 + m) h h are, respectively, the maximum and minimum values of r. The MOTION RELATIVE TO THE SUN. 45 points of the orbit, or trajectory of the body relative to the sun, corresponding to these values of r, are called the apsides; the former, the aphelion, and the latter, the perihelion. If we represent these values, respectively, by a (I + e) and a ( - e), we shall have h = ( a; 4f- ak2 (1 + m) (1 - e2) =k2p (1 + n), in which p a (1-e2). Introducing these values into the equation (24), it becomes l/p dr e dr dv - ri2r -2-p 1_ 1 1 ) the integral of which gives -1 1(p v==w-+cos - 1 e r co being an arbitrary constant. Therefore we shall have r - 1 cos (v - ) from which we derive 1 + e cos (v - )' which is the polar equation of a conic section, the pole being at the focus, p being the semi-parameter, e the eccentricity, and v - ( the angle at the focus between the radius-vector and a fixed line, in the plane of the orbit, making the angle wo with the semi-transverse axis a. If the angle v - o.is counted from the perihelion, we have ) = 0, and,= P (25) 1 + e cos v The angle v is called the true anonmaly. Hence we conclude that the orbit of a heavenly body revolving around the sun is a conic section wqith the sun in one of the foci. Observation shows that the planets revolve around the sun in ellipses, usually of small eccentricity, while the comets revolve either in ellipses of great eccentricity, in parabolas, or in hyperbolas, a circumstance which, as we shall have occasion to notice hereafter, greatly 46 THEORETICAL ASTRONOMY. lessens the amount of labor in many computations respecting their motion. Introducing into equation (23) the values of h and 4f2 already found, we obtain dt i/a rdr k 1/1 + i n I/a e2- (a — r) which may be written 3 a-\ - r~ __ -~ dt -- - a ae k ]/1 + l-t /] a r )' ae or dt a / e-4 ) de -)\ kV 1 I \ 1l (a - )2 - ( a-r 2 the integration of which gives t b 1/1' t 1 +4/ cos — Ce) ae- ) + C. (26) In the perihelion, r a (1 - e), and the integral reduces to t' - C; therefore, if we denote the time from the perihelion by t0, we shall have t-~1 -(-a cos — ( e- 1 - ) (27) ^ kVl-m\ \ inae -' ae' In the aphelion, r = a (1 + e); and therefore we shall have, for the time in which the body passes from the perihelion to the aphelion, to =- I, or 2cr - 2 a k / 1 -'mn T being the periodic time, or time of one revolution of the planet around the sun, a the semi-transverse axis of the orbit, or mean distance from the sun, and wT the semi-circumference of a circle whose radius is unity. Therefore we shall have = k 47 ( m (28) k2 (I + M)'> MOTION RELATIVE TO THE SUN. 47 For a second planet, we shall have a"3 4r72 _ 472 k2 (1 + A'); and, consequently, between the mean distances and periodic times of any two planets, we have the relation (1- -)- rn> ~r2 3 (1 + m') r"'3 (2 If the masses of the two planets m and m' are very nearly the same, we may take 1 + m -1 + m'; and hence, in this case, it follows that the squares of the periodic times are to each other as the cubes of the mean distances from the sun. The same result may be stated in another form, which is sometimes more convenient. Thus, since tab is the area of the ellipse, a and b representing the semai-axes, we shall have - =f areal velocity; and, since b2 = a e (I - e2), we have 3 1 1 3 Cra2a2 (I - e2) r //p f a - = - which becomes, by substituting the value of r already found, f= k /p (1 + m). (30) In like manner, for a second planet, we have I,- Tl, /p, (i + n); and, if the masses are such that we may take 1 + mn sensibly equal to 1 + in', it follows that, in this case, the areas described in equal times, in different orbits, are proportional to the square roots of their parameters. 17. We shall now consider the signification of some of the constants of integration already introduced. Let i denote the inclination of the orbit of m to the plane of xy, which is thus taken as the plane of reference, and let a be the angle formed by the axis of x and the line of intersection of the plane of the orbit with the plane of xy; then will the angles i and a determine the position of the plane of 48 THEORETICAL ASTRONOMY. the orbit in space. The constants c, c', and c", involved in the equation cz - c'y + Cx O, are, respectively, double the projections, on the co-ordinate planes, xy, xz, and yz, of the areal velocity f; and hence we shall have c = 2f cos i. The projection of 2f on a plane passing through the intersection of the plane of the orbit with the plane of xy, and perpendicular to the latter, is 2f sin i; and the projection of this on the plane of xz, to which it is inclined at an angle equal to 2, gives c' = 2f sin i cos g. Its projection on the plane of yz gives " == 2fsin i sin Q. Hence we derive z cosi - y sin i cos Q +- x sin i sin g = 0, (31) which is the equation of the plane of the orbit; and,.by means of the value of f in terms of p, and the values of c, c', c", we derive, also, dY dx _ — yd- kV/p (1 +qm) cos/, dz dx x-t - z- = k/p (1 + ) cos sini, (32) dz dy y-t Z -- k I/p (1 +- n) sin g2 sin i. These equations will enable us to determine Q, i, and p, when, for any instant, the mass and co-ordinates of m, and the components of its velocity, in directions parallel to the co-ordinate axes, are known. The constants a and e are involved in the value of p, and hence four constants, or elements, are introduced into these equations, two of which, a and e, relate to the form of the orbit, and two, g -and i, to the position of its plane in space. If we measure the angle v- to from the point in which the orbit intersects the plane of xy, the constant o will determine the position of the orbit in its own plane. Finally, the constant of integration C) in equation (26), is the time MOTION RELATIVE TO THE SUN. 49 of passage through the perihelion; and this determines the position of the body in its orbit. When these six constants are known, the undisturbed orbit of the body is completely determined. Let V denote the velocity of the body in its orbit; then will equation (20) become V2- k2 (1 + m) ( ) At the perihelion' r is a minimum, and hence, according to this equation, the corresponding value of V is a maximum. At the aphelion, V is a minimum. In the parabola, a = o, and hence V-= k i/l +m J2 \ r which will determine the velocity at any instant, when r is known. It will be observed that the velocity, corresponding to the same value of r, in an elliptic orbit is less than in a parabolic orbit, and that, since a is negative in the hyperbola, the velocity in a hyperbolic orbit is still greater than in the case of the parabola. Further, since the velocity is thus found to be independent of the eccentricity, the direction of the motion has no influence on the species of conic section described. If the position of a heavenly body at any instant, and the direction and magnitude of its velocity, are given, the relations already derived will enable us to determine the six constant elements of its orbit. But since we cannot know in advance the magnitude and direction of the primitive impulse communicated to the body, it is only by the aid of observation that these elements can be derived; and therefore, before considering the formulse necessary to determine unknown elements by means of observed positions, we will investigate those which are necessary for the determination of the heliocentric and geocentric places of the body, assuming the elements to be known. The results thus obtained will facilitate the solution of the problem of finding the unknown elements from the data furnished by observation. 18. To determine the value of k, which is a constant for the solar system, we have, from equation (28), 3 27r a2 k - 1/V +m 4 50 THEORETICAL ASTRONOMY. In the case of the earth, a — 1, and therefore 27 k - ___ In reducing this formula to numbers we should properly use, for r, the absolute length of the sidereal year, which is invariable. The effect of the action of the other bodies of the system on the earth is to produce a very small secular change in its mean longitude corresponding to any fixed date taken as the epoch of the elements; and a correction corresponding to this secular variation should be applied to the value of r derived from observation. The effect of this correction is to slightly increase the observed value of r; but to determine it with precision requires an exact knowledge of the masses of all the bodies of the system, and a complete theory of their relative motions,-a problem which is yet incompletely solved. Astronomical usage has, therefore, sanctioned the employment of the value of k found by means of the length of the sidereal year derived directly from observation. This is virtually adopting as the unit of space a distance which is very little less than the absolute, invariable mean distance of the earth from the sun; but, since this unit may be arbitrarily chosen, the accuracy of the results is not thereby affected. The value of r from which the adopted value of k has been computed, is 365.2563835 mean solar days; and the value of the combined mass of the earth and moon is 1 -354710' Hence we have log r 2.5625978148; log vl + m= 0.0000006122; log 27r =0.7981798684; and, consequently, log k 8.2355814414. If we multiply this value of k by 206264.81, the number of seconds of arc corresponding to the radius of a circle, we shall obtain its value expressed in seconds of arc in a circle whose radius is unity, or on the orbit of the earth supposed to be circular. The value of k in seconds is, therefore, log k = 3.5500065746. 2,r The quantity expresses the mean angular motion of a planet in a mean solar day, and is usually designated by,p. We shall, therefore, have MOTION RELATIVE TO THE SUN. 51 k~ 1 ~ m -- 1 +, (33) a2 for the expression for the mean daily motion of a planet. Since, in the case of the earth, 1/1 + m differs very little from 1, it will be observed that k very nearly expresses the mean angular motion of the earth in a mean solar day. In the case of a small planet or of a comet, the mass m is so small that it may, without sensible error, be neglected; and then we shall have -- -. (34) a" For the old planets whose masses are considerable, the rigorous expression (33) must be used. 19. Let us now resume the polar equation of the ellipse, the pole being at the focus, which is a(1 - e) r - + e cosv If we represent by ( the angle included between the conjugate axis and a line drawn from the extremity of this axis to the focus, we shall have sin v -= e; and, since a (1 - e2) is half the parameter of the transverse axis, which we have designated by p, we have r P 1 -- sin pO cos v The angle 5p is called the angle of eccentricity. Again, since p a (1 - e2) -= a cos2 (, we have a cos2 ( r= (35) 1 + sin CP cos v It is evident, from this equation, that the maximum value of r in an elliptic orbit corresponds to v = 1800, and that the minimum value of r corresponds to v = 0. It therefore increases from the perihelion to the aphelion, and then decreases as the planet approaches the perihelion. 52 THEORETICAL ASTRONOMY. In the case of the parabola, o - 90~, and sinp = e = 1; consequently, 1 -+ cos v But, since 1 + cos v = 2 cos2,v, if we put q -- p, we shall have r =cos, (36) in which q is the perihelion distance. In this case, therefore, when v ~ 1800, r will be infinite, and the comet will never return, but course its way to other systems. The angle (p cannot be applied to the case of the hyperbola, since in a hyperbolic orbit e is greater than 1; and, therefore, the eccentricity cannot be expressed by the sine of an arc. If, however, we designate by 4 the angle which the asymptote to the hyperbola makes with the transverse axis, we shall have e cos 4 - 1. Introducing this value of e into the polar equation of the hyperbola, it becomes p COS s p cos (7) 2 cos. (v +- ) cos (v-)' ( It appears from this formula that r increases with v, and becomes infinite when I + e cos v - 0, or cos v =- cos, in which case v - 180~ -- -: consequently, the maximum positive value of v is represented by 180~ -4, and the maximum negative value by - (180~ ~ ). Further, it is evident that the orbit will be that branch of the hyperbola which corresponds to the focus in which the sun is placed, since, under the operation of an attractive force, the path of the body must be concave toward the centre of attraction. A body subject to a. force of repulsion of the same intensity, and varying according to the same law, would describe the other branch of the curve. The problem of finding the position of a heavenly body as seen from any point of reference, consists of two parts: first, the determination of the place of the body in its orbit; and then, by means of this and of the elements which fix the position of the plane of the PLACE IN THE ORBIT. 53 orbit, and that of the orbit in its own plane, the determination of the position in space. In deriving the formulae for finding the place of the body in its orbit, we will consider each species of conic section separately, commencing with the ellipse. 20. Since the value of a - r can never exceed the limits - ae and + ae, we may introduce an auxiliary angle such that we shall have a - r - cos E. ae This auxiliary angle E is called the eccentric anomaly; and its geometrical signification may be easily known from its relation to the true anomaly. Introducing this value of into the equation ae (27) and writing t - T in place of to, T being the time of perihelion passage, and t the time for which the place of the planet in its orbit is to be computed, we obtain kill ~ m 3I + (t - T) -E - e sin E. (38) a But k1 I+ m But ~ — = mean daily motion of the planet ~- p; therefore a 4 (t - T) - E -e sinE. The quantity p (t- T) represents what would be the angular distance from the perihelion if the planet had moved uniformly in a circular orbit whose radius is a, its mean distance from the sun. It is called the mean anomaly, and is usually designated by M. We shall, therefore, have M= I (t — T), M= EE- e sin E. (39) When the planet or comet is in its perihelion, the true anomaly, mean anomaly, and eccentric anomaly are each equal to zero. All three of these increase from the perihelion to the aphelion, where they are each equal to 180~, and decrease from the aphelion to the perihelion, provided that they are considered negative. From the perihelion to the aphelion v is greater than E, and E is greater than lM. The same relation holds true from the aphelion to the perihelion, if we regard, in this case, the values of v, E, and M3 as negative. As soon as the auxiliary angle E is obtained by means of the mean motion and eccentricity, the values of r and v may be derived. For 54 THEORETICAL ASTRONOMY. this purpose there are various formulae which may be applied in practice, and which we will now develop. The equation a -r --- os CE, ae gives r a (1 - e cos E). (40) This also gives -- ae a cos E - ae, or - a cos E - ae, which, by means of equation (25), reduces to r cos v a cos E- ae. (41) If we square both members of equations (40) and (41), and subtract the latter result from the former, we get r2 sin2 v - a (1 - e2) sin2 E, or r sin v = a/1 e2 sin E =- b sin E. (42) By means of the equations (41) and (42) it may be easily shown that the auxiliary angle E, or eccentric anomaly, is the angle at the centre of the ellipse between the semi-transverse axis, and a line drawn from the centre to the point where the prolongation of the ordinate perpendicular to this axis, and drawn through the place of the body, meets the circumference of the circumscribed circle. Equations (40) and (41) give r(1 F cos v) = a(1 - e) (1 - cos E). By using first the upper sign, and then the lower sign, we obtain, by reduction, //r sin v - 1/a(1 - e) sin'E, 1/r7 cos v - /a(l -e) cos'E, (43) which are convenient for the calculation of r and v, and especially so when several places are required. By division, these equations give tan /1 - e tan V - tan E. (44) PLACE IN THE ORBIT. 55 Since e = sin q', we have 1 — e 1 - sin A(4 5o ~ tan'.(45~ - ^). I + e I + sin n Consequently, tan!E -- tan (45~ -. ) tan v.i (45) Again, 1/ -- e =-/1 -+ sin ( 1/1 - 2 sin cos (, which may be written V/1 + e = 1/sin2 - + cos2 q- + 2 sin 9 cos, or 1/1 e = sin s + cos 4. In a similar manner we find 1/1 - e - sin 1 D + cos. From these two equations we obtain 1/1 +- e + 1/1 -e=-2 cos v, 1/1 + e —1/1 - e 2 sin l, (46) which are convenient in many transformations of equations involving e or (p. Equation (42) gives r sin v p sin v sin E - - - -; b b (1 + e cos v) but p -a cos2 a, and b a cos D, hence sin v cos o sin v sin E=- (47) acos - 1 - e cosv Equation (41) gives r cosv - ae p cos v cos E ~ + e, oa a (- + e cos v) or -P cos v + ae + ae2 cos v cos --; a (1 + e cos v) and, putting a cos2 o instead of p, and sin (p for e, we get cos v +c cos E 1 cos (48) f we m y e cos v If we lmultiply the first of equations (43) by coslE, and the 56 THEORETICAL ASTRONOMY. second by sin E, successively add and subtract the products, and reduce by means of the preceding equations, we obtain sin 1 (v - E) =- - cos,Ip sin E, sin ~ (v -E)= - sin Lo sin E. (49) The perihelion distance, in an elliptic orbit, is given by the equation q = a ( - e). 21. The difference between the true and the mean anomaly, or v- M, is called the equaction of the centre, and is positive from the perihelion to the aphelion, and negative from the aphelion to the perihelion. When the body is in either apsis, the equation of the centre will be equal to zero. We have, from equation (39),;, E 1 M-+ e sinE. Expanding this by Lagrange's theorem, we get dF(M) ed 2 dF (M) e' F(E) - F(M) +- sin MdF (M - d( sin dF1M) ) e ^ 7 id, iMF(M) e +l d+ ( s dF( i +)3 +Mn M _ (50) +dM 2 diil 1d3d Let us now take, equation (40), a2 d F(E)- (1 — e cosE)2- r 2 and, consequently, F(M) = (1 - e cos M) Therefore we shall have -2 3 - (1 - e cos M) — 2e2 sin2 M(1 - e cos M) -e d (sin31(1- e os M) -.... Expanding these terms, and performing the operations indicated, we get ~2 2 = _ 1 + 2e cos M+ - (6 cos2 M- 4 sin2 M) (6 cos 36 sin2 M cos + (16 cos33 M 36 sin M cos M) +..., PLACE IN THE ORBIT. 57 which reduces to a2 e2 2-1+2e cosM+-+ (1+5 cos2M)+ (13 cos3M+3 cosM)+.... (51) r c 2 4 Equation (22) gives 2fdt r2 and, since f= kVp (1 + m), we have dv k/p(+ dt (52) ~~~~~or~~r or.2 dv- =/1 /- e. adt =/ e2 d. a2 But hc, and therefore a 2 d dv i (1 e2 dt /.) dM. 2 r By expanding the factor 1/l -- e2, we obtain /i-e 1 2_- 1e2 4i and hence dv (1. ~...) dM. Substituting for a its value from equation (51), and integrating, we get, since v = 0 when M=- 0, v- M=2e sin M-+ e sin 2M1+ 2 (13 sin 3M-3 sin M) +... (53) which is the expression for the equation of the centre to terms involving e3. In the same manner, this series may be extended to higher powers of e. When the eccentricity is very small, this series converges very rapidly; and the value of v- M for any planet may be arranged in a table with the argument M. For the purpose, however, of computing the places of a heavenly body from the elements of its orbit, it is preferable to solve the equations which give v and E directly; and when the eccentricity is 58 THEORETICAL ASTRONOMY. very great, this mode is indispensable, since the series will not in that case be sufficiently convergent. It will be observed that the formula which must be used in obtaining the eccentric anomaly from the mean anomaly is transcendental, and hence it can only be solved either by series or by trial. But fortunately, indeed, it so happens that the circumstances of the celestial motions render these approximations very rapid, the orbits being usually either nearly circular, or else very eccentric. If, in equation (50), we put F(E) -E, and consequently F(M) -, we shall have, performing the operations indicated and reducing, E = M + e sin M+ -e2 sin 211 + &c. (54) Let us now'denote the approximate value of E computed from this equation by E0, then will Eo + Eo - E, in which AEo is the correction to be applied to the assumed value of E. Substituting this in equation (39), we get M-= E + - aE, - e sinEo - e cos EoaEo; and, denoting by Mo the value of M corresponding to E0, we shall also have MO =Eo- e sin E. Subtracting this equation from the preceding one, we obtain 1 -e cos E It remains, therefore, only to add the value of aEo found from this formula to the first assumed value of E, or to B0, and then, using this for a new value of E,, to proceed in precisely the same manner for a second approximation, and so on, until the correct value of E is obtained. When the values of E for a succession of dates, at equal intervals, are to be computed, the assumed values of E0 may be obtained so closely by interpolation that the first approximation, in the manner just explained, will give the correct value; and in nearly every case two or three approximations in this manner will suffice. Having thus obtained the value of E corresponding to 31 for any instant of time, we may readily deduce from it, by the formulae already investigated, the corresponding values of r and v. In the case of an ellipse of very great eccentricity, corresponding to the orbits of many of the comets, the most convenient method of PLACE IN THE ORBIT. 59 computing r and v, for any instant, is somewhat different. The manner of proceeding in the computation in such cases we shall consider hereafter; and we will now proceed to investigate the formulae for determining r and v, when the orbit is a parabola, the formulse for elliptic motion not being applicable, since, in the parabola, a = o, and e 1. 22. Observation shows that the masses of the comets are insensible in comparison with that of the sun; and, consequently, in this case, nz-0 and equation (52), putting for p its value 2q, becomes k/ 2q dt - r2dv, or kl/2 dt= 22 dv which may be written k_ = t (I + tan2 v) sec2 ~vdv- (1 + tan2 -v) d tan Bv. 1/'2 qu Integrating this expression between the limits T and t, we obtain k(t-T) tanv + - tan3, v (55) i/2q2 which is the expression for the relation between the true anomaly and the time from the perihelion, in a parabolic orbit. Let us now represent by Tr the time of describing the arc of a parabola corresponding to v 90~; then we shall have ko 4 3 - 3' ~~~~3or 4 _3k 4 qf 1/2 To 3k Now, - is constant, and its logarithm is 8.5621876983; and if we take q -1, which is equivalent to supposing the comet to move in a parabola whose perihelion distance is equal to the semi-transverse axis of the earth's orbit, we find log r, 2.03987229, or r= 109.61558 days; that is, a comet moving in a parabola whose perihelion distance 60 THEORETICAL ASTRONOMY. is equal to the mean distance of the earth froin the sun, requires 109.61558 days to describe an arc corresponding to v- 90~. Equation (55) contains only such quantities as are comparable with each other, and by i t - T, the time from the perihelion, may be readily found when the remaining terms are known; but, in order to find v from this formula, it will be necessary to solve the equation of the third degree, tan 1v being the unknown quantity. If we put x =tan 1v, this equation becomes X-+ 3x - a -0, in which a is the known quantity, and is negative before, and positive after, the perihelion passage. According to the general principle in the theory of equations that in every equation, whether complete or incomplete, the number of positive roots cannot exceed the number of variations of sign, and that the number of negative roots cannot exceed the number of variations of sign, when the signs of the terms containing the odd powers of the unknown quantity are changed, it follows that when a is positive, there is one positive root and no negative root. When a is negative, there is one negative root and no positive root; and hence we conclude that equation (55) can have but one real root. We may dispense with the direct solution of this equation by forming a table of the values of v corresponding to those of t- T in a parabola whose perihelion distance is equal to the mean distance of the earth from the sun. This table will give the time corresponding to the anomaly v in any parabola, whose perihelion distance is q, by multiplying by q2, the time which corresponds to the same anomaly in the table. We shall have the anomaly v corresponding to the time t- T by dividing t - T by q', and seeking in the table the anomaly corresponding to the time resulting from this division. A more convenient method, however, of finding the true anomaly from the time, and the reverse, is to use a table of the form generally known as Barker's Table. The following will explain its construction: Multiplying equation (55) by 75, we obtain 75k 3 (t- T) - 75 tan -v + 25 tans Av. 1/ q2 Let us now put M= 75 tan ~v + 25 tan3 V, PLACE IN THE ORBIT. 61 75k and Co -= -, which is a constant quantity; then will v2 C (t- T) M. q The value of Co is log C - 9.9601277069. Again, let us take qC - which is called the mean daily motion in the parabola; then will M- m (t - T) = 75 tan ~v + 25 tan3 Mv. If we now compute the values of HM corresponding to successive values of v from v- 0~ to v 180~, and arrange'them in a table with the argument v, we may derive at once, from this table, for the time (t - T) either M when v is known, or v when M — m (t- T) is known. It may also be observed that when t- T is negative, the value of v is considered as being negative, and hence it is not necessary to pay any further attention to the algebraic sign of t - T than to give the same sign to the value of v obtained from the table. Table VI. gives the values of M for values of v from 0~ to 1800~, with differences for interpolation, the application of which will be easily understood. 23. When v approaches near to 180~, this table will be extremely inconvenient, since, in this case, the differences, between the values of MI for a difference of one minute in the value of v increase very rapidly; and it will be very troublesome to obtain the value of v from the table with the requisite degree of accuracy. To obviate the necessity of extending this table, we proceed in the following manler:Equation (55) may be written k (t- T) _= I tan8'v (1 + 3 cot2 Iv); 1/2 T and, multiplying and dividing the second member by (1 + cot2 ~v)% we shall have k (t- T ) 1 -+ 3 cot2 Iv V(tq T)__ I tan8 Iv (1 + cot2 -v)3 1 7- ctv)3 3 37^. 3 f' L cot, V)/ e~~~tt ~Z~y * 62 THEORETICAL ASTRONOMY. 2 But 1 + co sin v tan and consequently 2 sin v tan IV k (t- T) 8 1 + 3 cot2 2v 1/2 q 3 sin v ( l+cot2 v)' Now, when v approaches near to 180~, cot Iv will be very small, and the second factor of the second member of this equation will nearly 1. Let us therefore denote by w the value of v on the supposition that this factor is equal to unity, which will be strictly true when v 180~, and we shall have, for the correct value of v, the following equation: V W + A0, A being a very small quantity. We shall therefore have 3 tan I (w + A)+ tan' (w+ A), and, putting tan 1w -, and tan h = x, we get, from this equation, (1 + 0)33 0 + x (0 + x)3 03 1 - x (1 -- x)' Multiplying this through by 03 (1 - Ox), expanding and reducing, there results the following equation: 1 + 302= 30 (1 + 402 + 204 + 06) X - 302 (1 + 402 + 240 + 6) X2 + 03 (2 + 602 + 304 + 06) X3. Dividing through by the coefficient of x, we obtain 1 + 302 2 + 02 (2 + 682 + 30 + 06) x3 3~ (1 + 402 + 204 + 06) 3 (1 + 402 + 204 + 6) Let us now put 1 + 302 30 (1 + 402 + 204 + 0) y; then, substituting this inthe preceding equation, inverting the series and reducing, we obtain finally + (4 + 18 + 904 + 56) + &C - y + y +3 (1 + 402+ 2 + + o6) But tan A0 = x, therefore A, = 2x - x3 +..... PLACE IN THE ORBIT. 63 Substituting in this the value of x above found, and reducing, we obtain -- 2 - 32* -+- 166 t+ 100s AO (2y 2y2+ 32(1+402+ 160+ 0 y3 + &c. For all the cases in which this equation is to be applied, the third term of the second member will be insensible, and we shall have, to a sufficient degree of approximation, Ao = 2y + 20y2. Table VII. gives the values of a0, expressed in seconds of arc, corresponding to consecutive values of w from w = 155~ to w 180~. In the application of this table, we have only to compute the value of M precisely as for the case in which Table VI. is to be used, namely, M= m (t —T); then will w be given by the formula ^200 sin w -- M20' since we have already found k (t- T) 8 1/ 2 q 3sinw'.or i 8q V/2 200 sin w- 3 (t -T)k M' Having computed the value of w from this equation, Table VII. will furnish the corresponding value of a0; and then we shall have, for the correct value of the true anomaly, V -w +- A0, which will be precisely the same as that obtained directly from Table VI., when the second and higher orders of differences are taken into account. If v is given and the time t- T is required, the table will give, by inspection, an approximate value of A~ using v as argument, and then w is given by w =- V - Ao. 64 THEORETICAL ASTRONOMY. The exact value of A is then found from the table, and hence we derive that of w; and finally t- T from 200 q t - T CO sin3w 24. The problem of finding the time t - T when the true anomaly is given, may also be solved conveniently, and especially so when v is small, by the following process:Equation (55) is easily transformed into 3k (t- T) _ sin v (3 2 sin, -q= c-"OS'v ^ (3 - 2sin ^v), 2 qV2 Cos from which we obtain, since q =r cos2'v, 3^ (t~-T) _3( sin 2v ) 4 ( sin g 2 r0 V V~ Let us now put sin v sin x - ~-, and we have 3k (t T) _ - 3 sin x -4 sin3 x - sin 3x. 2 rConsequently, 2 3 3 t-T —k r sin 3x, which admits of an accurate and convenient numerical solution. To facilitate the calculation we put sin 3x N - sin v' the values of which may be tabulated with the argument v. When v = 0, we shall have N= -V' 2, and when v = 90, we have N- 1; from which it appears that the value of N changes slowly for values of v from 0~ to 90~. But when-v= 180, we shall have N= c, and hence, when v exceeds 90~, it becomes necessary to introduce an auxiliary different from N. We shall, therefore, put in this case, N' = N sin v = sin 3x; PLACE IN THE ORBIT. 65 from which it appears that N'- 1 when v =90~, and that N'= i/2 when v = 180~. Therefore we have, finally, when v is less than 90~, 2 3 t - T Nr2 sin v, 3k and, when v is greater than 90~, 2 3 t- T-= N'r2, 3k 2 in which log 3= 1.5883272995, from which t - T is easily derived t3iC when v is known. Table VIII. gives the values of N, with differences for interpolation, for values of v from v = 0~ to v 90~, and the values of N' for those of v from v- 90~ to v- 180~. 25. We shall now consider the case of the hyperbola, which differs from the ellipse only that e is greater than 1; and, consequently, the formulae for elliptic and hyperbolic motion will differ from each other only that certain quantities which are positive in the ellipse are negative or imaginary in the hyperbola. We may, however, introduce auxiliary quantities which will serve to preserve the analogy between the two, and yet to mark the necessary distinctions. For this purpose, let us resume the equation p cos % 2 cos - (v + A) cos 2 ( - )' When v 0, the factors cosI(v + A) and cos(v - ) in the denominator will be equal; and since the limits of the values of v are 180~-~ and -(180~ - ), it follows that the first factor will vanish for the maximum positive value of v, and that the second factor will vanish for the maximum negative value of v, and, therefore, that, in either case, r oo. In the hyperbola, the semi-transverse axis is negative, and, consequently, we have, in this case, p - a (e 1), or a p cot2. We have, also, for the perihelion distance, q a(e -1). Let us now put tanF= tan- + -,' (56) 5 66 THEORETICAL ASTRONOMY. which is analogous to the formula for the eccentric anomaly E in an ellipse; and, since e -, we shall have cos e -1 1 -cos 2 e +- 1 + cosa 2n and, consequently, tan -F = tan Iv tan 57. (57) We shall now introduce an auxiliary quantity a, such that 1 + tan;'F a tan (450~ +,F) - 1 +- tan. F' 1 - tan IF' whence we derive tan F- 1 (58) and also cos I (v - =OS. ~V w/ (59) cos ~ (v +,) This last equation shows that a- 1 when the comet is in its perihelion; a =oo when v =180~ —; and =Owhenv -(180~ - ). 2 tan'IF Since tan F -- t2 F' we shall have 1 - tan2 4F tan F= ( - (60),+1 Squaring this equation, adding 1 to both members, and reducing we obtain cos F= (61) Replacing a in this equation by its value from equation (59), we get 1 cos2 ~ (v + ) +os2 ( - ) cos F 2 cos (v + ) cos, (v-+)' or 1 1 + cos v cos _ (e + cos v) cos S cos F 2 cos (v + ) cos (v - ) 2 cos I (v + +) cos ~(v - which reduces to 1 __r(e+cos v) cosF-~ ~ (p2) PLACE -IN THE ORBIT. 67 If we add +T 1 to both members of this equation, we shall have 1 -t- cos F r(e - 1) (1 -+ cos v) cos F p Taking first the upper sign, and then the lower sign, and reducing, we get l/rsin v (e ) sin F, l/cos F Vr cos = f oV ) cos a 1. (63) These equations for finding r and v, it will be observed, are analogous to those previously investigated for an elliptic orbit. These equations give, by division, tan -! v -- tan -F, tang Je - 1 tan ~F, which is identical with the equation (56), and may be employed to verify the computation of r and v. j Multiplying the last of equations (63) by the first, putting for e2- 1 its value tan2 4, and reducing, we obtain r sin v atan, tan F==a tan —. (64) Further, we have p cos v ar (e + cos v) r cos v - = ae -- I + e cos v' p which, combined with equation (62), gives rcosv_ —a ) e -a( 2e- -- ~ (65) If we square these values of r sin v and r cos v, add the results together, reduce, and extract the square root, we find r —a -1 =Ja 2e - + — ~ (66) We might also introduce the auxiliary quantity a into the equations (63); but such a transformation is hardly necessary, and, if at all desirable, it can be easily effected by means of the formule which we have already derived. 68 THEORETICAL ASTRONOMY. 26. Let us now resume the equation cos (v - 4) cos (v + 4)' Differentiating this, regarding ~ as constant, we have sin dv -2 cos ( + ), and, dividing this equation by the preceding one, we get ds _ sin d_ a 2 cos - (v + ) cos (v-) But p cos 2 cos (v + 4) cos (v -)' consequently, da r tan 4 dv, which gives r2dv - pr da. a tan + Substituting this value of r2dv in equation (22), and putting instead of 2f its value kv/p, from equation (30), the mass being considered as insensible in comparison with that of the sun, we get klp dt pr d. atan Then, substituting for r its value from equation (66), and for p its value a tan2 ~, we have k/p dt == a2tan4 ( e (I + - da. Integrating this between the limits T and t, we obtain kl/ (t- T) atan ( -e ( —— I loge a), (67) in which loge a is the Naperian or hyperbolic logarithm of a. Since l/p -l/~a tan', if we put k = a PLACE IN THE ORBIT. 69 in which v is the mean daily motion; and if we also put - (t-T) - N0, in which No corresponds to the mean anomaly M in an ellipse, we shall have, from equation (67), N - (1 ( -!) loge1 a. (68) If we multiply both members of this equation by =0.434294482, the modulus of the common system of logarithms, and put y- -NA (t- T), a" we shall have x —;e -A A - log,; wherein log A = 9.6377843113, and log Ak - 7.8733657527. Let us now introduce F into this formula; and for this purpose we have tan Fi ^ ), and also log =- log tan (45~ + -F). Therefore we obtain N= eA tan F- log tan (450 +.F). (69) This equation will give, directly, the time t- T from the perihelion, when a, e, and Fare known; but, since it is transcendental, in the solution of the inverse problem, that of finding the true anomaly and radius-vector from the time, the value of F can only be found by successive approximations. If we differentiate the last equation, regarding N and F as variable we get dN= (e - cos F) dF. cos2F Hence, if we denote an approximate value of F by F,, and the corresponding value of N by N, the correction aF, to the assumed value of F may be computed by the formula F (N - N,) cos2F, (e - cosF,) 70 THEORETICAL ASTRONOMY. This correction being applied to F,, a nearer approximation to the true value of F will be obtained; and by repeating the operation there results a still closer approximation. This process may be continued until the exact value of F is found, and, when several successive places are required, the first assumed value may be estimated, in advance, so closely that a very few trials will suffice. In practice, however, cases will rarely occur in which this formula will be applied, since the probability of hyperbolic motion is small, and, whenever any positive indication of an eccentricity greater than I has been found to exist, it has only been after a very accurate series of observations has been introduced as the basis of the calculation. For a majority of the cases which do really occur, the most accurate and convenient method of finding r and v will be explained hereafter. 27. If we consider the equation M= E e sin E, we shall see that, when logarithms of six or seven decimals are used, the error which may exist in the determination of E when lMk and e are given, will increase as e increases, but in a much greater ratio; and, when the eccentricity becomes nearly equal to that of the parabola, the error may be very great. In the case of hyperbolic motion, also, the numerical solution of equation (69), when e — 1 is very small, and with the ordinary logarithmic tables, becomes very uncertain. This can only be remedied, when equations (39) and (69) are employed, by using more extended logarithmic tables; and when the orbit differs only in an extremely slight degree from a parabola, even with the most extended logarithmic tables which have been constructed, the error may be very large. For this reason we have recourse to other methods, which will give the required accuracy without introducing inconveniences which are proportionally great. We shall, therefore, now proceed to develop the formulae for finding the true anomaly in ellipses and hyperbolas which differ but little from the parabola, such that they will furnish the required accuracy, when the exact solution of equations (39) or (69) with the logarithmic tables in common use is impossible. For this purpose, let us resume equation (22), which, by substituting for 2f its value klVp, the mass of the comet being neglected in comparison with that of the sun, becomes k I/p dt == r2dv, PLACE IN THE ORBIT. 71 or k-V/P dt~( p~ ~dv k p dt - (1 + e cos )2' Let us now put i = tan v, and we shall have 1 -u2 d 2du cos -- u; dv 1 - Substituting these values in the preceding equation, and putting 1-e 1 q- e i, we get d- 2p2 (l + 2)du k pdt (1-+ e)2 (1 + i,2)2 or, since p=q (1 e), kl / + e dt (1 + u) du 2 - (1 + i2)' Let us now develop the second member into a series. This may be written thus: dut (1 + iu2) ( + j2)- 2; and developing the last factor into a series, we obtain (1 + iu2)- = 1- 2iu2 + 3i2u4 4-i36 + &c. Consequently, (1 -+ 2) (1 + iu2)-2 1 + u2 - 2i (+a2 + u') + 3i2 (u +- 6) - 4i (u6 t ) - +....; Multiplying this equation through by du, and integrating between the limits T and t, the result is k(t- T)l/l u +,U - 2i (1, + O5) + 3i2 (5 + i7) 2qg - 4i3 (u7 + I99) + &c. L. (70) In the case of the parabola, e = 1 and i =0, and this equation becomes identical with (55). Let us now put k (t -T) 1/ 1 + e U+ 3 (71) 2q - 3u (71) 2q 72 THEORETICAL ASTRONOMY. and also U= tan; then the angle Vwill not be the true anomaly in the parabola, but an angle derived from the solution of a cubic equation of the same form as that for finding the parabolic anomaly; and its value may be found by means of Table VI., if we use for M the value computed from 75k (t - T) <1 + e 2 2 Let U be expanded into a series of the form U = -+ i +- i2 + yr3 +.... which is evidently admissible, a,, r,.... being functions of u and independent of i. It remains now to determine the values of the coefficients A, r,, &c., and, in doing so, it will only be necessary to consider terms of the third order, or those involving i3, since, for nearly all of those cases in which the eccentricity is such that terms of the order i4 will sensibly affect the result, the general formula already derived, with the ordinary means of solution, will give the required accuracy. We shall, therefore, have U + U3 u, + ai + Xi2 + + ( + + + + + Y, ) or, again neglecting terms of the order 4, u+ 1 U3 == U + - u + i ( + u) a + 2 (uf2 + (1+ uf2) ) + i3 (i3 + 2uoaj + (1 + t2) r). But we have already found, (70), k(t -T)1/Te __ U u+ I U3 = u + l u -3 2i (Q + at&) 2qY + 3i2 (I u5 + Ju7) 4i3 (u 1 + _ 9). Since the first members of these equations are identical, it follows, by the principle of indeterminate coefficients, that the coefficients of the like powers of i are equal, and we shall, therefore, have (a + Mb2) ffi = 1a -d 1Q5) t2 + (1 + in+ ) 2P + 3 (5U5 + ue), 3' + 2uNfa + (1 + i)2). 4 (7 + 1 9). From the first of these equations we find PLACE IN THE ORBIT. 73 3 ~1 5 2 (u + U ) The second equation gives 3(L5 t+ 1,7) _ uQ2 1 +u2 or, substituting for a its value just found, and reducing, 3 (I5U + 3 —7 U- + 3 L9 + 547 11) (i +(1+u) We have also - 4 (-uI7 + -'1) - a ~3- 20,ue 1 + to and hence, substituting the values of a and 9 already found, and reducing, we obtain finally 4 (1u7 + 1 2 9 8I + 1 017 4 1 16t13 + T223 t15+ 2 7) r — (+1 -(+ t2)' Again, we have tan U- tan (I +- i -+ i2 + ri3). Developing this, and neglecting terms of the order i, we get tan' U= tan'u + (1+ ( +a + 2 (1 + -2)2 ('2i2 + 2afii3) 2 1 + (1 3 3i3' Now, since = tan v and U= tan g V, we shall have V — + 1+ (ai + (ii + ri3) ( + 2 i2 + 2cafis) + u- ) 3i3 or 2. 2/5 2 2U 7 -V + 1 + U2' + L2 (1+ ) ) (1 + U2 ( + i2)2 ( + 2))3 4 2r 4cL u 2('~1) Substituting in this equation the values of a, P, and r already found, and reducing, we obtain finally 4'U3 4t + + 4. 6 + TO ~ T 38 +1 I 3 V v -+ i +.3 1 0 V v — (1 +i) (1 + U) 8 7 + 52 8 1 2638411 4 13 5128 15+ 904 T 83 T ( 1 14 +I -- I 3 1 8 _ o 4787 5 3. (73) (] U26 74 THEORETICAL ASTRONOMY. This equation can be used whenever the true anomaly in the ellipse or hyperbola is given, and the time from the perihelion is to be determined. Having found the value of V, we enter Table VI. with the argument V and take out the corresponding value of 3I; and then we derive t - T from t _T__M 32 -- co \+ e in which log C0 - 9.96012771. For the converse of this, in which the time from the perihelion is given and the true anomaly is required, it is necessary to express the difference v - V in a series of ascending powers of i, in which the coefficients are functions of U. Let us, therefore, put u= U + ai + l'i2 + r'i3 + &c. Substituting this value of u in equation (70), and neglecting terms multiplied by i4 and higher powers of i, we get k(t- T)V1 +- e U+ U3+_ ('(1 + U2) _U3 2_ _U5)i 3 2q2 + (I'(1 + U2) + U' 2- 2 U2U ( + U2) + 3 U5 + 3 Ui) 2 + (r' (1 + U2) +' a3 + 2 Ua''+- 3U4a' (I + U2)- 2' U2 (1 + U2) -4 U3t2 - 2 Ud2- - 4 U - U9) i3. But, since the first member of this equation is equal to U I- U3, we shall have, by the principle of indeterminate coefficients,' (I + U) 2- U3 - U -O, (1 + U2)+ ud2-2 (+ U + U5 + 3 U' 0, 71 (I + U2) + I U~3 U2 (I + U2) r'(l + U2) + "'+ 2Ua1'' + 3U4'd(1 + U2) - 2'U2(1 + U2) -4 U3a' 2 Ua 2- 4 U7 4 U9 0. From these equations, we find 2 U3 +'U5 " + U' ( 1+ U2)3 29U + 1 7928 US 10328 U+" + U + U135 66 + U 17 _ 3 1 283 3 1 28 3a 15 JI f 48 Ua 1 57 r~ (1+ U2)5 If we interchange v and V in equation (72), it becomes, writing at, 3', r' for a, p, r, PLACE IN THE ORBIT. 75 ^,V 2ar2 ( 2/ 2d"U., V -+ U 1 U'I (1 + U2)2' 2'_ 4 aWi4/U +2 (U" — ) d~3) i.' I+ U ( (1 + U2)2 (1 / U2)' Substituting in this equation the above values of a', I', and y', and reducing, we obtain, finally, 4T3_+ 4 U5 22U 5+J 598U L 86 o U + 17 U 2 v 3 T-t- 3 ^ -a T5 ~.5rT T05, 2-175 -2' (1 + U')2' (1 + U2)4 f84 U7 9752 U9+ 37328 Ul+ 1 f48rU13+ 17 6 78 U15 184 U17 + 3 1a T 835 t 141T75 1 5756 I785 I 7875 3 1 (74) (1 +- U2)6 by means of which v may be determined, the angle Vbeing taken from Table VI., so as to correspond with the value of M derived from M=(t-T)q3 \ Equations (73) and (74) are applicable, without any modification, to the case of a hyperbolic orbit which differs but little from the parabola. In this case, however, e is greater than unity, and, consequently, i is negative. 28. In order to render these formulae convenient in practice, tables may be constructed in the following manner:Letx - v or V, and tan -x = 0, and let us put A 03+ 405 100(1 + 02)2 B 2 05+ 5 807 + %09 + _ T9 8 10000 (1 + 02)4 62 W5 + 984 6 6507 + "1109 + l+75 +'B5 5 3.1 85 0 11 1oooo (1- o+ 2)4 1000000 (I + 02)6 80' + 58 + 2638401 + 464 13+ 28015 + 9048 wherein s expresses the number of seconds corresponding to the length of arc equal to the radius of a circle, or logs 5.31442513. We shall, therefore, have: When x = V, v = + A (100i) + B (100i)2 + C(00i)3; 76 THEORETICAL ASTRONOMY. and, when x = v, v= v - A (100i) + B' (100) - C' (100i). Table IX. gives the values of A, B, B', C, and C' for consecutive values of x from x = 00 to x = 1490, with differences for interpolation. When the value of v has been found, that of r may be derived from the formula q(1 + e) 1 - e cos v Similar expressions arranged in reference to the ascending powers / /2 \2 of (1- e) or of (( ) 1 I ) may be derived, but they do not con2 ) verge with sufficient rapidity; for, although ( ( I -1 ) is less than i,yet the coefficients are, in each case, so much greater thah those of the corresponding powers of i, that three terms will not afford the same degree of accuracy as the same number of terms in the expressions involving i. 29. Equations (73) and (74) will serve to determine v or t - T in nearly all cases in which, with the ordinary logarithmic tables, the general methods fail. However, when the orbit differs considerably from a parabola, and when v is of considerable magnitude, the results obtained by means of these equations will not be sufficiently exact, and we must employ other methods of approximation in the case that the accurate numerical solution of the general formulae is still impossible. It may be observed that when E or F exceeds 50~ or 60~, the equations (39) and (69) will furnish accurate results, even when e differs but little from unity. Still, a case may occur in which the perihelion distance is very small and in which v may be very great before the disappearance of the comet, such that neither the general method, nor the special method already given, will enable us to determine v or t- T with accuracy; and we shall, therefore, investigate another method, which will, in all cases, be sufficiently exact when the general formulae are inapplicable directly. For this purpose, let us resume the equation k(t- T) - esinE, 8 PLACE IN THE ORBIT. 77 which, since q = a (- e), may be written k (t T) V/1- e 1 1 1 9e. 3 -- / _ (9E - sin E) + - (E - sinE). 2 10 1 If we put E-sin E ^9E - sinE' we shall have k (t -T)/1 — e 20i/A _ 1 1 +9e 2q2 * 9E- + sin E- 3'5 (1 e) Let us now put B 9E + sin E 201/A and tan'2w 1 + 9e_ A 5 (1- e) then we have k(t-T) 1/ (1 + 96) -~ ~T) - V T 1 9e) tan w + 1 tan' w. (75) 3 B T/2 When B is known, the value of w may, according to this equation, be derived directly from Table VI. with the argument M 75k(t T) /v(1 +9e) 3B 1/2 q and then from w we may find the value of A. It remains, therefore, to find the value of B; and then that of v from the resulting value of A. Now, we have 2 -tan 1E sin E -- 1 + tan"2E' and if we put tan2 E - T, we get sin E = 2 =_ 2r:2 ( T- + 2 — ~ + &c.). sin E=1 - We have, also, E =2 tan' =2 2 (1 - + 12 - T3 + &c.). 3 7 5? t ~j 78 THEORETICAL ASTRONOMY. Therefore, 15 (E - in E) 2+1 (10 - 6+0o2 + 9_ t _ + &c.), and 9E + sin E-= 2r~ (10 - 12r + 144T2 16 -+ 84-_ &c.) Hence, by division, 15 E- sinE A=r - 42 +43 _ 192r4 2 + 1744375 19Eq sinE --? 3 -c 1 0 8 99 8 85T6 + &C.; and, inverting this series, we get A 1 -_ ~ _ A + TA' +.A1 + _ 89fA4+ _ ~84 -&. &c. which converges rapidly, and from which the value of ~ may be found. Let us now put A I 2~ C2 then the values of C may be tabulated with the argument A; and, besides, it is evident that as long as A is small C2 will not differ much from 1 + 4A. Next, to find B, we have -A ^ (1 210 46-2 I041 - + 161 4 - &c.), and hence i(9E- +sinE) B 6 753r2 + 336874- &c.; -^-= 1 + T -,i +,i3 - &c.; V77 625 9007 1/A from which we easily find B = 1 + 3A+ a, + 3 337 A4 + &c. If we compare equations (44) and (56), we get tan -E=/= V 1 tan F. Hence, in the case of a hyperbolic orbit, if we put tan2 FF-=', we must write -' in place of r in the formulhe already derived; and, from the series which gives A in terms of r, it appears that A is in this case negative. Therefore, if we distinguish the equations for PLACE IN THE ORBIT. 79 hyperbolic motion from those for elliptic motion by writing A', B', and C' in place of A, B, and C, respectively, we shall have 1 A' t1 +4A'+ At 8 2.. 3+ 2 87 441 At+ &c., C~2 ~I =,at- ga'+,?gA~2 — L5.1,3 7't6 A~a 13 4 4 &C B' 1 + -f3 A2 -_ -2 HA3 +- 3-34 75 —14 - &c. Table X. contains the values of log B and log C for the ellipse and the hyperbola, with the argument A, from A= 0 to A- 0.3. For every case in which A exceeds 0.3, the general formulae (39) and (69) may be conveniently applied, as already stated. The equation tan v = 1-e+ e tan JE gives tan2 == _ e A C2 or, substituting the value of A in terms of w, tan v - Ctan C t1a 9e) (76) The last of equations (43) gives r cos v - q cos E +t +I -tan2 -E Hence we derive r - 77) (1 + AC2)cos o (77) The equation for v in a hyperbolic orbit is of precisely the same form as (76), the accents being omitted, and the value of A being computed from 5 (e —1) A- = 1 tan9 w. (78) For the radius-vector in a hyperbolic orbit, we find, by means of the last of equations (63), (1A AC2) cos2 v (79) When t T is given and r and v are required, we first assume B = I, and enter Table VI. with the argument M (t - T). (1 +9e) B' 80 THEORETICAL ASTRONOMY. in which log C0 = 9.96012771, and take out the corresponding value of w. Then we derive A from the equation A 5(1-e) tan 1 -+9e tanw in the case of the ellipse, and from (78) in the case of a hyperbolic orbit. With the resulting value of A, we find from Table X. the corresponding value of log B, and then, using this in the expression for M, we repeat the operation. The second result for A will not require any further correction, since the error of the first assumption of B 1 is very small; and, with this as argument, we derive the value of log C from the table, and then v and r by means of the equations (76) and (77) or (79). When the true anomaly is given, and the time t - T is required, we first compute r from 1 —e =. + e tan2 v, in the case of the ellipse, or from e-1 T =e l tan' v, e + 21 in the case of the hyperbola. Then, with the value of T as argument, we enter the second part of Table X. and take out an approximate value of A, and, with this as argument, we find logB and log C. The equation C2 will show whether the approximate value of A used in finding log C is sufficiently exact, and, hence, whether the latter requires any correction. Next, to find w, we have tan 1 1+9e tanlwz —.w t2 - C (1 be)' and, with w as argument, we derive M from Table VI. Finally, we have t-T= MBq (80) Co / -j (1 + 9e) by means of which the time from the perihelion may be accurately determined. POSITION IN SPACE. 81 30. We have thus far treated of the motion of the heavenly bodies, relative to the sun, without considering the positions of their orbits in space; and the elements which we have employed are the eccentricity and semi-transverse axis of the orbit, and the mean anomaly at a given epoch, or, what is equivalent, the time of passing the perihelion. These are the elements which determine the position of the body in its orbit at any given time. It remains now to fix its position in space in reference to some other point in space from which we conceive it to be seen. To accomplish this, the position of its orbit in reference to a known plane must be given; and the elements which determine this position are the longitude of the perihelion, the longitude of the ascending node, and the inclination of the plane of the orbit to the known plane, for which the plane of the ecliptic is usually taken. These three elements will enable us to determine the co-ordinates of the body in space, when its position in its orbit has been found by means of the formulae already investigated. The longitude of the ascending node, or longitude of the point through which the body passes from the south to the north side of the ecliptic, which we will denote by, is the angular distance of this point from the vernal equinox. The line of intersection of the plane of the orbit with the fundamental plane is called the line of nodes. The angle which the plane of the orbit makes with the plane of the ecliptic, which we will denote by i, is called the inclination of the orbit. It will readily be seen that, if we suppose the plane of the orbit to revolve about the line of nodes, when the angle i exceeds 180~, g will no longer be the longitude of the ascending node, but will become the longitude of the descending node, or of the point through which the planet passes from the north to the south side of the ecliptic, which is denoted by 3, and which is measured, as in the case of g, from the vernal equinox. It will easily be understood that, when seen from the sun, so long as the inclination of the orbit is less than 90~, the motion of the body will be in the same direction as that of the earth, and it is then said to be direct. When the inclination is 90~, the motion will be at right angles to that of the earth; and when i exceeds 90~, the motion in longitude will be in a direction opposite to that of the earth, and it is then called retrograde. It is customary, therefore, to extend the inclination of the orbit only to 90~, and if this angle exceeds a right angle, to regard its supplement as the inclination of the orbit, noting simply the distinction that the motion is retrograde. 6 82 THEORETICAL ASTRONOMY. The longitude of the perihelion, which is denoted by wr, fixes the position of the orbit in its own plane, and is, in the case of direct motion, the sum of the longitude of the ascending node and the angular distance, measured in the direction of the motion, of the perihelion from this node. It is, therefore, the angular distance of the perihelion from a point in the orbit whose angular distance back from the ascending node is equal to the longitude of this node; or it may be measured on the ecliptic from the vernal equinox to the ascending node, then on the plane of the orbit from the node to the place of the perihelion. In the case of retrograde motion, the longitudes of the successive points in the orbit, in the direction of the motion, decrease, and the point in the orbit from which these longitudes in the orbit are measured is taken at an angular distance from the ascending node equal to the longitude of that node, but taken, from the node, in the same direction as the motion. Hence, in this case, the longitude of the perihelion is equal to the longitude of the ascending node diminished by the angular distance of the perihelion from this node. It may, perhaps, seem desirable that the distinctions, direct and retrograde motion, should be abandoned, and that the inclination of the orbit should be measured from 0~ to 180~, since in this case one set of formulse would be sufficient, while in the common form two sets are in part required. However, the custom of astronomers seems to have sanctioned these distinctions, and they may be perpetuated or not, as may seem advantageous. Further, we may remark that in the case of direct motion the sum of the true anomaly and longitude of the perihelion is called the true longitude in the orbit; and that the sum of the mean anomaly and longitude of the perihelion is called the mean longitude, an expression which can occur only in the case of elliptic orbits. In the case of retrograde motion the longitude in the orbit is equal to the longitude of the perihelion minus the true anomaly. 31. We will now proceed to derive the formule for determining the co-ordinates of a heavenly body in space, when its position in its orbit is known. For the co-ordinates of the position of the body at the time t. we have x — r cos v, y - r sin v, POSITION IN SPACE. 83 the line of apsides being taken as the axis of x, and the origin being taken at the centre of the.sun. If we take the line of nodes as the axis of x, we shall have x r cos (v +- W), y = r sin (v+ W), w being the arc of the orbit intercepted between the place of the perihelion and df the node, or the angular distance of the perihelion from the node. No'w, we have o) — Tr- in the case of direct motion, and w= -ir in the case of retrograde motion; and hence the last equations become x = r cos (v z 7i =F 2) y = r sin (v ~ z =T ) the upper and lower signs being taken, respectively, according as the motion is direct or retrograde. The arc v +~ T - 7 - u is called the argunment of the latitude. Let us now refer the position of the body to three co-ordinate planes, the origin being at the centre of the sun, the ecliptic being taken as the plane of xy, and the axis of x, in the line of nodes. Then we shall have Xr = COS U, y' = = r sin u cos i, z' - r sin i sin i. If we denote the heliocentric latitude and longitude of the body, at the time t, by b and I, respectively, we shall have x' - r co os b cos (I ), y' = r cos b sin (I- g ), z' r sin b, and, consequently, cos u = cos b cos (I1- ), ~ sin u cos i. cos b sin (I -), (81) sin u sin i = sin b. From these we derive tan (I - ) -- = tan u cos i, tan b = ~ tan i sin ( — S ), (82) which serve to determine I and b, when 2, i, and i are given. Since 84 THEORETICAL ASTRONOMY. cos b is always positive, it follows that I - 2 and u must lie in the same quadrant when i is less than 90~; but if i is greater than 90~, or the motion is retrograde, 1 - Q and 360 - u will belong to the same quadrant. Hence the ambiguity which the determination of I~- g by means of its tangent involves, is wholly avoided. If we use the distinction of retrograde motion, and consider i always less than 90~, 1 - 2 and — u will lie in the same quadrant. 32. By multiplying the first of the equations (81) by sinu, and the second by cos u, and combining the results, considering only the upper sign, we derive cos b sin ( - 1+ - ) 2 sin u cos ut sin2 /i, or cos b sin (u -1 + 2 ) sin 2u sin2 /i. In a similar manner, we find cos b cos (u - 1-t + ) = cos2 + sin2u cos i, which may be written cos b cos (u - + g) -=(1 u)+1 (1 - cos 2u) cos i, or cos b cos (u - I+ ) = (1 + cos ) + (1 cos ) cos 2u; and hence cos b cos (u - 1 - + ) = cos2 i + sina i cos 2u. If we divide this equation by the value of cos b sin (u - 1 -+ g) already found, we shall have tan' Ai sin 2it tan (u -- + ) + t2 is 2. (83) +. I-ttan2 i cos2u( The angle u -1 + 2 is called the reduction to the ecliptic; and the expression for it may be arranged in a series which converges rapidly when i is small, as in the case of the planets. In order to effect this development, let us first take the equation n sin x tan y _ e t 1- n+ cos x Differentiating this, regarding y and n as variables, and reducing, we find dy sin x d — 1 + 2n1 cos x - n2' POSITION IN SPACE. 85 which gives, by division, or by the method of indeterminate coefficients, =d_ sin x - n sin 2x + n2 sin 3 - n3 sin 4x + &c. dn Integrating this expression, we get, since y = 0 when x = 0, y nsin X ~- n2 sin 2x -+ 13 nsin 3x - In4 sin 4x +...., (84) which is the general form of the development of the above expression for tan y. The assumed expression for tan y corresponds exactly with the formula for the reduction to the ecliptic by making n - tan2 i and x = 2X; and hence we obtain u - -]- = tan2 i sin 2u - tan4 i sin 4u + 1 tan6 1i sin 61 -4 tan8 -i sin 8u + 1 tan1~ Ii sin 10i - &c. (85) When the value of i does not exceed 100 or 12~, the first two terms of this development will be sufficient. To express u — 1 + in seconds of arc, the value derived from the second member of this equation must be multiplied by 206264.81, the number of seconds corresponding to the radius of a circle. If we denote by Re the reduction to the ecliptic, we shall have I - tu + ~ -R, v- +~ —Re. But we have v -= M the equation of the centre; hence I - M + - + equation of the centre - reduction to the ecliptic, and, putting L -1 + 7z = mean longitude, we get I L + equation of centre - reduction to ecliptic. (86) In the tables of the motion of the planets, the equation of the centre (53) is given in a table with M as the argument; and the reduction to the ecliptic is given in a table in which i and u are the arguments. 33. In determining the place of a heavenly body directly from the elements of its orbit, there will be no necessity for computing the reduction to the ecliptic, since the heliocentric longitude and latitude may be readily found by the formulae (82). When the heliocentric place has been found, we can easily deduce the corresponding geocentric place. Let x, y, z be the rectangular co-ordinates of the planet or comet referred to the centre of the sun, the plane of xy being in the ecliptic, 86 THEORETICAL ASTRONOMY. the positive axis of x being directed to the vernal equinox, and the positive axis of z to the north pole of the ecliptic. Then we shall have x-r cos b cos I, y r cos b sin 1, z - r sin b. Again, let X, Y, Zbe the co-ordinates of the centre of the sun referred to the centre of the earth, the plane of XY being in the ecliptic, and the axis of X being directed to the vernal equinox; and let 0 denote the geocentric longitude of the sun, R its distance from the earth, and Z its latitude. Then we shall have X-= R cos 2 cos 0, Y- R cos sin O, Z - R sin Z. Let x', y', zt be the co-ordinates of the body referred to the centre of the earth; and let a and 9 denote, respectively, the geocentric longitude and latitude, and J, the distance of the planet or comet from the earth. Then we obtain x'- cos ft cos X, y'- = cos f sin, (87) Z _ J sin f3. But, evidently, we also have x'=-x +X, y'=y + Y, z'=z + Z, and, consequently, A cos f1 cos r co cos b cos I -+c R cos cos 0, A cos 3 sin = r cos b sin I + R cos Z sin 0, (88) A sin f - r sin b + R sin 2. If we multiply the first of these equations by cos 0, and the second by sin 0, and add the products; then multiply the first by sin 0, and the second by cos 0, and subtract the first product from the second, we get cos t cos (A- 0) - r cos b cos ( - 0) + R cos 2, A cos ft sin ( - ) - r cos b sin (I - ), (89) A sin r sin b + R sin. It will be observed that this transformation is equivalent to the supposition that the axis of x, in each of the co-ordinate systems, is POSITION IN SPACE. 87 directed to a point whose longitude is 0, or that the system has been revolved about the axis of z to a new position for which the axis of abscissas makes the angle'( with that- of the primitive system. We may, therefore, in general, in order to effect such a transformation in systems of equations thus derived, simply diminish the longitudes by the given angle. The equations (89) will determine 2, /, and A when i, b, and I have been derived from the elements of the orbit, the quantities R, 0, and 2 being furnished by the solar tables; or, when A, /, and 2 are given, these equations determine 1, b, and r. The latitude 2 of the sun niever~ exceeds - 0'.9, and, therefore, it may in most cases be neglected, so that cos 2 1 and sin 2'-0, and the last equations become os cos s ( - 30) - r cos b cos ( - ) + R, cosS sin (A - )= r cos b sin (- 0), (90) A sill r sin b. If we suppose the axis of x to be directed to a point whose longitude is 2, or to the ascending node of the planet or comet, the equations (88) become J co Cs (- ) cosu + c eos ( ) c os + - 2 ), A cos i sill (A — ) =- r sin u cos i + R cos I sin (( - S ), (91) J sin /1 - r sin u sini + R sin 2, by means of which, and 2 may be found directly from 2, i, r, and u. If it be required to determine the geocentric right ascension and declination, denoted respectively by a and 8, we may convert the values of / and 2 into those of o( and 8. To effect this transformation, denoting by e the obliquity of the ecliptic, we have cos 8 cos a cos f cos A, cos S sin a -"y/cos f sin Ajcos — Sin sin e, sin c S cos n sin ) sin sin cos e. Let us now take r —-- n sin N - sin f, n cos N=- cos f sin, and we shall have cos 8 cos a COS / cos 2, cos 6 sin a = n cos (-N+ s), sin = - sin (N +- ). 88 THEORETICAL ASTRONOMY. Therefore, we obtain tan5 tac -cos(N — + t) Y t 1 an tan a s(N tan (92) sin A cos N tan s tan (N +- ) sin a. We also have cos (N + E) cos S sin a cos N cos i sin A' which will serve to check the calculation of a and 8. Since cos 8 and cos are always positive, cos a and cos must have the same sign, and thus the quadrant in which a is to be taken, is determined. For the solution of the inverse problem, in which o and 8 are given and the values of A and f are required, it is only necessary to interchange, in these equations, a and A, 8 and /, and to write - e in place of e. 34. Instead of pursuing the tedious process, when several places are required, of computing first the heliocentric place, then the geocentric place referred to the ecliptic, and, finally, the geocentric right ascension and declination, we may derive formula which, when certain constant auxiliaries have once been computed, enable us to derive the geocentric place directly, referred either to the ecliptic or to the equator. We will first consider the case in which the ecliptic is taken as the fundamental plane. Let us, therefore, resume the equations X - r cos u, y' -+ r sin u cos i, z' - r sin sin i, in which the axis of x is supposed to be directed to the ascending node of the orbit of the body. If we now pass to a new system x, y,, — the origin and the axis of z remaining the same,-in which the axis of x is directed to the vernal equinox, we shall move it back, in a negative direction, equal to the angle 2, and, consequently, = x' cos g - y' sin Q, y x' sin + -y' cos, Therefore, we obtain x r (cos u cos G + sin u cos i sin g ), y - r (+- sin u cos i cos + cos u sin ), (93) z -- r sin u sin i, POSITION IN SPACE. 89 which are the expressions for the heliocentric co-ordinates of a planet or comet referred to the ecliptic, the positive axis of x being directed to the vernal equinox. The upper sign is to be used when the motion is direct, and the lower sign when it is retrograde. Let us now put cos gQ -sin a sin A, -t cos i sin 2 - sin a cos A, sin a - sin b sin B, ~ cos i cos a - sin b cos B, in which sin a and sin b are positive, and the expressions for the coordinates become x r sin a sin (A + u), y r sin b sin (B + u), (95) z - r sin i sin u. The auxiliary quantities a, b, A, and B, it will be observed, are functions of 2 and i, and, in computing an e5la, i are constant so long as these elements are regarded as constant. They are called the constcats for the ecliptic. To determine them, we have, from equations (94), cot A- +- tan g, cos i, cot B =-~ cot g cos i, cos a sin s sin a in A sin b - sinB sinA smnB the upper sign being used when the motion is direct, and the lower sign when it is retrograde. The auxiliaries sin a and sin b are always positive, and, therefore, sin A and cos 2, sin B and sin 2, respectively, must have the same signs. The quadrants in which A and B are situated, are thus determined. From the equations (94) we easily find cos a - sini sin,, / cosb -- sin i cos g. (96) If we add to the heliocentric co-ordinates of the body the co-ordinates of the sun referred to the earth, for which the equations have already been given, we shall have x + X_= zd cos cos, y -- Y -- A cosj sin 2, (97) z + Z = J sin i, 90 THEORETICAL ASTRONOMY. which suffice to determine 2, &, and A. The values of a and 8 may be derived from these by means of the equations (92). 35. We shall now derive the formulae for determining a and 8 directly. For this purpose, let x, y, z be the heliocentric co-ordinates of the body referred to the equator, the positive axis of x being directed to the vernal equinox. To pass from the system of coordinates referred to the ecliptic to those referred to the equator as the fundamental plane, we must revolve the system negatively around the axis of x, so that the axes of z and y in the new system make the angle e with those of the primitive system, e being the obliquity of the ecliptic. In this case, we have x" x y" = y cos e - z sin e, " = y sin +- z cos e. Substituting for x, y, and z their values from equations (93), and omitting the accents, we get x - r cosu cos =F r sin u cos i sin., y r cossin cos r sin u (= cos i cos c cos+s - sin i sin e), (98) z =r cosu sin Q sin s+ r sinu (=t co i cos 2 sin -- sin i cos e). These are the expressions for the heliocentric co-ordinates of the planet or comet referred to the equator. To reduce them to a convenient form for numerical calculation, let us put cos a - sin a sin A, == cos i sin a = sin a cos A, sin a cos = sin b sin B,. (99) ~ cos i cos C cos e - sin i sin e - sin b cos B, sin a sin -- sin c sin C, - cos i cos a sin -- sin i cos e _ sin c cos C; and the expressions for the co-ordinates reduce to x - r sin a sin (A + u), y = r sin b sin (B + u), (100) z = r sin c sin (C + u). The auxiliary quantities, a, b, c, A, B, and C, are constant so long as a2 and i remain unchanged, and are called constants for the equator. It will be observed that the equations involving a and A, regarding the motion as direct, correspond to the relations between the parts of a quadrantal triangle of which the sides are i and a, the POSITION IN SPACE. 91 angle included between these sides being that which we designate by A, and the angle opposite the side a being 90~ - 2. In the case of b and B, the relations are those of the parts of a spherical triangle of which the sides are b, i, and 90~ -+ E B being the angle included by i and b, and 180~ - 2 the angle opposite the side b. Further, in the case of c and C, the relations are those of the parts of a spherical triangle of which the sides are c, i, and e, the angle C being that included by the sides i and c, and 180~ - 2 that included by the sides i and s. We have, therefore, the following additional equations: cos a - sin i sin R, cos b -- cos g sin i cos e - cos i sin e, (101) cos c - - cos a sin i sin +- cos i cos e. In the case of retrograde motion, we must substitute in these 180~ - i in place of i. The geometrical signification of the auxiliary constants for the equator is thus made apparent. The angles a, b, and c are those which a line drawn from the origin of co-ordinates perpendicular to the plane of the orbit on the north side, makes with the positive coordinate axes, respectively; and A, B, and C are the angles which the three planes, passing through this line and the co-ordinate axes, make with a plane passing through this line and perpendicular to the line of nodes. In order to facilitate the computation of the constants for the equator, let us introduce another auxiliary quantity E0, such that sin i - eo sin E,, ~cos i cos eo cos Eo, eo being always positive. We shall, therefore, have tan i tanE0 tan i cos Since both eo and sini are positive, the angle Eo cannot exceed 180~; and the algebraic sign of tan E0 will show whether this angle is to be taken in the first or second quadrant. The first two of equations (99) give cot A = - tan Q cos i; and the first gives cos g sin a -- -. sin A 92 THEORETICAL ASTRONOMY. From the fourth of equations (99), introducing e0 and E0, we get sin b cos B = eO cos Eo cos e - e sin Eo sine - e0 cos (Eo +- e). But sin b sin B = sin Q cos e; therefore cots eo cos (E,0+e) ^ 4 cos i cos (Eo + ) sin g cos e tan a cos Eo cose We have, also, sin QS cose sin b sinB In a similar manner, we find cot C= cos i sin (Eo + e) tan a cos Eo sin e and sin a sin e sin c -- sin C The auxiliaries sin a, sin b, and sin c are always positive, and, therefore, sill A and cos 2, sin B and sin g, and also sin C and'sin g., must have the same signs, which will determine the quadrant in which each of the angles A, B, and C is situated. If we multiply the last of equations (99) by the third, and the fifth of these equations by the fourth, and subtract the first product from the last, we get, by reduction, sin b sin c sin (C- B) =- sin i sin g~. But sin a cos A - + cos i sin g; and hence we derive sin b sine sin (C- B) --- +- tan i, sin a cos A which serves to check the accuracy of the numerical computation of the constants, since the value of tani obtained from this formula must agree exactly with that used in the calculation of the values of these constants. If we put A' = A - 7r, B' = B, and' = C_- f g2, the upper or lower sign being used according as the motion is direct or retrograde, we shall have POSITION IN SPACE. 93 - r sil a sin (A' + v), y - sin b sin (B' + v), (102) z r sin sin ( C' + v), a transformation which is perhaps unnecessary, but which is convenient when a series of places is to be computed. It will be observed that the formule for computing the constants a, b, c, A, B, and C, in the case of direct motion, are converted into those for the case in which the distinction of retrograde motion is adopted, by simply using 180~ - i instead of i. 36. When the heliocentric co-ordinates of the body have been found, referred to the equator as the fundamental plane, if we add to these the geocentric co-ordinates of the sun referred to the same fundamental plane, the sum will be the geocentric co-ordinates of the body referred also to the equator. For the co-ordinates of the sun referred to the centre of the earth, we have, neglecting the latitude of the sun, X R cos Q, Y — R sin ( cos e, Z, — R sin O sin z -- Y tan e, in which R represents the radius-vector of the earth, 0 the sun's longitude, and s the obliquity of the ecliptic. We shall, therefore, have x + X — J cos cos a, y+ y J cos sin, (103) z + Z = z sin S, which suffice to determine a, A, and J. If we have regard to the latitude of the sun in computing its geocentric co-ordinates, the formulhe will evidently become X- R cos O cos C, Y=- R sin 0 cos Z cos s -R sin 2 sin, (104) Z- R sin 0 cos 2 sin s -- R sin 2 cos e, in which, since Z can'niever exceeds - 0,.9, cos 2 is very nearly equal to 1, and sin 2 = 2. The longitudes and latitudes of the sun may be derived from a solar ephemeris, or from the solar tables. The principal astronomical ephemerides, such as the Berliner Astronomisches Jahrbuch, the Nautical Almanac, and the American Ephemeris and Nautical Al 94 THEORETICAL ASTRONOMY. manac, contain, for each year for which they are published, the equatorial co-ordinates of the sun, referred both to the mean equinox and equator of the beginning of the year, and to the apparent equinox of the date, taking into account the latitude of the sun. 37. In the case of an elliptic orbit, we may determine the coordinates directly from the eccentric anomaly in the following manner:The equations (102) give, accenting the letters a, b, and c, x - r cos v sin a' sin A' + r sin v sin a' cos A', y = cos v sin b' sin B' - r sin v sih b' cos B', z - r cos v sin c' sin C' + r sin v sin c' cos C'. Now, since r cos v=- a cos E- ae, and r sin v = a cos To sinE, we shall have x = a sin at sin A' cos E - ae sin a' sin A' + a cos p sin a' cos A' sin E, y a= c sin b' sin B' cos E -.ae sin b' sin B' + a cos sin b cos B' sin E, z = a sin C' sin C' cos E- ae sin c' sin C' + a cos ( sin c' cos C' sin E. Let us now put a cos c sin a' cos A' — A cos Lx, a sin a' sin sn L,, - ae sin a' sin A' - e2R sin Lx - v.; a cos no sin b' cos B' = cos Ly, a sin b sin B' - A sin Ly, -- ae sin b' sin B' - e- e sin Ly = vy; a cos ~ sin c' cos C' = A, cos Lz, a sin c' sin C' 2= sin L,, - ae sin c' sin C' - - e2z sin L. -- vY; in which sin a', sin b', and sin c' have the same values as in equations (102), the accents being added simply to mark the necessary distinction in the notation employed in these formulae. We shall, therefore, have x - Rx sin (Lx + E) + v, y = A, sin (+y +- E) + vy, (105) z - A sin (Lz + E) + vz. By means of these formulae, the co-ordinates are found directly from the eccentric anomaly, when the constants 2,, 2y, Z, Lx, Ly, L, vx, iy, and Pz, have been computed from those already found, or from a, 6, c, A, B, and C. This method is very convenient when a great POSITION IN SPACE. 95 number of geocentric places are to be computed; but, when only a few places are required, the additional labor of computing so many auxiliary quantities will not be compensated by the facility afforded in the numerical calculation, when these constants have been determined. Further, when the ephemeris is intended for the comparison of a series of observations in order to determine the corrections to be applied to the elements by means of the differential formulae which we shall investigate in the following chapter, it will always be advisable to compute the co-ordinates by means of the radius-vector and true anomaly, since both of these quantities will be required in finding the differential coefficients. 38. In the case of a hyperbolic orbit, the co-ordinates may be computed directly from F, since we have r cos v - a (e - sec F), r sin v _ a tan 4 tan F; and, consequently, x = ae sin a' sin A' - a sec F sin a' sin A' +- a tan 4 tan F sin a' cos A', y = ae sin b' sin B' - a sec F Y s b' sin B' + a tan 4 tan F sin b' cos B', z = ae sin c' sin C' - a sec F sin c' sin C' + a tan ~ tan F sin c' cos C'. Let us now put ae sin a' sin A' -A, - a sin a' sin A' p-, a tan 9 sin c' cos A' - vx; ae sin b' sin B' = y, - a sil b' sin B' = y, a tan 4 sin b' cos B' =v y; ae sin g sin C' - A, - a sin c' sin C' = -Z, a tan 4 sin c' cos C' --. Then we shall have x -- -+ l/, sec F + x tan F, y = Ay + -y sec F + v, tan F, (106) z -= A + z, sec F + Yz tan F. In a similar manner we may derive expressions for the co-ordinates, in the case of a hyperbolic orbit, when the auxiliary quantity a is used instead of F. 39. If we denote by,', 2', and i' the elements which determine the position of the orbit in space when referred to the equator as the 96 THEORETICAL ASTRONOMY. fundamental plane, and by to0 the angular distance between the ascending node of the orbit on the ecliptic and its ascending node on the equator, being measured positively from the equator in the direction of the motion, we shall have' U= — +-' -+ -Oo To find at and il, we have, from the spherical triangle formed by the intersection of the planes of the orbit, ecliptic, and equator with the celestial vault, cos i' cos i cos E - sin i sin e cos S, sin i' sin Q' = sin i sin g, a.-& c.9.. sin i' cos a' o s s i sin +- sin i cos e cos.'i Let us now put n sin N- cos i; n cos N — sin i cos 2,and these equations reduce to cos i' -- n sin (N - e),. sin i sin' =sin sin Q,'sin i' cos' = n cos (N- ); - from which we find tan N- cot i, cos N tan N co -^, tan os (V- tan, cos Q, cos (N - e) (107) coti' tan (N- e) cos g2'. (107) Since sin i is always positive, cos N and cos a must have the same signs. To prove the numerical calculation, we have sin i cos g cos N sin i' cos 2' cos (N )' the value of the second member of which must agree with that used in computing g'. In order to find wo, we have, from the same triangle, sin w0 sin i' = sin S sin e, Ho cos w sin i' = cos s sin i + sin e cos i cos g. Let us now take - m sin M= cos e, m cos M=- sin e cos S;; and we obtain POSITION IN SPACE. 97 cot M - tan s cos R, cos IM tan % (3G M tan, (108) 0 Cos (M i) and, also, to check the calculation, sin e cos cos M sin i' cosO — o cos (M- i) If we apply Gauss's analogies to the same spherical triangle, we get cos-.i' sin (g' + %W) sin cos. (i -), cos o) cos cos (' + ) _ cos cos + (109) i' sni' sin s (n' ~ ) - sin.} sin (i - ), sin ci' cos (~ Loo) = cos sin ( + ). The quadrant in which 2 ( + wo) or ( - o0) is situated, must be so taken that sin 2i' and cos -i' shall be positive; and the agreement of the values of the latter two quantities, computed by means of the value of li' derived from tan s', will serve to check the accuracy of the numerical calculation. For the case in which the motion is regarded as retrograde, we must use 180 — i instead of i in these equations, and we have, also,,o - =T -- + - to We may thus find the elements n', a i, and i', in reference to the equator, from the elements referred to the ecliptic; and using the elements so found instead of r, 2, and i, and using also the places of the sun referred to the equator, we may derive the heliocentric and geocentric places with respect to the equator by means of the formula already given for the ecliptic as the fundamental plane. If the position of the orbit with respect to the equator is given, and its position in reference to the ecliptic is required, it is only necessary to interchange Q and S', as well as i and 180~- i', e remaining unchanged, in these equations. These formulae may also be used to determine the position of the orbit in reference to any plane in space; but the longitude Q must then' be measured from the place of the descending node of this plane on the' ecliptic. The value of g,, therefore, which must be used in.the solution of the equations is, in this case, equal to the longitude of the ascending node of the orbit on the ecliptic diminished by the longitude of the descending node of the new plane of reference on the ecliptic. The quantities P', i' and wo will have the same signification in reference 7 98 THEORETICAL ASTRONOMY. to this plane that they have in reference to the equator, with this distinction, however, that a' is measured from the descending node of this new plane of reference on the ecliptic; and e will in this case denote the inclination of the ecliptic to this plane. 40. We have now derived all the formulae which can be required in the case of undisturbed motion, for the computation of the heliocentric or geocentric place of a heavenly body, referred either to the ecliptic or equator, or to any other known plane, when the elements of its orbit are known; and the formulae which have been derived are applicable to every variety of conic section, thus including all possible forms of undisturbed orbits consistent Avith the law of universal gravitation. The circle is an ellipse of which the eccentricity is zero, and, consequently, M —v - u, and - - a, for every point of the orbit. There is no instance of a circular orbit yet known; but in the case of the discovery of the asteroid planets between Mars and Jupiter it is sometimes thought advisable, in order to facilitate the identification of comparison stars for a few days succeeding the discovery, to compute circular elements, and from these an ephemeris. The elements which determine the form of the orbit remain constant so long as the system of elements is regarded as unchanged; but those which determine the position of the orbit in space, T, g, and i, vary from one epoch to another on account of the change of the relative position of the planes to which they are referred. Thus the inclination of the orbit will vary slowly, on account of the change of the position of the ecliptic in space, arising from the perturbations of the earth by the other planets; while the longitude of the perihelion and the longitude of the ascending node will vary, both on account of this change of the position of the plane of the ecliptic, and also on account of precession and nutation. If Tc, g, and i are referred to the true equinox and ecliptic of any date, the resulting heliocentric places will be referred to the same equinox and ecliptic; and, further, in the computation of the geocentric places, the longitudes of the sun must be referred to the same equinox, so that the resulting geocentric longitudes or right ascensions will also be referred to that equinox. It will appear, therefore, that, on account of these changes in the values of wT, 2, and i, the auxiliaries sina, sinb, sin c, A, B, and C, introduced into the formula for the coordinates, will. not be constants in the computation of the places for a series of dates, unless the elements are referred constantly, in the calculation, to a fixed equinox and ecliptic. It is customary, there POSITION IN SPACE. 99 fore, to reduce the elements to the ecliptic and mean equinox of the beginning of the year for which the ephemeris is required, and then to compute the places of the planet or comet referred to this equinox, using, in the case of the right ascension and declination, the mean obliquity of the ecliptic for the date of the fixed equinox adopted, in the computation of the auxiliary constants and of the co-ordinates of the sun. The places thus found may be reduced to the true equinox of the date by the well-known formule for precession and nutation. Thus, for the reduction of the right ascension and declination from the mean equinox and equator of the beginning of the year to the apparent or-true equinox and equator of any date, usually the date to which the co-ordinates of the body belong, we have a- =f+ g sin(G+ a)tan (, A= g cos (G + a), for which the quantitiesf, g, and G are derived from the data given either in the solar and lunar tables, or in astronomical ephemerides, such as have already been mentioned. The problem of reducing the elements from the ecliptic of one date t to that of another date t' may be solved by means of equations (109), making, however, the necessary distinction in regard to the point from which 2 and I2' are measured. Let 0 denote the longitude of the descending node of the ecliptic oft' on that of t, and let a denote the angle which the planes of the two ecliptics make with each other, then, in the equations (109), instead of ag we must write a - 0, and, in order that R' shall be measured from the vernal equinox, we must also write Q' - in place of 2'. Finally, we must write q instead of s, and ace for ow0 which is the variation in the value of co in the interval t' - t on account of the change of the position of the ecliptic; then the equations become cos i' sin (g' + A- ) - sin 1 (g - ) cos (i - ), Cos i' cos ~ (a' - Q + A@) = cos ( S - ) cos ~ ( +, sin -}i' sin 1(. Bo) A sin -1 ( 0) sin ] (i ), (111) sin,i' cos (a'-O- A c) = cos (2 -- O) sin. (i + ). These equations enable us to determine accurately the values of g', i', and Aco, which give the position of the orbit in reference to the ecliptic corresponding to the time t', when 0 and' are known. The longitudes, however, will still be referred to the same mean equinox as before, which we suppose to be that of t; and, in order to refer 100 THEORETICAL ASTRONOMY. them to the mean equinox of the epoch t', the amount of the precession in longitude during the interval' - t must also be applied. If the changes in the values of the elements are not of considerable magnitude, it will be unnecessary to apply these rigorous formulae, and we may derive others sufficiently exact, and much more convenient in application. Thus, from the spherical triangle formed by the intersection of the plane of the orbit and of the planes of the two ecliptics with the celestial vault, we get sin'C cos ( - 0) - cos i' sin i + sin i' cos i cos aw, from which we easily derive sin (' -i) = sin - cos ( 6 0) -+ 2 sin i cos i sin2 GAi. (112) We have, further, sin ao sin i' = sin 7 sin ( - ), or sin (g ) -10) sin Aw =sill sin i' (113) We have, also, from the same triangle, sin a^ cos i' -cos ( - ) sin (' -0) + sin (2 - ) cos (g' -a ) cos 7, which gives sin (2' - 2) - sin Aow cosi' - 2 sin (g2 - ) cos (' - a) sin2., or sin (~'- )=i n n sin (- ) cot i' - 2 sin ( -) cos (i'- ) sin2 -7. (114) Finally, we have - C - 2'- gr - + Ac. Since C is very small, these equations give, if we apply also the precession in longitude so as to reduce the longitudes to the mean equinox of the date', sin (2 -0) sin i i' =- i,- n+ cos ( a -- 0) +- 4 sin 2 i,'- +( —) 1 sin( — )coti' —sin2(g-), (115) - r + sin(-tan - s2( -; dl 2 = V_+0(t'-t) +7) sin(P2 - 6) tani i —O sin2(gb -6); dt 8 POSITION IN SPACE. 101 in which d is the annual precession in longitude, and in which s 206264".8. In most cases, the last terms of the expressions for i', 2', and w', being of the second order, may be neglected. For the case in which the motion is regarded as retrograde, we must put 1800 - i and 180 - i', instead of i and i', respectively, in the equations for acw i', and Q'; and for 7t', in this case, we have i' A ~ ^ = f' a ^ - ac, which gives a: -;. dl V" =r -+- (t' -t) d sin (g -0) tan i' —- sin 2(g -). dt' If we adopt Bessel's determination of the luni-solar precession and of the variation of the mean obliquity of the ecliptic, we have, at the time 1750 + v, - 50".21129 +- 0."0002442966r, dt - 0".48892- 0.000006143T, dt and, consequently, - = (0."48892 - 0."000006143r) ( - t); and in the computation of the values of these quantities we must put = (t + t) - 1750, t and t' being expressed in years., The longitude of the descending node of the ecliptic of the time t on the ecliptic of 1750.0 is also found to be 351~ 36' 10" - 5".21 (t - 1750), which is measured from the mean equinox of the beginning of the year 1750. The longitude of the descending node of the ecliptic of t' on that of t, measured from the same mean equinox, is equal to this value diminished by the angular distance between the descending node of the ecliptic of t on that of 1750 and the descending node of the ecliptic of t' on that of t, which distance is, neglecting terms of the second order, 5".21 (' - 1750); and the result is 351~ 36' 10" - 5".21 (t- 1750) - 5".21 (t' - 1750), or 351~ 36' 10" 10".42 (t - 1750) - 5".21 ('- t). 102. THEORETICAL ASTRONOMY. To reduce this longitude to the mean equinox at the time t, we must add the general precession during the interval t -1750, or 50".21 (t - 1750), so that we have, finally, 0 - 351~ 36' 10" + 39".79 (t - 1750) -- 5".21 (t' - t). When the elements w, Q, and i have been thus reduced from the ecliptic and mean equinox to which they are referred, to those of the date for which the heliocentric or geocentric place is required, they may be referred to the apparent equinox of the date by applying the..,nutation in longitude. Then, in the case of the determination of the right ascension and declination, using the apparent obliquity of the ecliptic in the computation of the co-ordinates, we directly obtain the place of the body referred to the apparent equinox. But, in computing a series of places, the changes which thus take place in the elements themselves from date to date'induce corresponding changes in the auxiliary quantities a, b, c, A, B, and C, so that these are no longer to be considered as constants, but as continually changing their values by small differences. The differential formule for the computation of these changes, which are easily derived from the equations (99), will be given in the next chapter; but they are perhaps unnecessary, since it is generally most convenient, in the cases which occur, to compute the auxiliaries for the extreme dates for which the ephemeris is required, and to interpolate their values for intermediate dates. It is advisable, however, to reduce the elements to the ecliptic and mean equinox of the beginning of the year for which the ephemeris is required, and using the mean obliquity of the ecliptic for that epoch, in the computation of the auxiliary constants for the equator, the resulting geocentric right ascensions and declinations will be referred to the same equinox, and they may then be reduced to the apparent equinox of the date by applying the corrections for precession and nutation. The places which thus result are free from parallax and aberration. In comparing observations with an ephemeris, the correction for parallax is applied directly to the observed apparent places, since this correction varies for different places on the earth's surface. The correction for aberration may be applied in two different modes. We may subtract from the time of observation the time in which the light from the planet or comet reaches the earth, and the true place for this reduced time is identical with the apparent place for the time NUMERICAL EXAMPLES. 103 of observation; or, in case we know the daily or hourly motion of the body in right ascension and declination, we may compute the motion during the interval which is required for the light to pass from the body to the earth, which, being applied to the observed place, gives the true place for the time of observation. We may also incl1de -th, aberration directly in the ephemeris by using the time t — ~497"-4: in computing the geocentric places for the time t, or by subtracting from the place free from aberration, computed for the time t, the motion in c and 8 during the interval 497.784, in which expression d is the distance of the body from the earth, and 497.78 the number of seconds in which light traverses the mean distance of the earth from the sun. It is customary, however, to compute the ephemeris free from aberration and to subtract the time of aberration, 497".78J, from the time of observation when comparing observations with an ephemeris, according to the first method above mentioned. The places of the sun used in computing its co-ordinates must also be free from aberration; and if the longitudes derived from the solar tables include aberration, the proper correction must be applied, in order to obtain the true longitude required. 41. EXAMPLES.-We will now collect together, in the proper order for numerical calculation, some of the principal formulae which have been derived, and illustrate them by numerical examples, commencing with the case of an elliptic orbit. Let it be required to find the geocentric right ascension and declination of the planet Eurynome @, for mean midnight at Washington, for the date 1865 February 24, the elements of the orbit being as follows:Epoch - 1864 Jan. 1.0 Greenwich mean time. M= 1~ 29' 40".21 - 44 20 33.09 Ecliptic and Mean _- - 206 42 40.13 2-206 42 40.13 Equinox, 1864.0. i- 4 36 50.51) -=- 11 15 51.02 log a 0.3881319 log — = 2.9678088 /i. 928".55745 When a series of places is to be computed, the first thing to be done is to compute the auxiliary constants used in the expressions for the co-ordinates, and although but a single pl:ace is required in the problem proposed, yet we will proceed in this manner, in order to 104 THEORETICAL ASTRONOMY. exhibit the application of the formulse. Since the elements a, Q, and i are referred to the ecliptic and mean equinox of 1864.0, we will first reduce them to the ecliptic and mean equinox of 1865.0. For this reduction we have t 1864.0, and t'= 1865.0, which give l= 50".239, 0- 352~ 51' 41", r 0".4882. dt - Substituting these values in the equations (115), we obtain i'- i- Ai 0".40, A -- + 53".61, As +- 50".23; and hence the elements which determine the position of the orbit in reference to the ecliptic of 1865.0 are - 44~ 21' 23".32, g -206~ 43' 33".74, i= 4~ 36' 50".11. For the same instant we derive, from the American Ephemeris and Nanltical Almanac, the value of the mean obliquity of the ecliptic, which is e 23~ 27' 24".03. The auxiliary constants for the equator are then found by means of the formulae tan i cot A -tang cos i, tan Eo - cos i o','!' cos i cos (o,+ E) cot B - tan a cos Eo cos cos i sin (E + e) cot C * tan Q cos E0 sine cos a sin C cos. sin a sine sin a sin b,i sin b - s in - sin A',inB' sin C The angle Eo is always less than 180~, and the quadrant in which it is to be taken, is indicated directly by the algebraic sign of tan E0. The values of sin a, sin b, and sin c are always positive, and, therefore, the angles A, B, and C must be so taken, with respect to the quadrant in which each is situated, that sin A and cos 2, sin B and sin 2, and also sin C and sin 2, shall have the same signs. From these we derive A - 296~ 39' 5".07, log sin a = 9.9997156, B =205 55 27.14, log sin b = 9.9748254, C= 212 32 17.74, log sin c 9.5222192. Finally, the calculation of these constants is proved by means of the formula NUMERICAL EXAMPLES. 105 sin b sin c sin ( C- B) tan i sin a cos A which gives log tan i 8.9068875, agreeing with the value 8.9068876 derived directly from i. Next, to find r and m.' The date 1865 February 24.5 mean time at Washington reduced to the meridian of Greenwich by applying the difference of longitude, 5h 8"" 118.2, becomes 1865 February 24.714018 mean time at Greenwich. The interval, therefore, from the epoch for which the mean anomaly is given and the date for which the geocentric place is required, is 420.714018 days; and multiplying the mean daily motion, 928".55745, by this number, and adding the result to the given value of VM, we get the mean anomaly for the required place, or M/=- 1~ 29' 40".21 + 108~ 30' 57".14 1100 0' 37".35. The eccentric anomaly E is then computed by means of the equation M E- e sin E, the value of e being expressed in seconds of arc. For Eurynome we have log sin p=- log e= 9.2907754, and hence the value of e expressed in seconds is log e = 4.6052005. By means of the equation (54) we derive an approximate value of E, namely, E, =119~ 49' 24", the value of e expressed in seconds being loge2=- 3.895976; and with this we get C.? C......i..- j:.: i M, — E e sin E, 110~ 6' 50". Then we have AE M- I- M 372".7 aE - cosE - -— 7- =l - 339~ 7, 1 -e osE0) 1.097 which gives, for a second approximation to the value of E, O 119~ 43' 44".3. This gives M3 = 1100 0' 36".98, and hence 0".37 AE- 1 - 197 +0".34. 1.097 106 THEORETICA.I ASTRONOMY. Therefore, we have, for a third approximation to the value of E, E- 119~ 43' 44".64, which requires no further correction, since it satisfies the equation between 3 and E. I - To find r and v, we have V r e)sin e) s, Vr cos, ~v 1/a(1 - e) cos ME. The values of the first factors in the second members of these equations are: log Va(l- + e) 0.2328104, and log /a(1 -e) 0.1468741; and we obtain v 129~ 3' 50".52, logr = 0.4282854. Since z - 197~ 37' 49/.58, we have u = v + - X a 326~ 41' 40".10. The heliocentric co-ordinates in reference to the equator as the fundamental plane are then derived from the equations x r sin a sin (A -+ t), y r sin b sin (B + u), z =r sin c sin(C + u), which give, for Eurynome, x — 2.6611270, y - +0.3250277, z = 0.0119486. The Americacn Nautical Almanac gives, for the equatorial co-ordinates of the sun for 1865 February 24.5 mean time at Washington, referred to the mean equinox and equator of the beginning of the year, X- + 0.9094557, Y=- 0.3599298, Z - 0.1561751. Finally, the geocentric right ascension, declination, and distance are given by the equations Y Y z Z+ z z -Z Z z- Z tan a Y ~ —, tan s -= sin a = - cos a, s = a x- + X' y+Y X sin 6' the first form of the equation for tan 3 being used when sin a is greater than cos a. The value of J must always be positive; and 8 cannot exceed -- 90~, the minus sign indicating south declination. Thus; we obtain NUMERICAL EXAMPLES. 107 -- 181~ 8' 29".29, a - 4~ 42' 21".56, log A = 0.2450054. To reduce a and 8 to the true equinox and equator of February 24.5, we have, from the Nautical Alcmnac, f= + 16".80, log g =1.0168, G =45~ 16'; and, substituting these values in equations (110), the result is Aa — + 17".42, -- 7".17. Hence the geocentric place, referred to the true equinox and equator of the date, is a 181~ 8' 46".71, 4~ 42' 28".73, log = 0.2450054. WVhen only a single place is required, it is a little more expeditious to compute r from r - a(1 - e cosE), and then v - E from sin - (v - E) - si sin si E. Thus, in the case of the required place of Eurynome, we get log r = 0.4282852, v - E — 9 20' 5".92, v - 129~ 3' 50".56, agreeing with the values previously determined. The calculation may be proved by means of the formula sin - (v + E) - \- cos (P sin E. In the case of the values just found, we have 1(v + E) - 124~ 23' 47".60, log sin (v + E)- 9.9165316, while the second member of this equation gives log sin - (v + E) -- 9.9165316. In the calculation of a single place, it is also very little shorter to compute first the heliocentric longitude and latitude by means of the equations (82), then the geocentric latitude and longitude by means of (89) or (90), and finally convert these into right ascension and declination by means of (92). When a large number of places are to be computed, it is often advantageous to compute the heliocentric 108 THEORETICAL ASTRONOMY. co-ordinates directly from the eccentric anomaly by means of the equations (105). The calculation of the geocentric place in reference to the ecliptic is, in all respects, similar to that in which the equator is taken as the fundamental plane, and does not require any further illustration. The determination of the geocentric or heliocentric place in the cases of parabolic and hyperbolic motion differs from the process indicated in the preceding example only in the calculation ofr and v. To illustrate the case of parabolic motion, let t - T 75.364 days; log q - 9.9650486; and let it be required to find r and v. First, we compute m from m -- in which log Co- 9.9601277, and the result is log m = 0.0125548. Then we find M from _-/ m (t T), which gives log M- 1.8897187. From this value of log M1I we derive, by means of Table VI., v - 79~ 55' 57".26. Finally, r is found from r — V Cos2 -eV which gives logr = 0.1961120. For the case of hyperbolic motion, let there be given t- T== 65.41236 days; = 370 35' 0".0, or log e 0.1010188; and log o - 0.6020600, to find r and v. First, we compute N from N — (- T), a2 in which log - 9.6377843, and we obtain log - 8.7859356; N= 0.06108514. The value of F must now be found from the equation N eA tan 4F- log tan (45~ + - F). NUMERICAL EXAMPLES. 109 If we assume F== 30~, a more approximate value may be derived from N+- log tan 60~ tanF,which gives F,= 280 40' 23", and hence N, 0.072678. Then we compute the correction to be applied to this value of F, by means of the equation AF -- (N — N,) cos2, ((e - cosF) 8, wherein s - 206264".8; and the result is AF,- 4.6097 (N- N,)s - 3 3' 43".0. Hence, for a second approximation to the value of F, we have F, = 25~ 36' 40".0. The corresponding value of Nis N,= 0.0617653, and hence AF, - 5.199 (N-,) s - 12' 9".4. The third approximation, therefore, gives F, -25~ 24' 30".6, and, repeating the operation, we get F — 25 24' 27".74. which requires no further correction. To find r, we have cos F ) which gives log r 0.2008544. Then, v is derived from tan vv - cot 1 tan F, and we find v 67~ 3' 0".0. When several places are req.uired, it is convenient to compute v and r by means of the equations 1/r sin sin- siF, V/cos F cr cos -IV ~ Cos F F. /cos F 110 THEORETICAL ASTRONOMY. For the given values of a and e we have log V/a(e + 1) = 0.4782649, log (/a(e - 1)- 0.0100829, and hence we derive v = 67~ 2' 59".92, log r = 0.2008545. It remains yet to illustrate the calculation of v and r for elliptic and hyperbolic orbits in which the eccentricity differs but little from unity. First, in the case of elliptic motion, let t- T= 68.25 days; e = 0.9675212; and log q- 9.7668134. We compute I from 2^ 2 M-= (t-T) O wherein log Co= 9.9601277, which gives log M 2.1404550. With this as argument we get, from Table VI., V= 101~ 38' 3".74, and then with this value of V as argument we find, from Table IX., A - 1540".08, B 9".506, C- 0".062. Then we have log i log 1 - 8.217680, and from the equation v - V-+ A (100i) + B (100i)2 + C(100i)3, we get v- V+ 42' 22".28 + 25".90 + 0".28 = 102~ 20' 52".20. The value of r is then found from q( +e) 1 + e cos v' namely, log r 0.1614051. We may also determine r and v by means of Table X. Thus, we first compute M from C (t - T). (1 + 9e) a'2 B Assuming B = 1, we get log M- 2.13757, and, entering Table VI. with this as argument, we find w -101~ 25'. Then we compute A from A 5(1 - te)an 1 + 9e NUMERICAL EXAMPLES. 111 which gives A 0.024985. With this value of A as argument, we find, from Table X., log B - 0.0000047. The exact value of M is then found to be log M= 2.1375635, which, by means of Table VI., gives w -101~ 24' 36".26. By means of this we derive A - 0.02497944, and hence, from Table X., log C= 0.0043771. Then we have tan 1v — C tan -w ( which gives v - 102~ 20' 52".20, agreeing exactly with the value already found. Finally, r is given by r C2= OIq (1- A G) cos2v' from which we get log r -0.1614052. Before the time of perihelion passage, t- T is negative; but the value of v is computed as if this were positive, and is then considered as negative. In the case of hyperbolic motion, i is negative, and, with this distinction, the process when Table IX. is used is precisely the same as for elliptic motion; but when table X. is used, the value of A must be found from 5 (e - 1) A = tau'1w (1 + 9e) and that of r from rr~q (1-A C2) cos2 v' the values of log B and log C being taken from the columns of the table which belong to hyperbolic motion. In the calculation of the position of a comet in space, if the motion 112 THEORETICAL ASTRONOMY. is retrograde and the inclination is regarded as less than 90~, the distinctions indicated in the formulae must be carefully noted. 42. When we have thus computed the places of a planet or comet for a series of dates equidistant, we may readily interpolate the places for intermediate dates by the usual formulae for interpolation. The interval between the dates for which the direct computation is made should also be small enough to permit us to neglect the effect of the fourth differences in the process of interpolation. This, however, is not absolutely necessary, provided that a very extended series of places is to be computed, so that the higher orders of differences may be taken into account. To find a convenient formula for this interpolation, let us denote any date, or argument of the function, by a + nqco and the corresponding value of the co-ordinate, or of the function, for which the interpolation is to be made, by f (a + no)). If we have computed the values of the function for the dates, or arguments, a -, a, a + w, a + 2-, &c., we may assume that an expression for the function which exactly satisfies these values will also give the exact values corresponding to any intermediate value of the argument. If we regard n as variable, we may expand the function into the series f(a + nw) =f(a) + An + Bn + C3 + &c. (116) and if we regard the fourth differences as vanishing, it is only necessary to consider terms involving n3 in the determination of the unknown coefficients A, B, and C. If we put n successively equal to -, 0, 1, and 2, and then take the successive differences of these values, we get I. Diff. II. Diff. III. Diff. f (a-) =f(a)-A -+B CA-B+ f(ao) =f(a) A -B+ 2B f(a+w) f(a)+A +B +C A+B+C 2B+6 f(a - 2) ==f (a) 2A + 4B + A8 C + + If we symbolize, generally, the difference f (a + no) (-f + (n- 1) w) by f' (a +- (n — ) w), the difference f (a + (n + ) w) -f' (a + (n — 2) w) by f" (a +- no), and similarly for the successive orders of differences, these may be arranged as follows: Argument. Function. I. Diff. II. Diff. III. Diff. a - f (a - o) a f(a)' (a -- 1) "f (a) a+u f+u) f ('+(a + ) f( ) f"'(a+iW) a +2 w f(a + 2)' (( + o) ) a + 2,, f/(a + 2o)' (f + i') INTERPOLATION. 113 Comparing these expressions for the differences with the above, we get C-f"' ( + 1), B - "(a), A.=f (a + -) f (a) - (a) - (a + s), which, from the manner in which the differences are formed, give C- (a" (a + ) -f" (a)), B = f" (a), A =f(a + w) -f (a) - f" (a) - (f" ( + w) -f (a)). To find the value of the function corresponding to the argument a -- -W, we have n =, and, from (116), f(a + J) =/(a) + -A + -B + I C. Substituting in this the values of A, B, and C, last found, and reducing, we get f( + - o) - (fa + a) +f (a)) - ( (f" (a + w) +- (a))), in which only fourth differences are neglected, and, since the place of the argument for n 0 is arbitrary, we have, therefore, generally, f( + (n + 1) ) -= (f(a + (n + 1) ) +f(a + nw)) ( (f" (a + (n + 1) w) -f" (a + nw))). (117) Hence, to interpolate the value of the function corresponding to a date midway between two dates, or values of the argument, for which the values are known, we take the arithmetical mean of these two known values, and from this we subtract one-eighth of the arithmetical mean of the second differences which are found on the same horizontal line as the two given values of the function. By extending the analytical process here indicated so as to include the fourth and fifth differences, the additional term to be added to equation (117) is found to be + 3 (f iv (a + (A + 1) a+) +f'i (a + nw))), and the correction corresponding to this being applied, only sixth differences will be neglected. It is customary in the case of the comets which do not move too rapidly, to adopt an interval of four days, and in the case of the asteroid planets, either four or eight days, between the dates for which the direct calculation is made. Then, by interpolating, in the case of an interval w, equal to four days, for the intermediate dates, we obtain a series of places at intervals of two days; and, finally, inter8 114 THEORETICAL ASTRONOMY. polating for the dates intermediate to these, we derive the places at intervals of one day. When a series of places has been computed, the use of differences will serve as a check upon the accuracy of the calculation, and will serve to detect at once the place which is not correct, when any discrepancy is apparent. The greatest discordance will be shown in the differences on the same horizontal line as the erroneous value of the function; and the discordance will be greater and greater as we proceed successively to take higher orders of differences. In order to provide against the contingency of systematic error, duplicate calculation should be made of those quantities in which such an error is likely to occur. The ephemerides of the planets, to be used for the comparison of observations, are usually computed for a period of a few weeks before and after the time of opposition to the sun; and the time of the opposition may be found in advance of the calculation of the entire ephemeris. Thus, we find first the date for which the mean longitude of the planet is equal to the longitude of the sun increased by 180~; then we compute the equation of the centre at this time by means of the equation (53), using, in most cases, only the first term of the development, or v - M — 2e sin M, e being expressed in seconds. Next, regarding this value as constant, we find the date for which L + equation of the centre is equal to the longitude of the sun increased by 180~; and for this date, and also for another at an interval of a few days, we compute u, and hence the heliocentric longitudes by means of the equation tan (I- ~ ) - tan u cos i. Let these longitudes be denoted by I and 1', the times to which they correspond by t and t', and the longitudes of the sun for the same times by ( and 0'; then for the time to, for which the heliocentric longitudes of the planet and the earth are the same, we have t —t + t(o- ) ( (e- t), or (118) i' - 180~ - 0' to - e + ( - - )- ) (t' — t), the first of these equations being used when 1 180~ - 0 is less TIME OF OPPOSITION. 115 than 1' - 180~ - ). If the time to differs considerably from t or t', it may be necessary, in order to obtain an accurate result, to repeat the latter part of the calculation, using to for t, and taking t' at a small interval from this, and so that the true time of opposition shall fall between t and t'. The longitudes of the planet and of the sun must be measured from the same equinox. When the eccentricity is considerable, it will facilitate the calculation to use two terms of equation (53) in finding the equation of the centre, and, if e is expressed in seconds, this gives 5 e' v -M -2e sinM+ - - sin 2M, 4 s s being the number of seconds corresponding to a length of arc equal to the radius, or 206264".8; and the value of v- M will then be expressed in seconds of arc. In all cases in which circular arcs are involved in an equation, great care must be taken, in the numerical application, in reference to the homogeneity of the different terms. If the arcs are expressed by an abstract number, or by the length of arc expressed in parts of the radius taken as the unit, to express them in seconds we must multiply by the number 206264.8; but if the arcs are expressed in seconds, each term of the equation must contain only one concrete factor, the other concrete factors, if there be any, being reduced to abstract numbers by dividing each by s the number of seconds in an arc equal to the radius. 43. It is unnecessary to illustrate further the numerical application of the various formulae which have been derived, since by reference to the formulae themselves the course of procedure is obvious. It may be remarked, however, that in many cases in which auxiliary angles have been introduced so as to render the equations convenient for logarithmic calculation, by the use of tables which determine the logarithms of the sum or difference of two numbers when the logarithms of these numbers are given, the calculation is abbreviated, and is often even more accurately performed than by the aid of the auxiliary angles. The logarithm of the sum of two numbers may be found by means of the tables of common logarithms. Thus, we have log (a + b) -log (+ a log b (+ If we put log tan x =- (log b - log a), 116 THEORETICAL ASTRONOMY. we shall have log (a + b) =log a - 2 log cos x, or log (a + b) log b - 2 log sin x. The first form is used when cos x is greater than sin x, and the second form when cos x is less than sin x. It should also be observed that in the solution of equations of the form of (89), after tan (2- )) —using the notation of this particular case-has been found by dividing the second equation by the first, the second members of these equations being divided by cos (2 - 0) and sin (2R- 0), respectively, give two values of J cos 3, which should agree within the limits of the unavoidable errors of the logarithmic tables; but, in order that the errors of these tables shall have the least influence, the value derived from the first equation is to be preferred when cos ( - o) is greater than sin (R-(0), and that derived from the second equation when cos (2i- ) is less than sin (2 - 0). The value of D, if the greatest accuracy possible is required, should be derived from dcos / when / is less than 450, and from A sin when /i is greater than 45~. In the application of numbers to equations (109), when the values of the second members have been computed, we first, by division, find tan ( q+o- W) and tan ( - w0); then, if sin ( + wo0) is greater than cos 1 (Q' + wo), we find cos ~i' from the first equation; but if sin (2/ + -oo) is less than cos (g + (w0), we find cos ~i' from the second equation. The same principle is applied in finding sin -i' by means of the third and fourth equations. Finally, from sin -i' and cos 1i we get tan ii', and hence i'. The check obtained by the agreement of the values of sin -i' and cos i', with those computed from the value of i' derived from tan i', does not absolutely prove the calculation. This proof, however, may be obtained by means of the equation sin i' sin Q' -sin i sin Q, or by sin i' sin o -= sin e sin a. In all cases, care should be taken in determining the quadrant in which the angles sought are situated, the criteria for which are fixed either by the nature of the problem directly, or by the relation of the algebraic signs of the trigonometrical functions involved. DIFFERENTIAL FORMULBE. 117 CHAPTER II. INVESTIGATION OF THE DIFFERENTIAL FORMULA WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATION OF THE ELEMENTS OF ITS ORBIT. 44. IN many calculations relating to the motion of a heavenly body, it becomes necessary to determine the variations which small increments applied to the values of the elements of its orbit will produce in its geocentric or heliocentric place. The form, however, in which the problem most frequently presents itself is that in which approximate elements are to be corrected by means of the differences between the places derived from computation and those derived from observation. In this case it is required to find the variations of the elements such that they will cause the differences between calculation and observation to vanish; and, since there are six elements, it follows that six separate equations, involving the variations of the elements as the unknown quantities, must be formed. Each longitude or right ascension, and each latitude or declination, derived from observation, will furnish one equation; and hence at least three complete observations will be required for the solution of the problem. When more than three observations are employed, and the number of equations exceeds the number of unknown quantities, the equations of condition which are obtained must be reduced to six final equations, from which, by elimination, the corrections to be applied to the elements may be determined. If we suppose the corrections which must be applied to the elements, in order to satisfy the data furnished by observation, to be so small that their squares and higher powers may be neglected, the variations of those elements which involve angular measure being expressed in parts of the radius as unity, the relations sought may be determined by differentiating the various formulae which determine the position of the body. Thus, if we represent by 0 any co-ordinate of the place of the body computed from the assumed elements of the orbit, we shall have, in the case of an elliptic orbit, 0 f (r, i, Mo,,,, ), 118 THEORETICAL ASTRONOMY. Mo being the mean anomaly at the epoch T. Let 0' denote the value of this co-ordinate as derived directly or indirectly from observation; then, if we represent the variations of the elements by AT, a^, Ai, &c., and if we suppose these variations to be so small that their squares and higher powers may be neglected, we shall have do do do do do do o A + d —a (1) ~ d1I dQ 0 do do The differential coefficients d,, &c. ust now be derived from d d must now b derived rom the equations which determine the place of the body when the elements are known. We shall first take the equator as the plane to which the positions of the body are referred, and find the differential coefficients of the geocentric right ascension and declination with respect to the elements of the orbit, these elements being referred to the ecliptic as the fundamental plane. Let x, y, z be the heliocentric co-ordinates of the body in reference to the equator, and we have 0 =f(x, y, z), or do do do do-d d + - dy + dz. dx dy dz Hence we obtain do do dx do dy do dz -d7r d. d7r.dy d7r dz'; and similarly for the differential coefficients of 0 with respect to the other elements. We must, therefore, find the partial differential coefficients of 8 with respect to x, y, and z, and then the partial differential coefficients of these co-ordinates with respect to the elements. In the case of the right ascension we put 0- a, and in the case of the declination we put 0 = 8. 45. If we differentiate the equations x + X — a cos 8 cos a, y -+ Y-= cos 8 sin a, z + Z = A sin 8, regarding X, Y, and Z as constant, we find DIFFERENTIAL FORMULLE. 119 dx - cos a cos 8 dJ -- J sin a cos S dCa - cos a sinl d, dy sin c cos 8 dA + A cos a cos 8 du, - A sin a sin 3 d3, dz - sin a dJ + J cos 3 dS. From these equations, by elimination, we obtain sin ac COS a cosa S d x+ - ady, (3) cos sin a sin cos 8 d8 - - dx dy + - dz. Therefore, the partial differential coefficients of a and ~ with respect to the heliocentric co-ordinates are du sin a da cos o sin a cos 6 ~ - -' dx a' dx A da cos a dd sin a sin ( cos ~ (4) dy A' dy () de dS cos cos 6 -w- -- 0,~ dz dz A Next, to find the partial differential coefficients of the co-ordinates x, y, z, with respect to the elements, if we differentiate the equations (100)1, observing that sin a, sin b, "sin c, A, B, C are functions of g and i, we get dx. dr + x cot(A + u) du + d + dx di, r dy di dy Y dr + y cot (B ud+ ) dd + d ddi, rd r -I- z cot (C+ u) du {+ I d - dIi. d Cdv dC dx d To find the expressions for -, d-I &c., we have the equations d2' di x - r cos u cos - r sin u sin Q cos i, y = r cos u sin Q cos e + r sin u cos Q cosi cos - r sin u sin i sin e, z = r cos u sin 2 sine - r sin u cos gQ cos i sine -- r sinu sin i cos, which give, by differentiation, dx - - =- r cos u sin 9 - r sin u cos cos i, d2 -- == r cos u cos O cos e - r sin u sin Q cos i cos, dg 120 THEORETICAL ASTRONOMY. dz = r cos u cos g sin - r sin u sin g2 cos i sin e, d2 dx -- r sin u sin sin i, dy i- r sin u cos g sin i cos e -r sin u cos i sin, dz. -- r sin u cos Rg sin i sin e -+- r sin u cos i cos e. di The first three of these equations immediately reduce to dx ddy dz d - YS x cos - z sine; (5) d2'' d2 ^ dQ ^ and since cos a - sin S sin i, cos b = - cos Q sin i cos e- cos i sin e, cos = - cos Q sin i sin e -- cos i cos, we have, also, dx dy dz — = r sin u cos a, - r sin u cos b, d r sin u cos c. di di di Further, we have du v - dv - ddQ, and hence, finally, dx =- dr + x cot(A + u) dv - x cot (A + u) dr + (- x cot(A + u) - y cos e - z sin e) d q + r sin u cos a di, dy - Ydr + y cot(B + u) dv + y cot(B- + u) dr ( r (6) + (- y cot(B + u) +x cos e) d2 -+r sinu cos b di, dz =- Zdr cot ( dv cot(C ) z ct(C + u) d, + (- z cot (C - +) - + x sin e) d - + r sin u cos di. These equations give, for the partial differential coefficients of the heliocentric co-ordinates with respect to the elements, dx dx dydy d= - = x cot (A + u), d -d =y cot(B + u), d7r dv cot (r dz dz cot(c+ U); dr - - DIFFERENTIAL FORMULE. 121 d- - cot (A+Zu)-y cos — zsine, d~ — y cot (B+-u)-x cos e, dz d = z cot(C + u) + x sin e; dg dx. dy i dz d — r -sin u cos a, - r sin u cos b, d - r sin u cos c; (7) di n di di dx x dy y dz z dr r dr r' dr r When the direct inclination is greater than 90~, if we introduce the distinction of retrograde motion, we have du = dv - dr+ d-2, and hence dx - dx ct, dy dy __ d — ~ -- - x cot (A 0+ u), dr- dv —y cot (B + u), da - dv da dv dz dz dz - -z cot(C+ t); (8) j= -- dv dx d dy dy dy dz d d2 ~ dv y cos - z sin d, + xco d dv+ sin d - - d - o, d -- dv - i dx dy dz The expressions dr' dr dr T he expressions for -y- d- and ~ remain unchanged; and we have, also, dx dy. dz -di -r rsincos a - rslntcosb, z- r slnu cosc. (9) ad^ snos di di It is advisable, in order to avoid the use of two sets of formule, in part, to regard the motion as direct and the inclination as susceptible of any value from 0~ to 1800. If the elements which are given are for retrograde motion, we take the supplement of i instead of i; and if we designate the longitude of the perihelion, when the motion is considered as being retrograde, by (wv), we shall have 7- 2 - (7r). If we introduce, as one of the elements of the orbit, the distance of the perihelion from the ascending node, we have du = dv + do, and, hence, dx dx dy dy dx dx _ cot(A + ) dy), — y cot (B 4- u), d d d co t ( u)dv do( dv dz dz do - d — zcot( C - ). (10) dw dv. 122 THEORETICAL ASTRONOMY. dx dy dz The values of -, -, and - must, in this case, be found by means dQ' dQ' d2 of the equations (5). By means of these expressions for the differential coefficients of the co-ordinates x,y, z, with respect to the various elements, and those given by (4), we may derive the differential coefficients of the geocentric right ascension and declination with respect to the elements 62, i, and rr or (o, and also with respect to r and v, by writing successively a and 8 in place of 0, and 2, i, &c., in place of z in the equation (2). The quantities r and v, however, are functions of the remaining elements p, M0, and p; and we have. d dr +dr dM + dr dv dv v I- dM, dA= d9p + dM dTo + d,~ Da. Therefore, the partial differential coefficients of x, with respect to the elements p:, Mo, and I, are dx. d dx dr d dv d'p dr d dv d' dx dx dr dx dv dMo ~dr dM, + dv'dM' (1) dx dx dr dx dv d- dr d+ v dv l The expressions for the partial differential coefficients in the case of the co-ordinates y and z are of precisely the same form, and are obtained by writing, successively, y and z in place of x. The values of dx dx dy dy dz dz d, -d' d -,d and - are given by the equations (7), and dr dv dr dv dr dv dr dv dr d dvdr dv when the expressions for d) d,and -d ave beend d d-' d-p dMo' dM-' dn dp found, the partial differential coefficients of the heliocentric co-ordinates with respect to the elements o(, HM, and / will be completely determined, and hence, by means of (2), making the necessary changes, the differential coefficients of a and a with respect to these elements. 46. If we differentiate the equation M E- e sin E, DIFFERENTIAL FORMULi.M 123 we shall have dM- dE(1 - e cos E) -cos op sin E dy. r r But, since 1 e cos E -, and cos ( sin E= - sin v, this reduces to'a'a r r dM- - dE - -sin v dr, a a or dE - dM - sin v dp. If we take the logarithms of both members of the equation tan Iv - tan.E tan (450 + ~ ), and differentiate, we find dv dE d__ 2 sin v cos v 2 sin Ecosos E 2 sin (45~ + - ) cos (450 + ) which reduces to sin v sin v dv - - dE+ dy'. sin E cos 5y Introducing into this equation the value of dE, already found, and. r sin vu.^ replacing sin E by r sn we get i a cos?a2= COS d sin v acos, ) dv F=~ -dM-{ ~+ 1 d.' r2 cos 5p r But since a cos2 == p, and - 1 + sin qp cos v, this becomes A2 COS r d cos + 2 + tan p cos v sin v dp. (12) If we differentiate the equation r a (1 - e cos E), we shall have - dr - da +- ae sin E dE - a V c cos E d; a and substituting for dE its value in terms of dMi and dp, the result is dr = - -da + a tan sinvdM+ (ae sinEsinv-acosy cosE)d(p. (13) a 124 THEORETICAL ASTRONOMY. sin v cos cos v + e lNow, since sin E; o, and cosE E= ~, we shall have 1 + e cos v, I - e cos v ae cos p5 sin2 v a cos p (cos v + e) ae sin E sin v a cos E cos E- c1 -+ e ecosv 1 cos which reduces to ae sin E sin v - a cos p cos E - a cos 9 cos v Hence, the expression for dr becomes dr =- da -+ a tan e sin v dM — a cos v cos v dp. (14) Further, we have H= =Mo +- (t- T), T being the epoch for which the mean anomaly is M, and kVll + m a2 Differentiating these expressions, we get dM=-dM, + (t - T) di, da 2 d/j a F and substituting these values in the expressions for dr and dv, we have, finally, dr = a tan S sin v dMo + (a tan so sin v (t T) ~- ) d/ -a cos cos v de, (15) d C acos a Ca cos_^,, 2 i tan< cosvsinvd. r a2 cos o From these equations for dr and dv we obtain the following values of the partial differential coefficients:dr dv 2 -- a cos cos v,sinv, C~~dsoT9^~~ d9'- cos p dr dv a2 cos st d-a tan sin v, (16) dM o dM0 2 dr 2r dv aa cos T d/ = a tan lo sin v (t- T) 3- 206264.8, d — r (t - T). dfi 3fiL d/~ r2 DIFFERENTIAL FORMULA.M 125 dr It will be observed that in the last term of the expression for we have supposed / to be expressed in seconds of arc, and hence the factor 206264.8 is introduced in order to render the equation homogeneous. 47. The formulae already derived are sufficient to find the variations of the right ascension and declination corresponding to the variations of the elements in the case of the elliptic orbit of a planet; but in the case of ellipses of great eccentricity, and also in the cases of parabolic and hyperbolic motion, these formulae for the differential coefficients require some modification, which we now proceed to develop. First, then, in the case of parabolic motion, sin ip - 1, and instead of MO and, we shall introduce the elements T and q, the differential coefficients relating to 7r, 2, and i remaining unchanged from their form as already derived. If we differentiate the equation k(t — T).I..t =/ ) - (tan v + i tan Iv), regarding T, q, and v as variable, we shall have d,T/ - / ~t (t — T) T)17) rV21/ 2q Instead of q, we may use logq, and the equation will, therefore, become k kT2q 3k T)(t - T)/2q d —-l/2qT (t- )/ d log q, (18) in which is the modulus of the system of logarithms. in which 20 is the modulus of th e system of logarithms. 126 TTHEORETICAL ASTRONOMY. If we take the logarithms of both members of the equation r q cos2 1vV and differentiate, we find r dr -= dq + r tan -v dv. q Introducing into this equation the value of dv from (17), we get 1 t t \ 1/2qtan1,^ dr= r- 3k-(t~ — T) tan' \dq tan dT. (19) q r2 2IIq r k_ (t - T) Now, since k(t ) q (tan 2v -+ tan3 ~v), and q = r cos2 v, we have v2q 1 3k (t - T) tan. 1 * 1- 3kt- T/anv2 _(1 + tan2 Iv -3 sin2 v - sin2 2v tan2 Iv) q r21/ 2q r cos V,. r We also have kV1/ 2q t 2q cos2 1v tan Iv k sin v tan - ~~r 4 1/~~~~ 2q Therefore, equation (19) reduces to k sin v dr - cos v dq dT. (20) 1/2q If we introduce d log q instead of dq, this equation becomes _cosv k sin v dr- os d log q -2_ dT. (21) From the equations (17), (18), (20), and (21), we derive dr ksin v dv kl/2q dT 1/ 2q dT r2 dr dv 3k (t - T) Cosv,~ (22) dq dq - 1/ 2 dr __ qcosv dv 3k (t -T) 1/ 2q dlog q o dlogq 2A r and then we have, for the differential coefficients of x with respect to T and q or log q, DIFFERENTIAL FORMULE. 127 dx dx dr dx d dx d dr dx dv dT dr dT dv dT' dq dr' dq + dv dq dx x dr dx dv dlogq- dr' dlogq v' dlogq' and similarly for the differential coefficients of y and z with respect to these elements. The expressions for the partial differential coefficients of x, y, and z, respectively, with respect to r and v are the same as already found in the case of elliptic motion. We shall thus obtain the equations which express the relation between the variations of the geocentric places of a comet and the variation of the parabolic elements of its orbit, and which may be employed either to correct the approximate elements by means of equations of condition furnished by comparison of the computed place with the observed place, or to determine the change in the geocentric right ascension and declination corresponding to given increments assigned to the elements. 48. We may also, in the case of an elliptic orbit, introduce T, q, and e instead of the elements po, Mo, and /u. If we differentiate the expression q= a( -e), we shall have da = - dq +- de. q q We have, also, M= l/1 + m a- (t - T), in which T is the time of perihelion passage, and dMi= -k / + a- dT — 1/ + m a- (t T) da. Hence we derive dMl - - kV/l - I m a d T- kl1 + m (t - T) dq Of -kq___ q k/l+ma(t T)de. q Substituting this value of dM in equation (12), replacing sin o( by e, and reducing, we get d kp (1 ) dTT-a kp (L -T) dq r2 o.2 -(~ k~7~ —~~r(t-T j _ i~ 1 ld (23) ( 2p (+ - - 1 sin v I ede. (23) qr 7) 128 THEORETICAL ASTRONOMY. In a similar manner, by substituting the values of da and dM in equation (14), and reducing, we find drz- kVl/ +m Cdr- e sin v dT Vp r kl/l +m(t-T) A 2.e sin)dq +-~1r ~',- \~'- \^~-e sin v dq + Pq — cos v -- k(1 (t - T) de. (24) + P q q e i —~,de' (24) These equations, (23) and (24), will furnish the expressions for the dv dv dv dr dr I dr partial differential coefficients dT d- d- - d-, and -de which are dT dq de' d T dq de required in finding the differential coefficients of the heliocentric coordinates with respect to the elements T, q, and e, these quantities being substituted for MI, A#, and (n, respectively, in the equations (11). 49. When the orbit is a hyperbola, we introduce, in place of /I0, 6u, and qp, the elements T, q, and ~. If we differentiate the equation N,- e tan F loge tan (450 +.F), we shall have dN, c o -F - I + tan F de, ~ ( cos F which is easily transformed into r dF, __tan _ dA==-' cos. + tan F t d, ~ cos F cos or dF a atan sinF r tan FdN r cos Let us now take the logarithms of both members of the equation tan F == tan Xv tan, and differentiate, and we shall have dF sin v dv - sin v s. d+. sin F sin dF Introducing into L;s equation the value of sin already found, we get =a sinv asinv tan { sin v dv a - ~ ~. +. d.i r tan ~ r cos t Sin DIFFERENTIAL FORMULAE. 129 But, since r sin v = a tan i tan F, and p =- a tan2 4, this reduces to Adv - a dNV - + 1-. (25) r2 TO I - r sinv r If we differentiate the equation r=a(~ ~ e 1 ) \cos F f we get r tan2 dF a tan, dr da + ae tan2F + d. a sin +- cos F cos dF7 Substituting in this equation the value of sin we obtain d r + a2e tan F dN a2etan2 F_ a\tan, a r r cosFF cos4d which is easily reduced to - sinuv r ae dr=r daa d +C c da ae).* a sin c ro c os F C F sin But, since r ae a cos F cos2 F cos this reduces to r, a sinv pa i \ d4 dr=- da d~N, + ~pa e a sin dN r cos Fjsin' or r sin v Cos V dr= -da + a -. dNo + p d%. (26) a sinm sin{ Now, since q = (e- 1), we have d q d - a tan d, dq _ - da d, a cos{ or a a da - dq - p' d{. q q cos We have, also, N k- (t - T), and hence dN - ka- 3 dT- aka — (t- T) da. By substituting the value of da, this becomes do kA T qm - kaq T _ _ka-(t- T ) T) dNO =.~ka~ ~2dT~ ~^~-dq -0~-~ ~ d9. q aq cos % 9 130 THEORETICAL ASTRONOMY. Substituting this value of dN0o in equation (25), and reducing, we obtain dv -- d T k p T) dq r2 qr ( qr2 (\r l /sins (27) In a similar manner, substituting in equation (26) the values of da and dNo, and reducing, we get dr l sinv 3 dT~(r 3k(t -T) sinv )d:- ~ —7' cos )dq Vp cos 4 2 V2q c 1 os4 /cos4/., p(t — T). sin v r )p) (28) C ( cos -v p\-~ (28) t~\ dl/c T iq cos q sin s'( The equations (27) and (28) will furnish the expressions for the partial differential coefficients of r and v with respect to the elements T, q, and 4, required in forming the equations for cos 8 doc and dS. It will be observed that these equations are analogous to the equations (23) and (24), and that by introducing the relation between e and 4, and neglecting the mass, they become identical with them. We might, indeed, have derived the equations (27) and (28) directly from (23) and (24) by substituting for e its value in terms of 4'; but the differential formulae which have resulted in deriving them directly from the equations for hyperbolic motion, will not be superfluous. 50. It is evident, from an inspection of the terms of equations (23), (24), (27), and (28) which contain de and de, that when the value of e is very nearly equal to unity, the coefficients for these differentials become indeterminate. It becomes necessary, therefore, to develop the corresponding expressions for the case in which these equations are insufficient. For this purpose, let us resume the equation k (t T) (I + - u 4-+ ~u - 2i(Iu' + q 5-u) + 3i2 (u5 ( + u) - &c, 2q p 1-e in which u =tan Av, and i = Then, since - i <(1 e) - (1- e)2 + — e &., e 1le) e)2 &ha we shall have DIFFERENTIAL FORMULaE. 131 k(t )- + T)(~ - t -t5) ( - e) 1/2 q+ (32u - o3' + 38u7) ( - e)2 + &c. (29) dv If it is required to find the expression for d in the case of the variation of the elements of parabolic motion, or when 1 - e is very small, we may regard the coefficient of 1 - e as constant, and neglect terms multiplied by the square and higher powers of 1 - e. By differentiating the equation (29) according to these conditions, and regarding u and e as variable, we get 0 = (1 +- u2) du- ( u- - Iu5) de; and, since du- (1 + qu) dv, this gives dv lu - 13 ~- ut5 de (1 + u2)2 The values of the second member, corresponding to different values of v, may be tabulated with the argument v; but a table of this kind dv is by no means indispensable, since the expression for de may be changed to another form which furnishes a direct solution with the same facility. Thus, by division, we have d - 2 - 9 +9 3 de — — ~ + -J - T (1 q- u2)2' and since, in the case of parabolic motion, k (t - T) I r2 = 2 + 2)2 3 +r2q2(1 +- u)2, this becomes dv 9 k(t-T) /- 2 tan v. (31) de o r2 If we differentiate the equation q (1 -e)3 1 + e cos v' regarding r, v, and e as variables, we shall have dr __ 2r in2 v re sin v dv _~ _ (i-i.- e21 + - ^ (i ^ g) ~ ^(32) de - q ( 1 q- e)2 - q(e)'de —' 132 THEORETICAL ASTRONOMY. In the case of parabolic motion, e- I, and this equation is easily transformed into dr- r tan v (tan + 2 (33) de de (33) Substituting for d its value from (31), and reducing, we get de dr o (t - T) sin - Ltan2Iv. (34) de^- =. sin v + -r tan (4 dv dr The equations (31) and (34) furnish the values of - and de to be used in forming the expressions for the variation of the place of the body when the parabolic eccentricity is changed to the value 1 + de. When the eccentricity to which the increment is assigned differs but dv little from unity, we may compute the value of - directly from d e equation (30). A still closer approximation would be obtained by dv using an additional term of (29) in finding the expression for -; but a more convenient formula may be derived, of which the numerical application is facilitated by the use of Table IX. Thus, if we differentiate the equation v V+ A (100i) + B (100o)2 + ( (00i)3, regarding the coefficients A, B, and C as constant, and introducing the value of i in terms of e, we have dv dV 200A 400B 0 600 de de s(1 + e)' s(1 - e)' s8(1 q- e) in which s- 206264.8, the values of A, B, and C, as derived from dV the table, being expressed in seconds. To find V, we have k (t- T)l/ 1 + etn V tan3 - - tan.~ +i tan 7 2q~ which gives, by differentiation, k (t-T) de dV 2q' c/ 1 + e cos4 V; and if we introduce the expression for the value of M used as the argument in finding V by means of Table VI., the result is DIFFERENTIAL FORMULIE. 133 dV Mcos4TV de 75(1 e)' Hence we have dv Mcos4V7 200A 400B (100i) 6000 ( de - 75 (1 + e) ~(l + e)2 s (l + e)2 s(1 + e) dv by means of which the value of de is readily found. When the eccentricity differs so much from that of the parabola that the terms of the last equation are not sufficiently convergent, dv the expression for, which will furnish the required accuracy, may dle be derived from the equations (75), and (76),. If we differentiate the first of these equations with respect to e, since B may evidently be regarded as constant, we get dw _ k (t -T) cos4 4w q' B+ (36) de-1" V/2qt *B1/i,(1 -9e) (6 If we take the logarithms of both members of equation (76),, and differentiate, we get dv dC dw 4de T+ - - __(37)' sinv C sin w (1 + e) (1 + 9 e) To find the differential coefficient of C with respect to e, it will be sufficient to take - 1- 4A C0 which gives dC - 2 C2 dA. The equation 5 (1 -- e) (1 - 9e) t gives 50 A_9_dw_ dA - - tan2 — w de + t and w; (1 + 9e) " tan wcos2 ^w and hence we obtain / dC 20 A.C dC — (12 9e) tan w de + -4 - dw. Substituting this value in equation (37), we get dv 20 C. C sin v dw 4 sin v d-e ~ (1 +9 )2 t + sinw * de ( +e)(l1 + 9e) 134 THEORETICAL ASTRONOMY. dw and substituting, finally, the value of dew- we obtain dv k (t — T) C2 sin v cos' uw 20 C. de 1/2 i (+1 - 9e) zw (1- 9e)2 4 sin v (1 + e) (1 + 9e)' which, by means of (76)1, reduces to dv 9 k(t- T) 2 sin v cos2 w 8 tan - (v de ~ 1/2 B/(1+9e) tan w (1 e) (1 9e) If we introduce the quantity M which is used as the argument in finding w by means of Table VI., this equation becomes dv 9 MGi cos2 sin 8 tan Iv Mco C2SV __ — _ (39) de 2 (I + 9e) 75 tan w (1e) 9 e)' This equation remains unchanged in the case of hyperbolic motion, the value of C being taken from the column of the table which cordv responds to this case.; and it will furnish the correct value of d in de all cases in which the last term of equation (23) is not conveniently dr applicable. The value of d is then given by the equation (32). When the eccentricity differs very little from unity, we may put B= 1, and tan Iw = tan 1 v (l + 9e), COos2 w =- C2 cos2 v. Then we shall have M cos2 w. 2k (t -- T) 4 ~^2 Csmv= ~, = cos4 1w. 75 tan 1 1/2 q3 The equation = (1 + A C2) coS2 V= (1 + A) cos2 I, gives r -- (1 + QA) cos4 w C cos4 w. r2 Hence we derive Mcos2w2 sn (t- T) p p C 2 e 75 tan -w r~ C (1 q ( e)' NUMERICAL EXAMPLES. 135 If we substitute this value in equation (39), and put C2 (1 + e)= 2, we get A __ 9 k ]c 8 tan __ ~dv 9 kl/2P (t) 8tanTv - (40) de 2(1+9e) r (1 + e) (1 + 9e)' and when e 1, this becomes identical with equation (31). 51. EXAMPLES.-We will now illustrate, by numerical examples, the formulae for the calculation of the variations of the geocentric right ascension and declination arising from small increments assigned to the elements. Let it be required to find for the date 1865 February 24.5 mean time at Washington, the differential coefficients of the right ascension and declination of the planet Eurynome ~ with respect to the elements of its orbit, using the data and results given in Art. 41. Thus we have =-181~ 8' 29".29, — 40 42' 21".56, log J= 0.2450054, log r 0.428285, v 129~ 3' 50".5, u = 326~ 41' 40".1, A - 2960 39' 5".0, B - 205~ 55' 27".1, C- 212~ 32' 17".7, log sin a - 9.999716, log sin b 9.974825, log sin c 9.522219, log x 0.425066,, log y - 9.511920, log z 8.077315, =23~ 27' 24".0, t - T 420.714018. First, by means of the equations (4), we compute the following values:log cos 8 d =- 8.054308, log d - 8.66899, dx dx l dct / log cos -- 9.75499, log -= 6.968348, dy n) dyI log - 9.753529. dz Then we find the differential coefficients of the heliocentric co-ordinates, with respect to iT, i, i, v, and r, from the formule (7), which give dx d 49199/, dy dy log d =7.876A3 log -8.830941, log 9.2229, dx dy_ dz log -di 8.726364, -lo - 9.687577, log -= 0.142443,, log dx d9.996780 log dz og dr 9.996780,, log 9.083635, log — 7.649030. 136 THEORETICAL ASTRONOMY. dx dy dz In computing the values of di -, and d-, those of cosa, cos b nld cos may generally be obtained with sufficient accuracy from sina, sin b, and sine. Their algebraic signs, however, must be strictly attended to. The quantities sin a, sin b, and sin c are always positive; and the algebraic signs of cos a, cos b, and cos c are indicated at once by the equations (101)1, from which, also, their numerical values may be derived. In the case of the example proposed, it will be observed that cos a and cos b are negative, and that cos c is positive. dla da To find the values of cos a d and d-, we have, according to equation (2), da dao dx da d__ dr dx d dd dy dr da da dx d8 dy d dz: d-' dx dy di d dr' which give du d: da d 8 cos -- cos -- - 1.42345, - d - 0.48900. dn dv dO dv In the case of, i, and r, we write these quantities successively in place of r in the equations (41), and hence we derive dcu da cos 8 d - = 0.03845, - - 0.09533, dQ d Z cos d da cos = 0di.2 -0 - 0.78993, du, da cos d- - 0.08020, d -+ 0.04873. dr dr Next, from (16), we compute the following values:dr dr dr log -= 0.179155, log dr = 9.577453, log d = 2.376581~, dv dv dv log = 0.171999, logd- 9.911247, log = 2.535234. dx dx We may now find We may now find d-, dloN &c. by means of the equations (11), du da and thence the values of cos a da d &c.; but it is most convenient d-p dy' do du da da to derive these values directly from cos 8 d, cos a-, -, and -, dr dv dr dv in connection with the numerical values last found, according to the NUMERICAL EXAMPLES. 137 equations which result from the analytical substitution of the expressions fordx dy dz. sios d - &c., in equation (2), writing successively p, Mo0, and i in place of 7r. Thus, we have dc da dr dc dv c os cos -- + cos a' ds dr dy dv do' da da dr +d dv (42) d* dr dr dv dv and similarly for Ml4 and,, which give cos - = + 1.99400, - 0.65307, dyo d cos -a = + 1.13004, d - 0.38023, dum, dM, cos 8 d - +507.264, d - 179.315. Therefore, according to (1), we shall have cos A - +1.42345Azr- 0.03845A ~ - 0.27641Ai +1.99400As + 1.13004lMO + 507.264a,u, a^ - 0.48900A7 - 0.09533^ ~ - 0.78993i - 0.65307Ao -0.38023AMo~- 179.315Aa. To prove the calculation of the coefficients in these equations, we assign to the elements the increments AMo + 10", AT-r -20", Aa -10" Ai-+ 10", a=, _ + 10", A L + -0".01, so that they become Epoch =1864 Jan. 1.0 Greenwich mean time. Mo- 1" 29' 50".21 r — 44 20 13.09) =-206 42 30.13 Mean Equinox 1864.0 i — 4 37 0.51 J -= 11 16 1.02 log a = 0.3881288 - 928.56745 With these elements we compute the geocentric place for 1865 February 24.5 mean time at Washington; and the result is -= 181~ 8' 34".81, = - 4~ 42' 30".58, log = 0.2450284, 138 THEORETICAL ASTRONOMY. which are referred to the mean equinox and equator of 1865.0. The difference between these values of a and ~ and those already given, as derived from the unchanged elements, gives a= =- + 5".52, cos 8 AM = + 5".50, - = 9".02, and the direct substitution of the assumed values of aWr, ad, Ai, &c. in the equations for cos 8 hao and AS, gives cos 8 Am - + 5".46, 9".29. The agreement of these results is sufficiently close to show that the computation of the differential coefficients has been correctly performed, the difference being due chiefly to terms of the second order. When the differential coefficients are required for several dates, if we compute their values for successive dates at equal intervals, the use of differences will serve to check the accuracy of the calculation; but, to provide against the possibility of a systematic error, it may be advisable to calculate at least one place directly from the changed elements. Throughout the calculation of the various differential coefficients, great care must be taken in regard to the algebraic signs involved in the successive numerical substitutions. In the example given, we have employed logarithms of six decimal places; but it would have been sufficient if logarithms of five decimals had been used; and such is generally the case. It will be observed that the calculation of the coefficients of AT, A 2, and Ai is independent of the form of the orbit, depending simply on the position of the plane of the orbit and on the position of the orbit in this plane. Hence, in the case of parabolic and hyperbolic orbits, the only deviation from the process already illustrated is in the computation of the coefficients of the variations of the elements which determine the magnitude and form of the orbit and the position of the body in its orbit at a given epoch. In all du, dC, d8 d_ cases, the values of cos,' cos,- and are determined as dv dr dv dr already exemplified. If we introduce the elements T, q, and e) we shall have d dda, dr dc dv cos 8 d-T -+ cos a.. d-os dT ldrd d+T dv drI d8 dJ dr dd dv, adT deren + dv eff' r and similarly for the differential coefficients with respect to q and e. NUMERICAL EXAMPLES. 139 dr cdv dr dv dr dv The mode of calculating the values of d-r, dv dr' d' and A depends on the nature of the orbit. In the case of passing from one system of parabolic elements to another system of parabolic elements, the coefficients of he vanish. dr dv To illustrate the calculation of - dT- &c. in the case of parabolic motion, let us resume the values t-T= 75.364 days, and logq = 9.9650486, from which we have found log r - 0.1961120, v = 790 55' 57".26. Then, by means of the equations (22), we find dr d., log _= 8.095802, log = 9.242547, dv dv log dT 7.976397, log -0.064602,. If, instead of dq, we introduce d log q, we shall have dr dv logd lg = 9.569812, log 0.391867 dlogq ~d log q From these, by means of (43), we obtain the differential coefficients of Oc and ~ with respect to T and q or log q. The same values are also used when the variation of the parabolic eccentricity is taken dv into account. But in this case we compute also - from equation dedr (31) and r from (33) or (34), which give, for v = 790 55' 571.3, de e log -8.147367., log -9.726869. dr dv In the case of very eccentric orbits, the values of d, T, &c. are found from dv k p dr k d-T 2~- 7~d T - _ ~ e sin v, (44) dT r2 dT - - dAv k/Pt dr.r k (t- T) ~dv=- ~P (t - T), _ 3 e sin v dq B qr2 dq q e sin dr r r2 e sinv dv dq + p'dq the mass being neglected. 140 THEORETICAL ASTRONOMY. To illustrate the application of these formula, let us resume the values, t- T= 68.25 days, e 0.9675212, and log q 9.7668134, from which we have found (Art. 41) v = 102~ 20' 52".20, log r 0.1614052. Hence we derive logp - 0.0607328, and dv dr log T 7.943137n, log =8.180711,n dv dr log - = 0.186517~, log d — 0186517. dq dq If we wish to obtain the differential coefficients of v and r with respect to log q instead of q, we have dv q d dr q dr dlogq O dq dlogq Aodq in which 20 is the modulus of the system of logarithms. dv Then we compute the value of de by means of the equation (30). (35), (39), or (40). The correct value as derived from (39) is dv -e- 0.24289. de The values derived from (35), omitting the last term, from (40) and from (30), are, respectively, - 0.24440, - 0.24291, and - 0.23531. The close agreement of the value derived from (40) with the correct value is accidental, and arises from the particular value of v, which is here such as to make the assumptions, according to which equation (40) is derived from (39), almost exact. dr Finally, the value of de may be found by means of (32), which gives dr =+ 0.70855. de, When, in addition to the differential coefficients which depend on the elements T, q, and e, those which depend on the position of the orbit in space have been found, the expressions for the variation of the geocentric right ascension and declination become NUMERICAL EXAMPLES. 141 dalda da dal cos a =os cS A,r- cos d A ^ +cos d, Ai + cos d- A T dc-r dg di dT dc du + cos o- q a+ cos - d e, A= A7T +- AR +d-g Ai+'^AT+q-Ag+-A e. d8r d2 dT dq de If we introduce log q instead of q, the terms containing q become du ~ d8 respectively cos d A log q and d A log q. It should be d log q d log q observed that if ATr, Ag, and ai are expressed in seconds, in order that these equations may be homogeneous, the terms containing AT, aq, and Ae must be multiplied by 206264.8; but if arr, ^2, and Ai are expressed in parts of the radius as unity, the resulting values of cos 8 AO and AJ must be multiplied by 206264.8 in order to express them in seconds of arc. The most general application of the equations for cos a ac and A8 in terms of the variations of the elements is for the cases in which the values of cos a Ao and of a8 are already known by comparison of the computed place of the body with the observed place, and in which it is required to find the values of ar,, A i, &c., which, being applied to the elements, will make the computed and the observed places agree. When the variations of all the elements of the orbit are taken into account, at least six equations thus derived are necessary, and, if more than six equations are employed, they must first be reduced to six final equations, from which, by elimination, the values of the unknown quantities aTr, A, t&c. may be found. In all such cases, the values of Ao and Ad, as derived from the comparison of the computed with the observed place, are expressed in seconds of arc; and if the elements involved are expressed in seconds of arc, the coefficients of the several terms of the equations must be abstract numbers. But if some of the elements are not expressed in seconds, as in the case of T, q, and e, the equations formed must be rendered homogeneous. For this purpose we multiply the coefficients of the variations of those elements which are not expressed in seconds of arc by 206264.8. Further, it is generally inconvenient to express the variations A T,, and ae in parts of the units of T, q, and e, respectively; and, to avoid this inconvenience, we may express these variations in terms of certain parts of the actual units. Thus, in the case of T, we may adopt as the unit of AT the nth part of a mean solar day, and the coefficients of the terms of the equations for cos 8 aa and Ad which involve aT 142 THEORETICAL ASTRONOMY. must evidently be divided by n. In the same manner, it appears that if we adopt as the unit of Aq the unit of the mth decimal place of its value expressed in parts of the unit of q, we must divide its coefficient by 10%, and similarly in the case of Ae, so that the equations become da da dac s dac cos 8 a -= Cos d Ar -+- cosa A -a + cosa d- i -A- - cos 8a A T dT dQ, di nit dT S d, s d + - cos ~ q+ 1-bcos - he, (45) dd ds da s ds da dA + d dT A dq s ds -- ~e, n which s 206264.8. When log q is introduced in place of q, the coefficients of A log q are multiplied by the same factor as in the case of aq, the unit of A log q being the unit of the mth decimal place of the logarithms. The equations are thus rendered homogeneous, and also convenient for the numerical solution in finding the values of the unknown quantities An, A, Ai, AT, &c. When AT, Aq, and corrections to be applied to the corresponding elements arein, Os, and 1e0u' In the same manner, we may adopt as the unknown quantity, instead of the actual variation of any one of the elements of the orbit, n times that variation, in which case its coefficient in the equations must be divided by n. The vale of ab, derived by taking the difference between the computed and the observed place, is affected by the uncertainty necessarily incident to the determination of a by observation. The unavoidable error of observation being supposed the same in the case of a as in e wen epreed n the case of when expressed in parts of the same unit, it is evident that an error of a given magnitude will produce a greater apparent error in a than in a, since in the case of a it is measured on a small circle, of which the radius is cos 8; and hence, in order that the difference between computation and observation in a and 8 may have the same influence in the determination of the corrections to be applied to the elements, we introduce cos 8t a instead of a s. The same principle is applied in the case of the longitude and of all corresponding spherical co-ordinates. DIFFERENTIAL FORMUL;E. 143 52. The formule already given will determine also the variations of the geocentric longitude and latitude corresponding to small increments assigned to the elements of the orbit of a heavenly body. In this case we put s 0, and compute the values of A, B, sin a, and sinb by means of the equations (94)1. We have also C= 0, sin c - sin i, and, in place of a and 8, respectively, we write 2 and /. But when the elements are referred to the same fundamental plane as the geocentric places of the body, the formulae which depend on the position of the plane of the orbit may be put in a form which is more convenient for numerical application. If we differentiate the equations x'r cosu cos r sinu sin m cosi, y' = r cos u sin Q + r sin u cos cos i, z _ r sin u sin i, we obtain x' dx' -=- dr - r (sin u cos 2 +- cos u sin Q cos i) du r (cosu sin g + sin u cos os s i) d q + r sin u sin 2 sin i di, dy' = cdr - r (sin u sin Q - cos u cos cos i) du + r (cos u cos - sin u sin 2 cos i) d2 - r sin u cos g sin i di, (46) dz' =- dr + r cos u sin i du + r sin u cos i di, r in which x', y', z' are the heliocentric co-ordinates of the body in reference to the ecliptic, the positive axis of x being directed to the vernal equinox. Let us now suppose the place of the body to be referred to a system of co-ordinates in which the ecliptic remains as the plane of xy, but in which the positive axis of x is directed to the point whose longitude is 2; then we shall have dx.= dx' cos Q + dy' sin Q, dy - dx' sin S, + dy' cos g^, dz - dz', and the preceding equations give dx - dr- r sin u du -r sin u cos i da, dy ==dr +rr cos u cos i du-+r cos u d - r sinu sini di, (47) dz -- dr + r cos u sin i du + r sin u cos i di. r 144 THEORETICAL ASTRONOMY. This transformation, it will be observed, is equivalent to diminishing the longitudes in the equations (46) by the angle 8g through which the axis of x has been moved. Let X,, Y,, Z, denote the heliocentric co-ordinates of the earth referred to the same system of co-ordinates, and we have x x,= J cos cos (A - 2), y Y, - cos sin ( - ), z + Z, = D sin fi, in which 2 is the geocentric longitude and P the geocentric latitude. In differentiating these equations so as to find the relation between the variations of the heliocentric co-ordinates and the geocentric longitude and latitude, we must regard PQ as constant, since it indicates here the position of the axis of x in reference to the vernal equinox, and this position is supposed to be fixed. Therefore, we shall have dx =cos p cos (A- ) da- sin f cos (A - ) df- J cos sin (A- ) dA, dy= cosfi sin ( -2) d- A sinf sin (A -g) d3 +- cosf cos(A —g) dA, dz -sin d dJ +- cosI dP, from which, by elimination, we find cos dA - sin (A- ) d cos ( -)dy, d A d sin cos ( -2 ) sin sin (A - ) cos ~~ d~x~ - dy- -dz. These equations give cd sin (A - 2) df sin f cos (A - ) cos = ~ cod'A dx - C d cos (Ay - ) df sin sin (-2) (48) dy d A dA dfp cos f cos f - 0, - cosdz~ dz D If we introduce the distance co between the ascending node and the place of the perihelion as one of the elements of the orbit, we have du - dv + dw, and the equations (47) give dx x dy y dz z d —-cosu, -— d sinu cosi, d=-= sinu sini; dr r dr r dr r dx dx d dy dy. d d - -= — -- r sin u, d - - rcos ucos, d - =- rcosu sin i dv dw dv dw dv do DIFFERENTIAL FORMULAE. 145 dx - dy dz d — =-r sin it cos i, d —r cos u,0 (49) d2 ~ ddb ~ d-0 dx dy dz — 0, d -r sin u sin i, - r sin u cos i. di di di If we introduce w, the longitude of the perihelion, we have du dv + di - d d, and hence the expressions for the partial differential coefficients of the heliocentric co-ordinates with respect to w and a become dx. dy dz d sn d- r os u cos, d r cos sin i; dir dr dr dx dy dz0) -g si = 2' sin si, d- 2r cos Si2, - = r cosu sin i When the direct inclination exceeds 90~ and the motion is regarded as being retrograde, we find, by making the necessary distinctions in regard to the algebraic signs in the general equations, dx. dy. dz di, - d r sin u sin i, d - r sin u cos; (51) dx dx dx dy and the expressions for d-' dv d', &c. are derived directly from (49) by writing 180~ - i in place of i. If we introduce the longitude of the perihelion, we have, in this case, du = -dv — d + d, and hence dx. dy dz d- r sin u, — =-r OS u Co, - - r cos d sin i; dv dy Ao (52) dx. 2.. dy n2 i dz dgQ d, 2r scosu sinin, d -r cosu sin i. But, to prevent confusion and the necessity of using so many formulae, it is best to regard i as admitting any value from 0~ to 180~, and to transform the elements which are given with the distinction of retrograde motion into those of the general case by taking 180~ - i instead of i, and 2g2 - n instead of T, the other elements remaining the same in both cases. 53. The equations already derived enable us to form those for the differential coefficients of A and: with respect to r, v, 2, i, and w or 7r, by writing successively i and f in place of 0, and 2, i, &c. in 10 146 THEORETICAL ASTRONOMY. place of 7r in equation (2). The expressions for the differential coefficients of r and v, with respect to the elements which determine the form of the orbit and the position of the body in its orbit, being independent of the position of the plane of the orbit, are the same as those already given; and hence, according to (42) and (43), we may derive the values of the partial differential coefficients of i and /9 with respect to these elements. The numerical application, however, is facilitated by the introduction of certain auxiliary quantities. Thus, if we substitute the values given by (48) and (49) in the equations d dA dx d2 dy cos f - d = — c os d d - +cosf d- d dv dx aa dv df_ df dx d3 dy d1 dz dv dx d dy dv dz dv' and put cos i cos (A- g) = A0 sin A, sin (A - A) = Ao cos A, sin i n sin (n5, -sin (A - 2) cos i -Z n cos N, in which AO and n are always positive, they become dA da r c os cos d = ~_ Ao sin (A -+ u), d- d3f r (sin f cos (A - ) sin u + n cos u sin (N + -) ). av - w J Let us also put n sin (N+ p) B, sin B, 54 sin A cos (2 - ~) = Bo cos B, and we have d2 d2 r cos -- = cos -- Ao sin (A + — ), dv dw ( dv d'- r Bo sin (B +- u). d dd 3 The expressions for cos d- and -d give, by means of the same auxiliary quantities, d- Ao cosd cos (A + u), ^d __ B (56) d — B cos (B + u). dr In the same manner, if we put DIFFERENTIAL FORMULE. 147 cos ( - g~)= C0 sin C, cos i sin (R- P2) =C Cos C;(57) cos i - Do sin D, sin ( - g) sin i = D cos D; we obtain d2 r cos id g- A C sin ( C + u), db _ Ao singi cos (A +- ); (58) dI r. cos sin i== — sin si cos (A- ), di d O d-l 4 ss sin (.D+p). If we substitute the expressions (55) and (56) in the equations dA dA dr dA dv cos /- Co = cos - * + cos P -, df dr dr d dv d dp dr d,p dv do dr do dv dso' and put dr - d _==f sin F = a cos (o cos v, (59) r - j cosF (- - + tan cos v r sin v, d14p " cos / we get cos/5 ( + cos d = f A sin (A+ F +u), t (60) dfBo d f, Bo sin (B - F+ q). In a similar manner, if we put dr d, = g sin G = - a tan s sin v, dv a2 cos P r-o =g cos G= Td0M r (61) - d - h sin H -( a tan sinv (t-T)- - 206264.8), r da =h cos r (t- T), ~dA^~ r 148 THEORETICAL ASTRONOMY. we obtain cos, dE g Ao sin (A + G + u- ), cos # - d- g- Bo sin (B + G + u); dH0 dMAz1 (62) cos d -- -- Ao sin (A + H+ u), df h Bo sin (B +- 1 + +). The quadrants in which the auxiliary angles must be taken are determined by the condition that A,, Bo Co f, g, and h are always positive. 54. If the elements T, q, and e are introduced in place of Mo, p, and j, we must put dr dv f sin F - d' fcos F- r -, dr dv gsin G= — d g cos G=r dT' (63) dr Av h sin _H -d hcos -=r -d dq' dq' and the equations become cos l d Ao sin (A+ F-+ -u), de -- -- B sin (B + F+ u); deA cos i.-g Ao sin (A + G + u), cost - d — s dq A 0 nA)(64) -p cs g- BA sin (A + + u6); dA h cod Bo sin (B + H-ff+- ) dq A In the numerical application of these formule, the values of the second members of the equations (63) are found as already exemplified for the cases of parabolic orbits and of elliptic and hyperbolic orbits in which the eccentricity differs but little from unity. In the same manner, the differential coefficients of i and f with respect to any other elements which determine the form of the orbit may be computed. NUMERICAL EXAMPLES. 149 In the case of a parabolic orbit, if the parabolic eccentricity is supposed to be invariable, the terms involving e vanish. Further, in the case of parabolic elements, we have dr k sin v dv gsin G —dT- 1/2q - - r tan.),vdT' dv gcos G =rdT dT which give tan G- -- tan'v. Hence there results G-180~ —'v, and g —=k., which is the expression for the linear velocity of a comet moving in a parabola. Therefore, cos fl d A -/2 A~ sin (A + ut - u v), _d _ k V2 (65) dT ~ Bo sin (B +u - ~v). For the case in which the motion is considered as being retrograde, 180~ - i must be used instead of i in computing the values of A0, A, n, N, CQ, and C, and the equations (55), (56), and the first two of (58), remain unchanged. But, for the differential coefficients with respect to i, the values of Do and D must be found from the last two of equations (57), using the given value of i directly; and then we shall have d_ r cos = - sin i sin u cos (A - ), (66) d - Do sin sin (D ). di A 55. EXAMPLES.-The equations thus derived for the differential coefficients of A and 9 with respect to the elements of the orbit, referred to the ecliptic as the fundamental plane, are applicable when any other plane is taken as the fundamental plane, if we consider A and f as having the same signification in reference to the new plane that they have in reference to the ecliptic, the longitudes, however, being measured from the place of the descending node of this plane on the ecliptic. To illustrate their numerical application, let it be required to find the differential coefficients of the geocentric right ascension and declination of Eurynome 0 with respect to the elements of its orbit referred to the equator, for the date 1865 February 24.5 mean time at Washington, using the data given in Art. 41. 150 THEORETICAL ASTRONOMI. In the first place, the elements which are referred to the ecliptic must be referred to the equator as the fundamental plane; and, by means of the equations (109),, we obtain Q'- 3530 45' 35".87, i'= 19~ 26' 25".76, wo 212~ 32' 17".71, and' -- w+ 0- = 50~ 10' 7".29, which are the elements which determine the position of the orbit in space when the equator is taken as the fundamental plane. These elements are referred to the mean equinox and equator of 1865.0. Writing a and a in place of 2 and A, and 2', i', wi in place of S, i, and w, respectively, we have Ao sinA cos (a,-') cos i', Ao cos A = sin (a -'); n sin N sini', n cosN- -cos i' sin (a-'); Bo sin B n sin (N + 3), B cos B z sin 8 cos (a~ -'); C sin C — cos ( - g'), Co cos C-sin (a -') cos'; Do sin D = cos i', Do cos D sin i' sin (a -'); f sin F = a cos' cos v, f cos F- + tan cosv) r sin v; g sin G= - a tan p sin v, a cos Gp - h sin H= ( a tan p sinv (t - T) - - 206264.8), a2 cos ( h cos H -- o (t - T). r The values of AO, n, B, (C,, Do, f, g, and h must always be positive, thus determining the quadrants in which the angles A, B, &c. must be taken; and these equations give log A,= 9.97497, A 262~ 10' 40", log BO 9.52100, B= 75 48 35, log C= 9.99961, C- 263 2 6, logDo= 9.97497, D= 92 35 47, logf =0.62946, F=339 14 0, log g =0.34593, G =350 11 16, log h =2.97759, H=- 14 30 48, u' v + -w'- 179~ 13' 58". NUMERICAL EXAMPLES. 151 Substituting these values in the equations (55), (58), (60), and (62), and writing a and 8 instead of 2 and j, and u' in place of u, we find cos a - - + 1.4235, d - 0.4890, dw' dw cos a d +- - 1.5098, d -+- 0.0176, da dd cos 8 I- + 0.0067,, + 0.0193, di' 0.0193, du d8 cos d -- + 1.9940, - 0.6530, d^ d d cos d -- +1.1300, d- - 0.3802, dlu4 dM8 cos d - + 507.25, - - 179.34; and hence cos 3 Aa = + 1.4235 Ao' + 1.5098 A'g' + 0.0067 Ai' + 1.9940 ao + 1.1300 AM +- 507.25 Aa/, a - - 0.4890 Aw' + 0.0176 A 2' + 0.0193 i' - 0.6530 A^ - 0.3802 AMo - 179.34 a/. If we put av'= - 6".64, A 2' - 14".12, i' - 8".86, -+ 10", M 10", +Am + 1- 0".01, we get cos 8 A +- 5".47, A - 9".29; and the values calculated directly from the elements corresponding to the increments thus assigned, are cos -- + 5".50, ^ - 9".02. The agreement of these results is sufficiently close to prove the calculation of the coefficients in the equations for cos 8 Aa and A^. When the values of Aoc, A a, and Ai' are small, the corresponding values of A^o, A, and Ai may be determined by means of differential formulae. From the spherical triangle formed by the intersection of the planes of the orbit, ecliptic, and equator with the celestial vault, we have cos s i cos' cos + sin i' sin e cos g', sin i cos g = - cos i' sin e + sin i' cos E cos', i sin i sin 2 == sin i' sin g', (67) sin i sin w =- sin' sin e, sin i cos wo =co s e sin i' - sin cos i' cos g, 152 THEORETICAL ASTRONOMY. from which the values of 2, i, and (o may be found from those of g' and i. If we differentiate the first of these equations, regarding e as constant, and reduce by means of the other given relations, we get di = cos wv di' - sin w sin i'd Q'. (68) Interchanging i and 180~ - i', and also a and t', we obtain di' = cos Wo di - sin w sin i d 2. Eliminating di from these equations, and introducing the value sin i' sin 2 sin i sin " the result is sin c, s di'. d - C.sin'os tDo d ~ ~ ~s (69) sin PI sinl If we differentiate the expression for cos w0 derived from the same spherical triangle, and reduce, we find dwo =- cos i dp - cos i' d'. Substituting for dQ its value given by the preceding equation, and reducing by means of sin 2' cos i' =sin a cos w0 cos i - cos g sin wo, we get sin w in %w dwto Sill cos 8g dP' -. cos i di'. (70) sinl sin i The equations (68), (69), and (70) give the partial differential coefficients of 2, i, and oo with respect to 2' and i', and if we suppose the variations of the elements, expressed in parts of the radius as unity, to be so small that their squares may be neglected, we shall have sin (% sin w ao - cos 2A'-. cosi Ai', sin' sni sin, sin., A~= sinb g S cost agg'....~Ai', (71) si 2 I "` Q' sin i Ai sin w0 sin' A Q' + cos w Ai', AZ t At- Atoo. If we apply these formulae to the case of Eurynome, the result is Aw o- 4.420A P' + 6.665Ai', A = — 3.488A 2' + 6.686Ai', Ai = - 0.179g' - 0.843Ai'; DIFFERENTIAL FORMULE. 153 and if we assign the values A _- 14".12, ai - 8".86, Aw' - 6".64, we get ao0 =- + 3".36, A - 10".0 i 0.0, i -]- 10.0, A 10.0, and, hence, the elements which determine the position of the orbit in reference to the ecliptic. The elements ow',', and i' may also be changed into those for which the ecliptic is the fundamental plane, by means of equations which may be derived from (109)t by interchanging 2 and 2' and 1800 -' and i. 56. If we refer the geocentric places of the body to a plane whose inclination to the plane of the ecliptic is i, and the longitude of whose ascending node on the ecliptic is Q, —which is equivalent to taking the plane of the orbit corresponding to the unchanged elements as the fundamental plane,-the equations are still further simplified. Let x', y', z' be the heliocentric co-ordinates of the body referred to a system of co-ordinates for which the plane of the unchanged orbit is the plane of xy, the positive axis of x being directed to the ascending node of this plane on the ecliptic; and let x, y, z be the heliocentric co-ordinates referred to a system in which the plane of xy is the plane of the ecliptic, the positive axis of x being directed to the point whose longitude is 2. Then we shall have dx' dx, dy' — dy cos i + dz sin i, dz' - dy sini -+ dz cos i. Substituting for dx, dy, and dz their values given by the equations (47), we get. X' dx' - dr- r sin u du- r sin u cos i d, r dy' - dr + r cos u du + r cos u cos i d2, dz' =- dr -r cos u sin i d2 +- r sin u di. It will be observed that we have, so long as the elements remain unchanged, X' = r cos u, y' =r sin u, 0, 154 THEORETICA.L ASTRONOMY. and hence, omitting the accents, so that x, y, z will refer to the plane of the unchanged orbit as the plane of xy, the preceding equations give dx - cos u dr- r sin u du- r sin u cos i dQ, dy = sin u dr + r cos u du -- r cos u cosi d 2, dz - r cosu sini dQ - r sin di. The value of w is subject to two distinct changes, the one arising from the variation of the position of the orbit in its own plane, and the other, from the variation of the position of the plane of the orbit. Let us take a fixed line in the plane of the orbit and directed from the centre of the sun to a point the angular distance of which, back from the place of the ascending node on the ecliptic, we shall designate by a; and let the angle between this fixed line and the semitransverse axis be designated by X. Then we have X/ - + 0. The fixed line thus taken is supposed to be so situated that, so long as the position of the plane of the orbit remains unchanged, we have a=- 2, — = 7 But if the elements which fix the position of the plane of the orbit are supposed to vary, we have the relations d6a cosid a, dw = dX - cos i dQ, (72) dn -= d + (1 - cos i) d =- dX + 2 sin's i d a. Now, since u v + w, we have u= - + -, and du - d + dv - d -_ dv + d% - cos i d. Substituting this value of du in the equations for dx, dy, dz, they reduce to dx cos u dr - r sin u dv- r sin u dz, dy - sin u dr + r cos u dv + r cos u dZ, (73) dz -- r cos u sin i d2 + r sin u di. The inclination is here supposed to be susceptible of any value from 0~ to 180~, and if the elements are given with the distinction of retrograde motion we must use 180~ - i instead of i. Let us now denote by 0 the geocentric longitude of the body measured in the plane of the unchanged orbit (which is here taken as the DIFFERENTIAL FORMULAE. 155 fundamental plane) from the ascending node of this plane on the ecliptic, and let the geocentric latitude in reference to the same plane be denoted by ^. Then we shall have x +- X- A cos C cos 0, y -- Y= J cos 7 sin 0, z +- Z = a sin 77, in which X, Y Z are the geocentric co-ordinates of the sun referred to the same system of co-ordinates as x, y, and z. These equations give, by differentiation, dx cos r cos 0 dDA A sin,^cos 0 dr - A cos -q sin 0 dO, dy= cos sin 0 dJ - A sin - sin 0 dv +- A cos - cos 0 dO, dz - sin f dJA - A cos V dr; and hence we obtain sin O cos 0 cos j dO - ~j dx.+ a - dy, sin cos 0 sin V sin 0 cos lr — dx- dy+ a dz. These give do sin o dO cos o do cos - - cos, cos - 0; dx J'dy A dz (74) dr, sin - cos dr sin sinO dv cos 7. dx u' d' d and from (73) we get dx dy s dz -C=GOSU d, - sin u,. = O; dr -cosu, dr dr dx dx dy dy dz dz...-.-... r sin, i -n- r cos u, =- 0; dv - d dv d dv d 7 (75) dx dy dz (75 -d d= 0, -- -r cos u sin i; ddb d2 dQ dx dy dz r sin. di di di r s Substituting the values thus found, in the equations do do dx dO dy cos = cos dx dv + cos f -d' dv dx dv dy dv drv d-r dx + dr dy dr d dv dx dv dy dv dz dv' 156 THEORETICAL ASTRONOMY. we get do do r cos d -- os ) - cos (O - t), d7 d7) r.(76) d-q d-q r. dvr - d ~ sin 7 sin (O -u). In a similar manner, we derive cos 7 sin ( ), d- _ 1 sin Cos (0 -~ z), dQ d^ ^, dr. dO d r cos = 0, d cos x sin cosu, (77) do dv r cos0 di -o d7 c+os s sin u. di' If we introduce the elements p, MO, and /, which determine r and v, we have, from dO do dr dO dv cos v -- cos d' cos - dSt dr dyp dv d. dr- drl dr d) dv d=o dr dp dv dA' if we introduce also the auxiliary quantitiesf and F, as determined by means of the equations (59), os f d cos (- -F), sin sin (0 -u —F). (78) ls cos do -s Finally, using the auxiliaries g, h, G, and H, according to the equations (61), we get Co. ( do C g sin V sin (0 - u - cos cos (G), dMo sin in ( uG), dMoo - - dM o - ~~dM~li0 zi~~ dlH0~~ ~(79) dO h dr) h.s( cos 0 d. cos (O - - H), d- =- sinm sin (O - u H). If we express r and v in terms of the elements T, q, and e, the values of the auxiliaries f, g, h, F, &c. must be found by means of,(64); and, in the same manner, any other elements which determine the form of the orbit and the position of the body in its orbit, may be introduced. The partial differential coefficients with respect to the elements having been found, we have do do d o do cos ] aO -= cos ) -- A2' q- cos 7 -~ zAx q- cos V 7j AM0 q- cos ) d7] 1, df) df) d7) d7) d7 i. d7A dQb di c-d di dM dAL DIFFERENTIAL FORMULaE. 157 from which it appears that, by the introduction of X as one of the elements of the orbit, when the geocentric places are referred directly to the plane of the unchanged orbit as the fundamental plane, the variation of the geocentric longitude in reference to this plane depends on only four elements. 57. It remains now to derive the formulae for finding the values of 7 and 0 from those of 2 and f. Let x, yo, z0 be the geocentric coordinates of the body referred to a system in which the ecliptic is the plane of xy, the positive axis of x being directed to the point whose longitude is Q; and let xO', yO, z0' be the geocentric co-ordinates of the body referred to a system in which the axis of x remains the same, but in which the plane of the unchanged orbit is the plane of xy; then we shall have Xw o J= = cos f cos ), cos cos 0, O = A cos f3 sin (A- ^), yo' = J cos 7 sin 0, z = A sin f,, z' = A sin V, and also Yo = yo cos i + zo sin i, o - yo sin i -+ z cos i. Hence we obtain cos 7 cos 0 = cos f cos (A - g ), cos v sin 0 cos f sin (A- ) cos i - sin f sin i, (80) sin ] - - cos sin ( - ) sin i + sin /i cos i. These equations correspond to the relations between the parts of a spherical triangle of which the sides are i, 900 -', and 90~ - f, the angles opposite to 90~ — ^ and 90~ - f being respectively 90~ + (2 - 2) and 90~ - 0. Let the other angle of the triangle be denoted by r, and we have cos sin -sin i cos (. - ), cos v cos - sin i sin (A - ) sin f + cos i cos. 8 The equations thus obtained enable us to determine r, 0, and r from 2 and f9. Their numerical application is facilitated by the introduction of auxiliary angles. Thus, if we put n sin N- sin fi, n cos N= cos f sin (A - ), 158 THEORETICAL ASTRONOMY. in which n is always positive, we get cos - cos 0 = cos cos (2 - ), cos ~ sin o n cos (N - i), (83) sin f = n sin (N - i), from which ^ and 0 may be readily found. If we also put n' sin N' - cos i, n' cos N' si i sin (A - ), () we shall have cot N' =tan i sin (A- g), -cos N' tan r (N' + i) cot (A- ). (85) If r is small, it may be found from the equation sini cos (t 2 -b) sin n cos - (86) cos -/ The quadrants in which the angles sought must be taken, are easily determined by the relations of the quantities involved; and the accuracy of the numerical calculation may be checked as already illustrated for similar cases. If we apply Gauss's analogies to the same spherical triangle, we get sin (450 - ~) sin (45~ - (0 + r)) cos (45~ + ( ))sin (45~ -- ( + i)), sin (450 - ) cos (45~ - (d + r)) sin (45~ + ~ (A - 2)) sin (45~ - - ( - i)), cos (45~ - ) sin (45~ - ( r)) (87) cos(45~ + - (A - )) cos (45~ - (- + i)), cos (45~ - V) cos (45~ - (o r)) sin (45~ + ( ( - 2)) cos (45~ - ( -)) from which we may derive I, 0, and r. When the problem is to determine the corrections to be applied to the elements of the orbit of a heavenly body, in order to satisfy given observed places, it is necessary to find the expressions for cos a AO and Ar in terms of cos / a/2 and A/. If we differentiate the first and second of equations (80), regarding 2 and i (which here determine the position of the fundamental plane adopted) as constant, eliminate the terms containing dq from the resulting equations, and reduce by means of the relations of the parts of the spherical triangle, we get NUMERICAL EXAMPLE. 159 cos d -= cos r cos 3 dA + sin r df. Differentiating the last of equations (80), and reducing, we find dv =- - sin r cos fp d) + cos r dfp. The equations thus derived give the values of the differential coefficients of 0 and V with respect to 2 and 9; and if the differences A/ and af/ are small, we shall have cos d A0 cos r cos aA -t silln A, A. =- - sin r cos P AA + cos r A. (8 The value of r required in the application of numbers to these equations may generally be derived with sufficient accuracy from (86), the algebraic sign of cos being indicated by the second of equations (81); and the values of j and 0 required in the calculation of the differential coefficients of these quantities with respect to the elements of the orbit, need not be determined with extreme accuracy. 58. EXAMPLE.-Since the spherical co-ordinates which are furnished directly by observation are the right ascension and declination, the formulae will be most frequently required in the form for finding V and 0 from o and 8. For this purpose, it is only necessary to write a and a in place of 2 and 9, respectively, and also 2', i', w', Zt, and u' in place of 2, i, w, Z, and u, in the equations which have been derived for the determination of f and 6, and for the differential coefficients of these quantities with respect to the elements of the orbit. To illustrate this clearly, let it be required to find the expressions for cos V aO and a^ in terms of the variations of the elements in the case of the example already given; for which we have 50~ 10' 7".29,' - 3530 45' 35".87, i — 190 26' 25".76. These are the elements which determine the position of the orbit of.Erynome @, referred to the mean equinox and equator of 1865.0. We have, further, logf= 0.62946, log g 0.34593, log h = 2.97759, F= 339~ 14' 0", G = 3500 11' 16", H-= 14~ 30' 48", u' — 1790 13' 58". In the first place, we compute (, 0, and r by means of the formulae 160 THEORETICAL ASTRONOMY. (83) and (85), or by means of (87), writing a, a, g', and i' instead of 2,, 2, and i, respectively. Hence we obtain 0 188~ 31'9", - - 1~ 59' 28", r- 19 17' 7". Since the equator is here considered as the fundamental plane, the longitude 0 is measured on the equator from the place of the ascending node of the orbit on this plane. The values of the differential coefficients are then found by means of the formulae COs7 O- - cos 0 sin'cos', dldb~' dg' do d_ r Cos di' -di' - cossin', dO r drj r csos dd- ft), sin sin (O - u'), cos 7 dO f cos (O - - F), d f sin sin ( -'- F), dcos d os ( t ) d - g sin sin (O -' - G), d M O J dM0 zi dO h dn h cos d cos (O - u'- H), d - sin sin (' - H), d~ A dtA. which give cos -dg, 0 d -+ 0.5072, C dos. di' -- - + 0.0204, cos d I, + 1.5051, d' =- + 0.0086, de' d,~ cos -r d - + 2.0978, d- + 0.0422, cos. d -4- 1.1922, d- - + 0.0143, dMp dyk do -1 cos d- + 538.00, dTherefore, the equations for cos ^A and ay become cos q =O + 1.5051 AX' + 2.0978 As + 1.1922 alM + 538.00 Alt, ^ =- + 0.0086 AX' + 0.0422 as + 0.0143 AMo - 1.71 ^A + 0.5072 A 2' + 0.0204 ai'. If we assign to the elements of the orbit the variations DIFFERENTIAL FORMULBE. 161 Aw= - 6".64, g - 14".12, i' -- -8".86, A -= + 10", A.0 = + 10", A = +r 0".01, we have A' = Aco' + cos i' A ~' - 19".96; and the preceding equations give cos hO + 8".24, = 6".96. With the same values of aw', A g', &c., we have already found cos 8 a c + 5".47, A a - 9".29, which, by means of the equations (88), writing a and a in place of A and I, give cos 0- + 8".23, n - 6".96. 59. In special cases, in which the differences between the calculated and the observed values of two spherical co-ordinates are given, and the corrections to be applied to the assumed elements are sought, it may become necessary, on account of difficulties to be encountered in the solution of the equations of condition, to introduce other elements of the orbit of the body. The relation of the elements chosen to those commonly used will serve, without presenting any difficulty, for the transformation of the equations into a form adapted to the special case. Thus, in the case of the elements which determine the form of the orbit, we may use a or loga instead of,/, and the equation kVl/ + m y - 3 /1- 0 a2 gives di Ada - = d log a, (89) in which 20 is the modulus of the system of logarithms. Therefore, the coefficient of AcZ is transformed into that of A log a by multiplying it by - -; and if the unit of the rnth decimal place of the logarithms is taken as the unit of A log a, the coefficient must be also multiplied by 10-m. The homogeneity of the equation is not disturbed, since, is here supposed to be expressed in seconds. If we introduce logp as one of the elements, from the equation p a cos' p 11 162 THEORETICAL ASTRONOMY. we get 2 d logp = d1. — 2Ao tan dsp, or do - I- 2 d logp - 3/ tan, d,. (90) 0 Hence it appears that the coefficients of A logp are the same as those of A log a, but since p is also a function of (p, the coefficients of o( are changed; and if we denote by cos 8 ( and ) the values of \ d(p \ the partial differential coefficients when the element / is used in connection with 5p, we shall have, for the case under consideration, d(d d fd s dA cos dco- 3-tan(p cos'.-y-, in which s - 206264".8. If the values of the differential coefficients with respect to / and yn have not already been found, it will be advantageous to compute the values of dr, d dr and dv by dip dp d logp d logp means of the expressions which may be derived by substituting in the equations (15) the value of d/u given by (90), and then we may __ dcd da ndd_ compute directly the values of cos 8, cos a -, and d (po d logp' d' d logp In place of M,, it is often convenient to introduce L,, the mean longitude for the epoch; and since Lo-Mo + 71 we have dL0 = dM3o +dr dM + dM aw dg, and, when X is used, dLo = dMo 4- d + (1 - cos i) dg2. Instead of the elements g and i which indicate the position of the plane of the orbit, we may use b = sini sin, C = sin i cos Q, and the expressions for the relations between the differentials of b and c and those of i and g are easily derived. The cosines of the angles which the line of apsides or any other line in the orbit makes with the three co-ordinate axes, may also be taken as elements of the DIFFERENTIAL FORMULAE. 163 orbit in the formation of the equations for the variation of the geocentric place. 60. The equations (48), by writing 1 and b in place of A and A, respectively, will give the values of the differential coefficients of the heliocentric longitude and latitude with respect to x, y, and z. Combining these with the expressions for the differential coefficients of the heliocentric co-ordinates with respect to the elements of the orbit, we obtain the values of cos b Al and Ab in terms of the variations of the elements. The equations for dx, dy, and dz in terms of du, d2, and di, may also be used to determine the corrections to be applied to the co-ordinates in order to reduce them from the ecliptic and mean equinox of one epoch to those of another, or to the apparent equinox of the date. In this case, we have du - dr - d g. When the auxiliary constants A, B, a, b, &c. are introduced, to find the variations of these arising from the variations assigned to the elements, we have, from the equations (99),, cot A -tan a cos i, cot B- cot a cos i -sini cosec a tane, cot C cot a2 cos i - sin i cosec g cot e, in which i may have any value from 0~ to 180~. If we differentiate these, regarding all the quantities involved as variable, and reduce by means of the values of sin a, sin b, and sin c, we get cos i sin A dA -in2 d - sin sin i di, sur2a sin a cos e dB = s (cos cos - sin i sin e cos g ) d sin b sin B sin i sin Q2 + s (cos sin i cos -cos i sin ) di d sin b sin' b sin e d C si2 c (cos i sin -- sin i cos e cos g ) dR ~~~~~sin c and these+ by means Sof (101)n reduce toSi d and these, by means of (101)i, reduce to 164 THEORETICAL ASTRONOMY. cos 7 dA- si d g — sin A cot a di, sina cos e Cos c cos a dB- =o b c 2 dg - sin B cotb di + ib de, (91) sin cos b cos a dC= dQ~.dg~ -~sin C cot c di+ de. sin c sin2 c Let us now differentiate the equations (101)1, using only the upper sign, and the result is da z - sin i sin A d -- cos A di, db - sin i sin B dQ + cos B di + cos ccosec b de, de -- - sin i sin C d -+ cos C di - co b cosec c de. If we multiply the first of these equations by cot a, the second by cot b, and the third by cot, and denote by 20 the modulus of the system of logarithms, we get d log sin a- -, sin i cot a sin A d + Ao cot a cos A di, cos b cos c d log sin b = - A, sin i cot b sin B dQ -+- A cot b cos B di + o b e de, sin b cos b cos c d log sin c — A sin i cot c sin C da + 0 cot c cos Cdi —a sin2 ds. (92) The equations (91) and (92) furnish the differential coefficients of A, B, C, log sin a, &c. with respect to 2, i, and e; and if the variations assigned to 2, i, and e are so small that their squares may be neglected, the same equations, writing AA, A, Ai, &c. instead of the differentials, give:'he variations of the auxiliary constants. In the case of equations (92), if the variations of, i, and e are' expressed in seconds, each term of the second member must be divided by 206264.8, and if the variations of log sin a, log sin b, and log sin are required in units of the rnth decimal place of the logarithms, each term of the second member must also be divided by 10. If we differentiate the equations (81),, and reduce by means of the same equations, we easily find cos b dl - cos i sec b du + cos b d - sin b cos (I- d) di, (93 db sin i cos( — ) du + sin(l ) di, which determine the relations between the variations of the elements of the orbit and those of the heliocentric longitude and latitude. By differentiating the equations (88), neglecting the latitude of DIFFERENTIAL FORMUL2E. 165 the sun, and considering 2, 9, J, and O as variables, we derive, after reduction, R cos f dA -= - cos (- ) do, 1? (94) df - a - sin f sin (A~- O) d0, which determine the variation of the geocentric latitude and longitude arising from an increment assigned to the longitude of the sun. It appears, therefore, that an error in the longitude of the sun will produce the greatest error in the computed geocentric longitude of a heavenly body when the body is in opposition. 166 THEORETICAL ASTRONOMY. CHAPTER III. INVESTIGATION OF FORMULE FOR COMPUTING THE ORBIT OF A COMET MOVING IN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION OF THE GEOCENTRIC DISTANCE. 61. THE observed spherical co-ofdinates of the place of a heavenly body furnish each one equation of condition for the correction of the elements of its orbit approximately known, and similarly for the determination of the elements in the case of an orbit wholly unknown; and since there are six elements, neglecting the mass,-which must always be done in the first approximation, the perturbations not being considered,-three complete observations will furnish the six equations necessary for finding these unknown quantities. Hence, the data required for the determination of the orbit of a heavenly body are three complete observations, namely, three observed longitudes and the corresponding latitudes, or any other spherical coordinates which completely determine three places of the body as seen from the earth. Since these observations are given as made at some point or at different points on the earth's surface, it becomes necessary in the first place to apply the corrections for parallax. In the case of a body whose orbit is wholly unknown, it is impossible to apply the correction for parallax directly to the place of the body; but an equivalent correction may be applied to the places of the earth, according to the formule which will be given in the next chapter. However, in the first determination of approximate elements of the orbit of a comet, it will be sufficient to neglect entirely the correction for parallax. The uncertainty of the observed places of these bodies is so much greater than in the case of well-defined objects like the planets, and the intervals between the observations which will be generally employed in the first determination of the orbit will be so small, that an attempt to represent the observed places with extreme accuracy will be superfluous. When approximate elements have been derived, we may find the distances of the comet from the earth corresponding to the three observed places, and hence determine the parallax in right ascension DETERMINATION OF AN ORBIT. 167 and in declination for each observation by means of the usual formulae. Thus, we have r p cos o' sin ( -- 0) aJ cos tan so' tan C = ta r cos ( -0)' 7p sin (,' sin ( -- 8) A sin r in which a is the right ascension, 8 the declination, J the distance of the comet from the earth, (n' the geocentric latitude of the place of observation, 6 the sidereal time corresponding to the time of observation, p the radius of the earth expressed in parts of the equatorial radius, and 7r the equatorial horizontal parallax of the sun. In order to obtain the most accurate representation of the observed place by means of the elements computed, the correction for aberration must also be applied. When the distance a is known, the time of observation may be corrected for the time of aberration; but if A is not approximately known, this correction may be neglected in the first approximation. The transformation of the observed right ascension and declination into latitude and longitude is effected by means of the equations which may be derived from (92), by interchanging a and i, a and 9, and writing -e instead of s. Thus, we have tan a tanN sin a cos (N- e) ) tan A ~ - tan a, (1) cos tan, tan - =tan (N- e ) sin, and also cos (N —- ) cos f sin A cos N cos S sin a which will serve to check the numerical calculation of A and /. Since cos / and cos 8 are always positive, cos, and cos a must have the same sign, thus determining the quadrant in which i is to be taken. 62. As soon as these preliminary corrections and transformations have been effected, and the times of observation have been reduced to the same meridian, the longitudes having been reduced to the 168 THEORETICAL ASTRONOMY. same equinox, we are prepared to proceed with the determination of the elements of the orbit. For this purpose, let t, t', t" be the times of observation, r, r', r" the radii-vectores of the body, and u, u', u" the corresponding arguments of the latitude, R, R', R" the distances of the earth from the sun, and 0, 0', 0, the longitudes of the sun corresponding to these times. Let [rr'] denote double the area of the triangle formed between the radii-vectores r, r' and the chord of the orbit between the corresponding places of the body, and similarly for the other triangles thus formed. The angle at the sun in this triangle is the difference between the corresponding arguments of the latitude, and we shall have [rr'] - rr' sin (u' - u), [rr"] rr" sin (a"' - ), (2) [r'r"] r'r" sin (" - a ), If we designate by x) y, z, x', y', z', x/, y", z" the heliocentric coordinates of the body at the times t, t', and t", we shall have IXY = r sin a sin (A -+ u),' - r' sin a sin (A + iu'), x"I - r" sin a sin (A + u"), in which a and A are auxiliary constants which are functions of the elements a and i and these elements may refer to any fundamental plane whatever. If we multiply the first of these equations by sin (u" —'), the second by - sin (" —tu), and the third by sin (a' - u), and add the products, we find, after reduction, x x' x" sin (" - u') -, sin (u"- ) +,- sin (u'- u) = 0, r' r ri which, by introducing the values of [rr'], [rr"], and [r' r"], becomes [r'r"] x - [rr"] x' + [rr'] " 0. If we put [r'r"] [rr"]' [r' ] (3) we get nx - x' + n"x"" 0. (4) In precisely the same manner, we find ny- y' + n'l"y = 0, nz' + n"z" =. (5) nz -- z' + nZ"1- - O. DETERMINATION OF AN ORBIT. 169 Since the coefficients in these equations are independent of the positions of the co-ordinate planes, except that the origin is at the centre of the sun, it is evident that the three equations are identical, and express simply the condition that the plane of the orbit passes through the centre of the sun; and the last two might have been derived from the first by writing successively y and z in place of x. Let A, A', A" be the three observed longitudes, /, j', f" the corresponding latitudes, and J, A', d" the distances of the body from the earth; and let J cos -=p, z' cos' — p, O z" cos P" =p", which are called curtate distances. Then we shall have x - p cos os - R cos, x' p' cos A' - R cos', y =p sin A - R sin 0, y' p' sin' - R' sin (0, z - p tan P, z' p' tan 1', x" =p" cos " - R" cos (",. y" = p sin'" - R" sinl ", z -= p" tan i", in which the latitude of the sun is neglected. The data may be so transformed that the latitude of the sun becomes 0, as will be explained in the next chapter; but in the computation of the orbit of a comet, in which this preliminary reduction has not been made, it will be unnecessary to consider this latitude which never exceeds 1", while its introduction into the formula would unnecessarily complicate some of those which will be derived. If we substitute these values of x, x., &c. in the equations (4) and (5), they become =- n (p cos - R cos 0) - (p' cos A' - R' cos') + n" (p" cos " - R" cos O"), 0 -= n (p sin - R sin ) - (p' sin' - R' sin Q') (6) + n" (p" sin " - R" sin 0 "), - 0 = np tan t - p' tan f' + n"p" tan fi". These equations simply satisfy the condition that the plane of the orbit passes through the centre of the sun, and they only become distinct or independent of each other when n and n"q are expressed in functions of the time, so as to satisfy the conditions of undisturbed motion in accordance with the law of gravitation. Further, they involve five unknown quantities in the case of an orbit wholly unknown, namely, in, in", p, o', and p"; and if the values of n and n1" are first found, they will be sufficient to determine p, p', and p". 170 THEORETICAL ASTRONOMY. The determination, however, of n and n1^ to a sufficient degree of accuracy, by means of the intervals of time between the observations, requires that p' should be approximately known, and hence, in general, it will become necessary to derive first the values of ni, n", and p'; after which those of p and p/l may be found from equations (6) by elimination. But since the number of equations will then exceed the number of unknown quantities, we may combine them in such a manner as will diminish, in the greatest degree possible, the effect of the errors of the observations. In special cases in which the conditions of the problem are such that when the ratio of two curtate distances is known, the distances themselves may be determined, the elimination must be so performed as to give this ratio with the greatest accuracy practicable. 63. If, in the first and second of equations (6), we change the direction of the axis of x from the vernal equinox to the place of the sun at the time t', and again in the second, from the equinox to the second place of the body, we must diminish the longitudes in these equations by the angle through which the axis of x has been moved, and we shall have o - n,(p cos(A- 0') —R os(' — 0)) -(p' cos (' - 0') -') + 1n" (p" cos (A" - 0')- I" cos (() " —')), 0 — n (p sin (-') + R sin ('- )) p' sin (A'- 0') + " (p" sin (-') — R" sin(0" — 0')), (7) O n (psin (A'- A) + J sin(O —')) - R'sin(0' —') - n" (p" sin (A" - 1') - R" sin (0"- A')), 0 - np tan 1 - p' tan i' + n"p" tan /". If we multiply the second of these equations by tan', and the fourth by - sin (2'- 0'), and add the products, we get 0; n"p" (tan /' sin (" - 0') - tan A" sin (A'- 0')) -n"R" sin ( "-') tan I'+ np (tan /' sin ( - 0') - tan A sin (A'-O')) + nR sin ( (0'-' ) tan'. (8) -Let us now denote double the area of the triangle formed by the sun and two places of the earth corresponding to R and R' by [RR'], and we shall have [RR'] -= RR'sin (0 — ), and similarly [RR"] - RR" sin (0"- 0), [R'R"] ='R" sin (0" —'). ORBIT OF A HEAAENLY BODY. 1.71 Then, if we put - lRII1 C-11 N — t[R'R"] " [RR'(9 [RB"]' R ](9) [RRS] I[RIR"]' we obtain R" sin (O" —') R sin ((' — )) N. Substituting this in the equation (8), and dividing by the coefficient of p", the result is tan /' sin (- 0') -tan p' sin (" - o') \XC t 7i tan /" sin (A'- 0')- tan /i' sin (A"-(0') - N \ __ R sin (~'- ) tan fi' ( n tan " sin (A' - (') - tan i' sin (A" -') Let us also put tan 9i' sin ( -') — tan sin (A' — 0') tan /" sin (' - 0)') - tan i' sin (2" - (')' ( - tan _ sin (G' — 0) tan i' - taln " sin (A'- )') - tan i' sin (l"- (')' and the preceding equation reduces to P__f, ]JI-='p +Jr ( s- jlI"R. (11) We may transform the values of 11' id Md" so as to be better adapted to logarithmic calculation with the ordinary tables. Thus, if w' denotes the inclination to the ecliptic of a great circle passing through the second place of the comet and the second place of the sun, the longitude of its ascending node will be 0', and we shall have sin (A'- 0') tan w' tan /'. (12) Let j0,,o" be the latitudes of the points of this circle corresponding to the longitudes A and 2/', and we have, also, tan io =sin ( -')tan w', (13) tan A/i" sin (A" --') tan w'. Substituting these values for tan f', sin(2 - 0) and sin(A" - 0') in the expressions for il' and M/", and reducing, they become sin (/o' - fi) cos /" cos o sin (i" - lio") cos OS' ( 4) (14) 1== ^ tanw zsc(0- os/" cos sin(Q- 0)sin(/"/i" 172 THEORETICAL ASTRONOMY. When the value of -, has been found, equation (11) will give the relation between p and p" in terms of known quantities. It is evident, however, from equations (14), that when the apparent path of the comet is in a plane passing through the second place of the sun, since, in this case, 3-=0 and f"= /9o", we shall have M'= 0 and M"= oo. In this case, therefore, and also when - /9 and " —- P0' are very nearly 0, we must have recourse to some other equation which may be derived from the equations (7), and which does not involve this indetermination. >i; It will be observed, also, that if, at the time of the middle observation, the comet is in opposition or conjunction with the sun, the values of M' and M" as given by equation (14) will be indeterminate in form, but that the original equations (10) will give the values of these quantities provided that the apparent path of the comet is not in a great circle passing through the second place of the sun. These values are M'- = sin(A- 0') " sin(('- 0) sin (A" - 0')' sin (A"- 0') Hence it appears that whenever the apparent path of the body is nearly in a plane passing through the place of the sun at the time of the middle observation, the errors of observation will have great: influence in vitiating the resulting values of 31' and Ml'; and to obviate the difficulties thus encountered, we obtain from the third of equations (7) the following value of p":_n Osin ('- A) p n' sin (A" - A') - Bsin(0 A') —A sin (0' — A) -+ I sin (0" A') sin (A"- ) We may also eliminate p between the first and fourth of equations (7). If we multiply the first by tan', and the second by -cos (A- 0), and add the products, we obtain 0 = n"p" (tan f' cos (A" -') - tan A" cos (' - 0')) - n"R" tan /' cos (0"- 0') + np (tan g' cos (A -0') - tan ft cos (A' —')) - nR tan A' cos ((0' — () + R' tan 3', from which we derive ORBIT OF A HEAVENLY BODY. 173 n tan 9' cos ( - GO') -tan cos (A'-') PP'-' tan f" cos( (- 0')- tan' cos (Af- 0') (16) n,' R" tan f' cos ( "- Q') +, R tan' cos ('- 0)), R'tan' tan ri cos (' -') -tan f' cos (A" - 0') Let us now denote by I' the inclination to the ecliptic of a great circle passing through the second place of the comet and that point of the ecliptic whose longitude is 0'- 90~, which will therefore be the longitude of its ascending node, and we shall have cos ( - (0') tan I' - tan f'; (17) and, if we designate by 3, and 3,, the latitudes of the points of this circle corresponding to the longitudes A and A", we shall also have tan, = cos ( - (') tan I', 18 tan A,, cos (" - ()') tan I'. Introducing these values into equation (16), it reduces to n sin (C, - f) cos i' cos l,, P" P -n" I sin (/" - /,,) cos A cos /, (19) tanll' cos/ os ( f cos(O')+ n r (,, + C (,) G)) n - sin(/n-)i) from which it appears that this equation becomes indeterminate when the apparent path of the body is in a plane passings through that point of the ecliptic whose longitude is equal to the longitude of the second place of the sun diminished by 90~. In this case we may use equation (11) provided that the path of the comet is not nearly in the ecliptic. When the comet, at the time of the second observation, is in quadrature with the sun, equation (19) becomes indeterminate in form, and we must have recourse to the original equation (16), which does not necessarily fail in this case. When both equations (11) and (16) are simultaneously nearly indeterminate, so as to be greatly affected by errors of observation, the relation between p and p" must be determined by means of equation (15), which fails only when the motion of the comet in longitude is very small. It will rarely happen that all three equations, (14), (15), and (16), are inapplicable, and when such a case does occur it will indicate that the data are not sufficient for the determination of the elements of the orbit. In general, equation (16) or (19) is to be used when the motion of the comet in latitude is considerable, and equation (15) when the motion in longitude is greater than in latitude. 174 THEORETICAL ASTRONOMY. 64. The formulae already derived are sufficient to determine the relation between p" and p when the values of n and n" are known, and it remains, therefore, to derive the expressions for these quantities. If we put k (t'-) =, k(t)" -'), (20) k (t" — t) - r', and express the values of x, y, x, y", y z" in terms of x', y', z' by expansion into series, we have C _Zx' 1Z" dX t T2 1 dt k + 1.2'dt2 k2 1.2.3 dt' k3 +.'" (21) dx' ~r 1 d2x' T2 1 d3x' 9 ( ) " + d t. kd + J.2 2. a. + 1.2.3 dt3. +&c., and similar expressions for y, y", z, and z". We shall, however, take the plane of the orbit as the fundamental plane, in which case z,', and z" vanish. The fundamental equations for the motion of a heavenly body relative to the sun are, if we neglect its mass in comparison with that of the sun, d2x' kx2. t2 + 7-0~ d2' k~ dt2 r,3 If we differentiate the first of these equations, we get d3x' 3k2x' dr' k2 dx' dt3 r'* dt r3' dt Differentiating again, we find d4' r' 12k2 1 dr' 2 3k2 d2r' 6k2 dr' dx' W dt4 in6g r \ dt r''dt" * dto2 f'4 d dt' dy Writing y instead of x, we shall have the expressions for - and dY4 Substituting these values of the differential coefficients in equadtO tions (21), and the corresponding expressions for y and y", and putting ORBIT OF A HEAVENLY BODY. 175 r-1 I i- 2 3 d4. 12. dr2 3 d2r... 22 r", ~ _Yk t 4r6 5( dt) k2r'4' dt2 /) T- T,3 f"1 dr' b kr'3 4k24 dt (22.y — q -. dy, -, y" a, y + b". ^-^.2 3' dr2 a"a -b d+ 1 1 dr,,, 3 d dtx From these equations we easily derive y' x -- - - - - x'dy'-y'dx' Tt' y -' -' y' = b dy'- yd (23) y' -x"y t-(ab" - a"b) x dy'-y'dx' The first members of these equations are double the areas of the triangles formed by the radii-vectores and the chords of,the orbit between the places of the comet or planet. Thus, y'x - x'y = [rr'], y"x' - x"y' -[rr"], y"x - x"y = [rr"], (24) and x'dy' - y'dx' is double the area described by the radius-vector xf d~_- -l yd'x during the element of time dt, and, consequently, dy - y'dx' dt double the areal velocity. Therefore we shall have, neglecting the mass of the body, x'dy' - y' dx' - dt =2f =oVr, in which p is the semi-parameter of the orbit. The equations (23), therefore, become [rr'] = bk 1/, [r',] = b"k 7, [rr"] = (ab" + a"b) k 1'/ Substituting for a, b, at, b" their values from (22), we find, since 73:7 -- T t, 176 THEORETICAL ASTRONOMY. *. /2 l'S" dr'd d [rr'] — =" p (1- " kr' dt ) 6 f3 4 4 V1 r%~34 kr'4 dt. 1' [rr"] - - + r t )' (25) Er"] -=' 1 6, + 4 kr' 4 T.... [r'r"] [r'] From these equations the values of n = [rr" and n" [ r] may be derived; and the results are. 1 ( + (T4-7) 1T (7T + T..-I) dr ) n + 6 rf3 4 Ikr" dt',(1A-6 r'~ kr'4 (26) -7' ~~-~1 (r + -T) r( (7T + r~ - r2) dr' 26 1t + 1 3 kr'4 d' which values are exact to the third powers of the time, inclusive. In the case of the orbit of the earth, the term of the third order, dR' being multiplied by the very small quantity dt' is reduced to a superior order, and, therefore, it may be neglected, so that in this case we shall have, to the same degree of approximation as in (26), ~~~~~N" -~~~~~(27) N =- t ( +6 ( I+, ) *) [r'r"] From the equations (26) or from (25), since-= -'] we find =rl -, r' + 12 r3r"3 +" dr' of 4, in the case of an orbit wholly unknown, can be determined 9l only by successive approximations. In the first approximation to the elements of the orbit of a heavenly body, the intervals between the observations will usually be small, and the series of terms of (28) will converge rapidly, so that we may take n - T n" I' ORBIT OF A HEAVENLY BODY. 177 and similarly N 1 NV" -- ~' Hence the equation (11) reduces to pt M'p. (29) It will be observed, further, that if the intervals between the observations are equal, the term of the second order in equation (28) vanishes, and the supposition that -,,, is correct to terms of the third order. It will be advantageous, therefore, to select observations whose intervals approach nearest to equality. But if the observations available do not admit of the selection of those which give nearly equal intervals, and these intervals are necessarily very unequal, it will be more accurate to assume n N n"- N" and compute the values of N and N" by means of equations (9), since, according to (27) and (28), if r' does not differ much from R', the error of this assumption will only involve terms of the third order, even when the Values of r and r" differ very much. Whenever the values of p and p" can be found when that of their ratio is given, we may at once derive the corresponding values of r and r", as will be subsequently explained. The values of r and r" may also be expressed in terms of r' by means of series, and we have dr' T 1 d2r' "2 r 7=v-d-t-k+^. 2.; —&c., r " r' +dt k. + M dt2 + &c. from which we derive r _ T+ " dr' k dt' neglecting terms of the third order. Therefore,?': k (r" — r~ =di~ T; (30) 12 178 THEORETICAL ASTRONOMY. and when the intervals are equal, this value is exact to terms of the fourth order. We have, also, -- r dr' r + r" - 2r' + dt which gives r' -- (r + r") -- (r" - r)-,. ~(31) Therefore, when r and r" have been determined by a first approxidr' mation, the approximate values of r' and dt are obtained from these n equations, by means of which the value of -, may be recomputed from equation (28). We also compute N R'R"sin (" - (') (32) N" - RR' sin ((' - 0)' on N and substitute in equation (11) the values of -, and No thus found. If we designate by M the ratio of the curtate distances p and p", we have P n W P In the numerical application of this, the approximate value of p will be used in computing the last term of the second member. In the case of the determination of an orbit when the approximate elements are already known, the value of -, may be computed from n /r" sin (v"~-v') n- rr' sin (v'- v)' (34) N and that of,N from (32); and the value of M derived by means of these from (33) will not require any further correction. 65. When the apparent path of the body is such that the value of M', as derived from the first of equations (10), is either indeterminate or greatly affected by errors of observation, the equations (15) and (16) must be employed. The last terms of these equations may be changed to a form which is more convenient in the approximations to ithe value of the ratio of p" to p. Let Y, Y', Y" be the ordinates of the sun when the axis of ORBIT OF A HEAVENLY BODY. 179 abscissas is directed to that point in the ecliptic whose longitude is A', and we have Y =-R sin(O -A'), Y' R' sin (O' - ), Y- =R" sin (0"- I'). Now, in the last term of equation (15), it will be sufficient to put nN n N"' and, introducing Y, Y', Y", it becomes ( N Y- n- Y' + Y" cosee (A" — ). (35) It now remains to find the value of,' From the second of equations (26) we find, to terms of the second order inclusive, =- 1 6,3 We have, also, R'3 and hence 6 7T~ ( +,, ) ( R') n"~ N" G ~\I Therefore, the expression (35) becomes N sin (A"-A') j(NY —Y' + N" Y"+'T (T'+ r") - ) Y But, according to equations (5), NY- Y' + N" Y" -0, and the foregoing expression reduces to ~+ rr+,,) ( 1 1 ) R' sin ("' —') since Y' = R' sin (0'- A'). Hence the equation (15) becomes, n sin (A' - A) rt 1 1 \R'sin (A'- ) p P l ^'sin(A -A) 6T - (I +T" )(3 B" ssin (-A') (36) n" sin (A" —)!)` 180 THEORETICAL ASTRONOMY. If we put n sin (A'- ) nA" sin (A"- A')' n sin(A' - ) R 1 )' we have -M MoF. (37) Let us now consider the equation (16), and let us designate by X, X', X" the abscissas of the earth, the axis of abscissas being directed to that point of the ecliptic for which the longitude is 0', then X R Jcos( -'), X' = R', X" - R" cos (O" - 0'). It will be sufficient, in the last term of (16), to put n N n"' N' and for,- its value in terms of N" as already found. Then, since NX- X' + N"X" 0, this term reduces to,~' (i+ ~,,) ( 1 1) R' tan/~' - r )' - an cos 0-0 tan P" cos ('-')- tan' os (A"-' and if we put,n tan f' cos ( --') -tan P cos (- i- Q') IO ~n" tan fI" cos (' — 0') - tan P3' cos (t" - 0')' (38) W_ i-' (I1 1\ _ tan fi' ^' (T + )r'3 R'3tan cos" 0')-tan' cos ((a'-O')'' the equation (16) becomes M- Mo'F'. (39) P In the numerical application of these formulae, if the elements are not approximately known, we first assume n Tr when the intervals are nearly equal, and ORBIT OF A HEAVENLY BODY. 181 n N n~ N' as given by (32), when the intervals are very unequal, and neglect the factors F and F'. The values of p and p" which are thus obtained, enable us to find an approximate value of r', and with this a more exact value of ~7 may be found, and also the value of F or F,. n " Whenever equation (11) is not materially affected by errors of observation, it will furnish the value of I1 with more accuracy than the equations (37) and (39), since the neglected terms will not be so great as in the case of these equations. In general, therefore, it is to be preferred, and, in the case in which it fails, the very circumstance that the geocentric path of the body is nearly in a great circle, makes the values of F and F' differ but little from unity, since, in order: that the apparent path of the body may be nearly in a great circle, r' must differ very little from R'. 66. When the value of M has been found, we may proceed to determine, by means of other relations- between p and p", the values of the quantities themselves. The co-ordinates of the first place of the earth referred to the third, are X,-=R cos " - R cos 0, (40) y,= R" sin 0"- R sin 0. If we represent by g the chord of the earth's orbit between the places corresponding to the first and third observations, and by G the longitude of the first place of the earth as seen from the third, we shall have x, g cos G, y, = g sin G, and, consequently, R" cos ("- 0)-R g- coso(G- 0), (41 R" sin (0" -0 ) -g sin (G-0 ). If 4 represents the angle at the earth between the sun and comet at the first observation, and if we designate by w the inclination to the ecliptic of a plane passing through the places of the earth, sun, and comet or planet for the first observation, the longitude of the ascending node of this plane on the ecliptic will be 0, and we shall have, in accordance with equations (81),, cos + -- cos 9 cos (A - 0), sin. cos w - cos f3 sin ( - 0), sin 4 sin w - sin f, 182 THEORETICAL ASTRONOMY. from which tan w tan sin ( - 0)' ~~~tan w-~ =,~^,(42) n tan (A -)) tan. -- cos w Since cos / is always positive, cos b and cos(A- 0) must have the same sign; and, further, 4 cannot exceed 180~. In the same manner, if w" and 4lf represent analogous quantities for the time of the third observation, we obtain tan f" tan w" tan " sin (" - 0")' tan(" — 0" tan ~tn( (43) cos W" cos "' = cos ft" cos (" - 0"). We also have r2 2 +_ R2 - 2R cos,, which may be transformed into 2- (p sec -- R co 4)2 + R2 sin2; (44) and in a similar manner we find'=~- (p"o sec R" - -" cos t")2 + Rl2 sin~' ". (45) Let x designate the chord of the orbit of the body between the first and third places, and we have 2 = (X"- )2 + (" -- y y)2 + (Z- z)2 But x p cos - R cos 0, y- =p sin A - R sin 0, z =p tan f, and, since p" = 2p, x" = Mp cos A" - R" cos 0", y"-l Mp sin " - R" sin 0", " - Mp tan A" from which we derive, introducing g and G, x" - = Mp c os 2"- p cos - G, "- y = Mp sin A"- p sin A - g sin G, z" z - = Mp tan " — p tan P. Let us now put ORBIT OF A HEAVENLY BODY. 183 Mp cos A" - p cos A ph c'os C cos H, Mp sin A"- p sin A - ph cos C sin H, (46) Mp tan P"- p tan = ph sin C. Then we have x"- x - ph cos C cos H- g cos G, y"- y ph cos Csin H- g sin G, z" z -= ph sin C. Squaring these values, and adding, we get, by reduction, 2 = h2 -2gph cos cos(G -H) + g2; (47) and if we put cos C cos (G - H) = cos (, (48) we have 2 (ph -- g cos )2 + g2 sin. (49) If we multiply the first of equations (46) by cos A" and the second by sin A", and add the products; then multiply the first by sin lA, and the second by cos A", and subtract, we obtain h cos C cos (H - A") M- cos (" - ), h cos C sin (H — ") = sin (A" - i), (50) h sin - M tan " tan i, by means of which we may determine h, C, and H. Let us now put' g sin =- A, R sin - B h cos - b, R" sin 4" - B", h s b", (51) g cos - bR cos - c, g cos v - b"R" cos 4" — c", ph - g cos (p = d, and the equations (44), (45), and (49) become ^^T/d+, 2+' r- d ( + B2, 2 (52) r'= 4(l $- c )ii + B2 2 The equations thus derived are independent of the form of the orbit, and are applicable to the case of any heavenly body revolving around the sun. They will serve to determine r and r" in all cases in which the unknown quantity d can be determined. If p is known, 184 THEORETICAL ASTRONOMY. d becomes known directly; but in the case of an unknown orbit, these equations are applicable only when p or d may be determined directly or indirectly from the data furnished by observation. 67. Since the equations (52) involve two radii-vectores r and rl and the chord x joining their extremities, it is evident that an additional equation involving these and known quantities will enable us to derive d, if not directly, at least by successive approximations. There is, indeed, a remarkable relation existing between two radiivectores, the chord joining their extremities, and the time of describing the part of the orbit included by these radii-vectores. In general, the equation which expresses this relation involves also the semitransverse axis of the orbit; and hence, in the case of an unknown orbit, it will not be sufficient, in connection with the equations (52), for the determination of cl, unless some assumption is made in regard to the value of the semi-transverse axis. For the special case of parabolic motion, the semi-transverse axis is infinite, and the resulting equation involves only the time, the two radii-vectores, and the chord of the part of the orbit included by these. It is, therefore, adapted to the determination of the elements when the orbit is supposed to be a parabola, and, though it is transcendental in form, it may be easily solved by trial. To determine this expression, let us resume the equations k(t- T) - = tan.v + - tan3 - v and, for the time t", k (t" — T) v"- ( = ) tan Iv v+ 1 tan3 Iv". Subtracting the former from the latter, and reducing, we obtain 3k(t -t) sin (v" — v) I r" cos- (v"- v) l/2 q - cq cos cos, cos and, since r q sec2 Iv, this gives 3k (t" -t) sin I (v"- v) /r r" + cos (V"-v )/'. ( ~r~=~'-_=~,/~^-~ r+r +cos~-(v /- r/ ". (53) 1/2 1q / But we have, also, from the triangle formed by the chord x and the radii-vectores r and r", x2 - r+ - rT2 - 2rr" cos (v"- v) = (r + r")2 4rr" cos2 I (" - v). PARABOLIC ORBIT. 185 Therefore, os (r(- v) + r ) (r + r" — x) (-21/rr Let us now put r + r" -+ = n, r + r" — X - n m and n being positive quantities. Then we shall have r+ r' = (2 + n2), 2 cos ~ (v" - v) l/rr" = — mn; and, since m and n are always positive, it follows that the upper sign must be used when v"- v is less than 180~, and the lower sign when v"-v is greater than 180~. Combining the last equation with (53), the result is 3k (t" t) =- si ("v) +(m+ n2 + n mn). (55) 1/2q Now we have sin -I- (v" v) sin v" cos - sin v" cos.cos Squaring this, and reducing, we get sin2 ~ (v" - v) -- cos2 -lv + c vos 2' cos cos (v v), or, introducing r and q, q q mn sin (v"- v)- q- q r r Therefore, 1/2q sin 1 (v "-v)2' 7 (m +- n). Introducing this value into equation (55), we find 6k (t" - t) =- m3 n. Replacing in and n by their values expressed in terms of r, r", and n, this becomes 6k(t" ) ( + r" + ) - (r + r" - t)h (56) the upper sign being used when v"-v is less than 180~. This equation expresses the relation between the time of describing any parabolic arc and the rectilinear distances of its extremities from each other and from the sun, and enables us at once, when three of these quantities are given, to find the fourth, independent of either the 186 THEORETICAL ASTRONOMY. perihelion distance or the position of the perihelion with respect to the arc described. 68. The transcendental form of the equation (56) indicates that, when either of the quantities in the second member is to be found, it must be solved by successive trials; and, to facilitate these approximations, it may be transformed as follows:Since the chord z can never exceed r + r", we may put + sin r', (57) r --- s" and, since x, r, and r" are positive, sin' must always be positive. The value of'r must, therefore, be within the limits 0~ and 180~. From the last equation we obtain cos2r' (r + r/)2 _ 2 and substituting for x2 its value given by -- (r + r")2 - 4rr" cos2 (v"- v), this becomes co 4rr" cos2.(v" -v) cos - ~(r + r ~)" Therefore, we have 21/rr1 cos' = cos ( "- v) r+, (58) and also tan /' - = (59) 21/rr" cos (v" - -v) Hence it appears that when v"- v is less than 180~, r' belongs to the first quadrant, and that when v" - v is greater than 180~, cos r' is negative, and r' belongs to the second quadrant. If we introduce?' into the expressions for m2 and n2 they become ma (r + r") (1 + sin -'), n2 (r + r") (1 - sin), which give m=a (r + Y") (cos l' + sin 1,r)2 n2 (r + r") (~ cos r' sin r')2; and, since r' is greater than 90~ when v"'-v exceeds 180~, the equation (56) becomes t +' = (cos.' + sin Ir,)3 (cos /r' -- sin,-')3. (r + r")Is PARABOLIC ORBIT. 187 From this equation we get 6rt - = 6 cos + 2' sin 2 si, (r + r")2 2 or 61rf - 6 sin My -4 sin3'; (r + r/ )2 and this, again, may be transformed into 6r =3( sin s ) 4(in)3 (60) Let us now put si sin x — / (61) 1/2 or sin /' =V/2 sin x, and we have 3~J ~(~ - r3 sin x - 4 sin3 x = sin 3. (62) V/2(r+ r")-2 When v" v is less than 180', r' must be less than 90~, and hence, in this case, sin x cannot exceed the value a, or x must be within the limits 00 and 30~. When v" -v is greater than 180~, the angle r' is within the limits 90~ and 180~, and corresponding to these limits, the values of sin x are, respectively, I and V/2. Hence, in the case that v" - v exceeds 180~, it follows that x must be within the limits 30~ and 45~. The equation 3 sin 3x 1/2 (i +r")i is satisfied by the values 3x and 180~ - 3x; but when the first gives x less than 15~, there can be but one solution, the value 180" -3x being in this case excluded by the condition that 3x cannot exceed 135~. When x is greater than 150, the required condition will be satisfied by 3x or by 180~ - 3x, and there will be two solutions, corresponding respectively to the cases in which v" -v is less than 180~, and in which v" -v is greater than 180~. Consequently, when it is not known whether the heliocentric motion during the interval t - t is greater or less than 180~, and we find 3x greater than 450, the same data will be satisfied by these two different solutions. In practice, however, it is readily known which of the 188 THEORETICAL ASTRONOMY. two solutions must be adopted, since, when the interval t" - t is not very large, the heliocentric motion cannot exceed 180~, unless the perihelion distance is very small; and the known circumstances will generally show whether such an assumption is admissible. We shall now put 2vt 2~~' (63) (r -+ r") 2 and we obtain sin 3x - (64) 1/8 We have, also, sin / -/2 sin x, and hence cos' =- i/ - 2 sin2 x = I/cos 2x. Therefore sin r'- 2] sin x V cos 2x, and, since =- (r + r") sin r', we have X = 23 (r + r") sin x 1/cos 2x. If we put 3 sin x /cos 2x, (65) sin 3 the preceding equation reduces to --- 2 I. (66) /' (r + r") From equation (64) it appears that 7] must be within the limits 0 and 1X/8. We may, therefore, construct a table which, with V as the argument, will give the corresponding value of y, since, with a given value of (, 3x may be derived from equation (64), and then the value of p. from (65). Table XI. gives the values of C corresponding to values of a from 0.0 to 0.9. 69. In determining an orbit wholly unknown, it will be necessary to make some assumption in regard to the approximate distance of the comet from the sun. In this case the interval t" — t will generally be small, and, consequently, x will be small compared with r and r". As a first assumption we may take r = I, or r + r" = 2, and z =1 and then find x from the formula X' 1/2. PARABOLIC ORBIT. 189 With this value of x we compute d, r, and r" by means of the equations (52). Having thus found approximate values of r and r", we compute z by means, of (63), and with this value we enter Table XI. and take out the corresponding value of /i. A second value for z is then found from (66), with which we recompute r and r", and proceed as before, until the values of these quantities remain unchanged. The final values will exactly satisfy the equation (56), and will enable us to complete the determination of the orbit. After three trials the value of r + r" may be found very nearly correct from the numbers already derived. Thus, let y be the true value of log (r + r"), and let Ay be the difference between any assumed or approximate value of y and the true value, or yo - y + Ay. Then if we denote by yo' the value which results by direct calculation from the assumed value y,, we shall have Yo' -Yo -f (YO) f(Y + AY). Expanding this function, we have yo' - Yo f(y) + A ay + B Ay2 + &c. But, since the equations (52) and (66) will be exactly satisfied when the true value of y is used, it follows that f(Y)- o0, and hence, when Ay is very small, so that we may neglect terms of the second order, we shall have yo' - yo A ay = A (yo - y). Let us now denote three successive approximate values of log (r + s") by yo; Yo', yoI, and let Yo -o a, y= - yo - c a'; then we shall have a A (y -), a' = A (yo' - Y). Eliminating A from these equations, we get y (a' - a) = a'yO -ay, from which Y - a = y a'" (67) a-a 190 THEORETICAL ASTRONOMY. Unless the assumed values are considerably in error, the value of y or of log (r + r") thus found will be sufficiently exact; but should it be still in error, we may, from the three values which approximate nearest to the truth, derive y with still greater accuracy. In the numerical application of this equation, a and a' may be expressed in units of the last decimal place of the logarithms employed. The solution of equation (56), to find t" -t when x is known, is readily effected by means of Table VIII. Thus we have -- = sin 3x, V/2 (9' + r")P and, when r' is less than 90~, if we put sin 3x N —,sin' we get'- = 1/2 Nsinr' (r+ r") (68) or C —' /2 f N r + r". When r' exceeds 900, we put N' - sin 3x, and we have T- v'/2 N' (r + r")- (69) in which log i/2 9.6733937. With the argument r' we take from Table VIII. the corresponding value of N or N', and by means of these equations T =- k (t" - t) is at once derived. The inverse problem, in which r' is known and x is required, may also be solved by means of the same table. Thus, we may for a first approximation put X- =T 1/2, and with this value of x compute d, r, and r". The value of r' is then found from sin r' - rr + r"' and, the table gives the corresponding value of N or N'. A second approximation to x will be given by the equation 3' + " N-r PARABOLIC ORBIT. 191 or by 3' sill sn 3 in which log ~-=0.3266063. Then we recompute d, r, and r", and proceed as before until x remains unchanged. The approximations are facilitated by means of equation (67). It will be observed that d is computed from d-= ~V/2 A2, and it should be known whether the positive or negative sign must be used. It is evident from the equation d = ph - g cos (p, since p, h, and g are positive quantities, that so long as sp (which must be within the limits 0~ and 180~) exceeds 90~, the value of d must be positive; and therefore o( must be less than 90~, and g cos So greater than ph, in order that d may be negative. The equation (4;) shows that when x is greater than g, we have g cos 9 < lpph, and hence d must in this case be positive. But when X is less than g, either the positive or the negative value of d will answer to the given value of (o, and the sign to be adopted must be determined from the physical conditions of the problem. If we suppose the chords g and x to be proportional to the linear velocities of the earth and comet at the middle observation, we have, the eccentricity of the earth's orbit being neglected, 2 which shows that x is greater than g, and that d is positive, so long as r' is less than 2. The comets are rarely visible at a distance from the earth which much exceeds the distance of the earth from the sun, and a comet whose radius-vector is 2 must be nearly in opposition in order to satisfy this condition of visibility. Hence cases will rarely occur in which d can be negative, and for those which do occur it will generally be easy to determine which sign is to be used. However, if d is very small, it may be impossible to decide which of the two solutions is correct without comparing the resulting elements with other and more distant observations. 192 THEORETICAL ASTRONOMY. 70. When the values of r and r" have been finally determined, as just explained, the exact value of d may be computed, and then we have d + g cos ( h' (70) p" = Mp, from which to find p and p". According to the equations (90)1, we have r cos b cos ( (0) = p cos (A -0)-R, r cos b sin (I - 0) = p sin (A- 0), (71) r sin b = p tan A, and also r" cos b" cos (I" - 0") - p" cos (A" - ") - -", r" cos b" sin (I"- 0") -p" sin (A" - 0"), (72) r" sin b" " tan A", in which i and I" are the heliocentric longitudes and b, b" the corresponding heliocentric latitudes of the comet. From these equations we find r, r", 1, 1", b, and b"; and the values of r and r" thus found, should agree with the final values already obtained. When 1" is less than I, the motion of the comet is retrograde, or, rather, when the motion is such that the heliocentric longitude is diminishing instead of increasing. From the equations (82),, we have -- tan i sin (l - ~) - tan b, - tan i sin (/" — ) = tan b", which may be written 4- tan i (sin ( - x) cos (x - a) + sin (x - ) cos /- x)) = tan b, ~ tan i (sin (l"- x) cos (x -- g) + sin (x - 2) cos ("- x)) = tan b. Multiplying the first of these equations by sin (" - x), and the second by - sin (1- x), and adding the products, we get -+- tan i sin (x - 2) sin(l" - 1) = tan b sin (" - x) - tan b" sin (I - x); and in a similar manner we find - tan i cos (x - ) sin (I"- l) - tan b" cos ( — x) - tan b cos (l"- x). Now, since x is entirely arbitrary, we may put it equal to I, and we have PARABOLIC ORBIT. 193 tan i sill (I - ) = tan b, tan b" - tan b cos(l"- ) (74) tan i cos (l — ) + sin-(- 1) the lower sign being used when it is desired to introduce the distinction of retrograde motion. The formnule will be better adapted to logarithmic calculation if we put x =(l" + 1), whence l- x (l"- ) and 1-x -2(1 1"); and we obtain tan i sin (41 (P+ 1)- ) - ~ )" -- -b) tn' 2 cos b cos b" cos (l" -1)' ( sin (b"- b) ~tani cos(l(" + - 1 )) - ) 2 cos b cos b" sin 1 ( )' These equations may also be derived directly from (73) by addition and subtraction. Thus we have ~ tan i (sin (l"- 2 ) + sin (l- )) tan b" + tan b, ~ tanl (sin (- ) -sin (- ) tan b" - tan b; and, since sin(l"- Q) + sinll( - 2) 2 sin ~ ("+ 1- 22) cos ( — 1), sin ("- ) - sin (I - ) - 2 cos ("+- 1 2 ) sinl (1- 1), these become tan i sin (1 (1+ 1) 2 ) 4 - +- tan 6) COS. __, A~ ~(tan V - tan 5) tan cos( (1" + ) ) - - sin "-l) sin ( -l which may be readily transformed into (75). However, since b and b" will be found by means of their tangents in the numerical application of equations (7v) and (72), if addition and subtraction logarithms are used, the equations last derived will be more convenient than in the form (75). As soon as; and i have been computed from the preceding equations, we have, for the determination of the arguments of the latitude u and u", tan (1 — g6) tan (l"-' ) utan ~ = u M tan e'=: -- -(77) cos i cos Now we have =- v +-, in which (o =n — 7 in the case of direct motion, and w- = - 13 194 THEORETICAL ASTRONOMY. when the distinction of retrograde motion is adopted; and we shall have Ut" -- U = - v v, and, consequently, x2 r2 + r"2 - 2rr" cos (u" - u), (78) or 2 -= (r" - r cos ('t" -_ ))2 + r2 sin2 ("tt- u). (79) The value of x derived from this equation should agree with that already found from (66). We have, further, r q sec - ( — w), r" = q sec2- (" - w), or COS (, COS (- ) = By addition and subtraction, we get, from these equations, - (cos I (" — ) + cos A (-w)) — + v_ (COS, ) COS Y(, I 1 1 (c (os(' -- COS (' - - )) -,/ ~ / ~, from which we easily derive vq 4 2_ 1 1 2 O co ls (4('~ +a) 1-0) COS 4-v- ( -t6)= J + X-, V;+1' (80) 2 _s 1 sin A ( (tu"+ qu) - W) sin 4 (u"- ) —But 1 1 / r" 4 jr \ l/Tr 1" t/rrr\ r \'r J and if we put tan (45~ +') - since 4 I~ will not differ much from 1,' will be a small angle; and we shall have, since tan (450 + 0') - cot (45~ + 0') = 2 tan 20', 4r 4 I r;vr'r, -=2 tan 20', + r"= + 4- 2 sec 20', PARABOLIC ORBIT. 195 Therefore, the equations (80) become 1 i (1("+ ) —) tan 20' sin ( 1(i("+ u) -- w) s / ~ Vq s (" - u) Vrr" (81) 1 sec 20' -cos +) ) -~+ U) -) 1/ q - cos (Vt - u) CVrr from which the values of q and 0o may be found. Then we shall have, for the longitude of the perihelion T- -+ a, when the motion is direct, and 7r - 0, when i unrestricted exceeds 90~ and the distinction of retrograde motion is adopted. It remains now to find T, the time of perihelion passage. We have V =- 1 w, V" - t - o. With the resulting values of v and v" we may find, by means of Table VI., the corresponding values of M (which must be distinguished from the symbol M already used to denote the ratio of the curtate distances), and if these values are designated by 31 and M3", we shall have M M" t T —M g T -Mor T t M t" M" m t C in which m 3=, and log C0 9.9601277. When v is negative, the qg corresponding value of 1Mis negative. The agreement between the two values of T will be a final proof of the accuracy of the numerical calculation. The value of T when the true anomaly is small, is most readily and accurately found by means of Table VIII., from which we derive the two values of N and compute the corresponding values of T from the equation 2 T= t -3 Nr sin v, in which log -= 1.5883273. When v is greater than 900, we de 196 THEORETICAL ASTRONOMY. rive the values of N' from the table, and compute the corresponding values of T from 2 T=t - N'r 71. The elements q and T may be derived directly from the values of r, r", and x, as derived from the equations (52), without first finding the position of the plane of the orbit and the position of the orbit in its own plane. Thus, the equations (80), replacing u and,u" by their values v + w and v +'-'; become 2 1 sin -1 (v" + v) sin 4(V"-V) =C4- 4 2^~ ~~~~~~~ ^ ^ ~ (82) S os 4 (V' + V) Cs ( ) = + Adding together the squares of these, and reducing, we get 1 1 2 1 + -~ _= COS V (V~ V) 1 i' - cos// "q sin2 ~ (v" - v) or rr" sin2 1 (v" - v) r" + r - 2 V/rr" cos A (v" - v) Combining this equation with (59), the result is rr" sin2 (v" v) r r+ "-x cot " and hence, since x = (r + r") sin r', q =-sin (v" v) cot r. (83h We have, further, from (78), 2 (r" - r)2 + 4rr" sin2 (v"' ), from which, putting sinv --, (84) we derive 21/rr" cos v T eer sin q (v"' v). (85) Therefore, the equation (83) becomes PARABOLIC ORBIT. 197 q -- (r + r") cos2 Iry' cos2, (86) by means of which q is derived directly from r, r", and x, the value of v being found by means of the formula (84), so that cos is positive. When T' cannot be found with sufficient accuracy from the equation sin r - r + r"' we may use another form. Thus, we have. r + r+r + - r+ ~" ~ - 1 + sin — r r- 1 - sin y'- which give, by division, tan (450 + -r') r + r" + x (87) r+ r~In a similar manner, we derive tan (450 + I) - + (r_ -r) (88) In order to find the time of perihelion passage, it is necessary first to derive the values of v and v". The equations (59) and (85) give, by multiplication, tan (v" - v) == tan r' cos v, (89) from which v" - v may be computed. From (82) we get IrJ/ tan i (v" + v) tan v (v" - v) - r + 1 If we put tan —, t (90) r' this equation reduces to tan 4 (v" + v) - tan (' - 45~) cot 4 (v" - v), (91) and the equations (81) give, also, tan I (v" + v) = cot 4 (v" - v) sin 2', either of which may be used to find v" + v. 198 THEORETICAL ASTRONOMY. From the equations cosv _ 1 cos'v" _ /q. r' /q If- __ by multiplying the first by sin tv" and the second by - sin v, ad'ding the products and reducing, we easily find sin - (v"- v) sin v cos (v" -v) 1 Hence we have sI. v cot (v"- v) 1 /q 2 - r /r" sin. (v"' —v)'( (92) COSa iV oo o- =, which may be used to compute q, v, and v" when v" -v is known. When (v" - v) and - (v" + v), and hence v" and v, have been determined, the time of perihelion passage must be found, as already explained, by means of Table VI. or Table VIII. It is evident, therefore, that in the determination of an orbit, as soon as the numerical values of r, r", and x have been derived from the equations (52), instead of completing the calculation of the elements of the orbit, we may find q and T, and then, by means of these, the values of r' and v' may be computed directly. When this has been effected, the values of n and n" may be found from (3), or that of -, from (34). Then we compute p by means of the first of equations (70), and the corrected value of M froni (33), or, in the special cases already examined, from the equations (37) and (39). In this way, by successive approximations, the determination of parabolic elements from given data may be carried to the limit of accuracy which is consistent with the assumption of parabolic motion. In the case, however, of the equations (37) and (39), the neglected terms may be of the second order, and, consequently, for the final results it will be necessary, in order to attain the greatest possible accuracy, to derive P from (15) and (16). When the final value of M has been found, the determination of the elements is completed by means of the formulae already given. PARABOLIC ORBIT. 199 72. EXAMPLE.-TO illustrate the application of the formulae for the calculation of the parabolic elements of the orbit of a comet by a numerical example, let us take the following observations of the Fifth Comet of 1863, made at Ann Arbor:Ann Arbor M. T. d 1864 Jan. 10 6h 57m 208.5 19^ 14m 4S.92 + 340 6' 27".4, 13 6 11 54.7 19 25 2.84 36 36 52.8, 16 6 35 11.6 19 41 4.54 +39 41 26.9. These places are referred to the apparent equinox of the date and are already corrected for parallax and aberration by means of approximate values of the geocentric distances of the comet. But if approximate values of these distances are not already known, the corrections for parallax and aberration may be neglected in the first determination of the approximate elements of the unknown orbit of a comet. If we convert the observed right ascensions and declinations into the corresponding longitudes and latitudes by means of equations (1), and reduce the times of observation to the meridian of Washington, we get Washington M. T.,B 1864 Jan. 10 7h 24m 3s 297~ 53' 7".6 + 550 46' 58".4, 13 6 38 37 302 57 51.3 57 39 35.9, 16 7 1 54 310 31 52.3 + 59 38 18.7. Next, we reduce these places by applying the corrections for precession and nutation to the mean equinox of 1864.0, and reduce the times of observation to decimals of a day, and we have t 10.30837, A = 297~ 52' 51".1, f + 55~ 46' 58".4, t' - 13.27682, 2' = 302 57 34.4, P' - 57 39 35.9, "- 16.29299,."- 310 31 35.0, " — + 59 38 18.7. For the same times we find, from the American Nautical Almanac, - 290 6' 27".4, log R 9.992763, 0' 293 7 57.1, log R' 9.992830, 0" -296 12 15.7, log R" - 9.992916, which are referred to the mean equinox of 1864.0. It will generally be sufficient, in a first approximation, to use logarithms of five decimals; but, in order to exhibit the calculation in a more complete form, we shall retain six places of decimals. Since the intervals are very nearly equal, we may assume 200 THEORETICAL ASTRONOMY. n T N Then we have ta _\ tn t' tan' sin ( O0') tan3 sn (2'-') t - f t tan i" sin (2'- 0') - tan i' sin (" - 0')' and g sin (G -0) R" sin (" - 0), g cos (G - 0) - R" cos ( " -- 0) - R; h cos: cos (H — A") i- - cos (A" 2), h cos sin (H- )") sin (A"- ), h sin C _ M tan fl" tan f1; from which to find M1, G, g, g,, and h. Thus we obtain log M —- 9.829827, H — 94~ 24' 1".8, G - 22~ 58' 1".7, — 40 28 21.9, log g 9.019613, log h 9.^8532. d" cos fi Since ~ — i- 0.752, it appears that the comet, at the time cos of these observations, was rapidly approaching the earth. The quadrants in which G - G and II- " must be taken, are determined by the condition that g and h cos C must always be positive. The value of 2M should be checked by duplicate calculation, since an error in this will not be exhibited until the values of 2' and -' are computed from the resulting elements. Next, from cos - os 1 Cos ( - ), cos CoS' cos (" — "), cos s -c cos cos (G - H), we compute cos 4, cos )", and cos (; and then from gsin 5 =- A, h cos i = b, h Cos f" Ri sin, - B, hc b9 PR" sin -" =- B", g cos p - bR cos, - c, g cos ~ - b"R" cos "' = c!, we obtain A, B, B", &c. It will generally be sufficiently exact to find sin 4 and sin 4" from cos 4 and cos "; but if more accurate values of 4 and 4" are required, they may be obtained by means of the equations (42) and (43). Thus we derive log A = 9.006485, log B _ 9.912052,." 9.933366, log b-= 9.438524, log b" 62387, =- 0.125067, e..150561 NUMERICAL EXAMPLE. 201 Then we have 22 17 v-7'+ d=r A/(-r':' d/r + r" 2' d -+/ f A2 r - + B'2 r,) + Bus, - b" +'btf from which to find, by successive trials, the values of r, r", and x, that of u being found from Table XI. with the argument ^. First, we assume log x log T' /2 9.163132, and with this we obtain log r = 9.913895, log r" = 9.938040, log (r + r") = 0.227165. This value of log (r + r") gives = 0.094, and from Table XI. we find log, 0.000160. Hence we derive log x 9.200220, log r = 9.912097, log r"- 9.935187, log(r + r") = 0.224825. Repeating the operation, using the last value 6f log(r + r"), we get log x 9.201396, log r 9.912083, log r" 9.935117, log (r + r") = 0.224783. The correct value of log(r + r") may now be found by means of the equation (67). Thus, we have, in units of the sixth decimal place of the logarithms, a -224825 - 227165 - 2340, a' = 224783 - 224825 = - 42, and the correction to the last value of log(r + r") becomes a} a" P- _ - -- -- -- 0.8. Therefore, log (r + r") = 0.224782, and, recomputing I, #A, X, r, and r", we get, finally, log x - 9.201419, log r = 9.912083, log r" 9.935116, log (r + r") = 0.224782. The agreement of the last value of log (r + r") with the preceding one shows that the results are correct. Further, it appears from the 202 THEORETICAL ASTRONOMY. values of r and r" that the comet had passed its perihelion and was receding from the sun. By means of the values of r and r" we might compute approxicdr mate values of 7r' and t from the equations (30) and (31), and then n N a more approximate value of,, from (28), that of N. being found from (32). But, since r' differs but little from R', the difference bewen an N between -~ and y'' is very small, so that it is not necessary to consider the second term of the second member of the equation (33); and, since the intervals are very nearly equal, the error of the assumption nt T is of the third order. It should be observed, however, that an error in the value of M affects H, C, h, and hence also A, b, b", c, and c", aind the resulting value of p may be affected by an error which considerably exceeds that of JM. It is advantageous, therefore, to select observations which furnish intervals as nearly equal as possible in order that the error of JM may be small, otherwise it may become necessary to correct JM and to repeat the calculation of r, r", and x. We may also compute the perihelion distance and the time of perihelion passage from r, r", and x by means of the equations (86), (89), and (91) in connection with Tables VI. and VIII. Then r' and v' may be computed directly, and the complete expression for Mi may be employed. In the first determination of the elements, and especially when the corrections for parallax and aberration have been neglected, it is unnecessary to attempt to arrive at the limit of accuracy attainable, since, when approximate elements have been found, the observations may be more conveniently reduced, and those which include a longer interval nmiy be used in a more complete calculation. Hence, as soon as r, r, -and z have. been found, the curtate distances are next determined, and then the elements of the orbit. To find p and o", we have d + 0.122395, the positive sign being used since x is greater than g, and the formula d +- g cos,, h' give log p = 9.480952, log p" - 9.310779. NUMERICAL EXAMPLE. 203 From these values of p and,op, it appears that the comet was very near the earth at the time of the observations. The heliocentric places are then found by means of the equations (71) and (72). Thus we obtain I = 106~ 40' 50".5, b -+ 330 1' 10".6, logr -9.912082, 1" 112 31 9.9, b"= + 23 55 5.8, logr" 9.935116. The agreement of these values of r and r" with those previously found, checks the accuracy of the calculation. Further, since the heliocentric longitudes are increasing, the motion is direct. The longitude of the ascending node and the inclination of the orbit may now be found by means of the equations (74), (75), or (76); and we get a = 304~ 43' 11".5, i 64~ 31' 21".7. The values of ue and i," are given by the formulae tan(l —) t ) tan (l"- ) tanu ~ - —, tai u'" cos i cos u and I - S being in the same quadrant in the case of direct motion. Thus we obtain u 1420 52' 12'.4, t"- 153~ 18' 49".4. Then the equation 2 (r" — r cos ('" - t))2 + r sin" (u" - u) gives log x - 9.201423, and the agreement of this value of x with that previously found, proves the calculation of Q, i, A, and u". From the equations 4 / r" tan (45~ + — 0') 1-,/ -1 - si ( u+ )a -tan 20' -- sin ( 1 (,a" + u) - w) - ( V/q ~sin ({u" —' u)V;r' Ic (( + ) sec 20' cos I (u'"- ) V rr' we get 0' -0~ 22' 47".4, = 115~ 40' 6".3, log q = 9.887378. Hence we have X- w - =+ 60~ 23' 17".8, 204 THEORETICAL ASTRONOMY. and v -- u- o 270 12' 6".1, v'" =- - wz= 37 38' 43".1. Then we obtain logm =- 9.9601277 - log q 0.129061 and, corresponding to the values of v and v", Table VI. gives log M — 1.267163, log M" = 1.424152. Therefore, for the time of perihelion passage, we have M T-t - t - 13.74364, and T — t" -- ='- 19.72836. m The first value gives T 1863 Dec. 27.56473, and the second gives T- Dec. 27.56463. The agreement between these results is the final proof of the calculation of the elements from the adopted value of pl P If we find T by means of Table VIII., we have log N= 0.021616, log N" - 0.018210, and the equation 2 2 T — t- 3 N3 r sin v=t" - 3 N"r"2 sin v", in which log - 1.5883273, gives for T the values Dec. 27.56473 and Dec. 27.56469. Collecting together the several results obtained, we have the following elements: T - 1863 Dec. 27.56471 Washington mean time. 60' 23' 17".8 -- 6304 43 1.85 Ecliptic and Mean 7- 304 43 11.5 64 31 21.7 Equinox 1864.0, / - 64 31 21.7) log q 9.887378. 3Motion Direct. 73. The elements thus derived will, in all cases, exactly represent the extreme places of the comet, since these only have been used in finding the elements after p and p" have been found. If, by means NUMERICAL EXAMPLES. 205 of these elements, we compute n and n", and correct the value of 3JT the elements which will then be obtained will approximate nearer the true values; and each successive correction will furnish more accurate results. When the adopted value of 31 is exact, the resulting elements must by calculation reproduce this value, and since the computed values of 2, A", P, and /" will be the same as the observed values, the computed values of )/ and 9' must be such that when substituted in the equation for M, the same result will be obtained as when the observed values of') and f' are used. But, according to the equations (13) and (14), the value of M depends only on the inclination to the ecliptic of a great circle passing through the places of the sun and comet for the time t', and is independent of the angle at the earth between the sun and comet. Hence, the spherical coordinates of any point of the great circle joining these places of the sun and comet will, in connection with those of the extreme places, give the same value of M, and when the exact value of M has been used in deriving the elements, the computed values of 2' and f' must give the same value for w' as that which is obtained from observation. But if we represent by a' the angle at the earth between the sun and comet at the time tt, the values of it derived by observation and by computation from the elements will differ, unless the middle place is exactly represented. In general, this difference will be small, and since w' is constant, the equations cos'= cos /' cos (' - 0), sin' cosw' - cos i' sin (A' - 0'), (93) sin 4' sin w' - sin f', give, by differentiation, cos' d-' = cos w' sec f' d'., d3' = sin w' cos (I' - )') d4'. (94) From these we get cos fi' di' tan (2' - 0!) dfi' sin'' which expresses the ratio of the residual errors in longitude and latitudce, for the middle place, when the correct value of M has been used. Whenever these conditions are satisfied, the elements will be correct on the hypothesis of parabolic motion, and the magnitude of the final residuals in the middle place will depend on the deviation of the actual orbit of the comet from the parabolic form. Further, 206 THEORETICAL ASTRONOMY. when elements have been derived from a value of M which has not been finally corrected, if we compute A' and 9' by means of these elements, and then tan w'- tan f' sin s - (D')' (95) the comparison of this value of tan w' with that given by observation will show whether any further correction of M is necessary, and if the difference is not greater than what may be due to unavoidable errors of calculation, we may regard M as exact. To compare the elements obtained in the case of the example given with the middle place, we find v' 32 31' 13".5, ui' 148~ 11' 19".8, log r' 9.922836. Then from the equations tan (I' — ) = cos itan iu', tan b' tan i sin (t'- g), we derive 1'- 109~ 46' 48".3, b' - 28~ 24' 56".0. By means of these and the values of 0' and R', we obtain' = 302~ 57' 41".1,' = 57 39' 37".0; and, comparing these results with the observed values of A' and /', the residuals for the middle place are found to be Comp. - Obs. cos i AA' -= + 3".6, A - + 1".1. The ratio of these remaining errors, after making due allowance for unavoidable errors of calculation, shows that the adopted'value of M is not exact, since the error of the longitude should be less than that of the latitude. The value of w' given by observation is log tan w' 0.966314, and that given by the computed values of i' and i' is log tan w' = 0.966247. The difference being greater than what can be attributed to errors of calculation, it appears that! the value of M requires further cor NUMERICAL EXAMPLES. 207 rection. Since the difference is small, we may derive the correct value of M by using the same assumed value of -, and, instead of the value of tanw' derived from observation, a value differing as much from this in a contrary direction as the computed value differs. Thus, in the present example, the computed value of log tan w' is 0.000067 less than the observed value, and, in finding the new value of M, we must use log tan w'- 0.966381 in computing P0 and Pri involved in the first of equations (14). If the first of equations (10) is employed, we must use, instead of tan' as derived from observation, tan [' =tan w' sin (A' -'), or log tan A' - 0.966381 - log sin ('- 0') 0.198559, the observed value of A' being retained. Thus we derive log M- 9.829586, and if the elements of the orbit are computed by means of this value, they will represent the middle place in accordance with the condition that the difference between the computed and the observed value of tan w' shall be zero. A system of elements computed with the same data from log 1M- 9.822906 gives for the error of the middle place, C. -O. cos pi a' - - 1' 26".2, a'- -40".1. If we interpolate by means of the residuals thus found for two values of M, it appears that a system of elements computed from log M- 9.829586 will almost exactly represent the middle place, so that the data are completely satisfied by the hypothesis of parabolic motion. The equations (34) and (32) give log - = 0.006955, log N -0.006831, and from (10) we get log M' = 9.822906, log M" = 9.663729,. 208 THEORETICAL ASTRONOMY. Then by means of the equation (33) we derive, for the corrected value of M, log M= 9.829582, which differs only in the sixth decimal place from the result obtained by varying tan w and retaining the approximate values ~t!- Nr"' 74. When the approximate elements of the orbit of a comet are known, they may be corrected by using observations which include a longer interval of time. The most convenient method of effecting this correction is by the variation of the geocentric distance for the time of one of the extreme observations, and the formula which may be derived for this purpose are applicable, without modification, to any case in which it is possible to determine the elements of the orbit of a comet on the supposition of motion in a parabola. Since there are only five elements to be determined in the case of parabolic motion, if the distance of the comet from the earth corresponding to the time of one complete observation is known, one additional complete observation will enable us to find the elements of the orbit. Therefore, if the elements are computed which result from two or more assumed values of D differing but little from the correct value, by comparison of intermediate observations with these different systems of elements, we may derive that value of the geocentric distance of the comet for which the resulting elements will best represent the observations. In order that the formulae may be applicable to the case of any fundamental plane, let us consider the equator as this plane, and, supposing the data to be three complete observations, let A, A', A" be the right ascensions, and D, D', D" the declinations of the sun for the times t, t', t". The co-ordinates of the first place of the earth referred to the third are x = R" cos D" cos A" - R cos D cos A, y R" cos D" sin A" - R cos D sin A, z = " sin D" -- R sin D. If we represent by g the chord of the earth's orbit between the places for the first and third observations, and by G and IK, respectively, the right ascension and declination of the first place of the earth as seen from the third, we shall have x g cos K cos G, y = g cos Ksin G, z = g sin K, VARIATION OF THE GEOCENTRIC DISTANCE. 209 and, consequently, g cos K cos (G - A) = R" cos D" cos (A"- A) - R cos D, g cos K sin (G - A) R" cos D" sin (A" - A), (96) g sin K R" sin D" - R sin D, from which g, K, and G may be found. If we designate by x,, y,, z, the co-ordinates of the first place of the comet referred to the third place of the earth, we shall have x, = cos ao - g cos K cos G, y, ~= D os sin g cos K sin G, z, = si g sin sinK. Let us now put x, - h' cos." cos H',, = h' cos V' sin H', -,- h' sin C', and we get 7' cos V' cos (H' G) = cos s cos(- G) + g cos K, h' cos' sin (H' — G) - cos 8 sin (a -G), (97) h' sin! = d sin +- g sin K, from which to determine H', g', and h'. If we represent by p'1 the angle at the third place of the earth between the actual first and third places of the comet in space, we obtain cos'= cos' cos H' cos a" co o s cos sin H' cos a" sin a"+- sin C' sin a", or cos' =- cos V' cos " cos ("' - H') + sin V' sin a"; (98) and if we put e sinf - sin 8", e cosf cos s 8 cos (a"' -') this becomes cos' = e cos (' -f). (99) Then we shall have 2 = h, + D42 2h_'" cOs' or -2 (" — h' cos'g)2 + h12 sin2, (100) in which J" is the distance of the comet from the earth corresponding to the last observation. We have, also, from equations (44) and (45), r2 _ (J -R cos 4)2 + R2 sin',, rl2 = ('" -- /" cos,"')2 + R'"2 sin12 4~, 14 210 THEORETICAL ASTRONOMY. in which ~ is the angle at the earth between the sun and comet at the time t, and 4"' the same angle at the time t". To find their values, we have cos = cos D cos 8 cos (a -A) + sin D sin, (102) cos "-= cos D" cos a" cos (a"- A") + sin D" sin s", which may be still further reduced by the introduction of auxiliary angles as in the case of equation (98). Let us now put h' sin' - C, h' cos ~' - c, R sin B=, R cos - = b, (103) B" sin 4"= B", " rcos "'= b", and we shall have X /(" -- c) + C2 -r; V - ( - b) + B, (104) r"= l/("~ — b')' + B"2 These equations, together with (56), will enable us to determine J" by successive trials when J is given. We may, therefore, assume an approximate value of a" by means of the approximate elements known, and find r" from the last of these equations, the value of r having been already found from the assumed value of J. Then x is obtained from the equation 2vT, V/r + r" p/ being found by means of Table XI., and a second approximation to the value of /" from a" = G /X2 _ C2. (105) The approximate elements will give a" near enough to show whether the upper or lower sign must be used. With the value of J" thus found we recompute r" and z as before, and in a similar manner find a still closer approximation to the correct value of J". A few trials will generally give the correct result. When A" has thus been determined, the heliocentric places are found by means of the formulae r cos b cos (I - A) =- cos 8 cos (a - A) - R cos D, r'cos b sin ( - A) = A cos 8 sin (a -- A), (106) r sin b =- A sin - R sin D; VARIATION OF THE GEOCENTRIC DISTANCE. 211 r" cos b" cos (" - A") a- S cos " cos (al - A") - R cos D", r" cos b" sin (I" - A") A" cos A" sin (a" - A"), (107) r" sin b" = i" sin " - R" sin D", in which b, b", 1, I" are the heliocentric spherical co-ordinates referred to the equator as the fundamental plane. The values of r and r" found from these equations must agree with those obtained from (104). The elements of the orbit may now be determined by means of the equations (75), (77), and (81), in connection with Tables VI. and VIII., as already explained. The elements thus derived will be referred to the equator, or to a plane passing through the centre of the sun and parallel to the earth's equator, and they may be transformed into those for the ecliptic as the fundamental plane by means of the equations (109),. 75. With the resulting elements we compute the place of the comet for the time t' and compare it with the corresponding observed place, and if we denote the computed right ascension and declination by a0' and 8', respectively, we shall have a +t = a, + d', -, in which a' and d' denote the differences between computation and observation. Next we assume a second value of J, which we represent by a +8J, and compute the corresponding system of elements. Then we have a' + a" = C't' + d" - 0', a" and d" denoting the differences between computation and observation for the second system of elements. We also compute a third system of elements with the distance A - 8A, and denote the differences between computation and observation by a and d; then we shall have a =f(d - ), a =f(D), a" =f(d +- ), and similarly for d, d', and d". If these three numbers are exactly represented by the expression' w h i + x in which- a - x is the general value of the argument, since the values of a, a', and a" will be such that the third differences may be neglected, this formula may be assumed to express exactly any value of the function corresponding to a value of the argument not differing 212 THEORETICAL ASTRONOMY. much from d, or within the limits x = — A and x + -, the assumed values d - 8/, a, and J + 8A being so taken that the correct value of A shall be either within these limits or very nearly so. To find the coefficients mn, n, and o, we have m - n + o ~ a, n -a', om + n +- o - a", whence n = c-, n = (a"- -a), o - (a" + a) -a. Now, in order that the middle place may be exactly represented in right ascension, we must have ( ) +n( - ) -m -O, from which we find x _ _ -_ (n - (Ve- 4mo) =p, di 20 or x -p8a = 0. In the same manner, the condition that the middle place shall be exactly represented in declination, gives x — p'ad - 0. In order that the orbit shall exactly represent the middle place, both conditions must be satisfied simultaneously; but it will rarely happen that this can be effected, and the correct value of x must be found from those obtained by the separate conditions. The arithmetical mean of the two values of x will not make the sum of the squares of the residuals a minimum, and, therefore, give the most probable value, unless the variation of cos 8' tao', for a given increment assigned to A, is the same as that of A8'. But if we denote the value of x for which the error in a' is reduced to zero by x', and that for which A- =0, by x", the most probable value of x will be n2X + -nf 2X" x_ + 12, (108) in which n (= (a - a) and n' = (d" — d). It should be observed that, in order that the differences in right ascension and declination shall have equal influence in determining the value of x, the values of a, a', and a" must be multiplied by cos o'. The value of 8A is most conveniently expressed in units of the last decimal place of the logarithms employed. NUMERICAL EXAMPLE. 213 If the elements are already known so approximately that the first assumed value of J differs so little from the true value that the second differences of the residuals may be neglected, two assumptions in regard to the value of J will suffice. Then we shall have o = 0, and hence mnz- a, n - a" a' The condition that the middle place shall be exactly represented, gives the two equations (a"- a') x + a'd O0, (d"- d')x + d'cJ =- 0. The combination of these equations according to the method of least squares will give the most probable value of x, namely, that for which the sum of the squares of the residuals will be a minimum. Having thus determined the most probable value of x, a final system of elements computed with the geocentric distance a + x, corresponding to the time t, will represent the extreme places exactly, and will give the least residuals in the middle place consistent with the supposition of parabolic motion. It is further evident that we may use any number of intermediate places to correct the assumed value of A, each of which will furnish two equations of condition for the determination of x, and thus the elements may be found which will represent a series of observations. 76. EXAMPLE.-The formulae thus derived for the correction of approximate parabolic elements by varying the geocentric distance, are applicable to the case of any fundamental plane, provided that a,, A, D, &c. have the same signification with respect to this plane that they have in reference to the equator. To illustrate their numerical application, let us take the following normal places of the Great Comet of 1858, which were derived by comparing an ephemeris with several observations made during a few days before and after the date of each normal, and finding the mean difference between computation and observation: Washington M. T. a d 1858 June 11.0 141~ 18' 30".9 + 24~ 46' 25".4, July 13.0 144 32 49.7 27 48 0.8, Aug. 14.0 152 14 12.0 + 31 21 47.9, which are referred to the apparent equinox of the date. These places are free from aberration. 214 THEORETICAL ASTRONOMY. We shall take the ecliptic for the fundamental plane, and converting these right ascensions and declinations into longitudes and latitudes, and reducing to the ecliptic and mean equinox of 1858.0, the times of observation being expressed in days from the beginning of the year, we get t - 162.0, A = 1350 51' 44".2, =+ 90 6' 57".8, t' 194.0, )' -137 39 41.2,' = 12 55 9.0, t" 226.0, " — 142 51 31.8, P " + 18 36 28.7. From the American Nautical Almanac we obtain, for the true places of the sun, o = 80~ 24' 32".4, logR 0.006774, 0' -110 55 51.2, log R' 0.007101, Q" = 141 33 2.0, log R" 0.005405, the longitudes being referred to the mean equinox 1858.0. When the ecliptic is the fundamental plane, we have, neglecting the sun's latitude, D- 0, and we must write 2 and P in place of o and A, and 0 in place of A, in the equations which have been derived for the equator as the fundamental plane. Therefore, we have g cos (G - ) R" cos (0" - 0) -- R, g sin (G - ) = R" sin(0" - 0); cos 4 =C c os f (- c 4 " os (A " -- O") R cos - b, R" cos A"= b", R sin - B=, " sin " = B", from which to find G, g, b, B, b", and Bi, all of which remain unchanged in the successive trials with assumed values of A. Thus we obtain G 2010 7' 57".4, log B -9.925092, b + 0.568719, log g = 0.013500, log B" = 9.510309, b" = 0.959342. Then we assume, by means of approximate elements already known, log A == 0.397800, and' from h' cos V' cos (H' - G) - cos C cos (A - G) + g, h' cos C' sin (H' - G) = cost sin (A - G), h' sin' = A sin f, we find H', C', and'. These give H' = 153~ 46' 20".5, = +- 7~ 24' 16".4, log h' 0.487484. NUMERICAL EXAMPLE. 215 Next, from cos P'- cos' os s " cos (A" - H') + sin' sin f", h' cos ('- c, h'sin'=' C, we get log C= 9.912519, c =-+ 2.961673; and from r = ( -- b)2 + B2, we find log r - 0.323446. Then we have a" = ~ V X2 C2 r"- (" - b) + B"2, 2~ 2~ ar' - k (e'l —), (r + ")' X 1 (r + r")/ r -t- r'; from which to find z", r", and x. First, by means of the approximate elements, we assume log 4" -0.310000, which gives log r" = 0.053000, and hence we have - 0.3783, log p- = 0.002706, log x = 0.090511. With this value of x we obtain from the expression for a", the lower sign being used, since A" is less than c, log " - 0.309717. Repeating the calculation of r", J,6 and x, and then finding d" again, the result is log D" = 0.309647. Then, by means of the formula (67), we may find the correct value. Thus we have, in units of the sixth decimal place, a - 309717 - 310000 - - 283, a' = 309647 - 309717 — 70, and for the correction to the last result for log A" we have ~/2 aa a -23. Therefore, log 4" = 0.309624. By means of this value we get log r" = 0.052350, log x = 0.090628, 216 THEORETICAL ASTRONOMY. and this value of x gives, finally, log i" -- 0.309623, log r" - 0.052348. The heliocentric places of the comet are now found from the equations (71) and (72), writing J cosf and A" cos fi for p and p", respectively. Thus we obtain I = 159~ 43' 14".2, b -+ 10~ 50' 14".0, log r 0.323447, r" 144 17 47.8, b" + 35 14 28.7, log r" =0.052347. The agreement of these results for r and r" with those already obtained, proves the accuracy of the calculation. Since the heliocentric longitudes are diminishing, the motion is retrograde. Then from (74) we get a _ 165~ 17' 30".3, i 63~ 6' 32".5; and from tan (1 tan (- Q) tan u" - tan t(l- an cos i cos i we obtain u = 12~ 10' 12".6, u" =40~ 18' 51".2, the values of - t and I1- 2 being in the same quadrant when the motion is retrograde. The equation (79) gives log - 0.090630, which agrees with the value already found. The formulae (81) give w 129~ 6' 46".3, log q 9.760326, and hence we have - u - - -116~ 56' 33".7, v" = u" - - 88~ 47' 55".1, from which we get T= 1858 Sept. 29.4274. From these elements we find log r' = 0.212844, v' =- 107~ 7' 34".0, u' = 21~ 59' 12".3, and from tan (I' - ) - cos i tan u', tan b'= - tan i sin (I'- ), we get i' 154~ 56' 33".4, b' - + 19~ 30' 22".1. NUMERICAL EXAMPLE. 217 By means of these and the values of 0' and R', we obtain'- =137~ 39' 13".3, f' - + 12~ 54' 45".3, and comparing these results with observation, we have, for the error of the middle place, C. -O. cos i' A' — 27".2, fP' - 23".7. From the relative positions of the sun, earth, and comet at the time t" it is easily seen that, in order to diminish these residuals, the geocentric distance must be increased, and therefore we assume, for a second value of /, log D = 0.398500, from which we derive H' - 153~ 44' 57".6, V' + 7~ 24' 26".1, log h' = 0.488026, log C- 9.912587, log c 0.472115, logr -0.324207, log A" - 0.311054, log r" = 0.054824, log X - 0.089922. Then we find the heliocentric places I -159~ 40' 33".8, b += 10~ 50' 8".6, logr -0.324207, 1" 144 17 12.1, b" =-+ 35 8 37.8, log r" =- 0.054825, and from these, = 165~ 15' 41".1, i 63~ 2' 49".2, ut 12 10 30.8, " — 40 13 26.0, w 128 54 44.4, logq 9.763620, T=- 1858 Sept. 29.8245, log r'-0.214116, v' - 106~ 55' 43".8, u' 21~ 59' 0".6, l'f- 154 53 32.3, b' = 19 29 31.9, i'= 137 39 39.7, - =+12 55 2.9. Therefore, for the second assumed value of d, we have C. O. cos A'' — 1".5, A4' -6".1. Since these residuals are very small, it will not be necessary to make a third assumption in regard to A, but w3 may at once derive the correction to be applied to the last assumed value by means of the equations (109). Thus we have a' -1.5, a" = - 27.2, d' - 6.1, d"- 23.7, ~ log z = - 0.000700, 218 THEORETICAL ASTRONOMY. and, expressing ~ log A in units of the sixth decimal place, these equations give 25.7x - 1050 - 0. 17.6x - 4270 - 0. Combining these according to the method of least squares, we get 105 X 2.57 + 427 X 1.76 _ (2.57)2 + (1.76)2 + Hence the corrected value of log J is log J = 0.398500 + 0.000106 - 0.398606. With this value of log J the final elements are computed as already illustrated, and the following system is obtained:T- 1858 Sept. 29.88617 Washington mean time. -= 36~ 22' 36".9 ) o-165 15 24.8 a 165 15 24.8 A Mean Equinox 1858.0. i 63 2 14.2 log q = 9.764142 Motion Retrograde. If the distinction of retrograde motion is not adopted, and we regard i as susceptible of any value from 0~ to 1800, we shall have - 294~ 8' 12".7, i=116 57 45.8, the other elements remaining the same. The comparison of the middle place with these final elements gives the following residuals:C.-O. cos f AA = + 0".2, Ai - 4".3. These errors are so small that the orbit indicated by the observed places on which the elements are based differs very little from a parabola. When, instead of a single place, a series of intermediate places is employed to correct the assumed value of X, it is best to adopt the equator as the fundamental plane, since an error in oc or a will affect both A and 9; and, besides, incomplete observations may also be used NUMERICAL EXAMPLE. 219 when the fundamental plane is that to which the observations are directly referred. Further, the entire group of equations of condition for the determination of x, according to the formulse (109), must be combined by multiplying each equation by the coefficient of x in that equation and taking the sum of all the equations thus formed as the final equation from which to find x, the observations being supposed equally good. 220 THEORETICAL ASTRONOMY. CHAPTER IV. DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF THE CONIC SECTION. 77. THE formule which have thus far been derived for the determination of the elements of the orbit of a heavenly body by means of observed places, do not suffice, in the form in which they have been given, to determine an orbit entirely unknown, except in the particular case of parabolic motion, for which one of the elements becomes known. In the general case, it is necessary to derive at least one of the curtate distances without making any assumption as to the form of the orbit, after which the others may be found. But, preliminary to a complete investigation of the elements of an unknown orbit by means of three, complete observations of the body, it is necessary to provide for the corrections due to parallax and aberration, so that they may be applied in as advantageous a manner as possible. When the elements are entirely unknown, we cannot correct the observed places directly for parallax and aberration, since both of these corrections require a knowledge of the distance of the body from the earth. But in the case of the aberration we may either correct the time of observation for the time in which the light from the body reaches the earth, or we may consider the observed place corrected for the actual aberration due to the combined motion of the earth and of light as the true place at the instant when the light left the planet or comet, but as seen from the place which the earth occupies at the time of the observation. When the distance is unknown, the latter method must evidently be adopted, according to which we apply to the observed apparent longitude and latitude the actual aberration of the fixed stars, and regard this place as corresponding to the time of observation corrected for the time of aberration, to be effected when the distances shall have been found, but using for the place of the earth that corresponding to the time of observation. It will appear, therefore, that only that part of the calculation of the DETERMINATION OF AN ORBIT. 221 elements which involves the times of observation will have to be repeated after the corresponding distances of the body from the earth have been found. First, then, by means of the apparent obliquity of the ecliptic, the observed apparent right ascension and declination must be converted into apparent longitude and latitude. Let'2 and p, respectively, denote the observed apparent longitude and latitude; and let () be the true longitude of the sun, 20 its latitude, and Ro its distance from the earth, corresponding to the time of observation. Then, if 2 and 13 denote the longitude and latitude of the planet or comet corrected for the actual aberration of the fixed stars, we shall have A - A - +- 20".445 cos (A - O) sec + - 0".343 cos (A - 281~) sec, (1) pf- -— = 20".445 sin (A - 0o) sin f - 0".343 sin (A - 281~) sin fl. In computing the numerical values of these corrections, it will be sufficiently accurate to use 20 and o instead of 2 and 9 in the second members of these equations, and the last terms may, in most cases, be neglected. The values of) and & thus derived give the true place of the body at the time t - 497s.784, but as seen from the place of the earth at the time t. When the distance of the planet or comet is unknown, it is impossible to reduce the observed place to the centre of the earth; but if we conceive a line to be drawn from the body through the true place of observation, it is evident that were an observer at the point of intersection of this line with the plane of the ecliptic, or at any point in the line, the body would be seen in the same direction as from the actual place of observation. Hence, instead of applying any correction for parallax directly to the observed apparent place, we may conceive the place of the observer to be changed from the actual place to this point of intersection with the ecliptic, and, therefore, it becomes necessary to determine the position of this point by means of the data furnished by observation. Let 00 be the sidereal time corresponding to the time to of observation, (p' the geocentric latitude of the place of observation, and po the radius of the earth at the place of observation, expressed in parts of the equatorial radius as unity. Then 00 is the right ascension and 5p' the declination of the zenith at the time t0. Let 1o and 0b denote these quantities converted into longitude and latitude, or the longitude and latitude of the geocentric zenith at the time to. The rectangular co-ordinates of the place of observation referred to the centre of the 222 THEORETICAL ASTRONOMY. earth and expressed in parts of the mean distance of the earth from the sun as the unit, will be -- po sin ro cos bo cos Io, yo =-Po sin r0 cos bo sin lo, o = Po sin r sin bo, in which ro 8t'.57116.1 Let Jo be the distance of the planet or comet from the true place of the observer, and J, its distance from the point in the ecliptic to which the observation is to be reduced. Then will the co-ordinates of the place of observation, referred to this point in the ecliptic, be x, - (A, - o) cos i cos ), y, = (d, - ) cos I sin A, (D, - do) sin, the axis of x being directed to the vernal equinox. Let us now designate by ( the longitude of the sun as seen from the point of reference in the ecliptic, and by R its distance from this point. Then will the heliocentric co-ordinates of this point be X — - R cos Q, Y —- R sin 0, Z=-0. The heliocentric co-ordinates of the centre of the earth are X-= - Ro cos 20 cos 00, Yo- - o cos 0 sin 0o, Zo = - Ro sin o0. But the heliocentric co-ordinates of the true place of observation will be X - x,, Y + y,, Z+ z,, or Xo +.0, YO + yO, Zo + zo, and, consequently, we shall have B cos 0 (, - Jo) cos f cos Rt = R cos 2O cos o - po sin r cos bo cos 1o, R sin ( - (, - o) cos sin A = Ro cos I, sin 30 - op sin r0 cos bo sin 1,, - (, - Jo) sin f = Ro sin A0 -po sin rO sin bo. If we suppose the axis of x to be directed to the point whose longitude is (O, these become DETERMINATION OF AN ORBIT. 223 R cos (0 - 0o) - (D, - o) cos A cos (s - o) - Ro cos Y - po sin cos cos (lo - o), R sin (( - 0() - (D, - 4) cos sin ( -- o) = (2) - p sin nr cos bo sin (1o - (o), - (D, - Do) sin 3 = R, sin -~ po sin r sin b,, from which R and D may be determined. Let us now put (J, -J) cos D; (3) then, since;, S, J and O - o 0 are small, these equations may be reduced to R = D cos (A - (o) -, p cos b, cos (lo - 0() + Ro, R (( - ) D sin ( - 0) — po cos bo sin (lo - 0), - D tan --, Po sin b, + Ro 0. Hence we shall have, if %O and -0 are expressed in seconds of arc,. o Posin bo, - Ro ocot, 206264.8 70p7 cos b cos (l —- O$) R = Ro + D cos (A - 0o)- "0 Po 0- (4) ~206264.8 ( 206264.8 D sin (A - (D) - o po cos bo sin (o — 0o) from which we may derive the values of ( and R which are to be used throughout the calculation of the elements as the longitude and distance of the sun, instead of the corresponding places referred to the centre of the earth. The point of reference being in the plane of the ecliptic, the latitude of the sun as seen from this point is zero, which simplifies some of the equations of the problem, since, if the observations had been reduced to the centre of the earth, the sun's latitude would be retained. We may remark that the body would not be seen, at the instant of observation, from the point of reference in the direction actually observed, but at a time different from to, to be determined by the interval which is required for the light to pass over the distance d, - Jo. Consequently we ought to add to the time of observation the quantity (J, - JO) 4978.78 = 4978.78 D sec f, (5) which is called the reduction of the time; but unless the latitude of the body should be very small, this correction will be insensible. The value of i derived from equations (1) and the longitude 0 224 THEORETICAL. ASTRONOMY. derived from (4) should be reduced by applying the correction for nutation to the mean equinox of the date, and then both these and the latitude j9 should be reduced by applying the correction for precession to the ecliptic and mean equinox of a fixed epoch, for which the beginning of the year is usually chosen. In this way each observed apparent longitude and latitude is to be corrected for the aberration of the fixed stars, and the corresponding places of the sun, referred to the point in which the line drawn from the body through the place of observation on the earth's surface intersects the plane of the ecliptic, are derived from the equations (4). Then the places of the sun and of the planet or comet are reduced to the ecliptic and mean equinox of a fixed date, and the results thus obtained, together with the times of observation, furnish the data for the determination of the elements of the orbit. When the distance of the body corresponding to each of the observations shall have been determined, the times of observation may be corrected for the time of aberration. This correction is necessary, since the adopted places of the body are the true places for the instant when the light was emitted, corresponding respectively to the times of observation diminished by the time of aberration, but as seen from the places of the earth at the actual times of observation, respectively. When 9 =- 0, the equations (4) cannot be applied, and when the latitude is so small that the reduction of the time and the correction to be applied to the place of the sun are of considerable magnitude, it will be advisable, if more suitable observations are not available, to neglect the correction for parallax and derive the elements, using the uncorrected places. The distances of the body from the earth which may then be derived, will enable us to apply the correction for parallax directly to the observed places of the body. When the approximate distances of the body from the earth are already known, and it is required to derive new elements of the orbit from given observed places or from normal places derived from many observations, the observations may be corrected directly for parallax, and the times corrected for the time of aberration. We shall then have the true places of the body as seen from the centre of the earth, and if these places are adopted, it will be necessary, for the most accurate solution possible, to retain the latitude of the sun in the formulae which may be required. But since some of these formulae acquire greater simplicity when the sun's latitude is not introduced, if, in this case, we reduce the geocentric places to the DETERMINATION OF AN ORBIT. 225 point in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane, the longitude of the sun will remain unchanged, the latitude will be zero, and the distance P will also be unchanged, since the greatest geocentric latitude of the sun does not exceed 1"'. Then the longitude of the planet or comet as seen from this point in the ecliptic will be the same as seen from the centre of the earth, and if J, is the distance of the body front this point of reference, and, its latitude as seen from this point, we shall have A, cos fi, - d cos f3, A, sin 3, -- A sin / - ~O sin z, from which we easily derive the correction 3, - 3, or AS, to be applied to the geocentric latitude. Thus, we find i 0 c 2o, A?=- D~ cosA, (6) ~o being expressed in seconds. This correction having been applied to the geocentric latitude, the latitude of the sun becomes z 0.. The correction to be applied to the time of observation (already diminished by the time of aberration) due to the distance A,- i will be afbsoltely insensible, its maximum value not exceeding 0'.002. It should be remarked also that before applying the equation (6), the latitude'0 should be reduced to the fixed ecliptic which it is desired to adopt for the definition of the elements which determine the position of the plane of the orbit. 78. When these preliminary corrections have been applied to the data, we are prepared to proceed with the calculation of the elements of the orbit, the necessary formula for which we shall now investigate. For this purpose, let us resume the equations (6)3; and, if we multiply the first of these equations by tan /9 sin )" - tan f" sin 2, the second by tan 13" cosI - tan 9 cos 2", and the third by sin (2 - 2"), and add the products, we shall have 0 nR (tan p" sin ( - 0) - tan fi sin (" - 0)) - p' (tan f sin ("-,') - tan fi' sin (" -- ) + tan f" sin (A' - 2)) -R' (tan f" sin (A - 0') tan fi sin (" -- 0')).7) t n"R" (tan [," sin (2 - 0") - tan fi sin ("- - 0")). It should be observed that when the correction for parallax is applied 15 226 THEORETICAL ASTRONOMY. to the place of the sun, pt is the projection, on the plane of the ecliptic, of the distance of the body from the point of reference to which the observation has been reduced. Let us now designate by Kthe longitude of the ascending node, and by I the inclination to the ecliptic, of a great circle passing through the first and third observed places of the body, and we h:l-ve tan fi = sin ( - K) tan,. (8) tan i" - sin (" - K) tan:L.- Introducing these values of tan: and tan:/1 into the equation (7), since sill (A - 0) sin (A" - K) - sin (A" - 0) sin (A - K) sin (" - A) sin (0 -K), sin (R' - ) sin (A" - K) + sin (A" - A' sin ( -- K) = + sin (A" - 2) sin (A' -- ), sin (A -') sin (A"- K)-sin (2"-0') sin ( -K) - sin (A" - A) sin (0' - K), sin (A - 0") sin (A" - K) - sin (A"- 0") sin (2 - K) = - sin (A" - A) sin (0"- K), we obtain, by dividing through by sin (2" - 2) tan I, 0 -- nR sin (( - K) + p' (sin ('- K) - tan i' cot) I) -' sin (0'- K) + ""R" sin (0" —K). Let. /0 denote the latitude of that point of the great circle passing through the first and third places which corresponds to the longitude 2', then tan i = sin (A' - K) tan I, and the coefficient of p' in equation (9) becomes sin (1 -') ) cos rio cos' tan I Therefore, if we put _sin (f'- io) a- cos i tan I' (10) we shall have R' sin(' —) _) Rsin(s -K) / sec - -+ n ( ao ao,,ao a0 (11) + R" sin ("- K) + 9~L.,! ao This formula will give the value of p', or of A', when the values of nW and n" have been determined, since a0 and K are derived from the data furnished by observation. DETERMINATION OF AN ORBIT. (227 To find K and 1, we obtain from equations (8) by a transformation precisely similar to that by which the equations (75)3 were derived, sin (Qi"+ f3) tan I sin Ge (A" + 2) -- K)2 cos' cOs sec 2 - ), tan I cos (- (A" + A) - K) 2 cs c ose We may also compute K and I from the equations which may be derived from (74)3 and (76)3 by making the necessary changes in the notation, and using only the upper sign, since T is to be taken always less than 9a~. Before proceeding further with the discussion of equation (11), let us derive expressions for p and p" in terms of o', the signification of p and p", when the corrections for parallax are applied to the places of the sun, being as already noticed in the case of p'. 79. If we multiply the first of equations (6)3 by sin 0" tan i", the second by - cos 0" tan i/', and the third by sin (" — O"), and add the products, we get 0=np (tan A" sin (0"-) )-tanf sin (0"- ")) —Rtan " sin (0"- 0) — p (tan i" sin ( " —')-tan f' sin ( " —")~)+R' tan A" sin (0"O'), (13) which may be written Omnp (tan l sin (A"- ")-tan " sin (2-GO "))nR tan i" sin (0"-0) + p' (tan r" sil ('- - 0") tan 0o sil (A" O ")) p' (tan f' - tan Si) sill (,"- (") +- R' tan f" sin (O"- 0'). Introducing into this the values of tall p, tall /9", and tan 90 in terms of I and K, and reducing, the result is 0 - np sin (2"- A) sin ( 0 "- K) - nR sin (O "- 0) sin (A"- K) - p' sin ("-2A') sil (O "- K) - p'a0 sec i' sin (A"- 0 ") - R' sin (O " — O') sin (A"- K). Therefore we obtain K (llp' " ) ase sin (A" - ") \ -~ sin (2" -) sin (2" -) ) sin ()" - K) sin (A" —K) R' sill (0 "- O') —n sin (O "-) ) s' sin (" - A) sin (0"- K) But, by means of the equations (9)3, we derive R' sin (0"-') - nR sin((0"- 30) = (N- n) R sin (0"- 0(), 228 THEORETICAL ASTRONOMY. and the preceding equation reduces to p'sin("-A') aosec' sin(A"- ) \ ) P - ~ e sin ("\ - A) si n (iA" — A) sin ((" - K) )(14 + 1 X R sin ((" — ) sin (" --- K) "( sil (A" sin (( ) sin( - K) To obtain an expression for p" in terms of pto if we multiply the first of equations (6)3 by sin (0 tan i, the second by - cos o tan, and the third by sin (A- 0), and add the products, we shall have O=n"p" (tan ft sin (2" —- ) —tan fi" sin (2 —0)) — n"R" tan sin (0"- 0) -p'(tan Psin (' — ) —tan' sin (A — ))+R'tan ( sin ('- 0). (15) Introducing the values of tan 9, tan j', and tan j9" in terms of K and I; and reducing precisely as in the case of the formula already found for p, we obtain,,__' ( sin' -) acsec' sin (A ) n"~ sin (A" - ) sin (A"- ) sin (( - K) /+ (1 N" I R" sin (0 "-i- ) sin( (- K)'\ n" 1 / sill ("- A) sin ( -K) Let us now put, for brevity, R sin (- K) R' sin ('K) b, C -- aO ao d " sin (" - K) see p' _ R" sin (E"- 0) a0~ ^' ^ sin("- 2)' a0 sin - A2) sin(2"-') rfR sin(2"- 0") sin ("- 2) - (17) si (n' - A) -R sin (2 -0 i)' sin (A"l- 2) V +sf(^-^ sin ~Q) sin( ~ b h sil ( —K) 1_ sinin (2 - K) 2 d 2 b and the equations (11), (14), and (16) become p' see f' -- c +- nb + n"d, pM- = i+M 1- 9l (18) If n and n" are known, these equations will, in most cases, be sufficient to determine p, p', and p". DETERMINATION OF AN ORBIT. 229 80. It will be apparent, from a consideration of the equations which have been derived for p, p', and p", that under certain circumstances they are inapplicable in the form in which they have been given, and that in some cases they become indeterminate. When the great circle passing through the first and third observed places of the body passes also through the second place, we have a- 0, and equation (11) reduces to n"Rs" sin (" - K) + nR sin ( - K) R' sin (' K). If the ratio of n" to n is known, this equation will determine the quantities themselves, and from these the radius-vector r' for the middle place may be found. But if the great circle which thus passes through the three observed places passes also through the second place of the sun, we shall have K= (', or K= 180~+ 0', and hence n"R" sin (0"- 0) - niR sin (0' 0) = 0, or n" R sin ((D' — O) n - R" sin ("-') from which it appears that the solution of the problem is in this case impossible. If the first and third observed places coincide, we have A = A" and 9 = " -, and each term of equation (7) reduces to zero, so that the problem becomes absolutely indeterminate. Consequently, if the data are nearly such as to render the solution impossible, according to the conditions of these two cases of indetermination, the elements which may be derived will be greatly affected by errors of observation. If, however, i is equal to A" and i"1 differs from i9, it will be possible to derive p', and hence o and p"; but the formulae which have been given require some modification in this particular case. Thus, when A =", we have K= A"' 2, I-=90~, and /0 900, and hence a, as determined by equation (10), becomes -. Still, in this case it is not indeterminate, since, by recurring to the original equation (9), the coefficient of p', which is - a0 sec t', gives a0 sin:' cot I — cos j' sin (2' - K), (19) and when A = ", it becomes simply a0o= - cos i' sin ('- K). 230 THEORETICAL ASTRONOMY. Whenever, therefore, the difference 2" — is very small compared with the motion in latitude, a0 should be computed by means of the equation (19) or by means of the expression which is obtained directly from the coefficient of o' in equation (7). When 2A-" =-K, the values of JiJ, JillM, JA2, and.-12" cannot be found by means of the equations (17); but if we use the original form of the expressions for p and p" in terms of p', as given by equations (13) and (15), without introducing the auxiliary angles, we shall have p' tan i' sin ("- 0") - tan i" sin (2' - 0") an tan 3" sin( ") tan sin ( -- 0") N/ L 1\ _ R tan Fi" sin ((D"- )) n tan Af sin (" - 0") - tan fA" sin ( - 0")', p"' tan / sin ('- 0) - tan ft' sin ( - 0) I " tan f sin ("- 0) - tan A" sin ( - 0) / N"\ R" tan f sin ("- 0) - 1n" tan i sin (A" 0) - tanl sin (- 0)' Hence tan /' sin (A" - (") - tan i" sin (2'- 0") tan i sin (,"- 0") tan f" sin ( - 0")' tan: sin (' — 0) -taln' sin (2 - 0) tan f5 sin (" 0) -- taln sin (2 0)' (20 _M-2 R tan /" sin (0" - 0) 2 tan f sin (A" - ") -- tan fi" sin (A - 0")' tn _ __ R" tan sin (0)"- 0) 5 tan sin (2" --- 0) - tal f" sin ( - 0)' are the expressions for It, 1M,", M2, and 312"' which must be used when 2 =2" or when 2 is very nearly equal to 2"; and then p and o" will be obtained from equations (18). These expressions will also be used when A" — A 180~, this being an analogous case. When the great circle passing through the first and third observed places of the body also passes through the first or the third place of the sun, the last two of the equations (18) become indeterminate, and other formula must be derived. If we multiply the second of equations (7)3 by tan 9" and the fourth by -sin(/l"- 0'), and add the products, then multiply the second of these equations by tan 9 and the fourth by -sin (A- 0'), and add, and finally reduce by means of the relation NR sin (0' - 0) _ N"R" sin (0"- 0'), we get DETERMINATION OF AN ORBIT. 231 p' tan fi" sin (' - 0') - tan if' sin (A" ~-') (n" Nil\ iR" tan i" sin (Q" — 0')'+\ - N - tan f sin (A 0(D') - tan ft sill ( V" Y 0')' p tan' sin (A - O') - tan f sin (' - 0') (21 - pn"' tan A" sin (A -') - tan l3 sin (A" - 0') t N R\ R tanf sin ((D' (- ) a " sN"' talln sin ( -')-tan sin((A" —') These equations are convenient for determining p ando p from p'; but they become indeterminate when the great circle passing through the extreme places of the body also passes through the second place of the sun. Therefore they will generally be inapplicable for the eases in which the equations (18) fail. If we eliminate p" from the first and second of the equations (6)3 we get 0 - np sin (" - A) - nR sin (" - 0) p' sin (A" - A') + R' sin (A" - O')-n"R" sin (A" 0"), from which we derive p' sin (-') (22).Y sin(A" - A) +nI sin (A" 0) R' sin (A" - G') + n"R" sin (A" - (") n sin (" - A) Eliminating o between the same equations, the result is p' sin (A'-A) " sin A) (23) ~0l —, sin (I" — ~) nR sin (A - 0) - R' sin (A - 0') + — n"R" sin (A - 0") in" sin (A" - A) These formulae will enable us to determine p and ot" from p' in the special cases in which the equations (18) and (21) are inapplicable; but, since they do not involve the third of equations (6)3, they are not so well adapted to a complete solution of the problem as the formula previously given whenever these may be applied. If we eliminate successively p"t and p between the first and fourth of the equations (7)3, we get _p' tan f" cos (A'-') - tan i' cos (A" - 0') n' tan i" cos (A (') -tan f cos (A"-') tan 1" nR cos (0' — Q )- R' + n"-R" cos (("- 0') n+ tan f" cos (A -') -tan f cos (A" - 0')',, p' tan j,' cos (A -') - tan i cos (A'- Q') n' tan fi" cos (A -') - ta.n 1 cos (A" (0') tan nR cos (' - 0) - R' +- "R"cos (" - 0') - n"' tan f" cos ( -') - tan fi cos (A" - 0') 232 THEORETICAL ASTRONOMY. which may also be used to determine p and p" when the equations (18) and (21) cannot be applied. When the motion in latitude is greater than in longitude, these equations are to be preferred instead of (22) and (23.) 81. It would appear at first, without examining the quantities involved in the formula for p', that the equations (26)3 will enable us to find n and n" by successive approximations, assuming first that and from the resulting value of' determining r and then carrying and from the resulting value of,o determining rt, and then carrying the approximation to the values of n and n" one step farther, so as to include terms of the second order with reference to the intervals of time between the observations. But if we consider the equation (10), we observe that a0 is a very small quantity depending on the difference / ~ —3, and therefore on the deviation of the observed path of the body from the arc of a great circle, and, as this appears in the denominator of terms containing n and n" in the equation (11), it becomes necessary to determine to what degree of approximation these quantities must be known in order that the resulting value of p' may not be greatly in error. To determine the relation of a. to the intervals of time between the observations, we have, from the coefficient of o' in equation (7), ao see c' - tan f sin (A" -') - tan A' sin (A" - 2) + tan:" sin (A' - A). We may put tan f = tan'- Ar" + Br"2...., tan = tan + AT + B"...., and hence we have ao sec l3' (sin (i" - ( 2) - sin (A" -A) + sin (A' - )) tan r' + (r sin (2' —) - -' sin (2" —A')) A-+(-tsin (A'2)' sin ((" —.')) B+.., which is easily transformed into ao sec r'- 4 sin. (A'- ) sin 2 (A" - 2') sin A (A"- ) tan r' (25) + ( (r ) —sin ('-)-" sin (A"'))A+(ir sin (A'- )+-" sin ("2'))B+.. If we suppose the intervals to be small, we may also put sin ( ( - A), and sin ( A) ~ Af - A, sin (A' - ) = A' -. DETERMINATION OF AN ORBIT. 233 Further, we may put A =' -' A'rf + B'fr2...., A - A + A'r + B'r' +.... Substituting these values in the equation (25), neglecting terms of the fourth order with respect to r, and reducing, we get ao — Tr'r" (A'" tan f' + A'B - AB') cos pi It appears, therefore, that a0 is at least of the third order with reference to the intervals of time between the observations, and that an error of the second order in the assumed values of n and n" may produce an error of the order zero in the value of p' as derived from equation (11) even under the most favorable circumstances. Hence, in general, we cannot adopt the values Ir A~t! n = ~,) n' -,' omitting terms of the second order, without affecting the resulting value of p' to such an extent that it cannot be regarded even as an approximation to the true value; and terms of at least the second order must be included in the first assumed values of n and n". The equation (28)3 gives -- 22 ^2__.i2 -— 1 + 61 T23 (26) n 2 \ rJ3 dr' omitting the term multiplied by dt, which term is of the third order with respect to the times; and hence in this value of -i only terms 7f of at least the fourth order are neglected. Again, from the equations (26)3 we derive, since r' -r - r", n- +n" 1 + 2r (27) in which only terms of the fourth order have been neglected. Now the first of equations (18) may be written: 91// b+-d / see 13' = (0i + n") - c, (28) n11+t in which, if we introduce the values of ~- and n + -nZ as given by (26) and (27), onl term o the fourt rde ith respet to the (26) and (27), only terms of the fourth order with respect to the 234 THEORETICAL ASTRONOMY. times will be neglected, and consequently the resulting value of o' will be affected with only an error of the second order when a, is of the third order. Further, if the intervals between the observations are not very unequal, ~T2 _'l2 will be a quantity of an order superior to T2, and when these intervals are equal, we have, to terms of the fourth order,,n T, n 7 The equation (27) gives 2r'3 (n + n"- ) - rt". Hence, if we put P~ -- a~~sn~~, ~(29) Q 2r'3 (n + n -- 1), we may adopt, for a first approximation to the value of p', ITt P =- Q — e (30) and p' will be affected with an error of the first order when the i-ntervals are unequal; but of the second order only when the intervals are equal. It is evident, therefore, that, in the selection of the observations for the determination of an unknown orbit, the intervals should be as nearly equal as possible, since the nearer they approach to equality the nearer the truth will be the first assumed values of P and Q, thus facilitating the successive approximations; and when a0 is a very small quantity, the equality of the intervals is of the greatest importance. From the equations (29) we get +P( +2'), (31) n" - nP; and introducing P and Q in (28), there results _ r i Q 9 \b+Pd P see i= I + 2)- - (32) This equation involves both o' and r' as unknown quantities, but by means of another equation between these quantities p' may be eliminated, thus giving a single equation from which r' may be found, after which p' may also be determined. DETEIRMINATION OF AN ORBIT. 235 82. Let,t' represent the angle at the earth between the sun and planet or comet at the second observation, and we shall have, from the equations (93)3, tan i tan w' -s (tasin ('-')' tan - (- ) (33) COS WO~ cos - = cos f' cos (' - 0'), by means of which we may determine', which cannot exceed 180. Since cos y' is always positive, cos 4' and cos(,'- 0') must have the same sign. We also have r2._ J2 + R12 2J'R' cos 4', which may be put in the form 2 (p' sec i' - R' cos 4)2 + R"2 sin2 2', from which we get p' sec 1i'= R' cos' l/1'r'"- R' sin2 4'. (34) Substituting for p' sec j' its value given by equation (32), we have ( I + 2 ) bPd c-R cos 4' / r'2- R' sin2. 2-~'3 1 q- 1 P For brevity, let us put b +Pd co — c - k, (35) - oQ -- 10, and we shall have ko - 3 R' cos 4/ 1/ r2 - R2 sin2'. (36) When the values of P and Q have been found, this equation will give the value of r' in terms of quantities derived directly from the data furnished by observation. We shall now represent by z' the angle at the planet between the sun and earth at the time of the second observation, and we shall have R' sin (' sin z' (37) 236 THEORETICAL ASTRONOMY. Substituting this value of r', in the preceding equation, there results I, sin4 z' (ko -- R' cos,') sin z' - R' sin' cos z' in ^ (38)'aI sina x~" and if we put'02 sin r = R' sin,', o cos - k - R' cos,', (39) O R'" sin'," m0 -- V0, sina/ the condition being imposed that m0 shall always be positive, we have, finally, sin (z' - 0) -m, sin4 z'. (40) In order that mr may be positive, the quadrant in which: is taken must be such that ^ shall have the same sign as 10, since sin y' is always positive. From equation (37) it appears that sin z' must always be positive, or z'< 180~; and further, in the plane triangle formed by joining the actual places of the earth, sun, and planet or comet corresponding to the middle observation, we have r' sin (z' + 4') R' sin (z'+ 4') sin' sin z' Therefore, A, sin (z' + ) osi (41) P sin z and, since p' is always positive, it follows that sin (z + b') must be positive, or that z' cannot exceed 1800 -'. When the planet or comet at the time of the middle observation is both in the node and in opposition or conjunction with the sun, we shall have /'=- 0,'-= 180~ when the body is in opposition, and' - 0~ when it is in conjunction. Consequently, it becomes impossible to determine r' by means of the angle z'; but in this case the equation (36) gives o - - R' + r', when the body is in opposition, the lower sign being excluded by the condition that the value of the first member of the equation must be positive, and for' = 0, the upper sign being used when the sun is between the earth and the DETERMINATION OF AN ORBIT. 237 planet, and the lower sign when the planet is between the earth and the sun. It is hardly necessary to remark that the case of an observation at the superior conjunction when' =0, is physically impossible. The value of r' may be found from these equations by trial; and then we shall have I' - R' when the body is in opposition, and p' = R' -r when it is in inferior conjunction with the sun-! For the case in which the great circle passing through the extreme observed places of the body passes also through the middle place, which gives a0co 0, let us divide equation (32) through by c, andl we have b d / Q \-G+ P 1p'seci' - \ +2r l-P 31 se c The equations (17) give b R sin ( - K) d R_" sin ()" — K) c R'sin ()'-K)' c R'sin (~'-~ K)' and if we put b d 1 + P l+P we shall have (1+ 2r"3) c,- 1, since c = co when ct 0. Hence we derive rf - (42) But when the great circle passing through the three observed places passes also through the second place of the sun, both c and CO become indeterminate, and thus the solution of the problem, with the given data, becomes impossible. 83. The equation (40) must give four roots corresponding to each sign, respectively; but it may be shown that of these eight roots at least four will, in every case, be imaginary. Thus, the equation may be written mn sin4' - sill' cos -T cos z' sin C, 238 THEORETICAL ASTRONOOMY. and, by squaring and reducing, this becomes o2 sin8 z' - 2mn cos C sin5 -'- sin z' sin2 =_ 0. When C is within the limits - 90~ and + 90~, cos g will be positive, and, In, being always positive, it appears from the algebraic signs of the terms of the equation, according to the theory of equations, that in this case there cannot be more than four real roots, of which three will be positive and one negative. When C exceeds the limits - 90~ and + 90~, cos C will be negative, and hence, in this case also, there cannot be more than four real roots, of which one will be positive and three negative. Further, since sin2 g is real and positive, there must be at least two real roots-one positive and the other negative -whether cos h be negative or positive. We may also remark that, in finding the roots of the equation (40), it will only be necessary to solve the equation sin (z' - ) = % sin4 z', (43) since the lower sign in (40) follows directly from this by substituting 180~ - z' in place of't; and hence the roots derived from this will comprise all the real roots belonging to the general form of the equation. The observed places of the heavenly body,,onlly give the direction in space of right lines passing through the places of the earth and the corresponding places of the body, and any three points, one in each of these lines, which are situated in a plane passing through the centre of the sun, and which are at such distances as to fulfil the condition that the areal velocity shall be constant, according to the relation expressed by the equation (30),, must satisfy the analytical conditions of the problem. It is evident that the three places of the earth may satisfy these conditions; and hence there may be one root of equation (43) which will correspond to the orbit of the earth, or give p- 0. Further, it follows from the equation (37) that this root must be z -- 180~ -~,'; and such would be strictly the case if, instead of the assumed values of P and Q, their exact values for the orbit of the earth were adopted, and if the observations were referred directly to the centre of the earth, in the correction for parallax, neglecting also the perturbations in the motion of the earth. DETERMINATION OF AN ORBIT. 239 In the case of the earth, n Nit_ RR' sin (' - 0) RR" sin (0"- 0)' X _N R'R" sin(0- 0') RR"I sin(0" — 0)' and the complete values of P and Q become RR' sin ((' - ) R'R" (sin (" — 0')' rr( Ro RI sill (O- 0)+ ~ KR" sill(0 "- ) 1); 9 2' ~(BR"^f sin("-0) ~ and since the approximate values TIt P —, Q rTT differ but little from these, as will appear from the equations (27)3, there will be one root of equation (43) which gives z' nearly equal to 1800 -'. This root, however, cannot satisfy the physical conditions of the problem, which will require that the rays of light in coming from the planet or comet to the earth shall proceed from points which are at a considerable distance from the eye of the observer. Further, the negative values of sin z' are excluded by the nature of the problem, since r' must be positive, or z' < 180~; and of the three positive roots which may result from equation (43), that being excluded which gives z' very nearly equal to 1800 - /, there will remain two, of which one will be excluded if it gives z' greater than 180~ - 4', and the remaining one will be that which belongs to the orbit of the planet or comet. It may happen, however, that neither of these two roots is greater than 180 4- 1, in which case both will satisfy the physical conditions of the problem, and hence the observations will be satisfied by two wholly different systems of elements. It will then be necessary to compare the elements computed from each of the two values of z' with other observations in order to decide which actually belongs to the body observed. In the other case, in which cos. is negative, the negative roots being excluded by the condition that r' is positive, the positive root must in most cases belong to the orbit of the earth, and the three observations do not then belong to the same body. However, in the case of the orbit of a comet, when the eccentricity is large, and the intervals between the observations are of considerable magnitude, if 240 THEORETICAL ASTRONOMY. the approximate values of P and Q are computed directly, by means of approximate elements already known, from the equations rr' sin (Vu'- u) - r'" sin ('/ -4I')' Q 2r, (rr sin ('- u) +- rr" sin ( u') rr" sin ("-) - - ) - 1 it may occur that cos g is negative, and the positive root will actually belong to the orbit of the comet. The condition that one value of z' shall be very nearly equal to 180~ - -', requires that the adopted values of P and Q shall differ but little from those derived directly from the places of the earth; and in the case of orbits of small eccentricity this condition will always be fulfilled, unless the intervals between the observations and the distance of the planet from the sun are both very great. But if the eccentricity is large, the difference may be such that no root will correspond to the orbit of the earth. 84. We may find an expression for the limiting values of mn and C, within which equation (43) has four real roots, and beyond which there are only two, one positive and one negative. This change in the number of real roots will take place when there are two equal roots, and, consequently, if we proceed under the supposition that equation (43) has two equal roots, and find the values of mn and: which will accord with this supposition, we may determine the limits required. Differentiating equation (43) with respect to z', we get cos (z' -) 4mn sin z' cos z; and, in the case of equal roots, the value of z' as derived from this must alo satisfy the original equation sin (z' - C) =- nm sin 4z'. To find the values of m0 and _ which will fulfil this condition, if we eliminate 0Q between these equations, we have sin z' cos (z' - C) 4 cos z' sin (z' - C), from which we easily find sin (2z' - C) = sin C. (45) This gives the value of g in terms of z' for which equation (43) has DETERMINATION OF AN ORBIT. 241 equal roots, and at which it ceases to have four real roots. To find the corresponding expression for mn, we have sin (z' - _) cos (z' C) % -- sin. -z 4 sin 3z' cos z" in which we must use the value of g given by the preceding equation. Now, since sin (2z' - ) must be within the limits — 1 and + 1, the limiting values of sin g will be + ] and- 3, or C must be within the limits + 360 52'.2 and - 36~ 52'.2, or 143~ 7'.8 and 216~ 52'.2. If C is not contained within these limits, the equation cannot have equal roots, whatever may be the value of z0, and hence there can only be two real roots, of which one will be positive and one negative. If for a given value of C we compute z' from equation (45), and call this z0, or sin (2z - C) -- sin C, we may find the limits of the values of mn within which equation (43) has four real roots. The equation for z0' will be satisfied by the values 2zo- b, 180~- (2Z, - ); and hence there will be two values of mo, which we will denote by mn, and m2, for which, with a given value of C, equation (43) will have equal roots. Thus we shall have sin (z' - ) 1 sin. 4 -Z and, putting in this equation 180~ — (2z' - C) instead of 2zox -, or 90~ (z0'- C) in place of zo', cos z 2cos4 (' - ) It follows, therefore, that for any given value of C, if m0 is not within the limits assigned by the values of m, and m2, equation (43) will only have two real roots, one positive and one negative, of which the latter is excluded by the nature of the problem, and the former may belong to the orbit of the earth. But if P and Q differ so much from their values in the case of the orbit of the earth that z' is not very nearly equal to 180~ - 4/, the positive root, when C exceeds the limits + 36~ 52'.2 and - 36~ 52'.2, may actually satisfy the conditions of the problem, and belong to the orbit of the body observed. 16 242 THEORETICAI ASTRONOMY. When C is within the limits 143~ 7/.8 and 216~ 52'.2, there will be four real roots, one positive and three negative, if mo is within the limits mn and mn2; but, if mn surpasses these limits, there will be only two real roots. Table XII. contains for values of g from - 36~ 52'.2 to + 36~ 52'.2 the values of on1 and 2,, and also the values of the four real roots corresponding respectively to mn1 and mn2. In every case in which equation (43) has three positive roots and one negative root, the value of mo must be within the limits indicated by m, and n2,, and the values of z' will be within the limits indicated by the quantities corresponding to m1 and m2 for each root, which we designate respectively by z/1, z/, z3', and z/,. The table will show, from the given values of m0 and 180~ - I, whether the problem admits of two distinct solutions, since, excluding the value of z', which is nearly equal to 180~- ~', and corresponds to the orbit of the earth, and also that which exceeds 1800, it will appear at once whether one or both of the remaining two values of z' will satisfy the condition that z' shall be less than 180~ - B'. The table will also. indicate an approximate value of z', by means of which the equation (43) may be solved by a few trials. For the root of the equation (43) which corresponds to the orbit of the earth, we have p' 0, and hence from (36) we derive a= -~B" Substituting this value for k0 in the general equation (32), we have p'sec i=10( - l, ); and, since o' must be positive, the algebraic sign of the numerical value of 10 will indicate whether r' is greater or less than R'. It is easily seen, from the formulae for lo, b, d, &c., that in the actual application of these formulhe, the intervals between the observations not being very large, 1o will be positive when P'- i0 and sin ( 0' — K) have contrary signs, and negative when i' - go has the same sign as sin (0' - K). Hence, when 0' - K is less than 180~, r' must be less than R' if i' - -o is positive, but greater than R' if /' - / is negative. When )' - K exceeds 180~, r' will be greater than R' if i~' - o is positive, and less than R' if' - /9 is negative: We may, therefore, by means of a celestial globe, determine by inspection whether the distance of a comet from the sun is greater or less than DETERMINATION OF AN ORBIT. 243 that of the earth from the sun. Thus, if we pass a great circle through the two extreme observed places of the comet, r' must be greater than R' when the place of the comet for the middle observation is on the same side of this great circle as the point of the ecliptic which corresponds to the place of the sun. But when the middle place and the point of the ecliptic corresponding to the place of the sun are on opposite sides of the great circle passing through the first and third places of the comet, r' must be less than R'. 85. From the values of p' and r' derived from the assumed values Tt P - and Q =-T", we may evidently derive more approximate values of these quantities, and thus, by a repetition of the calculation, make a still closer approximation to the true value of p'. To derive other expressions for P and Q which are exact, provided that r' and pt are accurately known, let us denote by s" the ratio of the sector of the orbit included by r and r' to the triangle included by the same radii-vectores and the chord joining the first and second places; by s' the same ratio with respect to r and r", and by s this ratio with respect to r' and r". These ratios s, s', s must necessarily be greater than 1, since every part of the orbit is concave toward the sun. According to the equation (30), we have for the areas of the sectors, neglecting the mass of the body,.':T 1 PV p Il and therefore we obtain s" [rr']- = " itp, s' r [rr"] =T, s r] rp. (46) Then, since lerr'']' in rr"]' we shall have 7 8' 7Tt 8t n=,. —, (47) and, consequently, P. 8 (48) _ST sS -& = = ~'-7rr"7? -Q r" s'/ 8' 8- ", 2r' (48) Substituting for s, s', and s" their values from (46), we have Q=2^' ]+3 [r'r"] + [rr'] - E[r"] ]7" -p [rr]. [rr"]. [r'r"] s' (49) 244 THEORETICAL ASTRONOMY. The angular distance between the perihelion and node being denoted by w, the polar equation of the conic section gives 7) P = 1 + e cos (t' - ), P 1= + e cos (V'-w), (50) r =- 1 + e cos (t" - W). If we multiply the first of these equations by sin (a" — u'), the second by - sin ('/" - u), and the third by sin (' - u), add the products and reduce, we get P sin (u"'i')- Psin (" sin ) + - sin (' - ) = sin (t" -' u')'' r' -sin ('" -,) + sin (tt) - s ); and, since sin (i"' - u') = 2 sin I ('" -') cos 4 (u" -- ~''), sin (a" — t) - sin (U' — u) - 2 sin! ('"-'') cos (it" + a'- 2u), the second member reduces to 4 sin - (u" -t) sin ('" -u ) sin - (t' - u). Therefore, we shall have 4rr'r" sin (" -t-') sin 1 (" - t) sin ~ (u'- ~u) 2 r'r" sin ('"- u') - rr" sin (u'" — ) + rr' sin (' — )' If we multiply both numerator and denominator of this expression by by 2rr' cos ('a" - a') cos O (" -') cos - (a' -' ), it becomes, introducing [rr'], [rr"], and [r'r"], _[r"']. [rr"]. [rr'] I -- [r'r"] + [rr'] - [rr"] 2rr'r" cos 1 (t'"-1_') coS ('~" —) Cos I (a' —)' Substituting this value of p in equation (49), it reduces to q~ ss, rr" cos " ( " -t) C Cos cos (-co I )' (5) 86. If we compare the equations (47) with the formula (28)3, we derive S = -,7 2 _ (f 3 +,.r3) d' -_,1 +_.1 13 +4 kr'4 dt (52) DETERMINATION OF AN ORBIT. 245 Consequently, in the first approximation, we may take st 8-1. If the intervals of the times are not very unequal, this assumption will differ from the truth only in terms of the third order with respect to the time, and in terms of the fburth order if the intervals are equal, as has already been shown. Hence, we adopt for the first approximation, P Q = TT the values of r and zT" being computed from the uncorrected times of observation, which may be denoted by to, to', and to". With the values of P and Q thus found, we compute r', and from this p', p, and p", by means of the formulae already derived. The heliocentric places for the first and third observations may now be found from the formula (71)3 and (72)3, and then the angle u" — u between the radii-vectores r and r" may be obtained in various ways, precisely as the distance between two points on the celestial sphere is obtained from the spherical co-ordinates of these points. When u" - u has been found, we have sin ('" — u') -, sin (le" - ua), it' (53) n'r"' sin (u' - u) = ~ sin ('" - u), from which u"l - u' and u'u - may be computed. From these results the ratios s and s' may be computed, and then new and more approximate values of P and Q. The value of u" -, found by taking the sum of ut" - t' and a' - as derived from (53), should agree with that used in the second members of these equations, within the limits of the errors which may be attributed to the logarithmic tables. The most advantageous method of obtaining the angles between the radii-vectores is to find the position of the plane of the orbit directly from 1, 1", b, and b", and then compute'a, u', and u" directly from n2 and i, according to the first of equations (82),. It will be expedient also to compute r', 1' and b' from o', i', and i', and the agreement of the value of r', thus found, with that already obtained from equation (37), will check the accuracy of part of the numerical 246 THEORETICAL ASTRONOMY. calculation. Further, since the three places of the body must be in a plane passing through the centre of the sun, whether P and Q are exact or only approximate, we must also have tan b' = tan i sin (l' -- ), and the value of b' derived from this equation must agree with that computed directly from p', or at least the difference should not exceed what may be due to the unavoidable errors of logarithmic calculation. We may now compute n and n" directly from the equations' r" sin ('"- u'),, rr' sin (u'- ) (54) rr" sin ("- ua)' rr sin ('a- ua) but when the values of u, U', and u'a are those which result from the assumed values of P and Q, the resulting values of n and n"l will only satisfy the condition that the plane of the orbit passes through the centre of the sun. If substituted in the equations (29), they will only reproduce the assumed values of P and Q, from which they have been derived, and hence they cannot be used to correct them. If, therefore, the numerical calculation be correct, the values of ai and n" obtained from (54) must agree with those derived from equations (31), within the limits of accuracy admitted by the logarithmic tables. The differences u, - u' and u' - u will usually be small, and hence a small error in either of these quantities may considerably affect the resulting values of n and n"'. In order to determine whether the error of calculation is within the limits to be expected from the logarithmic tables used, if we take the logarithms of both members of the equations (54) and differentiate, supposing only n, n", and u' to vary, we get d loge = -c cot ('a" - u') du', d loge n"- + cot (u' - u) dt'. Multiplying these by 0.434294, the modulus of the common system of logarithms, and expressing du' in seconds of arc, we find, in units of the seventh decimal place of common logarithms, d logn = - 21.055 cot ('" -- u') du', d log n" = + 21.055 cot (' -- ) du'. If we substitute in these the differences between log and log n" as found from the equations (54), and the values already obtained by DETERMINATION OF AN ORBIT. 247 means of (31), the two resulting values of du' should agree, and the magnitude of dc' itself will show whether the error of calculation exceeds the unavoidable errors due to the limited extent of the logarithmic tables. When the agreement of the two results for n and I" is in accordance with these conditions, and no error has been made in computing n and n" from P and Q by means of the equations (31), the,accuracy of the entire calculation, both of the quantities which depend on the assumed values of P and Q, and of those which are obtained independently from the data furnished by observation, is completely proved. 87. Since the values of n and n' derived from equations (54) cannot be used to correct the assumed values of P and Q, from which r, 9r', ua, U', &c. have been computed, it is evidently necessary to compute the values for a second approximation by means of the series given by the equations (26)3, or by means of the ratios s and s". The expressions for n and n" arranged in a series with respect to the time involve the differential coefficients of r' with respect to t, and, since these are necessarily unknown, and cannot be conveniently determined, it is plain that if the ratios s and s" can be readily found from r r, r", u,, U', and T, T, z", so as to involve the relation between the times of observation and the places in the orbit, they may be used to obtain new values of P and Q by means of equations (48) and (51), to be used in a second approximation. Let us now resume the equation M1/ E — e sin E, or k(t-T) E- e sin E, az and also for the third place k (t"-T) - E" - e sin E". al Subtracting, we get E" - E- 2e sin (E" E) cos (E" + E). (55) a2 This equation contains three unknown quantities, a, e, and the difference E" —E. We can, however, by means of expressions involving r, r",'t, and u", eliminate a and e. Thus, since p -a(1 -e2), we have i/p -- a2l/ - e (E"- E - 2e sin (E" -- E) cos 1 (E"+ E)). (56) 248 THEORETICAL ASTRONOMY. From the equations /r sin -v -/ a (1 + e) sin -E, V/r" sin v" -/Va (1 + e) sin AE", l/r cos v Va (1 -e) cos!E,:: /r" cos " = (1 e) cos E", since v"-v - u- u, we easily derive r sin (u" — u) a/1 - el sin s (E"- E), (57) and also a cos (E" — E) - ae cos - (E" + E) l/rr cos ~ (uo " - ), or l/rr" cos (58) e cos (E + E) ~ cos (E" - ) - " c (58) Substituting this value of e cos -(E" + E) in equation (56), we get r'Vap a'/1 - e (E" - E- si (E" - E)) + 2a/ - e2 sin (E" - E) cos (u" — u) I/rr", and substituting, in the last term of this, for aVl/- e2, its value from (57), the result is'1p = a'/1 - e' (E" - E - sin (E"- E)) + rr" sin (u"- u). (59) From (57) we obtain aV ~1 -- -2 l (/rr' sin - (u" - u)) e p sin3 1 (E"- E) or ____ / -rr" sin (u"- i) V i 21/rr" cOs u" u) psin (.E" E) Therefore, the equation (59) becomes 1 E"- E- sin (E"- E) (rr] ~ sin A (E" E) 21/rr" cos (t + rr" (60) Let x' be the chord of the orbit between the first and third places, and we shall have X12 (r + r"))2'- 4rr" cos2 (It" - ut). Now, since the chord xl can never exceed r + r", we may put' - (r + r") sin y', (61) and from this, in combination with the preceding equation, we derive 21rr" cos (u"- u) -= (r + r") cos r. (62) DETERMINATION OF AN ORBIT. 249 Substituting this value, and [rr"] -- / p, in equation (60), it reduces to E"- E — Esin (E"- E) t, 2 1 1 sin3 (E" -E) (r + r")3 os3 y + - (63) The elements a and e are thus eliminated, but the resulting equation involves still the unknown quantities E"-E and s'. It is necessary, therefore, to derive an additional equation involving the same unknown quantities in order that E"-E may be eliminated, and that thus the ratio s', which is the quantity sought, may be found. From the equations r a- ae cosE, r" a - aecos E", we get r" r - 2a 2ae cos (E" + E) cos (E E). Substituting in this the value of e cos (El" - E) from (58), we have r" + r = 2a sin2 (E"- E) + 21/ r" ) c (r" ) cos (E" - E), and substituting for sin (E" -- E) its value from (57), there results, "+ r 2rr" sin' r (" ~) + 2V /r" cos- ("- u) (1 —2 sin2 (E" —E)). But, since 2rr" sin2 1 (u" - )__ ([rr"])2 2'7 - I 2 p 2prr" cos2' ('t"- ) s 21/rr" cos 2 (u"-u) we have 2r'2 1 r r(r -") coS r + (r + r") co (1 -2 sin ( E"-E)), from which we derive 1 f2" sin2 -r' sin2(E" E)- (r "cosr cos (64) which is the additional equation required, involving E"- E and s' as unknown quantities. Let us now put sin (E" - E) E"- - E sin(E"- E) Tt2 (r + r") cos3"., sin';-' x' sin' ("- E), CO y 250 THEORETICAL ASTRONOMY. and the equations (63) and (64) become 1m' 1 1.- -— 1 yt > st3;+ 8t t (66) 8~2 When the value of y' is known, the first of these equations will enable us to determine s', and hence the value of x', or sin2- (E"- E), from the last equation. The calculation of r' may be facilitated by the introduction of an additional auxiliary quantity. Thus, let,r tan' =, (67) and from (62) we find cos cos - (u" - ) -- 2 cos - (u" - u) coS2Z' tanl', or cos -- sin 2' cos ( - ). (68) We have, also, x'2 (r + r")2 - 4rr" cOS2' (in" — i), which gives X^2 = (r - I-)2 + 4rr" sin2 1 (i Z). -: Multiplying this equation by cos2(tt-u) and the preceding one by sin2 1 (al" - ), and adding, we get - (r + r")2 sin2 " u) +- (r -- r")2 COS2 r (n"- n). (69) From (67) we get r r^ cos' - r, sin X= + rl" and, therefore, Ir - r"''COS 2Z -- r __ r' so that equation (69) may be written ( = ) sin/2 r' sin2 1 (r- _ u) + cos2 2 cOS2 (u" - U). (r + r')sln We may, therefore, put sin r' cos G-' sin - (i" - u), sin r' sin G' = cos -. ('" - u) cos 2', (70) cosr cos' (t" - i ) sin 2X', DETERMINATION OF AN ORBIT. 251 from which r' may be derived by means of its tangent, so that sin /' shall be positive. The auxiliary angle G' will be of subsequent use in determining the elements of the orbit from the final hypothesis for P and Q. 88. We shall now consider the auxiliary quantity y' introduced into the first of equations (66). For brevity, let us put - (E"- E), and we shall have sin g Y 2g - sin 2g This gives, by differentiation, dy__if 4 sin2g dg = 3 cotdg d ~ —, y 3 cot 2g - sin 2g or cy' dy -= 3y' cot g - 4y' cosec g. dg The last of equations (65) gives xI'- sin2 gg, and hence dq d- - 2 cosec g. dx' Therefore we have dy' 6y' cos g - 8y 3(1 - 2x') y'- 4y'2 dx' sin2 g 2x' (1 - x') It is evident that we may expand y' into a series arranged in reference to the ascending powers of', so that we shall have' +-tx'- r'2+ x'3 +'4 + 4'5-+ &c. Differentiating, we get dy' dL = ft^+ 2 34x' + + 4 &c., and substituting for dy' the value already obtained, there results 2/x' + (4r - 23) x" + (6S - 4r) x'3 + (8s - 6) x'4+ (10: 8) x'+ &c. _ (3a - 42) (3 - 6a - 8a) x' + (3r - 6 - 4/2 8ar) x" + (3 6r -- 8r 8c) x' + (3S - 6 - 4 — 8 8a~8^) x'4 + (3 - 6s - 8r 8 - /89 - 8aC) Xt5 + &C. Since the coefficients of like powers of x' must be equal, we have 3a - 42 Q 0, 3f - 6 - 8I -7 2, 3r - 6f~ - 4 ~2 - 8ar = 2 (2r - fi), &c.; 252 THEORETICAL ASTRONOMY. and hence we derive -l 3- ^-1 26 6__ 228 _265896 _ 19139024 - 336875 T18965 ~-g SI 17l 875 Therefore we have y' 2.~ + J~-(-X' 6 22 8'"-265896 -34-190 +175 1 It + 82a X8 3 75 336875 X 2-189 6 875 X5 _~ 19139024 - + 4 + & + 9 1 3 9 6 &c. (71) If we multiply through by U, and put it 1 2 1 5 2 X3 1 3_84 It 1_ 5 9088 X15 3X T 5 75 667 3 7 5 I 4379d375 +T38278048 16 + - 41389356 - +&c (72) we obtain 10o y l + x- * (73) Combining this with the second of equations (66), the result is y' +- - If we put 5+-f, (74) we shall have 1 -a,1 2 But from the first of equations (66) we get -= 12 ('- 1); y and therefore we have (S'- 1) s,.+ ~. (75) As soon as /' is known, this equation will give the corresponding value of s'. Since $' is a quantity of the fourth order in reference to the difference 1 (E" - E), we may evidently, for a first approximation to the value of ^', take nm' and with this find s' from (75), and the corresponding value of x' from the last of equations (66). With this value of x' we find the corresponding value of ~', and recompute ~', s', and x'; and, if the DETERMINATION OF AN ORBIT. 253 value of E' derived from the last value of x' differs from that already used, the operation must be repeated. It will be observed that the series (72) for A' converges with great rapidity, and that for E" - E= 940 the term containing x'6 amounts to only one unit of the seventh decimal place in the value of $'. Table XIV. gives the values of $' corresponding to values of x' from 0.0 to 0.3, or from E"'- E — to E" -E - 132~ 50'.6. Should a case occur in which E"'- E exceeds this limit, the expression sin'll (E" - E) Y E"-E - sin (E"- E) may then be computed accurately by means of the logarithmic tables ordinarily in use. An approximate value of x' may be easily found with which y' may be computed from this equation, and then $' from (73). With the value of i' thus found, I' may be computed from (74), and thus a more approximate value of x' is immediately obtained. The equation (75) is of the third degree, and has, therefore, three roots. Since s' is always positive, and cannot be less than 1, it follows from this equation that o' is always a positive quantity. The equation may be written thus: t3 -t2 tSt 1- 0and there being only one variation of sign, there can be only one positive root, which is the one to be adopted, the negative roots being excluded by the nature of the problem. Table XIII. gives the values of log 8/2 corresponding to values of' from i'=- 0 to'= 0.6. When ^' exceeds the value 0.6, the value of s' must be found directly from the equation (75). 89. We are now enabled to determine whether the orbit is an ellipse, parabola, or hyperbola. In the ellipse x sin2 4 (E - E) is positive. In the parabola the eccentric anomaly is zero, and hence x- 0. In the hyperbola the angle which we call the eccentric anomaly, in the case of elliptic motion, becomes imaginary, and hence, since sin I (E" - E) will be imaginary, x' must be negative. It follows, therefore, that if the value of x' derived from the equation - t2 is positive, the orbit is an ellipse; if equal to zero, the orbit is a parabola; and if negative, it is a hyperbola. 254 THEORETICAL ASTRONOMY. For the case of parabolic motion we have x' - 0, and the second of equations (66) gives,-.n ( 76) If we eliminate s' by means of both equations, since, in this case, y' -, we get r,~ =j, + 4j'3 Substituting in this the values of m andjgiven by (65), we obtain - =3 sin z' cos r' + 4 sin3 r, (r + r") which gives ~-, - 6 sin o' cos- y' + 2 sin' r'^ (r +-{ r") or =- (sill 1 + cos C")3 +- (sin }r' -- cos r)3. (r + r")> This may evidently be written (r + " = (i + sin ) ( sin r, the upper sign being used when ry is less than 90~, and the lower sign when it exceeds 900. Multiplying through by (r + r")^, and replacing (r +- r") sin r by x, we obtain 6r'= (r +'r" + X) (r + r" — x), which is identical with the equation (56)3 for the special case of parabolic motion. Since x' is negative in the case of hyperbolic motion, the value of' determined by the series (72) will be different from that in the case of elliptic motion. Table XIV. gives the value of' corresponding to both forms; but when x' exceeds the limits of this table, it will be necessary, in the case of the hyperbola also, to compute the value of $' directly, using additional terms of the series, or we may modify the expression for y' in terms of E" and E so as to be applicable. If we compare equations (44), and (56),, we get tan E 1/- 1 tan!F; DETERMINATION OF AN ORBIT. 255 and hence, from (58),, tan E=- V-1. We have, also, by comparing (65), with (41),, since a is negative in the hyperbola, 62 +1 cos E- 2 which gives sin E - 1/ 1. 26 Now, since cos E - V-1 sin E= eE/-i in which e is the base of Naperian logarithms, we have E 1/ — 1 logo (cos E + /- 1 sin E), which reduces to E 1/~- = loge-, or E= V- 1 loge. By means of these relations between E and a, the expression for y' may be transformed so as not to involve imaginary quantities. Thus we have E" - E = (loge " - loge ) 1/- 1 - V-1 loge 11 62 sin (E" - E) - sin E" cos E - cos E" sin E =' - 1/ 1. 266"f From the value of cos E we easily derive sin E 6 - ~ 1coSE - +2 Va 21V/~ and hence sin 2 (E" -E)= -2 - 1. 2 1/66 Therefore the expression for y' becomes =Y (/ lg (6 - - (. (1/6a )3 loge — 4 V 66"a (6" 2 a62) 256 THEORETICAL ASTRONOMY. Since the auxiliary quantity a in the hyperbola is always positive) let us now put - A2 and we have (A- )3 y'= — ~-~ (77) A2 A- -2 4 logeA from which y' may be derived when A is known. We have, further, sin2 (E" - E) - (I - cos (E" - E)) - (- 2 V6 ) and therefore' ( — 4V _, (78) 4 V' sa" 40' or ~' — ^(^ -71)- ( 1(79) These expressions for y' and x' enable us to find $' when x' exceeds the limits of the table. Thus, we obtain an approximate value of xz by putting, rn' ^ ~5 —+j' from which we first find s' and then x' from the second of equations (66). Then we compute A from the formula (79), which gives A =-1 2x'+ 21/x'" -', (80) y' fiom (77), and d' from (73). A. repetition of the calculation, using the value of t' thus found, will give a still closer approximation to the correct values of x' and s'; and this process should be continued until.' remains unchanged. 90. The formule for the calculation of s' will evidently give the value of s if we use., r' rt" u'd and u", the necessary changes in the notation being indicated at once; and in the same manner using r", r, r', u, and u', we obtain s". From the values of s and s" thus found, more accurate values of P and Q may be computed by means of the equations (48) and (51). We may remark, however, that if the times of the observations have not been already corrected for the DETERMINATION OF AN ORBIT. 257 time of aberration, as in the case of the determination of an unknown orbit, this correction may now be applied as determined by means of the values of o, p', and p" already obtained. Thus, if to, to', and to" are the uncorrected times of observation, the corrected values will be t - to - Cp sec, t' = to - p' sec /', (81) tr" — tol- Cp" see fl", in which log C- 7.760523, expressed in parts of a day; and from these values of t, t', t" we recompute T, t', and r", which values will require no further correction, since p, p', and p", derived from the first approximation, are sufficient for this purpose. With the new values of P and Q we recompute r, r'', r", and u, ut, ut as before, and thence again P and Q, and if the last values differ from the preceding, we proceed in the same manner to a third approximation, which will usually be sufficient unless the interval of time between the extreme observations is considerable. If it be found necessary to proceed further with the approximations to P and Q after the calculation of these quantities in the third approximation has been effected, instead of employing these directly for the next trial, we may derive more accurate values from those already obtained. Thus, let x and y be the true values of P and Q respectively, with which, if the calculation be repeated, we should derive the same values again. Let Ax and Ay be the differences between any assumed values of x and y and the true values, or x0-x + Ax, Y-Y + AY. Then, if we denote by x,', yo' the values which result by direct calculation from the assumed values x0 and yo, we shall have xo - X f (Xo, Yo) = f ( + Ax, y -+ Al). Expanding this function, we get xo-.,r =f (x, y) + AAx + Bay + CAX2 + DAx Ay + EAy +.., and if Ax and Ay are very small, we may neglect terms of the second order. Further, since the employment of x and y will reproduce the same values, we have f(x, y) = 0, and hence, since Ax - x0 — x and Ay y - y, xo'- xo A (x0- x) + B (y- y). 17 258 THEORETICAL ASTRONOMY. In a similar manner, we obtain yo'- Yo- A' (x - x) + B' (Yo - y). Let us now denote the values resulting from the first assumption for P and Q by P1 and Q1, those resulting from P1, Q, by P2, Q2, and from P2, Q2 by P3, Q3; and, further, let Pi~- P = P PI a, P - P2= = Q1- Q-b, Q2 — Q1 b', Q3- Q2 b". Then, by means of the equations for x'J - x and yo'- Yo, we shall have a =A(P -x) + B(Q -y), b = A'(P -x) + B'(Q -y), a' - A (P, - x) + B (Q1 - y), b' A' (P- x) - B' (Q- y), a" A (P, - x) + B (Q- y), b"- A' (P2- x) + B' ( Q- y). If we eliminate A, B, A', and B' from these equations, the results are P(a'b" — a"b') + P, (a"b - ab") + P, (ab' - a'b) (a'b" - a"b') +- (a"b - ab") - (ab' - a'b) Q (a - a"b') + Q, (a"b - ab") + Q, (ab'- a'b) - (a'b" — a"b') - (a"b - ab") - (ab'- a'b) from which we get (a" +- a') (a'b"- aC"b') + a' (a"b - ab") (a'"- a"b') - (a"b - ab") + (ab' - a'b)' (b" + b') (C'b" - a"b') + b" (a"b - ab") Y Q~- ^(a'b" — c"b') + (a"b - ab") + (ab'- a'b)' In the numerical application of these formule it will be more convenient to use, instead of the numbers P, P, P, 2,, Q, &c., the logarithms of these quantities, so that a- log P1- log P, b log Q~ log Q, and similarly for a', b', a", b", —which may also be expressed in units of the last decimal place of the logarithms employed,-and we shall thus obtain the values of log x and logy. With these values of log x and log y for log P and log Q respectively, we proceed with the final calculation of r, r, r, and u6, u', ut. When the eccentricity is small and the intervals of time between the observations are not very great, it will not be necessary to employ the equations (82); but if the eccentricity is considerable, and if, in addition to this, the intervals are large, they will be required. It may also occur that the values of P and Q derived from the last hypothesis as corrected by means of these formula will differ so DETERMINATION OF AN ORBIT. 259 much from the values found for x and y, on account of the neglected terms of the second order, that it will be necessary to recompute these quantities, using these last values of P and Q in connection with the three preceding ones in the numerical solution of the equations (82). 91. It remains now to complete the determination of the elements of the orbit from these final values of P and Q. As soon as Q, i, and u, u', u" have been found, the remaining elements may be derived by means of r, r', and u' —, and also from r', r, and "t'-'; or, which is better, we will obtain them from the extreme places, and, if the approximation to P and Q is complete, the results thus found will agree with those resulting from the combination of the middle place with either extreme. We must, therefore, determine s' and x' from r, r", and t"~- Ua by means of the formulae already derived, and then, from the second of equations (46), we have ( strr" sin (-"- u)2 (83) ^^sin^-^(~ - (83) from which to obtain p. If we compute s and s" also, we shall have (/ sr" n sin ("t - ) ) 2 s"rr' sin ('tt _) ) 2 and the mean of the two values of p obtained from this expression should agree with that found from (83), thus checking the calculation and showing the degree of accuracy to which the approximation to P and Q has been carried. The last of equations (65) gives sin I (E" - E) =/Vx', (84) from which E'"- E may be computed. Then, from equation (57), since e - sin ~p, we have sin (- ('-a" a) &.& cos e- sin" 2 rr~" (85) sin(E"- E) for the calculation of a cos (. But p a (1 - e2) a cos2 s, whence cos =a CS (86) a cos p which may be used to determine p when e is very nearly equal to unity; and then e may be found from e = 1 - 2 sin2(450 -~ ). 260 THEORETICAL ASTRONOMY. The equations (50) give e cos (u -o) - -1, r e cos (U"- c) P= -, and from these, by addition and subtraction, we derive 2e cos - (u" ) cos ( ("' + u) ) —+) + _ 2, (87) 2e sin (u" - ) sin (t ("+ u) - w) -P- P by means of which e and co may be found. Since r-" 2V/r/ cos 2X' r + r" sin 2,' = — rwe have p?P 2p 4-+ - 2 v,'2 2r V7 l/rr sin 2/ P P 2p cot 2X r r7 r" and from equations (70), cot 2z sin 1 (u" — u) tan G' cos r' cot 2X/ = -/,?' sin 2;' = co (" — )' cos cos7('~u) Therefore the formulae (87) reduce to e sin (w - (u" + u)) --, P tan G', ch may be deri. from which also e and o) may be derived. Then sin =- e, and the agreement of cos op as derived from this value of (p with that given by (86) will serve as a further proof of the calculation. The longitude of the perihelion will be given by R - - v+ a, or, when i exceeds 90~, and the distinction of retrograde motion is adopted, by r = - w. DETERMINATION OF AN ORBIT. 261 To find a, we have p (a cos 0)2 cos2 p p or it may be computed directly from the equation Tt2 a 4s2 rr" cos2 (u" t u) in2 (E" -E)' (89) which results from the substitution, in the last term of the preceding equation, of the expressions for a cos o and p given by (83) and (85). Then for the mean daily motion we have k as We have now only to find the mean anomaly corresponding to any epoch, and the elements are completely determined. For the true anomalies we have v= u -, v' v' -, 1,' v"-,tt and if we compute r, r', r" from these by means of the polar equation of the conic section, the results should agree with the values of the same quantities previously obtained. According to the equation (45), we have tan ME =tan (45~ - ~) tan,v, tan E' tan (450 - 2o) tan v', (90) tan ME" = tan (450 -~ s) tan l", from which to find E, E', and E". The difference E" - E should agree with that derived from equation (84) within the limits of accuracy afforded by the logarithmic tables. Then, to find the mean anomalies, we have M -E -esinE, M' =E'- e sin E', (91) M" - E"- e sin E"; and, if -1u0 denotes the mean anomaly corresponding to any epoch T, we have, also, M,= - - t (t - T) M'- (e - T) = 1M"- (t"- T), in the application of which the values of t, t', and t" must be those which have been corrected for the time of aberration. The agree 262 THEORETICAL ASTRONOMY. ment of the three values of Mlf will be a final test of the accuracy of the entire calculation. If the final values of P and Q are exact, this proof will be complete within the limits of accuracy admitted by the logarithmic tables. When the eccentricity is such that the equations (91) cannot be solved with the requisite degree of accuracy, we must proceed according to the methods already given for finding the time from the perihelion in the case of orbits differing but little from the parabola. For this purpose, Tables IX. and X. will be employed. As soon as v, v', and v" have been determined, we may find the auxiliary angle V for each observation by means of Table IX.; and, with V as the argument, the quantities M, Mi', M" (which are not the mean anomalies) must be obtained from Table VI. Then, the perihelion distance having been computed from P + 1 -p e' we shall have T _- p 2 _ Mt1- 2 M -- 2 ~'C,\l~e~ Co t ^e Cm 1 j _e' (92) in which log Co - 9.96012771 for the determination of the time of perihelion passage. The times t, t, t" must be those which have been corrected for the time of aberration, and the agreement of the three values of T is a final proof of the numerical calculation. If Table X. is used, as soon as the true anomalies have been found, the corresponding values of log B and log C must be derived from the table. Then w is computed from tan tan |v 1 +- 9e tan - C 5(I + e)' and similarly for w' and w"; and, with these as arguments, we derive M, M', M' from Table VI. Finally, we have T - MBq3 __ M'B'q M]"B" qa CVo y (1 + 9e) Co0/L (1 + 9e) Col/To (1+ 9e) (93) for the time of perihelion passage, the value of Co being the same as in (92). When the orbit is a parabola, e 1= and p = 2q, and the elements q and wo can be derived from r, r", u, and ut' by means of the equa DETERMINATION OF AN ORBIT. 263 tions (76), (83), and (88), or by means of the formulae already given for the special case of parabolic motion. 92. Since certain quantities which are real in the ellipse and parabola become imaginary in the case of the hyperbola, the formula already given for determining the elements from r, r" t, and iu' require some modification when applied to a hyperbolic orbit. When s' and x' have been found, p, e., and co may be derived from equations (83) and (87) or (88) precisely as in the case of an elliptic orbit. Since x' = sin2 (E" - E), we easily find sin (E" - E) - 2 V'x' x and equation (85) becomes sin 1 (i;"- ui) 1/Tr' a cos p - (94) 2 i/x'~ x' But in the hyperbola x' is negative, and hence /x' t- x' will be imaginary; and, further, comparing the values of p in the ellipse and hyperbola, we have cos2 - tan2, or cos?p =- — 1 Itan x. Therefore the equation for a cos

4 sec. The value of a may be found from a =p cot2 4= (a tan )2(9 P 264 THEORETICAL ASTRONOMY. or from 712 16s'2 rr" cos2 (tt" -- ) ('12- -')' which is derived directly from (89), observing that the elliptic semitransverse axis becomes negative in the case of the hyperbola. As soon as (o has been found, we derive from u, U', and u'" the corresponding values of v, v', and v", and then compute the values of F, F', and F" by means of the formula (57),; after which, by means of the equation (69),, the corresponding values of V, N', and N" will be obtained. Finally, the time of perihelion passage will be given by az.t a, T=t-A N= - t' N'- t Y NN" wherein log, k 7.87336575. The cases of hyperbolic orbits are rare, and in most of those which do occur the eccentricity will not differ much from that of the parabola, so that the most accurate determination of T will be effected by means of Tables IX. and X. as already illustrated. 93. EXAMPLE.-TO illustrate the application of the principal formule which have been derived in this chapter, let us take the following observations of Eurynome ~: Ann Arbor M.T. (79!) M 1863 Sept. 14 15h 53- 378.2 1h 0m 44".91 + 90 53' 30".8, 21 9 46 18.0 0 57 3.57 9 13 5.5, 28 8 49 29.2 0 52 18.90 +8 22 8.7. The apparent obliquity of the ecliptic for these dates was, respectively, 23~ 27' 20".75, 23~ 27' 20".71, and 23~ 27' 20".65; and, by means of these, converting the observed right ascensions and declinations into apparent longitudes and latitudes, we getAnn Arbor M. T. Longitude. Latitude. 1863 Sept. 14 15h 53- 37s.2 170 47' 37".60 + 30 8' 43".19, 21 9 46 18.0 16 41 36.20 2 52 27.46, 28 8 49 29.2 15 16 56.35 +2 32 42.98. For the same dates we obtain from the American Nautical Almanac the following places of the sun: NUMERICAL EXAMPLE. 265 True Longitude. Latitude. log Ro. 172~ 1' 42".1 - 0.07 0.0022140, 178 37 17.2 + 0.77 0.0013857, 185 26 54.8 + 0.67 0.0005174. Since the elements are supposed to be wholly unknown, the places of the planet must be corrected for the aberration of the fixed stars as given by equations (1). Thus we find for the corrections to be applied to the longitudes, respectively, -18".48, - 19".49, - 20"., and for the latitudes, + 0".47, + 0".30, + 0.14. When these corrections are applied, we obtain the true places of the planet for the instants when the light was emitted, but as seen from the places of the earth at the instants of observation. Next, each place of the sun must be reduced from the centre of the earth to the point in which a line drawn from the planet through the place of the observer cuts the plane of the ecliptic. For this purpose we have, for Ann Arbor, /' = 42~ 5'.4,. log p 9.99935; and the mean time of observation being converted into sidereal time gives, for the three observations, o0 3 29m 18, O' = 21" 48- 17J, 8 21P 18 55, which are the right ascensions of the geocentric zenith, of which sp' is in each case the declination. From these we derive the longitude and latitude of the zenith for each observation, namely, o0 60~ 33'.9,'0 - 347~ 0'.4, o1 —" 342~ 59'.2, bo-+22 25.0, bo' -+50 -15-8, bo- + -53 41.6. Then, by means of equations (4), we obtain A o - 18".92, A' = 36".94, A" = - 25".76, A log Ro - 0.0001084, A log Ro -0.0002201, A log R" = - 0.0002796. For the reduction of time, we have the values + 0S.15, + 0".28, and + 0".34, which are so small that they may be neglected. 266 THEORETICAL ASTRONOMY. Finally, the longitudes of both the sun and planet are reduced to the mean equinox of 1863.0 by applying the corrections -50".95, - 51".52, - 52".14; and the latitudes of the planet are reduced to the ecliptic of the same date by applying the corrections - 0".15, - 0".14, and - 0".14, respectively. Collecting together and applying the several corrections thus obtained for the places of the sun and of the planet, reducing the uncorrected times of observation to the meridian of Washington, and expressing them in days from the beginning of the year, we have the following data:to - 257.68079, 17~ 46' 28".17, f + 30 8' 43".51, to - 264.42570, )' 16 40 25.19, 13' - 2 52 27.62, 0" = 271.38625, A" - 15 15 44.03, i" +2 32 42.98, o - 172~ 0' 32".23, log R 0.0021056, 0' - 178 35 48.74, logR' 0.0011656, 0" 185 25 36.90, log R" 0.0002378. The numerical values of the several corrections to be applied to the data furnished by observation and by the solar tables should be checked by duplicate calculation, since an error in any of these reductions will not be indicated until after the entire calculation of the elements has been effected. By means of the equations R'R" sin (0"- 0') N BR' sin(0' - 0) RR" sin (0" — 0)' -RR" sin (0"- 0)' tan t' tan (' O') tan w s -- ~ tan Co ~, sin (A'-( i)') cos Wwe obtain log NV 9.7087449, log N" = 9.6950091, ~' -- 161~ 42' 13".16, log (R' sin,') - 9.4980010, log (R' cos,') = 9.9786355S. The quadrant in which 4/ must be taken is determined by the conditions that 4/ must be less than 180~, and that cos/' and cos ('- 0') must have the same sign. Then from NUMERICAL EXAMPLE. 267 sin (/s" + f ) tan Isin ((- (A" + A) - K)O- 2 (/ sec ('" - 2), 2 cos C3 cos pl tan Icos ((" +- 2) - K) - 2 cos cosee (" 2); 2 cos IOos 13S' sin (/-' - 1 tall o - sin (' - ) tan, a sin (Cos3 iotan b Rsin (O K)' sin (0'- K) in ( c CIo ao R" sin (e("-K) sec i' RR" sin (" — ) a csin ( )' ao sin (2" -') we compute K, I,, a, b, c, d, fand h. The angle ITmust be less than 90~, and the value of 0n must be determined with the greatest possible accuracy, since on this the accuracy of the resulting elements principally depends. Thus we obtain K — 4 47' 29".48, log tan I=- 9.3884640, 30 20 52' 59"1 5 log ao 6.8013583~, log b 2.5456342n, log c - 2.2328550,, log d - 1.2437914, logf = 1.3587437,, logh - 3.9247691. The formulae sin (" - 2') B" sin (2" - 0") -/11r sin (A"-2) d iM sin (A' - A) R sin (A -0) 1 sin (A" - A) b h sin (A"- K),,h sin ( - K) - d' b~' give log M1 - 9.8946712, log M" -- 9.6690383, log 31 = 1.9404111, log M" - 0.7306625,. The quantities thus far obtained remain unchanged in the successive approximations to the values of P and Q. For the first hypothesis, from T - (to —, tot) " = (to' - to) P, Q-", b +Pd bo P 1 + P o =o-c, lo 2o Q, ~0 sin 0 - R' sin', mo cos o - R' cos MO- R3 si 0 0 ~B ^^sin / 268 THEORETICAL ASTRON'OMY. we obtain log 9.0782249, log r" 9.0645575, log P- 9.9863326, log Q - 8.1427824, log cq = 2.2298567,, log o - 0.0704470, log lo - 0.0716091, log % =0 0.3326925, C 8~ 24' 49".74, logo, -- 1.2449136. The quadrant in which C must be situated is determined by the condition that ^q shall have the same sign as 10. The value of z' must now be found by trial from the equation sin (z' - CQ - m sin4 z'. Table XII. shows that of the four roots of this equation one exceeds 180~, and is therefore excluded by the condition that sin z' must be positive, and that two of these roots give z' greater than 1800 -- and are excluded by the condition that z' must be less than 180~ —'. The remaining root is that which belongs to the orbit of the planet, and it is shown to be approximately 10~ 40'; but the correct value is found from the last equation by a few trials to be Z - 9~' 22".96. The root which corresponds to the orbit of the earth is 18~ 20' 41", and differs very little from 180~ - /. Next, from SR sin, R' sin (z'+ 4) r' --, z' ~ cos ft, n - -- 1 Q n nP, f Nil P M. +M ), we derive log r' - 0.3025672, log p' 0.0123991, log n = 9.7061229, log in" - 9.6924555, log p = 0.0254823, log p" =0.0028859. The values of the curtate distances having thus been found, the heliocentric places for the three observations are now computed from NUMERICAL EXAMPLE. 269 r cos b cos (I - ) p cos ( - ) - R, r cos b sin (I - ) - p sin (A - ), r sinb -p tan f; r' cos b' cos (I' - 0') p' cos ('-') - R', r' cos b' sin (' 0')' sin (2'- O'), 9f sin b' = p' tan r'; r" cos b" cos (1" - 0") p" cos (" - Oi) - R", r" cos b" sin (I" - 0") p" sin (" - 0"), r" sin b" - p" tan f", which give I - 5~ 14' 39".53, log tan b 8.4615572, logr 0.3040994, I' - 7 45 11.28, log tan b' 8.4107555, log r' 0.3025673, 1" 10 21 34.57, log tan b" = 8.3497911, log r" -0.3011010. The agreement of the value of log r' thus obtained with that already found, is a proof of part of the calculation. Then, from tan b"- + tan b tan i sin (I (1" + 1) - -2 cos (1"-.)'. -tanb" -tanb tan i cos ( (l" + - ) ) -- 2.in b ( 1.~~IIID~VU C~O T Vj ~bl 2 sin (I" — )' tan u — = tan = tan = cost cosi cost we get = 2070 2' 38".16, i = 40 27' 23".84, u 158~ 8' 25".78,' = 1600 39' 18".13, u" = 163~ 16' 4".42. The equation tan b' — tan i sin (I' — f ) gives log tan 6b' 8.4107514, which differs 0.0000041 from the value already found directly from o'. This difference, however, amounts to only 0".05 in the value of the heliocentric latitude, and is due to errors of calculation. If we compute n and n" from the equations r'r" sill (V'"-'),, rr' sin (u' - u) rr" sn(- ) r" sin ('a"" - )r' the results should agree with the values of these quantities previously computed directly from P and Q. Using the values of u, u', and It" just found, we obtain log n = 9.7061158, log 2t" = 9.6924683, 270 THEORETICAL ASTRONOMY. which differ in the last decimal places from the values used in finding p and p". According to the equations d log n - 21.055 cot (u"- u') du', d log n" - + 21.055 cot (cu' - u) did', the differences of log n and log yi" being expressed in units of the seventh decimal place, the correction to u' necessary to make the two values of loga agree is -0".15; but for the agreement of the two values of log y", u' must be diminished by 0".26, so that it appears that this proof is not complete, although near enough for the first approxination. It should be observed, however, that a great circle passing through the extreme observed places of the planet passes very nearly through the third place of the sun, and hence the values of p and pl as determined by means of the last two of equations (18) are somewhat uncertain. In this case it would be advisable to compute p and p", as soon as pt has been found, by means of the equations (22) and (23). Thus, from these equations we obtain log p 0.0254918, log p" = 0.0028874, and hence 1 5~ 14' 40".05, log tan b - 8.4615619, log r = 0.3041042, I"= 10 21 34.)1, log tan b" 8.3497919, log r" = 0.3011017,' =207~ 2' 32".97, i - 4~ 27' 25".13, u = 158~ 8' 31".47, u'' 160~ 39' 23".31, at" 163~ 16' 9".22. The value of log tan b4 derived from A' and these values of Q and i, is 8.4107555, agreeing exactly with that derived from p' directly. The values of n and nt' given by these last results for u,'t' and l", are log n = 9.7061144, log n" 9.6924640; and this proof will be complete if we apply the correction de=d - 0".18 to the value of iu', so that we have't" u' -- 2~ 36' 46".09, t' - u - 2~ 30' 51".66. The results which have thus been obtained enable us to proceed to a second approximation to the correct values of P and Q, and we may also correct the times of observation for the time of aberration by means of the formulae t = to Cp see f, t' - t' - Cp' see p', t" to — Cop" see P", wherein log C= 7.760523, expressed in parts of a day. Thus we get t =257.67467, t'- 264.41976, t" - 271.38044, NUJMERICAL EXAMPLE. 271 and hence log T = 9.0782331, log r' = 9.3724848, log " - 9.0645692. Then, to find the ratios denoted by s and s", we have;Ia tanX - r, sin r cos G sin (" -'), sin r sin G - cos (u" - u') cos 2Z, cos r coo s (u" -') sin 2Z; \r tanX" -, sin r" cos G" - sin - (t' -- ), sin r" sin G" cos' (' - u) cos 2z", cos r" cos i (' — ~) sin 2/; = 2'. sin2'— (r'+r")3 cos' r cos r ~ ~2 2sin2 1, ~_ sin" ^^ m- (r + r')" cos3' os from which we obtain x - 440 57' 6".00, Z" 440 56' 57".50, r= 1 18 35.90, /' 1 15 40.69, log mn = 6.3482114, log mn" 6.3163548, logj - 6.1163135, logj" = 6.0834230. From these, by means of the equations mn. m 7 _ 6+jl x _ ff 2! using Tables XIII. and XIV., we compute s and s". First, in the case of s, we assume j += j -= 0.0002675, I +K and, with this as the argument, Table XIII. gives log s2 - 0.0002581. Hence we obtain x' 0.000092, and, with this as the argument, Table XIV. gives - = 0.00000001; and, therefore, it appears that a repetition of the calculation is unnecessary. Thus we obtain logs -0.0001290, logs"= 0.0001200. When the intervals are small, it is not necessary to use the formulae 272 THEORETICAL ASTRONOMY. in the complete form here given, since these ratios may then be found by a simpler process, as will appear in the sequel. Then, from 8 P-= *, -' 77TT rt2 Q ss" rr cos (u" -') os (( "- - )) cos (u — u)' we find log P - 9.9863451, log Q 8.1431341, with which the second approximation may be completed. We now compute c, ko, lo, z', &c. precisely as in the first approximation; but we shall prefer, for the reason already stated, the values of p and op" computed by means of the equations (22) and (23) instead of those obtained from the last two of the formula (18). The results thus derived are as follows:log c, 2.2298499, log ko - 0.0714280, log 1o 0.0719540, log 0 = 0.3332233, C 8~ 24' 12".48, log m,=- 1.2447277, z - 9 0' 30".84, log r' 0.3032587, log p' - 0.0137621, log n - 9.7061153, log n" —- 9.6924604, log -p 0.0269143, log" — 0.0041748, I = 5~ 15' 57".26, log tan b =8.4622524, logr =0.3048368, 1' - 7 46 2.76, log tan b' 8.4114276, log r' 0.3032587, l" 10 22 0.91, log tan b" 8.3504332, log r"= 0.3017481, a = 207~ 0' 0".72, i - 4~ 28' 35".20, = 1580 12' 19".54, u'= 160~ 42' 45".82, u" 163~ 19' 7".14. The agreement of the two values of log r' is complete, and the value of log tan b' computed from tan b' - tan i sin (T'- ), is log tan b' 8.4114279, agreeing with the result derived directly from p'. The values of n and nt obtained from the equations (54) are log n = 9.7061156, log n" = 9.6924603, which agree with the values already used in computing p and p", and the proof of the calculation is complete. We have, therefore,' -- u' = 2~ 36' 21".32, u't — u = 2~ 30' 26".28, u" — u - 5~ 6' 47".60. From these values of Iul - u' and u' - u, we obtain log s = 0.0001284, log s" - 0.0001193, NUMERICAL EXAMPLE. 273 and, recomputing P and Q, we get log P = 9.9863452, log Q = 8.1431359, which differ so little from the preceding values of these quantities that another approximation is unnecessary. We may, therefore, from the results already derived, complete the determination of the elements of the orbit. The equations tan X' 1-, sin r' cos G' sin 1 (u" t- ), sin' sin G' = cos - (a" - u) cos 2/, cos = cos ^ ('t" - u) sin 2'/,'2 2 "' sin2 r' m = (r +r" cos co give x' =440 53' 53".25, r' 20 33' 52".97, log tan' 8.9011435, log m' 6.9332999, logj 6.7001345. From these, by means of the formulae and Tables XIII. and XIV., we obtain log s'2 0.0009908, log x' = 6.5494116. Then from 8'rr" sin (u"- U) ) we get logp - 0.3691818. The values of logp given by = sr r" srr'sin (' sn ) )2 are 0.3691824 and 0.3691814, the mean of which agrees with the result obtained from u" - -u, and the differences between the separate results are so small that the approximation to P and Q is sufficient. The equations sin k (E"- E) - V', sin-! (Ot" - /a) cos p =_.- ~ -, a cos 18 274 THEORETICAL ASTRONOMY. give ] (E" - E) = 10 4' 42".903, log (a cos A)= 0.3770315, log cos -p = 9.9921503. Next, from e sin (w - (u + Iu))= ~' - tan G', cosr/rr e cos ( ~ (le + ))=- sec= (e - ), cos l/rr we obtain o = 190~ 15' 39".57, log e = log sin -= 9.2751434, s= -10 51 39.62, - o + g -- 37~ 15' 40".29. This value of p gives log cos S -9.9921501, agreeing with the result already found. To find a and u, we have p k a -- a y- 3X COS2 g a3 cos? x the value of k expressed in seconds of arc being log k = 3.5500066, from which the results are log a = 0.3848816, log4 -= 2.9726842. The true anomalies are given by V =- U w, V- =U' -w, v- - =u",_ according to which we have v - 327~ 56' 39".97,'- = 330~ 27' 6".25, v" = 333~ 3' 27".57. If we compute r, r', and r" from these values by means of the polar equation of the ellipse, we get logr - 0.3048367, log r' 0.3032586, log r" = 0.3017481, and the agreement of these results with those derived directly from p, p', and p" is a further proof of the calculation. The equations tan,E = tan (45~ - p) tan Iv, tan lE' = tan (45~ - ) tan Iv', tan E" = tan (45~- ~f) tan Iv" give E = 333~ 17' 28".18, E' = 335~ 24' 38".00, E" = 337~ 36' 19".78. NUMERICAL EXAMPLE. 275 The value of (E" - E) thus obtained differs only 0".003 from that computed directly from x'. Finally, for the mean anomalies we have M= E - e sin E, M' -E' e sin E', M" E" e sin E", from which we get M= 338~ 8' 36".71, M' - 339~ 54' 10".61, M" 341~ 43' 6".97; and if M. denotes the mean anomaly for the date T= 1863 Sept. 21.5 Washington mean time, from the formulse _A= d M, (t ~ T) M' (t'- T) M" - ( t" T), we obtain the three values 339~ 55' 25".97, 3390 55' 25".96, and 339" 55' 25".96, the mean of which gives M - 339~ 55' 25".96. The agreement of the three results for M0 is a final proof of the accuracy of the entire calculation of the elements. Collecting together the separate results obtained, we have the following elements: Epoch = 1863 Sept. 21.5 Washington mean time. M- 339~ 55' 25".96 -r= 37 15 40.29) = - 2097 0 0.72 Ecliptic and Mean i 4 28 35.20 Equinox 1863.0. i- 4 28 35.20 -=- 10 51 39.62 log a 0.3848816 log t= 2.9726842 - 939".04022. If we compute the geocentric right ascension and declination of the planet directly from these elements for the dates of the observations, as corrected for the time of aberration, and then reduce the observations to the centre of the earth by applying the corrections for parallax, the comparison of the results thus obtained will show how closely the elements represent the places on which they are based. Thus, we compute first the auxiliary constants for the equator, using the mean obliquity of the ecliptic, E = 23~ 27' 24".96, 276 THEORETICAL ASTRONOMY. and the following expressions for the heliocentric co-ordinates of the planet are obtained: x -r [9.9997272] sin (296~ 55' 46".05 + u), y = r [9.9744699] sin (206 12 42.79 + iu), z =r [9.5249539] sin (212 39 14.62 + u). The numbers enclosed in the brackets are the logarithms of sin a, sin b, and sin c, respectively; and these equations give the co-ordinates referred to the mean equinox and equator of 1863.0. The places of the sun for the corrected times of observation, and referred to the mean equinox of 1863.0, are True Longitude. Latitude. Log R. 172~ 0' 29".5 - 0".07 0.0022146, 178 36 4.5 + 0.77 0.0013864, 185 25 42.0 0.67 0.0005182. If we compute from these values, by means of the equations (104)1, the co-ordinates of the sun, and combine them with the corresponding heliocentric co-ordinates of the planet, we obtain the following geocentric places of the planet: - =15~ 10' 29".06, = + 9~ 53' 16".72, log A = 0.02726, a =14 15 0.22,' - 9 12 51.29, log J' = 0.01410, a" -13 3 49.47, 3" -+ 8 21 54.46, log J" = 0.00433. To reduce these places to the apparent equinox of the date of observation, the corrections - 48".14, + 48".54, + 48".91, must be applied to the right ascensions, respectively, and + 18".55, - 18".92, + 19".31, to the declinations. Thus we obtain: Washington M. T. Comp. a. Comp. d. 1863 Sept. 14.67467 1' 0" 45s.15 + 9~ 53' 35".3, 21.41976 0 57 3.25 9 13 10.2, 28.38044 0 52 18.56 -+8 22 13.8. The corrections to be applied to the respective observations, in order to reduce them to the centre of the earth, are +- 0.24, - 0.31, - 0.34 in right ascension, and + 4".5, + 4".8, + 5".1 in declination, so that we have, for the same dates, NUMERICAL EXAMPLE. 277 Observed ao. Observed J. 1 0' 45s.15 + 9~ 53' 35".3, 0 57 3.26 9 13 10.3, 0 52 18.56 +8 22 13.8. The comparison of these with the computed values shows that the extreme places are exactly represented, while the difference in the middle place amounts to only 08.01 in right ascension, and to 0".1 in declination. It appears, therefore, that the observations are completely satisfied by the elements obtained, and that the preliminary corrections for aberration and parallax, as determined by the equations (1) and (4), have been correctly computed. It cannot be expected that a system of elements derived from observations including an interval of only fourteen days, will be so exact as the results which are obtained from a series of observations or from those including a much longer interval of time; and although the elements which have been derived completely represent the data, yet, on account of the smallness of 9' - -, this difference being only 31'".893, the slight errors of observation have considerable influence in the final results. When approximate elements are already known, so that the correction for parallax may be applied directly to the observations, in order to take into account the latitude of the sun, the observed places of the body must be reduced, by means of equation (6), to the point in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane. The times of observation must also be corrected for the time of aberration, and the corresponding places of both the planet and the sun must be reduced to the ecliptic and mean equinox of a fixed epoch; and further, the reduction to the fixed ecliptic should precede the application of equation (6). If the intervals between the times of observation are considerable, it may become necessary to make three or more approximations to the values of P and Q, and in this case the equations (82) may be applied. But when approximate elements are already known, it will be advantageous to compute the first assumed values of P and Q directly from these elements by means of the equations (44) or by means of (48) and (51); and the ratios s and s" may be found directly from the equations (46). In the case of very eccentric orbits this is indispensable, if it be desired to avoid prolixity in the numerical calculation, since otherwise the successive approximations to P and Q will slowly approach the limits required. 278 THEORETICAL ASTRONOMY. The various modifications of the formulse for certain special cases, as well as the formulae which must be used in the case of parabolic and hyperbolic orbits, and of those differing but little from the parabola, have been given in a form such that they require no further illustration. 94. In the determination of an unknown orbit, if the intervals are considerably unequal, it will be advantageous to correct the first assumed value of P before completing the first approximation in the manner already illustrated. The assumption of Q -t is correct to terms of the fourth order with respect to the time, and for the same degree of approximation to P we must, according to equation (28)3, use the expression Ti!+ 6T2- ) TfT which becomes equal to only when the intervals are equal. The first assumed values P= Q=, furnish, with very little labor, an approximate value of r'; and then, with the values of P and Q, derived from P r l+ ), Q=", (98) the entire calculation should be completed precisely as in the example given. Thus, in this example, the first assumed values give log r' 0.30257, and, recomputing P by means of the first of these equations, we get log P = 9.9863404, log Q = 8.1427822, with which, if the first approximation to the elements be completed, the results will differ but little from those obtained, without this correction, from the second hypothesis. If the times had been already corrected for the time of aberration, the agreement would be still closer. The comparison of equations (46) with (25)3 gives, to terms of the fourth order, NUMERICAL EXAMPLE. 279 T 2 Tt2 1 -1 r 1 r3l+ - and, if the intervals are equal, this value of sa is correct to terms of the fifth order. Since loge s = loge (I + (s - 1)) = 8 - 1 - ( - 1)2 + &c., we have, neglecting terms of the fourth order, 10 r2: (99) logs -. r (99) in which log 20 —8.8596330. We have, also, to the same degree of approximation, log s' r3, log 5" = 6 03 (100) For the values log =- 9.0782331, log - =- 9.3724848, log r" = 9.0645692, log r' - 0.3032587, these formulse give log s 0.0001277, log s' = 0.0004953, log s" = 0.0001199, which differ but little from the correct values 0.0001284, 0.0004954, and 0.0001193 previously obtained. Since sec3' 1 + 6 sinU2 r' + &c., the second of equations (65) gives Tr2 6~ri =n' = + ~-,~ + 6p~ sin2 lr' + &c. + (r - )+ (r + r")3 s Substituting this value in the first of equations (66), we get s'" (s' - 1 r + )- + y (r + rr)3 y (sin2 + &e. If we neglect terms of the fourth order with respect to the time, it will be sufficient in this equation to put y' -, according to (71), and hence we have T2 S'2 (S _ 1)4= __ ____ 3 (r - r" )3 and, since s - 1 is of the second order with respect to r', we have, to terms of the fourth order, S2 (s'- 1) = logeo 8. 280 THEORETICAL ASTRONOMY. Therefore, T'2 log s'- 4 2o (101) which, when the intervals are small, may be used to find s' from r and r". In the same manner, we obtain og22 T2 log S = 4 Ao (r,+ r,)3, log SI' 4 r*o (102) For logarithmic calculation, when addition and subtraction logarithms are not used, it is more convenient to introduce the auxiliary angles X, X', and X', by means of which these formulae become z2osCs6 Z log s' 4Abl 2 Cos(V' lo eOTIN;s log s 3= o~ cos logs ^ o~,.c3 logs" -c s6 x (103) 3 r,3 3g, in which log2- = 9.7627230. For the first approximation these equations will be sufficient, even when the intervals are considerable, to determine the values of s and s" required in correcting P and Q. The values of t, r', r", and r'X above given, in connection with log r =0.3048368, log r" = 0.3017481, give log s 0.0001284, log s' 0.0004951, log s" = 0.0001193. These results for log s and log s" are correct, and that for log s' differs only 3 in the seventh decimal place from the correct value. ORBIT FROM FOUR OBSERVATIONS. 281 CHAPTER V. DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE. 95. THE formulae given in the preceding chapter are not sufficient to determine the elements of the orbit of a heavenly body when its apparent path is in the plane of the ecliptic. In this case, however, the position of the plane of the orbit being known, only four elements remain to be determined, and four observed longitudes will furnish the necessary equations. There is no instance of an orbit whose inclination is zero; but, although no such case may occur, it may happen that the inclination is very small, and that the elements derived from three observations will on this account be uncertain, and especially so, if the observations are not very exact. The difficulty thus encountered may be remedied by using for the data in the determination of the elements one or more additional observations, and neglecting those latitudes which are regarded as most uncertain. The formulae, however, are most convenient, and lead most expeditiously to a knowledge of the elements of an orbit wholly unknown, when they are made to depend on four observations, the second and third of which must be complete; but of the extreme observations only the longitudes are absolutely required. The preliminary reductions to be applied to the data are derived precisely as explained in the preceding chapter, preparatory to a determination of the elements of the orbit from three observations. Let t, t', t", t"' be the times of observation, r, r', r", r" the radiivectores of the body, tu, UC', ", Ui"' the corresponding arguments of the latitude, R, Rn R", R"' the distances of the earth from the sun, and 0', 0", 0"' the longitudes of the sun corresponding to these times. Let us also put [rVr"'] = rr"' sin ("' - u'), an[r"r'] r"r"' sin ("' - u"), and n'- n" [rTr-"' ] 282 THEORETICAL ASTRONOMY. Then, according to the equations (5)3, we shall have nx - x' -+ n"i" - 0, ny - + -"y" - O, n'x' -'f x 0 (2) n'y' - ~' + n"'y"' =- 0. Let 2, A', A", 2"' be the observed longitudes, i, 9', P", a"' the observed latitudes corresponding to the times t, t', t", t"', respectively, and d, Jz', a, A'mp the distances of the body from the earth. Further, let d"' cos A"' - p"', and for the last place we have x"' = C pw cos, cos ()' y s"' - p"' sin R in "'. Introducing these values of x"' and y"', and the corresponding values of x, x', x", y, y', y" into the equations (2), they become O = n (p cos A - R cos 0) - (p' cos A' -' cos 0') + ni" (p" cos A" R- " cos 0"), O - n (p sin A -R sin 0) -(p' sin A' - R' sin 0')' + -- " (P" sin A" -'" sin 0"), 0 n' (p' cos A' -' cos 0') - (p" cos A" - R" cos 0") (3) + n"' (p"' cos A"' - R"' cos ( "'), 0' (p' sin A' -' sin 0') (p" sin A" - R" sin 0") +- n"' (P"' sin "' - R"' sin G"'). If we multiply the first of these equations by sin 2, and the second by - cos 2, and add the products, we get 0 = nR sin (A - 0) - (' sin (A' - A) + R' sin (A -')) +'n (p" sin (A" - A) + " sin (A - ")); (4) and in a similar manner, from the third and fourth equations, we find 0 - n' (p' sil (A"' - A') - R' sin (A"' - 0')) (5) - (p" sin (A"'- A") -- R" sin('- ")) -n'"R'" si ("' — 0"'). Whenever the values of n,', n", and n"' are known, or may be determined in functions of the time so as to satisfy the conditions of motion in a conic section, these equations become distinct or independent of each other; and, since only two unknown quantities p' ORBIT FROMI FOUR OBSERVATIONS. 283 and p" are involved in them, they will enable us to determine these curtate distances. Let us now put cos j' sin (' — ) =A, cos A sin (" - ) B, cos P sin (A"'- ") - C, cos sin (A"'- 2') D- 1, and the preceding equations give Ap' sec'- Bn"p" sec " =nR sin (A -0) -' sin (- 0') + n"R" sin ( - 0"), Dn'p' sec'- Cp" sec i"- n'R' sin ("' — 0') - R" sin (A"'- 0") (7) + n"'R"' sin (A"' - 0"'). If we assume for n and n" their values in the case of the orbit of the earth, which is equivalent to neglecting terms of the second order in the equations (26).3 the second member of the first of these equations reduces rigorously to zero; and in the same manner it can be shown that when similar terms of the second order in the corresponding expressions for n' and n" are neglected, the second member of the last equation reduces to zero. Hence the second member of each of these equations will generally differ from zero by a quantity which is of at least the second order with respect to the intervals of time between the observations. The coefficients of p' and p" are of the first order, and it is easily seen that if we eliminate p" from these equations, the resulting equation for p' is such that an error of the second order in the values of n and n" may produce an error of the order zero in the result for p', so that it will not be even an approximation to the correct value; and the same is true in the case of p". It is necessary, therefore, to retain terms of the second order in the first assumed values for n n, n", and n"/; and, since the terms of the second order involve r' and r", we thus introduce two additional unknown quantities. Hence two additional equations involving r', r", p', p" and quantities derived from observation, must be obtained, so that by elimination the values of the quantities sought may be found. From equation (34)4 we have p' sec i' =- R' cos' i: V r'2 - R2 sin2', (8) which is one of the equations required; and similarly we find, for the other equation, p" sec i" = R" cos 4" + /'r"2 RB" sin2 4". (9) 284 TIEORETICAL ASTRONOMY. Introducing these values into the equations (7), and putting x' - =t 1/2"_ sin'2 4', x2 == Vr"~R/12 sin'' (10) x" — /r'a- R"I2 sin2 4", we get Ax' - Bn"x" = nR sin (2 - 0) - R' sin ( - 0') + nf"R" sin (2 -3 0") - AR' cos 4' + n"BR" cos 4",'x' - Cx" = l'R' sin ("' - 0(') - R sin (A"' - 0") +- n"'R"' sin (A"' - 0"') - n'DR' cos 4' + CR" cos ". Let us now put B h' Dhi A h', C - or, cos i' sin (A" -) h" cos /' sin (A"' —') cos /' sin (' - )' - os 3" sin (A"'- ")' R R' sin(A -') R' cos 4'+ ~A a, cos, - R" sin ("' - 0"),, hR" cos 4' R sin(2' R" sin ( -(DI) _, h'/~" cosa -" +- C-, cos -' sin (' - 0')h"R' cos ~~ ) -- e.R sin (A - 0) _ d R"' sin (2"' - O"') A C and we have x' = h'n"x" + nd'- a' + -n"e', - h"n'x' + n"'d" - a" + n'. (12) These equations will serve to determine x' and x", and hence r' and r", as soon as the values of n, n', n", and n"' are known. 96. In order to include terms of the second order in the values of n and "I, we have, from the equations (26)3, "(+ T d= i t6 r 3 ) =- 1 + 3 (r' + ) and, putting P' s q= (n + "- 1) r'3, (13) these give ORBIT FROM FOUR OBSERVATIONS. 285 = 7-2_1 2 _ l2 P) 7"\'-~Q6r" 13 (14) ^-^. Let us now put r"_-= k (t"' - "), r'- k (et — O, (15) and, making the necessary changes in the notation in equations (26)3, we obtain, r (+ ~ lr ( +~4 dt 7 - iI (1 + + r"' + 2 ( ~ + - L1 ) dr" 1 From these we get, including terms of the second order, a"' = t ^1 + 6- r"+ ) ( I N'\ (, I l r II+I', and hence, if we put 6 r,,1 - kr"' P _ (_-.' T " = (n' + "T' - 1) r"3, (17) we shall have, since r- = T + r'T", Qua'" ^ ^~ 11) (18)' When the intervals are equal, we have p 7. a,,, P' T.... ~{ = and these eressins m these e eindin te e of an unknown orbit, for the first approximation to th e quantities. The equations (13) and (17) give we shall have since' 1 + ( r3" ) 1"' z1'P" and, introducing these values, the equations (12) become ond the inrod vian thee values, the euavetioecm 236 THEORETICAL ASTRONOMY. + p I + -r )'(h'' + Pd' +c') - a', ~~~~~~~~1 /'\ (20) X 1 -+ P ( + rQ4 3 ) (h'. + P"d" + ")- a". Let us now put P'd' +- c' h' l+"i Uip,1~do -oi h+'i p(21) "d" -+ — c", C 7Ipq- P" -- P"' and we shall have x-(1 + ) (f'x"+ c') - a, (22) ~ -(i +;)(f I + Cot) " ~ We have, further, from equations (10), 13 (x,2 R 2 sin2')4 3 r (X 1 (23)- (x2, + R"2 sin2,")~, If we substitute these values of r'3 and r"3 in equations (22), the two resulting equations will contain only two unknown quantities x' and x", when P', P", Q', and Q" are known, and hence they will be sufficient to solve the problem. But if we effect the elimination of either of the unknown quantities directly, the resulting equation becomes of a high order. It is necessary, therefore, in the numerical application, to solve the equations (22) by successive trials, which may be readily effected. If z' represents the angle at the planet between the sun and the earth at the time of the second observation, and z" the same angle at the time of the third observation, we shall have, R' sin {' sin z (24) RB" sin 4" (24) sin z Substituting these values of r' and r" in equations (10), we get (25) x' =r cosz', (CO and hence ORBIT FROM FOUR OBSERVATIONS. 287, R' sin' tan z'' sin tan z -P x by means of which we may find z' and z" as soon as x' and x" shall have been determined; and then r' and r" are obtained from (24) or (25). The last equations show that when x' is negative, z' must be greater than 90~, and hence that in this case r' is less than R'. In the numerical application of equations (22), for a first approximation to the values of x' and x", since Q' and Q" are quantities of the second order with respect to r or r"', we may generally put'- o, q"- o; and we have xfx = f"- +- co- a"', or, by elimination, coI fj —'c0" ffail a' x' = - f'o~" — f" —a 1 -f'f" e Co +i ff" Co' —f" a' — all With the approximate values of x' and x" derived from these equations, we compute first r' and r" from the equations (26) and (24), and then new values of x' and xa from (22), the operation being repeated until the true values are obtained. To facilitate these approximations, the equations (22) give if _X + a' Cot -, f' (1 +')' x" + aff c" (27) fy T//3) Let an approximate value of x' be designated by xo0, and let the value of x" derived from this by means of the first of equations (27) be designated by x0". With the value of xa" for x" we derive a new value of x' from the second of these equations, which we denote by x,'. Then, recomputing x" and x', we obtain a third approximate value of the latter quantity, which may be designated by x2'; and, if we put x - xo' = ao, x2' 1 = ao' 288 THEORETICAL ASTRONOMY. we shall have, according to the equation (67)3, the necessary changes being made in the notation, x =1' a — x 0 X2 a (28) a,- a. o ao -- ao' The value of x' thus obtained will give, by means of the first of equations (27), a new value of x", and the substitution of this in the last of these equations will show whether the correct result has been found. If a repetition of the calculation be found necessary, the three values of x' which approximate nearest to the true value will, by means of (28), give the correct result. In the same manner, if we assume for x" the value derived by putting Q' = 0 and Q"= 0, and compute x', three successive approximate results for x" will enable us to interpolate the correct value. When the elements of the orbit are already approximately known, the first assumed value of x' should be derived from x='r" - r' R'2 sin2 2' instead of by putting Q' and Q" equal to zero. 97. It should be observed that when A2 =- or 2"' = ", the equations (22) are inapplicable, but that the original equations (7) give, in this case, either o" or p' directly in terms of n and n" or of n' and n"' and the data furnished by observation. If we divide the first of equations (22) by h', we have h' r'3 )h'' h') The equations (21) give p,d' f' 1 Co' P_ d + c ah -I + P" h P' - i P' and from (11) we get a' R' cos /'' sin ( -- 0')' A' h' B co R" sin(2 - 0~") ==- R cosB + R sin ( D" (29) hB d' R sin () - ~ h' B Then, if we put Cfr_ Pd'.' Gd = pi h' 3- hr~~~~ ORBIT FROM FOUR OBSERVATIONS. 289 c' d' its value may be found from the results for, and, derived by means of these equations, and we shall have a;' 1 XI72,'p~(Q- (x" at' -- (30) ht 1 +r P' ( XI r, + Co ) When' =, we have h'= oo, and this formula becomes 0-(lZ 1+ r3, ) (X + c')- (1 + P'), the value of t being given by the first of equations (29) This equation and the second of equations (22) are sufficient to determine x' and x" in the special case under consideration. The second of equations (22) may be treated in precisely the same manner, so that when )"' = -l, it becomes 0-( + Q (,+ C)- (i + ), and this must be solved in connection with the first of these equations in order to find x' and x". 98. As soon as the numerical values of x' and x"t have been derived, those of r' and r" may be found by means of the equations (26) and (24). Then, according to (41)4, we have,' sin (z' + c'), siln z' I, _ sin (z" + ) cos") p" - - cos fi~. sin z" The heliocentric places are then found from p' and p" by means of the equations (71),, and the values of r' and }r" thus obtained should agree with those already derived. From these places we compute the position of the plane of the orbit, and thence the arguments of the latitude for the times t' and t". The values of e', r", I', 6", n, n/, n', and n"'! enable us to determine r, r"', t, and t"'/. Thus, we have [rr'r] -r r" sin (a~" -'), and, from the equations (1) and (3),, 19 290 THEORETICAL ASTRONOMY, nt [ri"] - V- ['r"], f/ [r"r"']-n'' [r'r"], ~'~:],,W [r'r"]. Therefore, n" r in ( ) sin (u', - s ), r s sin'(i" si (' - ), (32) r" sin (u"'- 0') =? sin (,' ~ ), r"' sin (') - - sin ( ). From the first and second of these equations, by addition and subtraction, we get r sin ((' - u) + -( (" ~-')) r + Fr sin (U"-'), (33) r cos (('' -u') + 1 (t" - u')) = -- cos 2 ('" - u'), from which we may find r, ta' - u, and u =' - (u' - u). In a similar manner, from the third and fourth of equations (32), we obtain 9"' sin (('"' - u") + - (u"-')) " +' sinr (" ), f r (34) r' cos ((u"' - U") + (- t)) C,-'osr ( -l'), from which to find r"' and u"'. When the approximate values of r, rf, r", r"' and u, u', "/,'" have been found, by means of the preceding equations, from the assumed values of P', P", Q', and Q", the second approximation to the elements may be commenced. But, in the case of an unknown orbit, it will be expedient to derive, first, approximate values of r' and r', using r'g T~~T and then recompute P' and P" by means of the equations (14) and ORBIT FROM FOUR OBSERVATIONS. 291 (18), before finding u' and u". The terms of the second order will thus be completely taken into account in the first approximation. 99. If the times of observation have not been corrected for the time of aberration, as in the case of an orbit wholly unknown, this correction may be applied before the second approximation to the elements is effected, or at least before the final approximation is commenced. For this purpose, the distances of the body from the earth for the four observations must be determined; and, since the curtate distances p' and p" are already given, there remain only p and pi' to be found. If we eliminate p' from the first two of equations (3), the result is "n" sin (" — 2') P P sin (A ) (35) n sin ()/- A) nR sin (A'- 0) - R' sin (A'- 0') +- nm" R" sin (A' -0") n sin (A'- A) and, by eliminating p" from the last two of these equations, we also obtain P n'" sin (2"' —.") n' R' sin (A" - (') - R" sin (2" - (") + n"' R"' sin (A" - ("') "'1 sin (R"' — ") by means of which p and p"' may be found. The combination of the first and second of equations (3) gives p, cos (A') p cos () "- ) (37) nR cos (2 -- 0) -' cos (A- 0') + mR" " cos (A- 0") + n and from the third and fourth we get / "' (z " (tIm'I-' A' ", p"'- s ( Co "- A") P- cos (-"'-) (38) (Alt- ~ co s --,). n' R' cos ("'- (') -R" cos (A"' — ") + n"' R"' cos (A"'- "') + n"' Further, instead of these, any of the various formulae which have been given for finding the ratio of two curtate distances, may be employed; but, if the latitudes /, i', &c. are very small, the values of p and p"' which depend on the differences of the observed longitudes of the body must be preferred. 292 THEORETICAL ASTRONOMY. The values of p'and p"' may also be derived by computing the heliocentric places of the body for the times t and t"' by means of the equations (82)i, and then finding the geocentric places, or those which belong to the points to which the observations have been reduced, by means of (90)1, writing p in place of J cos f. This process affords a verification of the numerical calculation, namely, the values of, and 2"' thus found should agree with those furnished by observation, and the agreement of the computed latitudes d9 and "' with those observed, in case the latter are given, will show how nearly the position of the plane of the orbit as derived from the second and third observations represents the extreme latitudes. If it were not desirable to compute 2 and P'I in order to check the calculation, even when: and 3"' are given by observation, we might derive p and,o" from the equations p r sinusin i cot, p"' - r"' sin u"' sin i cot ", ( when the latitudes are not very small. In the final approximation to the elements, and especially when the position of the plane of the orbit cannot be obtained with the required precision from the second and third observations, it will be advantageous, provided that the data furnish the extreme latitudes B9 and i9"', to compute p and p"' as soon as p' and,o" have been found, and then find 1, 1"', b, and b"' directly from these by means of the formule (71)3. The values of a and i may thus be obtained from the extreme places, or, the heliocentric places for the times I' and t"F being also computed directly from p' and p", from those which are best suited to this purpose. But, since the data will be more than sufficient for the solution of the problem, when the extreme latitudes are used, if we compute the heliocentric latitudes b' and b-'from the equations tan b' = tan i sin (1' - ), tan b" = tan i sin (I"- g), they will not agree exactly with the results obtained directly from p' and p", unless the four observations are completely satisfied by the elements obtained. The values of r' and r', however, computed directly from p' and p" by means of (71)3, must agree with those derived from x' and x". The corrections to be applied to the times of observation on account ORBIT FROM FOUR OBSERVATIONS. 293 of aberration may now be found. Thus, if to, to', t", and to"' are the uncorrected times of observation, the corrected values will be t - to -Cp sec f, t' - to- Cp' see', t" - to-t CP" secf", ) t"' = to"- Cp"' see i"', wherein log C= 7.760523, and from these we derive the corrected values of r, r', T", Z"', and zT'. 100. To find the values of P', P", Q', and Q", which will be exact when r rrt, r"', and u,', Ull, u'll are accurately known, we have, according to the equations (47)4 and (51)4, since Q' = Q, P' —' Tr S' (41) 881' rr" cos t (t" - u'a) cos 2 (" -) cos, ('- )' In a similar manner, if we designate by s"' the ratio of the sector formed by the radii-vectores r" and r'" to the triangle formed by the same radii-vectores and the chord joining their extremities, we find r s"' T' _ T < (42) ~ s "' r'r"' cos (u'"' - u") cos ('a"' - t) cos - (Z" -')' The formule for finding the value of s"' are obtained from those for s by writing ZX"f "', lG"', &c. in place of X,, G, &c., and using r", r"', u"' - " instead of r', r", and u" - t', respectively. By means of the results obtained from the first approximation to the values of P', P", Q', and Q", we may, from equations (41) and (42), derive new and more nearly accurate values of these quantities, and, by repeating the calculation, the approximations to the exact values may be carried to any extent which may be desirable. When three approximate values of P' and Q', and of P" and Q", have been derived, the next approximation will be facilitated by the use of the formulae (82)4, as already explained. When the values of P', P", Q', and Q" have been derived with sufficient accuracy, we proceed from these to find the elements of the orbit. After Q, i, r, r',r", r"',', U','a", and u'a" have been found, the remaining elements may be derived from any two radii-vectores 294 THEORETICAL ASTRONOMY. and the corresponding arguments of the latitude. It will be most accurate, however, to derive the elements from r, r'", a, and a"'. If the values of P', P", Q', and Q" have been obtained with'great accuracy, the results derived from any two places will agree with those obtained from the extreme places. In the first place, from tan o = I 7' silno cos GO - sin ~ (u~"' - u), (43) sin y sin GQ = cos 1 (t"' - t) cos 2o, Cos y r cos - ("' - u) sin 2O we find ro and G0. Then we have ao — k (t'- t), ___________. sin (44) MO (r +- r"')3 sro cos Cos (44) o rn n M o ^~9o - +Jo + o 0 0 2, from which, by means of Tables XIII. and XIV., to find so and x0. We have, further, So rr"' sin (t"' - ) )2 and the agreement of the value of p thus found with the separate results for the same quantity obtained from the combination of any two of the four places, will show the extent to which the approximation to P', P", Q', and Q" has been carried. The elements are now to be computed from the extreme places precisely as explained in the preceding chapter, using r"' in the place of r" in the formulse there given and introducing the necessary modifications in the notation, which have been already suggested and which will be indicated at once. 101. EXAMPLE.-For the purpose of illustrating the application of the formulne for the calculation of an orbit from four observations, let us take the following normal places of Eurynome 0 derived by comparing a series of observations with an ephemeris computed from approximate elements. Greenwich M. T. a d 1863 Sept. 20.0 140 30' 35".6 + 90 23' 49".7, Dec. 9.0 9 54 17.0 2 53 41.8, 1864 Feb. 2.0 28 41 34.1 9 6 2.8, April 30.0 74 29 58.9 + 19 35 41.5. NUMERICAL EXAMPLE. 295 These normals give the geocentric places of the planet referred to the mean equinox and equator of 1864.0, and free from aberration. For the mean obliquity of the ecliptic of 1864.0, the American Nautical Almanac gives E-23~ 27' 24".49, and, by means of this, converting the observed right ascensions and declinations, as given by the normal places, into longitudes and latitudes, we get Greenwich M. T. P 1863 Sept. 20.0 16~ 59' 9".42 + 2~ 56' 44".58, Dec. 9.0 10 14 17.57 — 1 15 48.82, 1864 Feb. 2.0 29 53 21.99 2 29 57.38, April 30.0 75 23 46.90 -3 4 44.49. These places are referred to the ecliptic and mean equinox of 1864.0, and, for the same dates, the geocentric latitudes of the sun referred. also to the ecliptic of 1864.0 are + 0".60, +0".53, + 0".36, + 0".19. For the reduction of the geocentric latitudes of the planet to the point in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane, the equation (6)4 gives the corrections - 0".57, - 0".38, - 0".18, and - 0".07 to be applied to these latitudes respectively, the logarithms of the approximate distances of the planet from the earth being 0.02618, 0.13355, 0.29033, 0.44990. Thus we obtain t = 0.0, -16~ 59' 9".42, p ==+ 2~ 56' 44".01, t' 80.0, A' -10 14 17.57, P' =-1 15 49.20, t" = 135.0, A" =29 53 21.99, P" 2 29 57.56, t'" 223.0, A"' 75 23 46.90, i"' = -3 4 44.56; and, for the same times, the true places of the sun referred to the mean equinox of 1864.0 are O 177~ 0' 58".6, log R 0.0015899. 0' — 256 58 35.9, log R' 9.9932638, 0" - 312 57 49.8, log R" 9.9937748, 0"'- 40 21 26.8, log R"' " 0.0035149, 296 THEORETICAL ASTRONOMY. From the equations tan tan 3' tan (A' - ) t sin (A'-')' tan tn w" (tan tan tan (A" - 0") taln W' si. ~, tan " ~~ sin(,~ ) COS Wcos we obtain 4' — 113~ 15' 20".10, log(R' cost') 9.5896777, log(R' sin 4') = 9.9564624, 4"= 76 56 17.75, log (R" cos 4") 9.3478848, log (R" sin 4") - 9.9823904. The quadrant in which 4/ must be taken, is indicated by the condition that cos'5/ and cos(2'- 0') must have the same sign. The same condition exists in the case of /". Then, the formulae A -cos' sin (A' - ), B cos 1" sin (A"- ), C cos 3" sin (A"'- A"), D - cos s (' sin (A"'-'), B D - h', - -h" R' sin ( - 0') a' =- R' cos -'+ - a" " cos — B" sin (A"'- 0") it - ^ f cos4- ^^ C A B" sin ( - 0") c= h'R" cos 4~ + ~A ~ ) R' sin (A"' - (') C" - h"R' cos -/'R sin (A- ) "' sin (A"' - I) d — "A - C give the following results:log A = 9.0699254X, log C = 9.8528803, log B 9.3484939, log D 9.9577271, log h' - 0.2785685, log h -= 0.1048468, log a' 0.8834880,, log a" = 9.9752915, log c' = 0.9012910,, log c" = 9.7267348n, log d' = 0.4650841, log d" - 9.9096469. We are now prepared to make the first hypothesis in regard to the values of P', Q', P", and Q". If the elements were entirely unknown, it would be necessary, in the first instance, to assume for these quantities the values given by the expressions NUMERICAL EXAMPLE. 297 p,, rX QxA P -- =-w~ -, Q,,==' -,"; then approximate values of r' and r" are readily obtained by means of the equations (27), (26), and (24) or (25). The first assumed value of x' to be used in the second member of the first of equations (27), is obtained from the expression which results from (22) by putting Q' = 0 and Q"- 0, namely, c' - f'c'" — f'" - a' 1 -f'f" after which the values of x' and x" will be obtained by trial from (27). It should be remarked, further, that in the first determination of an orbit entirely unknown, the intervals of time between the observations will generally be small, and hence the value of x' derived from the assumption of Q' =0 and Q"- 0 will be sufficiently approximate to facilitate the solution of equations (27). As soon as the approximate values of r' and r" have thus been found, those of P' and P" must be recomputed from the expressions - (1 ) (6 ~ ~- ~ 7- 6 3 ) ~- 1 —~ r~, -- (1 ~ r" ~ With the results thus derived for P' and P", and with the values of Q' and Q" already obtained, the first approximation to the elements must be completed. When the elements are already approximately known, the first assumed values of P', P", Q', and Q" should be computed by means of these elements. Thus, from r'r" sin (v"-'),, r' sin (' - v) r" sin (v" v)' rr sin (v — v) r"r"' sin ("- "),,, rr" sin (v" - v') rr"'r sin (v"' v')' r'r"' sin (v"- v')' we find n, n', n", and n"'. The approximate elements of Eurynome give v -322~ 55' 9".3, logr -0.308327, v' =353 19 26.3, log r' 0.294225, v"= 14 45 8.5, log r" 0.296088, v'- 47 23 32.8, log r"' - 0.317278, 298 THEORETICAL ASTRONOMY. and hence we obtain log n = 9.653052, log n" = 9.806836, log n'= 9.825408, logrn"'= 9.633171. Then, from n P' = TQ n" P'=^' Q (n + " — 1) r", Win P" "" (n'+- n'"- 1)r we get log P' = 9.846216, log' = 9.840771, log P"- 9.807763, log Q"- 9.882480. The values of these quantities may also be computed by means of the equations (41) and (42). Next, from, P'd' + c' h Co +h % I- +P'' + P',, P"d"ff+ f ic"+ h" C~ 1 +P"' - 1+ P', we find log Coe 0.541344n, logf' - 0.047658n, log o" — 9.807665,, logf" 9.889385. Then we have St + C — - XI' +- a Co" f, + a'', sin a' R" sin 4" tan z' -, tan z" - R' sin 4' X',, R" sin 4" x" sin z' cos Z sin z" Cos "' from which to find r' and r". In the first place, from x' = V'1/r2 R2 sin 4', we obtain the approximate value log x' 0.242737. Then the first of the preceding equations gives log x" 0.237687. NUMERICAL EXAMPLE. 299 From this we get " - 29~ 3' 11".7, log r" = 0.296092; and then the equation for x' gives log x' - 0.242768. Hence we have z' 270 20' 59".6, log r' = 0.294249; and, repeating the operation, using these results for x' and r', we get log x" - 0.237678, log x' = 0.242757. The correct value of log xz may now be found by mieans of equation (28). Thus, in units of the sixth decimal place, we have a,= 242768 - 242737- + 31,' = 242757 - 242768- 11, and for the correction to be applied to the last value of log', in units of the sixth decimal place, Alog x'=-, +3. ac - a Therefore, the corrected value is log x'= 0.242760, and from this we derive log x" 70.237681. These results satisfy the equations for x' and xa, and give z' = 27~ 21' 1".2, log r' 0.294242, z" =29 3 12.9, logr" -0.296087. To find the curtate distances for the ist and second observations, the formulae are,' sin (z' +') 1 " sin (z" + ") f sin z' coS', p"' sin — " cos ", which give log p' = 0.133474, log p" = 0.289918. Then, by means of the equations 300 THEORETICAL ASTRONOMY. r' cos b' cos (I' 0') p' cos (' - 0') -R', r cos b' sin (I' - 0') p' sin (' - 0'), r' sin b' p' tan fi','r cos b cos (1" - 0") p" cos (A" ")- 0") - R", r" cos b" sin (" -- 0") - p" sin (A" - 0"), r" sin b" p" tan s", we find the following heliocentric places:' = 370 35' 26".4, log tanb' = 8.182861n, log r' 0.294243, " - 58 58 15.3, log tan b" = 8.6342091, log r" = 0.296087. The agreement of these values of log r' and log r" with those obtained directly from x' and x" is a partial proof of the numerical calculation. From the equations tan i sin ( " ( + -')- ) _ (tan b" tan b') sec 1 (" -'), tan i cos Q- (l" + 1') - ) (tan b tan') cosec (1"'), tan' -- tan (-') tan " tan (" - ) tan ttf tan u- cos i cos i we obtain _ -206" 42' 24".0, i _ 40 36' 47".2,,' - 190 55 6.6 u" —212 20 53.5. Then, from 1 1 + (~ ), ) a"' n'P", q- P " 1t -- P we get log n" - 9.806832, log n 9.653048, log i' = 9.825408, log n"' = 9.633171, and the equations r~n(t' 4)+1 (in" t6)) _ r+ si r n11 sin ((u'" - ) + (u" )) - (') sin (u- t' ), nr r cos ((n' - u) + - (it" -')) cos (i"'), r'" sin ((n"'" - u") + 1- (i" -')) - r" sin (" - in'), in', " Cos ((i"' - u") + I (i" - i')) = cos (i "- I), NUMERICAL EXAMPLE. 301 give logr 0.308379, u 160~ 30' 57".6, log r"' = 0.317273, "'= 244 59 32.5. Next, by means of the formule tan (I - Q ) - cos i tan u, tan b - tan i sin (I -- ), tan ("' - ) cos i tan "', tan b"' tan i sin (I"'- ), p os- ) r cos b cos (I - (3) + R, p sin (- 0) r cos b sin (I - 0), p tan f =r sin b; "' cos (A"' - 0"') -_ r" cos b"' cos (1"'- 0") + RI"', p" sin ("' - o"') = r"' cos b"' sin (1"', 0"'), p"' tan "' r"' sin b", we obtain I - 7 16' 51".8, "' =- 910 37' 40".0, b-+ 1 32 14.4, b"'= — 4 10 47.4, i - 16 59 9.0, i"' - 75 23 46.9, P-+ 2 5640.1, fi"' — 3 4 43.4, log p - 0.025707, log p"' 0.449258. The value of 2"' thus obtained agrees exactly with that given by observation, but A differs 0".4 from the observed value. This difference does not exceed what may be attributed to the unavoidable errors of calculation with logarithms of six decimal places. The differences between the computed and the observed values of 9 and i'^ show that the position of the plane of the orbit, as determined by means of the second and third places, will not completely satisfy the extreme places. The four curtate distances which are thus obtained enable us, in the case of an orbit entirely unknown, to complete the correction for aberration according to the equations (40). The calculation of the quantities which are independent of P', P", Q', and Q", and which are therefore the same in the successive hypotheses, should be performed as accurately as possible. The value of -,, required in finding x" from x', may be computed directly from o _ pd, t a od'C lbd' Gc' the values of hl and le being found by means of the equations (29); the valueso~fad bein 302 THEORETICAL ASTRONOMY. and a similar method may be adopted in the case of C' Further, f' in the computation of x' and x", it may in some cases be advisable to employ one or both of the equations (22) for the final trial. Thus, in the present case, x" is found from the first of equations (27) by means of the difference of two larger numbers, and an error in the last decimal place of the logarithm of either of these numbers affects in a greater degree the result obtained. But as soon as.r" is known Q" so nearly that the logarithm of the factor 1 + — 3 remains unchanged, the second of equations (22) gives the value of x" by means of the sum of two smaller numbers. In general, when two or more formulae for finding the same quantity are given, of those which are otherwise equally accurate and convenient for logarithmic calculation, that in which the number sought is obtained from the sum of smaller numbers should be preferred instead of that in which it is obtained by taking the difference of larger numbers. The values of r, r', r", r"', and u, t', u, ut"', which result from the first hypothesis, suffice to correct the assumed values of P', P",', and Q". Thus, from r-k ("- - t), - k- ( (t' — t), e- k (elOl- el), tan% = z tan tan! i_ ta tnX rZI sin r cos G sin (" — u'), sin y" cos G" _ sin 1 (L~'- i), sisin G'sin G cos. (u" — u') cos 2y, sin r" sin G" -cos 1 (u'-'u) cos 2", cos r cos - (t" — t') sin 2x, cos r" cos. (t'- u) sin 2/", sinl "' cos G"'= sin I ("' - u"), sin r"' sin G"' cos 2 (t"' -- Iu") cos 2/"' cos "' c=os- (t"' - u") sin 2%"'; 2 COS,/ c 7 Tcos, 1112 COS% 6 T s c3 COSr y' Cos3 y r r cos yr, sin it sinT" sind s"a cos' cos cos /' 9Th 9j 5 fl~sll) 5 9 jflt!b 9 t! _ _a __ _' S) ~! " ~__ _ ___:t8 in connection with Tables XIII. and XIV. we find s, ", and s"'. The results are NUMERICAL EXAMPLE. 303 log = 9.9759441, log r"- 0.1386714, log T"' 0.1800641, 450 3' 39".1, /"' = 440 32' 1".4, X"'= 450 41' 55".2, r 10 42 55.9, /'" 15 13 45.0, /"' 16 22 48.5, log m 8.186217, log n"'= 8.516727, log m"'= 8.590596, logj 7.948097, logj"- 8.260013, logj"'- 8.325365, log s = 0.0085248, log s"-= 0.0174621, log s"'=- 0.0204063. Then, by means of the formule P' —~r' s' r 2 TTQ t2 Q' ssO1 ~ rT" COS (a" -'a') COS - (,"'') cos s ('a- ~ )' r'r' Cos (t'"' - a"') cos (It"' - l') cos ('a"t - t')' we obtain log P' = 9.8462100, log Q' 9.8407536, log Pi" 9.8077615, log Q" 9.8824728, with which the next approximation may be completed. We now recompute co', c/", f', f", x', x", &c. precisely as already illustrated; and the results are log c0' = 0.5413485, log co = 9.8076649,, logf' = 0.0476614., logf" 9.8893851, log x' = 0.2427528, log x" 0.2376752, z' 27~ 21' 2".71, z" 29~ 3' 14".09, logr'= 0.2942369, logr" 0.2960826, log p'= 0.1334635, logp" 0.2899124, log n =9.6530445, log n" - 9.8068345, log n' = 9.8254092, log n'"' = 9.6331707. Then we obtain l' = 370 35' 27".88, log tan' = 8.1828572l, log r' = 0.2942369, 1" = 58 58 16.48, log tan b" - 8.6342073., log r" = 0.2960827. These results for log r' and log r" agree with those obtained directly from zt and z", thus checking the calculation of ~' and 4" and of the heliocentric places. Next, we derive -= 206~ 42' 25".89, i 4~ 36' 47".20,' = 190 55 6.27,'u"- 212 20 52.96, 304 THEORETICAL ASTRONOMY. and fronm u"- u, r, ry, l, n"i, n' and n, we obtain logr - 0.3083734, u = 160~ 30' 55".45, log r"'- 0.3172674, It"' 244 59 31.98. For the purpose of proving the accuracy of the numerical results, we compute also, as in the first approximation, l= 70 16' 51".54, l"'- 91~ 37' 41".20, b=-+ 1 32 14.07, b"' —- 4 10 47.36, A_ 16 59 9.38,'" - 75 23 46.99, i -+ 2 56 39.54, P"'- 3 4 43.33, log p -0.0256960, log p"' 0.4492539. The values of 2 and 2"' thus found differ, respectively, only 0".04 and 0".09 from those given by the normal places, and hence the accuracy of the entire calculation, both of the quantities which are independent of P', P", Q', and Q", and of those which depend on the successive hypotheses, is completely proved. This condition, however, must always be satisfied whatever may be the assumed values of P', P", Q, and Q". From r, r', U%, u', &c., we derive log s - 0.0085254, logs" = 0.0174637, log s"' = 0.0204076, and hence the corrected values of P', P", Q', and Q" become log P' - 9.8462110, log Q' = 9.8407524, log P" =9.8077622, log Q" 9.8824726. These values differ so little from those for the second approximation, the intervals of time between the observations being very large, that a further repetition of the calculation is unnecessary, since the results which would thus be obtained can differ but slightly from those which have been derived. We shall, therefore, complete the determination of the elements of the orbit, using the extreme places. Thus, from k (t"' t), tanXo 0 /, sin r cos Go sin 1 ('/"' - ), sin 7o sin G - cos - (u"'"- u) cos 2Z0, cos ro = cos ("' - u) sin 2X0, o _______ _o_, T2 sin2 l' ~(r + r"')3 c' co s ro _ mo + __o,m o - -_ - _' x ) 02 NUMERICAL EXAMPLE. 305 we get log r, = 0.5838863, log tan G = 8.0521953, O = 420 14' 30".17, log m0 9.7179026, log s2= 0.2917731, log x0= 8.9608397. The formula / Sorr"' sin ("'-'t) )2 gives logp — 0.3712401; and if we compute the same quantity by means of (8rI Isin (a sr'r" ) rr'sin (Vt'- _a)'2 ( s"'r"r"' sin (a"' u.") ) ~ ~ ) = ~ T/ TJthe separate results are, respectively, 0.3712397, 0.3712418, and 0.3712414. The differences between these results are very small, and arise both from the unavoidable errors of calculation and from the deviation of the adopted values of P', P", Q', and Q" from the limit of accuracy attainable with logarithms of seven decimal places. A variation of only 0".2 in the values of u' - and "-' - u" will produce an entire accordance of the particular results. From the equations sin j (E" E) (V O a cos s ('-E ) rr COS ~ P ~ a cosy we obtain (E"' E) _ 170 35' 42".12, log (a cos g) 0.3796883, log cos = 9.9915518. The formulae e sin(o -- ( ( "' +- u)) - tan Go, cos ro 1/rr' e cos w -' (t"' + u')) = sec I (u"' - u), cos ro 1/V "' give w = 197~ 38' 8".48, log e = log sin c - 9.2907881, p- 11~ 15' 52".22,, _ w + = 44~ 20' 34".37. This result for ( gives log cos (p= 9.9915521, which differs only 3 in, the last decimal place from the value found from p and a cos p. Then, from 20 830 THEORETICAL ASTRONOMIY. p- k Cb == ~ ~p. - -- ~ 0O 2 3 a3 COS a-? the value of k being expressed in seconds of arc, or log k= 3.5500066, we get log a = 0.3881359, log s = 2.9678027. For the eccentric anomalies we have tan ME - tan 1 (u - o) tan (45~ ~- ), tan -E' tan 1 (u' - w) tan (45~ -n), tan 4E" - tan I (u" - w) tan (45~ -), tan J-2E"' tan - (a"'- -W) tan (45~0- ), from which the results are E -329~ 11' 46".01, E" - 12~ 5' 33".63, E — 354 29 11.84, E"' 39 34 34.65. The value of (E"' - E) thus derived differs only 0".03 from that obtained directly from x0. For the mean anomalies, we have M f E - e sin E, M1" = E" - e sin E", M' - E' - e sin E', M"' - E"' - e sin E"', which give M = 334~ 55' 39".32, M" = 9~ 44' 52".82, M'= 355 33 42.97, M'"' 32 26 44.74. Finally, if JMo denotes the mean anomaly for the epoch T= 1864 Jan. 1.0 mean time at Greenwich, from M- - M~ (t -- T) = M' — (t' - T) = M" - /l (t" - T) M"' - t (t"' - T), we obtain the four values MO = 1~ 29' 39".40 39.49 39.40 39.40, the agreement of which completely proves the entire calculation of the elements from the data. Collecting together the several results, we have the following elements: NUMERICAL EXAMPLE. 307 Epoch =1864 Jan. 1.0 Greenwich mean time. M - 1~ 29' 39".42 7 — 44 20 34.37 Ecliptic and Mean Q_- 206 42 25 89 EP^ 2 - 206 42 25 8 Equinox 1864.0. i= 4 36 47.20 y I 11 15 52.22 log a - 0.3881359 log~ - 2.9678027 i - 928".54447. 102. The elements thus derived completely represent the four observed longitudes and the latitudes for the second and third places, which are the actual data of the problem; but for the extreme latitudes the residuals are, computation minus observation, a/ = ~- 4".47, A/' - + 1".23. These remaining errors arise chiefly from the circumstance that the position of the plane of the orbit cannot be determined from the second and third places with the same degree of precision as from the extreme places. It would be advisable, therefore, in the final approximation, as soon as p', p, ", n1, it', and nt"' are obtained, to compute from these and the data furnished directly by observation the curtate distances for the extreme places. The corresponding heliocentric places may then be found, and hence the position of the plane of the orbit as determined by the first and fourth observations. Thus, by means of the equations (37) and (38), we obtain log p = 0.0256953, log p"' = 0.4492542. With these values of p and p'", the following heliocentric places are obtained: I - 7~ 16' 51".54, log tan b 8.4289064, log r 0.3083732, 1"'- 91 37 40.96, log tan b"' 8.8638549,, logr"' 0.3172678. Then from tan sn sn ( (l"' + 1) - 2). 1 (tan b"' + tan b) see. (1"'- 1), tan i c 1) - an i"' - ) ( t an tan b) cosec ("' - 1), we get = 206~ 42' 45".23, 4~ 36' 49".76. For the arguments of the latitude the results are u = 160~ 30' 35".99, u"' - 244~ 59' 12".53. 308 THEORETICAL ASTRONOMY. The equations tan b' tan i sin (tI - ), tan b" = tan i sin (" - b ), give log tan b' = 8.1827129, log tan b"' 8.6342104, and the comparison of these results with those derived directly from p' and,op" exhibits a difference of + 1".04 in b', and of - 0".06 in 1". Hence, the position of the plane of the orbit as determined from the extreme places very nearly satisfies the intermediate latitudes. If we compute the remaining elements by means of these values of r, r't/, and u, um', the separate results are: log tan G = 8.0522282n, log on = 9.7179026, log s2 =0.2917731, log x, = 8.9608397, logp - 0.3712405, (E" - E) — 17~ 35' 42".12, log (a cos p) = 0.3796884, log cos o - 9.9915521, w = 197~ 37' 47".72, log e — 9.2907906, -= 11 15 52.46, log cos o = 9.9915520, log a -0.3881365, log,a 2.9678019, E= 329~ 11' 47".24, E"' 39~ 34' 35".70, 11=334 55 40.46, f"'- 32 26 45.49, i =o- 1 29 40.36, M 1 29 40.37. Hence, the elements are as follows: Epoch = 1864 Jan. 1.0 Greenwich mean time. M= 1~ 29' 40".36 44 20 32.95 Ecliptic and Mean d2 - 206 42 45.23 -206 42 45.23 Equinox 1864.0. i- 4 36 49.76?= 11 15 52.46 log a = 0.3881365 i - 928".5427. It appears, therefore, that the principal effect of neglecting the extreme latitudes in the determination of an orbit from four observations is on the inclination of the orbit and on the longitude of the ascending node, the other elements being very slightly changed. The elements thus derived represent the extreme places exactly, and if we compute the second and third places directly from these elements, we obtain JM' 355~ 33' 43".88, M" = 9~ 44' 53".73, E' - 354 29 12.93, E" 12 5 34.81, v' =353 16 59.07, v" -=14 42 45.96, NUMERICAL EXAMPLE. 309 log r' = 0.2942366, log r" 0.2960828, u' - 190~ 54' 46".79, u" 212~ 20' 33".68, 1' — 37 35 27.75, 1"- 58 58 16.50, b' - 0 52 21.25, b --- 2 27 59.06, A'-= 10 14 17.35, " — 29 53 21.99, i'- -1 15 47.67, 13" — 2 29 57.62, log p' 0.1334634, log p" = 0.2899122. Hence, the residuals for the second and third places of the planet areComp. - Obs. A/' - 0".22, aF' + 1".53, 2A)." 0.00, Al" - -0.06; and the elements very nearly represent the four normal places. Since the interval between the extreme places is 223 cays, these elements must represent, within the limits of the errors of observation, the entire series of observations on which the normals are based. It may be observed, also, that the successive approximations, in the case of intervals which are very large, do not converge with the same degree of rapidity as when the intervals are small, and that in such cases the mnmerical calculation is very much abbreviated by the determination, in the first instance, of the assumed values of P', P", Q', and Q" by means of approximate elements already known. For the first determination of an unknown orbit, the intervals will generally be so small that the first assumed values of these quantities, as determined by the equations fP' -. t (- Ci )_ \ i) = __i", P i__ (1 _ I2 & r) will not differ much from the correct values, and two or three hypotheses, or even less, will be sufficient. But when the intervals are large, and especially if the eccentricity is also considerable, several hypotheses may be required, the last of which will be facilitated by using the equations (82)4. The application of the formule for the determination of an orbit from four observations, is not confined to orbits whose inclination to the ecliptic is very small, corresponding to the cases in which the method of finding the elements by means of three observations fails, 310 THEORETICAL ASTRONOMY. or at least becomes very uncertain. On the contrary, these formulae apply equally well in the case of orbits of any inclination whatever, and since the labor of computing an orbit from four observations does not much exceed that required when only three observed places are used, while the results must evidently be more approximate, it will be expedient, in very many cases, to use the formula given in this chapter both for the first approximation to an unknown orbit and for the subsequent determination from more complete data. CIRCULAR ORBIT. 311 CHAPTER VI. INVESTIGATION OF VARIOUS FORBMITL FOR THE CORRECTION OF THE APPROXIMATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY. 103. IN the case of the discovery of a planet, it is often convenient, before sufficient data have been obtained for the determination of elliptic elements, to compute a system of circular elements, an ephemeris computed from these being sufficient to follow the planet for a brief period, and to identify the comparison stars used in differential observations. For this purpose, only two observed places are required, there being but four elements to be determined, namely, 2, i, a, and, for any instant, the longitude in the orbit. As soon as a has been found, the geocentric distances of the planet for the instants of observation may be obtained by means of the formule d - R cos ~ + q- 1 ~ RB sin" %, A" = R" cos " + - /c'a R, 2 sin2 4", the values of 4 and Q" being computed from the equations (42)3 and (43)3. For convenient logarithmic calculation, we may first find z and z" from R sin, R" sin," sin z 7 --, sin 2 s n," (2) a a since the formule will generally be required for cases such that these angles may be obtained with sufficient accuracy by means of their sines. Then we have R sin (z + 4) os R sin (Z" + os sin -z"1 cos 3", (3) sin z " sin z from which to find p and p". These having been found, we have tanQ-O^z psin(-~) tan (I- 0) == \ - p cos (A L 4- )-R p tanj i sin b t- ~, for the determination of I and b, and similarly for I' and b". The 312 THEORETICAL ASTRONOMY. inclination of the orbit and the longitude of the ascending node are then found by means of the formulae (75)3, and the arguments of the latitude by means of (77)3. Since tu"- u is the distance on the celestial sphere between two points of which the heliocentric spherical co-ordinates are 1, b, and 1', b", we have, also, the equations sin (u" - u) sin B -= cos b" sin (1"- 1), sin (au" - u) cos B = cos b sin b" - sin b cos b" cos (" - 1), cos (u" - u) - sin b sin b" -+ cos b cos b" cos (" - 1), for the determination of't"- iA, the angle opposite the side 90~ - b" of the spherical triangle being denoted by B. The solution of these equations is facilitated by the introduction of auxiliary angles, as already illustrated for similar cases. In a circular orbit, the eccentricity being equal to zero, ua - u expresses the mean motion of the planet during the interval t" — t, and we must also have t" —t -- (a" - u), (5) the value of k being expressed in seconds of arc, or log k 3.5500066. These formule will be applied only when the interval t" -- t is small, and for the case of the asteroid planets we may first assume a -2.7, which is about the average mean distance of the group. With this we compute p and p" by means of the equations (2) and (3), and the corresponding heliocentric places by means of (4). If the inclination is small, a u'- a will differ very little from 1"- 1. Therefore, in the first approximation, when the heliocentric longitudes have been found, the corresponding value of t"- t may be obtained from equation (5), writing " - 1 in place of "l - u. If this comes out less than the actual interval between the times of observation, we infer that the assumed value of a is too small; but if it comes out greater, the assumed value of a is too large. The value to be used in a repetition of the calculation may be computed from the expression log a - (log ( - t) + log k -- log (a - )), the difference ut"-u being expressed in seconds of arc. With this we recompute p, p", 1, and I", and find also b, b", 2, i,'t, and tL". Then, if the value of a computed from the last result for u"s- u differs from the last assumed value, a further repetition of the calcu CIRCULAR ORBIT. 313 lation becomes necessary. But when three successive approximate values of a have been found, the correct value may be readily interpolated according to the process already illustrated for similar cases. As soon as the value of a has been obtained which completely satisfies equation (5), this result and the corresponding values of 2, i, and the argument of the latitude for a fixed epoch, complete the system of circular elements which will exactly satisfy the two observed places. If we denote by u0 the argument of the latitude for the epoch T, we shall have, for any instant t, - uo +- t (t -T), L being the mean or actual daily motion computed from k - -3. az The value of u thus found, and r = a, substituted in the formulae for computing the places of a heavenly body, will furnish the approximate ephemeris required. The corrections for parallax and aberration are neglected in the first determination of circular elements; but as soon as these approximate elements have been derived, the geocentric distances may be computed to a degree of accuracy sufficient for applying these corrections directly to the observed places, preparatory to the determination of elliptic elements. The assumption of r'- a will also be sufficient to take into account the term of the second order in the first assumed value of P, according to- the first of equations (94. 104. When approximate elements of the orbit of a heavenly body have been determined, and it is desired to correct them so as to satisfy as nearly as possible a series of observations including a much longer interval of time than in the case of the observations used in finding these approximate elements, a variety of methods may be applied. For a very long series of observations, the approximate elements being such that the squares of the corrections which must be applied to them may be neglected, the most complete method is to form the equations for the variations of any two spherical co-ordinates which fix the place of the body in terms of the variations of the six elements of the orbit; and the differences between the computed places for different dates and the corresponding observed places thus furnish equations of condition, the solution of which gives the corrections to be applied to the elements. But when the observations do not in 314 THEORETICAL ASTRONOMY. elude a very long interval of time, instead of forming the equations for the variations of the geocentric places in terms of the variations of the elements of the orbit, it will be more convenient to form the equations for these variations in terms of quantities, less in number, from which the elements themselves are readily obtained. If no assumption is made in regard to the form of the orbit, the quantities which present the least difficulties in the numerical calculation are the geocentric distances of the body for the dates of the extreme observations, or at least for the dates of those which are best adapted to the determination of the elements. As soon as these distances are accurately known, the two corresponding complete observations are sufficient to determine all the elements of the orbit. The approximate elements enable us to assume, for the dates t and t", the values of A and A"; and the elements computed from these by means of the data furnished by observation, will exactly represent the two observed places employed. Further, the elements may be supposed to be already known to such a degree of approximation that the squares and products of the corrections to be applied to the assumed values of d and A" may be neglected, so that we shall have, for any date, coS A C os + cos d - A,- co Ad8 d3 " (6) doJ dJ" If, therefore, we compare the elements computed from A and A" with any number of additional or intermediate observed places, each observed spherical co-ordinate will furnish an equation of condition for the correction of the assumed distances. But in-order that the equations (6) may be applied, the numerical values of the partial differential coefficients of a and 8 with respect to A and A" must be found. Ordinarily, the best method of effecting the determination of these is to compute three systems of elements, the first from A and A", the second from A + D and A", and the third from A and A" + D", D and D" being small increments assigned to A and A" respectively. If now, for any date t', we compute a' and 8' from each system of elements thus obtained, we may find the values of the differential coefficients sought. Thus, let the spherical co-ordinates for the time t' computed from the first system be denoted by a' and 8'; those computed from the second system of elements, by a' + a sec 8' and A' + d: and those from the third system, by a'+ a" sec a' and J'+ d". Then we shall have VARIATION OF TWO GEOCENTRIC DISTANCES. 315 da' a d cos at' -, -a --- cd D' dzJD' (7) dJ'~D`' dA" D" and the equations (6) give a a"I C0os t, am A ^ A2, cib~ ci" - ~(8) d dl". aB - ^AJ + - ^A2. In the same manner, computing the places for various dates, for which observed places are given, by means of each of the three systems of elements, the equations for the correction of 4 and Ad", as determined by each of the additional observations employed, may be f'ormed. 105. For the purpose of illustrating the application of this method, let us suppose that three observed places are given, referred to the ecliptic as the fundamental plane, and that the corrections for parallax, aberration, precession, and nutation have all been duly applied. By means of the approximate elements already known, we compute the values of a and A" for the extreme places, and from these the heliocentric places are obtained by means of the equations (71)3 and (72)3, writing A cos / and A" cos /i" in place of p and,op. The values of'-, i, u, and utt will be obtained by means of the formule (76)3 and (77)3; and fiom r, r"t and u — u, the remaining elements of the orbit are determined as already illustrated. The first system of elements is thus obtained. Then we assign an increment to 4, which we denote by D, and with the geocentric distances a + D and A" we compute in precisely the same manner a second system of elements. Next,- we assign to A" an increment D", and from a and A"t -I D" a third system of elements is derived. Let the geocentric longitude and latitude for the date of the middle observation computed from the first system of elements be designated, respectively, by,'i and /9'; from the second system of elements, by 2.' and /^; and from the third system, by ),3 and /9'. Then from a = (' - 2') cos i,', d =P —, a" -- (['- ],') COS/ " -, (9) - we compute a, a", d, and dt, and by means of these and the values of D and D" we form the equations 316 THEORETICAL ASTRONOMY. Ad + a Ad" = cos i' a,', d d" D + D for the determination of the corrections to be applied to the first assumed values of d and D", by means of the differences between observation and computation. The observed longitude and latitude being denoted by 2' and 9', respectively, we shall have COS P' A'- (' -,') cos 13', for finding the values of the second members of the equations (10), and then by elimination we obtain the values of the corrections Ad and A/" to be applied to the assumed values of the distances. Finally, we compute a fourth system of elements corresponding to the geocentric distances A + ad and A" + ad" either directly from these values, or by interpolation from the three systems of elements already obtained; and, if the first assumption is not considerably in error, these elements will exactly represent the middle place. It should be observed, however, that if the second system of elements represents the middle place better than the first system, 2,' and 32 should be used instead of,,' and,9/' in the equations (11), and, in this case, the final system of elements must be computed with the distances A + D + AJ and d" + a"/. Similarly, if the middle place is best represented by the third system of elements, the corrections will be obtained for the distances used in the third hypothesis. If the computation of the middle place by means of the final elements still exhibits residuals, on account of the neglected terms of the second order, a repetition of the calculation of the corrections Ad and Azd, using these residuals for the values of the second members of the equations (10), will furnish the values of the distances for the extreme places with all the precision desired. The increments D and D" to be assigned successively to the first assumed values of A and A" may, without difficulty, be so taken that the true elements shall differ but little from one of the three systems computed; and in all the formulae it will be convenient to use, instead of the geocentric distances themselves, the logarithms of these distances, and to express the variations of these quantities in units of the last decimal place of the logarithms. These formulae will generally be applied for the correction of VARIATION OF TWO GEOCENTRIC DISTANCES. 317 approximate elements by means of several observed places, which may be either single observations or normal places, each derived from several observations, and the two places selected for the computation of the elements from A and A" should not only be the most accurate possible, but they should also be such that the resulting elements are not too much affected by small errors in these geocentric places. They should moreover be as distant from each other as possible, the other considerations not being overlooked. When the three systems of elements have been computed, each of the remaining observed places will furnish two equations of condition, according to equations (10), for the determination of the corrections to be applied to the assumed values of the geocentric distances; and, since the number of equations will thus exceed the number of unknown quantities, the entire group must be combined according to the method of least squares. Thus, we multiply each equation by the coefficient of Jd in that equation, taken with its proper algebraic sign, and the sum of all the equations thus formed gives one of the final equations required. Then we multiply each equation by the coefficient of.A" in that equation, taken also with its proper algebraic sign, and the sum of all these gives the second equation required. From these two final equations, by elimination, the most probable values of AJ and AJ" will be obtained; and a system of elements computed with the distances thus corrected will exactly represent the two fundamental places selected, while the sum of the squares of the residuals for the other places will be a minimum. The observations are thus supposed to be equally good; but if certain observed places are entitled to greater influence than the others, the relative precision of these places must be taken into account in the combination of the equations of condition, the process for which will be fully explained in the next chapter. When a number of observed places are to be used for the correction of the approximate elements of the orbit of a planet or comet, it will be most convenient to adopt the equator as the fundamental plane. In this case the heliocentric places will be computed from the assumed values of d and J/, and the corresponding geocentric right ascensions and declinations by means of the formule (106)3 and (107)3; and the position of the plane of the orbit as determined from these by means of the equations (76)3 will be referred to the equator as the fundamental plane. The formation of'the equations of condition for the corrections ad and AI"1 to be applied to the assumed values of the distances will then be effected precisely as in the case of A and fi, the 318 THEORETICAL ASTRONOMY. necessary changes being made in the notation. In a similar manner, the calculation may be effected for any other fundamental plane which may be adopted. It should be observed, further, that when the ecliptic is taken as the fundamental plane, the geocentric latitudes should be corrected by means of the equation (6)4, in order that the latitudes of the sun shall vanish, otherwise, for strict accuracy, the heliocentric places must be determined from A and A" in accordance with the equations (89)v. 106. The partial differential coefficients of the two spherical coordinates with respect to A and A" may be computed directly by means of differential formule; but, except for special cases, the numerical calculation is less expeditious than in the case of the indirect method, while the liability of error is much greater. If we adopt the plane of the orbit as determined by the approximate values of A and /A" as the fundamental plane, and introduce Z as one of the elements of the orbit, as in the equations (72)2, the variation of the geocentric longitude 0 measured in this plane, neglecting terms of the second order, depends on only four elements; and in this case the differential formulae may be applied with facility. Thus, if we express r and v in terms of the elements yp, 1M0, and /a, we shall have dr dr d d dr dM dr da dcv d + dMc d d' and dv dv do' dv dMoi dv dc, da dyd + dd -iMo da' J cd dd' or d (v-+ %) d% dv do dv dM, dv d, dA dJ dp d + di d d + did d' In like manner, we have dr" dr" dco dr" dMo dr" de. d- ~ d-' d- dMo da i df' d'd d (v" +:X) dv" d, d+ " d.0o dv" d+, d% D d d d dd d+ dol dJ' dr d (v - ) dr" d (v" -- x) As soon as the values of d — d~Z, dr- and d are dA dA dJ dA known, the equations necessary for finding the differential coefficients of the elements Z p, 1 ~, and / with respect to A are thus provided. In the case under consideration, when an increment is assigned to J, VARIATION OF TWO GEOCENTRIC DISTANCES. 319 the value of A" remaining unchanged, r" and v" + X are not changed, and hence ddD dA ~~ dr" d (v") I dJ -- O dJ -O To find - and d( Zfrom the equations J cos os s O= x + X, J cos sin 0 = y + Y, in which ^ is the geocentric latitude in reference to the plane of the orbit computed from A and A" as the fundamental plane, and X, Y the geocentric co-ordinates of the sun referred to the same plane, we get dx - cos. cos 0 dJ, dy = cos sin 0 dd, or, substituting for dx and dy their values given by (73),, cos;- cos 0 dA cos u dr - r sin u d (v +- ), cos v sin 0 dJ sinu dr + r cosu d (v + ). Eliminating, successively, d (v -+ ) and dr, we get dr d cos - cos (0.- u), (12) d (v+) 1 (1 d+) _ - Gcos n sin (0 - u). dJ r Therefore, we shall have dz dv dp cdv dMo dv d, 1 d.- d d -+ - +cos ~ sin (0 -- u), d d d dj dd lo d1 d d r ^J' ^ ^J ^^ c dJ dp. dJ r dZr d d dr dMo cdi dli d- -' d A + - T' +- ~\ d' - -= c o s'c o s(0 u ), d+ dv" d" dv" d il dv" d3) _ _1. _ -.. _. da de~d dr"F ddo dr" d dr" dio dr" d,/ d' d d di i da da ~ and if we compute the numerical values of the differential coefficients of r, r", v, and v" with respect to the elements po, 2Ml, and /z, these equations will furnish, by elimination, the values of the four un-.. cl d- dill d a known quantities d ) and diD Id ddi dd di In precisely the same manner we derive the following equations 320 TIHEORETICAL ASTRONOMY. for the determination of the partial differential coefficients of these elements with respect to d":dX dv dp dv cdlM. dv d, d'- d' dA " dM, dJ" d,,' da dr dp dr dMl dr dp,. - -0- + 0+ -o d li" d" d" d dlMo' d' d'(1 dc dv" dy dv" dMi dv" d,. 1.. d"+ dq dr " + dr" d" + dfji d- r" cos - in (O - n"). dr" dJ dr" dMl d " di'. d' +' di" ~ dL" li+ d,.' d - cos cos — co " Since the geocentric latitude V is affected chiefly by a change of the position of the plane of the orbit, while the variation of the longitude 6 is independent of q and i when the squares and products of the variations of the elements are neglected, if we determine the elements which exactly represent the places to which J and d" belong, as well as the longitudes for two additional places, or, if we determine those which satisfy the two fundamental places and the longitudes for any number of additional observed places, so that the sum of the squares of their residuals shall be a minimum, the results thus obtained will very nearly satisfy the several latitudes. Let 0' denote the geocentric longitude of the body, referred to the plane of the orbit computed from A and A" as the fundamental plane, for the date t' of any one of the observed places to be used for correcting these assumed distances. Then, to find the partial differential coefficients of 6' with respect to d and A", we have d o',dO' dz,do' d, do' dhii cos d - coos -. d — + cos -~-, + cos o' d d d/- d- dc ld' d>, dd do' d' +- Cos f' T d.' (15) do',dO' do,dO' d', do' d11 dco' d* d'l" + os dp d'i" + co/ dJoi' dli" dO' dp. and by means of the results thus derived, we form the equation cos7/ 9 0' = cos do' d dA cs' -d,-i" (16) A fourth observed place will furnish, in the same manner, the additional equation required for finding AJ and Ld". If more than two VARIATION OF TWO GEOCENTRIC DISTANCES. 321 observations are used in addition to the fundamental places on which the assumed elements as derived from A and A" are based, the several longitudes will furnish each an equation of condition, and the most probable values of AJ and AzJ will be obtained by combining the entire group of equations of condition according to the method of least squares. 107. In the actual application of these formule to the correction of the approximate elements, after all the preliminary corrections have been applied to the data, we select the proper observed places for determining the elements from the corresponding assumed distances d and 4", according to the conditions which have already been stated, and fronm these we derive the six elements of the orbit. Since the data furnished directly by observation are the right ascensions and the declinations of the body, the elements will be derived in reference to the equator as the plane to which the inclination and the longitude of the ascending node belong. These elements will exactly represent the two fundamental places, and, if the assumed distances 4 and Az are not much in error, they will also very nearly satisfy the remaining places. We now adopt as the fundamental plane the plane of the approximate orbit thus determined, and by means of the equations (83)2 and (85)2, or by means of (87)2, writing a, 8, A', and i' in place of,, 39, 9g, and i, respectively, we compute the values of 0, /, and r for the dates of the several places to be employed. Then the residuals for each of the observed places are found from the formula cos A SO sin r AS +- cosr cos a Aa, A = Cosry -sin rcos ^Ao (1 the values of AC and a8 for each place being found by subtracting from the observed right ascension and declination, respectively, the right ascension and declination computed by means of the elements derived from a and d". The values of 0, V, and r being required only for finding cos'q 0, A^q, and the differential coefficients of 0 and V, with respect to the elements of the orbit, need not be determined with great accuracy. dr d (v + z') Next, we compute d and ( d from equations (12), and from dr dr" dv dv" dr (16)2 the values of dr dr" d' d &c., by means of which, d-' d(p- d(p' d-p Mo, using the value of u in reference to the equator, we form the equations (13). The accent is added to X to indicate that it refers to the 21 322 THEORETICAL ASTRONOMY. equator as the plane for defining the elements. Thus we obtain four equations, from which, by elimination, the values of the differential coefficients of Z', (, Mo, and / with respect to J may be obtained. In the numerical solution, by subtracting the third equation from d"/ the first, the unknown quantity -d is immediately eliminated, so that we have three equations to find the three unknown quantities do, dMo di. dz dj' and -Jd These having been found, d may be obtained from the first or from the third equation. In the same manner we form the equations (14), and thence derive d-' dg dil d, the values of dl' d' d' and d-' Then, by means of the formule (76)2, (78)2, and (79)2, we compute for the date of each place to be employed in correcting the assumed distances the values of do' do' do' COS d,-' cos I' d-,, &c., and hence from (15) the values of cos B' do d/f do I, d/ and cos r da' The results thus obtained, together with the residuals dJl computed by means of the equations (17), enable us to form, according to (16), the equations of condition for finding the values of the corrections AJ and Ad". The solution of all the equations thus formed, according to the method of least squares, will give the most probable values of these quantities, and the system of elements which corresponds to the distances thus corrected will very nearly satisfy the entire series of observations. Since the values of cos' AO' are expressed in seconds of arc, the resulting values of AJ and aA" will also be expressed in seconds of arc in a circle whose radius is equal to the mean distance of the earth from the sun. To express them in parts of the unit of space, we must divide their values in seconds of arc by 206264.8. The corrections to be applied to the elements computed from d and /", in order to satisfy the corrected values A + AJ and. A" -- A/", may be computed by means of the partial differential coefficients already derived. Thus, in the case of X', we have dd dz, Az'Z- -Ad + A ja", from which to find AX'; and in a similar manner ay, A-1x, and Ad d (v - 7') d (v". - /) may be obtained. If, from the values of d ( ) and (- +,') ~~~w~~~~de compdute we compute VARIATION OF TWO GEOCENTRIC DISTANCES. 323 AeV" — (v" + z') Ad AX' AV d A A dA" and apply these corrections to the values of v and v" found from J and A", we obtain the true anomalies corresponding to the distances A + AJ and A" -1- AZi". The corrections to be applied to the values of r and r" derived from d and A" are given by dr acr Ar Ad, ar" ad" If AJ and Ad" are expressed in seconds of arc, the corresponding values of ar and ar" must be divided by 206264.8. The corrected results thus obtained should agree with the values of r and r" computed directly from the corrected values of v, vi', p, and e by means of the polar equation of the conic section. Finally, we have d - sin 7 dD, and similarly for dcz"; and the last of equations (73)2 gives r sin u Ai' - r cos it sin i' A g' - sin,7 A l, rf sin ai' - r" cos t sin i A' - sin ^^ Ai", (18) from which to find hi' and A', u and u" being the arguments of the latitude in reference to the equator. We have also, according to (72)2, Aw A/ C A am' - aC' -- cos i' a la I, A-' 7;A/' + 2sin 2i' a', from which to find the corrections to be applied to ow' and 7r'. The elements which refer to the equator may then be converted into those for the ecliptic by means of the formula which may be derived from (109), by interchanging 2 and I and 180~ - i' and i. The final residuals of the longitudes may be obtained by substituting the adopted values of AJ and adJ in the several equations of condition, or, which affords a complete proof of the accuracy of the entire calculation, by direct calculation from the corrected elements; and the determination of the remaining errors in the values of' will show how nearly the position of the plane of the orbit corresponding to the corrected distances satisfies the intermediate latitudes. Instead of po, M,, and p/, we may introduce any other elements which determine the form and magnitude of the orbit, the necessary 324 THEORETICAL ASTRONOMY. changes being made in the formule. Thus, if we use the elements T, q, and e, these must be written in place of Mo, i, and (o, respectively, in the equations (13), (14), and (15), and the partial differential coefficients of r, r", v, and v" with respect to these elements must be computed by means of the various differential formulae which have already been investigated. Further, in all these cases, the homogeneity of the formulae must be carefully attended to. 108. The approximate elements of the orbit of a heavenly body may also be corrected by varying the elements which fix the position of the plane of the orbit. Thus, if the observed longitude and latitude and the values of g and i are given, the three equations (91)i will contain only three unknown quantities, namely, J, r, and u, and the values of these may be found by elimination. When the observed latitude F/ is corrected by means of the formula (6)4, the latitudes of the sun disappear from these equations, and if we multiply the first by sin (0 - ) sin /, the second (using, only the upper sign) by - cos (0 - ~) sin f9, and the third by - sin (R - 0) cos /9, and add the products, we get sin sin (0 ~ 9 ) tanu — nn (19) cos i sin fe cos ( - ) sin i cos f sin (A - )' from which u may be found. If we multiply the second of these equations by sin /9, and the third by - cos / sin (A - 2), and add the products, we find r -R sin ( ~ - ) (20) sin u (sin i cot f sin (A - fi )- cos i(20) The expression for r in terms of the known quantities may also be found by combining the first and second, or by combining the first and third, of equations (91)1. If we put n cos N - sin A cos (O - ), n sin N= cos sin (A - 0), the formula for u becomes cos Ntan u -=os (N tan (( - ). (21) The last of equations (91)i shows that sin u and sin / must have the same sign, and thus the quadrant in which u must be taken is determined. Putting, also, m cos Mr sin u, m sin M=- sin u cot f sin (A - ), VARIATION OF THE NODE AND INCLINATION. 325 we have -= -_ cos 1 R sin(O - ) cos (M + i) sin t( When any other plane is taken as the fundamental plane, the latitude of the sun (which will then refer to this plane) will be retained in the equations (91)1 and in the resulting expressions for u and r. The value of u may also be obtained by first computing w and 4 by means of the equations (42)3, and then, if z denotes the angle at the planet or comet between the earth and sun, the values of u and z, as may be readily seen, will be determined by means of the relations of the parts of a spherical triangle of which the sides are 1800 - (z + 4), 180~ + 0 - ~, and tu, the angle opposite to the side u being that which we designate by w, and the side 180~ + (- 0 2 being included by this and the inclination i. Let S- 1800 -( + 4-), and, according to Napier's analogies, this spherical triangle gives tan - (S+ )- -) -i )cot( ) cos (i+,ow) 2 --, (23) sin. (i -- w') from which S and u are readily found. Then we have z - 180~ —, - S R sin (24) sin z to find r. If we assume approximate values of 2 and i, as given by a system of elements already known, the equations here given enable us to find r, u', ", and u" from 2,: and A", /, corresponding to the dates t and t" of the fundamental places selected, and from these results for two radii-vectores and arguments of the latitude, the remaining elements may be derived. From these the geocentric place of the body may be found for the date t' of any intermediate or additional observed place, and the difference between the computed and the observed place will indicate the degree of precision of the assumed values of S2 and i. Then we assign to a the increment 8, i remaining unchanged, and compute a second system of elements, and from these the geocentric place for the time t'. We also compute a third system from Q and i + Si, and by a process entirely analogous to that already indicated in the case of the variation of two geocentric 326 THEORETICAL ASTRONOMY. distances, we obtain the numerical values of the differential coefficients of At and /' with respect to 2 and i. Thus the equations COS A cos/' da' d + Cos' dA2 dg' di dg ^ + di for finding the corrections Ax and ai to be applied to the assumed values of these elements, will be formed; and each additional observation or normal place will furnish two equations of condition for the determination of these corrections. If the observed right ascensions and declinations are used directly instead of the longitudes and latitudes, the elements 2 and i must be referred to the equator as the fundamental plane, and the declinations of the sun will appear in the formula for u and r obtained from the equations (91)1, thus rendering them more complex. Their derivation offers no difficulty, being similar in all respects to that of the equations (19) and (20), and since they will be rarely, if ever, required, it is not necessary to give the process here in detail. In general, the equations (23) and (24) will be most convenient for finding r and u from the geocentric spherical co-ordinates and the elements 2 and i, since w,', w", and 4"/ remain unchanged for the three hypotheses. When the equator is taken as the fundamental plane, ~ is the distance between two points on the celestial sphere for which the geocentric spherical co-ordinates are A, D and a,', those of the sun being denoted by A and D. Hence we shall have sin. sin B cos 8 sin (a - A), sin. cos B = cos D sin - sin D cos a cos (a A), (26) cos n -= sin D sin a cos cos os S cos (a - A), from which to find ~ and B, the angle opposite to the side 90~ -- of the spherical triangle being denoted by B. Let K denote the right ascension of the ascending node on the equator of a great circle passing through the places of the sun and comet or planet for the time t, and let w0 denote its inclination to the equator; then we shall have sin w0 cos (A - K) = cos B, sin w sin (A - K) - sin B sin D, (27) cos WU = sin B cos D, from which to find w0 and K. In a similar manner, we may com VARIATION OF THE NODE AND INCLINATION. 327 pute the values of v" —, Qg, and i from the heliocentric spherical co-ordinates 1, b and 1", b". From the equations tan (& + ) COs i (i' —') cot K('- (2), tan(o - sin ~ (i' -- T sin (i' - w) o the accents being added to distinguish the elements in reference to the equator from those with respect to the ecliptic, the values of S, and ut (in reference to the equator) may be found. Let so denote the angular distance between the place of the sun and that point of the equator for which the right ascension is K, and the equation cot so cos wo cot (K- A) (29) gives the value of so, the quadrant in which it is situated being determined by the condition that cos s and cos(K- A) shall have the same sign. Then we have S = S0- s,, nd z 1800 - - - So+- S (30) R sin (30) r -. sin z from which to find r. 109. In both the method of the variation of two geocentric distances and that of the variation of g and i, instead of using the geocentric spherical co-ordinates given by an intermediate observation, in forming the equations for the corrections to be applied to the assumed quantities, we may use any other two quantities which may be readily found from the data furnished by observation. Thus, if we compute r' and u' for the date of a third observation directly from each of the three systems of elements, the differences between the successive results will furnish the numerical values of the partial differential coefficients of r' and u' with respect to J and d", or with respect to Q and i, as the case may be. Then, computing the values of r' and u' from the observed geocentric spherical co-ordinates by means of the values of Q and i for the system of elements to be corrected, the differences between the results thus derived and those obtained directly from the elements enable us to form the equations du' dau' ( d - + dz- -- lxu', dJ dJ/,dr' dr' (31) A zJ - WdJ A' Ar', dJ Wj I 328 THEORETICAL ASTRONOMY. or the corresponding expressions in the case of the variation of 2 and i, by means of which the corrections to be applied to the assumed values will be determined. In the numerical application of these equations, Au' being expressed in seconds of arc, ^rt should also be expressed in seconds, and the resulting values of AJ and Ad" will be converted into those expressed in parts of the unit of space by dividing them by 206264.8. When only three observed places are to be used for correcting an approximate orbit, from the values of r, r', r" and u,.', iu" obtained by means of the formulae which have been given, we may find p and a or - — the latter in the case of very eccentric orbits-from the first a - and second places, and also from the first and third places. If these results agree, the elements do not require any correction; but if a difference is found to exist, by computing the differences, in the case of each of these two elements, for three hypotheses in regard to J and S" or in regard to 2 and i, the equations may be formed by means of which the corrections to be applied to the assumed values of the two geocentric distances, or to those of a and i, will be obtained. 110. The formule which have thus far been given for the correction of an approximate orbit by varying the geocentric distances, depend on two of these distances when no assumption is made in regard to the form of the orbit, and these formula apply with equal facility whether three or more than three observed places are used. But when a series of places can be made available, the problem may be successfully treated in a manner such that it will only be necessary to vary one geocentric distance. Thus, let x, y, z be the rectangular heliocentric co-ordinates, and r the radius-vector of the body at the time t, and let X, Y, Z be the geocentric co-ordinates of the sun at the same instant. Let the geocentric co-ordinates of the body be designated by x0, y,, ZQ, and let the plane of the equator be taken as the fundamental plane, the positive axis of x being directed to the vernal equinox. Further, let p denote the projection of the radiusvector of the body on the plane of the equator, or the curtate distance with respect to the equator; then we shall have 0o p cosa, yo P in, z0o- tan (32) If we represent the right ascension of the sun by A, and its declination by D, we also have VARIATION OF ONE GEOCENTRIC DISTANCE. 329 X =_ R cos D cos A, Y —- R cos D sin A, Z — R sin D. (33) The fundamental equations for the undisturbed motion of the planet or comet, neglecting its mass in comparison with that of the sun, are d2x kx d2y kFy d z klz +' O,- - " dt2 r3 dt2 r3 dt + r 0; but since x- = = —yoSY Xo, O, - zZ, and, neglecting also the mass of the earth, d2X k2X d2 Y k2 y dZ k2Z ct2 + + A' d t'' + -- -o these become dXo 2 o I7,2 1 ~04 dt2 r3 (R3 r3 d'x -o+~x(I__-) —o dt2y + I Ok2RL 3 O,1=0 (34) dt2 r3 " r3 dt2 + -3 + kZ ( -r3-'o Substituting for x0, y,, and z0 their values in terms of a and 8, and putting (R3 r3)k ]Y(:3 r3)=)7 kZ(R3-r3)=' (33) we get d2x, +k2p d- + — cos + -=-O0, dlyo k'o ^+^ ^-+^0, (36) d sin c- + w 0, (36) d - - +p tan - +C 0. Differentiating the equations (32) with respect to t, we find dxo dp. dca * cos c ~ - p sin cc, cdt ct dt d sin a t- + p cosa d (37) dt= tan dt + P sec2 d 330 THEORETICAL ASTRONOMY. Differentiating again with respect to t, and substituting in the equations (36) the values thus found, the results are 2kp d'p( da\ dp da\ by- cos a, and add the products, we obtain dp_ _ - Coa - 2siP dat2 0(38) rdt t d dt dt k k;2p d2p d2CC 2,dd ddo 3 + tn tsil dl d-+ t + 2 se COS a (38) t sin cos a 2 (- R ~ - ) R cos D )2 sin (a Jps ), dt d da and the fpreceding s equation becomes by- cos c ), and a dd t he products, we obta in d2o. dp ( ( 2 -t + ddt2 - da (89) The value of -d- thus found is independent of the diffeential coa sin (a p A). Then, adding the products, since sin A = e cos A, the result is ~2 d( cot (a - A) d -- cot a sec- a - ] at \ dt dt d by sinA tan, the secnd by - cos A tan, and the third by dt- sin cot (a - A)dding the produc ts, since sin A cos A, from which we get da2 d2a C 2ecau"( dJ2 d - - cot (a - A) d + sec 2 + cot + ) cot dp dt' dt2 dt' dt' -T=p ca dd (40) cot ( - A) - - cot 8-sec" a dt dt VARIATION OF ONE GEOCENTRIC DISTANCE. 331 When the ecliptic is taken as the fundamental plane, the last term of the numerator of the second member of this equation vanishes, and the epuation may be written d, (41) dt the coefficient C being independent of p. 111. When the value of p is given, that of d will be determined in terms of the data furnished directly by observation and of the differential coefficients of a and ~ with respect to t from equation (39), or from (40), the latter being preferred when the motion of the body in right ascension is very slow. The value of d- having been found, we may compute the velocities of the body in directions parallel to the co-ordinate axes. Thus, since Xo x + X, o -- y + Y, z= + Z, the equations (37) give dx dp. da dX dtco dt sn -- ddt' dy dp duc dY dt -sin a + p cos dt dt (42) clt dt dZ dz _ dp dB dZ -- tan - +- p sec' 8 - dt dt dt dt dx dy dz by means of which d-x, d- and d may be determined. -y means of which dt dt dt dX dY dZ To find the values of dt- d) and -, the equations dt dt dt X R cos 0, Y- R sin 0 cos s, Z - R sin ( sin s, give, by differentiation, dX dR d -- cos 0 — ~ R sin 0 —, cit dt dt dY dR d (43) - sin cos e -d +Rcos 0 cose (43 dZ dR dO( dr= sin sin - + R cos 0 sins -- dt dt dt 332 THEORETICAL ASTRONOMY. Now, according to equation (52),, we have d k / (1 — e2) (1 + nl,) dt 112 (44) mn denoting the mass of the earth, and e0 the eccentricity of its orbit. The polar equation of the conic section gives dr r2e sin v dv dt p dt Let F denote the longitude of the sun's perigee, and this equation gives dR _2eosin (0-__d) d k l//1 m o d?- 2e0sin( - ) d k — /i + 0 eo sinu( -F). (45) dt 1 - eo2 dt V/1 -e2 If we neglect the square of the eccentricity of the earth's orbit, we have simply d~ kl- mo dR ~ dO 1+n ~ ~ - k 1/kV1 +- mo e sinll( - ). (46) dt -- dt aQ dR The values of d and dt having been found by means of these.t t.t r dX dY formulae, the equations (43) give the required results for -, -, and dZ dt dt d, and hence, by means of (42), we obtain the velocities of the comet or planet in directions parallel to the co-ordinate axes. 112. The values of x, y, and z may be derived by means of the equations x - - a cos a cos a X, y = cos 8 sin a - Y, z -- sin - Z, and from these, in connection with the corresponding velocities, the elements of the orbit may be found. The equations (32)1 give immediately the values of the inclination, the semi-parameter, and the right ascension of the ascending node on the equator. Then, the position of the plane of the orbit being known, we may compute r and u directly from the geocentric right ascension and declination by means of the equations (28) and (30). But if we use the values of the heliocentric co-ordinates directly, multiplying the first of equations (93), by cos 2, and the second by sin 2, and adding the products, we have VARIATION OF ONE GEOCENTRIC DISTANCE. 333 r sin u = z cosec i, r cos u x cos g +- y sin 2, from which r and 16 may be found, the argument of the latitude u being referred to the plane of xy as the fundamental plane. The equation r2 = x2 + y2 + z2 gives dr x dx y dy z dz dt r'dt r dt- r dt' (48 and, since dr re sin v dv dv k l/p dt p dt' dt r' we shall have Vp dr e sin v —,, esnv k dt (49) e cosv — 1, r from which to find e and v. Then the distance between the perihelion and the ascending node is given by t0 -- uv. The semi-transverse axis is obtained from p and e by means of the relation P a _ e2 Finally, from the value of v the eccentric anomaly and thence the mean anomaly may be found, and the latter may then be referred to any epoch by means of the mean motion determined from a. In the case of very eccentric orbits, the perihelion distance will be given by P q1 l+e; and the time of perihelion passage may be found from v and e by means of Table IX. or Table X., as already illustrated. The equation (21)1 gives, if we substitute for f its value in terms of p, denote by V the linear velocity of the planet or comet, and neglect the mass, V2r -_ r' = 2'p. dt' Let,0 denote the angle which the tangent to the orbit at the extremity of the radius-vector makes with the prolongation of this radius-vector, and we shall have 334 THEORETICAL ASTRONOMY. dr dx dy dz rVcos.0~o —r - x - y - y -Z, ^r^co^s -^ dt = dt dt+Zdt' so that the preceding equation gives k2p = V2r sin2 40. Hence we derive the equations Vr sin k0 kV/'p, dx y dz (50) Vr cos% —, x clt Y dt dt' from which Vr and,0 may be found. Then, since V2 k 2 ( 1.r a we shall have k2 2k2 r —V, (51) ca r by means of which a may be determined, and then e may be found by means of this and the value of p. The equations (49) and (50) give V2 e cos (u w - ) -= r sin2 0,, and, since V2 2 1 2 r a7 these are easily transformed into 2ae sin (ut - ) = (2a r) sin 2o,, 2ae cos ( - w) =- (2a - r) cos 2o, - r. If we multiply the first of these equations by - cos m and the second by sin m, and add the products; then multiply the first by sin e and the second by cos u, and add, we obtain 2ae sin w =- (2a - r) sin (2+ - + ) - r sin u, (52) 2ae cos w - (2a - r) cos (20, + u) - r cos t, These equations give the values of o and e. 113. We have thus derived all the formulae necessary for finding the elements of the orbit of a heavenly body from one geocentric distance, provided that the first and second differential coefficients of a and 8 with respect to the time are accurately known. It remains, VARIATION OF ONE GEOCENTRIC DISTANCE. 335 therefore, to devise the means by which these differential coefficients may be determined with accuracy from the data furnished by observation. The approximate elements derived from three or from a small number of observations will enable us to correct the entire series of observations for parallax and aberration, and to form the normal places which shall represent the series of observed places. We may now assume that the deviation of the spherical co-ordinates computed by means of the approximate elements from those which would be obtained if the true elements were used, may be exactly represented by the formula - A + Bh + Ch, (53) h denoting the interval between the time at which the deviation is expressed by A and the time for which this difference is AO. The differences between the normal places and those computed with the approximate elements to be corrected, will then suffice to form equations of condition by means of which the values of the coefficients A, B, and C may be determined. The epoch for which h 0 may be chosen arbitrarily, but it will generally be advantageous to fix it at or near the date of the middle observed place. If three observed places are given, the difference between the observed and the computed value of each right ascension will give an equation of condition, according to (53), and the three equations thus formed will furnish the numerical values of A, B, and C. These having been determined, the equation (53) will give the correction to be applied to the computed right ascension for any date within the limits of the extreme observations of the series. When more than three normal places are determined, the resulting equations of condition may be reduced by the method of least squares to three final equations, from which, by elimination, the most probable values of A, B, and C will be derived. In like manner, the corrections to be applied to the computed latitudes may be determined. These corrections being applied, the ephemeris thus obtained may be assumed to represent the apparent path of the body with great precision, and may be employed as an auxiliary in determining the values of the differential coefficients of a and 8 with respect to t. Let f(a) denote the right ascension of the body at the middle epoch or that for which h 0, and let f(a - no) denote the value of a for any other date separated by the interval nwo, in which (o is the interval between the successive dates of the ephemeris. Then, if we put n successively equal to 1, 2, 3, &c., we shall have 336 THEORETICAL ASTRONOMY. Function. I. Diff. II. Diff. III. Diff. IV. Diff: V. Diff. f(a f, ( - ~ f ) f (a-I f-) ( ) (a) f'"_(a f(- ) ) f+) (a q,o) "' fa (a ) f (a fv oa) f, (a )- w) f (a -., ) f( (-f", (a - 2 ) fiv( + ) f(a +) 3 ( ) manner in either direction. If we exp an f + n) into a series, the result is /S f~(a + ld) a+ + + &c f ( ) + - ++- + f++l- + or, putting for brevity A dt w, B —2 (t c2, &c., f(a +- n - (a + A +B2+ C + D 4+ &c. If we now put n successively equal to -4, -3, -2, -1, 0 +1, &c., we obtain the values of f(a - 4co), f (cta 3),...... f (a + 4o) in terms of A, B, C, &c. Then, taking the successive orders of differences and symbolizing them as indicated above, we obtain a e series of equations mans of which A, B, C, &.e will bte determined in terms of the successive orders of differences.I Finally, replacing A, B, C, &c. by the quantities which they represent, and putting f' + (a -.,w ) - lf' (a + ) - -f' (a), f f'" ) ('a -,) + f + -,) "' (a), &c., we obtain ) + i f -f c - f ( a) - o ()ft ( ta) - A/f" (a) + &c.), d'a 1 dt - (- (a)- - (a) + () viii (a) + &c.), ct3 = (fa (a) - f f(a) + Bfw" (a) - &.), d4w -o w (p v (a - f Vc(a) + eqfuii (a) - &c.), (54) d& e ( (a f (a - ).f vii (a) + &c(), dt o 1 ( t ( ) dts6e - (fvi() - m Vio (a) + &c.), dt - (fr(a) — ) )+&c.), (a) - ( a) -If.. - (a) + V(a - v( ) - -&C.), VARIATION OF ONE GEOCENTRIC DISTANCE. 337 by means of which the successive differential coefficients of a with respect to t may be determined. The derivation of these coefficients in the case of 8 is entirely analogous to the process here indicated for a. Since the successive differences will be expressed in seconds of arc, the resulting values of the differential coefficients of a and 8 withrespect to t will also be expressed in seconds, and must be divided by 206264.8 in order to express them abstractly. du cld 2 d8 cd2 We may adopt directly the values of, dt-' - and ddetermined by means of the corrected ephemeris, or, if the observed places do not include a very long interval, we may determine only the values dofa d. d d of d-7' do' &c. by means of the ephemeris, and then find cl- and dt directly from the normal places or observations. Thus, let A as, aft be three observed right ascensions corresponding to the times t, t' t", and we shall have d' d'' (t, t)icit d3 d4 t) & which give d' (t' —t)'~'d" a- a ddl' d t4 T- dt -'-i (t' — t). +' -> - t) -t -4 &c., u~t"dd' a'!"- / d',, d "4-,' +t l (t-, - - (') -~ -- ~'- t ~ 3~ — t ~') ~ -- c. t ~ dtZ dt4 These equations, being solved numerically, will give the values of cI2& dt and, and we may thus by triple combinations of the observed dt2 places, using always the same middle place, form equations of condition for the determination of the most probable values of these differential coefficients by the solution of the equations according to the method of least squares. In a similar manner the values of t- and -d may be derived. 114. In applying these formula to the calculation of an orbit, after the normal places have been derived, an ephemeris should be computed at intervals of four or eight days, arranging it so that one of the dates shall correspond to that of the middle observation or normal place. This ephemeris should be computed with the utmost 22 338 THEORETICAL ASTRONOMY. care, since it is to be employed as an auxiliary in determining quantities on which depends the accuracy of the final results. The comparison of the ephemeris with the observed places will furnish, by means of equations of the form A + Bh + C+h2 _ A,' A' + B'h + C'h- - a', h being the interval between the middle date t' and that of the place used, the values of A, B, C, A', &c.; and the corrections to be applied to the ephemeris will be determined by A + Bnw + Cnw2( = a-, A' -+ B'nw + C'n2o2-= A. The unit of h may be ten days, or any other convenient interval, observing, however, that c(o in the last equations must be expressed in parts of the same unit. With the ephemeris thus corrected, we dcu d2a d5 d8, compute the values of dt, dt-, and d as already explained. These differential coefficients should be determined with great care, since it is on their accuracy that the subsequent calculation principally dedX ddY dZ pends. We compute, also, the velocities dt - dt and dt by means of the formulae (43), d and being computed from (46). The dt dt quantities thus far derived remain unchanged in the two hypotheses with regard to A. Then we assume an approximate value of A, and compute p a cos; and by means of the equation (40) or (39) we compute the value of dp dt It will be observed that if we put the equation (40) in the form dp P P + cotS, the coefficient P remains the same in the two hypotheses. The three dp equations (38) may be so combined that the resulting value of d'a dt will not contain dt. This transformation is easily effected, and may d2Ca be advantageous in special cases for which the value of is very uncertain. The heliocentric spherical co-ordinates will be obtained from the RELATION BETWEEN TWO PLACES IN THE ORBIT. 339 assumed value of a by means of the equations (106)3, and the rectangular co-ordinates from x - r cos b cos I, y r cos b sin 1, z - r sin b. dx dy dz The velocities d-, d, and c will be given by (42), and from these and the co-ordinates x, y, z the elements of the orbit will be computed by means of the equations (32)1, (47), (49), &c. With the elements thus derived we compute the geocentric places for the dates of the normals, and find the differences between computation and observation. Then a second system of elements is computed from a + AD, and compared with the observed places. Let the difference between computation and observation for either of the two spherical co-ordinates be denoted by n for the first system of elements, and by n' for the second system. The final correction to be applied to a, in order that the observed place may be exactly represented, will be determined by ~ (n' — n) + n - O. (56) Each observed right ascension and each observed declination will thus furnish an equation of condition for the determination of AJ, observing that the residuals in right ascension should in each case be multiplied by cos A. Finally, the elements which correspond to the geocentric distance A + AD will be determined either directly or by interpolation, and these must represent the entire series of observed places. 115. The equations (52)3 enable us to find two radii-vectores when the ratio of the corresponding curtate distances is known, provided that an additional equation involving r, r", x, and known quantities is given. For the special case of parabolic motion, this additional equation involves only the interval of time, the two radii-vectores, and the chord joining their extremities. The corresponding equation for the general conic section involves also the semi-transverse axis of the orbit, and hence, if the ratio M of the curtate distances is known, this equation will, in connection with the equations (52)3, enable us to find the values of r and r" corresponding to a given value of a. To derive this expression, let us resume the equations 340 THEORETICAL ASTRONOMY. -E" - E- 2e sin (E" -E) cos (E"- E), (5 r + r" 2a - 2ae cos 4 (E" - E) cos (E" + E). For the chord x we have x (r + r")2 - 4rr" cos2 (" -r t)0, which, by means of (58)4, gives 2 = (1' + r//)2 -4a2 (co2 (E/ -E)- 2ecos(- (E/+ E) +-e2 cos (Ez-+E)); and, substituting for r + r" its value given by the last of equations (57), we get X2 42 sin2 (E" - ) (I 2 cos2-(E" + E)). (58) Let us now introduce an auxiliary angle h, such that cos h = e cos 4 (E" - E), the condition being imposed that h shall be less than 1800, and put g (E" — E); then the equations (57) and (58) become -3 2g -2 sing cosh, ay r + r" -_ 2a (1 -cos g cos h), (59) x 2a sin g sin h. Further, let us put h -g =, h -g= ~, and the last two of equations (59) give r + r" + - 4a sin' I,6 r + r" - 4a sin2 I. (60) Introducing 8 and e into the first of equations (59), it becomes ( - sin e) - ( -sin ). (61) a2 The formule (60) enable us to determine e and 8 from r + rl, x, and a, and then the time' =k (t" - t) may be determined from (61). Since, according to (58)4,'rr" cos (u"' u) -a (cos g - cos h) = 2Asin Z sin 8, RELATION BETWEEN TWO PLACES IN THE ORBIT. 341 and since sin s is necessarily positive, it appears that when u" -- exceeds 180~, the value of sin 18 must be negative, and when " — u- 180~, we have 8 -0; and thus the quadrant in which 8 must be taken is determined. It will be observed that the value of E, as given by the first of equations (60), may be either in the first or the second quadrant; but, in the actual application of the formulae, the ambiguity is easily removed by means of the known circumstances in regard to the motion of the body during the interval t~ - t. In the application of the equations (52)3, by means of an approximate value of x we compute d, and thence r and r". Then we compute e and a corresponding to the given value of a, and from (61) we derive the value of tj~k' If this agrees with the observed interval t - t, the assumed value of z is correct; but if a difference exists, by varying x we may,readily find, by a few trials, the value which will exactly satisfy the equations. The formulae (70)3 will then enable us to determine the curtate distances p and p", and from these and the observed spherical co-ordinates the elements of the orbit may be found. As soon as the values of u and u" have been computed, since e - = El- E, we have, according to equation (85)4, sin - (u"- u) cos ( P - ( ~I rr" a sin - (~ - ) which may be used to determine 5p when the orbit is very eccentric. To find p and q, we have p = a cos'2, q - 2a sin2 (450 -_ ); and the value of wo may be found by means of the equations (87)4 or (88),. 116. The process here indicated will be applied chiefly in the determination of the orbits of comets, and generally for cases in which a is large. In such cases the angles e and a will be small, so that the slightest errors will have considerable influence in vitiating the value of t" - t as determined by equation (61); but if we transform this equation so as to eliminate the divisor a' in the first member, the uncertainty of the solution may be overcome. The difference e -sine 342 THEORETICAL ASTRONOMY. may be expressed by a series which converges rapidly when z is small. Thus, let us put e - sin C = y sin sin, x =- sin2 I, and we have dy = 2 cosec -e — y cot -, de:z 2yc9t', de d- = 4 cosec -e. dx Therefore dy 8 -6y cos l 4- 3y ( -- 2x) dx sin2 -e 2x(1 x) If we suppose y to be expanded into a series of the form y = a- fx + rx + - x3 - &c., we get, by differentiation, dy dx= + 2rx + 3x+2 + &c., and substituting for d- the value already obtained, the result is 2jx + (4r - 21) x2 + (68 - 4r) x + &c. 4 - 3a + (6 - 31) x + (6 - 3) X2 + (6r - 3a) x + &c. Therefore we have 4 - 3a 0, 6 - 3f — 23, 63 - 3 -- 4r 23, 6r - 3 - 66 - 4r, from which we get -4, 46 4.6.8 4.6.8.10 "~ P-3.5' r"3.57' """7 579&c. i-ay 4 -3.5.. o. 7.9 & Hence we obtain e-sin e =Si + 4 Snine+ s. 6.8.1sie + c.) (62) and, in like manner, 6 8 4' 6.8.10 a-sin n'(1+sin + sin 4sin ( l ssin sin &c.), (63) 3 2 5 \ 5 T 7 4 5.7.9 4 which, for brevity, may be written e- sin e _ 4 Q sin3, I, sin Q'sin, z (64) -- sin a -- ~ Q' sin3,, RELATION BETWEEN TWO PLACES IN THE ORBIT. 343 Combining these expressions with (61), and substituting for sin 2s and sin 2 their values given by the equations (60), there results 6 — Q (r + r" + x)2 T Q' (r + r" - ), (65) the upper sign being used when the heliocentric motion of the body is less than 180~, and the lower sign when it is greater than 180~. The coefficients Q and Q' represent, respectively, the series of terms enclosed in the parentheses in the second members of the equations (62) and (63), and it is evident that their values may be tabulated with the argument s or 8, as the case may be. It will be observed, however, that the first two terms of the value of Q are identical with the first two terms of the expansion of (cos4s)-2 into a series of ascending powers of sin i, while the difference is very small between the coefficients of the third terms. Thus, we have (cos 2)- - - (1 - sin2 6) — 1 + sinC + s.11 sin 5= 215 1 sin ~ 6. 11. 16, & + 5 10 15sin + and if we put Bo -Q o (66) (cos i-) 512 (COS 4l)e we shall have B - + + si4 T+1 Sin6 I- + &c. (67) In a similar manner, if we put (Q = B', 12 (68) (cos Ia) 1 we find Bo' = 1 + 9 sin4 1 + 2142 sin6 S + &c. (69) Table XV. gives the values of Bo or Bo0 corresponding to s or a from 0~ to 60~. For the case of parabolic motion we have Q-=, Q' -, and the equation (65) becomes identical with (56)3. In the application of these formulhe, we first compute s and 8 by means of the equations (60), and then, having found B0 and B0' by means of Table XV., we compute the values of Q and Q' from (66) and (68). Finally, the time r' k(t"- t) will be obtained from (65), and the difference between this result and the observed interval will 344 THEORETICAL ASTRONOMY. indicate whether the assumed value of X must be increased or diminished. A few trials will give the correct result. 117. Since the interval of time t — t cannot be determined with sufficient accuracy from (65) when the chord z is very small, it becomes necessary to effect a further transformation of this equation. Thus, let us put Q - Q = 6P, x - sin ]e, x'= sin2, and we shall have — P a -- x') ( 1 + ( + x') + 7.9 (2 + xx' + X) + &c. ). Now, when X is very small, we may put cos I -= cos 8, and hence sin2 l sin12 x x' - sin' l -~ sin2 la- si n ~- - 4 cos2' which, by means of equations (60), becomes x - X 8a cos2 t Therefore we have, when x is very small, 40a cs ( + 1 sin2 2 + 8- sin4 + &c.) (70) If we put TO, (71) the equation (65) becomes, using only the upper sign, (r + r" + )- - (r + r" - ) - 6o, (72) which is of the same form as (56)3. Hence, according to the equations (63)3 and (66),, we shall have 2r<' x 2ro' /_, (73) the value of, being found from Table XI. with the argument 2=o - (74) yr -i — 3 RELATION BETWEEN TWO PLACES IN THE ORBIT. 345 It remains, therefore, simply to find a convenient expression for r0', and the determination of x is effected by a process precisely the same as in the special case of parabolic motion. Let us now put P c N q 40a cos'4 and we shall have C os, 1 2.8 3.8.10 4.8.10.12 6 &\ N_ 1+ 7sin"i+ -4sin4 +-7-_ Si4in Q 7. 7.9 "9.11 or, substituting for Q its value in terms of sin es, N- 1 + 3-a sin e + o 6sin 4 + 2o7 in6 e + &c. (75) Therefore, if we put a 40a r + -- cOS4lx, (76) the expression for r,' becomes T-0 Ar. (77) Table XV. gives the value of log N corresponding to values of e from -- 0 to - 60~. If the chord z is given, and the interval of time t1 - t is required, we compute Ar0' by means of (76), and, having found ro' from x, ^r + r" To - 2, as in the case of parabolic motion, we have t' t- Q (3 (1.i +&.- ) 3a (81) and similarly 2n 1/1 + 72i - 2 loge (1/1 +- n2 + n) 4^3^-1 3.1 213' (82) 4n3 1-3 -.12 + 324 Substituting these values in the equation (79), and denoting the series of terms enclosed in the parentheses by Q and Q, respectively, we get 6r' Q (r + r" + x)i + Q'(r + r"- x)- (83) which is identical with equation (65). If we replace m2 by - sin2 g and n2 by - sin2 8 in the expressions for Q and Q', as given by (81) and (82), we shall have the expressions for these quantities in terms of sin sE and sin -, respectively, instead of sin ~i and sin -8 as given by the equations (62) and (63), namely, Q-___1+_-3. 1sQin'T^ lp + 3 Sir in4 I-l Q = I:L~ s2. +1 3 n I5 sin &c., ( Q -2 1 a a, sin2 >d + 7 2-3 sin -, - 9 2 4 s ~ 2P3 1.3.5 si' sin4E ~.&.sin-4-2 +c For the case of an elliptic orbit it is most convenient to use the equations (66) and (68) in finding Q and Q'; but, since the cases of hyperbolic motion are rare, while for those which do occur the eccentricity is very little greater than that of the parabola, it will be sufficient to tabulate Q directly with the argument m. The same table, using n as the argument, will give the value of Q'. Table XVI. gives the values of Q corresponding to values of m from m n 0 to m 0.2. When the values of r + r", r', and a are given, and the chord x is required, we may compute Arot from (78), r0/ from (77), and finally x from (73). It may be remarked, also, that the formulae for the relation between Tr, r + r, x, and a suffice to find by trial the value of a when r + r" and x are given. Hence, in the computation of an orbit from assumed RELATION BETWEEN TWO PLACES IN THE ORBIT. 349 values of 4 and A", the value of x may be computed from r, r", and " - u, and then a may be found in the manner here indicated. If we substitute in the equations (84) the values of sin ~s and sin 18 in terms of r + r", a, and a, and then substitute the resulting values of Q and Q' in the equation (65), we obtain 3 51 6k (t"-t) = (r+r"+x) =F(r+r" ) i+ 3~ - ((r+r"+ x)_.(r+?r, "-) ) I 7 7 (85) + 8 4 + ((r + X + 2)' (r + r"- x)) + &c. the lower sign being used when ui" -t exceeds 180~. When the eccentricity is very nearly equal to unity, this series converges with great rapidity. In the case of hyperbolic motion, the sign of a must be changed. 119. The formulre thus derived for the determination of the chord x for the cases of elliptic and hyperbolic orbits, enable us to correct an approximate orbit by varying the semi-transverse axis a and the ratio M of two curtate distances. But since the formula will generally be applied for the correction of approximate parabolic elements, or those which are nearly parabolic, it will be expedient to use - and a M as the quantities to be determined. In the first place, we compute a system of elements from Ml and f —; and, for the determination of the auxiliary quantities preliminary to the calculation of the values of r, r", and x, the equations (41)3, (50)3, and (51)3 will be employed when the ecliptic is the fundamental plane. But when the equator is taken as the fundamental plane, we must first compute g, K, and G by means of the equations (96)3. Then, by a process entirely analogous to that by which the equations (47)a and (50)3 were derived, we obtain h cos C cos (H — a') - M- cos (a" - a), h cos C sin (H - a") - sin (a" -- ), (86) h sin C =- Mtan 8" - tan 8, from which to find H, C, and h; and also cos

-ls way we obtain normal places at convenient intervals] throughout the entire period during' which the body was observed. From three or more of these norlmal places, a new system of elements should be conmpuited by means of soine one of the niethods which have already been given; and these fundamental places beingl judiciouslly selected, the resultilni, elements will furnish a pretty cl1ose approximatio t to thei truth o tllht te residuals A hich are found bycomparing them with all the direct, observed places may be regarded as indicating very nearly the actual errors of those places. We 1ay thel proceed to inlvestigate the character of the observations mlore fully. But since the observations wxill have been made at iman!l diiferent places, bvy i-fferent observers,.with instrumlents of diiff1re!nt sizes, atIdnd unde'la variety of dissimilat attendantl circumsta.nl esl it may be easily.. understood that the investigation will involve m1uch thalt is vague and uncertain.. In the theory of errors which has been developed 1n fl s-chapter, it has been assumed that all consta nt errors have been duly eliminated, and that the only errors w cllieh remain are those accidental errors which must ever continue in a greater or less degree undetermined. The greater the number and 404 THEORETICA.L ASTRONOMIY. perfection of the observations employed, the more nearly will these errors be determined, and the more nearly will the law of their distribution conform to that which has been assumed as the basis of the method of least squares. When all known errors have been eliminated, there may yet remain constant errors, and also other errors whose law of distribution is peculiar, such as may arise from the idiosyncrasies of the different observers, from the systematic errors of the adopted star-places in the case of differential observations, and from a variety of other sources; and since the observations themselves furnish the only means of arriving at a knowledge of these errors, it becomes important to discuss them in such a manner that all errors which may be regarded, in a sense more or less extended, as regulcr may be eliminated. When this has been accomplished, the residuals which still remain will enable us to form an estimate of the degree of accuracy which may be attributed to the different series of observations, in order that they may not only be combined in the most advantageous manner, but that also no refinements of calculation may be introduced which are not warranted by the quality of the material to be employed. The necessity of a preliminary calculation in which a high degree of accuracy is already obtained, is indicated by the fact that, however conscientious the observer may be, his judgment is unconsciously warped by an inherent desire to produce results harmonizing well among themselves, so that a limited series of places may agree to such an extent that the probable error of an observation as derived from the relative discordances would assign a weight vastly in excess of its true value. The combination, however, of a large number of independent data, by exhibiting at least an approximation to the absolute errors of the observations, will indicate nearly what the measure of precision should be. As soon, therefore, as provisional elements which nearly represent the entire series of observations have been found, an attempt should be made to eliminate all errors which may be accurately or approximately determined. The places of the comparison-stars used in the observations should be determined with care from the data available, and should be reduced, by means of the proper systematic corrections, to some standard system. The reduction of the mean places of the stars to apparent places should also be made by means of uniform constants of reduction. The observations will thus be uniformly reduced. Then the perturbations arising from the action of the planets should be computed by means of formula which will be investigated in the next chapter, and the observed COMBINATION OF OBSERVATIONS. 405 places should be freed from these perturbations so as to give the places for a system of osculating elements for a given date. 147. The next step in the process will be to compare the provisional elements with the entire series of observed places thus corrected; and in the calculation of the ephemeris it will be advantageous to correct the places of the sun given by the tables whenever observations are available for that purpose. Then, selecting one or more epochs as the origin, if we compute the coefficients A, B, C in the equation A0 = A + Br + Cr2, (125) in the case of each of the spherical co-ordinates, by means of equations of condition formed from all the observations, the standard ephemeris may be corrected so that it may be regarded as representing the actual path of the body during the period included by the observations. When the number of observations is considerable, it will be more convenient to divide the observations into groups, and use the differences between computation and observation for provisional normal places in the formation of the equations of condition for the determination of A, B, and C. It thus appears that the corrected ephemeris which is so essential to a determination of the constant errors peculiar to each series of observations, is obtained without first having determined the most probable system of elements. The corrections computed by means of the equation (125) being applied to the several residuals of each series, we obtain what may be regarded as the actual errors of these observations. The arithmetical or probable mean of the corrected residuals for the series of observations made by each observer may be regarded as the average error of observation for that series. The mean of the average errors of the several series may be regarded as the actual constant error pertaining to all the observations, and the comparison of this final mean with the means found for the different series, respectively, furnishes the probable value of the constant errors due to the peculiarities of the observers; and the constant correction thus found for each observer should be applied to the corresponding residuals already obtained. In this investigation, if the number of comparisons or the number of wires taken is known, relative weights proportional to the number of comparisons may be adopted for the combination of the residuals for each series. In this manner, observations which, on account of the peculiarities of the observers, are in a certain sense heterogeneous, may be rendered homogeneous, being reduced to a. standard which 406 THEORETICAL ASTRONOMY. approaches the absolute in proportion as the number and perfection of the distinct series combined are increased. Wrhatever constant error remains will be very small, and, besides, will affect all places alike. The residuals which now remain must be regarded as consisting of the actual errors of observation and of the error of the adopted place of the comparison-star. Hence they will not give the probable error of observation, and will not serve directly for assigning the measures of precision of the series of observations by each observer. Let us, therefore, denote by E the mean error of the place of the comparison-star, by e, the mean error of a single comparison; then will -- be the mean error of In comparisons, and the mean error of the resulting place of the body will, according to equation (35), be given by co C + es2. (126) The value of %, in the case of each series, will be found by means of the residuals finally corrected for the constant errors, and the value of zs is supposed to be determined in the formation of the catalogue of star-places adopted. Hence the actual mean error of an observation consisting of a single comparison will be Vm,- ~/~(E2 _ E2). (127) The value of e, for each observer having been found in accordance with this equation, the mean error of an observation consisting of m comparisons will be The mean error of an observation whose weight is unity being denoted by e, the weight of an observation based on m comparisons will be p 2, (128) The value of e may be arbitrarily assigned, and we may adopt for it ~ 10" or any other number of seconds for which the resulting values of p will be convenient numbers. When all the observations are differential observations, and the stars of comparison are included in the fundamental list, if we do not take into account the number of comparisons on which each observed COMBINATION OF OBSERVATIONS. 407 place depends, it will not be necessary to consider e, and we may then derive E, directly from the residuals corrected for constant errors. Further, in the case of meridian observations, the error which corresponds to es will be extremely small, and hence it is only when these are combined with equatorial observations, or when equatorial observations based on different numbers of comparisons are combined, that the separation of the errors into the two component parts becomes necessary for a proper determination of the relative weights. According to the complete method here indicated, after having eliminated as far as possible all constant errors, including the corrections assigned by equation (125) to be applied to the provisional ephemeris, we find the value of a, given by the equation n^,2 [nvv] - [Mn] s, (129) in which in denotes the number of observations; i m, m n'n" &c. the number of comparisons for the respective observations; and v, v', v", &c. the corresponding residuals. Then, by means of equation (128), assuming a convenient number for E, we compute the weight of each observation. Thus, for example, let the residuals and corresponding values of m be as follows:AO m AO m + 2".0 5, - ".0 7, 1.8 5, + 1.5 5, - 0.4 10, + 4.1 8, -5.5 5, 0.0 5. Let the mean error of the place of a comparison-star be e - 2".0; then we have n - 8, and, according to (129), 8-,2 341.78 - 200.0, which gives e,- + 4".2. Let us now adopt as the unit of weight that for which the mean erroi is — 3".0; then we obtain by means of equation (128), for the weights of the observations, 2.5, 2.5, 5.1, 2.5, 3.6, 2.5, 4.1, 2.5, respectively. 408 THEORETICAL ASTRONOMY. In this manner the weights of the observations in the series made by each observer must be determined, using throughout the same value of s. Then the differences between the places computed from the provisional elements to be corrected and the observed places corrected for the constant error of the observer, must be combined according to the equations (123) and (125), the adopted values of p, p', p", &c. being those found from (128). Thus will be obtained the final residuals for the formation of the equations of condition from which to derive the most probable value of the corrections to be applied to the elements. The relative weights of these normals will be indicated by the sums formed by adding together the weights of the observations combined in the formation of each normal, and the unit of weight will depend on the adopted value of e. If it be desired to adopt a different unit of weight in the case of the solution of the equations of condition, such, for example, that the weight of an equation of average precision shall be unity, we may simply divide the weights of the normals by any number p0 which will satisfy the condition imposed. The mean error of an observation whose weight is unity will then be given by the value of being that used in the determination of the weights the value of e being that used in the determination of the weights 1, p'/ &c. 148. The observations of comets are liable to be affected by other errors in addition to those which are common to these and to planetary observations. Different observers will fix upon different points as the proper point to be observed, and all of these may differ from the actual position of the centre of gravity of the comet; and further, on account of changes in the physical appearance of the comet, the same observer may on different nights select different points. These circumstances concur to vitiate the normal places, inasmuch as the resulting errors, although in a certain sense fortuitous, are yet such that the law of their distribution is evidently different from that which is adopted as the basis of the method of least squares. The impossibility of assigning the actual limits and the law of distribution of many errors of this class, renders it necessary to adopt empirical methods, the success of which will depend on the discrimination of the computer. If e0 denotes the mean error of an observation based on n corm COMBINATION OF OBSERVATIONS. 409 parisons, and e, the mean error to be feared on account of the peculiarities of the physical appearance of the comet, 2 2+ ~o + e2 will express the mean error of the residuals; and if n of these residuals are combined in the formation of a normal place, the mean error of the normal will be given by -- [2 ] + 2. (130) The value of es may be determined approximately from the data furnished by the observations. Thus, if the mean error of a single comparison, for the different observers, has been determined by means of the differences between single comparisons and the arithmetical mean of a considerable number of comparisons, and if the mean error of the place of a comparison-star has also been determined, the equation (126) will give the corresponding value of e02; then the actual differences between computation and observation obtained by eliminating the error of the ephemeris and such constant errors as may be determined, will furnish an approximate value of es by means of the formula 2_ _ _ C 2 in which n denotes the number of observations combined. Sometimes, also, in the case of comets, in order to detect the operation of any abnormal force or circumstance producing different effects in different parts of the orbit, it may be expedient to divide the observations into two distinct groups, the first including the observations made before the time of perihelion passage, and the other including those subsequent to that epoch. 149. The circumstances of the problem will often suggest appropriate modifications of the complete process of determining the relative weights of the observations to be combined, or indeed a relaxation from the requirements of the more rigorous method. Thus, if on account of the number or quality of the data it is not considered necessary to compute the relative weights with the greatest precision attainable, it will suffice, when the discussion of the observations has been carried to an extent sufficient to make an approximate estimate of the relative weights, to assume, without considering the number of comparisons, a weight 1 for the observations at one observatory, a 410 THEORETICAL ASTRONOMY. weight I for another class of observations, 1 for a third class, and so on. It should be observed, also, that when there are but few observations to be combined, the application of the formulae for the mean or probable errors may be in a degree fallacious, the resulting values of these errors being little more than rude approximations; still the mean or probable errors as thus determined furnish the most reliable means of estimating the relative weights of the observations made by different observers, since otherwise the scale of weights would depend on the arbitrary discretion of the computer. Further, in a complete investigation, even when the very greatest care has been taken in the theoretical discussion, on account of independent known circumstances connected with some particular observation, it may be expedient to change arbitrarily the weight assigned by theory to certain of the normal places. It may also be advisable to reject entirely those observations whose weight is less than a certain limit which may be regarded as the standard of excellence below which the observations should be rejected; and it will be proper to reject observations which do not afford the data requisite for a homogeneous combination with the others according to the principles already explained. But in all cases the rejection of apparently doubtful observations should not be carried to a.ny considerable extent unless a very large number of good observations are available. The mere apparent discrepancy between any residual and the others of a series, is not in itself sufficient to warrant its rejection unless facts are known which would independently assign to it a low degree of precision. A doubtful observation will have the greatest influence in vitiating the resulting normal place when but a small number of observed places are combined; and hence, since we cannot assume that the law of the distribution of errors, according to which the method of least squares is derived, will be complied with in the case of only a few observations, it will not in general be safe to reject an observation provided that it surpasses a limit which is fixed by the adopted theory of errors. If the number of observations is so large that the distribution of the errors may be assumed to conform to the theory adopted, it will be possible to assign a limit such that a residual which surpasses it may be rejected. Thus, in a series of m observations, according to the expression (19), the number of errors greater than nr will be (1 2/ -t Qm 1 -- I Jetdt)' \ 1/7T ^0 / COMBINATION OF OBSERVATIONS. 411 and when n has a value such that the value of this expression is less than 0.5, the error nr will have a greater probability against it than for it, and hence it may be rejected. The expression for finding the limiting value of n therefore becomes 97 hr - e-. l-dt 1 2 (131) 77 o 2m' By means of this equation we derive for given values of m the corresponding values of nhr = 0.47694n, and hence the values of n. For convenient application, it will be preferable to use e instead of r, and if we put n' - 0.67449n, the limiting error will be n's, and the values of n' corresponding to given values of m will be as exhibited in the following table. TABLE. m it in 21? m m' 6 1.732 20 2.241 55 2.608 90 2.773 8 1.863 25 2.326 60 2.638 95 2.791 10 1.960 30 2.394 65 2.665 100 2.807 12 2.037 35 2.450 70 2.690 200 3.020 14 2.100 40 2.498 75 2.713 300 3.143 16 2.154 45 2.539 80 2.734 400 3.224 18 2.200 50 2.576 85 2.754 500 3.289 According to this method, we first find the mean error of an observation by means of all the residuals. Then, with the value of m as the argument, we take from the table the corresponding value of a', and if one of the residuals exceeds the value In'e it must be rejected. Again, finding a new value of e from the remaining n - 1 residuals, and repeating the operation, it will be seen whether another observation should be rejected; and the process may be continued until a limit is reached which does not require the further rejection of observations. Thus, for example, in the case of 50 observations in which the residuals -11".5 and +7".8 occur, let the sum of the squares of the residuals be [vv] - 320.4. Then, according to equation (30), we shall have - - 4- 2".56. 412 THEORETICAL ASTRONOMY. Corresponding to the value m 50, the table gives n'- 2.576, and the limiting value of the error becomes n' = 6".6; and hence the residuals -11".5 and +7".8 are rejected. Recomputing the mean error of an observation, we have 320.4 — 193.09 + "65.47 -~i- 1".65. In the formation of a normal place, when the mean error of an observation has been inferred from only a small number of observations, according to what has been stated, it will not be safe to rely upon the equation (131) for the necessity of the rejection of a doubtful observation. But if any abnormal influence is suspected, or if any antecedent discussion of observations by the same observer, made under similar circumstances, seems to indicate that an error of a given magnitude is highly improbable, the application of this formula will serve to confirm or remove the doubt already created. Much will therefore depend on the discrimination of the computer, and on his knowledge of the various sources of error which may conspire continuously or discontinuously in the production of large apparent errors. It is the business of the observer to indicate the circumstances peculiar to the phenomenon observed, the instruments employed, and the methods of observation; and the discussion of the data thus furnished by different observers, as far as possible in accordance with the strict requirements of the adopted theory of errors, will furnish results which must be regarded as the best which can be derived from the evidence contributed by all the observations. 150. When the final normal places have been derived, the differences between these and the corresponding places computed from the provisional elements to be corrected, taken in the sense computation minus observation, give the values of n, n', n", &c. which are the absolute terms of the equations of condition. By means of these elements we compute also the values of the differential coefficients of each of the spherical co-ordinates with respect to each of the elements to be corrected. These differential coefficients give the values of the coefficients a, b, c, a', b', &c. in the equations of condition. The node of calculating these coefficients, for different systems of co-ordinates, and the mode of forming the equations of condition, have been fully developed in the second chapter. It is of great import CORRECTION OF THE ELEMENTS. 413 ance that the numerical values of these coefficients should be carefully checked by direct calculation, assigning variations to the elements, or by means of differences when this test can be successfully applied. In assigning increments to the elements in order to check the formation of the equations, they should not be so large that the neglected terms of the second order become sensible, nor so small that they do not afford the required certainty by means of the agreement of the corresponding variations of the spherical co-ordinates as obtained by substitution and by direct calculation. As soon as the equations of condition have been thus formed, we multiply each of them by the square root of its weight as given by the adopted relative weights of the normal places; and these equations will thus be reduced to the same weight. In general, the numerical values of the coefficients will be such that it will be convenient, although not essential, to adopt as the unit of weight that which is the average of the weights of the normals, so that the numbers by which most of the equations will be multiplied will not differ much from unity. The reduction of the equations to a uniform measure of precision having been effected, it remains to combine them according to the method of least squares in order to derive the most probable values of the unknown quantities, together with the relative weights of these values. It should be observed, however, that the numerical calculation in the combination and solution of these equations, and especially the required agreement of some of the checks of the calculation, will be facilitated by having the numerical values of the several coefficients not very unequal. If, therefore, the coefficient a of any unknown quantity x is in each of the equations numerically much greater or much less than in the case of the other unknown quantities, we may adopt as the corresponding unknown quantity to be determined, not x but,x, v being any entire or fractional number such that the new coefficients, -, &c. shall be made to agree in magnitude with the other coefficients. The unknown quantity whose value will then be derived by the solution of the equations will be vx, and the corresponding weight will be that of vx. To find the weight of x from that of Zx, we have the equation P-^ __ 2 (132) In the same manner, the coefficient of any other unknown quantity may be changed, and the coefficients of all the unknown quantities may thus be made to agree in magnitude within moderate limits, the 414 THEORETICAL ASTRONOMY. advantage of which, in the numerical solution of the equations, will be apparent by a consideration of the mode of proving the calculation of the coefficients in the normal equations. It will be expedient, also, to take for v some integral power of 10, or, when a fractional value is required, the corresponding decimal. It may be remarked, further, that the introduction of v is generally required only when the coefficient of one of the unknown quanitiies is very large, as frequently happens in the case of the variation of the mean daily motion,. When the coefficients of some of the unknown quantities are extremely small in all the equations of condition to be combined, an approximate solution, and often one which is sufficiently accurate for the purposes required, may be obtained by first neglecting these quantities entirely, and afterwards determining them separately. In general, however, this can only be done when it is certainly known that the influence of the neglected terms is not of sensible magnitude, or when at least approximate values of these terms are already given. When we adopt the approximate plane of the orbit as the fundamental plane, the equations for the longitude involve only four elements, and the coefficients of the variations of these elements in the equations for the latitudes are always very small. Hence, for an approximate solution, we may first solve the equations involving four unknown quantities as furnished by the longitudes, and then, substituting the resulting values in the equations for the latitudes, they will contain but two unknown quantities, namely, those which give the corrections to be applied to 2 and i. 151. When the number of equations of condition is large, the computation of the numerical values of the coefficients in the normal equations will entail considerable labor; and hence it is desirable to arrange the calculation in a convenient form, applying also the checks which have been indicated. The most convenient arrangement will be to write the logarithms of the absolute terms n, n', In", &c. in a horizontal line, directly under these the logarithms of the coefficients a, aC, a", &c., then the logarithms of b, b', b", &c., and so on. Then writing, in a corresponding form, the values of log n, log n9', &c. on a slip of paper, by bringing this successively over each line, the sums [nn], [an], [bn], &c. will be readily formed. Again, writing on another slip of paper the logarithms of a, a', a", &c., and placing this slip successively over the lines containing the coefficients, we derive the values [aa], [ab], [ac], &c. The multiplication by b, c, d, CORRECTION OF THE ELEMENTS. 415 &c. successively is effected in a similar manner; and thus will be derived [bb], [be], [bd], &c., and finally [ff] in the case of six unknown quantities. In forming these sums, in the cases of sums of positive and negative quantities, it is convenient as well as conducive to accuracy to write the positive values in one vertical column and the negative values in a separate column, and take the difference of the sums of the numbers in the respective columns. The proof of the calculation of the coefficients of the normal equations is effected by introducing s, s', s", &c., the algebraic sums of all the coefficients in the respective equations of condition, and treating these as the coefficients of an additional unknown quantity, thus forming directly the sums [sn], [as], [bs], [cs], &c. Then, according to the equations (76) and (77), the values thus found should agree with those obtained by taking the corresponding sums of the coefficients in the normal equations. The normal equations being thus derived, the next step in the process is the determination of the values of the auxiliary quantities necessary for the formation of the equations (74). An examination of the equations (54), (55), &c., by means of which these auxiliaries are determined, will indicate at once a convenient and systematic arrangement of the numerical calculation. Thus, we first write in a horizontal line the values of [aca], [ab], [ac],... [as], [a], and directly under them the corresponding logarithms. Next, we write under these, commencing with [ab], the values of [bb], [be], [bd], F.. bs], [bn]; then, adding the logarithm of the factor ab] to the L a[aa] logarithms of [ab], [ac], &c. successively, we write the value of [ab] [] under [bb], that of [a] ac] under [be], and so on. Sub[aa] [a]ct tracting the numbers in this line from those in the line above, the differences give the values of [bb.1], [bc.1],... [bs.1], [bn.], to be written in the next line, and the logarithms of these we write directly under them. Then we write in a horizontal line the values of [cc], [cd],. [s], [en], placing [cc] under [bc.l], and, having added the logarithm of [a] to the logarithms of [ac], [ad], &c. in succession, [aca] we derive, according to the equations (55) and (58), the values of [CC.1], [cd.1],.. [cs.1], [cn.1], which are to be placed under the corresponding quantities [cc], [cd], &c. Next, we subtract from these, respectively, the products [be.ll] [bc.], [b.1] -[b.], [.1] [b], [bb.1]^, [bbd. ] [bs.l], [bb.1], [ub.] [bb.l]' [ bb.1 ] [ubS. 416 THEORETICAL ASTRONOMY. and thus derive the values of [cc.2], [cd.2],.. [cs.2], [cn.2], which are to be written in the next horizontal line and under them their logarithms. Then we introduce, in a similar manner, the coefficients [dd], [de],. [. dn], writing [dd] under [cd.2]; and from each of these in succession we subtract the products [ad] ], [ad] [a], [ad] [aa] aa] [aa] [aa ], thus finding the values of [dd.l], [de.1],.. [dn. 1]. From these we subtract the products [bd.1] [bd.1] [bd.i] [bb.1] [bb.1] [b b],..b [b. 1], respectively, which operation gives the values of [dd.2], [de.2],... [dn.2]. From these results we subtract the products [d.2] [ecd.2], [ed.2] [ [e.2], [ cn.2], [cc.2] [cc.2] [cc.2] and derive [dd.3], [de.3],.. [dn.3] under which we write the corresponding logarithms. Then we introduce [ee], f, ], [es], and [en], writing [ee.] under [de.3]. First, subtracting ] [ae], [] [af],. ~~~~~[ae] l~[aa] [aa] [a] [an], we get [ee.l], [ef.1], [es.1], and [en.1]; then subtracting [aaJ from these the products [b5 e. 1] [b] [be],** [be.1 ][b ] ^^ be~~~[b. 1], r. 111],. [bb.l] [bb.1] [bb.1] ^ we obtain the values of [ee.2], [ef.2], [es.2], and [en.2]. Again, subtracting [c.2] [e.2] [c2 f.2] ce. 2] [cn.2], [ce.2] [cc.2] [ec.2], we have the values of [ee.3], [ef.3], [es.3], [en.3]; and finally, subtracting from these the products [ d. [de. [de.r] rd [de.3ede [df.3],.. [d.3], [dd.3] [dd. dd] [dd.3] we derive the results for [ee.4], [ef.4], [es.4], and [en.4]; under which the corresponding logarithms are to be written. If there are six unknown quantities to be determined, we must further write in a horizontal line the values of [ff], [fs], and [fin], CORRECTION OF THE ELEMENTS. 417 placing [ff] under [ef.4], and by means of five successive subtractions entirely analogous to what precedes, and as indicated by the remaining equations for the auxiliaries, we obtain the values of [ff.5], [fs.5], and [fn.5]. The values of [bs.1], cs.l], [cs.2], &c. serve to check the calculation of the successive auxiliary coefficients. Thus we must have [bb.1] + [be.1] + [bd.1] + [be.1] + [bf.1] = bs.1] [be.1] + [c.1] + [cd.1] + [ce.1] + [cf.1] [cs.1], &c., [cc.2] + [cd.2] + [ce.2] + [cf.2] [s.2], [ed.2] + [dd.2] + [de.2] + [df.2] - [d.2], &c. Hence it appears that when the numerical calculation is arranged as above suggested, the auxiliary containing s must, in each line, be equal to the sum of all the terms to the left of it in the same line and of those terms containing the same distinguishing numeral found in a vertical column over the last quantity at the left of this line. There will yet remain only the auxiliaries which are derived from [sn] and [enn] to be determined. These additional auxiliaries will be found by means of the formulae [a][] [bn.1] [sn.i1]- [as, [Si. [s2] -[ 1]- [bs.1], [ac] [b.1] [sn.3] [sn.2] [c2] [sn.4] [n.3] - [d [ds.3], (133) Ice. 2] [dd.3] [sn.5] = [sn.4] [4] sG] [6] -n [n.5] [n.5] [ee.4] [es.4], [ff.5] [f.5], and the equations (81) and (83). The arrangement of the numerical process should be similar to that already explained. The values of [sn.l], [sn.2], &c. check the accuracy of the results for [bn.1], [cn.l], [cn.2], [dn.3], &c. by means of the equations [bn.1] + [e.1] + [dn.1] + [en.1] -F [n.1.]- [sn.1], [cn.2] + [dn.2] + [en.2] + [fit.2] -[sn.2], [d2.3] + [en.3] + [f0.3] = [sn.3], (134) [en.4] + [fn.4] [sn.4], [0f.5] [-s.5]. It appears further, that, in the case of six unknown quantities, since [fs.5] [ff.5], we have [sn.6] 0. Having thus determined the numerical values of the auxiliaries required, we are prepared to form at once the equations (74), by means of which the values of the unknown quantities will be determined 27 418 THEORETICAL ASTRONOMY. by successive substitution, first finding t from the last of these equations, then substituting this result in the equation next to the last and thus deriving the value of w, and so on until all the unknown quantities have been determined. It will be observed that the logarithms of the coefficients of the unknown quantities in these equations will have been already found in the computation of the auxiliaries. If we add together the several equations of (74), first clearing them of fractions, we get 0 = [a] x + ([ab] + [bb.1]) y + ([a] + [bc.1] + [cc.2]) + ([ad] + [bd.1] + [cd.2] + [dd.3]) u + ([ae] + [be.1] + [ce.2] + [de.3] + [ee.4])w (135) + ([af] + [bf.1] + [cf.2] + [df.3] + [efj4] + [ff.5])t + [an] + [b6.1] + [c.2] + [dn.3] + [en.4] + [fi.5]; and this equation must be satisfied by the values of x, y, z, &c. found from (74). 152. EXAMPLE.-The arrangement of the calculation in the case of any other number of unknown quantities is precisely similar; and to illustrate the entire process let us take the following equations, each of which is already multiplied by the square root of its weight:0.707x + 2.052y - 2.372z - 0.221u + 6".58 0, 0.471x + 1.347y - 1.715z - 0.085u- + 1.63 - 0, 0.260x + 0.770y - 0.356z + 0.483 - 4.40 - 0, 0.092 +- 0.343y + 0.235z + 0.469 - 10.21 - 0, 0.414x + 1.204y - 1.506z - 0.205tu - 3.99 0, 0.040x + 0.150y + 0.104z + 0.206u - 4.34 = 0. First, we derive [nn] - 204.313, [an] =+ 4.815, [aa] + 0.971, [bn] = - 12.961, [ab] + 2.821, [bb] = +8.208, [cn] - 25.697, [ac] - 3.175, [bc] - 9.168, [cc] = + 11.028, [cn] — 10.218, [ad] - 0.104, [bd] - 0.251, [cd] -+ 0.938, [dd] + 0.594, [si] - 18.139, [as] = 0.513, [bs] + 1.610, [cs] - 0.377, [ds] + 1.177. The values of [sn], [as], [bs], [cs], and [ds], found by taking the sums of the normal coefficients, agree exactly with the values computed directly, thus proving the calculation of these coefficients. The normal equations are, therefore, NUMERICAL EXAMPLE. 419 0.971x + 2.821y - 3.175z - 0.104- + 4.815 0, 2.821x + 8.208y - 9.168z - 0.251u + 12.961 0, - 3.175x - 9.168y + 11.028z + 0.938u - 25.697 0, -— 0.104x - 0.251y + 0.938z + 0.594u - 10.218 - 0. It will be observed that the coefficients in these equations are nu-.merically greater than in the equations of condition; andc this will generally be the case. Hence, if we use logarithms of five decimals in forming the normal equations, it will be expedient to use tables of six or seven decimals in the solution of these equations. Arranging the process of elimination in the most convenient form, the successive results are as follows:bb.1] = 0.0123, [bc.1] - 0.0562, [bd.1] = + 0.0511, [bs.1] = + 0.1196, [b?.l] =- 1.0278, [cc.l] = + 0.6463, [cd.1] = + 0.5979, [cs.1] = + 1.3004, [cn.l] =- 9.9528, [cc.2] = + 0.3895, [cd.2] = — 0.3644, [cs.2] = + 0.7539, [cn.2] = - 5.2567, [dd.1] = + 0.5829, [ds.l] = + 1.2319, [dn.l] =- 9.7023, [dd.2] = + 0.3706, [ds.2] = + 0.7350, [dn.2] =- 5.4323, [dd.3] = + 0.0297, [ds.3] = + 0.0297 [dn.3] = — 0.5143, [nn.l] = 180.436, [sn.1] = - 20.6828, [nn.2]= 94.552, [sn.2] - 10.6889, [nn.3]= 23.608, [sn.3]= - 0.5143, [nn.4] = 14.698, [sn.4] = 0. The several checks agree completely, and only the value of [mn.4] remains to be proved. The equations (74) therefore give x +- 2.9052y - 3.2698z- 0.1071u + 4.9588 = 0, y + 4.5691z + 4.1545u - 83.5610 = 0, z + 0.9356u - 13.4960 - 0, u - 17.3165 0, and from these we get + - 17".316, z_ - 2".705, y =+ 23".977, x = - 81".608. Then the equation (135) becomes 0 + 0.9710x + 2.8333y - 2.7293z + 0.3412u - 1.9838, which is satisfied by the preceding values of the unknown quantities. If we substitute these values of x, y, z, and mt in the equations of condition already reduced to the same weight by multiplication by the square roots of their weights, we obtain the residuals + 0".67, - 1".34, + 2".17, - 2".01, - 0".40, - 0".72, The sum of the squares of these gives [vv] =-[nn.4] -~ 11.672, and the difference between this result and the value 14.698 already 420 THEORETICAL ASTRONOMY. found is due to the decimals neglected in the computation of the numerical values of the several auxiliaries. The sum of all the equations of condition gives generally [a] x +- [b] y +- [l] z + [d] +... e + [$] = [v], (136) which may be used to check the substitution of the numerical values in the determination of v, v', &c. Thus, we have, for the values here given, 1.984x + 5.866y - 5.610z + 0.647t - 6.75 - [v] 1."63. It remains yet to determine the relative weights of the resulting values of the unknown quantities. For this purpose we may apply any of the various methods already given. The weights of u and z may be found directly from the auxiliaries whose values have been computed. Thus, we have p = [dd.3] - 0.0297, pZ [dd.3] [c.2] -0.0312. -[dd.2] If we now completely reverse the order of elimination from the normal equations, and determine x first, we obtain the values [bb.2] ~ + 0.0425, [aa.2] + 0.0033, [aa.3] - + 0.00056, [nn.4] 14.665, and also x ~ — 82."750, y -+ 24."365, z- 2."699, u + 17."272. The small differences between these results and those obtained by the first elimination arise from the decimals neglected. This second elimination furnishes at once the weights of x and y, namely, = [aa.3] 0.00056, p [aa.3] [bb.2] - 0.0072. ac [ca.2] We may also compute the weights by means of the equations (96). Thus, to find the'weight of y, we have [dd.2], - dd.1 - [.cd. [e] +- 0.02977, and hence [dd.3] [cc.2] p [da. a ] [cI.2] [bb.1] - 0.0074. [dd.21]b [cc.l1] The equations (103) and (108) are convenient for the determination of the values and weights of the unknown quantities separately. CORRECTION OF THE ELEMENTS. 421 Thus, by means of the values of the auxiliaries obtained in the first elimination, we find from the equations (100), (101), and (102), A' - 2.9052, A" += 16.5442, A"'" - 3.3012, B" -- 4.5691, B"' + 0.1202, C"' - 0.9356, and then the equations (103) and (108) give x - 81".609, y += 23".977, z - - 2".705, u -+ 17".316, pZ 0.00057, py - 0.0074, pZ - 0.0312, p.- 0.0297, agreeing with the results obtained by means of the other methods. The weights are so small that it may be inferred at once that the values of x, y, z, and u are very uncertain, although they are those which best satisfy the given equations. It will be observed that if we multiply the first normal equation by 2.9, the resulting equation will differ very little from the second normal equation, and hence we have nearly the case presented in which the number of independent relations is one less than the number of unknown quantities. The uncertainty of the solution will be further indicated by determining the probable errors of the results, although on account of the small number of equations the probable or mean errors obtained may be little more than rude approximations. Thus, adopting the value of [vv] obtained by direct substitution, we have [mm.4] F1.672 m -= -\A-/ ~ - ~ = = 2.416, and hence r -- 1".629, which is the probable error of the absolute term of an equation of condition whose weight is unity. Then the equations r r?' r= — r, r, give %r ~ 68".25, r = ~ 18".94, r = - 9".22, r =~ 9".45. It thus appears that the probable error of z exceeds the value obtained for the quantity itself, and that although the sum of the squares of the residuals is reduced from 204.31 to 11.67, the results are still quite uncertain. 153. The certainty of the solution will be greatest when the coefficients in the equations of condition and also in the normal equations 422 THEORETICAL ASTRONOMY. differ very considerably both in magnitude and in sign. In the correction of the elements of the orbit of a planet when the observations extend only over a short interval of time, the coefficients will generally change value so slowly that the equations for the direct determination of the corrections to be applied to the elements will not afford a satisfactory solution. In such cases it will be expedient to form the equations for the determination of a less number of quantities from which the corrected elements may be subsequently derived. Thus we may determine the corrections to be applied to two assumed geocentric distances or to any other quantities which afford the required convenience in the solution of the problem, various formulre for which have been given in the preceding chapter. The quantities selected for correction should be known functions of the elements, and such that the equations to be solved, in order to combine all the observed places, shall not be subject to any uncertainty in the solution. But when the observations extend over a long period, the most complete determination of the corrections to be applied to the provisional elements will be obtained by forming the equations for these variations directly, and combining them as already explained. A complete proof of the accuracy of the entire calculation will be obtained by computing the normal places directly from the elements as finally corrected, and comparing the residuals thus derived with those given by the substitution of the adopted values of the unknown quantities in the original equations of condition. If the elements to be corrected differ so much from the true values that the squares and products of the corrections are of sensible magnitude, so that the assumption of a linear form for the equations does not afford the required accuracy, it will be necessary to solve the equations first provisionally, and, having applied the resulting corrections to the elements, we compute the places of the body directly from the corrected elements, and the differences between these and the observed places furnish new values of n, n', n", &c., to be used in a repetition of the solution. The corrections which result from the second solution will be small, and, being applied to the elements as corrected by the first solution, will furnish satisfactory results. In this new solution it will not in general be necessary to recompute the coefficients of the unknown quantities in the equations of condition, since the variations of the elements will not be large enough to affect sensibly the values of their differential coefficients with respect to the observed spherical co-ordinates. Cases may occur, however, in which it may become necessary to recompute the coefficients of one CORRECTION OF THE ELEMENTS. 423 or more of the unknown quantities, but only when these coefficients are very considerably changed by a small variation in the adopted values of the elements employed in the calculation. In such cases the residuals obtained by substitution in the equations of condition will not agree with those obtained by direct calculation unless the corrections applied to the corresponding elements are very small. It may also be remarked that often, and especially in a repetition of the solution so as to include terms of the second order, it will be sufficiently accurate to relax a little the rigorous requirements of a complete solution, and use, instead of the actual coefficients, equivalent numbers which are more convenient in the numerical operations required. Although the greatest confidence should be placed in the accuracy of the results obtained as far as possible in strict accordance with the requirements of the theory, yet the uncertainty of the determination of the relative weights in the combination of a series of observations, as well as the effect of uneliminated constant errors, may at least warrant a little latitude in the numerical application, provided that the weights of the results are not thereby much affected. A constant error may in fact be regarded as an unknown quantity to be determined, and since the effect of the omission of one of the unknown quantities is to diminish the probable errors of the resulting values of the others, it is evident that, on account of the existence of constant errors not determined, the values of the variables obtained by the method of least squares from different corresponding series of observations may differ beyond the limits which the probable errors of the different determinations have assigned. Further, it should be observed that, on account of the unavoidable uncertainty in the estimation of the weights of the observations in the preliminary combination, the probable error of an observed place whose weight is unity as determined by the final residuals given by the equations of condition, may not agree exactly with that indicated by the prior discussion of the observations. 154. In the case of very eccentric orbits in which the corrections to be applied to certain elements are not indicated with certainty by the observations, it will often become necessary to make that whose weight is very small the last in the elimination, and determine the other corrections as functions of this one; and whenever the coefficients of two of the unknown quantities are nearly equal or have nearly the same ratio to each other in all the different equations of condition, this method is indispensable unless the difficulty is reme 424 THEORETICAL ASTRONOMY. died by other means, such as the introduction of different elements or different combinations of the same elements. The equations (113) furnish the values of the unknown quantities when we neglect that which is to be determined independently; and then the equations (114) give the required expressions for the complete values of these quantities. Thus, when a comet has been observed only during a brief period, the ellipticity of the orbit, however, being plainly indicated by the observations, the determination of the correction to be applied to the mean daily motion as given by the provisional elements, in connection with the corrections of the other elements, will necessarily be quite uncertain, and this uncertainty may very greatly affect all the results. Hence the elimination will be so arranged that ad shall be the last, and the other corrections will be determined as functions of this quantity. The substitution of the results thus derived in the equations of condition will give for each residual an expression of the following form:A — o + /As.. Therefore we shall have [vv] [v0v] + 2 [vr] aL + [rr] A/j2, (137) which may be applied more conveniently in the equivalent form [vv] [vo] [v ] [ + r] ( + [r ) (138) The most probable value of Al/ will be that which renders [vv] a minimum, or A-i- [krr] (139) and the corresponding value of the sum of the squares of the residuals is [vv] - [Vov] - [v~r]. (140) [rr] The correction given by equation (139) having been applied to t, the result may be regarded as the most probable value of that element, and the corresponding values of the corrections of the other elements as determined by the equations (114) having been also duly applied, we obtain the most probable system of elements. These, however, may still be expressed in the form 2 + Aoa i+, i + Boad, r + CGoa, &c. CORRECTION OF THE ELEMENTS. 425 the coefficients Ao, B0, Co, &c. being those given by the equations (114), and thus the elements may be derived which correspond to any assumed value of i differing from its most probable value. The unknown quantity A/l will also be retained in the values of the residuals. Hence, if we assign small increments to /, it may easily be seen how much this element may differ from its most probable value without giving results for the residuals which are incompatible with the evidence furnished by the observations. If the dimensions of the orbit are expressed by means of the elements q and e, it may occur that the latter will not be determined with certainty by the observations, and hence it should be treated as suggested in the case of p; and we proceed in a similar manner when the correction to be applied to a given value of the semi-transverse axis a is one of the unknown quantities to be determined. 42(6 THEORETICAL ASTRONOMY. CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. 155. WE have thus far considered the circumstances of the undisturbecl motion of the heavenly bodies in their orbits; but a complete determination of the elements of the orbit of any body revolving around the sun, requires that we should determine the alterations in its motion due to the action of the other bodies of the system. For this purpose, we shall resume the general equations (18),, namely, d', x d2 2 +k2(1~ +n) — 2(1 + m) d+' + k2(+ m) 72 k(1 + 2) d(1) dt2 r dy' d2Iz V dO. + k2(1 +m) -ik2 (1 +n) dz which determine the motion of a heavenly body relative to the sun when subject to the action of the other bodies of the system. We have, further, o-Z I+yy'+ \ n'' 1 xx\+? ~ ~ ~+ zz'' ) & which is called the iperturbinzg function, of which the partial differential coefficients, with respect to the co-ordinates, are dQ m! I x' x x'\ m" Z<' x jx") \ / +ep3 f3/ + + P/s-rt33+ &Cr,, dx 1 1\ m \r' d l+ m 3 p3y ) 1 +,I in 1-3 y )3 + &c., (2) dx I p3 14I' Im\ p r"' and in which iz', mn, &c. denote the ratios of the masses of the several disturbing planets to the mass of the sun, and mi the ratio of the mass of the disturbed planet to that of the sun. These partial differential coefficients, when multiplied by k;(1 + qn), express the PERTURBATTONS. 427 sum of the components of the disturbing force resolved in directions parallel to the three rectangular axes respectively. When we neglect the consideration of the perturbations, the general equations of motion become d2X0 + k (I + m)' 0, dt2 r'03 d + I (1 +. )) YO 0, (3) ci2 0 dt2 + k(1 +.) 0, the complete integration of which furnishes as arbitrary constants of integration the six elements which determine the orbitual motion of a heavenly body. But if we regard these elements as representing the actual orbit of the body for a given instant of time t, and conceive of the effect of the disturbing forces due to the action of the other bodies of the system, it is evident that, on account of the change arising from the force thus introduced, the body at another instant different from the first will be moving in an orbit for which the elements are in some degree different from those which satisfy the original equations. Although the action of the disturbing force is continuous, we may yet regard the elements as unchanged during the element of time dt, and as varying only after each interval dc. Let us now designate by to the epoch to which the elements of the orbit belong, and let these elements be designated by J, o, T 0,, i0, e0, and aO; then will the equations (3) be exactly satisfied by means of the expressions for the co-ordinates in terms of these rigorously-constant elements. These elements will express the motion of the body subject to the action of the disturbing forces only during the infinitesima.l interval dt, and at the time t + cdt it will commence to describe a new orbit of which the elements will differ from these constant elements by increments which are called the perturbations. According to the principle of the variation of parameters, or of the constants of integration, the differential equations (1) will be satisfied by integrals of the same form as those obtained when the second members are put equal to zero, provided only that the arbitrary constants of the latter integration are no longer regarded as pure constants but as subject to variation. Consequently, if we denote the variable elements by 31, w,, i, e, and a, they will be connected with the constant elements, or those which determine the orbit at the instant to, by the equations 428 THEORETICAL ASTRONOMY.'- 3 ( cdm -, r dQ dt,r 7 M o +lo f t',A -, + dt dt, a -- 20+q dt / 4 (4) dci de da i -= io J dt, e eo+ dt, a =a +J- dt c dM1 dr in which -, -, &c. denote the differential coefficients of the eledt dt' ments depending on the disturbing forces. When these differential coefficients are known, we may determine, by simple quadrature, the perturbations JM, J~r, &c. to be added to the constant elements in order to obtain those corresponding to any instant for which the place of the body is required. These differential coefficients, however, are functions of the partial differential coefficients of Q with respect to the elements, and before the integration can be performed it becomes necessary to find the expressions for these partial differential coefficients. For this purpose we expand the function Q into a converging series and then differentiate each term of this series relatively to the elements. This function is usually developed into a converging series arranged in reference to the ascending powers of the eccentricities and inclinations, and so as to include an indefinite number of revolutions; and the final integration will then give what are called the absolute or general perturbations. When the eccentricities and inclinations are very great, as in the case of the comets, this development and analytical integration, or quadrature, becomes no longer possible, and even when it is possible it may, on account of the magnitude of the eccentricity or inclination, become so difficult that we are obliged to determine, instead of the absolute perturbations, what are called the special perturbations, by methods of approximation known as mechanical quadrcatres, according to which we determine the variations of the elements from one epoch to to another epoch t. This method is applicable to any case, and may be advantageously employed even when the determination of the absolute perturbations is possible, and especially when a series of observations extending through a period of many years is available and it is desired to determine, for any instant to, a system of elements, usually called osculating elements, on which the complete theory of the motion may be based. Instead of computing the variations of the elements of the orbit directly, we may find the perturbations of any known functions of, these elements; and the most direct and simple method is to determine the variations, due to the action of the disturbing forces, of any system of three co-ordinates by means of which the position of PERTURBATIONS. 429 the body or the elements themselves may be found. We shall, therefore, derive various formulae for this purpose before investigating the formulae for the direct variation of the elements. 156. Let xz, y, zo be the rectangular co-ordinates of the body at the time t computed by means of the osculating elements M0,, To g0, &c., corresponding to the epoch to. Let x, y, z be the actual co-ordinates of the disturbed body at the time t; and we shall have x - X o- x, y -= + Sy, z =- +z, 8x, oy, and oz being the perturbations of the rectangular co-ordinates from the epoch to to the time t. If we substitute these values of x, y, and z in the equations (1), and then subtract from each the corresponding one of equations (3), we get d +xtk( +) ( -x~ )- k(l + ) d m) dlt2 ~IT r dx' dt2 + k ( + r3) ( ~o + k J ) 2 ( + m) d d2t2 + 9 (1 + +) ( 5 )_ + ) d'( dt - ( dyz d,~t + r(l+I) 9"3 9os d+ Let us now put r - r' + r'; then to terms of the order Or2, which is equivalent to considering only the first power of the disturbing force, we have -3 x 3 1 x-3x3,O r' r \ ro /YO + sy?/ 1( y -3Y~r 3 r 3 7'o O3 o3 1'' 3 r o z0 + az Z( _ \ r -, 3 Lo and hence d'2 2 d + (1 + )d+( +-)( dt2 k I r+ r - dt2 -k (1 + m)d ). S~r ~= S~x + + _ Sz. (7) Pt20 70 3 70 r._ x~0 + 7+ 0 430 THEORETICAL ASTRONOMY. The integration of the equations (6) will give the perturbations 8x, dy, and (z to be applied to the rectangular co-ordinates x0, Yo, zo computed by means of the osculating elements, in order to find the actual co-ordinates of the body for the date to which the integration belongs. But since the second members contain the quantities 8x, 8y, 8z which are sought, the integration must be effected indirectly by successive approximations; and from the manner in which these are involved in the second members of the equations, it will appear that this integration is possible. If we consider only a single disturbing planet, according to the equations (2), we shall have k2 (1 + m) d'P (8) k2 (1 + mn) d~- ( - ) 2 ^^ ^^-YP (8) and these forces we will designate by X, Y, and Z respectively; then, if in these expressions we neglect the terms of the order of the square of the disturbing force, writing w, yo, zo in place of x, y, z, the equations (6) become d28x l' (1 -- m) + dt2 XIo + 3 3~r - x, d Jy 3 7-(1 + m)(3y 8 a (9) dt2 -Y0+3r -, ( 9) d2eZ k (1 + M1,) (3 2o 8r- dt=2 - o 3-r- ro which are the equations for computing the perturbations of the rectangular co-ordinates with reference only to the first power of the masses or disturbing forces. We have, further, p (' - x) + ( - y) + (z' - Z', (10) in which, when terms of the second order are neglected, we use the values xo, yo, z for x, y, and z respectively. 157. From the values of 8x, 8y, and 8z computed with regard to the first power of the masses we may, by a repetition of part of the calculation, take into account the squares and products and even the higher powers of the disturbing forces. The equations (5) may be written thus: VARIATION OF CO-ORDINATES. 431 dt2 X+ 3 / )(( 1-3 d2y- - y k2(1 +'n)1 i \ ^\ y) dt2 r03 ) r dt2 -Z + P(i + (( n13) _Z S ) dt r703 r/ in which nothing is neglected. In the application of these formulae, as soon as ax, 8y, and 6z have been found for a few successive intervals, we may readily derive approximate values of these quantities for the date next following, and with these find x = Xo + ax, y - yo + ay, z = z + z, and hence the complete values of the forces X, Y, and Z, by means of the equations (8). To find an expression for the factor r3 r-3 which will be convenient in the numerical calculation, we have r2 = (o X)2 + (yo + 8y)2 + + + ( + )2 -= _2 x + 2 ys8x + 2yo + 2z0&4 + dX2 + ay2 + 8Z2, and therefore r2:1 2 (xo+ 2 x) 8x + (yo + ay) ay + (zo + Gz) 8z ro r2'0 Let us now put X0 o + ax q=i+ -- ~ + - 4- z (12) 702 r02'02 and fq —1 - 1 - (1+2)-; then we shall have /_o(1 5 5.7 5.7.9, - 3 -q+ 2 4q3 + &c.) (13) and the values off may be tabulated with the argument q. The equations (11) therefore become -- z+k-( in (14) d2 kT (f + az) t2 +(fq -- da), d2y k (1 + ) (fqy_ y) d z k ( ) (fz- z). 432 THEORETICAL ASTRONOMY. The coefficients of 8x, 8y, and 3z in equation (12) may be found at once, with sufficient accuracy, by means of the approximate values of these quantities; and having found the value of f corresponding to the resulting value of q, the numerical values of dt,'xdt~ and d 2Jz d'i d dt2 a dt~ which include the squares and products of the masses, will be obtained. The integration of these will give more exact values of 8x,'y, and Jz, and then, recomputing q and the other quantities which require correction, a still closer approximation to the exact values of the perturbations will result. Table XVII. gives the values of logf for positive or negative values of q at intervals of 0.0001 from q= 0 to q = 0.03. Unless the perturbations are very large, q will be found within the limits of this table; and in those cases in which it exceeds the limits of the table, the value of m3 fq -1may be computed directly, using the value of r in terms of r0 and ox, eay, 8z. In the application of the preceding formule, the positions of the disturbed and disturbing bodies may be referred to any system of rectangular co-ordinates. It will be advisable, however, to adopt either the plane of the equator or that of the ecliptic as the fundamental plane, the positive axis of x being directed to the vernal equinox. By choosing the plane of the elliptic orbit at the time to as the plane of xy, the co-ordinate z will be of the order of the perturbations, and the calculation of this part of the action of the disturbing force will be very much abbreviated; but unless the inclination is very large there will be no actual advantage in this selection, since the computation of the values of the components of the disturbing forces will require more labor than when either the equator or the ecliptic is taken as the fundamental plane. The perturbations computed for one fundamental plane may be converted into those referred to another plane or to a different position of the axes in the same plane by means of the formulae which give the transformation of the co-ordinates directly. 158. We shall now investigate the formulae for the integration of the linear differential equations of the second order which express the variation of the co-ordinates, and generally the formulae for finding the integrals of expressions of the form ff(x) dx and J jf(x) dx2 MECHANICAL QUADRATURE. 433 when the values of f(x) are computed for successive values of x increasing in arithmetical progression. First, therefore, we shall find the integral of f(x) dx within given limits. Within the limits for which x is continuous, we have f(X) -=a + + rx + 8x + x +....; (15) and if we consider only three terms of this series, the resulting equation f(x) - a + fx + rx2 is that of the common parabola of which the abscissa is x and the ordinate f(), and the integral of f(x) dx is the area included by the abscissa, two ordinates, and the included arc of this curve. Generally, therefore, we may consider the more complete expression for f(x) as the equation of a parabolic curve whose degree is one less than the number of terms taken. Hence, if we take n terms of the series as the value off(x), we shall derive the equation for a parabola whose degree is n - 1, and which has n points in common with the curve represented by the exact value of f(x). If we, multiply equation (15) by dx and integrate between the limits 0 and x', we get X' f(x)) dx = i +,"12 + rX3 + I Jt4 +. (16) o If now the values of f(x) for different values of x from 0 to x' are known, each of these, by means of equation (15), will furnish an equation for the determination of a, 9, r, &c.; and the number of terms which may be taken will be equal to the number of different known values of f(x). As soon as a, /, r, &c. have thus been found, the equation (16) will give the integral required. If the values of f(x) are computed for values of x at equal intervals and we integrate between the limits x 0, and x —=n, ax being the constant interval between the successive values ofx) and n the number of intervals from the beginning of the integration, we obtain 92Ax ff() dx an-x + _n2zx2 + l nrx3 + &c. Let us now suppose a quadratic parabola to pass through the points of the curve represented by f(x), corresponding to x = 0, x = Az 28 434 THEORETICAL ASTRONOMY. and x - 2Ax; then will the area included by the arc of this parabola, the extreme ordinates, and the axis of abscissas be 2Ax ff(x) dx Ax (2a + 2A^x + — rAx2). 0 The equation of the curve gives, if we designate thle ordinates of the three successive points by y, y,, and Y2, Y~ f t ^ ~- = = (YY A -2y, ~t yO), 2A1 2AX2 and hence we derive 2Ax ff(x) dx - 1 Ax (yo + 4Y1 + Y2)' 0 In a similar manner, the area included by the ordinates Y2 and y~,corresponding to x- = 2^x and x = 4ax,-the axis of abscissas, and the parabola passing through the three points corresponding to Y2, Y3, and y4, is found to be 4Ax ff(x) dx = ^x (Y2 + 4Y3 + Y4); 2Ax and hence we have, finally, fnAx ff(x) dx - 1 x (n -2 + 4yn-i + y?). (n- 2) Ax The sum of all these gives n^x ff(x) dx 0 (17) A ((yo + qy) + 4 (y, + Yy + Y5 +... y- _ ) + 2 (y + Y4 +... Y_ -2)), by means of which the approximate value of the integral within the given limits may be found. If we consider the curve which passes through four points corresponding to yo, y1 Y2, and y3, we have y =f(x) -+ix + a X2 + rx 3 + for the equation of the curve, and hence, giving to x the values 0, AX, 2Ax, and 3Ax, successively, we easily find MECHANICAL QUADRATURE. 435 Y = 0 1 / - (2y3 - 9y2 + 18y1 - ly), Y- 2^ (Y 3 + 4y - 5y1 + 2yo), =, (Y3 - 3Y2 + 31- Yo) Therefore we shall have 3Ax ff(x) dZ- = ax (yo + 3y1 + 3Y2 + y). (18) 0o In like manner, by taking successively an additional term of the series, we may derive 4Az f(x) dx- 2x (7yo + 32y1 + 12y2 + 32y3 + 7y,), 5A (19) J'(x) d -288 (19yo + 75y1 +0y + + 50y + 75y + 19y). 0 This process may be continued so as to include the extreme values of x for which f(x) is known; but in the calculation of perturbations it will be more convenient to use the finite differences of the function instead of the function itself directly. We may remark, further, that the intervals of quadrature when the function itself is used, may be so determined that the degree of approximation will be much greater than when these intervals are uniform. 159. Let us put Ax- o, and let the value of x for which n — 0 be designated by a; then will the general value be f(x) =f(a + nw), o being the constant interval at which the values of f(x) are given. Hence we shall have dx - owdn, f (x) dx -= f (a + no) dn. If we expand the function f(a -t- nw), we have f(a + nw) — f(a)) + n (a) + fd ) +'+ &c., (20) da 1.2 da 2 1.2.3 da3 436 THEORETICAL ASTRONOMY. and hence df (a) 2ddf(a) Jf(a + nw) dn - C + nf(a) + I 2, + I Wd2) da da 2 (21) C being the constant of integration. The equations (54)6 give'(22) da - () -'(a) + f (a) - -f () + a d2f(a)) (a) dc f (a) 1f V (a) + ovi (a).., d ci~f(c4) ca =fva (af a de? i- -/ f(a) - fvIII(a) +..., in which the functional symbols in the second members denote the different orders of finite differences of the function. Hence we obtain f f(a )+,,) dn- C + f(a) + b, (f' (a) - U1 f ( + _1- fv(), V (Vii(a) +..) +'3(f" (a) - fiv~ (C) + of V ()- _-_ fh (a) F...) + 4' (a - I(a () +- 7Of" () -....) +l5(fiV(a) _ iV (a) + if_ Vi(a) ) (23) + 14on (f (a) - (f" (a) +...) + - _,a (fvi (a) +- fv"' (a) +..) + 41-12 )8fvii ( + 4 If we take the integral between the limits — n and + ~', the terms containing the even powers of n disappear. Further, since the values of the function are supposed to be known for a series of values of r at intervals of a unit, it will evidently be convenient to determine the integral between the required limits by means of the sum of a series of integrals whose limits are successively increased by a unit, such that the difference between the superior and the inferior limit of each integral shall be a unit. Hence we take the first integral between the limits -- and +-2, and the equation (23) gives, after reduction, MECHANICAL QUADRATURE. 437 f(a + nu) dn =f(a) + -f" (a) - -l 7f (a) + g 67-f i (a,) (24) 4 6 4 8 5 of iii(a) + &c. It is evident that by writing, in succession, a + o, a + 2w,.... a + io in place of a, we simply add I to each limit successively, so that we have f(a + tnw) dn =fjf((a + iv) + (n - i) ) d(n -i) 417 6 i -— f(a+i)+i2f. (a+i)a)- f ivf (a+iJ)+L 967/fvi (a+iw) —&C. But since f(a +- nw) dn = f (a + nw) dn +J (a + n) dn.....+f (a+n) dn, - - - i — if we give to i successively the values 0, 1, 2, 3, &c. in the preceding equation, and add the results, we get i +- n=i n=i J a. + ^nw) dn = Vf(ac + mn) + 4 f (a + nU) -~ ~==o n=o ^o-2 n= O n - (25) z -i n=i 17 367 Vi - 1i fiv(a + }) + U7-80aS( + (a ) -+ &C. n==o0 n=o Let us now consider the functions f(a), f (a + no), &c. as being themselves the finite differences of other functions symbolized by'f, the first of which is entirely arbitrary, so that we may put, in accordance with the adopted notation, f(a)'f(a + w) - f(a- ), f(a +,o) -'f(a + o) - f(a + 1) f(a + n) - (a+ - (n + ) 0) - f(a + (n - )). Therefore we shall have n==i f (a + n,) -'f(a + (i + -) ) - Y(a' — -), n=O and also n=i if (a + nw) =f (a + (i + )) -f' (a - w), n==i n = if: (a + nw)=(a'" (a + + )-" - ), &c..6y=0f 2 n= = 6 438 THEORETICAL ASTRONOMY. Further, since the quantity'(ac- wo) is entirely arbitrary, we may assign to it a value such that the sum of all the terms of the equation which have the argument a - co shall be zero, namely,'f(a- ) -- 2 4(a~ —)o + -- ~ 6- 8f (a- )+&c. (26) Substituting these values in (25), it reduces to a + (i + ) i + f (x) dx - f f(a + nw) dn a-t -2 (27) — (r,'ff(a + (i + 3)(t)) + J-3-f(a + (i + ))..-. 5 0f"' (a+(i_+ 1 ) - )+_ vof' (a+ (i+) )-&c. } In the calculation of the perturbations of a heavenly body, the dates for which the values of the function are computed may be so arranged that for n - -, corresponding to the inferior limit, the integral shall be equal to zero, the epoch of f(a- wo) being that of the osculating elements. It will be observed that the equation (26) expresses this condition, the constant of integration being included in'f(ca - 1). If, instead of being equal to zero, the integral has a given value when n -, it is evidently only necessary to add this value to'f(a - -1o) as given by (26). 160. The interval o and the arguments of the function may always be so taken that the equation (27) will furnish the required integral, either directly or by interpolation; but it will often be convenient to integrate for other limits directly, thus avoiding a subsequent interpolation. The derivation of the required formulae of integration may be effected in a manner entirely analogous to that already indicated. Thus, let it be required to find the expression for the integral taken between the limits — ~ and i. The general formula (23) gives aff + nw) dab If f(a) + If' (a) + a-jsf" (a) - 3S-f"' (a) - - 5 "fIf (a) 0o + 61 3f v () + f 3 vi(a) -&c.; and since, according to the notation adopted, f' (a) - 2 (f' (a - o) +f' ( + v)) -f' ( + 1) - If" (a), (28) f"' (a) -f"' (a + o) - i (a) f v (af) f v (a+ at) f vi (a), &c., MECHANICAL QUADRATURE. 439 this becomes ff+(nw) dn=f (a)+if' (a+~w)- I4" (a)-+4.f"'. (a+2w) (29) 0 _ 11 ifiV (a)+(a + ) 19s fi(a) - &c. Therefore we obtain it'~~ (30) 11 v 7-4f... (a+(i+ 11 (+o)+ -If' (a+ito+) );4 _UfV(a7+(i) t Of- 960_ J' (a [ tc) - c. Now we have i i +- i +1 f (a +- nw) dn ff(a - nw) dn (a + no) dn; and if we substitute the values already found for the terms in the second member, and also fi(a + ito) f' (a) + (i + 1)w) -f" (a + (i - I)), f'v(a+ - i) = f ('a + (i + )o) - f(a + (i- )), (31 f Vi (a + ito) _f v ( a + (3 + 1 ) cw) - f V (a + (3- ) ") &G. we get f( (x) dx= )ff (ca +?w1) dn a -i) -_ i - 5'f'( + (i + ) ) + (-Y (i 2) ) + (3 -2),f) -+ ( + ) - a ( +(~ t-o))+T 440f +(a+ ) t)+44f (a+3+ +-) o) 2- orV (a + + 2) t)w - T191/ ofv ( + ( -) ) + &c. which is the required integral between the limits - 2 and i. 161. The methods of integration thus far considered apply to the cases in which but a single integration is required, and when applied to the integration of the differential equations for the variations of the co-ordinates on account of the action of disturbing bodies, they d~x d~y d~z will only give the values of dt-, d- and dt- and another integration becomes necessary in order to obtain the values of Ox, Sy, and Oz. We will therefore proceed to derive formule for the determination of the double integral directly. 440 THEORETICAL ASTRONOMY. For the double integralfff(x) dx we have, since dx" = wdw2, ffof( d2 =d ffr (a + 2W) dn2' The value of the function designated by f(a) being so taken that when n --, ff(a + nw) dn 0, the equation (23) gives o C=- ff(a + nw) dn. Therefore, the general equation is o Sf(a + nw) dn =ff(a ~ nw)dn -+ nf(a) -+ Pn + In- + r4n + l -n + &c. the values of a, t,,... being given by the equations (22). Multiplying this by dn, and integrating, we get o fff(a + nw) dn2 = C' + nf (a + no) dn + In2f(a) +, n- + pn4 + T -n + &c., C' being the new constant of integration. If we take the integral between the limits - 2 and +, we find +1 o ff(a + no) dn2 =ff(a + n) d + -a + + go + &c. From the equation (32) we get, for i 0, o f(a + n,) din - (a) -' (a) + f" (a) - f (a) + &c. (33) Substituting this value, and also the values of a, re, &c.,-which are given by the second niembers of the equations (22),-in the preceding equation, and reducing, we get f(a+~to) dn2='f (a) — f' (a) + — f"' (a) - 8 of (a)+&c (34) MECHANICAL QUADRATURE. 441 Hence i-+ jf(a + nw) dn2'f(a + iw) ~- f' (a + 0i) f"' (f + a W) - 367 (a + ) &C. and i + n-=z n =i ^fffa + n) dn2'f(a + n) - 4^ f' (a + mn) _- n= 0 nn=O (35) n = i n=i - + T -ZU s"' (a + n) - 36 6 f v (a+ 0) +.c. n=o0 = =0 We may evidently consider'f(a -- ),'f(ca + -w), &c. as the differences of other functions, the first of which is arbitrary, so that we have'f(a) -'f (a + -a,) + -- -(a - c) -- -"(a + w) -if"(a -'f(a + W,) ='f(a + 3w) + ~ff(a + -e) "- 7(a + 20) - f' (a) ('a + -no) (a + (n + + f( + ( ) ) r'f(a + (n+1+) w) -'(a + (n - 1) w). Therefore n = i'f (a+no) — i (a+(it+1)))+V' (a+i)- "f (a) "f (Ca-c), n==O n =- i n O n=i Vf _.v (a+nto) —- vfv (a+(i+l)()o)+- iv(a+i )- if (a)-_fv(a-,),&c n=O (a) + f(a - ) -2f"(a -f"' (a - ), &c., and that, since "f(a )lu is arbitrary, we may put "f - ) 4f(a) - a7 (2f" (a) + f" (a - )) + 963760 (3f'i (a) + 2f'v (a - c)) - &c., ( 442 THEORETICAL ASTRONOMY. the integral becomes a +- (i ) u ir f f(x) dx2 w2S fl(a + new) dn2 a- oa) "fc(a + (i + 1) )+ - f(a + io) - f(a + (i + 1) ) (37 4 8 (37) 31 I' 3 60 7!.. -4- +f(a-F)+w-8-F of (-+J(+1) ^+3-))- 08vf (a+o) - which is the expression for the double integral between the limits -2 and i+. The value of'"f(a - ) given by equation (36) is in accordance with the supposition that for n = - the double integral is equal to zero, and this condition is fulfilled in the calculation of the perturbations when the argument a - Co corresponds to the date for which the osculating elements are given. If, for in= — neither the single nor the double integral is to be taken equal to zero, it is only necessary to add the given value of the single integral for this argument to the value of'f(a - tw) given by equation (26), and to add the given value of the double integral for the same argument to the value of "f(t w) given by (36). 162. In a similar manner we may find the expressions for the double integral between other limits. Thus, let it be required to find the double integral between the limits -- and i. Between the limits 0 and 2 we have n- o yf (a + nw) dn f f(a + noW) dn + f (a) + -41 a 1 1 1 + 384P + 38-140r + 46008 + &C. which gives jff(a-+no) djn='2-,f(a) +f(a) — 1 f' (a)+, 4f" (a o () (a)(38) 0 1 i(a) 367 Fv (a)+3 78'7 -fvi (a)+&c.; and this again, by means of (28), gives f(a + nw) din = -fY(ac) + ( + ) f8(a + i) - -1-f (a + +) v) ~ 48 (i++a)}) + -384f" (a+iw) (a-( 0) +3 ) j 8of i+ (a +iow) 3 68 77_f v(a+(i+I) aS)H + si15 46 7o'ofvi (a+-iw)+&c. MECHANICAL QUADRATURE. 443 Therefore, since i i +- i F+ ff(a + _f) dn2 Sf(a n) d f( + + ) dn2, and'f(a + (i + ) W) -f(ca + (i + 1) w) - "f(a + iw), f'(a + (( + ^-) oi) - f(a + (i + 1) V) - f(a + w), f"' (a + (i + ) cl) -f" (a + (i + 1) w) -" (a + iw), &c. we shall have a + i i d f (x) dX2 - " rjf(a + tu) dn2' a-_ - (39) _ —2 S+I)(a-+i)+-If (Ca+'i,) -4 off" (a+-)+6io)-8-f"v (a+iw) —&c. I, which gives the required integral between the limits -- and i. 163. It will be observed that the coefficients of the several terms of the formulae of integration converge rapidly, and hence, by a proper selection of the interval at which the values of the function are computed, it will not be necessary to consider the terms which depend on the fourth and higher orders of differences, and rarely those which depend on the second and third differences. The value assigned to the interval w must be such that we may interpolate with certainty, by means of the values computed directly, all values of the function intermediate to the extreme limits of the integration; and hence, if the fourth and higher orders of differences are sensible, it will be necessary to extend the direct computation of the values of the function beyond the limits which would otherwise be required, in order to obtain correct values of the differences for the beginning and end of the integration. It will be expedient, therefore, to take w so small that the fourth and higher differences may be neglected, but not smaller than is necessary to satisfy this condition, since otherwise an unnecessary amount of labor would be expended in the direct computation of the values of the function. It is better, however, to have the interval o) smaller than what would appear to be strictly required, in order that there may be no uncertainty with respect to the accuracy of the integration. On account of the rapidity with which the higher orders of differences decrease as we diminish w, a limit for the magnitude of the adopted interval will speedily be obtained. The magnitude of the interval will therefore be suggested by the rapidity of the change of value of the function. In the comr 444 THEORETICAL ASTRONOMY. putation of the perturbations of the group of small planets between Mars and Jupiter we may adopt uniformly an interval of forty days; but in the determination of the perturbations of comets it will evidently be necessary to adopt different intervals in different parts of the orbit. When the comet is in the neighborhood of its perihelion, and also when it is near a disturbing planet, the interval must necessarily be much smaller than when it is in more remote parts of its orbit or farther from the disturbing body. It will be observed, further, that since the double integral contains the factor o2, if we multiply the computed values of the function by W2, this factor will be included in all the differences and sums, and hence it will not appear as a factor in the formule of integration. If, however, the values of the function are already multiplied by o2, and only the single integral is sought, the result obtained by the formula of integration, neglecting the factor 02, will be (o times the actual integral required, and it must be divided by ao in order to obtain the final result. 164. In the computation of the perturbations of one of the asteroid planets for a period of two or three years it will rarely be necessary to take into account the effect of the terms of the second order with respect to the disturbing force. In this case the numerical values of the expressions for the forces will be computed by using the values of the co-ordinates computed from the osculating elements for the beginning of the integration, instead of the actual disturbed values of these co-ordinates as required by the formulae (8). The values of the second differential coefficients of Jx, oy, and 6z with respect to the time, will be determined by means of the equations (9). If the interval ce is such that the higher orders of differences may be neglected, the values of the forces must be computed for the successive dates separated by the interval co, and commencing with the date t0 - co corresponding to the argument a - c, t being the date to which the osculating elements belong. Then, since the last terms dp'x d'ay d_,z of the formulae for -dt, dt2 and t involve 8x, 8y, and Jz, which are the quantities sought, the subsequent determination of the differential coefficients must be performed by successive trials. Since the integral must in each case be equal to zero for the date to, it will be admissible to assume first, for the dates to - ~c and to + -o corresponding to the arguments a - o and a, that 8x 0, y = 0, and 8z- 0 and hence that the three differential coefficients, for each VARIATION OF CO-ORDINATES. 445 date, are respectively equal to X0, Y0, and Z,. We may now by integration derive the actual or the very approximate values of the variations of the co-ordinates for these two dates. Thus, in the case of each co-ordinate, we compute the value of'f(a - 1w) by means of the equation (26), using only the first term, and the value of "f(a - w) from (36), using in this case also only the first term. The value of the next function symbolized by "f will be given by 7"(a) = "7(a- ) +'f(a- w). Then the formula (39), putting first i =-1 and then i = 0, and neglecting second differences, will give the values of the variations of the co-ordinates for the dates a - w and a. These operations will be performed in the case of each of the three co-ordinates; and, by means of the results, the corrected values of the differential coefficients will be obtained from the equations (9), the value of &r being computed by means of (7). With the corrected values thus derived a new table of integration will be commenced; and the values of'f(a- lo) and "f(a -- w) will also be recomputed. Then we obtain, also, by adding f(a - o) to f(a), the value of'f(a -+ w), and, by adding this to "f(a), the value of "f(a + w). An approximate value of f(a + w) may now be readily estimated, and two terms of the equation (39), putting i 1, will give an approximate value of the integral. This having been obtained for each of the co-ordinates, the corresponding complete values of the differential coefficients may be computed, and these having been introduced into the table of integration, the process may, in a similar manner, be carried one step farther, so as to determine first approximate values of Jx, 8y, and 8z for the date represented by the argument a + 2w, and then the corresponding values of the differential coefficients. We may thus by successive partial integrations determine the values of the unknown quantities near enough for the calculation of the series of differential coefficients, even when the integrals are involved directly in the values of the differential coefficients. If it be found that the assumed value of the function is, in any case, much in error, a repetition of the calculation may become necessary; but when a few values have been found, the course of the function will indicate at once an approximation sufficiently close, since whatever error remains affects the approximate integral by only onetwelfth part of the amount of this error. Further, it. is evident that, in cases of this kind, when the determination of the values of the differential coefficients requires a preliminary approximate inte 446 THEORETICAL ASTRONOMY. gration, it is necessary, in order to avoid the effect of the errors in the values of the higher orders of differences, that the interval m should be smaller than when the successive values of the function to be integrated are already known. In the case of the small planets an interval of 40 days will afford the required facility in the approximations; but in the case of the comets it may often be necessary to adopt an interval of only a few days. The necessity of a change in the adopted value of (w will be indicated, in the numerical application of the formul, by the manner in which the successive assumptions in regard to the value of the function are found to agree with the corrected results. The values of the differential coefficients, and hence those of the integrals, are conveniently expressed by adopting for unity the unit of the seventh decimal place of their values in terms of the unit of space. 165. Whenever it is considered necessary to commence to take into account the perturbations due to the second and higher powers of the disturbing force, the complete equations (14) must be employed. In this case the forces X, Y, and Z should not be computed at once for the entire period during which the perturbations are to be determined. The values computed by means of the osculating elements will be employed only so long as simply the first power of the disturbing force is considered, and by means of the approximate values of 8x, Jy, and 8z which would be employed in computing, for the next place, the last terms of the equations (9), we must compute also the corrected values of X, Y and Z. These will be given by the second members of (8), using the values of x, y, and z obtained from x -x= + 8x, - Yo + y, z -= zo + sz. We compute also q from (12), and then from Table XVII. find the d'8x d28y corresponding value of f. The corrected values of dta, dt~ y and d'6z dt' dt2'a dt2 will be given by the equations (14), and these being introduced, in the continuation of the table of integration, we obtain new values of Ax, By, and 8z for the date under consideration. If these differ much from those previously assumed, a repetition of the calculation will be necessary in order to secure extreme accuracy. In this repetition, however, it will not be necessary to recompute the coefficients of ax, ay, and Oz in the formula for q, their values being given with sufficient accuracy by means of the previous assumption; and gene VARIATION OF CO-ORDINATES. 447 rally a repetition of the calculation of X, Y, and Z will not be required. Next, the values of 8x, 8y, and 8z may be determined approximately, as already explained, for the following date, and by means of these the corresponding values of the forces X, Y. and Z will be found, and also f and the remaining terms of (14), after which the integration will be completed and a new trial made, if it be considered necessary. In the final integration, all the terms of the formulse of integration which sensibly affect the result may be taken into account. By thus performing the complete calculation of each successive place separately, the determination of the perturbations in the values of the co-ordinates may be effected in reference to all powers of the masses, provided that we regard the masses and co-ordinates of the disturbing bodies as being accurately known; and it is apparent that this complete solution of the problem requires very little more labor than the determination of the perturbations when only the first power of the disturbing force is considered. But although the places of the disturbing bodies as given by the tables of their motion may be regarded as accurately known, there are yet the errors of the adopted osculating elements of the disturbed body to detract from the absolute accuracy of the computed perturbations; and hence the probable errors of these elements should be constantly kept in view, to the end that no useless extension of the calculation may be undertaken. When the osculating elements have been corrected by means of a very extended series of observations, it will be expedient to determine the perturbations with all possible rigor. When there are several disturbing planets, the forces for all of these may be computed simultaneously and united in a single sum, so that in the equations (14) we shall have IX, 2 Y, and.SZ instead of X, Y, and Z respectively; and the integration of the expressions d2ax d12y d 2z for dt2, ~2 and dt will then give the perturbations due to the action of all the disturbing bodies considered. However, when the interval eo for the different disturbing planets may be taken differently, it may be considered expedient to compute the perturbations separately, and especially if the adopted values of the masses of some of the disturbing bodies are regarded as uncertain, and it is desired to separate their action in order to determine the probable corrections to be applied to the values of m, mn, &c., or to determine the effect of any subsequent change in these values without repeating the calculation of the perturbations. 448 THEORETICAL ASTRONOMY. 166. EXAMPLE.-TO illustrate the numerical application of the formulae for the computation of the perturbations of the rectangular co-ordinates, let it be required to compute the perturbations of Eurynome @ arising from the action of Jupiter' from 1864 Jan. 1.0 Berlin mean time to 1865 Jan. 15.0 Berlin mean time, assuming the osculating elements to be the following:Epoch 1864 Jan. 1.0 Berlin mean time. Mo= 10 29' 5".65 " 206_ 439 5.69 Ecliptic and Mean 0 24O-0 36 52 11 Equinox 1860.0 o - 4 36 52.11 o -= 11 15 51.02 log c0 0.3881319 o = 928".55745. From these elements we derive the following values:Berlin Mean Time. o yo zo log r 1863 Dec. 12.0 +'l.53616 -t 1.23012 -0.03312 0.294084, 1864 Jan. 21.0 1.15097 1.59918 0.07369 0.294837, March 1.0 0.69518 1.87033 0.10978 0.300674, April 10.0 +- 0.19817 2.03141 0.13936 0.310864, May 20.0 -0.31012 2.08092 0.16134 0.324298, June 29.0 0.80326 2.02602 0.17523 0.339745, Aug. 8.0 1.26055 1.87959 0.18122 0.356101, Sept. 17.0 1.66729 1.65711 0.17990 0.372469, Oct. 27.0. 2.01414 1.37473 0.17209 0.388214, Dec. 6.0 2.29597 1.04766 0.15870 0.402894, 1865 Jan. 15.0 -2.51077 + 0.68978 - 0.14066 0.416240. The adopted interval is w - 40 days, and the co-ordinates are referred to the ecliptic and mean equinox of 1860.0. The first date, it will be observed, corresponds to to - (o, and the integration is to commence at 1864 Jan. 1.0. The places of.Jupiter derived from the tables give the following values of the co-ordinates of that planet, with which we write also the distances of Eurynome from Jtupiter computed by means of the formula = (X - )2 + (yl- y) + (z'- z) Berlin Mean Time. X' y'' log r' log p 1863 Dec. 12.0 -4.09683 -3.55184 +0.10533 0.73425 0.86866, 1864 Jan. 21.0 3.89630 3.76053 0.10152 0.73368 0.86713, March 1.0 3.68416 3.95803 0.09744 0.73305 0.86292, April 10.0 — 3.46098 -4.14366 +0.09304 0.73237 0.85622, NUMERICAL EXAMPLE. 449 Berlin Mean Time. x/ z/ log log p 1864 May 20.0 -3.22739 -4.31684 +0.08839 0.73164 0.84732, June 29.0 2.98405 4.47693 0.08346 0.73086 0.83656, Aug. 8.0 2.73162 4.62343 0.07827 0.73003 0.82428, Sept. 17.0 2.47085 4.75576 0.07284 0.72915 0.81077, Oct. 27.0 2.20247 4.87345 0.06720 0.72823 0.79628, Dec. 6.0 1.92728 4.97606 6.06134 0.72726 0.78098, 1865 Jan. 15.0 -1.64600 -5.06301 0.05531 0.72625 0.76498. These co-ordinates are also referred to the ecliptic and mean equinox of 1860.0. If we neglect the mass of Eurynome and'adopt for the mass of Jupiter 1047.819' we obtain, in units of the seventh decimal place, 2,'k" =- 4518.27, and the equations (9) become 2'26 4l8 27(xf1xo X )i 0.47346 (3o r Ox) W2 dy = 4518.27( p -, 0.4 73,3 -3 ( r (40) dt\' ro " \ 0 ro W2 -' _ 4518.27( )+ 0.47346 r - ty92 dl83 4518-27 (- 3 ~r ) +( dt P3\ r1'3 \03 7 Substituting for the quantities in the first term of the second member of each of these equations the values already found, we obtain Argument. Date. w2XQ Y2 Yo 2Z a - w 1863 Dec.. 12.0 + 53.00 + 47.09 - 1.43, a 1864 Jan. 21.0 53.71 46.31 0.91, a + w March 1.0 54.23 45.18 - 0.37, a + 2w April 10.0 54.69 43.59. + 0.22, a + 3wo May 20.0 55.23 4151- 0.70, a + 40 June 29.0 56.06 -8.96 1.19, a + 5o Aug. 8.0 57.30 35.92 1.66, a + 6w Sept. 17.0 59.0O 32.47 2.08, a + 7w Oct. 27.0 61.55 28.60 2.43, a + 8w Dec. 6.0 64.85 24.34 2.69, a + 9w 1865 Jan. 15.0 + 69.09 + 19.78 + 2.83, which are expressed in units (' the seventh decimal place. We now, for a first approx:imation, regard the perturbations as 29 45I0 THEORETICAL ASTRONOMY. being equal to zero for the dates Dec. 12.0 and Jan. 21.0, and, in the case of the variation of x, we compute first'f(a _) w - - f' (a - )- - (53.71 - 53.00) - 0.03, 53.71 f(a- ) -f (a) - + + 2.24, and the approximate table of integration becomes f (a - ) + 53.00 f(a- ) — _ 0.03,f(a-w) + 2.24) f(a) + 53.71. " f() + 2.21. Then the formula (39), putting first i =- 1, and then i = 0, gives 53.00 Dec. 12.0 x = + 2.24 + 125 + 6.66, 53.71 Ja,. 21.0 x = + 2.21 + - + 6.69. In a similar manner, we find Dec. 12.0 + —. 5.85 z- 0.16, Jan. 21.0 + 5.82 = 0.14. By means of these results we compute the complete values of the second members of equations (40), Jr being found from'- Yo x + r Y + ZO:70 "o To and thus we obtain dax,d__y d2d Date. 02 dd2 ddy dt2 cdt dt' Dec. 12.0 + 53.86 -t 47.76 1.45 + 8.85, Jan. 21.0 -+ 54.23 + 47.25 -0.96 + 8.63. We now commence anew the table of integration, namely, 2x y. f'f f f f f f' "f +53.86_ 0.02 + 2.2C. +47.76 + 002+ 1.97, -1.45 -0 -0.04, +54.23 542 + 2.24, 4-47.25 +4727 + 1.99, -0.96 0.98 -0.06, +56.45, f49.26, -1o.04, the formation of which is made evilenti by what precedes. We may next assume for approximate values of the differential coefficients, for the date March:0, + 54.6, + 46.7, and — 0.5, respectively; and these give, for this date, NUMERICAL EXAMPLE. 451 54.6 x= + 56.45 -+- 5 = 4 + 61.00, 1-2i y + 49.26 + 46 + 53.15, 12 = + 53.15, a —- 1.04- 0.5- 1.08. 12 By means of these approximate values we obtain the following results:1864 March 1.0 dt - 55.01, - - 53.86, - -1.00, dtt2 dt 2 dt2 r -+- +71.03. Introducing these into the table of integration, we find, for the corresponding values of the integrals, =-1-t 61.03; y - +53.75, Oz= - 1.12. These results differ so little from those already derived from the assumed values of the function that a repetition of the calculation is unnecessary. This repetition, however, gives d' 3x d2sy d23Z d, -- + 55.04, + - 53.91, -- 1.00. d dtd d Assuming, again, approximate values of the differential coefficients for April 10.0, and computing the corresponding values of Ox, oy, and Jz, we derive, for this date, d~d~ dd3 d~z - 2 d = + 48.06, o2 -- 63.19, 2 2 dt2 dt2 dt2 Introducing these into the table of integration, and thus deriving approximate values of Sx, jy, and 8z for May 20, we carry the process one step further. In this manner, by successive approximations, we obtain the following results:dx d2jy 426 Date. 2 d 2 dy 2 d _dt2 dt2 dCt2 1863 Dec. 12.0 + 53.86 + 47.76 - 1.45, 1864 Jan. 21.0 54.23 47.25 0.96, March 1.0 55.04 53.91 1.00, April 10.0 48.06 63.19 1.54, May 20.0 32.85 65.40 2.07, June 29.0 16.74 54.48 1.75, Aug. 8.0 8.62 31.39 - 0.36, Sept. 17.0 -- 14.20 + 2.09 + 1.86, 452 THEORETICAL ASTRONOMY. Date. ~2 dld 02 ddY d2 ct2 dt2 dt2 1864 Oct. 27.0 + 34.84 - 26.32 + 4.44, Dec. 6.0 68.79 47.87 6.86, 1865 Jan. 15.0 + 112.64 - 58.39 + 8.68. The complete integration may now be effected, and we may use both equation (37) and equation (39), the former giving the integral for the dates Jan. 1.0, Feb. 10.0, March 21.0, &c., and the latter the integrals for the dates in the foregoing table of values of the function. The final results for the perturbations of the rectangular co-ordinates, expressed in units of the seventh decimal place, are thus found to be the following:Berlin Mean Time. 6x 6y 6z 1863 Dec. 12.0 + 6.7 + 5.9 -0.2, 1864 Jan. 1.0 0.0 0.0 0.0, 21.0 + 6.8 5.9 0.1, Feb. 10.0 27.1 23.5 0.5, March 1.0 61.0 53.7 1.1, 21.0 108.9 97.4 2.0, April 10.0 169.7 155.7 3.1, 30.0 242.7 229.9 4.7, May 20.0 325.7 320.3 6.7, Junel 9.0 417.1 427.2 9.3, 29.0 514.6 549.1 12.3, July 19.0 616.1 684.9 15.7, Aug. 8.0 720.8 831.4 19.5, 28.0 827.4 986.0 23.4, Sept. 17.0 936.8 1144.6 27.0, Oct. 7.0 1049.4 1303.8 30.2, 27.0 1168.2 1460.0 32.6, Nov. 16.0 1295.4 1609.4 33.9, Dec. 6.0 1435.6 1749.6 33.8, 26.0 1592.8 1877.6 32.0, 1865 Jan. 15.0 + 1772.6 + 1992.3 -28.2. During the interval included by these perturbations, the terms of the second order of the disturbing forces will have no sensible effect; but to illustrate the application of the rigorous formulae, let us commence at the date 1864 Sept. 17.0 to consider the perturbations of the second order. In the first place, the components of the disturbing force must be computed by means of the equations NUMERICAL EXAMPLE. 453 oJ2X_ oJ2'k2 __- oT( y) w2 Yw2m'k2 y y ), w3Z==m^^ k(-z - z) The approximate values of 8x, 3y, and 8z for Sept. 17.0 given immediately by the table of integration extended to this date, will suffice to furnish the required values of the disturbed co-ordinates by means of x =- o + 6x, y = + Yo - +Sy, z =z z - z; and to find p - Po + kp, we have ap —~ ~x- ~s ~ Y- y z 8_P P P or J log p - - ((.' -.) x + (y' - y) y + (z'- z) z), in which 2o is the modulus of the system of logarithms. Thus we obtain, for Sept. 17.0, ~ log p = +- 0.0000084, W2X= + 59.09, w2Y= - 32.48, w2Z — + 2.08, which rerequire no further correction. Next, we compute the values of o +- YO + a2 y Zo + ~ az po2'0r2 which also will not require any further correction, and thus we form, according to (12), the equation q = - - 0.29996ax + 0.29815ay - 0.03237z. The approximate values of tx, gy, and z being substituted in this equation, we obtain q -= + 0.0000061, corresponding to which Table XVII. gives Hence we derive (fqx - ax) - 44.87, - f (fy - y) - 30.40,'0 2'r0 ua2, (fqz-z) —0.21, whchalo il nt euie nyfuthr oretinan tusPefom 454 THEORETICAL ASTRONOMY. and the equations (14) give d"axod 2'ay dd2~x + 14.22, + 2.08, d 1.87. dt2 dt' +' dt' These values being introduced into the table of integration, the resulting values of the integrals are changed so little that a repetition of the calculation is not required. We now derive approximate values of Ax, 8y, and Jz for Oct. 27.0, and in a similar manner we obtain the corrected values of the differential coefficients for this date; and thus by computing the forces for each place in succession from approximate values of the perturbations, and repeating the calculation whenever it may appear necessary, we may determine the perturbations rigorously for all powers of the masses. The results in the case under consideration are the following:Date. 2 ddx 02 dy 2d2 dt2 dt2 dt2 1864 Sept. 17.0 + 14.22 + 2.08 + 1.87, Oct. 27.0 34.84 — 26.31 4.44, Dec. 6.0 68.77 47.86 6.86, 1865 Jan. 15.0 + 112.60 - 58.39 + 8.68. Introducing these results into the table of integration, the integrals for Jan. 15.0 are found to be x= + 1772.6, 8y + 1992.3, 8z - 28.2, agreeing exactly with those obtained when terms of the order of the square of the disturbing forces are neglected. If the perturbations of the rectangular co-ordinates referred to the equator are required, we have, whatever may be the magnitude of the perturbations, (x, = x, y, cos e ay - sin e sz, (41) Jz, = sin e 8y + cos e z, x,, y,, z, being the co-ordinates in reference to the equator as the fundamental plane. Thus we obtain, for 1865 Jan. 15.0, ax, = + 1772.6, ay, = + 1838.9,, =z + 767.2. These values, expressed in seconds of arc of a circle whose radius is the unit of space, are xr, - + 36".562, dy, = + 37".930, az, = + 15".825. VARIATION OF CO-ORDINATES. 455 The approximate geocentric place of the planet for the same date is a = 183~ 28', = - 5~ 39', log a = 0.3229, and hence, neglecting terms of the second order, we derive, by means of the equations (3)2, for the perturbations of the geocentric right ascension and declination, aa ~- 17".03, -+ =- 5".67. 167. The values of 8x, By, and Jz, computed by means of the coordinates referred to the ecliptic and mean equinox of the date t, must be added to the co-ordinates given by the undisturbed elements and referred to the same mean equinox. The co-ordinates referred to the ecliptic and mean equinox of t may be readily transformed into those referred to the ecliptic and mean equinox of another date t'. Thus, let0 denote the longitude of the descending node of the ecliptic of t' on that of t, measured from the mean equinox of t, and let' be the mutual inclination of these planes; then, if we denote by x', y', z' the co-ordinates referred to the ecliptic of t as the fundamental plane, the positive axis of x, however, being directed to the point whose longitude is 0, we shall have x' = x cos 8 + y sin 8, y' = —x sin + y cos 0, (42) 2f -2. Let us now denote by x", y", z" the co-ordinates when the ecliptic of i is the plane of xy, the axis of x remaining the same as in the system of x', y', z'. Then we shall have F, y" =y' cos - z' sin a, (43) z' - y' sin - + z' cos a. Finally, transforming these so that the axis of z remains unchanged. while the positive axis of x is directed to the mean equinox of t and denoting the new co-ordinates by x,, y,, z,, we get x, - x" cos (0 + p) ~ Y" sin (0 + p), y, x"" sin (o + p) + y" cos (0 + p), (44) Z, =- Z, in which p denotes the precession during the interval t'- t. Eliminating x", y", and z" from these equations by means of (43) and (42), observing that, since - is very small, we may put cos' = 1, we get 456 THEORETICAL ASTRONOIY. x, cosp - y sinCS + _ 1 sin ( -+ p), 8 y, x sinp + y cosp - z cos (~ + p), (45) z, —z - x sin 0 - + y os, 8s in which s =206264.8, V being supposed to be expressed in seconds of arc. If we neglect terms of the order p3, these equations become x,= x-x P x_ yP + (sin + p cos 0) z, P8 S S27 y, _y i P- y - (cos -_ sin ), (46) z, = z - xa sin 0+ -ycos0. S S These formulae give the co-ordinates referred to the ecliptic and mean equinox of one epoch when those referred to the ecliptic and mean equinox of another date are known. For the values of p, r, and 0, we have p (50".21129 + 0".0002442966r) (t'- ), = ( 0".48892 - 0".000006143T) (t'- t), o 351~ 36' 10" + 39".79 (t - 1750) - 5".21 (t' -— ), in which = -(t' - t) - 1750, t and t' being expressed in years from the beginning of the era. If we add the nutation to the value of p, the co-ordinates will be derived for the true equinox of t'. The equations (45) and (46) serve also to convert the values of 0x, Sy, and 8z belonging to the co-ordinates referred to the ecliptic and mean equinox of t into those to be applied to the co-ordinates referred to the ecliptic and mean equinox of t'. For this purpose it is only necessary to write 8x, 8y, and 8z in place of x, y, and z respectively, and similarly for x,, y,, z,. In the computation of the perturbations of a heavenly body during a period of several years, it will be convenient to adopt a fixed equinox and ecliptic throughout the calculation; but when the perturbations are to be applied to the co-ordinates, in the calculation of an ephemeris of the body taking into account the perturbations, it will be convenient to compute the co-ordinates directly for the ecliptic and mean equinox of the beginning of the year for which the ephemeris is required, and the values of 8x, Jy, and 8z must be reduced, by means of the equations (45), as already explained, from the ecliptic and mean equinox to which they belong, to the ecliptic and mean equinox adopted in the case of the co-ordinates required. VARIATION OF CO-ORDINATES. 457 In a similar manner we may derive formulae for the transformation of the co-ordinates or of their variations referred to the mean equinox and equator of one date into those referred to the mean equinox and equator of another date; but a transformation of this kind will rarely be required, and, whenever required, it may be effected by first converting the co-ordinates referred to the equator into those referred to the ecliptic, reducing these to the equinox of t' by means of (45) or (46), and finally converting them into the values referred to the equator of t'.' Since, in the computation of an ephemeris for the comparison of observations, the co-ordinates are generally required in reference to the equator as the fundamental plane, it would appear preferable to adopt this plane as the plane of xy in the computation of the perturbations, and in some cases this method is most advanta.geous. But, generally, since the elements of the orbit of the disturbed planet as well as the elements of the orbits of the disturbing bodies are referred to the ecliptic, the calculation of the perturbations will be most conveniently performed by adopting the ecliptic as the fundamental plane. The consideration of the change of the position of the fundamental plane from one epoch to another is thus also rendered more simple. Whenever an ephemeris giving the geocentric right ascension and declination is required, the heliocentric co-ordinates of the body referred to the mean equinox and equator of the beginning of the year will be computed by means of the osculating elements corrected for precession to that epoch, and the perturbations of the co-ordinates referred to the ecliptic and mean equinox of any other date will be first corrected according to the equations (46), and then converted into those to be applied to the co-ordinates referred to the mean equinox and equator. If the perturbations are not of considerable magnitude and the interval t' - t is also not very large, the correction of Ox, By, and az on account of the change of the position of the ecliptic and of the equinox will be insignificant; and the conversion of the values of these quantities referred to the ecliptic into the corresponding values for the equator, is effected with great facility. In the determination of the perturbations of comets, ephemerides being required only during the time of describing a small portion of their orbits, it will sometimes be convenient to adopt the plane of the undisturbed orbit as the fundamental plane. In this case the positive axis of x should be directed to the ascending node of this plane on the ecliptic, and the subsequent change to the ecliptic and equinox, whenever it may be required, will be readily effected. 458 THEORETICAL ASTRONOMY. 168. The perturbations of a heavenly body may thus be determined rigorously for a long period of time, provided that the osculating elements may be regarded as accurately known. The peculiar object, however, of such calculations is to facilitate the correction of the assumed elements of the orbit by means of additional observations according to the methods which have already been explained; and when the osculating elements have, by successive corrections, been determined with great precision, a repetition of the calculation of the perturbations may become necessary, since changes of the elements which do not sensibly affect the residuals for the given differential equations in the determination of the most probable corrections, may have a much greater influence on the accuracy of the resulting values of the perturbations. When the calculation of the perturbations is carried forward for a long period, using constantly the same osculating elements,-and those which are supposed to require no correction,-the secular perturbations of the co-ordinates arising from the secular variation of the elements, and the perturbations of long period, will constantly affect the magnitude of the resulting values, so that Ax, cy, and 8z will not again become simultaneously equal to zero. Hence it appears that even when the adopted elements do not differ much from their mean values, the numerical amount of the perturbations may be very greatly increased by the secular perturbations and by the large perturbations of long period. But when the perturbations d28x: d2oy are large, the calculation of the complete values of dt2' ~-' and d2sz dt2 (which is effected indirectly) cannot be performed with facility, dGb requiring often several repetitions in order to obtain the required accuracy, since any error in the value of the second differential coefficient produces, by the double integration, an error increasing proportionally to the time in the values of the integral. Errors, therefore, in the values of the second differential coefficients which for a moderate period would have no sensible effect, may in the course of a long period produce large errors in the values of the perturbations, and it is evident that, both for convenience in the numerical calculation and for avoiding the accumulation of error, it will be necessary from time to time to apply the perturbations to the elements in order that the integrals may, in the case of each of the co-ordinates, be again equal to zero. The calculation will then be continued until another change of the elements is required. CHANGE OF THE OSCULATING ELEMENTS. a59 The transformation from a system of osculating elements for one epoch to that for another epoch is very easily effected by means of the values of the perturbations of the co-ordinates in connection with the corresponding values of the variations of the velocities dx dy dz dt, - d- and -d. The latter will be obtained from the values of the second differential coefficients by means of a single integration according to the equations (27) and (32). Thus, in the case of the example given, we obtain for the date 1865 Jan. 15.0, by means of (32), in units of the seventh decimal place, dx day dz 40 dt + 385.9, 40 -- + 214., 40 9.7. -dt d4 — 2,t dThe velocities in the case of the disturbed orbit will be given by the formulae dx dxo dRx dy dyo day d d dz dzo dSz 47 dt dt + dt' dt dt + dt' dt = dt + dt - dt tdt To obtain the expressions for the components of the velocity resolved parallel to the co-ordinates, we have, according to the equations (6)2, dx dr dv = in a sin- r sinsi +) a cos (A + ) dt' at dt at dy., /-, dr dv dyt sin b sin (B +- u) d- + r sin b cos (B - u) —, dt t at dz dv = sin c sin (C + u) r + r sin c cos (C - ) u. These equations are applicable in the case of any fundamental plane, if fhe auxiliaries sin a, sin b, sin c, A, B, and C are determined in reference to that plane. To transform them still further, we have dr kl/ + m. -=, -- me sin (U -- ), dt l -p dv kl/p(l+m) k1/1 ~m ( rd_ p+)(1 _ e cos (u - )), dt r - 1/p in which w denotes the angular distance of the perihelion from the ascending node. Substituting.these values, we obtain, by reduction, 460 THEORETICAL ASTRONOMY. d- = i/ ((e cos w cos uo) cos A - (esin ~ si sin A)sin) ) i a, dy k1/1 + d —,_ ~ -p ((e cos -- cos u) cosB - (e sin o + sin u) sin B) sin b, dt v/p dz kl-+m dt- /- ((e cosw - + cos ic) cos C - (e sin - + siSn u) sin C) sin c. Let us now put 1/p + m (e sin (o + sin u) - Vsin U, /p;(48) ~k/1 +~m (e cos +- cos u) = Vcos U, and we have dx dt = Vsin a cos (A + U), dy dy Vsin b cos (B + U), (49) dt =Vsin c cos (C U). These equations determine the components of the velocity of a heavenly body resolved in directions parallel to the co-ordinate axes, and for any fundamental plane to which the auxiliaries A, B, &c. belong. When the ecliptic is the fundamental plane, we have sin c = sin i, C- 0. The sum of the squares of the equations (48) gives = 2k2(1m)(1 + e2+ cos( )) k2(+m) ( - I) and hence it appears that V is the linear velocity of the body. The determination of the osculating elements corresponding to any date for which the perturbations of the co-ordinates and of the velocities have been found, is therefore effected in the following manner:First, by means of the osculating elements to which the perturbations belong, we compute accurate values of r0, x0, y, z0, and by dx means of the equations (48) and (49) we compute the values of -dt dy, dz d and -d. Then we apply to these the values of the perturbations, and thus nd, dy d These having been tions, and thus find x, y, z, - -i, and d These having been CHANGE OF THE OSCULATING ELEiMENTS. 461 found, the equations (32), will furnish the values of 2, i, and p; and the remaining elements may be determined as explained in Art. 112. Thus, from Vr sin 4- ~kp (1 +- m), dx dy dz =Vr cos ^x dt + dtZ dt' we obtain Vr and 0o, and from r sin u - (- x sin + y cos 2) sec i, r cos u -- x cos +- y sin 2, we derive r and u; and hence Vfrom the value of Vr. When i is not very small, we may use, instead of the preceding expression for r sin u, r sin iu z z cosec i. Next, we compute a from r 2a-r=2 k'ld n)1 2 k 2 (Iq-)' r V2 and from 2ae sin ow - (2a - r) sin (2% -+ u) - r sin u, 2ae cos w - (2c -~ r) cos (2+ + t- ) - r cos u, we find co and e. The mean daily motion and the mean anomaly or the mean longitude for the epoch will then be determined by means of the usual formule. In the case of a very eccentric orbit, after r and ut have been found, dr dt will be given by equations (48)6, and the values of e and v will be given by the equations (49)6. Then the perihelion distance will be found from p q-1!+ e' and the time of perihelion passage will be found from v and e by means of Table IX. or Table X. dx dy In the numerical values of the velocities dt' d-, &c., more decimals t dt' must be retained than in the values of the co-ordinates, and enough must be retained to secure the required accuracy of the solution. If it be considered necessary, the different parts of the calculation may be checked by means of various formulae which have already been given. Thus, the values of 2g and i must satisfy the equation 462 THEORETICAL ASTRONOMY. z cos i -- y si i cos + + sin i sin -- 0. We have, also, _ dx 2 /d y \ \2 d 2 V2 (dt)+( dYt ) ( dt r2 2 + I-2 + 2-, z - r sin t sin i, which must be satisfied by the resulting values of V, r, and u; and the values of a and e must satisfy the equation p - a (1 -e2) a cos2 o. 169. When the plane of the undisturbed orbit is adopted as the fundamental plane, we obtain at once the perturbations S (r cos i), 8 (r sin u), az, and from these the perturbations of the polar co-ordinates are easily derived. There are, however, advantages which may be secured by employing formulae which give the perturbations of the polar co-ordinates directly, retaining the plane of the orbit for the date to as the fundamental plane. Let w denote the angle which the projection of the disturbed radius-vector on the plane of xy makes with the axis of x, and / the latitude of the body with respect to the plane of xy; then we shall have X = r cos t' cos w, y = r cos f sin w, (50) z - r sin f. Let us. now denote by X, Y, and Z, respectively, the forces which are expressed by the second members of the equations (1), and the first two of these equations give xd -d dy d ( Yx — y) dt + C, C being the constant of integration. The equations (50) give dx d(rcos f). d d- COS W r cos f sin wd- rco dt dt dt dy d (r cos f) dw dt= sin w dt - r cs cos w cwdt dt dt and hence dy dx 2 2 dw -dt -Y dt-r Cos dt VARIATION OF POLAR CO-ORDINATES. 463 Therefore we have r2 cos2 dj( Yx - Xy) dt + C. If we denote by S0 the component of the disturbing force in a direction perpendicular to the disturbed radius-vector and parallel with the plane of xy, we shall have X- -- SO sin w, Y= So cos w, and Yx - Xy = S r cos /f. Therefore r cos t 3 SO r cos f dt + C. In the undisturbed orbit we have 9 0, and dwo -kV/po (1 + n); and thus the preceding equation becomes r2 cosZ dwt — So r cos l dt + k o (-+ -n). (51) The equations (1) also give 1 xd2A + yd2 d2y' k(1 + nm) _ Z- (52) r d r2 r r If we denote by R the component of the disturbing force in the direction of the disturbed radius-vector, we have R= XY + Yy+Z (53) r r r We have, also, xd2x + yd2y + zd2z d (xdz + ydy + zdz) -- (dx2 + dy2 + dz2) = d (rdr) - (dr2 + r2dv2) - rd2r - rdv2, v denoting the true anomaly in the disturbed orbit, or, since dv2 = cos2 9 dW2 + d)2, xd2x + yd2y + zd2z = rd2r - r2 cos2 f dw2 - r2di2. Hence the equation (52) becomes d2r dw2 df2 k2 (1 ) + ) -- -r cos d - r -- ( = R. (54) d dt2 " dt2 r2 464 THEORETICAL ASTRONOMY. 170. The equations (51) and (54), in connection with the last of equations (1), completely represent the motion of a heavenly body about the sun when acted upon by disturbing forces, and, when completely integrated, they will give the values of w, r, and z for any point of the orbit; but, since they cannot be integrated directly, we must, as in the case of the rectangular co-ordinates, find the equations which give by integration the values of 8w, Jr, and z. In the case of the undisturbed orbit, we have dwo r I o (l+ (55) d2ro dWo2 k2(1+ m)_ (55) dt2 o dt2 + o2 If we denote by 8w the variation of w arising from the action of the disturbing force, we have aw= wo + -w; and hence we easily find, from (51), dsw 1 f' I 2\ p (1+ ) dt -r'cos2 pJ,Srcos dt-l- 2 4 (56)\ dt rc Cos fi dt 1 C — r. cos2 ~ We have, further, r2 - ro2 + 2rr + 8r2, which gives r _1+2r & r' ro2 Let us now put q c ( -o siOS2 i,' -- ai1 cos' (57) o2 c 8s2 —Co 2 2 and. we have 1' - (58) The equation (56), therefore, becomes dt r cos' j So r cosp f dt -g (59) in which we put dwo kVpo (1 +n) (60) go dt ro2 If we substitute r + 8r for r in equation (54), and combine the result with the second of equations (55), we get d2r dw22 dt2 + k2 ( r VARIATION OF POLAR CO-ORDINATES. 465 and if we put "ro + l r, r _q" - Or, fT- 1-__ (61) we have 1 + 2q" (62) and hence d'Jr k' (1 +- n) dbw dt - R + )+ f,,q, + go2fr + 2g dto %2 ftq dt (e 910 dt (63) -r sin 2i (go+dt ) + t r( dt Finally, we have, from the last of equations (1), d2z Ik2(1 + n) =Z- Z, (64) by means of which the value of z may be found, since, in the case of the undisturbed motion, we have z0 = 0. The values off' corresponding to different values of q' may be tabulated with the argument q', and, since the equation (62) is of the same form as (58), the same table will give the value of f" when q" is used as the argument. Table XVII. gives the values of logf or logfI corresponding to values of q' or q" from - 0.03 to + 0.03. Beyond the limits of this table the required quantities may be computed directly. 171. When we consider only terms of the first order with respect to the disturbing force, we have flqY 2ar ro, and the equations become d(t o -t rJ 0 d"6o d2 —- -R+ - fsorOdt + (2k. (1 + -n)) 3)r, (65) d2_z k2 (1 + nm) dt2 ro In determining the perturbations of a heavenly body, we first consider only the terms depending on the first power of the disturbing force, for which these equations will be applied. The value of dr 30 466 THEORETICAL ASTRONOMY. will be obtained from the second equation by an indirect process, as already illustrated for the case of the variation of the rectangular co-ordinates. Then aw will be obtained directly from the first equation, and, finally, z indirectly from the last equation. Each of the integrals is equal to zero for the date t,, to which the osculating elements belong. When the magnitude of the perturbations is such that the terms depending on the squares and products of the masses must be considered, the general equations (59), (63), and (64) will be applied. The values of the perturbations for the dates preceding that for which the complete expressions are to be used, will at once indicate approximate values of 8?w, dr, and z; and with the values r r -- 9 r d r, w = W -- + w, sin I -_, the components of the disturbing force will be computed. We compute also q' from the first of equations (57), and q" from the first of (61); then, by means of Table XVII., we derive the corresponding values of log f and logf". The coefficients of 8r in the expressions for q and q" will be given with sufficient accuracy by means of the approximate values of or and sin /9, and will not require any further correction. Then we compute S0r cos, and find the integral fSor cos H dt; dsw and the complete value of dt will be given by (59). The value of -ct2 will then be given by equation (63). The term r t ( ) will always be small, and, unless the inclination of the orbit of the disturbed body is large, it may generally be neglected. Whenever it shall be required, we may put it equal to ( - )' The corrected values of the differential coefficients being introduced into the table of integration, the exact or very approximate values of 8w, 8r, and z will be obtained. Should these results, however, differ much from the corresponding values already assumed, a repetition of the calculation may become necessary. In this manner, by computing each place separately, the terms depending on the squares, products, and higher powers of the disturbing forces may be included in the results. It will, however, be generally possible to estimate the values of Jw, &r, VARIATION OF POLAR CO-ORDINATES. 467 and z for two or three intervals in advance to a degree of approximation sufficient for the computation of the forces for these dates. In order that the quantity o, representing the interval adopted in the calculation of the perturbations, may not appear in the integration, we should introduce it into the equations as in the case of the variation of the rectangular co-ordinates. Thus, in the determinadcw tion of 8w we compute the values of o ot- and since the second member of the equation contains the integralfS0rcos dt, if we introduce the factor w2 under the sign of integration, this integral, omitting the factor w in the formule of integration, will become wofsor cos 9 dt, as required. The last term of the equation ~i-il be multiplied by co. In the case of Or, each term of the equation for must contain the factor 02. If the second of equations (65) is employed, the first and third terms of the second member will be multiplied by c'2; but since the value of S0 is supposed to be already multiplied by o2, the second term will only be multiplied by o. The perturbations may be conveniently determined either in units of the seventh decimal place, or expressed in seconds of arc of a circle whose radius is unity. If they are to be expressed in seconds, the factor s- 206264.8 must be introduced so as to preserve the homogeneity of the several terms, and finally 8r and 4z must be converted into their values in terms of the unit of space. 172. It remains yet to derive convenient formule for the determination of the forces So, R, and Z. For this purpose, it first becomes necessary to determine the position of the orbit of the disturbing planet in reference to the fundameital plane adopted, namely, the plane defined by the osculating elements of the disturbed orbit at the instant to. Let i' and 2' denote the inclination and the longitude of the ascending node of the disturbing body with respect to the ecliptic, and let I denote the inclination of the orbit of the disturbing body with respect to the fundamental plane. Further, let N denote the longitude of its ascending node on the same plane measured from the ascending node of this plane on the ecliptic or from the point whose longitude is ~0,, and let N' be the angular distance between the ascending node of the orbit of the disturbing body on the ecliptic and the ascending node on the fundamental plane adopted. Then, from the spherical triangle formed by the intersection of the plane of the 468 THEORETICAL ASTRONOMY. ecliptic, the fundamental plane, and the plane of the orbit of the disturbing body with the celestial vault, we have sin - I sin' (N + N') sin - (' a sin ( + sin 1Icos (N + N) cos (' - o) sin ( -i), cos I sin - (N N') sin (' — 0) co (' + i0), (. cos cos(- N') I- cos' (2' ~ o) cos I (i'- o), from which to find 2V, N', and I. Let f' denote the heliocentric latitude of the disturbing planet with respect to the fundamental plane, w' its longitude in this plane measured from the axis of x, as in the case of w, and ut' the argument of the latitude with respect to this plane. Then, according to the equations (82),, we have tan (w' - N) = tan uo' cos I, tan' = tan Isin (w' - N). If t' denotes the argument of the latitude of the disturbing planet with respect to the ecliptic, we have -' -' U - N'. (68) This formula will give the value of u,', and then wt and' will be found from (67). We have, also, cos Ut' = cos /' cos (w' - N), which will serve to indicate the quadrant in which w' - N must be taken. The relations here derived are evidently applicable to the case in which the elements of the orbits of the disturbed and disturbing planets are referred to the equator, the signification of the quantities involved being properly considered. The co-ordinates of the disturbing planet in reference to the plane of the disturbed orbit at the instant t0 as the fundamental plane will be given by' - r' cos 1 cos w', y' r' cos 3' sin w', (69) z' -r' sin f'. To find the force R, we have R=X-+ Y + Zr r r VARIATION OF POLAR CO-ORDINATES. 469 and ( p3n r3 3 )3 ( P3(3) P3 r 3 we get Rn'l (hr' cos'cosC ('- ) + hr' sinin sin' — ). (71) The equation S, r cos= Y-. XXy gives X- nSubs2 h r' cose' sin (w' - y w), (72) from which to find S. Finally, we have Z=n'k(/ r -- s )ri (73) from which to find Z. When we determine the perturbations only with respect to the first power of the disturbing force, the expressions for R, S0, and Z become R = n' (' hr' cos t cos (2- w) ~' ),, - m'k2 h r' cos f' sin (20' - wQ), Z -m'l2 h r' sin f'. To compute the distance p, we have 2 = (' - )2 + (y' y)2 + (Z'- Z)2 which gives p2 r= - _r_ 2r r' cos cos f' cos (w' - w) - 2r r' sin 3 sin', (75) and, if we neglect terms of the second order, we have po2 - r2 + r02 - 2ro r' cos i' cos (2' w o). (76) If we put cos y — cos. os s f cos (w' - w) + sin f sin i', (77) we have P2 r2 +- r2 - 2rr' cos y - r2 sin2r + (r - r' cos Y); 470 THEORETICAL ASTRONOMY. and hence we may readily find p from p sin n - r' sin r, p co n r - r' cos ry, the exact value of the angle n, however, not being required. Introducing r into the expression for R, it becomes R = m'k' h r' cos r- ) (79) by means of which 1R may be conveniently determined. 173. When we neglect the terms depending on the squares and higher powers of the masses in the computation of the perturbations, the forces R, S, and Z will be computed by means of the equations (74), po being found from (76) or from (78), when we put cos r cos f' cos (,w - WV). But when the terms of the order of the square of the disturbing force are to be taken into account, the complete equations must be used. Thus, we find p from (78), S0 from (72), Z from (73), and R from (71) or (79). The values of Jw, 8r, and z, computed to the point at which it becomes necessary to consider the terms of the second order, will enable us at once to estimate the values of the perturbations for two or three intervals in advance to a degree of approximation sufficient for the calculation of the forces; and the values of R, S0, and Z thus found will not require any further correction. When the places of the disturbing planet are to be derived from an ephemeris giving the heliocentric longitudes and latitudes, the values of a' and i' will be obtained from two places separated by a considerable interval, and then the values of u' will be determined by means of the first of equations (82), or by means of (85),. When the inclination i' is very small, it will be sufficient to take t' -'-' +- s tani2 i' sin 2 (1' - ), in which s — 206264.8. But when the tables give directly the longitude in the orbit, u' +- 2', by subtracting 2' from each of these longitudes we obtain the required values of iu. It should be observed, also, that the exact determination of the values of the forces requires that the actual disturbed values of r', w', and &' should be used. The disturbed radius-vector r' will be VARIATION OF POLAR CO-ORDINATES. 471 given immediately by the tables of the motion of the disturbing body, but the determination of the actual values of w' and /' requires that we should use the actual values of N', N, and I in the solution of the equations (68) and (67). Hence the disturbed values of g' and i' should be used in the determination of these quantities for each date by means of (66). It will, however, generally be the case that for a moderate period the variation of 2' and i' may be neglected; and whenever the variation of either of these has a sensible effect, we may compute new values of N, N', and I from time to time, by means of which the true values may be readily interpolated for each date. We may also determine the variations of V, N', and I arising from the variation of Q' and i', by means of differential formule. Thus the relations will be similar to those given by the equations (71)2, so that we have sin N' siln N' 8 sNr-' i —2g cos N' g - I cos Ii', sin(' - ) si O I sin N sin N' N - sin( )cos N'' — s-i', (80) sin(2'~-b) sQis 8I = sin N' sin i' s 2' + cosN' ai"', from which to find c81T, XN, and ~1. When the perturbations are computed only in reference to the first power of the mass, the change of 2' and i' may be entirely neglected; but when the perturbations are to be computed for a long period of time, and the terms depending on the squares and products of the disturbing forces are to be included, it will be advisable to take into account the values of 8NV, 8N', and 81, and, using also the value of u' in the actual orbit of the disturbing body, compute the actual values of w' and f'. In the case of several disturbing bodies, the forces will be determined for each of these, and then, instead of R, S0, and Z, in the formule for the differential coefficients, CR, 2'S, and 2Zwill be used. 174. By means of the values of 8wo, ar, and z, the heliocentric or the geocentric place of the disturbed planet may be readily found. Thus, let the positive axis of x be directed to the ascending node of the osculating orbit at the instant t0 on the plane of the ecliptic; then, in the undisturbed orbit, we shall have o o u denoting the argument of the latitude. Let x,, y,, z, be the co-or 472 THEORETICAL ASTRONOMY. dinates of the body referred to a system of rectangular co-ordinates in which the ecliptic is the plane of xy, and in which the positive axis of x is directed to the vernal equinox. Then we shall have x, = x cos g o -- y cos i sin g + z- sin i, sin go, y, x sin g o + y cos i cos ag0 - z sin i, cos a, z, - y sin i, + z cos, or, introducing the values of x and y given by (50), x, =- r cos f cos w cos 0 - r cos / sin w cos i sin + z si i sin, sin y, - r cos if cos w sin 0o +- r cos i sin w cos i cos -, z sin i, cos a o, (81) z, - r cos A sin w sin i + z cos i. Introducing also the auxiliary constants for the ecliptic according to the equations (94), and (96)i, we obtain x, -- r cos f sin a sin (A -- w) + z cos a, y, = r cos i sin b sin (B + w) + z cos b, (82) z, = r cos i sin i sin w + z cos i, by means of which the heliocentric co-ordinates in reference to the ecliptic may be determined. If the place of the disturbed body is required in reference to the equator, denoting the heliocentric co-ordinates by x,,, y,, z,,, and the obliquity of the ecliptic by s, we have XI,, - X =y, cos -s z, sin e, z,, y, sin e + z, cos e. Substituting for x,, y,, z, their values given by (81), and introducing the auxiliary constants for the equator, according to the equations (99), and (101), we get x,,= r cos f sin a sin (A + w) -]- z cos a, y,, = r cos f sin b sin (B + w) + z cos b, (83) z, - r cos f sin c sin ( C+ w) - z cos c. The combination of the values derived from these equations with the corresponding values of the co-ordinates of the sun, will give the required geocentric places of the disturbed body. These equations are applicable to the case of any fundamental plane, provided that the auxiliary constants a, A, b, B, &c. are determined with respect to that plane. In the numerical application of the formulae, the value of w will be found from w- = U -+- aw, VARIATION OF POLAR CO-ORDINATES. 473 Uo being the argument of the latitude for the fundamental osculating elements, and care must be taken that the proper algebraic sign is assigned to cos a, cos b, and cos c. If the values of wr, 20, and io used in the calculation of the perturbations are referred to the ecliptic and mean equinox of the date t0, and the rectangular co-ordinates of the disturbed body are required in reference to the ecliptic and mean equinox of the date t0or, the value of w must be found from w-o + V0 + a v, the value of oo referred to the ecliptic of to' being reduced to that of to", by means of the first of equations (115),. Then 0o and i0 should be r dfrm e reduced from the ecliptic and mean equinox of t to the ecliptic and mean equinox of to0 by means of the second and third of the equations (115),, and, using the values thus found in the calculation of the auxiliary constants for the ecliptic, the equations.(82) will give the required values of the heliocentric co-ordinates. If the coordinates referred to the mean equinox and equator of the date to" are to be determined, the proper corrections having been applied to o0 and i, the mean obliquity of the ecliptic for this date will be employed in the determination of the auxiliary constants a, A, &c. with respect to the equator, and the equations (83) will then give the required values of the co-ordinates. If we differentiate the equations (83), we obtain, by reduction, dxl, dw dr dx= r cos f sin a cos (A + w) d+ sec f sin a sin (A + w) d dt dt dt dz + (cos a - tan f sin a sin (A + w)) dt-, dy,, d d dyf i= r cos f sin b cos (B + w) d + see f sin b sin (B + w) dr dt d(t t 84 + (cos b - tan sin b sin (B +w)) -t, dz,, dw dr d = r cos f sin c cos ( C + w) t- + sec f sin c sin ( +- w) dt (t dt dz +(cos c - tan f sin c sin (C +w)) di' by means of which the components of the velocity of the disturbed body in directions parallel to the co-ordinate axes may be determined. d/6x do d23r d'z The values of -B and -d will be obtained from d2J and d2 by a single integration, and then we have single integration, and then we have 474 THEORETICAL ASTRONOMY. ldw kl/p'(l+rm) daw dr kl/1 - m dar ( dt ro2 + d' _- dt_ eo sinV0 + -t (85) dw dr from which to find -,t and d dt dt 175. EXAMPLE.-In order to illustrate the calculation of the perturbations of r, w, and z, let us take the data given in Art. 166, and determine these perturbations instead of those of the rectangular coordinates. In the first place, we derive from the tables of the motion of Jupiter the values' = 98~ 58' 22".7, i' = 1 8' 40".5, which refer to the ecliptic and mean equinox of 1860.0. We find, also, from the data given by the tables the values of u' measured from the ecliptic of 1860.0. Then, by means of the formulke (66), using the values of g0 and i0 given in Art. 166, we derive N- 194~ 0' 49".9, N' 301~ 38' 31".7, I- 5~ 9' 56".4. The value of u0' is given by equation (68), and then w' and fi are found from the equations (67). Thus we have Berlin Mean Time. log10^w0 WO= t0 log wo' 1863 Dec. 12.0, 0.294084 192~ 4/ 24.5 0.73425 140 18'54".6 -0~ 1'38/.1 1864 Jan. 21.0, 0.294837 207 40 52.2 0.73368 17 21 44.2 0'8 9.1 March 1.0, 0.300674 223 3 5.9 0.73305 20 25 5.2 0 34 39.9 April 10.0, 0.310864 237 51 38.3 0.73237 23 28 59.8 0 51 7.6 May 20.0, 0.324298 251 52 47.9 0.73164 26 33 32.1 1 7 29.7 June 29.0, 0.339745 264 59 30.0 0.73086 29 38 44.8 1 23 43.5 Aug. 8.0, 0.356101 277 10 24.6 0.73003 32 44 41.2 1 39 46.3 Sept. 17.0, 0.372469 288 28 4.1 0.72915 35 51 24.6 1 55 35.2 Oct. 27.0, 0.388214 298 57 16.3 0.72823 38" 58 57.5 2 11 7.5 Dec. 6.0, 0.402894 308 43 48.7 0.72726 42 7 23.3 2 26 20.3 1865 Jan. 15.0, 0.416240 317 53 39.1 0.72625 45 16 43.9 -2 41 10.6 The values of po may be found from (76) or (78)as already given in Art. 166. The forces R, So, and Z may now be determined by means of the equations (74), h being found from (70), and if we introduce the factor 02 for convenience in the integration, as already explained, we obtain the following results: Date. o2R (S0 2Z wSodt 1863 Dec. 12.0, + 1".4608 + 0".1476 + 0".0009 + 0".0282 1864 Jan. 21.0, + 1.4223 - 0.6757 -+0.0101 - 0.2361 NUMERICAL EXAMPLE. 475 Date. So2R 21.0'o (2 S8dt 1864 March 1.0, + 1".2616 - 1".4512 + 0".0190 - 1".3060 April 10.0, 1.0018 2.1226 0.0273 3.1035 May 20.0, 0.6760 2.6473 0.0347 5.5020 June 29.0, + 0.3179 2.9988 0.0406 8.3402 Aug. 8.0, -0.0452 3.1650 0.0449 11.4378 Sept. 17.0, 6.3944 3.1437 0.0470 14.6076 Oct. 27.0, 0.7180 2.9392 0.0466 17.6640 Dec. 6.0, 1.0097 2.5586 0.0432 20.4273 1865 Jan. 15.0, — 1.2674 - 2.0081 + 0.0362 - 22.7245 The integral oiSodrodt is obtained from the successive values of o2Soo0 by means of the formula (32). Next we compute the values of the differential coefficients by means of the formult (65). For the dates 1863 Dec. 12.0 and 1864 Jan. 21.0 we may first assume r — 0, and, by a preliminary integration, having thus derived very approximate values of 8r for these dates, the values of d~2 will be recomputed. Then, commencing anew the table of integration, we may at once derive an approximate value of Jr for the date March 1.0 with which the last term of the expression for ~-dt may be computed. Continuing this indirect process, as already illustrated in the case of the perturbations of the rectangular co-ordinates, we obtain the required values of the second d'2 differential coefficient. In a similar manner, the values of ct will dsw be obtained. The values of -d will then be given directly by means of the first of equations (65); and the final integration will furnish the perturbations required. Thus we derive the following results:dsw _d___ d, Date. 2dd d2 O2 d~z w o z dct dt2 dt" 1863 Dec. 12.0, -0".0423 +1".4509 +0".0009 -0".00 +0".18 +0".00 1864 Jan. 21.0, 0.1086 1.3405 0.0101 0.02 0.17 0.00 Mar. 1.0, 0.7162 +0.7829 0.0183 0.40 1.47 0.01 Apr. 10.0, 1.6114-0.0455 0.0251 1.55 3.53 0.04 May 20.0, 2.4795 0.9344 0.0300 3.61 5.54 0.09 June 29.0, 3.0807 1.7333 0.0326 6.42 6.62 0.18 Aug. 8.0, 3.2971 2.3752 0.0331 9.64 5.98 0.29 Sept. 17.0, 3.1080 2.8533 0.0311 12.88 +2.98 0.44 Oct. 27.0, —2.5425-3.1872+0.0265-15.73 -2.86 +0.62 476 THEORETICAL ASTRONOMY. Date. do 2 d2r 2 d w dt dt2 dt2 1864 Dec. 6.0, — 1".6443 — 3".4009 -0".0190 — 17".85 — 11".88 +0".83 1865 Jan. 15.0,-0.4511 -3.5334 +0.0079 -18.92-24.29 +1.05 It has already been found that, during the period included by these results, the perturbations arising from the squares and products of the disturbing forces are insensible, and hence the application of the complete equations for the forces and for the differential coefficients is not required. The equations (83) will give, by means of the results for w u0 + -w, r =- r + Jr, and z, the values of the heliocentric co-ordinates of the disturbed body, and the combination of these with the co-ordinates of the sun will give the geocentric place. When we neglect terms of the second order, we have, according to the equations (84), ax,, x= cot (A + w) aw + - ar + z cos a, ro aY,, y- cot (B ~ w) aw + Jr + z cos b, (86) r0 8z,, = z0 cot (C + w) 1W +-~r 0 + cos, the heliocentric co-ordinates x0, Y0, z being referred to the same fundamental plane as the auxiliary constants, a, b, A, &c. Thus, in the case of Eurynome, to find the perturbations of the rectangular co-ordinates, referred to the ecliptic and mean equinox of 1860.0, from 1864 Jan. 1.0 to 1865 Jan. 15.0, we have A 2960 34/ 37/.5, B -= 2060 43/ 34/.4, C= 0, log cos a 8.557354,, log cos b =8.856746, log cos log cos o = 9.998590, log x0 0.399807, log o 9.838709, log z == 9.148170,, w == wo + wt =- 317~ 53/ 20".2, and hence, by means of (86), we derive ax, + 36".559, ay, = + 41".083, az, - 0".588. If we express these in parts of the unit of space, and in units of the seventh decimal place, we obtain ax, + 1772.4, ay, = + 1991.8, 2, - 28.5, agreeing with the results already obtained by the method of the variation of rectangular co-ordinates, namely, 3ax2, - 1772.6, ay, + 1992.3, z,=- 28.2. CHANGE OF THE OSCULATING ELEMENTS. 477 176. By using the complete formule, the perturbations of r, w, and z may be computed with respect to all powers of the disturbing force, and for a long series of years, using constantly the same fundamental osculating elements. But even when these elements are so accurate as not to require correction, on account of the effect of the large perturbations of long period upon the values of 8w and or, the numerical values of the perturbations will at length be such that a change of the osculating elements becomes desirable, so that the integration may again commence with the value zero for the variation of each of the co-ordinates. This change from one system of elements to another system may be readily effected when the values of the perturbations are known. Thus, having found the disturbed values of r, w, and z, we have dv' dw2 di3 lhkp (1 + m) -r —- cos' + dt2 dt'2 +dt2 r4 p being the semi-parameter of the instantaneous orbit of the disturbed body. In the undisturbed orbit we have dv, kvpo (1 + m) o- dt 2 and hence we derive P ~2oro4 dt, dv - go 214dt Substituting for dt the value above given, there results / 21 1 dco \ 1 d1 — P -Po-,4 Cos + * dtj ). 2 d' (87),N — o dt rok2d "Q \ \ 90 d Io d' by means of which p may be determined. To find d-, we have dfi 1 dz tan i dr dt r cos dt r dt(88) We have, also, dr kl/l +m. kl/l -m. d8r d- -- _ - e sin v - /- e in o- d at and if we put - I+ a, r= ~p dar (89) 0o Vl+w dt 478 THEORETICAL ASTRONOMY. this equation becomes e sin v eo sin vo -- aeo sin o v- r. (90) We have, further, e cos v - - 1, and, putting P ro 1 -, (91) Po r we obtain e cos v - eo cos v + f. This equation, combined with (90), gives e sin (v - vo) - ae, sin v0 cos vo + r cos vO-0 f- sin vo ~ (92) e cos (v- v,) eo + ece sin2 v + r sin vo + Po cos v,, by means of which the values of e and v may be found, those of the auxiliaries o, 9, r, being found from (89) and (91). Then we have e - sin m, a -=p se2 s, p. =- +, tan E=- tan (45~ -') tan v, az M-1 E-e sin E, by means of which %, a, j,, and l1 may be determined. In the case of orbits of great eccentricity, we find the perihelion distance from P q — -b+e' and the time of perihelion passage will be derived from e and v by means of Table IX. or Table X. It remains yet to determine the values of g, i, and co or r. Let 00 denote the longitude of the ascending node of the instantaneous orbit on the plane of the osculating orbit, defined by 92 and i0, measured from the origin of w, and let ^o denote its inclination to this plane. Then we have tan %r sin (w - 0o) = tan f, di c^od~ (93) tan, cos (w - O) d sec2 ) ( and hence CHANGE OF THE OSCULATING ELEMENTS. 479 gd- dt dt by means of which 00 may be found. The quadrant in which o0 is situated is determined by the condition that sin(w - 00) and tan k must have the same sign. The value of ^ will be found from the first or the second of equations (93). If we denote by: the argument of the latitude of the disturbed body with respect to the adopted fundamental plane, we have tan tan (w -- (95) cOS )o and the angle ( must be taken in the same quadrant as w - 00. Then, from the spherical triangle formed by the intersection of the planes of the ecliptic and instantaneous orbit of the disturbed body, and the fundamental plane, with the celestial vault, we derive cos 1 i sin (u- ( ) + 1 (- 2o)) Sill O cos (i -0), cos M i cos( (- M0 + (-n- ~o)) - cos: co (96) sin i sin (i C)-. ) ) sill sin o) - sin - i cos ( (ut ) - 1 (2 - go)) cos 0so sin j(i + o). These equations will furnish the values of i, u -, and 2 - 0, and hence, since, and Q2 are given, those of a and a. The value of v having been already fonnd, we have, finally, V) - u 7 ==It- V + a, and the elements are completely determined. These elements will be referred to the ecliptic and mean equinox to which g20 and i0 are referred, and they may be reduced to the equinox and ecliptic of any other date by means of the formule which have already been given. The elements of the instantaneous orbit of the disturbed body may also be determined by first computing the values of x,,, y,, z,,, in reference to the fundamental plane to which a and i are to be referred, by means of the equations (83), and also those of dt', dt' d by means of (85) and (84), and then determining the elements from the co-ordinates and velocities, as already explained. It should be observed that when the factor (02, or the square of the 480 THEORETICAL ASTRONOMY. adopted interval, is introduced into the expressions for the forces and differential coefficients, the first integrals will be dsr dcw dz dt' dt' dt' and that when these quantities are expressed in seconds of arc, they must be converted into their values in parts of the unit of space whenever they are to be combined with quantities which are not expressed in seconds. In other words, the homogeneity of the several terms must be carefully attended to in the actual application of the formule. When the elements which correspond to given values of the perturbations have been determined, if we compute the heliocentric longitude and latitude of the body for the instant to which the elements belong, the results should agree with those obtained by computing the heliocentric place from the fundamental.osculating elements and adding the perturbations. 177. The computation of the indirect terms when the perturbations of the co-ordinates r, w, and z are determined, is effected with greater facility than in the case of the rectangular co-ordinates, although the final results are not so convenient for the calculation of an ephemeris for the comparison of observations. This indirect calculation, which, when the perturbations of any system of three coordinates are to be computed, cannot in any case be avoided without imlpairing the accuracy of the results, may be further simplified by determining, in a peculiar form, the perturbations of the mean anomaly, the radius-vector, and the co-ordinate z perpendicular to the fundamental plane adopted. Let the motion of the disturbed body be, at each instant, referred to the plane of its instantaneous orbit; then we shall have j= 0, and the equations (51) and (54) become r2 = r dt + ol + mk o + ), (97) dUr dW2+ k' (1 + (97) dt2 dt2 r2 in which R denotes the component of the disturbing force in the direction of the disturbed radius-vector, and S the component in the plane of the disturbed orbit and perpendicular to the disturbed radiusvector, being positive in the direction of the motion. The effect of VARIATION OF POLAR CO-ORDINATES. 481 the components R and S is to vary the form of the orbit and the angular distance of the perihelion from the node. If we denote by Z the component of the disturbing force perpendicular to the plane of the instantaneous orbit, the effect of this will be to change the position of the plane of the orbit, and hence to vary the elements which depend on the position of this plane. Let us take a fixed line in the plane of the instantaneous orbit, and suppose it to be directed from the centre of the sun to a point whose angular distance back from the place of the ascending node is a, and let the value of a be so taken that, so long as the position of the plane of the orbit is unchanged, we shall have The line thus taken in the plane of the orbit may be regarded as fixed during all changes in the position of this plane. Let Z denote the angle between this fixed line and the semi-transverse axis; then will X -- + - a, (98) and when the position of the plane of the orbit is unchanged, we have X -- 7r. But if, on account of the action of the component Z, the position of the plane of the orbit is changed, we have, according to the equations (72)2, the relations da -cosi d2, d dz - cosi dg, (99) dn = dX + (1 -cos i) d g =- d 2 + 2 sin'2 i d. We have, further,,I = v + X -', (100) v being the true anomaly in the instantaneous orbit. The two components of the disturbing force which act in the plane of the disturbed orbit will only vary X and the elements which determine the dimensions of the conic section. We have, therefore, in the case of the osculating elements, for the instant t0, Xo -- o + -o - o. Let us now suppose 2 to denote the true longitude in the orbit, so that we have =v + v7C +V + +2, 31 482 THEORETICAL ASTRONOMY. or = -v +X -- ( -- g); (101) then, since X is equal to t when the position of the plane of the orbit is unchanged, it follows that a- a represents the variation of the true longitude in the orbit arising from the action of the component Z of the disturbing force. The elements may refer to the ecliptic or the equator, or to any other fundamental plane which may be adopted. 178. For the instant t we have, in the case of the disturbed motion, the following relations:E- e sin E= M-~ ~ (t - to), r cos v - a cos E- ae, r sin v - at/l - e2 sin E, Let us first consider only the perturbations arising from the action of the two components of the disturbing force in the plane of the disturbed orbit, and let us put, =v +. (103) Further, let MH +, (t - t) + AM be the mean anomaly which, by means of a system of equations identical in form with the preceding, but in which the values of a., e, X0 are used instead of the instantaneous values a, e, and Z, gives the same longitude A,, so that we have E, - e, sin E, = Mo + L-o (t-to) -+ SM, r, cos, ao cos E, -- a(e, (104) r, sin v, - ao/1 - eo sin E, A, ==, + Xo V, + - o. If, therefore, we determine the value of 8M so as to satisfy the condition that i, =v + X, the disturbed value of the true longitude in the orbit, neglecting the effect of the component Z of the disturbing force, will be known. The value of r, will generally differ from that of the disturbed radius-vector r, and hence it becomes necessary to introduce another variable in order to consider completely the effect of the components R and S. Thus, we may put r r, (1 + (), (105) and v will always be a very small quantity. When IM and v have been found, the effect of the disturbing force perpendicular to the plane of the instantaneous orbit may be considered, and thus the complete perturbations will be obtained. VARIATION OF CO-ORDINATES. 483 dw In the equations (97), r2~ dt expresses the areal velocity in the instantaneous orbit, and it is evident that, since the true anomaly is not affected by the force Z perpendicular to the plane of the actual orbit, dr2 must also represent this areal velocity, and hence the equations idt become bdedt -Sfr dt + kl/pO ( + M) (10o6) d2r ( dv,\2 k2(i +M) ()1 df r -dt / + -- R. 179. If we differentiate each of the equations (104), we get dE d__ (1- e Cos E,) t, dr,. dv, dE, cos v, -- r, sin v, d- - a, sin E, dE dt d t (107) dr, vE,, / cs dE, sin - + r, cos vI -b a 1Ie dt, dv, dt dt From the second and the third of these equations we easily derive drv, edE, r, = (a/ - eo r, sin v, cos E,- ar, cos v, sin E,) d. dE, Substituting in this the values of r, sin,, r, cos v,, and dt"- and reducing, we get dr,.- sin L +- -i or dr, l/l + m., doM\ dt /- eosi n v, 1 dt dr, From the same equations, eliminating -t-, we get dv, __ 77dE rd -- (al/ - eo2 r, cos v, cos, + aor, sin v, sin E,) dE, which reduces to r -t o (1 + ) a (109) ---- P 484 THEORETICAL ASTRONOMY. or 2 dvf + /~^~~^ ^, ^~/^, I d8M\ r k Po m (1 + M) (m + V2\ ( 0 dt ) Combining this with the first of equations (106), we get dt (1 +) 1 + 1((~ ( )S/rd (1(10) from which i3f may be found as soon as v is known. The equation (105) gives dr dr, dv (I + V) + dt dtc d2r drr cr dv d'V (111) dt2 = (1 +.) 2r, +r 2 dr r dt2 dt9,- dt dt r,-dt. Differentiating equation (108) and substituting for dt its value already found, we obtain d2r, k2 (1+-m) e, cos, 1 daM\ k/l1 +me sinv, d2aM dt2 r,2 \' dt d l J /, dt2 and the last of the preceding equations becomes d2r d2V k+ (1 + m) eo cosV ( 1 d+M\ 2 ^ dt-_'dt2 r,2 o\+ dt~) kl/+m. (1 +v d2aM dv 2 dv d&M + d 2d + /- e0 in O dt2, dt The equation (110) gives 1 d2sM 2 dv 2 dy 1 r o' dc + (1 + v)' dt + ( + ) dt k' V1 (1 +) J 1 Sr (1 +)* kV/(1+m)' which is easily reduced to 1 + d2d M dv 2 dv d&M 1 Sr o dt dt+ 2T dt+ dt 1+ v k /p;o( + nt) and hence we derive d2r d2v k2 (1 + M) ecos (1+ 1\ dMt ) pesinv, The equation (109) gives The equation (109) gives VARIATION OF CO-ORDINATES. 485 dv,) 2 k2p0(1+m) (1~1 2^ d ~t r?~3 dt and, since r,- 1 - ecosv,, r, this becomes ( dv, )2 2 (1 + m) (+)(1+ di \ ( + ) r (1 + (113) + +r,2 [cosL+(+ -+. dt Combining equations (112) and (113) with the second of equations (106), we get d2i 1 + k 2(1 ( + m) ( + 1 d_ _ i 2 eo sin v, k2 (1 + m) (I + ) (1 Po r From (110) we derive (1+)4(1+1. d) t+_ 2 fSrdt (I+~)'j\ d+ dt Po (1 + m ) + ( m srdt): and the preceding equation becomes dt2 = By + 2' + 1 fSr dteo sin v, S w _h i t 3 e comp(le te) e r t I d m which is the complete expression for the determination of v. 180. It remains now to consider the effect of the component of the disturbing force which is perpendicular to the plane of the disturbed orbit. Let x,, y, z, denote the co-ordinates of the body referred to the fundamental plane to which the elements belong, and x, y the co-ordinates in the plane of the instantaneous orbit. Further, let a denote the cosine of the angle which the axis of x makes with that of %, and G the cosine of the angle which the axis of y makes with that of y, and we shall have z, -= ax +-y (116) If the position of the plane of the orbit remained unchanged, these 486 THEORETICAL ASTRONOMY. cosines a and: would be constant; but on account of the action of the force perpendicular to the plane of the orbit, these quantities are functions of the time. Now, the co-ordinate z, is subject to two distinct variations: if the elements remain constant, it varies with the time; and, in the case of the disturbed orbit, it is also subject to a variation arising from the change of the elements themselves. We shall, therefore, have indd _w I d \p "l dt dt )' + [ dt in which ( dt ) expresses the velocity resulting from the constant elements, and d j that part of the actual velocity which is due to the change of the elements by the action of the disturbing force. But during the element of time dt the elements may be regarded as dz, constant, and hence the velocity dt in a direction parallel to the axis of z, may be regarded as constant during the same time, and as receiving an increment only at the end of this instant. Hence we shall have dz, \ r_ -dF 0] dt dt ) L dtJ Differentiating equation (116), regarding a and. as constant, we get dz, dz dxdv dy (dIz) dId d +iy (117) {^k~~dt dt dt and differentiating the same equation, regarding x and y as constant, we get -- x d- y - 0o. (118) LdtJ dt Y-dt (8) Differentiating equation (117), regarding all the quantities involved as variable, the result is d2z, du dx d1i dy d Qx+ d2y dt2 dt' dt +dt' dt dt + dt2 ( 9)' Now, we have Z,. =- X+ ftY- Zcosi, (120) in which Z, denotes the component of the disturbing force parallel to the axis of z,, and i the inclination of the instantaneous orbit to VARIATION OF CO-ORDINATES. 487 the fundamental plane. Substituting for X and Y their values given by the equations (1), and reducing by means of (116), we obtain d X d~ 2 z.-~ y+ 2V+ (1 n+m)~ +Zcosi, Z c dt2 dt2,3 or d d x, Ad d2y dt2 dt2 + dt + Zco Comparing this with (119), there results da dx df Z dy_ os (1 _dt-' -d t- + =it-.dt Z COS i. (121) cidtct di dt 181. The equation (120) gives dlz, k'(1 + m) d2Z - k2 ( + M)' z,+Zcosi + ax+ fXy. (122) The component of the disturbing force perpendicular to the plane of the disturbed orbit does not affect the radius-vector r; and hence, when we neglect the effect of this component, and consider only the components R and S which act in the plane of the orbit, we have d -o, k(1+ + ) ~ +Pop Y1 (123) in which z0 denotes the value of z, obtained when we put Z= 0. Let us now denote by Jz, that part of the change in the value of z, which arises from the action of the force perpendicular to the plane of the disturbed orbit, so that we shall have zr-zo + azt, a = U0 + 8a,? - 0 -+- S. Substituting these in equation (122) and then subtracting equation (123) from the result, we get d28 - - r( + ) az + Zcos i + Xa + YAf (124) The equations (116) and (117) give dci, dx dy a%, = -s7$au + Ydfl am +. dt rdt - If we eliminate 89 between these equations, there results \ dt dt -- dt, dt' 488 THEORETICAL ASTRONOMY. and since the flctor of &a in this equation is double the areal velocity in the disturbed orbit, we have 1 (dy d~z l/p (1- ) - ( dt z Y t (125) k I-i +(l m)\dt I dt) Eliminating &oa from the same equations, we obtain, in a similar manner,.I (/ dsd,d dx d (126) k/p (-m) dt dt (126) Substituting these values in equation (124), it becomes d2__ k (I + m d?_2, 3 k8 1q-m z', + Z cos:/ dt2 r8 d;l~.. — ( ( X d _ )dr (Y(127) " kp(l1+) \m\i dt dt )' - dt ) If we introduce the components R and S of the disturbing force, we have X -R - s Y RY +S, r r r r and hence dy dx R kl/ ) dr d: d- r dt Yx Xy - Sr. Therefore the equation (127) becomes dz;~ — k2 (1 -), -Z +Zcosi dX~~t'2~ ~~- ~ r( ), O128) R S dr Sr dz ( ) Jk,+ kl/p(1~m)d (r kl/Vv (I +) rn )dt /p (1 +2 ) dt We have, further, dr = ( + + r, d dt dt dt which, by means of the equations (108) and (109), gives dr e( sin, -r2d + r, d p ) esin v, r, Substituting this value in the equation (128), we obtain VARIATION OF CO-ORDINATES. 489 d2_z, k2 (1 I + e) *_ IR eo sin v, \A, dt2 r3 Po / + v (129) Sr / daz, e z, dv \ ki /p(1 + ) \dt 1 + d' which is the complete expression for the determination of Jz,. 182. The equations (110), (115), and (129) determine the complete perturbations of the disturbed body. The value of v must first be obtained by an indirect process from the equation (115), and then JlIf is given directly by means of (110). The value of 8z will also be determined by an indirect process by means of (129). In order to obtain the expressions for the forces R, S, and Z, let w denote the longitude of the disturbed body measured in the plane of the instantaneous orbit from its ascending node on the fundamental plane to which g and i are referred, it being the argument of the latitude in the case of the disturbed motion. Let w' denote the longitude of the disturbing body measured from the same origin and in the plane of the orbit of the disturbed body, and let 9' denote its latitude in reference to this plane. Finally, let N, N', I, and 0' have the same signification in reference to the plane of the instantaneous orbit that they have in reference to the plane of the undisturbed orbit in the case of the equations (66). Then we shall have sin Isin - (N + N') sin 4 (' - g ) sin (i' + i), sin -1Icos (N — N') - (' - sin (i' -i), cos I sin (N- N') -sin.2 (2'-2 (130) cos —cos-cos( N')cos) cos ~ (it- ), from which to determine N, N', and I. We have, also, aO' i' - N', tan (w' - N) t tan I' cos 1, (131) tan 13' - tan Isin (w' - N), from which to find w' and i', u' being the argument of the latitude of the disturbing body in reference to the plane to which g and i are referred. Since, when the fYotion of the disturbed body is referred to the plane of its instantaneous orbit, ji- 0, the equations (71), (72), and (73) become R m' k2( h r' cos' cos (w' w)- ), (132) =- m'k2h r' cos i' sin (w' - w), Z = n'k2 h r' sin', 490 THEORETICAL ASTRONOMY. by means of which the required components of the disturbing force may be found, the value of h being given by 1 1 P3 3' To find p, we have p2 _ r2 + r2 _ 2rr' cos' cos (w' - w), (133) or, putting cos ry = cos b' cos (w'- w), the equations p sin n r' sill r, pcos n r — r' cos y(. 1 The values of r' and u' for the actual places of the disturbing body will be given by the tables of its motion, and the actual values of' and i' will also be obtained by means of the tables. The determination of the actual values of r and w requires that the perturbations shall be known. Thus, when 8IM and v have been found, we compute, by means of the mean anomaly MH + - (t - to) -4 8Ji and the elements a, e0, the values of v, and r,. Then, since v +-X =v, + T, we have, according to (100), w -= v, + -0 o-. (135) We have, also, r=(1 +) r,. In the case of the fundamental osculating elements, we have ^o =_ S o)y which may be used as an approximate value of a; but the complete determination of w requires that a 20 + -8a shall also be determined. The exact determination of the forces also requires that the actual values of 2 and i as well as those of a' and i', shall be used in the determination of N, N', and I for each instant. When these have been found, it will be sufficient to compute the actual values of N, N', and I at intervals during the entire period for which the perturbations are required, and to interpolate their values for the intermediate dates. The variations of these quantities arising from the variations of S, i, g2', and i' may also be determined by means of differential formulae. Thus, from the differential relations of the parts of the spherical triangle from which the equations (130) are derived, we easily find VARIATION OF CO-ORDINATES. 491 sini sinn' sin N dN' c s osN d(P2' - 2) - cos Idi' + sTdi, sin I sin I sin I d cos N' di'- cos NdI + sin - sin iNd ((' - ). When i and I are very small, it will be better to use sin i sin N' sin i' sin N sin sin(2' - g )' sin I sin(g'- g )' 137) in finding the numerical values of these coefficients. By means of these formulhe we may derive the values of AN, AN', and I1 corresponding to given values of OQ, bi, 2b', and ji'. The formulae by means of which ~(, 8n2, and Si may be obtained directly, will be presently considered. The results for ON, 6N', and AI being applied to the quantities to which they belong, we may compute the actual values of w' and I'. The value of r will be found from the given value of v, and that of w will be given by means of equation (135). Then, by means of the formulie (132), the forces R, S, and Zwill be obtained. The perturbations will first be computed in reference only to terms depending on the first power of the disturbing force, and, whenever it becomes necessary to consider the terms of the second order, the results already obtained will* enable us to estimate the values of the perturbations for two or more intervals in advance with sufficient accuracy for the determination of the three required components of the disturbing force; and when there are two or more disturbing bodies to be considered, the forces for each of these may be computed at once, and the values of each component for the several disturbing bodies may be united into a single sum, thus using 2R, IS, and 2Z in place of?, X, and Z respectively. The approximate values of the perturbations will also facilitate the indirect calculation in the determination of the complete values of the required differential coefficients. 183. When only the perturbations due to the first power of the disturbing force are required, the osculating elements 20 and i0 will be used in finding N, N', and I, and r0, w0 will be used instead of r and w in the calculation of the values of.R,, and Z. The equations for. the determination of the perturbations 8M, v, and Sz,, neglecting terms of the second order, are, according to the equations (110), (115), and (129), the following: — 492 THEORETICAL ASTRONOMY. d M 1 f dt-2 dt lo k/po (1+ ) fSro t- 2 (138) d2P R1 2k2 (1 + m) 1 eo sin v k2 (1 -) dt2 rO ro3 kl/0o (l + m) eS Po _r2 d2_, k2 (1+ m) d Z cos it -- s,. The value of v is first found by integration from the results given by the second of these equations, and then lM is found from the first equation. Finally, 8z, is found by means of the last equation. The integrals are in each case equal to zero for the dates to which the fundamental osculating elements belong, and the process of integration is analogous, in all respects, to that already illustrated in the case of the variation of the rectangular co-ordinates. It will be obd2y served, however, that the expression for - involves only one indirect term, the coefficient of which is small, and the same is true in d2J, d1IM the case of dt', while d is given directly. When the perturbations have been found for a few dates, the values for the following date can be estimated so closely that a repetition of the calculation will rarely or never be required; and the actual value of r may be used instead of the approximate value r0 in these expressions for the differential coefficients. Neglecting terms of the second order, we have log r - log r, + 2ov, wherein 20 denotes the modulus of the system of logarithms. We may also use v, instead of v.; but in this case, since r, and v, depend on AM, only the quantities required for two or three places may be computed in advance of the integration. A comparison of the equations (138) with the complete equations (110), (115), and (129) shows that, if the values of i' and w' are known to a sufficient degree of approximation, we may, with very little additional labor, consider the terms depending on the squares and higher powers of the masses. It will, however, appear from what follows, that when we consider the perturbations due to the higher powers of the disturbing forces, the consideration of the effect of the variation of z, in the determination of the heliocentric place of the disturbed body, becomes much more difficult than when the terms of the second order are neglected; and hence it will be found advisable to determine new osculating elements whenever the consideration of these terms becomes troublesome. VARIATION, OF CO-ORDINATES. 493 The results may be conveniently expressed in seconds of arc, and afterwards v and Az, may be converted into their values expressed in units of the seventh decimal place, or, giving proper attention to the homogeneity of the several terms of the equations, in the numerical operations, 86Al may be expressed in seconds of arc, while v and 8z, are obtained directly in units of the seventh decimal place. It will be advisable, also, to introduce the interval io into the formulae in such a manner that this quantity may be omitted in the case of the formulae of integration. 184. In the case of orbits of great eccentricity, the mean anomaly and the mean daily motion cannot be conveniently used in the numerical application of the formule. Instead of these we must employ the time of perihelion passage and the elements q and e. Thus, let T0 be the time of perihelion passage for the osculating elements for the date to, and let To -+- T be the time of perihelion passage to be used in the formula in the place of To and in connection with the elements q, and e0 in the determination of the values of r, and v,, so that we have V + X -, + -. In the case of parabolic motion we have, neglecting the mass of the disturbed body, (t - (To + T)) = tan -v, + I tan' v,, (139) the solution of which to find v, is effected by means of Table VI. as already explained. To find r,, we have r, q0 see2 "V,. For the other cases in which the elements M. and cannot be employed, the solution must be effected by means of Table IX. or Table X. Thus, when Table IX. is used, we compute M from M= (t-(To +T)) 1 -4 e o 2 wherein log Co = 9.9601277, and with this as the argument we derive from Table VI. the corresponding value of V. Then, having found i 1-+ e' by means of Table IX. we derive the coefficients required in the equation v, = V + A (1OOi) + B (100i)2 + C(100i)3, (140) 494 THEORETICAL ASTRONOMY. from which v, will be determined. Finally, r, will be found from r -(1 + eo) (141) 1 +1 60 COS V, When Table X. is used, we proceed as explained in Art. 41, using the elements T — To + T, q0, and eo, and thus we obtain the required values of v, and r,. It is evident, therefore, that, for the determination of the perturbations, only the formula for finding the value of 8M requires modification in the case of orbits of great eccentricity, and this modification is easily effected. The expression Mo + l, (t-to) + aM- M, gives _o (to - To) + Io (t - to) + aM= [o (t - ( + S T)), or, simply, sMl== - T, and the equation (110) becomes d 1~ 1 1 1 (Sr dt, (142) dt- (1 + v)2 (1+2 k'po (l+-m) by means of which the value 8T required in the solution of the equations for r, and v, may be found. If we denote by t, the time for which the true anomaly and the radius-vector computed by means of the fundamental osculating elements have the values which have been designated byv, and r,, respectively, we have +1 dM dt, 1 o C -* dt and the equation (110) becomes it, 1 2+ 1 1 S dtl (143) dt - (1 +v) (1 + v)2 k/i'o (1-+m) or, putting t, = t + At, d rt I 1 - f r dt (l+ 1 (+ (1+) (Sr ldt. (144) dt' (1 - +) 2 (1 - )2 kV'po ( 1f- d) If we determine 8t by means of this equation, the values of the radius-vector and true anomaly will be found for the time t+ 8t instead of t, according to the methods for the different conic sections, VARIATION OF CO-ORDINATES. 495 using the fundamental osculating elements. The results thus obtained are the required values of r, and v, respectively. 185. When the values of the perturbations v, az,, and OM, JT, or 8t have been determined, it remains to find the place of the disturbed body. The heliocentric longitude and latitude will be given by cos b cos (I - 2) cos ( - g), cos b sin (I - ) -sin ( - g) cos i, sin b - sin (A -- ) sin i, or, since -=, - + g, cos b cos (I - ) - cos (, -a), cos b sin (I - ) = sin (A, - ) cos i, (145) sin b sin (A, - a) sin i, in which,= v, -+ r. If we multiply the first of these equations by cos ( - h), and the second by -sin ( - h), in which h may have any value whatever, and add the results; then multiply the first by'sin(~ - h), and the second by cos (I - h), and add, we get cos b cos (l-h)-=cos (A, —) cos ( 2 -h)-sin (A,-A) sin ( -h) cos i, cos b sin ( —h)-cos (A, —) sin ( -h) +sin (A, —o) cos (~ -h) cos i sill b =in (2, —r) sin i. But, since 2,- a (2, - 0) - ( - o), these equations may be written cos b oss (- h) -cos ( ) os s( (a- 20 o)os s ( — h) +sin (6 — o) sin ( — h) cos i) +sin (A,- ) (sin ( — o)cos( 2 -h)-cos ( — o) sin (Q -h) cos ), cos b sin ( — ) (146) =cos (A,-s o) (cos (a- Ao) sin (2 -h)-sin ((- oC) cos ( 2 -h) cosi) +sin (,- o) (sin (a-, o) sin (Q2 -h)+cos (o-, o) cos (g -h) cos i), sinb —sin(A, —,) cos (a — ) sin i-cos (,,-' ) sin (6-g o) sin i. Let us now conceive a spherical triangle to be formed, of which two of the sides are --- o and a - h, respectively, and let the angle included by these sides be i. Since h is entirely arbitrary, we may assign to it a value such that the other angle adjacent to the side a - 2, will be equal to io. Let the third side be designated by ho- 20, and the angle opposite to a -- 2 by /'. The auxiliary triangle thus formed gives the following relations: — 496 THEORETICAL ASTRONOMY. cos (h, — g o)=cos (a- g o) cos (2 -h)-l-sin (a- o) sin ( s -h) cos i, sin (ho- g) sin i, -sin (g -h) sin i, (147) sin (h0 — o) cos i-sin (- ~, o) cos ( g -h)-cos (aT- o) sin ( -h) cos, sin (1ho — 0 ) cos v'-cos (o- g o) sin ( — h) —sin ( — ) cos (2 -h) cos i. Combining these with the preceding equations, we easily derive cos b cos ( —h) =cos (A,- g ) cos (h,- o) +sin (A, — ) sin (ho-,- o) cos i, cos b sin (1 —h) sin (A,- o) cos (ho- ) o) i,-coso (c, — o) sin (ho- o) +cos (, — a) sin (ho~ —2o) (1+cos V') (148) +sin (,- ) ((cos -cos i) cos (h — )+Ssin (- ) sin (2 -h) sin i), sin b -sin sin (A,- o) +-(cos (- g o) sin i-sin in) sin (A, - g ) cos (A,- g ) sin ( — g0o) sin i. Since the action of the component of the disturbing force perpendicular to the plane of the disturbed orbit does not change the radiusvector, we have r sin b - r sin io sin (A, —~o0) + 8z,, and hence the last of these equations gives =sin (A, - o) (cos (T - - o) s -s i - sin )( -cos (A, - o) sin (, - %o) sin i. From the relation of the parts of the auxiliary spherical triangle, we have sin i sin (a -- o) = sin -r sin (ho - 0o), sin i cos (o - no) - sin v' cos (ho - 2o) cos i + cos j' sin i0. Therefore, = sin(A, - o) (cos os io (ho - g2,) sin'-sin i(1 - os)), (150) - cos (A, - 2 o) sin (ho - o) sin n', and z, sin, = sin (A,- 0) (cos io cos (h,- 0o) (1+cos V')-sin i sin 7') - I cos' r^ ~~ I~~~os^~~ -qC'~(151) - cos (A,- 2o) sin (ho — o) (1 + cos ). We have, further, from the auxiliary spherical triangle, cos i = sin o sin V' cos (ho - o) - cos cos', from which we get cos i -cos io = sin i cos (ho- go) sin' - os io (1 + cos ). We have, also, sin (a - go) sin i sin -' sin (ho- o), sin ( - h) sin i sini, sin (ho — o), VARIATION OF CO-ORDINATES. 497 or sin ( - P o) sin (g - h) sin2 i sin2' (io - Po) sin io sin'. Hence we derive (cos i-cos i) cos (ho-. o) +sin (~ —g o) sin (2 -h) sin2 i-sin iosin 7' - (1+cos r') cos io cos (ho — o ). Combining this and the equation (151) with the equations (148), we obtain cob cos ( —h)=cos(2,,- ~o) cos (h- 0) +sin(2, — o)sinl (ho — o) cos i, cos b sin ( —h) sin (,- o)cos (ho — os) icoso (A,- g o) sin (ho- o) sin /' _z, - -cos' r sin b =sin (A,- o) sin i0+ -- If we multiply the first of these equations by cos (ho - Q2), and the second by -sin (h/0- 2), and add the results; then multiply the first by sin (ho - 2 0), and the second by cos (ho - a ), and add, we get sin -' 8z, cos b cos (1-2o-(h-ho))-cos (,- Qo)+sin (ho- o) l cs l, —, sin'2' (z, cos b sin (1- o-(h —ho))=sin (A,- So)cos-cos(ho — o) 1 -cos sin b =-sin (2, — o) sin i0o-+'- (152) Let us now put p = sin (e - 6'T) sin i,' = cos ( - go) sin i -sin, ( 3) and there results, from (149), - q' sin (, - o) -p' cos (A,- o). (154) Comparing this with equation (150), we observe that p' = sin' sin (^ho- o), q' = sin l' cos (ho - ao) cos i - sin i, (1 - cos'). Therefore, we have sin 7' _ p' -~ cost sin (ho -- 2o) = -cos ~" sin'2' __ q_ n,cos (^o - 0) =tan + (1 - c ) I - cos? ~/tl os (I~cos)32 32 498 THEORETICAL ASTRONOMY. and, if we put F= h- h,, the equations (152) become cosbcos(l — S r) —cos(A, — o)+ I -- cos. 1. -- eOS r'l r' (155) cosbs (inn ~o~Z')-sin (i,-~- o) cos io( tan i,+cos io ( ) C r, sin b =sin (A,- o) sin oi,+-. As soon as r, p', qt, and v' are known, these equations will furnish the exact values of I and b, those of i, and r being found by means of the perturbations v and lfM. 186. The value of F may be expressed in terms of p' and q' Thus, if we differentiate the first of equations (147) and reduce by means of the remaining equations of the same group, we get d (ho - a o) - cos' d ( - ) + cos i, da + sin i, sin ( - o0) di, and if we interchange a - h and h -- 20 in this equation, we must also interchange i and io, which are the angles opposite to these sides, respectively, in the auxiliary spherical triangle, so that we shall have d ( -- h) = cos' d (o - g2) + cos i do, io being constant. Adding these equations, observing that 20 is also constant, we get (1-cos -') d ( g -h+-o)=sin i sin (T — ) di+(cos i+cos io) d; (156) and since da = cos i d g, this becomes (1 - cos 7') d (h - ho) — sin i sin ( ~- Qgo)di + (sin2i - cos' -_ cosi cos i) d-, cos l' which, since cos 7' = sin i sin i, cos (a - P o) cos i cos io, (157) may be written (1-cos I') dF= - sin i sin ( — 2 o) di+tan i (sin i-sin io cos (o — g o)) da. (158) The differentiation of the equations (153) gives dp' sin (a - 2,o) cos i di + sin i cos (a - go0) d, dq' cos ( - go) cos i di - sin i sin (o - Qo) d<, from which we derive VARIATION OF CO-ORDINATES. 499 q'dp' —p'dq' = sin2 i da - sin i0 dp' - cos i (-sin io sin (~- g o) di+-tani (sin i-sin i cos ( — o)) d6). Combining this with equation (158), we get cos i (1 - cos?') dF == q'dp' — pdq', and hence r= - _.dtt dt t, (159) cos i (1- cos') the integral being equal to zero for the instant to which the fundamental osculating elements belong. It is evident from the equations (153) that p' and q' are of the order of the first power of the disturbing forces, and hence, since i^ differs but little from 180~-(i+io), it follows that, so long as i is not very large, ris at least of the second order. The last of equations (145) gives, - r sin i sin A, cos a - r sin i cos A, sin a, and since x z r cos,, y r sin X,, this becomes, -- x sin i sin C -- y sin i cos a. Comparing this with equation (116), it appears that -a =- sin i sin, - sin i cos a, (160) and hence, by means of (153), we derive p'=- a coso -- f sin o0, q' - - sin g o + f cos o ~- sin i, and also dp - d. d. == - COS g d - sin g0 ~ dt -dt dt - da d (161) d - sin g' dt + dCt dt t d- - From the equations (118) and (121), observing that dy dx d-Y dt kl/p (1 + m), we derive, by elimination, d r sin, cos i df _ c rs cs, os' 7. - - /_ (l.')'.' l+ 500 THEORETICAL ASTRONOMY. Therefore we shall have dp' r cos i sin (A, go) dt k (+m) (162) dq' r cos i cos (A, - Q o) Z dt kl (l+ m) by means of which p' and q'may be found by integration, the integral in each case being zero for the date to at which the determination of the perturbations begins. When the value of Oz, has already been found by means of the equation (129), if we compute the value of q', that of p' will be given by means of (154), or p = q' tan (~, 0) cs (, —0) and if p' is determined, q' will be given by q'- =p' cot (2,- d2) + - t I cot (A,-2 + r sin (i,- go) If both p' and q' are found from the equations (162), 8z, may be determined directly from (154); but the value thus obtained will be less accurate than that derived by means of equation (129). Since the formula for d, completely determines the perturbations due to the action of the component Z perpendicular to the plane of the instantaneous orbit, instead of determining p' and qt by an independent integration by means of the results given by the equations (162), it dsz, will be preferable to derive them directly from 8z, and - The equations (161) give = -- cos o 0 -- sin 0 i, q' -- sin Qo a +- cos o 3a. Substituting for ao and A3 their values given by (125) and (126), and putting x".x cos 20 + y sin o0, y"- - x sin 0 + y cos b0, we obtain I y,, diz, kV'p (1+ + y" ~'- ~) -\' 1 Il +dz, dx" (163) ki/p(1l + q) ( dt f dt ) VARIATION OF CO-ORDINATES. 501 Substituting further the values x" - r cos (, - o2), y" r sin (A — ), and also dA, kl/p (I -- n) dt r2 dr kV11 + n. ki/ (l + ) e sin v ~ —=^ ~-~ es~ sin v -- dt - s/p r -+- e cos v we easily find, since A,- v =,'=-(cos (A,-XO)+Hecos(x-G )z r sin/ (<, - (~) (d,6d ~' — (cos(0- g~o) + Gecos(- P/o)) — + -~ r ki//(l+ in) dt - r cos (, - ) dz, (164) - q=+ (sin (A,- Q0) + esin (X - /2o)) p + d )' - which may be used for the determination ofJp' andiq'. These equations require, for their exact solution, that the disturbed values e, X and p shall be known, but it is evident that the error will be slight, especially when e is small, if we use the undisturbed values e,, p,, and Xo ro. The actual values of A, and r are obtained directly from the values of the perturbations. When p' and q' have been found, it remains only to find cos i, and 1 - cos v', in order to be able to obtain Fby means of the equation (159). From (153) we get p' - q' in2 i -+ 2q' sin i0, and hence cos i 1/1 p'2 (q' ~ sin io,), (165) from which cos i may be found. The equation (157) gives 1 - cos' = cos i, (cos i, + cos i) - q' sin i,, (166) by means of which the value of 1 - cos v' will be obtained. If we substitute the values of pt, q, dt, and dt given by the equations (153) and (162) in (159), it is easily reduced to Zdt, (167) (1 - cos I') k/p (1 + m) which may be used for the determination of r. When we neglect terms of the order of the cube of the disturbing force, in finding 1I we may use po in place of p and put 1 - cos' =2 cos io, so that the formula becomes 502 THEORETICAL ASTRONOMY. ~r — f~ 1 - z, Zdt. (168) 2 cos2 i kl/o1 (1+r m) 187. By means of the formulae which have thus been derived, we may find the values of all the quantities required in the solution of the equations (155), in order to obtain the values of I and b for the disturbed motion. From r, 1, and b the corresponding geocentric place may be found. The heliocentric longitude and latitude may also be determined directly by means of the equations (145), provided that 2, a, and i are known; and the required formulae for the determination of these elements may be readily derived. Thus, the equations (160) give, by differentiation, da.. di do -d sin f cos dt sill i Cos dtdt at dt d3. di... da d-t = cos a cosi -- sin i sin ddt c t dt' whence. da d. d sin -cos -sin - dt dt dt' di da dfi cos d - — sin a - - cos - dt dt dt du du Introducing the values of dt and - already found into these equations, and putting a T=Of f==,- 8+ i+ 8= i - - a8 + 8, we obtain ci - m) cot i sin (A, -- ) rZ, di 1 (169) t kp(1 +~ cos (2,- ~) rZ, dt kl/p(l1 +- m) and also, since da - cos i d2, d _ 1 sin (A,- 6) rZ, (170) dt kVlp(1 -+) sini by means of which the variations of a, i, and a due to the action of the disturbing forces, may be determined. The integral is in each case equal to zero at the initial date to to which the fundamental osculating elements belong and at which the integration is to commence. VARIATION OF CO-ORDINATES. 503 If we find i, and then a- a from 6 _ ff - ~Sil (,i a) rZdt, (171) the true longitude in the orbit will be obtained from R-Ax,+ ~ 6 —. d~i d8a d 2g It is evident that since the expressions for d — dt and -t require, for an accurate solution, that the disturbed values i, a, and p shall be known, and require, besides, that three separate integrations shall be performed, unless the perturbations are computed only in reference to the first power of the disturbing force, in which case we use io, Po, and g20 in place of i, p, and a, respectively, in the equations (169) and (170), the action of the component Zcan be considered in the most advantageous manner by means of the variation of z, arising from this component alone; and even when only the perturbations of the first order are to be determined it will still be preferable to derive 8z, by the indirect process from the expression for d-F', and to determine the heliocentric place by means of the equations (155). When we neglect the terms of the second order, these equations become cosb cos ( - g20) - cos (A,- g0),.,Z cos b sin (I - Po) = sin (A, g~0) cos i, - tan 0' —, (172) s sin b = sin (,- g 0) sin i0+ - by means of which I and b are determined immediately from the perturbations 8s, v, and az,. The peculiar advantage of determining the effect of the action of the component Z by means of the partial variation of z, is apparent when we observe that the expressions for d~6 deg~ d and. -a involve sin i as a divisor; and in the case of orbits whose dt dc inclination is small, this divisor may be the source of a considerable amount of error. 188. The determination of the perturbations so as to include the higher powers of the masses is readily effected by means of the comd8M d'v dz,f plete expressions for t, and —, when the correct values of R, X, Z, i, and p are known. The corrected values of i and p 504 THEORETICAL ASTRONOMY. which are required only in the case of az,-may be easily estimated with sufficient accuracy, since we require only cosi, while V/p appears as the divisor of a term whose numerical value is generally insignificant. To obtain the actual values of R, S, and Z, the corrections to be applied to N, N', and I must first be determined by means of the formulae (136). The values of 8i' and 82' will be found by means of the data furnished by the tables of the motion of the disturbing body, and the corresponding corrections for N, N', and I having been found by means of the terms of (136) involving di' and dQg, there remain the corrections due to 6i and 802 to be applied. These may be found in terms of the quantities p' and q' already introduced. Thus, the equations dp' -- cos i sin (6 - o) di + sin i cos (6 - o) do, dq' =- cosi cos (6 -,) di - sin i sin (6 - d) da, give cos i di = sin (a - 0 o) dp' + cos ( - Q O) dlq', sin i d =- cos (6 - Q ) dp' - sin (a — 2 o) dq'. The equations (136) give, observing that da cos i d 2, d -- cos N di - tan i sin N d, sin NV tan i dN' -- + dit- cos Nda, sin I sin I and, substituting the preceding values of di and da, these become sin (N + a- Q) dp' cos (N + 6-,)d cos i cos d d NV'-_ cos (N+ 6- ao)d + sin (N + 6 ) - sin I cos i sin os dq. If we neglect the perturbations of the third order, these equations give 1 - sin N - cos N -, cos cos -cosi0 2 COSt0 (173) (N'= cosec I cos N - sin N2V, \ cos i cos to by means of which 8I and 8N may be determined, p' and q' being found by means of the equations (164), using e, 70, and p, in place of e, x, and p. The results for 8I and AN' obtained from (173) being applied to the values of I' and N' as already corrected on account of Ji' and 8 j', give the required values of these quantities. NUMERICAL EXAMPLE. 505 iWhen we consider only di and dQ, since sin i' cos' - cos i sin I sin i cIos N c, we easily find N= cos I aN' 8- a, (174) and if we add the quantity cos I N' to the value of N already corrected on account of 3i' and 820', and denote the result by N,, the required value of N will be N, - 8a. Then, according to (131), we may compute w' + Ja and d' by means of the formulae i' -t' N', tan ((w' + a,)- N,) tan u,' cos, (175) tan i' tan Isin ((w' +- ) - N,), using the values of N' and I as finally corrected. We have, further, according to (135), w - + = v, + =o - go, by means of which we may compute the value of w + Oa; then the value of w' - w required in the equations (132), and also in finding the value of p, will be given by w' -- = (w' + a) - (0 + 8a), and the forces R, S, and Z may be accurately determined. By thus determining the correct values of R, S, and Z from date to date, the perturbations AM, v, and 8z, may be determined in reference to the higher powers of the disturbing forces according to the process already explained. The only difficulty to be encountered is that which arises from the quantities r, p', and q', required in the determination of the heliocentric place of the disturbed body by means of the equations (155). If an exact ephemeris for a short period is required, by means of the complete perturbations we may determine new osculating elements, and by means of these the required heliocentric or geocentric places. 189. EXAMPLE.-We will now illustrate the application of the formulae for the determination of the perturbations JM, v, and 8z, by a numerical example; and for this purpose let it be required to determine the perturbations of Eurynomne @ arising from the action of Jupiter from 1864 Jan. 1.0 to 1865 Jan. 15.0, Berlin mean 506 THEORETICAL ASTRONOMY. time, the fundamental osculating elements being those given in Art. 166. In the first place, by means of the formula (130), using the values a - 2060 39' 5".7, i 4- 36' 52".1, dg'- 98 58 22.7, i'1= 18 40.5, which refer to the ecliptic and mean equinox of 1860.0, we obtain N= 194~ 0' 49".9, N'- 301~ 38' 31".7, - 5~ 9' 56".4. Then, by means of the data furnished by the Tables of Jupiter, we find the values of u', the argument of the latitude of Jupiter in reference to the ecliptic of 1860.0, and from the equations (131) we derive w' and i'. The values of r' are given by the Tables of Jupiter, and the values of r0 and %v are found from the elements given in Art. 166. The results thus obtained are the following:Berlin Mean Time. log o0 v0 log r w' W 1863 Dec. 12.0, 0.294084 354~ 26/ 18/.0 0.73425 14~18/ 54".6 -0~ 1 38/.1 1864 Jan. 21.0, 0.294837 10 2 45.7 0.73368 17 21 44.2 0 18 9.1 March 1.0, 0.300674 25 24 59.4 0.73305 20 25 5.2 0 34 39.9 April 10.0, 0.310864 40 13 31.8 0.73237 23 28 59.8 0 51 7.6 May 20.0, 0.324298 54 14 41.4 0.73164 26 33 32.1 1 7 29.7 June 29.0, 0.339745 67 21 23.5 0.73086 29 38 44.8 1 23 43.5 Aug. 8.0, 0.356101 79 32 18.1 0.73003 32 44 41.2 1 39 46.3 Sept. 17.0, 0.372469 90 49 57.6 0.72915 35 51 24.6 1 55 35.2 Oct. 27.0, 0.388214 101 19 9.8 0.72823 38 58 57.5 2 11 7.5 Dec. 6.0, 0.402894 111 5 42.2 0.72726 42 7 23.3 2 26 20.3 1865 Jan. 15.0, 0.416240 120 15 32.6 0.72625 45 16 43.9 -2 41 10.6 The value of w for each date is now found from wU —Vo + r- 0 -Qo + 197~ 38' 6".5, and the components of the disturbing force are determined by means of the formule (132), p being found from (133) or (134), and h from (70). The adopted value of the mass of Jupiter is 1047.879' and the results for the components R, S, and Z are expressed in units of the seventh decimal place. The factor (2 is introduced for convenience in the integration, (o being the interval in days between the successive dates for which the forces are to be determined. Thus we obtain the following results: NUMERICAL EXAMPLE. 507 Date. 2/R 2r5o o2Z coSo of Srodt 1863 Dec. 12.0, + 70.82 + 7.16 + 0.04 + 1.37 1864 Jan. 21.0, 68.95 - 32.76 0.49 - 11.45 March 1.0, 61.16 70.38 0.92 63.32 April 10.0, 48.57 102.91 1.32 150.48 May 20.0, 32.77 128.34 1.68 266.75 June 29.0, + 15.41 145.39 1.96 404.35 Aug. 8.0, - 2.19 153.44 2.17 554.54 Sept. 17.0, 19.12 152.41 2.29 708.21 Oct. 27.0, 34.81 142.50 2.25 856.39 Dec. 6.0, 48.95 124.04 2.09 990.36 1865 Jan. 15.0, - 61.45 - 97.36 + 1.75 - 1101.73 The single integration to find (wSrodt is effected by means of the formula (32). The equations for the determination of the required differential coefficients are W d8IM I d -- 2w w t ~^- -==-^ -,~=- ^ I S3t ~1 2 zd2~~X, w2k2 d 2 2 R 2P I Sil 2 2 k2 2 r, + k\,/p J P po < ~ clP",. -2 d ~ 2Zcos -- 9 z. Substituting in these the results already obtained, and also log Jo - 2.967809, logpo 0.371237, log e, - 9.290776, we obtain first, by an indirect process, as illustrated in the case of the direct determination of the perturbations of the rectangular cod'v d2Sff ordinates, the values of W2 dt' and 2 and having found dt2 then, having found v, w dt is given directly by the first of these equations. The integration of the results thus derived, by the formulae for mechanical quadrature, furnishes the required values of v, 8M, and cz,. The calculation of the indirect terms in the determination of v and Jz,, there being but one such term in each case, is, on account of the smallness of the coefficient, effected with very great facility. The final results are the following: 508 THEORETICAL ASTRONOMY. dl2 ~ d2v d2sz, Date. 2 d —- t2 2 d M v z, dt dt 2 dt 1863 Dec. 12.0, 0".028 + 36.16 + 0.04 + 0".01 + 4.41 + 0.02 1864 Jan. 21.0, 0.072 33.61 0.49 -0.01 4.31 0.04 March 1.0, 0.499 22.55 0.89 0.27 37.11 0.54 April 10.0, 1.213 + 5.58 1.21 1.11 91.96 1.93 May 20.0, 2.070 - 13.52 1.45 2.75 152.22 4.52 June 29.0, 2.902 31.59 1.53 5.24 199.05 8.54 Aug. 8.0, 3.546 46.65 1.60 8.49 214.54 14.10 Sept. 17.0, 3.858 57.88 1.52 12.22 183.69 21.24 Oct. 27.0, 3.723 65.19 1.28 16.05 + 95.29 29.90 Dec. 6.0, 3.056 68.83 0.92 19.49 - 58.00 39.82 1865 Jan. 15.0, - 1.800 - 69.19 + 0.40 -21.97 -279.84 +50.64 Since, during the period included by these results, the perturbations of the second order are insensible, we have, for the perturbations of Eurynome arising from the action of Jupiter from 1864 Jan. 1.0 to 1865 Jan. 15.0, sM-= - 21".97, - 0.00002798, z, 0.00000506. It is to be observed that Jz, is not the complete variation of the coordinate z, perpendicular to the ecliptic, but only that part of this variation which is due to the action of the component Z alone; and hence the results for 8z, differ from the complete values obtained when we compute directly the variations of the rectangular coordinates. Let us now determine the heliocentric longitude and latitude for 1865 Jan. 15.0, Berlin mean time, including the perturbations thus derived. From the equations M, - Mo + P0 (t- to) + 3M, E, e0 sin E, - M,, r, =a 0(1 - e cos E,), sin (v,- E,) -sin sin E, ao, r,, = v, -+, r - r, (1 + v), we obtain M, 99~ 29' 35".51, E,- 110~ 0'33".75, log r, 0.4162304, v, - 120 15 13.80, log r = 0.4162183, A, - 164 32 25.97. The calculation of the values of r, and v, from the values of M,, a0, and e,, may be effected by means of the various formulae for the NUMERICAL EXAMPLE. 509 determination of the radius-vector and true anomaly from given elements. If we substitute these results for ),, r, and 8z, in the equations (172), we get I 1640 37' 59".05, b - 3 5' 32".54, which are referred to the ecliptic and mean equinox of 1860.0, and from these we may derive the geocentric place of the disturbed body. If the place of the body is required in reference to the equinox and ecliptic of any other date, it is only necessary to reduce the elements %T, Qo, and i0 to the equinox and ecliptic of that date; and then, having computed i, and r, we obtain by means of the equations (172) the required values of I and b. In the determination of the perturbations it will be convenient to adopt a fixed equinox and ecliptic throughout the calculation; and afterwards, when the heliocentric or geocentric places are determined, the proper corrections for precession and nutation may be applied. In order to compare the results obtained from the perturbations M,; v, and 8z, with those derived by the method of the variation of rectangular co-ordinates, we have, for the date 1865 Jan. 15.0, o= - 2.5107584, -- + 0.6897713, zo - 0.1406590; and for the perturbations of these co-ordinates we have found'x - + 0.0001773, y -+ 0.0001992, az -0.0000028. Hence we derive z - 2.5105811, y - + 0.6899705, z — 0.1406618, and from these the corresponding polar co-ordinates, namely, log r 0.4162182, 1 164~ 37' 59".05, b - 30 5' 32".54, from which it appears that the agreement of the results obtained by the two methods is complete. 190. When the perturbations become so large that the terms of the second order must be retained, the approximate values which may be obtained for several intervals in advance by extending the columns of differences, will serve to enable us to consider the neglected terms partially or even completely, and thus derive the complete perturbations for a very long period. But on account of the increasing difficulties which present themselves, arising both from the consideration 510 THEORETICAL ASTRONOMY. of the perturbations due to the action of the component Z in computing the place of the body, and from the magnitude of the numerical values of the perturbations, it will be advantageous to determine, from time to time, new osculating elements corresponding to the values of the perturbations for any particular epoch, and thus commencing the integrals again with the value zero, only the terms of the first order will at first be considered, and the indirect part of the calculation will, on account of the smallness of the terms, be effected with great facility. The mode of effecting the calculation when the higher powers of the masses are taken into account has already been explained, and it will present no difficulty beyond that which is inseparably connected with the problem. The determination of F, p', rl' dap' dq' and q' may be effected from the results for -t dt, and by means of the formulae for integration by mechanical quadrature, as already illustrated, or we may find F by a direct integration, and the values of 1p and q' by means of the equations (164), dt being found from dt' by a single integration. The other quantities required for the complete solution of the equations for the perturbations will be obtained according to the directions which have been given; and in the numerical application of the formulae, particular attention should be given to the homogeneity of the several terms, especially since, for convenience, we express some of the quantities in units of the seventh decimal place, and others in seconds of arc. The magnitude of the perturbations will at length be such that, however completely the terms due to the squares and higher powers of the disturbing forces may be considered, the requirements of the numerical process will render it necessary to determine new osculating elements; and we therefore proceed to develop the formulhe for this purpose. Cd22v d'WZ 191. The single integration of the values of wt2 and W2 wdill th vade d& o6, dv d^z, give the values of ~-~d and o d-, and hence those of d and d-, dJ'M which, in connection with dt, are required in the determination of the new system. of osculating elements. Since r2 dv rersents double the areal velocity i the disturbed oents i we havedouble the areal velocity in the disturbed orbit, we have CHANGE OF THE OSCULATING ELEMENTS. 511 dv, kl/p (1 + mn) dt r2 The equation (109) gives dt, do' dt do, _ EVI (qI- mn)(1 -~ + dsM ) Hence, since r - r, (1 + v), we obtain p p(o l+ o1. dM ) (1+y), (176) by means of which we may derive p. This formula will furnish at once the value of p, which appears in the complete equation for dt2, and also in the equations (164); and the value of cos i may be determined by means of (165). In the disturbed orbit we have dr kVl + mn - - - e sin v, dt l/p and the equations (108) and (111)'give dr kil-. I d8M\ dv.d -.. e0sin v, 1+ - (1 + -) - + r, d d't d)' Therefore we obtain /o e sin v - 1/ e sin v, (1 )+ / +. dv /Po dt k/ l — m which, by means of (176), becomes.,I dM,1 +,) rj/p d (7 esin v = e sin v, 1 ( + dt + (177) p, Po f dt V(- I /lim dtThe relation between r and r, gives P Po 1 + e cos v 1 + eo cos v, and, substituting in this the value of p already found, we get e cos v = (1 +eo cosv,) (1 -1 d ) (1 + )31. (178) \ Po dt / Y) 512 THEORETICAL ASTRONOMY. Let us now put o dt r,Vp d' (179) k-/l + m dt a and f9 being small quantities of the order of the disturbing force, and the equations (177) and (178) become e sin v= e0 sin v, + aeo sin v, +-, e cos V = eo cos V, + ace COS V, +- a. These equations give, observing that r, (cos v, + eo) =po cos,, e sin (v,- v) =c a sin v, - f cos v,, a0o (180) e cos (v,- v) = e- P cos E, + ~ sin v,, from which e, v,- v, and v may be found; and thus, since - l0o + -(v - ), (181) we obtain the values of the only remaining unknown quantities in the second members of the equations (164). The determination of p' and q' may now be rigorously effected, and the corresponding value of cos i being found from (165), dt and dt will be given by (162). Then, having found also 1 cos i' by means of (166), r may be determined rigorously by the equation (159), and not only the complete values of the perturbations in reference to all powers of the masses, but also the corresponding heliocentric or geocentric places of the body, may be found. If we put - =ca sin v, - f cos v,,,' aPo cos E, + f sin v,, (182) and neglect terms of the third order, the equations (180) give t2 e - eo + + 2e-' r' r'a' (183) VV 2 - — 8 - s, e0 eo in which s= -206264"1.8. These equations are convenient for the CHANGE OF THE OSCULATING ELEMENTS. 513 determination of e and v, -v, and hence X by means of (181), when the neglected terms are insensible. The values of p, e, and v having been found, we have k/1 +- m sin -= e, a =p sec2, 3 -- (184) tan -1 E - tan (45~ - ) tan, v, M — E — e sin E, from which to find the elements p, a,,e, and M. The mean anomaly thus found belongs to the date t, and it may be reduced to any other epoch denoted by t, by adding to it the quantity / (t0 - t). When we neglect the terms of the -f4 order, we have sin vp - sin'o -— o - cos.o - ( -'o) sin go and if we substitute for sin c - sin o, - e - e the value given by the first of equations (183), the result is 2' sin o +- r'2'D 9~ 2 sin %oP cos g o-' sin go tan go from which we get a, 812 sin'o s 12' +- +. + s- + ~s - s s, (185) ~ cos o 2 cos 23so 2 sin cos'o by means of which (p may be found directly, terms of the third order being neglected. In the case of the orbits of comets for which e differs but little from unity, instead of OMf we compute by means of the formula (142) the value of AT, and since we have d8T 1 d8M dt /to dt the equation for p becomes P-p(l- daT) (1 + v)4; (186) and for a we have (- dt) (1 + ). (187) Then e, v, and q will be found by means of the equations 33 514 THEORETICAL ASTRONOMY. e sin (v, - v) = a sin v, - cos v,, e cos (v, - v) = e + a (cos v, + e) + 13 sin v,, (188) p l+e and the time of perihelion passage will be derived from e and v by means of Table IX. or Table X. There remain yet to be found the elements a, Qg, and i, which determine the position of the plane of the disturbed orbit in space. The values of p' and q' will be found from the equations (164), and F, whenever it may be required, will be determined as already explained. Then we shall have sin i sin (a - go) =p', (189) sin i cos (a - 20) q' +- sin i0, from which to find i and a. When we neglect the terms of the third order, these equations give sin i -sin io-q' si smn and hence p' p'q' a 0 + sin i, sin2 oi q' q" sin io P' + i -- io - c-o s si q- 32 c (190) cos i + 2 cos i, 2 sin i cos (g in which s -206264".8. The auxiliary spherical triangle which we have employed in the derivation of the equations (155) gives directly cos 1 (i + io) tan 2 (6 o) cos (i - i) tan ( -- h+ho go) and since h - h= r, we have t Cos (i-i) tan t- ( ~ — )- COS (i- o) tan (- go), (191) by means of which the value of 2 may be found. This equation gives, when we neglect terms of the third order, I — __o sin io g~ =- do + I + cos I co ( - io) (+ -- Po). (192) 0 00 0 cos ig'L 2 cos2 j Substituting in this the values of a- 20 and i —io given by (190), we get a = 1 - 3 sin io, -ini -- iio - ~- -pqs - /r, (193) sin i cos i sin.2 cos i CHANGE OF THE OSCULATING ELEMENTS. 515 r being expressed in seconds of arc. Finally, for the longitude of the perihelion, we have 7 X+ ~- a, (194) and the elements of the instantaneous orbit are completely determined. When we neglect terms of the third order, this equation, substituting the values given by (190) and (192), becomes = tan. is + tan' io (1 + 2 cos +. (195) cos 2c 2 cos' i, It should also be observed that the inclination i which appears in these formulae is supposed to be susceptible of any value from 0~ to 180~, and hence when i exceeds 90~ and the elements are given in accordance with the distinction of retrograde motion, they are to be changed to the general form by using 180 - i instead of i, and 2 -- instead of wr. The accuracy of the numerical process may be checked by computing the heliocentric place of the body for the date to which the new elements belong by means of these elements, and comparing the results with those obtained directly by means of the equations (155). We may remark, also, that when the inclination does not differ much from 90~, the reduction of the longitudes to the fundamental plane becomes uncertain, and r may be very large, and hence, instead of the ecliptic, the equator must be taken as the fundamental plane to which the elements and the longitudes are referred. 192. Although, by means of the formulae which have been given, the complete perturbations may be determined for a very long period of time, using constantly the same osculating elements, yet, on account of the ease with which new elements may be found from 8J1M, dJM d, d_, v, z,, d- - ~t and dt' and on account of the facility afforded in the calculation of the indirect terms in the equations for the differential coefficients so long as the values of the perturbations are small, it is evident that the most advantageous process will be to compute MM, v, and 8z, only with respect to the first power of the disturbing force, and determine new osculating elements whenever the terms of the second order must be considered. Then the integration will again commence with zero, and will be continued until, on account of the terms of the second order, another change of the elements is required. The frequency of this transformation will necessarily de 516 THEORETICAL ASTRONOMY. pend on the magnitude of the disturbing force; and if the disturbed body is so near the disturbing body that a very frequent change of the elements becomes necessary, it may be more convenient either to include the terms of the second order directly in the computation of the values of 8M, r, and 8z,, or to adopt one of the other methods which have been given for the determination of the perturbations of a heavenly body. In the case of the asteroid planets, the consideration of the terms of the second order in this manner will only require a change of the osculating elements after an interval of several years, and whenever this transformation shall be required, the equations for Ap, i, Q, and 7r, in which the terms of the third order are neglected, may be employed. It should be observed, however, that the perturbations of some of the elements are much greater than the perturbations of the co-ordinates, and hence when terms depending on the squares and higher powers of the masses have been neglected in the computation of these perturbations, it may still be necessary to include the values of the terms of the second order in the incomplete equations referred to. No general criterion can be given as to the time at which a change of the osculating elements will be required; but when, on account of the magnitude of the values of UM, v, and 8z,, it appears probable that the perturbations of the second order ought to be included in the results, by computing a single place, taking into account the neglected terms, we may at once determine whether such is the case and whether new elements are required. 193. We have already found the expressions for the variations of a and i due to the action of the disturbing forces, and we shall now consider those for the variation of the other'elements of the orbit directly. Let x, y, z be the co-ordinates of the body at any given time referred to any fixed system of co-ordinates. These will be known functions of the six elements of the orbit and of the time. If the body were not subject to the action of the disturbing forces, these six elements would be rigorously constant, and the co-ordinates would vary only with the time; but on account of the action of these forces the elements must be regarded as continuously varying in order that the relation between the elements and the co-ordinates at any instant shall be expressed by equations of the same form as in the case of the undisturbed motion. The co-ordinates will, therefore, in the disturbed motion, be subject to two distinct variations: that which results from considering the time alone to vary, and that which VARIATION OF CONSTANTS. 517 results from the variation of the elements themselves. Let these two kinds of partial variations be symbolized respectively by ( d) and dt, 2and similarly in the case of the other co-ordinates; then will the total variations be given by dx _ dx_ \ d rx dy_ dy \ _, d ~ dt ( dt dt ~d+ l _dt' dz _ dz \+ ~dz (196) dt \dt Ldt But if we differentiate twice in succession the equations which express the values of x, y, and z as functions of the elements and of the time, regarding both the elements and the time as variable, the substitution of the results in the general equations for the motion of the disturbed body will furnish three equations for the determination of the variations of the elements. There are, however, six unknown quantities to be determined; and hence we may assign arbitrarily three other equations of condition. The supposition which affords the required facility in the solution of the problem is that [do' [fd] 2 []-~' (197) and hence that dx dx \ dy dy\ dz dz \ dt dt }' dt \dt' dt \dt It thus appears that in order that the integrals of the equations (1) shall be of the same form as those of the equations (3),-the arbitrary constants of integration which result from the integration of the latter being regarded as variable when the disturbing forces are considered,-the first differential coefficients of the co-ordinates with respect to the time have the same form in the disturbed and undisdx dy dA turbed orbits. But since -, -d and are the velocities of the dt dt dt disturbed body in directions parallel to the co-ordinate axes respectively, it follows that during the element of time dt the velocity of the body must be regarded as constant, and as receiving an increment only at the end of this instant. The equations (197) show also that if we differentiate any co-ordinate, rectangular or polar, referred to a 518 THEORETICAL ASTRONOMY. fixed plane and measured from a fixed origin, with respect to the elements alone considered as variable, the first differential coefficient must be put equal to zero, and this enables us at once to effect the solution of the problem under consideration. It is to be observed, further, that the functions whose first differential coefficients with respect to the time when only the elements are regarded as variable are thus put equal to zero, must not involve directly the motion of the disturbed body, since the second differential coefficients of the coordinates have not the same form in the case of the disturbed motion as in that of the undisturbed motion. 194. If we suppose the disturbing force to be resolved into three components, namely, R in the direction of the disturbed radiusvector, S in a direction perpendicular to the radius-vector and in the plane of disturbed orbit, positive in the direction of the motion, and Z perpendicular to the plane of the instantaneous orbit, the latter will only vary 2 and i and the longitude of the perihelion so far as it is affected by the change of the place of the node, while the forces 1 and S will cause the elements I1, 7r, e, and a to vary without affecting 2 and i. Let us now differentiate the equation V2- k2 (1+ ) ( 2 ), regarding the elements as variable, and we get 2 rdr] - 1 da 2V dV ~2 L dt J - -2 dt k (I + m) dt' or da 2a V dV dt k2 (1 + m) dt dV The differential coefficient dt is here the increment of the accelerating force, in the direction of the tangent to the orbit at the given point, due to the action of the disturbing force; and if we designate the angle which the tangent makes with the prolongation of the radius-vector by O0, we shall have dV / -=R cos bS + sin 0. Substituting this value in the preceding equation, we obtain VARIATION OF CONSTANTS. 519 da 2a2 da = k2 (1 m) (R V cos (P0 + SV sin sb). dt k2 (1 — M) But we have, according to the equations (50)6, Vcos (d o - dt- H esnv, c /dr \ k~l + — m) Vsin -- =r( d~ t (I + )-' in which v denotes the true anomaly in the instantaneous orbit; and hence there results da 2a2 cdt kV Z (l (e sin v t P), (198) dt /p ( r by means of which the variation of a may be found. If we introduce the mean daily motion p, we shall have dg a / da dt~,t (199) dt a dt and hence d/2 3a/~ dl a - s ~((e sin vR + S), (200) dct k/p (1 + - m) r for the determination of 8u. The first of the equations (97) gives d / dv\ dt 2 -dt- Sr; and hence we obtain d (l/p) Sr dt kl/1 + m or -p -2pr S. (201) dt kl/1 + m The equation p = a (1 - e2) gives dp p da de dt a dt t2 Equating these values of d- and introducing the value of dalready found, we get de kpl/psin vR+ - e - (202)a dt k p(1+? e r a 520 THEORETICAL ASTRONOMY. and since P - 1 e cos v, r 1 e cos E, r a E being the eccentric anomaly in the instantaneous orbit, this becomes de 1 dt -- /1 (p sin vR + p (cos v cos E) S), (203) dt - k I/j (I T +m) which will give the variation of e. If we introduce the angle of eccentricity %, we shall have de dcs ~ - ==cos ) dt- r r sin c cos (C+ u) d If we multiply the first of these equations by Jx, the second by Sy, dx and the third by Jz; then multiply the first by d the second by dy dzdt second by -a d and the third by dt — and put dt/ dt P= sin a sin (A + u) Sx + sin b sin (B - u) ay + sin c sin (C +- u) Jz, Q sin a cos (A +- u) 8x + sin b cos (B+ + u) 8y + sin c cos ( C + u) az; dx dy (236) P' =sin a sin (A + u) d-t + sin b sin (B + u) 8 dt c~lt (idt + sin c sin (C + ( u) dt' ~~~dx Q' = =sa (A ) + in a cos (B + u) dy dt dt dz + sin c cos ( C + ut) 5 d dr k d ki/p we shall have, observing that dt = e sin v and that d — _ _ dt 1+p dt ra dx + dy dz k k sin P +l/Vp dx -^ - + Y -- ~z + " e sin vp + - +' dt dt Y t p r ""P (237) dx dx dy dy dz dz k k+ JQ/ dt q- t- ~da e sin vPJ +' dt ddt + t It dt dt ^/p r From the equations dr dx dy dz dt dt dt dt dX2 dy2 dZ2 dt- dt2+ dt2 PERTURBATIONS OF COMETS. 541 we get rdr d d d x dy dz dt dt + dt (dt dt dt dt dx dx dy dy dz dz dt t dt dt dt dt dt which by means of (237) become (rdr - - essinvP-~ - Q+P'r, \ dt i VP r / (238) V8VW-/= esinvP'+ P Q. 1p From the equation k2p - V2r. _ dr rdr2 we get rdr rdr \ 22pkk + kp = 2r2 V8 V + 2 V2rar - d 2 t. Substituting the values given by (238), observing also that P-=r, this becomes Sk +p V2r p re2 sin V e sin v and, since k2 V2 =-(1 + 2e cos v + e2) we obtain p e sin v _ k S(;)-= -pj Q + kQ'-/I 7 (239) by means of which the variation of /p may be found. The equation k2 2k2 - - - rV2 gives 1 2 2 VW 2 a)k a 2 k \r1 a k from which we derive a1 2 P- 2esin v 2 ( /'p+1 2 (- k) ( 240 a r r1 k (240) a ^ k/ rk Q' \- 2 ak 542 THEORETICAL ASTRONOMY. from which the new value of the semi-transverse axis a may be found. To find d/ we have = 1aa-8 + IJ (241) I ik or 3/1a 3tae sin v 31al/pQ (, _- 6 ) a * 63f= ap= a inVPIh — PQ~~(62) [Pp' 2. (242) Next, to find de, we have, from p a (1 - e2) de= —^A__ V () (243) 2e a ae or p cos E sinv sin v 1/ p P, e ^C ~~ P + ~ Q + ~n P t + -7 (cos v + cos E) Q -- C- -5 k 2p cos E (244) r k The equation (12)2 gives r22 sin V aM- r cos V s c-Jv (2 - e cos v) de, (245) ac cos 50 a2 Cos"-3 < and from - = 1 + e cos v we get (s 3V cos (1/-) (246) av — -+-~ — --- ~(). (246) e sin v r2e sin v re sin v Substituting this value of 8v in (245), and reducing, we find 6M (- cot ta sin vP tn (p cot v cos v — 2cos ) P/ 1 (P+ r) sin v ( cot tan 2r sin v k (247) -k tan~ l' + r + a k from which to derive the variation of the mean anomaly. 205. Let us now denote by x", y", z' the heliocentric co-ordinates of the comet referred to a system in which the plane of the orbit is the fundamental plane, and in which the positive axis of x is directed to the ascending node on the ecliptic. Let us also denote by x', y', z' the co-ordinates referred to a system in which the plane of the ecliptic is the plane of xy, and in which the positive axis of x is directed to the vernal equinox. Then we shall have PERTURBATIONS OF COMETS. 543 x" x' cos - y' sin g, y" = -' sin a cos i + y' cos g cos i + z' sin i, z" = x' sin g sin i - y' cos 2 sin i + z' cos i. If we transform the co-ordinates still further, and denote by x, y, z the co-ordinates referred to the equator or to any other plane making the angle e with the ecliptic, the positive axis of x being directed to the point from which longitudes are measured in this plane; and if we introduce also the auxiliary constants a, A, b, B, &c., we shall have 8x" sin a sin A 8x + sin b sin B ay - sin c sin C 8z, ay" = sin a cos A 8x + sin b cos B ay + sin c cos C 8z, (248) az' co- a cos+ oby + cos c 8z. Multiplying the first of these by -sinu, and the second by cosu, adding the results, and introducing Q as given by the second of equations (236), we get cos u ay" - sin u ax"r Q. Substituting for 8x" and Sy" the values given by the equations (73)2, the result is r (8v + 8) - Q, and, introducing the value of 8v given by (246), we obtain. cos 2 P 21/p Q = 2 _ + - (- -P ). r e sin v r"e sin v re sin v Substituting further for 8e, 8r, and 8 (/p) the values already obtained, and reducing, we find sin v cos E cos vl/p (p + r) sinv Q er er ek eki/ 2 sin v.k (249) by means of which XZ may be found. If we put cos a 8x + cos 6 y - cos cy + z s R, dx ^ dy, ds ^ (250) cos a a-d- + cos b -- + cos c ad- (250) the last of th atit gives the last of the equations (248) gives 544 THEORETICAL ASTRONOMY. z" =-= R; (251) and if we differentiate the equation dx dy dz cos a dt + cos b t + cos c —, which exists in the case of the unchanged elements, we shall have 0 =cosa d +- cos b 8 - + cos c 8 d dt dt dt dx dy. dz ~ sin a 8a - sin b 8b sin c ac. dt dt dt Substituting for Ja, bb, and 8c the values given in Art. 60, observing that r = 0, we have A-B'iL/(fk ~A, dy dz O R + sin a sin A -+ -t sin b sin B dt sin sin C sin i I\d t csn t dt I di dx dy Ad z.\' (252) \ d sin a cos A +- + sinin c cos B -sin cos C i. \ dt dt dt fI(C From the equations (100)1, observing that the relations between the auxiliary constants are not changed when the variable u is put equal to zero, or equal to 900, we get sin2 a sin2 A + sin2 b sin2 B + sin2 c sin2 C( 1, sin2 a cos2 A - sin' b cos2 B + sin2 c cos2 C -- 1, and from (235) we find sin2 a sin A cos A + sin' b sin B cos B + sin2 c sin Ccos C= 0. (254) dx dy dz Substituting in (252) for -t-, - and dt the values given by the dt dat dt equations (49), and reducing by means of (253) and (254), we get 0 R'- VsinUsin i 8 -V cosU 8i. (255) Substituting further for 8z" in (251) the value given by the last of the equations (73)2, there results 0 = R + r cos u sin i 8 - r sin u i. (256) From these equations we derive, by elimination, PERTURBATIONS OF COMETS. 545 e cos w -+ cosu R r sin u p sin ll V/p sin 27 P~iII~ kv'p SlUT (257) e sin wo + sin uz i cos, P kl/p by means of which 32g and Si may be found. To find (o and oir we have w = a - cos ia + , = a 3- 2 sin2 i, (258) 8X being found from equation (249). Neglecting the mass of the comet as inappreciable in comparison with that of the sun, the attractive force which acts upon the comet in the case of the undisturbed motion relative to the sun is P/, but in the case of the motion relative to the common centre of gravity of the sun and planet this force is P2 (1 + nz). Hence it follows that the increment of this force will be m'k2, and we shall have k - mInt (2b9) by means of which the value of this factor, which is required in the formulae for o ( I), -, &c., may be found. 206. The formulae thus derived enable us to effect the required transformation of the elements. In the first place, we compute the values of 8x, ay, z, d d, 8 ~-, and ~ dt by means of the formulae d' t cdt dto (234); then, by means of (236) and (250), we compute P, Q, R, P', Q', and R', the auxiliary constants a, A, &c. being determined in reference to the fundamental plane to which the co-ordinates are referred. When the fundamental plane is the plane of the ecliptic, or that to which 2 and i are referred, we have sin c sin i, C =0. The algebraic signs of cos c, cos b, and cos c, as indicated by the equations (101)i, must be carefully attended to. The formule for the variations of the elements will then give the corrections to be applied to the elements of the orbit relative to the sun in order to obtain those of the orbit relative to the common centre of gravity of the sun and planet. Whenever the elements of the orbit about the sun are again required, the corrections will be determined in the same manner, but will be applied each with a contrary sign. 35 546 THEORETICAL ASTRONOMY. Since the equations have been derived for the variations of more than the six elements usually employed, the additional formulae, as well as those which give different relations between the elements employed, may be used to check the numerical calculation; and this proof should not be omitted. It is obvious, also, that these differential formule will serve to convert the perturbations of the rectangular co-ordinates into perturbations of the elements, whenever the terms of the second order may be neglected, observing that in this case 8k = 0. If some of the elements considered are expressed in angular measure, and some in parts of other units, the quantity s 206264".8 should be introduced, in the numerical application, so as to preserve the homogeneity of the formulae. When the motion of the comet is regarded as undisturbed about the centre of gravity of the system, the variations of the elements for the instant t in order to reduce them to the centre of gravity of the system, added algebraically to those for the instant t' in order to reduce them again to the centre of the sun, will give the.total perturbations of the elements of the orbit relative to the sun during the interval' -t. It should be observed, however, that the value of 81i[ for the instant t should be reduced to that for the instant t', so that the total variation of M1 during the interval t'- t will be sM, + (t' -),,at, + aM,. In this manner, by considering the action of the several disturbing bodies separately, referring the motion of the comet to the common centre of gravity of the sun and planet whenever it may subsequently be regarlded as undisturbed about this point, and again referring it to the centre of the sun when such an assumption is no longer admissible, the determination of the perturbations during an entire revolution of the comet is very greatly facilitated. 207. If we consider the position and dimensions of the orbits of the comets, it will at once appear that a very near approach of some of these bodies to a planet may often happen, and that when they approach very near some of the large planets their orbits may be entirely changed. It is, indeed, certainly known that the orbits of comets have been thus modified by a near approach to Jupiter, and there are periodic comets now known which will be eventually thus acted upon. It becomes an interesting problem, therefore, to consider the formule applicable to this special case in which the ordinary methods of calculating perturbations cannot be applied. PERTURBATIONS OF COMETS. 547 If we denote by x', y', z', r', the co-ordinates and radius-vector of the planet referred to the centre of the sun, and regard its motion relative to the sun as disturbed by the comet, we shall have dt2 + r 1? r3 dc2y' k2 (1 + W-')' (' X ( dt2' 13 - 1! d2t2 + ( +') Z =r2( zZI Z Let us now denote by s, ^, t the co-ordinates of the comet referred to the centre of gravity of the planet; then will =-x x'- ^q = y y', = - - z'. Substituting the resulting values of x', y', z' in the preceding equations, and subtracting these from the corresponding equations (1) for the disturbed motion of the comet, we derive d(2 k2 (m + (''x) -' () dt2 + p ( +3" r3 dC2 + 12 (( + )) 2 z +-/ These equations express the motion of the comet relative to the centre of gravity of the disturbing planet; and when the comet approaches very near to the planet, so that the second member of each of these equations becomes very small in comparison with the second term of the first member, we may take, for a first approximation, d2 2 ({2 + m') + __ dt2 + P d7 (n + t n') __ (262) dt" " O 2'2 A2 (n + M') - dt + p3 p2 on the comet and of the reciprocal action of the comet on the planet, 548 THEORETICAL ASTRONOMY. these equations, being of the same form as those for the undisturbed motion of the comet relative to the sun, show that when the action of the disturbing planet on the comet exceeds that of the sun, the result of the first approximation to the motion of the comet is that it describes a conic section around the centre of gravity of the planet. Further, since — x', — y, - z are the co-ordinates of the sun referred to the centre of gravity of the planet, it appears that the second members of (261) express the disturbing force of the sun on the comet resolved in directions parallel to the co-ordinate axes respectively. Hence when a comet approaches so near a planet that the action of the latter upon it exceeds that of the sun, its motion will be in a conic section relatively to the planet, and will be disturbed by the action of the sun. But the disturbing action of the sun is the difference between its action on the comet and on the planet, and the masses of the larger bodies of the solar system are such that when the comet is equally attracted by the sun and by the planet, the distances of the comet and planet from the sun differ so little that the disturbing force of the sun on the comet, regarded as describing a conic section about the planet, will be extremely small. Thus, in a direction parallel to the co-ordinate e the disturbing force exercised by the sun is 2V I_ a I 2 I \ t ( 13 __ _ T3 3 ) and when the comet approaches very near the planet this force will be extremely small. It is evident, further, that the action of the sun regarded as the disturbing body will be very small even when its direct action upon the comet considerably exceeds that of the planet, and, therefore, that we may consider the orbit of the comet to be a conic section about the planet and disturbed by the sun, when it is actually attracted more by the sun than by the planet. 208. In order to show more clearly that the disturbing force of the sun is very small even when its direct action on the comet exceeds that of the planet, let us suppose the sun, planet, and comet to be situated on the same straight line, in which case the disturbing force of the sun will be a maximum for a given distance of the comet from the planet. Then will the direct action of the sun be r2, and that of the planet 2 The disturbing action of the sun will be p PERTURBATIONS OF COMETS. 549 1c2 c2 k2p 2r + p 2 - (r+ p)2 -' (r + p)2 which, since p is supposed to be small in comparison with r, may be put equal to 2k2p and hence the ratio of the disturbing action of the sun to the direct action of the planet on the comet cannot exceed 2p3 m32'r3 If the comet is at a distance, such that the direct action of the sun is equal to the direct action of the planet, we have and the ratio of the diirect action of the sun to its -disturbing action cannot in this case exceed 21/?/'. In the case of Jupiter this amounts to only 0.06. So long as p is small, the disturbing action of the planet is very nearly -2 in all positions of the comet relative to the planet, and hence the ratio of the disturbing action of the planet to the direct action of the sun cannot exceed M=rt T2 o At the point for which the value of p corresponds to R - RI, the comet, sun, and planet being supposed to be situated in the same straight line, it will be immaterial whether we consider the sun or the planet as the disturbing body; but for values of p less than this?t will be less than R', and the planet must be regarded as the controlling and the sun as the disturbing body. The supposition that R is equal to R' gives 2p3 n r2 n'3 2 and therefore p = r7r'2.. (263) Hence we may compute the perturbations of the comet, regarding the planet as the disturbing body, until it approaches so near the 550 THEORETICAL.ASTRONOMY. planet that p has the value given by this equation, after which, so long as p does not exceed the value here assigned, the sun must be regarded as the disturbing body. If ( represents the angle at the planet between the sun and comet, the disturbing force of the sun, for any position of the comet near the planet, will be very nearly 2k2p - COS V, and when this angle is considerable, the disturbing action of the sun will be small even when p is greater than r mi/'2. Hence we may colmmence to consider the sun as the disturbing body even before the comet reaches the point for which p ri/ "/}" and, since the ratio of the disturbing action of the planet to the direct action of the sun remains nearly the same for all values of s, when,o is within the limits here assigned the sun must in all cases be so considered. Corresponding to the value of p given by equation (263), we have R' = V4', and in the case of a near approach to Jupiter the results are p - 0.054r,'= 0.33. 209. In the actual calculation of the perturbations of any particular comet when very near a large planet, it will be easy to determine the point at which it will be advantageous to commence to regard the sun as the disturbing body; and, having found the elements of the orbit of the comet relative to the' planet, the perturbations of these elements or of the co-ordinates will be obtained by means of the formule already derived, the necessary distinctions being made in the notation. When the planet again becomes the disturbing body, the elements will be found in reference to the sun; and thus we are enabl,ed to trace the motion of the comet before and subsequent to its being considered as subject principally to the planet. In the case of the first transformation, the co-ordinates and velocities of the comet and planet in reference to the sun being determined for the instant at which the sun is regarded as ceasing to be the controlling body, we shall have PERTURBATIONS OF COMETS. 551: = x - x', j =_ y - y', z -- z2, d: dx dx' d' dy dy' d dz d dz' dt- dt dt' dt dt d' d ddt d t d' and from $,, (., -dt -d' and d- the elements of the orbit of the comet about the planet are to be determined precisely as the elements dx dy dz in reference to the sun are found from x, y, z, -dt -d-t and -, and as explained in Art. 168. Having computed the perturbations of the motion relative to the planet to the point at which the planet is again considered as the disturbing body, it only remains to find, for the corresponding time, the co-ordinates and velocities of the comet in reference to the centre of gravity of the planet, and from these the co-ordinates and velocities relative to the centre of the sun, and the elements of the orbit about the sun may be determined. As the interval of time during which the sun will be regarded as the disturbing body will always be small, it will be most convenient to compute the perturbations of the rectangular co-ordinates, in which case the d 4 drj d, values of $,, g -t -d, and dt will be obtained directly, and then, having found the corresponding co-ordinates x', y', z' and velocities di' dy' do' dt - -' dlt of the planet in reference to the sun, we have x- x'+-, y- Y' +-, z= z' -, dx dx' d$ dy dy' dz d z' d; + - -, i- d -d -' — dt dt dt dt dt dt' dt d t dt' by means of which the elements of the orbit relative to the sun will be found. If it is not considered necessary to compute rigorously the path of the comet before and after it is subject principally to the action of the planet, but simply to find the principal effect of the action of the planet in changing its elements, it will be sufficient, during the time in which the sun is regarded as the disturbing body, to suppose the comet to move in an undisturbed orbit about the planet. For the point at which we cease to regard the sun as the disturbing body, the co-ordinates and velocities of the comet relative to the centre of gravity of the planet will be determined from the elements of the orbit in reference to the planet, precisely as the corresponding quantities are determined in the case of the motion relative to the sun, the necessary distinctions being made in the notation. 552 THEORETICAL ASTRONOMY. 210. The results obtained from the observations of the periodic comets at their successive returns to the perihelion, render it probable that there exists in space a resisting medium which opposes the motion of all the heavenly bodies in their orbits; but since the observations of the planets do not exhibit any effect of such a resistance, it is inferred that the density of the ethereal fluid is so slight that it can have an appreciable effect only in the case of rare and attenuated bodies like the comets. If, however, we adopt the hypothesis of a resisting medium in space, in considering the motion of a heavenly body we simply introduce a new disturbing force acting in the direction of the tangent to the instantaneous orbit, and in a sense contrary to that of the motion. The amount of the resistance will depend chiefly on the density of the ethereal fluid and on the velocity of the body. In accordance with what takes place within the limits of our observation, we may assume that the resistance, in a medium of constant density, is proportional to the square of the velocity. The density of the fluid may be assumed to diminish as the distance from the sun increases, and hence it may be expressed as a function of the reciprocal of this distance. Let ds be the element of the path of the body, and r the radiusvector; then will the resistance be T - - (-) j > (264) K being a constant quantity depending on the nature of the body, and D( - ) the density of the ethereal fluid at the distance r. Since the force acts only in the plane of the orbit, the elements which define the position of this plane will not be changed, and hence we have only to determine the variations of the elements M, e, a, and Z. If we denote by ~0 the angle which the tangent makes with the prolongation of the radius-vector, the components R and S will be given by R = T cos s0,, S T sin b0, and, since Vcos0 o-,- e sin v, Vsin f0,- V d= we have Kl sin, SKjds) p ds;R==.~ Ky]o -~^esm- sin~-K \~]~v' (265) \r/ p dt Vr r dt RESISTING MEDIUM IN SPACE. 553 Substituting these values of R and S in the equation (205), it reduces to e d7 --- 2K ( ~ sin v ds. Now, since 1 Vz -k (1 + 2e cos v + e2), v/I we have 9.3"e2 2) ds =- Vdt - (1 + 2e cos v +- e2)dv, and hence e dy -- K:- -r2 (1 + 2e cos v + e2)2 sin v dv. (266) If we suppose the function IK ( r) (1 + 2e cos v + e2), the value of which is always positive, to be developed in a series arranged in reference to the cosines of v and of its multiples, so that we have K(p - r (1 - 2e cos v + e2)' = A + B cos v + C cos 2v + &c., (267) in which A, B, &c. are positive and functions of e, the equation (266) becomes 2 e c7z = (A +- B cos v +..) sin v dv. Hence, by integrating, we derive e =Z- = (A cos v os v cos 2v +...), (268) from which it appears that Z is subject only to periodic perturbations on account of the resisting medium. In a similar manner it may be shown that the second term of the second member of equation (210) produces only periodic terms in the value of 8Ij so that if we seek only the secular perturbations due to the action of the ethereal fluid, the first and second terms of the second member of (210) will not be considered, and only the secular perturbations arising from the variation of pe will be required. Let us next consider the elements a and e. Substituting in the 554 THEORETICAL ASTRONOMY. equations (198) and (202) the values of R and S given by (265), and reducing, we get 2c /1\ dc -= ~-~ K -(a, r2(1 + 2e cos v - e2)'dv, 2 1 1 %~i r/ (269) de _~~o -) r2 (1 + 2e cos v + e2). (e + cos v) dv. If we introduce into these the series (267), and integrate, it will be found that, in addition to the periodic terms, the expressions for 8a and 8e contain each a term multiplied by v, and hence increasing with the time. It is to be observed, further, that since A and B are positive, the secular variation of a, and also that of e, will be negative, and hence the resisting medium acts continuously to diminish both the mean distance and the eccentricity. 211. The magnitude of the disturbing force arising from the action of the resisting medium is so small that the periodic terms have no sensible influence on the place of the comet during the period in which it may be observed; and hence, since the effect of the resistance will be exhibited only by a comparison of observations made at its successive returns to the perihelion, the effect of the planetary perturbations being first completely eliminated, it is only necessary to consider the secular variations. Further, since X is subject only to periodic changes in virtue of the action of the resistance, and since the mean longitude is subjected to a secular change only through /i it will suffice to employ the formulse for A8/ and 8e or d85. The variations of these elements may be computed most conveniently by d/v. de do mechanical quadrature from given values of d and t- or -, al though their values for one complete revolution of the comet may be determined directly, the values of the coefficients A and B which appear in the series (267) being found by means of elliptic functions. The calculation of the effect of the resisting medium will be made in connection with the determination of the planetary perturbations, so that there will be no inconvenience in adding to the results the terms depending on this resistance. Since dp. 3 J da dp de dt 2 a dt' dt sec v -dt - the equations (269) give, putting — k2 U, RESISTING MEDIUM IN SPACE, 555 dt U (r) V dp 2k'p cos E (270) IoV. cl r cos \J r. It remains now to make an assumption in regard to the law of the density of the resisting medium. In the case of Encke's comet it has been assumed that'\ r r2 and this hypothesis gives results which suffice to represent the observations at its successive returns to the perihelion. Substituting for V its value in terms of r and a, the equations (270) thus become dit, a?~(2 1)-~, _ 3h,3 U ) clt 21 r a (1/1 373TT 1 \OS ~ COS E i 2r fa$, (271) d9- 2lo'U a c9 Os cos E( 2 1 by means of which 8, and oyv may be found; and from any given value of r?. we may derive the corresponding value of acc. The variation of JI, neglecting the periodic terms arising from the first and second terms of the second member of equation (210), will be given by.J' dut,t2 which will be integrated by mechanical quadrature so as to include the interval of an entire revolution of the comet. The quantity U has been determined, by means of observations of Encke's comet, to be U-894.892 This value may be corrected by introducing a term in the equations of condition precisely as in the case of the determination of the correction to be applied to the mass of a disturbing planet. Introducing U into the equation (264), and adopting the hypothesis that (1 ( ) 1= the expression for the action of the ethereal fluid becomes k2 U 2 T — 2y r 556 THEORETICAL ASTROiNOMY. Since the constant U depends on the nature of the comet, the value obtained in the case of Encke's comet may be very different from that in the case of another comet. Thus, in the case of Faye's comet the value has been found to be 10.232 and in the application of the formule to the motion of any particular body it will be necessary to make an independent determination of this constant. 212. The assumption that the density of the ethereal fluid varies inversely as the square of the distance from the sun, is that which appears to be the most probable, and the results obtained in accordance therewith seem to satisfy the data furnished by observation. It is true, however, that the whole subject is involved in great uncertainty as regards the nature of the resisting medium, so that the results obtained by means of any assumed law of density are not to be regarded as absolutely correct. From the formulae which have been given, it appears that, whatever may be the law of the density of the resisting fluid, the mean motion is constantly accelerated and the eccentricity diminished, and we may determine, by means of observations at the successive appearances of the comet, the amount of these secular changes independently of any assumption in regard to the density of the ether. Let x denote the variation of i during the interval r, which may be approximately the time of one revolution of the comet, and let4y denote the corresponding variation of po; then, after the lapse of anv interval t- T,, we shall have = o -t — T~x, 7- o - + -y, (272) and, since the average variation of a during the interval t — T is - T_0 _ -- Jro + ~o (t- To) (t- To) M= o, + P- (t - To) +( 2 ~ X. (273) If we introduce x and y as unknown quantities in the equations of condition for the correction of the elements by means of the differences between computation and observation, the secular variations of /l and ( may be determined in connection with the corrections to be RESISTING MEDIUM IN SPACE. 557 applied to the elements. For this purpose the partial differential coefficients of the geocentric spherical co-ordinates with respect to x and y must be determined. Thus, if we substitute the values of /f, A, and lV given by (272) and (273) in the equations (12)2 and (14)2, we obtain dr. (t- T,) 2r t - T -— = a tan p sin v - - - - o dx 2 r 63p. T dv cacosp (t - T0)2 dr t — T ~ ~ ^. ^ ~^=dx-y -y - -~ acos e cosv (274) dx 2 d d ( = co - + tan S cos v sinv7, dy cos y/ in which s 206264"/.8, i being expressed in seconds of arc. Combining the results thus obtained with the differential coefficients of the geocentric spherical co-ordinates with respect to r and v, as indicated by the equations (42),, we obtain the required coefficients of x and y to be introduced into the equations of condition. The solution of all the equations of condition by the method of least squares will then furnish the most probable values of y and x, or of the secular variations of the eccentricity and mean motion, without any assumption being made in reference either to the density of the ethereal fluid or to the modifications of the resistance on account of the changes in the form and dimensions of the comet, and the results thus derived may be employed in determining the values of 31, u, and so for the subsequent returns of the comet to the perihelion. In all the cases in which the periodic comets have been observed sufficiently, the existence of these secular changes of the elements seems to be well established; and if we grant that they arise from the resistance of an ethereal fluid, the total obliteration of our solar system is to be the final result. The fact that no such inequalities have yet been detected in the case of the motion of any of the planets, shows simply the immensity of the period which must elapse before the final catastrophe, and does not render it any the less certain. Such, indeed, appear to be the present indications of science in regard to this important question; but it is by no means impossible that, as in at least one similar case already, the operation of the simple and unique law of gravitation will alone completely explain these inequalities, and assign a limit which they can never pass, and thus afford a sublime proof of the provident care of the OMNIPOTENT CREATOR. TABLE I,. Angle of the Vertical and Logarithm of the Earth's Radius. Argument = Geographical Latitude. Compression - 299.15 299.15 1 2 ~ 34 33 54-430 34 33.907 o.68 0.678 ~ 35 34 54.266 35 34-904 0.70 0.698 2 C. c 36 35 54-I02 36 35.902 o.72 0.718 0 1 37 36 53-938 37 3699 0.74 73 0a a. 38 37 53.775 38 37.896 0.76 0.758 "': 39 38 53'61 39 38-894 0.78 0.778' - 4 40 39 53.447 40 39.891 0.80 0.798 0 ^.> 41 40 53.283 4 40.888 0.82 0.818 o 9 0 42 41 53.119 42 41.885 0.84 0.838.43 42 52.955 43 42.883 o.86 0.858 0 c o 44 43 52-792 44 43.880 0.88 0.878' - -: 45 44 52.628 45 44-877 0.90, 0.898 o || @ 46 45 52.464 46 45.874 0.92 0.917 " 47 46 52.300 47 46.872 0.94 0937 a ) ~ 48 47 52.136 48 47.869 0.96 0.957'o 5 49 48 51.972 49 48.866 0.98 0.977 So i 50 49 51.809 50 49.863 I.oo 0.997 0 Q 51 50 51.645 51 5o.861 2 q- 5a+ 51 5.1481 52 51.858 I * 53 52 51-317 53 52.855 ~ ~ 1 54 53 51-I53 54 53-853 1 ~55 54 50-990 55 54-850.d s 56 55 50.826 56 55.847.S 57 56 50.662 57 56.844 2g ~ 58 5 7 50.498 58 57.842 H- 59 58 50.334 59 58.839 6 59 50.170 60 59.836 564 TABLE IV. For converting Hours, Minutes, and Seconds into Decimals of a Day. Hours. Decimal. in. Decimal. Min. Decimal. Sec. Decimal. Sec. Decimal. I 0.0416 - 1.ooo694 + 31.021527- 1.0000116 31.0003588 2 o833 + 2.001388 + 32.022222+. 2.0000231 32.0003704 3 I.250 + 3.002083 + 33.0229I6- 3.0000347 33.0003819 4.I666+ 4.002777 -t 34.023611 4 ooo0000463 34.0003935 5.2083 - 5.003472 - 35.o243o5 - 5.0000579 35.0004051 6.2500 + 6.004166+d- 36.025000 + 6.0000694 36.0004167 7 0.2916 + 7.004861- 37.025694 + 7.oooo8io 37.0004282 8 -3333 + 8.005555 + 38.026388 + 8.oooo000925 38.0004398 9.3750-t 9.006250 + 39.027083 + 9.000I042 39.0004514 10.4I66 4- 10 o006944 + 40.027777- 10.0ooII57 40.o0o4630 11.4583++ 11.007638 + 41.028472- 11.0001273 41.0004745 12.5000+ 12.00833+ 42.029166+- 12.0001389 42.000486 13 0. 54 6 + 13.009027 + 43.029861 4- 13.000I505 43.0004977 14.5833) + 14.009722 1- 44.030555 + 14.000o 620 44-.o005093 15.620-o q 15-.0104I6 + 45.o3I250o 15.000o736 45.000o208 16.6666 -- 16.oiii+ 46.0o319444+ 16.O000852l 46.0005324 J17.7083 + 17.011805 +- 47.032638 -- 17.ooI968 47.0005440 18 7S300 - 18.012500 + 48 -033333 + 18.0002083 48.00oo556 19 0.7916 + 19.013194+ 49.034027 + 19.0002199gg 49.000ooo67 20 8333+ 20.013888 + 50.034722- + 20.0002315 50.0005787 21.8750 21.o45833546+ 21 1.00024.3 1.0005903 22.9166-1+ 22.015277 + 52.036 1- + 22.0002546 52.0006019 23 0o.9583 + 23.oI5972 + 53.036805 + 23.0002662 53.o0634 24 i.oooo- + 24.016666 + 54 -037500 + 24.0002778 54.ooo625 25 o01736I + 5 038I94- 25.ooo0002894 55.ooo6366 26.018055 + 56.038888+ 26.0003009 56.000o6481 27.18750-+ 57.039583- g 27.0003125 57.0006597 28.019444- 58.040277 + 28.0003241 58.0006713 29.020138 + 59.040972- + 29.0003356 59.ooo6829 30.o20833 + 60.041666- + 30.000347 60.0006944 The sign +, appended to numbers in this table, signifies that the last figure repeats to infinity. TABLE V. For finding the number of Days from the beginning of the Year. Date. Com. Bis. January o.o o o February o.o 31 31 March o.o 59 60 April o.o 90 91 May o.o 20 121 June o.o 15I 152 July o.o I8 182 August o.o 212 213 September o.o 243 244 October o.o 273, 274 November o.o 304 305 December o.o 334' 335 565 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 00 10 1 20 I3 M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". 0' o.oooooo 1 8.81 0.654532 181.83 1.309263 181.92 1.964393 182.05 1 0.0I0908 181.81 0.665442 181.83 1.320178 181.92 1.975316 I82.06 2 0.021817 181.8I 0.676352 I81.83 1.331093 181.92 1.986240 182.06 3 0.032725 I8I.8I 0.687262 181.84 1.342008 I81.92 1.997I64 182.06 4 0.043633 I81.8I 0.698172 181.84 1.352923 I81.92 2.008087 182.07 5 0.054542 i81.8I 0.709082 I81.84 1.363839 I81.93 2.019011 182.o7 6 o.o65450 181.8I 0.719993 181.84 1.374755 I81.93 2.029936 I82.07 7 0.076358 I81.8I 0.730903 I81.84 1-385670 I81.93 2.040860 I82.07 8 0.087267 181.81 0.741813 181.84 1-396586 I81.93 2.051785 I82.08 9 0.098175 I8I.8I 0.752724 181.84 1.407502 11-.93 2.062709 182.08 10 0.109083 181.81 0.763634 181.84 1.418418 181.94 2.073634 182.08 11 0.1 I9992 181.8 0.774545 I81.84 1.429334 181.94 2.084559 182.08 12 0.130900 I81.8I 0.785456 I81.84 1.44025 I181.94 2.095485 182.09 13 0.141808 181.81 0.796366 I81.8s 1.451167 I81.94 2.106410 I82.09 14 0.I527I7 I81.8I 0.807277 18I.85 1.462083 181.94 2.117335 I82.09 15 0.163625 181.81 0.8I8188 181.85 1.473000 181.95 2.128261 182.10 16 0.174534 I8I.8I 0.829099 181.85 1.483917 I8I.95 2.I39187 182.10 17 0.185442 I81.8I 0.840010 181.85 1.494834 181.95 2.150114 182.10 18 o. 96350 181.81 0.8509I2 I8I.85 1.50575I 181.95 2. I61040 I82.11 19 0.207259 I81.81 0.861832 I81.85 1.516668 181.95 2.171966 182.11 20 0.218167 I81.81 0.872743 181.85 1.527585 181.96 2.182894 182.11 21 0.229076 181.8I 0.883654 I81.86 1.538503 I81.96 2.193820 182.12 22 0.239984 I8I.81 0.894566 I81.86 1.549420 181.96 2.204747 182.I2 23 o.250893 181.81 0.905478 I8I.86 1.560338 181.96 2.215674 182.12 24 0.261801 I81.81 0.916389 181.86 1.571256 I8I.96 2.226602 182.13 25 0.272710 i81.81 0.927301 18i.86 1.582174 181.97 2.237529 182.13 26 0.283619 181.8i 0.938212 181.86 1.593092 181.97 2.248457 182.13 2 029457 i8I:81 0.949124 I81.86 1.604010 181.97 2.259385 182.I4 28 0.305436 I81.8I 0.960036 I81.86 I.614928 I81.97 2.270313 I82.I4 29 0.316345 181.81 0.970948 181.87 1.625847 181.97 2.281242 I82.I4 30 0.327253 181.81 0.981860 181.87 1.636766 181.98 2.292170 182.14 31 0. 338162 181.8 0.992772 I81.87 1.647684 181.98 2.303099 I82.15 32 0.349071 181.8I 1.003684 181.87 1.658603 181.98 2.314028 182.15 33 0.359980 8I.81 1I 014596 181.87 1.669522 I81.98 2.324957 182.I5 34 0.370888 181.81 1.025509 I81.87 1.680441 181.99 2.335887 182.16 35 0.381797 I81.81 1.036421 I81.87 1.691361 181.99 2.346816 182.16 36 0.392706 I81.8I 1.047334 I81.87 1.702280 I81.99 2-357746 I82.16 37 o.403615 181.81 1.058246 I81.88 1.713200 181.99 2.368676 182.17 38 o.414524 181.82 1.069159 181.88 1.724120 182.00 2.379606 182.17 39 0.425433 181.82 1.080072 181.88 1.735039 182.00 2.390536 182.17 40 0.436342 181.82 1.o0o985 181.88 1.745960 182.00 2.401467 I82.18 41 0.447251 181.82 1.101898 181.88 1.756880 182.oo 2.412398 182.18 42 0.458160 181.82 I.112811 181.89 1.767800 182.01 2.423329 182.18 43 0.469069 181.82 1.123724 18189 1.778721 182.01 2.434260 182.19 44 0-479979 I81.82 1.134637 181.89 1.789641 182.01 2.445191 182.I9 45 0.490888 181.82 1.145550 181.89 1.800562 i82.o0 2.456123 I82.9I 46 0.501797 181.82 1.156464 181.89 I.81I483 182.02 2.467055 182.20 47 0.512706 181.82 1.167377 181.89 I.822404 182.02 2.477987 I82.20 48 0.523616 I81.82 1.178291 181.89 1-833325 182.02 2.488919 182.20 49 0.534525 181.82 1.189205 181.90 1.844247 182.02 2.499851 182.21 50 0.545435 181.82 1.20011 181.90.855168 182.03 2.510784 182.21 51 0.556344 181.82 1.211033 I81.90 1.8660g 1I82.03 2.521717 182.22 52 0.567254 181.82 1.221947 I81.90 1.877012 182.04 2.532650 182.22 53 0.578163 181.83 1.232861 181.90 1.887934 I82.04 2.543583 182.22 54 0.589073 I81.83 1-243775 181.91 1.898856 182.04 2-554517 182.23 55 0.599983 181.83 1.254689 181.91 1.909779 182.o4 2.565450 182.23 56 0.610892 181.83 i.265604 181.91 1.920701 182.04 2.576384 182.23 57 0.621802 I81.83 1.276518 181.91 1.931624 182.05 2.587319 182.24 58 0.632712 181.83 1.287433 181.91 1.942547 182.05 2.598253 I82.24 59 0.643622 181.83 1.298348 181.91.1953470 182.05 2.609187 182.24 60 0.654532 181.83 1.309263 181.92 I.964393 182.o5 2.620122 182.25 566 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 40 50 6"~ 7~ /VI M. DDiff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". O' 2.620122 182.25 3.27665I 182.50 3.934182 I82.80 4.592917 I83.17 1 2.631057 182.25 3.287602 182.50 3.945151 I82.8I 4.603907 183.18 2 2.641993 182.26 3.298552 I82.51 3.956II9 I82.82 4.614898 I83.18 3 2.652928 182.26 3.309503 182.5I 3.967088 I82.82 4.625889 83.I19 4 2.663864 182.26 3.320454 182.52 3.978058 182.83 4-636880 I83.19 5 2.674800 182.27 3.331405 I82.52 3.989028 182.83 4.647872 183.20 6 2.685736 182.27 3.342356 I82.53 3.999998 I82.84 4.658864 I83.21 7 2.696672 182.27 3.353308 182.53 4.010968 182.84 4.669857 183.21 8 2.707609 182.28 3.364260 I82.54 4.021939 182.85 4.680850 183.22 9 2.7I8546 182.28 3.375212 i82.54 4.03291I I82.86 4.691843 I83.23 10 2.729483 182.29 3.386165. 182.55 4.043882 182.86 4.702837 183.24 11 2.740420 182.29 3-397118 I82.55 4.054854 182.87 4.713831 183.24 12 2.751358 182.29 3.408071 182.56 4.065826 182.87 4.724826 183.25 13 2.762295 182.30 3.419024 182.56 4.076799 182.88 4-735821 I83.25 14 2.773233 182.30 3.429978 182.57 4.087772 I82.88 4.746816 183.26 15 2.784I72 182.31 3-440932 182.57 4.098745 182.89 4.757812 183.27 16 2.795110 182.31 3.451887 182.58 4.109718 182.90 4.768809 183.27 17 2.806049 182.31 3.462841 182.58 4.I20692 182.90 4.779805 I83.28 18 2.816988 182.32 3.473796 182.59 4-I31667 182.91 4.790802 I83.28 19 2.827927 182.32 3.484752 182.59 4.I42641 I82.91 4.80I800 183-29 20 2.838867 182.33 3.495707 I82.60 4. I53616 182.92 4.812797 I83.30 21 2.849806 I82.33 3.506663 82.60 4. 64592 182.93 4.823796 I83.31 22 2.860746 I82.33 3.517619 182.61 4.175568 182.93 4.834795 183.32 23 2.871686 182.34 3.528575 I82.6I 4.186544 182.94 4.845794 183.32 24 2.882627 182.34 3-539532 182.61 4I.97520 182.94 4.856793 I83.33 25 2.893567 182.35 3.550489 182.62 4.208497 182.95 4.867793 183.34 26 2.904508 182.35 3.561447 182.62 4.219474 182.95 4-878793 183-34 27 2.915449 182.36 3.572404 182.63 4-230451 I82.96 4.889794 I83-35 28 2.926391 182.36 3.583362 182.63 4.241429 182.97 4-900795 183.36 29 2.937332 182.36 3.594320 I82.64 4.252408 182.97 4.911797 I 83.36 30 2.948274 182.37 3.605279 182.64 4.263386 182.98 4.922799 183.37 31 2.959217 182.37 3.6I6238 182.65 4.274365 182.99 4-93380I 183.38 32 2.970159 182.37 3.627197 182.65 4.285344 182.99 4-944804 I83.38 33 2.981102 182.38 3.638 56 I82.66 4.296324 183.00 4-955807 I83.39 34 2.992045 182.38 3.649116 182.66 4.307304 183.00 4.966811 183.40 35 3.002988 182.39 3.660076 182.67 4.318284 183.01 4.977815 I 83.4I 36 3.OI3931 1 82.39 3.671037 182.68 4-329265 183.01 4.988820 183.41 37 3.024875 182.39 3.681997 82.68 4-340246 183.02 4.999825 I83.42 38 3.035819 182.40 3.692958 182.69 4.351228 183.03 5.010830 I83.43 39 3.046763 182.40 3.703920 182.69 4.362210 183.03 5.02836 I 8 3.43 40 3.057707 82.41 3.714881 182.70 4.373192 I83.04. 5.032842 I83.44 41 3.068652 182.4I 3-725843 I82.70 4.384175 183.05 5.043849 183.45 42 3.079597 182.42 3.736806 182.71 4.395158 183.05 5.054856 183.46 43 3.090542 182.42 3.747768 182.71 4-40641 1 83.o6 5.065864. 183.46 44 3.101488 182.43 3-758731 182.72 4.417125 183.06 5.076872 83.47 45 3.1 243 3 182.43 3.769694 182.72 4.428109 183.07 5.087880 183.48 46 3-I23379 182.44 3.780658 182.72 4.439093 I83.o8 5.098889 183.48 47 3.134325 182.44 3.79I622 182.73 4.450078 183.08 5.109898 183.49 48 3.145272 182.44 3.802586 I82.74 4.461064 183.09 5.120908 I83.50 49 3.156219 182.45 3.813551 182.74 4.472049 183.10 5.131918 183.51 50 3.167I66 182.45 3-824515 182.75 4.483035 183.-1 5.142929 183.51 51 3.178113 182.46 3.835481 182.76 4.494022 183.11 5-153940 183.52 52 3I.89061 182.46 3.846446 182.76 4.505008 183.12 5.164951 183.53 53 3.200009 182.47 3.857412 I82.77 4.515995 183.12 5.175963 183-54 54 3.210957 182.47 3.868378 182.77 4.526983 183-13 5.186975 I83.54 55 3.221905 182.48 3.879345 182.78 4.537971 I83.14 5.197988 18 3.55 56 3.232854 182.48 3.890312 I82.78 4.548959,183.14 5.209002 183.56 57 3.243803 182.49 3.901279 182.79 4.559948 I83.15 5.2200I5 I 8357 58 3.254752 182.49 3.912246 182.79 4.570937 183-15 5.231029 I83.57 59 3.265702 182.49 3.923214 182.80 4.581927 183. 6 5.242044 183.58 60 3.276651 182.50 3.934182 182.80 4-592917 183.17 5-253059 183.59 567 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 8~ 90 100 11 V. M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". O' 5-253059 I83.59 5.9I4815 184.o6 6.578391 I84.60 7.243997 I85.19 1 5.264075 183.59 5.925859 184.07 6.589467 184.6I 7.255109 18s.2o 5.275090 183.60 5-936904 184.08 6.600544 184.62 7.266222 185.21 3 5.286107 183.6I 5'947949 184.09 6.611622 I84.63 7-277335 I85.22 4 5.297124 I83.62 5.958995 I84.Io 6.622700 184.64 7.288449 I85.23 5 5.308141 183.62 5.970041 184.11 6.633778 184.65 7-299563 185.25 6 5.319159 I83.63 5.98I087 184.11 6.644857 184.66 7.310678 185.26 7 5-330177 I83.64 5.992134 I84.12 6.655937 I84-67 7.321793 185.27 8 5-341195 I83-65 6.003182 184-I3 6.667017 184-67 7-332909 I85.28 9 5.352214 183.66 6.014230 184.14 6.678098 184.68 7-344026 I85.29 10 5.363234 183.66 6.025279 184-15 6.689179 184.69 7-355144 I85.30 11 5-374254 183-67 6.036328 I84.16 6.700261 184-70 7.366262 I85.31 12 5.385275 I83.68 6.047378 184.17 6.711343 184-71 7.377381 185.32 13 5.396296 I83.69 6.058428 184.18 6.722426 184.72 7.388500 185.33 14 5.407317 183.69 6.069479 848 6733510 I84.73 7-399620 I85.34 15 5.418339 183.70 6.080530 I84.19 6.744594 184.74 7-41074I 185.35 16 5.429361 I83.71 6.091582 184.20 6.755679 184.75 7.42I862 185.36 17 5.440384 183.72 6.102634 I84.21 6.766764 I84.76 7-432983 185.37 18 5.451407 183.73 6.113687 I84.22 6.777850 184.77 7-444I06 185-38 19 5.462431 I83.73 6I.24740 184.23 6.788937 184-78 7.455230 185.39 20 5-473455 I83.74 6.I35794 184-24 6.800024 184.79 j.466354 185.40 21 5-484480 183.75 6.I46849 184.25 6.811II2 I84.80 7.477478 185-41 22 5.495505 183-75 6-157904 184-25 6.822200 I84.81 7.488603 I85.42 23 5-506530 183.76 6.168959 184.26 6.833289 184.82 7.499729 185.43 24 5.517556 183-77 6.180015 184.27 6-844378 184.83 7.5I0855 185.44 25 5.528583 183.78 6.191072 184.28 6.855468 184.84 7.52I982 I85.46 26 5.539610 I83-79 6.202129 184.29 6.866559 184.85 7.53311 185.47 27 5.550637 183.79 6.2I3187 184-30 6.877650 I84.86 7.544239 185.48 28 5.56I665 183.80 6.224245 184-31 6.888742 I84.87 7.55368 185-49 2 5.572693 183.81 6.235304 184.32 6.899834 184.88 7.566497 1I8550 30 5.583722 183.82 6.246363 184.32 6. 90927 184.89 7.577628 185-51 31 5-594752 183.83 6.257422 184.33 6.922021 184.90 7.588759 I85.52 32 5.605782 183.83 6.268482 I84.34 6.933115 184.91 7-599890 185.53 33 5.616812 183-84 6.279543 I84.35 6.944210 184.92 7.611022 185.54 34 5.627843 183.85 6.290605 184.36 6.955305 I84.93 7.622155 185-55 35 5.638874 183.86 6.301667 184.37 6.966401 184.94 7.633289 185.57 36 5.6499-6 I83-87 6.312729 184-38 6.977498 184.95 7.644423 I85-58 37 5.660938 183.87 6.323792 184.39 6.988595 184.96 7.655558 I85.59 38 5.67197I I83.88 6.334855 184.40 6.999693 184.97 7.666694 185.60 39 5.683004 I83.89 6-345919 184-41 7.010791 184.98 7.677830 185.6I 40 5.694038 183.90 6.356984 18441I 7.021890 184.99 7.688967 185.62 41 5-705072 183.91 6.368049 184.42 7.032990 185.00 7.700104 185.63 42 5.7I6106 183.92 6.379115 I84-43 7.044090 185.0I 7.711242 I85.64 43 5.727141 183-92 6.390181 184.44 7.055191 185.02 7.722381 185-65 44 5.738177 183.93 6.4.I248 I84.45 7.066292 185.03 7-73352I 185.66'45 5:749213 183-94 6.412315 I84-46 7.077394 185.04 7.74466I 185.68 46 5.76025o I83.95 6.423383 I84-47 7.088497 185.05 7-755802 185-69 47 5.771287 183.96 6.434451 I84-48 7.099600 185.06 7.766943 I85.70 48 5.782325 I83.96 6.445520 184.49 7.110704 185.07 7.778085 I85-71 49 5.793363 I83-97 6.456590 184-50 7.121808 I85.08 7.789228 I85-72 50 5.804401 183.98 6.467660 184.5 7.132913 185.09 7.800372 185.73 51 5.815440 I83.99 6.47873 I84.52 7.144019 185.10 7.811516 185.74 52 5.826480 184.00 6.489802 184.52 7.155125 I85. I 7.822661 I85.75 53 5.837520 184.01 6.500874 I84.53 7-166232 185.12 7.833807 185.76 54 5.848561 184.01 6.511946 184.54 7.177340 185.13 7.844953 I85.78 5 585962 184.02 6.523019 184.55 7-188448 185I.4 7-856oo00 I8579 56 5.870644 184-03 6.534092 184-56 7-199557 185-I5 7.867247 185.80 57 5.881686 I84.04 6.545166 184.57 7.210666 I85.I6 7.878396 I85.81 58 5.892728 I84-05 6.556241 184.58 7.22I776 185.17 7.889545 185.82 59 5.903771 184.06 6.567316 1 I84-59 7.232886 185.18 7.900694 185-83 60 5.914815 184.06 6.578391 184.60 7.243997 185.19 7-911845 185.84 568 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 120 13 14~ 15" v.. MI. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". 0' 7.911845 185.84 8.582146 186.56 9.255120 187.33 9.930984 I88.i6 1 7.922995 185.86 8.593340 186.57 9.266360 I87.34 9.942274 188.I8 2 7.934147 185.87 8.6045 35 I86.58 9.277601 187.35 9.953565 I88.19 3 7.945300 I85'88 8.615730 I86.59 9.288842 I87.37 9.964857 188.2I 4 7.956453 185.89 8.626926 I86.6I 9.300085 187.38 9.976149 188.22 5 7.967606 185.90 8.638123 I86.62 9.311328 187.40 9.987443 I88.23 6 7.978761 I85.91 8.649320 I86.63 9,322572 187-41 9-998738 188.25 7 7.989916 185.92 8.660518 I86.64 9-333817 187.42 I0.010033 88.26 8 8.001072 185-93 8.671717 186.66 9-345063 I87.44 10.021329 188.28 9 8.012228 185.95 8.6829I7 186.67 9.3563IO I87.45 10.032626 188.29 10 8.023385 I85.96 8.694117 I86.68 9-367557 I87.46 IO.043924 188.31 11 8.034543 185.97 8.705318 186.69 9.378805 187.48 10.055223 188.32 12 8.045702 185.98 8.716520 186.71 9-390054 187.49 1O.066523 188.34 13 8.056861 185.99 8.727723 186.72 9.401304 I87-50 10.077823 188.35 14 8.06802I I86.oo 8.738927 I 86.73 9.412555 187-52 10.089125 188.37 15 8.079181 I86.02 8.750131 186.74 9.423806 I87.53 IO. 00427 188,38 16 8.090343 186.o3 8.76I336 186.76 9-435058 187.54 10. II730 188.39 17 8.1 1505 186.04 8.772542 186.77 9-446311 187-56 10.123035 188.41 18 8.xI2668 I86.o5 8.783748 186.78 9-457565 187-57 10.34340 188.42 19 8.123831 I86.06 8-794955 I86.79 9-468820 187.59 10.I45646 188.44 20 81 34995 186.07 8.806163 I86.8I 9.480076 187.60 0.1I56952 188.45 I21 8.146160 186.09 8-8I7372 186.82 9-491332 187.6I Io0.68260 188.47 22 8.157326 86. o 8.828582 186.83 9.502589 I87.63 10.179568 188.48 23 8.I68492 I86.I 8.839792 186.84 9.513847 187.64 Io.I90878 I88.50 24 8I.79659 I86.I2 8.851003 I86.86 9.525106 187.65 10.2088 188.51 25 8.I9o826 186.13 8.862215 186.87 9.536366 187.67 I0.213499 188.53 26 8.201995 186.15 8.873427 186.88 9.547626 187.68 10.2248I2 188.54 27 8.213164 86.16 8.884641 186.90 9.558888 187.70 Io.236125 I88.56 28 8.224334 186.17 8.895855 186.9I 9.570150 I87.71 I10247439 188.57 29 8.235504 186.18 8.907070 186.92 9.581413 187.72 10.258753 188.59 30' 8.246675 186.I9 8.918286 186.93 9.592676 I87.74 10.270069 I88.60 31 8.257847 186.20 8.929502 186.95 9.603941 187.75 10.281386 I88.62 3 8 269020 186.22 8.940719 186.96 9.615207 8777 I0.292703 188.63 33 8.280I93 186.23 8.951937 I86.97 9.626473 187.78 I0.30402I I88.65 34 8.291367 I86.24 8.963156 186.99 9-637740 187-79 10.3I5341 I88.66 35 8.302542 186.25 8.974376 187.00 9.649008 187.81 I0.32666I 188.68 36 8.313717 I86.26 8.985596 187.01 9-660277 I87.82 10.337982 188.69 37 8.324893 186.28 8.996817 187.02 9.671547 187.84 10.349304 I88.71 38 8.336070 186.29 9.008039 187.04 9.682817 187.85 10.360627 188.72 39.8.347248 I86.30 9.019262 187.05 9.694088 187.86 10.371951 188.74 40 8.358426 186.31 9.030485 187.06 9-705361 187.88 1O.383275 188.75 41 8.369605 I86.32 9.041709 I87.08 9.716634 187.89 I0.39460I 188.77 42 8.380785 I86.34 9-052934 1 87.09 9.727908 187.9I 10.405927 I88.78 43 8.391966 186.35 9.064160 187.10 9.739182 I87.92 10.417255 188.80 44 8.403147 186.36 9-075387 187.12 9-750458 18793 10.428583 I88.8I 45 8.414329 186.37 9.086614 187-13 9.761734 187-95 10.439912 188.83 46 8.425512 186.38 9.097842 187.14 9.773012 187.96 10.451242 188.84 47 8.436695 186.40 9.109071 187.16 9.784290 187.98 I0.462573 I88.86 48 8.447879 186,41 9.12030I 187.17 9-795569 187-99 10.473905 188.87 49 8.459064 186.42 9.131531 187I.8 9.806849 188.o00 0.485238 188.89 50 8.470250 186.43 9-142763 187.20 9.818129 I88.02 10.496572 188.90 51 8.481436 I86.45 9.153995 187.21 9.829410 I88.03 10.507907 188.92 52 8.492623 186.46 9.165228 187.22 9.840693 188.05 10.519242 188.93 53 8.5038II I86.47 9.176462 I87.23 9.851977 I88.o6 10.530579 188.95 54 8.515000 186.48 9.187696 187.25 9.86326I I88,o8 10.541916 188.97 55 8.526189 I86.49 9.198931 187.26 9.874546 i88.o9 10.553255 I88.98 56 8.537379 I86.5I 9.210167 187.27 9.885832 188.1o 10.564594 I89.00 57 8.548569 186.52 9.221404 187.29 9.897118 188.12 10.575934 189.01 58 8.559761 186.53 9.232642 187.30 9.908406 188.13 10.587276 189.03 59 8.570953 186-54 9-243880 187.31 9.-99694 188.15 Io.5986I8 I89.04 60 8.582146 i86.56 9.255120 187.33 9.930984 188.16 10.609961 189.06 569 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. _. _. _~if- ~- i 160 170 180 190 M. Diff. 1". Diff. 1". Diff. 1". ff. O' 10.60996I I89.06 11.292277 190.02 II.978I62 I91.04 12.667850 192. 13 1 10.621305 189.07 II.303679 I90.03 11.989625 191.06 I2.679379 I92.I5 2 10.632649 189.09 II.3I5082 190.05 I2.001089 19I.O8 I2.699098 192.17 3 10.643995 I89.I0 11.326485 I90.07 12.012554 I9I.09 12.702439 I92.19 4 10.655342 189I.2 11.337889 190.08 12.024021 I9I.I1 12.713970 I92.2I 5 Io.66669o 189-14 11.349295 I90.10 12.035488 I9I.I3 12.725503 192.22 6 10.678038 189.I5 1.360701 I90.I2 I2.046956 I9I.I5 12.737037 I92.24 7 o1.689388 189.I7 11372I09 I90.I3 12.058425 191I.6 12.748573 I92.26 8 Io.700738 189.18 11-383517 I90.I5 12.069896 I9I.I8 12.760109 I92.28 9 10.712090 189.20 II.394927 I90. 17 22.081367 19I.zo I2.771646 192.30 10 10.723442 189.2I 11.406337 I90.I8 12.092840 191.22 I2.783I85 192.32 11 10.734795 I89-23 11.417749 190.20 2.I104313 I9I.24 12.794724 I92.34 12 I0.746149 189.24 II.42916I I90.22 I2.115788 I9I.25 12.806265 I92.36 13 10.757505 89.26 II.44-0575 190.23 12.127264 191.27 I2.817807 I92-37 14 10.76886I 189.28 1I.451989 I90.25 I2.138741 19I.29 12.829350 I92.39 15 10.780218 I89.29 11.463405 190.27 I2.I502I9 I91.3I 12.840894 192.41 6I 10.791576 I89.3I 11.474821 190.28 12.161698 19I.32 12.852440 192.43 17 10.802935 189-32 II.486239 190.30 12-.73178 19.34 12.863986 I92.45 18 10.814295 189.34 11-497657 190.32 12.184659 I91.36 12.875534 192.47 19 Io.825655 I89.35 11.509077 190.33 I2.196I41 191.38 12.887082 I92.49 20 10.837017 189.37 11.520497 190.35 12.207624 I19I.40 12.898632 192.51 21 I0.848380 I89-39 11-53I9I9 I90-37 I2.219I08 191.41 12.910183 192.53 22 Io.859744 I89-40 II.543342 190.39 12.230594 19I.43 12.921736 192.55 23 10.871108 I89.42 11.554765 I90.40 12.242080 191.45 12.933289 I92.56 24 Io.882474 I89.43 II.566190 190.42 12.253568 191.47 I2.944843 192.58 25 Io.893840 189.45 11.577616 190.44 I2.265057 I91.49 I2.956399 I92.60 26 10.905208 189.47 11.589042 190.45 I2.276546 I91.50 I2.967956 I92.62 27 o1.916576 189.48 II.600470 190.47 I2.288037 19I.52 12.979514 I92.64 28 10.927946 189-50 1.61I899 190.49 I2.299529 I9I.54 12.991073 I92.66 29 O0.939316 189.5I 11.623328 I90.50 I2.311022 191.56 13.002633 I92.68 30 I0.950687 I89.53 I.634759 I90.52 I2.322516 I 91.58 13.014195 I92.70 31 0o.962059 I89.55 II.646191 190.54 12.334011 191.60 13.025757 192.72 32 10.973433 189-56 11.657624 190.56 I2.345508 I91.6I 13.03732I 192-74 33 Io.984807 189-58 1I.669057 190.57 I2.357005 I91.63 13.048886 I92.76 34 Io.996I82 I89.59 11.680492 190.59 12.368503 I91.65 13.060452 192.78 35 11.007558 I89.6I I1.69I928 190.61 12.380003 I91.67 13.072019 I92.80 36 1I.oI8935 I89-63 11.703365 I90.62 12.391504 I9I.69 13.083587 I92.82 37 1.-030313 I89-64 11.714803 i90.64 I2.403006 191.70 I3.095157 I92.83 38 II.041692 I89-66 11.726242 I90.66 12.414509 I91.72 13.106727 192.85 39 11.053072 I89.67 11.737682 I90.68 12.426013 191.74 13.118299 192.87 40 11.064453 I89.69 11-749123 I90.69 12.437517 191.76 13.-29872 I92.89 41 11.075835 I89-71 I1.760565 190.71 12.449023 191.78 13.141446 I92.91 42 11o.087218 I89.72 11.772008 I90.73 I2.46053I 19I.80 13.153022 192-93 43 II.098602 189-74 II.783452 190.74 12.472039 191.81 13.164598 192.95 44 II.I09987 189.76 11.794897 190.76 12.483548 I91.83 13.176176 I19297 45 11.121372 189.77 II.806344 I90.78 12.495059 191.85 13.187755 192.99 46 11.132759 189.79 II.8I7791 I90.80 12.506571 I91.87 I3-I99335 I93-0I 47 II.44I4 7 189-80 11.829239 I90.81 I2.5I8083 I91.89 I3.z209I6 193-03 48 11.155536 I89.82 11.840689 190-83 I2.529597 I91.91 13.222498 193-05 49 II.I66925 I89-84 11.852139 190.85 I2.541112 191.93 13.234082 193-07 50 11.178316 189.85 II.863590 190.87 12.552628 191.94 13-245667 I93.09 51 11.189708 189-87 11.875043 190.88 12.564145 191.96 13.257253 193-11 52 II.20IIoo I89.89 11.886496 190.90 I2.575664 I91-98 I3.268840 I93-13 53 11.212494 189.90 II.897951 I90.92 12.587183 192.00 13.0428 I93-15 54 11.223889 189.92 11.909407 190.94 12.598704 I92.02 13.292017 I93-I7 55 11.235284 189-93 II.920863 190.95 12.610225 192.04 13.303608 193.19 56 11.246681 189-95 II.932321 190.97 I2.621748 192.06 13.315200 193-2I 57 II.258078 189-97 II-943780 I90.99 12.633272 192.07 I3-326793 193-23 58 II.269477 I89.98 11.955239 I91.01 12.644797 192.09 13.338387 193-25 59 11.280876 190.00 II.966700 191.02 12.656323 I92.1I I3.349982 193.27 60 11.292277 r9o.oz II.978162 191.04 12.667850 192.13 13.361579 193.29 570 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 20~ 21~ 220 230 VI o M. Diff. 1". M. Dif 1. M. Diff. 1". M Diff. 1". O' 13.361579 I93.29 14-059591 I94.5I 14.762133 195-80 15-469459 1971I7 1 I3-373177 I93-3I 14-071262 194-53 I4-773882 195-83 15.48I290 I97.19 2 13.384776 193-33 14-082935 I94.55 14.785632 195.85 15-493I22 197.2I 3 13.396376 193.35 14.094608 194-57 14-79,7384 195.87 15.504956 197.24 4 13.407977 I93-37 14.106283 I94-59 14-809137 195.89 I5.516791 197.26 5 13.419580 193-39 14.II7960 I94.6I 14.820891 195.91 15.528627 197.28 6 13.431183 193-4I I4.I29637 194-64 I4-832647 195-94 I5-540465 I97-31 7 I3.442788 193-43 14-I41316 I94.66 14-844403 I95.96 I5.552304 I97.33 8 13.454394 193-45 I4I52996 I94-68 I4.856I6I I95.98 I5.564144 I97-35 9 13.466002 I93-47 14.164677 I94.70 14.867921 I96.oo 15.575986 I97.38 10 I3.4776I0 193-49 14.I76360 194.72 14.879682 I96.03 IS.587830 I97.40 1I 13.489220 I93.5I 14.I88044 194.74 14-891444 196.05 15.599675 I97-43 12 13.500831 I93-53 I4.I99729 194-76 I4.903208 I96.07 15.611521 I97-45 13 13.512443 193-55 14.214I15 194-78 14.914973 I96.09 I5.623369 197-47 14 13.524056 193-57 I4.223I03 194.8I I4.926739 I96.I2 15.635218 197-50 15 13.535671 193-59 I4.234792 I94.83 I4.938506 196.14 15.647068 I97-52 16 I3.547287 193-6I I4.246482 194-85 14.950275 I96.I6 15.658920 197.54 17 I3.558904 I93.63 14.258174 194.87 14-962045 I96.I8 15.670773 197,57 18 13.570522 I93.65 14.269867 194-89 I4-9738I7 196.20 15.682628 I97-59 19 I3.582141 I93.67 14.281561 194.91 I4.985590 196.23 15.694484 197.6I 20 I3.593762 I93.69 14.293256 I94-93 14.997365 I96.25 15.706342 197.64 21 I3.605383 193.71 14.304953 194.95 15.009140 I96.27 I5.718201 197.66 22 13.617006 193-73 I4.3I6651 194.98 15.020917 I96.30 I5.730061 197.69 23 13,628631 I93.75 14.328350 195.00 I5.032696 196.32 15.741923 197.71 24 I3.640256 193-77 14.340050 195-02 I5.044475 I96.34 I5-753786 I97-73 25 I3.651883 193-79 14.351752 I95.04 I5.056256 I96.36 15.765651 I97.76 2G 13.663511 I93.81 I4.363455 195.06 I5.068039 196.39 I5.777517 197.78 27 I3.675140 I93.83 14.375159 195-08 I5.079823 196.41 15.789385 197-80 28 I3.686770 I93.85 I4.386865 195.-0 15.09I608 I96.43 15.80I254 197.83 29 I3.698401 193.87 I4-398572 195.I3 I5.103394 196.45 I5.813I24 197.85 30 13.710034 193.89 14.410280 195.15 I5.II5182 I96.48 15.824996 197-88 31 13.72I668 193.9I 14.42I990 195.I7 I5.126971 I96.50 15.836870 197.90 32 13.733303 I93.93 14.433700 I95-I9 I5.I38762 196.52 15.848744 197.92 33 13.744940 I93-95 14.445412 195.2I 15.I50554 196.54 15.860620 197-95 34 13.756577 193.97 I4.457126 I95.23 I5.I62348 I96.57 15-872498 197.97 35 I3.768216 193.99 14.468841 195.26 15.174142 196.59 15.884377 I98.00 36 13.779856 I94.01 14.480557 I95-28 15.185938 I96.6I 15.896258 198.02 37 13.791498 194.03 14.492274 195.30 15.197736 I96.64 15.908140 198.04 38 I3.803140 I94.05 14.503992 195.32 15.209535 I96.66 I5.920023 198.07 39 13.814784 194-07 14.515712 195-34 15.221335 196.68 15.93190 198.09 40 13.826429 I94.09 I4.527434 195.36 I5.233137 I96.70 15.943794 198.12 41 13.838075 194.1I I4.539156 I95.39 I5.244940 I96.73 I5.955682 I98.14 42 13.849723 I94-I4 I4.550880 195.4I 15.256744 I96-75 15-96757I 198.17 43 13.861372 I94. 6 14.562605 I95-43 15.268550 I96.77 15-979462 198.19 44 13.873022 I94. I8 14.574331 195.45 I5.280357 I96.80 15.991354 198.21 45 13.884673 194.20 I4.586059 195.47 I5.292I65 I96.82 I6.003248 198.24 46 I3896325 194-22 14.597788 195.50 15.303975 I96.84 I6.0I5I43 I98.26 47 I3.907979 194.24 I4.609519 I95.52 15.315786 I96.87 I6.027039 I98.29 48 I3.919634 194-26 14.621250 195-54 I5.327599 I96.89 I6.038937 I98.31 49 13.93I290 194.28 14.632983 I95.56 I5.339413 I96.9I 16.050836 198.34 50 13.942948 I94.30 14.64 71I8 195.58 15.35I228 I96.94 16.o62737 I98.36 51 I3.954606 194.32 14.656453 195.60 15.363045 I96.96 I6.074639 198.38 52 13.966266 194-34 14.668I90 195.63 15.374863 I96.98 16.086543 198.41 53 I3-977927 194.36 I4.679929 I95.65 15.386683 197-00 16.098449 I98.43 54 13-989590 194-38 I4.69I668 I95.67 15.398504 197.03 16.110355 I98.46 55 14.001254 194.4I I4.703409 I95.69 I5.4I0326 197.05 I6.I22263 198.48 56 I4.012919 I94-43 14.715151 195.71 I5.422150 197.07 I6.134173 198.51 57 14.024585 194.45 14.726895 195.74 15.433975 197.I0 I6.146084 I98.53 58 14.036252 I94-47 I4.738640 195-76 15.445802 197.-2 I6.I57997 I98.56 59 14.047921 194-49 14-750386 195.78 I5.457630 197.I4 I6.I69911 198.58 60 14.05959I I94.51 14.762133 195.80 15.469459 197.I7 I6. 18826 198.60 571 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.. 240 250 26" 27o M. Diff. 1. M. | Diff. l. M. Diff. 1. iM. Diff. I". 0' I6.181826 198.60 I6.899499 2,00. I2 I7.622747 20I.70 18.351847 203-37 1 I6.193743 I98.63 I6.9I1507 200.14 17.634.850 20I.73 I8.364050 203.40 2 16.205662 198.65 I6.9235I6 200.I7 I7.646954 201.76 18.376255 203o42 3 16.217582 i98.68 16.935527 200.19 17.659060 201.78 18.388461 203-45 4 16.229503 198.70 16.947539 200.22 17.671168 201.81 I8.400669 203.48 a5 6.24I426 I98.73 I6.959553 200.24 17.683278 201.84 I8.412879 203.51 6 6.-253350 I98.75 I6.971568 200.27 17.695389 20I.87 I8.425090 203.54 7 16.265276 198.78 I6.983585 200.30 17.707502 201.89 18.437303 203-57 8 16.27720.4 I98.80 16.995604 200.32 17.7I9616 201.92 18-4495I8 203-59 9 I6.289133 198.83 17-007624 200.35 I7.731732 201.95 I8.461735 203.62 10 16.301063 I98.85 17.0I9646 200.37 I7-743850 201.97 8.-473953 203.65 11 16.312995 198.88 17.03I669 200.40 I7.755969 202.00 18.486173 203.68 12 16.324928 I98.90 17.043694 200.43 17.768090 202.03 I8.498395 203-.7 13 I6.336863 198.93 17-055720 200.45 17.780213 202.06 18.510618 2,03.74 14 16.348799 I98.95 17.067748 200.48 17.792337 202.08 I8.522843 203-77 15 I6.360737 198.97 17-079777 200.50 17.804462 202.1I 18.535070 203.80 16 16.372676 I99.00 17.091808 200.53 17.816590 202.14 18.547299 203.82 17 6.3846I7 I99.02 I17103841 200.56 17.828719 202.17 18.559529 203-85 18 16.396559 I99.05 i7.115875 200.58 17.840850 202.I9 I8.57176I 203.88 19 16.408503 I99.07 17.127911 200.61 17.852982 202.22 18.583995 203.9I 20 I6.420448 199.10 17. 39948 200.64 I7.865 16 202.25 18.596230 203.94 21 I6.432395 199. 2 17.151987 200.66 17.877252 202.28 18.608467 203.97 22 16.444343 I99.I5 17.164028 200.69 17.889389 202.30 I8.620706 204.00 23 16-456392 199-I7 17-176070 200.71 17.901528 202.33 18.632947 204-03 24 16.468243 I99.20 17.188114 200.74 17.913669 202.36 I8.645190 204.05 25 16.480196 I99.22 17.200159 200.77 17.92581 202.39 I8.657434 204.08 26 I6.492151 199.25 17.212206 200.79 17-937955 202.41 I8.669679 204.11 27 I6.504107 199.27 17.224254 200.82 17.950I10 202.44 I8.68I927 204.14 28 I6.516064 199.30 17.236304 200.85 I7.962248 202.47 I8.694177 204.I7 29 I6.,528022 199.33 7.248356 200.87 17-974397 202.50 18.706428 204.20 30 I 6.539983 I99-35 I7.260409 200.90 I7.986548 202.52 18.718680 204.23 31 16.551945 199.38 17-272464 200.93 I7.998700 202.55 18.730935 204.26 32 I6.563908 199.40 17.284520 200.95 I8.010854 202.58 18.743191 204.29 33 I6.575873 199.43 17.296578 200.98 18.023010 202.61 18.755449 204.32 34 6.587839 199-45 17-308637 201.00 18.035167 202.64 18.767709 204.35 35 6.599807 i99.48 I7.320698 201.03 18.047326 202.66 8.779971 204.37 36 16.611776 I99.50 17.33276I 201.06 18.059487 202.69 18.792234 204.40 37 I6.623747 I99.53 17.344825 201.08 18.071649 202.72 8.80o4499 204.43 38 I6.635719 199-55 17-356891 20.II 1 8.083813 202.75 I8.816767 204.46 39 6.647693 199-58 17-368959 201.14 18.095979 202.78 18.829036 204.49 40 I6.659669 I99.60 17.381028 201.16 18.IO8146 202.80 18.841305 204.52 41 16.671646 I99.63 17-393098 201.19 18.1203I5 202.83 I8.853577 204-55 42 I6.683624 199.65 17.405171 201.22 18.132486 202.86 I8.86585I 204.-58 43 I6.695604 I99.68 17-417245 201.24 18.144658 202.89 18.878127 204.61 44 16.707586 199-70 17.429320 201.27 18.156832 202.92 18.890404 204.64 45 I 6.79569 199-73 17-441397 201.30 I8. 69008 202.94 18.902684 204.67 46 16.731553 199-76 17.453476 201.32 18.181186 202.97 18.914965 204.70 47 16.743539 199-78 17.465556 201.35 I8.193365 203.00 18.927247 204.73 48 I6.755527 I99.81 17-477638 20I.38 18.205546 203-03 18.939532 204.76 49 I6.7675I6 199.83 17.489722 201.4I 18.217728 203.06 18.951818 204.79 50 16.779507 I99.86 17.501807 201.43 I8.22992z 203.08 18.964106 204.81 51 16.791499 199.88 17.513894 201.46 18.242098 203.I I8.976396 204-84 52 I6.803493 199.91 17.525982 201.49 18.254286 203-14 I8.988687 204-87 53 I6.815488 199-94 17.538072 201.5I I8.266475 203-17 19.00098I 204.90 54 I6.827485 I99.96 17.550163 201.54 I8.278666 203.20 19.013276 204.93 55 16.839484 199-99 17.562257 201.57 18.290859 203-23 I9.025573 204.96 56 16.851484 200.01 17.574352 201.59 I8.303053 203.25 I9.03787I 204.99 57 16.863485 200.04 17.586448 201.62 18.315249 203.28 I9.050I72 205.02 58 16.875488 200.06 I7.598546 201.65 I8.327447 203-31 19.062474 205.05 59 i6.887493 200.09 17610 39646 201.68 18.339646 203,34 19-074778 205.08 60 16.899499 200.12 17.622747 201.70 18.351847 203.37 19-087084 205.11 572 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 280 290 300 310 M. Diff. 1". M. Diff. 1". lolog. ff. 1. Diff. 1. O' I9.070874 205.11 19.828747 206.94 1.313 3849 44-.8 1.329 0430 42.92 1 I9.09939I 205.14 19.84II64 206.97.313 6493 44.06.329 3004 42-91 2 19.11170 2 95353 20517 9853 7.o00.313 9I36 44.04.329 5578 42.89 3 19.124012 205.20 19.866004 207.03.314 I778 44.02 -329 8I51 42.87 4 I9.I36325 205.23 19-878427 207.06.314 4419 44.00 -330 0723 42.85 5 I9.148639 205.26 19.890852 207.09 I1314 7058 43-98 I 330 3293 42.83 6 I9.160956 205.29 19.903279 207.13.314 9696 43.96.330 5862 42.81 7 19.173274 205.32 I9.915707 207.16.315 2333 43-94 -330 8431 42.80 8 I9. 85594 205.35 19.9z2837 207.19.315 4969 43.92.331 0998 42.78 9 19.1979i6 205.38 19.940569 207.22.315 7604 43.90.331 3564 42.76 10 I9.210240 205.41 19-953003 207-25 I.3I6 0237 43.88 I.33I 6129 42.74 I 19.222566 205.44. 9.965439 207.28.316 2869 43.86.331 8693 42.72 12 19.234893 205.47 19-977877 207.31.3I6 5500 43.84.332 1255 42.70 13 19.24-7222 205.50 I9.990o37 207.34.316 8130 43.82.332 3817 42.69 14 19.259553 205.53 20.002759 207.38.317 0759 43.80.332 6378 42.67 15 I9.27I885 205.56 20.015202 207.41 I-317 3386 43.78 1.332 8937 42.65 16 19.284220 205.59 20.027647 207.44.317 60I3 43-76.333 1496 42.63 1T7 I9.296556 205.62 20.040095 207.47.317 8638 43.74.333 4053 42.61 18 19.308894 205.65 20.052544 207.50,318 1262 43-72.333 6609 42.59 19 I9.321234 205.68 20.064995 207.53.318 3885 43-70.333 9I64 42.58 20 I9.333576 205.71 20.077448 207.57 I.318 65o6 43.68 I-334 I718 42.56 21 19.345920 205.74 20.089903 207.60.318 9127 43.67.334 4271 42.54 22 19.358265 205.77 20.102360 207.63.319 1746 43.65.334 6823 42.52 23 I9-370612 205.80 20. 11488 207.66.319 4364 43.63'334 9374 42.50 24 19.382961 205.83 20.127279 207.69.3I9 698I 43.6I.335 1924 42.49 25 19.395312 205.86 20.I39741 207.72 1.319 9597 43-59 1.335 4472 42.47 26 19.407665 205.89 20. 52206 207.76.320 2212 43-57 -335 7020 42.45 27 19.420019 205.92 20.164672 207.79.320 4825 43-55 -335 9567 42.43 28 19.432375 205.95 20.177I40 207.82.320 7438 43.53.336 211z 42.41 29 19.444734 205.98 20.189610 207.85.321 0049 43.51 -336 4656 42.40 30 I9.457094 206.01 20.202082 207.88 18.321 2659 43-49 1.336 7199 42.38 31 I9.469455 206.04 20.214556 207.91.321 5268 43-47 -336 9742 42.36 32 I9.481819 206.08 20.227032 207.95.321 7875 43-45 -337 2283 42.34 33 19-494184 206.11 20.239510 207.98.322 0482 43-43 -337 4823 42-33 34 I19506551 206.14 20.251989 208.01.322 3087 43.41'337 7362 42.31 35 19.518921 206.17 20.264471 208.04 1.322 5692 43.40 1.337 9900 42.29 36 19.531292 206.20 20.276954 208.07.322 8295- 43.38.338 2437 42.27 37 19.543664 206.23 20.289440 208.I.323 0897 43.36.338 4972 42.25 38 19.556039 206.26 20.301927 208.14,.323 3498 43-34 -338 7507 42.24 39 I9.5684I5 206.29 20.314416 208.17.323 6097 43-32.339 0041 42.22 40 19.580794 206.32 20.326907 208.20 1.323 8696 43.30 1.339 2573 42.20 41 9-.593174 206.35 20-339400 208.2 324 32 1294 43.28.339 5105 42.18 42 I9.605556 206.38 20.351895 208.27.324 3890 43.26.339 7635 42.17 43 I9.6I7939 206.41 20.364392 208.30.324 6485 43-24.340 OI65 42.15 44 19.630325 206.44 20.37689I 208.33 -324 9079 43-22.340 2693 42.13 45 9.642713 26.47 20.7389392 208.36 1.325 1672 43.21 1.340 5221 42.11 46 i9.655102 206.50 20.401895 208.39.325 4263 43.19 -340 7747 42.10 47 I9.667493 206.53 20.414399 208.43.325 6854 43-17.341 0272 42.08 48 19.679886 206.57 20.426906 208.46.325 9443 43.15.341 2796 42.06 49 I9.692281I 206.60 20.439415 208.49 -326 2032 43-13.341 5319 42.04 50 I9.704678 206.63 20.451925 208.52 1.326 4619 43.11 1.341 7841 42.03 51 19.717076 206.66 20.464437 208.56.326 7205 43-09 -342 0362 42.01 52 I9.729477 206.69 20.476952 208.59.326 9790 43-07.342 2882 41.99 53 I9.741879 206.72 20.489468 208.62.327 2374 43.05 -342 5401 41.97 54 19.754283 206.75 20.50I986 208.65 -327 4957 43.04 -342 7919 41.96 55 I9.766689 206.78 20.514506 208.69 1.327 7538 43.02 1.343 0436 41-94 56 19.779097 j206.81 20.527029 208.72.328 0119 43-00.343 2952 41.92 57 19.791507 206.84 20.539553 208.75.328 2698 42.98.343 5467 41-90 58 I9.803919 206.88 20.552079 208.78.328 5276 42.96.343 7980 41.89 59 19.816332 206.91 20.564607 208.82.328 7853 42.94 -344 0493 41.87 60 19.828747 2o6.94 20.577137 208.85 1-.39 043 42.92 1.344 3005 41-85 573 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 32~ 330 340 350 Vo log M. Diff. 1". log 3. Diff. 1". log M. Diff.". log 3. Diff. 1". 0' 1-344 3005 41.85 1.359 1859 40.86 -1373 7251 39-93 1-387 9418 39.06 1 -344 5515 41.84 -359 4310 40.84.373 9646 39-91.388 I76I 39.05 2 344 8025 41.82.359 6760 40.82.374 2041 39.90.388 4104 39-04 3 -345 0534 41.80.359 9209 40-81 -374 4434 39.88.388 6446 39.02 4 -345 304I 41.78 -360 1657 40.79 -374 6827 39.87.388 8787 39.01 5 I'345 5548 41.77 1.360 4104 40.78 1.374 92I8 39.85 1.389 1127 38.99 6.345 8053 41-75 -360 6550 40.76.375 1609 39.84.389 3466 38.98 7 -346 0558 41.73 -360 8995 40-74 -375 3999 39-82.389 5804 38.97 8.346 3061 41-72.36I I439 40-73.375 6388 39.8I.389 8142 38.95 9 -346 5564 41-70.36I 3883 40.71.375 8776 39-79.390 0479 38.94 10..346 8065 41.68 1.36I 6325 40.70 1.376 II64 39-78 1.390 2815 38-93 1.347 0565 41.66.361 8766 40.68.376 3550 39-77.390 5150 38.9I 12 -347 30646 5 4.6 362 1207 40.66.376 5935 39-75 -390 7484 38.90 13 347 5563 41.63 362 3646 40.65.376 8320 39-74 -390 9817 38.88 14.347 8060 41.6 36 6084 4.6. 3 377 0703 39-72.391 2150 38-87 15 1.348 0557 41.60 1.362 8522 40.62 1.377 3086 39-71 1.391 4482 38.86 16 -348 3052 41.58.363 0959 40.60.377 5468 39.69.391 6813 38.84 17 -348 5546 41.56.363 3394 40-59.377 7849 398 368 9143 38.83 18.348 8040 41.55.363 5829 40-57 378 0230 39.66.392 1472 38.82 19 349 0532 41.53.363 8263 40.56.378 2609 39.65.392 3801 38.80 20 1.349 3023 41.51 I.364 0696 40.54 1.378 4987 39.64 1.392 6128 38.79 21 -349 5513 41.50.364 3128 40.52.378 7365 39.62.392 8455 38.77 22 349 8003 41.48.364 5559 40-51.378 9742 39.6I.393 0781 38-76 23 350 049I 41-46.364 7989 40-49 -379 2II7 39-59 393 3107 38.75 24.350 2978 41.45 -365 0418 40.48.379 4492 39.58.393 5431 38-73 25 1.350 5464 41.43 1.365 2846 40.46 1.379 6866 39.56 I-393 7755 38.72 26.350 7950 41.41.365 5273 40.45.379 9240 39-55 -394 0078 38.71 27 35I 0434 41.40.365 7699 40.43.380 I612 39.53.394 2400 38.69 28.351 2917 41.38.366 0125 40.4.380 3983 39.52 -394 4721 38.68 29.351 5399 41-36.366 2549 40.40.380 6354 39-50.394 7041 38.67 30 1.351 7880 41-35 1.366 4973 40.38 1.380 8724 39.49 1.394 9361 38.65 31.352 0361 41.33.366 7395 40.37.38I 1093 39-47.395 I680 38.64 32.352 2840 41.3I.366 9817 40-35.381 346I 39.46.395 3998 38.63 33.352 5318 41.30.367 2238 40.34.381 5828 39-45.395 63I5 38.6I 34.352 7795 41.28.367 4657 40.32.381 8194 39-43.395 863I 38.60 35 I.353 0272 41.26 1I367 7076 40.31 1.382 0559 39.42 1.396 0947 38.59 36.353 2747 4I.25.367 9494 40-29.382 2924 39-40.396 3262 38.57 37 353 522I 41.23.368 I91 40.28.382 5288 39-39.39 6 5576 38.56 38 *353 7694 4I.2I.368 4327 40.26.382 7651 39.37.396 7889 38.55 39.354 0I67 4I.z0.368 6742 40.25.3830013 39-36.397 0201 38.53 40 1.354 2638 41.I8 I.368 9I57 40.23 1.383 2374 39-35 1-397 2513 38.52 41.354 5I08 41.16.369 1570 40.2I.383 4734 39-33 -397 4823 38-51 42 354 7578 4.15.369 3983 40.20.383 7093 39-32.397 7I33 38.49 43 355 oo0046 41.13.369 6394 40.-18 383 9.452 39.30.397 9442 38.48 44.355 2513 41.11 -369 8805 40.17 -384 1809 39.29 -398 1751 38-47 45 1.355 4980 41.10 1.370 1214 40.15 1.384 4I66 39.27 1.398 4058 38.45 46.355 7445 41.08.370 3623 40.14 -384 6522 39.26.398 6365 38 44 47 355 9909 41.07.370 6031 40.12.384 8878 39.25.398 8671 38-43 48.356 2373 41-05.370 8438 40.II.385 1232 39.23 *399 0976 38.4I 49.356 4836 41.03 -37I 0844 40.09.385 3585 39.22.399 3281 38.40 50 1.356 7297 41.02 1.371 3249 40.08 1.385 5938 39-20 1.399 5584 38.39 51.356 9758 41.00.37I 5654 40.06.385 8290 39.19.399 7887 38.37 52.357 22I7 40.98.371 8057 40.05.386 0641 39-18'400 0189 38.36 53.357 4676 40.97.372 0459 40.03.386 2991 39.16.400 249I 38.35 54.357 7134.4095.372 2861 40.02.386 5340 39-I5.400 4791 38.33 55 1.357 9590 40.94 1.372 5261 40.00 I.386 7689 39.13 I.400 709I 38.32 56.358 2046 40.92.372 7661 39-99.387 0036 39.I2.400 9390 38.31 57.358 450I 40.90.373 oo60 39-97 -387 2383 39-I11 40I I688 38.30 58.358 6954 40.89.373 2458 39.96.387 4729 39-09 -401 3985 38.28 59.358 9407 40.87.373 4855 39-94.387 7074 39-08.401 6282 38.27 60 1.359 1859 40.86 -1373 7251 39.93 1.387 94I8 39.06 1.401 8578 38.26 574 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 36" 37" 38~ 390 V.._ - log M. Diff. 1". log M. Diff. 1". log B. Diff. 1". lo M. Diff. 1". O' I.40I 8578 38.26 1.415 493 37.50 I1428 8662 36.80 1I44I 9943 36.14 1.402 0873 38.24.4I5 7180 37-49 -429 0869 36.79 -442 2II1 36.I3 2.402 3167 38.23.415 9429 37-47.429 3076 36.78.442 4279 36.12 3.402 5460 38.22.416 1678 37.46.429 5283 -36.77.442 6446 36.11 4 -402 7753 38.20 *4I6 3925 37-45 -429 7488 36.75.442 86I2 36.Io 5 1.403 0045 38.19 I.416 6172 37-44 I1429 9693 36-74 1-443 0778 36.09 6.403 2336 38.I8.416 8419 37-43 -430 1897 36.73 -443 2943 36.08 7.403 4626 38.I7.4I7 0664 37.41 -430 4IOI 36.72.443 5107 36.07 8.403 6916 38.15 -417 2909 37.40.430 6304 36.71.443 7271 36.06 9.403 9205 38.I4 -.47 5153 37-39 -430 8506 36.70.443 9434 36.05 10 1.404 I493 38I13 14I7 7396 37-38 1.43I 0708 36.69 1-444 I597 36.04 1 1 -404 3780 38.I2.I 96739 37-37.431 2909 36.68'444 3758 36.03 12.404 6067 38.10.418 i88I 37.36.431 5I09 36.66.444 5920 36.02 13.404 8352 38.09.4I8 4I22 37-35.431 7308 36.65 -444 8080 36.00 14.405 0637 38.08.418 6362 37-33.43I 9507 36.64.445 0240 35-99 15 1.405 2921 38.06 1.418 8602 37.32 1.432 1705 36.63 I.445 2400 35.98 16.405 5205 38.05.419 084I 37.31.432 3903 36.62.445 4558 35-97 17.405 7488 38.03.4I9 3079 37.30.432 6Ioo 36.61.445 6716 35.96 18.405 9769 38.02.419 53I7 37.29.432 8296 36.60.445 8874 35.95 19.406 2051 38.01.419 7554 37.27 -433 049I 36.59 -446 I031 35.94 20 1.406 4331 38.00 1.4I9 9790 37.26 1.433 2686 36.57 1.446 3187 35 93 21.406 6611 37-99 -420 2026 37.25 -433 4881 36.56,.446 5343 35.92 22.406 8889 37-97 -420 4260 37.24 -433 7074 36.55 -446 7498 35.91 23.407 1168 37.96.420 6494 37.23 -433 9267 36.54.446 9652 35-90 24.407 3445 37-95.420 8728 37.22.434 I459 36-53 -447 I806 35.89 25 1.407 5721 37-94 I.42 09 37.3720 1.434 3651 36.52 1.447 3959 35.88 26.407 7997 37.92.421 3192 37.I9 -434- 5842 36.51.447 612 35.87 27.408 0272 37.9I -421 5423 37.I8.4J4 8032 36.50.447 8263 35-86 28.408 2547 37.90.421 7654 37.I7'435 022I 36.49.448 04I5 35.85 29.408 4820 37.89.42I 9884 37.I6.435 2410 36.48.448 2565 35-84 30 1.408 7093 37.87 I.422 2113 37.I5 1.435 4598 36.47 1.448 4715 35.83 31.408 9365 37.86.422 434I 37.I3.435 6786 36.46.448 6865 35.82 32.409 1636 37.85.422 6569 37-.2.435 8973 36.44 -448 9014 35.8I 33.409 3907 37.84.422 8796 37.-I.436 1159 36.43.449 162 35.80 34.409 6177 37.82.423 1022 37.I10 436 3345 36.42.449 3309 35-79 35 1.409 8446 37.8I 1.423 3248 37.09 I.436 5530 36.4I I.449 5456 35.78 36.410 07I4 37.80.423 5473 37.08.436 7714 36.40.449 7603 35-77 37.410 2981 37.78.423 7697 37.06.436 9898 36.39.449 9749 35.76 38.410 5248 37-77 -423 9920 37.05.437 2081 36.38.450 1894 35-75 39.410 7514 37.76.424 2I43 37.04 -437 4263 36.37.450 4038 35.74 40 1.410 9780 37-75 1.424 4365 37-03 1-437 6445 36.36 1.450 6182 35-73 41.411 2044 37-74 -424 6586 37.02.437 8626 36.35 -450 8325 35.72 42.411 4308. 37-72.424 8807 37.0I -438 o806 36.34.451 0468 35.7I 43.411 6571 37.7I.425 1027 36.99.438 2986 36.32.45I 261O 35.70 44.4II 8833 37.70.425 3246 36.98.438 5165 36-3I -451 4752 35.69 45 1.412 1095 37.69 1.425 5465 36.97 1-438 7344 36-30 1.451 6893 35.68 46.412 3356 37.68.425 7683 36.96.438 9522 36.29.451 9033 35.67 47.412 56i6 37.66.425 9900 36.95 -439 1699 36.28.452 1173 35.66 48.412 7875 37.65.426 2117 36.94.439 3875 36.27 -452 3312 35.65 49.413 OI34 37.64.426 4333 36.92.439 605I 36.26.452 5450 35.64 50 1.413 2392 37.63 I.426 6548 36.91 1-439 8226 36.25 1.452 7588 35.63 51.413 4649 37.6I.426 8762 36.90.440 0401 36.24 -452 9725 35.62 52.43 6905 37.60.427 0976 36.89 -440 2575 36.23 -453 1862 35.61 53.413 9161 37-59.427 3189 36.88.440 4748 36.22.453 3998 35.60 54.414 1416 37.58.427 5402 36.87 -440 692I 36.20.453 6I34 35-59 55 1.414 3670 37.56 1.427 7613 36.86 1.440 9093 36.I9 I-453 8269 35.58 56.414 5924 37-55.427 9824 36-85.441 I264 36.I8 -454 0403 35-57 57.4I4 8176 37-54 -428 2035 36.83 -44I 3436 36.17 -454 2537 35-56 58.4I5 0429 37-53 -428 4244 36.82.441 5605 36.I6.454 4670 35-55 59.4I5 2680 37.5I -428 6453 36.8I.441 7774 36.15 -454 6802 35-54 60 I.415 4930 37-50 I.428 8662 36.80 1.441 9943 36.I4 1.454 8934 35.53 575 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 40O 41~ 42~ 430 log. Di. log. iff. 1. ff.. log M. Diff. ". log M. Diff. 1". 0' i454 8934 35-53 1-467 5782 34.95 1.480 0627 34-4I1 1492,3597 33-91 1.455 I065 35-52.467 7879 34-94.480 269I 34.40 -492 563I 33.90 2 455 3196 35-51.467 9976 34.93.480 4755 34.40.492 7665 33-89 3'45S 5326 35.50.468 2071 34.92.480 6819 34.39 -492 9698 33.88 4.455 7456 3549.468 4166 34.91.480 8882 34-38.493 1731 33.87 5 i.455 9585 35.48 1.468 6261 34.90 1.48I 0944 34-37 1.493 3764 33-87 6.456 1713 35-47.468 8355 34-90.48I 3006 34.36 -493 5796 33.86 7.456 3841 35.46.469 0448 34.89 -48I 5068 34-35.493 7827 33-85 8.456 5968 35.45.469 254I 34.88.48I 7129 34.34.493 9858 33.84 9.456 8094 35-44.469 4634 34.87.48I 9189 34-33 -494 I888 33.83 10 1.457 0220 35.43 1.469 6725 34.86 1.482 1249 34-33 1-494 3918 33-83 11 457 2346 35.42.469 8817 34.85.482 3308 34.32.494 5948 33.82 12.457 4470 35.41 -470 0907 34.84.482 5367 34.31.494 7977 33.81 13.457 6595 35.40.470 2998 34.83.482 7425 34-30.495 0005 33.80 14.457 8718 35.39.470 5087 34.82.482 9483 34-29 -495 2033 33.79 15 1.458 0841 35-38 1.470 7176 34.81 1.483 1540 34.28 1.495 4061 33I 79 16.458 2964 35-37.470 9265 34.80.483 3597 34.28.495 6088 33-78 17.458 5086 35.36.471 I353 34.79.483 5653 34.27.495 8114 33-77 18.458 7207 35-35'471 3440 34-79 -483 7709 34.26.496 OI40 33.76 19.458 9328 35-34 -471 5527 34.78.483 9764 34.25.496 2I66 33.75 20 1.459 1448 35.33 I.471 7613 34-77 1.484 I819 34-24 1I496 4191 33-75 21 459 3567 35.32.471 9699 34.76.484 3873 34.23.496 6216 33.74 22 459 5686 35.31.472 1784 34-75.484 5927 34.22.496 8240 33.73 23.459 7805 35.30.472 3869 34.74.484 7980 34.22.497 0264 33.72 24.459 9922 35.29.472 5953 34-73.485 0033 34.21.497 2287 33.71 25 I.460 2040 35.28 1.472 8037 34.73 1.485 2085 34.20 1.497 4310 33-7I 26.460 4156 35.27.473 OI20 34.72.485 4137 34.19 -497 6332 33.70 27.460 6272 35.26.473 2203 34.71 -485 6188 34.I8.497 8354 33-69 28.460 8388 35.25.473 4285 34.70.485 8239 34-I7 -498 0376 33.68 29.461 0503 35.24 -473 6366 34.69.486 0289 34.I6.498 2396 33.68 30 1.46I 2617 35.23 I.473 8447 34.68 1.486 2338 34.-6 1.498 4417 33.67 31 -461 4731 35.23.474 0527 34.67 -486 4388 34.I5.498 6437 33.66 32.46I 6844 35.22.474 2607 34.66.486 6436 34.I4.498 8456 33.65 33.46I 8957 35.21.474 4686 34.65.486 8484 34.13.499 0475 33.65 34.462 Io69 35.20.474 6765 34-64 -487 0532 34.12.499 2494 33.6435 1.462 3I80 35.19.1474 8843 34.63 1-487 2579 34.12 1.499 4512 33.63 36.462 5291 35.18.475 09I2 34.62.487 4626 34.II. 499 6530 33.62 37.462 740I 35.17.475 2998 34.6I.487 6672 34.I1.499 8547 33.62 38.462 9511 35.-6.475 5075 34.61.487 8718 34.09.500 0563 33.6I 39.463 I620 35.I5.475 7151 34.60.488 0763 34.08.500 2580 33.60 40 1.463 3729 35-I4 1-475 9227 34-59 1.488 2807 34.07 1.500 4595 33.59 41.463 5837 35.I3 -476 1302 34.58.488 4852 34.07.500 66II 33.58 42.463 7944 35.12.476 3376 34-57.488 6895 34-06.500 8625 33.58 43.464 0051 35.II -476 5450 34.56.488 8939 34.05.50I 0640 33.57 44.464 2158 35.IO.476 7524 34-55.489 0981 34-04 -50I 2654 33.56 45 1.464 4263 35-09 1476 9596 34-54 1.489 3023 34-03 1.501 4667 33.55 46.464 6369 35.08.477 I669 34-54.489 5065 34.02.501 6680 33-55 47 -464 8473 35.07 -477 3741 34.53 -489 7I06 34.02.501 8693 33.54 48.465 0577 35.o6.477 5812 34.52.489 9147 34.01.502 0705 33 53 49.465 2681 35.05.477 7883 34.5I.490 1187 34.00.502 27I6 33-52 50 1.465 4784 35-04 I-477 9953 34.50 1.490 3227 33-99 I.502 4727 33-5I 51.465 6886 35-04 -478 2023 34.49 -490 5266 33.98.502 6738 33.5I 52.465 8988 35.03 -478 4092 34-48.490 7305 33.97.502 8748 33.50 53.466 Io90 35.02.478 616I 34.47.490 9343 33.96.503 0758 33-49 54.466 3190 35-01.478 8229 34.46.491 I381 33-95.503 2767 33-48 55 1.466 5290 35-00 1.479 ~297 34.46 1.49I 3418 33.95 I.503 4776 33-48 56.466 7390 34.99 -479 2364 34-45.491 5455 33.94 -503 6784 33.47 57.466 9489 34.98.479 4430 34.44.49I 7491 33.93.503 8792 33-46 58.467 1587 34-97 -479 6496 34.43.49I 9527 33.92.504 0800 33-45 59.467 3685 34.96.479 8562 34.42.492 1562 33.91 -504 2807 33-44 60 1.467 5782 34-95 1'480 0627 34-41 1.492 3597 33.91 1504 4813 33-44 576 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. _.......... __ 44" 450 46" 470 V. ~ ~ - log M. Diff. 1". log. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 1.504 4813 33-44 I.5I6 4390 33-00 1.528 2435 32.59 1'539 9048 3-220 1.504 6819 33-43 -516 6370 32.99'528 4390 32.58.540 0980 32.20 2.504 8825 33.42.5i6 8349 32.98.528 6344 32-57.540 2912 32.19 3.505 o830 33.42 -517 0328 32.98 528 8299 32.57.540 4843 32.18 4.505 2835 33-41 -517 2306 32.97 -529 0252 32.56.540 6774 32.18 5 1.505 4839 33-40 1.517 4284. 32.96 1.529 2206 32-55 1.540 8705 32.17 6.505 6843 33-39 -517 6262 32.96.529 4159 32.55 -541 0635 32.17 7.505 8846 33.39.517 8239 32-95.529 6IIz 32.54.541 2564 32.16 8.506 0849 33.38.518 0216 32.94.529 8064 32.53 -541 4494 32.15 9.5o6 2852 33-37 -518 2192 6 32.3 53053.54I 6423 32.I5 10 1.506 4854 33-36 1.518 4I68 32-93 1.530 1967 32.52.1.541 8352 32.14 11.506 6855 33.36.5I8 6143 32.92.530 3918 32.51.542 0280 32.I4 12.5o6 8856 3335.8 8i8 32.91.530 5869 32.51.542 2208 32.13 13.507 0857 33.34 -519 0093 3291 530 789 3 0 5 542 4135 32-I2 4.507 2857 33-33 5I19 2067 32.90.530 9769 32.49 -542 6063 32.11 15 1.507 4857 33-33 1-519 4041 32.89 1-53I I7I9 32.49 I.542 7989 32.11 16.507 6856 33.32.519 6014 32.89.531 3668 32.48.542 9916 32.10 17.507 8855 33.31 -519 7987 32.88.531 5616 32.48.543 1842 32.10 18 5o08 0853 33-30.519 9960 32.87.53I 7565 32.47 -543 3768 32.09 19.508 2851 33.29 -520 I932 32.86.53I 9513 32.46 -543 5693 32.09 20 1.508 4849 33.29 1.520 3904 32.86 1.532 1460 32.46 1-543 76I8 32.08 21.5o8 6846 33.28.520 5875 32.85 -532 3407 32.45 -543 9543 32.08 22.508 8843 33.27 -520 7846 32.84 -532 5354 32-44.544 1467 32.07 23.509 0839 33-27 -520 9816 32.84 -532 7300 32-44 -544 339 2. 06 24.509 2835 33.26 521 1786 32.83.532 9246 32.43 -544 53I5 32.06 25 1.509 4830 33-25 1.521 3756 32.82 1-533 192 32.43 1-544 7238 32.05 26 -509 6825 33.24.521 5725 32.82.533 3137 32.42.544 916I 32.04 27.509 8819 33.24.521 7694 32.81 -533 5082 32.42.545 1083 32 04 28.510 08I3 33.23.521 9662 32.80.533 7027 32.41 -545 3005 32-03 29.50 287 33.22.522 1630 32.80.533 8971 32.40.545 4927 32.03 30 1.510 4800 33.21 1.522 3598 32.79 I1534 0914 32.39 1-545 6849 32.02 31.510 6792 33.21.522 5565 32.78.534 2858 32.39.545 8770 32.02 32.510 8785 33.20.522 753 32.78.534 4801 32.38.546 69go 32.01 33.511 0776 33.19 -522 9498 32.78.534 6743 32-37 -546 26I 32.00 34.51I 2768 33.18.523 I464 32-77.534 8685 32.37 -546 4531 32.00 35 1.511 4759 33.18 1.523 3429 32.76 1.535 0627 32-36 1.546 6450 31.99 36.51 6749 33.17 -523 5394 32.75 -535 2568 32.35.546 8370 31.98 37.5II 8739 33.I6.523 7359 32.74 -555 4509 32.35 -547 0289 31.98 38.512 0729 33.15 -523 9323 32-73.535 6450 32.34 -547 2207 3.197 39.512 2718 33.15.524 I287 32.73 -535 8390 32.33 547 4125 3I.97 40 1.512 4707 33.14 I.524 3251 32.72 1.536 0330 32.33 I1547 6043 31.96 41.512 6695 33.I3.524 5214 32.71 -536 2270 32.32.547 7961 31.96 42.512 8683 33.I3 -524 7176 32.71.536 4209 32.32.547 9878 31.95 43.513 o670 33.12.524 9I38 32-70.536 6148 32.31.548 1795 31.94 44.51I3 2657 33.I11 525 II00 32.70.536 8086 32.30.548 3711 31-94 45 1.513 4644 33.1 I-.525 3062 32.69 1-537 0024 32.30 1.548 5627 31-93 46.513 6630 33.10.525 5023 32.68.537 I962 32.29 -548 7543 31.93 47 3 865 33-09 525 6983 3.67 53 32.33899 32.28 548 9458 3192 48.514 060o 33.08.525 8944 32.67 -537 5836 32.28.549 1373 3I91I 49.514 2586 33.07.526 0903 32.66.537 7772 32.27 549 3288 31.9I 50 1.514 4570 33.07 I.526 2863 32.65 1-537 9708 32.26 1.549 5202 31-90 51.514 6554 33.06.526 4822 32.64.538 I644 32.26.549 7116 31.90 52.514 8537 33.05 -526 678079 32264 538 3579 32.25.549 9030 31.89 53 I515 0520 33.05.526 8739 32.63.538 5514 32.25.550 0943 31.88 54.515 2503 33-04.527 o696 32.62.538 7449 32-24.550 2856 31.88 55 I.5I5 4485 33.04 1.527 2654 32.62 1.538 9383 32.23 I.550 4769 3I.87 56.515 6467 33-03 -527 4611 32.61.539 1317 32-23 -550 668i 31.87 57.515 8449 33.02.527 6567 32.61.539 3250 32.22.550 8593 3I.86 58.5 6 0430 33.01.527 8524 32.60.539 5183 32.21.55I 0504 3I.86 59.5I6 2410 33.01.528 0479 32.60.539 716 3232I.551 24I6 31.85 60 i.516 4390 33.00 1.528 2435 32.59 1539 9048 32.20 1-551 4326 31.85 37 577 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 48 49~ 50~ 51"~ log M. Diff. 1V'. log M. Diff. 1". log M. Diff. I1". log M. Diff. 1.' 1.551 4326 31.85 1.562 8360 31.5I 1.574 1234 31.20 1.585 3031 30.91 1.55I 6237 3I.84.563 0250 31.51.574 3I06 31.20.585 4886 30-9I -55I 8147 31.83.563 2140 31.50.574 4977 31-19 -585 6740 30.90 3 552 0057 31.83.563 4030 31.50.574 684-9 3I.19.585 8594 30.90 4.552 I966 31.82.563 5920 31.49.574 8720 31.18.586 0448 30.89 5 1.552 3876 31.82 1.563 7809 31.48 1.575 0590 3I.I8 1.586 2302 30.89 6.552 5784 31-8I.563 9698 31.48.575 246I 31.17.586 4155 30.89 7.552 7693 31-80.564 1586 3I.47'575 4331 31-17.586 6008 30.88 8.552 9601 31.80.564 3475 3I147.575 620o 31.16.586 7859 30.87 9 *553 I508 31.79.564 5363 3'I.46.575 8070 31.16.586 9713 30.87 10 1.553 3416 31.79 1.564 7250 31.46 1.575 9939 3I.15 I.587 I565 30.87 11.553 5323 31.78.564 9138 31.45.576 i808 31-15.587 3417 30.86 12.553 7230 31.78.565 1025 31.45.576 3677 3I.14.587 5268 30.86 13.553 9136 31.77.565 2911 31.44.576 5546 3.1I4.587 7120 30.85 14 554 1042 3I.76 -565 4798 31.44.576 74I4 31.I3.587 8971 30.85 15 1.554 2948 3.176 1.565 6684 31.43 I.576 9281 3.13 1.588 0821 30.84 16.554 4853 31.75.565 8569 3.143 -577 II49 31.12.588 2672 30.84 17.554 6758 31.75.566 0455 31.42.577 30I6 31.12.588.4522 30.83 18.554 8663 31.74.566 2340 31.4I.577 4883 31.11.588 6372 30.83 19.555 0567 31.74.566 4225 31.41.577 6749 31.II.588 8222 30.83 20 1.555 2472 31.73 1.566 6109 31.40 1.577 8615 31.I0 1.589 0071 30.82 21 -555 4375 31.73.566 7993 31.40.578 0481 3I.10.589 1920 30.82 22.555 6279 31.72.566 9877 31.39.578 2347 31.09.589 3769 30.81 23 555 8182 3I.7I1 567 1761 3I-39.578 4213 31.09.589 5618 30.81 24.556 0084 31.7I.567 3644 3.138.578 6078 3I.08.589 7466 30.80 25 1.556 1987 3.170 1.567 5527 31.38 1.578 7942 31.08 1.589 9314 30.80 26.556 3888 31.70.567 7409 31.37.578 9807 3I.07.590 1162 30.79 27.556 5790 3I.69.567 9291 3.137.579 I671 3I.07 -590 3009 30.79 28 556 769I 31.68.568 1173 31.36 -579 3535 31.06.590 4857 30.78 29.556 9592 31.68.568 3055 31.36.579 5399 31.06.590 6704 30.78 30 1.557 1493 31.67 1.568 4936 31.35 1.579 7262 3I.06 1.590 8550 30.78 31.557 3393 31.67.568 6817 3I.35.579 9125 3I.105.591 0397 30.77 32 557 5293 31.66.568 8698 31.34.580 0988 3I.04 -591 2243 30.77 33.557 7193 31.66.569 0579 31.34.580 2851 31-04.591 4089 30.76 34.557 9092 31.65.569 2459 31.33.580 47I3 31.03 -591 5935 30-76 35 1.558 0991 31.65 1.569 4338 31.33 I.580 6575 31.03 I.591 7780 30.75 36.558 2890 31.64.569 6218 31.32.580 8436 31.03.591 9625 30.75 37.558 4788 31.64.569 8097 31.32.581 0298 3I.02.592 1470 30.75 38.558 6686 3I.63.569 9976 31.31.58I 2159 31.02.592 3315 30.74 39.558 8584 31.62.570 I854 3.130.581 4020 31.01.592 5159 30.74 40 1.559 0482 31.62 1.570 3733 31-30 1.58I 5880 3I.01 1.592 7003 30.73 41.559 2379 3I.6I.570 56II 31-29 -58I 7740 31.00.592 8847 30.73 42.559 4275 31.61.570 7488 3.129.58I 9600 31.00.593 0690 30.72 43.559 6172 31.60.570 9366 31.28.582 1460 30.99'593 2534 30.72 44.559 8068 31.60.571 I243 31.28.582 33I9 30-99 -593 4377 30.72 45 1-559 9963 31.59 I.57I 31I9 31.28 1.582 5179 30.98 I-593 62I9 30.71 46.560 1859 31.59 -57I 4996 31.27.582 7037 30-98.593 8062 30.71 47.560 3754 3.158.57I 6872 31.27.582 8896 30.97.593 9904 30.70 48.560 5648 31.57.57I 8748 31.26.583 0754 30.97.594 1746 30.70 49.560 7543 31.57.572 0623 31.26.583 26I2 30.96.594 3588 30.69 50 I.560 9437 31.56 1.572 2499 31.25 1.583 4470 30.96 1.594 5429 30.69 51.56I I331 31.56.572 4373 31.25.583 6327 30.95.594 7270 30.68 52.56I 3224 31.55 -572 6248 31.24.583 8I84 30-95.594 9III 30.68 53.56i 5117 31-55.572 8123 31.24.584 0041 30.94.595 0952 30.68 54.56I 7010 31.54.572 9997 31.23.584 1898 30.94 -595 2792 30.67 55 1.56i 8902 31.54 1.573 I870 31.23 1.584 3754 30-94 1.595 4633 30.67 56.562 0794 31.53 -573 3743 3I.22.584 56o1 30.93.595 6473 30.66 57.562 2686 31.53'573 56I6 3I.22.584 7466 30.93'595 83I2 30.66 58.562 4578 31.52.573 7489 31-2I.584 9321 30.92.596 0151 30.65 59.562 6469 31I52.573 9362 3121.585 1176 30.92.596 1990 30.65 60 1.562 8360 31-51 1-574 1234 31.20 1-585 3031 30.9I 1.596 3829 30.65 578 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 52- 53~ 54~ 55~ log M. iiff. 1". log M iff. ". log I. Diff. 1". log M. Diff. 1". 0' 1.596 3829 30.65 I-607 3703 30-40 I.618 2724 30.17 1.629 o959 29.96 1.596 5668 30.64.607 5527 30.39.6I8 4534 30.17.629 2757 29.96 2.596 7506 30.64.607 7350 30.39.6I8 6344 30.I6.629 4554 29.96 3 596 9344 30.63.607 9174 30.39.6I8 8I53 30.I6.629 6351 29.95 4.597 1182 30.63.6o8 0997 30.38.618 9963 30.x6.629 8148 29.95 5 1.597 3020 30.62 i.608 2820 30.38 1.619 1772 30.15 I.629 9945 29-95 6.597 4857 30.62.6o8 4642 30.38.619 3581 30.I5.630 1742 29.94 7.597 6694 30.62.608 6465 30-37.619 5390 30.15.630 3538 29.94 8 -597 8531 30.6I.608 8287 30.37.619 7199 30.14.630 5335 29.94 9.598 0368 30.61.609 OI09 30-36.619 9007 30-14.630 7131 23993 10 1.598 2204 30.60 1.609 1931 30.36 1.620 o8i6 30.14 1.630 8927 29*93 11.598 4040 30.60.609 3752 30.36.620 2623 30-13.63I 0722 29.93 12.598 5876 30.59.609 5573 30-35.620 4431 30.13.631 5 28 29.9 13.598 7711 30.59.609 7394 30.35.620 6239 30.12.631 4313 29.92 14.598 9547 30.59.609 9215 30-34.620 8046 30.12.631 6o08 29.92 15 1.599 I382 30-58 I.6IO 1036 30.34 I.620 9853 30.12 I.631 7903 29.9I 1G.599 32I7 30.58.6o1 2856 30-34.62I 1660 30.I.63I 9698 29.91 17 599 505I 30-57.610 4676 30-33.621 3467 30.1I.632 1492 29.9I 18.599 6885 30.57.610 6496 30.33.62I 5274 30.11.632 3286 29.90 19 -599 8719 30-57.6o10 8315 30-32.62I 7080 30.10.632 5081 29.90 20 1.600 0553 30.56 I.611 0135 30-32 1.621 8886 30.10 1.632 6875 29-90 21.600 2387 30.56.6 1 1954 30.32.622 0692 30.10.632 8668 29.89 22.600 4220 30.55.6 I 3773 30.31.622 2497 30.09.633 0462 29.89 23.600 6053 30-55 611I 5591 30-31.622 4303 30.09.633 2255 29.89 24.600 7886 30.55.622 7641 30.62 68 30.9 633 4048 29.88 25 I.6oo 9718 30-54 I.6TI 9228 30.30 1.622 7913 30.08 I.633 5841 29-88 26.60o 1551 30-54.612 1046 03030.6 978 3008 3 7634 2988 27.6o0 3383 30.53.612 2864 30.29.623 1523 30.08.633 9427 29.87 28.6o0 5214 30.53.612 4681 30.29.623 3327 30-07.634 I2I9 29-87 29.60o 7046 30-52.612 6499 30.29.623 5131 30-07.634 3011 29.87 30 I.60I 8877 30-52 1.612 8316 30.28 1.623 6935 30.06 I.634 4803 29.86 31.602 0708 30.52.613 0132 30.28.623 8739 30-06.634 6595 29.86 32.602 2539 30-51.613 1949 30.28.624 0543 30.06.634 8387 29.86 33.602 4370 30-51.613 3765 30.27.624 2346 30-05.635 OI78 29.86 34.602 6200 30.50.613 5582 30.27.624 4149 30.05.635 I969 29.85 35 1.602 8030 30.50 I.613 7398 30.26 1.624 5952 30.05 I1635 3760 29.85 36.602 9860 30.50.613 9213 30.26.624 7755 30-04.635 5551 29.85 37.603 1690 30.49.6 029 9557 0 3004.635 7342 29.84 38.603 3519 30.49.6 4 2844 30.25.625 1360 30-04.635 9132 29.84 39.603 5348 30.48.614 4659 30.25.625 3162 30.03.636 0922 29.84 40 i.603 7I77 30-48 1.6I4 6474 30.25 1.625 4964 30-03 I.636 2713 29-83 41.603 9005 30.47.614 8288 30.24.625 6765 30.03.636 4502 29-83 42.604 0834 3 30.47. 0103.65 8567 3002 636 6292 2983 43.604 2662 30.47 6I15 1917 30.23.626 0368 30.02.636 8082 29.82 44.604 4490 30-46.6I5 3731 30.23.626 2169 30.02.636 9871 29-82 45 1.604 6317 30-46 I.615 5545 30-23 1.626 3970 30.01 1.637 I660 29.82 46.604 8145 30-45.6I5 7358 30.22.626 5771 30.0I.637 3449 29.82 47.604 9972 30-45.6I5 9171 30-22.626 7571 30.01.637 5238 29-81 48.605 1799 30-45.6I6 0984 30.22.626 9372 30-00.637 7027 29.81 49.605 36267 30.44.676 2797 30.21.627 1172 30.00.637 8815 29.81 50 6 545 30.44.6x6 4610 30.214.627 2972 30.00 1.638 0603 29.80 51.605 7278 30.43.6 6 6422 30.20.627 4771 29-99.638 239I 29.80 52.605 9104| 30-43.6I6 8234 30-20.627 6571 29-99.638 4179 29.80 53.606 0930 3043 67 6 30.20.627 8370 29-99.638 5967 29-79 54 6 27.66 27558 3042 7 88 309.628 0169 29.98.638 7754 29-79 55 I.606 4581 30.42 I.617 3669 30-19 1.628 1968 29.98 1.638 9542 29.79 56.6o6 6406 30.42.617 5486 30319.628 3766 29-98.639 1329 29-78 57.606 8230 30-4I.67 729 308.68 5565 29-97.639 3116 29-78 58.607 0055 304..6 90 30.628 73 63 2997 639 4902 29-78 59.607 187.9 30.40.6I8 0913 30-17.628 9161 29-97.639 6689 29-77 60.607 370387 30.40 i.61 2734 30-I7 1.629 0959 29.96 1.639 8475 29-77 579 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. r 56 57~ 58" 59" log M5. Diff. 1D. logi 5~. Diff. 1". log M. Diff. 1". log 1. Diff. 1" 0' 1.639 8475 29.77 I.650 5336 29.60 I.66I I60o 29.44 I.671 733I 29.30 1.640 0262 29.77.650 7II2 29.60.66I 3368 29.44.67I 9089 29.30 2.640 2048 29.77.650 8887 29.59.66I 5134 29.44.672 0846 29.30 3.640 3833 29.76.65I 0663 29.59.66I 6900 29.43.672 2604 29.29 4.640 5619 29.76.651 2438 29.59.66 8666 29.43.672 4362 29.29 5 1.640 7405 29.76 1.65I 42I3 29.58 1.662 0432 29.43 1.672 6II9 29.29 6.640 9190 29.75.65I 5988 29.58.662 2197 29.43.672 7876 29.29 7.64I 0975 29.75.65I 7763 29-58.662 3963 29.42.672 9634 29.28 8.64I 2760 29.75.65I 9538 29.58.662 5728 29.42.673 1391 29.28 9.64I 4545 29.74.652 1312 29.57.662 7493 29.42.673 3147 29.28 10 i.64I 6329 29.74 I.652 3086 29.57 1.662 9258 29.42 1.673 4904 29.28 11.64I 8114 29.74.652 486I 29.57.663 1023 29.41.673 666I 29.28 12.64I 9898 29.74.652 6635 29.57.663 2788 29.41.673 8417 29.27 13.642 1682 29.73.652 8408 29.56.663 4553.29.41.674 0174 29.27 14.642 3466 29.73.653 0182 29.56.663 63I7 29.4i.674 1930 29.27 15 1.642 5250 29.73 I.653 1956 29.56 I.663 8082 29.40 1.674 3686 29.27 16.642 7033 29.72.653 3729 29.55.663 9846 29.40.674 5442 29.27 17.642 8816 29.72.653 5502 29.55.664 161o 29.40.674 7198 29.26 18.643 0599 29.72.653 7275 29-55 664 3374 29.40.674 8954 29.26 19.643 2382 29.71.653 9048 29.55.664 5137 29.39.675 0709 29.26 20 1.643 4165. 29.71 I.654 0821 29.54 1.664 690I 29.39 1.675 2465 29.26 21.643 5948 29.71.654 2593 29.54.664 8664 29.39.675 4220 29.25 22.643 7730 29.71.654 4366 29.54.665 0428 29.39.675 5975 29.25 23.643 9513 29.70.654 6138 29.54.665 2191 29.39.675 7730 29.25 24.644 1295 29-70.654 79IO 29.53.665 3954 29-38.675 9485 29.25 25 1.644 3077 29.70 I-654 9682 29.53 1.665 5717 29.38 1.676 1240 29.25 26.64. 4858 29.69.655 1454 29.53.665 7480 29.38.676 2995 29.24 27.644 6640 29.69 655 3225 29-53.665 9242 29.38.676 4749 29.24 28.644 8421 29.69 655 4997 29.52.666 I005 29.37.676 6504 29.24 29.645 0203 29.69.655 6768 29.52.666 2767 29.37.676 8258 29.24 30 1.645 1984 29.68 1.655 8539 29.52 I.666 4529 29.37 1.677 0012 29.24 31.645 3765 29.68.656 0310 29.51.666 6291 29.37.677 1766 29.23 32.645 5545 29.68.656 2081 29.51.666 8053 29.36.677 3520 29.23 33.645 7326 29.67.656 3852 29.5I.666 9815 29.36.677 5274 29.23 34..645 9106 29.67.656 5622 2 2951 67 577 36.677 7028 29.23 35 1.646 o886 29.67 1.656 7392 29.50 1.667 3338 29.36 I.677 8781 29.23 36.646 2666 29.67.656 9163 29.50.667 5100 29.35.678 0535 29.22 37.646 4446 29.66.657 0933 29.50.667 686I 29-.35.678 2288 29.22 38.64 66 622 6. 657 2703 29.50.667 8622 29.35.678 4041 29.22 39.646 8005 29.66.657 4472 29-49.668 0383 29.35.678 5794 29.22 40 I.646 9785 29.65 I.657 6242 29.49 i.668 2144 29.35 I.678 7547 29.22 41.647 1564 29.65.657 801I 29.49.668 3904 29.34.678 9300 29.21 42.647 3343 29.65.657 978I 29-49.668 5665 29.34.679 1053 29.21 43.647 5122 29..658 1550 29.48.668 7425 29-34.679 2806 29.21 44.647 6900 29.64.658 3318 29.48.668 9185 29.34.679 4558 29.21 45 1.647 8679 29.64 I.658 5087 29.48 1.669 0945 29.33 I.679 6310 29.20 46.648 0457 29.64.658 6855 29.48.669 2705 29.33.679 8063 29.20 47.648 2235 29.63.658 8624 29.47.669 4465 29.33.679 9815 29.20 48.648 4013 29.63.659 0393 29.47.669 6225 29.33.680 I567 29.20 49.648 5791 29.63.659 2161 29.47.669 7984 29.32.680 3319 29.20 50 1.648 7569 29.63 I.659 3929 29.47 1.669 9744 29.32 1.680 5070 29.19 51.648 9346 29.62.659 5697 29.46.670 1503 29.32.680 6822 29.19 52.649 1123 29.62.659 7465 29.46.670 3262 29.32.680 8574 29.I9 53.649 29o0 29.62.659 9232 29.46.670 502I 29.32.68I 0325 29.19 54.649 4677 29.6.6660 o10 29.46.670 6780 29.3I.68I 2076 29.19 55 1.649 6454 29.6I I.660 2767 29.45 1.670 8539 29.31 I.68I 3827 29.18 56.649 8231 29.61.660 4534 29-45.671 0298 29.31.681 5578 29.I8 57.650 0007 29.61.660 6301 29.45.671 2056 29.31.68I 7329 29.I8 58.650 1784 29.60.660 8068 29-45.671 3814 29-30.681 9080 29.18 59.650 3560 29.60.660 9835 29.44.671 5573 29.30.682 0831 29.18 60 I.650 5336 29.6.661 1601 29.44 i.671 7331 29.30 1.682 2581 29.17 580 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 60~ 61~ 62~ 630 log M. Diff. 1". lobg M. Diff. 1". log M. Diff. 1". log I. Diff. 1". 0' i.682 2581 29.17 1.692 7408 29.07 1.703 i866 28.97 I-713 6006 28.89 1.682 4332 29-I7 692 9I52 29.o06.703 3604 28.97 -713 7739 28.89 2.682 6082 29.17.693 0896 2g.o6.703 5342 28.97 -713 9473 28.89 3.682 7832 29-17.693 2640 29.06.703 7080 28.97 -714 i206 28.88 4.682 9582 29.-7.693 4383 29.o6.703 88I8 28.96.714 2939 28.88 5 i.683 1332 29.I6 1.693 6127 2g.06 1.704 0556 28.96 I.7I4 4672 28.88 6.683 3082 29.16.693 7870 29.05.704 2293 28.96.714 6405 28.88 7.683 4832 29.I6.693 9613 2.905 -704 4031 28.96.714 838 28.888.683 6581 29.16.694 1356 29-05.704 5768 28.96.714 9870 28.88 9.683 8331 29.16.694 3099 29.05.704 7506' 28.96.7I5 1603 28.88 10 I.684 0080 29.i6 1.694 4842 29.05 I-704 9243 28.96 1.715 3336 28.88 11.684 1830 29.15.694 6585 29.04.705 0981 28.95.715 5o68 28.88 12.684 3579 29-15.694 8328 29.04.705 2718 28.95.715 68o0 28.87 13.684 5328 29.15 695 0070 29.04.705 4455 -28.95.715 8533 28.87 14.684 7077 29-15.695 1813 29.04.705 6192 28.95.71 6 0266 28.87 15 1.684 8826 29.14 I.695 3555 29.04 1-705 7929 28.95 1.7I6 1998 28.87 16.685 0574 29.14;695 5298 29.04.705 9666 28.95 -76 3730 28.87 17.685 2323 29.14.695 7040 29.04.706 1402 28.95 -7I6 5462 28.87 18.685 4071 29.14.695 8782 29.03.706 3139 28.94.716 7194 28.87 19.685 5820 29.14.696 0524 29.03.706 4875 28.94.7I6 8926 28.87 20 1.685 7568 29.14 I.696 2266 29.03 I.706 66z2 28.94 1-7I7 o658 28.86 21.685 93I6 29.I3.696 4008 29.03.706 8348 28.94 -717 2390 28.86 22.686 1064 29.13.696 5750 29.03.707 0085 28.94 -717 4I22 28.86 23.686 2812 29.13.696 7491 29.03.707 1821 28.94 -717 5853 28.86 24.686 4560 29.13.696 9233 29.02.707 3557 28.94 -717 7585 28.86 25 I.686 6308 29.13 1.697 0974 29-02 1.707 5293 28.93 1.7I7 9317 28.86 26.686 8055 29.13.697 2716 29.02 -707 7029 28.93.7I8 1048 28.86 27.686 9803 29.12.697 4457 29.02.707 8765 28.93.718 2780 28.86 28.687 1550 29.12.697 6198 29.02.708 0501 28.93.718 4511 28.86 29.687 3297 29.12.697 7939 29.02.708 2237 28.93.718 6242 28.85 30 1.687 5044 29.12 I.697 9680 29.02 1.708 3972 28.93 1.718 7974 28.85 31.687 6791 29.12.698 1421 29.0I.708 5708 28.93.718 9705 28.85 32.687 8538 29.11.698 3162 29.01.708 7444 28.92.719 1436 28.85 33.688 0285 29.11.698 4902 29.01.708 9I79 28.92.719 3167 28.85 34.688 2032 29.11I.698 6643 29.01.709 0914 28.92.7I9 4898 28.85 35 I.688 3778 29.II 1.698 8383 29.01 1.709 2650 28.92 1.719 6629 28.85 36.688 5525 29.II.699 OI24 29.01.709 4385 28.92.7I9 8360 28.85 37.688 7271 29.10.699 1864 29.00.709 6I20 28.92.720 0090 28.85 38.688 9017 29.10.699 3604 29.00.709 7855 28.92.720 1821 28.84 39.689 0764 29.10.699 5345 29.00.709 9590 28.92.720 3552 28.84 40 I.689 2510 29.10 1.699 7085 29.00 I.710 1325 28.91 1.720 5282 28.84 41.689 4256 29.o.699 8824 29.00.710 3060 28.91.720 7013 28.84 42.689 6o00 29.09.700 0564 29.00.710 4794 28.91.720 8743 28.84 43.689 7747 29.09.700 2304 29.00.7o1 6529 28.91.721 0474 28.84 44.689 9493 29.09.700 4044 28.99.710 8263 28.91.721 2204 28.84 45 i.690 1238 29.09 1.700 5783 28.99 1.710 9998 28.91 1.721 3934 28.84 46.690 2984 29.09.700 7523 28.99.711 1732 28.91.721 5665 28.84 47.690 4729 29.09..700 9262 28.99.711 3467 28.90.72I 7395 28.84 48.690 6474 29.09.701 Ioo1 28.99.71 5201 28.90.72I 9125 28.83 49.690 82g9 29.08.701 2741 28.99.711 6935 28.90.722 0855 28.83 50 1.690 9964 29.08 1.701 4480 28.98 1.711 8669 28.90 1.722 2585 28.83 51.691 1709 29.08.70I 62I9 28.98.7I2 0403 28.90.722 43I5 28.83 52.691 3454 29.08.701 7958 28.98.7I2 2137 28.90.722 6044 28.83 53.691 5199 29.08.701 9697 28.98.712 3871 28.90.722 7774 28.83 54.69I 6943 29.o8.702 1435 28.98.712 5605 28.90.722 9504 28.83 55 i.691 8688 29.07 1.702 3174 28.98 1.712 7339 28.90 1.723 1233 28.83 56.692 0432 29.07 -702 4913 28.98.712 9072 28.89 -723 2963 28.83 57.692 2176 29.07.702 665I 28.97 -713 0806 28.89.723 4693 28.82 58.692 3920 29.07 -702 8389 28.97 -713 2539 28.89 -723 6422 28.82 59.692 5664 29.07.703 OI28 28.97'713 4273 28.89'723 815 28.82 60 1.692 7408 29.07 1.703 I866 23.97 1.713 6006 28.89 1.723 9881 28.82 581 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 64~ 65~ 66~ 67~ V. -_ log M. Diff. 1". logM.. Dif1. logM. Diff. I". log M. Diff. 1". 0' 1.723 9881 28.82 1.734 3539 28.77 1.744 7031 28.73 1.755 0405 28.70 1.724 I6Io 28.82.734 5265 28.77 -744 8755 28.73 -755 2127 28.70 2.724 3339 28.82.734 699I 28.77.745 0479 28.73.755 3849 28.70 3.724 5068 28.82.734 8718 28.77.745 2202 28.73 -755 5571 28.70 4 *724 6798 28.82.735 0444 28.77.745 3926 28.73.755 7293 28.70 5 1.724 8527 28.82 1I735- 2I69 28.76 1I745 5650 28.73 1.755 9015 28.70 6.725 0256 28.82.735 3895 28.76.745 7373 28.73.756 0737 28,70 7.725 1984 28.81.735 5621 28.76.745 9097 28.73.756 2459 28.70 8.725 3713 28.81.735 7347 28.76.746 0820 28.72.756 4181 28.70 9 -725 5442 28.81.735 9073 28-76.746 2544 28.72.756 5903 28.70 10 1.725 7171 28.81 1.736 0798 28.76 1.746 4267 28.72 1.756 7625 28.70 11.725 8900 28.81.736 2524 28.76.746 5991 28.72.756 9347 28.70 12.726 0628 28.8I.736 4250 28.76.746 7714 28.72.757 1069 28.70 13.726 2357 28.81.736 5975 28.76.746 9437 28.72.757 2791 28.70 14.726 4085 28.81.736 7701 28.76.747 1161 28.72.757 45I3 28.70 15 1.726 5814 28.8I 1.736 9426 28.76 1.747 2884 28.72 1I757 6235 28.70 16.726 7542 28.81.737 1152 28.76.747 4607 28.72.757 7957 28.70 17.726 9270 28.81.737 2877 28.76 *747 6330 28.72.757 9679 28.70 18.727 0999 28.80.737 4602 2876 747 805 28.7 72.758 I40I 28.70 19.727 2727 28.80.737 6328 28.75 -747 9777 28.72.758 3123 28.70 20 1.727 4455 28. 778053 28.75 1.748 1500 28.72 1.758 4844 28.70 21.727 6183 28.80.737 9778 28.75.748 3223 28.72.758 6566 28.70 22.727 791I 28.80.738 1503 28.75.748 4946 28.72.758 8288 28.70 23.727 9639 28.80.738 3228 2875.728 6669 2 8.7 759 2870 24.728 1367 28.80.738 4953 28.75.748 8392 28.72.759 1731 28.70 25 1.728 3095 28.80 1.738 6679 28.75 1.749 OII5 28.72 1.759 3453 28.70 26.728 4823 28.80.738 8404 28.75.749 1838 28.72.759 5I75 28.70 27.728 6551 28.80.739 0129 28.75.749 356I 28.72.759 6897 28.70 28.728 8279 28.80.739 1853 28.75.749 5284 28.72.759 8618 28.69 29.729 ooo6 28.79.739 3578 28.75.749 7007 28.71.760 0340 28.69 30 1.729 1734 28.79 1-739 5303 28.75 1.749 8730 2871 76 26 28.69 31.729 346I 28.79.739 7028 28.75.750 0453 28.71.760 3783 28.69 32.729 5I89 28.79.739 8753 28.75 -750 2176 28.71.760 5505 28.69 33.729 6916 28.79.740 0477 28.75 -750 3898 28.71.760 7227 28.69 34.729 8644 2879.79 40 2202 28.74.750 562I 28.71 -760 8948 28.69 35 1.730 0371 28.79 1.740 3927 28.74 1-750 7344 28.71 1.761 0670. 28.69 36.730 2099 28.79.740 5651 28.74.750 9067 28.71 -76I 2392 28.69 37.730 3826 28.79.740 7376 28.74.751 0789 28.71.761 4113 28.69 38.730 5553 28.79.740 9101 28.74.75I 2512 28.71.761 5835 28.69 39 -730 7280 28.79 -741 o825 28.74.751 4234 28.71.76I 7556 28.69 40 1.730 9007 28.78 I.741 2550 28.74 1.751 5957 28.7I 1.76I 9278 28.69 41.731 0735 28.78.74I 4274 287 75 76 28.74 7 760 0999 28.69 42.731 2462 2878.7741 5998 28.74 -75I 9402 28.71.762 2721 28.69 43.73I 4189 28.78.74I 7723 28.74.752 II25 28.71.762 4442 28.69 44.73I 5915 28.78.74I 9447 28.74 -752 2847 28.71.762 6I64 28.69 45 1.731 7642 28.78 1.742 II71 28.74 I.752 4570 28.71 1.762 7885 28.69 46.73I 9369 78 28.78 742 289 2 4.752 6292 28.71.762 9607 28.69 47.732 1096 28.78.742 4620 28.74 -752 801.5 28.71.763 I328 28.69 48.732 2823 28.78.742 6344 28.74 -752 9737 828.71.763 3050 28.69 49.732 4549 28.78.742 8068 28.74 -753 1460 28.71.763 477I 28.69 50 1.732 6276 28.78 1.742 9792 28.74 I-753 3I82 28.71 1.763 6493 28.69 51.732 8002 28.78.743 6 2873 753 4904 763 8214 28.69 52.732 9729 28.77 -743 3240 28.73 -753 6627 28.71.763 9936 28.69 53.733 7455 28.77.743 4964 28.73.753 34 28-.71 764 1657 28.69 54.733 3182 28.77.743 6688 28.73 -754 0071 28.70.764 3379 28.69 55 1.733 4908 28.77 1743 8412 28.73 1-754 1794 28.70 1.764 5I00 28.69 56 ~733 6635 28.77 -744 OI36 28.73.754 35I6 28.70.764 6821 28.69 57' 733 836I 28.77 744 I860 28.73 -754 5238 28.70.764 8543 28.69 58.734 0087 28.77' -744 3584 28.73 -754 6960 28.70.765 0264 28.69 59.734 1813 28.77 -744 5308 28.73.754 8682 28.70.765 1985 28.69 60 1.734 3539 28.77 1.744 7031 28.73 1.755 0405 28.70 1.765 3707 28.69 582 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. _ __..~~~. I log M. Diff. ". log M. Diff. 1". Diff. 1". I log. Diff. 1 0' 1.765 3707 28.69 I.775 6985 28.69 1.786 0284 28.70 1.796 3650 28.73.765 5428 28.69.775 8706 28.69.786 2006 28.70.796 5374 28.73 2.765 7150 28.69.776 0427 28.69.786 3728 28.70.796 7097 28.73 3.765 8871 28.69.776 2149 28.69.786 5450 28.70.796 882I 28.73 4.766 0592 28.69.776 3870 28.69.786 7172 28.70.797 0545 28-73 5 1.766 2314 28.69 1.776 559I 28.69 I.786 8894 28.70 1.797 2268 28.73 6.766 4035 28.69.776 7313 28.69.787 0617 28.70.797 3992 28.73 7.766 5756 28.69.776 9034 28.69 -787 2339 28.70.797 57I6 28.73 8.766 7478 28.69.777 0755 28.69 -787 4061 28.70.797 7440 28.73 9.766 9199 28.69.777 2477 28.69.787 5783 28.70.797 9164 28.73 10 1767 0920 28.69 1.777 4198 28.69 1.787 7506 28.70 1.798 0888 28.73 11.767 2642 28.69 -777 5920 28.69 -787 9228 28.71.798 26II 28.73 12 767 4363 28.69 -777 7641 28.69.788 0950 28.71.798 4335 28.73 13.767 6084 28.69.777 9363 28.69.788 2673 28.71.798 6060 28.73 14.767 7805 28.69.778 1084 28.69.788 4395 28.71.798 7784 28.73 15 1.767 9527 28.69.1778 2806 28.69 1.788 61I7 28.71 1.798 9508 28.73 16.768 1248 28.69.778 4527 2869 9 7887 7840 28.71.799 I232 28.74 17.768 2969 28.69.778 6248 28.69.788 9562 28.7I.799 2956 28.74 18.768 4691 28.69.778 7970 28.69.789 I284 28.71.799 4680 28.74 19.768 6412 28.69.778 969I 28.69 -789 3007 28.71 -799 6404 28.74 20 1.768 8133 28.69 1.779 I413 28.69 1.789 4730 28.71 1.799 8I28 28.74 21.768 9854 28.69.779 340. 28.69.789 6452 28.71.799 9853 28.74 22.769 1576 28.69.779 4862 28.69.789 8I75 28.71.800 1577 28.74 23 -769 3297 28.69.779 6578 28.69.789 9897 28.71.800oo 3301 28.74 24'769 5018 28.69.779 8299 28.69.790 I620 28.71.800 5026 28.74 25 1.769 6740 28.69 1.780 002I 28.69 1.790 3342 28.7I 1.800 6750 28.74 26.769 8461 28.69.780 1742 28.69.790 5065 28.7I.800 8475 28.74 27.770 0182 28.69.780 3464 28.69.790 6788 28.7I.80I 0199 28.74 28.770 I903 28.69.780 5185 28.69.790 8510 28.71.80I 1924 28.74 29.770 3625 28.69.780 6907 28.69 -791 0233 28.71.801 3648 28.74 30 1.770 5346 28.69 1.780 8629 28.69 I.79I 0956 28.71 1.801 5373 28.74 31.770 7067 28.69.781 0350 28.69.791 3678 28.71.801 7I07 28.74 32.770 8788 28.69.78I 2072 28.69.79I 5401 28.71.801 8822 28.74 33.771 050o 28.69.781 3793 28.69 -791 7I24 28.71.802 0547 28.75 34.771 223I 28.69.781 5515 28.69.791 8847 28.71.802 2271 28.75 35 1.771 3952 28.69 1.781 7237 28.69 1.792 0570 28.71 1.802 3996 28.75 36.77I 5673 28.69.781 8959 28.69.792 2293 28.71.802 5721 28-75 37.771 7395 28.69.782 o680 28.70.792 4016 28.72.802 7446 28.75 38.77I 9II6 28.69.78 2402 28.70.792 5738 28.72.802 9171 28.75 39.772 0837 28.69.782 4124 28.70.792 746I 28.72.803 0896 28.75 40 1.772 2559 28.69 1.782 5845 28.70 1.792 9184 28.72 1.803 2621 28.75 41.772 4280 28.69.782 7567 28.70.793 0907 28.72.803 4346 28.75 42.772 600oo 28.69.782 9289 2870.7 93 2630 2.8.72.803 6071 28.75 43.772 7722 28.69 -783 IOI 28.70.793 4354 28-72.803 7796 28.75 44.772 9444 28.69.783 2732 28.70.793 6077 28.72.803 9521 28.75 45 1.773 II65 28.69 I.783 4454 28.70 1.793 7800 28.72 1.804 1246 28.75 46.773 2886 28.69 -783 6176 28.70.793 9523 28.72.804 2971 28.75 47.773 4607 28.69.783 7898 28.70.794 1246 28.72.804 4697 28.75 48.773 6329 28.69.783 9620 28.70.794 2969 28.72.804 6422 28.76 49.773 8050 28.69.784 1342 28.70.794 4693 28.72.804 8147 28.76 50 1.773 9771 28.69 1.784 3064 28.70 1.794 6416 28.72 1.804 9873 28.76 51.774 I493 28.69.784 4786 28.70.794 8139 28.72.805 1598 28.76 52.774 3214 28.69.784 6508 28.70.794 9862 28.72.805 3324 28.76 53 -774 4935 28.69.784 8230 28.70.795 I586 28.72.8o5 5049 28.76 54 -774 6657 28.69.784 9952 28.70.795 3309 28.72.805 6775 28.76 55 1.774 8378 28.69 1.785 1674 28.70 1.795 5033 28.72 I.8o0 8500 28.76 56.775 0099 28.69.785 3396 28.70.795 6756 28.72.806 0226 28.76 57 -775 182 28.69.785 51I8 28.70.795 8480 28.72.806 I952 28.76 58.775 3542 28.69.785 6840 28.70.796 0203 28.73.806 3677 28.76 59.775 5263 28.69.785 8562 28.70.796 1927 28.73.8o06 5403 28.76 60 1.775 6985 28.69 1.786 0284 28.70 1.796 3650 28.73 1.8o6 7129 28.76 583 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 72 73~ 74~ 75~ V. log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. i Diff. 1". 0' i.806 7129 28.76 1.817 0765 28.81 1.827 4602 28.88 1.837 8686 28.95 1.806 8855 28.76.817 2494 28.8I.827 6335 28.88.838 0423 28.95 2.807 058I 28.77 -817 4222 28.82.827 8068 28.88.838 2160 28.95 3.807 2307 28.77.817 595I 28.82.827 9800 28.88.838 3898 28.95 4.807 4033 28.77.817 7680 28.82.828 1533 28.88.838 5635 28.96 5 1.807 5759 28.77 1.817 9410 28.82 1.828 3266 28.88 1.838 7372 28.96 6.807 7485 28.77.8I8 1139 28.82.828 4999 28.88.838 9II1 28.96 7.807 921I 28.77.8i8 2868 28.82.828 6732 28.88.839 0847 28.96 8.808 0937 28.77.818 4597 28.82.828 8465 28.88.839 2585 28.96 9.8o8 2663 28.77.818 6326 28.82.829 0198 28.89.839 4323 28.96 10 1.8o8 4389 28.77 I.818 8056 28.82 1.829 I931 28.89 1.839 6o60 28.96 11.808 6II6 28.77.818 9785 28.82.829 3665 28.89.839 7798 28.97 12.808 7842 28.77.819 1515 28.83..829 5398 28.89.839 9536 28.97 13.808 9568 28.77.819 3244 28.83.829 713I 28.89.8 o 1274 28.97 14.809 1295 28.77.819 4974 28.83.829 8865 28.89.840 3012 28.97 15 I.809 3021 28.78 1.819 6704 28.83 1.830 0599 28.89 1.840 4751 28.97 16.809 4748 28.78.819 8433 28.83.830 2332 28.89.840 6489 28.97 17.809 6474 28.78.820 0163 28.83.830 4066 28.90.840 8227 28.97 18.809 820I 28.78.820 1893 28.83.830 5800 28.90.840 9966 28.97 19.809 9928 28.78.820 3623 28..83 3 7533 28.90.84I I704 28.98 20 I.8Io i655 28.78 1.820 5353 28.83 1.830 9267 28.90 1-841 3443 28.98 21.810 3381 28.78.820 7083 28.83.83I IOOI 28.90.84I 5182 28.98 22.810 5108 28.78.820 88I3 28.84.831 2735 28.90.84I 6921 28.98 23.8Io 6835 28.78.821 0543 28.84.83I 4470 28.90.84I 8659 28.98 24.810 8562 28.78.821 2273 28.84.83I 6204 28.90.842 0398 28.98 25 I.81I 0289 28.78 1.821 4003 28.84 1.83I 7938 28.9I 1.842 2138 28.98 26.811 2016 28.78.821 5734 28.84.831 9672 28.91.842 3877 28.99 27.811 3743 28.78.82I 7464 28.84.832 I407 28.9I.842 5616 28.99 28.811 5470 28.79.821 9194 28.84.832 3141 28.9I.842 7355 28.99 29.8 1 7197 28.79.822 0925 28.84.832 4876 28.91.842 9095 28.99 30 1.811 8924 28.79 I.822 2656 28.84 1.832 6611 28.91 1.843 0834 28.99 31.812 0652 28.79.822 4386 28.84.832 8345 28.91.843 2574 28.99 32.812 2379 28.79.822 617 28.85.833 0080 28.92.843 4313 29.00 33.812 4I06 28.79.822 7848 28.85.833 1815 28.92.843 6053 29.00 34.8I2 5834 28.79.822 9578 28.85.833 3550 28.92.843 7793 29.00 35 I.812 7561 28.79 I.823 1309 28.85 1.833 5285 28.92 1.843 9533 29.00 36.812 9289 28.79.823 3040 28.85.833 7020 28.92.844 1273 29.00 37.813 io16 28.79'823 4771 28.85.833 8755 28.92.844 3013 29.00 38.8I3 2744 28.79.823 6502 28.85.834 049I 28.92.844 4753 29.00 39.8I3 4472 28.79.823 8233 28.85.834 2226 28.92.844 6494 29.01 40 1.813 6I99 28.80 1.823 9965 28.85 I.834 3961 28.92 1.844 8234 29.01 41.813 7927 28.80.824 1696 28.85.834 5697 28.93 -844 9974 29.01 42.8I3 9655 28.80.824 3427 28.86.834 7432 28.93.845 1715 29.01 43.8I4 1383 28.80.824 5159 28.86.834 9168 28.93.845 3456 29.01 44.814 3III 28.80.824 6890 28.86.835 0904 28.93.845 5196 29.01 45 I.814 4839 28.80 1.824 8622 28.86 1.835 2640 28.93 1.845 6937 29.01 46.814 6567 28.80.825 0353 28.86.835 4376 28.93.845 8678 29.02 47.814 8295 28.80.825 2085 28.86.835 612i 28.93.846 0419 29.02 48.8I5 0023 28.80.825 38I6 28.86.835 7848 28.93.846 2160 29.02 49.8I5 1751 28.80.825 5548 28.86.835 9584 28.94.846 3901 29.02 50 1.815 3479 28.80 1.825 7280 28.86 1.836 1320 28.94 1.846 5643 29.02 51.8I5 5208 28.81.825 9012 28.87.836 3056 28.94.846 7384 29.02 52.8I5 6936 28.81.826 0744 28.87.836 4792 28.94.846 9125 29.03 53.8I5 8664 28.81.826 2476 28.87 -836 6529 28.94.847 0867 29.03 54.8I6 0393 28.81.826 4208 28.87.836 8265 28.94.847 2609 29.03 55 I.8i6 2121 28.81 1.826 5940 28.87 I.837 0002 28.94 1.847 4350 29.03 56.8I6 3850 28.81.826 7673 28-87 -837 I739 28.95.847 6092 29.03 57.8I6 5578 28.81.826 9405 28.87.837 3475 28.95.847 7834. 29-03 58.8i6 7307 28.81.827 II37 28.87.837 5212 28.95.847 9576 29.03 59.8i6 9036 28.81.827 2870 28.87.837 6949 28.95.848 1318 29.04 60 1.817 0765 28.81 1.827 4602 28.88 1.837 8686 28.95 1.848 3060 29.04 584 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 760 __ 77~ 78~ 79~ V, V. - I.. log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 1.848 3060 29.04 1.858 7769 29.I4 1.869 2857 29.25 I.879 8369 29-37 1.848 4803 29.04.858 9517 29.14.869 4612 29.25.880 0I3I 29-37 2.848 6545 29.04.859 1266 29.14.869 6367 29.25.880 I894 29.38 3.848 8287 29.04.859 3014 29.14.869 8122 29.25.880 3656 29.38 4.849 0030 29.04.859 4763 29.15.869 9878 29.26.880 5419 29.38 5 1.849 I773 29.04 I.859 6512 29.15 1.870 1633 29.26 1.880 7182 29.38 6.849 3515 29.05.859 8260 29.15.870 3389 29.26.880 8945 29.38 7.849 5258 29.05.860 0009 29.15.870 5144 29.26.881 0708 29.39 8.849 7001 29.05.86o i758 29.I5.870 6900 29.26.881 2471 29-39 9.849 8744 29.05.860 3507 29-15.870 8656 29.26.881 4235 29-39 10 1.850 0487 29.05 1.860 5256 29.15 1.871 0412 29.27 1.88I 5998 29.39 11.850 2231 29.05.860 7006 29.16.871 2168 29.27.88I 7762 29.39 12.850 3974 29.06.860 8755 29.I6.87I 3924 29.27.88i 9526 29.40 13.850 5717 29.06.861 0505 29.16.87I 5681 29.27.882 1290 29.40 14.850 7461 29.06.86I 2254 29.16.871 7437 29.28.882 3054 29.40 15 1.850 9204 29.06 i.861 4004 29.I6 1.871 9194 29.28 1.882 4818 29.40 16.85I 0948 29.06.861 5754 29.16.872 0950 29.28.882 6582 29.41 17.85I 2692 29.06.861 7504 29.I7.872 2707 29.28.882 8347 29.4I 18.85I 4436 29.07.861 9254 29.17 -872 4464 29.28.883 OI12 29.41 19.85I 6i80 29.07.862 o004 29.17.872 6221 29.29.883 1876 29.41 20 1.85I 7924 29.07 I.862 2754 29.I7 1.872 7979 29.29 1.883 3641 29-42 21.85I 9668 29.07.862 4505 29.17.872 9736 29.29 -883 5406 29.42 22.852 1412 29.07.862 6255 29.18.873 1493 29.29.883 7171 29.42 23.852 3157 29.07.862 8006 29.18.873 325I 29.29.883 8937 29.42 24.852 4901 29.07.862 9756 29.I8.873 5008 29.30.884 0702 29.42 25 1.852 6646 29.08 1.863 1507 29.18 I-873 6766 29.30 1.884 2468 29-43 26.852 8391 29.08.863 3258 29.18.873 8524 29.30.884 4233 29-43 27.853 0135 29.08.863 5009 29.18.874 0282 29.30.884 5999 29-43 28.853 I880 29.08.863 6760 29.I9.874 2041 29.30.884 7765 29-43 29.853 3625 29.08.863 8512 29.19.874 3799 29.31.884 9531 29-44 30 1.853 5370 29.09 1.864 0263 29-19 1-874 5557 29.31 -.885 1297 29-44, 31.853 7115 29.09.864 2015 29.I9.874 7316 29.31.885 3064.29-44 32.853 886I 29.09.864 3766 29.I9.874 9074 29.3I.885 4830 29.44 33.854 0606 29.09.864 55I8 29.20.875 0833 29.31.885 6597 29-45 34.854 2351 29.09.864 7270 29.20.875 2592 29.32.885 8364 29.45 35 1.854 4097 29.09 1.864 9022 29.20 1-875 4351 29.32 1.886 013I 29.45 36.854 5843 29.Io.865 0774 29.20.875 6II 29.32.886 1898 29.45 37.854 7588 29.10.865 2526 29.20.875 7870 29.32.886 3665 29-45 38 -854 9334 29.IO.865 4278 29.20.875 9629 29.32.886 5432 29.46 39.855 o108 29.10.865 6030 29.21.876 1389 29-33.886 7200 29.46 40 1-855 2826 29.10 1.865 7783 29.2I 1.876 3148 29.33 i.886 8967 29.46 41.855 4572 29.IO.865 9536 29.2I.876 4908 29-33.887 0735 29.46 42.855 6319 29.II.866 I288 29.21.876 6668 29.33.887 2503 29-47 43.855 8065 29.11.866 3041 29.21.876 8428 29-33.887 4271 29-47 44 -855 9811 29.11.866 4794 29.22.877 OI88 29.34.887 6039 29.47 45 i.856 1558 29.II 1.866 6547 29.22 1.877 1949 29-34 1.887 7807 29-47 46.856 3305 29.11.866 8301 29.22.877 3709 29.34.887 9576 29.48 47.856 5052 29.I1.867 0054 29.22.877 5470 29.34.888 1344 29.48 48.856 6799 29.12.867 807 29.22.877 7230 29.34.888 3113 29.48 49.856 8546 29.12.867 3561 29.23.877 8991 29-35.888 4882 29.48 50 1.857 0293 29.12 1.867 5314 29.23 1.878 0752 29.35 i.888 6651 29.48 51.857 2040 29.12.867 7068 29.23.878 2513 29.35.888 8420 29.49 52.857 3787 29.I2.867 8822 29.23.878 4275 29.35.889 0189 29.49 53 -857 5534 29.I2.868 0576 29.23.878 6036 29.35.889 1959 29.49 54 -857 7282 29.13.868 2330 29.24.878 7797 29-36.889 3728 29-49 55 1-857 9030 29.I3 I.868 4084 29.24 1.878 9559 29.36 1.889 5498 29.49 56.858 0777 291.3.868 5839 29.24 -879 1321 29.36.889 72'68 29.50 57.858 2525 29-13.868 7593 29.24 -879 3082 29.36.889 8038 29.50 58.858 4273 29-13.868 9348 29.24.879 4844 29.36.890 0808 29.50 59.858 6o02 29-13.869 1102 29.25.879 6606 29.37.890 2578 29.5I 60.1858 7769 29.14 14-.869 2857 29.25 1.879 8369 29-37 1.890 4349 29.51 585 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. r 800 81 820 830 i _ 81~- _3~ 83o. log M. Diff. 1". log M. Diff. 1". log MI. Diff. 1". log M. Diff. 1".' 1.890 4349 29.5I 1.901 084I 29.66 I.9II 7893 29.82 1.922 5548 29.99 1.890 6II9 29.51.901 2621 29.66.9II 9682 29.82 922 7347. 29-99 2.890 7890 29.51.90I 4400 29.66.912 I471 29.82.922 9147 30.00 3.89o 9661 29:5I.g9o 6i80 29.66.912 3261 29.83.923 0947 30-00 4.891 1432 29.52.90I 7960 29.67.912 5050 29.83.923 2747 30.00 5 I.891 3203 29.52 I.901 9740 29.67 I.9I2 6840 29.83 I.923 4548 30.o0 6.891 4974 29.52.902 1521 29.67.9I2 8630 29.84.923 6348 30.01 7.891 6745 29.52.902 3301 29.67.9I3 0420 29.84.923 8I49 30.o0 8.891 8517 29.53.902 5o82 29.68.913 2211 29.84 -923 9950 30.02 9.892 0289 29.53.902 6862 29.68.913 400I 29-84.924 I751 30.02 10 i.8sz 2061 29.53 I.902 8643 29.68 1.913 5792 29.85 1.924 3552 30.02 11.892 3833 29.53.903'0424 29.69.913 7583 29.85 -924 5354 30-03 12.892 5605 29.54.903 2105 29.69.913' 9374 29.85.924 71I5 30.03 13.892 7377 29-54.903 3987 29.69.914 II65 29.85.924 8957 30-03 14.892 9149 29-54 -903 5768 29.69 91I4 2956 29.86.925 0759 30.03 15 1.893 0922 29.54 1-903 7550 29.70 I.94 4748 29.86 1.925 2561 30.04 16.893 2695 29.55 -903 9332 29.70.914 6540 29.86.925 4364 30-04 17.893 4467 29.55.904 1114 29.70.914 8331 29.87.925 6i66 30.04 18.893 6240 29-55.904 2896 29.70.9I5 0124 29.87.925 7969 30-05 19.893 8013 29.55 904 4678 29-71.915 I1916 29.87.925 9772 30.05 20 1.893 9787 29.56 1.904 6461 29.71 1.915 3708 29.87 1.926 1575 30-05 21.894 I56o 29.56.904 8243 29.71.915 5501 29.88.926 3378 30.06 22.894 3334 29.56.905 0026 29.7I.915 7294 29.88.926 5182 30.06 23.894 5o18 29.56.905 I809 29.72.915 9087 29.88.926 6986 30.06 24.894 6882 29-57'905 3592 29.72.916 o880 29.89.926 8789 30.07 25 1.894 8656 29.57 1.905 5376 29.72 I.916 2673 29.89 1.927 0593 30-07 26.895 0430 29.57.905 7159 29.73.9I6 4466 29.89.927 2398 30.07 27.895 2204 29-57.905 8943 29-73.916 6260 29.90.927 4202 30.08 28.895 3979 29.58.906 0726 29.73.916 8054 29.90.927 6007 30.08 29.895 5753 29.58.906 2510 29.73'916 9848 29.90.927 7811 30.08 30 1.895 7528 29.58 1.906 4294 29-74 1.917 1642 29.90.927 9616 30.08 31.895 9303 29-58.906 6079 29-74 -917 3436 29.91 I 928 1422 30.09 32.896 1078 29.59.906 7863 29.74'917 5231 29.91.928 3227 30.09 33.896 2854 29-59.906 9648 29.74.9I7 7025 29.91.928 5032 30.09 34.896 4628 29.59.907 I432 29-75.917 8820 29.92.928 6838 30.10 35 I.896 6404 29.59 1.907 3217 29-75 1.918,06I5 29.92 I.928 8644 300IO 36.896 8i80 29.60.907 5002 29-75 -918 24IO 29.92.929 0450 30.10 37.896 9955 29.60.907 6787 29-75.918 4206 29.92.929 2256 30.11 38.897 1732 29.60.907 8573 29.76.918 6o00 29-93.929 4063 30.I I 39.897 3508 29.60.908 0358 29.76.918 7797 29.93.929 5869 30. 11 40 r.897 5284 29.61 I.908 2144 29.76 1.918 9593 29-93 1.929 7676 30.12 41.897 7060 29.6I.908 3930 29.77.9I9 I389 29-94.929 9483 30.I2 42.897 8837 29.61.908 57I6 29.77 -919 3185 29-94.930 z19I 30.12 43.898 0614 29.61.908 7502 29.77 -919 4982 29.94.930 3098 30.13 44.898 2390 29.62.908 9288 29.77 i919 6778 29-94.930 4906 30.I3 45 1.898 4168 29.62 1.909 1075 29.78 I.919 8575 29-95 1.930 6713 30.13 46.898 5945 29.62.909 2862 29.78.920 0372 29.95 -930 8521 30.I3 47.898 7722 29.62.909 4648.29-78.920 2169 29.95 -931 0330 30.I4 48.898 9500 29.63.909 6436 29.78.920 3966 29.96.931 2138 30-I4 49.899 I277 29.63.909 8223 29-79.920 5764 29.96.931 3946 30.14 50 1.899 3055 29.63 1.9IO OOIO 29.79 1.920 7561 29.96 1.931 5755 30.I5 51.899 4833 29.63.9I0 I798 29.79.920 9359 29.97.93I 7564 30.I5 52.899 6611 29.64.910 3585 29.80.92I 1157 29.97.93I 9373 30.I5 53.899 8389 29.64 1.9I 5373 29.80.921 2956 29.97 -932 1183 30.16 54 1.900 O68 - 29.64.910 7161 29.80.92I 4754 29.98.932 2992 30.16 55 1.900 1946 29.64 I.9I0 8949 29.80 I.921 6552 29.98 I.932 4802 30.16 56.900 3725 29.65 -9II 0738 29.81.92I 8351 29.98.932 6612 30.17 57.900 5504 29.65.'91 2526 29.81.922 0150 29.98.932 8422 30.17 58.900 7283 29.65.9I1 4315 29.81.922 1949 29.99.933 0232 30.17 59.900 9062 29.66.9II 6104 29.82.922 3748 29.99 -933 2043 30.18 60 1.90I 0841 29.66 1.91 1 7893 29.82 1.922 5548 29.99 I-933 3853 30.I 8 586 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 84~ 850 86~ 870~ log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". O' -1933 3853 30.'8 I.944 2856 30.38 I-955 2602 30.59 1.966 3I40 30.82.933 5664- 30.18.944 4678 30.38.955 4438 30.60.966 4990 30.82 2.933 7475 30.I9.944 6502 30-39.955 6274 30.60.966 6839 30.83 3.933 9287 30-.9.944 8325 30-39.955 8IIO 30.60.966 8689 30.83 4.934 1098 30.I9.945 0148 30.39.955 9946 30.6I.967 0539 30.84 5 1.934 2910 30.20 1.945 I972 30-40 1.956 I783 30.6I 1.967 2389 30-84 6.934 4722 30.20.945 3796 30:40.956 3619 30.6I 967 4240 30.84 7.934 6533 30.20.945 5620 30.40.956 5456 30.62.967 6090 30.85 8.934 8346 30.21.945 7444 30.41.956 7294 30.62.967 794I 30.85 9 -935 OI58 30.21.945 9269 30.41.956 9131 30.63.967 9792 30.85 10 1.935 1971 30.21 1.946 1094 30.41 1.957 0969 30.63 1.968 I644 30.86 II.935 3784 30.22.946 2919 30.42.957 2807 30.63.968 3496 30.86 12 935 5597 30.22.946 4744 30.42.957 4645 30.64.968 5347 30.87 13.935 7410 30.22.946 6569 30.42.957 6483 30.64.968 7200 30.87 14.935 9223 30.22.946 8395 30-43.957 8322 30.64.968 9052 30.87 15 1.936 1037 30.23 1.947 0221 30.43 1.958 oi60 30.65 1.969 0905 30.88 16.936 2851 30.23.947 2047 30-44.958 I999 30.65.969 2757 30.88 17.936 4665 30.23.947 3873 30.44.958 3839 30.66.969 46I 30.89 18.936 6479 30.24 *947 5699 30.44.958 5678 30.66.969 6464 30.89 19.936 8293 30.24'947 7526 30.45.958 75I8 30.66.969 8317 30.89 20 1.937 OI08 30.24 I-947 9353 30.45 1.958 9358 30.67 1.970 OI7I 30.90 21.937 I922 30.25.948 II80 30.45.959 1198 30.67.970 2025 30.90 22.937 3737 30.25.948 3007 30.46.959 3038 30.67.970 3879 30.91 23.937 5553 30.25.948 4834 30.46.959 4879 30.68.970 5734 30.9I 24 -937 7368 30.26.948 6662 30.46.959 6720 30.68.970 7589 30.91 25 1.937 9184 30.26 1.948 8490 30-47 I.959 8561 30.69 1.970 9443 30.92 26.938 0999 30.26.949 0318 30.47.960 0402 30.69.971 I299 30.92 27.938 2815 30.27.949 2146 30.47.960 2243 30.69.971 3154 30-93 28.938 4632 30.27.949 3975 30.48.960 4085 30.70.97I 5010 30.93 29.938 6448 30.27.949 5804 30.48.960 5927 30.70.971 6866 30.93 30 1.938 8264 30.28 1.949 7633 30.48 I.960 7769 30.70 1.97I 8722 30.94 31.939 9462 30.849.96 9462 349.9 30.71.972 0578 30.94 32.939 1898 30.28.950 1291 30.49.96I 1454 30.71.972 2435 30-95 33.939 3715 30.29.950 3121 30.50.961 3297 30.71.972 4292 30.95 34.939 5533 30-29'950 4951 30.50.961 5140 30.72.972 6149 30.95 35 1.939 7350 30.29 I.950 6781 30.50 1.96I 6983 30.72 1.972 8006 30.96 36.939 9168 30.30.950 86II 30.5I.961 8827 30.73.972 9864 30.96 37.940 o986 30.30.951 0441 30.51.962 0671 30.73.973 I722 30.97 38.940 2804 30.30.951 2272 30.51.962 25I5 30.73'973 3580 30.97 39.940 4623 30.31.951 4103 30.52.962 4359 30.74 -973 5438 30-97 40 1.940 6441 30-31 1.951 5934 30.52 I.962 6203 30.74 1.973 7297 30.98 41.940 8260 30.31.951 7766 30.52.962 8048 30.75 -973 9156 30.98 42.941 0079 30.32.951 9597 30.53.962 9893 30.75.974 1015 30.99 43.94I 1898 30.32.952 1429 30.53.963 1738 30.75.974 2874 30-99 44.94I 3717 30.32.952 3261 30.53.963 3583 30.76.974 4734 30.99 45 I.94I 5537 30.33 1.952 5093 30.54 I.963 5429 30-76 1.974 6593 3I.00 46.941 7357 30-33.952 6925 30.54.963 7275 30-77.974 8454 31.00 47.941 9177 30.34.952 8758 30.55.963 9I2I 30-77.975 0314 31.01 48.942 0997 30.34 -953 059I 30.55.964 0967 30-77.975 2174 31.01 49.942 2817 30-34'953 2424 30.55.964 2814 30.78.975 4035 31I01 50 I.942 4638 30.35 I.953 4257 30.56' 1.964 4660 30.78 1.975 5896 31.02 51 -942 6459 30.35.953 609o 30.56.964 6507 30-78.975 7757 31.02 52 -942 8280 30.35 -953 7924 30.56.964 8354 30.79.975 9619 31.03 53.943 OIO1 30.36.953 9758 30.57.965 0202 30.79.976 148I 31.03 54.943 1923 30.36.954 I592 30.57.965 2050 30.80.976 3343 31.04 55 1.943 3744 30-36.954 34-27 30.57 I-965 3897 30-80 1.976 5205 31-04 56.943 5566 30.37.954 5262 30.58.965 5746 30.80.976 7067 31.04 57.943 7388 30.37.954 7096 30.58.965 7594 30-81.976 8930 31.05 58.943 9211 30.37.954 8931 30.59.965 9442 30-81.977 0793 31-05 59.944 1033 30.38 955 0766 30.59.966 I29I 30.81 *977 2656 31.06 60 1.944 2856 30.38 1.955 2602 30.59 1.966 3140 30.82 1.977 4520 31.06 587 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 880 89~ 900 91~ V. i log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1. 0' -1977 4520 3I.06 1.988 6789 31.31 2.000 0000 31-58 2.011 4203 31.87.977 6383 3I.06.988 8668 31.32.000 1895 31.59.0II 6115 31.87 2.977 8247 3I107.989 0548 31.32.000 3790 31.59.oiI 8027 31.88 3.978 oII2 31.07.989 2427 3I.33.000 5686 31.60.OII 9940 31.88 4.978 I976 31.08.989 4307 31.33.000 7582 31.60.012 I853 31.89 5 1.978 3841 31.08 1.989 6187 31.34 2.000 9478 31.60 2.012 3766 31.89 6.978 5706 3I.08.989 8067 31.34.ooI I375 31.61.OI2 5680 31.89 7.978 7571 3I.09.989 9948 31.34.001 3272 31.61.OI2 7594 31.90 8.978 9436 3i.o9.990 1829 3.135.OOI 5169 31.62.O12 9508 31.9o 9.979 1302 31.10.990 37IO 31.35.OOI 7066 31.62.013 1422 31.91 10 1.979 3168 3I.Io 1.990 5591 31.36 2.001 8963 3t.63 2.013 3337 31.9I 11.979 5034 31.11.990 7473 31.36.002 0861 31.63.oI3 5252 31.92 12.979 6901 31.11.990 9355 31-37.002 2759 31.64 I013 7167 31-92 13.979 8768 31.11.991 I237 31.37.002 4658 31.64.013 9083 31.93 14.980 0635 31.12.991 3119 31I38.002 6557 31.65.014 0999 31.93 15 1.980 2502 31-12 1.991 5002 31.38 2.002 8456 31.65 2.014 2915 31.94 16.980 4369 3I.I3.991 6885 31.38.003 0355 3I.66.14 4831 3I194 17.980 6237 3I.I3.991 8768 31.39.003 2254 3I.66.014 6748 31.95 18.980 8Io5 31.I3 -992 0651 31.39.003 4154 31.67.o04 8665 31.95 19.980 9973 31'I4 -992 2535 31.40.003 6054 31.67 015 0582 31.96 20 1.98I 1842 31.14 1.992 4419 31.40 2.003 7955 31.68 2.015 2500 31.96 21.981 3710 31.15.992 6304 31.4I.003 9855 3I.68.015 4418 31.97 22.981 5579 31.I5.992 8188 31.41.004 1756 31.68.015 6336 31.97 23.981 7449 3.-16.993 0073 31.42.004 3658 31.69.015 8255 31.98 24.981 9318 3I.I6.993 1958 31.42.004 5559 31.69.oi6 I174 31.98 25 1.982 1188 3.II6 1.993 3843 31.42 2.004 746I 31.70 2.016 2093 31.99 26.982 3058 31.17 -993 5729 31I43.004 9363 31.70.OI6 4012 31.99 27.982 4928 31.17.993 7615 31'43.005 1265 31.71.OI6 5932 32.00 28.982 6798 31.18.993 9501 31'44.005 3I68 31.71.OI6 7852 32.00 29.982 8669 31-18.994 1387 31.44.005 5071 31.72.016 9772 32.01 30 1.983 0540 31.18 1.994 3274 31'45 2.005 6974 31.72 2.017 I693 32.01 31.983 2411 31-19.994 5161 31.45.005 8878 31.73.017 3614 32.02 32.983 4283 31.I9.994 7048 31.46.oo6 0781 31.73.017 5535 32.02 33.983 6 155 3I.20.994 8936 31.46.006 2685 31.74.017 7456 32.03 34.983 8027 31.20.995 0823 31'46.oo6 4590 31.74.017 9378 32.03 35 1.983 9899 31.2I 1-995 27II 31.47 2.006 6494 31.75 2.018 1300 32.04 36.984 1772 31.21.995 4600 31-47.006 8399 3I.75.OI8 3223 32.04 37.984 3644 3I.22.995 6488 31-48.007 0304 31.76.018 5145 32.05 38.984 5517 31.22.995 8377 31.48.007 22IO 31.76.018 7068 32.05 39.984 7391 31.22.996 0266 31.49.007 4116 31.77.OI8 8992 32.06 40 1.984 9264 31.23 I.996 2155 3I.49 2.007 6022 31.77 2.019 0915 32.06 41.985 1138 3I.23.996 4045 31.50.007 7928 31.77.019 2839 32.07 42.985 3012 31.24.996 5935 31-50.007 9835 31-78.019 4763 32-07 43.985 4886 31.24.996 7825 3.5.00oo8 1742 3I.78.019 6688 32.08 44.985 6761 31.24.996 9716 31.51.oo8 3649 31.79.019 8613 32.08 45 1.985 8636 31.25 1.997 1606 31.5 200oo8 5556 31.79 2.020 0538 32.09 46.986 05II 31.25 -997 3497 352.oo8 7464 31.80.020 2463 32.09 47.986 2386 31.26.997 5389 31.52.008 9372 3.-80.020 4389 32.10 48.986 4262 31.26.997 7280 31.53.009 I280 31.81.020 6315 32.10 49.986 6138 3I.27.997 9172 31.53.009 3I89 31.81.020 824I 32.11 50 1.986 8014 31.27 1.998 1o64 31.54 2.009 5098 31.82 2.021 oI68 32.11 51.986 9890 31.28.998 2956 31.54.009 7007 31.82.021 2095 32.12 52.987 1767 31.28.998 4849 31.55 o009 8917 31.83.021 4022 32.12 53.987 3644 31.28.998 6742 3I.55.OIo 0826 31.83.021 5949 32.I3 54.987 552I 31.29.998 8635 31.56.oIO 2736 31.84.021 7877 32.I3 55 1.987 7398 31.29 1-999 0529 31.56 2.010 4647 31.84 2.021 9805 32.I4 56.987 9276 31.30.999 2422 31.56.oIO 6557 31.85.022 I734 32.14 57.988 1I54 3130.999 43I6 31.57.OIO 8468 31.85.022 3662 32.15 58 -988 3032 3I.3I.999 6211 3.157.OII 0380 31.86.022 5591 32.15 59.988 4911 3.3I.999 8I05 31.58.oII 2291 31.86.022 7521 32.16 60 1.988 6789 31.31 2.000 0000 31.58 2.011 4203 31.87 2.022 9450 32.16 588 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 92" 93o 94~ 95~ V. I log M. Diff. 1. log M.Diff. 1. M. Diff. ". 1 g M. Diff. 1". 0' 2.022 9450 32.I6 2.034 5797 32.48 2.046 3296 32.80 2.058 2005 33.I5 1.023 380 32.17.034 7745 32.48.046 5264 3z28I.o58 3994 33.15 2.023 33II 32.I7.034 9694 32.49.046 7233 32.82.058 5983 33.16 3.023 5241 32.I8.035 I644 32-49.046 9202 32.82.058 7973 33-I6 4.023 7172 32'.8.035 3593 32.50.047 II72 32.83 0.58 9963 33-I7 5 2.023 9103 32.I9 2.035 5543 32-50 2.047 3141 32.83 2.059 I953 33.I8 6.024 1035 32.I9.035 7494 32.51 -047 5I11 32.84.059 3944 33.18 7.024 2967 32.20.035 9444 32.51.047 7082 32.84.059 5935 33-19 8.024 4899 32.20.036 1395 32.52.047 9053 32.85.059 7927 33.I9 9.024 6831 32.21.036 3347 32.52.048 1024 32.85.059 9919 33.20 10 2.024 8764 32.2I 2.036 5298 32-53 2.048 2995 32.86 2.060 1911 33.21 11.025 0697 32.22.036 7250 32.53.048 4967 32.87.o60 3904 33.2I 12.025 2630 32.22.036 9202 32.54.048 6939 32.87.o60 5897 33.22 13.025 4564 32.23.037 II55 32.54.048 8912 32.88.o60 7890 33.22 14..025 6498 32.23.037 3I08 32.55.049 0884 32.88.o60 9884 33.23 15 2.025 8432 32.24 2.037 506i 32.55 2.049 2857 32.89 2.061 1878 33.24 16.026 0367 32.24.037 7015 32.56.049 4831 32.89.o6I 3872 33.24 17.026 2301 32.25.037 8969 32.57.049 6805 j32.90.06 5867 33.25 18.026 4236 32.26.038 0923 32-57.049 8879 32-90.06I 7862 33.25 19.026 6172 32.26.038 2877 32.58.050 0753 32.9I.o6I 9857 33.26 20 2.026 8Io8 32.27 2.038 4832 32.58 2.050 2728 32.92 2.062 1853 33.27 21.027 0044 32.27.038 6787 32.59.050 4703 32.92.062 3849 33-27 22.027 1980 32.28.038 8743 32.59.050 6679 32.93.062 5846 33.28 23.027 3917 32.28.039 0699 32.60.050 8655 32-93.062 7842 33.28 24.027 5854 32.29.o39 2655 32.61.051 0631 32.94.062 9840 33.29 25 2.027 779I 32.29 2.039 46II 32.61 2.05I 2608' 32.95 2.063 1837 33-30 26.027 9729 32.30.039 6568 32.62.051 4585 32.95.063 3835 33.30 27.o28 1667 32.30.039 8525 32.62.051 6562 32.96 -063 5833 33.3I 28.o28 3605 32.3I.040 0482 32.63.05I 8539 32-96.063 7832 33-3I 29.o28 5544 32.31.040 2440 32.63.052 0517 32-97 o063 983I 33.32 30 2.028 7483 32.32 2.040 4399 32.64 2.052 2496 32.97 2.064 1831 33-33 31.028 9422 32.32.040 6357 32.64.052 4474 32.98.064 3830 33-33 32.029 1361 32-33.040 83I6 32.65'.052 6453 32.98.064 5830 33.34 33.029 330I 32.33.041 0275 32.65.052 8432 32.99.064 7831 33.34 34.029 5241 32.34.041 2234 32.66.053 0412 33.00.064 9832 33.35 35 2.029 7182 32.34 2.04I 4194 32.67 2053 2392 33.00 2.o65 I833 33.36 36.029 9123 32.35.04I 6154 32.67.053 4372 33.I0.o65 3834 33.36 37.030 1064 32.35.04I 8114 32.68.053 6353 33-01.065 5836 33.37 38.030 3005 32.36.042 0075 32.68.053 8334 33.02.065 7839 33.37 39.030 4947 32.36.042 2036 32.69.054 0315 33.03.065 9841 33-38 40 2.030 6889 32-37 2.042 3998 32.69 2-054 2297 33.03 2.066 I844 33.39 41.030 8831 32.37.042 5960 32.70.054 4279 33-04.o66 3847 33-39 42.03I 0774 32.38.042 7922 32.70 ~054 6262 33.04.o66 585I 33-40 43.03I 2717 32.39.042 9884 32.71.054 8244 33.05.o66 7855 33-40 44.031 4660,32.39.043 I847 32.7I.055 0227 33.05.066 9860 33.4I 45 2.03I 6604 32.40 2.043 38Io 32.72 2.055 22II 33.o6 2.067 I865 33-42 46.031 8548 32.40.043 5773 32.73.055 4195 33.07 -067 3870 33.42 47 -032 0492 32.41.043 7737 32.73.055 6179 33-07.067 5875 33.43 48.032 2437 32.41.043 970I 32.74.055 8I63 33.08.067 7881 33-43 49.032 438'2 32.42.044 i665 32-74.056 0I48 33.08.067 9887 33-44 50 2.032 6327 32.42 2.044 3630 32.75 2.056 2133 33-09 2.068 1894 33.45 51.032 8272 32.43.044 5595 32.75.056 4119 33.10.068 90oI 33.45 52 -033 o218 32.43.044 756I 32.76.056 6o15 33.10.068 5908 33.46 53 -033 2164 32.44.044 9526 32.76.056 8091 33. 1.068 7916 33-47 54 -033 4111 32.44.045 1492 32.77.057 0078 33.11.o68 9924 33-47 55 2.033 6058 32.45 2-045 3459 32.78 2.057 2065 33.12 2.069 1933 33-48 56.033 8005 32.45.045 5426 32.78.057 4052 33.-2.069 3942 33-48 57 -033 9952 32.46.045 7393 32.79.057 6040 33.13.069 5951 33-49 58 -034 1900 32.47.045 9360 32.79.057 8028 33.14.o69 7960 33-50 59 -034 3848 32-47.o46 1328 32.80.058 oo0016 33-I4.069 9970 33.5 60 2.034 5797 32-48 2.046 3296 32.80 2.058 2005 33-15 2.070 1980 33.51 589 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 960 970 98~ _ 99 V.. - - log M. Diff. 1". log Diff. 1". log M. Diff. 1. llo M Di Dff. 1".' z.-070 1980 33.5I 2.08o 3282 33.88 2.o94 597I 34.28'2.IC7 oi09 34.69 1.070 3991 33.5I.o82 5316 33.89.094 8028 34.29.I07 2190 34.70 2.070 6002 33-52'.82 7349 33.90.095 0085 34.29.107 4272 34-70 3.070 8014 33-53.082 9383 33.90.095 2143 34.30.I07 6355 34-71 4.07I 0025 33-53.083 1418 33.91.095 4201 34-31.107 8437 34-72 5 2.071 2037 33-54 2.083 3453 33-92 2.095 6260 34.31 2.I08 0521 34.72 6.07I 4050 33.54.083 5488 33.92.095 8318 34.32.i08 2604 34.73 7 -071 6063 33.55.083 7523 33-93.096 0378 34-33.Io8 4689 31.74 8.0o7 8076 33.56.083 9559 33-94.096 2438 34-33.I08 6773 34-75 9.072 0090 33.56.084 I596 33-94 o096 4498 34-34.108 8858 34-75 10 2.072 2I04 33.57 2.084 3633 33-95 2.096 6558 34-35 2.109 0944 34.76 11.072 4118 33-58.084 5670 3396 096 8619 34-35.109 3029 34-77 12.072 6133 33.58 -084 7707 33.96.097 068I 34.36.1I9 116 34-77 13.072 8148 33-59.084 9745 33-97.097 2742 34-37.109 7202 34-78 14.073 oI63 33.59 -085 1783 33.98.097 4804 34-37.109 9289 34-79 15 2.073 2179 33.60 2.085 3822 33.98 2.097 6867 34.38 2.IIO 1377 34.80 16.073 4195 33.6I o85 5861 33-99.097 8930 34-39 IIO 3465 34.80 17.073 6212 33.61 -085 7901 33-99 -098 0993 34-39.IIO 5553 34.8I 18.073 8229 33.62.o85 9941 34-00.098 3057 34.40.IIO 7642 34.82 19.074 0246 33.63.o86 I98 34-01.098 5121 34.41.IIO 973I 34-82 20 2.074 2264 33.63 2.086 4021 34.01 2.098 7186 34-41 2.II 1821 34.83 21.074 4282 33.64.o86 6062 34.02.098 9251 34.4z2.II 39II 34.84 22.074 6301 33.64.o86 8I04 34-03.099 I3I6 34-43.III 6o00 34.85 23.074 8320 33.65.087 0146 34.03.099 3382 34-43.111 8092 34.85 24.075 0339 33.66.087 2188 34-04 -099 5449 34-44 -II2 0184 34-86 25 2.075 2358 33.66 2.087 4231 34-05 2I099 7515 34-45 2.12 2275 34.87 26.075 4378 33.67.087 6274 34.05 -099 9582 34.45 1II2 4368 34.87 27.075 6399 33.67.087 83I7 34.06.Ioo 1650 34.46.I12 6460 34.88 28.075 84I9 33.68.088 036i 34-07.IOO 3718 34-47.112 8553 34-89 29.076 040 369 8 240 307 786 3448.3 647 34-90 30 2.076 2462 33.69 2.088 4449 34.08 2.100 7855 34.48 2.113 2741 34.90 31.076 4484 33.70.o88 6494. 34.09.10o 9924 34.49 I.13 4835 34.9I 32.076 6507 33.71.o88 8540 34.09.IOI 1993 34.50.113 6930 34.92 33.076 8529 33.71.089 0586 34I10.'10 4063 34-50.113 9025 34-92 34.077 0552 33-72.089 2632 34.11.IOI 6134 34.51 -II4 II2I 34-93 35 2.077 2575 33-73 2.o89 4678 34.I0 2.101 8204 34.52 2.114 3217 34-94 36.077 4599 33.73.o89 6725 34.12.102 0276 34.52.114 5313 34-95 37.077 6623 33-74.089 8772 34.12.102 2347 34-53 -114 74IO 34-95 38.077 8647 33-74.090 0820 34.13.I02 4419 34-54 -114 9508 34-96 39.078 0672 33-75.o90 2868 34.14.102 6492 34-54.115 1605 34-97 40 2.078 2697 33.76 2.090 4917 34.15 2.102 8564 34-55 2.115 3704 34-97 41.078 4723 33.76.090 6966 34.15.103 0638 34.56.115 5802 34.98 42.078 6749 33-77.090 905 34.I6.103 2711 34-56 I I5 7901 34-99 43.078 8775 33.78.091 1065 34.17.103 4785 34-57.I 6 oo00 35.00 44 8 378.079 2 3115 34.17 03 6860 34-58.116 2101 35.00 45 2.079 2829 33-79 2.09I 5I65 34-18 2.103 8935 34-59 z2. 6 420I 35.01 46 -079 4857 33.80.09~ 72I6 34.19.104 IOIO 34-59. 16.6301 35.02 47.079 6885 33.80.09I 9268 34.19.104 3086 34.60.I 6 8403 35.02 48.079 8913 33.81.092 1319 34.20.104 5162 34.6I.117 0505 35.03 49.080 0942 33.81.092 337I 34.20.I04 7239 34.61.117 7 35-04 50 2.080 2971 33.82 2.092 5424 34.21 2.104 9316 34.62 2.117 4710 35-05 51.o80 5000 33.83.092 7477 34.22.105 1393 34.63.117 6813 35.05 52.080 7030 33-83.092 9530 34.22.I05 3471 34.63.117 896 35.o06 53.o80 9060 33-84.093 1584 34-23.105 5549 34.64.I-8 1020 35-07 54.o8I 109 1 33-85.093 3638 34.24.105 7628 34.65.II8 3124 35-08 55 2.081 3122 33-85 2.093 5692 34.25 2.105 9707 34.66 2.118 5229 35.08 56.o08 5153 33-86.093 7747 34-25.I06 1786 34.66.118 7334 35-09 57.081 7185 33-87.093 9803 34.26.Io6 3866 34-67 -.18 9440 35-I0 58 -08I 9217 33.87 o094 I858 34-27.I06 5947 34-68,119 I546 35-10 59.082 I249 33.88.094 3914 34.27.106 8027 34.68.119 3652 35-II 60 2.082 3282 33.88 2.094 5971 34.28 2.107 0109 34.69 2.119 5759 35.-2 590 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. _ 1001 1002 10 030 V. log M. Diff. 1". log M. Diff. 1. loDiff. 1 ". log M. Diff. 1... 0' 2.119 5759 35.12 2.I32 2989 35.57 2.145 I866 36.03 2.158 2460 36.52 1.119 7867 35.53.132 5223 35-57 -145 4028 36.04.158 4652 36.53 2.119 9974 35.13.I32 7258 35-58.I45 6191 36.05.I58 6844 36.54 3.120 2083 35.14.I32 9393 35-59.145 8354 36.06.I58 9036 36.55 4.120 4191 35.15.I33 1529 35.60.I46 05I8 36.07.159 1229 36.55 5 2.120 6301 35.16 2.I33 3665 35.6I 2.146 2682 36.07 2.159 3423 36.56 6.120 8410 35.I6.133 5802 35.61.146 4847 36.08.159 5617 36.57 7.121 0520 35.17 -133 7939 35.62.146 70I2 36.09.I59 7811 36.58 8.121 2630 35.18.I34 0076 35.63.I46 9178 36.1o.I60 0006 36.59 9.121 474I 35.19.134 22I4 35.64.I47 I344 36.II.i60 2202 36.60 10 2.121 6853 35.19 2.I34 4352 35.64 2.I47 3510 36.II 2.160 4398 36.60 11.121 8965 35.20 I.34 6491 35.65.147 5677 36.12.I60 6594 36.61 12.122 1077 35.21.I34 863I 35.66.I47 7845 36.13.I60 8791 36.62 13.122 3190 35.21.I35 0770 35.67.I48 0013 36.14.161 0989 36.63 14.122 5303 35.22.135 2910 35.67.148 2182 36.15,.16 3187 36.64 15 2.122 74I6 35-23 2.135 5051 35.68 2.148 435I 36.15 2.I6i 5385 36.65 16.122 9530 35.24.135 7192 35.69.I48 6520 36.16.I6I 7584 36.65 17.I23 1644 35.24.135 9334 35-70.148 8690 36.17.161 9784 36.66 18.123 3759 35.25.I36 1476 35.7I -149 o86I 36.18.162 1984 36.67 19.123 5875 35.26.136 36I9 35.71 -I49 3032 36.19.162 4185 36.68 20 2.123 7990 35*27 2.136 5762 35.72 2.149 5203 36.19 2.162 6386 36.69 21.124 0107 35.27.I36 7905 35.73.149 7375 36.20.162 8587 36.70 22.124 2223 35.28.I37 0049 35-74.I49 9547 36.21.163 0789 36.70 23 I.24 4340 35.29 -137 2193 35-74.150 I720 36.22.163 2992 36.7I 24.124 6458 35.30.137 4338 35 75.I50 3893 36.23.I63 5195 36.72 25 2.I24 8576 35.30 2.I37 6484 35.76 2.150 6067 36.23 2.163 7398 36.73 26.125 0694 35.31.I37 8630 35.77.150 8242 36.24.I63 9602 36.74 27.I25 2813 35.32.138 0776 35 77.151 0417 36.25.I64 I807 36.74 28.125 4933 35.33.138 2922 35.78.I51 2592 36.26 I.64 40I2 36.75 29.125 7052 35.33.138 5070 35-79 -I5I 4768 36.27.164 6218 36.76 30 2.125 9173 35.34 2.I38 7217 35.80 2.151 6944 36.28 2.164 8424 36.77 31.126 1293 35.35.138 9365 35-81.I51 9121 36.28.I65 0630 36.78 32.126 3414 35.35 -I39 1514 35.81.152 1298 36.29.I65 2837 36.79 33.126 5536 35.36.139 3663 35.82.152 3476 36.30.i65 5045 36.80 34.126 7658 35.37 -139 5813 35.83 1I52 5654 36-31.i65 7253 36.8I 35 2.126 9780 35.38 2.139 7963 35.84 2.152 7833 36.32 2.165 9462 36.81 36.I27 I903 35-39.I40 0113 35.84.153 0OI2 36.32.I66 i67I 36.82 37.127 4027 35-39.I40 2264 35.85.153 2192 36.33.I66 388I 36.83 38.127 615i 35.40.I40 4415 35.86.153 4372 36.34.i66 6091 36.84 39.I27 8275 35-4I -140 6567 35.87.I53 6552 36.35.I66 8301 36.85 40 2.128 0400 35.42 2.140 8720 35.88 2.153 8734 36.35 2.167 0513 36.86 41.128 2525 35.42.141 0873 35.88.154 09I5 36.36.167 2724 36.87 42.128 4650 35.43 -I41 3026 35.89 -154 3097 36.37'I67 4936 36.87 43.128 6776 35-44.I4I 5180 35.90.154 5280 36.38.167 7149 36.88 44.128 8903 35-45 -I41 7334 35.91.154 7463 36.39.167 9362 36.89 45 2.129 1030 35-45 2.141 9489 35.92 2.154 9647 36.40 2.168 1576 36.90 46.129 3157 35.46.142 I644 35.92.155 1831 36.4I.i68 3790 36.91 47.I29 5285 354-7.142 3799 35-93.155 4015 36.41.i68 6005 36.92 48.129 7414 35.48.142 5955 35-94.155 6200 36.42.168 8220 36.93 49.129 9542 35.48.142 8112 35.95 -155 8386 36.43.I69 0436 36.93 50 0.3o 1672 35.49' 2143 0269 35-96 2.156 0572 36.44 2.169 2652 36.94 51.130 3801 35.50.143 2427 35.96.I56 2759 36.45.I69 4869 36.95 52 -130 5931 3551.-I43 4585 35-97.156 4946 36.46.169 7087 36.96 53 i.30 8062 35-5.-143 6743 35.98.I56 7I33 36.46.169 9304 36.97 54 -13I 0193 35-52.143 8902 35-99.-56 932I 36.47.170 1523 36.98 55 2.131 2325 35-53 2.144 o062 36.00 2.157 1510 36.48 2.170 3742 36.99 56.131 4457 35-54 -144 3222 36.00.157 3699 36.49 I.70 596I 36.99 57.13I 6589 35.54 -144 5382 36.01.157 5889 36.50.I70 8181 37.00 58.131 8722 35.55 -I44 7543 36.02.157 8079 36.50.171 0401 37.01 59.132 0855 35-56.144 9704 36.03.I58 0269 36.51.171 2622 37.02 60 2.132 2989 35-57 2.145 I866 36.03 2.158 2460 36.52 2.171 4844 37-03 591 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. I 1040 105~ 1060 1070 log M. Diff. l". log M. Diff. 1". log M. Diff. 1". log Diff. 1. 0' 2.171 4844 37.03 2.184 9092 37.56 z2.198 5282 38.'11 2.212 3493 38.68 1.I7i 7066 37-04.185 i346 37.57.198 7568 38.12.212 5814 38.69 2.171 9288 37-.5..i85 3600 37-57.198 9856 38.13.22 8136 38.70 3.172 1511 37-05 -.85 5855 37-58.199 2144 38.I4.213 04-58 38.71 4.172 3735 37.o6.185 ilo8110 37-59 -199 4432 38-14'213 2781 38-72 5 2.172 5959 37-07 z2.86 0366 37.60 2.199 6721 38.15 2.213 5104 38.73 6.172 8184 37-08.186 2622 37.61.199 9010 38-16.213 7428 38.74 7.173 0409 37-09.i86 4879 37.62.200 1300 38-17.213 9753 38.75 8.173 2634 37-10.186 7137 37.63.200 3591 38.18 -214 2078 38.76 9.173 4860 37-11.186 9395 37.64.200 5882 38-I9.214 4404 3877 10 2.173 7087 37.I2 2.187 1653 37.65 2.200 8174 38.20 2.214 6730 38.78 11 -173 9314 37-12.187 3912 37.66.201 0467 38.21.214 9057 38.79 12.I74 1542 37-13 -187 6172 37.67.201 2760 38.22.215 1385 38.80 13.I74 3770 37.14 -187 8432 37.67 -201 5053 38.23.215 3713 38.81 14.174 5999 37.15.i88 0693 37.68.201 7347 38.24.215 6042 38.82 15 2.174 8228 37. 6 z2.188 2954 37.69 2.zo20I 9642 38.25 2.215 8371 38.83 16.175 0458 37-17.i88 5216 37-70.202 1937 38.26.216 0701 38.84 17.175 2688 37-18.188 7478 37.71.202 4233 38.27.216 3032 38.85 18.175 4919 37-I8 -I88 9741 37.72.202 6529 38-28.216 5363 38.86 19.175 7150 37-19.189 2005 37-73.202 88z6 38.29.216 7694 38.87 20 2.175 9382 37.20 2.189 4269 37-74 2.203 1123 38.30 2.217 0027 38.88 21.176 1615 37.21.189 6533 37.75.203 3421 38.31 -217 2360 38.89 22.176 3848 37.22.189 8798 37-76.203 5720 38.31.217 4693 38.90 23.176 6o81 37-23.190 1064 37-77 -203 8019 38.32.217 7027 38-91 24.176 8315 37.24 -190 3330 37-77 -204 0319 38.33.217 9362 38.92 25 2.177 0550 37-25 2.190 5597 37-78 2.204 2619 38.34 2.218 1697 38.93 26.177 2785 37.25 -190 7864 37-79.204 4920 38.35.218 4033 38.94 27.-177 5020 37.26.191 0132 37-80.204 7222 38.36.218 6369 38.95 28.177 7256 37-27.191 2401 37-81.204 9524 38.37.218 8706 38.96 29.177 9493 37.28.191 4670 37-82.205 1826 38.38.219 1044 38.97 30 2.178 1730 37-29 2.191 6939 37.83 2.205 4129 38.39 2.219 3382 38-98 31.178 3968 37-30.191 9209 37.84.205 6433 38.40.219 5721 38.99 32.178 6206 37-31' -192 1480 37-85.205 8737 38-41.219 8061 39-00 33.178 8445 37-32.192 3751 37-86.206 1042 38.42.220 0401 39-01 34.179 0684 37.33 1 92 6023 37.87.206 3348 38-43.220 2741 39-02 35 12179 2924 37-33 2.192 8295.37-88 2.206 5654 38-44 2.220 5082 39-03 36.-179 5164 37-34.193 0568 37-88.206 7961 38.45.220 7424 39.04 37.1I79 74-05 37-35 -193 2841 37-89.207 0268 38.46.220 9767 39-05 38.-179 9646 37-36.193 5115 37-90.207 2575 38.47.221 2110 39.06 39.-IS 1888 37.37.193 7389 37.91 207 4884 38.48.221 4453 39-07 40 2z.80 4131 37.38 2.193 9664 37-92 2.207 7193 38.49 [2.22z 6797 39-08 41.180 6374 37-39 -194 1940 37-93 -207 9502 38.50.221 9142 39.09 42.180o 8617 37-40.194 4216 37-94.208 1812 38.51.222 1488 39IO10 43.-81 o86i 37-41 194 6493 37-95.208 4123 38.52.222 3834 39-1 44.181 3106 37-41 -194 8770 37.96.208 6434 38.53.222 6i8o 39-12 45 -2.181 5351 37.42 2.195 1048 37-97 2.208 8746 38.54 [2.222 8528 39-13 46 -181 7597 37-43.95 3326 37-98 -209 1058 38.54 -223 0876 39-14 47.181 9843 37-44.195 56o5 37-99 -209 3371 38.55.223 3224 39.15 48.182 1o89 37-45 I195 7885 38.00.209 5685 38.56.223 5573 39.16 49.182 4337 37-46.196 oi65 38-00.209 7999 38.57 -.223 7923 39-17 50 2.182 6584 37-47 2.196 2445 38.01 2.210 0314 38.58 [2.224 0273 39-18 51I.182 8833 37-48.196 4726 38.02.210 2629 38.59.224 2624 39.19 52.183 1082 37-49.196 7008 38.03.210 4945 38.60.224 4975 39-20 53.183 3331 37-49.196 9290 38.04.210 7261 38.61.224 7327 39.21 54 -183 558i 37-50.197 1573 38-05.210 9578 38.62.224 9680 39.22 55 2.183 7831 37-51 2-197 3856 38.06 2.211 1896 38.63 2.225 2033 39.23 56.184 0082 37-52.197 6140 38-07.211 4214 38.64.225 4387 39-24 57 -184 2334 37.53.197 8425 38.08.211 6533 38.65.225 6741 39.25 58 -184 4586 37.54.198 0710 38-09 -211 8852 38.66.225 9096 39-26 59 -184 6839 37-55 -198 2995 38.10.212 1172 38.67.226 1452 39-27 60 2.184 9092 37-56 2.198 5282 38.11 2.212 3493 38.68 [2.226 3808 39.28 592 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 108~ 109~ 110 111ii V.. 1... log M. Diff. 1". Dif "log M. Diff. 1". log M. Diff. 1".' 2.226 3808 39.28 2.240 6314 39.90 2.255 IO99 40.54 -.269 8255 4I.2I 1.226 6165 39.29.240 8708 39.91.255 3532 40.55.270 0728 4I.23 2.226 8523 39.30.241 1103 39.92.255 5965 40.56.270 3202 41.24 3.227 088I 39.31.241 3498 39.93.255 8399 40.58.270 5676 41.25 4.227 324~ 39.32.24I 5894 39-94.256 0834 40.59.270 8152 41.26 5 2.227 5599 39-33 2.24I 8291 39-95 2.256 3270 40.60 2.271 0628 41.27 6.227 7959 39-34.242 0688 39.96.256 5706 40.6I.271 3104 41.28 7.228 0320 39-35.242 3086 39-97.256 I843 40.62.271 5582 41.29 8.228 268i 39.36.242 5485 39.98.257 0580 40.63.27I 8060 41.30 9.228 5043 39-37.242 7884 39-99.257 30I9 40.64.272 0538 4I.32 10 2.228 7405 39.38 2.243 0284 40.00 2.257 5458 40.65 2.272 30I8 41.33 11.228 9768 39-39.243 2685 40.0I.257 7897 40.66.272 5498 41-34 12.229 2131 39.40.243 5086 40.02.258 0337 40.68.272 7979 41I35 13.229 4496 39.41.243 7488 40.03.258 2778 40.69.273 0460 4I.36 14.229 6861 39.42.243 9890 40.05.258 5220 40.70.273 2942 41.38 15 2.229 9226 39.43 2.244 2293 40.06 2.258 7662 40.7I 2.273 5425 41-39 16.230 1592 39-44.'!44 4697 40.07.259 OI05 40-72.273 7909 41-40 17 -230 3959 39-45.244 7101 40.08.259 2548 40.73.274 0393 41-41 18.230 6326 39-46.204 9506 40.09.259 4992 40.74.274 2878 4I.42 19.230 8694 39-47 I 912 40.10.259 7437 40-75.274 5364 41-43 20 2.231 1063 39.48 2.245 4318 40.II 2.259 9883 40.76 2.274 7850 41.44 21.231 3432 39.49.245 6725 40.12.260 2329 40.78.275 0337 41.46 22.231 5802 39.50.245 9I32 40.I3.260 4776 40.79.275 2825 41.47 23.231 8172 39.51.246 I541 40.14.260 7223 40.80.275 5313 41.48 24.232 0543 39.52.246 3949 40.15.26o 967I 40.8I.275 7802 41.49 25 2.232' 2915 39-53 2.246 6359 40.16 2.261 2120 40.82 2.276 0292 4I.50 26.232 5287' 3954.246 8769 40.17.261 4570 40.83.276 2783 41-51 27.232 7660 39-55.247 II80 40.18.261 7020 40.84.276 5274 4I.53 28.233 0033 39.56.247 359I 40.I9.26I 9471 40.85.276 7766 41.54 29.233 2407 39-57.247 6003 40.21.262 1922 40.86.277 0258 41.55 30 2.233 4782 39.58 2.247 8416 40.22 2.262 4374 40.88 2.277 2752 41.56 31.233 7157 39-59.248 0829 40.23.262 6827 40.89.277 5246 41.57 32'.233 9533 39.60.248 3243 40.24.262 928I 40.90.277 7740 41.58 33 -234 1910 39.6I.248 5658 40.25.263 1735 40-9I.278 0236 4I.60 34.234 4287 39.63.248 8073 40.26.263 4190 40.92.278 2732 41.61 35 2.234 6665 39.64 2.249 0489 40.27 2.263 6645 40-93 2.278 5229 41.62 36.234 9043 39.65.249 2906 40.28.263 9I02 40.94.278 7726 41.63 37.235 1422 39.66.249 5323 40.29.264 1559 40.95.279 0224 41.64 38.235 3802 39.67.249 774I 40.30.264 4016 40.96.279 2723 41:65 39.235 6183 39.68.250 0159 40.31.264 6474 40.98.279 5223 41.67 40 2.235 8563 39.69 2.250 2578 40.32 2.264 8933 40.99 2.279 7723 41.68 41.236 0945 39.70.250 4998 40.34.265 1393 4I.00.280 0224 4I.69 42.236 3327 39.71.250 7419 40.35.265 3853 41.01.280 2726 41.70 43.236 57IO 39.72.250 9840 40.36.265 6314 41.02.280 5228 41.7I 44.236 8093 39-73.251 2262 40.37.265 8776 41-03.280 7731 4I.72 45 2.237 0478 39-74 2.251 4684 40.38 2.266 1238 4I.04 2.28I 0235 41.74 46.237 2862 39-75.251 7107 40.39.266 3701 4I.06.281 2740 41.75 47.237 5247 39.76.251 953I 40.40.266 6I65 4I.07.281 5245 41.76 48.237 7633 39-77.252 I955 40.41.266 8629 4I.08.281 7751 41.77 49.238 0020 39.78.252 4380 40.42.267 1094 4I.09.282 0258 41.78 50 2.238 2407 39-79 2.252 6806 40.43 2.267 3560 4I.10 2.282 2765 41.80 51.238 4795 39-80.252 9232 40.44.267 6026 4I.I1.282 5273 41.81 52.238 7284 39.8I.253 1659 40.46.267 8493 4I.I2.282 7782 4I.82 53.238 9573 39-82.253 4087 40.47.268 0961 4I.13.283 0291 41.83 54.239 1962 39.83.253 65I5 40.48.268 3430 4I.15.283 2801 4I.84 55 2.239 4353 39-84 2.253 8944 40-49 2.268 5899 4I.I6 2.283 5312 41.85 56.239 6744 39.86.254 I374 40.50.268 8369 41-17.283 7824 41.87 57.239 9235 39.87.254 3804 40.51.269 0839 41.18.284 0336 41.88 58.240 I528 39-88.254 6235 40-52.269 3310 4-II9.284 2849 4189 59.240 3921 39.89 -254 8666 40-53 269 5782 41.20.284 5363 4I.90 60 2.240 6314 39-90 2.255 1099 40.54 2.269 8255 41.21 2.284 7878 41.91 38 593 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 112~ 113~ 114~ 115~ V. log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 2.284 7878 4I.91 2.300 0067 42.64 2.3I5 4927 43-40 2.33I 2564 44.1I 1.285 0393 41.93.300 2626 42.65.315 7531 43.41.331 52I6 44.20 2.285 2909 41.94.300 5S86 42.67.316 0136 43.42'331 7868 44.2I 3.285 5425 4.195.300 7746 42.68.3I6 2742 43-44'332 0521 44.22 4.285 7943 4I.96 -301 0307 42.69.3I6 5348 43-45 -332 3175 44-24 5 2.286 0461 4.197 2.301 2869 42.70 2.316 7956 43.46 2.332 5830 44.25 6.286 2979 41.99.301 5431 42.72.317 0564 43-47.332 8485 44.26 7.286 5499 42.00 -301 7995 42.73 -317 3173 43-49 -333 I14I 44.28 8.286 80I9 42.o0.302 0559 42.74.317 5782 43-50 -.333 3799 44.29 9.287 0540 42.02.302 3123 42.75.317 8393 43.51.333 6456 44.3I 10 2.287 3062 42.03 2.302.5689 42.76 2.318 oo04 43.53 2.333 9II5 44.32 I1.287 5584 42.04 -302 8255 42.78.318 3616 43.54 -334 1775 44-33 12.287 8107 42.06.303 0822 42.79.3I8 6229 43.55 -334 4435 44-34 13.288 063I 42.07.303 3390 42.80.318 8842 43.56.334 7096 44.36 14.288 3155 42.08.303 5958 42.81.319 I456 43.58.334 9758 44.37 15 2.288 5680 42.09 2.303 8528 42.83 2.319 4072 43-59 2-335 242I 44.39 16.288 8206 42.10.304 1098 42.84.319 6687 43.60.335 5084 44-40 17.289 0733 42.12.304 3668 42.85 -319 9304 43.62.335 7749 44.4I 18.289 3260 42.I3.304 6240 42.86.320 1921 43.63 -336 o414 44.43 19.289 5788 42.14.304 8812 42.88 i32o 454 13.64 -336 3080 44-44 20 2.289 83I7 42.15 2.305 I385 42.89 2.320 7159 4366 2.336 5747 44-45 21.290 0847 42.I6.305 3959 42.90.320 9778 43.67.336 8414 44.47 22.290 3377 42.I8.305 6533 42.91.32I 2399 43.68.337 -083 44.48 23.290 5908 42.19 -305 9109 42.93.32I 5020 43.69.337 3752 44.49 24.290 8440 42.20.306 1685 42.94.32I 7642 43-70.337 6422 44.5I 25 2.291 0972 42.21 2.306 4261 42.95 2.322 0265 43.72 2.337 9093 44.52 26.291 3505 42.22.306 6839 42.96.322 2889 43-73.338 1765 44.53 27.291 6039 42.24.306 9417 42.98.322 5513 43-75 -338 4437 44.55 28.291 8574 42.25 -307 I996 42.99.322 8139 43.76.338 7111 44.56 29.292 1o09 42.26.307 4576 43.00.323 0765 43-77.338 9785 44.58 30 2.292 3645 42.27 2.307 7157 43.02 2.323 3391 43-79 2-339 2460 44-59 31.292 6i82 42.29 -307 9738 43.03.323 60I9 43.80.339 5I35 44.60 32.292 8719 42.30.308 2320 43.04.323 8647 43-8I.339 7812 44.62 33.293 I258 42.31.308 4903 43-05.324 I277 43.83.340 0490 44.63 34.293 3797 42.32.308 7486 43-07.324 3907 43-84.340 3168 44.64 35 2.293 6336 42.33 2.309 0071 43.08 2.324 6537 43.85 2.340 5847 44.66 36.293 8877 42.35.309 2656 43.09.324 9169 4387.340 8527 44.67 37.294 I418 42.36.309 5242 43.10.325 I80I 43.88.341 1207 44.69 38.294 3960 42.37 -309 7828 43.12.325 4434 43.89 -341 3889 44-70 39.294 6503 42.38.30 0o416 43.13 -325 7068 43.9I -341 6571 44.71 40 2.294 9046 42.40 2.310 3004 43.I4 2.325 9703 43.92 2.34I 9255 44-73 41.295 1590 42.4I.3I0 5593 43-15.326 2339 43-93 -342 1939 44-74 42.295 4135 42.42.310 8182 43.17.326 4975 43-94.342 4623 44-75 43.295 6680 42.43.3II 0773 43I18.326 7612 43.96.342 7309 44.77 44.295 9227 42.44.3II 3364 43-19.327 0250 43.97 -342 9995 44-78 45 2.296 I774 42.46 2.311 5956 43.2I 2.327 2889 43-98 2.343 2683 44.80 46.296 4321 42.47.311 8549 43.22.327 5528 44-00.343 5371 44.81 47.296 6870 42.48.3I2 1142 43.23.327 8I68 44.01.343 8060 44.82 48.296 9419 42.49.3I2 3736 43.24.328 0809 44.02.344 0750 44.84 49.297 I969 42.51.312 6331 43.26.328 3451 44.04.344 3440 44-85 50 2.297 4520 42.52 2.312 8927 43-27 2.328 6094 44-05 2.344 6132 44.86 51.297 7071 42-53 -3I3 1524 43.28.328 8737 44.06.344 8824 44.88 52.297 9623 42.54.313 4121 43.29.329 I382 44.08 -345 15I7 44.89 53.298 2I76 42.55 -3I3 6719 43.3I.329 4027 44-09.345 4211 44.9I 54.298 4730 42.57.313 9318 43.32.329 6672 44-10.345 6906 44.92 55 2.298 7284 42.58 2.314 1917 43-33 2.329 9319 44.12 2.345 960I 44.93 56.298 9839 42.59 *314 4518 43.35.330 I967 44.I3.346 2298 44.95 57.299 2395 42.60.3I4 719 43.36.330 4615 44.14.346 4995 44.96 58.299 4952 42.61.314 972I 43-37.330 7264 44. 6.346 7693 44.97 59 -299 7509 42.63.315 2323 43.38.330 9914 44.I7 -347 0392 44.99 60 2.300 oo067 42.64 2.315 4927 43.40 2.331 2564 44.18 2.347 3092 45-00 594 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 116o 1170 1180 1190 log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 2-347 3092 45-00 2.363 6626 45.86 2.380 3290 46.74 2-397 32Io 47.66 1 -347 5792 45-02.363 9378 45.87 -380 6095 46.76.397 6070 47.68 2.347 8494 45-03.364 213I 45.88 - 380 8901 46.77.397 8931 47-70 3.348 1196 45.04 -364 4885 45.90.381 I708 46.79.398 1794 47.71 4.348 3899 45.06.364 7639 45.91 -381 4515 46.80.398 4657 47.73 5 2.348 6603 45.07 2.365 0394 45-93 2.381 7324 46.82 2.398 752I 47-74 6.348 9308 45.09.365 3150 45-94.382 0133 46.83.399 0386 47.76 7.349 2014 45.IO.365 5907 45.96.382 2944 46.85 -399 3252 47.77 8.349 4720 45-11.365 8665 45.97.382 5755 46.86.399 6II9 47-79 9.349 7428 45.13.366 1423 45.99.382 8567 46.88.399 8987 47.81 10 2.350 0136 45.14 2.366 4183 46.00 2.383 1380 46.89 2.400 1856 47.82 11.350 2845 45.16.366 6944 46.01.383 4194 46.91.400 4725 47.84 12.350 5554 45.17.366 9705 46.03.383 7009 46.92.400 7596 47.85 13.350 8265 45.I8.367 2467 46.04.383 9825 46.94.401 0468 47.87 14.351 0977 45.20.367 5230 46.06.384 2642 46.95.401 3340 47.89 15 2.351 3689 45.21 2.367 7994 46.07 2.384 5460 46.97 2.401 6214 47.90 16.351 6402 45.23.368 0759 46.09.384 8278 46.98.401 9088 47.92 17.351 9116 45.24.68 3525 46.10.385 1098 46.99.402 1964 47-93 18.352 1831 45-25 629I 46.12.385 3918 47.01.402 4840 47.95 19 -352 4547 45.27 9059 46.13.385 6739 47.03 -402 7718 47.97 20 2.352 7263 45.28 2.369 1827 46.15 2.385 9562 47.05 2.403 0596 47.98 21 -352 9981 45.30.369 4596 46.16.386 2385 47.06.403 3475 48.00 22.353 2699 45.31.369 7367 46.I8.386 5209 47.08.403 6356 48.0 23.353 54I8 45.33 -370 0138 46.I9.386 8034 47.09.403 9237 48.03 24.353 8138 45-34.370 2909 46.21.387 0860 47.11.404 2119 48.04 25 2.354 0859 45.35 2.370 5682 46.22 2.387 3687 47.12 2.404 5002 48.06 26.354 3581 45.37.370 8456 46.24.387 6514 47.I4.404 7886 48.08 27.354 6303 45.38.371 I230 46.25.387 9343 47.15 -405 0771 48.09 28.354 9027 45.40.371 4006 46.26.388 2173 47.17.405 3657 48.11 29 -355 1751 45.41 -37I 6782 46.28.388 5003 47.I8.405 6544 48.12 30 2.355 4476 45.42 2.371 9559 46.29 2.388 7835 47.20 2.405 9432 48.14 31.355 7202 45.44.372 2337 46.3I -389 0667 47.21.406 232I 48.I6 32.355 9928 45.45 -372 516 46.32.389 3500 47.23 -406 5211 48.17 33.356 2656 45-47.372 7896 46.34.389 6335 47.24.406 8I02 48.I9 34.356 5385 45.48.373 0677 46.35.389 9170 47.26.407 0993 48.20 35 2.356 8114 45.50 2-373 3459 46.37 2.390 2006 47.28 2.407 3886 48.22 36.357 0844 45.51 -373 6241 46-38.390 4843 47.29.407 6780 48.24 37.357 3575 45-52.373 9024 46-40.390 7681 47-31 -407 9674 48.25 38.357 6307 45.54 -374 1809 46.41.39I 0519 47.32..408 2570 48.27 39 -357 9040 45.55.374 4594 46.43 -39I 3359 47-34.408 5467 48.28 40 2.358 1773 45.57 2-374 7380 46.44 2.391 6200 47.35 2.408 8364 48.30 41.358 4508 45.58.375 oI67 46.46.392 9042 47.37.409 1263 48.32 42.358 7243 45.60.375 2955 46.47.392 I884 47.38.409 4162 48.33 43.358 9979 45.61.375 5744 46.49 -392 4728 47.40.409 7063 48.35 44.359 2716 45.62 -375 8533 46.50.392 7572 47.41.409 9964 48.37 45 2.359 5454 45.64 2.376 1324 46.5 2.393 0417 47-43 2.410 2866 48.38 46.359 8193 45.65.376 4115 46.53.393 3264 47-45.410 5770 48.40 47.360 0933 45.67 -376 6908 46.55.393 61II 47.46.410 8674 48.4I 48.360 3673 45.68.376 9701 46.56.393 8959 47.48.41I 1579 48.43 49.360 6415 45.70.377 2495 46.58.394 1808 47-49.411 4486 48.45 50 2.360 9157 45.7I 2-377 5290 46.59 2.394 4658 47.51 2.411 7393 48.46 51.36I 1900 45.72.377 8086 46.60.394 7509 47-52.412 0301 48.48 52.361 4644 45-74.378 0883 46.62.395 036I 47-54.4I2 3210 48.49 53.361 7389 45.75.378 3681 46.64.395 3214 47-55.412 6120 48.5I 54.362 0134 45-77.378 6479 46.65.395 6067 47-57.412 9031 48.53 55 2.362 2881 45.78 2.378 9279 46.67 2-395 8922 47-59 2.413 1944 48.54 56.362 5628 45.80.379 2079 46.68.396 1778 47.60.413 4857 48.56 57.362 8376 45.81.379 4881 46.70.396 4634 47.62.413 7771 48.58 58 -363 1126 45.82.379 7683 46.71 -396 7492 47.63 -414 0686 48.59 59.363 3876 45-84.380 0486 46-73 -397 0350 47.65 -414 3602 48.61 60 2.363 6626 45.86 2.380 3290 46.74' 2-397 3210 47.66 2.414 6519 48.62 595 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1200 1210 122" 1230 VsVI log M. Diff. 1". log M. Diff. 1". log M. Diff. 1I" log M. Diff. 1". 0' z2.41i 6519 48.62- 2.432 3356 49.62 2.450.3868 5o.67 2.468 8205 5.75 1.414 9437 48.64.43z 6334 49.64.450 6908 50.68.469 1311 51.77 2.415 2356 48.66.432 9313 49.66 -450 9950 50~'70~ 469 4418 579 3.415 5276 48.67'433 2293 4968'451 2992 50.72.469 7526 517.8 4 i415 8197 48.69 -433 5274 49.69 -451 6036 50-74.470 0634 51.82 5'2.416 iii9 48.71 2.433 8257 49.71 2-45I 9081 50-75 2.470 3744 51.84 6.416 4042 48.72.434 1240 49.73.452 2127 50-77.470 6856 51.86 7.416 6965 48.74 -434 4224 49-74 -452 5174 50-79 -470 9968 51.88 8.416 9890 48.76.434 7209 49-76 -452 8222 50.81.471 3081 51-90 9.417 2816 -48.77 -435 0I95 49.78.453 1271 50.83 -471 6196 51.92 10 2.417 5743 48-79 2.435 3182 49.80 2.453 4321 50-84 2.471 9311 51.94 11.417 8671 48.8I.435 6171 49.81.453 7372 50-86.472 2428 51.95 12.418 i600 48.82.435 9160 49.83 -454 0424 50.88.472 5546 51.97 13.418 4529 48-84 -436 2150 49.85 -454 3477 50-90.472 8665 51.99 14.418 7460 48.85.436 5141 49-86.454 653z 50.92 -473 1785 52.01 15 z2.419 0392 48.87 2.436 8134 49.88 2.454 9587 50-93 2.473 4906 52.03 16.419 3325 48.89 -437 1127 49.90.455 2644 50-95 -473 8028 52.05 17.419 6258 48.90'437 4I22 49.92.455 5701 50.97 474 1152 52.07 18 1.419 9193 48-92.437 7117 49-93 -455 876o 50.99 -474 4276 52.09 19.420 2129 48-94.438 0114 49-95 -456 i8z.00.474 7402 52.10 20'2.420 5o66 48.95 2.438 3111 49-97 2.456 4881 51.o z 2.475 0529 52. 12 21.420 8003 48-97 -438 61Io 49.98.456 7943 51-04 -475 3657 52.14 22.421 0942 48-99 -438 9109 50.00.457 ioo6 51.06 [.475 6786 52.16 23.421 3882 49.00.439 2110 50.02.457 4070 51-08.475 99I6 52.1I8 24.421 6822 49.-02'439 5112 50.04.457 7135 51-09 -476 3047 52.20 25'2.4.21 9764 49.03 2.439 8II4 50.05 2.458 0201 51.11 2.476 618o 52.22 26.422 2707 49.05.440 1118 50.07 -458 3268 51-13 -476 9313 52.23 27.422 5650 49.07.440 4123 50.09 -458 6337 51-15.477 2448 52.25 28 [.422 8595 49.09.440 7129 50.11.458 9406 51-17.477 5584 52.27 29.423 1541 49.-10.44 01o36 50o. 2.459 2477 5-18.477 8721 52.29 30 2.423 4488 49.12 2.441 3143 50.14 2.459 5548 51.20 2.478 1859 52.31 31.423 7435 49-14 -441 6152 50.16.459 8621 51.22.4784998 52.33 32.424 0384 49.15.441 9162 50o.18.460 695 51-24 -478 8138 52.35 33.424 3334 49-17 -442 2173 50.-19.460 4770 51.26.479 1280 52.37 34.424 6284 49-19 -442 5185 50-21.460 7846 51.28.479 4422 52.39 35 2.424 9236 49-20 2.442 8199 50.23 2.461 0923 51.29 2.479 7566 52.40 36.425 2189 49-22.443 1213 50.24.461 4001 51-31.480 0711 52.42 37.425 5142 49-24 -443 4228 50.26.461 7080 51-.33 480 3857 52-44 38.425 8097 49.25.443 7244 50.28.462 oi6i 51-35.480 7004 52.46 39.426 1053 49-27 -444 0261 50.30.462 3242 51-37 -481 0152 52.48 40 2.426 4010 49.29 2.444 3280' 50.31 2.462 6325 51.38 2.481 3301 52.50 41.426 6967 4930.444 6299 50 -33.462 9408 51-40.481 6452 52.52 42.426 9926 49-32.444 9320 50.35 -463 2493 51.42.481 9604 52.54 43.427 2886 49-34 -445 2341 50.37 -463 5579 51.44.482 2756 52.56 44.427 5847 49-35 -445 5364 50.38.463 8666 51.46.482 5910 52.58 45 2.427 8808 49-37 2.445 8387 50.40 2.464 1754 51-48 2.482 9065 52.59 46.428 1771 49-39 -446 1412 50.42.464 4843 51-49 -483 2222 52.61 47.428 4735 49-40.446 4437 50.44 -464 7933 51.51 -483 5379 52.63 48.428 7700 49-42.446 7464 50-45 -465 1024 51-53 -483 8537 52.65 49. 429 0665 49-44 -447 0492 50.47.465 4116 51-55 -484 1697 52.67 50 2.429 3632 49-46 2.447 3521 50-49 2.465 7210 51-57 2.484 4858 52.69 51.429 6600 49.-47.447 6551 50.51.466 0305 51.59 -484 8020 52.71 52.429 9569 49-49 -447 9582 50.53.466 3400 51.60.485 1183 52.73 53.430 2539 49-51 -448 2614 50.54 -466 6497 51.62.485 4347 52.75 54.430 5510 49-52.448 5647 50.56.466 9595 51 -64'485 7513 52.77 55 [2.430 8482 49-54 2.448 8681 50.58 2.467 2694 5i.66 2.486 0679 52-78 56.431 1455 49-56.449 1716 5o.60.467 5794 51.68.486 3847 52.80 57.43I 4428 49.57.449 4753 5o.61.467 8895 51.70.486 7016 52.82 58.431 7403 49.59 -449 7790 50.63.468 1997 51.71 -487 oi86 52.84 59 -432 0379 49.61.450 08z8 50.65 -468 5101 51.73 -487 3357 52.86 60 B2.432 3356 49.62 2.450 3868 50.67 2.468 8205 51-75 2-487 6529 52.88 596 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1240 12 126 127 log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 2.487 6529 52.88 2.506 9006 4.06 2.526 5813 55-29 2546 7I35 56.57 1.487 9702 52.90.507 2251 54.08 -526 9131 55.31.547 530 56.59 2.488 2877 52.92.507 5496 54.10.527 2450 55.33 -547 3926 56.61 3.488 6053 52.94 -507 8742 54.I2.527 5771 55-35 547 7323 56.63 4.488 9230 52.96.508 I990 54-14 -527 9092 55-37 548 0722 56.65 5 2.489 2408 52.98 2.508 5239 54.I6 2.528 2415 55-39 2.548 4122 56.68 6.489 5587 53-00.508 8489 54.18.528 5739 55.41 -548 7523 56.70 7.489 8767 53.02.509 I741 54.20.528 9065 55-43.549 0926 56.72 8.490 I949 53.03 -509 4993 54.22.529 2391 55-45 -549 4330 56.74 9.490 5132 53-.5.509 8247 54.24.529 5719 55.48.549 7735 56.76 10 2.490 8315 530.7 2.510 1502 54.26 2.529 9048 55.50 2.550 II4I 56.79 11.491 1500 53.09.5I0 4758 54.28.530 2379 55-52.550 4549 56.8I 12.491 4686 53-II.510 8oi6 54.30.530 5710 55-54 -550 7958 56.83 13.491 7874 53.I3.511 1274 54-32.530 9043 55.56.551 1369 56.85 14.492'o063 53.15.5I 4534 54-34 -531 2378 55.58.55I 478I 56.87 15 2.492 4252 53.I7 2.5II 7795 54.36 2.531 5713 55.60 2.551 8194 56.90 16.492 7443 53.I9.512 I057 54.38.531 9050 55.62.552 I608 56.92 17.493 0635 53.21 -512 4321 54-40 -532 2388 55.64 -552 5024 56.94 18.493 3828.53.23 I2 7586 54.42.532 5727 55.67 -552 8441 56.96 19.493 7023 53.25 I3 0852 54-44.532 9068 55.69.553 I859 56.98 20 2.494 0218 53.27 2.5I3 4119 54.46 2.533 2410 55.71 2-553 5279 57.01 21.494 3415 53.29.513 7387 54.48.533 5753 55.73.553 8700 57.03 22.494 66I3 53.3I.514 0657 54.50.533 9097 55-75.554 2122 57.05 23.494 9812 53.33 -514 3927 54.52.534 2443 55-77'554 5546 57.07 24 -495 3012 53.35.514 7199 54-54 -534 5790 55.79 -554 8971 57.10 25 2.495 6213 53-37 2.5I5 0473 54.56 2.534 9138 55.8I 2.555 2398 57.I2 26.495 9416 53-39 -515 3747 54-58.535 2487 55.84 -555 5825 57.14 27.496 2619 53.4I.515 7023 54.60.535 5838 55.86.555 9254 57.16 28.496 5824 53.42.516 0300 54.63 -535 9190 55-88.556 2685 57.18 29.496 9030 53-44.i56 3578 54.65.536 2543 55.90.556 6 16 57.21 30 2.497 2238 53.46 2.516 6857 54.67 2.536 5898 55.92 2.556 9549 57.23 31.497 5446 53.48.517 0138 54.69.536 9254 55-94 -557 2984 57-25 32.497 8656 53.50.517 3420 54.7I.537 2611 55.96.557 6420 57.27 33.498 1867 53.52.517 6703 54-73.537 5970 55.98.557 9857 57.29 34.498 5079 53.54.517 9987 54.75 -537 9329 56.or.558 3295 57-32 35 2.498 8292 53.56 2.518 3273 54-77 2.538 2690 56.03 2.558 6735 57-34 36.499 15o6 53.58.5I8 6559 54-79 -538 6052 56.05.559 0176 57.36 37.499 4721 53.60.518 9847 54.81.538 9416 56.07 -559 36I8 57.38 38.499 7938 53.62.519 3137 54.83.539 2781 56.09.559 7062 57.41 39.500 II56 53.64.5I9 6427 54-85.539 6I47 56.II.560 0507 57-43 40 2.500 4375 53.66 2.519 9719- 54.87 2.539 9514 56.13 2.560 3953 57.45 41.500 7595 53.68.520 3012 54.89.540 2883 56.I5.560 7401 57.47 42.50I 0817 53-70.520 6306 54.I9.540 6253 56.18.56I 0850 57.50 43.501 4039 53-72.520 960I 54-93 -540 9625 56.20.561 4301 57-52 44.501 7263 53-74.52I 2898 54-95.541 2997 56.22.56I 7753 57.54 45 2.502 0488 53.76 2.52I 6196 54.97 2.54I 6371 56.24 2.562 2zo6 57.56 46.502 3714 53.78.52I 9495 54.99 -54I 9746 56.26.562 4660 57-59 47.502 6942 53.80.522 2795 55.02.542 3123 56.29.562 8116 57.61 48.503 0170 53.82.522 6097 55.04.542 6500 56.3I.563 1574 57.63 49.503 3400 53.84.522 9400 55.o6.542 9880 56.33.563 5032 57.65 50 2.503 663I 53.86 2.523 2704 55.08 2.543 3260 56.35 2.563 8492 57.68 51.503 9863 53.88.523 6009 55.I0.543 664I 56.37.564 I953 57-70 52.504 3096 53.90.523 93I6 55.12.544 0024 56.39.564 5416 57.72 53.504 6331 53.92.524 2624 55.I4 -544 3409 56.42.564 8880 57-74 54 -504 9567 53-94 -524 5933 55.-6.544 6794 56.44.565 2345 57-77 55 2.505 2804 53.96 2.524 9243 55.18 2.545 o0I8 56.46 2.565 5812 57.79 56.505 6042 53.98.525 2555 55.20.545 3569 56-48.565 9280 57.81 57.505 9282 54.00.525 5867 55.22.545 6959 56.50.566 2750 57.84 58.506 2522 54.02.525 9181 55-24.546 0350 56.52.566 622z 57.86 59.5o6 5763 54.04.526 2497 55.26.546 3742 56.55.566 9693 57.88 60 2.506 9006 54.06 2.526 5813 55-29 2.546 7I35 56.57 2.567 3166 57-90 597 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit... -128~ 129" 130~ 131~" log ML Dff.. Dff... - log M. Diff. 1'. log M. Diff. 1". 0' z.567 366 57-90 2.588 4112 59.30 2.6Io oi88 60.75 2.632 1622 62.28 1.567 664I 57-93.588 7670 59.32.6o0 3834 60.78.632 5360 62.30..568 0117 57'95.589 I230 59-35.6o 748I 60.80.632 9099 62.33 3.568 3595 57-97 -589 4792 59-37.6II II30 60.83.633 2839 62.35 4.568 7074 57-99 -589 8355 59-39.6II 4781 60.85.633 658i 62.38 5 2.569 0554 58.02 2.590 I9I9 59-4 2.6I I 8433 60.88 2.634 0325 62.4I G 569 4036 58.04.590 5485 59-44.612 2086 60.90.634 4070 62.43 7.569 7519 58.06.590 9052 59-47.6I2 574I 60.93.634 78I7 62-46 8.570 1004 58.09.591 2620 59-49.6I2 9397 60.95.635 1565 62.48 9.570 4490 58.II.591 6190 59.5I.613 3055 60.98.635 5315 62.51 10 2.570 7977 58.I3 2.59I 9762 59-54 2.613 6715 6I.oo 2635 9066 62.54 11.571 1465 58-15 -592 3335 59-56.6I4 0376 6I.03.636 2819 62.56 12.57I 4955 58.18.592 6909 59.58.6I4 4038 6I.o0.636 6573 62.59 1.3 -57I 8447 58.20 -593 0485 59.61.614 7702 6i.08.637 0329 62.61 14.572 1939 58.22.593 4062 59.63.6I5 1368 61.10.637 4087 62.64 15 2.572 5434 58.25 2.593 764I 59.66 2.6i5 5035 61.13 2.637 7846 62.67 16.572 8929 58.27 -594 I221 59.68.6I5 8703 6I.I5.638 1607 62.69 17 573 2426 58.29.594 4803 59-7o.616 2373 6I.I8.638 5369 62.72 18.573 5924 58.32'594 8386 59-73.616 6045 61.20.638 9133 62.75 19.573 9424 58.34'595 I970 59-75.6i6 97I8 161.23.639 2899 62.77 20 2.574 2925 58.36 2.595 5556 59.78 2.617 3392 61.25 2.639 6666 62.80 21.574 6427 58.38.595 9143 59.80.6I7 7068 61.28.640 0435 62.82 22.574 9931 58-41 -596 2732 59.82.618 0746 61.30.640 4205 62.85 23.575 3436 58.43 -596 6322 59.85.618 4425 61.33.640 7977 62.88 24.575 6943 58.45.596 9914 59.87.6I8 8105 61.36.641 1750 62.90 25 2.576 0451 58.48 2.597 3507 59.90 2.619 1787 61.38 2.641 5525 62.93 26.576 3960 58.5o.597 7102 59.92.619 5471 61.41.64I 9302 62.96 f 27 576 7471 58.52.598 0698 59-95.619 9156 61.44.642 3080 62.98 28.577 0983 58.55.598 4295 59.97.620 2843 6I.46.642 6860 63.01 29.577 4496 F 58.57.598 7894 59-99.620 653I 61.48.643 0641 63.04 30 2.577 801i 58.59 2.599 1494 60.02 2.621 0220 6.5I 2-.643 4424 63.06 31.578 I528 58.62.599 5096 60.04.621 391I 61.53.643 8209 63.09 32.578 5045 58.64 -599 8699 60.07.621 7604 6I.56.644 I995 63.12 33.578 8564 58.66.600 2304 60.09.622 298 61.58.644 5783 63.I4 34 -579 2085 58.69.600 591O 60.12.622 4994 6i.6i.644 9572 63.I7 35 2.579 5607 58.71 2.600 9518 60.14 2.622 8691 61.63 2.645 3363 63.I9 36'579 9I30 58.73.601 3127 60.16.623 2390 6I.66.645 7155 63.22 37.580 2655 58.76.6o0 6738 60.19.623 6091 6i.68.646 0949 63.25 38.580 6I8i 58.78.602 0350 60.21.623 9793 61.71.646 4745 63.27 39.580 9708 58.80.602 3963 60.24.624 3496 61.74.646 8542 633.0 40 2.581 3237 58.83 2.602 7578 60.26 2.624 720I 61.76 2.647 2341 63.33 41.58I 6768 58.85.603 1195 60.29.625 0907 G.679.647 6142 63.35 42.582 0299 58.87.603 4813 60.31.625 4615 61.81.647 9944 63.38 43 -582 3832 58.90.603 8432 60.34.625 8325 61.84.648 3748 63-41 44.582 8267 58.92.604 2053 60.36.626 2036 61.86.648 7553 63-44 45 2.583 0903 58-94 2.604 5675 60.38 2.626 5748 61.89 2.649 I360 63-46 46.583 4440 58-97.604 9299 60.41.626 9462 61.91.649 5i68 63-49 47 -583 7979 58-99.605 2924 60.43.627 3178 61.94.649 8978 63.52 48 -584 15I9 59.01.605 6551 60.46.627 6895 61.97.650 2790 63-54 49.584 506i 59-04.606 0179 60.48.628 0614 6I.99.650 6603 63.57 50 2.584 8604 59.o6 2.606 3809 60.5I 2.628 4334 62.02 2.65I 0418 63.60 51.585 2148 59-09.606 7440 60.53.628 8056 62.04.65I 4235 63.62 52 -585 5694 59.11.607 1073 60.56.629 1780 62.07.651 8053 63.65 53.585 9241 59.13.607 4707 60.58.629 5505 62.o09 652 I873 63.68 54.586 2790 59.16.607 8343 6o.61.629 9231 62.12.652 5695 63.70 55 2.586 6340 59. 8 2.608 1980 60.63 2.630 2959 62.15 2.652 9518 63-73 56.586 9891 59.20.608 5618 60.66.630 6689 62.17.653 3342 63.76 57.587 3444 59-23.608 9258 60.68.63I 0420 62.20.653 7168 63.79 58.587 6999 59.25.609 2901 60.70.63I 4I52 62.22.654 0996 63.81 59.588 0555 59.27.609 6544 60.73.63I 7887 62.25.654 4826 63.84 60 2.588 412 59-30 2.610 oi88 60.75 2.632 1622 62.28 2.654 8657 63-87 598 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1320 1330 134~ 1350 ve log M. Diff. 1". log 1. Diff. 1". log MI. Diff. 1". log M4. Diff. 1". 0' 2.654 8657 63.87 2.678 I547 65.53 2.702' 0562 67.27 2.726 5990 69.09 1.655 249o 63.89.678 5480 65.56.702 4600 67.30.7Z7 OI37 69.I2 2.655 6324 63.92.678 9414 65.59.702 8638 67.33.727 4285 69.I5 3.656 o16o 63.95.679 3350 65.6I.703 2679 67.36.727 8435 69.I9 4.656 3998 63.97.679 7288 65.64.7o3 672I 67.39.728 2587 69.22 5 2.656 7837 64.00 2.680 I227 65.67 27o04 0766 67.42 2.728 6741 69.25 6.657 1678 64.03.680 5i68 65.70.704 4812 67.45.729 0897 69.28 7.657 552I 64.06.68o 9I I 65.73.704 8860 67.48.729 5055 69.31 8.657 9365 64.08.68I 3056 65.76.705 2909 67.51.729 9215 69.34 9.658 32Ii 64. II.68I 7002 65.79.705 6961 67.54.730 3376 69.37 10 z.658 7058 64.I4 2.682 0950 65.8I 2.706 1014 67.57 2.730 7539 69.40 11.659 0907 64.1I7.682 490O 65.84.706 5069 67.60.731 I705 69-44 12.659 4758 64.19.682 885I 65.87.706 9I26 67.63 *73I 5872 69.47 13.659 86I 64.22.683 2804 65.90.707 3184 67.66.732' 004I 69.50 14.660 2465 64.25.683 6759 65.93.707 7244 67.69.732 42I2 69.53 15 2.660 6320 64.28 2.684 07I6 65.96 2.708 I307 67.72 2.732 8385 69.56 16.66I 0178 64.30.684 4674 65.99.708 5371 67.75.733 2559 69.59 17.66I 4037 64.33.684 8634 66.oI.708 9436 67.78.733 6736 69.62 18.661 7897 64.36.685 2596 66.o04 709 3504 67.81 -734 0914 69.66 19.662 I760 64.38.685 6559 66.07'709 7573 67.84.734 5094 69.69 20 2.662 5623 64.4I 2.686 o524 66.Io 2.7Io I645 67.87 2.734 9277 69.72 21.662 9489 64.44.686 449I 66.13.7Io 5718 67.90'735 346I 69.75 22.663 3356 64.47.686 8460 66.I6.710 9792 67.93.735 7647 69-78 23.663 7225 64.49.687 2430 66.19.7II 3869 67.96.736 I835 69.81 24.664 IO96 64.52.687 6402 66.22.7I 7947 67.99.736 6025 69.85 25 2.664 4968 64.55 2.688 0376 66.25 2.712 2028 68.02 2.737 0216 69.88 26.664 8842 64.57.688 4352 66.27.7I2 6IIO 68.05.737 4410 69.91 27.665 27I7 64.60.688 8329 66.30.713 OI94 68.og8 737 8605 69.94 28.665 6594 64.63.689 2308 66.33.7I3 4279 68.I.738 2803 69.97 29.666 0473 64.66.689 6289 66.36.713 8367 68.I4.738 7002 70.00 30?2.666 4354 64.69 2.690 o272 66.39 2.7I4 2456 68.17 2.739 I203 70.04 31.666 8236 64.72.690 4256 66.42.7I4 6547 68.20.739 5406 70.07 32.667 2120 64.74.690 8242 66.45.715 0640 68.23'739 9612 70.IO 33.667 6005 64.77.69gI 2230 66.48.715 4735 68.26.740 3819 70.I3 34.667 989z 64.80.691 6219 66.5 I 715 8832 68.29.740 8027 70.I6 35 2.668 3781 64.83 2.692 0210 66.54 2.7I6 2930 68.32 2.741 2238 70.20 36.668 7672 64.86.692 4203 66.56.7I6 703I 68.35.74I 645I 70.23 37.669 1564 64.88.692 8198 66.59.717 II33 68.38.742 o666 70.2z) 38.669 5457 64.9I.693 2194 66.62.717 5237 68.41.742 4882 70.29 39..669 9353 64.94.693 6193 66.65'717 9342 68.44'742 9IOI 70.32 40 2.670 3250 64.97 2.694 OI93 66.68 2.7I8 345o 68.48 2.743 3321 70.36 41.670 7149 65.oo.694 4194 66.71.7I8 7560 68.5I.743 7543 70.39 42.671 I050 65.o02.694 8I98 66.74.7I9 1671 68.54 -744 1768 70.42 43.67I 4952 65.o5.695 2203 66.77.719 5784 68.57 -744 5994 70.45 44.671 8856 65.08.69-5 62zo 66.80.7I9 9899 68.60.745 0222 70.48 45 2.672 276I 65.II 2.696 oz029 66.83 2.720 40I6 68.63 2.745 4452 70.52 46.672 6668 65.I3.696 4229 66.86.720 8I35 68.66.745 8684 70o55 47.673 0577 65.16.696 8242 66.89.721 2255 68.69.746 2918 70.58 48.673 4488 65.I9.697 2256 66.92.72I 6377 68.72.746 7154 70.6I 49.673 8400 65.22.697 6272 66.95.722 o502 68.75.747 139I 70.65 50 2.674 2314 65.25 2.698 o289 66.97 2.722 4628 68.78 2.747 563I 70.68 51.674 6230 65.28.698 4308 67.00.722 8756 68.8I.747 9873 70.71 52.675 0147 65.30.698 8330 67.03.723 2885 68.84.748 41I6 70-74 53.675 4066 65.33.699 2353 67.06.723 70I7 68.88.748 8362 70.78 54.675 7987 65.36.699 6377 67.09.724 II50 68.9I.749 2609 70.81 55 2.676 I909 65.39 2.700 0404 67.12 2.724 5286 68.94 2.749 6859 70.84 56.676 5833 65.42.700 4432 67.15.724 9423 68.97.750 IIIO 70.87 57.676 9759 65.44.700 8462 67.I8'725 3562 69.00.750 5364 70.90 58.677 3687 65.47.701 2494 67.21.725 7703 69.o3'750 96I9 70.94j 59.677 76I6 65.50.70I 6527 67.24 *726 1846 69.o6'75I 3876 70.97 o60 2.678 I547 65.53 12.702 0562 67.27 2.726 5990 69.o9 2.751 8135 71.00 599 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 136" 1370 138~ 139~ V. loM. Dff.. log M. Diff. 1". log M. D fDi.. log M. Diff. 1". 0' 2.75 8135 7I.00 2.777 7322 73.0I 2.804 3895 75.11 2.831 8224 77.32 1.752 2396 7I.03.778 1703 73-04.804 8403 75.I4.832 2864 77-35 2.752 6659 7I.07 -778 6087 73.07.805 2912 75.18.832 7506 77-39 3.753 0925 7I.10.779 0472 73I11.805 7424 75.2I.833 2151 77-43 4 -753 I592 71.13.779 4859 73.14.806 1938 75-25.833 6798 77-47 5 2.753 9461 71-17 2.779 9249 73.I8 2.806 6454 75.29 2.834 I447 77.50 6.754 3732 71.20.780 364I 73.21.807 0973 75.32.834 6098 77.54 7.754 o004 7.123.780 8034 73.24.807 5493 75.36.835 0752 77.58 8.755 2279 71.26.78I 2430 73.28.808 ooi6 75.40.835 5408 77.62 9.755 6556 7I-30.781 6828 73-3I.808 4541 75-43.836 oo0066 77.66 10 2.756 0835 71.33 2.782 1228 73.35 2.808 9068 75-47 2.836 4727 77.69 11.756 51I6 71.36.782 5630 73.38.809 3597 75-50.836 9390 77-73 12.756 9399 71.40.783 0034 73.42.809 8128 75-54.837 4055 77-77 13.757 3683 71-43.783 4440 73.45.8io 2662 75.58.837 8722 77.81 14.757 7970 71.46.783 8848 73.49.8Io 7I97 75.6I.838 3392 77.85 15 2.758 2259 7149 2-784 3258 73.53 2.811 I735 75.65 2.838 8064 77.89 16.758 6549 71.53.784 7671 73.56.8I 6275 75.69.839 2738 77.92 17.759 0842 71.56 -785 2085 73-59.812 0817 75.72.839 7414 77.96 18.759 5137 71.59.785 6502 73.63.8I2 5362 75.76.840 2093 78.00 19.759 9433 71.63.786 0920 73.66.8Iz 9908 75-79.840 6774 78.04 20 2.760 3732 71.66 2.786 534I 73.70 2.813 4457 75.83 2.841 1458 78.08 21.760 8032 7I.69.786 9764 73.73.8I3 9008 75.87.84I 6I44 78.1 22.761 2335 71.73.787 4189 73.76.814 356I 75.90.842 0832 78.15 23.76I 6639 71.76.787 8615 73.80.814 8117 75-94.842 5522 78.19 24.762 0946 71.79.788 3044 73.83.8I5 2674 75.98.843 0215 78.23 25 2.762 5255 7.183 2.788 7476 73.87 2.815 7234 76.01 2.843 4909 78.27 26.762 9565 71.86.789 I909 73.90.8I6 1796 76.05.843 9607 78.31 27.763 3878 71.89.789 6344 73-94.816 6360 76.09 -844 4306 78.35 28.763 8192 7193.790 078I 73.97.817 0927 76.12.844 9008 78.38 29.764 2509 71.96.790 522I 74.01.817 5495 76. 6.845 37I2 78.42 30 2.764 6827 71-99 2.790 9662 74-04 2.818 0066 76.20 2.845 8419 78.46 31.765 1148 72.03 -791 4I06 74.08.8I8 4639 76.23.846 3128 78.50 32.765 5470 72.06.791 8552 74.11.818 9214 76.27.846 7839 78.54 33.765 9795 72.09.792 3000 74.15.8I9 3792 76.31.847 2553 78.58 34.766 4121 72.13.792 7450 74-18.819 8371 76.34.847 7268 78,62 35 2.766 8450 72.16 2.793 1902 74.22 2.820 2953 76.38 2.848 I986 78.66 36.767 2781 72.19 -793 6356 74.25.820 7537 76.42.848 6707 78.69 37.767 7113 72.23 -794 0813 74.29.821 2I23 76.46.849 1430 78.73 38.768 1448 72.26.794 5271 74.32.821 6712 76.49.849 6155 78.77 39.768 5784 72.29 -794 973I 74-36.822 1302 76.53.850 0882 78.81 40 2.769 0123 72-33 2-795 4I94 74-40 2.822 5895 76.57 2.850 5612 78.85 41.769 4464 72.36.795 8659 74-43.823 0491 76.60.85 0344 78.89 42.769 8806 72.39.796 3126 74-47 -823 5088 76.64.851 5079 78.93 43.770 3151 72.43.796 7595 74.50.823 9688 76.68.85I 9816 78.97 44.770 7498 72.46.797 2066 74.54.824 4289 76.72.852 4555 79-01 45 2.77I 1846 72.50 2.797 6539 74.58 2.824 8894 76.75 2.852 9297 79.05 46.771 6I97 72.53.798 IO15 74.61.825 3500 76.79.853 4041 79.08 47.772 0550 72.56.798 5492 74.64.825 8Io8 76.83.853 8787 79.I2 48.772 4905 72.60.798 9972 74.68.826 2719 76.87 -854 3535 79.I6 49.772 9262 72.63.799 4454 74-71.826 7332 76.90.854 8286 79.20 50 2.773 3621 72.67 2.799 8938 74.75 2.827 1947 76.94 2.855 3040 79.24 51.773 7982 72.70.800 3424 74-79.827 6565 76.98.855 7795 79-28 52.774 2344 72-73.800 7912 74.82.828 Ix85 77.01.856, 2553 79.32 53.774 6709 72.77.801 2402 74.86.828 5807 77-05.856 7314 79-36 54 -775 1077 72.80.80o 6895 74-89 -829 0431 77-09 -857 2077 79-40 55 2.775 5446 72.84 2.802 I390 74-93 2.829 5058 77.13 2.857 6842 79.44 56.775 9817 72-87.802 5886 74-96.829 9686 77. 6.858 1609 79-48 57.776 4190 72.90.803 0385 75-00.830 4317 77.20.858 6379 79-52 58.776 8565 72.94.803 4886 75.04.830 8951 77.24.859 I151 79.56 59 777 2942 72.97 -803 9390 75-08.83I 3586 77.28.859 5926 79.60 60 2.777 7322 73.0I 2.804 3895 75.II 2.831 8224 77.32 2.860 0703 79.64 600 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. I 140~ 141~ 142~ 143~ Vo... lo M. Diff. 1". log M. Diff. 1". log M. Diff. 1" o. lo Diff. 1. 0' 2.860 0703 79.64 2.889 1754 82.08 2.9I9 1831 84.65 2.950 I420 87.37 1.860 5482 79.68.889 668o 82.12.919 6911 84.70.950 6664 87.41 2.86I 0264 79.72.890 1609 82.16.920 1994 84.74.951 I910 87-46 3.86I 5048 79.76.890 6540 82.20.920 7080 84.78.951 7159 87.50 4.86I 9835 79.80'.891 I473 82.25.92I 2I69 84.83.952 2411 87-55 5 2.862 4624 79.84 2.89I 6409 82.29 2.921 7260 84.87 2.952 7665 87.60 6.862 9415 79.88.892 1348 82.33.922 2353 84.92.953 2923 87.65 7.863 4209 79.92.892 6289 82.37.922 7450 84-96.953 8183 87.69 8.863 9005 79.96.893 1233 82.41.923 2549 85.01 -954 3446 87.74 9.864 3803 80.oo.893 6179 82.46.923 7650 85.05.954 8711 87.79 10 2.864 8604 80.04 2.894 1127 82.50 2.924 2755 85.Io 2.955 3980 87.83 11.865 3408 80.o8.894 6078 82.54.924 7862 85.I4 -955 9251 87.88 12.865 82I3 80.I2.895 1032 82.58.925 2972 85.18.956 4525 87.93 13.866 3021 80.I6.895 5989 82.63.925 8084 85.23.956 9802 87.97 14.866 7832 80.20.896 0948 82.67:926 3199 85.27.957 5082 88.02 15 2.867 2645 80.24 2.896 5909 82.7I 2.926 8317 85.32 2.958 0365 88.07 16.867 7460 80.28.897 0873 82.75.927 3437 85.36.958 565I 88. I 17.868 2278 80.32.897 5839 82.79.927 8560 85.41.959 0939 88.I6 18.868 7098 80.36.898 o808 82.84.928 3686 85.45.959 6230 88.21 19.869 I921 80.40.898 5780 82.88.928 8814 850 5 0.96 I524 88.26 20 2.869 6746 80.44 2.899 0754 82.92 2.929 3945 85.54 2.960 6821 88.30 21.870 1573 80.48.899 5730 82.96.929 9079 85.59.96I 2120 88.35 22.870 6403 80.52.900 0709 83.01.930 42I6 85.63.96I 7423 88.40 23.871 1235 80.56.900 5691 83.05 -930 9355 85.68.96~ 2728 88.45 24.871 6070 80.60.901 0675 83.09.931 4497 85.72.962 7036 88.49 252.872 0907 80.64 2.901 5662 83.13 2.93I 9641 85.77 2.963 3347 88.54 26.872 5747 80.68.902 o65I 83.18.932 4788 85.81.963 8661 88.59 27.873 0589 80.72.902 5643 83.22.932 9938 85.86.964 3978 88.64 28 -873 5433 80.76.903 o638 83.26.933 5091 85.91.964 9297 88.68 29.874 0280 80.80.903 5635 83.3I'934 0247 85.95.965 4620 88.73 30 2.874 5129 80.84 2.904 0635 83.35 2.934 5405 85.99 2.965 9945 88.78 31.874 9981 80.88.904 5637 83.39.935 0565 86.04.966 5273 88.83 32.875 4835 80.92.905 0642 83.43.935 5729 86.08.967 0604 88.87 33.875 9692 80.96.905 5649 83.48.936 0895 86.13.967 5938 88.92 34.876 455I 8I.oI.9o6 0659 83.52.936 6064 86.17.968 1275 88.97 35 2.876 94I3 81.o5 2.906 5672 83.56 2.937 I236 86.22 2.968 6615 89.02 36.877 4277 81.09.907 0687 83.61.937 64IO 86.26.969 1957 89.07 37.877 9143 81.13.907 5704 83.65 -938 1587 86.3I.969 7303 89.12 38.878 4012 81.I7.908 0725 83.69.938 6767 86.35.970 265I 89.I7 39.878 8883 81.21.908 5748 83.74.939 I950 86.40.970 8002 89.21 40 2.879 3757 81.25 2.909 0773 83.78 2.939 7135 86.45 2.971 3356 89.26 41.879 8633 81.29.909 580I 83.82.940 2323 86.49.971 8713 89.31 42.880 3512 81.33.910 0832 83.87.940 7514 86.54.972 4073 89.36 43.880 8393 81.37.910 5865 83.91.941 2708 86.58.972 9436 89.40 44.881 3277 81.42.911 0901 83.95.941 7904 86.63.973 4801 89.45 45 2.881 8163 81.46 2.91 5940 83.99 2.942 3103 86.67 2.974 0170 89.50 46.882 3052 8I.50.912 0981 84.04 -942 8305 86.72.974 5541 89.55 47.882 7943 81.54.912 6024 84.08.943 3510 86.77.975 0916 89.60 48.883 2837 81.58.9I3 1070 84.13 -943 8717 86.81.975 6293 89.65 49.883 7733 81.62.913 6II9 84.I7 -944 3927 86.86.976 1673 89.69 50 2.884 2631 8I.66 2.914 1171 84.22 2.944 9140 86.90 2.976 7056 89-74 51.884 7532 81.70.914 6225 84.26.945 4355 86.95.977 2442 89.79 52.885 2436 81.75.915 1282 84.30.945 9574 87.00.977 783I 89.84 53.885 7342 81.79.915 6341 84.34.946 4795 87.04.978 3223 89.89 54.886 2251 81.83.916 I403 84.39 -947 00I9 87.09.978 86x8 89.94 55 2.886 7162 8I.87 2.916 6468 84.43 2.947 5245 87.13 2-979 4015 89.99 56.887 2075 8I.91.917 I535 84.48.948 0475 87.18.979 9416 90.03 57.887 6991 81.95.917 6605 84.52.948 5707 87.23.980 4820 90.08 58.888 1910 81.99.918 1678 84.56.949 0942 87.27.98I 1226 90.13 59.888 6831 82.04.918 6753 84.61.949 6I80 87.32.981 6636 90.18 60 2.889 1754 82.o8 2.919 I83I 84.65 2.950 1420 87.37 2.982 1048 90.23 601 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. -1440 145" 146" 1470 f....... log M. Diff. 1". log M. Dif. 1". log I. Diff.. log M. Diff. i". O' 2.982 1048 90.23 3-I15 1281 93.26 3-049 2733 96.47 3-084 6070 99.87 1.982 6463 90.28.5I 6878 93.31.049 8522 96.52.085 2o64 09-92 2.983 I882 90.33.oi6 2478 93.36.050 43I5 96.58.085 8o6i 99.98 3.983 7303 0o.38.oi6 80z 93 05 82 96.63.o86 4062 Ioo.o4 4.984 2727 90.43 I017 3688 93-47.05I 59I 96.69.087 oo66 o100.1 5 2.984 8154 90.48 3.017 9298 93-52 3.052 I7I4 96.74 3.087 6073 Ioo.16 6 -985 3584 90.53.018 4911 93-57.052 7520 96.80.088 2085 100.22 7.985 9017 90.58.019 0526 93.62.053 3329 96.85.088 8099 I00.28 8.986 4453 90.63.o19 6145 93.68.053 9142 96.91.o89 4118 100.33 9.986 9892 90.67.020 1768 93-73.054 4959 96.96.090 0140 100.39 10 2.987 5334 90-72 3.020 7393 93.78 3-055 0778 97.01 3.090 6I65 100.45 1I.988 0779 90.77.021 3021 93.83.055 660I 97-07.091 2194 100.51 12.988 6227 90.82.021 8653 93.89.o56 2427 97.13.091 8226 I00.57 13.989 I678 90.87.022 4288 93.94.056 8256 97I19.092 4262 100.63 14.989 7132 90.92.022 9926 93.99.057 4089 97-24.093 0302 o00.69 15 2.990 2589 90.97 3.-23 5567 94.04 3.057 9925 97.30 3.093 6345 100.75 16,990 8049 91.02.024 1211 94.10.058 5765 97-35.094 2392 100.81 17.991 3512 91.07.024 6859 94-I5.059 I608 97.41.094 8442 100.87 18.991 8977 91.I2.o25 2509 94.20.059 7454 97-47 -095 4496 I00.93 19.992 4446 91-17.025 8163 94.26.o6o 3304 97.52.096 0553 100.98 20 2.992 9918 91.22 3.026 3820 94.31 30.60 9157 97.58 3.o96 6614 IOI.04 21.993 5393 91.27.026 9480 94.36.061 5013 97.63.097 2678 IOI.IO 22.994 0871 91.32.027 5143 94.41.062 0873 97.69.097 8746 ioi.i6 23.994 6351 91.37.028 o08I 94,47.062 6736 97-75.098 4818 I0.22 24.995 I835 91.42.028 6479 94-52.o63 2602 97.80.099 o893 101.28 25 2.995 7322 91.47 3.029 2152 94-57 3.063 8472 97.86 3.099 6972 10134 26.996 2812 91.52.029 7828 94.63.064 4345 97.91.Ioo 3054 101.40 27.996 8305 91.57.030 3507 94.68.065 0222 97-97.IOO 9140 101.46 28.997 3801 91.62.030 9190 94-73.065 6101 98.03.101 5230 101.52 29.997 9300 91.67.o31 4875 94-79.o66 1985 98.08.102 1323 101.58 30 2.998 4802 91.72 3.032 0564 94.84 3.o66 7872 98.14 3.I02 7420 101.64 31 -999 0307 91.77.032 6256 94.89.067 3762 98.20.103 3520 101.70 32.999 5815 91.82.033 1951 94-94.067 9655 98-25 I103 9624 101.76 33 3.000 1326 91.87.033 7650 95.00.068 5552 98-.31 104 5732 101.82 34.ooo 6840 91.93.034 335I 95-05 069 1453 98.37.I05 1843 101.88 35 3.001 2357 91.98 3.034 9056 95.11 3.069 7357 98.42 3.105 7958 101.94 36.o00 7877 92.03.035.4764 95-.6.070 3264 98.48.Io6 4076 102.00 37.002 3400 92.08.036 0475 95.22.070 9174 98.54.I07 OI98 102.07 38.002 8926 92.13.036 6190 95.27.071 5088 98.60.107 6324 102.13 39.003 4456 92.18.037 1908 95.32.072 Ioo6 98.65.-08 2454 102.I9 40 3.003 9988 92.23 3-037 7629 95.38 3.072 6927 98.71 3-108 8587 102.25 41.004 5523 92.28.038 3353 95-43 0.73 2851 98.77.109 4723 I02.31 42.005 1062 92.33.038 9080 95.48.073 8779 98.82.I11 0864 o12.37 43.005 6603 92.38.039 4811 95-54.074 4710 98.88.IIo 7008 102.43 44.oo6 2148 92.44.040 0545 95.60.075 0645 98.94.III 3155 102.49 45 3.006 7696 92.49 3-040 6282 95.65 3.075 6583 99.00 3.III 9306 102.55 46.007 3246 92.54.041 2023 95.70.076 2524 99.05.112 5461 102.61 47.oo7 8800oo 92.59.o41 7767 95.76.076 8469 99.11.113 1620 102.67 48.oo8 4357 92.64.042 3514 95.81.077 4418 99-17'113 7782 102.73 49.oo8 99I7 92.69.042 9264.95.86.078 0370 99.23.II4 3948 102.80 50 3-009 5480 92.74 3-043 5017 95.92 3.078 6325 99.28 3.115 o1I8 102.86 51.010 1046 92.79 -044 0774 95-97.079 2284 99-34'115 6291 I02.92 52.o0o 6615 92.85.044 6534 96.03.079 8246 99.40.116 2468 102.98 53.OII 2188 92.90.045 2297 96.08.o80 4212 99.46.116 8649 103.04 54.o I 7763 92-95.045 8064 96.I4.081 o181 99.52.117 4833 I03.10 55 3.012 3342 93-00 3.046 3834 96.19 3.08I 6154 99-57 3.18 1022 103.16 56.I12 8923 93.05.046 9607 96.25.082 2130 99.63.II8 7213 103.23 57.013 4508 93.10.047 5383 96.30.082 8IIo 99.69.119 3409 103-29 58.oI4 0096 93.16.048 1163 96-36.083 4093 99-75.II9 9608 103.35 59.014 5687 93.21.048 6946 96.41.084 0080 99.81.120 5811.03-41 60 3.015 Iz28 93-26 3.049 2733 96.47 3 084 6070 99.87 3.121 2018 103.48 602 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 148" 1490 1500 1510 V,. log M. Diff. 1". log M. Diff. 1". Dlogiff. 1". log M. Diff. 1". 0' 3.12I 2018 103-48 3-I59 I367 I07-31 3-I98 4984 111.41 3-239 3820 115-77 1 z.12 8228 I03.54.I59 7808 107.38.I99 1671 111.48.240 0768 115-85 2.122 4442 1 I03.60.i60 4253 107-45 199 836I 11.55.240 7722 I15-92 3.I23 0660 103.66.16I 0702 107-.51 -zo 5056 111.62.241 4680 II6.oo 4.123 6882 103-72.i6i 7154 107-58.201 1755 111.69.242 1642 II6.08 5 3.124 3107 103.79 3.I62 3611 I07.65 3.201 8459 III.76 3.242 8608 II6.I5 6.124 9336 I03-85.163 0072 I07-7I.202 5166 111.83.243 5580 II6.23 7.I25 5569 o03.9I.163 6536 107.78.203 1878 II.90o.244 2556 116.30 8.2z6 I805 I03.97.164 3005 107.85.203 8594 111.97 -244 9536 II6.38 9.126 8045 104.04.164 9478 107.91.204 5315 112.04.245 6521 II6.45 10 3.-27 4289 I04.10 3-I65 5955 107.98 3.205 2040 112.11 3.246 3511 1I6.53 11.128 0537 104.16.166 2435 108.04.205 8769 112.18.247 0505 II6.6I 12.128 6789 104.22.i66 8920 108.1.206 5502 112.26.247 7503 116.68 13.129 3044 104.29.i67 5409 I08.I8.207 2239 112.33.248 4507 II6.76 14 I.29 9303 104.35.i68 1901 108.25.207 898I 112.40.249 1515'16.84 15 3.130 5566 104.41 3.-68 8398 108.31 3.208 5727 112.47 3-249 8527 x16.91 16.131 1833 104.48.I69 4899 108.38.209 2478 II2.54 -250 5544 116.99 17.131 8103 104.54.170 1404 108.45.209 9232 112.61.251 2566 117.07 18.132 4377 104.60.170 7913 108.51.210 5991 112.69.251 9592 117.14 19.133 0655 104.67.171 4426 IO8.58.211 2755 112.76.252 6623 117.22 20 3.133 6937 104.73 3.172 0942 o08.65 3.-211 9522 112.83 3.253 3658 117.30 21.134 3223 104.79 -172 7463 108.72.212 6294 112.90.254 0698 117.37 2.134 9512 104.86.173 3988 108.78.213 3070 112.97.254 7743 117.45 23.135 5805 104.92.174 0517 108.85.213 9851 113-05.255 4792 117.53 24.136 2102 104.98.174 7051 108.92.214 6636 113.12.256 1846 117.60 25 3.I36 8403 105.05 3-175 3588 108.99 3.215 3425 113.19 3-256 8905 117.68 26.I37 4708 105.11.176 0129 109.06.2I6 o029 113.26.257 5968.117.76 27.138 o106 105.17.I76 6674 o19.12.z26 7017 113-34.258 3036 117.84 28.138 7329 I05.24.177 3224 109.19.217 38I9 113.41.259 0109 117.91 29.139 3645 105-30.177 9777 109.26.28 0626 113.48 259 7186 17.99 30 3.-39 9965 105-36 3.178 6335 109.33 3.218 7437 I13.55 3.260 4268 II8.07 31.140 6289 105.43.179 2897 109.40.219 4252 113.63.261 I354 118.15 32.141 2616 105.49.179 9462 o09.46.220 1072 113-70.261 8446 118.23 33.I41 8948 105.55.i8o 6032 109.53.220 7896 113.77.262 5542 118.30 34.142 5283 105.62.181 2606 109.60.221 4724 II3-84.263 2642 II8.38 35 3-143 I622 105.68 3.181 9184 109.67 3.222 1557 113.92 3.263 9747 118.46 36.143 7965 105.75.182 5766 I09.74.222 8395 II3.99.264 6857 118.54 37.144 4312 I05.81.183 2353 109.81.223 5236 114.06.265 3972 II8.62 38.145 0663 105.87 -183 8943 109.87.224 2082 114.14.266 1091 118.70 39.145 7018 105-94 -I84 5538 109.94.224 8933 II114.2I1.266 8216 118.77 40 3.146 3376 Io6.oo 3.185 2I36 IIo.oI 3.225 5788. 114.28 3.267 5345 1I8.85 41.146 9739 106.07 -185 8739 II.08.226 2647 II4-36.268 2478 II8.93 42.147 6105 0o6.14.I86 5346 IIo.15.226 9511 114.43.268 9616 19.OI 43.I48 2475 106.20.187 1957 110.22.227 6379 114.51.269 6759 119.09 44.148 8849 106.27.187 8572 110.29.228 3252 114.58.270 3907 119-17 45 3-149 5227 I06.33 3.188 5192 110.36 3.229 0129 114.65 3.27I o060 119.25 46.I50 I609 106.40.189 1815 II0.43.229 7010 114.73.271 8217 1I9.33 47.I50 7995 0 6.46.189 8443 IIo.o5.230 3896 114.80.272 5379 19.41 48.151 4385 I06.53.190 5075 110.57.231 0786 114.88.273 2546 119.49 49.152 0778 106.59.191 1711 II0.64.231 7681 I14-95.273 9717 II9-57 50 3.152 7176 Io6.66 3-191 8351 I10.71 3-232 4581 115.03 3-274 6894 119.65 51.I53 3577 I06.72.192 4996 110.77.233 I484 115.10.275 4075 119.73 52.153 9983 106.79.193 1644 110.84.233 8392 115.17.276 1261 119.81 53.154 6392 106.85.-93 8297 110.91.234 5305 II5-25.276 8452 119.89 54.155 2805 I06.92.194 4954 110.98.235 2222 115-32.277 5647 I19.97 55 3.155 9222 106.99 3-195 i615 111.05 3-235 9144 115-40 3.278 2848 120.05 56.156 5643 107.05 -195 828I 111.12.236 6070 115-47.279 0053 120.13 57.157 2068 I07-12.I96 4950 I111.9.237 3001 II5-55.279 7263 120.21 58.157 8497 107.18.197 I624 11I.26.237 9936 115.62.280 4477 120.29 59.158 4930 I07.25.197 8302 111.34.238 6876 115.70.281 I697 120.37 60 3-159 I367 107.31 3-198 4984 111I.41 3-239 3820 115-77 3-281 8921 120.45 603 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1520 1530 154~0 1550 log M. Diff. 1". log M. Diff. 1". 1logM. Diff. 1. lo. Diff. 1". 0' 3.281 8921 120.45 3.326 1448 I25.46 3.372 2684 130.85 3.420 4064 136.66 1.282 6151 I2o0.53.326 8978 125.55.373 0538 130-94.421 2266 136.76 2.283 3385 120.61.327 6513 I25.63.373 8397 131.04.422 0475 136.86 3.284- 0624 20.69.328 4054 I25.72.374 6262 131.-13.422 8690 136.96 4.284 7868 120.77.329 i6oo 125.81.375 4133 131.22.423 6910o 13706 5 3.285 5I16 120.85 3.329 9151 125.89 13376 2009 131-32 3'424 5137 137.16 6.286 2370 120.93.330 6707 125.98.376 9890 131-41.425 3370 137.26 7.286 9628 i21.01.331 4268 126.07.377 7778 131.50.426 160o 137-37 8.287 6891 121.10.332 1835 126.16.378 5671 131.60.426 9854 137.47 9.288 4160 121.18.332 9407 126.24.379 3570 131.69.427 8105 137-57 10 3.289 1433 121.26 3.333 6984 126.33 3.380 I474 131.79 3.428 6362 137.67 11.289 8711 121.34 -334 4567 126.42.380 9384 131-88.429 4626 137-77 12.290 5993 I21.42.335 2154 126.51.38I 7300 131.98.430 2895 137-88 13.291 3281 121.50.335 9747 126.59 -382 5221 132.07.431 1171 137.98 14.292 0574 121.59 -336 7346 I26.68.383 3148 132.16.431 9452 138.08 15 3.292 7872 121.67 3-337 4949 126.77 3.384 I08I 132.26 3.432 7740 138.18 16.293 5174 I21.75.338 2558 I26.86.384 9019 132.35.433 6034 138.29 17.294 2481 121.83'339 0172 26.95 -.385 6963 I32.45.434 4334 138-39 18.294 9794 121.91.339 7792 127.03.386 4913 132.54.435 2641 138-49 19.295 7111 122.00.340 5417 127.12.387 2869 132.64.436 0953 138.59 20 3.-296 4433 122.08 3-341 3047 127.21 3.388 0830 132.73 3.436 9272 138.70 31.297 1761 122.16.342 0682 127.30.388 8797 132.83.437 7597 138.80 22.297 9093 122.24.342 8323 127.39.389 6770 132.93.438 5928 138.90 23.298 6430 122.33.343 5969 127.48.390 4749 133.02.439 4266 139-01 24.299 3772 I22.41.344 3620 127.57.391 2733 133-12.440 2609 139-11 25 3.300 II1119 122.49 3.345 1277 I27.66 3.392 0723 133.22 3.441 0959 I39-22 26.300 8471 122.58.345 8939 127.75.392 8719 133.31.44-1 9315 139-32 27.301 5828 122,66.346 6606 127.84.393 6720 133.41.442 7677 13942 28.302 3190 I22.74.347 4279 I27-93.394 4728 133-50.44.3 6046 139.53 29.303 0557 122.83.348 1958 128.02.395 2741 133.60.444 4421 139-63 30 3.303 7929 122.91 3.348 9641 128.11 3.396 0760 133.70 3.445 2802 139-74 31.304 5306 122.99 -349 7330 128.19.396 8785 133.79.446 1189 139-84 32.305 2688 123.08.350 5024 128.28.397 6815 133.89.446 9583 139-95 33.306 0075 123.16.351 2724 128.37.398 4852 133.99.447 7983 140.05 34.306 7468 123.24 -352 0429 i28-.6.399 2894 134.09.448 6389 140-16 35 3-307 4865 123.33 3.352 8140 128.55 3.400 0942 134-.19 3.449 4802 140.26 36.308 2267 123.41 -353 5856 128.65.400 8996 I34.28.450 3221 140.37 37.308 9674 123.5o0 354 3577 128.74.401 7056 134.38.451 1646 140-47 38.309 7086 123.58.355 1304 128.83.402 5122 134-48.452 0077 140-57 39.310 4504 123.66.355 9037 128.92.403 3193 134-57.452 8515 140.68 40 3-311 1926 123.75 3.356 6774 129.01 3.404 1270 134.67 3.453 6959 140-79 41.-311 9354 123-83 -357 4517 129.10'404 9354 134-77.454 5410 140.90 4:2.312 6786 123.92.358 2266 129.19.405 7443 134.87.455 3867 141.00 43.313 4224 124.00.359 0020 129.28.406 5538 134.97.456 2330 141.II 44.314 1667 124.09.359 7780 129.37 -407 3639 I35.07.457 0800 141.21 45 3.314 9115 124.17 3-360 5545 I29.46 3.408 1746 135.16 3.457 9276 141.32 46 -315 6567 124.26.361 3316 129.56.408 9859 135.26.458 7759 141-43 47.316 4025 124-34.362 1092 12).65.409 7977 135.36.459 6248 141-54 48 -317 1489 124.43.362 8873 129.74.410 6102 135.46.460 4743 141.64 49.317 8957 124-51 363 6660 129.83 -411 4233 I35.56.461 3245 141-75 50 3.318 6430 124.60 3.364 4453 129.92 3.412 2369 135.66 3-462 1753 141.86 51 -319 3909 124.68.365 2251 130.01.413 0512 135.76.463 oz68 141-97 52 -320 1392 1 24.77.366 0055 130.11.413 8660 135.86.463 8789 142.07 53.320 888I 124.86.366 7864 130.20.414 6815 135.96.464 7317 142.18 54.321 6375 124.94 -367 5679 130.29.415 4975 136.06.465 5851 142.29 55 3-322z 3874 125.03 3.368 3499 130.38 3.416 3142 136.1I6 3.466 4392 142.40 56.323 1379 125.11.369 1325 I30.48.417 1314 136.26.467 2939 142.51 57'.323 8888 125.20.369 9156 130.57.417 9492 136.36.468 1492 142.61 58.324 6403 I25.29.370 6993 130.66.418 7677 136.46.469 0052 142.72 59.325 3923 I25-37.371 4836 130.76.419 5867 136.56.469 8619 142.83 60 3-326 1448 125.46 3-372 2684 130.85 3.-420 4064 136.66 3.470 7192 142.94 604 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1560 157~ 1580 159" log M. Dif f. 1"1". log I. iff. 1". log M. Diff. 1". 0' 3.470 7192 I42.94 3-523 3875 149-75 3.578 6154 157.17 3.636 6351 165.28 1.471 5772 143-05 -524 2864 149.87.579 5588 157.30.637 6272 I65.42 2.472 4358 I43.16 525 I860 149.99.580 5030 I57.43 -638 6202 165.56 3.473 295I 143.27.526 0863 150.11.581 4480 I57.56.639 6140 I65.7I 4.474 1550 143-38 526 9873 150.23.582 3937 157.69.640 6087 165.85 5 3.475 0I56 143-49 3.527 8890 150.35 3-583 3403 157.82 3.641 6042 165.99 6.475 8769 143.60.528 7915 150.47 -584 2876 I57-95.642 6006 166.I3 7.476 7388 143.71.529 6947 150.59 -585 2357 I58.08.643 5978 166.28 8.477 60o4 143-82.530 5985 150.71.586 1846 158.21.644 5959 I66.42 9 478 4646 I43-93.531 503I 150.83.587 1342 158.34.645 5948 166.56 10 3-479 3285 144-04 3-532 4085 150.95 3.588 0847 158.47 3-646 5946 I66.7I 11.480 1931 144.I5.533 3145 151.07.589 0359 158.61.647 5953 166.85 12.48I 0583 144.26.534 2213 5II.9.589 9880 158.74.648 5968 166.99 13.481 9242 144-37.535 1288 151.31.590 9408 I58.87.649 5992 I67.I4 14.482 7907 144.48.536 0370 I5I.43.591 8944 159.00.650 6025 167.28 15 3.483 6579 44-59 3-536 9459 15I.55 3-592 8488 I59.I3 3.651 6066 167.42 16.484 5258 I44.70.537 8556 I5I.67.593 8040 I59.26.652 6II6 I67.57 17.485 3944 144-81.538 7660 151I79 -594 7600 I59.40.653 6175 167.72 18.486 2636 144.93.539 6771 151.91.595 7I67 159.53.654 6242 167.86 19.487 1335 i45.04 -540 5890 152.04.596 6743 159.66.655 6318 i68.o0 20 3.488 0040 145-15 3-541 5015 I52.16 3.597 6327 I59-79 3.656 6403 I68.15 21.488 8752 145.26.542 4148 152.28.598 5919 159.93.657 6497 168.30 22.489 7472 145-37.543 3289 152.40.599 5518 i60.o6.658 6599 168.45 23.490 6198 145-49.544 2436 152.52.600 5126 160.19.659 6710 I68.59 24.491 4930 I45.60.545 1591 I52.65.6o0 4742 I60.33.660 6830 -68.74 25 3.492 3670 I45-71 3.546 0754 152.77 3.602 4365 160.46 3.661 6959 168.89 26.493 2416 145.82.546 9924 152.89.603 3997 I6o.6o.662 7096 169.03 27.494 ii68 145.94.547 9101 153.01.6-L 3637 160.73.663 7243 I69.18 28.494 9928 146.05.548 8285 I53.I4.605 3285 I60.87.664 7398 169.33 29.495 8695 146.16.549 7477 153-26.606 2941 I6I.oo.665 7562 I69.48 30 3.496 7468 146.28 3.550 6677 153.38 3.607 2605 161.14 3.666 7735 I69.62 31.497 6248 146.39.551 5883 I53.5I.608 2277 16I.27.667 7917 169.77 32.498 5035 146.50.552 5097 153.63.609 1957 I61.41.668 8io8 I69.92 33.499 3828 146.62.553 4319 153-75.6io 1646 I61.54.669 8308 I70.07 34.500 2629 146.73 -554 3548 153.88.611 I342 I6I.68.670 8516 170.22 35 3-501 1436 146.85 3-555 2785 5400oo 3.612 1047 161.8I 3.671 8734 170.37 36.502 0250 146.96.556 2029 I54.13.613 0760 161,95.672 8961 170.52 37.502 9071 I47-08.557 1280 154.25.6I4 0481 162.09.673 9I96 170.67 38.503 7899 147.19.558 0539 154.38.6I5 0210 162.22.674 944I 170.82 39.504 6734 147.31.558 9806 I54-50.6I5 9948 162.36.675 9694 I170.97 40 3.505 5576 I47-42 3.559 9080 154-63 3.616 9693 162.50 3.676 9957 I7I.I2 41.5o6 4425 147-54.560 8361 154-75.617 9447 162.63.678 0228 171.27 42.507 3280 147.65.56I 7650 154-88.6I8 9209 162.77.679 0509 17.-42 43.508 2143 147-77.562 6947 155.01.619 8980 162.91.680 0799 17-157 44.509 IOI2 147.88.563 6251 155.-3.620 8758 163.05.681 I098 I171.72 45 3.509 9889 148.00 3-564 5562 155.26 3.621 8545 163.18 3.682 1406 171.87 46.510 8772 148.11.565 4882 155.38.622 8340 163.32.683 1723 172.03 47.511 7662 I48.23.566 4209 155.51.623 8144 163.46.684 2049 172.18 48.512 656o 148.34.567 3543 155.64.624 7956 I163.60.685 2384 172.33 49 51 3 5464 I48.46.568 2885 155.76.625 7776 163.74.686 2728 172.48 50 3-514 4375 148.58 3.569 2235 155-89 3.626 7604 163.88 3.687 3082 172.64 51.515 3294 I48.70.570 I592 156.02.627 7441 164.oz.688 3445 172-79 52.516 2219 148.81.571 0957 56.15.628 7287 164.16.689 3817 J72.94 53.5I7 1151 148.93.572 0330 156.27.629 7140 164.30.690 4198 173.10 54.518 0090 149.05 -572 9710 156.40.630 7002 164.44.69I 4588 173.25 55 3-5I8 9037- 49-I7 3-573 9098 I56.53 3-631 6873 164.58 3.692 4988 173-40 56 -519 7990 149-28.574 8494 156.66.632 6751 164.72.693 5397 173-56 57.520 6951 149-40 575 7897 156-79.633 6638 164.86.694 5815 173-71 58.521 59I8 149.52.576 7308 156.92.634 6534 165.00.695 6243 173-87 59.522 4893 149.64 -577 6727 157-04.635 6438 165.14.696 6680 I74-02 60 3523 3875. 149-75 3-578 6154 157-17 3.636 6351 165.28 3.697 7126 174.18 605 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1600 1610 162 163~ V. log 1. Diff. 1". log M. Diff. 1". log M. Diff. 1. log M. Diff. 1". 0' 3.697 7I26 I74.18 3.762 I539 183.99 3.830 3147 194-87 3.902 6107 207.00 2.699 8046 174.49 -764 3639 I84.34.832 6554 195.25.905 0973 207.43 3.700 8520 I74.65.765 4704 I84-51.833 8275 195-44.906 3425 207.64 4.701 9003 174.80.766 5780 184.68 -835 0008 195.64.907 5890 207.86 5 3.702 9496 174-96 3.767 6867 184.86 3-836 1752 I95.83 3.908 8368 208.08 6.703 9999 I75-I2.768 7963 185-03 -837 3508 196.02.91I 0859 208.29 7.705 0511 175-28.769 9070 185.20.838 5275 196.22.911 3363 208.5I 8.706 1032 I75-43 -77I 0I87 185-38.839 7054 I96.4I.9gI 5880 208.72 9.707 1562 175.59.772 I315 i 85.55..840 8844 I96.60.913 8410 208.94 10 3.708 2102 175-75 3.773 2454 185.73 3-842 0646 196.80 3-915 0953 209.16 11.709 2652 175-91 -774 3603 185-90.843 2460 I96.99.916 3509 209.38 12.710 3211 176-07'775 4762 I86.o8.844 4286 I97.I9.917 6078 209.60 13.711 3780 176.22.776 5932 186.25.845 6123 197.38.g98 866I 209.8I 14.712 4358 I76.38.777 7112 I86.43.846 7972 197.58.920 1256 210.03 15 3.713 4946 I76.54 3.778 8303 186.60 3-847 9833 197-78 3.92I 3865 210.25 16.714 5543 I76-70.779 9505 I86.78.849 1705 I97-97.922 6487 210.48 17.715 6150 I76.86.78I 0717 186.96.850 3589 I98.I7.923 9122 210.70 18.716 6766 177.02.782 1940 187.14.85I 5486 198.37 -925 1770 210.92 19.7I7 7392 177.-8.783 3174 I87-3I.852 7394 I98.57.926 4432 21.I4 20 3.718 8028 177.34 3.784 4418 187.49 3.853 9314 198.76 3.927 7107 211.36 21.719 8673 177-50.785 5672 187.67 -855 1245 198.96.928 9795 2II.58 22.720 9328 177-66.786 6938 187.85.856 3189 I99 I6.930 2497 211.81 23.721 9993 177.83.787 8214 188.03.857 5145 199.36.931 5212 2I2.03 24.723 0668 178.00.788 9501 188.21.858 7112 199.56 -932 7940 212.25 25 3.724 1352 178-15 3-790 0799 188.39 3-859 9092 199.76 3-934 0682 212.48 26.725 2045 I78.3I.791 2I08 188.57.86I o084 199.96.935 3438 212.70 27.726 2749 178.47 -792 3427 I88.75.862 3087 z200.6.936 6207 212-93 28.727 3462 I78.63 -793 4757 188.93.863 5103 200.36.937 8989 2I3.I5 29.728 4185 I78.80 794 6098 189.11.864 7131 200.56.939 I785 213-38 30 3.729 49I8 178.96 3.795 7450 189.29 3.865 917I 200.77 3.940 4595 213.61 31.730 566i 179.13.796 88i2 189.47.867 1223 200.97 -941 7418 213.83 32.73I 6413 179.29.798 OI86 189.65.868 3287 201.17.943 0254 2I4.06 33.732 7I76 179-45.799 1571 189.83.869 5363 20I.37.944 3I05 214-29 34.733 7948 179.62.800 2966 I90.01.870 7452 201.58.945 5969 214.52 35 3-734 8730 179.78 3.801 4372 190.20 3-871 9552 201.78 3.946 8847 214.74 36 -735 9522 179.95.802 5790 190.38.873 I665 20I.98.948 I738 214.97 37.737 0324 I80.1.803 72I8 I90.56.874 379I 202.19 -949 4644 2I5.20 38 -738 1136 180.28.804 8657 190.65.875 5928 202.39.950 7563 215.43 39.739 1957 180.45.806 o008 190.93.876 8078 202.60.952 0496 215.66 40 3.740 2789 i80.61 3-807 1569 19I.11 3-878 0240 202.80 3-953 3443 2I6.90 41.74I 3631 I80.78.808 3041 I9I.30.879 2414 203.01.954 6403 2I6.13 42.742 4482 180.94.809 4525 191.48.880 4601 203-.2.955 9378 2I6.36 43.743 5344 I8I.II.8io 6020 19I.67.88x 6800 203.42.957 2366 216.59 44.744 6216 181.28.811 7525 191.86.882 g912 203.63 -958 5369 216.82 45 3.745 7097 181.45 3.812 9042 192.04 3-884 1236 203.84 3.959 8385 217.06 46.746 7989 181.6I.814 0570 192.23.885 3473 204.05.96I 1I46 217.29 47.747 8891 18I.78.8I5 2110 192.41.886 5722 204.26.962 4460 217.53 48.748 9803 I81.95.8I6 3660 192.60.887 7983 204.46.963 7519 217.76 49 -750 0725 182.12.8I7 5222 192.79.889 0257 204.67.965 0592 2I8.00 50 3.751 I657 I82.29 3.818 6795 192.98 3.890 2544 204.88 3.966 3678 2I1823 51.752 2599 I82.46.819 8379 193-.6.89I 4843 205.09.967 6779 218.47 52 -753 3552 182.63.820 9974 I93-35.892 7155 205.31.968 9895 218.70 53.754 4514 I82.80.822 1581 193.54.893 9480 205-52.970 3024 218.94 54.755 5487 I82.97.823 3I99 193.73.895 1817 205.73.97I 6i68 219.18 55 3-756 6470 I83 -4 3-824 4829 193.92 3.896 4167 205.94 3.972 9326 219.42 56 -757 7464 183-31.825 6470 194.11.897 6529 206.15.974 2498 2I9.66 57.758 8467 183-48.826 8122 194.30.898 8905 206.36.975 5684 2I9.90 58.759 9481 183.65.827 9785 194.49.900 1293 206.57.976 8885 220.13 59.76I 0505 183.82.829 1460 I94.68.90I 3694 206.79.978 2100 220.37 60 3.762 1539 I83.99 3.830 3147 194.87 3.902 6107 207.00 3-979 5330 220.61 606 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1640 165~ 1660 1670 " I -o.. ~ log M. Diff. i". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". ~0' 3.979 5330 220.62 4.061 6673 236.0I 4.I49 7198 253.57 14244 5537 273-78 I.980 8574 220.86.063 0842 236.28.151 2422 253.88.246 1975 274.I4 2.982 1833 221.10.064 5027 236.56.152 7664 254-19.247 8434 274.51 3.983 5I06 2zI.34.065 9229 236.83 I154 2925 254.51'249 49I6 274.87 r4.984 8394 221.58.067 3447 237.I.155 8205 254.83.251 1419 275.24 5 3.986 1696 221.83 4.068 7682 237-39 4.157 3504 255.14 4.252 7944 275.60 6.987 50I3 222.07.070 1933 237.66.158 8822 255.46.254 449I 275.97 7.988 8345 222.31.071 6201 237-94.i60 4159 255.78.256 o06i 276.34 8.990 1691 222.56.073 0486 238.22.161 9515 256.10.257 7652 276.71 9.991 5051 222.80.074 4787 238.50 I.63 489I 256.42.259 4266 277-08 10 3.992 8427 223.05 4-075 9I06 238.78 4-I65 0285 256.74 4.261 0902 277-45 11.994 1817 223.29.077 3441 239.06.166 5699 257.o6.262 7560 277.82 2 995 5222 223.54.078 7792 239-34.I68 1132 257-38.264 4240 278.20 13.996 8642 223.79.o8o 2161 239.62.169 6585 257-70.266 0943 278.57 14.998 2077 224.03.o8i 6546 239.90.171 2056 258.02.267 7669 278.95 15 3.999 5527 224.28 4-083 o948 240.18 4.172 7547 258.35 4.269 4417 279-32 16 4.000 8991 224.53.084 5368 240.46.174 3058 258.67.271 1I87 279.70 17.002 2471 224.78.085 9804 240.75 175 8588 259.00.272 7981 280.08 18.003 5965 225.03.087 4257 241.03.I77 4138 259.33.274 4797 280.46 19.004 9474 225.28.o88 8728 241.32.178 9707 259.65.276 1635 280.84 20 4.006 2999 225.53 4.090 3215 241.60 4.I80 5296 259.98 4.277 8497 281.22 21.007 6538 225.78.091 7720 241.89.182 0905 260.31.279 5381 281.60 22.009 0093 226.04.093 2242 242.08.183 6534 260.64.281 2289 281.98 23.oIo 3663 226.29.094 6781 242.56.185 2182 260.97.282 9219 282.36 24.oII 7248 226.54.096 I337 242-75.186 7850 261.30.284 6173 282.75 25 4-013 0848 226.79 4-097 5911 243-04 4-I88 3538 261.63 4.286 3149 283.14 26.oI4 4463 227.05.099 0502 243.33 I.89 9246 261.96.288 0149 283.52 27.015 8093 227.30.I00 5110 243.62. 19 4974 262.30.289 7172 283.91 28.017 1739 227.55.1II 9736 243.91.193 0722 262.63.291 4218 284.30 29.018 5400 227.81.103 4379 244.20 I.94 6490 262.97.293 1288 284.69 30 4.0I9 9077 228.06 4.104 9040 244-49 4.196 2278 263.30 4.294 8381 285.08 31.o02 2769 228.32.o16 3718 244.78.197 8086 263.64.296 5498 285.47 32.022 6476 228.58.107 8414 245-08.199 3915 263.98.298 2638 285.87 33.024 0199 228.84.109 3127 245-37'200 9764 264.32.299 9802 286.26 34.025 3937 229.09.110 7858 245.67.202 5633 264.66.30I 6990 286.66 35 4.026 7691 229.35 4.112 2607 245.96 4-204 1523 265.00 4-303 4201 287.05 36.028 1460 229.62.113 7374 246.26.205 743 265.34.305 1436 287.45 37.029 5245 229.88.115 2158 246.55.207 3363 265.68.306 8695 287.85 38.030 9045 230.14. ii6 6960 246.85.208 9314 266.02.308 5978 288.25 39.032 2861 230.40.Ix8 1780 247.15.210 5286 266.37.310 3285 288.65 40 4-033 6693 230.66 4-119 6618 247-45 4.212 1278 266.71 4.312 0616 289.05 41.035 0540 230.92.121 1474 247-75.2I3 7291 267.06.313 7971 289.45 42.036 4404 231.18.122 6348 248.05.215 3325 267.40.315 5350 289.86 43.037 8283 3I1-45.124 1239 248.35.216 9379 267.75.317 2753 290.26 44.039 2177 23I.71.I25 6149 248.65.218 5455 268.10.319 0181 290.67 45 4.040 6088 231.97 4.127 1077 248.95 4.220 I551 268.44 4.320 7633 291.07 46.042 OO15 232.24.128 6023 249.25.221 7668 268.79.322 5110 291.48 47.043 3957 232.51.130 0988 249.56.223 3806 269.14.324 2611 291.89 48.044 79I5 232.77 -I31 5970 249-86.224 9965 269.50.326 0137 292.30 49.046 1890 233.04.133 0971 250.17.226 6146 269.85 -327 7688 292.7I 50 4-047 5880 233.31 4.134 5990 250.47 4.228 2347 270.20 4-329 5263 293.I3 51.048 9887 233-57.136 I028 250.78.229 8570 270.55.331 2863 293.54 52.050 3909 233.84.137 6084 251.08.231 4814 270.91.333 0487 293.95 53.05I 7948 234.11.139 1158 251.39.233 1079 271.27.334 8137 294-37 54.053 2003 234.38.140 625I 251.70.234 7366 271.62.336 58I2 294.79 55 4.054 6074 234.65 4-142 1362 252.01 4.236 3674 271.98 4.338 351I 295.20 56.056 oi61 234.92.143 6492 252.32.238 0003 272.34.340 I236 295.62 57.057 4264 235.19.145 1641 252.63.239 6354 272.70.34I 8986 296.04 58.058 8384 235-46.146 6808 252.94.241 2727 273.06.343 6762 296.47 59.060 2520 235-73 -I48 1994 253.25.242 9121 273-42.345 4562 296.89 60 4.061 6673 236.01 4.149 7198 253-57 4-244 5537 273-78 4.347 2388 297.31 607 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. r 1680~ 169~" 170 1710 V. -- log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 4.347 2388 297.31 4.459 I242 325.07 4-581 9445 358-3I 4-7I7 9835 398-87 1.349 0240 297.74 46I 076I 325.57.584 96 32 589 720 379~ 399-6 2.350 8117 298.6.463 03II 326.08.586 2516 359.53.722 7790 400.38 3.352 6019 298.59.464 9891 326.59.588 41o6 360.I5.725 1835 40I.14 4.354 3948 299.02.466 950I 327.10.590 5734 360.76.727 5926 4o0.90 5 4-356 1902 299.45 4.468 9142 327.61 4-592 7398 361.38 4.730 0063 402.66 6 *357 9882 299.88.470 88I4 328.12.594 9I00 362.00.732 4245 403-43 7.359 7888 300.31.472 85I7 328.64.597 0838 362.62.734 8474 404.19 8 36I 59I9 300.75.474 8250 329.15.599 2615 363.25.737 2749 404.96 9.363 3977 30I.I18.476 8015 329.67.6o0 4428 363.88.739 7070 405.74 10 4-365 2061 301.62 4.478 7811 330-19 4.603 6280 364.50 4.742 1438 406.52 11.367 0171 302.05.480 7637 330.71.605 8169 365-14.744 5852 407.30 12.368 8308 302.49.482 7495 33I.23.608 0096 365.77.747 0314 408.08 13.370 6470 302.93.484 7385 331.75.610 2061 366.40.749 4822 408.87 14.372 4659 303-37.486 7306 332.28.612 4064 367.04.751 9378 409.66 15 4-374 2875 303-8I 4.488 7258 332.81 4.614 6o16 367.68 4.754 3981 4I0.45 16.376 III7 304.26.490 7242 333-33.616 8I86 368.32.756 8632 4I1.24 17.377 9386 304-70.492 7258 333.86.619 0304 368.96.759 3330 4I2.04 18.379 768I 305.I5.494 7306 334-40.62I 246I 369.61.76I 8077 4I2.84 19.381 6003 305-59'496 7386 334-93.623 4657 370.26.764 2872 413.65 20 4.383 4352 306.04 4.498 7498 335.46 4.625 6892 370.91 4.766 7715 414.46 21.385 2728 306.49.500 7642 336.00.627 9166 371.56.769 2606 415.27 22.387 1131 306.94.502 7818 336.54.630 1480 372.21.771 7547 4I6.08 23.388 9561 307.39.504 8026 337-08.632 3832 372.87.774 2536 4I6,9o 24.390 8019 307-85.5o6 8267 337-62.634 6224 373-53 -776 7574 417.72 25 4.392 6503 308.30 4.508 8541 338.I6 4.636 8656 374.I9 4.779 2662 418.54 26.394 5015 308.76.510 8847 338.71.639 II27 374-86.781 7799 419.37 27.396 3554 309.2I.512 9186 339-26.64I 3639 375-52.784 2986 420.20 28.398 2121 309.67.514 9558 339.80.643 6190 376.19.786 zz22 421.03 29.400 0715 310I.3.5I6 9962 340.35.645 8781 376.86.789 3509 421.86 30 4.401 9337 310.59 4-519 0400 340-91 4.648 1413 377.53 4-791 8846 422.70 31.403 7986 3II.06.52I 0871 341.46.650 4085 378.21 -794 4233 423.54 32.405 6663 311.52.523 1376 342.02.652 6798 378.89.796 9671 424-39 33.407 5368 311.99.525 1913 342.57.654 9552 379-57.799 5160 425.24 34.409 4102 312.45.527 2484 343.I3.657 2346 380.25.802 0700 426.09 35 4.411 2863 312.92 4.529 3089 343.69 4.659 5182 380.93 4-804 6291 426.95 36.4I3 I652 313.39.531 3728 344-26.66i 8059 381.62.807 1934 427.81 37.4 5 0469 313.86.533 4400 344-82.664 0977 382.31.809 7628 428.67 38.416 9315 3I4.33.535 5io6 345-39.666 3936 383.00.812 3374 429-53 39.418 8189 314-80.537 5846 345.95.668 6937 383.70.814 9172 430.40 40 4.420 709I 315.28 4.539 6620 346.52 4.670 9980 384.39 4-8I7 5022 431.28 41.422 6022 315.75 -541 7429 347.09.673 3064 385.09.820 0925 432.15 42.424 4982 3I6.23.543 8272 347.67.675 619I 385.80.822 688I 433-03 43.426 3970 316.71 -545 9149 348.24.677 9360 386-50.825 2889 433-9I 44.428 2987 317.19.548 oo6I 348.82.680 2571 387-2I.827 8950 434.80 45 4-430 2033 317.67 4-550 I007 349.40 4.682 5825 387.92 4-830 5065 435.69 46.432 II08 318.16.552 1989 349-98.684 9121 388.63.833 1234 436-59 47.434 0212 3I8.64.554 3005 350-56.687 2460 389.34 -835 7456 437-48 48 -435 9345 3I9.13'556 4056 35II5.689 5842 390.06.838 3732 438-38 49 *437 8507 319.61 -558 5143 351-73.691 9268 390.78.841 0062 439-29 50 4.439 7698 320.10 4-560 6264 352.32 4.694 2736 391I50 4.843 6446 440.20 51.441 69I9 320.59.562 7421 352.91.696 6248 392.23.846 2886 44I.II 52.443 6169 321.08.564 8614 353.50.698 9803 392-96.848 9380 442.03 53.445 5449 321.58.566 9842 354.IO.701 3402 393.68.85I 5929 442.95 54.447 4758 322.07.569 Io06 354.69.703 7046 394-42.854 2533 443-87 55 4.449 4097 322.57 4-571 2405 355.29 4.706 0733 395.15 4.856 9193 444.80 56 -45I 3466 323.06.573 3741 355.89.708 4464 395-89.859 5909 445,73 57.453 2865 323-56.575 5113 356-49.710 8240 396.63.862 2680 446.66 58.455 2294 324.06.577 652I 357.10.713 2060 397.38.864 9508 447.60 59.457 1753 324-56 -579 7965 357-70.715 5925 398-12.867 6392 448-54 60 4.459 1242 325.07 4-581 9445 358.31 4-717 9835 398.87 4.870 3333 449.49 608 TABLE VI. For finding tile True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1720 1730 1740 1 75" log M. Diff. 1. lo- M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 4.870 3333 449-49 5.043 3285 514.47 5-243 3165 601.00 5.480 1373 722.00 1.873 0331 450-44 046 4191 515.71.246 9276 602.69 -484 4765 724-42 -.875 7386 451.39 -049 5171 516.96.250 5488 604.38 -488 8304 726.87 3.878 4499 452.35 o052 6226 518.21.254 1802 606.8.493 1989 729-33 4.881 1668 453.31 -055 7356 519.47 -257 8218 607.80.497 5823 731.80 5.883 8896 454-28 5-058 8562 520.73 5.261 4738 609.53 5-501 9806 734-30 6.886 6182 455-25.o6i 9843 522.00.265 1361 611.26.506 3939 736.81 7.889 3526 456.23.065 1202 523.28.268 8089 613.00.510 8223 739.33 8.892 0929 457-20.068 2637 524.56.272 4922 614.75 -515 2659 741.87 9.894 8391 4581I9 -071 4149 525.85.276 i86o 616.52.519 7248 744-44 10 4.897 5912 459.17 15-074 5738 527.I4 5.279 8904 618.29 5.524 1992 747.02 11.900 3492 460.16.077 7406 528.44.283 6055 620.08.528 6890 749-61 12.903 1132 461.16.080 9151 529-75.287 3313 621.87 -533 1946 752.23 13.905 8831 462.16.084 0976 53.o06.291 0680 623.67.537 7158 754-86 14.908 6591 463.16.087 2879 532-38.294 8154 625.49 -542 2529 757.51 15 4.911 44JI 464-17 5.090 4862 533-71 5-298 5738 627.31 5-546 806o 760. I8 16.914 229I 465-I8.093 6924 535.04 -302 3432 629.15 -551 3751 762.87 17.917 0233 466.20.096 9067 536.38.306 I237 631.00.555 9605 765-58 18.919 8235 467.22.00oo 1290 537.73 309 9152 632.85 -560 5621 768.3I 19.922 6299 468.25 -103 3594 539.08.313 7179 634-72.565 1802 771-05 20 4.925 4425 469.28 5.Io06 5980 540.44 5-317 5319 636.60 5.569 8148 773-82 21.928 2612 470.3I I109 8447 54I1-I.321 357I 638.49.574 4661 776.61 22.931 0862 471.35 -113 0997 543-18 -325 1938 640.39 -579 134I 77941I 23.933 9174 472.39.ii6 3629 544.56.329 0418 642.30.583 8190 782.24 24.936 7549 473.44.119 6344 545-95 -332 9014 644-23.588 5210 785.08 25 4-939 5987 4744-9 5.-122 9143 547-34 5-336 7726 646.16 5.593 240I 787-95 26.942 4-489 475.55.126 2026 548.74.340 6554 648.11 -597 9764 790-84 27.945 3053 476.61.129 499z 550.15 -344 5499 650.07.602 7302 793-75 28.948 1682 477.68.132 804-4 551.57 -348 4562 652.04.607 5014 796.68 29.951 0375 478.75 -136 1181 552.99 -352 3744 654.02.612 2903 799.63 30 4.953 9132 479.83 5.139 44-03 554-.42 5.356 3045 656.01 5.617 0970 802.60 31.956 7954 480.91 -142 7711 555-86.360 2466 658.02.621 9216 805.60 32.959 6841 481.99.-46 iio6 557-30.364 2007 660.04.626 7642 808.62 33.962 5793 483-08.149 4588 558.75.368 1671 662.07.631 6250 811.66 34.965 4811 484-18 -152 8157 560.21.372 1456 664.11.636 5041 814.72 35 4.968 3894 485.28 5-156 1813 56i.68 5.376 1364 666.17 5.641 4017 817.81 36.971 3044 486.38.159 5558 563.16.380 1396 668.24.646 3179 820.92 37.974 2260 487.49.162 9392 564.64 -384 1553 670.32.651'2528 824.05 38.977 1543 488.61.i66 3315 566.13.388 1834 672.41.656 2065 827.21 39.980 0893 489.73.169 7328 567.63 -392 2242 674-52.66i 1793 830-39 40 4.983 0311 490.85 15-73 1431 569.13 5-396 2777 676.64 5.666 1713 833.60 41.985 9795 491.98.-176 5624 570.65.400 3439 678.77.671 1825 836.83 42.988 9348 493.-1 -179 9908 572.17.404 4229 680.92.676 2132 840.08 43.991 8970 494-26.183 4284 573.70.408 5149 683.08.68i 2635 84-3.3 44.994 8659 495-40.I86 8752 575-24.412 6199 685.25.686 3336 846.67 45 4-997 8418 496-55 5-190 3312 576.78 5.416 7379 687.44 5.691 4236 850.00 46 5.000 8246 497.71 -193 7966 578.34.420 8692 689.64.696 5337 853.36 47.003 8143 498.87.-97 2713 579.90.425 0136 691.85 -70o 6640 856.75 48.oo6 8111 500.04.200 7554 581-.47 -429 1714 694.08.706 8147 86 o. 16 49.009 8148 501.21.204 2489 583-05 -433 3427 696.33 -711 9860 863.60 50 5.012 8256 502.39 5.207 7520 584.64 5-437 5274 698.59 5.717 1779 867.06 51.015 8435 503.57.211 2646 586.23 -44I 7258 700.86.722 3908 870.56 52.o8. 8685 504-76.z214 7868 587.84 -445 9378 703-15 -727 6247 874-08 53.021 9006 505-95 -218 3186 589.45 -450 1636 705.45.732 8798 877.63 54.024 9399 507-15.221 8602 59.-07 -454 4032 707-77 -738 1563 881.21 55 5.027 9864 508.36 5.225 4116 592.71 5-458 6568 710.10 5.743 4544 884.82 56.031 0402 509-57.228 9727 594'35 -462 9244 712.45 -748 7742 888.46 57.034 1013 510-79.232 5437 596.00.467 2062 714-81.754 1159 892.13 58.037 1697 512.01.236 1247 597.66.471 5022 717.19.759 4798 895-83 59.040 2454 513.24.239 7156 599.32.475 8125 719-59 -764 8659 899.56 60 5.043 3285 514-.47 5.243 3165 6oi.oo 5.480 1373 722.00 5.770 2745 903-31 39 609 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 176~ 177~ 1780 179~ V. log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". O' 5.770 2745 903.3 6.I44 6289 I205.3 6.672 5724 I8o8.8 7.575 4640 3619 1.775 7058 907.I.I5I 8807 1212.0.683 4709 1824.0.597 3596 3680 2.78 I599 9IO.9.1I59 I733 I2I8.8.694 46I3 I839.5.6I9 6295 3744 3.786 6370 9I4.8.I66 5070 1225.7.705 5454 I155.3.642 2868 3809 4 -792 I374 9I8.7.I73 8823 I232.7.716 7248 I871.3.665 3452 3877 5'5797 6612 922.6 6.I8I 2997 I239.8 6.728 ooIo 1887.5 7.688 8I92 3948 6.803 2086 926.6.I88 7597 I246.9.739 3758 I904.1I.7I2 7239 402I 7.808 7798 930.6.I96 2628 I254-I.750 8509 192I.0.737 0756 4097 8.8I4 3751 934.6.203 8095 I261.4.762 4279 I938.2.76I 89I3 4176 9.8I9 9946 938.6.2II 4002 I268.8 *774 I~90 I955.6.787 I889 4257 10 5.825 6386 942.7 6.2I9 0354 I276.3 6.785 8958 I973.4 7.8I2 9876 4343 11.83I 3073 946.8.226 7I58 1283.8.797 7904 199I.5.839 3075 443I 12.837 0008 95.o0.234 44I9 I29I.5.809 7946 20I0.0.866 I702 4524 13.842 7195 955.2.242 2142 i299.2.82I 9o06 2028.8.893 5986 4620 14.848 4634 959.5.250 0333 I307.I.834 I404 2048.o.92I 6170 4720 15 5.854 2329 963.7 6.257 8997 I315.0 6.846 4863 2067.5 7.950 2513 4825 16.86o 0282 968.0.265 8139 1323.0,858 9503 2087.3 7.979 5292 4935 17.865 8495 972.4.273 7766 I33I.1.871 5348 2107.6 8.oog 4802 5050 18.87I 6970 976.8.281 7884 I339.4.884 2422 2I28.3.04o 1361 5170 19 g.77 57Io 981.2.289 8499 I347-7.897 0749 2149.4.0o7i; J309 5296 20 5.883 471I7 985.7 6.297 96I7 1356.2 6.9IO 0353 2170.9 8.o03 70II 5428 21.889 3993 990.2.306 I244 I364.7.923 I26I 2I92.8.I36 6857 5568 22.895 3542 994.8.314 3387 1373-3.936 3498 22I5.2.170 5274 57I4 23.901 3365 999.4.322 6052 1382.I1.949 7093 2238.0.205 2717 5869 24.907 34-65 I004.0.330 9247 i391.0.963 2073 226I.4.240 9679 6032 25 5.9I3 3845 I008.7 6.339 2977 1400.0 6.976 8466 2285.2 8.277 6700 6204 26.919 4507 10I3.4.347 7249 1409.1 6.990 6304 2309.6.315 436I 6387 27.925 5454 I'8.I.356 2072 I418.3 7.004 56i6 2334.3.354 3298 6580 28.93I 6688 O022.9.364 7451 I427.6.oI8 6437 2359.7.394 4205 6786 29.937 82I3 I027.8.373 3395 1437-I.032 8796 2385.7.435 7842 7004 30 5.944 0030 1032-7 6.381 99IO I446.7 7.047 2729 24I2.2 8.478 5044 7238 31.950 2I44 I037.6.390 7005 I456.4.0o6I 8271 2439.4.522 6731 7488 32.956 4556 I042.6.399 4687 I466.2.076 5458 2467.1.568 3920 7755 33.962 7269 I047.7.408 2965 1476.2.o9I 4329 2495.4..65I 7739 8042 34.969 0o287 I052.9.417 I846 I486.'4.io6 492I 2524.5.664 9442 8352 35 5.975 3613 I058.o 6.426 I337 I496.7 7.121 7276 2554.2 8.716 043I 9686 36.981 7249 I063.2.435 I449 1507.0.137 I434 2584.6.769 2286 9048 37.988 II98 I068.4.444 2I9 I5I17.6.I52 744~ 2615.8.824 6779 9441 38 5.994 5464 1073.7 *453 3569 1528.3.I68 5336 2647.6.882 5925 9870 39 6.ooi 0050 I079-.I 462 5594 1539.2.184 5171 2680.4.943 20I8 10340 40 6.007 4958 1084.5 6.47I 8275 1550.2 7.200 6993 2713.9 9.006 7690 10857 41.014 0192 I089.9.48I I620o 56I.3.217 0850 2748.3.073 5974 11429 42.020ozo 5756 1095.4.490 5641 I572.6.233 6796 2783-5.144 040I 12064 43.oz7 I652 IIOI.O.500 034-6 1584.1.250 4884 2819.7.218 5102 I2773 44.033 7885 IIo06.7.509 5746 1595.8.267 5170 2856.8.297 4963 13572 45 6.o40 4457 III2.-4 6.59 i85o0 I607.7 7.284 77I2 2894.8 9.38I 5820 14476 46.047 I372 1118.I.528 8669 I6I9.6.302 257I 2934.1.471 47II 155IO 47.053 8634 1123.9.538 62I6 I631.8.3I9 98IO 2974.2.568 0247 I6704 48.0o60 6246 1129.8.548 4499 I644.2'337 9494 30I5.6.672 3I06 I8096 49.067 4212 135.7.558 3530 i656.8.356 I692 3058.I.785 6758 19741 50 6.074 2535 1141.7 6.568 3320 I669.6 7.374 6475 3IOI.7 9.909 8535 217I5 51.08i 1219 1147.7.578 3881 I682.4.393 3918 3I46.8 10.047 1256 24127 52.o088 0269 II53.8.588 5227 I695.6.412 4099 3I93.0.200 5829 27I44 53.094 9687 ii6o.o.598 7368 1708.9.43I 7097 3240.7.374 5584 3I023 54.0Io 9479 II66.3.609 0317 1722.6.451 2999 3289.9.575 3986 36197 55 6.Io8 9647 1172.6 6.6I9 4086 I736.4 7.47I 1892 3340.3 10.812 9421 43450 56.ii6 oi0196 1179.0.629 8689 I750.3.49I 3870 3392.6 1.103 6719 57.I23 II3I 1185.4.640 4141 I764.5.5II 9029 3446.5 II.478 4880 58.130 2455 1192.0.65I 0455 1779.0.532 7472 3502.I I2.oo6 7617 59.I37 4173 1198.6.66I 7645 I793-8.553 9305 3559.6 12.909 8516 60 6.I44 6289 1205.3 6.672 5724 I808.8 7'575 4640 36I8.7 610 TABLE VII, For finding the True Anomaly in a Parabolic Orbit when v is nearly 180~. w AO Diff. w AO Diff. w AO Diff. 0 ] f 0 / o 0 i i // o I if II/ 155 0 3 23.09 0 0 0 6.70 165 0 0 1585 0. 5 1974 335 5 5. 3 7 10 1498 10 16.43 33 10 3.97 136 20 146 8 15 13.17 3-26 15 2,64 1'33 30 138 0.78 20 9.95 3.8 20 1.33. 40 12.3 075 25 677 3' x 25 0.04.26 50 11.9I 0.72 7 3.4. i 16 0 I. oil60.69 155 30 3 3.63 160 30 o 58.78 166 0 0 11.22 35 0.54 3 35.24 0 57.6 40 257 305 40 5631 1.23 20 95 45 54-41 3'~9 45 55. 1.18 30 9.36 50 297 50 5351 6 40 8.80 0.56 55 48.58 2'93 55 52.77 50 8.6 54 2.89 114 5 o8.2 156 0 2 45.69 8 161 0 o 51.63 I4 167 0 o 7.75 0.48 5 ^ - -8' I.' 0 1.-3 10 0.48 42.84 2.8 5 501.0 10 7.27 0.46 10 40.03 10 49 xo0 i' 20 6.8 44 15 37.26 2.77 15 43.oo6 430 596 04 20 34 2.73 20 476 40 56 25 30.83 2.70 25 46.zi2 I05 50 57 039 2.66 0.37 156 30 29.17 161 30 0o 45.19 168 0 o 5.20 o.36 35. 26.55 2.8 35 44.08 10 4.84 33 40 23.97 2.58 40 439 920 4 33 45 21.43 2.54 45 42.22 0.97 30 4.20 0.31 50 18.92 251 50 4.6 96 40 3.90 0.30 55 16.:44 248 55 40.33 0.93 50 3.62 0.28 2.44 0.92 0.36 157 0 I4.00 4oo 162 0 0 39.41 10 0 3.36 5 11.59 z-41 38-51 0.90 10 - 0.25 1 9.922 237 7.2 0.89 20 2.88 3 15 6.89 2.33 15 36.75 0.87 30 2.66 0.22 20 4.58 z.31 2 0 0.85 40 2.46 0^ 25 3 2.7 25 35.06 0.84 50 2,27 0'9 2.23 o.8z 157 30 2 o.o8 162 30 0 3424.8 10 2.09 35 I 57.89 5 43 10.92 o. 17 40 55. 40 3264 0.79 20 1.76 45 3.57 2.5 45.86 0.78 30.62 4 50 50I.46 2.1 50 31.1o 0 76 40 1.48 4 55 49.39 55 30.35 0.75 50 1.35 2.04 0.73 o. Iz 158 0 47.35 163 0 0o 29.672171 0 0 1.23 o 5 45.34. 5 28.90' 15..12 10 43.35 99 10 28.20o 0.70 20.02 15 4 139 96 15 0.69 30 0.93 09 xI~39 z7.51 Os 40 0.842 00 20 3.92 20 26.83 o.68 40 o.084 25 3 7 25 z6. i6 67 50 0.76 0 08 37.57 25 0.65 0 1.67 3 0.65 172. 08 158 30 I 35.70 1.83 163 30 2 1.63 0 0.68 35 33.87.835 24.8 063 10 o.6I 0.06 40 32.06 40 2 5 6 20 0.55 o.o 45 30.28 4578 2.6 o.6o 30 0.49 50 28.52. 76 50 o.6o 50 6852 7 50 23.04 400 0.44 0.05 5 5 26.8o 1.72 55 22.4 059 50 0.39 i..x70 0.57 ~'~4 159 0 I 25.0 164 0 0 21.88 173 0 o 0.35 5 23.43 i67 5 21.31 057 10 0.3 0.04 10 20.78'65 10 20.76 0.55 20 0.27 0.0 15 20o.6.6z 15 2 054 30 0.24 003 20 18.57 I'59 20 9.69 0~'53 40 0.21 0.03 25 700 57 25 9. 0'5 50.9 002 159 30 i 5.45 164 30 o 18.67 174 0 o o. 06 35 3.94 35 187 050 175 0 0.07 0 9 13'94 0.07 o.o 40 i2.44 0.50 40 7.69 0.48 176 0 0.02 0.0 45 10o.97 047 45 17.21 0.48 177 0 0o.o 50 9.53 1.44 50 x6.75 0'46 178 0 o.oo.00 55 8.0io'43 55 16.29 0.46 179 0 o. oo 1.40 0.44 0.00 160 0 x 6.70 165 0 o 15.85 180 0 o o.oo 611 TABLE VIII. For finding the Time from the Perihelion in a Parabolic Orbit. log N Diff. v log N Diff. v log N Diff. 0 0! o 0 1 30.02 5 5763 30 0 o.020 7913 6 30.008 64458 30 oZ5 5749 4 30,ozo 6 368 1545 30.008 8645 1 0.o25 5749 31 0.0 42 566 1 0.008 4277 30.025 5638 9 30 587 30.008 2103 2174 0.25 3 0 ozo 2 0.007 9934 6 30.025 54 124 3 019 9979 68 30.007 7774 2153 3 0 0.025 5266 33 0 0.019 8330 63 0 0.007 562 30.025 5087.ozo 32 30.02 87 30.oi 6662 688 30.007 3477 4 4 0.025 4881 zo6 34 0.019 4974 64 0.007 1343 30.025 4647 234 30.0oi9 3267 1707 30.006 gz920 2IZ3 5 0.25 36 35 0.019 1540 I727 65 0.006 7108 21 30.025 4097 29 30.oi8 9795 745 30.oo6 5008 2086 6 0 0.025 3781 36 0 o.oi18 8030 66 0 0.00oo6 2922 30.025 3437 344 30.18 6248 72 30.006 0849 2073 o.oz5 5266 oI344 380.ooo 6248 7 0.025 3066 37' 37 0.o18 4448 1 67 0.005 8792 2057 30.025 z668 398 30.o018 262 119 30.005 6750 2042 8 0.025 2243 5 38 0.o018 0794 I 35 68 0.005 4725 5.024 07947 30.025 179 452 30.017 8941 30.005 25717 200 9 0 0-0 —5 I3 480 3969 390.005 2 1988 9 0 0.025 3 o6 39 0 0.017 7072 88 69 0 0.005 0729 1969 30.025 0805 30 017 i86 30.004 8760 10 0.02 027 5 40 0.017 3283 1903 0 0 004 68i'949 30.024 9711 560 30.017 1365 1918 30.004 4884 1927 11 0.024 924 57 1 0.016 932'933 71 0.004 2980 1904 30.024 851 614 30.oi6 7483'949 30.004 iioo Iso'~25 -4641 61963 25 12 0 0.024 7869 6 42 0 o.oi6 5520 72 0 0.003. 9245 30.24 7201 30.o6 542 7 30.003 7416 829 13 0.024 6507 694 43 0.o16 1550 I992 73 0.003 5613 1803 30.024 5786 721 30.015 9545 2005 30.003 3839 774 14 0.024 5039 747 44 0.oi5 756 9 74 0.003 2094 I745 30.04 4266 773 30.015 5495 2031 30.003 0380 1714 800 2045 1682 15 0 0o.024 3466 45 0 0.015 3450 75 0 0.002 8698 649 30.024 z41 3 30 o 2056 30.002 7049 6 16 0.024 1789 46.014 9326 68 76 0.002 5433 30.024 0911 9 30.014 7247 2079 30.002 354 79 17 0.0 08 903 47 0.014 517 2090 77 0.002 2311 5.0205 929 30 6014.oo o8o6 61505 30.023 9079 0 30. 307 365. 86 954 2110 145 18 0 0.023 8125 48 0 0.014 0947 78 0 o.oo 9341 30.023 7145 980 30.o013 8827 2120 30.OOI 7917 19 0.023 6140 005 49 0 oi3 6698 9 79 0.ooi 6535 30.023 5IO9 1031 30.o013 4561 237 30.00oo 5196 339 20 0 023 4054 55 50 0.03 2416 2145 80 0.00 3903 293 30.03 2973 105 30 o013 0263 153 30.oo 2656 1247 2 01 0 0-03 i86. II1o405 2160 I1193 0 0.023 868 51 0 0.012 8103 81 0 o.oo00 1458 1149 30.023 0738 1130 30.01 5936 267 30.00oo1 0309 109 22 0.022 9584 54 0.012 3764 2172 82 0.000 9211 30.o022 84 79 30.012 i387 9 30.ooo 8166 05 23 0.022 7202 I03 53 0.oi0 940 213 83 0.000 7175 9 30.022 5975 1227 30.o0 7215 27 30.ooo 6240 1251. 29I3 I 876 24 0 0.022 4724 54 0 0.011 5024 84 0 o.ooo 5364 898 30.022 3449 12 30.o0 2829 295 30.00ooo 4546 25 0.022 151 298 55 0.011 0632 97 85 0.00ooo 3790 756 30.022 0829 132 30.oio 8432 2200 30.00ooo 3096 6 26 0.021 9484 345 56 0.oio 6231 220 86 0.000 2468 6 30.021 81i6 1368 30.oio 4029 2202 30.000 1906 5 1390 2202 493 27 0 0.021 6726 570 o.o010 1827 87 0 0o.ooo 413 30.021 5312 414 30.009 965 202 30.ooo 0990 3.o 502' 3876 1436 58 0.009 7424 88 0.000 0639 35' 30.021 2418 45 30.009 5225 30.00ooo 0363 276 29 0.Z 0938 59 0.o009 302 21 97 189.00o 0163 z 30.020 9436 02 30.009 0834 2I94 30 0000041 22'^ 1523 2190' 41 30 0 o. 7913 60 0 0.008 8644 9 0 o.ooo 0000oooo 612 TABLE VIII. For finding the Time from the Perihelion in a Parabolic Orbit. v log N' Diff. v log N' Diff. v log N' Diff. 0o o 0 90 0 o.ooo oooo 120 0 9.963 1069 150 0 9.889 0321 30 9.999 9876'24 30.962 0074 I0995 30.887 878 113 91 0 999 9507 6 121 0 96 8971 II03 151 0.886 7259 II479 30.999 8951 8, 30.959 7764 30.885 5887 11372 92 0.999 8039 54 I22 0.98 6454 11310 152 0.884 4627 30.999 6944 1095 30.957 5046 11408 30.883 3481 III46 93 0 9.999 563 123 0 9.956 3542 153 0 9.882 245 55 o 30.999 4046 8 30 955 1945 87 154 30.881 1552 0 94 0.999 26 124 0.954 0258 0.880775 30.999 2031 30 952 8483 775 30.879 09 46 30.998 5468 2487 30.950 4684 11940 30.876 9242 10374 96 0 9.998 2757 126 0 9.949 2666 156 0 9.875 90IO I0 zo 30.997 9824 2933 30.948 0573 12093 0.874 8922 97 0.997 6669 355 127 0.946 8408 12165 157 0.873 8984 9938 30 7 7 3372 30 945 674 12234 30.87 998 9786 98 0.91 197 3589 128 0.944 3875 299 158 0.871 9569 9629 30.996 5906 3802 30.9431513 36 30.871 0099 9470 4015 12421 9307 99 0 9.996 1891 129 0 9.941 9092 159 0 9.870 0792 30.995 7666 4225 30.940 66i5 I2477 30.869 1652 9140 100 0 995 3234 32 130 0.939 4085 2530 160 0.868 2683 8969 30'.994 8596 4 3 30.938 15o6 12579 30.867 3886 8797 101 0.994 3755 131 0.936 888 161 0.866 566 86 30.993 871 5043 30 935 6213 668 30.865 6827 439 5242 12707 85 102 0 9.993 3470 132 0 9.934 3506 162 0 9.864 8570 30 992 831 5439 30 933 0763 743 30.864 0500 8070 103 0.992 2397 5634 133 0 93 7987 2776 163 0.863 2620 7880 30.99i 670 527 30.930 5183 I2804 30.862 4932 7688 104 0.991 053 6017 134 0 9 3 12 30 164 0.861 7439 7493 9.996 354 120 9.929 2353 266 30.990 4347 30.927 9501 2 30.86i 0145 794 105 0 9.989 7956 ~ 135 0 9.926 6630 28 165 0 9.860 3053 30.99 180 6576 30.925 85 30.859 6164 6889 106l 0.988 4622 3758 136 0.9466 0.878 9482 668z 30.987 7685 6937 630.9 7913 905 30 858 30 647 107 0.987 0571 714 137 0.921 290 167 0.857 750 6 30.986 3281 7290 30.920 2126 12909 30 857 0704 7462 12906 5829 108 0 9.985 5819 6 138 0 9.918 9220 9 168 0 9.856 4875 9 30.94 86 76 33 30.917 30 855 9266 56 109 0.984 0385 7801 139 0.916 3433 28 169 0.855 3878 538 30.983 z2I8 7967 30.915 0559 I84 30.854 8714 5164 110 0.982 4288 8130 140 0.3 7703 125 1700.854 3775 4939 30 5996 30.912 470 33 30.853 9065 4710 111 0 9.980 7545 141 0 9.911 20zo62 171 0 9.853 4584 30.979 8938 8607 30.909 9283 12779 30.853 0335 4249 112 0 0177 876 142 0.98 6538'745 172 0.852 6319 406 30.8 1264 8913 30.07 3831 2 30.852 2538 3781 113 0 77 2202 906 143 0.906 164 7 173 0.851 8994 3544 30.976 2993 9209 30.904 854 62 30.8 5687 337 9353 12573 3067 114 0 9.975 3640 144 0 9.903 5969 174 0 9.851 2620 2826 30 97 45 995 30 9 3449 12520 30.850 9794 115 0.973 4510 93 145 0.901 0985 I2464 175 0.850 7209 2585 30 97 4739 977I 30.99 8582 2403 30.850 4868 2341 116 0.97i 4839 9906 146 0.898 6243 12339 176 0.850 2770 2098 30.970 4796 10037 30.897 3972 12271 30.850 0917 i6o8 10167 12198 117 0 9.969 4629 147 0 9.896 1774 177 0 9.849 93.09 30.968 430.894 9652 22 30 849 7948 1361 118 0.967 392 104I7 148 0.893 7610 12042 178 0.849 6833 113 30.966 3382 10538 30.892 5652 11958 30.849 5966 867 119 0.965 276 o656 149 0.891 372 11870 179 0.849 5346 62 30 0 I0885.9o3 ii6? 243 124 120 0 9.963 1069 150 0 9.889 0321 180 0 9.849 4850 3 613 TABLE IX, For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. x A Diff. B Diff. C B' Diff. C' o i f I 1 /1 nf 1o 2.oI 0.01 0.000 0.000 0.000 0.000o 3 0.o 0'04 0.000 0.000 0.000 0.000 4 0.12 0'07 0.000 0.000 0.000 0.000 0.11 5 0.23 o0.000 0.000 0.000 0.000ooo 039 o. 0.000 o.ooo 000 0.000 0.0 I o.ooo00 7.62 0.23 0.000.000 0.0ooo0 o.ooo 8 093 31 o.ooo 0.000ooo 0.000ooo o.ooo 9 133 040.000 0.000 0.000 o.ooo000 10 1.82 4 0.000 0.000,0.000 0.000ooo 1I 2.42 0.000 0.000 0.000 0.000 13 399 0. 5 0.000 0.000 0.000 0o.000 14 4.9 0.001 0.000.I 0.000ooo 15 6.13 0.001 1 0.000 0.001.0 0.000 16 7.4.3 O.OOI o'^'^ o o.ooo oooooo 16 7 143 I. 0.002 0.000 0.001.00 0.000 17 8.90 1.47 0.002.000 0' 0 o.ooo0 0.002 I 16 I.6 -OOI.000 18 10-.55 5 i. 0.003 0.000 0.002 0.000 19 12.4.0 185 0.004.001 0.000 0.003 o001 0.000 2.05.001oo.001 20 14-45 0.005 0.000 0.004 0.000 21 16.70 2.25 o.oo6.00 0.000 0.005.001 ooo 22 19.18 2.4 0.008.002 0.000 o.oo6.00'1 0.000 23 21.89 2.71 O;I o 002 000 ooo 0.008.002 0.000 24 24.83 294 0.012.002 0.000 0.010 0.02 3.20.002.002 25 28.03 0.014 0.000 0.012 o.ooo000 26 31.48 345 0.017 003 0o.oo000 0.014'002 o.oo 27 35.20 372 0.020.003 0.000.o017 003 o.ooo000 28 39-19 3.99 0.025.005 0.000 0.020.003 0.000 29 4. 28 0030.005 0.000 0.024.004 o0.000oo 4.57.005 004 30 48.04 o 0.035.o 000 0.02 0.000 31 5.91 8 0.041 o.ooo 0.033.00 0.000 *32 58.0 8 0.047.oo6 0.0oo 0.039 o 0.000 33~ 66529 5.83 0.055 0 o.ooo 0.045 ooo0.000.3 0.064 009 0.000 0.052.007 000o.ooo ~6.15'0.009.008 35 75.7 0.073 o.ooo. 0000 o.o6o oo 36 82.07 5 0.084 0'0I o.ooo /0.68 o0.000 37 88.92 6.8 o.o96.o. o.o0.oo 0.000 38 96.12 7-0 0.109' o.ooo012 0.088.010 oooo 39 0lo.68 7.56 0. 6. 0.1000 o 0.000 7-93.010.013 40 ii.6i 0.139 0.000 0. 13 0.000 41 119.92 8.3' 0.156.017 o.ooo000 0.12z7.014 o.ooo 42 128.62 8.70 0.175.0'9 o.ooo000 0.142 5 0.000 43 3770 98 196 o.ooo 0.000 0.159'017 o.ooo000 0.48.022 44 147.18 948 0.218. 0.000 0.177 0.000 45 157.05 10 0.243.026 0.000 0.197.022 0.000 46 167.34.29 0.269 0.000 0. 0.000oo 47 178.04 10.70 0.298.029 0.000 0.242 023 0.000 48 189.16 II.12 0.328.030 0.000 0.267.025 0.000oo 49 2007I 11.55 0.361'.033 0.000 0.294.027 0.000 11.98.036.029 50 212.69 0.397 0.000 0.323 0.000 51 225.10 2.4 0.436.039 o.ooo 0.354 03 o.ooo 12.85..0.7.04..oo 52 237.95 0477.034 0.000 53.251 -3 3 0o.oo2 0..044 0001 0.424.03 0.000 54 265.01 13-76 0.567.046 O.OOI 0.462.038. 14.20.050.040 55 2792l 4. 0.617 0.001 0.502 0.000 56 093.8 1.47 0.671.054 0.002 0.546.044 o.ooi 9388 15.6140.056 000 0.592.046 0.00 57 309.02 15-1 0.727'.06 0.0049 0.o.ooi 58 324.62 i6.oo 0.787 0.oo o.oo i05 59 340.70 i6 0.851 0.002 0.693 052 o.o 56.. 56o 0.o 06.04 056 60 357-26 0.919 0.003 0.749 0.002 614 TABLE IX. For finding the True Anomaly or the Time fromn the Perihelion in Orbits of great eccentricity. x A Diff. B Diff. C B' 1Diff. C/ o I! I f / 1/ n lI II! 0.002 60 357.26 0.99 0.003 0.749.058 0.00 374.30 0.990.003 0.807 0.002 62 391.4 754.o66 6 0.00o.3 0.869.o6 000 6 3 409.86 I. 145'07 0.004 o0935 0.002 64 4.8.38 52 i.23.85 0.004.9.o o.ooz I9.OZ.,8 8.073 65 447.40 1.3IS 0.004 1.077 0.003 66 466.92, 152.4i 93 0.005 1.154.07 0.003 676 486.96 20.04 Io'099 0.005 I.235 O d68 z6.6 55 i. 3 I.09 o.oo6 1.321 0 o.oo 6 50725I o 141.090 0004 69 528.58 2107 1.721I 0.006 1.4II o.oo4 21.59 -114.094 70 550.17 1.835 0.007 i.505 0.004 71 12.12.I~g ~605 2.002 57z.9 2265 1.954.19 0.007 i.6o.10 0.005 72 5 9494 2 2.078.124 0.00 1.704 73 61.2 20 28. z 18 09. 0.0099'.11 o.oo6'73.Ix3 74 64i.85 2373 345 0.009 1934 2 o.oo6 75 666.13 2.488 o.o4o 2.055 z6 0.007 76 690.96 24.83 2637'49 o.o01 2.181 0.007 77 6.4 25.38 2793.56 0.012 2.314.133 0.008 78 2,9U.956 o63. 3 2.453.139 78 742.29 295 2 0.013 2 0008 z6.2 z 3.146 79 768.81 27 3. 65 -I 0.0o4 599.153 0.009 27.09'177 I53 80 795.90 6 3.302 0.015 2.752 o.o00 81 82357 7.7 3.486 84 o.oi6 2.92 0i o.oi0 82 85884 2 3 27 677 3079 67 0.012 83 880.70 2.46 3878.200 o.oi8 3.255 0.013 84 x91o.16 2946 4.087 z9 0.020 3.439 -' o.oi 30.07.216.92 85 940.23 30.69 4303 0o.o021 3.63 o.o5 g6,3 z-6.zoz 970'92 14 5121 9 0.023 3.833 o.oii 87 1002.4. 3 4.764 235 0.024 4044. 0.018 88 10o34. 5.oo8.244 0.026 4.266.22z 0.019 ~ 320.z66. 3 5I 0..02 89 1068 365.z6z 2 54 0.028 4.498. 0.02I 33.27 265.243 90 1100oo.o8 557 0.030 4.743 0.023 91 194-. 33'94 5.801'274 0.03z 4.996'55 0.02 134-.02 - 34.zS6 4 99526.267 0.025 92 1 161 64 35-3 6.087 ~8 0.034 3.281 0.027 93 1203.9' 6.385 98 0.036 5544 0.029 94 1239. 36.75 6.694 309 0038 5.838.294 0.032 OQ I2 7 3^-75 -.322 -309 95 1276.72 7.01i6 0.041 6.147 0.035 96 114.21 37'49 7-350'334 0.044 6.471.324 0.038 97 11352-45 3 4 7698 3 o.o0047 6.2 341 0.041 98 139I.46 39- 8.o6o.36 0.050 7.171' 0.045 99 1431-27 39.1 8.437'377 0.053 7.549 0049 40.61.392 0397 100 0 1471.88 20.6 8.829 0.056 7.946 6 0.053 30 1492.50 9032.0 0.08 8.12 0.05 IzP 0 21 9.032.203 0.055 8 101 0 1513.3 9-238 o.o6o 8.364.212 o.o58 30,1534.38 9..05 4 9 ~44 0.062 8..28 o.o6o 102 0 1555.64 2 9.664 25 0.064 8.805 223 0.063 30 1577.1 z 48 9.883.29 o.o66 9.035.30 o.o66 21.70.225.236 103 0 1598.82'10.108 ] o0.o68 9.271 0.069 30 1620.75 1 10.337 9 0.070 9.513'.24 0.07 104 0 1642.91 21 o 20 33 0.072 9.761.4 0.075 30 1665.30 22. 39 1 0.09074 10.017 0.07 99639'I ~7 5 0. 0.077 20.082 105 0 1687.93 8'6 11-053 0.077 1.0 0.09 30 1710.80 87 11.302.249 0.079 10550.270 0.085 23~12.9557 106 0 1733-92 2336 11.557.260.286 0.089 30 1757.28 11.817 0.2 + I1.14 0.093 7 z8. 52o 3.62.266 0.08.29 0.0 107 0 1780.90 122083 0.087 11.408.294 30 1804.77 23. 12.354.271 0.090 11.711.303 o. 108 0 1828.90 24.13 1.632.278 0.093 12.022.311 30 1853.30 24.40 12.96.2 0096 12.343.32 o.11. 24.67 29.330 109 0 1877.97 13.207 0.099 12.673 o..17 615 TABLE IX. For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. x A Diff. B Diff. C Diff. B' Diff. C' Diff] 109 0 I877-97 13.2 07 0.099 12.673 0.117 a0 1902:9 I 24.94 13.504.297 0.003 130I.340 122 005 25.229.3040.004 I A 30.oo6 110 0 1928.13 5.22 13.808.3 0.10oio6.001 I13363.35 I 0.128'006 30 1953.64 35 141 1. 003:36 0.0o I ~i.~3~I' ~9 o. i'~~.o9~'~1 o,7.~ 0I3 O I 11 0 97944 5.0 14438 *319 0113 *004 0.371 341 30 zoo5.54 I 14~764.326 o.1i6' 1c) I4478 I38 o.383 48 i 26.40.326003 007 vz6-'40 / /.333 00. 396' 007 112 0 2031.94 15.o97.I20 14.874 0Io.155 30 2058.64 26.70 I15439'34.124.,I 5.282.408 I6. 007 113 0 2085.66 27.0 z 15~789.350 o.28S.004 I 570'.420 7' o8 15~7o2'.0 30 211300 2734 16.148 359 0.13.004 16.1 5 433 0.178 140 24.66 27.66 I.367.005 i6.5 3.448 o.IS7.oo0 1151 02197009o.6 6 1.035 0 1 6.85 I''/ |.9z 377 " 4 -_.005 2 46 o.'0.l9 30 21i68.66 386 6''~142'' ~ ~5"' I 115 0 2297.00 9 z28.69 17. 27 396 0.147 I7.52.010 17.674 0 152 30 z8.69 1767 96 05, 005 1 -493 ~.14 7 11t 0 2254.73 29.04 18.o8o.406 0157.005 I8. -24.509.7Oi:oo6 19.594 6z 0.251 30 34. 30. I963 439.17'674 4I~13'oo6 20156.56z 0.264 0I3 30 34406 30154 6 450.o6 582.I3 118 0 2374.6o I 9.'8'3 0 o. 73 1 0 o.z7 30 2405.54 30.94 20.276.463 o86.oo6 20.738.6oi 0.277 119 0 2436.88 Z3O 7 5 5.9 6.007 22.6' ~ 06 I', ~ ~ z~7'' q~ 2~5 9 oo 044.o12 i 30 2468.64. 23.240 475 200.007 21.626 64 32z 12 0 2O500-83 o. I' 4'50.007.667 -39.017 120 32.19 21.742.502 0"207 23.273.69 339 o8 30 3345 32.62 22.258.4.007 23.964 69 0.357 1 33.06 1 531 20. z.736 121 0 2566.51 33. 22. 0.222 0o8 24.680 0.376 30 26oo0.0 3 35 36 57 0.230 25422.742 0.396.021 122 0 2634.z 3 99 z3.98 56 o.239 009 26.9. 769 0.7 417 021 3 0 2668.49 3447 24477 0.248'009 36.988 797.022 14,o5~~~ ~~.579] ]l477 ~jg6 o~z.1 26.9 0.439 123 0 2703.46 34'97' 27 87 06.024 [!8 35347 25.687:614 o.z68.010 28.673 858 025 30 z6738.93 35 2..633.010.89 I 0.488 027 124 0 2774.91 26.320 6.278. 911 29.56 3 0.515 30 2118.43 36.52 26.973 563 0.289 30.489.9256 24 3.~~~~~3.961 0.544.029~~~~~~~~~.o3 125 0 2848.50 37.07 27.646 67 0.300. 300450 0.574.030 30 z2886.3 37.63 84I 695 0.3 01 48.998 o.6o6'0327 38~~~ ~ ~~~~~~~~~.20 1'176 0/I 3.:ti,2~~.6' 244.06 ~3 1.46) U 2924.33 ~38.zo ~ 7 032 813 1 037 0.640 034 138 29.057 0338 0~3 3 30 2963.2 389 29.797 740 0 0 3 6'078.036 ~47 o o~3 7 423 o45o 39.41 765 01. I22.039 127 0 3002.53 30.562.03 5 o.6 675 0.6715 I30 3042.56 4003 31.351 789 0o.32 356 0.757![ 8 Ollo8 123 B"-h7 31-35' g86 0.367 36.988.0 128 0 31o8,: 40.67 3z2.67 0.382 015 38.o67 1.215 0.800 043 3 0 4-1-34 -844 0.gz-o6 i.z64 4646 ~o( 30124357 3301 I 0.398 39.331 1.6 0.846.0o4 129 0 3i6. 4202 33885.874 0415 OI7 1.38.896 050 30 3209.31 42.72 34.789 904 043 3.019:5 13 3.43 05 09 43-45.9 36oI9 1-430o6 130 0 3252.76 35.725 642 0.452 43.452 0.987 1.005 20 3282.13 29.37 36.367 0465 OI3.040 40 331.85 6.37025.658 014 444139 I.o. 1045 0 9.7 ~ ~ o~ 45.4 04 3 118537 67.479 46.500 1.045 1087 04 03341.90 3005 37.699.6o 493075 1.130.0 20 37231 30,41 38 84839 I.8395 47575' 3403-09 39.097 708 0.523.015 868z 1.107 I.2237.725 o6 1.138.050 132 0 3434.z3 15 39 2 0.539.oi6 49.820 1.273 20 3465-74 3'5' 40.564.742 0555 50.992 1.172 325.052 40 349763 31.89 41.326.762 0572 5 ~2.99 2 7 o05 4 3Z.7,8 1-379 7 ~54 133 0 35299 32.2 42.108.782 0590.oI8 21243 I379 20 356260 3269 42.910.802 069 0 53442.28 1. 4.05 38'79I o.07 ] ~4.o36 40 3595.69 33.09.823 o6 Oz 49 5 63 351 43~733.843 g'.020 1.359 i.558.6 20 3663.13 3293 4 8 609.069 134 0 36z29. 3520 I 44.576 866 0.649 00 57.401 39 1.623 2 0 3 66. 3 3-93 45-442.86.669'020 5 I 2 I401 i.69z o~g ]11~7~~~~~~~343 8 35.6802 40 36970.50 34o7 46.33 o. 0691.22 60.247 1445 1.764 072 135 373.3 34 47.45 914 74.023 61.736 9 39 075 20 3767.58 35.27 48.183.9.9024 673 2173 I2537 7 4.3 07{ 40 3803.3I 35 49147.964 0763.025.857 1' 84 2.000.03 36.21 6 1634.087 t-2~9I.025 [ 13 0 3853952 50I3 o.088 66491' 2.087 / 616 TABLE IX. For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity x A Diff. B Diff. C Diff. B' Diff. C/ Duff. i I 0! I I /1!! 1!! f!! /; 1; 1! 136 0 3839.52 50.38 0.788 66.491 i68 2.087 20 3876.z 3669 51156 o.815 6.I78. 178 9 20 1..to I~047 o~ oz8 68.178 74 2.096 40 3913.4I 37' z 52.203 47 0.843 69.920.74.274 137 0 395. 37' 53.280 1.077 0873.030 7 I718 1.79 2.375 20 5.I 38-3 I.108 0.03I.g57 2'375 20 3989.35 543388 0.904.03 73575 2.480. I05 40 4028.11 38.76 55528 1.140 0.936 7549.032. 591 1380 0672 39.31 1.174.033 1.982.117 39~3 I 5' I, I74~ 03~I 138 4067. 3942 5702 08 o.969.035 77, 475 2.708 6'4702 x.208 0.969 79.52 2.0492-831 123 20 4Io7.Zg 39'86 i03 579t.oo4 z.83I.'" 40'44 1.244. 3 7923.II8.129 40 4147.7 4044 5 I244.4 037 81. 644 2.960 40 447.72 59'4 122'4-x.~3 383 x.zg2 ~9 139 0 4188.75 4103 60.436 1.8 1079.038 83.830.189 3.096.136 20 4230.38 41.63 6'757 I.32.119.040 86.094.4 3.239 40 4272.63 4'2.z5 63119.362 i.i6i.042 88.436 2.342 3.390 I5 42.89:.404.044!2.424.159 140 0 4315.52 64523 1.205 6 90.860 0 3.549 i68 0 43I5-5z 43'554 1.448.046 2-509. x68 20 4359.o6 43'54 I93.369 13.717 7 I.~4 o8 956 58 3'7r7 7 40 44403.1 44.20 674-65 I494 12:8 95967 259 3.89 3 87 141 0 4403.26 44.89 69007 1.542 1.350 05I 8.657 69 4080 69'o7I'35 ~.~ ~98 5 20 4493. 45.58 1.592.0.:54 9 278.o1 01.443Z. 4z7 I9 404.3 46.30 702599.6441404.o56.443 288 277 40 4540.03 7zz43 1.46~ 0~4.331 4-484 47.04.698.058 21993.224 142 0 4587.07. 73'941 8.518 107.34 4.704 10 4610.88 23.8 74811 1.549.03 io86i I537 4.8 9'I5 20 4634.88 24.00 75.695 0.884 1.580.03 1 10 I.5 4936.117 30'4659.07 76.595.2.9.5 30 465907 2419 76.595 0.900 1.6z12.032 112z.z022z.595.12zi 40 4683.46 24.39 77.509 0.914 i.645'033 I 3.646 1.624 5181.4 7~~~~~509X364 5'~57.I24 50 4708,o05 2459 78439 0.930.679.034 5.3 i65 5.39 Iz8 143047 4 24.79 0 7 946.035 i.685.1319 1 0 4757.84 79.385.962 1714 0 6.986 8 5.440 3 10 475784 00 80.347 0 I978'.749.05 8.704 178 5.575 I35 20 4783.05 25'z~' x1.3 5 0978.786'037 xzo.45z I'748 5.715 8r.3z5 o'996~~~~~~~~~~5..~ 75 30 4808.46 2541 8 0.3Zi 9 1.8z3 037 I20.45233 78 5.858 I43 25.641 82.3211.823.037 40 4834.10 z5664 83.333 I.01o2 1.862.039 I24249. i.Si6 6.005 147 ~ 4.~ 25.64 ~'~ 3 24.049.85~ 6o57 452 50 4859.95 5 2,85 84363'.030.9 o'039 125.8 99.8.152 26.07 1.048.041 i.886 I 567 144 0 4886.oz02 85'4 1.6 I494z 129z77 8.92 6.313 i6o 10 4912.31 z6.z9 86478 i'787 3.984.042 643 i 90(~~y18.81 ~~6.j~~ I.o~d ~8. z ~ i 6~.473 20 4938.83 z6.z 87564 i.o86.6.04 13.666.959 6.639 30 4965.58 26.75 88.668'.104 070 044 I33.663'997 6.809 170 40 4992.56 26.98 89793.z5 2. ii6 046 135.698 2.035 6.984 I75 50 501I 9.78 27.4z 95 79 3 I'I45 z.162 046 3 92.076 7.165 18 o27'45 9..65 37.774 2.116 7'x65 086 145 0 5047.23 2770 92.103.87 2.2 0o 0 139.890 2.I58 7.35I 9 10 27~~~~~~~~~~~~~~~~~~~.70.92 10 5074.93 27. 93.290,o8 2259.0543 20 5I029.88 95 944 91 4- 24:.zo 740'9 ~~230 5131.08 - 95 729 1.23I 16i 0 II44.249 2.245 79 30~ ~ ~ 2 2''O.3.361 102 46.494 21.203 40 5159.53 28.45 96982 1.253.053 I48.784 2.290 8.153 Z10 j0 I 5' 28.97 1.277 2.414.05 11102.33 8.369 50 5 I 8 8.7-4. z8-7 98-259 I-277 z-469.055 Iz. 2o~ 3 5.9188 7 9.300.057 2"383.223 146 0 5217.21 99.559 2.526 8 153503 8.592 10 5246.45 24 00.884 1.3z5 2.584.058 I9 24- 3I 882 30.o5 ~ ~ 43I 2.290.23o 20 5275.95 29.50 I02.34 1.350 2643.059 84 2481.238 29.-S ~~~~~o6i 1.1.6 30 5305.73 98 1036io 1.376 0 60947 2.5 32.244 40 30.06. 9 1.3402 2.704 63 I6.53 z584 9.304 5335~7 ~ ~ ~ ~ ~ ~ 249 10.055 40 5 3 3 5.79 I 1.429 2767.o66 3531 9555 260 50 5366.13 30-34 106.441 1.429 21833 06 i666668 2692.z 68 30.63.067 2.692 16.o5 7 I 2.o383., 147 0 5396.76.9 107.897 2.900.069 168.860 2748 10083.276 10 5427.67 109. 2.485.969 171.6 6.80 359 z86 20 5458.88 31.21 110.896 1.514 3.040.071 174.414 2.866 0. 645 30 49039 31.51 11 I2543 3 073 177.280 IO. 940 30 31.81 1141013 9 1.574 3.188.075 I7z180.206.2638 40 55zz2.zO I'3I.6o6.3 6'09 i8 I.2644.34 50 5554.33 23 I15.6o9.6o.078 I 1.7558 314 32.44 I.ogo 183.194 3.052.325 32~44~~~~~~~~~~2.3.244 148 0 5586.77 117.256 3.346.082 186.246 8 11.883 10 6 9.5 Z 32.75 ii8.9z6 i 4670 3.28 i8 9.64 3.118 I2.218 335 10 5619527 ~-x 26 4~ 1.670 3 06 8' 20. 5652.60 33.08 120.631 1.705 3.513.05' 49 3.185 12.564.346 35.o88 3'255 30 5686.0 33 I739 3.601 195804 325 2.921 357 122.370 3.326 3.70 40 5719.75 3374 124.144 1774 3.691'090 99g.130o 8 13291 370 55 3 34. 9 I.8 3.784.093'202528 339 13.673 5)738 340.3 1x.5.849.097 3.474.394 149 0 5788.26 27.804. 881 206.o002 14.067 017 TABLE X. For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits. Ellipse. Hyperbola. log B Diff. log loI.Diff. log B Diff. lo- C logI.Diff. alf halfII.logI. Diff. hlfII. Diff. 0.000 0.000 0.00oo oo o.ooo oooo0000 4.23990 1.778 oooo o.ooo oooo 0000 4.23982,,1 77.oi 0007.ooi 7432.24286 783 0007 9998 2688.23686.767.02 0030 3.003 4985.24583.78 0030 37.996 5493.23392.762 03 oo0067 3'005 2659.24885 -794 0067 i'.994 8414.23098.758.04 020 68.007 0457.25190.799 0118 66 993 1450.22807 -753 0.05 0188 0008 84 0.008 381 4.25497 1.805 0184 8 9-991 4599 4.22518z 1.748.o6 0272 99.oio 6432.25806.811 0265 94.989 7859.22230.743.07 0371 114 -012 4613.26116.86 0359 lo09.988 231.21943 -739.o08 0485 i30 -.04 2924.26427.821 0468 123.986 4711.21659 -734.09 0615 147.' 6 367.26741.827 0591 137.984 8298.21376 ~.730 o.io 0762 162'0.017 9945 4-27057 1.833 0728 529.'983 1992 4-21094n 1-725.11 0924 178'0o9 8659 -27376.839 0880 i65.981 5791.20815.720.12 102 94.021 7511.27697.845 1045 178'979 9694.20537.716.13 1296 211.023 6503.28020.851 1223 I93 -978 3699.2026o.711.14 1507 227'025 5637 -28344.857 1416 206.976 7805.19986.706 0.15 734 243 0.027 4916 4.28670 1.863 1622 220 9.975 2011 4.19712, 1.700.16 1977 z6i'029 4340.28999 1842 233.973 6316.19440.695.17 2238 77.031 3913 -29331.875 2075 246.972 0719.19170.690./8 2515 294.033 3636.29665.882 2321 260.970 5218.I8901.685.19 2809 31i.035 35I1.30001.888 2581 73.968 9813.18633 0.20 3120 g28 0.037 3542 4.30339 1.895 2854 286 9.967 4502 4-18 367% 1.672.21 3448 345.039 3730.30679 90 3140'965 9285.1810oz.666.22 3793 63'04 4077 31022.908 339 -964 59.17840.66 Ji 3 4156 3z5 I 96z 9I24 317579.23 4156 381.043 4585 -31368.915 371 962 24 17579 655.24 4537 398.045 5259.317216.922 4076.961 4.17319 649 0.25 4935 4I160047 6099 4-32066 1.929 4414 351 9959 9324 4'-7061, 1.643.26 5351 4.049 7109.32418.936 4765 363'958 4556.68o3 637.27 5785 452.051 8290.32773 -943 5128 376.956 9875.6547.631.28 6237 4i -053 9646.33I13I 951 5504 389 -955 5281.6292.625.29 6708 g s 056 117I.33492.958 5893 40 -954 0771.6038.6i8 0.30 7196 0.o058 2893 4'33856 1.966 6294 9-952 6346 4.157855 1.613 TABLE X. Part II Ellipse. H Iyperbola. Ellipse. Hyperbola. T ~~ T ~~ A Diff. A Diff. A Diff. A Diff. 0.00 0.00000 0.00000 o 0.20 0. 7266 0.23867.0.0099.oioo8 o5.2.x8oo8 742.25309 44.02.0169g 977. 02033.22.18740 732.26779'470.03.02930 03074 8 ~8.23.19462 72.2828~0 5I5.04.03 877 97.0432 1.24.20174 71 29813 564.o6 055 7264- 918 0.05209 0 25 o.20878 695 0.31377.07.06630576 904.06303 I94.26.21573 6.7.6630 I89.074.7 1 41.27.22258 677 9 -098 65.09o2 I.29 -3 o. o o.og263 0o. 0875 30 0.'24265 6.11.o10I6 853.1z 2069 2 I 694.31.24917 65.62.10956 840.13285 i 36. 2 5566i 61.13 -11783 8127 45z IZ37.33.2698 637.64 -.2599 85.15782.3.26826 6 0 0.5 o0.3404 0.17067 0.35 0.27447 61.i6.14I98 794 18375 1308.36.z8o6i 4.17 -I4981 783.9709 334.37.28668 607.28.15753 77 2 68 359.38..29268 oo.19.651 22454 762 38 9.2860 592 0.20 0.17266 0.23867 0.40 0.30446 618 TABLE XI. For the Motion in a Parabolic Orbit. log Diff. q log ) Diff. _ log fL Diff. 0.000 0.000 0000 o.oo.o o.o o06.65 o. 120 0.000 2617.OOI.000 0000 0.061.000 0674 22.12I.000 266I' 44 44.002.000 0001 I.o62.000 0697 23.122.000 2705 45.ooz 223.24.000 2795 46.oo003. 2.o0002 63.ooo 07Ig I23.ooo 2750 45.004.000ooo 0003 I.06.ooo 0742 2.I24.000 2795 46 o.oo o.ooo oo004 o0.o6 o.ooo 0766 0.125 o.ooo 284 1.oo6.ooo 0006 o66.ooo 0790 24.Iz6.ooo 2886 45.007.0 0.067.ooo000 0814 4.127.0 2933 46.008.00ooo 0012 3.o68.ooo00 0838.28.0oo 2979..009 0. 3.069.oo 0863 25.129.oo0 3026 47 3 25 48 0.010 0.000 oo08 0.070 0.000 o888 o 0.130 0.000 3074.011.000 0022 4.071.00o o914 26 I3I.000 3121 47.012.000 0026 4.2.000 0.132.000 3169 4.013'.ooo 0031.07.o000o 0966 7.I33.ooo 31 49.014.ooo 0035.074.0o 0993.134.0oo 3267 ~6 z~27 49 0.015 0.000 0041 0.075 0.000ooo 020.35 0.000 3316.016.ooo 0046.7.o oo000 1047 36.00oo 3365.1O7.000 oo52.077.ooo I075 28.37.000 34I5.o08.ooo 0059 7.078.000 II03 I38.0oo 3466 5.019.000 oo65 6 79.ooo000 II32 29.139.000 356 5 7 29 51 0.020 0.000 0072 o.o80 0.000 1161 0.140 0.000 3567.02.000 oo80.08I.000.0ooo. 36g1 8 1190.ooo 3679 52.022.000o oo88.082.000 IZI9 4. 3671 623 3~ 52.oz3.000 0096 8.083.ooo I249 3 I43 o000 373.0244.000 OO04.08. ooo 003775 53 05 0.0 00 0113 0.085 0.000 1311 I0-45 0.000 3828.026.00o 0122 9.o86.000 I342 3 146.000 3882 54 0.07 31 53.027.000 OI32.o 087 00oo I373'147.ooo 393 5.028.ooo 0142. o8. ooo I405 32.48.000 3989 5 IO.029 32 55.02.000ooo 0152..089 ooo 1437 3.149.ooo000 4044 II 33 5 0.030 o.ooo 0163 0.090 o.ooo 1470 0.150 0.000 4099 II 32 5.031.000 OI74.o09.000 502 3.11.00oo 4154.032.0ooo o85.092.ooo 1536 34.52.000 4209 56.033.0oo 0197.093.00oo 1569 33.I53.000 4265.034.ooo 0209.094.ooo I603 34.154.ooo 43 5 13 o35 o.o000 4378 56 0.035 0.00 0222 0.095 0o.ooo 1638 o.155 o.ooo 4378.036.ooo 0235 3.096.ooo 1673 35.56.000 4435.037.ooo 0248 3.097.ooo I708 35.57.ooo 4493'4.098.000 7433 58.038.ooo 0262 4.08 8 13. 3 158.ooo 4551 58.039.ooo 0275.5 099.000ooo 779 6.59.000 4609 59 0.04o o.ooo 0290 o. O 0.1 0.000 1815 o.I6o o.ooo 4668.04.ooo 0o304 I4.IOI.0ooo 852 37.16.ooo 4726.04z.oo0o 0320 o.102.0 889 37.6 4786 6o.043.ooo 0335 16.I03.000 I926 37 163.ooo 4846 60.044.ooo 035 6.104.000ooo 1964 164 000ooo 4906 6.045 o.ooo000 0367 I. 0 o15 o.oo0 2002 o. 65 o.ooo 4966 6.046.ooo00 0383.1o6 6.ooo 2040 3.000 5027 6.047.000 0400 107.00ooo 2079 39.167.o 5o88 62.048.ooo 04I 7.o08.ooo 2118 39.,68.ooo 550 62.o49.ooo 0435 I9.ooo 2 158 40.169 0 522 62.049.000 0435 I8.09 4000 5000 0.050 o.ooo 0453 o 0.110 o.ooo 2198 o. 170 o.ooo 5274 63.o05.ooo 04 I7.II2.ooo 2238 1.17I.oo0 5337 6.052.000 0490 I..000 279 7 000 5400 6.053.000 0509.113.000 2320 4I.173.000 5464 64.054.oooo 528 2.114.000 2361 |7. ooo 5528 64.054 5000058 20 I14.0 42 074 064 0.055 o.ooo 0548 0.II5 0.000 2403 o2.75 o.ooo 5592 6 42 65 |.o561.ooo 0568.I16.ooo 2445 | *I76.ooo 56s7 6.*057 o.0 05089 2'.117.00ooo 2487 4.177.ooo 5722 58.ooo 06.8.oo6o.2.0 530 43.178.ooo 5787 66.059.00ooo 0631 2.119.ooo 2573 4.79.ooo 5853 66 21 44 66 o.060 o.ooo 0652 0.120 0.0oo 2617 0.80 o.ooo0 5919 619 TABLE XI, For the Motion in a Parabolic Orbit. log I Diff. 7 log. Dff.. l log 1 Diff. 0.180 0.oo0 5919 6 0.20 oo 0603 0.300 o.ooI 6733.181.ooo 5986 6.241.ooi 0693 90.301.00 6848 "I5.12.000ooo 605 67.242.o00 0784.302.ooi 6963.183.o00 6120 6 7.243.001 0875 9'.303.00o 7079 i6.184.ooo 6188.244.oo 0966.30. 7195 92 0o.85 o.ooo 6256 6 0.245 0.00o 1058 0.305 0.001 7312.86.0ooo 6325 68.246.00 1150 92.306.001 7429 17.187.00ooo 6393.247.00oo 1242 9.307.307 0ol 7546 I17.188.0oo 6463 7.248.ooI 1335 93.308.oo0 7664.189.000 6532.oo 2.0 149 429 309 -001 7783, I o.190 0.000 6602.250 o.oo 1522 o0.310 0.001oo 7901.19I1 ooo 6673 7.251.001 1617 95.311.00 8020 119 72.252. 94.312 814.192.000 6744 7 251.I 1711'31 oo 8140 20.193.0oo 6815 7.253 oo i8o6 95.33 00oo 8260.194.000 6887 72.254.I 1901 95 -314 8381 121 6887 72.354. o o z 96 83I4 72 96 121 o.195 o.ooo 6959 0255 o0.001oo 1997 6o 0.315 o.oo0 8502.196.000.7031 72.256.ooi 2093.316.ooi 8623 12.197.000 7104 73.257.001 2190 97 317.00 8745 22.198.ooo 7177 73.258.ooi 2287 97.318 ooi 8867 122.199.ooo 7250 73.259.0 84 2 9 192 49.199.000 7z5O 7 3 Z59.001 Z3 84 97.319.001 8989 0.200 0,000 7324 o.26o 0o001 2482 0.320 0.001 9113.20.000ooo 7399 75.261.ooI 2580 98. oo 9236.23.202.000 7473 74 262.oo 2679 99.322 oo 9360 24.2033 000 7548 7 263 001 2778 99.323.oo 9484 124 203 76 zh3 ~ooI z?7X 99.324.00! 9609 125.204.ooo 7624 76.264 00oo1 2877 99.34.oo 9609 25 76 0100 125 0.205 o-000 7700 0.265 ~-ooi 2977 Too Q-325 "0I 9734 10.206.00ooo 7776 7.266.ooI 302977.326.ooi 986 i.207.oco 7853 77.267.ooi 3178'0.327.001 9986.208.ooo 793.268.00oo 3279 -328.002 0113 78.02.329.002 0240.209.000 8007 269.001 3381 329 002 0240 0.210 0.000 8085 0.270 0.001 3482 0.330 0.002 0367 128.211.000 863 78.271.001 3585 03.331.002 0495.212.000 8242 79.272.001 3688 103 332.002 0624 12.213.000 8321 79.273 oo00 3791 103 33 002 0752.214.000 8400.274.oo1 3894 103 -334.002 0882 I30 0.215 0o.oo000 8480 8o 0.275 o.oo0 3998 0.335 0o.002 oI 1.26.00ooo 8560 8.276.001 403 105 336.002 1141 13.217.000ooo 8641.277.ooI 4207 io -337.002 1272.218.000 8722 8i.278.001 4313 -338.0oo2 1403 I3 8'ooo 883.279.0oo 4418 132.2I9 -ooo 8803 s ^79 -Oi 4418.-339.002 1534 0.220 0.000 8885 8 0.280 o.oo0 4524 0-340 0.002 i666.221.000 8967.83 I ooi 42863 34.002 1799 13.222.000 9050 2.001 4738 07.34.002 1931.223.000 9132 8 283.00oo 4845 o8 343.002 2065 34.224.000 9216.284 0014953 34 02 1 0.225 o.ooo 9300 84 0.285 o.oo 5o61 o8 0.345 0.002 2333.226.ooo 938 84.286.ooi 69 346.oo002 2467.227 ooo 9468.287.oo 5278 9 260.228.000 9553 288.00! 5388 ^ oo0 2738 3.229.00ooo 9638 86.289.00oo 5497 -349 00 2874 0.230 o.ooo 9724 86 0.290 o.oo0 5608 0.350 o.002 3010.231.000 98o10.291.001 5718 1 35I.002 3147.23z2 000 9897 8.292.001 57829 -352.00 3284 3.23 3 -ooo 9984 87.293.001 5941 353.002 3422.234 0 0071 8.294 oo00i 6053 i2 -354.002oz 3560 13 0.235 o.oo00 0159 88 0.295 o.oo0 6165 0.355 0.002 3699.236.ooi 0247 88.296.ooi 6278'3.356.002 3838'39.237.001 0335 8 -297 oo00i 6391 13 -357.002 3977 39.238.ooi 0424 298.oo 6505 358.002 4117.239.oo0 0513 9.299.ooi 6619 4 -359.002 4258 148 90 114 141 0.240 o.oo0 0603 0.300 o.oo0 6733 0.360 0.002 4399 620 TABLE XI. For the Motion in a Parabolic Orbit. [ 1 logM. Diff. - log/x Diff. 77 log/x Diff. 0.360.002z 4399 0.420 0.003 3720 0.480 0o.oo004 4858.36.002 454 -42I.003 3890 170.48.004 5061 203.362 46z.002 42.422.003 4061 171.482.004 5263 202.363.0 482 42 389.483.004 5467 20.364.oo2 46824.423.003.4.004 5670 203.364 497 424.003 4404 17 -44 004 570 I~~43 I~~172 205 0.365 o-o.0 5110 4 0.425 0.003 4576 0.485 0-004 5875.366.02 5254.426.003 4749.486.004 6o8o0 20 1452 0543 453 4.42..367 ooz 5398 1 -427.34923 173 -437.004 6285 zO7.368.002 5543.42.003 5096 4 -88.004 6492 0 6.369.002 88 9 003 5271 75 489 004 6698 08 I4603 0I75 0.370 o.oo002 5834 46 0.430 0.003 5445 76 0.490 0.004 696 07.371.00o 61256 4 43.4 003 5621.491.00473 20.367.00o2 6196 47.432.003 5797 17.-492.004 7322 09.373 6273. -433 003 5973 -493 004 7531.374 6421.434 003 6150.494.004 7740 147 177 211 0.375 0.002 6568 4 0.435 0.003 6327 048 95 0.004 7951.376 ooz 6717. 436.003 6505 496.004 8161.377.002 6866 149.437.003 6683 178 -497.004 8373 22.378.o2 7015 49 -438.003 686 179 498.004 8585 2 379.002 765 50 -439 003 7042.499 004 8797 150 213 0.380 0.002 7315 0.440 0.003 7222 ISO 0.500 0.004 9010 2163.381.002 7466,.441.003 7402 -.51.005 1173.38.00276 17. 0 18 2224.382.002 7617 1z -442.003 7583 IS2 -52.005 3397, gl.383.002 7769 -443.003 7765 z -53.005 568X.384.002 7921 5 44.003 7947 8 -54.005 8029 2348 37 z 1 76"' I52 2412 0.385 0.002ooz 8073 0.445 0.003 8130 8 0.55 o.oo6 0441 24 386.00oo2 8226 3 446.003 8313.83.006 2919.387.002 83 54 47.003 8496 183..006 5464 2545.388.002 8 34..448.003 868o'4.58.006 8079 2.389 OOZ.002 8689.9 003 8865 I8 9.007 0765 276 0.390 0.002 8844 0450 0.003 9050 0.60 0.007 3525 836.391 002 8999 6 -451.003 9236.61.007 6361 2913.392.002 9155 156.452.003 9422.62.007 92741 z.393.002 9311 15 -453.003 9609 18.63.008 2268 299.394 002 9468 5.454 -003 9797 64.008 5 345 1 5 8 187 3163 0.395 0.002 9626 0.455 0.003 9984 0-65 0.008 850o8.396.oo2 9784 1.4569.004 73.66 009 1759 3251.396.002 9784 Iss 189 3344.397.002 9942 -.457.004 0362.67.009 5103.398.003 o1io 159.458.004 0551. 68.009 8542 3'59'o'3 399 003 0260 459 004 0741 I90.69.oio 2081 364 0.400 0.003 0420 0' 0.460 0.004 0932 0.70 o.o 5723 3.401.003 0580 I6.461.004 1123 719.71.oio 9473 3 181q Ir23 ~7I.o~o 9473 3863.402.003 0741 62 -462 004 15 192.72.OI1 3336 398.403.003 0903 2 -463.004 1507 192 73 -oi 7316.404.003 1064.-464.004 1700.7 4 21419 0 0.405 0.003 1227 I6 0.465 0.004. 1893 0.75 0o012 5652 406.003 1389.466.004 2087 194.76.013 0022 4370.407.003 1553 64 -467.004 2281 94.77 -013 4536 4566.408.003 1716 i3.468.004 2476'9.78.013 9202.409 -003 1881 165.469.004 2672 196.79 -014 4031 4829 i6r6~~~ ~14 615002 0.410 0.003 2045 i66 0.470 0.004 2868 0.80 0.014 9033 5186.411.003 2211.471.004 3064 81.015 4219.003 2376.472.004 326 197.82.015 9603.412.003 23765 198 -413.003 2543.66 -473 -004 3459 98.83.oi6 5202 5.414.003 2709 68 -474.004 3657 -84.0171033 0. 003 877 0475 0.00 3856 o985 031-7 7120.41.003 3044 -476.004 4055 99.86.018 3486 036.417.003 3213 i69'477.004 4255 2 -87.019 o165 6679.418.003 3381 9.478.004 4456 20..88.019 795 7030.4I9 ~003 o3550 479 004 467.89.020 4629 7434. 33 33 J / r"170 201 7900 0.420 0.003 3720 0.480 0.004 4858 0.90 0.021 2529 621 TABLE XII. Zl' Z2I Z Z4' r log mnl log 0m m1. |n, W12 Ml 1l 112 mn'2 W1l of o t o of o t o f o r o f o - 0 0. 0.0000 0 0 90 0 90 0 8 0 0 180 18 0 0 0 0 0 1 4.2976 9.99999 2 23 90 20 90 20 I78 40 178 40 I79 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 78 9 10 910 I76 o 176 o 177 o 357 0 357 I41 4 2.4938 9.9986 9 32 91 20 91 20 174 40 I74 40 176 o 356 o 356 I8 5 2.2044 9.9978 II 55 92 41 9I 41 I73 I9 173 I9 175 o 355 o 355 2.3 6 I.9686 9.9968 14 19 92 I 92 I 171 59 171 59 174 0 354 0 354 28 7 1.7698 9-9957 1 I6 42 92 22 92 22 170 38 I70 38 172 59 353 I 353 321 8 1-5981 9.9943 I9 7 92 42 92 42 I69 I8 I69 I8 171 59 352 I 352 37I 9 1.4473 9.9928 21 32 93 3 93 3 167 57' I67 57 170 58 351 2 351 42 1 0 1.3I30 9.99II 23 57 93 25 93 25 i66 35 i66 35 I69 57 350 3 350 471 11 1.1922 9.9892 26 23 93 46 93 46 i65 I4 i65 14 i68 55 349 4 349 52 12 1.0824 9.9871 28 50 94 8 94 8 163 52 163 52 I67 54 348 6 348 56 13 0.982I 9.9848 31 17 94 31 94 3I 162 29 162 29 i66 5 347 8 348 i 14 0.3898 9-9823 33 46 94 53 94 53 i6 6 7 6 7 65 48 346 II 347 6 15 o.8045 9-9796 36 5I 9 95 17 95 7 59 43 159 43 I64 44 345 14 346 I1 16 0.7254 9-9767 38 46 95 40 95 40 I58 20 I58 20 I63 40 344 I7 345 I6 17 0.6518 9.9736 4I I8 96 5 96 5 156 55 I56 55 162 34 343 21 344 21 18 0.5830 9.9702 43 5I 96 3 96 30 155 30 I55 30 I6I 27 342 27 343 27 I10 0.5185 9.9667 46 26 96 56 96 56 154 4 154 4 I60 I9 341 32 342 32 20 0.4581 9.9629 49 2 97 23 97 23 152 37 I52 37 159 9 340 38 341 37 21 0.4013 9,9588 51 41 97 50 97 50 151 o IO 51 io 157 58 339 45 340 43 22 0.3479 9.9545 54 22 98 I9 98 19 149 41 I49 41 156 45 338 53 339 49 23 0.2976 9.9499 57 5 98 49 98 49 I48 8 22 148 II 155 29 338 o 338 54 24 0.2501 9-945I 59 5I 99 20 99 20I 46 40 146 40 154 II 337 9 338 0 25 0.20o53 9.9400 62 40 99 53 99 53 145 7 I45 7 152 50 336 I9 337 6 |26 0.I631 9-9345 65 33 Ioo 28 Ioo 28 143 32 143 32 I5I 25 335 28 336 1I3' 27 0.1232 9.9287 68 30 IOI 5 IOI 5 I41 55 141 55 149 56 334 38 335 19 28 0.0857 9.9226 71 33 IOI 45 IOI 45 140 15 140 15 148 22 333 49 334 25 29 0.0503 9.9161 74 41 102 27 102 27 138 33 I38 33 146 42 333 i 333 32 30 0.0170 9.9092 77 58 I03 23 03 13 136 46 136 46 144 55 332 I2 332 39 S31 9-9857 9.90I9 81 23 I04 4 104 4 134 56 134 56I142 59 331 24 331 46' 32 9.9565 9-8940 85 O I05 I I05 I 132 59 I32 59 140 5I1330 3 73 330 54 3 33 9.9292 9.8856 88 54 106 6 1o6 6 130 54 130 54138 27 329 49 330 2 34 9-9040 9.8765 93 II 107 22 107 22 128 38 128 38 I35 39 3239 2 32 35 9.8808 9.8665 98 7 io8 8 58o8 58 126 2 126 2 I32 13 328 14 328 19 36 9.8600 9.8555 104 20 III 13 III 13 122 47 122 47 I27 29 327 27 327 28 -36 52.2 9.8443 9.8443 116 34 II6 34 II6 34!6 34 1I6 34 116 34 326 45 326 45 This table exhibits the limits of the roots of the equation sin (z' - C) -= o sin z', when there are four real roots. The quantities m, and m, are the limiting values of m,, and the values of z,, z2/, Z, and z4', qorresponding to each of these, give the limits of the four real roots of the equation. 622 TABLE XIII Z_! Z2/ Z3! Z4! ( log ml log m12 - - m,2 7;1 m mMl n, mM M, Ml n1 2 o o 0 o 0 o 0 o 0 o / o 0 o 0 o 0 -+ 0 oo 0.0000 0o o o o 0o 0o g0 9 o 180 18 0 80 0 1 4.2976 9.9999 I 0 I 20 I 20 89 40 8 40 I77 37 8So 55 I8I o 2 3.3950 9.9996 2 0 2 40 2 40 89 20 89,20 175 14 181 51 i82 3 2.8675 9.9992 3 0 4 0 4.o 89 0 89 0 172 52 182 46 I83 0 4 2.4938 9.9986 4 0 5 20 5 20 88 40 88 40 170 28 183 42 184 0o 5 2.2044 9.9978 5 0 6 41 641 88 I9 88 I9 i68 5 I84 37 I85 6 1.9686 9.9968 6 0 8 I 8 87 59 87 59 I65 41 I85 32 I86 o 7 1.7698 9.9957 7 922 9 2 87 38 87 38 I63 I8 I86 28 i86 59 8 15981 9.9943 8 I 10 42 10 42 87 18 87 18 60 53 187 23 187 59 9 1.4473 9.9928 9 2 I2 3 1 3 86 57 86 57 I58 28 88 I8 188 581 10 1.3130 9-9911 I0 3 13 25 13 25 86 35 86 35 I56 3 189 I3 I89 57 11 1.1922 9.9892 II 5 I4 46 14 46 86 4 86 I4 153 37 190 8 190 56 12 I.0824 9.9871 I2 6 16 8 I6 8 85 52 85 52 151 Io0 91 4 191 541 13 0.9821 9.9848 13 9 I7 31 I7 31 85 29 85 29 148 43 191 59 I92 521 14 0.8898 9-9823 14 I2 I8 53 IS 53 85 7 85 7 146 I4 192 54 193 491 15 0.8045 9.9796 I5 I6 20 17 20 17 84 43 84 43 143 45 193 4-9 I94 46 1I I 0.7254 9.9767 I6 20 21 40 21 40 84 20 84 20 I4I I4 I94 44 I95 43 17 o.65I8 9.9736 17 26 23 23 35 83 55 83 55 138 42 195 39 196 39 18 0.5830 9.9702 I8 33 24 30 24 30 83 30 83 30 136 9 196 33 197 33 19 0.5185 9.9667 I9 41 25 56 25 56 83 4 83 4 I33 34 197 28 198 28 20 0.4581 9.9629 205 1 27 23 27 23 82 37 8237 130 58 198 23 199 22 21 0.4013 9.9588 22 2 28 50 28 50 82 Io 82 IO 128 I9 I99 17 200 1 22 0.3479 9.9545 23 15 30 I9 30 I9 81 41 81 41 125 38 200 II 201 07 23 0.2976 9.9499 24 31 31 49 31 49 81 II 8i II 122 55 201 6 202 o 24 0.2501 9.9451 25 49 33 20 33 20 80 40 80 40 I20 9 202 0 202 51 25 0.2053 9.9400 27 I0 34 53 34 53 80 7 80 7 117 0 202 54 203 4 20 0.1631 9.9345 28 35 36 28 36 28 79 32 79 32 II4 27 203 47 204 32 2 7 0.1232 9-9287 30 4 38 5 38 5 78 55 78 55 III 30 204 41 205 2 28 0.0857 9.9226 31 38 39 45 39 45 78 15 78 I5 I08 27 205 35 206 ii 29 0.0503 9.9161 33 I8 41 27 4I 27 77 33 77 33 I05 19 206 28 206 59 30 0.0170 9.9092 35 5 43 13 43 13 76 47 76 47 102 3 207 2I 207 48 1 99857 9.9019 37 45 4 45 4 75 56 75 56 98 37 208 4 20836 32 9.9565 9-8940 39 9 47 47 I 74 59 74 59 95 0209 o6 209 23 |33 9.9292 9.88.56 41 33 49 6 49 6 73 54 73 54 91 6 209 58 2I0 II 34 9.9040 9.8765 44 21 51 22 51 22 738 72 38 86 49 210 50 2I0 58 3 5 9.8808 9.8665 4747 5358 53 58 71 2 7I 2 81 53 211 41 211 46 |36 9.8600 9.8555 52 31 57 I3 57 13 68 47 68 47 75 40 212 32 212 33 -+36 52.2 9.8443 9.8443 63 26 63 26 63 26 63 26 63 6 63 6 3 213 15 213 15 This table exhibits the limits of the roots of the equation sin (z' - ) m= sin4 z' when there are four real roots. The quantities m1 and m, are the limiting values of mn, and the values of z/, z2,/ z,', and z/', corresponding to each of these, give the limits of the four real roots of the equation. 623 TABLE XIII. For finding the Ratio of the Sector to the Triangle. I. log s9 Diff. Di log s Diff. o.oooo0000 o.oo0 00 o 0.00 5 o 7298 0.0120 O.OII 3417 6.ooo0.000 o965 965.oo6.0 3 012.o0 4343 92.0002.000 1930.oo62.005 9187 944.0122.01 I 5268 925.0003 ooo 2894 94.o063.oo6 O131 9.0123 I 6193 9.ooo4.ooo 388 964.6 944.oI2.0004..o 88 9.0064.006 1075.0124 011 7118 9 963 944 925 0.000 o.ooo000 4821 0.006 0.00oo6 2019 0.OI25 O.oI I 8043.0006.ooo 5784 963 oo66.oo6 262 943.o026.oil 8967 92.0007.ooo 6747 6.0067.oo6 3905 943.oII 9890 -0007 ooo 6747 963 09 923.ooo0008.ooo 77.oo68.oo6 4847 942.0128.012 0814 924.ooo.000 8672 962.oo6g.oo6 5790 943.0129.012 7 923.000962 ^ 942 923 o.oo o.ooo 9634 96 0.0070 o.00o6 6732 94I 0.O30 0.012 2660.0011.001 0595 96I.0071.oo6 7673 94.0131.I 3583 923..0 01 56 96.0072.oo6 8614.97.013.0124505 922.00I3.0oo 2517 96.0073 oo6 9555 94.33.OI 47 9.0014 - 3478 96.0074 007 0496 94.0134 0oI 6348 92I. 900049 940 921 0.0015 o.oo0 4438 6 0.0075 0.007 1436 0.0135 0.012 7269 gI.0oI6.oI 5398.0076.007 2376 940.036.012 8g190 9.0017.ooI 6357 959.0077.007 36 9012 9 92.0018.001 7316 959.0078.007 4255 939 I0138.013 0032 921.0019 -.00 8275 959.0079.007 5194 939 139.013 0952 92 959 939 919 0.0020 0.001 9234 9 0.0080 0.007 6I33 938 0.0140 0.13 187I.0021.002 0192 958.008.007 7071 8.0141.013 2791.0022 50.oo00.007 8009 98 04 3 371.0023 0083 7 007 8947.OI43.0I3 4629 9I.oo0024 002 3064 957 008.007 9884 937.0144 -3 5547 8 957 937 goi 547 0.0025 0.002 4021 96.oo85 0.008 082I 0.0145 0.013 6465.o26.002 4977 6.oo86.oo8 1758 937.046.013 7383 98 00 69 936I 9I7.0027 002 5933 h.0087.oo8 2694 6.0147 -013 8301.0028.002 6889 -oo88.oo8 3630 936.0148.013 9218 917.0029.002 7845 9).0089.008 4566 936.0149.014 0135 9I7 955 9I7 0.0030 0.002 800 0.0090 o.o008 5502 0.0150 0.014 I052 96.0031.o00 9755 95.0091.oo8 6437 935.0151 014 1968.0032.003 0709 54.0092. oo8 7372 935.052.014 2884.0033.003 1663 954.0093.oo008 8306 934.0.I 014 3800 96.0034.003 2617 95.oo94.oo8 9240 934.I54 OI14 47 96 0.0035 0.003 3570 0.0095 0.009 0174 0.0155 0.014 563I 95 0 / 934.0036.003 4523 953 o96.009 IIo08 934.0156.014 6546 9I5.0037.003 5476 97.097 009 2041 93.57.04 7460 914.0038 oo003 6428 9.0098.009 2974 93.I058.014 8374 9I.0039 -003 7380 95.0099.009 3906 932 0159.014 9288 9I4 952 932 914 0.0040 0.003 8332 9 0.0100.0009 4838 0.01i60 0.05 0202.0041.003 9284 952.oIoI.009 5770 932.o16.1o5 I15 913.0042.004 0235 95.o02.oo9 6702 932.0162.O5 2028 913.0043.o004 i86 95'.0103.009 7633 931.o63 6 015 2941 9.0044.004 2136 950.0104.009 8564 93'I oI64 -015 3854 913 950 93I g92 o.oo45 0.004 3086.o05 o.oo009 9495 o.oI65 o.oI5 4766.0046 4 436.004 40.oio6 I 0425 930.o66.0S5 5678 9.0047.004 4985 949 0107.00 1355 930.167.015 6589 g9.0048.004 593 949.o8.oIo 2285 93.0o68.I5 7500.0049.004 6883 9409 9.OO 3215 930.oi69.o15 8411 9II 949 929 gI9 0.0050 0.004 7832 948 o.oI IO o.o 4144 0.0170 0.015 9322.0051.004 8780 948.oiII.OIO 5073 928.017I.O6 oz32 gIo.00o5.004 9728 97.0112.oio 6001 98.0172.o6 1142 910.0053.005 0675 94.0113.010 6929 928 0173.oi6 2052 91.0054.005 I622 947.OI114 00 7857 8 0174 -o016 296I 909 947 6 909 0.0055 0.005 2569 0-.0115 o.o00 8785 o00175 o.oI6 3870.oo56.005 3515 946.oi6.oio 9712 927.76. oi 6 4779 909.0057 -005 446i 946.017.oI 0639 927 0177.o6 5688 90 ||.00*8.oo005 5407 946.0II8.oII 15 92.0178.oi6 6596 90.0059 -005 6353 945.011g9 24I 926 I79 oi6 7504 908 o.0060 0.005 7298 0.0120o 0.0o 34I7 0.0 6 81 624 TABLE XIII. For finding the Ratio of the Sector to the Triangle. -q logs2 sD Diff. log 2 log s Diff. 0o.o0so o.oi6 8412 0.0240 0.022 2330 890 0.0300 0.027 2zi8 8.o18I.oi6 93~9 907 -024 022 3220 89 030I 27 6091 873.0182.017 0226 907 zz.0302.027 6964 873.0183.017 233.0243.022 4998 8.0303.0277836.0184.017 2039 9. 5 887.0304.02z7 8708 872.o018.017 3^85I 9 Ozz 7664 III.o889.0630 872 90.o024 2 8889.0187.017 757.0247.022 8552 888.0307.028 1323 7'.0188.017 5662 905.0248.022 9440 888.0308.028 2194 871.0189.017 6567 905.0249.023 038 0309 028 365 87 904 887 8 871 0.0190 0.017 7471 0.0250 0.023 1215 887 0.0310 0.028 3936 8.0191.017 8376 905.0251.023 2102 886.0311.028 4806 89.0192.01oI7 9280 904.0252.023 2988 887.0312.028 5676 870.01o3.o018 0183.025 3.023 3875.0313.028 6546 870.0194.oi8 1087 904.0254.023 476.0314.028 745 869 0.0195 o.o08 2990 0.0255 0.023 5647 0 0-0315 0.02z8 8284.oi86 io3 885 ~9306.o28 o055 869.o796.018 2893 7.0256 02 6532.036.028 9153 869.0197.018 3796 903 257.023 74178.0317.029 0022.01I9.018 4698 902 o58 z3 8302 885.0318.29 0890 868.0199.o018 5600 902.0259.023 9187 885.03 9 1758 868 0299902 884.0319.029275886 0.0200oo o.oi8 6501oi 0.0260 0.024 0071 0.0320 0.029 262686.020.o018 7403 902.026.024 0956 885.032.02 3 68.oi8 gzog901 1839 884.0322.0203 9205.0263.024 2723 883.0323.o29 5228 0 7.0204.09 005.0264.024 30324.029 6095 866 0.0205 0.019 1005 0.8 265 0.028 4489 0-0325 o.o29 6961 866..006.019 1905 0.0266.024 5372 3.0326.029 7827.0207.oi0 2805 900.0267.024 654 88.0327 2 8693 866.0208.019 3704 899. 0268. 7136.0328.029 9559 86.0209.oi9 4603 899.269.024 8oI8 1.0329.030 0424 866 0.0210 0.019 5502 0.0270 0.024 8900 88 0.0330 0.030 1290 864.0211.019 6401 898.0271.024 9781 88I.0331.030 2154 865.0212.019 7299 98.0272.025 0662 88i 0332.030 3019 864.0213.019 8197 89.0273.025 1543 88.0333.030 3883 864.0214.019 9094 898.0274.025 2423 88o 0334.030 4747 864 0.0215 0.019 9992 0.0275 0.025 3303 3 o 0.0335 0.030 5611.ozi6.020 89 86.0276.025 4183 88.0336.030 6475 86.0217.020 1785 96 77.0257.5 63 7.0337.030 7338.0218.020 2682 9.0278.025 5942 80338 6030 820 3.021.o9.020 8 896.0258.o3 873.0338 863.ozI9.020 3578 96.0279.025 6 879.0339.0309064 863 0.0220 0.020 4474 o 0.0280 0.02oz5 7700 0.0340 0.030 9926 2.0221.020 5369 895.o8.025 8579 79.0341.03 0788 86.0222.020 6269 895.0282.025 9457 8.0342.031 1650 8.023.02 0 75.023 89.026 0335 878.0343.031 2522 86i -OzO 7159 8 - "0283 0335 88 868.0224.020 8054 09.0284.oz6 1213 8.0344.031 37 86 0.0225 0.020 8948 0.0285 0.026 2090 8 0.0345 0.03 4234 86.0226.020 9842 89.0286.026 2967 877 0346.0346 5095 86.0227.0210736 894.0287.026 3844 877.0347 0315956 86.0228.021 630 4.0288.026 4721 0348.03 68.0I2 13 974893.0289 ~ 876 86o.0229.021 2523 893.026 5597 0349.031 7676 86 0.0230 0.021 3416 89 0.0290 0.026 6473 6 0.0350 0.031 8536 86.0231.021 4309 892.0291.02oz6 7349 87.0351.03 9396.0232.021 5201.0292.oz6 8224 7.0352.032 0255 8.0233.021 6093 892.0293.026 ggg9099 85.0353.032 1124 859.0234.02 6985 89.0294.026 9974 875.0354.032 1973 85 0.0235 0.021 7876 o.o0295 0.027 0849 0.0355 0.032 2832.o2o6.36.4029 874 858. 0 9 592o 8 96.027'723 874.0356 032 3689 58.0237.0297.027 2597'8 0357.032 4547 856.0238.oz022 0549 89.0298.027 3471 874.0358.032 5405 857.ozo9.-i9 460039 6z9.0239.022 1440 89.0299.027 4345 873.0359.032 62 858 0.0240 0.022 2330 0.0300 0.027 5288 0.0360 0.032 7I20 40 625 TABLE XIII. For finding the Ratio of the Sector to the Triangle.. ) log s2 Diff. s/ log s2 Diff. log s Diff. 0.0-60 0.032 7120 856 o.o60 0.052 5626 6 O.I20 o.o96 8849 68.036.032 7976 856.o6 053 3602 797.02 97 5692 68.0362.032 8833 857.62 054 1556 7954.22.098 2520 68i 03.0o32 9689 8 5.o63.054 9488 7932.I3 098 933I *-364.033 46 855.64. 055' 7397 788.I24 -.9 7 80 o.o365 0.033 I401 856 0.065 0.056 5285.865 O.25 0.10 2907 6765.0366.o3 227 8 66.057 3157865.6.ioo 9672 6.0367 033.067.058 0994 4 I27 I 642 67 0368.033 3967 855.o68.o58 88I7 7823.I28.102 354 6733.0369 33 4822 855.069.059 6618 780.2 02 9873 677 0.0370 0.033 5677 854 0.070 o.o60 4398 0.130 0.103 6576 6688.0372.033 653I 854. o06I 257 7759.3.-04 3264 66 0372.033 7385 854.072.o06 9895 I32.04 9936 6678 ~0373854 07385.3 7622 77I7.0 668.0373.033 8239 854.073.062 762 776.133.05 6594 6643 0374.033 9092 854.074.063 5308 766.I34.o6 3237 6628 0.0375 0.033 9946 8 0.075 0.064 2984 o6.I 35 o.Io6 9865 663.0376.034 0799 852 076.065 o639 7655.I36.07 6478 598.0377.034 i65 852 077 065 8274 7635.7.8 3076 5.0378 034 2504 85.078.066 5888 4 I38.108 9660 6546'0379.034 3356 852.79 067 3482 7594.I39 I09 6229 65.0 34 3 7 85.0796554 0.0380 0.034 4208 8 o.o80 o0.68 1057 0.140 o.IIo 2783 6540.0381.034 5059 85'.68 8612 7555. 2i 2 iIo 9323 6526.0382.034 59II 8.082.069 6I46 7534.I42.III 5849 65x.0383.034 6762 8.083.070 3661 7515.I43.2 2360 64.0384.034 7613 85I.084.071 157 7496 44.I22 8857 6483 0.0385 0.034 8464 0.085 0.071 8633 o.I45 0o. 3 5340 646.o386 034 934.086.072 6090 7457.I46.I4 809 4 03 87 ~.o35' 6 7437.0388 85o.o 8 77440.0388 035oi4.0o07.073 352774 47.224 8264 6440.035 IoI4 85.o88.074 0945 7420.48.IS5 4704 6427.0389.035 I864 849.089 074 8345 7480.149.16 I3I 641 3 0.0390 0.035 2713 8 0.090 0.075 5725 62 o.i50 O.6 7544 6399 ~0391I 035 3562 9.09.076 3087 151.17 3943 68.0392.035 44I 8.092.077 043 7343.2.8 0329 637.0848 7324 88 670'0393.035 5259 8.093 077 7754 6'53.118 6701.0394.035 6Io8 849.094.078 5060o 78.154.I9 3059 634 8~48 ~0 7288 54.229 63457 0.0395 0.035 6956 0.095 0.079 2348 0.155 o. 19 9404 633.0396.035 7804 848.o96.079 967 7269.I56.20 5735 63 0397.035 865 848 097.o8o 6868 75.7.2 2053 630 o0398.o35 9499 8.8.o08I 410I 7233.I8.I21 8357.0 84034 q7 6 7215 6292 0399.036 0346 846.099.082 13I6 79.59.22 4649 627 0.040 0.036 1192 8 o.Ioo 0.082 8513 o.I6 0.1223 0927 626.04I.036 9646 8454.IoI.083 5693 7IX0.61.123 7192 6252.042.037 8075 8429.02.084 2854 7161.i62.24 3444 68.043.038 6478 848.03.84 9999 746'63 24 9682 6226.o44.039 4856 8353.04.08 7i2 0.6 I25 5908 62 8353 1I04 -085 7Iz5 7I'O 6213 0.045 0.040 3209 8 O.I05 o.o86 4235 0.I65 O.I26 212I.046.oI 537 28.Io6.087 I327 70.6 83 6200.047 041 984I 82.07.087 840 7074.67.I27 4508 6187.048.042 81 821.o.0 88 5459 7058.68.128 0683 6I.049.043 6376 8255.09.089 2500 704.69 8 6845 6 0823 2 7023 649 0.050 0.044 4607 82 O.II o.o0.089 9523 0oo.70 o.29 2994 637.oI.045 28i14 883.2II.090 6530 767.7I.29 9I3 137.052'046 0997 186o.12.o91 3520 6974 2 3 52556.053 046 9I57 1837.'13.092 0494 6957.73.31 I36.054 o47 72948123.II4.092 745 690.74 3 74 6087 0.055 0.048 5407 889 O.I5 0.093 4391 624 ~75 0.I32 3553 675.056.049 3496 09 i6.094 1315 2 76. 1 32 9628 60.057.050 1563.117.094 82 689.177 3 569 60.058 050 9607 8044.I 094 8223'77. 60 050 9607 8 II8.I 095 52I4 6876 *.78.134 2740 6o05.059.05 7628 98. og96 99 685 79.I34 7778 60 7998 8596oz6 o.o60 0.052 5626 o.zo 0.09o6 8849 0.180 0.I35 3804 626 TABLE XIII, For finding the Ratio of the Sector to the Triangle. _ log s Diff. log sa Diff. t log s2 Diff. 0.180 o.I35 3804 0o.240 o.I69 5092 o 0.300 0.200 2285.181.135 9818 6004.24i.70 0470 5368.30I.200 7157 4864.182.I36 582I 03.242.I70 5838 3O.302.201 202I.I83.137 I8II 599 243.I71 9735.303.20I 6878 4857.84.137 7789 596.244.17I 6547 350.304.202 177 484 5340 484 0.185.I 38 3755 0.245 0.172 1887 0.35 0.202 6569 34.I86.138 9710 5955.246.I72 728 5331.306.203 483.187.139 5653 59.247.73 2540 5322.307.203 6230.188.140 1585 593.248.173 7853 53 3.308.204 I05o 4820.189.40 7504 591.29.I74 3156 5303.309.204 5862 4805 594~ 5 508.5295 4805 0.190 0.141 3412 589 0.250 0.174 845I 0.310 0.205 0667.191.I41 9309 5889.251.75 3736 5285.3I.205 5464 4797.192.142 5194 58.252.175 9013 5277.312.206 0254 47 ~I93.I43 o68 5874.253.76 4280 5267 3.206 5037 47.I94 oI43 6931 58.254.176 9538 525.314.206 9813 47768 o.... 5g1!5z5O 9 4768 0.I95 0.144 272 8 0.255 0.177 4788 0.315 0.207 458I.196.144 8622 58z.256.I78 0029 524I.3I6.207 9342 97.145 4450 2 257.178 5261 5232.3.208 4096 475.98.i46 o268 8.258.I79 0484 5223.318.208 8843 4747.I99.146 6074 5806.259.179 5698 5214.319.209 3582 4739 5795 5205 4733 0.200 0.147 1869 o.260 o.80 0o903 0.320 0.209 8315.20I.I47 7653 5784.261.8o 6Ioo 5197.321.2I0 3040 4725.202.I48 3427 5774.262.I81 288 5188 322 2 7759 479.203.48 989 572.263.181 6467 5179.323.22 2470 47.204.149 4940 575I.264.I82 I638 5I6.324.21 7174 44 0.205 0.150 0681 0.265 o. 82 6800 0.325 0.2I2 1871.206.I50 6411 5730.266.183 1953 5153.326.212 6562 4691.207.15I 2130 579.267.183 7098 545.327.213 1245 468.208.15I 7838 5708.268.I84 2235 5137.328.213 5921 4676.209.52 3535 5697.269.184 7363 5228.329.2I4 0591 4670.209 55225687 51204662 0.2I0 O.152 9222 6 0.270 0.I85 2483 0.330 0z2I4 5253 4656.22.~253 4899 56.27.I85 7594 52.*33I.4 99~ 9.212.5 0565 6.i 272.186 2696 5102 332.25 455 4649.213.154 6220 565 273.86 779 5095 333.2 9200 464.2114.i55 865 5645.274.187 2877 5078 334.z26 3835 4629 5034 507 4629 0.2z5 O-I55 7499 5624 0.275 O.87 7955 69 0.335 o.2z6 8464.216.156 3123 6 i8'276.I1 88 3024 -?.2I7 3085 4623.226.156 3223 5624 52067 770 463.227.I56 8737 6.277.188 8085 337 217 7700.218.157 43 40.278.89 3138 5053.338.218 2308 460.2I9.157 9933 5583.279.8 883 339.28 690 4602 0.220 0.158 55I6 0.280 0.190 3220 02 34 0.29 2505 588.22z.159 1089 5573.281.o9 8249 509.34I.29 6093 5.222.I59 6652 5563.28.9 3269 5020 342.220 675 4582.2283 6 224 5552 3 9 828 502 343.22 525 4575 ~28.11 8281 343 5 4568.224.160 7747 5543.284.92 3286 5005 344.220 988 456 5532 4996 456 0.225 o.16 3279 0.285 0.192 8282 8 0.345 0.22 4380.226.161 8802 5523.286.193 3271 499 346.221 895 4555.227.i62 435 55I3.287.193 8251 490 347.222 43 4 54.228.6.194 3224 4973 34.222 025 4542.229.63 530 5493.289.194 8 88 496.349,223 2561 4536 543..9 02 4529 0.230 o.I64 0793 0.290 0.195 3145. 350 o.223 7090.231.164 6267 5464.291.195 8094 4949.351.224 I6I3 4523.232.165 1730 54 3.292.196 3035 494.352.224 6130 45I7.233.165 714 95454.293 6 7968 4933..225 060 451.234.166 262X8 54.294.5497 2894 4926.353 5 450.34 i66 5435.225 543 4503497 0.235 o.166 8o6 0.295 0.197 7811 0.355 0.225 9640 ~236.167 3481 5425.296.198 2721 4910.356.226 413 4491.237.167 8903 54I5.297 8 7624 4903 357.226 8615 4484 j238.268 4309 -.297 -198 7624 49 357 4478.28.i68 4309 54.23.199 25 8 4888.358.227 3093 447.239.168 9705 538.299.199 7406 87 359.227 7565 446 0.240 o.169 5092 0.300 0.200 2285 0.360 0.228 2031 627 TABLE XIII, For finding the Ratio of the Sector to the Triangle. -Y log s2 Diff. 7 log s2 Diff. log s Diff. 0.360 0.228 2031 0.420 0.253 953 I6 0.480 0.277 727 82.36.228 6490 4459 42 I.25 46 48 I.278 IO96 3824 ~362 4454I9.4482.278 4916.362.229 0943.422.254 7379 4 482 278 49 3 44907'24 7 39I 3826.363.229 5390 4447.423.255 1484 4I05.483.278 8732 386.364.229 983I 444 424.2 558 40 44.279 2543 3806 4434 4095'48 0.365 0.230 4265 0425 0.255 9679 0.485 0.279 6349.366.230 8694 4429.426.256 3769 4090.486.280 oI 380.367.23 316 44..26 783 4084.487.280 3949 379.368.23I 7532 44 428.257 932 4079.488.280 7743 794.369.232 942 44.429 -257 6o06 4074.489 281 532 378 4069 0.370 0.232 6346 0.-430 o.258 0075 46 0490 o.28I 5326 *37I.233 0743 4397.43I.258 4I39 4o4 49.28I 9096 3780 372.233 5135 4386.432.258 8198 4059 492.82 2872 3776 ~373 -233 952.4 33.259 2252 454 493.282 6644 3772 *374.234 3900 437 434.259 6300 4048.494.283 04 37 4044 3762 0-375 0.234 8274 37 0435 0.260 0344 0495 0.283 4173 3.376.235 2642 4368 436.260 4382 4038 496 283 793 3759 377 -25 7003 6 7.260 841 4033 497.284 1686 3754.27 4.35 4356 45 4029 34970 378.236 359 5 438.26 2444 4029.498.284 5436 3750.379.236 5709 43 439.26 6467 402 499.284 9I8I 374 4344 4029 3742 0.380 0.237 0053 0.440 0.262 0486 0.500 0.285 2923.381.237 439I 4338.44I.262 4499 40I3.50I.28 6660 3737.382.237 8723 4332.442.262 8507 4008 50.286 0392 3732.383 -238 3050 4327.443 263 25 4004 503.286 421 3729.384 -238 7370 430 444.263 6509 399.504.286 7845 3724' //4325 3994 3720 0.385 0.239 68543 0445 0.264 0503.o505 0.287 1565.386.239 5993 -446 264 4492 399.o6 287 528I 37.387 240 o2 96 433 447.264 8475 3983.07 287 8992 37II.388.240 4594 448.265 2454 3979.508.288 2700 3708.389.240 8885 4291 449 65 6428 39.4 509.288 6403 3703 4286 ~449 3969 3699 0.390 0.24I 317I 280 0.450 0.266 0397 6 0.510 0.289 0102.391.24I 7451.451.266 4362 39.5.289 3797 3695 392.242 I725 4274.452.266 8321 3959.2.289 7487 369 ~393 -242 5994 4269 -453.267 2276 3955.5I3.290 1174 387.394.243 0257 42.454.267 6226 3950.54 290 4856 3 4257 3945 3679 0.395 0.243 4514 0.455 0.268 I7I 0.515 0.290 8535.396.243 8766 4252 456.268 4I I 3940.6 229 3674 4246.29I 2209 3670.397.244 30I2 4 457.268 8046 3935.5I7.29I 5879 367.398.244 7252 4240.458.269 2977 39.8.29 9545 3666 4'23 5 9 1977 396. 6291 ~399 -245 1487 4229.459.269 5903 39 519 292 327 36 5716 0.269.5rg.292 3392 3657 0.4o0 o.245 5716 0.460 o.269 9824 0.520 0.292 6864.401.245 9940 424 46I.270 374I 39I7.52I.293 58 364.402.246 4I58 42.462.270 7652 39I.52.293 68 3650.403.2 4 6 3.27I 559 3907.523.293 783 3645.404.247 2578 4207.464 Z7I 5462 39 524 94 3642 420o.27 542 3898 524.294 455 3637 0.405 0.247 6779 1 -0.465 0.271 9360 o.525 0. 294 5092 3634.406.248 0975.466.272 3253 3888 526 94 72 3629.4o7.248 566 4.467 272 74I 388.527.295 2355 3626.408.248 935I 4I8 468.273 0Io5 8. 528 -295 5981.408.249 35x 418o 79.409.49 3531 4X74.469 273 4904 3874 529.295 9602 36i8 0.420 0.249.775 3732 0.4IO 0.249 7705 469 0.470 0.273 8778 0.530 o.296 3220 *4I1.2 50 I87 44164.471.274 2648 3870 53.296 833 36.42.250 603 8.472.274 6513 38'532 297 443.4X 3.ISX 0196 4158 ~386i'532'297 0443 36o6.4I3.252 0 1945.473 -275 0374 3856 533 -297 4049 360o ~-44 -25I 4349 4I47.474 -275 4230 3 534 -297 765 3598 0.415 0.251 8496 0.475 0.275 8082 0.535 0.298 I248.4I6.252 2638 4I42 476.276 92 3847 536.298 4842 3594.47.252 6775 43 477.276 577I 842 537.298 8432 59.4I8.253 o9o6 436.478.276 9609 38 538.299 2018 35.4I9 253 5032412I 479.277 3443 38 539.299 56o 358 0.420 0.253 9153/4Z 0.480 0.277 7272[ 0.540 0.299 9178 628 TABLE XIII. For finding the Ratio of the Sector to the Triangle. log 2 Diff. log s2 Diff.; log s2 Diff. 0.540 0.299 9I78 0.56o 0.306 9938 0.580 o.3I3 9215 6.541.300 2752 374.56.307 3437 3499 58I.34 264i 34.542.300 6323 35'.562.307 6931 3494.582.314 6064 3423.543.300 9890 3567 563.308 0422 341 583.314 9483 34I9.544.30o 3452 3.564.308 3910 3484.584 3I5 2898 34'5 ~54..315 283559 3+84 3415 o0545 0.30I 7011 0o565 0.308 7394 0 585 0.3I5 63 10.546.302 0566 3555.566.309 0874 34 86.315 9719 3409.547.302 4117..567 309 435o.587.316 3124 3405.548.302 7664 3547.568.309 7823 34.588.36 6525 34.549.303 208 344.569.3I 1292.589.36 9923 339 3540 3466 3395 0-550 0.303 4748 0.570 0.310 4758 3462 0.590 037 3318.551.303 8284.572I 31 82 458.591.317 6709 339'.552.304 864352.572.31 678.592.318 oo96 3387 ~553 ~304 5344-. -5z8 573 -311 5133 34-593 318.3480 3384.554.304 8869 35 574 3 11 8584 345.594.318 6861 338 3522 3447 3377 0.555 0.305 2390 3 0.575 0.312 203 0.595 0.319 0238 556 305 5907 35I7.576.312 5475 596.319 36I2 3374.557.305 9420.577.3I2 8915 34 597.39 6983 337.558.306 2930 350.578.313 2352 -598 320 0350 33.559.306 6436 3506.579.33 5785.599.32 374 336 3502 3430 ~593 ~320 3714 3360 0.56o 0.306 9938 0.580 0313 9215 0.600 0.320 7074 TABLE XIV. For finding the Ratio of the Sector to the Triangle. x i Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.ooo o.ooo 0000 0.000 0000 0.030 o.ooo 0523 o.ooo 0506.00 o.000 0002 Io oo I 36 I 33 |.oo0I|.o oooo i |.ooo000 ooo0.031.ooo000 05591 3.o 0539 36.002.000 0002.000 0002.032.00ooo o596.000 0575.003.ooo 0005.ooo000 05 3.033.ooo o634.3ooo 06i 3.004.0 000.o 0 0 oo 00674 40.oo00 0648 37 5 5 40 38 1 0.005 00ooo 0014 o.0ooo o014 6 035 o.ooo 0714 o.ooo o686.oo6.ooo 0021 7.ooo 0020.036.ooo 0756 42.0ooo o70 40.007.0o00o 28 7.ooo oo28 8.07. 0799 43.0oo00 0766 40.oo8.000 0037 9.000 0036.038 000ooo o0844 45.0o 807 4.009.000 0047.000 004 IO.39 ooo 0889 45.ooo 0850 43 -- x x - x ~ I-' 1' 47 44 o.oIO o.000 oo058 o.ooo 0057 0.040 o.ooo o936 8 o.ooo o894 o.01.000 0070 o.0 0o691 I.04I.0oo 0984 4.000 0938 44.012.000 083 3 0082.o 8 1 3.042.00ooo 33 49.000 0984 4.o 31.00o 97 1.o000 0096 I4.043.ooo 1084 5.000 1031 47.000 0213.000 15 551 4.014 000 o0113.000 01II |.044 000ooo 1135.ooo 79 I 7 16 " 53 49 0.0o5 0.000 0130 8 0.000 01o 27 8 0.045.000ooo 1188 0.000 1128.0I6.o00 0148.ooo 0145.046.0ooo 242 54.000 1178 50.017.000 oI67 19.000 064 29.047.000 298.000 1229.o18.o00 0187 2. o0 01 83.9.048.000 2354.000ooo 22 5 ll.OI8|.ooo o87 ~ |.ooo oi83//.ooo 9. |.ooo i28, 52.o19.0oo 0209.000 0204.049.000 1412.o000 334 5 22 22 z59 55 0.020 0.000 0231 o.ooo 0226 0.050 0.ooo 147I 6I 0.000 1389..o021 oo o25.000 02 49.05 024 0 153.0.000 I444 5.02.2000 2. 00 oo 24.o52oo 23593 63.o5000 500 58. 023.ooo o306 2.ooo 0298 25.53.ooo 1656 3 3.ooo 1558 58.024.000 0334 8.000 0325 27.05 000 4.ooo I7 065I6 5 0.025 0.000 0362 0o.0oo 0352 0.055 0o.o000 785 6 0o.ooo 675 6I.o26.ooo 0392 3.ooo 038 2.6. ooo 1852 6.0ooo 736 6.027.ooo 0423 3 1.oo 04 20 29.057.ooo 1920 6.ooo 10 798 6.028.0ooo 0455 3.000 0441 3 1. o08.ooo 18 9 ooo i860 2 -~ 32 ~3.~ 7 I 6 4.029.09.ooo489.000.00o 1924 0.030 0.000 0523 o.ooo o5o6 o.o60 o.ooo0 2I3 0.000 1988 629 TABLE XIV; For finding the Ratio of the Sector to the Triangle. x x Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.o6o 0.000 213 0.000 988 66 0.120 0.000 8 84 9 0.000 7698 1.o061.000 2204.000 20 7.822 12.062.000 278.ooo000 2I 67.122.000 9154 55.ooo000 7948 226 76 68 I57 iz6.ooo 2354.ooo 2189 68 I23 000 58.000 8202 128.063.000 23154 77 000 9 68'3 000 9311 15 000 874 12 78 57 70 I9469 159 I28 0.065 o.ooo 2509 o.ooo 2327 o.I25 o.ooo 9628 i o.ooo 8330.066.ooo 2688 8 I.ooo 2398 71.126.ooo 9789 6.000 8459 29.067.ooo 2669 82.oco 2470 7.127.000 9951.ooo000 8590 31 8z 7 131.ooo 236.o68.000 2751 8.000 2543 73.128 OO 0115 i6.000 872z 13.069 00oo 2834.00ooo 2617 74 129.00 0280 67.000 8853 I3 84 74 X67 9 33 0.070 o.ooo 2918 86 o.ooo 2691 6 0.130 0o.oo 0447 68 0.000 8986.071.ooo 3004.ooo 2767'131'oo 0615 i.000 9120 I4.072.000 3091.ooo 2844.132.oo 0784.ooo 9255.073.oo0 0 o65.ooo 980o.073.000 80.000 2922.133 001 0955 171.000 9390 135 ~07 o 69 79 73.ooI.oo 9 9 I37.074 3ooo 30 o 9 8.00 28ooo. 9527.79I'73 138 0.075 o.ooo000 3360 o.ooo000 3081 0.135.I 1301 176 o.ooo 9665.076.0003453.000 3162 82.136.001 1477.000 9803 38.077 -000 3546 93.000 3244.137 ooi 0011654.oo 9943 140.078.ooo 3641 95.ooo000 3327 3.138 ooi 1832 1.ooi 0083 140.079.000 3738 97 3 84.001 zIz I2.001 224.079'oo 373 97'o 85'139 ISI.ooi 224 14 0.080 o.ooo 3835 o.ooo 3496 86 0.140 o.OOI 2193 o.oo 0366.081.00ooo0 3934 99 000 35 8 14.o2376.00 0 309.082 04.ooo 34 102 8749.ooo 2360 o Q.oo 05093.083.0004136.000 83757.138.0o0 24o 1 85 945.08.000 4 103.000 3757 8'143 2745 18 0798'4.084 ooo 4239'104 000 3846 0.144'OOI 2933 188.00i 0944 0.085 0.000 4343 o.ooo000 3936 0.145 o.oo00 3121 o.oo0 1091.086.ooo 4448 0.000 4027 92 I46.001 33II 192.001 13 147.087.0oo 4555 o8'000 4119'147 o.00 3503 0 ooi 1387 I49.o088.ooo 4663 9 000 4212 93.148.oo 3696.ooi 1536 149.089.0004773.000 4306.149.001 3891 195 oo00 686 150 II: 95 196 152 0.090 0.000 4884 0.00 4401 0.150 0.001 4087 98 0.001 1838.091.000 4996.000 45936.152.001 4485 1.001 1990 15 115 4593.ooI 0 48 0 153.093.000 5224.000 469I 53.oo001 4684 20.001 2296 15'094.000 5341 000 4790.154 0014886 20.001 2451 0o.095 o.ooo 458 o.ooo 4890 0.155 0.001 500 o.oo 2607 096.00 5577.000 4991 05 25295 0 09 000 5697 120.000 101 O.001 5295 207.001 2763 156.097 1o2 5697 2 000 5092.157.ooI 5502 208.001 2921.098.000 5819.000 595 03.158.ooI 5710.001 3079 5.099 0 o oo 594 5299 104 159 0ooi 5920 ZO.OOI 3238 159 5 124 104 211 o6o 126 0.000 5403 0.160 o.ool 6131 o.oo0 3398 i6i.101.000 6192.000 5509 6.i6i.00oo 6344 2I.oo 3559 62.102.000 6319 27.000 o 566 7.162.oo0 6559 2 001 3721.1o3.ooo 6448 129.oo000 723 07.163 oo 6775.oo1 3883 13 68 130.ooo 0779 0 64.104 0ooo 6578 I3.ooo 5832 09.164.ooi 6992 217.00oo1 4047 164 0.105.o000 6709 0.000 594 0o.165 0o.001 7211 221 0.001 4211 66.106 ooo 6842 13.ooo 6052 II.166.oo 7432.00 4377 i66 I.7 00.0 6163 11 25.7.00ooo 6976 67.ooo 7654 ooi 4543 67.108 i3o 7111.68.oo 6275 7878 224.ooi 4710 68.109.000 7248 37.ooo 6389 4.9.ooi 8103 5 OOI 4878 2 1g138 114.9 227 00169 0.110 0.000 7386 14 0.000. 6503 115 0.170 0.001 8330 2 0.001ooI 5047 c..1II.000 7526 I 0 ooo 6.o71 6.0oo 868 3.oo 001 6 19.112.000 7667 14.oo000 6734 172.001 8788 23.001 5387 171.113'000 7809 142 000 6851 11.173 001 9020 232.001 5558 7 44.ooo 79 233 72.114.000 7953 000 6969.174.oo0 9253. oo 5730 145 119 234 173 0.115 0.000 8098 0.000 7088 0.175 0.001 9487 0.001 5903.116.000 8393 48 000 78 121.oo001 9724 237.001 6077 17.I1.000 8394 14.000 7329 1.177 ooi 9961 z37.00ooi 6252 175.119.000 8546 3 9.000 7451 12.178.002 oo0201 240.ooi 6428 176.119'ooo 58693 1' 7574 2.179.002 0442.ooi 6604 176 152 24 243 178 0.120I 0.000 8845 0.000 7698 o.80o 0.002 o685 0.001 6782 630 TABLE XIV. For finding the Ratio of the Sector to the Triangle. Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.180 o.oo2 0685 0.00ooI 6782 0.240 0.003 8289 0.002 8939..i oz.00 0929 246'9.242.0038983. 3 228.Iz.0oo02 0295 696 178.241.oo003 635 34.00 9166 22.182.002 11Z75. 7139 ^ 44'003918 350. 00 93 229.183.-00 1422.00I 7319 1 43.003 9333.02 93.184.002 1671 49.ooi 7500 8I.244.003 9685 352.002 9852 229 0.185 0-002 1922 0.00I 7681 I8 0.245 0.004 0039 0.003 0083 254I83. 0.03 0083.186.002 2174 252.0017864'.46.004 0394 358.003 0314 2.1887 OOZ 248 5. 0018047'.247 004 0752.o 003 0545 233.188.002 2683 25 00I 8231.248.004 1111. 003 0778 23.189 002 2941 001 84 1 8.249 004 472.003 IOII 23 0.190 0.002 3199 26I 0.OO 0 187 0.250 0.004 1835 364 0-003 1245.19gI.002 3460 26.001 8789.7 251.004 2199 67'003 1480 235.192.002 3722.001 8976 8.252.004 2566 368 003 176 236.193.002 3985 266 8ooi 65 98.253.004 2934.003 1952.194.002 4251 25.001 9354 I 254 -004 3305.003 231 6 4786 00 9735 56 004 4051'003 2666 39.197'002 5056 270.001 9926 ^.257 -004 4427'003 2905 239.198.002 5328 272.002 0119'93.258.004 4804 377 -003 3146 241.I99.oo002 0312 193.259.004 584 380.003 3387 4.X78 95 z382 241 0.200 0.002 5877 0.002 0507 0.260 0.004 5566 0.003 3628.201.002 6154 277.002 070 95.261.004 5949 383 003' 3871 243.0o7 195X. o o 3 8416 185 3 243.20zo.oo002 6433 2 002 0897 195.26.004 6334 38'.003 4314.203 6713 28.002 71094'9 263.004 6721 387.003 4358 2.204'002 6995. -002 1292 19,264.004 7111 39I.003 4603 245 283 1902 I292 f86 391 245 0.205 0.002 7278 zS6 0.002 1490 0.265 0.004 7502 0.003 4848.206.002 7564. 896'9.266.004 7894 392.003 5094 246.207 -002 7851 287.002 1889 200.267~ 004 8289 003 5341 2 28.4 20 294.208.002 425x.00I 93.209.002 9 290 002 2090 201.268.004 8686 397.003 5 589 249.209'002 8429.002 221 2.269.004 29085 399 003 5838 293 203 400 3 249 0.210 0.002 8722 0.002 2494 0.270 0.004 9485 0.003 6087.211.002 9015 93.o002oz 2697 203 271.004 9888 403.003 6337.I oz go~g 296 [91 374.003 666 2,3.212.002 9311.002 2901.272. 00 6587 250.213.002 9608 297.oo0 3io6 205.273 005 699 07.003 6839 252.214.002 9907.002 3311.274.005 1107.003 7091 300 207 4io 253 0.215 0.003 0207 0.002 3518 0.275 0.005 1517 0.003 7344.216.003 0509 30 002 3725 0 -276.005 1930 414.003 7598.217.003 0814 305.002 3932.277.005 2344 003 7852 254.218.003 1119 30.002 4142 210.278.005 2760 4I8.003 81O7 255.219.003 1427.oo2 4352.279 005 3178.003 8363 309 210 420 257 0.220 0.003 1736 0.002 4562 0.280 0.005 3598 0.003 8620.221.003 2047 3.002 4774 2.281.005 4020 42.003 8877 257.222.003 2359 312 002 4986 212.282.005 4444 424.003 9135 258.223.003 2674 315 002 5199 283.005 4870 426.003 9 259 316 002 2 3 28.005 5298 428.003 9394 26o.224.003 2990 31.002 54184 005 5298.003 9654 260 0.225 0.003 3308 0.002 5627.285 0.005 5728 0.003 9914 261.226.003 3627 002 5842 2I6.286.005 66o 432.004 0175 4.227.003 3949 1 002 6058.287.005 6594 43.004 0437 263.228.003 4272.002 6275 17.288.005 7030 43.004 0700 6.229 003 4597 325.002 6493 218.289.005 7468 438.004 0963 23 ~.209.00 45 327 218 440 264 0.230 0.003 4924 o 0.002ooz 6711 0.290 0.005 7908 0.004 1227 264.231 003 5252.002 6931 220 91.005 8350 744.00 491 26.232.003 5582 330.002 7151.292.005 8795 44.0041757 266 23 3 98 4436 566.233 -003 5914.002 7371.293.005 9241 448.004 2023 67.234.003 6248 334.002 7593.294.005 9689 4.004 2290 267 0.235 0.003 6584 0.002 7816 0.295 0.006 039 0.004 2557 269.236.003 6921 337.00 8039 223.296.006 0591 452.004 2826 237.003 7260 339.002 8263 224.297.oo6 1045 004 3095.238.003 7601 34 002 8487 224.298.006 1502 457 004 3364 269 3432z6 9.oo6 196 4861' 271.239.003 7944 3'002 87I3 z 6' 196.004 3635 27 0.240 0.003 8289 0.002 8939 I 0.o300 0.00oo6 2421 0.004 3906 631 TABLE XV. For Elliptic Orbits of great eccentricity. E or 8 log Bo or log N Diff. eor8 log B0 or log B Diff. log N Diff. o o 0 o.ooo oooo o.ooo oooo 30 o.ooo 0066 o.ooo 6400 436 1.ooo oooo o.oo000 o7 2 31.ooo 0075 I.000 6836 45 2.ooo oooo.ooo 0028 36.ooo oo86.oo 7286 464 3.ooo oooo.ooo 0064 33.ooo 0097 2.000 775 479 4.ooo oooo 0.00o 0113 49 34.ooo oIo9 I3.~ 8229 493 5.ooo 0000 0.ooo o 7 64 9 5 o.ooo oooo o o.ooo 0177 78 35 o.ooo o22 0.00 8722 508 6.000o oo.ooo 02 55 92 36.0ooo 037 i6.000 9230 523 7.ooo oooo o.ooo 0347 io7 37.000 OI53 18.ooo 9753 37 8.000 0000 I.ooo 0454 I2 38.ooo 0171 I9.0o 0290 552 9.o000 0oo1.000 0574 035 39.ooo oi90 20.00 0842 567 10. o.ooo 7 i o.0ooo 0709 840 o0.00 oIIo o.oo 149 58 1I.000 000I I.o00 08S8 i3 41.ooo 0232 3. I990 596 12.000 0002 0.ooo 1021 78 42.ooo 0255 6.o 2586 6 13.ooo 0002 I.0ooo 199 9I 43.ooo 028 27.ooI 3097 626 14.ooo 0003.0oo 390 206 44.ooo 0308 29.00I 3823 640 15 o.ooo 0004 I o.ooo 1596 220 45 o.ooo000 0337 3I o.ooI 4463 655 16.000 0005.ooo I86 5 46.00 0368 3.ooI 51I8 670 17.ooo 0007 2.ooo 205 248 47.000 0401 36.ooI 578 685 18.ooo o0009 2.o 2299 63 48.oo 0437 38 ooI 6473 7 19.ooo ooII Z.0o 2562 277 49.ooo 0475 40.OOI 7I73 715 20 0.000 0013 0o.oo000 2839 292 50 0.ooo000 055 4 0.00 7888 730 21.ooo 0016 3.ooo 33I 306 51.0o 0558 46.0 868 744 22.000 0ooI9.000 3437 320 52.ooo 0604 48.ooI 9362 760 23.ooo 0023.000 3757 ^ 53.ooo 0652 5.002 o22 775 24.ooo 0027 oo 4091 5.ooo 0703.ooo 0897 790 25 o.ooo 0032 5 o.ooo 4440 363 55 0.000 0757 58 0.002 I687 806 26.ooo 0037 6.000 4803 378 56.0ooo o85 o6.002 2493 820 27.ooo 0043 7 ooo 58I 57.ooo 0875 64.002 33I3 836 28.ooo 0050 7.ooo 5573 409 58.ooo 0939 68.0o 4149 85 29.ooo 0057 9 o 9 40 59.ooo 1007 71.00z 5000 866 30 o.ooo oo66.ooo 6400 60 0o.ooo 0078 0.00o 5866 TABLE XVI. For Hyperbolic Orbits. m or n log Q or log Q' log I. Diff. log halfII.Diff. m orn log Q or log Q' log I. Diff. log half II. Diff. 0.00 0.000 ooo 0 2. 1497n 0.10 9.998 7021 3-41256| 2.I046n.O 9.999 9870 2.4I597n 2.II461.11.998 4308 3.45326n 2.1025n.02.999 9479 2.71675n 2.II42n.12.998 1342 3.49028L 2.1003n 0.3.999 8828 2.89-2593 2.I137.-1 3.997 8I23 3-52423 2.09785 -04.999 79I7 3-01741 2.I 130n.I4 997 4654 3-55547n 2.0952n 0-05 9.999 6746 3.1141on 2.II2In 0.15 9.997 0936 3.58453n 2.0923n.o6.999 5316 3.I9290o 2. I On.6 96.96 97 36II54 2.0892n -07 -999 3628 38 25940 2.I0971.17.996 2760 3.63679 2.0860.08.999 I68z 3.3I687nt 2.I082a.18.995 8305 3.66048n 2.o0826n.09 -998 9479 3-36745n 2.Io65n.19 -995 3608 3.68276n 2.0790n 0o.i 9-998 7021 3.41256n 2.I046, 0.20 9-994 8671 3-70378n 2.0752.o 632 TABLE XVII. For special Perturbations. For positive values of the Argument. For negative values of the Argument. log f Diff. log/', logf" Diff. log f Diff. log f', logf/ Diff. 0.0000oooo 0477 1213 86 0.301 0300 86 0477 213 86 0.301 0300 86.0001 -477 1 1085.300 9431 868 477 2299 io86.30oi 169 868.0003 476 904 1.300 8563 868 477 3385 86 301 203.000o3 6 7957 085 300 7695 868 477 44I io8.301 3776 870.0004 476 6872 1085.300 687 868 477 58 7 8 0.0005 0.476 5787 0.300477 6645 Io8 0 301 4645 870.ooo6.4764702 o5.300 509 6.477 69.301 551.0007 476 368 1084.300 422 868 477 89 1087.]6384 869.0008.476 2534'1084 303 8 7 477 889 I8 75 870.0009.300 86 9906 Io8s 8 c -476 1450 83.300 2490 -478 0994 o.301 824 I083 867 477 4771 o.ooio 0.476 0367 o' 0.300 1623 0.478 2082 Io8 0.301 8995.0011.47 98 103 1.00 0756 47 867 1478 3170.3 9 870.00127 475 821 1o8.300 087 867.00 5 99 47 99 6 78 5 0 -302 0736 870 0013 475 7118 0 83.299 9023 9866.478 56348 1089 302 i6o6 870.0014.4755 6035 370.299 857 66 478 6437 i089 8.3 2 7 0.0015 0.475 4953 082 99 7291 866 0.478 7526 0.302 3348 872.oo00.475 3871 82.299 6425 866 47 8615.302 849.007 475 2789 o 99 5559 66 47.478 9705 090 30 5091.00 -475 I707 99 693 86 9 0795 090.30 5963 87 ~014 -.475 0626 108I.299 380.6 -479 988 109.302 6835 865 1090 872 0.0020 0.474 9545 io o 0.299 2963 0-479 2975 0.302 7707.0026.474 8464 o8s 2gq 2098 65 -479 4065 1.3032 8579 872 00 7001 865 4 1091.30z 945 8.00 ~474 73o83 l 865 zo88 873.002275.474 1080.299 1233 4.304 3115 87.0023 474 633 99 0368 86-.479 6247 303 032O4.8.0024.474 5223 Io8o.298 9504 865 479 7338 09.303 1196 73 0.0025 0.474 4143 8 0.29 8639 0.479 8430 0303 2069 873.0026.474 3063 og.298 7775.479 9522.303 2942 87.479 952 82Ol.99 983.3.0027.4741 15983.298 6911 864.480 614 9ogz 303 38 874.ooz -470904 1079.298 6047 Q64 -480 1706 1092.303 468.0029 -473 925 079.298 5I84 863.4802798 1092 -303 556 873 10.0029 79 864 29 1093 2 874 0.0030 0.473 8746 0.298 4320 3 0.480 3891 0.303 64'368.0032.473 6589 1078.298 85 863.303 37 7.0038.473 7667 1079.29748 2594 6.480 6077 1093.303 8i84 74.0033.473 5511 1078.298 1731 863:480 7170 1093 -303 9050 874.0034 -4723 947 3 1078.298 o6 86.480 164 094.303 9933 875.0034.473 4433 1078 868 3 4 094 874 0.0035 0.473 3935 1077 0.298 0005 862 0.480 9358 0.304 o8 8o 5.0036.473 2278.297 9143 8.481 2 045 95 304 68 875 12001.297 9143 863.481052.304 169 7.0037.473 2 7 8280 862 481 547 1095.304 2557 8.0038.473 0614 1077 297 7418.481 2641 109.30486 8 0.0039.-472 9047 0o67.'297 6556 86 8 76 1095 304 876 ^46.472o1516oi.297 0528 1^.482 1407 ^.3045963.00397 -471 869 1095 875 0.0040 0.472 797 076 0.297 5695 862 0.481 483 0.304 5183 876.0042.471 6 094.292 48367 -.481 5926 195 304 6059.004 -472 5818 176.297 4397 86.481 7022 9 0 6.304 6935 876 -00543 472 47429 51.297 340 862.481 8i8 096.304 7I 876.0044 472 3666 2975 2249 86.481 94 96.304 868 87 0.0045 0-472 2591 0.297 1388 86 0.48 0310 0.304 9563.0046.47 6 1075.297 0528.482 1407 1097 305 0440 877 0047.472 0 1075 9704 9667 86i. 3425 1097 137 877.00478 472 0366 1075:296 8807 86o -482 3650 1097.305 ZI94- 877.00489 471 936 1074.296 7946 863 o 482 4698 109 877.004^9 470 80292 8.9579 477067 3~ i 879 1074 6 1098 -305 307 87 0.0050 0~471 7218 o.z96 7086 860 0.482 5796 0305 3948.0051.470 6144 1074.296. 3 ~ 8 482 6894 58 0 877.0052 47I 5070 1074.96 367 859 8 7992 305 4825 878.0053 -471 3996 1074.296 4537 860 48 098 305 57 878.005.471 1073.296 3648 4859 1490 305. 80 8 0054.471 2923 1073 8 86 099 305 7459 878 0-0055 0~4471 1850 1 I073 o.296 2788 0.483 1287 0.305 8337 878 0.0056.47I 0777 1 96 859 43 286 2.305 01.0057.470 9704 I073 296 5070 858.483 3485 1099.306 0094 879.0058.470 8632 1072.296 oz01 85S.483 4584 1099.306 0973 879.295 9353 858.483 694 36. I85I.005I 0.474 41 724 i o 0~98 5 8798.o006o.470 6488 1072.95 8495 75 483 6784 110.306 2730 633 TABLE XVII. For special Perturbations. For positive values of the Argument. For negative values of the Argument. q, q', q" log f Diff. lo f', logf" Diff. log f Diff. log f', log f" Diff. 0.0060 0.470 6488 0.295 8495 o.483 6784 0.306 2730 80.oo61.470 54i6 072.295 7637 483 78 I.306 361o0.0062.47064345 1071.2 7637 858 306o44 89 8.006.470 4345 107 95 6779 858.483 8984 1101 306 4489.0063.470 374 101. 295 5921.484 0085 1.3o6 5369.0064.47 2203.295 5063 858.484 II86.306 6248 879 071 36 64Io 1 880 0.0065 0.470 1132 0.295 4205 0.484 2287 0. 306 7128 8.00o66 470 062 1070.295 3348 857.484 3388 110.306 8009 88.0067.469 8992 I070 2 8578889 080.0068.469 792 070.295 1634 857 484 5592 oz.306 9769 818.0069. 5469 65 1 070 9 0 857.484 6694 11Z.307 o65o 881 0.0070 0469 578 69 0.294 9920. 0484 7796 0.307 53 88.0073.469 2575 6g.294 7351 86.485 1104 I103.307 4174 882.0074.469 1506.294 6495.485 207 I0.307 56 1069 85 1104 307 5o56 88o 0.0075 0.469 0437 868 0.294 5640 0.485 3311 0.307 5938 882.0076.468 9369 o6.294 4784 856 485 445 II04 307 6820.0078.468 8301 IOGX 3lO7O.o66 9439.0077.468 830o io68:.29 928.48 55I9 307 7702 88.0078.468 733 6 24 73 85.485 6623 "o -307 8584 2.0079.468 6165 o.2894 2218 I 485 7728 1105 307 9466 883 io67 857 110o 883 0.0080 0.468 5098 O6 0.294 136390 0' 485 33 I 07 030 349 883.0081.468 4031 67.294 0508 855 0485 9938 1105.308 123.2 988.0082.468 2964 1067.293 9653 ^5.486 1043 o6.308 2115 88.0083.468 1897 067.293 8799.4 86 2.+ ".308 299 88.0083.468 083.293 7945 854.486 2 iio6 08 1 T lio66 3255 1 1 48 4 iio6.3087 64.0087.467 7633 o66.293 53o3 6 23 4.86 65 73:308 6532 88.oo0088 467 6567 o 9 4 2 854.486 7685 I07.308 746 84.0089.467 5502 1065.293 43675 85.486 8787 1107.308 8301 884.9 8265.85 6 1107 8 0.0090 0.467 4437 o6.293 2822 0.486 9894 0.308 985 88..0091.467 3372 293 1969 83 -487 iooi 1.309 0070 88'467 i5~6' 9 309 853954 884.0093.467 1243 106.293 16 853.-87 2109 o.309 0954 889.0092.10.291063 18.47 3217 309 1839 886.0094.467 0179 64.292 9411 853 487 4325 1108.309 2725 885 0.0095 0466 9 04 66 0.292 8558 0487 5433 0.309 36 88.0096.466 805 106.292 7706 852.487 654.309 44 886.0097.466 6988 263.2^2 6854 852.487 7651 1109.309 5381 886.0098.466 5925 63 2z 6002 52.487 8760 109 309 6267 88.0099.466 4862z Io63.92.487 9869 3 709.009 46 1063. 92 5150 852. 9869 1io.309 7153 886 o.o000oo 0.466 3799 63 0.29 4298 0.488 0979 0.309 8039.oioi.466 2736 106.2 3447.488 2089.309 8926 8.0102.466 1674 io6z.92 2595 85 488 3199 111.309 9012 887.01o03.466 061z io62.92 1744 81.488 4309.30I 0699 887.OI04.465 9550 1O62.292 89 85 4. 5428 I.3 10 586 887 00105 0.465 8488 62 0.292 0043 8 0.488 6531 0.310 2473 887.0106.465 74o27 O6.293 9192 85I.488 7642 1I.310 330 88.0107.465 6366 1o6i.29I 8341 ~ 48 8753 1o424'^IOS -465 5305 ioi.91 7491 1.488 9865 1.310 5136 8.010.465 4244.29 6641 489 0977 6023' 67699lO6693636'6 o6 7 110 3 884 8.488 87 1.310 6424 0.01 10 0'465 3183 i o0'.Z91579I 0 489 62057 112 0 3o 679i 884.0112 465 161 6.291 4029 854.48 314 76.3o8 16 88.oo8 6 530 3549 857 141 91 83o9. 465 5 o02 8.488 958.416 8317 5113 887.011.465 0033 io6.2913242 1.489 5427 1.310 9577 88.011.464 8943 059.291 2393 849.489 6540 1.3 o 105 88.014l.464 O849 6511 0. 3104 95 0.0115 0.464 7884 0.291 1044 0.489 7653 0.3II I C4.oii6.464 6825 1059 8. 849 8767 17 14 0.1~ 7 889.0117 464 5766 059.290 9846 89 9 988 1114.31 89.011.464 707 059.290 8997 490 2995.3II 4022 9.0119 464 2590 58.290 8149.48 21 9.311 4912 8.012o.4.64 -590 158.290 7300 4 490 3223 1II4.31 5802 9 634 TABLE XVII, For special Perturbations. For positive values of the Argument. For negative values of the Argument. q, 1', ~log f Diff. logf', logf" Diff. log f Diff. logf', logf" Diff. o.oIzo 0.464 2590 8 0.290 7300 848 0.490 3223 0.31 5802.0121.464 1532 I58.290 6452.490 4338.3 6692 890.464o047+ 105 8 848 1115 890.02.46 4 0 474.290490 545 3.311 7582 90.0123 463 94I6 290. 4756 18 4 0 8 " 4 1057.20 456 490 6568'5 311 8472.0124.463359 1057.290 3909 490 7689 4 ~3I 9363 891 1057 763 9 89x 0.0125 0.463 7302 0.290 3o61 0.490 88oo00 0.312 0254.0126.463 6245 1057.290 2214490 9916 6 312 114 89.0127.463 518 I1057.290 1367 491 1032 3 2 16 3 36 891.0128.463 4132 05 90 05 847 1117. 27 89.0129.463 3076 I056.29 9673 49I 3256 I17.112 3819 1056 847 1117 891 0.0130 0.463 2020 056.89 88z6 0.491 4383i 0.31 47110 1.0I31.463 0964 1056.-zs289 7980 846.491 5500.312 602 9.0132.461 9908o 05.8 7134 846.49i i668 ".1 6494.0133.462 8853 1055.289 6287 47.49i 7736.3i2 67 3 1055.289oCA 401118 829.0134.462 7798 055.29 544 84 -491 8854 I.312 8279 89 0.01oI35 0-462 6743 0.289 4596 846 0491 9972 0313 9172 892.036.462 5688 055.289 3750 6 492 109.3 0064.0117 462 4633 I055.289.492 2210'.313 0957 893.01 462 2525 054.289 059 845 492 3329 -313 1850 89.1054 2 4 845.449 4448.313 2744 893 0.0140 0.462 1471 0.289 0369 0.492 5567 o313 3637.0141.46z o1i7 1054.288 9524 845 -49z 6687 1120 894.oi04.462 0947 ^ 24 8 492 007 1.313 8.014Z.46i 9364 053.88 8679 49 7807 33 545 894.0143.461 Hit8311.2883 844 103 89.0144.46i 7258 1053.288 7835 8 492 89 17 1120.313 6319 894 440 461 758.288 6990 84.493 0047.313 7213 895 I053 6 44 I121 0.0145 0.461 6205 0.288 6146 08 493 ii68 00313 8108.0146.461 5153 1052.288 5302 84 493 8 112i.31 90 894.0147.461 4101 105.288 4458 44.93 34 2.313 9897 7 95.0148.461 3049 6105.288 3615'.493 4532.314z 895.0149.461997 77.288.771 8 493 54 54.311 797 1052 2771 1122845 3416 896 0o.o0150 046 0945 0.4688 928 43 0493 6776 0.194 283.oi 460 9894 1051.288 185 4 493. 7898 4 3478 896.0152.460 843 05I.288 0241 44 493 9021 3 4374 896.0153.460 7792 105.287 9399 842.494 0144 23 314 5270 6.01oi54.46 6741.05 87 8556 843.494 167 23.314 6i66 896 0.0155 0.460 5690 0.287 7713 0.494 290 I 03I4 7062.o0156.460 4640 1050.287 6871.314959 89.0118 3590 1050.287 29842 124 897.0158 460 2540 1050.287 5187.42 494 5762 1.314 95 897.oI3, g.460 I050.9 125 9752 897.01 469 1490 9.28 34 2 4 1124 315 649 7 o.oi6o 0.460 0441 0.287 3503 0.494 80o0io 03 15 154689.0161.459 9392 1049 287 2661 494 9135 5.315 2444 89.0162.459 8343 1049.287 1820 841.495 90260 35 2344 897.0163 459 1049.287 0979 1385.1 334 898 I049 7294 0998 41 95 I35 II25 315 439 89 0164.459 6245 049.287 0138 8.495 2510 315137 898 104 4 26 3 538457 8i9 o.o165 0.459 5197 I8 ~0.286 9297 841 0.495 366 o.31 36315 6 o.0166.459 449 8 6 8456 495 4762 1126.315 6934 88.0167.459 3101 104.286 7615 841.495 5888 117.35 7832.o0168 4 053 "0'.286 6775 40 495 7015 315 8731 899.0169.459 oo6 1047 8 4 11.499 6. 1 "411 04-7 406 5935 4 495 8142 ^' 3135 963.oI71 76 I 867 840 0396.3916 237 899.458 8912 1047.286 4255 840 -496 o396 ~316 14 899.0172.458 7865 86 3415 840 496 15 84 36 2z27 99.0173.458 6818 I04.286 2575 40.496 265. 16 3227 900 I 0.6575 839 -496 zZ Ilz8 316 3 900 ~I74.458 5772.286 36 8 496 3780 2-.6 2 900 00175 0.45,8 4726 1046 0.286 896 0 ~'0176 I 8 682 o.286 8396 0.496 4908 0.336 5o27 458 368o 046.285 92 839.496 6037 1129.316 5927 900.0177.458 2634 046 285 8 8 496 76 29 36 6827 900.0178.458 1589 1045.285 8380 38.496 8295 1129.316 7728 901.0179.458 0544 045.285 7541 39.496 9424 II9.36 8629 901.0180.457 045.285 6702 839 497 0554 1130 953 901 635 TABLE XVII. For special Perturbations. For positive values of the Argument. For negative values of the Argument. log f Diff. logf', logf" Diff. log f Diff. logf', logf" Diff. 0.0180 0.457 9499 0.285 6702 0.497 0554 0.316 9530.0181.457 8454 1045.285 5864 838 97 684 30.317 0431.018.-457 7409 1045 85 526.497 2814 II30.317 332 90I 8838 1131 902.0183.457 6365 I -285 4188 497 3944 I172234 0o184.457 5321 I4 -85 3350?8 -497 5075 j.317 335 4 0.0185 0.457 4277 0.2 85 2512 0.497 6206 0.317 4037.oi86.457 3233 1044.285 1675 837 7337 4939 90.0187.457 2189 104 -285 0838 13.497 8468.317 5841 90.o0188.457 1146 1043.285 0000 38.497 9600 I132 317 6744 903.0189.457 0103 1043.284 9163 837 498 0732 II32 317 7646 90 1043 837 II32 903 o.o90o 0.456 9060 0.284 8326 0.498 1864 0o317 8549.019I.456 8017 1043.284 7490 836.498 2996 1132 03.0192.456 6975 1042.284 6653 37.498 129.318 0355.0194.456 1933 I042.284 5817 836.498 5262 "33 318 1259 904.o193.456 491 04.284 491 836.498 6395 1133.318 2162 903 IO^ 42 836 X"-133 904 0.0195 0.456 3849 84 4145 8.498 752 3 o.318 3066 904.o9 456 1041.84 3309 836.498 8662 134.318 397 904.0197.456 1767 04 84 73 36 498 9796 II34 38 4874 904.0oI9.456 0726 1041.284 i637 8 499 0930 34.318 5778 904.0199 455 9685.284 08o 835.499 2064 11.3I8 583 905 1041 835 1135 905 0.0200 0.455 8644 0.283 9967 3 0.499 3199 0.318 7588.020 6o 040 x18 3 835 1135 8492 904.0201 455 7604 40.283 9132 35.499 4334.38 8492 9o4.0202.455 6564 283 8297 8 499 5469.318 9398 906.0203.455 5524 1040 -83 746.499 6604 113.319 0303 905 ~0oo4.455 4484 1040 -283 6627 499 7740 319 08 1040 834.499 7740 1136 906 0.0205 0.455 3444 o3 0.283 5793 0.499 8876 0.319 2114.0206.455 2405 9.283 4958 83.5000012 1.3193020 6.0207.455 I366 1039 1137 906.0207.455 366 1039.28 4124 834.500 1149 37 319 3926 o.0208 1039 7 83 3290 834.5002286 37 32 9.455 0327 I039. 329~ 31 3.0209 454 9288 039.283 2456 4.500 3423 II37.3I9 5738 2039 833 1137 907 0.ozIo 0.454 8249 0.283 1623 8 05004560 0.3i9 6645.0211.454 7211 1038.283 0789 834.500 5697 "137 7552 907 I03X.282 833 1138 ~3I9 9366 907.0212 454 6173 3 5 6835 138 3 8459 1907.0o213 454 35 38.282 123 833.500 7973 38 319 9366 07.021OZI4.454 4097 103 82 290 833.500 9111.320 0273 907 0 1 0'454 3oo 03 831139 9089 0.021 0.44 3060 0.282 7457 833 o0.501 0250 0.320 1181.0216 454 2023.28 6624 50 139 39.320 2088 907.0217 454 0986 1037 282 5791 32 2528.320 2996 908 1~ZI I037 8.I139 99 6 08.0218 453 9949.282 4959 83.50 3667 39 320 3904.0219.453 8912.28.2 4I7.501 4807 1140.320 4813 909 01'Z 045786 036 032 1140 908 0.0220 0.453 7876 6 0.282 3295 o 0.501 5947 0.320 5721.0221.453 6840 036 82 2463 032 350i 7087 "4.20 6630 909.0222.453 5804 I03 1z 1.3 50I 822~7 1140 20 759.0223.453 4768 36.282 080 831.501 9368 "4.320 8 909.0224 453 3733 1035 89968 3.502 0509.320 9357 99 O035 1141 909 0.0225 0.453 2698 0.281 9137 o 0.502 1650 0.321 0266.0226.453 663 035 281 836 31.02 27 "41 1141 3 76 910.0227 453 062 1035.28 7475 83.502 399 II42.321 2086 910.0228 4 9593 135.281~ 6644..502 5075 1142.321 2996.910.0229 452 8558 035.281 584 830.502 6217 II42 321 3906 910 0.0230 0.452 7564 9 0.281 4983 830 0.502 7360 0.321 4816.0231.4512 6490 81 4153 830.502 8503 4.321 5727 9.0232 452 5456 034.2813323 30 02 9646 3 6637.0233.45 4422 34.2812493 830.503 0789 1143.3217548 9".0 I0334 2 830 0343 3gI.0234.452 3309 I033.281 1663 8.503 1932 4.321 8460 91 I033 832 xI4"4 9' 0.0235 0.452 2356 0.28I00833.0503 3076 0.321 9371.0236.453 1033.280 295 o.o.322 0282 9'.023.4521323 033.281 0004 83.503 4220.2 457. o 1033.280 9174 829.5 364 "44.322 6I94 92 ~453 58 03.28I 8107.320 7539 9o,.0238 451 9258 -3 83451. 503 6508 "44.1 912.oz39 4511 8226 032.280 756 829.503 7653 45.322 3018 912 1032.28o 6687 829 8798 II4.322 3930.0240.45I 7194.283'5~7 0'722 593 12 636 TABLE XVII, For special Perturbations. For positive values of the Argument. For negative values of the Argument. log f Diff. logf', logf" Diff. log f Diff. logf', logf" Diff. 0.0240 0.451 7194 o.28o 6687 8 0-503 g79g 0.322 3930.o024.451 6i62 032.280 5858 2.503 9943 45.322 483.024.451 530 03.280 5030 504 o 8.1089 5756 913.0242.451 5130 8032 9 1146.322 6668 912.o243.451 4099 3.280 4201 82.504 2235 -3 668 9.024.45i 3068.2803373 8 58 3' II6.32 7581 913 Io31 II46 914 0.0245 0.451 2037 0.280 2545 8,8 0.504 4527 0.322 8495.0246.45 ioo6 103I.280 1717 828.504 5674 1147.322 9408 913.0247.450 9975 1030.280 0889.504 6821 7.323 0322 914.0248.450 8945 030.280 oo6z.504 7968.323 6 94.1030 27 1148 2 914 0.0250 0.450 6885 0.279 8407 0.505 0263 48 0.323 3064.0251.450 5855 1030.279 7580 827.505 1411 148.323 3978 914.0252.450 4825 030.279 6753 8 -505 2559 148 -323 4893 91.0253 -450 3796 1029 279 5926 27.505 3707.323 5808 915.0254.450 2767 279 5099 16.505 4856 "49.323 6723 915 0.0255 0.450 1738 0.279 4273 g 0.505 6005 0.323 7638.0256.450 0709 10.279 3446 827.505 7154 149.323 8553 915.0257.449 9681 Ioz8.279 2620 826.505 8303 149 -323 9469.0258.449 8653.279 1794 505 953. -324 0384 9.0259.449 7625 28 -279 968 25 506 0603.34 300 917 0.0260 0.449 6597 o1028 0.279 0143 826 0o506 153 150 0.324 227 6 916.026I.449 5569.7278 937 9 6506'9 3 243 1324 313.0262.449 4542 278 849 1 506 4054.324 4049.0263.449 355.278 7666 9 5o6 5205 70.324 4966 917.0264.449 2488 1027.278 6841 825.5o6 6356 1517.324 5883 917 4027 825 1152 917 0.0265 0.449 1461 1026 0.278 6o016 05o6 7508 0.324 68oo 91 4490435 75I9I824 -o6 866o 1152 91.0266.448 359 1026.278 5191 84.5o6 9813 53 324 86 918.0268.488383 6.278 3542.507 0965 152.324 9553 95 8.ozg9.448 7357 oz6 278 2718 82.5072117 15.3250470 97.0269 5 1026 824X 1153 9g9 0.0270 0.448 6331 1026 0.278 1894 0.507 3270 0.325 1389. 4 5002709 X50 28' 279 3446 8 53.220 91.0271.448 5305.225 78 1070 824 507 43 1154 325 2307 918.0272.448 4280 I025.278 0246 24.507 5577 325 375.0273 -448 3255 025.277 942 823 507 6731 54 325 444.0274 -448 2230 025 277 8599 8 -507 7885 II54 9325 5063 g9i 0.0275 0.448 1205 0.277 7775 0507 9039 0.325 5982.0276.448 0181 1024.277 86952 3.,0 04 II6.325 6901 99.0277 447 9157 1.277 6129 3.508 1349 -II.3257821 920I.0278.447 83 1024 -277 536 823.508 2504 1I5.325 8740 919.0279'447 719 1'277 43 83.508 3659 1155'325 66 966 79.447 270 140X 822 155 92 0.0280 0.447 6085 0.277 3661 8 0.508 4814 16 0.326 0580.0281.447 5062 1023.277 Z838 82 508 5970 1156.326 1500 920.0282.447 4039 1023.277 ZOI2016 ZZ.508 7126 115.326 2421 921.0283.447 3016 1.277 1194 822.5o8 282.26 334I 928.0284.447 2993 1023 277 0372.508 9439 57.326 4262 1023 5822 I 921'0.0285 0.447 0970 0.276 9550 0.509 0596 0.326 5183.0286.446 9948.IOZ.276 8728 821.509 1753.'57.326 6104 921.o26i 1157.5 ~6 98~3.0287.446 8926 22.26 7'07 50 25 9 o'5.326 7026 922.0288.446 7904 1022.276 86 821 509 4068 5.326 947 92.0289.446 6882 1022 276 6264 821.509 5226 115.36 8869 22 0.0290 0.446 5861 0.2765443 8z 0.509. 6384 0.326 9791.0291.446 4840 0.276 4622 8.509 7543 327 0713 9.02zz.446 3819 1021.276 3802 821.509 8702 59 327 163 922.0293.446 2798 1021 276 981 820.509 961 59.327 2558 923.0294.446 1777 02 276 2161 io 1020 1159.327 3481 923 202 6 8. 1159 923 0.o295 0.446 0756 1 0.27 1340 0.510 2179 6o 0.327 404 9.029 445 97 276 0520 820 510 3339 ir6o 327 5327.0297.445 8716 020.275 9700 824.510 4499 I.327 6250 923.0298 445 7696 020.27 880 82.50 5659 6o 37 7174 92.0299.445 6676 020 27 8061 8.50 6819 i 327 8097 923.0~003.^ 5 5657 ^^.^ 820 6924.0300 445 5657 275 7241.5O 7980.32 7 902 7 637 No. log q Motion. Computed by --: h ~ 8 o t It o i tt o t /t 66 Jan. 14, 5 o o 325 o o 3z 4~ o 4~ 3~ o T.O 9.6480 Retrograde. Hind. i 141 March29, z o o 25I55 o iz 5~ o I7 o o z.o 9.8573 t/, 240Nov. 10, o o o 27I o o I89 o o 44 o o i.o 9.5700 Direct. Burckhardt. 539 Oct. 20, 15 o o 313 3~ o 58~ or 2380 Io o o I.o 9.53307,, I 565 July 9~ o o o 88 o o I58 o o 62 o o x.o 9.85686 Retrograde. t/ 568 Aug. 28, 62849 31647 o 294 36 o 4 2 o I.o 9.949I Direct. Hind. 574 April 7, 64314 14339 o 128 x7 o 463t o i.o 9.9836,, = 770 June 6, x4 6 x 357 7 o 9~ 59 o 6x49 o t.o 9.807664 Retrograde. Laugier. g 837 March 1, o o o 289 3 o 206 33 o It o o i.o 9.763428, I~ingr~. 1 961 Dec. 30, 3 5~25 268 3 o 35~ 35 o 7933 o i.o 9.7418, Hind. ~ I 989 Sept. 12, o o o 264 o o 84 o o I7 o o i.o 9.7546, Burckhardt. ~ 1, o o o 26455 o 25 5~ o T7 O O T.O 9.8573 " Hind. li 1092 Feb. 15, o o o 156 zo o 125 4o o 2855 o 1.o 9.9676 Direct., ~' F —3 1097 Sept. 21, 2i2639 33z 3~ o 207 3~ o 733~ o 1.o 9.86832, Burekhardt. o 1231 Jan. 2]1066 April 30, 7 12 4~ 134. 48 o x3 3~ o 6 5 o x.o 9.9767 " Pingr& [..q? 12, 133t o 24t 38 o 157 4o o 35 5 o I.o 9.4938, Hock. ~ [-vj l~11264 July 1299 March31, 729 o 3 2o o Io7 8 o 6857 o 1.o 9.50233 Retrograde. Pingr,.. I 1337 June 15, t46 o 2 2o o 93 I o 4~28 o i.o 9.91815, Laugier. ~ ~.~ 1366 Oct. 13, o o o 66 o o 2x2 o o 6 o o 1.o 9.9814o Direct. Peirce. ~ F —q 1378 Nov. 8, T8X927 299 3x O 47I7 O 1756 O X.O 9.76604 Retrograde Laugier. ~=~ F-q ~? 1385 Oct. 16, 6 I4 25 1ox 47 o 268 31 o 52 I5 o 1.o 9.88860, ttind. =~ ~o 1433 Nov. 4, to 9 5t 28t 2 o 133 49 o 79 x o x.o 9.53079, Laugier.'~ 1456 June 8, 22 o 40 3oi o o 48 3~ o x7 56 o 1.o 9.76754, Pingr& =. 1468 Oct. 7, 949 4~ 356 3 o 6t x5 o 44 19 o I.O 9.931o9, Laugier. c0 1472 Feb. 28, 5 13 13 48 3 o 207 32 o x 55 o I.o 9.751718,, = O 1490 Dee. 24, 12 26 5~ 58 4~ o 288 45 o 51 37 o 1.o 9.8678 Direct. Hind. 1491 Jan. 4, zt 35 o 113 o o 268 o o 75 o o x.o 9.8780 Retrograde. Peirce..~ o~ 1506 Sept. 3, x552 34 25o 37 o I32 50 o 45 I o t.o 9.586565, Laugier. ~o 1531 Aug. 25, 19 o 4~ 3oI xz o 45 3~ o 17 o o 0.967391 9.763380, Halley. 3 1532 Oct. 19, ~453 o 135 44 o 119 8 o 4227 o ~.o 9.787141 Direct. Malehain. 1533 June 14, 2i IT 25 2I7 4~ O 299 t9 O 28t4 O 1.O 9.5t4362 " Olbers. 1556 April 22, o25 o 274 x5 o 175 26 o 3~ x2 o x.o 9.70323, /-End. i~ 1558 Aug. 10, i2 ~4 45 329 49 o 332 36 o 7329 o i.o 9.76~4o Retrograde. Olbers. 15'77 Oct 26, 22 44 36 I2~ 42 o 25 zo24 75 942 T.O 9.24920, Woldstedt. 1580 Nov. 28~ 13 6 39 Io8 29 20 19 725 643355 0.998631 9.77982 Direct. Schjellerup., i i..... i i i i i' i1 i ii i i i ill ii i ii ~,.i [ No.. T C Q l e 1og~ q Motion. Computed by h m s 0 0 1 0 f /f 0 36 1582 May 6, 9 51 22 256 15 18 229 ~IS 1 60 47 3 1.0 9.zz6i56 Retrograde. D'Arrest. 37 1585 Oct. 8, 0 38 44 8 8 z6 37 41 15 6 5 5z 1.0 0.0393531 Direct. Peters and Sawitsch. 38 [1590 Feb. 8, 0 39 4 217 57 1 z 65 36 56 29 29 44 1.0 9.7541386 Retrograde. Hind. 39 1593 July 18, 0339 o I76 09 o i64 05 o 87 58 0 1.0 8.949940 Direct. La Caille. 40 1596 July 25, 5 8 38 270 54 35 330 2o 49 58 9753704 Retrogra de. 41 1607 Oct. 26, 17 io 58 301 38 1o 48 40 28 17 12 17 0.9670888 9.769358 Bessel. 42 1618 Aug. 17, 3 2 0 318 20 0 293 25 0 2 is8 o 1.0 9.710100 Direct. Pingre. a 43 1618 Nov. 8, 825 I 3 5 21 75 44 10 37 I1 31 1.0 9.590556 it Bessel. 44 1-652 Nov. 12, 154x 0 28 I8 40 88 lo o 79 28 0 1.0 9.928140 H Halley. 45 1661 Jan. 26 21 i6 8 8 54 33 55 1.0 9.646131,I Machain. 46 1664 Dec. 4, xI 36 5 130 33 I5 8 1 5 52 21 I8 12 1.0 0.010949 Retrograde. Lindel6f. o 4:7 1665 April 24, 5 I6 o 7 14 o 228 2 o 76 5 0 1.0 9.027309 H Halley. 48 1668 Feb. 24, 1846 6 40 9 0 193 26 o 27 7 0 1.0 9.39990 Direct. Henderson. -' 4:9 1672 March 1, 838 0 46 59 30 297 30 30 8322 z o 1.0 9.843476 "/ Halley. 50 1677 May 6, 0 38 0 137 37 5 236 49 IO 79 3 15 1.0 9.448072 Retrograde., 51 16 78 Aug. 18, 7 34 0 322 47 37 163 20 0 2 5 0 0.626970 0.058919 Direct. Le Verrier. 52 1680 Dec. 17, 23 46 9 262 49 5 272 9 29 6o 40 i6 0.999985417 7.7939551 I/ Encke. 53 1682 Sept. 14, x9 453 301 55 37 5x xx8 17 44 45 o.96792019 9.7655898 Retrograde. Rosenberger. 54 1683 July 13, 17 25 I5 86 31 15 I73 1748 83 4746 0.9832470 9.7430I48 Clausen. }"'" 55 1684 June 88, o 17 0 238 52 0 268 15 0 65 48 40 1.0 9.982339 Direct. Halley. 56 [ 1686 Sept. 16, 434 0 77 0 30 350 34 40 31 21 40 i.0 9.511883,, 57 1689 Nov. 29, 4z48 269 41 0 0 25 0 59 5 o I.o 8.27720 Retrograde. Vogel. 58 1695 Nov. 9, 17 0 0 6o 0o z0 6 o o 22 0 0 1.0 9.9261 Direct. Burckhardt.:7 59 1698 Oct. 18, x6 58 0 27o 51 15 267 4415 iI 46 o 1.0 9.839660 Retrograde. Halley. 60 1699 Jan. 13, 8 23 0 212 3i 6 321 45 35 69 zo o 1.0 9.87I570, La Caille. 61 1701 Oct. 17, 9 5' 0 133 41 0 298 41 0 41 39 0 1.0 9.77278 ]3urckhardt. 62 1702 March13, 14 33 22 138 46 34 x88 59 10 4 24 44 1.0 9.8I0790 Direct. i.o 9.6 7 9 Sireyct. CD. 63 1706 Jan. 30, 4 57 0 72 36 25 13 II 23 55 14 5 1.0 9.63029i 64 1707 Dec. 11, 23 30 0 79 54 56 52 46 35 88 36 0 1.0 9.934368 La Caille. 65 1718 Jan. 14, 21 44 i6 Iz2 39 55 1z7 55 29 31 8 6 I.0 o.o0o908 Retrograde. Argelander. 66 1723 Sept. 27, 15 4 9 42 52 35 i4 14 17 50 0 IS 9.9994743 ii Spoerer. 67 1729 June 13, 6 19 27 320 31 22 310 38 0 77 5 i8 1.0050334 0.6067570 Direct. Burckhardt. 68 1737 Jan. 30, 8 21 0 325 55 0 226 22 o x8 2o 45 1.0 9.347960, Bradley. 69 1737 June 8, 7 39 0 z6z 36 39 123 53 43 39 14 5 1.o 9.93802 a Daussy. 70 1739 June 17, 1o 0 0 1oz 38 40 207 25 14 55 42z 44 1.0 9.828388 Retrograde. La Caille. N o. T ir T i e log q Motion. Computed by h m s / o o 0 i. 71 1742 Feb. 8, 4 ZI 14 217 33 44 185 34 45 67 4 iI.o 9.883976 Retrograde. Struyck. 72 1743 Jan. 10, zo 20o i6 92 57 51 67 31 57 2 i6 i6 i.o 9.923338 Direct. Olbers. 73 1743 Sept. 20, 14 II 13 247 15 37 6 15 29 45 38 Io I.o 9.719016 Retrograde. D'Arrest. 74 1744 March 1, 7 55 39 197 13 58 45 47 54 47 7 41 i.0 9.346842 Direct. Wolfers. 75 1747 March 3, 9 57 I9 277 2 5 147 18 42 79 6 45 1.0 0.342144 Retrograde. Maraldi. 0 76 1748 April 28, 19 25 24 215 0 50 23 52 i6 85 26 57 1.0 9.924626 77 1748 June 18, 21 18 i 278 47 10 33 8 29 67 3 28 i.o 9.976128 Direct. Bessel. 78 1757 Oct. 21, 7 54 39 122 58 o z14 12 50 12 50 20 i.o 9.528328 3 Bradley. - 79 1758 June 11, 3 17 39 267 38 0 230 50 o 68 19 o0.o 9.333148 Pingre. 80 1759 March12, 13 14 34 303 0io 28 53 50 27 17 36 52 0.96768436 9.7667989 Retrograde. Rosenberger. y 81 1759 Nov. 27, 0 33 58 53 38 4 139 40 15 79 3 19.0o 9.904218 Direct. Chappe. O 82 1759 Dec. 16, 12 48 51 139 3 52 79 20 24 4 42 io i.o 9.983064 Retrograde., 3 83 1762 May 28, 8 1 42 104 z o 348 33 5 85 38 I3 i.0 0.003912 Direct. Burckhardt. 84 1763 Nov. 1, 20 54 58 84 57 27 356 17 38 72 34 10 0.9954268 9.6974946 / Lexell. o 85 1764 Feb. 12, 13 42 15 15 14 52 120 4 33 52 53 31.0o 9.744462 Retrograde. Pingre.? L86 1766 Feb. 17, 8 41 0 143 15 25 244 io 50 40 50 20 1.0 9.703570 87 1766 April 26, 23 43 55 z51 13 0 74 11 0 8 I 45 0.864000 9.6009521 Direct. Burckhardt. ~ 4 88 1769 Oct. 7, 14 53 22 144 II 29 175 3 59 40 45 50 0.99924901 9.089039 Y Bessel. 89 1770 Aug. 14, 0 38 36 356 16 27 131 59 34 I 34 31 0.786839 9.8288597 Le Verrier. - }90 1770 Nov. 22, 5 39 o 208 22 44 0o8 42 10 31 25 55 1.0 9.722833 Retrograde. Pingre. 91 1771 April 19, 5 6 19 104 3 i6 57 51 55 II 15 19 1.0093698 9.9559104 Direct. Encke.: 92 1772 Feb. 16, 15 43 40 110 8 35 257 I5 38 17 3 8 0.724510 9.993890 D Hubbard. 93 1773 Sept. 5, 14 I 50 75 17 0 121 8 20 61i 5 I1 1.0024901 0.052420, Lexell. 94 1774 Aug. 15, 19 55 21 317 27 40 180 44 34 83 20 26.082955 0.1562065 Burckhardt. 95 1779 Jan. 4 2 4 20 87 14 27 25 4 10 32 30 57 1.0 9.853186 I Zach. o 96 1780 Sept. 30, 22 13 53 246 35 59 123 41 18 54 23 I2 0.9999460 8.9836418 Retrograde. Cliiver. 97 1780 Nov. 28, 20 21 o 246 52 0 142 o 72 3 30.o 9.712041 " Olbers. 3 98 1781 July 7, 4 31 59 239 11 25 83 0 38 81 43 26 i.o 9.889784 Direct. Mechain.. 99 1781 Nov. 29, 12 33 25 x6 3 7 77 22 55 27 z12 4.o 9.982723 Retrograde. Legendre. 100 1783 Nov. 19, Ii 50 50 50 3 8 55 45 20 44 53 24 0.5395345 o.1626829 Direct. Burckhardt. 101 1784 Jan. 21, 4 47 26. 80o 44 24 56 49 21 51 9 Iz i.o 9.849946 Retrograde. Mechain. 102 1785 Jan. 27, 7 48 43 109 51 56 264 12 15 70 14 I2 1.o 0.058198 Direct. / 103 1785 April 8, 8 58 51 297 29 33 64 33 36 87 31 54 1.0 9.630733 Retrograde. / 104 1786 Jan. 30, 20 57 5 15'I6 38 o 334 8 o 13 36 o 0.84836 9.524810 Direct. Encke. 105 1786 July 8, 13 37 io i58 38 30 195 23 32 50 58 33 1.0 9.595763 j " Reggio. N o. ~ ~ ~ ~ ~ --- I j i-r - No. T,, Q i e log q Motion. Computed by h m 0! 0! i 0 / /I 106 1787 May 10, I9 48 39 7 44 9 io6 5I 35 48 IS 5I 1.0 9.542714 Retrograde. Saron. 107 1788 Nov. 10, 7 25 26 99 8 7 156 56 43 12 27 40 1.0 0.026538 ni M~chain. 108 1788 Nov. 20, 7 15 39 22 49 54 352 24 26 64 30 24 1.0 9.879276 Direct. II 109 1790 Jan. 16, i8 58 9 58 2445 172 50 2 29 44 7 1.0 9.873516 Retrograde. S0ron. 110 1790 Jan. 28, 7 36 9 IiI 44 37 267 8 37 56 58 I3 I.0 0.026650 Direct. M~chain. 111 1790 May 21, 5 46 54 273 4327 33 II 2 63 52 27 1.0 9.901981 Retrograde. C 112 1792 Jan. 13, z12 50 i5 36 20 32 190o 42 9 39 45 47 1.0 o.III456 ni Zach. 113 1792 Dec. 27, 7 47 9 135 52 35 283 I4 44 49 7 14 1.o 9.985350 n Piazzi. 114 1793 Nov. 4, 2o 12 0 228 42 o 1o8 29 0 60 21 o 1.0 9.605736 n Saron. 0 115 1793 Nov. 20, 5 6 zi 7I 54 3 2 0 I2 51 31 10 0.9734211 o.I746744 Direct. DYArrest. 116 1795 Dec. 21, 1o 35 x I56 41 20 334 39 zz22 13 42 30 0.8488828 9.5243046, Encke. 117 1796 April 2, 19 47 42 192 44 13 17 2 I6 64 54 33 1.0 o.198151 Retrograde. Olbers. 118I 1797 July 9, 231 10 49 27 8 329 15 37 50 40 34.o 9.721489 I I 119 1798 April 4, II 58 16 0o5 657 zz2 12 21 43 44 42 1.0 9.685370 Direct. i 120 1798 Dec. 31, 13 17 3 34 27 27 249 30 30 42 26 4 1.0 9.891829 Retrograde. Burckhardt. 1211 1799 Sept. 7, 5 50 36 3 38 i6 99 23 3 51 2 27 1.o 9.924437 Wahl. L'i 122 1799 Dec. 25, 18 346 190 2246 326 30 18 77 5 4 1.o 9.795483 f n 123 1801 Aug. 8, 13 23 0 183 49 0 44 28 0 21 2 0 1.0 9.417804 Burckhardt.. 124: 1802 Sept. 9, 21 23 5 332 9 4 310 15 39 57 0 47 1.0 o.o39061 Olbers. n ~ 125 1804 Feb. 13, 14 6 55 148 44 51 176 47 8 56 28 40 1.0.oz9858 Direct. Gauss. 126 11805 Nov. 21, 11 59 50 156 47 24 334 20 o10 13 33 30 0.84617529 9.5320168 n Encke. 127 1806 Jan. 1, 22 I 10 109 28 25 251 i6 19 13 36 34 0.7457068 9.9576440 it Hubbard. 128 1806 Dec. 28, 22 9 2 97 3 24 322 23 16 35 2 33 1.OOI82 o.o34189 Retrograde. Hensel 129 1807 Sept. 18, 17 43 59 270 54 42 266 47 iI 63 Io 28 0.99548781 9.8103158 Direct. Bessel. 130 1808 May 12, 22 52 4 69 1257 322 58 36 45 43 7 I.o 9.5909 Retrograde. Encke. 131 1808 July 12, 4 o 58 252 38 50 24 11 15 39 i8 59 1.0 9.783870 1' Bessel. 132 1810 Sept. 29, 2 23 31 5z 44 42 31o 2z 2 6 II1 15 I.o 9.989355 Direct. Triesnecker. 133 1811 Sept. 12, 6 1o 32 75 o 34 14o 24 44 73 2 2z 0.99509330 o.o0151178 Retrograde. Argelander. 0 134 1811 Nov. 10, 23 46 17 47 27 27 93 I 52 31 17 II 0.98271088 o.I992359 Direct. Nicolai. 135 1812 Sept. 15, 7 31 31 92 18 44 253 I 2 72 57 3 0.9545412 9.8904995'I Encke. 136 1813 March 4, z12 38 o10 69 56 8 60 48 24 21 13 33 1.0 9.8445579 Retrograde. Nicollet. 137 1813 May 19; Io I 7 197 43 8 42 40 I5 81 2 1z 1.0 0.0849212 ii Gerling. 135 1815 April 25, 23 48 42 149 I 56 83 28 34 44 29 55 0.93121968 0.0838Io09 Direct. Bessel. 139 1816 March 1, 8 i8 o 267 35 33 323 I4 56 43 5 26 I.O0 8.685769 I' Burckhardt. 140 1818 Feb. 7, 55 95 7 250 4 2 24 9.865260 Pogson. I~~~~~~~~~~~~ 24 5 o 95856 7 ttO40 zOzz Pogson. No.- _________Tr - i e log q Motion. Com pnted by h m s............1 i isis1818, Feb. 25 13 0 49 182 45 2 0 70 26 II 89 43 48 1.0 0.0783711 Direct. Encke 142 1818 Dec. 4, 2 10 2 103 7 5 90 7 29 6z 40 50 1.0 9.92-8324 Retrograde. Bessel. 143 1819 Jan. 27, 6 8 53 I56 59 iz 334 33 19 I3 36 54 0.8485841 9.5253771 Direct. Encke. 144 1819 June 27, i6 52 9 287 5 5 273 43 44 80 45 53 1.0 9.5328194 n Brinkley. 145 1819 July 18, 21 3,6 I8 274 40 51 113'o 46 10 42 48 0.75519035 9.8885382 n Encke. 146 1819 Nov. 20, 5 53 34 67 IS 48 77 13 57 9 i i6 0.6867458 9.9506368 n 1/ 147 1821 March21, 12 52 39 239.29 25 48 40 56 73 33 7 I.o 8.9629523 Retrograde. Rosenberger. 148 1822 May 5, 13 34 52 192 47 45 177 25 4 53 35 34 1.0 9.7025976 n Gambart. 149 1822 May 23, 23 6 40 157 ii 44 334 25 9 13 20 17 0.8444643 9.5390382 Direct. Encke. 150 1822 July 16, 0 35 2 219 53 48 97 51 23 37 43 4 1.0 9.927430 Retrograde. v. Heiligenstein. 151 1822 Oct. 23, 28 28 29 271 40 17 92 44 42 5z 39 10 o.9963o21i 0.0588305 Encke. 15 2 1823 Dec. 9, 10 39 29 274 34 30 303 3 0 76 II 57 1.0 9.3550726 n 153 1824 July 11, I 2 8 40 260 I6 32 234 19 9 54 34 9 1.0 9.7717807 Riimker. g. 154: 1824 Sept. 29, 1 23 58 4 31 7 279 15 39 54 36 59 I.0017345 o.o2z2469 Direct. Encke. 155 1825 May 30, 13 6 39 273 55 I zo 6 8 56 41 6 1.0 9.9489616 Retrograde. Clausen. 156 1825 Aug. 18, 17 3 55 10 14 25 192 56 IO 89 41 47 1.0 9.9461924 Direct. i0 157 1825 Sept. 16, 6 33 I8 157 14 31 334 27 30 13 z1 24 0.8448885 9.5376348 n Encke. 158 1825 Dec. 10, I6 7 28 318 46 39 2z5 43 22 33 32 53 0.9954285 0.0937180 Retrograde. Hubbard. 159 1826 March18, 1o 43 9 109 48 47 25I 27 19 13 33 54 0.7466~z2 9.9554082 Direct. 160 1826 April 21, 23 27 46 117 II 14 197 30 19 39 57 24 1.0089597 0.3o16581 N Nicolai. 161 1826 April 29, 0 56 13 35 48 13 40 29 13 5 17 2 1.0 9.2744275 Retrograde. Cliiver. 16/ 1826 Oct. 8, zz 5z 14 57 48 24 44 6 28 25 57 z8.93085 Direct. Argelander. 163 1826 Nov. 18, 9 47 55 3I5 29 39 235 6 II 89 22 9 2.0 8.429581.. Retrograde. Gambart. 164: 1827 Feb. 4 22 7 4 33 30 i6 184 27 49 77 35 35 1.0 9.704600 C D.Heiligenstein. 165 1827 June 7, 20 o z 15 297 31 42 318 1028 43 38 45 1.0 9.907494 n 166 1827 Sept. 11, I6 37 44 250 57 12 149 39 II 54 4 42 0.99927305 9.1393857 Clver. 167 1829 Jan. 9, 17 54 7 157 17 53 334 29 32 13 20 34 0.8446245 9.5385038 Direct. Encke. 168 1830 April 9, 6 43 30 212 1I 38 206 21 36 21 i6 27 1.0 9.9644642 In Carlini. 169 1830 Dec. 27, 1550 58 310 59 19 337 53 7 44 45 30 1.0 9.0999822 Retrograde. Wolfers. 170 1832 May 3, 2 3 24 45 157 21 I 334 32 9 13 22 9 0.8454141 9.5358905 Direct. Encke. 171 1832 Sept. 25, 12 38 58 227 54 36 72 26 49 43 z8 41 1.0 0.0731607 Retrograde. Peters. 172 1832 Nov. 26, 9 36 44 109 56 24 248 II 49 13 II 48 0.7513780 9.9441275 Direct. Santini. 173 1833 Sept. 10, 9 29 30 224 21 23 323 28 17 7 IS 17 1.0 9.666836 Hartwig. 174 1834 April 2, 5 55 II 276 33 49 226 48 52 5 56 52 1.0 9.718304 n Petersen. 175 1835 March 30 I629 5I 206 9 242 i.0 o.312o691 Retrograde. Rfimker. No. T IT i e log q Motion. Comput-ed by ih m s o t' tt!! t 176 1835 Aug. 26, 8 39 32 157 23 29 334 34 59 13 zI 15 0.8450356 9-5371089 Direct. Encke. 177 1835 Nov. 15, 22 32 I 304 31 32 55 9 59 17 45 5 0.96739091 9.7683194 Retrograde. Westphalen. 178 1838 Dec. 19, o I7 38 157 27 4 334 36 41 13 2z i8 0.8451775 9.5366085 Direct. Encke. 179 1840 Jan. 4, Io 13 42 19i2 i 50 119 57 46 53 5 52 1.0002050 9.7913017, Peters and 0. Struve. 180 1840 March12, 23 46 32 80 8 io z36 49 6 59 f3 20 0.9978836 0.0868563 Retrograde. Plantamour. 0 181 1840 April 2, 11 53 27 324 12 27 i86 2 45 79 51 52 i.o 9.8740948 Direct. Rimker. 5 182 1840 Nov. 13, 15 27 55 22 31 40 248 56 22 57 57 23 0.96985265 0.1705070 / Goetze. 183 1842 April 12, o 26 9 157 29 27 334 39 io 13 20 26 0.8447904 9.5378361 " Encke. 184 1842 Dec. 15, 22 57 39 327 I6 13 207 49 1 73 33 37 i.o 9.7026605 Retrograde. Laugier. O 185 1843 Feb. 27, 9 51 9 278 40 17 I 14 55 35 40 39 0.9999157I7 7.7433765 i Hubbard. 186 1843 May 6, I 20 33 281 29 43 157 14 54 52 44 46 1.0001798 0.208536. Direct. Goetze. ~ 187 1843 Oct. 17, 3 33 46 49 33 52 209 29 36 II 22 33 0.5558997 0.2284632 ni Mller. - 188 1844 Sept. 2, 11 22 36 342 30 50 63 49 2 54 50 0.6176539 o.o742308 Brinnow.'. 189 1844 Oct. 17, 8 15 15 179 35 57 3 139 6 48 36 I 0.9996083 9.9321644 Retrograde. Plantamour. 190 1844 Dec. 13, i6 11 42 296 z i8 118 19 22 45 38 47 1.00035303 9.4009i26 Direct. Bofid. t:: 191 1845 Jan. 8, 3 58 z9 91 20 22 336 44 13 46 50 39 1.0 9.9567652 / Hind. I 192 1845 April 21, 0 44 37 192 33 19 347 6 45 56 23 36 1.o 0.0985330 f/ Faye. 193 1845 June 5, 16 9 44 262 z 56 337 48 56 48 41 59 0.9898742 9.603823 Retrograde. D'Arrest., > 194 1845 Aug. 9, 15 I 50 157 44 21 334 19 33 13 7 34 0.8474362 9.529oo1008 Direct. Encke. y h-_ 19,5 1846 Jan. 22, 2 15'i 89 6 zz22 i 8 26 47 26 6 0.9924026 0.1704680 /n Jelinek. 196 1846 Feb. 10, 22 1o 22 109 z 54 245 54 17 12 34 55 0.7566060 9.9327096 Ii Hubbard. 197 1846 Feb. 25, 8 58 39 116 28 i5 10z 40 58 30 55 53 0.7933880 9.8129825 Briinnow. - 198 1846 March 5, 13 5 18 90 27 0 77 33 33 85 5 42 0.96208911 9.8219813 Van Deinse. v 199 1846 May 27, 19 44 55 8z 39 20 i6i 18 29 57 36 24 i.o 0.1382020 Retrograde. Graham. ~ 200 1846 June 1, 5 5 53 240 7 35 260 28 59 30 24 24 0.7213385 0.1842997 Direct. C. H. F. Peters. 0 201 1846 June 5, 11 30 5 162 5 40 261 52 51 29 18 47 0.9899389 9.8oi08857 Retrograde. Oudemans. 202 1846 Oct. 29, 21 59 57 98 47 15 4 38 18 49 39 3 0.9933127 9.9187601 Direct. Quirling. 4 203 1847 March 30, 6 49 29 276 22 21 41 52 48 39 50 0.99991293 8.6z93024 Hornstein. 204 1847 June 12, 9 I 39 137 41 34 173 25 50 o80 6 57 1.o 0.3257617 Retrograde. D'Arrest. 205 1847 Aug. 9, 8 50 44 246 45 ii 338 i6 57 83 26 15 0.9985879 0.2470052 1/ Mauvais. 206 1847 Aug. 9, 6 13 10 21 20 41 76 42 10 32 38 24 0.9974348 0.1715154 I Schweizer. 207 1847 Sept. 9, 13 1 31 79 12 6 309 48 49 19 8 25 0.97z560 9.688z97 Direct. D'Arrest. 208 1847 Nov. 14, 9 36 39 274 12 57 190 49 53 71 50 56 1.0001326 9.5172334 Retrograde. Rfimker. 209 1848 Sept. 8, I 20 40 310 34 36 211 34 36 84 28 22 1.o 9.5048748 / 0 210 1848 Nov. 26, 2 44 10 157 47 8 334 22 12 13 8 36 0.8478280 9.5276718 Direct. Encke. No. T I Q ie log q Motion. Computed by h gn s 0 0 0 / f 0 f I 211 1849 Jan. 19, 8 31 37 63 14 56 215 12 51 85 2 13 1.0 9.9820756 Direct. Hensel. 212 1849 May 26, 12 37 26 235 45 15 202 33 27 67 7 50 0.9978863 0.06112,0 i/ Weyer. 213 1849 June 8, 4 53 15 267 6 8 30320 66 5 19 0.997830 9.951525 i D'Arrest. 214 1850 July 23, 12 40 16 273 25 5 92 53 28 68 1I 24 0.9988519 0.0340060 II Carrington. 215 1850 Oct. 19, 8 14 20 89 163 205 59 31 40 8 53 1.0 9.752749 1" Mauvais. 216 1851 April 1, 22 25 17 49 42 10o 209 31 6 ii 21 zI 38 0.5549601 0.2304281 II M611er. 217 1851 July 9, 2 39 15 322 55 55 148 24 51 13 55 8 0.6592674 0.0694270 /I Schuize. 218 1851 Aug. 26, 5 37 52 310 58 49 223 40 33 38 9 2 0.9968586 9.9931272 i Brorsen. 219 1851 Sept. 30, 19 8 58 338 46 26 44 21 30 73 58 37 1.0o 9.1521784 " Klinkerfues. 220 1852 Marchl4, 19 6 25 157 51 2 334 23 21 13 7 55 0.8476726 9.5282054 1 Encke. 221 1852 April 19, 15 15 6 278 42 18 317 29 30 49 11 8 1.0525041 9.9604040 Retrograde. Hartwig. 222 1852 Sept. 22, 22 38 25 109 8 16 245 51 28 i2 33 19 0.7558650 9.9348124 Direct. Hubbard. 223 1852 Oct. 12, 18057 431342 346 io 40 50 0.91891gI698 0.0968963 i Westphal. 224 1853 Feb. 23, 23 55 49 153 44 19 69 33 36 20 13 20 0.990412 0.0381820- Retrograde. Hartwig. o 225 1853 May 9, 19 39 59 201 44 37 40 57 37 57 49 3 0.9893194 9-9584172 V Riimker. ^ 226 1853 Sept. 1, 16 54 26 310 56 59 140 31 22 61 30 11 0.7294246 9.4871354 Direct. Stockwell. ^ 227 1853 Oct. 16, 14 31 44 302 14 53 220 5 52 6o 59 44 1.0012289 92372363 Retrograde. D'Arrest. 228 1854 Jan. 2, 17 g19 36 56 38 52 227 0 44 660 44 i.o 0.3108246 / Klinkerfues. C1 229 1854 March 24, 0 20 41 213 49 14 315 27 27 82 32 43 i10 9-4425551 " Mathieu. 230 1854 June 22, 2 43 272 58 6 347 48 45 71 8 21 1.0 9.8111244 " Bruhns. 231 1854 Oct. 27, 12 13 4 94 24 18 324 28 31 40 54 38 0.9933246 9.902384 Direct. Lesser. 232 1854 Dec. 15, 17 11 27 165 9 25 238 7 54 14 8 50 0.9864041 0.1327551 Adam. 233 1855 Feb. 5, I 8 11 226 37 34 189 43 33 51 24 19 0.9651850 0.3411427 Retrograde. Tiele. 234 1855 May 29, io 58 4 239 28 46 260 10 48 23 9 54 0.9039970 9-751970 Schulze. 0 235 1855 July 1, 4 400 157 53 12 334 26 24 13 8 9 0.8477869 9.5277600 Direct. Encke. 0 236 1855 Nov. 25, 9 8 58 862 13 51 34 31 10II 19 0.997255 0.090728 Retrograde. Hoek. 237 1857 March 21, 8 43 38 74 43 59 313 9 37 87 56 13 0.9992144 9.8878700 Direct. Schuize. 238 1857 March28, 16 4 19 115 46 25 ioi 45 15 29 48 53 0.8022946 9.7928091 Bruhns. 239 1857 July 17, 23 33 o10 249 36 234128 585751 0.9989984 9-5652331 Retrograde. Villarceau. 240 1857 Aug. 23, 23 54 59 2i 46 51 200 49 16 32 46 24 0.9803714 9.8732267 Direct. M6ller. 241 1857 Sept. 30, 21 7 5 250 7 38 14 57 48 56 3 2z 0.9969135 9.7504285 Retrograde. Linsser. 242 1857 Nov. 19, I 42 31 44 13 i6 139 18 42 37 48 55 0.9969918 0.003889 Auwers. 243 1857 Nov. 28, g19 36 14 323 3 9 148 27 7 13 56 0.6598094 0.0683373 Direct. Schuize. 244 1858 Feb. 23, 12 34 20 115 51 35 269 3 13 54 24 io 0.820903 0.010940 n Bruhns. 245 1 858 May 2, I 24 32 275 39 54 113 30 59 o10 47 55 0.7541036 9.8858281 //. Hansel. No. T 7r Q, i e log q MIotion. Computed by h m s 0' 0 f 0 1 i 246 1858 May 2, 7 42 37 195 58 44 170 42 56 22 59 49 1.0 0.0826760 Direct. Watson. 247 1858 June 5, 7 5 39 226 6 5 324 58 8 80 2 42 I.o 9.7358072 Retrograde. Auwers. 24S 1858 Sept. 29, 23 8 51 36 12 31 165 19 I3 63 1 49 0.99629326 9.7622804. Hill. 249 1858 Oct. 18, 8 41 33 157 57 30 334 28 34 13 4 15 0.8463915 9-5324034 Direct. Encke. 250 1858 Sept. 13, 21 26 37 49 51 54 209 40 2 II 22 II 0.5577360 0.2291239. M6ller. 251 1858 Oct. 12, 19 26 46 4 13 18 159 45 3 21 16 37 i.o 0.1544245 Retrograde. Weiss. g 252 1859 May 29, 5 25 38 75 20 31 357 20 44 83 31 45.0o 9.303265, Hertzsprung. 253 1860 Feb. 16, 16 9 30 173 45 21 324 3 z25 79 35 55 I.o 0.0782z9 Direct. Liais. a 2:54 1860 March 5, 17 12 25 50 16 5 8 56 9 48 13 4.0o o.1167062 / Seeling. o 255 1860 June 16, o 20 56 i61 31 o10 84 42 50 79 17 38 0.997240 9.465570. / Liais. 256 1860 Sept. 28, 6 49 0 ixI 59 o 104 14 0 28 14 0 i.o 9.9795 Retrograde. Valz. o 257 1861 June 3, 9 21 30 243 22 2 29 55 42 79 45 31 0.983463143 9.9641181 Direct. Oppolzer. 258 1861 June 11, 12 17 7 249 4 z7 278 57 59 85 26 28 0.9853832 9.9150740, Sawitsch. 259 1861 Dec. 7, 4 17 18 173 30 36 145 6 58 41 57 23.o0 9.9z3813 Retrograde. Pape. t 260 1862 Feb. 6, 4 7 49 I58 o 10 334 30 50 13 5 0 0.8467094 9.5313486 Direct. Encke. os 261 1862 June 22, o 43 59 229 20 27 326 32 53 7 54.'26 1.0 9.991818 Retrograde. Seeling. 0 262 1862 Aug. 22, 21 53 32 344 41 26 25 33 0.96I2708 9.9834648 " I Oppolzer. 3. 263 1862 Dec. 28, 8 33 28 125 9 43 355 44 58'42 22 53 i.o 9.904475, Engelmann. 264 1863 Feb. 3, II 47 16 191 22 45 116 55 33 85 21 56 0.9999470 9.9002349 Direct. i 265 1863 April 4, 21 42 13 247 15 25 251 15 35 67 22 13 1.0 0.0286067 Retrograde. Frischauf. _ 266 1863 April 20, 20 39 7 305 31 6 249 59 22 85 28 44.o0 9.798266 Direct. Karlinski. = 267 1863 Nov. 9, ii 35 I6 94 43 17 97 29 56 78 5 21 i.o 9.849171 Stampfer. 268 1863 Dec. 27, 189 44 60 24 28 304 43 26 64 28 46 1.o 9.887344, Weiss. 269 1863 Dec. 29, 4 0 45 183 7 18 105 1 24 83 19 17 1.0006499 0.1183045 / RBos6n. 270 1864 Jily 27, 19 50 29 I85 31 54 174 51 6 65 I 19 1.o 9.822162 Retrograde. Valentiner. 271 1864 Aug. 15, 13 46 54 304 11 52 95 14 27 I 52 10 0.9967771 9.9587003 Kowalczyk. 272 1864 Oct. 11, 9 41 54 159 i8 2 31 45 26 70 18 2 0.99995324 9.9690407 <' von Asten. 273 1864 Dec. 22, ix 7 31 321 42 31 203 13 12 48 52 20.o0 9.886982 Direct. Tietjen. 274 1864 Dec. 27, 17 16 2o i6z 23 36 16o 54 22 17 7 23 1.0 0.0471352 Retrograde. Valentiner. 275 1865 Jan. 14, 8 xo 23 141 15 37 253 3 15 87 32 20 i.o 8.4152071 I Tebbutt. 276 1866 Jan. 11, 3 12 47 6o 28 48 23 z26 3 17 i8 5 0.9054198 9.9896813 Oppolzer. 277 1866 Feb. 14, 0o 29 48 49 56 55 209 41 53 II 22 7 0.5575382 0.2258707 Direct. Moller. 278 1867 Jan. 19, 20 39 15 75 52 15 78 35 45 I8 12 35 0.8490551 0.1965869, Searle. 079 1867 Feb. 27, 20 17 25 1x62 40 17 i68 35 31 6 7 0 1.0 0.050900 f C. F. W. Peters. 280 Epoch and Mean Date of No. NaeIohadMa aeo Discoverer. No. Name. Equinox. M Q log a Discovery. iscoerer. Berlin Mean Time. 0 1 // 0 I 0! /' 0 1 I 0 /! 1 Cares. 1866 Jan. 21.0 337.10 35.7 148 22. 8.4 So 50 7.2 10 36 28.8 4 36 2.8 771.024I8 0.4419590 1801 Jan. 1 Piazzi. 2 Pallas. 1866 June 19.0 t 56 49.0 622 76.8 X72 43 55.6 34 42 38~8 13 53 17.5 769.80966 44245 1802 March2 Olbers. 3 Juno. 1865 Nov..3 11 56 49-0 12- 2 34 4 0-44'24 I 5 ~~~1802 Marchf28 01hers. 3 Juno. 1865 Nov. 3.0 329 20 8.9 54 56 31.9 170 49 36.8 I3 1 z2.3 I4 54 25.8 814.0068 0.4262524 1804 Sept. 1 Harding. 4Vesta. 1810 Jan. 0.0 2I6 42 25.8 249 19 28.6 103 II 22. I 7 8 5.0 5 5 36.3 977,6338563 0.3732203 1807 March29 Olbers. Astroeta. 1865 Sept. 8 Hencke 5 Astrsaa. 1865 Sept. 19.0 234 23 32.5 135 14 48.1 141 26 I8.3 5 19 9.0 10 48 30.I 857.60520 0.41I1462 1845 Dec. ecke. 6 Hebe. 1866 June 30.0 283 17 20.5 15 6 Iz2.7 138 39 I7.3 14 46 43.9 II 41 34.8 939.08225 0.3848687 1847 July 1 Heneke. 7 Iris. 1850 Jan. 0.0 066 7 9.0 41 23 21.1 259 47 5 5 28 3.0 13 20 50.2 962.580602 0.3777130 1847 Aug. 13 Hind. 8 Flora. 1848 Jan. 1.0 35 54 3.6 32 54 28.3 11i0 17 48.6 5 53 8.0 9 0 56.3 1086.33098 0.3426963 1847 Oct. 18 Hind. 9 Metis. 1858 June 30.0 57 4 34.7 71 3 52.1 68 31 35.2 5 36 0.3 7 2.4 962.33898 0.3777857 1848 April 25 Graham. 10 Hygeia. 1864 Feb. 22.0 3 220235 29.2 286 43 i.8 3 49 0.2 5 44 56.4 343 04984692 1849 April 12 (asparis. 199 13 22-0 235 10 z -44 564 634.3IIx8~11 Parthenope. 1865 March27.0 239 14 15.1 317 14 31.4 125 7 27.9 4 37 i.6 5 42 54.1 924.15366 0.3895083 1850 May 11 Gasparis. o 12 Victoria. 1851 Jan. 0.0 66 z 39.9 301 39 25.0 235 34 41-7 8 23 17-7 12 38 449 994.83472 0.368i389 1850 Sept. 13 Hind. 13 Egeria. 1866 Aug. 29.0 220 54 41.3 120 5 15.0 43 I5 56.3 I6 30 48.8 4 59 47 85787961 0.41054 1850 Nov. 2 Gasparis. 14 Irene. 1864 Nov. 28.0 134 55 9.z2 79 52 6.8 86 42 23.7 9 7 37-5 9 33 23.7 853.20824 0.4126344 1851 May 191 Hind. 15 Eunomia. 1854 Jan. 0.0 i22zz 5 31.5 27 52 0.5 293 52 14.5 II 44 174 10o 47 32.2 825.4550 0.422209 1851 July 29 Gasparis. O 16 Psyche. IS67 Jan.'1852 MarchPsyeasparis.. P 17 TXsyhet. 1867 Jan. 0.0 115 o10 46.6 15 26 27.0 I50 33 17.6 3 3 57.2 7 47 0.3 709.7603 0.465930 17 Thetis. 1866 July 1.5 177 17 24.1 260 24 17.6 125 23 4.2 5 36 6.2 7 20 I2.3 912.06563 03933 1852 April 17 Luther. 18 Melpomene. 1854 Jan. 0.0 o80 4 37.0 I5 5 310 1I50 3 49.7 10o 9 6.9 1z 34 20o.2 1OZO.1198 0.36903 1852 June 24 Hind. 19 Fortuna. 1863 June 24.0 258 15 3.3 30 57 54.2 2II 22 29.1 I 32 448 9 7 302 9302787 03875957 1852 Aug. 22 Hind. 20 Massalia. 1866 June 15.5 i6i 43 44.2 98 29 3.8 20zo6 45 6.7 0 4 9-5' 8 x5 13.7 948.55878 0.381I962 1852 Sept. 19 Gasparis. w 20 Massalia. 1866,June 15.5 I 61 43 44 852 ept.191 aspris. 21 Lutetia. 1853 Jan. 2.0 74 20 5.1 327 3 8.4 80 27 7.2 3 5 9.5 9 19 44.6 933.55438 0.3865780 1852 Nov. 15 Goldschmidt. 22 Calliope. 1866 Aug. 30.5 289 37 46.9 58 15 36.1 66 35 39.8 13 43 47-4 5 39 i8.6 714.51387 0.463998 1852 Nov. 16 Hind. 23 Thalia. 1867 Jan. 0.0 68 14 I6.3 123 49 4I.6 67 40 47.2 IO 13 26.6 3 25 8.833087 049537 1852 Dec. 15 Hind. 24 Themis. 1864 Aug. 20.0 40 14 0.7 140 8 26.5 36 12 12.6 o 48 52. 6 42 52.9 63.76345 04973523 1853 April 5 Gasparis. 25 Phocoea. 1865 Nov. 12.0 79 17 21.8 302 49 53-4 214 5 7.3 21 34 36.3 14 43 58.4 953.82327.3360 1853 Apil Chacornc. 26 Proserpina. 1853 June 11.0 351 5 55.6 236 25 15.0 45 54 59.3 3 35 47-7 5 0 37.3 89;68468 04242399 153 My 5 Luther. 27 Euterpe. 1866 May 26.5 149 7 51.3 87 35 3.6 93 48 1.5 I 35 29.8 9 58 29.2 986.9go48 0.370463 1853 Nov. 8 Hind. 28 Bellona. 1862 March24.0 303 8 27.8 122 55 29.6 144 41 9.9 9 zi 26.3 8 37 57.5 766.-x2228 04438057 1854 March 1 Luther. 29 Amphitrite. 1866 March 10.0 o04 21 32.3 56 56 1.8 356 30 5.z 6 7 49-3 4 14 35.3 8693 3444 0.4072I4 1854 March I Marth. 30 Urania. 1865 Aug. 18.0 306 32 25.0 31 28 57.9 308 9 39.2 2 6 6.9 7 i6 6.3 975.27438 0.373920 1854 July 22 Hind. 31 Euphrosyne. 1867 Jan. 0.0 8. 93 42 6.6 31 3 459 26 27 50 12 44 103 6338508 0.49868 1854 Sept. Feruson. 0-0 I 8 - 093 42. 6.6 31 31 45.9 32 Pomona. IS55 Jan. 5.0 222 54 24.0 194 21 32.1 220 48 14'5 5 29 5.0 4 43 43.7 852-5769 0.4128487 1854 Oct. 28 Goldschmidt. 33 Polyhymnia. 1866 Feb. 11.0 144 10o 45.7 342 31 6.7 9 5 54.9 I 56 I9.9 9 47 58.6 731.6869 0.45712I 1854 Oct. 28 Chacornac. 34 Circe. 1865 Aug. 20.0 170 13 2.5 150 3 19.2 I84 48 36.5 5 26 28.9 6 9 44.1 805.85537 0.429i663 1855 April 6 Chacornac. 35 Leucothea. 1866 June 22.0 47 39 33.7 201 40 29.0 355 55 35'8 8 io 47.6 z12 22 5.7 680.7841 0.477998 1855 April 19 Luther. Epoch and Mean Date of Discoerer. No. Name. Equinox. 1 I i log a Discovery. Berlin Mean Time.. o, o 0, o i 0 0 1 0 i o 1 i 36 Atalanta. 1866 Feb. 21.0 74 52 38.3 4Z 47 47-7 359 "I 14.9 8 42 14.8 17 31 53.2 779.6936 0.438721 1855 Oct. 5 Goldschmidt. 37 Fides. 1853 Oct. 5.0 z66 46 29.o 66 20 17.3 8 IZ 29-4 3 7 12.3 1o 10 46.1 826.54485 0.4218268 1855 Oct. 5 Luther. 38 Leda. 1856 Jan. 0.0 12 6 43-3 1lo 51 44-3 z96 27 34-9 6 58 25.3 8 56 30.7 782-2500 0-4377740 1856 Jan. 12 Chacornac. 39 Laetitia. 1866 May 2.0 231 39 4-8 2 30 27.3 157 21'II5 IO zz22 5.1 6 35 22 770.85681 0.4420219 1856 Feb. 8 Chacornac. 40 Harmonia. 1866 March 3.0 i6o 34 22.3 I z7 26.1 93 35 58-8 4 15 54-8 z 41 7.6 1039.45323 0.3554678 1856 March 31 Goldschmidt. 41 Daphne. 1866 July 29.5 55 45 I7.5220 I2 14.11 I79 6 58-7 15 59 12.1 15 25 19-7 769.99685 0.442346 1856 May 22 Goldschmidt. 42 Isis. 1860 Jan. 0.0 289 29 25.4 318 o 48.7 84 30 40.4 8 34 33-0 13 20.6 930.9057 0.3874006 1856 May 23 Pogson. 43 Ariadne. 1866 Jan. 1.0 184 54 151 277 48 9.6 26437439 3 27 40.5 9 38 37-8 1084.93658 0.3430683 1857 April 15 ogon. 1-0Ariadne. 15D6 Jan. 1.O/I84 5 4 115-277 48 9~3 15 Apri 175 Pegson. 44 Nysa. 1866 Oct. 9.0 283 21 50.5 112 5 31.5 131 3 31-2 3 41 57.6 8 40 17.9 94135966 0.3841674 1857 May 27 Goldschmidt. 45 Eugenia. 1866 June 4.0 19 22 i.6 230 50 34.9 148 6 37 65 25-0 4 35 2.2 790.4322 0.434762 1857 June 27 Goldschmidt. z 46 Hestia. 1.865 July 26.0 322 ii 46.6 354 io 34-9 I18 26 45-3 2 17 32z1 9 26 55.7 883.5638 0.4025124 1S57 Aug. 16 Pogson. 47 Aglaia. 1859 June 1.7.0 i6z 29 40.5 314 3 45.0 4 12 34.2 5 8.5 735 15-7 7254987 0.4595800 1857 Sept. 15 Luther. 48 Doris. 1862 July 25.0 235 II 27.8 74 20 42.41 85 5 29.6 6 29 28.2 4 23 42.9 64712954 0.4926769 1857 Sept. 19 Goldschidt 49 Pales. 1863 Nov. 14.0 o 30.8 32 14497 290 3 17.4 3 8 46.4 13 43 18-3 655.6289 0o.4889025 1857 Sept. 19 Goldschmidt. o 50 Virginia. 1863 Jan. 18.083 27 18. 9 953 21.4 173 31 59.2 2 47 484 6 40 22.5 822.94439 0.4230907 1857 Oct. 4 Ferguson. ~ h i 51 Nemausa. 1865 Jan. 17.0 316 39 29.6 174 52 0.6 175 43 6.3 9 56 52.8 3 47 40-7 975-13844 0.3739602 1858 Jan. 22 Laurent. Ct 52 Europa. 1858 Jan. 0.0 34 25 7.3 ioi 56 14.8 129 57 i6.0 7 24 41.0 5 49 14.3 650.0877 0.4913564 1858 Feb. 6 Goldschmidt. 53 Calypso. 1866 Jan. 4.0 7 11 44-0 92 53 30-3144 9-0 5 6 39-0 11 45 54-8 836.80511 o.4182540 1858 April 4 Luther. 54 Alexandra. 1863 Nov. 14.0 83 37 8.2 295 27 8.7 314 5 8.4 ii 46 41.9 1i 21 24.0 794.32164 0.4333401 1858 Sept. 10 Goldschmidt. C 55 Pandora. 1863 Oct. 25.0 35 42 11.7 11 9 47-8 io 52 9.6 7 13 49-8 8 19 19.2 774-2176 0.4407624 1858 Sept. 10 Searle. 56 Melete. 1865 June 20.0 344 40 12.6 293 29 25.0 194 27 23-7 8 I 40.9 13 44 9-5 848.33049 0.4142944 1857 Sept. 9 Goldschmidt. o 57 Mnemosyne. 1860 Jan. 1.0 335 30 22.2 53 7 9.9 200 5 31-5 15 8 8.6 5 58 17.1 632.68967 0.4992106 1859 Sept. 22 Luther. 58 Concordia. 1865 Jan. 7.0 2I 50 58.8188 41 55-0 161 19 35.6 5 I 53-2 2 26 15.2 799.63132 0.4314112 1860 March24 Luther. p 59 Elpis. 1865 Jan. 7.0 334 I8 42.6 18 18 54.2 170 20 28.8 8 37 3-5 6 44 1.3 793974093 0.4334669 1860 Sept. 12 Chacorna.. 60 Echo. 1866 Jan. 0.0 65 44 37-I 98 33 32.6 192 2 9.0 3 34 18.5 io 38 45.8 958.47412 0.3879508 1860 Sept. 15 Ferguson. 61 Danae. 1865 Aug. 19.0 345 54 41-.2 341 25 28.5 334 11 50.0 I8 15 25.6 9 17 59.3 688.-8150 0.4749112 1860 Sept. 19 Goldschmidt. 62 Erato. 1865 May 7.0279 40 20.8 34 8 29. 6 II 42.1 2 12 17.5 946 43 640.85910 495496 10 Sept. 14FoersterLesser. 63 Ausonia. 1865 April 17.0 307 24 5-0269 32 49-.0338 6 58.3 5 47 I6.3 7 13 45-3 957.32042 0.3792995 1861 Feb. 10 Gasparis..64 Angelina. 1865 Jan. 7.0 355 46 41.1 123 37 49.1 311 9 7-2 I 19 52.0 7 21 58.9 808.30600 0.428Z872 1861 March 4 Tempel. 65 Cybele. 1861 Jan. 0.0 281 57 347 258 20 36.9 I58 53 34-8 3 28 9.8 6 54 36.4 560.8775 0.534092 1861 March 8 Tempel. 66 Maia. 1865 Jan. 27.0 87 7 3-2 44 25 0o.6 8 15 23.7 3 4 15.1 9 5 46.9 821.9211 0.4234510 1861 April 9 Tuttle. 67 Asia. 1865 Jan. 7.0 296 2 14.0 306 8 6.9 2o02 43 29-0 5 59 35-9 10 39 58.6 941-4909 0.3841270 1861 April 17 Pogson. 68 Leto. 1863 Dec. 20.0 93 53 22.4 345 4 58.2 44 53 11-4 7 57 34-9 10 51 46.8 765.323 0.444108I 1861 April 29 Luther. 69 I-esperia. 1861 June 3.0 54 46 56.9 i9 6 25.41187 7-5 8 28 19.2 IO 0 38.3 692z.6300 0473004 1861 April 29 Schiaparelli. 70 Panopasa. 1861 May 28.0 308 41 11.5 300 3 30.3 48 14 42.6 Ii 38 30.2 10 33 7.5 839.90600 0.417184 1861 May 5 Goldschmidt. TABLE XIX. Elements of the Orbits of the Minor Planets. Epoch and Mean D of No. Name. Equinox. Iir log a ateof Discoverer. Berlin Mean Time. Discovery. o! 0 o0 o I / 0 I/ o0 f / 71 Niobe. 1864 Jan. 23.0 283 10 47.6 222 4 26.8 316 19 7.0 23 I8 51.2 10 0 15I1 775-73290 0.4401964 1861 Aug. 13 Luther. 72 Feronia. 1866 Jan. 0.0 31 17 25.1 307 54 49.5 207 44 59.6 5 23 54-5 6 5z 45-9 1040.14680 0-3552747 1861 May 29 Peters. 73 Clytia. 1864 Oct. 4.0 325 18 55.8 59 59 11i. 7 34 19.1 2 24 39-5 z 27 0-5 814.84338 0.425955 1862 April 7 Tuttle. 74 Galatea. 1866 Jan. 0.0 249 23 I2-1 7 22 10.2 197 58 59.3 3 58 54-.6 3 46 49.1 766.4390 0.4436860 1862 Aug. 29 Tempel. 75 Eurydice. 1864 Feb. 2.0 133 39 40.8 334 27 46.0 359 56 43.4 5 o 4-2 17 51.-1 812.9517 0.426628 1862 Sept. 22 Peters. 76 Freia. 1863 July 27.0 355 31 36.1 93 13 58.1 212 58 21.4 2 I 50.8 io 49 12.0 569.0505 0.5299038 1862 Oct. 21 D'Arrest. 77 Frigga. 1866 Jan. 0.0 228 36 I6.6 58 Ii 32.0 2 9 27.6 2 27 56.6 7 48 20.4 8I2.40096 0.4268241 1862 Nov. 12 Peters. 78 Diana. 1865 Oct. 4.0 256 20 50.5 121 42 47.5 333 55 48.4 8 38 39-9 ii 51 34-5 835-35315 0-4187577 1863 March15 Luther. 79 Eurynome. 1864 Jan. 1.0 I 30 56.7 44 17 58.i 206 42 42.6 4 36 46.5 II 14 53-1 929. 286 0.3879539 1863 Sept. 14 Watson. 80 Sappho. 1865 Oct. 7.0 50 Ii 5.7 355 5 12.5 2z8 31 45.0 8 36 51.3 II 33 5.6 1019.6804 0.3610284 1864 May 2 Pogson. 81 Terpsichore. 1864 Oct. 6.0 333 26 I8.1 43 33 7-9 2 32 x.6 7 55 40.8 12 13 31-5 735-0244 0.4558032 1864 Sept. 30 Tempel. 82 Alcmene. 1865 Feb. 16.0 332 33 22.9 131 I8 19.7 6 56 56.5 2 51 15-0 13 3 43.I 773-711 0.440952 1864 Nov. 27 Luther. 83 Beatrix. 1865 May 4.0 17 I 59.0 188 28 20.9 27 34 9-. 5 2 11.3 4 49 38.8 937.415 0-.385383 1865 April 26 Gasparis. 84 Clio. 1865 Nov. 13.0 14 36 45-5 339 12 o.I 327 22 1.5 9 22 I6.o 13 39 34-8 977-5422 0.3732474 1865 Aug. 25 Luther. 85 Io. 1866 Jan. 0.0 56 49 20.9 322 32 28.9 203 52 33.3 II 53 12.8 II o 53.1 820.7120 0.4238772 1865 Sept. 19 Peters. ^ I 86 Semele. 1866 Jan. 8.0 8 23 14.6 28 39 3-9 87 55 49.6 4 47 44-6 II 49 36.5 652.9848 0.490069 1866 Jan. 4 Tietjen. 87 Sylvia. 1866 May 16.5 274 4 2.3 337 21 48.6 76 23 59-0 10 51 22.0 4 39 22.6 543.5800 0.54316o20 1866 May 16 Pogson. 88 Thisbe. 1866 Aug. 4.5 356 5 1-4 308 55 30.5 277 44 78 5 14. 58-1 9 29 55-7 769-561 0.442509 1866 June 15 Peters. 89 1866 Sept. 1.0 339 44 19.2 349 30 29.8 311 31 7.5 i6 32 38.0 9 37 22.2 872.656 0.4o06o09 1866 Aug. 6 Stephan. 90 Antiope. 1866 Oct. 18.0 52 6 9.2 294 3 7-3 71 0 54-0 2 17 25-2 11 39 2.7 632.35913 0.4993618 1866 Oct. I Luther. 91 1866 Dec. 21.0 336 46 5.4 75 I6 23.5 II 19 IO.4 2 9 25.0 5 4 27.2 867.0876 0.4079624 1866 Nov. 4 Stephan. TABLE XX, Elements of the Orbits of the Major Planets. Epoch and Mean Name. Equinox. AT r A e, e a Greenwich Mean Time. o!! oI 0 If o 0 71/! i o 0 5 If Mercury. 1801 Jan. 1.0 i66 0 43.2 74 21 37-2 + 9 44 45 58 20.2-13 2 7 0 4-5 +18.1 0.2056003 +0.00000387 0.3870984 Venus. cc II 33 3.0128 43 53.1 - 4 28 74 54 12-9 - ii II 3 23 28.5 + 4-5 0.0068607 -o0.00006275 0.7233316 Earth. cc 100 39 10.2 99 30 5.0 + 19 41 0.0167836 - 0.00004359 i.0000000 Mars. " 64 22 55-5 332 23 56.6 + 26 22 48 0 3.5 — 38 49 I 51 6.2- 0.3 0.0933070 + 0.00009019 1.5236923 Jupiter. 112 15 23.0 II 8 34.6 +11 5 98 26 18.9 — 26 z I 18 5X..3 —22.6 0.048162z +0.000o 6036 5.2027760 Saturn. " 135 20 6.5 89 9 29.8 + 32 17 iII 56 37-4 -3222 2 29 35.7 — 15-5 0.0561505 — 0.00031240 9538786 Uranus. 177 48 23.0 167 31 16.1 + 4 0 72 59 35.3 — 59 59 0 46 28.o0+ 3.1 0.0466794 -0.00002521 19.1823900 Neptune. 1850 Jan. 0.0 335 5 38-9 43 17 30-3 130 7 3I.81 I 47 1.7 0.0084962 30.0705520 TABLE XXI, Constants, &c log Base of Naperian logarithms.. e. 2.71828183 0.43429448 Modulus of the common logarithms ~.. ~0 0.43429448 9.63778431 - 10 Radius of a Circle in seconds..... = 206264.806 5.31442513 It //t u minutes... r 3437.7468 3.53627388 I/ i IIo // degrees. r. 57.29578 1.75812263 Circumference of a Circle in seconds.. 1296000 6.11260500 II I It U when r - 1... 3.14159265 0.49714987 Sine of 1 second.... 0.000004848137 4.68557487 Equatorial horizontal parallax of the sun, according to Encke..... 8/.57116 0.9330396 Length of the sidereal year, according to Hansen and Olufsen........ 365.2563582 days 2.56259778 Length of the tropical year, according to Hansen and Olufsen...... 365.2422008 I/ 2.56258095 This value of the length of the tropical year is for 1850.0. The annual variation is 0.d0000000624. Time occupied by the passage of light over a distance equal to the mean distance of the earth from the sun, according to Struve... 497./827 2.6970785 Attractive force of the sun, according to Gauss. k 0.017202099 8.23558144 - 10 I! I/ / i It/ U/ / in seconds of arc..... 3548.18761 3.55000657 Constant of Aberration, according to Struve..... 20'.4451 t tU Nutation, eI I/ Peters....... 9".2231 Mean Obliquity of the ecliptic for 1750 + t, according to Bessel... 230 28' 18".00 - 0".48368t- 0".00000272295t2 Mean Obliquity of the ecliptic for 1800 + t, according to Struve and Peters. 23~ 27' 54".22- 0".4738t - 0".0000014t2 General Precession for the year 1750 + t, according to Bessel 50".21129 + 0".0002442966t tl l} l/ li I/ // Struve 50".22980 + 0".000226t MASSES OF THE PLANETS, THE MASS OF THE SUN BEING THE UNIT. 1 1 Mercury... m 1 Jupiter.m 1 7 —7, 4865751 1047.879 1 1 Venus......, Saturn... ~. ^ Venus * *390000' Saturn 3501.6 I 1 Earth.. Uranus Earth 3493'354936 raus24905 1 1 Mars.I.... Neptune. 2680637' 18780 649 EXPLANATION OF THE TABLES. TABLE I. contains the values of the angle of the vertical and of the logarithm of the earth's radius, with the geographical latitude as the argument. The adopted elements are those derived by Bessel. Denoting by p the radius of the earth, by op the geographical latitude, and by p' the geocentric latitude, we have' -s p - 11' 30".65 sin 2e -+ 1".16 sin 4 -- &c., log p = 9.9992747 + 0.0007271 cos 2SD - 0.0000018 cos 42- + &c., p being expressed in parts of the equatorial radius as the unit. These quantities are required in the determination of the parallax of a heavenly body. The formulae for the parallax in right ascension and in declination are given in Art. 61. TABLE II. gives the intervals of sidereal time corresponding to given intervals of mean time. It is required for the conversion of mean solar into sidereal time. TABLE III. gives the intervals of mean time corresponding to given intervals of sidereal time. It is required for the conversion of sidereal into mean solar time. TABLE IV. furnishes the numbers required in converting hours, minutes, and seconds into decimals of a day. Thus, to convert 13h 19mn 43.5s into the decimal of a day, we find from the Table 13h - 0.5416667 19n = 0.0131944 43s - 0.0004977 0.5s - 0.0000058 Therefore 13h 19m 43.5s - 0.5553646 651 652 THEORETICAL ASTRONOMY. The decimal corresponding to 0.5s is found from that for 5s by changing the place of the decimal point. TABLE V. serves to find, for any instant, the number of days from the beginning of the year. Thus, for 1863 Sept. 14, 15h 53m 37.2s, we have Sept. 0.0 = 243.00000 days from the beginning of the year. 14d 15h 53m 37.2s- 14.66224 Required number of days = 257.66224 TABLE VI. contains the values of M= 75 tan -v + 25 tan'3 v for values of v at intervals of one minute from 0~ to 180~. For an explanation of its construction and use, see Articles 22, 27, 29, 41, and 72. In the case of parabolic motion the formule are m== -,.1Jm==rm(t-T), wherein log C= 9.9601277. From these, by means of the Table, v may be found when t- T is given, or t - T when v is known. From v =30 to v- 180~ the Table contains the values of log Mi. TABLE VII., the construction of which is explained in Art. 23, serves to determine, in the case of parabolic motion, the true anomaly or the time from the perihelion when v approaches near to 180~. The formulse are ^o200,200 sin w -- =V + O- t T- 0'C 13 - M ~ Co siniw w being taken in the second quadrant. The Table gives the values of A0 with io as the argument. As an example, let it be required to find the true anomaly corresponding to the values t - T= 22.5 days and log q- 7.902720. From these we derive log M- 4.4582302. Table VI. gives for this value of log M, taking into account the second differences, v - 168~ 59' 32".49; but, using Table VII., we have w = 168~ 59' 29".11, a = 3".37, EXPLANATION OF THE TABLES. 653 and hence v =w + A, - 168~ 59' 32".48, the two results agreeing completely. TABLE VIII. serves to find the time from the perihelion in the case of parabolic motion. For an explanation of its construction and use, see Articles 24, 69, and 72. TABLE IX. is used in the determination of the true anomaly or the time from the perihelion in the case of orbits of great eccentricity. Its construction is fully explained in Art. 28, and its use in Art. 41. TABLE X. serves to find the value of v or of t - T in the case of elliptic or hyperbolic orbits. The construction of this Table is explained in Art. 29. The first part gives the values of log B and log C, with A as the argument, for the ellipse and the hyperbola. In the case of log C there are given also log I. Diff. and log half II. Diff.. expressed in units of the seventh decimal place, by means of which the interpolation is facilitated. Thus, if we denote by log (C) the value which the Table gives directly for the argument next less than the given value of A, and by AA the difference between this argument and the given value of A, expressed in units of the second decimal place, we have, for the required value, log C-_ log'(C) + AA X I. Diff. + AA X halfII. Diff. For example, let it be required to find the value of log C corresponding to A= 0.02497944, and the process will be:(1) (2) Arg. 0.02, log(C)= 0.0034986 logI.Diff.=4.24585 loghalfII.Diff. =1.778 (1)= 8770.6 logAA =9.69718 21og AA -9.394 AA = 0.497944, (2) = 14.8 3.94303 1.172 log C = 0.0043771 The second part of the Table gives the values of A corresponding to given values of r. TABLE XI. serves to determine the chord of the orbit when the extreme radii-vectores and the time of describing the parabolic arc are given. For an explanation of the construction and use of this Table, see Articles 68, 72, and 117. 654 THEORETICAL ASTRONOMY. TABLE XII. exhibits the limits of the real roots of the equation sin (z'- g) = m, sin4 z'. The construction and use of this table are fully explained in Articles 84 and 93. TABLES XIII. and XIV. are used in finding the ratio of the sector included by two radii-vectores to the triangle included by the same radii-vectores and the chord joining their extremities. For an explanation of the construction and use of these Tables, see Articles 88, 89, 93, and 101. TABLE XV. is used ini the deterlination of the chord of the part of the orbit described in a given time in the case of very eccentric elliptic motion, and in the determination of the interval of time whenever the chord is known. For an explanation of its construction and use, see Articles 116, 117, and 119. TABLE XVI. is used in finding the chord or the interval of time in the case of hyperbolic motion. See Articles 118 and 119 for an explanation of the use of the Table, and also the explanation of Table X. for an illustration of the use of the columns headed log I. Diff. and log half II. Diff. TABLE XVII. is used in the computation of special perturbations when the terms depending on the squares and higher powers of the masses are taken into account. For an explanation of its construction and use, see Articles 157, 165, 166, 170, and 171. TABLE XVIII. contains the elements of the orbits of the comets which have been observed. These elements are: T, the time of perihelion passage (mean time at Greenwich);;r, the longitude of the perihelion; 2, the longitude of the ascending node; i, the inclination of the orbit to the plane of the ecliptic; e, the eccentricity of the orbit; and q, the perihelion distance. The longitudes for Nos. 1, 2, 12, 16, 91, 92, 115, 127, 138, 155, 156, 159, 160, 162, 171, 173-175. 180, 181, 185, 191, 192, 195-199, 201, 203, 204, 207, 208, 212-215, 217-219, 221-228, 230, 233, 234, 237-248, 251-258, 261-267, 269-275, 277-279, are in each case measured from the mean equinox of the beginning of the year. In the case of Nos. 134, 146, 172, 182, 189, 190, 205, 231, 232, 236, 259, and 268, the longitudes are EXPLANATION OF THE TABLES. 655 measured from the mean equinox of the beginning of the next year. The longitudes for Nos. 19. and 27 are measured from the mean equinox of 1850.0; for No. 186, from the mean equinox of July 3; for No. 187, from the mean equinox of Nov. 9; for No. 200, from the mean equinox of July 1; for No. 202, from the mean equinox of Oct. 1; for No. 206, from the mean equinox of Oct. 7; for No. 211, from the mean equinox of 1848.0; for No. 216, from the mean equinox of Feb. 20; for No. 229, from the mean equinox of April 1; for No. 250, from the mean equinox of Oct. 1; and for No. 276, from the mean equinox of 1865 Oct. 4.0. Nos. 1, 2, 11, 12, 20, 23, 29, 41, 53, 80, and 177 give the elements for the successive appearances of Halley's comet; Nos. 104, 116, 126, 143, 149, 157, 167, 170, 176, 178, 183, 194, 210, 220, 235, 249, and 260, those for Encke's comet, the longitudes being measured from the mean equinox for the instant of the perihelion passage. Nos. 92, 127, 159, 172, 196, and 222 give the elements for the successive appearances of Biela's comet; Nos. 187, 216, 250, and 276, those for Faye's comet; Nos. 197 and 238, those for Brorsen's comet; Nos. 217 and 243, those for D'Arrest's comet; and Nos. 145 and 245, those for Winnecke's comet. For epochs previous to 1583 the dates are given according to the old style. This Table is useful for identifying a comet which may appear with one previously observed, by means of a similarity of the elements, its periodic character being otherwise unknown or at least uncertain. The elements given are those which appear to represent the observations most completely. For a collection of elements by various computers, and also for information in regard to the observations made and in regard to the place and manner of their publication, consult Carl's Repertorium der Cometen-Astronomie (Munich, 1864), or Galle's Cometen- Terzeichniss appended to the latest edition of Olbers's lilethode die Bahn eines Cometen zu berechnen. TABLE XIX. contains the elements of the orbits of the minor planets, derived chiefly from the Berliner Astronomisches Jahrbuch fir- 1868. The epoch is given in Berlin mean time; 3I denotes the mean anomaly, (p the angle of eccentricity, p the mean daily motion, and a the semi-transverse axis. The elements of Vesta, Iris, Flora, Metis, Victoria, Eunomia, Melpomene, Lutetia, Proserpina, and Pomona are mean elements; the others are osculating for the epoch. The date of the discovery of the planet, and the name of the discoverer, are also added. 656 THEORETICAL ASTRONOMY. TABLE XX. contains the mean elements of the orbits of the major planets, together with the amount of their variations during a period of one hundred years. The epoch is expressed in Greenwich mean time, and L denotes the mean longitude of the planet. TABLE XXI. gives the values of the masses of the major planets, and also various constants which are used in astronomical calculations. APPENDIX. A. Precession.-If we adopt the values for the precession and for the variation of the position of the plane of the ecliptic given in Art. 40, and put M= 171~ 36' 10" + 39".79 (t - 1750), the formulse for the annual precession in longitude (A) and latitude (p) become, for the instant t, -= 50".2113 + 0".0002443 (t - 1750) + (0".4889 - 0".00000614 (t - 1750)) cos ( - M) tan j, (1) d= _ (0".4889 - 0".00000614 (t - 1750)) sin (A - M). d(t If we denote the planetary precession by a, the luni-solar precession by 1,, and the obliquity of the fixed ecliptic, at the time 1750 + r, by s%, we have, according to Bessel, d = 0".17926 - 0".0005320786 r, dt dL = 50".37572 - 0".000243589 T, = 230 28' 18".0 + 0".0000098423 T2 and if we put dl, da. dl, cos o dM sin co d- - n, COS ~"dt dt dtm, the formulae for the annual precession in right ascension (a) and declination (8) become da d8 -t -- m + rn tan 8 sin, - n cos a (2) dt c7t 42 657 658 THEORETICAL ASTRONOMY. and the numerical values of qn and n are, for the instant t, m 46".02824 + 0".0003086450 (t - 1750), n 20".06442 - 0".0000970204 (t - 1750). To determine the precession during the interval t'- t, we compute the annual variation for the instant - (t +- t) and this variation multiplied by t'- t furnishes the required result. B. Nutation.-The expressions for the equation of the equinoxes and for the nutation of the obliquity of the ecliptic are, according to Peters, AS - 17".2405 sin Q + 0".2073 sin 2q - 0".2041 sin 2 ( + 0'".0677 sin ( - r") - 1".2694 sin 2(0 + 0".1279 sin (Q - r) - 0".0213 sin (( + r), Ae - + 9".2231 cos - 0".0897 cos 2Q + 0".0886 cos 2( (3) + 0".5510 cos 2Q + 0".0093 cos (Q + r), for the year 1800, and AX - 17".2577 sin 2 + 0".2073 sin 22 - 0".2041 sin 2 C + 0".0677 sin (C -') - 1".2695 sin 2( + 0".1275 sin (G - r) - 0".0213 sin (Q + r), Ae _ + 9".2240 cos - 0".0896 cos 2Q + 0".0885 cos 2(C + 0".5507 cos 2 +- 0".0092 cos (Q + r), for the year 1900. In these equations Q denotes the longitude of the ascending node of the moon's orbit, referred to the mean equinox, C the true longitude of the moon, 0 the ttue longitude of the sun, F the true longitude of the sun's perigee, and r the true longitude of the moon's perigee. The values of these quantities may be derived from the solar and lunar tables, and thus the required values of A2 and As may be found. The equations give the corrections for the reduction from the mean equinox to the true equinox. To find the nutation in right ascension and in declination, if we consider only the terms of the first order, we have dS d8 x Act =dA a +-7- - d de da ds (4) ha d dA A + d8 -1 The values of A) and ap are found from the preceding equations, and for the differential coefficients we have APPENDIX. 659 da,. d_ dA cos + - sin tan a sina, dA cos sin e, da do (5) d- - cos a tan d, d = sin a. d c a ds The terms of the second order are of sensible magnitude only when the body is very near the pole, and in this case by computing the second differential coefficients the complete values may be found. In the reduction of the place of a planet or comet from the mean equinox of one date t to the true equinox of another date t', the determination of the correction for precession and of that for nutation may be effected simultaneously. Thus, let T. denote the interval t'- t expressed in parts of a year, and the sum of the corrections for precession and nutation gives Aa =- J -r- A+ cos +- (n-r +- A sin E) sinl a tan - s cos a tan S, AS = (nr - + A sill ) cos a -I- A sin a. (6) Let us now put nMr +- A cos E =f, n, -- A), sin s g sin G, (7) -As g cos G, and the equations (6) become Aa f - g sin (G + a) tan,8 AO= gcos(G+ a),) as already given in Art. 40. The astronomical ephemerides give at intervals of a few days the values of the quantitiesf, g, and G for the reduction of the place of the body from the mean equinox of the beginning of the year to the true equinox of the date; and, in order to obtain uniformity and accuracy, the beginning of the year is taken at the instant when the mean longitude of the sun is 280~. When these tables are not available, the values of f, g, and G may be found directly by means of the equations (7). The reduction from the true equinox of' t to the mean equinox of t will be obtained by changing the signs of the corrections. C. Aberration.-The aberration in the case of the planets and comets may be considered in three different modes:1. If we subtract from the observed time the interval occupied by 660 THEORETICAL ASTRONOMY. the light in passing to the earth, the result will be the time for which the true place is identical with the apparent place for the observed time. 2. If we compute the time occupied by light in traversing the distance between the body and the earth, and, by means of the rate of the variation of the geocentric spherical co-ordinates, compute the motion during this interval, we may derive the true place at the instant of observation. 3. We may consider the observed place corrected for the aberration of the fixed stars as the true place at the instant when the light was emitted, but as seen from the place of the earth at the instant of observation. The formulae for the actual aberration of the fixed stars areaA =_- 20".4451 cos (A - 0) sec -- 0".3429 cos (A - F) sec f, Af = - 20".4451 sin ( - 0 ) sin f + 0".3429 sin (- I ) sin f), (9) in the case of the longitude and latitude, and Aa - - 20".4451 (cos 0 cos e cos -+ sin 0 sin a) see 8 - 0".3429 (cos F cos e cos a +- sin F sin a) sec 8, A -= + 20".4451 cos 0 (sin a sin 8 cos e - cos 3 sin e) (10) - 20".4451 sin 0 cos a sin a + 0".3429 cos F (sin a sin 8 cos s - cos a sin e) - 0".3429 sin F cos a sin 8, in the case of the right ascension and declination.. In these formulae Fdenotes the longitude of the sun's perigee, and they give the corrections for the reduction from the true place to the apparent place. D. Intensity of Light.-If we denote by r the distance of a planet or comet from the sun, by A its distance from the, earth, and by C a constant quantity depending on the magnitude of the body and on its capacity for reflecting the light, the intensity of the light of the body as seen from the earth will be r2 J When the constant C is unknown, we may determine the relative brilliancy of the comet at different times by means of the formula r2/2 B r,2- (12) APPENDIX. 661 In the case of the planets we adopt as the unit of the intensity of light the value of I when the planet is in opposition and both it and the earth are at their mean distances from the sun. Thus we obtain C- a (a - )2 and hence ~ (a - 1)2 Let us now denote by R the ratio of the intensities of the light for two consecutive stellar magnitudes; then, if we denote by 3I the apparent stellar magnitude of the planet when I= 1, and by m the magnitude for any value of I, we shall have and hence nM- log (14) log 11 By means of photometric determinations of the relative brilliancy of the stars, it has been found that R- 2.56, and hence we derive — M - 2.45 log I, (15) by means of which the apparent stellar magnitude of a planet may be determined, I being found by means of equation (13). The value of 1M must be determined for each planet by means of observed values of m. t EXAMPLE.-The value of 3M for Eurynome is 10.4; required the apparent stellar magnitude of the planet when log a=0.38795, log r 0.2956, and log a = 9.9952. The equation (13) gives log I= 0.5129, and from (15) we derive m=- 10.4 -1.3 - 9.1. For the values log r = 0.4338, log -= 0.2357, we obtain log I= 9.7555 - 10, and m - 10.4 + 2.45 X 0.2445 - 11.0. 662 THEORETICAL ASTRONOMY. E. Numerical Calculations.-The extended numerical calculations required in many of the problems of Theoretical Astronomy, render it important that a judicious arrangement of the details should be effected. The beginner will not, in general, be able to effect such an arrangement at the outset; and it would only confuse to attempt to give any specific directions. Familiarity with the formulae to be applied, and practice in the performance of calculations of this character, will speedily suggest those various devices of arrangement by which skillful computers expedite the mechanical part of the solution. There are, however, a few general suggestions which may be of service. Thus, it will always facilitate the calculation, when several values of a variable are to be computed, to arrange it so that the values of each function involved shall appear in the same vertical or horizontal column. The course of the differences will then indicate the existence of errors which might not otherwise be discovered until the greater part if not the entire calculation has been completed; and, besides, by carrying along the several parts simultaneously the use of the logarithmic and other tables will be facilitated. Numbers which are to be frequently used may be written on slips of paper and applied wherever they may be required; and by performing the addition or subtraction of two logarithms or of two numbers from left to right (which will be effected easily and certainly after a little practice), the sum or difference to be used as the argument in the tables may be retained in the memory, and theis the required number or arc may be written down directly. The number of the decimal figures of the logarithms to be used will depend on the character of the data as well as on the accuracy sought to be obtained, and the use of approximate formule will be governed by the same considerations. Whenever the formule furnish checks or tests of the accuracy of the numerical process, they should be applied; and whenever these are not provided, the use of differences for the same purpose should not be overlooked. By proper attention to these suggestions, much time and labor will be saved. The agreement of the several proofs will beget confidence, relieve the mind from much anxiety, and thus greatly facilitate the progress of the work. THE END.