John S. Prell . 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 FORMULA FOR THE CALCULATION OF THE GEOCENTRIC AND UELIO- CENTRIC PLACES, FOR THE DETERMINATION OF THE ORBITS OF PLANETS ANi COMETS, FOB THE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COMPUTATION OF SPECIAL PERTURBATIONS; TOGETHER WITH THE THEORY Of THE COMBI- NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. Utiih Jtummral femgte and JOHN S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. BY JAMES C. WATSON DIRECTOR OF THE OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF MICHIGAN PHILADELPHIA: J. B. LIPPINCOTT COMPANY. LONDON: 36 SOUTHAMPTON STREET, COVENT GARDEN. 1900. Entered, according to Act of Congress, in the year 1868 '-v J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. Copyright, 1896, by ANNETTE H. WATSON. GIFT 555 JOHN S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. PREFACE. THE discovery of the great law of nature, the law of gravitation, by NEWTON, prepared the way for the brilliant achievements which have distinguished the history of astronomical science. A first essential, how- ever, 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 character- izes 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 BERNOTJILLI, CLAIRATJT, and D'ALEMBERT deserve the most honorable mention as having contributed also, in a high degree, to give direction to the inves- tigations 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 planet- ary 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 LAGE-O.NGE 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 success- fully performed, served constantly to educe new discoveries, so that of all the problems relating to the mutual attraction of the several planets 3 M718385 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 intro- duce 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 para- bolic 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 appli cable 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 tc be applied to these ele- ments 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 Cwlestium, a most able work, in which he gave to the world, in a finished form, the results of many years of attention PRPJFACE. 5 to the subject of which it treats. His method for .determining all the elements directly from given observed places, as given in the Theoria 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 perturba- tions of these bodies from epoch to epoch, and it was not until the com- mencement of the brilliant researches by HANSEN that serious hopes were entertained of being able to compute successfully the general per- turbations of these bodies. By devising an entirely new mode of con- sidering 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 formulse 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 formulse for the determination of the perturbations of the latitude, the mean anomaly, and the loga- rithm 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 formulse for the perturbations of the latitude, 6 PREFACE. true longitude, and radius-vector, to be integrated in the same manner, were afterwards given by BRUNNOW. 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 formulae 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 observa- tions, 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 an 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 formulae 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 beau- tiful theorems from a consideration of the relations of some of these auxiliaries. For convenience, the formulae 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 atten tion to these problems, I have consulted the Astronomische Nachrichten, the Astronomical Journal, and other astronomical periodicals, in which 'is to be found much valuable information resulting from the experi- ence of those w r ho have been or are now actively engaged in astro- nomical pursuits. I must also express my obligations to the publishers, PREFACE. 7 Messrs. J. B. LIPPINCOTI & 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, of Vienna, 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 Correspondent, vol. xii., the tables for which were subsequently given by BRUNNOW 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 formulae and the other formulae for correcting approximate elements are given in a form convenient for application, and the formulas 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 treat- ing 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 fol lows 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 observa- tions, 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 recom- puted 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 intern; p- tions, and with other labors constantly pressing me, by which the progress of its publication has been somewhat delayed, even since the stereo- typing 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 de- velopment of the formulae, 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 hea- vens, the contemplation of which must ever serve to impress upon the mind the reality of the perfection of the OMNIPOTENT, the LIVING GOD ! OBSEBVATOKY, ANN ARBOR, June, 1867. CONTENTS. THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OP THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR- MULA FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COM- PUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. PAOB 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 Eight Ascension and Declination 90 .Reduction of the Elements from one Epoch to another 99 Numerical Examples 103 Interpolation < 112 Time of Opposition 114 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. PAGB Variation of the Eight 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 FORMULA 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 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. ll PAOK Fundamental Equations 225 Formulae 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 Posjtion indicated by the Curvature of the Observed Path of the Body 242 Formulae for a Second Approximation 243 Formulas for finding the Katio 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 Katio of the Sector to the Triangle 279 CHAPTER V. DETERMINATION OF THE ORBIT OP A HEAVENLY BODY FROM FOUR OBSERVA- TIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE. Fundamental Equations 282 Determination of the Curtate Distances , 289 Successive Approximations 293 Determination of the Elements of the Orbit 294 Numerical Example 294 Method for the Final Approximation 307 CHAPTER VI. INVESTIGATION OF VARIOUS FORMULAE FOR THE CORRECTION OF THE APPROXI- MATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY. Determination of the Elements of a Circular Orbit 311 Variation of Two Geocentric Distances 313 Differential Formulae 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 Velocities 332 Correction of the Ephemeris 335 Final Correction of the Elements 338 Kelation 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 Eatio of Two Curtate Distances 349 12 CONTENTS. PAQB Variation of the Geocentric Distance and of the Reciprocal of the Semi-Trans- verse Axis 352 Equations of Condition 353 Orbit of a Comet 355 Variation of Two Eadii-Vectores 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OP 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 Quanti- ties 386 Separate Determination of the Unknown Quantities and of their Weights 392 Eelation 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 418 Case of very Eccentric Orbits 423 CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULA 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 PAQB Mechanical Quadrature 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 2...... 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 Formulae 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 Keduction of the Elements to the Common Centre of Gravity of the Sun and Planet 538 Keduction 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 o* the Ether.... 5*6 14 CONTENTS. TABLES. PAOl I. Angle of the Vertical and Logarithm of the Earth's Radius ~ 561 H. 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 IX. 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 El- liptic and Hyperbolic Orbits 618 XI. For the Motion in a Parabolic Orbit 619 XII. For the Limits of the Boots of the Equation sin (z 1 )=m sin 4 y! ... 622 XIII. For finding the Eatio of the Sector to the Triangle 624 XIV. For finding the Eatio of the Sector to the Triangle 629 XV. For Elliptic Orbits of Great Eccentricity 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 OP THE TABLES 651 APPENDIX. Precession 657 Nutation 658 Aberration 659 Intensity of Light 660 Numerical Calculations .. 662 THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR- MULA FOB DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTA- TION FOR CASES OF ANY ECCENTRICITY WHATEVER. 1. THE study of the motions of the heavenly bodies does not re- quire 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 aggre- gate, a definite body whose relation to other bodies of the system it is required to investigate. On the contrary, in considering the ope- ration 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 mag- nitude 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. Ex- perience 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 16 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 com- parable 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 time, 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 repre- sented 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 ele- ment 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 accele- rated; and if we designate the acceleration due to the force by/, the unit of/ being the velocity generated in a unit of time, we shall have If, however, the force be variable, we shall have, at any instant, the relation efc / ~~ dt' the force being regarded as constant in its action during the element of time dt. The instantaneous value of v gives, by differentiation, dv _ d*s ~dt~~W and hence we derive so that, in varied motion, the acceleration due to the force is mea- sured 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 mat- ter. 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 action 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 Jf, 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 re- action ; 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 18 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 "7- , and the entire motive force F is expressed by M being the sum of all the elements, or the mass of the body. Since _ds_ this gives 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 d?s 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 uni- versality 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 them- selves, and of their aggregation. ATTRACTION OF SPHERES. 19 If we consider two bodies the masses of which are m and m', and whose magnitudes are so small, relatively to their mutual distance />, that we may regard them as material points, according to the law of gravitation, the action of m on each molecule or unit of m' will be , and the total force on m! will be m'. />' The action of m f on each molecule of m will be expressed by , and its total action by m' -r- The absolute or moving force with which the masses m and m f 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 exceed- ing that of the greater, and in the ratio of the masses moved. The expression for the velocity of m', 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 m at the distance p. For the acceleration due to the action of m r we derive, in a similar manner, 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, de- termine 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 dm 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 sup- pose 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 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

*, with respect to a, gives dp a r sin

dr d P Thia must be integrated between the limits tp = -f- fa an( ^ ^ = 22 THEORETICAL ASTRONOMY. but since p is a function of , we have 7 P 7 r cos

*. a J Integrating, finally, between the limits r and r = r,, we get r, being the radius of the sphere, and, if we denote its entire mass by m, this becomes F=-. a' Therefore, dV m da a? 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 and, denoting by m the mass of a spherical shell whose radii are r" and r f , this becomes 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. general equation; but, in the integration with reference to />, the limits will be p = r + a, and p = r a, which give V= and this being independent of a, we have A- - d -?=0 ^i- - 7 - \Jm 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 it relates to their action upon each other, is extremely small, and is in fact com- pensated 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 of 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 coLsideration, however, of the varia- tions 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 equi- librium. We are thus enabled to form at once the equations for the motion of a system of bodies. Let m, m', m", &c. be the masses of the several bodies of the system, and x, y, z, x' t y', z f , &c. their co- ordinates referred to any system of rectangular axes. Further, let the components of the total force acting upon a unit of the mass of m, or of the accelerating force, resolved in directions parallel to the co-ordinate axes, be denoted by X, F, and Z, respectively, then will mX, m Y, mZ, be the forces which act upon the body in the same directions. The velocities of the body m at any instant, in directions parallel to the co-ordinate axes, will be \ dx dy dz ~di' W dt' and the corresponding forces are dx dy dz m ~j7> m ~jl* m ~jT' dt dt dt By virtue of the action of the accelerating force, these forces for the next instant become *j + mXdt, .m+mYdt, m j + mZdt > which may be written respectively: MOTION OF A SYSTEM OF BODIES. 25 dz , dz .. dz The actual velocities for this instant are dx , dx dy dy dz dz and the corresponding forces are dx , dx dy dy dz . . dz - - 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 -j- -|- mXdt, -md^ + mYdt, (3) md-j--{-mZdt. In the same manner we find for the forces which will be destroyed in the case of the body m' : -m'd + m'X'dt, at 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 equi- librium for a system of points of invariable but arbitrary form, and subject to the action of forces directed in any manner whatever, are ZX, = 0, ZY, = 0, ZZ, = 0, S ( T> - X,y) = 0, Z (X t z - Z,x) = 0, Z (Z,y - I = , which X,, Y h Z,, denote the components, resolved parallel to the n 26 THEORETICAL ASTRONOMY. co-ordinate axes, of the forces acting on any point, and r, 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 de- stroyed in the case of a system of bodies, we shall have 2m r-5 2mX = 0, aP (4) 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 2mx 2my 2mz x, = ^ ) y. = ^ t z. = ^ > 2m 2m 2m we get d* 2m dP 2m Introducing these values into the first three of equations (4), they become d*x f 2mX d*y, 2m Y d*z f 2mZ t . "dP = '' ~^ y ~dP ~ ' ~2m' ~dP ~ ~2m ' 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 com- posed, 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 : = m'X', MOTION OF A SYSTEM OF BODIES. 27 and similarly for ^any two bodies ; and, consequently, so that equations (5) become Integrating these once, and denoting the constants of integration by c, c', c", we find, by combining the results, 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 co- ordinates. The action of m' on ra, according to the law of gravita- tion, is expressed by , in which p denotes the distance of m from m'. 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 m'(z'-z) ~~ Further, for the components of the accelerating force of m on m', we have m(x x') , m(y y f ) , m(g /) ""?"' ~7 ~7~ Hence we derive m ( Yx Xy} + m' ( TV Xy ) = 0, and generally 28 THEORETICAL ASTRONOMY. In a similar manner, we find Zm (Xi ZOD) = 0, (7) Im (Zq F) = 0. These relations will 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, %y=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 d*z ^ d*y the integration of which gives 2m {xdy ydx) = cat, Im (zdx xdz) = c'dt, (8; 2m (ydz zdy] = c"dt, c, c', and G" being the constants of integration. Now, xdy ydx is double the area described about the origin of co-ordinates by the pro- jection of the radius- vector, or line joining m with the origin of co-ordi- nates, 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 double 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 r , G fr , 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 pro- portional 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 the 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 , /9, and 7- be the angles which this principal plane makes with the 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 cos a dt, S cos p dt, S cos Y dt. Therefore, by means of equations (8), we have c = S cos a, c f S cos /?, c" = S cos p, and, since cos 2 a -f- cos 2 /? + cos 2 f = 1, 2 = c 2 -f c' 2 -f c" 2 . Hence we derive c' COS a = 1/V + c' 2 + c" 2 1/c 2 + c' 2 -j- c" 2 c" cos y = - These angles, being therefore constant and independent of the time, show that this principal plane of projection remains constantly par- allel 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 THEORETICAL ASTRONOMY. hence the values of , /?, and ?-, 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 /9 = and cos p = 0, and, therefore, c' 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 corre- sponding mass, is zero. It may also be observed that the value of 8 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 equation, 2 = c 2 -f c' 2 -f 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 n z, be the co- ordinates of the movable origin of co-ordinates, referred to a fixed point in space taken as the origin ; and let X Q , y , z , x f , yj, z/, &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 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 The condition of uniform rectilinear motion of the movable origin gives MOTION OF A SOLID BODY. 31 and the preceding equations become 0, (9) ZmZ=Q. Substituting the same values in the last three of equations (4), ob- serving that the co-ordinates x t , y,, z, are the same for all the bodies of the system, and reducing the resulting equations by means of equations (9), we get (/7 2 r f]i~ \ *. ^jr ~ *~jjr } - a* (A - 2O = 0, (10) - r =0. 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 c?m, the equations (5) for the motion of the centre of gravity of the body become -=zdm, (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 x , y Q) z w we have, by means of the equations (10), THEORETICAL ASTRONOMY. &X R d*y d\ \ . /-,__ _ N , . v ^ ~dl y ~di? } ~~J ' ~~ y *) ~ ' d' 2 x n d*z \ r z -^-~ x o-^-JdmJ(Xz Zx )dm = 0, d\\, Crr -%-$- )dm-J(Zy ~Yz )dm =0, dt 2 the integration with respect to dm 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. Equa- tions (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 0f the elements of its mass and forces directed to the origin of co- ordinates, 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. Wo 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 svstem of bodies on a MOTION OF A SOLID BODY. 33 distant mass, which we will denote by M. Let x 0) y w z w x f , 2/ ', z Q ', &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 lt 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 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 M y 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 and, therefore, that of the entire system on the element of M t resolved ; n the same direction, will be v mx r 5 "' We have also r* = (x, + x Q Y + (y, + 2/ ) 2 + 0, + *)', and, if we denote by r t the distance of the centre of gravity of the system from Jf, r, = *, + ?, + *,*, Therefore | = O, + *o) (r? + 2 (x, x + y, y + z, $ + r 2 ) 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 Jf, that terms of the order r 2 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 r 2 , we shall have _ _/ , ^o _ 3s, (Xa?o + y/ r 3 3 ' , 3 r 5 r, r f r t and -mx -m . 2mx n 3rc. , r == x , - + 3 -i (x,Smx Q + y,Zmy Q T r, r f i, I 34 THEORETICAL ASTRONOMY. But, since # , y , z , are the co-ordinates in reference to the centre of gravity of the system as the origin, we have 2mx = 0, Zmy Q = 0, 2wz == 0, and the preceding equation reduces to v w# 2m s -?=*'^f- In a similar manner, we find v my Im mz Im V =J V r 7 = V 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 Mj 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 m, m', 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 M 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 mag- nitude 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 diiferent 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, and 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 correspond- ing 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 COL ^deration 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 rota- tion 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 rela- tive 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 equa- tion. For the unit of space we may arbitrarily take the mean dis- tance 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 force - will be such that the units of both cannot be arbitrarily chosen. Thus, if we denote by / the acceleration due to the action of the mass m on a material point at the distance a, and by/' the accelera- tion corresponding to another mass m' acting at the same distance, we have the relation MOTION EELATIVE 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 of/, Integrating this, regarding /as constant, and the point to move from a state of rest, we get = i/P. (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, /=2; 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 dis- tance in the unit of time. Let the unit of time be a mean solar day; F the acceleration due to the force exerted by the mass of the sun at the unit of distance; and /the acceleration corresponding to the distance r; then will and k 2 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 k 2 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 #*; 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 unu of time, the space expressed by ft?. The acceleration due to the action of the sun at the unit of distance is designated by F, since the square root of this quantity appears frequently in the formulae 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), 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 *V=1; and hence 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 m 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 F, and that of the planet or comet by W& 2 . For a second body let the co-ordinates be x f , y f , z f ; the distance from the sun, r'; and the mass, ra'F; 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 , 77, , the co-ordinate planes being parallel to those of x, y, and z, respectively; then will the MOTION RELATIVE TO THE SUN. 39 mk* acceleration due to the action of m on the sun be expressed by . and the three components of this force in directions parallel to the co-ordinate axes, respectively, will be , , r 3 r 3 r 3 The action of ra' on the sun will be expressed by and hence the acceleration due to the combined and simultaneous action of the several bodies of the system on the sun. resolved par- allel to the co-ordinate axes, will be The motion of the centre of gravity of the sun, relative to the fixed origin, will, therefore, be determined by the equations = = , = (14) Let p denote the distance of m from m f ; p r its distance from w", adding an accent for each successive body considered; then will the action of the bodies m', m", &c. on m be of which the three components parallel to the co-ordinate axes, re- spectively, are The action of the sun on m, resolved in the same manner, is expressed by 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 othei bodies of the system on m are, therefore, 40 THEOKETICAL ASTRONOMY. Vx , wm'tf x) "T 5 " ~7 ' k *y ; X ^ + y, c + 2, the equations which determine the absolute motion are eP . (?a? ^ __ m' (20) h being an arbitrary constant. 44 THEORETICAL ASTRONOMY. If we add together the squares of the expressions for c, c', and c" and put c 2 + c' 2 + c" 2 = 4/ 2 , we shall have (x* -f f + * 2 ) (cfe* + dy* + <** 2 ) 0*k + ydy -f dz 2 + cos I being an arbitrary constant. Therefore we shall have I 1 1 = cos (v w), e \ r / from which we derive r ^^^ e cos (1; which is the polar equation of a conic section, the pole being at the focus, p being the semi -para meter, e the eccentricity, and v o the angle at the focus between the radius-vector and a fixed line, in the plane of the orbit, making the angle co with the semi-transverse axis a. If the angle v co is counted from the perihelion, we have a) = 0, and (25) P COS V The angle v is called the true anomaly. Hence we conclude that the orbit of a heavenly body revolving around the sun is a conic section with 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 cir- cumstance 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 4f already found, we obtain dt = - m j/aV (a r) 2 ' which may be written dt = or ^-(V)/ the integration of which gives J ae (26) In the perihelion, r = a (1 e), and the integral reduces to ' = C/ therefore, if we denote the time from the perihelion by t Q , we shall have o / -i/a r\ /~ (a r\*\ 7 / ^r == cos - ) e \l -- jfe|/l-f-w\ \ ae j \ \ ae ] ] (27) In the aphelion, r = a (1 -f- e) ; and therefore we shall have, for the time in which the body passes from the perihelion to the aphelion, t, = Jr, or 1 T- " r 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 dis- tance from the sun, and n the semi-circumference of a circle whose radius is unity. Therefore we shall have (28) MOTION RELATIVE TO THE SUN. 47 For a second planet, we shall have and, consequently, between the mean distances and periodic times of any two planets, we have the relation fft^ = - (29) If the masses of the two planets m and m r are very nearly the same, we may take 1 -f- m = 1 -f- m r 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 rcab is the area of the ellipse, a and b representing the semi-axes, we shall have =f= areal velocity; and, since b 2 = a 2 (1 e 2 ), we have T which becomes, by substituting the value of r already found, In like manner, for a second planet, we have and, if the masses are such that we may take 1 -f- m sensibly equal to 1 -f m', 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 con- stants 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 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 & 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 -\- c"x = 0, are, respectively, double the projections, on the co-ordinate planes, xy, xz, and yz, of the areal velocity// and hence we shall have e = 2/cosi. The projection of 2f on a plane passing through the intersection ot the plane of the orbit with the plane of XT/, and perpendicular to the latter, is 2/sini; and the projection of this on the plane of xz y to which it is inclined at an angle equal to &, gives c' = 2/ sin i cos 1. Its projection on the plane of yz gives Hence we derive z cos i y sin i cos Q> -f- x sin i sin = 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, dz dx , 73 ; T . . ,,, '-j z -j- = Jc i/p (1 + *) cos & sm l > (.&%) at at __ = cvpLm sn Cvv Ctv These equations will enable us to determine & , 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, > and i, to the position of its plane in space. If we measure the angle v a) from the point in which the orbit intersects the plane of xy, the con- stant co 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. 4S 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 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 = oo*, and hence 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 formulae necessary to determine unknown elements by means of observed positions, we will investi- gate those which are necessary for the determination of the helio- centric 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), 50 THEORETICAL ASTRONOMY. In the case of the earth, a = 1, and therefore *= fl/1 4- m 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 correr spending 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 cor- rection is slightly to increase the observed value of r; but to deter- mine 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 arbi- trarily chosen, the accuracy of the results is not thereby affected. The value of r from which the adopted value of k has been com- puted, is 365.2563835 mean solar days; and the value of the com- bined mass of the earth and moon is ~~ 354710* Hence we have log V = 2.5625978148; log j/1 + m = 0.0000006122; log 2x = 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. o The quantity expresses the mean angular motion of a planet in a mean solar day, and is usually designated by //. "We shall, therefore, have MOTION RELATIVE TO THE SUN. 51 ^, '" ' | C33) for the expression for the mean daily motion of a planet. Since, in the case of the earth, 1/1 -f- 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 ^ = 4 (M) a For the old planets whose masses are considerable, the rigorous ex- pression (33) must be used. 19. Let us now resume the polar equation of the ellipse, the pole being at the focus, which is 1 -f e cos v If we represent by

= e; and, since a(l e 2 ) is half the parameter of the transverse axis, which we have designated by p, we have . r 1 -}- sin (f> cos v The angle

) tan |v. (45) Again. V\-\- e = 1/1 -f sin y = 1/1 -|- 2 sin p cos fop, which may be written 1/1 -}- e I/sin 2 -|^ -f c s 2 i^ + 2 sin Jp cos Jp, or _ 1/1 -|- e = sin ^ -f cos ? In a similar manner we find 1/1 e = sin \

, and 6 = a cos ^, hence r sin v cos ^ sin v . _, sin E = 1 -f- e GOBV Equation (41) gives r cos v + ae p cos v or _, ^) cos v + ae ag2 cos v and, putting a cos 2 (p instead of _p, and sin

(sin 8 Jf(l e cos IT) ) dM v Expanding these terms, and performing the operations indicated, we get ^ = 1 + 2e cos M + - (6 xjos 2 M 4 sin 2 Jf) -f | (16 cos 8 M 36 sin 2 Jf cos Jf) -f- . . . , PLACE IN THE ORBIT. 57 which reduces to , (51) i Equation (22) gives . 2fdt = and, since /= }&|/p(l + m), we have dv = ^j- dt, (52) or ___ , &V 1 -f- Wl Cl 2 / But g = fa and therefore a? 2 2 CX By expanding the factor j/1 e 2 , we obtain and hence Substituting for its value from equation (51), and integrating, we get, since v = when M=0, vM=2e sin M +je 2 sin2Jf + -^- (13 sin 33f 3 sin Jf ) +. . . (53) which is the expression for the equation of the centre to terms involving e 3 . 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 THEOEETICAL 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 obtain- ing 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 celes- tial 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] = M, we shall have, performing the operations indicated and reducing, E = M -f e sin M + ^ sin 2 M + &c. (54) Let us now denote the approximate value of E computed from this equation by E ot then will in which &E Q is the correction to be applied to the assumed value of E. Substituting this in equation (39), we get M= E -f &E Q e sin E Q e cos E *E ; and, denoting by M Q the value of M corresponding to E w we shall also have M Q = E Q esmE . Subtracting this equation from the preceding one, we obtain M-M, - -- = = A A.. 1 e cos -c/ It remains, therefore, only to add the value of &E Q found from this formula to the first assumed value of E, or to E w 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 E may be ob- tained 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 M for any instant of time, we may readily deduce from it, by the formulse 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 CEBIT. 59 computing r and v, for any instant, is somewhat different. The manner of proceeding in the computation in such cases we shall con- sider 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 = , 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, m and equation (52), putting for p its value 2g, becomes k V2q dt = r*dv, or which may be written kdt 3-^ = 2 (1 -f- tan 2 J-y) sec 2 l>vdv = (!-}- tan 2 Jv) d tan 0. 1/2 g 1 Integrating this expression between the limits T and t, we obtain ^ = tan ^v + I tan 8 -^ v, (55) 1/2 g f 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 r the time of describing the arc of a parabola corresponding to v = 90 ; then we shall have 2 or T/2- 1 " 3 ' 3k __ 4 f ~ Now, ^ is constant, and its logarithm is 8.5621876983; and if we take q = l, 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 days = 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 from 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 it 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 \v being the unknown quantity. If we put x = tan Ji?, this equation becomes ar + 3z 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 correspond- ing to the anomaly v in any parabola, whose perihelion distance is q, by multiplying by q 2 , 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 (f y 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 gene- rally known as Barker's Table. The following will explain its con- struction : Multiplying equation (55) by 75, we obtain (t T) = 75 tan Jv + 25 tan 8 >. M=75 tan v -f 25 tan* |v, 1/2 < Let us now put PLACE IN THE ORBIT. 61 75k and C = - =r, which is a constant quantity ; then will v 2 The value of C is log <7 = 9.9601277069. Again. let us take -=? which is called the mean daily motion in the parabola; then will M = m (t T) == 75 tan v + 25 If we now compute the values of M corresponding to successive values of v from v = 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 neces- sary 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 If for values of v from to 180, 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 M 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 manner: Equation (55) may be written 1/2 q (1+3 cot 2 and, multiplying and dividing the second member by (1 + cot 2 Jt>)*, we shall ha\e F) ,. .,, ' _ = i tan- 4, (1 + cot' 62 THEORETICAL ASTRONOMY. 2 But 1 + cot 2 Jt; = - - and consequently sin v tan v 1/2 5* ~3sin 8 v (1 + cot 2 ^) 3 ' Now, when t? approaches near to 180, cot^v 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 + A , A O being a very small quantity. We shall therefore have and, putting tan %w = 0, and tan \ A O = x, we get, from this equation, """ 6 s 1 6x """ (1 dx} y Multiplying this through by # 3 (1 0#) 3 , expanding and reducing, there results the following equation : + 0* (2 + W + W + Dividing through by the coefficient of x, we obtain 1 + 30 _ 2 . 2 (2 + 60* + 04 1 /5fi N v "U/ o /- i A aZ T Let us now put 1 + 30* then, substituting this in the preceding equation, inverting the series and reducing, we obtain finally But tan ^A O = x, therefore PLACE IN THE ORBIT. 63 Substituting in this the value of x above found, and reducing, we obtain 2 -f 320* + 160 8 + 100 8 , , &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, Table VII. gives the values of A O , 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 vaiue 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 [206 -v-jT since we have already found or Having computed the value of w from this equation, Table VII. will furnish the corresponding value of A ; and then we shall have, for the correct value of the true anomaly, v w -f- A O , 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 A. 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 t-T- J? V J- - yy * GO Bin* to 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 ^- = 3v from which we obtain, since q = r cos 2 |v, ~2^~ \ 1/2 / ~ \ 1/2 / * Let us now put sin^v sm x = 7==, 1/2 and we have - - - = 3 sin x 4 sin's = sin 3#. 2r* Consequently, which admits of an accurate and convenient numerical solution. To facilitate the calculation we put ~- _ sin 3x J\ sin v the values of which may be tabulated with the argument v. "When v 0, we shall have N= fl/2, and when v = 90, we have N= 1 ; from which it appears that the value of ^V changes slowly for values of v from to 90. But when v = 180, we shall have N= * and hence, when v exceeds 90, it becomes necessary to introduce au auxiliarv different from N. We shall, therefore, put in this case, W = JV sin v = sin 3#; PLACE IN THE ORBIT. 65 from which it appears that N f = 1 when v = 90, and that JV' = Jv'iB when v = 180. Therefore we have, finally, when v is less than 90, 2 f t T=- 7 c 1 -Nr sinv, OK and, when v is greater than 90, o in which log = 1.5883272995, from which t T is easily derived oA/ when v is known. Table VIII. gives the values of N, with diiferences for interpola- tion, for values of v from v = to v = 90, and the values of N r 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 nega- tive 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 T = 2 cos J- (v + 4) cos J (v 4)' When v 0, the factors cos | fa + ^ an ^ cos(v ^) in the de- nominator will be equal ; and since the limits of the values of v are 180 4> 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, conse- quenlly, we have, in this case, p ==a (e 2 1), or a p cot* 4. We have, also, for the perihelion distance, q = a(e 1). Let us now put tan IF = tan ' (56) 66 THEOKETICAL ASTRONOMY. which is analogous to the formula for the eccentric anomaly E in an ellipse: and. since e= - , we shall have cos 4* _ e+l~l + COS+~ and, consequently, tan \F = tan ^v tan ^. (57) We shall now introduce an auxiliary quantity 2tan|^ , ,, , Since tan F= TTTr' we shall nave 1 tan' %F a l\ i 1 \ (60) - 1^=1] Wi/ Squaring this equation, adding 1 to both members, and reducing we obtain cos Replacing cosF ~~ 2cosi(^H- *) cos i (v 4) "~ 2 cos (v -f- 4) cos (v which reduces to 1 ~. f ~ I ^/vc,A (62) PLACE IN THE ORBIT. 67 If we add ip 1 to both members of this equation, we shall havp coaF p Taking first the upper sign, and then the lower sign, and reducing, we get VcosF 1/r cos %> = _--) cog , ^ (63) VcosF 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, ten v = which is identical with the equation (56), and may be employed to verify the computation of r and v. Multiplying the last of equations (63) by the first, putting for 1 its value tan 2 ij/, and reducing, we obtain r sin v = a tan 4/ tan F=a tan 4 1 ' + X) + &c - In the case of the parabola, e = 1 and i 0, and this equation becomes identical with (55). Let us now put 72 THEORETICAL ASTRONOMY. and also then the angle V will 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 com- puted from 75K^ _T) \T2 - 3 * \ 9 1/2 f Let U be expanded into a series of the form which is evidently admissible, a, /?, f y ____ being functions of u and independent of i. It remains now to determine the values of the coefficients a, /?, f, &c., and, in doing so, it will only be necessary to consider terms of the third order, or those involving i 3 , since, for nearly all of those cases in which the eccentricity is such that terms of the order i 4 will sensibly affect the result, the general formulae already derived, with the ordinary means of solution, will give the required accuracy. We shall, therefore, have U + I U 3 = U 4- ai + & 4 rP 4 l(u 4 ai 4- & + rW, or, again neglecting terms of the order i*, But we have already found, (70), k(tT)i/l + e = n + ^ = u + _ 2 . u , + ^ 2g f + 3? (iw 5 + > 7 ) 4i 3 (> 7 + > 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 ua * + (i _|_ w .) p = 4. 3 (>< -f 4r T ), 3 4- 2wa/9 4- (1 + w 2 ) r = 4 (> 7 4- ^U 9 ). From the first of these equations we find PLACE IN THE CEBIT. 73 The second equation gives 3(K + ju*)-n. l + u or, substituting for a its value just found, and reducing, g = 3 0* + f-H" 7 + jft * + jftu") (1 + **) We have also _ 4 (i^ 7 + X) j 3 2a^ti r " 1 + u 2 and hence, substituting the values of oc and ft already found, and reducing, we obtain finally (I -I- < Again, we have tan U tan (u { oil Developing this, and neglecting terms of the order i 4 , we get tan^U^f Now, since w = tan Ji? and U= tan \ F, we shall have or 2a . ,, (72) Substituting in this equation the values of a, ft, and ? already found, and reducing, we obtain finally (!+')' 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 F, we enter Table VI. with the argument F and take out the corresponding value of M; and l,hen we derive t T from t _ T= M< in which log C = 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 F in a series of ascending powers of i, in which the coefficients are functions of U. Let us, therefore, put u = U 4- o'i -f /3'i 2 -f r '$ + & c . Substituting this value of u in equation (70), and neglecting terms multiplied by i 4 and higher powers of i, we get _ 4 [73^2 _ 2 Ua' 2 4 U 7 1 17') t". But, since the first member of this equation is equal to U + J Z7 S , we shall have, by the principle of indeterminate coefficients, From these equations, we find s _ Iff W + ill! u 9 + VftV ff" + f If u 13 + fill + i-VV (i + tf 2 ) 6 If we interchange v and F in equation (72), it becomes, writing a ; , 0', f' for a, /9, ^, _ T7 , * PLACE IN THE ORBIT. 75 2/3' 2a' 2 i7 + /_^ ___ 4a^^7_ 2(t7^j) \ Ml + tf 2 (1 + tf 2 ) 2 (1 + tf 2 ) 3 / Substituting in this equation the above values of cc/, /?', and p', and reducing, we obtain, finally, _ F[ . , 22 . If I ^ T + JIM ^ 9 + f If f I ^ n + If If ^ 18 2 by means of which v may be determined, the angle V being taken from Table VI., so as to correspond with the value of M derived from 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, conse- quently, i is negative. 28. In order to render these formulae convenient in practice, tables may be constructed in the following manner: Let x = v or F, and tan Ja? = 6, and let us put <0*+|0s 100(1 -f- 2 ) 2 ' ^ 10000 (1 ii 10000 (1 + 2 ) 4 1000000(1 + 2 ) 6 + iiii^ 9 + flff^ 11 + If ^ 13 + jiff*" + 1000000 (1 + ^ 2 ) 6 wherein s expresses the number of seconds corresponding to the length of arc equal to the radius of a circle, or log s== 5.31442513. We shall, therefore, have : v=V+A (1000 + 5(10017+ 76 THEORETICAL ASTRONOMY. and, when x = v, V=v-A (lOOi) + & (lOOi) 2 - C' (lOOi) 8 . Table IX. gives the values of A, B, B', <7, and C' for consecu- tive values of x from x = to x = 149, with differences for inter- polation. When the value of v has been found, that of r may be derived from the formula r= g(l-M) ^ 1 -j- e cos v Similar expressions arranged in reference to the ascending powers of (1 e) or of I I ^ - 1 II may be derived, but they do not con- 2 \^ \ I 1 I is 1 -f~ & ' i than i, yet the coefficients are, in each case, so much greater than 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 impos- sible. 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 de- termine 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 = E e sin E t PLACE IN THE ORBIT. 77 which, since q = a(l e), may be written If we put we shall have ^e 201/1 \ 1 l + 9e " * 2 Let us now put 9E -f sin .E 201/2 and then we have When jB is known, the value of w may, according to this equation, be derived directly from Table VI. with the argument 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 and if we put tan 2 \E = r, we get Sn = == * -^ r We have, also, E = 2 tan" 1 r^= 2r^ (1 JT + Jt - ^r 8 + Ac.)- 78 THEOKETICAL ASTRONOMY. Therefore, 15 (E sin E) = 2t* (10r and 9E + sin E = 2r^ (10 ^ T + y r y r 8 + if i* &c.). Hence, by division, and, inverting this series, we get which converges rapidly, and from which the value of may be found. Let us now put A_ l_ r~0 2 ' then the values of O may be tabulated with the argument A; and, besides, it is evident that as long as A is small C 2 will not differ much from 1 -f- \A. Next, to find B, we have - 40.), and hence _ 62 . 9 ~~~ "" T ^ ~ 2525^+ 33eST6 T from which we easily find S = 1 + T | s ^ 2 + ,|^ s + riW,^ 4 + &c. If we compare equations (44) and (56), we get tan E = T tan J.P. Hence, in the case of a hyperbolic orbit, if we put tan 2 |.F r', we must write r' in place of r in the formulae 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 CEBIT. 79 Hyperbolic motion from those for elliptic motion by writing A', B f t and C f in place of A, B, and C, respectively, we shall have " - Ac. Table X. contains the values of log B and log C for the ellipse and the hyperbola, with the argument A, from A = 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 Hi Q tan |v = -v/ T 5 -- tan \E gives JL. & or, substituting the value of A in terms of w, tan^=(7tan>^^l^. (76) The last of equations (43) gives ~l+tan f j-tf Hence we derive 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 For the radius-vector in a hyperbolic orbit, we find, by means of the last of equations (63), T== (l When t T is given and r and v are required, we first assume B = 1 , and enter Table VI. with the argument i 1 B 80 THEORETICAL ASTRONOMY. in which log C Q = 9.96012771, and take out the corresponding value of w. Then we derive A from the equation 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 JB, and then, using this in the expression for My 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 T from in the case of the ellipse, or from in the case of the hyperbola. Then, with the value of r as argu- ment, we enter the second part of Table X. and take out an approxi- mate value of A, and, with this as argument, we find logJ? and log C. The equation 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 5(1 -H)' and, with w as argument, we derive M from Table VI. Finally, we have (80) 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 eccen- tricity and semi-transverse axis of the orbit, and the mean anomalv at a given epoch, or, what is equivalent, the time of passing thft 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 formula 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, & 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 t5, and which is measured, as in the case of & ? 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. 82 THEORETICAL ASTRONOMY. The I'jngitude of the perihelion, which is denoted by x, 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 dimi- nished 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 to 180, since in this case one set of formulae 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 per- petuated 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 ex- pression 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 formulae 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 s we have x = r cos v, 3 = 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 -f w), y = r sin (v -f- w), to being the arc of the orbit intercepted between the place of the perihelion and of the node, or the angular distance of the perihelion from the node. Ndw, we have co = 7r ft in the case of direct motion, and a) - ft TT in the case of retrograde motion ; and hence the last equations become x = r cos (v TT qp ft) y = r sin (v db ?r qp ft) the upper and lower signs being taken, respectively, according as the motion is direct or retrograde. The arc v7rq=ft=wis called the argument 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 #, in the line of nodes. Then we shall have x' = r cos u, y f = db r sin u cos i, z' r sin u 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 cos b cos (I ft ), y f = r cos b sin (I ft ), / = r sin b, and, consequently, cos u = cos b cos (I &), sin u cos i = cos b sin (7 ft), (81) sin w sin i = sin 6. From these we derive tan (I ft ) tan w cos i, tan 6 = db tan i sin (f ft ), (82) which serve to determine I and 6, when ft, w, and t are given. Since 84 THEORETICAL ASTRONOMY. cos b is always positive, it follows that I & 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, I ft and 360 u will belong to the same quadrant. Hence the ambiguity which the determination of / ft by means of its tangent involves, is wholly avoided. If we use the distinction of retrograde motion, and consider i always less than 90, I ft and u will lie in the same quadrant. 32. By multiplying the first of the equations (81) by sin u, and the second by cos u, and combining the results, considering only the upper sign, we derive cos b sin (u I -f- ft ) = 2 sin u cos u sin' ^*, or cos b sin (u I + ft ) = sin 2u sin 8 \i. In a similar manner, we find cos b cos (u l-\- ft ) = cos'it -J- sin 2 w cos i, which may be written cos b cos (u 1-\- ft ) = \ (1 -f- cos 2-w) + J (1 cos 2w) cos i, or cos b cos (u l-\- ft ) = (1 + cost) -j- i (1 cos i) cos 2u; and hence cos b cos (M / -f~ ft ) = cos 7 i -f- sin* Ji cos 2u. If we divide this equation by the value of cos b sin (u I + ft ) already found, we shall have f 7 * ^\ tan*J*sin2ti , QO , tan (u I + ft) = r L 2 , . . JT-. (83) 1 -f- tan 2 i cos 2it The angle u 1+ ft 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. a; tan v = j : 1 -f- n cos x Differentiating this, regarding y and n as variables, and reducing, we find dy sin a; ~dn 1 -J- 2n cos x -f- ^ POSITION IN SPACE. 85 which gives, by division, or by the method of indeterminate coefficients, -~ = sin x n sin 2x + n 2 sin 3# n 3 sin 4# -4- &c. dn Integrating this expression, we get, since y = when x = 0, y = n sin x \r? sin 2# -f- |n 3 sin 3# \n^ sin 4# -f- ____ , (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 = tan 2 Jt and x = 2u; and hence we obtain u I -j- & = tan 2 \i sin 2u ^ tan 4 i sin 4^ -f~ i tan 6 Ji sin Qu &c. ' (85) When the value of i does not exceed 10 or 12, the first two terms of this development will be sufficient. To express u I -f- & 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 jR e the reduction to the ecliptic, we shall have But we have v = M -f the equation of the centre ; hence 1 = M -f TT -f- equation of the centre reduction to the ecliptic, and, putting L = M -f- TT = mean longitude, we get I L -f- 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 geo- centric 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 I, 2 =r sin b. Again, let X y F, Z be the co-ordinates of the centre of the sun re- ferred to the centre of the earth, the plane of XY being in the eclip- tic, and the axis of X being directed to the vernal equinox ; and let denote the geocentric longitude of the sun, R its distance from the earth, and I its latitude. Then we shall have X=R cos S cos O, Y= R cos 2 sin O, Z = R sin 2. Let x'j y f , z' be the co-ordinates of the body referred to the centre of the earth ; and let X and ft denote, respectively, the geocentric longi- tude and latitude, and J, the distance of the planet or comet from the earth. Then we obtain x f = A cos ft cos A, y f =Jco8^Bin^ (87) z' = A sin p. But, evidently, we also have and, consequently, A cos p cos A = r cos b cos l-{- R cos 2 cos O > A cos p sin A = r cos 6 sin I -J- .R cos 2" sin O > (88) J sin /? = r sin b -\- R sin 2". If we multiply the first of these equations by cos Q, and the second by sin Q, and add the products; then multiply the first by sin Q, and the second by cos O , and subtract the first product from the second, we get A cos p cos (A Q) = r cos b cos (I Q) -J- R cos I, A cos/3sm(A 0) = rcos6sin(J O), (89) A sin p = r sin b -j- R sin 2". It will be observed that this transformation is equivalent to the sup- position that the axis of x, in each of the co-ordinate systems, is POSITION IN SPACE. 87 directed to a point whose longitude is > 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 Q 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 A, /9, and A when r. 6. and I have been derived from the elements of the orbit, the quantities J?, , and Z being furnished by the solar tables; or, when J, /9, and A are given, these equations determine /, 6, and r. The latitude 2 1 of the sun never exceeds 0".9, and, therefore, it may in most cases be neg- lected, so that cos 1 and sin = 0, and the last equations become A cos ft cos 0* Q ) = r cos b cos (I ) -j- R, A cos/5 sin (A 0) =r cosb sin (I ), (90) A sin /? =r sin b. If we suppose the axis of x to be directed to a point whose longi- tude is &, or to the ascending node of the planet or comet, the equa- tions (88) become A cos/? cos (A &) =r cos it -f- ^ cos -T cos (O &), A cos/? sin (A Q) = r smu cosi + R cos 2 sin (0 &), (91) A sin /? = r sin u sin i -f- R sin , by means of which /9 and ^ may be found directly from & , i, r, and u. If it be required to determine the geocentric right ascension and declination, denoted respectively by a and d, we may convert the values of f) and A into those of a and 8. To effect this transforma- tion, denoting by e the obliquity of the ecliptic, we have cos 8 cos a = cos /? cos A, cos 5 sin a = cos /? sin A cos e sin sin e, sin d = cos /5 sin A sin e -f- sin /5 cos e. Let us now take n sin JV= sin/?, w cos JV= cos /? sin A, and we shall have COS d COS tt = COS /? COS A, cos 5 sin a = n cos (JV+ 0> sin d =n sin ( JV -j- ) 88 THEORETICAL, ASTRONOMY. Therefore, we obtain (92) cos N tan 5 = tan (N + e) sin a. We also have cos (N -f- e ) _ cos <5 sin a cos N cos /? sin A ' which will serve to check the calculation of a and d. Since ccs d ana cos /? are always positive, cos a and cos A must have the same sign, and thus the quadrant in which oc is to be taken, is determined. For the solution of the inverse problem, in which cc and d are given and the values of X and /? are required, it is only necessary to interchange, in these equations, a and ^, 3 and /?, and to write s in place of e. 34. Instead of pursuing the tedious process, when several places are required, of computing first the heliocentric place, then the geo- centric place referred to the ecliptic, and, finally, the geocentric right ascension and declination, we may derive formulae which, when cer- tain 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' = d= r sin u cos i, z' = r sin u 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, z, 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 &, and, consequently, x = x f cos R> if sin y = x' sin Q -}- y' cos Therefore, we obtain fc = r (cos u cos & =P sin'it cos i sin & ), 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 Q = sin a sin A, ^F cos i sin & = sin a cos A, sin a = smbsmB, cos i cos & = sin b cos B, in which sin a and sin 6 are positive, and the expressions for the co- ordinates become x = r sin a sin (J. -f- u) t y = rsmb sin (B -f w), (95) z =r sin i sin it. The auxiliary quantities a, 6, J., and .B, it will be observed, are functions of & and i 9 and, in computing an ephemeris, are constant so long as these elements are regarded as constant. They are called the constants for the ecliptic. To determine them, we have, from equations (94), cot A = q= tan & cos i, cot B cot Q> cos i, cos & . , sin & sma= -p sm0 = ^-; sinJ. sin li 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 & , sin B and sin & , respectively, must have the same signs. The quadrants in which A and B are situated, are thus deter- mine^. From the equations (94) we easily find cos a = sin i sin & > cos b = sin t cos & . (96) If we add to the heliocentric co-ordinates of the body the co-ordi- nates of the sun referred to the earth, for which the equations have already been given, we shall have x -j- X= A cos ft cos A, y + Y= J cos /9 sin A, (97) 2 + Z = A sin /?, 90 THEORETICAL ASTRONOMY. which suffice to determine A, /?, and J. 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 d directly. For this purpose, let #, 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 co- ordinates 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, 2/" = y cos s z sin e, z" = y sin e -j- z cos e. Substituting for x, y, and z their values from equations (93), and omitting the accents, we get x = r cos u cos & qc r sin u cos i sin & , y r cos u sin & cos s-\-r sin u ( cos i cos & cos e sin i sin e), (98) z = r cos u sin & sin e-\-r sin u (db cos i cos & sin e -j- 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 con- venient form for numerical calculation, let us put cos & = sin a sin A, qp cos i sin & = sin a cos A, sin & cos e = sin b sin i?, cos i cos & cos e sin i sin e = sin b cos -B, sin 2 sin e = sin c sin 0, cos i cos & sin e -f sin i cos e = sin c cos (7; and the expressions for the co-ordinates reduce to x = r sin a sin (A -f- u), y = rsmb sin (. -f u), (100) 2 = r sin c sin ( (7 -f- u). The auxiliary quantities, a, 6, c, JL, jB, and C, are constant so long as & and i remain unchanged, and are called constants for th( equator. It will be observed that the equations involving a and A, regard- ing 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 Q> . In the case of b and jB, the relations are those of the parts of a spherical triangle of which the sides are 6, i, and 90 -f- e, B being the angle included by i and 6, and 180 & the angle opposite the side b. Further, in the case of c and (7, 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 & that included by the sides i and e. We have, therefore, the following additional equations : cos a = sin i sin & , cos b = cos & sin i cos s cos i sin e, (101) cos e = cos Q sin i sin s -f- 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 co- ordinate 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 E , such that sin i = e Q sin E Q , cos i cos Q = e Q cos E w e (} being always positive. We shall, therefore, have tani ~ cos Q,' Since both e Q and sini are positive, the angle E cannot exceed 180; and the algebraic sign of tan E Q will show whether this angle is to be taken in the first or second quadrant. The first two of equations (99) give cot A = HP tan & cosi; and the first gives cos & sm a = T . sin A 92 THEORETICAL ASTRONOMY. From the fourth of equations (99), introducing e Q and E ti , we get sin b cos B = e cos JE cos e e sin E Q sin e = e cos (J^ -{- e ) But sin 6 sin B = sin & cos e ; therefore sin & cos e tan & cos E Q ' cos e We have, also, . , sin cos sin b = . p . sm.B In a similar manner, we find cot C= C si sin (^Q + e ) ~ tan & cos EQ ' sin e and sin O sin e sine sin The auxiliaries sin a, sin 6, and sin c are always positive, and, there- fore, sin A and cos &, sini? and sin &, and also sin Q and sin &, 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 &. But sin a cos A = =F cos i sin & J and hence we derive sin b sin c sin ( C B} sin a cos A = tan i, which serves to check the accuracy of the numerical computation of the constants, since the value of tan i obtained from this formula must agree exactly with that used in the calculation of the values of these constants. If we put A' = A n T &, B' = B n q= a, and C' = C n ^F & , the upper or lower sign being used according as the motion is direct or retrograde, we shall have POSITION IN SPACE. 93 = r sin a sin (A' -f- v), y = r sin 5 sin (' -f *), (102) z = r sin c sin ( C' + v), a transformation which is perhaps unnecessary, but which is con- venient when a series of places is to be computed. It will be observed that the formula for computing the constants a, 6, c, A, By and (7, in the case of direct motion, are converted into those for the case in which the distinction of retrograde motion id 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 refeired 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= RcosQ, Y=- .Rsin O cose, Z = R sin O sin e = Ftan e, in which R represents the radius-vector of the earth, O the sun's longitude, and e the obliquity of the ecliptic. We shall, therefore, have x -}- JT = A cos cos a, y -f Y= A cos 8 sin a, (103) z -f Z= Jsind, which suffice to determine a, d, and J. If we have regard to the latitude of the sun in computing its geo- centric co-ordinates, the formulae will evidently become JT Ii cos O cos S, Y= R sin Q cos S cos e R sin 2 sine, (104) Z = H sin O cos sin e -\- JK sin S cos e, in which, since S can never exceed it 0".9, cos 2 is very nearly equal to 1, and sin I = I. 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 THEOKETICAL ASTRONOMY. manac, contaiD, 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 co- ordinates directly from the eccentric anomaly in the following manner : The equations (102) give, accenting the letters a, 6, and c, x = r cos v sin a' sin A' -j- r sin v sin a' cos A', y =r cos v sin b r sin B r -f- r sin v sin b' cos B', z = r cos v sin c r sin C' -[-r sin v sin c' cos C'. Now, since r cos v = a cos E ae, and r sin v = a cos

' sin B' = A y , a sin b f sin _B' = ^t y , a tan 4> sin b' cos B f = v y ; ae sin c' sin C" = >* a sin c' sin G r = /* a tan 4 sin c' cos C" = v z . Then we shall have x = A x -f fj. K sec .F + v x tan jP, v = A y 4- ^ sec jF + v y tan F, (106; 2 = A s -j- /7. z sec F -f- v z tan jP. 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 ;:', ', and V 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 o) 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 To find Q,' and i', we have, from the spherical triangle formed by the intersection of the planes of the orbit, ecliptic, and equator with the celestial vault, cos if = cos i cos e sin i sin e cos & , sin i r sin &' = sin i sin ^ , sin i' cos &' = cos t sin e -f- sin i cos e cos SI Let us now put n sin JV = cos i t n cos JV= sin i cos &, and these equations reduce to cos i' = n sin ( JV e), sin i' sin &' = sin i sin & , sin i' cos &' = n cos (JV e) ; from which we find HT cot i cos JV tan JV= , tan ' = ^ ^ tan &, cos 1 cos (JV e) cot i' = tan (JV e) cos &'. ( 107 ) Since sin i is always positive, cos JV and cos & must have the same signs. To prove the numerical calculation, we have sin i cos Q _ cos JV sin i' cos ' cos (JV e)' the value of the second member of which must agree with that used in computing & '. In order to find CD O , we have, from the same triangle, sin CM O sin i' = sin & sin e, cos a> n sin i' = cos e sin i -{- sin e cos i cos & . Liet us now take m sin Jlf = cos e, m cos Jlf = sin e cos & ; and we obtain POSITION IN SPACE. 97 cot M = tan e cos &, and, also, to check the calculation, sin e cos & cos Jf sin i' cos % cos (M i)' If we apply Gauss's analogies to the same spherical triangle, we get cos^' cos^ (' + > ) == cosi cosi (*' + s \ MAQ x sin It sin I (&' ) sin J& sin i (t e), The quadrant in which \ (&' + ^ ) or K^ ^o) ^ s situated, must be so taken that sin \i f and cos \V shall be positive ; and the agreement of the values of the latter two quantities, computed by means of the value of \i f derived from tan Ji', 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, We may thus find the elements TT', & ', and i f y in reference to the equator, from the elements referred to the ecliptic; and using the elements so found instead of it, &, 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 formulae 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 & and &', 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 & must then be measured from the place of the descending node of this plane on the ecliptic. The value of &, 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 & ', i', and w will have the same signification in reference 7 98 THEORETICAL ASTRONOMY. to this plane that they have in reference to the equator, with this dis- tinction, however, that & ' 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 helio- centric 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 with the law of uni- versal gravitation. The circle is an ellipse of which the eccentricity is zero, and, consequently, M=v = u, and r = 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 con- stant so long as the system of elements is regarded as unchanged ; but those which determine the position of the orbit in space, TT, & , 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 peri- helion 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 TT, &, 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 longi- tudes of the sun must be referred to the same equinox, so that the resulting geocentric longitudes or right ascensions will also be re- ferred to that equinox. It will appear, therefore, that, on account of these changes in the values of n, &, and i, the auxiliaries sin a, sin 6, sin c, A, J5, and C, introduced into the formulae for the co- ordinates, 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-ordinatey of the sun. The places thus found may be reduced to the true equinox of the date by the well-known formulae for precession and nutation. Thus, for the reduction of the right ascension and declina- tion 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<* =/+ g sin (G + a) tan d, for which the quantities /, 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 f may be solved by means of equations (109), making, however, the necessary distinction in regard to the point from which Q> and &>' are measured. Let d denote the longi- tude of the descending node of the ecliptic of t' on that of t y and let T} denote the angle which the planes of the two ecliptics make with each other, then, in the equations (109), instead of & we must write & 0, and, in order that Q, ' shall be measured from the vernal equinox, we must also write & ' 6 in place of Q, ' . Finally, we must write T] instead of e, and A os (' + Aw) = cosi (& 0) cos (i + 17), sin Jt* sin J (' Aw) = sin J (& 0) sin J (i 9 ), sin $i! cosi (' AW) = C os (& 0) sin J (t -f r,). These equations enable us to determine accurately the values of & ', i', and AW, which give the position of the orbit in reference to the ecliptic corresponding to the time t f , when d and y 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 pre- cession in longitude during the interval t f t must also be applied. If the changes in the values of the elements are not of consider- able magnitude, it will be unnecessary to apply these rigorous formulae, and -we may derive others sufficiently exact, and much more con- venient 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 7] cos (ft 0) = cos i' sin i -f- sin if cos i cos Aw, from which we easily derive sin (i r i) = sin TJ cos (ft 0) -f- 2 sin if cos i sin 2 JAW. (112) We have, further, sin Aw sin i' = sin f) sin (ft 0), or Shl(ft - 0) r+ + n\ sin AW = sm -TJ - ^-3 -. (113) sin* We have, also, from the same triangle, sin AW cos i' = cos (ft 0) sin (ft' 0) -f sh (ft 0) cos (ft' 0) cos 17, which givec sin (ft' ft) = sin Aw cos if 2 sin (ft 0) cos (ft' 0) sin 2 ^, or sin (ft' ft) = sin >? sin (ft 0)coti' 2 sin (ft 0) cos (ft' 0) sin 1 7. (114) Finally, we have TT' TT= ft f ft + Aw. Since ^ is very small, these equations give, if we apply also the pre- cession in longitude so as to reduce the longitudes to the mean equinox of the date *' smi i f = i_-^cos(ft 0) - 8 2 -0), (115) POSITION IN SPACE. 101 in which is the annual precession in longitude, and in which s = 206264". 8. In most cases, the last terms of the expressions for i f , Q'y and TT', being of the second order, may be neglected. For the case in which the motion is regarded as retrograde, we must put 180 i and 180 i', instead of i and i', respectively, in the equations for AO>, i f , and & ' ; and for TT', in this case, we have which gives *' = * + (*' 0-^7 7 sin (& 0) tan ' ^s Civ S If we adopt BesseFs determination of the luni-solar precession and of the variation of the mean obliquity of the ecliptic, we have, at the time 1750 + r, -^ = 50".21129 + O."0002442966r, at ^ = 0".48892 0/'000006143r, at and, consequently, 73 = (0."48892 O."000006143r) (f f) ; and in the computation of the values of these quantities we must put T \(t r 4" 1750, t and t 1 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 f on that of t f 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"31(f 1750); and the result is 351 36' 10" 5".21 (t 1750) 5".21 (if 1750), 351 36' 10" -10".42( 1750) 5".21(/ *). 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, e = 351 36' 10" + 39".79 (t 1750) 5".21 (if <). When the elements TT, &, 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 com- ^uting 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, 6, c, A, B, and CJ so that these are no longer to be considered as constants, but as continually changing their values by small differences. The differential formulae for the com- putation of these changes, which are easily derived from the equations (99), will be given in the next chapter; but they are perhaps unneces- sary, 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 preces- sion and nutation. The places which thus result are free from parallax and aberration. In comparing observations with an ephemeris, the correction for par- allax is applied directly to the observed apparent places, since this correction varies for different places on the earth's surface. The cor- rection 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 include the aberration directly in the ephemeris by using the time t 497 S .78^ in computing the geocentric places for the time t, or by subtracting from the place free from aberration, com- puted for the time t, the motion in a and d during the interval 497*.78 J, in which expression A is the distance of the body from tlu 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*.78 J, 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 aberra- tion; 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, com- mencing 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. 29' 40".21 20 33 .09^ Ecliptic and Mean Equinox, 1864.0. log a = 0.3881319 log fJL = 2.9678088 P = 928".55746 When a series of places is to be computed, the first thing to oe done is to compute the auxiliary constants used in the expressions for the co-ordinates, and although but a single pi ice is required in the problem proposed, yet we will proceed in this manner, in order to 104 THEORETICAL ASTRONOMY. exhibit the application of the formulae. Since the elements it, , 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 f = 1865.0, which give rll 5f = 50".239, 9 = 352 51' 41", 7 = 0".4882. at Substituting these values in the equations (115), we obtain i f i = At = 0".40, A& = + 53".61, ATT ^ + 50".23 ; and hence the elements which determine the position of the orbit in reference to the ecliptic of 1865.0 are TT = 44 21' 23".32, = 206 43' 33".74, i = 4 36' 50".ll. For the same instant we derive, from the American Ephemeris and Nautical 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 oi the formulas cot A = tan & cos i, tan E Q = -- , COS do cosi c tan & cos E Q cos e Coi0= tan & cos Q sin e cos Q> . , sin & cos e . sin & sm e sm a = r , sm b = - : ^ , sm c - : ~ . sm A SIHJD sm C The angle E is always less than 180, and the quadrant in which it is to be taken, is indicated directly by the algebraic sign of tan E Q . The values of sin a, sin 6, 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 &, sin B and sin &, and also sin C and sin & , 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 whic;i gives log tan i = 8.9068875, agreeing with the value 8.9068876 derived directly from i. Next, to find r and u. The date 1865 February 24.5 mean time at Washington reduced to the meridian of Greenwich by applying the difference of longitude, 5* 8 m 1P.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 mul- tiplying the mean daily motion, 928". 55745, by this number, and adding the result to the given value of M, we get the mean anomaly for the required place, or M = 1 29' 40".21 + 108 30' 57".14 = 110 0' 37".35. The eccentric anomaly E is then computed by means of the equation M=E esmE, the value of e being expressed in seconds of arc. For Eurynome we have log sin

= 37 35' 0".0, or log 6 = 0.1010188; and lug a = 0.6020600, to find r and v. First, we compute ^Vfrom " at in which log A = 9.6377843, and we obtain logN= 8.7859356; N= 0.06108514. The value of F must now be found from the equation N= el tan F log tan (45 + F). NUMERICAL EXAMPLES. 109 If we assume F= 30, a more approximate value may be derived from which gives F, = 28 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 (N-N,^o S *F ' l(e cosF,-) *' wherein s = 206264".8; and the result is *F, = 4.6097 (N N f ~) 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 AJ F, = 5.199 (N NJ s = 12' 9".4. The third approximation, therefore, gives F, = 25 24' 30".6, and, repeating the operation, we get J F T =2524'27".74. which requires no further correction. To find r, we have which gives log r = 0.2008544. Then, v is derived from tan v cot ^4 tan F, and we find v = 67 3' 0".0. When several places are required, it is convenient to compute and r by means of the equations 110 THEORETICAL ASTRONOMY. For the given values of a and e we have log V a(e + 1) = 0.4782649, logl / 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 M from wherein log C = 9.9601277, which gives log 3f= 2.1404550. \Yith 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. ^ _ e Then we have log i = log = 8.217680, and from the equation 1 -f- e v = V+ -4(1000 we get v = V+ 42' 22".28 -f 25".90 -f 0".28 = 102 20' 52".20. The value of r is then found from -f 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 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 5(l--e A ~ * 1 9e NUMERICAL EXAMPLES. Ill 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 which gives v = 102 20' 52".20, agreeing exactly with the value already found. Finally, r is given by " (1 -{-AC 2 ) cos 1 }*' 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 dis- tinction, 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) ^(f+ and that of r from (1 .AC 2 ) cos 8 in' 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 dis- tinctions 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 inter- polation, let us denote any date, or argument of the function, by a + na>> and the corresponding value of the co-ordinate, or of the function, for which the interpolation is to be made, by / (a -f nco). If we have computed the values of the function for the dates, or arguments, a to, a, a -f a>, a -f- 2w, &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 mav expand the function into the series f(a + no>) =/() + An + JSn 9 + On 9 -f &c. (116) and if we regard the fourth differences as vanishing, it is only neces- sary to consider terms involving n s in the determination of the unknown coefficients A, B, and C. If we put n successively equal to 1, 0, 1, and 2, and then take the successive differences of these values, we get I. Diff. II. Diff. III. Diff. /(<,-) =f(a)-A + B -C f(a + 2>) =/(o) + 2A + 4B + 8 C If we symbolize, generally, the difference f(a + nco) f(a -\-(n 1 ) ), the difference / (a + (n-+ J) a>) /( a + (n J) a) by /" (a -f- wa>), and similarly for the successive orders of differences^ these may be arranged as follows : Argument. Function. I. Diff. II. Diff. III. Diff. a a> /( a _a/) a /(a) /(-iK> f (a ) INTERPOLATION. 113 Comparing these expressions for the differences with the above, we get C=J/"(a + i), J = i/"(a), A =f (a + J) - if' (a) - J/ (a + , which, from the manner in which the differences are formed, give C= $ (/" (a 4- ,) -/" (a)), B = if" (a), A =/ (a + ) -/(a) - i/" (a) - J (/" (a + ,) -/" (a)). To find the value of the function corresponding to the argument a + Jw, we have n = , and, from (116), /(a + ) =/(a) + 14 + ^ + 4 a Substituting in this the values of A, B, and (7, last found, and re- ducing, we get /(a -H0 = i (/(a + +/()) ~ I (i (/" ( + ) +/" W)), in which only fourth differences are neglected, and, since the place of the argument for n = is arbitrary, we have, therefore, generally, /(a + (n + ) ) = i (/( + (* + !) ") +/(a + no,)) - 4 (i (/' (+( + 1) +/' ( + ))) (1.17) 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 arith- metical 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 + no,))), 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 co, equal to four days, for the intermediate dates, we obtain a series of places at intervals of two days ; and, finally, inter- 8 114 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 dif- ferences. 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 = e being expressed in seconds. Next, regarding this value as con- stant, we find the date for which L -J- 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 Q ) = tan u cos i. Let these longitudes be denoted by I and l f , the times to which they correspond by t and t' 9 and the longitudes of the sun for the same times by O and Q ' ; then for the time t , for which the heliocentric longitudes of the planet and the earth are the same, we have or (118) the first of these equations being used when I 180 O is less TIME OF OPPOSITION. 115 than V 180 O'. If the time t 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 t Q 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 calcula- tion to use two terms of equation (53) in finding the equation of the centre, and, if e is expressed in seconds, this gives _ M = 2e sin Jf + 5 . t. s i n 2M 9 4 S 8 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 loga- rithms 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 meana of the tables of common logarithms. Thus, we have If we put log tan x = J (log b log a), 116 THEORETICAL ASTRONOMY. we shall have log (a -f 6) = log a 2 log cos x, or log (a -f- 6) = 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 a; is less than sin a;. It should also be observed that in the solution of equations of the form of (89), after tan (A 0) 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 (A 0) and sin (^ ), respectively, give two values of J cos /?, 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 pre- ferred when cos (A ) is greater than sin (^ 0), and that derived from the second equation when cos (A 0) is less than sin (^ ). The value of J, if the greatest accuracy possible is required, should be derived from J cos /9 when /9 is less than 45, and from J sin /? when /? 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|(&' -f- ^ ) an( ^ tanj(&' fti ); then, if sin|(&' -f w ) is greater than cos|(&' + w ), we find cos^' from the first equation; but if sin f (&>' -f = sin e sin & . 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 FORMULAE. 117 CHAPTER II. INVESTIGATION OF THE DIFFERENTIAL FORMULAE 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 pro- duce 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 observa- tions 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 condi- tion 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 ele- ments, 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 6 any co-ordi- nate of the place of the body computed from the assumed elements of the orbit, we shall have, in the case of an elliptic orbit, 118 THEORETICAL ASTRONOMY. M Q being the mean anomaly at the epoch T. Let 6 f denote the value of this co-ordinate as derived directly or indirectly from observation ; then, if we represent the variations of the elements by ATI, 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 d A *+5a A a+-3r A< +27 do .... dO . The differential coefficients - 1 , ^ , &c. must now be derived from d* a$> the equations which determine the place of the body when the ele- ments 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 fun- damental plane. Let x, y, z be the heliocentric co-ordinates of the body in reference to the equator, and we have 9 =/(*, y, z), or do . , do , , do , dO = -j- dx -4- -j- dy 4- -,- dz. dx r dy y dz Hence we obtain dO _ d0_ dx dO_ dy_ dO_ dz ; ~dn dx ' d-K dy'dTtdz'dn* and similarly for the differential coefficients of with respect to the other elements. We must, therefore, find the partial differential co- efficients of d with respect to #, y, and z, and then the partial differen- tial coefficients of these co-ordinates with respect to the elements. In the case of the right ascension we put = oc, and in the case of the declination we put 6 = 3. 45. If we differentiate the equations x + X= A cos 8 cos a, y -{- Y= A cos d sin a, z -\- Z = A sin d, regarding X, F, and Z as constant, we find DIFFERENTIAL FORMULAE. 119 dx = cos u cos 8 d 4 A sin a cos 8 da, A cos a sin d dd, dy == sin a cos d d A -\- A cos a cos $ da, A sin a sin d dd, dz = sin d d A -j- A cos d dd. From these equations, by elimination, we obtain .. , sin a , cos a , /rk>l cos d da, = az -| -p cfy, (3) ,, cos a sin d sin a sin d cos<5 <** = dx ~ Therefore, the partial differential coefficients of a and d with respect to the heliocentric co-ordinates are da, sin a dS cos a sin <5 dx ~ A ' dx ~ A do, cos a a*<5 sin a sin 5 ... cos # -j- = -j , T- = -. , (4) dy A dy A da. dd cos# j = 0. j == ^ dz dz A Next, to find the partial differential coefficients of the co-ordinates x, y, Zj with respect to the elements, if we differentiate the equations (100)!, observing that sin a, sin 6, sin c, J., B, C, are functions of ft and i y we get dx = - dr -f # cot ( J. -|- w) efot -J- -j^- rfft -f -^r c^i, df2 = ? dr + 2 cot (0+ u) du + ^ rf^ + -jjjr di. To find the expressions for ~, -^, &c., we have the equations d/oo di' x = r cos w cos & r sin M sin & cos i, y r cos -w sin & cos e -j- r sin w cos ^ cos i cos e r sin u sin ^ sin e, 3 = r cos u sin & sin s -j- r sin u cos & cos i sin e -f r sin u sin i cos e. which give, by differentiation, -=^- = r cos u sin & r sin u cos & cos i, d& _ J^ = r cos w cos & cos e r sin w sin ft cos i cos e, 120 THEORETICAL ASTRONOMY. dz - = r cos u cos & sin e r sin u sin & cos i sin e. & dte - = r sm it sin $6 sin *> ew and remain unchanged; and we have, also, da; dy . dz , rtN T^- = rsmttcosa. -^n- = rsmwcoso. -77- = rsmttcosc. (9) di di di It is advisable, in order to avoid the use of two sets of formulae, in part, to regard the motion as direct and the inclination as susceptible of any value from to 180. 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 (TT), we shall have If we introduce, as one of the elements of the orbit, the distance of the perihelion from the ascending node, we have du = dv -f~ d* * and, hence, dx dx 122 THEORETICAL ASTKONOMY. The values of > - , and must, in this case, be found by means 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 geo- centric right ascension and declination with respect to the elements &, ij and n or o>, and also with respect to r and v, by writing suc- cessively a and d in place of 6, and &, i, &c., in place of TT in the equation (2). The quantities r and v, however, are functions of the remaining elements y>, M Q , and /*; and we have , dr , dr ,,.. dr , , dv j dv , f dv , dv =^ -j- d(p -\- - J1 r F dM Q -f- -j dp. d

, Jf , and //, are dx d

and have been aM d/j. a/j. found, the partial differential coefficients of the heliocentric co-ordj nates with respect to the elements tp, M w and // will be completely determined, and hence, by means of (2), making the necessary changes, the differential coefficients of a and d with respect to these elements. 46. If we differentiate the equation M=E DIFFERENTIAL FORMULA. 123 we shall have dM= dE(l e cos E) cos

we get J a cosy a? cos y j , , . sin v / dv = - - --- sn v a cos 2 ^ . \ , - + \\d =PJ and - = 1 + sin cos v, this becomes QJ COS (P ^ \ ? /< r*N dv = dM -4- 1 h tan y cos v] sm v c?^. (12) r 2 If we differentiate the equation r = a (1 e co .E), we shall have dr = - da -f ae sin .E , the result is .__ r l ^ a _}- a tan ^ sin-y dM-\- (ae sin ^J sin v a cos p cos E) dy. 124 THEORETICAL ASTRONOMY. X T . -n sin v cos

sin v dM a cos

. (14) Further, we have T being the epoch for which the mean anomaly is M w and A; 1/1 + m ^= - 3 -- a^ Differentiating these expressions, we get ), and q = r cos 2 Jv, we have 1 3& (* T) tan Jv 1 0-21 i x i_ =-(14- tan 2 \v 3 sm 2 \v sm 2 \v tan 2 q r*V 2q r __ cosv r We also have tan 4v Therefore, equation (19) reduces to (20) , ^ N (21) -. I/ 2q If we introduce cZ log q instead of dg, this equation becomes From the equations (17), (18), (20), and (21), we derive dr ksinv dv IcV 2q dT. ~ V5q 5? r 2 dr dv 3& (t T) V^V/O /. 7 y > V^^y dq dq dr_ _qcosv dv 3^ (t T} 1/2? " " " and then we have, for the differential coefficients of x with respect to T and q or log 7, DIFFEEENTIAL FOKMUL^. 127 dx _ dx dr dx dv dx dx dr dx dv dT == ~d^"dT + ~d^"dT' ~dq = ~dr ' ~efy + ~dv "dq dx dx dr dx dv ' d log q dr d log q dv ' d log q' and similarly for the differential coefficients of y and z with respect to these elements. The expressions for the partial differential co- efficients 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 fur- nished 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 ele- ments. 48. We may also, in the case of an elliptic orbit, introduce T, q, and e instead of the elements T / a sin v tan 4 sin v \ , ai? = ^,dJ\ n I 1 h : I 4' r tan F \ r cos ^ sin ^ / DIFFERENTIAL FORMULA. 129 Bufc, since r sin v = a tan $ tan F, and p a tan 2 ^ this reduces to * = ^v5i-(| + 1 )^d*. (25) If we differentiate the equation we get r , , T, d-F 7 a tan 4* T dr = - da -f ae tan* F ^ H -- ^ -- cfy. a smF cosF cos4- Trrr Substituting in this equation the value of ^ we obtain 7 r , , a?e tan F 7 . _ / a 2 e tan 2 jP a \ tan 4* dr = - da -\ -- dN tt I ---- =f ) a r \ r cos jP/ cos 4 which is easily reduced to r 7 sinv 7 , T . pi r ae dr = a da+a ^ dN + -r(^-^ But, since r ae a cos F ~~ cos 2 F ~~ cos F' this reduces to r or 7 r , sin v , , r . cos v , , n 'ia^- Si: 4sinw which, by means of (76)^ reduces to dv lctT ' ' 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 M cos 2 ^w . 8 tan Av f \6y) de ~ 2 (1 + 9e) " 75 tan^w v (1 + e) (1 + 9e)' This equation remains unchanged in the case of hyperbolic motion, the value of C being taken from the column of the table which cor- responds to this case-; and it will furnish the correct value of -=- in de all cases in which the last term of equation (23) is not conveniently C?7* applicable. The value of -- is then given by the equation (32). When the eccentricity differs very little from unity, we may put B = 1, and _ tan \w = tan v j/^ (i + g e ), cos 2 ^w = O 2 cos 2 %v. Then we shall have ^ 2 2k (t T) C 2 sm v .- , cos 4 4w. ; 1/2 g* The equation = (1 + A C") cos 2 Jw = (1 + i-^) cos 2 >, gives | l = (1 + p) cos 4 Jw = Ccos 4 w. Hence we derive T) NUMERICAL EXAMPLES. 135 [f we substitute this value in equation (39), and put C 2 (1 + e) 2, we get im r 2 (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 Feb- ruary 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 a = 181 8' 29".29, d = 4 42' 21".56, log J = 0.2450054, log r = 0.428285, v = 129 3' 50".5, u = 326 41' 40".l, A = 296 39' 5".0, B = 205 55' 27".l, C= 212 32' 17".7, log sin a = 9.999716, log sin b = 9.974825, log sin c = 9.522219, log x = 0.425066 n , log y = 9.511920, log z = 8.077315, e = 23 27' 24' 7 .0, tT= 420.714018. First, by means of the equations (4), we compute the following values: log cos d ~ = 8.054308, log -^ = 8.668959 W , log C os d ^ = 9.754919 , log -^ = 6.968348 , dy dy log ^ 9.753529. Then we find the differential coefficients of the heliocentric co-ordi- nates, with respect to TT, &, i, v, and r, from the formulae (7), which give io s = io = - 399496 - log -- = 7.876553, log -- = 8.830941, log -- = 9.222898., log -T- = 8.726364, log -- = 9.687577, log -- = 0.142443 ,,, log ~ =-- 9.996780 n , log -^- = 9.083635, log -^ = 7.649030. 136 THEORETICAL ASTRONOMY. , dx dy .. dz . In computing the values of -TT-, -^p and -TT, those of cos a, cos 6, and' cos c may generally be obtained with sufficient accuracy from Bin a, sin 6, and sine. Their algebraic signs, however, must be strictly attended to. The quantities sin a, sin 6, and sin c are always positive ; and the algebraic signs of cos a, cos 6, and cos c are indicated at once by the equations (101)i, 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. To find the values of cos d -j- and -7-, we have, according to equa- (ITC CfrTT tion (2), . da, ^ da dx , .dady , . ., N cos<5 = cos<5- -fcosd --, (41) dn dx dn dy d* which give dd _ dd dx dd dy dd dz dn ~ dx dn dy dn dz dn' = + 1.42345, ==_ 0.48900. n dv dn dv In the case of &, i, and r, we write these quantities successively in place of Tt in the equations (41), and hence we derive cos 8 4^ = - 0-03845, -^- = - 0.09533, dQ> "66 cos d -^- = 0.27641, -2L = 0.78993, ai cos d ~ = 0.08020, -^- = + 0.04873 Next, from (16), we compute the following values: log ^ = 0.179155, log ^- = 9.577453, log ~ = 2.376581,, G dy> dM dp log ^ = 0.171999, log -^r = 9.911247, log ~ = 2.535234. & d dM Q dfji j$y (j y \Ve may now find ^ , -^r^, &c. by means of the equations (11), and thence the values of cos d -= , -3, &c. : but it is most convenient d

-7-, and -7- 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 expres- sions for -j-, -j > -y-, &c., in equation (2), writing successively and fJL in place of TT. Thus, we have . da, .da,dr. .da, dv cos <5 - = cos d . -- k cos <5 -r- -r , a?' ar* a^ av a?> d<5 __ d dr dd dv d

+ 1.13004A3f -}- 507.264A/., A5 = 0.48900 ATT 0.09533 A & 0.78993Ai 0.65307 A 0.38023A^f 179.315A/Z. To prove the calculation of the coefficients in these equations, we assign to the elements the increments Ajf = + 10", ATT == - 20", A ft = - 10", Ai = + 10", A^ = + 10", A/ = + 0".01, BO that they become Epoch = 1864 Jan. 1.0 Greenwich mean time. M = 1 29' 50".21 TT = 44 20 13 .09 ^ ft = 206 42 30 .13 > Mean Equinox 1864.0 t = 4 37 .51 J V = 11 16 1 .02 log a = 0.3881288 L = 928".56745 With these elements we compute the geocentric place for 1865 Feb- ruary 24.5 mean time at Washington ; and the result is a = 181 8' 34".81, d = - 4 42' 30".58, log A = 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 d and those already given, as derived from the unchanged elements, gives Aa = + 5".52, COS * Aa = + 5".50, A coirections to be applied to the corresponding elements are , y^, and T7y^7- 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 value of Aa, 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 the case of d, 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 o, since in the case of a it is measured on a small circle, of which the radius is cos d and hence, in order that the difference between computation and observation in a and d may have the same influence in the determination of the corrections to be applied to the elements, we introduce COS^AOC instead of Aa. The same principle, is applied in the case of the longitude and of all corresponding spherical co-ordinates. DIFFERENTIAL FORMULAE. 143 52. The formulae already given will determine also the variations of the geocentric longitude and latitude corresponding to small in- crements assigned to the elements of the orbit of a heavenly body. In this case we put e = 0, and compute the values of A, I>, sin a, and sin b by means of the equations (94) r We have also (7=0, sin c = sin i, and, in place of a and d, respectively, we write ^ 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 f = r cos u cos ft r sin u sin ft cos i t y' = r cos u sin ft -f- r sin u cos ft cos *, z' = r sin u sin i, we obtain dx r = dr r (sin u cos ft -f- cos u sin ft cos i) du r (cos u sin ft -f sin u cos ft cos i) dft -f- r sin u sin ft sin i cfo', dtf = dr r (sin u sin & cos u cos & cos i) du -f r (cos tt cos ft sin w sin ft cos i) c?ft r sin w cos ft sin i eft, (46) dz 1 = - dr -{ r cos u sin idu -\- r sin it cos i eft, r in which #', /', 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 ft ; then we shall have dx = dx' cos & + dtf sin & , dy = dx' sin & + dtf cos ft , dz = dz', and the preceding equations give dx = - dr r sin u du r sin u cos i d& , T dy = - dr + r cos u cos i du -f- r cos u d& r sin it sin i di, (47) dz = _ dr -f r cos u sin idu ~{-r sin ?f cos i eft. 144 THEORETICAL ASTRONOMY. This transformation, it will be observed, is equivalent to diminishing the longitudes in the equations (46) by the angle & through which the axis of x has been moved. Let X n Y,, Z, denote the heliocentric co-ordinates of the earth referred to the same system of co-ordinates, and we have x -f- X, = A cos/3 cos 0* Q), V + F,= Jcos08in(J ), z -f Z, = A sin /?, in which ^ is the geocentric longitude and /9 the geocentric latitude. In differentiating these equations so as to find the relation between the variations of the heliocentric co-ordinates and the geocentric lon- gitude and latitude, we must regard Q, 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 cos sn dy= cos sin (A &)d/f A sin /5 sin (A )d-|- A cos/? cos (A cfe =sin /? dJ H- J cos y9 d/9, from which, by elimination, we find sin/? cos (A a) , sin /9 sin (A q) cos /? -- _ These equations give COS p = == r = -- -. - . j dx A dx Q ft cosj? -j- = 0, -,- C?3 ^2 If we introduce the distance o between the ascending node and the place of the perihelion as one of the elements of the orbit, we have du = dv -{- d<0, and the equations (47) give dx x dy y dz z . . . ~-=- = cosu, /- = -== smu cos i, = =- = smw smt; dr r dr r dr r dx dx dy dy . dz dz = = -_ = rsmu, . y = -. y - =r cos u cost, = = -= av aw dv dw dv dot DIFFERENTIAL FORMULA. 145 ^-=rcos, -*l=0j (49) dx A dy . . c?z 7 . = 0. =r- = ?* sin w sin ^. ^r- = r sin u cos i. at at di If we introduce TT, the longitude of the perihelion, we have du = dv -{- dit a* & , and hence the expressions for the partial differential coefficients of the heliocentric co-ordinates with respect to it and & become dx dy dz r= r sm w, 7 r cos u cos ^, , = r cos w sin i ; f f t () dx _ . , , . aw . . t . az - = 2r sm w sm 2 Jt, ~- = 2r cos w sm 2 J*, -7 = r cos w 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 -rr = 0, -T- = f sin u sin i, j-r = r sin u cos i ; (51) di di di dx dx dx dy , and the expressions for -7-, -j-, -TQ-> -f-, &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 oV + d Q , and hence dx dy dz . . /- = r cos i* cos i, -j = rcosttsmi; (52) -r~- = 2r sin w sin 2 i, -~- = 2r cos tt sin 2 Ji, -^ - = r cos w sin i. But, to prevent confusion and the necessity of using so many for- mulae, it is best to regard i as admitting any value from 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 2& n instead of TT, the other elements remaining the same in both cases. 53. The equations already derived enable us to form those for the differential coefficients of ^ and ft with respect to r, v, & , i, and co or T, by writing successively X and ft in place of 0, and &, i, &c. in 10 146 THEORETICAL, ASTRONOMY. place of it in equation (2). The expressions for the differential coeffi- cients 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 A 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 . c?A ctt d x . Q ctt dy cos /9 -j- = cos /? -=- -j f- cos /5- -/-, dv dx dv l dy dv &L&L ^L\~ ty-+dP- dv dx dv dy dv dz dv' and put cos i cos (A & ) = A Q sin A, sin (A & ) = A Q cos A, sin i = n sin N, sin (A 2 ) cos i = n cos N, in which A Q and n are always positive, they become dv dw A -~ = -== (sin /9 cos (A & ) sin u -f- n cos u sin (N + ft ). Let us also put n sin ( N + ft = -B g i n -^> sin /5 cos (A & ) = J? cos jB, and we have cos /? -j = cos /? -j = ^4 sin (A -\- u -, * ,! CM) 7 TO The expressions for cos ft -j- and -j- give, by means of the same auxiliary quantities, cos -= = ^ cos (J. + u), dr (56) f = -^eos(* + ). In the same manner, if we put DIFFERENTIAL FORMULA. 147 cos (/I )= C sin C, cos i sin (A & ) = (7, cos (7; 1 67) cos i = D sin Z> sin f A &) sin i = D cos D; we obtain -^- = = ^ (7 sin A r cos /9 JT- = -j sin i sin w cos (A & ), -2(L = -D fl sin w sin (D + ). Cd T If we substitute the expressions (55) and (56) in the equations fl . . fl cos /9 3 = cos /? -=- 3 -- ^- cos /? -j- 3 d? dr dy dv d and put __ _ ^._ d

sin v, dv ~ a 2 cos -- T -- = h sinH= 9r \ smv(t T) y- 206264.8 J, cy , ,-.. a* cos

, we must put =r-,, (63) -- -j-, - 5 - aq aq and the equations become = sn S 9 (64) ooi^-^iiaU + Jr+ii), 4^ = \ B sin (jB H- H+ u). In the numerical application of these formulae, the values of the second members of the equations (63) are found as already exem- plified 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 A and ft 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 ksmv dv = -- which give tan G = tan V2 -, which is the expression for the linear velocity of a comet moving in a parabola. Therefore, = 212 32' 17".71, and a/ == w + a> = 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 d in place of ^ and /9, and &', i f , w f in place of &, i, and co, respectively, we have A Q sin A cos (<*&') cos i r , A cos A = sin (a &') ; n sin N= sin i f , n cosN= cost' sin (a &'); J5 sin B = n sin (N + #) B Q cos -B = sin <5 cos (a &') ; <7 sin 0= cos (a &'), ^o c <> s C^sin (a &') cosi'; A sin Z> = cos i', D cos D = sin i' sin (a & ') ; / sin F=a cos ^ cos v, / 2 \ / cos F= 1 -- h tan ? cos v } r sin v; Vcos?' / gsmG = a tan f , sin i sin & = sin i' sin & f > sin i sin w = sin & ' sin e, sin i cos w = cos sin i' sin e cos i' cos & ', 152 THEORETICAL ASTRONOMY. from which the values of &, i, and co may be found from those of & ' and i r . 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 to di' -f- sin a> sin i' d & '. (68) Interchanging i and 180 i', and also & and &', we obtain di' = cos tt> di sin w sin idQ. Eliminating di from these equations, and introducing the value sin i' sin & sini sin&y' the result is sin Q sin to K Biny ami If we differentiate the expression for cos co derived from the same spherical triangle, and reduce, we find da> cos i dl cos i' dQ'. Substituting for dQ, its value given by the preceding equation, and reducing by means of sin & ' cos i' = sin & cos w cos i cos Q, sin w , we get sin tn ein m (70) The equations (68), (69), and (70) give the partial differential co- efficients of & , i, and w with respect to & ; and i', and if we sup- pose 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 oj n *^=Esf'" sin At = sin ta sin i' A Q f -f- cos % Ai r , If we apply these formulae to the case of Eurynome, the result is W O = 4.420A&' + & = _ 3.488A Q' + 6.686A/, Ai = 0.179A^' 0.843 AI* ; DIFFERENTIAL FORMULAE. 153 and if we assign the values A' = 14".12, Ai' = 8".86, AC// = 6".64, we get AOI O = + 3".36, A ^ = 10".0, Ai = + 10".0, A> = 10".0, and, hence, the elements which determine the position of the orbit in reference to the ecliptic. The elements a/, & ', and i f may also be changed into those for which the ecliptic is the fundamental plane, by means of equations which may be derived from (109)! by interchanging & and &' and 180 i'andt. 56. If we refer the geocentric places of the body to a plane whose inclination to the plane of the ecliptic is i y and the longitude of whose ascending node on the ecliptic is &, 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'j y'j z f 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 as- cending 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 *o the point whose longitude is Q . Then we shall have dtf = dy cos i -f- dz sin i t dz' = dy sin i -f- dz cos i. Substituting for dx, dy y and dz their values given by the equations (47), we get x r dx f = dr r sin u du r sin u cos i dQ, dy' = dr -f- r cos u du -\- r cos u cos i d& y dz' = dr r cos u sin i dQ> + r sul u di* It will be observed that we have, so long as the elements remain unchanged, a/ = r cos u, y f = r sin u, z' = 0, 154 THEOKETICAL ASTRONOMY. and hi.'nce, omitting the accents, so that x 9 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 -j- r cos u cos i dQ , dz r cos u sin ^ dQ> -f- r sin i* cfo*. The value of ^> is subject to two distinct changes, the one arising fiom 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 desig- nate by ff- and let the angle between this fixed line and the semi- transverse axis be designated by . Then we have X = a> -f ff. 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 But if the elements which fix the position of the plane of the orbit are supposed to vary, we have the relations dff = cosi dl, da> = d% cos i d& , (72) dn = dx + (1 cosi) d& = d x + 2 sm*i d&. Now, since u = v -f- to, we have u=v+X ff * and du dv -f- dx dff = dv -f- 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 dx, dy = sin u dr -f r cos u dv + r cos u d%, (73) dz = r cos u sin i dQ, -j- r sin u di. The inclination is here supposed to be susceptible of any value from 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 6 the geocentric longitude of the body mea- sured in the plane of the unchanged orbit (which is here taken as the DIFFERENTIAL FORMULA. 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 y. Then we shall have x -f- X A cos y cos 0, y + Y= A cos y sin 0, z + Z = A sin y, in which X, F, 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 y cos 6 d A A sin y cos dy A cos y sin dO t dy = cos y sin 6 d A A sin y sin dfy -f~ ^ cos ^ cos ^ ^> efe = sin y dA -\- A cos ^ cfy ; and hence we obtain 7 . sin , . cos cos y do = dx -\ -. dy, , _ sin y cos , __ sin 17 sin . (74) These give dO sin dO cos cos y -j- = . , eta ^ cos^ , = -. , dy A cfy sinvy cos 6 dy siny sin0 dx A dy A dd cos y -=- = ; c?>? cosi? and from (73) we get dx ~ = COS U, dr dx _ dx _ dv dy dz 'dr -0: " = 0> dx . dy -^- = sin, c?y c?v dz dz A -/- = -/- = r cos w, - j - ; -y- = ; (2v c?/ dfv ^/ ^2 -^ = r cos it SIP x ; ds - - = r sin w. d Substituting the values thus found, in the equations dO dO dx . dO dy cos y 3- = cos y -=- -j h cos 7 j~ ' j > dv ax av ay av dy dy dx dy dy dy dz dv dx dv dy dv dz d^ 156 THEORETICAL ASTRONOMY. we get do do r fn cos >? -=- = cos y -j- = -, cos (0 u) t dv d% A (76) In a similar manner, we derive do 1 . fa d^ 1 cos *] 3 = sin (0 u), ~ = sm y cos (0 u) t dO df) r cos V) -= = 0, -- = cos yj sm i cos u, (77 ) do di) . r COS Tj TV- = 0, yf- = -f- COS I] Sin W. CM Cw d If we introduce the elements ^, Jtf" , and //, which determine r and v, we have, from de dO dr , rf^ c?v cos i? -j = cos T? -j- -j h cos i? -j- -j , 1 d

[ ' dv dy df) _ df) dr df) dv d

? -- = ?cos(0 u G), ,, , i I (79) dO h , dr) h . f _. cos r) = cos (0 u L ). ^ = -j- sm TJ sm (0 u M ). d/j- A dp A If we express r and v in terms of the elements T, g, and e, the values of the auxiliaries /, g, A, 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 do dO COS JJ A0 = COS r) -- Ay 4- COS f] -7 ACP -f- COS r] -rTT A M n + COS Tt -7- &fJ. d% dy dM Q dp DIFFERENTIAL FORMULA. 157 from which it appears that, by the introduction of 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 T) and 6 from those of X and /?. Let X Q , y , z be the geocentric co- ordinates 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 & ; and let #/, y ', z ' be the geocentric co-ordi- nates 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 x A cos /? cos (A ft ), x ' = A cos i? cos 0, y o = J cos /? sin (A Q ), y ' = A cos T? sin 0, Z Q = J sin , z ' = A sin iy, and also < = *0 yo = y cos * -f z o sin * z ' = y Q sin i + z cos i. Hence we obtain COS r) COS = COS /? COS (A & ), cos T) sin = cos /? sin (A & ) cos i + sin sin t, (80 ) sin 17 = cos sin (A & ) sin i -\- sin /? cos i. These equations correspond to the relations between the parts of a spherical triangle of which the sides are i, 90 57, and 90 ft the angles opposite to 90 ^ and 90 ft being respectively 90 -f- (X ^) and 90 0. Let the other angle of the triangle be denoted by f, and we have cos 17 sin Y = sin * cos (A & ), cos T? cos r = sin i sin (A & ) sin ft -f- cos i cos /?. (81) The equations thus obtained enable us to determine y, 0, and f from I and /9. Their numerical application is facilitated by the intro- duction of auxiliary angles. Thus, if we put , , n cos ^= cos j9 sin (A ft), 158 THEORETICAL ASTRONOMY. in which n is always positive, we get cos TI cos 6 = cos /? cos (A & ), cos rj sin 6 = n cos (N t), (83) sin 7] = ra sin (JV i\ from which ^ and may be readily found. If we also put n' sin N' = cos i, n' cos JV = sin i sin (A ft ), we shall have cot N r - tan i sin (A ^ ), If f is small, it may be found from the equation sin* cos (A -S 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 fiin (45 - j,) sin (45 - J (0 + r )) = cos (45 -f i (A - )) sin (45 J ( + i)), sin (45 - j?) cos (45 - J (* + r )) = sin (45 + i (A - &)) sin (45 - J (0 0), cos (45 J?) sin (45 j (0 r)) = (87) cos (45 + i (A - 8)) cos (45 - i (p + t)), cos (45 - ^) cos (45 - J (? = sin Y cos /5 AA -f cos y A/1 The value of p required in the application of numbers to these equations may generally be derived with sufficient accuracy from (86), the algebraic sign of cosf being indicated by the second of equations (81) ; and the values of rj and 6 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 fur- nished directly by observation are the right ascension and declina- tion, the formulae will be most frequently required in the form for finding rj and from a and d. For this purpose, it is only necessary to write a and d in place of A and /9, respectively, and also & ', V, co f , %', and u' in place of &, i, a), %, and u, in the equations which have been derived for the determination of rj and 0, 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 rj A# and &rj in terms of the variations of the elements in the case of the example already given ; for which we have d \ d i s a/j. dd I dd\ nf i dd in which s = 206264".8. If the values of the differential coefficients with respect to fjt and

-7 ~n - > and -7-= - by d

) du -f- sin (I Q> ) 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) w neglecting the latitude of DIFFERENTIAL FORMULA. 165 the sun, and considering 1, /9, J, and O as variables, we derive, after reduction, r> cos /? dX = _ cos (A O)dO, R (94) dp = -- sin sin A O 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 OP FORMULAE 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-ordinates 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 alw r ays 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 longi- tudes and the corresponding latitudes, or any other spherical co- ordinates 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 i? 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 formula? which will be given in the next chapter. However, in the first determination of approximate ele- ments 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/> cos ' sin (o 0) A cos d tanr = COS (a 0)' sin and i, and these elements may refer to any fundamental plane whatever. If we multiply the first of these equations by sin (it" it'), the second by sin (u" it), and the third by sin (u f it), and add the products, we find, after reduction, - sin (u" u'} ~ sin (u" u) -f ^ sin (u' u) = 0, which, by introducing the values of [rr'], [rr"], and [r'r"], becomes [//'] x [rr"] x' + [rr'] x" = 0. If we put [r'r"] [rr'] , Q , n== D^J n FT weget > " "_ r^ In precisely the same manner, we find ny */' + n"f = 0, _ >' i ~"~"__n W DETERMINATION OF AN ORBIT. 169 ~ince the coefficients in these equations are independent of the posi- tions 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, ana 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', X" be the three observed longitudes, /9, /?', ft" the corre- sponding latitudes, and A, J', A" the distances of the body from the earth ; and let A cos p = P , A' cos p = f>', A" cos p' = //', which are called curtate distances. Then we shall have x = p cos A R cos O , d = p' cos A' R' cos Q ', y = p sin A R sin O , y' = p' sin X' R' sin O', z p tan ft, z' = p' tan p, of' = P " cos I" .fl"cosQ", 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 ex- plained 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 formulae would unnecessarily com- plicate 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 A R cos O ) (p r cos / R r cos O ') + n" (p" cos A" R" cos O"), = n (p sin A R sin O ) (p sin X' R' sin O') (6) + n" (p" sin A" R" sin "), = np tan /9 p' tan p -f ri'p" tan ft". 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" 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, w, n", p y p', and p" ; and if the values of n and n" are first found, they will be sufficient to determine p, p', and p". 170 THEOEETICAL ASTRONOMY. The determination, however, of n and n" to a sufficient degree of accuracy, by means of the intervals of time between the obsei vations, requires that p f should be approximately known, and hence, in general, it will become necessary to derive first the values of n, n" y and p r after which those of p and p tf 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, tho 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 deter- mined, 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' 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 = n(p cos(A 0') jRcos(O' O)) 0' cos (A' 0') #) + n" ( P " cos (A"_ 00 - R" cos (O" - 00), = 7i 0> sin (A 0')+jRsm(0' O)) /sin(A' 0') + n" (," sin (A" _ 00 -R" Bin(0"- ')), (7) = nO>sin(A' A) +jRsin(0 /)) R sin(O' A') - n" (p" sin (A" A') R" sin (O " *')), = np tan p p' tan p -f ri'p" tan p'. If we multiply the second of these equations by tan/3', and the fourth by sin (A' 0'), and add the products, we get = n" P " (tan p' sin (A" O tan 0" sin (A' O 0) n".R"sin(0" tan jfiH- n/ (tan ^ sin (A 0') tan/?sin(A' OO) + nJ2sin(' )tan/3 / . (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 f fry and we shall have [#]= KR f sm(Q' O), and similarly [ JH2"] = RR" sin ( O " O ), '] = R'R" sin(O" 00- ORBIT OF A HEAVENLY BODY. 17l Then, if we put _ , [#']' ~ IEK"Y we obtain Substituting this in the equation (8), and dividing by the coefficient of p", the result is _ Q _ ~ p n" ' tan ft" sin (A' Q ') tan ft' sin (A" Q ') V tan ft' sin (A Q ') tan ft sin (A' Q ') n (A' Q ') tan ft' sin (A" Q ' _!L _^M _ J?sin(0' Q)tanff' _ n" N" J tan /3" sin (A' - Q') tan /3' sin (A" 0')' Let us also put M' = tan ^ sin ( ; ~ Q ') ~ tan ^ sin (*' Q') ~~ tan ft" sin (A' 0') tan ft' sin (A" 0')' j^ __ _ sin(0'-0)tanf __ tan/5" sin (A' 0') tan/3' sin (A" 0')' and the preceding equation reduces to -*, JTR (11) "We may transform the values of 3/ r and M " 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 O', and we shall have sin (A' O') tan w' = tan ft'. (12) Let /9 , ft Q " be the latitudes of the points of this circle corresponding to the longitudes A and A", and we have, also, tan ft = sin (A 0') tan w', tan ft" = sin (A" Q') tan w'. Substituting these values for tan/9', sin (A O 7 ) and sin (A" O ; ) in the expressions for M ' and M ", and reducing, they become sin(/y- /?) cos p' cos ft " sin (ft" /3 ") ' cos ft cos ft ' =^ S in( '- 172 THEORETICAL ASTRONOMY. When the value of -77 has been found, equation (11) will give the relation between p and p" in terms of known quantities. It is evi- dent, 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, ft /? and ft" = /? ", we shall have M'= ^ and M fr = oo. In this case, theiefore, and also when /9 ft and ft" ft Q " 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. It will be observed, also, that if, at the time of the middle obser- vation, the comet is in opposition or conjunction with the sun, the values of M f and M " as given by equation (14) will be indeter- minate 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 ,_ sin (A 0') _ s in(Q' Q) " sin (A"')' " sm(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 M f and M" and to obviate the difficulties thus encountered, we obtain from the third of equations (7) the following value of p" : _ n sin (A' A) p ==p f ^E sin (G - A') - ^ # sin (' - AO + 12" sin (" - X) (15) We may also eliminate p between the first and fourth of equa- tions (7). If we multiply the first by tan/3', and the second by cos (A' '), and add the products, we obtain = n"p" (tan f cos (A" 0') tan /?" cos (A' ')) ri'R" tan p cos(O" 0') + w/> (tan /3' cos (A Q') tan/?cos(A' 00) nR tan fi cos (O' O) + R' tan /3', from which we derive ORBIT OF A HEAVENLY BODY. 173 "_ JL tan ^ cos (A Q ') tan ff cos (A' Q') p ~~ P n" ' tan /?" cos (A' 0') tan /5' cos (A" Q ') (16) R" tan/?' cos(O" 00 + 7 12 tan ^ COB (0' 0) R'tan/? tan /S" cos (A' Q ') tan $ cos (A" Q') 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 (A' 0') tan I' = tan p ; (17) and, if we designate by /9, and /? the latitudes of the points of this circle corresponding to the longitudes ^ and X", we shall also have tan /?, = cos (A 0')tan.T', tan /? = cos (A" 0') tan I'. Introducing these values into equation (16), it reduces to _ n sin (/9, /9) cos /3" cos /? ~ p ' sin (/5" /?) cos /5 cos /?, (19) tan r cos /5" cos/?,, / f ,. . n f . R' \ rinC9"- A( ) (^"^(e'-GO + ^^cosCG'-Q)-^). from which it appears that this equation becomes indeterminate when the apparent path of the body is in a plane passing 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 in- determinate, 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 //' and p when the values of n and n" are known, and it remains, therefore, to derive the expressions for these quan- tities. If we put t(f -f) = r", k (" - O = r, (20) and express the values of x, y, z, x lf y y", z" in terms of re', y f , z r by expansion into series, we have f _dx^ T" J_ dV r^_ __ 1_ dV r" 8 x ~ Substituting these values of the differential coefficients in equa- tions (21), and the corresponding expressions for y and y", and putting ORBIT OF A HEAVENLY BODY. 175 -i__i^_i_^L ^'+i/l 12/rf/y 3 dy\ 2>' 24- + 2'6-3 + ^~ *' T" r" 3 T"* dr r '~' _ 1 __, 1 __ , , 3 d *' 3 ~ t " 2 ' + ^''~ 5 " +4 " "' ___ _ . ~yfc 6 ^r' 3 nV* dt " " we obtaiD From these equations we easily derive > Civ ' ' (23) . . 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'z-s'y =|W], sV-a/y^ry], y"x - J'y = [n"], (24) and x'dy' y'dx' is double the area described by the radius-vector during the element of time dt> and, consequently, , - ia double the areal velocity. Therefore we shall have, neglecting the mass of the body, iii which p is the semi-parameter of the orbit. The equations (23 j, therefore, become [r/] = bk V>, [//'] = b"k V~p, [rr"] - (aW Substituting for a, 6, a", b" their values from (22), we find, since 176 THEORETICAL ASTRONOMY. ( 25) From these equations the values of n = ^ J an( ^ n " == r "n ma 7 be derived ; and the results are r" ? A;r'* ' dt '" 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 j- > is reduced to a dt 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 [r'r"~\ From the equations (26) or from (25), since , = TTTT, we find i 4 _ i i e -^T~ Since this equation involves r' and -3-, it is evident that the value dt of ~, in the case of an orbit wholly unknown, can be determined Tl 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 rap'dly, so that we may take n r n" ~ T"' ORBIT OF A HEAVENLY BODY, 177 and similarly W f = ^' Hence the equation (11) reduces to l>" = ~M'p. (29) It will be observed, further, that if the intervals between the observa- tions are equal, the term of the second order in equation (28) vanishes, and the supposition that , = , is correct to terms of the 71 T third order. It will be advantageous, therefore, to select observa- tions 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 JL- -JL n" ~~ N'" and compute the values of N and N ff 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 fr 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 j +* dy r " 2 _ & , dr' r dV r a from which we derive k w neglecting terms of the third order. Therefore dr'_k(r"-r) t ~dt~ ~T+V'~' 12 178 THEORETICAL ASTEONOMY. and when the intervals are equal, this value is exact to terms of the fourth order. We have, also, which gives r )!_ r . (31) Therefore, when r and r" have been determined by a first approxi- mation, the approximate values of r' and -^- are obtained from these equations, by means of which the value of may be recomputed from equation (28). We also compute N _#.B"sin(0"-0') (32) N" ' K'sm(C)' 0) and substitute in equation (11) the values of and -^ thus found. n" N designate by M the ratio of the cui we have If we designate by M the ratio of the curtate distances t o and //', NR 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 -77 may be computed from n __rV'sin(0" iQ ,, "~~ '' and that of -^ 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 f , as derived from the first of equations (10), is either indeter- minate or greatly aifected 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 the value of the ratio of p" to p. Let Y t Y f , 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 ^', and we have Y =E sin(O *'), Y' =E' sin (0' A'), Now, in the last term of equation (15), it will be sufficient to put n_ N n"~ N e " and, introducing Y, Y', Y", it becomes cosec - It now remains to find the value of -77- From the second of equa- tions (26) we find, to terms of the second order inclusive, We have, also, and hence Therefore, the expression (35) becomes But, according to equations (5), NYY'-\-N" and the foregoing expression reduces to sin (/'-/) ' since Y' = R f sin (O r A f ). Hence the equation (15) becomes 1 \^sin(/-0Q ( . -* (36) 180 THEORETICAL ASTRONOMY. If we put ... _ _n_ sin (A' ^) ~~"ri" *sin(/l" /)' JF-! _ . *" . ^ ' * sn - we have = M=M 9 F. (37) Let us now consider the equation (16), and let us designate by X, X' y X" the abscissas of the earth, the axis of abscissas being directed to that point of the ecliptic for which the longitude is O r , then X = JR cos (0 00, X'=K f , X"=JR"eos(0" 00. It will be sufficient, in the last term of (16), to put Jl JL n" ~ N" ' and for 77- its value in terms of N" as already found. Then, since this term reduces to -l^rfV+yW 1 -1\ _ ^ta B r n ^ * J\ pi E , 3 ] tan p cog y,_ , j _ and if we put __n_ tan /3' cos (A Q r ) tan /? cos (A ; Q ; ) ~~ "n 77 ' tan ft" cos (A' Q') tan p r cos (A" Q')' (38) tan^'cos (X 0') tan/3oos(X' Q') the equation (16) becomes ^F'. (39) In the numerical application of these formulae, if the elements are not approximately known, we first assume n when the intervals are nearly equal, and OKBIT OF A HEAVENLY BODY. 181 n" " N" ' as given by (32), when the intervals are very unequal, and neglect the factors F and F'. The values of t o and p fr which are thus ob- tained, enable us to find an approximate value of r f , and with this a more exact value of ,7 may be found, and also the value of F or F f . Whenever equation (11) is not materially affected by errors of observation, it will furnish the value of M 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 f . 66. When the value of M has been found, we may proceed to determine, by means of other relations between p and p tr 9 the values of the quantities themselves. The co-ordinates of the first place of the earth referred to the third, are x, R" cos Q " R cos Q , , .^ y, = E"smO" KsmQ. 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 longi- tude of the first place of the earth as seen from the third, we shall have x, = g cos G, y, = ff sin G, and, consequently, 12"cos(0" 0) jR = ?cos( O), jR"sin(" 0) =gsm(G'). If ^ 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 O, and we shall have, in accordance with equations (81)^ cos * = cos fi cos (A 0), sin 4 cos w = cos /? sin (A 0), sin 4 sin w sin & 182 THEORETICAL ASTRONOMY. from which tan/5 tan w = -T sin (A Q)' tan(A- G ) tan 4 = . costy Since cos ft is always positive, cos ^ and cos (A 0) must have the same sign; and, further, a// cannot exceed 180. In the same manner, if w" and ij/' represent analogous quantities for the time of the third observation, we obtain tan/5" tanw" = - sin (A" Q")' ^ (43) cos 4" = cos f cos (A" 0"). We also have which may be transformed into r a = G> sec R cos^ + .ft 2 sin**; (44) and in a similar manner we find r" 2 = (p" sec /3" #' cos V') 1 + #" sin 2 *". (45) Let K designate the chord of the orbit of the body between the first and third places, and we have X > = (*" _ ,.). _|_ (f _ y). + (2 " _ f ). But = p cos A R cos 0, 2/=i0smA E sin O, z = p tan /5, and, since />" = Mp, f = Mp sin X" jR" sin O", from which we derive, introducing g and (r, a/' # = JW/> cos A" /> cos A g cos $, y" V = Mp sm *-" jO sin A g sin G, z" z = Mp tan/5" p tan /?. Let us now put ORBIT OF A HEAVENLY BODY. 183 Mp cos A" p cos A = ph cos C cos H, Mp sin A" /> sin A = ph cos C sin .5", (46) Mp tan /3" p tan fi= ph sin C. Then we have #" x /oA cos C cos IT # cos 6r, 2/" y = ph cos C sin H g sin (2, z" z = ph sin C. Squaring these values, and adding, we get, by reduction, x 2 = /> 2 /i 2 2g ph cos C cos ( G .H") + f ; (47) and if we put cos C cos ( G H} = cos ?>, (48^ we have x 2 = (/>A y cos ^) 2 -f- <7 2 sin 2 y. (49) If we multiply the first of equations (46) by cos^", and the second by sin A", and add the products; then multiply the first by sin X", and the second by cos K r , and subtract, we obtain h cos C cos (H A") = M cos (A" A), h cos C sin (H - n = sin (/" A), (50) /i sin C =M tan /3" tan ft by means of which we may determine A, f , and H. Let us now put g sin ^ = Ay R sin 4* = B, h cos = b, R" sin 4" = JB", - 6", (51) # cos ?> 6jR cos ^ = c, g cos? b"R" cos 4" = c", jO/i gr cos p = d, and the equations (44), (45), and (49) become 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 r" and the chord x joining their extremities, it is evident that an addi- tional 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 radii- vectores, 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 semi- transverse 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 c?, 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 result- ing 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 sup- posed 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 lc(t-T} 1/2 jt and, for the time t" } = tan %v -j- I tan* ^v 1/2 gi Subtracting the former from the latter, and reducing, we obtain 1/2 gf ~~ cos J-y" cos ^v \ ~q cos ^v" cos i> q / and, since r = q sec 2 ^, this gives On/ 1 t ' C ) Sill 77 \ t/ i/ / I/ ** I r / i * s tt ~\ / /7 I /" r* r\ ^ = - = i: ^ 7=r^ 1 r -|- r'+ cos I (v vjvrr (53) 1/2 1/g \ / But we have, also, from the triangle formed by the chord vc and the radii-vectores r and r r/ , x 2 = r 2 -f- r" 2 2rr" cos (?/' v) = (r + r") 2 4rr" cos' J W v). PARABOLIC ORBIT. 185 Therefore, 21/W" Let us now put r + r" + x = m 3 , r + r" x = n, m and n being positive quantities. Then we shall have r + r"=J(m'+*), 2 cos J (v" v) Vrr" = 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 ^ign when v" v is greater than 180. Combining the last equation with (53), the result is 3k (f - = (m + m). (55) y 2g Now we have sin |- (v" v) = sin v" cos ^v cos ^v" sin ^v. Squaring this, and reducing, we get sin 2 \ (v" v) = cos 2 ^v -\- cos 2 v" 2 cos Jv" cos |v cos J (t^ v), or, introducing r and q, 9 1 s n \ Q , Bin 1 i (w " v) =1 -f Therefore, i " --7= introducing this value into equation (55), we find sin i (v" v) = --=% <> =1= n). Replacing m and n by their values expressed in terms of r, r", and x, this becomes 6k (if' -?> = (r + r" + x)t T (r + r" - x)t, (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 approxi- mations, it may be transformed as follows : Since the chord x can never exceed r + r", we may put 7Ippr = sin/, (57) and, since x, r, and r" are positive, sin f r must aiways be positive. The value of f must, therefore, be within the limits and 180. From the last equation we obtain cosV => + r") 2 -< and substituting for Jt 2 its value given by x 2 = (r + r") 2 4rr" cos 2 (v" v), this becomes * 4rr" cos 2 W v) (r + r ,,y Therefore, we have cos Y' = cos | (y" v) .,, (58) and also " r/ "" Hence it appears that when v" v is less than 180, f belongs to the first quadrant, and that when v" v is greater than 180, cosf is negative, and f' belongs to the second quadrant. Tf we introduce f' into the expressions for m 2 and n 2 , they becoma ^ = (r + ')(!- sin , which give m *= ( r + r") (cos tf + sin J/), W 2 = ( r 4. /') (-4- cos J/ ip sin yj ; and, since f i g greater than 90 when v" v exceeds 180, the equation (56) becomes = (cosy + sin /)- (cos J/-siny) (r PARABOLIC ORBIT. 187 From this equation we get = 6 cos 2 $ sin %/ -f- 2 sin* $/, or /> f ^TTT- = 6 sin !/ 4 sin 8 tf ; and this, again, may be transformed into 6r' (60) Let us now put sn " or sin \*f' = l/ 2 sin #, and we have = 3 sin # 4 sin 3 x = sin 3#. (62) is less than 180, f must be less than 90, and hence, in this case, sin x cannot exceed the value |, or x must be within the limits and 30. When v" v is greater than 180, the angle f is within the limits 90 and 180, and corresponding to these limits, the values of sin x are, respectively, \ and \ \/%. Hence, in the case that v rr v exceeds 180, it follows that x must be within the limits 30 and 45. The equation 1/20 is satisfied by the values 3# 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 15, the required condition will be satisfied by 3x or by 180 3#, and there will be two solutions, corresponding respectively to the cases in which v ff v is less than 180, and in which v" v is greater than 180. Consequently, wnen it is not known whether the heliocentric motion during the interval t rf t is greater or less than 180, and we find 3x greater than 45, the same data will be satisfied by these two different solutions. In practice, however, it is readily known which of the 188 THEOEETICAL 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 o/ n = -, (63) sin 3*= -L (64) T/8 and we obtain We have, also, sin ^/ = |/2 sin x, and hence cos \r' = i/l 2 sin 2 x = I/cos 2x. Therefore sin = 2t sin x V cos and, since JC = (r -|- r") sin f, we have x = 2% (r + r") sin a; I/cos 2x. If we put 3sin# / - ^- ,. x ^ = . o V cos 2x, (b5) sinoa; the preceding equation reduces to From equation (64) it appears that rj must be within the limits and Ji/8- We may, therefore, construct a table which, with ^ as the argument, will give the corresponding value of //, since, with a given value of 37, 3x may be derived from equation (64), and then the value of // from (65). Table XI. gives the values of // corre- sponding to values of r] 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 gene- rally be small, and, consequently, x will be small compared with r and r". As a first assumption we may take r = 1, or r 4 r" = 2, and fj. = 1, and then find x from the formula FiRABOLIC 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 y by means of (63), and with this value we enter Table XI. and take out the corresponding value of p. A second value for K is then found from (66), with which we recompute r and r" ', and proceed as before, until the values of these quantities remain un- changed. 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 -f r" may be found very nearly correct from the numbers already derived. Thus, let y be the true value of log (r -j- r"), and let A?/ be the difference between any assumed or approximate value of y and the true value, or 2/0 = y + Ay. Then if we denote by y f the value which results by direct calculation from the assumed value y Q , we shall have 2A>' y =/(y ) = .Expanding this function, we have 2/o' 2/o =/(y) + 4 *y -4- Ay* + Ac. But, since the equations (52) and (66) will be exactly satisfied when the true value of y is used, it follows that and hence, when AT/ is very small, so that we may neglect terms ot the second order, we shall have Vo' y, = 4 Ay = 4 (y y*). Let us now denote three successive approximate values of log (r + r") b 7 2A 2A/> 2/o"> and let y ' yc = a> y" &' = ' then we shall have a = A (y y), a' = -4 (&'-?). Eliminating A from these equations, we get y ( a f o) = a'y - - ay ', from which ' /' (67; 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 JC is known, is readily effected by means of Table VIII. Thus we have = sin 3#. l/2(r-f-*-") f and, when f 1 is less than 90, if we put . ---- . /~i sin / we get T' = -i 1/2 N sin r ' (r + r") t, (68) or When f exceeds 90, we put N' = sin 3s, and we have / = 4 1/2 JF(r + /')* in which log $ j/2 = 9.6733937. With the argument f we take from Table VIII. the corresponding value of N or JV 7 , and by means of these equations r' = k (t" t) is at once derived. The inverse problem, in which r' is known and K is required, may also be solved by means of the same table. Thus, we may for a first approximation put * = T ' 1/2. and with this value of K compute d, r, and r" . The value of f 1 is then found from and the table gives the corresponding value of N QT JV 7 . A second approximation to x will be given by the equation S_ 1/2 ' NVT- PAEABOLIC CEBIT. 101 or by 3 /sin/ 'v^'^'i/T+y' in which log = = 0.3266063. Then we recompute d, r, and r", v 2 and proceed as before until x remains unchanged. The approxima- tions are facilitated by means of equation (67). It will be observed that d is computed from and it should be known whether the positive or negative sign must be used. It is evident from the equation d = ph g cos y>, since /?, h, and g are positive quantities, that so long as

, 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, = S r V which shows that H is greater than g } and that d is positive, so long as r f 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. How- ever, 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 THEOKETICAL ASTKONOMY. 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

tan /?, and also r" cos b" cos (r O") = p" cos (A" O") R r , r" cos b" sin (r O") = p" sin (A" 0"), (72) r"sin&" =ytan/9", in which I and Z ;/ are the heliocentric longitudes and 6, b" the corre- sponding heliocentric latitudes of the comet. From these equations we find r, r", I, l ff , 6, and b" and the values of r and r" thus found, should agree with the final values already obtained. When I" 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) w we have tan i sin (I ) = tan 6, tan i sin (I" & ) = tan b", which may be written zt tan i(sin (I x) cos (x &) -f sin (re &) cos (I x)) = tan b, tan i (sin (" a?) cos (a; & ) -f sin (re ) cos (r #)) = tan b". Multiplying the first of these equations by sin (I" x) 9 and the second by sin (I a?), and adding the products, we get it tan i sin (x Q, ) sin (f f - - F) = tan b sin (I" x) tan b" sin (I a:) ; and in a similar manner we find it tan i cos (a; & ) sin (I" I) = tan 6" cos (I x) tan b cos (I" x). Now, since x is entirely arbitrary, we may put it equal to I, and we have PAKABOLIC CEBIT. 193 tan i sin (I ft ) = db tan b, . n ' tan 6" tan 6 cos (?' I) (74) the lower sign being used when it is desired to introduce the distinc- tion of retrograde motion. The formulae will be better adapted to logarithmic calculation if we put x = \(l"+ 0, whence l"x=l(l" l) and I x=\(l I")-, and we obtain taut sinQ(r-f Q ft) = : 5 - sin (^+6) - 2 cos 6 cos 6" cos J (/" I) ' ' ( jy \ tanicos(Kr + - O) = g CQ5i S C ' P 8 V, ~ |^. These equations may also be derived directly from (73) by addition and subtraction. Thus we have db tan i (sin (I" ft ) -f sin (I ft )) = tan b" -f tan b, tani(sin(r ft) sin (7 ft)) = tan&" tan6; and, since sin (r ft) -f sin (I ft) = 2 sin (f'+ ^ - 2ft ) cos i (r~ O, sin(r ft) sin^ ft) =2cosi(^'+ ^ 2ft) sin J(^' these become < d.(, ( r +l )- a )- "_. which may be readily transformed into (75). However, since b and b" will be found by means of their tangents in the numerical appli- cation of equations (71) and (72), if addition and subtraction loga- rithms are used, the equations last derived will be more convenient than in the form (75). As soon as ft and i have been computed from the preceding equa- tions, we have, for the determination of the arguments of the latitude u and u", COS I COS I Now we have u = v -f- o>, in which CO = TC ft in the case of direct motion, and CD = ft K 13 194 THEORETICAL ASTRONOMY. when the distinction of retrograde motion is adopted; and we shall have if.-ea-V' 4 and, consequently, x 2 = r 2 + r" 2 - 2rr" cos (u" u), (78) or x 2 = (/' r cos O" it)) 2 + r 8 sin 2 (" u). (79) The value of K derived from this equation should agree with that already found from (66). We have, further, r = q sec 2 (u ), **' = ) ==: - T=, -= COS i (w" >) = T/g 1/r 1/5 By addition and subtraction, we get, from these equations, ) + cos l(w o>)) = = 4- frum which we easily derive ~ cos HK" +)-) cos K"- ) = 4= 1/3 t/r But 1 1 1 / 4/7^ _ 4 T7 + ^~^\ X r and if we put 4 f7r since -yf will not differ much from 1, ^' will be a small angle; and we shall have, since tan (45 + 0') cot (45 + 6') = 2 tan 20', PARABOLIC ORBIT. 195 Therefore, the equations (80) become 1 . 1 e , f . . tan 2^ sin ' sin | (ti" u) Vrr 1 ' ' cos cos w - from which the values of q and w may be found. Then we shall have, for the longitude of the perihelion when the motion is direct, and * = & *, when i unrestricted exceeds 90 and the distinction of retrograde motion is adopted. It remains now to find T y the time of perihelion passage. We have V = U - 0>, 1/'=u" - to. With the resulting values of v and v" we may find, by means of Table VI., the corresponding values of M (which must be distin- guished from the symbol M already used to denote the ratio of the curtate distances), and if these values are designated by M and M ", we shall have * t-T= m m or m m f in which m = -f , and log C Q 9.9601277. When v is negative, the gi corresponding value of M is 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 ^V and compute the corresponding values of T from the equation 2 in which logg, = 1.5883273. When v is greater than 90, we de- 196 THEORETICAL ASTRONOMY. rive the values of N' from the table, and compute the corresponding values of T from 71. The elements q and Tmay 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 n by their values v -\- co and v + &>", become 4dni (^ + tO sinj (v" - v) = * V, Vr Vr Adding together the squares of these, and reducing, we get or Q _ " '^~'_~~ " " Combining this equation with (59), the result is " r -f- r" > and hence, since x = (r + r") sin 7-', 5 = sin 2 (i/' tO co We have, further, from (78), from which, putting s i nv = r ^H^ t (84) x we derive 2i v rr /// > foK."\ cos v = sin J (v v). (85; x Therefore, the equation (83) becomes PARABOLIC ORBIT. 197 = J (r + r") cos 2 tf cos'v, (86) by means of which q is derived directly from r, r", and K, the value of v being found by means of the formula (84), so that cosv ia positive. When f cannot be found with sufficient accuracy from the equa- tion we may use another form. Thus, we have which give, by division, tan(45-Hr') = \/r_ llllli (87) ~r~ -''--- X In a similar manner, we derive tan (45 + ") = \*_ S/Z ( 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 -/ cos v, (89) from which v" v may be computed. From (82) we get If we put tan /' = */_-., (9u) this equation reduces to tan $ (y" + v) = tan (/ 45) cot j (v" v), (91) and the equations (81) give, also, tan i (t/' + t>) = cot i (i/' v) sin 2^, either of which may be used to find v" -f- v. 198 THEORETICAL ASTRONOMY. From the equations cos |v _ 1 cos \v" 1 Vq Vr Vq Vr" by multiplying the first by sin \v" and the second by sin \v, add- ing the products and reducing, we easily find sin 4 (V' v) sin |t> cos \ (v" v) _ 1 Vq Vr Hence we have __ . , _ * T/r =COS^=-7=, Vq Vr which may be used to compute q, v, and v ff when v" v is known. When \(v" v) and J(v" + v), and hence t>" 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 Y. have been derived from the equations (52), instead of completing the calculation of the ele- ments of the orbit, we may find q and T, and then, by means of these, the values of r f and v f 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 from (33), or, in the special cases already examined, from the equations (37) and (39). In this way, by successive approximations, the determination of para- bolic 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 M=?"- 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 formula; 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. a 6 1864 Jan. 10 6* 57 m 20-.5 19* U m 4'.92 + 34 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 declina- tions 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. A (3 1864 Jan. 10 7 ft 24" 3' 297 53' 7".6 -f 55 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 pre- cession and nutation to the mean equinox of 1864.0, and reduce th<* times of observation to decimals of a day, and we have t = 10.30837, A = 297 52' 51".l, = + 55 46' 58".4, 1? = 13.27682, A' = 302 57 34 .4, /?' = 57 39 35 .9, tf' = 16.29299, A" = 310 31 35 .0, "= + 59 38 18 .7. For the same times we find, from the American Nautical Almanae, O =290 6' 27".4, log E =9.992763, 0' =293 7 57 .1, logJ?' =9.992830, O" = 296 12 15 .7, log #' = 9.992916, which are referred to the mean equinox of 1864.0. It will gene- rally 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"~r"~ N"' Then we have M== t"- t tan/5'sin(A O') tan/9sin(A' O') H t tan ?' sin (A' ') tan jf sin (A" O')' and g sin ( - O) = R" sin (0" - Q), g cos(G O) = R" cos(O" O) R; h cos C cos (H A") = Jf _ cos (X" A), A cos C sin (jff A") = sin (A" A), AsinC = M tan/5" tan/3; from which to find Jf, 6r, #, IT, C> an d ^ Thus we obtain log M= 9.829827, JET=r 94 24' 1".8, G = 22 58' 1".7, C = 40 28 21 .9, log g = 9.019613, log h = 9.688532. A" cos /? Since r- = M -- -^ = 0.752. it appears that the comet, at the time A cos p of these observations, was rapidly approaching the earth. The quadrants in which G O and H k" must be taken, are deter- mined by the condition that g and h cos must always be positive. The value of M should be checked by duplicate calculation, since an error in this will not be exhibited until the values of X f and f} f are computed from the resulting elements. Next, from cos 4. = cos /5 cos (A O), cos t" = cos /9" cos (A" 0";, cos

/ jy i \ \ sec 20' (I (t*" -f- ) ,) = - - - -j =, cos I (t* w) K rr we get 0' = 22' 47".4, w = 115 40' 6' ; .3, log q = 9.887378. Hence we have * = 01 -f = 60 23' 17". S, 204 THEORETICAL ASTRONOMY. and i = u > = 27 12' 6".l, v" = ti" = 37 38' 43 '.1. Then we obtain log m = 9.9601277 | log 5 = 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 and T=t = t 13.74364, m T=t' =f 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 M= f -. 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 Nr% sin v = if' 3^ N"i"* 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 fol- lowing elements: T 1863 Dec. 27.56471 Washington mean time. = 6023'17".8 _._ . , ic and Mean log q = 9.887378. Motion 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 n ', and correct the \alue of M, 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 M is exact, the result- ing elements must by calculation reproduce this value, and since the computed values of A, A", /9, and ft" will be the same as the observed values, the computed values of X and ft' must be such that when substituted in the equation for M, the same result will be obtained as when the observed values of X' and ft' 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 ', and is independent of the angle at the earth between the sun and comet. Hence, the spherical co- ordinates of any point of the great circle joining these places of the sun and comet wilj, 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 A' and ft' must give the same value for w r as that which is obtained from observa- tion. But if we represent by if/ the angle at the earth between the sun and comet at the time ', the values of if/ 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 4/ = cos p cos (A' O')> sin 4-' cos w' = cos ft' sin (A' '), (93) sin 4/ sin w' = sin ft, give, by differentiation, cos p dl' = cos w' sec p d$ r , dp = smufcoa(X O'HV- From these we get cosft'dA' tan(*' Q') dp sin p which expresses the ratio of the residual errors in longitude and latitude, 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 X r and $' by means of these elements, and then the comparison of this value of tan w f with that given by observa- tion will show whether any further correction of M is necessary, and if the difference is riot 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 f = 32 31' 13".5, u' = 148 IV 19".8, log / == 9.922836. Then from the equations tan (f Q> ) = cos i tan u', tan b f = tan i sin (l f & ), we derive I' = 109 46' 48".3, V = 28 24' 56".0. By means of these and the values of O' and R f , we obtain A' = 302 57' 41".l, p = 57 39' 37".0 ; and, comparing these results with the observed values of X' and /?', the residuals for the middle place are found to be Comp. Obs. cos p AA' = -f 3".6, A/9 = + I'M. 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 1 given by observation is log tan w' = 0.966314, and that given by the computed values of A' and ft' 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 Yb the value of tan&/ 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 f is 0.000067 less than the observed value, and, in finding the new value of Mj we must use log tan w' = 0.966381 in computing /? and /?/' involved in the first of equations (14). If the first of equations (10) is employed, we must use, instead of tan/9' as derived from observation, tan p = tan vf sin (A' 0'), or log tan f = 0.966381 + log sin (A' 0') = 0.198559, the observed value of X' 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 r shall be zero. A system of elements computed with the same data from log M= 9.822906 gives for the error of the middle place, C.-O. cos f A;/ = 1' 26".2, &p = 40".l. 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 -^ = 0.006831, / f\/ JLV 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 My log M= 9.829582, which differs only in the sixth decimal place from the result obtained by varying tanw' and retaining the approximate values t = -r f = j' 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 formulae 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 com- plete 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 A differing but little from the correct value, by comparison of intermediate observations with these different sys- tems 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 f , 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" EcosD sin A, z=R"smD" EsmD. 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 K, 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 K sin G, z = g sin K. VAKIATION OF THE GEOCENTKIC 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) = jR" cos D" sin (A" A), (96) g sin iT = .#" sin Z>" jR sin D, from which #, JT, and G may be found. If we designate by x n y n z, the co-ordinates of the first place of the comet referred to the third place of the earth, we shall havf x, = A cos 5 cos a -f g cos K cos G, y, A cos d sin a -f- g cos K sin (r, 2, = A sin 5 -f- # sin K. Let us now put x, = h' cos C' cos IT, ^ == A' cos :' sin #', z, = h' sin C', and we get h f cos % cos (.ff ' G) = A cos fl cos (a G) + cos JT, A' cos C' sin GET G) = J cos sin (a (f). C97) A.' sin C' = J sin '= cos C' cos H' cos <5" cos a"-\- cos C' sin JET' cos 5" sin a"-}- sin C' sin <5", or cos ?/ = cos ' cos ' = c, Rsm*=B, R cos * = b, (103) R" sin 4," = ", R" cos 4" = &", and we shall have = T/(J" c) 2 -f C 2 , &) 2 + 2 , (104) '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 T having been already found from the assumed value of J. Then x is obtained from the equation 2r' fj. being found by means of Table XI., and a second approximation to the value of J" from *. (105) The approximate elements will give A" near enough to show whether the upper or lower sign must be used. With the value of A" thus found we recompute r" and x as before, and in a similar manner find a still closer approximation to the correct value of A n '. A few trial? 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) = A cos d cos (a A) E cos D, r cos b sin (/ A) = A cos d sin (a A), r sin b = A sin <5 R sin D ; VARIATION OF THE GEOCENTRIC DISTANCE. 211 r" cos V cos <7' A") = A" cos V cos (a" - A") K" cos ", r" cos 6" sin (r A"} = A" cos d" sin (a" 4"), (107) r" sin 6" = J" sin 5" " sin Z>", in which 6, b" y I, I" are the heliocentric spherical co-ordinates re- ferred 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 re- ferred 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 f and compare it with the corresponding observed place, and if we denote the computed right ascension and declination by a/ and d Q ', respectively, we shall have a' + <*'=', *+* = .', in which a' and d' denote the differences between computation and observation. Next we assume a second value of J, which we repre- sent by J -f- d J, and compute the corresponding system of elements. Then we have a!' and d" denoting the differences between computation and obser- vation for the second system of elements. We also compute a third system of elements with the distance J J, and denote the differ- ences between computation and observation by a and d; then we shall have and similarly for c?, d f , and d". If these three numbers are exactly represented by the expression m in which J + x is the general value of the argument, since the values }f a, a', and a" will be such that the third differences may be neg- lected, 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 J, or within the limits x = dA and x = -f- <5J, the as- sumed values J d J, J, and J + J being so taken that the correct value of J shall be either within these limits or very nearly so. To find the coefficients m, n, and o, we have m n -f- o = a. m a', m -{- n-\- o a", whence ? = a', n = (a" a), o = (a" -f a) a'. Now, in order that the middle place may be exactly represented in right ascension, we must have which we find or In the same manner, the condition that the middle place shall be exactly represented in declination, gives 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 3 f AO/, for a given increment as- signed to J, is the same as that of &d'. But if we denote the value of x for which the error in a' is reduced to zero by x', and that for which &d' = 0, by x", the most probable value of x will be -- in which n = $(a ff a) and n f = \(d ff -- 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 9 the values of a, a', and a" must be multiplied by cos 8 f . The value of dA 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 A will suffice. Then we shall have o = 0, and hence m = a f , n = a" a'. The condition that the middle place shall be exactly represented, gives the two equations "' ' 0, Q. 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 #, a final system of elements computed with the geocentric distance A -f- #, 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 J, 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, d, 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. 1858 June 11.0 July 13.0 Aug. 14.0 a 141 18' 30".9 144 32 49 .7 152 14 12 .0 6 + 24 46' 25".4, 27 48 .8, -f 31 21 47 .9, which are referred to the apparent equinox of the date. These places are free from aberration. 214 THEOEETICAL ASTRONOMY. We shall take the ecliptic for the fundamental plane, and con- verting 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 ye^tr, we get t = 162.0, A = 135 51' 44".2, p = + 9 6' 57".8, If = 194.0, *' = 137 39 41 .2, p = 12 55 9 .0, *" = 226.0, A" = 142 51 31 .8, ^' = + 18 36 28 .7. From the American Nautical Almanac we obtain, for the true places of the sun, Q = 80 24' 32".4, logJ? =0.006774, O' =110 55 51 .2, log^R' =0.007101, 0" = 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 ^ and ft in place of a and <5, and O in place of A, in the equations which have been derived for the equator as the fundamental plane. Therefore, we have g cos (Q O) = R" cos (O" O) R, <7 sin ( - 0) = R" sin (O" - O) ; cos 4 = cos p cos (A O), cos 4/' = cos /3" cos (A" 0") R cos 4 = b, 12" cos *" = &", JB, 12"sin4/' = .B", from which to find G, g, b, B, b", and B ff , all of which remain unchanged in the successive trials with assumed values of J. Thus we obtain G = 201 T 57".4, log B = 9.925092, b = -f 0.568719, log g = 0.013500, log B" = 9.510309, b" = -f 0.959342. Then we assume, by means of approximate elements already known, log J = 0.397800, and from V cos :' cos (jff ' G) = J cos p cos (A G) + g, V cos ' sin (H' G) = Acosp sin (A G\ h r sin C' =4 sin p, we find U', C r , and A'. These give H' = 153 46' 20".5, C' = + 7 24' 16".4, log A' = 0.487484. NUMERICAL EXAMPLE. 215 Next, from cos = 116 56' 33".7, o" = u" a> = 88 47' 55".l, from which we get T= 1858 Sept. 29.4274. From these elements we find log r' == 0.212844, v f = 107 7' 34".0, u' = 21 59' 12".3, and from tan (f &) cos i tan u', tan b f = tan i sin (J! & ), we get I' = 154 56' 33".4, 6' = + 19 30' 22".l. NUMERICAL EXAMPLE. 217 By means of these and the values of O' and R f , we obtain A' = 137 39' 13".3, p = + 12 54' 45".3, and comparing these results with observation, we have, for the error of the middle place, C. O. / = 27".2, A/5' = 23".7. From the relative positions of the sun, earth, and cctnefc 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 J, log J = 0.398500, from which we derive H' = 153 44' 57".6, C' = -f 7 24' 26".l, log h' = 0.488026, log C= 9.912587, logc = 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, I" = 144 17 12 .1, b" = + 35 8 37 .8, log r" = 0.054825, and from these, = 165 15' 41".l, i = 63 2' 49".2, u = 12 10 30.8, w"=:40 13 26.0, ) cos sin ^ z, =(4 4>)sin/?, the axis of x being directed to the vernal equinox. Let us now designate by O 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 0, t T/" T> Q^ s~\ The heliocentric co-ordinates of the centre of the earth are X Q = R Q cos cos Q , Y Q = R cos T sin O , Z = R sin J . But the heliocentric co-ordinates of the true place of observation will be X+x,, Y+y n Z + z,, or and, consequently, we shall have R cos O (4, 4>) cos ^ cos A = /? cos r o cos Q p sin TT O cos 5 cos / , R sin O (4 ^ ) cos sin A = .R cos ^ sin O Po si n ^o cos ^o sm ^o> ( J, J ) sin /? = .Ro sin ^? sin ^ sin b . If we suppose the axis of x to be directed to the point whose longi- tude is O > these become DETERMINATION OF AN OKBIT. 223 R cos (0 O ) (4 4>) cos 13 cos (A O ) = E Q cos 2" p sin TT O cos 6 cos (7 O ), .# sin (O Go) (4 4.) cos /? sin (A O ) = (2) p sin TT O cos 6 sin (1 O )> ( A, J ) sin /? R Q sin 2" p Q sin TT O sin 6 , from which R and O may be determined. Let us now put Z>; (3) then, since ;r , S w and O O are small, these equations may be reduced to R = D cos (X O ) ^o PQ cos fy> cos (1 Q ) + R , R (O O ) = D sin (A Q ) 7r /? cos b sin (J Q 8 ), Dtan/? 7r ^o sin6 + ^ ^o- Hence we shall have, if TT O and 2" are expressed in seconds of arc, (4) 000 . o 206264.8 , 206264.8 D sin (A Q ) TT O /QO cos b sin (^ ) W W ~T~ " p from which we may derive the values of O 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 t Q , to be determined by the interval which is required for the light to pass over the distance A, J . Consequently we ought to add to the time of observation the quantity ( J, J ) 497'.78 = 497'.7S D sec /9, (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 A derived from equations (1) and the longitude O 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 /9 should be reduced by applying the correction for pre- cession 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 in- tersects 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 /? = 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 R 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 from this point of reference, and /9, its latitude as seen from this point, we shall have A, cos /?, = A cos /9, A t sin ft, = A sin /5 J? sin S , from which we easily derive the correction /?, /9, or A/9, to be applied to the geocentric latitude. Thus, we find A/9 = -^> cos /9, (6) ~t 2 Q being expressed in seconds. This correction having been applied to the geocentric latitude, the latitude of the sun becomes 2=0. The correction to be applied to the time of observation (already diminished by the time of aberration) due to the distance J, J will be absolutely insensible, its maximum value not exceeding O s .002. It should be remarked also that before applying the equa- tion (6), the latitude JT should be reduced to the fixed ecliptic which it is desired to adopt for the definition of the elements which deter- mine 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 formulae for which we shall now investi- gate. For this purpose, let us resume the equations (6) 3 ; and, if we multiply the first of these equations by tan /9 sin A" tan /9" sin A, the second by tan/3" cos A tan /9 cos A", and the third by sin (X )"\ and add the products, we shall have nR (tan /9" sin (A Q) tan /9 sin (A" Q)) p' (tan /? sin (A" A') tan jf sin (A" A) -f- tan $' sin (A' A)) R' (tan P" sin (A Q') tan ft sin (A" ')) + n"R" (tan ft" sin (A Q") tan /5 sin (A" 0")). It should be observed that when the correction for parallax is applied 15 226 THEORETICAL ASTRONOMY. to the place of the sun, p f 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 jfiTthe 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 have tan p sin (A K} tan J, tan 0" = sin (A" JE")tanJ.. Introducing these values of tan and tan ft" into the equation (7), since sin (A O) sin (A" K) sin (A" Q) sin (A JT) = sin (A" A) sin (Q JT), sin (A' A) sin (A" JT) + sin (A" A') sin (A JT) = -f- sin (A" A) sin (A' K\ sin (A O') sin (A" K) sin (A" G') sin (A JT) = sin (A" A) sin (G' JD, sin (A 0") sin (A" K) sin (A" 0") sin (A K) = sin (A" A) sin (G" K\ we obtain, by dividing through by sin (A" A) tan /, = nR sin (G -ET) + p' (sin (A' K} tan p f cot I) Let /9 denote the latitude of that point of the great circle passing through the first and third places which corresponds to the longitude ^', then tan ft = sin (A' K) tan J, and the coefficient of p r in equation (9) becomes sin (ft ff') cos ft cos p r tan / Therefore, if we put we shall have Bn - "= cos /. tan/' ,, This formula will give the value of p f t or of J', when the values of n and n" have been determined, since a and K are derived from the data furnished by observation. DETERMINATION OF AN OEBIT. 227 To find K and 7, we obtain from equations (8) by a transformation precisely similar to that by which the equations (75) 3 were derived, 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 I is to be taken always less than 90. 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 tf y when the corrections for parallax are applied to the places of the sun, being as already noticed in the case of p f . 79. If we multiply the first of equations (6) 3 by sin 0" tan/3", the second by cos 0" tan/3", and the third by sin (A" ")> and add the products, we get 0=w/>(tan"sin(0" A) tan/9sin(0" *")) njRtan/9"8in(0" 0) -V (tan p' sin (0" A') tan f sin (" A"))+# tan p sin (" Q'), (13) which may be written 0= np (tan /? sin (A" 0") tan /?" sin (X ")) rc^tan 0" sin (0" 0) -f ft (tan /?" sin (A' Q ") tan ft sin (A" ")) -VCtan/S' tanft)sin(A" Q") + # tan/9' ; sin(O r; 00- Introducing into this the values of tan ft, tan /3", and tan /3 in terms of 1 and jfif, and reducing, the result is Q = np sin (A" A) sin ( " K) nR sin ( " - ) sin (A" ") + R r sin (0" 00 sin W -K"). Therefore we obtain _^/sin(^ AQ q sec^ sin(^ Q^) \ = ~ w \ sin (A" A) + sin (A" A) ' sin (Q" K} / sn /i sin (A" A)sin(O" JE") But, by means of the equations (9) 3 , we derive " 0) = (N ri) Rsm(O"- 0), 228 THEORETICAL ASTRONOMY. and the preceding equation reduces to sin (A" 0") n\sin(A" A) ' sin(A" A)" s in(" _ y/ (u . N\Ksm(Q" Q)8m(A" K) "~rT/ sin (A" A)sin(O" K) ' To obtain an expression for p rf in terms of p r , if we multiply the first of equations (6) 3 by sin tan/9, the second by cos tan/9, and the third by sin (X ), and add the products, we shall have 0=V'(tan^sin(A' / ) tan /S" sin (A )) n"K"tansin(" ) *' ) tan/?' sin (A ))+ J R'tan/5sin(' ). (15) Introducing the values of tan /9, tan /9 r , and tan /9 r/ in terms of K and /, and reducing precisely as in the case of the formula already found for p, we obtain p' I sin (A' A) a sec j? sin (A ) \ n" \ sin (A" A) sin (A" A) sin ( K) / T^et us now put, for brevity, . Esm(Q-K) _R'sm(Q f K) o = - j c i a a ^_^sin(0 // g) sec/g f , _.Rfi"8in(0" 0) a J "" sin (A" A)' in(^ JQ , A sin (A .g) - and the equations (11), (14), and (16) become (18) If n and n" are known, these equations will, in most cases, be sufficient to determine p, //, and //'. DETEKMINATION OF AN OKBIT. 229 80. It will be apparent, from a consideration of the equations which have been derived for p, p f , and p lf t that under certain circum- stances 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 plaoe, we have a = 0, and equation (11) reduces to n"R" sin (Q" K) + nR sin (0 K} = R' sin (0' K). If the ratio of n" to n is known, this equation will determine the quantities themselves, and from these the radius-vector r 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== O', or K= 180 -j- O', and hence n"R" sin (O" 0') nR sin (' O) = 0, or n" Jg8in(0' 0) n ~.R"sm(O" O')' 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 ^ = X" and /9 = /3", 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 observa- tion. If, however, A is equal to A" and /9" differs from /9, it will be possible to derive p f , and hence p and p"; but the formulae which have been given require some modification in this particular case. Thus, when J = A", we have K=X' = 1 9 7=90, and ft =90, 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 f , which is a sec ft', gives a = sin p cot / cos /?' sin (/ K) t (19) and when A = A", it becomes simply = cos p sin (A' K). / N"\ \ n" / 230 THEORETICAL ASTRONOMY. Whenever, therefore, the difference A" A is very small compared with the motion in latitude, a should be computed by means of the equation (19) or by means of the expression which is obtained directly from the coefficient of p 1 in equation (7). When A = X" = K, the values of Jf w Jtf/', M 2 , and M 2 " 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 f , as given by equations (13) and (15), without introducing the auxiliary angles, we shall have = ft_ tan p sin (A" Q") tan p' sin (A' Q") n ' tan ft sin (A" 0") tan p' sin (A 0") / __N_\ _ jRtan"sin(0" Q) _ M n I tan ft sin (/I" 0") tan ft" sin (A - 0") 1 tan /? sin (A' Q) tan p sin (A Q) tan /9 sin (A" 0) tan p" sin (A 0) jR"tansin(0"-- 0) tan /? sin (A" 0) tan p' sin (A )' Hence _ tan /?' sin (X* ") tan p' sin (A' Q") 1 ~~ tan sin (A" 0") tan ft" sin (A 0") ' _ tan ft sin (A' 0) tan /?' sin (A Q) MI Z ' tan /S sin (A" - 0) - tan 0" sin (*)' , , _ J? tan ^ sin (0^0) _ 8 ~ tan /5 sin (A" 0") tan ft" sin (A ")' _ _ R" tan ft sin (Q" 0) __ a = = tan ft sin (A" 0) tan ft" sin (A ) ' are the expressions for M lt -Mi", M z , and M 2 " which must be used when X = X' r or when A is very nearly equal to A"; and then p and p rr will be obtained from equations (18). These expressions will also be used when A" X = 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 equa- tions (7) 3 by tan/3" and the fourth by sin (A" O r ), and add the products, then multiply the second of these equations by tan /? and the fourth by sin (A '), and add, and finally reduce by means of the relation NE sin (' - 0) = N"R" sin (0" - '), we get o DETERMINATION OF AN ORBIT. 231 tan p' sin (A' ') tan ft' sin (A" ') n tan ft" sin (A ') tan ft sin (A" ') .#" tan/S" sin (" 0') tan ft" sin (A ') tan ft sin (A" tan jt sin (A ') tan ft sin (A' ') tan ft" sin (A ') tan ft sin (A" ') R tan /5 sin (') (21) tan/S" sin (A 0') tan/? sin (A" 0') These equations are convenient for determining p and p' r from // ; 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 cases in which the equations (18) fail. If we eliminate p" from the first and second of the equations (6) 3 we get Q = np sin (A" A) nR sin (A" Q) p' sin (A" A') + R' sin (A" 0') ri'R" sin (A" 0"), from which we derive p > sin(A"-AQ -;Tsin(A"-A) nR sin (A" Q ) R sin (A" Q') + ri'R' sin (I" Q ") n sin (A" A) Eliminating p between the same equations, the result is P' sin (A'- A) p - n " sin (A" -A) nR sin (A Q) R' sin (A Q') -f- n"R" sin (A Q") n" sin (A" A) These formulae will enable us to determine p and p" from p 1 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 formulae previously given whenever these may be applied. If we eliminate successively p" and p between the first and fourth of the equations (7) 3 , we get tan P" cos (X Q ') tan p cos (X 1 Q ') _ p ~ _ P ~" n tan /3" cos (A Q') tan p cos (A" Q') tan/3" nR cos (0' Q ) R + n"R" cos(Q" Q') n tan /5" cos (A Q ') tan p cos (A" O') ' tan j3' cos (A Q') tan /9 cos (A' Q ') n"' tan ft" cos (A Q ') tan ft cos (A" Q') tan/9 nR cos (Q' Q) JT + ri'R" cos (0" 0') n" ' tan 0" cos (A ') tan/? cos (A" 0') ' 232 THEORETICAL ASTRONOMY. which may also be used to determine p and p ff 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 in- volved in the formula for p f , that the equations (26) 3 will enable us sec /?' = (n + n") - n - c, (28) in which, if we introduce the values of and n + n rr as given by fi (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 />' 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, r 2 r" 2 will be a quantity of an order superior to r 2 , and when these intervals are equal, we have, to terms of the fourth order, ?L = !L' n r' The equation (27) gives 2/ 8 (n + n" 1) = TT". Hence, if we put P- n - n' (29) we may adopt, for a first approximation to the value of p', and p f will be affected with an error of the first order when the in- tervals 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 in- tervals 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 , thus facilitating the successive approximations ; and when a is a very small quantity, the equality of the intervals is of the greatest importance. From the equations (29) we get 1 1/ _^ ( Tf-\ 1 * ~T ;\ t-t \\ /QIA and introducing P and Q into (28), there results This equation involves both p f and r' as unknown quantities, but by means of another equation between these quantities p f may be eliminated, thus giving a single equation from which r f may be found, after which p f may also be determined. DETERMINATION OF AN ORBIT. 235 82. Let ty 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/9 7 tan w r = sin (A' Q') -, (33^ cos *' = cos p cos (A' 0'), by means of which we may determine ij/, which cannot exceed 180. Since cos ft' is always positive, cos i// and cos (A' O ') must have the same sign. We also have /2 = j. + jgr. _ 2 A'R cos 4,', which may be put in the form r' 2 = 0>' sec /3' R cos 4? -f R 2 sin 2 4', from which we get P ' sec ,3' = R' cos 4' VV* jR"sin4/. (34) Substituting for /) r sec ^ r its value given by equation (32), we have For brevity, let us put 4, c = k m (35) ' J c oS = 4> and we shall have k ^ = R cos V 1/r' 2 jR' 2 sm 2 V. (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 sm z' 236 THEORETICAL ASTRONOMY. Substituting this value of r f , in the preceding equation, there results (& - E' cos 40 sin z' + K sin *' cos z' = j^jj^-,> C38 ) and if we put TJ Q sin C = K sin V, , o cos Z = lc Q R' cos *', (39) the condition being imposed that ra shall always be positive, we have, finally, sin (X q= C) = m sinV. (40) In order that m may be positive, the quadrant in which is taken must be such that % shall have the same sign as 1 , since sin i// is always positive. From equation (37) it appears that sin z' must always be positive, or 2' < 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 y / sin (X + 4/) K sin (z r + *Q sin 4/ sin z' Therefore, and, since p r is always positive, it follows that sin (z' + ty) must be positive, or that z 1 cannot exceed 180 ty. 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, '\J/ = 180 when the body is in opposition, and vJ/ = when it is in conjunction. Consequently, it becomes impos- sible to determine r r by means of the angle z r ; but in this case the equation (36) gives *.-=-#+.', 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 obser- vation at the superior conjunction when /9' = 0, is physically impos- sible. The value of r' may be found from these equations by trial ; and then we shall have when the body is in opposition, and / T}f fiJ 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 a = 0, let us divide equation (32) through by c, and we have *" c 1 p' sec /?' ~+7~ ~~T~' The equations (17) give and if we put c c _ r ~1 ; 7T ^>n- we shall have since c = GO when a = 0. Hence we derive (42) But when the great circle passing through the three observed places passes also through the second place of the sun, both c and C be- come 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 m sin 4 z' sin z' cos C + cos z' sin C, 238 THEORETICAL ASTRONOMY. and, by squaring and reducing, this becomes ra 2 sin 8 z' 2m cos C sin 5 / -f- sm2 *' sm2 C = 0. When is within the limits 90 and + 90, cos will be positive, and, m 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 exceeds the limits 90 and -f- 90, cos 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 sin 2 is real and positive, there must be at least two real roots one positive and the other negative whether cos 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 = ra sin* /, (43) since the lower sign in (40) follows directly from this by substituting 180 z r in place of 2'; 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 only 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) 1? 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 ,' = 0. Further, it follows from the equation (37) that this root must be and such would be strictly the case if, instead of the assumed values of P and , 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 perturbation* in the motion of the earth. DETERMINATION OF AN ORBIT. 239 In the case of the earth, sm(Q" Q)' sm(O" O') inCO" O)' and the complete values of P and Q become _ ^'sinCQ'-G) ~ f "" '' , 3 / ER' sin(0'- Q) + RR' B in(0"- Q') ~ ' \ '" ; and since the approximate values P=C Q = rr" differ but little from these, as will appear from the equations (27) 3 , there will be one root of equation (43) which gives z r nearly equal to 180 ij/. This root, however, cannot satisfy the physical con- ditions 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 1 are excluded by the nature of the problem, since r f 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 180 ij/, there will remain two, of which one will be excluded if it gives z' greater than 180 ^ r , 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 ^'j 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 Clements. It will then be necessary to compare the elements com- puted 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 p _ r/ sin (u r u) -/^sirTK^')' ^ _ sin (u f - u) + *V' sin (u"- u'} - 2r l it may occur that cos is negative, and the positive root will actually belong to the orbit of the comet. The condition that one value of z f 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 m c and , 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 m and which will accord with this supposition, we may determine the limits required. "Differentiating equation (43) with respect to z f , we get cos (d C) = 4m sin V cos z' ; and, in the case of equal roots, the value of z f as derived from this must also satisfy the original equation sin (z' C) = m sin V. To find the values of m and which will fulfil this condition, if we eliminate m between these equations, we have sin z' cos (z* C) = 4 cos z' sin (z' C), from which we easily find sin (2z' = 1 sin C. (45) This gives the value of in terms of z' for which equation (43) haa DETERMINATION OF AN ORBIT. 241 equal roots, and at which it ceases to have four real roots. To find the corresponding expression for m , we have _ sin (Y C) cos (Y C) ; sin 4 z' ~ 4 sin V cos / in which we must use the value of given by the preceding equation. Now, since sin (2z f ) must be within the limits 1 and -f- 1, the limiting values of sin will be + f and |, or must be within the limits + 36 52'.2 and 36 52'.2, or 143 7'.8 and 216 52'.2. If is not contained within these limits, the equation cannot have equal roots, whatever may be the value of ra , and hence there can only be two real roots, of which one will be positive and one negative. If for a given value of we compute z' from equation (45), and call this ZQJ or sin O ' C) = | sin C, we may find the limits of the values of m , within which equation (43) has four real roots. The equation for Z Q ' will be satisfied by the values 2* '-C, 180-(2z '-C); and hence there will be two values of m , which we will denote by m l and m 2 , for which, with a given value of , equation (43) will have equal roots. Thus we shall have m= sin(y-C) sin \ f ' and, putting in this equation 180 (2z/ ) instead of 2z ' , or 90 (V C) in place of z ', cos< It follows, therefore, that for any given value of f, if m is not within the limits assigned by the values of m x and m 2 , 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 1 is not very nearly equal to 180 tj/, the positive root, when exceeds the limits -f 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 THEORETICAL 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 m is within the limits m l and m 2 ; but, if m surpasses these limits, there will be only two real roots. Table XII. contains for values of from 36 52'.2 to + 36 52'.2 the values of m l and m 2 , and also the values of the four real roots corresponding respectively to m 1 and m 2 . In every case in which equation (43) has three positive roots and one negative root, the value of m must be within the limits indicated by m 1 and m 2 , and the values of z f will be within the limits indicated by the quantities corresponding to m l and m 2 for each root, which we designate respectively by z/, z/, z 3 ', and z/. The table will show, from the given values of m and 180 ij/, whether the problem admits of two distinct solutions, since, excluding the value of z', which is nearly equal to 180 ij/, and corresponds to the orbit of the earth, and also that which exceeds 180, it will appear at once whether one or both of the remaining two values of z 1 will satisfy the condition that z' shall be less than 180 <$,'. 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 f = 0, and hence from (36) we derive Substituting this value for k in the general equation (32), we have - and, since p r must be positive, the algebraic sign of the numerical value of 1 Q will indicate whether r f is greater or less than R f . It is easily seen, from the formulae for l w b, d, &c., that in the actual application of these formulae, the intervals between the observations not being very large, 1 will be positive when /9' /? and sin (O' K) have contrary signs, and negative when /9 r /9 has the same sign as sin (O' jfif). Hence, when O' K is less than 180, r r must be less than R r if ft' /9 is positive, but greater than R f if /?' /9 is negative. When O' K exceeds 180, r r will be greater than R' if /9' /? is positive, and less than &' 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 circ'e through the two extreme observed places of the comet, r f must be greater than R' when the place of the comet for the middle observa- tion 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 // and r' derived from the assumed values T " P = and Q = rr", we may evidently derive more approximate values of these quantities, and thus, by a repetition of the calcula- tion, make a still closer approximation to the true value of p r . To derive other expressions for P and Q which are exact, provided that r f and p r are accurately known, let us denote by s ff 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 f , s" must neces- sarily 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, and therefore we obtain s "[r/]=:TV> /[W'J^rV^ s[tV']=Ty$. (46) Then, since we shall have and, consequently, [r/] - .. T 8 f , _^ = =- (47) T-T" / oV p sn or _ . / rr"sin(tt" M) \ _ 1 = \2v / r/ 7 cosJ(w" u)l p sin* %(" Therefore, the equation (59) becomes 1 _ E- sin "- C ] ' ( J Let Jt r be the chord of the orbit between the first and third places, and we shall have x' 2 = (r + r") 2 4rr" cos 2 J (n" w). Now, since the chord #' can never exceed r + r /r , we may put x'=(r-fr")sin r ', (61) and from this, in combination with the preceding equation, we derive 7 cos J (M" M) = (r + /') cos /. (62) DETERMINATION OF AN ORBIT. 249 Substituting this value, and \rr"~\ = -, Vp, in equation (60), it reduces to E"-E- S m(E"-E) r" 1 1 _ sin 3 1 (E" E) "(r+ r") 8 cos 3 p * 7* "*" 7 The elements a and e are thus eliminated, but the resulting equation involves still the unknown quantities E" E and s f . It is neces- sary, therefore, to derive an additional equation involving the same unknown quantities in order that E" E maybe eliminated, and that thus the ratio s r , which is the quantity sought, may be found. From the equations r a ae cos E, r" = a ae cos E", we get r " + r = 2a 2ae cos J (E" -f E) cos | (E" E). Substituting in this the value of ecos%(E"-}- E) from (58), we have r " 4. r = 2 a s i n * j (E" E) + 21/77' cos j (w' ; t*) cos J (^" ^), and substituting for sin l(E tr E) its value from (57), there results But, since t*"~ u) (l-2sin r sin' j (u ;/ M) _ (K r ]) 2 = 2r y2 / _ 1 ^> ~ 2prr ff cos 2 ^ (t*" u)~ s' 2 \ 21/^y 7 cos we have from which we derive which is the additional equation required, involving E" j&and ' as unknown quantities. Let us now put (65) , sin'V J= "W F " 250 THEORETICAL ASTRONOMY. and the equations (63) and (64) become .- y f s' 3 '"' (66) When the value of y f is known, the first of these equations will enable us to determine s', and hence the value of #', or sin 2 (i" E} t from the last equation. The calculation of f ma 7 be facilitated by the introduction of an additional auxiliary quantity. Thus, let (67) and from (62) we find cos / = cos (u" u) ., = 2 cos | (u" u} cos'/ tan /, or cos / = sin 2/ cos (u" u). (68) We have, also, *' 2 = (r + r'J 4rr" cos 2 J (t*" ), which gives x ' 2 = ( r r") 2 + 4rr" sin a J (*" u}. Multiplying this equation by cos 2 J(it /r u) and the preceding one by sin 2 \ (u" u), and adding, we get x' 2 = (r + r") 2 sin 2 (w" u) + (r / r ) a cos 2 ^ (w v it). (69) From (67) we get and, therefore, cos 2/=L-=i r -f- r" so that equation (69) may be written _f 2 =: sin 2 / = sin 2 (w /; w) + cos 2 2/ cos 2 j (u" u). We may, therefore, put sin / cos G 1 = sin ^ (w" w), sin / sin G' = cos (w" u) cos 2/, (70) cos / = cos I (" w) sin 2/, DETERMINATION OF AN ORBIT. 251 from which p' may be derived by means of its tangent, so that sin f f 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 and we shall have ,_ sm 9 m 2g sin 2g This gives, by differentiation, dy' n 4 sin 2 q dq 4- = o cot q do -. g-i y ^9 sm *>9 or di/ -f- = 2>y cot g 4y 2 cosec g. dg The last of equations (65) gives a/ sin 2 #, and hence dg - = 2 cosec g. Therefore we have dy' _ Qy f cos g 8y fa = 3 (1 - 2aQ y' 4y" dx' ~ sin 2 g 2x' (1 #') It is evident that we may expand y f into a series arranged in refer- ence to the ascending powers of x 1 ', so that we shall have y' =. a, -f- fix' -f- ?%'* ~\- ^' 3 + ^'* -f- C' 5 -f~ &C. Differentiating, we get and substituting for -g* the value already obtained, there results afc-r + (4r 2/9) ^' 2 + (6* 4r) ^' 3 + (8e 65) ^ + (IOC = (3a 4a 2 ) + (3^ 6a 8a/9) a;' + (3r 6^ 4/5 2 + (35 6r 8/?r 8a^) x' 3 -f (3e 65 4f 8/?<5 8ae) ^ 4. (3C 6-- 8r<5 8/?e 8aC) x f& + &c. Since the coefficients of like powers of x' must be equal, we nave 3 tt _ 4 a 2 = o, 3{3 6a 8a = 2A 3r 6,5 4;5 2 8<* r = 2 (2 r ), &c. ; 252 THEORETICAL ASTRONOMY. and hence we derive = i, P = T 9 o, r=T-5, d = T cr _ 6228 f _ 265896 , _ 191390 e 336875* _ 265896 2TS3e573 Therefore we have If we multiply through by V, and put *" + dii-fts*' 5 <=., (72; we obtain Combining this with the second of equations (66), the result is VY+= +/+?'. If we put V = ____, (74) we shall have But from the first of equations (66) we get 7 -*(/-i;; and therefore we have ^^- (75) H- ^ As soon as 3/ is known, this equation will give the corresponding value of s'. Since ' is a quantity of the fourth order in reference to the differ- ence | (E" E\ we may evidently, for a first approximation to the value of y f , take and with this find s f from (75), and the corresponding value of x* from the last of equations (66). With this value of x f we find the corresponding value of r , and recompute ?/, s', and x f ; and, if tL DETERMINATION OF AN ORBIT. 253 value of ' derived from the last value of x f differs from that already used, the operation must be repeated. It will be observed that the series (72) for ' converges with great rapidity, and that for E" j=94 the term containing # /6 amounts to only one unit of the seventh decimal place in the value of c'. Table XIV. gives the values of ' corresponding to values of x f from 0.0 to 0.3, or from E" E=0 to E" E = 132 50'.6. Should a case occur in which E" E exceeds this limit, the expression * ~ 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 ' thus found, if 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 rf is always a positive quantity. The equation may be written thus : 8* a" ,Y 1,/=0, and 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 logs' 2 corresponding to values of if from j/=0 to J/=0.6. When r/ exceeds the value 0.6, the value of s f 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 = sin* \(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 \ (E" E) will be imaginary, x' must be negative. It follows, therefore, that if the value of x' derived from the equa- tion 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 r 0, and the second of equations (66) gives " = j' (76) If we eliminate s r by means of both equations, since, in this case, y' = f, we get m'l =/* -f- |/l Substituting in this the values of m and I given by (65), we obtain q f = 3 sin y cos r r + 4 sin 3 1/, which gives 6 ' = 6 sin i/ cos 2 tf + 2 sin" i/, or _f T , = (sin J/ + cos i/) 8 + (sin i/ - cos tf?. This may evidently be written = sn + - sn the upper sign being used when f is less than 90, and the lower sign when it exceeds 90. Multiplying through by (r -j- r /; )^, and replacing (r -f- r") sin ^ by X, we obtain 6r ' = ( r + r " + x)f qz (r + r" - x)f, which is identical with the equation (5G) 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 ' corre- sponding 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 ff and E so as to be applicable. If we compare equations (44)! and (56) 1? we get tan E= l/^I tan F; DETERMINATION OF AN ORBIT. 255 and hence, from (58) w We have, also, by comparing (65)! with (41 ) w since a is negative m the hyperbola, 2<7 ' which gives Now, since c< in which e is the base of Naperian logarithms, we have E 1/^1 = loge (cos E -f 1/^T sin E\ which reduces to or E= I/HI log. from which to obtain p. If we compute s and s" also, we shall have / sr'r" sin (u" u')\ 2 I s"rr f sin (u r u)\* and the mean of the two values of p obtained from this expression should agree with that found from (83), thus checking the calcula- tion and showing the degree of accuracy to which the approximation to P and Q has been carried. The last of equations (65) gives sm|(" Jg?) = T/^, (84) from which E" E may be computed. Then, from equation (57), since e = sin may be found. Since o , r r" - o f cos 2/ = ; n, sm 2y = ; j r + r ' r + r" we have g , 1> o^ _ 2 ^ o r T F V / r/ / sin2/ p p _ 2p cot 2/ " ' and from equations (70), . , sin i (it" w) tan 6r f -of cos / cot 2/ == - - / - , sm 2/ = - T7-77 - r. cos / cos ^ (u u) Therefore the formulae (87) reduce to * sin , - " t* = - == tan G' (88) cos (w (?/' + w)) = - x _ sec J (w v w;, cos/F rr" from which also 6 and co may be derived. Then sin

) 8 ~~ COS 2 ~ p or it may be computed directly from the equation 4s'W cos 2 ^ (u" u) sm*$ (E" JB which results from the substitution, in the last term of the preceding equation, of the expressions for a cos

, V f = U to, v" = u" ct> ' and if we compute r, r' 9 r" from these by means of the polar equa- tion of the conic section, the results should agree with the values of the same quantities previously obtained. According to the equation (45)j, we have tan \E = tan (45 ?) tan Jv, tan \E' = tan (45 J?) tan jv', (90) tan E" == tan (45 j?) tan W', from which to find E, E' y 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 eswE, M'=E'esmE', (91; M" = E"esmE"; and, if M Q denotes the mean anomaly corresponding to any epoch T t we have, also, ? n 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 M 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 accord- ing to the methods already given for finding the time from the peri- helion 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 f , 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 9 M f , M" (which are not the mean anoma- lies) must be obtained from Table VI. Then, the perihelion distance having been computed from P we shall have in which log (7 = 9.96012771, for the determination of the time of perihelion passage. The times t, t', t u 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 and similarly for w f and w" ; and, with these as arguments, we derive M, M r , M" from Table VI. Finally, we have MBq* _ (93) for the time of perihelion passage, the value of O being the same as in (92). When the orbit is a parabola, e = I and p 2g, and the elements Q and ' is in each case the declination. From these we derive the longitude and latitude of the zenith for each observation, namely, 4,= 6033'.9, 4'= 35035'.2, ^' = 342 59'.2, 6 = + 22 25.0, & '=4-50 50.9, & "= + 53 41.6. Then, by means of equations (4), we obtain = 18".92, A' = 36".94, A0" = 25".76, A log E Q = 0.0001084, A log Rj = 0.0002201, A log # " = 0.0002796. For the reduction of time, we have the values + OM5, -j~ 0*.28, and -f- 8 .34, which are so small that they may be neglected. 2G6 THEOEETICAL 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 O r/ .15, O r/ .14, and O r/ .14, respectively. Collecting together and applying the several corrections thus ob- tained for the places of the sun and of the planet, reducing the un- corrected times of observation to the meridian of Washington, and expressing them in days from the beginning of the year, we have the following data : t Q = 257.68079, A = 17 46' 28".17, = + 3 8' 43".51, tj = 264.42570, A' = 16 40 25 .19, f = 2 52 27 .62, C' = 271.38625, A" = 15 1544.03, P" = + 2 3242.98, O =172 0'32".23, log =0.0021056, O' =178 35 48 .74, logjR' =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 re- ductions will not be indicated until after the entire calculation of the elements has been effected. By means of the equations m(0" 0Q _ EE f sm(Q' Q) " RR" sin (O" O) ' RR" sin (" 0)' tan/3' tan (A' -Q') cosrf ' we obtain log N= 9.7087449, log N" = 9.6950091, 4/ = 16142'13".16, log (R' sin V) = 9.4980010, log (R cos 4/) = 9.9786355.. The quadrant in which -\}/ must be taken is determined by the con- ditions that i// must be less than 180, and that cos^' and cos (A' O') must have the same sign. Then from NUMEKICAL EXAMPLE. tan Jsin Q (/' + A) - K) = |^gf , sec J (/' - i), tanlcosQ (A" + A) - K) = " , cosec | - 267 JST) G" JjQ sec/3' jfcft" sm(G" 0) J~ sin (A" A)' a sin (/>/' A)~~ we compute K, I, $,, a , 6, c, c?, /, and h. The angle J must be less than 90, and the value of ft 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 1= 9.3884640, A 2 52' 59"f if , log a = 6.8013583 n , log 6 = 2.5456342 n , log c = 2.2328550 W , log d = 1.2437914, log/= 1.3587437 n , log h = 3.924769L The formulae , _ sin (A ; /I) ^ sin (/I Q) " J " gve sin (A" A) sin (/" K) log M, = 9.8946712, log Jf a = 1.9404111, hsm(l log Mf = 9.6690383, log M z " = 0.7306625 n . The quantities thus far obtained remain unchanged in the suc- cessive approximations to the values of P and Q. For the first hypothesis, from i? sin Z,=R sin 4/, T)Q cos C = & -R' cos 4*', 268 THEORETICAL ASTRONOMY. we obtain log r = 9.0782249, log T" = 9.0645575, log P = 9.9863326, log Q = 8.1427824, log c == 2.2298567 n , log k = 0.0704470, log 1 Q = 0.0716091, log 7y = 0.3326925, C = 8 24' 49".74, Iogm == 1.2449136. The quadrant in which must be situated is determined by the con- dition that ^ shall have the same sign as b The value of z 1 must now be found by trial from the equation sin (si C) = m sin 4 sf. Table XII. shows that of the four roots of this equation one exceeds 180, and is therefore excluded by the condition that sin/ must be positive, and that two of these roots give z f greater than 180 ij/, and are excluded by the condition that z r must be less than 180 -J/. 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 1'22".96. The root which corresponds to the orbit of the earth is 18 20' 41", and differs very little from 180 ty. Next, from we derive log r' = 0.3025672, log P ' = 0.0123991, '' log n = 9.7061229, log n" = 9.6924555, log p = 0.0254823, log f = 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 O) =p cos (A Q) R, r cos b sin (I Q) =^sin(A Q), r sin & = f> tan /? ; / cos 6' cos (V Q =y cos (A' ') #, / cos V sin (r .0') = P f sin (A' '), r' sin b' />' tan p ; r" cos 6" cos (r Q") = P " cos (A" 0") J2", r" cos 6" sin (f 0") = /e," sin (A" 0"), which give J = 5 14' 39".53, log tan b =8.4615572, logr =0.3040994, V = 7 45 11 .28, log tan b' =8.4107555, logr' =0.3025673, I" = 10 21 34 .57, log tan b" = 8.3497911, log r" = 0.3011010. The agreement of the value of logr' thus obtained with that already found, is a proof of part of the calculation. Then, from (\ nn . 7^ r^\ tan 6"-}- tan 6 nG(/ + - Q) = . . cos i cos ^ cos i we get ft = 207 2' 38".16, t = 4 27' 23".84, u = 158 8' 25".78, u f = 160 39' 18".13, u" = 163 16' 4".42. The equation tan b' = tan i sin (T ft ) gives log tan b r = 8.4107514, which differs 0.0000041 from the value already found directly from //. This difference, however, amounts to only O r/ .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" sin (u" u') rr r sin (u' u) n = . ; , n" = rr" sin (u" U*)' ~ rr" sin (" u) ' the results should agree with the values of these quantities previously computed directly from P and Q. Using the values of u, u', and u" just found, we obtain log n = 9.7061158, log n" = 9.6924683, 270 THEORETICAL ASTRONOMY. which differ in the last decimal places from the values used in finding p and p ff . According to the equations d log n = 21.055 cot (u" ') du', d log n" = -f 21.055 cot (u r u) du', the differences of logn and logn" being expressed in units of the seventh decimal place, the correction to u' necessary to make the two values of logn agree is 0".15; but for the agreement of the two values of log??/', u' must be diminished by 0".26, so that it appears that this proof is not complete, although near enough for the first approximation. 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 p" as determined by means of the last two of equations (18) are somewhat uncertain. In this case it would be advisable to com- pute p and p", as soon as p f has been found, by means of the equa- tions (22) and (23). Thus, from these equations we obtain log p = 0.025491 8, log p" = 0.0028874, and hence I = 514'40".05, log tan b =8.4615619, log r = 0.3041042, J"=10 2134.19, log tan b" =8.3497919, log/' =0.3011017, & == 207 2' 32".97, i = 4 27' 25".13, u = 158 8' 31".47, u' = 160 39' 23".31, u" = 163 16' 9".22. The value of log tan b f derived from X f and these values of Q, and i, is 8.4107555, agreeing exactly with that derived from p f directly. The values of n and n" given by these last results for u, u' and u" y are log n = 9.7061144, log n" = 9.6924640 ; and this proof will be complete if we apply the correction du f = 0".18 to the value of u', so that we have u" u' = 2 36' 46".09, u' 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 , and we may also correct the times of observation for the time of aberration by means of the formulae t = t Q Cp sec /?, t = t Q ' Cp' sec p, t" = t " Cp" sec p', wherein log C= 7.760523, expressed in parts of a day. Thus we get t = 257.67467, if = 264.41976, *" = 271.38044, NUMERICAL EXAMPLE. 271 and hence log r = 9.0782331, log r' = 9.3724848, log T" = 9.0645692. Then, to find the ratios denoted by s and s", we have sin Y cos Q = sin | (u" t/)> sin Y sin G = cos J (u" u') cos 2/, cos 7- = cos ^ (u" u') sin 2/ ; tan/' = sin f' cos G" = sin ^ (V w), sin r" sin G" = cos J (V t*) cos 2/', cos /' = cos (M' w) sin 2/' ; r 2 . sin 2 ") 3 cos 3 / from which we obtain y = 44 57 f 6".00, /' = 44 56 f 57".50, r= 1 18 35 .90, r "= 1 15 40 .09, logm = 6.3482114, logm' ; = 6.3163548, log.; = 6.1163135, log/' = 6.0834230. From these, by means of the equations m m V using Tables XIII. and XIV., we compute s and s". First, in the case of s, we assume y = _ = 0.0002675, and, with this as the argument, Table XIII. gives log s 2 = 0.0002581 . Hence we obtain x f = 0.000092, and, with this as the argument, Table XIV. gives c = 0.00000001 ; and, therefore, it appears that a repetition of the calculation is unnecessary. Thus we obtain logs =0.0001290, logs" =0.0001 200. 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 TT" r" ss" rr" cos J (it" w') cos J (it" tt) cos ^ (M' )' we find log P = 9.9863451, log Q = 8.1431341, with which the second approximation may be completed. We now compute c , 7c Q , 1 , z' ', &c. precisely as in the first approximation ; but we shall prefer, for the reason already stated, the values of p and p" computed by means of the equations (22) and (23) instead of those obtained from the last two of the formulae (18). The results thus derived are as follows : log c = 2.2298499 n , log Jc = 0.0714280, log 1 = 0.0719540, log i? = 0.3332233, C = 8 24' 12".48, log m = 1.2447277, z f = 9 0' 30".84, log / 0.3032587, log p' = 0.0137621, log n == 9.7061153, log n"= 9.6924604, log/> = 0.0269143, log/' = 0.0041748, I = 5M5'57".26, log tan b =8.4622524, logr =0.3048368, I' = 7 46 2.76, log tan V = 8.4114276, log/ = 0.3032587, l rf = 10 22 .91, log tan b" = 8.3504332, log r" = 0.3017481, a = 207 0' 0".72, * = 4 28 r 35' r .20, u = 158 12' 19".54, u' = 160 42' 45".82, u" = 163 19' 7".14. The agreement of the two values of logr' is complete, and the value of log tan b' computed from tan b' = tan i sin (l r Q> ), is log tan b f = 8. 41 1427 9, agreeing with the result derived directly from p r . The values of n and n" 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, tt "_ u > = 2 36' 21".32, u' u = 2 30' 26".28, u" u = 5& 47".60. From these values of ii n u r and u f u, we obtain log s = 0.0001284, logs" = 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 sin / cos G' = sin -^ (u" u\ sin / sin G r = cos J (u" u) cos 2/, cos / = cos (u" Uj sin 2/, m , = v _sm " 3 J "" (r-j-r") 3 cosV give / = 44 53' 53".25, / = 2 33' 52".97, log tan G' = 8.9011435 log m' = 6.9332999, log/ = 6.7001345. From these, by means of the formulae '_ m ' ' ""!+/+* ~*' a '^' and Tables XIII. and XIV., we obtain logs' 2< =: 0.0009908, loga/ = 6.549411U Then from we get logjs = 0.3691818. The values of logp given by s/r" sin (i*" w') \ 2 / s'Vr' sin (w f u) \" 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 P cos

= 190 15' 39".57, log e = log sin ? = 9.2751434, ?> == 10 51 39 .62, TT = , -f- = 37 15' 40".29. This value of ^ gives log cos

, v" = u" cw, according to which we have v = 327 56' 39".97, v' = 330 27' 6".25, i" = 333 3' 27".57. If we compute r, r f , and r" from these values by means of the polar equation of the ellipse, we get log r = 0.3048367, log r' = 0.3032586, log r" = 0.3017481, and the agreement of these results with those derived directly from p, p f , and p rr is a further proof of the calculation. The equations tan { E = tan (45 ?) tan v, tan ^E' = tan (45 1$0 tan ^, tan JJE" = tan (45 ?) tan W give E = 333 17' 28' .18, E' = 335 24' 38".00, E" = 337 36' 19".78. NUMERICAL EXAMPLE. 275 The \alue of J (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' y 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 Jf denotes the mean anomaly for the date T=1863 Sept. 21.5 Washington mean time, from the formulae M=M n(t r> we obtain the three values 339 55' 25".97, 339 55' 25".96, and 339 55' 25".96, the mean of which gives M 9 = 339 55' 25".96. The agreement of the three results for Jf is a final proof of the accuracy of the entire calculation of the elements. Collecting together the separate results obtained, we have the fol- lowing elements : Epoch = 1863 Sept. 21.5 Washington mean time. M = 339 55' 25".96 *= 37 15 40 .29) ft - 207 72 V Ech P tlc and Mean 06 * J " ' v v I *i / . . -or\ t= 4 28 35.20J Equinox 1863.0.

(99) in which log JA = 8.8596330. We have, also, to the same degree of approximation, !< '=71' to^ = -5> (100) For the values log r = 9.0782331, log r f = 9.3724848, log r" ** 9.0645692, log/ = 0.3032587, these formulae give log s = 0.0001277, log s' = 0.0004953, log " = 0.0001199, which differ but little from the correct values 0.0001284, 0.0004954, and 0.0001193 previously obtained. Since sec 8 / = 1 + 6 sin 2 tf + &c., the second of equations (65) gives sin ' &c - Substituting this value in the first of equations (66), we get If we neglect terms of the fourth order with respect to the time, it will be sufficient in this equation to put y r = f, according to (71), and hence we have 4 3 and, since s f 1 is of the second order with respect to r', we have, to terms of the fourth order, 280 THEORETICAL ASTRONOMY. Therefore, which, when the intervals are small, may be used to find s f from r and r". In the same manner, we obtain l^r,- (102) For logarithmic calculation, when addition and subtraction loga- rithms are not used, it is more convenient to introduce the auxiliary angles , r , and ", by means of which these formulae become K tW (103) in which log |^ = 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 r, r', r", and r" above given, in connection with log r = 0.3048368, log r" = 0.3017481, give log 5 = 0.0001284, log s f = 0.0004951, log s" = 0.0001193. These results for logs and logs" are correct, and that for logs' differs only 3 in the seventh decimal place from the correct value. ORBIT FEOM FOUK OBSERVATIONS. 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 ele- ments 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 diffi- culty 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 expe- ditiously 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 de- termination of the elements of the orbit from three observations. Let t, t', t", t'" be the times of observation, r, r', r", r'" the radii- vectores of the body, u, u f , u n ', u'" the corresponding arguments of the latitude, R, R', R", R" r the distances of the earth from the sun, and O, O', O", O'" the longitudes of the sun corresponding to these times. Let us also put = r'r'"sm(u'" u'\ [rV"] = rV" sin (u" r u"\ and 282 THEORETICAL ASTRONOMY. Then, according to the equations (5) 3 , we shall have nx x' -f- ri'x" = 0, ny _' cos A' R' cos O') + n"G = n sin A J2 sin O ) (f? sin A r R sin O' + w" = n ' G/ cos / - R cos O') (p" cos A" #' cos O") (3) -f n'" (p'" cos X'" R" cos 0'"), = w' (p r sin A' ^ sin O') (p" sin X" R" sin 0") 4- n'" (/>"' sin A'" R" sin 0'"). If we multiply the first of these equations by sin ^, and the second by cos ^, and add the products, we get = nR sin (A Q) (p* sin (X X) + R f sin (A 0')) + n" (P" sin (A" A) -f- #' sin (A ")) ; (4) and in a similar manner, from the third and fourth equations, we find = n' (p f sin (X" X ) R' sin (X" ')) (5j) - (p" sin (/" A") K' sin (/" ")) - n'"R" sin (A'" 0'"). Whenever the values of n, n' 9 n", and n //r 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 inde- Dendent of each other ; and, since only two unknown quantities p 1 OEBIT FROM FOUR OBSERVATIONS. 283 and p n are involved in them, they will enable us to determine these curtate distances. Let us now put cos p sin (/ /I) = A, cos ft" sin (A" X)=B, cos ft" sin (/'" A") = C, cos p sin (*'" X ) = D, and the preceding equations give Ap' sec ^ Bn" P " sec /5" = wJ5 sin (A ) R r sin (A Q') + n"J2"Biny 0"), !>>' sec /?' Q/' sec /9"= n'tf sin (/" ') R" sin (A'" Q") (7) + n'"R'" sin (/" Q'"). Tf 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 equa- tions reduces rigorously to zero ; and in the same manner it can be shown that when similar terms of the second order in the corre- sponding expressions for n f 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 // 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 f , so that it will not be even an approximation to the correct value ; and the same is true in the case of p rr . It is necessary, therefore, to retain terms of the second order in the first assumed values for n, n f , n", and ii' 1 ' \ and, since the terms of the second order involve r f and r f/ , we thus introduce two additional unknown quantities. Hence two additional equations in- volving r f , r", p'j 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 p = R' cos 4/ V r' 2 R r * sin 2 4/, (8) which is one of the equations required; and similarly we find, for the other eauation, P " sec ft" = R" cos V ' db y r'" 2 R" 2 sin 2 4". (9) 284 THEORETICAL ASTRONOMY. Introducing these values into the equations (7), and putting x' = l/V 2 -.fl' 2 sin 2 47, x''=V / r"* # 2 sin 2 4/', we get Ax' Bri'x" = nR sin (A O) R' sin (A 0') -f n"R" sin (A O") AR' cos V + n"BR" cos *", ZtoV Cx" = n'R' sin (X" 0') jR" sin (A"' 0") -j- n'"R"' sin (A'" 0'") n'DR' cos 4' + CR" cos 4,". Let us now put B -h' D V> A ht C =h > or , _ cos /3" sin (A" A) , _ cos ff sin (r r AQ " cos f sin (A' A) ' ~ cos jt' sin (A'" A")' and we have x' = Kn"x" -f ndT of + n'V, a;" = A" n y + n '"d" a" -f nV. These equations will serve to determine x r and x", and hence r f and r r/ , as soon as the values of n f n f , n ff , and n" 1 are known. 96. In order to include terms of the second order in the values of n and n", we have, from the equations (26) 3 , and, putting these give ORBIT FROM FOUR OBSERVATIONS. 285 Lot us now put T '" = k cr o, v = (*"' o and, making the necessary changes in the notation in equations (26) s , we obtain ,r-"(r ' + r) , T"' (T"*+ T"'T - 1) dr" \ ~* fn. -- - i^j --- ar*"f r ** at i ,. I . r ( r ' + ^'") , .r(r' + rr'"-r'"0 dr" fs ^^ ~& 7r ~ "df From these we get, including terms of the second order, and hence, if we put P" = ~, " = (' + '"-!)/', (17) we shall have, since r ' = r + r' /r , = n. When the intervals are equal, we have P' P" = 777 -pm and these expressions may be used, in the case of an unknown orbit, for the first approximation to the values of these quantities. The equations (13) and (17) give and, introducing these values, the equations (12) become 286 THEORETICAL ASTEOKOMY. * = W-pr ( 1 + % } (*V + P'd' + c') - 1 ff'\ x " = TTpr ( 1 + ;* ) (A ' V + p " d " + c " } Let us now put P'cf+c' An c" -f C 1 I TV/ J (21; 1 + P" 1 + and we shall have (22) +e ")-a". We have, further, from equations (10), r''=(*/'+P' 2 sinV)', r f; =(a/'+JB"sinV / ) f , If we substitute these values of r' 3 and r" z in equations (22), the two resulting equations will contain only two unknown quantities x' and x". when P', P", ', 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 , _ E' sin 4/ ~' (24) ,,_' sin z" Substituting these values of r f and r" in equations (10), we get *'=r>z> (25) x" = r" cos z ", and hence ORBIT FKOM FOUR OBSERVATIONS. 287 , R' sin 4,' tan z== -- -, - , (26) by means of which we may find z' and z" as ,soon as x f and x" shall have been determined ; and then r' and r" are obtained from (24) or (25). The last equations show that when x r is negative, z r must be greater than 90, and hence that in this case r' is less than R f . In the numerical application of equations (22), for a first approxi- mation to the values of x r and x", since Q' and Q" are quantities of the second order with respect to r or r r// , we may generally put and we have *' *" or, by elimination, , 1 -/'/" 1 /'/" With the approximate values of x' and x" derived from these equa- tions, we compute first r r and r" from the equations (26) and (24), and then new values of x' and x" from (22), the operation being repeated until the true values are obtained. To facilitate these ap- proximations, the equations (22) give ~/'(;+f ). >; */ = . " "^ Let an approximate value of x f be designated by # ', and let the value of x" derived from this by means of the first of equations (27) be designated by x ". With the value of x " for x" we derive a new value of x f from the second of these equations, which we denote by a?/. Then, recomputing x" and x', we obtain a third approximate value of the latter quantity, which may be designated by a?,'; and, if we put a?/ x ' = a Q , #,' xj ', 288 THEORETICAL ASTRONOMY. we shall have, according to the equation (67) 3 , the necessary changes being made in the notation, z' = < ?^- = x; PL-. (28) < o o o 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 f = and Q" = 0, and compute x f , 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 instead of by putting Q f and Q" equal to zero. 97. It should be observed that when A' = A or A'" = A", the equa- tions (22) are inapplicable, but that the original equations (7) give, in this case, either p" or p' directly in terms of n and n ff or of n' and n'" and the data furnished by observation. If we divide the first of equations (22) by h f , we have The equations (21) give f 1 /' P 17 ~r~ 17 h' ~~ 1 -I- P" A' ~ 1 + P' and from (11) we get of _ R' cos V B sin (I Q') A' ~ A' B 1 = tf'cos*" + ^" sin ^- Q "), (29; d' J?sin(A O) A'~ Then, if we put ORBIT FROM FOUR OBSERVATIONS. its value may be found from the results for and ^ derived by means of these equations, and we shall have When A' = I, we have h' == oo, and this formula becomes =( 1 + -J ) (*" + C ') - d + P'), the value of j-, being given by the first of equations (29) This equation and the second of equations (22) are sufficient to determine x f and x" in the special case under consideration. The second of equations (22) may be treated in precisely the same manner, so that when X ffr = X", it becomes o =( and this must be solved in connection with the first of these equations in order to find x f and x ff . 98. As soon as the numerical values of x r and x n have been derived, those of r f and r" may be found by means of the equations (26) and (24). Then, according to (41) 4 , we have X*& MO The heliocentric places are then found from p f and p" by means of the equations (71) 3 , 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 f and t". The values of r', r", u', u", n, n", n f , and n"' enable us to deter- mine r, r" r , u t and u" f . Thus, we have and, from the equations (1) and (3) 3 , 19 MJK) THEORETICAL ASTRONOMY. [rr"] = Therefore, n r sin (u f u) = r" sin (u" w') n r sin (u" u) = - r' sin (it" it'), (32) i>" sin ("' ") = 4? r' sin (" '), 71 r'" sin (t*'" *') = ^7 r" sin (u" ') From the first and second of these equations, by addition and sub- traction, we get r sin ((u f u) -J- 3 (w" w')) = sm 2 V w ') /- re 'V" (33) r cos ((i/ u) -f- 2 (w" i*')) = cos J (w" it'), 71 from which we may find r, u r it, and u = u r (u f it). In a similar manner, from the third and fourth of equations (32), we obtain r m sin ((u m t*") + J (it" i/)) = V sin J (i*" - w'), ,/ w (34) r'" cos ((i* w M ") + A (w" u')) = - m^- cos J (w" w'), 71 from which to find r'" and it'". When the approximate values of r, r f , r", r f ", and w, it', it", u fn have been found, by means of the preceding equations, from the assumed values of P', P", Q f , 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 and r", using p' _!_, p" = _L, and then recompute P f and P" by means of the equations (14) and ORBIT FROM FOUR OBSERVATIONS. 291 (18), before finding u f 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 com- menced. For this purpose, the distances of the body from the earth for the four observations must be determined ; and, since the curtate distances p f and p ff are already given, there remain only p and p' tf to be found. If we eliminate p f from the first two of equations (3), the result is nR sin (A' Q) R sin (A' Q') -f n" R" sin (X 0"). and, by eliminating p" from the last two of these equations, we also obtain , , n' sin (A" -A') ' : = / V'sm(A'"-/') n 1 R' sin (A" - Q') R" sin (A" Q ") -f ri" R" f sin (A" "') n'" sin (A'" /I") by means of which p and p" f may be found. The combination of the first and second of equations (3) gives COS (X nR cos a Q) R' cos (A 0') + n" R" cos (A 0") /o = ^ cos (A' A) 2_- cos (A" A) (37) n n + and from the third and fourth we get />'' cos( r'_A'o-$' n' R 1 cos (A'" 0') R' cos (A'" 0") -f n'" R'" cos (A'" Q '") .'" = ^7 cos (A'" - A") ^ cos (A'" A') (38) 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 /?, /9', &c. are very small, the values of p and p"' which depend on the differences of the observed longi- tudes of the body must be preferred. 292 THEORETICAL ASTRONOMY. T1ie values of p f and p rrf may also be derived by computing the heliocentric places of the body for the times t and t' n by means of the equations (82) 1? and then finding the geocentric places, or those which belong to the points to which the observations have been reduced, by means of (90) b writing p in place of Jcos/9. This process affords a verification of the numerical calculation, namely, the values of X and X' fr thus found should agree with those furnished by observation, and the agreement of the computed latitudes ft and ft'" 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 X and X" in order to check the calculation, even when ft and ft flf are given by observation, we mighi derive p and p" r from the equations p = r sin u sin i cot y5, p'" = r'" sin u" f sin i cot p", 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 ft and ft" f , to compute p and p f " as soon as p f and p ff have been found, and then find I, l f ", 6, and b'" directly from these by means of the formulae (71) 3 . The values of & and i may thus be obtained from the extreme places, or, the heliocentric places for the times t r and t'" 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 f and b'" from the equations tan b' = tan i sin (I' & ), tan b" = tan i sin (I" &), they will not agree exactly with the results obtained directly from p 1 and /?", unless the four observations are completely satisfied by the elements obtained. The values of r' and r n ', however, computed directly from p r and p" by means of (71) 3 , must agree with those derived from x r and x n . 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 t w t f , t Q ", and t Q '" are the uncorrected times of observation, the corrected values will be t = t Cp sec/5, wherein log C== 7.760523, and from these we derive the corrected values of r, r', r", r'", and r '. 100. To find the values of P', P", Q', and Q", which will be exact when r, r', r", r f// , and w, u' 9 u", u'" are accurately known, we have, according to the equations (47) 4 and (51) 4 , since Q f = |, _ _ _ ~" * ss" ' rr" cos J (t*" w') cos i (t*" M) cos j (' )' In a similar manner, if we designate by s fff the ratio of the sector formed by the radii-vectores r rr and r'" to the triangle formed by the same radii-vectores and the chord joining their extremities, we find _ i rr'" r^ V : ~ 2 ^777 ' r y/, C o S l (>'" M ") COS i (U'" t*') COS J (u" 1*')' The formulae for finding the value of s' rr are obtained from those for s by writing / /r/ , f ff , G" f , &c. in place of /, /-, (r, &c., and using r", r'", w //; u" instead of r', r", and it" M', respectively. By means of the results obtained from the first approximation to the values of P', P", Q', and ", 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 ', and of P" and ", 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 f , and Q" have been derived with sufficient accuracy, we proceed from these to find the elements of the orbit. After &, i, r, r f , r" , r'", u, u', u" ', and u f/f 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 /r/ , u, and u" f . 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 cos G = sin (u" f u), (43) sin YQ sin O = cos (V" w) cos 2/ , cos r = cos | (u'" u) sin 2/ , we find p and 6r . Then we have r '")S "to (44 ) from which, by means of Tables XIII. and XIV., to find s and a? . We have, further, 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 approxima- tion to P ; , P") Q f , 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 formula 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 formulae for the calculation of an orbit from four observations, let us take the following normal places of Eurynome derived by comparing a series of observations with an ephemeris computed from approximate elements. Greenwich M. T. a {, 1863 Sept. 20.0 14 30' 35".6 + 9 23' 49".7, Dec. 9.0 9 54 17 .0 2 53 41 .8, 1864 Feb. 2.0 28 41 34 .1 962 .8, April 30.0 74 29 58 .9 -f- 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 lati- tudes, we get Greenwich M. T. 1863 Sept. 20.0 Dec. 9.0 1864 Feb. 2.0 April 30.0 A 16 59' 9".42 10 14 17 .57 29 53 21 .99 75 23 46 .90 ft 4- 2 56' 44".58, 1 15 48 .82, 2 29 57 .38, 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, -f-0".53, -f-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 tc these latitudes respectively, the logarithms of the approximate dis- tances of the planet from the earth being 0.02618, 0.13355, 0.29033, 0.44990. Thus we obtain t = 0.0, A = 16 59' 9".42, ft = + 2 56' 44".01, if = 80.0, X =10 14 17 .57, f = 1 15 49 .20, " = 135.0, A" = 29 53 21 .99, f = 2 29 57 .56, f" = 223.0, r = 75 23 46 .90, jf n = 3 4 44 .56 ; and, for the same times, the true places of the sun referred to the mean equinox of 1864.0 are =177 0'58".6, logE =0.0015899. 0' =256 58 35.9, log# =9.9932638, 0" =312 57 49 .8, log-R" =9.9937748, 0'"= 40 21 26.8, log #" = 0.0035149, 296 THEORETICAL ASTRONOMY. From the equations tan/5' tan w f = . tan to" = . ,., -j^> sm(A' O) tan" . , ' , sm(A" O ) tan (A' Q') tan * = -- - - 7=^1 cosw/ tan(A" O") tan V = - - - \ cosw" we obtain 4/ = 113 15' 20".10, 4,"= 76 5617.75, cos V) = 9.5896777 n , log (R sin 4,') =9.9564624, log (#' cos 4") = 9.3478848, log (R" sin *") = 9.9823904. The quadrant in which ty must be taken, is indicated by the condi- tion that cosij/ and cos (A' O') must have the same sign. The same condition exists in the case of i//' Then, the formulaB A = cos ff sin (^ A), C = cos /9" sin (A'" A"), ^-A' A - h, B = cos p' sin (A" jl), Z> = cos f sin (A'" A'), ,, O) , ft " Q'") give the following results : log A = 9.0699254 n , log B = 9.3484939, log h' = 0.2785685 n , log a! = 0.8834880 n , log d = 0.9012910 n , log d' = 0.4650841, log C = 9.8528803, log D = 9.9577271, log h" = 0.1048468, log a" = 9.9752915 n , log c" = 9.7267348 w , log d" = 9.9096469 n . We are now prepared to make the first hypothesis in regard to the values of P', ', P f/ , and Q". If the elements were entirely un- known, it would be necessary, in the first instance, to assume for these quantities the values given by the expressions NUMERICAL EXAMPLE. 297 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 f = and Q" = 0, namely, x' = * after which the values of x f 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 ob- servations will generally be small, and hence the value of x f derived from the assumption of Q f = and Q" will be sufficiently ap- proximate to facilitate the solution of equations (27). As soon as the approximate values of r f and r" have thus been found, those of P' and P" must be recomputed from the expressions With the results thus derived for P' and P", and with the values of Q f 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 f/ should be computed by means of these elements. Thus, from _rV'smQ/' t/) r/sinQ/ 1>) ~ rr" sin (" v) ' ~ rr" sin (v" v) ' ,rV"sin(y" f/Q w _ rV ; sin(^ i;Q ~ r r sin (v v) TT sin (v v ) we find n, n' 3 n rf , and n"'. The approximate elements of Eurynom* givo v = 322 55' 9".3, logr =0.308327, it =353 19 26 .3, log/ =0.294225, v"= 14 45 8.5, log/' =0.296088, v'"= 47 2332.8, log /" = 0.317278, 298 THEORETICAL ASTRONOMY. and hence we obtain log n = 9.653052, log n" = 9.806836, log n 1 = 9.825408, logn'" = 9.633171. Then, from P" = , we get log P' = 9.846216, log Q f = 9.840771, log P"= 9.807763, log Qf' = 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 f C ~~ = 1 + P" * ~ 1 -f- P'' ,_Pd"+c" h" c o = = x + P n > J - fqrp> we find log c ' = 0.541344 n , log/ = 0.047658., log C Q " = 9.807665 n , log/" = 9.889385. Then we have ' *= *"+"" < , ,, tan / = --, tan z" = ,_R' sin V _ a/ ,,_jrsinV_ a" sin/ cos/' sin/ 7 cos/" from which to find r' and r rr . In the first place, from x' = v 1 jR"sinV, we obtain the approximate value log x f = 0.242737. Then the first of the preceding equations gives log *" = 0.237687. NUMERICAL EXAMPLE. 299 From this we get z" = 29 3' 11". 7, log r" = 0.296092 ; and then the equation for x' gives logo;' = 0.242768. Hence we have z' = 27 20' 59".6, log r' = 0.294249 ; and, repeating the operation, using these results for x r and r', we get log x" = 0.237678, log of = 0.242757. The correct value of log a?' may now be found by means of equation (28). Thus, in units of the sixth decimal place, we have = 242768 242737 = + 31, a Q ' = 242757 242768 = 11, and for the correction to be applied to the last value of log x', in units of the sixth decimal place, Therefore, the corrected value is log af= 0.242760, and from this we derive log a" = 0.237681. These results satisfy the equations for x' and x", and give z' = 27 21' 1".2, log/ = 0.294242, z" = 29 3 12 .9, log r" = 0.296087. To find the curtate distances for the first and second observations, the formulae are oo o ,, = - sin z' sm z" which give log p' = 0.133474, log p" = 0.289918. Then, by means of the equations 300 THEORETICAL ASTRONOMY. r' cos V cos (f 00 = p' cos (X K 9 r' cos V sin (? ') = p' sin (A' QO, / sin V = p' tan p, r" cos ft" cos (r Q'O = P " cos (A" 0") J2", r" cos b" sin (/" Q") = p" sin (A" "), r" sin ft" = //'tan/S", we find the following heliocentric places : r = 37 35' 26".4, log tan ft' = 8.182861 n , log / = 0.294243, r = 58 5815.3, logtanft" = 8.634209 n , 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 calcula- tion. From the equations tan i sin ( (I" -f /') ) = J (tan ft" + tan ftO sec j (r 7), tan i cos Q (J 1 -f I') & ) = J (tan ft" tan ftO cosec i (/" 0, tan (^ &0 tan(r ft) tan w = - - - r-^ , tan u = cos i cos t we obtain & = 206 42' 24".0, i = 4 36' 47".2, u r =190 55 6 .6 u" = 212 20 53 .5. Then, from we get log n" = 9.806832, logw =9.653048, log n' = 9.825408, log n'" = 9.633171, and the equations r sin ((u r - u) + i (u" - w')) = /+ J' V ' sin J (i*" - 0, r cos ((i*' - u) + i (w" - tO) = ^= cos i (u" - tif ), r"' sin ((*"' - w'O + J (u" - wO) - , sin j (t" - wO, r"' cos ((u'" - i/O -f i (u" - 1*0) = cos i (u" - NUMERICAL EXAMPLE. 301 give logr = 0.308379, u = 160 30 ; 57".6, log r"' = 0.317273, t*'"=244 5932.5. Next, by means of the formulae tan (I & ) = cos i tan u, tan 6 = tan i sin (/ ft ), tan (Y" ) = cos i tan w'", tan 6'" = tan i sin ('" ft ), ^ cos (A o ) = f cos 6 cos (7 0) + -K, jo sin (A Q ) = r cos 6 sin (7 O), /o tan /? = r sin & ; p ' C os (A"' O'") = r'" cos 6'" cos (f" "0 + 12"', p' s in (A'" O'") = r'" cos 6"' sin (f" 0'"), //"tan/5"' = r"'sin&'", we obtain J = 7 16' 51".8, r = 91 37' 40".0, b = + 1 32 14 .4, 6"' = 4 10 47 .4, A = 16 59 9 .0, /" 75 23 46 .9, /? = + 2 56 40 .1, /?'" = 3 4 43 .4, log P = 0.025707, log p'" = 0.449258. The value of X" f thus obtained agrees exactly with that given by observation, but / differs O r/ .4 from the observed value. This differ- ence 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 ft" 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 ff j Q f j and Q /f , and which are therefore the same in the successive hypotheses, should be performed as accurately as possible. The s* value of -^-> required in finding x" from x', may be computed directly from jf . the values of 77 and 77 being found by means of the equations (29); 302 THEORETICAL ASTRONOMY. and a similar method may be adopted in the case of j, r > Further, in the computation of x f 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 -f -^ 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 for- mulae 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 f , r /f , r'", and u, u f y u ff , u fff , which result from the first hypothesis, suffice to correct the assumed values of P f , P", ', and Q". Thus, from "r 77 " '/ , /r 777 " sin Y cos G = sin (u" u'), sin /' cos G" = sin A (u r u), sin f sin G = cos J (u" u') cos 2/, sin /' sin G" cos (u r u) cos 2/', cos f = cos | (u" u') sin 2/, cos f" = cos \ (V u) sin 2,%", sin f cos G'" = sin I (u" r w"), sin f sin G"' = cos (u" f u"} cos 2/" cos /" = cos J (u"' u") sin 2/" ; T 2 COS 6 / __ T" 2 COS 6 /' m = COS?' m r' /s cos 3 /' t+j + ? *+/'+* in connection with Tables XIII. and XIV. we find s, s", and '". The results are NUMEKICAL EXAMPLE. 303 log T = 9.9759441, / = 45 3'39".l, r=-10 42 55 .9, log m = 8.186217, log; = 7.948097, log s = 0.0085248, log r"= 0.1386714, 7"=4432' 1".4, /'IS 13 45 .0, log m"= 8.516727, log/'= 8.260013, log "== 0.0174621, log T"'= 0.1800641, /"== 45 41' 55".2, y"'=16 22 48 .5. log m'"= 8.590596, log/"= 8.325365, log s'"= 0.020406?. Then, by means of the formulae a -* V - 2 ,, y, = , nl' _ 2 ss "f r y // cog ^ (yftr u n^ cog ^ (jjin ^ CQg ^ ^/ % /y we obtain log P' = 9.8462100, log P" = 9.8077615, log ' = 9.8407536, log " = 9.8824728, with which the next approximation may be completed. We now recompute c ', c/', /', /", x f , x n ', &c. precisely as already- illustrated; and the results are log c ' = 0.5413485 n , log/ = 0.0476614 n , log x' = 0.2427528, z f = 27 21' 2".71, log/ =0.2942369, log />' = 0.1334635, log n = 9.6530445, log n' = 9.8254092, log c " = 9.8076649 n , log/" = 9.8893851, log x" = 0.2376752, z" = 29 3' 14".09, logr" =0.2960826, log p" = 0.2899124, log n" = 9.8068345, log ri" = 9.6331707. Then we obtain /' = 37 35' 27".88, I" 58 58 16 .48, log tan V = 8.1828572 n , log tan b" = 8.6342073 n , log/ = 0.2942369, log /' = 0.2960827. These results for log r f and log r" agree with those obtained directly from z f and z", thus checking the calculation of ty and ty f and of the heliocentric places. Next, we derive 206 42' 25".89, 55 6.27, t = 4 36' 47".20, 2052.96, 304 THEORETICAL ASTRONOMY. and from u ff u', r', r", n, n", n f , and n" f , we obtain logr =0.3083734, u = 160 30' 55".45, log/" =0.3172674, w'"=244 5931.98. For the purpose of proving the accuracy of the numerical results, we compute also, as in the first approximation, /= 716'51".54, /'" = 91 37' 41".20, b = + 1 32 14 .07, b'"= 4 10 47 .36, A= 16 59 9.38, A'" = 75 2346.99, = + 2 56 39 .54, p"= 3 4 43 .33, log P = 0.0256960, log p" r = 0.4492539. The values of ^ and \ tn thus found differ, respectively, only 0".04 and O r/ .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" ', ', and ", 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 f , and ". From TJ r'j u, it', &c., we derive log a = 0.0085254, log s" = 0.0174637, log s m = 0.0204076, and hence the corrected values of P', P", ', and Q" become logP' = 9.8462110, log q = 9.8407524, log P" = 9.8077622, log " = 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 deter- mination of the elements of the orbit, using the extreme places. Thus, from sin YQ cos G = sin J (u tn u), sin Y O sin G = cos J (u" f u) cos 2/ , cos YO cos (u"' u) sin 2/ , 5 i i ~r Jo ? - _ " m _ m a ~~ 7 ~ NUMERICAL EXAMPLE. 305 we get log r = 0.5838863, log tan G Q = 8.0521953 W> n = 42 14' 30".17, log m = 9.7179026, log s 2 = 0.2917731, log x = 8.9608397. The formula s rr m sm(u m t*)\ gves log p = 0.371 2401; and if we compute the same quantity by means of grV'sm(X' iQ \ 2 / g'Vr'sinK u) V I s'VV'sin (u'"- u"} \ I* the 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 x/ , Q', and Q" from the limit of accuracy attainable with logarithms of seven decimal places. A variation of only O r/ .2 in the values of u' u and u fff u ff wil? produce an entire accordance of the particular results. From the equations smi(V" u) /T,, C 8 p = sin K"' -" P cos

) tan (45 ?), t&n%E' = tani(X oO tan (45 ?>), tan IE" = tan J O" - o) tan (45 ' p), tan E'" = tan (u" r o) tan (45 ?>) from which the results are E = 329 11' 46".01, .E" 12 5' 33".63, ' = 354 29 11 .84, '" = 39 34 34 .65. The value of J (E" f E) thus derived differs only 0".03 from that obtained directly from x . For the mean anomalies, we have which give , M" = E" esmE", = E r e sin E', M'" = E'" e sin E'", M = 334 55' 39".32, M" = 9 44' 52".82, M' = 355 33 42 .97, JJf'" = 32 26 44 .74. Finally, if M Q denotes the mean anomaly for the epoch T= 1864 Jan. 1.0 mean time at Greenwich, from M Q = M fi (t T) =M t p(fT) = M" fJL (f T} = M'" p. (f" T), we obtain the four values Jf = l29'39".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 = ?= 11 15 52 .22 log a = 0.3881359 log ; = 2.9678027 A* = 928".54447. 102. The elements thus derived completely represent the four ob- served longitudes and the latitudes for the second and third places, which are the actual data of the problem ; but for the extreme lati- tudes 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 f , p", n, n' f , n f , and n rrr 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 //", the following heliocentric places are obtained : I = 716'51".54, log tan b =8.4289064, logr =0.3083732, l f " = 91 37 40 .96, log tan b'" = 8.8638549 n , log r"' = 0.3172678. Then from tan i sin (J (f" + 1) ft) = J (tan V" -f tan 6) sec J (f" Z), tan i cos Q (f" -f & ) = i (tan b"' tan 6) cosec J (f I), we get ^ = 206 42' 45".23, i = 4 36' 49".76. For the arguments of the latitude the results are u = 160 30' 35".99, u'" = 244 59' 12".53. 398 THEORETICAL ASTRONOMY. The equations tan V = tan i sin (I 1 Q ), tan V = tan i sin (I" ), give log tan &' = 8.1827129^ log tan b" = 8,6342104 n , and the comparison of these results with those derived directly from p' and p" exhibits a difference of -f 1".04 in &', and of 0".06 in 6". 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 f ", and u, u" r , the separate results are : log tan G = 8.0522282 n , log m = 9.7179026, log sj = 0.2917731, log x = 8.9608397, log j9 = 0.3712405, I (E" E) = 17 35' 42".12, log (a cos ?) = 0.3796884, log cos

9 i, u, and u". Then, if the value of a computed from the last result for u" u differs from the last assumed value, a further repetition of the calcu- CIKCULAB CEBIT. 313 latiori becomes necessary. But when three successive approximate values of a have been found, the correct value may be readily inter- polated 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 &, 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 u the argument of the latitude for the epoch T, we shall have, for any instant t, u being the mean or actual daily motion computed from Jc The value of u thus found, and r = a, substituted in the formulse for computing the places of a heavenly body, will furnish the approxi- mate ephemeris required. The corrections for parallax and aberration are neglected in the first determination of circular elements ; but as soon as these approxi- mate elements have been derived, the geocentric distances may be computed to a degree of accuracy sufficient for applying these cor- rections directly to the observed places, preparatory to the determi- nation of elliptic elements. The assumption of r f = 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 (98) 4 . 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 ele- ments 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 as- sumption 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 n , the values of A and J"; 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 A and A" may be neglected, so that we shall have, for any date, .. da, . da COS S Aa = COS d - A J -j- COS d -=--- rr A A , d d fR\ d* d3 ** = dA^ + ^F AJ ' If, therefore, we compare the elements computed from A and A" with any number of additional or intermediate observed places, each ob- served spherical co-ordinate will furnish an equation of condition for the correction of the assumed distances. But in order that the equa- tions (6) may be applied, the numerical values of the partial differen- tial coefficients of a and d with respect to A and d" 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 J", and the third from A and A" -f D", D and D" being small increments assigned to A and A" respectively. ] f now, for any date t' y we compute a/ and d f 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 f computed from the first system be denoted by a' and 3 f ; those computed from the second system of elements, by a/ -f a sec 3 f and 6' + d; and those from the third system, by a'+ a!' sec d' and d'-\- d". Then we shall have VARIATION OF TWO GEOCENTRIC DISTANCES. 315 .. da! a dd d ,,a__ -.. dA"~ D'" dA"~ D"' and the equations (6) give COS d r Aa' = ~ A A -f -^77 A J", d d" (8) 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 A and A" , as deter- mined by each of the additional observations employed, may be formed. 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 helio- centric places are obtained by means of the equations (71) 3 and (72) 3 , writing A cos /9 and A" cos ft" in place of p and p". The values of &, i, u, and u" will be obtained by means of the formulae (76) 3 and (77) 3 ; and from r, r" and u" u the remaining elements of the orbit are determined as already illustrated. The first system of ele- ments is thus obtained. Then we assign an increment to J, which we denote by D, and with the geocentric distances A -f- D and A" we compute in precisely the same manner a second system of ele- ments. Next, we assign to A" an increment D", and from A and A" + D" a third system of elements is derived. Let the geocentric longitude and latitude for the date of the middle observation com- puted from the first system of elements be designated, respectively, by V and /9/ ; from, the second system of elements, by V and /? 2 ' ; and from the third system, by ^/ and /9 3 '. Then from It ( -\ I i l\ of fjff O f O f \** s we compute a, a", d, and d" ', and by means of these and the values of D and D" we form the equations 316 THEORETICAL ASTRONOMY. = CM for the determination of the corrections to be applied to the f.rst assumed values of A and A n ', by means of the differences between observation and computation. The observed longitude and latitude being denoted by X and /?', respectively, we shall have COS f A/ = (/ A/) COS /?', ff = ff B' (11) for finding the values of the second members of the equations (10), and then by elimination we obtain the values of the corrections A J and AJ" to be applied to the assumed values of the distances. Finally, we compute a fourth system of elements corresponding to the geocentric distances A -f A A and A" -f A A" 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, A/ and /? 2 r 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 -f- D + A A and A" + A A". Similarly, if the middles place is best represented by the third system of elements, the cor- rections will be obtained for the distances used in the third hy- pothesis. If the computation of the middle place by means of the final ele- ments still exhibits residuals, on account of the neglected terms of the second order, a repetition of the calculation of the corrections A/f and AJ", using these residuals for the values of the second members of the equations (10), will furnish the values of the dis- tances 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, in- stead 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 T\VO 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 AJ 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 AJ" 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 AZ/ ;/ will be obtained ; and a system of elements computed with the distances thus corrected will exactly represent the two funda- mental 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 w* ! ll 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 A and J", and the corresponding geocentric right ascensions and declinations by means of the formulae (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 funda- mental plane. The formation of the equations of condition for the corrections AJ and A A" to be applied to the assumed values of the distances will then be effected precisely as in the case of I and /?, 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 phall vanish, otherwise, for strict accuracy, the heliocentric places must be determined from A and A" in accordance with the equations (39):- 106. The partial differential coefficients of the two spherical co- ordinates with respect to A and A" may be computed directly by means of differential formula; but, except for special cases, the numerical calculation is less expeditious than in the case of the indi- rect 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 / as one of the elements of the orbit, as in the equations (72) 2 , the variation of the geocentric longitude 6 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 ex- press r and v in terms of the elements ^, M Q) and /*, we shall have and or In like dr dA dv dA _dr dtp dv dtp dr dM dr h d,u dv dp. dp. ' dA 'dA -> dv dA ' dA' dp. dx dM Q d

'dA dM dA 1 dp. __ _ d? ' dA dM ' dA " dfi ' dA' dA ~ d

TT> an( i - n -- are dA dA dA dA known, the equations necessary for finding the differential coefficients of the elements , = o. dA aA To find -7-7- and - 7 . , from the equations dA dA A cos ^ cos = x -f- X, A cos >? sin = y -{- F", 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 7] cos 6 dA, dy = cos T? sin 6 dA, or, substituting for dx and dy their values given by (73) 2 , cos t] cos 6 dA = cos u dr r sin u d (v -f- /), cos r) sin O d A = sin udr -\- r cos it c? (v -f~ /). Eliminating, successively, c? (v + #) and c?r, we get dr , n x - = cos ^ cos (6 u), d I . , . = - cos ^ sm (6 u). 7 - dA r Therefore, we shall have dy , dv dv dv dip dA ' dM Q dA dp. dA ,_ d X _ _dif_ d?_ d^_ dM^,dv^_ ^_ dA H " ' dA + ^jf o jj ' ' ^ ' ^j _dr^_ ^__ ' ~ ^ dA dif dA dfji dA and if we compute the numerical values of the differential coefficients of r, r", v, and v tr with respect to the elements Sm(t U } ' dr" d, and , respectively, we compute the values of 0, 37, and f for the dates of the several places to be employed. Then the residuals for each of the observed places are found from the formulae cos f) A0 = sin Y A<5 -f- cos f cos 8 Aa, Aiy = cos Y A# sin f cos d Aa, the values of Aa and A 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 A". The values of 9 f], and f being required only for finding cos y A#, A/7, and the differential coefficients of d and ry, with respect to the elements of the orbit, need not be determined with great accuracy. Next, we compute -^ and - k , . from equations (12), and from ,+ n ^ .1 i dr dr" dv dv" dr , / i i (16), the values of , , , -= , -r^, &c,, by means of which, d, A!/ O , and may be obt we compute may be obtained. If, from the values of -- - and -p VARIATION OF TWO GEOCENTRIC DISTANCES. 323 and apply these corrections to the values of v and v" found from J and J", we obtain the true anomalies corresponding to the distances A -}- A J and J" + A J". The corrections to be applied to the values of r and r" derived from A and A" are given by If A J and AJ" 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" com- puted directly from the corrected values of v, v", p, and e by means of the polar equation of the conic section. Finally, we have dz = sin rj dd, and similarly for dz ff and the last of equations (73) 2 gives r sin u Ai' r cos u sin i' A & ' == sin 7 A J, r" sin u" Ai' r" cos u" sin i' A ft ' sin ^ A J", from which to find AI V and A & ', u and u" being the arguments of the latitude in reference to the equator. We have also, according to (72)* A a/ = A/ COS i' A & ', 4^ = A/ + 2 Sin 1 t* A &', from which to find the corrections to be applied to a) 1 and n f . 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 & and &' and 180 i' and i. The final residuals of the longitudes may be obtained by substi- tuting the adopted values of A J and A A" 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 rj will show how nearly the position of the plane of the orbit corresponding to the corrected distances satisfies the intermediate latitudes. Instead of 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 & and i, will be obtained. 110. The formulae which have thus far been given for the correc- tion 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 formulae 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, F, Z be the geocentric co-ordinates of the sun at the same instant. Let the geocentric co-ordinates of the body be designated by x 09 y Q , Z Q , 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 radius- vector of the body on the plane of the equator, or the curtate dis- tance with respect to the equator; then we-shall have x = p cos a, y = p sin a, z p tan d. (32) If we represent the right ascension of the sun by A, and its declina- tion by _D, we also have VARIATION OF ONE GEOCENTRIC DISTANCE. 329 Z=RsmD. (33) The fundamental equations for the undisturbed motion of the planet or comet, neglecting its mass in comparison with that of the sun, are but since and, neglecting also the mass of the earth, these become Substituting for a? , y , and 2; their values in terms of a and d, and putting we get + ^co Sa + , = 0, + sina + , = 0, ' (36) Differentiating the equations (32) with respect to t, we find dx n dp . da -d?= c f 31)0 THEORETICAL ASTRONOMY. Differentiating again with respect to t, and substituting in the equa tions (36) the values thus found, the results are If we multiply the first of these equations by sin a, and the second by cos a, and add the products, we obtain d' 2 a j gBin.-?COB.-/- dt 2 rff 57 Now, from (35) we get sin a ^ cos a = & 2 1 -= -- 3 I -K cos D sin (a A), and the preceding equation becomes The value of -vr thus found is independent of the differential co- dp efficients of d with respect to t. To find another value of -J-, using dt all three of equations (38), we multiply the first of these equations by sin A tan 8, the second by cos A tan , and the third by sin (a A). Then, adding the products, since sin A = TJ cos A, the result is from which we get % - cot (a - A) % + **( 2 ' + cot % ) + i cot* a/> _ , rr> and -77, the equations at at at X= Rcos Q, Y= jRsin O cose, Z = R sin O sin e, give, by differentiation, dX dR ^dO ^^eosQ.. _ jRsm0 _, ^= sin Q cose^ + R cos cose^ (43) at at at dZ dR . n .dQ -,. = sm O sm e ,, + R cos O sin e -=-. at at at 332 THEORETICAL ASTRONOMY. Now, according to equation (52) 1? we have dQ _ T(l-e 2 )(l + m ) ~dT z ~ir~ m Q denoting the mass of the earth, and e Q the eccentricity of its orbit. The polar equation of t^ie conic section gives dr _ r 2 e sin v dv ~dt~ p "ST Let -T denote the longitude of the sun's perigee, and this equation gives .sin(0-r, (45) If we neglect the square of the eccentricity of the earth's orbit, we have simply _ -7- (4<5) The values of -^- and -7- having been found by means of these dX dY formulae, the equations (43) give the required results for - , -, and dZ -j7, and hence, by means of (42), we obtain the velocities of the uit comet or planet in directions parallel to the co-ordinate axes. 112. The values of x, y, and 2 may be derived by means of the equations x = A cos <5 cos a X, y A cos d sin a Y, z = A sin d Z, and from these, in connection with the corresponding velocities, the elements of the orbit may be found. The equations (32), give im- mediately 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 equa- tions (93) x by cos &, and the second by sin &, and adding the pro- ducts, we have VAEIATION OF ONE GEOCENTRIC DISTANCE. 333 r sin u = z cosec i, r cos u = x cos & + y sin & > from which r and w may be found, the argument of the latitude u being referred to the plane of xy as the fundamental plane. The equation r 2 = x* -f if + s a gives dr_x dx.y dy , * dz cos 4 , F 2 e cos (u w) = - - r sin 2 4/ 1, and, since these are easily transformed into 2ae sin (w a>} = (2a r) sin 24 , 2ae cos (it w) = (2a r) cos 24> r. If we multiply the first of these equations by cos u and the second by sin w, and add the products; then multiply the first by sin it and the second by cos it, and add, we obtain 2ae sin w = (2a r) sin (2^ + w) r sin it, / ^N 2ae cos w = (2a r) cos (2^ + u) r cos u, These equations give the values of a) 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 d 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 obser- vation. 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 *0 = 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 A#. The differences between the normal places and those computed with the approximate elements to be corrected, will then suffice to form equa- tions of condition by means of which the values of the coefficients A, B, and C may be determined. The epoch for which h = 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 com- puted 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, J3, and C. These having been deter- mined, 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, j5, and C will bo 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 em- ployed as an auxiliary in determining the values of the differential coefficients of a and d with respect to t. Let f(a) denote the right ascension of the body at the middle epoch or that for which h = Q, and let /(a dz no)) denote the value of a for any other date separated by the interval na) 9 in which at 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. /(a 3a>) ff /(a -f 3a>) The series of functions and differences may be extended in the same manner in either direction. If we expand /(a + nco) into a series. the result is /(a-ftt0^:a-|-~nw + 2^ w2ty2 + ^ wVJ 4- A ^^* cy * + &c - or, putting for brevity A = -j-a) ) B = \ -^ - a) 2 , &c., /(a + nco) = a -f An -f 5^ 2 + Oi 3 -f Dn 4 + &c. If we now put n successively equal to 4, 3, 2, 1, 0, +1, &c., we obtain the values of /(a 4co),f(a 3w), ...... /(a -f 4a>) in terms of A, B, C, &c. Then, taking the successive orders of differences and symbolizing them as indicated above, we obtain a series of equations by means of which A, B, C, &c. will be deter- mined in terms of the successive orders of differences. Finally, re- placing Ay B, C, &G. by the quantities which they represent, and putting ir (<*-*) + if '(<*+jo =/'(), if'" ( - *-) + if '" ( + i) =/"' (a), &c. ( we obtain = A- (/'(a) - I/'" (a) + 3 > 5 /'(oO - Ti^f-(a) + &c.), = (/"() - */"() + *? 5 /"" () - *<>.), = (/"() -1/-W 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 d 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 d with respect to t will also be expressed in seconds, and must be divided by ^06264.8 in order to express them abstractly. We may adopt directly the values of -JT> -^p -yr. and -^ determined by means of the corrected ephemeris, or, if the observed places do not include a very long interval, we may determine only the values of -jp, -j, &G. by means of the ephemeris, and then find -=- and -^- directly from the normal places or observations. Thus, let a, a/, a" be three observed right ascensions corresponding to the times t, t', t", and we shall have which give These equations, being solved numerically, will give the values of -7- d*a and , and we may thus by triple combinations of the observed places, using always the same middle place, form equations of con- dition 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. 7 /72/^ In a similar manner the values of -^ and -jr may be derived. 114. In applying these formulae 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 338 THEORETICAL ASTRONOMY. care, since it is to be employed as an auxiliary in determining quan- tities on which depends the accuracy of the final results. The com- parison of the ephemeris with the observed places will furnish, by means of equations of the form A + Eh + Ch 2 = Aa', A _f- B'h + C'K = A*, k being the interval between the middle date t f and that of the place used, the values of A, J5, (7, A', &c.; and the corrections to be applied to the ephemeris will be determined by A -f Bna> -f Oi 2 " 8 = Aa, A' The unit of h may be ten days, or any other convenient interval, observing, however, that no) in the last equations must be expressed in parts of the same unit. With the ephemeris thus corrected, we da d?a dd _ d?8 compute the values of -j-, -^-, ^7, and j- as already explained, inese differential coefficients should be determined with great care, since it is on their accuracy that the subsequent calculation principally de- . . dX dY , dZ . pends. We compute, also, the velocities -jr t -rr, and rr by means 7x-v 7 -p dt at at of the formulaB (43), ^- and being computed from (46). The quantities thus far derived remain unchanged in the two hypotheses with regard to J. Then we assume an approximate value of J, and compute p J cos d ; and by means of the equation (40) or (39) we compute the value ol - It will be observed that if we put the equation (40) in the form dp P C _ = _, + _ p the coefficient -^ remains the same in the two hypotheses. The three equations (38) may be so combined that the resulting value of _J2 ^*^ will not contain -T~> This transformation is easily effected, and may (J tL 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 OEBIT. 339 assumed value of J by means of the equations (106) 31 and the rec- tangular co-ordinates from x = r cos b cos I, y = r cos b sin I, z ==r sin b. The velocities -57, ~, and -r- will be given by (42), and from these and the co-ordinates x, y, z the elements of the orbit will be com- puted 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 J 4- $d, 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 f for the second system. The final correction to be applied to J, in order that the observed place may be exactly represented, will be determined by ('_) + n-0. (56) Each observed right ascension and each observed declination will thus furnish an equation of condition for the determination of A J, observing that the residuals in right ascension should in each case be multiplied by cos d. Finally, the elements which correspond to the geocentric distance J -f- A A will be determined either directly or by interpolation, and these must represent the entire series of observed places. 115. The equations (52) s 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", K, 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 THEORETICAL ASTRONOMY. 4 = E" - E - 2e sin (E" - E} cos J (" + a? r -f r" = 2a 2ae cos (" ) cos J (" -f- JS). For the chord x we have x 1 = (r + r") 2 4rr" cos* J (u" w), which, by means of (58) 4 , gives cos 4 and, substituting for r + ^ r/ its value given by the last of equations (57), we get x 2 = 4a 2 sin 2 (" JE) (1 e 2 cos 2 &E' + E)). (58) J^et us now introduce an auxiliary angle ^, such that cos h = e cos CE" H- J5;), the condition being imposed that h shall be less than 180, and put then the equations (57) and (58) become = 2g 2 sin g cos h, r + r" = 2a (1 cos g x = 2a sin ^ sin h. " = 2a (1 cos g cos A), Further, let us put h ff = $, and the last two of equations (59) give Introducing and e into the first of equations (59), it becomes s = sin e) (<5 sin 5). (61) a? The formulae (60) enable us to determine e and d from r -f r", x, and a, and then the time r' = k (t ff t) may be determined from (61). Since, according to (58) 4 , Vrr" cos J (u" u) = a (cos ^ cos 7i) = 2 sin ^e Bin = r^-T - ~ V rr", 8) which may be used to determine

. In this case, and H will be referred to the equator as the fundamental plane. The angles ^ and i//' will be obtained from the equations (102) 3 , or from equations of the form 350 THEORETICAL ASTRONOMY. of (26), and finally the auxiliary quantities A, B, B", &c. will be obtained from (51) 3 , writing d and d /f in place of ft and ft", respect- ively. As soon as these auxiliary quantities have been determined, by means of (52) 3 the value of % must be found which will exactly satisfy equation (65). To effect this, we first compute e from sin ^e = and, if it be required, we also find 3 from using approximate values of r + r" and x. Then we find Q from (66), and Ar ' from (76) or from (78), the logarithms of the auxiliary quantities B Q and N being found by means of Table XV. with the argument e. The value of r/ having been found from (77), the equations (73) and (74), in connection with Table XI., enable us to obtain a closer approximation to the correct value of x. With this we compute new values of r and r ff , and repeat the determination of x. A few trials will generally give the correct result, and these trials may be facilitated by the use of the formula (67) 3 . It will be observed, also, that Q and Ar ' are very slightly changed by a small change in the values of r -j- r fl and x, so that a repetition of the calculation of these quantities pnly becomes necessary for the final trial in finding the value of x which completely satisfies the equa- tions (52) 3 and (65). When the value of a is such that the values of Q and ^V exceed the limits of Table XV., the equation (61) may be employed, and, in the case of hyperbolic motion, when Q and ' exceed the limits of Table XVI., we may employ the complete ex- pression for the time r' in terms of m and n as given by (79). The values of r, r /r , and x having thus been found, the equations will determine the curtate distances p and />". When the equator is the fundamental plane, we have p = A cos 9, p" = J" cos d". From p, p", and the corresponding geocentric spherical co-ordinates, the radii- vectores and the heliocentric spherical co-ordinates I, I", 6, and b" will be obtained, and thence &, i, u, u ff , and the remaining VARIATION OF THE SEMI-TRANSVERSE AXIS. 351 elements of the orbit, as already illustrated. In the case of elliptic motion, if we compute the auxiliary quantities e and d by means of the equations (60), we shall have e cos J (E" + J) = cos J (e + d), from which e and \(E fr -\~ E} may be found, and hence, since i(J" J) = i(e ), we derive .# and J0". The values of q and i? may then be found directly from these and quantities already obtained. Thus, the last of equations (43)! gives cos ^v cos \E cos ^v" cos ^E" V~q ' 1/r Vq 1/7' Multiplying the first of these expressions by sin Ju", and the second oy sin J0, adding the products, and reducing, we obtain smi(v" v)sin%v _ cos j (v" v) cos %E cos \E" l/q Vr V7' ' Therefore, we shall have _1_ g . n _ cosjJg cos E" V~q Sm 3V " ~ 1/r tan J (M" *) T/7' sin J (t*" uj 1 cos IE -= cos Iv = -, VQ V from which q and v may be found as soon as cos \E and cos \E" are known. In the case of parabolic motion the eccentric anomaly is equal to zero, and these equations become identical with (92) 3 . The angular distance of the perihelion from the ascending node will be obtained from (a = u V. Since r = a ae co$E, and q = a(l e), we have and hence 1-1 a (89) 352 THEORETICAL ASTRONOMY. When the eccentricity is nearly equal to unity, the value of q given by approximate elements will be sufficient to compute cos^ and cos^E" by means of these equations, and the results thus derived will be substituted in the equations (88), from which a new value of q results. If this should differ considerably from that used in com- puting cos \E and cos \E", a repetition of the calculation will give the correct result. In the case of hyperbolic motion, although E and E f> are imagi- nary, we may compute the numerical values of cos^ and cos^E'' from the equations (89), regarding a as negative, and the results will be used for the corresponding quantities in (88) in the computation of q and v for the hyperbolic orbit. Next, we compute a second system of elements from M and/ -f Sf, and a third system from M -f- dM and/, df and dM denoting the arbitrary increments assigned to / and M respectively. The com- parison of these three systems of elements with additional observed places of the comet, will enable us to form the equations of condition for the determination of the most probable values of the corrections &M and A/ to be applied to M and /respectively. The formation of these equations is effected in precisely the same manner as in the case of the variation of the geocentric distances or of & and i, and it does not require any further illustration. The final elements will be ob- tained from M-\- &M, and/-f- A/, either directly or by interpolation. We may remark, further, that it will be convenient to use log M as the quantity to be corrected, and to express the variations of log M in units of the last decimal place of the logarithms. When the orbit differs very little from the parabolic form, it will be most expeditious to make two hypotheses in regard to M, putting in each case = 0, and only compute elliptic or hyperbolic elements in the third hypothesis, for which we use J/and f=df. The first and second systems of elements will thus be parabolic. 120. Instead of M and - we may use A and - as the quantities to a a be corrected. In this case we assume an approximate value of J by means of elements already known, and by means of (96) 3 , (98) 3 , (102),, and (103) 3 , we compute the auxiliary quantities C, B, B", &c., re- quired in the solution of the equations (104) 3 . We assume, also, an approximate value of J r/ and compute the corresponding value of r", the value of r having been already found from the assumed value of J. Then, by trial, we find the value of x which, in connection witn EQUATIONS OF CONDITION. 353 the assumed value of -, will satisfy the equations (104) 3 and (65) or (61). The corresponding value of A" is given by J" = e i/x 2 C\ When A" has thus been determined, the heliocentric places will be obtained by means of the equations (106) 3 and (107) 3 , and, finally, the corresponding elements of the orbit will be computed. If the ecliptic is taken as the fundamental plane, we put D = Q, A = O, and write ^ and /9 in place of a and d respectively. If we now compute a second system of elements from A + d J and f = -, and a third system from J and /+ by ; then we shall have 1 , 1 y = i*~7 and the equations for the residuals are transformed into the form A f f\ I f \ f(\-\ \ A0 = (m nj ) -f- n j z*. (91) If we now assign to 2, successively, the values 1, 2, 3, &c., the re- siduals thus obtained will indicate the value of z which best satisfies the series of observations, and hence how many revolutions of the comet have taken place during the interval denoted by T. 122. In the determination of the orbit of a comet from three ob- served places, a hypothesis in regard to the semi-transverse axis may w r ith facility be introduced simultaneously with the computation of the parabolic elements. The numerical calculation as far as the form- ation of the equations (52) s will be precisely the same for both the parabolic and the elliptic or hyperbolic elements. Then in the one case we find the values of r, r f/ , and x which will satisfy equation (56) 3 , and in the other case we find those which will satisfy the equa- tion (65), as already explained. From the results thus obtained, the two systems of elements will be computed. Let /= -> then in the case of the system of parabolic elements Ave have/=0, and the com- parison of the middle place with these and also with the elliptic or hyperbolic elements will give the value of in which 6 l denotes the geocentric spherical co-ordinate computed from the parabolic elements, and 2 that computed from the other system of elements. Further, let A# denote the difference between computation and observation for the middle place, and the correction to be applied to /, in order that the computed and the observed values of 6 may agree, will be given by Hence, the two observed spherical co-ordinates for the middle place will give two equations of condition from which A/ may be found, 356 THEORETICAL ASTRONOMY. and the corresponding elements will be those which best represent the observations, assuming the adopted value of M to be correct. 123. The first determination of the approximate elements of the orbit of a comet is most readily effected by adopting the ecliptic as the fundamental plane. In the subsequent correction of these ele- ments, by varying - and M or J, it will often be convenient to use a the equator as the fundamental plane, and the first assumption in regard to M will be made by means of the values of the distances given by the approximate elements already known. But if it be desired to compute M directly from three observed places in reference to the equator, without converting the right ascensions and declina- tions into longitudes and latitudes, the requisite formula may be derived by a process entirely analogous to that employed when the curtate distances refer to the ecliptic. The case may occur in which only the right ascension for the middle place is given, so that the corresponding longitude cannot be found. It will then be necessary to adopt the equator as the fundamental plane in determining a system of parabolic elements by means of two complete observations and this incomplete middle place. If we substitute the expressions for the heliocentric co-ordinates in reference to the equator in the equations (4) 3 and (5) s , we shall have n (p cos a R cos D cos A} (p r cos a' R f cos U cos J/) -f n" (P" sin a" R" cos D" cos A"\ = n (p sin a R cos D sin A) (p f sin a' E' cos D' sin A') (92) -f n" (p" sin a" R" cos D" sin A"), Q = n(pttmd R sin D} (p tan S' R' sin Z>') -f- n" (p" tan a" R" sin IT), in which p y //, p tf denote the curtate distances with respect to the equator, A, A f , A" the right ascensions of the sun, and D, Z>', I/' its declinations. These equations correspond to (6) 3 , and may be treated in a similar manner. From the first and second of equations (92) we get = n (p sin (a' a) RcosD sin (ofA)) -f R cos D' sin (a' 4') n" (P" sin (a" a') -f R" cosZ>" sin (a' A")), and hence =? = ^ >4 na-a p n" Sin (a" a') ?i ft cos D sin (a' A) R r cosD' sin (a'- A'}-\-ri'R" cos D" sin (a' A") pri'sm(a" a') VARIATION OF TWO RADII- VECTORES. 357 This formula, being independent of the decimation d', may be used to compute M when only the right ascension for the middle place ia given. For the first assumption in the case of an unknown orbit, we take 1f sin (a' a) M= if t sin(a' / a')' and, by means of the results obtained from this hypothesis, the com- plete expression (93) may be computed. By a process identical with- that employed in deriving the equation (36) 3 , we derive, from (93), the expression p" = p--j$~$- ) (94) j TT f^\tt\i^- \ \R r cos D' sin (a' A!} -g^F^ ^ ; \r* #/ sin (a" a') and, putting ,_. n sin (a' a) n" ' sin (a" a') ' F I ' rr * (?' T ^ cos D' sin (a' A') ^/1__1_\ ^ g n ' r" ^ T sin (a' a) p \r' 3 R' 3 f' we have (95) The calculation of the auxiliary quantities in the equations (52) 3 will be effected by means of the formulae (96) 3 , (86), (87), (102) 3 , and (51) 3 . The heliocentric places for the times t and t" will be given by (106) 3 and (107) 3 , and from these the elements of the orbit will be found according to the process already illustrated. 124. The methods already given for the correction of the approxi- mate elements of the orbit of a heavenly body by means of additional observations or normal places, are those which will generally be applied. There are, however, modifications of these which may be advantageous in rare and special cases, and which will readily suggest themselves. Thus, if it be desired to correct approximate elements by varying two radii-vectores r and r", we may assume an approxi- mate value of each of these, and the three equations (88)j will con- tain only the three unknown quantities d, b, and /. By elimination, these unknown quantities may be found, and in like manner the S58 THEORETICAL ASTRONOMY. values of J", b"j and I". It will be most convenient to compute the angles ^ an< ^ Vj anc ^ then find z and z" from R sin * R" sin V sm z = - , sm z = - ^ , or, putting # 2 = r 2 J? 2 sin 2 ^, and x" 2 = r" 2 -R" 2 sin 2 ^", from sin 4 " sin V tan z = - , tan z = - . x x" The curtate distances will be given by the equations (3), and the heliocentric spherical co-ordinates by means of (4), writing r in place of a. From these u" u may be found, and by means of the values of r, r n ', and u" u the determination of the elements of the orbit may be completed. Then, assigning to r an increment dr, we com- pute a second system of elements, and from r and r" -f- dr ff a third system. The comparison of these three systems of elements with an additional or intermediate observed place will furnish the equations for the determination of the corrections Ar and Ar /r to be applied to r and r", respectively. The comparison of the middle place may be made with the observed geocentric spherical co-ordinates directly, or with the radius-vector and argument of the latitude computed directly from the observed co-ordinates; and in the same manner any number of additional observed places may be employed in forming the equa- tions of condition for the determination of Ar and Ar r/ . Instead of r and r", we may take the projections of these radii- vectores on the plane of the ecliptic as the quantities to be corrected. Let these projected distances of the body from the sun be denoted by r and r ", respectively ; then, by means of the equations (88)^ we obtain (96) from which I may be found ; and in a similar manner we may find I". If we put we have tan(;-,>) = * sin(A - 0) . (97, X Q Let 8 denote the angle at the sun between the earth and the place of the planet or comet projected on the plane of the ecliptic ; then we shall have VARIATION OF TWO RADII-VECTORES. 359 =180 + O I, ==: R8in(Z- L 0) (98) and p tan ft tan 6 = , (99) r o by means of which the heliocentric latitudes b and b" may be found. The calculation of the elements and the correction of r and r " are then effected as in the case of the variation of r and r" . In the case of parabolic motion, the eccentricity being known, we may take q and T as the quantities to be corrected. If we assume approximate values of these elements, r, r', r ff , and v, v f , v" will be given immediately. Then from r, r r , r' 1 and the observed spherical co-ordinates of the body we may compute the values of u" u' and u' u. In the same manner, by means of the observed places, wo compute the angles u" u' and u f u corresponding to q-\-dq and T y and to q and T-+- d T } 3q and dT denoting the arbitrary increments assigned to q and T, respectively. The comparison of the helio- centric motion, during the intervals t" t 1 and t' i, thus obtained, in the case of each of the three systems of elements, from the ob- served geocentric places with the corresponding results given by w" u' = v" v', enables us to form the equations by which we may find the cor- rections A^ and &T to be applied to the assumed values of q and T, respectively, in order that the values of u" u f and u' u computed by means of the observed places shall agree with those given by the true anomalies computed directly from q and T. 360 THEORETICAL ASTRONOMY. CHAPTER VII. METHOD OP LEAST SQUARES, THEORY OF THE COMBINATION OP OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OF OBSERVATIONS. 125. WHEN the elements of the orbit of a heavenly body are known to such a degree of approximation that the squares and products of the corrections which should be applied to them may be neglected, by computing the partial differential coefficients of these elements with respect to each of the observed spherical co-ordinates, we may form, by means of the differences between computation and observa- tion, the equations for the determination of these corrections. Three complete observations will furnish the six equations required for the determination of the corrections to be applied to the six elements of the orbit ; but, if more than three complete places are given, the number of equations will exceed the number of unknown quantities, and the problem will be more than determinate. If the observed places were absolutely exact, the combination of the equations of condition in any manner whatever would furnish the values of these corrections, such that each of these equations would be completely satisfied. The conditions, however, which present themselves in the actual correction of the elements of the orbit of a heavenly body by means of given observed places, are entirely different. When the observations have been corrected for all known instrumental errors, and when all other known corrections have been duly applied, there still remain those accidental errors which arise from various causes, such as the abnormal condition of the atmosphere, the imperfections of vision, and the imperfections in the performance of the instrument employed. These accidental and irregular errors of observation cannot be eliminated from the observed data, and the equations of condition for the determination of the corrections to be applied to the elements of an approximate orbit cannot be completely satisfied by any system of values assigned to the unknown quantities unless the number of equations is the same as the number of these unknown quantities. It becomes an important problem, therefore, to determine the par- ticular combination of these equations of condition, by means of which METHOD OF LEAST SQUARES. 361 the resulting values of the unknown quantities will be those which, while they do not completely satisfy the several equations, will afford the highest degree of probability in favor of their accuracy. It will be of interest also to determine, as far as it may be possible, the degree of accuracy which may be attributed to the separate results. But, in order to simplify the more general problem, in which the quantities sought are determined indirectly by observation, it will be expedient to consider first the simpler case, in which a single quantity is obtained directly by observation. 126. If the accidental errors of observation could be obviated, the different determinations of a magnitude directly by observation would be identical ; but since this is impossible when an extreme limit of precision is sought, we adopt a mean or average value to be derived from the separate results obtained. The adopted value may or may not agree with any individual result, since it is only necessary that the residuals obtained by comparing the adopted value with the observed values shall be such as to make this adopted value the most probable value. It is evident, from the very nature of the case, that we approach here the confines of the unknown, and, before we pro- ceed further, something additional must be assumed. However irregular and uncertain the law of the accidental errors of observation may be, we may at least assume that small errors are more probable than large errors, and that errors surpassing a certain limit will not occur. We may also assume that in the case of a large number of observations, errors in excess will occur as frequently as errors in defect, so that, in general, positive and negative residuals of equal absolute value are equally probable. It appears, therefore, that the relative frequency of the occurrence of an accidental error A in the observed value will depend on the magnitude of this error, and may be expressed by

(n' x) Q = (n x) 7 - ^rjr, -- ^ + (ri x) 7-? & \\, , , 4- &c., (n x) d (n x) (ri x) d (ri x) ' and the comparison of this with (6) shows that d log

(ri x) (n x)d(n x) (ri x)d(ri x) " k being a constant quantity. Hence we derive d loge 1 O ^ 1 2 S ' ^ 1 2 ^J 4 *' which converges rapidly when T is small. To find the value of T which corresponds to the value 0.44311 assigned to the integral, we compute the value of the series (17) for the values 0.45, 0.47, and 0.49 assigned to T, successively, and from the results thus obtained it is easily seen that when the sum of the terms of the series is 0.44311, we have T=Jir = 0.47694, or 0.47694 r = -j , (18) which determines the relation between the probable error and the measure of precision. The probability that the error of an observation, without regard to sign, does not exceed nr, is expressed by , > I <-/ wi/j ( J.t7 ) VTT^O and this integral, therefore, indicates the ratio of the number of obser- vations affected with an error which does not exceed nr to the whole number of observations. Hence, if we assign different values to n, the integral (19) computed for the several assumed values of nhr = 0.47694n will give the relative number of errors of a given magnitude. Thus, if we put n = \) we obtain 0.2385 -4: JV" eft = 0.264. from which it appears that in a series of 1000 observations there ought to be 264 observations in which the error does not exceed \r. It has been found, in this manner, that in the case of an extended series of observations the number of errors of a given magnitude assigned by theory agrees very closely with that actually given by the series of observations ; and hence we conclude that the error com- mitted in extending the limits of the summation in the expression (1) to oo and -f oo, instead of the finite limits which it is presumed that the actual errors cannot exceed, is very slight, so that the form 368 THEORETICAL ASTRONOMY. of the function (p (J) which has been derived may be regarded as that which best satisfies all the conditions of the problem. 129. The relative accuracy of different series of observations may also be indicated by means of what are called the mean error and the mean of the errors for each series, the former being the error whose square is equal to the mean of the squares of all the errors of the series, and the latter the mean of these errors without reference to their algebraic sign. Let denote the mean error ; then, since the number of observa- tions having the error J is m

V x m and we have In the case of a single observation, if P denotes the probability of the error zero, and P f the probability of the error d, we have Hence it appears that if h denotes the measure of precision of the arithmetical mean of m observations, the relation between h and A, the measure of precision of an observation, is given by h * = mh*; (27) and if r is the probable error of the arithmetical mean, and its mean error, we have, according to the equations (18) and (20), (28) These expressions determine the probable and the mean error of the arithmetical mean of a number of observations when these errors in the case of a single observation are known. 131. The expressions for the relation between the mean and pro- bable errors have been derived for the case of a very large number of observations, a number so great that the error of the arithmetical mean becomes equal to zero. In the case of a limited number of observed values of #, the residuals given by comparing the arith- metical mean with the several observations will not, in general, give the true errors of the observations ; but the greater the number of observations, the nearer will these residuals approach the absolute errors. If J, J r , A n ', &c. are the actual errors of the observations, and v, v f , v ff , &c. those which result from the most probable value of x, we shall have, denoting the arithmetical mean by x w and the true value by X Q -f 3, A=v S, A' = v' 3, A" = v " d, &c.; METHOD OF LEAST SQUABES. 371 hence me 2 = [ A J] = [ W ] + m<* 2 . (29) This equation will enable us to determine the mean error of an ob- servation when 3 is given ; but, since this is necessarily unknown, some assumption in regard to its value must be made. If we assume it to be equal to the mean error of the arithmetical mean, the re- maining error will be wholly insensible, and hence the equation (29) becomes we 2 = [iw] + me 2 = [vv~\ -f- 3 . Therefore, we shall have < and, according to (21), r = 0.6745 These equations give the values of the mean and probable errors of a single .observation in terms of the actual residuals found by com- paring the arithmetical mean with the several observed values. The probable and the mean error of the arithmetical mean will be given by ' ^ (32; r = 0.6745^ |^j_ . When the number of observations is very large, the probable error of an observation and also that of the arithmetical mean may be de- termined by means of the mean of the errors. If we suppose the number of positive errors to be the same as the number of negative errors, the mean of the errors without reference to the algebraic sign gives and hence we have, according to (23), r = 0.8453 &i (33) m For the mean error of an observation we have e = l/ii = 1.2533 4 (34) 372 THEORETICAL ASTRONOMY. If the number of observations is very great, the results given oy these equations will agree with those given by (30) and (31); but for any limited series of observed values, the results obtained by means of the mean error will afford the greatest accuracy. 132. The relative accuracy of two or more observed values of a quantity may be expressed by means of what are called their weights. If the observations are made under precisely similar circumstances, so that there is no reason for preferring one to the other, they are said to have the same weight. The weight must therefore depend on the measure of precision of the observations, and hence on their probable errors. The unit of the weight is entirely arbitrary, since only the relative weights are required, and if we denote the weight by p, the value of p indicates the number of observations of equal accuracy which must be combined in order that their arithmetical mean may have the same degree of precision as the observation whose weight is p. Hence, if the weight of a single observation is 1, the arithmetical mean of m such observations will have the weight m. Let the pro- bable error of an observation of the weight unity be denoted by r, and the probable error of that whose weight is p f by r' then, ac- cording to the first of equations (28), we shall have or For the case of an observation whose weight is p rr and whose pro- bable error is r", we have from which it appears that the weights of two observations are to each other inversely as the squares of their probable or mean errors, and f according to (18), directly as the squares of their measures of precision. Let us now consider two values of #, which may be designated by x r and a?", the mean errors of these values being, respectively, e' and e": then, if we put X=x'x" and suppose that both a;' and x tr have been derived from a large num- ber m of observations (and the same number in each case), so that the residuals v 9 v/, v n ', &c. in the case of x f and the residuals v,, v/, v/' t &c. in the case of x n may be regarded as the actual errors of obser- METHOD OF LEAST SQUARES. vatiou, the errors of the value of X t as determined from the several observations, will be V V,, V' V,', V" V,", &G. Let the mean error of X be denoted by E; then we have mE* =S(v v,y = [w] zh 2 [>,] + [v,v t ] ; and since the number of observed values is supposed to be so great that the frequency of negative products w, is the same as that of the similar positive products, so that [vvj] = 0, this equation gives or E 2 = e" -I- e" 2 . Combining X with a third value x" r whose mean error is e'", the mean error of x f x" x 1 " will be found in the same manner to be equal to e /2 + e //2 -j- s ///2 ; and hence we have, for the algebraic sum of any number of separate values, E = * + e' 2 + e" 2 + Ac., (35) and, according to the last of equations (21), R = ir 2 -hr' 2 + r" 2 + &c., (36) R being the probable error of the algebraic sum. If the probable errors of the several values are the same, we have and the probable error of the sum of m values will be given by E = rl/m. Hence tne probable error of the arithmetical mean of m observed values will be R r *= =77 m Vm which agrees with the first of equations (28). Let P denote the weight of the sum X, p f the weight of a?', and p /f that of x" ; then we shall have p _ " 374 THEORETICAL ASTRONOMY. from which we get Since the unit of weight is arbitrary, we may take and hence we have, for the weight of the algebraic sum of any number of values, = ^ == t /1 + r" 2 -f-r'" 2 -{-&c.' or, whatever may be the unit of weight adopted, . I * I __ I p' "" p" ' p'" ' In the case of a series of observed values of a quantity, if we designate by r f the probable error of a residual found by comparing the arithmetical mean with an observed value, by r the probable error of the observation, by X Q the arithmetical mean, and by n any observed value, the probable error of n=x +v, according to (36), will be .+**=; + ", r being the probable error of the arithmetical mean. Hence we derive m m 1 and if we adopt the value / = 0.8453 ^3 ; m' the expression for the probable error of an observation becomes r = 0.8453 M (40) l/m(m 1) in which [v] denotes the sum of the residuals regarded as positive, and m the number of observations. 133. Let n, n f , n", &c. denote the observed values of x, and let p, p', p ff } &c. be their respective weights ; then, according to the defi- METHOD OF LEAST SQUARES. 375 nition of the weight, the value n may be regarded as the arithmetical mean of p observations whose weight is unity, and the same is true in the case of n f , n f/ , &c. We thus resolve the given values into p + p f + p" + observations of the weight unity, and the arith- metical mean of all these gives, for the most probable value of x, _pn + p'ri + p"n" + Ac. _ [pn\ , ' ' &c. ' ' The unit of weight being entirely arbitrary, it is evident that the relation given by this equation is correct as well when the quantities P) p'y P"> & c - are fractional as when they are whole numbers. The weight of X Q as determined by (41) is expressed by the sum and the probable error of x is given by r ' (42) when r, denotes the probable error of an observation whose weight is unity. The value of r, must be found by means of the observa- tions themselves. Thus, there will be p residuals expressed by n x Q , p' residuals expressed by n' o? , and similarly in the case of n", n" r , &c. Hence, according to equation (31), we shall have r, = 0.6745 in which m denotes the number of values to be combined, or the number of quantities n, n f , n", &c. For the mean error of # , we have the equations (44; If different determinations of the quantity x are given, for which the probable errors are r, r', r", &c., the reciprocals of the squares of these probable errors may be taken as the weights of the respective values n, n', n rf , &c., and we shall have !L L ?L- I ^ L .,2 ~r ^/ 2 ~r v //2 T * 376 THEORETICAL ASTRONOMY. with the probable error = ' (46) The mean errors may be used in these equations instead of the pro- bable errors. 134. The results thus obtained for the case of the direct observa- tion of the quantity sought, are applicable to the determination of the conditions for finding the most probable values of several un- known quantities when only a certain function of these quantities is directly observed. In the actual application of the formulae it will always be possible to reduce the problem to the case in which the quantity observed is a linear function of the quantities sought. Thus, let V be the quantity observed, and , y, , &c. the unknown quan- tities to be determined, so that we have Let , y w f , &c. be approximate values of these quantities supposed to be already known by means of previous calculation, and let x, y, z, &c. denote, respectively, the corrections which must be applied to these approximate values in order to obtain their true values. Then, if we suppose that the previous approximation is so close that the squares and products of the several corrections may be neglected, we have _ T , dV , dV . dV v - v =d! x +^y+~dt z +--> and thus the equation is reduced to a linear form. Hence, in general, if we denote by n the difference between the computed and the ob- served value of the function, and similarly in the case of each obser- vation employed, the equations to be solved are of the following form : ax -f- by "h cz ~f~ du -\- ew -f- / -j~ n 0, a'x + b'y -f c'z + d'u + e'w +ft + ri = 0, (47) a " x 4. y'y 4- c " z 4. d"u -f e"w +ft + n" = 0, &c. &c. which may be extended so as to include any number of unknown quantities. If the number of equations is the same as the number of unknown quantities, the resulting values of these will exactly satisfy the several equations; but if the number of equations exceeds the number of unknown quantities, there will not be any system of METHOD OF LEAST SQUARES. 377 values for these which will reduce the second members absolutely to zero, and we can only determine the values for which the errors for the several equations, which may be denoted by v, v', v", &c., will be those which we may regard as belonging to the most probable values of the unknown quantities. Let J, J r , J", &c. be the actual errors of the observed quantities; then the probability that these occur in the case of the observations used in forming the equations of condition, will be expressed by and the most probable values of the unknown quantities will be those which make P a maximum. The form of the function

shall be a minimum. Hence it appears that when the observations are equally precise, the most probable values of the unknown quantities are those which render the sum of the squares of the residuals a minimum, and that, in general, if each error is multiplied by its measure of precision, the sum of the squares of the products thus formed must be a minimum. If we denote the actual residuals by v, v f , v", &c., and regard the observations as having the same measure of precision, the condition that the sum of their squares shall be a minimum gives dM =0 ^M = 0> ^W^o.&c., dx * dy dz 378 THEOEETICAL ASTRONOMY. or dv . ,dv' ,,dv" , V ~T- + v -J- + *> -T- + 0, eta ' eta n da? ~ dv . ,dv r dv" I p **/ . ff UU . - V ^ + V ^ + V dF +---- = > &c. &c. If we differentiate the equations ax -{-by + cz -f- du -j- ew -(-/ + n = v, a'z + 6'y -f- c'z -f d'w + e'w -\-ft + ^' =v', (49) a "a; 4. 5" y _j_ c " z 4_ rf" M + e "w; +f"t + n" = i;' r , &c. &c. with respect to x, y, z, &c., successively, we obtain dv _ dv' f dv" o (50) dv , dv f dv lf dy dy dy &c. &c. &c. Introducing these values into the equations (48), and substituting for 0, v r , v ff , &c. their values given by (49), we get [aa] x -j- [a&] y -f [ac] z -j- [ad] w + [ae] w + [a/] < + [a^] = 0, [a6] x + [65] y + [6c] 2 + [ W] w + \be] w + [6/] t + [6/1] = 0, [ac] a; + [6c] y + [cc] z -f [cd] u + [ce] ti; + [c/] * + [c/i] = 0, \_ad~] x + [6d] y + [cd] 3 + [dd] u + [de] w + [d/] * + [dw] = 0, [ae] a; + {be] y + [ce] 2 + \de\ u + [ee] w -f [e/] < + [en] = 0, [/]* -h P/]y + ['/]* + [d/] + [>/]* + [//] * + [>] - 0, in which [aa] = aa -f- a'a' -j- a" a" -f- . . . . [ac] =ac-f-aV+a'V' + .... [66] = 66 -f W -f V'b" + ____ &c. &c. The equations of condition are thus reduced to the same number as the number of the unknown quantities, and the solution of these will give the values for which the sum of the squares of the residuals will be a minimum. These final equations are called normal equations. When the observations are not equally precise, in accordance with the condition that AV + h /2 v' 2 -\- h tt2 v m -\- &c. shall be a minimum, METHOD OF LEAST SQUARES. 379 each equation of condition must be multiplied by the measure of precision of the observation; or, since the weight is proportional to the square of the measure of precision, each equation of condition must be multiplied by the square root of the weight of the observa- tion, and the several equations of condition, being thus reduced to the same unit of weight, must be combined as indicated by the equa- tions (51). 135. It will be observed that the formation of the first normal equation is eifected by multiplying each equation of condition by the coefficient of x in that equation and then taking the sum of all the equations thus formed. The second normal equation is obtained in the same manner by multiplying by the coefficient of y; and thus by multiplying by the coefficient of each of the unknown quantities the several normal equations are formed. These equations will gene- rally give, by elimination, a system of determinate values of the unknown quantities #, y, z, &c. But if one of the normal equations may be derived from one of the others by multiplying it by a con- stant, or if one of the equations may be derived by a combination of two or more of the remaining equations, the number of distinct rela- tions will be less than the number of unknown quantities, and the problem will thus become indeterminate. In this case an unknown quantity may be expressed in the form of a linear function of one or more of the other unknown quantities. Thus, if the number of independent equations is one less than the number of unknown quantities, the final expressions for all of these quantities except one, will be of the form X = a + Pt, y = a' -f fit, S = a" -f fi't, &C. (53^) The coefficients a, /9, a/, /9', &c. depend on the known terms and co- efficients in the normal equations, and if by any means t can be de- termined independently, the values of x, y, z, &c. become determinate. It is evident, further, that when two of the normal equations may be rendered nearly identical by the introduction of a constant factor, the problem becomes so nearly indeterminate that in the numerical appli- cation the resulting values of the unknown quantities will be very uncertain, so that it will be necessary to express them as in the equa- tions (53). The indeterrnination in the case of the normal equations results necessarily from a similarity in the original equations of condition, and when the problem becomes nearly indeterminate, the identity of 380 THEORETICAL ASTRONOMY. the equations will be closer in the normal equations than in the equa- tions of condition from which they are derived. It should be observed, also, that when we express x } y, z, &c. in terms of t, as in (53), the normal equation in t, which is the one formed by multiplying by the coefficient of t in each of the equations of condition, is not required. 136. The elimination in the solution of the equations (51) is most conveniently effected by the method of substitution. Thus, the first of these equations gives lab'] [ae] [ad] [ae] [a/] [an] . it/ - 7/ ~ "p ^r Z ' ~p - ^ Zv ~" "* r- -. IV ~^~* 1^ ^^ . [aa] [aa] [aa] \_aa\ \_aa] [aa] and if we substitute this for x in each of the remaining normal equa- tions, and put [66] - |g| [at] = [M.1], [fc] - [gj [ao] = [Jc.1], (54) r i r i [] - gj [ae] = [ce.ll, [ffl - jgj- [/] = [c/1] ; [dd] - [afl = [!] - 0. These equations are symmetrical, and of the same form as the normal equations, the coefficients being distinguished by writing the numeral 1 within the brackets. The unknown quantity x is thus eliminated, and by a similar pro- cess y may be eliminated from the equations (59), the resulting equa- tions being rendered symmetrical in form by the introduction of the numeral 2 within the brackets. Thus, we put . = [cc . 2 ], . = [ce.2], [/.!] - [i/.l] = . . and the equations become (.63) [cc.2] z + [crf.2] t* + [ce.2] ti; -f [c/.2] < 4- [cn.2] = 0, 1 [cd.2] z 4 [dd.2] M 4 [rfe-2] w 4- [rff.2] < 4 [d-2] = 0, [ce.2] z + [rfc.2] u 4 [ec.2] to 4 [e/.2] * 4 [cn-2] = 0, [c/.2] z 4 Hf-2] i* 4 [e/2] w 4- [//2] + [>.2] = 0. To eliminate z from these equations, we put 382 THEORETICAL ASTRONOMY. [ee.2] - l [e( ,2] = [ ee . 3 ], .2] = [dn.3], [en.2] - W21 [/-2]-g|-[e.2] = [/n. and we have [eta.3] M + [de.3] w -f [d/.3] < + [d.3] = 0, [efe.3] ti + [ee.3] w + [e/.3] + [en.3] = 0, (68) Again we put, in a similar manner, .3] - [dn.3] = [6n.4], (69) and the equations are + [671.4] = 0, . . . Finally, to eliminate w, we put = [/ re . 5 ] ) (71) and the resulting equation is 0, (72) which gives [/n.5] The value of t thus found enables us to derive that of w by means of the first of equations (70). The value of w being found, that of u will be obtained from the first of equations (68). In like manner, the remaining unknown quantities will be determined by means of the equations (64), (59), and (51). The determination of the unknown quantities is thus reduced to the solution of the following system of equations : METHOD OF LEAST SQUARES. 383 &&, + M M + M W +m t +M =0 [aa] r [aa] [aa] ' [aa] r [aa] 3+ "' f W+ <+ =0> [cc.2] [cc.2] [cc.2] ' the coefficients of which will have been found in the process of de- termining the several auxiliary quantities. It will be observed, further, that both in the normal equations and in those which result after each successive elimination, the coefficients which appear in a horizontal line, with the exception of the coefficient involving the absolute terms of the equations of condition, are found also in the corresponding vertical line. The form of the notation [66.1], [&c.l], &c. may be symbolized thus : = [fir. (ft + 1)1 (75) in which oc, /9, y, denote any three letters, and fj, any numeral. The equations (74) are derived for the case of six unknown quan- tities, which is the number usually to be determined in the correction of the elements of the orbit of a heavenly body; but there will be no difficulty in extending the process indicated to the case of a greater number of unknown quantities, except that the number of auxiliaries symbolized generally by (75) increases very rapidly when the number of unknown quantities is increased. 137. In the numerical application of the formulae, when so many quantities are to be computed, it becomes important to be able to check the accuracy of the calculation in its successive stages. First, then, to prove the calculation of the coefficients in the normal equa- tions, we put a+&+c+d-fe-f/=s, a! -|- j' _j_ c ' + d' + e r +/' = s f , &c. If we multiply each of the sums thus formed by the corresponding absolute term n, and take the sum of all the products, we have 384 THEORETICAL ASTRONOMY. [an] + [5n] + [en] + [an] + M + Lfii] = [m]. (76) In a similar manner, multiplying by each of the coefficients in the original equations of condition, we find [oa] + [aft] + M + [ad] + M + [of] = M, [a6] + [ftft] + [ftc] + [ftd] + [be] + [ft/] - [ft*], [ac] -f M + [cc] + [cd] + [ce] + [cf] = [,], [ad] + [ftd] + [cd] + [dd] -f [de] + [df ] = [da], [ae] + [fte] + M + [dc] + [ee\ + [e/] - [ea], [/] + P/1 + [cf] + Hf] + [/] + [//] - Hence it appears that if we compute the sums , s r , s r/ , s' /r , &c., and form [as], [6s], [cs], &c. simultaneously with the calculation of the coefficients in the normal equations, the equation (76) must be satis- fied when the absolute terms of the normal equations are correct; and the equations (77) must be satisfied when the coefficients of the unknown quantities in the normal equations are correct. The accuracy of the calculation of the auxiliary quantities sym- bolized by the equation (75) may be proved in a similar manner. Thus, we have which, by means of the first and second of equations (77), becomes = [W] _ [a4] + M _ gi [ac] or [ba.l] = [ftft.1] -f [6c.l] + [ftd.l] + [fte.l] + [ft/.l] ; (78) and similarly we derive the expressions for [cs.l], [cfe.l], &c. It is obvious, therefore, that the calculation of the coefficients in the equa- tions (59), (64), (68), and (70) will be checked as in the case of the coefficients in the normal equations, the auxiliaries depending on 8 being determined as if s, s f , s ff , &c. were the coefficients of an addi- tional unknown quantity in the several equations of condition. Hence we must have, finally, [*.5] - 1>5]. (79) If we multiply each of the equations (49) by its v, and take the sum of the several products, we get [av] x -f [bv] y + [cv] * + [dv] u -j- [ev] w -f- [/v] t + \vn\ = [w], METHOD OF LEAST SQUARES. 385 But, according to the equations (48) and (50), we have, for the most probable values of the unknown quantities, [av] = 0, |>] = 0, [eo] = 0, &c. ; and hence [vn~] = [w]. (80) If we multiply each of the equations (49) by its n, and take the sum of all the products thus formed, substituting [vv] for [wi], there re- sults [an] x -f- [bri] y -f- [en] + [dn] u -f- [eri] w -f- \Jri\ t ~j- [nn] = [vv]. Substituting in this the value of x given by the first normal equa- tion, it becomes [bn.Y] y + [cn.l] z + \_dn.Y] u + [en.l] w + [/n.l] < + [nn.l] = [w], in which [n.l] = [*m]-^[.3] t + [nn.3] = [w], [e.4]tc + [>.43<-{-[nn.4] = M, (82) |>.5]< + [nn.5] = M, [nn.6] = [vv]. The expressions for the auxiliaries [ww.2], [nn.3], &c. are [.2] = [.!] - M [in.1], [nn.8] = [n.2] - M [m.2], [nn.4] = [nn.8] - [dn.3], [n.5] = [nn.4] - [on.4], .5]. (83) The process here indicated may be readily extended to the case of a greater number of unknown quantities, and we have, in general, when u denotes the number of unknown quantities, [vv] = [nn.fi"]. (84) 25 386 THEORETICAL ASTRONOMY. This equation affords a complete verification of the entire numerical calculation involved in the determination of the unknown quantities from the original equations of condition. Thus, after the elimination has been completed, we substitute the resulting values of x, y y 2, &c. in the equations of condition, and derive the corresponding values of the residuals v, v f , v", &c. Then, taking the sum of the squares of these, the equation (84) must be satisfied within the limits of the unavoidable errors of calculation with the logarithmic tables em- ployed. If this condition is satisfied, it may be inferred that the entire calculation of the values of the unknown quantities from the given equations of condition is correct. 138. If the values of x, y, z, &c. thus found were the absolutely exact values, the residuals v, v', v", &c. would be the actual errors of observation. But since the results obtained only furnish the most probable values of the unknown quantities, the final residuals may differ slightly from the accidental errors of observation. Further, it is evident that the degree of precision with which the several unknown quantities may be determined by means of the data of the problem may be very different, so that it is desirable to be able to determine the relative weights of the different results. It will be observed that the expressions for either of the unknown quantities resulting from the elimination of the others is a linear "unction of n, n f , n", &c., so that we have x + an + a!n' + a"n" + a" V" + ....== 0, (85) in wLich the coefficients a, a/, a/', &c. are functions of the several coefficients of the unknown quantities in the equations of condition. If we now suppose the equations of condition to be reduced to the same unit of weight, the mean error of the several absolute terms of the equations will be the same, and will be the mean error of an observation whose weight is unity. Thus, if e denotes the mean error of an observation of the weight unity, the mean error of an will be ae, that of a'n' will be o/e', and similarly for the other terms of (85, ; and, according to the equation (35), the mean error of x will bt) z x = ]/a 2 -f- a' 2 -f a" 2 -f- &C. = l/[aa]. (86) Hence the weight of x will be expressed by ' METHOD OF LEAST SQUARES. 387 Let x, denote the true value of x, namely, that which would be obtained if the true values of v, v f , v", &c. were retained in the second members of the equations of condition instead of putting them equal to zero ; then it is evident that the expression for x, must be that which would result by substituting n v in place of n in the formulae for the most probable value as determined from the actual data. Hence we have Xf + n ( n v -) + a>'-t/) + . . . . = 0, and comparing this with the expression (85), we obtain Substituting in this the values of v, v f , v", &c. given by the equations (49), there results x, = x+ [aa] X, -f [06] y, + [ac] z, + [ad] u, -f [oe] w, -f [a/] t, + [aw], and since, according to (85), x -j- [an] = 0, in order to satisfy this expression for x n we must evidently have [aa] = l, [06] =0, [ac]=0, [ad] = 0, [ae] = 0, [a/] = 0. (88) Since the values of the unknown quantities as determined by the normal equations must be the same by whatever mode the elimination may have been performed, let us suppose the method of indeterminate multipliers to be applied for the determination of a?, and let these multipliers be designated by 9, q f , q", &c. ; then, the values of these factors are determined by the condition that the coefficient of x in the final equation shall be unity, and that the coefficients of the other unknown quantities shall be zero. Hence we shall have [aa] q + [oft] q f + [ac] g" + [ad] q'" + .... = 1, lab] q + [56] ^ + [be] q" + [bd] q'" +.... = 0, (89) [ac] q + [be] q' + M 5" + M 5"' -f . . . . = 0, &c. &c. and also, retaining the residuals v, v f , v", &c. in the formation of the normal equations, Therefore, since Xf + [an] = [av], and since the first member of this equation must be identical with the first member of (90), we have 388 THEORETICAL ASTRONOMY. which gives, by expanding the several sums, aq + bq' + cq" + dq m +.... = , a'q -f Vq' + e'q" + d'q'" + .... = ', (91) a"? + &'Y + *'Y' + <*'Y" 4- = a", &c. &c. Multiplying each of these equations by its cc, and adding the pro- ducts, the result is [a] q + [06] q f + [c] 3 " + [ad] 9 '" + ....== [aa], which, by means of the equations (88), reduces to 9 4- (92) Hence it appears that the eliminating factor q is the reciprocal of the weight of x, and, since the coefficients of q, q r , q ff , &c. in the equa- tions (89) are the same as those of x, y, z, &c. in the normal equa- tions, that if we put [an] = 1, \bn\ = 0, [en] = 0, &c., in the normal equations, the resulting value of x will be the reciprocal of the weight of the most probable value of this quantity. The equation (90) shows that if, in the general elimination, by whatever method it may have been effected, we write [av], [bv], &c. instead of zero in the second members of the normal equations re- spectively, the coefficient of [at?] is the reciprocal of the weight of x. It is obvious that it will not be necessary to know the numerical values of [av\ 9 [bv], &c., since only the coefficient q is required. The most probable value of x is found from (90) by the condition of a minimum of the squares of the residuals, namely, that [av] = 0, [bv] = 0, [co] = 0, &c. The process here indicated for the determination of the weight of the final value of x is general, and applies to the case of any other unknown quantity provided that the necessary changes are made in the notation. Thus, the reciprocal of the weight of y is determined by writing, in the normal equations, 1 in place of [bri], and putting [an], [en], &c. equal to zero, and completing the elimination. It is also the coefficient of [bv] in the value of y when the elimination is effected with the symbols [av], [bv], &c. retained in the second members of the normal equations. 139. It may be easily shown that when the elimination is effected by the method of successive substitution, as already explained, the METHOD OF LEAST SQUARES. 3] = I>-1] = l>-2] = |>.3] = |>.4] = [/n.5] = 1, and hence, according to (72), for the reciprocal of the weight of t, which gives (93) The weight of t is therefore equal to its coefficient in the final equa- tion which results from the elimination of the other unknown quan- tities by successive substitution. Hence, by repeating the elimination, successively changing the order of the quantities, so that each of the unknown quantities may have the last place, the weights will be determined independently, and the agreement of the several sets of values for the unknown quantities will be a proof of the accuracy of the calculation. It is not necessary, however, to make so many repetitions of the elimination, since, in each case, the weights of two of the unknown quantities will be given by means of the auxiliaries used in the elimination. Thus, the reciprocal of the weight of w is obtained by putting [eri\ = 1, and the other absolute terms of the normal equations equal to zero, and finding the corresponding value of w. This operation gives Hence the equation (73) becomes t = =- and substituting this value of t in the last of equations (70), we get f.41 ' or <*.4], (94) 390 THEOEETICAL ASTKONOMY. which gives the weight of w in terms of the auxiliary quantities required in the determination of its most probable value. If the order of elimination is now completely reversed, so that x is made the last in the elimination, the weights of x and y will be determined by the equations p.==[oa.5], ^=M [M . 4] . * [aa.4] L A third elimination, in which z and u are the unknown quantities first determined, will give the weights of these determinations. It appears, therefore, that when only four unknown quantities are to be found, a single repetition of the elimination, the order of the quan- tities being completely reversed, will furnish at once the weights of the several results, and check the accuracy of the calculation. When there are only two unknown quantities, the elimination gives directly the values of these quantities and also of their weights. 140. In the case of three or more unknown quantities, the weights of all the results may be determined without repeating the elimina- tion when certain additional auxiliary quantities have been found. The weights of the two which are first determined are given in terms of the auxiliaries required in the elimination, that of the quantity which is next found will require the value of an additional auxiliary quantity, the succeeding one will require two additional auxiliaries, and so on. The equations (74) show that when the substitution is effected analytically the final value of x will have the denominator D = [ad] [66.1] [cc.2] [eW.3] [ee.4] [jgf.5], and this denominator, being the determinant formed from all the coefficients in the normal equations, must evidently have the same value whatever may be the order in which the unknown quantities are eliminated. Let us now suppose that each of the unknown quantities is, in succession, made the last in the elimination, and let the auxiliaries in each elimination be distinguished from those when t is last eliminated by annexing the letter which is the coeffi 3ient rf the quantity first determined ; then we shall have D = [ad] [66.1] [cc.2] [dd.3] \_eeA~] [ff.5] = M, [W.H [<*.2]. [ by means of which the weights of the six unknown quantities may be determined. The process here indicated may be readily extended to the case of a greater number of unknown quantities. The equa- tion for p w is identical with (94), the expression for p u introduces the new auxiliary quantity [,^.4]^, and that for p s introduces two new auxiliaries. The expressions for the new auxiliaries \JfA~\ d ^ [jf/".4] c , [ee.3] c , &c. are easily formed by observing that all the auxiliaries as far as those which are designated by the numeral 4 are not affected by putting e or /last, that, as far as those which contain the numeral 3, it makes no difference whether d, e, or / is placed last, that those distinguished by the numerals 1 and 2 are not affected by making c, c/, e, or /the last, and that those designated by the numeral 1 are unchanged unless a is made the last. Thus, we obtain = [ ^-p]^ 3] ' (97) J92 THEORETICAL ASTRONOMY. and, also, Egl = [ee.2] - In like manner we may derive the expressions for the new auxiliaries introduced into the equations for p y and p x . It will be expedient, however, in the actual application of the formulae, to eliminate first in the order a?, y, z, u, w, t, and the weights of the results for u, w, and t will be obtained by means of the first three of equations (96), the single additional auxiliary required being found by means of (97). Then the elimination should be performed in the order t, w, u, Zj y, x, and we shall have (99) by means of which the weights of #, y, and 2 will be determined. The agreement of the two sets of values of the unknown quantities will prove the accuracy of the numerical calculation in the process of elimination. 141. The weights of the most probable values of the unknown quantities may also be computed separately when certain auxiliary factors have been found, and these factors are those which are intro- duced when the equations (74) are solved by the method of inde- terminate multipliers instead of by successive substitution. Thus, in order to find x, let the first of these equations be multiplied by 1, the second by A', the third by A n ', the fourth by A'", and so on, and let the sum of all these products be taken ; then the equations of condition for the determination of the several eliminating factors will be (ioo) _ , A> , - An , - , jiv - [aa] + t bb.l \ A f [cc.2] * *" ^ 1 ' _ Ca/1 , [V-l] ., , [#2] ,, U "" [aa] + IWj + [cc.2] ^ ^ [dd.3] [ee.4] METHOD OF LEAST SQUARES. 393 To determine y from the last five of equations (74), let the eliminating factors be denoted by B ', "', B* 9 and . v , and we shall have [6e.l] R ,, " _ . [eeJ.2] - + ^ ? > [C6.2] , [fo.3] , , f * ' , , [6/4] Iv fj6 fjE In a similar manner, we obtain the following equations for the de- termination of the eliminating factors necessary for finding the values of the remaining unknown quantities : [ce.2] [rfe.3] ~,,, " [6/4] 1T " (102) [.4J The expressions for the values of the unknown quantities will there- fore become _ - [an] [6ro.l] ,, [ot.2] , [dro.3] . w [en.4] ., " 1 " ""^ [66.1] ' [cc.2] r 7 en r AT rf\~\ (1^3) \_dn.6} \enA\ ^. v L/^AI ^ r LW1 394 THEOKETICAL ASTRONOMY. The tiit of these equations will give the reciprocal of the weight of x, when we put [an] = 1, and the other absolute terms of the normal equations equal to zero; the second will give the reciprocal of the weight of y by putting [bn] = - 1, and the other absolute terms of the normal equations equal to zero ; and, continuing the process, finally the last equation will give the reciprocal of the weight of t when we put fn = 1, and [an], [bn], [en], &c. equal to zero. It remains, therefore, to determine the particular values of [6n.l], [en. 2], &c., and the expressions for the weights will be complete. If we multiply the first of equations (100) by [an], it becomes \bn.l-] = [an-] A' + [bn]. 104) Multiplying the second of equations (100) by [an], and the first of (101) by [bn], adding the products, and introducing the value of [6n.l] just found, we get [en-] - [cn.l] + j^l [bn.l] + [an] A" + [bn] B" = 0, which reduces to [an~] A" + [bn-] B" -f [en] = [cn.2]. (105) Multiplying the third of equations (100) by [an], the second of (101) by [bn], and the first of (102) by [en], adding the products, and re- ducing by means of (104) and (105), we obtain = Idn] - [] = [en.4], [an] A* + [bn] B v + [en] <7 V + \dn\ D v -f [en] E* + [>] = [/7i.5]. ^ The equations (104), (105), (106), and (107), enable us to find the particular values of [6n.l], [cn.2], &c. required in the expressions for the reciprocals of the weights. Thus, for the weight of x, we have Ian] = 1, [bn] = [m] = \dn\ = [en] = [fn] = ; METHOD OF LEAST SQUARES. 395 and these equations give [^.1] = _ A' t [en.2] = A", [dn.3] = - A'", [6W.4] = A iv , L/w.5] = A\ For the case of the weight of y, we have [bn] = 1, [cm] = [CTI] = [drc] = [en] = [/w] = 0, and the same equations give p ?l .l] = 1, [cn.2] = J3", [dn.3] = J3'", [en.4] = B iv , L/k5] = v . We have, also, for the weight of z, |_cn.2] = 1, [cfri.3]= C"", [en.4] = C iv , |>.5] = C\ for the weight of w, [dn.3] = 1, [m.4] = D ly , [./h.5] = 7) T ; for the weight of w, [i.4] = -l, [/n.5] = -^; and finally, for the weight of , [/.5]= -1. Introducing these particular values into the equations (103), the cor- responding values of the unknown quantities are the reciprocals of the weights of their most probable values, respectively; and hence we derive 1 AA ' A " A ' A '" A> " ^ 1Vj[iV AVA * ]7 ~ [oa] ^ [56.1] "*" [cc.2] n " [dd.3] T [ee.4] T [jfif.5] ' 1 _J __ jy^ y B'^^^ B*B* y ~ [SO] + [^2j "" [dd.3] + [e6.4] " 1 1 C'" C'" C IV C IV C V C V [SSI H ' LPT 1 _ 1 E V E V ~Wl 1 1 The equations (103) and (108) will serve to determine separately the value of each unknown quantity and also that of its weight, the 396 THEORETICAL ASTRONOMY. auxiliary factors A' ', A", B", &c. having been found from the equa- tions (100), (101), and (102). If we reverse the operation and re- compose the equations (74) by means of the expressions for the un- known quantities given by (103), the conditions which immediately follow furnish another series of equations for the determination of the auxiliary factors. The ea nations thus derived will give first the values of A', B", v ; and so on. They are equally as convenient as those already given, provided that the values of all the unknown quantities are required as well as their respective weights. 142. The formulae already given for the relations between the data of the problem and the weights of the most probable values of the unknown quantities, are those which are of the greatest practical value. It will be apparent from what has been derived that there must be a variety of methods which may be applied, but that all of these methods involve essentially the same numerical operations. The peculiar symmetry of the normal equations affords also a variety of expressions applicable to the different phases under which the problem presents itself. According to the general theory of elimination, the expression for any unknown quantity, as determined from the normal equations, may be put in the form *= -|[]-f [i]-^M-&c, (109) in which D is the determinant formed from all the coefficients of tho unknown quantities in the normal equations, and in which A, A', A", &c. are the partial determinants required in the elimination. Thus, A is the determinant formed from the coefficients of all the unknown quantities except x, in all the equations except the first; A" is the determinant formed from the coefficients of y, z, &c. in all the equa- tions except the second ; and the values of A n r , A f ", &c. are formed in a similar manner. Now, since the value of x which results when we put [an] 1, and the other absolute terms of the normal equations equal to zero, is the reciprocal of the weight of the most probable value of this unknown quantity as given by (109), we have In like manner, the expression for the most probable value of y will be METHOD OF LEAST SQUARES, 397 y= -M- [^]-M-&c., (ill) B, B', B", &c. being the partial determinants formed when the co- efficients of y are omitted; and for its weight we have The formulae for the most probable value of z and for its weight are entirely analogous to those for x and y y so that the process here indi- cated may be extended to the case of any number of unknown quan- tities. It appears, therefore, that the weight of the most probable value of any unknown quantity is found by dividing the complete determinant of all the coefficients by the partial determinant formed when we omit the normal equation corresponding particularly to this unknown quantity, and when we omit also the coefficients of this quantity in the remaining normal equations. The peculiar arrangement of the coefficients in the normal equa- tions abbreviates somewhat the expressions for the several determi- nants. Thus, in the case of three unknown quantities, we have A = [66] [cc] [6c] 2 , B f = [ad] [cc] [ac] 2 , C" = [aa] [66] [a6] 2 , D = [aa] [66] [cc] + 2[a6] [be] [ac] [ad] [6c] 2 [66] [ac] 2 [cc] [a6] 2 , which are all the quantities required for finding simply the weights of the most probable values of a?, y, and z. The expression for the weight of z is D When there are but two unknown quantities, we have A = [66], B' = [ad], D = [aa] [66] [a6] 2 , and hence _ [aa] [66] [a6] 2 _ [oa] [66] [a6] 2 Px ~ [66] P *~ [aa] When the number of unknown quantities is increased, the expressions for the determinants necessarily become much more complicated, and hence the convenience of other auxiliary quantities is manifest. 143. The case has been already alluded to in which the determina- tion of the values of the unknown quantities is rendered uncertain by the similarity of the signs and coefficients in the normal equations, 398 THEORETICAL ASTRONOMY. and in which the problem becomes nearly indeterminate. Sometimes it will be possible to overcome the difficulty thus encountered by a suitable change of the elements to be determined ; but, generally, for a complete and satisfactory solution, additional data will be required. It often happens, however, that several of the unknown quantities may be accurately determined from the given equations when the values of the others are known, but that the certainty of the deter- mination of the same quantities is very greatly impaired when all the unknown quantities are derived simultaneously from the same equations. Let us suppose that one of the unknown quantities is, from the very nature of the problem, not susceptible of an accurate determination from the data employed. The equations will then present themselves in a form approaching that in which the number of independent relations is one less than the number of unknown quantities, so that it will be necessary to determine the other unknown quantities in terms of that whose value is necessarily uncertain. In this case the elimination should be so arranged that the quantity which is regarded as uncertain is that whose value would be first determined. Then, if its coefficient in the final equation, corre- sponding to (72), is very small, a circumstance which indicates at once the existence of the uncertainty when it is not otherwise sus- pected, the process of elimination should not be completed, and the auxiliary quantities should be determined only as far as those re- quired in the formation of the equation which corresponds to the first of (70). Thus, let t be the uncertain quantity, and we have [6/4] len.4] W = - - - t - p - -pr-, [ee.4] [ee.4] which must be substituted for w in the first of equations (68). AVe thus obtain w, u, z, y, and x as functions of t. If the solution is effected by means of the equations (103), let x w y w Z Q , &c. denote the values of these unknown quantities when we put = 0; and then we shall have _ [an] _ [bnA.~\ [c?i.2] , [cfoi.3] ., [en A] , lv [cm] [55.1] [ce.2] [cW.3] [ee.4] [fokl] [e*.2] [d-3] , [en.4] v riij , 2/0 ~ ~ \m\~teS\* ~\ddX\" [66.4]*' l>*.2] [.4] [ (115) in which (e x ), (e y ), &c. denote the mean errors of x w y w &c. These formulae show, also, that when one of the variables is neglected, the equations assign too great a degree of precision to the results thus obtained. When there are two or more unknown quantities which cannot be determined from the data with sufficient certainty, the problem must be treated in a manner entirely analogous to that here indicated; but, since cases of this kind will rarely, if ever, occur, it is not necessary to pursue the subject further. 144. The weights which are obtained for the most probable values of the unknown quantities enable us to find the mean and probable errors of these values. Let s denote the mean error of an observa- tion whose weight is unity; then the mean error of x will be (116) and, in like manner, the expressions "for the mean errors of y, z, u, &c. will be e =" '. = -?- . = -=,&c. (117) 1/P, Vp. Vp, It remains, therefore, to determine the value of e by means of the final residuals obtained by comparing the observed values of the function with those given by the most probable values of the va- 400 THEORETICAL ASTRONOMY. riables. If these residuals were the actual fortuitous errors of obser- vation, the mean error of an observation would be M m being the number of equations of condition. This value is evi- dently an approximation to the correct result; but since by supposing the residuals v, v f , v", &c. to be the actual errors of the several ob- served values of the function, we assign too high a degree of pre- cision to the several results, the true value of e must necessarily be greater than that given by this equation. Let the true values of the unknown quantities be x -f- A#, y -f- Ay, z -f- AZ, &c., the substitution of which in the several equations of condition would give the residuals J, J r , J", &c. ; then we shall have &A?/ -f- cAz -f- d&u ---- -f v = A, ' ' &c. &c. If we multiply each of these equations by its J, and take the sura of all the products, we get [o J] *x -f [6 J] Ay -f [cJ] A^ + [dJ] AW + .... -f [vJ] = [J J]. But if we multiply each of the same equations by its v, take the sum of the products, and reduce by means of (48) and (50), we obtain _ = [>*]; and hence we derive [J J] = [VV] 4 [O J] AX -f [6 J] Ay + [C J] A3 + [d J] AW + .... (119) If we form the normal equations from (118), it will be observed that they are of the same form as the normal equations formed from the original equations of condition, provided that we write A in place of n; and hence, according to (85), we have A* = aJ + a'J' + a"J'' + ..... We have, also, [o J] = aA + a' A' 4 a" A" 4 , and the product of these equations gives [aJ] AX aa J 2 4- aVJ' 2 4 a"a" J" 2 + /. . . 4-aa'JJ'-f aa"JJ"4-.... The mean value of the terms containing JJ', Jd",.&c. is zero, and COMBINATION OF OBSERVATIONS. 40J for the mean values of J 2 , J' 2 , J" 2 , &c. we must, in each case, write e 2 . Hence the mean value of the product [a J] A# will be and this, by means of the first of equations (88), is further reduced to [a J] &x 2 . In a similar manner, we obtain the value e 2 for the mean value of each of the products [ftzfJA?/, [cJJA2, &c. Now, the terms added to [vv] in the second member of the equation (119) are necessarily very small, and, although their exact value cannot be determined, we may without sensible error adopt the mean values of the several terms as here determined, so that the equation becomes [ J J] = [tw] + pe*, (120) u being the number of unknown quantities. Therefore, since [ J J] = me 2 , we shall have m fj. * m /j. by means of which the mean error of an observation whose weight is unity may be determined. When p = 1, this equation becomes identical with (30). For the determination of the probable errors of the final values of the unknown quantities, if r denotes the probable error of an obser- vation of the weight unity, we have the following equations: r = 0.67449 r = i/? 145. The formulae which result from the theory of errors according to which the method of least squares is derived, enable us to combine the data furnished by observation so as to overcome, in the greatest degree possible, the effect of those accidental errors which no refine- ment of theory can successfully eliminate. The problem of the cor- rection of the approximate elements of the orbit of a heavenly body by means of a series of observed places, requires the application of nearly all the distinct results which have been derived. The first approximate elements of the orbit of the body will be determined om three or four observed places according to the methods which 26 102 THEORETICAL ASTRONOMY. have been already explained. In the case of a planet, if the inclina- tion is not very small, the method of three geocentric places may be employed, but it will, in general, afford greater accuracy and require but little additional labor to base the first determination on four observed places, according to the process already illustrated. In the case of a comet, the first assumption made is that the orbit is a parabola, and the elements derived in accordance with this hypothesis may be successively corrected, until it is apparent whether it is ne- cessary to make any further assumption in regard to the value of the eccentricity. In all cases, the approximate elements derived from a few places should be further corrected by means of more extended data before any attempt is made to obtain a more complete determi- nation of the elements. The various methods by which this pre- liminary correction may be effected have been already sufficiently de- veloped. The fundamental places adopted as the basis of the correction may be single observed places separated by considerable intervals of time; but it will be preferable to use places which may be regarded as the average of a number of observations made on the same day or during a few days before and after the date of the average or normal place. The ephemeris computed from the approximate elements known may be assumed to represent the actual path so closely that, for an interval of a few days, the difference between computation and observation may be regarded as being constant, or at least as varying proportion- ally to the time. Let n, n', n", &c. be the differences between com- putation and observation, in the case of either spherical co-ordinate, for the dates t, t f , t n ', &c., respectively; then, if the interval between the extreme observations to be combined in the formation of the normal place is not too great, and if we regard the observations as equally precise, the normal difference n Q between computation and observation will be found by taking the arithmetical mean of the several values of n, and this being applied with the proper sign to the computed spherical co-ordinate for the date , which is the mean of t, I', t", &c.j will give the corresponding normal place. But when different weights p, p', p ff , &c. are assigned to the observations, the value of n must be found from _ np -f n'p' + n"p" + . . . . > and the weight of this value will be equal to the sum P+p' COMBINATION OF OBSERVATIONS. 403 The date of the normal place will be determined by _pt+ P -e + prr + .... * P+P'+P" + .... ' ]f the error of the ephemeris can be considered as nearly constant, it is not necessary to determine with great precision, since any date not differing much from the average of all may be adopted with suf- ficient accuracy. It should be observed further that, in order to obtain the greatest accuracy practicable, the spherical co-ordinates of the body for the date t should be computed directly from the elements, so that the resulting normal place may be as free as possible from the effect of neglected differences in the interpolation of the ephemeris. When the differences between the computed and the observed places to be combined for the formation of a normal place cannot be considered as varying proportionally to the time, we may derive the error of the ephemeris from an equation of the form of (53) 6 , namely, the coefficients J., jB, and C being found from equations of condition formed by means of the several known values of A# in the case of each of the spherical co-ordinates. 146. In this way we obtain normal places at convenient intervals throughout the entire period during which the body was observed. From three or more of these normal places, a new system of elements should be computed by means of some one of the methods which have already been given; and these fundamental places being judi- ciously selected, the resulting elements will furnish a pretty close approximation to the truth, so that the residuals which are found by comparing them with all the directly observed places may be regarded as indicating very nearly the actual errors of those places. We may then proceed to investigate the character of the observations more fully. But since the observations will have been made at many dif- ferent places, by different observers, with instruments of different sizes, and under a variety of dissimilar attendant circumstances, it may be easily understood that the investigation will involve much that is vague and uncertain. In the theory of errors which has been developed in this chapter, it has been assumed that all constant errors have been duly eliminated, and that the only errors which remain are those accidental errors which must ever continue in a greater or less degree undetermined. The greater the number and 404 THEORETICAL ASTRONOMY. perfection of the observations employed, the more nearly will these errors be determined, and the more nearly will the law of their dis- tribution 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 regular 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 reduc- tion 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 pro- visional elements with the entire series of observed places thus cor- rected; and in the calculation of the ephemeris it will be advan- tageous 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 + Cr\ (125) in the case of each of the spherical co-ordinates, by means of equa- tions 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 obser- vations. When the number of observations is sonsiderable, 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 cor- rections 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 pro- bable mean of the corrected residuals for the series of observations made by each observer may be regarded as the average error of obser- vation 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 pro- bable 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. Whatever 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 4=- be the mean error of m comparisons, and the mean error of V m the resulting place of the body will, according to equation (35), be given by . = + '.'. (126) The value of e , in the case of each series, will be found by means of the residuals finally corrected for the constant errors, and the value of e s is supposed to be determined in the formation of the catalogue of star-places adopted. Hence the actual mean error of an observa- tion consisting of a single comparison will be e, = l/m(> 2 -e, 2 ). (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 Sf Vm The mean error of an observation whose weight is unity being de- noted by e, the weight of an observation based on m comparisons will be , = (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 t) and we may then derive e, directly from the residuals corrected for constant errors. Further, in the case of meridian observations, the error which corre- sponds to e s will be extremely small, and hence it is only when these are combined with equatorial observations, or when equatorial obser- vations 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 correc- tions assigned by equation (125) to be applied to the provisional ephemeris, we find the value of e, given by the equation ne,* = [row] [m] e, 2 , (129) in which n denotes the number of observations; m, m', m", &c. the number of comparisons for the respective observations; and v, v', v ff , &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 : Ad m Ad m + 2".0 5, 1".0 7, 1 .8 5, + 1 .5 5, .4 10, +4 .1 8, 5 .5 5, .0 5. Let the mean error of the place of a comparison-star be then we have n ~ 8, and, according to (129), 8s, 2 =341.78 200.0, which gives *,= :4".2. Let us now adopt as the unit of weight that for which the mean erroi is 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 e. Then the differences between the places computed from the provisional elements to be corrected and the observed places cor- rected for the constant error of the observer, must be combined ac- cording to the equations (123) and (125), the adopted values of p, p', p"j &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 de- sired 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 p which will satisfy the condition imposed. The mean error of an observation whose weight is unity will then be given by the value of e being that used in the determination of the weights p, p', &o. 148. The observations of comets are liable to be affected by other errors in addition to those which are common to these and to planet- ary 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 fur- ther, 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 dis- tribution of many errors of this class, renders it necessary to adopt empirical methods, the success of which will depend on the discrimi- nation of the computer. If e denotes the mean jerror of an observation based on m com- COMBINATION OF OBSERVATIONS. 409 parisons, and e c the mean error to be feared on account of the pecu- liarities of the physical appearance of the comet, 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 e n ' = M -{-./. (130) The value of e c 2 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 2 ; 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 e c by means of the formula in which n denotes the number of observations combined. Sometimes, also, in the case of comets, in order to detect the opera~ tion 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 observa- tions 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 appro- priate modifications of the complete process of determining the rela- tive weights of the observations to be combined, or indeed a relaxa- tion 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 | for another class of observations, J for a third class, and so on. It should be observed, also, that when there are but few obser- vations 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 any 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 pre- cision. 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 pro- vided that it surpasses a limit which is fixed by the adopted theory of errors. If the number of observations is so large that the dis- tribution 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 observa- tions, according to the expression (19), the number of errors greater than nr will be COMBINATION OF OBSERVATIONS. 4J1 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 2m (131) By means of this equation we derive for given values of m the cor- responding 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 f = 0.67449w, the limiting error will be n'e, and the values of n' corresponding to given values of m will be as exhibited in the following table. TABLE. m ' m n' m n' m ' | i 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 obser- vation by means of all the residuals. Then, with the value of m as the argument, we take from the table the corresponding value of n', and if one of the residuals exceeds the value n'e it must be rejected, Again, finding a new value of e from the remaining m 1 residuals, and repeating the operation, it will be seen whether another observa- tion should be rejected ; and the process may be continued until a limit is reached which does not require the further rejection of ob- servations. 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 [w] = 320.4. Then, according to equation (30), we shall have * = 2".56. 412 THEORETICAL ASTRONOMY. Corresponding to the value m = 50, the table gives n f -- 2.576, and the limiting value of the error becomes and hence the residuals ll /7 .5 and + 7".8 are rejected. Recom- puting the mean error of an observation, we have 320-4 -193.0S = 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 observa- tions, 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 doubt- ful 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 con- tinuously or discontinuously in the production of large apparent errors. It is the business of the observer to indicate the circum- stances peculiar to the phenomenon observed, the instruments em- ployed, and the methods of observation; and the discussion of the data thus furnished by different observers, as far as possible in ac- cordance 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 differ- ences 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 f , 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, 6, c, a', &', &c. in the equations of condition. The mode of calculating these coefficients, for different systems of co-or- dinates, 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 care- fully checked by direct calculation, assigning variations to the ele- ments, 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 equa- tions will thus be reduced to the same weight. In general, tho numerical values of the coefficients will be such that it will be con- venient, 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 equa- tions, 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 vz, i/ being any entire or fractional number such that the new coefficients -, T , &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 vx, we have the equation ?. = >*-. (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, th* 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 calcula- tion 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 quantities 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 funda- mental plane, the equations for the longitude involve only four ele- ments, 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, substi- tuting 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 & 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 f , n", &c. in a horizontal line, directly under these the logarithms of the coefficients a, a', a", &c., then the logarithms of 6, &', 6", &c., and so on. Then writing, in a corresponding form, the values of logn, logn r , &c. on a slip of paper, by bringing this successively over each line, the sums [nri], [an], [6n], &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], [a&], [ac], &c. The multiplication by 6, c, d t CORRECTION OF THE ELEMENTS. 415 &c. successively is effected in a similar manner; and thus will be derived [66], [6c], [bcl], &c., and finally \_ff] in the case of six un- known 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], [6s], [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 [aa], [a6], [ac], . . . [as], [an], and di- rectly under them the corresponding logarithms. Next, we write under these, commencing with [a6], the values of [66], [6c], [bd], . . [6s], [6n] ; then, adding the logarithm of the factor = - to the \_aa\ logarithms of [a6], [ac], &c. successively, we write the value of p=r [a6] under [66], that of p =r [ac] under [6c], and so on. Sub- [oa] L J [aa] L J tracting the numbers in this line from those in the line above, the differences give the values of [66.1], [6c.l], . . . [6s.l], [6n.l], 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], . . [cs], [en], placing [cc] under [6c.l], and, having added the logarithm of ^ - to the logarithms of [ac], [ad], &c. in succession, we derive, according to the equations (55) and (58), the values of [cc.l], [coM], . . [cs.l], [cn.l], which are to be placed under the cor- responding quantities [cc], [cd], &c. Next, we subtract from these, respectively, the products t [6c - 1] r6on [6e - 1] rwn [ic - 1] rfcn [6 ^r6n f6UJ [6 [663] Lftrf - 1J> ' ' [663] L * S ' 1J ' f 66.11 L ' 416 THEORETICAL ASTRONOMY. and thus derive the values of [cc.2], [cd2], . . [es.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] y . . [dn], writing \_dd~] under [ccZ.2] ; and from each of these in succession we subtract the products thus finding the values of [ we g e ^ [ ee '^]j [^/-l]) [ 6S '1]> an( i [0W-.1]; then subtracting from these the products j ' ' [66.1] we obtain the values of [ee.2], [e/.2], [es.2], and [en.2]. Again, subtracting we have the values of [ee.3], [^f.3], [es.3], [cn.3]; and finally, sub- tracting from these the products j ' ' E3Z3] ' > we derive the results for [66.4], [e/.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 [jj/*], [/s], and [/n], CORRECTION OF THE ELEMENTS. 41* placing [jgf] under [e/.4], and by means of five successive subtrac- tions entirely analogous to what precedes, and as indicated by the remaining equations for the auxiliaries, we obtain the values of [j^.5], [/a.5], and [/n.5]. The values of [fo.l], [cs.l], [cs.2], &c. serve to check the calcula- tion of the successive auxiliary coefficients. Thus we must have [56.1] + [6c.l] -f- [fcd.1] + [6e.l] + [6/.1] = [6s.l] [6c.l] -f [cc.l] + [cdl] -f [ce.l] -f [c/.l] = [.l], &c., [cc.2] + [ed.2] -f [ce.2] -f [c/.2] = [cs.2], [cd.2] + [eta.2] + [efe.2] + [d/.2] = [cfo.2], Ac. 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 [nri] to be determined. These additional auxiliaries will be found by means of the formulae . , - . .3] = [m.2] - [es . 2] , [m . 4] = [m.3] - [efe.3], (133) 0.5] = [w.4] - [.4], [w.8] = [w.5] - and the equations (81) and (83). The arrangement of the numerical process should be similar to that already explained. The values of [sw.l], [sw.2], &c. check the accuracy of the results for [6n.l], [cn.l], [cn.2], [e?n.3], &c. by means of the equations [bn.l] + [cn.l] -f \dn.V\ + [67i.l] + [>.l] = [i.l], [c7i.2] + [dn.Z] -f- [e^.2] -f [/7i.2] == [sw.2], [dw.3] -}- [e.3] + LM] = [m.3], (134; It appears further, that, in the case of six unknown quantities, sinct [/s.5] = [jgr.5], we have [n.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 equa- tions, 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 loga- rithms of the coefficients of the unknown quantities in these equa- tions will have been already found in the computation of the aux- iliaries. If we add together the several equations of (74), first clearing them of fractions, we get = [ad] x + ([06] + [56.1]) y + ([oc] + [6c.l] -f [ee.2]) z + (M + [W-l] + + (M + [&e-i] + -f ([/] + P/1] + + [aw] + [6n.l] + [m.2] -+ [d.3] + [m.4] + [>.5] ; and this equation must be satisfied by the values of #, 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: 0.707z + 2.052y 2.3723 0.221w + 6".58 = 0, 0.471z + 1.347y 1.715s 0.085u + 1 .63 = 0, 0.260* + 0.770y 0.356z -f 0.483w 4 .40 = 0, 0.092* + 0.343?/ -f 0.2352 -f 0.469w 10 .21 = 0, 0.414* -f 1.204y 1.5063 0.205w + 3 .99 = 0, 0.040* -f 0.150# + 0.1042 + 0.206w 4 .34 = 0. First, we derive [nn] = 204.313, [ol = + 4.815, [oa]^ + 0.971, [6n] = + 12.961, [06] = + 2.821, [66] = + 8.208, [en] - 25.697, [oc]= 3.175, [6c] = 9.168, [cc] = + 11.028, [dn]= -10.218, [od] = 0.104, [W] = 0.261, [cd] = + 0.938, [dd] = + 0.594, [w] =-18.139, [as] = + 0.513, [6s] = + 1.610, [c] = 0.377, [as] =+1.177. The values of [sn], [as], [6s], [cs], and [cfe], found by taking the sums of the normal coefficients, agree exactly with the values com- puted directly, thus proving the calculation of these coefficients. The normal equations are, therefore, NUMERICAL EXAMPLE. 419 0.971* -f 2.821y 3.1752 0.104w -f 4.815 = 0, 2.821ar -f- 8.208y 9.168s O.'Zdlu -f 12.961 = 0, 3.175* 9.168y + 11.0282 -f- 0.938it 25.697 = 0, - 0.104* 0.251y + 0.9382 -f- 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; and 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 : = + 0.0123, [fcc.l] = + 0.0562, [6*1] = + 0.0511, [bs.l] = + 0.1196, [fenl] = 1.0278, [cc.l] = -f 0.6463, [ccZ.l] = + 0.5979, [cs.l] = + 1.3004, [cn.l] = 9.9528, [cc.2] = + 0.3895, [cd.2] = + 0.3644, [cs.2] = + 0.7539, [cn.2] = - 5.2567, [dd.l] = + 0.5829, [ds.l] = + 1.2319, [dn.l] = 9.7023, [dd.2] = + 0.3706, [ds.2] = + 0.7350, [dre.2] = 5.4323, [cW.3] = + 0.0297, [ds.3] = -f- 0.0297 [dn.3] = 0.5143, [nw.l] = 180.436, [sn.l] = 20.6828, [nn.2]= 94.552, [sn.2] = 10.6889, [nn.3]= 23.608, [i.3]= 0.5143, [wi.4]= 14.698, [sn.4] = 0. The several checks agree completely, and only the value of [rw.4] remains to be proved. The equations (74) therefore give * -f 2.9052y 3.26982 0.1071w + 4.9588 == 0, y + 4.56912 + 4.1545w 83.5610 = 0, z + 0.9356w 13.4960 = 0, u 17.3165 = 0, and from these we get u = -f 17".316, z = 2".705, y = + 23".977, * = 81".608. Then the equation (135) becomes r= + 0.9710* + 2.8333y 2.72932 + 0.3412w 1.9838, which is satisfied by the preceding values of the unknown quantities. If we substitute these values of x, y, z, and u in the equations of condition already reduced to the same weight by multiplication by the square roots of their weights, we obtain the residuals i Q" gj -^f g^ i 2" 17 2" 01 0".40 0".72 The sum of the squares of these gives [w] = {nnA'] = 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 to* + l^y + M* + ld]u + ....+ [n] - 0], (136) which may be used to check the substitution of the numerical values in the determination of v, v' 9 &c. Thus, we have, for the values here given, 1.984s + 5.866y 5.610z + 0.647w 6.75 = [>] = l."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. = [*-3] 0.00056, p 9 = |^|| [66.2] 0.0072. We may also compute the weights by means of the equations (96). Thus, to find the weight of y, we have [cW.21 = [cM.l] - - [crf.l] = + 0.02977, and hence 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" = -f 16.5442, A'" = 3.3012, #' = 4.5691, JB"' = + 0.1202, C"" = 0.9356, and then the equations (103) and (108) give e = 81".609, y = + 23".977, z = 2".705, u =-- + 17".316 f Pm = 0.00057, p v = 0.0074, A = 0.0312, Pv = 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 deter- mining 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 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 = . , r, = , > &c., * Vpl ' VP, ' Vp. give r x = 68".25, r y = 18".94, r z = 9".22, r u = 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 coef- ficients in the equations of condition and also in the normal equation** 422 THEORETICAL ASTRONOMY. differ very considerably both in magnitude and in sign. In the cor- rection of the elements of the orbit of a planet when the observa- tions 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 formulae 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 uncer- tainty 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 calcula- tion 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 mag- nitude, 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 cor- rections 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 f , 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 vhich 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 suffi- ciently accurate to relax a little the rigorous requirements of a com- plete solution, and use, instead of the actual coefficients, equivalent numbers which are more convenient in the numerical operations re- quired. 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 deter- mination 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 esti- mation of the weights of the observations in the preliminary combi- nation, 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 coeffi- cients 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 indi- cated by the observations, the determination of the correction to be applied to the mean daily motion as given by the provisional ele- ments, 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 A// 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 : Therefore we shall have [>] = [ Vo ] + 2 [ V ] AM -f which may be applied more conveniently in the equivalent form M = [ Vo ] - d [v] + M AM + [ '. (138) The most probable value of A/Z will be that which renders [vv] a minimum, or (139) and the corresponding value of the sum of the squares of the residuals is l- (140) The correction given by equation (139) having been applied to /*, the result may be regarded as the most probable value of that ele- ment, 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 -f QAM, &c. CORRECTION OF THE ELEMENTS. 425 the coefficients A QJ B w C , &c. being those given by the equations (114), and thus the elements may be derived which correspond to any assumed value of p differing from its most probable value. The unknown quantity A/* will also be retained in the values of the residuals. Hence, if we assign small increments to //, it nrjy easily be seen how much this element may differ from its mo^fi probable value without giving results for the residuals which are i/v,ompatible with the evidence furnished by the observations. If the dimensions of the orbit are expressed by mearx.* of the ele- ments 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 //; and we proceed in a similar manner when the correction to be applied to a given value of tL; semi-transverse axis a is one of the unknown quantities to be dete mined. 426 THEORETICAL ASTRONOMY, CHAPTER VIII. INVESTIGATION OP VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. 155. WE have thus far considered the circumstances of the undis- turbed 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) 1? namely, m) , 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, m' /I xaf + yy + t \ , m" / 1 xx"+ yf+ zz" \ '-lm\ r' 3 J^l m \' r "* P which is called the perturbing function, of which the partial differen- tial coefficients, with respect to the co-ordinates, are dQ_ m' Ix'-x ^\ m " lx"-x x" fa-l+m\ ff r'*l + l + m\ p" r"> dQ _ m' [z' z J_\ , m" jz" z 4' \ , ^ S"~rfm\ P 3 r^/^l+ml ?'* r"* / " and in which m', m r/ , &c. denote the ratios of the masses of the several disturbing planets to the mass of the sun, and m the ratio of the mass of the disturbed planet to that of the sun. These partial differential coefficients, when multiplied by & 2 (l-f-w), express thn PERTURBATIONS. 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 dt* /7/2 ' ^ ' J ,, 3 ' \*^y 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 dt. Let us now designate by t the epoch to which the elements of the orbit belong, and let these elements be designated by M w TT O , & , i w e , and a ; 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 sub- ject to the action of the disturbing forces only during the infinitesimal interval dt, and at the time t Q + dt it will commence to describe a new orbit of which the elements will differ from these constant ele- ments 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 M, TT, &, i, e, and a, they will be connected with the constant elements, or those which determine the orbit at the instant , by the equations 428 THEORETICAL ASTRONOMY. di (4) in which 7-, -r- . &c. denote the differential coefficients of the ele- at at ments depending on the disturbing forces. When these differential coefficients are known, we may determine, by simple quadrature, the perturbations dM, dn, &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 con- verging series and then differentiate each term of this series relatively to the elements. This function is usually developed into a converg- ing series arranged in reference to the ascending powers of the eccen- tricities 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 approxima- tion known as mechanical quadratures, according to which we deter- mine the variations of the elements from one epoch t Q to another epoch t. This method is applicable to any case, and may be advan- tageously 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 , 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 deter- mine 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, there- fore, derive various formulae for this purpose before investigating th< formulae for the direct variation of the elements. 156. Let X Q , y , z be the rectangular co-ordinates of the body at the time t computed by means of the osculating elements M w TT O , Q , &c., corresponding to the epoch t . Let x, y, z be the actual co-ordi- nates of the disturbed body at the time t; and we shall have dx, dy, and dz being the perturbations of the rectangular co-ordinates from the epoch t to the time t. If we substitute these values of x, y, and z in the equations (1), and then subtract from each the corre- sponding one of equations (3), we get Let us now put r = r -f- dr; then to terms of the order ^r 2 , which is equivalent to considering only the first power of the disturbing force, we have and hence We have also from r'-z' + ^ neglecting terms of the second order, (7) 430 THEORETICAL ASTRONOMY. The integration of the equations (6) will give the perturbations dx, dy, and dz to be applied to the rectangular co-ordinates x w y w z com- puted 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 dx 9 %, dz 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 inte- gration is possible. If we consider only a single disturbing planet, according to the equations (2), we shall have *' \ 7')' 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 .r , y , Z Q in place of x, y, z, the equations (6) become df r, (9) which are the equations for computing the perturbations of the rec- tangular co-ordinates with reference only to the first power of the masses or disturbing forces. We have, further, , = (gf _ .,.) + ft - y ) -I- (/ _ z y, (10) in which, when terms of the second order are neglected, we use the values x Q , y , z for x, y, and z respectively. 1 57. From the values of 8x, dy, and dz 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: VAEIATION OF CO-OKDINATES. 431 d? in which nothing is neglected. In the application of these formulae, as soon as dx, dy, and $2 have been found for a few successive inter- vals, we may readily derive approximate values of these quantities for the date next following, and with these find x = x -\-dx, y = 2/ + dy, Z = Z Q -\- dz, 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 -5f which will be convenient in the numerical calculation, we have and therefore . = x 2 g * (y 2 ~T~ Let us now put ?= and /}=i then we shall have and the values of/ may be tabulated with the argument q. The equations (11) therefore become (14) 432 THEOKETICAL ASTEONOMY. The coefficients of 8x, dy, and dz 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 / corresponding to the resulting value of g, the numerical values of , 2 > , f > and d*8z -ITT, which include the squares and products of the masses, will be obtained. The integration of these will give more exact values of dx, dy, and dz, 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 log/ for positive or negative values of q at intervals of 0.0001 from q = 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 may be computed directly, using the value of r in terms of r Q and dx, dy, dz. In the application of the preceding formulae, 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 funda- mental plane, the positive axis of x being directed to the vernal equinox. By choosing the plane of the elliptic orbit at the time t as the plane of xy, the co-ordinate z will be of the order of the per- turbations, and the calculation of this part of the action of the dis- turbing force will be very much abbreviated; but unless the inclina- tion is very large there will be no actual advantage in this selection, since the computation of the values of the components of the dis- turbing 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 J f(x)dx and JJ J(x)dx* MECHANICAL QUADRATURE. 433 when the values of f(x) are computed for successive values of x in- creasing 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 + {3x + r x* + W+ex*+....; (15) and if we consider only three terms of this series, the resulting equa- tion f(x) ^a + px + rx* is that of the common parabola of which the abscissa is x and the ordinate /(#), and the integral of f(x) dx is the area included by the abscissa, two ordinates, and the included arc of this curve. Gene- rally, 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 of/(#), 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 and x', we get dx = If now the values of f(x) for different values of x from to x f are known, each of these, by means of equation (15), will furnish an equation for the determination of a, /?, 7-, &c. ; and the number of terms which may be taken will be equal to the number of different known values of /(a?). As soon as a, /?, 7-, &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 inter- vals and we integrate between the limits x = 0, and x = n&x, &x being the constant interval between the successive values of x y and n the number of intervals from the beginning of the integration, we obtain x Cf(x) dx = Let us now suppose a quadratic parabola to pass through the points of the curve represented by f(x), corresponding to x = 0, x = &x, 28 434 THEORETICAL ASTRONOMY. and x = 2&x; then will the area included by the arc of this parabola, the extreme ordinates, and the axis of abscissas be 'Ibx J/(aO'dn, I f(x) dx = (o I /(a -j- TWO) dn. If we expand the function /(a + no)), we have na> f . ., (20) 436 THEOKET1CAL ASTRONOMY. and hence //(a + n) dn = C + n/(a) + in- ^ + JnV C being the constant of integration. The equations (54) 6 give * ^ =/() - ir w + w() - T -' =/"() - A/*() + */"() - */"() + . =/" (a) - \r () + T !*/' u ()-..., (22) =/ " (a) - * /VI(a) + ' / "" (a) ~ =/' () -if'" ()+.-., iii which the functional symbols in the second members denote the diiferent orders of finite differences of the function. Hence we obtain + no*) cfo = + nf(a) + in 1 (/(a) - I/" (a) + A/* (a) - T ^/ vU (a) + . . .) + Jn(/"(a) - A/ 1 ' (a) + ^/"(a) - v hf^(a) + ...) Ef we take the integral between the limits n r and +n', the terms containing the even powers of n disappear. Further, sinc the values of the function are supposed to be known for a series of values of n 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 J and +J, and the equation (23) gives, after reduction, MECHANICAL QUADRATURE. 43? -f * f/(a + n0 dn -/(a) + /"() ~ s4Is/ iv () + J -i (24) -4 g f!fl!^/ Tiil W + &c. It is evident that by writing, in succession, a -\- a), a + 2o>, .... a -f- ia> in place of a, we simply add 1 to each limit successively, so that we have % < + 1 + * J/(a + na) cfo ==(/(( + + ( i) ) d (n t) But since 1 *+* -* -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 -f i n = i n = i J/(a + n>) dn = ^f(a + n0 -f- ^ ? ^/' (a + na>) t n = i rCa + ) -I- ^W 5 ^/ vl ( + n) - Ac. n=0 n=0 Let us now consider the functions /(a), /(a + not), &c. as being themselves the finite differences of other functions symbolized by '/, the first of which is entirely arbitrary, so that we may put, in accord- ance with the adopted notation, /O) = '/( -f M - '/( - J0, /(a + ) = '/(a + j,) - '/(a + J), /(a + tia,) = '/(a + ( n + J '/(a + ( - J) >). Therefore we shall have n = t Y/O + no) = '/(a + (i + i) ,) - '/(a - i), n = and also n = i ) =/ (a + (t + J) ) -/ (a - ^), 433 THEORETICAL ASTRONOMY. Further, since the quantity f f(a \a>) 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 \a> shall be zero, namely, (26) Substituting these values in (25), it reduces to I f(x) dx to I /(a -|~ nw) efoi a-fc. -i (27) = {'/( + (* + i + &/( + (*' + J 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 = J, corresponding to the inferior limit, the integral shall be equal to zero, the epoch of f(a \co) being that of the osculating elements. It will be observed that the equation (26) expresses this condition, the constant of integration being included in '/(a \to). If, instead of being equal to zero, the integral has a given value when n = J, it is evidently only necessary to add this value to f f(a Jo>) as given by (26). 160. The interval co 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 inter- polation. The derivation of the required formulae of integration may be effected in a manner entirely analogous to that already indi- cated. Thus, let it be required to find the expression for the integra 1 taken between the limits \ and i. The general formula (23) gives J, " (a)- and since, according to the notation adopted, / () = 1 CT ( - i") +/'( + i-)) = f( + i-) -iT(), (28) /"()=ro+i) -*/"(), /'() =/'( + i0 - if" (), &e., MECHANICAL QUADRATURE. 439 this becomes i //(a+n-O cfo=l/(a)+ J/ (a+ j0-&r (a)-,!,/'' (+ (29 > Therefore we obtain < + * /(a-j-nw) dn=^f(a+ic Now we have \ f( a + nuj ) dn= I /(a -|- nai] dn ( /(a -f- na>) dn ; -i -i and if we substitute the values already found for the terms in the second member, and also /" ( + fa) = f( a + (, + !))_ f( a + (i-l) a,), /( + i.) =/'" (a + (t + i) -/"' (< + (t - i) ), we get a + i -TT-> and ;-> and another integration becomes necessary in order to obtain the values of dx, dy, and dz. We will therefore proceed to derive formulae for the determination of the double integral directly. 440 THEORETICAL ASTRONOMY. For the double integral j J f(x) dx 2 we have, since dx* = a) 2 dn 2 , The value of the function designated by /(a) being so taken that when n = \, Cf(a + n0 dn 0, the equation (23) gives Therefore, the general equation is o J/(a -f nJ o>i ; = ^ J(CL -\- nw) 2? ^ / ( a ~t~ ^ w ) A/ >^ *i ,17 Q n= \_, (35) -f-&c. n=0 =0 We may evidently consider '/(a Ja>), '/(a + Jw), &c. ae the differ- ences of other functions, the first of which is arbitrary; 40 that we have 7 (a) = f(a + -J0 + i'/O - i) - i"/(a + ) - if ( - ), '/(a + *) = J'/(a + i) + i'/(a + ^) - J"/(a + 2) - ^/" (a), -f i'/( + (^-i)) = i7(a + ( - r/(a + (?l _l )w) . Therefore n = Substituting these values in equation (35), and observing that 700 + 7( - ) = 2"/( - ) + '/( - -^)> /(a) + /(a - a,) = 2/(a) - / (a - Jo,), /' W +/" () 2/' (a) -/"' (a - i0, Ac., and that, since "f(a to) is arbitrary, we may put "/(a - ,) = () + 2/" (a )) - &o., (36) 442 THEORETICAL ASTRONOMY. the integral becomes * + + *><> r / + * JJ /(*) dx* = > 2 JJ /(a + no,) dn* l)>) (37) which is the expression for the double integral between the limits \ and i + J. The value of "/(a to) 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 pertur- bations when the argument a \a> corresponds to the date for which the osculating elements are given. If, for n = J, neither the single nor the double integral is to be taken equal to zero, it is only neces- sary to add the given value of the single integral for this argument to the value of '/(a Jai) given by equation (26), and to add the given value of the double integral for the same argument to the value of "/(a ai) 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 J and i. Between the limits and we have jjf(a which gives dn + J/(a) + &. + ssW + 1*4*3* + Ac. GO (38 ) and this again, by means of (28), gives ** MECHANICAL QUADRATURE. 443 Therefore, since CC rr rr JJ /O -f no) dn* =JJ f(a + no) drc, 2 JJ /( a + n0 d, -* -t and '/( + (* + i) = 7( + (* + 1; ) 7( + '), A* -K* + *>)== /(a + (t + l /(<* + *), /'" (a + (i + i) a,) =/" (a + (i + 1) w) -f" (a -f ,), &c. we shall have a -f i w ^ (^) da? = *0/(a + nai) (fo 8 *w -i (39) which gives the required integral between the limits J 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 a) 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 (o so small that the fourth and higher differences may be neglected, but not smaller than is necessary to satisfy this condition, since other- wise an unnecessary amount of labor would be expended in the direct computation of the values of the function. It is better, how- ever, to have the interval at 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 to, a limit for the magnitude of the adopted interval will speedily be obtained. The magnitude of the interval will therefore be suggested by tne rapidity of the change of value of the function. In the coin- 444 THEORETICAL ASTRONOMY. ptitation 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 evi- dently 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 neces- sarily 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 o> 2 , if we multiply the computed values of the function by o> 2 , this factor will be included in all the differences and sums, and hence it will not appear as a factor in the formulae of integration. If, however, the values of the function are already multiplied by w 2 , and only the single integral is sought, the result obtained by the formula of integration, neglecting the factor o> 2 , will be to times the actual integral required, and it must be divided by a) 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 dx, %, and dz with respect to the time, will be determined by means of the equations (9). If the interval CD is such that the higher orders of differences may be neg- lected, the values of the forces must be computed for the successive dates separated by the interval o>, and commencing with the date t Q \co corresponding to the argument a co, t Q being the date to which the osculating elements belong. Then, since the last terms .. . . . , of the formulae for ^~, jp and 7^- involve ox, oy, and 02, which are the quantities sought, the subsequent determination of the differ- ential coefficients must be performed by successive trials. Since the integral must in each case be equal to zero for the date t Q , it will be admissible to assume first, for the dates t \co and t Q -f- \co corre- sponding to the arguments a a> and a, that dx 0, dy = 0, and dz = 0, and hence that the three differential coefficients, for each VARIATION OF CO-ORDINATEb. 445 date, are respectively equal to X w Y Q , and Z . We may now by inte- gration 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 f(a \co) by means of the equation (26), using only the first term, and the value of "f(a to) from (36), using in this case also only the first term. The value of the next function symbolized by "f will be given by Then the formula (39), putting first * = 1 and then i = 0, and neglecting second differences, will give the values of the variations of the co-ordinates for the dates a co 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 coeffi- cients will be obtained from the equations (9), the value of 3r being computed by means of (7). With the corrected values thus derived a new table of integration will be commenced ; and the values of r f(a \co) and "f(a to) will also be recomputed. Then we obtain, also, by adding '/(a ;) to /(a), the value of f f(a -f- Jw), and, by adding this to "f(a), the value of "f(a -f to). An approximate value of f(a + to) may now be readily estimated, and two terms of the equation (39), putting i= 1, will give an ap- proximate 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 approxi- mate values of 3x, 8y, and dz for the date represented by the argu- ment a + 2, and then the corresponding values of the differential coefficients. We may thus by successive partial integrations deter- mine the values of the unknown quantities near enough for the cal- culation of the series of differential coefficients, even when the inte- grals 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 what- ever error remains affects the approximate integral by only one- twelfth 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. gratiou, it is necessary, in order to avoid the effect of the errors in the values of the higher orders of differences, that the interval (o 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 approxi- mations; 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 co will be indicated, in the numerical applica- tion of the formula?, by the manner in which the successive assump- tions 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 Jf, Y 9 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 dx, dy, and dz which would be employed in computing, for the next place, the last terms of the equations (9), we must compute also the cor- rected values of JT, F, and Z. These will be given by the second members of (8), using the values of x, y, and z obtained from We compute also q from (12), and then from Table XVII. find the corresponding value of /. The corrected values of , 2 , ,, 2 > and Ej- will be given by the equations (14), and these being introduced, in the continuation of the table of integration, we obtain new values of fix, %, and dz 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 repe- tition, however, it will not be necessary to recompute the coefficients of dx, dy, and dz 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 dx, %, and dz may be determined approxi- mately, as already explained, for the following date, and by mean* of these the corresponding values of the forces Jf, F, and Z will be found, and also/ and the remaining terms of (14), after which the integration will be completed and a new trial made, if it be con- sidered necessary. In the final integration, all the terms of the for- mulae of integration which sensibly aifect 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-or- dinates 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 cor- rected 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 ZX, SY, and 2Z instead of X, Y, and Z respectively; and the integration of the expressions fJ^dx d^ftii d^ds for fif-, ~j and ^- will then give the perturbations due to the Cit Cut ut action of all the disturbing bodies considered. However, when the interval to for the different disturbing planets may be taken differently, it may be considered expedient to compute the perturbations sepa- rately, 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, m f , &c., or to determine the effect of any subsequent change in these values without repeating the cal- culation 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. M 9 = 1 29' 5".65 7r = 44 17 12 .17) T r\ of\a oo K a I Ecliptic and Mean &6 = ZUO oa O .oy * i,= 4 36 52.11. Po = 11 15 51 .02 log a = 0.3881319 Equinox 1860.0 From these elements we derive the following values : Berlin Mean Time. x y z Ios;r 1863 Dec. 12.0 + 1.53616 + 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 + 68978 0.14066 0.416240. The adopted interval is a) = 40 days, and the co-ordinates are re- ferred to the ecliptic and mean equinox of 1860.0. The first date, it will be observed, corresponds to t Q \to y 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 Jupiter computed by means of the formula ,. = (of - x? + Aug. 8.0 57.30 35.92 1.66, a + 6> Sept. 17.0 59.09 32.47 2.08, a + 7" Oct. 27.0 61.55 28.60 2.43, a-{-8a> Dec. 6.0 64.85 24.34 2.69, a + 9w 1865 Jan. 15.0 + 69.09 + 19.78 -f 2.83, which are expressed in units of the seventh decimal place. "We now, for a first approximation, regard the perturbations as 450 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 '/(a - K> = - A/' (a j0 = - A (53.71 - 53.00) = - 0.03, and the approximate table of integration becomes /( ~ = + 53.00 m _ >} __ 03 "/( - *) = /() = + 53.71 K "/(a) -= Then the formula (39), putting first i = 1, and then i = 0, gives Dec. 12.0 to= + 2.24 + -^p = + 6.66, Jan. 21.0 dx= + 2.21 + -^ = + 6.69. In a similar manner, we find Dec. 12.0 Sy = + 5.85 dz = 0.16, Jan. 21.0 fy = + 5.82 & = 0.14. By means of these results we compute the complete values of the second members of equations (40), dr being found from and thus we obtain (PJoj cPtJy .cfrte Pate " "^ "^ W ^ Dec. 12.0 -|- 53.86 + 47.76 1.45 -f 8.85, Jan. 21.0 -f 54.23 + 47.25 0.96 + 8.63. We now commence anew the table of integration, namely, / 7 7 / '/ 7 / '/ 7 +53.86 _ 002 + 2.26, +47.76 , 02 + 1.97, -1.45 _ Q2 -0.04, +54.23 5421 + 2.24, +47.25 ^ ' + 1.99, -0.96 _ Q98 -0.06, +56.45, +49.26, -1.04. the formation of which is made evident by what precedes. We may next assume for approximate values of the differential coefficients, for the date March 1.0, +54.6, +46.7, and 0.5, respectively; and these give, for this date, NUMERICAL EXAMPLE. 451 dx = -f 56.45 + ~ = + 61.00, 1Z fy = + 49.26 + ^. = + 53.15, fe = l.04~4^ = 1.08. By means of these approximate values we obtain the following results : 1 864 March 1.0 " 2 = + 55.01, " 2 r= + 53.86, ^~ = - 1.00, 3r = + 71.03. Introducing these -into the table of integration, we find, for the corre- sponding values of the integrals, tx = + 61.03, fy = + 53.75, da = 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 55.04, Assuming, again, approximate values of the differential coefficients for April 10.0, and computing the corresponding values of dx, dy, and dz, we derive, for this date, = + 48.06, = + 63.19, Introducing these into the table of integration, and thus deriving approximate values of dx, dy, and dz for May 20, we carry the pro- cess one step further. In this manner, by successive approximations, we obtain the following results : Date. 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. * 1864 Oct. 27.0 + 34.84 Dec. 6.0 68.79 1865 Jan. 15.0 + 112.64 26.32 47.87 58.39 w2 ^ + 4.44, 6.86, + 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 9* 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, June 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 + 1^72.6 + 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 com- mence 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 *Z= rfm'tf ( Z -^ 4, V \ ^s 575 j The approximate values of Sx, dy, and dz for Sept. 17.0 given imme- diately by the table of integration extended to this date, will suffice to furnish the required values of the disturbed co-ordinates by means of and to find p = p Q + dp y we have or in which ^ is the modulus of the system of logarithms. Thus we obtain, for Sept. 17.0, d log ^ = + 0.0000084, * Y= + 32.48, >*Z = + 2.08, which require no further correction. Next, we compute the values of which also will not require any further correction, and thus we form, according to (12), the equation q = 0.29996&C + 0.29815fy 0.03237&. The approximate values of dx, dy, and dz being substituted in thia equation, we obtain q= + 0.0000061, corresponding to which Table XVII. gives log/= 0.477115. Hence we derive 7.2 /M 2 P (f qx -3x*) = - 44.87, S3* (fqy - 9y) = - 30.40, 454 THEORETICAL ASTRONOMY. and the equations (14) give = + 14.22, = + 2.08, 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 dx, %, and dz for Oct. 27.0, and in a similar manner we obtain the corrected values of the differ- ential 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 follow- ing: J^/CttC/* dP dp dt z 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 fa = + 1772.6, dy = -f 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, dx, = dx, dy f = COS e dy sin e dz, (41 ) dz f = sin e dy -f- cos e dz, x n y,, z, being the co-ordinates in reference to the equator as the fun damental plane. Thus we obtain, for 1865 Jan. 15.0, dx, = + 1772.6, dy, = + 1838.9, 8z, == + 767.2. These values, expressed in seconds of arc of a circle whose radius is the unit of space, are dx= + 36".562, dy, = + 37".930, dz, = + 15".825. VARIATION OF CO-ORDINATES. 455 The approximate geocentric place of the planet for the same date is a = 183 28', S = 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, A* = + 5".67. 167. The values of dx, dy, and dz, computed by means of the co- ordinates 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, let # denote the longitude of the descending node of the ecliptic of t f on that of , measured from the mean equinox of t, and let TJ be the mutual inclination of these planes; then, if we denote by a?', y f , z r the co-ordinates referred to the ecliptic of t as the fundamental plane, the positive axis of x 9 however, being directed to the point whose longitude is 6, we shall have x' = x cos -f- y sin 0, (42) Let us now denote by a?", y", z" the co-ordinates when the ecliptic of t is the plane of xy, the axis of x remaining the same as in the system of x', y f , z'. Then we shall have y" = y f cos i) z sin 17, (43) 2" = y' sin f) -f- d cos ^. 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 a?,, y,, z n we get . x, = x" cos (0 + p) f sin (0 + p\ y, = x" sin (0 + p) + y" cos (0 + p\ (44) ,=", in which p denotes the precession during the interval t' t. Elimi- nating x", y", and z" from these equations by means of (43) and (42), observing that, since y is very small, we may put cos 7 = 1, we get 456 DHEOKETICAL ASTRONOMY. x, x cosp y sin p -f- - z sin (0 -f~ P)> 8 y, xsmp-\-y cosp - z cos (0 + P)> (45) s z, =z - x sin 6 -f - y cos 0, 9 S in which s = 206264.8, rj being supposed to be expressed in seconds of arc. If we neglect terms of the order j? 3 , these equations become A^2 (V\ y* x, = x ^x - y + - ( s i n ^ + P cos 0) 2 > "3 S S & = y i|rY+f*- 7(008* .p sin*)*, (46) o S o 2, = 2 - x sin + - y cos 0. 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, y, and 0, we have p = (50".21129 + 0".0002442966r) (f t\ TJ=( (T.48892 0".000006143r) (*'-*), = 351 36' 10" + 39".79 (t 1750) 5".21 (if 0, in which r = \(t f 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 dx y %, and dz belonging to the co-ordinates referred to the ecliptic and mean equinox of t into those to be applied to the co-ordinates re- ferred to the ecliptic and mean equinox of t' . For this purpose it is only necessary to write dx, <%, and dz in place of x, y y and z re- spectively, and similarly for x,, y n 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 equi- nox and ecliptic throughout the calculation ; but when the perturba- tions 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 dx, dy, and dz 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 1 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 advan- tageous. But, generally, since the elements of the orbit of the dis- turbed 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 ren- dered more simple. Whenever an ephemeris giving the geocentric right ascension and declination is required, the heliocentric co-ordi- nates 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 con- siderable magnitude and the interval t' t is also not very large, the correction of dx, dy y and dz 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 posi- tive 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 deter- mined rigorously for a long period of time, provided that the oscu- lating 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 observa- tions 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 ele- ments which do not sensibly affect the residuals for the given differ- ential 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 per- turbations 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 fix, Sy, and $2 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 are large, the calculation of the complete values of , , 2 > , 2 > and j- (which is effected indirectly) cannot be performed with facility, requiring often several repetitions in order to obtain the required accuracy, since any error in the value of the second differential coeffi- cient produces, by the double integration, an error increasing propor- tionally to the time in the values of the integral. Errors, therefore, in the values of the second differential coefficients which for a mode- rate 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. *59 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 -T-, -J-, and -rr- The latter will be obtained from the values of the at at at second differential coefficients by means of a single integration ac- cording 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, = + 385.9, 40^ = + 214.6, 40^ = + 9.7. at at The velocities in the case of the disturbed orbit will be given by the formulae dx_ _dx d8x dy dy d$y dz _ dz ddz ( . "dt~~~3r~T"dT ~dt~~~dt~~~dt' ~dt '"" "dt '" W ^ ' To obtain the expressions for the components of the velocity resoHed parallel to the co-ordinates, we have, according to the equa- tions (6) 2 , dx . . f . , dr . f . , dv -j- = sma sm ( A -f u) -f- r sin a cos ( A -f- u) -j-, at at at -~ sin b sin (5 -J- u) r- -f- r sin b cos (L -f- u) =- at at at dz . rsi . \dr * ff , \ dv -jr = sm c sm ( C + w) -TT + r sm e cos ( C -f- u) -j-- These equations are applicable in the case of any Fundamental plane, if the auxiliaries sin a, sin 6, sin c, A, B, and C are determined in reference to that plane. To transform them still further, we have rfr dt = dt in which to denotes the angular distance of the J erihelion from the ascending node. Substituting these values, we ot tain, by reduction, 460 THEORETICAL ASTRONOMY. dx /- ((e cos -f- cos u) cosB (e sin w -f- sin u) sin 5) sin 6, efe kv\ -\-iMf, N /~ / -f- sin w) = Fsin U, (48) (e cosa> -f- cos if) = Fcos i7, KJ and we have = Fsin a cos (J. -f U"), -|- = Fsin b cos (5 + CO, (49) dt ^=Fsinccos(a4- CO- These equations determine the components of the velocity of a hea- venly 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, (7=0. The sum of the squares of the equations (48) gives (*-)' P and hence it appears that F 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 veloci- ties have been found, is therefore effected in the following manner : First, by means of the osculating elements to which the perturba- tions belong, we compute accurate values of r , a? , y Q , z , and by means of the equations (48) and (49) we compute the values of -~, -jp and -7' Then we apply to these the values of the perturba- tions, and thus find x, y, z, -J-, ~jt, and -^-- These having been CHANGE OF THE OSCULATING ELEMENTS. 461 found, the equations (32)! will furnish the values of ft, i, and p; and the remaining elements may be determined as explained in Art* 112. Thus, from Vr sin * = kVp (1 dx . dy , dz we obtain Vr and and from r sin u = ( x sin ft + y cos ft) sec t, r cos it = a; cos ft we derive r and u; and hence Ffrom the value of Vr. When i is not very small, we may use, instead of the preceding expression for r sin u, r sin u = z cosec i. Next, we compute a from 2a r = ; and from 2ae sin o> = (2a r) sin (2^ -}- w) r sin w, 2ae cos a> = (2a r) cos (2^ + M ) r cos w > we find to 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 formulae. In the case of a very eccentric orbit, after r and u have been found, -r- 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 and the time of perihelion passage will be found from v and e by means of Table IX. or Table X. In the numerical values of the velocities -rr -77, &c., more decimals at at 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 formulse which have already been given. Thus, the values of ft and i must satisfy the equation 462 THEORETICAL ASTRONOMY. z cos i y sin i cos & -f x sin i sin & = 0. We have, also, r'^ + ^-f z\ z = r sin it sin i, which must be satisfied by the resulting values of F, r, and u; and the values of a and e must satisfy the equation p = a (1 e 2 ) a cos* ?>. 169. When the plane of the undisturbed orbit is adopted as the fundamental plane, we obtain at once the perturbations d (r cos u), 8 (r sin u), fa, 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-or- dinates directly, retaining the plane of the orbit for the date t 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 /9 the latitude of the body with respect to the plane of xy; then we shall have x = r cos ft cos w, y = r cos ft sin w, (50) z = r sin ft. Let us now denote by X, Y y and Zj respectively, the forces which are expressed by the second members of the equations (1), and the first two of these equations give C being the constant of integration. The equations (50) give dx d(rcosft} Q . dw - = cos w j. - r cos ft sm w -r dt dt at dy . d(rcoaft) . Q dw ~- = smw j. - -f r cos ft cosw -=-, dt at at and hence dy dx . dw - - VARIATION OF POLAR CO-ORDINATES. 463 Therefore we have r'cos 2 ^T= (Yx Xy)dt+ C. ut %/ If we denote by S the component of the disturbing force in a direc- tion perpendicular to the disturbed radius-vector and parallel with the plane of xy, we shall have X = S sin iv, Y= S cos w, and Therefore r 2 cos 2 /5^-= Cs 9 r f'g"=l-^, (61) we have '"=rw- (63) and hence Finally, we have, from the last of equations (1), db *'(l+m) 3? = *- -? -- *> (64) by means of which the value of z may be found, since, in the case of the undisturbed motion, we have z = 0. The values of/' corresponding to different values of q' may be tabulated with the argument 2 under the sign of integration, this integral, omitting the factor at in the formulae of integration, will become (*)J S r cos /9 dtj as required. The last term of the equation will be multiplied by a). In the case of dr, each term of the equation for - must contain (Jut the factor co 2 . If the second of equations (65) is employed, the first and third terms of the second member will be multiplied by co 2 -, but since the value of S Q is supposed to be already multiplied by o> 2 , the second term will only be multiplied by a). 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 dr and dz must be con- verted into their values in terms of the unit of space. 172. It remains yet to derive convenient formula for the deter- mination of the forces S , 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 fundamental plane adopted, namely, the plane defined by the osculating elements of the disturbed orbit at the instant 1 Q . Let V and & ' denote the inclination and the longitude of the ascending node of the disturbing body with respect to the ecliptic, and let /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 & , and let N f be the angular distance between the as- cending 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 THEORETICAL ASTRONOMY. ecliptic, the fundamental plane, and the plane of the orbit of the dis- turbing body with the celestial vault, we have sin ^Isin J (.y + N') = sin (&' - ) sin J (i' + i ), from which to find N 9 N f , and /. Let ft' denote the heliocentric latitude of the disturbing planet with respect to the fundamental plane, w r its longitude in this plane measured from the axis of x, as in the case of w, and u Q f the argu- ment of the latitude with respect to this plane. Then, according to the equations (82) w we have tan (V N) = tan u ' cos I, If u r denotes the argument of the latitude of the disturbing planet with respect to the ecliptic, we have u ' = u' N'. (68) This formula will give the value of u f , and then w f and ft' will be found from (67). We have, also, cos u Q r = cos ft cos (w' N\ which will serve to indicate the quadrant in which w r J\Tmust 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 t as the fundamental plane will be given by x' = r' cos ft cos w'j i/ = r' cospsinw', (691 To find the force R, we have R = X-+ Y y - + Z-. r ' r ' r VARIATION OF POLAK CO-ORDINATES. 469 and Substituting in these the values of x' t y f , z' given by (69), and the corresponding values of x, y y z given by (50), and putting * = ^~' (70) we get . R = m'k* I h r' cos /3' cos /? cos (w w) + /i r' sin sin /5' -^ ). (71) The equation S rcosj3= Yx- Xy gives S Q = m 'k* h r' cos f sin (w r w), (72) from which to find S . Finally, we have Z=m f k* lhr f sin p - 8 ), (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, S Q , and Z become jB = w'* i ( hr' cospcosW wJ^}, \ Po I (74) S = m'k* h r' cos jf sin (w f wj, Z = To compute the distance p, we have , = (-,/ _ .) + (y' - which gives ^j _ r ' _j_ r 2 _ 2 r / cos ^9 cos ft cos (/ w) 2r r' sin sin ^, (75) and, if we neglect terms of the second order, we have P * = r' 2 + r 2 2r / cos p cos (w f w ). (76) If we put cos r = cos cos ' cos (w' w) + sin sin f, (77) we have 2rr'cosr + (r r' cos r) 8 ; 470 THEOEETICAL ASTKONOMY. and hence we may readily find p from ^ ' the exact value of the angle n, however, not being required. Introducing f into the expression for R, it becomes (79) by means of which R 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 jR, S Q , and Z will be computed by means of the equations (74), p Q being found from (76) or from (78), when we put cos f = cos p cos (w f w ). 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), S from (72), Z from (73), and R from (71) or (79). The values of dw, dr, 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, 8 Q , and Z thus found will not require any further cor- rection. When the places of the disturbing planet are to be derived from an ephemeris giving the heliocentric longitudes and latitudes, the values of & ' and i r will be obtained from two places separated by a considerable interval, and then the values of u f will be determined by means of the first of equations (82) L or by means of (85) x . When the inclination V is very small, it will be sufficient to take u' = l'R' + 8 tan 2 Jt* sin 2 (l r - ft'), in which s = 206264.8. But when the tables give directly the lon- gitude in the orbit, u f + &', by subtracting &' from each of these longitudes we obtain the required values of u r . It should be observed, also, that the exact determination of the values of the forces requires that the actual disturbed values of r', 10', 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 ft' re- quires that we should use the actual values of N', N y and I in the solution of the equations (68) and (67). Hence the disturbed values of & ' and i f should be used in the determination of these quantities for each date by means of (66). It will, however, generally be tho case that for a moderate period the variation of &' and i f may be neglected; and whenever the variation of either of these has a sensi- ble effect, we may compute new values of N 9 N r , and / 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 N, N f , and / arising from the variation of &' and i 1 , by means of differential formulae. Thus the relations will be similar to those given by the equations (71) 2 , so that we have sin-ZV' . ,_., - ; sin(' Q.) smJ 31 = sin N' sin i' $&' -f cos N' 8i r , fh>m which to find dN', 3N, and 81. When the perturbations are computed only in reference to the first power of the mass, the change of Q f and i f may be entirely neg- lected; 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 dN, 3N f , and dl, and, using also the value of u' in the actual orbit of the disturbing body, compute the actual values of w' and ft'. In the case of several disturbing bodies, the forces will be deter- mined for each of these, and then, instead of R, S , and Z, in the formulae for the differential coefficients, 2R, 28 W and 2 Z will be used. 174. By means of the values of dw, fir, and z, the heliocentric or (he 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 t on the plane of the ecliptic; then, in the undisturbed orbit, we shall have u denoting the argument of the latitude. Let # 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 & y cosi sin & + z sin i Q sin , y, = x sin & + y cosi Q cos & * sin i Q cos & , z,=y sin i + z cos t , or, introducing the values of x and y given by (50), Xf=r cos /9 cos w cos & r cos ft sin w cos i sin ^ -f- 2 sin i sin ^ , y, =r cos /? cos w sin Q -f- r cos /5 sin w cos i cos & z sin i cos & , (81) z t =r cos /? sin w sin i -f- 2 cos i . Introducing also the auxiliary constants for the ecliptic according to the equations (94^ and (96) w we obtain x t = r cos /? sin a sin ( J. -(- w) -f- 2 cos a, y, =r cos /3 sin 6 sin (^ -f w) -f z cos 6, (82) 2, = r cos /? sin i sin w + 2 cos i , by means of which the heliocentric co-ordinates in reference to the ocliptic may be determined. If the place of the disturbed body is required in reference to the equator, denoting the heliocentric co-ordinates by x fn y,, y z,,, and the obliquity of the ecliptic by e, we have x n = x, y,,=y f cose z, sine, z n = y f sin e -f- z, cos e. Substituting for x n y n z, their values given by (81), and introducing the auxiliary constants for the equator, according to the equations '99) x and (101) w we get x n = r cos /? sin a sin (A -f- w} -\- z cos a, y lt = r cos /3 sin b sin (B -f- w) -f- z cos b, (83) z t , = r cos ft sin c sin ( (7 -j- 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, jB, &c. are determined with respect to that plane. In the numerical application of the formulae, the value of w will be found from w = it, -f- dw t VAEIATION OF POLAR CO-ORDINATES. 473 U Q 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 TT O , & , and i Q used in the calculation of the per- turbations are referred to the ecliptic and mean equinox of the date t ', and the rectangular co-ordinates of the disturbed body are required in reference to the ecliptic and mean equinox of the date t ", the value of w must be found from the value of W Q referred to the ecliptic of t f being reduced to that of ", by means of the first of equations (115)^ Then & and i Q should be reduced from the ecliptic and mean equinox of t ' to the ecliptic and mean equinox of t Q " 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 co- ordinates referred to the mean equinox and equator of the date t Q " are to be determined, the proper corrections having been applied to Q Q and i w the mean obliquity of the ecliptic for this date will be employed in the determination of the auxiliary constants a, J., &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, -^ = r cos ft sin a cos ( A -f w) rr -f- sec ft sin a sin ( A -j- w) j- -f (cos a tan ft sin a sin (A + w)) -rrt ^- = r cos ft sin b cos (B -j- w) -7- -j- sec ft g i n & sm C^ + tu) -jr dz (84) -f (cos b tan ft sin b sin (J5 +w)) -77. A' =rcos/5sinccos((7-f w)- + sec ft sin c sin ( C + w) -j- dt at -|-(cos c tan ft sin c sin ( C + to)) -?-, by means of which the components of the velocity of the disturbed body in directions parallel to the co-ordinato axes may be determined. The values of -^ and -^ will be obtained from -^ and -^ by a single integration, and then we have 474 THEORETICAL ASTRONOMY. dw &V / pT(l-hm) , d$w dr k\/l-\-m , I ~ - -3P I -^T -di' (85) from which to find =- and :- at at 175. EXAMPLE. In order to illustrate the calculation of the per- turbations of r, w, and z, let us take the data given in Art. 166, and determine these perturbations instead of those of the rectangular co- ordinates. In the first place, we derive from the tables of the motion of Jupiter the values ' = 98 58' 22".7, i' = 1 18' 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 formulae (66), using the values of & and i given in Art. 166, we derive N= 194 0' 49".9, N' = 301 38' 31".7, 1= 5 9' 56".4. The value of u f is given by equation (68), and then w f and $' are found from the equations (67). Thus we have Berlin Mean Time. log r w = UQ log r wf ft' 1863 Dec. 12.0, 0.294084 192 4' 24".5 0.73425 14 18' 54".6 V 38".l 1864 Jan. 21.0, 0.294837 207 40 52 .2 0.73368 17 21 44 .2 18 9 .1 March 1.0, 0.300674 223 3 5 .9 0.73305 20 25 5 .2 34 39 .9 April 10.0, 0.310864 237 51 38 .3 0.73237 23 28 59 .8 51 7 .6 May 20.0, 0.324298 251 52 47 .9 0.73164 26 33 32 .1 17 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 p may be found from (76) or (78) as already given in Art. 166. The forces R, S , and Z may now be determined by means of the equations (74), h being found from (70), and if we introduce the factor (o 2 for convenience in the integration, as already explained, we obtain the following results : Date. 6> 2 .K u*S r u*Z 1863 Dec. 12.0, + 1".4608 + 0".1476 4 0".0009 + 0".0282 1864 Jan. 21.0. + 1 .4223 .6757 4- .0101 - .2361 NUMERICAL EXAMPLE. 475 Date. IR rfS r tfZ u\ S r dt 1864 March 1.0, + 1".2616 1".4512 -} 0".0190 1".3060 April 10.0, 1 .0018 2 .1226 .0273 3 .1035 May 20.0, .6760 2 .6473 .0347 5 .5020 June 29.0, + .3179 2 .9988 .0406 8 .3402 Aug. 8.0, .0452 3 .1650 .0449 11 .4378 Sept. 17.0, .3944 3 .1437 .0470 14 .6076 Oct. 27.0, .7180 2 .9392 .0466 17 .6640 Dec. 6.0, 1 .0097 2 .5586 .0432 20 .4273 1865 Jan. 15.0, 1 .2674 - 2 .0081 + .0362 22 .7245 The integral wl S Q r dt is obtained from the successive values of Q)*S Q r 9 by means of the formula (32). Next we compute the values of the differential coefficients by means of the formulae (65). For the dates 1863 Dec. 12.0 and 1864 Jan. 21.0 we may first assume dr = Q, and, by a preliminary inte- gration, having thus derived very approximate values of dr for these dates, the values of ^- will be recomputed. Then, commencing anew the table of integration, we may at once derive an approximate value of dr for the date March 1 .0 with which the last term of the expression for 7 may be computed. Continuing this indirect pro- cess, as already illustrated in the case of the perturbations of the rec- tangular co-ordinates, we obtain the required values of the second differential coefficient. In a similar manner, the values of -TZ will be obtained. The values of =r 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 :- 1863 Dec. 12.0, 0".0423 -j-l".4509 +0".0009 0".00 + 0".18 +0".00 1864 Jan. 21.0, .1086 1 .3405 .0101 .02 .17 .00 Mar. 1.0, .7162 +0 .7829 .0183 .40 1 .47 .01 Apr. 10.0, 1.61140.0455 0.0251 1.55 3.53 0.04 May 20.0, 2 .4795 .9344 .0300 3 .61 5 .54 .09 June 29.0, 3 .0807 1 .7333 .0326 6 .42 6 .62 .18 Aug. 8.0, 3 .2971 2 .3752 .0331 9 .64 5 .98 .29 Sept. 17.0, 3 .1080 2 .8533 .0311 12 .88 +2 .98 .44 Oct. 27.0, 2 .5425 3 .1872 4-0 .0265 15 .73 2 86 +0 .62 476 THEORETICAL ASTRONOMY. 1864 Dec. 6.0, 1".6443 3".4009 -f 0".0190 17".85 11".88 +0".83 1865 Jan. 15.0, .45113 .5334 -fO .007918 .9224 .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 u -f- dw, r = r -f- dr, and z, the values of the helio- centric 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), y dx lf = X Q cot ( A -j- w] dw -f _ dr -f z cos a, r o dy,, = y cot (B -f w) dw -f ?? dr -f z cos b, (86) r o dz n = z cot (C -{- w) dw -{- $r -\- z cos c, r o the heliocentric co-ordinates x 0) y 0) Z Q being referred to the same fun- damental plane as the auxiliary constants, a, 6, A, &c. Thus, in the case of Eurynome, to find the perturbations of the rectangular co-or- dinates, referred to the ecliptic and mean equinox of 1860.0, from 1864 Jan. 1.0 to 1865 Jan. 15.0, we have A = 296 34 X 37 x/ .5, B = 206 43 7 34".4, C=Q, log cos a = 8.557354n, log cos b = 8.856746, log cos c = log cos i = 9.998590, log X Q = 0.399807 n , log y = 9.838709, log z = 9.148170,, w = w + 6 W = 317 53 r 20 // .2, and hence, by means of (86), we derive 3x, = + 36".559, fy, = + 41".083, dz, = 0".588. If we express these in parts of the unit of space, and in units of the seventh decimal place, we obtain dx, = + 1772.4, dy, = + 1991.8, to, = 28.5, agreeing with the results already obtained by the method of the va- riation of rectangular co-ordinates, namely, 9x, = + 1772.6, fy, = f 1992.3, to, = 28.2. CHANGE OF THE OSCULATING ELEMENTS. 477 176. By using the complete formulae, 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 fun- damental 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 dw and dr, 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 ele- ments 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* .-did* , dp* Vp(l + ro) _ = cos . /? _ + _ == __ -- , p being the semi-parameter of the instantaneous orbit of the disturbed body. In the undisturbed orbit we have _ dv ffo ~ dt and hence we derive dv* Substituting for -5- the value above given, there results 1 ddw J n by means of which p may be determined. To find --^ we have d3 1 dz ian/3 dr ~dt rcos/5 ~df r dt We have, also, dr kVT+^n, . &l/l-f-m . dfr - _ e sm v = -- / e sin v + -^-, dt i/p Vp Q and if we put (89) ( ; p, 478 THEORETICAL ASTRONOMY. this equation becomes e sin v = e n sin v -f- ae Q sin v -f- Y. (90) We have, further. ecosv = - 1, r and, putting *?-!+* () we obtain e cos r = e cos v -}- /5 . r o This equation, combined with (90), gives e sin (v v ) = ae n sin v cos v -f- f cos v $ sin v , p (92) e cos (v v ) = e -f- ae sin 2 v + r sin v + - ft cos v , r o by means of which the values of e and v may be found, those of the auxiliaries a, ft f t being found from (89) and (91). Then we have e = sin , a=p sec* , p = ^ 1/1 + m , tan $E = tan (45 ?) tan |v, a^ M=E esinE, by means of which ^, a, //, and Jf may be determined. In the case of orbits of great eccentricity, we find the perihelion distance from 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 2, i, and a> or it. Let 6 C denote the longitude of the ascending node of the instantaneous orbit on the plane of the osculating orbit, defined by & and i Q) mea- sured from the origin of w, and let ^ denote its inclination to this plane. Then we have tan fj sin (w ) = tan /?, . , dw O dp (93) tan i? cos (w 0^ = sec' /? -^, and hence CHANGE OF THE OSCULATING ELEMENTS. 479 .dfiw g + ~df tan (w O = ^sin 2/9 ^ , (94) "dT by means of which may be found. The quadrant in which is situated is determined by the condition that sin (w ) and tan /3 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 cos>? (95) and the angle must be taken in the same quadrant as w . 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 \ i sin ( (u C) -f- A (& & )) = sin J)) cos 2^0 cos i (*o ~f~ ^o); sin ^ i sin (^(u C) i ( & &)) = sin A0 sin ^ (t ^ ), sin \ i cos {^(u C) ^ ( & & )) :r= cos l^o s ^ n i (*o ~f~ 7o)' These equations will furnish the values of i, u f , and ^ hence, since and & are given, those of R> and u. The value of v having been already found, we have, finally, a)=U V, and the elements are completely determined. These elements will be referred to the ecliptic and mean equinox to which & and i Q are referred, and they may be reduced to the equinox and ecliptic of any other date by means of the formulae 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 #, y,,, z, n in reference to the fundamental plane to which & and i are to be re- ferred, by means of the equations (83), and also those of -|p -,'-', -~ 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 w 2 , 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 dtiw dz * "IT "dp 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 ex- pressed in seconds. In other words, the homogeneity of the several terms must be carefully attended to in the actual application of the formulae. When the elements which correspond to given values of the per- turbations have been determined, if we compute the heliocentric longitude and latitude of the body for the instant to which the ele ments belong, the results should agree with those obtained by com- puting the heliocentric place from the fundamental osculating ele- ments and adding the perturbations. 177. The computation of the indirect terms when the perturba- tions 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 cal- culation, which, when the perturbations of any system of three co- ordinates are to be computed, cannot in any case be avoided without impairing 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 /9 = 0, and the equations (51) and (54) become T> ^ = far dt + kl/p^l + m), at J ,0- d?r dw* , &'(! +m) _ dt* T dt 3 H r* ' 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 radius- vector, 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 eifect 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 0, 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 denote the angle between this fixed line and the semi-transverse axis ; then will * = + ', (98) and when the position of the plane of the orbit is unchanged, we have 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 &, (99) dn =dz + (l wai)dtt =dfc + 28in f lida. We have, further, 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 / and the elements which deter- mine the dimensions of the conic section. We have, therefore, in the case of the osculating elements, for the instant t , Let us now suppose /I to denote the true longitude in the orbit, so that we have J = t;-f7r = v+a>-t-a, 31 482 THEORETICAL ASTRONOMY. or * = +* ('); (101) then, since is equal to TZ when the position of the plane of the orbit is unchanged, it follows that 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 + ft (t y, r cos v = a cosE ae, fl 02") r sin v = al/1 e 2 sin E, *=*+* o a). Let us first consider only the perturbations arising from the action of the two components of the disturbing force in the plane of the dis- turbed orbit, and let us put (103) Further, let Jf + /* (t Q -f- 8M 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 Q , are used instead of the instanta- neous values a, 6, and , gives the same longitude ^,, so that we have r, cos v, = a cos E, a e , r, sin v, = a T/l e 2 sin E f If, therefore, we determine the value of dM so as to satisfy the con- dition that ^, = v + , the disturbed value of the true longitude in the orbit, neglecting the effect of the component ^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,(l + 0, (105) and v will always be a very small quantity. When dM 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 In the equations (97), ^~JT expresses the areal velocity in the in- stantaneous orbit, and it is evident that, since the true anomaly is not affected by the force ^perpendicular to the plane ol the actual orbit, \r^ -jj- must also represent this areal velocity, and hence the equations become ^ = f Srdt , W r \~dt}^ ~^~ 179. If we differentiate each of the equations (104), we get dr. . dv, . ,., dE, cos v, - r,smv,-j- = a a sm E, -= t * * ' (107) sin v, -j- + r, cos v, - = a T/l e * cos .E, =--', (it dt dt ~dt~' = ~di' From the second and the third of these equations we easily derive cosE-ar cos* sm) '. ' ' dt it 77* Substituting in this the values of r, sin v n r, cos v,, and ^p and re- ducing, we get r dr L _ or W = JTV dr, From the same equations, eliminating , we get r, -^r = (a l^l e * r > cos v t cos ^ + a o r ^ sm v / sm -^/) -gj-' which reduces to + -~i (109) . /- , 1 ^Jf\ /1AQ> . m v, 1 H ^T- . (108) \ H Q dt 1 484 THEORETICAL ASTRONOMY. Combining this with the first of equations (106), we get from which dM may be found as soon as v is known. The equation (105) gives dr dr, dv = (l+v)^ + 2^. + r . (Li) Differentiating equation (108) and substituting for -^ its value already found, we obtain ~d& = ~~^ \ + v '~dr) + and the last of the preceding equations becomes ^ = r,j H >< e cosv, ^ + ^ . . , rfv 2 ___ eoSm __,__ + 2 __ + - The equation (110) gives 2 dv 2 s , ,. " which is easily reduced to ~fT d?~^~ < *~dt ~^v~ ' ~dt'~dT' = l + v* and hence we derive The equation (109) gives VARIATION OF CO-ORDINATES. 485 dv,\*_ ffip (l + m) I 1 d3M\* and, since this becomes , m 3,0081),,, ~~ , (H3) Combining equations (112) and (113) with the second of equations (106), we get ^v_l + v p , = "-*- From (110) we derive and the preceding equation becomes ^ or at 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 n y n z f 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 x n and /9 the cosine of the angle which the axis of y makes with that of y,, and we shall have z, = ax + {3y. (116) If the position of the plane of the orbit remained unchanged, these 486 THEORETICAL ASTRONOMY. cosines a and /9 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 dis- tinct 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 dt ~\ dt in which I -^ 1 expresses the velocity resulting from the constant elements, and - 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 constant, and hence the velocity -~ in a direction parallel to the axis of z f 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 d* L _(dz L \ dt~\dt] Differentiating equation (116), regarding a and /9 as constant, we get dz, \ dz, dx dy and differentiating the same equation, regarding x and y as constant, we get (118) Differentiating equation (117), regarding all the quantities involved as variable, the result is d*z,_da dx d? dy d*x Eliminating da from the same equations, we obtain, in a similar manner, (126) Substituting these values in equation (124), it becomes . _ 1 ll^dy v dx\, , H -- / I X- Y-rr ] dz, -f- (Fa; A;l/l m\ dt dt] . dt If we introduce the components R and S of the disturbing force, we have r r and hence r Yx Xy =Sr. Therefore the equation (127) becomes tffc, & 2 (l+mV dr\ Sr p 5 dr\ \ r jfei/p(l+m) ' * / Z ' We have, further, which, by means of the equations (108) and (109), gives dr e sinv, 9 dv, . dv ^ Substituting this value in the equation (128), we obtain VARIATION OF CO-ORDINATES. 489 Sr Iddz, 3z, (129) \ ^ " 1 + ' ' which is the complete expression for the determination of dz t . 182. The equations (110), (115), and (129) determine the complete perturbations of the disturbed body. The value of v must first bo obtained by an indirect process from the equation (115), and then dM is given directly by means of (110). The value of dz will also be Determined by an indirect process by means of (129). In order to obtain the expressions for the forces It, 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 & and i are referred, it being the argument of the latitude in the case of the disturbed motion. Let w f denote the lon- gitude of the disturbing body measured from the same origin and in the plane of the orbit of the disturbed body, and let $' denote its latitude in reference to this plane. Finally, let N, N', /, and u ' have the same signification in reference to the plane of the instanta- neous orbit that they have in reference to the plane of the undisturbed orbit in the case of the equations (66). Then we shall have = cos -- sn * ^Jsini (N N') = sin J (ft' from which to determine N, N f , and /. We have, also, tan (w r N) tan u 9 ' cos I, (131) tan p = tan /sin (uf JV), from which to find w f and /?', u f being the argument of the latitude of the disturbing body in reference to the plane to which & and i are referred. Since, when the motion of the disturbed body is referred to the plane of its instantaneous orbit, /? = 0, the equations (71), (72), and (73) become R = m'telhi* cosfl cos(w f w) - t }, * p ' (132) 8 = m'tfh r' cos ft sin (w f w), Z =m'k 2 hr f sin/5', 490 THEORETICAL ASTRONOMY. by means of which the required components of the disturbing force may be found, the value of h being given by To find p, we have f = r' 2 + r 8 2rr' cos p cos (yf 10), (133) or, putting cos Y cos f? cos (w f w), the equations P sin w = / sin ?, p cos n = r r' cos y. 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 V will also be obtained by means of the tables. The de- termination of the actual values of r and w requires that the pertur- bations shall be known. Thus, when dM and v have been found, we compute, by means of the mean anomaly M Q -f- f2 (t t ) + dM and the elements a , e 0) the values of v, and r,. Then, since v -f- % = v, -f- TT O , we have, according to (100), W = V, + 7T ff. (135) We have, also, Tn the case of the fundamental osculating elements, we have which may be used as an approximate value of and i as well as those of Q, ' 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'j and Jat intervals during the entire period for which the per- turbations are required, and to interpolate their values for the inter- mediate dates. The variations of these quantities arising from the variations of Q , i, & ', and i f 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) aie derived, we easily find VARIATION OF CO-ORDINATES. 491 ,, T , smi , ^., ~x sin .AT ,. sinN dN := - sini' wjfnt ~N sinJV' ., , sinA r (136) d/ = cos JV di' cos -ZVdi -}- sin sin Nd (ft' ft ). When i and / are very small, it will be better to use sin i sin N' sin i f sin JV sin/ sin(ft' ft)' sin/ sin(ft' ft)' (137) in finding the numerical values of these coefficients. By means of these formulae we may derive the values of dN, dN', and dl corre- sponding to given values of ft, di, ft', and di 1 . The formulae by means of which da, $ft, and di may be obtained directly, will be presently considered. The results for dN, dN', and di being applied to the quantities to which they belong, we may compute the actual values of w r and /9'. 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 formulae (132), the forces R, S, and Z will be obtained. The perturbations will first be computed in reference only to terms de- pending 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, 2S, and 2Z in place of R, S, and Z respectively. The approximate values of the perturbations will also facilitate the indirect calculation in the deter- mination of the complete values of the required differential coeffi- cients. 183. When only the perturbations due to the first power of the disturbing force are required, the osculating elements ft and i will be used in finding N 9 N', and /, and r , w will be used instead of r and w in the calculation of the values of R, S, and Z. The equations for the determination of the perturbations dM, v, and dz, 9 neglecting terms of the secotfi order, are, according to the equations (110), (115), and (129), the following: 492 THEORETICAL ASTRONOMY ddM 1 frq+m), -5 dz >- The value of v is first found by integration from the results given by the second of these equations, and then dM is found from the first equation. Finally, dz, 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 integra- tion is analogous, in all respects, to that already illustrated in the case of the variation of the rectangular co-ordinates. It will be ob- d*v served, however, that the expression for -^- involves only one indi- rect term, the coefficient of which is small, and the same is true in . d?dz, ddM . the case of ~^r> while j^- is given directly. When the perturba- tions 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 r in these expressions for the differential coefficients. Neglecting terms of the second order, we have = logr, -f V, wherein ^ denotes the modulus of the system of logarithms. We may also use v f instead of V Q ; but in this case, since r, and v, depend on dM, 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 /3' 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 con- sideration of these terms becomes troublesome. VARIATION OF CO-ORDINATES. 493 The results may be conveniently expressed in seconds of arc, and afterwards v and Sz, 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, SM may be expressed in seconds of arc, while v and 8z f are obtained directly in units of the seventh decimal place. It will be advisable, also, to introduce the interval CD into the formulse 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 nu- merical application of the formula?. Instead of these we must employ the time of perihelion passage and the elements q and e. Thus, let T Q be the time of perihelion passage for the osculating ele- ments for the date t , and let T + 8T be the time of perihelion pas- sage to be used in the formulae in the place of T and in connection with the elements q and e in the determination of the values of r, and v, 9 so that we have In the case of parabolic motion we have, neglecting the mass of the disturbed body, =s (139) the solution of which to find v, is eifected by means of Table VI. as already explained. To find r,, we have r, = q sec 2 %v t . For the other cases in which the elements M Q and fjt cannot be em- ployed, the solution must be effected by means of Table IX. or Table X. Thus, when Table IX. is used, we compute M from wherein log <7 = 9.9601277, and with this as the argument we derive from Table VI. the corresponding value of V. Then, having found t = |rr-^ by means of Table IX. we derive the coefficients required 1 H~ e n in the equation v, = V + A (1000 + B (1000* + (7(1000', (140) 494 THEORETICAL ASTRONOMY. from which v, will be determined. Finally, r, will be found from When Table X. is used, we proceed as explained in Art. 41, using the elements T T + ST, q w and e w and thus we obtain the required values of v, and r,. It is evident, therefore, that, for the determination of the pertur- bations, only the formula for finding the value of dM requires modi- fication in the case of orbits of great eccentricity, and this modifica- tion is easily effected. The expression gves ^ ft ) cos (ft h) cos (> ft ) sin (ft A) cost), cos 6 sin (Z A) (146) cos (A . ft ) (cos (o)=cos(ff & )cos(& A)4-sin( & )sin (& A) sin (*, & )cos(A & ) cos t' cos (>*,& ) sin (h Q & ) H-cos(/l, )sin(A ^ )(l-f-cosV) (148) 4-sin (A, & ) ((cos^ cosi ) cos (h ^ )-fsin (% and hence the last of these equations gives -^ = sin (A, ) (cos (ff & ) sini sin i ) cos (J, ^ ) sin ()sin(A & ) (l + cosi/)- We have, further, from the auxiliary spherical triangle, cos i = sin i sin >/ cos (A ^ ) cos to cos if, from which we get cos i cos t' = sin t* cos (A & ) sin if cos t* (1 + cos if). We have, also, sin (a & ) sin t = sin rj sin ( AQ & o)> sin ( & A) sin ?' = sin i sin (Ao & ), VARIATION OF CO-ORDINATES. 497 or sin ( sin if. Hence we derive (cos i cos i ) cos (h Q> o) -f sin () cos (ho & ) cosi cos (A, & ) sin (h^ & sin t{ Sz, sin b =sm (A, & ) sin i^ -\ -. r 1 COST/ r If we multiply the first of these equations by cos(A & ), and the second by sm(/i & ), and add the results; then multiply the first by sin (h & )> and the second by cos (h & ), and add, we get cosfc cos(Z ^o (& Ao))=cos(A, ^ )+sin(^o S^o) ~' * * -~, T sin b =sin (I, & ) sin ^^ '- (152) Let us now put y= S in( sn o c[ = sin V cos (&o ^o) cos ^ sin to (1 cos >?') Therefore, we have cos ~ o> = tan * 32 498 THEORETICAL ASTRONOMY. and, if we put F= h h , the equations (152) became cos b cos Mo _ r) =cos ft -a J + _J^_, - cosfcsin -- - - sin 6 =sin (A, & ) sin i -] - As soon as 7 1 , p', 5', and ^' are known, these equations will furnish the exact values of I and 6, those of X, and r being found by means of the perturbations v and dM. 186. The value of F may be expressed in terms of p f and 0 ~df~ Sm >0 ~dt' From the equations (118) and (121), observing that we derive, by elimination, da _ r sin A, cos i d __ r cos A ; cos i , ~~ ~ : ""' 500 THEOEETICAL ASTRONOMY. Therefore we shall have dp' = rcosi sin (A, & ) ^ * ~ Al/pCl + m) r COS * COS (^' &o) oy means of which p f and 5' may be found by integration, the inte- gral in each case being zero for the date t at which the determina- tion of the perturbations begins. When the value of dz, has already been found by means of the equation (129), if we compute the value of q f , that of p f will be given by means of (154), or and if p f is determined, q' will be given by / I J. f 1 /-V \ I > If both p f and q' are found from the equations (162), dz, may be de- termined directly from (154); but the value thus obtained will be less accurate than that derived by means of equation (129). Since the formula for ~ 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 q f by an independent integration by means of the results given by the equations (162), it will be preferable to derive them directly from dz, and -^-'- The equations (161) give p' = cos & da sin & d{3, q' = sin * + cos & dp. Substituting for a and dfl their values given by (125) and (126), and putting x" x cos &, + y sin , y" = x sin & + y cos & , we obtain / ,,dtz, . dx"\ r dt ' & 'W./- VAKIATION OF CO-OKDINATES. 501 Substituting further the values x" = r cos (J, & ), y" = r sin (J, and also (W, _ ~dt -\-m . _ &lp (1 -f- m) e sin v = - 6 Sin V - - j 1 -j- e cos v we easily find, since X, v = %, , O n ) d JP = cos ijp , . f . ,. ^.x . . / _ v.fe, A v f 5' = + (Bin (^- ft ) + esm (/- ))- il/p (1 + m) a* which may be used for the determination of p f and q f . These equa- tions require, for their exact solution, that the disturbed values e, %, 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 Q) p 09 and = TT O . The actual values of X, and r are obtained directly from the values of the perturbations. When p f and q f have been found, it remains only to find cos i, and 1 cos r/, in order to be able to obtain F by means of the equation (159). From (153) we get p' y -f- 5" = sin 2 i sin 2 i 2q f sin i , and hence cos i = 1/1 p' 2 (^ + sini ) 2 , (165) from which cos* may be found. The equation (157) gives 1 cos if = cos i Q (cos i Q + cos i) q f sin i , (166) by means of which the value of 1 cos rf will be obtained. If we substitute the values of p f , q f , -Jp and -~ given by the equations (153) and (162) in (159), it is easily reduced to = f J **' Zdt, (167; (1 cos V) kVp (1 + m) which may be used for the determination of P. When we neglect terms of the order of the cube of the disturbing force, in finding P we may use p in place of p and put 1 cos rf = 2 cos 2 i 0) so that the formula becomes 502 THEORETICAL ASTRONOMY. Zdt. (168) Cdz, 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, I, 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 &, 0, and i are known; and the required formula for the deter- mination of these elements may be readily derived. Thus, the equa- tions (160) give, by differentiation, da, . di dff -TT = sin ff cos i ^ sin i cos a rr, at at at d/3 . di , . . dff whence cos ff cos i -=- sin i sm ff ,. at at at . . dff da, . d0 sm i -=- = cos ff -=T sm ff -=-, at at at . di da . dp cosi-j- = sm ff -=- -j- cosff-j-. dt dt dt Introducing the values of ~rr and -=7- already found into these equa- tions, and putting we obtain d** 1 ^ 7- = cot i sin U, * (169) cos . and also, since c? -7-, and ;? re- eft d eft quire, for an accurate solution, that the disturbed values i, 0, 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 i w p , and & in place of i, p, and ' 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 y N f t and I having been found by "means of the terms of (136) involving di f and d&', there remain the corrections due to di and S&> to be applied. These may be found in terms of the quantities p f and q' already introduced. Thus, the equations dp' = cos i sin (' 4- by means of which we may compute the value of w -}- d0; then the value of w f w required in the equations (132), and also in finding the value of /?, will be given by ' w = (w r -j- da) ( w and the forces R, 8 9 and Z may be accurately determined. By thus determining the correct values of H, S, and Z from date to date, the perturbations 3M, v, and dz, may be determined in refer- ence 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 7 1 , p' 9 and q f , 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 3M, v, and 8z, by a numerical example; and for this purpose let it be required to determine the perturbations of Eurynome @ 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 formulae (130), using the values = 206 39' 5".7, '= 98 58 22 .7, i=4 36'52'M, i' = l 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, 1= 5 9' 56".4. Then, by means of the data furnished by the Tables of Jupiter, we find the values of u f , the argument of the latitude of Jupiter in refer- ence to the ecliptic of 1860.0, and from the equations (131) we derive w' and f) f . The values of r' are given by the Tables of Jupiter, and the values of r and v are found from the elements given in Art. 166. The results thus obtained are the following: Berlin Mean Time. Iogr *V> logr 1 ft 1863 Dec. 12.0, 0.294084 354 26' 18 /x .O 0.73425 14 18' 54".6 l/38".l 1864 Jan. 21.0, 0.294837 10 2 45 .7 0.73368 17 21 44 .2 18 9 .1 March 1.0, 0.300674 25 24 59 .4 0.73305 20 25 5 .2 34 39 .9 April 10.0, 0.310864 40 13 31 .8 0.73237 23 28 59 .8 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 1365 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 w = v. + r - = v + 197 38' 6".5, and the cojmponents of the disturbing force are determined by means of the formulae (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 k* 1 r e sinv Q 20 w'P -j^: = -- -- 1 --- T= w I $.S -- r v, d? r r 3 J r* , = wZ cos l Substituting in these the- results already obtained, and also log fi = 2.967809, log Po = 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 co- ordinates, the values of w 2 -^ and a)2 ~Jp L > an ^ then, having found v, w rr- is given directly by the first of these equations. The integra- tion of the results thus derived, by the formulae for mechanical quad- rature, furnishes the required values of v, dM, and dz,. The calcula- tion of the indirect terms in the determination of v and dz,, 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. Date. it dSM dt &v ^w (W, ^-W dM V *, 1863 Dec. 12.0, 0' '.028 + 36.16 + 0.04 + 0' '.01 + 4.41 4- 0.02 1864 Jan. 21.0, .072 33.61 0.49 .01 4.31 0.04 March 1.0, .499 22.55 0.89 .27 37.11 0.54 April 10.0, 1 .213 -f 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 + 0.4021 .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, 3 M = 21".97, = 0.00002798, , = + 0.00000506. It is to be observed that dz, is not the complete variation of the co- ordinate z f 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 dz, differ from the complete values obtained when we compute directly the variations of the rectangular co- ordinates. 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 E, e sin E, = M,, r, =a (l e cosJ,), sin J (v, E,) = sin ? sin E, we obtain M, = 99 29' 35".51, log r, = 0.4162304, log r = 0.4162183, ,= 110 0'33".75, v, = 120 15 13 .80, J = 164 32 25 .97. The calculation of the values of r, and v, from the values of M f9 a , 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 dz, in the equa- tions (172), we get I = 164 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 x , QQ, and i to the equinox and ecliptic of that date; and then, having computed X, and r, we obtain by means of the equations (172) the required values of I and 6. In the determination of the pertur- bations 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 8M, v, and dz, with those derived by the method of the variation of rectangular co-ordinates, we have, for the date 1865 Jan. 15.0, x = 2.5107584, y = + 0.6897713, z = 0.1406590 ; and for the perturbations of these co-ordinates we have found 8x = + 0.0001773, 3y = + 0.0001992, dz = 0.0000028. Hence we derive x = 2.5105811, y= + 0.6899705, z = 0.1406618, and from these the corresponding polar co-ordinates, namely, log r = 0.4162182, I = 164 37' 59".05, b = 3 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 perturba- tions for a very long period. But on account of the increasing diffi- culties which present themselves, arising both from the consideration 510 THEORETICAL ASTRONOMY. of the perturbations due to the action of the component Z in com- puting the place of the body, and from the magnitude of the numeri- cal 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 com- mencing 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 in- separably connected with the problem. The determination of F 9 p f , and q r may be effected from the results for -77-, -r-, and -~r by means dt dt dt J of the formulae for integration by mechanical quadrature, as already illustrated, or we may find F by a direct integration, and the values of p f and q f by means of the equations (164), ^- being found from -^- 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 formulae for this purpose. 191. The single integration of the values of a) 2 -p- and co 2 ^- will give the values of o> -57- and CD jr* and hence those of -j and -rr> which, in connection with ^ , are required in the determination of the new system of osculating elements. Since r 2 -^ represents double the areal velocity in the disturbed orbit, we have CHANGE OF THE OSCULATING ELEMENTS. 511 dv, _ TcVp(l-\-m) dt ~ ~7 The equation (109) gives dv, _kV Po (l+m) I 1 d9M\ 'dt ~ r? \ %/ dt r Hence, since r = r, (1 + v), we obtain by means of which we may derive p. This formula will furnish at once the value of p, which appears in the complete equation foi TOT* and also in the equations (164); and the value of cost may be determined by means of (165). In the disturbed orbit we have dr kyU 4- m and the equations (108) and (111) give dr Therefore we obtain dv which, by means of (176), becomes The relation between r and r, gives P __ __. Po / I 1 -f e cos v 1 + e cos v, and, substituting in this the value of p already found, we get , cos.,,) ( 1 + 1 . ^ J (1 + v)' - 1. (178) e cos v 512 THEORETICAL ASTRONOMY. Let us now put (179) a and /5 being small quantities of the order of the disturbing force, and the equations (177) and (178) become e sin v = e sin v t -\- ae sin v, -f- /?, e cos v = e cos v, -f- ae cos v, -f- a. These equations give, observing that r, (cos v, -f- e ) =p cos .# e sin (v, v) = a sin v, ft cos v n e cos (v, v) = e -f cos E,-\- P sin v,, ^*/ from which e, v, v, and ?; may be found; and thus, since * = * + (,), (181) we obtain the values of the only remaining unknown quantities in the second members of the equations (164). The determination of p f and (f may now be rigorously effected, and the corresponding value of cosi being found from (165), -- and -g- will be given by (162). Then, having found also 1 cos 37' by means of (166), F 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, hut also the corresponding heliocentric or geocentric places of the body, may be found. If we put Y* = a sin v, p cos v n and neglect terms of the third order, the equations (180) give r' * v v = s s, e, ej in which s = 206264".8. These equations are convenient for the CHANGE OF THE OSCULATING ELEMENTS. 513 determination of e and v, v, and hence 7. by means of (181), when the neglected terms are insensible. The values of p, e, and v having been found, we have m , a 2 tan A E = tan (45 ?) tan % v, M= Eesin E, from which to find the elements

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 dM we compute by means of the formula (142) the value of dT, and since we have ddT _ _!_ dSM dt n dt the equation for p becomes (jxm \2 l-^f) (! + ")'; (186) and for a we have Then e, t/, and q will be found by means of the equations 514 THEORETICAL ASTRONOMY. e sin (Vf v) = a sin v, /5 cos v,, e cos (v, v) = e Q -f a (cos v, + e ) + /5 sin v,, (188) P =IT7 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 (191) by means of which the value of & may be found. This equation gives, when we neglect terms of the third order, = '+ r +^ + 2^-^-.). ^2) Substituting in this the values of ff & and i i given by (190), we get 1 ~ 3 iri2/1 (193) sin i cos i, sm z I Q -~~ a - " * CHANGE OF THE OSCULATING ELEMENTS. 515 r being expressed in seconds of arc. Finally, for the longitude of; the perihelion, we have *=*+*, (194)' and the elements of the instantaneous orbit are completely deter- mined. When we neglect terms of the third order, this equation, substituting the values given by (190) and (192), becomes It should also be observed that the inclination i which appears in these formulae is supposed to be susceptible of any value from 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& TT instead of TT. The accuracy of the numerical process may be checked by com- puting 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 F 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 dM, . , f . f .. . , , . v, oz,, -ji- -ji and fj-> and on account of the facility afforded in. the calculation of the indirect terms in the equations for the differen- tial coefficients so long as the values of the perturbations are small, it is evident that the most advantageous process will be to compute 8M, 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 Hgain 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- 5io 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 dM, v y and dz n 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 consider- ation of the terms of the second order in this manner will only require a change of the osculating elements after an interval of seve- ral years, and whenever this transformation shall be required, the equations for and ?'. Let us now differentiate the equation regarding the elements as variable, and we get 2rdrl_ _1_ da 2V l r L dt J " a 2 ' dt + k 2 (1 + m) ' dt T dt tf (1 + m) dt The differential coefficient is here the increment of the accele- dt rating 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 ^ , we shall have _ = E cos 4' + S sin 4' v dt -Substituting this value in the preceding equation, we obtain VARIATION OF CONSTANTS. 519 But we have, according to the equations (50) 6 , dv in wnich v denotes the true anomaly in the instantaneous orbit; and hence there results e sn by means of which the variation of a may be found. If we introduce the mean daily motion ft, we shall have ^r = ^'-^p ( 199 ) and hence ? sin vR + 2- ), (200) for the determination of dp. The first of the equations (97) gives and hence we obtain d (i/p) = (205) from which the value of ~- may be derived. at If we introduce the element co, or the angular distance of the peri- helion from the ascending node, it will be necessary to consider also the component Z; and, since co = X, and hence (206) In the case of the longitude of the perihelion, we have dt dt dt and therefore dn 1 1 ( p cos vR + (p sin 2 Ji^. (207) The first of the equations (15) 2 gives dt*, de in which M Q denotes the mean anomaly at the epoch, which is usually adopted as one of the elements in the case of an elliptic orbit. Sub- stituting for -=r and -=- the values already found, we get dt dt dM L o { (p cot

) It f (2 cos 2 v cos v cos E} cot y S] (t O-> /> sin v ^ dt or dM 1 - . = (( p cot

cos vR d/ -f cos

Sfi. ftjf 1863 Dec. 12.0, + 0' '.01 0" .00 + 8".43 + 0" .12 - f 0".0007 5".48 1864 Jan. 21.0, .04 .01 8 .49 .38 .0040 + 5 .72 March 1.0, .24 .03 26 .78 1 .80 .0216 19 .15 April 10.0, .66 .06 48 .01 3 .88 .0502 37 .11 May 20.0, 1 .35 .08 72 .82 6 .27 .0875 60 .91 June 29.0, 2 .28 .10 100 .83 8 .61 .1299 90 .73 Aug. 8.0, 3 .40 .09 130 .93 10 .65 .1751 125 .79 Sept. 17.0, 4 .63 .07 161 .66 12 .26 .2200 164 .79 Oct. 27.0, 5 .84 .03 191 .48 13 .48 .2624 206 .19 Dec. 6.0, 6 .93 4-0 .03 219 .27 14 .44 .3004 248 .72 1865 Jan. 15.0, y .81 -f .10 244 .24 15 .37 + .3323 + 291 .33 Applying the variations of the elements thus obtained to the oscu- lating elements for 1864 Jan. 1.0, as given in Art. 166, the osculating elements for the instant 1865 Jan. 15.0 are found to be the following: Epoch = 1865 Jan. 15.0 Berlin mean time. M= 99 34' 48".81 TT = 44 13 7 .93 i = 4 36 52. 21 ? = 11 15 35 .65 log a = 0.3880283 ft = 928".8897. 1860.0. NUMERICAL, EXAMPLE. 531 In order to compare the results thus derived with the nerturbations computed by the other methods which have been given, let us com- pute the heliocentric longitude and latitude, in the case of the dis- turbed orbit, for the date 1865 Jan. 15.0, Berlin mean time. Thus, by means of the new elements, we find M= 99 34' 48".81, E= 110 5' 14".15, logr= 0.4162182, v = 120 19 18 .01, I = 164 37' 59".04, I = 3 5 32 .54, agreeing completely with the results already obtained by the other methods. The heliocentric place thus found is referred to the ecliptic and mean equinox of 1860.0, to which the elements ;r, 2, and i are referred ; and it may be reduced to any other ecliptic and equinox by means of the usual formulae. Throughout the calculation of the per- turbations it will be convenient to adopt a fixed equinox and ecliptic, the results being subsequently reduced by the application of the cor- rections for precession and nutation. In the determination of dM, if we denote by AM the value which 7 Ttr is obtained when we neglect the last term of the equation for jr> we shall have which form is equally convenient in the numerical calculation. Thus, for 1865 Jan. 15.0, we find AM = + 234".74, and from the several values of 1600^- we obtain, for the same date v by means of the formula for double integration, Hence we derive dM = + 234".74 + 56".59 = + 291".33, agreeing with the result already obtained. If we compute the variation of the mean anomaly at the epoch, by means of equation (209), we find, in the case under consideration, dM = + 165".29, 532 THEORETICAL ASTKONOMY. and since the place of the body in the case of the instantaneous orbit is to be computed precisely as if the planet had been moving con- stantly in that orbit, we have, for 1865 Jan. 15.0, and hence dM = 3M + (* 4,) 8?= + 9.91" M. The error of this result is 0".23, and arises chiefly from the in- crease of the accidental and unavoidable errors of the numerical cal- culation by the factor t 1 09 which appears in the last term of the equation (209). Hence it is evident that it will always be preferable to compute the variation of the mean anomaly directly; and if the variation of the mean anomaly at a given epoch be required, it may easily be found from dM by means of the equation If the osculating elements of one of the asteroid planets are thus determined for the date of the opposition of the planet, they will suffice, without further change, to compute an ephemeris for the brief period included by the observations in the vicinity of the opposition, unless the disturbed planet shall be very near to Jupiter, in which case the perturbations during the period included by the ephemeris may become sensible. The variation of the geocentric place of the disturbed body arising from the action of the disturbing forces, may be obtained by substituting the corresponding variations of the ele- ments in the differential formulae as derived from the equation (1) 2 , whenever the terms of the second order may be neglected. It should be observed, however, that if we substitute the value of dM directly in the equations for the variations of the geocentric co-ordinates, the coefficient of d/j. must be that which depends solely on the variation of the semi-transverse axis. But when the coefficient of dp. has been computed so as to involve the effect of this quantity during the in- terval t t w the value of dM Q must be found from dM and substi- tuted in the equations. 200. It will be observed that, on account of the divisor e in the expressions for -> -TT-, and -^p these elements will be subject to large perturbations whenever e is very small, although the absolute effect on the heliocentric place of the disturbed body may be small ; and on VARIATION OF CONSTANTS. 533 account of the divisor sin i in the expression for 5 the variation of & will be large whenever i is very small. To avoid the difficul- ties thus encountered, new elements must be introduced. Thus, in the case of &, let us put a" = sin i sin & , 0" = sin i cos & ; (224) then we shall have da" . di . d 7 = sm & cos i-j- -f- sin t cos & 7 > at ac at - cos a cos t- - sm * sin Introducing the values of -^- and given by the equations (212), and introducing further the auxiliary constants a, b, A, and B com- puted by means of the formulae (94) x with respect to the fundamental plane to which & and i are referred, we obtain 7 tl -j rZsin a cos (.4 + w). (225) 6 cos (B + w), (1 -f m) by means of which the variations of a." and /9" may be found. If the integrals are put equal to zero at the beginning of the integration, the values of da" and dp" will be obtained, so that we shall have sin i sin & = sin i sin & -f- a", sin i cos & = sin i cos & -f &&' * or sin i sin ( & ) = cos #" sin & <5y?", sin i cos (& $^ ) = sin i + sin ^ ^a" -J- cos ^ dp", (226) by means of which i and & & may be found. In the case of #, let us put y" = e sin /, C" = e cos /, (227 ) and we have de . d x d?' de d x 534 THEORETICAL ASTRONOMY. Substituting for -=- and -j- the values given by the equations (203) and (205), and reducing, we obtain -{- m) * -f-ersin/j/S), d" 1 I (228) p Sin + ^ ^ + ^ + ') CQS ercos/j/Sl, by means of which the values of dy" and ^f" may be found. Then we shall have e sin x = e sin TT O -f &/', e cos / = e cos TT O -f- <5C", or e sin (/ TT O ) = cos TT O &/' sin TT O (*> and, also, dy m' dtf ~dt = 1-fm' ' "3T m' fr 1 -f- m' ' eft ' If, therefore, from the elements of the orbit of the disturbing planet we compute the auxiliary constants for the adopted fundamental plane by means of the equations (94) x or (99) 1? and also V and U r from V P ' (e r sin a/ + sin w') =7' sin J7', l m ' (e r cos / + cos ti') = F cos U', ' Vp' the equations (100)! and (49), in connection with (232) and (233), give 9x = ' r , r' sin a' sin (A* + i*'), (234^ PERTURBATIONS OF COMETS. 539 dy = f r ' sin b f sin ( f -f u') t m f Sz = -T r r sm c sm ( (J -f- w') ; X -J- Wi *" m ' i' + U'), (234) "* j,., . , ^ ~, TT f \ If we add the values of dx 9 %, z, 3-j-> J-TT and (5 -yr- to the cor- w* wt dt responding co-ordinates and velocities of the comet in reference to the centre of gravity of the sun, the results will give the co-ordinates and velocities of the comet in reference to the common centre of gravity of the sun and disturbing planet, and from these the new elements of the orbit may be determined as explained in Art. 168. The time at which the elements of the orbit of the comet may be referred to the common centre of gravity of the sun and planet, can be readily estimated in the actual application of the formulae, by means of the magnitude of the disturbing force. In the case of Mer- cury as the disturbing planet, this transformation may generally be effected when the radius-vector of the comet has attained the value 1.5, and in the case of Venus when it has the value 2.5. It should be remarked, however, that the distance here assigned may be in- creased or diminished by the relative position of the bodies in their orbits. The motion relative to the common centre of gravity of the sun and planet disregarding the perturbations produced by the other planets, which should be considered separately may then be re- garded as undisturbed until the comet has again arrived at the point at which the motion must be referred to the centre of the sun, and at which the perturbations of this motion by the planet under consider- ation must be determined. The reduction to the centre of the sun will be effected by means of the values obtained from (234), when the second member of each of these equations is taken with a contrary sign. 204. In the cases in which the motion of the comet will be referred to the common centre of gravity of the sun and disturbing planet, the resulting variations of the co-ordinates and velocities will be so smalJ that their squares and products may be neglected, and, there- 540 THEORETICAL ASTRONOMY. fore, instead of using the complete formulae in finding the new ele- ments, it will suffice to employ differential formulae. The formulas (100), give dx . . f . dr . . , A dv -Tr = 8iiia sin ( A -f u) -%- -f r sm a cos (A + u) -JT* Hj- = sin b sin (B + u) ^ + r sin b cos ( + u) ^-, (235) (it ut (it dz . . , dr . f ~ . dv -jr = sm c sm ( C + w ) -57 + r sm c cos ( (7 -f- w) -TI- at at at If we multiply the first of these equations by 3x, the second by dy } and the third by dz; then multiply the first by d> the second by 7 7 Ctf ^-^Y* and the third by S-j-> and put at dt P= sin a sin (A -{-u) 3x -{- sin 6 sin (JB -}- w -f- sin c sin ( C -}- w) ^2, Q = sin a cos ( J. + w) 5 -f sin 6 cos (JB -f w 4- sin c cos ((7 -f- u) dz; P' = sin a sin (4 + i*) *- + sin 6 sin (B + ti) d-- (236) g' = sin a cos (J. + w) d-^ + sin 6 cos (B + w) J-~ -f- sin c cos ( C + u) djr* we shall have, observing that - = -^= e sin t? and that -^ = -^r- -dt -dT--dr -df- r -dt v ^t From the equations dr dx _ ~ df ^ dt' ^ de' PERTUKBATIONS OF COMETS. 541 we get which by means of (237) become '+ ! +v (238) =-= esinvP'H- ^ O^ Xf> From the equation V-W-TF * we get Substituting the values given by (238), observing also that P=dr, this becomes dk dp __ V*r re* sin 2 v esi T' h ^ ^ J "7T- and, since F 8 = -(l + 2ecosv+e), we obtain flr-., (239, by means of which the variation of \/p may be found. The equation ^. = 2P_ F a t gives from which we derive 542 THEORETICAL ASTRONOMY. from which the new value of the semi-transverse axis a may be found. To find dp we have 1 3k dfj. = Ifiad J- /A -, (241 ") a k or Next, to find 8e, we have, from p = a (1 e 2 ), **! '<** (243 > or jocose smv smv~l/p' f , ^P , to = -j P H H r-^ 1 -P H 77- (cos v + cos E} The equation (12) 2 gives (2 + e cos i>) to, (245) v 2 a 2 cos? a'cos'p and from = 1 + e cos v we get e sin i; re sm v re sin -y Substituting this value of dv in (245), and reducing, we find (24 2 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, 6, B, &c., we shall have dx" = sin a sin A8x-\- sin b sin B dy -f sin c sin C 8z, dy" = sin a cos A ox -f- sin b cos B dy -f- sin c cos C dz, (248) dz" cos a dx -\- cos 6 fy -f- cos c -f sin w _ ft = - .# -f- by means of which 8Q and 8i may be found. To find dot and d;r we have &y = and d-jr by means of the formulae Ctv Ctt> Civ (234) ; then, by means of (236) and (250), we compute P, , R, P' , Q f , and jR r , the auxiliary constants a, A, &c. being determined in reference to the fundamental plane to which the co-ordinates are re- ferred. When the fundamental plane is the plane of the ecliptic, or that to which & and i are referred, we have sinc = sini, 0=0. The algebraic signs of cos a, cos 6, and cos c, as indicated by the equa- tions (101) 17 must be carefully attended to. The formulae 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 em- ployed, may be used to check the numerical calculation; and this proof should not be omitted. It is obvious, also, that these differen- tial formulae 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 pertur- bations of the elements of the orbit relative to the sun during the interval t f t. It should be observed, however, that the value of dM for the instant t should be reduced to that for the instant t f , so that the total variation of M during the interval t 1 t will be 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 regarded as undisturbed about this point, and again referring it to the centre of the sun when such an assumption is no longer admissi- ble, the determination of the perturbations during an entire revolu- tion 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 con- sider the formulse applicable to this special case in which the ordinary methods of calculating perturbations cannot be applied. PEKTUKBATIONS OF COMETS. 547 If we denote by x f , y r , 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 fflaf dP r" f + - 1 ^ s m) ^ = mp(^-|), (260) W + ~ ~7*~ Let us now denote by , 57, the co-ordinates of the comet referred to the centre of gravity of the planet; then will Substituting the resulting values of #', y f , z r in the preceding equa- tions, and subtracting these from the corresponding equations (1) for *-he disturbed motion of the comet, we derive 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, (fy k* (m -f- m') I _ (262) dt* "* >* _ A and, since * ^ - is the sum of the attractive force of the planet 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 f , y f , z f are the co-ordinates of the sun re- ferred 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 dis - turbed 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 the disturbing force exercised by the sun is l*_ r' 3 fa I / o I '" I /j and when the comet approaches very near the planet this force will be extremely small. It is evident, further, that the action of the gun 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 1 the planet. Then will the direct action of the sun be , and that m'Tc* of the planet ^-- The disturbing; action of the sun will be PERTURBATIONS OF COMETS. 549 which, since p is supposed to be small in comparison with r, may btj put equal to and hence the ratio of the disturbing action of the sun to the direct action of the planet on the comet cannot exceed 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 p* = m 'r\ and the ratio of the direct action of the sun to its disturbing action cannot in this case exceed 21/m'. 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 m'k* nearly 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 At the point for which the value of p corresponds to R = R' y 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 R will be less than R f , and the planet must be regarded as the con- trolling and the sun as the disturbing body. The supposition that R is equal to R r gives and therefore p = rS/X 5 . (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 (p 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 cos and when this angle is considerable, the disturbing action of the sun will be small even when p is greater than rl/^m' 2 . Hence we may commence to consider the sun as the disturbing body even before the comet reaches the point for which 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 ^, when p 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 and in the case of a near approach to Jupiter the results are P = 0.054 r, # = 0.33. 209. In the actual calculation of the perturbations of any particu- lar 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 gun as the disturbing body; and, having found the elements of the prbit of the comet relative to the planet, the perturbations of these elements or of the co-ordinates will be obtained by means of the formulse 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 enabled to trace the motion of the comet before and subsequent to it*s 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 d__dx__dx' di) _ dy dif dZ __ dz cM , ~dt~~~dt~~~dt' ~dt~~~dt~~~dt* ~dt" ~dt~~~dt' and from , 57, , -^-, -7-, and -7% the elements of the orbit of the comet about the planet are to be determined precisely as the elements in reference to the sun are found from x, y y z, yp j-, 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 in- terval of time during which the sun will be regarded as the disturb- ing body will always be small, it will be most convenient to compute the perturbations of the rectangular co-ordinates, in which case the values of , y, , -TJ-> -^-> and -rr will be obtained directly, and then, having found the corresponding co-ordinates x r , y', z' and velocities rr> -Jr 7T of the planet in reference to the sun. we have at at at ___ _ _ \ I ~dt ~~ ~dt "" ~di' dt ~~ dt " dt ' dt " = dt h 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 corre- sponding quantities are determined in the case of the motion relative to the sun, the necessary distinctions being made in the notation. 552 THEOEETICAL 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 in- ferred 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 direc- tion 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 con- stant 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 radius- vector; then will the resistance oe K being a constant quantity depending on the nature of the body, and

= Vp Jc Jct/p ds V cos (f> = -7- e sin v, Fsin ^ = - -, V = -rr> r \ve have EESISTING MEDIUM IN SPACE. 553 Substituting these values of R and 8 in the equation (205), it reduces to = 2Ky ( - 1 sin v ds. Now, since we have V= -- (I -f- 2e cos v + e, Vp ds = Vdt = (l + 2e cos v and hence ' X \ f* ' V^OOJ If we suppose the function 2e cos v -f- e*) 3 , ( i j r 3 (1 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 ^ ( 7 ) ** C 1 + 2e cos v + e rf = A + ^ cos v + c cos 2v + &c -> ( 267 > in which A, B, &c. are positive and functions of e, the equation (266) becomes 2 ed% = -- ( A -j- -B cos v -f- . . . .) sin v dv. Hence, by integrating, we derive + ---- ), (268) from which it appears that / 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 dM, 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 // 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 E and S given by (265), and reducing, we get 2o / 1 \ 2 .. | , da = r- K

n\ \ r I (270) 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 and this hypothesis gives results which suffice to represent the obser- vations at its successive returns to the perihelion. Substituting for F its value in terms of r and a, the equations (270) thus become dt r'r a - OM TT - LKr \J a COS COS E r dt v* l 2 * 1 --- \ r a by means of which djy. and d

-j- = a cos

cos v I sin v in which s = 206264". 8, /* being expressed in seconds of arc. Com- bining the results thus obtained with the differential coefficients of the geocentric spherical co-ordinates with respect to r and v, as indi- cated by the equations (42) 2 , 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 #, or of the secular variations of the eccentricity and mean motion, without any assump- tion being made in rei jrence either t ) 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 J!f, //, and

f 9476 9 99 50 2.15 '5 J 3 4807 40 12 13 14 15 16 17 4 40.06 5 1-85 5 23.28 5 44-33 6 4.95 6 25.14 21.79 21.43 21.05 20.62 20.19 19.72 9-999 9377 9271 9'57 935 8905 8768 106 114 122 I 3 137 144 37 10 20 30 40 50 II 3.28 4-39 5-47 6-54 7-58 8-59 .11 .08 .07 .04 I.OI 1. 00 9-999 4767 4726 4686 4645 4604 45 6 3 41 40 4 1 4 1 4 1 4' 18 19 20 21 22 23 6 44.86 7 4.09 7 22.80 7 40.99 7 58.61 8 15.66 19.23 18.71 18.19 17.62 17.05 16.44 9.999 8624 8472 8314 8149 7977 7799 165 172 I 7 8 185 38 10 20 30 40 50 ii 9.59 10.56 11.51 12.44 13-34 14.22 0.97 0.95 0.93 0.90 0.88 0.86 9-999 45" 4481 4440 4399 4358 43'7 4 1 4 1 4 1 4 1 24 25 26 27 28 8 32.10 8 47.93 9 3- 12 9 17.65 9 3 x -5 I5-83 15.19 14-53 I3-85 9-999 7614 7424 7228 7027 6820 190 196 201 20 7 39 10 20 30 40 ii 15.08 15.92 16.73 17.52 18.29 0.84 0.8 1 0.79 0.77 9-999 4276 4234 4*93 4152 4110 42 42 29 9 44.66 13.16 12.46 6608 212 216 50 19.04 0.75 0.72 4069 42 30 10 20 30 40 50 9 57.12 9 59- 12 10 I. II 3-7 5.02 6.94 2.00 1-99 1.96 1.95 1.92 1.91 9-999 6392 6 355 6319 6282 6245 6208 P 37 37 37 37 40 10 20 30 40 50 ii 19.76 20.46 21.13 21.79 22.42 23.02 0.70 0.67 0.66 0.63 0.60 9.999 4027 3985 3944 3902 3860 42 4 1 42 42 4 1 42 31 10 20 30 40 50 10 8.85 10.73 12.59 14.44 16.26 1 8.06 1.88 1.86 I.g 5 1.82 1.80 1.78 9.999 6171 6096 6059 6021 5984 I 38 3 8 41 10 20 30 40 50 ii 23.61 24.17 24.70 25.22 25.71 26.18 0.56 o-53 0.52 0.49 0.47 0.44 9-999 3777 3735 3 6 93 3651 3609 3567 42 42 42 42 42 42 32 10 20 30 40 50 10 19.84 21. 60 23-34 25.05 26.75 28.43 1.76 1.74 1.71 1.70 1.68 1.65 9-999 5946 5908 5870 5832 5794 5755 38 38 38 38 39 3 8 42 10 20 30 40 50 ii 26.62 27.04 27.44 27.82 28.17 28.50 0.42 0.40 0.38 o-35 o-33 0.30 9-999 3525 3483 3399 3357 33'5 42 42 42 42 42 33 10 20 cO 40 10 30.08 33-32 34-9J 36.48 1.63 1.61 i-59 i-57 9-999 5717 5678 5640 5601 39 38 39 39 43 10 20 30 40 ii 28.80 29.08 29-34 29.58 29.79 0.28 0.26 0.24 0.21 9-999 3273 3 270 3188 3H6 3*4 43 42 42 42 42 50 38.03 "55 1.52 5523 39 39 50 29.98 o.i 6 3062 T-* 43 34 10 20 30 40 50 10 39-55 41.06 42.54 44.00 JIU 1.51 1.48 1.46 i-44 1.42 I. -JO 9-999 5484 5445 5406 53 6 7 5327 5288 39 39 39 40 39 4 44 10 10 SO 40 50 ii 30.14 30.29 30.41 30.50 3-57 30.62 0.15 0.12 0.09 0.07 0.05 0.03 9-999 3 OI 9 2977 2935 2892 2850 2808 42 42 43 42 42 42 35 10 48.25 J 7 9-999 5248 45 ii 30.65 9.999 2766 661 TABLE I, Angle of the Vertical and Logarithm of the Earth's Eadius, = Geocentric Latitude. P = Earth's Eadius. 0-0' Diff. logp Diff. 0-0' Diff. logp Diff. O / 45 10 20 30 40 50 II 30.65 30.65 30.63, 30.58 30.42 0.00 0.02 O.O5 0.07 0.0 9 Oil 9.999 2766 2723 2681 2639 2596 43 42 42 43 42 O / 55 10 20 30 40 50 10 49.74 48.36 46.97 45-55 2% II 1.38 1.39 1.42 1.44 1.46 9-999 *75 0235 0195 0153 OIK 0076 40 40 40 39 40 42 1.49 39 46 10 20 30 40 50 II 30.31 30.17 30.01 29.82 29.61 29.38 0.14 0.16 0.19 0.21 0.23 O.26 9.999 2512 2470 2427 2385 2343 2300 42 43 42 42 43 42 56 10 20 30 40 50 10 41.16 39-65 38.13 36.5! 35-oJ 33-4 1 1.51 1.52 'I 7 1. 60 1.61 9-999 37 9.998 9998 9958 9919 9880 9841 39 40 39 39 39 39 4T 10 20 30 II 29.12 28.85 28.54 28.22 0.27 0.31 0.32 9.999 225* 2216 2174 2132 42 42 42 57 10 20 30 10 31.80 30.16 28.50 26.83 1.64 1.66 1.67 9.998 9802 9764 9725 9686 38 39 39 40 50 27.87 27.50 -35 0-37 0.40 2089 2047 43 42 42 40 50 23.40 1.70 1.73 1.74 9648 9610 39 48 10 20 II 27.10 26.69 26.24 0.41 0.45 9.999 2005 1963 1921 42 42 58 10 20 10 21.66 19.90 18.11 1.76 9.998 9571 9533 9495 1 30 40 50 25.78 25.29 24.78 0.40 0.49 0.51 0.54 1879 1837 '795 42 42 42 42 30 40 50 16.31 14.48 12.63 I'll 1.85 1.86 9457 9419 9382 | 49 10 20 30 40 50 ii 24.24 23.69 23.11 22.50 21.87 21.22 0.55 0.58 0.61 0.63 0.65 0.67 9-999 753 1711 1669 1627 1586 1544 42 42 42 42 42 59 10 20 30 40 50 10 10.77 8.88 6.97 5.04 3-o8 IO I. II 1.89 1.91 1.93 1.96 1.97 1.99 9.998 9344 9307 9269 9232 9158 11 37 37 37 37 50 10 20 30 40 50 II 20.55 19.85 18.39 17.63 16.84 0.70 0.72 0.74 0.76 0.79 0.82 9.999 1502 1460 1419 '377 ^335 1294 42 42 42 4' 42 60 61 62 63 64 65 9 59-i* 9 46.74 9 33-65 9 19.85 9 5-36 8 50.21 12.38 13.09 13.80 14.49 15.15 15.81 9.998 9121 8902 8688 8479 8275 8077 219 214 209 204 198 51 10 20 30 40 50 II l6.02 15.19 14-33 13-45 12.55 11.62 0-.83 0.86 0.90 -93 0.95 9.999 1252 I2II II 7 1128 1087 1046 42 4 1 4 1 66 67 68 69 70 71 8 34-40 8 17-97 8 0.92 7 43- 2 9 7 25.08 7 6.33 16.43 17.05 17.63 18.21 18.75 19.27 9.998 7884 7697 75'7 734* 7174 7013 187 1 80 III 161 J 54 52 10 20 30 II 10.67 9.70 8.71 7.69 0.97 0.99 .02 9-999 1005 0963 0922 O88l 42 72 73 74 75 6 47.06 6 27.28 6 7.03 5 46-33 19.78 20.25 20.70 9.998 6859 6713 6573 6441 146 140 132 40 50 6.66 5.60 3 .09 0840 0800 40 4 1 76 77 5 3-67 21.13 2I -53 21.90 6317 6201 Til 1 08 53 10 20 30 40 50 ii 4.51 3.40 2.27 II 1. 12 10 59.94 58.74 .11 13 .1C .18 .20 .22 9-999 759 0718 0677 0637 0596 0556 40 4i 40 4 1 78 79 80 81 82 83 4 41-77 4 19-53 3 56.96 3 34-io 3 10.98 2 47.63 22.24 22.57 22.86 23.12 2 3-35 23.56 9.998 6093 5993 5901 5818 5676 100 ii 57 54 10 20 30 40 50 10 57.52 56.28 55.02 53-73 52.42 51.09 .29 33 35 9.999 0515 0475 435 395 355 03*5 40 40 4 40 40 4 84 85 86 87 88 89 2 24.07 2 0.33 I 36.44 I 12.43 o 4.8.3! o 24.18 23:89 24.01 24.09 24.16 24.18 9.998 5619 5570 553 5498 5476 5463 49 40 3* 22 5 55 10 49.74 J J 9.999 0275 90 O O.OO 9-998 5458 662 TABLE II. For converting intervals of Mean Solar Time into equivalent intervals of Sidereal Hours. Minutes. Seconds. Decimals. Mean T. Sidereal Time. Mean T. Sidereal Time. lean T. Sidereal Time. Mean T. Sidereal Time. h A m s m m s g 1 s s I I o 9.856 I I 0.164 I 1.003 0.02 0.02O 2 2 19.713 2 2 0.329 2 2.005 0.04 0.04.0 3 3 o 29.569 3 3 0-493 3 3.008 0.06 O.OOO 4 4 o 39.426 4 4 0.657 4 4.011 0.08 0.080 c 5 o 49.282 5 0.821 5 5.014. 0.10 O. IOO i 6 o 59-139 6 6 0.986 6 6.016 0.12 O.I 20 1 7 8.995 8 18.852 I 7 -15 8 -3H i 7.019 8.022 O.IJ o.i 6 0.140 O.I 60 9 9 28.708 9 9 .478 9 9.025 0.18 0.180 10 10 38.565 10 10 643 10 10.027 0.2O 0.201 ii n 48.421 ii ii .807 ii I 1.030 0.22 0.221 12 12 58.278 12 12 .971 12 12.033 0.24 0.241 '3 4 13 2 8.134 14 2 17.991 14 13 2.136 I 4 2.300 13 4 I3-036 14.038 0.26 0.28 0.26l 0.28l i5 15 2 27.847 '5 '5 2.464 15.041 0.3.0 0.301 16 16 2 37.704 if 16 2.628 1 6 16.044 O.J2 0.321 18 17 2 47.560 1 8 2 5,7.416 18 17 2.793 18 2.957 ii 17.047 18.049 0-34 0.36 0.341 0.361 19 19 3 7.273 19 19 3.121 19 19.052 0.38 0.381 20 20 3 17.129 20 20 3.285 20 20.055 0.40 0.401 21 21 3 26.986 21 21 3.450 21 21.057 0.42 0.421 22 22 3 36.842 22 22 3.614 22 22.000 o-44 0.441 23 24 23 3 46.699 *4 3 56.555 2 4 24 3-943 23 24 24.066 . f O 0.46 o. 4 S 0.461 0.481 11 25 4.107 26 4-271 11 25 ooo 26.071 0.50 0.52 0.501 0.521 oJ 27 27 4-435 27 27.074 o-54 0.541 Q C 28 28 4.600 28 28.077 0.56 0.562 EH 8 2 9 30 29 4-764 30 4.928 2 9 3 29.079 30.082 0.58 0.60 0.582 0.602 g i 3 1 31 5.092 3 1 31.085 0.62 O.622 rQ fl 32 S 2 5-257 32.088 0.64 0.642 OQ bfi 33 33 5-421 33 33.090 0.66 O.662 2 .5 o5 34 34 5-585 34 34-093 0.68 0.682 8 15 35 35 5-750 35 35.096 0.70 0.702 S fj S 6 3 6 5-9H 3 6 36.099 0.72 0.722 ill 37 37 6.078 37 37.101 0-74 0.742 UU 4) 2 PJ rfi H 38 ' 38 6.242 38 38.104 0.76 0.762 * 1 s 39 39 6.407 39 39.107 0.78 0.782 40 40 6.571 40 40.110 0.80 0.802 H a "So 4 1 6-735 41.112 0.82 0.822 *o .5 a) 42 42 6.899 42 42.115 0.84 0.842 Ii! 43 44 43 7-064 44 7.228 43 44 43.118 44.120 086 0.88 0.862 0.882 ill O i, c3 46 45 7-392 46 7-557 Jl 45.123 46.126 0.90 0.92 0.902 0.923 . 47 47 7-721 47 47.129 0.94 0.943 Ill 48 48 7.885 48 48.131 0.96 0.963 5-1 O O *J3 ^ 49 49 8-049 49 49.134 0.98 0.983 5 50 8.214 5 50.137 1. 00 1.003 3 * i 51 8.378 5 1 51.140 <3 5* 52 8.542 52 52.142 j S 53 53 8.707 53 53-145 .s 54 54 8.871 54 54.148 <2 ^ II 55, 57 55 9-035 5 6 9 '99 57 9-364 I? 57 55-I5I 56.153 57.156 e 58 58 9.528 58 58.159 M I 9 60 59 9.692 60 9.856 59 60 59.162 60.164 563 TABLE III. For converting intervals of Sidereal Time into equivalent intervals of Mean Solar Time, Hours. Minutes. Seconds. Decimals. Sid. T. Mean Time. Sid. T. Mean Time. Sid. T. Mean Time. Sid. T. Mean Time. h Am m m 5 s 8 s s I o 59 50.170 I o 59.836 I 0.997 O.02 0.020 a i 59 40.341 a I 59.672 2 1.995 0.04 0.040 3 a 59 30.511 3 a 59.509 3 2.992 O.O6 0.060 4 3 59 20.682 4 3 59-345 4 3-989 0.08 0.080 5 4 59 IO - 8 5 2 5 4 59.181 5 4.986 O.IO O.I 00 6 5 59 i- oa 3 6 5 59017 6 5.984 0.12 0.120 7 6 58 51.193 7 6 58.853 7 6.981 0.14 0.140 8 7 58 4i-3 6 3 8 7 58.689 8 7-97 8 0.16 0.160 9 8 58 31.534 9 8 58.526 9 8.975 0.18 0.180 10 9 58 21.704 10 9 58-362 10 9-973 0.20 0.199 ii 10 58 11.875 ii 10 58.198 ii 30.970 0.22 0.219 la ji 5$ 2.045 la ii 58.034 12 11.967 0.24 0.239 J 3 la 57 52.216 '3 12 57.870 13 12.964 0.26 0.259 H 13 57 42.386 4 13 57.706 H 13.962 0.28 0.279 II H 57 3 2 -557 15 57 22.727 11 *4 57-543 15 57-379 !i 14-959 I5-956 0.30 0.32 0.299 0.319 17 16 57 12.897 17 16 57.215 17 16.954 0-34 o-339 18 17 57 3.068 18 17 57.051 18 17.953 0.36 o-359 19 18 56 53.238 '9 18 56.887 19 18.948 0.38 0:379 ao 19 5 6 43-49 20 19 56.723 20 19.945 0.40 o-399 ai 20 56 33.579 21 20 56.560 21 20.943 0.42 0.419 aa 21 56 23.750 22 21 56.396 22 21.940 o-44 0.439 a 3 22 56 13.920 a 3 22 56.232 2 3 22.937 0.46 0,459 24 43 5 6 4-09 1 24 23 56.068 2 4 a 3-934 0.48 0-479 a 5 24 55.904 2 5 24.932 0.50 0.499 26 25 55.740 26 25.929 0.52 0.519 C) a 7 26 55.577 27 26.926 0.54 0.539 a a8 27 55-4I3 28 27.924 0.56 0.558 EH a 9 28 55.249 29 28.921 0.58 0-578 83 73 30 2 9 55.085 3 29.918 0.60 0-598 'o g 3 1 30 54.921 3 s 30.915 0.62 0.618 rj 3 a 3 1 54-758 32 3 I -9 I 3 0.64 0.638 o3 "OD 33 32 54.594 3.3 32.910 0.66 0.658 S C? S 34 33 54-43 34 33.907 0.68 0.678 Vn '& 3S 34 54.266 35 34.904 0.70 0.698 C 8 73 36 35 54.102 36 35-9 oa 0.72 0.718 i II H 36 53.938 37 53-775 P 36.899 37.896 0.74 0.76 0-738 0.758 -5 50 51.645 50.861 a ~l~ 5 a 51 51.481 52 51.858 *.d 53 5 a S'-S 1 ? 53 52.855 .a^g 54 53 51.153 54 53-853 3 1 55 54 50.990 55 54.850 5 S3 5 6 55 50.826 5 6 55- 8 47 .2 8 57 56 50.662 57 56.844 ~ s 58 57 50-498 5 8 57.842 *^ 59 5 8 50-334 59 58.839 60 59 S -^ 60 59.836 564 TABLE IV. For converting Hours, Minutes, and Seconds into Decimals of a Day. Hours. Decimal. Min. Decimal. Min. Decimal. Sec. Decimal. Sec. Decimal. 1 0.0416 + 1 .000694 + 31 .021527 + 1 .0000116 31 .0003588 2 833 + 2 .001388 + 32 .022222 + 2 .0000231 32 .0003704 3 .1250 + 3 .002083 + 33 .022916 + 3 .0000347 33 .0003819 4 .1666 + 4 .002777 + 34 .023611 + 4 .0000463 34 .0003935 5 .2083 + 5 .003472 + 35 .024305 + 5 .0000579 35 .0004051 6 .2500 + 6 .004166 + 36 .025000 + 6 .0000694 36 .0004167 7 0.2916 + 7 .004861 + 37 .025694 + 7 .0000810 37 .0004282 8 3333 + 8 .005555 + 38 .026388+ 8 .0000925 38 .0004398 9 375 + 9 .006250 + 39 .027083 + 9 .0001042 39 .0004514 10 .4166 + 10 .006944 + 40 .027777+ 10 .0001157 40 .0004630 11 .4583 + 11 .007638 + 41 .028472 + 11 .0001273 41 .0004745 12 .5000 + 12 008333 + 42 .029166 + 12 .0001389 42 .0004861 13 0.5416 + 13 .009027 + 43 .029861 + 13 .0001505 43 .0004977 14 .5833 + 14 .009722 + 44 030555+ 14 .0001620 44 .0005093 15 .6250 + 15 .010416 + 45 .031250 + 15 .0001736 45 .0005208 16 .6666 + 16 .01 II I I + 46 .031944 + 16 .0001852 46 .0005324 17 7083 + 17 .011805 + 47 .032638+ 17 .0001968 47 .0005440 18 .7500 + 18 .012500 + 48 033333+ 18 .0002083 48 .0005556 19 0.7916 + 19 .013194 + 49 .034027 + 19 .0002199 49 .0005671 20 8333 + 20 .013888 + 50 .034722 + 20 .0002315 50 .0005787 21 .8750 + 21 .014583 + 51 .035416 + 21 .0002431 51 .0005903 22 .9166 + 22 .015277 + 52 .036111 + 22 .0002546 52 .0006019 23 0.9583 + 23 .015972 + 53 .036805 + 23 .0002662 53 .0006134 24 1.0000 + 24 .016666 + 54 .037500 + 24 .0002778 54 .0006250 25 .017361 + 55 .038194 + 25 .0002894 55 .0006366 26 .018055 + 56 .038888+ 26 .0003009 56 .0006481 27 .018750 + 57 039583+ 27 .0003125 57 .0006597 28 .019444 + 58 .040277 + 28 .0003241 58 .0006713 29 .020138 + 59 .040972 + 29 .0003356 59 .0006829 30 .020833 H" 60 .041666 + 30 .0003472 60 .0006944 The sign +, appended to numbers in this table, signifies that the last figure repeats to fcfinity TABLE V. For finding the number of Days from the beginning of the Y< Date. Com. Bi8. January o.o February o.o March o.o 3 1 59 11 April o.o 90 9' May o.o 120 121 June o.o !5' 152 July o.o 181 182 August o.o 212 2I 3 September o.o 243 244 October o.o *73 274 November o.o 34 35 December o.o 334 335 565 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 1 2 3 M. Diff. 1". M. Difif. 1". M. Diff. 1". M. Diff. 1". 0' 1 o.oooooo 0.010908 181.81 181.81 0.654532 0.665442 181.83 181.83 1.309263 1.320178 181.92 181.92 1.964393 1.975316 182.05 182.06 2 0.021817 181.81 0.676352 181.83 1.331093 181.92 1.986240 182.06 3 0.032725 181.81 0.687262 181.84 1.342008 181.92 1.997164 182.06 4 0.043633 181.81 0.698172 181.84 1.352923 181.92 2.008087 182.07 5 0.054542 181.81 0.709082 181.84 1.363839 181.93 2.019011 182.07 6 0.065450 181.81 0.719993 181.84 1-374755 181 93 2.029936 182.07 7 0.076358 181.81 0.730903 181.84 1.385670 181 93 2.040860 182.07 8 0.087267 181.81 0.741813 181.84 1.396586 181.93 2.051785 182.08 9 0.098175 181.81 0.752724 181.84 1.407502 181.93 2.062709 182.08 10 0.109083 181.81 0.763634 181.84 1.418418 181.94 1.073634 182.08 11 o.i 19992, 181.81 0-774545 181.84 1-4*9334 181.94 2.084559 182.08 12 0.130900 181.81 0.785456 181.84 1.440251 181.94 2.095485 182.09 13 0.141808 181.81 0.796366 181.85 1.451167 181.94 2.106410 182.09 14 0.152717 181.81 0.807277 181.85 1.462083 181.94 2.117335 182.09 15 0.163625 181.81 0.818188 181.85 1.473000 181.95 2.128261 182.10 16 0-174534 181.81 0.829099 181.85 1.483917 181.95 2.139187 182.10 17 0.185442 181.81 0.840010 181.85 1.494834 181.95 2.1501 14 182.10 18 19 0.196350 0.207259 181.81 181.81 0.850921 0.861832 181.85 181.85 1-50575 1 1.516668 181.95 181.95 2.161040 2.171966 181 ii 182.11 20 0.218167 181.81 0.872743 181.85 1.527585 181.96 2.182894 182.11 21 0.229076 181.81 0.883654 181.86 i-53 8 53 181.96 2.193820 182.12 22 0.239984 181.81 0.894566 181.86 1.549420 181.96 2.204747 182.12 23 0.250893 181.81 0.905478 181.86 1.560338 181.96 2.215674 182.12 24 0.261801 181.81 0.916389 181.86 1.571256 181.96 2.226602 182.13 25 26 0.272710 0.283619 181.81 181.81 0.927301 0.938212 181.86 181.86 1.582174 1.593092 181.97 181.97 2.237529 2.248457 182.13 182.13 27 0.294527 181.81 0.949124 181.86 1.604010 181.97 *-*593 8 5 182.14 28 0.305436 181.81 0.960036 181.86 1.614928 181.97 2.270313 182.14 29 0.316345 181.81 0.970948 181.87 1.625847 181.97 2.281242 182.14 30 0.327253 181.81 0.981860 181.87 1.636766 181.98 2.292170 182.14 31 0.338162 181.81 0.992772 181.87 1.647684 181.98 2.303099 182.15 32 0.349071 181.81 1.003684 181.87 1.658603 181.98 2.314028 182.15 33 0.359980 181.81 1.014596 181.87 1.669522 181.98 2.324957 182.15 34 0.370888 181.81 1.025509 181.87 1.680441 181.99 *-335 88 7 182.16 35 0.381797 181.81 1.036421 181.87 1.691361 181.99 2.346816 182.16 36 0.392706 181.81 1.047334 181.87 1.702280 181.99 2.357746 182.16 37 0.403615 181.81 1.058246 181.88 1.713200 181.99 2.368676 182.17 38 0.414524 181.82 1.069159 181.88 1.724120 182.00 2.379606 182.17 39 -4*5433 181.82 1.080072 181.88 1.735039 182.00 2.390536 182.17 40 0.436342 181.82 1.090985 181.88 1.745960 182.00 2.4^1467 182.18 41 0.447251 181.82 1.101898 181.88 1.756880 182.00 2.411 ^98 182.18 42 0.458160 181.82 1.112811 181.89 1.767800 182.01 2.425329 182.18 43 0.469069 181.82 1.123724 181.89 1.778 21 182.01 2.434260 182.19 44 0.479970 181.82 1.134637 181.89 1.789(41 182.01 2.445191 182.19 45 46 0.490888 0.501797 181.82 181.82 1.145550 1.156464 181.89 181.89 1.800562 1.811483 182.01 182.02 2.456123 2.467055 182.19 182.20 47 0.512706 181.82 1.167377 181.89 1.822404 182.02 2477987 182.20 48 0.523616 181.82 1.178291 181.89 1.833325 182.02 2.488919 182.20 49 0-5345*5 181.82 1.189205 181.90 1.844247 182.02 2.499851 182.21 50 0-545435 181.82 1.200119 181.90 1.855168 182.03 2.510784 182.21 51 0.556344 181.82 1.211033 181.90 1.866090 182.03 2.521717 182.22 52 0.567254 181.82 1.221947 181.90 1.877012 182.04 2.532650 182.22 53 0.578163 181.83 1.232861 181.90 1.88 79 34 182.04 *-5435 8 3 182.22 54 0.589073 181.83 i-*43775 181.91 1.898856 182.04 2.554517 182.23 55 -5999 8 3 181.83 1.254689 181.91 1.909779 182.04 2.565450 182.23 56 0.610892 181.83 1.265604 181.91 1.920701 182.04 2.576384 182.23 57 0.621802 181.83 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 182.24 59 0.643622 181.83 1.298348 181.91 1-953470 182.05 2.609187 182.24 60 0.654532 181.83 1.309263 181.92 I-9 6 4393 182.05 2.620122 182.25 566 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 4 5 6 7 M. Diflf. V. If, Diflf. 1". M. Diflf. 1". M. Diflf. 1". 2.620122 182.25 3.276651 182.50 3.934182 182.80 4.592917 183.17 1 2.631057 182.25 3.287602 182.50 3-945J5 1 182.81 4.603907 183.18 2 2.641993 182.26 3.298552 182.51 3.956119 182.82 4.614898 183.18 3 2.652928 182.26 3-309503 182.51 3.967088 182.82 4.625889 183.19 4 2.663864 182.26 3.320454 182.52 3.978058 182.83 4.636880 183.19 5 2.674800 182.27 3-33I405 182.52 3.989028 182.83 4.647872 183.20 6 2.685736 182.27 3.342356 182.53 3.999998 182.84 4.658864 183.21 7 2.696672 182.27 3-353308 182.53 4.010968 182.84 4.669857 183.21 8 2.707609 182.28 3.364260 182.54 4.021939 182.85 4.680850 183.22 9 2.718546 182.28 3,375212 182.54 4.032911 182.86 4.691843 183.23 10 2.729483 182.29 3-386165 182.55 4.043882 182.86 4-70*837 183.24 11 2.740420 182.29 3.397118 182.55 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 14 2.762295 2.773233 182.30 182.30 3.419024 3.429978 182.56 182.57 4.076799 4.087772 182.88 182.88 4.735821 4.746816 183.25 183.26 15 2.784172 182.31 3.440932 182.57 4.098745 182.89 4.757812 183.27 16 2.795110 182.31 3.45x887 182.58 4.109718 182.90 4.768809 183.27 17 2.806049 182.31 3.462841 182.58 4.120692 182.90 4.779805 183.28 18 2.816988 182.32 3-473796 182.59 4.131667 182.91 4.790802 183.28 19 2.827927 182.32 3.484752 182.59 4.14264! 182.91 4.801800 183.29 20 2.838867 182.33 3.495707 182.60 4.153616 182.92 4.812797 183.30 21 2.849806 182.33 3.506663 182.60 4.164592 182.93 4.823796 183.31 22 2.860746 182.33 3.517619 182.61 4.175568 182.93 183.32 23 2.871686 182.34 3-5*8575 182.61 4.186544 182.94 4.845794 183.32 24 2.882627 182.34 3.539532 182.61 4.197520 182.94 4.856793 183-33 25 2.893567 182.35 3.550489 182.62 4.208497 182.95 4.867793 l8 3-34 26 2.904508 182.35 3.56i447 182.62 4.219474 182.95 4.878793 183.34 27 2.915449 182.36 3.572404 182.63 4.230451 182.96 4.889794 183.35 28 2.926391 182.36 3.583362 182.63 4.241429 182.97 4.900795 183.36 29 2.937332 182.36 3-5943* 182.64 4.252408 182.97 4.911797 183.36 30 2.948274 182.37 3.605279 182.64 4.263386 182.98 4.922799 183-37 31 2.959217 182.37 3.616238 182.65 4.274365 182.99 4.933801 183.38 32 2.970159 182.37 3.627197 182.65 4.285344 182.99 4.944804 183.38 33 2.981102 182.38 3.638156 182.66 4.296324 183.00 4.955807 183-39 34 2.992045 182.38 3.649116 182.66 4.307304 183 oo 4.966811 183.40 35 3.002988 182.39 3.660076 182.67 4.318284 183.01 4.977815 183.41 36 3.013931 182.39 3.671037 182.68 4.329265 183.01 4.988820 183.41 37 3.024875 182.39 3.681997 182.68 4.340246 183.02 4.999825 183.42 38 39 3.035819 3.046763 182.40 182.40 3.692958 3.703920 182.69 182.69 4.351228 4.362210 183.03 183.03 5.010830 5.021836 183-43 18 3-43 40 3.057707 182.41 3.714*81 182.70 4-373'9* 183.04 5.032842 183.44 41 3.068652 182.41 3-7*5843 182.70 4-3 8 4 T 75 183.05 5.043849 183.45 42 43 3-079597 3.090542 182.42 182.42 3.736806 3.747768 182.71 182.71 4-395I58 4.406141 183.05 183.06 5.054856 5.065864 183.46 183.46 44 3.101488 182.43 3-75873I 182.72 4.417125 183.06 5.076872 183.47 45 1 46 ; 47 48 3.112433 3.123379 3-'343*5 182.43 182.44 182.44 182.44 3.769694 3.780658 3.791622 3.802586 182.72 182.72 182.73 182.74 4.428109 4-439093 4.450078 4.461064 183.07 183.08 183.08 183.09 5.087880 5.098889 5.109898 5.120908 183.48 185.48 183.49 183.50 i 49 3.156219 182.45 3.813551 182.74 4.472049 183.10 5.131918 183.51 50 3.167166 182.45 3.824515 182.75 4-483035 183.10 5.142929 183.51 51 3.178113 182.46 3.835481 182.76 4.494022 183.11 5- I 5394 183.52 52 3.189061 182.46 3.846446 182.76 4.505008 183.12 5.164951 183-53 53 3.200009 182.47 3.857412 182.77 4-5I5995 183.12 5- J 759 6 3 183.54 54 3.210957 182.47 3.868378 182.77 4.526983 183.13 5.186975 183.54 55 3.221905 182.48 3-879345 182.78 4-53797 1 183.14 5.197988 !8 3 . 5 5 56 3,232854 182.48 3.890312 182.78 4.548959 183.14 5.209002 183.56 57 3.243803 182.49 3.901279 182.79 4^59948 183.15 5.220015 183.57 58 3.254752 182.49 3.912246 182.79 4-570937 183.15 5.231029 *83-57 59 3.265702 182.49 3.923214 182.80 4.581927 183.16 5.242044 183.58 60 3.276651 182.50 3.934182 182.80 4.592917 183.17 5-*5359 183.59 567 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 8 9 10 11 M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". 0' 5.253059 183.59 5.914815 184.06 6.578391 184.60 7-243997 185.19 1 5.264075 183.59 5.925859 184.07 6.589467 184.61 7.255109 185.20 2 5 275090 183.60 5.936904 184.08 6.600544 184.62 7.266222 185.21 3 5.286107 183.61 5-947949 184.09 6.611622 184.63 7-277335 185.22 4 5.297124 183.62 5.958995 184.10 6.622700 184.64 7.288449 185.23 5 5.308141 183.62 5.970041 184.11 6.633778 184.65 7.299563 185.25 6 5.319159 183.63 5.981087 184.11 6.644857 18466 7.310678 185.26 7 5 33oi77 183.64 5.992134 184.12 6.655937 184 67 7.321793 185.27 8 5.341195 183.65 6.003182 184.13 6.667017 184.67 7.332909 185.28 9 5.352214 183.66 6.014230 184.14 6.678098 184.68 7.344026 185.29 10 5.363234 183.66 6.025279 184.15 6.689179 184.69 7-355*44 185.30 11 5.374254 183.67 6.036328 184.16 6.700261 184.70 7.366262 185.31 12 13 5-385275 5.396296 183.68 183.69 6.047378 6.058428 184.17 184.18 6.711343 6.722426 184.71 184.72 7.377381 7.388500 185.32 185-33 14 S-WS 1 ? 183.69 6.069479 184.18 6.733510 184.73 7.399620 185-34 15 5-4 l8 339 183.70 6.080530 184.19 6-744594 184.74 7.410741 185.35 16 5.429361 183.71 6.091582 184.20 6.755679 184.75 7.421862 185.36 17 5.440384 183.72 6.102634 184.21 6.766764 184.76 7.432983 185.37 18 5.451407 183.73 6.113687 184.22 6.777850 184.77 7.444106 185.38 19 5.462431 183-73 6.124740 184.23 6.788937 184.78 7-455230 185.39 20 5-473455 183.74 6-135794 184.24 6.800024 184.79 7-466354 185.40 21 5.484480 183.75 6.146849 184.25 6. 811112 184.80 7.477478 185.4! 22 5-49555 183.75 6.157904 184.25 6.822200 184.81 7.488603 185.42 23 24 5-506530 5-5I7556 18376 183.77 6.168959 6.180015 184.26 184.27 6.833289 6.844378 184.82 184.83 7.499729 7.510855 185.43 185.44 25 5.528583 183.78 6.191072 184.28 6.855468 184.84 7.521982 185.46 26 27 5.539610 5.550637 183.79 183.79 6.202129 6.213187 184.29 184.30 6.866559 6.877650 184.85 184.86 7.533110 7-544239 185.47 185.48 28 5.561665 183.80 6.224245 184.31 6.888742 184.87 7-555368 185.49 29 5-572693 183.81 6.235304 184.32 6.899834 184.88 7.566497 185.50 30 5.583722 183.82 6.246363 184.32 6.910927 184.89 7.577628 185.51 31 5-594752 183.8; 6.257422 184.33 6.922021 184.90 7.588759 185.52 32 5.605782 183.8; 6.268482 184.34 6.933115 184.91 7-59989 185.53 33 5.616812 183.84 6-279543 l8 4-35 6.944210 184.92 7.611022 185.54 34 5.627843 183.85 6.290605 184.36 6.9553 5 184.93 7.622155 185.55 35 5.638874 183 86 6.301667 184.37 6.966401 184.94 7.633289 185.57 36 37 5.6499 6 5.660938 183.87 183.87 6.312729 6.323792 184.38 184.39 6.977498 6.988595 184.95 184.96 7.644423 7.655558 185.58 185.59 38 5.671971 183.88 6-334855 1 84.40 6.999693 184.97 7.666694 185.60 39 5.683004 183.89 6.345919 184.41 7.010791 184.98 7.677830 185.61 40 5.694038 183.90 6.356984 184.41 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 5.716106 183.91 6.379115 184.43 7.044090 185.01 7.711242 185.64 43 5.727141 183.92 6.390181 1 84.44 7.055191 185.02 7.722381 185.65 44 5.738177 183-93 6.401248 184.45 7.066292 185.03 7-733521 185.66 45 5.749213 183.94 6.412315 184.46 7-077394 185.04 7.744661 185.68 46 5.760250 18395 6.423383 184.47 7.088497 185.05 7.755802 185.69 47 5.771287 183.96 6.434451 184.48 7.099600 185.06 7.766943 185.70 48 5.782325 183.96 6.445520 1 84.49 7.1 10704 185.07 7.778085 185-71 49 5-793363 183.97 6.456590 184.50 7.121808 185.08 7.789228 185.72 50 51 5.804401 5.815440 183.98 183.99 6.467660 6.478731 184.51 184.52 7.132913 7.144019 185.09 185.10 7.800372 7.811516 185.73 185.74 52 5.826480 184.00 6.489802 184.52 7.155125 185.11 7.822661 185.75 53 5.837520 184.01 6.500874 184-53 7.166232 185.12 7.833807 fH 54 5.848561 184.01 6.511946 184.54 7.177340 185.13 7.844953 185.78 55 5.859602 184.02 6.523019 184.55 7.188448 185.14 7.856100 J o 5 -z 9 56 (5.870644 184.03 6.534092 184.56 7-199557 185.15 7.867247 185.80 57 5.881686 184.04 6.545166 184-57 7.210666 185.16 7.878396 185. Si 58 59 5.892728 5.903771 184.05 184.06 6.556241 6.567316 184.58 184.59 7221776 7.232886 185.17 185.18 7.889545 7.900694 185.82 185.83 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 Perihelior in a Paralolic Orbit. V. 12 13 14 15 M. 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 188.16 1 7.922995 185.86 8.593340 186.57 9.266360 187.34 9.942274 188.18 2 7-934I47 185.87 8.604535 186.58 9.277601 187.35 9-953565 188.19 3 7.945300 185-88 8.615730 186.59 9.288842 187.37 9.964857 188.21 4 7.956453 185.89 8.626926 186.61 9.300085 187.38 9.976149 188.22 5 7.967606 185.90 8.638123 186.62 9.311328 187.40 9.987443 188.23 6 7.978761 185.91 8.649320 186.63 9.322572 187.41 9.998738 188.25 7 7.989916 185.92 8.660518 186.64 9-3338I7 187.42 10.010033 188.26 8 8.001072 l8 5-93 8.671717 186.66 9.345063 187.44 10.021329 188.28 9 8.012228 185-95 8.682917 186.67 9.356310 187.45 10.032626 188.29 10 8.023385 185.96 8.694117 186.68 9-3 6 7557 187.46 10.043924 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 10.066523 188.34 13 8.056861 185.99 8.727723 186.72 9.401304 187.50 10.077823 188.35 14 8.068021 186.00 8.738927 186.73 9-41*555 187.52 10.089125 188.37 15 8.079181 186.02 8.750131 186.74 9.423806 187.53 10.100427 188.38 16 8.090343 186.03 8.761336 186.76 9.435058 187.54 10.1 1 1730 '^ 39 17 8.101505 186.04 8.772542 186.77 9.446311 187.56 10.123035 188.41 18 8.112668 186.05 8.783748 186.78 9-4575 6 5 187.57 10.134340 188.42 19 8.123831 186.06 8 -794955 186.79 9.468820 187.59 10.145646 188.44 20 8.134995 186.07 8.806163 186.81 9.480076 187.60 10.156952 188.45 21 8.146160 186.09 8.817372 186.82 9.491332 187.61 10.168260 188.47 22 8.157326 186.10 8.828582 186.83 9.502589 187.63 10.179568 188.48 23 8.168492 186.11 8.839792 186.84 9.513847 187.64 10.190878 188.50 24 8.179659 186.12 8.851003 186.86 9.525106 187.65 10.202188 188.51 25 8.190826 186.13 8.862215 186.87 9.536366 187.67 10.213499 188.53 26 8.201995 186.15 8.873427 186.88 9547626 187.68 10.224812 188.54 27 8.213164 186.16 8.884641 186.90 9.558888 187.70 10.236125 188.56 28 8.224334 186.17 8.895855 186.91 9.570150 187.71 10.247439 X ^- S7 29 8.235504 186.18 8.907070 186.92 9.581413 187.72 10.258753 188.59 30 31 8.246675 8.257847 186.19 186.20 8.918286 8.929502 186.93 186.95 9.592676 9.603941 187.74 187.75 10.270069 10.281386 188.60 188.62 32 8 269020 186.22 8.940719 186.96 9.615207 187.77 10.292703 188.63 33 8.280193 186.23 8 -95i937 186.97 9.626473 187.78 10.304021 188.65 34 8.291367 186.24 8.963156 186.99 9.637740 187.79 10.315341 188.66 35 8.302542 186.25 8.974376 187.00 9.649008 187.81 10.326661 188.68 36 8.313717 186.26 8.985596 187.01 9.660277 187.82 10.337982 188.69 37 38 8.324893 8.336070 186.28 186.29 8.996817 9.008039 187.02 187.04 9.671547 9.682817 187.84 187.85 10.349304 10.360627 188.71 188.72 39 8.347248 186.30 9.019262 187.05 9.694088 187.86 10.371951 188.74 40 41 8.358426 8.369605 186.31 186.32 9.030485 9.041709 187.06 187.08 9.705361 9.716634 187.88 187.89 10.383275 10.394601 188.75 188.77 42 8.380785 186.34 9.052934 187.09 9.727908 187.91 10.405927 188.78 43 44 8.391966 8.403147 186.35 186.36 9.064160 9.075387 187.10 187.12 9.739182 9.750458 187.92 18793 10.417255 10.428583 188.80 188.81 45 8.414329 186.37 9.086614 187-13 9.761734 187.95 10.439912 188.83 I 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 10.462573 188.86 48 49 8.447879 8.459064 186.41 186.42 9.120301 9.131531 187.17 187.18 9795569 9.806849 187.99 188.00 10.473905 10.485238 188.87 188.89 50 8.470250 186.43 9.142763 187.20 9.818129 188.02 10.496572 188.90 51 52 8.481436 8.492623 186.45 186.46 9-J53995 9.165228 187.21 187.22 9.829410 9.840693 188.03 188.05 10.507907 10.519242 188.92 188.93 53 8.503811 186.47 9.176462 187.23 9.851977 188.06 10.530579 188.95 54 8.515000 186.48 9.187696 187.25 9.863261 18808 10.541916 188.97 55 8.526189 186.49 9.198931 187.26 9.874546 188.09 10-553*55 188.98 56 8-537379 186.51 9.210167 187.27 9.885832 188.10 10.564594 189.00 57 8.548569 186.52 9.221404 187.29 9.897118 188.12 10-575934 189.01 58 59 8.559761 8'570953 186.53 186.54 9.232642 9.243880 187.30 187.31 9.908406 9.919694 188.13 188.15 10.587276 10.598618 189.03 189.04 60 8.582146 186.56 9.255120 187.33 9.930984 188.16 10.609961 189.06 669 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 16 17 18 19 M. Diff. I". M. Diff. 1". M. Diff. 1". M. Diff. I". 0' 10.609961 189.06 11.292277 190.02 11.978162 191.04 12.667850 192.13 I 2 10.621305 10.632649 189.07 189.09 11.303679 11.315082 190.03 190.05 11.989625 12.001089 191.06 191.08 12.679379 12.690908 192.15 192.17 3 10.643995 189.10 11.326485 190.07 12.012554 191.09 12.702439 192.19 4 10.655342 189.12 11.337889 190.08 12.024021 191.11 12.713970 192.21 5 10.666690 189.14 11.349295 190.10 12.035488 191.13 12.725503 192.22 6 10.678038 189.15 11.360701 190.12 12.046956 191.15 12.737037 192.24 7 10.689388 189.17 11.372109 190.13 12.058425 191.16 I2 -748573 192.26 8 10.700738 189.18 11.383517 190.15 12.069896 191.18 12.760109 192.28 9 10.712090 189.20 11.394927 190.17 12.081367 191.20 12.771646 192.30 10 10.723442 189.21 11.406337 190.18 12.092840 191.22 12.783185 192.32 11 12 10.734795 10.746149 189.23 189.24 11.417749 11.429161 190.20 190.22 12.104313 12.115788 191.24 191.25 12.794724 12.806265 192.34 192.36 13 10.757505 189.26 11.440575 190.23 12.127264 191.27 12.817807 14 10.768861 189.28 11.451989 190.25 12.138741 191.29 12.829350 192.39 15 10.780218 189.29 "4 6 3405 190.27 12.150219 191.31 12.840894 192.41 16 10.791576 189.31 11.474821 190.28 12.161698 191.32 12.852440 192.43 17 10.802935 189.32 11.486239 190.30 12.173178 191.34 12.863986 192.45 18 10.814295 189.34 11.497657 190.32 12.184659 191.36 12.875534 192.47 19 10.825655 I89-35 11.509077 190.33 12.196141 191.38 12.887082 192.49 20 10.837017 189.37 11.520497 190.35 12.207624 191,/j.o 12.898632 192.51 21 10.848380 189.39 11.531919 190.37 12.219108 191.41 12.910183 192.53 22 10.859744 189.40 "543342 190.39 12.230594 191.43 12.921736 192.55 23 10.871108 189.42 "554765 190.40 12.242080 191.45 12.933289 192.56 24 10.882474 189.43 11.566190 190.42 12.253568 191.47 12.944843 192.58 25 10.893840 189.45 11.577616 190.44 12.265057 191.49 12.956399 192.60 26 10.905208 189.47 1 1.589042 190.45 12.276546 191.50 12.967956 192.62 27 10.916576 189.48 11.600470 190.47 12.288037 191.52 12.979514 192.64 28 10.927946 189.50 11.611899 190.49 12.299529 191.54 12.991073 192.66 29 10.939316 189.51 11.623328 190.50 12.311022 191.56 13.002633 192.68 30 10.950687 189-53 11.634759 190.52 12.322516 191.58 13.014195 192.70 31 10.962059 189.55 11.646191 190.54 12.334011 191.60 13.025757 192.72 32 10973433 189.56 11.657624 190.56 12.345508 191.61 13.037321 192.74 33 34 10.984807 10.996182 189.58 189.59 11.669057 11.680492 190.57 190.59 12.357005 12.368503 191.63 191.65 13.048886 13.060452 192.76 192.78 35 11.007558 189-61 11.691928 190.61 12.380003 191.67 13.072019 192.80 36 11.018935 189.63 11.703365 190.62 12.391504 191.69 13.083587 192.82 37 11.030313 189.64 11.714803 190.64 12.403006 191.70 13.095157 192.83 38 11.041692 189.66 11.726242 190.66 12.414509 191.72 13.106727 192.85 39 11.053072 189.67 11.737682 190.68 12.426013 191.74 13.118299 192.87 40 11.064453 189.69 11.749123 190.69 12-4375 1 ? 191.76 13.129872 192.89 41 11.075835 189.71 11.760565 190.71 12.449023 191.78 13.141446 192.91 42 11.087218 189.72 11.772008 190.73 12.460531 191.80 13.153022 192.93 43 11.098602 189.74 11.783452 190.74 12.472039 191.81 13.164598 192.95 44 11.109987 189.76 11.794897 190.76 12.483548 191.83 13.176176 192.97 45 11.121372 189.77 11.806344 190.78 12.495059 191.85 i3- l8 7755 192.99 46 11.132759 189.79 11.817791 190.80 12.506571 191.87 i3-'99335 193.01 47 11.144147 189.80 11.829239 190.81 12.518083 191.89 13.210916 193.03 48 11.155536 189.82 11.840689 190.83 12.529597 191.91 13.222498 193.05 49 11.166925 189.84 11.852139 190.85 12.541112 191.93 13.234082 193.07 50 11.178316 189.85 11.863590 190.87 12.552628 191.94 13.245667 193.09 51 11.189708 189.87 11.875043 190.88 12.564145 191.96 13.257253 193.11 52 II.20IIOO 189.89 11.886496 190.90 12.575664 191.98 13.268840 193.13 53 11.212494 189.90 11.897951 190.92 12.587183 192.00 13.280428 193.15 54 11.223889 189.92 11.909407 190.94 12.598704 192.02 13.292017 I93- X 7 55 11.235284 189.93 11.920863 190.95 12.610225 192.04 13.303608 193.19 56 11.246681 189.95 11.932321 190.97 12.621748 192.06 13.315200 193.21 57 11.258078 189.97 11.943780 190.99 12.633272 192.07 13.326793 193.23 58 11.269477 189.98 191.01 12.644797 192.09 13-338387 193.25 ! 59 11.280876 190.00 11.966700 191.02 12.656323 192.11 13.349982 193.27 60 11.292277 190.02 11.978162 191.04 12.667850 192.13 13.361579 193-29 570 TABLE VI. For finding tne True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 20 21 22 23 M. Diff. 1". K, Diff. 1". M. Diff. 1". M. Diff. 1". 0' 13.361579 193.29 14.059591 194-51 14.762133 195.80 15.469459 197.17 1 13-373177 193-3I 14.071262 194-53 14.773882 195-83 15.481290 197.19 2 13-384776 193-33 14.082935 194-55 14.785632 195.85 15.493122 197.21 3 13.396376 193-35 14.094608 194-57 14.797384 195-87 15-504956 197.24 4 13.407977 193-37 14.106283 194.59 14.809137 195.89 15-516791 197.26 5 13.419580 193-39 14.117960 194.61 14.820891 I95-9I 15.528627 197.28 6 13.431183 193-4I 14.129637 194.64 1483*647 19594 15.540465 197-31 i 7 13.442788 193-43 14.141316 194.66 14.844403 195.96 15.552304 197.33 8 13-454394 193-45 14.152996 194.68 14.856161 195-98 I5-564H4 '97-35 9 13.466002 193-47 14.164677 194.70 14.867921 196.00 15.575986 197.38 10 13.477610 193-49 14.176360 194.72 14.879682 196.03 15.587830 197-4 11 13.489220 I93-5I 14.188044 19474 14.891444 196-05 15-599675 197-43 12 13.500831 193-53 i4-i997*9 194.76 14.903208 196.07 15.611521 197-45 13 I 3.5 12443 193-55 14.211415 194.78 I4-914973 196.09 15.623369 '97-47 14 13.524056 19357 14.223103 194.81 14.926739 196.12 15.635218 197.50 15 16 13.535671 13.547287 193-59 193.61 14.234792 14.246482 194.83 194.85 14.938506 14.950275 196.14 196.16 15.647068 15.658920 197.52 197-54 17 13.558904 193-63 14.258174 194.87 14.962045 196.18 15.670773 197-57 18 13.570522 14.269867 194-89 14.973817 196.20 15.682628 197-59 19 13.582141 193.67 14.281561 194.91 14.985590 196.23 I5-694484 197.61 20 13.593762 193-69 14.293256 194-93 14-997365 196.25 15.706342 197.64 21 22 13.605383 13.617006 I93-7I 193-73 14.304953 14.316651 194.95 194.98 15.009140 15 020917 196.27 196.30 15.718201 15.730061 197.66 197.69 23 13.628631 193-75 14.3*835 195.00 15.032696 196.32 15.741923 I97-71 24 13.640256 193-77 14.340050 195.02 15-044475 196.34 I5-753786 197-73 25 13.651883 193-79 14.351752 195.04 15.056256 196.36 15.765651 197.76 26 13-663511 193.81 14.363455 195.06 15.068039 196.39 15.777517 197.78 27 13.675140 193-83 H-375'59 195.08 15.079823 196.41 I5-789385 197.80 28 29 13.686770 13.698401 I93-85 I93-87 14.386865 14.398572 195.10 '95-13 15.091608 i5-i3394 196.43 196.45 15.801254 15.813124 197.83 197-85 30 13.710034 I93-89 14.410280 195.15 15.115182 196.48 15.824996 197.88 31 13.721668 193.91 14.421990 195.17 15.126971 196.50 15.836870 197-9 32 13-733303 193-93 14-433700 I95-19 15.1 38762 196.52 15-848744 197.92 33 13.744940 193-95 14.445412 195.21 15.150554 196.54 15.860620 197-95 34 13.756577 193-97 14.457126 195.23 15.162348 196.57 15.872498 197-97 35 36 37 13.768216 13.779856 13.791498 19399 194.01 194.03 14.468841 14.480557 14.492274 195.26 195.28 195-3 15.174142 15.185938 15.197736 196.59 196.61 196.64 r ff I5-884377 15.896258 15.908140 198.00 198.02 198.04 38 13.803140 194-05 14.503992 195-3* 15.209535 196.66 15.920023 198.07 39 13.814784 194.07 14.515712 195-34 15.221335 196.68 15.931908 198.09 40 13.826429 194.09 14.527434 195-36 15.233137 196.70 15-943794 198.12 41 13.838075 194.11 14.539156 195-39 15.244940 196.73 15-955682 198.14 42 13.849723 194.14 14.550880 195.41 15.256744 196.75 15.967571 198.17 43 13.861372 194.16 14.562605 195-43 15.268550 196.77 15.979462 198.19 44 13.873022 194.18 14-574331. 195-45 15.280357 196.80 15-991354 198.21 45" i 46 47 I 48 13.884673 13-8963*5 13-907979 13-919634 194.20 194.22 194.24 194.26 14.586059 14.597788 14.609519 14.621250 195-47 195.50 195.52 195-54 15.292165 15-303975 15-315786 15-3*7599 196.82 196.84 196.87 196.89 16.003248 16.015143 16.027039 16.038937 198.24 198.26 198.29 198.31 49 13.931290 194.28 14.632983 I95-56 I5-339413 196.91 16.050836 198.34 50 51 52 53 54 13.942948 13.954606 13.966266 13.977927 13.989590 194-30 194-3* 194-34 194.36 I94-38 14.644718 14.656453 14.668190 14.679929 14.691668 195.58 195.60 195-63 I95-65 195.67 15.351228 15-363045 15.374863 15.386683 15.398504 196.94 196.96 196.98 197.00 197.03 16.062737 16.074639 16.086543 16.098449 16.110355 198.36 198.38 198.41 198-43 198.46 55 14.001254 194.41 14.703409 195-69 15.410326 197.05 16.122263 198.48 56 14.012919 194-43 14-715151 195.71 15.422150 197.07 10.134173 198.51 57 58 59 14.024585 14.036252 14.047921 194.45 194.47 194.49 14.726895 14.738640 14.750386 195-74 195.76 195.78 15-433975 15.445802 15.457630 197.10 197.12 197.14 16.146084 16.157997 16.16991 1 198-53 198.56 198.58 60 14.059591 194.51 14.762133 195.80 15-469459 197.17 16.181826 198.60 571 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 24 25 26 27 M. Diff. 1". M. Diff. I". M. Diff. 1". M. Diff. 1". 0' 16. 181826 198.60 16.899499 200.12 17.622747 201.70 18.351847 203.37 1 16.193743 198.63 16.911507 200.14 17.634850 201.73 18.364050 203.40 2 16.205662 198.65 16.923516 200.17 17.646954 201.76 18.376255 203.42 3 16.217582 198.68 16.935527 200.19 17.659060 201.78 18.388461 203.45 j 4 16.229503 198.70 16.947539 200.22 17.671168 201. 8l 18.400669 203.48 5 16.241426 198.73 16959553 200.24 17.683278 201.84 18.412879 203.51 6 16.253350 198.75 16.971568 200.27 17.695389 201.87 18.425090 203.54 7 16.265276 198.78 16.983585 200.30 17.707502 201.89 18.437303 203.57 8 16.277204 198.80 16.995604 200.32 17.719616 201.92 18.449518 203.59 9 16.289133 198.83 17.007624 200.35 17.731732 201.95 18.461735 203.62 10 16.301063 198.85 17.019646 200.37 17.743850 201.97 18.473953 203.65 11 16.312995 198.88 17.031669 200.40 17.755969 202.00 18.486173 203.68 12 16.324928 198.90 17.043694 200.43 17.768090 2O2.O3 18498395 203.71 13 16.336863 198.93 17.055720 200.45 17.78021 3 202.06 18.510618 203.74 14 16.348799 198.95 17.067748 200.48 17.792337 202.08 18.522843 203.77 15 16.360737 198.97 17.079777 200.50 17.804462 202.11 18.535070 203.80 16 16.372676 199.00 17.091808 200.53 17.816590 202.14 18.547299 203.82 17 16.384617 199.02 17.103841 200.56 17.828719 202.17 18.559529 203.85 18 16.396559 199.05 17.115875 200.58 17.840850 2O2.I9 18.571761 203.88 19 16.408503 199.07 17.127911 200.61 17.852982 202.22 18.583995 203.91 20 16.420448 199.10 17.139948 200.64 17.865116 2O2.25 18.596230 203.94 21 16.432395 199.12 17.151987 200.66 17.877252 202.28 18.608467 203.97 22 16.444343 199.15 17.164028 200.69 17.889389 202.30 18.620706 204.00 23 16.456292 199.17 17.176070 200.71 17.901528 202-33 18.632947 204.03 24 16.468243 199.20 17.188114 200.74 17.913669 202.36 18.645190 204.05 25 16.480196 199.22 17.200159 200.77 17.925811 202.39 18.657434 204.08 26 16.492151 199.25 17.212206 200.79 J7-937955 202.41 18.669679 204.11 27 16.504107 199.27 17.224254 200.82 17.950101 202.44 18.681927 204.14 28 16.516064 199.30 17.236304 200.85 17.962248 202.47 18.694177 204.17 29 16.528022 '99-33 17.248356 200.87 17-974397 202.50 18.706428 204.20 30 16.539983 199-35 17.260409 200.90 17.986548 202.52 18.718680 204.23 31 l6 -55 J 945 199.38 17.272464 200.93 17.998700 202.55 18.730935 204.26 32 16.563908 199.40 17.284520 200.95 18.010854 202.58 18.743191 204.29 33 l6 -575 8 73 199-43 17.296578 200.98 18.023010 202.6l 18.755449 204.32 34 16.587839 199.45 17.308637 201.00 18.035167 202.64 18.767709 204.35 35 16.599807 199.48 17.320698 2OI.O3 18.047326 202.66 18.779971 204.37 36 16.611776 199.50 17.332761 201.06 18.059487 202.69 18.792234 204.40 37 16.623747 199-53 17.344825 2OI.O8 18.071649 202.72 18.804499 204.43 38 16.635719 J 99-55 17.356891 201. II 18.083813 202.75 18.816767 204.46 39 16.647693 199.58 17.368959 201.14 18.095979 202.78 18.829036 204.49 40 16.659669 199.60 17.381028 201. l6 18.108146 202.80 18.841305 204.52 41 16.671646 199.63 17.393098 201.19 18.120315 202.83 18.853577 204.55 42 16.683624 199.65 17.405171 201.22 18.132486 202.86 18.865851 204.58 43 16.695604 199.68 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 16.719569 J99-73 17.441397 201.30 18.169008 202.94 18.902684 204.67 46 * 6 -73 f 553 199.76 17.453476 201.32 18. 181186 202.97 18.914965 204.70 47 l6 -743539 199.78 17.465556 201-35 18.193365 203.00 18.927247 204.73 48 16.755527 199.81 17.477638 201.38 18.205546 203.03 18.939532 204.76 49 16.767516 199.83 17.489722 201.41 18.217728 203.06 18.951818 204.79 50 16.779507 199.86 17.501807 201.43 18.229912 203.08 18.964106 204.81 51 16.791499 199.88 17.513894 201.46 18.242098 203.11 18.976396 204.84 52 16.803493 199.91 17.525982 201.49 18.254286 203.14 18.988687 204.87 53 16.815488 199.94 17.538072 201.51 18.266475 203.17 19.000981 204.90 54 16.827485 199.96 17.550163 201.54 18.278666 203.20 19.01 3276 204.93 55 16.839484 199.99 17.562257 201.57 18.290859 203.23 19.025573 204.96 56 16.851484 200.01 17.574352 201.59 18.303053 203.25 19.037871 204.90 57 16.863485 200.04 17.586448 201.62 18.315249 203.28 19.050172 205.02 58 16.875488 2OO.O6 17.598546 201.65 18.327447 203.31 19.062474 205.05 59 16.887493 200.09 17.610646 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. V. 28 29 30 31 M. Diff. 1". M. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 19.087084 205.11 19.828747 206.94 1.313 3849 44.08 1.329 0430 42.92 I 19.099391 205.14 19 841164 206.97 .313 6493 44.06 .329 3004 42.91 2 19.111701 205.17 I9-853583 207.00 .313 9136 44.04 329 5578 42.89 3 19 124012 205.20 19.866004 207.03 .314 1778 44.02 .329 8151 42.87 4 19.136325 205.23 19.878427 207.06 .314 4419 44.00 .330 0723 42.85 5 19 148639 205 26 19 890852 207.09 1.314 7058 43.98 1.330 3293 42.83 6 19.160956 205.29 19.903279 207.13 .314 9696 43.96 .330 5862 42.81 7 19.173274 205.32 19.915707 207.16 3 X 5 2333 43-94 33 8431 42.80 8 19.185594 205-35 19.928137 207.19 .315 4969 43.92 .331 0998 42.78 j 9 19.197916 205.38 19.940569 207.22 .315 7604 43-90 33 1 3564 42.76 10 19.210240 205.41 19.953003 207.25 1.316 0237 43.88 1.331 6129 42.74 11 19 222566 205.44 19.965439 207.28 .316 2869 43.86 .331 8693 42.72 12 19.234893 205.47 19.977877 207.31 .316 5500 43-84 332 1255 42.70 13 19.247222 205.50 19.990317 207.34 .316 8130 43.82 332 3817 42.69 14 !9-259553 205-53 20.002759 207.38 .317 0759 43.80 332 6378 42.67 15 19.271885 205.56 20.015202 207.41 1.317 3386 43.78 1.332 8937 42.65 16 19.284220 205.59 20.027647 207.44 .317 6013 43.76 333 H96 42.63 17 18 19.296556 19.308894 205.62 205.65 20.040095 20.052544 207.47 207.50 .317 8638 .318 1262 43-74 43-72 333 453 -333 6609 42.61 42.59 19 19.321234 205.68 20.064995 207.53 .318 3885 43-70 333 9 l6 4 42.58 20 19.333576 205.71 20.077448 207.57 1.318 6506 43.68 1.334 1718 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 19 370612 205.80 20.114818 207.66 .319 4364 43.63 334 9374 42.50 24 19.382961 205.83 20.127279 207.69 .319 6981 43.61 335 1924 42-49 25 19.395312 205.86 20.139741 207.72 I -3 I 9 9597 43-59 1-335 4472 42.47 20 19.407665 205.89 20.152206 207.76 .320 2212 43-57 -335 7020 42.45 27 19.420019 205.92 20.164672 207.79 .320 4825 43-55 335 95 6 7 42.43 28 19432375 205.95 20.177140 207.82 .320 7438 43-53 .336 2112 42.41 29 1 9-4447 34 205.98 20.189610 207.85 .321 0049 435 1 .336 4656 42.40 30 19.457094 206.01 20.202082 207.88 I.32I 2659 43-49 1.336 7199 42.38 31 19.469455 206.04 20.214556 207.91 3*1 5268 43-47 .336 9742 42.36 32 19.481819 206.08 20.227032 207.95 .321 7875 43-45 -337 2283 42-34 33 19.494184 206. 1 1 20.239510 207.98 .322 0482 43-43 337 4823 42.33 34 19.506551 206.14 20.251989 208.01 .322 3087 43-41 337 73 6 2 42.31 35 19.518921 206.17 20.264471 208.04 1.322 5692 43.40 1-337 99 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. II .323 0897 43-36 .338 4972 42.25 38 19.556039 206.26 20.301927 208.14 3*3 3498 43-34 338 757 42.24 39 19.568415 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 43-3 i-339 2573 42.20 41 19.593174 206.35 20.339400 208.24 .324 1294 43.28 339 5 I0 5 42.18 42 19.605556 206.38 20.351895 208.27 .324 3890 43.26 339 7635 42.17 43 19.617939 206.41 20.364392 208.30 .324 6485 43.24 .340 0165 42.15 44 19.630325 206.44 20.376891 208.33 .324 9079 43.22 340 2693 42.13 45 19.642713 206.47 20.389392 208.36 1.325 1672 43.21 1.340 5221 42.11 46 47 19.655102 19.667493 206.50 206.53 20.401895 20.414399 208.39 208.43 .325 4263 .325 6854 43.19 43-17 .340 7747 .341 0272 42.10 42.08 48 19.679886 206.57 20.426906 208.46 325 9443 43-15 .341 2796 42.06 19 19.692281 206.60 20.439415 208.49 .326 2032 43-13 -341 53'9 42.04 50 19.704678 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 19.729477 206.69 20.476952 208.59 .326 9790 43-07 .342 2882 41.99 53 19.741879 206.72 20.489468 208.62 .327 2374 43-05 .342 5401 41-97 54 19.754283 206.75 20.501986 208.65 3*7 4957 43.04 .342 7919 41.96 55 19.766689 206.78 20.514506 208.69 1.327 7538 43.02 1-343 43 6 41.94 56 19.779097 206.81 20.527029 208.72 .328 0119 43-0 343 2952 41.92 57 19.791507 206.84 20.539553 208.75 .328 2698 42.98 343 5467 41.90 58 19.803919 206.88 20.552079 208.78 .328 5276 42.96 343 798o 41.89 59 19.816332 206.91 20.564607 208.82 .328 7853 42.94 344 493 41.87 60 19.828747 206.94 20.577137 208.85 1.329 0430 42.92 1.344 3005 41.85 573 TABLE VI. b or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 32 33 34 35 logM. Diflf. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.344 3005 41.85 1.359 1859 40.86 1-373 7*5i 39-93 1.387 9418 39.06 1 -344 55*5 41.84 359 43*o 40.84 373 9 6 46 39.91 .388 1761 39-5 2 .344 8025 41.82 359 6 76o 40.82 -374 *4i 3990 .388 4104 39.04 3 -345 0534 41.80 359 9*9 40.81 374 4434 39.88 .388 6446 39.02 4 345 34i 41.78 .360 1657 40.79 374 6827 39.87 .388 8787 39.01 5 1.345 5548 41-77 1.360 4104 40.78 1.374 9218 39-85 1.389 1127 38-99 6 345 8053 4*-75 .360 6550 40.76 375 l6 9 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 .361 1439 4-73 .375 6388 39.81 .389 8x42 38.95 9 .346 5564 41.70 .361 3883 40.71 .375 8776 39-79 .390 0479 38.94 10 1.346 8065 41.68 1.361 6325 40.70 1.376 1164 39-78 1.390 2815 3893 11 347 05 6 5 41.66 .361 8766 40.68 .376 3550 39-77 39 5*5 38.91 12 347 3 6 5 41.65 .362 1207 40.66 376 5935 39-75 .390 7484 38.90 13 347 55 6 3 41.63 .362 3646 40.65 .376 8320 39-74 39 9817 38.88 14 .347 8060 41.61 .362 6084 40.63 377 703 39-7* .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 35* 41.58 .363 0959 40.60 377 54 68 39- 6 9 .391 6813 38.84 17 .348 5546 41.56 3 6 3 3394 40.59 377 7849 39.68 .391 9143 38.83 18 .348 8040 4!-55 .363 5829 40.57 378 0230 39.66 .392 1472 38.82 19 349 53* 4'-53 .363 8263 40.56 .378 2609 39- 6 5 -39* 3801 38.80 20 1.349 3023 41.51 1.364 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 39* 8455 38.77 22 349 8o 3 41.48 3 6 4 5559 40.51 .378 9742 39.61 393 78i 38.76 23 .350 0491 41.46 .364 7989 40.49 379 *"7 39-59 393 3 I0 7 38.75 24 .350 2978 41.45 .365 0418 40.48 379 449* 39.58 393 543 1 38.73 25 1.350 5464 41.43 1.365 2846 40.46 1.379 6866 39-56 1-393 7755 38.72 26 35 7950 41.41 3 6 5 5*73 40.45 379 9*4 39-55 394 78 38.71 27 .351 0434 41.40 .365 7699 40.43 .380 1612 39-53 394 *4 38.69 28 .351 2917 41.38 .366 0125 40.41 .380 3983 39-5* 394 47*i 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 4'-33 .366 7395 4-37 .381 1093 39-47 .395 1680 38.64 1 32 .352 2840 41.31 .366 9817 4-35 .381 3461 39-4 6 395 3998 38-63 I 33 35* 53'8 41.30 .367 2238 40.34 .381 5828 39-45 395 6 3'5 38.61 34 35* 7795 41.28 .367 4657 40.32 .381 8194 39-43 -395 8631 38.60 35 1.353 o*7* 41.26 1.367 7076 40.31 1.382 0559 39-4* 1.396 0947 38.59 36 353 *747 41.25 3 6 7 9494 40.29 .382 2924 39.40 .396 3262 38-57 37 353 5**i 41.23 .368 1911 40 28 .382 5288 39-39 .396 5576 38.56 38 353 7 6 94 41.21 .368 4327 40.26 .382 7651 39-37 .396 7889 38.55 39 354 Ol6 7 41.20 .368 6742 40.25 .383 0013 39-3 6 .397 0201 38.53 40 1.354 2638 41.18 1.368 9157 40.23 1.383 2374 39-35 1.397 2513 38.52 41 354 5 IQ 8 41.16 .369 1570 40.21 383 4734 39-33 397 4823 38-51 42 354 7578 41.15 3 6 9 39 8 3 40.20 383 793 39.32 397 7133 38.49 43 355 4 6 41-13 3 6 9 6 394 40.18 383 94-5* 39-3 397 944* 38.48 44 355 *5'3 41.11 .369 8805 40.17 .384 1809 39-*9 .398 1751 38-47 45 1.355 49 8 41.10 1.370 1214 40.15 1.384 4166 39.27 1.398 4058 38.45 46 355 7445 41.08 .370 3623 40.14 .384 6522 39.26 398 6365 38.44 47 355 999 41.07 .370 6031 40.12 .384 8878 39-*5 .398 8671 38-43 48 .356 2373 41.05 .370 8438 40.11 .385 1232 39*3 -399 976 38.41 49 .356 4836 41.03 .371 0844 40.09 385 3585 39.22 -399 3*8i 38.40 50 1.356 7297 41.02 1.371 3249 40.08 1.385 5938 39.20 1.399 5584 38-39 51 35 6 9758 41.00 371 5 6 54 40.06 .385 8290 39-19 399 7887 38.37 52 357 **'7 40.98 .371 8057 40.05 .386 0641 39.18 .400 0189 38.36 53 357 4 6 76 40.97 37* 0459 40.03 .386 2991 39.16 .400 2491 38.35 54 357 7134 40.95 .372 2861 40.02 .386 5340 39-15 .400 4791 38.33 55 i-357 959 40.94 1.372 5261 40.00 1.386 7689 39-13 1.400 7091 38.32 56 .358 2046 40.92 .372 7661 39-99 .387 0036 39.12 .400 9390 38-31 57 .358 4501 40.90 .373 0060 39-97 -387 *383 39-" .401 1688 .38.30 58 .358 6954 40.89 373 *458 39.96 387 47^9 39.09 .401 3985 38.28 59 .358 9407 40.87 373 4 8 55 39-94 .387 7074 39.08 .401 6282 38.27 60 1.359 1859 40.86 1.373 7251 39-93 1.387 9418 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. V. 36 37 38 39 | logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 1.401 8578 38.26 1.415 4930 37-5 1.428 8662 36.80 1.441 9943 36.14 1 .402 0873 38.24 .415 7180 37-49 .429 0869 36-79 .442 21 1 I 36.13 2 .402 3167 38-23 .415 9429 37-47 .429 3076 36.78 .442 4.279 36.12 3 .402 5460 38.22 .416 1678 37-46 .429 5281 36-77 .442 6446 36.11 4 .402 7753 38.20 .416 3925 37-45 .429 7488 36-75 .442 86l2 36.10 5 1.403 0045 38.19 1.416 6172 37-44 1.429 9693 36.74 1-443 0778 36 co 6 .403 2336 38.18 .416 8419 37-43 .430 1897 36.73 443 2943 36.08 7 .403 4626 38.17 .417 0664 37-4 1 .430 4101 36.72 .443 5107 36.07 8 .403 6916 38.15 .417 2909 37.40 43 6304 36-71 443 7271 36.06 9 .403 9205 38.14 417 5153 37-39 .430 8506 36-7 443 9434 36.05 10 1.404 1493 38.13 1-4*7 7396 37-38 1.431 0708 36.69 1.444 1597 36-04 11 .404 3780 38.12 4i7 963.9 37-37 431 2909 36.68 444 3758 36-03 12 .404 6067 38.10 .418 1881 37-36 .431 5109 36.66 444 5920 36.02 13 .404 8352 38.09 .418 4122 37-35 .431 7308 36.65 .444 8080 36.00 14 .405 0637 38.08 .418 6362 37-33 431 957 36.64 .445 0240 35-99 15 1.405 2921 38.06 1.418 8602 37.32 1.432 1705 36.63 1.445 2400 35-98 16 17 .405 5205 .405 7488 38.05 38.03 .419 0841 .419 3079 37-31 37.3 -432 393 .432 6100 36.62 36.61 445 4558 445 6716 35-97 35-96 18 .405 9769 38.02 419 5317 37-29 .432 8296 36.60 .445 8874 35-95 19 .406 205 i 38.01 .419 7554 37-27 .433 0491 36.59 .446 1031 35-94 20 1.406 4331 38.00 1.419 9790 37.26 1.433 2686 36.57 1.446 3187 3593 21 .406 66 1 1 37-99 .420 2026 37-25 .433 4881 36.56 446 5343 3592 22 .406 8889 37-97 .420 4260 37-24 -433 774 36-55 .446 7498 35-91 23 .407 1 1 68 37.96 .420 6494 37-23 -433 9267 36.54 .446 9652 35-90 24 407 3445 37-95 .420 8728 37.22 .434 1459 36.53 .447 1806 35-89 25 1.407 5721 37-94 1.421 0960 37-20 1.434 3651 36-52 1-447 3959 35-88 26 .407 7997 37-92 .421 3192 37-19 434 5842 36-51 .447 6112 35.87 27 .408 0272 37-91 .421 5423 37-i8 434 8032 36.50 .447 8263 35.86 28 .408 2547 37.90 .421 7654 37.17 .435 0221 36.49 .448 0415 35.85 29 .408 4820 37.89 .421 9884 37.16 435 2410 36.48 .448 2565 35-84 30 1.408 7093 37.87 1.4*2 2113 37.15 1-435 4598 36.47 1.448 4715 35-83 31 32 33 .408 9365 .409 1636 .409 3907 37-86 37.85 37.84 .422 A 3 4i .422 6569 .422 8796 37-13 37-12 37-11 .435 6786 435 8973 43 6 H59 36-46 36.44 36.43 .448 6865 .448 9014 .449 1162 35-82 35.81 35.80 34 .409 6177 37.82 .423 1 022 37.10 43 6 3345 36-42 449 339 35-79 35 1.409 8446 37.81 1.423 3248 37-09 1.436 5530 36-41 1-449 5456 35-78 36 .410 0714 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 2143 37.04 437 4263 36.37 .450 4038 35-74 40 1.410 9780 37-75 1.424 4365 37-03 i-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 43 .411 4308 .411 6571 37-72 37-7 1 .424 8807 .425 1027 37.01 36.99 .438 0806 .438 2986 36-34 36.32 .451 0468 .451 2610 35-71 35-70 44 .411 8833 37-7 .425 3246 36-98 .438 5165 3 6 .31 .451 4752 35-69 45 1.412 1095 37- 6 9 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 36.29 451 933 35.67 47 t s- .412 5616 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 0134 37-64 .426 4333 36-92 .439 6051 36.26 452 5450 35-64 50 1.413 2392 37-63 1.426 6548 36-91 1.439 8226 36.25 1.452 7588 35-63 51 .413 4649 37.61 .426 8762 36.90 .440 0401 36-24 .452 9725 35-62 52 53 54 .413 6905 .413 9161 .414 1416 37.60 37-59 37.58 .427 0976 427 3 l8 9 .427 5402 36.89 36.88 36.87 .440 2575 .440 4748 .440 6921 36.23 36.22 36.20 .453 1862 453 3998 453 6134 35.60 35-59 55 56 1.414 3670 37.56 37-55 1.427 7613 .427 9824 36.86 36-85 1.440 9093 .441 1264 36-19 36.18 1.453- 8269 .454 0403 35-57 57 .414 8176 37.54 .428 2035 36-83 441 3436 36.17 454 2537 35-56 58 59 .415 0429 .415 2680 37-53 37-51 .428 4244 .428 6453 36.82 36.81 .441 5605 441 7774 36.16 36.I5 454 4670 .454 6802 35-55 35-54 60 1.415 4930 37-5 1.428 8662 36.80 1.441 9943 36.14 1-454 *934 35-53 075 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 40 41 42 43 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 1.454 8934 35-53 1.467 5782 34-95 1.480 0627 34-41 T -492 3597 33-9 1 1 455 1065 35-51 .467 7879 34-94 .480 2691 34.40 .492 5631 33-90 2 455 3*9 6 35-51 .467 9976 34-93 .480 4755 34.40 .492 7665 3389 3 .455 5326 35-5 .468 2071 34.92 .480 6819 34-39 .492 9698 33-88 4 455 7456 35-49 .468 4166 34- 9 l .480 8882 34-38 493 i73i 33-87 5 T -455 95 8 5 35.48 1.468 6261 34.90 1.481 0944 34-37 r -493 3764 33-87 6 .456 1713 35-47 .468 8355 34-9 .481 3006 34-36 493 5796 33.86 | 7 .456 3841 35-46 469 44 8 34.89 .481 5068 34-35 493 7827 33-85 8 .456 5968 35-45 .469 2541 34.88 .481 7129 34-34 493 9858 33-84 1 9 .456 8094 35-44 .469 4634 34.87 .481 9189 34-33 .494 1888 33-83 10 1-457 0220 35-43 1.469 6725 34.86 1.482 1249 34-33 1.494 39 l8 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 33-8i 13 457 6 595 35-40 .470 2998 34.83 .482 7425 34-3 .495 0005 33.80 14 457 8718 35-39 .470 5087 34.82 .482 9483 34-29 495 233 33-79 15 1.458 0841 35.38 1.470 7176 34.81 1.483 1540 34.28 1.495 46i 33-79 16 .458 2964 35-37 .470 9265 34.80 483 3597 34.28 .495 6088 33-78 17 .458 5086 35-3 6 471 1353 34-79 483 5 6 53 34-27 -495 8114 33-77 18 .458 7207 35-35 .471 3440 34-79 483 779 34.26 .496 0140 33.76 19 .458 9328 35-34 471 55*7 34.78 .483 9764 34-25 .496 2166 33-75 20 1.459 H4 8 35-33 1.471 7613 34-77 1.484 1819 34-24 1.496 4191 33-75 21 459 35 6 7 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 35-3 472 3 86 9 34-74 .484 7980 34-22 .497 0264 33-72 24 459 99" 35-^9 472 5953 34-73 .485 0033 34.21 497 2287 33.71 25 1.460 2040 35.28 1.472 8037 34-73 1.485 2085 34.20 1.497 4310 33-71 26 .460 4156 35.27 .473 0120 34-72 485 4'37 34-19 497 6332 33.70 27 .460 6272 35.26 .473 2203 34.71 .485 6188 34.18 497 8354 33-69 28 .460 8388 35-25 473 4285 34-7 485 8239 34-17 .498 0376 33-68 29 .461 0503 35-24 473 6366 34.69 .486 0289 34.16 .498 2396 33.68 30 1.461 2617 35-23 1.473 8447 34.68 1.486 2338 34.16 1.498 4417 33.67 31 .461 4731 35-23 474 5 2 7 34-67 .486 4388 3415 .498 6437 33.66 32 .461 6844 35-22 .474 2607 34-66 .486 6436 34-14 .498 8456 33.65 33 .461 8957 35-21 .474 4686 34.65 .486 8484 34.13 499 475 33-65 34 .462 1069 35.20 474 6765 34.64 487 <>53 2 34.12 .499 2494 33-64 35 1.462 3180 35- J 9 1.474 8843 34-63 1.487 2579 34.12 1.499 4512 33-63 36 .462 5291 35-i8 .475 0921 34.62 .487 4626 34- " 499 6 53o 33.62 37 .462 7401 35- J 7 475 2998 34.61 .487 6672 34.10 499 8547 33.62 38 .462 9511 35-i6 -475 575 34.61 .487 8718 34.09 .500 0563 33.61 39 .463 1620 35-15 475 7 J 5i 34.60 .488 0763 34.08 .500 2580 33.60 40 1.463 3729 35-H 1.475 9227 34-59 1.488 2807 34.07 1.500 4595 33-59 41 4 6 3 5 8 37 35- x 3 .476 1302 34.58 .488 4852 34.07 .500 6611 33.58 42 4 6 3 7944 35- 12 .476 3376 34-57 .488 6895 34.06 .500 8625 33-58 43 .464 0051 35- 11 .476 5450 34-56 .488 8939 34.05 .501 0640 33-57 44 .464 2158 35-i .476 7524 34-55 .489 0981 34-4 .501 2654 33-56 45 1.464 4263 35-9 1.476 9596 34-54 1.489 3023 34-03 1.501 4667 33-55 46 .464 6369 35.08 477 1669 34-54 489 5 6 5 34.02 .501 6680 33-55 47 .464 8473 35.07 477 374i 34-53 .489 7106 34.02 .501 8693 33-54 48 465 577 35.06 477 5812 34-5* .489 9147 34.01 .502 0705 33-53 49 .465 2681 35-5 477 7883 34-5 i .490 1187 34.00 .502 2716 33-5 a 50 1.465 4784 35-4 1-477 9953 34-5 1.490 3227 33-99 1.502 4727 33-5 1 51 .465 6886 35-4 .478 2023 34-49 .490 5266 33.98 .502 6738 33-5 1 52 .465 8988 35-3 .478 4092 34-48 49 735 33-97 .502 8748 33-5 53 .466 1090 35.02 .478 6161 34-47 .490 9343 3396 53 758 33-49 54 .466 3190 35> 01 .478 8229 34.46 .491 1381 33-95 .503 2767 33-48 55 1.466 5290 35.00 1.479 0297 34-46 1.491 3418 33-95 1.503 4776 33-48 56 .466 7390 34-99 479 2364 34-45 49 1 5455 33-94 .503 6784 33-47 57 .466 9489 34.98 .479 4430 34-44 49 * 749 * 33-93 53 8792 33-46 58 .467 1587 34-97 479 6 496 34-43 .491 9527 3392 .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 !-49 a 3597 33.91 1.504 4813 33-44 576 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Oibit. V. 44 45 4:6 47 log M. Diff. 1". logM. Diff. I". logM. Diff. I". logM. Diff. 1". 0' 1.504 4813 33-44 1.516 4390 33.00 1.528 2435 3*59 *-539 94 8 32.20 1 .504 6819 33-43 .516 6370 32.99 .528 4390 32.58 .540 0980 32.20 2 .504 8825 33 4* .516 8349 32.98 .528 6344 32.57 .540 2912 32.19 3 .505 0830 33-4^ .517 0328 3298 .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 6 1.505 4839 .505 6843 3340 33-39 1.517 4284 .517 6262 32.96 32.96 1.529 2206 .529 4159 3M5 3255 1.540 8705 .541 0635 32-I7 32.17 7 .505 8846 33-39 .517 8239 32-95 .529 6112 32-54 .541 2564 32.16 8 .506 0849 33-3^ .518 0216 3*94 .529 8064 32.53 .541 4494 32.15 9 .506 2852 33 37 .518 2192 32-93 .530 0016 32-53 .541 6423 32.15 10 11 1.506 4854 .506 6855 33-36 33-3 6 1.518 4168 .518 6143 32.93 3292 1.530 1967 .530 3918 32.52 32-51 1.541 8352 .542 0280 32.14 32.14 12 .506 8856 33-35 .518 8118 32.91 .530 5869 32-51 .542 2208 32.13 13 .507 0857 33-34 .519 0093 32.91 53 78*9 32.50 542 4135 32.12 14 .507 2857 33-33 .519 1067 32.90 .530 9769 32.49 .542 6063 32.11 15 1.507 4857 33-33 1.519 4041 32.89 1.531 1719 32.49 1.542 7989 32.11 16 .507 6856 33-32 .519 6014 32.89 .531 3668 32.48 542 99 l6 32.10 17 .507 8855 33-3 1 .519 79 8 7 32.88 .531 5616 32.48 543 l8 42 32.10 18 .508 0853 33-3 .519 9960 32- 8 7 53 1 75 6 5 32.47 543 376 8 32.09 19 .508 2851 33.29 .520 1932 32.86 53 1 95 J 3 32.46 543 5 6 93 32.09 20 1.508 4849 13.29 1.520 3904 3286 1.532 1460 32.46 1.543 76i 8 32.08 21 .508 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- 2 7 .54.0 9816 32.84 .532 7300 3244 544 339i 32.06 24 .509 2835 33.26 .521 1786 32.83 .532 9246 32-43 544 53*5 32.06 25 1.509 4830 33-25 1.521 3756 32.82 1.533 "92 32.43 1.544 7238 32.05 26 .509 6825 33- 1 4 .521 5725 32.82 533 3 J 37 32-42 544 9*6i 32.04 27 .509 8819 33 24 .521 7694 32.81 533 5*2 32.42 545 ' 8 3 32.04 28 .510 0813 33,23 .521 9662 32.80 533 70 2 7 32.41 545 35 32.03 29 .510 2807 33.22 .522 1630 32.80 533 8 97i 32.40 545 4927 32.03 30 1.510 4800 33,21 1.522 3598 32-79 1.534 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 7531 32.78 534 4801 32.38 .546 0690 32.01 33 .511 0776 33-*9 .522 9498 32.78 534 6743 32-37 .546 2611 32.00 34 .511 2768 i3 .i8 .523 1464 3*-77 .534 8685 32.37 54 6 453 1 32.00 35 1.511 4759 33.18 1.523 3429 32.76 1.535 0627 32.36 1.546 6450 3*-99 36 .511 6749 33-*7 523 5394 32.75 535 2568 32-35 .546 8370 31.98 37 .511 8739 33.16 5*3 7359 32.74 555 459 32-35 .547 0289 31.98 38 .512 0729 33- I 5 5*3 93*3 32-73 535 6450 32-34 547 2207 31.97 39 .512 2718 33-15 .524 1287 32.73 535 8 39 32.33 547 4"5 3 J 97 40 1.512 4707 33-H 1.524 3251 32.72 1.536 0330 32.33 1.547 6043 31.96 41 .512 6695 33-!3 .524 5214 32.71 .536 2270 32.32 .547 7961 31.96 42 .512 8683 33-n .524 7176 32.71 .536 4209 32.32 547 9 8 7 8 31-95 43 .515 0670 33- 12 .524 9138 32.70 .536 6148 32.31 .548 1795 3i-94 44 .513 2657 33-" .525 noo 32.70 .536 8086 32.30 .548 3711 3'-94 45 1.513 4644 33.11 1.525 3062 32.69 1.537 0024 32.30 1.548 5627 31-93 46 s r .513 6630 33.10 .525 5023 32.68 537 i9 62 32.29 54 8 7543 3*-93 47 .513 8615 33.09 .525 6983 32.67 -537 3899 32.28 .548 9458 31.92 48 .514 0601 3308 .525 8944 32.67 537 5 8 3 6 32.28 549 H73 3*-9i 49 .514 2586 33-7 .526 0903 3266 537 7772 32.27 549 S 288 3 l '9 l 50 51 1.514 4570 .514 6554 33.07 33.06 1.526 2863 .526 4822 32.65 32.64 1.537 9708 .538 1644 32.26 32.26 1.549 5202 549 7"6 31.90 31.90 52 .514 8537 33-5 .526 6780 32.64 53 8 3579 32.25 549 93 31.89 53 .515 0520 33-5 .526 8739 32.63 538 55H 32.25 .550 0943 31.88 rt r> 54 5*5 2503 33-4 .527 0696 32.62 53 8 7449 32.24 .550 2856 31.88 55 1.515 4485 33-4 1.527 2654 32.62 i-53 8 93 8 3 32.23 1.550 4769 31.87 56 57 .515 6467 .515 8449 33-3 33.02 .527 4611 .527 6567 32.61 32.61 539 i3'7 539 325 32.23 32.22 .550 6681 .550 8593 31.87 31.86 58 .516 0430 33 oi .527 8524 32.60 539 5 l8 3 32.21 .551 050A 31.86 59 .516 2410 33.01 .528 0479 32.60 539 7" 6 32.21 .551 2416 3i- 8 5 60 1.516 4390 33.00 1.528 2435 3 2 -59 1.539 9048 32.20 1.551 4326 3I-85 577 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 48 49 50 51 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. l". 0' 1 1.551 4326 .551 6237 31.85 31.84 1.562 8360 .563 0250 HI! 1.574 1234 574 3 T 6 31.20 31.20 .585 4886 30.91 30.91 2 . 55 i 8i 4 7 3I-83 .563 2140 3 I -5 574 4977 3 I - I 9 585 6740 30.90 3 .552 0057 .563 4030 31.50 574 68 49 3 I - I 9 .585 8594 30.90 4 .552 1966 31.82 .563 5920 3M9 .574 8720 31.18 .586 0448 30.89 5 1.552 3876 31.82 1.563 7809 31.48 1.575 0590 31.18 1.586 2302 30.89 6 .552 5784 31.81 .563 9698 31.48 575 2461 31.17 .586 4155 30.89 7 .552 7693 31.80 .564 1586 3 '-47 575 4331 31.17 .586 6008 30.88 9 .552 9601 553 '5 08 31.80 5 6 4 3475 5 6 4 53 6 3 3M7 31.46 .575 6201 575 8070 31.16 31.16 .586 7859 .586 9713 30.87 30.87 10 1-553 34 l6 31.79 1.564 7250 31.46 J-575 9939 31.15 1.587 1565 30-87 ; 11 553 5323 3I-78 .564 9138 345 .576 1808 31.15 587 3417 30.86 12 553 723 31.78 .565 1025 3-45 .576 3677 3 I - I 4 .587 5268 30.86 13 553 9*3 6 3'-77 .565 2911 3i-44 .576 5546 3 I - I 4 .587 7120 30.85 14 .554 1042 31.76 .565 4798 3i-44 .576 7414 3 I - 1 3 .587 8971 30.85 15 1.554 2 94 8 31.76 1.565 6684 3i-43 1.576 9281 31.13 1.588 0821 30.84 16 554 4853 .565 8569 577 H49 31.12 .588 2672 30.84 17 554 6 75 31-75 .566 0455 31.42 -577 3 l6 31.12 .588 4522 30-83 18 554 8663 3 I -74 .566 2340 31.41 577 4 88 3 31.11 .588 6372 30.83 19 555 5 6 7 3'-74 ' .566 4225 3J-4 1 ' 577 6749 31.11 .588 8222 20 1.555 2472 3 J -73 1.566 6109 31.40 1.577 8615 31.10 1.589 0071 30.82 21 555 4375 .566 7993 31.40 .578 0481 31.10 .589 1920 30.82 22 555 6279 31.72 - .566 9877 31.39 57 8 2347 31.09 .589 3769 30.81 23 555 8182 31.71 .567 1761 3 r -39 .578 4213 31.09 .589 5618 30.81 24 .556 0084 31.71 : 5 6 7 3 6 44 31.38 .578 6078 31.08 .589 7466 30.80 25 1.556 1987 31.70 i 1.567 5527 31.38 1.578 7942 31.08 1.589 9314 30.80 26 .556 3888 31.70 .567 7409 3i-37 578 9807 31.07 .590 1162 30.79 27 55 6 579 31.69 .567 9291 3*-37 .579 1671 31.07 .590 3009 30-79 28 .556 7691 31.68 .568 1173 31.36 579 3535 31.06 59 4857 30.78 29 .556 9592 31.68 .568 3055 31.36 i 579 5399 31.06 59 6 74 30.78 30 i-557 H93 31.67 i 1.568 4936 31-35 1.579 7262 31.06 1.590 8550 30.78 31 557 3393 31.67 .568 6817 579 9 I2 5 3 I>O 5 .591 0397 3-77 32 557 5293 31.66 .568 8698 3'-34 .580 0988 31.04 .591 2243 30-77 33 557 7i93 31.66 .569 0579 3>-34 .580 2851 31.04 .591 4089 30.76 34 557 9092 3 I - 6 5 .569 2459 31-33 .580 4713 3 J -3 ,59i 5935 30.76 35 1.558 0991 3*- 6 5 1.569 4338 3 J -33 1.580 6575 31.03 1.591 7780 30-75 36 .558 2890 31.64 .569 6218 31.32 i .580 8436 .591 9625 30-75 37 558 4788 31.64 .569 8097 31.32 , .581 0298 31.02 .592 1470 30-75 38 .558 6686 31.63 .569 9976 .581 2159 31.02 592 33*5 30-74 39 558 8584 31.62 .570 1854 31.30 ( .581 4020 31.01 592 5'59 30.74 40 1.559 0482 31.62 i-570 3733 31.30 1.581 5.8*0 31.01 1.592 7003 30-73 41 559 2379 31.61 .570 5611 31.29 .581 7740 31.00 .592 8847 3-73 42 559 4275 31.61 .570 7488 31.29 .581 9600 31.00 .593 0690 30.72 43 559 6172 31.60 .570 9366 31.28 .582 1460 3-99 593 2534 30.72 44 .559 8068 31.60 .571 1243 31.28 .582 3319 30.99 593 4377 30-72 45 r -559 99 6 3 3'59 1.571 3119 31.28 1.582 5179 30.98 1.593 6219 30.71 46 .560 1859 3'-59 .571 4996 31.27 -582 7037 3098 .593 8062 30.71 47 5 6 3754 31.58 .571 6872 31.27 .582 8896 3-97 593 994 30.70 48 49 .560 5648 .560 7543 $*-57 3i-57 .571 8748 .572 0623 31.26 31.26 -583 754 .583 2612 30.97 30.96 594 1746 594 3588 30.70 30.69 50 1.560 9437 3 J -5 6 1.572 2499 31.25 1.583 4470 30.96 1.594 5429 30.69 51 .561 1331 31.56 572 4373 31.25 .583 6327 3-95 594 7270 30.68 52 .561 3224 3 J -55 .572 6248 31.24 .583 8184 3-95 594 9 111 30.68 53 .561 5117 355 .572 8123 31.24 .584 0041 3-94 595 952 30.68 54 .561 7010 3'-54 572 9997 31.23 .584 1898 30.94 595 2792 30.67 55 1.561 8902 3*-54 1.573 1870 31.23 1-584 3754 30.94 i-595 4633 30.67 56 57 58 .56* 0794 .562 2686 .562 4578 si-si J-53 573 3743 573 5 6 6 573 7489 31.22 31.22 31.21 .584 5610 .584 7466 584 93 21 3-93 30-93 30.92 595 6473 595 8312 .596 0151 30 66 30.66 30.65 59 .562 6469 S'-S* 573 93 6z 31.21 .585 1176 30.92 >$g6 1990 30.65 60 1.562 8360 M-S- 1.574 1234 31.20 i-5 8 5 303 1 30.91 1.596 3829 30.65 578 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 52 53 54 55 logM. Diff. 1". logM. Diff. 1". logM. Diff. I". logM. Diff. 1". O' 1.596 3829 30.65 1.607 3703 30.40 1.618 2724 30.17 1.629 959 29.96 1 .596 5668 30.64 .607 5527 30-39 .618 4534 30.17 .629 2757 29.96 2 .596 7506 30.64 .607 7350 3-39 .618 6344 30.16 .629 4554 29.96 3 59 6 9344 30.63 .607 9174 30-39 .618 8153 30.16 .629 6351 29.95 4 .597 1182 30.63 .608 0997 30.38 .618 9963 30.16 .629 8148 29.95 5 1.597 3020 30.62 1.608 2820 30.38 1.619 !77 Z 30.15 1.629 9945 29.95 6 7 597 4857 597 6694 30.62 30.62 .608 4642 .608 6465 30-38 30-37 .619 3581 .619 5390 3 -I5 30.15 .630 1742 .630 3538 29.94 29.94 8 597 8531 30.61 .608 8287 30-37 .619 7199 30.14 .630 5335 29.94 9 .598 0368 30.61 .609 0109 30.36 .619 9007 30-I4 .630 7131 29.93 10 1.598 2204 30.60 1.609 1931 30.36 1.620 0816 30.14 1.630 8927 29.93 11 .598 4040 30.60 .609 3752 3-3 6 .620 2623 30.13 .631 0722 29.93 12 .598 5876 30.59 .609 5573 30-35 .620 4431 30.I3 .631 2518 29.92 13 .598 7711 3-59 .609 7394 3-35 .620 6239 30.12 63 1 43 J 3 29.92 14 .598 9547 3-59 .609 9215 30-34 .620 8046 30.12 .631 6108 29.92 15 1.599 '382 30.58 1.610 1036 30-34 1.620 9853 30.12 1.631 7903 29.91 16 599 3217 30.58 .610 2856 3 .34 .621 1660 30.11 .631 9698 29.91 17 599 5051 3-57 .610 4676 30-33 .621 3467 30.11 .632 1492 29.91 18 .599 6885 3-57 .610 6496 30.33 .621 5274 30.11 .632 3286 29.90 19 599 8 7i9 3-57 .610 8315 30-32 .621 7080 30.10 .632 5081 29.90 20 i. 600 0553 30.56 1.611 0135 30.32 1.621 8886 30.10 1.632 6875 29.90 21 .600 2387 30.56 .611 1954 30.32 .622 0692 3P.IO .632 8668 29.89 22 .600 4220 3-55 .611 3773 30-3 1 .622 2497 30.09 .633 0462 29.89 23 .600 6053 30.55 .611 5591 3-3' .622 4307 3^-09 .633 22 5 C 29.89 24 .600 7886 30-55 .611 7410 30.31 .622 6108 30.09 .633 4048 29.88 25 1.600 9718 3-54 1.611 9228 30-3 1.622 7917 30.08 1.633 5841 29.88 26 .601 1551 3-54 .612 1046 30.30 .622 9718 30.08 .633 7634 29.88 27 .601 3383 3-53 .612 2864 30.29 .623 1523 30.08 .633 9427 29.87 28 .601 5214 3-53 .612 4681 30.29 .623 3327 3.0.07 .634 1219 29.87 29 .601 7046 30.52 .612 6499 30.29 -623 5131 30.07 .634 3011 29.87 30 1.601 8877 30.52 i. 612 8316 30.28 *-62 3 6935 30.06 1.634 4803 29.86 31 .602 0708 3 -52 .613 0132 30.28 .623 8739 30.06 -634 6 595 29.86 32 .602 2539 30-5 1 .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 0178 29.86 34 .602 6200 30.50 .613 5582 30.27 .624 4149 30,05 .635 1969 29.85 35 1.602 8030 30.50 1.613 7398 30.26 i.6z 4 5952 30.05 1.635 376o 29.85 36 .602 9860 30.50 .613 9213 30.26 , .624 7755 30.04 635 555i 29.85 37 38 39 .603 1690 .603 3519 .603 5348 30.49 30.49 30.48 .614 1029 .614 2844 .614 4659 30.26 30.25 30.25 .624 9557 .625 1360 .625 3162 30.04 30.04 30.03 .635 7342 .635 9132 .636 0922 29.84 29.84 29.84 40 1.603 777 30.48 1.614 6474 30.25 1.625 4964 30.03 1.636 2713 29.83 41 42 43 44 .603 9005 .604 0834 .604 2662 .604 4490 3-47 30.47 30.47 30.46 .614 8288 .615 0103 .615 1917 615 373 1 30.24 30.24 30-23 30-23 .625 6765 .625 8567 .626 0368 .626 2169 30.03 30.02 30.02 30.02 .636 4502 .636 6292 .636 8082 .636 9871 29.83 29.83 29.82 29.82 45 1.604 6317 30.46 1-615 5545 30.23 1.626 3970 30.01 1.637 1660 29.82 46 .604 8145 3-45 .615 7358 30.22 .626 5771 30.01 637 3449 29.82 47 48 .604 9972 .605 1799 3-45 3-45 .615 9171 .6i v 6 0984 30.22 30.22 .626 7571 .626 9372 30.01 30.00 .637 5238 .637 7027 29.81 29.81 49 .605 3626 3-44 , .616 2797 30.21 .627 1172 30.00 .637 8815 29.81 50 1.605 5452 3-44 1.616 4610 30.21 1.627 2972 30.00 1.638 0603 29.80 51 .605 7278 3-43 .616 6422 30.20 .627 4771 29.99 .638 2391 29.80 52 .605 9104 3-43 .616 8234 30.20 .627 6571 29.99 .638 4179 29.80 53 .606 0930 30.43 , .617 0046 30.20 .627 8370 29.99 .638 5967 29-79 54 .606 2755 30.42 .617 1858 30.19 .628 0169 29.98 .638 7754 29.79 55 56 57 58 59 i. 606 4581 .606 6406 .606 8230 .607 0055 .607 1879 30.42 3 -42 30.41 30.41 30.40 1.617 3669 .617 5481 .617 7292 .617 9102 .618 0913 30.19 30.19 30.18 30.18 30.17 1.628 1968 .628 3766 .628 5565 .628 7363 .628 9161 29.98 2998 29.97 29.97 29.97 1.638 9542 .639 1329 .639 3116 .639 4902 .639 6689 29.79 29.78 29.78 29.78 29.77 60 1.607 3703 30.40 1.618 2724 30.17 1.629 959 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. V. 56 57 58 59 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.639 8475 29.77 1.650 5336 49.60 1.661 1601 49.44 1.671 7331 4 93 o 1 .640 0262 29.77 .650 7112 49.60 .661 3368 49.44 .671 90^9 49.30 2 .640 2048 29.77 .650 8887 29.59 .661 5134 49.44 .674 0846 49.30 3 .640 3833 29.76 .651 0663 29.59 .661 6900 29.43 .674 4604 29.29 4 .640 5619 29.76 .651 4438 49.59 .661 8666 49.43 .672 4362 29.29 5 1.640 7405 29.76 1.651 4213 49.58 1.662 0432 29.43 1.672 6119 29.29 6 .640 9190 29.75 .651 5988 49.58 .662 2197 29-43 .672 7876 29.29 1 7 .641 0975 29.75 .651 7763 29.58 .662 3963 29.42 .672 9634 29.28 8 .641 2760 29.75 65 1 953 8 49.58 .662 5728 29.42 .673 1391 29.28 9 .641 4545 29.74 .652 1312 29.57 .662 7493 29.42 - 6 73 3*47 29.28 10 1.641 6329 29.74 1.652 3086 29-57 1.662 9258 49.42 1.673 4904 29.28 11 .641 8114 29.74 .652 4861 49.57 .663 1023 29.41 .673 6661 29.28 12 .641 9898 29.74 .652 6635 49.57 .663 2788 29.41 .673 8417 29.27 13 .642 1682 29.73 .652 8408 29.56 66 3 4553 29.41 .674 0174 29.27 14 .642 3466 29-73 .653 0182 29.56 .663 6317 49.41 .674 1930 29.47 15 1.642 5250 29.73 1.653 1956 49.56 1.663 8o8 2 49.40 1.674 3 686 49.47 16 .642 7033 29.72 .653 3729 29-55 .663 9846 49.40 .674 5442 49.47 17 .642 8816 29.72 .653 5502 29-55 .664 1610 49.40 .674 7198 29.26 18 .643 0599 29.72 .653 7275 29-55 .664 3374 49.40 .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 1.654 0821 49.54 1.664 6901 29-39 1.675 2465 29.26 21 .643 5948 29.71 .654 2593 49.54 .664 8664 29-39 .675 4220 2925 22 .643 7730 29.71 .654 4366 29.54 .665 0428 29-39 6 75 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 7910 29-53 .665 3954 49.38 .675 9485 29.25 25 1.644 377 29.70 1.654 9682 29-53 1.665 5717 49.38 1.676 1240 29.25 26 .644 4858 49:69 655 H54 29.53 .665 7480 49.38 .676 2995 29.24 27 .644 6640 29/69 6 55 3225 29-53 .665 9242 49.38 .676 4749 29.24 28 .644 8421 29.69 6 55 4997 49.54 .666 1005 49.37 .676 6504 29.24 29 .645 0203 29.69 .655 6768 29.52 .666 2767 29-37 .676 8258 29.24 30 1.645 J 9^4 29.68 *- 6 55 ^539 49.54 1.666 4529 29-37 1.677 0012 29.24 31 645 3765 29.68 .656 0310 49.51 .666 6291 29-37 .677 1766 29.23 32 645 5545 29.68 .656 2081 4951 .666 8053 49.36 .677 3520 29-23 33 .645 7326 29.67 .656 3852 49.51 .666 9815 29.36 .677 5274 29.23 34 .645 9106 29,67 .656 5644 49.51 .667 1577 2936 .677 7028 29.23 35 1.646 0886 49.67 1.656 7394 49.50 1.667 3338 29.36 1.677 8 7 8 I 29.23 36 .646 2 6 #6 29.67 .656 9163 49.50 .667 5100 29-35 .678 0535 29.22 37 .646 4446 2,9.66 *57 933 29.50 .667 6861 29-35 .678 2488 29.22 38 .646 6226 29.66 .657 4703 29.50 .667 8622 29-35 .678 4041 29.22 39 .646 8005 49.66 .657 4474 29.49 .668 0383 29-35 6 7 8 5794 29.22 40 1.646 9785 29.65 1.657 6242 49.49 1.668 2144 29-35 1.678 7547 29.22 41 .647 1564 29.65 .657 8011 49.49 .668 3904 29.34 .678 9300 29.21 42 6 47 3343 29.65 .657 9781 29.49 .668 5665 29.34 .679 1051 29.21 43 .647 5122 29.65 .658 '1550 29.48 .668 7425 29-34 .679 2806 29.21 44 .647 6900 29.64 .658 3318 49.48 .668 9185 29.34 .679 4558 29.21 45 1.647 8679 29.64 1.658 5087 49.48 1.669 945 29-33 1.679 6310 29.20 46 .648 0457 29.64 .658 6855 49.48 .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 49.47 .669 6225 29-33 .680 1567 29.20 49 .648 5791 29.63 .659 2161 49.47 .669 7984 29.32 .680 3319 29.20 50 1.648 7569 29.63 1.659 3929 49.47 1.669 9744 29.32 i. 680 5070 29.19 51 .648 9346 29.62 .659 5697 49.46 .670 1503 29.32 .680 6822 29.19 52 .649 1123 29.62 .659 7465 49.46 .670 3262 29.32 .680 8574 29.19 53 .649 2901 29.62 .659 9232 49.46 .670 5021 29.32 .681 0325 29.19 54 .649 4677 29.61 .660 1000 49.46 .670 6780 29.31 .681 2076 29.19 55 1.649 6454 29.61 1. 660 2767 49.45 1.670 8539 29.31 1.681 3827 29.18 56 .649 8231 29.61 .660 4534 49.45 .671 0298 49.31 .681 5578 29.18 57 .650 0007 29.61 .660 6301 49.45 .671 2056 49.31 .681 7329 29.18 58 .650 1784 29.60 .660 8068 49.45 .671 3814 49.30 .681 9080 29.18 59 .650 3560 29.60 .660 9835 29.44 671 5573 49.30 .682 0831 29.18 60 1.650 5336 29.60 1.661 1601 49.44 1.671 7331 49.30 1.684 4581 29.17 580 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 60 61 62 63 log M. Diff. 1". logM Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 1.682 2581 29.17 1.692 7408 29.07 1.703 1866 28.97 1.713 6006 28.89 1 2 .682. 4332, .682 6082 29.17 29.17 .692 9152 .693 0896 29.06 29.06 .703 3604 .703 5342 28.97 28.97 713 7739 713 9473 28.89 28.89 3 .682 7832 29.17 .693 2640 29.06 .703 7080 28.97 .714 1206 28.88 4 .682 9582 29.17 6 93 43 8 3 29.06 .703 8818 28.96 .714 2939 28.88 5 1.683 1332 29.16 1.693 6127 29.06 1.704 0556 28.96 1.714 4672 28.88 6 .683 3082 29.16 .693 7870 29.05 .704 2293 28.96 .714 6405 28.88 7 .683 4832 29.16 .693 9613 29.05 .704 4031 28.96 .714 8138 28.88 8 .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 .715 1603 28.88 10 1.684 0080 29.16 1.694 4842 29.05 1.704 9243 28.96 1.715 3336 28.88 11 .684 1830 29.15 .694 6585 29.04 .705 0981 28.95 .715 5068 28.88 12 .684 3579 29.15 .694 8328 29.04 .705 2718 28.95 .715 6801 28.87 13 .684 5328 29.15 .695 0070 2904 75 4455 28.95 7*5 8 533 28.87 14 .684 7077 29.15 .695 1813 29.04 .705 6192 28.95 .716 0266 28.87 15 1.684 8826 29.14 ' 6 95 3555 29.04 1.705 7929 28.95 1.716 1998 28.87 16 .685 0574 29.14 .695 5298 29.04 .705 9666 28.95 .716 3730 28.87 17 .685 2323 29.14 .695 7040 29.04 .706 1402 28.95 .716 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 .716 8926 28.87 20 1.685 7568 29.14 1.696 2266 29.03 1.706 6612 28.94 1.717 0658 28.86 21 .685 9316 29.13 .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 4122 28.86 23 .686 2812 29.13 .696 7491 29.03 .707 1821 28.94 7 X 7 5 8 53 28.86 24 .686 4560 29.13 .696 9233 29.02 707 3557 28.94 .717 7585 28.86 25 1.686 6308 29.13 1.697 0974 29.02 1.707 5293 28.93 1.717 9317 28.86 26 .686 8055 29.13 .697 2716 29.02 .707 7029 28.93 .718 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 1.697 9680 29.02 1.708 3972 28.93 1.718 7974 28.85 31 .687 6791 29.12 .698 1421 29.01 .708 5708 2893 .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 9179 28.92 .719 3167 28.85 34 .688 2032 29.11 .698 6643 29.01 .709 0914 28.92 .719 4898 28.85 35 1.688 3778 29.11 1.698 8383 29.01 1.709 2650 28.92 1.719 6629 28.85 36 .688 5525 29.11 .699 0124 29.01 .709 4385 28.92 .719 8360 28.85 37 .688 7271 29.10 .699 1864 29.00 .709 6120 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 1.689 2510 29.10 1.699 7085 29.00 1.710 1325 28.91 1.720 5282 28.84 41 .689 4256 29.10 .699 8824 29.00 .710 3060 28.91 .720 7013 28.84 42 .689 6001 29.09 .700 0564 29.00 .710 4794 28.91 .720 8743 28.84 43 .689 7747 29.09 .700 2304 29.00 .710 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 1.690 1238 29.09 1.700 5783 28.99 1.710 9998 28.91 1.721 3934 28.84 46 47 .690 2984 .690 4729 29.09 29.09 .700 7523 .700 9262 28.99 28.99 .711 1732 .711 3467 28.91 28.90 .721 5665 .721 7395 28.84 28.84 48 .690 6474 29.09 .701 looi 28.99 .711 5201 28.90 .721 9125 28.83 49 .690 8219 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 .701 6219 28.98 .712 0403 28.90 .722 4315 28.83 52 .691 3454 29.08 .701 7958 28.98 .712 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 .691 6943 29.08 .702 1435 28.98 .712 5605 28.90 .722 9504 28.83 55 1.691 8688 29.07 1.702 3174 28.98 1.712 7339 28.90 1.723 1233 2 ^ 3 56 .692 0432 29.07 .702 4913 28.98 .712 9072 28.89 .723 2963 28.83 57 .692 2176 29.07 .702 6651 28.97 .713 0806 28.89 .723 4693 28.82 58 .692 3920 29.07 .702 8389 28.97 713 2 539 28.89 .723 6422 28.82 59 .692 5664 29.07 .703 0128 28.97 713 4*73 28.89 .723 8151 28.82 60 1.692 7408 29.07 1.703 1866 22.97 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. V. 64 65 66 67 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.723 9881 28.82 1-734 3539 28.77 1.744 731 28.73 1-755 45 28.70 1 .724 1610 28.82 734 5265 28.77 744 8755 28.73 755 2127 28.70 2 724 3339 28.82 734 6 99i 28.77 745 479 28.73 755 3849 28.70 3 .724 5068 28.82 734 8718 28.77 .745 2202 28.73 755 557i 28.70 4 .724 6798 28.82 735 444 28.77 745 39 26 28.73 755 7293 28.70 5 1.724 8527 28.82 1.735 2169 28.76 1.745 5 6 5o 28.73 1.755 9 i5 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 28.76 745 997 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 28.81 735 973 28.76 .746 2544 28.72 756 593 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 75 6 9347 28.70 12 .726 0628 28.81 .736 4250 28.76 .746 7714 28.72 757 1069 28.70 13 .726 2357 28.81 73 6 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 4513 28.70 15 1.726 5814 28.81 1.736 9426 28.76 1.747 2884 28.72 1.757 6235 28.70 16 .726 7542 28.81 737 "5 2 28.76 .747 4607 28.72 757 7957 28.70 17 .726 9270 28.81 737 2877 28.76 747 6330 28.72 757 9 6 79 28.70 18 .727 0999 28.80 .737 4602 28.76 .747 8054 28.72 .758 1401 28.70 19 .727 2727 28.80 .737 6328 28.75 747 9777 28.72 .758 3123 28.70 20 1-727 4455 28.80 1.737 8053 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 7911 28.80 738 i53 28.75 .748 4946 28.72 .758 8288 28.70 23 .727 9639 28.80 .738 3228 28.75 .748 6669 28.72 .759 ooio 28.70 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 "5 28.72 '759 3453 28.70 26 27 .728 4823 .728 6551 28.80 28.80 .738 8404 .739 0129 28.75 28.75 749 l8 3 8 749 35 61 28.72 28.72 759 5175 759 6897 28.70 28.70 28 .728 8279 28.80 739 1857 28.75 749 5284 28.72 .759 8618 28.69 29 .729 0006 28.79 739 3578 28.75 749 707 28.71 .760 0340 28.69 30 1.729 1734 28.79 1-739 533 28.75 1-749 873 28.71 1.760 2062 28.69 31 .729 3461 28.79 739 7 28 28.75 75 453 28.71 .760 3783 28.69 32 .729 5189 28.79 739 8753 28.75 .750 2176 28.71 .760 5505 28.69 33 .729 6916 28.79 .740 0477 28.75 75 3898 28.71 .760 7227 28.69 34 .729 8644 28.79 .740 2202 28.74 .750 5621 28.71 .760 8948 28.69 35 1.730 0371 28.79 1-74 3927 28.74 1.750 7344 28.71 1.761 0670 28.69 36 .730 2099 28.79 74 5 6 5i 28.74 .750 9067 28.71 .761 2392 28.69 37 .730 3826 28.79 .740 7376 28.74 .751 0789 28.71 .761 4113 28.69 38 73 5553 28.79 .740 9101 28.74 .751 2512 28.71 .761 5835 28.69 39 .730 7280 28.79 .741 0825 28.74 751 4234 28.71 .761 7556 28.69 40 1.730 9007 28.78 1.741 2550 28.74 1-75' 5957 28.71 1.761 9278 28.69 41 .731 0735 28.78 .741 4274 28.74 .751 7680 28.71 .762 0999 28.69 42 .731 2462 28.78 .741 5998 28-74 .751 9402 28.71 .762 2721 28.69 43 .731 4189 28.78 .741 7723 28.74 .752 1125 28.71 .762 4442 28.69 44 73 1 5915 28.78 74 1 9447 28.74 .752 2847 28.71 .762 6164 28.69 45 1.731 7642 28.78 1.742 1171 28.74 1.752 4570 28.71 1.762 7885 28.69 I 46 .731 9369 28.78 .742 2896 28.74 .752 6292 28.71 .762 9607 28.69 47 .732 1096 28.78 .742 4620 28.74 .752 8015 18.71 .763 1328 28.69 48 .732 2823 28.78 .742 6344 28.74 752 9737 a8. 7 i .763 3050 28.69 49 732 4549 28.78 .742 8068 28.74 753 H 6 28.71 .763 4771 28.69 50 1.732 6276 28.78 1.742 9792 28.74 1-753 3 l8 2 28.71 1.763 6493 28.69 51 .732 8002 28.78 743 *5i6 28.73 753 494 28.71 .763 8214 28.69 1 52 .732 9729 28.77 743 324 28.73 753 66 27 28.71 .763 9936 28.69 53 733 "455 28.77 743 49 6 4 28.73 753 8349 28.71 .764 1657 28.69 54 733 3'82 28.77 .743 6688 28.73 754 7i 28.70 7 6 4 3379 28.69 55 1-733 49 8 28.77 1.743 8412 28.73 1-754 1794 28.70 1.764 5100 28.69 56 733 6635 28.77 .744 0136 28.73 754 35 l6 28.70 .764 6821 28.69 57 58 733 8 36i 734 8 7 28.77 28.77 .744 1860 .7^4 3584 28.73 28.73 754 5238 .754 6960 28.70 28.70 .764 8543 .765 0264 28.69 28.69 59 734 l8l 3 28.77 744 53 8 28.73 .754 8682 28.70 .765 1985 28.69 60 1-734 3539 28.77 1.744 7 3! 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. 68 69 70 71 logM. Diflf. 1". logM. Diflf. 1". logM. Diflf. 1". logM. Diflf. 1". 1.765 3707 28.69 775 6985 28.69 1.786 0284 28.70 1.796 3650 28.73 1 .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 8821 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 .776 5591 28.69 1.786 8894 28.70 1.797 2268 28.73 6 .766 4035 28.69 .776 7313 28.69 .787 0617 28.70 797 399 2 28.73 7 .766 5756 28.69 .776 9034 28.69 787 2339 28.70 797 576 28.73 8 .766 7478 28.69 777 0755 28.69 .787 4061 28.70 797 744 28.73 9 .766 9199 28.69 777 *477 28.69 787 5783 28.70 797 9 l6 4 28.73 10 1.767 0920 28.69 1.777 4198 28.69 1.787 7506 28.70 1.798 0888 28.73 i 11 .767 2642 28.69 777 59 20 28.69 .787 9228 28.71 .798 2611 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 93 6 3 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 1.778 2806 28.69 1.788 6117 28.71 1.798 9508 28.73 16 .768 1248 28.69 .778 4527 28.69 .788 7840 28.71 .799 1232 28.74 17 .768 2969 28.69 .778 6248 28.69 .788 9562 28.71 799 2 95 6 28.74 18 .768 4691 28.69 .778 7970 28.69 .789 1284 28.71 .799 4680 28.74 19 .768 6412 28.69 778 9 6 9' 28.69 .789 3007 28.71 .799 6404 28.74 20 I. 7 68 8133 28.69 1.779 1413 28.69 1.789 4730 28.71 1.799 8128 28.74 21 .768 9854 28.69 779 3*4 28.69 .789 6452 28.71 799 9853 28.74 22 .769 1576 28.69 .779 4862 28.69 789 8175 28.71 .800 1577 28.74 23 .769 3297 28.69 779 6 57 8 28.69 789 9897 28.71 .800 3301 28.74 24 .769 5018 28.69 779 82 99 28.69 .790 1620 28.71 .800 5026 28-74 25 26 1.769 6740 .769 8461 28.69 28.69 1.780 OO2I .780 1742 28.69 28.69 1.790 3342 .790 5065 28.71 28.71 1.800 6750 .800 8475 28.74 28.74 27 .770 0182 28.69 .780 3464 28.69 .790 6788 28.71 .801 0199 28.74 28 .770 1903 28.69 .780 5185 28.69 79 8510 28.71 .801 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 1.791 1956 28.71 1.801 5373 28.74 31 .770 7067 28.69 .781 0350 28.69 .791 3678 28.71 .801 7107 28.74 32 .770 8788 28.69 .781 2072 28.69 .791 5401 28.71 .801 8822 28.74 33 .771 0510 28.69 .781 3793 28.69 .791 7124 28.71 .802 0547 28.75 34 .771 2231 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 37 .771 5673 77* 7395 28.69 28.69 .781 8959 .782 0680 28.69 28.70 .792 2293 .792 4016 28.71 28.72 .802 5721 .802 7446 28.75 28.75 38 .771 9116 28.69 .782 2402 28.70 .792 5738 28.72 .802 9171 28.75 39 .772 0837 28.69 .782 4124 28.70 .792 7461 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 97 28.72 .803 4346 28.75 42 .772 6001 28 69 .782 9289 28.70 793 26 3 2.8.72 .803 6071 28.75 43 .772 7722 28.69 .783 ion 28.70 793 4354 28.72 .803 7796 28.75 44 .772 9444 28.69 .783 2732 28.70 793 6 77 28.72 .803 9521 28.75 45 1.773 1165 28.69 1-783 4454 28.70 1.793 7800 28.72 1.804 1246 28.75 46 .773 2886 28.69 .783 6176 28.70 793 95 2 3 28.72 .804 2971 28.75 47 1 48 773 4607 773 6329 28.69 28.69 .783 7898 .783 9620 28.70 28.70 .794 1246 794 2 9 6 9 28.72 28.72 .804 4697 .804 6422 28.75 28.76 49 773 8 5 28.69 .784 1342 28.70 .794 4693 28.72 .804 8147 28.76 50 51 1773 977 1 .774 1493 28.69 28.69 1.784 3064 .784 4786 28.70 28.70 1.794 6416 794 8139 28.72 28.72 1.804 9877 .805 1598 28.76 28.76 52 53 .774 3214 774 4935 28.69 28.69 .784 6508 .784 8230 28.70 28.70 794 9862 795 1586 28.72 28.72 -805 3324 805 5049 28.76 28.76 54 774 66 57 28.69 .784 9952 28.70 795 339 28.72 805 6775 28.76 55 56 1-774 8 37 8 .775 0099 28.69 28.69 1.785 1674 785 339 6 28.70 28.70 1-795 533 795 6 75 6 28.72 28.72 1.805 8500 .806 0226 28.76 28.76 57 .775 1821 28.69 .785 5118 28.70 .795 8480 28.72 .806 1952 28.76 58 775 354 2 28.69 .785 6840 28.70 .796 0203 28.73 .806 3677 28.76 1 59 775 5 26 3 28.69 .785 8562 28.70 .796 1927 28.73 .806 5403 28.76 60 1.775 6 9 8 5 28.69 1.786 0284 28.70 1.796 3650 28.73 i. 806 7129 28.76 583 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 72 73 74 75 log It, Difif. 1". logM. Diff. 1". logM. Diff. 1". logM. 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 28.81 .827 6335 28.88 .838 0423 28.95 2 .807 0581 28.77 .817 4222 28.82 .827 8068 28.88 .838 2160 28.95 3 .807 2307 28.77 .817 5951 28.82 .827 9800 28.88 .838 3898 28.95 4 .807 4033 28.77 .817 7680 28.82 .828 1533 28.88 8 3 8 5635 28.96 5 ' 8 7 5759 28.77 1.817 94 10 28.82 1.828 3266 28.88 1.838 7372 28.96 6 .807 7485 28.77 .818 1139 28.82 .828 4999 28.88 .838 9110 28.96 7 .807 9211 28.77 .818 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 .808 2663 28.77 .818 6326 28.82 .829 0198 28.89 .839 4323 28.96 , 10 i. 808 4389 28.77 1.818 8056 28.82 1.829 1931 28.89 1.839 6060 28.96 11 .808 6116 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 7131 28.89 .8^0 1274 28.97 14 .809 1295 28.77 .819 4974 28.83 .829 8865 28.89 .840 3012 28.97 15 1.809 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 8201 28.78 .820 1893 28.83 .830 5800 28.90 .840 9966 28.97 19 .809 9928 28.78 .820 3623 28.83 8 3 7533 28.90 .841 1704 28.98 20 1.810 1655 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 .831 looi 28.90 .841 5182 28.98 22 .810 5108 28.78 .820 8813 28.84 8 3 ! 2 735 28.90 .841 6921 28.98 23 .810 6835 28.78 .821 0543 28.84 .831 4470 28.90 .841 8659 28.98 24 .810 8562 28.78 .821 2273 28.84 .831 6204 28.90 .842 0398 28.98 25 1.811 0289 28.78 1.821 4003 28.84 I-83 1 793 8 28.91 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 .821 7464 28.84 .832 1407 28.91 842 5616 28.99 28 .811 5470 28.79 .821 9194 28.84 .832 3141 28.91 8 4* 7355 28.99 29 .811 7197 28.79 .822 0925 28.84 .832 4876 28.91 .842 9095 28.99 30 1.811 8924 28.79 1.822 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 6117 28.85 .833 0080 28.92 8 43 43*3 29.00 33 .812 4106 28.79 .822 7848 28.85 .833 1815 28.92 .843 6053 29.00 34 .812 5834 28.79 .822 9578 28.85 8 33 355 28.92 8 43 7793 29.00 35 i. 812 7561 28.79 1.823 1309 28.85 1.833 5285 28.92 i- 8 43 9533 29.00 36 .812 9289 28.79 .823 3040 28.85 .833 7020 28.92 .844 1273 29.00 37 .813 1016 28.79 .823 4771 28.85 8 33 8 755 28.92 .844 3013 29.00 38 .813 2744 28.79 .823 6502 28.85 .834 0491 28.92 8 44 4753 29.00 39 .813 4472 28.79 .823 8233 28.85 .834 2226 28.92 .844 6494 29.01 40 1.813 6199 28.80 1.823 99 6 5 28.85 1.834 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 43 8i3 9655 .814 1383 28.80 28.80 .824 3427 .824 5159 28.86 28.86 8 34 743 2 .834 9168 2893 28.93 .845 1715 .845 3456 29.01 29.01 44 .814 3111 28.80 .824 6890 28.86 .835 0904 28.93 .845 5196 29.01 1 45 1.814 4 8 39 28.80 1.824 8622 28.86 1.835 2640 28.93 1.845 6937 29.01 46 .814 6567 28.80 82 5 353 28.86 .835 4376 28.93 .845 8678 29.02 47 .814 8295 28.80 .825 2085 28.86 .835 6112 28.93 .846 0419 29.02 48 .815 0023 28.80 .825 3816 28.86 .835 7848 28.93 .846 2160 29.02 49 .815 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 C2 51 .815 5208 28.81 .825 9012 28.87 .836 3056 28.94 .846 7384 2O O2 52 .815 6936 28.81 .826 0744 28.87 .836 4792 28.94 .846 9125 29.03 53 .815 8664 28.81 .826 2476 28.87 .836 6529 28.94 .847 0867 29.03 54 .816 0393 28.81 .826 4208 28.87 .836 8265 28.94 .847 2609 29.03 55 1.816 2121 28.81 1.826 5940 28.87 1.837 0002 28.94 1.847 4350 29.03 56 .816 3850 28.81 .826 7673 28.87 .837 1739 28.95 .847 6092 29.03 57 58 .816 5578 .816 7307 28.81 28.81 .826 9405 .827 1137 28.87 28.87 8 37 3475 .837 5212 28.95 28.95 .847 7834 .847 9576 29.03 29.03 59 .816 9036 28.81 .827 2870 28.87 .837 6949 28.95 .848 1318 29.04 60 1.817 0765 a8.8i 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 76 77 78 79 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.848 3060 29.04 1.858 7769 29.14 1.869 2857 29.25 1.879 8369 29-37 1 .848 4803 29.04 858 9517 29.14 .869 4612 29.25 .880 0131 29.37 2 .848 6545 29.04 .859 1266 29.14 .869 6367 29.25 .880 1894 29.38 3 4 .848 8287 .849 0030 29.04 29.04 8 59 3 OI 4 859 4763 29.14 29.15 .869 8122 .869 9878 29.25 29.26 .880 3656 .880 5419 29.38 29.38 5 1-849 '773 29.04 1.859 6512 29.15 1.870 1633 29.26 1. 880 7182 29.38 6 849 35*5 29.05 .859 8260 29.15 .870 3389 29.26 .88c 8945 29.38 7 8 .849 5258 .849 7001 29.05 29.05 .860 0009 .860 1758 29-I5 29.15 .870 5144 .870 6900 29.26 29.26 .881 0708 .881 2471 29-39 1 29.39 1 9 .849 8744 29.05 .860 3507 29.15 .870 8656 29.26 .881 4235 29-39 10 1.850 0487 29.05 i. 860 5256 29.15 1.871 0412 29.27 1.881 5998 29.39 11 .850 2231 29.05 .860 7006 29.16 .871 2168 29.27 .881 7762 29-39 12 .850 3974 29.06 .860 8755 29.16 .871 3924 29.27 .881 9526 29.40 13 .850 5717 29.06 .861 0505 29.16 .871 5681 29.27 .882 1290 29.40 14 .850 7461 29.06 .861 2254 29.16 .871 7437 29.28 .882 3054 29.40 15 1.850 9204 29.06 1.861 4004 29.16 1.871 9194 29.28 1.882 4818 29.40 16 .851 0948 29.06 .861 5754 29.16 .872 0950 29.28 .882 6582 29.41 17 .851 2692 29.06 .861 7504 29.17 .872 2707 29.28 .882 8347 29.41 18 .851 4436 29.07 .861 9254 29.17 .872 4464 29.28 .883 0112 29.41 19 .851 6180 29.07 .862 1004 29.17 .872 6221 29.29 .883 1876 29.41 20 1.851 7924 29.07 1.862 2754 29.17 1.872 7979 29.29 1.883 3641 29.42 21 .851 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 "493 29.29 .883 7171 29.42 23 .852 3157 29.07 .862 8006 29.18 .873 3251 29.29 .883 8937 29.42 24 .852 4901 29.07 .862 9756 29.18 .873 5008 29.30 .884 0702 29.42 , 25 1.852 6646 29.08 1.863 1507 29.18 1.873 6 7 66 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 OI 35 29.08 .863 5009 29.18 .874 0282 29.30 .884 5999 29.43 28 .853 1880 29.08 .863 6760 29.19 .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 I-853 537 29.09 1.864 0263 29.19 1-874 5557 29.31 1.885 1297 29.44 31 853 7^5 29.09 .864 2015 29.19 .874 7316 29.31 .885 3064 29.44 32 .853 8861 29.09 .864 3766 29.19 .874 9074 29.31 .885 4830 29.44 33 .854 0606 29.09 .864 5518 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 1-875 435^ 29.32 1.886 0131 29.45 36 854 5843 29.10 .865 0774 29.20 .875 6m 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 29.10 .865 4278 29.20 .875 9629 29.32 .886 5432 29.46 39 .855 1080 29.10 .865 6030 29.21 .876 1389 29.33 .886 7200 29.46 40 41 1.855 2826 .855 4572 29.10 29.10 1.865 7783 .865 9536 29.21 29.21 1.876 3148 .876 4908 29-33 29.33 1.886 8967 .887 0735 29.46 29.46 42 .855 6319 29.11 .866 1288 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 0188 29-34 .887 6039 29.47 45 1.856 1558 29.11 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 29.11 .867 0054 29.22 .877 5470 29-34 .888 1344 29.48 48 .856 6799 29.12 .867 1807 29.22 .877 7230 29-34 .888 3113 29.48 I 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 1.888 6651 29.48 51 .857 2040 29.12 .867 7068 29.23 .878 2513 29.35 .888 8420 29-49 52 857 3787 29.12 .867 8822 29.23 .878 4275 29-35 .889 0189 29.49 53 857 5534 29.12 .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 56 1.857 9030 858 0777 29.13 29.13 1.868 4084 .868 5839 29.24 29.24 1.878 9559 879 1321 29.36 29.36 1.889 5498 .889 7268 29-49 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 6021 29.13 .869 1102 29.25 .879 6606 29-37 .890 2578 29.51 60 1.858 7769 29.14 1.869 2857 29.25 1.879 8369 29-37 1.890 4349 293 32.01 31 .983 2411 31.19 994 5' 61 31-45 .005 8878 3 I -73 .017 3614 32.02 32 33 .983 4283 .983 6155 31.19 31.20 994 7048 994 8 93 6 31.46 31.46 .006 0781 .006 2685 31-73 3*-74 -017 5535 .017 7456 32.02 32.03 34 .983 8027 31.20 995 8i 3 31.46 .006 4590 3!-74 .017 9378 3*-3 35 1.983 9899 31.21 1.995 2711 3M7 2.006 6494 31-75 2.018 1300 32.04 36 .984 1772 31.21 .995 4600 31-47 .006 8399 31-75 .018 3223 32.04 37 .984 3644 31.22 .995 6488 31.48 .007 0304 31.76 .018 5145 32.05 38 .984 5517 31.22 995 8 377 31.48 .007 22 I O 31.76 .018 7068 32.05 39 .984 7391 31.22 .996 0266 31.49 .007 4116 3-77 .018 8992 32.06 40 1.984 9264 31.23 1.996 2155 3 J -49 2.007 6022 3'-77 2.019 9 I 5 32.06 41 .985 1138 31.23 .996 4045 3!-5 .007 7928 3'-77 .019 2839 32.07 42 .985 3012 31.24 .996 5935 3 l -S Q .007 9835 3I-7 8 .019 4763 32.07 43 .985 4886 31.24 .996 7825 31.51 .008 1742 31.78 .019 6688 32.08 44 .985 6761 31.24 .996 9716 3!'5i .008 3649 31.79 .019 8613 32.08 : 45 1.985 8636 31.25 1.997 1606 31.51 2.008 5556 31.79 2.020 0538 32.09 46 .986 0511 3 r - 2 5 997 3497 31.52 .008 7464 31.80 .020 2463 32.09 47 .986 2386 31.26 997 53 8 9 3 T -5 2 .008 9372 31.80 .020 4389 32.10 48 .986 4262 31.26 .997 7280 3 ! -53 .009 I28o 31.81 .O2O 6315 32.10 49 .986 6138 31.27 997 9 J 7* 3'-53 .009 3189 31.81 .020 8241 32.11 50 1.986 8014 31.27 1.998 1064 31-54 2.009 59 8 31.82 2.021 Ol68 32.11 51 .986 9890 31.28 .998 2956 3'-54 .009 7007 31.82 .021 2095 32.12 52 .987 1767 31.28 .998 4849 3'-55 .009 8917 3i- 8 3 .021 4022 32.12 53 .987 3644 31.28 .998 6742 3'-55 .010 0826 31.83 .021 5949 32.13 54 .987 5521 31.29 .998 8635 3'-5 6 .010 1736 31.84 .021 7877 3*-3 55 1.987 7398 31.29 1.999 0529 3i-5f 2.OIO 4647 31.84 2.021 9805 32.14 56 .987 9276 31.30 .999 2422 31.56 .OIO 6557 31.85 .022 1734 32.14 57 .988 1154 31.30 999 43' 6 31-57 .010 8468 31-85 .022 3662 32.15 58 .988 3032 31.31 .999 6211 31-57 .on 0380 31.86 .022 5591 J*.i| 59 .988 4911 31.31 999 8l 5 3'-5 8 .on 2291 31.86 .022 7521 32.16 60 1.988 6789 S 1 ^ 1 2.000 0000 31.58 2.01 1 4103 31.87 2.O22 945O 32.16 -^J 588 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 92 93 94 95 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". 1 gM. Diff. 1". 0' .022 9450 32.16 1-034 5797 32.48 2.046 3296 32.80 2.058 2005 33-15 1 .023 1380 32.17 34 7745 32.48 .046 5264 32.81 .058 3994 33-15 2 .023 33 II 32.17 .034 9694 32-49 .046 7233 32.82 .058 5983 33-i6 3 .023 5241 32.18 .035 1644 32.49 .046 9202 32.82 .058 7973 33.16 4 .023 7172 32.18 035 3593 32.50 .047 1172 32.83 .058 9963 31-17 5 2.023 9103 32.19 035 5543 32.50 2.047 3141 32.83 2.059 1953 33-i8 6 .024 1035 32.19 .035 7494 3^-51 .047 5111 32-84 59 3944 33-i8 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 33-19 9 .024 6831 32.21 .036 3347 32.52 .048 1024 32.85 .059 9919 33-20 10 2.024 8764 32.21 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 .060 3904 33-21 12 .025 2630 32.22 .036 9202 32.54 .048 6939 32-87 .060 5897 33.22 13 .025 4564 32.23 .037 1155 32-54 .048 8912 32.88 .060 7890 33-22 14 .025 6498 32.23 .037 3108 32.55 .049 0884 32.88 .060 9884 33-23 15 2.025 8432 32.24 2.037 5061 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 .061 3872 33-24 17 .026 2301 32.25 .037 8969 32.57 .049 6805 32.90 .061 5867 33-25 18 .026 4236 32.26 .038 0923 32.57 .049 8879 32.90 .061 7862 33-25 19 .026 6172 32.26 .038 2877 32-58 .050 0753 32.91 .061 9857 33.26 20 2.026 8lo8 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 .039 2655 32.61 .051 0631 32.94 .062 9840 33-29 25 2.027 7791 32.29 2.039 4611 32.61 2.051 2608 32.95 2.063 1837 33-3 26 27 .027 9729 .028 1667 323 32.3 .039 6568 .039 8525 32.62 32.62 .051 4585 .051 6562 32.95 32.96 .063 3835 .063 5833 33-3 33-31 28 .028 3605 32-31 .040 0482 32.63 .051 8539 32.96 .063 7832 33-31 29 .028 5544 32.31 .040 2440 32.63 .052 0517 32.97 .063 9831 33.32 30 2.028 7483 32-3* 2.040 4399 32.64 2.052 2496 32.97 2.064 l8 3i 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 8316 32.65 .052 6453 32.98 .064 5830 33-34 33 .029 3301 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 2.041 4194 32.67 2053 2392 33.00 2.065 l8 33 33-3 6 36 37 .029 9123 .030 1064 32.35 32.35 .041 6154 .041 8114 32.67 32.68 53 4372 .053 6353 33-oi 33.01 .065 3834 .065 5836 33-36 33-37 38 .030 3005 32.36 .042 0075 32.68 53 8 334 33.02 .065 7839 33-37 39 .030 4947 32.36 .042 2036 32.69 .054 0315 33-3 .065 9841 33-38 40 2.030 6889 32-37 2.042 3998 32.69 2.054 2297 33-03 2.066 1844 33-39 41 .030 8831 32-37 .042 5960 32.70 .054 4279 33-4 .066 3847 33-39 42 .031 0774 32.38 .042 7922 32.70 .054 6262 33-4 .066 5851 33-4 43 .031 2717 32.39 .042 9834 32.71 .054 8244 33-05 .066 7855 33-40 44 .031 4660 32.39 .043 1847 32.71 .055 0227 33-5 .066 9860 33.41 45 2.031 6604 32.40 2.043 3810 32.72 2.055 221 1 33.06 2.067 1865 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 9701 32.74 .055 8l6l 33.08 .067 7881 33-43 49 .032 4382 32.42 .044 1665 32.74 .056 0148 33.08 .067 9887 33-44 50 2.032 6327 32.42 2.044 3630 32-75 2.056 2133 33-9 2.068 1894 33-45 51 52 53 .032 8272 .033 0218 .033 2164 32.43 32-43 32-44 044 5595 .044 7561 .044 9526 32-75 32.76 32.76 .056 4119 .056 6105 .056 8091 33.10 33.10 33-" .068 3901 .068 5908 .068 7916 33-45 33-46 33-47 54 .033 4111 32-44 .045 1492 32.77 .057 0078 33-" .068 9924 33-47 55 2.033 6058 32.45 2.045 3459 32.78 2.057 2065 33.12 2.069 1933 3348 56 .033 8005 32-45 .045 5426 32.78 .057 4052 33.12 .069 3942 33-48 57 33 9952 32.46 45 7393 32.79 .057 6040 33.13 .069 5951 33-49 58 .034 1900 32-47 .045 9360 32-79 .057 8028 33-14 .069 7960 33.50 59 .034 3848 32.47 .046 1328 32.80 .058 00l6 33-H .069 9970 33.50 60 2.034 5797 3^.48 2.046 3296 32.80 2.058 2005 33-*5 2.070 1980 33-51 589 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 96 97 98 99 logic. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Dtff. l". 0' 2.070 1980 33-51 2.082 3282 33-88 2.094 5971 34.28 2.107 0109 34.69 1 .070 3991 33-51 .082 5316 33-89 .094 8028 34.29 .107 2190 34-70 2 .070 6002 33-52 .082 7349 33-9 .095 0085 34.29 .107 4272 34.70 3 .070 8014 33-53 .082 9383 33-90 .095 2143 34-3 .107 6355 34-71 4 .071 0025 33-53 .083 1418 33-9 1 .095 4201 34.31 .107 8437 34-72 5 2.071 2037 33-54 2-083 3453 33-92 2.095 6260 34-31 2.108 0521 34-72 6 .071 4050 33-54 .083 5488 33-92 .095 8318 34-32 .108 2604 34-73 i 7 .071 6063 33-55 .083 7523 33-93 .096 0378 34.33 .108 4689 34-74 8 .071 8076 33-5 6 .083 9559 33-94 .096 2438 34-33 .108 6773 34-75 9 .072 0090 33.56 .084 1596 33-94 .096 4498 34-34 .108 8858 34-75 10 2.072 2104 33-57 2.084 3633 33-95 2.096 6558 34-35 2.109 0944 34-76 11 .072 4118 33.58 .084 5670 33-96 .096 8619 34-35 .109 3029 34-77 12 .072 6133 33-58 .084 7707 33-96 .097 0681 34-36 .109 5116 34-77 13 -.072 8148 33-59 .084 9745 33-97 .097 2742 34-37 .109 7202 34.78 14 .073 0163 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 33.61 .085 5861 33-99 .097 8930 34-39 .no 3465 34.80 17 .073 6212 33.61 .085 7901 33-99 .098 0993 34-39 "o 5553 34.81 18 .073 8229 33.62 .085 9941 34.00 .098 3057 34-4 .no 7642 34-82 19 .074 0246 33- 6 3 .086 1981 34.01 .098 5121 34-41 .no 9731 34.82 20 2.074 2264 33- 6 3 2.086 4021 34.01 2.098 7186 34-41 2. ni 1821 34.83 21 .074 4282 33- 6 4 .086 6062 34.02 .098 9251 3442 .in 3911 34.84 22 .074 6301 33- 6 4 .086 8104 34-03 .099 1316 34-43 .in 6001 34.85 23 .074 8320 33- 6 5 .087 0146 34-03 .099 3382 34-43 .in 8092 34-85 24 75 339 33.66 .087 2188 34-04 99 5449 34-44 .HZ 0184 34-86 25 2.075 2358 33.66 2.087 4231 34-05 2.099 7515 34-45 2.II2 2275 34-87 26 .075 4378 33- 6 7 .087 6274 34.05 .099 9582 34-45 .112 4368 34-87 27 .075 6399 33-67 .087 8317 34-o6 .100 1650 34-46 .112 6460 34-88 28 .075 8419 33-68 .088 0361 34-07 .100 3718 34-47 .112 8553 34.89 29 .076 0440 33-69 .088 2405 34.07 .100 5786 34.48 .113 0647 34-90 30 2.076 2462 33.69 2.088 4449 34.08 2.100 7855 34-48 2.II3 2741 34.90 31 32 .076 4484 .076 6507 33-7 33.71 .088 6494 .088 8540 34.09 34-09 .100 9924 .101 1993 34-49 34.50 .113 4835 .113 6930 34-9 1 34-92 33 .076 8529 33-71 .089 0586 34.10 .101 4063 34.50 .113 9025 34-92 34 .077 0552 33-72 .089 2632 34-n .101 6134 34-51 .114 II2I 34-93 35 2.077 2575 33-73 2.089 4678 34-" 2.IOI 8204 34-52 2.114 3*!7 34-94 36 .077 4599 3J-73 .089 6725 34.12 .102 0276 34-52 "4 53 J 3 34-95 37 .077 6623 33-74 .089 8772 34.12 .102 2347 34-53 .114 7410 34-95 38 .077 8647 33-74 .090 0820 34-13 .102 4419 34-54 .114 9508 34.96 39 .078 0672 33-75 .090 2868 34.14 .IO2 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 9015 34.16 .103 2711 34.56 .115 7901 34-99 43 .078 8775 33-78 .091 1065 34-17 .103 4785 34-57 .116 oooi 35-0 44 .079 0802 33-78 .091 3115 34-17 .103 6860 34.58 .Il6 2101 35-oo 45 2.079 2829 33-79 2.091 5165 34.18 2.103 8935 34-59 2.116 4201 35-oi 46 .079 4857 33.80 .091 7216 34.19 . 1 04 i o i o 34-59 .116 6301 35-02 47 .079 6885 3380 .091 9268 34.19 .104 3086 34.60 .116 8403 35-02 48 .079 8913 33.81 .092 1319 34-20 .104 5162 34.61 .117 0505 35-03 49 .080 0942 33-8i .092 3371 34-20 .104 7239 34.61 .117 2607 35-04 50 2.080 2971 33-82 2.092 5424 34.21 2.104 93 J 6 34.62 2.117 4710 35-05 51 .080 5000 3383 .092 7477 34-22 .105 1393 34-63 .117 6813 35-05 1 52 .080 7030 33-83 .092 9530 34-22 .105 3471 34.63 .117 8916 35.06 53 .080 9060 33-84 .093 1584 34-23 -105 5549 34-64 .Il8 1020 35-07 54 .081 1091 33-85 .093 3638 34.24 .105 7628 34-65 .Il8 3124 35.08 55 2.081 3122 33-85 2.093 5692 34.25 2.105 977 34-66 2.118 5229 35.08 56 .081 5153 33-86 .093 7747 34.25 .106 1786 34-66 .118 7334 35-09 57 .081 7185 33-87 .093 9803 34-26 .106 3866 34.67 .118 9440 35- 10 58 .081 9217 33-87 .094 1858 34-27 .106 5947 34-68 ,119 1546 35- 10 59 .082 1249 33-88 .094 3914 34-27 .106 8027 34.68 .119 3652 35-" 60 2.082 3282 33-88 2.094 5971 34.28 2.107 0109 34.69 2.119 5759 35-ia 590 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 100 101 102 103 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff I''. 2.119 5759 35-12 2.132 2989 35-57 2.145 *866 36.03 2.158 2460 36.52 1 .119 7867 35-13 .132 5123 35-57 .145 4028 36.04 .158 4652 36.53 2 .119 9974 35-13 .132 7258 35.58 .145 6191 36.05 .158 6844 36.54 3 .120 2083 35-H 132 9393 35-59 H5 8354 36.06 .158 9036 36.55 4 .120 4191 35-15 .133 1529 35.60 .146 0518 36.07 .159 I22 9 36.55 ' 5 2.120 6301 35.16 2-133 3 66 5 35-6i 2.146 2682 36.07 2.159 3423 36.56 6 .120 8410 35-16 .133 5802 35-6i .146 4847 36.08 .159 5617 36-57 7 .121 0520 35-17 !33 7939 35-62 .146 7012 36.09 .159 7 8ll 36.58 8 .121 2630 .134 0076 35-63 .146 9178 36.10 .160 0006 36.59 9 .121 4741 35-19 .134 2214 35-64 .147 1344 36.11 .l6o 2202 36.60 10 2. 121 6853 35-19 2.134 4352 35-64 2.147 3510 36.11 ^.160 4398 36.60 11 .121 8965 35.20 .134 6491 35.65 .147 5677 36.12 .l6o 6594 36.61 12 .122 1077 35-21 .134 8631 35-66 .147 7845 36.13 .l6o 8791 36.62 13 .122 3190 .135 0770 35-67 .148 0013 36.14 ,l6l 0989 36.63 14 .122 5303 35-22 .135 2910 35-67 .148 2182 36.15 .l6l 3187 36.64 15 2.122 7416 35-23 2.135 5051 35-68 2.148 4351 36.15 2.161 5385 36.65 16 .122 9530 35-24 .135 7192 35.69 .148 6520 36.16 .161 7584 36.65 17 .123 1644 35-24 *35 9334 35.70 .148 8690 36.17 .161 9784 36.66 18 .123 3759 35-25 .136 1476 35.71 .149 0861 36.18 .162 1984 36.67 19 .123 5875 35.26 .136 3619 35-71 .149 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 .136 7905 35-73 H9 7375 36.20 .162 8587 36.70 22 .124 2223 35-28 .137 0049 35-74 .149 9547 36.21 .163 0789 36.70 23 .124 4340 35-29 .137 2193 35-74 .150 1720 36.22 .163 2992 36.71 24 .124 6458 35-3 *37 4338 35-75 .150 3893 36-23 .163 5195 36.72 25 2.124 8576 35-3 2.137 6484 35-76 2.150 6067 36.23 2.163 7398 36.73 26 .125 0694 35-3 1 .137 8630 35-77 .150 8242 36-24 .163 9602 36.74 27 .125 2813 35-32 .138 0776 35-77 .151 0417 36.25 .164 1807 36.74 28 29 .125 4933 .125 7052 35-33 35-33 .138 2922 .138 5070 35-78 35-79 .151 2592 .151 4768 36.26 36.27 .164 4012 .164 6218 36.75 36.76 30 2.125 9173 35-34 2.138 7217 35.80 2.151 6944 36.28 2.164 8424 36.77 31 .126 1293 35-35 138 9365 .151 9121 36.28 .165 0630 36.78 32 .126 3414 35-35 .139 1514 35'8i .152 1298 36.29 .165 2837 36.79 33 .126 5536 35-36 .139 3663 35-82 .152 3476 36.30 .165 5045 36.80 34 .126 7658 35 37 .139 5813 35-83 .152 5654 36.31 .165 7253 36.81 35 36 2.126 9780 .127 1903 35-38 35-39 2.139 7963 .140 0113 35-84 35.84 2.152 7833 .153 OO I 2 36.32 36.32 2.165 9462 .166 1671 36.81 36.82 37 .127 4027 35-39 .140 2264 35.85 .153 2I 9 2 36.33 .166 3881 36-83 38 .127 6151 35-4 .140 4415 35-86 153 4372 36.34 .166 6091 36.84 39 .127 8275 35-4 1 .140 6567 35-87 153 6552 36.35 .166 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 42 43 44 .128 2525 .128 4650 .128 6776 .128 8903 35.42 35-43 35-44 35-45 .141 0873 .141 3026 .141 5180 .141 7334 35-88 35.89 35-9 35.91 .154 0915 .154 3097 .154 5280 .154 7463 36.36 36.37 36.38 .167 2724 .167 4936 .167 7149 .167 9362 36.87 36.87 36.88 36.89 45 46 47 48 19 2.129 1030 .129 3157 .129 5285 .129 7414 .129 9542 35-45 35-46 35-47 3548 35.48 2.141 9489 .142 1644 .142 3799 .142 5955 .142 8112 35.92 35-93 35-94 35-95 2.154 9 6 47 .155 1831 .155 4015 .155 6200 .155 8386 36.40 36.41 36-41 36.42 36.43 2.168 1576 .168 3790 .168 6005 .168 8220 .169 0436 36.90 36.91 36.92 36.93 36.93 50 2.130 1672 35-49 2.143 0269 35-96 2.156 0572 3 6 .44 2.169 2652 36.94 51 52 53 54 .130 3801 13 593 1 .130 8062 .131 0193 35-5 35-51 35-5i 35-52 .^43 2427 -143 4585 -143 6743 .143 8902 35-96 35-97 35.98 35-99 .156 2759 .156 4946 .156 7133 .156 9321 36.45 3646 36.47 .169 4869 .169 7087 .169 9304 .170 1523 36.96 36.97 36.98 55 56 57 2.131 2325 131 4457 .131 6589 35-53 35-54 35-54 2.144 1062 .144 3222 .144 5382 36.00 36.00 36.01 2.157 i5 10 .157 3699 .157 5889 36.48 36.49 36.50 1.170 3742 .170 5961 .170 8181 36.99 36-99 37.00 58 59 .131 8722 .132 0855 35-55 35-56 .144 7543 .144 9704 36.02 36-03 .157 8079 .158 0269 36.50 36.51 .171 OAOI .171 2622 37.01 37.02 60 2.132 2989 35-57 2.145 1866 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. r. 104 105 106 107 logM. Diff. 1". logM. Diff. 1". logM. Diff. I". logM. Diff. 1". 0' 1 2 2.171 4844 .171 7066 .171 9288 37.03 37-04 37-05 .184 9092 .185 1346 .185 3600 37-56 37-57 37-57 .198 5282 .198 7568 .198 9856 38.11 38.12 38.13 .212 3493 .212 5814 .212 8136 38.68 38.69 38.70 3 4 .172 1511 .172 3735 37-05 37.06 .185 5855 .185 8110 37.58 37-59 .199 2144 .199 4432 38.14 38.14 .213 0458 .213 2781 38-7I 38.72 5 2.172 5959 37.07 2.186 0366 37.60 2.199 6721 38.15 2.213 5104 38.73 6 .172 8184 37.08 .186 2622 37-6i .199 9010 38.16 .213 7428 38.74 7 .173 0409 37.09 .186 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 38.19 .214 4404 38.77 10 2.173 70 8 7 37.12 2.187 1653 37.65 2.200 8174 38.20 2.214 673 38-78 11 173 93*4 37.12 .187 3912 37-66 .201 0467 38.21 .214 9057 38.79 12 .174 1542 37-13 .187 6172 37.67 .2OI 2760 38.22 .215 1385 38.80 13 .174 3770 37-H .187 8432 37.67 .201 5053 38.23 .215 3713 38.81 14 .174 5999 37.15 .188 0693 37.68 .201 7347 38.24 .215 6042 38.82 15 2.174 8228 37.16 2.188 2954 37.69 2.2OI 9642 38.25 2.215 8371 38.83 16 .175 0458 37-17 .188 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 37.18 .188 9741 37-72 .202 6529 38.28 .216 5363 38.86 19 .175 7150 37-19 .189 2005 37-73 .202 8826 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 .217 4693 38.90 23 .176 6081 37-23 .190 1064 37-77 .203 8019 38*32 .217 7027 38.91 24 .176 8315 .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 27 .177 5020 37.26 .191 0132 37.8o .204 7222 38.36 .218 6369 38-95 28 .177 7256 37.27 .191 2401 37.8i .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 2.2O5 4.129 38.39 2.219 3382 38-98 31 .178 3968 373 .191 9209 37.84 .205 6433 38.40 .219 5721 38.99 32 .178 6206 37-3 1 .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 .192 6023 37.87 .206 3348 38.43 .220 2741 39.02 35 2 179 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 39-4 37 .179 7405 37-35 .193 2841 37.89 .207 O268 38.46 .220 97^7 39-05 38 .179 9646 37.36 .193 5115 37-9 -207 2575 38.47 .221 2110 39.06 39 .180 1888 37-37 .193 7389 37-9 1 .207 4884 38.48 221 4453 39.07 40 2.l8o 4131 37.38 2.193 9664 37-92 2.207 7 J 93 38.49 2.221 6797 39.08 41 .l8o 6374 37-39 .194 1940 37-93 .207 9502 38.50 .221 9142 39.09 42 .l8o 8617 37-4 .194 4216 37-94 .208 1812 38.51 .222 1488 39.10 43 .l8l o86l 37-4 1 .194 6493 37-95 .208 4123 222 3834 39- 11 44 .l8l 3106 37.41 .194 8770 37-96 .208 6434 38-53 .222 6l8o 45 2.181 5351 37.42 2.195 1048 37-97 2.208 8746 38.54 2.222 8528 39-'3 46 .181 7597 37-43 .195 3326 37-98 .209 1058 38.54 .223 0876 39-H 47 .181 9843 37-44 .195 5605 37-99 .209 3371 38.55 .223 3224 39-'5 48 .182 1089 37-45 .195 7885 38.00 .209 5685 38.56 .223 5573 39,16 49 -182 4337 37.46 .196 0165 38.00 209 7999 38-57 .223 7923 5O 2.182 6584 37-47 2.196 2445 38.01 2.210 0314 38.58 2.224 0273 39.18 51 .182 8833 37.48 .196 4726 38.02 .2IO 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 54 .183 3331 .183 5581 37-49 37.50 .196 9290 .197 1573 38.04 38.05 .210 7261 .210 9578 38.61 38.62 .224 7327 .224 9680 39.21 39-22 55 2.183 7831 37.51 2.197 3856 38.06 2.21 I 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 3925 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 39.28 692 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 108 109 110 111 logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". 1 .226 3808 .226 6165 39.28 39- 2 9 .240 6314 .240 8708 39.90 39.91 .255 1099 2 55 353* 40.54 40.55 .269 8255 .270 0728 41.21 2 .226 8523 39-3 .241 1103 39-9* .255 5965 40.56 .270 3202 41.24 3 .227 0881 39-31 .241 3498 3993 .255 8399 40.58 .270 5676 41.25 4 .227 3240 39.32 .241 5894 39-94 .256 0834 40.59 .270 8152 41.26 5 .227 5599 39-33 2.241 8291 39-95 .256 3270 40.60 .271 0628 41.27 6 .227 7959 39-34 .242 0688 39-96 .256 5706 40.61 .271 3104 41.28 7 .228 0320 39-35 .242 3086 39-97 .256 8143 40.62 .271 5582 41.29 8 .228 2681 39-36 -242 5485 39'98 .257 0580 40.63 .271 8060 41.30 9 .228 5043 39-37 .242 7884 39-99 .257 3019 40.64 .272 0538 41.32 10 2.228 7405 39.38 2.243 0*84 40.00 2.257 5458 40.65 2.272 3018 41-33 11 .228 9768 39-39 .243 2685 40.01 .257 7897 40.66 .272 5498 41-34 12 .229 2131 39-4 .243 5086 40.02 .258 0337 40.68 .272 7979 41-35 13 .229 4496 39.41 .243 7488 40.03 .258 2778 40.69 .273 0460 41.36 14 .229 6861 39.42 .243 9890 40.05 .258 5220 40.70 .273 2942 41.38 15 2.229 9 2 *6 39-43 2.144 2293 40.06 2.258 7662 40.71 2.273 54*5 41-39 16 .230 1592 39-44 .244 4697 40.07 .259 0105 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 39-4 6 .244 9506 40.09 .259 4992 4-74 .274 2878 41.42 19 .230 8694 39-47 .245 1912 40.10 2 59 7437 40.75 .274 5364 41.43 20 2.231 1063 39-48 2.245 4318 40.11 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 39-5 .245 9132 40.13 .260 4776 40.79 .275 2825 41.47 23 .231 8172 39-5 1 .246 1541 40.14 .260 7223 40.80 -275 53*3 41.48 24 .232 0543 39.52 .246 3949 40.15 .260 9671 40.81 .275 7802 41.49 25 2.232 2915 39-53 2.246 6359 40.16 2.26l 2120 40.82 2.276 0292 41.50 26 .232 5287 39-54 .246 8769 40.17 .26l 4570 40.83 .276 2783 27 .232 7660 39-55 .247 1 1 80 40.18 .26l 7020 40.84 .276 5274 41-53 28 .233 0033 39-56 *47 359 1 40.19 .26l 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 *75 2 41.56 31 233 7*57 39-59 .248 0829 40.23 .262 6827 40.89 .277 5246 41-57 32 2 33 9533 39.60 .248 3243 40.24 .262 9281 40.90 .277 7740 41.58 33 .234 1910 39.61 .248 5658 40.25 .263 1735 40.91 .278 0236 41.60 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 .249 2906 40.28 . .263 9102 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 7741 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 77 2 3 41.68 41 .236 0945 39.70 .250 4998 40.34 .265 1393 41.00 .280 0224 41.69 42 .236 3327 39-71 .250 7419 40.35 .265 3853 41.01 .280 2726 41.70 43 .236 5710 39-7 2 .250 9840 40.36 .265 6314 41.02 .280 5228 41.71 44 .236 8093 39-73 .251 2262 4037 .265 8776 41.03 .280 7731 41.72 45 2.237 0478 39-74 2.251 4684 40.38 2.266 1238 4 I. OA 2.281 0235 41.74 46 .237 2862 39-75 251 7107 40.39 .266 3701 41.06 .281 2740 4*-75 47 .237 5247 39-76 .251 9531 40.40 .266 6165 41.07 .281 5245 41.76 48 .237 7633 39-77 .252 1955 40.41 .266 8629 41.08 .281 7751 41-77 49 .238 OO2O 39.78 .252 4380 40.42 .267 1094 41.09 .282 0258 41.78 50 51 52 2.238 2407 2 38 4795 .238 7284 39-79 39.80 39.81 2.252 6806 .252 9232 253 l6 59 4-43 40.44 40.46 2.267 3560 .267 6026 .267 8493 41.10 41.11 41.12 2.282 2765 .282 5273 .282 7782 41.80 41.81 41.82 53 2 3 8 9573 39.82 253 4 8 7 40.47 .268 0961 41.13 .283 0291 41.83 54 .239 1962 .253 6515 40.48 .268 3430 4 I.I5 .283 2801 41.84 55 2 - 2 39 4353 39-84 2.253 8944 40.49 2.268 5899 4I.l6 2.283 53'* 41.85 56 .239 6744 39.86 40.50 .268 8769 41.17 .283 7824 41.87 57 58 59 .239 9235 .240 1528 .240 3921 39-87 39-88 39-89 .254 3804 .254 6235 .254 8666 40.51 40.52 40.53 .269 0839 .269 3310 .269 5782 4I.I8 41.19 41.20 .284 0336 .284 2849 .284 5363 41.88 41.89 41.90 60 2.240 6314 39.90 2-255 I0 99 40.54 2.269 8255 4 I.2I 2.284 7878 41.91 38 593 TABLE VI. finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V, 112 113 114 115 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 2.284 7878 41.91 2.300 0067 42.64 2.315 4927 43.40 2.331 2564 44.18 1 .285 0393 41.93 .300 2626 42.65 315 7531 43.41 .331 5216 44.20 2 .285 2909 41.94 .300 5186 42.67 .316 0136 43.42 .331 7868 44.21 3 .285 5425 41-95 .300 7746 42.68 .316 2742 43-44 .332 0521 44.22 4 285 7943 41.96 .301 0307 42.69 .316 5348 43-45 33 2 3*75 44.24 5 2.286 0461 41.97 2.301 2869 42.70 2.316 7956 43.46 2.332 5830 44.25 6 .286 2979 41.99 3 01 543 1 42.72 .317 0564 43-47 .332 8485 44.26 7 .286 5499 42.00 .301 7995 42.73 3*7 3'73 43-49 333 "4i 44.28 8 .286 8019 42.01 .302 0559 42.74 .317 5782 43-5 333 3799 44.29 9 .287 0540 42.02 .302 3123 42.75 3i7 8 393 43-51 333 6456 44.31 10 2.287 3062 42.03 2.302 5689 42.76 2.318 1004 43-53 2-333 9"5 44.32 11 .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 .318 6229 43-55 334 4435 44-34 13 .288 0631 42.07 .303 3390 42.80 .318 8842 43.56 334 79 6 44.36 14 .288 3155 42.08 33 595 8 42.81 .319 1456 43-58 334 9758 44-37 15 2.288 5680 42.09 2.303 8528 42.83 2.319 4072 43-59 2.335 2421 44-39 16 .288 8206 42.10 .304 1098 42.84 .319 6687 43.60 335 5 8 4 44.40 17 18 .289 0733 .289 3260 42.12 42.13 .304 3668 .304 6240 42.85 42.86 .319 9304 .320 1921 43.62 43- 6 3 335 7749 .336 0414 44.41 44-43 19 .289 5788 42.14 .304 8812 42.88 .320 4540 43- 6 4 .336 3080 44.44 20 2.289 8317 42.15 2-35 J 3 8 5 42.89 2.320 7159 43-66 2.336 5747 44-45 21 .290 0847 42.16 35 3959 42.90 .320 9778 43.67 .336 8414 44-47 22 .290 3377 42.18 35 6 533 42.91 .321 2399 43-68 337 1-083 44.48 23 .290 5908 42.19 .305 9109 42.93 .321 5020 43-69 337 3752 44-49 24 .290 8440 43.20 .306 1685 42.94 .321 7642 43-70 .337 6422 44-51 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 1996 42.99 .322 8139 43-76 .338 7111 44-56 29 .292 1109 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 2 46o 44-59 31 .292 6182 42.29 .307 9738 43-03 .323 6019 43.80 339 5*35 4460 32 .292 8719 4230 .308 2320 43 -4 .323 8647 43.81 339 7812 44.62 33 .293 1258 4*. 3 * .308 4903 43.05 .324 1277 43-83 34 49 44.63 34 293 3797 42.32 .308 7486 43-7 .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 4*- 3 5 .309 2656 43.09 .324 9169 43-87 .340 8527 44 h 7 37 .294 1418 42.36 .309 5242 43.10 .325 1801 43-88 .341 1207 44.69 38 .294 3960 42.37 .309 7828 43.12 3*5 4434 43-89 .341 3889 44.70 ( 39 .294 6503 42.38 .310 0416 43- ' 3 .325 7068 43-91 .341 6571 44-71 j 40 2.294 9046 42.40 2.310 3004 43-H 2.325 9703 43-9* 2-341 9255 44-73 41 .295 1590 42.41 3 10 5593 43- 1 5 .326 2339 43-93 .342 1939 44-74 42 295 4135 42.42 .310 8182 43- J 7 .326 4975 43-94 .342 4623 44-75 43 .295 6680 42.43 .311 0773 43.18 .326 7612 43.96 .342 7309 44-77 44 .295 9227 42.44 3" 33 6 4 43- J 9 .327 0250 43-97 342 9995 44.78 45 2.296 1774 42.46 2.311 5956 43.21 2.327 2889 43-98 2-343 2683 44.80 46 .296 4321 42.47 .311 8549 43.22 .327 5528 44.00 343 537i 44.81 47 .296 6870 42.48 .312 1142 43.23 .327 8168 44.01 .343 8060 44.82 48 .296 9419 42.49 .312 3736 43-24 .328 0809 44.02 344 0750 44.84 49 .297 1969 42.51 .312 6331 43.26 .328 3451 44.04 344 344 44.85 50 2.297 4520 42.52 2.312 8927 43.27 2.328 6094 44.05 2-344 6132 *- 51 .297 7071 42. 5 3 .313 1524 43.28 .328 8737 44.06 .344 8824 44.88 52 .297 9623 42.54 .313 4121 43.29 .329 1382 44.08 345 MI7 44.89 53 .298 2176 42-55 .313 6719 43-3 1 .329 4027 44.09 345 4211 44.91 54 .298 4730 42-57 3i3 93 l8 43-32 .329 6672 44.10 .345 6906 44.92 55 56 2.298 7284 .298 9839 42.58 42.59 2.314 1917 .314 4518 43-33 43-35 2.329 9319 .330 1967 44.12 44- ' 3 2.345 9601 .346 2298 44-93 44-95 57 .299 2395 42.60 .314 7119 43-36 33 4615 44.14 .346 4995 44.96 58 .299 4952 42.61 .314 9721 43-37 .330 7264 44.16 346 7693 44-97 59 .299 7509 42.63 .315 2323 43.3 8 33 99 J 4 44.17 347 0392 44-99 60 2.300 0067 42.64 2.315 4927 43.40 2.331 2564 44.18 2-347 3092 45.00 694 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 116 117 118 119 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 2.347 3092 45.00 2.363 6626 45.86 2.380 3290 46.74 1.397 3210 47.66 1 347 5792 45.02 3 6 3 9378 45.87 .380 6095 46.76 .397 6070 47.68 2 347 8494 45-3 .364 2131 45.88 .380 8901 46.77 397 8931 47.70 1 3 .348 1196 45.04 .364 4885 45.90 .381 1708 46.79 .398 1794 47-71 4 348 3 8 99 45.06 .364 7639 45.91 381 45*5 46.80 398 4657 47-73 5 2.348 6603 45-07 2-3 6 5 394 45-93 1.381 7324 46.82 2.398 7521 47-74 6 .348 9308 45-09 3 6 5 3^5 45-94 .382 0133 46.83 399 3 86 47.76 7 .349 2014 45.10 -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 6119 47-79 9 349 7428 45-13 .366 1423 45-99 .382 8567 46.88 399 8987 47.81 10 2350 0136 45.14 2.366 4183 46.00 2.383 1380 46.89 2.400 1856 47-82 11 35 2845 45.16 .366 6944 46.01 .3 8 3 4i94 46.91 .400 4725 47.84 12 35 5554 45-17 .366 9705 46.03 .383 7009 46.92 .400 7596 47.85 13 .350 8265 45 i 8 .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 2351 3689 45.21 2,367 7994 46.07 2.384 5460 46.97 2.401 6214 47.90 16 .351 6402 45-^3 .368 0759 46.09 .384 8278 46.98 .401 9088 47.92 17 .351 9116 45.24 .368 3525 46.10 .385 1098 46.99 .402 1964 47-93 18 .352 1831 45-25 .368 6291 46.12 .385 3918 47.01 .402 4840 47-95 19 35* 4547 45.27 .368 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 45-3 .369 4596 46.16 .386 2385 47.06 43 3475 48.00 22 353 2699 45.31 .369 7367 46.18 .386 5209 47.08 43 6356 48.01 23 353 54 l8 45-33 .370 0138 46.19 .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 8 59 45-35 2.370 5682 46.22 2.38-7 3687 47.12 2.404 5002 48.06 2G 354 358i 45-37 .370 8456 46.24 3 8 7 6514 47.14 .404 7886 48.08 27 354 6 33 45-38 .371 1230 46.25 -3 8 7 9343 47-15 45 O77i 48.09 28 .354 9027 45.40 .371 4006 46.26 .388 2173 47-*7 45 3657 48.11 29 355 '75 1 45-41 .371 6782 46.28 .388 5003 47.18 .405 6544 48.12 30 2-355 4476 45-42 2-37' 9559 46.29 2-388 7835 47.20 2.405 9432 4 8.H 31 .355 7202 45-44 .372 2337 46.31 .389 0667 47.21 .406 2321 48.16 32 355 9928 45-45 372 5 Ij6 46.32 389 35 47-23 .406 5211 48.17 33 .356 2656 45-47 -372 7 8 9 6 4 6 -34 3 8 9 6 335 47.24 .406 8102 48.19 34 356 53 8 5 45-48 .373 0677 46.35 389 9*70 47.26 47 0993 48.20 35 2.356 8114 45-5 2-373 3459 4 6 -37 2.390 2006 47.28 2.407 3886 48.22 36 357 8 44 45-51 373 624* 46.38 .390 4843 47.29 .407 6780 48.24 37 357 3575 45-52 373 9024 46.40 .390 7681 47-3 1 .407 9674 48.25 38 357 6307 45-54 .374 1809 46.41 .391 0519 47-32 .408 2570 4*'*l 39 357 9040 45-55 374 4594 46.43 391 3359 47-34 .408 5467 48.28 40 2.358 1773 45-57 2-374 73 8 46.44 2.391 6200 47-35 2.408 8364 48.30 41 .358 4508 45.58 375 Ol6 7 46.46 ".392 9042 47-37 .409 1263 48.32 42 .358 7243 45.60 375 2955 46.47 .392 1884 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 8 533 46.50 .392 7572 47.41 .409 9964 48.37 45 2-359 5454 45.64 2.376 1324 46.51 *-393 4 J 7 47-43 2.410 2866 48.38 46 359 8l 93 45.65 .376 4115 46.53 393 3264 47-45 .410 5770 48.40 47 .360 0933 45.67 .376 6908 46.55 393 6l11 47-46 .410 8674 48.41 , 48 .360 3673 45.68 .376 9701 46.56 393 8959 47.48 .411 1579 48.43 49 .360 6415 45.70 377 2495 4 6.. 5 8 .394 1808 47-49 .411 4486 48-45 50 2.360 9157 45-7 1 2-377 529 46.59 2-J94 465 8 47.51 2.411 7393 48.46 51 .361 1900 45-72 .377 8086 46.60 394 759 47.52 .412 0301 48.48 52 .361 4644 45-74 .378 0883 46.62 -395 3 61 47-54 .412 3210 48.49 53 54 .361 7389 .362 0134 45-75 45-77 .378 3681 378 6479 46.64 46.65 395 3214 395 6067 47-55 47-57 .412 0120 .412 9031 48.51 48.53 55 2.362 2881 45-78 2-378 9279 46.67 2-395 8922 47-59 2.413 1944 4 s' 5 * 56 .362 5628 45.80 379 2079 46.68 .396 1778 47.60 .413 4857 48-56 57 .362 8376 45.81 379 4 881 46.70 .396 4634 47.62 .413 7771 48.58 58 59 .363 1126 .363 3876 45.82 45.84 379 76 8 3 .380 0486 46.71 46-73 39 6 7492 397 0350 47- 6 3 47.65 .414 0686 .414 3602 48-59 48.61 60 2.363 6626 45-86 2.380 3290 46.74 2.397 3210 47.66 2.414 6519 48.62 695 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 120 121 122 123 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 2.414 6519 48.62 M-3* 3356 49.62 2.450 3868 50.67 2.468 8205 5'-75 1 .414 9437 48.64 43* 6 334 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 51-79 3 .415 5276 48.67 433 2293 49.68 45 * 2992 50.72 .469 7526 5 l.8l 4 .415 8197 48.69 433 5*74 49.69 .451 6036 5-74 .470 0634 51.82 5 2.416 1119 48.71 2-433 8257 49.71 2.451 9081 5-75 2.470 3744 51.84 6 .416 4042 48.72 .434 1240 49-73 .452 2127 50.77 .470 6856 51.86 7 8 .416 6965 .416 9890 48.74 48.76 434 42*4 .434 7209 49-74 49.76 45* 5174 .452 8222 50.79 50.81 47 99^8 .471 3081 51.88 51.90 9 .417 2816 48.77 435 i95 49.78 453 1271 50.83 .471 6196 51.92 10 2.417 5743 48.79 2.435 3182 49.80 2.453 432i 50.84 2.471 9311 51.94 11 .417 8671 48.81 435 6171 49.81 453 7372 50.86 .472 2428 5*-95 12 .418 1600 48.82 435 916 49.83 454 04M 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 6532 50.92 473 J 7 8 5 52.01 15 2.419 0392 48-87 1.436 8134 49.88 2.454 9587 5-93 2.473 49 6 52-03 16 4i9 3325 48.89 *437 "27 49.90 .455 2644 50.95 .473 8028 52-05 17 .419 6258 48.90 .437 4122 49.92 455 57i 50.97 474 "5* 52-07 18 .419 9193 48.92 437 7ii7 49-93 .455 8760 50.99 .474 4276 52.09 19 .420 2129 48.94 .438 0114 4995 .456 1820 51.00 474 742 52.10 20 4.420 5066 48.95 2.438 3111 49-97 2.456 4881 51.02 2.475 0529 52.12 21 .420 8003 48.97 .438 6110 49.98 .456 7943 51.04 475 3 6 57 52.14 22 .421 0942 48.99 .438 9109 50.00 .457 1006 51.06 .475 6786 52.16 23 .421 3882 49.00 .439 21 10 50.02 .457 4070 51.08 475 99 l6 52.18 24 .421 6822 49.02 439 5"2 50.04 457 7135 51.09 .476 3047 52.20 25 2.421 9764 49-3 2.439 8114 50.05 2.458 0201 51.11 2.476 6180 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 45 8 6 337 51.15 .477 2448 52.25 28 .422 8595 49.09 .440 7129 50.1 1 .458 9406 51.17 477 55 8 4 52.27 29 .423 1541 49.10 .441 0136 50.12 459 2 477 51.18 .477 8721 52.29 30 2.423 4488 49.12 2.441 3143 50.14 2.459 5548 51.20 2.478 1859 52-3 1 31 .423 7435 49.14 .441 6152 50.16 .459 8621 51.22 .478 4998 52-33 32 .424 0384 49.15 .441 9162 50.18 .460 1695 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 5M 1 .480 0711 52.42 37 .425 5142 49.24 .443 4228 50.26 .461 7080 5'-33 .480 3857 52.44 38 .425 8097 49.25 .443 7244 50.28 .462 0161 51-35 .480 7004 52.46 39 .426 1053 49.27 .444 0261 50.30 .462 3242 5'-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 49-3 .444 6299 5-33 .462 9408 51.40 .481 6452 52.52 42 .426 9926 49.32 .444 9320 5-35 .463 2493 51.42 .481 9604 52-54 43 .427 1886 49-34 445 2 34i 5037 4 6 3 5579 5M4 .482 2756 52.56 44 .427 5847 49-35 445 53 6 4 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 4 6 4 7933 5*-5 4 8 3 5379 52.63 ! 48 .428 7700 49.42 .446 7464 50.45 .465 1024 5'-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 4.447 3521 50.49 2.465 7210 51-57 2.484 4858 52.69 51 .429 6600 49-47 447 6 55i 50.51 .466 0305 51-59 .484 8020 52.71 52 .429 9569 49-49 ,447 9582 5-53 .466 3400 51.60 .485 1183 52.73 53 .430 2539 49-5 1 .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 51.66 2.486 0679 52.78 56 .431 1455 49.56 449 i7i 6 50.60 .467 5794 51.68 .486 3847 52.80 57 .431 4418 49-57 .449 4753 50.61 .467 8895 51.70 .486 7016 52.82 58 .431 7403 49-59 449 779 50.63 .468 1997 51.71 .487 0186 52.84 59 .432 0379 49.61 .450 0828 50.65 .468 5101 5'-73 487 3357 52.86 60 2.432 3356 49.62 2.450 3868 50.67 2.468 8205 51-75 2.487 6529 1 52.88 596 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit V. 124 125 126 127 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 2.487 6529 52.88 2.506 9006 54.06 2.526 5813 55-29 2.546 7135 56-57 1 .487 9702 52.90 .507 2251 54.08 526 9131 55-31 547 0530 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 54.12 527 577 1 55-35 547 7323 56-63 4 .488 9230 52.96 .508 1990 54.14 .527 9092 55-37 548 0722 56-65 5 2.489 2408 52.98 2.508 5239 54.16 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 1741 54.20 .528 9065 55-43 .549 0926 56.72 8 .490 1949 53.03 59 4993 54-22 .529 2391 55-45 549 433 56.74 9 .490 5132 53-05 .509 8247 54.24 529 579 55.48 549 7735 56.76 10 2.490 8315 53.07 2.510 1502 54-26 2.529 9048 55-50 2.550 1141 56.79 11 491 1500 53-9 .510 4758 54.28 53 2379 55-52 55 4549 56.81 12 .491 4686 53.11 .510 8016 54-30 -53 57io 55-54 55 7958 56.83 13 .491 7874 53-13 .511 1274 54-32 53 9043 55-56 .551 1369 56.85 14 .492 1063 53-15 5 11 4534 54-34 .531 2378 55-58 55 1 478i 56.87 15 2.492 4252 53-17 2.511 7795 54.36 2-53 1 5713 55.60 2.551 8194 56.90 1C .492 7443 53-19 .512 1057 54-38 .531 9050 55.62 .552 1608 56.92 17 493 6 35 53-21 .512 4321 54.40 .532 2388 55.64 .552 5024 56.94 18 .493 3828 53-^3 .512 7586 54-42 532 5727 55-67 552 8441 56.96 19 493 7023 53-^5 .513 0852 54-44 .532 9068 55-69 553 1859 56-98 20 2.494 0218 53.27 2.513 4119 54.46 2-533 2410 55-71 2-553 5279 57.01 21 494 3415 53-29 5i3 73 8 7 54.48 533 5753 55-73 553 8700 57.03 22 494 66l 3 53-31 .514 0657 54-50 533 997 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 3i2 53-35 .514 7199 54-54 534 579 55-79 554 8971 57.10 25 2.495 62 i3 53-37 2.515 0473 54.56 2.534 9138 55.81 2-555 2398 57-12 2G .495 9416 53-39 5*5 3747 54-58 535 2487 55.84 555 5825 57.H 27 28 29 .496 2619 .496 5824 .496 9030 53-4i 53.42 53-44 .515 7023 .516 0300 .516 3578 54.60 54- 6 3 54-65 535 5838 535 9*9 53 6 2543 55.86 55-88 55.90 555 9254 .556 2685 .556 6116 57.16 57.18 57-21 30 2.497 2238 53-46 2.516 6857 54.67 2.536 5898 55-92 2-55 6 9549 57-23 31 .497 5446 53.48 .517 0138 54.69 .536 9254 55-94 557 2984 57-25 32 .497 8656 53-5 .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 56.01 558 3295 57.32 35 2.498 8292 53-5 6 2.518 3273. 54-77 2.538 2690 56.03 2.558 6735 57-34 36 .499 1506 53-58 .518 6559 54-79 .538 6052 56-05 559 OI 7 6 57.36 37 .499 4721 53.60 .518 9847 54-8i .538 9416 56.07 559 3 6lg 57.38 38 499 793 8 53.62 5*9 3*37 54.83 539 2781 56.09 559 7062 57.41 39 .500 1156 53- 6 4 .519 6427 54.85 539 6i47 56.11 .560 0507 57-43 40 2.500 4375 53.66 2.519 9719 54.87 2-539 95H 56.13 2.560 3953 57-45 41 .500 7595 53.68 .520 3012 54.89 .540 2883 5 5- I5 > .560 7401 57-47 42 .501 0817 53-70 .520 6306 54-9 1 .540 6253 56.18 .561 0850 57-50 43 .501 4039 53-72 .520 9601 54-93 .540 9625 56.20 .561 4301 57-52 44 .501 7263 53-74 .521 2898 54-95 .541 2997 56.22 .561 7753 57-54 45 46 47 2.502 0488 .502 3714 .502 6942 53-76 53-78 53.80 2.521 6196 .521 9495 .522 2795 54-97 54-99 55.02 2.541 6371 .541 9746 542 3*23 56.24 56.26 56.29 2.562 1206 .562 4660 .562 8116 57-56 57-59 57-6i 48 .503 0170 53.82 .522 6097 55-04 542 6500 56.31 563 '574 57-63 49 .503 3400 53-84 .522 9400 55.06 .542 9880 56.33 563 5032 57.65 50 2.503 6631 53.86 2.523 2704 55.08 2-543 3260 56.35 2.563 8492 57.68 51 .503 9863 53.88 .523 6009 55.10 543 6641 56-37 564 !953 57-70 52 53 54 .504 3096 504 6331 .504 9567 53-9 53.92 53-94 .523 9316 .524 2624 524 5933 55-12 55-14 55.16 .544 0024 544 349 544 6794 56.39 56.42 56.44 564 54i6 .564 8880 565 2345 57-72 57-74 57-77 55 56 2.505 2804 .505 6042 53-96 53.98 2.524 9243 525 2555 55.18 55.20 2.545 0181 545 3569 56.46 56.48 2.565 5812 .565 9280 57-79 57-81 57 58 59 .505 9282 .506 2522 .506 5763 54.00 54.02 54.04 -525 5867 .525 9181 .526 2497 55-22 55.24 55-26 545 6959 .546 0350 .546 3742 56.50 56-52 56-55 .566 2750 .566 6221 .566 9693 57.84 57-86 57-88 6O 2.506 9006 54.06 2.526 5813 55-29 2-546 7135 5 6 -57 2.567 3166 57.90 597 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 128 129 130 131 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 2.567 3166 57.90 2.588 4112 59-3 2.610 0188 60.75 2.632 1622 62.28 1 .567 6641 57-93 .588 7670 59-3* .610 3834 60.78 .632 5360 62.30 2 .568 0117 57-95 .589 1230 59-35 .610 7481 60.80 .632 9099 62.33 3 .568 3595 57-97 .589 4792 59-37 .611 1130 60.83 .633 2839 62.35 4 .568 7074 57-99 .589 8355 59-39 .611 4781 60.85 .633 6581 62.38 5 6 2.569 0554 .569 4036 58.02 58.04 2.590 1919 .590 5485 59.42 59-44 2.611 8433 .612 2086 60.88 60.90 2.634 0325 .634 4070 62.41 62.43 7 .569 7519 58.06 .590 9052 59-47 .612 5741 60.93 634 78l7 62.46 8 .570 1004 58.09 .591 2620 59-49 .612 9397 60.95 .635 1565 62.48 9 .570 4490 58.11 .591 6190 59-51 .613 3055 60.98 635 5315 62.51 10 2.570 7977 58.13 2.591 9762 59-54 2.613 6715 6 1. oo 2.635 966 62.54 11 12 .571 1465 571 4955 58-15 58.18 59* 3335 .592 6909 59.56 59-58 .614 0376 .614 4038 61.03 61.05 .636 2819 .636 6573 62.56 62.59 13 .571 8447 58.20 593 4 8 5 59.61 .614 7702 61.08 .637 0329 62.61 14 .572 1939 58,22 .593 4062 .615 1368 61.10 .637 4087 62.64 15 *-57* 5434 58.25 2.593 7641 59.66 2.615 5035 61.13 2.637 7846 62.67 16 .572 8929 58.27 .594 1221 59.68 .615 8703 61.15 .638 1607 62.69 17 .573 2426 58.29 594 4803 59.70 .616 2373 61.18 638 5369 62.72 18 573 59*4 58.32 594 8386 59-73 .616 6045 61.20 638 9133 62.75 19 573 94*4 58.34 595 '97 59-75 .616 9718 61.23 .639 2899 62.77 20 2.574 2925 58.36 *-595 555 6 59-78 2.617 3392 61.25 2.639 6666 62.80 21 574 6427 58-38 595 9H3 59.80 .617 7068 61.28 .640 0435 62.82 22 23 574 993 1 575 3436 58.41 58.43 .596 *73* .596 6322 59.82 59.85 .618 0746 .618 4425 61.30 61.33 . .640 4205 .640 7977 62.85 62.88 24 575 6943 58.45 .596 9914 59.87 .618 8105 61.36 .641 1750 62.90 25 2.576 0451 58-48 2.597 3507 59-9 2.619 *787 61.38 2.641 5525 62.93 26 .576 3960 58.50 .597 7102 59.92 .619 5471 61.41 .641 9302 62.96 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 61.46 .642 6860 63.01 29 577 4496 1 58.57 .598 7894 59-99 .620 6531 61.48 .643 0641 63.04 30 2.577 8011 58.59 2.599 J 494 60.02 2.621 0220 61.51 2.643 44*4 63.06 31 .578 1528 58.62 599 59 6 60.04 .621 3911 61.53 .643 8209 63.09 32 578 545 58.64 .599 8699 60.07 .621 7604 61.56 .644 1995 63.12 33 578 8564 58.66 .600 2304 60.09 .622 1298 61.58 .644 5783 63.14 34 .579 2085 58.69 .600 5910 60.12 .622 4994 61.61 .644 9572 63.17 35 2.579 5607 58.71 2.600 9518 60.14 2.622 8691 61.63 2 - 6 45 3363 63.19 36 579 9 I 3 58.73 .601 3127 60. 1 6 .623 2390 61.66 .645 7155 63.22 37 .580 2655 58.76 .601 6738 60.19 .623 6091 61.68 .646 0949 63.25 38 .580 6181 58-78 .602 0350 60.21 6*3 9793 61.71 .646 4745 63.27 39 .580 9708 58.80 .602 3963 60.24 .624 3496 61.74 .646 8542 63.30 40 2.581 3237 58.83 2.602 7578 60.26 2.624 7* 01 61.76 2.647 2341 6 3-33 41 .581 6768 58.85 .603 1195 60.29 .625 0907 61.79 .647 6142 63.35 42 43 .582 0299 .582 3832 58-87 58.90 .603 4813 .603 8432 60.31 60.34 .625 4615 .625 8325 61.81 61.84 .647 9944 .648 3748 63.38 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 J 36o 63.46 46 .583 4440 58.97 .604 9299 60.41 .626 9462 61.91 .649 5168 63.49 47 5 8 3 7979 58.99 .605 2924 60.43 .627 3178 61.94 .649 8978 63.52 48 .584 1519 59.01 .605 6551 60.46 .627 6895 61.97 .650 2790 63.54 49 .584 5061 59.04 .606 0179 60.48 .628 0614 61.99 .650 6603 50 2.584 8604 59.06 2.606 3809 60.51 2.628 4334 62.02 2.651 0418 63.60 51 .585 2148 59.09 .606 7440 60.53 .628 8056 62.04 .651 4235 63.62 1 52 585 5 6 94 59.11 .607 1073 60.56 .629 1780 62.07 .651 8053 63-65 53 .585 9241 .607 4707 60.58 6*9 555 62.09 .652 1873 63.68 54 .586 2790 59.16 607 8343 60.61 .629 9231 62.12 .652 5695 63.70 55 2.586 6340 59.18 2.608 1980 60.63 2.630 2959 62.15 2.652 9518 6 1-73 56 .586 9891 59.20 .608 5618 60.66 .630 6689 62.17 653 334* 6j.70 57 587 3444 59-*3 .608 9258 60.68 .631 0420 62.20 .653 7168 63.79 58 .587 6999 59-*5 .609 2901 60.70 .631 4152 62.22 .654 0996 63.81 59 -588 0555 59-*7 .609 6544 60.73 .631 7887 62.25 .654 4826 63.84 60 2.588 4112 59-3 2.610 0188 60.75 2.632 i6i2 62.28 2.654 8657 63.87 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 132 133 134 135 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". o .654 8657 63.87 2.678 1547 65-53 2.702 0562 67.27 2.726 5990 69.09 1 .655 2490 63.89 .678 5480 65.56 .702 4600 67.30 .727 0137 69.12 2 3 .655 6324 .650 oioo 63.92 63-95 .678 9414 679 335 65.59 65.61 .702 8638 .703 2679 67-33 67.36 .727 4285 7^7 84^5 69.15 69.19 4 .656 3998 63.97 .679 7288 65.64 .703 6721 67-39 .728 2587 69.22 5 1.656 7837 64.00 2.680 1227 65.67 2.704 0766 67.42 2.728 6741 69.25 6 .657 1678 64.03 .680 5168 65.70 .704 4812 67.45 .729 0897 69.28 7 8 .657 5521 .657 9365 64.06 64.08 .680 9111 .681 3056 65.73 65.76 .704 8860 .705 2909 67.48 67.51 .729 5055 .729 9215 69.31 69-34 9 .658 3211 64.11 .681 7002 65-79 .705 6961 67.54 .730 3376 69.37 10 i 11 2.658 7058 .659 0907 64.14 64.17 2.682 0950 .682 4900 65.81 65.84 2.706 1014 .706 5069 67.57 67.60 2-73 7539 .731 1705 69.40 69.44 12 .659 4758 64.19 .682 8851 65.87 .706 9126 67-63 .731 5872 69.47 13 .659 8611 64.22 .683 2804 65.90 .707 3184 67.66 .732 0041 69.50 14 .660 2465 64.25 .683 6759 65.93 .707 7244 67.69 .732 4212 69-53 15 2.660 6320 64.28 2.684 0716 65.96 2.708 1307 67.72 2.732 8385 69.56 16 .661 0178 64.30 .684 4674 65.99 .708 5371 67.75 733 *559 69.59 17 .661 4037 6 4-33 .684 8634 66.01 .708 9436 67.78 733 6736 69.62 18 .661 7897 64.36 .685 2596 66.04 .709 3504 67.81 734 0914 69.66 19 .662 1760 64.38 .685 6559 66.07 709 7573 67.84 734 594 69.69 20 2.662 5623 64.41 2.686 0524 66.10 1.710 1645 67.87 2.734 9277 69.72 21 .662 9489 64.44 .686 4491 66.13 .710 5718 67.90 735 346i 69.75 22 23 663 3356 .663 7225 64.47 64.49 .686 8460 .687 2430 66.16 66.19 .710 9792 .711 3869 67.93 67.96 735 7647 73 6 1835 69.78 69.81 24 .664 1096 64.52 .687 6402 66.22 .711 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 .712 6110 68.05 737 44io 69.91 27 .665 2717 64.60 .688 8329 66.30 .713 0194 68.08 737 8605 69.94 28 .665 6594 64.63 .689 2308 66.33 .713 4279 68.11 .738 2803 69.97 29 .666 0473 64.66 .689 6289 66.36 .713 8367 68.14 738 7002 70.00 30 2.666 4354 64.69 2.690 0272 66.39 2.714 2456 68.17 2.739 I 2 3 70.04 31 .666 8236 64.72 .690 4256 66.42 .714 6547 68.20 739 54o6 70.07 32 .667 2120 64.74 .690 8242 66.45 .715 0640 68.23 739 961* 70.10 33 .667 6005 64.77 .691 2230 66.48 .715 4735 68.26 .740 3819 70.13 34 .667 9892 64.80 .691 6219 66.51 .715 8832 68.29 .740 8027 70.16 35 2.668 3781 64.83 2.692 0210 66.54 2.716 2930 68.32 2.741 2238 70.20 36 .668 7672 64.86 .692 4203 66.56 .716 7031 68.35 .741 6451 70.23 37 .669 1564 64.88 .692 8198 66.59 77 "33 68.38 .742 0666 70.26 38 .669 5457 64.91 .693 2194 66.62 .717 5*37 68.41 .742 4882 70.29 39 -669 9353 64.94 .693 6193 66.65 .717 9342 68.44 .742 9101 70.32 40 2.670 3250 64.97 2.694 0193 66.68 2.718 3450 68.48 2.743 33* 1 70.36 41 .670 7149 65.00 .694 4194 66.71 .718 7560 68.51 743 7543 70-39 42 .671 1050 65.02 .694 8198 66.74 .719 1671 68.54 744 1768 70.42 43 44 .671 4952 .671 8856 65.05 65.08 .695 2203 ,695 6210 66.77 66.80 .719 5784 .719 9899 68.57 68.60 .744 5994 .745 0222 70.45 70.48 45 2.672 2761 65.11 2.696 O2I9 66.83 2.720 .4016 68.63 *-745 44p 70.52 46 .672 6668 65.13 .696 4229 66.86 .720 8135 68.66 .745 8684 70.55 47 .673 0577 65.16 .696 8242 66.89 .721 2255 68.69 .746 2918 70.58 48 .673 4488 65.19 .697 2256 66.92 .721 6377 68.72 .746 7154 70.61 49 .673 8400 65.22 .697 6272 66.95 .722 0502 68.75 747 '39 1 70.65 . 5O 2.674 2314 65.25 2.698 0289 66.97 2.722 4628 68.78 2.747 5631 70.68 51 .674 6230 65.28 .698 4308 67.00 .722 8756 68. 81 747 9873 70.71 52 i 53 54 ..675 0147 .675 4066 .675 7987 65.30 65-33 65-36 .698 8330 .699 2353 .699 6377 67.03 67.06 67.09 .723 2885 .723 7017 .724 1150 68.84 68.88 68.91 .748 4116 .748 8362 .749 2609 70.74 70.78 70.81 55 2.676 1909 65.39 2.700 0404 67.12 2.724 5286 68.94 2.749 6859 70.84 56 57 .676 5833 .676 9759 65.42 65.44 .700 4432 .700 8462 67.15 67.18 .724 9423 .725 3562 68.97 69.00 .750 mo 75 53 6 4 70.87 70.90 58 59 .677 3687 .677 7616 6^.47 65.50 .701 2494 .701 6527 67.21 67.24 7*5 773 .726 1846 69.03 69.00 .750 9619 .751 3876 70.94 70.97 60 2.678 1547 65-53 2.702 0562 67.27 |2.726 5990 69.09 2.751 8135 71.00 599 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 136 137 138 139 log Hi Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 2.751 8135 71.00 2.777 7322 73.01 2.804 3 8 95 75.11 2.831 8224 77-32 1 75* 2396 71.03 .778 1703 73-4 .804 8403 75-14 .832 2864 77-35 2 .752 6659 71.07 .778 6087 73.07 .805 2912 75.18 .832 7506 77-39 3 753 925 71.10 779 472 73-" .805 7424 75.21 .833 2151 77-43 4 753 519* 7LI3 779 4 8 59 73- J 4 .806 1938 75-25 .833 6798 77-47 5 2.753 94 6 i 71.17 2.779 9249 73.18 2.806 6454 75.29 2.834 1447 77.50 6 754 3732 71.20 .780 3641 73.21 .807 0973 75-32 .834 6098 77-54 7 754 *4 71.23 .780 8034 73-24 .807 5493 75-36 8 35 0752 77-5 8 8 755 2279 71.26 .781 2430 73.28 .808 0016 75.40 .835 5408 77.62 9 755 6 55 6 71.30 .781 6828 73-3 1 .808 4541 75-43 .836 0066 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 5116 71.36 .782 5630 73.38 .809 3597 75-5 .836 9390 77-73 12 75 6 9399 71.40 .783 0034 73.42 .809 8128 75-54 8 37 4055 77-77 13 757 3 68 3 71.43 .783 4440 73-45 .810 2662 75-5 8 .837 8722 77.81 14 757 797 71.46 .783 8848 7349 .810 7197 75.61 .838 3392 77- 8 5 15 2.758 2259 71.49 2.784 3258 73-53 2.811 1735 75-65 2.838 8064 77.89 16 .758 6549 7L53 .784 7671 73.56 .811 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 5*37 71-59 .785 6502 73-63 .812 5362 75.76 .840 2093 78.00 19 759 9433 71.63 .786 0920 73.66 .812 9908 75-79 .840 6774 78.04 20 2.760 3732 71.66 2.786 5341 73-7 2.813 4457 75- 8 3 2.841 1458 78.08 21 22 .760 8032 .761 2335 71.69 71.73 .786 9764 .787 4189 73-73 73.76 .813 9008 .814 3561 75- 8 7 75.90 .841 6144 .842 0832 78.0 78.15 23 .761 6639 71.76 .787 8615 73.80 .814 8117 75-94 .842 5522 78.19 24 .762 0946 71.79 .788 3044 73.83 .815 2674 75.98 .843 0215 78.23 25 2.762 5255 -71.83 2.788 7476 73.87 2.815 7234 76.01 2.843 499 78.27 26 .762 9565 71.86 .789 1909 73-9 .816 1796 76.05 .843 9607 78.31 27 .763 3878 71.89 .789 6344 73-94 .816 6360 76.09 .844 4306 7 8 -35 28 .763 8192 71.93 .790 0781 73-97 .817 0927 76.12 .844 9008 78.38 29 .764 2509 71.96 .790 5221 74.01 .817 5495 76.16 .845 3712 78.42 30 2.764 6827 71.99 2.790 9662 74.04 2.818 0066 76.20 2.845 8 4*9 78.46 31 .765 1148 72.03 .791 4106 74.08 .818 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-'5 .819 3792 76.31 8 47 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 1 9* 74.22 2.820 2953 76.38 2.848 1986 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 2123 7646 .849 1430 7 8 -73 38 .768 1448 72 26 794 5271 74.32 .821 6712 76.49 .849 6155 7 8 -77 39 .768 5784 72.29 794 9731 74- 3 6 .822 1302 76.53 .850 0882 78.81 40 2.769 0123 7^-33 2-795 4i94 74.40 2.822 5895 76-57 2.850 5612 7 !-! 5 41 .769 4464 72.36 795 86 59 74-43 .823 0491 76.60 .851 0344 78.89 42 .769 8806 72-39 .796 3126 74-47 .823 5088 76.64 .851 5079 78.93 1 43 .770 3151 72.43 .796 7595 74.50 .823 9688 76.68 .851 9816 78. 97 1 44 .770 7498 72.46 .797 2066 74-54 .824 4289 76.72 8 5 2 4555 79.01 45 2.771 1846 72.50 2-797 6539 74-5 8 2.824 88 94 76-75 2.852 9297 79.05 46 .771 6197 7^-53 .798 1015 74.61 .825 3500 76.79 .853 4041 79.08 47 .772 0550 72.56 .798 5492 74.64 .825 8108 7683 .853 8787 79.12 48 .772 4905 72.60 .798 9972 74.68 .826 2719 76.87 8 54 3535 79.16 49 .7-2 9262 72.63 799 4454 74-7 .826 7332 76.90 .854 8286 7920 : so 2.7-3 3621 72.67 2-799 8 93 8 74-75 2.827 1947 76.94 2.855 3 4o 79.24 51 773 7982 72.70 .800 3424 74-79 .827 6565 7698 8 55 7795 79.28 52 .774 2344 7 2 -73 .800 7912 74.82 .828 1185 77.01 8 5 6 2553 i 79.32 53 .774 6709 72.77 .801 2402 74.86 .828 5807 77-5 .856 73x4 79-3& 54 775 i77 72.80 .801 6895 74.89 .829 0431 77.09 .857 2077 79.40 55 2.775 544 6 72.84 2.802 1390 74-93 2.829 5058 77.13 2.857 6842 79-44 56 775 9 8l 7 72.87 .802 5886 74-96 .829 9686 77.16 .858 1609 79.48 57 .776 4190 72.90 .803 0385 75.00 .830 4317 77.20 .858 6379 79.52 58 59 .776 8565 .777 2942 72.94 72.97 .803 4886 .803 9390 75.04 75.08 .830 8951 .831 3586 77.24 77.28 .859 1151 .859 5926 79.56 79.60 60 2.777 7322 73.01 2.804 3895 75.11 2.831 8224 77-32 2.860 0703 79.64 600 TABLE VI. ]? or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 140 141 142 143 logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". O' 2.860 0703 79.64 2.889 1754 82.08 2.919 1831 84.65 2.950 1420 87-37 1 .860 5482 79.68 .889 6680 82.12 .919 6911 84.70 .950 6664 87-41 2 .861 0264 79.72 .890 1609 82.16 .920 1994 84.74 .951 1910 87.46 3 .861 5048 79.76 .890 6540 82.20 .920 7080 84.78 95 1 7159 87.50 4 .861 9835 79.80 .891 1473 82.25 .921 2169 84.83 .952 2411 87-55 5 2.862 4624 79.84 2.891 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 2 9*3 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 I2 33 82.41 .923 2549 85.01 .954 3446 87.74 9 .864 3803 80.00 893 6l 79 82.46 .923 7650 85.05 954 8711 87.79 10 2.864 8604 80.04 2.894 1127 82.50 2.924 2755 85.10 2.955 398o 87.81 11 .865 3408 80.08 .894 6078 82.54 .924 7862 85.14 955 9*51 87.88 12 .865 8213 80.12 .895 1032 82.58 .925 2972 85.18 .956 4525 87.93 13 .866 3021 80.16 .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 58^ 88.02 15 2.867 2645 80.34 2.896 5909 82.71 2.926 8317 85.32 2.958 0365 88.07 16 .867 7460 80.28 .897 0873 82.75 .927 3437 85.36 .958 5651 88.11 17 .868 2278 80.32 .897 5839 82.79 .927 8560 85.41 959 939 88.16 18 .868 7098 80.36 .898 0808 82.84 .928 3686 85.45 .959 6230 88.21 19 .869 1921 80.40 .898 5780 82.88 .928 8814 85.50 .960 1524 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 .961 2I2O 88-35 22 .870 6403 80.52 .900 0709 83.01 .930 4216 85.61 .961 7423 88.40 23 .871 1235 80.56 .900 5691 83.05 93 9355 85.68 .962 2728 88.45 24 .871 6070 80.60 .901 0675 83.09 .931 4497 85.72 .962 7036 88.49 25 2.872 0907 80.64 2.901 5662 83.13 2.931 9641 85.77 *-9 6 3 3347 88.54 26 .872 5747 80.68 .902 0651 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 0638 83.26 933 S^ 1 85.91 .964 9297 88.68 29 .874 0280 80.80 .903 5635 83.31 934 M7 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 5 6 5 86.04 .966 5273 88.83 32 875 4835 80.92 .905 0642 83-43 935 57^9 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 4551 81.01 .906 0659 83.52 .936 6064 86.17 .968 1275 88.97 35 36 2.876 9413 .877 4277 81.05 81.09 2.906 5672 .907 0687 83.56 83.61 2.937 1236 937 6410 86.22 86.26 2.968 6615 .969 1957 89.02 89.07 37 .877 9143 81.13 .907 5704 83.65 .938 1587 86.31 .969 7303 89.12 38 .878 4012 81.17 .908 0725 83.69 .938 6767 86.35 .970 2651 89.17 39 .878 8883 81.21 .908 5748 83.74 .939 1950 86.40 .970 8002 89.21 40 *- 8 79 3757 81.25 2.909 0773 83.78 2.939 7135 86.45 2.971 3356 89.26 41 .879 8633 81.29 .909 5801 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 2.911 5940 8399 2.942 1103 86.67 2.974 0170 89.50 46 .882 3052 81.50 .912 0981 84.04 .942 8305 86.72 974 5541 89.55 47 .882 7943 81.54 .912 6024 84.08 943 35io 86.77 975 9 l6 89.60 48 .883 2837 81.58 .913 1070 84.13 943 8717 86.81 975 6293 89.65 49 883 7733 81.62 .913 6119 84.17 944 39 2 7 86.86 .976 1673 89.69 50 2.884 2631 81.66 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 53 .885 2436 .885 7342 81.75 81.79 .915 1282 .915 6341 84.30 84-34 945 9574 .946 4795 87.00 87.04 977 7831 .978 3223 a a o 89.84 89.89 54 .886 2251 81.83 .916 1403 84.39 947 OOI 9 87.09 .978 8618 89.94 55 2.886 7162 81.87 2.916 6468 84.43 2.947 5245 87-13 1.979 4015 89.99 56 57 58 59 .887 2075 .887 6991 .888 1910 .888 6831 81.91 81.95 81.99 82.04 .917 1535 .917 6605 .918 1678 .918 6753 84.48 84.52 84.56 84.61 .948 0475 .948 5707 .949 0942 949 6180 87.18 87.23 87.27 8 7 . 3 z 979 94i6 .980 4820 .981 1226 .981 6636 90.01 90.08 90.13 90.18 60 2.889 1754 82.08 2.919 1831 84.65 2.950 1420 87-37 2.982 1048 90.*^ 601 TABLE VI. Far finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 144 145 146 147 logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 2.982 1048 90.23 3.015 1281 93.26 3.049 2733 96.47 3.084 6070 99.87 1 .982 6463 90.28 .015 6878 93-3 1 .049 8522 96.52 .085 2064 99.92 2 .983 1882 9-33 .016 2478 93.36 .050 4315 96.58 .085 8061 99.98 3 .983 7303 qo. 3 S .016 8082 93.42 .051 on 2 96-63 .086 4062 100.04 4 .984 2727 90.43 .017 3688 93-47 .051 5911 96.69 .087 0066 100.10 1 5 2.984 8154 90.48 3.017 9298 93.52 3.052 1714 96.74 3.087 6073 ioo. 16 6 .985 3584 9-53 .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 100.28 8 .986 4453 90.63 .019 6145 93.68 .053 9142 96.91 .089 4118 100.33 9 .986 9892 90.67 .020 1768 93-73 54 4959 96.96 .090 0140 100.39 1O i-9 8 7 5334 90.72 3.020 7393 93-7 8 3.055 0778 97.01 3.090 6165 100.45 11 .988 0779 90.77 .021 3021 93.83 .055 6601 97.07 .091 2194 100.51 12 .988 6227 90.82 .O2I 8653 93.89 .056 2427 97.I3 .091 8226 100.57 13 .989 1678 90.87 .022 4288 93-94 .056 8256 97.19 .092 4262 100.63 14 .989 7132 90.92 .022 9926 93-99 .057 4089 97.24 .093 0302 100.69 15 2.990 2589 90.97 3.023 5567 94.04 3.057 9925 97.30 3.093 6345 100.75 16 .990 8049 91.02 .024 121 I 94.10 .0.58 5765 97-35 .094 2392 I00.8I 17 .991 3512 91.07 .024 6859 94.15 .059 1608 97.41 .094 8442 100.87 18 .991 8977 91.12 .025 2509 94.20 .059 7454 97-47 .095 4496 100.93 19 .992 4446 91.17 .025 8163 94.26 .060 3304 97.52 .096 0553 100.98 20 A. 992 9918 91.22 3.026 3820 94- 3 1 3.060 9157 97.58 3.096 6614 101.04 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 101.16 23 994 6351 9'-37 .O28 o8lO 94-47 .062 6736 97-75 .098 4818 101.22 24 995 I& 35 91.42 .028 6479 94.52 .063 2602 97.80 .099 0893 101.28 25 1-995 73" 91.47 3.029 2152 94-57 3.063 8472 97.86 3.099 6972 101.34 26 .996 2812 91.52 .029 7828 94-63 .064 4345 97.91 .100 3054 101.40 27 .996 8305 91-57 .030 3507 94.68 .065 O222 97-97 .100 9140 101.46 28 997 3 801 91.62 .030 9190 94-73 .065 6101 98.03 .101 5230 101.52 29 997 93 91.67 .031 4875 94-79 .066 1985 98.08 .102 1323 101.58 30 31 2.998 4802 999 37 91.72 91.77 3.032 0564 .032 6256 94.84 94.89 3.066 7872 .067 3762 98.14 98.20 3.102 7420 .103 352O 101.64 101.70 32 999 5 8l 5 91.82 .033 1951 94-94 .067 9655 98.25 .103 9624 101.76 33 3.000 1326 91.87 .033 7650 95.00 .068 5552 98.31 .104 5732 101.82 34 .000 6840 91.93 034 3351 95.05 .069 1453 98.37 .105 1843 101.88 35 3.001 2357 91.98 3.034 9056 95.11 3.069 7357 98.42 3- 10 5 795 8 101.94 36 .001 7877 92.03 .035 4764 95.16 .070 3264 98.48 .106 4076 102.00 37 .002 3400 92.08 .036 0475 95.22 .070 9174 98.54 .107 0198 IO2.O7 38 .002 8926 92.13 .036 6190 95.27 .071 5088 98.60 .107 6324 102.13 39 .003 4456 92.18 .037 1908 95-3 2 .072 1006 98.65 .108 2454 102.19 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 3 8 3353 95-43 .073 2851 98.77 .109 4723 102.31 42 .005 1062 9L33 .038 9080 95.48 .073 8779 98.82 .no 0864 102.37 43 .005 6603 92.38 .039 4811 95-54 .074 4710 98.88 .no 7008 102.43 44 .006 2148 92.44 .040 0545 95.60 .075 0645 98.94 "' 3*55 102.49 45 3.006 7696 92.49 3.040 6282 95-65 3.075 6583 99.00 3.111 9306 102-55 46 47 .007 3246 .007 8800 92.54 92.59 .041 2023 .041 7767 95.70 95.76 .07* 2524 .076 8469 99.05 99.11 .112 5461 .113 l620 102.61 102.67 48 .ooS 4357 92.64 .042 3514 95.81 .077 4418 99.17 .113 7782 102.73 49 .008 9917 92.69 .042 9264 95.86 .078 0370 99-13 .114 3948 102.80 50 3.009 5480 92.74 3.043 5017 95-91 3.078 6325 99.28 3.115 0118 102.86 51 .010 1046 92.79 .044 0774 95-97 .079 2284 99-34 .115 6291 102.92 52 .010 6615 92.85 .044 6534 96.01 .079 8246 99.40 .116 2468 102.98 53 .on 2188 92.90 .045 2297 96.08 .080 4212 99.46 .116 8649 103.04 54 .011 7763 92.95 .045 8064 96.14 .081 0181 99.52 .117 4833 103.10 55 3.012 3342 93.00 3.046 3834 96.19 3.081 6154 99-57 3.118 1022 103.16 56 .012 8923 93-05 .046 9607 96.25 .082 2130 99.63 .118 7213 103.23 57 .013 4508 93.10 47 53 8 3 96.30 .082 8no 99.69 .119 3409 103.29 58 .014 0096 93.16 .048 1163 96.36 .083 4093 99-75 .119 9608 103.35 59 .014 5687 93.21 .048 6946 96.41 .084 0080 99.81 .120 5811 103.41 60 3.015 I28l 93.26 3.049 2733 96.47 3.084 6070 99.87 3.I2I 20l8 103.48 602 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 148 149 150 151 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' .121 20l8 103.48 .159 1767 107.31 .198 4984 111.41 .239 3820 115.77 1 .121 8228 103.54 .159 7808 107.38 .199 1671 111.48 .240 0768 115.85 2 3 .122 4442 .127 OOOO 103.60 103.66 .160 4253 .161 0702 107.45 107.51 .199 8361 .200 5056 " X -SS 111.62 .240 7722 .241 4680 115.92 116.00 4 .123 6882 103.72 .161 7154 107.58 .201 1755 111.69 .242 1642 116.08 5 .124 3107 103.79 .162 3611 107.65 .201 8459 111.76 3.242 8608 116.15 6 .124 9336 103.85 .163 0072 107.71 .202 5166 111.83 .243 5580 116.23 7 .125 5569 103.91 .163 6536 107.78 .203 1878 111.90 .244 2556 Il6.70 8 9 .126 1805 .126 8045 103.97 104.04 .164 3005 .164 9478 107.85 107.91 .203 8594 .204 5315 111.97 112.04 .244 9536 .245 6521 116.38 116.45 10 11 12 .127 4289 .128 0537 .128 6789 104.10 104.16 104.22 3-i 6 5 5955 .166 2435 .166 8920 107.98 108.04 108.11 .205 2040 .205 8769 .206 5502 112. II 112.18 112.26 3.246 3511 .247 0505 .247 7503 116.53 116.61 116.68 13 .129 3044 104.29 .167 5409 108.18 .207 2239 112.33 .248 4507 116.76 14 .129 9303 104.35 .168 1901 108.25 .207 8981 1 1 2.40 .249 1515 116.84 15 .130 5566 104.41 3.168 8398 108.31 3.208 5727 112.47 3.249 8527 116.91 16 .131 1833 104.48 .169 4899 108.38 .209 2478 112.54 .250 5544 116.99 17 .131 8103 104.54 .170 1404 108.45 .209 9232 112.61 .251 2566 117.07 18 19 .132 4377 133 6 55 104.60 104.67 .170 7913 .171 4426 108.51 108.58 .210 5991 .211 2755 112.69 112.76 .251 9592 .252 6623 117.14 117.22 20 21 3.133 6937 .134 3223 104.73 104.79 3.172 0942 .172 7463 108.65 108.72 3-2II 9522 .212 6294 112.83 112.90 3.253 3658 .254 0698 117.30 "7-37 22 .134 9512 104.86 .173 3988 108.78 .213 3070 112.97 *54 7743 117.45 23 24 135 5o5 .136 2IO2 104.92 104.98 .174 0517 .174 7051 108.85 108.92 .213 9851 .214 6636 113.05 113.12 .255 4792 .256 1846 117.53 117.60 25 3.136 8403 105.05 3-J75 3588 108.99 3.215 3425 113.19 3.256 8905 117.68 26 .137 4708 105.11 .176 0129 109.06 .2l6 0219 113.26 .257 5968 117.76 27 .138 1016 105.17 .176 6674 109.12 .2l6 7017 "3-34 .258 3036 117.84 28 29 .138 7329 .139 3645 105.24 105.30 .177 3224 .177 9777 109.19 109.26 .217 3819 .218 0626 113.41 113-48 .259 0109 .259 7186 117.91 117.99 30 3.139 9965 105.36 S-^ 8 6 335 109.33 3.218 7437 "3-55 3.260 4268 118.07 31 .140 6289 i5-43 .179 2897 109.40 .219 4252 113.63 .261 1354 118.15 32 .141 2616 105.49 .179 9462 109.46 .220 1072 113.70 .261 8446 118.23 33 34 .141 8948 .142 5283 105.55 105.62 .180 6032 .181 2606 i9-53 109.60 .220 7896 .221 4724 "3-77 113.84 .262 5542 .263 2642 118.30 118.38 35 3.143 1622 105.68 3.181 9184 109.67 3.222 1557 113.92 3.263 9747 118.46 36 143 79 6 5 105.75 .182 5766 109.74 .222 8395 113.99 .264 6857 "8.54 37 38 .144 4312 .145 0663 105.81 105.87 183 2353 .183 8943 109.81 109.87 .223 5276 .224 2082 114.06 114.14 265 3972 .266 1091 118.62 118.70 39 .145 7018 105.94 .184 5538 109.94 .224 8933 114.21 .266 8216 118.77 40 41 3.146 3376 .146 9739 106.00 106.07 3.185 2136 .185 8739 IIO.OI 110.08 3.225 5788 .226 2647 114.28 114.36 3-^7 5345 .268 2478 118.85 118.93 42 .147 6105 106.14 .186 5346 110.15 .226 9511 "4-43 .268 9616 119.01 43 .148 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 106.33 3.188 5192 110.36 3.229 0129 114.65 3.271 1060 119.25 46 .150 1609 106.40 .189 1815 110.43 .229 7010 114.73 .271 8217 "9-33 47 .150 7995 1 06.4(1 .189 8443 110.50 .230 3896 114.80 .272 5379 119.41 48 .151 4385 106.53 .190 5075 110.57 .231 0786 114.88 .273 2546 119.49 49 .152 0778 106.59 .191 1711 110.64 .231 7681 114.95 .273 9717 119.57 50 3.152 7176 106.66 3.191 8351 110.71 3.232 4581 115.03 3.274 6894 119.65 51 52 153 3577 .153 9983 106.72 106.79 .192 4996 .193 1644 110.77 110.84 .233 1484 .233 8392 115.10 115.17 .275 4075 .276 1261 "9-73 119.81 53 .154 6392 106.85 .193 8297 110.91 34 535 115.25 .276 84.52 119.89 54 .155 2805 106.92 .194 4954 110.98 .235 2222 "5-3* .277 5647 119.97 55 56 57 3.155 9222 .156 5643 .157 2068 106.99 107.05 107.12 3.195 1615 .195 8281 .196 4950 111.05 III. 12 III.I9 3.235 9144 .236 6070 .237 3 OOI 115.40 115.47 "5-55 3.278 2848 .279 0053 .279 7263 120.05 120.13 120.21 58 59 .157 8497 .158 4930 107.18 107.25 .197 1624 .197 8302 III.26 111.34 .237 9936 .238 6876 115.62 115.70 .280 4477 .281 1697 120.29 120-37 GO 3-i<;Q 1367 107.31 3.198 4984 III.4I 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. V. 152 153 154 155 logM. Diff. 1". logM. Diff. 1". logM. DiF 1". logM. Diff. 1". O' 3.281 8921 120.45 3.326 1448 125.46 3.372 2684 130.85 3.420 4064 136.66 1 .282 6151 120.53 .326 8978 1^5-55 373 53 8 130.94 .421 2266 136.76 2 3 .283 3385 .284 0624 120.61 120.69 .327 6513 .328 4054 125.63 125.72 373 8397 .374 6262 131.04 131.13 .422 0475 .422 8690 136.86 136.96 4 .284 7868 120.77 .329 1600 125.81 375 4133 131.22 .423 6910 137.06 5 3.285 5116 120.85 3.329 9151 125.89 3.376 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 121.01 .331 4268 126.07 377 7778 131.50 .426 1609 '37-37 ; 8 .287 6891 121. 10 .332 1835 126.16 378 5 6 7i 131.60 .426 9854 J37-47 | 9 .288 4160 I2I.I8 .332 9407 126.24 379 3570 131.69 .427 8105 137-57 1O 11 3.289 1433 .289 8711 121 26 121.34 3-333 6 984 334 4567 126.33 126.42 3.380 1474 .380 9384 131.79 131.88 3.428 6362 .429 4626 137.67 137.77 12 .290 5993 121.42 335 2154 126.51 .381 7300 131.98 .430 2895 137.88 13 .291 3281 121.50 335 9747 126.59 .382 5221 132.07 .431 1171 I37-98 14 .292 0574 121.59 33 6 734 6 126.68 383 3H8 132.16 .431 9452 138.08 15 3292 7872 121.67 3-337 4949 126.77 3.384 1081 132.26 3.432 7740 138.18 16 293 5 I 74 121.75 .338 2558 126.86 .384 9019 132.35 433 6034 138.29 17 .294 2481 121.83 339 OI 72 126.95 385 6963 132.45 434 4334 '38.39 18 .294 9794 121.91 339 779 2 127.03 .386 4913 132.54 .435 2641 138.49 19 .295 7111 122.00 34 5417 127.12 .387 2869 132.64 43 6 953 138.59 20 3.296 4433 122.08 3-341 3047 127.21 3.388 0830 132.73 3.436 9272 138.70 21 .297 1761 I22.I6 .342 0682 127.30 .388 8797 132.83 437 7597 138.80 ! 22 .297 9093 122.24 .342 8323 127-39 .389 6770 J 32-93 .438 5928 138.90 ; 23 .298 6430 122.33 343 59 6 9 127.48 -39 4749 133.02 .439 4266 139.01 24 .299 3772 122.41 344 3620 127.57 .391 2733 133.12 .440 2609 139.11 25 3.300 1119 122.49 3-345 I2 77 127.66 3.392 0723 133.22 3.441 0959 139.22 26 .300 8471 122.58 345 8 939 '27-75 .392 8719 '33-31 .441 9315 139.32 27 .301 5828 122.66 .346 6606 127.84 393 6 72o 133.41 .442 7677 13942 28 .302 3190 122.74 347 4279 127-93 394 4728 '33-5 .443 6046 '39-53 29 .303 0557 122.83 .348 1958 128.02 395 274 1 133.60 .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 122.99 349 733 128.19 .396 8785 J33-79 .446 1189 139.84 32 .305 2688 123.08 35 524 128.28 397 6815 133-89 44 6 9583 139-95 ! 33 .306 0075 123.16 .351 2724 128.37 .398 4852 1 33-99 447 7983 140.05 34 .306 7468 123.24 .352 0429 128.46 399 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 123.41 353 585 6 128.65 .400 8996 134.28 45 3221 140.37 37 38 .308 9674 .309 7086 123.50 123.58 354 3577 355 '34 128.74 128.83 .401 7056 .402 5122 134-38 134.48 .451 1646 452 0077 140.47 14057 39 .310 4504 123.66 355 937 128.92 .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 3 11 9354 123.83 357 45i7 129.10 .404 9354 134-77 454 541 140.90 42 .312 6786 123.92 .358 2266 129.19 45 7443 I34-87 455 3867 141.00 43 3*3 4224 124.00 .359 0020 129.28 .406 5538 134-97 .456 2330 141.11 44 .314 1667 124.09 359 778o 129.37 .407 3639 I35-07 .457 0800 141.21 45 3-3H 9H5 124.17 3.360 5545 129.46 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 i 47 .316 4025 124.34 .362 1092 129.65 .409 7977 '35-36 .459 6248 141.54 ! 48 -317 1489 1 24-43 .362 8873 129.74 .410 6102 I35-46 .460 4743 141.64 49 -3 1 ? 8 957 124.51 .363 6660 129.83 .411 4233 I35-5 6 .461 3245 141.75 50 3-3 l8 6 43 124.60 3-3 6 4 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 0268 141.97 52 .320 1392 124.77 .366 0055 130.11 .413 8660 135.86 .463 8789 142.07 53 .320 8881 124.86 .366 7864 130.20 .414 6815 135.96 .464 7317 I42.I8 54 .321 6375 124.94 .367 5679 130.29 4*5 4975 136.06 465 5851 142.29 55 3.322 3874 125.03 3.368 3499 130.38 3.416 3142 136.16 3.466 4392 142.40 56 .323 1379 125.11 .369 1325 130.48 .417 1314 136.26 467 2939 142.51 57 .323 8888 125.20 .369 9156 *3-57 .417 9492 136.36 .468 1492 I42.6l 58 .324 6403 125.29 .370 6993 130.66 .418 7677 136.46 .469 0052 142.72 59 .325 3923 125-37 .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. V. 156 157 158 159 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 3.470 7192 142.94 3.523 3875 '49-75 3-578 6154 157.17 3.636 6351 165.28 1 2 3 47' 5772 .472 4358 473 2951 '43-5 143.16 143.27 .524 2864 .525 1860 .526 0863 149.87 149.99 150.11 579 5588 .580 5030 .581 4480 157.30 '57-43 '57-56 .637 6274 -638 6202 .639 6140 '6542 165.56 4 .474 1550 I43-38 .526 9873 150.23 S 8 ^ 3937 157.69 .640 6087 165.85 5 6 3.475 0156 475 8769 143.49 143.60 3.527 8890 .528 7915 '50-35 150.47 3.583 3403 .584 2876 157.82 '57-95 3.641 6042 .642 6006 165.99 166.13 7 .476 7388 '43-7 1 .529 6947 150.59 585 2357 158.08 .643 5978 166.28 8 .477 6014 143-82 .530 5985 150.71 .586 1846 158.21 .644 5959 166.42 9 .478 4646 '43-93 53' 53' 150-83 587 '342 '58-34 645 5948 166.56 10 3.479 3285 144.04 3-532 4085 150.95 3.588 0847 158.47 3.646 5946 166.71 11 12 .480 1931 .481 0583 144-15 144.26 533 3'45 .534 2213 151.07 151.19 589 0359 .589 9880 158.61 158.74 647 5953 .648 5968 166.85 166.99 13 .481 9242 '44-37 .535 1288 151.31 .590 9408 158.87 .649 5992 167.14 14 .482 7907 144-48 536 0370 151.43 .591 8944 159.00 .650 6025 167.28 15 3.483 6579 '44-59 3.536 9459 '5'-55 3.592 8488 '59-'3 3.651 6066 167.42 16 .484 5258 144.70 537 8556 151.67 593 8040 159.26 .652 6116 167.57 17 485 3944 144-81 .538 7660 151.79 594 7600 159.40 .653 6175 167.72 18 19 .486 2636 487 '335 '44-93 145.04 539 6771 .540 5890 151.91 152.04 595 7167 596 6743 '59-53 159.66 .654 6242 655 6318 167.86 168.01 20 21 22 3.488 0040 .488 8752 489 7472 145.15 145.26 '45-37 3-54' 5'5 .542 4148 543 3289 152.16 152.28 152.40 3-597 6327 .598 5919 599 55'8 '59-79 '59-93 160.06 [.656 6403 657 6497 .658 6599 168.15 168.30 168.45 23 .490 6198 '45-49 .544 2436 152.52 .600 5126 160.19 659 6710 168.59 24 49' 493 145.60 545 '59' 152.65 .601 4742 160.33 .660 6830 168.74 25 3.492 3670 145.71 3.546 0754 152.77 3.602 4365 160.46 ;.66i 6959 168.89 26 493 2416 145-82 .546 9924 152.89 .603 3997 160.60 .662 7096 169.03 27 .494 1 1 68 145-94 .547 9101 153.01 .604 3637 160.73 .663 7243 169.18 28 .494 9928 146.05 .548 8285 153-14 .605 3285 160.87 .664 7398 169.33 29 495 8695 146.16 549 7477 153.26 .606 2941 161.00 .665 7562 169.48 30 .496 7468 146.28 3.550 6677 '53.38 1-607 2605 161.14 3.666 7735 169.62 31 .497 6248 146.39 .551 5883 '53-5' .608 2277 161.27 .667 7917 169.77 32 .498 5035 146.50 552 597 153-63 609 1957 161.41 .668 8108 169.92 33 499 3828 146.62 553 43'9 '53-75 .610 1646 161.54 .669 8308 170.07 34 .500 2629 146.73 554 3548 153.88 .611 1342 161.68 .670 8516 170.22 35 .501 1436 146.85 555 2785 154-00 1-612 1047 161.81 .671 8734 170.37 36 .502 0250 146.96 .556 2029 '54-'3 .613 0760 161.95 .672 8961 170.52 37 38 .502 9071 .503 7899 147.08 147.19 .557 1280 558 0539 154-25 .614 0481 .615 0210 162.09 162.22 .673 9196 .674 9441 170.67 170.82 39 .504 6734 147-3' .558 9806 154.50 .615 9948 162.36 .675 9 6 94 170.97 40 .505 5576 147-42 -559 98o 154-63 .6l6 9693 162.50 .676 9957 171.12 41 .506 4425 '47-54 .560 8361 '54-75 .617 9447 162.63 .678 0228 171.27 42 .507 3280 '47-65 .561 7650 154.88 .618 9209 162.77 .679 0509 171.42 43 .508 2143 147.77 .562 6947 155.01 .619 8980 162.91 .680 0799 '71-57 44 .509 IOI2 147-88 .563 6251 '55-'3 .620 8758 163.05 .681 1098 171.72 45 .509 9889 148.00 .564 5562 155.26 .621 8545 163.18 .682 1406 171.87 46 .510 8 77 2 148.11 .565 4882 155.38 .622 8340 163.32 .683 1723 172.03 I 47 .511 7662 148.23 .566 4209 '55-5' .623 8144 163.46 .684 2049 172.18 - 48 49 .512 6560 5 '3 5464 148.34 148.46 5 6 7 3543 .568 2885 I55-64 155.76 .624 7956 .625 7776 163.60 163.74 .685 2384 .686 2728 172-33 172.48 50 5'4 4375 148.58 .569 2235 155.89 .626 7604 163.88 .687 3082 172.64 51 5'5 3294 148.70 .570 1592 156.02 .627 7441 164.02 .688 3445 172.79 52 .516 2219 148.81 57' 0957 156.15 .628 7287 164.16 .689 3817 172.94 53 5'7 "5i 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 .691 4588 '73-25 55 .518 9037 149.17 573 9098 156.53 .631 6873 164.58 .692 4988 I7340 56 5'9 799 149.28 574 8494 156.66 .632 6751 164.72 .693 5397 173.56 57 58 59 .520 6951 .521 5918 .522 4893 149.40 149.52 149.64 575 7897 576 738 577 6727 156.79 156.92 '57-4 633 6638 634 6534 .635 6438 164.86 165.00 165.14 694 5815 .695 6243 .696 6680 '73-7' 173.87 174.02 60 523 3875 '49-75 .578 6154 I57-I7 .636 6351 165.28 .697 7126 174.18 605 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 160 161 162 163 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 3.697 7126 174.18 3.762 1519 183.99 3.830 3147 194.87 3.902 6107 207.00 1 2 .698 7581 .699 8046 '74-34 174.49 .763 2584 .764 3639 184.16 I84-34 .831 4845 .832 6554 195.06 195.25 .903 8534 .905 0973 207.21 207.43 3 .700 8520 174.65 .765 4704 184.51 833 8275 '95-44 .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 195.83 3.908 8368 208.08 i 6 .703 9999 175.12 .768 7963 185.03 .837 3508 196.02 .910 0859 208.29 7 .705 0511 175.28 .769 9070 185.20 .838 5275 196.22 .911 3363 208.51 8 .706 1032 '75-43 .771 0187 185.38 839 754 196.41 .912 5880 208.72 9 .707 1562 '75-59 .772 1315 I85.55 .840 8844 196.60 .913 8410 208.94 10 3.708 2102 '75-75 3-773 2 454 185.73 3.842 0646 196.80 3-9'5 953 209.16 11 12 13 .709 2652 .710 3211 7" 378o 175-9' 176.07 176.22 774 3 6 3 775 4762 .776 5932 185.90 186.08 186.25 .843 2460 .844 4286 .845 6133 196.99 197.19 '97-3 8 .916 3509 .917 6078 .918 8661 209.38 209.60 209.81 14 .712 4358 176.38 777 7"2 186.43 .846 7972 197.58 .920 1256 210.03 15 16 3.713 4946 7'4 5543 176.54 176.70 3.778 8303 779 955 186.60 186.78 3-847 9833 .849 1705 197.78 197.97 3.921 3865 .922 6487 210.25 210.48 17 .715 6150 176.86 .781 0717 186.96 .850 3589 198.17 .923 9122 210.70 18 .716 6766 177.02 .782 1940 187.14 .851 5486 198-37 .925 1770 210.92 19 .717 7392 177-18 783 3'74 187.31 .852 7394 198.57 .926 4432 211.14 20 21 3.718 8028 .719 8673 '77-34 177.50 3.784 4418 .785 5672 187.49 187.67 3-853 93'4 .855 1245 198.76 198.96 3.927 7107 .928 9795 211.36 211.58 22 .720 9328 177-66 .786 6938 187.85 .856 3189 199.16 93 2 497 211. 8l 23 .721 9993 177-83 .787 8214 188.03 857 5'45 199.36 .931 5212 212.03 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 27 .725 2045 .726 2749 178.31 178.47 .791 2108 .792 3427 188.57 188.75 .861 1084 .862 3087 199.96 200.16 935 3438 .936 6207 212.70 212.93 28 29 .727 3462 .728 4185 178.63 178.80 793 4757 .794 6098 188.93 189.1! .863 5103 .864 7131 200.36 200.56 937 8989 939 '785 213.15 213.38 30 3.729 4918 178.96 3-795 745 189.29 3.865 9171 200.77 3.940 4595 213.61 31 .730 5661 179.13 .796 8812 189.47 .867 1223 200.97 .941 7418 213.83 32 .731 6413 179.29 .798 0186 189.65 .868 3287 201.17 943 02 54 214.06 33 .732 7176 '79-45 799 '57' 189.83 869 5363 201.37 944 3'5 214.29 34 733 7948 179.62 .800 2966 190.01 .870 7452 201.58 945 59 6 9 214.52 35 3.734 8730 I79-78 3.801 4372 190.20 3-871 955* 201.78 3.946 8847 214.74 36 735 95^2 '79-95 .802 5790 190.38 .873 1665 201.98 .948 1738 214.97 37 .737 0324 180.11 .803 7218 190.56 874 379' 202.19 .949 4644 215.20 38 738 H3 6 180.28 .804 8657 190.65 .875 5928 202.39 .950 7563 215.43 39 739 '957 180.45 .806 0108 190.93 .876 8078 202.60 .952 0496 215.66 40 3.740 2789 180.61 3.807 1569 191.11 3.878 0240 202.80 3-953 3443 216.90 41 42 74' 3 6 3' .742 4482 180.78 180.94 .808 3041 .809 4525 191.30 191.48 .879 2414 .880 4601 203.01 203.22 954 6403 955 9378 216.13 216.36 43 743 5344 i8i.ii .810 6020 191.67 .881 6800 203.42 .957 2366 216.59 44 .744 6216 181.28 .811 7525 191.86 .882 9012 203.63 958 53 6 9 216.82 45 3.745 7097 181.45 3.812 9042 192.04 3.884 1236 203.84 3-959 8385 217.06 46 .746 7989 181.61 .814 0570 192.23 .885 3473 204.05 .961 1416 217.29 47 747 8891 181.78 .815 2110 192.41 .886 5722 204.26 .962 4460 217.53 48 748 9803 181.95 .8l6 3660 192.60 887 7983 204.46 .963 7519 217.76 49 .750 0725 182.12 .817 5222 192.79 .889 0257 204.67 .965 0592 218.00 50 3.751 1657 182.29 3.818 6 795 192.98 3.890 2544 204.88 3.966 3678 218.23 51 .752 2599 182.46 .819 8379 193.16 .891 4843 205.09 .967 6779 218.47 j 52 753 355* 182.63 .820 9974 '93-35 .892 7155 205.31 .968 9895 218.70 53 .754 4514 182.80 .822 1581 193-54 .893 9480 205.52 .970 3024 218.94 54 755 5487 182.97 .823 3199 '93-73 .895 1817 205.73 .971 6168 219.18 55 56 57 3.756 6470 757 7464 758 8467 183.14 183.31 183.48 3.824 4829 .825 6470 .826 8122 193.92 194.11 '94-3 3.896 4167 .897 6529 .898 8905 205.94 206.15 206.36 3.972 9326 .974 2498 975 5684 219.42 219.66 219.90 58 59 759 948i .761 0505 183.65 183.82 .827 9785 .829 1460 194.49 194.68 .900 1293 .901 3694 206.57 206.79 .976 8885 .978 2100 220.13 220.37 60 3.762 1539 183.99 3.830 3147 194.87 3.902 6107 207.00 3-979 533 220.61 606 TABLE VI. For findin e the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 164 165 166 167 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. l". o 1 3-979 533 -980 8574 220.62 220.86 4.061 6673 .063 0842 236.01 236.28 4.149 7198 .151 2422 253-57 253.88 4-244 5537 .246 1975 27A.I4 2 3 4 .982 1833 .983 5106 .984 8394 221.10 221-34 221.58 .064 5027 .065 9229 .067 3447 236.56 236.83 237.11 .152 7664 .154 2925 .155 8205 254.19 254.51 247 8434 .249 4916 .251 1419 274.51 5 6 7 8 9 3.986 1696 .987 5013 9*8 8345 .990 1691 .991 5051 221.83 222.07 222.31 222.56 222.80 4.068 7682 .070 1933 .071 6201 .073 0486 .074 4787 237-39 237.66 237-94 238.22 238.50 4-157 354 .158 8822 .160 4159 .161 9515 .163 4891 255.14 255.46 255.78 256.10 256.42 4.252 7944 .254 4491 .256 1061 .257 7652 .259 4266 275.60 275-97 276.34 276.71 10 11 3.992 8427 994 1817 223.05 223.29 4.075 9106 .077 3441 238.78 239.06 4.165 0285 .166 5699 256.74 257.06 4.261 0902 .262 7560 277-45 277.82 12 .995 5222 223.54 .078 7792 239-34 .168 1132 .264 4240 ' L 278.20 13 14 .996 8642 .998 2077 223.79 224.03 .080 2161 .081 6546 239.62 239.90 .169 6585 .171 2056 257.70 258.02 .266 0943 .267 7669 278.95 15 16 17 3-999 5527 I..OOQ 8991 .002 2471 224.28 224-53 224.78 ^.083 0948 .084 5368 .085 9804 240.18 240.46 240.75 4.172 7547 .174 3058 .175 8588 258.35 258.67 259.00 4.269 4417 .271 1187 .272 7981 279.70 280.08 18 .003 5965 225.03 .087 4257 241.03 .177 4138 259 33 .274 4797 280.46 19 .004 9474 225.28 .088 8728 241.32 .178 9707 259.65 .276 1635 280.84 20 4.006 2999 22553 ..090 3215 241.60 ^.180 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 24 .010 3663 .on 7248 226.29 226.54 .094 6781 .096 1337 242.56 242.75 .185 2182 .186 7850 260.97 261.30 .282 9219 .284 6173 282.36 25 4.013 0848 226.79 4.097 5911 243.04 4.188 3538 261.63 ..286 3149 283.14 26 .014 4463 22 7 .05 .099 0502 243 33 .189 9246 261.96 .288 0149 283.52 27 .015 8093 227.30 .100 5110 243.62 .191 4974 262.30 .289 7172 283.91 28 29 .017 1739 .018 5400 227.55 227.81 .101 9736 .103 4379 243-91 244.20 .193 0722 .194 6490 262.63 262.97 .291 4218 .293 1288 284.30 284.69 30 31 4.019 9077 .021 2769 228.06 228.32 .104 9040 .106 3718 244.49 244.78 4.196 2278 .197 8086 263.30 263.64 4.294. 8 3 8l .296 5498 285.08 285.47 32 .022 6476 228.58 .107 8414 245.08 1 99 39*5 263.98 .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 .no 7858 245.67 202 5633 264.66 .301 6990 286.66 35 .026 7691 229.35 .112 2607 245.96 4.204 1523 265.00 4.303 4201 287.05 ' 36 37 .028 1460 .029 5245 229.62 229.88 JI 3 7374 .115 2158 246.26 246.55 205 7433 .207 3363 265-34 265.68 .305 1436 .306 8695 287.45 287.85 38 .030 9045 230.14 .116 6960 246.85 .208 9314 266.02 .308 5978 288.25 39 .032 2861 230.40 .118 1780 247.15 .2IO 5286 266.37 .310 3285 288.65 40 .033 6693 230.66 .119 6618 247.45 .212 1278 266.71 .312 0616 289.05 41 .035 0540 230.92 .121 1474 247-75 .213 7291 267.06 .313 7971 289-45 42 .036 4404 231.18 .122 6348 248.05 .215 3325 267.40 3'5 535 289.86 43 .037 8283 231.45 .124 1239 248-35 .216 9379 267-75 3'7 2753 290.26 44 .039 2177 231.71 .125 6149 248.65 .218 5455 268.10 .319 0181 290.67 45 .040 6088 231.97 .127 1077 248-95 .220 1551 268.44 .320 7633 291.07 46 .042 0015 232.24 .128 6023 249.25 .221 7668 268.79 .322 5110 291.48 ; 47 1 48 43 3957 .044 7915 232.51 232.77 .130 0988 .131 5970 249.56 249.86 .223 3806 .224 9965 269.14 269.50 .324 2611 .326 0137 291.89 I 292.30 49 .046 1890 233.04 .133 0971 250.17 .226 6146 269.85 .327 7688 292.71 50 .047 5880 233-3 1 .134 5990 25047 .228 2347 270.20 .329 5263 293-13 51 .048 9887 233-57 .136 1028 250.78 .229 8570 270.55 .331 2863 293-54 52 .050 3909 233.84 .137 6084 251.08 .231 4814 270.91 333 4 8 7 293-95 53 .051 7948 234.11 .139 1158 25I-39 .233 I0 79 271.27 334 8137 294.37 54 053 2003 234-38 140 6251 251.70 .234 7366 271.62 336 5812 294.79 55 054 6074 234.65 .142 1362 252.01 236 3674 271.98 .338 3511 295.20 56 056 0161 234.92 143 6492 252.32 238 0003 272.34 .340 1236 295.62 57 057 4264 235-19 145 1641 252.63 239 6354 272.70 .341 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 I 4 8 I 994 253-25 242 9121 273.42 345 4562 296.89 6O ..061 6673 236.01 I 49 7198 253.57 244 5537 273-78 347 *388 297.31 607 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 168 169 170 171 logM. Diff. I". logM. | Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 4-347 *388 297.31 4.459 1242 325.07 4.581 9445 358.31 4.717 9835 398.87 j 1 2 .349 0240 .350 8117 *97-74 298.16 .461 0761 .463 0311 3*5-57 326.08 .584 0962 .586 2516 358.92 359-53 .720 3790 .722 7790 399.62 ; 400.38 3 .352 6019 298.59 .464 9891 326.59 .588 4106 360.15 7*5 1835 401.14 4 354 3948 299.02 .466 9501 327.10 59 5734 360.76 .727 5926 401.90 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 8814 328.12 594 910 362.00 .732 4245 403-43 I 7 .359 7888 300.31 .472 8517 328.64 597 0838 362.62 734 8474 44 ^9 i 8 .361 5919 300.75 474 8250 3*9-i5 599 *6i5 3 6 3-*5 737 *749 404.96 . 9 .363 3977 301.18 .476 8015 329.67 .601 4428 363.88 739 77 405.74 1O 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 33 -7i .605 8169 365-14 744 585* 407.30 12 .368 8308 302.49 .482 7495 33'-*3 .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 2 8 75 303.81 4.488 7258 33*-8i 4.614 6106 367.68 4-754 398i 410.45 16 .376 1117 304.26 .490 7242 333-33 .616 8186 368.32 75 6 8632 411.24 17 377 93 86 304.70 .492 7258 333.86 .619 0304 368.96 759 333 412.04 18 379 7 68 i 35-!5 494 73 6 334-4 .621 2461 369.61 .761 8077 412.84 19 .381 6003 35-59 .496 7386 334-93 .623 4657 370.26 .764 2872 4I3-65 20 4-383 435* 306.04 4.498 7498 335-46 4.625 6892 370.91 4.766 7715 414.46 21 385 *7*8 306.49 .500 7642 336.00 .627 9166 37I-56 .769 2606 415.27 22 -387 "3 1 306.94 .502 7818 336.54 .630 1480 372.21 771 7547 416.08 23 .388 9561 37-39 .504 8026 337.08 .632 3832 37*-87 774 *53 6 416 90 24 .390 8019 307.85 .506 8267 337.62 .634 6224 373-53 776 7574 417.72 25 4.392 6503 308.30 4.508 8541 338.16 4.636 8656 374-19 4.779 2662 418.54 I 26 394 5 OI 5 308.76 .510 8847 338.71 .639 1127 374-86 .781 7799 4^9-37 27 39 6 3554 309.21 .512 9186 339.26 .641 3639 375-5* .784 2986 420.20 28 .398 2121 309.67 5'4 9558 339.80 .643 6190 376.19 .786 8222 421.03 29 .400 0715 310.13 .516 9962 34-35 .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 311.06 .521 0871 341-46 .650 4085 378.21 794 4*33 4*3-54 32 .405 6667 311.52 .523 1376 342.02 .652 6798 378-89 .796 9671 424.39 33 .407 5368 3H-99 .525 1913 34*-57 .654 9552 379-57 .799 5160 425.24 34 .409 4102 311-45 .527 2484 343-J3 .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 .413 1652 3'3-39 .531 3728 344.26 .661 8059 381.62 807 1934 427.81 37 .415 0469 313-86 533 44 344.82 .664 0977 382.31 .809 7628 428.67 38 .416 9315 3'4-33 535 5 106 345-39 .666 3936 383.00 .812 3374 4*9-53 39 .418 8189 3H-8o 537 5 8 4 6 345-95 .668 6937 383.70 .814 9172 430.40 40 4.420 7091 315-28 4-539 6620 346.52 4.670 9980 38439 4.817 5022 431.28 41 .422 6022 3'5-75 .541 7429 347-09 .673 3064 385-09 .820 0925 432.15 42 .424 4982 316.23 543 8272 347-67 .675 6191 385.80 .822 6881 433.03 43 .426 3970 316.71 545 9H9 348.24 677 936 386.50 .825 2889 433-9 1 44 .428 2987 317-19 .548 0061 348.82 .680 2571 387.21 .827 8950 434.80 45 4.430 2033 317.67 4.550 1007 349.40 4.682 5825 387-9* 4.830 5065 435.69 46 .432 1108 318.16 .552 1989 349.98 .684 9121 388.63 833 !*34 436.59 47 .434 0212 318.64 554 35 350-56 .687 2460 389-34 835 7456 437.48 48 435 9345 3 I 9- I 3 .556 4056 35i.i5 .689 5842 390.06 838 373* 438-38 49 437 8507 319.61 55 8 5H3 351-73 .691 9268 390.78 .841 0062 439.29 50 4-439 7698 320.10 4.560 6264 35*-3* 4.694 2736 391.50 4.843 6446 440.20 : 51 .441 6919 320.59 .562 7421 352.91 .696 6248 392.23 .846 2886 441.11 52 443 6l6 9 321.08 .564 8614 353-5 .698 9803 392.96 .848 9380 442.03 53 445 5449 321.58 .566 9842 354-io .701 3402 393-68 .851 5929 442.95 54 447 4758 322.07 .569 1106 354-69 .703 7046 394.42 8 54 *533 443.87 55 4.449 4097 322.57 4.571 2405 355-*9 4.706 0733 395-15 4.856 9193 444.80 56 .451 3466 323.06 573 374i 355-89 .708 4464 395.89 .859 5909 44 IH 57 453 *865 323.56 575 5 IJ 3 356.49 .710 8240 396-63 .862 2680 446.66 58 .455 2294 324.06 577 6521 357-io .713 2060 3973 .864 9508 447.60 59 457 753 324.56 579 79 6 5 357.70 7i5 59*5 398.12 .867 6392 448.54 60 4.459 1242 3*5-07 4.581 9445 358.31 4.717 9835 398.87 4.870 3333 449-49 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 172 173 174 175 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 4-870 3333 449-49 5.043 3285 5H-47 5.243 3165 601.00 5.480 1373 722.00 1 .873 0331 450.44 .046 4191 5I5-7I .246 9276 602.69 .484 4765 7*4-4* 2 .875 7386 451-39 .049 5171 516.96 .250 5488 604.38 .488 8304 726.87 3 .878 4499 45*-35 .052 6226 518.21 .254 1802 606.08 .493 1989 7*9-33 4 .881 1668 453-31 055 7356 5I9-47 .257 8218 607.80 497 5823 731.80 5 6 4.883 8896 .886 6182 454.28 455-*5 5.058 8562 .061 9843 5*o-73 522.00 5.261 4738 .265 1361 609.53 611.26 5.501 9806 .506 3939 734-3 736.81 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 458.19 .071 4149 5*5-85 .276 i860 616.52 519 7*4 744-44 10 4.897 5912 459-17 5.074 5738 5*7-14 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 5*9-75 .287 3313 621.87 533 1946 75*-*3 13 .905 8831 462.16 .084 0976 531.06 .291 0680 623.67 537 7158 754.86 14 .908 6591 463.16 .087 2879 53*-38 .294 8154 625.49 54* *5*9 757-51 15 4.911 4411 464.17 5.090 4862 533-71 5.298 5738 627.31 5.546 8060 760.18 16 .914 2291 465.18 .093 6924 535-04 .302 3432 629.15 55i 375i 762.87 17 .917 0233 466.20 .096 9067 536-38 .306 1237 631.00 555 9605 765.58 18 .919 8235 467.22 .100 1290 537-73 39 915* 632.85 .560 5621 768.31 19 .922 6299 468.25 i3 3594 539.08 313 7179 634.72 .565 1802 771.05 20 21 4-9*5 44*5 .928 2612 469.28 470. 3 i 5.106 5980 .109 8447 540.44 541.81 5.317 5319 .321 3571 636.60 638.49 5.569 8148 574 4661 773-8* 776.61 22 .931 0862 471-35 .113 0997 543-18 3*5 1938 640.39 579 1341 779-41 23 933 9174 472.39 .116 3629 544.56 .329 0418 642.30 .587 8190 782.24 24 .936 7549 473-44 .119 6344 545-95 33* 9014 644.23 .588 5210 785.08 25 4-939 5987 474-49 5.122 9143 547-34 5.336 7726 646.16 5.593 2401 787-95 26 .942 4489 475-55 .126 2026 548.74 340 6554 648.11 597 9764 790.84 27 945 353 476.61 .129 4992 550-15 344 5499 650.07 .602 7302 793-75 28 .948 1682 477-68 .132 8044 551-57 .348 4562 652.04 .607 5014 796.68 29 95* 375 478.75 .136 1181 55*-99 35* 3744 654.02 .612 2903 799.63 30 4-953 9i3* 479-83 5.139 4403 554-4* 5-35 6 345 656.01 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 .146 1 1 06 557-3 .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 5 6'o. 2 1 .372 1456 664.11 .636 5041 814.72 35 36 4.968 3894 .971 3044 485.28 486.38 5.156 1813 159 5558 561.68 563.16 5.376 1364 .380 1396 666.17 668.24 5.641 4017 .646 3179 817.81 820.92 37 .974 2260 487.49 .162 9392 564.64 384 1553 670.32 .651 2528 824.05 38 977 1543 488.61 .166 3315 566.13 .388 1834 672.41 .656 2065 827.21 39 .980 0893 489.73 .169 7328 567-63 .392 2242 674.52 .661 1793 830.39 40 41 4.983 0311 985 9795 490.85 491.98 5-173 *43i .176 5624 569.13 570.65 5.396 2777 .400 3439 676.64 678.77 5.666 1713 .671 1825 833.60 836.83 42 .988 9348 493.12 .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 .681 2635 843.36 44 994 8659 495.40 .186 8752 575-*4 .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 .197 2713 579.90 4*5 OI 3 6 691.85 .701 6640 856.75 48 19 .006 8m .009 8148 500.04 501.21 .200 7554 .204 2489 581.47 583-05 .429 1714 433 34*7 694.08 696.33 .706 8147 .711 9860 860.16 863.60 50 5.012 8256 502.39 5.207 7520 584.64 5-437 5*74 698.59 5-717 1779 867.06 51 .015 8435 503-57 .211 2646 586.23 .441 7258 700.86 .722 3908 870.56 52 .018 8685 * J J ' f 504.76 .214 7868 587.84 445 9378 703.15 .727 6247 874.08 53 .021 9006 55-95 .218 1l86 589.45 .450 1636 705.45 -73* 8798 I 77 ' 6 * 54 *4 9399 .221 86O2 591-07 .454 4032 707.77 .738 1563 881.21 55 5.027 9864 508.36 5.225 4116 59*-7i 5.458 6568 710.10 5.743 4544 884.82 56 .031 0402 509-57 .228 9727 59435 .462 9244 712.45 748 774* 888.46 57 .034 1013 510.79 .232 5437 596.00 .467 2062 714.81 754 "59 892.13 58 .037 1697 512.01 .236 1247 597.66 .471 5022 717.19 .759 4798 895-81 59 .040 2454 513.24 .239 7156 599.32 475 81*5 719.59 .764 8659 899.56 60 5.043 3285 5I4-47 5.243 3165 601.00 5.480 1373 722.00 5.770 2745 903.31 609 TABLE VI. I or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 176 177 178 179 logM. Diff. 1". logM. Diff. V. logM. Diff. 1". logM. Diff. 1". 0' 5.770 2745 903.3 6.144 6 *89 1205.3 6.672 5724 1808. 8 7-575 464 3619 1 775 75 8 907.1 .151 8807 I2I2.0 .683 4709 1824.0 597 359 6 3680 2 781 1599 910.9 59 1733 1218.8 .694 4613 1839.5 .619 6295 3744 3 .786 6370 914.8 .166 5070 1225.7 75 5454 1855.3 .642 2868 3809 4 .792 1374 918.7 .173 8823 1232.7 .716 7248 1871.3 .665 3452 3877 5 5.797 6612 922.6 6.181 2997 1239.8 6.728 ooio 1887.5 7.688 8192 3948 6 .803 2086 926.6 .188 7597 1246.9 739 3758 1904.1 .712 7239 4021 7 .808 7798 930.6 .196 2628 1254.1 .750 8509 1921.0 737 0756 4097 8 .814 3751 934.6 .203 8095 1261.4 .762 4279 1938.2 .761 8913 4176 i 9 .819 9946 938.6 .211 4002 1268.8 774 I0 9 1955.6 .787 1889 4257 10 5.825 6386 942.7 6.219 354 1276.3 6.785 8958 1973-4 7.812 9876 4343 11 83 1 373 946.8 .226 7158 1283.8 797 794 1991.5 839 375 443i 12 .837 0008 951.0 .234 4419 1291.5 .809 7946 2OIO.O .866 1702 4524 13 .842 7195 955-2 .242 2142 1299.2 .821 9106 2028. g .893 5986 4620 14 .848 4634 959-5 .250 0333 1307.1 .834 1404 2048.0 .921 6170 4720 15 5.854 2329 963.7 6.257 8997 1315.0 6.846 4863 2067.5 7.950 2513 4825 16 .860 0282 968.0 .265 8139 1323.0 .858 9503 2087.3 7.979 5292 4935 17 .865 8495 972.4 .273 7766 1331.1 .871 5348 2107.6 8.009 4802 5050 18 .871 6970 976.8 .281 7884 J339-4 .884 2422 2128.3 .040 1361 5170 19 .877 5710 981.2 .289 8499 1 347-7 .897 0749 2149.4 .071 5309 5296 20 5.883 4717 985-7 6.297 9617 1356.2 6.910 0353 2170.9 8.103 7 TI 5428 21 .889 3993 990.2 .306 1244 1364.7 .923 1261 2192.8 .136 6857 5568 22 .895 3542 994.8 3 J 4 33 8 7 J 373-3 .936 3498 2215.2 .170 5274 57H 23 .901 3365 999-4 .322 6052 1382.1 949 793 2238.0 .205 2717 5869 24 .907 3465 1004.0 33 9*47 1391.0 .963 2073 2261.4 .240 9679 6032 25 5.913 3845 1008.7 6-339 2977 1400.0 6.976 8466 2285.2 8.277 6700 6204 26 .919 4507 1013.4 .347 7249 1409.1 6.990 6304 2309.6 .315 4361 6387 27 925 5454 1018.1 .356 2072 1418.3 7.004 5616 2334-3 354 3 2 9 8 6580 28 .931 6688 1022.9 .364 7451 1427.6 .018 6437 2359-7 394 4205 6786 29 937 8213 1027.8 373 3395 H37-I .032 8796 2385-7 .435 7842 7004 30 5.944 0030 1032.7 6.381 9910 1446.7 7.047 2729 2412.2 8.478 5044 7238 31 .950 2144 1037.6 .390 7005 1456.4 .061 8271 2439.4 .522 6731 7488 32 .956 4556 1042.6 399 4 68 7 1466.2 .076 5458 2467.1 .568 3920 7755 33 .962 7269 1047.7 .408 2965 1476.2 .091 4329 2495.4 .615 7739 8042 34 .969 0287 1052.9 .417 1846 1486.4 .106 4921 2524.5 .664 9442 8352 35 5-975 3613 1058.0 6.426 1337 1496.7 7.121 7276 2554.2 8.716 0431 8686 36 .981 7249 1063.2 435 J 449 1507.0 137 1434 2584.6 .769 2286 9048 37 .988 1198 1068.4 .444 2191 1517.6 .152 7440 2615.8 .824 6779 944 i 38 5.994 5464 1073.7 453 35 6 9 1528.3 .168 5336 2647.6 .882 5925 9870 39 6.001 0050 1079.1 .462 5594 1539.2 .184 5171 2680.4 .943 2018 10340 40 6.007 4958 1084.5 6.471 8275 1550.2 7.200 6993 2713-9 9.006 7690 10857 41 .014 0192 1089.9 .481 1620 1561.3 .217 0850 2748.3 .073 5974 11429 42 .020 5756 1095.4 .490 5641 1572.6 .233 6796 2783.5 .144 0401 12064 43 .027 1652 IIOI.O .500 0346 1584.1 .250 4884 2819.7 .218 5102 12773 44 .033 7885 1106.7 .509 5746 1595.8 .267 5170 2856.8 297 4963 13572 45 6.040 4457 1112.4 6.519 1850 1607.7 7.284 7712 2894.8 9.381 5820 14476 46 .047 1372 1118.1 .528 8669 1619.6 .302 2571 2934-1 .471 4711 15510 47 .053 8634 1123.9 .538 6216 1631.8 .319 9810 2974.2 .568 0247 16704 48 .060 6246 1129.8 .548 4499 1644.2 337 9494 3015.6 .672 3106 18096 49 .067 4212 "35-7 558 353 1656.8 .356 1692 3058.1 .785 6758 19741 50 6.074 2535 1141.7 6.568 3320 1669 6 7.374 6475 3101.7 9.909 8535 21715 51 .081 1219 1147.7 .578 3881 1682.4 393 39 l8 3146.8 10.047 1256 24127 52 .088 0269 1153.8 .588 5227 1695.6 .412 4099 3193.0 .200 5829 27144 53 .oyj. 9687 1 160.0 .598 7368 1708.9 .431 7097 3240.7 374 5584 31023 54 .101 9479 1166.3 .609 0317 1722.6 .451 2999 3289.9 575 3986 36197 55 6.108 9647 1172.6 6.619 486 1736.4 7.471 1892 334-3 10.812 9421 4345 56 .116 0196 1179.0 .629 8689 1750.3 .491 3870 3392.6 11.103 6719 57 .123 1131 1185.4 .640 4141 1764.5 .511 9029 3446.5 11.478 4880 58 .130 2455 1192.0 .651 0455 1779.0 .532 7472 3502.1 12.006 7617 59 137 4173 1198.6 .661 7645 1793-8 553 935 3559-6 12.909 8516 60 6.144 6289 1205.3 6.672 5724 1808. 8 7-575 4640 3618.7 610 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. / / // n f i n II O 1 / // II 155 5 10 15 20 25 3 23-9 19.74 16.43 13.17 9-95 6.77 3-35 3-3' 3.26 3.22 3.18 3.14 160 5 10 15 20 25 I 6.70 5-33 3-97 2.64 i-33 0.04 :% 33 3' 2 ? 26 165 10 20 30 40 50 o 15.85 14.98 14.16 13.38 12.63 11.91 0.87 0.82 0.78 0.75 0.72 0.69 155 30 35 40 45 50 55 3 3- 6 3 0.54 2 57-49 54.48 5i-5i 48.58 3.09 3-5 3.01 2.97 2.93 2.89 160 30 35 40 45 50 55 o 58.78 57-54 56-3 1 55-" 53-93 52.77 .24 23 .20 .18 .16 .14 166 10 20 30 40 50 11.22 10.57 9-95 9.36 8.80 8.26 0.65 0.62 0.59 1 0.56 ; 0.54 ' 0.51 156 5 10 15 20 25 2 45.69 42.84 40.03 37.26 34-53 3i-83 2.85 2.8 1 2.77 2.73 2.70 2.66 161 5 10 15 20 25 o 51.63 50.50 49.40 48.32 47.26 46.21 13 .10 .08 .06 .05 .02 167 10 20 30 40 50 o 7.75 l:ll 6.37 5.96 5-57 0.48 0.46 0.44 0.41 0-39 0.37 156 30 35 40 45 50 55 2 29.17 26.55 23-97 21.43 18.92 16.44 2.62 2.58 2.54 2.51 2.48 2.44 161 30 35 40 45 50 55 o 45.19 44.18 43-*9 42.22 41.26 4-33 1. 01 0.99 0.97 0.96 o-93 0.92 168 10 20 30 40 50 o 5.20 4.84 4.51 4.20 3-9 3-62 0.36 -33 0.31 0.30 0.28 0.26 157 5 2 14.00 I I.'sO 2.41 162 5 o 39.41 ^8.ci o 90 169 10 o 3-36 3.11 0.25 10 15 20 25 11 Ox 9.22 6.89 4 .58 2.31 2.37 2-33 2.31 2.27 2.23 10 15 20 25 J 3 37.62 3 6 -75 35-9 35.06 0.89 0.87 0.85 0.84 0.82 20 30 40 50 2.88 2.66 2.46 2.27 0.23 0.22 0.20 0.19 0.18 157 30 35 40 45 50 55 2 0.08 i 57- 8 9 55-71 53-57 51.46 49-39 2.19 2.17 2.15 2. II 2.07 2.04 162 30 35 40 45 50 55 o 34.24 33-43 32.64 31.86 31.10 3-35 0.8 1 0.79 0.78 0.76 0.75 0.73 170 10 20 30 40 50 o 2.09 1.92 1.76 1.62 1.48 1.35 0.17 0.16 0.14 0.14 0.13 O.IZ 158 5 10 15 20 25 i 47-35 45-34 43-35 41-39 39-47 37-57 2.01 1-99 1.96 1.92 I.gO 1.87 163 5 10 15 20 25 o 29.62 28.90 28.20 27.51 26.83 26.16 0.72 0.70 0.69 0.68 0.67 0.65 171 10 20 30 40 50 o 1.23 1. 12 1.02 0.93 0.84 0.76 O.I I O.IO 0.09 0.09 0.08 0.08 158 30 35 40 45 50 55 i 3570 33.87 32.06 30.28 28.52 26.80 / 1.8 3 1.81 1.78 1.76 1.72 1.70 163 30 35 40 45 50 55 o 25.51 24.88 24.25 23.64 23.04 22.45 0.63 0.63 0.61 0.60 0.59 0.57 172 10 20' 30 40 50 o 0.68 0.6 1 0.55 0.49 0.44 0.39 0.07 0.06 0.06 0.05 0.05 0.04 159 5 10 15 20 25 i 25.10 23-43 21.78 20.16 18.57 17.00 1.67 1.65 1.62 1.59 i-57 I.CC 164 5 10 15 20 25 o 21.88 21.31 20.76 20.22 I9.6 9 I9.l8 0.57 0.55 0.54 -53 0.51 0.51 173 10 20 30 40 50 o 0.35 0.31 0.27 0.24 0.21 0.19 0.04 0.04 0.03 0.03 O.O2 0.03 159 30 35 40 45 50 55 i 15.45 13.94 12.44 10.97 9-53 8.10 J J 1.51 1.50 1.47 1.44 1.43 1.40 164 30 35 40 45 50 55 o 18.6-7 18.17 17.69 17.21 16.75 16.29 0.50 0.48 0.48 0.46 0.46 0.44 174 175 176 177 178 179 o o.i 6 0.07 O.O2 O.OI 0.00 0.00 0.09 0.05 O.OI O.OI O.OO o.oc 160 i 6.70 165 o 15-85 180 O.OO fill TABLE Vfll. For finding the Time from the Perihelion in a Parabolic Orbit. V log N Diff. V log J\" Diff. V log^T Diff. / / O / 30 1 30 2 30 0.025 5763 .025 5749 .025 5707 .025 5638 .025 5542 .025 5418 J? 96 I2 4 152 30 30 31 30 32 30 0.020 7913 .O2O 6368 .020 4802 .O2O 3215 .020 1607 .019 9979 '545 1566 1587 1608 1628 1649 60 30 61 30 62 30 0.008 8644 .008 6458 .008 4277 .008 2103 .007 9934 .007 7774 2186 2181 2174 2169 2160 I 2153 3 , 30 4 30 5 30 0.025 5*66 .025 5087 .025 4881 .025 4647 .025 4386 .025 4097 179 2OO *34 261 289 316 33 30 34 30 35 30 0.019 833 .019 6662 .019 4974 .019 3267 .019 1540 .018 9795 1668 1688 1707 1727 1745 1765 63 30 64 30 65 30 0.007 5621 .007 3477 .007 1343 .006 9220 .006 7108 .006 5008 2144 2134 2123 2112 2IOO 2086 6 30 7 30 8 0.025 378 1 02 5 3437 .025 3066 .025 2668 .025 2243 344 371 398 425 36 30 37 30 38 0.018 8030 .018 6248 .018 4448 .018 2629 .018 0794 1782 1800 1819 JJJ35 66 30 67 30 68 0.006 2922 .006 0849 .005 8792 .005 6750 .005 4725 2073 2057 2042 2025 2008 30 .025 1791 452 30 .017 8941 1869 30 .005 2717 1988 9 30 0.025 i5 1 1 .025 0805 506 39 30 0.017 7072 .017 5186 1886 69 30 0.005 07*9 .004 8760 1969 10 .025 0271 534 40 .017 3283 1903 70 .004 68 i i 1949 30 .024 9711 560 30 .017 1365 1918 30 .004 4884 1927 11 .024 9124 1*7 41 .016 9432 '933 71 .004 2980 1904 - O O _ 30 .024 8510 614 641 30 .016 7483 1949 1963 30 .004 1 1 oo I ooO 1855 12 30 13 80 14 30 0.024 7869 .024 7201 .024 6507 .024 5786 .024 5039 .024 4266 668 694 721 747 773 800 42 30 43 30 44 30 0.016 5520 .016 3542 .016 1550 .015 9545 .015 7526 .015 5495 1978 1992 2005 2019 2031 2045 72 30 73 30 74 30 0.003. 9*45 .003 7416 .003 5613 .003 3839 .003 2094 .003 0380 1829 1803 1774 1745 1714 1682 15 30 16 30 17 30 0.024 3466 .024 2641 .024 1789 .024 091 1 .024 0008 .023 9079 825 852 878 9 3 929 954 45 30 46 30 47 30 0.015 3450 .015 1394 .014 9326 .014 7247 .014 5157 .014 3057 2056 2068 2079 2090 2100 2110 75 30 76 30 77 30 O.002 8698 .002 7049 .002 5433 .002 3854 .002 2311 .002 0806 1649 1616 1543 1505 1465 18 30 19 30 20 30 0.023 8125 .023 7145 .023 6140 .023 5109 .023 4054 .023 2973 980 1005 1031 ! I0 55 1081 I 1105 48 30 49 30 50 30 0.014 0947 .013 8827 .013 6698 .013 4561 .013 2416 .013 0263 2120 2129 2137 2145 2153 2IDO 78 30 79 30 80 30 o.ooi 9341 .001 7917 .001 6535 .001 5196 .001 3903 .001 2656 1424 1382 1339 1293 1247 1198 21 30 22 0.023 1868 .023 0738 .022 9584 1130 1154 51 30 52 0.012 8103 .012 5936 ,OI2 7764 2167 2I 7 2 81 30 82 o.ooi 1458 .001 0309 .000 9211 1149 1098 30 23 30 .022 8405 .022 72O2 .022 5975 1179 1203 1227 30 53 30 J / T .012 1585 .on 9402 .on 7215 2179 2183 2187 30 83 30 .000 8l66 .000 7175 .000 6240 1045 991 935 xvn 1251 2191 oy D 24 30 25 O.O22 4724 .022 3449 .022 2151 1298 54 30 55 o.on 5024 .on 2829 .on 0632 2195 2197 84 30 85 o.ooo 5364 .000 4546 .000 3790 818 756 30 .022 0829 1322 30 .010 8432 22OO 30 .000 3096 h 26 .021 9484 I3 tl 56 .010 6231 22OI 86 .000 2468 O2S 30 .021 8ll6 1368 30 .010 4029 22O2 30 .000 1906 562 1390 2202 493 27 0.021 6726 57 o.oio 1827 87 o.ooo 1413 30 28 30 29 30 .O2I 5312 .021 3876 .021 2418 .021 0938 .020 9436 1414 I43 6 1458 1480 1502 30 58 30 59 30 .009 9625 .009 7424 .009 5225 .009 3028 .009 0834 22O2 2201 2I 99 2197 2I 94 2190 30 88 30 89 30 .000 0990 .000 0639 .000 0363 .000 0163 .000 0041 423 276 200 122 4 1 30 0.020 7913 60 0.008 8644 90 O.OOO 0000 612 TABLE VIII. For finding the Time from the Perihelion in a Parabolic Orbit. V log N' Difif. V log N' Diflf. V log N' Diff. O 1 1 o t 90 30 91 0.000 0000 9.999 9876 999 957 124 3 6 9 120 30 121 9.963 1069 .962 0074 ,960 8971 10995 11103 150 30 151 9.889 0321 .887 8738 .886 7259 11583 11479 30 92 30 999 88 93 999 8 39 999 6 944 614 854 1095 J 33 X 30 122 30 959 7764 .958 6454 .957 5046 1 1 207 11310 11408 11504 30 152 30 .885 5887 .884 4627 .883 3481 11372 11260 . 11146 11026 93 30 94 30 1 95 30 9-999 5613 999 44 6 .999 2246 .999 0215 99 8 7955 .998 5468 1567 1800 2031 2260 2487 2711 123 30 124 30 125 30 9.956 3542 955 '945 954 0*58 .952 8483 .951 6624 .950 4684 11597 11687 11775 11859 11940 12018 153 30 154 30 155 30 9.882 2455 .881 1552 .880 0775 .879 0129 .877 9616 .876 9242 10903 10777 10646 10513 10374 10232 96 30 97 30 98 30 9.998 2757 .997 9824 .997 6669 997 3*97 .996 9708 .996 5906 2933 3155 337^ 3589 3802 4015 126 30 127 30 128 30 9.949 2666 948 0573 .946 8408 945 6174 944 3875 943 15*3 12093 12165 12234 12299 12362 12421 156 30 157 30 158 30 9.875 9010 .874 8922 873 8984 .872 9198 .871 9569 .871 0099 10088 9938 9786 9629 9470 9307 99 30 100 30 101 30 9.996 1891 .995 7666 995 3*34 994 59 6 994 3755 993 8712 4225 J$ 4841 543 5242 129 30 130 30 131 30 9.941 9092 .940 6615 939 4 8 5 .938 1506 .936 8881 935 6213 12477 12530 12579 12625 12668 12707 159 30 160 30 161 30 9.870 0792 .869 1652 .868 2683 .867 3886 .866 5266 .865 6827 9140 8969 8797 8620 8439 8257 103 30 103 30 104 30 9-993 347 .992 8031 .992 2397 .991 6570 .991 0553 99 4347 5439 5 6 34 5827 6017 6206 6391 132 30 133 30 134 30 9-934 35 6 933 0763 .931 7987 .930 5183 .929 2353 .927 9501 12743 12776 12804 12830 12852 12871 162 30 163 30 164 30 9.864 8570 .864 0500 .863 2620 .862 4932 .861 7439 .861 0145 8070 7880 7688 7493 7294 7092 105 30 106 30 107 30 9.989 7956 .989 1380 .988 4622 .987 7685 987 0571 .986 3281 6576 6758 6937 7114 7290 7462 135 30 136 30 137 30 9.926 6630 925 3745 .924 0848 .922 7943 .921 5035 .920 2126 12885 12897 12905 12908 12909 12906 165 30 166 30 167 30 9.860 3053 .859 6164 .858 9482 .858 3010 .857 6750 .857 0704 6889 6682 j>47 O20O 6046 582 9 108 30 109 30 110 30 9.985 5819 .984 8186 .984 0385 .983 2418 .982 4288 .981 5996 7633 7801 7967 8130 8292 84.0 138 30 139 30 140 30 9.918 9220 .917 6321 .916 3433 .915 0559 .913 7703 .912 4870 12899 12888 12874 12856 12833 12808 168 30 169 30 170 30 9.856 4875 .855 9266 855 3878 854 8714 854 3775 .853 9065 ! 5164 4939 4710 4481 111 30 112 30 113 30 9.980 7545 979 8938 979 0177 .978 1264 .977 2202 .976 2993 TJ 8607 8761 8913 9062 9209 Q"K1 141 30 142 30 143 30 9.911 2062 .909 9283 .908 6538 .907 3831 .906 1164 .904 8542 12779 12745 12707 12667 12622 12 573 171 30 172 30 173 30 9.853 4584 853 335 .852 6319 .852 2538 .851 8994 .851 5687 4249 4016 378i 3544 3307 ' 3067 114 30 115 30 116 30 9-975 3 6 4 974 4H5 973 45io 972 4739 .971 4833 .970 4796 7 J J 3 9495 9635 9771 9906 10037 10167 144 30 145 30 146 30 9.903 5969 .902 3449 .901 0985 .899 8582 .898 6243 897 397 2 12520 12464 12403 12339 12271 12198 174 30 175 30 176 30 9.851 2620 .850 9794 .850 7209 .850 4868 .850 2770 .850 0917 2826 2585 2341 2098 1851 - 1608 j 117 30 118 30 119 30 9.969 4629 .968 4337 .967 3920 .966 3382 .965 2726 .964 1954 10292 10417 '538 10656 10772 10885 147 30 148 30 149 30 9.896 1774 .894 9652 .893 7610 .892 5652 .891 3782 .890 2004 I2I22 J2042 II958 II870 JI778 n6?3 177 30 178 30 179 30 9.849 9309 .849 7948 .849 6833 .849 5966 .849 5346 .849 4974 1361 1115 ; 867 620 37* ' 124 120 9.963 1069 150 9.889 0321 180 9.849 4850 613 TABLE IX. F>r finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity X A Difif. B Difif. c B' Difif. C' n n it If II II It O.OO 0.000 0.000 o.ooo o.ooo 1 O.OO O.OO 0.000 o.ooo 0.000 0.000 2 0.0 1 O.OI 0.000 0.000 0.000 o.oco 3 0.05 0.04 o.ooo o.ooo o.ooo o.ooo 4 0.12 0.07 0.000 o.ooo 0.000 o.ooo O.I I 5 0.23 o. 1 6 o.ooo 0.000 0.000 o.ooo 6 0.35 0.000 o.ooo o.ooo 0.000 7 0.02 0.23 o.ooo 0.000 0.000 o.ooo 8 o-93 0.31 0.000 o.ooo o.ooo o.ooo 1 9 1.77 0.40 o.ooo 0.000 0.000 o.ooo J J 0.49 1 10 1.82 o.ooo 0.000 0.000 o.ooo 11 2.42 o.6c 0.000 0.000 0.000 0.000 12 7.14 0.72 o.ooo 0.000 o.ooo o.ooo 13 J T^ 3-99 0.85 0.000 0.000 o.ooo 0.000 14 4-99 .00 .14 O.OO I 0.000 0.001 o.ooo 15 6.13 0.001 o.ooo O.OOI o.ooo 16 7.4-1 .30 O.OO2 .001 0.000 0.00 1 .000 o.coo 17 /TO 8.90 f 0.002 .000 o.ooo O.OO2 .001 o.ooo 18 19 10.55 12.40 6 5 .85 2.05 0.003 0.004 .001 .001 .001 0.000 o.ooo 0.002 0.003 .000 .001 .001 0.000 o.oco 20 21 14.45 l6.7O 2.25 0.005 0.006 .001 0.000 o.ooo 0.004 0.005 .001 0.000 o.ooo 22 I9.l8 2.48 0.008 .002 0.000 0.006 .001 0.000 23 21.89 2.71 O.OIO .002 o.ooo 0.008 .002 o.ooo 24 X 24.81 2.94 O.OI 2 .002 0.000 O.OIO .002 0.000 T J 3.20 .002 .002 25 28.07 0.014 0.000 0.012 o.ooo 26 27 m ** J 31.48 35.20 3-45 3-72 T. 0.017 0.020 .003 .003 0.000 0.000 0.014 0.017 .002 .003 o.ooo 0.000 28 29 39-'9 43-47 3-99 428 4-57 0.025 0.030 .005 .005 .005 0.000 0.000 O.O2O 0.024 .003 .004 004 o.ooo 0.000 30 31 32 33 34 48.04 52.91 58.09 6 3-59 69.42 4.87 5.18 5-5 5.83 6.15 0.035 0.041 0.047 0.055 0.064 .006 .006 .008 .009 .009 0.000 0.000 0.000 0.000 o.ooo 0.028 0.033 0.039 0.045 0.052 .005 .006 .006 .007 .008 o.ooo 0.000 o.ooo 0.000 0,000 35 36 37 38 75-57 82.07 88.92 96.12 6.50 6.85 7.20 0.073 0.084 0.096 0.109 .01 1 .012 .013 0.000 o.ooo 0.000 o.ooo 0.060 0.068 0.078 0.088 .008 .010 .010 o.ooo 0.000 o.ooo 0.000 39 103.68 7.56 7-93 0.123 .014 .016 0.000 O.1OO .012 .013 o.ooo 40 41 43 43 44 in. 61 119.92 128.62 137.70 147.18 8.31 8.70 9.08 9.48 0.139 0.156 0.175 0.196 0.218 .017 .019 .021 .022 0.000 o.ooo o.ooo o.ooo o.ooo 0.113 0.127 0.142 0.159 0.177 .014 .015 .017 .018 0.000 o.ooo 0.000 0.000 o.ooo 9.87 .025 .020 45 46 47 48 49 157.05 167.34 178.04 189.16 200.71 10.29 10.70 II. 12 11-55 11.98 0.243 0.269 0.298 0.328 0.361 .026 .029 .030 .033 .036 o.ooo 0.000 o.ooo 0.000 o.ooo 0.197 0.219 0.242 0.267 0.294 .022 .023 .025 .027 .029 0.000 o.ooo 0.000 o.ooo o.ooo 5O 51 52 53 54 212.69 225 ic 237-95 251.25 265.01 12.41 12.85 13.30 13.76 14.20 Q-397 0.436 0.477 0.521 0.567 39 .041 .044 .046 .050 o.ooo 0.000 O.OOJ 0.00 1 O.OO I 0.323 0-354 0.388 0.424 0.462 .031 .034 .036 .038 .040 0.000 o.ooo o.ooo 0.000 o.ooo j 55 56 57 58 59 279.21 293.88 309.02 324.62 340.70 14.67 I5.I 4 15.60 16.08 16.56 0.617 0.671 0.727 0.787 0.851 .054 .056 .060 .064 .068 O.OO I 0.002 O.OO2 0.002 O.OO2 0.502 0.546 0.592 0.641 0.693 .044 .040 .049 .052 .056 o ooo 0001 O OOI C OOI O.OOI 60 357-^6 0.919 0.003 0.749 O.OO2 614 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 1 Diff. C' n ,/ n II n // n 60 61 62 357-26 374-3 391.84 17.04 17-54 18.02 0.919 0.990 .066 .071 .076 0.003 0.003 0.003 0.749 0.807 0.869 .058 .062 .066 0.002 0.002 O.OO2 63 64 409.86 428.38 18.52 19.02 '45 -230 SI .088 0.004 0.004 0.935 1.004 .069 73 0.002 O.OO2 65 66 67 68 69 447.40 466.92 486.96 507.51 528.58 19.52 20.04 2-55 21.07 21.59 .318 .411 .510 .613 .721 .093 .099 .103 .108 .114 0.004 0.005 0.005 0.006 0.006 .077 .154 235 .321 .411 .077 .081 .086 .090 .094 0.003 o 003 0.003 0.004 0.004 70 71 72 73 74 550.17 572.29 59494 618.12 641.85 22.12 22.65 23.18 23.73 24.28 .835 954 2.078 2.209 2-345 .119 .124 '.\\6 H3 0.007 0.007 0.008 0.009 0.009 .505 .605 .709 .819 934 .100 .104 .110 .115 .121 0.004 0.005 0.005 0.006 0.006 75 666.13 _ 2.488 0.0 1 2.055 ,-f. 0.007 76 690.96 24.83 , f O 2.637 .149 O.OII 2.181 .1 2O 0.007 77 78 79 716.34 742.29 768.81 25.38 25.95 26.52 27.09 2-793 2.956 3.125 .163 .169 .177 0.012 0.013 0.014 2.314 2-453 2-599 139 .146 153 0.008 0.008 0.009 80 81 82 83 84 795-9 8 5 i'.8 4 880.70 910.16 27.67 28.27 28.86 29.46 30.07 3.302 3.486 3.678 3.878 4.087 .184 .192 .200 .209 .216 0.015 0.016 0.017 0.018 0.020 2.752 2.912 3-79 3-255 3-439 .l6o S3 .184 .192 0.010 O.OII O.OI2 0.013 0.014 85 86 87 88 89 940.23 970.92 1002.24 1034.20 1066. 81 30.69 31.96 32.6l 33-27 4-33 4.529 4.764 5.008 5.262 .226 235 244 .254 .265 0.021 0.023 0.024 O.O26 0.028 3.631 3-833 4.044 4.266 4.498 .Z02 .211 .222 .232 .243 0.015 0.016 0.018 0.019 0.021 90 1100.08 5 527 0.030 4.741 ire 1 0.023 91 92 93 94 1134.02 1168.64 1203.95 ~ 1239.97 33-94 34.62 35-31 36.02 3 6 -75 5.801 6.087 6.385 6.694 .298 39 .322 0.032 0.034 0.036 0.038 4.996 5.263 5-544 5.838 255 .267 .281 .294 39 O.O25 0.027 0.029 0.032 95 98 97 1276.72 1314.21 37-49 38.24 7.016 7.350 7.698 334 .348 0.041 0.044 0.047 6.147 6.471 6.812 .324 341 "9 C" A 0.035 0.038 0.041 98 1391.46 39.01 8.060 .362 0.050 7.171 359 178 0.045 99 1431.27 40.61 8.437 377 392 0.0 5 3 7-549 3/ 397 0.049 100 30 101 30 102 30 1471.88 1492.50 I534-38 1555.64 1577.12 20.62 20.83 21.05 21.26 21.48 21.70 8.829 9.032 9.238 9-449 9.664 9-883 S3 .211 .215 .219 .225 0.056 0.058 0.060 0.062 0.064 0.066 7.946 8.152 8.364 8.582 8.805 9.035 .206 .212 .218 .223 .230 0.053 0.055 0.058 O.o6o 0.063 O.o66 i 103 1598.82 10.108 0.068 9.271 0.069 30 104 1620.75 1642.91 21.93 22.16 10.337 10.570 .229 233 0.070 0.072 9-5I3 9.761 :^8 .2;6 0.072 0.075 30 105 30 1665.30 1687.93 1710.80 22.39 22.63 22.87 23. 12 10.809 11.053 11.302 239 .244 .249 0.074 0.077 0.079 10.017 10.280 10.550 .2% .270 .2 7 8 0.078 O.o82 0.085 106 30 , 107 30 1 108 30 I733-92 1757.28 1780.90 1804.77 1828.90 1853.30 23.36 23.62 23.87 24.13 24.40 24.67 "557 11.817 12.083 12.354 12.632 12.916 .260 .266 .271 .2 7 8 .284 .291 0.082 0.084 0.087 0.093 o 096 10.828 11.114 1 1.408 11.711 12.022 12-343 .286 .294 33 .311 .321 .330 0.089 0.093 0.098 0.102 O.IO7 O.I I 2 109 1877.97 _f i 13.207 0.099 12.673 O.II7 615 TABLE IX. For tindm fe the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity A Diff. B Diff. c Diff. B' Diff. C" Diff. / n n n n 11 n II n 109 30 110 30 111 30 1877.97 1902.91 1928.13 1953.64 1979.44 2005.54 24.94 25.22 25.51 25.80 26.10 26.40 13.207 13.504 13.808 14.119 14.438 14.764 297 -34 311 .319 .320 -333 0.099 0.102 0.106 0.109 0.113 0.116 .003 .004 .003 .004 .003 .004 12.673 13.013 13-363 I3-724 14.095 14.478 34 .350 .361 .371 .383 .396 0.117 0.122 0.128 0.134 0.141 0.148 ,005 .006 .006 .007 .007 .007 112 30 113 2031.94 2058.64 2085.66 26.70 27.02 15.097 15-439 15-789 .342 35 O.I 20 0.124 0.128 .004 .004 14.874 15.282 15.702 .408 .420 0.155 O.l62 0.170 -.1:1 30 114 30 2113.00 2140.66 2168.66 27-34 27.66 28.00 28.34 16.148 16.515 16.892 359 .367 :]ll 0.132 0.137 0.142 .004 .005 .005 .005 16.135 16.583 17.045 433 .448 .462 -477 0.178 0.187 o 196 .008 .009 .009 .010 115 30 116 3D 117 30 2197.00 2225.69 2254.73 2284.13 2313.91 2344.06 28.69 29.04 29.40 29.78 3-l5 3-54 17.278 17.674 18.080 18.496 18.924 19-363 .396 .406 .416 .428 .439 45 0.147 0.152 0.157 0.162 0.168 0.174 .005 .005 .005 .006 .006 .006 17.522 18.015 18.524 19.050 19-594 20.156 -493 59 .526 -544 .562 .582 0.206 0.2 1 6 0.227 0.239 0.251 0.264 .010 .Oil .012 .012 .013 .013 118 30 119 30 120 30 2374.60 2405.54 2436.88 2468.64 2500.83 2533-45 3-94 31-34 31.76 32.19 32.62 33-o6 19.813 20.276 20.751 21.240 21.742 22.258 4 6 3 475 489 .502 .516 531 0.180 0.186 0.193 O.200 0.207 O.2I4 .006 .007 .007 .007 .007 .008 20.738 21-339 21.962 22.606 23.273 23.964 .601 .623 .644 .667 .691 .716 0.277 0.291 0.306 0.322 0-339 0-357 .014 .015 .Ol6 .017 .018 .019 121 30 122 30 123 30 2566.51 2600.03 2634.02 2668.49 2703.46 2738.93 33-52 3399 34-47 34-97 35-47 35-98 22.789 23.336 23.898 24.477 25.073 25.687 -547 .562 579 .596 .614 6 33 0.222 0.230 0.239 0.248 0.258 0.268 .008 .009 .009 .010 .010 .010 24-680 25.422 26.191 26.988 27.815 28.673 .742 .769 -797 .827 .858 .891 0.376 0.396 0-417 0-439 0.463 0.488 .020 .O2I .022 .024 .025 .027 124 30 125 30 126 30 2774.91 2811.43 2848.50 2886.13 2924.33 2963.12 36.52 37.07 37-63 38.20 38.79 39-4 1 26.320 26.973 27.646 28.341 29.057 29.797 .653 .673 .695 .716 .740 -765 0.278 0.289 0.300 0.312 0.325 0.338 .Oil .Oil .012 .013 .013 .014 29.564 30.489 3!-45o 3M48 33.485 34-563 .925 .961 .998 037 .078 .ri2 -5i5 0.544 0.606 0.640 0.676 .029 .030 .032 034 .036 .039 127 30 128 30 129 30 3002.53 3042.56 3083.23 31*4-57 3166.59 3209.31 40.03 40.67 4L34 42.02 42.72 43-45 30.562 31-351 32.167 33-on 33.885 34-789 .789 .816 .844 -874 .904 93 6 0-352 0.367 0.382 0.398 0-415 0-433 .015 .015 .Ol6 .017 .018 .019 35.685 36.852 38.067 39-33 1 40.649 42.022 .167 .215 .264 . 3 !8 -373 .430 0-715 0-757 0.800 0.846 0.896 0.949 .042 .043 .046 .050 -053 .050 130 20 40 131 20 40 3252.76 3282.13 3311-85 3341.90 3372.31 3403.09 29.37 29.72 30.05 30.41 30.78 3i.!4 35-725 36.367 37.025 37.699 38.389 39.097 .642 .658 .674 .690 .708 .725 0.452 0.465 0.479 0-493 0.508 0.523 .013 .014 .014 .015 .015 .Ol6 43.452 44-439 45-455 46.500 47-575 48.682 0.987 .016 .045 075 .i7 .138 .005 045 .087 130 175 .223 .040 .042 043 -045 .048 .050 132 20 40 , 133 20 40 3434-23 3465.74 3497-63 3529.91 3562.60 3595- 6 9 31-51 31-89 32.28 32-69 33-09 33-51 39822 40.564 41-326 42.108 42.910 43-733 .742 .762 .782 .802 .823 .843 0.539 0-555 0-572 0.590 0.609 0.629 .Ol6 .017 .018 .019 .020 .O2O 49 820 50.992 52.199 53-442 54-723 56.042 .172 .207 :3? 319 359 .273 -325 379 .436 495 558 .052 054 .057 .059 .063 .005 134 20 40 135 20 40 3629.20 3663.13 3697.50 3732.31 3767.58 3803.31 33-93 34-37 34.81 35.27 35-73 36.21 44.576 45.442 46-331 47.245 48.183 49.147 .866 .889 .914 .964 .991 0.649 0.669 0.691 o.7H 0-738 0763 .020 .022 .023 .024 .025 .025 57.401 58.802 60.247 61.736 63.273 64.857 .401 445 .489 .537 r 4 .634 .623 .692 -764 839 .917 2.000 .069 .072 .075 .078 .083 .087 136 3839.52 50.138 0.788 66.491 2.087 616 TABLE IX, f 01 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. 1 if n II n n 136 20 40 137 20 40 3839-5* 3876.21 3913.41 3951.12 3989.35 4028.11 36.69 37.20 37-71 38.23 38.76 39-3 1 50.138 51.156 52.203 53.280 54.388 55.528 1.018 1.047 .077 .108 .140 .174 0.788 0.815 0.843 0.873 0.904 0.936 .027 .028 .030 .031 .032 33 66.491 68.178 69.920 71.718 73-575 75-493 .687 74* 798 .857 .918 .982 2.087 2.178 2.274 *-375 2.480 2.591 .091 .096 .101 .105 .Hi .117 138 20 40 139 20 40 4067.42 4107.28 4147.72 4188.75 4230.38 4272.63 39-86 40.44 41.03 41.63 42.25 42.89 56.702 57.910 59-'54 60-436 61.757 63.119 .208 .244 .282 .321 .362 .404 0.969 .004 .041 .079 .119 .161 .035 .040 .042 .044 77-475 79-5*3 81.641 81.830 86.094 88.436 2.048 2.118 2.189 2.264 2.342 2.424 2.708 2.831 2.960 3.096 3.239 3-39 .123 .129 , .136 .143 .151 .159 140 20 40 141 20 40 4315.52 4359.06 4403.26 4448.15 4493-73 4540.03 43-54 44.20 44.89 45.58 46.30 47.04 64.523 65.971 67.465 69.007 70.599 72.243 .448 494 54* .592 .644 .698 .205 .251 .299 350 .404 .460 .046 .048 .051 .054 .056 .058 90.860 93-369 95.967 98.657 101.443 104.331 2.509 2.598 2.690 2.786 2.888 2.993 3-549 3.717 3-893 4.080 4.277 4-484 .168 .176 .187 .197 .207 .220 142 10 20 30 40 50 4587.07 4610.88 4634.88 4659.07 4683.46 4708.05 23.81 24.00 24.19 24.39 24.59 24.79 73-941 74.811 75-695 76-595 77.509 78.439 0.870 0.884 0.900 0.914 0.930 0.946 .518 549 .580 .612 .645 .679 .031 .031 .032 .033 34 35 107.124 108.861 110.427 I I 2. 022 113.646 II5.30I 595 .624 655 .685 4.704 4.819 4.936 5-057 5.181 5.309 .115 .117 .121 .I2A .128 .131 143 10 20 30 40 50 4732.84 4757.84 4783.05 4808.46 4834.10 4859-95 25.00 25.21 25.41 25.64 25.85 26.07 79-385 80.347 81.325 82.321 83-333 84.363 0.962 0.978 0.996 .012 .030 .048 .714 749 .786 .823 .862 .901 035 .037 .037 39 .039 .041 116.986 118.704 120.452 122.233 124.049 125.899 .718 .748 .781 .816 .850 .886 5.440 5-575 5-7I5 5-858 6.005 6.157 .140 '43 .147 '\\l 144 10 20 30 40 50 4886.02 4912.31 4938.81 4965.58 4992.56 5019.78 26.29 26.52 26.75 26.98 2.7.22 85.411 86.478 87.564 88.668 89-793 90.938 .067 .086 .104 .125 .145 .165 94* .984 2.026 2.070 2.116 2.162 .042 .042 .044 .046 .046 127.785 129.707 131.666 133.661 135.698 137-774 .922 959 997 035 .076 .116 6.313 6.473 6.639 6.809 6.984 7.165 .l6o .166 .170 !i86 145 10 20 30 40 50 5047.23 5074.93 5102.88 5131.08 5*59-53 5188.24 27.70 27.95 28.20 28.45 28.71 28.97 92.103 93.290 94.498 95.729 96.982 98.259 .187 .208 .231 *53 *77 .300 2.210 2.259 2.309 2.361 2.414 2.469 .049 .050 .052 053 .055 .057 139.890 142.048 144.249 146.494 148.784 151.120 2.158 2.2OI 2.245 2.290 2.316 2.383 7-351 7-543 7.740 8.'?5 3 3 .192 .197 .203 .210 .216 .223 146 10 20 30 40 50 5217.21 5246.45 5*75-95 535-73 5335-79 5366.13 29.24 29.50 29.78 30.06 3-34 10.61 99-559 100.884 102.234 103.610 105.012 106.441 3*5 350 376 .402 4*9 .456 2.526 2.584 2.643 2.704 2.767 2.833 .058 .059 .061 .063 .066 .067 I53-503 155-934 158.415 160.947 163.531 I66.I68 2.431 2.4I 2.512 2.584 2.637 2.692 8.592 8.822 9.060 9.304 9-555 9.815 .230 .238 .244 ! 2 68 147 10 20 30 40 50 5396.76 5427.67 545888 5490.39 5522.20 5554-33 j j 30.91 31.21 3 /.8i 32.13 32.44 107.897 109.182 110.896 112.439 114.013 115.619 .485 .514 543 637 2.900 2.969 3.040 3- IJ 3 3.188 3.266 .069 .071 073 .075 .078 .080 168.860 171.608 174.414 177.280 180.206 183.194 2.748 2.806 2.866 2.926 2.988 3.052 10.083 10.359 10.645 10.940 11.244 11.558 .276 .286 - .295 .304 .314 3*5 148 10 20 30 40 50 149 5586.77 5619.52 5652.60 5686.01 57I9-75 5753-83 5788.26 33-o8 33-74 34.08 34-43 117.256 118.926 120.631 122.370 124.144 125.955 127.804 .670 .705 739 774 .811 .849 3-346 3-4*8 3-5I3 3.601 3.691 3.881 .082 .085 .088 .090 .093 .097 186.246 189.364 192.549 195.804 199.130 202.528 2O6.OO2 3.118 3-185 3-*55 3-3*6 3-398 3-474 11.883 12.218 12.564 12.921 13.291 13.673 14.067 335 .346 -357 .370 .382 394 617 TABLE X. For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits. A Ellipse. Hyperbola. log B Diff. log log I. Diff. log half II. Diff. logJB Diff. logtf log I. Diff. log half II. Diff. 0.000 0.000 o.oo oooo 7 0.000 OOOO 4.23990 1.778 oooo o.ooo oooo 4.2 3 98 2 n 1.771 .01 0007 / 27 .001 7432 .24286 .783 0007 27 9.998 2688 .23686 .767 .02 0030 3 77 .003 4985 .24583 .788 0030 "3 77 .996 5493 23392 .762 .03 0067 31 e-i .005 2659 .24885 794 0067 3 / r I .994 8414 .23098 758 .04 0120 68 .007 0457 .25190 799 0118 66 993 J 45 .22807 753 &.05 0188 84. 0.008 8381 4.25497 1.805 0184 81 9.991 4599 4.225i8 M 1.748 .06 0272 T 00 .010 6432 .25806 .811 0265 989 7859 .22230 -743 .07 0371 y s I 14 .012 4613 .26116 .816 359 IOQ .988 1231 .21943 739 .08 .09 0485 0615 *-t 130 147 .014 2924 .016 1367 .26427 .26741 .821 .827 0468 0591 * W 7 123 .986 4711 .984 8298 .21659 .21376 734 .730 O.IO 0762 162 0.017 9945 4.27057 I-833 0728 I C2 9.983 1992 4-21094* 1.725 .11 0924 178 .019 8659 .27376 839 0880 I6 5 .981 5791 .20815 .720 .12 1 102 IQ4. .021 7511 .27697 845 1045 979 9 6 94 .20537 .716 '3 .14 1296 1507 7T 211 227 .023 6503 .025 5637 .28020 .28344 .851 .857 1223 1416 193 2O6 978 3699 .976 7805 .20260 .19986 .706 ^ 1734 1977 243 26l 0.027 49 r 6 .029 4340 4.28670 .28999 1.863 .869 1622 1842 220 277 9.975 201 1 973 6316 4.19712,, .19440 1.700 695 .19 2238 2515 2809 277 294 3 II 03 1 39*3 .033 3636 .035 3511 .29331 .29665 .30001 .875 .882 .888 2075 2321 2581 v 246 260 273 .972 0719 .970 5218 .968 9813 .19170 .18901 .18633 .690 .685 .679 O.2O .21 3120 3448 328 0.037 3542 .039 3730 4-3339 .30679 1.895 .901 2854 3140 286 9.967 4502 .965 9285 4.18367, .18102 1.672 .666 .22 3793 4156 363 78l .041 4077 .043 4585 .31022 .31368 .908 .915 3439 375i 312 .964 4159 .962 9124 .17840 17579 .661 .655 .24 4537 J" * 398 .045 5259 .31716 .922 4076 338 .961 4180 .17319 .649 'H 4935 535 1 4 l6 4.74. 0.047 6099 .049 7109 4.32066 .32418 1.929 936 4414 4765 767 9.959 9324 958 455 6 .16803" 1.643 637 .27 5785 T JT 4.^2 .051 8290 32773 943 5128 376 95 6 9875 .16547 .631 .28 6237 TO 4.71 .053 9646 .951 554 955 5281 .16292 .625 .29 6708 TV 488 .056 1179 33492 .958 5893 401 954 77i .16038 .618 0.30 7196 0.058 2893 4.33856 1.966 6294 9.952 6346 4-i5785n 1.613 TABLE X, Part II, r Ellipse. Hyperbola. T Ellipse. Hyperbola. A Diff. A Diff. A Diff. A Diff. o.oo .01 .02 .03 .04 O.OOOOO .00992 .01969 .02930 .03877 992 977 961 947 971 o.ooooo .01008 .02033 .03074 .04132 1008 1025 1041 1058 1077 0.20 .21 .22 .24 0.17266 .18008 .18740 .19462 .20174 742 732 722 712 704 0.23867 .25309 .26779 .28280 .29813 1442 1470 1501 1533 1564 0.05 .06 .09 0.04808 .05726 .06630 .07521 .08398 oo rh M r^ w H O ON t^so ON ONOO OO OO 0.05209 .06303 .07417 .08550 .09702 1094 1114 "33 1152 "73 0.25 .26 .27 .28 .29 0.20878 21573 .22258 .22935 .23604 695 685 677 669 661 0.31377 O.IO .11 .12 .I 4 0.09263 .10116 .10956 .11783 .12599 853 840 827 816 805 0.10875 .12069 .13285 .14522 .15782 1194 1216 1237 1260 1285 0.30 32 33 34 0.24265 .24917 .25561 =26198 .26826 652 644 637 628 621 O.I| .16 17 .18 0.13404 .14198 .14981 J 5753 794 783 772 0.17067 18375 .19709 .21068 1308 1334 1359 T 1%f\ | 0.27447 .28061 .28668 .29268 614 607 600 i .19 .16515 762 75 1 .22454 1413 39 .29860 592 586 0.20 0.17266 0.23867 0.40 0.30446 618 TABLE XL For the Motion in a Parabolic Orbit. 1? log/* Diff. , lOg /A Diff. lOg/4 Diff. 0.000 .001 0.000 OOOO .OOO OOOO 0.060 .061 o.ooo 0652 .000 0674 22 0.120 .121 o.ooo 2617 .000 2661 44 .002 .000 0001 I .062 .000 0697 23 .122 .000 2705 44 .oo 3 .000 0002 I J .063 .000 0719 22 .123 .000 2750 45 .004 .000 0003 I .064 .000 0742 24 .124 .000 2795 4 6* O.005 .oo6 o.ooo 0004 .000 0006 2 0.065 .066 o.ooo 0766 .000 0790 24 0.125 .126 o.ooo 2841 .000 2886 45 .007 .008 .000 0009 .000 0012 3 3 .067 .068 .000 0814 .000 0838 24 24 .127 .128 .000 2933 .OOO 2Q7Q 46 .009 .000 0015 3 3 .069 .000 0863 25 .129 S I 7 .OOO 3O26 s O.OIO o.ooo 0018 0.070 o.ooo 0888 0.130 o.ooo 3074 T .Oil .OOO OO22 4 .071 .000 0914 tft '3 1 .000 3121 47 .012 .000 0026 4 5' .072 .000 0940 20 .132 .000 3169 48 .0! 3 .000 0031 .073 .000 0966 20 .000 3218 49 .014 .000 0035 I .074 .000 0993 27 27 '34 .000 3267 49 49 O.OI5 o.ooo 0041 0.075 0.000 1020 o 135 o.ooo 3316 .Ol6 .000 0046 I .076 .000 1047 27 -0 .136 .000 3365 49 .017 .000 0052 .077 .000 1075 2o -0 .137 .000 3415 5 .018 .000 0059 7 5 .078 .000 1103 2o .138 .000 3466 5 1 .019 .000 0065 7 .079 .000 1132 29 29 '39 .000 3516 5 51 0.020 o.ooo 0072 0.080 o.ooo 1161 0.140 o.ooo 3567 J .021 .022 .000 0080 .000 0088 8 .081 .082 .000 1190 .000 1219 29 29 .141 .142 .000 3619 .000 3671 5 2 5* .023 .000 0096 g .083 .000 1249 3 .000 3723 5 2 .024 .000 0104 9 .084 .000 1280 3 1 3 1 .144 .000 3775 53 0.025 o.ooo 0113 0.085 o.ooo 1311 0.145 o.ooo 3828 .026 .000 0122 9 .086 .000 1342 3 1 .146 .000 3882 54 .027 .000 0132 IO .087 .000 1373 3 1 .147 .000 3935 53 .028 .029 .000 0142 .000 0152 IO 10 II .088 .089 .000 1405 .000 1437 3 2 3 2 33 .148 .149 .000 3989 .000 4044 54 55 cc 0.030 o.ooo 0163 I I 0.090 o.ooo 1470 A -> 0.150 o.ooo 4099 J J .031 .000 0174 .091 .000 1502 3 2 .151 .000 4154 55 .032 .000 0185 1 1 1 2 .092 .000 1536 34 - .152 .000 4209 11 33 .000 0197 93 .000 1569 33 J 53 .000 4265 5 34 .000 0209 I 2r '3 .094 .000 1603 34 35 .154 .000 4322 56 0.035 O.OOO O222 T ^ 0.095 o.ooo 1638 * 0.155 o.ooo 4378 .036 .000 0235 J 3 .096 .000 1673 35 .156 .000 4435 -O 037 .038 .000 0248 .000 0262 14 .097 .098 .000 1708 .000 1743 35 35 '57 .158 .000 4493 .000 4551 5 s $ .039 .000 0275 3 .099 .000 1779 g .159 .000 4609 5 8 15 3 59 0.040 o.ooo 0290 0.100 o.ooo 1815 0.160 o.ooo 4668 .0 .041 .000 0304 J 4 I .101 .000 1852 37 .161 .000 4726 r .042 .000 0320 o .102 .000 1889 37 .162 .000 4786 Oo (\r\ .043 .000 0335 *5 .103 .000 1926 11 .163 .000 4846 OO fin .044 .000 0351 I O 16 .104 .000 1964 3 38 .164 .000 4906 OO 60 0.045 .046 .047 o.ooo 0367 .000 0383 .000 0400 16 17 0.105 .106 .107 0.000 2002 .000 2040 .000 2079 3 39 0.165 .166 .167 o.ooo 4966 .000 5027 .000 5088 61 I 61 .048 .000 0417 Q .108 .000 21 1 8 39 .168 .000 5150 ft .049 .000 0435 15 18 .109 .000 2158 4 4 .169 .000 5212 62 0.050 o.ooo 0453 I O.IIO o.ooo 2198 0.170 o.ooo 5274 fi .051 .000 0471 I o .III .000 2238 4 .171 .000 5337 ?3 .052 53 .054 .000 0490 .000 0509 .000 0528 19 19 20 .112 .113 .114 .ooc 2279 .000 2320 .000 2361 H IH M f< .172 .173 .174 .000 5400 .000 5464 .000 5518 i* 64 6 4 0.055 o.ooo 0548 0.115 o.ooo 2403 0.175 o.ooo 5592 , .056 .000 0568 20 .116 .000 2445 42 .176 .000 5657 57 .000 0589 21 .117 .000 2487 42 .177 .000 5722 ftC .000 0610 21 .118 .000 2530 43 .178 .000 5787 05 ftfi .059 .000 0631 21 21 .119 .000 2573 43 44 .179 .000 5853 OO 66 0.060 o.ooo 0652 0.120 o.ooo 2617 0.180 o.ooo 5919 619 TABLE XI. For the Motion in a Parabolic Orbit. log/* Diff. n >o g . Diff. , * Diff. o.i8o .181 o.ooo 5919 .000 5986 6? 0.240 .241 o.ooi 0603 .001 0693 90 0.300 .301 o.ooi 6733 .001 6848 "5 .182 .000 6053 07 .242 .001 0784 9 1 .302 .001 6963 115 1 1 6 .183 .184 ..ooo 6120 .000 6l88 68 68 243 .244 .001 0875 .001 0966 9 1 91 92 33 .304 .001 7079 .001 7195 116 117 0.185 .186 .187 .188 .189 o.ooo 6256 .000 6325 .000 6393 .000 6463 .000 6532 69 68 7 69 7 0.245 .246 .247 .248 .249 o.ooi 1058 .001 1150 .001 1242 .001 1335 .001 1429 92 92 93 94 93 0.305 .306 37 .308 .309 o.ooi 7312 .001 7429 .001 7546 .001 7664 .001 7783 117 117 118 119 118 0.190 o.ooo 6602 0.250 o.ooi 1522 n e 0.310 o.ooi 7901 .191 .192 193 .194 .000 6673 .000 6744 .000 6815 .000 6887 72 72 .251 .252 253 .254 .001 1617 .001 1711 .001 1806 .001 1901 95 94 95' 96 3" .312 3'3 .314 .001 8020 .001 8140 .001 8260 .001 8381 119 120 120 121 121 0.195 o.ooo 6959 0.255 o.ooi 1997 06 -3'5 o.ooi 8502 121 .196 .000 7031 mm .256 .obi 2093 yu .316 .001 8623 I 22 .197 .198 .199 .000 7104 .000 7177 .000 7250 73 73 73 74 .257 .258 .259 .001 2190 .001 2287 .001 2384 97 97 98 .317 .318 .319 .001 8745 .001 8867 .001 8989 122 122 124 0.200 .201 o.ooo 7324 .000 7399 75 0.260 .261 o.ooi 2482 .001 2580 98 0.320 .321 o.ooi 9113 .001 9236 123 .202 .000 7473 74 .262 .001 2679 99 .322 .001 9360 124 .203 .000 7548 11 .263 .001 2778 99 .001 9484 124 .204 .000 7624 70 76 .264 .001 2877 99 100 3*4 .001 9609 125 125 O.205 o.ooo 7700 76 0.265 o.ooi 2977 IOO 0.325 o.ooi 9734 126 .206 .207 .208 .000 7776 .000 7853 .000 7930 / u 77 77 *7*t .266 .267 .268 .001 3077 .001 3178 .001 3279 101 101 I 02 .326 3*7 .328 .001 9860 .001 9986 .002 0113 126 127 .209 .000 8007 78 .269 .001 3381 101 3*9 .002 0240 127 O.2IO .211 .212 o.ooo 8085 .000 8163 .000 8242 78 79 0.270 .271 .272 o.ooi 3482 .001 3585 .001 3688 103 103 T Of 0.330 33 1 33* O.OO2 0367 .002 0495 .002 0624 128 129 .213 .214 .000 8321 .000 8400 79 11 .273 .274 .001 3791 .001 3894 103 103 T r*A 333 334 .002 0752 .002 0882 130 oO 104 129 O.2I5 .216 o.ooo 8480 .000 8560 80 0.275 .276 o.ooi 3998 .001 4103 I0 5 T C\A 0-335 .336 0.002 1 01 1 .002 1141 I 3 .217 .000 8641 T .277 .001 4207 104 337 .002 1272 131 .218 .000 8722 o I 81 .278 .001 4313 .338 .002 1403 131 .219 .000 8803 o I 82 .279 .001 4418 "I 339 .002 1534 132 O.220 .221 .222 o.ooo 8885 .000 8967 .000 9050 82 ll 0.280 .281 .282 o.ooi 4524 .001 4631 .001 4738 107 107 T 0*7 0.340 .341 .342 O.OO2 l666 .002 1799 .002 1971 133 132 T * A .223 .224 .000 9132 .000 9216 oZ 84 84 .283 .284 .001 4845 .001 4953 107 108 108 343 344 .002 2065 .002 2198 1 34 133 0.225 .226 .227 .228 o.ooo 9300 .000 9384 .000 9468 .000 9553 84 8 4 5 < 0.285 .286 .287 .288 o.ooi 5061 .001 5169 .001 5278 .001 5388 1 08 109 no o-345 34 6 347 348 0.002 2333 .OO2 2467 .002 2602 .002 2738 134 '35 136 .229 .000 9638 ll .289 .001 5497 109 in 349 .002 2874 l\6 O.230 o.ooo 9724 86 0.290 o.ooi 5608 I JO 0.350 0.002 3010 .231 .232 .233 .234 .000 9810 .000 9897 .000 9984 .001 0071 3 ll .291 .292 .293 .294 .001 5718 .001 5829 .001 5941 .001 6053 in 112 112 112 35 1 352 353 354 .002 3147 .002 3284 .OO2 3422 .002 3560 137 138 138 139 0.235 .236 .237 o.ooi 0159 .001 0247 .001 0335 88 88 0.295 .296 .297 o.ooi 6165 .001 6278 .001 6391 II 3 "3 *$ 357 0.002 3699 .002 3838 .002 3977 139 139 .238 .239 .001 0424 .001 0513 89 9 .298 .299 .001 6505 .001 6619 II 4 114 358 359 .002 4117 .002 4258 141 141 0.240 o.ooi 0603 0.300 o.ooi 6733 0.360 0.002 4399 620 TABLE XI. For the Motion in a Parabolic Orbit. , log/* Diff. , *, Diff. , log ft Diff. o n co tj- so so so so so CO CO CO to CO d 0.002 4399 .002 4540 .002 4682 .002 4824 .002 4967 141 142 142 143 143 0.420 .421 .422 4*3 .424 0.003 37 20 .003 3890 .003 4061 .003 4232 .003 4404 170 171 171 I 7 2 172 0.480 . 4 8l .482 483 484 0.004 4858 .004 5061 .004 5263 .004 5467 .004 5670 203 202 204 20 3 205 0.365 .366 O.002 5IIO .002 5254 144 0.425 .426 0.003 4576 .003 4749 173 0.485 .486 0.004 5875 .004 6080 20 5 .367 368 .369 .002 5398 .002 5543 .002 5688 144 '45 '45 146 .427 .428 .429 .003 4923 .003 5096 .003 5271 173 175 J 74 .487 .488 489 .004 6285 .004 6492 .004 6698 205 20J 206 208 0.370 0.002 5834 0.430 0.003 5445 0.490 0.004 6906 371 .372 .002 5980 .002 6l26 146 43 1 .432 .003 5621 .003 5797 ! 7 6 f nf\ .491 .492 .004 7113 .004 7322 207 209 373 .002 6273 Hg 433 003 5973 170 493 .004 7531 209 374 .002 6421 434 .003 6150 177 177 494 .004 7740 209 211 0-375 376 0.002 6568 .002 6717 149 y A f\ o-435 .436 0.003 6327 .003 6505 I 7 8 1 7% o-495 496 0.004 7951 .004 8161 210 377 .378 .002 6866 .002 7015 1 49 149 rr\ 437 438 .003 6683 .003 6862 178 III 497 498 .004 8373 .004 8585 212 379 .002 7165 150 150 439 .003 7042 I oO 180 499 .004 8797 212 213 O M N COrJ- OO 00 00 00 00 CO CO CO CO CO d 0.002 7315 ,OO2 7466 .002 7617 .OO2 7769 .002 7921 151 151 152 152 152 0.440 .441 .442 443 444 0.003 7 22a .003 7402 .003 7583 .003 7765 .003 7947 180 181 182 183 0.500 5 1 .52 53 54 0.004 9010 .005 1173 .005 3397 .005 5681 .005 8029 2163 2224 2284 2348 2412 0.385 .3^6 387 388 389 0.002 8073 .002 8226 .002 8380 .002 8574 .002 8689 '53 154 '55 155 o-445 446 447 .448 449 0.003 8130 .003 8313 .003 8496 .003 8680 .003 8865 183 183 184 185 185 59 0.006 0441 .006 2919 .006 5464 .006 8079 .007 0765 2478 *545 2615 2686 2760 0.390 .391 0.002 8844 .OO2 8999 J 55 0.450 451 0.003 95 .003 9236 186 186 0.60 .61 0.007 3525 .007 6361 2836 201 1 39* 393 394 .002 9155 .002 9311 .002 9468 ^56 157 158 .452 453 454 .003 9422 .003 9609 003 9797 187 188 187 .62 I 3 .64 .007 9274 .008 2268 .008 5345 *7 5 2994 377 3163 o-395 396 397 398 O.OO2 9626 .002 97 g 4 .002 9942 .003 oioi 158 158 i59 0-455 45 6 457 458 0.003 9984 .004 0173 .004 0362 .004 0551 189 189 189 0.65 .66 6 7 .68 0.008 8508 .009 1759 .009 5103 .009 8542 3*51 3344 3439 3C 1 O 399 .003 0260 *59 160 459 .004 0741 191 .69 .010 2081 33V 3642 0.400 0.003 0420 0.460 0.004 9 3 2 0.70 o.oio 5723 17CO .401 .402 43 .404 .003 0580 .003 0741 .003 0903 .003 1064 161 162 161 163 .461 .462 .463 .464 .004 1123 .004 1315 .004 1507 .004 1700 192 192 193 193 .72 73 74 .010 9473 .on 3336 .on 7316 .012 1419 *-> 3863 3980 4103 4*33 0.405 .406 .407 .408 .409 0.003 1227 .003 1389 .003 1553 .003 1716 .003 1881 162 164 163 '65 164 [467 .468 .469 0.004 1893 .004 2087 .004 2281 .004 2476 .004 2672 194 194 '95 196 196 0.75 .76 77 .78 79 0.012 5652 .013 0022 .013 4536 .013 9202 .014 4031 4370 45 H 4666 4829 5002 0.410 .411 .412 413 .414 0.003 2045 .003 221 I .003 2376 .003 2543 .003 2709 166 165 167 166 168 0.470 .471 .472 473 474 0.004 2868 .004 3064 .004 3261 .004 3459 .004 3657 196 197 198 198 199 0.80 .81 .82 0.014 9033 .015 4219 .015 9603 .Ol6 5202 .017 1033 5186 5384 5599 ' 5831 | 6087 0.415 .416 .417 .418 .419 0.003 ^877 .003 3044 .003 3213 .003 3381 .003 3550 167 169 168 169 170 0-475 476 477 478 479 0.004 3856 .004 4055 .004 4255 .004 4456 .004 4657 199 200 201 201 201 0.85 .86 87 .88 .89 O.OI7 7I 10 .Ol8 3486 .019 0165 .019 7195 .020 4629 6366 6679 7030 7434 7900 0.420 0.003 37 20 0.480 0.004 4858 0.90 0.021 2529 621 TABLE XII, 9 t if i . .4 3' 4 A S log wij log TO 2 m l m, m, TO! i m 3 7H 2 m i 00 0.0000 90 o 90 o 180 o 180 o 180 o 1 4.2976 9.9999 2 2 3 90 20 90 20 178 40 178 40 179 o 359 o 359 5' 2 3-395 9.9996 44 6 90 40 90 40 177 20 177 20 178 o 358 o 358 9 3 2.8675 9.9992 7 8 91 o 91 o 176 o 176 o 177 o 357 o 357 H 4 2.4938 9.9986 9 3 2 91 20 91 20 174 40 174 40 176 o 356 o 356 18 5 2.2044 9.9978 " 55 9 I 41 91 41 173 19 173 19 175 o 355 355 Z 3 6 .9686 9.9968 14 19 9 2 I 9 2 I 171 59 171 59 174 o 354 o 354 28 7 .7698 9-9957 1 6 42 92 22 92 22 170 38 170 38 172 59 353 * 353 3^ 8 .S98i 9-9943 19 7 92 42 92 42 169 18 169 18 171 59 35* i 35 2 37, 9 4473 9.9928 ai 32 93 3 93 3 167 57 167 57 170 58 35i 2 35 1 4a; 10 .3130 9.9911 23 57 93 2 5 93 *5 166 35 166 35 169 57 35 3 35 47! 11 .1922 9.9892 26 23 93 46 93 46 165 14 165 14 1 68 55 349 4 349 5 2 ; 12 .0824 9.9871 28 50 94 8 94 8 163 52 163 52 167 54 348 6 348 56: 13 0.9821 9.9848 3 1 i? 94 3 1 94 3i 162 29 162 29 166 51 347 8 348 i 14 0.8898 9.9823 33 46 94 53 94 53 161 7 161 7 165 48 346 ii 347 6: 15 0.8045 9.9796 36 15 95 *7 95 '7 '59 43 '59 43 164 44 345 H 346 ii 16 0.7254 9.9767 38 46 95 40 95 4 158 20 158 20 163 40 344 17 345 '6 17 0.6518 9.9736 41 18 96 5 96 5 156 55 156 55 162 34 343 ai 344 21 18 0.5830 9.9702 43 5i 96 30 96 30 155 30 '55 3 161 27 342 27 343 2 7 19 0.5185 9.9667 46 26 96 56 96 56 154 4 154 4 160 19 341 32 342 32 20 0.4581 9.9629 49 2 97 23 97 a3 "5a 37 'Sa 37 '59 9 340 38 34i 37 21 0.4013 9.9588 51 41 97 5 97 5 151 10 151 10 157 58 339 45 34 43> 22 0.3479 9-9545 54 98 19 98 19 149 41 149 41 156 45 338 53 339 49 23 0.2976 9-9499 57 5 98 49 98 49 148 ii 148 ii 155 29 338 o 338 54| 24 0.2501 9-945 1 59 5 1 99 20 99 20 146 40 146 40 154 ii 337 9 338 o 25 0.2053 9.9400 62 40 99 53 99 53 H5 7 H5 7 152 50 336 19 337 6' 26 0.1631 9-9345 6 5 33 100 28 100 28 143 32 H3 3 2 I5 1 2 5 335 28 336 13 27 0.1232 9.9287 68 30 ioi 5 ioi 5 '4i 55 HI 55 149 56 334 38 335 '9 28 0.0857 9.9226 7i 33 101 45 ioi 45 140 15 140 15 148 22 333 49 334 25 29 0.0503 9.9161 74 4i 102 27 102 27 138 33 138 33 146 42 333 i 333 3 2 30 0.0170 9.9092 77 58 I0 3 I 3 I0 3 I 3 136 46 136 46 '44 55 332 12 33* 39 31 9-9857 9.9019 8x 23 104 4 104 4 134 5 6 134 5 6 142 59 331 24 331 46 32 9.95 6 5 9.8940 85 o 105 i 105 i '3 2 59 '3* 59 140 51 33 37 330 54 33 9.9292 9.8856 88 54 106 6 106 6 '3 54 13 54 138 27 3 2 9 49 33 2 34 9.9040 9-8765 93 " IO7 22 107 22 128 38 128 38 135 39 329 2 329 10 35 9.8808 9.8665 98 7 108 58 108 58 126 2 126 2 132 13 328 14 328 19 36 9.8600 9-8555 104 20 III 13 III 13 122 47 122 47 127 29 327 27 327 28 36 52.2 9- 8 443 9.8443 116 34 116 34 116 34 116 34 116 34 116 34 3*6 45 3^6 45 This table exhibits the limits of the roots of the equation sin (/ C) = mo sin 4 z', when there are four real roots. The quantities m l and w 2 are the limiting values of ra , and the values of z/, z 2 ' 9 z/, and z/, corresponding to ea(;h of these, give the limits of the four real roots of the equation. 622 TABLE XII, Y z i' z / 1 8 * / 4 b log TO! log ?M m, m, m l ro, n 2 ! mi m 3 + 00 00 0.0000 O 90 o 90 180 o 180 o 180 o 1 4.2976 9-9999 I I 20 I 20 89 40 89 40 177 37 180 55 181 o! 2 3-395 9.9996 2 O 2 4 2 40 89 20 89 20 175 14 181 51 182 o' 8 2.8675 9.9992 3 o 4 o 4 o 89 o 89 o 172 52 182 46 183 o 4 2.4938 9.9986 4 o 5 20 5 20 88 40 88 40 170 28 183 4* 184 o 5 2.2044 9.9978 5 o 6 41 6 41 88 19 88 19 168 5 184 37 185 o 1 6 .9686 9.9968 6 o 8 i 8 i 87 59 87 59 165 41 185 32 186 o 7 .7698 9-9957 7 i 9 22 9 22 87 38 87 38 163 18 186 28 186 59, 8 .5981 9-9943 8 i 10 42 10 42 87 18 87 18 160 53 187 23 187 59 9 4473 9.9928 9 * i* 3 12 3 86 57 86 57 158 28 188 18 188 58! 10 .3130 9.9911 10 3 13 25 13 25 86 35 86 35 i5 6 3 189 13 189 57 11 .1922 9.9892 ii 5 14 4 6 14 46 86 14 86 14 153 37 190 8 19 56: 12 .0824 9.9871 12 6 16 8 16 8 85 52 85 52 151 10 191 4 191 54; 13 0.9821 9.9848 13 9 17 31 17 3 1 85 29 85 29 148 43 191 59 192 52 14 0.8898 9.9823 14 12 18 53 18 53 85 7 85 7 146 14 192 54 193 49 15 0.8045 9.9796 15 16 20 17 20 17 84 43 84 43 H3 45 193 49 194 46 16 0.7254 9.9767 I 6 20 21 40 21 40 84 20 84 20 141 H 194 44 195 43, 17 0.6518 9.9736 17 26 ^3 5 2 3 5 83 55 83 55 138 42 195 39 196 39 J 18 0.5830 9.9702 18 33 24 30 24 30 83 3 83 30 136 9 196 33 197 33i 11) 0.5185 9.9667 19 41 25 56 25 56 83 4 83 4 133 34 197 28 198 28 20 0.4581 9.9629 20 51 27 2 3 27 23 82 37 82 37 130 58 198 23 199 22 21 0.4013 9.9588 22 2 28 50 28 50 82 10 82 10 128 19 199 17 200 15 22 0-3479 9-9545 23 15 3 19 30 19 81 41 8 1 41 125 38 200 II 201 07 23 0.2976 9.9499 24 31 3i 49 3i 49 81 ii 81 ii 122 55 201 6 202 24 0.2501 9-945 i 25 49 33 ^o 33 20 80 40 80 40 120 9 202 202 51 25 0.2053 9.9400 27 10 34 53 34 53 80 7 80 7 117 20 202 54 20 3 41 26 0.1631 9-9345 28 35 36 28 36 28 79 3 2 79 3 2 114 27 203 47 204 32 27 0.1232 9.9287 3 4 38 5 38 5 78 55 78 55 III 30 204 41 205 22 1 28 0.0857 9.9226 3i 38 39 45 39 45 78 15 78 15 108 27 205 35 206 II 29 0.0503 9.9161 33 18 4i 27 4i 27 77 33 77 33 i5 19 206 28 206 59 30 0.0170 9.9092 35 5 43 1 3 43 n 76 47 76 47 102 3 207 a i 207 48 1 31 9-9 8 57 9.9019 37 i 45 4 45 4 75 5 6 75 5 6 98 37 208 14 208 36 32 9.95 6 5 9.8940 39 9 47 * 47 i 74 59 74 59 95 209 06 209 23 33 9.9292 9.8856 4i 33 49 6 49 6 73 54 73 54 91 6 209 58 210 II 34 9.9040 9.8765 44 21 5 I 22 51 22 72 38 72 38 86 49 210 50 210 58 35 9.8808 9.8665 47 47 53 58 53 58 71 2 71 2 81 53 211 41 211 46 36 9.8600 9.8555 5 2 3i 57 13 57 13 68 47 68 47 75 40 212 32 212 33 +36 52.2 9- 8 443 9.8443 63 26 63 26 63 26 63 26 63 26 63 26 213 15 2I 3 15 This table exhibits the limits of the roots of the equation sin (d C) = m sin* 2', when there are four real roots. The quantities m l and m^ are the limiting values of m , and the values of z/> z z'> z s> 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. n log*S Diff. logs* Diff. i to* Diff. o.oooo .0001 .0002 .0003 .0004 O.OOO 0000 .000 0965 .000 1930 .000 2894 .000 3858 965 965 964 964 963 0.0060 .0061 .0062 .0063 .0064 0.005 7*98 .005 8243 .005 9187 .006 0131 .006 1075 945 944 944 944 944 0.0120 .OI2I .0122 .0123 .0124 o.on 3417 .on 4343 .on 5268 .on 6193 .on 7118 926 925 925 925 925 0.0005 o.ooo 4821 0.0065 0.006 2019 O.OI25 o.on 8043 .0000 .0007 .0008 .0009 .000 5784 .000 6747 .000 7710 .000 8672 9 6 3 963 963 962 962 .0066 .0067 .0068 .0069 .006 2962 .006 3905 .006 4847 .006 5790 943 943 942 943 942 .0126 .0127 .0128 .0129 .on 8967 .01 1 9890 .012 0814 .012 1737 924 ! 923 924 923 923 O.OOI .001 1 .0012 .0013 .0014 o.ooo 9634 .001 0595 .001 1556 .001 2517 .001 3478 961 961 961 961 960 0.0070 .0071 .0072 .0073 .0074 0.006 6732 .006 7673 .006 8614 .006 9555 .007 0496 941 941 941 941 940 0.0130 .0131 .0132 .0133 .0134 0.012 2660 .012 3583 .012 4505 .012 5427 .012 6348 9*3 922 922 921 921 0.0015 .0016 .0017 o.ooi 4438 .001 5398 .001 6357 960 959 0.0075 .0076 .0077 0.007 *436 .007 2376 .007 3316 940 94 0.0135 .0136 .0137 0.012 7269 .012 8190 .012 9III 921 921 i .0018 .0019 .001 7316 .001 8275 959 959 .0078 .0079 .007 4255 .007 5194 939 939 n^ n .0138 .0139 .013 0032 .013 0952 921 920 n T n 959 y3:/ y*y O.OO2O .0021 .0022 .0023 .0024 o.ooi 9234 .002 0192 .002 1150 .002 2107 .002 3064 958 958 957 957 957 0.0080 .0081 .0082 .0083 .0084 0.007 6133 .007 7071 .007 8009 .007 8947 .007 9884 93 8 93 8 93 8 937 937 0.0140 .0141 .0142 .0143 .0144 0.013 I ^7 I .013 2791 .013 3710 .013 4629 .013 5547 920 919 9*9 ! 918 918 O.0025 .0026 .0027 .O028 .OO29 0.002 4021 .002 4977 .002 5933 .002 6889 .002 7845 956 956 956 956 955 0.0085 .0086 .0087 .0088 .0089 0.008 0821 .008 1758 .008 2694 .008 3630 .008 4566 937 936 936 936 936 0.0145 .0146 .0147 .0148 .0149 0.013 6465 .013 7383 .013 8301 .013 9218 .014 0135 918 918 917 917 917 O.0030 .0031 .0032 0033 .0034 0.002 8800 .002 9755 .003 0709 .003 1663 .003 2617 955 954 954 954 953 0.0090 .0091 .0092 .0093 .0094 0.008 5502 .008 6437 .008 7372 .008 8306 .008 9240 935 935 934 934 934 0.0150 .0151 .0152 153 .0154 0.014 I0 5 2 .014 1968 .014 2884 .014 3800 .014 4716 916 916 916 916 0.0035 0.003 3570 n f * 0.0095 0.009 OI 74 O -5 A 0.0155 0.014 563! .0036 .0037 .0038 .0039 .003 4523 .003 5476 .003 6428 .003 7380 y.>3 953 952 952 952 .0096 .0097 .0098 .0099 .009 i i 08 .009 2041 .009 2974 .009 3906 y 53- 933 933 932 932 .0156 !o! 5 8 .0159 .014 6546 .014 7460 .014 8374 .014 9288 914 914 914 914 0.0040 0.003 8332 A ? 1 O.OIOO 0.009 4838 C\ 1 1 0.0160 0.015 202 .0041 .003 9284 952 .0101 .009 5770 y^- 6 .0161 .015 1115 9 3 .0042 .004 0235 95 1 .0102 .009 6702 932 r\ * T .0162 .015 2028 n T -9 .0043 .0044 .004 i i 86 .004 2136 950 95 .0103 .0104 .009 7633 .009 8564 93 1 931 93 I .0163 .0164 .015 2941 015 3 8 54 y 1 3 913 912 0.0045 .0046 .0047 .0048 .0049 0.004 3086 .004 4036 .004 4985 .004 5934 .004 6883 950 949 949 949 949 0.0105 .0106 .0107 .0108 .0109 0.009 9495 .010 0425 .010 1355 .010 2285 .010 3215 93 93 93 93 929 0.0165 .0166 .0167 .0168 .0169 0.015 4766 .015 5678 .015 6589 .015 7500 .015 8411 912 911 911 911 911 0.0050 .0051 .0052 .0053 .0054 0.004 7832 .004 8780 .004 9728 .005 0675 .005 1622 948 948 947 947 947 O.OIIO .0111 .0112 .0113 .0114 o.oio 4144 .010 5073 .010 6001 .010 6929 .010 7857 929 928 928 928 928 0.0170 .0171 .0172 .0173 .0174 0.015 9322 .016 0232 .016 1142 .016 2052 .016 2961 910 910 910 909 909 0.0055 .0056 .0057 .0058 .0059 0.005 ^569 005 3515 .005 4461 .005 5407 .005 6353 946 946 946 946 945 O.OII5 .0116 .0117 .0118 .OII9 o.oio 8785 .010 9712 .on 0639 .on 1565 .on 2491 927 927 926 926 926 0.0175 .0176 .0177 .0178 .0179 0.016 3870 .016 4779 .016 5688 .016 6596 .016 7504 909 909 908 908 908 O.0060 0.005 7298 0.0120 o.on 3417 0.0180 0.016 8412 624 TABLE XIII For finding the Eatio of the Sector to the Triangle. n logs* Diff. 1? log si Diff. >? logs* Diff. 0.0180 .0181 .0182 0.016 8412 .016 9319 .017 0226 907 907 0.0240 .0241 .0242 0.022 2330 .022 3220 .022 4109 890 889 oo' 0.0300 .0301 .0302 0.027 5 2 *8 .027 6091 .027 6964 873 873 .0183 .0184 .017 1133 .017 2039 907 906 906 .0243 .0244 .022 4998 .022 5887 009 889 889 .0303 .0304 .027 7836 .027 8708 872 872 872 0.0185 .0186 .0187 0.017 ^945 .017 3851 .017 4757 906 906 0.0245 .0246 .0247 0.022 6776 .022 7664 .022 8552 888 888 ooo 0.0305 .0306 .0307 0.027 9580 .028 0452 .028 1323 872 I 71 .0188 .0189 .017 5662 .017 6567 95 905 904 .0248 .0249 .022 9440 .023 0328 oofl 888 887 .0308 .0309 .028 2194 .028 3065 871 871 871 0.0190 .0191 .0192 .0193 .0194 0.017 7471 .017 8376 .017 9280 .018 0183 .018 1087 905 904 903 904 93 O.O25O .0251 .0252 .0253 .0254 0.023 I2I 5 .023 2IO2 .023 2988 .023 3875 .023 4761 887 886 887 886 886 0.0310 .0311 .0312 .0313 .0314 0.028 3936 .028 4806 .028 5676 .028 6546 .028 7415 870 870 870 869 869 0.0195 .0196 .0197 .0198 .0199 0.018 1990 .018 2893 .018 3796 .018 4698 .018 5600 93 903 902 902 901 0.0255 .0256 .0257 .0258 .0259 0.023 5 6 47 .023 6532 .023 7417 .023 8302 .023 9187 885 885 885 885 884 0.0315 .0316 .0317 .0318 .0319 0.028 8284 .028 9153 .029 0022 .029 0890 .029 I 75 8 869 869 868 868 868 0.0200 .0201 .0202 .0203 .0204 0.018 6501 .018 7403 .018 8304 .018 9205 .019 0105 902 901 901 900 900 0.0260 .026l .0262 .0263 .0264 0.024 7 I .024 0956 .024 1839 .024 2723 .024 3606 885 883 884 883 883 0.0320 .0321 .0322 .0323 .0324 0.029 2626 .029 3494 .029 4361 .029 5228 .029 6095 868 867 867 867 866 0.0205 .0206 0.019 1005 .019 1905 900 0.0265 .0266 0.024 4489 .024 5372 883 00^ 0.0325 .0326 0.029 ^961 .029 7827 866 O /I .0207 .0208 .0209 .019 2805 .019 3704 .019 4603 QOO 8 99 8 99 899 .0267 .0268 .0269 .024 6254 .024 7136 .024 8018 o o 2 882 882 882 .0327 .0328 .0329 .029 8693 .029 9559 .030 0424 oDO 866 865 866 0.0210 .0211 0.019 5502 .019 6401 III 0.0270 .0271 0.024 8900 .024 9781 881 OO f 0.0330 .0331 0.030 1290 .030 2154 864 Of. \ .0212 .019 7299 oQo 8n, .0272 .025 0662 OO 1 00, .0332 .030 3019 SOS Of. .0213 .019 8197 090 807 .0273 .025 1543 oo 1 880 333 .030 3883 a 04 864 .0214 .019 9094 7 / 898 .0274 .025 2423 880 334 .030 4747 864 0.0215 .0216 .0217 .0218 .0219 0.019 999 2 .020 0889 .020 1785 .020 2682 .020 3578 897 896 897 896 896 0.0275 .0276 .0277 .0278 .0279 0.025 333 .025 4183 .025 5063 .025 5942 .025 6821 880 880 879 879 879 0.0335 0336 0337 .0338 339 0.030 5611 .030 6475 .030 7338 .030 8201 .030 9064 861 86 3 863 86 3 862 O.0220 0.020 4474 o_ O.O28O 0.025 7700 9.Tn 0.0340 0.030 9926 8fi-7 .O22I .0222 .0223 .0224 .020 5369 .020 6264 .020 7159 .020 8054 895 895 895 8 95 894 .0281 .0282 .0283 .0284 .025 8579 .025 9457 .026 0335 .026 1213 079 878 878 878 877 .0341 .0342 343 .0344 .031 0788 .031 1650 .031 2512 03 1 3373 OO2 862 862 861 861 O.0225 .0226 .0227 .0228 .0229 0.020 8948 .020 9842 .021 0736 .021 1630 .O2I 2523 8 94 894 894 893 893 0.0285 .0286 .0287 .0288 .0289 0.026 2090 .026 2967 .026 3844 .026 4721 .026 5597 877 877 877 876 876 0.0345 .0346 347 .0348 .0349 0.031 4234 .031 5095 031 5956 .031 6816 .031 7676 861 861 860 860 860 0.0230 .0231 .0232 0233 .0234 0.021 3416 .O2I 4309 .021 5201 .O2I 6093 .021 6985 893 8 9 2 892 892 891 0.0290 .0291 .0292 .0293 .0294 0.026 6473 .026 7349 .026 8224 .026 9099 .026 9974 876 875 875 875 875 0.0350 .0351 .0352 353 .0354 0.031 8536 .031 9396 .032 0255 .032 1114 .032 1973 860 859 859 Hi 0.023 5 .0236 .0237 .0238 .0239 0.021 7876 .021 8768 .021 9659 .022 0549 .022 1440 8 9 2 891 890 891 890 0.0295 .0296 .0297 .0298 .0299 0.027 0849 .027 1723 .027 2597 .027 3471 .027 4345 874 874 874 874 873 o-0355 .0356 .0357 .0358 .0359 0.032 2831 .032 3689 .032 4547 .032 5405 .032 6262 858 858 858 857 8 5 8 0.0240 0.022 2330 0.0300 0.027 5218 0.0360! 0.032 7120 40 625 TABLE XIII, For finding the Ratio of the Sector to the Triangle. i log> Diff. *, Diff. < log* Diff. 0.0360 .0361 .0362 .0363 .0364 0.032 7120 .032 7976 .032 8833 .032 9689 .033 0546 856 857 856 857 855 0.060 .061 .062 .063 .064 0.052 5626 .053 3602 .054 1556 .054 9488 055 7397 7976 7954 7932 7909 7888 0.120 .121 .122 .123 .124 0.096 8849 .097 5692 .098 2520 .098 9331 .099 6127 6843 6828 6811 6796 6780 mvO t^-OO ON CO CO CO CO CO O O O O O ' 0.03^3 1401 .033 2257 .033 3112 .033 3967 .033 4822 856 855 855 855 855 0.065 .066 .067 .068 .069 0.056 5285 .057 3150 .058 0994 .058 8817 .059 6618 7865 7844 7823 7801 7780 0.125 .126 .127 .128 .129 o.ioo 2907 .100 9672 .101 6421 .102 3154 .102 9873 6765 > 6749 6733 6719 , 6703 0.0370 .0371 .0372 0373 .0374 0.033 5677 .033 6531 .033 7385 .033 8239 .033 9092 854 854 854 853 854 0.070 .071 .072 .073 .074 0.060 4398 .061 2157 .061 9895 .062 7612 .063 5308 7759 7738 7696 7676 0.130 .131 .132 '33 '34 O.IO3 6576 .104 3264 .104 9936 .105 6594 .106 3237 6688 6672 ! 6658 ! 6643 6628 0.0375 .0376 .0377 0378 0379 0.033 9946 .034 0799 .034 1651 .034 2504 .034 3356 852 853 852 852 0.075 .076 .077 .078 79 0.064 2984 .065 0639 .065 8274 .066 5888 .067 3482 7655 7635 7614 7594 7575 0.135 .136 137 .138 '39 0.106 9865 .107 6478 .108 3076 .108 9660 .109 6229 6613 6598 6584 6569 6 554 O M O rorj- oo oo oo oo oo CO CO CO CO CO q q q q q d 0.034 4208 .034 5059 .034 5911 .034 6762 .034 7613 851 852 851 851 851 0.080 .081 .082 .083 .084 0.068 1057 .068 8612 .069 6146 .070 3661 .071 1157 7555 7534 7496 7476 0.140 .141 .142 143 .144 o.no 2783 .no 9323 .III 5849 .112 2360 .112 8857 6540 6526 6511 6497 6483 vnvo r--oo ON 00 00 00 00 00 CO CO CO CO CO O 0.034 8464 34 93H .035 0164 .035 1014 .035 1864 850 850 850 850 849 0.085 .086 .087 .088 .089 0.071 8633 .072 6090 073 3527 .074 0945 74 8345 7457 7437 7418 7400 738o 0.145 .146 .147 .148 .149 0.113 534 .114 1809 .114 8264 .115 4704 .116 1131 6469 6455 6440 6427 6413 0.0390 .0391 0.035 2713 035 35 6 2 849 0.090 .091 0-075 5725 .076 3087 7362 0.150 .151 0.116 7544 "7 3943 6399| .0392 .0393 0394 035 44" 035 5259 .035 6108 848 849 848 .092 93 .094 .077 0430 .077 7754 .078 5060 7343 7324 7306 7288 .152 153 '54 .118 0329 .118 6701 .119 3059 6372 6358 6 345 i 0.0395 .0396 397 .0398 399 0.035 6956 .035 7804 .035 8651 .035 9499 .036 0346 848 847 848 847 846 0.095 .096 .097 .098 .099 0.079 2348 .079 9617 .080 6868 .081 4101 .082 1316 7269 7251 7233 7215 7197 0.155 .156 '57 .158 '59 0.119 944 120 5735 .121 2053 .121 8357 .122 4649 6331 6318 6304 | 6292 6278 | 0.040 .041 .042 43 44 0.036 1192 .036 9646 037 8075 .038 6478 .039 4856 8454 8429 8403 8378 8353 O.I 00 .101 .102 .103 .104 0.082 8513 .083 5693 .084 2854 .084 9999 .085 7125 7180 7161 7H5 7126 7110 0.160 .161 .162 .163 .164 0.123 927 .123 7192 .124 3444 .124 9682 .125 5908 6265 6252 6238 6226 6213 0.045 .046 .047 .048 049 0.040 3209 .041 1537 .041 9841 .042 8121 .043 6376 8328 8304 8280 8255 8231 0.105 .IO6 .107 .108 .109 0.086 4235 .087 1327 .087 8401 .088 5459 .089 2500 7092 7074 7058 7041 7023 0.165 .166 .167 .168 .169 0.126 ai2i .126 8321 .127 4508 .128 0683 .128 6845 6200 ; 6187 ! 6175 6162 6149 0.050 .051 .052 .053 .054 0.044 4 6 7 .045 2814 .046 0997 .046 9157 .047 7294 8207 8183 8160 8113 O.IIO .III .112 "3 .114 0.089 9523 .090 6530 .091 3520 .092 0494 .092 7451 7007 6990 6974 6957 6940 0.170 .171 .172 .173 .174 0.129 2994 .129 9131 13 5255 .131 1367 .131 7466 6137 6124 6112 6099 6087 0.055 .056 .057 .058 .059 0.048 5407 049 3496 .050 1563 .050 9607 .051 7628 8089 8067 8044 8021 7998 0.115 .116 .117 .118 .119 0.093 439 1 .094 1315 .094 8223 .095 5114 .096 1990 6924 6908 6891 6876 6859 0.175 .176 .177 .178 .179 0-132 3553 .132 9628 .133 5690 .134 1740 134 7778 6062 6050 6038 6026 O.O60 0.052 5626 0.120 0.096 8849 0.180 0.135 3804 626 TABLE XIII. For finding the Ratio of the Sector to the Triangle. * logs* Diff. * logs* Diff. i toff* Diff. o.i8o .181 .182 .183 .184 0.135 3804 .135 9818 .136 5821 .137 1811 .137 7789 6014 6003 5990 5978 5966 O.240 .241 .242 *43 .244 0.169 509* .170 0470 .170 5838 .171 1197 .171 6547 5378 5368 5359 5350 534 0.300 .301 .302 .303 34 0.200 2285 .200 7157 .201 2021 .201 6878 .202 1727 4872 4864 4857 4849 4842 0.185 .186 .187 .188 .189 0-138 3755 .138 9710 !39 5 6 53 .140 1585 .140 7504 5955 5943 593* 5019 5908 0.245 .246 .247 .248 .249 0.172 1887 .172 7218 .173 2540 .173 7853 .174 3156 15322 |53'3 15303 5*95 0.305 .306 37 .308 .309 O.2O2 6569 .203 1403 .203 6230 .204 1050 .204 5862 4834 4827 4820 4812 4805 0.190 .191 0.141 3412 .141 9309 5897 c88 c 0.250 .251 0.174 8451 .175 3736 5285 O.3IO .311 0.205 0667 .205 5464 4797 .192 193 .194 .142 5194 .143 1068 .143 6931 5874 5863 5851 .252 *53 .254 .175 9013 .176 4280 .176 9538 5*77 5267 5258 5*5 .312 3'3 3'4 .206 0254 .206 5037 .206 9813 4790 4783 4776 4768 0.195 .196 .197 .198 .199 0.144 2782 .144 8622 .145 4450 .146 0268 .146 6074 5840 5828 5818 5806 5795 0.255 .256 .257 .258 .259 0.177 4788 .178 0029 .178 5261 .179 0484 .179 5698 5*3* 5**3 5214 5*5 -3'5 .316 :\\l 3*9 O.207 4581 .207 9342 .208 4096 .208 8843 .209 3582 4761 4754 4747 4739 4733 O.200 .201 0.147 *86g .147 7653 5784 0.260 .261 0.180 0903 .180 6100 5*97 0.320 3*i 0.209 8315 .210 3040 47*5 .202 .203 .204 .148 3427 .148 9189 .149 4940 5774 5762 575 1 574* .262 .263 .264 .181 1288 .181 6467 .182 1638 5188 5*79 5171 5162 .322 3*3 3*4 .210 7759 .211 2470 .211 7174 4719 4711 4704 4697 O.20C 0.150 0681 0.265 0.182 6800 0.325 O.2I2 1871 .206 .150 6411 5730 .266 .183 1953 5153 .212 6562 4691 .207 .208 .209 .151 2130 .151 7838 '5* 3535 57 X 9 5708 5697 5687 .267 .268 .269 .183 7098 .184 2235 .184 7363 5*45 5'37 5128 5120 3*7 3*8 3*9 .213 1245 .213 5921 .214 0591 4676 4670 4662 0.210 .211 .212 .213 .214 0.152 9222 .153 4899 .154 0565 .154 6210 .155 1865 5677 5666 5655 5645 5634 0.270 .271 .272 *73 .274 0.185 2483 .185 7594 .186 2696 .186 7791 .187 2877 5111 5102 5086 5078 0.330 33* 33* 333 334 0*214 5*53 .214 9909 .215 4558 .215 9200 .216 3835 4656 4649 4642 4635 4629 O.2I5 .216 .217 .218 .219 o-i55 7499 .156 3123 .156 8737 .157 4340 '57 9933 5624 5614 5603 5593 5583 0.275 .276 .277 .278 .279 0.187 7955 .188 3024 .188 8085 .189 1138 .189 8183 5069 5061 553 545 537 0-335 336 337 338 339 0.216 8464 .217 3085 .217 7700 .218 2308 .218 6910 4621 4615 4608 4602 0.220 .221 .222 .223 .224 0.158 5516 .159 1089 .159 6652 .160 2204 .160 7747 5573 5563 555* 5543 553* 0.280 .281 .282 .283 .284 0.190 3220 .190 8249 .191 3269 .191 8281 .192 3286 5029 5020 5012 5005 4996 0.340 341 34* 343 344 0.219 1505 .219 6093 .220 0675 .220 5250 .220 9818 4588 4582 4575 4568 4562 0.225 .220 .227 .228 .229 0.161 3279 .161 8802 .162 4315 .162 9817 .163 5310 55*3 5513 5502 5493 5483 0.285 .286 .287 .288 .289 0.192 8282 .193 7271 .193 8251 .194 3224 .194 8188 4989 4980 4973 4964 4957 0-345 .346 347 .348 349 0.221 4380 .221 8915 .222 7483 .222 8025 ,223 2561 4555 4548 4542 4536 0.230 .231 .232 *33 *34 0.164 0793 .164 6267 .165 1710 .165 7184 .166 2628 5474 5463 5454 5444 5435 0.290 .291 .292 .293 .294 0.195 3145 .195 8094 .196 3035 .196 7968 .197 2894 4949 494 1 4933 (.926 0.350 35' 35* 353 354 0.223 7090 .224 1613 .224 6130 .225 0640 .225 5143 45*3 45^7 4510 453 4497 0.235 .236 0.166 8063 .167 3488 54*5 0.295 .296 0.197 7811 .198 2721 1.910 0-355 356 0.225 9 6 4 .226 4131 449 > 4484 *37 .167 8903 r^nfi .297 .198 7624 I-903 A 9 nA 357 .226 8615 .238 .168 4309 400 .298 .199 2518 1.094 l888 358 .227 3093 AA72 .239 .168 9705 5396 387 .299 .199 7406 4879 359 .227 7565 4--j- / * 44 66 0.240 0.169 59* 0.300 0.200 2285 0.360 0.228 2031 627 TABLE XIII, For finding the Katio of the Sector to the Triangle. * logss Diff. < log " Diff. * logs* Diff. 0.360 . 3 6i .362 .363 3 6 4 0.228 2031 .228 6490 .229 0943 .229 5390 .229 9831 4459 4453 4447 4441 4434 0.420 .421 .422 4*3 4*4 -*53 9 T 53 .254 3269 *54 7379 .255 1484 *55 5584 4116 4110 4105 4100 4095 0.480 . 4 8i .482 .483 .484 0.277 7*7* .278 1096 .278 4916 .278 8732 *79 *543 3824 3820 i 3816 3811 3806 U^SO t^OO O SO so SO so VO CO CO CO CO CO 6 * 0.230 4265 .230 8694 .231 3116 .231 7532 .232 1942 4429 4422 4416 4410 444 0.425 .426 4*7 .428 4*9 0.255 9679 .256 3769 .256 7853 .257 1932 .257 6006 4090 4084 4079 4074 4069 0.485 .486 .487 .488 489 0.279 6349 .280 0151 .280 3949 .280 7743 .281 1532 3802 379 8 3794 3785 3784 0.370 .371 37* 0.232 6346 *33 743 *33 5135 4397 439* Atxn 0430 431 43* 0.258 0075 .258 4139 .258 8198 4064 4059 0.490 .491 .492 0.281 5316 .281 9096 .282 2872 378o 3776 373 374 .233 9521 .234 3900 4300 4379 4374 433 434 .259 2252 .259 6300 4054 4048 4044 493 494 .282 6644 .283 0411 377* 3767 3762 0.375 376 377 .378 379 0.234 8274 .235 2642 .235 7003 .236 1359 .236 5709 4368 4361 4356 435 0.435 436 437 438 439 0.260 0344 .260 4382 .260 8415 .261 2444 .261 6467 4038 4033 4029 4023 o-495 496 497 .498 499 0.283 4173 .283 7932 .284 1686 .284 5436 .284 9181 3759 3754 ! 375 i 3745 4344 4019 374* O M M COrJ- OO 00 00 00 00 CO tO CO CO CO o 0.237 0053 .237 4391 .237 8723 .238 3050 .238 7370 4338 433* 43 2 7 4320 0.440 44 1 44* 443 444 0.262 0486 .262 4499 .262 8507 .263 2511 .263 6509 4013 4008 4004 3998 0.500 .501 .502 .503 .504 0.285 *9*3 .285 6660 .286 0392 .286 4121 .286 7845 3737 373* I 37^9 37^4 . 43* 5 3994 3720 tJ->sO t--OO O oo oo oo oo oo CO CO CO tO tO 6 * 0.239 *685 *39 5993 .240 0296 .240 4594 .240 8885 4308 433 4298 4291 4286 0.445 446 447 448 449 0.264 53 .264 4492 .264 8475 .265 2454 .265 6428 3989 3983 3979 3974 39 6 9 0.505 .506 507 .508 .509 0.287 ^65 .287 5281 .287 8992 .288 2700 .288 6403 3716 37" ; 3708 i 3703 3699 1 0.390 391 39* 393 394 0.241 3171 .241 7451 .242 1725 .242 5994 .243 0257 4280 4*74 4269 4*63 4*57 0.450 45 * 45* 453 454 0.266 0397 .266 4362 .266 8321 .267 2276 .267 6226 3965 3959 3955 395 3945 0.510 .511 .512 .514 0.289 OIO2 .289 3797 .289 7487 .290 1174 .290 4856 3695 3690 ; 3687 i 3682 3679 0.395 396 397 398 399 0.243 45H .243 8766 .244 3012 .244 7252 .245 1487 4*5* 4246 4240 4*35 4229 0.455 456 457 458 459 0.268 0171 .268 4111 .268 8046 .269 1977 .269 5903 3940 3935 393 1 3926 3921 0.515 .516 517 .518 .519 0.290 8535 .291 2209 .291 5879 .291 9545 .292 3207 3674 3670 3666 3662 ; 3657 0.400 .401 .402 403 0.245 57 J 6 .245 9940 .246 4158 .246 8371 4224 4218 0.460 .461 .462 463 0.269 9824 .270 3741 .270 7652 .271 1559 39*7 39" 3907 0.520 .521 .522 5*3 0.292 6864 .293 0518 .293 4168 .293 7813 3654 365 : 3645 404 .247 2578 4207 4201 .464 .271 5462 3898 5*4 .294 1455 3642 3637 0.405 .406 .407 .408 409 0.247 6779 .248 0975 .248 5166 .248 9351 .249 3531 4196 4191 4*85 4180 4*74 0.465 .466 467 .468 .469 0.271 9360 .272 3253 .272 7141 .273 1025 .273 4904 mi 3884 3879 3874 0-5*5 .526 5*7 .528 5*9 0.294 59* .294 8726 .295 2355 .295 5981 .295 9602 3634 36291 3626 3621 ; 3618 0.410 .411 .412 .413 .414 0.249 775 .250 1874 .250 6038 .251 0196 .251 4349 4169 4164 4158 4*53 4*47 0.470 .471 47* 473 474 0.273 8778 .274 2648 .274 6513 *75 374 .275 4230 3870 3865 3861 3856 385* 0-53 531 53* 533 534 0.296 3220 .296 6833 .297 0443 *97 449 .297 7650 3613 3610 i 3606 3601 3598 0.415 .416 .417 .418 .419 0.251 8496 .252 2638 .252 6775 .253 0906 .253 5032 4142 4*37 4131 4126 4121 0.475 476 477 .478 479 0.275 8082 .276 1929 .276 5771 .276 9609 .277 3443 3847 3842 3838 3834 3829 0-535 536 537 .538 539 0.298 1248 .298 4842 .298 8432 .299 2018 .299 5600 3594 359 3586 358* 3578 0.420 0.253 9*53 0.480 0.277 7*7* 0.540 0.299 9*78 628 TABLE XIIL For finding the Katio of the Sector to the Triangle. 1? logs* Diff. * log*a Diff. * log Diff. 0.540 54 54* 543 0.199 9178 .300 2752 .300 6323 .300 9890 3574 35 6 7 0.560 .561 .562 563 0.306 9938 37 3437 .307 6931 .308 0422 3499 3494 0.580 .581 .582 583 0.313 9215 .314 26^.1 .314 6064 .314 9481 34*6 34*3 3419 544 .301 3452 3559 .564 .308 3910 *484 584 .315 2898 3412 0-545 b$4* 547 .548 549 0.301 7011 .302 0566 .302 4117 .302 7664 .303 1208 3555 355 1 3547 3544 3540 0.565 .566 .567 .568 .569 0.308 7394 .309 0874 .309 4150 .309 7823 .310 1292 3480 3476 3473 3469 3466 0.585 .586 .587 .588 .589 0.315 6310 3'5 9719 .316 1124 .316 6525 .316 9923 3409 3405 3401 3398 3395 0.550 55 1 55* 553 554 0-303 4748 .303 8284 .304 1816 .304 5344 .304 8869 353 6 353* 35*8 35*5 0.570 571 57* 573 574 0.310 4758 .310 8220 .311 1678 3 11 5i33 3462 3458 3455 345 i 3447 0.590 .591 59* 593 594 0.317 3318 .317 6709 .318 0096 .318 3480 .318 6861 338 7 3384 3381 3377 0-555 556 557 558 559 0.305 2390 .305 5907 .305 9420 .306 2930 .306 6436 35'7 35*3 3510 3506 3502 0-575 576 577 .578 579 0.312 2031 3 1 * 5475 .312 8915 .313 2352 3'3 5785 3444 3440 3437 3433 343 0.595 596 597 598 599 0.319 0238 .319 3612 .319 6983 .320 0350 3*o 37H 3374 33 6 7 3364 3360 0.560 0.306 9938 0.580 0.313 9215 0.600 0.320 7074 TABLE XIV. For finding the Ratio of the Sector to the Triangle. X t X f Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.ooo 0.000 0000 o.ooo oooo I 0.030 o.ooo 0523 76 o.ooo 0506 11 .001 .000 0001 1 .000 0001 J .031 .000 0559 3 .000 0539 76 .002 .OOO OOO2 .OOO OOO2 .032 .000 0596 8 .000 0575 11 .003 .000 0005 3 .000 0005 3 033 .000 0634 3 .000 06 1 1 3 U MM .004 .000 0009 4 5 .000 0009 4 5 .034 .000 0674 4 40 .000 0648 3? O.005 o.ooo 0014 o.ooo 0014 0.035 o.ooo 0714 o.ooo 0686 .006 .OOO OO2I 7 .OOO OO2O .036 .000 0756 42 .000 0726 40 .007 .000 0028 7 .000 0028 037 .000 0799 43 .000 0766 4 .008 .009 .000 0037 .000 0047 9 10 II .000 0036 .000 0046 10 II .038 039 .000 0844 .000 0889 45 45 47 .000 0807 .000 0850 4 1 43 44 O.OIO .Oil o.ooo 0058 .000 0070 12 o.ooo 0057 .000 0069 12 0.040 .041 o.ooo 0936 .000 0984 48 o.ooo 0894 .000 0978 .012 .013 .000 0083 .000 0097 i! .000 0082 .000 0096 *3 14 .042 .043 .000 1033 .000 1084 49 51 .000 0984 .000 1031 $ i .014 .000 0113 ID 17 .OOO 01 I I 11 044 .000 1135 53 .000 1079 49 0.015 ! .Ol6 o.ooo 0130 .000 0148 18 o.ooo 0127 .000 0145 IJB 0.045 .046 o.ooo 1 1 88 .000 1242 It o.ooo 1128 .000 1178 5 ; .017 .018 .019 .000 0167 .000 0187 .000 0209 19 20 22 22 .000 0164 .000 0183 .000 0204 19 19 21 22 .047 .048 .049 .000 1298 .000 1354 .000 1412 56 56 58 59 .000 1229 .000 1281 .000 1334 5 1 5* 53 55 0.020 .021 .022 .023 .024 o.ooo 0231 .000 0255 .000 0280 .000 0306 .000 0334 24 \\ 28 28 o.ooo 0226 .000 0249 .000 0273 .000 0298 .000 0325 23 24 27 27 0.050 .051 .052 053 .054 o.ooo 1471 .000 1532 .000 1593 .000 1656 .000 1720 OS ON ON ON ON in 4* OJ HI M o.ooo 1389 .000 1444 .000 1500 .000 1558 .000 1616 VNVOOOOO ON w> 10 u-> 1/1 iri 0.025 .026 .027 .028 .029 o.ooo 0362 .000 0392 .000 0423 .000 0455 .000 0489 3 31 32 34 34 o.ooo 0352 .000 0381 .000 0410 .000 0441 .000 0473 29 *9 31 3* 33 0.055 .056 .057 .058 059 o.ooo 1785 .000 1852 .000 1920 .000 1989 .000 2060 6 7 68 69 o.ooo 1675 .000 1736 .000 1798 .000 i860 .000 1924 61 62 62 64 64 0.030 o.ooo 0523 o.ooo 0506 0.060 | o.ooo 2131 o.ooo 1988 629 TABLE XIV. For finding the Katio of the Sector to the Triangle. X * X e Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.o6o .061 .062 o.ooo 2131 .000 2204 .000 2278 73 a o.ooo 1988 .000 2054 .000 2 1 21 66 67 0.120 .121 .122 o.ooo 8845 .000 8999 .000 9154 154 155 o.ooo 7698 .000 7822 .000 7948 124 126 _ f .063 .064 .000 2354 .000 2431 70 .OOO 2189 .000 2257 Do 68 70 .123 .124 .000 9311 .000 9469 J 57 158 '59 .000 8074 .000 8202 120 128 128 0.065 .066 .067 .068 .069 o.ooo 2509 .000 2588 .000 2669 .000 2751 .000 2834 79 81 82 83 84 o.ooo 2327 .000 2398 .oco 2470 .000 2543 .000 2617 71 72 73 74 74 0.125 .126 .127 .128 .129 o.ooo 9628 .000 9789 .000 9951 .001 0115 .001 0280 161 162 164 '65 167 o.ooo 8330 .000 8459 .000 8590 .000 8721 .000 8853 129 131 132 133 0.070 .071 .072 o.ooo 2918 ,000 3004 .000 3091 86 87 o.ooo 2691 .000 2767 .000 2844 76 7 7 8 0.130 .131 .132 o.ooi 0447 .001 0615 .001 0784 168 169 o.ooo 8986 .000 9120 .000 9255 134 .073 .074 .000 3180 .000 3269 89 9 1 .000 2922 .000 3001 1 79 80 J33 .134 .001 0955 .001 1128 171 173 J 73 .000 9390 .000 9527 *35 137 0.075 .076 o.ooo 3360 .000 3453 93 o.ooo 3081 .000 3162 Si 82 .136 o.ooi 1301 .001 1477 176 o.ooo 9665 .000 9803 138 077 .000 3546 93 .000 3244 137 .001 1654 III .000 9943 140 .078 .000 3641 95 .000 3327 0^ .138 .001 1832 178 IJ?0 .001 0083 140 .079 .000 3738 97 97 .000 3411 85 .139 .OOI 2OI2 i oo 181 .001 0224 141 142 0.080 .081 o.ooo 3835 .000 3934 99 o.ooo 3496 .000 3582 86 8- 0.140 .141 o.ooi 2193 .001 2376 183 o.ooi 0366 .001 0509 143 .082 .000 4034 IOO .000 3669 y 88 .142 .001 2560 184 .001. 0653 144 .083 .084 .000 4136 .000 4239 I 02 103 104 .000 3757 .000 3846 89 90 .143 .144 .001 2745 .001 2933 188 188 .001 0798 .001 0944 '45 146 147 0.085 .086 .087 .088 o.ooo 4341 .000 4448 .000 4555 .000 4663 105 107 108 o.ooo 3936 .000 4027 .000 4119 .000 4212 9 1 92 93 0.145 .146 .147 .148 o.ooi 3121 .001 3311 .001 3503 .001 3696 190 192 193 o.ooi 1091 .001 1238 .001 1387 .001 1536 147 149 149 .089 .000 4773 no in .000 4306 94 95 .149 .001 3891 195 196 .001 1686 150 152 0.090 .091 o.ooo 4884 .000 4996 112 o.ooo 4401 .000 4496 95 0.150 .151 o.ooi 4087 .001 4285 198 o.ooi 1838 .001 1990 152 .092 93 .094 .000 5109 .000 5224 .000 5341 "5 117 117 .000 4593 .000 4691 .000 4790 97 98 99 IOO .152 .154 .001 4484 .001 4684 .001 4886 199 2OO 2 O2 204 .001 2143 .001 2296 .001 2451 *53 153 155 156 0.095 .096 .097 .098 .099 o.ooo 5458 .000 5577 .000 5697 .000 5819 .000 5942 119 120 122 I2 3 124 o.ooo 4890 .000 4991 .000 5092 .000 5195 .000 5299 101 101 103 104 104 0.155 .156 J 57 .158 .159 o.ooi 5090 .001 5295 .001 5502 .001 5710 .001 5920 205 207 208 2IO 211 o.ooi 2607 .001 2763 .001 2921 .001 3079 .001 3238 156 158 158 III O.IOO .101 .102 .103 .104 o.ooo 6066 .000 6192 .000 6319 .000 6448 .000 6578 126 I2 7 I2 9 130 131 o.ooo 5403 .000 5509 .000 5616 .000 5723 .000 5832 1 06 107 107 109 109 o.i 60 .161 .162 .163 .164 o.ooi 6131 .001 6344 .001 6559 .001 6775 .001 6992 213 215 216 217 219 o.ooi 3398 .001 3559 .001 3721 .001 3883 .001 4047 161 162 162 164 164 O.I05 .106 .107 ,108 o.ooo 6709 .000 6842 .000 6976 .000 7111 133 134 135 o.ooo 5941 .000 6052 .000 6163 .000 0275 III III 112 0.165 .166 .167 .168 o.ooi 7211 .001 7432 .001 7654 .001 7878 221 222 224 o.ooi 4211 .001 4377 .001 4543 .001 4710 166 ' 166 167 _ fa .109 .000 7248 137 138 .000 6389 114 .169 .001 8103 225 227 .001 4878 IOO 169 O.I 10 .III o.ooo 7386 .000 7526 140 o.ooo 6503 .000 66l8 115 0.170 .171 o.ooi 8330 .001 8558 228 o.ooi 5047 .001 5216 i 9 .IJ2 "3 .114 .000 7667 .000 7809 .000 7953 141 I 4 2 144 45 .000 6734 .000 6851 .000 6969 1 1 6 117 118 119 .172 .173 .174 .001 8788 .001 9020 .001 9253 230 232 233 234 .001 5387 .001 5558 .001 5730 ' / * 171 172 173 0.115 .116 o.ooo 8098 .000 8245 147 _ . 6 o.ooo 7088 .000 7208 120 0.175 .176 o.ooi 9487 .001 9724 237 o.ooi 5903 .001 6077 *74 .117 .118 .119 .000 8393 .000 8542 .000 8693 148 149 152 .000 7329 .000 7451 .000 7574 121 122 123 124 .177 .178 .179 .001 9961 .002 0201 .002 0442 237 240 241 343 .001 6252 .001 6428 .001 6604 176 176 178 0.120 o.ooo 8845 o.ooo 7698 o.i 80 0.002 0635 o.ooi 6782 630 TABLE XIY. For finding the Katio of the Sector to the Triangle. X S X Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. 0.180 .181 .182 .183 .184 0.002 0685 .002 0929 .002 1175 .002 I42Z .002 1671 244 246 247 249 "7 C T o.ooi 6782 .001 6960 .001 7139 .001 7319 .001 7500 178 179 180 181 181 0.240 .241 .242 *43 .244 0.003 8289 .003 8635 .003 8983 .003 9333 .003 9685 346 348 35 35* 0.002 8939 .002 9166 .002 9394 .002 9623 .002 9852 227 228 229 229 451 354 231 0.185 .186 .187 .188 .189 O.002 1922 .OO2 2174 .002 2428 .OO2 2683 .002 2941 252 254 255 2 5 8 2 5 8 o.ooi 7681 .001 7864 .001 8047 .001 8231 .001 8416 183 183 184 185 186 0.245 .246 .247 .248 .249 0.004 39 .004 0394 .004 0752 .004 ii ii .004 1472 355 358 359 361 363 0.003 83 .003 0314 .003 0545 .003 0778 .003 ion 231 231 *33 233 234 0.190 .191 .192 .193 .194 O.OO2 3199 .002 3460 .OO2 3722 .002 3985 .002 4251 26l 262 263 266 267 o.ooi 8602 .001 8789 .001 8976 .001 9165 .001 9354 187 187 189 189 190 0.250 .251 .252 *53 .254 0.004 1835 .004 2199 .004 2566 .004 2934 .004 3305 364 36? 368 371 372 0.003 i*45 .003 1480 .003 1716 .003 1952 .003 2189 235 236 236 *37 238 0.195 .196 .197 .198 .199 0.002 45l8 .002 4786 .002 5056 .002 5328 .002 5602 268 2 7 272 274 275 o.ooi 9544 .001 9735 .001 9926 .002 0119 .002 0312 191 191 '93 193 J 95 0.255 .256 .257 .258 .259 0.004 3677 .004 4051 .004 4427 .004 4804 .004 5184 374 376 377 380 382 0.003 2427 .003 2666 .003 2905 .003 3146 .003 3387 239 239 241 241 241 0.200 .201 .202 .203 .204 OOO2 5877 .002 6154 .002 6433 .002 6713 .002 6995 277 279 280 282 283 0.002 0507 .002 0702 .002 0897 .OO2 1094 .OO2 1292 '95 195 197 198 198 0.260 .261 .262 .263 .264 0.004 5566 .004 5949 .004 6334 .004 6721 .004 7111 383 385 387 390 391 0.003 3628 .003 3871 .003 4114 .003 4158 .003 4603 243 243 244 245 245 0.205 .206 .207 .208 .209 0.002 7278 .002 7564 .002 7851 .002 8139 .002 8429 286 287 288 290 293 O.O02 1490 .002 1689 .002 1889 .OO2 2090 .002 2291 199 200 201 2OI 203 0.265 .266 .267 .268 .269 0.004 7502 .004 7894 .004 8289 .004 8686 .004 9085 392 395 397 399 400 0.003 4848 .003 5094 .003 5341 .003 5589 .003 5838 246 247 248 249 249 O.2IO .211 .212 .213 .214 O.002 8722 .002 9015 .002 9311 .002 9608 .002 9907 a 93 296 297 299 300 0.002 2494 .002 2697 .OO2 2901 .002 3106 .OO2 3311 20 3 204 205 205 20 7 0.270 .271 .272 .273 .274 0.004 9485 .004 9888 .005 0292 .005 0699 .005 1107 43 404 407 408 410 0.003 6087 .003 6337 .003 6587* .003 6839 .003 7091 250 250 252 252 *53 O.2I5 .216 .217 .218 .219 0.003 0207 .003 0509 .003 0814 .003 III9 .003 1427 302 305 35 308 309 O.002 3518 .002 3725 .002 3932 .OO2 4142 .002 4352 207 207 2IO 210 210 0.275 .276 .277 .278 .279 0.005 1517 .005 1930 .005 2344 .005 2760 .005 3178 413 414 416 418 420 0.003 7344 .003 7598 .003 7852 .003 8107 .003 8363 254 254 *55 256 257 O.22O .221 .222 .223 .224 0.003 1736 .003 2047 .003 2359 .003 2674 .003 2990 3" 312 3'5 316 318 O.OO2 4562 .002 4774 .002 4986 .002 5199 .OO2 5412 212 212 213 2I 3 215 0.280 .281 .282 .283 .284 0.005 3598 .005 4020 .005 4444 .005 4870 .005 5298 422 4*4 426 428 430 0.003 8620 .003 8877 .003 9135 .003 9394 .003 9654 % III 260 O.225 .226 .227 .228 .229 0.003 3308 .003 3627 .003 3949 .003 4272 .003 4597 3*9 322 3*3 3*5 327 O.002 5627 .002 5842 .OO2 6058 .002 6275 .002 6493 "I 216 217 2Ig 218 0.285 .286 .287 .288 .289 0.005 5728 .005 6160 .005 6594 .005 7030 .005 7468 432 434 436 438 44 0.003 9914 .004 0175 .004 0437 .004 0700 .004 0963 261 262 263 263 264 0.230 .231 .232 .233 234 0.003 4924 .003 5252 .003 5582 .003 5914 .003 6248 328 330 33* 334 336 0.002 6711 .OO2 6931 .002 7151 .002 7371 .002 7593 22O 220 220 222 223 0.290 .291 .292 .293 .294 0.005 7908 .005 8350 .005 8795 .005 9241 .005 9689 44* 445 446 448 45 0.004 1227 .004 1491 .004 1757 .004 2023 .004 2290 26^ 266 266 ^ \ 267 0.235 .236 *37 .238 .239 0.003 65^4 .003 6921 .003 7260 .003 7601 .003 7944 337 339 341 343 7 AC 0.002 7816 .002 8039 .002 8263 .002 8487 .002 8713 223 224 224 226 226 0.295 .296 .297 .298 .299 0.006 0139 .006 0591 .006 1045 .006 1502 .006 1960 452 454 457 458 461 0.004 2557 .004 2826 .004 3095 .004 3364 .004 3635 269 269 269 \ 271 , 271 0.240 0.003 8289 J i J 0.002 8939 0.300 0.006 2421 0.004 3906 631 TABLE XV. For Elliptic Orbits of great eccentricity. c or 3 log .B or log .B ' Diff. log^V Diff. eorS logj&oorlog.Bo' Diff. log N Diff. o.ooo oooo o.ooo oooo 30 o.ooo 0066 o.ooo 6400 A >& 1 .000 oooo o Q .000 0007 7 2 1 31 .000 0075 9 J J .000 6836 436 2 .000 OOOO .000 0028 tft 32 .000 0086 .000 7286 450 3 .000 oooo o o .000 0064 3 b 40 33 .000 0097 I I 12 .000 7750 464 4 .000 OOOO o .000 0113 64 34 .000 0109 13 .000 8229 493 5 o.ooo oooo o o.ooo 0177 78 35 0.000 0122 15 o.ooo 8722 508 6 .000 OOOO o .000 0255 Q2 36 .000 0137 .000 9230 J 7 .000 OOOO o .000 0347 y 107 37 .000 0153 18 .000 9753 537 8 .000 oooo .000 0454 1 20 38 .000 0171 IQ .001 0290 J J / 5r 2 9 .000 OOOI o .000 0574 39 .000 0190 * 7 20 .001 0842 * 567 1 10 O.OOO OOOI o o.ooo 0709 149 40 O.OOO O2 1 22 o.oo i 1409 581 11 12 .000 OOOI .000 0002 I .000 0858 .000 I 02 I 163 178 41 42 .000 0232 .000 0255 11 .001 1990 .001 2586 596 611 13 .OOO OOO2 I .000 1199 43 .000 0281 27 .001 3197 626 14 .000 0003 I .000 1390 206 44 .000 0308 / 29 .001 3823 640 15 16 o.ooo 0004 .000 0005 I 2 o.ooo 1596 .000 1816 220 27C 45 46 o.ooo 0337 .000 0368 3 1 77 o.oo i 4463 .001 5118 655 670 17 .000 0007 2 .000 2051 JJ 248 47 .000 0401 36 .001 5788 685 18 .000 0009 2 .000 2299 Tr 263 48 .000 0437 38 .001 6473 J 700 19 .000 00 1 1 2 .000 2562 277 49 .000 0475 J 40 .001 7173 / w 715 20 21 22 23 24 o.ooo 0013 .000 0016 .000 0019 .000 0023 .000 0027 3 3 4 4 5 o.ooo 2839 .000 3131 .000 3437 .000 3757 .000 4091 292 306 320 334 349 50 51 52 53 54 o.ooo 0515 .000 0558 .000 0604 .000 0652 .000 0703 3 48 51 54 o.ooi 7888 .001 86l8 .001 9362 .002 0122 .002 0897 73 744 760 775 79 25 26 27 28 o.ooo 0032 .000 0037 .000 0043 .000 0050 1 7 7 o.ooo 4440 .000 4803 .000 5181 .000 5573 367 378 392 4 7 55 56 57 58 o.ooo 0757 .000 0815 .000 0875 .000 0939 64 68 0.002 1687 .002 2493 .002 3313 .002 4149 806 820 836 851 29 .000 0057 9 .000 5980 420 59 .000 1007 .002 5000 866 30 o.ooo 0066 o.ooo 6400 60 o.ooo 1078 0.002 5866 TABLE XVI. For Hyperbolic Orbits. m or ?! log Q or log Qf log I. Diff. log half II. Diff. m orra log Q or log Q' log I. Diff. log half II. Diff. 0.00 .01 .02 .04 0.000 OOOO 9-999 9 8 7 999 9479 .999 8828 999 79 J 7 2.41597* 2.71675* 2.89259* 3.01741* 2-1149* 2.1146* 2.1130* O.IO .11 .12 '3 .14 9.998 7021 .998 4308 .998 1342 997 8123 997 4654 3.41256* 3.45326* 3.49028* 3.52423* 3-55547 2.1046* a. 1 025* 2.1003* 2.0978* 2.0952* 0.05 .06 .07 .08 .09 9 999 6746 999 53i6 .999 3628 .999 1682 .998 9479 3.19290* 3.31687* 3-3 6 745 2.II2I* 2.1 1 IO* 2.1097* 2.I082* 2.1065* 0.15 .17 .19 9.997 0936 .996 6971 .996 2760 995 8305 995 3 6 8 3-5 8 453 3.61154* 3.67679* 3.66048* 3.68276* 2.0923* 2.0892* 2.0860* 2.0826* 2.0790* O.IO 9.998 7021 3.41256* 2.1046* O.2O 9.994 8671 3-737 8 n 2.0"C2 n 632 TABLE XVII, For special Perturbations. q, " For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. 0.0180 .0181 .0182 .0183 .0184 0.457 9499 457 8454 457 749 457 6 3 6 5 457 53* 1 1045 1045 1044 1044 0.285 6702 .285 5864 .285 5026 .285 4188 * 8 5 335 838 838 838 Q~Q 0.497 0554 .497 1684 .497 2814 497 3944 497 575 1130 1130 1130 1131 0.316 9530 .317 0431 .317 1332 .317 2234 3*7 3*35 901 901 902 901 1044 o ^o 902 0.0185 .0186 .0187 .0188 .0189 0-457 4*77 457 3*33 457 *i8g .457 1146 457 OI0 3 1044 1044 1043 1043 10 43 0.285 2512 .285 1675 .285 0838 .285 oooo .284 9163 837 837 838 837 837 0.497 6206 497 7337 .497 8 46 8 497 9600 .498 0732 1131 1131 1132 1132 "3* 0.317 4037 3*7 4939 .317 5841 .317 6744 .317 7646 902 902 93 902 903 0.0190 .0191 .0192 .0193 .0194 0.456 9060 .456 8017 .456 6975 45 6 5933 .456 4891 1043 1042 1042 1042 1042 0.284 8326 .284 7490 .284 6653 .284 5817 .284 4981 836 837 836 836 836 0.498 1864 .498 2996 .498 4129 .498 5262 .498 6395 1132 "33 "33 "33 "33 0.317 8549 3*7 945* .318 0355 .318 1259 .318 2162 93 93 904 93 904 0.0195 .0196 .0197 .0198 .0199 0.456 3849 .456 2808 .456 1767 .456 0726 455 9 68 5 1041 1041 1041 1041 1041 0.284 4'45 .284 3309 .284 2473 .284 1637 .284 0802 836 836 836 835 835 0.498 7528 .498 8662 .498 9796 499 0930 499 * 6 4 "34 "34 "34 "34 "35 0.318 3066 .318 3970 .318 4874 .318 5778 .318 5683 904 904 904 905 905 O.O2OO .0201 .O2O2 .0203 .0204 0.455 8644 .455 7604 455 6564 455 55*4 .455 4484 1040 1040 1040 1040 1040 0.283 9967 .283 9132 .283 8297 .283 7462 .283 6627 835 835 835 835 834 0.499 3199 499 4334 .499 5469 .499 6604 499 774 "35 "35 1135 1136 1136 0.318 7588 .318 8492 .318 9398 .319 0303 .319 1208 904 900 905 905 906 0.0205 .O2O6 .0207 .0208 .0209 0.455 3444 455 *4Q5 455 '366 455 3*7 .454 9288 1039 1039 1039 1039 1039 0.283 5791 .283 4958 .283 4124 .283 3290 .283 2456 834 834 834 833 0.499 8876 .500 OOI2 .500 1149 .500 2286 .500 3423 1136 "37 "37 "37 "37 0.319 2114 .319 3020 .319 3926 .319 4832 3'9 573 8 906 906 906 906 97 0.0210 .0211 .O2I2 .0213 .0214 0.454 8 *49 .454 7211 454 6173 454 5*35 454 4097 1038 1038 1038 1038 1037 0.283 1623 .283 0789 .282 9956 .282 9123 .282 8290 834 833 833 8 33 833 0.500 4560 .500 5697 .500 6835 .500 7973 .500 9111 "37 1138 1138 1138 "39 0.319 6645 -V9 755* .319 8459 .319 9366 .320 0273 907 907 907 907 908 O.02I5 .02l6 .0217 .0218 1 .0219 0.454 3060 454 **3 .454 0986 453 9949 453 8912 1037 1037 1037 1037 1036 0.282 7457 .282 6624 .282 5792 .282 4959 .282 4127 833 832 8 33 832 832 0.501 0250 .501 1389 .501 2528 .501 3667 .501 4807 "39 "39 "39 1140 1140 t>.32o 1181 .320 2088 .320 2996 .320 3904 .320 4813 907 908 908 909 908 0.0220 .0221 .0222 .0223 .O224 0.453 7876 453 68 4 453 5 8 4 453 478 453 3733 1036 1036 1036 I0 35 0.282 3295 .282 2463 .282 1631 .282 0800 .281 9968 832 832 831 832 831 0.501 5947 .501 7087 .501 8227 .501 9368 .502 0509 1140 1140 1141 1141 1141 0.320 5721 .320 6630 3* 7539 320 8448 3*o 9357 909 909 909 909 909 0.0225 .0226 .O227 .0228 .0229 0.453 * 6 9 8 453 l66 3 .453 0628 45* 9593 45* 8558 1035 1035 I0 35 1035 1034 0.281 9137 .281 8306 .281 7475 .281 6644 .281 5814 831 8 3 ! 830 831 0.502 1650 .502 2791 .502 3933 .502 5075 .502 6217 1141 1142 1142 1142 "43 0.321 0266 .321 1176 .321 2086 .321 2996 .321 3906 910 910 910 910 910 O.0230 .0231 .0232 o*33 .0234 0.452 7524 .452 6490 45* 5456 .452 4422 45* 3389 "034 I0 34 i34 1033 1033 0.281 4983 .281 4153 .281 3323 .181 2493 .281 1663 830 830 830 830 830 0.502 7360 .502 8503 .502 9640 53 0789 .503 1932 "43 "43 "43 "43 "44 0.321 4816 .321 5727 .321 6637 .321 7548 .321 8460 9" 910 9" 912 9" 0.0235 .0236 o*37 .0238 .0239 .0240 0.452 2356 45* 13*3 .452 0290 .451 9258 .451 8226 .451 7194 1033 1032 1032 1032 0.281 0833 .281 0004 .280 9174 .280 8345 .280 7516 .280 6687 829 830 829 829 829 0.503 3076 .503 4220 53 53 6 4 .503 6508 503 7653 503 8798 "44 "44 "44 "45 "45 0.321 9371 .322 0282 .322 1194 .322 2106 .322 3018 .322 3930 9" 912 912 912 912 636 TABLE XVII. For special Perturbations. q, t^. t-^ O\ O\ O\ ON ON ONONONONON oo >* ON IN O t^ d co to O * O vo covo co f> oo co IN NVn t^rt-ON O\OO ^- VOON OooOvnTf- vnMvnOco IN r^ H d O d d M d co co t^d^ OtnONdON O M vn t~x rj- ON ONOO vnQco Mt^-ddO MdcoM ddcoMM gQOOOO ON-* SOOOOO ooco TH O? TH C u-> m <$ o vo O vo <* m <*- O O t^ t-> wrowosf-s COMVOMM Os m Os ^ O so Os rj- Os m ON O ON O Osoo Os OO so O t^ Os x Osso O O ro O M to 00 O so t^ 00 Os Osso "6 t-. voso rt 00 sovo rx i-~ rt m ON *> r-~. ON O m ON M rt O O ^ O Osso f^oo oo o ONSO TJ-OO M r-x c< m r~* t-^ M m m M OsO^hrort O > l~> t~~ t^. ON vo rt C^vOO OO t^.OO SO Oj\ O O^^ ^ O^OO ot^osONON osoo'ososON o^ONO^ONO* O\O*O\ONO\ 00 O vo O t^ f^oo rt f Os Os Os rt rt Osso oo q q q q q a\ q q q q q q q q q vo o> cr\ ON q q q q q q q q o o o o q o o O O ^-^- 0000 OO OO J- VO _ OsOOst~xM t-^i-it-^ONrO MSOt^fOOs HOt^covo MONrtMON M "so rtoovoMrtmt^rortt^rtoot^ SOMOOSO covortMso rf a Os f- vo rf o oo m vooo M M oo m ^so to rt M tooo O O so t^ M O rt cosO SO CO ^ ON fxOO OO OOrtONOstOsOf^vo t^sO vo Os M \O rt rt MMCO rt rt 1-1 rt rt rt M rt to rt rt f OOOoo voOOO 1 '* >O oo r~vO O M oo tooo vo On O vo M t^ t^ O M HM mt^ H rtMrtmco MM rtrtMsooo sOOrt tooo rtONM rt sot>.M comr-t^rt ^h vort votovoco ^ *s|^I f|J5 tsjili f | 'B . - T2 . d'B S . 3 3 P c S c3 g llflH-IFlllllilllll .-3045 3 o> .J3 o t^coo vovoclooN M M HI H M M T}- rj- ^ vo 1VO VO CO ON d co M ON O vo t^OO vo c4 VO H M CO CO $ VOOO Ht^-CO COON^-T^-IO COM Tj-VO OO rj- O vo t^ t^ MfxdvOO VOCOQOOM rJ-vO'tj-voO OMt-^MOO rHrH C<1 r-l Cq rH Illil rj "HI bo t> *S rj -S bb -S "^ Q^lrJ r-s o CL.^ A 3 5 SO PH-O; C. 00 1C OS O O rH >> > > $&%&& O O rH r- 1 CO oo oo oo oo oo I l>. IN. i i>. i>i>. c> !> QO QCaDCCQOQID OO 00 00 QO Ct 640 otctot ooooo TABLE XV1IL Elements of the Orbits of Comets which have been observed. "a 1 .S .S *-'! s'l Encke. Gibers. // // Burckhardt. Wahl. // Burckhardt. Gibers. Gauss. c5 I'- ll 1|1 Bessel. Triesnecker. Argelander. Nicolai. Encke. "H 1 fa* i Hill Motioo. I 1- r 'c\r -vO M M VO VO "4- O w oo O t> ON M ONVO M ON O* ON ON O* vo ^ i ON O ON O vooo r-^ r> CO M T^- COOO Tj-00 M VO M H ON H OO ON VO *H t^x VO OO ON O* ON ON ON t~s CO ^- M OO cooo O vo vo ** r}-oo O oo ^}- vo r-x ON ON H ON M co H ON f^ rj- O O ON ON ON O o" 00 O 00 VO Tj- ON vo O vo M CO ON H r-- TJ- o o CO VO CO >H ON vo a\ o oo vo ON ON O* ON ON oo ON vo O vo t^ vo ON t^ vo M co ON OO CO M C ^- CO ON VO ON O OO oo M ON ON t~~ ON O M OO ON ON o' O' ON ON t ON l-^ M O ON O vo c M vo VO vo ONOO t~- t r}- ^ co 10 vo ^ M o ^* r^ to t- l^rh co^-c* * * 3- O <* cooo f* ON vo M M to CO C< VOVO Tt" s. "*" M H CO Tj- VO d VO t^ W M VO Tj- C< tO t> rj- O 'J-VO rt vo O O oo co-1o"2? OO M cJ t^ C^ MM M VO co rt ON vo H 3~ M VO t^^ VO co -O VO M M COVO rj- tovo r-to^ c* oo ^ ** tJ ^ vo covo c^ t^ CO ^- rt CO ^ M vo rj- O 00 ON^-Ot, M M * ON O d VO t~x M O H w co rt co cooo O ONOO M CO VO ONVO M VO iT rt ^ Jo ** 4- vo T}-VO O f M CO VO CS VO VO H VO VO VO H t t^ O o vo vo t-^vo to O cooo c< CO ONOO O M Cl 1-1 ^J- t~ ON rt ON CO M c< C^ ^" to co M tJ ONVO Tj- O vo ON r> 'i- M f- CO CO M Tj- M f* VO t< co vo r* vo r) to 6) co rJ co ^h O O to to H HH rj- ON vo CO M C4 O c* co co O VO TJ-00 H vo It ft ON ts. Tj- vo 1-. VO Tj- CO TOO ON Tj- rj- t ON ci oo w o ON rt vo M t^ cl vo O to C< CO CO co O ?< f* rj- h t* vo Tt- vo COVO VOOO 11 rt" ^ nT c* "^ 8 cooo t^ l-^ M VO tl VO H ON VO rj" vo ON T|- O to VO vo O -^- w 00 rt ON ON r- CO rl rj- T}- to O co ^ oo ONOO co rj- M M CO M t-^00 co rt- t* <*- H 10 M vo ON t-* O ON o O ON t^vo VO T- CO rt 4- oo ^ O i~^oo CO ^" H ^ vo vo r-s r}- ON 00 00 VO CO O VO CO vo to M to t^ vo ^J- co ON t-~ ON f- vo vo ON Tj-vo ON *> ONVO ON ON ON CO C CO S 00 vo VOOO VO rj- vo ON O 1 VO M H vo o r^* rt vo n H O vo co vo t~ w OO f^ vo vo O vo vo CO rj- VO O to co covo O O H ON rj- 10 M VO ON M ON co r* VO CO CO M CO O to O M O 1-- d O O oo M oo oo vo < ON t~ f>.OO f^ vo t l^ O w> O ON t M co VOOO CO M TJ- H, f* f t- t> 5j- rt VO co t^ t O coao O ^ f^\ & l~5 ^ CO 00 00 OS OS CO rH CO CO CO CO CO CO OS OS OS OS CO CO io o t^ co co OS OS OS OS OS iMslM OS OS O O O t~- t^ CO CO OO jllft CO OO CO 00 OO S co SSi5 00 CO CO OO OO lllll CO CO lO CO CO oo oo co no co OOOO'H i-l d W *# i 222i SUSSw gJSSSw 1-1 SI 95 rfl CO CO CO CO CO 09 WCO CO-* iHlHrHlHiH 41 641 TABLE XVIIL Elements of the Orbits of Comets which have been observed. " ver. geland mbart. Heilig 1= rs^ -g 5 s^ o-3O > owo^w R ONOO vo f-oo O co co cooo to vo ON to O "<$ ooootodoo OddONt-~ r~ d d cooo tovo O co d O ON to tooo ON ON t-~- to ON O" ON ON ON ON ONOO' ON ON O*N tovo l~- ONVO *3-oo O d M to d OdQvOi-i d^i-oooooo t^ d M co r~-oo Tj-vo ON co w o to d tooo OO O t^ d ON M vO t^ Tj" vo T$- OO tr OQtOMMOO VO t-^ CO to M ^- O O ^- co ON to O ON to O O\ co O' ON ON O' ON ON ON O>' ON O* ON ONOO* ON ON t-.00 d d to t- rj- M rj- to co *J- d O O f--vO O ON ONOOOvDOQONVOdro covo $ CO to rj- ONOO 1-1 M OO OO O t^ ONOO ** ON to co ^vo M d **. ^ ^t" ^ ^ M . *"? ^ . *? . ^^ D . ^ *** ONO>ONONON o'oNoVoNO* to M M co oo ro M 'i- O o rj- d 00 ON rj- VO O to M t^ T*- co OO to VO TJ- VO rl- to OO TJ- ON OOooOt-^voOOooO ONOO oo oo M ON co rj- T^- oo d O to t-~- d M oo rj-vo ON d vo ^ TJ- tovo oo ON r*- to TJ- ON rj- O ON ^ rj- OOOONt--O OOOOO ONOO O O 00 MMOMO OMMOU OMMMM MOOOM MMMMM O O M O MQMMM CO W u-i o O t-~ ON ONVO t~^ ^* M to to ^J- d ON M T$-VO M MM ** t^ o c CO d co Ovotod ooMoovod d M T^" d M M M tO Tj-vO t^-VO rj- ON co d O ON ONOO TJ- rj- to o d ^*- co coO co O to to M rj-coo\ tovo NO T^- M M Tt~ d d tO ^" M M tO to ^J-OO ON rJ-VO f^ cooo M rj- co O co rl M co M to d co MdCOl-IT^- MtOCOw tOt^-ONtOO OOONt-^MCO O Tj-VO M O Th TJ-VO OO M d covo r~*J- t^ONdt^ON MTj-o^hco O r^oo ON t^ to t~> to co t-x co to M co ON co d Ot^-dOt-^ f^ONTj-vovo totoMMto dOdt~^O d HI d co M d M d d d Soooodvo cod M co to T*- oo v d co M co r^-vo vo O co eo s cTi-rcrcr CO"O > O N OO* V ^" (N i I (M CO i-lrHr-(r--O?OCD rl r-l ^ Jli ^H ^-t CM CNJ (M CN (M CN ^ CM C^ (N CN CM CM CM cooooocooo oooooocooo ooooooooco c-oooooooco ^ ^^1 ^_ ^ >* -*4 > " >!H ^ ? s^ o CO CO CO CO CO CO CO COOOOOOOOO OOOOOOOOOO OOOOOOCO'CO 642 TABLE XVIII, Elements of the Orbits of Comets which have been observed. Encke. Westphalen. Encke. Peters and O. S Plantarnour. g.S Dir ii n Re Di Direct. Retrograde. // Direct. ONTj-ioj^coooOHivotovodooTj-vo * O ooo OO ONOO n vo -^h t---vo Qvo M co O *j- d vococoQOO O HI o O to ON O co\O r-~ covo covo HI vo co d O vo M covo cooo O vooo voco to^-dnON r-. tooo M T^- t^oo vo M o rf-Ot---dcoooooTt-dO voooroONO covo co ONOO l~x r^ co O 3- O d l-- co o vo ON O d t-~ tor~.toi>.q oo M to t^. t--. d d O ON * O^ O vo to M o\ 6\ o\ o\ d o\ o* o\ ON t>. o o" o o\ o\ o\ o o\ o\ o' H ON co ON d M d ON toodnto OOOOOOON oo vo O vo o t^ ON ON d d oo c~^ co r*** o vo d ddnoorj- wOOONiOt^. MOO co d cooo o M d d -* r^oo OjvOO oo HI HI oo ONVO co d M vO ON ON ON O O* ON O\oo' O O O ON ON o"s ON t^ vo r^ ^- r^ w o rt vo ON to O vo vo T*- vo O l-^ to co d O CO ON r^ O OO to ON O co M d oo oo t^ to r^. to O r^ ON T}- TJ-VO Tj- O ON vo ^J- oo ONOO O ON O ONOO O O* O O HI* o* M O* O* HI" t^ oo t--. ON co O vo ON ON cooo HI r^ ON to O ON HI oo vo vo ON O vo t-^ ON ON O to HI ON ON O vovo ON co d d vo O O w rj-vo d vo oo ON t>- co o O oo oo oo >* T!- vo co o ON t-^ d vo co d OO Tj- ON vo ONVO O O ONOO ON t~- t^ ON O 00 to ON t^ ON X3 oo d d CO CO M HI co ON co ON HI ON co ON d OO ON ON ON ON t^ t^ ON ON 0-. O ON ONO> vo rj- O H COVO CO -J- U1 H, O HiOOOni nHiQOO OOO>iO OOOHiQ OOMHiO vo vooo d O d covo t^ ON vo co O M t^ ONVO ON HI Hivod voddcoco rj-covo T}- cocovo ^ M to M IO CO Hlt^. S c^g- $ d MTj-tocod wvocodto n 10 to to a * vo Tj- ONVO VO 00 O COlOCOHi CO OO^t- co co M d HI d VOQVO COM CO^-COCOM Hid HI cocococOHt 4- 10 U-, w CO ^ M dcodHiM Hicod vo o vo ONOO d d to d t^ HICOMM CO MCOd M ON d ONVO HI d d 00 * ^- t-^ ON ON ONVO dMoodd t^vooNONON covo vo to d ON r>- CO COrJ-COdVO CO CO^-COHITJ-HICO ONMVO OOOVO^-VO OONO>MO co o vo to cf ^*i _OO OOC^C^CO COCO^Tflrtl lOiO>OvOO COCOOCO?O COCOr^t~-t-> h'.t^.t^-OOQO OO OO CO OO CO CO CO GO 0000000000 GO 00 00 OO CO CO 00 00 00 00 CO CO CO 00 QO 00 00 CO 00 00 v*G*C6'*#*6 S3t- GCQCaDODOO QCQCQCQOO ?Df>.Gt CtO QCQCQCQCiO C&CtCtC^Ci C6 d Gi Ct O -,- , ^_ __ fHfHrHTHOl ooooo TABLE XYIII. Elements of the Orbits of Comets which have been observed. * s S | iiia-s *:s 3 w2w 53 00 . g J O> 03 ( i C JH r* . ^ Q} t ^C2 -S '3^^ r =^c; CTli^ g 1^3^ ^^,5^^ | 1=J= b^^rCO L-3 ^4 2 X K*i h"! 7-5^ O"! h> ^ ^^i^T-S^r^M -^ H isfcstst ftfeftfci +> Re Di / Re H vovo ON w o Th oo f-^OO vo O <* to 6 c5 T*- c* vo c i- O HI ONOO HI cocOt>vot> OO vo rj- t>-v rf-OO OO HI Tfr- IH rt OO vo HI to t-~ HI ON ON O 6. .. .. rt t>. O H * <* HI M H CJ ^" HI rj- rt O ON CO cJ OO O "^ VO OO OO r^ t^OclHl C)C)HIHH t^^ O r^ t^ vo CO OO VO vo CO vo tOVO ONVOt* VOtOONtOvooOCOi-iTj-i-i OcOr}-votJ ONOO ONVO t~^ ONOONOt^- rJOONHivo O\O\OOON ^-t)to rj-oo ONi- 00 Tj- ONVO _._._. VOVOON , O 0\ ON ON O vovo ON O OO O t>- ON ON ON HI* O* O" O* HI' o o' o' w o* H* o' O' O' O* vo HI O ON ^t- vo ON H ^i" vovo cJ ON rj- oo r^ O vo HI M w ti cl vo vooo ON TJ- to TJ- H r-- H VOOO O ON ON HI vo vo w ONOO H O VO M O O ON T-VO i*- -4- vooo <$- vo Tj- T^- vo r^-vo vo T}- rt-oo HI to ON co co M O OO ONOO vowONONr- HiONOOO cOTt-wONt^ HHHONCO ON ONOO ON >-i CO>votOC^ t^ONrtOOO VDVOONO'i- VO O * ON ON O ONOO ON ON vo M vo O O O O ON 0\ Ov ONOO ON OsOO ON ON ON C^vO OO M M w* O* O* O* O' O* O* O* O* 0* O* O* O* O* O* tO O ON rt- CO 00 00 00 ON O O vo HI HI coox) coco ONHOOt^ HiOvOc^Hi O^-HICOCO Ol~^ ONOO t vo t-~- co rj- o M M tO f^ HI ^* M T^ H VO VO VO VO OO t^ ^ M VO 61 HI hH OO tJ vo CO VO CO -( vo IM HI OOOOVOf-. t COC4 COCO Clvo^cJ^J- COVOCO^J-H COCOwHHI TJ-Tl- vo t>^ co O ON t~ ON to co in O vo covo r^oo c* co ON ON rt O 00 . ^f^HICOVO t^O't'VOO HI COCOCOM C M gi>H fe Ret Direct. Retrog O vo Tj- <* ON VO d O co co TJ- vo ONVO O r^OooOd d vo M o t-^ vo oo d rj- M ^t-ddt^vo d vo d d ON TJ- cooo vo vo oo covo co d vo O f-- w vo O t^- t*^ vo d M co o w ^J* o" c?\ o\ o\ o" o oV o" o* o\ i O VO oo ON r-N oo ^~ cooo oo T^ vo ^ vo M t^% M ^- M vo t-^ co O vo M Ooo co oo ^-<4-dvo ONTJ-VOCOM M COTJ-OOO t^vo i-i d co ONOO O O d OvONONOjNVO ONOJNO\O\O ON ON ON ON ON ON ON ON ON O vo M T- - TJ- f^ ON OOvOMwd VO ONVO OO OO OO VO M ONOO TJ-rj-Tj-coTi- cp co cp cp T}- r p rr| '*"'^"' : J" rj- co cp cp co c P T ^" T }" T i" r P 66666 66666 66666 66666 66666 ON f^ r^- co ON CO vo -rt- O OOOMVOOO covo OMd oo *!- d vo ON d rj-oo d ON vo oo M >H ON rJ-Ocor^co co d r~- ON r> d t^ *}- O t- ON M vo d t-~ ^J" cp TJ- TJ- cp r J" T *" T t" r t" T l" 66666 66666 VO oo vo vo O M VD OO OO d T}- ONVO CO vo d O O coo q oo q vo vo M ONTJ- " vooooooo vodMTl- d vo ON ONOO vo t^-vo d O d O O oo M co T$- ONOO vo CO OO CO CO II VO CO r*.* O VO Ovococoro M oo oo d ^h co oo oot~ vor~>oooooo rj-oo t^ covo oo t^. r^ cooo r>--i-d vorj-d^j-co OOONONCOM O VO ONOO OO rl- COOO COCO ^-OdTj-Tj- OVOVOVOTJ- vo vo ** t^ vo vo M o vo d oo ON d co t*^ vo t^*co vooo r^* OMd vo vo vo M f^oo vo ON M co d oo vovo oo t^ M l-^. vo ON N vo d -4- COVO OO VO CO OsONO ONVO T$- T}- c> ON CO VO vo vo d ON ONOO OO OO ONd O O oo O w d COrJ- t^ ONO ON ON CO rj- COVO CO CO 11 CO CO VO ON f~-oo vo ON ONVO VOONVO codMVoO MCOVOVOt^ COVOCOQOO OO ON t^OO O> VO OO t^-OO VO OO VOOO COM OOCNjCOrl-Tl- MONM d t^ vovo 6 4- O vo d vo M d CO CO CO IO VO OO O d d d co 4-oo oo d 06 M d COM ^4- M VOVO VO CO CO CO I COOO Tj- M Q co rj- vo O M co ON t-^ vo o d Tj- ON O i O O ON O O co d d d ONOO vo vo T}-VO M cp ON l^-vo d vo t--OO CO CO O\ ^cT^d 5^" q q > vo TJ- vo vo vo co t^ co O I s * "*J" co d co ^ <1-00 vo ONM CO vo O M O VOVOM d vo oo M co cooo vo d oo t^ vovo d OO t^ vooo vo w vocodM Mvorhco I -4- t~~ <* ON ONVO t~- ON t^ d cp vo Cfr d d ON M ON O ON ON vo ON vooo . vo O d vo r^oo M O ON M oo vooo M O vo O w vo vo d vo M O O vo r^vo <* 00 vo VO CO w d VO CO rj-VO OO rl- ON rt- vo O M CO CO M O ONTj-VO co d oo vo d M co tJ-OO ONVO w 00 VO M 00 00 M cod -* i- q q oo vo M VO VOVO 6 co d M vocoti-coO T}-ONVOI d w co vo co vo rf- vo t~~- ON rj- M VO VO Tj- 00 vo M vd CO OVOVOOOON vOMt^dO to vo ONOO ON OO d rj- ON vo VO M N M VO W TJ- CO t^ CO H vo O vo O oo M VO n CO ON d t^OO coO d d vod rj-0 tO M IH CO vo t^ d vo M coco d vo co d M CO r}- d O M ON ON T*- voO M CO M d t^ O ONOO vo VOO*NOO vod co TJ- d co _ t^ M ONVO Tj- O m M d M co vo q vo t^ q 6 6* co 4- d H VO 11 MONCpclvo vOMQcpd MO_NCO t^oo vo cooo cp O O O vo ON M ON M vo 4" l"^ co 4" vovo vo O M vo M r^ d vo d 4" llCOrl- CO T}-dCO rj- rh d T}- vo vo O t r}- vo co = 1 e-* il -sf.pl HI'L 1 3 m'j 0^ "S'djajJ ifq PL|HS^S H^OHHPn lMl.2 Ii II Is 3 S* tn P- S ^> 3 3 c o .- Ol^0004O r* 646 J. xi JJ J_LU WiYV i Elements of the Orbits of the Minor Planets. i 1: 1 1 I aa a a d . a a a I *i . d a -g^SS-g -goc-g-g a ri S $ i! j 4 I IB i 1 Ss" c -3 "3 g.'o TO O C 5 OT O O X W O M Jg TO 0> O 3 TO . * 0. O ( - g 2-52 2 Softs 2 So:222? 25213 2:2:2 S ** Seg-gg SS)5|2 00000 0000 nJOi-^Oi CO T-l ,d t- CO iO OS OS T* iii-31 SSJrS* 9fttt O O i-s fM S S ^ <; S >-s - d vo f^oo t^ O OQ M t-^ N U~, O VO CO T^- vo O oo r- d <* o vo vo vo COTJ-O M t-. d r-- co rj- T|- rt-00 rt-00 CO TJ- CO CO CO 5J- 66666 vo vovO ON O SON N OO CO vo ONOO d r}-vo d ONOO d M vo d ON M M vo vo M ON ONOO ON dd^J--^-ON ONTj-ddO <4- ON M co t~- Tt- vo ONOO Tj- M ON co cooo t~- ON t^ d CO ; . , , T *" T *"i" r *' t r T *"i" r r' t *" i o ddddd ddddd ddddd o o M t^OO ^ ^J- vo d O O oo T}" M M O M co rj- rj- co r> r4 oo T*- t M <* "* d c d d d VOMCOVOOOVO ^ONON vo oo O oo d oo f-- vovo d oo t^ vooo co CO <$- O vo CO vo vovo ON d COOO ON O ** ft ON 4 vo vo vo ONO co vo co vo ON d d ** VOVOdOO-^- ONONONCOTj- VOTJ-MVOON ONVO dOON ONO4-MO covot^-voci t^ d oo t^- co vo cooo xj- ON oo d TJ- vo d r-^oo t^-r^O t^ONOONt^. oo t-~vo vo oo TJ- t^ M VO VO Tj-VO CO O M cooo O d M cooo co f>- t^ M o oo co d covo vo ON T!- O O d O O VOMTj-QVO MON OO MONQVOt^ MOCOQVO oovodOr^ dONdcoo O 00 co cooo ON Tj- covo ON i d ON .VOCOONS. TJ-COONONVO oo^J-voOvb d -"t-vo ON co ONVO OOt>t^ OOVOt^t^ON vovo ONOO vo OO ON f-vo OO ON cooo ON ON vo O r^oo O MM -)vo d rj-' M r-~ d vo r--vo ooONd t-^t^-ONcovo VO M c< COOO . O oo vo d vo co ONOO tj- ON f^ ^- covo ONVO OO co vo to co cp ON rt- ON 4 VOOO VO VO 00 vo 00 I VO Tj- VO CO Tj- VO <$- CO M Tj-M d VO ONO OOO 00 CO CO M 00 MO vpvo O 4-dvovo4 - dcodt~-vo MMd VO MCO^t-VOd M vo d 'T rj- co TJ- TJ- ONVO dVOVO VOVpCpOOO MONONdd d oo co cooo vo r-'-vo d ON vo vo 4 ON 6 rf- VO ^4 M d M M VO M CO CO M CO ^ d Tj- 00 COVO O VOOO CO COVO d VOVO COd ONl*VOMt^ OOVO VOOO CO OOdvOMCO COVO t^OO M 847 1 jj -^ ** a *> 9 IUJI O c- 3 o a s i S is 9 I/ate of )iscovery. CO OS i OS C0 01 ec 0 OrH <* r- r-t O CO CO CO CO CO CO CO OO CO CO CO CO CO CO co co oo co co CO CO CO CO CO oo co oo co co CO CO CO CO CO CO co oo co oo co co 8 S VD rh vovo OO ON r~- vooo N M N ONVO vo O vo vr> covo Tfr-voN rj- N O O O O O* 00 M fx ONTj- CO TJ- t~^ COOO O N vo vo N ONOO t-^ ONO ONVO OO t^. M N N M OO vo VO Tt- Tj- CO CO 0* co N co t^. t^. O vooo TJ- t^ oo ON co N oo vo O vo co co vo r)-OO t--. N T|- -^- co co TJ- 66666 ONN ON ON M N vo vo O O vo vo O ** vo M co ON O co N vo ON r ON T$- T^ O ON O 6 o o o o o 1 a. O O 00 ONOO CO O fx N VO CO ON M corl- -4- co vo 5 t-. M OO -^ ON vo vo vo ON M vo -4- O O cooo O vo O vo N oo O rt- COM vo -4- c? r}-Hi VON d O t-- TJ- VO t> CO 00 O M vo ^- O >H VD ON t^ oo oo vo vo vooo ON vo vovo co O 1 vo O -i-vo d t^ ^J" M VO M I--* O oo t-,oo ON N vo ON ON vo M CO N HI vooo OO ON O vo co t~- t^ O cot^ co f^ N t>- t^ONONOO N co ON N d t^ vo Tj-vo r^ covo VO vo t^OO VO 00 s o M ON VO M M O ^h vo M vo VOM 00 00 M vovo f-x N t^ N o 1 U-i e. 5. vo vo O ON N vo N r)- vo - Ovo d co t^ O M MM N O ^ co vo i^ N co vo ONOO M rj- CO rt- ^- vo M co t 5 8 . "* vo vo co vo v 00 CO Tj-00 O M N Cj VO O co vo N co vo O vo ONVO M vo vo co TJ- vo M t^oo vo vo N CO CO CO O VOM vo N VO M N O CO vo vo N vo t^. C voONM rj- N OO OO vo vo rj- N vo CO N N r- HI ^- N t^. ON TJ- VO M CO M rj- O vovo N H 9 *s 03 O vo M co^f Tj-VD rt-vO O vo vo M vo co vo O oo voO * , Element CS % f ON ON ON CO vo M vo rj- ^ ON T}- T}-OO VO M ^" CO VO VO _ VO t-^ t^ t-x ON M O ONVO co N M co M t^oo N vo 00 ON VON HI VO vo ^- CO N N covo oo M CO O M cJ co N N M M ON M CO vo co N vo r}- N H CO VO CO N VO dvo t^ t-x co O c d O coN ON ON fx. t^ Tj- O Tj- VO VO M vo co T}- M o ON \fl N *^- co M ^xVO f>. M M M N co Element? M M W t-q pq h 5. vo ON M O vo v "4- ^ ON N fx vo vo N d N O vo co N co co 00 N t~-00 N vo CO rj- vo M CO M N t^* VO COOO M T$- vo ON vo N r}- vo M CO t^ ON O O 00 cooo OO N N co M N M co COM 00 ONN TJ- COOO CO N M M CO CO cooo O ON f^ co TJ- coN N ON M vo O covo co N vo co M 00 f-00 ON T- vo C* CO O r}- ON t^ co co CON M M H i--l pq 00 CO 00 CO 00 CO | . o I III II i .2 a .% vVQoT'H <3 j i* s* n ** *n OJf-QOOO sssss Cf) 00 00 3D Q 05 e Tf-vo O co o M o O OOMONvovDOO ON co O ON C-^oo ON vo O coOvD t~~t^covo t^ co O co N oo N O OO N O N O cooo t^ cot^O voN VOM O OOMHIVOONONO M CO < t* vo ON ONVO O HI OO r^- vo M ro rt- N co N co O O N vo O VO rf- ONVO M N O O O O HI co O 6666666 + 1 1 ++ 1 1 m CO f-AO O M VO TJ- N 8O co t^~ N O CNVO vo OO O vo vo t-^ ON vooo r-.coM MVO ^i- vovO vo COOO VO vo 00 O O M ONrJ-VOrJ-O NOOOOOOO OOOOOOOO M vo covo vo M < % oo rj- O d vo co M N M ++ 111+ vo vo N co r^ o t- % r}-oo vo vooo M N vo CO N ^ O co M oo ONVO f^ N VOM N 5*-Tj- Ot^CO M M N O M a < N HI ON M N ON 5 M tl- N d vo COM 00 vo H ON M C" CO N CO VO II 1 1 1 1 1 cl ON vo ON !- cooo a 4 O M cooo I--VOM rl M M co co co ^ooTh OVOVOONI>. vo vo N vo vo ^VOTj" OOOOMNO ^'t^ Tj-ONMt^CO N < ^ -<*-00 M N vo t^ O * 't-N T-N H, * O^ ^ ONVO M N rt- M N M CO ts NniOvovoooMco * t^ co vovo 4- oXvo d covo vo co N M co M COO COOO ONM t^ c <* CON <*">HI _ Tj-oo ON N HI O 1 * t^ co t-NONcoMOOVO'<4- M CO M NONvoOvoOON ^ < CO CO O VO COVO COOO T$- M VON c ro O COONN VOOOO VO co co N M d ^ vOMOrJ-Nvor^vo o vo ** O vo ** co t 1 ^ co M M M HI M CO Epoch and Mean Equinox. Greenwich Mean Time. 558 8S5 | i I 6 to l^4iil 6 Si Si 1-1 Z 8- S>HS^PJ25 1 ^^^^^^^^^^^M 648 TABLE XXI, Constants, &c, log of Naperian logarithms e 2.71828183 0.43429448 Modulus of the common logarithms . . A = 0.43429448 9.63778431 10 Radius of a Circle in seconds r = 206264.806 5.31442513 '/ '/ a // minutes r = 3437.7468 3.53627388 " // // degrees r = 57.29578 1.75812263 Circumference of a Circle in seconds .... 1296000 6.11260500 " // // whenr=l. . . . TT = 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 // 2.56258095 This value of the length of the tropical year is for 1850.0. The annual variation is 0/0000000624. 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 // // // // // // // in se- conds of arc 3548.18761 3.55000657 Constant of Aberration, according to Struve 20".4451 // a Nutation, // // Peters 9".2231 Mean Obliquity of the ecliptic for 1750 -f t, according to Bessel .... 23 28' 18".00 0".48368* 0".00000272295< Mean Obliquity of the ecliptic for 1800 + t, according to Struve and Peters . . 23 27' 54".22 0".4738< 0".0000014* 2 General Precession for the year 1750 + t, according to Bessel 50".21129 + 0".00024429f>6< /, // u it 'i ii Struve 50".22980 + 0".000226* MASSES OF THE PLANETS, THE MASS OP THE SUN BEING THE UNIT. TO , Jupiter _ 1 4865751 ~ 1047.879' 1 1 VWlllH , Saturn 390000 * * " ' 3501.6 1 1 TT.artTi 354936 * * * ' ' 24905 me 1 1 Mars . . . 3107713 * - 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. De- noting by p the radius of the earth, by

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 formula 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 t)f 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 I3h 19m 43.5s into the decimal of a day, we find from the Table 13A =0.5416667 19m =0.0131944 43s = 0.0004977 0.5s = 0.0000058 Therefore ISA 19m 43.5s = 0.5553646 861 652 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. Ud 15/i 53m 37.2s = 14.66224 Required number of days = 257.66224 TABLE VI. contains the values of M = 75 tan %v -f 25 tan 3 \v for values of v at intervals of one minute from to 180. For an ex- planation of its construction and use, see Articles 22, 27, 29, 41, and 72. In the case of parabolic motion the formulae are m== f M mO 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 = !SO the Table contains the values of log M. 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 formulae are 8 /200 =Y- w being taken in the second quadrant. The Table gives the values of A O with w 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 g = 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, i> = 16859'32".49; but, using Table VII., we have w = 168 59' 29".ll, A O = 3".37, EXPLANATION OF THE TABLES. 653 and hence v = w + A O = 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 01 the time from the perihelion in the case of orbits of great eccen- tricity. 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 ex- plained in Art. 29. The first part gives the values of log B and log C 9 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. Diif., 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 &A 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 0= log (0) + Aj. X I. Diff. + AJ. 2 X half II. Diff. For example, let it be required to find the value of log C correspond- ing to A = 0.02497 944, and the process will be: (1) (2) Arg. 0.02, log (C) = 0.0034986 log I. Diff. =4.24585 log half II.Diff. = 1.778 ^ _ 8770.6 log* A =9.69718 21ogA4 9.394 A4 = 0.497944, (2)= 14.8 3.94303 1.172 log = 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 (/ C) = m sin* /. 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 in the determination 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 construc- tion 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 construc- tion 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 peri- helion passage (mean time at Greenwich) ; TT, the longitude of the perihelion; &, the longitude of the ascending node; i, the inclina- tion 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 pleasured from the mean equinox of the beginning of the next year. The longitudes for Nos. 19 and 27 are measured from the me*\n 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 equi- nox of Feb. 20 ; for No. 229, from the mean equinox of April 1 ; foi 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 ap- pearances 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 ele- ments, its periodic character being otherwise unknown or at least un- certain. The elements given are those which appear to represent the observations most completely. For a collection of elements by vari- ous 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-Verzeichniss appended to the latest edition of Olbers's Meihode die Bahn eines Cometen zu berechnen. TABLE XIX. contains the elements of the orbits of the minor planets, derived chiefly from the Berliner Aslronomisches Jahrbiivh fur 1868. The epoch is given in Berlin mean time ; M denotes the mean anomaly,

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