THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA IRVINE GIFT OF Mrs. John Williams THEORETICAL ASTRONOMY RELATING TO THE MOTIONS OF THE HEAVENLY BODIES REVOLVING AROUND THE SUN IN ACCORDANCE WITH THE LAW OF UNIVERSAL GRAVITATION EMBRACING SYSTEMATIC DERIVATION OP THE FORMULS FOR THE CALCULATION OF THE GEOCENTRIC AXD HEU CENTRIC PLA7E3, FOR THE DETERMINATION' OF THE ORBITS OF PLANETS AND COMETS, FOE THE CORRECTION OF APPROXIMATE ELEMENTS, ANT FOR THE COMPUTATION- OF NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. iili numerical (Examples and ^uxili BY JAMES C. WATS OX PHILADELPHIA: J. B. LIPPINCOTT COMPANY. LONDON : 10 HENRIETTA ST., COVENT GARDEN. 1892. Entered, according to Act of Congress, in the year 1868, by J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Bustern District of Pennsylvania. PREFACE THE discovery of the great law of nature, the law of gravitation, by NEWTOX, 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 BERXOUILLI, CLAIRAUT, 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 LAGK^XGE 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 4 PREFACE. but litlle 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 LAGKANGE 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 to 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 Ccelestium, a most able work, in which he gave to the world, in a finished form, the results of many years of attention PREFACE. 5 to the subject of which it treats. Ills method for determining all the elements directly from given observed places, as given in the Theorla Mohis, and as subsequently given in a revised form by ENCKE, 1< a yes 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 formulae which are required for the cases actually occurring. The refined and difficult analysis and the laborious calculations involved were such that, even after HANSEN'S methods were made known, astronomers still adhered to the method of special perturbations by the variation of constants as developed by LAGRANGE. The discovery of Astrcea by HENCKE was speedily followed by the discovery of other planets, and fortunately indeed it so happened that the subject of special perturbations was to receive a new improvement. The discovery by BOND and ENCKE of a method by which we determine at once the variations of the rectangular co-ordinates of the disturbed body by integrating the fundamental equations of motion by means of mechanical quadrature, directed the attention of HANSEN to this phase of the problem, and soon after he gave formulae 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 formulae for the perturbations of the latitude, 6 PREFACE. true longitude, and radius-vector, to be integrated in the same manner, were afterwards given by BRIJNNOW. 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, togethei 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 who 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. LIPPINCOTT & Co., for the generous interest which they have manifested in the publication of the work, and also to Dr. B. A. GOULD, of Cambridge, Mass., and to Dr. OPPOLZER, 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 Correspondenz, 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 formulae 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 interrup- 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 whicli 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 ! OBSERVATORY, ANN ARBOR, June, 1867. CONTENTS. 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 HEAVENLT BODY, ADAPTED TO NUMERICAL COM- PUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. PAGE Fundamental Principles 15 Attraction of Spheres 19 Motions of a System of Bodies 23 Invariable Plane of the System 29 Motion of a Solid Body 31 The Units of Space, Time, and Mass 36 Motion of a Body relative to the Sun 38 Equations for Undisturbed Motion 42 Determination of the Attractive Force of the Sun 49 Determination of the Place in an Elliptic Orbit 53 Determination of the Place in a Parabolic Orbit 59 Determination of the Place in a Hyperbolic Orbit 65 Methods for finding the True Anomaly and the Time from the Perihelion in the case of Orbits of Great Eccentricity- 70 Determination of the Position in Space 81 Heliocentric Longitude and Latitude 83 Reduction to the Ecliptic 85 Geocentric Longitude and Latitude 86 Transformation of Spherical Co-ordinates 87 Direct Determination of the Geocentric Right Ascension and Declination 90 .Reduction of the Elements from one Epoch to another 99 Numerical Examples 103 Interpolation 112 Time of Opposition 114 9 10 CONTENTS. CHAPTER II. INVESTIGATION OP THE DIFFERENTIAL FORMULA WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATIONS OF THE ELEMENTS OF ITS ORBIT. PAOB 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 MOVTNO 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 Katio of Two Curtate Distances 178 Determination of the Curtate Distances 181 Kelation 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. 11 PAOt 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 Position indicated by the Curvature of the Observed Path of the Body 242 Formulae for a Second Approximation 243 Formulae for finding the Ratio of the Sector to the Triangle ... 247 Final Correction for Aberration 257 Determination of the Elements of the Orbit 259 Numerical Example 264 Correction of the First Hypothesis 278 Approximate Method of finding the Ratio of the Sector to the Triangle ~ 279 CHAPTER V. DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR 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 FORMULA 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 Relation between Two Places in the Orbit 339 Modification when the Semi-Transverse Axis is very large 341 Modification for Hyperbolic Motion 346 Variation of the Semi-Transverse Axis and Ratio of Two Curtate Distances... ~ 349 12 CONTENTS. PAGJI Variation of the Geocentric Distance and of the Eeciprocal of the Semi-Trans- verse Axis 352 Equations of Condition 353 Orbit of a Comet 355 Variation of Two Kadii-Vectores 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OP THE COMBINATION OP OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OP 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 x . 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 Kesulting Values of the Unknown Quanti- ties 386 Separate Determination of the Unknown Quantities and of their Weights 392 Relation between the Weights and the Determinants 396 Case in which the Problem is nearly Indeterminate 398 Mean and Probable Errors of the Results 399 Combination of Observations 401 Errors peculiar to certain Observations 408 Rejection of Doubtful Observations 410 Correction of the Elements 412 Arrangement of the Numerical Operations 415 Numerical Example 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 42? CONTENTS. 13 PAOK 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 i74 Change of the Osculating Elements 477 Variation of the Mean Anomaly, the Eadius- Vector, and the Co-ordinate z 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 Reduction of the Elements to the Common Centre of Gravity of the Sun and Planet 538 Reduction by means of Differential Formulae 540 Near Approach of a Comet to a Planet 546 The Sun may be regarded as the Disturbing Body 548 Determination of the Elements of the Orbit about the Planet 550 Subsequent Motion of the Comet 551 Effect of a Resisting Medium in Space 552 Variation of the Elements on account of the Resisting Medium 554 Method to be applied when no Assumption is made in regard to the Density of the Ether 556 14 CONTENTS. TABLES. PAOl I. Angle of the Vertical and Logarithm of the Earth's Kadius 561 IL For converting Intervals of Mean Solar Time into Equivalent Intervals of Sidereal Time 563 in. 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 XL For the Motion in a Parabolic Orbit 619 XII. For the Limits of the Boots of the Equation sin (z' f) = n^ sin 4 z' ... 622 XIII. For finding the Katio 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 OF THE TABLES 651 APPENDIX. Precession 657 Nutation 658 Aberration 659 Intensity of Light 660 Numerical Calculations.... ... fif.2 THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL, EQUATIONS OF MOTION, AND OF THE FOR- MULAS FOR 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 15 16 THEORETICAL ASTRONOI.il'. 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 nlso 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 '* ~ dt' which will vary from one instant to another. FUXDAME:N*TAL PRINCIPLES. 1? 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 V =ft. If, however, the force be variable, we shall have, at any instant, the relation 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~~dl? 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 tbat 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 2 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 , and the entire motive force F is expressed by = Hi> M being the sum of all the elements, or the mass of the body. Since ds V== W 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 d?' and this, therefore, expresses that part of the intensity r>f 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 Jaw 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 f , 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 P* .m m . ? The action of m' on each molecule of m will be expressed by , and its total action by m' "7* The absolute or moving force with which the masses m and m! tend toward each other is, therefore, the same on each body, which result is a necessary consequence of the equality of action and reaction. The velocities, however, with which these bodies would approach each other must be different, the velocity of the smaller mass 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 m 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' 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 /> its distance from the point attracted; then will dm ~P r 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. 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 y cos 0, y = r cos

do df= f- If we suppose the axis of z to be directed to the point attracted, the co-ordinates of this point will be a being the distance of the point from the centre of the sphere, and, since />' = (x - ao- + (y - 2 , with respect to a, gives do a r sin -f- = = cos r. da p Therefore, if we denote the attraction of the sphere by A } we shall have, by means of the values of df and cos ^, .. r* cos

, and 6 are independent of a, and hence ji* cos dr , if we differentiate the expression for p 2 with respect to (p, we have r cos

we have p = a r, and p = a -f- r , and taking the integral with respect to /> between these limits, we obtain 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 TO, this becomes m Y == a Therefore, dV m da 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', 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 p, the limits will be p = r -f- a, and p = r a, which give and this being independent of a, we have ^=-=- 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 ins! ant. 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' } y f , z', &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, Y, and Z y respectively, then will mX, mY, 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 W dP ~dt' and the corresponding forces are m m By virtue of the action of the accelerating force, these forces for the next instant become m^+mXeft, m* + at at which may be written respectively: MOTION OF A SYSTEM OF BODIES. 25 The actual velocities for this instant are dx , dx dy dy dz dz -dt+ d W ~di + d -di' -dt+ d ^di' 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 md-j^ + mXdt, -md + mYdt, tf) at In the same manner we find for the forces which will be destroyed in the case of the body m! : m'd-- 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 IX, = 0, IY, = 0, IZ, = 0, 0, Z (Z,y - YJ = 0, in which X,, Y,, Z,, denote the components, resolved parallel to the 26 THEORETICAL ASTRONOMY. co-ordinate axes, of the forces acting on any point, and , rr, 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 force <=> dU Btroyed in the case of a system of bodies, we shall have at which are the general equations for the motions of a system of bodies. 8. Let x,, y,, z t) 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 _ Imx _ Imy _ Zmz X >-~2^' y '--Zm~' Z '2^> we get v &* v d _ m (y yO , _ m(z jO ~7~~~' / ' ~? Hence we derive m (Yx - Xy~) + m' (Y'x' X'^j = 0, and generally 28 THEORETICAL ASTRONOMY. In a similar manner, we find Zm(Xz Zx) = Q, (7) Im (Zy Yz) = 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 t Xz = Zx, Zy=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 *(-)-* the integration of which gives Im (xdy ydx) = cat, Im(zdxxdz}=c'dt, (8) Im (ydz zdy} = c"dt, c, c', and c" being the constants of integration. Now, xdy ydx is double the area described about the origin of co-ordinates by the 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', c", express the same sum with reference to the co-ordinate planes xz and yz respectively. Hence the sum of the areal velocities of the several bodies of the system about the origin of co-ordinates, each multiplied by the corresponding mass, is constant; and the sum of the areas traced, each multiplied by the corresponding mass, is 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 a, /?, and f be the angles which this principal plane makes with the co-ordinate planes xy, xz, and yz, respectively; and let 8 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 /5 dt, S cos f dt. Therefore, by means of equations (8), we have c = S cos a, c' = S cos /3, c" = S cos r, and, since cos 2 a + cos 2 /? + cos 2 f = 1, Hence we derive COS a = , . .. =r, COS /? = VV + c" 1 + c" 2 ' l/V + c ' 2 + c"> 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 30 THEORETICAL ASTRONOMY. hence the values of a, /?, and f y 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 f = 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 S is constant whatever may be the position of the co-ordinate planes, and that its value is necessarily greater than that of either of the quantities in the second member of the equation, except when two of them are each equal to zero. It is, therefore, a maximum, and the invariable plane is also the plane of maximum areas. 10. If we suppose the origin of co-ordinates itself to move with uniform and rectilinear motion in space, the relations expressed by equations (8) will remain unchanged. Thus, let x,, y,, z, be the co- ordinates of the movable origin of co-ordinates, referred to a fixed point in space taken as the origin; and let x , y , z , x f , y ', z f , &c. be the co-ordinates of the several bodies referred to the movable origin. Then, since the co-ordinate planes in one system remain always parallel to those of the other system of co-ordinates, we shall have * = x, + ar , y = y, -f y , z = z,+ z , nnd 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. 3] and the preceding equations beo>me (9) - dr Substituting the same values in the last three of equations (4), ob- serving that the co-ordinates x,, y,, z, are the same for all the bodies of the system, and reducing the resulting equations by means of equations (9), we get C10) 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 dm, the equations (5) for the motion of the centre of gravity of the body become m^=fzdm, (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 w z w we have, by means of the equations (10), THEORETICAL ASTRONOMY. the integration with respect to c?m 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 of 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 system of bodies on a MOTION OF A SOLID BODY. 33 distant mass, which we will denote by M. Let X Q , y , z , x ', y ' } z f , &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 n y n and z, the co-ordinates of the centre of gravity of the system referred to the centre of gravity of the body M. The co-ordinates of the body m, of the system, referred to this origin, will therefore be x = x, -f x ot y = y f + y * = */ + %> 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, 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 -pr, and, therefore, that of the entire system on the element of Jf, resolved ; n the same direction, will be S-. We have also r = (x, + x Y + (y, + y Y + 0, + *)', and, if we denote by r, the distance of the centre of gravity of the system from M, r? = x? + y? + z?. Therefore - = (X, + X Q ~) (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 M, 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, (a?,a? + y,y + g,s ) r" r/ "T r , r * and ^mx -m . Imx 3#, , s -^ = * r * + 3 TT ( 3 34 THEORETICAL ASTRONOMY. But, since x , y , z , are the co-ordinates in reference to the centre of gravity of the system as the origin, we have Imx u = 0, Zmy Q = 0, and the preceding equation reduces to y^ x "~^- x> r In a similar manner, we find 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 M, 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 tho 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 different systems from each other and from the other planets, that they act upon these, and are reciprocally acted upon, in nearly the same manner as if the masses of the secondary systems were united at their common centres of gravity, respectively. The motion of the centre of gravity of a system consisting of a planet and its satellites is not affected by the reciprocal action of the bodies of that system, and hence it may be considered independently of this action. The difference of the action of the other planets on a planet and its satellites will simply produce inequalities in the relative motions of the latter bodies as determined by their mutual action alone, 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 consideration of the motion of rotation of the several bodies of the solar system about their respective centres of gravity, is also independent of the motion of translation. If the resultant of all the forces which act upon a planet passed through the centre of gravity, the motion of rotation would be undisturbed; and, since this resultant in all cases very nearly satisfies this condition, the disturbance of the motion of 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, 30 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 in 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 1 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 wi". 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 RELATIVE TO THE SOX. 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/, dfc J ~ df Integrating this, regarding /as constant, and the point to move from a state of rest, we get = $&. (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 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; W 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 1? 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 12 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 F; 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 F. 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 oi time, the measure of the acceleration corresponding to this unit is 2s, and this being the unit of force, we have JW = 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 k 2 , and that of the planet or comet by mk 2 . For a second body let the co-ordinates be x*, y', z'; the distance from the sun, r'j and the mass, m'& 2 ; 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 , 7, , the co-ordinate planes being parallel to those of x, y, and z, respectively; then will the MOTIOX RELATIVE TO THE SUN. acceleration due to the action of TO on the sun be expressed by . and the three components of this force in directions parallel to the co-ordinate axes, respectively, will be . The action of m' 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 ,, v wur my mz ? ?' -pr The motion of the centre of gravity of tjie sun, relative to the fixed origin, will, therefore, be determined by the equations ~dT- If we multiply these three equations respectively by z, y, anct x, and add the products, we obtain cz c'y + c"x = 0. This, being the equation of a plane passing through the origin of co-ordinates, shows that the path of the body relative to the sun is a plane curve, and that the plane of the orbit passes through the centre of the sun. Again, if we multiply the first of equations (19) by 2dx, the second by 2dy, and the third by 2dz, take the sum and integrate, we shall find > But, since r 2 = a 2 + f + z 2 , we shall have, by differentiation, rdr = xdx -f- ydy -\- zdz. Therefore, introducing this value into the preceding equation, we obtain ^+^f+if_af(l + m) (2Q) ar r 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* -\- y* -f a 2 ) (cfo 2 -f eft/ 2 -f c?2 2 ) (a:c?a; -}- ycZy -f z ^ 2 ) 2 Af*. M+df+d* r*dr*_ eft 2 df J ' If we represent by dv the infinitely small angle contained between tw consecutive radii- vectores r and r + dr, since dar 2 + d^/ 2 + d# is the square of the element of path described by the body, we shall have da? + df + dz* = dr* + rW. Substituting this value in the preceding equation, it becomes i*dv = 2fdt. (22) The quantity r*dv is double the area included by the element of path described in the element of time dt, and by the radii-vectores r and r + dr; and/, therefore, represents the areal velocity, which, being a constant, shows that the radius-vector of a planet or comet describes equal areas in equal intervals of time. From the equations (20) and (21) we find, by elimination, dt = - -^L- (23) 1/2r& 2 (1 + m) Ar 2 4/ 2 Substituting this value of dt in equation (22), we get *=- -3*- (24) V2r& 2 (1 + TO) Ar 2 4/' which gives, in order to find the maximum and minimum values of r, dr rl/2rtf (1 + m) Ar 2 4/ 2 ~dv= -2T~ or 2r& 2 (1 -f m) Ar 2 4/ 2 = 0. Therefore P(l-fm) , I 4/ 2 ~ "A +V-1T and P(l+m)_ / 4/ 2 h \ are, respectively, the maximum and minimum values of r. The MOTION EELATIVE TO THE SUN. points of the orbit, or trajectory of the body relative to the sun, cor- responding to these values of r, are called the apsides; the former, the aphelion, and the latter, the perihelion. If we represent these values, respectively, by a(l + e) and d(l e\ we shall have in which p = a (1 e 2 }. Introducing these values into the equation (24), it becomes = r- \ a the integral of which gives (o being an arbitrary constant. Therefore we shall have I 1 I = cos (v w\ e \ r from which we derive p 1 -f- e cos (y w) which is the polar equation of a conic section, the pole being at the focus, p being the semi-parameter, e the eccentricity, and v to the angle at the focus between the radius-vector and a fixed line, in the plane of the orbit, making the angle to with the semi-transverse axis a. If the angle v ft) is counted from the perihelion, we have (o = 0, and P -\-e cos v (25) 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 4/ 2 already found, we obtain I/a rdr = k^/Y^m ' v/aV (a ^rf 2 ' which may be written dt== the integration of which gives J In the perihelion, r a (1 e), and the integral reduces to i' = C; therefore, if we denote the time from the perihelion by t 0) we shall have _.i _?=r. (27) In the aphelion, r a (1 + e) ; and therefore we shall have, for the time in which the body passes from the perihelion to the aphelion, < = or 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 re 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 f =jr ( 29 > If the masses of the two planets TO and TO' are very nearly the same, we may take 1 + TO = 1 -j- TO'; 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 ~ab is the area of the ellipse, a and b representing the semi-axes, we shall have ~ab , 11. - =/= areal velocity; and, since b 2 = a? (1 e*), we have y which becomes, by substituting the value of r already found, -m). (30) 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 -H TO', 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 t denote the inclination of the orbit of TO 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 SI 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 -J- 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 The projection of 2/ on a plane passing through the intersection of the plane of the orbit with the plane of xy, and perpendicular to the latter, is 2/sini; and the projection of this on the plane of xz, to which it is inclined at an angle equal to ft, gives c' 2/sinicos ft. Its projection on the plane of yz gives c" 2/smtsin SI. Hence we derive z cos i y sin i cos ft -f- x sin i sin SI = 0, (31) which is the equation of the plane of the orbit; and, by means of the value of / in terras of p, and the values of c, c', c", we derive, also, m) cos ft sini, (32; * dt y dt~ dz_ z dx__ y~ ~ z -Ji = ki/p(l-\-m) sin ft sin t. These equations will enable us to determine ft, 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 eleinents, are introduced into these equations, two of which, a and e, relate to the form of the orbit, and two, ft and i, to the position of its plane in space. If we measure the angle v to from the point in which the orbit intersects the plane of xy, the con- stant to 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 F 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 F is a maximum. At the aphelion, F 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 formula 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 Ic, which is a constant for the solar system, we have, from equation (28), 60 THEORETICAL ASTRONOMY. Tn the case of the earth, a = 1, and therefore 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 corre- sponding 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 T 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 Hence we have log T = 2.5625978148; log j/l+m = 0.0000006122; log 2n = 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. The quantity - expresses the mean angular motion of a planet in a mean solar day, and is usually designated by p. We shall. therefore, have MOTION RELATIVE TO THE SUN. 51 n = kVl + m , (33) a* for the expression for the mean daily motion of a planet. Since, in the case of the earth, V\ + 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 - or (34) 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 r= (1 -e^ 1 + 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 1 + sin cos v The angle

+ 40 cos i ( v ~~ 40* tn ^ s S^ ves ^+y (37) It appears from this formula that r increases with v, and becomes in- finite when 1 + e cosv = 0, or cost; = cos 4, in which case v = 180 4 : consequently, the maximum positive value of v is represented by 180 4> an d the maximum negative value by (180 40- Further, it is evident that the orbit will be that branch of the hyper- bola which corresponds to the focus in which the sun is placed, since, under the operation of an attractive force, the path of the body must be concave toward the centre of attraction. A body subject to a force of repulsion of the same intensity, and varying according to the same law, would describe the other branch of the curve. The problem of finding the position of a heavenly body as seen from any point of reference, consists of two parts: first, the deter- mination of the place of the body in its orbit; and then, by means >f this and of the elements which fix the position of the plane of the PLACE IN THE ORBIT. 53 orbit, and that of the orbit in its own plane, the determination ot the position in space. In deriving the formulae for finding the place of the body in its orbit, we will consider each species of conic section separately, com- mencing with the ellipse. 20. Since the value of a r can never exceed the limits ae and -j- ae, we may introduce an auxiliary angle such that we shall have a r = cos E. This auxiliary angle E is called the eccentrie anomaly; and its geo metrical signification may be easily known from its relation to the true anomaly. Introducing this value of - - into the equation (27) and writing t T in place of t , T being the time of perihelion passage, and t the time for which the place of the planet in its orbit is to be computed, we obtain (38) But -- j - = mean daily motion of the planet = p. ; therefore The quantity [t(t T) represents what would be the angular distance from the perihelion if the planet had moved uniformly in a circular orbit whose radius is a, its mean distance from the sun. It is called the mean anomaly, and is usually designated by M. We shall, there- fore, have M=(t-T), M=EesmE. (39) When the planet or comet is in its perihelion, the true anomaly, mean anomaly, and eccentric anomaly are each equal to zero. All three of these increase from the perihelion to the aphelion, where they are each equal to 180, and decrease from the aphelion to the peri- helion, provided that they are considered negative. From the peri- helion to the aphelion v is greater than E, and E is greater than M. The same relation holds true from the aphelion to the perihelion, if ve regard, in this case, the values of v, E, and M as negative. As soon as the auxiliary angle E is obtained by means of the mean motion and eccentricity, the values of r and v may be derived. For 54 THEORETICAL ASTRONOMY. this purpose there are various formulae which may be applied in practice, and which we will now develop. The equation ^-'-00.* gives -). (40) This also gives a r ae = a cos E ae, or =. a cos E ae, which, by means of equation (25), reduces to r cos v = a cos E ae. (41^ If we square both members of equations (40) and (41), and subtract the latter result from the former, we get r sin v = oj/1 e* sin E = b sin E. (42) By means of the equations (41) and (42) it may be easily shown that the auxiliary angle E, or eccentric anomaly, is the angle at the centre of the ellipse between the semi-transverse axis, and a line drawn from the centre to the point where the prolongation of the ordinate perpendicular to this axis, and drawn through the place of the body, meets the circumference of the circumscribed circle. Equations (40) and (41) give r(l=pcost;) = a(l:te) (1 + coaE). By using first the upper sign, and then the lower sign, we obtain, by reduction, Vr sin \v = i/o(l + e) sin $E, Vr cos & = Va(l e) cos $E, (43) which are convenient for the calculation of r and v, and especially so when several places are required. By division, these equations give PLACE IN THE ORBIT. 55 Since e sin y>, we have 1 -f e 1 Consequently, tan $E = tan (45 ^) tan $>o. (45) Again. VI -f- e = 1/1 + sin

-f cos 2?>. In a similar manner we find 1/1 e = sin \

, and 6 = a cos instead of p, and sin (p for e, we get (48) If wo multiply the first of equations (43) by cos 2?, and the 56 THEORETICAL ASTRONOMY. second by sin^E, successively add and subtract the products, and reduce by means of the preceding equations, we obtain sin \ (v + E} */- cos ^

and 1? directly; and when the eccentricity is 58 THEORETICAL ASTRONOMY. very great, this mode is indispensable, since the series will not in that ease 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 + e sin M + ^ sin 2 M + &c. (54) Let us now denote the approximate value of E computed from this equation by E , then will in which A-E" is the correction to be applied to the assumed value of E. Substituting this in equation (39), we get M = E + A-E" e sin E e cos E bE ; and, denoting by M the value of M corresponding to E , we shall also have M = E e sin E . Subtracting this equation from the preceding one, we obtain _. - _^. 1 ecos E It remains, therefore, only to add the value of &E found from this formula to the first assumed value of E, or to E , and then, using this for a new value of E w 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 formulae already investigated, the corresponding values of r and v. In the case of an ellipse of very great eccentricity, corresponding to the orbits of many of the comets, the most convenient method of PLACE IS THE ORBIT. 59 computing r and v, for any instant, is somewhat different. The manner of proceeding in the computation in such cases we shall 3011- sider hereafter; and we will now proceed to investigate the formula for determining r and t?, when the orbit Ls a parabola, the formulae for elliptic motion not being applicable, since, in the parabola, a = co , 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 2q, becomes or which may be written -^-j = (1 + tan 1 v) sec 1 $vdv = (1 + tan 1 $v) d tan t?. 1/2 3* Integrating this expression between the limits T and t, we obtain k( t T J = tan Aw + { tan 8 v, (55) 1/2 9* 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 t; = 90 ; then we shall have 3k V* tyt Now, - is constant, and its logarithm is 8.5621876983; and if we take <7 = 1, which is equivalent to supposing the comet to move in a parabola whose perihelion distance is equal to the semi-transverse axis of the earth's orbit, we find log r ** = 2.03987229, or r, = 109.61558 days ; that is, a comet moving in a parabola whose perihelion distance 60 THEORETICAL ASTRONOMY. is equal to the mean distance of the earth 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 \v, this equation becomes a* 4. 3 X 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 )*, we shall have 62 THEORETICAL ASTRONOMY. 2 But 1 + cot 2 \v = and consequently 1 + 3 cot 2 fo 3sm s t;~ (1-f cot 2 ^) 8 ' Now, when v 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 A O ), and, putting tan \w = 0, and tan \ A O = x, we get, from this equation, (! + *) + x (o + xy s l 0x ~r (1 eY Multiplying this through by 0* (I Ox) 3 , expanding and reducing, there results the following equation : 30 2 = 30 (1 -f 40* + 20* +P)x 30 2 (1 + 40' + 20* -f 0) a? + s (2 + 60' + 30* + Dividing through by the coefficient of x t we obtain 2 (2 + 60* + 30* + g )a. > 3 (1 _j_ 40* . 1 4- 30' _ 30 (1 + 40' + 20* + 0) ~~ y; then, substituting this in the preceding equation, inverting the series and reducing, we obtain finally y + oy + "3 (i _j_ 4 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 3 [200 = ^~M' since we have already found k(t-T} = 8 V 2 q% ^ sin'tt' or (200 002 y Z " ZUU Sin W = \ jrjl ^y-y; = \ff' Having computed the value of w from this equation, Table VII. will furnish the corresponding value of A O ; and then we shall have, for the correct value of the true anomaly, 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 O , using v as argument, and then w is given by w = v A n . 64 THEORETICAL ASTRONOMY. The exact value of A O is then found from the table, and hence we derive that of w; and finally t T from ~~ C sin 8 w* 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 =J un_ , I/ 2 g* cos & from which we obtain, since q = r Let us now put siniv sin x = =, 1/2 and we have 3 ^~ T ) = 3 sin x 4 sin 8 * = sin 3z. 2r* Consequently, *-r=~r ! sin3a;, which admits of an accurate and convenient numerical solution. To facilitate the calculation we put ,., sin 3* " == ~~- - > sin v the values of which may be tabulated with the argument v. "When v = 0, we shall have ^V= fl/2, and when v = 90, we have N= 1 ; from which it appears that the value of N changes slowly for values of v from to 90. But when v = 180?, we shall have N=] and hence, when v exceeds 90, it becomes necessary to introduce an auxiliarv different from N. We shall, therefore, put in this case, PLACE IN THE ORBIT. 65 from which it appears that N'= 1 when v = 90, and that ^' when v = 180. Therefore we have, finally, when v is less than 90, and, when v is greater than 90, 2 in which log = 1.5883272995, from which t T is easily derived when v is known. Table VIII. gives the values of N, with differences for interpola- tion, for values of v from v = to v = 90, and the values of N' for those of v from t; = 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 j 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 p cos 4- 2 cos (v + 4) cos (v 4)' When v - 0, the factors cos \ (v + 4) and cos \(v 4) in the de- nominator will be equal; and since the limits of the values of v are 180 4- and (180 4), it follows that the first factor will vanish for the maximum positive value of r, and that the second factor will vanish for the maximum negative value of v, and, therefore, that, in either case, r=cc. In the hyperbola, the semi-transverse axis is negative, and, conse- quently, we have, in this case, p = a(e t 1), or a =p cot* 4. We have, also, for the perihelion distance, = ae-l. Let us now put (56) 66 THEORETICAL ASTRONOMY. which is anal .>gous to the formula for the eccentric anomaly E in an ellipse; and, since e = - , we shall have _- 1 - and, consequently, tan %F = tan %v tan $+. (57) We shall now introduce an auxiliary quantity a, such that whence we derive tanlJP^l^ (58) we shall have e l \ i -i \ (60) Squaring this equation, adding 1 to both members, and reducing we obtain Replacing + +) cos ^ (w 4) ' or 1 _ 1 -f- cos v cos 4> _ ( e -\- cos v) cos 4 cos F ~ 2 cos + 4) c o s 2 ( 4) ~~ 2 cos ^ (v + 4) cos | (v -- which reduces to 1 r(e-fcos7;) (62) PLACE IN THE CEBIT. 6? If we add ^ 1 to both members of this equation, we shall havp p Taking first the upper sign, and then the lower sign, and reducing, we get VcosF V~r cos 3,v = , 6 ~ 1) cos F. (63) I/cos F 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, 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 e 2 1 its value tan 2 oj/, and reducing, we obtain r sin v = a tan 4 tan F = \a tan 4. 1 a -I, (64) Further, we have p cos v ar (e + cos v) r cos v = 3-^j- = ae , 1 + e cos v p which, combined with equation (62), gives r cos v = a I e = I = la I 2e a - I. (65) \ cos F } 2 \ a / If we square these values of r sint? and r cosv, add the results to- gether, reduce, and extract the square root, we find (66) cosF We might also introduce the auxiliary quantity a into the equations (63); but such a transformation is hardly necessary, and, if at all desirable, it can be easily effected by means of the formulae which we have already derived. gjj THEORETICAL ASTRONOMY. 26. Let us now resume the equation _ cos ^ (v 4) ~~ cos $ (v + 4)' Differentiating this, regarding $ as constant, we have sin 4- , d " = 2 cos' i (.+*)*' and, dividing this equation by the preceding one, we get d*_ _ sin 4 _ dv ff ~ 2 cos (v + 4) cos (v 4) But _ _ j? cos 4 _ ~~ 2 cos K + 4) cos ( *)' consequently, c?r _ r tan 4 p which givea i*dv = da. a tan 4 Substituting this value of r*dv in equation (22), and putting instead of 2/ its value kVp, from equation (30), the mass being considered as insensible in comparison with that of the sun, we get tan 4 Then, substituting for r its value from equation (66), and for p its value a tan 2 T^, we have = a? tan 4 ( \e (\ + 1) 1 j d*. Integrating this between the limits T and t, we obtain i) log e< rj, (67) in which log, , and we shall have 1 ti' 2du cos v = ; - ; ; dv = T ; 1 -f- tt 2 1 + W* Substituting these values in the preceding equation, and putting r* = *> we get or, since p = q (I + e), (1 + O du Let us now develop the second member into a series. This may be written thus: and developing the last factor into a series, we obtain (1 + M*)~* = 1 2tii + 3tV 4tV + &c. Consequently, (1 + M 1 ) (1 + Mi 1 )"' = 1 + M f 2l(' + Multiplying this equation through by du, and integrating between the limits T and t, the result is k(t- (70) In the case of the parabola, e = 1 and t = 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 _ Let U be expanded into a series of the form which is evidently admissible, a, /9, f, . . . . being functions of u and independent of i. It remains now to determine the values of the coefficients a, ft, 7-, &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* 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 17' = u + ai + & + rf + I (u + ai + /9t or, again neglecting terms of the order i 4 , U+ I U 9 = u + J u* + i (1 -f O a + i 2 (o 2 + (1 But we have already found, (70), k(t- + 3? 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 tt .) p = + 3 ( i W 6 + ^ ti l ) r = 4 (> T + X) From the first of these equations we find PLACE IN THE ORBIT. 73 1+"' The second equation gives or, substituting for a its value just found, and reducing, We have also and hence, substituting the values of a and ft already found, and reducing, we obtain finally _ Again, we have Developing this, and neglecting terms of the order i 4 , we get tan" 1 U= t&n~\u + cu + ., l Now, since w = tanp and U= tan J F, we shall have or 2a Substituting in this equation the values of a, /9, and f already found, and reducing, we obtain finally frp+l*. - 1 71 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 Fand take out the corresponding value of Jfy and l.hen we derive t T from = iii 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 + o!i + p? -f r V + &c. Substituting this value of u in equation (70), and neglecting terms multiplied by i 4 and higher powers of i, we get V O9'(l _ 4 ov 2 2 C7a' 2 4 W | U 9 ) ?. But, since the first member of this equation is equal to Z7+ JZ7 3 , we shall have, by the principle of indeterminate coefficients, /(!+ W) |Z7 S C7 6 = 0, ' 2 217V (1 + E7') + |?7 5 + ^CP= 0, 3 40V* 2l7' i $17 fC*==$ From these equations, we find (1 + 1/ 2 ) 8 If we interchange v and F in equation (72), it becomes, writing a', P, f for a, 0, r , PLACE IN THE ORBIT. 75 JP a + u*y a + Substituting in this equation the above values of a f , fl f , and f, and reducing, we obtain, finally, |ff s +|t7. , fl^+lff^+ " (1 + t/*) 2 (1 + , Ifl^+igil^+Hfff^+lflf^+^. (1 + C7 2 ) 6 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 100(1 10000 (1 + 2 ) 4 y = t* + itt*+Wi 10000 (1 -f 2 )* 1000000 (1 + ^ 2 ) If I* 13 1000000 (1 + *) wherein s expresses the number of seconds corresponding to the length of arc equal to the radius of a circle, or logs = 5.31442513. We shall, therefore, have: When x =V, v = F+ A (lOOt) + (1000'+ (7(100;)'; 76 THEORETICAL ASTRONOMY. and, when x v, V=v-A (1000 + & (1000 2 - C' (100i) s . Table IX. gives the values of A, B, B' } C, 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 1 + e cos v Similar expressions arranged in reference to the ascending powers of (1 e) or of I I I 1 ) may be derived, but they do not con- / / 2 V* \ verge with sufficient rapidity ; for, although ( I - I 1 I is less \ \ 1 ~~|~ & I i than t, 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 PLACE IX THE ORBIT. 77 which, since q = a (I e), may be written If v\ e put ^^ ' we shall have k(t T}VY=~e 201/1 .1 1 l+9g .f 2 | '~ + * Let us now put 2 35(1 9E -f sin E 20v/2 ' and 1 then we have When jB is known, the value of w may, according to this equation, l>e derived directly from Table VI. with the argument and then from tc we may find the value of A. It remains, therefore, to find the value of B; and then that of t; from the resulting value of^l. Xow, we have and if we put tan z |^= r, we get sin E = jHj- = 2r* (1 r -f r 1 T + &c.). We have, also, = 2 tan" 1 T*= 2r* (1 Jr + p ir + Ac.)- 78 THEORETICAL ASTRONOMY. Therefore, 15 (E - sin E} = 2r* (lOr - 6 ^ + V" - l ^ + Ac-), and sin E = 2r* (10 - L 2 r + *f* - V* + V^ - Ac.). 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 1 then the values of 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 + \A. Next, to find B, we have A k = r* (i - IT + T < 7 y - til** + WVV&V 4 - Ac.), and hence from which we easily find 5 = 1 + T f 5^ + i3^ 8 + W3^ + AC. If we compare equations (44) and (56), we get tan \E t/^T tan \F. Hence, in the case of a hyperbolic orbit, if we put tan 2 |.F= r', we must write T' in place of r in the formulae already derived; and, from the series which gives A in terms of T, it appears that A is in this case negative. Therefore, if we distinguish the equations for PLACE IN THE DEBIT. 79 Hyperbolic motion from those for elliptic motion by writing A', B f , and C" in place of A, B, and C, respectively, we shall have + &c., 3 3s 4 - Ac. Table X. contains the values of log jB 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 gves or, substituting the value of A in terms of w, ~s\ (76) The last of equations (43) gives Hence we derive -' (77) The equation for v in a hyperbolic orbit is of precisely the same form as (76), the accents being omitted, and the value of A being computed from 4 = tan ' iw - (78) For the radius-vector in a hyperbolic orbit, we find, by means of the last of equations (63), r (1 _^^ cos n 1> - (79) "When t T is given and r and v are required, we first assume B = 1 , and enter Table VI. with the argument M= 80 THEORETICAL ASTRONOMY. in which log C = 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 B, and then, using this in the expression for M t we repeat the operation. The second result for A will not require any further correction, since the error of the first assumption of B = 1 is very small ; and, with this as argument, we derive the value of log C from the table, and then v and r by means of the equations (76) and (77) or (79). When the true anomaly is given, and the time t T is required, we first compute r from in the case of the ellipse, or from '-5+1*"* in the case of the hyperbola. Then, with the value of T 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 log B 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 ".orrection. Next, to find w, we have C \5(l+e) ; anil, with w as argument, we derive M from Table VI. Finally, we have t-T= _ * . (80) by means of which the time from the perihelion may be accurately determined. POSITION IX 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 anomaly at a given epoch, or, what is equivalent, the time of passing tht 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 ft, 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, SI 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 ft, and which is measured, as in the case of ft, 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 Imgitude of the perihelion, which is denoted by K, 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. we have x = r cos v, POSITION IX 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 + <")> y = r sin (v + ), a) 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. Now, we have co = 7r SI in the case of direct motion, and at - SI ~ in the case of retrograde motion; and hence the last equations become x = r cos (v JT qr ft) the upper and lower signs being taken, respectively, according as the motion is direct or retrograde. The arc t7~=Fft=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 x, in the line of nodes. Then we shall have x' = r cos u, i/ = 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 6 and Z, respectively, we shall have x' = r cos b cos (I ft ), j/ = r cos b sin (7 SI )> d = r sin b, and, consequently, cos M = cos b cos (I ft), sin u cos i = cos 6 sin (I ft), (81) sin u sin i = sin b. From these we derive tan (I ft ) = db tan u cos i, tan b = =b tan i sin (J ft), (82) which serve to determine I and 6, when ft, u, and i are given. Since 84 THEORETICAL ASTRONOMY. cos b is always positive, it follows that I Q and u must lie iu the same quadrant when i is less than 90 ; but if i is greater than 90, or the motion is retrograde, I & and 360 u will belong to the same quadrant. Hence the ambiguity which the determination of I & 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 Q, and u will lie in the same quadrant. 32. By multiplying the first of the equations (81) by sinw, and the second by cos u, and combining the results, considering only the upper sign, we derive cos b sin (u I + & ) = 2 sin u cos u sin 2 %i, or cos b sin (u I + & ) = sin 2w sin 1 {. In a similar manner, we find cos b cos (it I + ^ ) = COS'M -f sin'w cos i, which may be written cos&cos(w 1+ &) = (! + eos2w) + (l C os2ti)cost, dr cos b cos ( i + &) = (1 + cost) + 2 C 1 cost) cos 2w; and hence cos 6 cos (it i + ) = cos* i + sin 1 cos 2. If we divide this equation by the value of cos b sin (u I + already found, we shall have (83) The angle u 1+ & is called the reduction to the ecliptic; and the expression for it may be arranged in a series which converges rapidly when * is small, as in the case of the planets. In order to effect this development, let us first take the equation l +ncosx Differentiating this, regarding y and n as variables, and reducing, we find dy _ sin a; dn ~ 1 -f 2n cos af +~n v POSITION IN SPACE. 85 which gives, by division, or by the method of indeterminate coefficients, ~ = sin x n sin 2x 4- n* sin &c n* sin 4c -f- &c. an Integrating this expression, we get, since y = when x = 0, y = n sin * \n* sin 2# -f- | s sin 3z |n 4 sin 4z + ...., (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 \i and x - 2u; and hence we obtain u I + SI = tan* \i sin 2u \ tan* i sin 4w -f I tan* i sin 6w i tan 8 \i sin 8u + \ tan 10 i sin lOw Ac. (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 + & 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 R e the reduction to the ecliptic, we shall have l = u + & R e = v-}-K R f . But we have v = M + the equation of the centre ; hence 1 = M -\- if -f equation of the centre reduction to the ecliptic, and, putting L = M~\- 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 = rcosb sin I, z = r sin b. Again, let X, Y, 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 O denote the geocentric longitude of the sun, R its distance from the earth, and 2 its latitude. Then we shall have Let x' y y f , z' be the co-ordinates of the body referred to the centre of the earth ; and let ^ and /9 denote, respectively, the geocentric longi- tude and latitude, and J, the distance of the planet or comet from the earth. Then we obtain x' = A cos ft cos A, if = A cos ft sin A, (87) z' = A sin ft. But, evidently, we also have and, consequently, A cos ft cos A = r cos b cos I -f- R cos S cos Q , A cos ft sin A = r cos 6 sin J + R cos 2" sin Q, (88) A sin = r sin b -\- R sin 2". If we multiply the first of these equations by cos O, and the second by sin O, and add the products; then multiply the first by sin Q, and the second by cos, and subtract the first product from the second, we get A cos/9 cos(J O) = r cos& cos( O) +jR cosT, A cos/? sin (A O) = r cosfc sin (I Q), (89) ^ sin ft =rsinb + 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 Q, 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 )^ /9, and J when r. 6. and I have been derived from the elements of the orbit, the quantities R, Q, and ~ being furnished by the solar tables; or, when J, /9, and / are given, these equations determine I, b, and r. The latitude I of the sun never exceeds 0".9, and, therefore, it may in most cases be neg- lected, so that cos I = 1 and sin 2 = 0, and the last equations become A cosy? cos (A Q) = rcosb cos(J Q) + R, A cos $ sin (I O ) = r cos I sin (I Q ), (90) A sin ft =r sin 6. If we suppose the axis of a; to be directed to a point whose longi- tude is SI , or to the ascending node of the planet or comet, the equa- tions (88) become A cos ft cos (J &) = r cos u + R cos S cos (O SI ), A cos ft sin (^ &) r sin u cos i + R cos - sin (O SI ), (91) J sin ft = r sin w sin i -f- .# sin -T, by means of which fi and ). may be found directly from Q, , i, r, and u. If it be required to determine the geocentric right ascension and declination, denoted respectively by a and 3, we may convert the values of and A into those of a and d. To effect this transforma- tion, denoting by e the obliquity of the ecliptic, we have cos 3 cos a cos /5 cos i, cos 8 sin a = cos ft sin ^ cos s sin ft sin e, sin 5 = cos ft sin ^ sin -j- sin ,5 cos e. Let us now take n sin N=sinft, n cosN= cos ft sin ^, and we shall have cos d cos a = cos /? cos -*, cos 5 sin a = n cos (-ZV+ e), sin d =n sin (2V -f- e ). 88 THEORETICAL ASTRONOMY. Therefore, we obtain (92) sin A cos IV tan 5 = tan (N+ sin o. We also have cos (N + e) _ cos 3 sin a cos-ZV ~~ cos/? sinA' which will serve to check the calculation of a and d. Since cos d atut cos ft are always positive, cos a and cos A must have the same sign, and thus the quadrant in which a is to be taken, is determined. For the solution of the inverse problem, in which a and 3 are given and the values of A and ft are required, it is only necessary to interchange, in these equations, a and X, d and ft, and to write e in place of e. 34. Instead of pursuing the tedious process, when several places are required, of computing first the heliocentric place, then the 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, tf = r sin u cos i, z 1 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 ft, and, consequently, x = x f cos ft i/ sin ft, y = x' &in ft +?/'cos ft, z=z'. Therefore, we obtain x = r(cos u cos ft T sin u cos i sin ft ), y = r( sin u cosi cos ft + cos u sin ft), (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 SI sin a sin A, =F coa i sin & = sin a cos A, 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 ( A -j- u), y = rsinb sin (B -f u~), (95) 2 r sin i sin u. The auxiliary quantities a, 6, A, and B, it will be observed, are functions of SI and i, 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 = =F tan & cos i, cotB = cot & cos i, cos & . , sin & smo r , ^ sin A sinB 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 SI, sinB and sin &, respectively, must have the same signs. The quadrants in which A and B are situated, are thus deter- mined. From the equations (94) we easily find cos a = sin i sin &, cos b = sin i 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 -\- X= J cos 13 cos A, y+ Y= J cos /9 sin A, (97) 2 -f Z = A sin ft 90 THEORETICAL ASTRONOMY. which suffice to determine I, ft, 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 x, y, z be the heliocentric co-ordinates of the body referred to the equator, the positive axis of x being directed to the vernal equinox. To pass from the system of 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, being the obliquity of the ecliptic. In this case, we have x" = x, y" = y cos e z sin e, z" = y sin e -f- z cos e. Substituting for x, y, and z their values from equations (93), and omitting the accents, we get x = r cosu cos SI =fi r sin u cos i sin ft, y = rcossmft cose + rsinw ( cost cos ft cose sinisine), (98) z r cos w sin ft sin e -f- r sin u ( cos i cos ft sin e -f- sin i cos e)- These are the expressions for the heliocentric co-ordinates of tl. planet or comet referred to the equator. To reduce them to a con- venient form for numerical calculation, let us put cos ft = sin a sin A, ip cos i sin ft = sin a cos .4, sin SI cos e = sin b sin B, db cos i cos ft cos e sin i sin e = sin b cos B, sin ft sin e = sin c sin G, cos i cos ft sin e -f sin i cos e sin c cos C; and the expressions for the co-ordinates reduce to x = r sin a sin (A -f- u), y = rsinb sin (B + u), (100) z = r sin c sin ( G + u). The auxiliary quantities, a, 6, c, .A, .B, and C, are constant so long as ft and i remain unchanged, and are called constants for ih( 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 IX SPACE. 91 angle included between these sides being that which we designate by A, and the angle opposite the side a being 90 ft. In the case of b and B, the relations are those of the parts of a spherical triangle of which the sides are b, i, and 90 + e, B being the angle included by i and 6, and 180 ft the angle opposite the side 6. Further, in the case of c and C, the relations are those of the parts of a spherical triangle of which the sides are c, i, and e, the angle C being that included by the sides and e, and 180 ft that included by the sides i and e. We have, therefore, the following additional equations : cos a sin i sin ft, cos 6 = cos ft sin i cos e cos i sin e, (101) cos c = cos ft sin i sin e -f- cos i cos s. In the case of retrograde motion, we must substitute in these 180 i in place of t. 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 t = e sin E c , cos i cos ft = e cos E , being always positive. We shall, therefore, have Since both e and sini are positive, the angle E cannot exceed 180; and the algebraic sign of tan E will show whether this angle is to be taken in the first or second quadrant. The first two of equations (99) give cot .4 = H= tan ft cosi; and the first gives cos ft sin a = -^f. sin .4 92 THEOEETICAL ASTRONOMY. From the fourth of equations (99), introducing e and E m we get sin b cos B = e cos E cos e e sin E sin s = e cos (E -f- e). But sin 6 sin 5 = sin & cos e; therefore B _ _6o_ cosQEo+e) _ cost cosCE -fe) ~~ sin ft * " cos e tan ft cos E ' cos e We have, also, . , sin ft cos e sm 6 = f 5 -^ . smB In a similar manner, we find cot C = tan ft cos and sin ft sin e smc = FT: . sin C The auxiliaries sin a, sin 6, and sin c are always positive, and, there- fore, sin A and cos ft, sin B and sin ft, and also sin C and sin ft, 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 6 sin c sin (C B) = sin i sin ft. But sin a cos A = q= cos i sin ft ; 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 ic =F ft, ' = B n =p ft, and C f = C n =F ft, the upper or lower sign being used according as the motion is direct or retrograde, we shall have POSITION IX SPACE. 03 t = r sin a sin ( A' -f- v), y = r sin 6 sin (& -\- v), (102) 2 = r sin c sin ( (7 + 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 formulae for computing the constants a, 6, c, A, B, and (7, in the case of direct motion, are converted into those for the case in which the distinction of retrograde motion is adopted, by simply using 180 i instead of i. 36. When the heliocentric co-ordinates of the body have been found, referred to the equator as the fundamental plane, if we add to these the geocentric co-ordinates of the sun referred to the same fundamental plane, the sum will be the geocentric co-ordinates of the body 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, Y J?sin O cos z, Z = R sin O sin = = 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 -f- X= A cos <5 cos o, y+ Y= J cos 5 sin o, (103) z + Z Jsin- centric co-ordinates, the formulae will evidently become X=RcosQ cos J, Y It sin O cos cos R sin Tsin e, (104) Z= R sin O cos -sin e -{- R sin -cos e, in which, since 2 can never exceed O r '.9, cos I is very nearly equal to 1, and sin 2. = 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 Jahrbuck, the Nautical Almanac, and the American Epliemeris and Nautical Al- 94 THEORETICAL ASTRONOMY. tnanac, contain, for each year for which they are published, the equatorial co-ordinates of the sun, referred both to the mean equinox and equator of the beginning of the year, and to the apparent equinox of the date, taking into account the latitude of the sun. 37. In the case of an elliptic orbit, we may determine the co- ordinates directly from the eccentric anomaly in the following manner : The equations (102) give, accenting the letters a, b, and c, x = r cos v sin a' sin A' + r sin v sin a! cos A', y = r cos v sin b' sin B' -j- f sin v sin b' cos I?, z rcosv sin c' sin C' -f- r sin v sin c' cos C". Now, since r cos v = a cos E ae, and r sin v = a cos

V By means of these formula, the co-ordinates are found directly from the eccentric anomaly, when the constants A x , J y , J,, i x , i y , i,, y x , v y , and v z , have been computed from those already found, or from a, b, c, A, B, and C. This method is very convenient when a groat POSITION IX SPACE. 95 number of geocentric places are to be computed ; but, when only a few places are required, the additional labor of computing so many auxiliary quantities will not be compensated by the facility afforded in the numerical calculation, when these constants have been deter- mined. Further, when the ephemeris is intended for the comparison of a series of observations in order to determine the corrections to be applied to the elements by means of the differential formula? which we shall investigate in the following chapter, it will always be ad- visable to compute the co-ordinates by means of the radius-vector and true anomaly, since both of these quantities will be required in finding the differential coefficients. 38. In the case of a hyperbolic orbit, the co-ordinates may be com- puted directly from F, since we have r cos v a (e sec _F), r sin v = a tan 4 tan F; and, consequently, x = ae sin a' sin A' a sec F sin a' sin A' -j- a tan 4 tan F sin a' cos A ', y = ae sin b' sin I? a sec F sin b' sin B 1 -\- a tan 4 tan F sin b' cos B, z =. ae sin c' sin C" a sec F sin c' sin C' -\- a tan 4 tan F sin c' cos C". Let us now put ae sin a' sin A' = * a sin a' sin A = yu x , a tan 4. sin a' cos A = v x ; ae sin b' sin B = A,, a sin b' sins' = n,, a tan 4. sin b' cos B' = v y ; ae sin c' sin C' = >* a sin c' sin C' = /* a tan 4 sin c' cos C' = v t . Then we shall have x ^ + n f sec F -f v,. tan .F, y = ^ + j^ sec .F + *, tan J 1 , (106) z = i, + JM, sec F + v t tan F. In a similar manner we may derive expressions for the co-ordinates, in the case of a hyperbolic orbit, when the auxiliary quantity a is used instead of F. 39. If we denote by ~', &', 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 a) 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 Sl f and i f , we have, from the spherical triangle formed by the intersection of the planes of the orbit, ecliptic, and equator with the celestial vault, cos i' = cos i cos e sin i sin e cos SI > sin i' sin &' = sin i sin SI, sin i' cos &' = cos i sin e -f- sin i cos e cos SI . Let us now put n sin N== cost, n cosN= sin i cos &, and these equations reduce to cos i' = n sin (N e), sin i' sin &' = sin i sin SI, sin t' cos ' = n cos (-AT e) ; from which we find cot i' = tan (.tf- e) cos & '. ( 107 ) Since sin i is always positive, cos N and cos & must have the signs. To prove the numerical calculation, we have sin i cos cos N 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 sin i' = sin sin e, cos cos (M i)' If we apply Gauss's analogies to the same spherical triangle, we get cosii' sin^ (ft' + o ) = sin ^ft cos (i e), cosli' cos^ (ft' + >) = cos^ft cos (* + 0, sin K sin i (ft' < )=8inasin(* e), sin i' cos (ft' o ) = cos ^ ft sin ^ (i + e). The quadrant in which \ (ft' -f- o> ) or (ft w ) is situated, must be so taken that sin^i' and cos^i' shall be positive; and the agreement of the values of the latter two quantities, computed by means of the value of \i' derived from tan \i l ', 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, 7r' = 7T ft + ft' . We may thus find the elements n', ft ', and i f , in reference to the equator, from the elements referred to the ecliptic ; and using the elements so found instead of TT, ft, 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 formulfe 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 ft and ft r , as well as i and 180 i', e remaining unchanged, in these equations. These formulaB may also be used to determine the position of the orbit in reference to any plane in space; but the longitude ft must then be measured from the place of the descending node of this plane on the ecliptic. The value of ft, 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 ft', i', and a) Q will have the same signification in reference 7 98 THEOKETICAL 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 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, ?r, &, 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, SI, and i, the auxiliaries sin a, sin 6, sin c, A, B, 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-ordinates 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 =/ + 9 s in ( & > 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 i to that of another date t' may be solved by means of equations (109), making, however, the necessary distinction in regard to the point from which & and &' are measured. Let 6 denote the longi- tude of the descending node of the ecliptic of t' on that of t, and let rj denote the angle which the planes of the two ecliptics make with each other, then, in the equations (109), instead of & we must write & 6, and, in order that &' shall be measured from the vernal equinox, we must also write &>' 6 in place of &'. Finally, we must write 37 instead of e, and bio for o> , which is the variation in the value of ot in the interval t' t on account of the change of the position of the ecliptic ; then the equations become cos ^' sin $(&' -\- Aw) = s ini(& (?) cos \(i owi^eoslCQ' + A0 = eos(Q *) cos (* sin tf sin (' 9 Aw) = sin |( 0) sin l (i sin \y cos^ (&' e A/) = cos (& 0) sin -' (i -f r y ). These equations enable us to determine accurately the values of &', i'j and AW, which give the position of the orbit in reference to the ecliptic corresponding to the time t', when and TJ 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 10 J THEORETICAL ASTRONOMY. them to the mean equinox of the epoch t', the amount of the pre- cession in longitude during the interval tf 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 , cos (ft 0) = cos i' sin i -f- sin i' cos i cos AW, from which we easily derive sin (i' t) = sin , cos (ft 0) -f- 2 sin i' cos i sin 2 ^Aw. (112) We have, further, sin Aw sin i' = sin , sin ( ft 0), or sin Aw = sin , -^ r , -* (113) We have, also, from the same triangle, sin AW cos H = cos (ft 0) sin (ft' 0) -f su (ft 0) cos (ft' 0) cos ,, which gives sin (ft' ft) = sin Aw cost 2 sin (ft 0) cos (ft' 0) sin 2 &, or sin (ft' ft ) = sin , sin (ft 0) cot i' 2 sin (ft 0) cos (ft' - 0) sin*^,. (114) Finally, we have *' *= ft' ft -MW. 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 t', 8 * M - r~ 1* ~ 0-^- i?sm(ft 0)cot*' ilsm2(ft 0), (115) 0-^-4-7 sin(ft0)t a n^'^sin2(ft0); POSITION IX SPACE. 101 in which is the annual precession in longitude, and in which g = 206264 / '.8. In most cases, the last terms of the expressions for i', SI', and -', 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 AOJ, i f , and &'; and for -', in this case, we have ,/.-*=a' a A-, which gives Tf we adopt Bessel's determination of the luni-solar precession and of the variation of the mean obliquity of the ecliptic, we have, at the time 1750 + r, -^ = 50".21129 + O."0002442966r, at ^ = 0".48892 O."000006143r, and, consequently, r, = (0."48892 O."000006143r) (Y 5 and in the computation of the values of these quantities we must put r = \(t r + t) 1750, t and i' being expressed in years. The longitude of the descending node of the ecliptic of the time t on the ecliptic of 1750.0 is also found to be 351 36' 10" 5".21 (t 1750), which is measured from the mean equinox of the beginning of the year 1750. The longitude of the descending node of the ecliptic of t' on that of i, measured from the same mean equinox, is equal to this value diminished by the angular distance between the descending node of the ecliptic of t on that of 1750 and the descending node of the ecliptic of t' on that of t, which distance is, neglecting terms of the second order, 5".21(f 1750); and the result is 351 36' 10" - 5".21 (t 1750) 5".21 (f 1750), 351 36' 10" 10".42 (t 1750) 5".21 (f i). 1Q2 THEORETICAL ASTROXOMY. 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 (* 1750), so that we have, finally, = 351 36' 10" + 39".79 (t - 1750) - 5".21 (t 0- When the elements n, , and t 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- puting a series of places, the changes which thus take place in the elements themselves from date to date induce corresponding changes in the auxiliary quantities a, b, c, A, B, and C, so that these are no longer to be considered as constants, but as continually changing their values by small differences. The differential formula 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 tht 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 i 497".78 J 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 3 during the interval 497 S .78 J, in which expression J is the distance of the body from tht 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. M= 1 29' 40".21 TT = 44 20 33 .09 ^ = 206 42 40 .13 > i= 4 36 50 .51 J Ecliptic and Mean Equinox, 1864.0. f = 11 15 51 .02 log a = 0.3881319 log /* = 2.9678088 fi = 928".55745 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 K e 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 TT, ft. and i are referred to the ecliptic and mean equinox of 1864.0, we will first reduce them to the ecliptic and mean equinox of 1865.0. For this reduction we have t = 1864.0, and t'= 1865.0, which give = 50".239, = 352 51' 41", ^ = 0".4882. dt Substituting these values in the equations (115), we obtain i' i = Ai = 0".40, A & = -f 53".61, A* 1= + 50".23 ; and hence the elements which determine the position of the orbit in reference to the ecliptic of 1865.0 are * = 44 21' 23".32, ft = 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 ol the formulas cot A = tan ft cos i, tan E = _ F . , tan ft cos E cos e V tan ft cos E sin e The angle # 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 . 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 ft, sin B and sin ft, and also sin Cand sin ft, 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 JS) tan i ; -: sin a cos A whicn 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 appl) ing the difference of longitude, 5* 8 m 11*.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 H= 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=EesinE, the value of e being expressed in seconds of arc. For Eurynome we have log sin tp = log e = 9.2907754, and hence the value of e ex- pressed in seconds is log e = 4.6052005. By means of the equation (54) we derive an approximate value of E, namely, ^=119 49' 24", the value of e 2 expressed in seconds being log (?= 3.895976; and with this we get Jf = E -e sin E = 110 6' 50". Then we have M-M. 372".7 _ A A. = : j=r = -\ (\nT ' * 5c ' y ' 1 ecosJ 1.097 which gives, for a second approximation to the value of .E t E = 119 43' 44".3. This gives M Q = 110 0' 36".98, and hence "' 37 IOC THEORETICAL ASTRONOMY. Therefore, we have, for a third approximation to the value of E, =11943'44".64, which requires no further correction, since it satisfies the equatior between M and E. To find r and v, we have V~r sin \v = Va(l + e) sin ^E, Vr cos \v = l/a(l e) cos ^ JS The values of the first factors in the second members of these equations are: log Va(l + e) = 0.2328104, and logl/a(l e) = 0.1468741; and we obtain v = 129 3' 50". 52, log r = 0.4282854 Since ?r - B = 197 37' 49' / .58, we have = ,,-}-* ft = 326 41' 40".10. The heliocentric co-ordinates in reference to the equator as the fun damental plane are then derived from the equations x = r sin a sin (A -f- u\ y = r sin b sin ( + u\ z = r sin c sin ( C -\- it), which give, for Eurynome, x = 2.6611270, y = -f 0.3250277, z = + 0.0119486. The American Nautical Almanac gives, for the equatorial co-ordi- nates of the sun for 1865 February 24.5 mean time at Washington, referred to the mean equinox and equator of the beginning of the year, X= -f 0.9094557, Y= 0.3599298, Z= 0.1561751. Finally, the geocentric right ascension, declination, and distance are given by the equations -- the first form of the equation for tan 3 being used when sin a i? greater than cos a. The value of J must always be positive; and S cannot exceed 90, the minus sign indicating south declination. Thus, we obtain NUMERICAL EXAMPLES. 107 a = 181 8' 29".29, 3 = - 4 42' 21".56, log A = 0.2450054. To reduce a and d to the true equinox and equator of February 24.5, we have, from the Nautical Almanac, /= + 16".80, log^ = 1.0168, G = 45 16'; and. substituting these values in equations (110), the result is Aa = +17".42, &S = 7".l7. Hence the geocentric place, referred to the true equinox and equator of the date, is a = 181 8' 46".71, <5 = 4 42' 28".73, log A = 0.2450054. When only a single place is required, it is a little more expeditious to compute r from r = a (1 e cos E), and then v E from v E} = "Y- si sin ( Thus, in the case of the required place of Eurynome, we get log r = 0.4282852, v E = 9 20' 5".92, v = 129 3' 50".56, agreeing with the values previously determined. The calculation may be proved by means of the formula -\- E) = \- cos 4f sin E. In the case of the values just found, we have %(v + E) = 124 23' 47".60, log sin %(y + E) = 9.9165316. while the second member of this equation gives log sin .J (v + E} = 9.9165316. Tn the calculation of a single place, it is also very little shorter to compute first the heliocentric longitude and latitude by means of the equations (82), then the geocentric latitude and longitude by means of (89) or (90), and finally convert these into right ascension and declination by means of (92). When a large number of places are to be computed, it is often advantageous to compute the heliocentrio 108 THEORETICAL ASTRONOMY. co-ordinates directly from the eccentric anomaly by means of the equations (105). The calculation of the geocentric place in reference to the ecliptic- is, in all respects, similar to that in which the equator is taken as the fundamental plane, and does not require any further illustration. The determination of the geocentric or heliocentric place in the cases of parabolic and hyperbolic motion differs from the process indicated in the preceding example only in the calculation of r and v. To illustrate the case of parabolic motion, let t T= 75.364 days; log q = 9.9650486; and let it be required to find r and v. First, we compute m from in which log C = 9.9601277, and the result is log ro = 0.01 25548. Then we find M from M=m(t-T\ which gives log M= 1.8897187. From this value of log M we derive, by means of Table VI., v = 79 55' 57".26. Finally, r is found from * <*#& which gives log r = 0.1961120. For the case of hyperbolic motion, let there be given t T= 05.41236 days; 4--37 35' 0".0, or log = 0.1010188; and loga = 0.6020600, to find r and v. First, we compute N from in which log,* = 9.6377843, and we obtain log N= 8.7859356 ; N= 0.06108514. The value of F must now be found from the equation N= el tan F log tan (45 + -> jp). NUMERICAL EXAMPLES. 105) If we assume F= 30, a more approximate value may be derived from ,., JT-f- log tan 60 tan JT , = - : - , eA 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-NJcofF, '~ ' wherein s = 206264".8 ; and the result is AjF, = 4.6097 (NN,~)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, = 0.061 7653, and hence A.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 .F=25 24'27".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. AVhen several places are required, it is convenient to compute v and r by means of the equations I/cos F 110 THEORETICAL ASTRONOMY. For the given values of a and e we have log V / a(e + 1) 0.4782649, log Va(eT) = 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 M= 2.1404550. \Vith this as argument we get, from Table VI., V= 101 38' 3".74, and then with this value of Fas argument we find, from Table IX., A ^ 15 40".08, B = 9".506, C = 0".062. Then we have log i = log j-j-^ = 8.217680, and from the equation v = V+ A (lOOi) + B (1000 2 + CXlOOi) 1 , we get v = V+ 42' 22".28 + 25".90 -f 0".28 = 102 20' 52".20. The value of r is then found from e cos v' namely, log r = 0.1614051. We may also determine r and v by means of Table X. Thus, first compute M from Assuming B=l, we get log M= 2.13757, and, entering Table VI. with this as argument, we find w = 101 25'. Then we compute A from JOJMEEICAJL EXAMPLES. HI which gives A = 0.024985. With this value of A as argument, we Qnd, from Table X., log = 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, / is given by (1 -rAC 1 ) cos'Av from which we get log r = 0.1614052. Before the time of perihelion passage, t T is negative ; but the value of t? 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 and that of r from 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 ASTEONOMY. 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 + n(o, and the corresponding value of the co-ordinate, or of the function, for which the interpolation is to be made, by / (a + mo). If we have computed the values of the function for the dates, or arguments, a (a, a, a -f- a>, a + 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 may expand the function into the series /(a + no,) =/(a) + An + En* + Cn 3 + &c. (116) and if we regard the fourth differences as vanishing, it is only neces- sary to consider terms involving n 3 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. /(a) =f(a)-A + B -C /GO =/(a) +B + C If we symbolize, generally, the difference /(a + wa>) f( a + (n 1) a>) ] >7 /' ( + ( n 1) *>), the difference / (a + (n + i) to) f (a + (n J) to) by /'(a + no)}, and similarly for the successive orders of differences, these may be arranged as follows : Argument. Function. I. Diff. II. Diff. Ill Diff. -" /(a-) a /(a) /( 3") y"( a ) INTEEPOL ATION. 113 Comparing these expressions for the differences with the above, we get ff=if"(a + i0, 5 = if '(a), A =f (a + i) - if' (a) - if" (a + $), which, from the manner in which the differences are formed, give C= ft (/" (a + ) -/" (a)), B = if (), 4 -/(a + ) -/(a) - if' (a) - ft (/" (a + ) -/" (a)). To find the value of the function corresponding to the argument a + \(o } we have n = \, and, from (116), /(a + =/( a ) + * J. Substituting in this the values of -4, B, and (7, last found, and re- ducing, we get /(a + = i (/(a + ) +/()) - I (' (/" (a + ) + /" (a))), 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 + - I (^ (/" ( + (n + 1) ) +r ( + ))) 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 + ( + 1) +/ T (a + "))), 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 o) f 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 al 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 + 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 M, and hence the heliocentric longitudes by means of the equation tan (I & ) tan u cos i. Let these longitudes be denoted by I and I', the times to which they correspond by t and t', and the longitudes of the sun for the same times by O and 0'; then for the time t , for which the heliocentric longitudes of the planet and the earth are the same, we have the first of these equations being used when I 180 is less TIME OF OPPOSITION. 115 thau V 180 0'. If the time i^ differs considerably from t or t' t it may be necessary, in order to obtain an accurate result, to repeat the latter part of the calculation, using t for t. and taking t' at a small interval from this, and so that the true time of opposition shall fell 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 v - M= 2e s being the number of seconds corresponding to a length of arc equal to the radius, or 206264".8 ; and the value of v J/ 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 formula 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 means of the tables of common logarithms. Thus, we have If we put log tan x = i (log b log a), 116 THEORETICAL ASTRONOMY. we shall have log (a + i) = log a 2 log cos x, or log (a + 5) = 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 O) 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 (^ ) and sin (^ Q ), respectively, give two values of A cos /9, 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 O) is greater than sin (^ O), and that derived from the second equation when cos (\ O) is less than sin (\ Q). The value of J, if the greatest accuracy possible is required, should be derived from J cos /9 when ft is less than 45, and from J sin /9 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 \ (SI ' + > ) and tan ( SI ' o> ) ; then, if sin \ ( ' + to ) is greater than cos(' + w ), we find cos^' from the first equation; but if sin \ (SI ' + o ) is less than cos \ (& ' + w ), we find cos \i' from the second equation. The same principle is applied in finding sin \i' by means of the third and fourth equations. Finally, from sin \i' and cos \i' we get tan Ji', and hence V. The check obtained by the agreement of the values of sin \i' and cos \i' t with those computed from the value of V derived from tan \i' t does not absolutely prove the calculation. This proof, however, may be obtained by means of the equation sin i' sin &' = sin t sin SI, or by sin i' sin ta = sin e sin SI 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 FORMULA. 117 CHAPTER II. INVESTIGATION OF THE DIFFERENTIAI, FORMULA WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATION OF THE ELEMENTS OF ITS ORBIT. 44. IN many calculations relating to the motion of a heavenly body, it becomes necessary to determine the variations which small increments applied to the values of the elements of its orbit will 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 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. JHf being the mean anomaly at the epoch T. Let 0' denote the value of this co-ordinate as derived directly or indirectly from observation ; then, if we represent the variations of the elements by ATT, A&, At, &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 ^ i " + rf8 4 +*- a< do .do . The differential coefficients -=-, ^ , &c. must now be derived from 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 *=/(, y,), or , a dO dO . dO , do = -j- dx + -r- dy + -, dz. dx * dy y ' dz Hence we obtain dO__dO_ dx_ dO_ dy_ dO_ dz dn ~ dx ' ~ an d -i are given by the equations (7), and dr dv dr dv dr dv f dr dv dr dv dr . dv , when the expressions for -T-, -r-, -rrr> -TTT -r~ and have been dy d

= tan ^.Etan (45 -f ?), and differentiate, we find dv dE _ _ d

cos E} dip. (IB") 124 THEORETICAL ASTKONOMY. sin v cos? , -r, cosv+e , , Isow, since sm E= , and cos K=^, - we shall have 1 + e cos v I -f- e cos v _, ae cos sin 2 v a cos

. (l4t) a Further, we have M= T being the epoch for which the mean anomaly is M w and _ Differentiating these expressions, we get and substituting these values in the expressions for dr and dv, we have, finally, dr = a tan

an d ~r> which are rfT dg de dT dq de required in finding the differential coefficients of the heliocentric co- ordinates with respect to the elements T, q, and e, these quantities being substituted for M , p, and S + / 5 O (1 - )' + Ac. (29) If it is required to find the expression for -7- in the case of the variation of the elements of parabolic motion, or when 1 e is very small, we may regard the coefficient of 1 e as constant, and neglect terms multiplied by the square and higher powers of 1 *- e. By differentiating the equation (29) according to these conditions, and regarding u and e as variable, we get = (1 + O du Q X iw 6 ) de; and, since du = |(1 + w 2 ) dv, this gives The values of the second member, corresponding to different values of Vy may be tabulated with the argument v; but a table of this kind is by no means indispensable, since the expression for -j- may be changed to another form which furnishes a direct solution with the same facility. Thus, by division, we have and since, in the case of parabolic motion, this becomes (31) If we differentiate the equation ^ 1 -{- e cos i/ regarding r, v, and e as variables, we shall have dr 2r* sinVv r'esint; dv 132 THEORETICAL ASTRONOMY. In the case of parabolic motion, e = l, and this equation is easily transformed into 7 / 7. \ (33) Substituting for -- its value from (SI), and reducing, we get (34) The equations (31) and (34) furnish the values of and to be used in forming the expressions for the variation of the place of the body when the parabolic eccentricity is changed to the value 1 -f de. When the eccentricity to which the increment is assigned differs but little from unity, we may compute the value of -7- directly from equation (30). A still closer approximation would be obtained by using an additional term of (29) in finding the expression for -=- ; but a more convenient formula may be derived, of which the numerical application is facilitated by the use of Table IX. Thus, if we differ- entiate the equation v = V+ A (lOOi) + B (1000 2 + (7(100*7, regarding the coefficients A, B, and C as constant, and introducing the value of i in terms of e, we have dt> = dF__200J_ ._ 4 iA_ (100i) _ 600(7 in which s . 206264.8, the values of A, B, and C, as derived from the table, being expressed in seconds. To find , we have de k(i-T)VT+l which gives, by differentiation, k(t p de dV and if we introduce the expression for the value of M used as the argument in finding V by means of Table VI., the result is DIFFERENTIAL FORMULA. 133 Hence we have dv JfcoafjF 200,1 400J? 6000 (1 ~- (10(>l/ ' ( by means of which the value of is readily found. When the eccentricity differs so much from that of the parabola that the terms of the last equation are not sufficiently convergent, the expression for , which will furnish the required accuracy, may be derived from the equations (75) 1 and (76) r If we differentiate the first of these equations with respect to e, since B may evidently be regarded as constant, we get dw__ 9 de~ ' If we take the logarithms of both members of equation (76) n and differentiate, we get dv _ dC . dw _ 4de _ ,g-^ smv ~" ~C + slnw ~~ (1 + e) (I -f- 9e)' To find the differential coefficient of C with respect to e, it will be sufficient to take which gives The equation gives cos'^w= O'cos'^v. Then we shall have The equation ? = (1 + A C") cos 1 = gives ^ = (1 + H) cos* > 1= (7 cos 4 w. Hence we derive , . c iinv = NUMERICAL EXAMPLES. 135 If we substitute this value in equation (39), and put C 2 (1 -f e) = 2, are get ^?_ _ ?_ kl/p 8 tan | de 2(1 + 9e)' r* (l + e)(l + 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, 3 = 4 42' 21".56, log J = 0.2450054, logr = 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., logy = 9.511920, logs = 8.077315, e = 23 27' 24".0, tT= 420.714018. First, by means of the equations (4), we compute the following values- log cos 8 -^ = 8.054308, log ~ = 8.668959., log cos 8 jL = 9.754919., log ^ = 6.968348., log ~ = 9.753529. Then we find the differential coefficients of the heliocentric co-ordi- nates, with respect to -, ft, f, v, and r, from the formulae (7), which give log -^ = log -^ = 9.491991,., log ^ = log ^ = 0.399496., log ^~ = log |i= 9.950466., = 7.876553, log -jj- = 8.830941, log l og _ = 8.726364, log -- = 9.687577, log - log _L == 9.996780., log -- = 9.083635, log -- = 7.649030. 136 THEORETICAL ASTRONOMY. In computing the values of -rr; -TT> and -p-, those of cos a, cos 6, *nd cos c may generally be obtained with sufficient accuracy from ein a, sin b, and sin c. 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 8-r- and -T-, we have, according to equa- tion (2), da, da, dx . .da, dy cos <5 -r- = cos 8 t- cos 8 - . ^-, (41 ) a* dx d* dy dx which give d8__ dd_ dx_ dd dy dd dz dx: dx' dx ~du ' ~dK ~*~ ~dz ' Ifa' cos 1L = cos 9%- = + 1.42345, d8 = d8_ = _ an av an dv In the case of &, i, and r, we write these quantities successively in place of ic in the equations (41), and hence we derive cos 8 -^~ = 0.03845, ~- = 0.09533, cos 8 ^T = - ' 27641 ' -^T = - 0.78993, COS *~W = ~ - 08020 > ^ = + -04873. Next, from (16), we compute the following values: log % =: ' 179155 ' l S ^ = 9 - 57 ?453, log * = 2.376581., log * = 0.171999, log ^ = 9.911247, log * = 2.535234. We may now find ~, -^, & c . by means of the equations (11), and thence the values of cos 8^, ~ & c .; but it is most convenient to derive these values directly from cos<5-^-, cos<5 , , and , in connection with the numerical values last found, according to the NUMERICAL EXAMPLES. 137 equations which result from the analytical substitution of the expres- ., dx dy dz sions lor -^-, ^- ^ , &c., in equation (2), writing successively (p, 3/ , and n in place of TT. Thus, we have , da .dadr, t da dv cos 5 = cos S u cos 3 d dr d dv d d

= + 507.264, -|L = - 179.315. Therefore, according to (1), we shall have coeaA=-J-1.42345Aff0.03845A 0.27641 At +1.99400^^ + 1.13004A3/ + 507.264A.a, A Mean Equinox 1864.0 t = 4 37 .51 J

log ~ = 0.186517., log ^ = 0.186517.. If we wish to obtain the differential coefficients of v and r with respect to log 3 instead of q, we have dv q dv dr q d r dlogq A O dq dlogq ^ dq in which ). is the modulus of the system of logarithms. Then we compute the value of -j- by means of the equation (30). (35), (39), or (40). The correct value as derived from (39) is = 0.24289. de The values derived from (35), omitting the last term, from (40) and from (30), are, respectively, 0.24440, 0.24291, and 0.23531. The close agreement of the value derived from (40) with the correct value is accidental, and arises from the particular value of v, which is here such as to make the assumptions, according to which equation (40) is derived from (39), almost exact. Finally, the value of -^- may be found by means of (32), whicJi gives J = + 0.70855. ae When, in addition to the differential coefficients which depend on the elements T, q, and e, those which depend on the position of the orbit in space have been found, the expressions for the variation of the geocentric right ascension and declination become NUMERICAL EXAMPLES. 141 COS <5 Aa = COS S -? A* + COS 3 ~ A & 4- COS 5 -? Ai + COS 5 A I 7 a- d& dt dT _ c?a c?o -f- COS A cosi)dSl + rs i nwsm & sinidi, dy 1 = dr r (sin u sin & cos u cos SI cos f) du -\- r (cos u cos SI sm u sin & cos i) dSl r sin u cos & sin i di, (46) d/ = - dr + r cos u sin i dw -f r sin w cos i di, in which #', y', z' are the heliocentric co-ordinates of the body in reference to the ecliptic, the positive axis of x being directed to the vernal equinox. Let us now suppose the place of the body to be referred to a system of co-ordinates in which the ecliptic remains as the plane of xy, but in which the positive axis of x is directed to the point whose longitude is & ; then we shall have dx = dx' cos SI + di/ sin SI , dy = dx' sin & + dy' cos SI, dz = dz', and the preceding equations give dx = -dr r sin u du r sin u cos i d SI , r dy = y - dr + r cos u cos i du -4- r cos u d& r sin u sin i di, (47) dz = - dr -\- r cos u sin i du + r sin u cos i di. M4 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 n Z, denote the heliocentric co-ordinates of the earth referred to the same system of co-ordinates, and we have x + X, = A cos /9 cos (A ), z + Z, = /f sin/9, in which ^ is the geocentric longitude and /? 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 A sin/9cos(A &)d/5 A cos/? sin (A -^ r =- - sin w cost, -=_ = Bin sint; dx _dx _ . dy dy dz dz ~~ Smu ' ~ ===rCOSUCOsi ~ = = DIFFERENTIAL FORMULA. 145 = -rs n cos, ;-=> CO.* =0 ; (49) dx A dy . . dz -^T- = 0, -^- = r sinwsmi, p-=r smw cost. If we introduce >r, the longitude of the perihelion, we have du = dv -f- d* dft, and hence the expressions for the partial differential coefficients of the heliocentric co-ordinates with respect to TT and SI become dx dy dz r- = rsmw, f- = r cost* cost, j = r cosw sint; dx dr. dx ' dx dy n dz ~ - 2r sm u sin* U, - T ^ r = 2r cos u sin 1 it, -y - = r cos w sin t. aft aft 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, -^ = 0, -~ = r sin u sin i, -p- = rsinwcosi; (51) d% di di and the expressions for -p, -7^, -j, -^r, &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 and hence dx dy dz . . = rsmw. -r- = r cos u cos i, ^ = rcoswsmt; * * * (52) -- = 2r sin u sin 2 , -- = 2r cos u sin 2 \i, -- = rcosu 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 t, 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 A and with respect to r, t>, ft, i, and a> or T, by writing successively A and ft in place of 0, and ft, t, &c. in 10 146 THEORETICAL ASTRONOMY. place of TT 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 ^ and /9 with respect to these elements. The numerical application, however, 18 facilitated by the introduction of certain auxiliary quantities. Thus, if we substitute the values given by (48) and (49) in the equations dl _ (ft dx - eM dy cos ft -r- = cos p -, =- + cos 6 A dv dx dv ~ dy dv dp__d]3_ dx_ d dy_ d dz_ dv dx dv dy dv dz ' dv' and put cos i cos 0* &) = A sin A, Bin (J- 8) = 4, COB 4, ami = nainN, sin (A % ) cos i = n cos N, in which A and n are always positive, they become /7/9 *7# ~dv = d^ = -^ (sin cos (A )sinw -f n cos u sin (JV + /9) ). Let us also put n sin (N + /9) = 5 sin ^, , _., sin /? cos (-i Q, ) = cos B, and we have ~~dv~ = The expressions for cos/9 and ^j- give, by means of the same auxiliary quantities, In the same manner, if we put DIFFERENTIAL FORMULA. 147 cos (A ft) = C sin C, cos i sin (X ft) = C cos C; cos i = D sin D, sin 0* ft) sin i = D cos D; we obtain ett r (58 > T^ = -jsini smtt cos(A ft), If we substitute the expressions (55) and (56) in the equations eW O dl dr , O dl dv cos /? -y = cos /? -j- -= -- f- cos y5 -= -j , d? dr dy ' dw d^ ^. = ^..^?L_i_^l *!. d^ ~ ^ dy dv d, we must put dr dv de' de dr __ * (63) dr j dv -r-, hcosH=r-j-, dq' dq and the equations become 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 ^ and /9 with respect to any other elements which determine the form of the orbit may be computed. NUMERICAL EXAMPLES. 148 In the case of a parabolic orbit, if the parabolic eccentricity is supposed to be invariable, the terms involving e vanish. Further, in the case of parabolic elements, we have . ~ dr k sin v , dv g sin Cr = ~j~m == - = T tan ~>v , d J? ~Y 2o ~ d T dv which give tan G = tan ^v. Hence there results G = 18Q%v, and g = k\~, which is the expression for the linear velocity of a comet moving in a parabola. Therefore, fcl/2 , For the case in which the motion is considered as being retrograde, 180 i must be used instead of i in computing the values of A w A, n, N, C , and (7, and the equations (55), (56), and the first two of (58), remain unchanged. But, for the differential coefficients with respect to i, the values of D and D must be found from the last two of equations (57), using the given value of i directly ; and then we shall have cos /9 -TT = sin t sin u cos (A & ), 55. EXAMPLES. The equations thus derived for the differential coefficients of A and ft with respect to the elements of the orbit, referred to the ecliptic as the fundamental plane, are applicable when any other plane is taken as the fundamental plane, if we consider ^ and /9 as having the same signification in reference to the new plane that they have in reference to the ecliptic, the longitudes, however, being measured from the place of the descending node of this plane on the ecliptic. To illustrate their numerical application, let it be required to find the differential coefficients of the geocentric right ascension and declination of Eurynome with respect to the ele- ments of its orbit referred to the equator, for the date 1865 February 24.5 mean time at Washington, using the data given in Art. 41. 150 THEOEETICAL ASTRONOMY In the first place, the elements which are referred to the ecliptic must be referred to the equator as the fundamental plane ; and, by means of the equations (109)^ we obtain ' = 353 45' 35".87, **= 19 26' 25".76, > 9 = 212 32' 17".71, and <' = + = 50 I cos 5-- = + 507. 25, -- = 179.34: and hence cos <5 Aa = -f 1.4235 Aw' + 1.5098 Aft' + 0.0067 Ai' + 1.9940 Af + 1.1300 AJ/ O + 507.25 AJU, A di sin u> sin i dft . Eliminating di from these equations, and introducing the value sin i' _ sin ft sini sin ft'' the result is If we differentiate the expression for cos a> derived from the same spherical triangle, and reduce, we find d cos i cos ft sin (o^ The equations (68), (69), and (70) give the partial differential co- efficients of ft, i, and w with respect to ft 7 and V, 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 AI = sin w sin i' A ft ' -f cos u> AI', &to Ao/ Atw . If we apply these formulae to the case of Eurynome, the result is A. O = 4.420Aft' -f 6.665 Ai', A & = 3.488A ft' -f 6.686Ai', At = 0.179Aft' 0.843Ai' ; DIFFERENTIAL FORMULAE. 153 and if we assign the values A &' = - 14".12, A* = - 8".86, W = - 6".64, we get AOI O = + 3".36, A & = 10".0, Ai == + 10".0, AW = 10".0, and, hence, the elements which determine the position of the orbit in reference to the ecliptic. The elements a/, &', and V 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 SI and &' and 180 V and i. 56. If we refer the geocentric places of the body to a plane whose inclination to the plane of the ecliptic is i } and the longitude of whose ascending node on the ecliptic is SI, 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 e , y', z' be the heliocentric co-ordinates of the body referred to a system of co-ordinates for which the plane of the unchanged orbit is the plane of xy, the positive axis of x being directed to the 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 to the point whose longitude is SI- Then we shall have dz' = dx, dy 1 = dy cos i + dz sin i, dz' = dy sin i -f- dz cos i. Substituting for dx, dy, and dz their values given by the equations (47), we get dy! = dr r sin u du r sin u cos i d&, t/ dtf = L dr + r cos u du -f r cos u cos i d&, dz' = - dr r cos u sin i d& -j- r sin u di. It will be observed that we have, so long as the elements remain unchanged, a/ = r cos u, y l = rsinu, sf = 0, 154 THEORETICAL ASTRONOMY. and h.jnce, omitting the accents, so that x, y, z will refer to the plane of the unchanged orbit as the plane of xy, the preceding equations give dx = cos u dr r sin u du r sin u cos i dl , dy = sin u dr -f- r cos u d u -f- r cos w cos i dSl , dz = r cos u sin i d& + r sin w <&. The value of < is subject to two distinct changes, the one arising fioin 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 ?- S T L =; d, * d, d^ * (?5) _^L = 0, -S- = 0, ^-^ rcoswsbx; . " -=- = 0, 77- = r sm w. di di di Substituting the values thus found, in the equations dd do dx , d0 dy dy dx dy dy dy dz dx dv dy dv dz d*' 1 156 THEORETICAL ASTRONOMY. we get ^^ n , sn dv dx d In a similar manner, we derive ~ = -isin(0- W ), -^L = - (77) , 0, A-= + If we introduce the elements dr d dv dy>' if we introduce also the auxiliary quantities /and F, as determined by means of the equations (59), -- t* .F). (78) d

f n TT-N dlj h . , -f-f^. cos ij ; = -7 cos (0 u H), r- = -T sin w sm (0 u H ). d/j- a o/t J If we express r and v in terms of the elements T, q, and e, the values of the auxiliaries /, g, h, F, &c. must be found by means of (64); and, in the same manner, any other elements which determine the form of the orbit and the position of the body in its orbit, may be introduced. The partial differential coefficients with respect to the elements having been found, we have dO do dO do COS 1) A0 = COS i) -j- *X + COS 1? -r- A? -f COS 1} AM. + COS TJ A/, dx df dM dfj. 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 r t and d from those of X and ,3. Let x 0f y w 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 ft ; and let x r , y f , z f 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 a- = J cos p cos 0* ft ), ' = -1 cos r t cos 9, y,= d cos ,3 sin (<* ft ), y ' = A cos r t sin 0, z, = J sin ,3, z,' = J sin r h and also z ' = y sin i + z cos i. Hence we obtain COST) COS = COS, 3 COS 0* ft), cos 7 sin = cos /5 sin (^ ft ) cos i + sin ,3 sin ?', (80 ; sin ij = cos ,3 sin (J ft ) sin i + sin ,3 cos i. These equations correspond to the relations between the parts of a spherical triangle of which the sides are i, 90 7, and 90 /9, the angles opposite to 90 r t and 90 being respectively 90 -f (J ft) and 90 d. Let the other angle of the triangle be denoted by 7-, and we have cos T) sin r = sin * cos (J ft), ,gj cosi7cosr = i3inisin(>l ft ) sin p -f cos i cos p. The equations thus obtained enable us to determine 57, 6, and f from i and /?. Their numerical application is facilitated by the intro- duction of auxiliary angles. Thus, if we put -ax < 82) 158 THEORETICAL ASTRONOMY. in which n is always positive, we get cos Jj cos e = cos 13 cos (A ft), cos i) sin 6 = n cos ( N t), (8 3) =nsin(Ni\ from which ^ and may be readily found. If we also put n' sin N' = cost, ,,, n' cos ^T = sin i sin (A ft ), we shall have cot N' = tan i sin 0* ft), If f is small, it may be found from the equation 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 rin (45 - &) sin (45 - $ (0 + r )) = cos(45 H- i (i - ft)) sin (45 - W + i);, sin (45 - ,)' cos (45 - (0 + r )) = sin (45 + (* ft)) sin (45 - (j9 t)), cos (45 - ,) sin (45 - (* r)) = (87) cos (45 + { (X - ft )) cos (45 $(fi + i)), cos (45 - ^) cos (45 $ (0 r )) = sin (45 + ^C; - ft )) cos (45 - (0 - 0), from which we may derive 37, 0, and f. When the problem is to determine the corrections to be applied to the elements of the orbit of a heavenly body, in order to satisfy givei. observed places, it is necessary to find the expressions for cosjy A0 and &q in terms of cos/9 A^ and A/9. If we differentiate the first and second of equations (80), regarding ft and i (which here determine the position of the fundamental plane adopted) as con- stant, eliminate the terms containing dy from the resulting equations, and reduce by means of the relations of the parts of the spherical triangle, we get NUMERICAL EXAMPLE. 151 cos TJ dd = cos f cos ,9 dl -\- sin 7 d/9. Differentiating the last of equations (80), and reducing, we find dy = sin f cos /9 dl -f cos f dp. The equations thus derived give the values of the differential co- efficients of 6 and y with respect to ^ and p ; and if the differences A/ and & t 3 are small, we shall have cos TJ A0 = cos ^ cos ft AA -f sm 7" A & Aiy = sin 7- cos ,9 AA -(- cos f A,9. The value of f 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 r t 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 37 and 6 from a and S. For this purpose, it is only necessary to write a and 3 in place of A and ,?, respectively, and also Q, ', ', ft/, %', and u' in place of , i, a, %, and u, in the equations which have been derived for the determination of ^ 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 37 A# and &y in terms of the variations of the elements in the case of the example already given; for which we have / = 5010'7".29, ' = 35345'35".87, t*=19 26' 25".76. These are the elements which determine the position of the orbit of Eurynome . referred to the mean equinox and equator of 1865.0. We have, further, log/= 0.62946, Iog0 = 0.34593, log h = 2.97759, F=33914'0", G = 350 11' 16", H = 14 30' 48", u' = 179 13' 58". In the first place, we compute >;, 6, and f by means of the formulae 160 THEORETICAL ASTRONOMY. (83j and (85), or by means of (87), writing a, d, ft', and i' instead of )>, , ft, and i, respectively. Hence we obtain * = 188 31' 9", i) = 1 59' 28", r = 19 17' 7". Since the equator is here considered as the fundamental plane, the longitude d is measured on the equator from the place of the ascend- ing node of the orbit on this plane. The values of the differential r>oefficients are then found by means of the formulas d6 d-n r ... . 7 = 0, -7^7 = cos TI sin i cos u , d0 o ^ I - di' di' A do r f ,, di] r (tit a f , .^T. di) 008?-^ = * cos (* ' GO, -r~ = - CLJtl A uM which give do di) cos i) -- = 0, -^- = 4- 0.0204, cos i) ~ = 4- 1.5051, -^- = 4- 0.0086, cos >j - = + 2.0978, -^- = 4- 0.0422. cos 7 - = + 1.1922, -^- = 4- 0.0143, cos i} -^- -f 538.00, -&. = 1.71. Therefore, the equations for cos TJ A^ and A^ become cos 13 A0 = 4- 1.5051 A/ + 2.0978 A? 4- 1.1922 Ajtf + 538.00 AJ AT = 4- 0.0086 A/ 4- 0.0422 A? 4- 0.0143 *M 9 1.71 A/I 4- 0.6072 Aft' 4- 0.0204 Ai'. If we assign to the elements of the orbit the variations DIFFERENTIAL FORMULA. 161 A/ = -6".64, Aft' = -14".12, At*:=-8".86, Af = + 10", Ajf. = -f 10", A/* = + 0.01, we have A/ = Ao*' + cos H A ft' = 19".96 ; and the preceding equations give cos i) A0 = + 8".24, A, = 6".96. With the same values of AO/, Aft', &c., we have already found cos 3 Aa = + 5".47, A<5 = 9".29, which, by means of the equations (88), writing a and 3 in place of >l and /9, give cos 7 A0 = -f 8".23, A 7 = 6".96. 59. In special cases, in which the differences between the calcu- lated and the observed values of two spherical co-ordinates are given, and the corrections to be applied to the assumed elements are sought, it may become necessary, on account of difficulties to be encountered in the solution of the equations of condition, to introduce other ele- ments of the orbit of the body. The relation of the elements chosen to those commonly used will serve, without presenting any difficulty, for the transformation of the equations into a form adapted to the special case. Thus, in the case of the elements which determine the form of the orbit, we may use a or log a instead of p, and the equation kVl -f m ^ '~^~ gives (89) in which 1 is the modulus of the system of logarithms. Therefore, the coefficient of A^ is transformed into that of A log a by multiply- ing it by f j; and if the unit of the mth decimal place of the loga- rithms is taken as the unit of A log a, the coefficient must be also multiplied by 10~ m . The homogeneity of the equation is not disturbed, since p. is here supposed to be expressed in seconds. If we introduce logp as one of the elements, from the equation p = a cos" f 11 162 THEORETICAL ASTRONOMY. we get ^ ogp- i- P- o an? y>, or dfj. = | d logp 3/x tan

, the coefficients of A^> are changed ; and if we denote by cos 3 \ -^- } and I ~ } the values of \ a

I the partial differential coefficients when the element // is used in con- nection with (p, we shall have, for the case under consideration, .da / da \ M , da cos 8 -7 = cos d I -7 1 3 - tan

I s dfj. in which s = 206264".8. If the values of the differential coefficients with respect to // and

and -p: by d

l and p, respectively, will give the values of the differential coefficients of the heliocentric longitude and latitude with respect to x, y, and 2. Combining these with the expressions for the differential coefficients of the heliocentric co-ordinates with respect to the elements of the orbit, we obtain the values of cos 6 A and A& in terms of the varia- tions of the elements. The equations for dx, dy, and dz in terms of du, eZft, and di, may also be used to determine the corrections to be applied to the co-or- dinates in order to reduce them from the ecliptic and mean equinox of one epoch to those of another, or to the apparent equinox of the date. In this case, we have du = d* dft. When the auxiliary constants A, B, a, 6, &c. are introduced, to find the variations of these arising from the variations assigned to the elements, we have, from the equations (99) u cot A = tan ft cos i, cot B = cot ft cos i sin i cosec ft tan e, cot C = cot ft cos i -f- sin i cosec ft cot e, in which i may have any value from to 180. If we differentiate these, regarding all the quantities involved as variable, and reduce by means of the values of sin a, sin b, and sin c, we get cosi ,_ sin .4 . ,. . . ,. dA = . , c?ft . sin ft sm i di, sin a sma dB = . , . (cos i cos e sin i sin e cos ft ) dft sin 2 6 . sinl? , ^ ,. . sinisinft , - (cos ft sin ^ cos e -j- cos ^ sin e) di -\ r-^-r - as, dC= . , (cos i sin e + sin i cos e cos ft ) rfft sm 2 c sin C , , j. . sin i sin H -- -. - (cos ft sin i sm e cos i cos e) d^ -\ -- t sin c and these, by means of (101)^ reduce to 104 THEORETICAL ASTRONOMY. = ^~ d& sin A cot a di, sin* a . (91) Let us now differentiate the equations (101) w using only the upper sign, and the result is da = sin i sin A d& + cos A di, db = sin i sin JB d& -f- cos B di -f cos c cosec b ds, dc = sin i sin C d& -f- cos C di cos b cosec c ds. If we multiply the first of these equations by cot a, the second by cot 6, and the third by cote, and denote by ^ the modulus of the system of logarithms, we get dlogsina= A sin i cot a sin A d& -f <* cot a cos A di, d log sin b = ^ sin i cot b sin dQ -{- -* cot b cos B di -f A - r-^r de, d log sin c = A sin i cot c sin C dQ + ^o c t c cos Gdi A - r- 2 - de. (92) The equations (91) and (92) furnish the differential coefficients of A, B, C, log sin a, &c. with respect to , *, and e; and if the varia- tions assigned to SI, i, and e are so small that their squares may be neglecitd, the same equations, writing &A, A&, At, &c. instead of the differentials, give the variations of the auxiliary constants. In the case of equations (92), if the variations of &, i, and e are ex- pressed in seconds, each term of the second member must be divided by 206264.8, and if the variations of log sin a, log sin 6, and log sine are required in units of the mth decimal place of the logarithms, each term of the second member must also be divided by 10. If we differentiate the equations (81) 1} and reduce by means of the same equations, we easily find cos b dl = cosi sec b du -f cos b d& sin b cos (I &) di, /QOX db = sini cos(l 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 FORMULAE. 165 the sun, and considering X, /?, J, and O as variables, we derive, after reduction, cosfidl = ^ cos (A O) dQ, R (94) dp = j sin sin (A Q ) d0 , which determine the variation of the geocentric latitude and longitude arising from an increment assigned to the longitude of the sun. Ii appears, therefore, that an error in the longitude of the sun will produce the greatest error in the computed geocentric longitude of a heavenly body when the body is in opposition. 166 THEORETICAL ASTRONOMY. CHAPTER III. INVESTIGATION OF FORMTJUE FOB 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 always be done in the first approximation, the perturbations not being considered, three complete observations will furnish the six equations necessary for finding these unknown quantities. Hence, the data required for the determination of the orbit of a heavenly body are three complete observations, namely, three observed 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 _ xp cos ?>' sin (o 0) A cos d tan

' sin )' R sin 0', z=/otan/9, J=p't&np, x" = p" cos x" R" cos ", f = p"sml"-R"sinQ", 2"= /'tan ,5", 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 f , &c. in the equations (4) and (5), they become = n (p cos A" R cos ) (p cos /' R 1 cos ') + n"G>"cosA" E"cos"), = n(>smA- sinO) (//sin A' #sin') (6) -t-n'V'Binr jr'sinO"), = np tan ft p' tan ff + ri'p" tan ". 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, n, n", p, 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 THEORETICAL ASTRONOM*. The determination, however, of n and n" to a sufficient degree of accuracy, by means of the intervals of time between the observations, requires that p' should be approximately known, and hence, in general, it will become necessary to derive first the values of n, n", and p' ' ', after which those of p and p" 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 f , 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 = wOcos(A O') #cos(O' 0)) 0' cos(A' O') #) + n"( P " cos (A"- Q') - R" cos(0"- ')), 0:=n(>sin(A O')+^sin(O' -O)) /sin(A' ') + n" ( P " sin (A" - Q ') _ R" sin ( Q " O ')), (7) = n(p sin (A' A) + R 8 in(0 A')) R' sin (O' A') n" ( P " sin (A" A') R' sin ( Q " A')), = np tan /9 p' tan ft -\- n"p" tan ft'. If we multiply the second of these equations by tan/3', and the fourth by sin (A' O'), and add the products, we get = ri'p" (tan ft sin (A" Q ') tan /9" sin (A' Q ')) O')tan^+n/>(tan/9'sm(A O') tan/3sin(A' O')) (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' b;r [_RR'~\, and we shall have ' O), and similarly [_RR"-\ = RR' gin (O " O ), = RR" sin(0"- O'). ORBIT OF A HEAVENLY BODY. 17l Then, if we put w= l*T. *-=&$. (9) we obtain ^"sin(0"-0')-: J Rsm(0'-0)^ / . Substituting this in the equation (8), and dividing by the coefficient of p", the result is _ n tan p sin (A Q ') tan ft sin (A' ') P ~ P n" ' tan p' sin (A' ') tan/5' sin (A" ') JL_^\ _ _ n" jV" / tan /S" sin (A' ') tan /S' sin (A"')' Let us also put tan p sin (A Q ') tan /9 sin (A' Q') tan ft" sin (A' Q') _ tan p sin (A" 0')' 8in(0 y _ tan A" sin (A' Q') tan p sin (A" Q') ' and the preceding equation reduces to (n) We may transform the values of J/' 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' 0') tan w' tan p. (12) Let /9 , /?/' be the latitudes of the points of this circle corresponding to the longitudes A and X", and we have, also, tan /? = sin (A ') tan w', Qgx tan fl," = sin (A" ') tan w'. Substituting these values for tan/9', sin (A O') and sin (A" in the expressions for M ' and M ", and reducing, they become ,_ sin (/?/?) cos p' cos /?" " sin (p r /V) ' cos /9 cos /? ' (u ^ cosp' 172 THEORETICAL ASTRONOMY. When the value of has been found, equation (11) will give the relation between p and p" in terms of known quantities. It is 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, P = ft and jl"= j9 ", we shall have H'= ^ and M"= CD. In this case, therefore, and also when /3 /3 and /5" /9 " 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' 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 Q* 00 = sin(Q' Q) sin (A" 0')' sin (A" ') ' 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' 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) sin (X 1 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' 0'), and add the products, we obtain == n"p" (tan p cos (A" ') _ tan /?" cos (A' ')) n'72"tan/S'cos(0" ') +n/>(tan/9'cos(A ') tan cos (A' 00) nR tan p cos (' ) + E' tan ^, from which we derive ORBIT OF A HEAVENLY BODY. 173 p P n" tan ft" cos (A' Q') tan ft' cos (/' O ') (16) tan ft' cos (A Q ') tan ft cos (A' Q') tan ft" cos (A' Q') tan ft' cos (A" Q ' ^"tan/3'cos(0" 0') + ^7^tan/S / co8(0'-0)-^ 7 tan ft" cos (A' Q ') tan ft' cos (A" ') Let us now denote by /' 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 O' 90, which will therefore be the longitude of its ascending node, and we shall have cos (A' Q') tan /' = tan ft' ; (17) and, if we designate by /3, and ft,, the latitudes of the points of this circle corresponding to the longitudes \ and X", we shall also have tan ft, = cos (A Q') tan I', (lg) tan ft,, = cos (A" Q') tan /'. Introducing these values into equation (16), it reduces to _ n_ sin (ft, ft) cos ft" cos ft,, P ~ P n" ' sin (ft" /?) ' cos ft cos ft f (19) 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 ft" 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 k(f f)=*v,' (20) and express the values of x, y, z, x", y", z" in terms of x f , y', z' by expansion into series, we have dx' r" , 1 fflaf r"* 1 dV r" 8 , * = x --dt'-k + iL2'W!s-T33'-dJ'- + &c " dx' r 1 c2V r 1 dV and similar expressions for y, y", z, and z". We shall, however, take the plane of the orbit as the fundamental plane, in which case 2, z' t and z" vanish. The fundamental equations for the motion of a heavenly body relative to the sun are, if we neglect its mass in comparison with that of the sun, If we differentiate the first of these equations, we get W = "r 7 *" "dt~ r' 3 ' ~di' Differentiating again, we find Writing y instead of x, we shall have the expressions for -jt and 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 r" r" r"* we obtain =*-*% ,=^-6^, From these equations we easily derive jx -x'y = The first members of these equations are double the areas of the triangles formed by the radii-vectores and the chords of the orbit between the places of the comet or planet. Thus, y' x _ a / y=[ ^ l jf' a >- a >y=[r'r"], y" x - x "y = lrr"l (24) and sidy 9 y'dx' is double the area described by the radius-vector during the element of time dt, and, consequently, - -j- - is double the areal velocity. Therefore we shall have, neglecting the mass of the body, in which p is the semi-parameter of the orbit. The equations (23 ), therefore, become [rV] = bk y'p, [tV] = b"k Vp, [r/'] = (ob" + o"6) k i/p Substituting for a, 6, a", b" their values from (22), we find, since 176 THEORETICAL ASTRONOMY. T" r" 3 dr' From these equations the values of n ^ TFT and n" = ,. -. may he derived ; and the results are + C+"H-^''"-' ^ \ zvi jT- 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. T TV being multiplied by the very small quantity j-, is reduced to u 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), (27) From the equations (26) or from (25), since , = . we find -r"* ^ r"' dr' Since this equation involves r' and -^-, it is evident that the value n of -TJ, in the case of an orbit wholly unknown, can be determined 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 ORBIT OF A HEAVENLY BODY 177 and similarly N r W'^7'' Hence the equation (11) reduces to P" = ~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 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 n N and compute the values of N and N" by means of equations (9), since, according to (27) and (28), if r f does not differ much from E' t the error of this assumption will only involve terms of the third order, even when the values of r and r" differ very much. Whenever the values of p and p" can be found when that of their ratio is given, we may at once derive the corresponding values of r and r", as will be subsequently explained. The values of r and r" may also be expressed in terms of r' by means of series, and we have from which we derive -. k dt' neglecting terms of the third order. Therefore dr' = k (r" r} . dt r+r" '* 12 (30) 178 THEORETICAL ASTEONOMY. and when the intervals are equal, this value is exact to terms of the fourth order. We have, also, which gives (31) Therefore, when r and r" have been determined by a first approxi- mation, the approximate values of r' and -57 are obtained from these equations, by means of which the value of may be recomputed from equation (28). We also compute N - ,. ' and substitute in equation (11) the values of and ==?, thus found. If we designate by M the ratio of the curtate distances p and p" t we have n 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 f ~. and that of j=> 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 7 , as derived from the first of equations (10), is either indeter- minate or greatly affected by errors of observation, the equations (15) and (16) must be employed. The last terms of these equations may be changed to a form which is more convenient in the approximations to the value of the ratio of p" to p. Let F, Y' t 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 =K sin(0 -A'), Y' =R' sin(' /O, Now, in the last term of equation (15), it will be sufficient to put n N n" and, introducing F, Y f , F", it becomes Y ~ Y ' + Y " cosec (A "~ v. 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), NY Y' and the foregoing expression reduces to since F' = 5' sin (O r ^)- Hence the equation (15) becomes Ql'-Q') ' (36) n sin (*' *) ^'./l 1 \#sinQl'-Q') 180 THEORETICAL ASTRONOMY. If we put n sn (*'-0') Rll we have (37) Let us now consider the equation (16), and let us designate by X, X', X" the abscissas of the earth, the axis of abscissas being directed to that point of the ecliptic for which the longitude is O', then X = yri _ TV It will be sufficient, in the last term of (16), to put n __ N n" ~ N" ' and for its value in terms of N" as already found. Then, since this term reduces to .^ (1 1\ s T" ^ J \7* R 3 / tan ft" cos (/ ') tan p cos (A" ') ' and if we put = _re_ tan /?' cos (7i Q') tan ^ cos (K Q') ^ ~ n" ' tan p" cos (^-' Q') tan /?' cos (A" Q')' (38) F' i i^l r 'j. "i/-!- -^^ _ tan/3' _ J?' ' n ' r" ^ r " 1 " ' Hr' s ~ E' 3 ) tan/3'cos (A Q') tan /3 cos (A' Q') "7^ the equation (16) becomes (39) In the numerical application of these formulae, if the elements are not approximately known, we first assume n r ^' = 7 when the intervals are nearly equal, and ORBIT OF A HEAVENLY BODY. 181 n__ N ri''N*'' as given by (32), when the intervals are very unequal, and neglect the factors F and F f . The values of p and p" which are thus ob- tained, enable us to find an approximate value of r', and with this a more exact value of may be found, and also the value of F or 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 r . 66. When the value of M has been found, we may proceed to determine, by means of other relations between p and />", the values of the quantities themselves. The co-ordinates of the first place of the earth referred to the third, are x, = R" cos 0" R cos Q, ,.Q. y, = jR" sin 0" ^sinQ. 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 t = g sin G, and, consequently, .R"cos(O" 0) E = gcos(G 0), (41 ^ E" sin (0" 0) = g sin ( G 0). 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 4 cos /9 cos (A 0), sin 4. cos w = cos p sin (A O), sin 4 sin w = sin & 182 THEORETICAL ASTRONOMY. from which tan/9 tan 4- = sin 01 0)' ( tanOl-0) Since cosj9 is always positive, cos-v// and cos (A 0) must have th same sign; and, further, ^ cannot exceed 180. In the same manner, if w" and fy r represent analogous quantities for the time of the third observation, we obtain 3 in (/' _ 0")' cos V = cos ft' cos (A" "). We also have r= which may be transformed into r f =0'sec/9 JBcos^' + JPsin^; (44) and in a similar manner we find r" 2 = 0" sec /9" 12" cos +")* -f U" 2 sin 2 4". (45) Let x designate the chord of the orbit of the body between the first and third places, and we have X' = (* - x? + (f - But 2 = /> COS A .R COS , jr = /> sin A J? sin , z = p tan /?, and, since ^''^ Jtf/>, x"= M P cos A" ST cos w , y* = Mp sin A" 12" sin ", from which we derive, introducing g and G. d' x = Mp cos A" /) cosA y" y = Mp sin A" /> sin A g sin G, s!' z=Mp tan /S" /> tan /9. Let us now put OEBIT OF A HEAVENLY BODY. 183 Mp cos x" pco8l = ph cos C cos H, Mp sin x" p sin -i = ph cos C sin 2T, (46) Jtf/> tan/S" p tan/5= ^A sin C. Then we have tf' x = ph cos C cos IT $r cos G, y" y = ph cos C sin .H" <7 sin G, z" z = /A sin C. Squaring these values, and adding, we get, by reduction, x> = pW 2 g phcos:cos(G H) + g>; (47) and if we put cos : cos ( G jff) = cos f , (48) we have x 1 = (ph g cos p) 1 + 0* sin 1 p. (49) If we multiply the first of equations (46) by cos A", and the second by sin A"", and add the products ; then multiply the first by sin A", and the second by cos A", and subtract, we obtain h cos C cos (H /') = M cos (x" x), A cos C sin (jff x") = sin (x" ;.), (50) h sin C = Jf tan j?' tan /9, by means of which we may determine h, , and .ff. Let us now put R sin 4 = #, A cos ,3 = 6, TV/ h COS p ... ,_ 13", = 6", (ol) g cos b R cos 4 = c, g cos 9? Z>"jR" cos 4." = c", /oA g cos f = d, and the equations (44), (45), and (49) become -viT^r+K < =vf ,, ./|'1_^\ 2 , TV,, r = v I m I H- XT . 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 n 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 d, 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 and, for the time t" } Subtracting the former from the latter, and reducing, we obtain Bk(t"_ <) = B in^(t/ / tp/r" cosK^" 0) r\ 1/2 q? cos a v " cos i v \ ~?~ cos ;X' cos i* S / and, since r = q sec 2 t?, this gives . (53) But we have, also, from the triangle formed by the chord >e and the radii-vectores r and r", x = r + r"* - 2rr" cos ( y<>"> and let y.' & = , 2/0" yo= a ' then we shall have a = A (y y), a' = ^(y '-y). Eliminating A from these equations, we get y ( a ' a) = a'y - - ay,', from which , CM' a" /R7-, v = v' -- -. - =V -- -> - vP*' y y a' a y a' a 190 THEORETICAL ASTRONOMY. Unless the assumed values are considerably in error, the value ul* y or of log (r + r") thus found will be sufficiently exact ; but should it be still in error, we may, from the three values which approximate nearest to the truth, derive y with still greater accuracy. In the numerical application of this equation, a and a' may be expressed in units of the last decimal place of the logarithms employed. The solution of equation (56), to find t" t when x is known, is readily effected by means of Table VIII. Thus we have 3r' ~ - 5 = sin 3ar, 1/2 (r + r") 1 and, when f is less than 90, if we put _ ~~ we get T' = 1 1/2 Nsiu / (r + r") f (68) or When f exceeds 90, we put N' = sin 3z, and we have r' = J 1/2 #'(r + /')*> (69) in which log } j/2 = 9.6733937. With the argument f we take from Table VIII. the corresponding value of N or N f , 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 and with this value of x compute eZ, r, and r". The value of f is then found from x sin?- = r"' and the table gives the corresponding value of N or N*. A second approximation to x will be given by the equation PARABOLIC ORBIT. 10] or by in which log = = 0.3266063. Then we recompute rf, r, and r", 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 and p". According to the equations (90) u we have r cos b cos (I O) = P cos (A 0) R, r cos 6 sin (I 0) = p sin (A Q), (71) r sin 6 = p tan /?, and also r" cos 6" cos (r - O") = P" cos (A" - 0") #', . r" cos ft" sin (I" ") = i*" sin (/' O"), (72) r" sin ft" =/0"tan/9", in which and I" are the heliocentric longitudes and 6, 6" the corre- sponding heliocentric latitudes of the comet. From these equations we find r, r", I, I", 6, and b n '; 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) x , we have tan i sin (I SI ) = tan ft, /wo\ tantsin(r ft) = tan 6", which may be written tan t(sin (I x) cos (x ft ) -f- sin (s ft ) cos (I x)) tan ft, tan i (sin (r x) cos (x ft ) + sin (x ft ) cos (*" *)) = tan ft". Multiplying the first of these equations by sin(Z" *), and the second by sin (I x), and adding the products, we get tan i sin (a? ft ) sin (f l) = tan ft sin (/" ar) tan ft" sin (l x): and in a similar manner we find tan i cos ( ft ) sin (?' = tan ft" cos (l x) tan 6 cos (r *). Now, since & is entirely arbitrary, we may put it equal to I, and we have PARABOLIC ORBIT. 193 tan i siii (I ) = tan b, tan i cos tf- Q) = 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"-\- 1\ whence l"x = \(l"-t) and lx=\(l-l")- t and we obtain tantcos(K*" + - ft) = o jTJt"^,,, 7 2 cos 6 cos 6" sm | (" These equations may also be derived directly from (73) by addition and subtraction. Thus we have tan i (sin (I" SI ) + sin (I )) = tan b" -f tan 6, tan i (sin (I" R ) sin (I SI )) = tan 6" tan b ; and, since sin (I" SI) + sin (i ft) = 2 sin (?'+ i 2ft) cos (/" 0, sin (I" Q, ) sin (I SI ) = 2 cos (/"+ J 2 ft) sin 5 (f I), these become (76) which may be readily transformed into (75). However, since b and 6' r 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 SI and i have been computed from the preceding equa- tions, we have, for the determination of the arguments of the latitude , teaM '-= . (77) cos i Now we have u = v -\- to, in which co = TT Si in the case of direct motion, and a) == SI if 13 194 THEORETICAL ASTRONOMY. when the distinction of retrograde motion is adopted ; and we shall have u u = 1 /' V) and, consequently, x* = r 2 -f r" 1 2rr" cos (u" u), (78) x' = (/' r cos (u" u)y + r 4 sin 2 (u" u). (79) The value of x derived from this equation should agree with that already found from (66). We have, further, r = qsec^(u a,), r" = q sec' ^(u" ), or 1 .., ,1 1 , , , 1 7= COS 4 (W ", become 2 8in i (,,' + v) sin i ("_) = _!__ -- !_, *< ^ *? (82) _ COS ^' + *;cos| (" v and hence, since jc = (r + "") sin /, We have, further, from (78), x' = (r" r) J + 4rr" sin' ^ (i/' v\ from which, putting sinv = ^=^, (84) we derive cos v = - - sin ^ (t/' r). (85) Therefore, the equation (83) becomes PARABOLIC ORBIT. 197 ? = ( + O cos* y cos'v, (86) by means of which g is derived directly from r, r", and x, the value of v being found by means of the formula (84), so that cosv is positive. When f 1 cannot be found with sufficient accuracy from the equa- tion we may use another form. Thus, we have which give, by division, tan (45 + $/) = JlI__* (87) v r -f- r x In a similar manner, we derive tan (45 -f & = x--r, \ x _ ( r " _ r ) In order to find the time of perihelion passage, it is necessary first to derive the values of v and v". The equations (59) and (85) give, by multiplication, tan ^ (v" v) = tan f' cos v, (89) from which v" v may be computed. From (82) we get (90) this equation reduces to tan \ (v" + ) = tan (/ 45) cot \ (" - r), (91) and the equations (81) give, also, tan I (v" + ) = cot | (v" v) sin 2^, either of which may be used to find v" + v. 198 THEORETICAL ASTRONOMY From the equations COB$V _ 1 by multiplying the first by sin \v" and the second by sin \v, add- ing the products and reducing, we easily find sin \ (tf' cos v _ l/r I//' Hence we have l/r ,/f* sin* (*'-*)' = cos iv = 7=, l/q Vr which may be used to compute q, v, and v" when v" v is known. When(t/' v) and \(v" -{- v\ and hence v" and v, have been determined, the time of perihelion passage must be found, as already explained, by means of Table VI. or Table VIII. It is evident, therefore, that in the determination of an orbit, as soon as the numerical values of r, r ff , and x 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 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 OEBIT. 199 72. EXAMPLE. To illustrate the application of the formula 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- 20'.5 19 14- 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. 3 1864 Jan. 10 7* 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 -f 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, * = 297 52' 51".l, ft = + 55 46' 58".4, f = 13.27682, X = 302 57 34 .4, jt = 57 39 35 .9, T = 16.29299, /I" = 310 31 35 .0, ft" = + 59 38 18 .-7. For the same times we find, from the American Nautical Almanac, O =290 6'27".4, log.R =-9.992763, ' =293 7 57 .1, logtf =9.992830, O" == 296 12 15 .7, log E" = 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. Then we have M*- tan p sin (X 0') tan ft sin (V Q') ^ ~~ ? t' tan/9" sin (A' 0') tan/3 7 sin (A" ')' and # sin ( G Q) = R" sin (0" Q), ^ cos(G -)==#' cos(0"-0)-; h cos C cos GET A") = M cos (A" A), A cos C sin (H A") = sin (A" A), ft sin: = Jftan/S" tan/9; from which to find M, G, g, H, , and h. Thus we obtain log M= 9.829827, H= 94 24' 1".8, G = 22 58' 1".7, C = 40 28 21 .9, log ^ = 9.019613, log A = 9.688532. Since r- Jif - 777 = 0.752, it appears that the comet, at the time A cos ft of these observations, was rapidly approaching the earth. The quadrants in which G O and H X" 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' and ft 1 are computed from the resulting elements. Next, from cos 4- = cos 13 cos (A 0), cos 4" = cos ft" cos (A" Q"), cos = cos C cos ( G If), we compute cos ij/, cos ^", and cos

p =Mp, give log p = 9.480952, log p" = 9.310779. NUMERICAL EXAMPLE. 203 From these values of p and //', it appears that the comet was very near the earth at the time of the observations. The heliocentric places are then found by means of the equations (71) and (72). Thus we obtain I = 106 40' 50".5, b = + 33 1' 10".6, logr = 9.912082, r=112 31 9.9, b"= + 23 55 5.8, logr" =9.935116. The agreement of these values of r and r" with those previously found, checks the accuracy of the calculation. Further, since the heliocentric longitudes are increasing, the motion is direct. The longitude of the ascending node and the inclination of the orbit may now be found by means of the equations (74), (75), or (70); and we get & == 304 43' 11".5, i = 64 31' 21". 7. The values of u and u" are given by the formulae tan(J &) tail (7" &) tan u = -- - r-^A tan u' ^^-, cos i cos i u and I & being in the same quadrant in the case of direct motion Thus we obtain u = 142 52' 12".4, ' u"= 153 18' 49".4. Then the equation x s = (r" r cos (tt" u))* + i 3 sin' (u" u) gives log x = 9.201423, and the agreement of this value of X with that previously found, proves the calculation of SI, i, u, and u". From the equations 7=r sin i (i (u" + u) a>) = */?}' Vq sm \ (u" u) /rr" we get tf = 22' 47".4, a, = 115 40' 6".3, log g = 9.88737B. Hence we have TT = w + ^ = 60 23' 17".8, 204 THEORETICAL ASTRONOMY. and i = u * = 27 12' 6".l, t/ 7 = u" < = 37 38' 43 M. Then we obtain log m = 9.9601277 f log q = 0.129061, and, corresponding to the values of v and t>", Table VI. gives log Jf = 1.267163, log M" = 1.424152. Therefore, for the time of perihelion passage, we have T=t M- = t 13.74364, m and M" T=if'=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 Jf= 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 Nr* sin v = if' N"il 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^ 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 ff , and correct the value of Jf, 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 x 1 , x", /9, and ft" will be the same as the observed values, the computed values of // and ft' must be such that when substituted in the equation for J/, the same result will be obtained as whon 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 t', 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 will, in connection with those of the extreme places, give the same value of J/, and when the exact value of M has been used in deriving the elements, the computed values of x' and ft' must give the same value for w' as that which is obtained from observa- tion. But if we represent by ij/ the angle at the earth between the sun and comet at the time t', the values of i// 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 V = cos $ cos (/' O') sin 4/ cos w' = cos ,? sin (A' '), (93) sin V sin id = sin $, give, by differentiation, cos ft dX = cos w' sec ft d$', . . d? = sin w' cos (A' ') d*'. From these we get cosj5'dA'__tan(A'' ') d? sin ff 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' and /9' by means of these elements, and then the comparison of this value of tan w' with that given by observa- tion will show whether any further correction of M is necessary, and if the difference is not greater than what may be due to unavoidable errors of calculation, we may regard M as exact. To compare the elements obtained in the case of the example given with the middle place, we find t/ = 32 31' 13".5, u' = 148 11' 19".8, log / = 9.922836. Then from the equations tan (/ Q, ) = cos i tan u', tan b' = tan i sin (T SI ), we derive t = 109 46' 48".3, b' = 28 24' 56".0. By means of these and the values of O' and R f , we obtain ;' = 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*X= + 3".6, A/5 = + 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' given by observation is log tan /== 0.966314, and that given by the computed values of /(' 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 rectiou. Since the difference is small, we may derive the correct value of 31 by using the same assumed value of ,, and, instead of the value of tan w' derived from observation, a value differing as much from this in a contrary direction as the computed value differs. Thus, in the present example, the computed value of log tan w' is 0.000067 less than the observed value, and, in finding the new value of M, we must use log tan uf = 0.966381 in computing $, and /9 " involved in the first of equations (14). If the first of equations (10) is employed, we must use, instead of tan^S' as derived from observation, tan jf = tan u/ sin (/' O')> or log tan ft = 0.966381 -f- log sin (// ') = 0.198559, the observed value of /' being retained. Thus we derive log M= 9.829586, and if the elements of the orbit are computed by means of this value, they will represent the middle place in accordance with the condition that the difference between the computed and the observed value of tan w' shall be zero. A system of elements computed with the same data from log M 9.822906 gives for the error of the middle place, C.-O. cos jf A// = 1' 26".2, A,? = 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 -jp = 0.006831, and from (10) we get log M' = 9.822906, log M" = 9.663729.. 208 THEORETICAL ASTRONOMY. Then by means of the equation (33) we derive, for the corrected value of M. log M =9.829582, which differs only in the sixth decimal place from the result obtained by varying tanw' and retaining the approximate values p = ^/ == w/' 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', A" be the right ascensions, and _D, D' } D" the declinations of the sun for the times t, t', t". The co-ordinates of the first place of the earth referred to the third are x =. R" cos D" cos A" R cos D cos A, y R" cos D" sin A" R cos D sin A, z = R" sin D" R sin D. If we represent by g the chord of the earth's orbit between the places for the first and third observations, and by G and K, respectively, the right ascension and declination of the first place of the earth as seen from the third, we shall have x = g cosK cos G, y = g cosK sin G, z = g sin K, VARIATION OF THE GEOCENTRIC DISTANCE. 209 and, consequently, g cos K cos ( G A) = R' cos D" cos (A" A) R cos D, g cos K sin ( G A} = R' cos W sin (A" A\ (96) gsmK = K' sin Z>" R sin D, from which ^, IT, and G may be found. If we designate by x n y,, z, the co-ordinates of the first place of the comet referred to the third place of the earth, we shall have x, = J cos 8 cos a -}- g cos K cos O t y, = J cos 5 sin a + g cos JT sin G, z, = J sin 3 -f- <7 sm -K, Let us now put x, = h' cos * cos H', y t = h' cos ? sin IT, z, h' sin C 7 , and we get h' cos f cosCET GT) = A cost cos(o GO + ? cos^", A' cos r sin (^ G) = ^J cos 3 sin (a (?), (97) A' sin C' = J sin 3 -|- ^ sin K, from which to determine H f , ', and A'. If we represent by ^ ' the angle at the third place of the earth between the actual first and third places of the comet in space, we obtain cos ?'= cos ? cos H' cos 5" cos o"+ cos *' sin H' cos 3" sin o" -f sin C' sin o", or cos / = cos r cos d" cos (a" "') + sin r sin 3" ; (98) and if we put e sin/ sin <5", e cos/= cos 3" cos (a" H') this becomes cos ? ' = ecos(r-/). (99) Then we shall have x 1 = h'* + J'" 2A' J" cos ?' x' = ( J" A' cos ?')* + A" sin 1 /, (100) in which 4" is the distance of the comet from the earth correspond- ing to the last observation. We have, also, from equations (44) and (45), r =(J tfcos*) 1 + 7? sin 1 4-, QOV, /' = ( J" #' cos V') 1 u 210 THEORETICAL ASTRONOMY. in which ^ is the angle at the earth between the sun and comet at the time t, and 4/' the same angle at the time t". To find their values, we have cos 4- = cos D cos S cos (a A) + sin D sin <$, cos *"= cos D" cos S" cos (a" 4") + sin H' sin d", which may be still further reduced by the introduction of auxiliary angles as in the case of equation (98). Let us now put h'- sin 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 dJ and x + 84, the as- sumed values A <5J, J, and J + d A 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 + o = a. m = a', m-\-n + o = a", whence m = a', n =l(a"a), o = $(a"+ a) a'. Now, in order that the middle place may be exactly represented in right ascension, we must have ()'+"()+-='> *Vora which we find a: 9< = 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 d f A', for a given increment as- signed to J, is the same as that of A<5'. But if we denote the value of x for which the error in a' is reduced to zero by x', and that for which A<5' = 0, by x", the most probable value of x will be in which n = \ (a" a) and n r = \ (d" d). It should be observed that, in order that the differences in right ascension and declination shall have equal influence in determining the value of x, the values of a, a', and a" must be multiplied by cos d'. The value of d J in 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', n = a" a'. The condition that the middle place shall be exactly represented, gives the two equations (a" a') x -f a' 3 J = 0, (d"-d')x + d'dJ = 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 x, a final system of elements computed with the geocentric distance J -f- .r, 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 formula? 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, 8, 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 epheineris with several observations made during a few days before and after the date of each normal, and finding the mean difference between computation and observation : Washington M. T. a 6 1858 June 11.0 141 18' 30".9 + 24 46' 25".4, July 13.0 144 32 49 .7 27 48 .8, Aug. 14.0 152 14 12 .0 + 31 21 47 .9, which are referred to the apparent equinox of the date. These places are free from aberration. 214 THEORETICAL ASTRONOMY. Wo 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 yenr, we get t =162.0, vl = 135 51' 44".2, ft = + 9 6' 57".8, t' = 194.0, / = 137 39 41 .2. p = 12 55 9 .0, f!" = 226.0, A" = 142 51 31 .8, ft' = + 18 36 28 .7. From the American Nautical Almanac we obtain, for the true places of the sun, = 80 24' 32".4, log E = 0.006774, 0' =110 55 51 .2, Iog.R' =0.007101, "=141 33 2.0, log #' = 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 /9 in place of ex and , and in place of A, in the equations which have been derived for the equator as the fundamental plane. Therefore, we have sin (-)=#' sin ("-); cos * = cos /? cos (J 0), cos V = cos ft" cos (A" 0") E sin 4 = B, K' sin 4" = ", from which to find G, g, 6, B, b ff , and B", 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^ = 0.013500, log B" = 9.510309, b" = + 0.959342. Then we assume, by means of approximate elements already known, log A = 0.397800, and from A' cos C' cos (H r (?) = A cos /? cos (A GO + g, h' cos ? sin (IT (?) = J cos /? sin (Jl G). A' sin C' = J sin /?, we find IT 7 , ', and h f . These give 5"' = 153 46' 20".5, :' = + 7 24' 16".4, log h' = 0.487484. NUMERICAL EXAMPLE. Next, from cos and p", respectively. Thus we obtain / = 159 43' 14".2, b = + 10 50' 14".0, logr = 0.323447, J" = 144 1747.8, b" = + 35 1428.7, log r" = 0.052347. The agreement of these results for r and r" with those already obtained, proves the accuracy of the calculation. Since the helio- centric longitudes are diminishing, the motion is retrograde. Then from (74) we get ft = 165 17' 30".3, i = 63 6' 32".5 ; and from tan u = _ tan u n = _ tan (r- we obtain u = 12 10' 12".6, u" = 40 18' 51". 2, the values of u and I ft being in the same quadrant when the motion is retrograde. The equation (79) gives log x = 0.090630, which agrees with the value already found. The formulae (81) give w = 129 6' 46".3, log q = 9.760326, and hence we have v = u u> = 116 56' 33".7, v" = u" u> = 88 47' 55".l, from which we get T= 1858 Sept. 29.4274. From these elements we find log r 1 = 0.212844, v' = 107 7' 34".0, u' = ?1 59' 12".3, and from tan (JF ft) = cos i tan u', tan b' = tan i sin (F ft ), we get r = 154 56' 33".4, V= + 19 30' 22".l. NUMERICAL EXAMPLE. 217 By means of these and the values of O' and R', we obtain A' = 137 39' 13".3, & = -f 12 54' 45".3, and comparing these results with observation, we have, for the error of the middle place, C.-O. cos p AA' = 27".2, A^ = 23".7. From the relative positions of the sun, earth, and ccmet 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 A, log A = 0.398500, from which we derive H' = 153 44' 57".6, C' = + 7 24' 26".l, log h' = 0.488026, log C= 9.912587, logc = 0.472115, logr = 0.324207, log A" = 0.311054, logr" = 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, T=144 1712.1, b" = + 35 837.8, log r" = 0.054825, and from these, = 165 15' 41".l, i = 63 2' 49".2, u = 12 10 30 .8, u" = 40 13 26 .0, a, = 128 54 44 .4, log q = 9.763620, T= 1858 Sept. 29.8245, log r' = 0.214116, v ' = 106 55' 43".8, u'= 21 59' 0".6, V = 154 53 32 .3, b' = + 19 29 31 .9, A' = 137 3939.7, /3' = + 12 55 2.9. Therefore, for the second assumed value of J, we have C.-O. cos ft AA' = 1".5, A,9' = 6".l. Since these residuals are very small, it will not be necessary to make a third assumption in regard to J, but we may at once derive the correction to be applied to the last assumed value by means of the equations (109). Thus we have ' = 1.5, a '=-27.2, d' = 6.1, t q = 165 15 24 .8 Mean E 1 uinOX 1858 -' t= 63 2 14.2 log ? = 9.764142 Motion Retrograde. If the distinction of retrograde motion is not adopted, and we regard i as susceptible of any value from to 180, we shall have TT = 294 8' 12".7, i = 116 57 45 .8, the other elements remaining the same. The comparison of the middle place with these final elemenis gives the following residuals : C.-O. cos /? AA = + 0".2, A/9 = 4".3. These errors are so small that the orbit indicated by the observed places on which the elements are based differs very little from a parabola. When, instead of a single place, a series of intermediate places is employed to correct the assumed value of J, it is best to adopt the equator as the fundamental plane, since an error in a or d will affect both X and /?; and, besides, incomplete observations may also be used NUMERICAL EXAMPLE. 219 when the fundamental plane is that to which the observations are directly referred. Further, the entire group of equations of con- dition for the determination of x, according to the formula? (109), must be combined by multiplying each equation by the coefficient of x in that equation and taking the sum of all the equations thus formed as the final equation from which to find x, the observations being supposed equally gocd. 220 THEORETICAL ASTRONOMY. CHAPTER IV. DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF THE CONIC SECTION. 77. THE formulae which have thus far been derived for the deter- mination of the elements of the orbit of a heavenly body by means of observed places, do not suffice, in the form in which they have been given, to determine an orbit entirely unknown, except in the particular case of parabolic motion, for which one of the elements becomes known. In the general case, it is necessary to derive at least one of the curtate distances without making any assumption as to the form of the orbit, after which the others may be found. But, preliminary to a complete investigation of the elements of an un- known orbit by means of three complete observations of the body, it is necessary to provide for the corrections due to parallax and aber- ration, so that they may be applied in as advantageous a manner as possible. When the elements are entirely unknown, we cannot correct the observed places directly for parallax and aberration, since both of these corrections require a knowledge of the distance of the body from the earth. But in the case of the aberration we may either correct the time of observation for the time in which the light from the body reaches the earth, or we may consider the observed place corrected for the actual aberration due to the combined motion of the earth and of light as the true place at the instant when the light left the planet or comet, but as seen from the place which the earth occu- pies at the time of the observation. When the distance is unknown, the latter method must evidently be adopted, according to which we apply to the observed apparent longitude and latitude the actual aberration of the fixed stars, and regard this place as corresponding to the time of observation corrected for the time of aberration, to be effected when the distances shall have been found, but using for the place of the earth that corresponding to the time of observation. It will appear, therefore, that only that part of the calculation of the DETERMINATION OF AX ORBIT. 221 elements which involves the times of observation will have to be re- peated after the corresponding distances of the body from the earth have been found. First, then, by means of the apparent obliquity of the ecliptic, the observed apparent right ascension and declination must be converted into apparent longitude and latitude. Let >1 and $,, respectively, denote the observed apparent longitude and latitude; and let O be the true longitude of the sun, 2' its latitude, and R n its disfiuce from the earth, corresponding to the time of observation. Then, if ^ and /9 denote the longitude and latitude of the planet or comet corrected for the actual aberration of the fixed stars, we shall have ;- - x = + 20".445 cos (A - ) ?ec ,3 + 0".343 cos (A - 281) sec ? . ,3 /?,= 20".445 sin (A Q ) sin ,3 0".343 sin (A 281) sin ,3. ^ In computing the numerical values of these corrections, it will be sufficiently accurate to use ^ and ^ instead of / and ; 3 in the second members of these equations, and the last terms may, in most cases, be neglected. The values of / and ,3 thus derived give the true place of the body at the time t 497*.78 J, but as seen from the place of the earth at the time t. When the distance of the planet or comet is unknown, it is impos- sible to reduce the observed place to the centre of the earth ; but if we conceive a line to be drawn from the body through the true place of observation, it is evident that were an observer at the point of intersection of this line with the plane of the ecliptic, or at any point in the line, the body would be seen in the same direction as from the actual place of observation. Hence, instead of applying any correc- tion for parallax directly to the observed apparent place, we may conceive the place of the observer to be changed from the actual place to this point of intersection with the ecliptic, and, therefore, it be- comes necessary to determine the position of this point by means of the data furnished by observation. Let d be the sidereal time corresponding to the time ^ of obsei vat ion, fp' the geocentric latitude of the place of observation, and fJ the radius of the earth at the place of observation, expressed in pails of the equatorial radius as unity. Then is the right ascension and cos C E sin O (4 J ) cos ^ sin A = R cos i' sin O P sin ^ cos b sin /, ( A, J ) sin /5 = B sin J p sin ?: sin i . If we suppose the axis of x to be directed to the point whose longi- tude is O , these become DETERMINATION OF AN OEBIT. 223 H cos (O GO) (4 4.) cos ft cos (/I ) = E cos 2; Po sin TT O cos & cos (7 .), # sin (G - ) (J, - 4,) cos sin (A ) = (2; f> ? sin TT O cos 6 sin (1 9 Q ), ( J, J ) sin P = R sin sin TT O sin b , from which J2 and G may be determined. Let us now put (J, J.)cos/9 = D; (3) then, since TT O , reduced to E = D cos 0* Go) * Po cos fy> cos 01, G ) + R > ^ (G - Go) = D sin (i - ) - /> cos 6 sin (^ - 0.), = Z>tan/9 ^ /> sin 6 + J? J . Hence we shall have, if /r and 2*,, are expressed in seconds of arc, _ 206264.8 - Q ) - _ _ , 206264.8 D sin (A - Q ) - * Po cos & sin (/ - Q ) G - Go + , from which we may derive the values of G 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 instani of observation, from the point of reference in the direction actually observed, but at a time different from t , to be determined by the interval which is required for the light to pass over the distance J, J . Consequently we ought to add to the time of observation the quantity (j, _ j o ) 497'.78 = 497M8 D sec ft (5; which is called the reduction of the time; but unless the latitude of the body should be very small, this correction will be insensible. The value of /I derived from equations (1) and the longitude 224 THEORETICAL ASTRONOMY. derived from (4) should be reduced by applying the correction fop 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 /9 = 0, the equations (4) cannot be applied, and when the latitude is so small that the reduction of the time and the correction to be applied to the place of the sun are of considerable magnitude, it will be advisable, if more suitable observations are not available, to neglect the correction for parallax and derive the elements, using the unconnected 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 formula? which may be required. But since some of these formula 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 I". 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 ft, its latitude as seen from this point, we shall have J, cos 0, = J cos /?, A, sin 0, = A sin R am S v from which we easily derive the correction ft, ft, or A/3, to be applied to the geocentric latitude. Thus, we find (6) -T being expressed in seconds. This correction having been applied to the geocentric latitude, the latitude of the sun becomes =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 0*.002. It should be remarked also that before applying the equa- tion (6), the latitude -T 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 tho 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 ft sin X" tan ft" sin )., the second by tan/9" cos A tan ft cos X", and the third by sin (A ^"), and add the products, we shall have = nR (tan [?' sin (I ) tan sin (A" Q)) P ' (tan /9 sin (I" A') tan p sin (A" A) + tan ft' sin (A' - A)) R (tan /9" sin (A ') tan /9 sin (A" ')) -f ri'R' (tan 0" sin (A Q") tan 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 K the longitude of the ascending node, and by J the inclination to the ecliptic, of a great circle passing through the first and third observed places of the body, and we have tan/5 = sin (A JT) tan J, tan ,9" = sin (A" JZO tan 7. Introducing these values of tan /? and tan ft" into the equation (7), since sin 01-0) sin 01" JQ sin (*" 0) sin 0* - K} = sin 01" X) sin (0 K), sin (A' A) sin (A" K) + sin (/' A') sin (A K) = H-sin(A" Jl)8inOl' K), sin (A 0') sin (A" .BT) sin (A" sin 01 -K") = _ gin (A" A) sin (' JD, sin (A W ) sin (/' J5T) sin (A" 0") sin (J JT) = sin (X" /I) sin (" JTi, we obtain, by dividing through by sin (X" X) tan 7, = nR sin ( K) + />' (sin (T ^) tan & cot /) # sin (' K) + ri'R' sin (" K}. Let ^9 denote the latitude of that point of the great circle passing through the first and third places which corresponds to the longitude X, then tan = sin (/ K} tan I, and the coefficient of // in equation (9) becomes sin(/3 -/5Q cos /S cos ft tan/ Therefore, if we put '~ cos /9 tan/' we shall have This formula will give the value of //, or of J', when the values of n and w r/ have been determined, since a and ./Tare derived from the data furnished by observation. DETERMINATION OF AN ORBIT. 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 / 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 //, the signification of p and p", when the corrections for parallax are applied to the places of the sun, being as already noticed in the case of p'. 79. If we multiply the first of equations (6) 3 by sin " tan/9", the second by cos " tan/3", and the third by sin (A" "), and add the products, we get 0=n,o (tan /9" sin (" A) tan/?sin(" A")) ri.Rtan/9"sm(" ) p' (tan /3" sin ("A') tan /3' sin (" A"))+# tan /?" sin ("'), (13) which may be written 0=n /0 (tan^sin(A"") tan /9" sin (A")) wJJtan/S" sin (") + p' (tan ft" sin (A' ") tan /S sin (A" ")) p' (tan /? tan /9 ) sin (A" ") -f R' tan /9" sin (" '). Introducing into this the values of tan ft, tan /9", and tan /9 in terms of 1 and K, and reducing, the result is = np sin (A" A) sin ( " K} nR sin ( " ) sin (A" K} p' sin (A" A') sin ( " K) p'a sec p sin (A" "^ + K' sin ("- ') sin (A"- K). Therefore we obtain _y/sin(A" A') a sec/3' sin (A" Q") \ P ~ n \ sin (A" A) + sin (A" A) * sin (" K) / sin (A" K} #sin(Q" ') n.Rsin(Q" Q) n sin (A" A) sin ( " K) But, by means of the equations (9) 3 , we derive E' sin ("') nR sin (" ) = (N n) Rsm(Q"- ), 228 THEORETICAL ASTRONOMY. and the preceding equation reduces to _/ /sin (/'-/) q sec/y sinQl" Q") P n\sinq" " i ~sin(/'-;) I N\R Bin(Q" Q) sm(/' - K) "*"\ / sin (A" >l)sin(0" -K") ' To obtain an expression for p" in terms of p f , if we multiply the first of equations (6) 3 by sin Q tan /3, the second by cos tan ft, and the third by sin (A O), and add the products, we shall have 0=n"p" (tan /9 sin (A" ) tan/9" sin (A 0)) w"^" tan /3sin (0" ) / (tail /3 sin (A' O) tan/5' sin (A ))+.#' tan /?sin (0' Q). (15) Introducing the values of tan/3, tan/3', and tan/3" in terms of K &ud I, and reducing precisely as in the case of the formula already found for p, we obtain // / sin (A' A) a sec ft sin (A ) \ ~^Un(/'-/0~sm(A"-r>*sin(0-J!0/ / JV" \ 7?" sin f(7\" ^ dn (I TT\ + ( 1 ~ *" A) sin (; T^et us now put, for brevity, A = a a '-) and the equations (11), (14), and (16) become p' sec p = c + nb + w"d, ,=.;-- + *; (i-f). as) If n and n'' are known, these equations will, in most cases, be sufficient to determine /?, //, and p". DETERMINATION OF AN ORBIT. 22D 80. It will be apparent, from a consideration of the equations which have been derived for />, />', and p", 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 If the ratio of n" to n is known, this equation will determine the quantities themselves, and from these the radius-vector r 1 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 + O', and hence n"J2"sin(O" O') nJ?sin(0' O) = 0, or n" _ JgsinCQ' Q) "n ~~#'sin(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 / /" and ft = ft", 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, X is equal to /" and $" differs from ,3, it will be possible to derive p r , and hence p and p"; but the formulae which have been given require some modification in this particular case. Thus, when l = X", we have K =)>" = *, 1= 90, and &=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', which is a sec ft', gives a = sin ? cot I cos ft sin (A' K) t (19) and when /I = X", it becomes simply a, = cos $ sin (/' JT). 230 THEORETICAL ASTRONOMY. Whenever, therefore, the difference X" X 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' in equation (7). When X = X" = K, the values of M v Jtf"/', l/ 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', as given by equations (13) and (15), without introducing the auxiliary angles, we shall have _p' tan ft' sin (A" Q") tan ft" sin (A' Q") p ~ n ' tan ft sin (A" ") tan ft" sin (A ") / N \ E tan ft" s + \ rT/tan/?sin(A" ") p' tan ft sin (A' ) tan ft' sin (A ) n" tan ft sin (A" ) tan ft" sin (A ) n" I tan ft sin (A" ) tan ft" sin (A ) Hence _ tan ft sin (A" Q") tan ft" sin (A' Q") 1 ~ tan ft sin (A" Q") tan ft" sin (A ") ' , tan ft sin (A' Q) tan ft' sin (A ) tan ft sin (A" ) tan ft" sin (A ) ' * tan ft sin (A" ") tan ft" sin (A ")' ,, = K' tan/? sin (") a ~~ tan ft sin (A" ) tan ft" sin (A ) ' are the expressions for M v M^, M 2 , and M 2 " which must be used when X = X" or when X is very nearly equal to X" ; and then p and p" will be obtained from equations (18). These expressions will also be used when X" 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 formulae must be derived. If we multiply the second of equa- tions (7) 3 by tan/9" and the fourth by sin (A" '), and add the products, then multiply the second of these equations by tan /? and the fourth by sin (X '), and add, and finally reduce by means of the relation NE sin (' ) = N"R" sin (" '), we get DETERMINATION OF AN ORBIT. 231 = / tan p' sin (X Q ') tan ft' sin (A" 0') ~~ n 'tan /5" sin (A ') tan /9 sin (A" ') N"\ #' tan /S" sin ("') 2V / tan ft" sin (A ') tan /? sin (A" ')' p^ tan /? sin (A Q ') tan ft sin (/ Q ') : ri' ' tan /3" sin (A ') tan ;3 sin (A" ') ^L -^ \ w 77 ~~ ^" / .R tan /9 sin (') tan /S" sin (A ') tan /? sin (A" / / These equations are convenient for determining p and p" from |', that the equations (26) 3 will enable us to find n and n" by successive approximations, assuming first that " T " n = r n = 7 , and from the resulting value of p' determining r 1 ', and then carrying the approximation to the values of n and n" one step farther, so an to include terms of the second order with reference to the intervals of time between the observations. But if we consider the equation (10), we observe that is a very small quantity depending on the difference ft' /9 , and therefore on the deviation of the observed path of the body from the arc of a great circle, and, as this appears in the denominator of terms containing n and n" in the equation (11), it becomes necessary to determine to what degree of approxi- mation these quantities must be known in order that the resulting value of p' may not be greatly in error. To determine the relation of to the intervals of time between the observations, we have, from the coefficient of p f in equation (7), a sec ff = tan sin (A" A') tan ft sin (A" A) + tan/S" sin (A' A). W r e may put tan/9 =tan/9' AT" + BT"* ...., tan ,3" = tan /9' + AT -f- BT* + ...., and hence we have a c sec p = (sin (A" A') sin (A" A) + sin (A' A)) tan p + (T sin (A' A) T" sin (A" A')) ^+(r 2 sin (A' A)-f T"* sin (A" A')) B+. ., which is easily transformed into a sec p = 4 .BUI (A' A) sin (A" A') sin | (A" A) tan p (25 ) -f (r sin (A'-A) r" sin (A A'))4+( T * sin (A'-A)+r"* sin (A"-A'))5+. . . . If we suppose the intervals to be small, we may also put emK^-^-K^-^, and sin (A" A) . I" A, sin (A' A) = A' A. DETERMINATION OF AX ORBIT. 233 Further, we may put * = *' A'r" /' = *' -f A'r -f JBV -f ..... Substituting these values in the equation (25), neglecting terms of the fourth order with respect to r, and reducing, we get a == rrY' QA* tan ff + ^L' ^#) cos p. It appears, therefore, that a is at least of the third order with reference to the intervals of time between the observations, and that an error of the second order in the assumed values of n and n" mav produce an error of the order zero in the value of // as derived from equation (11) even under the most favorable circumstances. Hence, in general, we cannot adopt the values omitting terms of the second order, without affecting the resulting value of // to such an extent that it cannot be regarded even as an approximation to the true value ; and terms of at least the second order must be included in the first assumed values of n and n". The equation (28) 3 gives omitting the term multiplied by -7-* which term is of the third order with respect to the times ; and hence in this value of , only terms of at least the fourth order are neglected. Again, from the equations (26), we derive, since r' T -4- T", (27) in which only terms of the fourth order have been neglected. the first of equations (18) may be written: d it in which, if we introduce the values of and n + n" as given by n (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 p' 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 m 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, n^_r" n r ' The equation (27) gives n" !) = TT". Hence, if we put P = n - n' (29) we may adopt, for a first approximation to the value of p', and p' 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 Q, thus facilitating the successive approximations ; and when is a very small quantity, the equality of the intervals is of the greatest importance. From the equations (29) we get (, , Q \ (31) n" = nP; and introducing P and Q into (28), there results c. (32) This equation involves both p r and r' as unknown quantities, but by means of another equation between these quantities p' may be eliminated, thus giving a single equation from which r' may be found, after which p' may also be determined. DETERMINATION OF AN ORBIT. 235 82. Let i|/ 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/? tan w' = sin (/ 07 n ~, (33^ cos V = cos ft cos (A' ')> by means of which we may determine $', which cannot exceed 180. Since cos /9' is always positive, cos $/' and cos ()J O ') must have the same sign. We also have r" = J" + R* ZA'R cos *'. which may be put in the form r" = 0' sec ft IS cos V) 2 + R" sin 2 4', from which we get P ' sec ft = Rcos*'V r' 2 R* sin 2 4'. (31) Substituting for p' sec /9' its value given by equation (32), we have For brevity, let us put _l + Pd C ~ 1 + P' e o -. c = k , (35) -H = 4. and we shall have lc ^ = R cos V vV s - #* sin* 4'. (36) When the values of P and 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 f the angle at the planet between the sun and earth at the time of the second observation, and we shall have / = 236 THEORETICAL ASTRONOMY. Substituting this value of r', in the preceding equation, there results (k - K cos V) sin *' =F K sin 4' cos *' = ^ 1 and if we put 7o sinC = 12' sin V, = *o -#' cos V, the condition being imposed that m shall always be positive, we have, finally, sin (z r =F :) = m sin*/. (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 0) since sin ty r is always positive. From equation (37) it appears that sin z' must always be positive, orz'<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 _ sin 4.' sin 2' Therefore, gsiny+V) sm z and, since // is always positive, it follows that sin (z' -f ^') must be positive, or that z' cannot exceed 180 fy. 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 ft' = 0, 4 // ~ 180 when the body is in opposition, and | 4/ = when it is in conjunction. Consequently, it becomes impos- sible to determine r' by means of the angle z' 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 4/ = 0, Hie upper sign being used when the sun is between the earth and the DETEKMrXATIOX OF AN OBBIT. 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 p' = 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 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 The equations (17) give " K) and if we put we shall have since c = oo when o = 0. Hence we derive r/= '\/i i r% o - < 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 77i 8 sin* z sin / cos C = =F cos / sin T, 238 THEORETICAL ASTRONOMY. and, by squaring and reducing, this becomes w 2 sin 8 z' 2w cos sin 5 z' -f sin" z' sin 2 C = 0. When C 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 + 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 = m sin* z', (43) since the lower sign in (40) follows directly from this by substituting 180 z' in place of z'; 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) u must satisfy the analytical conditions of the problem. It is evident that the three places of the earth may satisfy these conditions ; and hence there may be one root of equation (43) which will correspond to the orbit of the earth, or give P ' = Q. Further, it follows from the equation (37) that this root must be 2 ' = 180 V; and such would be strictly the case if, instead of the assumed values of P and Q, their exact values for the orbit of the earth were adopted, and if the observations were referred directly to the centre of the earth, in the correction for parallax, neglecting also the perturbations in the motion of the earth. DETERMINATION OF AX ORBIT. 239 In the case of the earth, n" = N" = BR *(& O) RR'sm(Q" 0)' . = ' " O)' and the complete values of P and Q become p = '(8111(0" O')' /fl#Bin(0'- Q) + ##'sin(G"- ~ ^ and since the approximate values differ but little from these, as will appear from the equations (27) s , there will be one root of equation (43) which gives z' nearly equal to 180 ty. 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' are excluded by the nature of the problem, since r' must be positive, or z' < 180 ; and of the three positive roots which may result from equation (43), that being excluded which gives z' very nearly equal to 180 ^/, there will remain two, of which one will be excluded if it gives z' greater than 180 ^', 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 ^/, 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 r/ sin (u 1 u) W ')' ( . rr' sin (u'- u) + rV' sin (u"- u'} \ n"Bin(tt" M) ~ L J' 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' shall be very nearly equal to 180 4/, 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 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 2', we get cos (V C) = 4w sin V cos 2' ; and, in the case of equal roots, the value of z' as derived from this must also satisfy the original equation sin (/ = 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 of cos (/ C) = 4 cos z' sin (/ C), from which we easily find sin (2/ = | sin C. (45) This gives the value of in terms of z' for which equation (43) has DETERMINATION OF AN ORBIT. 241 equal roots, and at which it ceases to have four real roots. To find the corresponding expression for m , we have m = 8iny Q = cos( 2 '-C) sinV 4 sin y cos 2^ in which we must use the value of given by the preceding equation. Now, since sin (2z r ) must be within the limits 1 and -f 1, the limiting values of sin will be -f f and f, 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 m , 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 ', or sin (2z ' 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 ' will be satisfied by the values 2< C, 180 (2z '-0; and hence there will be two values of m , which we will denote by m l and ra 2 , for which, with a given value of , equation (43) will have equal roots. Thus we shall have and, putting in this equation 180 (2 ' ) instead of 2z f , or 90 (V ) in place of z f , cos z f m *~ cos'Oo' C)' It follows, therefore, that for any given value of , if m is not within the limits assigned by the values of m l 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' is not very nearly equal to 180 ty, the positive root, when exceeds the limits + 36 52'.2 and 36 52'.2, may actually satisfy the conditions of the problem, and belong to the orbit of the body observed. 16 242 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 w 2 ; but, if m surpasses these limits, there will be only two real roots. Table XII. contains for values of C 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 ra 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, and m 2 , and the values of z' 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 2 ', 2 3 ', and z/. The table will show, from the given values of ra and 180 a//', whether the problem admits of two distinct solutions, since, excluding the value of z', which is nearly equal to 180 ty, 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' 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 /sec ,5' = -; and, since p' must be positive, the algebraic sign of the numerical value of 1 will indicate whether r' is greater or less than R'. It is easily seen, from the formulae for 1 , 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' /9 and sin (' K] have contrary signs, and negative when ft' /? has the same sign as sin ('' K). Hence, when 0' K is less than 180, r' must be less than R' if fl' /9 is positive, but greater than R' if /?' /? is negative. When 0' K exceeds 180, r' will be greater than R f if ft' /9 is positive, and less than R' if ft' $, 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 cir^e through the two extreme observed places of the comet, r f must be greater than R 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 p r and r' derived from the assumed values 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 //. To derive other expressions for P and Q which are exact, provided that T* and p' are accurately known, let us denote by s" the ratio of the sector of the orbit included by r and r' to the triangle included by the same radii-vectores and the chord joining the first and second places ; by s r the same ratio with respect to r and r", and by s this ratio with respect to r' and r". These ratios s, s', s" must neces- sarily be greater than 1, since every part of the orbit is concave toward the sun. According to the equation (30) t , we have for the areas of the sectors, neglecting the mass of the body, and therefore we obtain s"lrr^ = r" l /p ) s' [rr"] = r' } /p, s [r'r"]=r v /p. (46) Then, since n = W 7 l' n " = Wl we shall have r s' ,, 1" f! ,,-^ "=?T "=7-^ ^ 47) and, consequently, p __ J_ 8_ (48) ss \ TT rr rr Substituting for s, s f , and s" their values from (46), we have 244 THEORETICAL ASTRONOMY. The angular distance between the perihelion and node being denoted by a), the polar equation of the conic section gives = 1 -f e cos (u w), 4 = 1 + e cos <>' ,), (50) . If we multiply the first of these equations by sin (u n u'}, the second by sin (u" M), and the third by sin (u r tt), add the products and reduce, we get sin (M" M') sin (u" w) + sin (tt f M) = sin (tt" w') sin (w" w) -f- sin (tt' w) ; and, since sin (u" M') = 2 sin 4 (u" if) cos ^ (tt" tt') sin (" w) sin (tt' w) = 2 sin (" ') cos i (" -f u' 2u), the second member reduces to 4 sin ^ (M" M') sin ^ (tt" w) sin ^ (M' \ Therefore, we shall have 4rr'r" sin ^ (M" tt') sin j (u" u) sin ^ (u' ~ ) " w ') rr " s i n ( w "^ w ) _^_ rr' sin (w' M)' If we multiply both numerator and denominator of this expression ty 2rr'r" cos (" ') cos | (tt" ) cos -^ (w' w), it becomes, introducing [rr r ], [rr''], and [rV], . [rr"].[rr'] __ 1 _ ~~ [Vr"] + [r/] [rr"] ' 2rrV cosi (w" ') cos (tt" ) cos (w' )' Substituting this value of p in equation (49), it reduces to rr" r"' rr" cos (tt" tt') cos (w" w) cos ^ (tt' u)' 86. If we compare the equations (47) with the formula (28) 3 , we derive DETERMINATION OF AN ORBIT. 245 Consequently, in the first approximation, we may take If the intervals of the times are not very unequal, this assumption will differ from the truth only in terms of the third order with respect to the time, and in terms of the fourth order if the intervals are equal, as has already been shown. Hence, we adopt for the first approximation, the values of T and T" being computed from the uncorrected times of observation, which may be denoted by t , t f , and t ". With the values of P and Q thus found, we compute r r , and from this p' } />, and p", by means of the formulae already derived. The heliocentric places for the first and third observations may now be found from the formulae (71) 3 and (72) 3 , and then the angle u" u between the radii-vectores r and r" may be obtained in various ways, precisely as the distance between two points on the celestial sphere is obtained from the spherical co-ordinates of these points. When u" u has been found, we have sin (u" ') = ^ sin (u" - ), n'Y' (53) sin (u' u) = j- sin (u" u), from which u" u' and u' u may be computed. From thest results the ratios s and s" may be computed, and then new and more approximate values of P and Q. The value of u" u, found by taking the sum of u" u f and u' u as derived from (53), should agree with that used in the second members of these equations, within the limits of the errors which may be attributed to the logarithmic tables. The most advantageous method of obtaining the angles between the radii-vectores is to find the position of the plane of the orbit directly from /, I", 6, and 6", and then compute u, u', and u" directly from ft and i, according to the first of equations (82) t . It will be expedient also to compute r', V and &' from />', //, and /?', and the agreement of the value of r', thus found, with that already obtained from equation (37), will check the accuracy of part of the numerical 24P THEORETICAL ASTRONOMY. calculation. Further, since the three places of the body must be in a plane passing through the centre of the sun, whether P and Q are exact or only approximate, we must also have tan b' = tan i sin (I' & ), and the value of 6' derived from this equation must agree with that computed directly from p' } or at least the difference should not exceed what may be due to the unavoidable errors of logarithmic calcula- tion. We may now compute n and n" directly from the equations r'r" sin (u" u'} rr' sin (u' u) , , 71 = 77 ; ; TI ' N~> Ti 7 Ti - \ > rr sm (u u) rr sin (u u) but when the values of u, u f , and u" are those which result from the assumed values of P and Q, the resulting values of n and n" will only satisfy the condition that the plane of the orbit passes through the centre of the sun. If substituted in the equations (29), they will only reproduce the assumed values of P and Q, from which they have been derived, and hence they cannot be used to correct them. If, therefore, the numerical calculation be correct, the values of n and n" obtained from (54) must agree with those derived from equa- tions (31), within the limits of accuracy admitted by the logarithmic tables. The differences u" u' and u' u will usually be small, and hence a small error in either of these quantities may considerably affect the resulting values of n and n". In order to determine whether the error of calculation is within the limits to be expected from the logarithmic tables used, if we take the logarithms of both members of the equations (54) and differentiate, supposing only n, n", and u' to vary, we get cot (u" u'} du', d \og e n" = + cot (u' u) du'. Multiplying these by 0.434294, the modulus of the common system of logarithms, and expressing du' in seconds of arc, we find, in units of the seventh decimal place of common logarithms, d log n = 21.055 cot (u" i//) du', d log n" = + 21.055 cot (u r u) du'. If we substitute in these the differences between log n and log n" as found from the equations (54), and the values already obtained by DETEEMINATION OF AN ORBIT. 247 means of (31), the two resulting values of du r should agree, and the magnitude of du' itself will show whether the error of calculation exceeds the unavoidable errors due to the limited extent of the logarithmic tables. When the agreement of the two results for n and n" is in accordance with these conditions, and no error has been made in computing n and n" from P and Q by means of the equa- tions (31), the accuracy of the entire calculation, both of the quan- tities which depend on the assumed values of P and Q, and of those which are obtained independently from the data furnished by observa- tion, is completely proved. 87. Since the values of n and n" derived from equations (54) cannot be used to correct the assumed values of P and Q, from which r, r f , u, u', &c. have been computed, it is evidently necessary to compute the values for a second approximation by means of the series given by the equations (26) 3 , or by means of the ratios s and s". The expressions for n and n" arranged in a series with respect to the time involve the differential coefficients of r' with respect to t, and, since these are necessarily unknown, and cannot be conveniently determined, it is plain that if the ratios s and s" can be readily found from r, r', r", u, u', u", and r, r', r", so as to involve the relation between the times of observation and the places in the orbit, they may be used to obtain new values of P and Q by means of equations (48) and (51), to be used in a second approximation. Let us now resume the equation a? and also for the third place T) . - - = E"e sin E ". Subtracting, we get * = E" E 2e sin \ (E" E} cos |- (E" + E\ (55,) a? This equation contains three unknown quantities, a, e, and the dif- ference E" E. We can, however, by means of expressions in- volving r, r", u, and u", eliminate a and e. Thus, since p =a (1 e 2 ), we have = aVT^l? (E" J 2e sin (E" E) cos (E" + )). (56) 248 THEORETICAL ASTKONOMY. From the equations Vr sin & = Va(l + e) sin %E, V^_ sin v" = T/o(l + e) sin \E\ 1/r cos v = l/o (I e) cos , 1/r" cos t/' = l/a(l e) cos .E", since w" v = u"u, we easily derive K tt) = al/1 e 2 sin (" E}, (57) and also a cos (.E" - .E) - ae cos (E" + E) = V~ri' cos (" tt), "- 10 - C58) Substituting this value of e cos^(^ /7 + ^) in equation (56), we get = aVl^e" 2 (-E" E sin (" )) + 2al/T^7 s sin ^ (JB" E) cos (w" ) and substituting, in the last term of this, for al/1 e 2 , its value from (57), the result is T' i/p = aVI^e 2 (E" E sin (' ; )) + rr" sin (" ). (59) From (57) we obtain sn - 8 , 2l/W" cos | (M" t) / p sin 3 ^ (E" E) Therefore, the equation (59) becomes ^ , ( p sin s i (" ) \ 2v/rr" cos (ii" ) Let x' be the chord of the orbit between the first and third places, and we shall have x' 2 = (r + r") 2 4rr" cos 8 -'- (u" u). Now, since the chord x f can never exceed r + r", we may put and from this, in combination with the preceding equation, we derive 21/rr 77 cos (u" u) = (r + r") cos /. (t2) DETER3IIXATIOX OF AX ORBIT. 249 Substituting this value, and [rr"] = -, Vp, in equation (60), it reduces to E"-E-*\n(E"-E} T* l.l-i rfi o, ~~ 7 " sin(jE" E) (r The elements a and e are thus eliminated, but the resulting equation involves still the unknown quantities E" bands'. 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 *', which is the q\iantity sought, may be found. From the equations r = a aecosE, r" = a ae cos E", we get r" + r = 2a 2ae cos (E" + ) cos 4 (E" E). Substituting in this the value of eco&\(E"+E) from (58), we have r" + r = 2a sin' i (") + 2 VrP"' cos 4 (u" w) cos \(E" E\ and substituting for sm\(E" E} its value from (57), there results But, since 2rr" sin' $ (u" ' u) = ([rr"])' = 2g I 1 ^> 2prr /r co&^ ^ (u" u) s* \ '2V rr" cos A (ii" u) /' we have from which we derive which is the additional equation required, involving E" E and J as unknown quantities. Let us now put (G5) - sin .7 = 250 THEORETICAL ASTRONOMY. and the equations (63) and (64) become When the value of y' is known, the first of these equations will enable us to determine ', and hence the value of x f , or sin 2 ("" JE"), from the last equation. The calculation of f' ma y be facilitated by the introduction of an additional auxiliary quantity. Thus, let X (67) and from (62) we find cos / = cos (u" w) _J = 2 cos ^ (w" it) cos 1 / tan /, or cos / = sin 2/ cos | (u" u). (68) We have, also, *" = (r + r") 2 4rr" cos 2 (t*" it), which gives x" = (r r") s + 4rr" sin a (t*" M). Multiplying this equation by cos 2 (w r/ u) and the preceding one by sin 2 \ (u" w), and adding, we get x" = (r + r'J sin 2 \ (u" u) + (r r") 2 cos 2 ^ (u" w). (69) From (67) we get cos V = r _^ / /> sin'/ = y + ^ and, therefore, so that equation (69) may be written (y^'^y = sin2 ^' = sin2 2 (^" - ) + cos 2 2^' cos 1 \ (u" ). We may, therefore, put sin / cos G' = sin ^ (u" ), sin / sin G' = cos ^ (u" u) cos 2/, ( 70) cos f = cos ^ (" w) sin 2/, DETERMINATION OF AN ORBIT. 251 from which f' 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 g = ^(E"-E\ and we shall have 2<7 sin 2g This gives, by differentiation, or -- = By' cot g 4/ 2 cosec g. The last of equations (65) gives x' sin 2 %g, and hence | = 2 cosec g. Therefore we have dy' Qy' cos g By' 1 3 (1 2aQ y' 4y" It is evident that we may expand y' into a series arranged in refer- ence to the ascending powers of x', so that we shall have y' = Differentiating, we get :*'* + &c., and substituting for -^ the value already obtained, there results 2/5) a/ J + (Qd 4 r ) = (3 _ 4a 2 ) + (3/3 6a 8a/?) ^ + (3f 6/? 4/5 2 + (3- _ Q, S r d 8j3e 80:) a;' 5 + &c. Since the coefficients of like powers of x' must be equal, we nave 3 tt 4o s = 0, 3/5 6a 8o/5 = 2/J, 3 r _ 6,3 4/5 2 8r = 2 (2/- /3), &c. ; 252 THEORETICAL ASTRONOMY. and hence we derive = !, = -ft, r=i%-s> . = ,1111,, c = 3 ||$m* Therefore we have (71) If we multiply through by y, and put 4- iff f If Hf*^ *** ( 72 ) we obtain ^- (73) Combining this with the second of equations (66), the result is VY + ^ = !+/ + ' If we put ^f-q^+7>' ( 74 ) we shall have m' ^V* Ay=^T7 But from the first of equations (66) we get ftfiV-iii and therefore we have v-^l a.) As soon as jy' 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 7}', take , _ m' '-FF7 and with this find ' from (75), and the corresponding value of x' from the last of equations (66). With this value of x' we find the corresponding value of c', and recompute r/ } s', and x' ; and, if tho DETERMINATION OP AX ORBIT. 253 value of ' derived from the last value of x' differs from that already used, the operation must be repeated. It will be observed that the series (72) for ' converges with great rapidity, and that for E" ^=94 the term containing x' 6 amounts to only one unit of the seventh decimal place in the value of '. Table XIV. gives the values of ' corresponding to values of x' from 0.0 to 0.3, or from E" E=0 to .E" =132 50'.6. Should a case occur in which E" E exceeds this limit, the expression sin^CE"-ff) y E"E sin(E"E) may then be computed accurately by means of the logarithmic tables ordinarily in use. An approximate value of x' may be easily found with which y' may be computed from this equation, and then ' from (73). With the value of ' thus found, r/ 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 8 1 is always positive, and cannot be less than 1, it follows from this equation that r/ is always a positive quantity. The equation may be written thus : 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 r/ from r/=Q to r/=0.6. When r/ exceeds the value 0.6, the value of s' must be found directly from the equation (75). 89. We are now enabled to determine whether the orbit is an ellipse, parabola, or hyperbola. In the ellipse x = sm z \(E" E) is positive. In the parabola the eccentric anomaly is zero, and hence x = Q. 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 1 derived from the equa- tion is positive, the orbit is an ellipse ; if equal to zero, the orbit i parabola ; and if negative, it is a hyperbola. 254 THEORETICAL ASTBOXOMY. For the case of parabolic motion we have x f = 0, and the second of equations (66) gives * = j- (76) If we eliminate s' by means of both equations, since, in this <>asc, Substituting in this the values of m and I given by (65), we obtain = 3 sin ir' cos r ' + 4 sin' /. which gives 2 sin ^, } -,=(sm This may evidently be written (r + / , }! = (1 + Bin rO* =*=(!- sin /)!, the upper sign being used when 7^ is less than 90, and the lower sign when it exceeds 90. Multiplying through by (r + r")*, and replacing (r -f- r") sin f by X, we obtain 6/ = (r + r" + x)f zp (r + r"- )*, which is identical with the equation (56) 3 for the special case of parabolic motion. Since a/ 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 XIY. gives the value of ' corre- sponding to both forms; but when x r exceeds the limits of this table, it will be necessary, in the case of the hyperbola also, to compute the value of c' directly, using additional terms of the series, or we may modify the expression for y* in terms of E' and E so as to be applicable. If we compare equations (44) t and (56) u we get tan \E = 1/^ DETERMINATION OF AN ORBIT. 255 and hence, from (58)j, We have, also, by comparing (65) : with (41) u since a is negative in ihe hyperbola, which gives Now, since cosE -{- V 1 sinJ^=: e^~ l , in which e is the base of Naperian logarithms, we have E 1/^1 = log e (cos E + l/^T sin E\ which reduces to or JE= l/^~l log. (78) 4 V ffff" or / ._ 1 \ (79) These expressions for y' and x' enable us to find ' when x' exceeds the limits of the table. Thus, we obtain an approximate value of x' by putting , __ m' *-*+? from which we first find s' and then x' from the second of equations (66). Then we compute A from the formula (79), which gives A = 1 2z' + 2lx'* x', (80) y' from (77), and ' from (73). A repetition of the calculation, using the value of ' thus found, will give a still closer approximation to the correct values of x' and s' ; and this process should be continued until ' remains unchanged. 90. The formulae for the calculation of s' will evidently give the value of s if we use r, r f , r", u f , and u", the necessary changes in the notation being indicated at once; and in the same manner using r", r, r f , u, and u f , we obtain s". From the values of 8 and s" thus found, more accurate values of P and Q may be computed by means of the equations (48) and (51). We may remark, however, that if the times of the observations have not been already corrected for the DETERMINATION OF AN CEBIT. 257 time of aberration, as in the case of the determination of an unknown orbit, this correction may now be applied as determined by means of the values of />, />', and p" already obtained. Thus, if t w ', and t^" are the unconnected times of observation, the corrected values will be t =t a - (81) m which log C= 7.760523, expressed in parts of a day; and from these values of t, t', t" we recompute r, r', and r", which values will require no further correction, since p, p', and p", deriv r ed from the first approximation, are sufficient for this purpose. With the new values of P and Q we recompute r, r', r", and u, u', u" as before. and thence again P and $, an( i if the last values differ from the pre- ceding, we proceed in the same manner to a third approximation, which will usually be sufficient unless the interval of time between the extreme observations is considerable. If it be found necessary to proceed further with the approximations to P and Q after the calculation of these quantities in the third approximation has been effected, instead of employing these directly for the next trial, we may derive more accurate values from those already obtained. Thus, let x and y be the true values of P and Q respectively, with which, if the calculation be repeated, we should derive the same values again. Let &x and Ay be the differences between any assumed values of x and y and the true values, or x = x + AZ, y e = y + Ay. Then, if we denote by x r , y ' the values which result by direct cal- culation from the assumed values x and y , we shall have Expanding this function, we get *; *=/(*, y) + A^x + Biy and if &x and Ay are very small, we may neglect terms of the second order. Further, since the employment of x and y will reproduce the same values, we have /(*,y) = 0, and hence, since AZ = x x and Ay = y y, 258 THEOEETICAL ASTEONOMY. In a similar manner, we obtain Let us now denote the values resulting from the first assumption for P and Q by P, and Q v those resulting from P,, Q l by P 2 , Q v and from P 2 , Q 2 by P 3 , 3 ; and, further, let Then, by means of the equations for X 'x and y ' y , we shall have y), b = A' (P -x) + B' (Q -y\ y\ b' = A'(P-x) [f we eliminate A, B, A', and B' from these equations, the results _ P(a!b" a"6') -f P, (o"6 a6") (a'b" a"6') + (a"b a&") + (ab r a'6) _ Q (a'W a"V} -f Q l (a"b ab") + & (aV afb) y (a! l" _ a "6') _|. ( a "fc _ a ft") ^. ( a 6' a'b) (6" + 6') (a'b" ~" 1 ^ ' v"~"* ~*"^ from which we get ~" s (a'b" a"b') + (a"b a&") -f- (06' a'6)' In the numerical application of these formulae it will be more con- venient to use, instead of the numbers P, P p P 2 , Q, Q l} &c., the loga- rithms of these quantities, so that a = log P l log P,b = log Q l log Q, und similarly for a', b', a", b", which may also be expressed in units of the last decimal place of the logarithms employed, and we shall thus obtain the values of logo; and logy. With these values of log x and log y for log P and log Q respectively, we proceed with the final calculation of r, r', r ff , and u, u' } u". When the eccentricity is small and the intervals of time between the observations are not very great, it will not be necessary to employ the equations (82) but if the eccentricity is considerable, and if, in addition to this, the intervals are large, they will be required. It may also occur that the values of P and Q derived from the last hypothesis as corrected by means of these formulae, will differ so DETERMINATION OF AX ORBIT. 259 much from the values found for x and y, on account of the neglected terms of the second order, that it will be necessary to recompute these quantities, using these last values of P and Q in connection with the three preceding ones in the numerical solution of the equations (82). 91. It remains now to complete the determination of the elements of the orbit from these final values of P and Q. As soon as &, t, and u, u', u" have been found, the remaining elements may be de- rived by means of r, r', and u' u, and also from r', r", and " u'; or, which is better, we will obtain them from the extreme places, and, if the approximation to P and Q is complete, the results thus found will agree with those resulting from the combination of the middle place with either extreme. We must, therefore, determine s' and x r from r, r", and u" u, by means of the formulae already derived, and then, from the second of equations (46), we have from which to obtain p. If we compute s and s" also, we shall have (sr'r" sin (u" i/) \ l / s"rr sin (u' 11) \ * -r- -} = \ 7' ")' 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 ) = T/a?, (84) from which E" E may be computed. Then, from equation (57), since e = sin (p, we have sin A (u" w) /r. , Q ->. a cos y = . //_,.. =pr KTT" (80) sin A (" ) for the calculation of a cos (p. But _p = a (1 e 2 )^^ cos 2

^2/^^qpr, we have _o _ p _ 2pcot2/ and from equations (70), sin i (-u" ) tan 6?' . , cos / cot 2/ = - -r - , sin 2y' = - j-f-v - r. cos/ cos^Cw" u) Therefore the formulae (87) reduce to * sin ( - i (w" + )) = -; tan " cos (88) e cos o> tt w = - / sec 2 w w ^ cos Y Vn' from which also e and to may be derived. Then sin <{> = e, and the agreement of cos y as derived from this value of y with that given by (86) will serve as a further proof of the calculation. The longitude of the perihelion will be given by or, when i exceeds 90, and the distinction of retrograde motion is adopted, by n = & w. DETERMINATION OF AN ORBIT. 261 To find a, we have p (a cos ^p) 1 cos*?' p ' or it may be computed directly from the equation T" 4s *rr" cos 1 \ (u" u) sin^ (E" E)' (89) which results from the substitution, in the last term of the preceding equation, of the expressions for a cos

l) we compute K, I, ,3 , a , 6, c, c/, /, and A. The angle I must be less than 90, and the value of ^9 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, ,e o = 2 52' 59"f if, log a = 6.8013583,,, log b = 2.5456342., log c = 2.2328550., ' log d = 1.2437914, log/= 1.3587437., log h = 3.9247G9L The formulae __ 8in(x" , / +/ sin(/"-/) _ sin (V 1} E sin (A - Q) 1 ~sin(A" x) J b give log MI = 9.8946712, log J//' = 9.6690383, log M a = 1.9404111, log J/," = 0.7306625.. The quantities thus far obtained remain unchanged in the suc- cessive approximations to the values of P and Q. For the first hypothesis, from ij sin C = R sin 4.', 1, cos C == *, J^ cos Vt 268 THEORETICAL ASTRONOMY. we obtain logr =9.0782249, log T" = 9.0645575, log P = 9.9863326, log Q = 8.1427824, log c = 2.2298567 B , log k = 0.0704470, log / == 0.0716091, log i? = 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 ^. The value of z' must now be found by trial from the equation sin (/ C) = m sin 4 if. Table XII. shows that of the four roots of this equation one exceeds 180, and is therefore excluded by the condition that sin z' must be positive, and that two of these roots give z r greater than 180 i]/, and are excluded by the condition that z' must be less than 180 ty. 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' = Q V 22".96. The root which corresponds to the orbit of the earth is 18 20' 41", and differs very little from 180 ij/. Next, from : (i + -2- 1+P\ ' 2r we derive log r' = 0.3025672, log / = 0.01 23991, log n = 9.7061 229, log n" = 9.6924555, log P = 0.0254823, log p" = 0.0028859. The values of the curtate distances having thus been found, the heliocentric places for the three observations are now computed from NUMERICAL EXAMPLE. 269 r cos b cos (I O) p cos (A Q) R, r cos 6 sin (7 O) = />sm(A Q), r sin 6 = /> tan /? ; / cos 6' cos (? O ') = /' cos (/ O') ', / cos b' sin (f O') = /o'sm(/l' O')> / sin 6' = p' tan /S' ; r" cos 6" cos (r O") = P" cos (A" Q") ", r" cos 6" sin (" O") = p" sin (X" Q"), r"sin&" = />"tan/S", which give J = 5 14' 39".53, log tan b =8.4615572, logr =0.3040994, I' = 1 45 11 .28, log tan b' = 8.4107555, log /= 0.3025673, I" = 10 21 34 .57, log tan b" =8.3497911, log r" = 0.3011010. The agreement of the value of logr 7 thus obtained with that already found, is a proof of part of the calculation. Then, from firm i 7\ ^-v tan 6" 4- tan 6 tan , sm (} (I" + - 8) = 2co3 ,^_,y i j\ ^\ tan 6" tan b we get & = 207 2' 38".16, i = 4 27' 23".84, u = 158 8' 25".78, u' = 160 39' 18".13, u" = 163 16' 4".42. The equation tan b' = tan i sin (^' ^ ) gives log tan 6' = 8.4107514, which differs 0.0000041 from the value already found directly from p'. This difference, however, amounts to only O'^OS 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' sin (u f u) ~ rr" sin (u" u)' ~ rr" sin (u" 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 ASTEONOMY. which differ in the last decimal places from the values used in finding p and p". According to the equations d log n = 21.055 cot (ti" ') du', d log n" = -f 21.055 cot (u' it) 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 logw agree is 0".15; but for the agreement of the two values of log n", 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' 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 21 34.19, log tan b" =8.3497919, log r"= 0.3011017, & = 207 2' 32".97, i = 4 27' 25".13, u = 158 8' 31".47, u' = 160 39' 23".31, u" = 163 16' 9".22. The value of log tan b f derived from X' and these values of & and i, is 8.4107555, agreeing exactly with that derived from p' directly. The values of n and n" given by these last results for u, u' and u", are log n = 9.7061144, log n" = 9.6924640 ; and this proof will be complete if we apply the correction du'= 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 Q, and we may also correct the times of observation for the time of aberration by means of the formulae t = t C P sec ft if = t ' C P ' sec p, i" = f " Cp" sec /?", wherein log C= 7.760523, expressed in parts of a day. Thus we gfit t = 257.67467, t = 264.41976, if' = 271.38044, NUMERICAL EXAMPLE. 271 and hence log T = 9.0782331, log T> = 9.3724848, log r" = 9.0645692. Then, to find the ratios denoted by s and s", we have sin f cos G = sin $ (u" '), sin f sin G = cos (w" w') cos 2/, cos p = cos \ (u" u') sin 2/ ; tan/'^l, sin Y" cos G" = sin \ (u' u), sin Y" sin G" = cos ^ (u r u) cos 2/', cos /' = cos A (u' it) sin 2/' ; from which we obtain / = 44 57' 6".00, x " = 44 56' 57".50, r= 1 18 35 .90, r " =1 15 40 .69, log m = 6.3482114, log m" = 6.3163548, logy = 6.1163135, log/' = 6.0834230. From these, by means of the equations using Tables XIII. and XIV., we compute s and s". First, in the case of s, we assume ij = y-^-: = 0.0002675, s ~r J and, with this as the argument, Table XIII. gives log s 2 = 0.0002581 . Hence we obtain x' = 0.000092, and, with this as the argument, Table XIV. gives = 0.00000001 ; and, therefore, it appears that a repetition of the calculation is unnecessary. Thus we obtain logs =0.0001290, logs" =0.0001 200. When the intervals are small, it is not necessary to use the formulas 272 THEOEETICAL ASTKONOMY. 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 _ _ ~~ n" cos (u" u') cos (u" u) cos (' w)' \ve find log P = 9.9863451, log Q = 8.1431341, with which the second approximation may be completed. We now compute c , k , 1 , z', &c. precisely as in the first approximation ; but we shall prefer, for the reason already stated, the values of f) and p" computed by means of the equations (22) and (23) instead of those obtained from the last two of the formulse (18). The results thus derived are as follows : log c = 2.2298499 n , log k = 0.0714280, log 1 = 0.0719540, log 7] = 0.3332233, C = 8 24' 12".48, log m = 1.2447277, z' = 9 0' 30".84, log / == 0.3032587, log p' = 0.0137621, log n = 9.7061153, log n"= 9.6924604, lo sf > = 0.0269143, logp"= 0.0041748, I = 515'57".26, log tan b =8.4622524, logr =0.3048368, I' = 7 46 2.76, log tan b' =8.4114276, log /= 0.303258.7, J" = 10 22 0.91, log tan b" = 8.3504332, log i" 0.3017481, ft = 207 0' 0".72, i = 4 28' 35".20, u = 158 12' 19".54, u' = 160 42' 45".82, u" = 163 19' 7".14. The agreement of the two values of log r' is complete, and the value of log tan b f computed from tan b' = tan i sin (V Q ), is log tan b f = 8.41 14279, agreeing with the result derived directly from //. 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, u" u' = 2 36' 21".32, u' u = 2 30' 26".28, u" u = 5& 47".60. From these values of u"u' and u' u, we obtain log s = 0.0001284, log s" = 0.0001193, NUMERICAL, EXAMPLE. 273 and, recomputing P and Q, we get log P = 9.9863452, log Q = 8.1431359, which differ so little from the preceding values of these quantities that another approximation is unnecessary. We may, therefore, from the results already derived, complete the determination of the elements of the orbit. The equations sin / cos G' = sin -^ (u" w), sin / sin G' = cos ^ {u" u) cos 2/', cos y' = cos \ (u" w) sin 2/, give x ' = 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 and Tables XIII. and XIV., we obtain logs' 2 = 0.0009908, log a/ = 6.5494116. Then from _/s'r/'sin(tt" tf)\' p ~\ * r we get log j9 = 0.3691818. The values of log p given by s/r" sin (u" u'} \ 2 / s'W sin (V w) 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 -E) = V7, rini (?) a cos

, M" = E' e sin JE". from which we get M= 338 8' 36".71, IT = 339 54' 10".61, M" = 341 43* 6".97 ; and if M denotes the mean anomaly for the date T=1863 Sept. 21.5 Washington mean time, from the formulae = tf !*(? T) = M" r*(tr T\ we obtain the three values 339 55' 25".97, 339 55' 25".96, and 339 55' 25".96, the mean of which gives J/ = 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) Q = 207 7*> > Ech P tlc and Mean i= 4 28 35:20J Eq-ox 1863.0.

(99) in which log fa = 8.8596330. We have, also, to the same degree of approximation, ioy=i.^ log " = (ioo) For the values log r = 9.0782331, 1(^=9.3724848, log r"= 9.0645692, log/ = 0.3032587, these formulae give log* = 0.0001277, logs' = 0.0004953, log *" = 0.0001199, which differ but little from the correct values 0.0001284, 0.0004954, and 0.0001193 previously obtained. Since secV = 1 + 6 sm 2 tf -f &c., the second of equations (65) gives 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' = f , according to (71), and hence we have and, since s' l 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' from r and r". In the same manner, we obtain (102j 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 . , ,. T' 2 cos 6 / . ., r" 2 cosV' log * = & - T*- in which log |A = 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 s = 0.0001284, log s' = 0.0004951, log s" = 0.0001193. These results for log 8 and log s" are correct, and that for log s' differs only 3 in the seventh decimal place from the correct value. CEBIT FROM FOUR OBSERVATIONS. 281 CHAPTER V. DETERMINATION OF THE OBBIT OF A HEAVENLY BODY FROM FOUB 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 w r ill 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", u'" the corresponding arguments of the latitude, 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 [rV"] = rV" sin ('" ") and ' m= 282 THEORETICAL ASTRONOMY. Then, according to the equations (5) 3 , we shall have nx x' + ri'x" = 0, ny -tf +n"f = 0, W V x" ri"x'" = Q, , X', X", I'" be the observed longitudes, ft ', ", /?'" the ob- served latitudes corresponding to the times t, t f , t", t"', respectively, and A, A', A", A'" the distances of the body from the earth. Further, let A'" cos ?" = it" t and for the last place we have x'" == //" cos A'" R" cos "', y"' = /'" sin x"' #" sin 0'". Introducing these values of x'" and y"' t and the corresponding values of x, x' } x", y, y f , y" into the equations (2), they become = n (p cos A R cos Q) (/ cos A' R cos 0') + n" G>" cos A" tf'cosQ"), = n OB sin A .R sin ) (p 1 sin A' R sin ') + w" (p" sin A" J2" sin "), = w,' (/>' cos /' ft cos ') OB" cos /" J?" cos ") (3) + '" (p"' cos A'" #" cos '"). = n' (p' sin A' ^' sin ') (/>" sin A" 12" sin ") + n'" (p'" sin A'" - R" sin "'). If we multiply the first of these equations by sin I, and the second by cos ^, and add the products, we get = nR sin (x ) (/ sin (A' A) + R sin (A ')) -f n" ( P " sin (A" - A) + R' sin (A - ")) ; (4) and in a similar manner, from the third and fourth equations, we find = n' (p' sin (A'" A') R sin (A'" ')) (5) (p" sin (A'" A") R" sin (A'" ")) ri"R" sin (A'" '") Whenever the values of n, n r , n", and n'" are known, or may be determined in functions of the time so as to satisfy the conditions of motion in a conic section, these equations become distinct or inde- Dendent of each other ; and, since only two unknown quantities p 1 ORBIT FROM FOUR OBSERVATIONS. 283 and p" are involved in them, they will enable us to determine these curtate distances. Let us now put cos p sin (X A) =A, cos p' sin (A" A) =B, cos p' sin (A'" A") = C, cos /? sin (X" A') = D, ( 6 ) and the preceding equations give Ap' sec ,? Bri'p" sec ,3" = nR sin (A ) R f sin (A Q') H- ri'R" sin (JL "), sec /? CP" sec j9"= w'Jr sin (/" ') R" sin (/" 0") (7) If we assume for n and w" 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' 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 f and p" are of the first order, and it is easily seen that if we eliminate p" from these equations, the resulting equation for p' is such that an error of the second order in the values of n and n" may produce an error of the order zero in the result for p', so that it will not be even an approximation to the correct value ; and the same is true in the case of p". It is necessary, therefore, to retain terms of the second order in the first assumed values for w, w', n", and n'"; and, since the terms of the second order involve r r and r", we thus introduce two additional unknown quantities. Hence two additional equations in- volving r', r", p', p" and quantities derived from observation, must be obtained, so that by elimination the values of the quantities sought may be found. From equation (34) 4 we have p' sec p = K cos V vV* R'* sin 2 *', (8) which is one of the equations required ; and similarly we find, for the other eauation. P" sec P' = R" cos V Vi"* R"* sin* 4". (9) 284 THEORETICAL ASTRONOMY. Introducing these values into the equations (7), and putting *'=Vr'*-R"^, x ! '=V / r" i -R" 2 sin 2 4", we get Ax' - Bn"x" = nR sin (A - ) R' sin (A ') + ri'R" sin (A ") AR 1 cos 4' + n"BR" cos 4", ZtoY OB" = n'R sin (A'" ') R" sin (A'" ") + n'"R'" sin (A'" '") n'lXR' cos 4' + CR" cos 4". Let us now put or ,_cos/5"sin(A" /I) cos/3'sinQl'" A') ~~ cos yS' sin (/ A) ' A ~ cos /9" sin (A'" /')' . .. _. . R' sin (A'" ') h R cos 4 -7^ = e , R sin (A _ Q) , jR"' sin (A'" '") 1 = i ?^ = d, -a. O and we have x" = h"n!x' + ri"d" a" + n'c". ^ These equations will serve to determine x' and x", and hence r' and r", as soon as the values of n, n f , n", and n'" 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 e=w'. Lot us now put t"' = *("'-"), r '=k(r-t'\ (is\ and, making the necessary changes in the notation in equations (26) s , we obtain , i* , ^r- , = 1 + i " -r^ dr" From these we get, including terms of the second order, and hence, if we put P" = ^ "=(n' + n'"-l)r" 8 , (17) we shall have, since r ' = r + r r// , When the intervals are equal, we have and these expressions may be used, in the case of an unknown orbit, tor the first approximation to the values of these quantities. The equations (13) and (17) give = n"P'; i / of\ aw J-TTJ-V+W* and, introducing these values, the equations (12) become 286 THEORETICAL ASTRONOMY. Let an approximate value of x' be designated by a? ', and let the value of z" 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' 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 288 THEORETICAL ASTRONOMY. we shall have, according to the equation (67) 3 , the necessary changes being made in the notation, < ~ R " cos (*" '- Q") + n '" R '" cos (*" '~ Further, instead of these, any of the various formulte which have been given for finding the ratio of two curtate distances, may be employed; but, if the latitudes 0, /?', &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. TJie values of ft' and />"' may also be derived by computing the heliocentric places of the body for the times t and t'" by means of the equations (82) t , and then finding the geocentric places, or those which belong to the points to which the observations have been reduced, by means of (90) u writing ft in place of J cos /9. This process affords a verification of the numerical calculation, namely, the values of ^ and X'" thus found should agree with those furnished by observation, and the agreement of the computed latitudes /3 and $'" with those observed, in case the latter are given, will show how nearly the position of the plane of the orbit as derived from the second and third observations represents the extreme latitudes. If it were not desirable to compute 1. and X'" in order to check the calculation, even when ft and ft'" are given by observation, we might derive p and p" f from the equations p = r sin u sin i cot /?, / '" = r"'8intt'"sintcot0" f , 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 /9 and fi" f , to compute p and p'" as soon as p' and ft" have been found, and then find I, V", 6, and &"' directly from these by means of the formulae (71) 3 . The values of ft and i may thus be obtained from the extreme places, or, the heliocentric places for the times t' and t'" being also computed directly from ft' 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 &' and b 1 " from the equations tan b' = tan i sin (P ft), tan b" = tan i sin (I" ft ), they will not agree exactly with the results obtained directly from p f and p", unless the four observations are completely satisfied by the elements obtained. The values of r 7 and r", however, computed directly from p f and p" by means of (71) 3 , must agree with those derived from x' and x". The corrections to be applied to the times of observation on account ORBIT FROM FOUR OBSERVATIONS. 293 of aberration may now be found. Thus, if t w t f , t ", and t "' are the uncorrected times of observation, the corrected values will be t =t a Cf> sec/9, (40) wherein log O= 7.760523, and from these we derive the corrects! values of r, r', r n ', T'", and r '. 100. To find the values of P', P", Q', and Q", which will be exact when r, r', r", r" f , and u, u', u", u" r are accurately known, we have, according to the equations (47) 4 and (51) 4 , since Q' = \Q, (*) COS 2 (u" u') COS j (u" It) COS 3 (u' u)' In a similar manner, if we designate by s'" the ratio of the sector formed by the radii-vectores r" and r'" to the triangle formed by the same radii-vectores and the chord joining their extremities, we find (42) rV" cos 2 (M'" u"} cos 3 (w'" w') cos 2 CM" u')' The formulae for finding the value of s'" are obtained from those for s by writing %'", f rn ', Cr'", &c. in place of /, 7-, G 1 , &c., and using r", r'", u"' u" instead of r f , r", and u" u', respectively. By means of the results obtained from the first approximation to the values of P', P", Q', and Q", we may, from equations (41) and (42), derive new and more nearly accurate values of these quantities, and, by repeating the calculation, the approximations to the exact values may be carried to any extent which may be desirable. When three approximate values of P' and Q f , and of P" and Q", have been derived, the next approximation will be facilitated by the use of the formulae (82) 4 , as already explained. When the values of P', P", Q', and Q" have been derived with sufficient accuracy, we proceed from these to find the elements of the orbit. After , i, r, r', r", r'", u, u', u", and u"' 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'", u, and u'". If the values of P', P", Q f , 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 I?* sin ro cos G, = sin (u" r u), (43) sin YO sin G = cos ^ (u'" u) cos 2/ , cos YQ = cos * 2 (u'" u) sin 2/ , we find ft, and G . Then we have sin from which, by means of Tables XIII. and XIV., to find s and x . We have, further, _ ' 8 rr'" sin (u'" ) \ 8 and the agreement of the value of j9 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' } 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. o f, 1863 Sept. 20.0 14 30' 35".6 -f- 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 + 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 Nautionl 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 \ 16 59' 9".42 10 14 17 .57 29 53 21 .99 75 23 46 .90 + 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, +0".53, +0".36, + 0".19. For the reduction of the geocentric latitudes of the planet to the point in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane, the equation (6) 4 gives the corrections -0".57, 0".38, -0".18, and 0".07 to be applied 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, p = + 2 56' 44".01, f = 80.0, X = 10 14 17 .57, P = 1 15 49 .20, t' = 135.0, A" = 29 53 21 .99, ?' = 2 29 57 .56, *'"= 223.0, A'" = 75 23 46 .90, /?"' = 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, log-R =0.0015899. Q' =256 58 35.9, log If =9.9932638, Q" =312 57 49 .8, logP" =9.9937748, O'"= 40 21 26.8, log B" = 0.0035149, 296 THEORETICAL, ASTROXOMY. From the equations tan/? . tan (/- tan(A"-Q") W== S in(A"-Q")' = we obtain 4' = 113 15' 20".10, log(jR' log (K sin V) =9.9564624, 4"= 76 5617.75, log CR"cosV') = 9.3478848, log (JR" sin V') = 9.9823904. The quadrant in which fy must be taken, is indicated by the condi- tion that cos'*// and cos(^' O') must have the same sign. The same condition exists in the case of ^". Then, the formulae A = cos /?' sin (A' A), B = cos $' sin (X" A), C= cos p' sin (/" A"), J5 = cos ' sin (/"- A'), ,, - , TT" give the following results : log A = 9.0699254,,, log C = 9.8528803, log B = 9.3484939, log D = 9.9577271, log h' = 0.2785685 n , log h" = 0.1048468, log a' = 0.8834880 n , log a" = 9.9752915 n , log d = 0.9012910 n , log c" = 9.7267348^, log d' = 0.4650841, log d" = 9.9096469 n . "We are now prepared to make the first hypothesis in regard to the values of P', Q f , P", 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 EXAMPLE. 297 T T then approximate values of T' 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, a? = after which the values of x' and x" will be obtained by trial from (27). It should be remarked, further, that in the first determination of an orbit entirely unknown, the intervals of time between the ob- servations will generally be small, and hence the value of x' 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' 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" should be computed by means of these elements. Thus, from _rV'sin(t/' tO rr'sin(V v) ~ rr"sin(t>" v)' ~ rt" sin (v" v) ' , _ rV"sm(i/" t/') , _ rV'sinCt/' tQ ~ rV"sin(i/" tO' - /r'" sin (t/" - t/)' we find n, n' } n", and n'". The approximate elements of Ewynonf givo v =322 55' 9".3, logr =0^08327, t/ =353 19 26 .3, log/ =0.294225, v"= 14 45 8.5, logr" =0.296088, v"'= 47 2332.8, log r'" = 0.317278. 298 THEOEETICAL ASTRONOMY. and hence we obtain log n 9.653052, log ri= 9.825408, log n" = 9.806836, logn'" = 9.633171. Then, from we get = ( n '+ n'" 1) r" 8 , logP' = 9.846216, log Q = 9.840771, ' log P" = 9.807763, log Q" = 9.882480. The values of these quantities may also be computed by means of the equations (41) and (42). Next, from we find log c r = 0.541344 B , log c " = 9.807665 n , Then we have log/' = 0.047658., log/" = 9.889385. , tan / = ._ : in sin^ tan 2" , / sin/' from which to find r 1 and r". In the first place, from we obtain the approximate value log a/ = 0.242737. Then the first of the preceding equations gives log x" = 0.237687. NUMERICAL EXAMPLE. 299 From this we get 4' = 29 3' 11". 7, log r" = 0.296092 ; and then the equation for x f gives log af = 0.242768. Hence we have J = 27 20' 59".6, ' log / = 0.294249 ; and, repeating the operation, using these results for x 1 and r', we get log *" = 0.237678, log x' = 0.242757. The correct value of log x' may now be found by means of equation (28). Thus, in units of the sixth decimal place, we have o. = 242768 242737 = + 31, a,' = 242757 242768 = 11, and for the correction to be applied to the last value of log x', in units of the sixth decimal place, A log 3^ = -^!_ = + 3. ' Therefore, the corrected value is log i! = 0.242760, and from this we derive log x" = 0.237681. These results satisfy the equations for x' and x", and give /=2721' 1".2, log/ =0.294242, t" = 29 3 12 .9, log r" = 0.296087. To find the curtate distances for the first and second observations, the formulae are t sin z sin z which give log P ' = 0.133474, log P " = 0.289918. Then, by means of the equations 300 THEORETICAL ASTRONOMY. r' cos V cos (7 0') = p' cos (A' ') K f , r' cos b' sin (F O') = p' sin (/ '), r' sin 6' = p' tan /5', r" cos b" cos (r Q") = p" cos (A" ") -R", r" cos b" sin (r O") = p" sin (/' O"), r"sin&" =p"t&nl?', we find the following heliocentric places : f --=-. 37 P5' 26".4, log tan b' = 8.182861 n , log/ = 0.294243, T' = 58 58 15 .3, log tan b" = 8.634209^, 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 cal cita- tion. From the equations tan i sin ( (r -f- O - ft ) = (tan 6" + tan 6') sec (r - l'\ tan i cos (| (I" + /') ) = (tan 6" tan 6') cosec (" Of we obtain ft = 206 42' 24".0, i = 4 36' 47".2, u' =190 55 6 .6 w"=212 20 53 .5. Then, from we get log n" = 9.806832, logn =9.653048, log n' = 9.825408, log '" = 9.633171, and the equations r sin ((u' - ) + i (u" - tO) = ^ sin -J (" - u'), r cos ((w' tt) + i (w" tt')) = r> ~ nV ' cos (u" tt'), r'" sin (('" - tt") + ^ (M" - w')) = ~~ sin ^ (ft" - tt'), r"' COB((U"' - 1") + i (w" - u')) = r "~r V co 8 J (v" - tt') f NUMERICAL EXAMPLE. 301 give logr =0.308379, u = 160 30' 57".6, log r'" = 0.317273, u'" = 244 59 32 .5. Next, by means of the formula tan (7 ) = cost tan u, tan b = tan i sin (I ), tan (F" SI ) = cos i tan u'", tan b'" = tan i sin (/" & ), />cos(-l O) = r cos 6 cos (J O)+-R, /o sin (^ O ) r cos 6 sin (7 Q), /> tan /9 = r sin b ; ,'" cos (A'" - O'") = r'" cos b'" cos (r"- '") + R", p'" sin (X" O'") = r'" cos V" sin (/"' "'), /"tan ,5"' =r"'sin&"', we obtain J == 7 16' 51".8, r = 91 37' 40".0, b = + 1 32 14 .4, b'" = 4 10 47 .4, I = 16 59 9 .0, X" = 75 23 46 .9, /3 = + 2 5640.1, ?" = 3 443.4, log p = 0.025707, log p'" = 0.449258. The value of /'" thus obtained agrees exactly with that given by observation, but / differs 0".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 ,? and fl" show that the position of the plane of the orbit, as determined by means of the second and third places, will not completely satisfy the extreme places. The four curtate distances which are thus obtained enable us, in the case of an orbit entirely unknown, to complete the correction for aberration according to the equations (40). The calculation of the quantities which are independent of P', P", Q f , and Q", and which are therefore the same in the successive hypotheses, should be performed as accurately as possible. The value of jp required in finding x" from x' y may be computed directly from the values of and being found by means of the equations (29); 302 THEORETICAL ASTRONOMY. c" and a similar method may be adopted in the case of 7/7- Further, in the computation of x' and x", it may in some cases be advisable to employ one or both of the equations (22) for the final trial. Thus, in the prasent 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 -\- -^ 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", r" f , and u, u f , u", u fn ', which result from the first hypothesis, suffice to correct the assumed values of P' } P", Q', and Q". Thus, from T = k(f' O, T" = k(t' t\ r'" = k (If" O, sin f cos G = sin T} (u" u'\ sin /' cos G" = sin ^ (u' w), sin f sin (? = cos (" w') cos 2%, sin /' sin 6 s " = cos \ (u f it) cos 2/', cosy =cos|(w" u')sin2/, cos/' = cos(ti' it) sin2/', sin /" cos G'" = sin (w'" w"), sin /" sin G'" = cos (tt w w") cos 2/'" cos /" == cos A (tt w w") sin 2/" ; r 2 cos 6 / T" 2 cos 6 /' , r"' 3 cos e /" -^^' -T-TT* m =-- Fr3 ^ fff , j a _^_jn f jt> __ P1 " nr -nt __ sin'' 1 \f'" cosy' J cos f" ' J cos /" ' m'" in connection with Tables XIII. and XIV. we find s, s", and s f ". The results are NUMEKICAL EXAMPLE. 303 log r = 9.9759441, j = 45 3'39".l, r = .10 42 55 .9, log m = 8.186217, log; = 7.948097, log s = 0.0085248, log T"= 0.1386714, /'= 44 32' 1".4, /'= 15 13 45 .0, log m"= 8.51 6727, log/'= 8.260013, log s"= 0.0174621, Then, by means of the formulae log r'"= 0.1800041, /"=4541'55".2, r"'= 16 22 48 .5. log m'"= 8.590596, log/"= 8.325365, log s"'= 0.0204063. sy i ^ ^_ V ~~ 2 ss" ' rr" cos ^ (u" u'} cos \ (u" u) cos (u' w)' ' /r'" cos (w'" u") cos O' " it') cos (w" ') we obtain logP' =9.8462100, log P" = 9.8077615, log Q' == 9.8407536, log ft' = 9.8824728, with which the next approximation may be completed. We now recompute c ', c ", /', f", x f , x ff , &c. precisely as already illustrated ; and the results are Iogc ' = 0.5413485 n , log/' = 0.0476614 n , log x' = 0.2427528, / = 27 21' 2".71, log/ =0.2942369, log/ =0.1334635, logn =9.6530445, log n'= 9.8254092, Then we obtain Iogc " =9.8076649 n , log/" =9.8893851, logo/' =0.2376752, z" = 29 3' 14".09, log/' =0.2960826, logp" =0.2899124, logn" =9.8068345, log ri" = 9.6331707. I' = 37 35' 27".88, T=58 58 16 .48, log tan V = 8.1828572 n , log tan b" =8.6342073., log/ =0.2942369, log/' =0.2960827. These results for logr' and logr" agree with those obtained directly from z' and z", thus checking the calculation of a// and ty' and of the heliocentric places. Next, we derive = 206 42' 25".89, i = 4 36' 47".20, u' 190 55 6 .27, u" = 212 20 52 .96, 304 THEORETICAL ASTRONOMY. and from u" ' u f , r', r", n, n", n', and n" r , 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, 1= 716'51".54, l'"= 91 37' 41".20, & = + 1 32 14 .07, b'"= 4 10 47 .36, 1= 16 59 9.38, *"'== 75 2346.99, = + 2 56 39 .54, F"= 3 4 43 .33, log P = 0.0256960, log //" = 0.4492539. The values of I and V" thus found differ, respectively, only 0".04 and 0".09 from those given by the normal places, and hence the accuracy of the entire calculation, both of the quantities which are independent of P', P", Q', and Q", and of those which depend on the successive hypotheses, is completely proved. This condition, however, must always be satisfied whatever may be the assumed values of P', P", Q', and Q". From r, r', w, u f , &c., we derive log = 0.0085254, log s" = 0.0174637, log s'" = 0.0204076, and hence the corrected values of P', P", Q', 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 T. = *("'-- 0, sin Y O cos G Q = sin -\ (u'" u), sin YO sin G = cos ^ (u'" u) cos 2/ , cos y = cos ^ (it'" it) sin 2/ , (r + r'") s cos* YQ ^ cos ^ 6 ' '^ITSH? - fc- r ^ w > NUMERICAL EXAMPLE. 305 we get log r = 0.5838863, log tan G 9 = 8.0521953,, r. = 42 14' 30".17, log m, = 9.7179026, log *.* = 0.2917731, log * = 8.9608397. The formula _/*orr"'sin(u"' tt)\ gives log p = 0.371 2401; and if we compute the same quantity by means of _ / *rV'sin(M" Q \'_ / s"rr'8m(V u) \_ / <"VV" 8 m(ti"' M") \ P ~\ r )-\ r" |-\ ?* -J, 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", Q', and Q" from the limit of accuracy attainable with logarithms of seven decimal places. A variation of only 0".2 in the values of u' u and u'" u" wil 1 produce an entire accordance of the particular results. From the equations sin !('" -) = >/*., we obtain P cos

' a$ ' the value of k being expressed in seconds of arc, or logfc = 3.5500066, we get log a = 0.3881359, log ft 2.9678027. For the eccentric anomalies we have w) tan (45 ?), E f = tan(u' ) tan (45 ?), tan ^E" = tan (M" o) tan (45 p), tan J ,E'" = tan (tt w ) tan (45 ?), from which the results are E = 329 11' 46".01, " = 12 5' 33".63, ' = 354 29 11 .84, '" = 39 34 34 .65. The value of J (E'" E} thus derived differs only 0".03 from that obtained directly from x . For the mean anomalies, we have M =E esinE, M" =E" M' = E'e sin E', M'" = E'" e sin E'", which give M = 334 55' 39".32, M" = 9 44' 52".82, M' = 355 33 42 .97, .'" = 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 = M p.(t T) =M' fi(1fT) = M"-n (If' T~) = M'" - ft ({" - T), we obtain the four values M = l 29'39".40 39 .49 39 .40 39 .40, the agreement of which completely proves the entire calculation of the elements from the data. Collecting together the several results, we have the following elements : NUMERICAL EXAMPLE. 307 Epoch = 1864 Jan. 1.0 Greenwich mean time. M = 1 29' 39".42 = 4 36 47. 20 V = 11 15 52 .22 log a = 0.3881 359 log j = 2.9678027 n = 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/9 = 4".47, Ay9'" = -f 1".23. These remaining errors arise chiefly from the circumstance that the position of the plane of the orbit cannot be determined from the second and third places with the same degree of precision as from the extreme places. It would be advisable, therefore, in the final approximation, as soon as p', p", n, n", n f , and n" f are obtained, to compute from these and the data furnished directly by observation the curtate distances for the extreme places. The corresponding heliocentric places may then be found, and hence the position of the plane of the orbit as determined by the first and fourth observations. Thus, by means of the equations (37) and (38), we obtain log p = 0.0256953, log p'" = 0.4492542. With these values of p and p'", the following heliocentric places are obtained : I = 7 16' 51".54, log tan b =8.4289064, logr =0.3083732, r = 91 37 40 .96, logtan&'" = 8.8638549 n , log r"' = 0.3172678. Then from tan.t sin ( (?" + f) ) = \ (tan b'" + tan 6) sec (f" I), tan i cos ( (V" + ) = 2 (tan b"' tan 6) cosec^ (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. 308 THEORETICAL ASTRONOMY. The equations tan b' = tan i sin (/ SI )> tan b" = tan i sin (I" r- ft), give log tan 5' = 8.1827129 n , log tan b" = 8.6342104,, and the comparison of these results with those derived directly from ft' and p" exhibits a difference of + 1".04 in b', and of 0".06 in b". Hence, the position of the plane of the orbit as determined from the extreme places very nearly satisfies the intermediate latitudes. If \ve compute the remaining elements by means of these values of r, r'", and u, u" f , the separate results are : log tan G = 8.0522282 n , log m = 9.7179026, log = 0.2917731, log x = 8.9608397, logp = 0.3712405, i (E" E) = 17 35' 42".12, log (a cos = 197 37' 47".72, log e 9.2907906, dr dM dr dp ~dA~~dj'dA + ~dM ' ~dA ~r~dH"dA' and dv _ dv d

. dv (Of, . rfv d;j. 1 . , fl '+' + ' = ' ' ( _ _ = dJ + ^ ' d A "*" dlf ' rf J "*" ^ ' d J dr" ?, 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 formula? cos y A0 sin Y A<5 -f- cos f cos 8 Aa, Aij = cos f A<5 sin f cos , and the differential coefficients of and q, with respect to the elements of the orbit, need not be determined with great accuracy. Next, we compute -^ and ^ J * from equations (12), and from (16), the values of * ^' * $C =, Ac., by means of which, dy d d d

'. If AJ 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", p, and e by means of the polar equation of the conic section. Finally, we have dz = sin T) dA, and similarly for dz" and the last of equations (73) 2 gives rsinwAi' rcoswsint' A&' =sini?AJ, r" sin u" Ai' r" cos u" sin i' A & ' = sin r t " A J", (18) from which to find AI V and A & ', u and w" being the arguments of the latitude in reference to the equator. We have also, according to (72), Aw = A/ COSt' A&', A-' = A/ -f- 2 sin* ^ A ', from which to find the corrections to be applied to a> f and ~ r . The elements which refer to the equator may then be converted into those for the ecliptic by means of the formulae which may be derived from (109^ by interchanging SI and &' and 180 i' and t. The final residuals of the longitudes may be obtained by substi- tuting the adopted values of A J and AJ" in the several equations of condition, or, which affords a complete proof of the accuracy of the entire calculation, by direct calculation from the corrected elements ; and the determination of the remaining errors in the values of ^ will show how nearly the position of the plane of the orbit corresponding to the corrected distances satisfies the intermediate latitudes. Instead of

, respect- ively, in the equations (13), (14), and (15), and the partial differential coefficients of r, r", v, and v" with respect to these elements must be computed by means of the various differential formulae which have already been investigated. Further, in all these cases, the homo- geneity of the formulae must be carefully attended to. 108. The approximate elements of the orbit of a heavenly body may also be corrected by varying the elements which fix the position of the plane of the orbit. Thus, if the observed longitude and lati- tude and the values of SI and i are given, the three equations (91), will contain only three unknown quantities, namely, J, r, and u, and the values of these may be found by elimination. When the observed latitude /9 is corrected by means of the formula (6) 4 , the latitudes of the sun disappear from these equations, and if we multiply the first by sin (O SI) sin/9, the second (using only the upper sign) by cos ( O SI ) sin /9, and the third by sin (^ O) cos /9, and add the products, we get ~~ cos i sin /3 cos (O SI ) sin i cos ft sin (>i O)' from which u may be found. If we multiply the second of these equations by sin /9, and the third by cos /9 sin (A SI), and add the products, we find (20) sin u (sin i cot /3 sin U Q ) cos i) The expression for r in terms of the known quantities may also be found by combining the first and second, or by combining the first and third, of equations (91) r If we put n cosN= sin /? cos (O SI), nsmN= cos /3 sin 0* O). the formula for u becomes cos N The last of equal ions (91) t shows that sin u and sin/9 must have the same sign, and thus the quadrant in which u must be taken is deter- mined. Putting, also, m cos M = sin u, msin H ;= sin w cot /3 sin (>l SI), VARIATION OF THE NODE AND INCLINATION. 25 we have __cos^_ JZejn_(0--jB) cos (If +0" sinu When any other plane is taken as the fundamental plane, the latitude of the sun (which will then refer to this plane) will be re- tained in the equations (91)! and in the resulting expressions for u and r. The value of u may also be obtained by first computing w and -^ by means of the equations (42) 3 , and then, if z denotes the angle at the planet or comet between the earth and sun, the values of u and z, as may be readily seen, will be determined by means of the rela- tions of the parts of a spherical triangle of which the sides are 180 (z + ^), 180 -f O SI, and u, the angle opposite to the side u being that which we designate by tr, and the side 180 -f O & being included by this and the inclination i. Let S= 180 (z and, according to Napier's analogies, this spherical triangle gives _ ' from which S and u are readily found. Then we have z = 180 4 S, snz to find r. If we assume approximate values of R and /, as given by a system of elements already known, the equations here given enable us to find r, U, r", and u" from A, /9 and /", /3", corresponding to the dates t and t" of the fundamental places selected, and from these results for two radii-vectores and arguments of the latitude, the remaining elements may be derived. From these the geocentric place of the body may be found for the date t' of any intermediate or additional observed place, and the difference between the computed and the observed place will indicate the degree of precision of the assumed values of ft and L Then we assign to ft the increment , w", and ty' remain unchanged for the three hypotheses. When the equator is taken as the fundamental plane, ^ is the distance between two points on the celestial sphere for which the geocentric spherical co-ordinates are A, D and a, 3, those of the sun being denoted by A and D. Hence we shall have sin 4 sin B = cos d sin (a A), sin 4 cos B = cos D sin 3 sin D cos <5 cos (a A), (26) cos 4 = sin D sin 8 -j- cos D cos S cos (a A), from which to find ^ and B, the angle opposite to the side 90 d of the spherical triangle being denoted by B. Let K denote the right ascension of the ascending node on the equator of a great circle passing through the places of the sun and comet or planet for the time t, and let w denote its inclination to the equator; then we shall have sin w cos ( A K) = cos B, sin w sin (A K} = sin B sin D, (27) cos w = sin B cos D, from which to find w and K. In a similar manner, we may com- VARIATION OF THE NODE AND INCLINATION. 327 pute the values of u" u, ft, and i from the heliocentric spherical co-ordinates I, b and I", b". From the equations the accents being added to distinguish the elements in reference to the equator from those with respect to the ecliptic, the values of S and u (in reference to the equator) may be found. Let denote the angular distance between the place of the sun and that point of the equator for which the right ascension is K, and the equation cot = cos w cot (K A) (29) gives the value of * , the quadrant in which it is situated being deter- mined by the condition that cos and cos(^T A) shall have the same sign. Then we have S = , and z = 180 4, & + s , r= * < 30 > sin 2 from which to find r. 109. In both the method of the variation of two geocentric dis- tances and that of the variation of Q> and i, instead of using the geocentric spherical co-ordinates given by an intermediate observa- tion, in forming the equations for the corrections to be applied to the assumed quantities, we may use any other two quantities which may be readily found from the data furnished by observation. Thus, if we compute r' and u' for the date of a third observation directly from each of the three systems of elements, the differences between the successive results will furnish the numerical values of the partial differential coefficients of r' and u' with respect to J and J", or with respect to & and t, as the case may be. Then, computing the values of r' and u' from the observed geocentric spherical co-ordinates by means of the values of SI and i for the system of elements to be corrected, the differences between the results thus derived and those obtained directly from the elements enable us to form the equations ** + >,=*. 328 THEORETICAL ASTRONOMY. or the corresponding expressions in the case of the variation of & and t, by means of which the corrections to be applied to the as- sumed values will be determined. In the numerical application of these equations, AW' being expressed in seconds of arc, Ar' should also be expressed in seconds, and the resulting values of AJ and A A" will be converted into those expressed in parts of the unit of space by dividing them by 206264.8. When only three observed places are to be used for correcting an approximate orbit, from the values of r, r', r" and u, u', u" obtained by means of the formulae which have been given, we may find p and a or -- the latter in the case of very eccentric orbits from the first a and second places, and also from the first and third places. If these results agree, the elements do not require any correction; but if a difference is found to exist, by computing the differences, in the case of each of these two elements, for three hypotheses in regard to J and J" or in regard to & 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 SI and i, will be obtained. 1.10. 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 formula apply with equal facility whether three or more than three observed places are used. But when a series of places can be made available, the problem may be successfully treated in a manner such that it will only be necessary to vary one geocentric distance. Thus, let x, y, z be the rectangular heliocentric co-ordinates, and r the radius-vector of the body at the time t, and let X, Y, Z be the geocentric co-ordinates of the sun at the same instant. Let the geocentric co-ordinates of the body be designated by x , y w z , 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 = pcosa., ?/ = /> sin a, z = pta l nd. (32) If we represent the right ascension of the sun by A, and its declina- tion by Z), we also have VARIATION OF ONE GEOCENTRIC DISTANCE. 329 X= RcosDcosA, Y=RcosDsmA, Z=RsinD. (33) The fundamental equations for the undisturbed motion of the planet or comet, neglecting its mass in comparison with that of the sun, are d*x k*x d*y .k 2 y d*z 1?z but since -*r .. .. ~y 17 and, neglecting also the mass of the earth, tfX k*X _ ^YjL^Z d * Z & z _ these become C84) Substituting for a? , y w and z their values in terms of a and S, and putting we get rf*x n . ^V i * A ^ + - ? fcosa + ,=0, rin. + , = 0, (36) Differentiating the equations (32) with respect to t, we find dx dp 330 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 dt~^ da ' dt Now, from (35) we get $ sin a TI cos a P I - I M cos D sin (a A), and the preceding equation becomes The value of -jr thus found is independent of the differential co- at T efficients of d with respect to t. To find another value of -J-, using all three of equations (38), we multiply the first of these equations by sin A tan d, the second by cos A tan d, and the third by sin (a ^4). Then, adding the products, since sin A = y cos A t the result is from which we get t g-co t (.- *" VARIATION OF ONE GEOCENTRIC DISTANCE. 331 When the ecliptic is taken as the fundamental plane, the last term of the numerator of the second member of this equation vanishes, and the equation may be written Ihc coefficient C being independent of p. 111. When the value of p is given, that of ~ will be determined in terms of the data furnished directly by observation and of the differential coefficients of a and d with respect to t from equation (39), or from (40), the latter being preferred when the motion of the body in right ascension is very slow. The value of -7- having been found, we may compute the velocities of the body in directions parallel to the co-ordinate axes. Thus, since the equations (37) give dx dp . da dX dt= =cosa dt- p8ma dt--dt' dy dp , da dY - --- = a - p a ---, (42) dz dp . ..dS dZ by means of which -^ ~, and - may be determined. * at at dt J T7" 7 T7" J 17 To find the values of -, -, and -, the equations Y= ^sinO cose, Z = It sin O si* 1 e, give, by differentiation, dX ^dR _ dQ -^smO-^-, + J2 cos oos., (43) ^ _ = COSO _ - dt 332 THEORETICAL ASTRONOMY. Now, according to equation (52) u we have dt m denoting the mass of the earth, and e the eccentricity of its orbit. The polar equation of Ue conic section gives dr _ r 2 e sin v dv ~dt ~ ~~p W Let r denote the longitude of the sun's perigee, and this equation gives _ ^ jg = g^rin(0-.r) t j0 == *!/! + : iu(0 _ r) . (45) dt 1 e* dt V I e* If we neglect the square of the eccentricity of the earth's orbit, we have simply dR 7 /^ ; . ,_ m .... -^ = kVl+m e sm(Q T). (4b) The values of r- and rr having been found by means of these dX dY formulae, the equations (43) give the required results for -3-, -5-1 and dZ ~p and hence, by means of (42), we obtain the velocities of the comet or planet in directions parallel to the co-ordinate axes. 112. The values of x, y, and z may be derived by means of the equations x = A cos 3 cos a X, y = J cos 3 sin a Y, z = A sin 5 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) t by cos &, and the second by sin &, and adding the pro- ducts, we have VARIATION OF ONE GEOCENTRIC DISTANCE. 333 r sin u = z cosec i, r cos u = x cos & + # sin & , ( from which r .and u may be found, the argument of the latitude u being referred to the plane of xy as the fundamental plane. The equation r* = z> + f + z* gives dr_x dx y dy z dz dt~r'dt ^r' dt^r'dt' and, since dr r'esinv dv dv kl/p dt p dt' dt ' r* we shall have V~ dr p e cosv = - 1, r from which to find e and v. Then the distance between the peri helion and the ascending node is given by cw = U V. The semi-transverse axis is obtained from p and e by means of the relation Finally, from the value of v the eccentric anomaly and thence the mean anomaly may be found, and the latter may then be referred to any epoch by means of the mean motion determined from a. In the case of very eccentric orbits, the perihelion distance will be given by 9 =r-b ; and the time of perihelion passage may be found from v and e by means of Table IX. or Table X., as already illustrated. The equation (21)! gives, if we substitute for / its value in terms of p, denote by V the linear velocity of the planet or comet, and neg- lect the mass, Let o^o denote the angle which the tangent to the orbit at the ex- iremity of the radius-vector makes with the prolongation of this radius-vector, and we shall have 334 THEORETICAL ASTRONOMY. dr dx . dy . dz - BO that the preceding equation gives Vp= Hence we derive the equations *!/, dx , dy . dz x-^ + y-^ + z w , from which Vr and o^o ma y be found. Then, since we shall have M OU _ = F 2 > (51) a r by means of which a may be determined, and then e may be found by means of this and the value of p. The equations (49) and (50) give F 2 e sin (u /) = -p- r sin 4 cos 4 , F 2 e cos (u u>) = -p- r sin 1 4- 1, and, since II 2 _ 1 A 1 ~~r a these are easily transformed into 2ae sin (w >) == (2a r) sin 2-4/ , 2ae cos (u 4/ r. If we multiply the first of these equations by cos u and the second by sinw, and add the products; then multiply the first by sin it and the second by cos u, and add, we obtain 2ae sin 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 3 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 coefficient 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 A0 = 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 A0. 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 = Q 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, B, 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, B, and C will be derived. In like manner, the corrections to be applied to the computed latitudes may be determined. These corrections being applied, the ephemeris thus obtained may be assumed to represent the ap|>arent 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 /(a) denote the right ascension of the body at the middle epoch or that for which h = 0, and let /(a not) denote the value of a for any other date separated by the interval nta, in which to 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. Diffi The series of functions and differences may be extended in the same manner in either direction. If we expand f(a + nu>) into a series, the result is or, putting for brevity A = -j- w, B = \ -=- co?, &c., /(a -f wo) = a + An -f Bn* + Cn s + Dn* + &c. If we now put n successively equal to 4, 3, 2, 1, U, + 1, &c., we obtain the values of f(a 4w),/(a 3o>), ...... /(a + 4>) in terms of A, _B, (7, &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 A, S, C, &c. by the quantities which they represent, and putting we obtain W = V (/ ' (a) ~ *' '" (a) + ^/ T () - T^/ TU () + &C-). I? = -^ (/" () - -W (a) + ^/ Tl (a) - ,^/ Ttu (a) + Ac.), 3? = ^ (/"' () - l/ v () + T W" () - &C.), = (/' V ^> ~ 1/ H ^ a ) + 2!u/ vi " () ~ Ac.), (54 j y--3r(T l CiO-ir" ^ = ^T CT () - Ac.), g = -1 (/-(a) - &c.), VARIATION OF ONE GEOCENTRIC DISTANCE. 337 by means of which the successive differential coefficients of a with respect to t may be determined. The derivation of these coefficients in the case of 3 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 3 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 ~, -^, -^, 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, &c. 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 %-i%-$=;-w-v+* ' These equations, being solved numerically, will give the values of -7- and , and we may thus by triple combinations of the observed ctt 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. In a similar manner the values of -,- and -5- 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 22 338 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 h being the interval between the middle date t' and that of the place used, the values of A, B, C, A', &c.; and the corrections to be applied to the ephemeris will be determined by A -f Bna> -f 2 = 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 compute the values of -77 > -T=, -77, and -7 as already explained. These ctt ctt cLt dt> differential coefficients should be determined with great care, since it is on their accuracy that the subsequent calculation principally de- , TTT , , dX dY , dZ . pends. We compute, also, the velocities -r-> -77, and -rr by means TX-X fij? at at at of the formula? (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 ft = A cos 8 ; 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 p the coefficient -^ remains the same in the two hypotheses. The three equations (38) may be so combined that the resulting value of -~ d?a, will not contain -7 This transformation is easily effected, and may be advantageous in special cases for which the value of -^ is very uncertain. The heliocentric spherical co-ordinates will be obtained from the RELATION BETWEEN TWO PLACES IN THE ORBIT. 339 assumed value of J by means of the equations (106) 3 , and the rec- tangular co-ordinates from x = r cos b cos I, y = rcosb sin /, z = r sin b. The velocities ^-, ~, and -^ 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) u (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 d + <5J, and compared with the observed places. Let the difference between computation and observation for either of the two spherical co-ordinates be denoted by n for the first system of elements, and by n' for the second system. The final correction to be applied to J, in order that the observed place may be exactly represented, will be determined by .(n'-n) + = 0. (56) Each observed right ascension and each observed declination will thus furnish an equation of condition for the determination of AJ, observing that the residuals in right ascension should in each case be multiplied by cos d. Finally, the elements which correspond to the geocentric distance J + A J will be determined either directly or by interpolation, and these must represent the entire series of observed places. 115. The equations (52) 3 enable us to find two radii-vectores when the ratio of the corresponding curtate distances is known, provided that an additional equation involving r, r", X, and known quantities is given. For the special case of parabolic motion, this additional equation involves only the interval of time, the two radii-vectores, and the chord joining their extremities. The corresponding equation for the general conic section involves also the semi-transverse axis of the orbit, and hence, if the ratio M of the curtate distances is known, this equation will, in connection with the equations (52)j, 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 I J40 THEORETICAL ASTRONOMY. -,= E" - E - 2e sin (E" - E} cos 4 (E" + ), (5?) r + r" = 2a 2ae cos (" E} cos | (" -f E\ For the chord x we have x' = (r + r") 1 4rr" cos 1 (" -u), which, by means of (58) 4 , gives and, substituting for r + r ;/ its value given by the last of equation? (57), we get x s = 4a' sin 2 -\ (E" E}(1 e* cos'^E" + .E)). (58) Let us now introduce an auxiliary angle A, such that cos h = e cos \ (E" + E\ the condition being imposed that h shall be less than 180, and put then the equations (57) and (58) become = 2^ 2 sin g cos h, x = '2a sin # sin h. Further, let us put and the last two of equations (59) give Introducing 3 and e into the first of equations (59), it becomes 4 = (e sin ) (* sin 5). (61) a* The formulae (60) enable us to determine e and # from r -f ?'", x, and a, and then the time r f = k (t" t) may be determined from (61). Since, according to (58) 4 , Vrr" cos (" w) = a (cos $r cos 7i) = 2 sin ^e ein ^5, RELATION BETWEEN TWO PLACES IN THE ORBIT. 341 and since sin \z is necessarily positive, it appears that when u" u exceeds 180, the value of sin Id must be negative, and when u" u = 180, we have 3 = 0-, and thus the quadrant in which <5 must be taken is determined. It will be observed that the value of \s, as given by the first of equations (60), may be either in the first or the second quadrant ; but. in the actual application of the formulae, the ambiguity is easily removed by means of the known circumstances in regard to the motion of the body during the in- terval t" t. In the application of the equations (52) 3 , by means of an approxi mate value of x we compute d, and thence r and r". Then we com- pute e and d corresponding to the given value of a, and from (61) we derive the value of ' F If this agrees with the observed interval t" t, the assumed value of K is correct; but if a difference exists, by varying x we may readily find, by a few trials, the value which will exactly satisfy the equations. The formulae (70) 3 will then enable us to determine the curtate distances p and p", and from these and the observed spherical co-ordinates the elements of the orbit may be found. As soon as the values of u and u" have been computed, since e d = E" E, we have, according to equation (80)4, sin A (u" u) / r , cos

, q = 2a sin 1 (45 ^?) ; and the value of to may be found by means of the equations (87) 4 or (88) 4 . 116. The process here indicated will be applied chiefly in the de- termination of the orbits of comets, and generally for cases in which a is large. In such cases the angles and d will be small, so that the slightest errors will have considerable influence in vitiating the value of t" t as determined by equation (61); but if we transform this equation so as to eliminate the divisor a* in the first member, the uncertainty of the solution may be overcome. The difference sine '342 THEORETICAL ASTRONOMY. may be expressed by a series which converges rapidly when e is small, Thus, let us put e sin = y sin* ^e, a? = sin 2 -je > and we have -~ = 2 cosec \s | y cot ^e, -j = 4 cosec -^e. Therefore dy _ 8 6y cos ^e _ 4 3y (1 2a?) _ _ da; ~ sin^e 2*(1 x) If we suppose y to be expanded into a series of the form we get, by differentiation, and substituting for -~ the value already obtained, the result is 2#e + (4 r 2/3) x 1 + (65 4 r ) a? + &c. = 4 3a + (60 3,9) x + (6/9 - 3 r ) ^ l + (6r - 35) * + &c. Therefore we have 4 3a:=0, 60 3^9=2/9, 6 3r = 4r 2/9, 6r 35 = 65 4r, from which we get 4.6 4.6.8 4.6.8.10 . >&C " - 37ff 3T5T7' 3.5.7.9 Hence we obtain and, in like manner, which, for brevity, may be written e sin e = | Q sin* e, RELATION BETWEEN TWO PLACES IN THE ORBIT. 343 Combining these expressions with (61), and substituting for sin s and sin %d their values given by the equations (60), there results 8t'=g(r + f" + x)t=F #0 + r "_x)f, (65) the upper sign being used when the heliocentric motion of the body is less than 180, and the lower sign when it is greater than 180. The coefficients Q and Q' represent, respectively, the series of terms enclosed in the parentheses in the second members of the equations (62) and (63), and it is evident that their values may be tabulated with the argument e or d, as the case may be. It will be observed, however, that the first two terms of the value of Q are identical with the first two terms of the expansion of (co$\s)~ into a series of ascending powers of sin Je, while the difference is very small between the coefficients of the third terms. Thus, we have (cos Je)- = (1 sin 2 JO'* =1 + 1 sin 2 ^ + g--~ sin* $e 6 . 11 . 16 . + 5Tl the expression for r/ becomes T ' = T ^-* T o'' (77) Table XY. gives the value of log N corresponding to values of e from e = to e = 60. If the chord x is given, and the interval of time t" t is required, we compute Ar/ by means of (76), and, having found r/ from r " as in the case of parabolic motion, we have It should be observed that although equation (76) is derived for the case of a small value of x, yet it is applicable whenever the differ- ence d is very small, whatever may be the value of x. For orbits which differ but little from the parabolic form, it will in all cases be sufficient to use this expression for Ar/; and for cases in which the difference between and d is such that the assumption of cos \s = cos 8, x -f- x' = 2x, &c., made in deriving equation (70), does 346 THEORETICAL ASTRONOMY. not afford the required accuracy, we may compute both Q and Q r directly, and then we have + r "_ x )f. (78) (/y \ 1 ~ \ may be tabulated directly with as the vertical argument and -j- as the horizontal argument; but for the few cases in which the value of N given by the equation (75) is not sufficiently accurate, it will be easy to compute Q and Q' by means of the formulae. (66) and (68), and then find Ar ' from (78). Further, when there is any doubt as to the accuracy of the result given by (76), for the final trial in finding x from r + f" and r by means of the equations (73) and (74), it will be advisable to compute Ar ' from (78). It appears, therefore, that for nearly all the cases which actually occur the determination of the value of x, corresponding to given values of a and M = > is reduced by means of the equation (72) to the method which is adopted in the case of parabolic orbits. The calculation of the numerical values of r + r"-\- x and r + r" x will be most conveniently effected by the aid of addition and sub- traction logarithms. If the tables of common logarithms are used, we may first compute and then we have r -f r" + x = 2 (r + r") sin 2 (45 + r") cos 2 (45 + 118. In the case of hyperbolic motion, the semi-transverse axis is negative, and the values of sin \s and sin \8 given by the equations (60) become imaginary, so that it is no longer possible to compute the interval of time from r -f- r" and K by means of the auxiliary angles e and d. Let us, therefore, put sin* e = m 1 , sin 2 ^ = n f ; then, when a is negative, m and n will be real. Now we have e = sin ~ l V m 2 , 8 = sin "' 1/ n*, and | e \/^l = log. (cos ^ RE1ATIOX BETWEEN TWO PLACES DT THE ORBIT. 347 Hence we derive e == 2 sin - 1 V'^tf = = log. (l/l + m* + m), <* = 2 sin ~* l/=r^i = _!== bg. (i/l + n' + n). Substituting these values in the equation (61), and writing a in- stead of a, since sin e = 2ml/ 1- 1/1 + m*, we shall have ^ = 2m l/r+^ - 2 lo ge (VT+1? + m ) ( +V - 2 log. (l/l+T f + )), the upper sign being used when the heliocentric motion is less than 180, and the lower sign when it is greater than 180. Therefore, if we compute m and n from (80) regarding the hyperbolic semi-transverse axis a as positive, the for- mula (79) will determine the interval of time T' = k (t" t). The first two terms of the second member of equation (79) may be expressed in a series of ascending powers of m, and the last two terms in a series of ascending powers of n. Thus, if we put lg (v'l + m* + m) = om + /3m* + Y? -f ' + 3^| m <_- &c .), ( and similarly In 1/1 + n 1 2 log e (I/ 1 + ft 1 + n) = (82) Substituting these values in the equation (79), and denoting the series of terms enclosed in the parentheses by Q and Q f , respectively, we get 6r' = Q (r + r" + x)* + ' (r + r" x)l (83) which is identical with equation (65). If we replace m 2 by sin 2 s and n 2 by sin 2 ^ in the expressions for Q and Q', as given by (81) and (82), we shall have the expressions for these quantities in terms of sin \s and sin \d, respectively, instead of sin \ and sin \d as given by the equations (62) and (63), namely, 1.3.5 (84) For the case of an elliptic orbit it is most convenient to use the equations (66) and (68) in finding Q and Q f ; but, since the cases of hyperbolic motion are rare, while for those which do occur the eccen- tricity is very little greater than that of the parabola, it will be suf- ficient to tabulate Q directly with the argument m. The same table, using n as the argument, will give the value of Q'. Table XVI. gives the values of Q corresponding to values of m from m = to m = 0.2. When the values of r -f- r", r', and a are given, and the chord x is required, we may compute Ar ' from (78), r a f from (77), and finally x from (73). It may be remarked, also, that the formulae for the relation between r', r-\-r n ', X, and a suffice to find by trial the value of a when r + r" and x are given. Hence, in the computation of an orbit from assumed RELATION BETWEEN TWO PLACES IN THE ORBIT. T, 40 values of J and A", the value of X may be computed from r, r", and u" - u, and then a may be found in the manner here indicated. If we substitute in the equations (84) the values of sin %s and sin \d in terms of r -f r", x, and a, and then substitute the resulting values of Q and Q' in the equation (65), we obtain + sfg ^ ((r + r" + x)* rp (r + r" - x) J) + &c. f the lower sign being used when u" u exceeds 180. When the eccentricity is very nearly equal to unity, this series converges with great rapidity. In the case of hyperbolic motion, the sign of a must be changed. 119. The formulae thus derived for the determination of the chord x for the cases of elliptic and hyperbolic orbits, enable us to correct an approximate orbit by varying the semi-transverse axis a and the ratio M of two curtate distances. But since the formulae will gene- rally be applied for the correction of approximate parabolic elements, or those which are nearly parabolic, it will be expedient to use - and M as the quantities to be determined. In the first place, we compute a system of elements from J/ and /=-; and, for the determination of the auxiliary quantities pre- liminary to the calculation of the values of r, r", and x, the equa- tions (41) 3 , (50) 3 , and (51) 3 will be employed when the ecliptic is the fundamental plane. But when the equator is taken as the funda- mental plane, we must first compute g, K, and G by means of the equations (96) 3 . Then, by a process entirely analogous to that by which the equations (47) 3 and (50) 3 were derived, we obtain h cos C cos (-H" a") = M cos (a" a), A cos: sin Off a") = sin (a" a), (86) h sin C = M tan d" tan = 0, A = O, and write A and /9 in place of a and ft respectively. If we now compute a second system of elements from J + 3 J and / = -> and a third system from J and / + df, the comparison of the three systems of elements with additional observed places will furnish the equations of condition for the determination of the corrections A J and A/ to be applied to J and - respectively. When the eccentricity is very nearly equal to unity, we may as- sume / for the first and second hypotheses, and only compute elliptic or hyperbolic elements for the third hypothesis. 121. The comparison of the several observed places of a heavenly body with one of the three systems of elements obtained by varying the two quantities selected for correction, or, when the required dif- ferential coefficients are known, with any other system of elements such that the squares and products of the corrections may be neg- lected, gives a series of equations of the form mx -f ny =p, m'x -f n'y =p', &c., in which x and y denote the final corrections to be applied to the two assumed quantities respectively. The combination of these equations which gives the most probable values of the unknown quantities, is effected according to the method of least squares. Thus, we multiply each equation by the coefficient of x in that equation, and the sum of all the equations thus formed gives the first normal equation. Then we multiply each equation of condition by the coefficient of y in that equation, and the sum of all the products gives the second normal equation. Let these equations be expressed thus : [mm] x -f [mn] y = [mp], [mn] x + [nn] y = [np], 23 354 THEORETICAL ASTRONOMY. in which [mra]==m 2 +m' 2 +m' /2 -{-&c., [mn]=mn+m r n f +m"n"+&c., and similarly for the other terms. These two final equations give, by elimination, the most probable values of x and y, namely, those for which the sum of the squares of the residuals will be a minimum. It is, however, often convenient to determine x in terras of y, or y in terms of x, so that we may find the influence of a variation of one of the unknown quantities on the diiferences between computation and observation when the most probable value of the other unknown quantity is used. Thus, if it be desired to find x in terms of y } the most probable value of x will be x = \mp]_ _ [mn] [mm] [mtri] ** If we substitute this value of x in the original equations of condition, the remaining differences between computation and observation will be expressed in terms of the unknown quantity y, or in the form A0 = m + n y. (90) Then, by assigning different values to y, we may find the correspond- ing residuals, and thus determine to what extent the correction y may be varied without causing these residuals to surpass the limits of the probable errors of observation. In the determination of the orbit of a comet there must be more or less uncertainty in the value of a, and if y denotes the correction to be applied to the assumed value of -, we may thus determine the probable limits within which the true value of the periodic time must be found. In the case of a comet which is identified, by the similarity of elements, with one which has previously appeared, if we compute the system of elements which will best satisfy the series of observations, the supposition being made that the comet has per- formed but one revolution around the sun during the intervening interval, it will be easy to determine whether the observations are better satisfied by assuming that two or more revolutions have been completed during this interval. Thus, let T denote the periodic time assumed, and the relation between T and a is expressed by ' k ' in which it denotes the semi-circumference of a circle whose radius ORBIT OF A COMET. 355 is unity. Let the periodic time corresponding to - -f y be denoted T by ; then we shall have * z and the equations for the residuals are transformed into the form A0 = (m nj} + njzi. (91) If we now assign to z, 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 with facility be introduced simultaneously with the computation of the parabolic elements. The numerical calculation as far as the form- ation of the equations (52) 3 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", 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 we 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 O l denotes the geocentric spherical co-ordinate computed from the parabolic elements, and 2 that computed from the other system of elements. Further, let A0 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 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 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 formulae 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) 3 , we shall have = n (p cos a RcosD cos A) (// cos a' R' cos U cos J/) -f n" ( P " sin a" R" cos D" cos A"), Q = n(psma RcosD sin A) (/>' sin a' R' cos D' sin A') (92) + n" ( P " sin a" R" cos D" sin A"), Q = n(pt&n8 R sin D) (p' tan # R' sin IX) + n" (P" tan *" R" sin D"), in which />, p', p" denote the curtate distances with respect to the equator, A, A', A" the right ascensions of the sun, and J), Z)', L' f 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) R cos D sin (a,' A)) + R' cos IX sin (a' A') n" (p" sin (a" a') + R" cosD" sin (a'- 4")), and hence M P "~ n "('') ,v ** ~ ~n " n i \ . p n" Sin (a" a') nR cos D sin (a'A^R' cos D' sin (a'- A^+ri'R' cos D" sin (a f A") pn" sin (a" a') VARIATION OF TWO RADII-VECTORES. 357 This formula, being independent of the declination S', may be used to compute M when only the right ascension for the middle place is given. For the first assumption in the case of an unknown orbit, we take M= t't Sin (a' a) H 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),, we derive, from (93), the expression n sin (a' a) " -_ ( 94 ) 1 \ R cos iy sin (a A'} sin (a" a') and, putting Mo'- n^ sin (a' a) n" ' sin (a" a') ' we have The calculation of the auxiliary quantities in the equations (52) 3 will be effected by means of the formulas (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 l>e 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)! will con- tain only the three unknown quantities J, 6, and I. By elimination, these unknown quantities may be found, and in like manner the 358 THEORETICAL ASTRONOMY. values of J", b", and I". It will be most convenient to compute the angles i// arid 4/', and then find z and z" from . . sin 2 = - -, sm r = r" or, putting a 2 = r 2 - E 2 sin 2 ^, and x m = r m R" 2 sin 2 -^", from E sin 4 R" sin 4" tan z = - , tan z" = ^, . 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", and u" u the determination of the elements of the orbit may be completed. Then, assigning to r an increment r,.we com- pute a second system of elements, and from r and r" -f- dr" 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" 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". 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 . f n j O) ,r.K\ sin (I -I) = - , (96) r o from which I may be found ; and in a similar manner we may find I". If we put 3^ = ^ JZ'sin'O* 0), we have G) . (97; 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 RADn-VECTOREB. 359 S=1SQ + Q I, _ Rsiu(l 0) (98) " sin(f ^) ' and tan6 = >-^, (99) r o by means of which the heliocentric latitudes 6 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", and r, r', v" will be given immediately. Then from r, r*, r" 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, we compute the angles u" u' and u' u corresponding to q + oq and T, and to q and T -\- dT, 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' and t' t, 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 u" u' = v" if, u ' u = v v, enables us to form the equations by which we may find the cor- rections Ag and AT 7 to be applied to the assumed values of q and T, respectively, in order that the values of u" u' and u f 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 OP THE MOST PROBABLE SYSTEM OP ELEMENTS FROM A SERIES OP 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 J in the observed value will depend on the magnitude of this error, and may be expressed by ( A). This function will also express the probability of an error J in an observed value. At the limit beyond which an error of the magnitude J can never occur, we must have y(J) 0: when J = 0, the value of

( J') dA' Let 7i, n', n", &c. be the observed values of x, and m the number of observations ; then we have A = n x, A' = ri x, A" = n" x, &c., and hence dA _ dA' dA" j -j -j = 1. ax ax ax Therefore the equation (4) becomes Cn x) dlog?(n' x) d( n -x) , This equation will serve to determine the value of x as soon as the form of the function symbolized by is known. It becomes neces- sary, therefore, to make some further assumption in regard to the errors J, J', J", dy = tdz; and hence Then, since we have, in general, . the preceding equation gives in which TT denotes the semi-circumference of a circle whose radius is unity. Therefore we have (11) and the equation (10) gives e = i- (12) V* Hence, the expression for m Vm which agrees with the first of equations (28). Let P denote the weight of the sum X, p f the weight of x f , and p" that of x" \ then we shall have 374 THEORETICAL ASTRONOMY. from which we get Since the unit of weight is arbitrary, we may take P f = ^ P" = JV & c -5 nnd hence we have, for the weight of the algebraic sum of any number of values, P = ~W = r' 1 + r"* + r'"* + &c.' or. whatever may be the unit of weight adopted, P = (39) In the case of a series of observed values of a quantity, if we designate by r r 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 the arithmetical mean, and by n any observed value, the probable error of n = x + v, according to (36), will be r Q being the probable error of the arithmetical mean. Hence we derive 7* """"; and if we adopt the value r' = 0.8453^, m the expression for the probable error of an observation becomes r = 0.8453 - J2L=, (40) in which [v] denotes the sum of the residuals regarded as positive, and m the number of observations. 133. Let n, n', n" } &c. denote the observed values of x, and let^p, p',p", &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', n", &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'n' + p"n" + &c. _ [ pn\ "~ ' " ~ ' 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', p", &c. are fractional as when they are whole numbers. The weight of x as determined by (41) is expressed by the sum P +P' +p"+P m + &c- and the probable error of ar is given by (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 w p' residuals expressed by n' x w and similarly in the case of n", n'", &c. Hence, according to equation (31), we shall have ,, = 0.8745^ (43) in which m denotes the number of values to be combined, or the number of quantities n, n', n", &c. For the mean error of x , we have the equations t . I/O] n* -DO]' If different determinations of the quantity x are given, for which the probable errors are r, r 7 , r", &c., the reciprocals of the squares of these probable errors may be taken as the weights of the respective values n, n ; , n", &c., and we shall have 376 THEORETICAL ASTRONOMY. with the probable error f =^ + 4,+j, + ...; (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 formulas 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 , ^, , &c. the unknown quan- tities to be determined, so that we have Let , %, , &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 F - F '=**+-3i*+*' + -" . 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 -\- by -f- cz -\- du -f- ew -\- ft -f- n = 0, a'x + b'y + c'z + d'u + e'w +/' -f n' = 0, (47) a"x + b"y + c"z + d"u + 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 wLich 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', 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

/]+ [on] = 0, [06] + [M] y + M 2 + [MJ u + [6e] to + [&/] < + [6] = 0, [ac>-f [6c]y + [cc] + [cci] + [ce] w + [c/] < + [en] = 0, [ad] re + [ W] y 4- [cd] 2 + [dd] M + [de] 10 + [d/] + [d] 0, ^ O1 ; M x + [be] y + [_ce] z + [de] u + [ee~] w + [e/] < + [en~] = 0, in which [aa] = aa -{- a'a' -j- a"a" -{" [a&]=a& + a'6' + a"6" + .... [oc] = ac + oV + a"c" + . . . . [66] = 66 4- 6'6' 4- b"b" 4- .... &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 + A'V 2 + h m v" 2 + Ac. shall be u 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 effected 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 x, 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'+l?t, z = a" + fi't, &C. (53) The coefficients a, , a', ft', &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 [06] [ac] [ad] [ae] [of] [an], [aa] f [aa] [aa] [aa] [aa] [aa] ' and if we substitute this for x in each of the remaining normal equa- tions, and put (54) fcflrn= [i ^ [ee] B=t [oe] = [ ee .l], [ e /] _ t^i [ /] : M ' (57) - [an] = [en .l], (58) we obtain METHOD OF LEAST SQUARES. 381 [16.1] y + [6c.l] z + [6rf.l] u + [6e.l] w + [6/.l]< + [6n.l] = 0, [6c.l] y + [cc.l] 2 + [crf.l] u + [ce.l] w + [c/.l] t + [cn.l] = 0, [6d.l] y + [cd.l] 2 + [dd.l] w + [cfe.l] 10 + [d/.l] * + [d.l] = 0, (59) [6.l] y + [?.!] + [.2] - 0. To eliminate z from these equations, we put 382 THEORETICAL ASTRONOMY. [cc.2] L ^ [dn.2] - j^ [c.2] = [dn.3], [n.2] -^|1 [cn.2] = [en.8], and we have [tta.3] u + [rfe.3] w + [rf/3] < + [d.3] = 0, [de.3] + [ee.3] w + [e/.3] * + l>.3] = 0, (68; + [6/3] w + [//.3] t + O.3] - 0, Again we put, in a similar manner, [ee.3] - [d.3] - [ee.4], [/3] - [rf/3] = [e/.4], [d/3] = [//.4], [en.3] - S [dn.3] = [en.4], (69) and the equations are [ee.4] w + [e/4] + [*.4] - 0, [e/4] W +[//-4]<+[/n.4] = 0. Finally, to eliminate w, we put [>.4]-g|i[6n.4] = [>.5], (71) and the resulting equation is ] = 0, (72) which gives The value of ^ 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 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], [6c.l], &c. may be symbolized thus : (75) in which a, /9, f, denote any three letters, and // 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 +6-fe+d+e-{-/=, a ' _(_ b' + c' + d' + e' +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 ASTEOXOMY. [an] + [6n] + [en] + [an] + [en] + [/] - M. (76) In a similar manner, multiplying by each of the coefficients in the original equations of condition, we find [oo] + [a6] + [ac] -f [ad] + [ae] + [a/] = [a*], [a6] + [66] + [6c] + [6d] + [6e] + [&/] = [6s], [ac] + [6c] + [cc] + [cd] + [] + [.5]. (79) If we multiply each of the equations (49) by its v, and take the sum of the several products, we get M x + [6v] y + [cw] z + [d] u 4- [ew] w + [>] t + [ Wl ] - [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, and hence W = [w]. (80) If we multiply each of the equations (49) by its n, and take the sum of all the products thus formed, substituting [yv] for [vn], there re- sults [cm]* + My + [cn]z+ [dn]u + [en] w + !>]<+ [nn] = [w]. Substituting in this the value of x given by the first normal equa- tion, it becomes [6n.l] y + [cn.l] + [dn.l] tt + [en.l] w + [/.!] * + [nn.l] = [w], in which n]. (81) Substituting, further, for y its value given by the first of equations (59), and continuing the process as in the elimination of the unknown quantities by successive substitution, we obtain the following equa- tions : [cn.2] z + [dn.2] u + O&.2] w + Q/n.2] < + [nn.2] = [w], [dn.3] M + [en.3] w + [/n.3] < + [nn.3] = [w], [en.4] w + [/n.4] < + [nre.4] = [w], (82) [/n.5] + [nn.5] = [twj, [nn.6] = [w]. The expressions for the auxiliaries [rm.2], [nn.3], &c. are [im.2] = [nn.1] - [^ ] [6.l], [nn.3] = [nn.2] - ^ [m.2], [nn.4] = [nn.3] - ^ [dn . 3 ] f [nn.5] - [nn.4] - [^ [en.4], [nn.6] = [nn.5] - [/n.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, [w] = [nn.^. (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, z, &c. in the equations of condition, and derive the corresponding values of the residuals v, v', 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 value?, 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', n", &c., so that we have x + an + oV + a."n" + a" V" + .... = 0, (85) in which 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 denotes the mean error of an observation of the weight unity, the mean error of an will be ae, that of a'n f will be aV, and similarly for the other terms of (85, ; and, according to the equation (35), the mean error of x will bo e m = e I/ a* + a" + a" 2 + &c. = e V/M- (86) Hence the weight of x will be expressed by (87) 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', 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 formula for the most probable value as determined from the actual data. Hence we have x, + a( n v) + '( w ' tO + .... = 0, and comparing this with the expression (85), we obtain *,=* + [WJ. Substituting in this the values of v, v', v", &c. given by the equations (49), there results X, = Z+ [oo] X, + [aft] y, + [ac] 2, + lad] u, + [ae] w, + [a/] t, + [a], and since, according to (85), x + [an] = 0, in order to satisfy this expression for x,, we must evidently have [aa] = l, [aft]=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 x, and let these multipliers be designated by q, q', 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 [oa] q + [aft] q' + [ac] q" + [ad] q"' + .... = 1, [aft] q + [ftft] q' + [be] q" + [bd] q'" +.... = 0, (89) [ac]o + [bc]j + Wq" + [_cd]q'" + .... -0, &c. Ac. and also, retaining the residuals v, v', v", &c. in the formation of the normal equations, Therefore, since x, + [] = M> and since the first member of this equation must be identical with the first member of (90), we have W q + [ftf] q' + [w] 3" + . . . = ai; + V + V + . . . , 388 THEORETICAL ASTRONOMY. which gives, by expanding the several suras, aq + bq' + cq" + dq"' +.... = a, a'q + V] = 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 [bn], 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 svmbols [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. 3rf9 coefficient of the unknown quantity which is made the last in the elimination, in the final equation for its determination, is equal to the weight of the resulting value of that quantity. Thus, in the case of the equations for six unknown quantities, since the reciprocal of the weight of the most probable value of i is the value of t obtained from the normal equations by putting [>] = 1, and [an], [6n], [en], &c. equal to zero, the equations (63), (67), (69), and (71) show that we have = [>.2] = [/.3] = [>.4] = [/.5] = - 1, and hence, according to (72), for the reciprocal of the weight of t, which gives A =[#5]- (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 [en\ = 1, and the other absolute terms of the normal equations equal to zero, and finding the corresponding value of w. This operation gives [en.4] = - 1, [>.4] = 0, [>.5] = &Q. Hence the equation (73) becomes and substituting this value of t in the last of equations (70), we get ~ 390 THEORETICAL ASTKOXOMY. wliich 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 = [aa.5], ..... 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 oi 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] [dd.3] [ee.4] jjfif.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 coefficient rf the quantity first determined; then we shall have D = [oo] [66.1] [cc.2] [oU3] [ee.4] [#5] = [oa], [66.1], [ee.21. [dd.3], [jQT.4], [ee.5] = [] [W.lUcc.2-1, [ee.3] d [#41, [3] _ [06.33 [a6 . 3] , 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 #, 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, z, y, x, and we shall have (99) by means of which the weights of a;, y, and z 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", 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 0-W + *, [oo] T _acl L^.J^ [aaj '' 1 METHOD OF LEAST SQUARES. 393 To determine y from the last five of equations (74), let the eliminating factors be denoted by B ', '", ^ v , and ^, and we shall have [fel] = - QJ.2] * " 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 : (102) The expressions for the values of the unknown quantities will there- fore become [en.2] [dn.3] _. . .^ ~ "" [ee.4] "^ 394 THEORETICAL ASTRONOMY. The fiit 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 [6n] = 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], [bri], [en], &c. equal to zero. It remains, therefore, to determine the particular values of [6-n.l], [cn.2], &c., and the expressions for the weights will be complete. If we multiply the first of equations (100) by [an], it becomes [bn.Y] = [an] A' + [bri]. 104) Multiplying the second of equations (100) by [an], and the first of (101) by [bri], adding the products, and introducing the value of [6n.l] just found, we get [en] - [cn.l] + [6n.l] + [an] A" + [bn] B" = 0, which reduces to [an] A" + [bri] B" + [en] = [cn.2]. (105) Multiplying the third of equations (100) by [an], the second of (101) by [bri], and the first of (102) by [en], adding the products, and re- ducing by means of (104) and (105), we obtain -= [dn] _ [dn.l] + frlr LCC.ZJ which, by means of the expressions for the auxiliaries, is further re- duced to [an] A'" + [6n] ff" + [en] C"' + [rfn] = [a 7 n.3]. (106) In a similar manner we find, from the remaining equations of (100), (101), and (102), the following expressions : [an] A* + [bn-] B*+ [en] C + [dn] D"+ [] = [i.4], [an] ^ T + [6n] B* + [en] T + [dn] D T + [en] T + [>] = |>-5]. U The equations (104), (105), (106), and (107), enable us to find the particular values of [6w.l], [cn.2], &c. required in the expressions for the reciprocals of the weights. Thus, for the weight of x, we have [an] = - 1, [6t] = [en] == [dn] = [] = [/n] = ; METHOD OF LEAST SQUARES. 395 and these equations give [6n.l] = - A', [cn.2] = - A", [dn.3] = - A'", - For the case of the weight of y, we have [&] = _ 1, [an] = [en] = [dn] == [en] = [/n] = 0, and the same equations give [&.!] = _ 1, [ C71 . 2 ] = - B" t [dn.3] = - 3"'. [ en .4] = _ R* } [/ n .o] = &. We have, also, for the weight of z, [en.2] = -l f [efa.3]= C"", [en.4] = -C' T , [/n.5] = for the weight of M, [d-t.3] = 1, [en.4] = i> T , [/n.5] = 7)' ; for the weight of w, [n.4] = -l, [/n.5] = - T ; and finally, for the weight of t, 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 A A' A"A" A'"A'" A"A* A*A r [, B = [aa] [cc] - [ae], C" = [aa] [66] - [06]', D = [aa] [66] [cc] + 2 [a6] [be] [ac] - [aa] [6c]* - [66] [ae] - [cc] [a6]', which are all the quantities required for finding simply the weights of the most probable values of x, y, and z. The expression for the weight of z is D *.=-V 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 ora 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 f , n", &c. be the differences between com- putation and observation, in the case of either spherical co-ordinate, for the dates t, t', t", &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 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 t w which is the mean of t, t', t", &c., will give the corresponding normal place. But when different weights p, p', p", &c. are assigned to the observations, tho value of n must be found from . P +/+/' + .... and the weight of this value will be equal to the sum COMBINATION OP OBSERVATIONS. 403 The date of the normal place will be determined by _*+& + ?<+... p+rr+rr + ..~ If 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, 10 = A + r + Or 1 , the coefficients A, B, and C being found from equations of condition formed by means of the several known values of A0 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 those 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, aud 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 formulae 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, , C in the equation *0 = A + Br -f 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 considerable, it will be more convenient to divide the observations into groups, and use the differences between computation and observation for provisional normal places in the formation of the equations of condition for the determination of A, B, and (7. 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 THEOBETICAL 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 s, the mean error of a single comparison; then will 4=- be the mean error of m comparisons, and the mean error of Vm the resulting place of the body will, according to equation (35), be given by '' = ^ + ''- (126) The value of , in the case of each series, will be found by means of the residuals finally corrected for the constant errors, and the value of 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/ = i/m(e '-e.'). (127) The value of , for each observer having been found in accordance with this equation, the mean error of an observation consisting of m comparisons will be The mean error of an observation whose weight is unity being de- noted by e, the weight of an observation based on m comparisons will be P = ^ (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 plane depends, it will not be necessary to consider e,, and we may then derive s, directly from the residuals corrected for constant errors. Further, in the case of meridian observations, the error which corre- sponds to , 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 become* 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 m,'=[mvv-]-[m-]e t >, (129) in which n denotes the number of observations; m, m r , m", &c. the number of comparisons for the respective observations; and v, v', v", &c. the corresponding residuals. Then, by means of equation (128), assuming a convenient number for e, we compute the weight of each observation. Thus, for example, let the residuals and corresponding values of m be as follows : A0 m &0 m + 2".0 5, 1".0 7, - 1 .8 5, + 1 .5 5, .4 10, +4 .1 8, 5 .5 5, .0 5. I^et the mean error of the place of a comparison-star be e.= 2".0 ; then we have n ~ 8, and, according to (129), 8s, 1 =341.78 200.0, which gives e,= 4".2. Let us now adopt as the unit of weight that for which the mean erroi i 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", &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 s. 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 1 ', &c. 148. The observations of comets are liable to be affected by other errors in addition to those which are common to these and to 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 denotes the mean error of an observation based on m com- COMBINATION OF OBSERVATIONS. 4QD parisons, and s e 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 '-=^+''- ( 13 > 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 bv 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 l>een 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, \ 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 O x /.<" COMBINATION OF OBSERVATIONS. 411 and when n has a value such that the value of this expression is less than 0.5, the error nr will have a greater probability against it than for it, and hence it may be rejected. The expression for finding the limiting value of n therefore becomes 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' = 0.67449n, 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 n' m n' m n' m , ] 6 1.732 20 2.241 55 2.608 90 2.773 | 8 1.863 25 2.326 60 2.638 95 2.791 10 1.960 30 2.394 65 2.665 100 2.807 12 2.037 35 2.450 70 2.690 200 3.020 14 2.100 40 2.498 75 2.713 300 3.143 16 2.154 45 2.539 80 2.734 400 3.224 18 2.200 50 2.576 85 2.754 500 3.289 According to this method, we first find the mean error of an 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't it must be rejected Again, finding a new value of 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 e = 2".56. 412 THEORETICAL ASTHOXOMY. Corresponding to the value m 50, the table gives n' -- 2.576, and the limiting value of the error becomes n's = 6".6 ; and hence the residuals 11 ".5 and +7".% are rejected. Recom- puting the mean error of an observation, we have 193.09 \ 47 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', n ff , &c. which are the ubsolute 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', 6', &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 ar 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, th^ 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 vx, v bein any entire or fractional number such that the new coefficients " ^ &c. shall be made to agree in magnitude with the other coefficients. The unknown quantity whose value will then be derived by the solution of the equations will be we, and the corresponding weight will be that of i/x. 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 [JL. 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 ff , &o. in a horizontal line, directly under these the logarithms of the coefficients a, a', a", &c., then the logarithms of 6, b', 6", &c., and so on. Then writing, in a corresponding form, the values of logw, logn', &c. on a slip of paper, by bringing this successively over each line, the sums [nn], [an], [bri], &c. will be readily formed. Again, writing ou 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 b, c, d, COREECTION OP THE ELEMENTS. 415 &c. successively is effected in a similar manner; and thus will be derived [66], [6c], [bd~], &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 [ad], [a6], [ac], . . . [as], [an], and di- rectly under them the corresponding logarithms. Next, we write under these, commencing with [6], the values of [66], [6c], [bd], - [bs], \bn\ ; then, adding the logarithm of the factor ^ ^ to the logarithms of [a6], [ac], &c. successively, we write the value of ^1 [06] under [66], that of jSj- [ac] under [6c], and so on. Sub- tracting the numbers in this line from those in the line above, the differences give the values of [66.1], [6c.l], . . . [6s.l], [6)U], 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], [ccT|, . . [cs], [en], placing [cc] under [6c.l], and, having added the logarithm of jMr to the logarithms of [ac], [ad], &c. in succession, we derive, according to the equations (55) and (58), the values of [cc.l], [cdl], . . [cs.l], [c.l], which are to be placed under the cor- responding quantities [cc], [cd], &c. Next, we subtract from these, respectively, the products 416 THEOEETICAL ASTRONOMY. and thus derive the values of [cc.2], [cc?.2], . . [cs.2], [cn.2], which are to be writcen in the next horizontal line and under them their logarithms. Then we introduce, in a similar manner, the coefficients [dd], [de\, . . [dn], writing [dd~] under [od.2]j and from each of these in succession we subtract the products [od] r thus finding the values of [c?d.l], [cZe.l], . . [dn . 1]. From these we subtract the products respectively, which operation gives the values of [ LAJ? an( l C/ M ] CORRECTION OF THE ELEMENTS. 41" placing [jf] 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], O5], and [/n.5]. The values of [6*.l], [cs.l], [c*.2], &c. serve to check the calcula- tion of the successive auxiliary coefficients. Thus we must have -f + [ec.2] + [ed.2] + [ee.2] + [c/.2] = [.2], [ed.2] + [ + 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 F(l+wi), express th* 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 (3) 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 J/ , - , ft , /, 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 + 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, ~, ft, *, e, and a, they will be connected with the constant elements, or those which determine the orbit at the instant f , by the equations 428 THEORETICAL ASTRONOMY. in which -7-, -'-, &c. denote the differential coefficients of the ele- dt at ments depending on the disturbing forces. When these differential coefficients are known, we may determine, by simple quadrature, the perturbations SM, d~, &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 (? 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 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 t , 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 uny system of three co-ordinates by means of which the position of PERTURBATIONS. 429 the body or the elements themselves may be found. "NYe shall, there- fore, derive various formulae for this purpose before investigating thi- formulae for the direct variation of the elements. 156. Let x , y , z be the rectangular co-ordinates of the body at the time t computed by means of the osculating elements 3/ , ~ , & , &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, %, and dz being the perturbations of the rectangular co-ordinates from the epoch t to the time t. If we substitute these values of ar, 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 + dr; then to terras of the order or 2 , which is equivalent to considering only the first power of the disturbing force, we have + to_gk I/, X _ 3 *o 3 \ r 3 r s r s \ r f and hence We have also from r' neglecting terms of the second order (7) 430 THEORETICAL ASTRONOMY. The integration of the equations (6) will give the perturbations 3x, 8y, and dz to be applied to the rectangular co-ordinates x , y , 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, dy, 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 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 x , y w z in place of x, y, z, the equations (6) become 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, ,), (10) in which, when terms of the second order are neglected, we use the values x , y , z for x, y, and z respectively. 157. From the values of dx, 8y, 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: VARIATION OF CO-ORDINATES. 431 In which nothing is neglected. In the application of these formulae, as soon as 3x, %, and 3z 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 * = * -f Sx > y = y<>+ s y, * = -f 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 which will be convenient in the numerical calculation, we have r 2 = (x + dx)' -f (y + fy)' -f (, + &)' = r -f 2 te + 2y fy + 2 *2 + <5a; 2 + <5y 2 + W, and therefore r 2 _ , 0(^0+ jte) to + (y. + ^y) ^y + (g n + &0 ^ ~^~ Let us now put g = iJ*! to + &i4* + Z 4^ te> (12) r o r o r o and /3 = l-^ = l-(l + 2g)- | ; then we shall have sind the values of/ may be tabulated with the argument q. The equations (11) therefore become 432 THEORETICAL ASTRONOMY. The coefficients of dx, 3y, 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 q, the numerical values of ~JTT ~3Jn an( ^ j-j-, which include the squares and products of the masses, will be (It obtained. The integration of these will give more exact values of dx, Sy, and 8z, 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 tho table, the value of may be computed directly, using the value of r in terms of r 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 /(a?) dx* MECHANICAL QUADRATURE. 133 when the values of f(x) are computed for successive values of x in- creasing in arithmetical progression. First, therefore, we shall finci the integral of f(x) dx within given limits. Within the limits for which x is continuous, we have /O) = a + #B + ?x* + dx 3 + ex* + ....; (15) and if we consider only three terms of this series, the resulting equa- tion 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 /(#). If we multiply equation (15) by dx and integrate between the limits and x', we get J/(*) dx = *x' + ifix'* + ir*'* + W +. (16) o If now the values of f(x) for different values of x from to a:' are known, each of these, by means of equation (15), will furnish an equation for the determination of a, ft, ?, &c. ; and the number of terms which may be taken will be equal to the number of different known values of f(x). As soon as a, ft, f, &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, and n the number of intervals from the beginning of the integration, we obtain 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, 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 A* J/O) dx = A* (2a + 2/5A* + The equation of the curve gives, if we designate the ordinates of the three successive points by y , y u and y v -=2/o, = 2^(0. - and hence we derive In a similar manner, the area included by the ordinates y 2 and y^ corresponding to x = 2&x and x=4&x, the axis of abscissas, and the parabola passing through the three points corresponding to y v y y and y v is found to be and hence we have, finally, i A* (y n _ 2 + 4y n _ The sum of all these gives n\x f/C*)* J (17) by means of which the approximate value of the integral within the given limits may be found. If we consider the curve which passes through four points corre- sponding to 7/ , y u 7/2, and y a , we have for the equation of the curve, and hence, giving to x the values 0, /vr, 2A.r, and SAZ, successively, we easily find MECHANICAL QUADRATURE. 435 Therefore we shall have ) d* = I A* (y. + 3y, + 3y, + y,). (18) In like manner, by taking successively an additional term of the series, we may derive ) dx = (7y + 32ft + 12y f + 32y 3 + 7y 4 ), (19) dc = (19y + 75^ + 50/ 3 + 50y 3 + 75y 4 This process may be continued so as to include the extreme values of x for which /(x) is known; but in the calculation of perturbations it will be more convenient to use the finite differences of the function instead of the function itself directly. We may remark, further, that the intervals of quadrature when the function itself is used, may be so determined that the degree of approximation will be much greater than when these intervals are uniform. 159. Let us put &x = to, and let the value of x for which n = l>e designated by a; then will the general value be /O) =/( + no.), to being the constant interval at which the values of f(x) are given. Hence we shall have dx = wdn, J/O) dx = J/(o + n) dn. If we expand the function /(a -f ruo), we have , . ., (20) 436 THEORETICAL ASTRONOMY. and hence //( + ) dn = O + /() + in'. ^ + JnV being the constant of integration. The equations (54) 6 give =r GO - i/ T () + Ti*r u () - , (22) in which the functional symbols in the second members denote the different orders of finite differences of the function. Hence we obtain J/O + no) dn = C + n/(a) + ^n-Cf (a) - J/" (a) + ^T( a ) _ ^^(a) + - - .) + *(/"() - T^/' T () + AT () - o^/ TU1 () - h ) If we take the integral between the limits n r and +w', the terras 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 -f i, and the equation (23) gives, after reduction, MECHANICAL QUADRATURE. 437 J/(a + n0 dn =/(a) + &f (a) - (24) It is evident that by writing, in succession, a + , ____ a -}- ito in place of a, we simply add 1 to each limit successively, so that we have < + * +* if (a + ru) dn = (/((< + {.) + (n - 1) <*)<*(>- 1) i-t -t = /( a+ ^ ) + 5 ._ r ( a+{w )_^7 u/ .r (a+ ^ )+?gS7 6 g 7 g ^ (a+iw) _ Ac> But since /' + * r J * + J /(a + n) dn =J /(a + iw) dn +J /(a + n) dn ..... +J /(o+n) dn, -i -i t i-t if we give to i successively the values 0, 1, 2, 3, &c. in the preceding equation, and add the results, we get t + n = i n = J/(a -f- n) dn = ^/(a + no.) -f ^ ^/" (a + no,) n = o ^^ (25) n = ii = 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, /() ='/( + -'/(a -.), /(a + ) = '/(a + |0 - '/( + , /(a + n) = '/(a + (n + ) ) - '/(a -f (n - ) *>). Therefore we shall have " V( + -) = '/( + (* + 1) ")-'/(- and also n = t 2 n = ^/" (a + n0 =/ (a + ( + i) ) / ( ~ I"). 433 THEORETICAL ASTRONOMY. Further, since the quantity 'f(a \io) 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 \to shall be zero, namely, (26) Substituting these values in (25), it reduces to f/() dx = J/( + no.) dn (27) In the calculation of the perturbations of a heavenly body, the dates for which the values of the function are computed may be so arranged that for n = , corresponding to the inferior limit, the integral shall be equal to zero, the epoch of f(a \io) 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 \io). If, instead of being equal to zero, the integral has a given value when n = \, it is evidently only necessary to add this value to 'f(a \io) as given by (26). 160. The interval a> 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 integral taken between the limits \ and i. The general formula (23) gives * J/(a + n) dn = J/(a) + if (a) + &f (a) - ,fr/"' (a) - y^W* () + ildliF/'Ca) + TiiSWGO -&c.: and since, according to the notation adopted, / (a) = h (f ( - i0 +/ ( + *)) = r(a + i0 -if, (28) /"()=/"( + *) -if" (a), MECHANICAL QUADRATURE. 439 this becomes r J /(a+n-0 ) d ; and if we substitute the values already found for the terras in the second member, and also / Tl (a + t'a) = / v (a + (i -f- i)w) / T (a -f (i ^)o), &c. we get + t i x = io ( f(a + na>) rfn (32) which is the required integral between the limits \ and i. 161. The methods of integration thus far considered apply to the cases in which but a single integration is required, and when applied to the integration of the differential equations for the variations of the co-ordinates on account of the action of disturbing bodies, they dSx dtiy , d$z will only give the values of -571 -rp and j-> and another integration becomes necessary in order to obtain the values of 3x, 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 jjftydx* we have, since da? = cuPdn*, The value of the function designated by /(a) being so taken that when 7i = the equation (23) gives -i Therefore, the general equation is o J/(a -f n) dn =J/(a + u) <*n + n/(a) the values of a, ^9, 7-, ... being given by the equations (22). Multi- plying this by dn, and integrating, we get o + n) <*n = C" + -i C' being the new constant of integration. If we take the integral between the limits and + , we find + * o ff/fa + *)<*n* - f/(a + n^dn + ^a + T V 2IJ r + &vv' + A*- Jt i* ^J From the equation (32) we get, for f = 0, o f/fa + n) rfn = '/(a) - ^ (a) + Jkf" (a) - f \\ W (a) + Ac. (33) J Substituting this value, and also the values of a, 7-, e, &c., which are given by the second members of the equations (22), in the pre- ceding equation, and reducing, we get ff/Ya-f-nOrfn'='/( a ) 2 ) j/(a)-f- 5 ^ c /'"(a) *J?|io.T()+Ac. (34) // MECHANICAL QUADRATURE. 44] Hence t + t f< i-t + lilc/' "( + TsWs/"( a + *0 -h Ac, and I I f( a + na> ) ^ w * :z= 7( a + * ttf ) 2^/' ( a + *"*) *-* and i' + t = f n = f Jj/(a + n) d/i 1 - ^ '/(a + na>) - A ^/' , (35; 4- &c. "We may evidently consider '/(a $*), r /(a + |i), &c. a? the differ- ences of other functions, the first of which is arbitrary, ao that we have '/(a) = '/( + J) + i'/( ~ ) = 57( + ) - 1/" ( ~ ), '/(a + ,) = i'/( a + j) + ^'/(a + i) - T/(a + 2) - i/" (a), Therefore n = ' . (o+n)=J/ Substituting these values in equation (35), and observing that "/(a) + "/(a - ) = 27(a - ) + '/(a - ^), /(a) + /(a - = 2/(a) - / (a - , /" (a) +f (a - *) = 2/" (a) -/'" (a - >), &c., and that, since "/(a ai) is arbitrary, we may put '/(a - ) = A/Ca) ~ ^In(2/"() +/"( ~ )) (36) u 2' T a-. - Ac., 442 THEOKETICAL ASTRONOMY. the integral becomes ++- <+* (x) JJ f(x) da? = * JJ /(a + no,) d (37) which is the expression for the double integral between the limits }andt + |> The value of "/(a ) given by equation (36) is in accordance with the supposition that for n = \ the double integral is equal to zero, and this condition is fulfilled in the calculation of the pertur- bations when the argument a \co corresponds to the date for whicli the osculating elements are given. If, for n = , 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 'f(a o>) given by equation (26), and to add the given value of the double integral for the same argument to the value of "/(a ft*) given by (36). 162. In a similar manner we may find the expressions for the double integral between other limits. Thus, let it be required to find the double integral between the limits \ and i. Between the limits and \ we have * o Jj/(a + no,-} dn* = $ff(a + n) dn + i/(a) + - 4 ' s a o -j which gives O+i tf D/'" GO (38) and this again, by means of (28), gives MECHANICAL QUADRATURE. 443 Therefore, since n* =jfp"(a + no,) efa 2 jff/(a + no.) dn, * i i and '/( + (* + = "/(a + (+!;)_ "/(a + t0, /'(-}-(* + = /( + (i + !),)_ /(a + i0, /'" ( + (* + ) <") =/ " ( + (i + 1) ) -/" (a + to,), &c. we shall have a + iia j (ar) <& = ,' f f/(a + n.) d -* (39) which gives the required integral between the limits } and i. 163. It will be observed that the coefficients of the several terms of the formulae of integration converge rapidly, and hence, by a proper selection of the interval at which the values of the function are computed, it will not be necessary to consider the terms which depend on the fourth and higher orders of differences, and rarely those which depend on the second and third differences. The value assigned to the interval co 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 to 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 to 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 >, 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. putation of the perturbations of the group of small planets between Mars and Jupiter we may adopt uniformly an interval of forty days; but in the determination of the perturbations of comets it will 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 2 , if we multiply the computed values of the function by (a 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 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 at 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, 8y, and dz with respect to the time, will be determined by means of the equations (9). If the interval to 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 to, and commencing with the date t \). An approximate value of f(a -f- 10) may now be readily estimated, and two terms of the equation (39), putting i=l, 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 dx, %, and dz for the date represented by the argu- ment a + 2w, 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 May 20.0 55.23 41.51 et -f- 4> June 29.0 56.06 38.96 a + 5w Aug. 8.0 57.30 35.92 a + 6* Sept. 17.0 59.09 32.47- a + 7a Oct. 27.0 61.55 28.60 a + 8w Dec. 6.0 64.85 24.34 1865 Jan. 15.0 +69.09 +19.78 which are expressed in units of the seventh decimal place. We now, for a first approximation, regard the perturbations ais 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 >f( a _;!,) = _ f ( 0.) = & (53.71 53.00) = 0.03, 53.71 and the approximate table of integration becomes _ i ,-) = _o 03 "-ft = 4- 2.24, 7(o) = + 2.21. Then the formula (39), putting first i 1, and then i 0, gives Dec. 12.0 dx = + 2.24 + ^^- = + 6.66, Jan. 21.0 dx = + 2.21 + -^- = + 6.69. In a similar manner, we find Dec. 12.0 8y = + 5.85 dz = 0.16, Jan. 21.0 fy == + 5.82 8z == 0.14. By means of these results we compute the complete values of the second members of equations (40), dr being found from X Wo Z r o r o r o and thus we obtain d?$x dpfiy d?&z Dec. 12.0 + 53.86 + 47.76 1.45 + 8.85, Jan. 21.0 + 54.23 + 47.25 0.96 + 8.63. We now commence anew the table of integration, namely, / "f 7 / '/ 7 / '/ 7 +53.86 ft no + 2.26, +47.76 , n n9 + 1.97, -1.45 ft no -0.04, 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 9x= + 56.45 + ^ = + 61.00, ifi 7 ty = + 49.26 + ~ = + 53.15, fe = - l. 04 _-M-:=_ 1.08. By means of these approximate values we obtain the following results : 1864 March 1.0 ,'= + 55.01, <"'-= + 53.86, __i.oo f Sr = + 71.03. Introducing these -into the table of integration, we find, for the corre- sponding values of the integrals, 9x = + 61.03, dy = + 53.75, fe = 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, ^= + 53.91, ^=-1.00. Assuming, again, approximate values of the differential coefficients for April 10.0, and computing the corresponding values of ox, oy, and Sz, we derive, for this date, ^ = + 43.06, ,'^ = + 63.19, '^ = -l.M. Introducing these into the table of integration, and thus deriving approximate values of 3x, 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. ut df* -3 ,:Wt '-Jf 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. w'^f 1864 Oct. 27.0 + 34.84 Dec. 6.0 68.79 1865 Jan. 15.0 + 112.64 2^6.32 + 4.44, 47.87 6.86, 58.39 + 8.68. The complete integration may now be effected, and we may use both equation (37) and equation (39), the former giving the integral for the dates Jan. 1.0, Feb. 10.0, March 21.0, &c., and the latter the integrals for the dates in the foregoing table of values of the function. The final results for the perturbations of the rectangular co-ordinates, expressed in units of the seventh decimal place, are thus found to be the following: Berlin Mean Time. fe Sy fc 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 9SM 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 + 1772.6 + 1992.3 28.2. During the interval included' by these perturbations, the terms of the second order of the disturbing forces will have no sensible effect; but to illustrate the application of the rigorous formulae, let us 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 The approximate values of 3x, dy, and 3z 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 + 8p, we have or in which ^ is the modulus of the system of logarithms. Thus we obtain, for Sept, 17.0, 8 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^ -f 0.29815% 0.03237&. The approximate values of dx, 8y, and dz being substituted in this equation, we obtain q = -f- 0.0000061, corresponding to which Table XVII. gives log/= 0.477115. Hence we derive ^ Of, -to) = -44.87, ^C%-^)=- 30.40, r r 454 THEOEETICAL ASTEONOMY. and the equations (14) give 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 3x, dy, and 8z 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: * ^ * 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 3x = + 1772.6, Sy = + 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, fa, = Sx, dy, = cos e dy sin 3z, (41 ; Sz f = sin e dy -\- cos e dz, x,, y,, z, being the co-ordinates in reference to the equator as the fun damental plane. Thus we obtain, for 1865 Jan. 15.0, 8x, = + 1772.6, dy, = -f- 1838.9, dz, = + 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-ORDIXATES. 455 The approximate geocentric place of the planet for the same date is a = 183 28', 8 = - 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 = 4- 5".67. 167. The values of dx, dy, and 8z, 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 6 denote the longitude of the descending node of the ecliptic of t' on that of t, measured from the mean equinox of t, and let y be the mutual inclination of these planes; then, if we denote by x', y f , z' the co-ordinates referred to the ecliptic of t as the fundamental plane, the positive axis of x, however, being directed to the point whose longitude is 0, we shall have x = x cos -|- y sin 6, y r = x8m0 + ycos0, (42) /=*. Let us now denote by x", 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' } z'. Then we shall have y" = i/ cos TJ z sin ij, (43) 2" = y' sin y -|- / 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 x,, y,, z,, we get x, = x" cos (0 + p) y" sin (0 + p\ y, = x" sin (9 + p) + y" cos (8 + p\ (44) in which p denotes the precession during the interval i't. Elimi- nating x", y", and z" from these equations by means of (43) and (42), observing that, since TJ is very small, we may put cos y = 1, we get 456 THEORETICAL ASTRONOMY. x, = x cosp y sinp -f- - z sin 5 y, = x sinp -f- ?/ cos^> - z cos (0 -4- j?), (45 > z,=z - a; sin -4- - y cos 0, 5* S in wliich s 206264.8, y being supposed to be expressed in seconds of arc. If we neglect terms of the order p 3 , these equations become x, x .x -y + - (sin 6 + p cos (?) z, ~ 8 S S (46) z, = z - x sin + - y cos (9. 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, 37, and 0, we have p = (50".21129 + 0".0002442966r) (g f), y = ( 0".48892 0".000006143r) (t' -- t), = 351 36' 10" + 39".79 (t 1750) 5".21 (f i), in which r \(t' t) 1750, t and t' being expressed in years from the beginning of the era. If we add the nutation to the value of p, the co-ordinates will be derived for the true equinox of t f . The equations (45) and (46) serve also to convert the values of dx, Sy, 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, dy, and dz in place of ar, y, and z re- spectively, and similarly for # 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 OP 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 f 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 Ixxlies 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, 3y, and oz 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 dx, 3y, and dz 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 jfr> ~^a~' an ^ 77- (which is effected indirectly) cannot be performed with facility, dt 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-, -T-, and -7-- 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, 40^ = +385.9, 40^ = + 214.8, 40^ = + 9.7. at at at The velocities in the case of the disturbed orbit will be given by the formula? dx _dx dox dy__d% , <% dz^ _dz^ d^z ~dt~ dt + dt' dt ~ dt ~*~ dt' dt ~ dt + dt' { ' To obtain the expressions for the components of the velocity resolved parallel to the co-ordinates, we have, according to the equa- tions (6) 2 , -j- = sin a sin ( A + u) -^ + r sin a cos ( A + u) -^, -^- = sin b sin (B + u) - + r sin b cos (J> + u~) -, dt at -- = sin c sin ( C + u) -- + r sin c cos(C'+ w)--- 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 dr Jbl/l+m . , , - = - -J - e sin (u to), dt in which 10 denotes the angular distance of the ] terihelion from the ascending node. Substituting these values, we ol tain, by reduction, THEOHETICAL ASTRONOMY. - ((e cos * + cos w) cos A (e sin w -J- sin w) sin J.) sin a, at i/p JL = T m ((e cos * -f- cos it) cos 2? (e sin a* -(- sin u) sin.B) sin 6, a i/n - ((e cos * + cos u ) cos C' (e sin u* -f- sin u) sin C) sin c. (e sin * + sin w) = Fsin U, JP (48) -^t (e cos ft* -f- cosw) = Fcos U, Vp and we have = Fsin 6 cos (B + C7), (49) U). 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 sinc = sini, C 0. The sum of the squares of the equations (48) gives 7 2 = 21A_E^ (1 _(_ e _j_ 2e cos (M *)) = k* (1 + r 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 , x , y , z , and by means of the equations (48) and (49) we compute the values of ~, ~-j- and ft- Then we apply to these the values of the perturba- tions, and thus find x, y, z, -J-, -j-, and --. These having been CHANGE OF THE OSCULATING ELEMENTS. 461 found, the equations (32) t 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 4 = kVp (1 + m), we obtain Vr and ^ , and from r sin u = ( a; sin ft -|- y cos ft) sect, r cosw x cos ft + 1/ sin ft, we derive r and w; 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 = " ^ - \* I "/ -i and from 2ae sin ca = (2a r) sin (2^ -f u) r sin u, 2ae cos (a = (2a r) cos (2^ + u~) r cos u, 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, -jr 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 -rp -rr, &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 formulae 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 1 -}- x sin i sin 2 = 0. We have, also, z = r sn w sn i, which must be satisfied by the resulting values of F, r, and w; 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 * 8 (r cos w), 8 (r sin u), 8z, 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 ft the latitude of the body with respect to the plane of xy; then we shall have x = r cos /? cos w, y = r cos ft sin w, (50) z = r sin /?. Let us now denote by X, F, and Z, 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 c?(rcos/9) . dw = cos w ^j-. -- r cos p sin w -r-> at at d (r cos /?) , dw ^ J -f r cos /9 cos w -rr, at VARIATION OF POLAR CO-ORDINATES. 463 Therefore we have 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 = sn w, and Yx Xy = Therefore r 1 cos 1 ,3 =s o r cos p dt -f C. In the undisturbed orbit we have /? = 0, and and thus the preceding equation becomes r* cos 1 ,9 ^ =fs t r cos ,9 dt + l/p.(l + w). (51 ) The equations (1) also give + r + (52) If we denote by R the component of the disturbing force in direction of the disturbed radius- vector, we have We have, also, = X- + Y- + Z-. (53) zffix + y Finally, we have, from the last of equations (1), ffiz P(lm by means of which the value of z may be found, since, in the case of the undisturbed motion, we have z a = 0. The values of/' corresponding to different values of q' may be tabulated with the argument q', and, since the equation (62) is of the same form as (58), the same table will give the value of/" when q" is used as the argument. Table XVII. gives the values of log/ or log/' corresponding to values of q' or q" from 0.03 to + 0.03. Beyond the limits of this table the required quantities may be com- puted directly. 171. When we consider only terms of the first order with respect to the disturbing force, we have and the equations become (65) ffz _ P(l+m) ~de"~ ~~^T~ In determining the perturbations of a heavenly body, we first con- sider only the terms depending on the first power of the disturbing force, for which these equations will be applied. The value of dr 30 466 THEORETICAL ASTRONOMY. will be obtained from the second equation by an indirect process, as already illustrated for the case of the variation of the rectangular co-ordinates. Then dw will be obtained directly from the first equation, and, finally, z indirectly from the last equation. Each of the integrals is equal to zero for the date t Q , to which the osculating elements belong. When the magnitude of the perturbations is such that the terms depending on the squares and products of the masses must be con- sidered, the general equations (59), (63), and (64) will be applied. The values of the perturbations for the dates preceding that for which the complete expressions are to be used, will at once indicate approximate values of dw, dr, and z; and with the values r = r -J- dr, w = w -{- dw, sin /? -, the components of the disturbing force will be computed. "We compute also q f from the first of equations (57), and q" from the first of (61); then, by means of Table XVII., we derive the corresponding values of log/' and log/". The coefficients of dr in the expressions for q and q" will be given with sufficient accuracy by means of the approximate values of dr and sin /9, and will not require any further correction. Then we compute S r cos/9, and find the integral C aiid the complete value of ^- will be given by (59). The value of - will then be given by equation (63). The term r ( -J- I will always be small, and, unless the inclination of the orbit of the dis- turbed body is large, it may generally be neglected. Whenever it shall be required, we may put it equal to - 1 -rr I The corrected values of the differential coefficients being introduced into the table of inte- gration, the exact or very approximate values of dw, dr, and z will be obtained. Should these results, however, differ much from the corresponding values already assumed, a repetition of the calculation may become necessary. In this manner, by computing each place separately, the terms depending on the squares, products, and higher powers of the disturbing forces may be included in the results. It will, however, be generally possible to estimate the values of dw, dr, VARIATION OF POLAR CO-ORDINATES. 467 and 2 for two or three intervals in advance to a degree of approxi- mation sufficient for the computation of the forces for these dates. In order that the quantity a, representing the interval adopted in the calculation of the perturbations, may not appear in the integra- tion, we should introduce it into the equations as in the case of the variation of the rectangular co-ordinates. Thus, in the determina- tion of Sw we compute the values of to r-, and since the second member of the equation contains the integral CSj-cosftd'., if we introduce the factor to 2 under the sign of integration, this integral, omitting the factor to in the formulae of integration, will become MJ S r cos /9 dt, 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 the factor of. If the second of equations (65) is employed, the first and third terms of the second member will be multiplied by tar; but since the value of 8 is supposed to be already multiplied by a?, the second term will only be multiplied by to. 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 formulae for the deter- mination of the forces S w 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 t Q . Let i' and ft' denote the inclination and the longitude of the ascending node of the disturbing body with respect to the ecliptic, and let J 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 ft , and let N' 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 4fi8 THEORETICAL ASTRONOMY. ecliptic, the fundamental plane, and the plane of the orbit of the dis- turbing body with the celestial vault, we have sin l8in (2V + 2V') = sin $ (ft' ft c ) sin (i' + i \ sin $JCOB CflT + 2V') = coaKft' - 8.) sin i (* - s), ,x cos^Jsin K2V -ZV') = sin (&' - ) cos from which to find 2V, 2V', and /. Let ft' denote the heliocentric latitude of the disturbing planet with respect to the fundamental plane, w' its longitude in this plane measured from the axis of x, as in the case of w, and u r the argu- ment of the latitude with respect to this plane. Then, according to the equations (82) u we have tan (w' 2V) = tan u f cos I, = tan /sin (w' 2V). If u f 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 w ', and then w' and ft' will be found from (67). We have, also, cos u ' = cos /?' cos (w r JV), which will serve to indicate the quadrant in which w' 2V must be taken. The relations here derived are evidently applicable to the case in which the elements of the orbits of the disturbed and disturbing planets are referred to the equator, the signification of the quantities involved being properly considered. The co-ordinates of the disturbing planet in reference to the plane of the disturbed orbit at the instant t as the fundamental plane will be given by a/ = / cos /5' cos w', tf = r' cos /9' sin w', (69) Z '=r'sin/3'. To find the force R, we have R = X X -+ Y y - + Z-. r r r T r VARIATION OP POLAR CO-ORDINATES. 469 and r"/' Substituting in these the values of x f , y', z' given by (69), and the corresponding values of x, y, z given by (50), and putting h = ^~^ (70) we get R = m 'Jf I h r' cos ff cos ,3 cos (to w) + h / sin ? sin ? I V (71 ) \ /o 1 / The equation S e rcos,9= Yx Xy gives ^ = m'^Ar'cos / 5'sin(M;' w), (72) from which to find $ . Finally, we have Z=m'v(hrs\n? -A (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 0) and Z become K = m'v( A/ cos,? cos (vf wj -^ ), ^ ' (74) S, = m'P h r' cos ,? sin (vf tr a ), To compute the distance /o, we have which gives /> = r' 1 + r 1 2rr' cos ^S cos ? cos (/ w) 2r r' sin /9 sin/S', (75) and, if we neglect terms of the second order, we have />.' = r' 1 + r. 1 2r. r' cos ^ cos (vf w,). (76) If we put cosr = cos^cos/9'cosOo' 10) -}- sin sin *, (77) we have ^^r/' + r 1 2iycosr = r" sinV + (r / cosr) 9 ; 470 THEORETICAL ASTRONOMY. and hence we may readily find p from P sin n = r' sin r, .. pcoan = r /cos r, the exact value of the angle n, however, not being required. Introducing f into the expression for R, it becomes K = m'k*(hr'coar L .\, (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 R, S , and Z will be computed by means of the equations (74), /> being found from (76) or from (78), when we put cos f = cos ft cos (vf 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), Zfrora (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, S , and Zthus 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' 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) t or by means of (85) t . When the inclination V is very small, it will be sufficient to take u' = I' - &' + s tan* K sin 2 (I' '), in which s = 206264.8. But when the tables give directly the lon- gitude in the orbit, u f -f &', by subtracting SI' from each of these longitudes we obtain the required values of u'. It should be observed, also, that the exact determination of the values of the forces requires that the actual disturbed values of r', tc', and ft' should be used. The disturbed radius-vector r 1 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 r and fi re- quires that we should use the actual values of N' t N, and I in the solution of the equations (68) and (67). Hence the disturbed values of ft ' and i' 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 ft ' and ' may bo neglected ; and whenever the variation of either of these has a sens* blc effect, we may compute new values of N, N f , and I from time to time, by means of which the true values may be readily interpolated for each date. We may also determine the variations of N, JV 7 , and / arising from the variation of ft' and ', by means of differential formulae. Thus the relations will be similar to those given by the equations (71) 2 , so that we have -'*. *. < 31 = sin N' sin if y,=r cos /5 cos w sia & + r cos /? sin w cos i cos Q, z sin t cos & , (81) z f =r cos sin w sin t -f 2 cos v Introducing also the auxiliary constants for the ecliptic according to the equations (94) t and (96) 1? we obtain x, r cos ft sin a sin (A -\- w) -\- z cos a, y,=r cos /9 sin 6 sin (B -{-w) -\- z cos 6, (82) z, = r cos /? sin i sin w -f 2 cos i , by means of which the heliocentric co-ordinates in reference to the ^cliptic may be determined. If the place of the disturbed body is required in reference to the equator, denoting the heliocentric co-ordinates by x,,, y, t) z,,, and the obliquity of the ecliptic by e, we have x n = x, y = y,cose z, sine, z,, = y, sin e -f- z, cos e. Substituting for x,, y n z, their values given by (81), and introducing the auxiliary constants for the equator, according to the equations '99)! and (101) w we get x n = r cos ft sin a sin ( A + w) -\- z cos a, y,, = rcosft sin b sin (5 + w} -f- z cos b, (83) 2,, = r cos /? sin c sin ( G -f- w) + 2 cos c. The combination of the values derived from these equations with the corresponding values of the co-ordinates of the sun, will give the required geocentric places of the disturbed body. These equations are applicable to the case of any fundamental plane, provided that the auxiliary constants a, A, b, B, &c. are determined with respect to that plane. In the numerical application of the formulae, the value of w will be found from w = w, -f- <*w> VARIATION OF POLAR CO-ORDINATES. 473 u 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 6, and cos c. If the values of /T O , , and i used in the calculation of the per- turbations are referred to the ecliptic and mean equinox of the date ', 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 (0 referred to the ecliptic of t f being reduced to that of ", by means of the first of equations (115)i. Then & and i should be reduced from the ecliptic and mean equinox of t ' to the ecliptic and mean equinox of t " 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 " are to be determined, the proper corrections having been applied to & and i , the mean obliquity of the ecliptic for this date will be employed in the determination of the auxiliary constants a, A, &c. with respect to the equator, and the equations (83) will then give the required values of the co-ordinates. If we differentiate the equations (83), we obtain, by reduction, -p = r cos ft sin a cos (A -f- w) -^ -f- sec ft sin a sin (A + w)-^ -f- (cos a tan ft sin a sin (A + 0) -^- a llL r C o S ft s in b cos (B + w) - + sec ft sin b sin (B + to) -^ dz (84) + (cos b tan ft sin b sin (B +to)) -^-, -f(cos c tan ft sin c sin ( C -f- w)) -^-, by means of which the components of the velocity of the disturbed body in directions parallel to the co-ordinate 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 Jcl/pT( 1 + m) , ddw dr IcVl+m , dSr /OCN i= -^r"' 8 " 11 ' ^ from which to find -37 and -37- a eft 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 f measured from the ecliptic of 1860.0. Then, by means of the formula? (66), using the values of & an( i *o 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 r is given by equation (68), and then w r and /3' are found from the equations (67). Thus we have Berlin Mean Time. Iogr W O = UQ logr* vf ft' 1863 Dec. 12.0, 0.294084 192 4'24".5 0.73425 1418'54".6 l'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 /> 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 to 2 for convenience in the integration, as already explained, we obtain the following results : Date. tfR tflSp 9 uPZ 1863 Dec. 12.0, -f 1".4608 + 0".1476 -f 0".0009 + 0".0282 1864 Jan. 21.0. + 1 .4223 .6757 + .0101 .2361 NUMERICAL EXAMPLE. 475 Date. 2 jR "'Vo u*Z <->fV 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 .6070 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 x The integral = l + A (91) we obtain e cos v = e cos v -f- /9 This equation, combined with (90), gives cos v y3 sin v , (92) e sin (v v ) = a,e a sin v cos t? + f cos v e cos (v ) == e + ae sin 2 v + y sin v + - /3 cos , r o by means of which the values of e and v may be found, those of the auxiliaries a, /9, 7-, being found from (89) and (91). Then we have a=p sec 2 f, tan = tan (45 ?) tan w, 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 &, i, and to or TT. Let 6 C denote the longitude of the ascending node of the instantaneous orbit on the plane of the osculating orbit, defined by & and , mea- sured from the origin of w, and let :y denote its inclination to this plane. Then we have tan y sin (w ) = tan ft, (93) a . tan y cos (w } = and hence CHANGE OF THE OSCULATING ELEMENTS. 479 tan O - ) = sin 2/3 *-, (94) by means of which may be found. The quadrant in which is situated is determined by the condition that sin (w 6 ) 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 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 , , Qr in i i sin Q- (_:)_(&_ )) = sin < sin (i - ,), ^ in icosG( - Kft - &)) --=cos^ 8in J ( These equations will furnish the values of i, u , and & & , and hence, since and & are given, those of & and u. The value of v having been already found, we have, finally, and the elements are completely determined. These elements will be referred to the ecliptic and mean equinox to which & and i 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 x, n y,,, z,,, in reference to the fundamental plane to which SI and i are to be re- ferred, by means of the equations (83), and also those of -^-, -?, -~ 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 o> 2 , or the square of the 480 THEORETICAL ASTRONOMY. adopted interval, is introduced into the expressions for the forces aud differential coefficients, the first integrals will be d3r ddw dz 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 /5 = 0, and the equations (51) and (54) become ffir dv? P(l+m) 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 8 is to vary the form of the orbit and the angular distance of the perihelion from the node. If we denote by Z the component of the disturbing force perpendicular to the plane of the instantaneous orbit, the effect of this will be to change the position of the plane of the orbit, and hence to vary the elements which depend on the position of this plane. Let us take a fixed line in the plane of the instantaneous orbit, ami 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 ) 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 w Let us now suppose Jt to denote the true longitude in the orbit, so that we have 482 THEORETICAL ASTRONOMY. aoi) then, since is equal to TT 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 + PL (t t \ r cos v = a cos E ae, (1 02") 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 *, = v+ x . (103) Further, let M + f* (t ) ~J~ ^M 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 , / are used instead of the instanta- neous values a, e, and j, gives the same longitude ^,, so that we have E, e a sin E, = M + ^ (t + dM, r, cos v, = a cos E, a e Q , ^ r, sin v t = a V/l e * sin E, *,=Vi+Z = V,+ o- If, therefore, we determine the value of dM so as to satisfy the con- dition that I, = v + , the disturbed value of the true longitude in the orbit, neglecting the effect of the component Z of the disturbing force, will be known. The value of r, will generally differ from that of the disturbed radius-vector r, and hence it becomes necessary to introduce another variable in order to consider completely the effect of the components R and 8. Thus, we may put r = r,(l + v), (105) and v will always be a very small quantity. When 8M 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), \r-j- expresses the areal velocity in the in- stantaneous orbit, and it is evident that, since the true anomaly is not. affected by the force Z perpendicular to the plane of the actual orbit, $r* -p must also represent this areal velocity, and hence the equations become ** w = S Sr dt tr ' 179. If we differentiate each of the equations (104), we get ^dE, , dSM (i e COS.E;) - = fi + --, dr, dv, . -dE, cosv, -T- r, sm v, -jr = a a sin E, -?-, (107) dr, , dv, /, sm v, -- + r, cos v, - = a V 1 From the second and the third of these equations we easily derive r,- = (a l/'l J r > sin v ' cos E > V> cos v ' sin E ) ' < Substituting in this the values of r, sin v,, r, cos and -^, and re- ducing, we get dr, / . d5M\ dr, , From the same equations, eliminating -^-, we get e *r ; cos v, cos ^ + a r, sin v, sin U,) -', which reduces to 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 f + ' * dr, . dv _ = (1+ ,)_ + r ,_, d'r . ,dV, , O dr, dv ffv Differentiating equation (108) and substituting for its value already found, we obtain dV,_fe 2 (l+m)e co8t;,/ 1 d8M\* lcl+ e sin v, &9M dt* ~ r, \ + /z ' dt I 4 oT /^ df ' and the last of the preceding equations becomes The equation (110) gives 1 MM 2 d>, 2 which is easily reduced to , dt 1 -f dv 2 (! + ")' Jfel/p (l + m)' 1 Sr and hence we derive The equation (109) gives VARIATION OF CO-ORDDsATES. 485 and, since this becomes , (113) Combining equations (112) and (113) with the second of equations (106), we get .V*T--VV*+) From (110) we derive m dt and the preceding equation becomes 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,, z, denote the co-ordinates of the body referred to the fundamental plane to which the elements belong, and ar, 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 fj and we shall have (116) If the position of the plane of the orbit remained unchanged, these 486 THEORETICAL ASTRONOMY. cosines a and /? would be constant; but on account of the action of the force perpendicular to the plane of the orbit, these quantities are functions of the time. Now, the co-ordinate z, is subject to two 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 -~ I expresses the velocity resulting from the constant elements, and -^- that part of the actual velocity which is duo 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, may be regarded as constant during the same time, and as receiving an increment only at the end of this instant. Hence we shall have dz, _idz,\ ~dt -\~dtj Differentiating equation (116), regarding a and /3 as constant, we and differentiating the same equation, regarding x and y as constant, we get Differentiating equation (117), regarding all the quantities involved as variable, the result is Now, we have Z, = aX + ft Y + Z cos i, (120) in which Z, denotes the component of the disturbing force parallel to the axis of z,, and i the inclination of the instantaneous orbit to VARIATION OP CO-ORDINATES. 487 the fundamental plane. Substituting for X said Y their values given by the equations (1), and reducing by means of (116), w^ obtain or Comparing this with (119), there results T-5 + M- 181. The equation (120) gives (122; The component of the disturbing force perpendicular to the plane of the disturbed orbit does not affect the radius-vector r ; and hence, when we neglect the effect of this component, and consider only the components R and 8 which act in the plane of the orbit, we have d\ - in which z denotes the value of z, obtained when we put Z=Q. Let us now denote by dz, that part of the change in the value of z, which arises from the action of the force perpendicular to the plane of the disturbed orbit, so that we shall have 2, = z -f- Sz,, a = o -f- So,, j3 = P + 5/3. Substituting these in equation (122) and then subtracting equation (1 23) from the result, we get = _ da + w (124) The equations (116) and (117) give If we eliminate o/3 between these equations, there results 488 THEORETICAL ASTRONOMY. and since the factor of da in this equation is double the areal velocity in the disturbed orbit, we have Eliminating da from the same equations, we obtain, in a similar manner, (126) / kl/p(l+m) Substituting these values in equation (124), it becomes = il/p(l+m)\\ dt dt If we introduce the components R and S of the disturbing force, we have r r r r and hence dy_ _ dx_ _ It , / (14 _ N _ o^ Fx Xy = >Sr. Therefore the equation (127) becomes d*5z, P(l+w), ~-~ = -T ^2, + ZCOS I * , fe , ( 128 ) + We have, further, which, by means of the equations (108) and (109), gives dr _ e c sinv, .<&>, dv _ ^ljt>(l + m) . dv ~~ +r '~~ e " smv ' + r '-- Substituting this value in the equation (128), we obtain VARIATION OF CO-ORDINATES. 489 (129) which is the complete expression for the determination of dz,. 182. The equations (110), (115), and (129) determine the complete perturbations of the disturbed body. The value of v must first be obtained by an indirect process from the equation (115), and then 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 R, S, and Z, let w denote the longitude of the disturbed body measured in the plane of the instantaneous orbit from its ascending node on the fundamental plane to which & and i are referred, it being the argument of the latitude in the case of the disturbed motion. Let w' 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 ft' denote its latitude in reference to this plane. Finally, let N, N f , /, and u f 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 sn - sn cosK&'-a)sinK*'--*) sin K&'~ 8) cos^' + *), cos Jcos \ ( N N') = cos K a ' ft ) cos i (i' i), from which to determine N, N', and /. We have, also, < =u' N', tan (w' JV) tan ' cos J, (131) tan p tan I sin (w N), from which to find w' and ft' , u f being the argument of the latitude of the disturbing body in reference to the plane to which R and i are referred. Since, when the motion of the disturbed body is referred to the plane of its instantaneous orbit, {3 = 0, the equations (71), (72), and (73) become It = m'k* I h r' cos /?' cos (yf w) - s ), p ' (132) S = m'tfh r' cos (r sin (w' to), 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* 2rr' cos jf cos (w' w\ (133) or, putting COS f = COS f? COS (w' W), , the equations ^sinw^/sinr, (l .. p cos w = r r' cos p. 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 SI ' and i' 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 + // ( ) -J- dM and the elements , e , the values of v, and r,. Then, since v + / = v, -f- TT O , we have, according to (100), w = v, + TT O ff. (135) We have, also, r = (l+v)r,. Tn the case of the fundamental osculating elements, we have which may be used as an approximate value of a; but the complete determination of w requires that ff= & -f- da shall also be deter- mined. The exact determination of the forces also requires that the actual values of SI and i as well as those of SI ' and i', shall be used in the determination of N t N f , and I for each instant. When these have been found, it will be sufficient to compute the actual values of N, N', 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 SI, i, Sl f , and i' may also be determined by means of differential formula?. 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 ** - d/ = cos N' di' cosNdi -f sin i sia JVd(' ft). When i and /are very small, it will be better to use sin i sin N' sin t' sin N sin/ sin(ft' ft)' sin/ sin(ft' ft)' U37) in finding the numerical values of these coefficients. By means of these formulae we may derive the values of dN, dN f , and di corre- sponding to given values of ft, di, <5ft', and di'. The formulae by means of which da, dft, 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' and /?'. 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 diiferential 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, 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,, neglecting terms of the second order, are, according to the equations (110), (115), and (129), the following: 492 THEORETICAL ASTRONOMY c\ OQ^ ddM 1 r JT-= jo ; I & r o dt ^/v> * fcl/> (l+m)J Po The value of v is first found by integration from the results given by the second of these equations, and then 8M 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- db served, however, that the expression for -5 involves only one indi- rect term, the coefficient of which is small, and the same is true in the case of -^-, while -^- 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 log r log r, + V, wherein ^ denotes the modulus of the system of logarithms. We may also use v, 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 /?' 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-ORDIXATES. 493 The results may be conveniently expressed in seconds of are, and afterwards v and dz, 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, dM may be expressed in seconds of arc, while v and dz, are obtained directly in units of the seventh decimal place. It will be advisable, also, to introduce the interval to into the formulae in such a manner that this quantity may be omitted in the case of the formulae of integration. 184. In the case of orbits of great eccentricity, the mean anomaly and the mean daily motion cannot be conveniently used in the nu- merical application of the formulae. Instead of these we must employ the time of perihelion passage and the elements q and e. Thus, let T be the time of perihelion passage for the osculating ele- ments for the date t , and let T + $T 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,, so that we have v + X = v, + JT O . In the case of parabolic motion we have, neglecting the mass of the disturbed body, = ^ (139) the solution of which to find v, is effected by means of Table VI. as already explained. To find r,, we have r, = q sec 2 Av,. For the other cases in which the elements 1/ and f* 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 C = 9.9601277, and with this as the argument we derive from Table VI. the corresponding value of V. Then, having found i ^= l ~ e , by means of Table IX. we derive the coefficients required in the equation Vt = V + A (1000 + B (1000' + (7(1000", (140) 494 THEORETICAL ASTRONOMY. from which v, will be determined. Finally, r, will be found from g.a+o. (14 i) 1 + e cos v, When Tut>le X. is used, we proceed as explained in Art. 41, using the elements T= T -j- dT, q , and e , 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 gives or, simply, dM and the equation (110) becomes (142) by means of which the value dT required in the solution of the equa- tions for r, and v, may be found. If we denote by t, the time for which the true anomaly and the radius-vector computed by means of the fundamental osculating ele- ments have the values which have been designated by v, and r t) re- spectively, we have _! d3M_ dt, and the equation (110) becomes dtf 1 + M ! * fSr dt, (143) (! + ")' jfevV.n-4-mW dt (1+x) 2 ' (l + v) or, putting t, = t -f- dt, dSt 1 11 r e = 1 + TI ; T= 7 I flr ai. (144) dt (1 + O 2 (1 + ^i/j-j (1+m) / If we determine dt by means of this equation, the values of the radius-vector and true anomaly will be found for the time t -f dt instead of t, according to the methods for the different conic sections, VARIATION OF COORDINATES. 495 using the fundamental osculating elements. The results thus obtained are the required values of r, and v, respectively. 185. When the values of the perturbations v, dz t) and 8M, 8T, or dt have been determined, it remains to find the place of the disturbed body. The heliocentric longitude and latitude will be given by cosbcos(l ft) = cos(A ft), cos b sin (I ft ) = sin (A ft ) cos i, sin 6 = sin (A ft ) sin i, or, since ^ = ^ a -f ft, cos b cos (I ft ) = cos (A, ff~), cos 6 sin (7 ft ) = sin (A, *, ft ) sin (A ^ ) +cos(A-^ )sin(A -^ )(l+cosr/) (148) H sin (^, ^ ) ((cost cos i ) cos (A ^ )+sin (/ cos (h ft ) cos to cos 5', from which we get cos t cos to = sin t' cos (^ ft o) sin r/ cos ^ (1 + cos We have, also, sin (ff ft ) sin t = sin / sin ( h* ft ), sin ( ft A) sin t = sin t' sin (h^ ft ), VAEIATION OF CO-OHDINATES. 497 or sm(ff ft )sin(ft A) sin 1 i = sin j (A ft ) sin^sim/. Hence we derive (cosi cos to) cos fa ft )4-sin (a ft ) sin (ft &) sin 1 1=8^ sin if (1+cos V) cos to cos (V-fto). Combining this and the equation (151) with the equations (148), we obtain cos b cos (I fi)=cos 0*, ft ) cos C^, ft ) +sin(A, ft ) sin (h^ ft ) cos 4, cos b sin (I h) =sin (<*, ft ) cos (^ ft ) cos ^ cos (A, ft ) sin (^> ft ) sin i}' 8z f ~l cosV"' sin b =sin (A, ft ) sin ^ -| '- If we multiply the first of these equations by cos (h ft ), and the second by sin (h ft ), and add the results; then multiply the first by sin(Ao ft ), and the second by cos^ ft ), and add, we get cos 6 cos(J fto (A ^))=cos(>i, ft )+sin(A <) ft ) S1 J , . ^-, cos6 sin (I ft (h At,))=sin (-*, ft )cos4 cos(^ ft ) sin b Let us now put sin b =sin(A, ft )sinioH . (152) p' = sin (ff ft ) sin i, (f = cos (a ft ) sin i sin i* (153) and there results, from (149), = q sin (/., ft ) p' cos (J, ft ). (154) Comparing this with equation (150), we observe that p' = sin TJ' sin (Ao fto). q' = sin V cos (A,, ft) cos 4 sin 4 (1 cos 7'). Therefore, we have S2 498 THEORETICAL ASTRONOMY. and, if we put F= h h , the equations (152) became sin b =sin(A, & )sintoH '-. As soon as F, p r , q', and if are known, these equations will furnish the exact values of I and 6, those of \, and r being found by means of the perturbations v and dM. 186. The value of F may be expressed in terms of p' and q'. Thus, if we differentiate the first of equations (147) and reduce by means of the remaining equations of the same group, we get d(h Ro) = cosy'd(& A) + cosi dff -f- smt sin( di at at From the equations (118) and (121), observing that we derive, by elimination, d _ r sin J, cost df __ r cos ^ cost "~ ~ ' 500 THEORETICAL ASTRONOMY. Therefore we shall have r cosi sin (A, & ) kVp(l+m) d(f _ r cosi cos (^a,)^ d , ' kVp (1 + m) oy means of which p' and q f 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' y that of p' will be given by means of (154), or and if p' is determined, q' will be given by rsin(/l, & ) If both p' 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' 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 -57-'. The equations (161) give p' = cos & So, sin & $P, q' = sin & 3a -f- cos & 8,3. Substituting for da and S{3 their values given by (125) and (126), and putting . - x" = xcos& 9 + y sin , y" = x sin & + y cos > we obtain 1 / ,.d8g. . W\ p dx"'\ (163) .*]' VARIATION OF CO-ORDINATES. 501 Substituting further the values x" = r cos (I, - a.), y" = r sin (A, - ,), and also \\ <^V , ** sin (^. Q n ) d P = (cos (^ a ) + e cos (* Q )) - . , . ,. _. , . . f ^z, v ' = + (sin (A, & ) + esm ( X Q.))-^ JP which may be used for the determination of p' and q'. 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 , /> , and <> "o- The actual values of X, and r are obtained directly from the values of the perturbations. When p' and q' have been found, it remains only to find cos ?', and 1 cos r/j in order to be able to obtain F by means of the equation ~(159). From (153) we get P'* ~f ?'* s i n * * sm * ? 'o 2g' sin i , and hence _ cos i = 1/1 p' 2 (q' -}- sin i ) 2 , (165) from which cost may be found. The equation (157) gives 1 cos V = cos i (cos i + cos f) c[ sin i , (166) by means of which the value of 1 cos r/ will be obtained. If we substitute the values of p', q', -, and -jj given by the equations (153) and (162) in (159), it is easily reduced to =f J 1 (1 cosi/) kVp(\ + m) which may be used for the determination of f. When we neglect terms of the order of the cube of the disturbing force, in finding r we may use p Q in place of p and put 1 cos r/ = 2 cos 2 ^ so that the formula becomes 502 THEORETICAL ASTRONOMY. r= - * Cdz,zdt. (168) 2cos*i k\/p (l+m)J 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 6 for the disturbed motion. From r, I, and 6 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 SI, a, and i are known; and the required formulae for the deter- mination of these elements may be readily derived. Thus, the equa- tions (160) give, by differentiation, whence da, . . di . . d -= and F re- dt at at quire, for an accurate solution, that the disturbed values i, ', there remain the corrections due to di and 3& 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 (a & ) di -\- sin i cos (o &) dff, dc[ = cos i cos (ff & ) di sin i sin (a & ) da, give cost di = sin (a & ) dp' -\- cos(-aj d.(j-+>-aj nnZcost sin /cost If we neglect the perturbations of the third order, these equation?. COSt COSt I i \ -r rinJT-2-rl, < COS 1 / by means of which di and dN may be determined, p f and g' being found by means of the equations (164), using e w TT O , and p in place of e, /, and p. The results for di and ^^V' obtained from (173) being applied to the values of I' and N f as already corrected on account of di f and 32', give the required values of these quantities. NUMERICAL EXAMPLE. 5 W 5 When we consider only di and d&, since sin i' cos N' = cos i sin J -f- sin i cos J cos N, we easily find d N = COS I dN' are found from the elements given in Art. 166. The results thus obtained are the following: Berlin Mean Time. log r v logr' w' /3 1863 Dec. 12.0, 0.294084 354 26' 18' 'jfl 0.73425 14 18' 54' '.6 l'38".l 1864 Jan. 21.0, 0.294837 10 2 45 .7 0.73368 17 21 44 a 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 & 0.73237 23 28 59 J8 51 7 .6 May 20.0, 0.324298 54 14 41 A 0.73164 26 33 32 .1 1 7 29 .7 June 29.0, 0.339745 67 21 23 J6 0.73086 29 38 44 J 1 23 43 .5 Aug. 8.0, 0.356101 79 32 18 a 0.73003 82 44 41 J 1 39 46 .3 Sept. 17.0, 0.372469 90 49 57 .6 0.72915 86 51 24 .6 1 55 35 .2 Oct. 27.0, 0.388214 101 19 9 & 0.72823 88 58 57 .5 2 11 7 .5 Dec. 6.0, 0.402894 111 5 42 .2 0.72726 42 7 23 .8 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 + JT O = + 197 38' 6".5, and the components of the disturbing force are determined by means of the formulae (132), ft being found from (133) or (134), and h from (70). The adopted value of the mass of Jupiter is 1 1047.879 and the results for the components R, S, and Z are expressed in units of the seventh decimal place. The factor of is introduced for conve- nience in the integration, 10 being the interval in days between the successive dates for which the forces are to be determined. Thus we obtain the following results: NUMERICAL EXAMPLE. 507 Date. Bfi u 2 Sr wlZc and then, having found v, a) -jj- 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 f> 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 ffv *& d?6z, "~M SM V an d *o t 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 oM, 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 Q = 2.5107584, y = + 0.6897713, z = 0.1406590; and for the perturbations of these co-ordinates we have found dx = + 0.0001773, 3y = + 0.0001992, 3z = 0.0000028. Hence we derive a; = 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, 6 = 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 610 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 smalluess 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 } p' f and q' may be effected from the results for -57, -^-. and -4r by means of the formulEe for integration by mechanical quadrature, as already illustrated, or we may find F by a direct integration, and the values of p' and q' 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 < z -jp and o> 2 -^ will dv . dfa, . . . dv ddz. give the values ol < a ~jT and W ~JT~' and hence those of -r and -57- , which, in connection with ~TJ~> 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 The equation (109) gives dv, kl/p (1 -f- m) / 1 dSM "dt~ ~^? l 1+ /T -~dT Hence, since r = r, (1 -f 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 J^T. and a ls in the equations (164); and the value of cosi may be determined by means of (165). In the disturbed orbit we have dr kVl+m . -- = - ~ e sm v, and the equations (108) and (111) give dr kVY+m - , 1 Therefore we obtain which, by means of (176), becomes , - . The relation between r and r, gives P _ P* xj _^_ y ^ 1 + e cos v 1 -f- e cos v, and, substituting in this the value of p already found, we get + v)-l. (178) 512 THEORETICAL ASTRONOMY. Let us now put _ . _ (179) ~~ ' ' a and ft being small quantities of the order of the disturbing force, and the equations (177) and (178) become e sin v = e sin v, -f- *e sm v i + ft ecosv = e cos i>, -j- ae cos v, -f- a. These equations give, observing that r, (cos v, -f- e ) =p cos jE",, e sin (v, v) = a sin v, /? cos r,, , op (180) e cos (v, v) = e -f- -*= cos E,-\- P sin V,, from which e, v, v, and v may be found ; and thus, since / = fo + (v/-), (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 q' may now be rigorously effected, and the corresponding value of cos* being found from (165), -jjr and -^ will be given by (162). Then, having found also 1 cos if 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, but also the corresponding heliocentric or geocentric places of the body, may be found. If we put f' = a sin v, ft cos v,, and neglect terms of the third order, the equations (180) give e e e a in which s=206264".8. These equations are convenient for the CHANGE OF THE OSCULATING ELEMENTS. 513 determination of e and , v, and hence f. by means of (181), when the neglected terms are insensible. The values of p, e, and v having been found, we have tan i J5= from which to find the elements sin (p = e e the value given bv the first of equations (183), the result is 2^ sin

(192) Substituting in this the values of a ft and i i given by (190), we get a = ft, + -^_ - r , - l ~? sm l l .p'q's + 1\ (193) 1 sin i t cos i '^^ CHANGE OF THE OSCULATING ELEMENTS. 51 5 T 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 ?, and 2& x 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 3M, .. d$M dv , dftz. , v, oz,, JT~> -ji> and 57-' 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 dz, 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, and Sz,, 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 hence = (e sin v R -\- ), (200) dt kvp (1 + w) for the determination of dp. The first of the equations (97) gives and hence we obtain Sr Jtl/l + ro ^ = ^^=S. (201) dt ytl/1 4- m The equation p = a (I e 2 ) gives Equating these values of -^ and introducing the value of -^- already found, we get de 1 dt kVp(l+m) (202) 520 THEORETICAL ASTRONOMY. and since = 1 4- e cos v, = 1 e cos E, r a E being the eccentric anomaly in the instantaneous orbit, this becomes ~ = * (p sin , (a cos p sin v5 + a cos ? (cos v + cos -E) ^>'). (204) . f A;!/^ (1 + m) 195. When we consider only the components R and S of the dis- turbing force, the longitude in the orbit will be We have, therefore, JL the differentiation of which, regarding the elements as variable, gives Therefore . : r cos v -= h r sin v -^- at at cosi; and, since j) cos E=r (cos t? + e), we have j? (1 cos v cos E}=r sin* v, so that the equation becomes VARIATION OP CONSTANTS. 521 ' (205) from which the value of ~ may be derived. at If we introduce the element to, or the angular distance of the peri- helion from the ascending node, it will be necessary to consider also the component Z; and, since a)=Z a, we shall have du> d% dff d% . dl -dt--df--dt = -df- cosl -dr' and hence -coB<2. (206) In the case of the longitude of the perihelion, we have dx da> dR and therefore I i e . (207) The first of the equations (15) 2 gives in which M 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 -- _ (2 cos 1 v cos v cos E) cot ?S\ (t f.)- ((T) cot y cos v 2r cos y-) Jg (j> + cot

S) , -(*-4)^ 0208) The equation (205) gives 522 THEORETICAL ASTRONOMY. 1 P cot P cos v -fr kVp(l + m) ~~dt' by means of which (208) reduces to _ d* 2r COS y I? /, , N cos r --- ==== K (t r c ; -j-i at which will determine the variation of the mean anomaly at the epoch. Since the equations for the determination of the place of the body in the case of the disturbed motion are of the same form as those for the undisturbed motion, the mean anomaly at the time t will be given by Jf = Jf. + 8M. + (t- < ) O + ^), in which // denotes the mean daily motion at the instant / . There- fore we shall have Jf = Jf. + * + , (* - O + (< - the integrals being taken between the limits t and <. The quantity expresses the mean anomaly at the time in the undisturbed orbit ; and if we designate by dM the correction to be applied to this in order to obtain the mean anomaly in the disturbed orbit, so that we shall have and hence Differentiating this with respect to t, we get VARIATION OF CONSTANTS. Substituting iu this the value of ^~ from (209), the result is which does not involve the factor t 1 explicitly, and by means of which the mean anomaly in the disturbed orbit, at any instant t, may be found directly from that for the same instant in the undisturbed orbit. To find the variation of the mean longitude L, we have dL dM d* d x .d -df == -dr + ^t = -dt+-d and therefore To find the variations of SI and i, since u denoting the argument of the latitude in the disturbed orbit, we have, according to the equations (169) and (170), * r sin u ^ sin * (212) .:= _ - - dt kVp (1 + m) The inclination t, may have any value from to 180 ; and when- ever the elements are given in accordance with the distinction of re- trograde motion, they must be converted into those of the general form by taking 180 i in place of the given value oft, and 2 x in place of the given value of ic, before applying the formula? which involve these elements. 196. In the case of the orbits of comets in which the eccentricity differs but little from that of the parabola, the perturbations of the perihelion distance q and of the time of perihelion passage T will be determined instead of those of the elements M and a or //. The equation j=3(l-M) gives 524 THEORETICAL ASTRONOMY. dq _ 1 dp q de ~dt ~ 1+6 ' ~dt 1 + e ' ~dt' and substituting in this the value of -^ already found, and neglect- ing the mass of the comet, which is always inconsiderable, we get -- by means of which the variation of q may be found. In the case of elliptic motion the value of -^ may be found by means of (202) or -(203); but in the case of hyperbolic motion the equation (202) will be employed. It should be observed, also, that when the general formulse for the ellipse are applied to the hyperbola, the semi- transverse axis a must be considered negative. When the orbit is a parabola, the equation (202) becomes ~ = - (p sin vR + 2p cos' -M), (214) and for the value of -jr we have at It remains now to find the formula for the variation of the time of perihelion passage. The relation between T and M is expressed by the differentiation of which gives and, substituting for -~ the value given by equation (209), we ge* dT 2ar aVp d x 1 dn Substituting further the values of -37- and JT given by the equations (205) and (199), the result is VARIATION OF CONSTANTS. 525 dT aR p 3k (t T} > which may be employed to determine the variation of T whenever the eccentricity is not very nearly equal to unity. It is obvious, however, that when a is very large this equation will not be con- venient for numerical calculation, and hence a further transformation of it is desirable. Thus, if we derive the expressions for -j- and -y- from the equations (24) 2 and (23) 2 , we easily obtain 2p dr p 3&(* T) . * . -j- = a (2r cos v -- ^-^ - e sin v) -\ 1 + e de e y p 2p r dv lp + r . 3k(tT) p\ p 1 1. r\ . - r - = al y smv -- ^ = - - 1 + - sin v. de \ e \/ r f e (1 -f ef \ n p / I + e de e \/ p r f e (1 -f ef p By means of these results the equation (216) is transformed into which may be used for the determination of -Tr> the values of -7- and -- being found by means of the various formulae developed in etc Art. 50. When a is very large, its reciprocal denoted by / may often be conveniently introduced as one of the elements, and, for the deter- mination of the variation of/, we derive from equation (198) $- = -- L= (e sin vR + 2- S\ (218) dt k In the case of parabolic motion we have e = l, and p = 2q; and if we substitute in (217) for - and - the values given by the equa- tions (33) 2 and (30) 2 , the result is dt 1 -f- tan 2 : CT \ (219) 526 THEORETICAL ASTRONOMY. 197. Instead of the elements usually employed, it may be desirable in rare and special cases, to introduce other combinations of the ele- ments or constants which determine the circumstances of the undis- turbed motion, and the relation between the new elements adopted and those for which the expressions for the differential coefficients have been given, will furnish immediately the necessary formulae. In the case of the periodic comets, it will often be desired to deter- mine the alteration of the periodic time arising from the action of the disturbing planets. Let us, therefore, suppose that a comet has been identified at two successive returns to the perihelion, and let r denote the elapsed interval. The observations at each appearance of the comet, however extended they may be, will not indicate with certainty the semi-transverse axis of the orbit, and hence the periodic time. But when r is known, by eliminating the effect of the disturbing forces, we may determine with accuracy the value of the semi-trans- verse axis a at each epoch, and, from this and the observed places, the other elements of the orbit according to the process already explained. Let // be the mean daily motion at the first epoch, and we shall have in which TT denotes the semi-circumference of a circle whose radius is unity. Hence we obtain *-/"* . (220) by means of which to determine f* . Then, to find the mean daily motion p at the instant of the second return to the perihelion, we have (221) the integral being taken between the limits and r. The provisional value of the mean motion as given by the observed interval r will be sufficiently accurate for the calculation of the variations of M and fi during this interval. The semi-transverse axis will now be derived by means of the formula VARIATION OF CONSTANTS. 527 from the values of p for the two epochs. Let r' denote the interval which must elapse before the next succeeding perihelion passage of the comet, and we have and consequently T> = ^ M "'. ( 222 ) M the integral being taken between the limits t = 0, corresponding to the beginning of the interval, and t = T'. We have, therefore, for the change of the periodic time due to the action of the disturb- ing forces. 198. The calculation of the values of the components R, S, and Z of the disturbing force will be effected by means of the formula given in Art. 182. It will be observed, however, that not only these components of the disturbing force, but also their coefficients in the expressions for the differential coefficients, involve the variable ele- ments, and hence the perturbations which are sought. But if we consider only the perturbations of the first order, the fundamental osculating elements may be employed in place of the actual variable elements, and whenever the perturbations of the second order have a sensible influence, the elements must be corrected for the terms of the first order already obtained. Then, commencing the integration anew at the instant to \fhich the corrected elements belong, the calculation may be continued until another change of the elements becomes necessary. The several quantities required in the computation of the forces may also be corrected from time to time as the elements are changed. The frequency with which the elements must be changed in ordei to include in the results all the terms which have a sensible influence in the determination of the place of the disturbed body, will depend entirely on the circumstances of each particular case. In the case of the asteroid planets this change will generally be required only after an interval of about a year; but when the planet approaches very near to Jupiter, the interval may necessarily be much shorter. The 528 THEORETICAL ASTRONOMY. magnitude of the resulting values of the perturbations will suggest the necessity of correcting the elements whenever it exists; and if we apply the pioper corrections and commence anew the integration for one or more intervals preceding the last date for which the per- turbations of the first order have been found, it will appear at once, by a comparison of the results, whether the elements have too long been regarded as constant. The intervals at which the differential coefficients must be com- puted directly, will also depend on the relation of the motion of the disturbing body to that of the disturbed body; and although the in- terval may be greater than in the case of the variations of the co- ordinates which require an indirect calculation, still it must not be so large that the places of both the disturbing and the disturbed body, as well as the values of the several functions involved, cannot be inter- polated with the requisite accuracy for all intermediate dates. In the case of the asteroid planets a uniform interval of about forty days will generally be preferred; but in the case of the comets, which rapidly approach the disturbing body and then again rapidly recede from it, the magnitude of the proper interval for quadrature will be very different at different times, and the necessity of shortening the inter- val, or the admissibility of extending it, will be indicated, as the numerical calculation progresses, by the manner in which the several functions change value. If we compute the forces for several disturbing bodies by using IR, IS, and IZ in the formula in place of R, S, and Z, respect- ively, the total perturbations due to the combined action of all of these bodies may be computed at once. But, although the numerical process is thus somewhat abbreviated, yet, if the adopted values of the masses of some of the disturbing bodies are uncertain, and it is desired subsequently to correct the results by means of corrected values of these masses, it will be better to compute the perturbations due to each disturbing body separately, and, since a large part of the numerical process remains unchanged, the additional labor will not bo very considerable, especially when, for some of the disturbing bodies, the interval of quadrature may be extended. The successive correction of the elements in order to include in the results the per- turbations due to the higher powers of the masses, must, however, involve the perturbations due to all the disturbing bodies considered. The differential coefficients should be multiplied by the interval (o, so that the formulae of integration, omitting this factor, will furnish directly the required integrals; and whenever a change of the inter- NUMERICAL EXAMPLE. 529 val is introduced, the proper caution must be observed in regard to the process of integration. The quantity 8 206264".8 should be introduced into the formulae in such a manner that the variations of the elements which are expressed in angular measure will be obtained directly in seconds of arc; and the variations of the other elements will be conveniently determined in units of the nth decimal place. It should be observed, also, that if the constants of integration are put equal to zero at the beginning of the integration, the integrals obtained will be the required perturbations of the elements. 199. EXAMPLE. We shall now illustrate the calculation of the perturbations of the elements by a numerical example, and for this purpose we shall take that which has already been solved by the other methods which have been given. From 1864 Jan. 1.0 to 1865 Jan. 15.0 the perturbations of the second order are insensible, and hence during the entire period it will be sufficient to use the values of r, v, and E given by the osculating elements for 1864 Jan. 1.0. The calculation of the forces jR, S, and Z is effected precisely as already illustrated in Art. 189, and from the results there given we obtain the following values of the forces, with which we write also the values of E : Berlin Mean Time. 4QR 40S 40Z JEb 1863 Dec. 12.0, + 0' '.0365 + 0".0019 + 0".00002 355 26' 8".2 1864 Jan. 21.0, .0356 .0086 .00025 8 14 57 .8 March 1.0, .0315 .0182 .00047 20 57 55 .1 April 10.0, .0250 .0259 .00068 33 26 47 .6 May 20.0, .0169 .0314 .00087 45 3f> 25 .3 June 29.0, + .0079 .0343 .00101 57 20 3 .8 Aug. 8.0, .0011 .0349 .00112 68 39 14 .6 Sept. 17.0, .0099 .0333 .00117 79 33 13 .1 Oct. 27.0, .0179 .0301 .00116 90 3 23 .2 Dec. 6.0, .0252 .0253 .00108 100 11 49 .1 1865 Jan. 15.0, -0 .0317 .0193 + o .00090 110 54 .3 We compute the values of the required differential coefficients by means of the equations dt r sin u dSi ~dt = 7= r cos u Z, Wp dfa "dT sin ? 530 THEORETICAL. ASTRONOMY. ~- = -= (a cos

(cos v -f cos E} we shall have which form is equally convenient in the numerical calculation. Thus, for 1865 Jan. 15.0, we find J3/= + 234".74, and from the several values of 1600-^- we obtain, for the same date> by means of the formula for double integration, Hence we derive 3M = + 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, 8M = -f 165".29, 532 THEORETICAL ASTRONOMY. 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, (t g 8ft. = + 126".27, and hence 6M = 8M + (t < ) Sfi = + 291".56. 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 , 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 formula 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 8M 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 d/t has been computed so as to involve the effect of this quantity during the in- terval t 1 0) the value of dM 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 -~> -> and -g- 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 -~ the variation of ft 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 ft, let us put a" = sin i sin ft , p' = sin i cos ft ; (224) then we shall have da" . di , ^ dft -jj- = sin ft cos i- -f sm i cos ft-^- dp' di dft -^- = cos ft cos t^- - sm ^ sm ft-gp Introducing the values of -7- and - given by the equations (212), and introducing further the auxiliary constants a, 6, A, and B com- puted by means of the formulae (94) t with respect to the fundamental plane to which ft and i are referred, we obtain r = -- , rZsin a cos (A -f- w), /) (225) rZsin 6 cos (B + w), x < kVp (I + m) by means of which the variations of a" and ft" may be found. If the integrals are put equal to zero at the beginning of the integration, the values of da" and dft" will be obtained, so that we shall have sin i sin ft = sin i sin ft -j- cos ^' + = ~ TT^ 7 ' Sin C ' COS (C " + ^' If we add the values of dx, %, dz, d^ d-jr> and d-jr to the cor- 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 sim 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 small 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 formulae (lOOX give -j- = sin a sin (A + ) -~ + r sin a cos (A + ) -g-i ?JL = sin b sin (B + ) ^- + r sin 6 cos (5 + ti) ^-, (235) the second by , , J-|f and the third by $--> and put dt. at P= sin a sin (A -\- u~) 8x -}- sin 6 sin (B -f- w) 5y -f- sin c sin ( C -\- u) dz, Q = sin a cos (^1 +u')dx-\- sin 6 cos (5 + = sin a sin (A + ) ^ + sin b sin T5 + u) t^L -f sin c sin (C + w ) 5 ~jp ^ = sin a cos (4 + u) 8~ + sin 6 cos (B + w) 5-^- + sin c cos ( C + w) 5 -3r we shall have, observing that -^ = = e sin v and that -^- = ~^- dz . Tc n . TcVp ^ From the equations PERTURBATIONS OP COMETS. 541 we get which by means of (237) become * ..in. . ,/ (238) + M - From the equation F* F== we get 27Wr- 2 s~ Substituting the values given by (238), observing also that P = ^r, this becomes j^ ^_FV re'sin't; esinj; _j^ i+^-l^p^ y .p + V and, since F' = - (1 + 2e cos t; -f- e 1 ), we obtain by means of which the variation of > x p may be found. The equation ^ = ^_F- a gives from which we derive 542 THEORETICAL ASTRONOMY. from which the new value of the semi-transverse axis a may be found. To find d/j. we have dp = |//a ?^e sin v re sin v Substituting this value of 3v in (245), and reducing, we find from which to derive the variation of the mean anomaly. 205. Let us now denote by x", y", z" the heliocentric co-ordinates of the comet referred to a system in which the plane of the orbit is the fundamental plane, and in which the positive axis of x is directed to the ascending node on the ecliptic. Let us also denote by x', y', z' the co-ordinates referred to a system in which the plane of the ecliptic is the plane of xy, and in which the positive axis of x is directed to the vernal equinox. Then we shall have PERTURBATIONS OF COMETS. 543 y" = z' sin cos t -f- y cos & cos -f z' sin , z" = z' sin & sin i y' cos & sin + z' cos i. If we transform the co-ordinates still further, and denote by x, y, 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, b, B, &c., we shall have 8x" = sin a sin A Sx -f sin 6 sin B Sy -f sin e sin C fe, fy" = sin a cos A 3x -{- sin b cos B Sy -f sin e cos C k, (248) 9J' = cos a fa -\- cos 6 Sy -f- cos e fa. Multiplying the first of these by sin u, and the second by cos u, adding the results, and introducing Q as given by the second of equations (236), we get cos u fy" sin u fa" = Q. Substituting for dx" and 8y" the values given by the equations (~3^j, the result is and, introducing the value of 3v given by (246), we obtain _ J __ * r e sin v r*e sin v ^ re sin Substituting further for 3e, dr, and o(\ p] the values already ob- tained, and reducing, we find g sint? p _ cos E , cos vVp p , (p + r)^in v by means of which 8% may be found. If we put cos a 8x + cos b 8y + cos e 8z = R, dx L dy A* _ix C 250 ) cos a *-j;- -f cos 6 o-^- + "5os e o^- K* the last of the equations (248) gives 544 THEORETICAL ASTRONOMY. Sz" = E', (251) and if we differentiate the equation dx . , dy . clz cos a-j- -f- cos b-jj- -J- cos e-j- = 0, which exists in the case of the unchanged elements, we shall have ^dx , ,dy .dz = cos a 8 -f cos o 8-~ -f- cos c 8 dx . dy . , .. dz . -- j- sin a Sa -- sin b db -- r- sin c dc. at at at Substituting for da, db, and dc the values given in Art. 60, observing that ds = 0, we have = R' 4- { -=- sin a sin A 4- ~- sin b sin B 4- -j- sin c sin C I sin i SQ \ dt dt dt [dx . ,<**., -o , dz . -,\ .. I -j- sm a cos^i + -^ sm 6 cos B -f- -^ sin c cos C I d. From the equations (100)^ observing that the relations between the auxiliary constants are not changed when the variable u is put equal to zero, or equal to 90, we get sin* a sin s A + sin 2 b sin 2 B -\- sin 2 c sin 2 (7=1, C253") sin 2 a cos 2 JL -f sin 2 b cos 2 .B -f- sin 2 c cos 2 (7=1, and from (235) we find sin 1 a sin A cos J. -f sin 2 b sin 5 cos B -f- sin 2 c sin (7 cos (7= 0. (254) Substituting in (252) for -,- - and the values given by the equations (49), and reducing by means of (253) and (254), we get (255) Substituting further for dz" in (251) the value given by the last of the equations (73) 2 , there results = R -f- r cos u sin i dQ r sin u = d x cos id , &r = $r -f- 2 sin 1 $i8Q , (258) <5/ being found from equation (249). Neglecting the mass of the comet as inappreciable in comparisou with that of the sun, the attractive force which acts upon the cornel in the case of the undisturbed motion relative to the sun is Jt 2 , but in the case of the motion relative to the common centre of gravity of the sun and planet this force is # (1 + m'). Hence it follows that the increment of this force will be w'F, and we shall have 3k by means of which the value of this factor, which is required in the formulae for ^(|/p)> $' <&#> ma y be found. 206. The formulae thus derived enable us to effect the required transformation of the elements. In the first place, we compute the values of dx, Sy } dz, 8, <^-IT> and d-^- by means of the formulae (234) ; then, by means of (236) and (250), we compute P, Q, E, P', Q f , and 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 SI and i are referred, we have sin e = sint, (7=0. The algebraic signs of cos a, cos 6, and cos c, as indicated by the equa- tions (101) w 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. 35 516 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 dk - 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' t. It should be observed, however, that the value of dM for the instant t should be reduced to that for the instant t' } so that the total variation of M during the interval t' t will be dM t + (t 3fi t -f dM f . 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 formulae applicable to this special case in which the ordinary methods of calculating perturbations cannot be applied. PERTURBATIONS OF COMETS. 547 If we denote by x', y', z', r', the co-ordinates and radius-vector of the planet referred to the centre of the sun, and regard its motion relative to the sun as disturbed by the comet, we shall have Let us now denote by f, ^, the co-ordinates of the comet referred to the centre of gravity of the planet; then will e = x af, 1i =y-y', C = J. Substituting the resulting values of x' } y', z' 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, (m + m')g F (m + m f ) 7 _ (262) (>> and since ** ( + ^ is the sum of the attractive force of the planet r on the comet and of the reciprocal action of the comet on the planet, 548 THEORETICAL ASTRONOMY. these equations, being of the same form as those for the undisturbed motion of the comet relative to the sun, show that when the action of the disturbing planet on the comet exceeds that of the sun, the result of the first approximation to the motion of the comet is that it describes a conic section around the centre of gravity of the planet. Further, since x', y', z' are the co-ordinates of the sun 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 /*' *'+g\_ p /*' xi Mr" f* /"Mr" r 3 / and when the comet approaches very near the planet this force will be extremely small. It is evident, further, that the action of the sun regarded as the disturbing body will be very small even when its direct action upon the comet considerably exceeds that of the planet, and, therefore, that we may consider the orbit of the comet to be a conic section about the planet and disturbed by the sun, when it is actually attracted more by the sun than by the planet. 208. In order to show more clearly that the disturbing force of the sun is very small even when its direct action on the comet exceeds that of the planet, let us suppose the sun, planet, and comet to be situated on the same straight line, in which case the disturbing force of the sun will be a maximum for a given distance of the comet from k* the planet. Then will the direct action of the sun be -, and that m'Jr* *" of the planet j The disturbing action of the sun will be PERTURBATIONS OF COMETS. 549 which, since /> is supposed to be small in comparison with r, may be 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 j-=' rn'i* If the comet is at a distance, such that the direct action of the sun is equal to the direct action of the planet, we have and the ratio of the direct action of the sun to its disturbing action cannot in this case exceed 2l/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 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', 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', 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' gives 2/> _ mV mV ~~ p* ' and therefore p = rl/~^T\ (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 and when this angle is considerable, the disturbing action of the sun will be small even when p is greater than n/^m'^. 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 t/>, 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 K = 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 sun as the disturbing body; and, having found the elements of the orbit of the comet relative to the planet, the perturbations of these elements or of the co-ordinates will be obtained by means of the formulae 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 its being considered as subject principally to the planet. In the case of the first transformation, the co-ordinates and velocities of the comet and planet in reference to the sun being determined for the imtant at which the sun is regarded as ceasing to be the controlling body, we shall have PERTURBATIONS OF COMETS. 551 $ = X x', r,=y y f , r = z _ ^ d = dx__d^ cb L _dy__cty_ d: _dz a* & dt dt' dt dt dt' ~dt~~dt~~dT ; and from f, 7, , -^- -J-> and -^-, 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, z, -^, A and ^-, and dt dt dt 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 , 37, , -^, ~, and ~ will be obtained directly, and then, having found the corresponding co-ordinates x f , y f , z' and velocities ~dt' ~dt' ~dt ^ *^ e P^ ane * * n re f erence to the sun, we have _ _ _ _ __ dt dt dt' dt ~ dt dt' dt ~ dt ~~ 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 THEORETICAL ASTRONOMY. 210. The results obtained from the observations of the periodic comets at their successive returns to the perihelion, render it probable that there exists in space a resisting medium which opposes the motion of all the heavenly bodies in their orbits; but since the observations of the planets do not exhibit any effect of such a resistance, it is 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 (p I I the density of the ethereal fluid at the distance r. Since the force acts only in the plane of the orbit, the elements which de- fine the position of this plane will not be changed, and hence we have only to determine the variations of the elements M, e, a, and . If we denote by ^ the angle which the tangent makes with the prolon- gation of the radius-vector, the components R and 8 will be given by E = T cos , S = T sin in V ' -^5 v~ ~l/p e * ~ r ~ dt' we have ^esinv-j-, S= J 'p dt RESISTING MEDIUJa IX SPACE. *g& Substituting these values of R and S in the equation (205), it reduces to Now, since F=^(l + 2e COB we have and hence ed X = -K?(-}r'(l + 2ecosv + e t ')*swvdv. (266) If we suppose the function 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 (l + 2ecost> + e')^=4 + cost;+ecos2i--f &c., (267) in which A } B, &c. are positive and functions of e, the equation (266) becomes 2 edz = -- (A + B cos v + ---- ) 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 dMj 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 p will be required. Let us next consider the elements a and e. Substituting in the 554 THEORETICAL ASTRONOMY. equations (198) and (202) the values of R and 8 given by (265), and reducing, we get da = ^K. j> \ r / If we introduce into these the series (267), and integrate, it will be found that, in addition to the periodic terms, the expressions for da and de contain each a term multiplied by v, and hence increasing with the time. It is to be observed, further, that since A and B are posi- tive, the secular variation of a, and also that of e, will be negative, and hence the resisting medium acts continuously to diminish both the mean distance and the eccentricity. 211. The magnitude of the disturbing force arising from the action of the resisting medium is so small that the periodic terms have no sensible influence on the place of the comet during the period in which it may be observed; and hence, since the effect of the resist- ance will be exhibited only by a comparison of observations made at its successive returns to the perihelion, the effect of the planetary per- turbations being first completely eliminated, it is only necessary to consider the secular variations. Further, since / is subject only to periodic changes in virtue of the action of the resistance, and since the mean longitude is subjected to a secular change only through //, it will suffice to employ the formulae for dp and de or d; then, after the lapse of any interval t T m we shall have . tT . t T , v = ^o H *> ? =

may be determined in connection with the corrections to b< RESISTING MEDIUM IN SPACE. 657 applied to the elements. For this purpose the partial differential co- efficients of the geocentric spherical co-ordinates with respect to x and y must be determined. Thus, if we substitute the values of ji, for the subsequent returns of the comet to the perihelion. In all the cases in which the periodic comets have been observed sufficiently, the existence of these secular changes of the elements seems to be well established; and if we grant that they arise from the resistance of an ethereal fluid, the total obliteration of our solar system is to be the final result. The fact that no such inequalities have yet been detected in the case of the motion of any of the planets, shows simply the immensity of the period which must elapse before the final catastrophe, and does not render it any the less certain. Such, indeed, appear to be the present indications of science in re- gard to this important question; but it is by no means impossible that, as in at least one similar case already, the operation of the simple and unique law of gravitation will alone completely explain these inequalities, and assign a limit which they can never pass, and thus afford a sublime proof of the provident care of the OMNIPOTENT CREATOR. TABLES. 5M TABLE I, Angle of the Vertical and Logarithm of the Earth's Radius, Argument f = Geographical Latitude. Compression = * *-*' Diff: logp Diff. * *-*' Diff. logp ruff. O 1 2 3 4 5 o o.oo o 24.02 o 48.02 I 11.95 I 35.80 i 59 54 24.02 24.00 23.93 23.85 *3-74 23.58 0.000 0000 9-999 9996 9961 9930 9891 4 21 3' / 35 10 20 30 40 50 10 48.25 49.63 50.98 52.31 53.62 54.90 I. 3 8 1-35 i-33 I'iS 1.26 9-999 5*48 5208 5169 5129 5089 5049 4 39 40 40 i 40 1 40 6 7 8 9 10 11 2 23.12 2 46.54 3 9-76 3 55-47 4 7-9* 23.42 23.22 22.98 22.73 22.45 22.14 9-999 9 8 43 9786 9721 9566 9476 II 11 90 44 36 10 20 30 40 50 10 56.16 57-4' 58.63 10 59.82 II 1.00 2.15 1.25 1.22 I.I9 1.18 1.15 1.13 9-999 5009 4969 49*9 4888 4848 4807 40 40 -- 4' 40 12 13 14 15 16 17 4 40.06 5 1.85 5 *3-*8 5 44-33 6 4-95 6 25.14 21.79 21.43 21.05 20.62 20.19 19.72 9-999 9377 Io6 957 JJ* il 37 10 20 30 40 50 II 3.28 4-39 5-47 6.54 7.58 8-59 i. ii i. 08 1.07 1.04 I.OI 1.00 ,9994767 4 4 686| 40 4645, ' 4604! J 4563! J, 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 I8 5 38 10 20 30 40 50 ii 9.59 10.56 1.51 *-44 3-34 4.22 0.97 0.95 0.93 0.90 0.88 0.86 9-999 45**, 444 *t 435^ ^, 43'T 4 , 24 ! 25 26 ! 27 28 ; 29 8 32.10 8 47.93 9 3.12 9 17-65 9 3i-5o 9 44.66 15.83 15.19 14-53 13.85 13.16 12.46 9-999 7614 74*4 7228 7027 6820 6608 I 9 I 9 6 201 ::- :i - 216 39 10 20 30 40 50 ii 5.08 5.92 16.73 17.52 18.29 19.04 0.84 ' 0.79 0.77 0.75 . 0.72 9-999 4176 4*34 *. 4'93 J, 4 I > 2 la 4" Ji 4 6 9, ^ a 30 10 20 9 57-" 9 59-" IO I. II 2.00 1.99 9-999 639* 6319 36 40 10 20 II 19.76 20.46 21.13 0.67 : 0.66 '""$; t; 3944 Tj 30 40 3.07 5.02 1.96 i-95 6282 37 6245 \l 30 40 21.79 22.42 0.63 390*: ^ 50 6-94 1.92 1.91 6208 37 50 23.02 j -59 38i9| 4 j 31 10 20 30 40 10 8.85 10.73 12.59 its 1.88 1.86 1.85 1.82 9-999 6171 ^34! \l 6096, 3* 6 59j \l 6021 ?_ 41 C 3d 40 24.70 j 0.49 9-999 3777 3735 3693 3651 3609 4* 4* 4* 42 J.2 50 1 8.06 i. 80 i 78 5984 S 50 26.18 ! o!^ 3567 4Z : 32 10 10 19.84 2 1. 60 l./O 1.76 9-999 5946 " 9 598 .s 42 10 ii 26.62 I 1 ' 04 ' n "* 9-999 35*5 4Z 20 1.74 5870 * 20 ~S 344' , 2 30 25.05 1.71 583* II 30 27.8: -3" 3399 4* 40 50 26.75 28.43 \'S 1.65 5794 L 5755 $ 40 50 IMC i o.\l 3357 3315 4* 33 lo 30.08 1.63 9-999 57i7 43 1(1 " jf'oi! 0.28 9-999 3*73 . 3230 4 f 1C 31.71 1.61 5678 i ^f 2(1 y- 1 0.26 4* 20 ~C 33-3^ MS c6oi 30 *9-5' i -*-1 3146 AT. 4C 34-9' 36.48 556*' H 40 29-7< ' 1C 3'=4 ;; 50 38.03 *-s; 55*3 S 50 29.98 i 0.16 3062 43 34 10 20 30 40 50 10 39.55 41.06 42.54 44.00 #8 46 1.44 1.42 9-999 5484 1 |i 39 39 39 ir 44 10 10 ro 40 50 11 3-i4 OI . 9-999 3019 *977 *93S 2892 2850 2808 42 43 r. 4 i : 35 10 48.25 1.39 9-999 5*48| 45 II 30.65 9.999 2766! TABLE I, Angle of the Vertical and Logarithm of the Earth's Eadius, 0' = Geocentric Latitude. p = Earth's Eadius. * *-*' Diff. logp Diff. * .-* Diff. logp Diff. / ~ / / // 45 10 20 30 40 50 II 30.65 30.65 30.63 30.58 30.51 30.42 0.00 0.02 0.05 0.07 0.09 O.I I 9-999 2766 2723 2681 2639 2596 43 42 42 43 42 4 2 55 10 20 30 40 50 10 49.74 48.36 46.97 45-55 44.11 42.65 1.38 i-39 1.42 '49 9-999 275 0235 0195 0155 0116 0076 40 40 40 39 40 i 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 36.58 35-' 33-4' 1.51 1-52 '55 '57 i. 60 1.61 9-999 37 9.998 9998 9958 9919 9880 9841 39 40 39 39 39 39 47 10 20 30 40 50 II 29.12 28.85 28.54 28.22 27.87 27.50 0.27 0.31 0.32 0.35 0-37 O.4O 9-999 2258 2216 2174 2132 2089 2047 42 42 42 43 42 42 57 10 20 30 40 50 10 31.80 30.16 28.50 26.83 23.40 1.64 1.66 1.67 1.70 1.73 1.74 9.998 9802 9764 9725 9686 9648 9610 38 39 39 38 38 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 1.76 i-79 i. 80 9.998 9571 9533 9495 38 30 40 50 25.78 25.29 24-78 0.46 0-49 0.51 1879 1837 42 42 42 30 40 50 16.31 14.48 12.63 l:il 9457 9419 9382 I 8 37 38 49 10 20 30 40 50 ii 24.24 23.69 23.11 22.50 21.87 21.22 O.54 0-55 0.58 0.6 1 0.63 0.65 0.67 9-999 '753 1711 1669 1627 1586 '544 42 42 42 4 1 42 42 59 10 20 30 40 50 10 10.77 8.88 6.97 5.04 3.08 10 I. II i.8 9 1.91 '93 1.96 i-97 1.99 9.998 9344 9307 9269 9232 9'95 9158 H 37 37 37 37 50 10 20 30 40 50 II 20.55 19.85 19.13 18.39 17.63 16.84 0.70 0.72 0.74 0.76 o-79 0.82 9.999 1502 1460 1419 1377 '335 1294 42 42 42 42, 60 61 62 63 64 65 9 59-'2 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 i 51 10 20 30 40 50 52 10 20 30 II 16.02 15.19 14-33 13-45 12.55 11.62 II 10.67 9.70 8.71 7.69 0.83 0.86 0.88 0.90 0.93 0.95 0.97 0.99 .02 9.999 1252 121 1 1170 1128 I08 7 1046 9.999 1005 096-: 0922 O88l 4 1 4 1 42 4 1 4 J 4' 42 66 67 68 69 7O 71 72 73 74 75 8 3440 8 17-97 8 0.92 7 43.29 7 25.08 7 6.33 6 47.06 6 27.28 6 7.03 5 46.33 '6.43 17.05 17.63 18.21 18.75 19.27 19.78 20.25 20.70 9.998 7884 7697 75'7 7342 7174 7013 9.998 6859 67,3 6573 6441 ill 161 '54 146 140 132 40 50 6.66 5-6o .06 0840 0800 4' 40 76 77 5 25.20 5 3-67 21.51 21.90 6317 6201 1 16 1 08 53 ii 4.51 .09 9-999 759 4 1 78 4 41-77 22.24 9-998 6093 IOO 10 3.40 . 1 1 0718 79 4 '9-53 5993 Q2 20 2.27 .11 0677 4 1 80 3 56-96 22.57 22.86 Ts 83 30 40 50 II 1. 12 10 59.94 58.74 .20 .22 0637 0596 0556 4 40 41 81 82 83 3 34-' 3 10-98 2 47.63 23.12 23-35 23.56 5818 5743 5676 11 57 54 10 20 30 40 50 10 57.52 56.28 55.02 53-73 52.42 51.09 .24 .26 .29 3 1 33 9-999 OS'S 047 043 39 035 031 40 40 40 4 4 40 84 85 86 87 88 89 2 24.07 2 0.33 I 36.44 I 12.43 o 48.3, o 24.1! 23.74 23.89 24.01 24.09 24.16 24.18 9.998 5619 5570 553 5498 5476 5463 49 40 31 22 '3 5 55 10 49.74 35 9-999 27 90 O O.OC 9.998 5458 TABLE II, For converting intervals of Mean Solar Time into equivalent intervals of Sidereal Time Hours. Minutes. Seconds. Decimals. Mean T. Sidereal Time. eanT. Sidereal Time. ean T. idereal Time. eanT. Sidereal Time. 7i A s TO TO S s 1 j 1 I I 9.856 I I 0.164 1.003 0.02 O.O2O 2 2 19.713 2 2 0.329 2 2.00C O.OA 0.040 3 3 3 o-493 3 3.008 O.O6 0.060 4 4 39.426 4 4 o-657 4 4-OII 0.08 O.o8o 5 5 49.282 5 5 0.821 5 5.014 O.IO O.I 00 6 6 59-139 6 6 0.986 6 6.0l6 O.I2 0. 1 2O 7 7 8.995 7 7 1-15 7 7.019 0.14 0.140 8 8 18.852 8 8 1.314 8 8.022 0.16 O.I 60 9 9 28.708 9 9 I-478 9 9.025 0.18 0.180 10 10 38.565 10 10 1.643 10 10.027 O.2O O.2OI 1 1 ii 48.421 ii II 1.807 ii 11.030 O.22 O.22I 12 12 58.278 12 12 I. 97 I 12 12.033 0.24 0.241 *3 13 8.134 '3 13 2.136 '3 13.036 0.26 o 261 14 I 4 17.991 4 2.300 14.038 0.28 0.28l 15 I 5 27.847 15 5 2.464 15 15.041 0.30 0.301 '6 16 37-704 16 6 2.628 16 16.044 0.32 0.321 17 17 47-56o 1 7 7 2.793 17 17-047 0.34 0.341 18 18 57.416 18 8 2.957 18 18.049 0.36 0.361 I9 19 3 7.273 19 9 3-'2i 19 19.052 0.38 0.381 20 20 3 17.129 20 o 3.285 20 20.055 0.40 0.401 21 21 3 26.986 21 i 3.450 21 21.057 0.42 0.421 22 22 3 36.842 22 2 3.614 22 22.O6O o-44 0.441 23 2 4 23 3 46-699 24 3 56.555 23 24 3 3-778 4 3-943 23 24 23.063 24.066 0.46 0. 4 S 0.461 0.481 11 5 4- I0 7 6 4.271 3 25 068 26.071 0.52 0.521 27 7 4-435 27 27.074 o-54 0.541 S a 28 28 4.600 28 28.0 77 0.56 0.562 H 29 29 4.764 29 29.079 0.58 0.582 3 30 4.928 30.082 0.60 0.602 % i 3 1 31 5.092 3 1 31.085 0.62 O.622 S 60 3 .9 a5 32 33 34 S 2 5-257 33 5-42i 34 5-585 32 33 34 32.088 33.090 34-093 0.64 0.66 0.68 0.642 O.662 0.682 35 35 5-750 35 35.096 0.70 0.702 111 36 3 6 5-9'4 36 36.099 0.72 0.722 .S * S 37 37 6.078 37 37.101 -74 0.742 ll I 39 38 6.242 39 6.407 38 39 38.104 39.10 7 0.76 0.78 0.762 0.782 111 "S J 40 4 1 42 40 6.571 41 6.735 42 6.899 4 4 1 42 40. 1 1 o 41.112 42.115 0.80 0.82 0.84 0.802 0.822 0.842 111 43 43 7-o64 43 43.118 086 0.862 44 44 7.228 44 44.120 0.88 0.882 *TI ^~ 45 45 7-392 45 45.123 0.90 0.902 c ^8 .2 46 4 6 7-557 46 46.126 0.92 0.923 II > 47 47 7.721 47 47.129 o-94 0-943 Jg T3 '^ 48 48 7.885 48 48.131 0.96 0.963 b .a 49 49 8.049 49 49.134 0.98 0.983 < g< J; 5 50 8.214 5 5- I 37 1. 00 1.003 "a % T 5 51 8.378 5' 51.140 s S 5* S 2 8.542 52 52.142 D S 53 53 8-707 53 53-H5 .1 Er 54 54 8.871 54 54.I48 11 jj 55 9-35 55 55.151 v ^6 56 9 199 56 56.153 ~t r2 57 57 9-3 6 4 57 57-I56 gco 58 8 58 9.528 59 9-692 60 9.856 58 8 58.159 59-162 60.164 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 A m , m m s s f 1 s o 59 50.170 1 o 59.836 I 0.997 O.O2 O.O2O z i 59 40.341 2 I 59.672 Z 1.995 0.04 0.040 3 2 59 30.511 3 2 59.509 3 2.992 0.06 O.o6o 4 3 59 20.682 4 3 59-345 4 0.08 O.oSo 5 4 59 10.852 5 4 59-i8i 5 986 0.10 O.I 00 6 5 59 1-023 6 5 59-oi7 6 5-984 0.12 O.I 20 7 6 58 51.193 7 6 58.853 7 6.981 0.14 0.140 8 7 58 41.363 8 7 58.689 8 7-978 o.i 6 O.I 60 9 8 58 31-534 9 8 kjg 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 1 1 10 58.198 ii 10.970 0.22 0.219 IZ li 53 2.045 12 ii 58.034 IZ 11.967 0.24 0.239 J3 iz 57 52.216 3 57 42-386 '3 12 57.870 13 57.706 H 12.964 13.962 0.26 0.28 0.259 0.279 15 14 57 32.557 15 H 57-543 15 14.959 0.30 0.299 16 15 57 22.727 16 15 57-379 16 15.956 0.32 0.319 17 16 57 12.897 17 16 57.215 '7 16.954 0-34 0-339 18 17 57 3.068 18 17 57.051 18 17.951 0.36 o-359 19 18 56 53.238 19 18 56.887 19 18.948 0.38 0.379 20 19 56 43.409 20 19 56.723 20 19.945 0.40 0-399 21 20 56 33.579 21 20 56.560 21 20.943 0.42 0.419 22 21 56 23.750 22 21 56.396 22 21.940 0.44 o-439 23 2 4 22 56 13.920 23 56 4.091 2 3 24 22 56.232 23 56.068 23 24 22.937 23.934 0.46 0.48 0.459 0.479 11 24 55.904 25 55.740 3 24.932 25.929 0.50 0.52 0.499 0.519 ' -:--:; ,86.71 ,86.72 9.390054 9.40,304 187-49 ,87.50 ,0.066523 ,0.077823 188.34 14 8.068021 186.00 8.738927 j ,86.73 9.412555 187-52 ,0.089125 i88'.37 15 8.07918, i 186.02 8.750,3, j ,86.74 9.423806 ,87.53 ,0.100427 188.38 16 1? 8.090343 8.10,505 186.03 ,86.04 8.76,336 1 8-772542 ! ,86.76 ,86.77 9.435058 9.446311 ,87.54 18756 ,o.i 1 1730 ,0.123035 188.39 188.41 18 8.1,2668 186.05 8.783748 ,86.78 ig 7 . 57 10.134340 188.42 ! 19 8.12383, j 186.06 8-794955 ,86.79 9.468820 ,87.59 10.145646 188.44 20 8.134995 i ,86 07 8.806,63 ,86.?i 9.480076 ,87.60 ,0.156952 '"45 21 8.146,60 ,86.09 8.817372 ,86.82 9.491332 ,8 7 6i 10.168260 188 47 i 22 23 8.157326 8.168492 ,86.,o ,86.,, 8.828582 8.839792 ,86.83 ,86.84 9.502589 9.513847 187-6+ 10.179568 10.190878 188.48 188.50 24 8.179659 ,86.12 8.85,003 ,86.86 9.525106 ,87.65 I0.2C2188 188.5, 25 26 27 8.190826 i 8.201995 8.213164 186.13 ,86.15 ,86.16 8.8622,5 8.873427 8.88464, ,86.87 ,86.88 ,86.90 9.536366 9 547626 9558888 ,87.67 ,87.68 187.70 10.213499 10.224812 io 236125 ,88.53 ,88.54 188.56 28 8.224334 i 8.235504 ] ,86., 7 ,86.18 8-895855 8.907070 ,86.9, ,86.92 9.5-0150 9.581413 ,87.71 ,87.72 10.247439 10.258753 1 188.57 188.59 30 31 32 8.246675 8 257847 ' 8 269020 ,86.19 ,86.20 ,86 22 8.918286 8.929502 8.9407,9 ,86.93 ,86.95 ,86.96 9 592676 9.603941 9.615207 ,87.74 187-75 ,87.77 10.270069 10.281386 10.292703 188.60 188.62 188.63 33 8.280193 ,86.23 8.95,937 ,86 97 9.626473 ,87.78 ,0.304021 ,88.65 , 34 8.291367 ,86.24 8963,56 ,86.99 9.637740 ,87.79 ,0.31534, 188.66 35 36 37 38 39 8.302542 8.313717 8.324893 8.336070 8.347248 ,86.25 ,86.26 ,86.28 ,86.29 ,86.30 8.974376 8-985596 8.9968,7 9.008039 9.019262 ,87.00 ,87.0, ,87.02 ,87.04 187-05 9.649008 9.660277 9.671547 9.682817 9.694088 ,87.8, ,87.82 187.84 ,87.85 187.86 ,0.326661 ,0.337982 10.349304 10.360627 10.37,95, 188.68 188.69 188.7, ! ,88.72 188.74 i 40 41 42 ! 43 44 8.358426 8.369605 8.380785 8.391966 8.403147 ,86.3, ,86.32 186.34 9-030485 9041709 9.052934 9.064160 9.075387 ,87.06 ,87.08 ,87.09 ,87.10 ,87.12 9.705361 9 716634 9 727908 9.739,82 9.750458 187.88 ,87.89 187-9. ,87.92 18793 ,0.383275 ,0.39460, ,0.405927 0.417255 0.428583 188.75 188.77 ; 188.78 | 188.80 ,88.8, 45 46 4? 48 49 8.414329 8.4255,2 8.436695 8-H7879 8.459064 ,86.37 ,86.38 ,86.40 ,86.4, ,86.42 9.086614 9 097842 M 09071 9.12030, 187-13 187-14 ,87.16 ,87.17 ,87.18 9773012 9 784290 9795569 9.806849 ,87.95 ,8796 ,87.98 187-99 ,88.00 0.439912 0.451242 0-462573 ,0.473905 10.485238 ,88.83 188.84 188.86 ,88.87 i 188.89 50 51 52 53 51 8^81436 8.492623 8.5038,1 8.515000 ,86.43 ,86.45 ,86.46 ,8647 ,86.48 9.142763 9-'53995 9.165228 9.176462 9.187696 ,87.20 ,87.2, ,87.22 ,87.23 187-25 9.818129 9.829410 9.840693 9-851977 9.86326, 188.02 188.03 ,88.05 ,88.06 ,8808 ,0.496572 jo 507907 10.519242 10.530579 ,0.54,9,6 ,88.90 188.92 ,88.93 ! 188.95 - ,88. 97 55 56 57 58 59 8.526189 8-537379 8.548569 8.559761 8.570953 ,86.49 ,86.5, ,86.52 ,86.53 ,86.54 9.19893, 9.2,0,67 922,404 9.232642 9.243880 ,8726 187-27 ,87.29 ,87.30 187-31 9.874546 9.885832 9897,18 9.908406 9919694 ,88.09 ,88.10 ,88 12 ,88.13 ,88.15 ,0.553255 ,0.564594 10-575934 ,0.587276 ,0.5986,8 ,88.98 ,89.00 189.01 ]$.ll GO 8.582146 ,86.56 9.255120 187.33 9.930984 188.16 ,0.60996, 189.06 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. v 16 17 J 18 19 o M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". i O' 10 609961 189.06 .292277 190.02 1 1.978162 191.04 12.667850 192.13 1 10.621305 189.07 .303679 190.03 1 1.989625 191.06 12.679379 192.15 2 10.632649 189.09 .3 15082 190.05 12.001089 191.08 2.69090^ 192.17 j 3 10.643995 189 10 .326485 190.07 12.012554 191.09 2.702439 192.19 4 10.655342 189.12 .337889 190.08 12.024021 191.11 2.713970 192.21 5 10 666690 189.14 .349295 190 10 12.035488 191.13 2.725503 192.22 6 10.678038 189.15 .360701 190.12 i2.O4b956 191.15 2737037 192.24 7 10689388 189.17 .372109 190.13 12.058425 191.16 2.748573 192.26 8 10.700738 189.18 383517 190.15 12.069896 191.18 2.760109 192.28 i 9 10.712090 189.20 .394927 190.17 12.081367 191.20 2.771646 192.30 ' 10 10.723442 189.21 .406337 190.18 12.092840 191.22 2.783185 192.32- 11 10.734795 189.23 .417749 190.20 12.104313 191.24 2.794724 192.34 12 10.746149 18924 .429161 190.22 12.115788 191.25 2.806265 192.36 13 10.757505 189.26 440575 190.23 12.127264 191.27 2.817807 J 9 2 -37 ' 14 10.768861 189.28 .451989 190.25 12.138741 191.29 2.829350 192.39 15 10.780218 18929 11.463405 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 18934 11.497657 190.32 12.184659 191.36 12.875534 192.47 19 10.825655 189.35 11.509077 190.33 12.196141 191.38 12.887082 192.49 2O 10837017 189.37 11.520497 190.35 12.207624 191.40 12.898632 192.51 21 10.848380 189.39 11.531919 190.37 12 219108 191.41 12.910183 192.53 22 10.859744 18940 11.543342 190.39 12.230594 191.43 2.921736 192.55 23 10.871108 189.42 "554765 190.40 12.242080 191.45 2.933289 192.56 24 10.882474 18943 11.566190 190.42 12.253568 191.47 2.944843 192.58 25 10.893840 18945 11.577616 190.44 12.265057 191.49 2.956399 192.60 26 27 10 905208 10916576 189.47 189.48 11.589042 11.600470 190.45 19047 12.276546 12.288037 191.50 191.52 2.967956 2.979514 192.62 192.64 28 10.927946 18950 11.611899 19049 12.299529 I9 i. 54 2.991073 192.66 29 10939316 189.51 11.623328 190.50 12.31 IO22 191.56 3.002633 192.68 30 10.950687 189.53 II 634759 190.52 12.322516 191.58 13.014195 192.70 31 10 962059 18955 11.646191 190.54 I2.334.0II 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 1 1.680492 19057 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 19064 12.403006 191.70 13.095157 192.83 38 1 1.041692 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 I3.Il82 9 9 192.87 10 41 11.064453 11.075835 189.69 189.71 11.749123 11.760565 190 69 190.71 12.437517 12.449023 191.76 191.78 13.129872 13.141446 192.89 192.91 42 11.087218 i8 9 . 72 11.772008 190.73 12.460531 191.80 13.153022 192.93 43 44 11.098602 11.109987 189.74 189.76 II. 7831.52 11.794897 190.74 190.76 12.472039 12.483548 191.81 191.83 I 3 164598 13.176176 192.95 192.97 45 1 1.121372 189.77 11.806344 190.78 12.495059 191.85 13.187755 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 I2.54III2 191.93 13.234082 193.07 , 50 11.178316 189.85 11.863590 190.87 12.552628 191.94 13.245667 I93-09 j 51 11.189708 189.87 11.875043 190.88 12.564145 191.96 13.257253 193.1! 52 II.20IIOO 189.89 11.886496 190.90 12.575664 191.98 13 268840 193.13 I 53 11.212494 189 90 11.897951 190.92 12.587183 192 oo 13.280428 193.15 54 11.223889 189 92 11.909407 190.94 12.598704 192.02 13.292017 193.17 55 56 11.235284 11.246681 189.93 18995 11.920863 11.932321 190.95 190.97 I2.6lO225 12.621748 192.04 192.06 13.303608 13.315200 I 93- I 9 193.21 1 57 11.258078 189.97 11.943780 190.99 12.633272 192.07 13.326793 i93- 2 3 58 11.269477 189.98 11.955239 191 01 12.644797 192.09 13.338387 i93- 2 5 59 11.280876 190.00 1 1.966700 191.02 12.656323 I 9 2.H 13.349982 193.27 60 11.292277 190.02 11.978162 191.04 12.667850 I 9 2.I3 13.361579 193.29 TABLE VI, For finding tne True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 20 21 22 23 M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1 '. 13.361579 193.29 14.059591 194.51 14.762133 195.80 15.469459 197.17 1 13.373177 193.31 14.071262 '94-53 14.773882 195.83 15.4!; 1290 197.19 2 3 13.384776 13.396376 '9333 193-35 14.082935 14.094608 '9455 19457 14.785632 14.797384 195.85 195.87 '5-4931** 15.504956 19724 i 4 I 3.407977 193.37 14.106283 194-59 14.809137 195.89 15516791 197.16 5 13.419580 193.39 14.117960 194.61 14.820891 195.91 15.528627 19728 6 13.431183 193.41 14.129637 194.64 14831647 '9594 15.540465 '97 3' 8 13442788 '3-454394 '93-43 '93-45 14.141316 14.152996 194.66 194.68 14.844403 14856161 195.96 19598 15.552304 15 564144 '97-33 197-35 9 13.466002 '93-47 14.164677 194.70 14.867911 196.00 15.575986 197.38 10 13.477610 193.49 14.176360 194.71 14.879681 196.03 15.587830 197.40 11 13 489220 I93-51 14.188044 19474 14.891444 196.05 15.599675 197-43 1x4 13.500831 '93-53 14.199729 194.76 14.903208 196.07 15 611521 '97-45 13 13 512443 '93-55 14.211415 194.78 14914973 19609 15.623369 '97-47 14 13.524056 '93-57 14.223103 194.81 14.926739 196.12 15.635218 197.50 15 16 13.535671 13.547287 '9359 '93 6l 14.234792 14.246482 194.83 194.85 14.938506 14.950275 196.14 196.16 15.647068 15.658920 197.51 '97-54 17 13.558904 193.63 14.258174 I94-87 14962045 196.18 15.670773 '97-57 ! 18 19 13.570522 13582141 193.65 193.67 14.269867 14.281561 194.89 I94-91 14.973817 14.985590 196 20 196.23 15.682628 15.694484 197-59 197.61 20 13.593762 193.69 14.293256 '94-93 14.997365 196.25 15.706342 197.64 21 13.605383 193.71 14.304953 '9495 15.009140 196.27 I 5.718201 197.66 22 13.617006 '93-73 14.316651 194.98 15 020917 196.30 15.730061 19769 23 13.628631 193-75 14.328350 195.00 15.032696 196.32 i5-74!9*3 i977i 24 1 3.640256 193.77 14.340050 195.01 15.044475 196.34 15.753786 197-73 25 13 651883 '9379 14.351751 195.04 15.056256 196.36 15.765651 197.76 26 27 28 29 13.663511 13.675140 13.686770 13.698401 193.81 193.85 193.87 '4-3 6 3455 I4-375I59 14.386865 14398572 195.06 195.08 195.10 195.13 15.068039 15.079823 15.091608 15.103394 196.39 196.41 196.43 196.45 i5-7775'7 15.789385 15.801254 15.813124 197-78 197.80 197.83 197.85 30 31 33 34 13.710034 13.721668 13733303 '3-74494 13756577 I93-89 193 91 '9393 '9395 '93-97 14.410280 14.421990 14.433700 14.445412 14.457126 I95-I5 195.17 195.19 195.21 195.23 15.115182 15.126971 15.138762 15.150554 15.161348 196.48 196.50 196.52 196.54 196.57 15.824996 15 836870 15.848744 15.860620 15.872498 ,97.88 197.90 197.91 '97-95 197-97 35 36 37 38 39 13 768216 13.779856 13.791498 13.803140 13.814784 '93 99 194.01 194.03 194.05 19407 14.468841 14.480557 14.492274 14.503992 14.515711 195.26 195.28 195.30 195.32 '95-34 15.174142 15.185938 15.197736 15.209535 15.221335 196.59 196.61 196.64 196.66 196.68 15.884377 ,5.896258 15.908140 15.92002? 15.931908 198.00 198.01 198.04 198.07 198.09 40 41 42 43 44 13.826429 13.838075 13.849723 13.861372 13.873022 194.09 194.11 194.14 194.16 194.18 14.527434 14.539156 14.550880 14.562605 I4-574331 195.36 '95-39 195.41 195-43 195-45 15.244940 15.256744 15.268550 15.280357 196.70 196.73 196.75 196.77 196.80 15943794 15.955682 15.967571 15 979462 I599I354 ,98.11 198.14 198.17 198.19 198.11 45 46 47 ! 48 49 13.884673 13 896325 13.907979 13.919634 13.931290 194.10 19422 194.24 194.26 194.28 14.586059 14.597788 14.609519 14.621250 14.632983 '95-47 195.50 195.52 195-54 195.56 15.292165 I5-3 3975 15.3,5786 15.327599 I5-3394I3 196.82 196.84 196.87 196.89 196.91 16.003248 16.015143 16.027039 16.038937 16.050836 198.14 198.16 198.19 198.31 198.34 5O 51 5Z 53 54 13.942948 13.954606 13.966266 139779*7 13.989590 194.30 194.32 '94-34 194.36 194.38 14.644718 14.656453 14.668190 14.679929 14.691668 195.58 195.60 195 63 195.65 195.67 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 ,9838 198.4, 198-43 55 56 57 58 59 14.001254 14.012919 14.024585 14.036252 14.047921 19441 19443 19445 19447 194.49 14.703409 14.715151 14.726895 14.738640 14.750386 195.69 I957I '9574 195.76 195.78 15.410326 15.422150 '5-433975 15.445802 15457630 197.05 197.07 197.10 197.12 197.14 16.122263 ,6.134173 16.146084 16.157997 16.169911 198.48 198.5, 198.53 198.56 198.58 60 ___ 14.059591 . 194.51 i 14.761133 = 195.80 ; - K.7 1 15469459 ^- '- 197.17 . 16.181826 - ,98.60 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 24 25 o 26 27 v M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. 1". 16.181826 198.60 16.899499 2OO. 12 17.622747 20 .70 18.351847 203.37 1 16.193743 198.63 16.911507 2OO.I4 17.634850 20 .73 18.364050 203.40 2 16.205662 198.65 16.923516 200.17 17.646954 20 .76 18.376255 203.42 3 4 16.217582 16.229503 198.68 198.70 16.935527 16-947539 200.19 200.22 17.659060 17.671168 20 .78 20 .81 18.388461 18.400669 203.45 , 203.48 5 6 16.241426 16.253350 198.73 198.75 16959553 16.971568 200.24 2OO.27 17.683278 17695389 2O .84 20 .87 18.412879 18.425090 203.51 203.54 7 16.265276 198.78 16.983585 200.30 17.707502 20 .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 1O 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 20200 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 17.780213 202.o6 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 2O2.I I 18.535070 203.80 16 16.372676 199.00 17.091808 20053 17.816590 2O2.I4 18.547299 203.82 IT 16.384617 I 99 .oa 17.103841 200.56 17.828719 2O2.I7 18559529 203.85 18 16.396559 199.05 17.115875 200.58 17 840850 202.19 18.571761 203.88 1 19 16.408503 199.07 17.127911 200.6l 17.852982 2O2.22 18.583995 203.91 20 16.420448 199 10 17139948 200.64 17.865116 2O2.25 18.596230 203.94 21 16.432395 199.12 17.151987 2OO.66 17.877252 202.28 18.608467 203.97 22 16444343 199.15 17.164028 200.69 17.889389 202.30 18.620706 204.00 23 24 16.456292 16.468243 199.17 199.20 17.176070 17.188114 2OO.7I 200.74 17.901 528 17.913669 202-33 202.36 18.632947 18.645190 204.03 204.05 25 16.480196 199.22 17.200159 200-77 17.925811 202.3 9 18.657434 204.08 26 16.492151 199.25 17.212206 200-79 17-937955 202.41 18.669679 204.11 27 16.504107 199.27 17.224254 200.82 17.950101 202.44 18.681927 204.14 28 29 16.516064 16.528022 199.30 199.33 17.236304 17.248356 200.85 200.87 17.962248 17-974397 202.47 2O2.5O 18.694177 18.706428 204.17 204.20 30 16.539983 *99-35 17.260409 200.90 17.986548 202.52 18.718680 204.23 31 16.551945 199.38 17.272464 200.Q3 17.998700 2O2-55 18.730935 204.26 32 16.563908 19940 17.284520 200-95 18.010854 202.58 18.743191 204.29 33 16.575873 J 99-43 17.296578 2OO 98 18.023010 2O2.6l 18.755449 204.32 34 16.587839 199.45 17.308637 2OI.OO 18.035167 202.64 18.767709 204.35 35 16.599807 19948 17.320698 201.03 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 201.08 18.071649 202.72 18.804499 204.43 38 16.635719 '99-55 17.356891 20 1. 1 I 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 '99-73 17.441397 2OI.3O 18.169008 202.94 18.902684 204.67 46 l6 -73i553 199.76 17.453476 201.32 18.181186 202.97 18.914965 204.70 47 16.743539 199.78 I7-465556 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 18.976-596 204.84 52 16.803493 199.91 17.525982 201-49 18.254286 203 14 18.988687 204.87 53 16.815488 19994 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 57 16.851484 16.863485 2OO.OI 2OO.O4 J7.574352 17.586448 201.59 201.62 18.303053 18.315249 203.25 203.28 19.037871 19.050172 204.90 205.02 58 16.875488 2OO.O6 17.598546 2OI.65 18.327447 203.31 19.062474 205.05 59 16.887493 2OO.O9 17.610646 201.68 18.339646 203.34 19.074778 205.08 60 16.899499 2OO.I2 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. I". M. Diff. 1". JogM. Diff. 1". logM. Diff. l". 19 087084 205.11 19,328747 206.94 1.313 3849 44.08 1.329 0430 42.92 1 19.099391 205.14 19841164 206.97 .313 6493 44.06 .329 3004 42.91 2 19.111701 205.17 19.853583 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 19890852 207.09 1.314 7058 4398 1.330 3293 42.83 6 19.160956 205.29 19 903279 207.13 .314 9696 43.96 .330 5862 42.8, 7 19.173274 205 32 19.915707 207 1 6 3'5 2333 4394 .330 8431 42.80 8 9 19.185594 19.197916 205 35 205.38 19 928137 19.940569 207.19 207.22 .315 4969 .315 7604 43.92 43.90 .331 0998 33 1 3564 42.78 j 42.76 10 19.210240 205.41 19953003 207.25 1.316 0217 43.88 1.331 6129 42-74 11 19 222566 205.44 19.965^39 207.28 .316 2869 43-86 331 8693 42.72 12 19.234893 205.47 l9-977 8 77 207.31 .316 5 5 oo 43.84 332 1255 42.70 13 19.247222 205.50 19.990317 207.34 .316 8130 43-82 332 3817 42.69 14 19.259553 205.53 20.002759 207.38 .317 0759 43.80 332 6378 42.67 15 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 '496 42.63 17 19.296556 205.62 20.040095 207.47 .317 8638 4374 333 4053 42.61 18 19.308894 205.65 20.052544 207.50 .318 1262 43 72 333 66 9 42-59 19 19.321234 205.68 20.064995 207.53 .318 3885 43.70 333 9^4 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- 6 7 334 4271 42-54 22 19.358265 205.77 2O 102360 207.63 .319 1746 A7.6C 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 335 1924 42.49 25 19.395312 205.86 20.139741 207.72 1.319 9597 43-59 J-335 4472 42-47 26 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 "9432375 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 435i .336 4656 42.40 30 19.457094 206.01 2O.2O2O82 207.88 I.32I 2659 43-49 1.336 7199 42.38 31 19.469455 206.04 20.214556 207.91 .321 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 34 19.494184 19.506551 206. 1 1 206.14 20.239510 2O 251989 207.98 208.01 .322 0482 .322 3087 43-43 43.41 337 4823 337 73 6a 42.33 42-31 35 19.518921 206.17 20.264471 208.04 1.322 5692 43.40 1-337 990 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 43-3 6 .338 4972 42.25 38 19.556039 206.26 20.301927 208.14 323 3498 43-34 338 757 42.24 39 19568415 206.29 20.314416 208.17 .323 6097 43-32 .339 0041 42.22 40 19.580794 206.32 20.326907 208.20 1.323 8696 43.30 J-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 20351895 208.27 324 3890 43.26 339 7635 42.17 43 44 19.617939 19.630325 206.41 206.44 20.364392 20.37689! 208.30 208.33 324 M5 .324 9079 4324 43.22 .340 0165 .340 2693 42.15 42.13 45 ' 46 ! 47 48 19 642713 19.655102 19.667493 19.679886 206.47 206.50 20653 206.57 20.389392 2O.4OI 895 20.414399 20.426906 208.36 208.39 208.43 208.46 1.325 1672 .325 4263 .325 6854 325 9443 43.21 43.19 43-17 43-M 1.340 5221 34 7747 .341 0272 .341 2796 42.11 42.10 42.08 42.06 ; 49 19.692281 206.60 20439415 208.49 .326 2032 43-13 34 S3I9 42.04 5O 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 0762 42.01 52 53 54 19.729477 19.741879 19.754283 206.69 206.72 206.75 20.476952 204894.68 20.501986 208.59 208.62 208.65 .326 9790 .327 2374 327 4957 43-07 43-05 43.04 .342 2882 .342 5401 .342 7919 41.99 4'-97 41.96 55 56 57 58 59 19.766689 19.779097 19.791507 19.803919 19.816332 206.78 206.81 206.84 206.88 206.91 20.514506 20.527029 20.539553 20.552079 20.564607 208.69 208.72 208.75 208.78 208.82 1-127 7S3 8 .328 0119 .328 2698 .328 5276 .328 7853 43.01 43.00 42.98 42.96 42.94 1.343 0436 343 2952 343 54 6 7 -343 798o 344 493 4'94 41.92 41.90 41.89 41.87 GO 19.82^747 206.94 20.577137 208.85 1.329 0430 42.92 1.344 3005 41.85 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 32 33 34 35 log M. Diff. 1". log M. Diff. 1". logM. Diff. 1". logM. Diff. I". 1.344 35 4I-85 1.359 l8 59 40.86 1-373 7*5i 39-93 1.387 9418 39.06 1 344 55"5 41.84 359 431 40.84 373 9M 39.91 .388 1761 39.05 2 344 8025 41.82 .359 6760 40.82 374 2 4! 3990 .388 4104 39.04 3 345 534 41.80 359 9 2 9 40.81 374 4434 39-88 .388 6446 39.02 4 345 34i 41.78 .360 1657 40.79 .374 6827 3987 .388 8787 39.01 5 J-345 5548 4 J -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 1609 39-84 -389 3466 38.98 7 .346 0558 4*-73 .360 8995 4074 375 3999 39.82 .389 5804 38.97 8 .346 3061 41.72 .361 1439 40.73 375 6388 39.81 .389 8142 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 4070 1.376 1164 3978 1.390 2815 3893 11 347 5 6 5 41.66 .361 8766 40.68 376 355 39-77 .390 5150 38.91 12 347 3 6 5 41.65 .362 1207 40.66 376 5935 39-75 .390 7484 38 9 13 347 5563 41.63 .362 3646 40.65 .376 8320 39-74 .390 9817 38.88 14 .347 8060 41.61 .362 6084 40.63 377 0703 3972 .391 2150 38.87 15 1.348 0557 41.60 1.362 8522 40.62 1.377 3086 397i 1.391 4482 38.86 10 34 8 35 J 41.58 .363 0959 40.60 377 5468 39.69 .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 41-55 .363 5829 4 57 378 0230 39.66 .392 1472 38.82 19 349 53^ 41-53 .363 8263 40.56 .378 2609 39.65 .392 3801 38.80 20 1.349 3 02 3 41.51 1.364 0696 4-54 1.378 4987 39.64 1.392 6128 38.79 21 349 55J3 41.50 .364 3128 40.52 .378 7365 39.62 .392 8455 38-77 22 349 8o 3 41.48 3 6 4 5559 4051 378 974 2 39.61 393 78i 38.76 23 .350 0491 41.46 .364 7989 40.49 379 "7 3959 393 3io7 38.75 24 .350 2978 41.45 .365 0418 40.48 379 449* 39.58 393 543i 38-73 25 1.350 5464 41.43 1.365 2846 40.46 1.379 6866 39.56 !-393 7755 38.72 26 35 795 41.41 365 5*73 40.45 379 9H 39-55 .394 0078 38.71 27 .351 0434 41.40 .365 7699 4-43 .380 1612 39-53 .394 2400 38.69 28 .351 2917 41.38 .366 0125 40.41 .380 3983 39.52 .394 4721 38.68 29 35i 5399 41.36 .366 2549 40.40 .380 6354 39-50 394 7041 38.67 30 1.351 7880 4 J -35 1.366 4973 40.38 1.380 8724 39-49 1.394 9361 38.65 31 .352 0361 4'-33 .366 7395 40.37 .381 1093 39-47 .395 1680 38.64 1 32 .352 2840 41.31 .366 9817 40.35 .381 3461 39.46 395 3998 38.63 i 33 .352 5318 41.30 .367 2238 40.34 .381 5828 39-45 395 6315 38.61 34 35^ 7795 41.28 3 6 7 4657 40.32 .381 8194 39-43 395 8631 38.60 35 1-353 01 7 2 41.26 1.367 7076 40.31 1.382 0559 39-4 2 1.396 0947 38.59 36 353 2 747 41.25 .367 9494 40.29 .382 2924 39-4 .396 3262 38.57 37 353 5"i 41.23 .368 1911 4028 .382 5288 39-39 396 5576 38.56 38 353 7694 41.21 .368 4327 40.26 .382 7651 39-37 .396 7889 38.55 39 354 0167 41.20 .368 6742 40.25 .383 0013 39.36 .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 5io8 41.16 .369 1570 40.21 3 8 3 4734 39-33 397 4823 38-51 42 354 7578 41.15 3 6 9 3983 40.20 383 793 39.32 397 7133 38.49 I 43 355 0046 41.13 .369 6394 40.18 383 91-5 2 39-3 397 944* 38.48 44 355 *5'3 41.11 .369 8805 40.17 .384 1809 39.29 398 1751 3 8 -47 45 i-355 498o 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 9909 41.07 .370 6031 40.12 .384 8878 39.25 .398 8671 38.43 1 48 .356 2373 41.05 .370 8438 40.11 385 i*3 2 39-*3 399 97 6 38.41 49 .356 4836 41.03 .371 0844 40.09 3 8 5 355 39.22 399 3 2g i 38.40 50 I-356 7^97 41.02 1.371 3249 40.08 1.385 5938 39.20 1.399 5584 38.39 51 356 975 s 41.00 371 5654 40.06 .385 8290 39.19 399 7887 38-37 52 357 "i7 40.98 .371 8057 40.05 .386 0641 39.18 .400 0189 38.36 53 357 4676 40.97 .372 0459 40.03 .386 2991 39.16 .400 2491 38.35 54 357 7i34 40.95 .372 2861 40.02 .386 5340 39- J 5 .400 4791 38-33 55 1.357 959 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 3 6 39-i* .400 9390 38-31 57 .358 4501 40.90 .373 0060 39-97 .387 2383 39.11 .401 1688 38.30 58 35 8 6 954 40.89 373 2 458 3996 387 47*9 39.09 .401 3985 38.28 59 358 947 40.87 373 4855 39-94 .387 7074 39.08 .401 6282 38-^7 60 1.359 1859 40.86 1-373 7 2 5 J 39-93 1.387 9418 39.06 1.401 8578 38.26 TABLE VI, For finding the True Anomaly or the Tiise from the Perihelion in a Parabolic Orbit. V. 36 37 38 39 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". .401 8578 38.26 415 4930 37.50 .428 8662 36.80 .441 9943 36.14 1 .402 0873 38.24 .415 7180 37-49 .429 0869 3679 .442 21 1 1 36.13 2 .402 3167 38-23 .415 9429 37-47 4*9 3 76 36.78 .442 4279 36.11 3 .402 5460 38.22 .416 1678 37.46 429 5285 36.77 44* 6446 36.11 4 .402 7753 38.20 .416 3925 37-45 .429 7488 36.75 .442 8612 36.10 5 .403 0045 38.19 .416 6172 37-44 429 9693 36.74 .443 0778 3609 6 .403 2336 38.18 .416 8419 37-43 43 l8 97 36.73 443 *943 36.08 7 .403 4626 38.17 .417 0664 37-4' .430 4101 36.72 443 5'7 36.07 8 .403 6916 38.'5 .417 2909 37.40 .430 6304 3671 443 7271 36.06 9 .403 9205 38.14 4'7 5'53 | 37-39 .430 8506 36.70 443 9434 36.05 10 .404 1493 38.13 .417 7396 37.38 1.431 0708 36.69 .444 1597 36.04 11 .404 3780 38.12 .417 9639 37-37 .431 2909 36.68 444 375* 36.03 12 .404 6067 38.10 .418 1881 37-36 -43' 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 9507 36.64 .445 0240 35-99 15 .405 2921 38.06 .418 8602 37.32 1.432 1705 36.63 .445 2400 35.98 16 .405 5205 3805 .419 0841 37-3 1 432 393 36.62 445 4558 35-97 17 18 .405 7488 .405 9769 38.03 38.02 .419 3079 .419 5317 37-3 37.29 .432 6100 .432 8296 36-61 36.60 .445 6716 -445 8874 35.96 35-95 19 .406 2051 38.01 4'9 7554 37.27 433 49' 36.59 .446 1031 35-94 2O 21 .406 4331 .406 66 i i 38.00 37 99 4'9 9790 .420 2026 37.26 37-*5 1.433 2686 433 488i 36-57 36.56 .446 3187 446 5343 3593 359* 22 .406 8889 37 97 .420 4260 37-24 433 774 3M5 .446 7498 35-9' 23 .407 i i 68 37.96 .420 6494 37-23 433 9267 36.54 .^46 9652 3590 24 47 3445 37-95 .420 8728 37.22 434 *459 36.53 .447 1806 35-89 25 1.407 5721 37-94 1.421 0960 37.20 '434 3 6 5' 36-5* 1.447 3959 35.88 26 27 .407 7997 .408 0272 37-9* 37.91 4*' 3'9* .421 5423 37-19 37.18 434 5842 .434 8032 3651 36.50 .447 6112 447 8263 35-87 3586 28 29 .408 2547 .408 4820 37.90 37.89 .421 7654 .421 9884 37-17 37.16 .435 0221 .435 2410 36.49 36.48 .448 0415 448 2565 35-85 35.84 30 31 32 33 34 1.408 7093 .408 9365 .409 1636 .409 3907 .409 6177 37-86 37.85 37-84 37.82 i 42 2113 .422 4341 .422 6569 .422 8796 .423 1022 37-'5 37-13 37-12 37.11 37.10 1.435 4598 435 67*6 -435 8973 43 6 "59 436 3345 36.47 36.46 36.44 36.43 36.42 1.448 4715 .448 6865 448 9 OI 4 .449 1162 449 339 35.83 35.82 35.81 35-8o 35-79 35 36 37 1.409 8446 .410 0714 .410 2981 37.81 37.80 37-78 1.423 3248 4*3 5473 .423 7697 37.09 37.08 37.06 I-43 6 553 .436 7714 .436 9898 36.41 36.40 '449 5456 449 7603 449 9749 35-77 35-76 38 39 .410 5248 .410 7514 37-77 37.76 .423 9920 4*4 *'43 37-05 37.04 .437 208! 437 4263 36.37 .450 1894 .450 4038 35-75 35-74 40 41 42 43 44 1.410 9780 .411 2044- .411 4308 .411 6571 .411 8833 37-75 37-74 377* 37-7' 37.70 1.424 4365 .424 6586 .424 8807 .425 1027 .425 3246 37.03 37.02 37.01 36.99 36.98 1.437 6445 437 8626 .438 0806 .438 2986 .438 5165 36.36 36.35 36.34 36.32 36.31 1.450 6182 .450 83*2 .451 0468 .451 2610 -45' 475* 35-73 35-72 35-7' 35-70 35.69 45 46 47 48 49 1.412 1095 .412 3356 .412 5616 .412 7875 .413 0134 37.69 3768 37.66 37-64 1-4*5 5465 -4*5 7683 .425 9900 .426 2117 426 4333 36.97 36.96 36.95 36.94 36.92 1.438 7344 .438 9522 439 ' 6 99 439 3875 439 6051 36.30 36-29 36.28 36.27 36.26 1.451 6893 .451 9033 45* "73 45* 33'2 .452 5450 3568 35.66 35-65 35.64 50 51 52 53 54 1.413 2392 .413 4649 .413 6905 .413 9161 .414 1416 37.61 37.60 3759 37-58 1.426 6548 .426 8762 .427 0976 .427 3189 -4*V 54* 36.91 36.90 36-89 36'.8 7 1.439 8226 .440 0401 .440 2575 .440 4748 .440 6921 36.25 36.24 36.23 36.22 36.20 i 452 7588 45* 97*5 .453 1862 453 3998 453 6134 35.62 35.61 35.60 3559 55 56 57 58 59 1.414 3670 .414 59*4 .414 8176 .415 0429 .415 2680 37-56 37-55 37-54 37-53 37-5 1.427 7613 4*7 9^*4 .428 2035 .428 4244 4*8 6453 36.86 36.85 36.83 36.82 36.81 1.440 9093 .441 1264 .441 3436 44' 5605 .441 7774 36.10 36.18 36.17 36.16 36.15 1.453 8269 .454 0403 454 *537 454 467 .454 6802 35-57 35-56 35-55 35-54 60 1.415 4930 37-5 1.428 8662 36.80 1.441 9943 36.14 J-454 *934 35-53 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 40 41 42 43 logM. Diff. I". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 1.454 8934 35-53 1.467 5782 34-95 1.480 0627 34.41 1-492 3597 33.91 1 455 1065 35-52 .467 7879 34-94 .480 2691 344 492 5631 33-90 2 455 3196 35-5i .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 4 455 745 6 35-49 .468 4166 34.91 .480 8882 34-3* 493 i73i 33-87 5 6 1.455 9585 .456 1713 35.48 35-47 1.468 6261 .468 8355 34.90 34.90 1.481 0944 .481 3006 34-37 34-36 1.493 3764 493 579 6 33-87 33-86 7 .456 3841 35.46 .469 0448 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 9 .456 8094 35-44 .469 4634 34-87 .481 9189 34-33 .494 1888 33-83 10 1-457 O22O 3543 1.469 6725 34-86 1.482 1249 34-33 1.494 3918 33-83 11 457 234 6 35-4* .469 8817 34-85 .482 3308 34.32 494 594 8 33.82 12 457 447 35-41 .470 0907 34-84 .482 5367 34-3' 494 7977 33.81 i 13 457 6595 35.40 .470 2998 34-83 .482 7425 34.30 .495 0005 33-8o 14 457 8718 35-39 .470 5087 34.82 .482 9483 34.29 495 2033 33-79 15 1.458 0841 35-38 1.470 7176 34-8i 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 35-36 471 1353 34-79 .483 5653 34-27 495 8114 33-77 18 .458 7207 35-35 .471 3440 34-79 .483 7709 34.26 .496 0140 33.76 19 .458 9328 35-34 471 5527 34-78 4 8 3 9764 34-25 .496 2166 33-75 20 1.459 1448 35-33 1.471 7613 34-77 1.484 1819 34-24 1.496 4191 33-75 21 459 3567 353* .471 9699 34-76 484 3873 34-23 .496 6216 33-74 22 .459 5686 35-31 472 1784 34-75 .484 5927 34-22 .496 8240 33-73 23 459 7805 35-30 .472 3869 34-74 .484 7980 34.22 .497 0264 33-72 24 459 99" 35.29 472 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 4137 34.19 497 6332 33.70 27 .460 6272 35.26 .473 2203 34-71 .485 6,88 34.18 497 8354 33-69 28 29 .460 8388 .461 0503 35-25 35.24 473 4285 .473 6366 34-70 34.69 .485 8239 .486 0289 34-17 34.16 .498 0376 .498 2396 33-68 33-68 30 1.461 2617 35-23 1.473 844.7 34.68 1.486 2338 34.16 1.498 4417 33.67 31 .461 4731 35-23 474 527 34.67 .486 4388 34-15 .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- 6 5 .486 8484 34.I3 499 475 33-65 34 .462 1069 35-2 474 6765 34-64 .487 0532 34-12 499 2494 33- 6 4 35 1.462 3180 35-19 1.474 8843 34-63 1.487 2579 34.12 1.499 4512 33-63 36 .462 5291 35.18 475 92i 34-62 .487 4626 34.11 499 6 530 33.62 37 .462 7401 35-17 475 2998 34-6i .487 6672 34.10 499 8547 33-62 38 .462 9511 3516 475 575 34.61 .487 8718 34.09 .500 0563 33.61 39 .463 1620 35-iS 475 7Si 34.60 .488 0763 34.08 .500 2580 33.60 40 1.463 3729 35-14 1.475 9227 34-59 1.488 2807 34.07 1.500 4595 33-59 41 42 .463 5837 4 6 3 7944 35-13 " 35.12 .476 1302 .476 3376 34.58 3457 .488 4852 .488 6895 34.07 34.06 .590 6611 .500 8625 33-58 33-58 43 .464 0051 35-" .476 5450 34-56 488 8939 34.05 .501 0640 33-57 44 .464 2158 35.10 .476 7524 34-55 .489 0981 34.04 .501 2654 33.56 45 1.464 4263 35.09 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 5065 34-02 .501 6680 33-55 47 .464 8473 35-7 477 3741 34-53 .489 7106 34-02 .501 8693 33-54 48 .465 0577 35.06 .477 5812 34-52 489 9'47 34.01 .502 0705 33-53 49 .465 2681 35-5 477 7883 345i .490 1187 34-oo .502 2716 33-52 50 .465 4784 35-4 J-477 9953 34.50 1.490 3227 33-99 1.502 4727 335i 51 .465 6886 35 4 .478 2023 34-49 .490 5266 33.98 .502 6738 33-5' 52 .465 8988 35.03 .478 4092 34-48 .490 7305 33-97 .502 8748 33.50 53 .466 1090 35.02 .478 6161 34-47 49 3 9343 3396 .503 0758 33-49 54 .466 3190 35.01 .478 8229 34-46 .491 1381 33-95 .503 2767 33.48 55 .466 5290 35.00 1.479 297 34.46 1.491 3418 33-95 1.503 4776 33-48 56 .466 7390 34-99 479 2364 34-45 49i 5455 33-94 .503 6784 33-47 57 .466 9489 34-98 479 443 34-44 .491 7491 33-93 .503 8792 33-46 58 .467 1587 34-97 479 6 49 6 34-43 .491 9527 33-92 .504 0800 33-45 59 .467 3685 34.96 479 8562 34.42 .492 1562 33.91 .504 2807 33-44 60 .467 5782 34-95 1.480 0627 34-41 1.492 3597 33.91 54 4813 33-44 576 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V 44 45 46 47 logM. Diff. 1" logM. Diff. 1' logM. Diff. l' logM. Diff. 1". \ 1.504 4813 .504 6819 .504 8825 .505 0830 505 2835 33-44 33-43 3342 33-42 33-41 1.516 4390 .516 6370 .516 8349 .517 0328 .517 2306 33.00 32.99 32.98 3298 32-97 1.528 2435 .528 4390 528 6344 .528 8299 .529 0252 3259 32-58 32-57 32.57 32.56 1.539 9048 .540 0980 .540 2012 .540 4843 .540 6774 32.20 32.20 32.10 32.18 32.18 5 6 8 9 1.505 4839 .505 6843 .505 8846 .506 0849 .506 2852 33.40 33-39 33-39 33.38 33 37 1.517 4284 .517 6262 .517 8239 .518 0216 .518 2192 32.96 32.96 32-95 32.94 32.93 1.529 2206 529 4159 .529 6112 .529 8064 .530 0016 32-55 3255 32-54 32-53 32.53 1-540 8705 .541 0635 -541 2564 54 1 4494 .541 6423 32.17 32.17 32.16 S2.I5 32.15 10 11 12 13 14 1.506 4854 .506 6855 .506 8856 .507 0857 .507 2857 33-35 33-34 33-33 1.518 4168 .518 6143 .518 8118 519 0093 .519 1067 32.93 3292 32.91 32.91 32.90 1.530 1967 53 39'8 53 5869 .530 7819 53 9769 32-52 12.51 32.51 32.50 32-49 i 541 8352 .542 0280 .542 2208 .542 4135 542 6063 32.14 32.14 32.13 32.12 32.11 15 16 17 18 19 1.507 4857 .507 6856 -507 8855 .508 0853 .508 2851 33-33 33-32 33-3i 33-30 33-29 1.519 4041 .519 6014 .519 7987 .519 9960 .520 1932 32.89 32-89 32.88 S2-87 32.86 1.531 1719 .531 3668 .531 5616 53' 75 6 5 531 95'3 32.49 32-48 32.48 32-47 32.46 1.542 7989 .542 9916 543 1842 543 3768 543 5693 32.11 32.10 32.10 32.09 32-09 2O 21 22 23 24 1.508 4849 .508 6846 .508 8843 .509 0839 .509 2835 33-29 33.28 33-27 33-27 33.26 1.520 39x34 .520 5875 .520 7846 .520 9816 .521 1786 3286 3285 32.84 32.84 3283 1.532 1460 532 347 532 5354 .532 7300 .532 9246 32.46 32-45 32.44 3244 32-43 1.543 7618 543 9543 .544 1467 544 339i 544 53'5 32.08 32.08 | 32. o-* 32.06 32.06 25 26 27 1.509 4830 .509 6825 .509 8819 33.25 3324 1.521 3756 .521 5725 .521 7694 32.82 32.82 32.81 1.533 "92 533 3'37 533 5082 32.43 32.42 32.42 -544 7238 544 9'6i 545 '083 32.05 32.04 3204 i 29 .510 0813 .510 2807 33.23 33-22 .521 9662 .522 1630 32.80 32.80 533 7027 533 8971 32.41 32.40 545 35 545 4927 32.03 , 32.03 30 510 4800 33-21 522 3598 32-79 1.534 0914 32.39 545 6849 32.02 31 32 510 6792 510 8785 33-21 33-20 522 5565 522 7531 3278 32-78 534 2858 .534 4801 32.39 32 38 545 8770 546 0690 32.02 32.01 33 511 0776 33.I9 522 9498 32.78 534 6 743 32-37 546 2611 32.00 34 511 2768 33-iS 523 1464 32.77 534 8685 32.37 546 4531 32.00 35 5" 4759 33.18 5*3 3429 32.76 535 0627 32.36 546 6450 3'-99 36 511 6749 33-17 523 5394 32.75 535 2568 32-35 546 8370 31.98 37 511 8739 33.16 523 7359 32-74 555 459 32.35 547 0289 31.98 38 512 0729 33-15 523 9323 32-73 535 6 45 32-34 547 2207 3'-97 39 512 2718 33-15 524 1287 32.73 535 8390 32-33 547 4>25 3i'97 40 41 512 4707 512 6695 33-'4 3 i i -5 524 3251 C2J. C2IJ. 3272 32.71 536 0330 32-33 547 6043 31.96 31 nfi 42 512 8683 5 J' * 3 33-13 5 A T J * *T 524 7176 * / * 32.71 536 4209 32.32 547 9878 1.90 3'-95 43 513 0670 33-12 524 9138 32.70 536 6148 32.31 548 1795 3'-94 I 44 513 2657 33.11 525 i i oo 32-70 536 8086 32-30 548 3711 3'-94 45 513 4644 33.11 525 3062 32.69 537 0024 32.30 548 5627 31-93 46 513 6630 33.10 525 5023 32.68 537 1962 32.29 548 7543 3 J 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 5836 32.28 549 1373 31.91 49 514 2586 33.07 526 0903 32.66 537 7772 32-27 549 3288 31.91 50 514 4570 33-7 526 2863 32.65 537 978 32.26 549 5202 31.90 51 514 6554 33.06 526 4822 32.64 S3 8 1644 32.26 549 7116 31.90 52 5H 8537 33-05 526 6780 32.64 S3 8 3579 32.25 549 9030 31-89 53 515 0520 33.05 526 8739 32.63 S3 8 55H 32.25 550 0943 31.88 54 55 2503 33-04 527 0696 32.62 538 7449 32.24 550 2856 31.88 55 515 4485 33-04 527 2654 32.62 538 9383 32.23 550 4769 3J-J7 56 515 6467 33-3 527 4611 |2.6l 539 1317 32-23 550 6681 57 515 8449 33-02 527 6567 32.61 539 3250 32.22 550 8593 31.86 58 59 516 0430 516 2410 3301 33-oi 527 8524 528 0479 2.60 2.60 539 5183 539 7" 32.21 32,21 551 0503. 551 2416 l\:ll 1 60 516 4390 33.00 528 2435 32-59 539 9048 32.20 551 4316 31.85 j For TABLE VI, the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 48 49 50 51 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". I-55I 4326 .551 6237 31.85 31.84 1.562 8360 .563 0250 3I-5I S'-S 1 1.574 1234 574 3 106 31.20 31.20 '5 8 5 303 1 .585 4886 30.91 30.91 2 .551 8l 47 31.83 .563 2140 31.50 574 4977 3I-I9 .585 6740 30.90 3 .552 0057 31.83 .563 4030 31.50 574 6849 31.19 5 8 5 8 594 30.90 4 .552 1966 31.82 .563 5920 31-49 574 872 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 55 Z 57 8 4 3 I.8l .563 9698 31.48 575 2461 31.17 .586 4155 30.89 7 55 a 7 6 93 31.80 .564 1586 31-47 575 433i 31.17 .586 6008 30.88 8 .552 9601 31.80 5 6 4 3475 31-47 .575 6201 31.16 .586 7859 30.87 9 553 i5 8 31.79 564 53 6 3 31.46 575 8 7 31.16 .586 9713 30.87 10 1.553 34'6 31-79 i-5 6 4 7*5 31.46 1-575 9939 3I-I5 1.587 1565 30.87 11 12 553 53 2 3 553 7 2 3 31.78 31.78 .564 9138 .565 1025 31-45 31-45 .576 1808 .576 3677 31.15 3I-I4 5 8 7 34'7 .587 5268 30.86 30.86 13 553 9'3 6 31.77 .565 2911 31-44 576 5546 SI-H .587 7120 30.85 14 554 1042 31.76 .565 4798 31.44 .576 7414 3I-I3 .587 8 971 30.85 15 1.554 2 94 8 31.76 1.565 6684 31-43 1.576 9281 3I-I3 1.588 0821 30.84 16 554 4 8 53 3''75 .565 8569 3J-43 577 "49 31.12 .588 2672 30.84 17 554 6 75 8 31-75 .566 0455 31.42 577 3i6 31.12 .588 4522 30.83 18 554 866 3 3!-74 .566 2340 31.41 577 4 88 3 31.11 .588 6372 30.83 19 555 5 6 7 31-74 .566 4225 3i-4i 577 6749 31.11 .588 8222 30.83 20 i-555 ^472 3'-73 1.566 6109 31.40 1.577 8615 31.10 1.589 0071 30.82 21 555 4375 31-73 .566 7993 31.40 .578 0481 31.10 .589 1920 30.82 22 555 62 79 31.72 .566 9877 3'-39 .578 2347 31.09 .589 3769 30.81 23 .555 8182 31.71 .567 1761 3 ! -39 .578 4213 31.09 .589 5618 30.81 24 .556 0084 31.71 .567 3644 31.38 .578 6078 31.08 .589 7466 30.80 25 1.556 1987 31.70 1.567 5527 31.38 1.578 7942 31.08 1.589 9314 30.80 26 .556 3888 31.70 .567 7409 31-37 .578 9807 31.07 .590 1162 3-79 27 .556 5790 31.69 .567 9291 V-37 579 l6 7' 31.07 59 39 30.79 28 29 .556 7691 .556 9591 31.68 31.68 .568 1173 568 355 3i-36 31.36 579 3535 579 5399 31.06 31.06 59 4 8 57 .590 6704 30.78 30.78 30. '557 H93 31.67 1.568 4936 3'3S 1.579 7262 31.06 1.590 8550 30.78 31 -557 3393 31.67 .568 6817 3*-35 579 9^5 31.05 .591 0397 30.77 32 557 5^93 31.66 .568 8698 3i-34 .580 0988 31.04 .591 2243 3-77 33, 557 7*93 31.66 .569 0579 3'34 .580 2851 31.04 .591 4089 30.76 34: 557 9092 31.65 .569 2459 3'-33 .580 4713 31.03 59' 5935 30.76 35 1.558 0991 31.65 1.569 4338 3M3 1.580 6575 31.03 1.591 7780 30.75 36 ..558 2890 3 J - 6 4 .569 6218 3'-3 2 .580 8436 31.03 .591 9625 30.75 37 38 -558 47 88 .558 6686 31.64 31.63 .569 8097 .569 9976 31.32 3i-3i .581 0298 .581 2159 31.02 31.02 .592 1470 592 33'5 30.75 30.74 39 .558 85-84 31.62 .570 1854 31.30 .581 4020 31.01 59* 5i59 30.74 40 1.559 0482 31.62 1-570 3733 31.30 1.581 5880 31.01 1.592 7003 30.73 41 559 2 379 31.61 .570 5611 31.29 .581 7740 31.00 .592 8847 30.73 42 559 4 2 75 31.61 .570 7488 31-29 .581 9600 31.00 .593 0690 30.72 43 559 6172 31.60 57 93 66 31.28 .582 1460 30.99 593 2534 30.72 44 .559 o68 31.60 .571 1243 31.28 5 8 * 33i9 30.99 593 4377 30.72 45 "559 99 6 3 3M9 1.571 3119 31.28 1.582 5179 30.98 1.593 6219 30.71 46 .560 1859 31-59 .571 4996 31.27 .582 7037 3098 .593 8062 30.71 47 5 6 3754 31.58 .571 6872 31.27 .582 8896 30.97 593 994 30.70 1 ** .560 5648 3^57 .571 8748 31.26 5 8 3 754 30.97 594 1746 30.70 49 .560 7543 31-57 , .572 0623 31.26 .583 2612 30.96 594 35 88 30.69 50 1.560 9437 31.56 , 1.572 2499 31.25 1.583 4470 30.96 1.594 54 2 9 30.69 51 .561 1331 31.56 t .572 4373 31.25 .583 6327 3-95 594 7 2 7 30.68 52 .561 3224 3J-55 .572 6248 31.24 .583 8184 3-95 594 9"' 30.68 53 561 5"7 3i-55 .572 8123 31.24 .584 oo 4I 30.94 595 95 2 30.68 54 .561 7010 3*-54 .572 9997 31.23 .584 1898 30.94 595 2792 30.67 55 1.561 8902 3M4 1.573 1870 31.23 1.584 3754 30.94 1.595 4633 30.67 56 .501 0794 3i-53 573 3743 31.22 .584 5610 3-93 595 6 473 30.66 57 .561 2686 3'-53 573 S66 31.22 .584 7466 30.93 595 8 3'2 30.66 58 .562 4578 3*-5 a 573 74 8 9 31.21 .584 9321 30.92 .596 0151 30.65 59 .562 6469 3i-5i 573 93 6z 31.21 .585 1176 30.92 .596 1990 30.65 60 1.562 8360 31.5,1 1.574 .1234 31.20 1.585 3031 30.91 1.596 3829 30.65 TABLE 71. For finding the True Anomaly or the Time from the Perihelion in a Parabolic vjrbit. 52 o 53 54 o 55 1 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". O' 1 2 3 1.596 3829 .596 5668 .596 7506 596 9344 30.65 30.64 30.64 30.63 1.607 3703 .607 5527 .607 7350 .607 9174 30.40 30.39 30.39 30.39 1.618 2724 .618 4534 .618 6344 .618 8153 30.17 30.17 30.16 30.16 1.629 0959 .629 2757 .629 4554 .629 6351 29.96 29.96 29.96 29.95 4 .597 1182 30.63 .608 0997 30.38 .618 9963 30.16 .629 8148 5 1.597 3020 30.62 1.608 2820 30.38 1.619 I 77 1 30.15 1.629 9945 29 9- 6 597 4857 30.62 .608 4642 30-38 .619 3581 30.15 .630 1742 29 94 7 .597 6694 30.62 .608 6465 30-37 .619 5390 30.15 .630 3538 29.94 8 597 8531 30.61 .608 8287 3-37 .619 7199 30.14 .630 5335 29.94 i 9 .598 0368 30.61 .609 0109 30.36 .619 9007 30.14 .630 7131 ^9-93 . 10 11 1.598 2204 .598 4040 30.60 30.60 1.609 1931 .609 3752 30.36 30.36 1.620 0816 .620 2623 30.14 30.13 1.630 8927 .631 0722 29.93 29.93 12 .598 5876 30-59 -609 5573 30-35 .620 4431 30.13 .631 2518 29.92 13 .598 7711 30.59 .609 7394 30-35 .620 6239 30.12 .631 4313 29.92 11 598 9547 30.59 .609 9215 30-34 .620 8040 30.12 .631 6108 29.92 15 1.599 *3 ? 2 30.58 1.610 1036 30.34 1.620 9853 30.12 1.631 7903 29.91 16 5)9 3217 30.58 .610 2856 30.34 .621 1660 30.11 .631 9698 29.91 17 599 55! 30-57 .610 4676 30.33 .621 3467 30.11 .632 1492 29.91 18 .599 6885 30.57 .610 6496 30.33 .621 5274 30.11 .632 3286 29.90 19 599 8719 30.57 .610 8315 30.32 .621 7080 30.10 .632 5081 29.90 ; 20 i. 600 0553 30.56 1.61 0135 30.32 1.621 8886 30.10 1.632 6875 29.90 21 .600 2387 30.56 .61 1954 30.32 .622 0692 30.10 .632 8668 29.89 22 .600 4220 3-55 .61 3773 30.31 .622 2497 30.09 .633 0462 29.89 23 .600 6053 30-55 .61 5591 30.31 .622 4303 30.09 .633 2255 29.89 24 .600 7886 30-55 .61 7410 30.31 .622 6108 30.09 .633 4048 29.88 25 1.600 9718 3-54 1.611 9228 30.30 1.622 7913 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 94*7 29.87 28 .601 5214 3-53 .612 4681 30.29 .623 3327 30.07 .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 1.623 6935 30.06 1.634 4803 29.86 31 32 .602 0708 .602 2539 30.52 30.51 .61 3 01 32 .613 1949 30.28 30.28 .623 8739 .624 0543 30.06 30.06 .634 6595 .634 8387 29.86 29.86 33 31 .602 4370 .602 6200 30.51 30.50 .613 3765 .613 5582 30.27 30.27 .624 2340 .624 4149 30.05 30.05 .635 0178 .635 1969 29.86 29.85 33 1.602 8030 30.50 i 613 7398 30.26 1.624 5952 30.05 1.635 3760 29.85 36 .602 9860 30.50 .613 9213 30.26 624 7755 30.04 635 5551 29.85 37 38 .603 1690 .603 3519 30.49 30.49 .614 1029 .614 2844 30.26 30.25 .624 9557 .625 1360 30.04 30.04 635 7342 .635 9132 29.84 29-84 39 .603 5348 30.48 .614 4659 30.25 .625 3162 30.03 .636 0922 29.84 40 43 ! 43 44 1.603 7177 .603 9005 .604 0834 .604 2662 .604 4490 30.48 30-47 30-47 30.47 30.46 1.614 6474 .614 8288 .615 0103 .615 1917 .615 3731 30.25 30.24 3024 30.23 30-23 1.625 4964 .625 6765 .625 8567 .626 0368 .626 2169 30.03 30.03 30.02 30.02 30.02 1.636 2713 .636 4502 .636 6292 .636 8082 .636 9871 29.83 29.83 29.83 i 29.82 29.82 45 46 17 48 49 1.604 6317 .604 8145 .604 9972 .605 1799 .605 3626 30.46 3-45 30.45 3-45 30.44 1.615 5545 .615 7358 .615 9171 .616 0984 .616 2797 30.23 30.22 30.22 30.22 30.21 1.626 3970 .626 5771 .626 7571 .626 9372 .627 1172 30.01 30.01 30.01 30.00 30.00 1.637 1660 -637 3449 .637 5238 .637 7027 .637 8815 29.82 29.82 29.81 29.81 29.81 50 51 52 53 1.605 5452 .605 7278 .605 9104 .606 0930 .606 2755 344 30.43 30.43 30-43 30.42 1.616 4610 .616 6422 .616 8234 .617 0046 .617 1858 30.21 30.20 30.20 30.20 30.19 1.627 2971 .627 4771 .627 6571 .627 8370 .628 0169 30.00 29.99 29.99 29.99 29.98 1.638 0603 .638 2391 .638 4'79 .638 5967 .638 7754 29.80 29.80 29.80 29-79 29.79 55 30 57 58 59 1.606 4581 .606 6406 .606 8230 .607 0055 .607 1879 30.42 30.42 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 29.70 29.78 29.78 29.78 60 1.607 3703 30.40 1.618 2724 30-17 1.629 0959 29.96 1.639 8475 9-77 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. f*y 56 57 58 59 log M. Diff. 1". logM. Diff. l". log M. Diff. 1". logM. Diff. 1". 1.639 8475 29.77 1.650 5336 29.60 1.661 1601 29.44 I.6 7 I 7331 29 30 i .640 0262 29.77 .650 7112 29.60 .661 3368 29.44 .671 9089 29.30 9 .640 2048 29.77 .650 8887 29.59 .661 5134 29.44 .672 0846 29.30 3 .640 3833 29.76 .651 0663 29.59 .661 6900 29.43 .672 2604 29.29 4 .640 5619 29.76 .651 2438 29.59 .661 8666 29-43 .672 4362 29.29 5 1.640 7405 29.76 1.651 4213 29.58 1.662 0432 29.43 1.672 6119 29.29 6 .640 9190 29.75 .651 5988 29.58 .662 2197 29-43 .672 7876 29.29 7 8 .641 0975 .641 2760 29.75 29.75 .651 7763 .651 9538 29.58 29.58 .662 3963 .662 5728 29.42 29.42 .672 9634 .673 1391 29.28 29.28 9 .641 4545 29.74 .652 1312 29-57 .662 7493 29.42 673 3H7 29.28 10 1.641 6329 29.74 1.652 3086 29.57 1.662 9258 29.42 1.673 4904 29.28 11 .641 8114 29.74 .652 4861 29-57 .663 1023 29.41 .673 6661 29.28 12 .641 9898 29.74 .652 6635 29.57 .663 2788 29.41 .673 8417 29.27 13 .642 1682 29.73 .652 8408 29.56 .663 4553 29.41 .674 0174 29.27 14 .642 3466 29-73 .653 0182 29.56 .663 6317 29.41 .674 1930 29.27 15 1.642 5250 29.73 1.653 1956 29.56 1.663 8og 2 29.40 1.674 3 686 29.27 16 .642 7033 29.72 .653 3729 29-55 .663 9846 29.40 .674 5442 29.27 17 .642 8816 29.72 .653 5502 29-55 .664 1610 29.40 .674 7198 29.26 18 .643 0599 29.72 .653 7275 29-55 .664 3374 29.40 .674 8954 29.26 19 .643 2382 29.71 .653 9048 29-55 .664 5137 29.39 .675 0709 29.26 20 1.643 4165 29.71 1.654 0821 29.54 1.664 6901 29.39 1.675 2465 29.26 21 .643 5948 29.71 .654 2593 29.54 .664 8664 29.39 .675 4220 29.25 22 .643 7730 29.71 .654 4366 29.54 .665 0428 29-39 675 5975 29.25 23 .643 9513 29.70 .654 6138 29.54 .665 2191 29.39 .675 7730 29.25 24 .644 1295 29.70 .654 7910 29-53 .665 3954 29.38 .675 9485 29.25 25 1.644 3077 29.70 1.654 9682 29-53 1.665 5717 29.38 1.676 1240 29.25 26 .644 4858 29.69 .655 1454 29.53 .665 7480 29.38 .676 2995 29.24 27 .644 6640 29.69 .655 3225 29.53 .665 9242 29.38 .676 4749 29.24 28 .644 8421 29.69 6 55 4997 29.52 .666 1005 29.37 .676 6504 29.24 29 .645 0203 29.69 .655 6768 29.52 .666 2767 29-37 .676 8258 29.24 30 1.645 1984 29.68 J - 6 55 8 539 29.52 1.666 4529 29.37 1.677 OOI2 29.24 31 6 45 37 6 5 29.68 .656 0310 29.51 .666 6291 29.37 .677 1766 29.23 32 645 5545 29.68 .656 2081 2951 .666 8053 29.36 .677 3520 29.23 33 .645 7326 29.67 .656 3852 29.51 .666 9815 29.36 .677 5274 29.23 34 .645 9106 29.67 .656 5622 29.51 .667 1577 2936 .677 7028 29.23 35 1.646 0886 29.67 1.656 7392 29.50 1.667 3338 29.36 1.677 8781 29.23 36 .646 2666 29.67 .656 9163 29.50 .667 5100 29.35 .678 0535 29.22 37 .646 4446 29.66 .657 0933 29.50 .667 6861 29-35 .678 2288 29.22 38 .646 6226 29.66 .657 2703 29.50 .667 8622 29.35 .678 4041 29.22 39 .646 8005 29.66 .657 4472 29.49 .668 0383 29-35 .678 5794 29.22 40 1.646 9785 29.65 1.657 6242 29.49 1.668 2144 29.35 1.678 7547 29.22 41 .647 1564 29.65 .657 8011 29.49 .668 3904 29.34 .678 9300 29.21 42 647 3343 29.65 .657 9781 29.49 .668 5665 29.34 .679 1053 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 29.48 .668 9185 29.34 .679 4558 29.21 45 1.647 8679 29.64 1.658 5087 29.48 1.669 945 29-33 1.679 6310 29.20 46 .648 0457 29.64 .658 6855 29.48 .669 2705 29.33 .679 8063 29.20 47 .648 2235 29.63 .658 8624 29.47 .669 4465 29-33 .679 9815 29.20 48 .648 4013 29.63 .659 0393 29.47 .669 6225 29.33 .680 1567 29.20 49 .648 5791 29.63 .659 2IOI 29.47 .669 7984 29.32 .680 3319 29.20 50 1.648 7569 29.63 1.659 3929 29.47 1.669 9744 29.32 i. 680 5070 29.19 51 .648 9346 29.62 .659 5697 29.46 .670 1503 29.32 .680 6822 29.19 52 .649 1123 29.62 .659 7465 29.46 .670 3262 29.32 .680 8574 29.19 53 .649 2901 29.62 .659 9232 29.46 .670 5021 29.32 .681 0325 29.19 54 .649 4677 29.61 .660 1000 29.46 .670 6780 29.31 .681 2076 29.19 55 i 649 6454 29.61 1. 660 2767 29.45 1.670 8539 29.31 1.681 3827 29.18 56 .649 8231 29.61 .660 4534 29.45 .671 0298 29.31 .681 5578 29.18 57 .650 0007 29.61 .660 6301 29.45 .671 2056 29.31 .681 7329 29.18 58 .650 1784 29.60 .660 8068 29.45 .671 3814 29.30 .681 9080 29.18 59 .650 3560 29.60 .660 9835 29.44 671 5573 29.30 .682 0831 29.18 60 1.650 5336 29.60 1.661 1601 29.44 1.671 7331 29.30 1.682 2581 29.17 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit V 60< > : 61 62< t 63 C log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 682 2581 29.17 692 7408 29.07 703 1866 28.97 713 6006 2889 I 2 3 682 4332 682 6082 682 7832 29.17 29.17 29.17 6 9 2 9152 693 0896 693 2640 29.06 29.06 29.06 703 3604 .703 5342 .703 7080 28.97 28.97 28.97 713 7739 7'3 9473 .714 1206 28.89 28.89 28.88 4 682 9582 29.17 .693 4383 29.06 .703 8818 28.96 .714 2939 28.88 5 683 1332 29.16 .693 6127 29.06 .704 0556 28.96 .714 4672 28.88 6 683 3082 29.16 .6 93 7870 29.05 .704 2293 28.96 .714 6405 28.88 7 .683 4852 29.16 .693 9613 29.05 .704 4031 28.96 .714 8138 28.8? 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 .684 0080 29.16 .694 4842 29.05 .704 9243 28.96 7'5 333 6 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 : i 13 14 .684 5328 .684 7077 29.15 29.15 .695 0070 .695 1813 2904 29.04 75 4455 .705 6192 28.95 28.95 715 8533 .716 0266 28.87 ' 28.87 , 15 .684 8826 29.14 695 3555 29.04 .705 7929 28.95 .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 2O 21 .685 7568 .685 9316 29.14 29.13 .696 2266 .696 4008 29.03 29.03 1.706 6612 .706 8348 28.94 28.94 .717 0658 .717 2390 28.86 28.86 ; 22 23 .686 1064 .686 2812 29.13 29.13 .696 5750 .696 7491 29.03 29.03 .707 0085 .707 1821 28.94 28.94 .717 4122 7i7 5853 28.86 28.86 24 .686 4560 29.13 .696 9233 29.02 77 3557 28.94 7'7 7585 28.86 25 .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 28 29 .686 9803 .687 1550 .687 3297 29.12 29.12 29.12 .697 4457 .697 6198 .697 7939 29.02 29.02 29.02 .707 8765 .708 0501 .708 2237 28.93 28.93 28.93 . 71 8 2780 .718 4511 .718 6242 28.86 28.86 , 28.85 ; 30 31 1.687 5044 .687 6791 29.12 29.12 1.697 9680 .698 1421 29.02 29.01 1.708 3972 .708 5708 28.93 2893 1.718 7974 .718 9705 28.85 28.85 32 .687 8538 29.11 .698 3162 29.01 .708 7444 18.9= .719 1436 llil 33 34 .688 0285 .688 2032 29.1! 29.1 1 .698 4902 .698 6643 29.01 29.01 .708 9179 .709 0914 28.92 28.92 .719 3167 .719 4898 2.X. 5 28.85 35 36 37 38 39 1.688 3778 .688 5525 .688 7271 .688 9017 .689 0764 29.11 29.11 29.10 29.10 29.10 1.698 8383 .699 0124 .699 1864 .699 3604 699 5345 29.01 29.01 29.00 29.00 29.00 1.709 2650 .709 4385 .709 6120 .709 7855 .709 9590 28.92 28.92 28.92 28.92 28.92 1.719 6629 .719 8360 .720 0090 .720 1821 .720 3552 28.85 28.85 ! 28.85 28.84 ! 28.84 40 41 42 43 44 1.689 2510 .689 4256 .689 6001 .689 7747 .689 9493 29.10 29.10 29.09 29.09 29.09 1.699 7085 .699 8824 .700 0564 .700 2304 .700 4044 29.00 29.00 19.00 29.00 28.99 1.710 1325 .710 3060 .710 4794 .710 6529 .710 8263 28.91 28.91 28.91 28.91 28.91 1.720 5282 .720 7013 .720 8743 .72 0474 .72 2204 28.84 28.84 28.84 28.84 28.84 45 1 46 47 i 48 49 1.690 1238 .690 2984 .690 4729 .690 6474 .690 8219 29.09 29.09 29.09 29.09 29.08 1.700 5783 .700 7523 .700 9262 .701 icoi .701 2741 28.99 28.99 28.99 28.99 28.99 1.710 9998 .711 1732 .711 3467 .711 5201 .711 6935 28.91 28.91 28.90 28.90 28.90 1.72 3934 .72 5665 7* 7395 .721 9125 .722 0855 28.84 28.84 - 28.84 28.83 28.83 50 51 52 53 54 1.690 9964 .691 1709 .691 3454 .691 5199 .691 6943 29.08 29.08 29.08 29.08 29.08 1.701 4480 .701 6219 .701 7958 .701 9697 .702 1435 28.98 28.98 28.98 28.98 28.98 1.711 8669 .712 0403 .712 2137 .712 3871 .712 5605 28.90 28.90 28.90 28.90 28.90 1.722 2585 .722 4315 .722 6044 .722 7774 .722 9504 28.83 2883 28.83 28.83 28.83 55 56 57 58 59 1.691 8688 .692 0432 .692 2176 .692 3920 .692 5664 29.07 29.07 29.07 29.07 29.07 1.702 3174 .702 4913 .702 6651 .702 8389 .703 0128 28.98 28.98 28.97 28.97 28.97 1.712 7339 .712 9072 .713 0806 .713 2539 .713 4273 28.90 28.89 28.89 28.89 28.89 1.723 1233 .723 2963 .723 ^693 .723 6422 .723 8151 28.83 28.83 28.82 28.82 28.82 i 60 1.692 7408 29.07 1.703 1866 22.97 1.713 6006 28.89 1.723 9881 28.82 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. '?' 64 o 65 o 66 o 67 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.7*3 9881 28.8 J-734 3539 28.77 1.744 73! 28.73 1.755 45 28.70 1 .724 1610 8.8 734 5 26 5 28.77 744 8755 28.73 755 2127 28.70 2 724 3339 8.8 734 6 99 X 28.77 745 0479 28.73 755 3849 28.70 3 .724 5068 8.8 734 8718 28.77 .745 2202 28.73 755 557i 28.70 4 .724 6798 8.8 735 444 28.77 745 3926 28.73 755 7293 28.70 5 1.724 8527 8.8 1.735 2169 28.76 1.745 565 28.73 '755 9 OI 5 28.70 6 .725 0256 8.8 735 3895 28.76 745 7373 28.73 .756 0737 Z8.70 7 .725 1984 8.81 735 5 6 2i 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 28.8! .736 4250 28.76 .746 7714 28.72 757 1069 28.70 13 .726 2357 28.81 736 5975 28.76 .746 9437 28-72 757 2791 28.70 14 .726 4085 28.81 .736 7701 28.76 .747 1161 28.72 757 4513 28.70 15 r-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 "52 28.76 747 4607 28.72 757 7957 28.70 17 .726 9270 28.81 737 2877 28.76 747 6330 28.72 757 9^79 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 1503 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 i73i 28.70 25 1.728 3095 28.80 1.738 6679 28.75 1.749 "5 28.72 '759 3453 28.70 26 .728 4823 28.80 .738 8404 28.75 749 '838 28.72 759 5175 28.70 27 .728 6551 28.80 .739 0129 28.75 749 35 6 i 28.72 759 6897 28.70 28 .728 8279 28.80 739 1853 28.75 749 5284 28.72 .759 8618 28.69 29 .729 0006 28.79 739 357 28.75 749 77 28.71 .760 0340 28.69 30 1.729 1734 28.79 !-739 533 28.75 1.749 873 28.7, 1.760 2062 28.69 31 .729 3461 28.79 .739 7028 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 .750 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 1.740 3927 2874 i-750 7344 28.71 1.761 0670 28.69 36 .730 2099 28.79 .740 5651 28.74 .750 9067 28.71 .761 2392 28.69 37 .730 3826 28.79 74 737 6 28.74 .751 0789 28.71 .761 4113 28.69 38 73 S5S3 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 i-75i 5957 28.71 1.761 9278 28.69 41 73 1 735 28.78 .741 4274 28.74 .751 7680 28.71 .762 0999 28.69 43 .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 .731 5915 28.78 .741 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 28.71 .763 1328 28.69 ! 48 .732 2823 28.78 .742 6344 28.74 .752 9737 28.71 .763 3050 28.69 49 73* 4549. 28.78 .742 8068 28.74 753 H 60 28.71 .763 4771 28.69 50 1.732 6276 28.78 1.742 9792 28.74 1.753 3^2 28.71 1.763 6493 28.69 51 .732 8002 28.78 743 J 5 l6 2873 753 494 28.71 .763 8214 28.69 52 i 53 .732 9729 733 H55 28.77 28.77 743 324 .743 4964 28.73 28.73 753 662 7 753 8349 28.71 28.71 .763 9936 .764 1657 28.69 28.69 54 733 3182 28.77 .743 6688 28.73 754 0071 28.70 .764 3379 28.69 55 1.733 498 28.77 1.743 8 4^ 28.73 1.754 1794 28.70 1.764 5100 28.69 56 733 6635 28.77 .744 0136 28.73 754 35'6 28.70 .764 6821 28.69 57 733 8361 28.77 .744 1860 28.73 754 5 2 38 28.70 .764 8543 28.69 58 734 0087 28.77 7-14 3584 28.73 .754 6960 28.70 .765 0264 28.69 59 734 1813 28.77 744 53 8 28.73 .754 8682 28.70 .765 1985 28.69 60 '734 3539 28.77 1.744 7031 28.73 1.755 45 28.70 1.765 3707 28.69 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 68 69 70 71 logM. I Diff.l". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 2 3 4 1.765 3707 .765 5428 .765 7150 .765 8871 .766 0592 28.69 28.69 28.69 28.69 28.69 1-775 6985 775 8706 .776 0427 .776 2149 .776 3870 28.69 28.69 28.69 28.69 28.69 1.786 0284 .786 2006 .786 3728 .786 5450 .786 7172 28.70 28.70 28.70 28.70 28.70 1.796 3650 796 5374 79 7097 .796 8821 797 545 28.73 28.73 28.73 28.73 28.73 5 6 7 8 9 1.766 2314 .766 4035 .766 5756 .766 7478 .766 9199 28.69 28.69 28.69 28.69 28.69 1.776 5591 .776 7313 .776 9034 777 0755 777 *477 28.69 28.69 28.69 28.69 28.69 1.786 8894 .787 0617 .787 2339 .787 4061 .787 5783 28.70 28.70 28.70 28.70 28.70 1.797 2268 797 399* 797 57'6 797 744 797 9*64 28.73 28.73 28.73 28.73 28.73 10 11 12 13 14 .767 0920 .767 2642 .767 4363 .767 6084 .767 7805 28.69 28.69 28.69 28.69 28.69 1.777 4198 777 59 20 777 7641 777 9363 .778 1084 28.69 28.69 28.69 28.69 28.69 1.787 7506 .787 9228 .788 0950 .788 2673 .788 4395 28.70 28.71 28.71 28.71 28.71 1.798 0888 .798 2611 798 4335 .798 6060 798 7784 28.73 28.73 28.73 28.73 28.73 15 16 17 18 19 1.767 9527 .768 1248 .768 2969 .768 4691 .768 6412 28.69 28.69 28.69 28.69 28.69 1.778 2806 .778 4527 .778 6248 .778 7970 .778 9691 28.69 28.69 28.69 28.69 28.69 1.788 6117 .788 7840 .788 9562 .789 1284 .789 3007 28.7. 28.71 28.71 28.71 28.71 1.798 9508 799 I2 3* 799 *9)6 .799 4680 799 6404 28.73 28.74 28.74 28.74 28.74 20 21 22 23 1.768 8133 .768 9854 .769 1576 .769 3297 28.69 28.69 28.69 28.69 1.779 H3 779 3'4 779 462 779 6578 28.69 28.69 28.69 28.69 1.789 4730 .789 6452 .789 8175 .789 9897 28.71 28.71 28.71 28.71 1.799 8128 799 9*53 .800 1577 .800 3301 28.74 28.74 28.74 28.71 24 .769 5018 28.69 779 8299 28.69 .790 1620 28.71 .800 5026 28.74 25 26 27 1.769 6740 .769 8461 .770 0182 28.69 28.69 28.69 1.780 0021 .780 1742 .780 3464 28.69 28.69 28.69 1.790 3342 .790 5065 .790 6788 28.71 28.71 28.71 .800 6750 28.74 .800 8475 i 28.74 .801 0199 : 28.74 28 .770 1903 28 69 .780 5185 28.69 .790 8510 28.71 .801 1924 28.74 29 .770 3625 28.69 .780 6907 28.69 .791 0233 28.71 .801 3648 28.74 3O 1.770 5346 28.69 .780 8629 28.69 .791 1956 28.71 .801 53-3 28.74 31 .770 7067 28.69 .781 0350 28.69 .791 3678 28.71 .801 7107 28.74 i 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 1 34 .771 2231 28.69 .781 5515 28.69 .791 8847 28.71 .802 2271 28.75 35 .771 395* 28.69 .781 7237 28.69 .792 0570 28.71 .802 3996 28.75 36 .771 5673 28.69 .781 8959 28.69 .792 2293 i 28.71 .802 5721 28.75 37 771 7395 28.69 .782 0680 28.70 .792 4016 28.72 .802 7446 28.75 38 .771 9116 28.69 .782 2402 28.70 79 2 5738 28.72 .802 9171 28.75 39 77* 0837 28.69 .782 4124 28.70 .792 7461 28.72 .803 0896 28.75 40 .772 2559 28.69 .782 5845 28.70 79* 9 l8 4 28.72 .803 2621 *8.75 41 .772 4280 28.69 .782 7567 28.70 793 0907 28.72 .803 4346 28.75 f 42 .772 6001 2869 .782 9289 28.70 .793 2630 28.72 .803 6071 28.75 43 .772 7722 28.69 .783 ion 28.70 793 4354 28.72 803 7796 *8-75 44 .772 9444 28.69 .783 2732 28.70 793 6077 28.72 803 9521 28.75 45 773 " 6 5 28.69 783 4454 28.70 .793 7800 28.72 804 1246 *-75 46 .773 2886 28.69 .783 6176 28.70 793 95 2 3 28.72 804 2971 28.75 i 47 773 4607 28.69 .783 7898 28.70 .794 1246 28.72 804 4697 28.75 , 1 48 .773 6329 28.69 .783 9620 28.70 794 *9 6 9 28.72 804 6422 28.76 49 .773 8050 28.69 .784 1342 28.70 794 4 6 93 28.72 804 8147 28.76 50 773 977 1 28.69 784 3 6 4 28.70 794 6 4! 6 28.72 804 9873 28.76 51 774 '493 28.69 .784 4786 28.70 794 8139 28.72 805 1598 28.76 52 .774 3214 28.69 .784 6508 28.70 794 9862 28.72 805 3324 28.76 53 .774 4935 *8 69 784 8230 28.70 795 1586 28.72 805 5049 28.76 54 .774 6657 i 28 69 .784 9952 28.70 795 339 28.72 805 6775 28.76 55 .774 8378 28.69 785 1674 28.70 795 533 28.72 805 8500 28.76 56 .775 0099 28.69 785 3396 28.70 795 6 75 6 28.72 806 0226 28.76 57 775 1821 28.69 .785 5.18 28.70 795 8480 28.72 806 1952 28.76 j 58 59 775 354* 775 5* 6 3 28.69 28.69 .785 6840 785 8562 28.70 28.70 796 0203 796 1927 28.73 28.73 806 3677 806 5403 28.76 28.76 60 775 69 8 5 28.69 786 0284 18.70 796 3650 28.73 806 7129 8. 7 6 TABLE VI, ifot finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. /* 72 73 74 o 75 o logM. Diff. 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 a .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 .838 5635 28.96 5 1.807 5759 28.77 1.817 9410 28.82 1.828 3266 28.88 1.838 7372 28.96 6 .807 7485 28.77 .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 8 39 43 2 3 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 .RIO 1274 28.97 14 .809 1295 28.77 .819 4974 28.83 .829 8865 28.89 .840 3012 28.97 15 16 1.809 3021 .809 4748 28.78 28.78 1.819 6704 .819 8433 28.83 2883 1.830 0599 .830 2332 28.89 28.89 1.840 4751 .840 6489 28.97 28.97 17 .809 6474 28.78 .8 o 0163 28.83 .830 4066 28.90 .840 8227 28.97 j 18 .809 8201 28.78 .8 o 1893 28.83 .830 5800 28.90 .840 9966 28.97 19 .809 9928 28.78 .8 o 3623 28.83 83 7533 28.90 .841 1704 28.98 2O 1.810 1655 28.78 1.8 o 5353 28.83 1.830 9267 28.90 1.841 3443 28.98 21 .810 3381 28.78 .8 o 7083 28.83 .831 loot 28.90 .841 5182 28.98 22 .810 5108 28.78 .8 o 8813 28.84 .831 2735 28.90 .841 6921 28.98 23 .810 6835 28.78 .8 i 0543 28.84 .831 4470 28.90 .841 8659 28.98 24 .810 8562 28.78 .8 i 2273 28.84 .831 6204 28.90 .842 0398 28.98 25 1.811 0289 28.78 1.8 i 4003 28.84' 1.831 7938 28.91 1.842 2138 28.98 26 .811 2016 28.78 8 i 5734 28.84 .831 9672 28.91 .842 3877 28.99 27 .811 3743 28.78 .8 i 7464 28.84 .832 1407 28.91 .842 5616 28.99 28 .811 5470 28.79 .8 i 9194 28.84 .832 3141 28.91 .842 7355 28.99 29 .811 7197 28.79 .8 2 0925 28.84 .832 4876 28. 9 I .842 9095 28.99 30 1.811 8924 28.79 1.8 2 2656 28.84 1.832 6611 28.91 1.843 0834 28.99 31 .812 0652 28.79 .8 2 4386 28.84 .832 8345 28.91 .843 2574 28.99 32 33 34 .812 2379 .812 4106 .812 5834 28.79 28.79 28.79 .8 2 6117 .8 2 7848 .8 2 9578 28.85 28.85 28.85 .833 0080 .833 1815 8 33 355 28.92 28.92 28.92 843 43'3 .843 6053 843 7793 29.00 29.00 29.00 35 1.812 7561 28.79 1.8 3 1309 28.85 1.833 5285 28.92 1-843 9533 29.00 36 .812 9289 28.79 .8 3 3040 28.85 .833 7020 28.92 .844 1273 29.00 37 38 .813 1016 .813 2744 28.79 28.79 .8 3 4771 .8 3 6502 28.85 28.85 833 8755 .834 0491 28.92 28.92 844 3i3 8 44 4753 29.00 29.00 39 .813 4472 28.79 .823 8233 28.85 .834 2226 28.92 8 44 6 494 29.01 40 1.813 6199 28.80 1.823 9965 28.85 1.834 39 61 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 .813 9655 28.80 .824 3427 28.86 .834 7432 2893 .845 1715 29.01 43 .814 1383 28.80 .824 5159 28.86 .834 9168 28.93 .845 3456 29.01 44 .814 3111 28.80 .824 6890 28.86 .835 0904 28.93 .845 5196 29.01 1 45 1.814 4839 28.80 1.824 8622 28.86 1.835 26 4 28.93 1.845 6937 29.01 46 .814 6567 28.80 .825 0353 28.86 835 4376 28.93 .845 8678 29.02 47 .814 8295 28.80 .825 2085 28.86 .835 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 2Q C2 51 .815 5208 28.81 .825 9012 28.87 .836 3056 28.94 .846 7384 20 O2 52 CO .815 6936 28.81 .826 0744 28.87 .836 4792 28.94 .846 9125 8j.7 O867 29.03 *>*> 54 55 .816 0393 .8l6 2121 28.81 .826 4208 28.87 !836 8265 28.94 28.94 28 04. .847 2609 I 8A7 A1CO 29.03 29.03 2.O O1 56 .8l6 3850 28.81 .826 7673 28.87 .837 1739 zo.y^. 28.95 TT/ - TJ.) U .847 6092 i y* u 3 29.03 57 .816 5578 28.8x .826 9405 28.87 837 3475 28.95 847 7834 29.03 58 .8l6 7307 28.8! .827 1137 28.87 .837 5212 28.95 .847 9576 29.03 59 .8l6 9036 28.81 .827 2870 28.87 .837 6949 28.95 .848 1318 29.04 60 .817 0765 28.81 1.827 4^02 28.88 1.837 8686 28.95 1.848 3060 29.04 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V 76 77 78 C 79 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". 848 3060 29.04 838 7769 2914 869 2857 29.25 879 8369 19.37 g 848 4803 29.04 858 9517 29.14 869 4612 29.25 880 0131 29.37 2 3 848 6545 848 8287 29.04 2904 859 1266 859 3H 29.14 29.14 869 6367 869 8122 29.25 29.25 880 1894 880 3656 29.38 29.38 4 849 0030 29.04 859 4763 29.15 869 9878 29.26 880 5419 29.38 5 849 1773 29.04 859 6512 29.15 870 1633 29.26 880 7182 29.38 6 849 35i5 29.05 859 8260 29.15 870 3389 29.26 880 8945 2938 7 849 5158 29.05 860 0009 29.15 870 5144 29.26 88 0708 29.39 8 849 7001 29.05 860 1758 29.15 870 6900 29.26 88 2471 29.39 | 9 849 8744 29.05 860 3507 29.15 870 8656 29.26 88 4235 29.39 1O 850 0487 29.05 860 5256 29.15 871 0412 29.27 .88 5998 29.39 11 850 2231 29.05 860 7006 29.16 871 2168 29.27 88 7762 29.39 ! 12 850 3974 29.06 860 8755 29.16 .871 3924 29.27 88 9526 29.40 ! 13 850 5717 29.06 .861 0505 29.16 871 5681 29.27 .88 1290 29.40 ' 14 850 7461 29.06 .861 2254 29.16 .871 7437 29.28 .882 3054 29.40 15 850 9204 29.06 .861 4004 29.16 .871 9194 29.28 .882 4818 29.40 16 8-1 6 4 * 29.06 -861 5754 29.16 .872 0950 29.28 .882 6582 .882 8347 29.41 17 18 851 4436 29.07 .861 9254 29.17 .872 4464 2928 .883 0112 29.41 19 29.07 .862 1004 29.17 .872 6221 29.29 .883 l8 7 6 29.41 20 .851 7924 29.07 .862 2754 29.17 .872 7979 29.29 .883 3641 29.42 21 .851 9668 29.07 .862 4505 29.17 .872 9736 29.29 .883 54 06 29.42 ' 22 .852 1412 29 07 .862 6255 29.18 873 '493 29.29 .883 7 I 7 I 29.42 23 .852 3157 29.07 .862 8006 29.18 .873 3251 29.29 .883 8 937 29.42 34 .852 4901 29.07 .862 9756 29.18 .873 5008 29.30 .884 C702 29.42 25 26 .852 6646 .852 8391 29.08 29.08 1.863 1507 .863 3258 29.18 29.18 1.873 6766 -873 85*4 29.30 29.30 1.884 2468 .884 4233 29.43 29.43 27 .853 0135 29.08 .863 5009 29.18 .874 oz82 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 31 32 '853 537 .853 7115 .853 8861 29.09 29.09 29.09 1.864 0263 .864 2015 .864 3766 29.19 29.19 29.19 1-874 5557 .874 7316 .874 9074 29.31 29.31 29.31 1.885 1297 .885 3064 .885 4830 29.44 29.44 ! 29.44 , 33 .854 0606 29.09 .864 5518 29.20 .875 0X33 29.31 .885 6597 19-45 34 .854 2351 29.09 .864 7270 29.20 .875 2592 29.32 .885 8364 29.45 35 1.854 4097 29.09 1.864 9022 29.20 1.875 4351 29.32 1.886 0131 19-45 ! 36 37 -854 5843 854 7588 29.10 29.10 .865 0774 .865 2526 29.20 29.20 .875 6m .875 7870 29.32 29.32 .886 1898 .886 3665 29.45 19-45 38 39 .854 9334 .855 1080 29-10 29.10 .865 4278 .865 6030 29.20 29.21 .875 9629 .876 1389 29.32 29.33 .886 5432 .886 7200 29.46 , 29.46 40 41 42 43 44 1.855 2826 -855 457i 855 6319 .855 8065 .855 9811 29.10 29.10 29.11 29.11 29.11 1.865 7783 .865 9530 .866 1288 .866 3041 .866 4794 29.21 29.21 29.21 29.21 29.22 1.876 3148 .876 4908 .876 6668 .876 8428 .877 0188 29.33 29.33 29.33 29.34 1.886 8967 .887 0735 .887 2503 .887 4271 .887 6039 29.46 29.46 : 29.47 29.47 29.47 45 46 47 48 49 1.856 1558 .856 3305 .856 5052 .856 6799 .856 8546 29.11 29.11 29.11 29.12 29.12 1.866 6547 .866 8301 .867 0054 .867 1807 .867 3561 29.22 29.22 29.22 29.22 29.23 1.877 1949 .877 3709 .877 547 .877 7230 .877 8991 29.34 29.34 29.34 19 34 1.887 7807 .887 9576 III ;sj .888 4882 29.47 29.48 19.48 2948 29.48 50 51 52 53 54 1.857 0293 .857 2040 857 3787 857 5534 .857 7282 29.12 29.12 29.12 29.12 29.13 1.867 5314 .867 7068 .867 8822 .868 0576 .868 2330 29.23 29.23 29.23 2923 29.24 1.878 0752 .878 2513 .878 4175 .878 6036 -878 7797 29.35 19-35 29.35 19-35 29.36 i 888 6651 .888 8420 .889 0189 .889 1959 .889 3728 29.48 29.49 29.49 2949 19-49 55 56 57 58 59 1.857 9030 -858 0777 .858 2525 .858 4273 .858 6021 29.13 29.13 29.13 29.13 29.13 1.868 4084 .868 5839 .868 7593 .868 9348 .869 IIO2 29.24 29.24 29.24 29.24 29.25 1.878 9559 .879 1321 .879 3082 .879 4844 .879 6606 29.36 29.36 2936 29.36 29.37 1.889 5498 .889 7268 .889 8038 .890 0808 .890 2578 29.49 29.50 29.50 29.50 29.51 60 1.858 7769 29.14 1.869 2857 29.2.5 1.879 8369 29.37 1.890 4349 i9.1I TABLE VI, For finding the True Anomaly or the Time from the , erihelion in a Parabolic Orbit. V, 80 > 81 o 82 o 83 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". o 1.890 4349 29.51 1.901 0841 29.66 1.911 7893 29.82 1.922 5548 29.99 1 .890 6119 29 51 .901 2621 29.66 .911 9682 29.82 .922 7347 29.99 2 .890 7890 29.51 .901 4400 29.66 .912 1471 29 82 .922 9147 30.00 3 .890 9661 29.51 .901 6180 29 66 .912 3261 2983 9 i 3 947 30.00 4 .891 I 43 z 29.52 .901 7960 29.67 .912 5050 29.83 .923 2747 30.00 5 6 1.891 3203 .891 4974 29.52 29.52 1.901 9740 .902 1521 29.67 29.67 1.912 6840 .912 8630 29.83 29.84 1.923 4548 .923 6348 30.01 30.01 7 .891 6745 29.52 .902 3301 29.67 .913 0420 29.84 .923 8149 30.01 8 .891 8517 29.53 .902 5082 29.68 .913 2211 29.84 .923 9950 30.02 9 .892 0289 29-53 .902 6862 29.68 .913 4 OOI 2984 .924 1751 30.02 10 1.892 2061 29.53 1.902 8643 29.68 I.9I3 5792 29.85 1.924 3552 30.02 11 .892 3833 29-53 .903 0424 29.69 9i3 75 8 3 29.85 924 5354 30.03 12 .892 5605 29.54 .903 2105 29.69 913 9374 29.85 924 7'55 30.03 13 .892 7377 29.54 .903 3987 29.69 .914 1165 29.85 924 8957 30.03 14 .892 9149 29.54 .903 5768 29.69 .914 2956 29.86 .925 0759 30.03 15 1.893 0922 29.54 1-903 755 29.70 1.914 4748 29.86 1.925 2561 30.04 16 .893 2695 ^9-55 .903 9332 29.70 .914 6540 29.86 925 43 6 4 30.04 17 .893 4467 29-55 .904 1114 29.70 .914 8331 29.87 .925 6166 30.04 18 .893 6240 29-55 .904 2896 29.70 .915 0124 29.87 925 7969 30.05 19 .893 8013 29-55 .904 4678 29.71 .915 1916 29-87 .925 9772 30.05 20 1.893 9787 29.56 .904 6461 29.71 1.915 3708 29.87 1.926 1575 30.05 21 .894 1560 29.56 .904 8243 29.71 .915 5501 29.88 .926 3378 30.06 22 894 3334 29.56 .905 0026 29.71 .915 7294 29.88 .926 5182 30.06 23 .894 5108 29.56 .905 1809 29.72 .915 9087 29.88 .926 6986 30.06 24 .894 6882 29-57 95 3592 29.72 .916 0880 29.89 .926 8789 30.07 25 !.8 9 4 8656 29.57 .905 5376 29.72 1.916 2673 29.89 1.927 0593 30.07 26 .895 0430 29-57 .905 7159 29.73 .916 4466 29.89 .927 2398 30.07 27 .895 2204 29.57 .905 8943 29.73 .916 6260 29.90 .927 4202 30.08 28 .895 3979 29.58 .906 0726 29.73 .916 8054 29.90 .927 6007 30.08 29 895 5753 29.58 .906 2510 29.73 .916 9848 29.90 .927 7811 30.08 30 .895 7528 29.58 .906 4294 29.74 1.917 1642 29.90 1.927 9616 30.08 31 .895 9303 29.58 .906 6079 29.74 .917 3436 29-9I .928 1422 30.09 32 .896 1078 29.59 .906 7863 29.74 .917 523! 29.91 .928 3227 30.09 33 .896 2854 2959 .906 9648 29.74 .917 7025 29.91 .928 5032 30.09 34 .896 4628 29.59 .907 1432 29-75 .917 8820 29.92 .928 6838 30.10 35 .896 6404 29.59 .907 3217 29-75 1.918 0615 29.92 1.928 8644 30.10 36 .896 8180 29.60 .907 5002 29.75 .918 2410 29.92 .929 0450 30.10 37 .896 9955 29.60 .907 6787 29.75 .918 4206 29.92 .929 2256 30.11 38 .897 1732 29.60 .907 8573 29.76 .918 6001 29.93 .929 4063 30.11 39 .897 3508 29.60 .908 0358 29.76 .918 7797 2993 .929 5869 30.11 40 .897 5284 29.61 .908 2144 29.76 1.918 9593 29-93 .929 7676 30.12 41 .897 7060 29.61 .908 3930 29.77 .919 1389 29.94 .929 9483 30.12 42 .897 8837 29.61 .908 5716 29.77 .919 3185 29.94 .930 1291 30.12 43 .898 0614 29.61 .908 7502 29.77 .919 4982 29.94 .930 3098 30.13 44 .898 2390 29.62 .908 9288 29.77 .919 6778 29.94 .930 4906 30.13 45 .898 4168 29.62 .909 1075 29.78 1.919 8575 29.95 .930 6713 30.13 46 .898 5945 29.62 .909 2862 29.78 .920 0372 29-95 .930 8521 30.13 47 .898 7722 29.62 .909 4648 29.78 .920 2169 29.95 .931 0330 30.14 48 .898 9500 29.63 .909 6436 29.78 .920 3966 29.96 .931 2138 30.14 49 .899 1277 29.63 .909 8223 29.79 .920 5764 29.96 93 * 394 6 30.14 50 .899 3055 29.63 .910 ooio 29.79 .920 7561 29.96 931 5755 30.15 51 .899 4833 29.63 .910 1798 29.79 .920 9359 29.97 .931 7564 30.15 52 .899 6611 29.64 .910 3585 29.80 .921 1157 29.97 .931 9373 30.11; 53 .899 8389 29.64 91 5373 29.80 .92 2956 29.97 .932 1183 30.1(1 54 .900 0168 29.64 .910 7161 29.80 92 4754 29.98 .932 2992 30.16 55 .900 1946 29.64 .910 8949 29.80 .92 6552 29.98 .932 4802 30.16 56 .900 3725 29.65 .911 0738 29.81 .92 8351 29.98 .932 6612 30.17 57 .900 5504 29.65 .911 2526 29.81 .922 0150 29.98 .932 8422 30.17 58 .900 7283 29-65 .911 4315 29.81 .922 1949 29.99 .933 0232 30.17 59 .900 9062 29.66 .911 6104 29.82 922 3748 29-99 933 2043 30.18 60 .901 0841 29.66 .911 7893 29.82 .922 5548 29.99 933 3853 30.18 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 84 85 86 87 log M. Mff. 1". log M. Diff. 1". logM. DUT. i". logM. DMT. 1*. 1 933 3 8 53 933 5 66 4 30.18 30.18 944 z8 5 6 944 4678 30.38 30.38 .955 2602 955 443 8 30.59 30.60 .966 3140 .966 4990 30.82 30.82 2 933 7475 30.19 944 6502 3-39 955 6274 30.60 .966 6839 30.83 3 933 9287 30.19 944 8 3*5 30-39 .955 8110 30.60 .966 8689 30.83 4 934 1098 30.19 945 OI 4 8 3-39 955 9946 30.61 .967 0539 30.84 5 .934 2910 30.20 .945 1972 30.40 .956 1783 30.61 .967 2389 30.84 6 7 934 47" 934 6 533 30.20 30.20 945 3796 945 5620 30.40 30.40 .956 3619 .956 5456 30.61 30.62 .967 4240 .967 6090 30.84 30.85 8 934 8 34 6 30.21 945 7444 30.41 .956 7294 30.62 .967 7941 30.85 9 935 oi5 8 30.21 945 9269 30.41 .956 9131 30.63 967 9792 30.85 10 935 1 97 I 30.21 .946 1094 30.41 957 0969 30.63 .968 1644 30.86 11 935 37 8 4 30.22 .946 2919 30.42 957 28 o7 30.63 .968 3496 30.86 12 13 14 935 5597 935 741 935 9"3 30.22 30.22 30.22 946 4744 .946 6569 .946 8395 30.42 30.42 30.43 957 4645 957 M3 957 8322 30.64 30.64 30.64 .968 5347 .968 7200 .968 9052 30.87 30.87 30.87 15 .936 1037 30.23 .947 0221 3-43 .958 0160 30.65 .969 0905 30.88 16 .936 2851 30.23 947 *47 30.44 .958 1999 30.65 .969 2757 30.88 18 .936 4665 .936 6479 30.23 30.24 947 3 8 73 947 5699 30.44 30.44 .958 3839 .958 5678 30.66 30.66 .969 4610 .969 6464 30.89 i 30.89 ; 19 .936 8293 30.24 947 75*6 30-45 .958 7518 30.66 .969 8317 30.89 2O .937 0108 30.24 947 9353 30.45 .958 9358 30.67 .970 0171 30.90 21 .937 1922 30.25 .948 1180 3-45 959 "9 8 30.67 .970 2025 30.90 22 23 937 3737 937 5553 30.25 30.25 .948 3007 .948 4834 30.46 30.46 959 33 8 959 4 8 79 30.67 30.68 .970 3879 970 5734 30.91 30.91 24 937 73 68 30.26 .948 6662 30.46 959 6 7 20 30.68 .970 7589 30.91 25 .937 9184 30.26 .948 8490 30-47 1.959 8561 30.69 1.970 9443 30.92 i 26 .938 0999 30.26 949 3 l8 30.47 .960 0402 30.69 .971 1299 30.92 27 .938 2815 30.27 949 2I 4 6 30-47 .960 2243 30.69 971 3'54 3-93 28 29 .938 4632 .938 6448 30.27 30.27 949 3975 .949 5804 30.48 30.48 .960 4085 .960 5927 30.70 30.70 .971 5010 .971 6866 30.93 30.93 30 31 1.938 8264 .939 0081 30.28 30.28 1-949 7633 949 94 62 30.48 30.49 1.960 7769 .960 9612 30.70 30.71 1.971 8722 .972 0578 30-94 30.94 32 .939 1898 30.28 .950 1291 30.49 .961 1454 30.71 .972 2435 395 33 939 37 r 5 30.29 .950 3121 30.50 .961 3297 30.71 .972 4292 30.95 34 939 5533 30.29 95 495 1 30.50 .961 5140 30.72 .972 6149 30.95 35 36 37 38 39 i'939 735 939 9 l68 .940 0986 .940 2804 .940 4623 30.29 30.30 30.30 30.30 30.31 1.950 6781 .950 8611 .951 0441 .951 2272 .951 4103 30.50 30.51 30.51 30.51 30.52 1.961 6983 .961 8827 .962 0671 .962 2515 .962 4359 30.72 30.73 30-73 3-73 30.74 1.972 8006 .972 9864 973 17" 973 35 8 973 543 8 30.96 30.96 30-97 30.97 30.97 40 41 42 43 44 1.940 6441 .940 8260 .941 0079 .941 1898 .941 3717 30.31 30.31 30.32 30.32 30.32 1-95' 5934 .951 7766 95 1 9597 .952 1429 .952 3261 30.52 30.52 3-53 30-53 30-53 1.962 6203 .962 8048 .962 9893 .963 1738 .963 3583 30.74 30-75 30-75 30-75 30.76 1.973 7297 973 9>5 6 974 I01 5 .974 2874 974 4734 30.98 30.98 30.99 30.99 30.99 45 ! 46 , 47 48 49 i-94i 5537 94 1 7357 .941 9177 .942 0997 .942 2817 3-33 3-33 3-34 30.34 30.34 1.952 5093 .952 6925 .952 8758 953 059 1 953 2 4 2 4 30.54 30.54 30-55 30-55 30-55 1.963 5429 .963 7275 .963 9121 .964 0967 .964 2814 30.76 3-77 30.77 30.77 30.78 1.974 6593 974 8 454 975 0314 975 2I 74 .9-75 4035 31.00 31.00 31.01 31.01 31.01 50 51 52 53 54 1.942 4638 .942 6459 .942 8280 .943 oioi 943 *9 2 3 30.35 30.35 30-35 30.36 30.36 J-953 4 2 57 .953 6091 953 79 2 4 953 975 8 954 159* 30.56 30.56 30.56 30-57 30-57 1.964 4660 .964 6507 .964 8354 .965 O2O2 .965 2050 30.78 30.78 30.79 30.79 30.80 1.975 5896 975 7757 .975 9619 .976 1481 976 3343 31.02 31.02 31.03 31.03 31.04 55 56 57 58 59 1-943 3744 943 55 66 943 73 88 943 9 211 .944 1033 30.36 30.37 30.37 30.37 30.38 1.954 3427 954 5* 62 954 7090 .954 8 93' 955 7 66 30-57 30.58 30-59 30.59 1.965 3897 .965 5746 9 6 5 7594 .965 9442 .966 1291 30.80 30.80 30.81 30.8, 30.81 1.976 5205 .976 7067 .976 8930 977 0793 .977 2656 31.04 31.04 31.05 31.05 31.06 60 1.944 2856 j 30.38 1.955 2602 30.59 1.966 3140 30.82 1-977 45 ao 31.06 TABLE VI For finding the True Anomaly or the Time from the Perihelion in a Parabolic Oibit. v 88 o 89 o 90 o 91 O logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". 1.977 4520 31.06 1.988 6789 31-31 2.OOO OOOO 31-58 2. Oil 4203 31.8 7 1 977 6383 31.06 .988 8668 31.32 .OOO 1895 31-59 .on 6115 31.87 2 977 8247 31.07 .989 0548 31.32 .OOO 3790 31-59 .on 8027 31.88 3 .978 0112 31.07 .989 2427 31-33 .000 5686 31.60 .on 9940 31.88 4 .978 1976 31.08 .989 4307 31-33 .000 7582 31.60 .012 1853 31.89 5 1.978 3841 31.08 1.989 6187 3'-34 2. OOO 9478 31.60 2.OI2 3766 31.89 6 .978 5706 31.08 .989 8067 3'-34 .001 1375 31.61 .012 5680 31.89 7 .978 7571 31.09 .989 9948 31-34 .001 3272 31.61 .012 7594 31.90 8 .978 9436 31.09 .990 1829 31-35 .001 5169 31.62 .012 9508 31.90 9 979 J 3* 31.10 .990 3710 31-35 .001 7066 31.62 .013 1422 3J-9 1 1O 1.979 3168 31.10 1.990 5591 31.36 2.OOI 8963 31.63 2.013 3337 31.91 11 979 534 31.11 .990 7473 31.36 .002 O86l 31.63 .013 5252 31.92 12 .979 6901 3 I.II .990 9355 31-37 .002 2759 31.64 .013 7167 31.92 13 .979 8768 31.11 .991 1237 31 37 .002 4658 31.64 .013 9083 3'-93 14 .980 0635 31.12 .991 3119 31.38 .002 6557 31.65 .014 0999 31.93 15 1.980 2502 31.12 1.991 5002 31.38 2.002 8456 31.65 2.014 2915 3'-94 16 .980 4369 3I-I3 .991 6885 3i-38 .003 0355 3 t.66 .014 4831 3'-94 17 .980 6237 3I-I3 .991 8768 3'-39 .003 2254 31.66 .014 6748 3'-95 18 .980 8105 3I-I3 .992 0651 31.39 .003 4154 31.67 .014 8665 3 J -95 19 .980 9973 31.14 .992 2535 31.40 .003 6054 31.67 .015 0582 31.96 20 1.981 1842 31.14 1.992 4419 31.40 2.003 7955 31.68 2.015 2 5 31.96 21 .981 3710 3I-IS .992 6304 31.41 .003 9855 31.68 .015 4418 31.97 22 .981 5579 31.15 .992 8188 31.41 .004 1756 31.68 .015 6336 3 J -97 23 .981 7449 31.16 993 73 31.42 .004 3658 31.69 .015 8255 31.98 24 .981 9318 31.16 993 '958 31.42 .004 5559 31.69 .016 0174 31.98 25 1.982 1188 31.16 1-993 3843 31.42 2.004 7461 31.70 2.016 2093 31.99 26 .982 3058 31.17 993 5729 31-43 .004 9363 31.70 .016 4012 3 J -99 27 28 .982 4928 .982 6798 3 3 S 993 7 6l 5 993 95 01 31-43 Ji-44 .005 1265 .005 3168 3I-7I 31.71 .016 5932 .016 7852 3*.* 32.00 29 .982 8669 3 I.l8 994 i3 8 7 31.44 .005 5071 31.72 .016 9772 32.01 30 1.983 0540 31.18 1-994 3 2 74 3i-45 2.005 6 974 31.72 2.017 l( >93 32.01 31 .983 2411 31.19 994 5i6i 31-45 .005 8878 31.73 .017 3614 32.02 32 .983 4283 31.19 994 7048 31.46 .006 0781 3-73 017 5535 32.02 33 .983 6155 31.20 994 8936 31.46 .006 2685 3-74 .017 7456 32.03 34 .983 8027 31.20 995 0823 31.46 .006 4590 31-74 .017 9378 32.03 35 1.983 9899 31.21 1.995 2711 3i-47 2.006 6494 31-75 2.018 1300 32.04 36 .984 1772 31.21 995 4 6o 3i-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 8377 31.48 .007 22IO 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 3i-49 2.007 6022 3-77 2.019 0915 32.06 41 .985 1138 3 J - 2 3 .996 4045 31.50 .007 7928 31.77 .019 2839 32.07 42 .985 3012 31.24 .996 5935 31.50 .007 9835 31.78 .019 4763 32.07 43 .985 4886 31.24 .996 7825 31.51 .OO8 1742 31.78 .019 6688 32.08 44 .985 6761 31.24 .996 9716 31.51 .OO8 3649 31.79 .019 8613 32.08 45 1.985 8636 3J-25 1.997 1606 31.51 2.008 5556 3 J -79 2.O2O 0538 32.09 46 .986 0511 31-25 997 3497 31.52 .008 7464 31.80 .O2O 2463 32.09 47 .986 2386 31.26 997 5389 3M2 .008 9372 31.80 .O2O 4389 3210 1 48 .986 4262 31.26 .997 7280 3M3 .009 J28o 31.81 .O2O 6315 32.10 49 .986 6138 31.27 997 9'72 3i-53 .009 3189 31.81 .020 8241 32.11 50 1.986 8014 31.27 1.998 1064 3 x -54 2.009 598 31.82' 2.021 Ol68 32.11 51 .986 9890 31.28 .998 2956 3M4 .009 7007 31.82 .021 2095 32.12 ! 52 .987 1767 31.28 .998 4849 31-55 .009 8917 31.83 .021 4022 32.12 ! 53 .987 3644 31.28 .998 6742 31-55 .010 0826 3i-83 .021 5949 32-13 54 .987 5521 31.29 .998 8635 31.56 .010 2736 31.84 .021 7877 32.13 '5 1.987 7398 31.29 i-999 529 31.56 2.0 4647 31.84 2.021 9805 32.14 56 .987 9276 31.30 .999 2422 31.56 .0 o 6557 31.85 .022 1734 32.14 57 .988 1154 31.30 999 43 l6 31-57 .0 8468 31-85 .022 3662 32.15 58 .988 3032 31.31 .999 6211 31-57 .0 I 0380 31.86 .022 5591 32.15 59 .988 4911 31.31 999 8105 3I-58 .0 I 2291 31.86 .022 7521 3 2.!6 60 1.988 6789 31.31 2.OOO OOOO 31.58 2.OII 4203 31-87 2.O22 9450 32.16 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. i". logM. Diff. 1". logM. Diff. 1". IgM. Diff. 1". 2. 022 9450 .023 I 3 80 .023 33 II .023 5241 .323 7172 32.16 32.17 32.17 32.18 32.18 2-34 5797 34 7745 .034 9694 .035 1644 35 3593 32.48 32.48 32-49 32.49 32.50 2.046 3296 .046 5264 .046 7233 .046 9202 .047 1172 32.80 32-81 32.82 32.82 32.83 2.058 2005 .058 3994 .058 5983 .058 7973 .058 9963 3315 ' 33-15 , 33.16 33.16 33-17 5 2.023 9 I0 3 32.19 2-035 5543 32.50 2.047 3141 32.83 2.059 1953 33.18 6 .024 1035 32.19 35 7494 32.51 .047 5111 32-84 .059 3944 33.18 8 9 .024 2967 .024 4899 .024 6831 32.20 32.20 32.21 .035 9444 .036 1395 .036 3347 32.51 32-52 32.52 .047 7082 .047 9053 .048 1024 32.84 32.85 32.85 59 5935 .059 7927 .059 9919 33.19 33-'9 ; 33.20 10 2.024 8764 32.21 2.036 5298 32.53 2.048 2995 32.86 2.060 1911 33-21 11 12 13 .025 0697 .025 2630 .025 4564 32.22 32.22 32.23 .036 7250 .036 9202 .037 1155 32-53 32.54 32-54 .048 4967 .048 6939 .048 8912 32.87 32.87 32.88 .060 3904 .060 5897 .060 7890 33-21 33-22 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 17 18 19 .026 0367 .026 2301 .026 4236 .026 6172 32.24 32-25 32.26 32.26 37 7015 .037 8969 .038 0923 .038 2877 32.56 32.57 32.57 32.58 .049 483! .049 6805 .049 8879 .050 0753 32-89 32.90 32.90 32.91 .061 3872 .061 5867 .061 7862 .061 9857 33-24 ' 3325 3325 33.26 20 21 22 2.026 8108 .027 0044 .027 1980 32.27 32-27 32.28 2.038 4832 .038 6787 .038 8743 32.58 32.59 32.59 2.050 2728 .050 4703 .050 6679 32.92 32-92 3 2 -93 2.062 1853 .062 3849 .062 5846 33.27 : 33.27 ' 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 .062 9840 33.29 25 2.027 7791 32.29 2.039 46" 32.61 2.051 2608 32-95 2.063 1837 33.30 26 27 28 .027 9729 .028 1667 .028 3605 3230 32.30 32-31 .039 6568 .039 8525 .040 0482 32.62 32.62 32.63 .051 4585 .051 6562 .051 8539 32-95 32.96 32-96 .063 3835 .063 5833 .063 7832 33.30 33-31 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 a 2.040 4399 32.64 2.052 2496 32-97 2.064 1831 33-33 ! 31 .028 9422 32.32 -040 6357 32.64 .052 4474 32.98 .064 3830 33-33 32 .029 1361 32.33 .040 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 4 !2 33.00 .064 9832 33-35 35 2.029 7 J 82 32.34 2.041 4194 32.67 2053 2392 33.00 2-065 1833 33.36 36 37 .029 9123 .030 1064 32-35 32-35 .041 6154 .041 8114 32-67 32.68 053 4372 53 6333 33-oi 33.01 065 3834 .065 5836 33.36 33 37 ! 38 .030 3005 32.36 .042 0075 32.68 053 8334 33-02 .065 7839 33 37 39 .030 4947 32.36 .042 2036 32.69 .054 0315 33.03 .065 9841 33-38 40 2.030 6889 32.37 2.042 3998 32.69 2.054 "97 33-03 2.066 1844 33-39 41 .030 8831 32-37 .042 5960 32.70 .054 4279 33.04 .066 3847 33-39 i 42 .031 0774 32.38 .042 7922 32-70 .054 6262 33-4 .066 5851 33.40 43 44 .031 2717 .031 4660 32-39 32.39 .042 9834 .043 1847 32.71 32.71 .054 8244 .055 0227 33.05 33.05 .066 7855 .066 9860 33.40 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 32.4 043 5773 32.73 .055 4195 33-07 .067 3870 33-42 47 .032 0492 32-41 .043 7737 32.73 .055 6179 33-7 .067 5875 33-43 48 .032 2437 32-41 .043 9701 32-74 .055 8l67 33.08 .067 7881 33-43 49 .032 438-2 32.42 .044 1665 32.74 .056 0148 33.08 .067 9887 33-44 50 2.032 6327 32.42 2.044 3630 3 2 -75 2.056 2133 33.09 2.068 1894 33-45 51 .032 8272 32-43 044 5595 32.75 .056 4119 33.10 .068 3901 33-45 52 53 .033 0218 .033 2164 32-43 32.44 44 756i .044 9526 32.76 32.76 .056 6105 .056 8091 33-io 33-ii .068 5908 .068 7916 33.46 33-47 54 033 4"i .045 1492 32.77 .057 0078 33.11 .068 9924 33-47 55 2.033 6058 32.45 2-045 3459 32.78 .057 2065 33.12 .069 1933 33-48 56 .033 8005 32-45 045 5426 32.78 .057 4052 33-12 .069 3942 33-48 57 58 .033 9952 .034 1900 32.46 32-47 .045 7393 .045 9360 32.79 32.79 .057 6040 .057 8028 33-13 33-H .069 5951 .069 7960 33-49 33-50 59 .034 3848 3M7 .046 1328 32.80 .058 00l6 33-14 .069 9970 33.50 60 2-034 5797 32.48 .046 3296 32.80 .058 2005 33-15 .070 1980 33-51 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. *v 96 o 97 o 98 99 o logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". 2.070 1980 33-51 2.082 3282 33-88 2.094 5971 34.28 2.107 OIC 9 34-69 1 .070 3991 33-5' .082 5316 33.89 .094 8028 34.29 .107 2190 34-70 2 .070 6002 33-5* .082 7349 33.90 .095 0085 34.29 .107 4272 34-70 3 .070 8014 33-53 .082 9383 33-9 95 2I 43 34.30 .107 6355 34-71 4 .071 0025 33-53 .083 1418 33.91 .095 4201 34-31 .107 8437 34.72 : 5 2.071 2037 33-54 *-8 3 3453 33.92 2.095 6260 34-3' 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 33-56 .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 o68l 3436 .109 5116 34-77 13 .072 8148 33-59 .084 9745 33-97 .097 2742 34-37 .109 7202 34.78 14 .073 0163 3359 .085 1783 33-98 .097 4804 34-37 .109 9289 34-79 15 2.073 "79 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 .no 5553 34.81 18 .073 8229 33.62 .085 9941 34.00 .098 3057 34-40 .no 7642 34.82 19 .074 0246 33-63 .086 1981 34.01 .098 5121 34-41 .no 9731 34-82 20 2.074 2264 33.63 2086 4021 34-oi 2.098 7186 34-4 1 2. Ill l82I 34-83 21 .074 4282 33- 6 4 .086 6062 34-02 .098 9251 34.42 .III 3911 34-84 22 .074 6301 33 64 .086 8104 34-03 .099 1316 34-43 .III 6001 34-85 23 .074 8320 33- 6 5 .087 0146 34-03 .099 3382 34-43 .III 8092 34.85 24 .075 0339 33.66 .087 2188 34-04 99 5449 34-44 .112 0184 34.86 25 2.075 *358 33.66 2.087 4231 34-05 2-099 75'5 34-45 2. 1 12 2275 34-87 26 .075 4378 33.67 .087 6274 34-05 .099 9582 34-45 .112 4368 34.87 27 .075 6399 33.67 .087 8317 34.06 .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 1.076 2462 3369 2.088 4449 34.08 2. 00 7855 34-48 2.II3 274 1 34.90 31 .076 4484 33.70 .088 6494 34-09 . oo 9924 34-49 "3 4835 34-91 32 .076 6507 33.71 .088 8540 34.09 J 993 34.50 .113 6930 34-92 33 .076 8529 33.71 .089 0586 34.10 . o 4063 34-50 .113 9025 34-92 34 .077 0552 33-7* .089 2632 34.11 .0 6134 34-51 .114 II2I 34-93 35 2.077 2 575 33-73 2.089 4678 34 ii 2. O 8204 34-52 2.114 3 ai 7 34-94 36 77 4599 33-73 .089 6725 34.12 . o 0276 34-52 .114 5313 34-95 37 .077 6623 33-74 .089 8772 34-12 2347 34-53 .114 7410 34-95 38 .077 8647 33-74 .090 0820 34-13 . o 4419 34-54 .114 9508 34-96 39 .078 0672 33-75 .090 2868 34-14 . o 6492 34-54 .115 1605 34-97 40 2.078 2697 33-76 2.090 4917 34-15 2. 02 8564 34-55 2.115 3704 34-97 41 .078 4723 33.76 .090 6966 34-15 . 03 0638 34-56 .115 5802 34.98 42 .078 6749 33-77 .090 9015 34.16 . 03 2711 34-56 .115 7901 34-99 43 .078 8775 33-78 .091 1065 34-17 3 4785 34-57 .1 16 oooi 35.00 44 .079 0802 33.78 .091 3115 34-17 .103 6860 34.58 .Il6 2101 35.00 45 2.079 2829 33-79 2.091 5165 34.18 2. 03 8935 34-59 2.116 4201 35-oi 46 .079 4857 33.80 .091 7216 34-19 . 04 1010 34-59 .116 6301 35-02 47 .079 6885 33.80 .091 9268 34-19 .104 3086 34.60 .116 8403 35.02 48 .079 8913 33-8i .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 z.o8o 2971 33.82 2.092 5424 34.21 2.104 93 1 6 34.62 2.117 4710 35-05 , 51 .080 5000 3383 .092 7477 34.22 .105 1393 3463 .117 6813 35-05 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 .118 1020 35-07 54 .081 1091 33-85 .093 3638 34-24 .105 7628 34.65 .118 3124 35.08 55 2.081 3122 33-85 2.093 5692 34-25 2.105 9707 34.66 2.118 5229 35.08 56 .081 5153 33.86 .093 7747 34-25 .106 1786 34-66 "8 7334 35-09 57 .081 7185 33-87 .093 9803 34.26 .106 3866 34- 6 7 .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 35.11 60 2.082 3282 33-88 2.094 597 1 34.28 2.107 0109 34.69 2.119 5759 35.11 590 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit V. 101 102 103 logM. Diff. 1" log M. Diff.l" logM. Diff. 1" logM. DiffV. o 1 2 3 4 2.119 5759 .119 7867 .119 9974 .12 2083 .12 4191 35-12 35-13 ! 35-13 35-H 35-iS 2.132 2989 .132 5123 .132 7258 .132 9393 .133 1529 35-57 35-57 35-58 35-59 35.60 2.145 866 .145 4028 .145 6191 .145 8354 .146 0518 36.03 36.04 36.05 36.06 36.07 2.158 2460 .158 4652 .158 6844 .158 9036 .159 1229 36.52 36.53 3654 3655 36.55 5 6 7 8 2 2 6301 . 2 8410 . 2 0520 . 2 2630 35.16 35.16 35-17 35.18 2.133 3665 .133 5802 '33 7939 .134 0076 35.61 35.61 35.62 35-63 2.146 2682 .146 4847 .146 7012 .146 9178 36.07 36.08 36.09 ! 36.10 2.159 3423 .159 5617 .159 7811 .160 0006 36.56 36.57 36.58 76 co ' 9 2 4741 35-19 .134 2214 35-64 H7 1344 36-11 .l6o 2202 ' 36.60 10 212 6853 35-19 2.134 4352 35-64 2.147 35 10 36-11 ~i6o 4398 36.60 11 .12 8965 35.20 .134 6491 35.65 147 5 b 77 36.12 .l6o 6594 ' 36.61 12 13 14 .12 1077 .12 3190 .12 5303 35 21 35-21 35.22 .134 8631 35.66 .135 0770 35.67 .135 2910 j 35.67 .147 7845 1 36-13 .148 0013 ! 36-14 .148 2182 1 36.15 .l6o 8791 ; 36.62 .l6l 0989 36.63 .l6l 3187 36.64 I ~) 2.12 7416 35-23 2-135 5051 i 35-68 2.148 4351 1 36.15 2.161 5385 36.65 16 IT : 18 .12 9530 .123 1644 .123 3759 35-24 35-24 35-25 .135 7192 135 9334 .136 1476 ; 35.69 35.70 35-71 .148 6520 36.16 .148 8690 | 36.17 .149 0861 36.18 .161 7584 ! 36.65 .161 9784 36.66 .162 1984 36.67 19 .123 5875 35.26 .136 3619 .149 3032 36.19 .162 4185 36.68 : 20 2.123 7990 3527 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 85^7 36.70 22 .124 2223 3528 .137 0049 35-74 49 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 35-30 .137 4338 3575 .150 3893 36.23 l6 3 5>95 36.72 25 26 2.124 8576 .125 0694 35-30 35-3' 2 137 6484 .137 8630 35-76 35-77 2.150 6067 I 36.23 .150 8242 | 36.24 2.163 T393 36 73 .163 9602 36.74 27 .125 2813 35-32 .138 Z--- ;; -- .151 0417 36.25 .164 1807 36.74 : 28 .125 4933 35 33 .138 2922 35.78 .151 2592 j 36.26 .164 4012 36.75 , 29 .125 7052 35-33 .138 5070 35 79 .151 4-68 : 36.27 .164 6218 36.76 30 .125 9173 35-34 2.138 7217 35.80 2.151 6944 36.28 2.164 8424 36.77 1 31 .126 1293 35-35 .138 9365 35-81 .151 9121 36.28 .165 0630 36.78 32 .126 3414 35-35 .139 1514 35.81 .152 1298 ; 36.29 .165 2837 ! 36.79 i 33 34 .126 5536 .126 7658 35-36 3537 .139 3663 .139 5813 35-82 3583 .152 3476 i 36.30 .152 5654 36.31 '65 5045 | 36.80 .165 7253 36.81 35 .126 9780 35-38 .139 7963 3 -"* 4 .152 7833 ; 36.32 .165 9462 36.81 36 .127 1903 35-39 .140 0113 .153 OOI2 36.32 .166 1671 36.82 37 38 39 .127 4027 .127 6151 .127 8275 35-39 35-4 .140 2264 .140 4415 .140 6567 114 35.^6 35-87 .153 2I 9 2 3633 153 43/2 ! 36-34 .153 6552 36.35 .166 3881 ; 36.83 .166 6091 ! 36.84 .166 8301 36.85 40 128 0400 3542 .140 8720 35.88 -'53 8734 1 36-35 167 0513 36.86 1 41 128 2525 3542 1^1 0871 35.88 .154 0915 i 36.36 .167 2724 36.87 42 128 4650 35-43 .141 3 025 35.89 .154 3097 36 37 167 4936 36.87 i 43 128 6776 35-44 .141 5180 35.90 154 5280 36.38 167 7149 i 36.88 44 128 8903 35-45 Hi 7334 35-9 1 154 7463 36-39 167 9362 36.89 45 129 1030 35-45 141 9489 35-92 154 9647 36.40 i 68 1576 36.90 46 129 3157 35-46 142 1644 35-92 155 1831 36.41 168 3790 1 36.91 47 129 5285 35-47 142 3799 3593 155 4015 36.41 168 6005 i 36.92 ; 48 129 7414 35.48 142 5955 35-94 155 6200 36-42 168 8220 36.93 49 129 9542 35.48 142 8112 35-95 155 8386 36-43 169 0436 36.93 50 130 1672 35-49 143 0269 3596 156 0572 36.44 169 2652 36.94 51 ! 52 53 130 3801 130 5931 130 8062 35-50 35-5 1 143 2427 143 4585 '43 6743 35.96 35 97 35.98 156 2759 156 4946 156 7133 36.45 36.46 36.46 169 4869 169 7087 169 9304 36.96 3697 54 131 0193 35-5* 143 8902 35-99 156 9321 3 6 .47 170 1523 36.98 55 131 2325 35-53 144 1062 36.00 157 1510 36.48 170 3742 36.99 56 *3* 4457 35-54 144 3222 36.00 157 3699 36.49 170 5961 36-99 57 131 6589 35-54 144 5382 36.01 157 5889 36-50 170 8181 37-00 58 131 8722 35.55 144 7543 36-02 157 8079 36.50 171 0401 37-oi 59 132 0855 35.56 144 9704 36-03 158 0269 36.51 171 2622 37.01 60 132 2989 35-57 145 1866 36.03 158 2460 36.52 i?' 4844 3703 591 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 104 105 106 107 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". 2.171 4844 37-03 2.184 992 37.56 2.198 5282 38.11 2.212 3493 38.68 1 .171 7066 37.04 .185 1346 37-57 .198 7568 38.12 .212 5814 38.69 2 .171 9288 37-05 .185 3600 37-57 .198 9856 38.13 .212 8136 38.70 3 .172 1511 37-05 .185 5855 37.58 .199 2144 38.14 .213 0458 38.71 4 .172 3735 37.06 .185 8110 37-59 .199 4432 38.14 .213 2781 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 37-61 .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 2075 9 .173 4860 37.11 .186 9395 37.64 .200 5882 38.19 .214 4404 3^77 10 2.173 7087 37.12 2.187 1653 37.65 2.2OO 8174 38.20 2.214 6730 38.78 11 J73 93 J 4 37-12 .187 3912 37.66 .2OI 0467 38.21 .214 9057 38.79 12 .174 1542 37-13 .187 6172 37.67 .201 2760 38.22 .215 1385 38.80 13 .174 3770 37.14 .187 8432 37-67 .201 5053 38.23 215 37"3 38.81 14 174 5999 37.I5 .188 0693 37-68 .201 7347 38.24 .215 6042 38.82 15 16 2.174 8228 .175 0458 37.16 37-17 2.188 2954 .188 5216 37.69 37.70 2.2OI 9642 .202 1937 38.25 38.26 2.215 8371 .216 0701 38.83 38-84 17 .175 2688 37.18 .188 7478 37-71 .202 4233 38.27 .216 3032 38.85 18 .175 4919 37-i8 .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 II2 3 38.30 2.217 002 7 38.88 21 .176 1615 37-2i 189 6533 37-75 .203 3421 38.31 .217 2360 38-89 22 .176 3848 37-22 .189 8798 37-76 .203 5720 38.31 .217 4693 38-90 ! 23 .176 6081 37-23 .190 1064 37-77 .203 8019 38.32 .217 7027 38-91 24 .176 8315 37.24 .190 3330 37-77 .204 0319 38.33 .217 9362 38.92 25 2.177 55 37.25 2.190 5597 37.78 2.204 2619 38.34 2.218 1697 38-93 ! 26 .177 2785 37.25 .190 7864 37-79 .204 4920 38.35 .218 4033 38-94 27 .177 5020 37.26 .191 0132 37.80 .204 7222 38.36 .218 6369 38.95 28 .177 7256 37-27 .191 2401 37-8i .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 2.205 AI29 38.39 2.219 3382 38.98 31 .178 3968 3730 .191 9209 37.84 205 6433 38.40 .219 5721 38.99 32 .178 6206 37.31 .192 1480 .205 8737 38.41 .219 8061 39.00 33 34 .178 8445 .179 0684 37-32 37-33 .192 3751 .192 6023 37-87 .206 1042 .206 3348 38.42 38.43 .220 0401 .220 2741 39.01 39.02 35 2 179 2924 37-33 2.192 8295 37-88 2.206 5654 38.44 a.22o 5082 39.03 36 .179 5164 37-34 .193 0568 37-88 .206 7961 38.45 .220 7424 39.04 37 .179 7405 37-35 .193 2841 37-89 .207 0268 38.46 .220 9767 39.05 38 .179 9646 .193 5115 37.90 .207 2575 38.47 .221 2110 39.06 39 .l8o 1888 37-37 .193 7389 37.91 .20 7 4884 38.48 .221 4453 39-07 40 2.l8o 4131 37.38 2.193 9664 37-92 2.207 7193 38.49 2.221 6797 39.08 41 .180 6374 37-39 .194 1940 37-93 .207 9502 38.50 .221 9142 39-9 42 .l8o 8617 37-4 .194 4216 37-94 .208 1812 38.51 .222 1488 39.10 43 .l8l o86l 37-41 .194 6493 37-95 .208 4123 38.52 .222 3834 39- 11 44 .l8l 3106 37-41 .194 8770 .208 6434 38.53 .222 6l8o 39.12 45 2.181 5351 37.42 2.195 1048 37-97 2.208 8746 38-54 2.222 8528 39- r 3 46 .181 7597 37-43 .195 3326 37-98 .209 1058 38.54 .223 0876 39-'4 47 .181 9843 37-44 .195 5605 37-99 .209 3371 .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 39.17 50 2.182 6584 37-47 2.196 2445 38.01 2.210 0314 38.58 2.224 0273 39'8 51 .182 8833 37-48 .196 4726 38.02 .210 2629 38.59 .224 2624 39-J9 52 .183 1082 37-49 .196 7008 38.03 .210 4945 38.60 .224 4975 39.20 53 .183 3331 37-49 .196 9290 38.04 .210 7261 38.61 .224 7327 39-21 54 .183 5581 37.50 197 1573 38.05 .210 9578 38.62 .224 9680 39.22 55 2.183 7831 37-51 2.197 3856 38.06 2. 21 I 1896 38.63 2.225 2033 3923 56 .184 0082 37-52 .197 6140 38.07 .211 A2I 4 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.l". logM. Diff. i". logM. Diff. 1". logM. j Diff.l". ! o 1 2 3 4 2.226 3808 .226 6165 .226 8523 .227 0881 .227 3240 i 39- 2 9 39.30 39-3 1 39-32 2.240 6314 .240 8708 .241 1103 .241 3498 .241 5894 39.90 39-91 39-92 3993 39-94 2.255 I0 99 255 3532 255 59 6 5 .255 8399 .256 0834 40.54 40.55 40.56 40.58 40.59 2.269 8255 .270 0728 -270 3202 .270 5676 .270 8152 41.11 41.23 41.24 41.25 41.26 5 6 7 8 9 2-227 5599 .227 7959 .228 0320 .228 2681 .228 5043 39-33 39-34 39-35 39.36 39-37 2.241 8291 .242 0688 .242 3086 .242 5485 .242 7884 39-95 39.96 39-97 39-98 39-99 2.256 3270 .256 5706 .256 8143 .257 0580 .257 3019 40.60 40.61 40.62 40.63 40.64 1.271 0628 41.27 .271 3104 i 41.28 .271 5582 ; 41.29 .271 8060 : 41.30 .272 0538 j 41.32 10 11 12 13 14 2.228 7405 .228 9768 .229 2131 .229 4496 .229 6861 39.38 39-39 39.40 39.42 2.243 0284 .243 2685 .243 5086 .243 7488 .243 9890 40.00 40.01 40.02 40.03 40.05 2.257 5458 .257 7897 .258 0337 .258 2778 -258 5220 40.65 40.66 40.68 40.69 40.70 2.272 3 oi8 i 41.33 .272 5498 41.34 .272 7979 41.35 .273 0460 41.36 .273 2942 | 41.38 15 16 17 18 19 2.229 9226 .230 1592 23 3959 .230 6326 .230 8694 39-43 39-44 39-45 39.46 39-47 2.244 2293 .244 4697 244 7ioi .244 9506 .245 1912 40.06 40.07 40.08 40.09 40.10 2.258 7662 .259 0105 .259 2548 .259 4992 2 59 7437 40.71 40.72 40.73 40.74 40-75 2-273 5425 41-39 2 73 799 4'-4 .274 0393 | 41.41 .274 2878 41.42 .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 22 23 .231 3432 .231 5802 .231 8172 3949 39-5 39-51 245 6 7 2 5 .245 9132 .246 1541 40.12 40.13 40.14 .260 2329 40.78 .260 4776 40.79 .260 7223 40.80 - 2 75 337 41-46 .275 2825 41.47 .275 5313 41.48 24 .232 0543 3952 246 3949 40.15 .260 9671 40.81 .275 7802 , 41.49 25 26 2.232 2915 .232 5287 39-53 39-54 2.246 6359 .246 8769 40.16 40.17 2.26l 2120 40.82 .26l 4570 40.83 2.276 0292 41.50 .276 2783 | 41.51 27 .232 7660 39-55 .247 1180 40.18 .26l 7020 40.84 .276 5274 ; 41.53 28 .233 0033 39-56 247 359' 40.19 .261 9471 40.85 .276 7766 ; 41.54 29 .233 2407 39-57 .247 6003 40.21 .262 1922 40.86 .277 C2j8 ; 41.55 30 2.233 4782 39.58 2.247 8416 40.22 2.262 4374 40.88 2.277 2752 41.56 31 32 .233 7157 2 33 9533 39-59 39.60 .248 0829 .248 3243 40.23 40.24 .262 6*27 .262 9281 40.89 40.90 .277 5246 i 41.57 .277 7740 41.58 33 34 .234 1910 .234 4287 39.61 39-63 .248 5658 .248 8073 40.25 40.26 .263 1735 .263 4190 40.91 40.92 .278 0236 .278 2732 41.60 4 I.6l 35 2.234 6665 39.64 2.249 0489 40.27 2.263 6645 40.93 2.278 5229 41.62 36 .234 9043 39.65 .249 2906 40.28 .263 9102 40.94 .278 7726 41.63 37 .235 1422 39.66 249 53 2 3 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 .279 7723 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 39.72 .250 9840 40.36 .265 6314 | 41.02 .280 5228 41.71 44 .236 8093 39-73 .251 2262 40 37 .265 8776 j 41.03 .280 7731 41.72 45 2.237 0478 39-74 2.251 4684 40.38 2.266 1238 1 41.04 .281 0235 41-74 46 .237 2862 39-75 .251 7107 40.39 .266 3701 41.06 .28l 2740 41-75 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 0020 39.78 .252 4380 40.42 .267 1094 41.09 .282 0258 4 1. 7 8 50 .238 2407 3979 2.252 6806 40.43 2.267 3560 41.10 .282 276^ 41.80 51 .238 4795 39.80 .252 9232 40.44 .267 6026 41.11 .282 5273 4 I.8l 52 .238 7284 39.81 253 l6 59 40.46 .267 8493 41.12 .282 7782 4 1.82 , 53 .238 9573 39.82 .253 4087 40.47 .268 0961 41.13 .283 0291 4183 54 .239 1962 39.83 253 6515 40.48 .268 3430 41.15 .283 2801 41.84 55 239 4353 39.84 .253 8944 40.49 .268 5899 41.16 .283 S3 I2 41.85 56* .239 6744 39.86 .254 1374 40.50 .268 8369 41.17 .283 7 82 4 41.87 57 239 9 2 35 39-87 .254 3X04 40.51 .269 0839 41.18 .284 0336 41.88 58 .240 1528 39.88 .254 6235 40.52 .269 3310 41.19 284 28 49 41.8 9 59 .240 3921 39.89 .254 8666 40.53 .269 5782 41.20 .284 5363 41.90 6O .240 6314 39.90 255 i99 40.54 .269 8255 41.21 284 7878 41.91 TABLE VI. For 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". O' 2.284 7878 41.91 2.300 0067 42.64 2.315 4927 43.40 2.331 2564 4418 1 .285 0393 41.93 .300 2626 42.65 .315 7531 43.41 .331 5216 44.20 a .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 .332 3175 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 .301 5431 42.72 .317 0564 43-47 .332 8485 44.26 7 .286 5499 42.00 .301 7995 42-73 3*7 3 J 73 43-49 333 "4i 44.28 8 .286 8019 42.01 .302 0559 42.74 .317 5782 43.50 333 3799 44.29 9 .287 0540 42.02 .302 3123 42.75 3 I 7 8 393 43-51 333 6456 44-31 1O 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 5958 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 5084 44.40 17 .289 0733 42.12 .304 3668 42.85 .319 9304 43.62 335 7749 44.41 18 .289 3260 42.13 .304 6240 42.86 .320 1921 43.63 .336 0414 44-43 19 .289 5788 42.14 .304 8812 42.88 .320 4540 43.64 .336 3080 44-44 20 2.289 8317 42.15 2.305 1385 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 .305 6533 42.91 .321 2399 43.68 337 1083 4448 23 .290 5908 42.19 .305 9109 42.93 .321 5020 43.69 337 3752 4449 24 .290 8440 42.20 .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 7 1 " 44.56 29 .292 1109 42.26 .307 4576 43.00 .323 0765 43-77 .338 9785 44.58 3O 2.292 3645 42.27 2.307 7157 43.02 2.323 3391 43-79 2-339 2460 44-59 31 .292 6182 42.29 .307 9738 43.03 .323 6019 4380 339 5 J 35 4460 32 .292 8719 42 30 .308 2320 43.04 .323 8647 43- 8 ' 339 7812 44.62 33 .293 1258 42.31 .308 4903 43.05 .324 1277 43.83 .340 0490 44- 6 3 : 34 293 3797 42.32 .308 7486 43.07 324 397 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^-35 .309 2656 43.09 .324 9169 43 8 7 .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 43.12 325 4434 43- 8 9 .341 3889 44-70 : 39 .294 6503 42.38 .310 0416 43-13 .325 7068 43.91 .341 6571 44-71 40 2.294 94 6 42.40 2.310 3004 43-14 2.325 9703 43.92 2.341 9255 44-73 41 .295 1590 42.41 .310 5593 43-15 .326 2339 43-93 .342 1939 44-74 42 295 4i35 42.42 .310 8182 43-17 .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 .311 3364 43.19 .327 0250 43-97 342 9995 44.78 45 2.296 1774 42.46 2.311 5956 43.21 2.327 2889 43.98 2343 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 49 .296 9419 .297 1969 42.49 42-5I .312 3736 .312 6331 43-24 43.26 .328 0809 .328 3451 44.02 44.04 344 75 344 344 44.84 44.85 50 2.297* 4520 42.52 2.312 8927 43.27 2.328 6094 44.05 2344 6132 44.86 51 .297 7071 42.53 .313 1524 43.28 .328 8737 44.06 344 8824 44.88 52 .297 9623 4254 .313 4121 43.29 .329 1382 44.08 345 '5'7 44.89 53 .298 2176 42.55 .313 6719 43-31 .329 4027 44.09 345 4211 44.91 54 .298 4730 42-57 .313 9318 43.32 .329 6672 44.10 345 6 9 6 44-92 55 2.298 7284 42.58 2.314 1917 43-33 2.329 9319 44.12 2.345 9661 44-93 56 .298 9839 42.59 .314 4518 43-35 .330 1967 44-13 .346 2298 44- 9 5 57 .299 2395 42.60 .314 7119 43.36 .330 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 43.38 33 99H 44-17 347 0392 44-99 60 a. 300 0067 42.64 2.315 4927 43.40. 2.331 2564 44.18 2-347 392 45.00 594 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 116 117 118 119 log M. Diff. I". log M. Diff. 1". log M. Diff. i". logM. Diff. 1". 1 2 3 4 2.347 392 347 5792 347 8494 .348 1196 348 3899 45.00 45-02 45.03 45.04 45.06 2.363 6626 .363 9378 .364 2131 .364 4885 .364 7639 45-86 45^88 45.90 45-9' 2.380 3290 .380 6095 .380 8901 .381 1708 .381 4515 46.74 46.76 46-77 46.79 46.80 2.397 3210 397 6070 397 8931 .398 1794 398 4657 47-66 47-68 47.70 47-71 47-73 5 6 7 8 9 2.348 6603 .348 9308 349 2014 349 4720 349 7428 45-7 45.09 45.10 45.11 45.13 2.365 0394 .365 3150 3 6 5 597 .365 8665 .366 1423 45-93 45-94 45.96 45-97 45-99 2.381 7324 .382 0133 .382 2944 382 5755 .382 8567 46.82 46.83 46.85 46.86 46.88 2.398 7521 399 3 86 399 3252 399 6119 399 8987 47 ' 7 26 .422 2707 49.05 .440 1118 50.07 .458 3268 51.13 .476 9313 52.23 i 27 28 .422 5650 .422 8595 49.07 49-09 .440 4123 .440 7129 50.09 50.1 1 .458 6337 .458 9406 51.15 51.17 477 2448 477 5584 52.25 52-27 29 .423 1541 49.10 .441 0136 50.12 459 2477 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 52.31 31 423 7435 49.14 .441 6152 50.16 .459 8621 51.22 478 4998 5 2 -33 i 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 i 35 2.424 9236 49.20 2.442 8199 50.23 2.461 0923 51.29 2-479 75 66 52.40 36 .425 2189 49.22 443 !2i3 50.24 .461 4001 51.31 .480 0711 52-42 37 38 425 5H2 425 8097 49.24 49.25 .443 4228 .443 7244 50.26 50.28 .461 7080 .462 0161 5'-33 5^35 .480 3857 .480 7004 52.44 52.46 39 .426 1053 49.27 .444 0261 50.30 .462 3242 5i-37 .481 0152 52-48 40 2.426 4010 49.29 2-444 3280 5-3 T 2.462 6325 51.38 2.481 3301 52.50 41 .426 6967 49 3 444 6 299 50.33 .462. 9408 51.40 .481 6452 52.52 : 42 .426 9926 49.32 444 93 20 50-35 .463 2493 51.42 .481 9604 52-54 43 .427 2886 49-34 445 2341 5037 4 6 3 5579 5 ! -44 .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 i 46 .428 1771 49-39 .446 1412 50.42 .464 4843 51.49 .483 2222 52.61 47 428 4735 49.40 .446 4437 50.44 464 7933 51.51 483 5379 52.63 18 .428 7700 4942 .446 7464 50.45 .465 1024 5'-53 483 8537 52.65 49 .429 0665 49-44 447 0492 50.47 .465 4116 5'-55 .484 1697 52.67 5O 2.429 3632 49.46 2.447 3521 50.49 2.465 7210 51-57 2.484 4858 52.69 51 .429 6600 49-47 447 6551 50.51 .466 0305 .484 8020 52.71 52 .429 9569 49-49 447 9582 50.53 .466 3400 51.60 .485 1183 52.73 53 .430 2539 49.51 .448 2614 50-54 .466 6497 51.62 485 4347 52.75 54 .430 5510 49.52 448 5647 50.56 .466 9595 51.64 .485 7513 52.77 55 2.430 8482 49-54 2.448 8681 50.58 2.467 2694 51.66 2.486 0679 52.78 56 .431 1455 49.56 .449 1716 50.60 4 6 7 5794 51.68 .486 3847 52.80 57 .431 4428 49-57 449 4753 50.61 .467 8895 51.70 .486 7016 52.82 58 43 J 743 49-59 .449 7790 50.63 .468 1997 51.71 .487 0186 52.84 59 .432 0379 49.61 .450 0828 50.65 .468 5101 51-73 487 3357 52.86 60 2.432 3356 49.62 2.450 3868 50.67 2.468 8205 5^-75 2.487 6529 52.88 596 TABLE VI, Foi finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit V* 124 125 126 127 log M. Diff. I". logM. Diff. 1". log M. Diff. l". logM. Diff. 1". .487 6529 52.88 1.506 9006 54.06 .526 5813 55-29 546 7135 56.57 1 487 972 5290 .507 2251 54.08 526 9131 55-31 -547 0530 5659 2 .488 2877 52.92 57 5496 54.10 .527 2450 55-33 547 3926 56.61 3 .488 6053 52-94 .507 8742 54.12 527 5771 55-35 547 7323 56.63 4 .488 9230 52.96 .508 1990 54.14 .527 9092 55-37 548 0722 56-65 5 .489 2408 52.98 1.508 5239 54.16 .528 2415 55-39 .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 .509 4993 54-22 .529 2391 55-45 549 4330 56.74 9 490 5132 53-05 .509 8247 54.24 529 5719 55.48 549 7735 56-76 10 .490 8315 53.07 510 1502 54.26 2-529 94 8 55-50 2.550 1141 56.79 11 .491 1500 53.09 .510 4758 54.28 .530 2379 55-52 .550 4549 56.81 12 .491 4686 53.11 .510 8016 54.30 53 57 IQ 55-54 .550 7958 56.83 13 .491 7874 53-13 .511 1274 54.32 .530 9043 55-56 .551 1369 56.85 14 .492 1063 53-15 5" 4534 54-34 .531 2378 55.58 .551 4781 56.87 15 .492 4252 53-17 5 11 7795 54.36 2-531 57i3 55.60 2.551 8194 56.90 16 .492 7443 53-19 .512 1057 54.38 .531 9050 55.62 .552 1608 56-92 17 493 6 35 S3- 21 .512 4321 54.40 .532 2388 55.64 .552 5024 56.94 18 19 493 3828 493 7023 53-23 53-25 .512 7586 .513 0852 54.42 54-44 -532 5727 .532 9068 55-67 55.69 552 8441 -553 1859 56.96 56.98 20 .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 7387 54.48 533 5753 55-73 553 8700 57.03 22 494 6613 53-3 1 .514 0657 54.50 533 9097 55-75 .554 2122 57-05 23 .494 9812 53-33 5H 3927 54-52 534 2443 55-77 554 5546 57-07 24 495 3 012 53-35 .514 7199 54-54 534 579 55-79 554 8971 57.10 25 2-495 6213 53-37 25 r 5 473 54.56 2-534 9138 55-8i 2-555 2398 57-12 26 495 94 l6 53-39 515 3747 54.58 535 2487 55.84 -555 5825 57-14 27 28 29 .496 2619 .496 5824 .496 9030 53-41 53-42 53-44 .515 7023 .516 0300 .516 3578 54.60 54.65 535 5838 535 9190 S3 6 2543 55-88 55.90 555 9254 .556 26X5 .556 6116 57.16 57-i8 57-21 30 31 32 33 2.497 2238 497 5446 .497 8656 .498 1867 5346 53.48 53-5 53-52 2.516 6857 .517 0138 .517 3420 .517 6703 54-67 54.69 54-7 1 54-73 2.536 5898 536 9254 .537 2611 537 5970 55-92 55-94 55-96 55.98 2-556 9549 557 2984 557 6420 iS7 9857 57-23 57-25 57-27 57-29 34 .498 5079 53-54 .517 9987 54-75 537 9329 56.01 558 3295 57-32 35 36 37 2.498 8292 .499 1506 .499 4721 53-56 53.60 2.518 3273 .518 6559 .518 9847 54-77 54-79 54.81 2.538 2690 .538 6052 538 94i6 56.03 56.05 56.07 2.558 6735 . 559 0176 559 3618 57-34 57-36 57.38 38 39 499 7938 .500 1156 53.62 53-64 5"9 3137 .519 6427 54.83 54.85 539 2781 539 6l 47 56.09 56.11 559 7062 .560 0507 57-4 1 57-43 40 41 42 43 44 2.500 4375 .500 7595 .501 0817 .501 4039 .501 7263 53-66 53.68 5370 53-72 53-74 2-519 9719 .520 3012 .520 6306 .520 9601 .521 2898 54^9 54-9 1 5493 54-95 2-539 95H .540 2883 54 6253 54 9625 .541 2997 56-13 56-15 56.18 56.20 56.22 2-560 3953 .560 7401 .561 0850 .561 4301 .561 7753 57-45 57-47 57-5 57.52 57-54 45 46 47 48 49 2.502 0488 .502 3714 .502 6942 .503 0170 .503 3400 53.76 53-78 53.80 53-82 53.84 2 521 6196 .521 9495 .522 2795 .522 6097 .522 9400 54-97 54-99 55-02 55-04 55.06 2.541 6371 .541 9746 .542 3123 .542 6500 .542 9880 56.26 56.29 56-31 56-33 2.562 1206 .562 4660 .562 8116 5 6 3 '574 563 5032 57.56 57-59 57-63 50 i 51 52 , 53 54 2.503 6631 .503 9863 .504 3096 54 6331 .504 9567 53.86 53-88 53.90 53.92 5394 2.523 2704 .523 6009 .523 9316 .524 2624 524 5933 55-o8 55-io 55 12 55-14 55.16 2-543 3260 543 664! 544 0024 544 3409 -544 6 794 56.35 56.37 56.39 56.42 56.44 2.563 8492 564 1953 .564 5416 .564 8880 .565 2345 57-68 57-7 57-72 57-74 57-77 55 56 57 58 59 2.505 2804 .505 6042 .505 92X2 .506 2522 .506 5763 53-96 53.98 54.00 54.02 54.04 2.524 9243 525 2555 .525 5867 .525 9181 .526 2497 55-18 5520 55-22 55-24 55.26 2.545 0181 545 3569 545 6 959 546 0350 .546 3742 56.46 56.48 56-5 56.52 5 6 -55 2.565 5812 .565 9280 .566 2750 .566 6221 .566 9693 57-79 57.81 sa 57.88 60 2.506 9006 54.06 2.526 5813 55.29 2.546 7135 56-57 2.567 3166 57.90 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. IAS. I". logM. Diff. 1". log M. Diff. 1". 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 59-32 .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 5 8 <> 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 2.569 0554 58.02 2.590 1919 59.42 2.6II 8433 60.88 2.634 0325 62.41 6 7 .569 4036 569 75'9 58.04 58.06 .590 5485 .590 9052 59-44 59-47 .612 2086 .612 5741 60.90 60.93 .634 4070 634 78i7 62-43 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 59-5 1 .613 3055 60.98 635 53'5 62.51 10 2.570 7977 58-13 2.591 9762 59-54 2.613 6715 6 1. oo 2 635 9066 62.54 ! 11 .571 1465 58.15 592 3335 .614 0376 61.03 .636 2819 62.56 12 571 4955 58.18 .592 6909 59-58 .614 4038 61.05 .636 6573 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 6264 15 2-572 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 5924 58.32 .594 8386 59-73 .616 6045 61.20 .638 9133 62.75 19 573 9424 58.34 595 1970 59-75 .616 9718 61.23 .639 2899 62.77 20 2.574 2925 58.36 2-595 5556 59.78 2.617 3392 61.25 2.639 6666 62.80 21 .574 6427 58.38 595 9'43 59.80 .617 7068 61.28 .640 0435 62.82 22 574 993 1 58.41 .596 2732 59.82 .618 0746 61.30 .640 4205 62.85 23 575 343 6 58.43 .596 6322 59-85 .618 4425 61.33 .640 7977 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 2-597 357 59.90 2.619 1787 61.38 2.641 5525 62.93 2G .576 3960 58.50 597 7 IQ 2 59.92 .619 5471 61.41 .641 9302 62.96 1 27 576 747i 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 58.57 .598 7894 59-99 .620 6531 61.48 .643 0641 63.04 30 2.577 8011 58.59 2.599 1494 60.02 2.621 O22O 61.51 2.643 4424 63.06 31 .578 1528 58.62 599 59 6 60.04 .621 3911 6i-53 .643 8209 63.09 32 .578 5045 58.64 599 86 99 60.07 .621 7604 61.56 644 J 995 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 5 6 7 58.71 2.600 9518 60.14 2.622 8691 61.63 1.645 33 6 3 63.19 36 579 9 1 3 .601 3127 60.16 .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 .623 9793 61.71 .646 4745 63-27 39 .580 9708 58.80 .602 3963 60.24 .624 3496 61.74 .646 8542 63.3.0 40 2.581 3237 5883 2.602 7578 60.26 2.624 7201 61.76 2.647 2341 63.33 41 .581 6768 58-85 .603 1195 60.29 .625 0907 61.79 .647 6142 63-35 42 .582 0299 58.87 .603 4813 60.31 .625 4615 61.81 .647 9944 63.38 43 .582 3832 58.90 .603 $432 60.34 .625 8325 61.84 .648 3748 63.41 44 .582 8267 58.92 .604 2053 60.36 .626 2036 61.86 .648 7553 63.44 45 2.583 0903 58.94 2.604 5675 60.38 2626 5748 61.89 2.649 1360 63.46 46 .583 4440 58.97 .604 9299 60.41 .626 9462 61.91 .649 5168 63-49 47 583 7979 58-99 .605 2924 60.43 .627 3178 61.94 .649 8978 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 63.57 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 52 585 5694 59.11 .607 1073 60.56 .629 1780 62.07 .651 8053 63-65 i 53 .585 9241 59- J 3 .607 4707 60.58 .629 5505 62.09 .652 1873 63.68 54 .586 2790 59.16 .607 8343 60.6 1 .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'!-73 56 .586 9891 59.20 .608 5618 60.66 630 6689 62.17 6 53 3342 1.76 57 587 3444 59.23 .608 9258 60.68 .631 0420 62.20 .653 7168 63.79 58 587 6999 59.25 .609 2901 60.70 .631 4152 62.22 .654 0996 63. 81 59 .588 0555 59-27 .609 6544 60.73 .631 7887 62.25 .654 4826 63.84 60 2.588 4112 59.30 2.610 0188 60.75 2.632 1622 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. 133 134 135 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 2 3 4 2.654 8657 .655 2490 .655 6324 .656 oioo .656 3998 63.87 63.89 63.92 63-95 63.97 2.678 1547 .678 5480 .678 9414 .679 3350 .679 7288 65.53 65.56 65-59 65.61 6564 2.702 0562 .702 4600 .702 8638 .703 2679 .703 6721 67.27 67.30 8:51 67.39 2.726 5990 .727 0137 .727 4285 .727 8435 .728 2587 69.09 69.12 69.15 69.19 69.22 5 G 7 8 2.656 7837 .657 1678 .657 5521 6S7 93 6 5 .658 3211 64.00 6 4 4 : l 64.08 64.11 2.68o 1227 .680 5168 .680 9111 .681 3056 .681 7002 65.67 65.70 65-73 65.76 65.79 2.704 0766 .704 4812 .704 8860 .705 2909 .705 6961 67.42 67.45 67.48 67.51 67.54 2.728 6741 .729 0897 .729 5055 .729 9215 .730 3376 69.25 69.28 69.31 69.34 69.37 10 1 n 12 13 14 z.6 S 8 7058 .659 0907 .659 475 8 .659 8611 .660 2465 64.14 64.17 64.19 64.22 64.25 2.682 0950 .682 4900 .682 8851 .683 2804 .683 6759 65.81 65.84 65.87 65.90 65-93 2.706 1014 .706 5069 .706 9126 .707 3184 .707 7244 67-57 67.60 67.63 67.66 67.69 2.730 7539 .731 1705 .731 5872 .732 0041 .732 4212 69.40 69.44 69.47 69.50 69-53 I w 16 17 18 19 2.660 6320 .661 0178 .661 4037 .661 7897 .662 1760 64.28 64.30 64.33 64.36 64.38 2 684 0716 .684 4674 .684 8634 .685 2596 .685 6559 65.96 65.99 66.01 66.04 66.07 2.708 1307 .708 5371 .708 9436 .709 3504 79 7573 67.72 6 7-75 67.78 67.81 67.84 2.732 8385 733 *559 733 6 73 6 734 09H 734 594 69.56 69-59 69.62 69.66 69.69 20 21 22 23 24 2.662 5623 .662 9489 .663 3356 .663 7225 .664 1096 6 4 . 4 i 64.44 64.47 64.49 64.52 2.686 0524 .686 4491 .686 8460 .687 2430 .687 6402 66.10 66.13 66.16 66.19 66.22 2.710 1645 .710 5718 .710 9792 .711 3869 .711 7947 67.87 67.90 6 7-93 67.96 67-99 2.734 9 2 77 735 346i 735 7647 .736 1835 .736 6025 69.72 69.75 69.78 69.81 69.85 25 2.664 49 68 64.55 2.688 0376 66.25 2.712 2028 68.02 2.737 0216 69.88 26 .664 8842 6 4-57 .688 4352 66.27 .712 6110 68.05 .737 4410 69.91 27 28 29 .665 2717 .665 6594 .666 0473 64.60 64.63 64.66 .688 8329 .689 2308 .689 6289 66.30 6633 66.36 .713 0194 .713 4279 .713 8367 68.08 68.11 68.14 737 8605 .738 2803 .738 7002 69.94 69.97 70.00 30 31 32 33 34 2.666 4354 .666 8236 .667 2120 .667 6005 .667 9892 64.69 64.72 64.74 64.77 64.80 2.690 0272 .690 4256 .690 8242 .691 2230 .691 6219 66-39 66.42 66.45 66.48 66.51 2.714 2456 .714 6547 .715 0640 7i5 4735 .715 8832 68.17 68.20 68.23 68.26 68.29 2.739 1^03 .739 5406 739 9 6iz .740 3819 .740 8027 70.04 70.07 70.10 70.13 70.16 35 36 37 2.668 3781 .668 7672 .669 1564 6483 64.86 64.88 2.692 O2IO .692 4203 .692 8198 66.54 66.56 66.59 2.716 2930 .716 7031 .717 1133 68.32 68.35 68.38 2.741 2238 .741 6451 .742 0666 70.20 70.23 70.26 38 .669 5457 64.91 .693 2194 66.62 .717 5237 68.41 .742 4882 70.29 39 .669 9353 64.94 .693 6193 66.65 .717 9342 68.44 .742 9101 70.32 40 2.670 3250 64.97 2.694 '93 66.68 .718 3450 68.48 2.743 33 21 70.36 41 42 .670 7149 .671 1050 65.00 65.02 .694 4194 .694 8198 66.71 66.74 .718 7560 .719 1671 68.51 68.54 743 7543, .744 1768 70.39 70.42 43 .671 4952 65.05 .695 2203 66.77 .719 5784 68.57 .744 5994 70-45 44 .671 8856 65.08 .695 6210 66.80 .719 9899 68.60 .745 0222 70.48 45 672 2761 65.1 1 .696 0219 66.83 .720 4016 68.63 .745 4452 70.52 46 .672 6668 6S-I3 .696 4229 66.86 .720 8135 68.66 .745 8684 7-55 47 .673 0577 65.16 .696 8242 66.89 .721 2255 68.69 .746 2918 70.58 i 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 I39 1 70.65 . 50 .674 2314 65.25 .698 0289 66.97 .722 4628 68.78 747 5 6 3' 70.68 i 51 .674 6230 65.28 .698 4308 67.00 .722 8756 68.81 747 9 8 73 70.71 52 .675 0147 65.30 .698 8330 67.03 .723 2885 68.84 .748 4116 70.74 , 53 54 .675 4066 .675 7987 65-33 65.36 .699 2353 .699 6377 67.06 67.09 .723 7017 .724 1150 68.88 68.91 .748 836* .749 2609 70.78 70.81 55 .676 1909 65.39 .700 0404 67.12 .724 5286 68.94 749 6859 70.84 56 676 5833 65.42 .700 4432 67.15 .724 9423 68.97 .750 mo 70.87 57 676 9759 65.44 .700 8462 67.18 725 3562 69.00 75 5364 70.90 58 59 .677 3687 .677 7616 65.47 65.50 .701 2494 .701 6527 67.21 67.24 725 7703 726 1846 69.03 69.06 750 9619 75! 3876 70.94 70.97 60 .678 1547 65-53 .702 0562 67.27 J2.726 5990- 69.09 75* 8135 71.00 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 136 137 138 139 logM. Diff. I". logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". 1.751 8135 71.00 2.777 73" 73.01 2.804 3895 75-11 2.831 8224 77.32 1 .752 2396 71.03 .778 1703 73.04 .804 8403 75-H .832 2864 77-35 2 .752 6659 71.07 .778 6087 73.07 .805 2912 75.18 .832 7506 77-39 3 753 9*5 71.10 .779 0472 73.11 .805 7424 75.21 .833 2151 77-43 4 753 5i9* 71.13 779 4859 73-14 .806 1938 75-25 .833 6798 77-47 5 2.753 94 61 71.17 2.779 9249 73.18 2.806 6454 75.29 2.834 H47 77-5 6 754 3732 71.20 .780 3641 73.21 .807 0973 75.32 .834 6098 77-54 7 .754 boo 4 71.23 .780 8034 73.24 .807 5493 75.36 .835 0752 77-58 8 755 2*79 71.26 .781 2430 73.28 .808 0016 75.40 .835 5408 77.62 9 755 6556 71.30 .781 6828 73-31 .808 4541 75-43 .836 0066 77.66 10 1.756 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 75-50 .836 9390 77-73 12 .756 9399 71.40 .783 0034 73.42 .809 8128 75-54 .837 4055 77-77 13 757 3^3 71-43 .783 4440 73-45 .810 2662 75-58 .837 8722 77-81 14 757 797 71.46 .783 8848 7349 .810 7197 75-6i .838 3392 77-85 15 2.758 2259 71.49 2.784 3258 73-53 2.811 1735 75-65 2.838 8064 77-89 16 .758 6549 7M3 .784 7671 73-5 6 .811 6275 75.69 839 2738 77.92 IT 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 73.70 2-813 4457 75-83 2.841 1458 78.08 21 .760 8032 71.69 .786 9764 73-73 .813 9008 75-87 .841 6144 78.11 22 .761 2335 71.73 .787 4I 8 9 73.76 .814 3561 75.90 .842 0832 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 73.90 .816 1796 76.05 .843 9607 78-31 ! 27 .763 3878 71.89 .789 6344 73-94 .816 6360 76.09 .844 4306 78-35 28 .763 8192 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 8419 78.46 31 .765 1148 72.03 .791 4106 74.08 .818 4639 76.23 .846 3118 78.50 3i .765 5470 72.06 791 8552 74.11 .818 9114 76.27 .846 7839 78.54 33 765 9795 72.09 .792 3000 74-15 .819 3792 76.31 .847 2553 78.58 34 .766 4121 72.13 .792 7450 74.18 .819 8371 76.34 .847 7268 78.62 35 2.766 8450 72.16 2.793 190^ 74.22 2.820 2953 76.38 1.848 1986 78.66 36 .767 2781 72.19 793 6356 74- 2 5 .820 7537 76.42 .848 6707 78.69 37 .767 7113 72.23 .794 0813 74.29 .821 2123 76.46 .849 1430 78.73 38 .768 1448 7226 .794 5271 7432 .821 6712 76.49 .849 6155 78.77 39 .768 5784 72.29 794 973 1 74- 3 6 .822 1302 76.53 .850 0882 78.81 40 1.769 0123 72.33 2.795 4194 74.40 2.822 5895 76.57 2.850 5612 78-85 41 .769 4464 72.36 795 8659 7443 .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 77 3'S 1 72.43 79 6 7595 74.50 .823 9688 76.68 .851 9816 78.97 1 " .770 7498 72.46 .797 2066 74-54 .824 4289 76.72 852 4555 79.01 45 2.771 1846 72.50 2 -797 6539 74-58 2.824 8894 76.75 2.852 9297 79.05 46 .771 6197 72-53 .798 1015 74.61 .825 3500 76.79 .853 4041 79.08 47 .772 0550 72.56 .798 5492 74.64 .825 8108 76.83 853 8787 79.12 48 .772 4905 72.60 .798 9972 74-68 .826 2719 76-87 854 3535 79.16 49 .772 9262 72.63 799 4454 74-7* .826 7332 76.90 .854 8286 7920 5O 51 2.773 36n 773 7982 72.67 72.70 2.799 8 93 8 .800 3424 74-75 74-79 2.827 J947 .827 6565 76.94 76.98 2.855 3040 855 7795 79.24 79.28 52 774 2344 72.73 .800 7912 74.82 .828 1185 77.01 856 2553 79.32 i 53 774 6709 72.77 .801 2402 74-86 .828 5807 77-5 .856 7314 79.36 54 775 I0 77 72.80 .801 6895 74.89 .829 0431 77.09 857 2077 79.40 55 2.775 5446 72.84 2.802 1390 74-93 2.829 5058 77.13 2.857 6842 79-44 56 775 9817 72.87 .802 5886 74.96 .829 9686 77 l6 .858 1609 79-48 57 .776 4190 72.90 .803 0385 75.00 .830 4317 77.20 .858 6379 79.52 58 .776 8565 72.94 .803 4886 75.04 .830 8951 77-24 .859 1151 79.56 59 777 294* 72.97 .803 9390 75.08 .831 3586 77.28 .859 5926 79.60 60 1.777 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 log M. Diff. 1" logM. Diff. 1" logM. Diff. 1" logM. Diff. 1". 1 2 3 4 2.86o 0703 .860 5482 .861 0264 .861 5048 .861 9835 79.64 79-68 79.72 79.76 79.80 2.889 1754 .889 6680 .890 1609 .890 6540 .891 1473 82.08 82.12 82.16 82.20 82.25 2.919 1831 919 6911 .920 1994 .920 7080 .921 2169 84.65 84.70 84.74 84.78 84-83 2.950 1420 .950 6664 .951 1910 .951 7159 .952 2411 87-37 87.41 87.46 87.50 87-5S 5 6 7 8 9 2.862 4624 .862 9415 .863 4209 .63 9005 .864 3803 79.84 79.88 79.92 79.96 80.00 2.891 6409 .892 1348 .892 6289 .893 1233 893 6179 82.29 82.33 82.37 82.41 82.46 2.921 7260 .922 2353 .922 7450 9*3 *549 9*3 7650 84.87 84.92 84.96 85.01 85.05 2.952 7665 953 *9*3 953 8183 954 3446 954 87" 87.60 87.65 87.69 87.74 87.79 10 11 12 13 14 2.864 8604 .865 3408 .865 8213 .866 3021 .866 7832 80.04 80.08 80.12 80. 1 6 80.20 2.894 *7 .894 6078 .895 1032 .895 5989 .896 0948 82.50 82.54 82.58 82.63 82.67 2-9*4 *755 .924 7862 9*5 *97* .925 8084 .926 3199 85.10 p fell 2.955 398o 955 9*5i 956 45*5 .956 9802 957 5082 87.83 87.88 87.93 87.97 88.02 15 17 ! 18 19 2.867 2645 .867 7460 .868 2278 .868 7098 .869 1921 80.24 80.28 80.32 80.36 80.40 2.896 5909 .897 0873 .897 5839 .898 0808 .898 5780 82.71 82.75 82.79 82.84 82.88 2.926 8317 9*7 3437 .927 8560 .928 3686 .928 8814 85.32 85.36 85.41 85-45 85.50 2.958 0365 958 565' 959 939 959 6230 .960 1524 88.07 88.11 88.16 88.21 88.26 20 21 22 23 24 2.869 6746 .870 1573 .870 6403 .871 1235 .871 6070 80.44 80.48 80.52 80.56 80.60 2.899 754 .899 5730 .900 0709 .900 5691 .901 0675 82.92 82.96 83.01 83.05 83.09 2.929 3945 9*9 979 .930 4216 93 9355 93' 4497 85-54 85.59 85.63 8 5 .68 85.72 2.960 6821 .961 2I2O .961 7423 .962 2728 .962 7036 88.30 88.35 88.40 88-45 88.49 25 i 26 2.872 0907 .872 5747 80.64 80.68 2 901 5662 .902 0651 83.13 83. IX 2.931 9641 .932 4788 85-77 Si ;.8 i 2.963 3347 .963 66i as 27 .873 0589 80.72 .902 5643 83.22 93* 9938 1 85.86 .964 3978 -. . . , i 28 873 5433 80.76 .903 0638 83.26 933 59' 85.91 .964 9297 88^68 29 .874 0280 80.80 .903 5635 83-3I .934 0247 85.95 .965 4620 88-73 30 31 32 33 .874 5129 .874 9981 .875 4835 875 9 6 9* 80.84 80.88 80.92 80.96 .904 0635 .904 5637 .905 0642 .905 5649 83.35 83.39 8343 83.48 *-934 545 935 5 6 5 935 57*9 .936 0895 85-99 86.04 86.08 86.13 9 6 5 9945 .966 5273 .967 0604 .967 5938 88.78 88.83 88.87 88.92 34 876 455i 81.01 .906 0659 83.52 .936 6064 86.17 .968 1275 88.97 35 .876 9413 81.05 .906 5672 83.56 2.937 1236 86.22 .968 6615 89.02 36 .877 4277 81.09 .907 0687 83.61 .937 6410 86.26 969 1957 89.07 37 .877 9'43 81.13 .907 5704 83.65 .938 1587 86.31 9 6 9 733 89.12 i 38 .878 4012 81.17 .908 0725 83.69 .938 6767 86-35 970 2651 89.17 39 .878 8883 81.21 .908 5748 83.74 939 195 86.40 .970 8002 89.21 40 879 3757 81.25 .909 0773 83.78 939 7135 86.45 971 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 97* 4073 89.36 43 .880 8393 81.37 910 5865 83.91 .941 2708 86.58 97* 943 s 89-40 44 .881 3277 81.42 911 0901 83.95 .941 7904 86.63 973 4801 89.45 ' 45 .881 8163 81.46 9 n 5940 s 8399 .942 3103 86.67 974 0170 89.50 ! 46 .882 3052 81.50 912 0981 I 84.04 94* 8305 86.72 974 554' 89-55 47 .882 7943 81.54 912 6024 84.08 943 35' 86-77 975 0916 89.60 48 .883 2837 81.58 913 1070 84.13 943 *77 86.81 975 6293 89.65 49 .883 7733 81.62 913 6119 84.17 944 39*7 86.86 976 1673 89.69 5O .884 2631 81.66 914 1171 84.22 944 9!4 86.90 976 7056 89.74 51 884 753* 81.70 914 6225 84.26 945 4355 86.95 977 *44* 89.79 ' 52 .885 2436 81.75 915 1282 84.30 945 9574 87.00 977 7831 89.84 , 53 54 885 734* 886 2251 81.79 Si 915 6341 916 1403 84.34 84.39 946 4795 947 0019 87.04 87.09 978 3223 978 8618 89.89 89.94 55 886 7162 81.87 916 6468 84-43 947 5*45 87.13 979 4i5 89.99 56 887 2075 81.91 917 1535 84.48 948 0475 87.18 979 9416 90.03 57 887 6991 81.95 917 6605 84.52 948 577 87.23 980 4820 90.08 58 888 1910 81.99 918 1678 84.56 949 0942 87.27 981 1226 90.13 59 888 68 3I 82.64 918 6753 84.61 949 6180 87.32 981 6636 90.18 00 889 1754 82.08 919 1831 84.65 950 1420 87-37 982 1048 90-M TABLE VI, F jr finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 144 145 146 147 log M. Diff. 1". logM. Diff. 1". log M. Diff. 1". log M. Diff. 1". 2.982 1048 9023 3.015 1281 93.26 3049 2733 96.47 3.084 6070 99.87 1 .982 6463 90.28 .015 6878 93.31 .049 8522 96.52 .085 2064 99.92 2 .983 1882 90.33 .016 2478 93-3 6 .050 4315 96.58 .085 8061 99.98 3 .983 7303 00.38 .016 8082 93.42 .051 Oil 2 96.63 .086 4062 100.04 4 .984 2727 90.43 .017 3688 93-47 .051 5911 96.69 .087 0066 100.10 5 2.984 8154 90.48 3.017 9298 93-52 3.052 1714 96.74 3.087 6073 100.16 6 .985 3584 9-53 .018 4911 9357 .052 7520 96.80 .088 2085 IOO.22 7 .985 9017 90.58 .019 0526 93.62 53 33 2 9 96.85 .088 8099 100.28 | 8 .986 4453 90.63 .019 6145 9368 .053 9142 96.91 .089 4118 100.33 9 .986 9892 90.67 .020 1768 93-73 .054 4959 96.96 .090 0140 100.39 10 2.987 5334 90.72 3.020 7393 93.78 3.055 0778 97.01 3 090 6165 100.45 11 .988 0779 90.77 .021 3021 9383 .055 6601 97.07 .091 2194 100.51 12 .988 6227 90.82 .021 8653 93.89 .056 2427 97-13 .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 1589 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 .058 5765 97-35 .094 2392 I00.8l 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 59 7454 97-47 .095 4496 100.93 19 99 2 444 91.17 .025 8163 94.26 .060 3304 97-5* .096 0553 100.98 20 -.4.992 9918 91.22 3.026 3820 94.31 3.060 9157 97.58 3.096 6614 IOI.O4 21 993 5393 91.27 .026 9480 94.36 .061 5013 97.63 .097 2678 IOI.IO 22 .994 0871 91.32 .027 5143 94-4 ! .062 0873 97.69 .097 8746 101.16 23 994 6 35! 9*-37 .028 O8l0 9447- .062 6736 97-75 .098 4818 IOI.il 24 995 1^35 91.42 .028 6479 94.52 .063 2602 97.80 .099 0893 101.28 25 2.995 73" 91.47 3.029 2152 94-57 3.063 8472 97.86 3.099 6972 IOL 34 26 .996 2812 91.52 .029 7828 9463 .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 2.998 4802 91.72 3.032 0564 94.84 3.066 7872 98.14 3-102 7420 101.64 31 999 0307 91.77 .032 6256 94.89 .067 3762 98.20 .103 3520 101.70 32 999 5 8l 5 91.82 .033 1951 94-94 .067 9655 98.25 .103 9624 101.76 33 3 ooo i 326 91.87 .033 7650 95.00 .068 5552 98.31 .104 5732 101.82 34 .000 6840 91.93 .034 3351 95-5 .069 1453 98.37 .105 1843 101.88 35 3.001 2357 91.98 3.034 9056 95.11 3.069 7357 98.42 3.105 7958 101.94 36 .001 7877 92.03 35 47 6 4 95.16 .070 3264 98.48 .106 4076 IO2.OO 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 .038 3353 95-43 .073 2851 98.77 .109 4723 102.31 42 .005 1062 92.33 .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 I" 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 .007 3246 92.54 .041 2023 95.70 .076 2524. 99.05 .112 5461 102.61 47 .007 8800 92.59 .041 7767 95.76 .076 8469 99.11 .113 l62O 102.67 ' 48 .008 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 99.23 .114 3948 102.80 50 3.009 5480 92.74 3.043 5017 95.92 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 96.03 .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 .on 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 3.1l8 IO22 103.16 56 .012 8923 93-5 .046 9607 96,25 .082 2130 99.63 .Il8 7213 103.23 57 .013 4508 93.10 47 53 8 3 96.30 .082 8110 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 6O 3.015 1281 93.26 3.049 2733 96.47 3.084 6070 99.87 3-I2I 20l8 103.48 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. I". logM. Diff. 1". logM. Diff. 1". iai 2018 103.48 159 1367 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 2 122 4442 103.60 160 4253 107.45 199 8361 111.55 240 7722 115.92 3 123 0660 103.66 161 0702 107.51 200 5056 111.62 241 4680 116.00 4 123 6882 103.72 161 7154 107.58 .201 1755 111.69 .242 1642 1 1 6.08 5 124 3107 103.79 .162 3611 107.65 201 8459 1 11.76 .242 8608 116.15 124 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 116.30 1 8 9 126 I80 S 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 10 127 4289 104.10 165 5955 107.98 .205 2040 I I 2. 1 1 .246 3511 116.53 11 128 0537 104.16 .166 2435 108.04 .205 8769 112.18 .247 0505 116.61 12 128 6789 104.22 .166 8920 io8.n .2O6 55O2 112.26 .247 7503 116.68 13 129 3044 104.29 .167 5409 108.18 .207 2279 112-33 .248 4507 116.76 14 I2 9 9303 104.35 .168 1901 108.25 .207 8981 I I 2.40 .249 1515 116.84 15 .130 5566 104.41 .168 8398 108.31 .208 5727 112-47 .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. 6l .251 2566 117.07 18 .132 4377 104.60 .170 7913 108.51 .210 5991 112.69 .251 9592 117.14 19 133 6 55 104.67 .171 4426 108.58 .211 2755 112.76 .252 6623 117.22 20 *33 6 937 i4-73 .172 0942 108.65 3 .2II 9522 112.83 .253 3658 117.30 21 .134 3223 104.79 .172 7463 108.72 .212 6294 112.90 .254 0698 117.37 22 .134 9512 104.86 .173 3988 108.78 .213 3070 112.97 254 7743 117.45 23 .135 5805 104.92 .174 0517 108.85 .2I 3 9851 113.05 .255 4792 "7-53 24 .136 2102 104.98 .174 7051 108.92 .214 6636 113.12 .256 1846 1 17.60 25 26 27 .136 8403 .137 4708 .138 1016 105.05 105.11 105.17 3.175 3588 .176 0129 .176 6674 108.99 109.06 109.12 3- 2I 5 3425 .216 0219 .216 7017 113.19 113.26 "3-34 3.256 8905 .257 5968 .258 3036 117.68 117.76 117.84 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 3.178 6335 109.33 3.218 7437 "3-55 3.260 4268 118.07 31 32 33 34 .140 6289 .141 2616 .141 8948 .142 5283 105.43 105.49 105.55 105.62 .179 2X97 .179 9462 .180 6032 .181 2606 109.40 109.46 109.53 109.60 .219 4252 .220 1072 .220 7896 .221 4724 113.63 113.70 113.77 113.84 .261 1354 .261 8446 .262 5542 .263 2642 118.15 118.23 118.30 118.38 35 3.143 1622 105.68 3.181 9184 109.67 3-222 1557 113.92 3.263 9747 118.46 36 .143 7965 105.75 .182 5766 109.74 .222 8395 "3-99 .264 6857 118.54 37 38 39 .144 4312 .145 0663 .145 7018 105.81 105.87 105.94 .183 2353 .183 8943 .184 5538 109.81 109.87 109.94 .223 5236 .224 2082 .224 8933 114.06 114.14 114.21 .265 3972 .266 1091 .266 8216 1 18.62 118.70 118.77 40 41 42 43 44 3.146 3376 .146 9739 .147 6105 .148 2475 .148 8849 106.00 106.07 106.14 106.20 106.27 3.185 2136 .185 8739 .186 5346 .187 1957 .187 8572 IIO.OI no.oS 110.15 IIO.22 IIO.29 3.225 5788 .226 2647 .226 9511 .227 6379 .228 3252 114.28 114.36 114.43 114.51 "458 3.267 5345 .268 2478 .268 9616 .269 6759 .270 3907 ,18.85 118.93 119.01 119.09 119.17 45 46 47 48 49 3.149 5227 .150 1609 .150 7995 .151 4385 .152 0778 106.33 106.40 106.45 106.53 106.55 3.188 5192 .189 1815 .189 8443 .190 5075 .191 1711 110.36 110.43 I 10.50 110-57 1 10. 6. 3.229 0129 .229 7010 .230 3896 .271 0706 .231 7681 114.65 114.73 114 80 ,4.88 114.95 3.271 1060 .271 8217 .272 5379 .273 2546 .273 9717 119.25 119.33 119.41 119.49 119.57 50 51 52 53 54 3.152 7176 153 3577 153 9983 .154 6392 .155 2805 106.66 106.72 106.79 106.85 106.92 3.191 8351 .192 4996 .193 1644 .193 8297 .194 4954 110.71 110.77 110.84 1 10.91 110.98 3.232 4 58l .233 1484 .233 8392 .234 5305 .235 2222 115.03 115.10 115.17 115.25 115.32 3.274 6894 .275 4075 .276 1261 .276 8452 .277 5647 119.65 119.73 119.81 119.89 119.97 55 56 57 58 59 3.155 9222 .156 5643 .157 2068 .157 8497 .158 4930 106.99 107.05 107.12 107.1! 107.25 3.195 1615 .195 8281 .196 4950 .197 1624 .197 8302 111.05 II 1.1 : 1 1 1. 1 1 111.26 111.34 3.235 9144 .236 6070 .237 3001 .237 9936 .238 6876 115.40 115.47 "5-55 115.62 115.70 3.278 2848 .279 0053 .279 7263 .280 4477 .281 1697 120.05 120.1 3 120.21 120.29 120-37 GO 3-icq 1367 107.31 3.198 4984 111.41 3.239 3820 115.77 3.281 8921 12045 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". log M. Diff. 1". log M. Diff. 1". log M. 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 125.55 373 0538 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 4'33 131.22 .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 I 31.50 .426 1609 137.37 8 .287 6891 121. IO 332 1835 126.16 .378 5671 131.60 .426 9854 H7-47 9 .288 4160 121. l8 .332 9407 126.24 379 3570 131.69 .427 8105 '37-57 : 1O 3.289 1433 121 26 3-333 6 984 126.33 3.380 1474 131.79 3 428 6362 I37-67 ! 11 .289 8711 121.34 334 4567 126.42 380 9384 131.88 .429 4626 '37-77 12 .290 5993 121.42 335 2154 126.51 .381 7300 131.98 43 2895 137-88 13 .291 3281 121.50 335 9747 126.59 .382 5221 132.07 43' "7' I37-98 14 292 0574 121.59 .336 7346 126.68 383 3H8 132.16 43' 9452 138.08 15 3 292 7872 121.67 3-337 4949 126.77 3.384 1081 132.26 3.432 7740 138.18 ' 16 .293 5174 121.75 338 2558 126.86 .384 9019 '32.35 433 6034 13829 \ 17 .294 2481 121.83 339 0172 126.95 .385 6963 434 4334 '38-39 | 18 294 9794 121.91 339 7792 127.03 .386 4913 132.54 435 2641 19 .295 7111 122. OO 34 54'7 127.12 .387 2869 132.64 .436 0953 138.^9 20 3.296 4433 122.08 3 34' 347 127.21 3.388 0830 13273 3.436 9272 138.70 21 .297 1701 122. l6 .342 0682 127.30 -388 8797 132.83 437 7597 138.80 ' 22 .297 9093 122.24 .342 8323 127.39 389 6770 132.93 .438 5928 138.90 23 .298 6430 122 33 343 59 6 9 12748 390 4749 133.02 439 4266 139.01 24 .299 3772 122.41 344 3 6 2o 127-57 39' 2733 133.12 .440 2609 139.11 25 3 300 1119 122.49 3-345 '277 127.66 3.392 0723 133.22 3.441 0959 139.22 26 .300 8471 122.58 345 8 939 127-75 .392 8719 '33-3' 44' 93 '5 139.32 ! 27 .301 5828 122.66 .346 6606 127.84 .393 6720 '33-4' .442 7677 13942 28 .302 3190 122.74 347 4279 127.93 .394 4728 133-50 443 6046 '39-53 29 33 557 122.83 .348 1958 128.02 395 *74' 133.60 444 442i 139.63 30 3 303 7929 122.91 3.348 9641 128.11 3.396 0760 133-70 3.445 2802 '39-74 31 .304 5306 122.99 349 7330 128.19 .396 8785 '33-79 .446 1189 139-84 32 .305 2688 I 23.08 35 5024 128.28 397 6815 133.89 .446 9583 '39-95 33 .306 0075 123.16 35' 2724 128.37 .398 4852 133-99 447 7983 140.05 34 .306 7468 123.24 .352 0429 128.46 399 2894 134.09 .448 6389 140.16 35 3.307 4865 '23-33 3.352 8140 128.55 1.400 0942 134.19 3 449 4802 140.26 36 .308 2267 123.41 353 5856 128.65 .400 8996 134.28 .450 3221 140.37 ! 37 .308 9674 123.50 354 3577 128.74 .401 7056 I34-38 .451 1646 140.47 38 .309 7086 123.58 355 1304 128.83 .402 5122 134.48 452 0077 14057 39 .310 4504 123.66 355 937 128.92 .403 3193 '34-57 .452 8515 140.68 40 3311 1926 123-75 3.356 6774 129.01 3.404 1270 134.67 3-453 6 959 140.79 41 3" 9354 123.83 357 45'7 129.10 404 9354 '34-77 454 54io 140.90 42 .312 6786 123.92 .358 2266 129 19 405 7443 134-87 455 3867 141.00 43 .313 4224 124.00 .359 0020 129.28 .406 5538 '34-97 .456 2330 141.11 44 .314 1667 124.09 359 778o 129.37 .407 3639 135.07 .457 0800 141.21 45 3-3'4 9"5 12417 3-36o 5545 129.46 .408 1746 135.16 3.457 9276 141.32 46 .315 6567 124.26 36' 33'6 129.56 .408 9859 135.26 458 7759 141.43 , 47 48 .316 4025 .317 1489 124.34 124.43 .362 1092 .362 8873 129.65 129.74 .409 7977 .410 6102 135.36 135.46 459 6248 .460 4743 141.54 141.64 i 49 3i7 8957 124.51 .363 6660 129.83 4" 4233 135.56 .461 3245 141.75 50 3.318 6430 124.60 364 4453 129.92 .412 2369 135.66 3.462 1753 141.86 51 319 3909 124.68 .365 2251 130.01 .413 0512 I35-76 .463 0268 '4' 97 52 .320 1392 124.77 .366 0055 130 ii .413 8660 '35-86 .463 8789 142.07 , 53 .320 8881 12486 .366 7864 I 30.20 .414 6815 135.96 4 6 4 73'7 142.18 54 321 6375 124.94 .367 5679 130.29 415 4975 136.06 .465 5851 142.29 55 3.322 3874 125.03 .368 3499 130.38 .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 12520 .369 9156 '3-57 .417 9492 136.36 .468 1492 142.61 58 .324 6403 125 29 37 6993 130.66 .418 7677 136.46 .469 0052 142.72 59 325 39*3 125.37 .371 4836 130.76 .419 5867 136.56 .469 8619 142.83 60 3.326 1448 125.46 .372 2684 130.85 .420 4064 136.66 .470 7192 142.94 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit V. 156 157 158 159 logM. Diff. l" logM. Diff. 1" log M. Diff. 1" logM. | Diff. 1". 1 2 3 4 3-470 7192 471 5772 .472 4358 473 2951 474 '550 142.94 143.05 143.16 143.27 '43-38 3-5*3 3875 .524 2864 .525 1860 .526 0863 5*6 9873 '49-75 149.87 149.99 150.11 150.23 3.578 6154 579 5588 .580 5030 .581 4480 .582 3937 '57-'7 157.30 '57-43 157.56 157.69 3.636 6351 -637 6271 .638 6202 .639 6140 .640 6087 165.28 165.42 165.56 | 165.71 165.85 , 5 6 7 8 9 3-475 '5 6 .475 8769 .476 7388 .477 6014 .478 4646 '43-49 143.60 '437' 143-82 '43-93 3.527 8890 .528 7915 .529 6947 .530 5985 53' 503' 150.35 150.47 150.59 150.71 150.83 3-583 343 .584 2876 585 2357 .586 1846 587 134* 157.82 '57-95 158.08 158.21 158.34 3.641 6042 .642 6006 .643 5978 644 5959 .645 5948 166^8 166.42 166.56 10 11 12 13 14 3-479 3*85 .480 1931 .481 0583 .481 9242 .482 7907 144.04 '44-'5 144-26 '44-37 144-48 3.532 4085 533 3'45 534 2213 .535 1288 53 6 0370 150.95 151.07 151-19 151.31 151.43 3-588 0847 589 0559 .589 9880 .590 9408 .591 8944 158.47 158.61 158.74 158.87 159.00 3.646 5946 647 5953 .648 5968 649 599* .650 6025 166.71 i 166.85 166.99 167.14 167.28 15 16 17 18 19 3.483 6579 .484 5258 485 3944 .486 2636 .487 1335 '44-59 144.70 144.81 '44-93 145.04 3-536 9459 537 8556 .538 7660 539 6771 .540 5890 '5'-55 151.67 151.79 151.91 152.04 3-59* 8488 .593 8040 .594 7600 595 7167 .596 6743 159.13 159.26 159.40 '59-53 159.66 3.651 6066 .652 6116 .653 6175 .654 6242 .655 6318 167.42 167.57 167.72 167.86 168.01 20 21 22 23 3.488 0040 .488 8752 .489 7472 .490 6198 '45-'5 145.26 '45-37 '4549 3-54' 5015 .542 4148 543 3*89 544 *43 6 152.16 152.28 152.40 152.52 3-597 63*7 598 59'9 599 55i8 .600 5126 '59-79 '5993 160.06 160.19 .656 6403 .657 6497 .658 6599 .659 6710 168.15 168.30 168.45 l68.ro 24 .491 4930 145.60 545 '59' 152.65 .601 4742 160.33 .660 6830 168.74 25 3.492 3670 145.71 3.546 0754 '52-77 3.602 4365 160.46 3.661 6959 168.89 26 493 2416 145-82 546 99*4 152.89 .603 3997 160.60 .662 7O96 ; I69.O3 27 .494 i i 68 '45-94 .547 9101 153.01 .6oj. 3637 160.73 .663 7243 ; 169.18 28 .494 9928 146.05 .548 8285 153.14 .605 3285 160.87 .664 7398 j 169.33 29 495 8695 146.16 549 7477 153.26 .606 2941 161.00 .665 7562 \ 169.48 30 31 32 .496 7468 .497 6248 .498 5035 146.28 146.39 146.50 .550 6677 .551 5883 .552 5097 153-38 i53-5i 153.63 3.607 2605 .608 2277 .609 1957 161.14 161.27 161.41 3.666 7735 1 169.62 .667 7917 169.77 .668 8108 i6q.Q2 33 34 .499 3828 .500 2629 146.62 146.73 553 43'9 554 3548 '53-75 153-88 .610 1646 .611 1342 161.54 161.68 .669 8308 .670 8516 170.07 170.22 35 .501 1436 146.85 555 *785 15400 .612 1047 ifl.M .671 8734 170.37 36 .502 0250 146.96 .556 2029 '54-'3 .613 0760 161.95 .672 8961 170.52 37 .502 9071 147-08 557 1280 '54-25 .614 0481 162.09 .673 9196 170.67 38 503 7899 147.19 558 0539 154.38 .615 02 1 o 162.22 .674 9441 170.82 39 .504 6734 '47-3' 558 9806 154.50 .615 9948 162.36 .675 9694 170.97 40 -55 5576 147.42 559 9080 154.63 .616 9693 162.50 .676 9957 171.11 41 .506 4425 '47-54 560 8361 '54-75 .617 9447 162.63 678 0228 171.27 4VJ .507 3280 147.65 561 7650 154.88 618 9209 162.77 679 0509 171.42 43 508 2143 '47-77 562 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 8772 148.11 565 4882 155.38 622 8340 163.32 683 1723 172.03 48 49 511 7662 512 6560 5'3 5464 148.23 148.34 148.46 566 4209 567 3543 568 2885 '555' 155.64 155.76 623 8144 624 7956 625 7776 163.46 163.60 163.74 684 2049 685 2384 686 2728 172-15 i i7*-33 172.48 50 5'4 4375 148.58 569 **35 155.89 626 7604 163.88 687 3082 172.64 51 52 5'5 3294 516 2219 148.70 148.81 570 1592 57' 0957 156.02 156.15 627 7441 628 7287 164.02 164.16 688 3445 689 3817 172.79 172.94 53 5'7 "51 148.93 57* 0330 156.27 629 7140 164.30 690 4198 173.10 54 518 0090 149.05 57* 97'0 156.40 630 7002 164.44 691 4588 '73-*5 55 5'8 937 '49 '7 573 9098 156.53 631 6873 164.58 692 4988 17340 56 5'9 799 149.28 574 8494 1 56.66 632 6751 164.72 6 93 5397 173.56 57 58 59 520 6951 521 5918 522 4893 14940 149 52 149.64 575 7897 576 7308 577 6727 156.79 156.92 157.04 633 6638 634 6534 635 6438 164.86 165.00 165.14 694 58.5 695 6243 696 6680 173.71 173.87 174.02 60 5*3 3875 '49-75 578 6154 '57-7 636 6351 165.28 697 7126 174.18 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. v 16C ) 161 o 162 162 ; log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". 0' 3.697 7126 174.18 3.762 1539 183.99 3.830 3147 194.87 3.902 6107 207.00 1 .698 7581 174-34 .763 2584 184.16 .831 4845 195.06 .903 8534 207.21 2 .699 8046 174.49 .764 3639 184.34 .832 6554 195.25 .905 0973 207.43 3 .700 8520 174.65 .765 4704 184.51 .833 8275 195.44 .906 3425 207.64 4 .701 9003 I 74 .8o .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 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 175-43 .771 0187 185.38 .839 7054 196.41 .912 5880 208.72 , 9 .707 1562 '75-59 .772 1315 '85-55 .840 8844 196.60 .913 8410 208.94 : 1O 3.708 2IO2 175-75 3.773 2454 185.73 3.842 0646 196.80 39'5 953 209.16 11 .709 2652 175.91 774 3603 185.90 .843 2460 196.99 .916 3509 209.38 12 .710 3211 176.07 .775 4762 186.08 .844 4286 197.19 .917 6078 209.60 | 13 .711 3780 176.22 .776 5932 186.25 .845 6123 197.38 .918 8661 209.81 14 .712 4358 176.38 777 7" 2 186.43 .846 7972 197.58 .920 1256 210.03 15 3.713 4946 176.54 3-778 8303 186.60 3-847 9833 197.78 3.921 3865 210.25 16 7'4 5543 176.70 779 955 186.78 .849 1705 197.97 .922 6487 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 .85, 5486 I98-37 .925 1770 210.92 19 .717 7392 177.18 .783 3174 187.31 .852 7394 198.57 .926 4432 211.14 20 3.718 8028 '77-34 3-784 44'8 187.49 3-853 93'4 198.76 3.927 7107 211.36 21 .719 8673 177.50 .785 5672 187.67 855 "45 198.96 .928 9795 211.58 22 .720 9328 177-66 .786 6938 187.85 .856 3189 199.16 .930 2497 2II.81 1 23 .721 9993 177.83 .787 8214 188 03 857 5'45 199.36 .931 5212 212.03 i 24 .723 0668 178.00 .788 9501 188.21 .858 7112 199.56 .932 7940 212.25 25 3.724 1352 178.15 3.790 0799 188.39 3.859 9092 199.76 3.934 0682 212.48 26 .725 2045 178.31 .791 2108 188.57 .861 1084 199.96 935 3438 212.70 27 .726 2749 178.47 .792 3427 188.75 .862 3087 200.16 .936 6207 21293 28 .727 3462 178.63 793 4757 188.93 .863 5103 200.36 937 8989 213.15 29 .728 4185 178.80 .794 6098 189.11 .864 7131 200.56 939 1785 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 J 79-45 799 i57i 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 179.78 3.801 4372 190.20 3-871 955* 201.78 3.946 8847 214.74 36 37 735 95" 737 3 a 4 179-95 i8o.n .802 5790 .803 7218 190.38 190.56 .873 1665 .874 3791 201.98 202.19 948 1738 949 4 6 44 214.97 215.20 38 .738 1136 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 .741 3631 180.78 .808 3041 191.30 .879 2414 203.01 954 6 43 216.13 42 .742 4482 180.94 .809 4525 191.48 .880 4601 203.22 955 9378 216.36 43 743 5344 181.11 .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 5369 216.82 45 3-745 7097 181.45 3.812 9042 192.04 3.884 1236 203.84 3-959 8385 217.06 46 .746 7989 181.61 .814 0570 192.23 885 3473 20405 .961 1416 217.29 1 47 48 747 8891 .748 9803 18 .78 18 .95 .815 2110 .8l6 3660 192.41 192.60 .886 5722 .887 7983 204.26 204.46 .962 4460 .963 7519 217.53 217.76 j 49 .750 0725 18 .12 .817 5222 192.79 .889 0257 204.67 .965 0592 218.00 50 3-751 1657 18 .29 3.8l8 6 795 192.98 3.890 2544 204.88 3966 3678 218.23 51 .752 2599 1 8 .46 .819 8 379 193.16 .891 4843 205.09 .967 6779 218.47 52 753 3552 18 .63 .820 9974 '93-35 .892 7155 205.31 .968 9895 218.70 53 754 45 H 18 .80 .822 1581 '93-54 .893 9480 205.52 .970 3024 218.94 54 755 5487 18 .97 .823 3199 '9373 .895 ,817 205.73 .971 6168 219.18 55 3.756 6470 183.14 3.824 4829 193.92 3.896 4167 205.94 3.972 9326 219.42 56 757 74 6 4 183.31 .825 6470 194.11 .897 6529 206.15 974 H9 8 219.66 57 .758 8467 183.48 .826 8122 194.30 .898 8905 206.36 975 5684 219.90 58 759 948i 183.65 .827 9785 194.49 .900 1293 206.57 .976 8885 220.13 59 .761 0505 183.82 .829 1460 194.68 .901 3694 206.79 .978 2IOO 220.37 60 3.762 1539 183.99 3.830 3147 194.87 3.902 6107 207.00 3-979 533 220.6l TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. '_ 164 165 166 167 logM. | Diff.l" logM. Diff. 1" logM. i Diff. 1" logM. Diff. l'. o 3 979 533<> .980 8574 .982 1833 .983 5106 .984 8394 220.62 27.0.81 221. IO 221.34 221. 5 8 4.061 6673 -063 0842 .064 5027 .065 9229 .067 3447 236.01 236.28 236.56 236.83 237.11 4-149 798 .151 2422 .152 7664 .154 2925 -155 8205 *53-57 253.88 254.19 25+^3 4-244 5537 .246 1975 *47 8434 *49 49 6 .251 1419 273.78 2-^.14 274.51 275.24 I 1 8 9 3.986 1696 987 5>I3 9*8 8345 .990 1691 .991 5051 221.83 222.O7 222.31 222.56 i 222.80 4.068 7682 .070 1933 .071 6201 .:-: 34X6 .074 4787 2 3739 237.66 2 37-94 238.22 238.50 4-157 354 .158 8822 .160 4159 .161 95,5 .163 4891 *55->4 255.46 255.78 256.10 256.42 4 2 5 2 7944 2 54 449i .256 ,06 i 2 57 7652 .259 4266 27560 *7597 *7*34 276.71 277.08 It 11 12 13 14 3.992 8427 994 1817 .995 5222 .996 8642 .998 2077 I 223.05 223.20 223.54 223.79 224.03 4-075 9106 :-- :-,! .:- -- .1 .080 2161 .081 6546 238.78 23906 2 39-34 239.62 239.90 4.165 0285 ..66 5699 .168 1132 .169 6585 .171 2056 i:- JA ! 257.0* 257.70 258.02 4.261 0902 .262 7560 .264 4240 .266 0943 .267 7669 *77-45 278.20 278.95 15 16 | 3-999 55 2 7 - : : ; *"' j 224.28 2 2 453 4.083 0948 .084 5363 i_ : : 240.46 4-172 7547 .174 3058 258.35 258.67 1-269 4417 .27, ,187 279 32 279.70 ' "* * 224.78 .085 9804 240.75 . I ~ ; ' ; * 25900 .272 7981 ' 280.08 IS .003 5965 .004 9474 225 03 1 1 : 1 ; .087 4257 .088 8728 241.03 241.32 -177 413* .178 9707 2 5933 .:-- --.- .276 1635 280.46 280.84 20 ** l^l 11 : -: 4.090 3215 241.60 4.180 5296 25998 4.277 8497 281.22 '41 . - , ': ; : v 225.78 -091 7720 241.89 .182 0905 260.31 .279 5381 281 60 "4'4 43 44 .009 0093 .OIO 3663 .on 7248 226.04 226.29 226.54 C 93 22 4 2 .094 6781 .096 1337 242.08 242.56 2 4 2 -75 .185 2182 .186 7850 260.64 ' 260.97 : 261.30 .281 2289 .282 9119 .284 6,73 281.98 282.36 282.75 25 JOI] :*_* 226.79 4.097 5911 243.04 4-188 3538 I 261.63 4-286 3149 j 283.14 26 .014 4463 11-.: ; .099 0502 2 43 33 .189 9246 261.96 .288 0149 '47 .015 8093 11-.:: .100 5110 243.62 .191 4974 262 30 .289 TI-Z 2839* : 28 . : : - i - : . 11- .-; .101 9736 243.91 .193 0722 262.63 .291 4118 284.30 4'J .018 5400 227.81 .::; _:- . 244.20 .194 6490 I 262. Q7 .293 1288 284.69 30 31 *!o2? 2769 228.06 ***-3* 4.104 9040 .106 3718 2 44-49 244.78 4.196 2278 .197 8c86 263.30 ! 265.64 4.294 8381 .296 5498 285.08 285.47 34 .022 6476 228.58 .107 8414 245.08 .199 3915 26398 .298 2638 285.87 33 .024 CI 99 228.84 .109 3127 H5-37 .200 9764 26.132 .299 9801 286.26 34 .025 3937 22909 .11: 24567 .202 5633 26466 .301 6990 | 286.66 3o 30 37 ; 38 4-026 7691 .028 1460 029 5245 030 9045 229.35 229.62 229.88 230.14 .112 2607 ' "3 7374 .115 2158 .116 6960 245.96 246.26 246.55 246.85 4-204 1523 205 7435 .20^ 3363 .208 9314 265 oo i 2b 5-34 265.68 266.cs 4.303 4201 i .305 1436 ; .306 8695 .308 5978 287.05 I'-.C 287.85 288.25 3'J 032 2861 230.40 .!<] i-- : 247-15 .210 5286 266.37 .310 3285 i 288.65 40 033 6693 230.66 .II 9 6618 247-45 4-212 1278 ; 266.71 4.312 0616 1 289.05 41 C 35 54 230.92 .121 1474 247-75 .213 7291 267.06 3'3 797 i 289.45 4-4 43 036 4. f ::- flSj i: ; i * 231.45 122 6348 .124 1239 248.05 248.35 2 '5 3325 .216 9379 26740 267.75 3'5 535 ' 3'7 2753 289.86 290.26 44 039 2177 231.71 .125 6149 248.65 .218 5455 268.10 319 0181 290.6 7 4.5 040 6088 231.97 127 1077 : 248-95 .220 1551 268.44 320 7633 291.07 4t', 042 0015 ! 232.24 128 6023 249.25 221 7668 268.79 322 5110 291.48 i 47 43 3957 i 232.51 130 0988 24956 223 3806 26914 324 26,, ! 49 44 7915 3-77 046 1890 233.04 131 5970 133 0971 249.86 250.17 226 6146 269.50 269.85 326 0177 ( 3*7 7688 j 292 30 92.7I 50 047 5880 2 333' 134 5990 25047 228 2347 270.20 329 5263 i *93-'3 i 51 048 9887 2 33-57 136 1028 25078 22 9 8 7 270-55 331 2863 i 2 93-54 I 5'4 050 3909 233.84 137 6084 251.08 231 4814 270.91 333 0487 *93-95 i 53 .34 051 7948 053 2003 234.11 234.38 I 39 II 5 8 ! 140 6251 2 5>-39 251.70 233 1079 2 34 7366 271.27 271.62 334 8137 336 5812 29437 294-79 55 054 6074 _! 234.65 .142 1362 252.01 236 3674 271 98 338 35" 295.20 58 056 0161 234.92 057 4264 235.19 058 8384 235.46 143 6492 >45 l6 4* 146 6808 25232 252.63 25294 :: MIOJ 2 39 6354 241 2727 272.34 272.70 273.06 340 ,236 341 8986 I 343 6762 , 295.62 296.0^ 59 060 2520 235.73 148 1994 242 9121 273.42 345 4562 296.89 60 061 6673 236.01 149 7198 253.57 244 5537 27378 347 1388 297.31 i 607 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 168 169 170 171 log M. Diff. I". logM. | Diff. 1". log M. Diff. 1". logM. Diff. 1". 4-347 2388 297.31 4-459 i*4* 325.07 4.581 9445 358.31 4.717 9835 398.87 ; 2 3 , 4 .349 0240 .350 8117 .352 6019 354 3948 297-74 298.16 298.59 299.02 .461 0761 .463 0311 .464 9891 .466 9501 3^5-57 326.08 326.59 327 10 .584 0962 .586 2516 .588 4106 590 5734 358.92 359-53 36o.i| 360.76 .720 3790 .722 7790 .725 1835 .727 5926 399.62 400.38 | 401.14 ; 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 9'oo 362.00 .732 4245 403.43 . 1 7 359 7888 300.31 .472 8517 328.64 597 0838 362.62 734 8474 404 19 i 8 .361 5919 3-75 .474 8250 329.15 599 2615 36325 737 2749 404.96 , 9 3 6 3 3977 301.18 .476 8015 329.67 .601 4428 363.88 739 77 405.74 10 4.365 2061 301.62 4.478 7811 330.19 4.603 6280 364.50 4.742 1438 406.52 11 .367 0171 302.05 .480 7637 330.71 .605 8169 365.14 744 5852 407.30 : 12 13 .368 8308 .370 6470 30249 302.93 .482 7495 484 7385 33I-23 33 J -75 .608 0096 .610 2061 365.77 366.40 747 3J4 749 4822 408.08 ! 408.87 i 14 37* 4 6 59 303.37 .486 7306 332.28 .612 4064 367.04 .751 9378 409.66 15 4-374 2875 303.81 4.488 7258 332.81 4.614 6106 367.68 4-754 398i 410.45 16 .376 1117 304.26 .490 7242 333-33 .616 8186 368.32 .756 8632 411.24 17 377 9386 304.70 .492 7258 333.86 .619 0304 368.96 759 333 412.04 18 379 7 6 8i 35- I 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 21 4-383 4352 .385 2728 306.04 306.49 4.498 7498 .500 7642 335-46 336.00 4.625 6892 .627 9166 370.91 37I-56 .769 2606 414.46 415.27 22 387 "3i 306.94 .502 7818 33 6 -54 .630 1480 372.21 771 7547 416.08 23 .388 9561 37-39 .504 8026 337-08 .632 3832 372.87 774 2536 4l6 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 4l8.5 4 26 394 5 OI 5 308.76 .510 8847 338-7I .639 1127 374-86 781 7799 4'9-37 27 28 396 3554 .398 2121 309.21 30967 .512 9186 5'4 9558 339.26 339.80 .641 3639 .643 6190 375-52 376.19 .784 2986 .786 8222 420.20 421.03 29 .400 0715 310.13 .516 9962 340.35 .645 8781 376.86 789 3509 421.86 30 4.401 9337 310.59 4.519 0400 340.91 4.648 1413 377-53 4.791 8846 422.70 31 .403 7986 311.06 .521 0871 34I-4 6 .650 4085 378.21 794 4233 423 54 32 .405 6667 311.52 .523 1376 342.02 .652 6798 378.89 .796 9671 424.39 33 .407 5368 311.99 .525 1913 342-57 .654 9552 379-57 .799 5160 425.24 34 .409 4102 3^-45 .527 2484 343-'3 657 2346 380.25 .802 0700 426.09 35 4.411 2863 3129* 4.529 3089 343.69 4.659 5182 380.93 4.804 6291 426.95 36 37 .413 1652 .415 0469 3I3-39 313.86 .531 3728 533 44 344.26 344.82 .661 8059 .664 0977 381.62 382.31 .807 1934 .809 7628 427.81 428.67 38 .416 9315 3H-33 535 5>o6 345-39 .666 3936 383.00 .812 3374 429-53 39 .418 8189 314.80 537 5846 345-95 .668 6937 383-70 .814 9172 430.40 40 4.420 7091 315.28 4.539 6620 346-52 4.670 9980 38439 4.817 5022 431.28 41 .422 6022 3I5-75, .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 9360 386.50 .825 2889 433-91 44 .428 2987 3I7-I9 .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 387.92 4.830 5065 435.69 46 .432 1108 318.16 .552 1989 349.98 .684 9121 388.63 .833 1234 436.59 47 .434 0212 318.64 554 35 350-56 .687 2460 389-34 835 7456 437.48 48 435 9345 319.13 .556 4056 35i-i5 .689 5842 390.06 .838 3732 438.38 49 437 8507 319.61 558 5H3 351-73 .691 9268 390.78 .841 0062 439.29 50 4-439 7 6 98 320.10 4.560 6264 352-32 4.694 2736 391.50 4.843 6446 440.20 i 51 .441 6919 320.59 .562 7421 352.91 .696 6248 392.23 .846 2886 441.11 52 443 6169 321.08 .564 8614 353-50 .698 9803 392.96 .848 9380 442.03 53 445 5449 321.58 .566 9842 354-io .701 3402 393-68 .851 5929 442-95 54 447 4758 322.07 .569 1106 354-69 .703 7046 394.42 .854 2533 443-87 55 4.449 4097 32*-57 4.571 2405 355-29 4.706 0733 395-15 4.856 9193 44480 56 .451 3466 323.06 573 374i 355-89 .708 4464 395-89 859 599 44573 57 453 2865 323 56 575 5"3 356.49 .710 8240 396.63 .862 2680 446.66 58 455 "94 324.06 577 6521 357-io .713 2060 39738 .864 9508 447.60 59 457 '753 324.56 579 7965 357.70 715 5925 398.12 .867 6392 448.54 60 4.459 1242 325.07 4.581 9445 358.31 4-7I7 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. IT. ITc 17< E? 17{ ) logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 2 3 4 4.870 3333 .873 0331 .875 7386 :Si! nil 449.49 450.44 451-39 452-35 453-3' 5.043 3285 .046 4191 .049 5171 .052 6226 55 7356 5 '4-47 5'5-7' 516.96 518.21 5'9-47 5.243 3165 .246 9276 .250 5488 .254 1802 .257 8218 601.00 602.69 604.38 606.08 607.80 5.480 1373 484 4765 .488 8304 493 1989 497 5823 722.00 724-42 726.87 72933 731.80 5 6 7 8 9 4.883 8896 .886 6182 .889 3526 .892 0929 .894 8391 454.28 455-25 456.23 457-20 458.19 5.058 8562 .061 9843 .065 I2O2 .068 2637 071 4'49 520.73 522.00 52328 524-56 525-85 5.261 4738 .265 1361 .268 8089 .272 4922 .276 1860 69-53 611.26 613.00 614.75 616.52 5.501 9806 .506 3939 .510 8223 5'5 2659 .519 7248 734-3 736.81 739-33 741.87 744-44 10 11 12 13 14 4.897 5912 .900 3492 .903 1132 905 8831 .908 6591 459-17 460.16 461.16 462.16 463.16 5-074 5738 .077 7406 .080 9151 .084 0976 .087 2879 527.14 528.44 532^8 5.279 8904 .283 6055 .287 3313 .291 0680 .294 8154 618.29 620.08 621.87 623.67 625-49 5.524 1992 .528 6890 533 1946 537 7158 542 2529 747.02 749-6i i 752-23 i 75486 j 757-5' 15 4.911 4411 464.17 5.090 4862 533-7' 5.298 5738 627.31 5.546 8060 760.18 16 17 18 19 .914 2291 .917 0233 .919 8235 .922 6299 465.18 466.20 467.22 468.25 .093 6924 .096 9067 .100 1290 i3 3594 535-04 536.38 537-73 539.08 .302 3432 .306 1237 .309 9152 .313 7179 629.15 631.00 632-85 634.72 55' 375' 555 9 6 5 .560 5621 .565 1802 762.87 ! 765.58 | 768.3' ! 771.05 20 4.925 4425 469.28 5.106 5980 540.44 5-3'7 53'9 636.60 5.569 8148 773.82 21 .928 2612 470.31 .109 8447 541.81 .321 3571 638.49 .574 4661 776.61 i 22 .931 0862 471-35 .113 0997 325 1938 640.39 579 i34i 779-4' 23 933 9'74 472.39 .116 3629 544-56 .329 0418 642.30 .583 8190 782.24 , 24 .936 7549 473-44 .119 6344 545-95 .332 9014 644-23 .588 5210 785.08 25 4-939 5987 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 34 6554 648.11 597 9764 790.84 | 27 28 29 945 353 .948 1682 95' 0375 476.61 477-68 478.75 .129 4992 .132 8044 .136 1181 550.15 551-57 552.99 344 5499 348 4562 352 3744 650.07 652.04 654.02 .602 7302 .607 5014 .612 2903 793-75 j 796.68 799.63 3O 4-953 9'32 479-83 5.139 4403 554-42 5-356 3045 656.01 5.617 0970 802.60 31 .956 7954 480.91 .142 7711 555-86 .360 2466 658.02 .621 9216 805.60 32 959 6841 48i.99 .146 1106 557-3 .364 2007 660.04 .626 7642 808.62 33 9 6 2 5793 483.08 .149 4588 558.75 .368 1671 662.07 .631 6250 811.66 34 .965 4811 484.18 .152 8157 560.21 372 1456 664.! I .636 5041 814.72 35 4.968 3894 485.28 5.156 1813 561.68 5.376 1364 666.17 5.641 4017 817.81 ' 36 .971 3044 486.38 '59 5558 563.16 .380 1396 668.24 .646 3179 820.92 i 37 .974 2260 487-49 .162 9392 564.64 384 '553 670.32 .651 528 824.05 1 38 977 '543 488.61 .166 3315 566.13 388 1834 672.41 .656 065 827.21 39 .980 0893 489-73 -'69 7328 567.63 .392 2242 674.52 .661 793 830.39 40 4.983 0311 490.85 .173 1431 569.13 .396 2777 676.64 5.666 713 833.60 41 985 9795 491.98 .176 5624 570.65 .400 3439 678.77 .671 825 836-83, 42 .988 9348 493.12 .179 9908 572.17 .404 4229 680.92 .676 2132 840.08 43 .991 8970 494.26 .183 4284 573-7 .408 5149 683.08 .681 2635 843-36 ! 44 994 8659 495.40 .186 8752 575-24 .412 6199 685.25 .686 3336 846.67 45 46 47 .997 8418 .000 8246 .003 8143 496.55 497-7' 498.87 .190 3312 .193 7966 .197 2713 576.78 578.34 579.90 .416 7379 .420 8692 .425 0136 687.44 689.64 691.85 5.691 4236 .696 5337 .701 6640 850.00 853.36 856.75 48 .006 8m 500.04 .200 7554 581.47 429 '7'4 694.08 .706 8147 860.16 19 .009 8148 501.21 .204 2489 58305 433 3427 696.33 .711 9860 863.60 5O .012 8256 502.39 .207 7520 584.64 437 5274 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 504.76 .214 7868 587-84 445 9378 703.15 .727 6247 87408 53 .021 9006 55-95 .218 3186 589.45 .450 1636 705-45 .732 8798 877.63 54 .024 9399 507-15 .tzi 8602 591.07 454 43* 707.77 738 1563 881.21 55 .027 9864 508.36 .225 4116 59271 .458 6568 7 1 o. i o 743 4544 8^4.82 56 .031 0402 59 57 228 9727 59435 .462 9244 712.45 .748 7742 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 7I7-I9 759 4798 895-83 59 .040 2454 513.24 239 7156 599.32 475 8125 7'9-59 .764 8659 899-56 6O .043 3285 5'4-47 243 3165 601.00 480 1373 722.00 .770 2745 903.31 609 TABLE VI, I or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. v 17 i 177 17 1 17 > log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 5.770 2745 903.3 6.144 6239 1205.3 6.672 5724 1808. 8 7.575 4640 3619 1 775 7058 907.1 .151 8807 I2I2.O .683 4709 1824.0 597 3596 3680 2 .781 1599 910.9 59 1733 I2I8.8 .694 4613 1839.5 .619 6295 3744 3 4 .786 6370 .792 1374 914.8 918.7 . 66 5070 . 73 8823 1225.7 1232.7 75 5454 .716 7248 1855.3 1871.3 .642 2868 .665 3452 3809 3877 5 5.797 6612 922.6 6. 8 1 2997 1239.8 6.728 ooio 1887.5 7.688 8192 3948 6 1 .803 2086 808 7708 926.6 930.6 . 88 7597 Q6 2.628 1246.9 739 3758 1904.1 .712 7239 4021 8 .814 3751 934.6 . 03 8095 I2|4.I 1261.4 *75 5 9 .762 4279 1938.2 .761 8913 4097 4176 9 .819 9946 938.6 . 1 1 4002 1268.8 .774 1090 1955.6 .787 1889 4257 10 5.825 6386 942.7 6. 19 0354 1276.3 6.785 8958 1973-4 7.812 9876 4343 11 .831 3073 946.8 . 26 7158 1283.8 797 794 1991.5 .839 3075 443i 12 .837 0008 951.0 34 44'9 I2 9 1.5 .809 7946 2OIO.O .866 1702 4524 13 .842 7195 955- 2 . 42 2142 1299.2 .821 9106 2028.8 .893 5986 4620 14 .848 4634 959-5 5 333 1307.1 .834 1404 2048.0 .921 6170 4720 15 5.854 2329 963.7 6. 57 8997 I3I5.0 6.846 4863 2067.5 7.950 2513 4825 16 .860 0282 968.0 . 65 8139 1323.0 .858 9503 2087.3 7-979 5292 4935 17 .865 8495 972.4 - 73 7766 I33I.I .871 5348 2107.6 8.009 4802 5050 18 .871 6970 976.8 . 81 7884 J 339-4 .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 7 011 5428 21 .889 3993 990.2 .306 1244 1364.7 .923 1261 2192.8 .136 6857 5568 22 .895 3542 994.8 .314 3387 1373-3 .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 9247 1391.0 .963 2073 2261.4 .240 9679 6032 25 S-9'3 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 ^334-3 354 3 2 9 8 6580 28 .931 6688 1022.9 3 6 4 745 1 1427.6 .018 6437 2359.7 394 4 20 5 6786 29 937 8213 1027.8 373 3395 1437.1 .032 8796 2385-7 435 784* 7004 30 5.944 0030 1032.7 6.381 9910 1446.7 7.047 2729 24.12.2 8.478 5044 7238 31 .950 2144 1037.6 .390 7005 1456.4 .061 8271 2439.4 .522 6731 7488 32 33 .956 4556 .962 7269 1042.6 1047.7 399 4687 .408 2965 1466.2 1476.2 .076 5458 .091 4329 2467.1 2495.4 .568 3920 615 7739 7755 8042 34 .969 0287 1052.9 .417 1846 1486.4 .106 4921 2524.5 .664 9442 8352 35 5-975 3 6 i3 1058.0 6.426 1337 1496.7 7.121 7276 2554.2 8.716 0431 8686 36 .981 7249 1063.2 .435 1449 1507.0 137 H34 2 5 8 4 .6 .769 2286 9048 37 .988 1198 1068.4 .444 2191 1517.6 .152 7440 2615.8 .824 6779 9441 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 42 .014 0192 .020 5756 1089.9 1095.4 .481 1620 .490 5641 1561.3 1572-6 .217 0850 .233 6796 2748.3 2783.5 .073 5974 .144 0401 11429 12064 43 44 .027 1652 .033 7885 IIOI.O 1106.7 .500 0346 .509 5746 1584.1 1595.8 .250 4884 .267 5170 2819.7 2856.8 .218 5102 .297 4963 12773 13572 45 6.040 4457 1 112.4 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 1 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 3530 1656.8 .356 1692 3058.1 .785 6758 19741 50 6074 2535 1141.7 6.568 3320 16696 7-374 6475 3IOI.7 9909 8535 21715 51 .081 1219 1147.7 .578 3881 1682.4 393 39i8 3146.8 10.047 1256 24127 52 .088 0269 1153.8 .588 5227 1695.6 .412 4099 3193.0 .200 5829 27144 ; 53 .094 9687 1 160.0 .598 7368 1708.9 43 * 797 3 2 4-7 .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 344 6 -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 TABLE VII, For finding the True Anomaly in a Parabolic Orbit when v is nearly 180. VJ AO Diff. to AO Diff. w AO Wff. 1 / / i H 1 , 155 5 10 15 20 25 3 23-09 19-74 16.43 13.17 9-95 6.77 3-35 3.31 3.26 3.22 3.18 3.14 160 5 10 15 20 25 I 6.70 5-33 ua 1-33 0.04 $ i-33 1.31 1.29 1.26 165 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 j 0.69 155 30 35 40 45 50 55 3 3-63 0.54 2 57-49 54.48 51.51 48.58 3.09 3.05 3.01 2.97 III 160 30 35 40 45 50 55 o 58.78 57-54 56.31 55.11 53-93 52.77 1.24 1.23 1.20 1.18 1.16 1.14 166 10 20 30 40 50 11.22 10.57 9-95 9.36 8.80 8.26 0.65 0.62 0.59 ' 0.56 0.54 0.51 156 5 10 15 20 25 2 45.69 42.84 40.07 37.26 34-53 3i-83 55? 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 1.13 1. 10 1.08 i. 06 1.05 1.02 167 10 20 SO 40 50 7-75 7.27 6. 8 1 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 a 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 40.33 I.OI 0.99 0.97 0.96 0.93 0.92 168 10 20 30 40 50 o 5.20 484 4-5 4.20 3.90 3.62 0.36 0.33 0.31 0.30 0.28 1 0.26 157 5 10 15 20 25 2 14.00 11.59 9.22 6.89 4.58 2.31 2.41 2.37 2-33 2.31 2.27 2.23 162 5 10 15 20 25 o 39.41 38.51 37.62 3 6 -75 35.90 35.06 o 90 0.89 0.87 0.85 0.84 0.82 169 10 20 30 40 50 o 336 3.11 1.88 2.66 2.46 2.27 0.25 0.23 0.22 0.20 1 O.I 9 0.18 157 30 35 40 45 50 55 2 0.08 I 57.8 9 55-7* 53-57 51.46 49-39 2.19 2.17 2.15 2. 1 1 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 .09 : F .62 .48 35 0.17 \ 0.16 0.14 0.14 . 0.13 O.I 2 158 5 10 i 47-35 45-34 43-35 2.01 99 an 163 5 10 o 29.62 28.90 28.20 0.72 0.70 0.69 171 10 20 30 o -23 .12 1.02 O.I I 0.10 0.09 15 20 25 41-39 39-47 37-57 .90 . 9 2 .90 8? 15 20 25 27.51 26.83 26.16 0.68 0.67 0.65 40 50 0.93 0.84 0.76 0.09 0.08 0.08 158 30 35 40 45 50 55 i 35.70 33-87 32.06 30.28 28.52 26.80 .0 / .83 .81 .78 .76 72 163 30 35 40 45 50 55 o 25.51 24.88 24.25 23.64 23.04 22.45 0.63 0.63 0.6 1 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 i 0.06 0.05 0.05 0.04 159 5 10 15 20 i 25 159 30 35 40 45 50 55 i 25.10 ^3-43 21.78 20.16 18.57 17.00 i 15.45 1 3-94 12.44 10.97 9-53 8.10 7 .67 f 5 .62 59 -57 55 5' 5 1.47 1.44 1.43 164 5 10 15 20 25 164 30 35 40 45 50 55 o 21.88 21.31 20.76 20.22 19.69 19.18 o 18.6" 18.17 17.69 17.21 16.75 16.29 0.57 0.55 0.54 0.53 0.51 0.51 0.50 0.48 0.48 0.46 0.46 0.44 173 10 20 30 40 50 174 175 176 177 178 179 o 0.35 0.31 0.27 0.24 0.21 0.19 o 0.16 0.07 0.02 O.OI o.oo 0.00 0.04 0.04 0.01 o.oi 0.02 o.o: 0.09 0.01 O.OI O.OI 0.00 o.oc 160 i 6.70 1.40 165 o 15.85 180 o o.oo TABLE VIII, For finding the Time from the Perihelion in a Parabolic Orbit. t> log A T Diff. V log A* Diff. t, log N Diff. / 1 / 30 1 0.025 57 6 3 .025 5749 .025 5707 f 30 30 31 O.O2O 7913 .020 6368 .020 4802 '545 1566 60 30 61 0.008 8644 .008 6458 .008 4277 2186 2181 30 2 30 .025 5638 .025 5542 .025 5418 69 96 I2 4 152 30 32 30 .020 3215 .020 1607 .019 9979 1587 1608 1628 1649 30 62 30 .008 2103 .007 9934 .007 7774 2174 2169 2160 2153 1 3 i 30 0.025 5266 .025 5087 179 2O6 33 30 0.019 8330 .019 6662 1668 63 30 0.007 5621 .007 3477 2144 4 .025 4881 34 .019 4974 l688 64 .007 1343 21 34 30 5 .025 4647 .025 4386 ll\ 30 35 .019 3267 .019 1540 1707 1727 30 65 .006 9220 .006 7108 2123 21 12 30 .025 4097 III 30 .018 9795 '745 1765 30 .006 5008 21OO 2086 6 30 7 30 8 30 0.025 3781 02 5 3437 .025 3066 .025 2668 .025 2243 .025 i 791 344 371 398 425 & 36 30 37 30 38 30 0.018 8030 .018 6248 .018 4448 .018 2629 .018 0794 .017 8941 1782 1800 1819 1835 1853 1869 66 30 67 30 68 30 0.006 2922 .006 0849 .005 8792 .005 6750 .005 4725 .005 2717 2073 2057 2042 2O25 2OO8 1988 9 30 10 o 30 0.025 1311 .025 0805 .025 0271 .024 9711 506 534 560 39 30 40 30 0.017 77 2 ' .017 5186 .017 3283 .017 1365 1886 1903 1918 69 30 70 30 0.005 7 i 9 .004 8760 .004 6811 .004 4884 1969 '949 1927 11 .024 9124 IS 41 .016 9432 '933 71 .004 2980 !*8o 30 .024 8510 614 641 30 .016 7483 1949 1963 30 .004 noo looO 1855 12 30 13 SO 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 9245 .003 7416 .003 5613 .003 3839 .003 2094 .003 0380 1829 1803 '774 '745 1714 1682 15 30 16 0.024 3466 .024 2641 .024 1789 825 8 5 8 45 30 46 0.015 3450 o'5 '394 .014 9326 2OC6 2068 75 30 76 0.002 8698 ,OO2 7049 .002 5433 1649 1616 30 17 .024 091 i .024 0008 903 30 47 .014 7247 .014 5157 2079 2090 30 77 .002 3854 .002 2311 '579 '543 30 .023 9079 929 954 30 .014 3057 2IOO 21IO 30 .002 0806 1505 1465 18 30 19 30 20 30 0.023 8125 .023 7145 .023 6140 .023 5109 .023 4054 .023 2973 980 1005 1031 1055 1081 1105 48 30 49 30 50 30 0.014 0947 .013 8827 .013 6698 .013 4561 .013 2416 .013 0263 2I2O 2129 2137 2I 4S 2153 2IOO 78 30 79 30 80 30 o.ooi 9341 .001 7917 .001 6535 .001 5196 .001 3903 .001 2656 143.4 1382 '339 1293 1247 1198 21 30 22 0.023 '868 .023 0738 .022 9584 1130 "54 51 30 52 0.012 8103 .012 5936 .OI2 3764 2167 2172 81 30 82 o.ooi 1458 .001 0309 .000 9211 "49 1098 30 .022 8405 "79 30 .012 1585 2179 30 .000 8l66 1045 23 30 .022 7202 .022 5975 1203 1227 1251 53 30 .OH 9402 .on 7215 2183 2187 2191 83 30 .000 7175 .000 6240 99' m 24 30 25 30 0.022 4724 .022 3449 .022 2151 .022 0829 1275 1298 1322 54 30 55 30 o.oi i 5024 .on 2829 .OH 0632 .010 8432 2195 2197 22OO 84 30 85 30 o.ooo 5364 .000 4546 .000 3790 .000 3096 818 756 * 9 f 26 30 .021 9484 .021 8116 1368 1390 56 30 .010 6231 .010 4029 22OI 2202 2202 86 30 .000 2468 .000 1906 02o 562 ! 493 27 0.021 6726 57 o.oio 1827 87 o.ooo 1413 30 28 30 29 30 .021 5312 .O2I 3876 .021 2418 .021 0938 .020 9436 1414 1436 1458 1480 1502 1523 30 58 30 59 30 .009 9625 .009 7424 .009 5225 .009 3028 .009 0834 22O2 2201 2199 2197 2194 2190 30 88 30 89 30 .000 0990 .000 0639 .000 0363 .000 0163 .000 0041 423 35' 276 200 122 41 30 O.O2O 7913 60 0.008 8644 90 0.000 0000 l TABLE For finding the Time from the Perihelion in a Parabolic Orbit V log A" Diff. log A'' Diff. * log A"' I 'iff. 90 30 91 o.ooo oooo 9-999 9876 .999 9507 124 120 30 121 9.963 1069 .962 0074 .960 8971 10995 11 103 150 30 151 9.889 0321 .887 8738 .886 7*59 1,583 11479 30 92 30 999 8893 .999 8039 999 6944 854" 1095 1331 30 122 30 959 77H 958 6454 .957 5046 11207 11310 11408 11*04 30 152 30 .885 5887 .884 4627 .883 3481 11372 11260 11146 11026 93 30 94 30 95 30 9.999 5613 999 44 6 .999 2246 999 0215 998 7955 .998 5468 1800 2031 2260 2487 2711 123 30 124 30 125 30 9956 3542 955 '945 954 0*58 .952 8483 .951 6624 .950 4684 "597 11687 11775 11859 11940 12018 153 30 1 154 oj 30 155 30 9-882 2455 1 .88, ,552 .880 0775 .879 0129 .877 9616 .876 9242 10903 10777 10646 10513 '374 I 10232 96 30 97 30 98 30 9.998 2757 .997 9824 .997 6669 997 3*97 .996 9708 .996 5906 2933 3'55 337* 3589 3802 4015 126 9.949 2666 30 .948 0573 127 ! .946 8408 30 .945 6174 128 .944 3875 30 .943 1513 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 947 9307 99 30 9.996 1891 995 7666 4225 129 | 9 941 9092 30 .Q4.Q 66 1 ^ 1*477 159 30 9.870 0792 .869 16^2 100 30 101 30 995 P34 994 8596 994 3755 993 871* 443* 4638 4841 5043 5*4* 130 30 131 30 939 4 8 5 .938 1506 .936 8881 935 6213 12530 I2 579 12625 12668 12707 160 30 161 30 .868 2683 ! .867 3886 ' *797 .866 5266 I 620 ' .865 6827 *439 102 30 103 30 104 30 9.993 5470 .992 8031 .992 2397 .991 6570 .991 0553 .990 4347 5439 5634 5827 6017 6206 6-01 132 30 133 30 134 30 9934 35 6 933 0763 .931 7987 .930 5183 9*9 *353 9*7 95 01 12743 12776 12804 12830 12852 12871 162 30 163 30 164 u 30 9.864 8570 .864 0500 .863 2620 7jo .862 4932 861 79 Ull .861 0145 ' 2 94 105 9.989 7956 U 3V A -- 135 9.926 6630 12885 165 9-860 3053 ; 30 0570 30 9*5 3745 . 5_ - 30 *59 oi"4 i AAC., 1O6 .988 4622 6 75 8 136 .924 0848 12^97 166 o .858 9482 ! , 30 107 987 7685 .987 0571 6937 7114 30 137 .922 7943 9*i 535 1 2905 1290!! I 2.OOQ 30 167 :^ 7 675 6260 . 30 .986 3281 7290 30 .920 2126 1 iyw^ 12906 30 .857 0704 ^g^ : 108 30 109 30 110 30 9.985 5819 .984 8186 .984 0385 .983 2418 .982 4288 .981 5996 7402 7633 7801 7967 8130 8292 O. r . 138 30 139 30 140 30 9.918 9220 .917 6321 9 l6 3433 .915 0559 .913 7703 .912 4870 I289 9 12888 12874 12856 12833 12808 168 30 169 30 17O 30 9.856 4875 .855 9 266 .855 3878 .854 8714 854 3775 853 9065 5609 5388 | 5164 4939 4710 448, 111 30 112 30 113 30 9.980 7545 979 8938 .979 0177 .978 1264 .977 2202 .976 2993 5451 8607 8761 8 9 13 9O62 9209 141 30 142 30 143 30 9.911 2062 .909 9283 .908 6538 .907 3831 .906 1164 .904 8542 12779 1*745 12707 12667 12622 12571 171 30 172 30 173 30 9.853 4584 .853 335 .852 6319 .852 2538 .851 8994 .851 5687 4249 4016 378i i 3544 i 3307 3067 114 30 115 30 116 30 9.975 3640 .974 4145 973 451 97* 4739 .971 4833 .970 4796 9353 9495 9635 977' 9906 10037 144 30 145 30 146 30 9.903 5969 .902 3449 .901 0985 .899 8582 .898 6243 .897 3972 12520 12464 12401 1*339 12271 12198 174 30 175 30 176 30 9 851 2620 850 9794 .850 7209, .850 4868 .850 2770 .850 0917 2826 2 5 *1 117 30 118 30 ; 119 30 9.969 4629 .968 4337 .967 3920 .966 3382 .965 2726 .964 1954 10167 10292 10417 10538 10656 10772 - -.00. 147 30 148 30 149 30 9.896 1774 .894 9652 .893 7610 .892 5652 .891 3782 I .890 2004 I2I22 12042 II058 11870 \\lll 177 30 178 30 179 30 9.849 9309 .849 7048 .849 6*33 .849 5966 .849 5346 .849 4974 1361 1115 867 620 124 120 9.963 1069 10005 150 9.889 0321 180 9.849 4850 TABLE IX, F >r finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity X A Diff. B Diff. c B' Diff. C' o.oo " o.ooo " 0.000 o.ooo " o.ooo 1 o.oo 0.00 o.ooo 0.000 0.000 o.ooo 2 O.OI O.OI 0.000 o.ooo 0.000 o.oco 3 0.05 0.04 0.000 o.ooo o.ooo 0.000 4 0.12 0.07 0.000 o.ooo o.ooo 0.000 O.I I 5 0.23 , 0.000 0.000 000 0.000 6 o-39 ' o.ooo o.ooo o.ooo o.ooo 7 0.62 0.23 o.ooo o.ooo o.ooo 0.000 8 0.93 0.31 o.ooo o.ooo o.ooo o.ooo 1 9 1.33 0.40 o.ooo o.ooo o.ooo o.ooo 0.49 10 1.82 o.oo 0.000 o.ooo o.ooo 0.000 11 2.42 o.ooo o.ooo o.ooo 0.000 12 3.14 0.72 o.ooo o.ooo o.ooo o.ooo 13 3-99 0.85 o.ooo o.ooo o.ooo o.ooo 14 4-99 I.OO 1.14 O.OO I o.ooo 0.001 o.ooo 15 6.13 O.OO I o.ooo O.OO I o.ooo 16 7-43 1.30 O.OO2 .001 o.ooo O.OO I .000 o.ooo 17 8.90 '47 O.OO2 .000 o.ooo O.OO2 .001 o.ooo 18 19 10.55 12.40 i 65 1.85 2.05 0.003 0.004 .001 .001 .001 o.ooo o.ooo O.OO2 0.003 .000 .001 .001 o.ooo o.oco 20 14.45 0.005 o.ooo 0.004 o.ooo 21 16.70 2.25 0.006 .001 o.ooo 0.005 .001 o.ooo 22 19.18 2.48 0.008 .002 o.ooo 0.006 .001 o.ooo 23 21.89 2.71 O.OIO .002 o.ooo 0.008 .002 o.ooo 24 24.83 2.94 3.20 O.OI 2 .002 .002 0.000 O.OIO .002 .002 o.ooo 25 28.03 0.014 o.ooo O.OI2 o.ooo 26 27 28 29 3-48 35.20 39.19 43-47 3-45 3.72 3-99 428 4-57 0.017 O.O2O 0.025 0.030 .003 .003 .005 .005 .005 o.ooo o.ooo o.ooo 0.000 0.014 0.017 0.020 0.024 .002 .003 .003 .004 004 o.ooo o.ooo o.ooo o.ooo 3O 48.04 0.035 .006 o.ooo 0.028 o.ooo 31 52.91 8 0.041 .006 o.ooo 0.033 .005 .006 o.ooo 32 33 34 58.09 63.59 69.42 ft 0.047 0.055 0.064 .008 .009 .009 o.ooo 0.000 0.000 0.039 0.045 0.052 .006 .007 .008 o.ooo o.ooo 0,000 35 36 75-57 82.07 6.50 0.073 0.084 .Oil o.ooo o.ooo 0.060 0.068 .008 o.ooo o.ooo 37 38 39 j 88.92 96.12 103.68 6.85 7.20 7.56 7-93 0.096 0.109 0.123 .012 .013 .014 .016 o.ooo 0.000 0.000 0.078 0.088 O.I 00 .010 .010 .012 .013 o.ooo o.ooo 0.000 40 41 ! 42 43 44 .1.11.61 119.92 28.62 37.70 47,18 8.31 8.70 9.08 9.48 9.87 0.139 0.156 0.175 0.196 0.2 1 8 .017 .019 .021 .022 .025 o.ooo o.ooo 0.000 0.000 0.000 0.113 0.127 0.142 0.159 0.177 .014 .015 .017 .018 .020 o.ooo 0.000 0.000 0.000 0.000 45 46 47 57-oj 67.34 78.04 10.29 10.70 0.243 0.269 0.298 .026 .029 o.ooo o.ooo 0.000 0.197 0.219 0.242 .022 .023 0.000 0.000 o.ooo 48 89.16 II. 12 0.328 .030 0.000 0.267 .025 o.ooo 49 00.71 11.98 0.361 .033 .036 o.ooo 0.294 .027 .029 o.ooo 1 50 51 52 53 54 12.69 25 10 37-95 51.25 65.01 I2. 4 I 12.85 t3.30 13.76 I4.2O 0-397 0.436 0.477 0.521 0.567 .039 .041 .046 .050 0.000 0.000 001 o.oo i O.OO I 0.323 0-354 0.388 0.424 0.462 .031 .034 .036 .038 .040 o.ooo o.ooo o.ooo 0.000 0.000 55 56 57 58 59 279.21 293.88 309.02 324.62 340.70 14.67 15.14 I5.6O 16.08 16.56 0.617 0.671 0.727 0.787 0.851 .054 .056 .060 .064 .068 0.00 1 O.OO2 O.OO2 0.002 0.002 0.502 0.546 0.592 0.641 0.693 .044 .046 049 .052 .056 o ooo 0001 C 001 C 001 O.OO I 60 357.26 0.919 0.003 0-749 O.OO2 TABLE IX, For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. X A Diff. B Diff. C ff Diff. C' 60 61 62 63 64 357-26 374-30 391 8* 409.86 428.38 17.04 17-54 18.02 18.52 19.02 0.919 0.990 1. 066 1.145 1.230 .071 .076 .079 .085 .088 0.003 0.003 0.003 0.004 0.004 0-749 I'.sll 0.935 1.004 35 .066 .069 .073 O.OO2 O.OO2 0.002 0.002 O.OO2 65 66 67 447.40 466.92 486.96 19.52 20.04 1.318 1.411 1.510 .093 .099 0.004 0.005 0.005 1.077 1.154 1.235 .077 .081 .086 0.003 o 003 0.003 68 507.51 20.55 1.613 103 0.006 1.321 0.004 69 21.07 1.721 .IO5 0.006 1.411 .090 0.004 21.59 .1 14 .094 70 71 72 55o-'7 572.29 59494 22.12 22.65 27 l8 1.835 1-954 2.078 .119 .124 0.007 0.007 0.008 1.505 1.605 1.709 .100 .104 0.004 0.005 0.005 73 74 618.12 641.85 23-73 24.28 2.209 2-345 '\\6 .143 0.009 0.009 1.819 -934 .121 0.006 0.006 75 666.13 2.488 0.010 2.055 .126 0.007 76 77 690.96 716.34 25'38 2.637 2-793 .149 O.OII O.OI2 2.181 2.3H [33 0.007 0.008 78 742.29 25-95 2.956 .103 0.013 2-453 6 0.008 79 768.81 27.09 3-!25 .177 0.014 2-599 'S3 0.009 80 81 82 83 84 795-90 823.57 851.84 880.70 910.16 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 o.o 1 8 0.020 2.752 2.912 3.079 3-255 3-439 .160 .167 .176 .184 .192 O.OIO O.OI I 0.012 0.013 0.014 85 86 87 88 89 940.23 970.92 1002.24 1034.20 1066.81 30.69 31.32 31.96 32.6l 33.27 4-303 4.529 4.764 5.008 5.262 .226 235 244 254 .265 O.O2I 0.023 0.024 0.026 0.028 3.631 4.044 4.266 4.498 .202 .211 .222 .232 .243 0.015 0.016 o.oi 8 0.019 O.O2I 90 1 100.08 5 5*7 0.030 4.741 0.023 91 1 134.02 33-94 5.801 'lit 0.032 4.996 .255 0.025 92 1168.64 34.62 6.087 .280 2()8 0.034 5.263 .28l 0.027 93 94 1203.95 1239.97 36.02 36.75 6.385 6.694 .309 .322 0.036 0.038 5-544 5-M 294 0.029 0.032 95 93 97 98 99 1276.72 1314.21 1352.45 1391.46 1431.27 37-49 38.24 39.01 39.81 40.61 7.016 7.350 7.698 8.060 8.437 334 .348 .362 377 392 0.041 0.044 0.047 0.050 0.053 6.147 6.471 6.812 7.171 7-549 .324 341 359 .378 397 0.035 0.038 0.041 0.045 0.049 100 30 101 30 1O2 30 103 30 104 30 105 30 106 30 , 107 30 1 108 ( 30 1471.88 1492.50 '.slJIi 1555.64 1577.12 1598.82 1620.75 1642.91 1665.30 1687.93 1710.80 1733.92 1757.28 1780.90 1804.77 1828.90 1853.30 20.62 20.83 21.05 21.26 21.48 21.70 21-93 22.16 22.39 22.63 22.87 23.12 23.36 23.62 23.87 24.13 24.40 24.67 8.829 9.032 9-238 9-449 9.664 9.883 10.108 10.337 10.570 10.809 11.053 11.302 "557 11.817 12.083 12.354 12.632 12.916 23 .211 .215 .219 .225 .229 233 239 .244 249 255 .260 .266 2?! .284 .291 0.056 0.058 O.o6o 0.062 0.064 0.066 0.068 0.070 0.072 0.074 0.077 0.079 O.O82 0.084 0.087 0.090 0.093 o 096 7.946 8.152 8.364 8.582 8.805 9-035 9.271 9-5'3 9.761 10.017 10.280 10.550 10.828 n. 1 14 11.408 11.711 12.022 J2-343 .206 .212 .218 .223 .230 .236 .1142 .248 .256 .263 .270 .278 .286 .294 .303 .311 .321 .330 0.053 0.055 0.058 0.060 0.063 0.066 0.069 0.072 0.075 0.078 0.082 0.085 0.089 0.093 0.098 O.I O2 0.107 o.i i a 1877.97 13.207 0.099 12.673 0.117 TABLE IX, For rindiirj, the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. I * f Diff. B Diff. C Diff. B' Diff. C" Diff. 109 30 110 30 111 30 1877.97 1902.91 1928.13 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 .304 .311 .319 .326 333 0.099 0.102 O.I 06 O.IO9 O.II3 0.116 .003 .004 .003 .004 .003 .004 12.673 13.013 I3-363 13.724 14.095 14.478 340 .350 .361 371 .383 .396 0.117 0. 122 0.128 0.134 0.141 0.148 Ioo6 .007 .007 .007 112 30 113 3D 114 o 30 2031.94 2058.64 2085.66 2113.00 2140.66 2168.66 26.70 27.02 27.34 27.66 28.00 28.34 15.097 15-439 15.789 16.148 16.515 16.892 342 .350 O.I 2O 0.124 0.128 0.132 0.137 0.142 .004 .004 .004 .005 .005 .005 14.874 15.282 15.702 16.135 16-583 17.045 .408 .420 433 .448 .462 477 0.155 O.l62 0.170 0.178 0.187 o 196 .007 .008 .008 .009 .009 .010 115 30 116 30 117 30 2197.00 2225.69 2254-73 2284.13 2313.91 2344.06 28.69 29.04 29.40 29-78 30.15 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 19594 20.156 493 .509 .526 544 .562 .582 O.2O6 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 30.94 !$ 32.19 32.62 33.06 19.813 20.276 20.751 21.240 21.742 22.258 .463 475 .489 .502 .516 53 1 0.180 0.186 0.193 0.200 0.207 0.214 .006 .007 .007 .007 .007 .008 20.738 21.339 21.962 22.606 23.273 23.964 .601 .623 .644 .667 .691 . 7 t6 0.277 0.291 0.306 0.322 o-339 o-357 .014 .016 .017 .018 .019 121 30 122 30 123 30 2566.51 2600.03 2634.02 2668.49 2703.46 2738.93 33.52 33-99 34-47 34-97 35-47 35.98 22.789 23.336 23.898 24-477 25.073 25.687 547 .562 579 .596 .614 633 O.222 0.230 0.239 0.248 0.258 0.268 .008 .009 .009 .010 .OIO .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 .021 .022 .024 .025 .027 124 30 125 2774.91 2811.43 2848.50 36.52 37-07 26.320 26.973 27.646 653 .673 0.278 0.289 0.300 .01 1 .01 1 29.564 30.489 31.450 .925 .961 -5'5 o-544 0-574 .029 .030 30 126 30 2886.13 2924.33 2963.12 38.20 38.79 39.41 28.341 29.057 29.797 .695 .716 74 .765 0.312 0.325 0.338 .012 .013 .013 .014 32448 34-563 .998 1.037 1.078 1. 122 0.60^ 0.640 0.676 .032 034 .036 039 127 30 128 30 129 30 3002.53 3042.56 3083.23 3124-57 3166.59 3209.31 40.03 40.67 41-34 42.02 42.72 43-45 30.562 31.351 32.167 33.011 33-885 .789 .816 -844 .874 .904 .936 0.352 0.367 0.382 0.398 0.415 -433 .015 .015 .016 .017 .018 .019 35.685 36.852 38.067 39-331 40.649 42.02Z 1.167 I.2I 5 1.264 1.318 1-373 1.430 0.715 0.757 0.800 0.846 0.896 0.949 .042 .043 .046 .050 .053 .056 130 20 40 131 20 40 3252.76 3 282.! 3 3311.85 3341.90 3372.31 3403.09 29.37 29.72 30.05 30.41 30.78 35-725 36.367 37.025 37-699 38.389 39.097 .642 .658 .674 .690 .708 .725 0.452 0.465 o-479 0-493 0.508 0-523 .013 .014 .0 4 .0 5 ol 43-452 44-439 45-455 46.500 47-575 48.682 0.987 i.oio 1.045 1.075 1.107 1.138 .005 .045 .087 .130 .223 .040 .042 .043 .045 .048 .050 ' 132 20 40 3434.23 34 6 5-74 3497.63 31.51 31-89 39822 40.564 41.326 742 .762 -X' -> o-539 0.555 0.572 .0 6 .0 7 o 8 49.820 50.992 52.199 1.172 1.207 .273 325 379 .052 054 133 20 40 352 9 . 9 I 3562.60 3595-69 32.69 33-9 33-5 1 42.108 42.910 43-733 .782 .802 .823 .843 0.590 0.609 0.629 .0 9 .020 .020 53-442 54-723 56.042 1.319 '359 -436 495 .558 .057 .059 .063 .065 134 20 3629.20 3663.13 33-93 44.576 45.442 .866 0.649 0.669 .020 57.401 58.802 1.401 .623 .692 .069 40 135 20 40 3697.50 3732.31 3767.58 3803.31 3437 34-8i 35-27 35-73 36.21 46.331 47- 2 45 48.183 49.147 .889 .914 .938 .964 .991 0.691 0.714 0.738 0763 .022 .023 .024 .025 .025 60.247 61.736 63.273 64.857 1.445 1.489 i-537 1.584 1.634 -764 .839 .917 2.OOO .072 .075 .078 .083 .087 136 3839-52 50.138 0.788 66.491 2.087 616 TABLE IX, Foi flnding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity X A Diff. B Diff. C Diff. ff Diff. C' Diff. 138 20 40 137 20 40 3876.21 3913.41 3951.12 3989-35 4028.11 36.69 37.20 ji-i; 38.76 39.31 50.138 51.156 52.203 53.280 54-388 55-5*8 1.018 1.047 1.077 1.108 1.140 1.174 0.788 0.815 0.843 0.873 0.90} 0.936 .027 .028 .030 .031 .032 33 66.491 68.178 69.920 71.718 73-575 75-493 1.687 1.742 1.798 1.857 1.918 1.982 2.087 2.178 2.274 *-375 2.480 2.591 .091 .096 .101 .105 "7 138 20 4067.42 4107.28 39.86 56702 57.910 1.208 0.969 1.004 .035 77-475 79-5*3 2.048 2.1 1 8 2.708 2.831 .123 40 139 20 40 4147.72 4188.75 4230.38 4272.63 40.44 41.03 41.63 42.25 42.89 59'54 60.436 61.757 63.119 1.321 1.362 1.404 1.041 1.079 1.119 I.IOI .037 .038 .040 .042 .044 81.641 83-830 86.094 88.436 2.189 2.264 2.342 2.424 2.960 3.096 3.239 3.390 .129 .136 '43 ! .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 70599 72.243 1.448 '494 1.542 1.592 1.644 1.698 1.105 1.251 1.299 1.350 1.404 1.460 .046 .048 .051 .054 .056 .058 90.860 93369 95.967 98657 01.443 04.331 2.509 2.598 2.690 2.786 2.888 2.993 3-549 3-7'7 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 73-94' 74.811 75-695 76-595 77.509 0.870 0.884 0.900 0.914 0.930 0946 1.518 1-549 1.580 1.612 1.645 1.679 .031 .031 .032 .033 .034 .035 07.324 08.861 10.427 I 2. 022 13.646 15.301 '-595 i 624 1.655 1.6X5 4.704 4.819 4.936 5-057 5.181 5.309 .115 .117 .121 .124 .128 .131 143 10 20 30 40 50 4732.84 4757 84 4783.05 4808.46 483410 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 0978 0.996 I.OI2 1.030 1.048 1.714 1.749 1.786 1.823 1.862 1.901 .035 .037 .037 .039 .039 .041 16.986 18.704 20.452 22.233 124049 I2 5 .8 9 9 1.718 1.748 1.781 1.816 1.850 1.886 5.440 5-575 1:$ '35 .140 '43 i .147 '5* .156 144 10 20 4886.02 4912.31 26.29 26.52 85.411 86.478 87 <;64 1.067 1.086 1.942 1.984 2.026 .042 .042 127.785 129.707 131.666 1.922 '959 6.313 M73 6.639 .160 .166 30* 40 50 4965'5^ 4992.56 5019.78 26.75 26.98 27.22 27.45 88^668 89.793 90.938 I.I04 I.I25 I.I45 1.165 2.070 2.116 2.162 .044 .046 .046 '37-774 1.997 2.035 2.076 2.116 6.809 6.984 7.165 .170 35 .186 145 10 20 30 40 50 5047.23 5074.93 5102.88 5131.08 5I59-53 5188.24 27.70 27.95 28.20 28.45 28.7, 28.97 92.103 93.290 94.498 95-7*9 96.982 98.259 1.187 1. 208 1231 1.253 1.277 1.300 2.2IO 2.259 2.309 2 361 2.414 2.469 49 .050 .052 .053 .055 .057 139.890 142.048 144.249 146.494 148.784 151.120 2.158 2.201 2.245 2.290 *-33 6 2.383 7-35' 7-543 7.740 7-943 8-'53 8.369 .192 .197 .203 .210 .216 .223 146 10 20 30 40 50 5217.21 5246.45 5*7595 5305-73 5335-79 5366.13 29.24 29.50 29.78 30.06 30.34 30.63 99-559 00.884 02.234 03.610 05.012 06.441 1.325 1-35 1.376 1.402 1.429 1.456 2.526 2.584 2.643 2.704 2.767 2.833 .058 .059 .061 .063 .066 .067 I53.503 '55-934 158-415 160947 '63.531 166.168 1.431 2.48. 2.532 2.584 2.637 2.692 8.592 8.822 9.060 9-34 9-5S5 9.815 .230 .238 .244 *5' .260 .268 147 10 20 30 40 50 5396.76 5427.67 545888 5490.39 5522.20 5554-33 30.91 31.21 31.51 31.81 32.13 J2, A\ 07.897 09.382 10.896 112.439 114.013 115.619 I. 4 85 1.514 '543 J -574 i. 606 1.637 2.900 2.969 3.040 I'.lee .069 .071 .073 .075 .078 .080 168.860 171.608 174.414 177.280 180.206 183.194 1.748 2.806 2.866 2.926 1.988 3.051 10.083 10.359 10.645 10.940 11.244 11.558 .276 .286 .295 .304 3'4 3*5 148 10 20 30 40 50 149 558677 5619.52 5652.60 5686.01 57I9-75 5753-83 5788.26 J ip T-T 31.75 33.08 33-4' 33-74 34.08 34-43 117.256 118.926 120.631 122.370 124.144 127.804 1.670 1.705 1.739 '774 i.8n 1.849 3.346 3.428 3-513 3.601 3.691 3-784 3.88. .082 .085 .088 .090 .093 .097 186.246 189.364. 192.549 195.804 199.130 202.528 206.002 3.118 3.185 3-*55 3.326 3.398 3-474 11.883 12.218 12.564 12.921 13.291 13.673 14.067 'III 357 .370 .382 394 GIT TABLE X, For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits. A Ellipse. Hyperbola. log* Diff logC log I. Diff. log half II. Diff. logtf Diir. logt? log I. Diff. lialflLDiff. o.ooo 0.000 0.00 oooo o.ooo oooo 4.23990 1.778 oooo 0.000 OOOO 4-239 8 2 n 1.771 .01 0007 7 .001 7432 .24286 .783 0007 7 27 9.998 2688 .23686 .767 .02 0030 1 7 .003 4985 .24583 .788 0030 * 3 37 996 5493 .23392 .762 .03 0067 3 / .005 2659 .24885 794 0067 ; I .994 8414 .23098 .758 .04 OI2O 68 .007 0457 .25190 799 0118 66 993 145 .22807 753 5.05 0188 84 0.008 8381 4.25497 1.805 0184 81 9.991 4599 4.22518,, 1.748 .06 0272 99 .010 6432 .25806 .811 0265 94 .989 7859 .22230 743 .07 0371 .012 4613 .26116 .816 0359 109 .988 1231 .21943 739 .08 0485 i -20 .014 2924 .26427 .821 0468 I 2T .986 4711 .21659 734 .09 0615 1 3 U 147 .016 1367 .26741 .827 o 59I 1 ^ J I 37 .984 8298 .21376 .730 0.10 0762 162 0.017 9945 4.27057 1-833 0728 9.983 1992 4.21094,, 1.725 .11 0924 178 .019 8659 .27376 .839 0880 \l] .981 5791 .20815 .720 .12 1 1 02 I O J. .021 7511 .2 7 b 97 .845 1045 979 9 6 94 .20537 .716 .13 1296 1 /-T 21 I .023 6503 .28020 .851 1223 IO1 978 3 6 99 .20260 .711 .14 1507 227 .025 5637 .28344 .857 1416 206 .976 7805 .19986 .706 0.15 1734 1977 III 0.027 49 '6 .029 4340 4.28670 .28999 1.863 .869 1622 1842 220 2 1 1 9.975 2011 973 6316 4.19712, .19440 1.700 .695 '7 2238 277 .031 3913 .29331 .875 2075 246 .972 0719 .19170 .690 .18 2515 .033 3636 .29665 .882 2321 26O .970 5218 .18901 .685 .19 2809 3i .035 3511 .30001 .888 2581 273 .968 9813 .18633 .679 O.2O .21 3120 3448 0.037 3542 .039 3730 4-3339 .30679 1.895 .901 2854 3140 286 2QO 9.967 4502 .965 9285 4.18367, .18102 1.672 .666 .22 .23 3793 4156 363 3X1 .041 4077 .043 4585 .31022 .31368 .908 .915 3439 3751 ^ V J 312 32 s .964 4159 .962 9124 .17840 17579 .661 .655 .24 4537 .045 5259 .31716 .922 4076 *3 .961 4180 .17319 .649 0.25 4935 416 0.047 6099 4.32066 1.929 4414 1 C I 9-959 9324 417061, 1.643 .26 i i\ .049 7109 .32418 .936 4765 III .958 4556 .16803 .637 .27 .28 5785 6237 T .3-T 452 .051 8290 .053 9646 32773 943 .951 5128 554 li .956 9875 955 5281 .16547 .16292 .631 .625 .29 6708 4SS .056 1179 33492 .958 5893 401 954 77! .16038 .618 0.30 7,96 0.058 2893 4.33856 1.966 6294 9.952 6346 4.15785, 1.613 TABLE X, Part II, T Ellipse. Hyperbola. T Ellipse. Hyperbola. A Diff. A Diff. A Diff. A Diff. o.oo .01 .02 .04 o.ooooo .00992 .01969 .02930 .03877 992 977 961 947 931 o.ooooo .01008 .02033 .03074 .04132 1008 1025 1041 1058 1077 O.2O .21 .22 .23 .24 0.17266 .18008 .18740 .19462 .20174 742 732 722 712 704 0.23867 .25309 .26779 .28280 .29813 1442 1470 1501 '533 1564 3 a .09 0.04808 .05726 .06630 .07521 .08398 918 004 91 877 865 0.05209 .06303 .07417 .08550 .09702 1094 114 133 152 173 :3 .29 0.20878 .21573 .22258 .22935 .23604 695 685 677 669 661 0.31377 O.IO .1 1 .12 I ->3 .14 0.09263 .10116 .10956 .11783 .12599 853 840 827 816 805 0.10875 .12069 .13285 .14522 .15782 194 216 237 260 285 0.30 32 33 34 0.24265 .24917 26198 .26826 652 644 637 628 621 0.15 .'18 0.13404 .14198 .1498! '5753 794 $ 0.17067 .18375 .19709 .21068 308 334 ill 0.35 36 0.27447 .28061 .28668 .29268 6,4 607 600 I .19 .16515 751 .22454 4 J 3 39 .29860 592 586 | 0.20 0.17266 0.23867 0.40 0.30446 TABLE XI, For the Motion in a Parabolic Orbit 1) kg* Diff. 1 log M Diff. , log. Diff. 0.000 o.ooo oooo 0.060 o.ooo 0652 0.120 o.ooo 2617 .OOI .000 oooo o .061 .000 0674 22 .121 .000 2661 44 .002 .000 0001 I .062 .000 0697 23 .122 .000 2705 44 .003 .000 0002 1 .063 .000 0719 22 .123 .000 2750 45 .004 .000 0003 I I .064 .000 0742 23 .124 .000 2795 46 0.005 o.ooo 0004 0.065 0.000 0766 0.125 o.ooo 2841 .006 .000 0006 2 .066 .000 0790 ^4 .126 .000 2886 45 .007 .000 0009 3 '.067 .000 0814 24 .127 .000 2933 3 .008 .000 0012 3 .068 .000 0838 2.4 .128 .000 2979 4 .009 .000 0015 3 3 .069 .000 0863 *5 .129 .000 3026 47 48 0.0 10 o.ooo 0018 0.070 o.ooo 0888 26 0.130 o.ooo 3074 .Oil .OOO OO22 4 .071 .000 0914 26 .131 .000 3121 is .012 .000 0026 4 .072 .000 0940 26 .132 .000 3169 4* .013 .014 .000 0031 .000 0035 5 t .073 .074 .000 0966 .000 0993 27 27 '33 .134 .000 3218 .000 3267 49 . 49 0.015 o.ooo 0041 0.075 O.OOO IO2O 0.135 o.ooo 3316 AO .016 .000 0046 I .076 .000 1047 28 .136 .000 3365 T 1 " CO .017 .018 .019 .000 0052 .000 0059 .000 0065 D I 1 .077 .078 .079 .000 1075 .000 1103 .000 1132 28 29 29 .139 .000 3415 .000 3466 .000 3516 J 5 5 5i O.O2O o.ooo 0072 0.080 o.ooo 1161 0.140 o.ooo 3567 52 * .021 .000 0080 .081 .000 1190 29 .141 .000 3619 .022 .000 0088 8 .082 .000 1219 29 .142 .000 3671 5 * .023 .000 0096 8 .083 .000 I 249 3 .143 .000 3723 .024 .000 0104 9 .084 .000 1280 3 1 3 1 .144 .000 3775 53 0.025 .026 o.ooo 0113 .OOO OI22 9 0.085 .086 0.000 1311 .000 1342 3 1 ;$ o.ooo 3828 .000 3882 54 53 .027 .000 0132 IO .087 .000 1373 3 1 .147 .000 3935 CJ. .028 .000 0142 IO .088 .000 1405 3 2 .148 .000 3989 JT r r .029 .000 0152 IO II .089 .000 1437 33 .149 .000 4044 j J 55 O.O3O .031 .032 .033 .034 o.ooo 0163 .000 0174 .000 0185 .000 0197 .000 0209 II 1 1 12 12 0.090 .091 .092 .093 .094 o.ooo 1470 .000 1502 .000 1536 .000 1569 .000 1603 34 33 34 35 0.150 .151 .152 154 o.ooo 4099 .000 4154 .000 4209 .000 4265 .000 4322 55 II n 0.035 .036 O.OOO O222 .000 0235 1 3 13 0.095 .096 o.ooo 1638 .000 1673 35 ;IP o.ooo 4378 .000 4435 37 .000 0248 13 .097 .000 1708 35 .157 .000 4493 r8 .o 3 3 8 .039 .000 0262 .000 0275 14 13 .098 .099 .000 1743 .000 1779 36 .159 .000 4551 .000 4609 58 59 0.040 .041 .042 .043 .044 o.ooo 0290 .000 0304 .000 0320 .000 0335 .000 0351 1 5 51 16 O.I 00 .101 .102 .10^ o.ooo 1815 .000 1852 .000 1889 .000 1926 .000 1964 37 37 I O.I 60 .161 .162 .163 .164 o.ooo 4668 .000 4726 .000 4786 .000 4846 .000 4906 & 60 60 60 0.045 .046 .047 .048 .049 0.000 0367 .000 0383 .000 0400 .000 0417 .000 0435 16 17 18 0.105 .106 .107 .108 .109 O.OOO 2OO2 .000 2040 .000 2079 .000 21 I 8 .000 2158 38 39 39 4 4 0.165 .166 .167 .168 .169 o.ooo 4966 .000 5027 .000 5088 .000 5150 .000 5212 6l 61 62 62 62 0.050 .051 .052 .053 .054 o.ooo 0453 .000 0471 .000 0490 .000 0509 .000 0528 DNO>ONO 00 ' O.I 10 .III .112 .113 .1 li o.ooo 2198 .000 2238 .000 2279 .000 2320 .000 2361 40 4 1 4 a 0.170 .171 .172 173 .174 o.ooo 5274 .000 5337 .000 5400 .000 5464 .000 5518 I 0.055 .056 .057 .059 o.ooo 0548 .000 0568 .000 0589 .000 0610 .000 0631 20 21 21 21 21 0.115 .1 li .1 17 .11! .119 o.ooo 2403 .000 2445 .000 2487 .000 2530 .000 2573 4* 4* 43 43 44 0.175 .176 .177 .178 .179 o.ooo 5592 .000 5657 .000 5722 .000 5787 .000 5853 P u 66 0.060 o.ooo 0652 0.120 o.ooo 2617 0.180 o.ooo 5919 TABLE XI, For the Motion in a Parabolic Orbit. log,* Diff. q log/a Diff. , lopr/x Diff. o.i8o .181 o.ooo 5919 .000 5986 67 67 0.240 .241 o.ooi 0603 .001 0693 90 91 1 0.300 .301 o.ooi 6733 .001 6848 "5 IIC .182 .000 6053 67 .242 .001 0784 .302 .001 6963 n6 .183 .184 .000 6l20 .000 6l88 07 68 68 .243 .244 .001 0875 .001 0966 9 1 9 1 92 .303 .304 .001 7079 .001 7195 116 117 0.185 o.ooo 6256 60 0.245 o.ooi 1058 0.305 o.ooi 7312 .186 .000 6325 68 .246 .001 1150 92 .306 .001 7429 1 17 .187 .188 .189 .000 6393 .000 6463 .000 6532 Do 70 6 9 7 .247 .248 .249 .001 1242 .001 1335 .001 1429 92 93 94 93 .307 .308 .309 .001 7546 .001 7664 .001 7783 117 118 119 118 0.190 .191 .192 '93 o.ooo 6602 .000 6673 .000 6744 .000 6815 7 1 7' 7' 0.250 .251 .252 o.ooi 1522 .001 1617 .001 1711 .001 1806 95 94 95 0.310 3" .312 o.ooi 7901 .001 8020 .001 8140 .001 8260 119 120 120 '94 .000 6887 72 72 .254 .001 1901 95 96 .314 .001 8381 121 121 0.195 o.ooo 6959 72 0.255 o.ooi 1997 06 0.315 o.ooi 8502 121 .196 .000 7031 .256 .001 2093 y .316 .001 8623 .197 .000 7104 73 .257 .001 2190 97 .001 8745 I 22 .198 .199 .000 7177 .000 7250 73 73 74 .258 .259 .001 2287 .001 2384 97 97 98 'l\l .319 .001 8867 .001 8989 I 22 122 124 0.200 o.ooo 7324 0.260 o.ooi 2482 0.320 o.ooi 9113 .201 .000 7399 75 .261 .001 2580 90 .321 .001 9236 123 .202 .000 7473 74 .262 .001 2679 99 .322 .001 9360 124 .203 .204 .000 7548 .000 7624 H .263 .264 .001 2778 .001 2877 99 99 IOO .324 .001 9484 .001 9609 124 I2 S 125 0.205 .206 .207 .208 o.ooo 7700 .000 7776 .oco 7853 .000 7930 77 77 77 0265 .266 .267 .268 o.ooi 2977 .001 3077 .001 3178 .001 3279 IOO 101 101 IO2 0.325 .326 3 2 7 .328 c.ooi 9734 .001 9860 .001 9986 .002 0113 126 126 I2 7 127 .209 .000 8007 78 .269 .001 3381 IOI .329 .002 0240 127 0.210 .211 .212 .213 o.ooo 8085 .000 8163 .000 8242 .000 8321 78 79 79 0.270 .271 .272 .273 o.ooi 3482 .001 3585 .001 3688 .001 3791 103 103 103 0.330 331 .332 333 O.OO2 0367 .002 0495 .OO2 0624 .OO2 0752 128 129 128 .214 .000 8400 79 80 .274 .001 3894 103 104 334 .002 0882 130 129 0.215 0.000 8480 80 0.275 o.ooi 3998 0.335 0.002 ion .216 .000 8560 81 .276 .001 4103 105 33 6 .002 1141 130 .217 .000 8641 Q .277 .001 4207 4 337 .002 1272 131 .218 .000 8722 I 81 .278 .001 4313 1 06 338 .OO2 1403 131 .219 .000 8803 82 .279 .001 4418 log 339 .002 1534 131 132 0.220 o.ooo 8885 0.280 o.ooi 4524 0.340 0.002 l666 .221 .222 .223 .000 8967 .000 9050 .000 9132 2 83 82 .281 .282 .283 .001 4631 .001 4738 .001 4845 107 107 107 108 .341 .342 343 .002 1799 .002 1931 .002 2065 132 '34 .224 .000 9216 84 .284 .001 4953 108 344 .002 2198 133 '35 0.225 .226 .227 .228 .229 o.ooo 9300 .000 9384 .000 9468 .000 9553 .000 9638 1 86 0.285 .286 .287 .288 .289 o.ooi 5061 .001 5169 .001 5278 .001 5388 .001 5497 log 109 no 109 in 0.345 .346 347 348 349 0.002 2333 .002 2467 .002 2602 .002 2738 .002 2874 '34 '35 136 '36 I 136 0.230 .231 .232 o.ooo 9724 .000 9810 .000 9897 86 87 87 0.290 .291 .292 o.ooi 5608 .001 5718 .001 5829 I IO in 112 0.350 .352 0.002 3010 .002 3147 .002 3284 '37 .233 .234 .000 9984 .001 0071 ii .293 .294 .001 5941 .001 6053 112 112 353 354 .OO2 3422 .OO2 3560 138 139 0.235 .236 .237 o.ooi 0159 .001 0247 .001 0335 88 88 80 0.295 .296 .297 o.ooi 6165 .001 6278 .001 6391 "3 II 3 Q-355 .356 357 O.OO2 3699 .002 3838 .002 3977 139 '39 .238 .001 0424 09 . .298 .001 6505 1 14 .358 .002 4117 140 .239 .001 0513 9 .299 .001 6619 114 "4 359 .002 4258 141 141 0.240 o.ooi 0603 0.300 o.ooi 6733 0.360 o.ooi 4399 TABLE XI, For the Motion in a Parabolic Orbit. Diff. , Diff , I.iff. 0.360 . 3 6i .362 .363 .364 0.002 4399 .002 4540 .002 4682 .002 4824 .002 4967 141 142 142 '43 143 0.420 .421 .422 423 .424 0.003 3720 .003 3890 .003 4061 .003 4232 .003 4404 170 171 171 172 172 0.480 .481 .482 .483 .484 0.004 4858 .004 5061 .004 5263 .004 5467 .004 5670 203 202 204 20 3 !3 3 66 0.002 5110 .002 5254 144 144 0.425 .426 0.003 4576 .003 4749 :$l 0.004 5875 .004 6080 *5 w 5 ' : 3 3 68 7 .369 .002 5398 .002 5543 .002 5688 45 45 146 427 .428 .429 .003 4921 .003 5096 .003 5271 173 174 487 .488 489 .004 6285 .004 6492 .004 6698 207 206 208 , 0.370 .371 .372 373 374 0.002 5834 .002 5980 .002 6126 .002 6273 .002 6421 146 146 148 0.430 43' .432 433 434 0003 5445 .003 5621 .003 5797 .003 5973 .003 6150 176 ,76 176 177 177 0.490 491 492 493 -494 0.004 6906 .004 7113 .004 7322 4 753' .004 7740 207 209 209 209 : 211 ' 0.375 .376 377 .378 0.002 6568 .OO2 6717 .002 6866 .002 7015 149 149 149 0.435 .436 437 .438 0.003 ^327 .003 6505 .003 6683 .003 6862 178 178 III 0.495 .496 497 .498 0.004 7951 .004 8161 .004 8373 .004 8585 2IO 212 212 379 .002 7165 150 150 439 .003 7042 IoO 1 80 499 .004 8797 212 0.380 .381 .382 .383 .384 0.002 7315 .002 7466 .002 7617 .002 7769 .002 7921 5i 52 52 0.440 441 .442 443 444 0.003 7222 .003 7402 .003 7583 .003 7765 .003 7947 180 181 182 182 183 0.500 Si .52 53 54 0.004 9010 .005 1173 .005 3397 .005 5681 .005 8029 213 2163 2224 2284 2348 2412 1i 0.002 8073 .002 8226 .002 8380 153 J54 I CJ. 0.445 .446 447 0.003 8130 .003 8313 .003 8496 183 ! 0.006 0441 .006 2919 .006 5464 2 47 8 Ittl .388 .002 8534 * JT .448 .003 8680 _ - 5 j} .006 8079 2D1 5 .389 .002 8689 *55 '55 449 .003 8865 i5 185 59 .007 0765 2686 2760 0.390 .391 0.002 8844 .002 8999 155 0.450 .451 0.003 95 .003 9236 1 86 186 0.60 .61 0.007 3525 .007 6361 2836 392 393 394 .002 9155 .002 93 II .002 9468 '56 '57 158 452 453 454 .003 9422 .003 9609 .003 9797 187 188 187 .62 i 3 .64 .007 9274 .008 2268 .008 5345 2913 2994 377 3163 -395 O.OO2 9626 ,-g 0.455 0.003 9984 180 0.65 0.008 8508 .396 397 .398 .002 9784 .002 9942 .003 oioi 1 5 158 159 .456 457 .458 .004 0173 .004 0362 .004 0551 169 189 189 67 .68 .009 1759 .009 5103 .009 8542 3344 3439 399 .003 0260 '59 1 60 459 .004 0741 190 191 .69 .010 2081 3539 3642 0.400 .401 0.003 420 .003 0580 1 60 161 0.460 .461 0.004 93 a .004 1123 191 0.70 o.oio 5723 .010 9473 375 .402 .403 .404 .003 0741 .003 0903 .003 1064 162 161 163 .462 .004 1315 .004 1507 .004 1700 192 192 '93 193 .72 73 74 .on 3336 .on 7316 .012 1419 3863 398o 4103 4233 0.405 i .406 .407 .408 0.003 1227 .003 1389 .003 1553 .003 1716 162 164 163 T ft - : 4 4 6l .467 .468 0.004 1893 .004 2087 .004 2281 .004 2476 194 '94 '95 0.75 .76 77 .78 O.OI2 5652 .013 OO22 .013 4,36 .013 9202 4370 iF .409 .003 1881 105 l6 4 .469 .004 2672 190 196 79 .014 4031 5002 0.410 .411 .412 .413 .414 0.003 2045 .003 221 1 .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 '97 198 198 199 0.80 .81 0.014 9033 .015 4219 .015 9603 .Ol6 52O2 .017 1033 5,86 534 5599 a; 0.415 .416 .417 0.003 2877 .003 3044 .003 3213 167 0.475 .476 477 0.004 3856 .004 4055 .004 4255 199 200 87 O.OI7 7I1O .018 3486 .019 0165 6366 6679 .418 .419 .003 3381 .003 3550 169 170 .478 479 .004 4456 | .004 4657 201 201 .88 .89 .019 7195 .O2O 4629 7030 7434 7900 ' 0.420 0.003 3720 0.480 0.004 4858 0.90 O.O2I 2529 1 TABLE XII, f logBl! logm a *i' Z 2 Z 3 *' Wlj VI 3 / Wj I / / / / / 1 / 00 0.0000 90 o 90 o 180 o 180 o 180 o o o 1 1 4.2976 9-9999 2 2 3 90 20 90 20 178 40 178 40 179 o 359 359 5 2 3-395 9.9996 446 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 '4 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 91 41 9 I 41 173 I 9 173 I 9 175 355 355 2 3 6 .9686 9.9968 14 19 92 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 32 8 .5981 9-9943 19 7 92 42 92 42 169 1 8 169 18 | 171 59 352 i 35 2 37 9 4473 9.9928 21 3 2 93 3 93 3 167 57 167 57 170 58 351 2 35i 4*j 10 .3130 9.9911 *3 57 93 ^5 93 ^5 166 35 166 35 169 57 35 3 350 47 11 .1922 9.9892 26 23 93 46 93 46 165 14 165 14 .68 55 349 4 349 5* 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 31 17 94 31 94 3 1 162 29 162 29 1 66 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 i7 95 '7 *59 43 159 43 164 44 345 H 346 ii 16 0.7254 9.9767 38 46 95 40 95 40 158 20 158 20 163 40 344 17 345 16 17 0.6518 9.9736 41 18 96 5 96 5 156 55 *5 6 55 162 34 343 *i 344 21 j 18 0.5830 9.9702 43 5 1 96 30 96 30 155 3 '55 3 1 61 27 342 27 343 2 7 19 0.5185 9.9667 46 26 96 56 96 56 154 4 154 4 1 60 19 341 32 34^ 3*: 20 0.4581 9.9629 49 2 97 23 97 23 152 37 152 37 159 9 340 38 34i 37' 21 0.4013 9.9588 51 41 97 5 97 50 151 10 151 10 157 58 339 45 340 43' 22 0-3479 9-9545 54 22 98 19 98 19 149 41 149 41 156 45 338 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 l 59 5' 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 7 145 7 152 50 33 6 19 337 6 2G 0.1631 9-9345 65 33 100 28 100 28 H3 3* 143 32 151 25 335 *8 336 13 27 0.1232 9.9287 68 30 ioi 5 ioi 5 HI 55 HI 55 i49 5 6 334 38 335 19 28 0.0857 9.9226 7' 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 13 103 I 3 136 46 136 46 H4 55 332 12 33* 39 31 9-98S7 9.9019 81 23 104 4 104 4 134 56 134 56 142 59 33i Z 4 33i 4 6 32 9.9565 9.8940 85 o 105 i 105 i 13* 59 132 59 140 51 33 37 330 54 33 9.9292 9.8856 88 54 106 6 106 6 13 54 13 54 138 27 3 2 9 49 330 2 34 9.9040 9.8765 93 ii 107 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 in 13 122 47 122 47 127 29 327 27 327 28 36 52.2 9.8443 9.8443 116 34 116 34 116 34 116 34 116 34 116 34 3*6 45 326 45 This table exhibits the limits of the roots of the equation sin (/ C) = wio sin 4 2 '> when there are four real roots. The quantities m l and m 2 are the limiting values of m , and the values of z/, z/, z s ', and z/, corresponding to each of these, give the limits of the four real roots of the equation. 422 TABLE XII, f log*! log , gf *' ' *' * M, i i "i + 00 00 :.:::: 90 o 90 o 180 o 180 o 1 80 4.1976 9-9999 i I 20 I 20 89 40 8940 177 37 180 55 181 2 3-395 9.9996 2 40 a 40 89 20 89 20 175 H 181 51 182 3 1.8675 9.9992 3 4 o 4 o 89 o 89 o 172 52 182 46 .83 4 2.4938 9.9986 4 5 20 5 20 88 40 88 40 170 28 183 4.2 184 5 2.2044 9.9978 5 641 6 41 88 19 88 ,9 168 5 184 37 185 6 1.9686 9.9968 6 8 i 8 i 87 59 87 59 165 41 185 3* 186 7 1.7698 9.9957 1 7 9 22 9 22 87 38 87 38 163 18 186 28 186 59 8 9 1.5981 9.9943 i 8 i J-4473 9-99 2 9 * 10 42 12 3 10 42 12 3 87 18 86 57 87 18 8657 160 53 187 23 158 28 188 18 187 59 188 58 10 11 1.3130 9.9911 1.192219.9892 10 3 ii 5 13 *5 14 46 13 *5 14 46 8635 86 14 86 35 86 , 4 156 3 153 37 189 '3 190 8 189 57 190 56 12 1.0824 9.9871 ii 6 16 8 16 8 85 52 85 V 151 10, 191 4 191 54 13 0.9821 19.9848: 13 9 17 31 17 3 1 85 29 85 *9 148 43 191 59 192 52 U 0.8898 i 9.9823 I 14 12 18 53 18 53 85 7 85 7 146 14 192 54 193 49 15 0.8045 9.9796 15 16 20 I 7 20 17 84 43 8443 143 45 ' 193 49 194 46 16 3.7*54 9-9767 16 20 21 40 21 40 84 20 84 20 141 14 194 44 '95 43 17 0.6518 9.9736 17 26 2 3 5 *3 5 83 55 83 55 138 4* 195 39 196 39 18 0.5830 9.9702 18 33 24 30 24 30 83 30 83 30 136 9 196 33 "97 33 19 0.5185 9.9667 19 4i 25 56 25 56 83 4 83 4 133 34 97 *8 198 28 20 0.4581 9.9629 20 51 27 23 27 23 82 37 82 37 130 58 198 23 199 22 21 0.4013 9-9588 22 2 28 50 28 50 1 82 10 | 82 10 128 19 199 17 200 15 22 0-3479 9-9545 2 3 IS 30 19 30 19 Si 41 81 41 125 38 200 II 2OI O7 ; 23 0.2976 9-9499 24 31 3 1 49 3i 49 81 ii 81 ii 122 55 201 6 201 24 0.2501 9-945' *5 49 33 20 33 *> 80 40 80 40 120 9 201 ! 201 51 25 0.2053 9.9400 27 10 34 53 34 53 80 7 80 7 117 20 202 54 203 41 26 0.1631 9-9345 28 35 36 28 36 28 79 3- 79 3- I! 4 -7 -~> 4' 204 32 27 0.1232 9.9287 30 4 3 5 38 5 78 55 j 78 55 i" 3'*>4 41 20, 22 28 29 0.0857 0.0503 9.9226 9.9161 3i 38 33 18 39 45 4i *7 39 45 | 41 7 78 15 77 33 78 15 77 33 loS 27 205 35 105 19 206 28 2O6 1 1 206 59 ' 30 1 31 ! 32 0.0170 9-9857 9-95 6 5 9.9092 9.9019 9.8940 35 5 37 i 39 9 43 13 45 4 47 i 43 '3 45 4 47 i 7647 75 56 74 59 76 47 102 3 75 56 i 98 37 74 59 95 1 207 21 208 14 209 06 207 48 208 361 209 23 33 34 9.9292 9.9040 9.8856 9.8765 4i 33 44 21 49 6 5 I 22 49 6 51 22 73 54 72 38 73 54 9i 6 72 38 86 49 209 58 210 50 2IO 1 1 210 58! | 35 36 -36 52." 9.8808 9.8600 9.8443 9.8665 9.8555 9-8443 47 47 52 31 63 26 53 5 57 13 63 26 53 58 57 13 63 26 7 I 2 68 47 63 26 71 2 68 47 63 26 S, 53 75 40 63 26 211 41 212 32 213 15 211 46 212 33 l" !ls This table exhibits the limits of the roots of the equation sin (/ :)= 7/i sin 4 z', when there are four real roots. The quantities 7/h and tn, are the limiting values of i , and the values of /, z,', z s ', and z/, corresponding to each of these, give the limits of the four real roots of the equation. 023 TABLE XIII, For finding the Ratio of the Sector to the Triangle. logs* Diff. logs* Diff. 1 log** Diff. o.oooo .0001 .0002 .0003 .0004 0.000 0000 .OOO 0965 .OOO I93O .OOO 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 O.O I 2O .01 i .OI 2 .01 3 .01 4 o.oi i 3417 .01 4343 .0 5268 .0 6193 .0 7118 926 925 925 925 925 0.0005 .0006 .0007 .0008 .0009 o.ooo 4821 .000 5784 .000 6747 .000 7710 .000 8672 963 963 963 962 962 0.0065 .0066 .0067 .0068 .0069 0.006 2019 .006 2962 .006 3905 .006 4847 .006 5790 943 943 942 943 942 o.oi 5 .01 6 .01 7 .01 8 .0129 o.o 8043 .0 8967 .0 9890 .0 0814 .0 1737 924 923 924 923 923 0.0010 .0011 .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 94' 94i 941 940 o.oi 30 .0131 .0132 .0133 OI 34 o.o 2660 o 3583 .0 4505 .0 5427 .0 6348 923 922 922 921 921 0.0015 .0016 .0017 .0018 .0019 o.ooi 4438 .001 5398 .001 6357 .001 7316 .001 8275 960 959 959 959 959 0.0075 .0076 .0077 .0078 .0079 0.007 1436 .007 2376 .007 3316 .007 4255 .007 5194 940 940 939 939 939 0.0135 .01 36 .0137 .0138 .0139 o.o 7269 .0 8190 .0 9111 .0 3 0032 .0 3 0952 so SO SO SO SO ' so O IH 1-1 i- 0.0020 .0021 .0022 .0023 o.ooi 9234 .002 0192 .002 1150 .002 2107 958 958 957 0.0080 .0081 .0082 .0083 0.007 6133 .007 7071 .007 8009 .007 8947 938 938 938 0.0140 .0141 .0142 .0143 0.013 l8 7i .013 2791 .013 3710 .013 4629 920 919 919 l8 .0024 .002 3064 957 957 .0084 .007 9884 937 937 .0144 .013 5547 918 9 l8 O.OO25 .OO26 .0027 .0028 .0029 O.OO2 4O2I .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 9 I8 9 l8 | 917 917 917 O.OO3O O.OO2 88OO 0.0090 0.008 5502 0.0150 0.014 1052 016 .0031 .002 9755 955 .0091 .008 6437 935 .0151 .014 1968 910 .0032 .0033 .0034 .003 0709 .003 1663 .003 2617 954 954 954 .0092 .0093 .0094 .008 7372 .008 8306 .008 9240 935 934 934 .0152 .0153 .0154 .014 2884 .014 3800 .014 4716 910 916 916 953 934 9*5 0.0035 .0036 0.003 3570 .003 4523 953 0.0095 .0096 0.009 OI 74 .009 i i 08 934 0.0155 .0156 0.014 563' .014 6546 9 1 ? .0037 .003 5476 953 .0097 .009 2041 933 .0157 .014 7460 914 .0038 .003 6428 952 .0098 .009 2974 933 .0158 .014 8374 914 .0039 .003 7380 952 952 .0099 .009 3906 932 932 .0159 .014 9288 914 914 O.OO4O .0041 0.003 8332 .003 9284 952 O.OIOO .OIOI 0.009 4838" .009 5770 93 2 o.o 1 60 .0161 O.OI5 2O2 .015 me 913 .0042 .004 0235 95 1 .0102 .009 6702 932 .0162 .015 2028 913 .0043 .0044 .004 1 1 86" .004 2136 95 1 95 95 .0103 .0104 .009 7633 .009 8564 931 931 93 1 .0163 .0164 .015 2941 .015 3854 913 913 912 0.0045 .0046 .0047 0.004 386 .004 4036 .004 4985 950 949 0.0105 .0106 .0107 0.009 9495 .010 0425 .010 1355 930 93 0.0165 .0166 .0167 0.015 4766 .015 5678 .015 6589 912 911 .0048 .004 5934 949 .0108 .010 2285 93 .0168 .015 7500 911 .0049 .004 6883 949 .0109 .010 3215 93 .0169 .015 8411 911 949 929 911 0.0050 0.004 7832 QJ.8 O.OI IO o.oio 4144 0.0170 0.015 9322 .0051 .0052 .004 8780 .004 9728 94 948 .0111 .0112 .010 5073 .010 6001 929 928 .0171 .0172 .016 0232 .016 1142 910 910 .0053 .005 0675 947 .0113 .010 6929 928 .0173 .016 2052 910 .0054 .005 1622 947 947 .0114 .010 7857 III .0174 .016 2961 909 909 0.0055 .0056 .00 S 7 .0058 .0059 0.005 2 5^9 .005 3515 .005 4461 .005 5407 .005 6353 946 946 946 946 945 0.0115 .0116 .0117 .0118 .0119 o.oio 8785 .010 9712 .on 0639 .on 1565 .on 2491 OSO SO vo so ' OX ON OX-J ^1 0.0175 .0176 .0177 .0178 .0179 0.016 3870 .016 4779 .016 5688 .016 6596 .016 7504 . OX ONOO OC 00 O O O O O , OXON ON Ov O.OO6O 0.005 7 2 98 O.OI2O o.on 3417 0.0180 o.o 1 6 8412 A24 TABLE XIH For finding the Ratio of the Sector to the Triangle. * * Diff. * tog* Diff. i log* Din o.oi 8o .0181 00l6 8412 .Ol6 9319 907 0.0240 .0241 j 0.022 2330 .022 3220 890 880 O.O3OO .O3OI 0.027 5 218 .027 6091 873 .0182 .0183 .0184 .017 0226 .017 1133 .017 2039 907 907 906 906 .0242 .0243 .0244 .022 4,09 .022 499 8 .022 5887 509 889 889 889 .O3O2 .0303 .0304 .027 6964 .027 7836 .027 8708 872 872 872 0.0185 .0186 .0187 .0188 .0189 0.017 2 945 .017 3851 .017 4757 .017 5662 .017 6567 9 06 906 905 905 904 0.0245 .0246 .0247 .0248 .0249 0.022 6776 .022 7664 .022 8552 .022 9440 .023 0328 888 888 888 888 887 0.0305 .0306 .0307 .0308 .0309 0.027 9580 .028 0452 .028 1323 .028 2194 .028 3065 8 ?2 87! 87, 871 0.0190 .0191 .0192 .0193 .0194 0.017 747 * .017 8376 .017 9280 .018 0183 .018 1087 905 904 903 904 903 0.0250 .0251 .0252 .0253 0.023 12I 5 .023 2IO2 .023 2988 .023 3875 .023 4761 887 886 887 886 886 0.0310 .0311 .0312 3 '3 .0314 0.028 3936 .028 4806 .028 5676 .028 6546 .028 7415 870 870 870 869 869 0.0195 .0196 0.018 1990 .018 2893 903 0.0255 .0256 0.023 5 6 47 .023 6532 885 88? 0.0315 .0316 0.028 8284 .028 9153 869 860 .0197 .0198 .0199 .018 3796 .018 4698 .018 5600 903 902 9 02 901 .0257 .0258 .0259 .023 7417 .023 8302 .023 9187 005 885 885 884 .0317 .0318 .0319 .029 OO22 .029 0890 .029 1758 009 868 868 868 O.O2OO .O2OI .O2O2 .O2O1 .020.4 0.018 6501 .018 7403 .018 8304 .018 9205 .019 0105 9 02 9 OI 9 OI 9 00 900 0.0260 .0261 .0262 .0261 .026.4 0.024 007 1 .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 5128 .029 6095 868 867 867 867 866 i 0.0205 0.019 1005 0.0265 0.024 44 8 9 gg- 0.0325 0.029 6961 .0206 .0*07 .019 1905 .019 2805 900 9 00 .0266 .0267 .024 5372 .024 6254 OO j 882 882 .0326 .0327 .029 7827 ; * .029 8693 ^, .O2OJ .019 3704 899 .0268 .024 7136 882 .0328 *9 9559 86 .O2OC .019 4603 o" 899 .0269 .024 8018 882 .0329 .030 0424 g6 > 0.0210 .0211 0.019 55 OZ .019 6401 899 00 0.0270 .0271 0.024 8900 .024 9781 881 881 0.0330 .0331 0.030 1290 ; , .030 2154 |J4 .0212 .019 7299 9 .0272 .025 0662 881 .0332 .030 3019 -, 5 .0213 .019 8197 892 .0273 .025 1543 880 .0333 .030 3883 "4 .021* .019 9094 8?8 .0274 .025 2423 s 8 E 334 3 4747 H* j O.O2I i .02 1 i 0.019 999 2 .020 0889 897 0.0271 .0276 0.025 333 .025 4183 880 880 0.0335 .0336 0.030 5611 i _, 1 -306475. 6 fi i ' .0217 .020 1785 896 .0277 .025 5063 870 .03371 .030 7338 jjjj3 .02li .020 2682 897 .0278 .025 5942 079 870 .0338 .030 8201 P .0219 .020 3578 896 896 .0279 .025 6821 079 879 .0339 .030 9064 3 f 0.0220 0.020 4474 0.0280 0.025 7700 a 0.0340! 0.030 9926 862 .022 .O222 .022' .020 5369 .020 6264 .020 7159 895 895 8 95 .028 .0282 .028 .025 8579 .025 9457 .026 0335 $ 878 Q-.Q .0341 .0342 343 i .031 0788 .031 1650 .031 2512 862 862 861 .022; .020 8054 895 89^ .0284 1 .026 ,213 jj^ 344 3 3373 861 0.0225 0.020 8948 0.0285 0.026 2090 - 0.0345 0.031 4234 861 .O22I .O2O 9842 8 94 .0286 .026 2967 0-- .0346 .031 5095 861 .0227 .0228 .021 0736 .021 1630 894 894 .0287 .0288 .026 3844 .026 4721 *77 876 347 .0348 .031 5956 .031 68,6 860 860 .0229 .021 2523 89: .0289 .026 5597 oyu 876 .0349 .031 7676 860 0.0230 .023 .023 .023 .023 0.021 3416 .021 4309 .021 5201 .021 6093 .021 6985 ? Os Os OS OS 0.0290 .0291 .0292 .0293 .0294 ! 0.026 6473 .026 7349 .026 8224 i .026 9099 .026 9974 876 875 li 0.0350 .0351 .0352 353 0354 0.031 8536 .031 9396 .032 0255 .032 11,4 .032 1973 860 iH 0.023 .023 .023 .023 .023 0.021 7876 .021 8768 .021 9659 .022 0549 .022 1440 892 8 9 i 8 9 o 8 9 i 0.0295 .0296 .0297 .029? .0299 0.027 0849 .027 1723 .027 2597 .027 3471 .027 4345 III 874 874 87; 0.0355 .0356 357 .0358 0359 0.032 2831 .032 3689 .032 4547 .032 5405 .032 6262 in ft m 0.024 0.022 2330 0.0300 0.027 5218 0.0360] 0.032 7120 40 TABLE XIII, For finding the Ratio of the Sector to the Triangle. n logs* Diff. 17 logs" Diff. i? log** Diff. 0.0360 .0361 .0362 .0363 0.032 7120 .032 7976 .032 8833 .032 9689 8 5 6 857 856 r-7 0.060 .061 .062 .063 0.052 5626 .053 3602 .054 1556 .054 9488 7976 7954 7932 0.120 .121 .122 .123 0.096 8849 .097 5692 .098 2520 .098 9331 6843 6828 6811 f\ini .0364 .033 0546 8 57 855 .064 .055 7397 7909 7888 .124 .099 6127 0790 6780 0.0365 .0366 .0367 .0368 .0369 0.033 1401 .033 2257 .033 3112 .033 3967 .033 4822 8 5 6 855 855 855 855 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 :3 .129 o.ioo 2907 .100 9672 .101 6421 .102 3154 .102 9873 6765 6749 6733 6719 6703 0.0370 .0371 , .0372 .0373 374 0.033 5677 .033 6531 .033 7385 .033 8239 .033 9092 8 54 8 54 854 853 854 0.070 .071 .072 73 .074 0.060 4398 .061 2157 .061 9895 .062 7612 .063 5308 7759 7738 7717 7696 7676 0.130 .131 .132 .133 134 0.103 657<> .104 3264 .104 9936 .105 6594 .106 3237 6688 6672 6658 6643 6628 0.0375 .0376 .0377 .0378 379 0.033 994 6 .034 0799 .034 1651 .034 2504 34 3356 853 8 S 2 853 8 5 2 8 5 2 0.075 .076 .077 .078 .079 0.064 2984 .065 0639 .065 8274 .066 5888 .067 3482 7655 7635 7614 7594 7575 0.135 .136 '37 .138 .139 0.106 9865 .107 6478 .108 3076 .108 9660 .109 6229 6613 6598 6584 6569 554 0.0380 .0381 .0382 .0383 .0384 0.034 4208 .034 5059 .034 5911 .034 6762 .034 7613 8 S I 8 5 2 8 5 I 8 5 , 8 5 I 0.080 .081 .082 .083 .084 0.068 1057 .068 8612 .069 6146 .070 3661 .071 1157 7555 7534 75'5 7496 7476 0.140 .141 .142 .143 .144 o.no 2783 .no 9323 .III 5849 .112 2360 .112 8 57 654 6526 6511 6497 6483 0.0385 .0386 .0387 .0388 .0389 0.034 8464 .034 9314 .035 0164 .035 1014 .035 1864 8 5 850 850 8 5 849 0.085 .086 .087 .088 .089 0.071 8633 .072 6090 .073 3527 .074 0945 .074 8345 7457 7437 7418 7400 7380 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 2 7i3 .035 3562 849 Q . 0.090 .091 0.075 5725 .076 3087 7362 0.150 .151 0.116 7544 "7 3943 6399 6186 .0392 .035 4411 111 .092 .077 0430 7343 .152 .118 0329 .0393 .0394 35 5*59 .035 6108 849 848 .093 .094 V7 7754 .078 5060 7324 7306 7288 '53 '54 .118 6701 .119 3059 6372 6358 6 345 i 0.0395 0.035 6 956 848 0.095 0.079 Z 34 8 - 0.155 0.119 9404 .0396 .035 7804 Q . _ .096 .079 9617 7209 .156 .120 5735 6^8 397 .0398 .0399 .035 8651 .035 9499 .036 0346 6 47 848 847 846 .097 .098 .099 .080 6868 .081 4101 .082 1316 7251 7233 7215 7197 57 .158 .159 .121 2053 .121 8357 .122 4649 6318 6304 6292 6278 j 0.040 .041 .042 0.036 1192 .036 9646 .037 8075 8454 8429 O.I 00 . OI . 02 0.082 8513 .083 5693 .084 2854 7180 7161 0.160 .161 .162 0.123 9 2 7 .123 7192 .124 3444 6265 6252 c*o 43 .044 .038 6478 .039 4856 H3 8378 8353 . 03 . 04 .084 9999 .085 7125 7H5 7126 71 10 .163 .164 .124 9682 .125 5908 0230 6226 6213 0.045 .046 .047 .048 .049 0.040 3209 .041 1537 .041 9841 .042 8121 .043 6376 8328 8304 8280 8255 8231 o. 05 . 06 07 .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 1121 .126 8321 .127 4508 .128 0683 .128 6845 6200 6187 6175 6162 6149 0.050 .051 .052 .053 .054 0044 4607 045 2814 046 0997 046 9157 047 7294 8207 8183 8160 8137 8113 O.I IO .III .112 .113 .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 '3 5255 .131 1367 .131 7466 6137 6124 6112 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 53 .119 0.093 4391 .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 675 6062 6050 6038 6026 0.060 Q.Q$3. 5626 O.I 2O 0.096 8849 O.I 80 0.135 3804 TABLE XIII, For finding the Ratio of the Sector to the Triangle. ' log* Diff 1 log* Di , log* Diff. I 0.1*0 .i8i .182 .183 i4 0.135 3o .135 981 .136 582 .137 181 .137 778 60 1 600 599 597 596 0.240 .241 .242 243 .244 0.169 5 92 .170 0470 .170 5838 .171 1197 537 536 535 0.300 .301 .302 .303 34 0.200 2285 -200 7157 .201 2021 .201 6878 .202 1727 4872 4864 4f57 4849 0.185 $ ::" 0.138 375 .138 971 139 565 .140 158 .140 750 595 594 593 501 590 .249 0.172 1887 .172 7218 533 532 |53* 529 0.305 .306 .307 .308 .309 0.202 6569 .203 1401 .203 6230 .204 1050 .204 5862 4842 4834 4827 4820 4812 j -9 .19 .19 '1 9 9 0.141 341 .141 930 .142 519 .143 106 .143 693 585 0.250 .251 .252 253 254 0.174 8451 '75 3736 .175 9013 .176 4280 .176 9538 528 527 526 525 525 0.310 .311 .312 3'3 .314 0.205 0667 .205 5464 .206 0254 .206 5037 .206 9813 4797 4790 4783 1 4770 4.768 1 0.195 .196 1 -97 j -198 .199 0.144 278 .144 862 .145 4450 .146 0268 .146 6074 fii 581 580 579 0.255 .256 .257 .258 259 0.177 4788 .178 0029 .178 5261 .179 0484 .179 5698 524 523 522 5214 0-3I5 .316 23 .319 O.2O7 458l .207 9342 .208 4096 .208 8843 .209 3582 4700 4761 4754 4747 i 4739 1 0.200 1 .201 0.147 1869 .147 765 5784 0.260 .261 o.i 80 0903 .180 6100 520 0.320 .321 0.209 8315 .2IO 3040 4733 4725 1 .202 .203 .20. .148 3427 .148 9189 .149 4940 577' 5762 575 574 .262 .263 .264 .181 128! .181 6467 .182 1638 5188 5'79 5i7 .322 323 324 .210 7759 .211 2470 .211 7174 4719 47" 4704 1 M-,l- 0.205 1 .206 .207 .208 .209 0.150 0681 .150 6411 .151 2130 .151 7838 '5* 3535 5730 57'9 5708 5697 5687 0.265 .266 .267 .268 .269 0.182 6800 .183 1953 .183 7098 .184 2235 .184 7363 5153 5'45 5137 5128 5I2.O 1 AVJ 0.325 .326 .327 .328 .329 0.212 1871 .212 6562 .213 1245 .213 5921 .214 0591 497 J68J 4676 I 4670 i 4.662 o.aio 1 :l\l -4 0.152 9222 .153 4899 .154 0565 .154 6220 .155 1865 5677 666 655 645 634 0.270 .271 .272 .273 .274 0.185 2483 185 7594 .186 2696 .186 7791 .187 2877 5111 5102 5095 5086 5078 0.330 331 .332 333 334 0.214 5253 .214 9909 .215 4558 .215 9200 .216 3835 4001 i 4656 4649 4642 4635 4629 1 0.215 si! .219 0-155 7499 .156 ,123 .156 737 J 57 434 '57 9933 624 614 603 593 0.275 .276 .277 .278 .279 0.187 7955 .188 3024 .188 8085 .189 3138 .189 8183 5069 5061 5053 5045 5037 0.335 336 337 .338 339 0.216 8464 .217 3085 .217 7700 .218 2308 .218 6910 462, 46 15 > j 460$ J 4602 4595 | 0.220 .221 1 .222 I -223 .224 0.158 5516 .159 1089 .159 6652 .160 2204 .160 7747 573 563 552 543 532 0.280 .281 .282 .283 .284 0.190 3220 .190 8249 .191 1269 .191 8281 .192 3286 5029 5020 5012 5005 499" 0.340 .341 .342 343 344 0.219 '55 .219 6093 .220 0675 .220 5250 .220 9818 4588 4582 4562 1 0.161 3279 .161 8802 523 0.285 .286 0.192 8282 .193 3271 4989 Q8O 0-345 .346 O.22I 4380 .221 8915 .227 .162 4115 M3 .287 .193 8251 347 .222 5483 454^ 1 .228 .162 9817 502 .288 .194 3224 973 .348 .222 8025 454^ 1 .229 .163 5310 493 483 .289 .,94 *i88 I-9&4 957 349 ,113 2561 4529 1 0.230 .231 .232 0.164 793 .164 6267 .165 1730 474 463 0.290 .291 .292 0.195 1145 .195 8094 .196 3035 949 941 .350 35i 352 0.223 7090 .224 1613 .224 6130 4523 1| 233 .165 7184 1-54 293 .196 7968 933 353 .225 0640 45" 4503 1 .234 .166 2628 ^44 435 .294 .197 2894 917 354 .225 5143 0.235 .236 .237 .238 .239 .166 8063 .167 3488 .167 903 .168 4309 .168 9705 25 96 8 7 0.295 .296 .297 .298 .299 0.197 7811 .198 2721 .198 7624 .199 2518 .199 7406 ^ 910 003 894 888 879 355 .356 357 .358 359 0.225 9 6 4 .226 4131 .226 8615 .227 3093 .217 7565 i 4478 j 4472 4466 1 0.240 .169 5092 .300 0.2OO 2285 .360 0.228 2031 j 627 TABLE XIII, For finding the Ratio of the Sector to the Triangle. 1 log*' Diff. 1 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 0.253 9153 .254 3269 *54 7379 .255 1484 *55 5584 4116 4110 4105 4100 495 0.480 .48! .482 483 .484 0.277 717* .278 1096 .278 4yl6 .278 8732 .279 2543 3824 3820 3816 3811 3806 .369 0.230 4265 .230 8694 .231 3116 .231 7532 .232 1942 44*9 44** 4416 4410 4404 0.425 .426 4*7 .428 4*9 0.255 9679 .256 3769 256 7853 *57 193* .257 6006 4090 4084 4079 4074 4069 :J86 487 .488 .489 0.279 6349 .280 0151 .280 3949 .280 7743 .281 1532 3802 3798 3794 3785 3784 0.370 .371 37* 0.232 6346 .233 0743 .233 5135 4397 llll 0430 43 * 43* 0.258 0075 .258 4139 .258 8198 4064 4059 0.490 .491 49* 0.281 5316 .281 9096 .282 2872 3776 373 374 .233 9521 .234 3900 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 4344 0.435 .436 437 .438 439 0.260 0344 .260 4382 .260 8415 .261 2444 .261 6467 4038 4033 4029 4023 4019 0.495 .496 497 .498 499 0.283 4173 .283 7932 .284 1686 .284 5436 .284 9181 3759 3754 375 3745 374* 0.380 $ 1! 0.237 0053 *37 4391 .237 8723 .238 3050 .238 7370 433 433* 43*7 43*o 43'5 0.440 44 * 44* 443 444 0.262 0486 .262 4499 .262 8507 .263 2511 .263 6509 4011 4008 4004 3998 0.500 .501 .502 53 .504 0.285 *9*3 .285 6660 .286 0392 .286 4121 .286 7845 3737 373* 3729 37*4 0.385 386 0.239 *685 .239 5993 .240 0296 4308 4303 0.445 446 447 0.264 53 .264 4492 .264 8475 3989 3983 0.505 .506 .507 0.287 ^65 .287 5281 .287 8992 3716 37" 1 --Q J 388 .240 4594 4290 448 .265 2454 3979 .508 .288 2700 37O8 .389 .240 8885 4291 4286 449 .265 6428 3974 3969 .509 .288 6403 373 i 3699 | 0.390 .391 39* 393 394 0.241 317! .241 7451 .242 1725 .242 5994 .243 0257 4280 4*74 4269 4263 4*57 0.450 .451 45* 453 454 0.266 0397 .266 4362 .266 8321 .267 2276 .267 6226 3965 3959 3955 395 3945 0.510 .511 .512 5'3 .514 0.289 0102 .289 3797 .289 7487 .290 1174 .290 4856 3 6 95 3690 3687 : 3682 ! 3679 0.395 .396 397 0.243 45H .243 8766 .244 3012 4*4 6 0.455 456 457 0.268 0171 .268 4111 .268 8046 3940 3935 -5i5 .516 5'7 0.290 8535 .291 2209 .291 5879 3674 3670 | ifififi 398 399 .244 7252 .245 1487 4240 4*35 4229 .458 459 .269 1977 .269 5903 3926 39*' .518 .519 .291 9545 .292 3207 3000 3662 ; 3 6 57 0.400 .401 .402 .403 .404 0.245 57 J 6 .245 9940 .246 4158 .246 8371 *47 *578 4224 4218 4213 4207 4201 0.460 .461 .462 ? 0.269 9824 .270 3741 .270 7652 .271 1559 .271 5462 3917 39" 3907 39? 3898 0.520 .521 .522 5*3 5*4 0.292 6864 .293 0518 .293 4168 .293 7813 .294 1455 3654 3650 3 6 45 3642 3637 0.405 .406 .407 .408 .409 0.247 6779 .248 0975 .248 5166 .248 9351 *49 353' 4196 4191 4185 4180 ll .468 469 0.271 9360 .272 3253 .272 7141 .273 1025 .273 4904 3893 7888 3884 3879 3874 0.525 .526 5*7 .528 5*9 0.294 59* .294 8726 *95 *355 .295 5981 .295 9602 3634 36*9 1 3626 3621 3618 0.410 .411 0.249 775 .250 1874 4169 0.470 .471 0.273 8778 .274 2648 3870 3865 3861 0.530 53* 0.296 3220 .296 6833 .297 0443 3613 3610 3606 .250 6038 .274 6513 .413 .414 .251 0196 *5i 4349 4'53 4*47 473 474 .275 0374 .275 4230 3856 385* 533 534 .297 4049 .297 7650 3601 3598 0.415 .416 .417 .418 .419 0.251 8496 .252 2638 *5* 6775 .253 0906 *53 S3* 4142 437 4131 4126 4121 0.475 .476 477 .478 479 0.275 8082 .276 1929 .276 5771 .276 9609 .277 3443 3847 384* 3838 3834 3829 o-535 .536 537 .538 539 0.298 1248 .298 4842 .298 8432 .299 2018 .299 5600 3594 359 3586 3582 3578 0.420 0.253 9>53 0.480 0.277 7*7* 0.540 0.299 9178 TABLE XHL For finding the Ratio of the Sector to the Triangle. 1) log.* Diff. , log** Diff. i log** Diff. 0.540 54 54* 543 544 o.*99 9178 .300 2752 .300 6323 .300 9890 .301 3452 3574 3567 3562 3559 0.560 .561 .562 .563 564 0.306 9938 307 3437 .307 6931 .308 0422 .308 3910 3499 3494 349' 3488 0.580 .58! .582 0.313 9215 .314 2641 .314 6064 .314 9481 .315 2898 34*6 34*3 34'9 34'5 3412 0-545 . S4 6 547 548 549 0.301 7011 .302 0566 .302 4117 .302 7664 .303 1208 3555 355' 3547 3544 ' S6 J .566 :JS .569 0.308 7394 .309 0874 .309 4350 .309 7823 .310 1292 3480 3476 3473 3469 3466 0.585 .586 587 .588 589 0.315 6310 .315 9719 .316 3124 .316 6525 .316 9923 3409 3405 | 3401 1 3398; 3395 o-55o 55* 55* i -553 554 "3038^353; 0.570 571 57* 573 574 0.310 4758 .310 8220 .311 1678 3" 5'33 -3 54 3462 3458 3455 345' 3447 0.590 .591 59* 593 594 0.317 3318 .317 6709 .318 0096 .318 3480 .318 6861 33 3 8 7 3384! 3381 ; 3377 0-555 556 557 .558 559 0.305 2390 .305 5907 '35*7 .305 9420 3> 3 .306 2970 35 1 " 36 6436 \\H o-575 .576 577 .578 579 0.312 2031 3'* 5475 .312 8915 .313 2352 313 575 3444 3440 3437 3433 343 o-595 596 597 .598 599 0.319 0238 .319 3612 .319 6983 .320 0350 .320 3714 3374 337i 3367 ' 33 6 4 7l6o ' 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. j : Ellipse. Diff. Hyperbola. Diff. x Ellipse. Diff. Hyperbola. Diff. O.OOO O.OOO OOOO o.ooo oooo 0.030 o.ooo 0523 16 o.ooo 0506 .001 .OOO OOO I 1 .OOO OOO I 1 .031 .000 0559 3 .000 0539 36 .ooa .003 .004 .OOO OOO2 .000 0005 .000 0009 3 4 5 .OOO OOO2 .000 0005 .000 0009 3 4 5 .032 .033 .034 .000 0596 .000 0634 .000 0674 40 40 .000 0575 .000 06 I I .000 0648 i 0.005 .006 o.ooo 0014 .OOO OO2I 7 o.ooo 0014 .OOO OO2O 6 0.035 .036 o.ooo 0714 .000 0756 4* o.ooo 0686 .000 0726 4 .007 .000 0028 7 .000 0028 8 O77 .000 0799 43 .000 0766 4 .008 ! .009 .000 0037 .000 0047 9 10 II .000 0036 .000 0046 8 10 1 1 .039 .000 0844 .000 0889 45 45 47 .000 0807 .000 0850 4 1 43 44 o.o o o.ooo 0058 o.ooo 0057 0.040 o.ooo 0936 4.8 o.ooo 0894 : .0 I .000 0070 I 2 .000 0069 .041 .000 0984 4- .000 0938 h .O 2 .0 3 .000 0083 .000 0097 '3 \l .000 0082 .000 0096 '3 14 .042 043 .000 o-t 3 .000 084 49 5' 51 1 .000 0984 .000 1031 J5 .0 4 .000 0113 17 .000 01 1 1 11 .044 .000 135 53 .ooc 1079 49 o.o 5 .0 6 ; .0 7 .0 8 .0 9 o.ooo 0130 .000 0148 .000 0167 .000 0187 .000 0209 18 19 20 22 22 o.ooo 0127 .000 0145 .000 0164 .000 0183 .000 0204 18 19 '9 21 22 0.045 .046 33 049 o.ooo 1 88 .000 242 .000 1298 .000 1354 .000 1412 S 8 59 o.ooo 1128 .000 1178 .000 1229 .000 1281 .000 1334 50 5' 5* 53 55 o.o o o.ooo 0231 o.ooo 0226 0.050 o.ooo 1471 61 o.ooo 1389 C C .0 I .022 .000 0255 .OOO O28O 11 .000 0249 .000 0273 23 .051 .052 .000 1532 .000 593 61 .000 1444 .000 1500 .023 .024 .000 0306 .000 0334 20 28 28 .000 0298 .000 0325 *7 .053 .054 .000 650 .000 720 6J .000 1558 .000 1616 58 59 0.025 .026 .027 .028 .029 o.ooo 0362 .000 0392 .000 0423 .000 0455 .000 0489 30 31 3* 34 34 o.ooo 0352 .000 0381 .000 0410 .000 0441 .000 0473 29 29 3 1 3* 33 0.055 .056 .057 .058 .059 o.ooo 785 .000 852 .000 1920 .000 1989 .000 2060 69 7i 71 o.ooo 1675 .000 1736 .000 1798 .000 i860 .000 1924 61 62 62 64 64 0.030 oooo 0523 o.ooo 0506 0.060 o.ooo 2131 o.ooo 1988 TABLE XIV, For finding the Ratio of the Sector to the Triangle. i i Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. oo6o .061 .062 .063 .064 o.ooo 2131 .000 2204 .000 2278 .000 2354 .000 2431 73 F o.ooo 1988 .000 2054 .000 2 I 21 .OOO 2189 .OOO 2257 66 67 68 68 70 O.I 2O .121 .122 12 3 .124 o.ooo 8845 .000 8999 .000 9154 .000 9311 .000 9469 '54 '55 '57 158 '59 o.ooo 7698 .000 7822 .000 7948 .000 8074 .000 8202 124 126 126 128 128 0.065 .066 .068 .069 oooo 2509 .000 2588 .000 2669 .000 2751 .000 2834 K 82 H o.ooo 2327 .000 2398 .oco 2470 .coo 2543 .000 2617 72 73 74 74 O.I25 .126 .127 .128 .129 o.ooo 9628 .000 9789 .000 9951 .001 0115 .001 0280 161 162 164 '65 167 c.ooo 8330 .000 8459 .000 8590 .000 8721 .000 8853 129 '3' 131 132 133 0.070 o.ooo 2918 86 o.coo 2691 76 0.130 o.ooi 0447 168 o.ooo 8986 .071 .072 .000 3004 .000 3091 87 80 .000 2767 .000 2844 77 78 .132 .001 0615 .001 0784 169 .000 9120 .000 9255 '34 '35 .073 .000 3180 8 9 80 .000 2922 7 '33 .001 0955 '7' .000 9390 '35 .074 .000 3269 6 9 9' .000 3001 80 '34 .001 1128 '73 '73 .000 9527 138 0.075 .076 .077 .078 .079 o.ooo 3360 .000 3453 .000 3546 .000 3641 .000 3738 93 93 95 97 97 o.ooo 3081 .000 3162 .000 3244 .000 3327 .000 3411 1 85 0.135 .136 .137 .138 '39 o.ooi 1301 .001 1477 .001 1654 .001 1832 .OOI 2OI2 176 '77 178 180 181 o.ooo 9665 .000 9803 .000 9943 .001 0083 .001 0224 ,38 140 140 141 142 0.080 .081 .082 .083 o ooo 3835 .000 3934 .000 4034 .000 4136 99 IOO 102 o.ooo 3496 .000 3582 .000 3669 .000 3757 86 88 go 0.140 .141 .142 .143 o.ooi 2193 .001 2376 .001 2560 .001 2745 183 184 185 188 o.ooi 0366 .001 0509 .001 0653 .001 0798 '43 '44 III .084 .000 4239 103 I0 4 .000 3846 7 9 .144 .001 2933 188 .001 0944 140 '47 0.085 .086 o ooo 4343 .000 4448 10 5 o.ooo 3936 .000 4027 9' 'J 4 6 o.ooi 3121 .001 3311 190 o.ooi 1091 .001 1238 '47 .087 .088 .000 4555 .000 4663 I0 7 1 08 .000 4119 .000 4212 92 93 :5J! .001 3505 .001 3696 192 '93 .001 1387 .001 1536 '49 '49 .089 .000 4773 I IO III .000 4306 94 95 .149 .001 3891 '95 196 .001 1686 150 152 0.090 oooo 4884 o.ooo 4401 0.150 o.ooi 4087 108 o 001 1838 .091 .092 .093 .094 .000 4996 .000 5109 .000 5224 .000 5341 "3 "5 117 "7 .000 4496 .000 4593 .000 4691 .000 4790 95 97 98 99 IOO .151 .152 '53 .154 .001 4285 .001 4484 .001 4684 .001 4886 198 199 200 202 204 .001 1990 .001 2143 .001 2296 .001 2451 152 '53 '53 '55 156 0.095 .096 .097 .098 o.ooo 5458 .000 5577 .000 5697 .000 5819 "9 I2O 122 123 o.ooo 4890 .000 4991 .000 5092 .000 5195 101 101 103 1 04 0.155 .156 '57 .158 o.ooi 5090 .001 5295 .001 5502 .001 5710 20 5 207 208 2IO o.ooi 2607 .001 2763 .001 2921 .001 3079 156 158 158 .099 .000 5942 124 .000 5299 104 .159 .001 5920 211 .001 3238 160 O.I 00 .101 . 02 3 . 04 o ooo 6066 .000 6192 .000 6319 .coo 6448 .000 6578 126 i*7 129 130 131 o.ooo 5403 .000 5509 .000 5616 .000 5723 .000 5832 106 107 107 109 109 o.i 60 .161 .162 .163 .164 o.ooi 6131 .001 6344 .001 6559 .001 6775 .001 6992 2I 3 III 217 2I 9 o.ooi 3398 .001 3559 .001 3721 .001 3883 .001 4047 161 162 162 ,64 164 o. 05 . 06 07 . 08 o.ooo 6709 .000 6842 .000 6976 .000 7111 133 '34 '35 o.ooo 5941 .000 6052 .000 6163 .000 0275 III III 2 0.165 .166 .167 .168 o.ooi 7211 .001 743* .001 7654 .001 7878 221 222 224 o.ooi 4211 .001 4377 .001 4543 .001 4710 166 166 167 . 09 .000 7248 '37 138 .000 6389 4 4 .169 .001 8103 225 227 .001 4878 168 169 O. IO . 1 1 o.ooo 7386 .000 7526 140 o.ooo 6503 .000 6618 5 0.170 .171 o.ooi 8330 .001 8558 228 o.ooi 5047 .001 5216 *5 f . 12 .113 .114 .000 7667 .000 7809 .000 7953 141 142 144 '45 .000 6734 .000 6851 .000 6969 o J 9 .172 '73 .174 .001 8788 .001 9020 .001 9253 230 232 233 234 .001 5387 .001 5558 .001 5730 171 172 '73 o 115 .116 .117 .118 .119 o ooo 8098 .000 8245 .000 8393 .000 .8 542 .000 8693 '47 148 '49 152 o.ooo 7088 .000 7208 .000 7329 .000 7451 .000 7574 20 i 2 3 124 0.175 .176 .177 .178 .179 o.ooi 9487 .001 9724 .001 9961 .OO2 O2OI .002 0442 237 237 240 241 243 o.ooi 5903 .001 6077 .001 6252 .001 6428 .001 6604 '74 '75 176 176 178 0.120 o.ooo 8845 o.ooo 7698 o.i 80 O.OO2 0635 o.ooi 6782 TABLE XIV, For finding the Ratio of the Sector to the Triangle. X * r Ellipse. Diff. Hyperbola. I Diff. Ellipse. Diff. Hyperbola. Diff. o.iSo .181 .182 3j O.OO2 0685 .002 0929 .002 1175 .OO2 1422 .002 1671 244 246 247 249 251 0.001 6782 .001 6960 .001 7139 .001 7319 .001 7500 178 179 1 80 181 181 0.240 .241 .242 .243 .244 0.003 8289 .003 8635 .003 8983 .003 9333 .003 9685 346 348 350 35* 354 0.002 8939 .002 9166 .002 9394 .002 9623 .002 9852 228 229 229 231 0.185 .186 O.OO2 1922 .002 2174 252 o.ooi 7681 .001 7864 :U 0.245 .246 0.004 39 .004 0394 I 0.003 83 .003 0314 231 .187 .188 .OO2 2428 .OO2 2683 254 .001 8047 .001 8231 ill .247 .248 .004 0752 .004 mi 35 359 3 A* .003 0545 .003 0778 233 .189 .OO2 2941 i .001 8416 185 186 .249 .004 1472 J 1 363 .003 ion 233 a 34 0.190 .191 .192 .193 0.002 3199 .OO2 3460 .OO2 3722 .OO2 3985 261 262 263 266 0.001 8602 .001 8789 .001 8976 .001 9165 187 187 189 I 8r\ 0.250 .251 .252 0.004 1835 .004 2199 .004 2566 .004 2934 364 is 0.003 Ia 45 .003 1480 .003 1716 .003 1952 235 236 236 .194 .OO2 4251 267 .001 9354 109 190 .254 .004 3305 37 ! 372 .003 2189 237 238 0.195 .196 .197 .198 .199 0.002 4518 .002 4786 .002 5056 .002 5328 .002 5602 268 2 7 o 272 274 275 o.ooi 9544 .001 9735 .001 9926 .002 0119 .002 0312 191 191 193 193 195 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 *39 ' 239 *4 241 241 O.2OO .201 0.002 5877 .002 6154 277 O.OO2 0507 .OO2 0702 '95 O.26o .26l 0.004 5566 .004 5949 33 0.003 3628 .003 3871 .202 .203 .204 .002 6433 .002 6713 .002 6995 279 280 282 283 .002 0897 .002 1094 .002 1292 '95 197 198 198 .262 .263 .264 .004 6334 .004 6721 .004 7111 390 391 .003 4114 .003 4358 .003 4603 243 244 245 t 245 O.2O5 .206 .207 .208 .209 0.002 7278 .OO2 7564 .002 7851 .002 8139 .002 8429 286 287 288 2 9 o 293 O.OO2 I49O .OO2 1689 .002 1889 .OO2 2090 .OO2 2291 199 200 20 1 2OI 203 o 265 .266 .26 7 .268 .269 0.004 75* .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 ; 2$ 249 i 249 O.2IO .211 .212 .213 .214 O.OO2 8722 .OO2 9015 .002 9311 .002 9608 .002 9907 2 93 296 297 299 300 O.OO2 2494 .OO2 2697 .OO2 2901 .OO2 3106 .002 3311 20 3 204 20 5 20 5 207 0.270 .271 .272 .273 .274 0.004 9485 .004 9888 .005 0292 .005 0699 .005 1107 403 404 407 408 410 0003 6087 .003 6337 .003 6587 .003 6839 .003 7091 250 ; 250 252 252 | 253 0.215 .216 .217 .218 0.003 O2O7 .003 0509 .003 0814 .003 III9 .003 1427 302 305 305 3 o8 39 O OO2 351? .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 4'3 414 416 418 420 0.003 7344 .003 7598 .003 7852 .003 8107 .003 8363 *54 i 254 *S5 256 *57 0.220 .221 .222 .223 .224 O.OO3 1736 .003 2047 .003 2359 .003 2674 .003 2990 311 312 3'5 3j6 O.OO2 4562 .002 4774 .002 4986 .002 5199 .002 5412 212 212 213 2I 3 215 0.280 .281 .282 .283 .284 0.005 3598 .005 4020 .005 4444 .005 4870 .005 5298 422 426 428 43 0.003 862 .003 8877 .003 9135 .003 9394 .003 9654 257 258 259 260 260 0.225 .226 .227 .228 .229 0.003 3308 .003 3627 .003 3949 .003 4272 .003 4597 319 322 323 3 2 5 O.OO2 5627 .002 5842 .002 6058 .002 6275 .002 649 \\l 217 III 0.285 .286 .28 7 .288 .28 9 0.005 57*8 .005 6160 .ooj 6594 .005 7030 .005 7468 43* 434 436 438 440 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 33 332 334 i ?f> 0.002 6711 .002 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 442 445 446 448 45 0.004 1227 .004 1491 .004 1757 .004 2023 .004 2290 266* ' 266 26? 0.235 .236 .237 .238 .239 0.003 6584 .003 6921 .003 7260 .003 7601 .003 7944 33 337 339 34' 343 O.OO2 7816 .002 8039 .002 ^263 .002 8487 .002 8713 223 224 224 226 226 0.2 9 5 .2 9 6 .2 9 7 .298 .299 0.006 0139 .006 0591 .006 1045 .006 1502 .006 1960 45* 454 457 0.004 2557 .004 2826 .004 3095 .004 3364 .004 3635 269 269 269 271 271 345 0.300 0.006 2421 0.004 3906 TABLE XV, For Elliptic Orbits of great eccentricity. cor 3 log .Bo or log -B ' Diff. log tf Diff. tort log B or log B ' Diff. log JV Diff. o o.ooo oooo o.ooo oooo 30 o.ooo 0066 o.ooo 6400 1 .000 oooo .000 0007 7 31 .000 0075 9 1 1 .000 6836 43" 2 .000 oooo .000 0028 16 32 .000 0086 .000 7286 450 3 .000 oooo .000 0064 3 33 .000 0097 I 2 .000 7750 A7O 4 .000 OOOO .000 0113 64 34 .000 0109 13 .000 8229 T-/V 493 5 6 o.ooo oooo .000 oooo o.ooo 0177 .000 0255 78 92 35 36 0.000 0122 .000 0137 11 o.ooo 8722 .000 9230 508 5 2 3 7 .000 oooo .000 0347 107 37 .000 0153 18 .000 9753 c-37 8 .000 oood .000 0454 1 20 38 .000 0171 19 .001 0290 J 3 / C C2 9 .000 OOOI .000 0574 135 39 .000 0190 2.0 .001 0842 567 10 O.OOO OOOI o.ooo 0709 40 0.000 0210 22 o.ooi 1409 eg! 11 12 .000 OOOI .000 0002 .000 0858 .000 I 02 I !t| 178 41 42 .000 0232 .000 0255 11 .001 1990 .001 2586 596 611 13 14 .OOO OOO2 .000 0003 .000 1199 .000 1390 '9i 206 43 44 .000 0281 .000 0308 27 29 .001 3197 .001 3823 626 640 15 16 o.ooo 0004 .000 0005 0.000 1 596 .000 1816 220 45 46 o.ooo 0337 .000 0368 31 7-3 o.ooi 4463 .001 5118 655 670 17 .000 0007 .000 2051 248 47 .000 0401 36 .001 5788 685 18 19 .000 0009 .OOO OOII .000 2299 .000 2562 26^ 277 48 49 .000 0437 .000 0475 38 4 .001 6473 .001 7173 700 715 20 21 22 23 24 o.ooo 0013 .OOO OOIO .000 0019 .000 0023 .000 0027 3 3 4 4 5 o.ooo 2839 .000 3131 .000 3437 .000 3757 .000 4091 2 9 2 306 320 334 349 50 51 52 53 54 o.ooo 0515 .000 0558 .000 0604 .000 0652 .000 0703 H 51 54 o.ooi 7888 .001 86l8 .001 9362 .002 0122 .002 0897 73 744 760 775 790 25 o.ooo 0032 r 0.000 4440 36l 55 o.ooo 0757 eg O.OO2 1687 806 26 27 28 29 .000 0037 .000 0043 .000 0050 .000 0057 6 7 7 9 .000 4803 .000 5181 .000 5573 .000 5980 378 392 407 420 56 57 58 59 .000 0815 .000 0875 .000 0939 .000 1007 60 64 68 71 .002 2493 .002 3313 .002 4149 .002 5000 820 836 851 866 30 o.ooo 0066 0.000 6400 60 o.ooo 1078 0.002 5866 TABLE XVI. For Hyperbolic Orbits. w orn log Cor log Qf log I. Diff. log half II.Diff. m orn log Q or log 9453 1 i^-y 1149 1150 j A-n 3*3 8553 l\l 3*3 9469 I.. 3*4 0384 Z,l 0*59 .449 7625 IO2o 1028 .279 0968 825 .506 0603 1 F5 r-> t*^ - ^ t*-- t-*-OO vO oom*o vot^mo^-ooo m^ ON O "* OO ^ OO ro^O m in vo t-^ OO vO OO *O m o O r< ?s,*s,-s. Jn R^ H" 1^ oV o\ oV o\ oV ooooo ooooo ooooo OOOOO OOOOO OOOONO \0 O O O O ON tOOOOO OOOOO OOOOO ooooo ooooo ooooo rj_ jn .*t*sj ** a = Rttnc- ral-tS- Islll oo 0^ rn jn^- a tOOO^O OOOOO OOOOO H ooooo ooooo ooooo OOOMOI-. ^^g^^Ji, w SN?i 00 * jn ON NO VO O l-> 5 OOOOO OOOOO OOOOO ooooo ooooo ooooo OOOOO i b uo ~, *n mr * ^* r ^ * * cn $ * * OH-^t^i-iOO ^h J^OO ^O vO vO t^-t ri r^ Th \OON OooomrJ- in^inoro 3?S?2 ON*- jn O; O JT trw5-5-r & ^^. c< * in rt M ^H*- *^ ww ~ rt 2 1 MM rJ M 2 ~ r< Mr )M>H H r) JS P "S ~ "^ ^S^O^ -<*Jj^gP CB^I^K^ o^f*fsoto ccr^t^-so^ cc^c^Cico I-H ea us to to 10 t^ oo o o o t rH SO i-l t-l i-l ^ c3 (M " C4 S i^ S ^ C <5 O 55 ^ O ^H O t-s JE -s ~~ _-,:r-.7 reso -*1M^>* t-xiO ^sn^'* 55 TABLE XVIII, Elements of the Orbits of Comets which have been observed. ll|3|f 2a^5j3 ni ill 111 -S *i . > -P . -o . ~ . | .5 a ^-s-f . s ,. -:5 fr-s 5 ftl 7 !^ gjsiT! P3 * leg 3 a =-^ a, * a-s : o 6w3 3^ |j.ONONON ONOO' ONONON ONONONoVo" O\O*ONoVo\ q q q q q o^ q q q q q q q q q vo ON ON ON o o o o q q q q o o o q o o o ^OO t^O^Qvo dOOOvo > ONOO oo doOMooO oovnr^clco 00000 O^voovooo >- vo o vo O ON ^- TJ-VO oo O H * M CO cl M CO " O t~~00 ON vo ON 00 O O VOclVOOON OONHtt^ 1VO VO CO Tj- ON t^OO 00 00 ^ vooo t^ OS ^ oo O vooo vo co^-ONONt^- t^ONVOwrl O M Oi M VOVO^-ON C!M vovo 00 co d co cT "*" cl cl * co vo S oo ON ONOO O clvo ON vovovooo a vo vo oo oo **-vo it^ T}-OOOOOCO MCOI^O* ^-ON>ONO 2? liTi-i 00 m Hrii-i i<5fe^; i 11114 lllll Jl^l )* ^.^^*-*^ TABLE XVIIL Elements of the Orbits of Comets which have been observed. . ardt rdt ier. *Ai$ 4**\ t. I--* I--'!* t! ef-d i C 2 H 8 C S i IK B K fi S O f O ON vo oo ro o to q q q q q q q q q ON q q q ON q q oo ON r^. q q t^- q q q ON q q q in q q q oo q s i- vo o inrtH in wwito rl^-mtoin M Mrtm ntoH VO M t^ 2 8, O t^J ON ^*5: AS ^3\3-: <*- O in t-~ t^ vo vo ONVO O O ON t^vo oo O ~ r^oo ^Tj-to-,inooOONww rloom omnvoo toQONtno votoo in u-inm ri^-inu-, to M to o t>oo oo t^ rv '8? ^dilltl OO rHC^CO^CJi OOi-I^CO ^lOlOOO TABLE XVIII, Elements of the Orbits of Comets which have been observed. ^ 1 Saron. Mechain. Saron. Mechain. ** ; ,; j d a P5 P P3 P L. 14...., it x 1 , "S ,5n fc "S * ^5 .in * * "S .3 "qj ,!i "S .3 * 1y .^ * fc P5 Q PH -1 O pq Q pq Q 3 Q CM I *OO^VO 0^ d vo ON r*lNO vr> O oo oo O 0\ 0\ 0% 0* 9tO^^O ONOC^C^^\ CT\^C^OO cf\CNOC^CN ^ CN O O CN CNOO*OOO\ O O O O O M O 00 00 OOOOO\ooOOOO OOOOO oot^ooo OO S^^CN <> O O SJ O O .- H*M - romm* ** mto^M WH mwH^j cs * 5^ H" & VO VO f tf t^ O ^> w f^vo ' ^s| SS THH *!?**-"& k t^ ON r 00 M ON r ui M l*^ fl v^ O t*^ O r*^oo t^t^vovOO*i *^n*rtt^ OH*t^*ooooNOroo WVO tr,00 VOHONW* coOrortOO vOONt^OON Mrw-vr~f ONt-^ONt^w S'^^Trt'^irS^ " w'S^IwMw r rt -rt - S oo_ w woo^ c ON t~ t~.oo t~. & 55 S^^H?^ t^ 00 00 O O OO 00 00 O> OJ ^sU-* s^s,isr s!*??* a-*?& "?sis! i-^-ssj a 5f 85*^ a"**" -^a" ^ 00 " or 2 sf^s tf tf gfs" ^-^ v |i s i s gt^ts |-j ^fi | ij*l -sf III Jlll-s ' gnjp^;^ PO SSt^-QCCSO * - *J**2!!2 Sl^2S2 TABLE XVIII, Elements of the Orbits of Comets which have been observed. J& 1 1 c 1 1 J* 5 . si . |1||| J||IlJ|j ^CQt^Q2taC] C O W > W M M ^ * iJ- ,;, .H ISllffl Motion. a> o3 73 13 1 22 2 +S bO^S tJO^^S) ^ ^ ^ +S tt> *T3 no H3 T3 ^ n3 222 22 2 ^ pd O p Q o3 ftP^ S (3 S 3 3 3 * ^ ^ ^ g.^ ^ o^jrvg^ oV^ c 2 c o c io $" 8 5; oT^vc'cS ON S^&vg" I JT^ 8,1 g 1? ^;^H <>**** oVoc-oVoXoX doVoXooX oVoVdoo oVoVoo-oVoX oVoVoVoVcf, o***S ^. -. O vt * r * 00 ON ^ vf> o ro *4- *j~i oo ^ ON O OOooOt-^vOOOooO ONOOOO OOt>T-tON tO ^ 5-00 00 -^-vO CT\ tTxO 5- ro ^ON-^-O O^- Tt-J^ oo^vo iToo 2 ^ tC w ^T ro SN*- * S'Si J*Br^wrM*ya*s-Jff- M M ** ,, * _ __ _ _* ~ **_ , cs O^C^^roro r-o^ t;- ^ rj^ ro ^- ON O r.ro^-M wr ,ro^-^co ^^^^H b 0^-00 ONJ, -^--5; 0,J 05 B-g.*^ OOOOrvO rorl ^-O to oo ONOO rovo ^5-?^^-= Ss?s!J^s : ss ; Illlf llllllllll GO 00 GO CO CO CO OO OO CO CO CO OO GO CO CO njitljij4t tpt^ t^i-t] <1ccQS*9r-i~!9 5i9>9>9? -5^;^;^^ ^?g^5r^ t^C^rit^ti TABLE XVIII, Elements of the Orbits of Comets which have been observed. a g Cjg .1 .ll fe- lt-pi J wJc|S ! Ill lilil Illlj |lsj ri liljj I-SJ*' 53H OSKr-iM Wfe.QW^ W>OU GO'S 25 2 cfc^ l.f f..r NO H oo T*O HO ooo NO m to o l"^ l~* HI ^ t-^ * ^- ^-oooo O HI o O *n ON O toNO t^ HI toNO tooo O inoo NO to 1-^ oo NO H, NO -4-or^rltooooOTl-rio NO oo to ON 6 rt H ~ oo <*- ~ooONnr- ~ >x i ~ + t toNO to ONOO t^r^-too^- OHt^too mONOrr^ tot< tooo O * H t* rj- t^oo - O ti ON ON ON ON O' ON o" ON ON tA o' o' o' ON oV oV o' ON ON o' ON ON oV o' o" ON oVoo' o' o' O* oV oV oX oV ON r^ o to M t <*rNOOO- inONt--ON ONOO NO o rj-NO MNOOOON oooonn i^Thon oo Oo^ooON t^t^-CNqt^. OSS>ONOCN ONCNOOOO ' O O' O o' o' O' "i* O' O' O' O' HI' o' O' O' H,' o $ rt *t-NC oo O to HI t-^NO ^- in < ON ONNO NO OO OO O OO i ONONHINOVO mHOv> -4-^0 O NO r co in-.o toNO t^oo toONON MOOHOt- O ON to NO rt ONNO r~-ON HNOONONT!- ^1- i toin^ mto^-Hi HI m Ti- ONNO NO oo <*- r- HI t~ iintOHitooo^-too m ON r * O O t-. HI ^mininM HniinHH "" to HI f~ M 00 H HI ONNO O ONfOOWrt Oto Tt-NO d t^*t--rtO 't'rtt^ t^.00 M ON H ONNO w r< H t^ OS ONNO OHO H ' in O in ONOO rTtinrr^.oO'*'-t--ON ON ONNO moo O - ONOO * NO + HI H, NO t~~NO I f-NO HI ON * O -T? " ^S" W H H t-~ irh to rt &$ ~ ~ ON r O HI ON O ( ^^=1 titg-i Nfi?y ?!IN -s^lii hlii =iiii -.t>;x sr&xy.x xv.Kir.s* es ss st st ct ^-?::; H < 91 >! 5i 11 i TABLE XVIII. Elements of the Orbits of Comets which have been observed. < fc 1 .5 > v 6x3 B B =3 .11 rid s ,JS8 8|S S-3 ^-3 li-sl-i -Ss-silF! 1*11 M^-V! 33e,SrS H,^^^ ri-SS^M >2 W W 15 '"I. o Oo Oo 0=0 2 * -g S* p p3'q p5Q pq p VOO OONMQ'H'J- M^-?rJ-rl OONCOHi oo vo 10 co to coso ON to ON O ON O t^. r O ON M_ ONOONOON OOONON t-,VO M M t^ tO CO M ON covo oo ON r- O oo Ocor HoocorJ-H O Tl-vo oooo t^t^-OclM rtHMMC^% Ol^eJtoco o cooo o to vOroONCOtoOOCOMT^-M Oco-^-vorl ONOO ON^O t^ to O vo 1-1 oo ONONOOON Tt-clco^oo ON M CO f.. to OOOt--tooo t>-OOOoO ON ON O O ON Ov ON O oV ON ON d O ON ON O oV ON ON ON ON O O O ON O vo w w M O O ON *J- 10 r-^vo 00 ONOO M Ov O Q VO vo "ON O 5- M M H t- 00 ONOOCO M^j-COM OOOON M M M-^-^-^J-61 COtOM ^ X ct T"^- Mt~,oooM VOMCOOM oooovot^ dr>*r--to M^- cooo Th M t-> moo vo oo M t-^ CO ON 5- c3 S ^^cf t-x tovo ON 5- 5-S?r^^ ^ M CO -3-VO M so i^oo u^ m o n O-NVO H oo vo r}- i^isO v^vO rl u^oo vo w c^ OO .J'&^S, ? ON t^OO CO CONO 2 o^" co co d MTj-Q OOOOONOON ii-i-^ uru >OO r-i i-H 1-1 1-1 (M (M -. >> >S^>~.oi) )^>>j3>-> llg Jim^ ct^^ll OiOiO lCOiOiOO iOiOiOOO 00 OO 00 00 CO 00 00 OO 00 00 CO OO CO 1 91 (N 91 9^ *i i si H '* *1 .S * tv a * * P PH & .S * .tJ * * * Q O S * vgfL^a^^toONvgo cSjmSoo^-to? f - ro r- ON to r>- - O vo to o oo t^oo i H rt Ov <* rooo vo to vOrort lOOl-^nvo O ON ON CN O O ON O O* ON ON ON ON O\ ON ON oV ON O\ O ON ON ON O ON ON oV ON O oo ON O O O III I I' vS- 2 ON ON OOC^oov> O^ O O O CN OCNCNOoo OC\OC\O OOOOO ONONOOO O^ ^>oo" O MnQOO w M o M o O *^ O wQ'-'O 1 ^ WWNM o O* *- -' -* O O* O oC\Nr^j^^- t^.r^. io\o w^i-Tj-t^ 1 ^ ti^c*^^- O C\*O 0020 o^ciO^vO^- t^>O cl 1-1 - Mi-ic4c*i'->e*c*i'-ii-!Hc* to^m M ot^O\i-t cot^ vooo O H-IVO O O t^oo ON O r oo vO ro t^vC J^O T*-T*--^O^ ^oo tri irt S H tooo ^o vovic?\rto or^t^ 2?SS2S SoS 355p SSSpo ggggg si^^^ ^ .,.* , ..",*.. IA kA kA kA M sr;s""' 3 ? ? ? f* rt^t>rr^ r r i> r* QC 2 91 9V 91 5J 91 9191919191 91 91 91 91 -'l 9191919191 9191919191 9191919191 9191919191 645 TABLE XIX Elements of the Orbits of the Minor Planets. 1 .5! . . S . . 111 . I U^JI INlll lllll ll-g-sl lllll |Hll | lie EoKoW W K B o^SSd SoSStS ! ij ilili r-l IM * r- -0 00 00 00 00 ftsl slls3 O ^O O iO O 00 00 00 00 00 c M ~ l-l 00 00 COO O ON o O f> f* NO t^OO VO T*- d d ONOO vo in -rj-\o -" co t^oo co co o* O O ON ONNO *^* 'r*!! 11 r^oo t^- 1^ ^-NO'NO' 4 ON CONO ON d 11 " ^"d vo^co ^ d "d 5^ ^^^ ^ CO'* - tod co * CO tO^J- Tt-Hl Tt CO tO d d d vod M C0 - " " " - d d d NO oo co s l^-voNo 1 cio6 cpoo vo d oo ONt^ro^uo vo d t^ n r^ riooN^Oto cocnc^r'f^ vo 4" -4-ND vo d JVOd O d O COM 00 0\O 00 NO CO vo 11 NO 00 VO VO CONO CO d co TJ-OO ON gill 5S.J 5sr s 1. cod oo v ~-g,J-8K S-ff-R ^d^NOONON IS*?? ^^Jo^^ t^ M O ON t^ t^OO COOd NOt^dNOM co^m 11 co cT 00 M * C^TJ-P & ON rj- VO O m *-J- rt to s^ w sa VOO-M ONJ^ ^.ct^-^ -^5 ^cTJoc? j^S gSJS? ~ss S5S S^cJ MM HI -j- * ONOO O ON w ON CO ^-NO = 3* U ||l cS^di . . . \\"st 'I*s* 55S !s 1 a aa .ss.s ZZ&3& r-5 1-5 r-5 I-J fM JS^fcH? t-s i-a "-j 1-5 i-j <** ^I|S<3 ^4^<^ 1* 1 sssss SSS^g SS^SS SSSSS SSSSS SSfeooS S S fe 5 " c) "* id|! d lllll a a 1 c5 '5. o o! '! j lllll ifHi "s'S^-jSja Sa'oSC h^OHHiii fl, WS

- J>cXlC5O FM ^ n-4 m 2J;228 -HjNjM'tirs e t- x ss o N f? J IB 646 TABLE XIX, Elements of the Orbits of the Minor Planets. iftft! I Hill !!!!! tttti iilfi irill Hill i|i|| . .$ ;g g gg 3 ~ ~ 1J|1 |rf|| c - Jll ~1J . 3 C J I IIIJI Illll Hill |||ll lllll ll^sa 2&5|s 0^000 gft^c-BO PH^OOEg >3o^6cg oJ^Ofe oloHH H^^lo SJS '& 3-^^5.5. ^5,3-vgjc : 2J!S rZ&Z ZSg^ g&2- Issl ISllJ 51:-** * H^ ?1^?^ S?i?d |?JH ^Jr^ S^H^ IH5^ -^^^^^5- ^?^^ ^^St^ sr^rss; w ?;^ ^ w O VO VN M If) c< ( 1--OOOVO r mt> *r> m M oo MO ONO q H ; u ^ r ;'?" T f ooqq ovx> o-so M WM^ \oincpqoo ON CN' n M "*" t ^^* 4 - -^ ^oj^^ 2, ooc 2-s. c s. Jiisi^ 4* ?.?*? ^^^o\ ^^^.^d 8 "** OOtp t^OO OCvOrJvO iri O - ON vOONN^cp -*OOCt^ WOMATi-fJ OOOvOv ONvO O VO OO' lA O co rj- r) inwiro N H -4- ^)- m tnuro ON t^ - oo PO woo c t^. O in< o . *. T 30 t> ^^O'J^ v ? t ! ^ v ?" ? ^ 9 " ^ * o * ON % ooONt'i--r ti^dM vo'o't^ciod oXt^- 4-oo ripioo'cir^. diA,-4- r^4-r-i Hj I-, -ajSlIl-sW, HjHjQl-jS lilll 11111 ilili liiii liiil iiiii iliii Jiill y A ijui j}fl| jJlii nil iiifj IM >0 * rt O tooo -rj-t^* vovo O O O ^ r*-trtlTj-f4 drtHiocvo ^n tJ-OO f^ H ON ^- rj- O ON < O Ooo vo n ONOO mot^. u-jOsi-ivOTf- rj- t*O 'co^ . O 00 t-OO 1000 I 4-covOMU-iMMt^ W ONOO WThcO COCOONONO ONONONf~.< IO ^^lOWCO w ^-CO ^-COMCO< M IO ^ovo M OOvoOMncpooOcoqoo voOwOMO inmcom mioco^-io -^iTw^w rJ-M 'vrfm cT cT j _ O voinwincovoqooinqrh v ON C?v co w t^-oo Min wwONwco Ovavt^-t^4-d toincoriin OH^t- ON w d vd OO M 1^00 M t^. ON O O OO cooo O ON t^ CO ON w mO rovo H t^ c4 O * >r^r^ wONOmONvocoTt-MN-* v O f^OO COON wvoOOw vocowvoO> COTJ-IO *vo VO ts G -** a & ^ c-*Jc M 02 CO H < I Si oc x ac i oo oo oc < 64S __ M 00 M O co o O O O D vo vo O M Q O f O ON t^oo ON to 00 M O N O cooo r^ d d W W InONONO oo rC ON ONVO O w Kw jj;i-i < 11 if! II O O o o o o o + 1 1 ++ 1 1 ?S O vo ^O O i^ rj- H ^o%2 S,^ * o* r) si^^tl d d o o o o o o w in covo m w < t oo 4- d cj into ~ ^ to vo w inoo w in co M W 00 ONVO t^. 0w H rh^- cs ^ CO w OOvO M ON <] tOM to 10 1 1 1 1 1 1 * 82 ^colQco a ^S H mm o $ ^-ONW ^CO N - CNXJ- 2*8 = 5,* + 1 r) M O vo vo oo w co 10^5 4- ONVO O * to m b " M 5- O COOO ON t~- COM tow I??? sgs^? ! 9 O ^n covo f^>oo - -* M V^C C* tO os: gl-s^s i " ? 8 SM I 1 ! I" 252X2^ 1 o i g llllll TABLE XXI, Constants, &c, Base of Naperian logarithms e = 2.71828183 0.43429448 Modulus of the common logarithms . . ^ = 0.43429448 9.63778431 10 Radius of a Circle in seconds r = 206264.806 5.31442513 " " " " minutes r = 3437 .7468 3.53627388 " " " " degrees r = 57.29578 1.75812263 Circumference of a Circle in seconds .... 1296000 6.11260500 " " a whenr=l. . . . it = 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 // // u it it it ii in se- conds of arc 3548.18761 3.55000657 Constant of Aberration, according to Struve 20".4451 '/ // Nutation, // u Peters 9".2231 Mean Obliquity of the ecliptic for 1750 + t, according to Bessel .... 23 28' 18".00 0".48368i 0".00000272295< Mean Obliquity of the ecliptic for 1800 + t, according to Struve and Peters . . 23 27' 54".22 0".4738* 0".0000014< s General Precession for the year 1750 + t, according to Bessel 50".21129 + 0".00024429f>6< a it it ii 'i ii Struve 50".22980 + 0".000226< MASSES OF THE PLANETS, THE MASS OP THE SUN BEING THE UNIT. Mercury ..... m= ' Jupiter ' ' ' ' m = Satura Uranus ..... 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 /> the radius of the earth, by y the geographical latitude, and by sin ~dt =n ' the formula? for the annual precession in right ascension (a) and declination (<5) become ~ = m + n tan 8 sin o, -^- = n cos o, (2; 42 57 658 THEORETICAL ASTRONOMY. and the numerical values of m and n are, for the instant t, m = 46".02824 -f 0".0003086450 (t 1750), n = 20".06442 0".0000970zu* (* i ou;. To determine the precession during the interval t' t, we compute the annual variation for the instant J (t f + t) and this variation mul- tiplied by t' t furnishes the required result. B. Nutation. The expressions for the equation of the equinoxes and for the nutation of the obliquity of the ecliptic are, according to Peters, AA = 17".2405 sin ft + 0".2073 sin 2ft 0".2041 sin 2 + 0".0677 sin ( (C F) 1".2694 sin 20 + 0".1279 sin (0 r) 0".0213sin(O + r), Ae = + 9".2231 cos ft 0".0897 cos 2ft + 0".08S6 cos 2 + 0".5510 cos 20 + 0".0093 cos (0 + T), for the year 1800, and A* = _ 17".2577 sin ft + 0".2073 sin 2ft 0".2041 sin 2 + 0".0677 sin ( I") 1".2695 sin 20 + 0".1275 sin (0 r) 0".0213sin(O + r), Ae = + 9".2240 cos ft 0".0896 cos 2ft + 0".0885 cos 2 The values of A/I and AS are found from the preceding equations, and for the differential coefficients we have APPENDIX. 659 ? 7 -^- = cos e -f- sm e tan 5 sin o, = cos a sin s, da (5) The terms of the second order are of sensible magnitude only when the body is very near the pole, and in this case by computing the second differential coefficients the complete values may be found. In the reduction of the place of a planet or comet from the mean equinox of one date t to the true equinox of another date t', the determination of the correction for precession and of that for nutation may be effected simultaneously. Thus, let r denote the interval t' t expressed in parts of a year, and the sum of the corrections for precession and nutation gives Aa == mr -f- AA cos -}- (nr -f- AA sin e) sin o tan 3 AS cos <* tan