> / ^^W%^ / THEORETICAL ASTRONOMY RELATING TO THE MOTIONS OF THE HEAVENLY BODIES REVOLVING AROUND THE SUN IN ACCORDANCE WITH THE LAW OF UNIVERSAL GRAVITATION EMBRACING A SYSTEMATIC DERIVATION OF THE FORMULA FOR THE CALCULATION OF THE GEOCENTRIC AND CENTRIC PLACES, FOR THE DETERMINATION OF THE ORBITS OF PLANETS AND COMETS, FOB THE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COMPUTATION OF SPECIAL PERTURBATIONS; TOGETHER WITH THE THEORY OF THE COMBI- NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. Wtiili Uunwwtl feunjjte mul ^uriliarg i BY JAMES C. WATSON DIRECTOR OF THE OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF MICHIGAN PHILADELPHIA J. B. LIPPINCOTT & CO. LONDON: TRUBNER & CO. 1868 ASTRONOMY UBRAR* / Entered, according to Act of Congress, in the year 1868, by J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Eastern District of Pennsylvania. W3 PREFACE. THE discovery of the great law of nature, the law of gravitation, by NEWTON, prepared the way for the brilliant achievements which have distinguished the history of astronomical science. A first essential, how- ever, to the solution of those recondite problems which were to exhibit the effect of the mutual attraction of the bodies of our system, was the development of the infinitesimal calculus ; and the labors of those who devoted themselves to pure analysis have contributed a most important part in the attainment of the high degree of perfection which character- izes the results of astronomical investigations. Of the earlier efforts to develop the great results following from the law of gravitation, those of EULER stand pre-eminent, and the memoirs which he published have, in reality, furnished the germ of all subsequent investigations in celestial mechanics. In this connection also the names of BERNOUILLI, 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 LAGRANGE and LAPLACE. The genius and the analytical skill of these extraordinary men gave to the progress of Theoretical Astronomy the most rapid strides ; and the intricate investigations which they 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 little more remained to be accomplished by their successors than to develop and simplify the methods which they made known, and to intro- duce such modifications as should be indicated by experience or rendered possible by the latest discoveries in the domain of pure anatysis. The problem of determining the elements of the orbit of a comet moving in a parabola, by means of observed places, which had been considered by NEWTON, EULER, BOSCOVICH, LAMBERT, and others, received from LAGRANGE and LAPLACE the most careful consideration in the light of all that had been previously done. The solution given by the former is analytically complete, but far from being practically complete; that given by the latter is especially simple and practical so far as regards the labor of computation; but the results obtained by it are so affected by the unavoidable errors of observation as to be often little more than rude approximations. The method which was found to answer best in actual practice, was that proposed by OLBERS in his work entitled Leichteste und bequemste Methode die Bakn 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. His method for determining all the elements directly from given observed places, as given in the Theoria Motus, and as subsequently given in a revised form by ENCKE, leaves scarcely any thing to be desired on this topic. In the same work he gave the first explanation of the method of least squares, a method which has been of inestimable service in investigations depending on observed data. The discovery of the minor planets directed attention also to the methods of determining their perturbations, since those applied in the case of the major planets were found to be inapplicable. For a long time astronomers were content simply to compute the special perturba- tions of these bodies from epoch to epoch, and it was not until the com- mencement of the brilliant researches by HANSEN that serious hopes were entertained of being able to compute successfully the general per- turbations of these bodies. By devising an entirely new mode of con- sidering the perturbations, namely, by determining what may be called the perturbations of the time, and thus passing from the undisturbed place to the disturbed place, and by other ingenious analytical and mechanical devices, he succeeded in effecting a solution of this most difficult problem, and his latest works contain all the 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 formula for the determination of the perturbations of the latitude, the mean anomaly, and the loga- rithm of the radius-vector, which are exceedingly convenient in the process of integration, and which have been found to give the most satisfactory results. The formulse for the perturbations of the latitude, 6 PREFACE. true longitude, and radius-vector, to be integrated in the same manner, were afterwards given by BRUNNOW. Having thus stated briefly a few historical facts relating to the problems of theoretical astronomy, I proceed to a statement of the object of this work. The discovery of so many planets and comets has furnished a wide field for exercise in the calculations relating to their motions, and it has occurred to me that a work which should contain a development of all the formulae required in determining the orbits of the heavenly bodies directly from given observed places, and in correcting these orbits by means of more extended discussions of series of observa- tions, including also the determination of the perturbations, together with a complete collection of auxiliary tables, and also such practical directions as might guide the inexperienced computer, might add very materially to the progress of the science by attracting the attention of a greater number of competent computers. Having carefully read the works of the great masters, my plan was to prepare a complete work on this subject, commencing with the fundamental principles of dynamics, and systematically treating, from one point of view, all the problems presented. The scope and the arrangement of the work will be best understood after an examination of its contents ; and let it suffice to add that I have endeavored to keep constantly in view the wants of the computer, providing for the exceptional cases as they occur, and giving all the formulae which appeared to me to be best adapted to the problems under consideration. I have not thought it worth while to trace out the geometrical signification of many of the auxiliary quantities introduced. Those who are curious in such matters may readily derive many beau- tiful theorems from a consideration of the relations of some of these auxiliaries. For convenience, the formula) are numbered consecutively through each chapter, and the references to those of a preceding chapter are defined by adding a subscript figure denoting the number of the chapter. Besides having read the works of those who have given special 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 which astronomers are enabled to arrive at so many grand results connected with the motions of the heavenly bodies, and by which the grandeur and sublimity of creation are unveiled. The labor of the preparation of the work will have been fully repaid if it shall be the means of directing a more general attention to the study of the wonderful mechanism of the hea- vens, the contemplation of which must ever serve to impress upon the mind the reality of the perfection of the OMNIPOTENT, the LIVING GOD ! OBSERVATORY, ANN ARBOR, June, 1867. CONTENTS. THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR- MULAE FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COM- PUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. 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 Eeduction to the Ecliptic 85 Geocentric Longitude and Latitude 86 Transformation of Spherical Co-ordinates 87 Direct Determination of the Geocentric Eight Ascension and Declination 90 Reduction of the Elements from one Epoch to another 99 Numerical Examples 103 Interpolation , 112 Time of Opposition 114 9 10 CONTENTS. CHAPTEE II. INVESTIGATION OF 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. PAGE Variation of the Right Ascension and Declination 118 Case of Parabolic Motion 125 Case of Hyperbolic Motion 128 Case of Orbits differing but little from the Parabola 130 Numerical Examples 135 Variation of the Longitude and Latitude 143 The Elements referred to the same Fundamental Plane as the Geocentric Places 149 Numerical Example 150 Plane of the Orbit taken as the Fundamental Plane to which the Geocentric Places are referred 153 Numerical Example 159 Variation of the Auxiliaries for the Equator 163 CHAPTER III. INVESTIGATION OF FORMULA FOR COMPUTING THE ORBIT OF A COMET MOVING IN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION OF THE GEOCENTRIC DISTANCE. Correction of the Observations for Parallax 167 Fundamental Equations 169 Particular Cases 172 Ratio of Two Curtate Distances 178 Determination of the Curtate Distances 181 Relation between Two Radii-Vectores, the Chord joining their Extremities, and the Time of describing the Parabolic Arc 184 Determination of the Node and Inclination 192 Perihelion Distance and Longitude of the Perihelion '. 194 Time of Perihelion Passage 195 Numerical Example 199 Correction of Approximate Elements by varying the Geocentric Distance 208 Numerical Example 213 CHAPTER IV. DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF THE CONIC SECTION. Reduction of the Data 220 Corrections for Parallax .. 223 CONTENTS. 11 PAGE 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 Katio 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 3TL 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 Formula 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 Katio of Two Curtate Distances 349 12 CONTENTS. PAGE Variation of the Geocentric Distance and of the Reciprocal of the Semi-Trans- verse Axis 352 Equations of Condition 353 Orbit of a Comet 355 Variation of Two Eadii-Vectores 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OF THE COMBINATION OF OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OF OBSERVATIONS. Statement of the Problem 360 Fundamental Equations for the Probability of Errors 362 Determination of the Form of the Function which expresses the Probability ... 363 The Measure of Precision, and the Probable Error 366 Distribution of the Errors 367 The Mean Error, and the Mean of the Errors 368 The Probable Error of the Arithmetical Mean 370 Determination of the Mean and Probable Errors of Observations 371 Weights of Observed Values , 372 Equations of Condition 376 Normal Equations 378 Method of Elimination 380 Determination of the Weights of the Resulting Values of the Unknown 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 FORMULAE FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. Fundamental Equations 426 Statement of the Problem 428 Variation of Co-ordinates 429 CONTENTS. 13 PAGE Mechanical Quadrature 433 The Interval for Quadrature 443 Mode of effecting the Integration 445 Perturbations depending on the Squares and Higher Powers of the Masses 446 Numerical Example 448 Change of the Equinox and Ecliptic 455 Determination of New Osculating Elements 459 Variation of Polar Co-ordinates 462 Determination of the Components of the Disturbing Force 467 Determination of the Heliocentric or Geocentric Place 471 Numerical Example 474 Change of the Osculating Elements 477 Variation of the Mean Anomaly, the Kadius- 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. PAGE I. Angle of the Vertical and Logarithm of the Earth's Radius 561 II. For converting Intervals of Mean Solar Time into Equivalent Intervals of Sidereal Time 563 III. For converting Intervals of Sidereal Time into Equivalent Intervals of Mean Solar Time 564 IV. For converting Hours, Minutes, and Seconds into Decimals of a Day... 565 V. For finding the Number of Days from the Beginning of the Year 565 VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit 566 VII. For finding the True Anomaly in a Parabolic Orbit when v is nearly 180 611 VIII. For finding the Time from the Perihelion in a Parabolic Orbit 612 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 Eoots of the Equation sin (z' ) = m sin 4 z' ... 622 XIII. For finding the Ratio of the Sector to the Triangle 624 XIV. For finding the Ratio of the Sector to the Triangle 629 XV. For Elliptic Orbits of Great Eccentricity 632 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 XXL Constants, &c 649 EXPLANATION OF THE TABLES 651 APPENDIX. Precession 657 Nutation 658 Aberration 659 Intensity of Light 660 Numerical Calculations 662 THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL, EQUATIONS OF MOTION, AND OF THE FOR- MULAE FOB DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTA- TION FOR CASES OF ANY ECCENTRICITY WHATEVER. 1. THE study of the motions of the heavenly bodies does not re- quire that we should know the ultimate limit of divisibility of the matter of which they are composed, whether it may be subdivided indefinitely, or whether the limit is an indivisible, impenetrable atom. Nor are we concerned with the relations which exist between the separate atoms or molecules, except so far as they form, in the aggre- gate, a definite body whose relation to other bodies of the system it is required to investigate. On the contrary, in considering the ope- ration of the laws in obedience to which matter is aggregated into single bodies and systems of bodies, it is sufficient to conceive simply of its divisibility to a limit which may be regarded as infinitesimal compared with the finite volume of the body, and to regard the mag- nitude of the element of matter thus arrived at as a mathematical point. An element of matter, or a material body, cannot give itself motion; neither can it alter, in any manner whatever, any motion which may have been communicated to it. This tendency of matter to resist all changes of its existing state of rest or motion is known as inertia, and is the fundamental law of the motion of bodies. Ex- perience invariably confirms it as a law of nature; the continuance of motion as resistances are removed, as well as the sensibly unchanged motion of the heavenly bodies during many centuries, affording the 15 16 THEORETICAL ASTRONOMY. most convincing proof of its universality. Whenever, therefore, a material point experiences any change of its state as respects rest or motion, the cause must be attributed to the operation of something external to the element itself, and which we designate by the word force. The nature of forces is generally unknown, and we estimate them by the effects which they produce. They are thus rendered com- parable with some unit, and may be expressed by abstract numbers. 2. If a material point, free to move, receives an impulse by virtue of the action of any force, or if, at any instant, the force by which motion is communicated shall cease to act, the subsequent motion of the point, according to the law of inertia, must be rectilinear and uniform, equal spaces being described in equal times. Thus, if s, v, and t represent, respectively, the space, the velocity, and the time, the measure of v being the space described in a unit of time, we shall have, in this case, s = vt. It is evident, however, that the space described in a unit of time will vary with the intensity of the force to which the motion is due, and, the nature of the force being unknown, we must necessarily compare the velocities communicated to the point by different forces, in order to arrive at the relation of their effects. We are thus led to regard the force as proportional to the velocity; and this also has received the most indubitable proof as being a law of nature. Hence, the principles of the composition and resolution of forces may be applied also to the composition and resolution of velocities. If the force acts incessantly, the velocity will be accelerated, and the force which produces this motion is called an accelerating force. In regard to the mode of operation of the force, however, we may consider it as acting absolutely without cessation, or we may regard it as acting instantaneously at successive infinitesimal intervals repre- sented by dt, and hence the motion as uniform during each of these intervals. The latter supposition is that which is best adapted to the requirements of the infinitesimal calculus; and, according to the fundamental principles of this calculus, the finite result will be the same as in the case of a force whose action is absolutely incessant. Therefore, if we represent the element of space by ds, and the ele- ment of time by dt, the instantaneous velocity will be which will vary from one instant to another. FUNDAMENTAL PRINCIPLES. 17 3. Since the force is proportional to the velocity, its measure at any instant will be determined by the corresponding velocity. If the accelerating force is constant, the motion will be uniformly accele- rated; and if we designate the acceleration due to the force by/, the unit of/ being the velocity generated in a unit of time, we shall have If, however, the force be variable, we shall have, at any instant, the relation /= dt the force being regarded as constant in its action during the element of time dt. The instantaneous value of v gives, by differentiation, dv _ d*s ~dt == ~di? and hence we derive d * s so that, in varied motion, the acceleration due to the force is mea- sured by the second differential of the space divided by the square of the element of time. 4. By the mass of the body we mean its absolute quantity of mat- ter. The density is the mass of a unit of volume, and hence the entire mass is equal to the volume multiplied by the density. If it is required to compare the forces which act upon different bodies, it is evident that the masses must be considered. If equal masses receive impulses by the action of instantaneous forces, the forces acting on each will be to each other as the velocities imparted ; and if we consider as the unit of force that which gives to a unit of mass the unit of velocity, we have for the measure of a force F, denoting the mass by M, F = Mo. This is called the quantity of motion of the body, and expresses its capacity to overcome inertia. By virtue of the inert state of matter, there can be no action of a force without an equal and contrary re- action ; for, if the body to which the force is applied is fixed, the equilibrium between the resistance and the force necessarily implies the development of an equal and contrary force ; and, if the body be free to move, in the change of state, its inertia will oppose equal and 18 THEORETICAL ASTRONOMY. contrary resistance. Hence, as a necessary consequence of inertia, it follows that action and reaction are simultaneous, equal, and contrary. If the body is acted upon by a force such that the motion is varied, the accelerating force upon each element of its mass is represented by -7-, and the entire motive force F is expressed by clt 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 and this, therefore, expresses that part of the intensity of the motive force which is impressed upon the unit of mass, and is what is usually called the accelerating force. 5. The force in obedience to which the heavenly bodies perform their journey through space, is known as the attraction of gravitation ; and the law of the operation of this force, in itself simple and unique, has been confirmed and generalized by the accumulated researches of modern science. Not only do we find that it controls the motions of the bodies of our own solar system, but that the revolutions of binary systems of stars in the remotest regions of space proclaim the uni- versality of its operation. It unfailingly explains all the phenomena observed, and, outstripping observation, it has furnished the means of predicting many phenomena subsequently observed. The law of this force is that every particle of matter is attracted by every other particle by a force which varies directly as the mass and inversely as the square of the distance of the attracting particle. This reciprocal action is instantaneous, and is not modified, in any degree, by the interposition of other particles or bodies of matter. It is also absolutely independent of the nature of the molecules them- selves, and of their aggregation. ATTRACTION OF SPHERES. 19 If we consider two bodies the masses of which are m and m', and whose magnitudes are so small, relatively to their mutual distance />, that we may regard them as material points, according to the law of gravitation, the action of m on each molecule or unit of m' will be in , and the total force on m! will be ,m m . f>' 2 The action of m' on each molecule of m will be expressed by -, and its total action by m' The absolute or moving force with which the masses m and m f tend toward each other is, therefore, the same on each body, which result is a necessary consequence of the equality of action and reaction. The velocities, however, with which these bodies would approach each other must be different, the velocity of the smaller mass exceed- ing that of the greater, and in the ratio of the masses moved. The expression for the velocity of m', which would be generated in a unit of time if the force remained constant, is obtained by dividing the absolute force exerted by m by the mass moved, which gives m 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 THEOEETICAL ASTEONOMY. of the mass of the sphere, and p its distance from the point attracted ; then will dm express the action of this element on the point attracted. If we sup- pose the density of the sphere to be constant, and equal to unity, the element dm becomes an element of volume, and will be expressed by dm = dx dy dz ; x, y, 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 (p cos 0, y = r cos sin &, z =r sin ?>, the expression for dm becomes dm = r 2 cos

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

, if we differentiate the expression for p 2 with respect to m-jt+mYdt, m ~ -f mZdt , which may be written respectively: MOTION OF A SYSTEM OF BODIES. 25 dx dx dx The actual velocities for this instant are 5+4 1+4- +4> and the corresponding forces are dx . jdx dy . jdy 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 -=- -\- mXdt, -md^jL + mYdt, (3) md r- -f mZdt. In the same manner we find for the forces which will be destroyed in the case of the body m! : fJr' -m'd^ + m'X'dt, -m'd^jt + m'Y'dt, m'd^+m'Z'dt; dt and similarly for the other bodies of the system. According to the general principle above enunciated, the system under the action of these forces alone, will be in equilibrium. The conditions of 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, 2Y t = 0, SZ, = 0, Q, Z(X,z-Zp) = 0, Z (Z ,y - Y,z~) = ; in which X,, Y,, Z,, denote the components, resolved parallel to the 26 THEORETICAL ASTEONOMY. co-ordinate axes, of the forces acting on any point, and x 9 y, z, the co-ordinates of the point. These equations are equally applicable to the case of the equilibrium at any instant of a system of variable form ; and substituting in them the expressions (3) for the forces de- stroyed in the case of a system of bodies, we shall have ^ m ~jr 2 2mX= 0, (4) which are the general equations for the motions of a system of bodies. 8. Let x t) y h 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 2mx 2my 2mz Xf = ~Zm' y ' = ~Zm~' *' ^ ~Ym' we get v ffix v d*y v d*z " - df 2m , dt 2 Sm dp 2m Introducing these values into the first three of equations (4), they become fe, __ ImX tfy, _ZmY tfz, __ ImZ , ~dP ~ ' Zm' ~di? ~ ' 2m' ~dP ~~ ~2m ' from which it appears that the centre of gravity of the system moves in space as if the masses of the different bodies of which it is com- posed, were united in that point, and the forces directly applied to it. If we suppose that the only accelerating forces which act on the bodies of the system, are those which result from their mutual action, we have the obvious relation : = ro'JT, mY= m'Y', mZ= m'Z', MOTION OF A SYSTEM OF BODIES. 27 and similarly for any two bodies ; and, consequently, 2mX = 0, SmY= 0, ZmZ= ; so that equations (5) become *5 = o, ^t = o, **' = o. dt* dff d? Integrating these once, and denoting the constants of integration by c, c', c" ', we find, by combining the results, and hence the absolute motion of the centre of gravity of the system, when subject only to the mutual action of the bodies which compose it, must be uniform and rectilinear. Whatever, therefore, may be the relative motions of the different bodies of the system, the motion of its centre of gravity is not thereby affected. 9. Let us now consider the last three of equations (4), and suppose the system to be submitted only to the mutual action of the bodies which compose it, and to a force directed toward the origin of co- ordinates. The action of m' on m, according to the law of gravita- tion, is expressed by , in which p denotes the distance of m from m'. To resolve this force in directions parallel to the three rectangular axes, we must multiply it by the cosine of the angle which the line joining the two bodies makes with the co-ordinate axes respectively, which gives m'(af x) v m'tf y) m'(z'-z) A = - = - , JL = - r - , A = -- - . p 3 p* p 3 Further, for the components of the accelerating force of m on m r , we have , m (x of) m(y y f ) m(z Q ' ~~ ' ~ * Hence we derive m(Yx Xy) + m' (FV Xy) = 0, and generally Q. (6) 28 THEORETICAL ASTRONOMY. In a similar manner, we find 2m (Xz Zx) = 0, (7) 2m (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, 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 2m (xdy ydx] = cdt, 2m (zdx xdz) = c'dt, (8) 2m (ydz zdy} c"dtj c, c', and c" being the constants of integration. Now, xdy ydx is double the area described about the origin of co-ordinates by the projection of the radius-vector, or line joining m with the origin of co-ordinates, on the plane of xy during the element of time dt; and, further, zdx xdz and ydz zdy are respectively double the areas described, during the same time, by the projection of the radius-vector on the planes of xz and yz. The constant c, therefore, expresses the sum of the products formed by multiplying the areal velocity of each body, in the direction of the co-ordinate plane xy, by its mass; and c', c r/ , 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 INVAKIABLE PLANE. 29 resulting from the mutual action of the bodies which compose the system, this result is correct whatever may be the point in space taken as the origin of co-ordinates. The areas described by the projection of the radius-vector of each body on the co-ordinate planes, are the projections, on these planes, of the areas actually described in space. We may, therefore, conceive of a resultant, or principal plane of projection, such that the sum of the areas traced by the projection of each radius-vector on this plane, when projected on the three co-ordinate planes, each being multiplied by the corresponding mass, will be respectively equal to the first members of the equations (8). Let , /9, and y be the angles which this principal plane makes with the co-ordinate planes xy, xz, and yz 9 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 /? dt, S cos Y dt. Therefore, by means of equations (8), we have c = S cos a, c r = S cos /?, c" = S cos Y, and, since cos 2 a + cos 2 /5 -f cos 2 7- = 1, Hence we derive cos a = , cos /5 = _ / i '21 '/2 c" cos Y = These angles, being therefore constant and independent of the time, show that this principal plane of projection remains constantly par- allel to itself during the motion of the system in space, whatever may be the relative positions of the several bodies; and for this reason it is called the invariable plane of the system. Its position with reference to any known plane is easily determined when the velocities, in directions parallel to the co-ordinate axes, and the masses and co-ordinates of the several bodies of the system, are known. The values of c, c r , c" are given by equations (8), and 30 THEORETICAL ASTRONOMY. hence the values of a, /9, and 7-, 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 /? = 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- vector es 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 equatity. 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 n y n z, be the co- ordinates of the movable origin of co-ordinates, referred to a fixed point in space taken as the origin ; and let X Q , y Q9 z , a? ', y ' 9 z/, &c. be the co-ordinates of the several bodies referred to the movable origin. Then, since the co-ordinate planes in one system remain always parallel to those of the other system of co-ordinates, we shall have x x, + X Q , y = y, J ry Q , * = *, + z and similarly for the other bodies of the system. Introducing these values of x 9 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. 33 and the preceding equations become o, (9) ZmZ =0. Substituting the same values in the last three of equations (4), ob- serving that the co-ordinates x h y,, z, are the same for all the bodies of the system, and reducing the resulting equations by means of equations (9), we get (r7 2 r rf 2 ? \ *, ^W~ x ^ } ~ 2m (** - 2O = 0, (10) 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 (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 Q , y Q , Z Q , we have, by means of the equations (10), 32 THEORETICAL ASTRONOMY. dm -f ( Yx ~ x%) dm = - dm - (Xz - Zx } dm = - y. - - *, dm -(% - K O ) ^ = o, the integration with respect to dm being taken so as to include the entire mass of the body. These equations, therefore, determine the motion of rotation of the body around its centre of gravity regarded as fixed, or as having a uniform rectilinear motion in space. Equa- tions (11) determine the position of the centre of gravity for any instant, and hence for the successive instants at intervals equal to dt; and we may consider the motion of the body during the element of time dt as rectilinear and uniform, whatever may be the form of its trajectory. Hence, equations (11) and (12) completely determine the position of the body in space, the former relating to the motion of translation of the centre of gravity, and the latter to the motion of rotation about this point. It follows, therefore, that for any forces which act upon a body we can always decompose the actual motion into those of the translation of the centre of gravity in space, and of the motion of rotation around this point ; and these two motions may be considered independently of each other, the motion of the centre of gravity being independent of the form and position of the body about this point. If the only forces which act upon the body are the reciprocal action 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. We may observe, also, that on account of the invariability of the mutual distances of the elements of the mass, according to equations (8), the motion of rotation must be uniform. 12. We shall now consider the action of a system of bodies on a MOTION OF A SOLID BODY. 33 distant mass, which we will denote by M. Let x w y Q , Z Q , x f , y ', z ', &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, 9 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, + a? , y = y, -f y fl , z = z,-\- z , and similarly for the other bodies of the system. If we denote by r the distance of the centre of gravity of m from that of 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 y will be mx and, therefore, that of the entire system on the element of M 9 resolved in the same direction, will be We have also r 2 = (x, + x Q Y + (y, + 2/ ) 2 + (*/ + z ) 2 and, if we denote by r, the distance of the centre of gravity of the system from M, r, = *, + y, + i,, Therefore x S - = (X, + X Q } (r; + 2(^ + Mo + z,z ) + r 2 ) . We shall now suppose the mutual distances of the bodies of the system to be so small in comparison with the distance r, of its centre of gravity from that of M 9 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 r 3 ~~ r/ r, 3 and , , =x t - -~ 3 L (x,Zmx Q + y,Zmy Q r f r, i f 3 34 THEORETICAL ASTRONOMY. But, since o? , y , z w are the co-ordinates in reference to the centre of gravity of the system as origin, we have Imx Q = 0, Smy Q = 0, Sm& = 0, and the preceding equation reduces to mx 2m s ^= x '^- In a similar manner, we find my Sm mz _ Zm ' q~~ \J 9 q * / * r 3 r, 3 r 3 r, 3 The second members of these equations are the expressions for the total accelerating force due to the action of the bodies of the system on M 9 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 w, m', m", &c. to be the elements of the mass of a single body, the form of the equations remains unchanged; and hence it follows that the mass M is acted upon by another mass, or by a system of bodies, as if the entire mass of the body, or of the system, were collected at its centre of gravity. It is evident, also, that reciprocally in the case of two systems of bodies, in which the mutual distances of the bodies are small in comparison with the distance between the centres of gravity of the two systems, their mutual action is the same as if all the several masses in each system were collected at the common centre of gravity of that system ; and the two centres of gravity will move as if the masses were thus united. 13. The results already obtained are sufficient to enable us to form the equations for the motions of the several bodies which compose the solar system. If these bodies were exact spheres, which could be considered as composed of homogeneous concentric spherical shells, the density varying only from one layer to another, the action of MOTION OF A SYSTEM OF BODIES. 35 each on an element of the mass of another would be the same as if the entire mass of the attracting body were concentrated at its centre of gravity. The slight deviation from this law, arising from the ellipsoidal form of the heavenly bodies, is compensated by the mag- nitude of their mutual distances; and, besides, these mutual distances are so great that the action of the attracting body on the entire mass of the body attracted, is the same as if the latter were concentrated at its centre of gravity. Hence the consideration of the reciprocal action of the single bodies of the system, is reduced to that of material points corresponding to their respective centres of gravity, the masses of which, however, are equivalent to those of the corresponding bodies. The mutual distances of the bodies composing the secondary systems of planets attended with satellites are so small, in comparison with the distances of the 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, 36 THEORETICAL ASTRONOMY. been rigidly demonstrated that the axis of rotation of the earth rela- tive to the body itself is fixed, so that the poles of rotation and the terrestrial equator preserve constantly the same position in reference to the surface; and that also the velocity of rotation is constant. This assures us of the permanency of geographical positions, and, in connection with the fact that the change of the length of the mean solar day arising from the variation of the obliquity of the ecliptic and in the length of the tropical year, due to the action of the sun, moon, and planets upon the earth, is absolutely insensible, amounting to only a small fraction of a second in a million of years, assures us also of the permanence of the interval which we adopt as the unit of time in astronomical investigations. 14. Placed, as we are, on one of the bodies of the system, it is only possible to deduce from observation the relative motions of the different heavenly bodies. These relative motions in the case of the comets and primary planets are referred to the centre of the sun, since the centre of gravity of this body is near the centre of gravity of the system, and its preponderant mass facilitates the integration of the equations thus obtained. In the case, however, of the secondary systems, the motions of the satellites are considered in reference to the centre of gravity of their primaries. We shall, therefore, form the equations for the motion of the planets relative to the centre of gravity of the sun; for which it becomes necessary to consider more particularly the relation between the heterogeneous quantities, space, time, and mass, which are involved in them. Each denomination, being divided by the unit of its kind, is expressed by an abstract number ; and hence it offers no difficulty by its presence in an equa- tion. For the unit of space we may arbitrarily take the mean dis- tance of the earth from the sun, and the mean solar day may be taken as the unit of time. But, in order that when the space is expressed by 1, and the time by 1, the force or velocity may also be expressed by 1, if the unit of space is first adopted, the relation of the time and the mass which determines the measure of the force will be such that the units of both cannot be arbitrarily chosen. Thus, if we denote by / the acceleration due to the action of the mass m on a material point at the distance a, and by/' the accelera- tion corresponding to another mass m! acting at the same distance, we have the relation MOTION KELATIVE TO THE SUN. 37 and hence, since the acceleration is proportional to the mass, it may be taken as the measure of the latter. But we have, for the measure of/, ^ J dV Integrating this, regarding /as constant, and the point to move from a state of rest, we get s = $fi*. (13) The acceleration in the case of a variable force is, at any instant, measured by the velocity which the force acting at that instant would generate, if supposed to remain constant in its action, during a unit of time. The last equation gives, when t = 1, /=2; and hence the acceleration is also measured by double the space which would be described by a material point, from a state of rest, during a unit of time, the force being supposed constant in its action during this time. In each case the duration of the unit of time is involved in the measure of the acceleration, and hence in that of the mass on which the acceleration depends ; and the unit of mass, or of the force, will depend on the duration which is chosen for the unit of time. In general, therefore, we regard as the unit of mass that which, acting constantly at a distance equal to unity on a material point free to move, will give to this point, in a unit of time, a velocity which, if the force ceased to act, would cause it to describe the unit of dis- tance in the unit of time. Let the unit of time be a mean solar day; ]& 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 T; then will and F 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 1& 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 unit of time, the space expressed by /v 2 . The acceleration due to the action of the sun at the unit of distance is designated by k? } since the square root of this quantity appears frequently in the formulae which will be derived. If we take arbitrarily the mass of the sun as the unit of mass, the unit of time must be determined. Let t denote the number of mean solar days which must be taken for the unit of time when the unit of mass is the mass of the sun. The space which the force due to this mass, acting constantly on a material point at a distance equal to the mean distance of the earth from the sun, would cause the point to describe in the time t, is, according to equation (13), But, since t expresses the number of mean solar days in the unit of time, the measure of the acceleration corresponding to this unit is 2s, and this being the unit of force, we have W = 1 ; and hence -=! Therefore, if the mass of the sun is regarded as the unit of mass, the number of mean solar days in the unit of time will be equal to unity divided by the square root of the acceleration due to the force exerted by this mass at the unit of distance. The numerical value of k will be subsequently found to be 0.0172021, which gives 58.13244 mean solar days for the unit of time, when the mass of the sun is taken as the unit of mass. 15. Let x, y, z be the co-ordinates of a heavenly body referred to the centre of gravity of the sun as the origin of co-ordinates; r its radius-vector, or distance from this origin; and let m denote the quotient obtained by dividing its mass by that of the sun; then, taking the mean solar day as the unit of time, the mass of the sun is expressed by F, and that of the planet or comet by mk z . For a second body let the co-ordinates be a/, y r , z' ; the distance from the sun, r' ; and the mass, m'k 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 , 57, , the co-ordinate planes being parallel to those of x, y, and 2, respectively; then will the MOTION RELATIVE TO THE SUN. 39 acceleration due to the action of m on the sun be expressed by ^~, and the three components of this force in directions parallel to the co-ordinate axes, respectively, will be m tf-, mtf-V-, mJc 2 . r 3 r 3 r 3 The action of m f 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 mx my mz "73"' " ^ > L ^*' The motion of the centre of gravity of the sun, relative to the fixed origin, will, therefore, be determined by the equations Let p denote the distance of in from m r p f its distance from m" y adding an accent for each successive body considered; then will the action of the bodies m', m", &c. on m be of which the three components parallel to the co-ordinate axes, re- spectively, are The action of the sun on m, resolved in the same manner, is expressed by Vx tf tfz which are negative, since the force tends to diminish the co-ordinates Xj y, and z. The three components of the total action of the other bodies of the system on m are, therefore, 40 THEORETICAL ASTRONOMY. & x _j_ 7. 2 v ra'? + y, c -f ^ the equations which determine the absolute motion are <* cfcc F* _ 7<2V mV x) dt* + ^ H " r 3 " ' _ y ^ " ' the symbol of summation in the second members relating simply to the masses and co-ordinates of the several bodies which act on m, exclusive of the sun. Substituting for -j^, , and -~ their values O/t (Mi ut given by equations (14), we get Since x, y y z are the co-ordinates of m relative to the centre of gravity of the sun, these equations determine the motion of m relative to that point. The second members may be put in another form, which greatly facilitates the solution of some of the problems relating to the motion of m. Thus, let us put m' t\ xx '+yy' +z ^\ m " II xx"+yy"+2z" (17) and we shall have for the partial differential coefficient of this with respect to x, ^\_L/_ i*_^.\ +j5L/_ i*:_^.\ 4 & c dxl l + m\ p* dx r' s !^l + m\ p'* dx r" 3 P MOTION EELATIVE TO THE SUN. 41 But, since we have dp _ x r x dp' _ x" x dx p dx f> and hence we derive (d^\_ m' Ix' x x' \ m" lx"x x" or We find, also, in the same manner, for the partial differential coeffi- cients with respect to y and z, The equations (16), therefore, become - - It will be observed that the second members of equations (16) ex- press the difference between the action of the bodies m', m", &c. on m and on the sun, resolved parallel to the co-ordinate axes respect- ively. The mutual distances of the planets are such that these quan- tities are generally very small, and we may, therefore, in a first approximation to the motion of m relative to the sun, neglect the second members of these equations; and the integrals which may then be derived, express what is called the undisturbed motion of m. By means of the results thus obtained for the several bodies succes- sively, the approximate values of the second members of equations (16) may be found, and hence a still closer approximation to the actual motion of m. The force whose components are expressed by the second members of these equations is called the disturbing force ; 42 THEOEETICAL ASTKONOMY. and, using the second form of the equations, the function , which determines these components, is called the perturbing function. The complete solution of the problem is facilitated by an artifice of the infinitesimal calculus, known as the variation of parameters, or of constants, according to which the complete integrals of equations (16) are of the same form as those obtained by putting the second mem- bers equal to zero, the arbitrary constants, however, of the latter integration being regarded as variables. These constants of integra- tion are the elements which determine the motion of m relative to the sun, and when the disturbing force is neglected the elements are pure constants. The variations of these, or of the co-ordinates, arising from the action of the disturbing force are, in almost all cases, very small, and are called the perturbations. The problem which first presents itself is, therefore, the determination of all the circumstances of the undisturbed motion of the heavenly bodies, after which the action of the disturbing forces may be considered. It may be further remarked that, in the formation of the preceding equations, we have supposed the different bodies to be free to move, and, therefore, subject only to their mutual action. There are, in- deed, facts derived from the study of the motion of the comets which seem to indicate that there exists in space a resisting medium which opposes the free motion of all the bodies of the system. If such a medium actually exists, its effect is very small, so that it can be sen- sible only in the case of rare and attenuated bodies like the comets, since the accumulated observations of the different planets do not exhibit any effect of such resistance. But, if we assume its existence, it is evidently necessary only to add to the second members of equa- tions (16) a force which shall represent the effect of this resistance, which, therefore, becomes a part of the disturbing force, and the motion of m will be completely determined. 16. When we consider the undisturbed motion of a planet or comet relative to the sun, or simply the motion of the body relative to the sun as subject only to the reciprocal action of the two bodies, the equations (16) become g + *(!+ *) = <), g + *(! + ) 1 = 0, (19) MOTION RELATIVE TO THE SUN. 43 The equations for the undisturbed motion of a satellite relative to its primary are of the same form, the value of k 2 , however, being in this case the acceleration due to the force exerted by the mass of the primary at the unit of distance, and m the ratio of the mass of the satellite to that of the primary. The integrals of these equations introduce six arbitrary constants of integration, which, when known, will completely determine the undisturbed motion of m relative to the sun. If we multiply the first of these equations by y y and the second by a?, and subtract the last product from the first, we shall find, by inte- grating the result, xdy ydx _ ~~ ~ c being an arbitrary constant. In a similar manner, we obtain xdz zdx f ydz zdy ~ :c > ~~ G ' If we multiply these three equations respectively by z, y, and x, and add the products, we obtain ez c'y -f 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 = x 2 + y* -\- z 2 , we shall have, by differentiation, rdr = xdx -j- ydy -f- zdz. Therefore, introducing this value into the preceding equation, we obtain m) j ft = ^ Civ 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 ' _ 4f2 . J ' df df or -^= 4 ^- If we represent by dv the infinitely small angle contained between two consecutive radii- vectores r and r + dr, since doc 2 -\- dy 2 -f- dz 2 is the square of the element of path described by the body, we shall have da? + dy 2 -f dz 2 = dr 2 + r 2 dv\ Substituting this value in the preceding equation, it becomes r 2 dv = 2fdt. (22) The quantity r 2 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 -f- 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 = rdr (23) l/SwF (1 + m) hr 2 4/ 2 Substituting this value of dt in equation (22), we get ^ r ^ (24) 2 (1 -f m) /tr 2 4/ which gives, in order to find the maximum and minimum values of r, dr _ rVZrk 2 (1 -f- m) hr 2 4/ 2 _ eft; ~ "IT" : > or Therefore ff(l+m) and - m) / 4/ 2 *(! + m) 2 -V- JT T y * ^ are, respectively, the maximum and minimum values of r. The MOTION RELATIVE TO THE SUN. 45 points of the orbit, or trajectory of the body relative to the sun, 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 (I + e) and a(l e), we shall have in which > a (1 e 2 ). Introducing these values into the equation (24), it becomes tg. j/jp dr the integral of which gives v = CD -f cos -1 I 1 1, e\ r I to being an arbitrary constant. Therefore we shall have I(^_ 1 ) =cog( ,_ fl , ) , e \ r I from which we derive r= P , 1 -f- e cos (v >) which is the polar equation of a conic section, the pole being at the focus, p being the semi-parameter, e the eccentricity, and v 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 CD is counted from the perihelion, we have o = 0, and 1 -f- e cos v The angle v is called the true anomaly. Hence we conclude that the orbit of a heavenly body revolving around the sun is a conic section with the sun in one of the foci. Observation shows that the planets revolve around the sun in ellipses, usually of small eccentricity, while the comets revolve either in ellipses of great eccentricity, in parabolas, or in hyperbolas, a cir- cumstance which, as we shall have occasion to notice hereafter, greatly 46 THEOKETICAL 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 ! 1 _l-<-V V 1 l^r) the integration of which gives / -i/a r\ L la rV\ . ff)fi . t=~ ... , ( cos - -| e\ 1 i- - I TV. (26) ^1/1 +7^\ \ ae / \ \ ae } ] ' In the perihelion, r = a (1 g), and the integral reduces to t f = C; therefore, if we denote the time from the perihelion by t QJ we shall have -f m\ \ ae I \ \ ae (27) In the aphelion, r = a (1 + &) ', an d therefore we shall have, for the time in which the body passes from the perihelion to the aphelion, t = \r, or i r _ a ^ ky'l -\-m r being the periodic time, or time of one revolution of the planet around the sun, a the semi-transverse axis of the orbit, or mean dis- tance from the sun, and n the semi-circumference of a circle whose radius is unity. Therefore we shall have 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 m')r' 2 If the masses of the two planets m and m' are very nearly the same, we may take 1 -f m = 1 -j- m' and hence, in this case, it follows that the squares of the periodic times are to each other as the cubes of the mean distances from the sun. The same result may be stated in another form, which is sometimes more convenient. Thus, since nab is the area of the ellipse, a and 6 representing the semi-axes, we shall have -=/= areal velocity; and, since b 2 = a 2 (1 e 2 ), we have TO 'g'(l-^)' = which becomes, by substituting the value of r already found, F1S). (30) In like manner, for a second planet, we have and, if the masses are such that we may take 1 + m sensibly equal to 1 -f- m', it follows that, in this case, the areas described in equal times, in different orbits, are proportional to the square roots of their parameters. 17. We shall now consider the signification of some of the con- stants of integration already introduced. Let i denote the inclination of the orbit of m to the plane of xy, which is thus taken as the plane of reference, and let & be the angle formed by the axis of x and the line of intersection of the plane of the orbit with the plane of xy; then will the angles i and & determine the position of the plane of 48 . THEORETICAL ASTRONOMY. the orbit in space. The constants c, c', and c", involved in the equation cz c'y -\- c"x = 0, are, respectively, double the projections, on the co-ordinate planes, xy } xz, and yz, of the areal velocity /; and hence we shall have = cos . 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 &, gives c' = 2/ sin i cos &. Its projection on the plane of yz gives c" = 2/sin i sin & . Hence we derive z cos i y sin i cos 1 -\- x sin i sin = 0, (31) "which is the equation of the plane of the orbit; and, by means of the value of / in terms of p, and the values of c, c', c", we derive, also, + m) cos a sin *' (82) sn These equations will enable us to determine &, i, and p, when, for any instant, the mass and co-ordinates of m, and the components of its velocity, in directions parallel to the co-ordinate axes, are known. The constants a and e are involved in the value of p, and hence four constants, or elements, are introduced into these equations, two of which, a and e, relate to the form of the orbit, and two, 1 and i, to the position of its plane in space. If we measure the angle v co from the point in which the orbit intersects the plane of xy, the con- stant co will determine the position of the orbit in its own plane. Finally, the constant of integration C, in equation (26), is the time MOTION EELATIVE TO THE SUN. 49 of passage through the perihelion ; and this determines the position of the body in its orbit. When these six constants are known, the undisturbed orbit of the body is completely determined. Let 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 &, which is a constant for the solar system, we have, from equation (28), VI 4 50 THEORETICAL ASTEONOMY. In the case of the earth, a = l, and therefore rl/1 + m In reducing this formula to numbers we should properly use, for r, the absolute length of the sidereal year, which is invariable. The eifect 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 r derived from observation. The eifect of this cor- rection is to slightly 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 aifected. 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 m = 354710 Hence we have log r = 2.5625978148; log j/1 +m = 0.0000006 122; log 271 = 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 &1/1 4- m * = f, (33) a 2 for the expression for the mean daily motion of a planet. Since, in the case of the earth, V\ -f- m differs very little from 1, it will be observed that k very nearly expresses the mean angular motion of the earth in a mean solar day. In the case of a small planet or of a comet, the mass m is so small that it may, without sensible error, be neglected; and then we shall have M = 4- (34) a 2 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 a(l-e 2 ) " ^ e cos v If we represent by

cos v The angle (p is called the angle of eccentricity. Again, since p = a (1 e 2 ) = a cos 2 cos v (35) It is evident, from this equation, that the maximum value of r in an elliptic orbit corresponds to v 180, and that the minimum value of r corresponds to v = 0. It therefore increases from the perihelion to the aphelion, and then decreases as the planet approaches the peri- helion. 52 THEORETICAL ASTRONOMY. In the case of the parabola,

, if we put 5 = |_p, we shall have (36) in which q is the perihelion distance. In this case, therefore, when v = 180, r will be infinite, and the comet will never return, but course its way to other systems. The angle cannot be applied to the case of the hyperbola, since in a hyperbolic orbit e is greater than 1 ; and, therefore, the eccen- tricity cannot be expressed by the sine of an arc. If, however, we designate by ^ the angle which the asymptote to the hyperbola makes with the transverse axis, we shall have e cos 4 = 1. Introducing this value of e into the polar equation of the hyperbola, it becomes p cos ^ cost; -j- cos 4* But, since cos v + cos ^ = 2 cos } (v + ^) cos \(v $), this gives = _ ff cos4 _ ( . ' It appears from this formula that r increases with v, and becomes in- finite when 1 + e cosv = 0, or cosv = cos^, in which case v = 180 J/ : consequently, the maximum positive value of v is represented by 180 ^, and the maximum negative value by (180--^). 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 of this and of the elements which fix the position of the plane of the PLACE IN THE ORBIT. 53 orbit, and that of the orbit in its own plane, the determination of the position in space. In deriving the formulae for finding the place of the body in its orbit, we will consider each species of conic section separately, com- mencing with the ellipse. 20. Since the value of a r can never exceed the limits ae and + ae, we may introduce an auxiliary angle such that we shall have a r - = cos E. ae This auxiliary angle E is called the eccentric anomaly; and its geo- metrical signification may be easily known from its relation to the true anomaly. Introducing this value of - into the equation ctC' (27) and writing t T in place of t w 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 a But - -- = mean daily motion of the planet = // ; therefore a The quantity fjt (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=v(t-T), M=EesinE. (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 we 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 THEOEETICAL ASTRONOMY. this purpose there are various formulae which may be applied in practice, and which we will now develop. The equation ae gives a r ^ = cos E, This also gives ae = a cos E ae, or P r T^ E - a cos E ae, e / 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 or _ r sin v = a-j/1 e 2 sin E = b sin E. (42) By means of the equations (41) and (42) it may be easily shown that the auxiliary angle E y or eccentric anomaly, is the angle at the centre of the ellipse between the semi-transverse axis, and a line drawn from the centre to the point where the prolongation of the ordinate perpendicular to this axis, and drawn through the place of the body, meets the circumference of the circumscribed circle. Equations (40) and (41) give r (1 HH cos v) = a(l e) (1 q= cos E). By using first the upper sign, and then the lower sign, we obtain, by reduction, 1/r sin ^v = l/a(l -f- e} sin \E, Vr cos -f- cos ^9?. * In a similar manner we find 1/1 e = sin \

. (47) _, cos J5J = x a a (1 + e cos v ) or cos v -4- ae 4- ae 2 cos v cos^=^ ; a(l -}- e cos v) and, putting a cos 2 ^ instead of p, and sin ^ for e, we get cosv -f-e S AQ ^ (48) 1 4- e cos i; If we multiply the first of equations (43) by cos^, and the 56 THEORETICAL ASTRONOMY. second by sin|22, successively add and subtract the products, and reduce by means of the preceding equations, we obtain sin J (v -f E} -J- cos \

, and e = \. 22. Observation shows that the masses of the comets are insensible in comparison with that of the sun ; and, consequently, in this case, m and equation (52), putting for p its value 2g, becomes kV2q dt = or which may be written JO*L =4(1 + tan 2 t>) sec 2 %vdv = (1 + tan 2 M d tan fr. 1/2 2* Integrating this expression between the limits T and t, we obtain = tan it; + J tan 3 >, (55) which is the expression for the relation between the true anomaly and the time from the perihelion, in a parabolic orbit. Let us now represent by r the time of describing the arc of a parabola corresponding to v = 90 ; then we shall have Jcr 4 or Now, -- is constant, and its logarithm is 8.5621876983; and if we take (1 + 3 cot 2 4t>) ; and, multiplying and dividing the second member by (1 -f- cot 2 |v) 3 , we shall have k (t ~ P = I tan 3 & (1 + cot 2 ^) 8 JJ 1/2 g* 62 THEOEETICAL ASTRONOMY. 2 But 1 -f cot 2 iv = - and consequently sin v tan %v k(tT}_ 8 l 1/2 gt ~~3snr (1 + 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 and, putting tan %w = 0, and tan|A = x, we get, from this equation, 03 "I to ' (l-ex) y Multiplying this through by 3 (1 0#) 3 , expanding and reducing, there results the following equation : 1 4- 30 2 = 30 (1 -f 40 2 -f- 20* -f 6 ) x 30 2 (1 + 40 2 + 20 4 4- s ) x 2 4- s (2 4- 60 2 4 30* 4- s ) a?. Dividing through by the coefficient of x, we obtain 30 (1 4- 40^+20* ~+0) = x W-{- 3 ^ _|_ 402 _j_ 2^4 _|_ ^) Let us now put 14-30 2 30(l-f 40 2 + 20* + 6 )~~ then, substituting this in the preceding equation, inverting the series and reducing, we obtain finally But tan |A O = x, therefore PLACE IN THE ORBIT. 63 Substituting in this the value of x above found, and reducing, we obtain For all the cases in which this equation is to be applied, the third term of the second member will be insensible, and we shall have, to a sufficient degree of approximation, Table VII. gives the values of A O , expressed in seconds of arc, corresponding to consecutive values of w from w = 155 to w = 180. In the application of this table, we have only to compute the value of M precisely as for the case in which Table VI. is to be used, namely, M=m(t T~); then will w be given by the formula 3 1 200 sin w = \| , since we have already found 8 3sm 3 i(/ or 200 r Having computed the value of w from this equation, Table VII. will furnish the corresponding value of A ; and then we shall have, for the correct value of the true anomaly, V = W -f- A , which will be precisely the same as that obtained directly from Table VI., when the second and higher orders of differences are taken into account. If v is given and the time t T is required, the table will give, by inspection, an approximate value of A, using v as argument, and then w is given by w = v A n . 64 THEOBETICAL 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 a 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 , Q . _-. 3 = V (3 2 sm'itr). 2q? cos 3 iv from which we obtain, since q = r cos 2 Jv, 1/2 / Let us now put sin 4 -y sm # = r=, 1/2 and we have 3& (i T} = 3 sm x 4 sin 3 re = sin 3, Consequently, which admits of an accurate and convenient numerical solution. To facilitate the calculation we put sin-o the values of which may be tabulated with the argument v. When v = 0, we shall have N= fv'2, and when v 90, we have N=l; from which it appears that the value of ^changes slowly for values of v from to 90. But when v = 180, we shall have ^V oo; and hence, when v exceeds 90, it becomes necessary to introduce an auxiliary different from N. We shall, therefore, put in this case, N' = N sin v sin 3z; PLACE IN THE ORBIT. 65 from which it appears that N'=\ when v = 90, and that N' = $\/2 when v = 180. Therefore we have, finally, when v is less than 90, t T=-JN and, when v is greater than 90, in which log = 1.5883272995, from which t T is easily derived OA/ when v is known. Table VIII. gives the values of N, with differences for interpola- tion, for values of v from v = to v = 90, and the values of N f for those of v from v = 90 to v = 180. 25. We shall now consider the case of the hyperbola, which differs from the ellipse only that e is greater than 1 ; and, consequently, the formulae for elliptic and hyperbolic motion will differ from each other only that certain quantities which are positive in the ellipse are nega- tive or imaginary in the hyperbola. We may, however, introduce auxiliary quantities which will serve to preserve the analogy between the two, and yet to mark the necessary distinctions. For this purpose, let us resume the equation _ p cos 4 2 cos ~i (v -f- 4) cos \ (v 4)' When v = 0, the factors cos^fv + a]/) and cos(v 40 in the de- nominator will be equal ; and since the limits of the values of v are 180 ^ and (180 -J/), it follows that the first factor will vanish for the maximum positive value of v, and that the second factor will vanish for the maximum negative value of v, and, therefore, that, in either case, r = oo. In the hyperbola, the semi-transverse axis is negative, and, conse- quently, we have, in this case, p = a(e 2 1), or a =p cot 2 4. We have, also, for the perihelion distance, q a(e 1). Let us now put tan F = tan 66 THEORETICAL ASTRONOMY. which is analogous to the formula for the eccentric anomaly E in an ellipse: and. since e = --- , we shall have cos V e 1 _ 1 cos 4 _ 2 . ~~ ~ and, consequently, tan ^F = tan %v tan *. (57) We shall now introduce an auxiliary quantity 1 ff - I. (64) Further, we have p cos v ar (e 4- cos v) T 1 -f- e cos v p which, combined with equation (62), gives (65) If we square these values of r sin v and r cos v, add the results to- gether, reduce, and extract the square root, we find (66) We might also introduce the auxiliary quantity ff 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. 68 THEORETICAL ASTRONOMY. 26. Let us now resume the equation _ cos %(v 4) ~~ cos -J (v -j- 4)* Differentiating this, regarding fy as constant, we have sin 4 7 dff = ~ j-y-r -p r dv, and, dividing this equation by the preceding one, we get dff sin 4 T ~ ~~ 2 cos ^(v + 4) cos (v 4) But p cos 4 7* = consequently, ff p which gives r tan 4 _ dv, tan 4 Substituting this value of r z dv in equation (22), and putting instead of 2/ its value feVjpj from equation (30), the mass being considered as insensible in comparison with that of the sun, we get . ff tan 4 Then, substituting for r its value from equation (66), and for p its value a tan 2 ^, we have Ue (l +-3} ---Jc?(r. Integrating this between the limits T and , we obtain k\/p (t T) = a? tan 4 ( \e ( ff ^\ log e 5 + |a 3 + 2wa + (1 + O r = 4 (>' 4- From the first of these equations we find PLACE IN THE ORBIT. 73 1-f-w 2 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 /? already found, and reducing, we obtain finally r ~ "(1 + u'V" Again, we have 1 TT 1 x . , - . , . tan U = tan (u -\- ai -f- pi* -f- ^i ). Developing this, and neglecting terms of the order i 4 , we get Now, since u = tan Jv and Z7= tan | F, we shall have or 2a . U - ( , Substituting in this equation the values of a, /?, and f already found, and reducing, we obtain finally _ f (i ^+ttft^+j^^^ (73) 74 THEORETICAL ASTRONOMY. This equation can be used whenever the true anomaly in the ellipse or hyperbola is given, and the time from the perihelion is to be determined. Having found the value of V, we enter Table VI. with the argument Fand take out the corresponding value of M; and then we derive t T from r Mq* | : ~CT\T in which log C == 9.96012771. For the converse of this, in which the time from the perihelion is given and the true anomaly is required, it is necessary to express the difference v V in a series of ascending powers of i, in which the coefficients are functions of U. Let us, therefore, put u = U + o!i + /3V + r 'i* + &c. Substituting this value of u in equation (70), and neglecting terms multiplied by i* and higher powers of i, we get + G3'(l + U*) + Ua' 2 2 ffV (1 + U 2 ) + | U 5 -f | J7 7 ) i 2 Z7 2 ) + JV 3 + 2Ua'/S'-f- 3C7V(1 -f U^ 2pU 2 (l + C7 2 ) 4 C7 3 a' 2 2 W 2 4 ^ _ 4 f/9) ^ But, since the first member of this equation is equal to U-\- %U 3 , we shall have, by the principle of indeterminate coefficients, r ' (1 + ?7 2 ) + i a' 3 + 2 Z7d'/5' + 3 C7*a' (1 + C7 2 ) 2/5' C7 2 (1 + C7 2 ) 4J 7 = =0< From these equations, we find .^ , _ If! ^ T + 3111 (i -f- c/" 2 ) 5 If we interchange v and V in equation (72), it becomes, writing a', /?', r ' for a, ft r , PLAOE IN THE ORBIT. 75 v ~ VJr i+u* l ~^U+ U* (1 -f C7 2 ) 2 , 4.^17 2(t7'-j) (1 + *7 2 ) 2 (1 + -j) \ IP) 8 / Substituting in this equation the above values of a/, /3', and f, and reducing, we obtain, finally, = v , iu* + 1 rp , i fi^ + iHt/'+^^+T^ "" _l_tii^ (1 + C7 2 ) by means of which v may be determined, the angle F 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 \x = 6, and let us put 100(1 -f 2 ) 2 ' 10000 (1 + 10000 (1-M 2 ) 4 1000000 (1 + ^ 2 ) 6 r _ ^ + iill^ 9 + flf ?|0 n + If l^ 13 4- 4IIF 15 + jjfrV" 1000000 (1 + 2 ) 6 wherein s expresses the number of seconds corresponding to the length of arc equal to the radius of a circle, or logs = 5.31442513. We shall, therefore, have: WhenzF, v=V+A (1000 76 THEORETICAL ASTRONOMY. and, when x = v, V=v-A (lOOi) + S (10(K) 2 C" (lOOi) 3 . Table IX. gives the values of J., B, B f , C, and C f for consecu- tive values of x from # = 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 _ ^ \-\-e cos v" Similar expressions arranged in reference to the ascending powers of (1 e) or of I ( - - J 11 may be derived, but they do not con- verge with sufficient rapidity ; for, although I ( - - 1 1 I is less than ij 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 formula 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 IN THE ORBIT. 77 which, since q = a(l e), may be written Ic (t T) l/l e 1 , ft 1 1 -f 9e , . If we put A- 15 E 9E we shall have ^~e 20T/2 \ 1 l + 9e .f 2 | "9^+sm^~ f 3*5(1 e) ' Let us now put 201/A and 4-<-% -w-1.2 1 .. __ 1 then we have -Htan 3 -|w. (75) When B is known, the value of w may, according to this equation, be derived directly from Table VI. with the argument 75k(t-r) ~ and then from w we may find the value of A. It remains, therefore, to find the value of IB ; and then that of v from the resulting value of A. Now, we have 2 tan -\E Sm ^=l + tanV and if we put tan 2 ^ r, we get s i n E = j^- = 2ri (1 r -f r 2 r 3 + &c.). We have, also, E= 2 tan" 1 ^ 2r*(l JT + ^r 2 4T 3 -f- &c.). 78 THEORETICAL ASTRONOMY. Therefore, 15 (E- Sin E) = 2r^(10r - ^r 2 + yi* - if^r* + &c.), and 9E + sin = 2r* (10 - V r + VT Hence, by division, and, inverting this series, we get A which converges rapidly, and from which the value of may be found. Let us now put A 1 T-. (78) For the radius-vector in a hyperbolic orbit, we find, by means of the last of equations (63), T = (1 ^0 2 )cos 2 ^' (79) When t T is given and r and v are required, we first assume B = 1, and enter Table VI. with the argument 80 THEORETICAL ASTRONOMY. in which log C = 9. 96012771, and take out the corresponding value of w. Then we derive A from the equation 5(1 e) 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 9 we repeat the operation. The second result for A will not require any further correction, since the error of the first assumption of J5 = 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 _eJ L 21 in the case of the hyperbola. Then, with the value of r as argu- ment, we enter the second part of Table X. and take out an approxi- mate value of A 9 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 correction. Next, to find w, we have and, with w as argument, we derive M from Table VI. Finally, we have (80) by means of which the time from the perihelion may be accurately determined. POSITION IN SPACE. 81 30. We have thus far treated of the motion of the heavenly bodies, relative to the sun, without considering the positions of their orbits in space ; and the elements which we have employed are the eccen- tricity and semi-transverse axis of the orbit, and the mean anomaly at a given epoch, or, what is equivalent, the time of passing the perihelion. These are the elements which determine the position of the body in its orbit at any given time. It remains now to fix its position in space in reference to some other point in space from which we conceive it to be seen. To accomplish this, the position of its orbit in reference to a known plane must be given ; and the elements which determine this position are the longitude of the perihelion, the longitude of the ascending node, and the inclination of the plane of the orbit to the known plane, for which the plane of the ecliptic is usually taken. These three elements will enable us to determine the co-ordinates of the body in space, when its position in its orbit has been found by means of the formulae already investigated. The longitude of the ascending node, or longitude of the point through which the body passes from the south to the north side of the ecliptic, which we will denote by &, is the angular distance of this point from the vernal equinox. The line of intersection of the plane of the orbit with the fundamental plane is called the line of nodes. The angle which the plane of the orbit makes with the plane of the ecliptic, which we will denote by i, is called the inclination of. the orbit. It will readily be seen that, if we suppose the plane of the orbit to revolve about the line of nodes, when the angle i exceeds 180, & 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 5 , and which is measured, as in the case of & , from the vernal equinox. It will easily be understood that, when seen from the sun, so long as the inclination of the orbit is less than 90, the motion of the body will be in the same direction as that of the earth, and it is then said to be direct. When the inclination is 90, the motion will be at right angles to that of the earth ; and when i exceeds 90, the motion in longitude will be in a direction opposite to that of the earth, and it is then called retrograde. It is customary, therefore, to extend the inclination of the orbit only to 90, and if this angle exceeds a right angle, to regard its supplement as the inclination of the orbit, noting simply the distinction that the motion is retrograde. 82 THEOEETICAL ASTRONOMY. The longitude of the perihelion, which is denoted by TT, 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 formula would be sufficient, while in the common form two sets are in part required. However, the custom of astronomers seems to have sanctioned these distinctions, and they may be per- petuated or not, as may seem advantageous. Further, we may remark that in the case of direct motion the sum of the true anomaly and longitude of the perihelion is called the true longitude in the orbit; and that the sum of the mean anomaly and longitude of the perihelion is called the mean longitude^ an ex- pression which can occur only in the case of elliptic orbits. In the case of retrograde motion the longitude in the orbit is equal to the longitude of the perihelion minus the true anomaly. 31. We will now proceed to derive the formulae for determining the co-ordinates of a heavenly body in space, when its position in its orbit is known. For the co-ordinates of the position of the body at the time t s we have x = r cos v y y r sin v, POSITION IN SPACE. 83 the line of apsides being taken as the axis of x, and the origin being taken at the centre of the sun. If we take the line of nodes as the axis of x, we shall have x = r cos (v -f~ >), y = r sin (y -f- 0, 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. Ntfw, we have a) = rc ft in the case of direct motion, and co = ft it in the case of retrograde motion ; and hence the last equations become x = r cos (v =b TT qz ft) y = r sin (v K ^ ft) the upper and lower signs being taken, respectively, according as the motion is direct or retrograde. The arc v TT ip ft = uis 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 t y' = r sin u cos i, z f r sin u sin i. If we denote the heliocentric latitude and longitude of the body, at the time t } by 6 and I, respectively, we shall have x f = r cos b cos (I ft ), y f = r cos b sin (I ft ), z' = r sin 6, and, consequently, cos u = cos b cos (I ft), rfc sin it cos i = cos b sin (i ft), (81) sin u sin = sin 6. 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, ^, and i are given. Since 84 THEORETICAL ASTRONOMY. cos b is always positive, it follows that I & and u must lie in the same quadrant when i is less than 90 ; but if i is greater than 90, or the motion is retrograde, I & 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 & and u will lie in the same quadrant. 32. By multiplying the first of the equations (81) by sin u, and the second by cos u, and combining the results, considering only the upper sign, we derive cos b sin (u I + ) = 2 sin u cos u sin 2 i, or cos b sin (u I -f- & ) = sin 2u sin 2 ^i. In a similar manner, we find cos b cos (u I -j- & ) = cos 2 it -j- sin 2 it cos i, which may be written cos b cos (u 1-\- Q ) = J (1 -f- cos 2it) -j- i (1 cos 2u) cos i, or cos b cos (u l-\- & ) = ^ (1 + cos i) + 2 (1 cos i) cos 2w; and hence cos 5 cos (u I -f & ) = cos 2 ^i -f sin 2 \i cos 2w. If we divide this equation by the value of cos b sin (u I + & ) already found, we shall have tan(M _ f + ) = J'^f! n *U (83) 1 -f- tan 2 h cos 2tt The angle w 1-\- & is called the reduction to the ecliptic; and the expression for it may be arranged in a series which converges rapidly when i is small, as in the case of the planets. In order to effect this development, let us first take the equation ?i sin # tan y = , 1 -j- n cos x Differentiating this, regarding y and n as variables, and reducing, we find dy sin x dn 1 -f- 2n cos x -f- ri* POSITION IN SPACE. 85 which gives, by division, or by the method of indeterminate coefficients, Cfl/ = sin x n sin 2x -j- n z sin 3# n* sin 4x -4- &c. an Integrating this expression, we get, since y = when x = 0, y = n sin x ^?i 2 sin 2x -j- ^n 3 sin 3x ^n 4 " sin 4# -f~ . . . . , (84) which is the general form of the development of the above expression for tan y. The assumed expression for tan y corresponds exactly with the formula for the reduction to the ecliptic by making n = tan 2 \i and x = 2u; and hence we obtain u I -j- & = tan 2 %i sin 2u tan 4 ^i sin 4u -j- | tan 6 i sin 6u - 1 tan 8 i sin Su + J tan 10 i sin lOw &c. (85) When the value of i does not exceed 10 or 12, the first two terms of this development will be sufficient. To express u 1-\- & .in seconds of arc, the value derived from the second member of this equation must be multiplied by 206264.81J the number of seconds corresponding to the radius of a circle. If we denote by R Q the reduction to the ecliptic, we shall have I = u -f R Q = v -f TT J2 e . But we have v = M-\- the equation of the centre ; hence l=M-}-'K-\- equation of the centre reduction to the ecliptic, and, putting L = M-}-n = mean longitude, we get I = L -f- equation of centre reduction to ecliptic. (86) In the tables of the motion of the planets, the equation of the centre (53) is given in a table with M as the argument ; and the reduction to the ecliptic is given in a table in which i and u are the arguments. 33. In determining the place of a heavenly body directly from the elements of its orbit, there will be no necessity for computing the reduction to the ecliptic, since the heliocentric longitude and latitude may be readily found by the formulae (82). When the heliocentric place has been found, we can easily deduce the corresponding geo- centric place. Let x, y, z be the rectangular co-ordinates of the planet or comet referred to the centre of the sun, the plane of xy being in the ecliptic, 86 THEORETICAL ASTRONOMY. the positive axis of x being directed to the vernal equinox, and the positive axis of z to the north pole of the ecliptic. Then we shall have x = r cos b cos I, y = r cos b sin /, z = r sin b. Again, let X, F, Z be the co-ordinates of the centre of the sun re- ferred to the centre of the earth, the plane of XY being in the eclip- tic, and the axis of X being directed to the vernal equinox ; and let denote the geocentric longitude of the sun, E its distance from the earth, and 2 its latitude. Then we shall have Z = E sin I. Let x'j y', z' be the co-ordinates of the body referred to the centre of the earth ; and let X and ft denote, respectively, the geocentric longi- tude and latitude, and J, the distance of the planet or comet from the earth. Then we obtain a/ = A cos /? cos ^, if = A cos /5 sin A, (87) z' = A sin /?. But, evidently, we also have and, consequently, A cos /? cos A = r cos b cos / -f- R cos 2 cos Q , A cos /5 sin A = r cos b sin I -f E cos S sin O , (88) A sin /5 = r sin b -\- R sin I". If we multiply the first of these equations by cos Q, and the second by sin Q? and add the products; then multiply the first by sin O, and the second by cos Q 9 and subtract the first product from the second, we get A cos /? cos (A O ) f cos b cos (I O ) + R cos ^, A cos /? sin (A Q ) = r cos b sin (7 O ), (89) A sin /5 = r sin b -\- R sin I*. 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 A, /9, and A when r, 6, and I have been derived from the elements of the orbit, the quantities R, Q , and 2 being furnished by the solar tables ; or, when J, /?, and / are given, these equations determine , 6, and r. The latitude 2 of the sun never exceeds 0".9, and, therefore, it may in most cases be neg- lected, so that cos 2 T 1 and sin 2' = 0, and the last equations become A cos /5 cos (A Q ) = r cos b cos (I O ) + R, A cos ft sin (A Q ) = r cos b sin (I Q ), (90) A sin ft ==r sin b. If we suppose the axis of x to be directed to a point whose longi- tude is &, or to the ascending node of the planet or comet, the equa- tions (88) become A cos ft cos (A ) = r cos u -j- R cos S cos (O &)> A cos /? sin (A ^) = r sin u cos i -\- R cos S sin (O & ) (91) /J sin /? == . r sin w sin i -\- R sin 2", by means of which /9 and X may be found directly from & , i, r, and u. If it be required to determine the geocentric right ascension and declination, denoted respectively by a and d, we may convert the values of /? and X into those of a and d. To eifect this transforma- tion, denoting by the obliquity of the ecliptic, we have cos d cos a cos ft cos A, cos d sin a cos ft sin A cos sin ft sin e, sin <5 = cos /? sin A sin e -f- sin ft cos e. Let us now take n sin N = sin /5, ra cos N= cos /5 sin A, and we shall have COS d COS a = COS ft COS A, cos d sin a = w cos (JV+ 0> sin sin e -f- r sin u (=fc cos i cos & sin -j- sin i cos e). These are the expressions for the heliocentric co-ordinates of the planet or comet referred to the equator. To reduce them to a con- venient form for numerical calculation, let us put cos & = sin a sin A, qp cos i sin & = sin a cos J., sin & cos sin b sin B, zfc cos i cos & cos sin i sin = sin b cos 5, sin & sin = sin c sin (7, rb cos i cos & sin -}- sin i cos = sin c cos C; and the expressions for the co-ordinates reduce to x r sin a sin (A -j- w)> y = r sin 6 sin (5 -f- u), (100) 2 = r sin c sin ( C -j- w). The auxiliary quantities, a, 6, c, J., J2, and (7, are constant so long as Q> and i remain unchanged, and are called constants for the equator. It will be observed that the equations involving a and J., regard- ing the motion as direct, correspond to the relations between the parts of a quadrantal triangle of which the sides are i and a, the POSITION IN SPACE. 91 angle included between these sides being that which we designate .by Ay and the angle opposite the side a being 90 - & . In the case of b and J5, the relations are those of the parts of a spherical triangle of which the sides are 6, i, and 90 -f- e, B being the angle included by i and 6, and 180 - - & the angle opposite the side 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 i and c, and 180 - - ^ that included by the sides i and e. We have, therefore, the following additional equations : cos a = sin i sin & , cos b = cos & sin i cos e cos i sin s, (101) cos c = cos & sin i sin e -{- cos i cos e. In the case of retrograde motion, we must substitute in these 180 im place of i. The geometrical signification of the auxiliary constants for the equator is thus made apparent. The angles a, 6, 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 9 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 w such that cos i cos & = e cos E Q , e Q being always positive. We shall, therefore, have _, tan E n = -f ~cos Since both e and sinz are positive, the angle E Q cannot exceed 180; and the algebraic sign of tan E Q will show whether this angle is to be taken in the first or second quadrant. The first two of equations (99) give cot A = + tan & cos i ; and the first gives cos & sm a = . . gin .4 92 THEOEETICAL ASTKONOMY. From the fourth of equations (99), introducing e Q and E w we get sin b cos B = e cos E cos e Q sin E Q sin e = e Q cos (E -(- e). But sin b sin ^ = sin & cos e ; therefore sin Q cos e tan &6 cos Jb Q cos e We have, also, . , sin & cos sin b = : 5 smB In a similar manner, we find cot C ^^ =r . Sm . , tan Q cos A sin e and sin O sin sin C The auxiliaries sin a, sin 6, and sin c are always positive, and, there- fore, sin A and cos & , sin B and sin & , and also sin C and sin & , must have the same signs, which will determine the quadrant in which each of the angles A, B, and C is situated. If we multiply the last of equations (99) by the third, and the fifth of these equations by the fourth, and subtract the first product from the last, we get, by reduction, sin b sin e sin ( C B) = sin i sin &. But sinacosJ. = =F cos i sin &; and hence we derive sin b sin c sin ( C J3) sin a cosJ. = 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 TT q= a, B' = B n q= , and C' = C n =F Sly the upper or lower sign being used according as the motion is direct or retrograde, we shall have POSITION IN SPACE. 93 x = r sin a sin (A' -f- v), y = r sin b sin (B 1 -j- v), (102) z = r sin c sin ( C" -j- v), a transformation which is perhaps unnecessary, but which is con- venient when a series of places is to be computed. It will be observed that the formula? for computing the constants a, 6, c, A, By and (7, in the case of direct motion, are converted into those for the case in which the distinction of retrograde motion is adopted, by simply using 180 -i instead of i. 36. When the heliocentric co-ordinates of the body have been found, referred to the equator as the fundamental plane, if we add to these the geocentric co-ordinates of the sun referred to the same fundamental plane, the sum will be the geocentric co-ordinates of the body referred also to the equator. For the co-ordinates of the sun referred to the centre of the earth, we have, neglecting the latitude of the sun, X=Kcos, Y= R sin O cos e, Z = JR sin O sin e = Ktan e, in which R represents the radius-vector of the earth, O the sun's longitude, and s the obliquity of the ecliptic. We shall, therefore, have x -{- X=,A cos S cos a, y -f Y= A cos 8 sin a, (103) z -f- Z = A sin d, which suffice to determine a, d, and J. If we have regard to the latitude of the sun in computing its geo- centric co-ordinates, the formulae will evidently become Y= E sin O cos S cos e E sin S sin e, (104) Z = R sin O cos S sin e -\- JR sin S cos e, in which, since S can never exceed 0".9, cos S is very nearly equal to 1, and sin I = 2. The longitudes and latitudes of the sun may be derived from a solar ephemeris, or from the solar tables. The principal astronomical ephemerides, such as the Berliner Astronomisches Jahrbuch, the Nautical Almanac, and the American Ephemeris and Nautical Al- 94 THEORETICAL ASTRONOMY. raanac, 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, 6, and c, x r cos v sin a! sin A' -f- r sin v sin a' cos A', y = r cos v sin b' sin B r -\- r sin v sin b f cosJB', z = r cos v sin c' sin C' -}- r sin v sin c' cos C'. Now, since r cos v = a cos E ae, and r sin v = a cos

ae sin >' sin _B' -f- a cos 9? sin b' cos 1?' sin i, z = a sin c' sin C' cos .E ae sin c' sin 6" -f- a cos ^ sin c' cos (7 sin _Z Let us now put a cos

sin i' sin &' = sin i sin & , sin i' cos SI ' = cos i sin e -j- sin i cos e cos SI Let us now put wsin JV=cosi, n cosN= sint cos Sit and these equations reduce to cos i' = n sin (N e), sin i' sin &' = sin i sin SI , sin i' cos SI' = n cos (N e) ; from which we find cot i* = tan (N e) cos &'. ( 107 ) Since sin i is always positive, cos N and cos & must have the same signs. To prove the numerical calculation, we have sin i cos & _ cos N sin i' cos SI' cos (N e)' the value of the second member of which must agree with that used in computing SI '. In order to find CO Q) we have, from the same triangle, sin W Q sin i' = sin & sin e, cos fo n sin i' = cos e sin i -f sin e cos i cos & . Let us now take m sin M= cos e, m cos J[f = sin e cos & ; and we obtain POSITION IN SPACE. 97 cot M = tan e cos ft , and, also, to check the calculation, sin e cos ft cos M sinicosw cos (M i) If we apply Gauss's analogies to the same spherical triangle, we get cos-K' sin^ (ft' -{- w ) =: sin Jft cos ^(i e), cosii'cos^(ft' -f w ) = cos^ft cos(i + e), sin K' sin J (ft' o* ) = sin ft sin 1 (i e), sin j|i' cos ( ft ' w ) cos ^ ft sin ^ (i -f- e). The quadrant in which J (ft' + ft> ) or J (ft . We may thus find the elements ?r', ft ', and i', 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 formulae already given for the ecliptic as the fundamental plane. If the position of the orbit with respect to the equator is given, and its position in reference to the ecliptic is required, it is only necessary to interchange ft and ft', as Avell as i and 180 ^, e remaining unchanged, in these equations. These formula 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 r , i', and tt> will have the same signification in reference 7 98 THEOKETICAL ASTRONOMY. s to this plane that they have in reference to the equator, with this dis- tinction, however, that &' is measured from the descending node of this new plane of reference on the ecliptic ; and e will in this case denote the inclination of the ecliptic to this plane. 40. We have now derived all the formulae which can be required in the case of undisturbed motion, for the computation of the helio- centric or geocentric place of a heavenly fyody, 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 9 and r = a, for every point of the orbit. There is no instance of a circular orbit yet known ; but in the case of the discovery of the asteroid planets between Mars and Jupiter it is sometimes thought advisable, in order to facilitate the identification of comparison stars for a few days succeeding the discovery, to compute circular elements, and from these an ephemeris. The elements which determine the form of the orbit remain con- stant so long as the system of elements is regarded as unchanged ; but those which determine the position of the orbit in space, TT, &, and i 9 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 TT, &, and i, the auxiliaries sin a, sin 6, sin c, A, J5, and (7, introduced into the formula? 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 formula? for precession and nutation. Thus, for the reduction of the right ascension and declina- tion from the mean equinox and equator of the beginning of the year to the apparent or true equinox and equator of any date, usually the date to which the co-ordinates of the body belong, we have + a ) tan 3 a), for which the quantities/, g, and Gr are derived from the data given either in the solar and lunar tables, or in astronomical ephemerides, such as have already been mentioned. The problem of reducing the elements from the ecliptic of one date t to that of another date t f may be solved by means of equations (109), making, however, the necessary distinction in regard to the point from which & and & ' are measured. Let 6 denote the longi- tude of the descending node of the ecliptic of t' on that of t, and let -f} 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 & ' d in place of Q, ' . Finally, we must write y instead of e, and AO> for CO Q , which is the variation in the value of co in the interval t r t on account of the change of the position of the ecliptic ; then the equations become cos U' sin (' + Aw) = sin(& 0) cos (1 17), cosU' cosi (' + A0 = cos (& 0) cosi (i + 7 ), sin $ sin l (' Aw) = sin J ( 0) sin (i 7 ), ' AW) = cos^(& 0) sin % (i -f- ^). 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 d and 37 are known. The longitudes, however, will still be referred to the same mean equinox as before, which we suppose to be that of t; and, in order to refer 100 THEORETICAL ASTKONOMY. them to the mean equinox of the epoch t' ', the amount of the pre- cession in longitude during the interval t r 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 formula, 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 T? cos (& 0) = cos i r sin i -\- sin i r cos i cos Aw, from which we easily derive sin (i' i) sin y cos ( 0) -f- 2 sin i' cos i sin 2 ^Aw. (112) We have, further, sin AW sin i' = sin y sin (& 0), . (113) We have, also, from the same triangle, sin AW cos i' = cos (^ 0) sin (&' 0) -j- sin (& 0) cos (^' 0) cos 7, which gives sin (ft' ft) = sin Aw cos i' 2 sin (ft 0) cos (&' 0) sin 2 Jiy, q or sin(&' &) = sin 7 sin (& 0) coti' - 2 sin (a *) cos (' 0) sin 2 ^. (114) Finally, we have Since 37 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 ', sint 8'= 8 + tan & cos E cos e tan & cos E Q sin e cos & sin & cos e sin & sin e sm a = -p sin b = - -. - - sm c = - ~- . sm A sm B sm C The angle E is always less than 180, and the quadrant in which it is to be taken, is indicated directly by the algebraic sign of tan E Q . The values of sin a, sin 6, and sin c are always positive, and, therefore, the angles A, jB, and C must be so taken, with respect to the quadrant in which each is situated, that sin A and cos &, sin B and sin &, and also sin C and sin & , shall have the same signs. From these we derive A = 296 39' 5".07, log sin a = 9.9997156, B = 205 55 27 .14, log sin b = 9.9748254, C 212 3217.74, log sin e = 9.5222192. Finally, the calculation of these constants is proved by means of the formula NUMERICAL EXAMPLES. 105 sin b sin e sin ( C -B) tan^ = sin a cos A which gives log tan i = 8.9068875, agreeing with the value 8.9068876 derived directly from i. Next, to find r and u. The date 1865 February 24.5 mean time at Washington reduced to the meridian of Greenwich by applying the difference of longitude, 5 h S m 1T.2, becomes 1865 February 24.714018 mean time at Greenwich. The interval, therefore, from the epoch for which the mean anomaly is given and the date for which the geocentric place is required, is 420.714018 days; and mul- tiplying the mean daily motion, 928".55745, by this number, and adding the result to the given value of M, we get the mean anomaly for the required place, or M= 1 29' 40".21 + 108 30' 57".14 = 110 0' 37".35. The eccentric anomaly E is then computed by means of the equation M=EesmE, the value of e being expressed in seconds of arc. For Eurynome we have log sin

Then we have log i = log ^- - = 8.217680, and from the equation i -f- e v = F-f A (lOOt) + (100i) 2 + C(1000 8 , we get v= F+ 42' 22".28 + 25".90 -f- 0".28 = 102 20' 52".20. The value of r is then found from r=: 1 -(- e cos v' namely, log r = 0.1614051. We may also determine r and v by means of Table X. Thus, we first compute M from - Assuming jB = 1, we get log M= 2.13757, and, entering Table VI. with this as argument, we find w = 101 25 r . Then we compute A from NUMERICAL EXAMPLES. Ill which gives A = 0.024985. With this value of A as argument, we find, 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 G =0.0043771. Then we have tan ^v G tan ^w \ -< , Q , which gives v = 102 20' 52".20, agreeing exactly with the value already found. Finally, r is given by from which we get log r = 0.1614052. Before the time of perihelion passage, t T is negative ; but the value of v is computed as if this were positive, and is then considered as negative. In the case of hyperbolic motion, i is negative, and, with this dis- tinction, the process when Table IX. is used is precisely the same as for elliptic motion; but when table X. is used, the value of A must be found from and that of r from f ~ AC 2 ) cos*|v' the values of log B and log C being taken from the columns of the table which belong to hyperbolic motion. In the calculation of the position of a comet in space, if the motion 112 THEORETICAL ASTRONOMY. is retrograde and the inclination is regarded as less than 90, the 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 + nct)j and the corresponding value of the co-ordinate, or of the function, for which the interpolation is to be made, by / (a '-{- no)). If we have computed the values of the function for the dates, or arguments, a ) =/(a) + An -f Bn z + On* -f &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 _ w ) =f(a)-A +B -C A _ +c /( + ) =fW + A +B +C /(a -f 2oi) =/(a) + 2 If we symbolize, generally, the difference f(a + n), the difference / (a + (n + i) a>) f (a + (n 1) a) by f (a + na>) 9 and similarly for the successive orders of differences, these may be arranged as follows : Argument. Function. I. Diff. II. Diff. III. Diff. a /(a + 2a) / (a + ^ INTERPOLATION. 113 Comparing these expressions for the differences with the above, we get c=tr(+i-), -B=if(), A=f(a + - - if (a) - If' (a + -, which, from the manner in which the differences are formed, give C= J (/" (a + >) -/" ), ^ - /" (a), J. = /( a + ) _/( ) _ -i/- ( a ) - J (/" ( a + ) _/" ( a) ). To find the value of the function corresponding to the argument a -f- |w, we have n = ^, and, from (116), /(a + >) =/(a) + 14 + iJJ + a Substituting in this the values of J., j5, and (7, last found, and re- ducing, we get f(a + i0 = i (/( + + /()) ~ I (J (/" ( + + /" to)), in which only fourth differences are neglected, and, since the place of the argument for n = is arbitrary, we have, therefore, generally, - j a (r ( -t> + 1) 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 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 CD, equal to four days, for the intermediate dates, we obtain a series of places at intervals of two days ; and, finally, inter- 8 114 THEORETICAL ASTRONOMY. polating for the dates intermediate to these, we derive the places at intervals of one day. When a series of places has been computed, the use of differences will serve as a check upon the accuracy of the calculation, and will serve to detect at once the place which is not correct, when any discrepancy is apparent. The greatest discordance will be shown in the differences on the same horizontal line as the erroneous value of the function ; and the discordance will be greater and greater as we proceed successively to take higher orders of dif- ferences. In order to provide against the contingency of systematic error, duplicate calculation should be made of those quantities in which such an error is likely to occur. The ephemerides of the planets, to be used for the comparison of observations, are usually computed for a period of a few weeks before and after the time of opposition to the sun ; and the time of the opposition may be found in advance of the calculation of the entire ephemeris. Thus, we find first the date for which the mean longitude of the planet is equal to the longitude of the sun increased by 180 ; then we compute the equation of the centre at this time by means of the equation (53), using, in most cases, only the first term of the development, or v M 2esin M, e being expressed in seconds. Next, regarding this value as con- stant, we find the date for which L -j- equation of the centre is equal to the longitude of the sun increased by 180 ; and for this date, and also for another at an interval of a few days, we compute Uj and hence the heliocentric longitudes by means of the equation tan (I & ) = tan u cos i. Let these longitudes be denoted by I and /', the times to which they correspond by t and t f , and the longitudes of the sun for the same times by O and O ' ; then for the time t w for which the heliocentric longitudes of the planet and the earth are the same, we have or (113) the first of these equations being used when 180 O is less TIME OF OPPOSITION. 115 than V 180 O'. If the time t Q differs considerably from t or t', it may be necessary, in order to obtain an accurate result, to repeat the latter part of the calculation, using t Q for t, and taking t r at a small interval from this, and so that the true time of opposition shall fall between t and t f . 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 5 e z v M= 2e sin if + - - sin 2M, 4 8 s being the number of seconds corresponding to a length of arc equal to the radius, or 206264".8 ; and the value of v M will then be expressed in seconds of arc. In all cases in which circular arcs are involved in an equation, great care must be taken, in the numerical application, in reference to the homogeneity of the different terms. If the arcs are expressed by an abstract number, or by the length of arc expressed in parts of the radius taken as the unit, to express them in seconds we must multiply by the number 206264.8 ; but if the arcs are expressed in seconds, each term of the equation must contain only one concrete factor, the other ' concrete factors, if there be any, being reduced to abstract numbers by dividing each by s the number of seconds in an arc equal to the radius. 43. It is unnecessary to illustrate further the numerical application of .the various formulae which have been derived, since by reference to the formulae themselves the course of procedure is obvious. It may be remarked, however, that in many cases in which auxiliary angles have been introduced so as to render the equations convenient for logarithmic calculation, by the use of tables which determine the logarithms of the sum or difference of two numbers when the 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 = ^ (log b log a), 116 THEORETICAL ASTRONOMY. we shall have log (a -f- 6) = log a 2 log cos x, or log (a -f- 6) = log 6 2 log sin x. The first form is used when cos x is greater than sin x, and the second form when cos x is less than sin x. It should also be observed that in the solution of equations of the form of (89), after tan (X ) 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 (X 0) and sin (X ), 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 ) is greater than sin (A ), and that derived from the second equation when cos (A O) is less than sin (A 0). The value of J, if the greatest accuracy possible is required, should be derived from J cos /9 when /9 is less than 45, and from A sin ft when /3 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 tanJ(&'H-fl> ) and tan j (' fl> ); then, if sin|(&'-f w ) is greater than cosJ(&'-[- ) * s ^ ess tnan cos J(&' + w o)> we ^ nc ^ cos Ji' from the second equation. The same principle is applied in finding sin \%' by means of the third and fourth equations. Finally, from sin $ and cos \V we get tan \V ^ and hence i'. The check obtained by the agreement of the values of sin \i r and cos %i f , with those computed from the value of i f derived from tan \i f , does not absolutely prove the calculation. This proof, however, may be obtained by means of the equation sin i' sin &' = sin i sin & , or by sin i' sin w = sin e sin & In all cases, care should be taken in determining the quadrant in which the angles sought are situated, the criteria for which are fixed either by the nature of the problem directly, or by the relation of the algebraic signs of the trigonometrical functions involved. DIFFERENTIAL FORMULAE. 117 CHAPTER II. INVESTIGATION OF THE DIFFERENTIAL FORMULAE WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATION OF THE ELEMENTS OF ITS ORBIT. 44. IN many calculations relating to the motion of a heavenly body, it becomes necessary to determine the variations which small increments applied to the values of the elements of its orbit will pro- duce in its geocentric or heliocentric place. The form, however, in which the problem most frequently presents itself is that in which approximate elements are to be corrected by means of the differences between the places derived from computation and those derived from observation. In this case it is required to find the variations of the elements such that they will cause the differences between calculation and observation to vanish ; and, since there are six elements, it follows that six separate equations, involving the variations of the elements as the unknown quantities, must be formed. Each longitude or right ascension, and each latitude or declination, derived from observation, will furnish one equation ; and hence at least three complete observa- tions will be required for the solution of the problem. When more than three observations are employed, and the number of equations exceeds the number of unknown quantities, the equations of condi- tion which are obtained must be reduced to six final equations, from which, by elimination, the corrections to be applied to the elements may be determined. If we suppose the corrections which must be applied to the ele- ments, in order to satisfy the data furnished by observation, to be so small that their squares and higher powers may be neglected, the variations of those elements which involve angular measure being expressed in parts of the radius as unity, the relations sought may be determined by differentiating the various formulae which determine the position of the body. Thus, if we represent by 6 any co-ordi- nate of the place of the body computed from the assumed elements of the orbit, we shall have, in the case of an elliptic orbit, 118 THEORETICAL ASTRONOMY. M Q being the mean anomaly at the epoch T. Let 6' 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&, A^, &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 de dM n A3f o + -j- *r- (1) The differential coefficients -= , -= -, &c. must now be derived from dj: d& 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 or dd dO , dO = -=- dx 4- -y- dy -4- -=- dz r y * Hence we obtain dO = dx dd_ dx ' dy dz dO_ dy dO_ dz dy dx dz ' dn (2) and similarly for the differential coefficients of 6 with respect to the other elements. We must, therefore, find the partial differential co- efficients of d with respect to x, y, and z, and then the partial differen- tial coefficients of these co-ordinates with respect to the elements. In the case of the right ascension we put 6 = a, and in the case of the declination we put 6 = 3. 45. If we differentiate the equations X -f- X== A COS d COS a, y -f Y= A cos d sin a, z -J- Z = A sin d, regarding X, Y, and Z as constant, we find DIFFERENTIAL FORMULAE. 119 dx = cos a cos d d A A sin a cos d da A cos a sin 8 dd, dy = sin a cos d d A -\- A cos a cos d da, A sin a sin d dd, dz = sin d d A -\- A cos d dd. From these equations, by elimination, we obtain sin a 7 , COS a , cos d da = -- dx -\ -- dy, (3) cos a sin d _ sin a sin d cos 8 . dd = --- - - dx --- - - dy -f- az. A A /j Therefore, the partial differential coefficients of a and d with respect to the heliocentric co-ordinates are da sin a eZfl cos a sin d cos o r~ = -- 7. -j = --- ~ A - > dx A ' dx A da cosa dd sinasind .. Next, to find the partial differential coefficients of the co-ordinates Xy y, z y with respect to the elements, if we differentiate the equations (100)!, observing that sin a, sin 6, sin c, A, B, C, are functions of & and i y we get dx = - dr -{- x cot ( A -{- u) du -{- ~r=- dQ -j- -^ c?i, 7* ft 55 Ml' co c?2 = - - dr -j- 2 cot ( C -f- w) du + TQ ^^ + -p- ^' To find the expressions for -= , -p, &c., we have the equations d 66 ft^ 1 a; rr= r cos it cos S7 f sin it sin Q cos i, y = r cos w sin S^ cos e -{- r s i n w cos S7 cos i cos e r sin u sin i sin e, 2 = r cos u sin &7 sin e -f- f sin w cos Q cos t sin e -j- r sin it sin i cos e, which give, by differentiation, dx -7 = r cos w sin Q r sin w cos & cos t, rtS7 - 7 $- = r cos u cos & cos e r sin it sin & cos i cos e, 120 THEORETICAL ASTRONOMY. dz ...... - = r cos u cos & sin e r sm u sin & cos i sin e, d& dx . . . -=r- = r sm w sm & sin *, cfi> dy - = r sin i* cos & sin i cos e r sm w cos i sm e, at rv = r sin it cos & sin i sin e -f- r sin it cos i cos e. CM The first three of these equations immediately reduce to dx . dy dz , K , = yeoss 2 sine, ^-^ajcose, ^^zsme; (5) and since cos a = sin & sin i, cos 6 = cos & sin i cos e cos i sin e, cos c cos & sin i sin e -j- cos i cos e, we have, also, dx dy dz -jr = r sm u cos , -- = r sm w cos o, -JT- = r sm u cos c. cfo c?i di Further, we have du dv-\-dn dQ , and hence, finally, 3T dx = - dr + x cot (J. + ) cfo + cot (A + w) G?TT -f- ( * cot (A-{-u) y cos e 2! sin e) d& -f- r sin it cos a di,' y = -dr-\-y cot (-B -f- w) c?v -f 2/ cot (B + w ) ^ , fi \ ( y cot (5 + w ) + x cos e) c?^ -\-rsmu cos 6 di, ?2 = - dr + 2 cot ( C -f w) dv + 2 cot ( C -f- w) d;r -f ( 2 cot ( (7 -f M) -j- a; sin e) d& + r sin w cos c These equations give, for the partial differential coefficients of the heliocentric co-ordinates with respect to the elements, dx dx = dz dz DIFFERENTIAL FORMULAE. 121 -T^ = - x cot (A+u)y cose z sine, -^-= y cot (B+u)+x cos e, tt&6 "d6 -y = z cot ( (7 + u) + sin e ; da? . , and, hence, dx dx . /A , dy dy , ,, , -^ -j- = x cot (A +1*), -/- = -/- = y cot (B 4- u), dot dv da) dv 122 THEORETICAL ASTRONOMY. The values of > , and - must, in this case, be found by means of the equations (5). By means of these expressions for the differential coefficients of the co-ordinates x, y, z, with respect to the various elements, and those given by (4), we may derive the differential coefficients of the geo- centric right ascension and declination with respect to the elements &, i, and TT or o>, and also with respect to r and v t by writing suc- cessively a and d in place of 6, and &, i, &c., in place of x in the equation (2). The quantities r and v, however, are functions of the remaining elements

dv dy ' dx dx dr dx dv == ~j~ ' ~JT/f I dM ~~ dr dM dv dM } dx dx dr dx dv dfj. dr dfj, dv dfj. The expressions for the partial differential coefficients in the case of the co-ordinates y and z are of precisely the same form, and are ob- tained by writing, successively, y and z in place of x. The values of dx dx dy dy dz n dz . , ^ /m > r- } ~-> r-> r-> and are given by the equations (7), and dr dv dr dv dr dv dr dv dr dv dr , dv , when the expressions lor , -7 > > , , and -7 have been d

fV dM= - dE - sin v dy, a a or dE = - dM 4- sin v dy. T If we take the logarithms of both members of the equation tan v tan ji;tan (45 -f y), and differentiate, we find dv dE dy 2 sin v cos -\v 2 sin E cos %E ' 2 sin (45 -f y) cos (45 -f JpJ which reduces to snv , snv . . smE cosy Introducing into this equation the value of dE, already found, and ._. , r sin v replacing sm E by -- , we get J a cosy a? cos y , , , sin v I a cos 2 y \ cZv = --- = d M -\ -- -- \-l\cUp. i* cos?>\ r I (Y\ But since a cos 2

. (13) a 124 THEOEETICAL ASTRONOMY. -r, cosv-\-e , ,, , [Now, since sin E= > and cos E= , we shall have 1 -j- e cos v 1 H- e cos v ae cos

cos v c?^. (14) a Further, we have M=M + fJ .(t-T-), T being the epoch for which the mean anomaly is Jf , and Jfel/1 +m " = -JT Differentiating these expressions, we get dM= da substituting these values in the expressions for dr and dv, we have, finally, (2r\ a tan

cos v d(f> f (15) , a 2 cos

~dq = ^75=^ W _dr_ _ q cos v dv 3& (t T) ~~ A ' dlogq~ and then we have, for the differential coefficients of x with respect to T and q or log q, DIFFERENTIAL FORMULA. 127 dx _ dx dr dx dv dx _ dx dr dx dv dT~~dr~'dT + ~dv~'dT' ~d^ = 'dr'"d^^~~dv"df dx dx dr dx dv d log q dr d log q dv ' d log q and similarly for the differential coefficients of y and z with respect to these elements. The expressions for the partial differential co- efficients of x, y, and z, respectively, with respect to r and v are the same as already found in the case of elliptic motion. We shall thus obtain the equations which express the relation between the variations of the geocentric places of a comet and the variation of the parabolic elements of its orbit, and which may be employed either to correct the approximate elements by means of equations of condition fur- nished by comparison of the computed place with the observed place, or to determine the change in the geocentric right ascension and declination corresponding to given increments assigned to the ele- ments. 48. We may also, in the case of an elliptic orbit, introduce I 7 , q, and e instead of the elements , that when the value of e is very nearly equal to unity, the coefficients for these differentials become indeterminate. It becomes necessary, therefore, to develop the corresponding expressions for the case in which these equations are insufficient. For this purpose, let us resume the equation - T) (1 + 6)1 J _ Q in which u = tan Jw, and i = . Then, since I -(- e we shall have DIFFERENTIAL FOKMTJXJE. 131 + (Aw - / 2 u 3 + 2 X) (1 - *) 2 + 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 + w 2 ) du (\u > 3 > 5 ) de; and, since du = J(l + u 2 ) dv, this gives dv _ u lu* lu* de ~ (1 + u^ The values of the second member, corresponding to different values of v, may be tabulated with the argument v; but a table of this kind is by no means indispensable, since the expression for -7- may be changed to another form which furnishes a direct solution with the same facility. Thus, by division, we have de~ and since, in the case of parabolic motion, T^r="+^' -'=9* d this becomes (31) If we differentiate the equation 1 -fecosv' regarding r, v, and e as variables, we shall have dr e)~ 2 g (1 + e) ' ~de 132 THEORETICAL ASTRONOMY. In the case of parabolic motion, e = l, and this equation is easily transformed into (33) Substituting for -=- its value from (31), and reducing, we get CtC 1 %- = 2 n * (< ~ r) sin , + T y tan* Jt>. (34) V 2q The equations (31) and (34) furnish the values of and to be de de used in forming the expressions for the variation of the place of the body when the parabolic eccentricity is changed to the value 1 -j- de. When the eccentricity to which the increment is assigned differs but little from unity, we may compute the value of - directly from equation (30). A still closer approximation would be obtained by di) 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 (1000 + B (lOOi) 2 + O(100i) 8 , regarding the coefficients A, B, and (7 as constant, and introducing the value of i in terms of e, we have dv__d_V 200^ 4005 6000 de ~ de ~ s(l + e)> ~ s7T+ e) U (l+e)' U in which s 206264.8, the values of A, B, and C, as derived from dV the table, being expressed in seconds. To find , we have O/G ~ which gives, by differentiation, k(t T) 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 FORMULAE. 133 dV de " Hence we have 20(U _4005 600C s(l + e)* s(l + e)^ ^T+ej* ( )} by means of which the value of is readily found. do When the eccentricity differs so much from that of the parabola that the terms of the last equation are not sufficiently convergent, dv the expression for , which will furnish the required accuracy, may be derived from the equations (75) 1 and (76)j. If we differentiate the first of these equations wifch respect to e, since B may evidently be regarded as constant, we get , _ _ 9 IV A f we take the logarithms of both members of equation (76) w and differentiate, we get dv dC . dw 4de C """sinw (1 + e) (1 + 9)' (37) To find the differential coefficient of C with respect to e } it will be sufficient to take 1 which gives -1 ~*~ The equation gives j A 50 i j dA = ,^ . n ^ tan 2 -Aw ae -f - - - -- r-: (1 -f- 9e) 2 tan w cos 2 ^w and hence we obtain dC 200 2 tan -77 7T- 7T 2 O (1 + Oe) 2 sin w Substituting this value in equation (37), we get dv _ 20 C 2 2 1 . ^ 2 sin v ^ w de~ ~ 2S1 "'" 134 THEORETICAL ASTRONOMY. and substituting, finally, the value of -y-, we obtain C 2 smv cos'JUfl 20 O a . S1 4 sin?; ~ (1 + e) (1 + 9e)' which, by means of (76) 1? reduces to cos 2 |w Stanjv ' ' If we introduce the quantity M which is used as the argument in finding w by means of Table VI., this equation becomes 9e)75tan> (1 -f- e) (1 + 9e>' This equation remains unchanged in the case of hyperbolic motion, the value of C being taken from the column of the table which cor- d^o responds to this case-: and it will furnish the correct value of -7- in ae all cases in which the last term of equation (23) is not conveniently d/T applicable. The value of ~ is then given by the equation (32). a/ c> When the eccentricity differs very little from unity, we may put jg 1, and tan Jii> = tan Jv y \^ (\ cos 2 w = JO 2 cos 2 v. Then we shall have 2 ^^ 2k(tT) 2 sin v = -- -=; cos 4 75 tan % The equation ? = (1 + A C 2 ) cos 2 v = (1 -f iJL) cos* Jw, gives ^ = (1 + P) cos* Jw = Ccos 4 iw. Hence we derive I '\ NUMERICAL EXAMPLES. 135 If we substitute this value in equation (39), and put C 2 (1 + e) = 2, we get _ de 2(l-i-9e)" r 2 (1 + e) (1 + 9e)' and when e 1, this becomes identical with equation (31). 51. EXAMPLES. We will now illustrate, by numerical examples, the formula 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 =t 181 8' 29".29, d = 4 42' 21".56, log A = 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 n , log y = 9.511920, log z = 8.077315, e = 23 27' 24".0, t T= 420.714018. First, by means of the equations (4), we compute the following values : log cos d ~ = 8.054308, log ^ = 8.668959 n , log cos d = 9.754919 n , log - == 6.968348 log = 9.753529. dz Then we find the differential coefficients of the heliocentric co-ordi- nates, with respect to TT, ^, i, v, and r, from the formulae (7), which give log ^ = log ^ = 0.399496 , ait dv log = log = 9.491991 n , dit dv log -- = 7.876553, log -- = 8.830941, log ~ = 9.222898., ^d6 "66 W$g log -^- = 8.726364, log -^- = 9.687577, log -~ = 0.142443 n , log ^j- = 9.996780 n , log -^- = 9.083635, log -^~ = 7.649030. 136 THEORETICAL ASTRONOMY. In computing the values of -jr> -p-> and -yr, those of cos a, cos b f and cos c may generally be obtained with sufficient accuracy from sin a, sin 6, and sine. Their algebraic signs, however, must be strictly attended to. The quantities sin a, sin 6, and sin c are always positive ; and the algebraic signs of cos a, cos 6, and cos c are indicated at once by the equations (101)!, from which, also, their numerical values may be derived. In the case of the example proposed, it will be observed that cos a and cos b are negative, and that cos c is positive. To find the values of cos d -y- and -r-> we have, according to equa- tion (2), . da da dx , .da dv COS d - = COS d ~- . - -\- COS d - 21, (41) dx dx dr. dy dit which give JL _,_ _ dx dit dy'dn: dz dn' cos *L = cos %- = + 1.42345, *L = *= - 0.48900. arr dv dr. dv In the case of &, i, and r, we write these quantities successively in place of TT in the equations (41), and hence we derive cos d -A- = - 0.03845, J^ = ~ - 09533 > cos 8 ~ = 0.27641, ~ = 0.78993, ai ai cos d ~ = 0.08020, ~ = + 0.04873. dr dr Next, from (16), we compute the following values: log |L = 0.179155, log ^L = 9.577453, log ^ = 2.376581 n , log L = 0.171999, log L ^ 9.911247, log = 2.535234. r/7* fi T "We may now find ^-, ^r, &c. by means of the equations (11), and thence the values of cos d -y-, -y-, &c. : but it is most convenient d

- r , and -j- are determined as dv dr dv dr already exemplified. If we introduce the elements T, q, and e, we shall have da da, dr , .da dv dd _ dd dr dd dv dT-~d^'dT^~dv"~di' and similarly for the differential coefficients with respect to q and e. NUMERICAL EXAMPLES. 139 , , , . dr dv dr dv dr dv The mode of calculating the values of -7, -r=, - r , -=-, -j-, and -=- dT dT dq dq de de depends on the nature of the orbit. In the case of passing from one system of parabolic elements to another system of parabolic elements, the coefficients of Ae vanish. To illustrate the calculation of -7, -7, &c. in the case of parabolic motion, let us resume the values t T= 75.364 days, and log q = 9.9650486, from which we have found log r = 0.1961120, v = 79 55' 57".26. Then, by means of the equations (22), we find log ~ = 8.095802,,, log = 9.242547, rAji r/7? log jj- = 7.976397 M , log j- = 0.064602 n . If, instead of dq, we introduce d log q, we shall have log -7^- = 9.569812, log ~- = 0.391867 . fo d log q & d log q From these, by means of (43), we obtain the differential coefficients of a and d with respect to T and q or log q. The same values are also used when the variation of the parabolic eccentricity is taken /y/y into account. But in this case we compute also j- from equation /7w (31) and ^ from (33) or (34), which give, for v = 79 55' 57".3, log ~ = 8.147367 n , log ^ == 9.726869. U6 U6 In the case of very eccentric orbits, the values of -T~, -7, &c. are found from dv kV dr k .... dq q qi / p dr r . r 2 e sin v dv dq q p dq the mass being neglected. 140 THEORETICAL ASTRONOMY. To illustrate the application of these formulae, let us resume the values, *T= 68.25 days, e = 0.9675212, and log q = 9.7668134, from which we have found (Art. 41) v = 102 20' 52".20, log r = 0.1614052. Hence we derive = 0.0607328, and log^=7.943137 n , log ^ = 0.186517., log ~ = 0.186517.. aq uq If we wish to obtain the differential coefficients of v and r with respect to log q instead of g, we have dv _ q dv dr _q dr dlogq A fl ' dq d logq I dq in which ^ is the modulus of the system of logarithms. Then we compute the value of -7- by means of the equation (30). d/6 (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. diT Finally, the value of -7- may be found by means of (32), which CLG gives = + 0.70855. de When, in addition to the differential coefficients which depend on the elements T, q, and e, those which depend on the position of the orbit in space have been found, the expressions for the variation of the geocentric right ascension and declination become NUMERICAL EXAMPLES. 141 ^da , da ^da da COS d Act = COS d ATT -f- COS 3 A & + cos S T^ Al + cos ^ T7^ A * ofTT agg cu aj[ R dfa da -f- COS A*/ -|~ COS -7- A and -^,> In the same manner, we may adopt as the unknown quantity, instead of the actual variation of any one of the elements of the orbit, n times that variation, in which case its coefficient in the equations must be divided by n. The value of ACC, derived by taking the difference between the computed and the observed place, is affected by the uncertainty necessarily incident to the determination of oc by observation. The unavoidable error of observation being supposed the same in the case of a as in the case of d, when expressed in parts of the same unit, it is evident that an error of a given magnitude will produce a greater apparent error in a than in S, since in the case of a it is measured on a small circle, of which the radius is cos d ; and hence, in order that the difference between computation and observation in a and d may have the same influence in the determination of the corrections to be applied to the elements, we introduce cos d AOC instead of AOC. The same principle is applied in the case of the longitude and of all corresponding spherical co-ordinates. DIFFERENTIAL FORMULA. 143 52. The formulae already given will determine also the variations of the geocentric longitude and latitude corresponding to small in- crements assigned to the elements of the orbit of a heavenly body. In this case we put e = 0, and compute the values of A, B, sin a, and sin 6 by means of the equations (94) r We have also (7=0, sin c = sin i, and, in place of a and d, respectively, we write A and ft. But when the elements are referred to the same fundamental plane as the geocentric places of the body, the formulae which depend on the position of the plane of the orbit may be put in a form which is more convenient for numerical application. If we differentiate the equations x' = r cos u cos & r sin u sin Q cos i, y' = r cos u sin & -j- r smw cos & cost *> z' rrrrsmwsini, we obtain x' dx' = dr r (sin u cos & -j- cos u sin & cos i) du r r (cos u sin & -j- sin w cos & cos i) d& -f r sin u sin & sin i di, dy' = dr r (sin u sin & cos u cos & cos i) du -{- r (cos u cos & sin u sin & cos i) d& r sin w cos & sin i di, (46) dz f =-dr -}-r cos w sin i du -j- r sin it cos i di, in which x', y 1 ', 2' 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 Q> ; then we shall have dx = dx f cos & -f- dy' sin & , dy = dx' sin & -f dy' cos Q> , dz = dz 1 , and the preceding equations give OC dx = -dr r sin u du r sin u cos i d& , T dy = y dr + r cos u cos i du -j- r cos u d Q r sin u sin i di, (47) dz = - dr + r cos u sin idu-\-r smu cos i di. 144 THEORETICAL ASTRONOMY. This transformation, it will be observed, is equivalent to diminishing the longitudes in the equations (46) by the angle ft through which the axis of x has been moved. Let X n F,, Z, denote the heliocentric co-ordinates of the earth referred to the same system of co-ordinates, and we have x + X, = A cos /? cos (A ft), y+ F, = Jcos/5sin(A ft), z--\- Z, = A sin /?, in which I is the geocentric longitude and ft 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 ft 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 s(A ft)d/J J sin cos (A ft)d/5 A cos /9 sin (A ft ) ax _ . , 1 . dy _ ' "2! -T7T- = 2?- sm w sm 2 51. -."L = 2r cos u sm* *t. 7 ^ = r cos w sin i. dQ dQ> -j~, &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 dn -f d& , and hence dx . dy dz rsmu f -~- = r cos u cos i, ^ = rcosusim; (52) dx . . . dy . , , . dz -?7^- = 2r sin u sin 2 At, -y~- = 2r cos u sm 2 ii, -y - = r cos u sin ^. W a^ rt^ But, to prevent confusion and the necessity of using so many for- mulae, it is best to regard i as admitting any value from to 180, and to transform the elements which are given with the distinction of retrograde motion into those of the general case by taking 180 i instead of i, and 2& TT instead of TT, the other elements remaining the same in both cases. 53. The equations already derived enable us to form those for the differential coefficients of ^ and /? with respect to r, v, & , z, and at or TT, by writing successively ^ and ft in place of d, and &, i, &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 A and /9 with respect to these elements. The numerical application, however, is facilitated by the introduction of certain auxiliary quantities. Thus, if we substitute the values given by (48) and (49) in the equations . dk ctt dx Q dA dy cos /5 = cos /? -^ --- j \- cos /?-= --- , l ^ j - -= dv dx dv l dy dv and put df^_dp_ dx_ d{3 dy_ dft dz dv dx dv dy dv dz dv ' cos i cos (A & ) = A Q sin A, sin (A & ) = 4 cos ^, sin i = n sin JV, sin (A & ) cos i = n cos JV, in which A and n are always positive, they become 8 r A ' (A -J- d/3 d/3 r -7- = i == -7 (sin /? cos (A O ) sin u -4- n cos w sin av au) A Let us also put n sin (JV + /?) = 5 sin 5, , ,^ sin y? cos (A ^ ) = ^ cos B, and we have c?A _ dk r t The expressions for cos/9-^- and -^ give, by means of the same auxiliary quantities, fjl A cos/3-5- = 2*L cos (A + u), * In the same manner, if we put DIFFERENTIAL FORMULA. 147 cos (A &)= <7 sin C, cos i sin (A O ) = (7. cos (7; (57) cos i = D Q sin Z), sin (A & ) sin i = D Q cos D; we obtain dft r . = -j ^ sin ft cos ( J. -f- d& cos /? ,. = -7 sin i sin -w cos (A & ), ^/3 A = D sin w sin (jD -f- /5). cw 7* If we substitute the expressions (55) and (56) in the equations ^A dk dr dk dv - = cos p -=- f- cos /5 -= 7, a^> ^r d d

cZv a 2 cos $0 2r \ (61) tan

sin u sin (D + 0). 55. EXAMPLES. The equations thus derived for the differential coefficients of ^ 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 A and ft 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 THEORETICAL 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) D we obtain &' = 353 45' 35".87, t' = 19 26' 25".76, e = 212 32' 17".71, and a>' = w -f w = 50 10' 7".29, which are the elements which determine the position of the orbit in space when the equator is taken as the fundamental plane. These elements are referred to the mean equinox and equator of 1865.0. Writing a and d in place of A and /9, and &', i 1 ', w f in place of &, i, and a), respectively, we have A sin A = cos (a &') cos i' t A cos A = sin (a &') ; n smN=smi f , n cosN= cosi'sin(a &'); B Q smJB n sin (N + <0, -B cos B == sm ^ cos ( a &') J <7 sin (7= cos (a &'), C cos C= sin (a &') cosi'; D sin D = cos i f , D Q cos D = sin i' sin (a Q ') ; / sin .F= a cos

ao> aw =+0.0178, rfft cos d -^ = + 0.0067, -^r = + 0.0193, cos d ^~ = + 1.9940, ~- = 0.6530, -^=+1.1300, ^r- - 8802 ' cos d -^- = -f- 507.25, 4 = 179.34 ; qp d[j. and hence cos (5 Aa = + 1.4235 A^' + 1.5098 Aft' + 0.0067 At* + 1.9940 A? + 1.1300 *M + 507.25 AA*, A = 0.4890 AO/ + 0.0176 Aft' + 0.0193 At* 0.6530 A? 0.3802 A^f 179.34 AA*. If we put A? = + 10", ' Ajf - + 10", Ai' = 8".86, AAX = + 0".01, we get cos (5 Aa = + 5".47, A<5 = 9".29 ; and the values calculated directly from the elements corresponding to the increments thus assigned, are cos d Aa = + 5".50, A<5 = 9".02. The agreement of these results is sufficiently close to prove the cal- culation of the coefficients in the equations for cos d AOC and A. When the values of AW', A ft ', and Ai r are small, the correspond- ing values of AW, Aft, and A^ may be determined by means of differential formula?. From the spherical triangle formed by the intersection of the planes of the orbit, ecliptic, and equator with the celestial vault, we have cos i = cos i f cos e -j- sin i' sin e cos ft ', sin i cos ft = cos i' sin e -j- sin i' cos e cos ft', sin i sin ft = sin i' sin ft', (67) sin i sin o> = sin ft ' sin e, sin i cos u> = cos e sin i' sin e cos i' cos ft ', 152 THEORETICAL ASTRONOMY. from which the values of ft, i, and co may be found from those of ft ' and V . If we differentiate the first of these equations, regarding e as constant, and reduce by means of the other given relations, we get di = cos a> di' -j- sin a> Q sin i' d ft '. (68) Interchanging i and 180 i', and also ft and ft', we obtain di' = cos Q di sin % sin i dft - i Eliminating di from these equations, and introducing the value sin i f _ sin ft sini sin &'' the result is If we differentiate the expression for cos = cos i c?ft cos i' dQ,'. Substituting for dft its value given by the preceding equation, and reducing by means of sin ft' cos i' = sin ft cos % cos i cos ft sin Q sin i' A &' + cos w fl Ai', Aw = Ao>' ^^o- If we apply these formulae to the case of Eurynome, the result is AO O = 4.420A^' + 6.665 Ai', = 3.488A ft' -f 6.686Ai', = 0.179A&' 0.843Ai r ; DIFFERENTIAL, FORMULA. 153 and if we assign the values A ' = 14".12, Ai' = 8".86, W = 6".64, we get AW O = -f 3".36, A & = 10".0, A^ = + 10".0, A 10 = 10".0, and, hence, the elements which determine the position of the orbit in reference to the ecliptic. The element^ a/, & ', and i f may also be changed into those for which the ecliptic is the fundamental plane, by means of equations which may be derived from (109)! by interchanging & and &' and i' audi. 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 &, 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' 9 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 Q> . Then we shall have dx' = dx, dtf = dy cos i -f- dz sin i, dz' = dy sin i -j- dz cos i. Substituting for dx, dy, and dz their values given by the equations (47), we get x' dx f = dr r sin u du r sin u cos i d&, dy' = dr -j- r cos u du -f r cos u cos i d& , dz' = - dr r cos u sin i dQ, -f- r sin u di. It will be observed that we have, so long as the elements remain unchanged, af ==r cos u, y' = r sin u, z' 0, 154 THEOEETICAL ASTKONOMY. and hence, omitting the accents, so that x, y, z will refer to the plane of the unchanged orbit as the plane of xy, the preceding equations give dx = cos u dr r sin u du r sin u cos i dR>, dy = sin u dr ~\- r cos u du -j- r cos u cos i dQ, dz = r cos u sin i dQ, -j- r sin w di. The value of < is subject to two distinct changes, the one arising from the variation of the position of the orbit in its own plane, and the other, from the variation of the position of the plane of the orbit. Let us take a fixed line in the plane of the orbit and directed from the centre of the sun to a point the angular distance of which, back from the place of the ascending node on the ecliptic, we shall desig- nate by a; and let the angle between this fixed line and the semi- transverse axis be designated by . Then we have X = a) -f ff . The fixed line thus taken is supposed to be so situated that, so long as the position of the plane of the orbit remains unchanged, we have But if the elements which fix the position of the plane of the orbit are supposed to vary, we have the relations da = d? sin cfy + ^ cos 7 cos ^ ^> dz sin fj dA -\- A cos iy d^ ; ind hence we obtain cos T] sin cos = -- - dx -\ ^ dy, sin i? cos , sin ^ sin = - -dx -- lese give de sin0 sin t] cos d0 cos 6 COS7 )-j-= -T-i do _ cos TI -j- ; az (74) dx~ A id from (73) we get dx dr rr= COS 1*. dx dx = -= = r sin u, dv dx dx d& dx r = 0, dy dr = sin u y dz "dr = 0; dy dy dz dz rt L dz V 01 is u si ~d x n ?!. "j (75) dz = r sin w. 7 v 7 V) 7 * eu a^ d^ ibstituting the values thus found, in the equations COS 1] -j- = 7 dv _d0 _d^_ da; dv da; dd dy COS 7) ,- -j-, 1 ..-, _ -^ ^ i ^ t dy . dy dz dv da; ' dv dy ' dv dz dv' 156 THEOKETICAL ASTRONOMY. we get do dd r ,. , COS 7} -r- == COS Tf) -J- = -r COS (0 - U), d dx J (76) dv d% In a similar manner, we derive do 1 . , a N cos >y -- = sm (0 it), dO dr) . r cos r) JT- = 0, -JT- = = + j- cos ^ sm w - (77) If we introduce the elements A Finally, using the auxiliaries #, h, 6r, and H, according to the equa- tions (61), we get dO h . N dt\ h . . , n cos i) = cos (0 u H), ~- = -j- sm TJ sm(0 u JET). If we express r and v in terms of the elements T, q, and e, the values of the auxiliaries /, g, Ti y 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 d6 dd COS 1? A0 = COS TJ A/ -{- COS ?) -, &

cos f\ sin = cos sin (A & ) cos i -j- sin /? sin i, (80) sin >? = cos ft sin (A & ) sin i + sin /5 cos i. These equations correspond to the relations between the parts of a spherical triangle of which the sides are i, 90 27, and 90 /9, the angles opposite to 90 fj and 90 /5 being respectively 90 -f (; a) and 90 6. Let the other angle of the triangle be denoted by p, and we have cos *) sin Y = sin i cos (A & ), cos >? cos 7- = sin i sin (A & ) sin -f- cos i cos /?. The equations thus obtained enable us to determine y, d, and 7- from A and /9. Their numerical application is facilitated by the intro- duction of auxiliary angles. Thus, if we put n sin N= sin/?, n cos^= cos sin (A ft), 158 THEORETICAL ASTRONOMY. in which n is always positive, we get cos f] cos cos ft cos (A ft), cos 77 sin = n cos (N i), (83) sin 7} =n sin (N i), from which y and may be readily found. If we also put n' sin N' = cos i, n' cos N' = sin i sin (A ft ), we shall have cot N f = tan i sin (A ft ), cot ( ;-a). (85) If Y is small, it may be found from the equation sintcosO^ The quadrants in which the angles sought must be taken, are easily determined by the relations of the quantities involved ; and the accuracy of the numerical calculation may be checked as already illustrated for similar cases. If we apply Gauss's analogies to the same spherical triangle, we get sin (45 - J,) sin (45 - (0 + r )) = cos (45 + i (A - ft)) sin (45 - (/? + 0), sin (45 &) cos (45 (0 + 7-)) = ~ siu (45 + 4 (A - ft)) sin (45 - J Q9 - i)), cos (45 ii?) sin (45 J (0 r)) =f (87) cos (45 + i (A _ ft )) cos (45 - J ( + i)), cos (45 ^) cos (45 \ (9 r }} = sin (45 + (A - ft )) cos (45 - (0 - 1)), from which we may derive ^, 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 given observed places, it is necessary to find the expressions for cos 5? A# and A^ in terms of cos ft AA 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. 159 cos V] dd cos Y cos /? oU -f- sin y c?/3. Diiferentiating the last of equations (80), and reducing, we find dy = sin Y cos ft cW -j- cos The equations thus derived give the values of the differential co- efficients of and 57 with respect to A and /5 ; and if the differences A^ and A/3 are small, we shall have cos TI A0 = cos Y cos /? AA -f sin p A/9, A^ = sin Y cos /? AA -f cos y A/5. The value of 7- required in the application of numbers to these equations may generally be derived with sufficient accuracy from (86), the algebraic sign of cos Y being indicated by the second of equations (81) ; and the values of 37 and d 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 y and 6 from a and 3. For this purpose, it is only necessary to write a and d in place of A and /9, respectively, and also & ', i f , ft/, /', and u' in place of &, i, a), , and u, in the equations which have been derived for the determination of y and 6, and for the differential coefficients of these quantities with respect to the elements of the orbit. To illustrate this clearly, let it be required to find the expressions for cos rj A# and A/? in terms of the variations of the elements in the case of the example already given ; for which we have aj' = 5Q 10' 7".29, ' = 353 45' 35".87, i' = 19 26' 25".76. These are the elements which determine the position of the orbit of Eurynome (79), referred to the mean equinox and equator of 1865.0. We have, further, log/= 0.62946, log# = 0.34593, log ft = 2.97759, F= 339 14' 0", G = 350 11' 16", H= 14 30' 48", u' = 179 13' 58". In the first place, we compute /, 6, and Y by means of the formulae 160 THEORETICAL ASTKONOMY. (83) and (85), or by means of (87), writing a, d, &', and i' instead of Aj ft) Q>, and i, respectively. Hence we obtain = 188 31' 9", = 159'28", = 19 17' Since the equator is here considered as the fundamental plane, the longitude 6 is measured on the equator from the place of the ascend- ing node of the orbit on this plane. The values of the differential coefficients are then found by means of the formula? COB " = do do r fa ,, COS Ti . = COS (0 IT I, d% A cos 77 y = L cos (0 u'F"), dtp A do q cos "n -J^T = -: cos (0 u' G\ dO h - = -cos(0 u dri , , = cos yj sin i cos u , dy , r > t ~ = + cos ^ sm u = 4 sin y sin (O u' F), A -T^T = -^ sin 7 sin (0 u 1 G\ aM Q A dr) h . . f , , = sm ~n sm (0 u H). dv. A which give dB COST; do V, *$. - -p V.UV< Zi, _ i n AOA/I di' do 1 = -U 1 5051 dt . _i_ OOKfi do <"!-^r = + 2.0978, -/-- = + 1.1922, cos)) -^- = + 538.00, d^ dtp dp. = + 0.0422, = + 0.0143, = 1.71. Therefore, the equations for cos 37 A0 and A7y become cos ri Ld = + 1.5051 A/ + 2.0978 A^ + 1.1922 Ajf + 538.00 A//, A>? = 4- 0.0086 A/ -f- 0.0422 A? + 0.0143 Aj^ 1.71 A/* + 0.5072 A a' + 0.0204 Ai'. If we assign to the elements of the orbit the variations DIFFERENTIAL FORMULAE. 161 AW' = 6".64, A a' = 14".12, Ai' = _ 8".86, A? = -f- 10", A Jf g + 10", A/< = + 0".01, we have A/ == AO/ + cos i v A' = 19".96 ; and the preceding equations give cos 7? A0 = + 8".24 3 AT? == 6".96. With the same values of AO/, A & ', &c., we have already found cos d Aa = -f 5".47, A<5 = 9".29, which, by means of the equations (88), writing a and d in place of A and /?, give cos 7? A0 = + 8".23, i) = 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 /*, and the equation jfei/l + m gives = 3 - da = | ^d log a, (89) "a / in which ^ 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 y- ; and if the unit of the mth decimal place of the loga- ^0 ritlmis 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 fj. is here supposed to be expressed in seconds. If we introduce logp as one of the elements, from the equation p = a cos 3

, the coefficients of A^> are changed ; and if we denote by cos d ( -= 1 and I -j- I the values of the partial differential coefficients when the element fj. is used in con- nection with s , dp. dd tdd\ fi dd j-= -5- 3 - tan

\d^ / r c?0 in which s = 206264". 8. If the values of the differential coefficients with respect to // and (p have not already been found, it will be ad- dr dv dr dv , vantageous to compute the values or -r > -7? -n - , and -=-1 - by d d

+ d&, and, when ^ is used, d x + (1 COST Instead of the elements ft and i which indicate the position of the plane of the orbit, we may use b = sin i sin ft, c = sin i cos ft, and the expressions for the relations between the differentials of b and c and those of i and ft are easily derived. The cosines of the angles which the line of apsides or any other line in the orbit makes with the three co-ordinate axes, may also be taken as elements of the DIFFERENTIAL FORMULAE. 163 orbit in the formation of the equations for the variation of the geo- centric place. 60. The equations (48), by writing I and 6 in place of X and ft, respectively, will give the values of the differential coefficients of the heliocentric longitude and latitude with respect to x, y, and z. Combining these with the expressions for the differential coefficients of the heliocentric co-ordinates with respect to the elements of the orbit, we obtain the values of cos b &l and A& in terms of the varia- tions of the elements. The equations for dx, dy, and dz in terms of du, dQ, 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 = dx d Q . 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)^ cot A = tan & cos i, cot B = cot & cos i sin i cosec & tan e, cot C cot & cos i -}- sin i cosec 2 cot , 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 6, and sin c, we get cos^ sin A dA . d& --- : - sm & sin ^ di, sm 2 a 7 T, COS , ^ N T ^ dB = . (cos i cos e sin i sm e cos & ) dl sin o sin B , . . . . , . sin i sin & , -j : 7- (cos & sm i cos e -j- cos % sm t) di -j -- r-jr de, 7 ~ sin , dC= . 2 (cosi sine -f sin i cos cos &)d& sin o . sin C , , sin i sin 7 H -- : - (.cos & sini sin cos i cos 9) di -\ -- 2 -- as ; sin o sin c and these, by means of (101) D reduce to 164 THEORETICAL ASTRONOMY. dA = - - d& sin A cot a di, sm 2 a cose cose 7 ^ . -r, , 7 7 . . cos a sine cos & , . ~ ,. cosa smc smc Let us now differentiate the equations (101 ) 1? using only the upper sign, and the result is da = sin i sin A d& -f- cos A di, db = sin i sin B dQ> -j- cos B di -{- cos c cosec b ds, dc = sin i sin C dQ -f~ cos C di cos b cosec c ds. If we multiply the first of these equations by cot a, the second by cot b, and the third by cot c, and denote by ^ the modulus of the system of logarithms, we get (Hog sin a = A sin i cot a sin A dl -j- A cot a cos A di, d log sin b = A sin i cot 6 sin B dQ -f- ^ co * & cos Bdi-\-k Q - r-yy ds, sin o ~ ,^ .^ ,. . cos b cos c 7 a log sine = / sin % cot c sin G aQ -f- / cote cos Udi / - r 2 - ds. Sill (92) The equations (91) and (92) furnish the differential coefficients of A 9 By C, log sin a, &c. with respect to &, i, and e; and if the varia- tions assigned to &, i, and are so small that their squares may be neglected, the same equations, writing A^., A&, A*, &c. instead of the differentials, give the variations of the auxiliary constants. In the case of equations (92), if the variations of &, *, and s 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) D and reduce by means of the same equations, we easily find cos b dl = cos i sec b du -\- cos b d & sin b cos (I & ) di, db = sin i cos (I & ) du + sin (I Q> ) di, which determine the relations between the variations of the elements of the orbit and those of the heliocentric longitude and latitude. By differentiating the equations (88)^ neglecting the latitude of DIFFERENTIAL FORMULJE. 165 the sun, and considering ^, /?, J, and O as variables, we derive, after reduction, T) cos /? ctt = -r cos (A O ) dQ , B d{3 = -j- sin ft sin (A O ) d) , which determine tlie variation of the geocentric latitude and longitude arising from an increment assigned to the longitude of the sun. It appears, therefore, that an error in the longitude of the sun will produce the greatest error in the computed geocentric longitude of a heavenly body when the body is in opposition. 166 THEORETICAL ASTRONOMY. CHAPTER III. INVESTIGATION OF FOBMULJE FOB COMPUTING THE ORBIT OF A COMET MOVING IN A PABABOLA, AND FOB COEBECTING APPEOXIMATE ELEMENTS BY THE VABIATION OF THE GEOCENTBIC 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 is impossible to apply the correction for parallax directly to the place of the body; but an equivalent correction may be applied to the places of the earth, according to the 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 th< distances of the comet from the earth corresponding to the thi observed places, and hence determine the parallax in right ascensioi DETERMINATION OF AN ORBIT. 167 and in declination for each observation by means of the usual formulae. Thus, we have Ttp cos sin ?'] denote double the area of the triangle formed between the radii-vectores r, r f and the chord of the orbit between the corre- sponding places of the body, and similarly for the other triangles thus formed. The angle at the sun in this triangle is the difference between the corresponding arguments of the latitude, and we shall have [rr'~\ = rr f sin (u' u), [r/'J =rr"sm(M"tO, (2) If we designate by x } y } z } x f , y f , z', x", y", z" the heliocentric co- ordinates of the body at the times t, t 1 ', and t", we shall have x' = r sin a sin '(A -f- w), x' = r' sin a sin ( A -f- u'), x" = r" sin a sin (A + "), in which a and A are auxiliary constants which are functions of the elements & and i, and these elements may refer to any fundamental plane whatever. If we multiply the first of these equations by sin (u lf u r ) 9 the second by sin (u ff u), and the third by sin (u f u), and add the products, we find, after reduction, x x' x" - sin (u" u'} - t sin (u" u) -\ Tl sin (u r u) = 0, which, by introducing the values of [rr r ], [Vr"], and [V r"], becomes [r'r"~\ x [r/'] x' -f- [rr'~\ x" = 0. If we put we get =&?T tt " ;= [i^r * In precisely the same manner, we find 2-l'^n"l" = 0. < DETERMINATION OF AN ORBIT. 169 Since the coefficients in these equations are independent of the posi- tions of the co-ordinate planes, except that the origin is at the centre of the sun, it is evident that the three equations are identical, and express simply the condition that the plane of the orbit passes through the centre of the sun ; and the last two might have been derived from the first by writing successively y and z in place of x. Let A, A', A" be the three observed longitudes, /9, /?', ft" the corre- sponding latitudes, and J, //', A" the distances of the body from the earth ; and let J cos /? = ,,, J'cos^ = /o', A"cQ8p' = P ", which are called curtate distances. Then we shall have x = p cos A R cos Q , x' = p' cos A' R' cos 0', y = p sin A R sin , y' = p' sin A' R' sin ', z = p tan /5, z' = p' tan {?, jR"cos0", 12" sin 0", 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 ) (// cos A' R r cos ') + n"(p"cosl" 12" cos 0"), = n (p sin A E sin ) (// sin A' R' sin 0') (6) + n" (p" sin A" 12" sin 0"), = np tan /? p' tan p -f- ri'p" tan 0". 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, //, and p" and if the values of n and n" are first found, they will be sufficient to determine p, p f , and p". 170 THEORETICAL ASTRONOMY. 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, the effect of the errors of the observations. In special cases in which the conditions of the problem are such that when the ratio of two curtate distances is known, the distances themselves may be 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 ', and again in the second, from the equinox to the second place of the body, we must diminish the longitudes in these equations by the angle through which the axis of x has been moved, and we shall have = n(p cos(A 00 jRcos(' Q)) GO' cos (A' 00120 + n"(p" cos (A" GO R" cos(" 0), = n (p sin (A -f- R sin ( ' )) p' sin (A' '\ + ^" GO" sin (A"- -R' sin(0"- 0), (7) = n (p sin (A' A) -f- R sin ( /)) R' sin ( ' AO - n" (p" sin (A" AO R" sin ( " AO), == np tan ft p' tan jf -f- ri'p" tan /?". If we multiply the second of these equations by tan/9 7 , and the fourth by sin (A' 0, and add the products, we get = ri'p" (tan /?' sin (A" tan /5" sin (A' 0) n"E"sin(" Otan/5 r -f n/>(tan/5'sin(A tan/?sin(A' O; >' )tan/S / . Let us now denote double the area of the triangle formed by the sun and two places of the earth corresponding to E and E f by [RR'~ and we shall have \_RR] = ' sin ('), and similarly [RR" J = RR" sin ( " Q ), '] = R'R" sin(O" 00- ORBIT OF A HEAVENLY BODY. 171 Then, if we put " we obtain Substituting this in the equation (8), and dividing by the coefficient of p", the result is _ ^_ tan fi' sin (A - ') tan /? sin (A' ') ~ P W tan ft" sin (A' Q ') tan p sin (A" Q ') JL _j^\ _ Jgsin(' Q)tan/3' _ " ~ A^' /tan ft" sin (/' - ') tan/5' sin (A" 0')' Let us also put , _ tan ft sin (A ') tan /9 sin (/ 0') tan /5" sin (/ Q r ) tan p sin (A" ')' _ tan ft" sin (A' 0') tan /?' sin (A" ') ' and the preceding equation reduces to "R. (11) We may transform the values of M ' 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 /?'. (12) Let /? , /9 /r be the latitudes of the points of this circle corresponding to the longitudes A and A /r , and we have, also, tan/3 =sin(A ')tanw', ,_ Substituting these values for tan/9 7 , sin (A ; ) and sin(A r/ ') in the expressions for M ' and M", and reducing, they become M'= fln ' cos cos " sin (p> _ ft ") ' C o S ft cos /5 ' 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, ft = ft Q and ft"=ft Q ", we shall have M f = ^ and M rf =00. In this case, therefore, and also when /9 ft and ft" ft Q " are very nearly 0, we must have recourse to some other equation which may be derived from the equations (7), and which does not involve this indetermination. It will be observed, also, that if, at the time of the middle obser- vation, the comet is in opposition or conjunction with the sun, the values of M f and M" as given by equation (14) will be indeter- minate in form, but that the original equations (10) will give the values of these quantities provided that the apparent path of the comet is not in a great circle passing through the second place of the sun. These values are sin (A QQ _ sin(Q' Q) sm (A"_G')' sin (A"- GO ' Hence it appears that whenever the apparent path of the body is nearly in a plane passing through the place of the sun at the time of the middle observation, the errors of observation will have great influence in vitiating the resulting values of M f and Jf"; and to obviate the difficulties thus encountered, we obtain from the third of equations (7) the following value of p" : _ n sin (A' A) P "^'sinCA" A') ^sin(O A') ^7#'sm(G' A') +" sin(G" A') ^ sin (A" A') We may also eliminate p between the first and fourth of eqw tions (7). If we multiply the first by tan/9', and the second cos (A' G')i an d add tne products, we obtain = n"p" (tan p cos (A" 0') tan P" cos (A' 0')) ?i"E"tan,S'cos(O" G') +^(tan/5'cos(A 0') tan /5 cos (A' 0')] nR tan p cos (G' G) + R' tan p, from which we derive DEBIT OF A HEAVENLY BODY. 173 tan ff cos (A Q tan j3 cos (A' 0Q osO*' 0') tan /3' cos (/' 0') (16) tan /S" cos (/ 0') tan I? cos (A" 0') Let us now denote by I' the inclination to the ecliptic of a great circle passing through the second place of the comet and that point of the ecliptic whose longitude is 0' 90, which will therefore be the longitude of its ascending node, and we shall have cos (A' 0') tan I' = tan f ; (17) and, if we designate by /9, and $ the latitudes of the points of this circle corresponding to the longitudes A and A", we shall also have tan /5, = cos (A tan /', ~ g x tan ft, = cos (A" 0') tan/'. Introducing these values into equation (16), it reduces to _ n sin (/3, /?) cos $' cos /? ^ ~ p ^T ' sin 09" /?) ' cos /5 cos /?, (19) sin (/?" /?) from which it appears that this equation becomes indeterminate when the apparent path of the body is in a plane passing through that point of the ecliptic whose longitude is equal to the longitude of the second place of the sun diminished by 90. In this case we may use equation (11) provided that the path of the comet is not nearly in the ecliptic. When the comet, at the time of the second observation, is in quadrature with the sun, equation (19) becomes indeterminate in form, and we must have recourse to the original equation (16), which does not necessarily fail in this case. When both equations (11) and (16) are simultaneously nearly in- determinate, so as to be greatly affected by errors of observation, the relation between p and p" must be determined by means of equation (15), which fails only when the motion of the comet in longitude is very small. It will rarely happen that all three equations, (14), (15), and (16), are inapplicable, and when such a case does occur it will indicate that the data are not sufficient for the determination of the elements of the orbit. In general, equation (16) or (19) is to be used when the motion of the comet in latitude is considerable, and equation (15) when the motion in longitude is greater than in latitude. 174 THEORETICAL ASTRONOMY. 64. The formulae already derived are sufficient to determine the relation between //' and p when the values of n and n" are known, and it remains, therefore, to derive the expressions for these quan- tities. If we put k(t-f) = *', t') = r, (20) and express the values of x, y, z, x", y", z' f in terms of x f , y', z f by expansion into series, we have X = X '~~^'i; + 13'W'l?~T33''~d?'~W + &C '' x " = xr +^'J + ^'W'^ + l^'W'^ + &C '> and similar expressions for y, y", z, and z". We shall, however, take the plane of the orbit as the fundamental plane, in which case z, 2', and z" vanish. The fundamental equations for the motion of a heavenly body relative to the sun a*re, if we neglect its mass in comparison with that of the sun, If we differentiate the first of these equations, we get W = : ~r*~ '~dt~r' 3 ' ~di' Differentiating again, we find r' 5 Writing y instead of #, we shall have the expressions for -^ and d*ii' 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 j'i r "3 J r f /I 10 / /7 r ' \2 Q //V \ a __ i T I I_ i T f . _ _|_ __i_ | -_ __ _ir_( ar j _|_ ^ . CT r ) r" 4 . . . r" r" 3 r"* dr' a ff - ^ .1 T _|_ i . j_ _.i_ J Jl _ "*"" I "" j _i_ .. ^ . . ^ ' . | r 4 . . . we obtain From these equations we easily derive ) (23) he first members of these equations are double the areas of the iangles formed by the radii-vectores and the chords of the orbit tween the places of the comet or planet. Thus, x - x'y = [r/], y"x' - x"y' - [//'], y"x - x"y = [r/'J, (24) and x'dy' y f dx f is double the area described by the radius-vector X '^J _ y'ftx 1 during the element of time dt, and, consequently, : - ia 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 [r/] = bk t/>, [rV] = b"k i/p, [rr"] = (ab" + a"6) Substituting for a, 6, a /r , 6 /r their values from (22), we find, since 176 THEORETICAL ASTRONOMY. r" 2 r" 3 dr' \ -i^-ifes-* ..... )' From these equations the values of n = be derived ; and the results are [//'] p ^ LTT J (25) and n ff = [r/] -^ ^ L rr J may (26) 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, being multiplied by the very small quantity 7-, is reduced to a superior order, and, therefore, it may be neglected, so that in this case we shall have, to the same degree of approximation as in (26), (27) n [//'] From the equations (26) o*r from (25), since , = - TT. we find - - ( 1 - *" "\ n r \ + r" 3 dr^ kr" "dt dr' Since this equation involves r' and j-, it is evident that the value of , 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 rapidly, so that we may take n' CEBIT OF A HEAVENLY BODY. 177 and similarly *L -1 N" ~~ T"' Hence the equation (11) reduces to . (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) 7? T vanishes, and the supposition that , = , is correct to terms of the r f\j T third order. It will be advantageous, therefore, to select observa- tions whose intervals approach nearest to equality. But if the observations available do not admit of the selection of those which give nearly equal intervals, and these intervals are necessarily very unequal, it will be more accurate to assume n_ N^ n" ~ N'" and compute the values of N and N" by means of equations (9), since, according to (27) and (28), if r f does not differ much from _R', the error of this assumption will only involve terms of the third order, even when the values of r and r" differ very much. Whenever the values of p and p" can be found when that of their ratio is given, we may at once derive the corresponding values of r and r h ', as will be subsequently explained. The values of r and T" may also be expressed in terms of r' by means of series, and we have , M r" dy r" 2 r ^'- --* from which we derive T + T" dl> f' - rf - - ' . _ k dV neglecting terms of the third order. Therefore : ; (30) 12 178 THEORETICAL ASTRONOMY. 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 ff have been determined by a first approxi- mation, the approximate values of r f and -=- are obtained from these equations, by means of which the value of -^ may be recomputed ft from equation (28). We also compute _ N"~~ JR#Bin(0' 0)' and substitute in equation (11) the values of -77 and -^ thus found. If we designate by M the ratio of the curtate distances p and p", we have - . (33) 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 ^7 elements are already known, the value of -77 may be computed from n" ~ rr' sin (v f v) ' N and that of -^ from (32) ; and the value of M derived by means of these from (33) will not require any further correction. 65. When the apparent path of the body is such that the value of M', 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 Y y Y f , Y" be the ordinates of the sun when the axis of ORBIT OF A HEAVENLY BODY. 179 abscissas is directed to that point in the ecliptic whose longitude is A', and we have Y = R sin(Q A'), Y' = R r sin(O'--A'), r"=JB"sin(0" A'). Now, in the last term of equation (15), it will be sufficient to put n_ _N_ n"~ N r " and, introducing Y, Y f , Y", it becomes oosec (r ~ /} - (35) It now remains to find the value of 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'+N"Y"=0, and the foregoing expression reduces to since F' = E' sin(0' A r ). Hence the equation (15) becomes n sin (A- -A) ^ /I 1 sn- ~^^" sin(r-A' 5^C^- ;-^ sin A" A' ' ( 180 THEOKETICAL ASTKCXNOMY. If we put n sin (A' A) M " = ^" sin (jf'-xy ,., 1 l n" rJ_, . . ,,s sin (A' 0') R' [ 1 ' sin (A' A) i we have o" r Tl/T H/T 7? /'Q7^ = IrJL = -"*o-^ W ' / ^ Let us now consider the equation (16), and let us designate by X, X f , X" the abscissas of the earth, the axis of abscissas being directed to that point of the ecliptic for which the longitude is 0', then X =R cos (0-00, X' =R, X"=R" cos(" 0'). It will be sufficient, in the last term of (16), to put n_ _N_ n" ~~ N" ' and for ^ its value in terms of N tf as already found. Then, since this term reduces to 5 T" ' \ r 75 R' z I tan ft" cos (A r ') tan /S' cos (A" 0') ' and if we put n tan/3'cos (A 0') tan/? cos (A' 0') ~~n"' tan /3" cos (*/ Q') tan p' cos (X" 0')' (38) F , = 1 _i^l Hl rr j T ,,x/_L M tan/3^ Ef_ * n ' r" ^ ^ ; \r' 3 E' 3 /tan/3'cos(A 0') tan/3cos(A' 0') p ' the equation (16) becomes n (39) In the numerical application of these formulae, if the elements are not approximately known, we first assume n r W f= 7 7 when the intervals are nearly equal, and ORBIT OF A HEAVENLY BODY. 181 JL N n" "" 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 tf which are thus ob- tained, enable us to find an approximate value of r' ', and with this a 77 more exact value of ^ may be found, and also the value of F or F f . Tif 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'. 66. When the value of M has been found, we may proceed to determine, by means of other relations between p and p ff y the values of the quantities themselves. The co-ordinates of the first place of the earth referred to the third, are x, = R" cos Q" R cos O, (AK\ y, = .R"sin0" .RsinO. 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, = g sin G, and, consequently, .#"cos(O" O) R = gcos(G O), an "si n (O" O) = sin (00). 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) D cos 4/ = cos /? cos (A O), sin 4/ cos w cos ft sin (A Q), sin 4* sin w = sin p f 182 THEORETICAL ASTRONOMY. from which tan/9 tan w = -r tan (A O) (42) tan 4 = cosw Since cos/9 is always positive, cos^ and cos (A Q) must have the same sign; and, further, fy cannot exceed 180. In the same manner, if w" and oj/' represent analogous quantities for the time of the third observation, we obtain v= ta ^" sin (A" Q")' "-0") cos4"=cos/3"cos(A" "; We also have 2 2J.Rcos4, which may be transformed into r 2 = GO sec /9 R cos 4) 2 -f R* sin 2 4 ; (44) and in a similar manner we find r" 2 = (p" sec /9" JT cos 4/') 2 + ^' 2 sin 2 4". (45) Let K designate the chord of the orbit of the body between the first and third places, and we have x 2 - (*" - *) 2 + cos A" .#" cos O", f = Mp sin ^' R" sin O", from which we derive, introducing g and Gr, a?" x = Mp cos A" p cos A y cos G, y" y = Mp sin A" p sin A g sin (2, z" ~z = Mp tan/5" ^ tan /?. Let us now put ORBIT OF A HEAVENLY BODY. 183 Mp cos A" p cos A ph cos C cos J?, JKf/3 sin A" /> sin A = ph cos C sin 17", (46) J^f/9 tan /5" /o tan /? = ph sin C. Then we have x" x />/i cos C cos IT # cos 6r, y" y = ph cos C sin H g sin 6r, z" z= ph sin C. Squaring these values, and adding, we get, by reduction, x 2 = pW 2g ph cos cos ( G JET ) + g 2 ; (47) and if we put cos C cos ( G jH") = cos ?>, (48) we have x 2 = (/>& ^ cos ^) 2 -f- # 2 sin 2 /2 Hence, in the case that v ff v exceeds 180, it follows that x must be within the limits 30 and 45. The equation = = sin 3x 1/2 (r + is satisfied by the values 3# and 180 3x ; but when the first gives x less than 15, there can be but one solution, the value 180 3x being in this case excluded by the condition that 3x cannot exceed 135. When x is greater than 15, the required condition will be satisfied by 3x or by 180 3#, and there will be two solutions, corresponding respectively to the cases in which v" v is less than 180, and in which v ff v is greater than 180. Consequently, when it is not known whether the heliocentric motion during the interval t" t is greater or less than 180, and we find 3x greater than 45, the same data will be satisfied by these two different solutions. In practice, however, it is readily known which of the 188 THEOEETICAL ASTEONOMY. two solutions must be adopted, since, when the interval t" t is not very large, the heliocentric motion cannot exceed 180, unless the perihelion distance is very small ; and the known circumstances will generally show whether such an assumption is admissible. We shall now put aj 1 = -- ? (63) sin 3s = (64) v 8 and we obtain We have, also, sin ^/ = 1/2 sin x, and hence cos j/ = i/l 2 sin 2 x = I/ cos 1x. Therefore sin / =P= 2^ sin x V cos 2#, and, since K (r + r") sin p', we have x = 2^ (r -f- ^") sin a; y cos 2a;. If we put 3^^^^ (65) sm3a; the preceding equation reduces to * = - * (66) From equation (64) it appears that ^ must be within the limits ( and \ |/g. We may, therefore, construct a table which, with 37 a the argument, will give the corresponding value of /*, since, with given value of 37, 3# may be derived from equation (64), and thei the value of // from (65). Table XI. gives the values of /JL corre sponding to values of r] from 0.0 to 0.9. 69. In determining an orbit wholly unknown, it will be necessary to make some assumption in regard to the approximate distance oi the comet from the sun. In this case the interval t" t will gene rally be small, and, consequently, x will be small compared with r and r ff . As a first assumption we may take r = 1, or r -f- r" = 2, and fi. = 1, and then find K from the formula PAEABOLIC ORBIT. 189 "With this value of K we compute d, r, and r" by means of the equations (52). Having thus found approximate values of r and r", we compute y by means of (63), and with this value we enter Table XI. and take out the corresponding value of //. A second value for K is then found from (66), with which we recompute r and r", and proceed as before, until the values of these quantities remain un- changed. The final values will exactly satisfy the equation (56), and will enable us to complete the determination of the orbit. After three trials the value of r -f r" may be found very nearly correct from the numbers already derived. Thus, let y be the true value of log (r -f- r") 9 an( i ^ A # be the difference between any assumed or approximate value of y and the true value, or y = y + A 2/- Then if we denote by y Q r the value which results by direct calculation from the assumed value y Q) we shall have Expanding this function, we have But, since the equations (52) and (66) will be exactly satisfied when the true value of # is used, it follows that and hence, when &y is very small, so that we may neglect terms of the second order, we shall have Let us now denote three successive approximate values of log (r -f r") b 7 2/o> 2A/> 2/o"> and let then we shall have a = A (y Q y\ Eliminating A from these equations, we get y (a' a) = a'y ay ', trom which f ttft ^ ft ffi7^ 190 THEORETICAL ASTRONOMY. Unless the assumed values are considerably in error, the value of y or of log (r + r") thus found will be sufficiently exact ; but should it be still in error, we may, from the three values which approximate nearest to the truth, derive y with still greater accuracy. In the numerical application of this equation, a and a' may be expressed in units of the last decimal place of the logarithms employed. The solution of equation (56), to find t" t when K is known, is readily effected by means of Table VIII. Thus we have = sin 3#. 1/2 ( and, when ? r is less than 90, if we put si /v . i-T ' _ t sm / we get J = i 1/2 N sin / (r + r") f , (68) or When f r exceeds 90, we put N' = sin 3a?, and we have in which log $ i/% = 9.6733937. 'With the argument f we take from Table VIII. the corresponding value of N or N', and by means of these equations r' = k (t rr t) is at once derived. The inverse problem, in which T' 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 % compute d, r, and r". The value of f is then found from and the table gives the corresponding value of N or N f . A second approximation to x will be given by the equation 3 T> v vT PARABOLIC ORBIT. 191 or by 3 r'sin/ ^Ti'-^TF^ in which log = 0.3266063. Then we recompute d, r, and r", and proceed as before until u remains unchanged. The approxima- tions are facilitated by means of equation (67). It will be observed that d is computed from d = i/x' JL', and it should be known whether the positive or negative sign must be used. It is evident from the equation d = ph g cos , since p, h, and g are positive quantities, that so long as

" cos (A" 0") #', r" cos 6" sin (/" Q") = ?" sin (A" 0"), (72) r"sin6" = in which Z and Z" are the heliocentric longitudes and 6, 6" the corre- sponding heliocentric latitudes of the comet. From these equations we find T y r ff , I, I", 6, and b" ; and the values of r and r" thus found, should agree with the final values already obtained. When I" is less than I, the motion of the comet is retrograde, or, rather, when the motion is such that the heliocentric longitude is diminishing instead of increasing. From the equations (82) 1? we have tan i sin (I & ) = tan 6, /Q\ tanisin(r ft)=-" which may be written tani(sin(Z x ) cos (a; &) + sin (a ) cos(Z a;)) = tan 5, tan i (sin (I" x) cos (re & ) + sin -*&) cos (Z ;/ )) = tan b". Multiplying the first of these equations by sin(Z" a;), and the second by sin( x\ and adding the products, we get tan i sin (x & ) sin (t' 1) = tan 6 sin (r x) tan 6" sin (I x) ; and in a similar manner we find tan i cos (x Q ) sin (I" l)= tan 6" cos (J a) tan 6 cos (r a). Now, since x is entirely arbitrary, we may put it equal to I, and we have PARABOLIC ORBIT. 193 tan i sin (I ft ) = tan b, tan b" tan 6 cos (I" I) (74) tan i cos (7 ft ) = sin(r 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" -}- I), whence I" x=\(l ff Z) and I x=$(l I"); and we obtain tan i sin ( (I" -j- /) ft ) = =h ~ , Sm lff , ,, ^, 2 cos b cos b" cos (I I) .__, / 7 a } N V ' Oy tan i cos (J- (Z" + ft ) = r , 7 \,, ~ ,,-, TV 2 cos b cos 6 sm | (I" I) These equations may also be derived directly from (73) by addition and subtraction. Thus we have tan i (sin (I" & ) + sin (J ft )) = tan 6" -f tan 6, tan t (sin (Z" ft) sin(Z ft)) = tan &" tan 6; and, since sin(r- ft) -f sin (/- ft) = 2 siny"+ ^- 2ft) cosi(^- 0, siiiC/' ft) sin (I ft) = 2cos(J"+ ^ 2Q) sin J(^~ 0, these become i (tan 6" + tan 6) = ^ ? . which may be readily transformed into (75). How r ever, since 6 and 6" will be found by means of their tangents in the numerical appli- cation of equations (71) and (72), if addition and subtraction loga- rithms are used, the equations last derived will be more convenient than in the form (75). As soon as ft and i have been computed from the preceding equa- tions, we have, for the determination of the arguments of the latitude u and u", cos ^ cos i Now we have u = v -f- w > in which to = TT ft in the case of direct motion, and co = ft TT 13 194 THEORETICAL ASTRONOMY. when the distinction of retrograde motion is adopted; and we shall have U "U = V " V} and, consequently, x 2 = r 2 -j- r" 2 2rr" cos (u" u), (78) x 2 = (r" r cos (t/' ))"+ r 2 sin 2 (it" w). (79) The value of K derived from this equation should agree with that already found from (66). We have, further, r = q sec 2 (u to), r" = q sec 2 (u" w), or = COS l(u >) T^> 7= COS 4 (lt' r to) = Vq Vr Vq Vi" By addition and subtraction, we get, from these equations, - (cos 30*" ) + cos J( )) 7=- (COS J K o) COS J (l* o)) = -/== -/=, y q Vr Vr from which we easily derive -?=- cos J (J (" +)-) cos i (" - f.) = -7= + 4^' v ^ sin But 1 1 l / t (7 r _ /T"\ ^T^v^ V- :: \-' and if we put tan (45 if? 7 " since ^| will not diifer much from 1, 0' will be a small angle; and we shall have, since tan (45 + 6'} cot (45 + 6') = 2 tan 20', V q cos | (w w) V r/' from which the values of q and o> may be found. Then we shall have, for the longitude of the perihelion = + , when the motion is direct, and when i unrestricted exceeds 90 and the distinction of retrograde motion is adopted. It remains now to find T, the time of perihelion passage. We have V = U - >, tf'=u" - U>. With the resulting values of v and v" we may find, by means of Table VI., the corresponding values of M (which must be distin- guished from the symbol M already used to denote the ratio of the curtate distances), and if these values are designated by M and M " y we shall have t-T=~, r-r= m ' m ' or m m C in which m = f , and log C = 9.9601277. When v is negative, the 9* corresponding value of M is negative. The agreement between the two values of T will be a final proof of the accuracy of the numerical calculation. The value of T when the true anomaly is small, is most readily and accurately found by means of Table VIII., from which we derive the two values of ^V and compute the corresponding values of T from the equation 2 T=t TN 2 in which logjr, = 1.5883273. When v is greater than 90, we de- 196 THEORETICAL ASTRONOMY. rive the values of N f from the table, and compute the corresponding values of T from 71. The elements q and T may be derived directly from the values of r, r", and x, as derived from the equations (52), without first finding the position of the plane of the orbit and the position of the orbit in its own plane. Thus, the equations (80), replacing u and u" by their values v + CD and v -{- CD", become A sin J 0" + v) sini (I/'-*) =4= -- 1,, Vq Vr Vr" 2 11 tM) -7= cosi (i," + v ; cos| (" - 1;) - 4= + -7=-- Vq Vr Vr" Adding together the squares of these, and reducing, we get 1 q ~ sin 2 J (v" v) or _ ~r" + r 2l/^ 77 cos^(v' / vj Combining this equation with (59), the result is _rr"sm^(v' f y) V- r + /' xcoty" and hence, since X (r + r") sin^', 5 = ~ sin 2 1 (v" v) cot y . We have, further, from (78), x 2 = (/' r ) 2 + 4rr" sin 2 J (v' 7 v ) from which, putting r" r smv= =^-> we derive 2l/rr"~ (85) Therefore, the equation (83) becomes PARABOLIC ORBIT. 197 g^Kr + r'Ocos'-i/cos'v, (86) by means of which q is derived directly from r, r", and x y the value of v being found by means of the formula (84), so that cos v is positive. When f f cannot be found with sufficient accuracy from the equa- tion we may use another form. Thus, we have r + /' + x r + r" x i + *r = r + / , l-sm/, ; + / , , which give, by division, tan (45 + i r ') *= J r + r " + * (87) x r -f r" x In a similar manner, we derive tan (45 + |v) = x -f- ^r ty. (88) * x (/' 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 tan i (v" -f v) tan j (v" v) = XL . L// If we put tan/ = Apl', (90) this equation reduces to tan i (v" +v)=* tan (/ 45) cot | (v" v), (91) and the equations (81) give, also, tan I (v" -f v) = cot | (t 1 " v) sin 2^, either of which may be used to find v ff + v. 198 THEORETICAL ASTRONOMY. From the equations cos |v _ 1 cos %v" __ 1 V~q ~~Vr Vq V7 r ' by multiplying the first by sinjv" and tne second by sin Jv, add- ing the products and reducing, we easily find sin \ (v ff v) sin v cos \ (v" v) _ 1 Hence we have = sin \v 1 1 = COS %V = ;=, Vq Vr which may be used to compute q, v, and v" when v" v is known. When \ (v" v) and \ (v ff + 0), and hence v" and v, have been determined, the time of perihelion passage must be found, as already explained, by means of Table VI. or Table VIII. It is evident, therefore, that in the determination of an orbit, as soon as the numerical values of r, r" , and x have been derived from the equations (52), instead of completing the calculation of the 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). W^hen the final value of If has been found, the determination of the elements is completed by means of the formulae already given. PARABOLIC CEBIT. 199 72. EXAMPLE. To illustrate the application of the formulae for the calculation of the parabolic elements of the orbit of a comet by a numerical example, let us take the following observations of the Fifth Comet of 1863, made at Ann Arbor: Ann Arbor M. T. a 6 1864 Jan. 10 6* 57 m 20'.5 19* 14" 4 8 .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. 13 1864 Jan. 10 7 h 24 3' 297 53' 7".6 -}- 55 46' 58".4, 13 6 38 37 302 57 51 .3 57 39 35 .9, 16 7 1 54 310 31 52 .3 + 59 38 18 .7. Next, we reduce these places by applying the corrections for pre- cession and nutation to the mean equinox of 1864.0, and reduce the times of observation to decimals of a day, and we have t = 10.30837, I = 297 52' 51".l, /? = + 55 46' 58".4, t' = 13.27682, A' = 302 57 34 .4, p = 57 39 35 .9, " = 16.29299, A" = 310 31 35.0, /5"=-j-59 38 18.7. For the same times we find, from the American Nautical Almanac, Q =290 6' 27".4, log.R =9.992763, O' 293 7 57 .1, logE f =9.992830, Q" = 296 1215.7, log #' = 9.992916, which are referred to the mean equinox of 1864.0. It will gene- rally be sufficient, in a first approximation, to use logarithms of five decimals ; but, in order to exhibit the calculation in a more complete form, we shall retain six places of decimals. Since the intervals are very nearly equal, we may assume 200 THEORETICAL ASTRONOMY. JL L *L ri~'~~r"~ N"' Then we have lf_t' t tan i? sin (A Q ') tan ft sin (;/ Q ') ~ t' t' tan/S"sin(A' ') tan p sin (A" ')' and g sin ( G 0) = R" sin (" 0), ^ cos( Q) = R" cos(Q" O) R; h cos C cos (IT A") = Jf cos (A" A), h cos C sin (# A") == sin (A" A), h sin C = M tan/5" tan /5; from which to find Jf, G, g, H, , and h. Thus we obtain log M= 9.829827, #= 94 24' 1".8, == 22 58' 1".7, C = 40 28 21 .9, log^ = 9.019613, log h = 9.688532. Since r = J!f - 777 = 0.752, it appears that the comet, at the time A cos /9" of these observations, was rapidly approaching the earth. The quadrants in which G O and H A" 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 A r and $' are computed from the resulting elements. Next, from i cos * = cos /? cos (A O), cos *" = cos ft" cos (A" "), cos

, r", and x, that of /. being found from Table XI. with the argument y. First, we assume log x = log r'i/2 = 9.163132, and with this we obtain log r = 9.913895, log r" = 9.938040, log (r -f r") =*= 0.227165. This value of log(V + r"} gives ^ 0.094, and from Table XI. we find log/Jt = 0.000160. Hence we derive log x = 9.200220, log r == 9.912097, log r" = 9.935187, log (r + r") = 0.224825. Repeating the operation, using the last value of log(r + r"), we get log x = 9.201396, log r = 9.912083, log r" = 9.935117, log (r + /') = 0.224783. The correct value of log(r -}- r r/ ) may now be found by means of the equation (67). Thus, we have, in units of the sixth decimal place of the logarithms, a = 224825 227165 == 2340, a' = 224783 224825 = 42, and the correction to the last value of log(r + r"} becomes a a Therefore, log (r + r") = 0.224782, and, recomputing ?y, //, >c, r, and r", we get, finally, log x = 9.201419, logr = 9.912083, logr" = 9.935116, log (r -f r") = 0.224782. The agreement of the last value of log(r + "") with the preceding one shows that the results are correct. Further, it appears from the 202 THEORETICAL ASTRONOMY. values of r and r" that the comet had passed its perihelion and was receding from the sun. By means of the values of r and r" we might compute approxi- dr' mate values of r f and -rr from the equations (30) and (31), and then n N a more approximate value of 7l from (28), that of -^ being found /I/ -i-V from (32). But, since r f differs but little from R f y the difference 77 ?V between ^ and -^77 is very small, so that it is not necessary to con- YL -A.V sider the second term of the second member of the equation (33); and, since the intervals are very nearly equal, the error of the as- sumption n r is of the third order. It should be observed, however, that an error in the value of M affects H, , 7i, and hence also A 9 b, b", c, and c /r , and the resulting value of p may be affected by an error which con- siderably exceeds that of M. It is advantageous, therefore, to select observations which furnish intervals as nearly equal as possible in order that the error of M may be small, otherwise it may become necessary to correct M and to repeat the calculation of r, r r/ , and x. We may also compute the perihelion distance and the time of peri- helion passage from r, r", and K by means of the equations (86), (89), and (91) in connection with Tables VI. and VIII. Then r' and v f may be computed directly, and the complete expression for M may be employed. In the first determination of the elements, and especially when the corrections for parallax and aberration have been neglected, it is un- necessary to attempt to arrive at the limit of accuracy attainable, since, when approximate elements have been found, the observations may be more conveniently reduced, and those which include a longer interval may be used in a more complete calculation. Hence, as soon as r, r", and K have been found, the curtate distances are next deter- mined, and then the elements of the orbit. To find p and p", we have d = + 0.122395, the positive sign being used since x is greater than g, and the formulae _ d + gcosy P- ^ -- , 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, 6 = + 33 1' 10".6, log r = 9.912082, r=112 31 9.9, 6"= + 23 55 5.8, log r" = 9.935116. The agreement of these values of r and r rr with those previously found, checks the accuracy of the calculation. Further, since the heliocentric longitudes are increasing, the motion is direct. The longitude of the ascending node and the inclination of the orbit may now be found by means of the equations (74), (75), or (76); and we get & = 304 43' 11".5, i = 64 31' 21".7. The values of u and u" are given by the formulae tan = 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 2 = (r" r cos (u" u))' 2 -f r 2 sin 2 (u" u) gives log x = 9.201423, and the agreement of this value of x with that previously found, proves the calculation of &, i, u, and u" '. From the equations tan (45 -j- 0') = , , , , f ,,. . ,=- sin -i (i (u r> -f u) w) = V ^ ( i (u" + u) w) = tan 20' q = Vq we get d' = 22' 47".4, w = 115 40' 6".3, log q 9.887378. Hence we have TT = u> -f- ^ = 60 23' 17".8, 204 THEOKETICAL ASTEONOMY. and v = u w = 27 12' 6".l, v" = u"w = 37 38' 43".l. Then we obtain log m = 9.9601277 f log q = 0.129061, and, corresponding to the values of v and v", Table VI. gives log M = 1.267163, log M" = 1.424152. Therefore, for the time of perihelion passage, we have T=t = t 13.74364, m and T = t" =f 19.72836. m The first value gives T= 1863 Dec. 27.56473, and the second gives T= Dec. 27.56463. The agreement between these results is the final proof of the calculation of the elements from the adopted value of M= p -. p If we find T by means of Table VIII., we have log N = 0.021616, log N" = 0.018210, and the equation T= t 3 Nr* sin v = t" g JVV't 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) ^ o qru 43 1 1 K V ^ cll P tlc an d Mean f=6431 3 217/ E ^x 1864 .0, 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" ', and correct the value of M, the elements which will then be obtained will approximate nearer the true values ; and each successive correction will furnish more accurate results. When the adopted value of M is exact, the result- ing elements must by calculation reproduce this value, and since the computed values of A, A", /9, and ft" will be the same as the observed values, the computed values of X f and /9' must be such that when substituted in the equation for M, the same result will be obtained as when the observed values of A' 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 r , 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 M t and when the exact value of M has been used in deriving the elements, the computed values of X and /9 r 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 r , 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 4/ cos ft cos (A' 0'), sin 4/ cos w' = cos ft sin (A' 0'), (93) sin 4' sin w' = sin ft, give, by differentiation, %i cos ft dX = cos w' sec ft c?4/> dft = sin w' cos (A' Q') d*'. From these we get tan (A' 0') dp sin ft 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 tan /5' /ncr\ tan w = YTr rVY (&&) 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 v' = 32 31' 13".5, u' = 148 11' 19".8, log / = 9.922836. Then from the equations tan (l r & ) = cos i tan u', tan V = tan i sin (V & ), we derive I = 109 46' 48".3, V == 28 24' 56".0. By means of these and the values of O' and R f , we obtain A' = 302 57' 41".l, p = 57 39' 37".0 ; and, comparing these results with the observed values of X f and /?', the residuals for the middle place are found to be Comp. Obs. cos p AA' == -f 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 r given by observation is log tan w' = 0.966314, and that given by the computed values of X' and /9' is log tan w' = 0.966247. The difference being greater than what can be attributed to errors of calculation, it appears that the value of M requires further cor- NUMERICAL EXAMPLES. 207 rection. Since the difference is small, we may derive the correct value of M by using the same assumed value of -,, and, instead of Ti the value of tan w f derived from observation, a value differing as much from this in a contrary direction as the computed value differs. Thus, in the present example, the computed value of log tan w f is 0.000067 less than the observed value, and, in finding the new value of M 9 we must use log tan w' = 0.966381 in computing /9 and /9 " involved in the first of equations (14). If the first of equations (10) is employed, we must use, instead of tan/3' as derived from observation, tan {? = tan w' sin (A' Q')> or log tan p = 0.966381 + log sin (A' 0') = 0.198559, the observed value of X' being retained. Thus we derive log M= 9.829586, and if the elements of the orbit are computed by means of this value, they will represent the middle place in accordance with the condition that the difference between the computed and the observed value of tan w f 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, a o. cos p A;/ = V 26".2, A/3' = 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 ^=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 -jt = 0.006831, Yl J\ and from (10) we get log M' = 9.822906, log M" = 9,663729 n . 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 ^ = ^' = ^ 74. When the approximate elements of the orbit of a comet are known, they may be corrected by using observations which include a longer interval of time. The most convenient method of effecting this correction is by the variation of the geocentric distance for the time of one of the extreme observations, and the formula? which may be derived for this purpose are applicable, without modification, to any case in which it is possible to determine the elements of the orbit of a comet on the supposition of motion in a parabola. Since there are only five elements to be determined in the case of parabolic motion, if the distance of the comet from the earth corresponding to the time of one complete observation is known, one additional 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 J 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 Z>, D', D" the declinations of the sun for the times , 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" RcosD sin A, z=R"smD" -RsmD. If we represent by g the chord of the earth's orbit between the places for the first and third observations, and by G and K, respectively, the right ascension and declination of the first place of the earth as seen from the third, we shall have x = g cos K cos G, y = gcosK 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 D" sin ( J." A\ (96) # sin K = R" sin D" R sin Z), from which g, K, and 6r may be found. If we designate by x n y n z, the co-ordinates of the first place of the comet referred to the third place of the earth, we shall have x, = A cos d cos a -J- <7 cos K cos Oy y f = A cos d sin a -f- g cos _T sin 6r, z, = A sin d -\- g sin .fiT. Let us now put x, = Ji' cos ' cos if', y f A' cos C' sin if', 2, = /*/ sin C', and we get A' cos:' cos (If' #) = J cos<5cos( 6?) + $rcos.K; A' cos C' sin ( JT (?) = J cos <5 sin (a ), (97) A' sin C' = ^ sin d -j- ^ sin jfiT, from which to determine H f , f , and h f . If we represent by is the angle at the earth between the sun and comet at the time t, and i//' the same angle at the time t". To find their values, we have cos ^ = cos D cos 8 cos (a A) -f- sin D sin 5, cos 4,"= cos Z>" cos <*" cos (a" 4") + sin D" 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 r sin f the declination of the zenith at the time t Q . Let 1 Q and b Q denote these quantities converted into longitude and latitude, or the longitude and latitude of the geocentric zenith at the time t . The rectangular co-ordinates of the place of observation referred to the centre of the 222 THEORETICAL ASTRONOMY. earth and expressed in parts of the mean distance of the earth from the sun as the unit, will be X Q = p sin - cos b cos l w y = /> sin TT O cos b Q sin 1 , z = PQ sin TT O sin b Q , in which n Q = 8".57116. Let J be the distance of the planet or comet from the true place of the observer, and J, its distance from the point in the ecliptic to which the observation is to be reduced. Then will the co-ordinates of the place of observation, referred to this point in the ecliptic, be s, = (J, J ) cos /5 cos A, y, = (A f J ) cos /? sin A, s,= (J, J ) sin/*, the axis of x being directed to the vernal equinox. Let us now designate by O the longitude of the sun as seen from the point of reference in the ecliptic, and by R its distance from this point. Then will the heliocentric co-ordinates of this point be X= It cos Q, The heliocentric co-ordinates of the centre of the earth are X Q = E Q cos -T cos Q , But the heliocentric co-ordinates of the true place of observation will be X+x n Y+y n Z+z,, or X Q + x m F + y , Z Q + Z , and, consequently, we shall have R cos O ( A, J ) cos /5 cos A = R Q cos S cos O /o sin ^o cos b cos 1 , It sin O ( J, J ) cos /3 sin A = R Q cos 2 Q sin O P sin TT O cos b sin 4, - ( J , ^ ) sin /3 = ^ sin 2' - Po sin TT O sin b . If we suppose the axis of x to be directed to the point whose longi- tude is O , these become DETERMINATION OF AN ORBIT. 223 J? cos (Q Q ) ( J, J ) cos cos (A Q ) = R cos Z 9 f> sin - cos b Q cos (7 Q ), E sin (O o) (4 4) cos /9 sin (A Q ) == (2) ^ sin TT O cos 6 sin (J O )> (4 4>) sin = ^o sin 2 o Po sin ^o sin * > from which R and O may be determined. Let us now put D; (3) then, since TT O , Jf , and O O are small, these equations may be reduced to R = D cos (A O ) ^oft cos 6 cos (7 O ) + -R > R (O GO) = D sin (A O ) T O JO cos 6 sin (7 Q 8 ), = D tan /S - TT O Po sin 6 + .R 2 V Hence we shall have, if TT O and 2" are expressed in seconds of arc, _ 000 206264.8 p p _i_ n nna r; /^ > K Q p Q w$b Q cQ$(l Q Q . , , = ^ + Z>< (A-Q )- 206264.8 -- ' , 206264.8 D sin (A Q ) TT O ,0, cos b sin (^ Q ) W W T " T> > from which we may derive the values of Q and R which are to be used throughout the calculation of the elements as the longitude and distance of the sun, instead of the corresponding places referred to the centre of the earth. The point of reference being in the plane of the ecliptic, the latitude of the sun as seen from this point is zero, which simplifies some of the equations of the problem, since, if the observations had been reduced to the centre of the earth, the sun's latitude would be retained. We may remark that the body would not be seen, at the instant of observation, from the point of reference in the direction actually observed, but at a time different from , to be determined by the interval which is required for the light to pass over the distance 4 J . Consequently we ought to add to the time of observation the quantity ( J, J ) 497'.78 = 497'.78 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 A derived from equations (1) and the longitude O 224 THEOKETICAL ASTEONOMY. derived from (4) should be reduced by applying the correction for nutation to the mean equinox of the date, and then both these and the latitude /9 should be reduced by applying the correction for pre- cession to the ecliptic and mean equinox of a fixed epoch, for which the beginning of the year is usually chosen. In this way each observed apparent longitude and latitude is to be corrected for the aberration of the fixed stars, and the corresponding places of the sun, referred to the point in which the line drawn from the body through the place of observation on the earth's surface in- tersects the plane of the ecliptic, are derived from the equations (4). Then the places of the sun and of the planet or comet are reduced to the ecliptic and mean equinox of a fixed date, and the results thus obtained, together with the times of observation, furnish the data for the determination of the elements of the orbit. When the distance of the body corresponding to each of the observations shall have been determined, the times of observation may be corrected for the time of aberration. This correction is necessary, since the adopted places of the body are the true places for the instant when the light was emitted, corresponding respectively to the times of observation diminished by the time of aberration, but as seen from the places of the earth at the actual times of observation, respectively. When [3 = 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 formulae acquire greater simplicity when the sun's latitude is not introduced, if, in this case, we reduce the geocentric places to the DETERMINATION OF AN OKBIT. 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 V f . Then the longitude of the planet or comet as seen from this point in the ecliptic will be the same as seen from the centre of the earth, and if J, is the distance of the body from this point of reference, and /9, its latitude as seen from this point, we shall have A t cos /?, = A cos ft, J, sin ft, = A sin ft R sin 2 Q , from which we easily derive the correction /?, /9, or A/9, to be applied to the geocentric latitude. Thus, we find (6) Q being expressed in seconds. This correction having been applied to the geocentric latitude, the latitude of the sun becomes 2=0. The correction to be applied to the time of observation (already diminished by the time of aberration) due to the distance J, J will be absolutely insensible, its maximum value not exceeding O s .002. It should be remarked also that before applying the equa- tion (6), the latitude I Q should be reduced to the fixed ecliptic which it is desired to adopt for the definition of the elements which deter- mine the position of the plane of the orbit. 78. When these preliminary corrections have been applied to the data, we are prepared to proceed with the calculation of the elements of the orbit, the necessary formulae for which we shall now investi- gate. For this purpose, let us resume the equations (6) 3 ; and, if we multiply the first of these equations by tan /9 sin A" tan ft" sin A, the second by tan/3" cos A tan/9 cos A", and the third by sin (X A"), and add the products, we shall have = nR (tan ft" sin (X 0) tan ft sin (A" 0)) - p' (tan ft sin (A" A') tan ft' sin (A" A) -f tan ft" sin (A' A)) - R' (tan ft" sin (A ') tan ft sin (A" Q')) -f ri'R" (tan ft" sin (A 0") tan ft 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 I the inclination to the ecliptic, of a great circle passing through the first and third observed places of the body, and we have tan p = sin (A K) tan J, tan 0" = sin (A" JT)tanJ. Introducing these values of tan ft and tan ft" into the equation (7), since sin (A O) sin (A" K) sin (A" Q) sin (A K) = -sin (A" A)sin(O K\ sin (A' X) sin (A" K) + sin (X" A') sin (A JT) = + sin (A" A) sin (X K) 9 sin (A O') sin (A" K) sin (A" Q') sin (A JT) = - sin (A" A) sin (' 7T), sin (A O") sin (A" K) sin (A" Q") sin (A K) = sin (A" A) sin (Q" K\ we obtain, by dividing through by sin (X" X) tan 7, = nR sin (Q K} -f P ' (sin (A' K) tan p cot J) - R sin (Q' JT) + n"R" sin (O" #> Let ft denote the latitude of that point of the great circle passing through the first and third places which corresponds to the longitude A', then tan ft = sin (A' JT) tan _Z, and the coefficient of p' in equation (9) becomes sin (ft /?) cos ft cos p' tan/ Therefore, if we put sin (f- ft) a '-^tanT' we shall have , a n 1 ^ This formula will give the value of p', or of A', when the values of n and ?i" 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, tan/sin ( K r + A) - g) = ^ +^, sec -| (/'-.), tan 7cos (i (A" + A) - JT) = ~ cosec J (A" - A). We may also compute K and / 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 0" tan/9", the second by cos 0" tan/3", and the third by sin(/" 0"), and add the products, we get Q=n P (tan ft' sin (" X) tan /? sin (0" A")) wtan/8"sin (0" 0) p' (tan &' sin (O" A') tan p sin (0" *"))+# tan/5" sin (Q" 00, (13) which may be written 0=n^(tan/9sin(r 0") tan/3" sin (A 0")) wJStan"sin(0" 0) + P f (tan /5" sin (A' ") tan ft sin (A" ")) -j/Oan/S' tan ft) sin (A" 0") + #tan,9"sin(0" 0'). Introducing into this the values of tan /9, tan /9", and tan /9 in terms of I and K, and reducing, the result is == npsm (A" A) sin ( Q " K) nE sin ( " - ) sin (A" IT) - /o' sin (A" A') sin ( Q " K) p'a, sec jf sin (A" "} + E' sin (0 " ') sin (A" K). Therefore we obtain ^/ s jn(A" A') q sec/S' sin(^ ; Q \ 10 n \ sin (A" A) + sin (A" A) " sin (Q" JT) / sin (A" K) E'sm("O f )nJRsin(O"Q) n sin (A" A) sin (Q" K) But, by means of the equations (9) 3 , we derive jR'sin(O" 0') wJRsin(0" 0) = (N n*) K S m(Q" 0), 228 THEORETICAL ASTRONOMY. and the preceding equation reduces to _//sin(r AQ a^ecF sin(A" - Q") \ P ~~ n \ sin (A" A) + sin (A" A) ' sin (0" K) } ( . ., / N\ R sin (0" 0) sin (A" K) +V ~n) sin (A" A) sin (O 7 ' K) To obtain an expression for p" in terms of p f , if we multiply the first of equations (6) 3 by sin tan /?, the second by cos tan /9, and the third by sin (^ O), and add the products, we shall have 0=71 Y' (tan/? sin (A" ) tan /5" sin (A )) n"#"tan/5sin(" ) P 1 (tan /5 sin (A' ) tan/S'sin (A 0))-j-jR'tan /5sin (O' O). (15) Introducing the values of tan /9, tan /9 ; , and tan /9" in terms of K and /, and reducing precisely as in the case of the formula already found for p, we obtain _ p' I sin (A' A) a sec ft sin (A Q ) \ P ~ ^IsinCA" A) ~~sin(A" A) ' sin(O K) } / N"\R'sm(&' Q)sin(A f \ n" ) sin (A" A)sin(0 K} Let us now put, for brevity, ?t _ K) _ o - c d K'amW K) sec^ KB" sm(Q" Q) a ^ ~~ sin (A" A)' a sin (A" A) Q^) , _ sin (A' -A) _ .RsinCA Q) 1 "~sm(A /; A) J b 7i sin (A" JT) A sin (A and the equations (11), (14), and (16) become p' sec /5' = c + w6 + ri'd, (18) n If n and TI" are known, these equations will, in most cases, be sufficient to determine />, p r , and p". DETERMINATION OF AN ORBIT. 229 80. It will be apparent, from a consideration of the equations which have been derived for p, p', and p ff , 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 place, we have a = 0, and equation (11) reduces to n"R" sin (0" K) + nR sin ( J5Q = K sin (0' K). If the ratio of n" to n is known, this equation will determine the quantities themselves, and from these the radius-vector r f for the middle place may be found. But if the great circle which thus passes through the three observed places passes also through the second place of the sun, we shall have K ', or K= 180 -f O', and hence n"R" sin(O" 00 nR sin(' ) = 0, or tf_ __ Rsm(Q r 0) n ~~S" sin (" ')' from which it appears that the solution of the problem is in this case impossible. If the first and third observed places coincide, we have A = X" and /9 /9", and each term of equation (7) reduces to zero, so that the problem becomes absolutely indeterminate. Consequently, if the data are nearly such as to render the solution impossible, according to the conditions of these two cases of indetermination, the elements which may be derived will be greatly affected by errors of observa- tion. If, however, X is equal to X" and /9 X/ differs from /9, it will be possible to derive />', and hence p and p" ; but the formulae which have been given require some modification in this particular case. Thus, when A = >*", we have K=X' = i, 1= 90, and /? =90, and hence , 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/?', gives a Q = sin p cot I cos ,5' sin (A' K\ (19) and when X A r/ , it becomes simply a = cos j? sin (A' K). 230 THEORETICAL ASTRONOMY. Whenever, therefore, the difference X" A is very small compared with the motion in latitude, a should be computed by means of the equation (19) or by means of the expression which is obtained directly from the coefficient of p f in equation (7). When ; = A ;/ = JBT, the values of M lt If/', M 2 , and M 2 " cannot be found by means of the equations (17); but if we use the original form of the expressions for p and p" in terms of p 1 , as given by equations (13) and (15), without introducing the auxiliary angles, we shall have _/ tan jf sin (X' Q") tan ft" sin (A' Q") P ~ n ' tan ft sin (A" 0") tan p' sin (A 0") fL tan ft sin (X 1 0) tan /5' sin (A 0) 10 n" ' tan/? sin (A" 0) tan ft" sin (A 0) ' / __N_\ _ jRtan/5"sin(0" 0) _ M " n } tan ft sin (A" 0") tan ft" sin (A - 0")' (X 1 0) A" 0) / J\T"\ f " \ n" / tan /? sin (A" ) tan p' sin (A )' Hence _ tan p sin (A" 0") tan ft" sin (A' 0") 1 ~~ tan ft sin (A" 0") tan p" sin (A ") ' ,, _ tan ft sin (A' 0) tan ft' sin (A 0) 1 : " tan ft sin (A" 0) tan ft" sin (A ) ' .Rtan/S"sin(0" ) /Ll - .-, __ 2 tan ft sin (A" - - ") tan ft" sin (A ")' M _ _ J?"tan/3sm(0"--0) __ a " " tan ft sin (A" ) tan ft" sin (A ) ' are the expressions for M 19 M^', M 2 , and M 2 " which must be used when A = A" or when A is very nearly equal to A /r ; and then p and p tf will be obtained from equations (18). These expressions will also be used when A /r ^ = 180, this being an analogous case. When the great circle passing through the first and third observed places of the body also passes through the first or the third place of the sun, the last two of the equations (18) become indeterminate, and other formula? must be derived. If we multiply the second of equa- tions (7) 3 by tan/9" and the fourth by sin(A' r 0'), and add the products, then multiply the second of these equations by tan /9 and the fourth by sin (A r ), and add, and finally reduce by means of the relation NR sin (' ) = N"R" sin (0" '), we get DETERMINATION OF AN ORBIT. 231 = P L tan ft ff sin (A' Q ') tan ft' sin (A" 0') P ~ n ' tan ft" sin (A Q') tan ft sin (A" ') " ^"\ R" tan 0" sin (0" 0') - \ " N") tan/3" sin (A 0') tan ft sin (A" 0')' '-O) ,,__i^_ tan /5' sin (A Q') tan ft sin (/' Q') , . , n" ' tan /S" sin (A tan /3 sin (A" O') tan/5" sin (A ') tan/3 sin (A" ') These equations are convenient for determining p and //' from p' ; but they become indeterminate when the great circle passing through the extreme places of the body also passes through the second place of the sun. Therefore they will generally be inapplicable for the cases in which the equations (18) fail. If we eliminate p" from the first and second of the equations (6) 3 we get = np sin (A" A) nE sin (A" Q) p 1 sin (A" A') + B sin (A" OO ri'R 1 sin (A" 0"), from which we derive _./ sin(A"-AO p -n' sin (A" -A) nR sin (A" Q) R sin (A" - .0') -f n"R" sin (A" Q ") n sin (A" A) Eliminating p between the same equations, the result is / sin (/ -A) P ~ ?7' "sin (A" -A) , nE sin (A 0) R sin (A 0') -f n"E" sin (A Q") n" sin (A" A) These formulae will enable us to determine p and p" from p f in the special cases in which the equations (18) and (21) are inapplicable; but, since they do not involve the third of equations (6) 3 , they are not so well adapted to a complete solution of the problem as the formulae previously given whenever these may be applied. If we eliminate successively p" and p between the first and fourth of the equations (7) 3 , we get tan /?" cos (A' Q ') tan ? cos (A" Q Q = _ p ~ _ n tan ft" cos (A 0') tan ft cos (A" 0') ten^ ^cos(0' 0) # + n"jR"cos(0" 0') n tan /5" cos (A ') tan /3 cos (A" O') tan ft' cos (A 0') tan ft cos (A' 0') tan ft" cos (A 0') tan Jcos (A" ') _tan/3 nE cos (0' 0) R'+ri'R" cos (0" 00 "n" ' tan? 7 cos"(A 0') tan ft cos (A" 0') ' 232 THEORETICAL ASTRONOMY. which may also be used to determine p and p" when the equations (18) and (21) cannot be applied. When the motion in latitude is greater than in longitude, these equations are to be preferred instead of (22) and (23.) 81. It would appear at first, without examining the quantities in- volved in the formula for p', that the equations (26) 3 will enable us to find n and n" by successive approximations, assuming first that n = T - r , n" = , and from the resulting value of p r determining r f y and then carrying the approximation to the values of n and n ff one step farther, so as to include terms of the second order with reference to the intervals of time between the observations. Bat if we consider the equation (10), we observe that a Q is a very small quantity depending on the difference /?' /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 a to the intervals of time between the observations, we have, from the coefficient of p f in equation (7), a Q sec ft = tan /9 sin (A" A') tan /?' sin (A" A) -f tan/S" sin (A' A). We may put tan/5 = tan/S' At" + Br" 2 ...., tan ,5" = tan/S' -f AT -f Br* -f . . . . , and hence we have a sec /?' = (sin (A" A') sin (A" A) -f sin (A' A)) tan /5' + (r sin (A' A) r" sin (A" A')) ^-j-(r 2 sin (A'-A)+r" 2 sin (A"-A')) J5-f . ., which is easily transformed into a sec p = 4 sin J (A' A) sin I (A" A') sin (A" A) tan p (25) + (^ sin (A'-A) r" sin (A" A'))^+(^ 2 sin (A' A)+r" 2 sin (A"-A'))+. . . , If we suppose the intervals to be small, we may also put smA(^'-A)=:l(A''-A), and sin (A" A) = A" A, s i n (A' A) = A' A. DETERMINATION OF AN OEBIT. 233 Further, we may put X" = X' 4- A'r -f jB'r 2 -f ..... Substituting these values in the equation (25), neglecting terms of the fourth order with respect to r, and reducing, we get a = TT'T" ($A r * tan ft' -f A'B AB') 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" may produce an error of the order zero in the value of p f as derived from equation (11) even under the most favorable circumstances. Hence, in general, we cannot adopt the values T n = , = , = omitting terms of the second order, without affecting the resulting value of p f 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' f . The equation (28) 3 gives n omitting the term multiplied by -77, which term is of the third order n 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) 3 we derive, since r' = r -f- r", + " = l + J. (27) in which only terms of the fourth order have been neglected. Now the first of equations (18) may be written : sec =* in which, if we introduce the values of - - and n + n tf as given by TL (26) and (27), only terms of the fourth order with respect to the 234 THEOEETICAL ASTEONOMY. 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" 2 will be a quantity of an order superior to r 2 , and when these intervals are equal, we have, to terms of the fourth order. The equation (27) gives 2/3 ( w _|_ n n 1) == TT". Hence, if we put P (29) Q = 2r' 3 (n + n" 1), , we may adopt, for a first approximation to the value of p f , P=C = ", (30) and p r will be affected with an error of the first order when the in- tervals are unequal ; but of the second order only when the intervals are equal. It is evident, therefore, that, in the selection of the observations for the determination of an unknown orbit, the in- tervals should be as nearly equal as possible, since the nearer they approach to equality the nearer the truth will be the first assumed values of P and , thus facilitating the successive approximations ; and when a is a very small quantity, the equality of the intervals is of the greatest importance. From the equations (29) we get n = P\ r 2r' 3 /' (31) n" = nP; and introducing P and Q in (28), there results i?-* (32) This equation involves both p' and r' as unknown quantities, but by means of another equation between these quantities p f may be eliminated, thus giving a single equation from which r' may be found, after which p' may also be determined. DETERMINATION OF AN OEBIT. 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 ft tan w = - - -TT-, - TT, sm (A O ) (33 ) cosw cos 4/ = cos ft cos (/' by means of which we may determine ij/, which cannot exceed 180. Since cos /3' is always positive, cos fy and cos (K O ') must have the same sign. We also have which may be put in the form r' 2 = (p r sec ft K cos V) 2 + R' 2 sin 2 *', from which we get p' sec ft == R' cos*' VV 2 .R' 2 sin 2 4'. (34) Substituting for p f sec /?' its value given by equation (32), we have For brevity, let us put _5J-Ptf C O-TH^P' c c = *o, (35) -Hft-^t and we shall have k Q l ^ = R' cos V 1/r' 2 ^sm 2 ^. (36) When the values of P and Q have been found, this equation will give the value of r' in terms of quantities derived directly from the data furnished by observation. We shall now represent by z' the angle at the planet between the sun and earth at the time of the second observation, and we shall have / = ^SUH/_ } smz 236 THEORETICAL ASTRONOMY. Substituting this value of r', in the preceding equation, there results 7 oi n * 2 ' (k - R' cos 4') sin z' + R sin 4' cos z' = ^-rj7> (38) and if we put rj Q sin C = R f sin 4', (39) the condition being imposed that m Q shall always be positive, we have, finally, sin (z' qp C) = m sin* 2'. (40) In order that m may be positive, the quadrant in which f is taken must be such that y shall have the same sign as 1 Q , since sin ty f is always positive. From equation (37) it appears that sin z f 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 sn + 4') R sin (z f + 4') sin 4/ sin z' Therefore, R sin (z' + 4') P= sin / -cos and, since ^o' is always positive, it follows that sin (z' -f- ^ r ) must be positive, or that z' cannot exceed 180 ty. When the planet or comet at the time of the middle observation is both in the node and in opposition or conjunction with the sun, we shall have /3' = 0, 4^ 180 when the body is in opposition, and 'vj/ = when it is in conjunction. Consequently, it becomes impos- sible to determine r' by means of the angle z f but in this case the equation (36) gives -^= -R' + r', when the body is in opposition, the lower sign being excluded by the condition that the value of the first member of the equation must be positive, and for -J/ = 0, the upper sign being used when the sun is between the earth and the DETERMINATION OF AN ORBIT. 237 planet, and the lower sign when the planet is between the earth and the sun. It is hardly necessary to remark that the case of an obser- vation at the superior conjunction when /3' = 0, is physically impos- sible. The value of r f 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 sum. 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 b d c 1 p sec p T J- ' ~~ ' 2r' 3 / 1 The equations (17) give and if we put b +p d ' c l+p we shall have since c co when a = 0. Hence we derive '^4- (42) ^0 But when the great circle passing through the three observed places passes also through the second place of the sun, both c and C be- come indeterminate, and thus the solution of the problem, with the given data, becomes impossible. 83. The equation (40) must give four roots corresponding to each sign, respectively; but it may be shown that of these eight roots at least four will, in every case, be imaginary. Thus, the equation may be written m sin 4 z' sin z' cos = = cos z' sin C, 238 THEORETICAL ASTRONOMY. and, by squaring and reducing, this becomes m 2 sin 8 z' 2m cos C sin 5 z f -\- sin 2 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 f exceeds the limits 90 and -f 90, cos will be negative, and hence, in this case also, there cannot be more than four real roots, of which one will be positive and three negative. Further, since sin 2 is real and positive, there must be at least two real roots one positive and the other negative whether cos be negative or positive. We may also remark that, in finding the roots of the equation (40), it will only be necessary to solve the equation sin (z = m sin 4 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) 17 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' = V. Further, it follows from the equation (37) that this root must be and such would be strictly the case if, instead of the assumed values of P and , their exact values for the orbit of the earth were adopted, and if the observations were referred directly to the centre of the earth, in the correction for parallax, neglecting also the perturbations in the motion of the earth. DETERMINATION OF AN ORBIT. 239 In the case of the earth, in(0"-- O)' _' ~ " sin (0" O)' and the complete values of P and Q become in(0' Q) " O')' sin(Q'- Q) + #ff' sin(Q" 0Q " sn - and since the approximate values differ but little from these, as will appear from the equations (27) 3 , there will be one root of equation (43) which gives z f nearly equal to 180 ^/. 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 f are excluded by the nature of the problem, since r r must be positive, or z r < 180 ; and of the three positive roots which may result from equation (43), that being excluded which gives z' very nearly equal to 180 or f must be within the limits -f 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 f from equation (45), and call this z ', or sin(2z ' C) = | sinC, we may find the limits of the values of m , within which equation (43) has four real roots. The equation for z f will be satisfied by the values 2<-C, 180 - (2*; -0; and hence there will be two values of m , which we will denote by m x and m 2 , for which, with a given value of , equation (43) will have equal roots. Thus we shall have sin 4 z ' and, putting in this equation 180 (2z Q r f) instead of 2z f , or 90 (V ) in place of z ', It follows, therefore, that for any given value of , if m is not within the limits assigned by the values of m^ 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 f is not very nearly equal to 180 i//, the positive root, when exceeds the limits + 36 52'.2 and 36 52 r .2, may actually satisfy the conditions of the problem, and belong to the orbit of the body observed. 16 242 THEOEETICAL ASTEONOMY. 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^ and m 2 ; but, if m surpasses these limits, there will be only two real roots. Table XII. contains for values of from 36 52'. 2 to + 36 52'.2 the values of m 1 and m 2 , and also the values of the four real roots corresponding respectively to m l and m 2 . In every case in which equation (43) has three positive roots and one negative root, the value of m must be within the limits indicated by m x and w 2 , and the values of z f will be within the limits indicated by the quantities corresponding to m l and m 2 for each root, which we designate respectively by z/, z. 2 f , z B f , and /. The table will show, from the given values of m and 180 tj/, whether the problem admits of two distinct solutions, since, excluding the value of z f , which is nearly equal to 180 ij/, and corresponds to the orbit of the earth, and also that which exceeds 180, it will appear at once whether one or both of the remaining two values of z' will satisfy the condition that z r shall be less than 180 ^'. The table will also indicate an approximate value of z r , 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 Q in the general equation (32), we have and, since p f must be positive, the algebraic sign of the numerical value of 1 will indicate whether r f is greater or less than R f . It is easily seen, from the formulae for l w 6, c?, &c., that in the actual application of these formulae, the intervals between the observations not being very large, 1 Q will be positive when ft' ft and sin (O' K) have contrary signs, and negative when ft' ft has the same sign as sin (O' K). Hence, when O' K is less than 180, r' must be less than R f if ft' ^ is positive, but greater than R f if /9 r ft is negative. When ; K exceeds 180, r f will be greater than R' if /?' /9 is positive, and less than R' if /9 r 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 circle through the two extreme observed places of the comet, r' must be greater than R' when the place of the comet for the middle 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 f . 85. From the values of o' and r f derived from the assumed values T" P = and Q = TT", we may evidently derive more approximate values of these quantities, and thus, by a repetition of the calcula- tion, make a still closer approximation to the true value of p'. To derive other expressions for P and Q which are exact, provided that r f and p f are accurately known, let us denote by s" the ratio of the sector of the orbit included by r and r f to the triangle included by the same radii-vectores and the chord joining the first and second places ; by s f the same ratio with respect to r and r r/ , and by s this ratio with respect to r f 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) we have for the areas of the sectors, neglecting the mass of the body, and therefore we obtain s"[rr f ]=r"- ] /p, s' [r/'] = r' yft 8 [//'] = T j/p. (46) Then, since _ = we shall have r s' r" s' ,._ = -- n ff = -.- n) (47) T S TO and, consequently, P = Substituting for s, s', and s" their values from (46), we have -' rr" 244 THEORETICAL ASTRONOMY. The angular distance between the perihelion and node being denoted by to, the polar equation of the conic section gives = 1 -f- e cos (u w), r 4 = 1 + e cos (u f >), (50) r Ijf = 1 + e cos (u" 01). If we multiply the first of these equations by sin (u rf u'\ the second by sin (u ff u), and the third by sin (u 1 u), add the products and reduce, we get - sin (u" u') ^ sin (u" w) + 4 sin (u f u) = sin (u" u') sin (u" u) + sin (u' u) ; and, since sin (u" u') = 2 sin j (u" u'') cos (u" u'\ sin (u" u) sin (u r u) = 2 sin A (u" u') cos ^ (u" + u' 2w), the second member reduces to 4 sin ^ (u" u') sin |- (u" u) sin J (u' u). Therefore, we shall have 4rr'r" sin \ (u" u') sin ^ (u" u) sin J (u' u) P r y s i n tyi u '^ rr sm ( u " u ^ _[_ rr f gm (yf u y If we multiply both numerator and denominator of this expression by 2rr'r" cos J (u" w') cos -| (u" VL) cos J (u r u), it becomes, introducing [rr r ], [rr r/ ], and [rV r/ ], [r'r"] ^ [ r /'] . [ rr 'j 1 = [rV']-h[rr'] [rr"] ' 2rrV' cos (u"tf) cos J (M" M) cos J (*' M)' Substituting this value of _p in equation (49), it reduces to rr" r" 2 f\ * * _ f^l^ ss" ' rr" cos J (u" u'} cos (u" 11) cos J (u r w)' 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 r and r" being computed from the uncorrected times of observation, which may be denoted by t w and t Q ". With the values of P and Q thus found, we compute r', and from this p', p, and p", by means of the formulae already derived. The heliocentric places for the first and third observations may now be found from the formulae (71) 3 and (72) s , 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 / tf t~\ ft i / // \ sin (u u ) - sin (u tt), n'V (53) sin (u r u) = sin (u" u), r from which u 1 ' u f and u f u may be computed. From these results the ratios s and s r/ 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' 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, l n ', 6, and b ff , and then compute u, u f , and u" directly from & and i, according to the first of equations (82)^ It will be expedient also to compute r', V and b r from p', ^', 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 246 THEOKETICAL ASTKONOMY. 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 (f Q, ), and the value of b' derived from this equation must agree with that computed directly from p', or at least the difference should not exceed what may be due to the unavoidable errors of logarithmic calcula- tion. We may now compute n and n" directly from the equations _rV'siny-tQ rr' sm(u'-u) . ~ rr" sin (u" u) ' 7 rr" sin (u' r u) ' but when the values of u, u f , and u" are those which result from the assumed values of P and , 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 (it" u')du' t d log e n" = -f cot <>' 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' 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 DETERMINATION OF AN ORBIT. 247 means of (31), the two resulting values of did should agree, and the magnitude of du f 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 , 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 TJ r f , u, u r , &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 f 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 M=E esinE, or k(tT} . ^ 5 J - = E esmE, a* and also for the third place a? Subtracting, we get ll = E" E 2e sin i (E" E) cos J (E" + E). (55; aa 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 & aVl^^ (E" E2esmi (E" E) cos .J (E" + E)). (56) 248 THEORETICAL ASTRONOMY. From the equations Vr sin -y = I/a (1 + e) sin %E, t/r" sin ^v" I/a (1 -j- e) sin $E" t V~r cos Av i/o (1 e) cos ^-E 1 , 1/r" cos X = I/a (1 e) cos 1", since v"v = u" u, we easily derive 1/rT 7 sm (u" u) = al/T^* sin (E" JE), (57) and also a cos J (jEJ" E} ae cos (JE" + JK) = 1/rr 77 cos i (w" w), or " ^. (58) Substituting this value of e cos|(^ r/ + ^J) in equation (56), we get - e 2 (E" --Esm (E" - - JBJ)) e 2 sin (jB" JE) cos ^ (it" w) T/rr", and substituting, in the last term of this, for al/1 e 2 , its value from (57), the result is r'Vp = a 2 vT=7 2 (E" E sm (E" E)) + rr" sin (u" u}. (59) From (57) we obtain or _ / rr^sinC^ u) \ 3 1 = \ 21/r? 7 cos J (*" w) / jp sin 3 J (^" - Therefore, the equation (59) becomes Let x' be the chord of the orbit between the first and third places, and we shall have x' 2 = (r -f- r") 2 4rr" cos 2 -J (u" u). Now, since the chord x f can never exceed r -\- r ff , we may put and from this, in combination with the preceding equation, we derive 21/n 7 "' cos J (u" - u ~) = (r + r") cos r'. (62) DETERMINATION OF AN ORBIT. 249 T' Substituting this value, and [rr ff ~\ -, Vp, in equation (60), it reduces to E"-E-*m(E"-E) T" 1 1_ * / , //\q q / * ~75" ~| ~f * V^^y (r -\- r ') cos 3 Y g^ s sn The elements a and e are thus eliminated, but the resulting equation involves still the unknown quantities E ff E and s f . It is neces- sary, therefore, to derive an additional equation involving the same unknown quantities in order that E" E may be eliminated, and that thus the ratio s', which is the quantity sought, may be found. From the equations r = a ae cos E, r" = a ae cos E", we get r" + r = 2a 2ae cos J (E" + E} cos (E" E~). Substituting in this the value of e cos \(E' f -\- E) from (58), we have r" + r = 2a sin 2 (E" E ) -f 21/^V 7 cos K u) cos (E"E), and substituting for sinj(_E/ r/ E) its value from (57), there results cosi (u"-u) (l-2sin 2 | (E?' P But, since ^> 2prr" cos 2 ^ (u" u) s' 2 \ 2l/rr' r cos i (w^ w) / ' we have T '2 from which we derive -' 2 < 2 Jr (64) which is the additional equation required, involving E" E and s f as unknown quantities. Let us now put (05) E" E sin (E" E}' 250 THEOKETICAL ASTKONOMY. and the equations (63) and (64) become (66) When the value of y' is known, the first of these equations will enable us to determine s', and hence the value of x f , or sin 2 |(^ r/ E\ from the last equation. The calculation of f ma y be facilitated by the introduction of an additional auxiliary quantity. Thus, let (67) and from (62) we find cos / = cos (u" u) rr ,, = 2 cos (u" u) cos 2 / tan /, or cos r' = sin 2%' cos J (u" u). (68) We have, also, % " = ( T -f r '7 4rr" cos 2 -'- (u" u), which gives x' 2 = (r r") 2 + 4rr" sin 2 ,} (u" u). Multiplying this equation by cos 2 %(u" u) and the preceding one by sin 2 J(t&" u), and adding, we get " = ( r + /')* S i n 2 j ( u '/ _ w ) + ( r _ r ^)2 C0g2 1 y, __ u ^ From (67) we get and, therefore, -%'=^, so that equation (69) may be written x' 2 (r + //)a - sin 2 / = sin 2 J ( M _ W ) -|- cos 2 2/ cos 2 J (w" -~ it). We may, therefore, put sin / cos G' = sin ^ (w /r M), sin / sin G' = cos (" M ) cos 2/, (70) cos r' = cos J (^' u) 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 and we shall have 2g sin 2g This gives, by differentiation, dy' , 4 sin y' 2g sin 2g' or -2- = 3y r cot g 4?/' 2 cosec g. The last of equations (65) gives a/' = sin 2 J an< ^ P ut ?'= ,V 2 + if !*" + + iffliflHf.*" + &c -> (72) we obtain yy- 1 +*'=?' (73) Combining this with the second of equations (66), the result is If we put we shall have But from the first of equations (66) we get and therefore we have As soon as r/ 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 fr E), we may evidently, for a first approximation to the value of y 1 ', take +/ and with this find s' from (75), and the corresponding value of x' from the last of equations (66). With this value of x' we find the corresponding value of ', and recompute 37', s f , and x' ; and, if the DETERMINATION OF AN 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 f from 0.0 to 0.3, or from E" E=0 to E" E=132 50'.6. Should a case occur in which E ff E exceeds this limit, the expression sin 3 * (" ) * ~~ E" E sin (E" E) may then be computed accurately by means of the logarithmic tables ordinarily in use. An approximate value of x' may be easily found with which y' may be computed from this equation, and then ' from (73). With the value of ' thus found, if may be computed from (74), and thus a more approximate value of x' is immediately obtained. The equation (75) is of the third degree, and has, therefore, three roots. Since s f is always positive, and cannot be less than 1, it follows from this equation that if is always a positive quantity. The equation may be written thus : S '3_ /2 _ 7yV _1 V = 0j and there being only one variation of sign, there can be only one positive root, which is the one to be adopted, the negative roots being excluded by the nature of the problem. Table XIII. gives the values of logs' 2 corresponding to values of if from y'=Q to ?/=0.6. When if exceeds the value 0.6, the value of s f must be found directly from the equation (75). 89. We are now enabled to determine whether the orbit is an ellipse, parabola, or hyperbola. In the ellipse x = sm 2 {(E" E) is positive. In the parabola the eccentric anomaly is zero, and hence x = 0. In the hyperbola the angle which we call the eccentric anomaly, in the case of elliptic motion, becomes imaginary, and hence, since sin \ (E" E) will be imaginary, x f must be negative. It follows, therefore, that if the value of x f derived from the equa- tion m ' is positive, the orbit is an ellipse ; if equal to zero, the orbit is a parabola ; and if negative, it is a hyperbola. 254 THEOKETICAL ASTKONOMY. For the case of parabolic motion we have x r = 0, and the second of equations (66) gives s" = j- (76) If we eliminate s' by means of both equations, since, in this case, y' |j we get Substituting in this the values of m and I given by (65), we obtain q I ^-j = 3 sin / cos / -f- 4 sin 3 ^/, (r -f- which gives fi ' - = 6 sin i/ cos 2 \^ + 2 sin 3 ^', or 6r ' = (sin I/ + cos I/) 3 + (sin & - cos J/)'. 04-r"> This may evidently be written ft the upper sign being used when f is less than 90, and the lower sign when it exceeds 90. Multiplying through by (r + r f/ )%, and replacing (r -\~ r ff ) sin f by x, we obtain which is identical with the equation (56) 3 for the special case of parabolic motion. Since x' is negative in the case of hyperbolic motion, the value of ' determined by the series (72) will be different from that in the case of elliptic motion. Table XIV. gives the value of ' 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 ' 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)j and (56) 1? we get tan E =1/^1 tan F DETERMINATION OF AN OEBIT. 255 and hence, from (58),, We have, also, by comparing (65)! with (41 ) w since a is negative in the hyperbola, * 2 + l 2(7 ' which gives . Now, since cos E + l/^l sin E = e EV ~\ in which e is the base of Naperian logarithms, we have E l/-^~l = log e (cos E + 1/^T sin #), which reduces to or By means of these relations between E and ' 2/o, we shall have a =4(P-aO + JB(e y), 6 = 4'(P-aO + JS'(e !0 If we eliminate A, P, A', and B' from these equations, the results are (a'b" a"b') -j- (a"b ab") -\- (ab r a'b) y= (a'b" a"b') + (a"b ab") -j- (ab' a'b) ' from which we get (a" -f a') (a'b" a"b f ) -f a" ' (a"b ab") (82) (b -4- b ) \a b a b ) \~ b (a b ab ) ( a b a b ) j~ (a b ab ) -4 (ab a b) In the numerical application of these formulae it will be more- con- venient to use, instead of the numbers P, P lt P 2 , Q, ft, &c., the loga- rithms of these quantities, so that a = log P l log P,b = log ft lg ft and similarly for a', b', a" ', b ff , which may also be expressed in units of the last decimal place of the logarithms employed, and we shall thus obtain the values of log x and log y. 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' y r", 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 AN ORBIT. 259 much from the values found for x and y, on account of the neglected terms of the second order, that it will be necessary to recompute these quantities, using these last values of P and Q in connection with the three preceding ones in the numerical solution of the equations (82). 91. It remains now to complete the determination of the elements of the orbit from these final values of P and Q. As soon as , i 9 and u y u f j u" have been found, the remaining elements may be de- rived by means of r, r f y and u f u, and also from r f , r n ', and u" u r ; 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 f and x' from ?, r fl , and u fr u, by means of the formula? 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" u') \ 2 _ / s' f rr' sin (u f u)\* - _ / --\ and the mean of the two values of p obtained from this expression should agree with that found from (83), thus checking the calcula- tion and showing the degree of accuracy to which the approximation to P and Q has been carried. The last of equations (65) gives from which E ff E may be computed. Then, from equation (57), since e = sin

') = - f7 1, and from these, by addition and subtraction, we derive 2e cos (u" u) cos (4 (w" + u) *>) - + -77 2, (87) T> P 2e sin i (w" w) sin (J (w' r + w) w) = - -77, by means of which e and a) may be found. Since r-r" . , 21/rr" we have _ __ _ r^? 7 i/rr"sin2/ ^> ^ 2pcot2/ r"? 7 and from equations (70), -. . cos r ' Therefore the formulae (87) reduce to e sin (, - 4 (t*" + i*)) = = tan " e cos (o> 4 (w" + w)) = - sec w cos (88) from which also e and w may be derived. Then sin ? = e, and the agreement of cos

)' 2 CL COS' 5 p or it may be computed directly from the equation r' 2 4s' 2 rr" cos 2 J fa" w which results from the substitution, in the last term of the preceding equation, of the expressions for a cos

(1 -j- 9e) Ql/A (1 + 9e)' (93) for the time of perihelion passage, the value of <7 being the same as in (92). When the orbit is a parabola, e = 1 and p = 2q, and the elements Q and CD can be derived from r, r", tt, and u" by means of the equa- DETERMINATION OF AN OEBIT. 263 tions (76), (83), and (88), or by means of the formulae already given for the special case of parabolic motion. 92. Since certain quantities which are real in the ellipse and para- bola become imaginary in the case of the hyperbola, the formulae already given for determining the elements from r, r n ', u, and u" require some modification when applied to a hyperbolic orbit. When s' and x' have been found, p, e, and w may be derived from equations (83) and (87) or (88) precisely as in the case of an elliptic orbit. Since x r = sin 2 J (E fr E\ we easily find sin i (E" E)=2 V x' x'\ and equation (85) becomes " )VW -. (94 ) But in the hyperbola x f is negative, and hence V x' x n will be imaginary ; and, further, comparing the values of p in the ellipse and hyperbola, we have cos 2 ^ = tan 2 ^/, or cos = V 1 tan 4/. Therefore the equation for a cos (p becomes if a is considered as being positive, from which a tan ^ may be obtained. Then, since p = a tan 2 ^ we have tan 4 = ^ , (96) atan^/ for the determination of ij/, and the value of e computed from e = sec 4 = 1/1 +tan 2 4 should agree with that derived from equation (88). When e differs but little from unity, it is conveniently and accurately computed from e = 1 -f- 2 sin 2 ^ sec 4. The value of a may be found from (atan-4/) 2 , Q - a =p cot 2 4 = , (97) 264 THEORETICAL ASTRONOMY. or from a = 16s' 2 rr" cos 2 (u" u) (a/ 2 x'J which is derived directly from (89), observing that the elliptic semi- transverse axis becomes negative in the case of the hyperbola. As soon as to has been found, we derive from u, u', and u" the corresponding values of v, v', and v", and then compute the values of F 9 F'j and F" by means of the formula (57)! ; after which, by means of the equation (69) w the corresponding values of N, N f , and N" will be obtained. Finally, the time of perihelion passage will be given by T = t -*N=t-N> = t'-N" l Q k IJc IJc wherein log^fc 7.87336575. The cases of hyperbolic orbits are rare, and in most of those which do occur the eccentricity will not differ much from that of the para- bola, so that the most accurate determination of T will be effected by means of Tables IX. and X. as already illustrated. 93. EXAMPLE. To illustrate the application of the principal for- mula which have been derived in this chapter, let us take the follow- ing observations of Eurynome : Ann Arbor M. T. @o ? = 0.3326925, C = 8 24' 49".74, Iogm == 1.2449136. The quadrant in which f must be situated is determined by the con- dition that J? shall have the same sign as 4,. The value of z f must now be found by trial from the equation sin (z f C) = m sin 4 /. Table XII. shows that of the four roots of this equation one exceeds 180, and is therefore excluded by the condition that sins' must be positive, and that two of these roots give z' greater than 180 ( 4/, and are excluded by the condition that z 1 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' = 9 1' 22".96. The root which corresponds to the orbit of the earth is 18 20' 41 ", and differs very little from 180 ty. Next, from sin 2 smz _ p" M" 4- M i 1 \ MI n"^ 2 \ n" r we derive logr' = 0.3025672, log/ = 0.0123991, log n = 9.7061229, 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 b sin (I O) = j0sin(A Q), r sin b = p tan /9 ; / cos b' cos (V ') =10' cos (A' O') -#', / cos V sin (/' O') = p' sin (A' '), /sin 6' = /tan/5'; /' cos V cos (/" 0") = P" cos (A" 0") R", /' cos V sin (" 0") = p" sin (A" 0"), /'sin 6" = ,o"tan/3", which give / = 514'39".53, log tan b =8.4615572, logr =0.3040994, r = 7 '45 11 .28, log tan b' = 8.4107555, log/ =0.3025673, I" = 10 21 34 .57, log tan b" = 8.3497911, log /' = 0.3011010. The agreement of the value of log r f thus obtained with that already found, is a proof of part of the calculation. Then, from /-i nit . 7\ ^\ tan 6" -f- tan 6 tan t sm (J (* +/)-) = tan 6" tan 6 tan cos (1 ( + - 8) = 2sin . (r _ . cos ^ cos i cos i 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 (l r & ) gives log tan b' = 8.4107514, which differs 0.0000041 from the value already found directly from p f . This difference, however, amounts to only 0".05 in the value of the heliocentric latitude, and is due to errors of calculation. If we compute n and n" from the equations //' sin (u" u'*) rr' sin (u' u) n = n : 7 r, =r> n = 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 11 just found, we obtain log n = 9.7061158, log n" = 9.6924683, 270 THEORETICAL ASTRONOMY. which differ in the last decimal places from the values used in finding p and p". According to the equations d log n = 21.055 cot (u" u') du r , d log n" = -}- 21.055 cot (u' u) du', the differences of logn and logn" being expressed in units of the seventh decimal place, the correction to u' necessary to make the two values of logn agree is 0".15; but for the agreement of the two values of logn", u f must be diminished by 0".26, so that it appears that this proof is not complete, although near enough for the first approximation. It should be observed, however, that a great circle passing through the extreme observed places of the planet passes very nearly through the third place of the sun, and hence the values of p and p" as determined by means of the last two of equations (18) are somewhat uncertain. In this case it would be advisable to com- pute p and p", as soon as p f has been found, by means of the equa- tions (22) and (23). Thus, from these equations we obtain log p = 0.025491 8, log p" = 0.0028874, and hence I = 514'40".05, log tan b =8.4615619, log r = 0.3041042, r=10 2134.19, log tan b" =8.3497919, log /' = 0.3011017, = 207 2' 32".97, i = 4 27' 25".13, u = 158 8' 31".47, u' = 160 39' 23".31, u" = 163 16' 9".22. The value of log tan b' derived from X' and these values of Q, 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 n \ are log n = 9.7061144, log n" = 9.6924640 ; and this proof will be complete if we apply the correction du f = O'MS to the value of u', so that we have u" u' = 2 36' 46".09, u' u = 2 30' 51".66. The results which have thus been obtained enable us to proceed to a second approximation to the correct values of P and , and we may also correct the times of observation for the time of aberration by means of the formulae t==t Q Cp sec /?, t = t Q ' Cft sec p, t" = t " C P " sec /?", wherein log C= 7.760523, expressed in parts of* a day. Thus we get t == 257.67467, t r = 264.41976, t" = 271.38044, NUMERICAL EXAMPLE. 271 and hence log r = 9.0782331, log r' = 9.3724848, log r" = 9.0645692. Then, to find the ratios denoted by a and s", we have 17' sin f cos G = sin J (u" w'), sin Y sin G = cos \ (u" u') cos cos Y = cos 2 (u" it') sin ! tan/' == sin /' cos G" = sin (it' w), sin /' sin G" = cos ^ (it' u) cos 2y 7 , cos /' = cos J (it' it) sin 2;/' ; r 2 sin 2 = r" 2 m = from which we obtain x = 44 57' 6".00, /" 44 56' 57".50, Y = 1 18 35 .90, /' =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 m m r!' = using Tables XIII. and XIV., we compute a and s". First, in the case of Sj we assume 7 = - : = 0.0002675, 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 log s = 6.0001290, logs" = 0.0001200. When the intervals are small, it is not necessary to use the formula? 272 THEOKETICAL ASTEONOMY. 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 P r " s p = ~ T 7 "' _^_ ' * ~~ ss " ' rr" cos |- (u" u') cos -J (u" u) cos (u r u)' we find log P = 9.9863451, log Q = 8.1431341, with which the second approximation may be completed. We now compute c , Tc w l w z f , &c. precisely as in the first approximation ; but we shall prefer, for the reason already stated, the values of p and p" computed by means of the equations (22) and (23) instead of those obtained from the last two of the formulae (18). The results thus derived are as follows : log c = 2.2298499 n , log Jc = 0.0714280, log 1 = 0.0719540, log % = 0.3332233, C = 8 24' 12".48, log m = 1.2447277, z' = 90'30".84, log / = 0.3032587, log p r = 0.0137621, log n = 9.7061153, log n"= 9.6924604, logp = 0.0269143, log p" = 0.0041748, I = 5 15' 57".26, log tan b =8.4622524, logr =0.3048368, /' = 7 46 2,76, log tan V =8.4114276, log/ == 0.3032587, J" = 10 22 0.91, log tan b" = 8.3504332, log r" = 0.3017481, ^ = 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 f is complete, and the value of log tan b r computed from tan b' = tan i sin (I 1 & ), is log tan b' = 8.4114279, agreeing with the result derived directly from p f . 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 6' 47".60. From these values of u"u r and u r u, we obtain log s = 0.0001284, log s" = 0.0001193, NUMEKICAL EXAMPLE. 273 and, recomputing P and , 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 n u), sin / sin G' = cos | (u" u) cos 2/, cos / cos (u" u} sin 2/, r' 2 . sin 2 _ (r-fr") 3 cosV "" cos/' give / = 44 53' 53".25, / = 2 33' 52".97, log tan G' = 8.9011435, log m' = 6.9332999, log/ = 6.7001345. From these, by means of the formula _ - and Tables XIII. and XIV., we obtain log s' 2 = 0.0009908, log of = 6.5494116. Then from s'rr" sin w" we get = 0.3691818. The values of logp given by /grV'siny ^)\ 2 _ / s"rr' sin (u f u) \ 2 p ~-\ r ; = -\ '>' -/ are 0.3691824 and 0.3691814, the mean of which agrees with the result obtained from u ff u, and the differences between the separate results are so small that the approximation to P and Q is sufficient. The equations sin 4 ( Hi J] = a cos ^ 18 274 THEOKETICAL ASTRONOMY. give i (E" E) = l 4' 42".903, log (a cos ?) = 0.3770315, log cos = 9.9921503. Next, from e sin (> - $ (u" + u)) = - f/= tan (7, cos/V rr" e cos ( - i (w" + t*)) = cQg f^x-,7 ~ sec 2 fa" - M), we obtain ^ = 190 15' 39".57, log e = log sin ? = 9.2751434, P = 10 51 39 .62, TT = = 9.9921501, agreeing with the result already found. To find a and //, we have Tc the value of k expressed in seconds of arc being log k = 3.5500066, from which the results are log a = 0.3848816, log ft = 2.9726842. The true anomalies are given by V = U W, t/ = U f - W, I/' = U ff - W, according to which we have v = 327 56' 39".97, v' = 330 27' 6".25, v" = 333 3' 27".57. If we compute r, r f , and r" from these values by means of the polar equation of the ellipse, we get log r = 0.3048367, log / = 0.3032586, log r" = 0.3017481, and the agreement of these results with those derived directly from p, p f , and p" is a further proof of the calculation. The equations tan $E = tan (45 ?) tan %v, tan bE = tan (45 jp) tan jt/, tan E" = tan (45 J ? ) tan Jt/' give E = 333 17' 28".18, E' = 335 24' 38' ; .00, E" = 337 36' 19".78. NUMERICAL EXAMPLE. 275 The value of \ (E" E) thus obtained differs only 0".003 from that computed directly from x f . Finally, for the mean anomalies we have M= E e sin E, M ' = E' e sin E', M" = E" e sin E", from which we get M = 338 8' 36".71, M' = 339 54' 10".61, M" = 341 43' 6".97 ; and if M denotes the mean anomaly for the date T=1863 Sept. 21.5 Washington mean time, from the formulae M =M fit T we obtain the three values 339 55' 25".97, 339 55' 25".96, and 339 55' 25".96, the mean of which gives M = 339 55' 25".96. The agreement of the three results for M Q 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 n = 37 15 40 .29) = 207 72 V Ecll P tlc and Mean < = 4 28 35. 20 J Equinox 1863.0. 2 T "2 and, if the intervals are equal, this value of s' is correct to terms of the fifth order. Since we have, neglecting terms of the fourth order, in which log^ 8.8596330. We have, also, to the same degree of approximation, For the values log r = 9.0782331, log r' = 9.3724848, log r" = 9.0645692, log/ = 0.3032587, these formulae give log s = 0.0001277, log s f = 0.0004953, log " = 0.0001199, . which differ but little from the correct values 0.0001284, 0.0004954, and 0.0001193 previously obtained. Since sec 3 / = 1 + 6 sin 2 tf + Ac., the second of equations (65) gives r' 2 6r' 2 = ( 4- "V ~^~ (' -L "\* sm I* "h & Ct Substituting this value in the first of equations (66), we get r' 2 6r' 2 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 f 1 is of the second order with respect to r', we have, to terms of the fourth order, 280 THEORETICAL ASTRONOMY. Therefore, which, when the intervals are small, may be used to find s f from r and r". In the same manner, we obtain =i tin* lo g s "=3%7s- (102) For logarithmic calculation, when addition and subtraction loga- rithms are not used, it is more convenient to introduce the auxiliary angles , /', and #", by means of which these formulae become (103) in which log J^ = 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 ff 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 FKOM FOUR OBSERVATIONS. 281 CHAPTER V. DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE. 95. THE formulae given in the preceding chapter are not sufficient to determine the elements of the orbit of a heavenly body when its apparent path is in the plane of the ecliptic. In this case, however, the position of the plane of the orbit being known, only four ele- ments remain to be determined, and four observed longitudes will furnish the necessary equations. There is no instance of an orbit whose inclination is zero ; but, although no such case may occur, it may happen that the inclination is very small, and that the elements derived from three observations will on this account be uncertain, and especially so, if the observations are not very exact. The diffi- culty thus encountered may be remedied by using for the data in the determination of the elements one or more additional observations, and neglecting those latitudes which are regarded as most uncertain. The formulae, however, are most convenient, and lead most expe- ditiously to a knowledge of the elements of an orbit wholly unknown, when they are made to depend on four observations, the second and third of which must be complete ; but of the extreme observations only the longitudes are absolutely required. The preliminary reductions to be applied to the data are derived precisely as explained in the preceding chapter, preparatory to a de- termination of the elements of the orbit from three observations. Let t, t', t", t'" be the times of observation, r, r', r" , r"' the radii- vectores of the body, u, u', u" ', u fff 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" sin (u m '), [rV"] = rV" ito. u"), and (1) 282 THEORETICAL ASTRONOMY. Then, according to the equations (5) 3 , we shall have nx x' + n"x" = 0, ny -y f +n'Y = 0, ' "''"'" = Let ;, A', X", X'" be the observed longitudes, ft /?', /9", /3'" the ob- served latitudes corresponding to the times t, t', t fr , t ln ', respectively, and J, A', A", A'" the distances of the body from the earth. Further, let ^008^=^ and for the last place we have of" = p" r cos A'" R" cos 0'", r cos A' # cos O') -f "0>"cosA" jR"cosO"), = n sin A ^ sin Q) (/>' sin A' J?' sin Q') + n"0>"sinA" JB" sin 0"), = ' 0>' cos / ^ cos O') (/>" cos A" R" cos 0") (3) + n'" V cos /" JR"' cos O'"), = ri (/>' sin A' ' sin O') (?" sin A" R" sin ") + n'" G/" sin A w 12"' sin Q r "). If we nfultiply the first of these equations by sin ^, and the second by cos ^, and add the products, we get = nR sin (A Q) ( P f s i n (X 1 X) + K sin (A Q')) H- n" 0>" sin (A" - A) + R" sin (A - ")) ; (4) and in a similar manner, from the third and fourth equations, we find = n' O/ sin (A'" A') R s i n (A'" Q ')) (5) - 0>" sin (A'" A") U" S i n (A'" 0")) W '".R"' sin (/'" '"). Whenever the values of w, n', TI /; , and n" f 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- pendent of each other ; and, since only two unknown quantities p' OEBIT FROM FOUR OBSERVATIONS. 283 and p ff are involved in them, they will enable us to determine these curtate distances. Let us now put cos p sin (/ X) =A, cos ?' sin (A" A) = J5, cos /5" sin (A'" A") = C, cos jf sin (X" A') = D, and the preceding equations give Ap' sec p Bn"p" sec 0" = nR sin (A Q ) R' sin (A 0') + w"U" sin (A 0"), J>&y sec ft' Cp" sec 0"= n'R' sin (A'" Q ') R" sin (A'" 0") (7) + n'"jR"'sm(;i'" 0'"). If we assume for n and n" their values in the case of the orbit of the earth, which is equivalent to neglecting terms of the second order in the equations (26) 3 , the second member of the first of these equa- tions reduces rigorously to zero ; and in the same manner it can be shown that when similar terms of the second order in the corre- sponding expressions for n f and n" are neglected, the second member of the last equation reduces to zero. Hence the second member of each of these equations will generally differ from zero by a quantity which is of at least the second order with respect to the intervals of time between the observations. The coefficients of p 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 1 , so that it will not be even an approximation to the correct value ; and the same is true in the case of p". It is necessary, therefore, to retain terms of the second order in the first assumed values for n, n', n", and n f// ; and, since the terms of the second order involve r' and r ff , we thus introduce two additional unknown quantities. Hence two additional equations in- volving r', r", p'j p rf and quantities derived from observation, must be obtained, so that by elimination the values of the quantities sought may be found. From equation (34) 4 we have p' sec p = R' cos *' Vr' 2 J^'sin 1 *', (8) which is one of the equations required ; and similarly we find, for the other equation, p" sec 0" = R" cos 4/' d= V r m R" 2 sin 2 4". (9) 284 THEORETICAL ASTRONOMY. Introducing these values into the equations (7), and putting x' = =b vV' jff'Bin'V, a^zfcvV" " 2 sin 2 4/', we get Ax' Bn"x" = nR sin (A Q) # sin (A 0') -|- n"R" sin (A 0") AK cos V + n"BR" cos V, D*V - Cx" = n'R' sin (A'" - Q') - R" sin (A'" - 0") + ri"R'" sin (A'" 0'") n'DR' cos 4*' + CR" cos V'. Let us now put A = K ' C =r> or i, cos /3" sin (A" A) cos fi' sin (A'" A') ~ cos p sin (A' A) ' - cos ft" sin (A'" A")' _ and we have x' = Kn"x" + nd r - a! + nV, " == A"wV + w'"d" a" + ^c' r . These equations will serve to determine x f and x n ', and hence r r and ^ ;/ , as soon as the values of n, n 1 ', n'', and 7i r// are known. 96. In order to include terms of the second order in the values of n and n /r , we have, from the equations (26) 3 , and, putting these give ~ e = (w + n"-l)r, (13) ORBIT FKOM FOUR OBSERVATIONS. 285 Let us now put and, making the necessary changes in the notation in equations (26) 3 , we obtain '"_^/1 ,*"'W + T) , ~ f5 ~ "* rr rr-r * From these we get, including terms of the second order, ^-^d i r " and hence, if we put P = 5-, Q'= (' + ' -!)/", (17) 7&" we shall have, since r/ r + r ;// , / 2 -.'"2 \ Pit I -| 1 \ ' fff \ 6 /v,"3 /' ,,_1 T \ (18) When the intervals are equal, we have P' P" -* /^ > -* /// ? and these expressions may be used, in the case of an unknown orbit, for the first approximation to the values of these quantities. The equations (13) and (17) give (19) and, introducing these values, the equations (12) become 286 THEORETICAL ASTRONOMY. (20) Let us now put P'd'+c' , J > _ I P" -P' __ ' r ^ * and we shall have (22) We have, further, from equations (10), If we substitute these values of r /3 and r //3 in equations (22), the two resulting equations will contain only two unknown quantities x r and a/', when P', P /r , r , and r/ are known, and hence they will be sufficient to solve the problem. But if we effect the elimination of either of the unknown quantities directly, the resulting equation becomes of a high order. It is necessary, therefore, in the numerical application, to solve the equations (22) by successive trials, which may be readily effected. If z f represents the angle at the planet between the sun and the earth at the time of the second observation, and z" the same angle at the time of the third observation, we shall have Substituting these values of r r and r" in equations (10), we get and hence (25) CEBIT FKOM FOUR OBSERVATIONS. 287 R' sin V : - -j, "" (26) by means of which we may find z r and z" as soon as x f and x" shall have been determined ; and then r f and r rr are obtained from (24) or (25). The last equations show that when x f is negative, z f must be greater than 90, and hence that in this case r f is less than R f . In the numerical application of equations (22), for a first approxi- mation to the values of x f and x rf , since Q f arid Qf r are quantities of the second order with respect to r or r ;// , we may generally put # = 0, e" = 0; and we have x > =f'x" +<-a', *"=/'V+c "-a", or, by elimination, v_ i-//" I-/'/" With the approximate values of x f and x" derived from these equa- tions, we compute first r f and r" from the equations (26) and (24), and then new values of x f and x" from (22), the operation being repeated until the true values are obtained. To facilitate these ap- proximations, the equations (22) give (27) Let an approximate value of x f be designated by # x t x i a o'> 288 THEORETICAL ASTRONOMY. we shall have, according to the equation (67) 3 , the necessary changes being made in the notation, ^, ,__^^ ,__jC_. (28) The value of x r thus obtained will give, by means of the first of equations (27), a new value of x", and the substitution of this in the last of these equations will show whether the correct result has been found. If a repetition of the calculation be found necessary, the three values of x f which approximate nearest to the true value will, by means of (28), give the correct result. In the same manner, if we assume for x fr the value derived by putting Q' = and Q" = 0, and compute x', three successive approximate results for x" will enable us to interpolate the correct value. When the elements of the orbit are already approximately known, the first assumed value of x f should be derived from instead of by putting ' and Q" equal to zero. 97. It should be observed that when A' = X or )J rr X n ', the equa- tions (22) are inapplicable, but that the original equations (7) give, in this case, either p ff or p 1 directly in terms of n and n" or of n' and n" f and the data furnished by observation. If we divide the first of equations (22) by /*/, we have h r The equations (21) give h'~ 1 + P' and from (11) we get rf __R cosV , R'sm(X 0') h'~ ~hT ~W ' + X'^-W (29) Q) h'~ Then, if we put n> -p'd' c ' C '- P h' + W ORBIT FROM FOUR OBSERVATIONS. 289 c' $ . its value may be found from the results for , and , derived by means of these equations, and we shall have n-a')-r, (30) When A' = A, we have A/ = oo, and this formula becomes the value of ^ being given by the first of equations (29) This equation and the second of equations (22) are sufficient to determine x' and x" in the special case under consideration. The second of equations (22) may be treated in precisely the same manner, so that when X'" = A", it becomes o=(i+- ( iD ', u", u'" are accurately known, we have, according to the equations (47) 4 and (51) 4 , since / = i> r /V _ 1 " " 2 177" ' rr" cos fa" t*') cos fa" w) cos j fa' u)' In a similar manner, if we designate by s f " the ratio of the sector formed by the radii- vectores r" and T'" to the triangle formed by the same radii-vectores and the chord joining their extremities, we find (42) ss"' r'r'" cos j (u'" u") cos J (u" f u') cos J (u" u'}' The formulse for finding the value of s f " are obtained from those for s by writing j r// , f rf , G'", &c. in place of , f y G, &c., and using r f/ , r fff , u" r u" instead of r f t r" ', and u" it', respectively. By means of the results obtained from the first approximation to the values of P', P r/ , Q f y 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 r and r , and of P' 1 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", ', and Q" have been derived with sufficient accuracy, we proceed from these to find the elements of the orbit. After &, *, r, r', r' 1 ', r'", u, v/, u n ', and u'" have been found, the remaining elements may be derived from any two radii-vectores v 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 /ff . If the values of P', P", ', 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 sin^ cos 6r = sin (u'" u), (43) sin Y Q sin 6r a = cos (u m - u) cos 2/ , cos r = cos | (u" r u) sin 2/ , we find and Gr. Then we have from which, by means of Tables XIII. and XIV., to find S Q and a? . We have, further, s rr'"sm(u'" and the agreement of the value of p thus found with the separate results for the same quantity obtained from the combination of any two of the four places, will show the extent to which the approxima- tion to P', P", Q', 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 rff in the place of r" in the formulae there given and introducing the necessary modifications in the notation, which have been already suggested and which will be indicated at once. 101. EXAMPLE. For the purpose of illustrating the application of the formulae for the calculation of an orbit from four observations, let us take the following normal places of Eurynome @ derived by comparing a series of observations with an ephemeris computed from approximate elements. Greenwich M. T. a 6 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 -f 19 35 41 .5. NUMERICAL EXAMPLE. 295 These normals give the geocentric places of the planet referred to the mean equinox and equator of 1864.0, and free from aberration. For the mean obliquity of the ecliptic of 1864.0, the American Nautical Almanac gives e = 23 27' 24".49, and, by means of this, converting the observed right ascensions and declinations, as given by the normal places, into longitudes and lati- tudes, we get Greenwich M. T. 1863 Sept. 20.0 Dec. 9.0 1864 Feb. 2.0 April 30.0 a 16 59' 9".42 10 14 17 .57 29 53 21 .99 75 23 46 .90 P + 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, -fO".53, + 0".36, + 0".19. For the reduction of the geocentric latitudes of the planet to the point in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane, the equation (6) 4 gives the corrections 0".57, 0".38, 0".18, and 0".07 to be applied to these latitudes respectively, the logarithms of the approximate dis- tances of the planet from the earth being 0.02618, 0.13355, 0.29033, 0.44990. Thus we obtain t = 0.0, ; == 16 59' 9".42, /? = + 2 56' 44".01, H = 80.0, A' 10 14 17 .57, /?' == 1 15 49 .20, f = 135.0, A" = 29 53 21 .99, /?" = 2 29 57 .56, *'"= 223.0, A'" = 75 23 46 .90, P" = 3 444.56; and, for the same times, the true places of the sun referred to the mean equinox of 1864.0 are Q =177 0'58".6, logJR 0.0015899. Q' =256 58 35.9, log.R' =9.9932638, Q" =312 57 49 .8, log^R" =9.9937748, O'" = 40 21 26.8, log #" = 0.0035149, 296 THEORETICAL ASTRONOMY. From the equations tan/3' tan W = sin (A' -07 tan/5" we obtain 4,' = 113 15' 20".10, *" = 76 5617.75, tan (/QO cost!/ ' tan(A"-0") log OR' cos V) == 9.5896777., log (E f sin 4/) =9.9564624, log (J2" cos 4,") = 9.3478848, log (12" sin V') = 9-9823904. The quadrant in which tj/ must be taken, is indicated by the condi- tion that cos i// and cos(A' 0') must have the same sign. The same condition exists in the case of $". Then, the formula A = cos /?' sin (A' A), == cos /?" sin (A"' A"), B B = cos /5" sin (A" A), D = cos /?' sin (A'" A'), D " sin (A 0") r"r m sin(v"'v") , _ rV'sinQ/' v'} r'r'" sin (v'" v')' ~ r'r"' sin (v" f i/ we find n, n f } n", and n f ". The approximate elements of Eurynome give v =322 55' 9".3, logr =0.308327, v' =353 19 26 .3, log/ =0.294225, v"= 14 45 8.5, log/' =0.296088, t/"= 47 23 32 .8, log /" = 0.317278, 298 THEORETICAL ASTRONOMY. and hence we obtain log n = 9.653052, log n" = 9.806836, log n'= 9.825408, log n'" = 9.633171. Then, from " = (n'+ n'" 1) r" 3 , n we get log P' = 9.846216, log Q f = 9.840771, log P" = 9.807763, log " == 9.882480. The values of these quantities may also be computed by means of the equations (41) and (42). Next, from C ' = : 1 + P'' f = FT? 7 ' n P d -\- G ft ~ -i I -pti ' j ~~ -i 1 p//' we find log c ' = 0.541344 n , log/ = 0.047658 n , log c ;/ = 9.807665 n , log/" = 9.889385. Then we have if.r^ rf , tan 2' = - , tan z" = , = = = sin4/; = ^ sin d ~ cos 2/' sin 2" ~ cos 3'" from which to find r r and r". In the first place, from of = 1r' 2 J^Bin 1 *', we obtain the approximate value log x' = 0.242737. Then the first of the preceding equations gives log a" = 0.237687. NUMERICAL EXAMPLE. 299 From this we get z" = 29 3' 11" .7, log r" = 0.296092 ; and then the equation for x f gives log x' ==0.242768. Hence we have z' = 27 20' 59".6, log / = 0.294249 ; and, repeating the operation, using these results for x f and r f , we get log x" = 0.237678, log x r = 0.242757. The correct value of log x f may now be found by means of equation (28). Thus, in units of the sixth decimal place, we have o = 242768 242737 ~ -f 31, a f = 242757 242768 = 11, and for the correction to be applied to the last value of log x f , in units of the sixth decimal place, Therefore, the corrected value is log x' = 0.242760, and from this we derive log s" = 0.237681. These results satisfy the equations for x' and x rf , and give 2' =27 21' 1".2, log/ =0.294242, z" = 29 312 .9, log r" == 0.296087. To find the curtate distances for the first and second observations, the formulae are which give log p r = 0.133474, log p" = 0.289918. Then, by means of the equations 300 THEORETICAL ASTRONOMY. / cos V cos (J 0') = p' cos (A' 0') R, r' cos V sin (f - 0') = p f sin (A' '), / sin 6' = p' tan /S', r" cos V cos (^ - 0") = p" cos (A" - 0") - R', r" cos b" sin (/" ") = p" sin (A" "), we find the following heliocentric places : I' = 37 35' 26".4, log tan 6' === 8.182861 n , log r' = 0.294243, r = 58 5815.3, logtan&":=8.634209 n , log r" = 0.296087. The agreement of these values of log r f and log r" with those obtained directly from x r and x" is a partial proof of the numerical calcula- tion. From the equations tan i sin ( J (I" + /') ft ) = % (tan 6" -f tan &') sec (I" Z'), tan i cos ( (^ + ft ) = i (tan b" tan 6') cosec J (" O, COS 1 COS I we obtain ^ = 206 42' 24".0, w' = 190 55 6 .6 i = 4 36' 47".2, u" = 212 20 53 .5. Then, from we get log n" = 9.806832, log w' = 9.825408, =9.653048, log w'" = 9.633171, and the equations r sin ((u f -u) + % (u" - u')) r cos ((u f tt ) + J- (u" gn = cos i (u" sin (("' - ") + ' (" _ ' cos ((w w - 1*) -f i (^ _ w ' sn - - ~ cos \ (u" - u'\ NUMERICAL EXAMPLE. 301 give logr =0.308379, u = 160 30' 57".6, log/" = 0.317273, u'" = 24 5932.5. Next, by means of the formulae tan (I & ) = cos i tan u, tan b = tan i sin ( &), tan (/'" SI ) = cos i tan w'", tan V" = tan i sin (l" r & ), />cos(A Q) = rcos&cos(J 0)+-^, p sin (A O ) =r cos 6 sin (7 0), p tan /? = r sin 6 ; ,'" C os (A'" 0'") = r'" cos b m cos (r 0'") + #", />'" sin (X" 0'") = r'" cos &'" sin (r '"), //"tan/3"' = r"'8in&'", we obtain I = 7 16' 51".8, l m = 91 37' 40".0, b = + 1 32 14 .4, 6'" = - 4 10 47 .4, *== 16 59 9 .0, A'" == 75 23 46 .9, /9 = + 2 5640.1, j3'" = 3 443.4, log p = 0.025707, log ?"' = 0.449258. The value of X rrr thus obtained agrees exactly with that given by observation, but / differs /r .4 from the observed value. This differ- ence does not exceed what may be attributed to the unavoidable errors of calculation with logarithms of six decimal places. The differences between the computed and the observed values of /9 and /9 r/ 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 r/ , Q f , and Q", and which are therefore the same in the successive hypotheses, should be performed as accurately as possible. The Q ' value of -> required in finding x" from x f , may be computed directly from SL P> ^ 4- - f ~ h' "*" h'' d f e f the values of p and jj being found by means of the equations (29) ; 302 THEORETICAL ASTRONOMY. c" and a similar method may be adopted in the case of ~. 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 present case, x" is found from the first of equations (27) by means of the difference of two larger numbers, and an error in the last decimal place of the logarithm of either of these numbers affects in a greater degree the result obtained. But as soon as r" is known Q" so nearly that the logarithm of the factor 1 -f- -^ remains unchanged, the second of equations (22) gives the value of x" by means of the sum of two smaller numbers. In general, when two or more for- mula for finding the same quantity are given, of those which are otherwise equally accurate and convenient for logarithmic calculation, that in which the number sought is obtained from the sum of smaller numbers should be preferred instead of that in which it is obtained by taking the difference of larger numbers. The values of r, r f , r f> ', r f ", and u, u', u n r , u" f , which result from the first hypothesis, suffice to correct the assumed values of P' y P" , Q', and Q". Thus, from r = Jc (t" O, r" = Jc(t' 0, T'" = Jc (H" *"), l~7 r \~r' tan* = X / -p tan/' ~ 1, tan/" = sin r cos G = sin ^ (u n u'\ sin /' cos G" = sin J (u r u\ sin r sin G = cos i (u" u'} cos 2/, sin /' sin (r" = cos (' w) cos 2/', cos r = cos J (u" u') sin 2/, cos /' = cos ^ (w' w) sin 2/", sin /" cos G" = sin J (u" f w"), sin /" sin G" = cos i (u" f w") cos 2/" cos r"' = cos A (V " tt ") sin 2/" ; T 2 COS 6 / r" 2 COS 6 /' T'" 2 COS 6 /" W = ~ m = ^" * = -77^ ^77> r 3 cos 3 /" sin 2 iy' 7 ' J '"' == cos/"' m ;/ '?'" l-f^-Ff " 5 +/'-f^ m in connection with Tables XIII. and XIV. we find *, s", and The results are NUMERICAL EXAMPLE. 303 log r = 9.9759441, x = 45 3' 39".l, ?- = 10 42 55 .9, logm = 8.186217, log,/ = 7.948097, log == 0.0085248, log T"= 0.1386714, /"= 44 32' 1".4, y"= 15 13 45 .0, logm"= 8.516727, log/'= 8.260013, log s"= 0.0174621, log r'"= 0.1800641, /"'= 45 41' 55".2, r'"= 16 22 48 .5, logm"'= 8.590596, log/"= 8.325365, log "'== 0.0204063. Then, by means of the formulae __ . ~ r" rr" = * 77 r' 2 rr" cos (%" ti') cos -J (M" w) cos J (w' w)' P--L ^ ~ T'" ' 8 ' r'r'" cos i (it'" u") cos O' " u r ) cos J (it" uj we obtain log P' = 9.8462100, log P" = 9.8077615, with which the next approximation We now recompute c ', CQ",/',/ illustrated ; and the results are log c ' =s 0.5413485 n , log/' = 0.0476614 n , log a/ = 0.2427528, z' = 27 21' 2".71, log / = 0.2942369, log P ' =0.1334635, log n = 9.6530445, log n ' = 9.8254092, log ' == 9.8407536, log " =i 9.8824728, may be completed. , ^ r , oj r/ , &c. precisely as already log e " = 9.8076649 n , log/" = 9.8893851, log af' = 0.2376752, 4' = 29 3' 14".09, log r" = 0.2960826, log/' =0.2899124, log n" = 9.8068345, log n'" = 9.6331707. Then we obtain V == 37 35' 27".88, r=58 5816.48, log tan V = 8.1828572 w , logtan6"=8.6342073 H , log / = 0.2942369, log /'= 0.2960827. These results for log r f and log r" agree with those obtained directly from z r and z", thus checking the calculation of ty and ty r and of the heliocentric places. Next, we derive ft = 206 42' 25".89, u' = 190 55 6 .27, i = 4 36' 47".20, u" = 212 20 52 .96, 304 THEORETICAL ASTRONOMY. and from u"u', r', r", n, n", n 1 ', and n'", we obtain logr =0.3083734, u = 160 30' 55".45, log/" =0.3172674, it'" =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, b = + 1 32 14 .07, b'"= 4 10 47 .36, A= 16 59 9 .38, *'"= 75 23 46 .99, /?= + 2 56 39 .54, fi'"= 3 4 43 .33, log p = 0.0256960, log p'" = 0.4492539. The values of A and A //r 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", ', and Q ff , and of those which depend on the successive hypotheses, is completely proved. This condition, however, must always be satisfied whatever may be the assumed values of P', P", Q f , and Q". From r, r f , u, u r , &c., we derive log s = 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 r =k(1f ff -^t') ) tan/ = sin Y Q cos Cr = sin -J- (u m u), sin YQ sin G = cos -^ (u" r u) cos 2/ , cos Y Q = cos J (u m u) sin 2/ , (f _I_ w'"\* r*c\ = we obtain I (E'" E) = 17 35' 42".12, log (a cos ?) = 0.3796883, log cos

'" + w)) = ^ tan G , COS ft Yff ftt e cos (a, i (u" f + M)) = ^ = sec 1 ( M w u), cos y Vvf" give ai = 197 38' 8".48, log e = log sin ? = 9.2907881, ? s 11 15' 52".22, TT = 01 + ^ = 44 20' 34".37. This result for

P- = r> cos 2 , tan lE" f = tan J- (M'" ") tan (45 ^), from which the results are E = 329 ll r 46".01, E" = 12 5' 33".63, E' = 354: 29 11 .84, ^ r " = 39 34 34 .65. The value of J (J& /;/ J&) thus derived differs only 0".03 from that obtained directly from a? . For the mean anomalies, we have which give , Jtf" =E" = E f e sin J5', Jf' " = E'" e sin E'", M = 334 55' 39".32, M" = 9 44' 52".82, M r = 355 33 42 .97, M'" = 32 26 44 .74. Finally, if M denotes the mean anomaly for the epoch T 1864 Jan. 1.0 mean time at Greenwich, from M Q = M (Ji.(t T) =M' v.(t'T) = M" [i. (t" T} = M'" v (f" T), we obtain the four values M = 129'39".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 = 44 20 34 .37 1 Ecliptio and Me an ?= 11 15 52 .22 log a = 0.3881359 log fi = 2.9678027 At = 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, 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 ff , n, n f/ , 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 />'" = 0.4492542. With these values of p and p" f , the following heliocentric places are obtained : I = 7 16' 51". 54, log tan b =8.4289064, logr =0.3083732, r = 91 3740.96, logtan&'" = 8.8638549 M , log/" = 0.3172678. Then from tan i sin Q (l" r -f ) = 1 (tan V" -f tan 6) sec J (f" Q, tan i cos (I (I'" + ) = $ (tan V" tan 6) cosec J (f" I), we get = 206 42' 45".23, i = 4 36' 49".76. For the arguments of the latitude the results are u = 160 30' 35".99, u'" = 244 59' 12".53. 308 THEORETICAL ASTRONOMY. The equations tan b' = tan i sin (I' & ), tan b" tan i sin (I" & ), give log tan b' = 8.1827129 n , log tan b" = 8,6342104 n , and the comparison of these results with those derived directly from p' and p" exhibits a difference of -f l /r .04 in b r , 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 we compute the remaining elements by means of these values of r, r'" , and u, u' n ', the separate results are : log tan Q = 8.0522282 n , log w = 9.7179026, log s 2 = 0.2917731, log x = 8.9608397, logjp = 0.3712405, \ (E" E) = 17 35' 42".12, log (a cos ?) = 0.3796884, log cos

' d A ~ dM Q ' dA ' dn ' d A , dr d(v + y) dr" As soon as the values of TT , - ~~^-, jj, and - ,, are dA dA dA dA known, the equations necessary for finding the differential coefficients of the elements ^, , M 0) and p with respect to A are thus provided. In the case under consideration, when an increment is assigned to J, VARIATION OF TWO GEOCENTRIC DISTANCES. 319 the value of A" remaining unchanged, r" and v" + are not changed, and hence . . dA dA To find -7-7- and , , from the equations A cos f] cos B = x -f- -3T, J cos >? sin = y -\- Y, in which 57 is the geocentric latitude in reference to the plane of the orbit computed from A and A" as the fundamental plane, and X y Y the geocentric co-ordinates of the sun referred to the same plane, we get dx = cos y cos d dA, dA, or, substituting for dx and dy their values given by (73),, cos ?) cos d A = cos u dr r sin u d (v -f- /), cos fj sin 6 d A = sin udr -\- r cos u d (v -f- /). Eliminating, successively, d (v -f- ^) and c?r, we get dr - = COS f) COS (0 U) t i = - cos T? sm (0 it). - Therefore, we shall have dy , dv d

' , dr d

-rr 1 * and -j-r- dJ c?J dJ dJ In precisely the same manner we derive the following equations 320 THEORETICAL ASTRONOMY. for the determination of the partial differential coefficients of these elements with respect to A" : d% dv d

and i when the squares and products of the variations of the elements are neglected, if we determine the elements which exactly represent the places to which A and A" belong, as well as the longitudes for two additional places, or, if we determine those which satisfy the two fundamental places and the longitudes for any number of additional observed places, so that the sum of the squares of their residuals shall be a minimum, the results thus obtained will very nearly satisfy the several latitudes. Let 6 f denote the geocentric longitude of the body, referred to the plane of the orbit computed from A and A" as the fundamental plane, for the date t r of any one of the observed places to be used for cor- recting these assumed distances. Then, to find the partial differential coefficients of 6' with respect to A and 4", we have , dtf _ ^ ,dO' dx ,dO' d

\ ^ -i n dr dr" dv dv" dr , / i i (16) 2 the values of , -: , , - , , &c,, by means of which, dtp dy> d d(f> dM Q using the value of u in reference to the equator, we form the equa- tions (13). The accent is added to % to indicate that it refers to the 21 322 THEORETICAL ASTRONOMY. equator as the plane for defining the elements. Thus we obtain four equations, from which, by elimination, the values of the differential coefficients of #', anc ^ Jj 77 ' Then, by means of the for- mula (76) 2 , (78) 2 , and (79) 2 , we compute for the date of each place to be employed in correcting the assumed distances the values of cos j/--,, cos r/ ,, &c., and hence from (15) the values of COST/-J- and cos if j t - The results thus obtained, together with the residuals Cfr^J computed by means of the equations (17), enable us to form, accord- ing to (16), the equations of condition for finding the values of the corrections AZ/ and &A" . The solution of all the equations thus formed, according to the method of least squares, will give the most probable values of these quantities, and the system of elements which corresponds to the distances thus corrected will very nearly satisfy the entire series of observations. Since the values of cos rf A#' are expressed in seconds of arc, the resulting values of A J and A A" will also be expressed in seconds of arc in a circle whose radius is equal to the mean distance of the earth from the sun. To express them in parts of the unit of space, we must divide their values in seconds of arc by 206264.8. The corrections to be applied to the elements computed from A and A", in order to satisfy the corrected values A -\- A A and A" 4- A A", may be computed by means of the partial differential coefficients already derived. Thus, in the case of ', we have from which to find AJ(' ; and in a similar manner tup, Alf c , and may be obtained. If, from the values of ^.t* and ^ d A we compute VARIATION OF TWO GEOCENTRIC DISTANCES. 323 and apply these corrections to the values of v and v" found from A and A" ^ we obtain the true anomalies corresponding to the distances A -f A A and A" -\- A J". The corrections to be applied to the values of r and r" derived from A and A" are given by dr ar" If AJ and A A" are expressed in seconds of arc, the corresponding values of Ar and Ar /r must be divided by 206264.8. The corrected results thus obtained should agree with the values of r and r" com- puted directly from the corrected values of V, v ff , p, and e by means of the polar equation of the conic section. Finally, we have dz = sin TJ dA, and similarly for dz" ; and the last of equations (73) 2 gives T sin u Ai' r cos u sin i' A ' = sin 17 A J, i' r" cos it" sin i' A &' = sin V from which to find A^ V and A ', it and it" being the arguments of the latitude in reference to the equator. We have also, according to (?2) 2 , Aw' = A/ COS i' A&', ATT' = A/ + 2 Sin 2 ^' A &', from which to find the corrections to be applied to co f and TT'. The elements which refer to the equator may then be converted into those for the ecliptic by means of the formula which may be derived from (109)! by interchanging & and &' and 180 V and i. The final residuals of the longitudes may be obtained by substi- tuting the adopted values of A A and A A" in the several equations of condition, or, which affords a complete proof of the accuracy of the entire calculation, by direct calculation from the corrected elements ; and the determination of the remaining errors in the values of y will show IIOAV nearly the position of the plane of the orbit corresponding to the corrected distances satisfies the intermediate latitudes. Instead of (p, M w and //, we may introduce any other elements which determine the form and magnitude of the orbit, the necessary 324 THEORETICAL ASTRONOMY. changes being made in the formulae. Thus, if we use the elements T, q, and e, these must be written in place of Jf , //, and sin B = cos d sin (a A), sin 4 cos B = cos D sin d sin D cos 8 cos (a A), (26) cos 4- = sin D sin d -j- cos D cos d cos (a A), from which to find $ and J5, 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 Q cos (A K) = cos B, sin W Q sin (A K) = sin B sin Z>, (27) cos w = sin B cos D, from which to find IV Q and K. In a similar manner, we may com- VARIATION OF THE NODE AND INCLINATION. 327 pute the values of u" u, &, 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 8 and u (in reference to the equator) may be found. Let s 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 s = cos W Q cot (JBT A) (29) gives the value of s 09 the quadrant in which it is situated being deter- mined by the condition that coss and cos(K A) shall have the same sign. Then we have 8 = 8 S Q , and z = 180 4, & + 8 0) E sin 4, ^30) sin z from which to find r. 109. In both the method of the variation of two geocentric dis- tances and that of the variation of & 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 r and u f 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 r and u' with respect to A and A" , or with respect to Q, and i, as the case may be. Then, computing the values of r f and u' from the observed geocentric spherical co-ordinates by means of the values of Q, and i for the system of elements to be corrected, the differences between the results thus derived and those obtained directly from the elements enable us to form the equations du' du' , dA dA m-t \ (31) 328 THEORETICAL ASTEONOMY. or the corresponding expressions in the case of the variation of & and i, by means of which the corrections to be applied to the as- sumed values will be determined. In the numerical application of these equations, AM' being expressed in seconds of arc, A?-' should also be expressed in seconds, and the resulting values of A A 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' 9 r" and u, u 1 ', 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 A and A" 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 & and i, will be obtained. 110. The formulae which have thus far been given for the correc- tion of an approximate orbit by varying the geocentric distances, depend on two of these distances when no assumption is made in regard to the form of the orbit, and these 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 Jf, Y 9 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 w 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 = p cos a, y Q = p sin a, z = p tan d. (32) If we represent the right ascension of the sun by A, and its declina- tion by D, we also have VARIATION OF ONE GEOCENTRIC DISTANCE. 329 sA, Y=RcosDsmA, Z=EsmD. (33) The fundamental equations for the undisturbed motion of the planet or comet, neglecting its mass in comparison with that of the sun, are but since X X Q X, y^y^Y, and, neglecting also the mass of the earth, dP + ^ 3 ~~ ' d? K* ~' dt* these become (8*) Substituting for ic , 2/ , and 2 their values in terms of a and (?, and putting v ^ 3 -^) = C, (35) we get 2/0 I ^ I A f'Qfi^ Jj, i > ; rt/m 3 DAJJ. ** I / T ^^ x*"' x Differentiating the equations (32) with respect to t, we find dx n dp . da and -=- may be determined. ell/ CLL a/L To find the values of -37 > r-> and -77, the equations at at at X=RcosO, Y=Rsin O cose, Z = R sin O sin e, give, by differentiation, dX ^dR _ dO ___ cos0 ___ jRsm0 _, dY dR . dQ , A0 . -- = sin O cos s -=7- + -R cos O cos e j-, (43) at at at dZ . dR . D v . dO -JT = sm Q sin e -- - -j- R cos O sin e -jr at at at 332 THEORETICAL ASTRONOMY. Now, according to equation (52) w we have m denoting the mass of the earth, and e the eccentricity of its orbit. The polar equation of the conic section gives dr r 2 e sin v dv ~di~ p "dt' Let F denote the longitude of the sun's perigee, arid this equation gives dR J? 2 e sin(Q -T) dQ _kV / l + m n . ., . . -^ _- . . - "" ~ -\ _ -~- Or* &i.ui v \i/ * j* \ y dt 1 e 2 tW Vl e 2 If we neglect the square of the eccentricity of the earth's orbit, we have simply dt ~ R* dt The values of ^7 and -^r having been found by means of these fj JT d Y formula?, the equations (43) give the required results for , i and 7 f7 COv Civ -T-, 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 y and z may be derived by means of the equations x = A cos <5 cos a X, y = A COS d sin a Y t and from these, in connection with the corresponding velocities, the elements of the orbit may be found. The equations (32)j 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)! 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 & -J- ?/ sin & , from which r and w may be found, the argument of the latitude u being referred to the plane of xy as the fundamental plane. The equation gives ~di == r'di~^^'di~^r"di' and, since dr r*e sin v dv dv k Vp di~ ~^p dt } di~ r 3 ' we shall have Vp dr " F '* > (49) from which to find e and v. Then the distance between the peri- helion and the ascending node is given by (it =11 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 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 (2l\ 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// denote the angle which the tangent to the orbit at the ex- tremity of the radius-vector makes with the prolongation of this radius-vector, and we shall have 334 THEOEETICAL ASTRONOMY. dr dx dy dz - so that the preceding equation gives Vp= FV Hence we derive the equations dx , dy , dz - from which Fr and ^ may be found. Then, since 'p= we shall have (51) 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 (o) = r sin 4/ cos 4 , F 2 e cos (i* a*) = -75 - r sin 2 4 1, A/ and, since F_2_l "F ~~ r ~ a these are easily transformed into 2ae sin (u o) = (2a r) sin 24 , 2ae cos (u to) = (2a r) cos 24/ r. If we multiply the first of these equations by cos u and the second by sin u, and add the products ; then multiply the first by sin u and the second by cos u, and add, we obtain 2ae sin to = (2a r) sin (2^ -f- u) r sin u, /^ 2ae cosw = (2a r) cos (24/ -f- w) r cost*, These equations give the values of 01 and e. 113. We have thus derived all the formulae necessary for finding the elements of the orbit of a heavenly body from one geocentric distance, provided that the first and second differential coefficients of a and d with respect to the time are accurately known. It remains, VARIATION OF ONE GEOCENTRIC DISTANCE. 335 therefore, to devise the means by which these differential coefficients may be determined with accuracy from the data furnished by obser- vation. The approximate elements derived from three or from a small number of observations will enable us to correct the entire series of observations for parallax and aberration, and to form the normal places which shall represent the series of observed places. We may now assume that the deviation of the spherical co-ordinates computed by means of the approximate elements from those which would be obtained if the true elements were used, may be exactly represented by the formula 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 A#. The differences between the normal places and those computed with the approximate elements to be corrected, will then suffice to form equa- tions of condition by means of which the values of the coefficients A, B, and C may be determined. The epoch for which h = may be chosen arbitrarily, but it will generally be advantageous to fix it at or near the date of the middle observed place. If three observed places are given, the difference between the observed and the com- puted value of each right ascension will give an equation of condition, according to (53), and the three equations thus formed will furnish the numerical values of A, J5, and (7. 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 } I>, and C will be derived. In like manner, the corrections to be applied to the computed latitudes may be determined. These corrections being applied, the ephemeris thus obtained may be assumed to represent the apparent path of the body with great precision, and may be em- ployed as an auxiliary in determining the values of the differential coefficients of a and 3 with respect to t. Let f(a) denote the right ascension of the body at the middle epoch or that for which h = Q, and let /(a nco) denote the value of a for any other date separated by the interval rwo, in which at is the interval between the successive dates of the ephemeris. Then, if we put n successively equal to 1, 2, 3, &c., we shall have 336 THEORETICAL ASTRONOMY. Function. I. Diff. II. Diff. III. Diff. IV. Diff V. Diff. P^SPsfciS-^i g^|li^lisii^*f The series of functions and differences may be extended in the same manner in either direction. If we expand f(a + not) into a series, the result is f(ct -4- nut} = a -I z VHD -4- 4 -^ ?i 2 w 2 -I- i - ?i 3 o> 3 -4- Tf^r -rr-7- 7i 4 a>* -4- &c.. ^ V^ I '" J " \ Ji i - ^2 I a Jtf l iJ4 ,7M or, putting for brevity A==-^ra) 9 B = ^~rp w 2 , &c., j( a _|_ nw ) = a _j_ An -f ?i 2 + (7?i 3 + D?i 4 + &c. If we now put % successively equal to 4, 3, 2, 1, 0, -f-1, &c., we obtain the values of f(a 4(o),f(a 3 TS~ & c - by means of the ephemeris, and then find -7- and -^ dr at' at dr directly from the normal places or observations. Thus, let a, a/, a" be three observed right ascensions corresponding to the times t y t f , t ff 9 and we shall have which give These equations, being solved numerically, will give the values of -77 f/^rt and , and we may thus by triple combinations of the observed ctz 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 -3- may be derived. at at 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 A + Bh + Ch? = AC/, A' -f B'h + C'K = A<5', h being the interval between the middle date t f and that of the place used, the values of A, JB, C, A f , -f CnW = Aa, A' -j- The unit of h may be ten days, or any other convenient interval, observing, however, that nco in the last equations must be expressed in parts of the same unit. With the ephemeris thus corrected, we compute the values of -=-, -^-, 37, and -j- as already explained. These Ctv (A/L CLv CLu differential coefficients should be determined with great care, since it is on their accuracy that the subsequent calculation principally de- pends. We compute, also, the velocities -37-, -^-, and -3- by means 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 ^ = A cos d ; and by means of the equation (40) or (39) we compute the value of 3j- It will be observed that if we put the equation (40) in the form d/>_P c p the coefficient -^ remains the same in the two hypotheses. The three equations (38) may be so combined that the resulting value of ~ will not contain ^. 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 A by means of the equations (106) 3 , and the rec- tangular co-ordinates from x = r cos b cos I, y = r cos b 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) w (47), (49), &c. "With the elements thus derived we compute the geocentric places for the dates of the normals, and find the differences between computation and observation. Then a second system of elements is computed from A -f- dJ f and compared with the observed places. Let the difference between computation and observation for either of the two spherical co-ordinates be denoted by n for the first system of elements, and by n f for the second system. The final correction to be applied to J, in order that the observed place may be exactly represented, will be determined by ^-(n'-n) + n = 0. (56) Each observed right ascension and each observed declination will thus furnish an equation of condition for the determination of A J, observing that the residuals in right ascension should in each case be multiplied by cos d. Finally, the elements which correspond to the geocentric distance J -f A// 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 fl ', x, and known quantities is given. For the special case of parabolic motion, this additional equation involves only the interval of time, the two radii-vectores, and the chord joining their extremities. The corresponding equation for the general conic section involves also the semi-transverse axis of the orbit, and hence, if the ratio M of the curtate distances is known, this equation will, in connection with the equations (52) 3 , enable us to find the values of r and r" corresponding to a given value of a. To derive this expression, let us resume the equations 340 THEOKETICAL ASTRONOMY. 4 = E" - E - 2e sin.i (E" - E) cos -J (E" + E\ ,_ a^ r + /' = 2a 2ae cos (E" E} cos | (E" + -#) For the chord K we have x = (r + r") 2 4rr" cos 2 j (u" u), which, by means of (58) 4 , gives K 2 = (r + r") 2 - 4a 2 (cos 2 i (&'E)-to cos J (^/ E) cos| (^^+^J)+e 2 cos 2 and, substituting for r -\- r ff its value given by the last of equations (57), we get x 2 = 4a 2 sin 2 -J (" E) (1 e 2 cos 2 i(JE" + ^)). (58) Let us now introduce an auxiliary angle h, such that the condition being imposed that h shall be less than 180, and put then the equations (57) and (58) become = 2<7 2 sin g cos /&, r" = 2a(l cos ? cos A), x = 2a sin may be found by means of the equations (87) 4 or (88), 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 e 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 e sine 342 THEORETICAL ASTRONOMY. may be expressed by a series which converges rapidly when e is small. Thus, let us put e sin e = y sin 3 ^e, x = sm 2 -\s, and we have rJL 2 COS6C ^ - |y COt |e, cfe _ Therefore dfy _8 6ycos^e_4 3y(l 2aQ cfo sin 2 e 2# (1 x) If we suppose y to be expanded into a series of the form y = a + fa + r& + **+ &c., we get, by differentiation, and substituting for -- the value already obtained, the result is 2,3* -f (4 r 2/5) ^ + (6<5 4 r ) a 8 + &c. = 4 3a + (60 3/3) x -f (6/3 3r) ic 2 + (6r 3d) ^ + &c. Therefore we have 4 3a:=0, 6a 3/3=2/3, 6/3 3 r = 4 r 2/3, 6 r 35 = 65 4r, from which we get 4.6 4.6.8 4.6.8.10, >&C ' 35' 7 3.5.7.9 Hence we obtain and, in like manner, ^BWs*^B^^ which, for brevity, may be written sind = JC' sin 8 i, RELATION BETWEEN TWO PLACES IN THE ORBIT. 343 Combining these expressions with (61), and substituting for sin^s and sin ^3 their values given by the equations (60), there results 6r' = Q (r + r" + x)* + q (r + r" - x)t, (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 f 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 (cosje)"^ 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 (cosle)-V = (1 _ sin'ie)-* = 1 + f sm'le + |^ sin^e , 6 . 11 . 16 . + 5.10.15 Sm * + &C " and if we put Q= , *\... (66) (cos Je) * we shall have JS = 1 + T f 5 sm* ie -f- 2 V 2 3 5 sin 6 Je -f Ac. (67) In a similar manner, if we put g= *'i.. (68) (cos|<5)s we find -Bo' = 1 -f T?5 sin 4 i* + 3 V& sin 6 ^ + Ac. (69) Table XV. gives the values of J5 or JB ' corresponding to e or d from to 60. For the case of parabolic motion we have and the equation (65) becomes identical with (56) 3 . In the application of these formulae, we first compute and 3 by means of the equations (60), and then, having found ^ and BJ by means of Table XV., we compute the values of Q and Q f from (66) and (68). Finally, the time T '=k(t"t) will be obtained from (65), and the difference between this result and the observed interval will 344 THEOEETICAL ASTRONOMY. indicate whether the assumed value of x must be increased or di- minished. A few trials will give the correct result. 117. Since the interval of time t" t cannot be determined with sufficient accuracy from (65) when the chord JC is very small, it becomes necessary to effect a further transformation of this equation. Thus, let us put Q q = 6P, x = sin 2 ie, x r = sin 2 \d, and we shall have Now, when K is very small, we may put COS^e COS \d, and hence / ... . , sn a 4e sin 2 ^d x x' = sin 2 4e sm 2 4 the equation (65) becomes, using only the upper sign, (r + r" + x)i - (r + r" - x)t = 6r ', (72) which is of the same form as (56) 3 . Hence, according to the equa- tions (63) 3 and (66) 3 , we shall have X =vl|^> < 73 > the value of p. being found from Table XI. with the argument 1 = -. 2T , a- (74) RELATION BETWEEN TWO PLACES IN THE ORBIT. 345 It remains, therefore, simply to find a convenient expression for r/, and the determination of K is effected by a process precisely the same as in the special case of parabolic motion. Let us now put P_ _x_ N ~Q ~ 40a ' costs' and we shall have . 2.8 . 3.8.10 . 4.8.10.12 . or, substituting for Q its value in terms of sin Je, N= 1 + A sin 2 |e + ^ sin 4 ^e + gj^\ sin 6 Je -f &c. (75) Therefore^ if we put the expression for r ' becomes V = T ^-Ar '. (77) Table XV. gives the value of log N corresponding to values of e from = to e = 60. If the. chord K is given, and the interval of time t" t is required, we compute AT/ by means of (76), and, having found r 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 -f r" K 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 + r" -f x = 2 (r + r") sin 2 (45 + /), r -f r" x = 2 (r + r") cos 2 (45 + /). 118. In the case of hyperbolic motion, the semi-transverse axis is negative, and the values of sin e and sin J# 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 x by means of the auxiliary angles e and 3. Let us, therefore, put sin 2 e = m 2 , sin 2 ^ == ri* ; then, when a is negative, m and n will be real. Now we have s = sn m* t id = sin ~ and RELATION BETWEEN TWO PLACES IN THE ORBIT. 347 Hence we derive e = 2 sin V m 2 = / _ . log e (l/l + m 2 -f m), = 2 sin ~ l-^tf = Substituting these values in the equation (61), and writing a in- stead of a, since sin e == 2m 1/^T - l/l + m*, we shall have = 2m i/l + m * - 2 log e (l/I+^ -f m) (79) +7? - 2 lo ge (1r+TT 2 + n)), 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 regarding the hyperbolic semi-transverse axis a as positive, the for- mula (79) will determine the interval of time T' = Jc (t ff 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 log e (/I + m 2 + m) = am -f /?m 2 + ?-m 3 + (5m 4 -f- &c., we get, by differentiation, 3/m 2 -f 4 3 + 1 |m 5 4 ^m T + Ac. 348 THEORETICAL ASTRONOMY. We have, also, 2m l/r+m = 2m + m 3 \m b + ^ m 7 &c. Therefore, 2m i/l 4- m 2 2 log e (l/l + m 2 + m) = 4 m (l A.'m' + * m*&C \ ^) and similarly 1/1 + n* 2 loge (1/1 + n* + n) = Substituting these values in the equation (79), and denoting the series of terms enclosed in the parentheses by and Q f , respectively, we get 6r' = Q (r + r" + x)i + ' (r + r" - x)f (83) which is identical with equation (65). If we replace m 2 by sin 2 ^e and n 2 by sin 2 ^5 in the expressions for Q and ', as given by (81) and (82), we shall have the expressions for these quantities in terms of sin |e and sin | and a, and then substitute the resulting values of Q and Q' in the equation (65), we obtain 1 7 7 (85) + sis -, ((? + r" + x)* =p (r + r" - x)*) + &c., 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 K for the cases of elliptic and hyperbolic orbits, enable us to correct an approximate orbit by varying the semi-transverse axis a and the ratio M of two curtate distances. But since the formula? will 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 M 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), h cos C sin (H a") =? sin (a" a), (86) h sin C = M tan 8" tan 8, from which to find H 9 , and h ; and also cos

, p' t p" denote the curtate distances with respect to the equator, A, A', A" the. right ascensions of the sun, and D, D', D" 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 (of A)) + R' cos D' sin (a' A*) - n" (p" sin (a!' a') + R" cosD" sin (a' 4")), and hence nR cos D sin (a' A)- R' cos D' sin (a!A'}-\-n"R" cos D" sin (a! A") pn" sin (a" a') ~ VARIATION OF TWO RADII-VECTORES. 357 This formula, being independent of the declination S f , 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 " if sin (a' a) M= if t ' sin (a" a')' and, by means of the results obtained from this hypothesis, the com- plete expression (93) may be computed. By a process identical with that employed in deriving the equation (36) 3 , we derive, from (93), the expression _i r ^_( f , ^x/ 1 1 \ -R' cos D' sin (a' A') *?TV* ;\/3 jpjj sin (a" a') and, putting -.. n sin (a' a) ~~^ 77 ' Sin (a" a')' __ 1 _ 1 ^ IlV ' i "-) cos D' sin (a/ A') R_l^ 1_\ 5 n 'T" ^ sin(a' a) ' p \r' 3 K'*}' we have M= P ~==M F. (95) The calculation of the auxiliary quantities in the equations (52) 3 will be effected by means of the formula (96) 3 , (86), (87), (102) 3 , and (51) 3 . The heliocentric places for the times t and t" will be given by (106) 3 and (107) 3? and from these the elements of the orbit will be found according to the process already illustrated. 124. The methods already given for the correction of the approxi- mate elements of the orbit of a heavenly body by means of additional observations or normal places, are those which will generally be applied. There are, however, modifications of these which may be advantageous in rare and special cases, and which will readily suggest themselves. Thus, if it be desired to correct approximate elements by varying two radii-vectores r and r fr , 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 THEORETICAL ASTRONOMY. values of J", b", and I". It will be most convenient to compute the angles $ and $", and then find z and z" from E sin $ ,, R" sin 4" sin 2 = - ? sin z = - Tt - r T or, putting a? = r* R 2 sin 2 ^, and x" 2 = r" 2 R" 2 sin 2 4/', from R sin * R" sin 4" tan z = , tan z" - , . x x The curtate distances will be given by the equations (3), and the heliocentric spherical co-ordinates by means of (4), writing r in place of a. From these u" u may be found, and by means of the values of r, r ff y and u" u the determination of the elements of the orbit may be completed. Then, assigning to r an increment dr, we com- pute a second system of elements, and from r and r" -f dr" 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 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 riri-.Q = - Brin <'- g (96) from which I may be found ; and in a similar manner we may find I". If we put * 2 = r 2 -^sm'(A-0), we have tan(;-A) = * sin ( A - ). (97) X Q Let S denote the angle at the sun between the earth and the place of the planet or comet projected on the plane of the ecliptic ; then we shall have VARIATION OF TWO RADII-VECTORES. 359 =180-}- Q I, *(l 0) (98) P = sin (I and tenb^-^Z, (99) r o by means of which the heliocentric latitudes b and b" may be found. The calculation of the elements and the correction of r Q and r " are then effected as in the case of the variatioVi 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 , r /r , and #, v 1 , v" will be given immediately. Then from r, r', r ff and the observed spherical co-ordinates of the body we may compute the values of u 1 ' u f and u r u. In the same manner, by means of the observed places, we compute the angles u" u f and u' u corresponding to q-\-dq and T, and to q and T -\- ST y dq and dT denoting the arbitrary increments assigned to q and T, respectively. The comparison of the helio- centric motion, during the intervals t" t f 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 enables us to form the equations by which we may find the cor- rections Ag and AT to be applied to the assumed values of q and T, respectively, in order that the values of u ff u r 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 THEOEETICAL ASTRONOMY. CHAPTER VIL METHOD OF LEAST SQUARES, THEORY OF THE COMBINATION OF OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OF OBSERVATIONS. 125. WHEN the elements of the orbit of a heavenly body are known to such a degree of approximation that the squares and products of the corrections which should be applied to them may be neglected, by computing the partial differential coefficients of these elements with respect to each of the observed spherical co-ordinates, we may form, by means of the differences between computation and observa- tion, the equations for the determination of these corrections. Three complete observations will furnish the six equations required for the determination of the corrections to be applied to the six elements of the orbit; but, if more than three complete places are given, the number of equations will exceed the number of unknown quantities, and the problem will be more than determinate. If the observed places were absolutely exact, the combination of the equations of condition in any manner whatever would furnish the values of these corrections, such that each of these equations would be completely satisfied. The conditions, however, which present themselves in the actual correction of the elements of the orbit of a heavenly body by means of given observed places, are entirely different. When the observations have been corrected for all known instrumental errors, and when all other known corrections have been duly applied, there still remain those accidental errors which arise from various causes, such as the abnormal condition of the atmosphere, the imperfections of vision, and the imperfections in the performance of the instrument employed. These accidental and irregular errors of observation cannot be eliminated from the observed data, and the equations of condition for the determination of the corrections to be applied to the elements of an approximate orbit cannot be completely satisfied by any system of values assigned to the unknown quantities unless the number of equations is the same as the number of these unknown quantities. It becomes an important problem, therefore, to determine the par- ticular combination of these equations of condition, by means of which METHOD OF LEAST SQUARES. 361 the resulting values of the unknown quantities will be those which, while they do not completely satisfy the several equations, will afford the highest degree of probability in favor of their accuracy. It will be of interest also to determine, as far as it may be possible, the degree of accuracy which may be attributed to the separate results. But, in order to simplify the more general problem, in which the quantities sought are determined indirectly by observation, it will be expedient to consider first the simpler case, in which a single quantity is obtained directly by observation. 126. If the accidental errors of observation could be obviated, the different determinations of a magnitude directly by observation would be identical ; but since this is impossible when an extreme limit of precision is sought, we adopt a mean or average value to be derived from the separate results obtained. The adopted value may or may not agree with any individual result, since it is only necessary that the residuals obtained by comparing the adopted value with the observed values shall be such as to make this adopted value the most probable value. It is evident, from the very nature of the case, that we approach here the confines of the unknown, and, before we pro- ceed further, something additional must be assumed. However irregular and uncertain the law of the accidental errors of observation may be, we may at least assume that small errors are more probable than large errors, and that errors surpassing a certain limit will not occur. We may also assume that in the case of a large number of observations, errors in excess will occur as frequently as errors in defect, so that, in general, positive and negative residuals of equal absolute value are equally probable. It appears, therefore, that the relative frequency of the occurrence of an accidental error J in the observed value will depend on the magnitude of this error, and may be expressed by

(d) must be the same. Hence, in a given series of observations, the number m of observations being supposed to be large, the number of times in which the error J occurs will be expressed by my ( J), and the number of times in which the error A' occurs will be expressed by m

( J) -f- m? ( J') -f- my ( J") -f- &c., 362 THEORETICAL ASTRONOMY. or J ? (J) = 1. The sum I must be taken between the limits for which the accidental errors of observation are considered possible ; but since the assignment of these limits is, in a certain sense, arbitrary, we must evidently have A=+co O>) = 1, (l) the value of tp (A) being absolutely zero for the limits 4- oo and oo. Within any given limits there are an infinite number of values, any one of which may possibly be the true value of J, and hence the number of the functions expressed by tp (A) must be infinite. The probability of an error A is expressed by tp ( J), and will be the same as the probability that the error is contained within the limits A and A -f dA. The latter is expressed by the sum of all the functions tp (A) between the limits A and A -f- dA, or by , We conclude, therefore, that the probability that an error falls between the limits a and b is expressed by the integral and this integral, taken so as to include all possible accidental errors of observation, is, according to equation (1), (2) According to the theory of probabilities, the probability that the errors A, A', &c. occur simultaneously is equal to the continued pro- duct of the probabilities of the occurrence of these errors separately. Let P denote the probability that these errors occur at the same time in the given series of observed values, and we have P=9>GO.pCd').?0*") ..... (3) The most probable value of the quantity sought, which we will de- note by x, must evidently be that which makes P a maximum. If METHOD OF LEAST SQUARES. 363 we take the logarithms of both members of equation (3), and differ- entiate, the condition of a maximum gives . &c- 4) dA dx dA' dx Let n, n f , n", &c. be the observed values of x, and m the number of observations ; then we have and hence ___ _i dx dx ~ dx Therefore the equation (4) becomes = d log f (n - s) d log p (n' - ) &g c? (n a;) d (X $) This equation will serve to> determine the value of x as soon as the form of the function symbolized by tp is known. It becomes neces- sary, therefore, to make some further assumption in regard to the errors J, J', A n ', &c., in order that the form of this function may be determined; and, although the hypothesis which presents itself gives directly the most probable value of x, since the function

(J):=:&JdJ, the integration of which gives log e c being the constant of integration. From this equation there results , ., ^feA2 , >. ^ ( J) = ce , (8) in which e is the base of Naperian logarithms. Since

?. METHOD OF LEAST SQUAEES. 369 130. Let us denote by v, v f , v", &c. the differences between any assumed value of x and the observed values for a given series of observations, the number of observations being denoted by m; then, if we put [w] = v* + v' 2 + v" 2 + Ac., (24) and similarly in the case of the sum of any other series of similar terms, we shall have for the probability of the value x, 9 3 [w] . _ _. and this probability will be a maximum when [wi] is a minimum. Now we have v = n x n v f n' x n v" = n" x n &c., n, n', n h ', &c. being the observed values of x, and hence [wj] = [rm] 2 [n] x, -\- mx? It appears, therefore, that '[wi] will be a minimum when *, = M (26) and this is a necessary consequence of the assumption that the arith- metical mean of the observations gives the most probable value of a?, according to which the form of the function

and, according to (21), r == 0.6745 \-~^U (31) These equations give the values of the mean and probable errors of a single observation in terms of the actual residuals found by com- paring the arithmetical mean with the several observed values. The probable and the mean error of the arithmetical mean will be given by pZ (32) r o = - 6745 V m (^l 1 y When the number of observations is ve^y large, the probable error of an observation and also that of the arithmetical mean may be de- termined by means of the mean of the errors. If we suppose the number of positive errors to be the same as the number of negative errors, the mean of the errors without reference to the algebraic sign gives and hence we have, according to (23), r== 0.8453 E4 (33) m For the mean error of an observation we have e = ^i/ji:= 1.2533^1 (34) m 372 THEOEETICAL ASTKONOMY. If the number of observations is very great, the results given by these equations will agree with those given by (30) and (31); but for any limited series of observed values, the results obtained by means of the mean error will afford the greatest accuracy. 132. The relative accuracy of two or more observed values of a quantity may be expressed by means of what are called their weights. If the observations are made under precisely similar circumstances, so that there is no reason for preferring one to the other, they are said to have the same weight. The weight must therefore depend on the measure of precision of the observations, and hence on their probable errors. The unit of the weight is entirely arbitrary, since only the relative weights are required, and if we denote the weight by p, the value of p indicates the number of observations of equal accuracy which must be combined in order that their arithmetical mean may have the same degree of precision as the observation whose weight is p. Hence, if the weight of a single observation is 1, the arithmetical mean of m such observations will have the weight m. Let the pro- bable error of an observation of the weight unity be denoted by r, and the probable error of that whose weight is p r by r f -, then, ac- cording to the first of equations (28), we shall have or For the case of an observation whose weight is p ff and whose pro- bable error is r" 9 we have from which it appears that the weights of two observations are to each other inversely as the squares of their probable or mean errors, and, according to (18), directly as the squares of their measures of precision. Let us now consider two values of x, which may be designated by x r and x", the mean errors of these values being, respectively, e' and e" then, if we put X=x'x" and suppose that both x' and x" have been derived from a large num- ber m of observations (and the same number in each case), so that the residuals v,, v/, v'", &c. in the case of x r and the residuals v,, v/, v,", &c. in the case of x" may be regarded as the actual errors of obser- METHOD OF LEAST SQUARES. 373 vation, the errors of the value of X, as determined from the several observations, will be v v f , v f v,', v" v", &c. Let the mean error of X be denoted by E; then we have S( V v y = |>] 2 [w,] + [ W ] ; and since the number of observed values is supposed to be so great that the frequency of negative products vv, is the same as that of the similar positive products, so that [vv,~] = 0, this equation gives or E 2 s' 2 -f- e" 2 . Combining X with a third value x" f whose mean error is s r// , the mean error of x f x" x'" will be found in the same manner to be equal to e /2 -f- e //2 + e ///2 ; and hence we have, for the algebraic sum of any number of separate values, E = l/ 2 -f-e' 2 + e" 2 -J-&c., (35) and, according to the last of equations (21), R = Tr 2 + r' 2 + r" 2 + &c., (36) R being the probable error of the algebraic sum. If the probable errors of the several values are the same, we have and the probable error of the sum of m values will be given by jR = rl/ra. Hence the probable error of the arithmetical mean of m observed values will be r = - = r m v in which agrees with the first of equations (28). Let P denote the weight of the sum X, p r the weight of x' t and p lf that of x n ; then we shall have p __ r ' 2 + r" 2 p ~ r '/2 ; 374 THEORETICAL ASTRONOMY. from which we get (37) Since the unit of weight is arbitrary, we may take and hence we have, for the weight of the algebraic sum of any number of values, P ~~" W = r' 2 + r" 2 + r'" 2 + &c.' or, whatever may be the unit of weight adopted, P=I r-^i - : (39) . J __ __ i -- __ L p' " p" ~T~ _p x// ~! In the case of a series of observed values of a quantity, if we designate by 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 according to (36), will be ^*+*$** r Q being the probable error of the arithmetical mean. Hence we derive m I 7 and if we adopt the value r' = 0.8453 &2 m the expression for the probable error of an observation becomes r = 0.8453 M (40) l/m(m 1) in which [v] denotes the sum of the residuals regarded as positive, and m the number of observations. 133. Let n, n', 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 n ', &c. We thus resolve the given values into P -f~ p f -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. _ f . ' ' 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'J P") <& c - are fractional as when they are whole numbers. The weight of X Q as determined by (41) is expressed by the sum p+p'+p"+p"'+&c., and the probable error of x is given by r, r, (42) when r, denotes the probable error of an observation whose weight is unity. The value of r, must be found by means of the observa- tions themselves. Thus, there will be p residuals expressed by n x w p r residuals expressed by n r o? , and similarly in the case of n", n'", &c. Hence, according to equation (31), we shall have r, = 0.6745 \/!^l (43) * m 1 in which m denotes the number of values to be combined, or the number of quantities n, n', n n ', &c. For the mean error of X Q , we have the equations (44; If different determinations of the quantity # are given, for which the probable errors are r, r', r", &c., the reciprocals of the squares of these probable errors may be taken as the weights of the respective values n, n f , n n ', &c., and we shall have (45; - * r 2 -r r / 2 -r p?2 376 THEOKETICAL ASTRONOMY. with the probable error /I , 1 , 1 , v^+^+^+---- (46) The mean errors may be used in these equations instead of the pro- bable errors. 134. The results thus obtained for the case of the direct observa- tion of the quantity sought, are applicable to the determination of the conditions for finding the most probable values of several un- known quantities when only a certain function of these quantities is directly observed. In the actual application of the formulae it will always be possible to reduce the problem to the case in which the quantity observed is a linear function of the quantities sought. Thus, let V be the quantity observed, and , y, , &c. the unknown quan- tities to be determined, so that we have Let , J^o, , &c. be approximate values of these quantities supposed to be already known by means of previous calculation, and let x, y, 2, &c. denote, respectively, the corrections which must be applied to these approximate values in order to obtain their true values. Then, if we suppose that the previous approximation is so close that the squares and products of the several corrections may be neglected, we have T . __ dV , dV . dV V ~ V = dr+d5 y+ dC f + -" 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 -f du -\- ew -{-ft -f- n = 0, a'x + b'y -f c'z -f d'u + e'w +ft + n' = 0, (47) a"x + V'y + c"z -f d"u + e"w +f't + n"= 0, &c. &c. which may be extended so as to include any number of unknown quantities. If the number of equations is the same as the number of unknown quantities, the resulting values of these will exactly satisfy the several equations; but if the number of equations exceeds the number of unknown quantities, there will not be any system of METHOD OF LEAST SQUARES. 377 values for these which will reduce the second members absolutely to zero, and we can only determine the values for which the errors for the several equations, which may be denoted by v, v f , v", &c., will be those which we may regard as belonging to the most probable values of the unknown quantities. Let J, A', 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

J " ~~~J ax ay az 378 THEOEETICAL ASTRONOMY. or dv <. ,dv f ,,dv" V-j- 4- tf-y- +V-J- + ---- = dv . ,dv' ,,dv" A , A(:i ^ + +" + ---- =0 ' (48) dz dz Ac. Ac. If we differentiate the equations -}- by -\- ez ~\- du -{- ew -\-ft -\- n = v, -|- V y _{- c 'z + d'w -f- c]z+ [bd]u+ \be]w+ [&/] + [ft/i] = 0, [ac] x + [6c] y + [cc] z + [cd] u + [ce] w + [c/] < + [c?i] = 0, [ad] x + [ftd] y + [cd] z + [dd] u -f [de] w + [df ] * + [^] = 0, l " [ae] ^ -{- [5e] y -j- [ce] z -|- [de] it -f~ L ee ] w ~i~ [ e /] ^ ~\~ L en ] == ^j in which [aa] = aa + a'a' + a"a" + [aft] =06 + '&' + "&" + -... [ac] = ac + aV + a' r c" + . . . . \bb~] =bb + b'b' + b"b" -f . . . . &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 h?v 2 + h' 2 v /2 + h" 2 v" 2 -f &c. shall be a minimum, METHOD OF LEAST SQUARES. 379 each equation of condition must be multiplied by the measure of precision of the observation; or, since the weight is proportional to the square of the measure of precision, each equation of condition must be multiplied by the square root of the weight of the observa- tion, and the several equations of condition, being thus reduced to the same unit of weight, must be combined as indicated by the equa- tions (51). 135. It will be observed that the formation of the first normal equation is 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 flJ = a + #, y = ' + &, = o"+/J"$,&C. (53) The coefficients a, /?, a', /?', &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 #, 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 indetermination 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 #, y, z, &c. in terms of t, as in (53), the normal equation in /, 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 [ad] y [ad] [ad] [ad] [ad] [ad]' and if we substitute this for x in each of the remaining normal equa- tions, and put = E6.a, (54) [cc] - [ac] = - [a/] = [e/.l], (57) aw = ^.l], [e]-.[ att]== [ OT .l], (58) M -[5 M==C/ "- 1:i ' we obtain METHOD OF LEAST SQUARES. 381 [65.1] y + [6c.l] 2 + \bd.l~\ u -f [6e.l] u> -f [6/.1] < + [6n.l] = 0, [6c.l] y + [cc.l] 2 + [cd.l] u + [ce.l] w -f [c/.l] + [cn.l] aa 0, \bd.V\ y + [cd.l] 2 + [dd.l] M + [cfe.l] w + [d/.l] < + \_dn.l~] = 0, (59) + [ce.l] 2 + [.i] - o. These equations are symmetrical, and of the same form as the normal equations, the coefficients being distinguished by writing the numeral 1 within the brackets. The unknown quantity x is thus eliminated, and by a similar pro- cess y may be eliminated from the equations (59), the resulting equa- tions being rendered symmetrical in form by the introduction of the numeral 2 within the brackets. Thus, we put [Jo.1] = [cc.2], [L1] - [M.1] = [ed.2], M [fc.1] = [ce.2], [c/.l] - M [M.1] = [AJ.2], Qfa.1] - = [cn.2], [fa.l] - [in.1] = [dn.2], . and the equations become [cc.2] 2 + [cd.2] w + [ce.2] w + [c/.2] < + [m.2] = 0, [cd.2] 2 + [drf.2] w + [de.2] w + [d/.2] < + [dn.2] = 0, [ce.2] 2 + [de.2] u + [ee.2] w -f [e/.2] < + [e.2] = 0, 2 + [d/.2] t* + [e/.2] w + [//.2] < + |>.2] - 0. To eliminate 2 from these equations, we put [crf.2] = [.3] = 0, Again we put, in a similar manner, [".3] - [-4]^0. Finally, to eliminate w, we put [/.4]-l [e , l . 4 ] = [/ re .5], (71) and the resulting equation is [#5]* + [>.5] = 0, (72) which gives DPT 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 tee ' 2] 4- [c/2] t I 0^-0 [^2l W f [cc.2] f [ce.2] -"' [333} [] <+ [jPT =0> 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 : [/?r.(A + D], (75) in which a, /5, 7-, denote any three letters, and ft 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 formula, when so many quantities are to be computed, it becomes important to be able to check the accuracy of the calculation in its successive stages. First, then, to prove the calculation of the coefficients in the normal equa- tions, we put a -f b +c +d +e +/=*, a' + V + c' + d' + e r -f /' = s' y &c. If we multiply each of the sums thus formed by the corresponding absolute term n, and take the sum of all the products, we have 384 THEORETICAL ASTRONOMY. [an] + [bvi] -f [cii] + \dn\ + M + |>] = M- (76) In a similar manner, multiplying by each of the coefficients in the original equations of condition, we find [ad] + lab'] + [ac] -f [ad] + [oe] + [a/] = [as], [a6] + [55] + M + Pfl + [&] + Kf] = M, [ac] + [6c] + [cc] + [cd] + [ce] + [cf] = M, , [ad] + [bd] 4- [cd] + [eta] + M + W\= [ac] + [6c] + [ce] + [de] + [ee] ] 4- Hence it appears that if we compute the sums s, s', s", s" f , &c., and form [as], [bs], [cs], &c. simultaneously with the calculation of the coefficients in the normal equations, the equation (76) must be satis- fied when the absolute terms of the normal equations are correct; and the equations (77) must be satisfied when the coefficients of the unknown quantities in the normal equations are correct. The accuracy of the calculation of the auxiliary quantities sym- bolized by the equation (75) may be proved in a similar manner. Thus, we have which, by means of the first and second of equations (77), becomes or [fo.l] = [66.1] + [6c.l] + [6dl] + [66.1] + [6/.1] ; (78) and similarly we derive the expressions for [cs.l], [ds.l], &c. It is obvious, therefore, that the calculation of the coefficients in the equa- tions (59), (64), (68), and (70) will be checked as in the case of the coefficients in the normal equations, the auxiliaries depending on s being determined as if s, s r , s", &c. were the coefficients of an addi- tional unknown quantity in the several equations of condition. Hence we must have, finally, CA5] = [//.5], [n.5] = [>.5]. (79) If we multiply each of the equations (49) by its v, and take the sum of the several products, we get [av] x + [6r] y -f [cv] z + [dv] u + [ev\ w + [/v] < + [vn] = [w], METHOD OF LEAST SQUARES. 385 But, according to the equations (48) and (50), we have, for the most probable values of the unknown quantities, lav] = 0, Ibv] = 0, lev] = 0, &c. ; and hence \m\ = [w]. (80) If we multiply each of the equations (49) by its n, and take the sum of all the products thus formed, substituting [vv\ for [vri] 9 there re- sults Ian] x -|- \bn\ y -f [cri] z -\- \dn] u -f \_eri] w -f [jri] t -f- [nri] = [vv\. Substituting in this the value of x given by the first normal equa- tion, it becomes |>i.l] y + [CTI.I] z -f [dn.V] u + \_en.l~] w + [/ra.l] t + [rw.l] == [vv], in which [ re .l] = []-gl[a re ]. (81) Lo-j 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 + [e.2] w + Q/H.2] < + [nn.2] = [w], [c^w.3] u + [i.3] w + [>.3] < + [wn.3] = [tw], [eri.4] w -f L/H.4] < + [.4] = [w], (82) The expressions for the auxiliaries [ww.2], [nw.3], &c. are [nn.2] = [nn.1] - [g~ ] [*n.l], [nn.8] = [nn.2] - M [m . 2]> [nn.4] = [nn.3] - [dn.3], [.5] = [nn.4] - jgg [n.4], /.5]. (83) The process here indicated may be readily extended to the case of a greater number of unknown quantities, and we have, in general, when fj. denotes the number of unknown quantities, Ivv] = Inn-.fj."]. (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 n ', &c. Then, taking the sum of the squares of these, the equation (84) must be satisfied within the limits of the unavoidable errors of calculation with the logarithmic tables em- ployed. If this condition is satisfied, it may be inferred that the entire calculation of the values of the unknown quantities from the given equations of condition is correct. 138. If the values of x, y, z, &c. thus found were the absolutely exact values, the residuals v, v', v", &c. would be the actual errors of observation. But since the results obtained only furnish the most probable values of the unknown quantities, the final residuals may differ slightly from the accidental errors of observation. Further, it is evident that the degree of precision with which the several unknown quantities may be determined by means of the data of the problem may be very different, so that it is desirable to be able to determine the relative weights of the different results. It will be observed that the expressions for either of the unknown quantities resulting from the elimination of the others is a linear function of n, n f , n", &c., so that we have x + an + M -f o V + o!"n'" +....=30, (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 .!] = [>.2] = |>.3] = \JnA-] = [>.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] Hence the equation (73) becomes [ee.4] and substituting this value of t in the last of equations (70), we get or .4 (94) 390 THEORETICAL ASTRONOMY. which gives the weight of w in terms of the auxiliary quantities required in the determination of its most probable value. ' If the order of elimination is now completely reversed, so that x is made the last in the elimination, the weights of x and y will be determined by the equations f) = [aa.5], (95) A third elimination, in which z and u are the unknown quantities first determined, will give the weights of these determinations. It appears, therefore, that when only four unknown quantities are to be found, a single repetition of the elimination, the order of the quan- tities being completely reversed, will furnish at once the weights of the several results, and check the accuracy of the calculation. When there are only two unknown quantities, the elimination gives directly the values of these quantities and also of their weights. 140. In the case of three or more unknown quantities, the weights of all the results may be determined without repeating the elimina- tion when certain additional auxiliary quantities have been found. The weights of the two which are first determined are given in terms of the auxiliaries required in the elimination, that of the quantity which is next found will require the value of an additional auxiliary quantity, the succeeding one will require two additional auxiliaries, and so on. The equations (74) show that when the substitution is effected analytically the final value of x will have the denominator D == [oo] [66.1] [ce.2] [eta.3] [ee.4] [jfiT.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 of the quantity first determined; then we shall have = [aa] [66.1] [cc.2] [eta.3] [ee.4] [#5] = [aa] e [66.1] o [cc.2] e [dd.3 = [aa] d [66.1]Jcc.2] d [ee.3] d [# = [aa] B [66.1] c [dd.2] B [ee.3] c [jf/.4] c [cc.5] = M, [ec.l] [- 4 J C^- 3 ] [^2] [66.1] "'"" " r/V4i r^ QI ' r^7/7 on ' [ cc .l] [5^] by means of which the weights of the six unknown quantities may be determined. The process here indicated may be readily extended to the case of a greater number of unknown quantities. The equa- tion for p w is identical with (94), the expression for p u introduces the new auxiliary quantity [,]Qf.4] d , and that for p s introduces two new auxiliaries. The expressions for the new auxiliaries [jOf.4] d , [j(/".4] c , [ee.3] c , &c. are easily formed by observing that' all the auxiliaries as far as those which are designated by the numeral 4 are not affected by putting e or /last, that, as far as those which contain the numeral 3, it makes no difference whether d, e, or / is placed last, that those distinguished by the numerals 1 and 2 are not affected by making c, c?, e, or /the last, and that those designated by the numeral 1 are unchanged unless a is made the last. Thus, we obtain C//-4L = DP] - 392 THEORETICAL ASTRONOMY. and, also, (98) 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 formula, to eliminate first in the order x, 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 , w, u, z, y, x, and we shall have [aa.5] [aa.5] by means of which the weights of x, 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 #, let the first of these equations be multiplied by 1, the second by A f , 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 METHOD OF LEAST SQUARES. 393 To determine y from the last five of equations (74), let the eliminating factors be denoted by B" y B f ", B iv , and B v , and we shall have 01 OC. _L I . r \r f = =L _L If' [bb.i-] ^ ' Q L^^'^J _i_ L C ^'^J jyit I -ntn fbeV '21 W/31 (101) Lr^J I L OP -^J r | L^ C -J T>tn i TO V u rz,z, -IT i r on - "I r J^ OT - T~ - > _ . . . w . ~ [6O3 + [oc.2] f [rfd.3] f [ee.4] ^ 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 : 0.2] [rfe.3] + C " (102) e. " [rfd.3] * The expressions for the values of the unknown quantities will there- fore become - x = ^3 4- I^J .A'4- ^""^A" ^ dn '^ A'" 4- ^ enA ^A v ^ n '^ A* _ [6m.11 [c.2] [rfm.3] , [em.4] [/.5] ~~ [66.1] + [cc.2] ^ f f - 6 f ' _ 5 , n. en , - [cc.2] * [3d.3] * [eo.4] ^ [jflT.6] ' _n. e. , - + f ' 394 THEORETICAL ASTRONOMY. The first of these equations will give the reciprocal of the weight of x, when we put [an] = 1, and the other absolute terms of the normal equations equal to zero; the second will give the reciprocal of the weight of y by putting [bn] = - 1, and the other absolute terms of the normal equations equal to zero ; and, continuing the process, finally the last equation will give the reciprocal of the weight of t when we put fn = 1, and [an], [bn], [cn], &c. equal to zero. It remains, therefore, to determine the particular values of [6n.l], [CM. 2], &c., and the expressions for the weights will be complete. If we multiply the first of equations (100) by [an], it becomes [bn.l] = [an]A + [bn]. 104) Multiplying the second of equations (100) by [an], and the first of (101) by [bri], adding the products, and introducing the value of [6?i.l] just found, we get [cn^ - [cn.l] + [bn.l] + [an] A" + [bn] B" = 0, which reduces to [an] A" -f [bn] .B" -f [en] = [cn.2]. (105) Multiplying the third of equations (100) by [an~\, the second of (101) by [bn], and the first of (102) by [cn], adding the products, and re- ducing by means of (104) and (105), we obtain which, by means of the expressions for the auxiliaries, is further re- duced to [an] A" + [bn] B'" + [cn] C'" + [dn] = [dn.Z]. (106) In a similar manner we find, from the remaining equations of (100), (101), and (102), the following expressions : [an] A v -f [bn] B'"+ [cn] C* + [dn] D iv + [en] = [enA], [an] A v + [bn] B* + [cn] <7 V + [dn] Z> v + [en] E v + [fn] == [/n.5]. (107) The equations (104), (105), (106), and (107), enable us to find the particular values of [6n.l], [cn.2], &c. required in the expressions for the reciprocals of the weights. Thus, for the weight of x, we have [an] = 1, [bn] = [cn] = [dn] = [en] = [>] = ; METHOD OF LEAST SQUARES. 395 and these equations give [bn 11 = A 1 [fn 2] ^= 4." [dn 31 = A'" \_enA~] = A lv . [/ft-5] A 1 . For the case of the weight of y, we have [bn] == 1, [cm] = [en] = [dn\ = [eri] = \Jri\ 0, and the same equations give [6w.l] = 1, [c/i.2] =1 B", [eZw.3] = B" f , We have, also, for the weight of 2, for the weight of u, [d-1.3] 1, [e?i.4] D (v , [/w.5] _Z) V ; for the weight of w, 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 _L _j_ AA ' 4- A " A " _i A '" A> " . ^ iv ^ iv , AVA! Y x ~~ [aa] + [66.1] + [cc.2] H ~ [dd.3] + [ee.4] + [jfjf.5] ' j^~ z= [66JJ + [c^2j + "^"^"^ + ~~^ J I _J PH [^. 1 1 C'" C'" C [V C [V C V C V 1 The equations (103) and (108) will serve to determine separately the value of each unknown quantity and also that of its weight, the 396 THEORETICAL ASTRONOMY. auxiliary factors A', A", B", &c. having been found from the equa- tions (100), (101), and (102). If we reverse the operation and re- compose the equations (74) by means of the expressions for the un- known quantities given by (103), the conditions which immediately follow furnish another series of equations for the determination of the auxiliary factors. The equations thus derived will give first the values of A', B", C'", D iv , and v ; then, those of A", B" f , C iv , D v ; and so on. They are equally as convenient as those already given, provided that the values of all the unknown quantities are required as well as their respective weights. 142. The formuise already given for the relations between the data of the problem and the weights of the most probable values of the unknown quantities, are those which are of the greatest practical value. It will be apparent from what has been derived that there must be a variety of methods which may be applied, but that all of these methods involve essentially the same numerical operations. The peculiar symmetry of the normal equations affords also a variety of expressions applicable to the different phases under which the problem presents itself. According to the general theory of elimination, the expression for any unknown quantity, as determined from the normal equations, may be put in the form x = - jj [an-] - jj [bri] - ^ [>] - &c., (109) in which D is the determinant formed from all the coefficients of the unknown quantities in the normal equations, and in which A, A r , A fl ', &c. are the partial determinants required in the elimination. Thus, A is the determinant formed from the coefficients of all the unknown quantities except x, in all the equations except the first; A" is the determinant formed from the coefficients of y, z 9 &c. in all the equa- tions except the second; and the values of A" , A' n ', &c. are formed in a similar manner. Now, since the value of x which results when we put [an] = 1, and the other absolute terms of the normal equations equal to zero, is the reciprocal of the weight of the most probable value of this unknown quantity as given by (109), we have p. = 2- (no) In like manner, the expression for the most probable value of y will be METHOD OF LEAST SQUARES. 397 9= -[.] -f [*] -5 [o-&e- an) 5, J? r , .B", &c. being the, partial determinants formed when the co- efficients of y are omitted; and for its weigh^we have *>, = (112) The formulae for the most probable value of z and for its weight are entirely analogous to those for x and y, so that the process here indi- cated may be extended to the case of any number of unknown quan- tities. It appears, therefore, that the weight of the most probable value of any unknown quantity is found by dividing the complete determinant of all the coefficients by the partial determinant formed when we omit the normal equation corresponding particularly to this unknown quantity, and when we omit also the coefficients of this quantity in the remaining normal equations. The peculiar arrangement of the coefficients in the normal equa- tions abbreviates somewhat the expressions for the several determi- nants. Thus, in the case of three unknown quantities, we have A = [66] [cc] [6c] 2 , E' = [ad] [cc] ~ [ac] 2 , C" = [ad] [66] [a6] 2 , D = [aa] [66] [cc] -f 2[a6] [be] [ac] [aa] [6c] 2 [66] [ac] 2 [cc] [a6] 2 , which are all the quantities required for finding simply the weights of the most probable values of x, y, and z. The expression for the weight of z is When there are but two unknown quantities, we have A = [66], B' = [aa], D = [aa] [66] [a6] 2 , and hence [aa] [66] - [a6] 2 _ [aa] [66] - [a6] 2 [66] ** ~ [aa] When the number of unknown quantities is increased, the expressions for the determinants necessarily become much more complicated, and hence the convenience of other auxiliary quantities is manifest. 143. The case has been already alluded to in which the determina- tion of the values of the unknown quantities is rendered uncertain by the similarity of the signs and coefficients in the normal equations, 398 THEORETICAL ASTRONOMY. and in which the problem becomes nearly indeterminate. Sometimes it will be possible to overcome the difficulty thus encountered by a suitable change of the elements to be determined; but, generally, for a complete and satisfactory solution, additional data will be required. It often happens, however, that several of the unknown quantities may be accurately determined from the given equations when the values of the others are known, but that the certainty of the deter- mination of the same quantities is very greatly impaired when all the unknown quantities are derived simultaneously from the same equations. Let us suppose that one of the unknown quantities is, from the very nature of the problem, not susceptible of an accurate determination from the data employed. The equations will then present themselves in a form approaching that in which the number of independent relations is one less than the number of unknown quantities, so that it will be necessary to determine the other unknown quantities in terms of that whose value is necessarily uncertain. In this case the elimination should be so arranged that the quantity which is regarded as uncertain is that whose value would be first determined. Then, if its coefficient in the final equation, corre- sponding to (72), is very small, a circumstance which indicates at once the existence of the uncertainty when it is not otherwise sus- pected, the process of elimination should not be completed, and the auxiliary quantities should be determined only as far as those re- quired in the formation of the equation which corresponds to the first of (70). Thus, let be the uncertain quantity, and we have , \eeA which must be substituted for w in the first of equations (68). We thus obtain w, u, z, y, and x as functions of t. If the solution is effected by means of the equations (103), let x 09 y w z w &c. denote the values of these unknown quantities when we put = 0; and then we shall have x = _ W _ [foi.l] _ [en.2] , _ [efoi.3] O.4] ~ A = _ c^ ci. , [w.i] [CC.2J 1 333]* ~&A] B > _ _ [enA\ [cc.2] [V> C - (O 2 + E V E\ 2 , in which (ej, (e y ), &c. denote the mean errors of x , y w &c. These formulae show, also, that when one of the variables is neglected, the equations assign too great a degree of precision to the results thus obtained. When there are two or more unknown quantities which cannot be determined from the data with sufficient certainty, the problem must be treated in a manner entirely analogous to that here indicated; but, since cases of this kind will rarely, if ever, occur, it is not necessary to pursue the subject further. 144. The weights which are obtained for the most probable values of the unknown quantities enable us to find the mean and probable errors of these values. Let e denote the mean error of an observa- tion whose weight is unity; then the mean error of x will be (116) and, in like manner, the expressions for the mean errors of y, z, u, &c. will be It remains, therefore, to determine the value of e by means of the final residuals obtained by comparing the observed values of the function with those given by the most probable values of the va- 400 THEOKETICAL ASTRONOMY. riables. If these residuals were the actual fortuitous errors of obser- vation, the mean error of an observation would be r J m being the number of equations of condition. This value is evi- dently an approximation to the correct result; but since by supposing the residuals v, v', v", &c. to be the actual errors of the several ob- served values of the function, we assign too high a degree of pre- cision to the several results, the true value of must necessarily be greater than that given by this equation. Let the true values of the unknown quantities be x -j- A#, y -f- AT/, z -f- AS, &c., the substitution of which in the several equations of condition would give the residuals J, J', J", &c. ; then we shall have -f- b&y 4- c &z ~h d&u . . . . -\- v A, 4- b'y 4- C'AZ -f- d'*u . . . . + v f = A', &c. &c. If we multiply each of these equations by its J, and take the sum of all the products, we get ~[>J]AZ4 [&J]A 2/ -f-[ C J]A2+ [d/J]AW-j- ____ 4- [ V J] = [JJ]. But if we multiply each of the same equations by its v, take the sum of the products, and reduce by means of (48) and (50), we obtain M Wif and hence we derive [ J J] = [ W ] + [a J] X + [6J] AT/ + [ C J] A3 + [d J] AW -f- .... (119) If we form the normal equations from (118), it will be observed that they are of the same form as the normal equations formed from the original equations of condition, provided that we write J in place of n ; and hence, according to (85), we have Arc = a J -f- a'J' 4 a" A" 4- ..... We have, also, [oJ] =aA + a! A' 4. a" J" + ..... , and the product of these equations gives [a J] AZ = aa J 2 4- a' a' A"* -f a" a " J" 2 + ____ + aa'JJ'-}-aa"JJ"-{-.... The mean value of the terms containing JJ', JJ", &c. is zero, and COMBINATION OF OBSERVATIONS. 401 for the mean values of J 2 , J /2 , J //2 , &c. we must, in each case, write e 2 . Hence the mean value of the product [a J] &x will be and this, by means of the first of equations (88), is further reduced to [a J] Az i= e 2 . In a similar manner, we obtain the value s 2 for the mean value of each of the products [6^] AT/, [cJJAz, &c. Now, the terms added to [vv] in the second member of the equation (119) are necessarily very small, and, although their exact value cannot be determined, we may without sensible error adopt the mean values of the several terms as here determined, so that the equation becomes [JJ] = [>] + !^\ (120) u. being the number of unknown quantities. Therefore, since [ J J] = we 2 , we shall have t== m ft m by means of which the mean error of an observation whose weight is unity may be determined. When /* = 1, this equation becomes identical with (30). For the determination of the probable errors of the final values of the unknown quantities, if r denotes the probable error of an obser- vation of the weight unity, we have the following equations : r = 0.67449 (122) r r e r = -=, r v = ---=, &c. 145. The formulae which result from the theory of errors according to which the method of least squares is derived, enable us to combine the data furnished by observation so as to overcome, in the greatest degree possible, the effect of those accidental errors which no refine- ment of theory can successfully eliminate. The problem of the cor- rection of the approximate elements of the orbit of a heavenly body by means of a series of observed places, requires the application of nearly all the distinct results which have been derived. The first approximate elements of the orbit of the body will be determined from three or four observed places according to the methods which 26 402 THEOEETICAL 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, tjie 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 ff , &c. be the differences between com- putation and observation, in the case of either spherical co-ordinate, for the dates t, t f , t n ', &c., respectively; then, if the interval between the extreme observations to be combined in the formation of the normal place is not too great, and if we regard the observations as equally precise, the normal difference n 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 f , p", &c. are assigned to the observations, the value of n must be found from _np + n'p' + n"p"+.... ~7+i>' +*" + .... ' and the weight of this value will be equal to the sum COMBINATION OF OBSERVATIONS. 403 The date of the normal place will be determined by '+.... ' If the error of the ephemeris can be considered as nearly constant, it is not necessary to determine t Q 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 Q 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, A0 = A + Br + CV 2 , the coefficients A, B, and C being found from equations of condition formed by means of the several known values of A# in the case of each of the spherical co-ordinates. 146. In this way we obtain normal places at convenient intervals throughout the entire period during which the body was observed. From three or more of these normal places, a new system of elements should be computed by means of some one of the methods which have already been given; and these fundamental places being judi- ciously selected, the resulting elements will furnish a pretty close approximation to the truth, so that the residuals which are found by comparing them with all the directly observed places may be regarded as indicating very nearly the actual errors of those places. We may then proceed to investigate the character of the observations more fully. But since the observations will have been made at many dif- ferent places, by different observers, with instruments of different sizes, and under a variety of dissimilar attendant circumstances, it may be easily understood that the investigation \vill involve much that is vague and uncertain. In the theory of errors which has been developed in this chapter, it has been assumed that all constant errors have been duly eliminated, and that the only errors which remain are those accidental errors which must ever continue in a greater or less degree undetermined. The greater the number and 404 THEORETICAL ASTRONOMY. perfection of the observations employed, the more nearly will these errors be determined, and the more nearly will the law of their dis- tribution conform to that which has been assumed as the basis of the method of least squares. When all known errors have been eliminated, there may yet remain constant errors, and also other errors whose law of distribution is peculiar, such as may arise from the idiosyncrasies of the different observers, from the systematic errors of the adopted star-places in the case of differential observations, and from a variety of other sources; and since the observations themselves furnish the only means of arriving at a knowledge of these errors, it becomes important to discuss them in such a manner that all errors which may be regarded, in a sense more or less extended, as regular may be eliminated. When this has been accomplished, the residuals which still remain will enable us to form an estimate of the degree of accuracy which may be attributed to the different series of observations, in order that they may not only be combined in the most advantageous manner, but that also no refinements of calculation may be introduced which are not warranted by the quality of the material to be employed. The necessity of a preliminary calculation in which a high degree of accuracy is already obtained, is indicated by the fact that, however conscientious the observer may be, his judgment is unconsciously warped by an inherent desire to produce results harmonizing well among themselves, so that a limited series of places may agree to such an extent that the probable error of an observation as derived from the relative discordances would assign a weight vastly in excess of its true value. The combination, however, of a large number of independent data, by exhibiting at least an approximation to the absolute errors of the observations, will indicate nearly what the measure of precision should be. As soon, therefore, as provisional elements which nearly represent the entire series of observations have been found, an attempt should be made to eliminate all errors which may be accurately or approximately determined. The places of the comparison-stars used in the observations should be determined with care from the data available, and should be reduced, by means of the proper systematic corrections, to some standard system. The reduc- tion of the mean places of the stars to apparent places should also be made by means of uniform constants of reduction. The observations will thus be uniformly reduced. Then the perturbations arising from the action of the planets should be computed by means of formula? which will be investigated in the next chapter, and the observed COMBINATION OF OBSERVATIONS. 405 places should be freed from these perturbations so as to give the places for a system of osculating elements for a given date. 147. The next step in the process will be to compare the pro- visional elements with the entire series of observed places thus cor- rected; and in the calculation of the ephemeris it will be advan- tageous to correct the places of the sun given by the tables whenever observations are available for that purpose. Then, selecting one or more epochs as the origin, if we compute the coefficients A, J3, C in the equation A0 = A -f 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 C. It thus appears that the corrected ephemeris which is so essential to a determination of the constant errors peculiar to each series of observations, is obtained without first having determined the most probable system of elements. The cor- rections computed by means of the equation (125) being applied to the several residuals of each series, we obtain what may be regarded as the actual errors of these observations. The arithmetical or pro- bable mean of the corrected residuals for the series of observations made by each observer may be regarded as the average error of obser- vation for that series. The mean of the average errors of the several series may be regarded as the actual constant error pertaining to all the observations, and the comparison of this final mean with the means found for the different series, respectively, furnishes the pro- bable value of the constant errors due to the peculiarities of the observers; and the constant correction thus found for each observer should be applied to the corresponding residuals already obtained. In this investigation, if the number of comparisons or the number of wires taken is known, relative weights proportional to the number of comparisons may be adopted for the combination of the residuals for each series. In this manner, observations which, on account of the peculiarities of the observers, are in a certain sense heterogeneous, may be rendered homogeneous, being reduced to a standard which 406 THEOEETICAL ASTRONOMY. approaches the absolute in proportion as the number and perfection of the distinct series combined are increased. Whatever constant error remains will be very small, and, besides, will affect all places alike. The residuals which now remain must be regarded as consisting of the actual errors of observation and of the error of the adopted place of the comparison-star. Hence they will not give the probable error of observation, and will not serve directly for assigning the measures of precision of the series of observations by each observer. Let us, therefore, denote by e, the mean error of the place of the comparison-star, by e, the mean error of a single comparison; then will , be the mean error of m comparisons, and the mean error of V m the resulting place of the body will, according to equation (35), be given by m The value of e w 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 a is supposed to be determined in the formation of the catalogue of star-places adopted. Hence the actual mean error of an observa- tion consisting of a single comparison will be e, = l/m(e 2 e/). (127) The value of e, for each observer having been found in accordance with this equation, the mean error of an observation consisting of m comparisons will be e, Vm The mean error of an observation whose weight is unity being de- noted by s, the weight of an observation based on m comparisons will be *= (128) The value of e may be arbitrarily assigned, and we may adopt for it 1 10" or any other number of seconds for which the resulting values of p will be convenient numbers. When all the observations are differential observations, and the stars of comparison are included in the fundamental list, if we do not take into account the number of comparisons on which each observed COMBINATION OF OBSERVATIONS. 407 place depends, it will not be necessary to consider e a) and we may then derive e, directly from the residuals corrected for constant errors. Further, in the case of meridian observations, the error which corre- sponds to e a will be extremely small, and hence it is only when these are combined with equatorial observations, or when equatorial obser- vations based on different numbers of comparisons are combined, that the separation of the errors into the two component parts becomes necessary for a proper determination of the relative weights. According to the complete method here indicated, after having eliminated as far as possible all constant errors, including the correc- tions assigned by equation (125) to be applied to the provisional ephemeris, we find the value of e, given by the equation nsf = [mw] [m] e/, (129) in which n denotes the number of observations; m, m', m", &c. the number of comparisons for the respective observations; and v, v' } v ff , &c. the corresponding residuals. Then, by means of equation (128), assuming a convenient number for e, we compute the weight of each observation. Thus, for example, let the residuals and corresponding values of m be as follows : A0 m A0 m + 2".0 5, -1".0 7, - 1 .8 5, + 1 .5 5, - .4 10, +4 .1 8, - 5 .5 5, .0 5. Let the mean error of the place of a comparison-star be then we have n = 8, and, according to (129), 8e,'= 341.78 200.0, which gives e,= 4".2. Let us now adopt as the unit of weight that for which the mean erroi is 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 e. If it be de- sired to adopt a different unit of weight in the case of the solution of the equations of condition, such, for example, that the weight of an equation of average precision shall be unity, we may simply divide the weights of the normals by any number p Q which will satisfy the condition imposed. The mean error of an observation whose weight is unity will then be given by Vp. the value of e being that used in the determination of the weights p, p', &c. 148. The observations of comets are liable to be affected by other errors in addition to those which are common to these and to 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. 409 parisons, and e c the mean error to be feared on account of the pecu- liarities of the physical appearance of the comet, will express the mean error of the residuals; and if n of these residuals are combined in the formation of a normal place, the mean error of the normal will be given by e^M + e/. (130) TV The value of e c 2 may be determined approximately from the data furnished by the observations. Thus, if the mean error of a single comparison, for the different observers, has been determined by means of the differences between single comparisons and the arithmetical mean of a considerable number of comparisons, and if the mean error of the place of a comparison-star has also been determined, the equation (126) will give the corresponding value of 2 ; then the actual differences between computation and observation obtained by eliminating the error of the ephemeris and such constant errors as may be determined, will furnish an approximate value of e c by means of the formula in which n denotes the number of observations combined. Sometimes, also, in the case of comets, in order to detect the opera- tion of any abnormal force or circumstance producing different effects in different parts of the orbit, it may be expedient to divide the observations into two distinct groups, the first including the observa- tions made before the time of perihelion passage, and the other including those subsequent to that epoch. 149. The circumstances of the problem will often suggest appro- priate modifications of the complete process of determining the rela- tive weights of the observations to be combined, or indeed a relaxa- tion from the requirements of the more rigorous method. Thus, if on account of the number or quality of the data it is not considered necessary to compute the relative weights with the greatest precision attainable, it will suffice, when the discussion of the observations has been carried to an extent sufficient to make an approximate estimate of the relative weights, to assume, without considering the number of comparisons, a weight 1 for the observations at one observatory, a 410 THEORETICAL ASTRONOMY. weight | for another class of observations, for a third class, and so on. It should be observed, also, that when there are but few obser- vations to be combined, the application of the formulae for the mean or probable errors may be in a degree fallacious, the resulting values of these errors being little more than rude approximations ; still the mean or probable errors as thus determined furnish the most reliable means of estimating the relative weights of the observations made by different observers, since otherwise the scale of weights would depend on the arbitrary discretion of the computer. Further, in a complete investigation, even when the very greatest care has been taken in the theoretical discussion, on account of independent known circumstances connected with some particular observation, it may be expedient to- change arbitrarily the weight assigned by theory to certain of the normal places. It may also be advisable to reject entirely those observations whose weight is less than a certain limit which may be regarded as the standard of excellence below which the observations should be rejected; and it will be proper to reject observations which do not afford the data requisite for a homogeneous combination with the others according to the principles already explained. But in all cases the rejection of apparently doubtful observations should not be carried to any considerable extent unless a very large number of good observations are available. The mere apparent discrepancy between any residual and the others of a series, is not in itself sufficient to warrant its rejection unless facts are known which would independently assign to it a low degree of pre- cision. A doubtful observation will have the greatest influence in vitiating the resulting normal place when but a small number of observed places are combined ; and hence, since we cannot assume that the law of the distribution of errors, according to which the method of least squares is derived, will be complied with in the case of only a few observations, it will not in general be safe to reject an observation pro- vided that it surpasses a limit which is fixed by the adopted theory of errors. If the number of observations is so large that the dis- tribution of the errors may be assumed to conform to the theory adopted, it will be possible to assign a limit such that a residual which surpasses it may be rejected. Thus, in a series of m observa- tions, according to the expression (19), the number of errors greater than nr will be COMBINATION OF OBSEEVATIONS. 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 nhr r e - df== i_ */ _. 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' 0.67449rt, the limiting error will be n f 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 n' 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 f e it must be rejected. Again, finding a new value of e from the remaining m 1 residuals, and repeating the operation, it will be seen whether another observa- tion should be rejected; and the process may be continued until a limit is reached which does not require the further rejection of ob- servations. Thus, for example, in the case of 50 observations in which the residuals 11". 5 and + 7".8 occur, let the sum of the squares of the residuals be O] = 320.4. Then, according to equation (30), we shall have e = 2".56. 412 THEORETICAL ASTRONOMY. Corresponding to the value m == 50, the table gives n r = 2.576, and the limiting value of the error becomes n'e = 6".6; and hence the residuals 11 ".5 and -f 7".8 are rejected. Kecom- puting the mean error of an observation, we have = J320.4- 193.09 ==1 ,, 65- \ 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 f , n ff , &c. which are the absolute terms of the equations of condition. By means of these elements we compute also the values of the differential coefficients of each of the spherical co-ordinates with respect to each of the elements to be corrected. These differential coefficients give the values of the coefficients a, 6, c, a', b f , &c. in the equations of condition. The mode of calculating these coefficients, for different systems of co-or- dinates, and the mode of forming the equations of condition, have been fully developed in the second chapter. It is of great import- CORRECTION OF THE ELEMENTS. 413 ance that the numerical values of these coefficients should be care- fully checked by direct calculation, assigning variations to the ele- ments, or by means of differences when this test can be successfully applied. In assigning increments to the elements in order to check the formation of the equations, they should not be so large that the neglected terms of the second order become sensible, nor so small that they do not afford the required certainty by means of the agreement of the corresponding variations of the spherical co-ordinates as obtained by substitution and by direct calculation. As soon as the equations of condition have been thus formed, we multiply each of them by the square root of its weight as given by the adopted relative weights of the normal places; and these equa- tions will thus be reduced to the same weight. In general, the numerical values of the coefficients will be such that it w T ill 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 being any entire or fractional number such that the new coefficients -> , &c. shall be made to agree in V V magnitude with the other coefficients. The unknown quantity whose value will then be derived by the solution of the equations will be vx, and the corresponding weight will be that of vx. To find the weight of x from that of vx, we have the equation P.=!P- ( 132 ) In the same manner, the coefficient of any other unknown quantity may be changed, and the coefficients of all the unknown quantities may thus be made to agree in magnitude within moderate limits, the 414 THEORETICAL ASTRONOMY. advantage of which, in the numerical solution of the equations, will be apparent by a consideration of the mode of proving the calcula- tion of the coefficients in the normal equations. It will be expedient, also, to take for v some integral power of 10, or, when a fractional value is required, the corresponding decimal. It may be remarked, further, that the introduction of v is generally required only when the coefficient of one of the unknown quantities is very large, as frequently happens in the case of the variation of the mean daily motion //. When the coefficients of some of the unknown quantities are extremely small in all the equations of condition to be combined, an approximate solution, and often one which is sufficiently accurate for the purposes required, may be obtained by first neglecting these quantities entirely, and afterwards determining them separately. In general, however, this can only be done when it is certainly known that the influence of the neglected terms is not of sensible magnitude, or when at least approximate values of these terms are already given. When we adopt the approximate plane of the orbit as the funda- mental plane, the equations for the longitude involve only four ele- ments, and the coefficients of the variations of these elements in the equations for the latitudes are always very small. Hence, for an approximate solution,. we may first solve the equations involving four unknown quantities as furnished by the longitudes, and then, substi- tuting the resulting values in the equations for the latitudes, they will contain but two unknown quantities, namely, those which give the corrections to be applied to & and i. 151. When the number of equations of condition is large, the computation of the numerical values of the coefficients in the normal equations will entail considerable labor; and hence it is desirable to arrange the calculation in a convenient form, applying also the checks which have been indicated. The most convenient arrangement will be to write the logarithms of the absolute terms n, n f , n n ', .5]. The values of [&s.l], [cs.l], [cs.2], &c. serve to check the calcula- tion of the successive auxiliary coefficients. Thus we must have [66.1] 4- |>.l] 4- [bd.l-\ + [>.l] + [6/.1] = j>.l] [cc.l] 4- [cd.l] + I>.1] 4- [c/.l] - |>.l], Ac., [cc.2] + [ed.2] + [ce.2] + 0/-2] - [cs.2], [ed.2] -I- [dd.2] + [de.2] 4- [d/.2] [efe.2], Ac. Hence it appears that when the numerical calculation is arranged as above suggested, the auxiliary containing s must, in each line, be equal to the sum of all the terms to the left of it in the same line and of those terms containing the same distinguishing numeral found in a vertical column over the last quantity at the left of this line. There will yet remain only the auxiliaries which are derived from [srf] and [nn\ to be determined. These additional auxiliaries will be found by means of the formulae , . . ., [es . 2]> [sn .4] = [.8] - [A.8], (133) 0.5] = [w.4] - ~ [.4], [.l], [e.2] 4- [rfn.2] 4- [erc.2] 4- [}n.2] = [sn.2], [.4] == [<.4], It appears further, that, in the case of six unknown quantities, since [/s.5] == |jgr.5], we have [n.6] = 0. Having thus determined the numerical values of the auxiliaries required, we are prepared to form at once the equations (74), by means of which the values of the unknown quantities will be determined 27 418 THEORETICAL ASTRONOMY. by successive substitution, first finding t from the last of these equa- tions, then substituting this result in the equation next to the last and thus deriving the value of w, and so on until all the unknown quantities have been determined. It will be observed that the loga- rithms of the coefficients of the unknown quantities in these equa- tions will have been already found in the computation of the aux- iliaries. If we add together the several equations of (74), first clearing them of fractions, we get = [aa] x + ([aft] + [55.1]) y + (M + [6c.l] + [ec.2]) z + ([ad] + [6aM] + [ed.2] + [aU3]) u 4- (M + [&!] + [ce-2] 4- [de.3] + [ee.4])w (135) + ([a/] 4- Rf.l] 4- [cf.2] + [d/.3] + [e/.4] + [//.5])* + Ian-] + [6n.l] + [m.2] + [dn.3] + [e.4] + |>.5] ; and this equation must be satisfied by the values of x, y, z, &c. found from (74). 152. EXAMPLE. The arrangement of the calculation in the case of any other number of unknown quantities is precisely similar ; and to illustrate the entire process let us take the following equations, each of which is already multiplied by the square root of its weight: 0.707* + 2.052y 2.3720 0.221w + 6".58 = 0, OA71x + 1.347y 1.7150 0.085w -f 1 .63 ^ 0, 0.260^ -f 0.770y ~ 0.3560 -f 0.483w 4 .40 = 0, 0.092z -f 0.343y + 0.2350 + 0.469w 10 .21 = 0, OAUx -f 1.204y 1.5060 0.205w 4- 3 .99 = 0, 0.040^ + 0.150^ 4- 0.1040 4- 0.206tt 4 .34 = 0. First, we derive [ran] = 204.313, [an] = + 4.815, [aa] = + 0.971, [6n] = + 12.961, [06] = + 2.821, [66] = + 8.208, [en] = -25.697, [ac] = -3.175, [6c] =-9.168, [ce] = + 11.028, [dn] = - 10.218, [ad~] =-0.104, [bd] = 0.251, [cd] = + 0.938, [cW] = + 0.594, -18.139, [] = + 0.513, [bs] =4-1-610, [a] =-0.377, [&] = + 1.177. The values of [n], [as], [6s], [c.s], and [&], found by taking the sums of the normal coefficients, agree exactly with the values com- puted directly, thus proving the calculation of these coefficients. The normal equations are, therefore, NUMERICAL EXAMPLE. 419 0.9713 -f 2.821y 3.1752 O.K^i -f 4.815 = 0, 2.821a; -f 8.208y 9.168z 0.251w + 12.961 == 0, - 3.175a; 9.168y + 11.0283 -f- 0.938w 25.697 = 0, _ o.!04a; 0.251y + 0.9382 + 0.594it 10.218 = 0. It will be observed that the coefficients in these equations are nu- merically greater than in the equations of condition; and this will generally be the case. Hence, if we use logarithms of five decimals in forming the normal equations, it will be expedient to use tables of six or seven decimals in the solution of these equations. Arranging the process of elimination in the most convenient form, the successive results are as follows : [66.1] = + 0.0123, [6c.l] = + 0.0562, [6eZ.l] = + 0.0511, [6s.l] sp + 0.1196, [6n.l] B 1.0278, [oc.1] = + 0.6463, [cd.l] = + 0.5979, [cs.l] -= + 1.3004, [cn.l] = 9.9528, [cc.2] = + 0.3895, [cd.2] = -f 0.3644, [C8.2] := + 0.7539, [cn.2] B. 5.2567, [eZdf.l] = + 0.5829, [tfe.1] a + 1.2319, [eZn.1] a 9.7023, [cM.2] = + 0.3706, [d.2] == + 0.7350, [d.2J a 5.4323, [cW.3] = + 0.0297, [ds.3] am -f 0.0297 [rf.3] = 0.5143, [nn.l] = 180.436, [WI.1] SB 20.6828, [nn.2] -= 94.552, [sn.2] a. 10.6889, [nn.3] 23.608, [sn.3] aa 0.5143, [nn.4] 14.698, [sn.4] 0. The several checks agree completely, and only the value of [?in.4] remains to be proved. The equations (74) therefore give x + 2.9052?/ 3.2698s 0.1071w + 4.9588 = 0, y + 4.5691s + 4.1545w 83.5610 == 0, z -f- 0.9356w 13.4960 = 0, u 17.3165 = 0, and from these we get u== + 17".316, z = - 2".705, y = + 23".977, x = 81".608. Then the equation (135) becomes = + 0.9710^ -f 2.8333y 2.7293* + 0.3412 1.9838, which is satisfied by the preceding values of the unknown quantities. If we substitute these values of x, y, z, and u in the equations of condition already reduced to the same weight by multiplication by the square roots of their weights, we obtain the residuals + 0".67, -1".34, +2".17, -2".01, -0".40, -0".72, The sum of the squares of these gives [w] = [wi.4] = 11.672, and the difference between this result and the value 14.698 already 420 THEORETICAL ASTRONOMY. found is due to the decimals neglected in the computation of the numerical values of the several auxiliaries. The sum of all the equations of condition gives generally M* + My + LC]Z + id]u + .... + M = M, (136) which may be used to check the substitution of the numerical values in the determination of v, v f , &c. Thus, we have, for the values here given, 1.984a? + 5.866y 5.610z + 0.647w 6.75 = [>] = l."63. It remains yet to determine the relative weights of the resulting values of the unknown quantities. For this purpose we may apply any of the various methods already given. The weights of u and z may be found directly from the auxiliaries whose values have been computed. Thus, we have p. = [cW.3] = 0.0297, p, = [.2] = 0.0312. If we now completely reverse the order of elimination from the normal equations, and determine x first, we obtain the values [66.2] = + 0.0425, [oo.2] = + 0.0033, [ao.3] = -f 0.00056, [)w.4] = 14.665, and also x= -82/750, 2/ = + 24."365, a = 2."699, w = + 17."272. The small differences between these results and those obtained by the first elimination arise from the decimals neglected. This second elimination furnishes at once the weights of x and y, namely, Px = [oo.3] = 0.00056, p = j^lj [66.2] = 0.0072. [ttd.^J We may also compute the weights by means of the equations (96). Thus, to find the weight of y, we have _ + 0.02977, j i . and hence The equations (103) and (108) are convenient for the determination of the values and weights of the unknown quantities separately. CORRECTION OF THE ELEMENTS. 421 Thus, by means of the values of the auxiliaries obtained in the first elimination, we find from the equations (100), (101), and (102), A' = 2.9052, A" = + 16.5442, A"' = 3.3012, JB" = 4.5691, B"'=+ 0.1202, C'" = 0.9356, and then the equations (103) and (108) give x'= 81".609, y = + 23".977, z = 2".705, u = + 17".316, p x = 0.00057, p y = 0.0074, #, = 0.0312, ^ = 0.0297, agreeing with the results obtained by means of the other methods. The weights are so small that it may be inferred at once that the values of x, y, z, and u are very uncertain, although they are those which best satisfy the given equations. It will be observed that if we multiply the first normal equation by 2.9, the resulting equation will differ very little from the second normal equation, and hence we have nearly the case presented in which the number of independent relations is one less than the number of unknown quantities. The uncertainty of the solution will be further indicated by deter- mining the probable errors of the results, although on account of the small number of equations the probable or mean errors obtained may be little more than rude approximations. Thus, adopting the value of [vv] obtained by direct substitution, we have = 2.416, 'Hi p. * t t and hence which is the probable error of the absolute term of an equation of condition whose weight is unity. Then the equations ff.-f.....p X I/A* y vp ^ P* give r x = 68".25, r y = 18".94, r. = 9".22, r u = 9".45. It thus appears that the probable error of z exceeds the value obtained for the quantity itself, and that although the sum of the squares of the residuals is reduced from 204.31 to 11.67, the results are still quite uncertain. 153. The certainty of the solution will be greatest when the coef- ficients in the equations of condition and also in the normal equations 422 THEORETICAL ASTRONOMY. differ very considerably both in magnitude and in sign. In the cor- rection of the elements of the orbit of a planet when the observa- tions extend only over a short interval of time, the coefficients will generally change value so slowly that the equations for the direct determination of the corrections to be applied to the elements will not afford a satisfactory solution. In such cases it will be expedient to form the equations for the determination of a less number of quantities from which the corrected elements may be subsequently derived. Thus we may determine the corrections to be applied to two assumed geocentric distances or to any other quantities which afford the required convenience in the solution of the problem, various formula? for which have been given in the preceding chapter. The quantities selected for correction should be known functions of the elements, and such that the equations to be solved, in order to combine all the observed places, shall not be subject to any uncer- tainty in the solution. But when the observations extend over a long period, the most complete determination of the corrections to be applied to the provisional elements will be obtained by forming the equations for these variations directly, and combining them as already explained. A complete proof of the accuracy of the entire calcula- tion will be obtained by computing the normal places directly from the elements as finally corrected, and comparing the residuals thus derived with those given by the substitution of the adopted values of the unknown quantities in the original equations of condition. If the elements to be corrected differ so much from the true values that the squares and products of the corrections are of sensible mag- nitude, so that the assumption of a linear form for the equations does not afford the required accuracy, it will be necessary to solve the equations first provisionally, and, having applied the resulting cor- rections to the elements, we compute the places of the body directly from the corrected elements, and the differences between these and the observed places furnish new values of n, n', n", &c., to be used in a repetition of the solution. The corrections which result from the second solution will be small, and, being applied to the elements as corrected by the first solution, will furnish satisfactory results. In this new solution it will not in general be necessary to recompute the coefficients of the unknown quantities in the equations of condition, since the variations of the elements will not be large enough to affect sensibly the values of their differential coefficients with respect to the observed spherical co-ordinates. Cases may occur, however, in which it may become necessary to recompute the coefficients of one CORRECTION OF THE ELEMENTS. 423 or more of the unknown quantities, but only when these coefficients are very considerably changed by a small variation in the adopted values of the elements employed in the calculation. In such cases the residuals obtained by substitution in the equations of condition will not agree with those obtained by direct calculation unless the corrections applied to the corresponding elements are very small. It may also be remarked that often, and especially in a repetition of the solution so as to include terms of the second order, it will be ciently accurate to relax a little the rigorous requirements of a plete solution, and use, instead of the actual coefficients, equivalent numbers which are more convenient in the numerical operations re- quired. Although the greatest confidence should be placed in the accuracy of the results obtained as far as possible in strict accordance with the requirements of the theory, yet the uncertainty of the deter- mination of the relative weights in the combination of a series of observations, as well as the effect of uneliminated constant errors, may at least warrant a little latitude in the numerical application, provided that the weights of the results are not thereby much affected. A constant error may in fact be regarded as an unknown quantity to be determined, and since the effect of the omission of one of the unknown quantities is to diminish the probable errors of the resulting values of the others, it is evident that, on account of the existence of constant errors not determined, the values of the variables obtained by the method of least squares from different corresponding series of observations may differ beyond the limits which the probable errors of the different determinations have assigned. Further, it should be observed that, on account of the unavoidable uncertainty in the esti- mation of the weights of the observations in the preliminary combi- nation, the probable error of an observed place whose weight is unity as determined by the final residuals given by the equations of condition, may not agree exactly with that indicated by the prior discussion of the observations. 154. In the case of very eccentric orbits in which the corrections to be applied to certain elements are not indicated with certainty by the observations, it will often become necessary to make that whose weight is very small the last in the elimination, and determine the other corrections as functions of this one; and whenever the coeffi- cients of two of the unknown quantities are nearly equal or have nearly the same ratio to each other in all the different equations of condition, this method is indispensable unless the difficulty is reme- 424 THEOKETICAL ASTRONOMY. died by other means, such as the introduction of different elements or different combinations of the same elements. The equations (113) furnish the values of the unknown quantities when we neglect that which is to be determined independently; and then the equations (114) give the required expressions for the complete values of these quantities. Thus, when a comet has been observed only during a brief period, the ellipticity of the orbit, however, being plainly indi- cated by the observations, the determination of the correction to be applied to the mean daily motion as given by the provisional ele- ments, in connection with the corrections of the other elements, will necessarily be quite uncertain, and this uncertainty may very greatly affect all the results. Hence the elimination will be so arranged that A, shall be the last, and the other corrections will be determined as functions of this quantity. The substitution of the results thus derived in the equations of condition will give for each residual an expression of the following form : Therefore we shall have M = KV] + 2 [ V ] AM + [77] AM*, (137) which may be applied more conveniently in the equivalent form M = [v.1 - ^ Cvl + M ( <* + [ ^j )' (138) The most probable value of A// will be that which renders [vv] a and the corresponding value of the sum of the squares of the residuals is M-Cvd-^Evl. (140) The correction given by equation (139) having been applied to /*, the result may be regarded as the most probable value of that ele- ment, and the corresponding values of the corrections of the other elements as determined by the equations (114) having been also duly applied, we obtain the most probable system of elements. These, however, may still be expressed in the form & -f A ty, i -f B^IJL, TT -- C*t* &c. CORRECTION OF THE ELEMENTS. 425 the coefficients A 09 S 0) C w &c. being those given by the equations (114), and thus the elements may be derived which correspond to any assumed value of // differing from its most probable value. The unknown quantity A^ will also be retained in the values of the residuals. Hence, if we assign small increments to /*, it may easily be seen how much this element may differ from its most probable value without giving results for the residuals which are incompatible with the evidence furnished by the observations. If the dimensions of the orbit are expressed by means of the ele- ments q and e, it may occur that the latter will not be determined with certainty by the observations, and hence it should be treated as suggested in the case of //; and we proceed in a similar manner when the correction to be applied to a given value of the semi-transverse axis a is one of the unknown quantities to be determined. 426 THEOKETICAL ASTRONOMY. CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. 155. WE have thus far considered the circumstances of the undis- turbed motion of the heavenly bodies in their orbits; but a complete determination of the elements of the orbit of any body revolving around the sun, requires that we should determine the alterations in its motion due to the action of the other bodies of the system. For this purpose, we shall resume the general equations (18) 1? namely, d*z ,72,. x z , .dQ w + k\l + m)- = #(1 + m) -^, which determine the motion of a heavenly body relative to the sun when subject to the action of the other bodies of the system. We have, further, n which is called the perturbing function, of which the partial differen- tial coefficients, with respect to the co-ordinates, are c?fl _ m' Ix' x rf\ m," x"x x" dx _ m' Ix' x rf_\ m," lx"x x" \ ~l+m\ f / 3 / + l+7 l \~ 7r ~~/ 7i "/ + ' " (f-y f\+^ ,<* ^ \~7i~ -^) +&C> ' z' z z' \ , m" tz" z z" dz and in which ra', m", &c. denote the ratios of the masses of the several disturbing planets to the mass of the sun, and m the ratio of the mass of the disturbed planet to that of the sun. These partial differential coefficients, when multiplied by F(l -f m), express the PERTURBATIONS. 427 surn 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 eifect 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 diiferent from the first will be moving in an orbit for which the elements are in some degree diiferent 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 y 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 If , TT O , & , i w e Q , and G O ; then will the equations (3) be exactly satisfied by means of the expressions for the co-ordinates in terms of these rigorously-constant elements. These elements will express the motion of the body sub- ject to the action of the disturbing forces only during the infinitesimal interval dt, and at the time t Q + dt it will commence to describe a new orbit of which the elements will cliffer 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, x, &, i, e, and a, they will be connected with the constant elements, or those which determine the orbit at the instant t Q) by the equations 428 THEORETICAL ASTRONOMY. in which , , &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 dM, dx, &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 # , a system of elements, usually called osculating elements, on which the complete theory of the motion may be based. Instead of computing the variations of the elements of the orbit directly, we may find the perturbations of any known functions of these elements; and the most direct and simple method is to deter- mine the variations, due to the action of the disturbing forces, of any system of three co-ordinates by means of which the position of PERTURBATIONS. 429 the body or the elements themselves may be found. We shall, there- fore, derive various formulae for this purpose before investigating the formulae for the direct variation of the elements. 156. Let XQ, 2/ , Z Q be the rectangular co-ordinates of the body at the time t computed by means of the osculating elements M w TT O , & , &c., corresponding to the epoch t Q . 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 x, y, and z in the equations (1), and then subtract from each the corre- sponding one of equations (3), we get Let us now put r = r Q + dr; then to terms of the order ^r 2 , which is equivalent to considering only the first power of the disturbing force, we have r 3 and hence to s dQ k? (1 4- M) I n x n \ Ar(l 4- *)-; ; -I 3 J or ^o; I, (^^C 7* It* / C/f (Xu Cf^I 7*rt We have also from neglecting terms of the second order, dr = dx -f fy + ^ (7) 430 THEOEETICAL ASTRONOMY. The integration of the equations (6) will give the perturbations Sx, dy, and dz to be applied to the rectangular co-ordinates X Q , y w z com- puted by means of the osculating elements, in order to find the actual co-ordinates of the body for the date to which the integration belongs. But since the second members contain the quantities dx, 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, F, 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 Q 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, P* = (x' - x)> + #- 2/) 2 + dn, f(x) dx w/0* + nto) dn. \ If we expand the function f(a -j- nco\ we have /(a + M ,)^) + ^ + 436 THEOKETICAL ASTRONOMY. and hence ff(a (21) C being the constant of integration. The equations (54) 6 give . ^- =/' (a) - if" () + tW () (22) in whioh the functional symbols in the second members denote the different orders of finite differences of the function. Hence we obtain = C+ nf(a) + X (/"() - A/ |T W + A/* 00 - 5lo/ viii W h - ) + An* (TOO - if () + ilo/ vii () - ) If we take the integral between the limits n' and -f-n', the terms containing the even powers of n disappear. Further, since the values of the function are supposed to be known for a series of values of 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 \ and +i, and the equation (23) gives, after reduction, MECHANICAL QUADRATURE. 437 x -I a (24) It is evident that by writing, in succession, a -\- co, a + 2o>, .... a -f tw in place of a, we simply add 1 to each limit successively, so that we have I f( a + na) ) dn= I /((a* -j- ito) -\- (n z) u>) d (n i) i-t -I ~/<, /( a _|_ a,) = '/(a + |ai) - '/(a + ^), /(a + -no,) = '/( + ( n + i ) ai) - '/(a + (w - J) ). Therefore we shall have and also "( + n) =/'"( + (i + i) -)'-/'" (a - >), Ac. 438 THEORETICAL ASTRONOMY. Further, since the quantity e f(a Ja>) is entirely arbitrary, we may assign to it a value such that the sum of all the terms of the equation which have the argument a \a) shall be zero, namely, W^ (26) Substituting these values in (25), it reduces to /x f(x) dx = I /(a -f nw) 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 = J, corresponding to the inferior limit, the integral shall be equal to zero, the epoch of f(a Jw) being that of the osculating elements. It will be observed that the equation (26) expresses this condition, the constant of integration being included in f f(a Jo;). If, instead of being equal to zero, the integral has a given value when n = J, it is evidently only necessary to add this value to 'f(a \co) as given by (26). 160. The interval to 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 + n0 dn = J/(a) + J/' (a) -f &f" W - *!*/" (a) - T AW* () + ?ii*f to + T *JSW Tl to - &c -; and since, according to the notation adopted, /' to = i (/' ( - ->) +/' (a -f 50) = /'(a + l w ) -i/" (a), /"=/"> + i0 -iTto, / v to=r(a + -/*(), &c., MECHANICAL QUADRATURE. 439 this becomes i J/(a+w.) d=i/(a)+J/ (a+i)-&r ()-.fe/" G*+i) (29) + rli^W + *Mij/'( + i0 - TWW() ~ Ac. Therefore we obtain r J /(a Now we have i + k i + i I /(a -|- ww ) dn I /(a -j- ww) cfo I /(a -j- na>) dta ; -* and if we substitute the values already found for the terms in the second member, and also we get a + t'w ) dx = to\ f(a (32) ="ima+(i+^")+-w(a+( i -^")-* i J f (<*>+(i+D<) + -i) ) + &c.|, 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 _ ddx ddy d8z .. will only give the values of -^-> - and -5-, and another integration becomes necessary in order to obtain the values of dx, %, and 8z. We will therefore proceed to derive formulae for the determination of the double integral directly. 440 THEORETICAL ASTKONOMY. For the double integral jJ/W dx * we have > since dx " = Mn2 > a + wo) dn\ The value of the function designated by f(a) being so taken that when n = , Cf(a + na>)dn = 0, the equation (23) gives o -4 Therefore, the general equation is o Cf(a -f no,) dn =ff(a + w0 dn + nf(a) the values of a, /9, ^, . . . being given by the equations (22). Multi- plying this by dn, and integrating, we get o CCf(a + not') dri* = C' + n Cf(a + new) c?n + ^n 2 f(a) &c., O 7 being the new constant of integration. If we take the integral between the limits \ and + %, we find rr r JJ f^ + m< "- ) ^ = J ^ + W "- ) d + 3 ' J<1 +TW5r + 322IS(! + &c - i 5 From the equation (32) we get, for i 0, o J/(a 4- n) - r/( + -) - ir c - ), Therefore ^ / v,o-t* > ^ j i-f-vt ^(*4^^ Substituting these values in equation (35), and observing that 700 + 70 ~^ /(a) + /(a - ) = 2/(a) - / (a - ^), /" (a) +/' (a - a,) = 2f (a) -/" (a - -, Ac., WJ^^>.^A/) is arbitrary, we may put "/(a _ ) = /() - , ' J, (2/ (a) + /" (a - )) " - - 442 THEORETICAL ASTRONOMY. the integral becomes ff/O) dx 2 = > 2 f f /( + w) dn * = <* 2 { i "/(a + (i + 1) a,) + i"/(a + wO ,V/(a + tf + 1) ai) (37) which is the expression for the double integral between the limits \ and i + J. The value of "f(a to) given by equation (36) is in accordance with the supposition that for n = | the double integral is equal to zero, and this condition is fulfilled in the calculation of the pertur- bations when the argument a \o> corresponds to the date for which 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 %(*>) given by equation (26), and to add the given value of the double integral for the same argument to the value of "/(a to) 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 i o 1 1 f( a H~ nw ) dn* = i I f( a + n^^ 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 CD 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 o> smaller than what would appear to be strictly required, in order that there may be no uncertainty with respect to the accuracy of the integration. On account of the rapidity with which the higher orders of differences decrease as we diminish co, a limit for the magnitude of the adopted interval will speedily be obtained. The magnitude of the interval will therefore be suggested by the rapidity of the change of value of the function. In the com* 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 a) 2 , if we multiply the computed values of the function by G> 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 o> 2 , and only the single integral is sought, the result obtained by the formula of integration, neglecting the factor a?, will be a) times the actual integral required, and it must be divided by CD 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, dy, and dz with respect to the time, will be determined by means of the equations (9). If the interval o) 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 co, and commencing with the date t Q \a) corresponding to the argument a co, t being the date to which the osculating elements belong. Then, since the last terms , - d*dx d*dy ., cRz of the formulae for -^~, -gp and =-- involve dx, dy, and dz, which are the quantities sought, the subsequent determination of the differ- ential coefficients must be performed by successive trials. Since the integral must in each case be equal to zero for the date t , it will be admissible to assume first, for the dates t to and t + \u corre- sponding to the arguments a to and , that dx = 0, dy = 0, and dz = 0, and hence that the three differential coefficients, for each VARIATION OF CO-ORDIXATES. 445 date, are respectively equal to X , Y Q , and Z Q . We may now by inte- gration derive the actual or the very approximate values of the variations of the co-ordinates for these two dates. Thus, in the case of each co-ordinate, we compute the value of '/(a Ja>) by means of the equation (26), using only the first term, and the value of "f(a coy from (36), using in this case also only the first term. The value of the next function symbolized by ff f will be given by Then the formula (39), putting first i = 1 and then i = 0, and neglecting second differences, will give the values of the variations of the co-ordinates for the dates a co and a. These operations will be performed in the case of each of the three co-ordinates; and, by means of the results, the corrected values of the differential coeffi- cients will be obtained from the equations (9), the value of dr being computed by means of (7). With the corrected values thus derived a new table of integration will be commenced; and the values of 'f(a \co) and "f(a to) will also be recomputed. Then we obtain, also, by adding f f(a Jo;) to f(a\ the value of 'f(a -f- Jo>), and, by adding this to "f(a\ the value of "f(a -\- o>). An approximate value of /(a + co) may now be readily estimated, and two terms of the equation (39), putting i = 1, will give an ap- proximate value of the integral. This having been obtained for each of the co-ordinates, the corresponding complete values of the differential coefficients may be computed, and these having been introduced into the table of integration, the process may, in a similar manner, be carried one step farther, so as to determine first approxi- mate values of dx, dy, and dz for the date represented by the argu- ment a 4* 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. Tf 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 to should be smaller than when the successive values of the function to be integrated are already known. In the case of the small planets an interval of 40 days will afford the required facility in the approxi- mations; but in the case of the comets it may often be necessary to adopt an interval of only a few days. The necessity of a change in the adopted value of CD will be indicated, in the numerical applica- tion of the formulae, by the manner in which the successive assump- tions in regard to the value of the function are found to agree with the corrected results. The values of the differential coefficients, and hence those of the integrals, are conveniently expressed by adopting for unity the unit of the seventh decimal place of their values in terms of the unit of space. 165. Whenever it is considered necessary to commence to take into account the perturbations due to the second and higher powers of the disturbing force, the complete equations (14) must be employed. In this case the forces X, Y, and Z should not be computed at once for the entire period during which the perturbations are to be determined. The values computed by means of the osculating elements will be employed only so long as simply the first power of the disturbing force is considered, and by means of the approximate values of ox, dy, and 3z which would be employed in computing, for the next place, the last terms of the equations (9), we must compute also the cor- rected values of X, Y, and Z. These will be given by the second members of (8), using the values of x, y, and z obtained from ^ = X Q + dx, y = y +dy, z = Z -f dz. We compute also q from (12), and then from Table XVII. find the corresponding value of /. The corrected values of -r^-, rj- , and j_ . Clt Cit -jjp- will be given by the equations (14), and these being introduced, in the continuation of the table of integration, we obtain new values of dx, dy, and dz for the date under consideration . If these differ much from those previously assumed, a repetition of the calculation will be necessary in order to secure extreme accuracy. In this repe- tition, however, it will not be necessary to recompute the coefficients of dx, dy, and dz in the formula for q, their values being given with sufficient accuracy by means of the previous assumption ; and gene- VARIATION OF CO-ORDINATES. 447 rally a repetition of the calculation of X, Y, and Z will not be required. Next, the values of dx, dy, and dz may be determined approxi- mately, as already explained, for the following date, and by means of these the corresponding values of the forces X, Y, and Z will be found, and also/ and the remaining terms of (14), after which the integration will be completed and a new trial made, if it be con- sidered necessary. In the final integration, all the terms of the for- mula? of integration which sensibly affect the result may be taken into account. By thus performing the complete calculation of each successive place separately, the determination of the perturbations in the values of the co-ordinates may be effected in reference to all powers of the masses, provided that we regard the masses and co-or- dinates of the disturbing bodies as being accurately known ; and it is apparent that this complete solution of the problem requires very little more labor than the determination of the perturbations when only the first power of the disturbing force is considered. But although the places of the disturbing bodies as given by the tables of their motion may be regarded as accurately known, there are yet the errors of the adopted osculating elements of the disturbed body to detract from the absolute accuracy of the computed perturbations; and hence the probable errors of these elements should be constantly kept in view, to the end that no useless extension of the calculation may be undertaken. When the osculating elements have been cor- rected by means of a very extended series of observations, it will be expedient to determine the perturbations with all possible rigor. When there are several disturbing planets, the forces for all of these may be computed simultaneously and united in a single sum, so that in the equations (14) we shall have 2X y 2Y, and 2Z instead of X, Y, and Z respectively; and the integration of the expressions d*dx d?dy _ d?Sz for ry-, -Tjp and ^- will then give the perturbations due to the action of all the disturbing bodies considered. However, when the interval co for the different disturbing planets may be taken differently, it may be considered expedient to compute the perturbations sepa- rately, and especially if the adopted values of the masses of some of the disturbing bodies are regarded as uncertain, and it is desired to separate their action in order to determine the probable corrections to be applied to the values of m, m', &c., or to determine the effect of any subsequent change in these values without repeating the cal- culation of the perturbations. 448 THEORETICAL ASTRONOMY. 166. EXAMPLE. To illustrate the numerical application of the formulae for the computation of the perturbations of the rectangular co-ordinates, let it be required to compute the perturbations of Eurynome @ arising from the action of Jupiter from 1864 Jan. 1.0 Berlin mean time to 1865 Jan. 15.0 Berlin mean time, assuming the osculating elements to be the following : Epoch = 1864 Jan. 1.0 Berlin mean time. M n = 1 29' 5".65 I Ecliptic and Mean 4 36 52 .111 Equinox 1860.0 LI 15 51 .02 17 12 .17 39 5 .69 log a = 0.3881319 j u = 928".55745. From these elements we derive the following values : Berlin Mean Time. X Q y z Iogr 1863 Dec. 12.0 + 1.53616 + 1.23012 0.03312 0.294084, 1864 Jan. 21.0 1.15097 1.59918 0.07369 0.294837, March 1.0 0.69518 1.87033 0.10978 0.300674, April 10.0 + 0.19817 2.03141 0.13936 0.310864, May 20.0 - 0.31012 2.08092 0.16134 0.324298, June 29.0 0.80326 2.02602 0.17523 0.339745, Aug. 8.0 1.26055 1.87959 0.18122 0.356101, Sept. 17.0 1.66729 1.65711 0.17990 0.372469, Oct. 27.0 2.01414 1.37473 0.17209 0.388214, Dec. 6.0 2.29597 1.04766 0.15870 0.402894, 1865 Jan. 15.0 2.51077 + 0.68978 0.14066 0.416240. The adopted interval is co = 40 days, and the co-ordinates are re- ferred to the ecliptic and mean equinox of 1860.0. The first date, it will be observed, corresponds to t \a), and the integration is to commence at 1864 Jan. 1.0. The places of Jupiter derived from the tables give the following values of the co-ordinates of that planet, with which we write also the distances of Eurynome from Jupiter computed by means of the formula Berlin Mean Time. 1863 Dec. 12.0 1864 Jan. 21.0 March 1.0 April 10.0 a/ y f / logr' logP -4.09683 3.55184 +0.10533 0.73425 0.86866, 3.89630 3.76053 0.10152 0.73368 0.86713, 3.68416 3.95803 0.09744 0.73305 0.86292, -3.46098 4.14366 +0.09304 0.73237 0.85622, NUMERICAL EXAMPLE. 449 Berlin Mean Time. x' y' z' logr' logp 1864 May 20.0 3 .22739 4 .31684 4 0.08839 0. 73164 0.84732, June 29.0 2 .98405 4 .47693 0.08346 0. 73086 0.83656, Aug. 8.0 2 .73162 4 .62343 0.07827 0.73003 0.82428, Sept. 17.0 2 .47085 4 .75576 0.07284 0. 72915 0.81077, Oct. 27.0 2 .20247 4 .87345 0.06720 0. 72823 0.79628, Dec. 6.0 1 .92728 4 .97606 0.06134 0. 72726 0.78098, 1865 Jan. 15,0 1 .64600 5 .06301 4 0.05531 0. 72625 0.76498. These co-ordinates are also referred to the ecliptic and mean equinox of 1860.0. If we neglect the mass of Eurynome and adopt for the mass of Jupiter 1047.819' we obtain, in units of the seventh decimal place, atm'k* = 4518.27, and the equations (9) become Substituting for the quantities in the first term of the second member of each of these equations the values already found, we obtain Argument. Date. w 2 X w 2 F w% a to 1863 Dec. 12.0 4 53.00 4 47.09 - 1.43, a 1864 Jan. 21.0 53.71 46.31 0.91, a 4 w March 1.0 54.23 45.18 - 0.37, a 4 2w April 10.0 54.69 43.59 4- 0.22, a + 3a> May 20.0 55.23 41.51 0.70, a 4 4w June 29.0 56.06 38.96 1.19, a 4 5* Aug. 8.0 57.30 35.92 1.66, a46w Sept. 17.0 59.09 32.47 2.08, a 4" 7&> Oct. 27.0 61.55 28.60 2.43, a 4 8w Dec. 6.0 64.85 24.34 2.69, a 4 9w 1865 Jan. 15.0 4 69.09 4 19.78 4 2.83, which are expressed in units of the seventh decimal place. We now, for a first approximation, regard the perturbations as 450 THEORETICAL ASTRONOMY. being equal to zero for the dates Dec. 12.0 and Jan. 21.0, and, in the case of the variation of x, we compute first y (a _ i ,) = &f (a = A (53.71 53.00) = 0.03, /(a_0 =j and the approximate table of integration becomes Then the formula (39), putting first i = l, and then i = 0, gives Dec. 12.0 to = + 2.24 + ^ = + 6 .66, Ko 71 Jan. 21.0 to = + 2.21 + ^- = + 6.69. In a similar manner, we find Dec. 12.0 Sy = + 5.85 dz = 0.16, Jan. 21.0 fy = + 5.82 & = 0.14. By means of these results we compute the complete values of the second members of equations (40), dr being found from *=**** + **.#**> r Q r r and thus we obtain Dec. 12.0 + 53.86 -f 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, x y z / '/ 7 / y j / r f 7 +53.86 _ OQ2 + 2.26, +47.76 , 02 + 1.97, -1.45 _ Q Q2 -0.04, +54.23 ' + 2.24, +47.25 J^' + 1.99, -0.96 __' -0.06, ' +56.45, +49.26, -1.04, the formation of which is made evident by what precedes. We may next assume for approximate values of the differential coefficients, for the date March 1.0, + 54.6, + 46.7, and 0.5, respectively; and these give, for this date, NUMEEICAL EXAMPLE. 451 8x = + 56.45 + ^j- = + 61.00, fy = + 49.26 + ^- = + 53.15, A K te= 1.04 -^- = 1.08. -L-^ By means of these approximate values we obtain the following results : 1864 March 1.0 = + 55.01, = + 53.86, Sr = + 71.03. Introducing these -into the table of integration, we find, for the corre- sponding values of the integrals, te = + 61.03, fy = + 53.75, fc = 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 = + 5,04, = + 53.91, Assuming, again, approximate values of the differential coefficients for April 10.0, and computing the corresponding values of dx 9 %, and 8z, we derive, for this date, = + 48.06, = + 68.19, < Introducing these into the table of integration, and thus deriving approximate values of dx, %, 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. ~W 6)2 ~M " 2 dt* 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 THEOKETICAL ASTRONOMY. Date. .d*dx 1864 Oct. 27.0 + 34.84 Dec. 6.0 68.79 1865 Jan. 15.0 + 112.64 , 2 ^J/ 26.32 47.87 58.39 u * + 4.44, 6.86, + 8.68. The complete integration may now be effected, and we may use both equation (37) and equation (39), the former giving the integral for the dates Jan. 1.0, Feb. 10.0, March 21.0, &c., and the latter the integrals for the dates in the foregoing table of values of the function. The final results for the perturbations of the rectangular co-ordinates, expressed in units of the seventh decimal place, are thus found to be the following: Berlin Mean Time. dx 6y 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 27.1 23.5 0.5, March 1.0 61.0 53.7 1.1, 21.0 108.9 97.4 2.0, April 10.0 169.7 155.7 3.1, 30.0 242.7 229.9 4.7, May 20.0 325.7 320.3 6.7, June 9.0 417.1 427.2 9.3, 29.0 514.6 549.1 12.3, July 19.0 616.1 684.9 15.7, Aug. 8.0 720.8 831.4 19.5, 28.0 827.4 986.0 23.4, Sept. 17.0 936.8 1144.6 27.0, Oct. 7.0 1049.4 1303.8 30.2, 27.0 1168.2 1460.0 32.6, Nov. 16.0 1295.4 1609.4 33.9, Dec. 6.0 1435.6 1749.6 33.8, 26.0 1592.8 1877.6 32.0, 1865 Jan. 15.0 + 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 formulse, 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 dx, dy, and dz for Sept. 17.0 given imme- diately by the table of integration extended to this date, will suffice to furnish the required values of the disturbed co-ordinates by means of X = X 9 + dx, y = y Q +fy, z = Z + dz; and to find p = p Q + dp, we have x' x* i/ y ft z' 2 , OP = -- jte-^ay -- _ fe , or *lo g/0 = -^ 2 ((a/ -a?) dx + O/' -y) fy + 0'-*) H in which ^ is the modulus of the system of logarithms. Thus we obtain, for Sept. 17.0, d log p = + 0.0000084, w*X=-}- 59.09, /* F= + 32.48, w 2 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&& + 0.29815fy 0.03237&. The approximate values of dx, dy, and dz being substituted in this equation, we obtain q = + 0.0000061, corresponding to which Table XVII. gives log/= 0.477115. Hence we derive ,.,27.2 27.2 -f (fp, - to) = - 44.87, -f (fqy- *) = - 30.40, 454 THEORETICAL ASTEONOMY. and the equations (14) give f/v, rj 2 dll (J?&2 + -I A OO 2 9 I O ClQ. 2 | -1 Qfr 14.22, ~dj~~~ " ^' ~W = " These values being introduced into the table of integration, the resulting values of the integrals are changed so little that a repetition of the calculation is not required. We now derive approximate values of dx, %, and dz for Oct. 27.0, and in a similar manner we obtain the corrected values of the differ- ential coefficients for this date ; and thus by computing the forces for each place in succession from approximate values of the perturbations, and repeating the calculation whenever it may appear necessary, we may determine the perturbations rigorously for all powers of the masses. The results in the case under consideration are the follow- ing: Date. u*n, ^ 2 ~rr " 2 -rr dt* dt 2 dt 2 1864 Sept. 17.0 + 14.22 + 2.08 -f 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 dx = + 1772.6, dy = + 1992.3, dz = 28.2, agreeing exactly with those obtained when terms of the order of the square of the disturbing forces are neglected. If the perturbations of the rectangular co-ordinates referred to the equator are required, we have, whatever may be the magnitude of the perturbations, dx f = dx, dy, = cos edy sin e dz, (41) dz, sin e dy + cos e dz, x n y z, being the co-ordinates in reference to the equator as the fun- damental plane. Thus we obtain, for 1865 Jan. 15.0, te, = + 1772.6, dy, = + 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 f = + 37".930, dz, = + 15".825. VARIATION OF CO-ORDINATES. 455 The approximate geocentric place of the planet for the same date is a == 183 28', d = 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 J == + 5".67. 167. The values of dx, %, and dz, computed by means of the co- ordinates referred to the ecliptic and mean equinox of the date t, must be added to the co-ordinates given by the undisturbed elements and referred to the same mean equinox. The co-ordinates referred to the ecliptic and mean equinox of t may be readily transformed into those referred to the ecliptic and mean equinox of another date t 1 '. Thus, let d denote the longitude of the descending node of the ecliptic of t' on that of , measured from the mean equinox of t, and let TJ be the mutual inclination of these planes; then, if we denote by x', y r , 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 6, we shall have x' = x cos -f- y sin 0, y' = x sin + y cos 0, (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 f . Then we shall have x" = x', y" = y' cos if) z sin i), (43) z" = y' sin >? -j- z' 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 f) y,, z n we get x, = x" cos (0 -f p) y" sin (0 -f p), y, = x" sin (0 + p) + y" cos (0 -f- p\ (44) in which p denotes the precession during the interval t' t. Elimi- nating x" 9 y ff , and z" from these equations by means of (43) and (42), observing that, since ^ is very small, we may put cos ^ = 1, we get 456 THEORETICAL ASTRONOMY. x, = x cosp y sinp + - z sin (0 + j>), s y, = x sin^ + y cosp - z cos (0 + jp), (45) s 2, = z ? sin + - y cos 0, s s in which s = 206264.8, y being supposed to be expressed in seconds of arc. If we neglect terms of the order p B , these equations become x, = x 1^ x | y + 1 (sin + p cos 0) 2, S o o y, == y \ P ly + ^z 2 (costf p sin 0) s, (46) 8 S S 2, = z - x sin -4- -V cos 0. s s y 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, ?], and 0, we have p = (50".21129 + 0".0002442966r) (f t), 7 = ( 0".48892 0".000006143r) (f t), 6 = 351 36' 10" + 39".79 (t 1750) 5".21 (t r f), in which r = \(t f t) 1750, t and t' being expressed in years from the beginning of the era. If we add the nutation to the value of p, the co-ordinates will be derived for the true equinox of t'. The equations (45) and (46) serve also to convert the values of dx, dy, 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 &; in place of x, y, and z re- spectively, and similarly for a?,, y, 9 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 9 8y, and dz must be reduced, by means of the equations (45), as already explained, from the ecliptic and mean equinox to which they belong, to the ecliptic and mean equinox adopted in the case of the co-ordinates required. VARIATION OF CO-ORDINATES. 457 In a similar manner we may derive formulse for the transformation of the co-ordinates or of their variations referred to the mean equinox and equator of one date into those referred to the mean equinox and equator of another date; but a transformation of this kind will rarely be required, and, whenever required, it may be effected by first converting the co-ordinates referred to the equator into those referred to the ecliptic, reducing these to the equinox of t 1 by means of (45) or (46), and finally qonverting them into the values referred to the equator of t' . Since, in the computation of an ephemeris for the comparison of observations, the co-ordinates are generally required in reference to the equator as the fundamental plane, it would appear preferable to adopt this plane as the plane of xy in the computation of the perturbations, and in some cases this method is most advan- tageous. But, generally, since the elements of the orbit of the dis- turbed planet as well as the elements of the orbits of the disturbing bodies are referred to the ecliptic, the calculation of the perturbations will be most conveniently performed by adopting the ecliptic as the fundamental plane. The consideration of the change of the position of the fundamental plane from one epoch to another is thus also ren- dered more simple. Whenever an ephemeris giving the geocentric right ascension and declination is required, the heliocentric co-ordi- nates of the body referred to the mean equinox and equator of the beginning of the year will be computed by means of the osculating elements corrected for precession to that epoch, and the perturbations of the co-ordinates referred to the ecliptic and mean equinox of any other date will be first corrected according to the equations (46), and then converted into those to be applied to the co-ordinates referred to the mean equinox and equator. If the perturbations are not of con- siderable magnitude and the interval t' t is also not very large, the correction of dx, %, and dz on account of the change of the position of the ecliptic and of the equinox will be insignificant; and the conversion of the values of these quantities referred to the ecliptic into the corresponding values for the equator, is effected with great facility. In the determination of the perturbations of comets, ephemerides being required only during the time of describing a small portion of their orbits, it will sometimes be convenient to adopt the plane of the undisturbed orbit as the fundamental plane. In this case the posi- tive axis of x should be directed to the ascending node of this plane on the ecliptic, and the subsequent change to the ecliptic and equinox, whenever it may be required, will be readily effected. 458 THEORETICAL ASTRONOMY. 168. The perturbations of a heavenly body may thus be deter- mined rigorously for a long period of time, provided that the oscu- lating elements may be regarded as accurately known. The peculiar object, however, of such calculations is to facilitate the correction of the assumed elements of the orbit by means of additional observa- tions according to the methods which have already been explained; and when the osculating elements have, by successive corrections, been determined with great precision, a repetition of the calculation of the perturbations may become necessary, since changes of the ele- ments which do not sensibly affect the residuals for the given differ- ential equations in the determination of the most probable corrections, may have a much greater influence on the accuracy of the resulting values of the perturbations. When the calculation of the perturbations is carried forward for a long period, using constantly the same osculating elements, and those which are supposed to require no correction, the secular per- turbations of the co-ordinates arising from the secular variation of the elements, and the perturbations of long period, will constantly affect the magnitude of the resulting values, so that 8x, 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 7. 2 > ~j^~ an ^ jp- (which is effected indirectly) cannot be performed with facility, requiring often several repetitions in order to obtain the required accuracy, since any error in the value of the second differential coeffi- cient produces, by the double integration, an error increasing propor- tionally to the time in the values of the integral. Errors, therefore, in the values of the second differential coefficients which for a mode- rate period would have no sensible effect, may in the course of a long period produce large errors in the values of the perturbations, and it is evident that, both for convenience in the numerical calculation and for avoiding the accumulation of error, it will be necessary from time to time to apply the perturbations to the elements in order that the integrals may, in the case of each of the co-ordinates, be again equal to zero. The calculation will then be continued until another change of the elements is required. CHANGE OF THE OSCULATING ELEMENTS. <*59 The transformation from a system of osculating elements for one epoch to that for another epoch is very easily effected by means of the values of the perturbations of the co-ordinates in connection with the corresponding values of the variations of the velocities -7r, -rr, and -^r- The latter will be obtained from the values of the dt dt dt 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^ = + 214.6, 40^= + 9.7. at at at The velocities in the case of the disturbed orbit will be given by the formula? dx dx Q ddx dy dy Q ddy dz dz ddz . . , ~dt~~~di dt' ~dt~~~dt"~dt' ~dt == ~di^~~di' ^ ' 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 , dx . . , . , dr . . , dv -7- = sin a sin (A + u) -j- -4- r sin a cos ( A -f- u) -y-, at at at dy . 7 . /T , , , dr , s-n t \ dv -j?r = smb sin (B + u) -=- + r sm b cos (jB + <*) -jr, at at at dz . - f n \ \ ar \ f rt i \ dv = sin c sin ( C + u ) -=- -4- r sin c cos ( C + u ) =-. at at at These equations are applicable in the case of any fundamental plane, if the auxiliaries sin a, sin 6, sin c, A, I> } and C are determined in reference to that plane. To transform them still further, we have dr &l/l-fra . , , u - = = in which w denotes the angular distance of the perihelion from the ascending node. Substituting these values, we obtain, by reduction, 460 THEOEETICAL ASTKONOMY. dx_ __ kvl-j-m cQg ^ _j_ cQg ^ cog ^ _ ( e si n w _f- sm u) sin J.) sin a, dt \/p m (( 6 cos j j Civ ~dt' an( * ~d' ^ en we a PPty to tne se the values of the perturba- tions, and thus find x, y, z, 2L, , and These having been CHANGE OF THE OSCULATING ELEMENTS. 461 found, the equations (32)! will furnish the values of &, i, and p; and the remaining elements may be determined as explained in Art. 112. Thus, from Vr sin 4 = Tcp (1 -f m), dx . dy . dz we obtain Vr and ^/ , and from r sin u = ( x sin & + 2/ cos &) sec i, r cos i* x cos & -f- 2/ sm & i we derive r and it; 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 y r sin u = z cosec i. Next, we compute a from r F 2 and from 2ae sin w = (2a r) sin (24/ + w) r sin u, 2ae cos 01 = (2a r) cos (2<4/ -\- u) r cos w, we find co and e. The mean daily motion and the mean anomaly or the mean longitude for the epoch will then be determined by means of the usual formulae. In the case of a very eccentric orbit, after r and u have been found, -j7 will be given by equations (48) 6 , and the values of e and v will dt be given by the equations (49) 6 . Then the perihelion distance will be found from and the time of perihelion passage will be found from v and e by means of Table IX. or Table X. In the numerical values of the velocities -rr> -571 &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 & and i must satisfy the equation 462 THEORETICAL ASTRONOMY. z cos i y sin i cos + x sin i sin & = 0. We have, also, V, / 2 r sn w sn , which must be satisfied by the resulting values of F, r, and u; and the values of a and e must satisfy the equation p = a(l e 2 ) = a cos 2 . 169. When the plane of the undisturbed orbit is adopted as the fundamental plane, we obtain at once the perturbations 8 (r cos it), <5 (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 Q 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 9 and fi the latitude of the body with respect to the plane of xy; then we shall have x = r cos /9 cos w, y = r cos /? 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 d(rcos/3) a . dw - = cos w -=- r cos ft sin w -rp dt dt dt dy . d(rcosft} , dw ~rr = sin w -=2 h r cos /3 cos w =7-? etc at and hence VARIATION OF POLAR CO-ORDINATES. 463 Therefore we have r 2 cos 2 /5 = J"( FZ Xy) dt + (7. If we denote by S the component of the disturbing force in a direc- tion perpendicular to the disturbed radius-vector and parallel with the plane of xy, we shall have X = S Q sin w, Y= S cos w, and Yx Xy = S rcosfi. Therefore r 2 cos 2 /9 ^ == ffl r cos 'o and the equations become dt Z 5 Z. 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 so 466 THEOKETICAL 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 , 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, Sr, and z; and with the values r = r Q -f 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 XVIL, we derive the corresponding values of log/' and log/ /r . The coefficients of dr in the expressions for q and q rf will be given with sufficient accuracy by means of the approximate values of or and sin /9, and will not require any further correction. Then we compute 8 Q r cos/9, and find the integral CS Q and the complete value of rr- will be given by (59). The value of T-^J- will then be given by equation (63). The term fl -jgr 1 w^l 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 - ( -jr \ - The corrected values of the differential coefficients being introduced into the table of inte- gration, the exact or very approximate values of Sw, Sr, and z will be obtained. Should these results, however, diifer 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 POLAE CO-ORDINATES. 467 and z 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 co, 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 dw we compute the values of co _, , and since the second member of the equation contains the integral j /S^ cos /9 eft, if we introduce the factor co 2 under the sign of integration, this integral, omitting the factor co in the formulae of integration, will become co I 8 Q r cos /9 eft, as required. The last term of the equation will be multiplied by co. In the case of dr, each term of the equation for ^- must contain the factor co 2 . If the second of equations (65) is employed, the first and third terms of the second member will be multiplied by co 2 ; but since the value of $ is supposed to be already multiplied by co 2 j the second term will only be multiplied by co. The perturbations may be conveniently determined either in units of the seventh decimal place, or expressed in seconds of arc of a circle whose radius is unity. If they are to be expressed in seconds, the factor s 206264.8 must be introduced so as to preserve the homogeneity of the several terms, and finally dr and dz must be con- verted into their values in terms of the unit of space. 172. It remains yet to derive convenient formula? for the deter- mination of the forces S Q) 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 r and & ' denote the inclination and the longitude of the ascending node of the disturbing body with respect to the ecliptic, and let I denote the inclination of the orbit of the disturbing body with respect to the fundamental plane. Further, let N denote the longitude of its ascending node on the same plane measured from the ascending node of this plane on the ecliptic or from the point whose longitude is & , and let N f be the angular distance between the as- cending node of the orbit of the disturbing body on the ecliptic and the ascending node on the fundamental plane adopted. Then, from the spherical triangle formed by the intersection of the plane of the 468 THEORETICAL ASTRONOMY. ecliptic, the fundamental plane, and the plane of the orbit of the dis- turbing body with the celestial vault, we have sin -i/sin (N+ N f ) = sin A & & sn cosUsin J (JV ^') = sin $ (&' ) cos J (z' + i ), cosi/cosiCZV -tf') = cosKfc' 8 ) cos i (** V>> from which to find JV, JV, and J. Let /3' 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)j, we have tan (w' N) tan u r cos I, tan jt = tan Jsin (w r JV). If u r denotes the argument of the latitude of the disturbing planet with respect to the ecliptic, we have UQ > = u ' N'. (68) This formula will give the value of u f , and then w f and /9' will be found from (67). We have, also, cos U Q ' = cos /?' cos (w f JV), which will serve to indicate the quadrant in which w r N must be taken. The relations here derived are evidently applicable to the case in which the elements of the orbits of the disturbed and disturbing planets are referred to the equator, the signification of the quantities involved being properly considered. The co-ordinates of the disturbing planet in reference to the plane of the disturbed orbit at the instant as the fundamental plane will be given by a/ = / cos j cos w', 2/=rr'cos/5'sinw', (69) z' = r f sin ft. To find the force R, we have VARIATION OF POLAR CO-ORDINATES. 469 and Substituting in these the values of #', y 1 ', z r given by (69), and the corresponding values of x, y, z given by (50), and putting we get R = m'k* i h / cos ft cos ft cos (w 1 iv*) + h r' sin /3 sin /3' --A (71) The equation S r cos = Fa; JTy gives == m '& 2 A r' cos /5' sin (w' w), (72) from which to find $ . Finally, we have Z= m'tfihr'smp -A (73) from which to find When we determine the perturbations only with respect to the first power of the disturbing force, the expressions for R, S Qt and Z become R = m'P ( h r' cos jf cos (w' w ) -^ ) , )o / $ o = ?n'^ 2 h r' cos /?' sin (w f IV Q \ Z =m'tfhr'smft'. To compute the distance p, we have which gives p i _ r '2 _|_ r 2 _ 2r / cos 13 cos ' cos (w' w) 2r r' sin /9 sin /9', (75) and, if we neglect terms of the second order, we have p* = r' 2 + r 2 2r / cos p cos (w/ w ). (76) If we put cos Y cos ft cos p cos (w' to) + sin /9 sin /5 ; , (77) we have pi _ r ' 2 _j_ r 2 _ 2rr ' cos r = / 2 sinV -j- (r / cos ^) 3 ; 470 THEOKETICAL ASTEONOMY. and hence we may readily find p from p sin n r sin 7% p cos n = r r' cos y, the exact value of the angle w, however, not being required. Introducing f into the expression for _B, it becomes 9 , (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), p Q being found from (76) or from (78), when we put COS f = COS ft COS (w' WQ). 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), 8 Q from (72), Z from (73), and E from (71) or (79). The values of dw, dr, and 2, 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, $ , and Z thus found will not require any further cor- rection. When the places of the disturbing planet are to be derived from an ephemeris giving the heliocentric longitudes and latitudes, the values of & ' and V 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) 1 or by means of (85) r When the inclination V is very small, it will be sufficient to take tf = l'Q' + 8 tan 2 tf sin 2 (I' ft'), in which s = 206264.8. But when the tables give directly the lon- gitude in the orbit, u' + ft', by subtracting ft' from each of these longitudes we obtain the required values of u f . It should be observed, also, that the exact determination of the values of the forces requires that the actual disturbed values of r', w f , and /3' should be used. The disturbed radius-vector r f will be VARIATION OF POLAR CO-ORDINATES. 471 given immediately by the tables of the motion of the disturbing body, but the determination of the actual values of w' and ft' re- quires that we should use the actual values of N', N, and I in the solution of the equations (68) and (67). Hence the disturbed values of & ' and V should be used in the determination of these quantities for each date by means of (66). It will, however, generally be the case that for a moderate period the variation of &' and i' may be neglected; and whenever the variation of either of these has a sensi- ble effect, we may compute new values of N, N f , and / from time to time, by means of which the true values may be readily interpolated for each date. We may also determine the variations of N 9 N f , and / arising from the variation of &' and i', by means of differential formulae. Thus the relations will be similar to those given by the equations (71) 2 , so that we have sin N 1 sin N' sin(a'-a o ) sin/ ., r sin JV , , sin^V' , 9N = . 7 ; -r cos N <5Q ' : di, (80) sin(Q &6 ) sin 1 dl sin N' sin i' dQ' -f cos N' Si', from which to find dN', dN, and dl. When the perturbations are computed only in reference to the first power of the mass, the change of &' and i' may be entirely neg- lected; but when the perturbations are to be computed for a long period of time, and the terms depending on the squares and products of the disturbing forces are to be included, it will be advisable to take into account the values of dN, dN', and dl, and, using also the value of u' in the actual orbit of the disturbing body, compute the actual values of w' and ft'. In the case of several disturbing bodies, the forces will be deter- mined for each of these, and then, instead of R, 8 Q , and Z, in the formulae for the differential coefficients, 2R, 2S Q9 and 2 Z will be used. 174. By means of the values of dw, dr, and z, the heliocentric or the geocentric place of the disturbed planet may be readily found. Thus, let the positive axis of x be directed to the ascending node of the osculating orbit at the instant on the plane of the ecliptic; then, in the undisturbed orbit, we shall have W Q = u , u denoting the argument of the latitude. Let x,, y,, z, be the co-or- 472 THEORETICAL ASTRONOMY. dinates of the body referred to a system of rectangular co-ordinates in which the ecliptic is the plane of xy, and in which the positive axis of x is directed to the vernal equinox. Then we shall have x, = x cos & y cosi sin & -j- z sin i sin & , y, = zsin & -f- 2/cosi cos& zsini cos& , Zf=y sin i -j- z cos i , or, introducing the values of x and y given by (50), x, = r cos ft cos w cos & r cos ft sin w cos i Q sin & -f 2 sin ^ sin & , y, =r cos /5 cos w sin & + r cos sm w cos ^o cos &o z sm h cos &o> (81) z, =r cos /? sin w sin i -f- 2 cos i . Introducing also the auxiliary constants for the ecliptic according to the equations (94) t and (96) D we obtain x f = r cos ft sin a sin (A -\- w) -\- z cos a, y,=r cos sin 6 sin (B -f it;) -j- z cos 5, (82) z, r cos /5 sin i sin w -\- z cos i , by means of which the heliocentric co-ordinates in reference to the ecliptic may be determined. If the place of the disturbed body is required in reference to the equator, denoting the heliocentric co-ordinates by x,,, y, n z m and the obliquity of the ecliptic by e, we have x tl = x f y,,=y,cose z,$me, z n = y f sin s -j- z, cos e. Substituting for x n y,, z, their values given by (81), and introducing the auxiliary constants for the equator, according to the equations (99) x and (101) w we get x ff = rcosft sin a sin (A -f- w) -J- z cos a, . y tr = rcosft sin b sin (B -f- w) -f- 2 cos 6, (83) 2,, = r cos /9 sin c sin ( G -J- w) -}- z cos c. The combination of the values derived from these equations with the corresponding values of the co-ordinates of the sun, will give the required geocentric places of the disturbed body. These equations are applicable to the case of any fundamental plane, provided that the auxiliary constants a, A, b, B, &c. are determined with respect to that plane. In the numerical application of the formulae, the value of w will be found from W = U Q -f- dw, VARIATION OF POLAE 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 TT O , & , and i used in the calculation of the per- turbations are referred to the ecliptic and mean equinox of the date t f , and the rectangular co-ordinates of the disturbed body are required in reference to the ecliptic and mean equinox of the date /', the value of w must be found from the value of to Q referred to the ecliptic of t Q ' being reduced to that of ", by means of the first of equations (115)!. Then & and i should be reduced from the ecliptic and mean equinox of t Q f to the ecliptic and mean equinox of " by means of the second and third of the equations (115)^ and, using the values thus found in the calculation of the auxiliary constants for the ecliptic, the equations (82) will give the required values of the heliocentric co-ordinates. If the co- ordinates referred to the mean equinox and equator of the date t Q " are to be determined, the proper corrections having been applied to & 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, dx,. , . , dw , _ . . , . , dr -Z. = r cos p sin a cos (A -+- w) jr 4- sec p sin a sin (A -f- w) =- dt at at -J- (cos a tan /? sin a sin (A -f- w)) -77, Jjg- = r cos /5 sin b cos (J5 4- w) -yr 4- sec /5 sin b sin (B -j- w) jr (cos b tan /? sin b sin (J5 + w )) -JT & (84) -JT dz., ^ dw , , . ./>-., \ dr l' = r cos /9 sin c cos ( C -4- w} ~ 4- sec /? sm c sm ( C -j- w) j- dt dt at -(-(cos c tan /5 sin c sin ( C -f- w)) =r- t by means of which the components of the velocity of the disturbed body in directions parallel to the co-ordinate axes may be determined. d$r 1 dz d*Sr , d*z The values of -rr and -jr will be obtained from 7^- and -j^ by a dt dt dt* dt* single integration, and then we have 474 THEOKETICAL ASTRONOMY. dw ft/frd + ifij ddw ^_l! m ,, + (85) dt ~ r 2 dt ' dt Vp Q dt , , dw , dr from which to find -^ and ~^- 175. EXAMPLE. In order to illustrate the calculation of the per- turbations of r, w, and 2, 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 formulae (66), using the values of &o an d * given in Art. 166, we derive N= 194 0' 49".9, N' = 301 38' 31",7, 1= 5 9' 56".4. The value of u f is given by equation (68), and then w' and /3' are found from the equations (67). Thus we have Berlin Mean Time. Iogr WQ = U O logr' w' ft' 1863 Dec. 12.0, 0.294084 192 24".5 0.73425 14 18' 54".6 V 38".l 1864 Jan. 21.0, 0.294837 207 40 52 .2 0.73368 17 21 44 .2 18 9 .1 March 1.0, 0.300674 223 3 5 .9 0.73305 20 25 5 .2 34 39 .9 April 10.0, 0.310864 237 51 38 .3 0.73237 23 28 59 .8 51 7 .6 May 20.0, 0.324298 251 52 47 .9 0.73164 26 33 32 .1 17 29 .7 June 29.0, 0.339745 264 59 30 .0 0.73086 29 38 44 .8 1 23 43 .5 Aug. 8.0, 0.356101 277 10 24 .6 0.73003 32 44 41 .2 1 39 46 .3 Sept, 17.0, 0.372469 288 28 4 .1 0.72915 35 51 24 .6 1 55 35 .2 Oct. 27.0, 0.388214 298 57 16 .3 0.72823 38 58 57 .5 2 11 7 .5 Dec. 6.0, 0.402894 308 43 48 .7 0.72726 42 7 23 .3 2 26 20 .3 1865 Jan. 15.0, 0.416240 317 53 39 .1 0.72625 45 16 43 .9 2 41 10 .6 The values of p Q may be found from (76) or (78) as already given in Art. 166. The forces R, S , and Z may now be determined by means of the equations (74), h being found from (70), and if we introduce the factor (o 2 for convenience in the integration, as already explained, we obtain the following results : Date. tf]R tfS r 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 Bate. wlR &Sf 6 tfZ o>\ S r dt 1864 March 1.0, + 1".2616 1".4512 + 0".0190 1".3060 April 10.0, 1 .0018 2 .1226 .0273 3 .1035 May 20.0, .6760 2 .6473 .0347 5 .5020 June 29.0, + .3179 2 .9988 .0406 8 .3402 Aug. 8.0, .0452 3 .1650 .0449 11 .4378 Sept. 17.0, .3944 3 .1437 .0470 14 .6076 Oct. 27.0, .7180 2 .9392 .0466 17 .6640 Dec. 6.0, 1 .0097 2 .5586 .0432 20 .4273 1865 Jan. 15.0, 1 .2674 - 2 .0081 + .0362 22 .7245 The integral (o\ S Q r d dt is obtained from the successive values by means of the formula (32). Next we compute the values of the differential coefficients by means of the formulae (65). For the dates 1863 Dec. 12.0 and 1864 Jan. 21.0 we may first assume Sr Q, and, by a preliminary inte- gration, having thus derived very approximate values of 8r for these dates, the values of , will be recomputed. Then, commencing (MI anew the table of integration, we may at once derive an approximate value of dr for the date March 1.0 with which the last term of the expression for -^ may be computed. Continuing this indirect pro- cess, as already illustrated in the case of the perturbations of the rec- tangular co-ordinates, we obtain the required values of the second differential coefficient. In a similar manner, the values of -^ will be obtained. The values of j will then be given directly by means etc/ of the first of equations (65); and the final integration will furnish the perturbations required. Thus we derive the following results : 1863 Dec. 12.0, 0".0423 -fl".4509 +0".0009 0".00 +0".18 -f-0".00 1864 Jan. 21.0, .1086 1 .3405 .0101 .02 .17 .00 Mar. 1.0, .7162 +0 .7829 .0183 .40 1 .47 .01 Apr. 10.0, 1 .61140 .0455 .0251 1 .55 3 .53 .04 May 20.0, 2 .4795 .9344 .0300 3 .61 5 .54 .09 June 29.0, 3 .0807 1 .7333 .0326 6 .42 6 .62 .18 Aug. 8.0, 3 .2971 2 .3752 .0331 9 .64 5 .98 .29 Sept. 17.0, 3 .1080 2 .8533 .0311 12 .88 +2 .98 .44 Oct. 27.0, 2 .54253 .1872+0 .026515 .73 2 .86+0 .62 476 THEORETICAL ASTRONOMY. d$w ,d'tJr , cPz Date - "^dT *-& "V 6w 6r 1864 Dec. 6.0, 1".6443 3".4009 -|- 0".0190 17".85 11".88 +0".83 1865 Jan. 15.0, .45113 .5334+0 .007918 .9224 .29+1 .05 It has already been found that, during the period included by these results, the perturbations arising from the squares and products of the disturbing forces are insensible, and hence the application of the complete equations for the forces and for the differential coefficients is not required. The equations (83) will give, by means of the results for w = u + Sw, r = r Q -\- 3r, and z, the values of the helio- centric co-ordinates of the disturbed body, and the combination of these with the co-ordinates of the sun will give the geocentric place. When we neglect terms of the second order, we have, according to the equations (84), dx tf = X Q cot ( A + w) Sw + dr + z cos a, r o dy n = y Q cot (B + w) dw + V* 3r + z cos b, (86) r o dz tf = Z Q cot ( C + w) dw + dr + z cos e, r o the heliocentric co-ordinates # , y w Z Q being referred to the same fun- damental plane as the auxiliary constants, a, 6, A, &c. Thus, in the case of Eurynomey to find the perturbations of the rectangular co-or- dinates, referred to the ecliptic and mean equinox of 1860.0, from 1864 Jan. 1.0 to 1865 Jan. 15.0, we have A = 296 34' 37 x/ .5, B = 206 43' 34".4, (7=0, log cos a = 8.557354n, log cos 6 = 8.856746, log cos c = log cos ?' = 9.998590, log x = 0.399807 n , log y = 9.838709, log z = 9.148170 n , w = w + 6w = 317 53 X 20".2, and hence, by means of (86), we derive dx, = + 36".559, fy, = + 41".083, dz, = 0".588. If we express these in parts of the unit of space, and in units of the seventh decimal place, we obtain fa, = + 1772.4, dy, = + 1991.8, 8z, = 28.5, agreeing with the results already obtained by the method of the va- riation of rectangular co-ordinates, namely, fa, = + 1772.6, fy, = + 1992.3, dz, = 28.2. CHANGE OF THE OSCULATING ELEMENTS. 477 176. By using the complete formulae, the perturbations of r, w, and z may be computed with respect to all powers of the disturbing force, and for a long series of years, using constantly the same fun- damental osculating elements. But even when these elements are so accurate as not to require correction, on account of the effect of the large perturbations of long period upon the values of dw and dr, the numerical values of the perturbations will at length be such that a change of the osculating elements becomes desirable, so that the integration may again commence with the value zero for the variation of each of the co-ordinates. This change from one system of ele- ments to another system may be readily effected when the values of the perturbations are known. Thus, having found the disturbed values of r 9 w, and z, we have dv* _ q dw> dp _ tfp (1 + m) ~ " sl ~ " p being the semi-parameter of the instantaneous orbit of the disturbed body. In the undisturbed orbit we have _ dv kv'po (1 "j"*0 ^ 0= W = r 2 and hence we derive Substituting for -j- the value above given, there results (Mi ' ,.-, rt ' R by means of which p may be determined. To find -rr, we have d/3 _ 1 dz tan/? dr "df ~ fcosj* " "3T ~ ~T~'~dt' We have, also, dr k\/l 4- m IcVl+m . d3r ^ = ~^ and if we put , P, Vl (89) 478 THEORETICAL ASTRONOMY. this equation becomes e sin v = e Q sin V Q -f- e sin v -f ^. (90) We have, further, cos/y _.P _1 ... r and, putting P 1*0 11/3 /Q1 ^ ' ~ :r ^ : J- ~T~ Pi \v-Lj we obtain ecosv = e Q cos -y + This equation, combined with (90), gives Pa o ' e sin (v VQ) = ae sin v cos V Q -j- y cos v p sin i; , 1 02) e cos (-U v ) = e + ae sin 2 v + y sin v + /? cos v ot by means of which the values of e and v may be found, those of the auxiliaries a, /9, f, being found from (89) and (91). Then we have e = sin ^, a=p sec 2 ?>, /* = ^ , tan i E = tan (45 p) tan -J v, M=E esmE, by means of which ^>, a, //, and If may be determined. In the case of orbits of great eccentricity, we find the perihelion distance from g= P 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 co or x. Let denote the longitude of the ascending node of the instantaneous orbit on the plane of the osculating orbit, defined by & and i , mea- sured from the origin of w, and let y denote its inclination to this plane. Then we have tan y sin (w ) = tan p, ( a \ dw a dP (93) and hence CHANGE OF THE OSCULATING ELEMENTS. 479 tan O ) = ^sin 2/3 di , (94) by means of which may be found. The quadrant in which # is situated is determined by the condition that sin (w # ) and tan /9 must have the same sign. The value of fj Q will be found from the first or the second of equations (93). If we denote by f the argument of the latitude of the disturbed body with respect to the adopted fundamental plane, we have COS rj Q (95) and the angle must be taken in the same quadrant as w; . Then, from the spherical triangle formed by the intersection of the planes of the ecliptic and instantaneous orbit of the disturbed body, and the fundamental plane, with the celestial vault, we derive cos %i sin ( (u C) -f K& &)) = sin i^o cos i (% ~ 7o) cos A i cos (J (u C) -f i (ft ft,)) = cos cos -J (i -f- ^ ), sin i i sin (A (w - C) - i (ft 8 }) = sin 10 sin (i - 7o ), ( sin J t cos (i(w C) | (Q ft )) = cos p sin (i + 7o ). These equations will furnish the values of i, u f , and & 2 , and hence, since and & are given, those of & and u. The value of v having been already found, we have, finally, w = u v, x = u v -f- &> 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 formula 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 fn y ff , z, n in reference to the fundamental plane to which & and i are to be re- ferred, by means of the equations (83), and also those of -~-' ^, -^ by means of (85) and (84), and then determining the elements from the co-ordinates and velocities, as already explained. It should be observed that when the factor w 2 , or the square of the 480 THEORETICAL ASTRONOMY. adopted interval, is introduced into the expressions for the forces and differential coefficients, the first integrals will be d$r ddiv 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 /9 0, and the equations (51) and (54) become r^=fSrcft + ^ (l + m), (97^ dV __ duj ^(1 -fm) __ in which R denotes the component of the disturbing force in the direction of the disturbed radius-vector, and 8 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, and suppose it to be directed from the centre of the sun to a point whose angular distance back from the place of the ascending node is dz cos i d& , (99) du = d%-{-(l We have, further, v being the true anomaly in the instantaneous orbit. The two components of the disturbing force which act in the plane of the disturbed orbit will only vary and the elements which deter- mine the dimensions of the conic section. We have, therefore, in the case of the osculating elements, for the instant t , Let us now suppose ), to denote the true longitude in the orbit, so that we have A = V -f- 7T = V -fttl + &, 31 482 THEORETICAL ASTRONOMY. (*-); (101) then, since # is equal to K 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 r cos v = a cos E ae, (102) r sin v = al/1 e 2 sin E, Let us first consider only the perturbations arising from the action of the two components of the disturbing force in the plane of the dis- turbed orbit, and let us put *, = *+* (103) Further, let if + f*o(t t ) + 8M be the mean anomaly which, by means of a system of equations identical in form with the preceding, but in which the values of a , e 09 fa are used instead of the instanta- neous values a, e, and , gives the same longitude ^,, so that we have E, e Q smE, = M + v (t tj + dM, r, cosv, = a Q cos E, a Q e Q , r, sin v, = a l/l e 2 sin E f If, therefore, we determine the value of dM so as to satisfy the con- dition that A, = 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 dM and v have been found, the effect of the disturbing force perpendicular to the plane of the instantaneous orbit may be considered, and thus the complete perturbations will be obtained. VARIATION OF CO-ORDINATES. 483 In the equations (97), fr 2 -=- 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 2 -jj- must also represent this areal velocity, and hence the equations become ^ =Sr dt + */Po(l+"0 (106) _ ri _ dP \ dt I r 2 179. If we differentiate each of the equations (104), we get d8M _ o _ o __, dr, dv, dE. SV '-dt~ r ' SmV '^t = -^^'-SP dr, . dv f /- - - dE, sm v, - + r, cos v, - = a Q V le* cos E, - ^,_^ ^ " dt' From the second and the third of these equations we easily derive e 2 r, sin v, cos E, a Q r, cos v, sin E r ) -^-'. Substituting in this the values of r, sin v,, r, cos v,, and -p, and re- ducing, we get dr, or * fct/TX^ / + I^\ (108) From the same equations, eliminating -^, we get r, 2 ^ = (a l/l e 2 r ; cos v, cos ^ -f- a r, sin v ; sin E t ) -^, which reduces to (109 ) 484 THEOKETICAL ASTRONOMY. or Combining this with the first of equations (106), we get dSM_ dt ~~ from which dM may be found as soon as v is known. The equation (105) gives dr f-\ \ \^ r ' \ ^L dt dt ' dt' fw Differentiating equation (108) and substituting for -g its value already found, we obtain JcVl-\-me smv, dt n and the last of the preceding equations becomes dV d*v I? (1 -{- m) e cos v, . The equation (110) gives i 2 ^, 2 ^ 1 o ,. " 2 f ' f 1 ' ' " which is easily reduced to o^.4.^ J* dt^' dt dt and hence we derive ^ The equation (109) gives r, and, since this becomes VARIATION OF CO-ORDINATES. 485 / dv,V_ ffptt(l + m) / 1 ,1 ddM\* '\~5~r ~^r \ *v * ; PQL + mHcoBiw / 1 cMfV* r, 2 ; \ "% * / Combining equations (112) and (113) with the second of equations (106), we get _ p , " r r 3 Po From (110) we derive + ( . l \kVp 9 (l and the preceding equation becomes -( F= CSrdtY, \ lev p Q (1 -f- m) *^ ' which is the complete expression for the determination of y. 180. It remains now to consider the effect of the component of the disturbing force which is perpendicular to the plane of the disturbed orbit. Let x n y n z, denote the co-ordinates of the body referred to the fundamental plane to which the elements belong, and x, y the co-ordinates in the plane of the instantaneous orbit. Further, let a denote the cosine of the angle which the axis of x makes with that of x,, and ft the cosine of the angle which the axis of y makes with that of y,, and we shall have t z t = ax + py. (116) If the position of the plane of the orbit remained unchanged, these 486 THEORETICAL ASTRONOMY. cosines a and /9 would be constant; but on account of the action of the force perpendicular to the plane of the orbit, these quantities are functions of the time. Now, the co-ordinate z, is subject to two dis- tinct variations : if the elements remain constant, it varies with the time; and, in the case of the disturbed orbit, it is also subject to a variation arising from the change of the elements themselves. We shall, therefore, have dz L _idz L \ dt ~\ dt I dt iii which I -jr I expresses the velocity resulting from the constant elements, and -~ that part of the actual velocity which is due to the change of the elements by the action of the disturbing force. But during the element of time dt the elements may be regarded as constant, and hence the velocity -j- in a direction parallel to the axis of z f may be regarded as constant during the same time, and as receiving an increment only at the end of this instant. Hence we shall have dt \ dt Differentiating equation (116), regarding a and ft as constant, we get dz, \__dz, dx . dy 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 A = ^a^ + ^5^^^ , n9 . Now, we have Z, = aX+ /3F+ Zcosi, (120) in which Z, denotes the component of the disturbing force parallel to the axis of z n and i the inclination of the instantaneous orbit to VARIATION OF CO-ORDINATES. 487 the fundamental plane. Substituting for X and Y their values given by the equations (1), and reducing by means of (116), we obtain or d 2 z, d*x Comparing this with (H9) 7 there results da dx , dB dy drift + IT 1=* COSJ - (121 > 181. The equation (120) gives The component of the disturbing force perpendicular to the plane of the disturbed orbit does not aifect the radius-vector r; and hence, when we neglect the effect of this component, and consider only the components R and S which act in the plane of the orbit, we have d\ 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 f which arises from the action of the force perpendicular to the plane of the disturbed orbit, so that we shall have 2, = Z + **,> a = a + to, /? = ft -j- 3{3. Substituting these in equation (122) and then subtracting equation (123) from the result, we get (124) , The equations (116) and (117) give If we eliminate dp between these equations, there results . d y *. ., dda! , 488 THEOKETICAL ASTEONOMY. 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 ,. ro)\ dt dt '/ Substituting these values in equation (124), it becomes tfdz, #(l + m), - ( 1 26) , 1 / / dy v dx\ fv v . ddz, \ -\ -- . I X-JT Y-j- \dZf-}- (Yx Xv) ~ I. ^l/p(l + m)\\ dt dt f dt I' If we introduce the components R and 8 of the disturbing force, we have r r r r and hence ydy y dx __ It / dr Yx -Xy = Sr. Therefore the equation (127) becomes dz, -f- Zcosi / 7? We have, further, (128) dr dr, . dv ~ which, by means of the equations (108) and (109), gives dr - gpsin^ dv, dv Substituting this value in the equation (128), we obtain VARIATION OF CO-OKDINATES. 489 dz, df ~ r 3 ' ~ If * + r * 2rr f cos /5' cos (w r ui), (133) or, putting COS f = COS p COS (W W), the equations p sin n = 1*810.7, = r / cos f. The values of r f and u 1 for the actual places of the disturbing body will be given by the tables of its motion, and the actual values of & ' and V will also be obtained by means of the tables. The de- termination of the actual values of r and w requires that the pertur- bations shall be known. Thus, when SM and v have been found, we compute, by means of the mean anomaly M Q -f- fJL (t t ) + dM and the elements a , e w the values of v, and r,. Then, since v -f % = v , + TT O , we have, according to (100), w = v, -f TT O 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 logr = logr, + 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 dMj 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 /9 r and w' are known to a sufficient degree of approximation, we may, with very little additional labor, consider the terms depending on the squares and higher powers of the masses. It will, however, appear from what follows, that when we consider the perturbations due to the higher powers of the disturbing forces, the consideration of the effect of the variation of z, in the determination of the heliocentric place of the disturbed body, becomes much more difficult than when the terms of the second order are neglected; and hence it will be found advisable to determine new osculating elements whenever the con- sideration of these terms becomes troublesome. VARIATION OF CO-ORDINATES. 493 The results may be conveniently expressed in seconds of arc, and afterwards v and 3z, 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 .3z, are obtained directly in units of the seventh decimal place. It will be advisable, also, to introduce the interval to into the formula 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 + 8T be the time of perihelion pas- sage to be used in the formulre in the place of T and in connection with the elements q Q and e in the determination of the values of r, and v,j so that we have ^ + / = v f + K O - In the case of parabolic motion we have, neglecting the mass of the disturbed body, the solution of which to find v, is effected by means of Table VI. as already explained. To find r n we have r, = g Q sec 2 k,. For the other cases in which the elements M Q and p cannot be em- ployed, the solution must be effected by means of Table IX. or Table X. Thus, when Table IX. is used, we compute M from wherein log (7 = 9.9601277, and with this as the argument we derive from Table VI. the corresponding value of V. Then, having found 1 _ e i = 3 -- -, by means of Table IX. we derive the coefficients required 1 ~r e o in the equation (lOOi) + B (1000 2 + (7(100*7, (140) 494 THEORETICAL ASTRONOMY. from which v, will be determined. Finally, r, will be found from 1 -f- e cos v, When Table X. is used, we proceed as explained in Art. 41, using the elements T= T Q + dT, q 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 gves or, simply, dM and the equation (110) becomes ' 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,, re- spectively, we have *M=p(tt) II 1 ddM = dif and the equation (110) becomes **L- 1 l 1 Cv * dt - (!+) + ( T+^ ' ai/ro+S) J Sr dt ' or, putting t, = t + dt, ddt 1 & (144) If we determine 8t by means of this equation, the values of the radius- vector and true anomaly will be found for the time t + dt instead of t, according to the methods for the different conic sections, I VARIATION OF CO-ORDINATES. 495 using the fundamental osculating elements. The results thus obtained are the required values of r, and v, respectively. 185. When the values of the perturbations V, 8z,, and dM, ST, or 8t have been determined, it remains to find the place of the disturbed body. The heliocentric longitude and latitude will be given by cos b cos (I ft) = cos (A ft), cos b sin (I ft ) = sin (A ft ) cos i, sin b = sin (A ft ) sin i t or, since A = A, a -f- ft, cos b cos (I ft ) = cos (A, s *o, cos b sin (7 A)=sin (A, ft ) cos (A ft ) cos i cos (A, & ) sin (A ^ ) +cosa-^ )sin(A -^ )(l-fcosV) (148) +sin (;, ) ((cosi cosi ) cos (A ^o)4-sin ( and the second by sin (^ & ), and add the results; then multiply the first by sin(/i & ), and the second by cos(/i & ), and add, we get cos6 cos (7 ^o (h W)=cos(A ; ^ )+sin(^ &Q) 1 , , J. - COS Tj T cosb sin (7 ^ (h /& )) sin (*, ^ )cosi cos(A ^ ) * - , '-> X' cos ^9 ^r sin b =sin (A, ^ ) sin ^-j '-. (152) Let us now put g f = cos (o "37' dt dt dt From the equations (118) and (121), observing that dy dx ~dt~ y ~dt we derive, by elimination, da _ r sin A, cos i dp _ r cos A, cos i 500 THEOEETICAL ASTRONOMY. Therefore we shall have dp r cos i sin (A, & ) r cos i cos (>*, by means of which p' and g' 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 f will be given by means of (154), or and if p f is determined, q f will be given by If both p f and q f are found from the equations (162), dz, may be de- termined directly from (154); but the value thus obtained will be less accurate than that derived by means of equation (129). Since the formula for ~ completely determines the perturbations due to the action of the component Z perpendicular to the plane of the instantaneous orbit, instead of determining p' and q f by an independent integration by means of the results given by the equations (162), it will be preferable to derive them directly from dz, and -^-'. The equations (161) give p' = cos da sin dp, q f = sin & da -f cos 80. Substituting for da and 3ft their values given by (125) and (126), and putting x" = x cos -f y sin & , y" = x sin + y cos > we obtain d8z, df \ " .irr '^/' dx dt~ dZ '~dt VARIATION OF CO-ORDINATES. 501 Substituting further the values x" = r cos 0*, & ), y" = r sin (A, ), and also dt r 2 dr &t/l -f- m Jcyp (1 4- m) e sin v -TT = 7^ esmv = -r r , rfi l/p r 1 -j- e cos v we easily find, since ^, v = , Tl . (164) which may be used for the determination of p f and q f . These equa- tions require, for their exact solution, that the disturbed values e, %, and p shall be known, but it is evident that the error will be slight, especially when e is small, if we use the undisturbed values e , p , and TT O . The actual values of X, and r are obtained directly from the values of the perturbations. When p f and q r have been found, it remains only to find cos i, and 1 cos r/, in order to be able to obtain F by means of the equation (159). From (153) we get o sn Q9 and hence _ cos i = 1/1 p' 2 (q r + sini ) 2 , (165) from which cosi may be found. The equation (157) gives 1 cos ff = cos i Q (cos i Q + cos i) and -^p when the correct values ol R, S, Z } i, and p are known. The corrected values of i and p 504 THEORETICAL ASTRONOMY. which are required only in the case of dz, may be easily estimated with sufficient accuracy, since we require only cos i, while Vp ap- pears as the divisor of a term whose numerical value is generally insignificant. To obtain the actual values of R, S, and Z, the cor- rections to be applied to N, N', and / must first be determined by means of the formulae (136). The values of 3i' and are found from the elements given in Art. 166. The results thus obtained are the following: Berlin Mean Time. log r t>o logr' w' p 1863 Dec. 12.0, 0.294084 354 26' 18".0 0.73425 14 IS' 54".6 l'38".l 1864 Jan. 21.0, 0.294837 10 2 45 .7 0.73368 17 21 44 .2 18 9 .1 March 1.0, 0.300674 25 24 59 .4 0.73305 20 25 5 .2 34 39 .9 April 10.0, 0.310864 40 13 31 .8 0.73237 23 28 59 .8 51 7 .6 May 20.0, 0.324298 54 14 41 .4 0.73164 26 33 32 .1 1 7 29 .7 June 29.0, 0.339745 67 21 23 .5 0.73086 29 38 44 .8 1 23 43 .5 Aug. 8.0, 0.356101 79 32 18 .1 0.73003 32 44 41 .2 1 39 46 .3 Sept. 17.0, 0.372469 90 49 57 .6 0.72915 35 51 24 .6 1 55 35 .2 Oct. 27.0, 0.388214 101 19 9 .8 0.72823 38 58 57 .5 2 11 7 .5 Dec. 6.0, 0.402894 111 5 42 .2 0.72726 42 7 23 .3 2 26 20 .3 1865 Jan. 15.0, 0.416240 120 15 32 .6 0.72625 45 16 43 .9 2 41 10 .6 The value of w for each date is now found from w = v + TT O = VQ + 197 38' 6".5, and the components of the disturbing force are determined by means of the formulae (132), p being found from (133) or (134), and h from (70). The adopted value' of the mass of Jupiter is 1047.879 and the results for the components E, S, and Z are expressed in units of the seventh decimal place. The factor co 2 is introduced for conve- nience in the integration, w being the interval in days between the successive dates for which the forces are to be determined. Thus we obtain the following results : NUMEKICAL EXAMPLE. 507 Date. tfR u 2 Sr tfZcosi ujSr dt 1863 Dec. 12.0, + 70.82 + 7.16 + 0.04 + 1.37 1864 Jan. 21.0, 68.95 - 32.76 0.49 - 11.45 March 1.0, 61.16 70.38 0.92 63.32 April 10.0, 48.57 102.91 1.32 150.48 May 20.0, 32.77 128.34 1.68 266.75 June 29.0, + 15.41 145.39 1.96 404.35 Aug. 8.0, - 2.19 153.44 2.17 554.54 Sept. 17.0, 19.12 152.41 2.29 708.21 Oct. 27.0, 34.81 142.50 2.25 856.39 Dec. 6.0, 48.95 124.04 2.09 990.36 1865 Jan. 15.0, 61.45 97.36 + 1.75 1101.73 The single integration to find colSr dt is effected by means of the formula (32). The equations for the determination of the required differential coefficients are ddM "-dT d 2 v w 2 jR 2rf 1 r e smv 2Cr w 2 & 2 w 2 -y.r - - H -- = --- -7= "> \ &rj& -- - -t 2 S -- r v, dt 2 T Q r Q * kVp. J Po T* , d*dz, "W , ~ = C S h ~ ~ r '' Substituting in these the' results already obtained, and also log ,J. Q = 2.967809, log^ = 0.371237, log e = 9.290776, we obtain first, by an indirect process, as illustrated in the case of the direct determination of the perturbations of the rectangular co- ordinates, the values of 2 -^~, and then, having found v, to j-- is given directly by the first of these equations. The integra- tion of the results thus derived, by the formulae for mechanical quad- rature, furnishes the required values of v, dM, and dz,. The calcula- tion of the indirect terms in the determination of v and dz,, there being but one such term in each case, is, on account of the smallness of the coefficient, effected with very great facility. The final results are the following : 508 THEORETICAL ASTRONOMY. Date. o-jj- ^ 2 ^r 2 -^f' < 1863 Dec. 12.0, 0"-.028 -f- 36.16 + 0.04 + 0".01 + 4. 41 + 0.02 1864 Jan. 21.0, .072 33.61 0.49 .01 4. 31 0.04 March 1.0, .499 22.55 0.89 .27 37. 11 0.54 April 10.0, 1 .213 + 5.58 1.21 1 .11 91. 96 1.93 May 20.0, 2 .070 - 13.52 1.45 2 .75 152. 22 4.52 June 29.0, 2 .902 31.59 1.53 5 .24 199. 05 8.54 Aug. 8.0, 3 .546 46.65 1.60 8 .49 214. 54 14.10 Sept. 17.0, 3 .858 57.88 1.52 12 .22 183. 69 21.24 Oct. 27.0, 3 .723 65.19 1.28 16 .05 + 95. 29 29.90 Dec. 6.0, 3 .056 68.83 0.92 19 .49 58. 00 39.82 1865 Jan. 15.0, 1 .800 69.19 + 0.40 21 .97 279.84 -j-50.64 Since, during the period included by these results, the perturbations of the second order are insensible, we have, for the perturbations of Eurynome arising from the action of Jupiter from 1864 Jan. 1.0 to 1865 Jan. 15.0, dM = 21".97, v = 0.00002798, dz, = + 0.00000506. It is to be observed that dz, is not the complete variation of the co- ordinate z, perpendicular to the ecliptic, but only that part of this variation which is due to the action of the component Z alone; and hence the results for dz, differ from the complete values obtained when we compute directly the variations of the rectangular co- ordinates. Let us now determine the heliocentric longitude and latitude for 1865 Jan. 15.0, Berlin mean time, including the perturbations thus derived. From the equations M, = M Q + p. Q (t-Q + SM, E, e sin E, = M n r = al ecosE , sin in I 0, E,) = sin J

and -J- by means of the formulae for integration by mechanical quadrature, as already illustrated, or we may find P by a direct integration, and the values of p r and q' by means of the equations (164), ~- being found from -j^- 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 formula? for this purpose. 191. The single integration of the values of u> 2 -Tp and w 2 -^ will ,1 dv _ ddz. _ dv , ddz, give the values of co -^ and a) -~, and hence those of j and -jg-'j which, in connection with -^-, are required in the determination of the new system of osculating elements. Since r 2 -~ represents double 0JV the areal velocity in the disturbed orbit, we have CHANGE OF THE OSCULATING ELEMENTS. 511 dv, Wp (1 + m) dt ~ r 2 The equation (109) gives dv f _ klT^m / 1 ddM\ dt ~ r? V-- 1 "*' dt I Hence, since r = r,(l-\- i>), we obtain ^A < 176) by means of which we may derive p. This formula will furnish at once the value of p, which appears in the complete equation for d 2 dz TJ> and also in the equations (164); and the value of cosi may be dt determined by means of (165). In the disturbed orbit we have dr ~7- = - 7== e sm v, dt and the equations (108) and (111) give dr Jcl/T+^i Therefore we obtain which, by means of (176), becomes r,Vp 3 The relation between r and r, gives , 1 -f e cos v 1 -f e cos v i and, substituting in this the value of p already found, we get ecosv = (l + e cosi;,) 1+-.^ (i + y )s i. (178) 512 THEORETICAL ASTRONOMY. Let us now put r,Vp dv f (179) ~ ' ' a and /9 being small quantities of the order of the disturbing force, and the equations (177) and (178) become e sin v = e Q sin v, -}- a ^ sin ?;, -f & e cos v = e cos v f -j- ae o cos v / 4~ " These equations give, observing that r, (cos v, + c ) > cos E n e sin (i>, v) = a sin v, /5 cos v,, e cos (v f v) = e + cos j&, + T f from which e, v, v, and v may be found; and thus, since e cos (v f v) = e + cos j&, + /5 sin v,, T f (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 f may now be rigorously effected, and the corresponding value of cosi being found from (165), -jrr and Hte- 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 Y' a sin v t ft cos v n * = 2icoe, + /JBin / , (182) ^V and neglect terms of the third order, the equations (180) give in which s = 206264".8. These equations are convenient for the CHANGE OF THE OSCULATING ELEMENTS. 513 determination of e and v, v, and hence X by means of (181), when the neglected terms are insensible. The values of p, e, and v having been found, we have + m tan i E = tan (45 ^ ?) tan v, M=E e sin j, from which to find the elements- ^>, a, //, and Jf. The mean anomaly thus found belongs to the date t y and it may be reduced to any other epoch denoted by t by adding to it the quantity fj. (t Q t). When we neglect the terms of the third order, we have sin?- sin ? cos ? ^ (f Q d r sin

-ji and JT-' 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 dM, 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 again commence with zero, and will be continued until, on account of the terms of the second order, another change of the elements is required. The frequency of this transformation will necessarily de- 516 THEORETICAL ASTRONOMY. pend on the magnitude of the disturbing force; and if the disturbed body is so near the disturbing body that a very frequent change of the elements becomes necessary, it may be more convenient either to include the terms of the second order directly in the computation of the values of 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 = a (1 e 2 ) gives . dp _ p da ~ de ~dt~^~a"~dt ae ~dt' Equating these values of -~> and introducing the value of -^r already found, we get (202) 520 THEORETICAL ASTRONOMY. and since A* _ = 1 -J- e cos v, = 1 e cos .J, ^ being the eccentric anomaly in the instantaneous orbit, this becomes = _ x * (p sin vR+p (cos v + cos E) S), (203) rft which will give the variation of e. If we introduce the angle of eccentricity tp 9 we shall have _^_ _ cog &V _ a COS 2 dt dt and hence r7, 1 = (a cos ^> sin v It -f~ a cos ^ (cos v -|- cos JE7) /S). (204) 195. When we consider only the components R and 8 of the dis- turbing force, the longitude in the orbit will be We have, therefore, = + e cos A, the differentiation of which, regarding the elements as variable, gives dp p[ dr^\ de on and, since pcosEr (cos v + e), we have p (1 cos v cos E) = r sin 2 so that the equation becomes VARIATION OF CONSTANTS. 521 ) > (205) dy from which the value of ~- may be derived. ut If we introduce the element a), or the angular distance of the peri- helion from the ascending node, it will be necessary to consider also the component Z\ and, since co = % (1-|- (j. JN J5t /'9^^ \ O/ 7 . ^^v/O^ The equation (205) gives 522 THEOKETICAL ASTEONOMY. _______ (j9 _|_ r ) C ot

--> at by means of which (208) reduces to "%^--g^. (209) dt dt 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 M= M, + dM + (t- t ) fa + ty), in which p Q denotes the mean daily motion at the instant / . There- fore we shall have M= M + dt + /, (t - Q + {*- . dt, the integrals being taken between the limits t and t. The quantity expresses the mean anomaly at the time i 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 dM __dM dfi . ran at --dr + -^ VARIATION OF CONSTANTS. 523 Substituting in this the value of ~~TT from (209), the result is dM dy 2r cos y D which does not involve the factor t t Q 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*_dy dM . w-~-~dT'-~dt-~dt~ i ~^dr^ (l and therefore - 2 sin' Jf * + 2 sin' *i** -- ***-. R + f * dt, (211) dt dt dt kVp(\ + m) J dt To find the variations of & and i since u denoting the argument of the latitude in the disturbed orbit, we have, according to the equations (169) and (170), dQ, _ 1 r siuu dt ~ kVp (1 + m) * sin * di 1 (212) -jr = r r cos uZ. dt JcVp (1 + m) The inclination i 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 of*, and 2& n in place of the given value of JT, before applying the formulae 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 p = q(l+e) gives 524 THEORETICAL ASTRONOMY. dq _ 1 dp q de_ ~dt ~~ 1 + e ' ~dt 1 + e ' ~dt' and substituting in this the value of -^r already found, and neglect- ing the mass of the comet, which is always inconsiderable, we get dq _ 2qr q de_ ~dt~~ kl/p MM dt 9 by means of which the variation of q may be found. In the case of elliptic motion the value of -rr 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 formulae 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 de = 1_ sm ^_i 2 cosMfS) dt kl/p S and for the value of -~ we have It remains now to find the formula for the variation of the time of perihelion passage. The relation between T and Jf is expressed by 360-ir =:KT-g, the differentiation of which gives d and, substituting for - the value given by equation (209), we get Substituting further the values of -^ and ^~ given by the equations (205) and (199), the result is VARIATION OF CONSTANTS. 525 dJT _ aE __ p^ _ 3fc (t T) . dt e V P (216) + r . Zlc(tT} OJ.J.J. t/ , * Vp 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 --r- and -, from the equations (24) 2 and (23) 2 , we easily obtain 2p dr p . - -j- a (2r cos v -- ^-7-* - e sin v) -\ -- -rf-. - cos v, 1 -f e de e V> e U ~h e) 2 dv /^ + r . 3k(t T) p\ p 2 I. r\ . -j- = a 7 --- sin v --- v .- - - -- -. - 1 + - sm v. de \ e y rj e(l + e) 2 \ { pl 1-t-e By means of these results the equation (216) is transformed into dT qR . n dr q which may be used for the determination of -rr> the values of - T - J dt de dv and -T- being found by means of the various formula? developed in 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) (218) In the case of parabolic motion we have e = l, and p = 2q; and if we substitute in (217) for -j- and -?- the values given by the equa- tions (33) 2 and (30) 2 , the result is dT ~dt + -J[- (4 tan -J v f tan 5 A v) ) (219) 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 fJL Q be the mean daily motion at the first epoch, and we shall have in which x denotes the semi-circumference of a circle whose radius is unity. Hence we obtain dM C J ^_ dt dt (220) by means of which to determine p Q . Then, to find the mean daily motion // at the instant of the second return to the perihelion, we have 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 p during this interval. The semi-transverse axis will now be derived by means of the formula ' VARIATION OF CONSTANTS. 527 from the values of // for the two epochs. Let r f denote the interval which must elapse before the next succeeding perihelion passage of the comet, and we have dt and consequently (222) the integral being taken between the limits t = 0, corresponding to the beginning of the interval, and t = r'. 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 which 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 order 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 THEOKETICAL 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 proper 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 2R, 28, and IZ in the formulae in place of R, 8, and Z y 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 be 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 CD, 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 s = 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 A 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 R, 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 Q : Berlin Mean Time. 40JS 40S 40Z EQ 1863 Dec. 12.0, -fcO ".0365 + 0' '.0019 - f 0' '.00002 355 26' 8".2 1864 Jan. 21.0, .0356 -0 .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 35 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, .0317 .0193 - f o .00090 110 54 .3 We compute the values of the required differential coefficients by means of the equations -^rr = ;= r cos u Z. dt sin i dt "1 / " kVp'\ sin

we shall have dt which form is equally convenient in the numerical calculation. Thus, for 1865 Jan. 15.0, we find JJf = + 234".74, and from the several values of 1600^ we obtain, for the same date, by means of the formula for double integration, = 4- 56".59. dt 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, 3M = + 165".29, 532 THEOEETICAL 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, and hence 3M = 3M + (t t ) 8fi= + 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 formulae as derived from the equation (1) 2 , whenever the terms of the second order may be neglected. It should be observed, however, that if we substitute the value of dM directly in the equations for the variations of the geocentric co-ordinates, the coefficient of Sp must be that which depends solely on the variation of the semi-transverse axis. But when the coefficient of dp has been computed so as to involve the effect of this quantity during the in- terval t 1 09 the value of dM Q must be found from o M and substi- tuted in the equations. 200. It will be observed that, on account of the divisor e in the expressions for -^-, -^-, and ~> 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 VAEIATION OF CONSTANTS. 533 account of the divisor sin i in the expression for = the variation of SI 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 , /3" sin i cos ft ; (224) then we shall have da! f .di , dft = sm ft cos i-j- -f- sm i cos ft dt dt dt dfi" .di dft cos ft cos i-=- sin ^ sin ft = at at at Introducing the values of -jr an( l " r given by the equations (212), ctv ctt> and introducing further the auxiliary constants a, 6, A, and com- puted by means of the formulae (94) t with respect to the fundamental plane to which ft and i are referred, we obtain i = r^sin a cos (J. -f- it), *v>(l+n5 (225) d/3" _ 1 ^ &l/p (1 -{- 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 dfl" will be obtained, so that we shall have sin i sin ft = sin i sin ft -f + cos u '} = V cos U', the equations (100) : and (49), in connection with (232) and (233), give dx= -*'- (234) PERTURBATIONS OF COMETS. 539 m' , . oy = , r sin 6 sin (H + 1*) m ' 7 r' sine' sin (C" + w')i I 1-f m 1 ^ m ' "' ' ' os (4' + 17), (234) m If we add the values of dx, %, #z, ^-77' ^~TT/ ' anc ^ ^~^~ * tne cor ~ dt ut dt responding co-ordinates and velocities of the comet in reference to the centre of gravity of the sun, the results will give the co-ordinates and velocities of the comet in reference to the common centre of gravity of the sun and disturbing planet, and from these the new elements of the orbit may be determined as explained in Art. 168. The time at which the elements of the orbit of the comet may be referred to the common centre of gravity of the sun and planet, can be readily estimated in the actual application of the formulae, by means of the magnitude of the disturbing force. In the case of Mer- cury as the disturbing planet, this transformation may generally be effected when the radius-vector of the comet has attained the value 1.5, and in the case of Venus when it has the value 2.5. It should be remarked, however, that the distance here assigned may be in- creased or diminished by the relative position of the bodies in their orbits. The motion relative to the common centre of gravity of the sun and planet disregarding the perturbations produced by the other planets, which should be considered separately may then be re- garded as undisturbed until the comet has again arrived at the point at which the motion must be referred to the centre of the sun, and at which the perturbations of this motion by the planet under consider- ation must be determined. The reduction to the centre of the sun will be effected by means of the values obtained from (234), when the second member of each of these equations is taken with a contrary sign. 204. In the cases in which the motion of the comet will be referred to the common centre of gravity of the sun and disturbing planet, the resulting variations of the co-ordinates and velocities will be so 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 (100)j give - sin a sin ( A -f u) ~ + r sin a cos (A + u) -3--, at at at ^- = sinb sin (B + w) -^- -f r sin 6 cos (B + w) ~L (235) at at at dz x V . , ~ , cfo -y- = sm c sin ( -f- w) -vr + r sm e cos ( C + *) -57* dt at at If we multiply the first of these equations by dx, the second by dy, and the third by dz; then multiply the first by d-> the second by , , at d-j-i and the third by -TT J and put dt J dt P sin a sin (A -j- u) dx -j- sin 6 sin (jB -|- u) dy -j- sin c sin ( C -f- w) &z, Q =; sin a cos (A -j- w) ^ -j- sin 6 cos (jB -f- w) 5i/ -j- sin c cos ( C -f- u) dz ; (236) / A \ * . .,./, N sin a sm ( J. + u) 8 -j- sm 6 sm fJ? + it) r > etc -j- sin c sin ( O -f- w ) ^ r > ^ = sin a cos (A + w) 3-r- -f sin b cos (J5 + u) d-j- at at -j- sin c cos ( (7 + u) 8-^- t at we shall have, observing that -jj- = e sin v and that -^- = -~/-> ^ /} j_ ^ ^ 3 ^ ^P r (237) da; ^ dx dy dy dz dz _ I From the equations PERTURBATIONS OF COMETS. 541 we get ./ rdr\ dx , dy . , dz dx , . dy , ^ dz S\ ;- } = -^r 8x + -r 3y + -j- 8z 4- xd -4- yd-- -f 2<5-r-, \ dt I dt ~ dt y dt dt ' y dt dt dx dx , dy ^dy , dz c?z which by means of (237) become Jc = = e sin 1/p From the equation we get 2^^ + If dp = Substituting the values given by (238), observing also that P = 8r, this becomes dk dp __ FV p re 2 sin 2 v p _ e sin v ~ r ~ ~~ and, since F 2 = - (1 + 2e cos v + e 2 ), we obtain + -V--' (239, by means of which the variation of i/p may be found. The equation v_ = _v* a ~ ~ r gives from which we derive (240) 542 THEORETICAL ASTRONOMY. from which the new value of the semi-transverse axis a may be found. To find 8p we have Next, to find 8e, we have, from p = a (I e 2 ), or ~ (242) (243) e a ae or 2p cos r The equation (12) 2 gives dM= - ^ - ( 2 + e cos v a 2 cos ?> a 2 cos 3 p v and from = 1 + e cos v we get cos v P : ^e -f - ^-. 5r --- r e sin v r*e sin v re sin v Substituting this value of dv in (245), and reducing, we find 6M= lcot 1 + tanl\ . mvp a / ^ r from which to derive the variation of the mean anomaly. 205. Let us now denote by x ff , 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 f , z f the co-ordinates referred to a system in which the plane of the ecliptic is the plane of xy, and in which the positive axis of x is directed to the vernal equinox. Then we shall have PERTURBATIONS OF COMETS. 543 x" = x' cos Q + y' sin & , y" = x' sin & cos i -\- y' cos & cos i -f- 2' sin i, z" = x' sin & sin i y' cos & sin i -|- z' cos i. If we transform the co-ordinates still further, and denote by x, y, z the co-ordinates referred to the equator or to any other plane making the angle with the ecliptic, the positive axis of x being directed to the point from which longitudes are measured in this plane; and if we introduce also the auxiliary constants a, A, 6, J5, &c., we shall have dx" = sin a sin A fix -j- sin b sin B dy -[- sin c sin C $z, dy" = sin a cos A dx -f- sin b cos B dy -f- sin c cos C Sz, (248) #z" = cos a dx -f- cos b 8y -\- cos c dz. 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 dy" sin u dx" = Q. Substituting for dx" and dy" the values given by the equations (73) 2 , the result is r (3v + %) - ft and, introducing the value of dv given by (246), we obtain s Q_cos jLse _^__ s ji4L, ( - r esmv reainv resmv v P J Substituting further for de, 8r, and d((/p) the values already ob- tained, and reducing, we find sin v cosff cos vVp , (p -f r) sin v , '"- --Q- -~~ by means of which <5^ may be found. If we put cos a dx -j- cos 5 dy -f cos cdz = B, ( 25 ) cos a oj- dt the last of the equations (248) gives 544 THEORETICAL ASTRONOMY. dz" = R; (251) and if we differentiate the equation dx . ,dy. dz _ cos a-,- -f cos b -~ -f- cos c-^- = 0, which exists in the case of the unchanged elements, we shall have dx . , dy . , , dz . -- j- sm a da -- -~- sin b db -- -=- sm c dc. at at at Substituting for da, db, and dc the values given in Art. 60, observing that de = 0, we have =E' -f ( -IT- sin a sin J. + -j- sin 6 sin ^ + ~ji s ^ n c sin (7 1 sin i I -r- sin a cos A -\ - sin b cos 5 -j jr sin c cos (7 1 di. \ at at at / (252) From the equations (100) D 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 2 a sin 2 A + sin 2 b sin 2 B -f- sin 2 c sin 2 C= 1, (253") sin 2 a cos 2 J. + sin 2 b cos 2 -B -f sin 2 c cos 2 (7=1, and from (235) we find sin 2 a sin A cos J. -f sin 2 b sin 5 cos B -f sin 2 c sin Ccos 0. (254) Substituting in (252) for -.*, -^, and * the values given by the at dt at equations (49), and reducing by means of (253) and (254), we get = R FsmJJsin i d& VcosUdi. (255) Substituting further for dz" in (251) the value given by the last of the equations (73) 2 , there results jR-frcosiismi and &' may be found. To find dco and 8n we have da> = 8 x cos i<5& , ^TT = 3% -f 2 sin 2 i<5 , (258) d% being found from equation (249). Neglecting the mass of the comet as inappreciable in comparison with that of the sun, the attractive force which acts upon the comet in the case of the undisturbed motion relative to the sun is k 2 , but in the case of the motion relative to the common centre of gravity of the sun and planet this force is k 2 (1 -f- m r ). Hence it follows that the increment of this force will be m'P, and we shall have ai> -f = W, (269) by means of which the value of this factor, which is required in the formulae for d(]/p) } d> &c., may be found. 206. The formulae thus derived enable us to effect the required transformation of the elements. In the first place, we compute the values of dx, %, dz, -?- ^~5* ' an( ^ ^~j7 by means of the formulae Ctu Ctt> CtL (234) ; then, by means of (236) and (250), we compute P, Q, R, P', Q r , and P', 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 Q> and i are referred, we have sin G = sin i t C==Q. The algebraic signs of cos a, cos 6, and cos c, as indicated by the equa- tions (101) D must be carefully attended to. The formulae for the variations of the elements will then give the corrections to be applied to the elements of the orbit relative to the sun in order to obtain those of the orbit relative to the common centre of gravity of the sun and planet. Whenever the elements of the orbit about the sun are again required, the corrections will be determined in the same manner, but will be applied each with a contrary sign. 35 546 THEOEETICAL ASTEONOMY. 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 1 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 r t. It should be observed, however, that the value of 3M 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 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 1 ', y f y z f , r f , 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 dV , I x x' x \ = mk 2 ; r , \ / r 3 / Let us now denote by , -^, the co-ordinates of the comet referred to the centre of gravity of the planet; then will Substituting the resulting values of a/, y f , z f in the preceding equa- tions, and subtracting these from the corresponding equations (1) for the disturbed motion of the comet, we derive (m + m') _ / x f x' -f "~ fc ^ 2 (in (261) 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, d 2 7? #* (m -f m'} f] _ ft (262) W^ ~7~ d 2 : W (m + mQ C _ A and, since m j- is the sum of the attractive force of the planet P on the comet and of the reciprocal action of the comet on the planet, 548 THEORETICAL ASTKONOMY. 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 1 ', 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 and when the comet approaches very near the planet this force will be extremely small. It is evident, further, that the action of the sun regarded as the disturbing body will be very small even when its direct action upon the comet considerably exceeds that of the planet, and, therefore, that we may consider the orbit of the comet to be a conic section about the planet and disturbed by the sun, when it is actually attracted more by the sun than by the planet. 208. In order to show more clearly that the disturbing force of the sun is very small even when its direct action on the comet exceeds that of the planet, let us suppose the sun, planet, and comet to be situated on the same straight line, in which case the disturbing force of the sun will be a maximum for a given distance of the comet from the planet. Then will the direct action of the sun be , and that '72 7* 2 of the planet - The disturbing action of the sun will be PERTURBATIONS OF COMETS. 549 which, since p is supposed to be small in comparison with r, may 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 R = r-? 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 ZVm'. 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 mV 2 At the point for which the value of p corresponds to R R f , 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 f gives and therefore P = rVTpiF. (263 ) Hence we may compute the perturbations of the comet, regarding the planet as the disturbing body, until it approaches so near the 550 THEOEETICAL ASTEONOMY. planet that p has the value given by this equation, after which, so long as p does not exceed the value here assigned, the sun must be regarded as the disturbing body, If $ represents the angle at the planet between the sun and comet, the disturbing force of the sun, for any position of the comet near the planet, will be very nearly cos and when this angle is considerable, the disturbing action of the sun will be small even when p is greater than rV\m n . Hence we may commence to consider the sun as the disturbing body even before the comet reaches the point for which and, since the ratio of the disturbing action of the planet to the direct action of the sun remains nearly the same for all values of $, when p is within the limits here assigned the sun must in all cases be so considered. Corresponding to the value of p given by equation (263), we have K = t/4m f , and in the case of a near approach to Jupiter the results are /> = 0.054 r, jR'^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 instant at which the sun is regarded as ceasing to be the controlling body, we shall have PERTURBATIONS OF COMETS. 551 = x x', f j = y y' J ^ ==z z ' ) dz _ dx dx r df) _ dy dy' d dz dz' ~dt == ~dt~~~~dt' ~dt == ~dt dt' ~dt == ~dt~~~d' and from , 27, f , -^-, -57-, and -TT> 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 a?, y, z, jr, -, and -g-i and as explained in Art. 168. Having computed the perturbations of the motion relative to the planet to the point at which the planet is again considered as the disturbing body, it only remains to find, for the corresponding time, the co-ordinates and velocities of the comet in reference to the centre of gravity of the planet, and from these the co-ordinates and velocities relative to the centre of the sun, and the elements of the orbit about the sun may be determined. As the in- terval of time during which the sun will be regarded as the disturb- ing body will always be small, it will be most convenient to compute the perturbations of the rectangular co-ordinates, in which case the values of , /;, , -^-> -rr> and jjr will be obtained directly, and then, having found the corresponding co-ordinates a/, y f , z' and velocities ~dt' -rr> 77- of the planet in reference to the sun, we have dx _ dx' d_ dy _ dy' d^ dz __ dz' dZ ~dt ~~ ^t " W ~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 be T =- K 4\Yw (264) K being a constant quantity depending on the nature of the body, and

/S= -fi^l 1 sr- (265) r ' 1/p W \ r / r eft RESISTING MEDIUM IN SPACE. 553 Substituting these values of R and S in the equation (205), it reduces to e d% = 2K I - J sin v ds. Now, since F= (l + 2ecos<; + e 2 )S VP we have r 2 ds = Vdt = (1 -4- 2e cos v 4- tftrdv. P and hence If we suppose the function + 2e cos v - 1 - *^ j the value of which is always positive, to be developed in a series arranged in reference to the cosines of v and of its multiples, so that we have K

. Hence, by integrating, we derive e d x = - (4 cos v + i ^ cos 2v + ), (268) from which it appears that j 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 o Jf, 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

the expression for the action of the ethereal fluid be- comes 556 THEORETICAL ASTEONOMY. Since the constant ([/"depends on the nature of the comet, the value obtained in the case of Encke's comet may be very different from that in the case of another comet. Thus, in the case of Faye's comet the value has been found to be n 1 - ~ 10.232' and in the application of the formula to the motion of any particular body it will be necessary to make an independent determination of this constant. 212. The assumption that the density of the ethereal fluid varies inversely as the square of the distance from the sun, is that which appears to be the most probable, and the results obtained in accord- ance therewith seem to satisfy the data furnished by observation. It is true, however, that the whole subject is involved in great uncer- tainty as regards the nature of the resisting medium, so that the results obtained by means of any assumed law of density are not to be regarded as absolutely correct. From the formulae which have been given, it appears that, whatever may be the law of the density of the resisting fluid, the mean motion is constantly accelerated and the eccentricity diminished, and we may determine, by means of observations at the successive appearances of the comet, the amount of these secular changes independently of any assumption in regard to the density of the ether. Let x denote the variation of ft during the interval r, which may be approximately the time of one revolution of the comet, and let y denote the correspond- ing variation of

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 559 TABLE I, Angle of the Vertical and Logarithm of the Earth's Radius, i Argument Geographical Latitude. Compression 299.15 ,-*- Diff. logp Diff. 0-0' Diff. logp Diff. o / 1 2 o o.oo o 24.02 o 48.02 n 24.02 24.00 o.ooo oooo 9-999 9996 9982 4 O / 35 10 20 10 48.25 49- 6 3 50.98 n 1.38 9-999 5248 5208 5 l6 9 40 39 3 I 11.95 23-93 9961 2 I 30 52.31 I- 33 4 4 1 35-80 23.85 993 3 1 40 53.62 I -3 I 5089 4 5 1 59-54 23-74 23.58 9891 39 48 50 54-9 1^26 549 40 40 6 7 8 9 10 11 2 23.12 2 46.54 3 9.76 3 32.74 3 55-47 4 17-92 23.42 23.22 22.98 22.73 22.45 22.14 9-999 9843 9786 972i 9648 9566 9476 ii 9 o 99 36 10 20 30 40 50 10 56.16 57-4 1 58.63 10 59.82 II I.OO 2.15 1.25 1.22 I.I9 1.18 1.15 1.13 9-999 59 4969 4929 4888 4848 4807 40 40 4 1 40 40 12 13 14 15 16 17 4 40.06 5 1.85 5 23.28 5 44-33 6 4.95 6 25.14 21.79 21.43 21.05 20.62 20.19 19.72 9-999 9377 9271 9'57 935 8905 8768 106 114 122 I 3 137 144 37 10 20 30 40 50 II 3.28 4-39 5-47 6-54 7.58 8-59 i. ii 1.08 1.07 1.04 I.OI I.OO 9-999 4767 4726 4686 4645 4604 45 6 3 4 1 40 4 1 4i 18 19 20 21 22 23 6 44.86 7 4-9 7 22.80 7 40.99 7 58.61 8 15.66 19.23 l8. 7 I 18.19 17.62 16.44 9.999 8624 8472 8149 7977 7799 152 165 172 I 7 8 185 38 10 20 30 40 50 ii 9.59 10.56 11.51 12.44 13-34 14.22 0.97 0.95 o-93 0.90 0.88 0.86 9-999 4522 4481 4440 4399 4358 4 1 4 1 4 1 4 1 24 25 26 27 28 29 8 32.10 8 47-93 9 3- 12 9 17-65 9 31-5 9 44.66 I5-83 15.19 14-53 I3-85 I 3 .l6 12.46 9-999 7614 7424 7228 7027 6820 6608 190 196 201 20 7 212 216 39 10 20 30 40 50 ii 15.08 15.92 16.73 17.52 18.29 19.04 0.84 0.8 1 0.79 0.77 o-75 0.72 9.999 4276 4234 4'93 4152 4110 4069 42 42 42 30 10 20 30 40 50 9 57-12 9 59- 12 10 I. II 3-7 5.02 6.94 2.00 .99 . 9 6 95 .92 .91 9-999 6392 6355 6319 6282 6245 6208 H 37 37 37 37 40 10 20 30 40 50 ii 19.76 20.46 21.13 21.79 22.42 23.02 0.70 0.67 0.66 0.63 0.60 0.59 9-999 4027 3985 3944 3902 3860 3819 42 42 42 42 31 10 20 30 40 50 10 8.85 10.73 12.59 14.44 16.26 18.06 .88 .86 85 .82 .80 .78 9.999 6171 6096 6059 6021 5984 P 41 10 20 30 40 50 ii 23.61 24.17 24.70 25.22 25-71 26.18 0.56 o-53 0.52 0.49 0.47 0.44 9-999 3777 3735 3693 3609 35 6 7 42 42 42 42 42 42 32 10 20 30 40 50 10 19.84 21.60 23-34 25.05 26.75 28.43 .76 74 .70 .68 .65 9-999 5946 598 5870 5832 5794 5755 38 38 38 38 39 38 42 10 20 30 40 50 ii 26.62 27.04 27.44 27.82 28.17 28.50 0.42 0.40 0.38 o-35 -33 0.30 9-999 3525 3483 3399 3357 42 42 42 42 42 42 33 10 20 30 40 50 10 30.08 33-32 34-9J 36.48 38.03 63 .61 59 57 55 .52 9-999 57i7 5678 5640 5601 5562 5523 39 38 39 39 39 39 43 10 20 30 40 50 ii 28.80 29.08 29-34 29.58 29.79 29.98 0.28 0.26 0.24 0.21 0.19 0.16 9-999' 3273 3230 3188 3146 3104 3062 43 42 42 42 42 43 34 10 10 39-55 41.06 s 9-999 5484 5445 39 44 10 ii 30.14 30.29 0.15 O. I 2. 9-999 3'9 2977 42 42 20 30 40 50 42-54 44.00 45-44 46.86 .48 .46 44 .42 39 5406 53 6 7 5327 5288 39 39 40 39 4 20 30 40 50 30.41 30.50 3-57 30.62 0.09 0.07 0.05 0.03 2935 2892 2850 2808 43 42 42 42 35 10 48.25 9-999 5248 45 ii 30.65 9.999 2766 561 TABLE I, Angle of the Vertical and Logarithm of the Earth's Eadius, 0' = Geocentric Latitude. P = Earth's Kadius. 0-0' Diff. logp Diff. 0-0' Diff. logp Diff. r 45 10 20 30 40 50 ii 30.65 30.65 30-63 30.58 30.51 30.42 O.OO 0.02 0.05 0.07 0.09 O.I I 9.999 2766 2723 2681 2639 2596 2554 43 42 42 43 42 42 55 10 20 30 40 50 10 49.74 48.36 46.97 45-55 44.11 42.65 -38 39 .42 .44 .46 49 9-999 0275 0235 0195 oi55 0116 0076 40 40 40 39 40 39 46 10 20 30 40 50 ii 30.31 30.17 30.01 29.82 29.61 29.38 0.14 o.i 6 0.19 0.21 0.23 O.2D 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 01 33-4 1 5 1 -52 -55 !6i 9-999 37 9.998 9998 9958 9919 9880 9841 39 40 39 39 39 39 47 10 20 30 ii 29.12 28.85 28.54 28.22 0.27 0.31 0.32 9-999 2258 2216 2174 2132 42 42 42 57 10 20 30 10 31.80 30.16 28.50 26.83 .64 .66 .67 9.998 9802 9764 9725 9686 38 39 39 40 50 27.87 27.50 o-35 o-37 0.40 2089 2047 43 42 42 40 50 23.40 7 73 74 9648 9610 39 48 10 20 30 40 50 ii 27.10 26.69 26.24 25.78 25.29 24.78 0.41 0.45 0.46 0.49 0.51 o-54 9-999 2005 1963 1921 1879 1837 1795 42 42 42 42 42 42 58 10 20 30 40 50 10 21.66 19.90 18.11 16.31 14.48 12.63 .76 79 .80 83 85 .86 9-998 9571 9533 9495 9457 9419 9382 38 38 38 38 38 49 10 20 30 40 50 ii 24.24 23.69 23.11 22.50 21.87 21.22 0.55 0.58 0.6 1 0.63 0.65 0.67 9-999 J 753 1711 1669 1627 1586, 1544 42 42 42 42 42 59 10 20 30 40 50 10 10.77 6.97 5-04 3.08 10 I. II 89 .91 .96 97 -99 9-998 9344 937 9269 9232 9'95 9158 37 37 37 37 50 10 20 30 40 50 II 20-55 19.85 !! r .|9 17.63 16.84 0.70 0.72 0.74 0.76 0.79 0.82 9.999 1502 1460 1419 1377 '335 1294 42 42 42 4 1 42 60 61 62 63 64 65 9 59-12 9 46.74 9 33-65 9 19-85 9 5-36 8 50.21 12.38 13.09 13.80 14.49 15.15 15.81 9.998 9121 8902 8688 8479 8275 8077 219 214 209 204 198 '93 51 10 20 30 40 50 ii 16.02 15-19 14-33 13-45 12.55 11.62 0.83 0.86 0.88 0.90 o-93 0.95 9.999 1252 I2II II 7 1128 1087 1046 4 1 42 66 67 68 69 70 71 8 34.40 8 17.97 8 0.92 7 43.29 7 25.08 7 6.33 16.43 17.05 17.63 18.21 18.75 19.27 9.998 7884 7697 7517 7342 7174 7013 187 180 168 161 J 54 52 10 20 30 ii 10.67 9.70 8. 7 i 7.69 0.97 0.99 1.02 9-999 1005 0963 0922 O88l 42 72 73 74 75 6 47.06 6 27.28 6 7.03 19.78 20.25 20.70 9.998 6859 6713 6 573 6441 146 140 132 40 50 6.66 5.60 I.Og 0840 0800 4i 40 4 1 76 77 5 25.20 5 3-67 21.13 21-53 21.90 6317 6201 124 116 1 08 53 10 ii 4.51 3-4 I. II 9-999 759 7 l8 78 79 4 41.77 22.24 9.998 6093 5993 100 20 30 40 50 2.27 II 1. 12 10 59.94 58.74 I.I5 1.18 1.20 1.22 0677 0637 59 6 0556 40 4 1 40 80 81 82 83 3 56-96 3 34-10 3 10.98 2 47.63 22.57 22.86 23.12 23.35 23.56 59 01 5818 5676 92 83 57 54 10 20 30 40 50 10 57.52 56.28 55.02 53-73 52.42 51.09 1.24 1.26 1.29 '33 9-999 0515 475 435 395 355 0315 40 40 40 40 40 4 84 85 86 87 88 89 2 24.07 2 0.33 I 36.44 I 12.43 o 48.34 o 24.18 23.74 23.89 24.01 24.09 24.16 24.18 9.998 5619 5570 553 5498 5476 5463 49 40 32 22 13 5 55 10 49.74 9-999 0275 90 o o.oo 9.998 5458 562 TABLE H. For converting intervals of Mean Solar Time into equivalent intervals of Sidereal Time. Hours. Minutes. Seconds. Decimals. Mean T. Sidereal Time. Mean T. Sidereal Time. Mean T. Sidereal Time. Mean T. Sidereal Time. h h m s m m s s s s I I o 9.856 I I 0.164 I 1.003 0.02 0.020 2 2 19.713 2 2 0.329 2 2.005 0.04 0.040 3 3 o 29.569 3 3 o-493 3 3.008 0.06 0.060 4 4 o 39.426 4 4 0.657 4 4.011 0.08 0.080 5 o 49.282 6 o 59.139 5 0.821 6 0.986 I 5- OI 4 6.016 0.10 0.12 O.I 00 0.120 7 7 8.995 7 7 -15 7 7.019 0.14 0.140 8 8 18.852 8 8 -314 8 8.022 0.16 O.I 60 9 9 28.708 9 9 -478 9 9.025 0.18 0.180 10 10 38.565 10 10 .643 10 10.027 0.20 0.201 ii ii 48.421 ii ii .807 ii 11.030 0.22 0.221 12 12 58.278 12 12 .971 12 12.033 0.24 0.241 J 3 13 2 8.134 13 I 3 2.136 I 3 13.036 0.26 0.26l 14 14 2 17.991 14 14 2.300 14 14.038 0.28 0.281 15 15 2 27.847 15 ic 2.464 15 15.041 0.30 0.301 16 16 2 37.704 16 16 2.628 16 16.044 0. 3 2 0.321 17 17 2 47.560 17 17 2.793 17 17.047 0-34 0.341 18 18 2 57.416 18 18 2.957 18 18.049 0.36 0.361 '9 19 3 7.273 19 19 3.121 19 19.052 0.38 0.38l 20 20 3 17.129 20 20 3.285 20 20.055 0.40 0.401 21 21 3 26.986 21 21 3.450 21 21.057 0.42 0.421 22 22 3 36.842 22 22 3.614 22 22.o6o 0.44 0.441 23 23 3 46-699 23 23 3.778 23 23.063 0.46 0.461 24 24 3 56.555 24 24 3-943 2 4 24.066 0.48 0.481 25 26 25 4.107 26 4.271 11 2C.068 20.071 0.50 0.52 0.501 0.521 o5 27 27 4-435 27 27.074 o-54 0.541 a a 28 28 4.600 28 28.077 0.56 0.562 H 8 2 9 29 4-764 29 29.079 0.58 0.582 'cS C 3 30 4.928 3 30.082 0.60 0.602 1 | 3 1 31 5.092 3 1 31.085 0.62 0.622 T3 S 32 32 5-257 32 32.088 0.64 0.642 '^ bC 33 33 5-421 33 33.090 0.66 O.662 3 .S o3 34 34 5-585 34 34.093 0.68 0.682 .s 'g .a 35 35 5-750 35 35.096 0.70 0.702 ^ ^ 'a 36 3 6 5-9 J 4 36 36.099 0.72 0.722 oB ^ 37 37 6.078 37 37.101 0.74 0.742 29 38 38 6.242 38 3 8.104 0.76 0.762 5 '-^ g ^ g 39 39 6.407 39 39.107 0.78 0.782 ^ > 40 40 6.571 40 40.110 0.80 0.802 u- a "fib 4 1 , 4i 6.735 4 1 41.112 0.82 0.822 . 0> 42 42 6.899 42 42.115 0.84 0.842 1 1 5 43 43 7-064 43 43.118 0.86 0.862 44 44 7.228 44 44.120 0.88 0.882 & T3 a Jl 45 7-392 46 7-557 45 4 6 45.123 46.126 0.90 0.92 0.902 II . 47 47 7-721 47 47.129 0-94 0.943 -a ^ ?rl 48 48 7-885 48 48.131 0.96 0.963 tO *3 OJ 49 49 8.049 49 49.134 0.98 0.983 5 50 8.214 5 5- I 37 1. 00 1.003 fM | 5 1 51 8.378 51 51.140 4> ' 52 52 8.542 S 2 52.142 . 53 53 8.707 53 53-145 .2 H 54 54 8.871 54 54.148 3 I 55 55 9-035 55 55-J5 1 j2 ^ 5 6 5 6 9- J 99 56 56.153 rH "- / 57 57 9-364 57 57.156 8 58 58 9.528 58 58.159 c^ 59 59 9-692 59 59.162 60 60 9.856 60 60.164 563 TABLE III, For converting intervals of Sidereal Time into equivalent intervals of Mean Solar Time. Hours. Minutes. Seconds. Decimals. Sid. T. Mean Time. Sid. T. Mean Time. Sid. T. Mean Time. Sid. T. Mean Time. h h m s m m s 5 * s g I o 59 50.170 I o 59.836 0.997 0.02 0.020 2 I 59 40-34 2 I 59.672 2 1.995 0.04 0.040 3 2 59 30.51 3 2 59-509 3 2.992 0.06 O.o6o 4 3 59 20.682 4 3 59-345 4 0.08 0.080 4 59 IO - 8 52 5 4 59.181 5 4.986 0.10 O.I 00 6 5 59 1.02 6 5 59- OJ 7 6 5.984 0.12 0.120 I 6 58 5i-i93 7 58 4 I -3 6 3 8 6 58.853 7 58.689 I 6.981 7.978 0.14 o.i 6 0.140 0.160 9 8 58 3'-534 9 8 58.526 9 8-975 0.18 0.180 10 9 58 21.70. 10 9 58-362 10 9-973 0.20 0.199 ii 12 10 58 11.875 ii 58 2.045 ii 12 10 58.198 ii 58.034 ii 12 10.970 11.967 0.22 0.24 0.219 0.239 13 4 12 57 52.216 13 57 42.386 '3 14 12 57.870 13 57.706 5J 12.964 13.962 0.26 0.28 0.259 0.279 li H 57 32-557 15 57 22.727 15 16 H 57-543 15 57-379 15 16 14.959 I5-956 0.30 0.32 0.299 0.319 17 16 57 12.897 52 16 57.215 17 16.954 o-34 o-339 18 17 57 3.068 18 17 57.051 18 17.951 0.36 -359 '9 18 56 53.238 19 18 56.887 19 18.948 0.38 -379 20 21 22 23 24 19 56 43.409 20 5 6 33-579 21 56 23.750 22 56 13.920 23 5 6 4-09 1 20 21 22 23 2 4 19 56.723 20 56.560 21 56.396 22 56.232 23 56.068 20 21 22 23 24 19-945 20.943 21.940 22.937 23-934 0.40 0.42 -44 0.46 0.48 . o-399 0.419 0-439 0.459 o-479 2 5 24 55-904 25 24.932 0.50 -499 26 25 55-740 26 25.929 0.52 0.519 oJ 27 26 55-577 27 26.926 o-54 -539 JH 28 27 55-413 28 27-924 0.56 o-558 H 2 9 28 55-249 29 28.921 0.58 0-578 J"i 3 29 55.085 3 29.918 0.60 0.598 s 31 3 54-921 3 1 50.915 0.62 0.618 c rs 3 2 3i 54-758 32 31.913 0.64 0.638 * 36.899 0.74 0.738 S 8 37 53-775 38 37.896 0.76 0.758 *O ^ C 39 38 53-6n 39 38-894 0.78 0.778 K "3 g 40 39 53-447 40 39.891 0.80 0.798 *o Q 'So 4 1 40 53-283 40.888 0.82 o.8i8 gl^ 42 41 53-119 42 41.885 0.84 0.838 2 e "** 43 42 52.955 43 42.883 0.86 0.858 K S ~ | - S 44 43 52-792 44 52.628 44 45 43.880 44.877 0.88 0.90 0.878 0.898 o< S 45 52-464 46 52.300 47 52-136 46 47 48 45-874 46.872 47.869 0.92 o-94 0.96 0.917 o-937 0-957 *H 3 6 s r^ ill 49 50 48 5I-972 49 51.809 50 51.645 49 5 48.866 49.863 50.861 0.98 I.OO 0.977 0-997 . 52 51 51.481 52 51.858 tn ;_ 53 52 51-317 53 52.855 ' J3 54 53 5i.i53 54 53-853 ? * 55 54 50.990 55 54.850 *- 56 55 50-826 56 55-847 13 *si 57 56 50.662 57 56.844 EH 5 8 57 50.498 58 57.842 59 58 50.334 58.839 ====================== 60 === 59 50-170 59.836 - mm^mammmmmmmmamm 564 TABLE IV. For converting Hours, Minutes, and Seconds into Decimals of a Day. Hours. Decimal. Min. Decimal. Min. Decimal. Sec. Decimal. Sec. Decimal. 1 0.0416 -\- 1 .000694 + 31 .021527 + 1 .0000 i i 6 31 .0003588 2 0833 + 2 .001388 + 32 .O22222 + 2 .0000231 32 .0003704 3 .1250 + 3 .002083 + 33 .022916 + 3 .0000347 33 .0003819 4 .1666 + 4 .002777 + 34 .023611 + 4 .0000463 34 .0003935 5 .2083 + 5 .003472 + 35 .024305 + 5 .0000579 35 .0004051 6 .2500 + 6 .004166 + 36 .025000 + 6 .0000694 36 .0004167 7 0.2916 + 7 .004861 + 37 .025694 + 7 .0000810 37 .0004282 8 3333 + 8 5555 + 38 .026388 + 8 .0000925 38 .0004398 9 375 + 9 .006250 + 39 .027083 + 9 .0001042 39 .0004514 10 . 4166 + 10 .006944 + 40 .027777 + 10 .0001157 40 .0004630 11 4583 + 11 .007638 + 41 .028472 + 11 .0001273 41 .0004745 12 .5000 + 12 oo8333 + 42 .029166 + 12 .0001389 42 .0004861 13 0.5416 + 13 .009027 + 43 .029861 + 13 .0001505 43 .0004977 14 5833 + 14 .009722 + 44 03 555 + 14 .0001620 44 .0005093 15 .6250 + 15 " .010416 + 45 .031250 + 15 .0001736 45 .0005208 16 .6666 + 16 .OIHII + 46 .031944 + 16 .0001852 46 .0005324 17 7083 + 17 .011805 + 47 .032638 + 17 .0001968' 47 .0005440 18 .7500 + 18 .012500 + 48 033333 + 18 .0002083 48 .0005556 19 0.7916 + 19 .013194 + 49 .034027 + 19 .0002199 49 .0005671 20 8333 + 20 .013888 + 50 .034722 + 20 .0002315 50 .0005787 21 .8750 + 21 -.014583 + 51 .035416 + 21 .0002431 51 .0005903 22 .9166 + 22 .015277 + 52 .036111 + 22 .0002546 52 .0006019 23 0.9583 + 23 .015972 + 53 .036805 + 23 .0002662 53 .0006134 24 1. 0000 + 24 .016666 + 54 .037500 + 24 .0002778 54 .0006250 25 .017361 + 55 .038194 + 25 .0002894 55 .0006366 26 .018055 + 56 .038888 + 26 .0003009 56 .0006481 27 .018750 + 57 395 8 3 + 27 .0003125 57 .0006597 28 .019444 + 58 .040277 + 28 .0003241 58 .0006713 29 .020138 + 59 .040972 + 29 .0003356 59 .0006829 30 .020833 + 60 .041666 + 30 .0003472 60 .0006944 The sign +, appended to numbers in this table, signifies that the last figure repeats to infinity. TABLE V, For finding the number of Days from the beginning of the Year. Date. Com. Bis. January o.o o February o.o 31 3 1 March o.o 59 60 April o.o 90 9 1 May o.o 120 121 June o.o '5 1 I 5 2 July o.o 181 182 August o.o 212 213 September o.o 243 244 October o.o 273 274 November o.o 304 35 December o.o 334 335 565 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. , V. - ~ 1 rss=^== 2 3 M. Diff. 1". M. Diff. 1". M. Diff. 1". M. Diff. I". 0' 1 2 3 4 o.oooooo 0.010908 0.021817 0.032725 0.043633 181.81 181.81 181.81 181.81 181.81 0.654532 0.665442 0.676352 0.687262 0.698172 181.83 181.83 181.83 181.84 181.84 1.309263 1.320178 1.331093 1.342008 1.352923 181.92 181.92 181.92 181.92 181.92 1.964393 1.975316 1.986240 1.997164 2.008087 182.05 182.06 182.06 182.06 182.07 5 6 7 8 9 0.054542 0.065450 0.076358 0.087267 0.098175 181.81 181.81 181.81 181.81 181.81 0.709082 0.719993 0.730903 0.741813 0.752724 181.84 181.84 181.84 181.84 181.84 1.363839 1-374755 1.385670 1.396586 1.407502 181.93 181.93 181 93 181.93 181.93 2.019011 2.029936 2.040860 2.051785 2.062709 182.07 182.07 182.07 182.08 182.08 10 11 12 13 0.109083 0.119992 0.130900 0.141808 181.81 181.81 181.81 181.81 0.763634 0-774545 0.785456 0.796366 181.84 181.84 181.84 181.85 1.418418 1-4^9334 1.440251 1.451167 181.94 181.94 181.94 181.94 2.073634 2.084559 2.095485 2.106410 182.08 182.08 182.09 182.09 14 T^ 0.152717 181.81 0.807277 181.85 1.462083 181.94 2.117335 182.09 15 16 17 0.163625 0.174534 0.185442 181.81 181.81 181.81 0.818188 0.829099 0.840010 181.85 181.85 181.85 1.473000 1.483917 1.494834 181.95 181.95 181.95 2.128261 2.139187 2.150114 182.10 182.10 182.10 18 19 0.196350 0.207259 181.81 181.81 0.850921 0.861832 181.85 181.85 I-50575 1 1.516668 181.95 181.95 2.161040 2.171966 182. II 182.11 20 21 22 23 24 0.218167 0.229076 0.239984 0.250893 0.261801 181.81 181.81 181.81 181.81 181.81 0.872743 0.883654 0.894566 0.905478 0.916389 181.85 181.86 181.86 181.86 181.86 1.527585 i-53 8 53 1.549420 1.560338 1.571256 181.96 181.96 181.96 181.96 181.96 2.182894 2.193820 2.204747 2.215674 2.226602 182.11 182.12 182.12 182.12 182.13 25 0.272710 181.81 0.927301 181.86 1.582174 181.97 2.237529 182.13 26 0.283619 181.81 0.938212 181.86 1.593092 181.97 2.248457 182.13 27 28 0.294527 0.305436 181.81 181.81 0.949124 0.960036 181.86 181.86 1.604010 1.614928 181.97 181.97 2.259385 2.270313 182.14 182.14 29 0.316345 181.81 0.970948 181.87 1.625847 181.97 2.281242 182.14 ! 30 ! 31 0.327253 0.338162 181.81 181.81 0.981860 0.992772 181.87 181.87 1.636766 1.647684 181.98 181.98 2.292170 2.303099 182.14 182.15 32 0.349071 181.81 1.003684 181.87 1.658603 181.98 2.314028 182.15 33 0.359980 181.81 1.014596 181.87 1.669522 181.98 2.324957 182.15 34 0.370888 181.81 1.025509 181.87 1.680441 181.99 2.335887 182.16 35 36 0.381797 0.392706 181.81 181.81 1.036421 1.047334 181.87 181.87 1.691361 1.702280 181.99 181.99 2.346816 2.357746 182.16 182.16 | 37 0.403615 181.81 1.058246 181.88 1.713200 181.99 2.368676 182.17 38 0.414524 181.82 1.069159 181.88 1.724120 182.00 2.379606 182.17 < 39 0-425433 181.82 1.080072 181.88 1.735039 182.00 2.390536 182.17 40 0.436342 181.82 1.090985 181.88 1.745960 182.00 2.401467 182.18 ! 41 0.447251 181.82 1.101898 181.88 1.756880 182.00 2.412398 182.18 42 0.458160 181.82 1.112811 181.89 1.767800 182.01 2.423329 182.18 43 0.469069 181.82 1.123724 181.89 1.778721 182.01 2.434260 182.19 44 0.479979 181.82 1.134637 181.89 1.789641 182.01 2.445191 182.19 4f 0.490888 181.82 1.145550 181.89 1.800562 182.01 2.456123 182.19 46 0.501797 181.82 1.156464 181.89 1.811483 182.02 2.467055 182.20 47 0.512706 181.82 1.167377 181.89 1.822404 182.02 2.477987 182.20 48 0.523616 181.82 1.178291 181.89 i- 8 333 2 5 182.02 2.488919 182.20 49 0.534525 181.82 1.189205 181.90 1.844247 182.02 2.499851 182.21 50 0-545435 181.82 1.200119 181.90 1.855168 182.03 2.510784 182.21 51 0.556344 181.82 1.211033 181.90 1.866090 182.0^ 2.521717 182.22 52 0.567254 181.82 1.221947 181.90 1.877012 182.0^ 2.532650 182.22 53 0.578163 181.83 1.232861 181.90 1.887934 182.0^ 2.5435 8 3 182.22 54 0.589073 181.83 'M3775 181.91 1.898856 182.04 2-5545*7 182.23 55 56 0.599983 0.610892 181.83 i8i.8i 1.254689 1.265604 181.91 181.91 1.909779 1.920701 182.0; 182.0; 2.565450 2.576384 182.23 182.23 57 0.621802 i8i.8 : 1.276518 181.91 1.931624 182.05 2.587319 182.24 58 59 0.632712 0.643622 181.83 181.83 1-^87433 1.298348 181.91 181.91 1.942547 1-95347 182.05 182.05 2.598253 2.609187 182.24 182.24 60 0.654532 181.83 1.309263 181.92 I -9 6 4393 182.05 2.620122 182.25 56t) TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 4 5 6 7 M. Diff. I". M. Diff. 1". M. Diff. 1". M. Diff. 1". 0' 2.620122 182.25 3.276651 182.50 3.934182 182.80 4.592917 183.17 1 2.631057 182.25 3.287602 182.50 3-945J5 1 182.81 4.603907 183,18 2 2.641993 182.26 3.298552 182.51 3.956119 182.82 4.614898 183.18 3 2.652928 182.26 3-309503 182.51 3.967088 182.82 4.625889 183.19 4 2.663864 182.26 3.320454 182.52 3.978058 182.83 4.636880 183.19 5 2.674800 182.27 3-33H05 182.52 3.989028 182.83 4.647872 183.20 6 2.685736 182.27 182.53 3.999998 182.84 4.658864 183.21 7 2.696672 182.27 3-3533 8 182.53 4.010968 182.84 4.669857 183.21 8 2.707609 182.28 3.364260 182.54 4.021939 182.85 4.680850 183.22 9 2.718546 182.28 3.375212 182.54 4.032911 182.86 4.691843 !8 3 .2 3 10 2.729483 182.29 3.386165 182.55 4.043882 182.86 4.702837 183.24 11 2.740420 182.29 3.397118 182.55 4.054854 182.87 4.713831 183.24 12 182.29 3.408071 182.56 4.065826 182.87 4.724826 !8 3 .2 5 13 2.762295 182.30 3.419024 182.56 4.076799 182.88 4-735821 183.25 14 2.773233 182.30 3.429978 182.57 4.087772 182.88 4.746816 183.26 15 2.784172 182.31 3.440932 182.57 4.098745 182.89 4.757812 183.27 16 2.795110 182.31 3.451887 182.58 4.109718 182.90 4.768809 183.27 17 2.806049 182.31 3.462841 182.58 4.120692 182.90 4.779805 183.28 18 2.816988 182.32 3-473796 182.59 4.131667 182.91 4.790802 183.28 19 2.827927 182.32 3.484752 182.59 4.142641 182.91 4.801800 183.29 20 2.838867 182.33 3.495707 182.60 4.153616 182.92 4.812797 183.30 21 2.849806 182.33 3.506663 182.60 4.164592 182.93 4.823796 183.31 22 2.860746 182.33 3.517619 182.61 4.175568 182.93 4.834795 183.32 23 2.871686 182.34 182.61 4.186544 182.94 4.845794 183.32 24 2.882627 182.34 3-539532 182.61 4.197520 182.94 4-856793 183.33 25 2.893567 182.35 3-550489 182.62 4.208497 182.95 4.867793 183.34 26 2.904508 182.35 3.561447 182.62 4.219474 182.95 4.878793 183.34 27 2.915449 182.36 3.572404 182.63 4.230451 182.96 4.889794 183.35 28 2.926391 182.36 3-583362 182.63 4.241429 182.97 4.900795 183.36 29 2.937332 182.36 3.594320 182.64 4.252408 182.97 4.911797 183.36 30 2.948274 182.37 3.605279 182.64 4.263386 182.98 4.922799 183.37 31 2.959217 182.37 3.616238 182.65 4.274365 182.99 4.933801 183.38 32 2.970159 182.37 3.627197 182.65 4.285344 182.99 4.944804 183.38 33 2.981102 182.38 3.638156 182.66 4.296324 183.00 4.955807 183.39 34 2.992045 182.38 3.649116 182.66 4.307304 183.00 4.966811 183.40 35 3.002988 182.39 3.660076 182.67 4.318284 183.01 4.977815 183.41 36 3.013931 182.39 3.671037 182.68 4-329265 183.01 4.988820 183.41 37 38 3.024875 182.39 182.40 3-68I997 3.692958 182.68 182.69 4.340246 4.351228 183.02 183.03 4.999825 5.010830 183.42 I8343 39 3.046763 182.40 3.703920 182.69 4.362210 183.03 5.021836 1 8 3'43 40 41 3.057707 3.068652 182.41 182.41 3.71488! 3-725843 182.70 182.70 4.373!92 4.384175 183.04 183.05 5.032842 5.043849 183.44 183.45 42 3.079597 182.42 3.736806 182.71 4.395158 183.05 5.054856 183.46 43 3.090542 182.42 3.747768 182.71 4.406141 183.06 5.065864 183.46 44 3.101488 182.43 3-75873I 182.72 4.417125 183.06 5.076872 183.47 45 46 47 48 3-H2433 3- I2 3379 3.145272 182.43 182.44 182.44 182.44 3.769694 3.780658 3.791622 3.802586 182.72 182.72 182.73 182.74 4.428109 4-439093 4.450078 4.461064 183.07 183.08 183.08 183.09 5.087880 5.098889 5.109898 5.120908 183.48 183.48 183.49 183.50 49 3.156219 182.45 3-8i355i 182.74 4-472049 183.10 5.131918 183.51 50 3.167166 182.45 3.824515 182.75 4-483035 183.10 5.142929 183.51 51 3.178113 182.46 3.835481 182.76 4.494022 183.11 5- i 5 394 183.52 52 3.189061 182.46 3.846446 182.76 4.505008 183.12 5.164951 183-53 53 3.200009 182.47 3.857412 182.77 4-5*5995 183.12 5> I 759 6 3 183.54 54 3.210957 182.47 3.868378 182.77 4.526983 183.13 5.186975 183.54 55 3.221905 182.48 3-879345 182.78 4-537971 183.14 5.197988 183.55 56 3,232854 182.48 3.890312 182.78 4.548959 183.14 5.209002 183.56 57 3.243803 182.49 3.901279 182.79 4.559948 183.15 5.220015 183.57 58 182.49 3.912246 182.79 4-570937 183.15 5.231029 l8 3-57 59 3.265702 182.49 3.923214 182.80 4.581927 183.16 5.242044 183.58 60 3.276651 182.50 3.934182 182.80 4.592917 183.17 5-253 59 183.59 567 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit, V. 8 9 10 11 M. Diff. 1". M. Diff. I". M. Diff. 1". M. Diff. 1". 0' 5253059 183.59 5.914815 184.06 6.578391 184.60 7.243997 185.19 1 (5.264075 183.59 5-925859 184.07 6.589467 184.61 7.255109 185.20 2 3 J T / J 5.275090 183.60 183.61 5.936904 5.947949 184.08 184.09 6.600544 6.611622 184.62 184.63 7.266222 7-277335 185.21 185.22 4 5.297124 183.62 5.958995 184.10 6.622700 184.64 7.288449 185.23 5 6 7 8 9 5.308I4I 5.3I9I59 5-330I77 5-34II95 5.352214 183.62 183.63 183.64 183.65 183.66 5.970041 5.981087 5.992134 6.003182 6.014230 184.11 184.11 184.12 184.13 184.14 6.633778 6.644857 6.655937 6.667017 6.678098 184.65 184.66 184.67 184.67 184.68 7.310678 7.32I793 7.332909 7.344026 185.25 185.26 185.27 185.28 185.29 10 11 12 13 5-363234 5.374254 5.38|275 5.396296 183.66 183.67 183.68 183.69 6.025279 6.036328 6.047378 6.058428 184.15 184.16 184.17 184.18 6.689179 6.700261 6.711343 6.722426 184.69 184.70 184.71 184.72 7-355144 7.366262 7.37738i 7.388500 185.30 185.31 185.32 185.33 14 5-473 I 7 183.69 6.069479 184.18 6.733510 184.73 7.399620 185-34 15 16 5-418339 183.70 183.71 6.080530 6.091582 184.19 184.20 6-744594 6.755679 184.74 184.75 7.410741 7.421862 185.35 185.36 17 5.440384 183.72 6.102634 184.21 6.766764 184.76 7-432983 185.37 18 5.451407 183.73 6.113687 184.22 6.777850 184.77 7.444106 185-38 19 5.462431 183-73 6.124740 184.23 6.788937 184.78 7-455230 185.39 20 5.473455 183-74 6.135794 184.24 6.800024 184.79 7.466354 185.40 21 5.484480 183.75 6.146849 184.25 6.8111 12 184.80 7.477478 185.41 22 5.495505 183.75 6.157904 184.25 6.822200 184.81 7.488603 185.42 23 24 5.506530 5.5I755 6 183.76 183.77 6.168959 6.180015 184.26 184.27 6.833289 6.844378 184.82 184.83 7.499729 7.510855 I85-43 185.44 25 5.528583 183.78 6.191072 184.28 6.855468 184.84 7.521982 185.46 26 5.539610 183.79 6.202129 184.29 6.866559 184.85 7.5331 10 185.47 27 5.55 637 183.79 6.213187 184.30 6.877650 184.86 7.544239 185.48 28 5.561665 183.80 6.224245 184.31 6.888742 184.87 7-555368 185-49 29 5-572693 183.81 6.235304 184.32 6.899834 184.88 7.566497 185.50 30 5-583722 183.82 6.246363 184.32 6.910927 184.89 7.577628 185.51 31 32 5-594752 5.605702 183.83 183.83 6.257422 6.268482 184.33 184.34 6.922021 6.933115 184.90 184.91 7-588759 7.599890 185.52 185.53 33 5.616812 183.84 6-279543 l8 4-35 6.944210 184.92 7.611022 185.54 34 5.627843 183.85 6.290605 184.36 6.955305 184.93 7.622155 185.55 35 5-638874 183.86 6.301667 184.37 6.966401 184.94 7.633289 185-57 36 5.6499:6 183.87 6.312729 184.38 6.977498 184.95 7.644423 185-58 37 5.660938 183.87 6.323792 184.39 6.988595 184.96 7.655558 185.59 38 39 5.671971 5.683004 183.88 183.89 6.334855 6.345919 184.40 184.41 6.999693 7.010791 184.97 184.98 7.666694 7.677830 185.60 185.61 40 5.694038 183.90 6.356984 184.41 7.021890 184.99 7.688967 185.62 41 5.705072 183.91 6.368049 184.42 7.032990 185.00 7.700104 185.63 42 5.716106 183.92 6.379115 184.43 7.044090 185.01 7.711242 185.64 43 44 5.727141 5-738I77 183.92 183.93 6.390181 6.401248 184.44 184.45 7.055191 7.066292 185.02 185.03 7.722381 7.733521 185.65 185.66 45 5-7492I3 183.94 6.412315 184.46 7.077394 185.04 7-74466i 185.68 46 5.760250 183-95 6-423383 184.47 7.088497 185.05 7.755802 185.69 47 5.771287 183.96 6.434451 184.48 7.099600 185.06 7.766943 185.70 48 5.782325 183.96 6.445520 184.49 7.110704 185.07 7.778085 185.71 49 5-793363 183.97 6.456590 184.50 7.121808 185.08 7.789228 185.72 50 51 52 53 54 5.804401 5.815440 5.826480 5.837520 5.848561 183.98 183.99 184.00 184.01 184.01 6.467660 6.478731 6.489802 6.500874 6.511946 184.51 184.52 184.52 184-53 184.54 7.132913 7.144019 7.155125 7.166232 7.177340 185.09 185.10 185.11 185.12 185.13 7.800372 7.81 1516 7.822661 7.833807 7-844953 185.73 185.74 185.75 185.76 185.78 55 56 57 5.859602 5.870644 5.881686 184.02 184.03 184.04 6.523019 6.534092 6.545166 184-55 184.56 184.57 7.188448 7-199557 7.210666 185-15 185.16 7.856100 7.867247 7.878396 185.79 185.80 185.81 58 59 5.892728 5.903771 184.05 184.06 6.556241 6.567316 184.58 184.59 7.221776 7.232886 185.17 185.18 7.889545 7.900694 185.82 185-83 60 5.914815 184.06 6.578391 184.60 7.243997 185.19 7.911845 185.84 568 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 12 13 14 15 M. Diff. I". M. Diff. 1". M. Diff. 1". M. Diff. I". 0' 7.911845 185.84 8.582146 186.56 9.255120 187-33 9.930984 188.16 1 7.922995 185.86 8.593340 186.57 9.266360 187.34 9.942274 188.18 2 7.934147 185.87 8.604535 186.58 9.277601 187.35 9.9535 6 5 188.19 3 7.945300 185-88 8.615730 186.59 9.288842 187-37 9.964857 188.21 4 7.956453 185.89 8.626926 186.61 9.300085 187.38 9.976149 188.22 5 7.967606 185.90 8.638123 186.62 9.311328 187.40 9.987443 188.23 6 7.978761 185.91 8.649320 186.63 9.322572 187.41 9.998738 I88.2 5 7 ' 7.989916 185.92 8.660518 186.64 9-3338I7 187.42 10.010033 1 188.26 8 8.001072 l8 5-93 8.671717 186.66 9-345 6 3 187.44 10.021329 188.28 9 8.012228 185-95 8.682917 186.67 9.356310 187.45 10.032626 188.29 1O 8.023385 185.96 8.694117 186.68 9-3 6 7557 187.46 10.043924 188.31 11 8-034543 185.97 8.705318 186.69 9.378805 187.48 10.055223 188.32 12 8.045702 185.98 8.716520 186.71 9.390054 187.49 10.066523 188.34 13 8.056861 185.99 8.727723 186.72 9.401304 187.50 10.077823 188.35 14 8.068021 186.00 8.738927 186.73 9.412555 187.52 10.089125 188.37 15 8.079181 186.02 8.750131 186.74 9.423806 187.53 10.100427 188.38 16 8.090343 186.03 8.761336 186.76 9.435058 187.54 10.111730 l ll- 17 8.101505 186.04 8.772542 186.77 9.446311 187.56 10.123035 188.41 18 8.II2668 186.05 8.783748 186.78 9-457565 1.87.57 10.134340 188.42 19 8.123831 186.06 S-794955 186.79 9.468820 187.59 10.145646 188.44 20 8.134995 186.07 8.806163 186.81 9.480076 187.60 10.156952 188.45 21 8.146160 186.09 8.817372 186.82 9.491332 187.61 10.168260 188.47 22 8.157326 186.10 8.828582 186.83 9.502589 187.63 10.179568 188.48 23 8.168492 i86.n 8.839792 186.84 9.513847 187.64 10.190878 188.50 24 8.179659 186.12 8.851003 186.86 9.525106 187.65 I0.202l88 188.51 25 8.190826 186.13 8.862215 186.87 9.536366 187.67 10.213499 188.53 26 8.201995 186.15 8.873427 186.88 9.547626 187.68 10.224812 188.54 27 8.213164 186.16 8.884641 186.90 9.558888 187.70 10.236125 188.56 i 28 8.224334 186.17 8.895855 186.91 9.570150 187.71 10.247439 '^ 57 29 8.235504 186.18 8.907070 186.92 9.581413 187.72 10.258753 188.59 30 8.246675 186.19 8.918286 186.93 9.592676 187.74 10.270069 188.60 31 8.257847 186.20 8.929502 186.95 9.603941 187.75 10.281386 188.62 32 8 269020 186.22 8.940719 186.96 9.615207 187.77 10.292703 '88.63 33 8.280193 186.23 8 -95i937 186.97 9.626473 187.78 10.304021 188.65 34 8.291 367 186.24 8.963 156 186.99 9.637740 187.79 I0.3I534 1 188.66 35 36 8.302542 8.313717 186.25 186.26 8.974376 8.985596 187.00 187.01 9.649008 9.660277 187.81 187.82 10.326661 10.337982 188.68 188.69 37 8.324893 186.28 8.996817 187.02 9.671547 187.84 10.349304 188.71 38 39 8.336070 8.347248 186.29 1.86.30 9.008039 9.019262 187.04 187.05 9.682817 9.694088 187.85 187.86 10.360627 10.371951 188.72 188.74 40 8.358426 186.31 9.030485 187.06 9.705361 187.88 10.383275 188.75 41 8.369605 186.32 9.041709 187.08 9.716634 187.89 10.394601 188.77 o o o 42 8.380785 186.34 9.052934 187.09 9.727908 187.91 10.405927 188.78 43 8.391966 186.35 9.064160 187.10 9.739182 187.92 10.417255 188.80 44 8.403147 186.36 9.075387 187.12 9.750458 18793 10.428583 188.81 45 8.414329 186.37 9.086614 187.13 9.761734 187.95 10.439912 188.83 46 8.425512 186.38 9.097842 187.14 9.773012 187.96 10.451242 188.84 47 8.436695 186.40 9.109071 187.16 9.784290 187.98 10.462573 188.86 48 49 8.447879 8.459064 186.41 186.42 9.120301 9- I 3 I 53 I 187.17 187.18 9.795569 9.806849 187.99 188.00 10.473905 10.485238 188.87 188.89 50 8.470250 186.43 9.142763 187.20 9.818129 188.02 10.496572 188.90 51 8.481436 186.45 9- r 53995 187.21 9.829410 188.03 10.507907 188.92 o o 52 8.492623 186.46 9.165228 187.22 9.840693 188.05 10.519242 188.93 53 54 8.503811 8.515000 186.47 186.48 9.176462 9.187696 187.23 187.25 9.851977 9.863261 188.06 188.08 10.530579 10.541916 188.95 188.97 55 56 57 58 59 8.526189 8-537379 8.548569 8.559761 8.570953 186.49 186.51 186.52 186.53 186.54 9.198931 9.210167 9.221404 9.232642 9.243880 187.26 187.27 187.29 187.30 187.31 9.874546 9.885832 9.897118 9.908406 9.919694 188.09 188.10 188.12 188.13 188.15 10.553255 10.564594 10-575934 10.587276 10.598618 188.98 189.00 189.01 189.03 189.04 60 8.582146 186.56 9.255120 187.33 9.930984 188.16 10.609961 189.06 569 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 16 17 18 19 M. Diff. 1". M. Diff. 1". M. Diff. I". M. Diff. I". 0' I 2 10.609961 10.621305 10.632649 189.06 189.07 189.09 11.292277 11.303679 11.315082 190.02 190.03 190.05 x 11.978162 11.989625 12.001089 191.04 191.06 191.08 12.667850 12.679379 12.690908 192.13 192.15 192.17 3 10.643995 189.10 11.326485 190.07 12.012554 191.09 12.702439 192.19 4 10.655342 189.12 11.337889 190.08 12.024021 191.11 12.713970 192.21 5 10.666690 189.14 11.349295 190.10 12.035488 191.13 12.725503 192.22 6 7 10.678038 10.689388 189.15 189.17 11.360701 11.372109 190.12 190.13 12.046956 12.058425 191.15 191.16 12.737037 12.748573 192.24 192.26 8 10.700738 189.18 n-3 8 35 I 7 190.15 12.069896 191.18 12.760109 192.28 9 10.712090 189.20 11.394927 190.17 12.081367 191.20 12.771646 192.30 10 10.723442 189.21 11.406337 190.18 12.092840 191.22 12.783185 192.32 11 10.734795 189.23 11.417749 190.20 12.104313 191.24 12.794724 192.34 12 10.746149 189.24 1 1.429161 190.22 12.115788 191.25 12.806265 192.36 13 10.757505 189.26 11.440575 190.23 12.127264 191.27 12.817807 192.37 14 10.768861 189.28 11.451989 190.25 12.138741 191.29 12.829350 192.39 15 10.780218 189.29 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 189.34 11.497657 190.32 12.184659 191.36 12-875534 192.47 19 10.825655 l8 9-35 11.509077 l9 -33 12.196141 191.38 12.887082 192.49 20 10.837017 189-37 11.520497 l9 -35 12.207624 191.40 12.898632 192.51 21 10.848380 189.39 11.531919 190.37 12.219108 191.41 12.910183 *92-53 22 10.859744 189.40 11.543342 190.39 12.230594 i9 J -43 12.921736 I92-55 23 10.871108 189.42 11.554765 190.40 12.242080 191.45 12.933289 192.56 24 10.882474 189.43 11.566190 190.42 12.253568 191.47 12.944843 192.58 25 10.893840 189.45 11.577616 190.44 12.265057 191.49 12.956399 192.60 26 10.905208 189.47 11.589042 190.45 12.276546 191.50 12.967956 192.62 27 10.916576 189.48 1 1.600470 190.47 12.288037 191.52 12.979514 192.64 28 10.927946 189.50 11.611899 190.49 12.299529 191.54 12.991073 192.66 29 10.939316 189.51 11.623328 190.50 12.311022 191.56 13.002633 192.68 30 10.950687 189-53 11.634759 190.52 12.322516 191.58 13.014195 192.70 31 10.962059 189.55 11.646191 190.54 12.334011 191.60 I3-025757 192.72 32 10.973433 189.56 11.657624 190.56 12.345508 191.61 13.037321 192.74 33 10.984807 189.58 11.669057 190.57 12.357005 191.63 13.048886 192.76 34 10.996182 189.59 11.680492 190.59 12.368503 191.65 13.060452 192.78 35 11.007558 189.61 11.691928 190.61 12.380003 191.67 13.072019 192.80 36 11.018935 189.63 11.703365 190.62 12.391504 191.69 13.083587 192.82 37 11.030313 189.64 11.714803 190.64 12.403006 191.70 i3-95 I 57 192.83 38 11.041692 189.66 11.726242 190.66 12.414509 191.72 13.106727 192.85 39 11.053072 189.67 11.737682 190.68 12.426013 191.74 13.118299 192.87 40 11.064453 189.69 11.749123 190.69 12.437517 191.76 13.129872 192.89 41 11.075835 189.7.! 11.760565 190.71 12.449023 191.78 13.141446 192.91 42 11.087218 189.72 11.772008 190.73 12.460531 191.80 13.153022 192.93 43 11.098602 189.74 11.783452 190.74 12.472039 191.81 13.164598 192.95 44 11.109987 189.76 11.794897 190.76 12.483548 191.83 13.176176 192.97 45 11.121372 189.77 11.806344 190.78 12.495059 191.85 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 i93- 3 48 "155536 189.82 11.840689 190.83 12.529597 191.91 13.222498 i93- 5 49 11.166925 189.84 11.852139 190.85 12.541 112 I9I-93 13.234082 193.07 50 11.178316 189.85 11.863590 190.87 12.552628 191.94 13.245667 193.09 51 11.189708 189.87 11.875043 190.88 12.564145 191.96 13.257253 193.11 52 II.20IIOO 189.89 11.886496 190.90 12.575664 191.98 13.268840 i93- I 3 53 11.212494 189.90 11.897951 190.92 12.587183 192.00 13.280428 i93-!5 54 11.223889 189.92 11.909407 190.94 12.598704 192.02 13.292017 193.17 55 11.235284 189.93 11.920863 190.95 I2.6I0225 192.04 13.303608 i93-'9 56 11.246681 189.95 11.932321 190.97 12.621748 192.06 13.315200 193.21 57 11.258078 189.97 11.943780 190.99 12.633272 192.07 13.326793 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 11.966700 191.02 12.656323 192.1 1 13.349982 I93-27 60 11.292277 190.02 11.978162 191.04 12.667850 192.13 13.361579 193.29 570 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V 20 21 22 23 u M. Diff. I". M. Diff. I". M. Diff. 1". M. Diff. 1". 0' i3-36i579 193.29 14.059591 194.51 14.762133 195.80 15.469459 197.17 i '3-373'77 '93-3' 14.071262 194-53 14.773882 I95-83 15.481290 197.19 2 13.384776 '93-33 14.082935 '94-55 14.785632 195.85 15.493122 197.21 3 13.396376 193-35 14.094608 '94-57 14.797384 195.87 15.504956 197.24 4 13.407977 193-37 14.106283 1 94-59 14.809137 195.89 15.516791 197.26 5 13.419580 193-39 14.117960 194.61 14.820891 195.91 15.528627 197.28 6 i3-43" 8 3 I93-4I 14.129637 194.64 14.832647 195.94 15.540465 197.31 7 13.442788 '93-43 14.141316 194.66 14.844403 195.96 I5-552304 197.33 8 1 3-454394 193-45 14.152996 194.68 14.856161 195.98 15.564144 '97-35 9 13.466002 '93-47 14.164677 194.70 14.867921 196.00 15.575986 '97-3 8 10 13.477610 193-49 14.176360 194.72 14.879682 196.03 '5-5 8 7 8 3 197.40 11 13.489220 1 93-5 I 14.188044 194.74 14.891444 196.05 i5-599 6 75 '97-43 12 13.500831 '93-53 14.199729 194.76 14.903208 196.07 15.611521 '97-45 13 I3-5I2443 '93-55 14.211415 194.78 '4-9'4973 196.09 15.623369 '97-47 14 13.524056 193-57 14.223103 194.81 14.926739 196.12 15.635218 197.50 15 i3.535 6 7i '93-59 14.234792 194.83 14.938506 196.14 15.647068 197.52 16 13.547287 193.61 14.246482 194.85 14.950275 196.16 15.658920 '97-54 17 13.558904 193.63 14.258174 194.87 14.962045 196.18 15.670773 '97-57 18 13.570522 i93- 6 5 14.269867 194.89 14.973817 196.20 15.682628 '97-59 19 13.582141 193.67 14.281561 194.91 14.985590 196.23 15.694484 197.61 20 13.593762 193.69 14-293256 194-93 i4-9973 6 5 196.25 15.706342 197.64 21 22 13.605383 13.617006 '93-7' 193-73 i4-3 4953 14.316651 194.95 194.98 15.009140 15.020917 196.27 196.30 15.718201 15.730061 197.66 197.69 23 13.628631 193-75 14.328350 195.00 15.032696 196.32 '5-74'9 2 3 197.71 24 13.640256 193.77 14.340050 195.02 15.044475 196.34 15.753786 '97-73 25 13.651883 193-79 H.35I752 195.04 15.056256 196.36 15.765651 197.76 26 13.663511 193.81 14.363455 195.06 15.068039 196.39 15.777517 197.78 27 13.675140 193.83 I 4-375 I 59 195.08 15.079823 196.41 '5-7 8 93 8 5 197.80 28 13.686770 '93- 8 5 14.386865 195.10 15.091608 196.43 15.801254 '97- 8 3 29 13.698401 193.87 14.398572 ^"S '5-'3394 196.45 15.813124 197.85 30 13.710034 193.89 14.410280 195.15 15.1 15182 196.48 15.824996 197.88 31 13.721668 '93-9' 14.421990 195.17 15.126971 196.50 15.836870 197.90 32 '3-73333 193-93 14.433700 195.19 15.138762 196.52 15.848744 197.92 33 1 3 -74494 '93-95 14.445412 195.21 i5.i5554 196.54 15.860620 '97-95 34 '3-756577 193-97 14.457126 195.23 15.162348 196.57 15.872498 197.97 35 13.768216 '93-99 14.468841 195.26 15.174142 196.59 15.884377 198.00 36 13.779856 194.01 14.480557 195.28 15.185938 196.61 15.896258 198.02 37 13.791498 194.03 14.492274 195-3 15.197736 196.64 15.908140 198.04 38 13.803140 194.05 14.503992 I95-3 2 15.209535 196.66 15.920023 198.07 39 13.814784 194.07 14.515712 '95-34 15.221335 196.68 15.931908 198.09 40 13.826429 194.09 14.527434 195-36 '5-233'37 196.70 '5-943794 198.12 41 13.838075 194.11 I 4-539 I 5 6 '95-39 15.244940 196.73 15.955682 198.14 42 13.849723 194.14 14.550880 195.41 15.256744 196.75 15.967571 198.17 43 13.861372 194.16 14.562605 '95-43 15.268550 196.77 15.979462 198.19 44 13.873022 194.18 '4-57433' 195-45 15.280357 196.80 '5-99'354 198.21 45 13.884673 194.20 14.586059 '95-47 15.292165 196.82 16.003248 198.24 46 I 3- 8 9 6 3 2 5 194.22 14.597788 195-5 i5-3 3975 196.84 16.015143 198.26 1 47 ' 3-97979 194.24 14.609519 '95-52 I5-3I5786 196.87 16.027039 198.29 | 48 13.919634 194.26 14.621250 195-54 I5-327599 196.89 16.038937 198.31 49 13.931290 194.28 14.632983 195.56 I5-3394I3 196.91 16.050836 198.34 50 51 13.942948 13.954606 194.30 '94-32 14.644718 14.656453 195.58 195.60 15.351228 15-363045 196.94 196.96 16.062737 16.074639 198.36 198.38 52 13.966266 '94-34 14.668190 '95-63 15.374863 196.98 16.086543 198.41 53 54 13.977927 13.989590 194.36 194.38 14.679929 14.691668 195.65 195.67 15.386683 15.398504 197.00 197.03 16.098449 16.110355 198.43 198.46 55 14.001254 194.41 14.703409 195.69 15.410326 197.05 16.122263 198.48 56 14.012919 '94-43 14.715151 195.71 15.422150 197.07 16.134173 198.51 57 14.024585 194.45 14.726895 195-74 '5-433975 197.10 16.146054 198.53 58 14.036252 194.47 14.738640 195.76 15.445802 197.12 16.157997 198.56 59 14.047921 194.49 14.750386 195.78 15.457630 197.14 16.16991 1 198.58 60 14.059591 194.51 14.762133 195.80 15.469459 197.17 16.181826 198.60 571 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 24 25 26 27 M. Diflf. I". M. Difif. 1". M. Diff. I". M. Diff. I". O' 16.18182,6 198.60 16.899499 2OO. 12 17.622747 201.70 18.351847 203.37 1 2 3 4 16.193743 16.205662. 16.217582 16.229503 198.63 198.65 198.68 198.70 16.911507 16.923516 16.935527 16.947539 200.14 200.17 200.19 2OO.22 17.634850 17.646954 17.659060 17.671168 201.73 201.76 201.78 201. 8l 18.364050 18.376255 18.388461 18.400669 203.40 203.42 203.45 j 203.48 5 6 16.241426 16.253350 198.73 198.75 16.959553 16.971568 2OO.24 200.27 17.683278 17.695389 201.84 201.87 18.412879 18.425090 203.51 203.54 7 16.265276 198.78 16.983585 200.30 17.707502 201.89 l8.4373 3 203.57 8 9 16.277204 16.289133 198.80 198.83 16.995604 17.007624 200.32 200.35 17.719616 17.731732 201.92 201.95 18.449518 18.461735 203.59 203.62 10 11 12 16.301063 16.312995 16.324928 198.85 198.88 198.90 17.019646 17.031669 17.043694 200.37 200.40 200.43 17.743850 17.755969 17.768090 201.97 202.00 202.03 18.473953 18.486173 18.498395 203.65 203.68 203.71 13 14 16.336863 16.348799 198.93 198.95 17.055720 17.067748 200.45 200.48 17.78021 3 17.792337 2O2.O6 202.08 18.510618 18.522843 203.74 203.77 15 16.360737 198.97 17.079777 200.50 17.804462 202.11 18.535070 203.80 16 16.372676 199.00 17.091808 200-53 17.816590 202.14 18.547299 203.82 17 16.384617 199.02 17.103841 200.56 17.828719 2O2.I7 18.559529 203.85 18 19 16.396559 16.408503 199.05 199.07 17.115875 17.127911 200.58 2OO.6l 17.840850 17.852982 202.19 2O2.22 18.571761 18.583995 203.88 203.91 20 16.420448 199.10 17.139948 200.64 17.865116 202.25 18.596230 203.94 21 16.432395 199.12 17.151987 2OO.66 17.877252 2O2.28 18.608467 203.97 22 16.444343 i99-!5 17.164028 200.69 17.889389 202.30 18.620706 204.00 23 16.456292 199.17 17.176070 200.71 17.901528 202.33 18.632947 204.03 24 16.468243 199.20 17.188114 200.74 17.913669 202.36 18.645190 204.05 25 16.480196 199.22 17.200159 200.77 17.925811 202.39 18.657434 204.08 26 16.492-151 199.25 17.212206 200.79 17-937955 202.41 18.669679 204.11 27 16.504107 199.27 17.224254 2OO.82 17.950101 202.44 18.681927 204.14 28 16.516064 199.30 17.236304 200.85 17.962248 202.47 18.694177 204.17 29 16.528022 J 99-33 17.248356 200.87 17-974397 202.50 18.706428 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 200.93 17.998700 202.55 l8.73 935 204.26 32 16.563908 199.40 17.284520 200.95 18.010854 202.58 18.743191 204.29 33 i 6 -575873 199.43 17.296578 200.98 18.023010 202.6l 18.755449 204.32 34 16.587839 199.45 17.308637 2OI.OO 18.035167 202.64 18.767709 204.35 35 36 16.599807 16.611776 199.48 199.50 17.320698 17.332761 201.03 2OI.O6 18.047326 18.059487 202.66 202.69 18.779971 18.792234 204.37 204.40 37 16.623747 '99-53 I7-344825 201.08 18.071649 202.72 18.804499 204.43 38 16.635719 '99-55 17.356891 201. II 18.083813 202.75 18.816767 204.46 39 16.647693 199.58 17.368959 201.14 18.095979 202.78 18.829036 204.49 40 16.659669 199.60 17.381028 201.16 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 199.73 17.441397 201.30 18.169008 202.94 18.902684 204.67 46 l6 -73 J 553 199.76 17.453476 201.32 18. 181186 202.97 18.914965 204.70 47 l6 -743539 199.78 17.465556 201-35 18.193365 203.00 18.927247 204.73 48 16.755527 199.81 17.477638 201.38 18.205546 203.03 18.939532 204.76 49 16.767516 199.83 17.489722 201.41 18.217728 203.06 18.951818 204.79 50 51 16.779507 16.791499 199.86 199.88 17.501807 17.513894 201.43 201.46 18.229912 18.242098 203.08 203.1 1 18.964106 18.976396 204.81 204.84 52 16.803493 199.91 17.525982 201.49 18.254286 203.14 18.988687 204.87 53 16.815488 199.94 17.538072 201.51 18.266475 203.17 19.000981 204.90 54 16.827485 199.96 17.550163 201.54 18.278666 203.20 19.013276 204.93 55 16.839484 199.99 17.562257 201.57 18.290859 203.23 19.025573 204.96 56 16.851484 200.01 17.574352 201.59 18.303053 203.25 19.037871 204.99 57 16.863485 2OO.O4 17.586448 201.62 18.315249 203.28 19.050172 205.02 58 16.875488 200.06 17.598546 201.65 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 200.12 17.622747 201.70 18.351847 203.37 19.087084 205.11 572 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 28 29 30 31 M. Diff. 1". M. Diff. 1". log M. Diff. 1". logM. Diff. I". O' 19.087084 205.11 19.828747 206.94 1.313 3849 44.08 1.329 0430 42.92 1 19.099391 205.14 19.841164 206.97 .313 6493 44.06 .329 3004 42.91 2 19.1 1 1701 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 3'4 44'9 44.00 .330 0723 42.85 5 19.148639 205.26 19.890852 207.09 1.314 7058 43.98 1.330 3293 42.83 6 19.160956 205.29 19.903279 207.13 .314 9696 43.96 33 5862 42.81 7 19.173274 205.32 19.915707 207.16 3'5 2333 43-94 .330 8431 42.80 8 19.185594 205-35 19.928137 207.19 .315 4969 43-92 33' 0998 42.78 9 19.197916 205.38 19.940569 207.22 .315 7604 43.90 33' 3564 42.76 10 19.210240 205,41 19.953003 207.25 1.316 0237 43-88 1.331 6129 42.74 ' 11 19.222566 205.44 I9-965439 207.28 .316 2869 43.86 .331 8693 42.72 12 19.234893 205.47 19.977877 207.31 .316 5500 43.84 332 1255 42.70 13 19.247222 205.50 19.990317 207.34 .316 8130 43.82 .332 3817 42.69 14 '9-259553 205-53 20.002759 207.38 .317 0759 43.80 332 6378 42.67 15 16 19.271885 19.284220 205.56 205.59 20.015202 20.027647 207.41 207.44 1.317 3386 .317 6013 43-78 43-76 1-332 8937 333 '496 42.65 42.63 17 19.296556 205.62 20.040095 207.47 .317 8638 43-74 333 4053 42.61 18 19.308894 205.65 20.052544 207.50 .318 1262 43-72 333 6609 42-59 19 19.321234 205.68 20.064995 207.53 .318 3885 43-70 333 9' 6 4 42.58 20 I9-333576 205.71 20.077448 2P7.57 1.318 6506 43-68 1.334 1718 42.56 21 19.345920 205.74 20.089903 207.60 .318 9 !2 7 43.67 .334 4271 42-54 22 19.358265 205.77 20.102360 207.63 .319 1746 43.65 .334 6823 42.52 23 19.370612 205.80 20.114818 207.66 .319 4364 43-63 334 9374 42.50 24 19.382961 205.83 20.127279 207.69 .319 6981 43.61 335 1924 42.49 25 19.395312 205.86 20.139741 207.72 '3'9 9597 43-59 '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 9567 42.43 28 '9-432375 205.95 20.177140 207.82 .320 7438 43-53 .336 2112 42.41 29 '9-444734 205.98 20.189610 207.85 .321 0049 43-51 .336 4656 42.40 30 19.457094 206.01 20.202082 207.88 I.32I 2659 43-49 1.336 7199 42.38 31 19.469455 206.04 20.214556 207.91 .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 19.494184 206. 1 1 20.239510 207.98 .322 0482 43-43 337 4823 42.33 34 19.506551 206.14 20.251989 208.01 .322 3087 43-41 337 73 6 2 42.31 35 19.518921 206.17 20.264471 208.04 1.322 5692 43.40 '337 99o 42.29 36 19.531292 206.20 20.276954 208.07 .322 8295 338 2437 42.27 37 19.543664 206.23 20.289440 208. II .323 0897 43-36 .338 4972 42.25 38 19.556039 206.26 20.301927 208.14 323 3498 43-34 338 757 42.24 39 19.568415 206.29 20.314416 208.17 .323 6097 43-32 339 0041 42.22 40 19.580794 206.32 20.326907 208.20 1.323 8696 43-30 '339 2573 42.20 41 '9-593 I 74 206.35 20.339400 208.24 .324 1294 43.28 339 5'5 42.18 42 19.605556 206.38 20.351895 208.27 .324 3890 43.26 339 7635 42.17 43 I9.6i7939 206.41 20.364392 208.30 .324 6485 43-24 .340 0165 42.15 44 19.630325 206.44 20.376891 208.33 .324 9079 43.22 .340 2693 42.13 45 19.642713 206.47 20.389392 208.36 1.325 1672 43-21 1.340 5221 42.11 46 19.655102 206.50 20.401895 208.39 325 4263 43.19 .340 7747 42.10 47 19.667493 206.53 20.414399 208.43 .325 6854 43-'7 .341 0272 42.08 48 19.679886 206.57 20.426906 208.46 325 9443 43-'5 .341 2796 42.06 49 19.692281 206.60 20.439415 208.49 .326 2032 43-'3 34' 5319 42.04 50 19.704678 206.63 20.451925 208.52 1.326 4619 43-n 1.341 7841 42.03 51 19.717076 206.66 20.464437 208.56 .326 7205 .43.09 .342 0362 42.01 52 19.729477 206.69 20.476952 208.59 .326 9790 43.07 .342 2882 41.99 53 19.741879 206.72 20.489468 208.62 327 2374 43.05 .342 5401 4'-97 54 19.754283 206.75 20.501986 208.65 327 4957 43-04 .342 7919 41.96 55 19.766689 206.78 20.514506 208.69 1.327 7538 43.02 1.343 0436 41.94 56 19.779097 206.81 20.527029 208.72 .328 0119 43.00 343 2952 41.92 57 19.791507 206.84 20-539553 208.75 .328 2698 42.98 343 54 6 7 41.90 58 19.803919 206.88 20.552079 208.78 .328 5276 42.96 343 798o 41.89 59 19.816332 206.91 20.564607 208.82 .328 7853 42.94 344 493 41.87 60 19.828747 206.94 20.577137 208.85 1.329 0430 42.92 1.344 3005 41.85 573 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 32 33 34 35 log M. Diff. 1". logM. Diff. 1". log M. Diff. 1". log M. Diff. I". O' 1 1.344 3005 344 55 : 5 41-85 41.84 1.359 1859 359 431 40.86 40.84 1.373 7251 .373 9646 39-93 39-9 1 1.387 9418 .388 1761 39.06 39-5 2 .344 8025 41.82 .359 6760 40.82 374 2041 39-90 .388 4104 39-4 3 345 534 41.80 359 9209 40.81 374 4434 39.88 .388 6446 39-02 4 345 3041 41.78 .360 1657 40.79 374 6827 39-87 .388 8787 39.01 5 6 1-345 554 8 345 8 53 41-77 4 r -75 1.360 4104 .360 6550 40.78 40.76 1.374 9218 375 1609 39-85 39-84 1.389 1127 389 3466 38-99 38.98 7 .346 0558 4 J -73 .360 8995 40.74 375 3999 39.82 389 5804 38.97 8 .346 3061 41.72 .361 1439 4-73 .375 6388 39.81 .389 8142 38.95 9 .346 5564 41.70 .361 3883 40.71 375 8776 39-79 39 479 38.94 10 1.346 8065 41.68 1.361 6325 40.70 1.376 1164 39-78 1.390 2815 38.93 11 347 5 6 5 41.66 .361 8766 40.68 376 355 39-77 39 5 T 5 38.91 12 347 3 6 5 41.65 .362 1207 40.66 376 5935 39-75 .390 7484 38.90 13 347 55 6 3 41.63 .362 3646 40.65 .376 8320 39-74 39 9817 38.88 14 .347 8060 41.61 .362 6084 40.63 377 73 39-72 .391 2150 38.87 15 1.348 0557 41.60 1.362 8522 40.62 1.377 3086 39-71 1.391 4482 38.86 16 .348 3052 41.58 3 6 3 959 40.60 377 5468 39- 6 9 .391 6813 38-84 17 34 8 5546 41.56 3 6 3 3394 40.59 377 7849 39-68 39 1 9*43 38.83 18 .348 8040 4'-55 .363 5829 40.57 378 0230 39-66 .392 1472 38.82 19 349 532 41-53 .363 8263 40.56 .378 2609 39-65 .392 3801 38.80 20 1.349 3023 41.51 1.364 0696 40.54 1.378 4987 39- 6 4 1.392 6128 38.79 21 349 55*3 41.50 .364 3128 40.52 .378 7365 39.62 .392 '8455 38.77 22 349 8003 41.48 3 6 4 5559 40.51 .378 9742 39.61 393 78i 38-76 23 35 049 1 41.46 .364 7989 40.49 379 2II 7 39-59 393 3 I0 7 38.75 24 35 2 978 41.45 .365 0418 40.48 379 4492 39-58 393 543i 38.73 25 1.350 5464 41.43 1.365 2846 40.46 1.379 6866 39-56 J-393 7755 38.72 26 .350 7950 41.41 3 6 5 5*73 40.45 379 9 2 4o 39-55 .394 0078 38-71 27 351 0434 41.40 3 6 5 7 6 99 40.43 .380 1612 39-53 394 240 38.69 28 .351 2917 41.38 .366 0125 40.41 .380 3983 39-52 394 472i 38.68 29 351 5399 41.36 .366 2549 40.40 .380 6354 39-5 394 74! 38-67 30 1.351 7880 4M5 1.366 4973 40.38 1.380 8724 39-49 1.394 9361 38.65 31 .352 0361 41-33 .366 7395 4-37 .381 1093 39-47 .395 1680 38.64 32 .352 2840 4 I -3i .366 9817 40.35 .381 3461 39-46 395 3998 38.63 j 33 .352 5318 41.30 .367 2238 4-34 .381 5828 39-45 -395 6315 38-61 34 .352 7795 41.28 .367 4657 40.32 .381 8194 39-43 -395 8631 38.60 35 1.353 0272 41.26 1.367 7076 40.31 1.382 0559 39-42 1.396 0947 38.59 36 353 2747 41.25 3 6 7 9494 40.29 .382 2924 39-4 .396 3262 38.57 37 353 52*1 41.23 .368 1911 40.28 .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 oi 6 7 41.20 .368 6742 40.25 .383 0013 39-36 .397 0201 38.53 40 1.354 2 6 3 8 41.18 1.368 9157 40.23 1.383.2374 39-35 1.397 2513 38.52 41 354 5i8 41.16 .369 1570 40.21 383 4734 39-33 397 48^3 38-51 42 354 7578 41.15 3 6 9 3983 40.20 383 7093 39-32 397 7133 38.49 43 355 4 6 41.13 .369 6394 40.18 383 9452 39-3 397 9442 38.48 44 355 2513 41.11 .369 8805 40.17 .384 1809 39-29 398 I75 1 38.47 45 1.355 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 48 .356 2373 41.05 .370 8438 40.11 385 1232 39-23 399 97 6 38.41 49 .356 4836 41.03 .371 0844 40.09 385 3585 39-22 -399 3281 38.40 50 1.356 7297 41.02 1.371 3249 40.08 1-385 5938 39.20 1-399 55 8 4 38.39 51 35 6 9758 41.00 371 5 6 54 40.06 .385 8290 39.19 399 7887 38.37 52 .357 2217 40.98 .371 8057 40.05 .386 0641 39.18 .400 0189 38-36 53 357 4 6 76 40.97 .372 0459 40.03 .386 2991 39.16 .400 2491 38.35 54 357 7134 40.95 .372 2861 40.02 .386 5340 39- I 5 .400 4791 38.33 55 1.357 9590 40.94 1.372 5261 40.00 1.386 7689 39- J 3 1.400 7091 38-32 56 .358 2046 40.92 .372 7661 39-99 .387 0036 39.12 .400 9390 38-31 57 .358 4501 40.90 .373 0060 39-97 .387 2383 39- 11 .401 1688 38.30 58 358 6954 40.89 373 2458 39-9 6 387 4729 39-9 .401 3985 38.28 59 358 947 40.87 373 4855 39-94 .387 7074 39.08 .401 6282 38-27 60 1.359 l8 59 40.86 1.373 7251 39-93 1.387 94I 8 39.06 1.401 8578 38.26 574 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 36 37 38 39 log M. Diff. 1". logM. Diff. I". log M. Diff. 1". logM. Diff. 1". 1.401 8578 38.26 1.415 4930 37-5 1.428 8662 36.80 I -44 I 9943 36.14 1 .402 0873 38.24 .415 7180 37-49 .429 0869 36.79 .442 21 I I 36.13 2 .402 3167 38.23 .415 9429 37-47 .429 3076 36.78 .442 4279 36.12 3 .402 5460 38.22 .416 1678 37.46 429 5283 3 6 -77 442 6446 36.11 4 402 7753 38.20 .416 3925 37.45 .429 7488 36.75 .442 86l2 36.10 5 1.403 0045 38.19 1.416 6172 37-44 1.429 9693 36-74 1-443 0778 36.09 6 .403 2336 38.18 .416 8419 37-43 .430 1897 36.73 .443 2943 36.08 7 .403 4626 38.17 .417 0664 37.41 .430 4101 36.72 .443 5107 36.07 8 .403 6916 38.15 .417 2909 37.40 .430 6304 36.71 443 7271 36.06 9 .403 9205 38.14 .417 5153 37-39 .430 8506 36.70 443 9434 36.05 10 1.404 1493 38.13 1.417 7796 37.38 1.431 0708 36-69 1.444 1597 36.04 11 .404 3780 38.12 .417 9639 37-37 .431 2909 36.68 .444 3758 36.03 12 .404 6067 38.10 .418 1881 37.36 .431 5109 36.66 .444 5920 36.02 13 .404 8352 38.09 .418 4122 37-35 .431 7308 36.65 .444 8080 36.00 14 .405 0637 38.08 .418 6362 37.33 431 957 36-64 .445 0240 35-99 15 1.405 2921 38.06 1.418 8602 37-3* 1.432 1705 36-63 1.445 2400 35-98 16 .405 5205 38.05 .419 0841 37-3 1 432 393 36.62 445 4558 35-97 17 .405 7488 38.03 .419 3079 37-3 .432 6100 36.61 .445 6716 35-96 18 .405 9769 38.02 419 5317 37-29 .432 8296 36.60 -445 8874 35-95 19 .406 2051 38.01 419 7554 37-27 433 0491 36.59 .446 1031 35-94 2O 1.406 4331 38.00 .419 9790 37.26 1.433 2686 3 6 -57 1.446 3187 35 93 21 .406 66 1 1 37-99 .420 2026 37.25 433 4881 36.56 446 5343 35 92 22 .406 8889 37-97 .420 4260 37.24 -433 774 36.55 .446 7498 35-9 1 23 .407 1168 37.96 .420 6494 37-23 433 9267 36.54 .446 9652 35-90 24 47 3445 37-95 .420 8728 37.22 434 H59 36.53 .447 1806 35.89 25 1.407 5721 37-94 .421 0960 37-20 .434 3651 36.52 M47 3959 35-88 26 .407 7997 37-92 .421 3192 37.19 434 5842 36-51 .447 6112 35.87 27 .408 0272 37.91 .421 5423 37-18 4^4 8032 36.50 447 8263 35.86 28 .408 2547 37-90 .421 7654 37-17 435 0221 36-49 .448 0415 35.85 29 .408 4820 37.89 .421 9884 37.16 435 2410 36.48 .448 2565 35-84 30 .408 7093 37-87 .422 2113 37.I5 435 4598 36.47 1.448 4715 35.83 31 .408 9365 37-86 .422 4341 37.13 435 6786 36.46 .448 6865 35-82 32 .409 1636 37-85 .422 6569 37.12 435 8973 36.44 .448 9014 35-8i 33 .409 3907 37.84 .422 8796 37.11 .436 1159 36.43 .449 1162 35.80 34 .409 6177 37.82 .423 1 022 37.10 436 3345 36.42 449 339 35-79 35 .409 8446 37.8i .423 3248 37-09 436 553 36.41 449 5456 35.78 36 .410 0714 37.80 4 2 3 5473 37.08 436 77H 36.40 449 7603 35-77 37 .410 2981 37.78 .423 7697 37-06 436 9898 36.39 449 9749 35-76 38 .410 5248 37-77 .423 9920 37-05 .437 2081 36.38 .450 1894 35-75 39 .410 7514 37-76 .424 2143 37.04 437 4263 36.37 .450 4038 35-74 4O 41 .410 9780 .411 2044 37-75 37-74 .424 4365 .424 6586 37-03 37.02 437 6445 .437 8626 36.36 36.35 .450 6182 .450 8325 35-73 35-72 42 .411 4308 37-72 - .424 8807 37.01 .438 0806 3 6 .34 .451 0468 35-71 43 .411 6571 37.71 .425 1027 36.99 .438 2986 36.32 .451 2610 35.70 44 .411 8833 37-7 .425 3246 36.98 .438 5165 36.31 451 4752 35.69 45 .412 1095 37.69 .425 5465 36.97 43 8 7344 36-30 .451 6893 35-68 46 .412 3356 37.68 .425 7683 36-96 .438 9522 36.29 451 933 35-67 47 .412 5616 37.66 .425 9900 36.95 439 1699 36.28 .452 1173 35.66 48 .412 7875 37-65 .426 2117 36.94 439 3875 36.27 -452 3312 35-65 49 413 OI 34 37-64 426 4333 36.92 439 6051 36.26 452 5450 35.64 5O .413 2392 37.63 .426 6548 36.91 .439 8226 36.25 452 7588 35-63 51 .413 4649 37.61 .426 8762 36.9 .440 0401 36.24 .452 9725 35.62 52 .413 6905 37.60 .427 0976 36.89 .440 2575 36.23 .453 1862 35-6i 53 .413 9161 37-59 .427 3189 36.88 .440 4748 36.22 453 3998 35-6o 54 .414 1416 37-58 .427 5402 36-87 .440 6921 36.20 453 6134 35-59 55 .414 3670 37-56 .427 7613 36.86 44 9093 36.19 453 8269 35-58 56 .414 5924 37-55 .427 9824 36.85 .441 1264 36.18 454 43 35-57 57 .414 8176 37-54 .428 2035 36.83 .441 3436 36.17 454 2537 35-56 58 .415 0429 37-53 .428 4244 36.82 .441 5605 36.16 .454 4670 35-55 59 .415 2680 37-51 .428 6453 36.81 .441 7774 36.15 .454 6802 35-54 60 .415 4930 37-5 .428 8662 36.80 .441 9943 36.14 454 8934 35-53 575 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". log M. Diff. 1". log M. Diff. 1". 0' 1.454 8934 35-53 1.467 5782 34-95 1.480 0627 34-41 I -49 a 3597 33-91 1 .455 1065 35-52 467 7879 34-94 .480 2691 34-4 .492 5631 33-90 2 455 3*96 35-5 1 467 9976 34-93 .480 4755 34-40 .492 7665 33-89 3 45.1 5326 35-5 .468 2071 34-92 .480 6819 34-39 .492 9698 33-88 4 455 745 6 35-49 .468 4166 34- 9 i .480 8882 34-38 493 *73 Z 33-87 5 *-455 95 8 5 35-4 8 1.468 6261 34-9 1.481 0944 34-37 x -493 3764 33-87 6 .456 1713 35-47 -468 8355 34-9 .481 3006 34-3 6 493 5796 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 0220 35-43 1.469 6725 34-86 1.482 1249 34-33 1.494 39 l8 33-83 11 .457 2346 35-42 .469 8817 34-85 .482 3308 34-32 494 5948 33-82 12 457 4470 35-4i .470 0907 34-84 .482 5367 34-3 1 494 7977 33-8i 13 457 6595 35-40 .470 2998 34-83 .482 7425 34-3 .495 0005 33.80 14 457 8718 35-39 .470 5087 34.82 .482 9483 34- 2 9 495 2033 33-79 15 1.458 0841 35-38 1.470 7176 34.81 1.483 1540 34.28 1.495 46i 33-79 16 .458 2964 35-37 .470 9265 34.80 483 3597 34.28 .495 6088 33-78 17 .458 5086 35-3 6 47 1 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 34-78 483 9764 34-35 .496 2166 33-75 20 1.459 H4 8 35-33 1.471 7613 34-77 1.484 1819 34-24 1.496 4191 33-75 21 459 35 6 7 35-3* .471 9699 34-76 484 3873 34-23 .496 6216 33-74 22 459 5686 35-3 1 472 1784 34-75 .484 5927 34.22 .496 8240 33-73 23 459 7 8 5 35.30 472 3869 34-74 .484 7980 34-22 497 0264 33-72 24 459 99 i2 35-29 472 5953 34-73 485 o33 34-21 .497 2287 33-71 25 1.460 2040 35-28 1.472 8037 34-73 1.485 2085 34.20 1.497 4310 33-71 26 .460 4156 35.27 473 0120 34-72 485 4*37 34-19 497 6332 33-70 27 .460 6272 35.26 473 220 3 34-71 .485 6188 34.18 497 8354 33-69 28 .460 8388 35-25 473 4285 34-70 .485 8239 34-17 498 0376 33-68 29 .461 0503 35-24 473 6366 34-69 .486 0289 34.16 .498 2396 33-68 30 1.461 2^17 35-23 1.473 8447 34-68 1.486 2338 34.16 1.498 4417 33-67 31 .461 4731 35-23 474 5 2 7 34-67 .486 4388 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 34-65 .486 8484 34-13 499 475 33-65 34 .462 1069 35-2Q 474 6765 34-64 .487 0532 34.12 499 2494 33-64 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-i8 475 0921 34.62 .487 4626 34- IJ 499 6530 33.62 37 .462 7401 35-17 .475 2998 34.61 .487 6672 34.10 499 8547 33-62 38 .462 9511 35-i6 -475 575 34.61 .487 8718 34.09 .500 0563 33.61 39 .463 1620 35-15 475 7i5i 34.60 .488 0763 34.08 .500 2580 33.60 40 1.463 3729 35-H 1.475 9"7 34-59 1.488 2807 34-07 1.500 4595 33-59 41 .463 5837 35-13 .476 1302 34-58 .488 4852 34.07 .500 6611 33.58 42 463 7944 35- 12 476 3376 34-57 .488 6895 34.06 .500 8625 33-58 43 .464 0051 35-" 476 545 34-56 488 8939 34-05 .501 0640 33-57 44 .464 2158 35-i .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 35-07 .477 3741 34-53 .489 7106 34-02 .501 8693 33-54 48 465 0577 35.06 .477 5812 34.52 .489 9147 34.01 .502 0705 33-53 49 .465 2681 35-5 477 7883 34-51 .490 1187 34.00 .502 2716 33-5 2 50 1.465 4784 35-4 '477 9953 34.50 1.490 3227 33-99 1.502 4727 33-5 1 51 .465 6886 35.04 .478 2023 34-49 .490 5266 33-98 .502 6738 33-5 1 52 .465 8988 35.03 .478 4092 34-48 49 735 33-97 .502 8748 33-5 53 .466 1090 35-02 .478 6161 34-47 .490 9343 33-96 53 758 33-49 54 .466 3190 35-01 .478 8229 34.46 .491 1381 33-95 .503 2767 33-48 . 55 1.466 5290 35,00 1.479 0297 34-4 6 1.491 3418 33-95 1.503 4776 33-48 56 .466 7390 34-99 479 2364 34-45 49 1 5455 33-94 .503 6784 33-47 57 58 .466 9489 .467 1587 34-98 34-97 .479 4430 .479 6496 34-44 34-43 .491 7491 .491 9527 33-93 33-92 .503 8792 .504 0800 33-46 33-45 59 .467 3685 34.96 .479 8562 34.42 .492 1562 33.91 .504 2807 33-44 60 1.467 5782 34-95 1.480 0627 34-41 '492 3597 33-9 1 1.504 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 log M. Diff. 1". logM. Diff. I". logM. Diff. 1". logM. Diff. 1". O' 1.504 4813 33-44 1.516 4390 33-oo 1.528 2435 32.59 1.539 9048 32.20 1 .504 6819 33-43 .516 6370 32-99 .528 439 32.58 .540 0980 32.20 2 .504 8825 33-42 .516 8349 32.98 .528 6344 32.57 .540 2912 32.19 3 .505 0830 33-42 .517 0328 32.98 .528 8299 32.57 54 4843 32.18 4 .505 2835 33-41 .517 2306 32-97 .529 0252 32.56 .540 6774 32.18 5 1.505 4839 33-40 1.517 4284 32.96 1.529 2206 32-55 1.540 8705 32.17 6 .505 6843 33-39 .517 6262 32.96 529 4159 32.55 54 1 6 35 32.17 7 .505 8846 33-39 .517 8239 32.95 .529 6112 32.54 .541 2564 32.16 8 .506 0849 33-38 .518 0216 32-94 .529 8064 32.53 .541 4494 32-15 9 .506 2852 33-37 .518 2192 32.93 .530 ooio 32.53 .541 6423 32.15 10 1.506 4854 33-36 1.518 4168 32.93 1.530 1967 32.52 I -54 I 8352 32-14 11 .506 6855 .518 6143 32.92 .530 3918 32.51 .542 0280 32.14 12 .506 8856 33-35 .518 8118 32.91 53 5869 32.51 .542 2208 32-13 13 .507 0857 33-34 .519 0093 32.91 .530 7819 32-50 .542 4135 32.12 14 .507 2857 33-33 .519 2067 32.90 53 9769 32.49 .542 6063 32.H 15 I -57 4857 33-33 1.519 4041 32.89 1.531 1719 32.49 1.542 7989 32.11 16 .507 6856 33-32 .519 6014 32.89 .531 3668 32-48 .542 9916 32.10 17 .507 8855 33-3i .519 7987 32.88 .531 5616 32.48 32.10 18 .508 0853 33-30 .519 9960 32.87 53 1 7565 32.47 543 37 6 8 32.09 19 .508 2851 33-29 .520 1932 32.86 53 1 95*3 32-46 543 5693 32.09 20 1.508 4849 33-29 1.520 3904 32.86 1.532 1460 32.46 1.543 7618 32.08 21 .508 6846 33-28 520 5875 32-85 532 347 32.45 543 9543 32.08 22 .508 8843 33-27 .520 7846 32.84 532 5354 32.44 544 ^67 32.07 23 .509 0839 33-27 .520 9816 32-84 532 73 32.44 544 339 1 32.06 24 .509 2835 33-26 .521 1786 32-83 .532 9246 32.43 544 53 J 5 32.06 25 1.509 4830 33-25 1.521 3756 32.82 1.533 IJ 92 32-43 !-544 7238 32.05 26 .509 6825 33-24 521 5725 32.82 533 3*37 32.42 .544 9161 32.04 27\ .509 8819 33-24 .521 7694 32.81 533 5082 32.42 .545 1083 32.04 28; 29 .510 0813 .510 2807 33-23 33-22 .521 9662 .522 1630 32.80 32.80 533 7027 533 8971 32.41 32.4 545 3005 545 4927 32.03 32.03 30 .510 4800 33-21 1.522 3598 32.79 1.534 0914 32.39 545 6849 32.02 31 .510 6792 33-21 522 5565 32-78 534 2858 32.39 545 8770 32.02 32 .510 8785 33-20 .522 7531 32.78 534 4801 32-38 .546 0690 32.01 33 .511 0776 33-19 .522 9498 32-78 534 6743 32.37 .546 2611 32.00 34 .511 2768 33-i8 .523 1464 32.77 534 8685 32.37 .546 4531 32.00 35 1.511 4759 33.18 .523 3429 32-76 535 0627 32-36 .546 6450 31.99 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 6450 32.34 547 2207 3 J -97 39 .512 2718 33-15 .524 1287 32.73 535 8390 32.33 547 4 I2 5 31.97 40 .512 4707 33-H .524 3251 32.72 S3 6 0330 32.33 .547 6043 31.96 41 .512 6695 33-13 .524 5214 32.71 .536 2270 32.32 547 796i 31.96 42 .512 8683 33-13 524 7176 32.71 .536 4209 32-32 547 9878 3 J -95 43 .513 0670 33-12 524 9138 32.70 536 6148 32-31 .548 1795 31-94 44 .513 2657 33-" - .525 1 100 32.70 .536 8086 32.3 .548 3711 3L94 45 .513 4644 33. H 525 3062 32.69 .537 0024 32.30 .548 5627 3*-93 46 .513 6630 33.10 525 5023 32.68 537 1962 32.29 548 7543 31.93 47 .513 8615 33-9 525 6983 32.67 537 3899 32.28 548 9458 31.92 48 .514 0601 33.08 .525 8944 32.67 537 5836 32.28 549 1373 49 .514 2586 33.07 .526 0903 32.66 537 7772 32.27 549 3288 3 I -9 I 50 51 .514 4570 5*4 6 554 33-07 33.06 .526 2863 .526 4822 32.65 32.64 537 9708 .538 1644 32.26 32.26 .549 5202 .549 7116 31.90 31.90 52 5*4 8 537 33-05 .526 6780 32.64 538 3579 32.25 .549 9030 31.89 53 .515 0520 33.05 .526 8739 32-63 538 55'4 32-25 .550 0943 31.88 54 5!5 2503 33-04 .527 0696 32.62 538 7449 32-24 .550 2856 31.88 55 5i5 4485 33-04 527 2654 32.62 538 9383 32.23 550 4769 31.87 56 .515 6467 33-3 .527 4611 32.61 539 !3*7 32.23 .550 6681 3 I - 8 7 57 5*5 8449 33.02 .527 6567 32.61 539 3250 32.22 55 8593 31.86 58 .516 0430 33- 01 .527 8524 32.60 539 5183 32.21 551 0504 31.86 59 .516 2410 33.01 .528 0479 32.60 539 7 IJ 6 32,21 .551 2416 31.85 60 .516 4390 33.00 .528 2435 32-59 539 9048 32.20 .551 4326 3<-*5 577 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. .48 49 50 51 logM. Diff. 1". log M. Diff. 1". log M. Diff. 1". logM. Diff. 1". 0' 1.551 4326 3I-85 1.562 8360 S 1 ^ 1 1.574 1234 31.20 I-585 3031 30.91 1 .551 6237 31.84 .563 0250 31.51 574 3 106 31.20 .585 4886 30.91 2 .551 8147 3I-83 563 2140 31-50 -574 4977 31.19 .585 6740 30.90 3 .552 0057 31-83 .563 4030 31.50 574 6849 3 I - I 9 .585 8594 30.90 4 .552 1966 31.82 563 5920 3M9 574 8720 31.18 .586 0448 30-89 5 1.552 3876 31.82 1.563 7809 31.48 1-575 59 31.18 1.586 2302 30.89' 6 -552 5784 31.81 .563 9698 31.48 .575 2461 31.17 .586 4155 30-89 7 .552 7693 31.80 .564 1586 575 433 1 31.17 .586 6008 30.88 8 9 .552 9601 553 '508 31.80 3'-79 -564 3475 564 53 6 3 3M7 31.46 .575 6201 .575 8070 31.16 31.16 .586 7859 .586 9713 30.87 30.87 10 J -553 34 l6 3 J -79 1.564 7250 31.46 1-575 9939 31.15 1.587 1565 30.87 11 553 5323 3I-78 .564 9138 3i-45 .576 1808 31.15 587 3417 30.86 12 553 723 3I-78 .565 1025 3i-45 .576 3677 31.14 .587 5268 30.86 13 553 9*36 3 J -77 565 2911 576 5546 31.14 .587 7120 30.85 14 554 1042 31.76 565 4798 31.44 .576 7414 3i-i3 .587 8971 30.85 15 1.554 2948 31.76 1.565 6684 3i-43 1.576 9281 31.13 1.588 0821 30.84 16 554 4853 565 8569 3i-43 577 "49 31.12 .588 2672 30-84 17 554 6758 31.75 566 0455 31.42 577 3 l6 31.12 .588 4522 30-83 18 554 8663 3 J -74 .566 2340 31.41 577 4883 31.11 588 6372 30.8 3 19 555 0567 3^-74 .566 4225 3 I -4 I 577 6749 31.11 .588 8222 30-83 20 1.555 2472 31-73 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 6279 31.72 .566 9877 3!-39 .578 2347 31.09 589 3769 30.81 23 555 8182 .567 1761 3 r -39 .578 4213 31.09 .589 5618 30.8l 24 .556 0084 3 J -7i 567 3644 3I-38 .578 6078 31.08 5 8 9 7466 30.80 25 1.556 1987 31.70 1.567 5527 3I-38 1.578 7942 31.08 1.589 9314 30.80 26 .556 3888 31.70 .567 7409 3 J -37 .578 9807 31.07 .590 1162 30.79 27 556 579 31.69 567 929 1 .579 1671 31.07 590 3009 30-79 28 .556 7691 31.68 .568 1173 31.36 -579 3535 31.06 59 4857 30-78 29 556 9592 31.68 568 355 31.36 -579 5399 31.06 .590 6704 30.78 30 1-557 H93 31.67 1.568 4936 31-35 1.579 7262 31.06 1.590 8550 30.78 31 557 3393 31.67 .568 6817 3'-35 -579 9 I2 5 31.05 .591 0397 30.77 32 557 5293 31.66 .568 8698 3 J -34 .580 0988 31.04 .591 2243 30-77 33 557 7193 31.66 569 579 31.34 .580 2851 31.04 .591 4089 30.76 34 557 9 92 3 I>6 5 569 2459 31.33 580 4713 3 I -3 59 1 5935 30.76 35 1.558 0991 3 x - 6 5 1-569 4338 3!-33 1.580 6575 31.03 1.591 7780 3-75 36 .558 2890 31.64 .569 6218 3 I -3 2 .580 8436 3 I -3 .591 9625 30.75 37 .558 4788 31.64 .569 8097 .581 0298 31.02 .592 1470 3-75 38 .558 6686' 31.63 .569 9976 Si-3 1 .581 2159 31.02 592 33 J 5 30.74 39 558 8584 31.62 .570 1854 31.30 .581 4020 31.01 592 5'59 30-74 40 1.559 0482 31.62 1-570 3733 31.30 1.581 5880 31.01 1.592 7003 30-73 41 559 2379 31.61 .570 5611 31.29 .581 7740 31.00 .592 8847 3-73 42 559 4275 31.61 .570 7488 31.29 .581 9600 31.00 .593 0690 30.72 43 559 6172 31.60 57 93 6 6 31.28 .582 1460 30.99 593 2534 30.72 44 .559 8068 31.60 .571 1243 31.28 .582 3319 30-99 593 4377 30.72 45 r -559 9963 3'-59 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 30.98 .593 8062 30.71 47 56o 3754 31-58 .571 6872 31.27 .582 8896 30-97 593 994 30.70 48 .560 5648 .571 8748 31.26 583 754 30.97 .594 1746 30.70 49 560 7543 31-57 .572 0623 31.26 .583 2612 30.96 594 3588 30.69 50 1.560 9437 3 J -5 6 1.572 2499 31-25 1.583 4470 30.96 1.594 5429 30.69 51 .561 1331 31-56 572 4373 31-25 .583 6327 30-95 594 7270 30.68 52 .561 3224 3'-55 .572 6248 31.24 .583 8184 30-95 594 9 111 30.68 53 .561 5117 3'-55 .572 8123 31.24 .584 0041 30-94 595 0952 30.68 54 .561 7010 3*-54 .572 9997 31.23 .584 1898 3-94 595 2792 30.67 55 1.561 8902 31-54 1.573 J 870 31-23 I -5 8 4 3754 30.94 1-595 4 6 33 30.67 56 57 .562 0794 .562 2686 31-53 3'-53 573 3743 573 5616 31.22 31.22 .584 5610 .584 7466 30-93 30-93 -595 6473 595 8312 30.66 30.66 58 .562 4578 31-52 573 7489 31.21 .584 9321 30.92 .596 0151 30.65 59 .562 6469 S'-S* 573 93 6 2 31.21 .585 1176 30.92 596 199 30.65 60 1.562 8360 31.51 1.574 1234 31.20 1.585 3031 30.91 1.596 3829 30.65 578 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 52 53 54 55 logM. Diff. V. logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 1.596 3829 30.65 1.607 3703 30.40 1.618 2724 30.17 1.629 959 29.96 1 .596 5668 30.64 .607 5527 3-39 .618 4534 30.17 .629 2757 29.96 2 .596 7506 30.64 .607 7350 30-39 .6:8 6344 30.16 .629 4554 29.96 3 596 9344 30.63 .607 9174 30-39 .618 8153 30.16 .629 6351 29-95 4 .597 1182 30.63 .608 0997 30.38 .618 9963 30.16 .629 8148 29.95 5 1.597 3020 30.62 1. 608 2820 30^38 1.619 *77 2 30-I5 1.629 9945 29.95 6 597 4^57 30.62 .608 4642 30.38 .619 3581 3 .I5 .630 1742 29.94 7 597 6694 30.62 .608 6465 3-37 .619 5390 3 -I5 .630 3538 29.94 8 597 8531 30.61 .608 8287 30-37 .619 7199 30-I4 6 3 5335 29.94 9 .598 0368 30.61 .609 0109 3-3 6 .619 9007 30.14 .630 7131 29-93 10 1.598 2204 30.60 1.609 1931 3-3 6 1.620 0816 30.14 1.630 8927 29-93 11 .598 4040 30.60 -609 3752 3-3 6 .620 2623 30-13 .631 0722 29.93 12 .598 5876 3-59 6 9 5573 30-35 .620 4431 30.13 .631 2518 29.92 13 .598 7711 3-59 .609 7394 30-35 .620 6239 30.12 631 43*3 29.92 14 598 9547 30.59 .609 9215 30-34 .620 8046 30.12 .631 6108 29.92 15 1.599 '382 30.58 1.610 1036 30-34 1.620 9853 30.12 1.631 7903 29.91 16 599 3217 30.58 .610 2856 30-34 .621 1660 30.11 .631 9698 29.91 17 599 55i 3-57 .610 4676 30.33 .621 3467 30.11 .632 1492 29.91 18 .599 6885 3-57 .610 6496 3-33 .621 5274 30.11 .632 3286 29.90 19 599 8719 3-57 .610 8315 30.32 .621 7080 30.10 .632 5081 29.90 20 i. 600 0553 30.56 i. on 0135 30.32 1.621 8886 30.10 1.632 6875 29.90 21 .600 2387 30.56 .611 1954 30-32 .622 0692 30.10 .632 8668 29.89 22 .600 4220 3-55 .611 3773 30-3 1 .622 2497 30.09 .633 0462 29.89 23 .600 6053 30-55 .611 5591 30-3 1 .622 4303 30.09 6 33 2255 29.89 24 .600 7886 30-55 .611 7410 30-3I .622 6108 30.09 .633 4048 29.88 25 i. 600 9718 3-54 1.611 9228 3-3 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 9427 29.87 28 .601 5214 30-53 .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 .601 8877 3 -52 i. 612 8316 30.28 1.623 6935 30.06 1.634 4803 29.86 31 .602 0708 30.52 .613 0132 30.28 .623 8739 30.06 634 6595 29.86 32 .602 2539 30-5 1 .613 1949 30.28 .624 0543 30.06 .634 8387 29.86 33 .602 4370 ,3-5 i .613 3765 30.27 .624 2346 30.05 .635 0178 29.86 34 .602 6200 30.50 .613 5582 30.27 .624 4149 30.05 .635 1969 29-85 35 .602 8030 30.50 1.613 7398 30.26 1.624 5952 30.05 1.635 3760 2 9 .8 5 36 .602 9860 30.50 .613 9213 30.26 .624 7755 30.04 6 35 555 1 29.85 37 .603 1690 3-49 .614 1029 30.26 .624 9557 30-04 635 7342 29.84 38 .603 3519 30.49 .614 2844 30.25 .625 1360 30.04 .635 9132 29.84 39 .603 5348 30.48 .614 4659 30-25 .625 3162 30.03 .636 0922 29.84 40 .603 7177 30.48 1.614 ^474 30.25 1.625 4964 30.03 1.636 2713 29.83 41 .603 9005 3-47 .614 8288 30.24 .625 6765 30.03 .636 4502 29.83 42 .604 0834 30.47 .615 0103 30.24 .625 8567 30.02 .636 6292 29.83 43 .604 2662 30-47 .615 1917 30-23 .626 0368 30.02 .636 8082 29.82 44 .604 4490 30.46 615 373i 30.23 .626 2169 30.02 .636 9871 29.82 45 .604 6317 30.46 1-615 5545 30.23 1.626 3970 30.01 .637 1660 29.82 46 .604 8145 3-45 -615 735 8 30.22 .626 5771 30.01 637 3449 29.82 47 .604 9972 30-45 .615 9171 30.22 .626 7571 30.01 .637 5238 29.81 48 .605 1799 30-45 .616 0984 30.22 .626 9372 30.00 .637 7027 29.81 49 .605 3626 30-44 .616 2797 30.21 .627 1172 30.00 .637 8815 29.81 50 .605 5452 30.44 1.616 4610 30.21 1.627 2972 30.00 .638 0603 29.80 51 .605 7278 30-43 .616 6422 30.20 .627 4771 29.99 .638 2391 29.80 52 .605 9104 3-43 .616 8234 30.20 .627 6571 29.99 .638 4179 29.80 53 .606 0930 30-43 .617 0046 30.20 .627 8370 29-99 638 5967 29-79 54 .606 2755 30.42 .617 1858 30.19 .628 0169 29.98 .638 7754 29-79 55 .606 4581 30.42 .617 3669 30.19 .628 1968 29.98 .638 9542 29-79 56 .606 6406 30.42 .617 5481 30.19 .628 3766 29.98 .639 1329 29.78 57 .606 8230 30.41 .617 7292 30.18 .628 5565 29-97 639 3116 29.78 58 .607 0055 30.41 .617 9102 30.18 .628 7363 29.97 .639 4902 29.78 59 .607 1879 30.40 .618 0913 30.17 .628 9161 29-97 .639 6689 29-77 60 .607 3703 30.40 .618 2724 30.17 .629 0959 29.96 .639 8475 29.77 579 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 56 57 58 59 V* log M. Diff. 1". log M. Diff. V. log M. Diff. 1". log M. Diff. 1". 0' 1 1.639 8475 .640 0262 29.77 29.77 1.650 5336 .650 7112 29.60 29.60 1.661 1601 .661 3368 29.44 29.44 1.671 7331 .671 9089 29.30 29.30 2 3 4 .640 2048 .640 3833 .640 5619 29.77 29.76 29.76 .650 8887 .651 0663 .651 2438 29.59 29.59 29.59 .661 5134 .661 6900 .661 8666 29.44 29.43 29-43 .672 0846 .672 2604 .672 4362 29.30 29.29 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 .641 0975 29.75 .651 7763 29.58 .662 3963 29.42 .672 9634 29.28 8 .641 2760 29.75 .651 9538 29.58 .662 5728 29.42 .673 1391 29.28 9 .641 4545 29.74 .652 1312 29.57 .662 7493 29.42 .673 3147 29.28 10 11 1.641 6329 .641 8114 29.74 29.74 1.652 3086 .652 4861 29.57 29.57 1.662 9258 .663 1023 29.42 29.41 1.673 4904 .673 6661 29.28 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 19.41 .674 1930 29.27 15 1.642 5250 29.73 1.653 1956 29.56 1.663 8082 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 6 53 552 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 6 75 5975 29.25 23 .643 9513 29.70 .654 6138 29-54 .665 2191 29-39 .675 7730 29.25 24 .644 1295 29.70 .654 7910 29-53 .665 3954 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 6 55 3225 29-53 .665 9242 29.38 .676 4749 29.24 28 .644 8421 29.69 .655 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 !9 8 4 29.68 I-655 8539 29.52 1.666 4529 29-37 1.677 0012 29.24 . 31 .645 3765 29.68 .656 0310 29.51 .666 6291 29.37 .677 1766 29.23 32 645 5545 29.68 .656 2081 29.51 .666 8053 29.36 .677 3520 29.23 33 .645 7326 29.67 .656 3852 29.51 .666 9815 29.36 .677 5274 29.23 34 .645 9106 29.67 .656 5622 29.51 .667 1577 29.36 .677 7028 29.23 35 1.646 0886 29.67 1.656 7392 29.50 1.667 3338 29.36 1.677 8 7 g l 29.23 36 .646 2666 29.67 .656 9163 29.50 .667 5100 29-35 .678 0535 29.22 37 .646 4446 29.66 657 933 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 64? 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 6 59 393 29.47 .669 6225 29-33 .680 1567 29.20 49 .648 5791 29.63 .659 2161 29.47 .669 7984 29.32 .680 3319 29.20 5O 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 1.649 6 454 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 29.17 580 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 60 61 62 63 log M. Diff. I". log It Diff. I". logM. Diff. 1". log M. Diff. 1". O' 1.682 2581 29.17 1.692 7408 29.07 1.703 1866 28.97 1.713 6006 28.89 1 .682 4332 29.17 .692 9152 29.06 .703 3604 28.97 713 7739 28.89 2 .682 6082 29.17 .693 0896 29.06 .703 5342 28.97 713 9473 28.89 3 .682 7832 29.17 .693 2640 29.06 .703 7080 28.97 .714 1206 28.88 4 .682 9582 29.17 693 43 8 3 29.06 .703 8818 28.96 .714 2939 28.88 5 1.683 J 33 2 29.16 1.693 6127 29.06 1.704 0556 28.96 1.714 4672 28.88 6 .683 3082 29.16 .693 7870 29.05 .704 2293 28.96 .714 6405 28.88 7 .683 4832 29.16 .693 9613 29.05 .704 4031 28.96 .714 8138 28.88 8 .683 6581 29.16 .694 1356 29.05 .704 5768 28.96 .714 9870 28.88 9 .683 8331 29.16 .694 3099 29.05 .704 7506 28.96 .715 1603 28.88 10 1.684 0080 29.16 1.694 4842 29.05 1.704 9243 28.96 *'7iS 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 13 .684 5328 29.15 .695 0070 29.04 75 4455 28.95 .715 8533 28.87 14 .684 7077 29.15 .695 1813 29.04 .705 6192 28.95 .716 0266 28.87 15 1.684 8826 29.14 1-695 3555 29.04 1.705 7929 28.95 1.716 1998 28.87 16 .685 0574 29.14 .695 5298 29.04 .705 9666 28.95 .716 3730 28.87 17 .685 2323 29.14 .695 7040 29.04 .706 1402 28.95 .716 5462 28.87 18 .685 4071 29.14 .695 8782 29.03 .706 3139 28.94 .716 7194 28.87 19 .685 5820 29.14 .696 0524 29.03 .706 4875 28.94 .716 8926 28.87 20 1.685 7568 29.14 1.696 2266 29.03 1.706 6612 28.94 1.717 0658 28.86 21 .685 9316 29.13 .696 4008 29.03 .706 8348 28.94 .717 2390 28.86 22 .686 1064 29.13 .696 5750 29.03 .707 0085 28.94 .717 4122 28.86 23 .686 2812 29.13 .696 7491 29.03 .707 1821 28.94 7 J 7 5 8 53 28.86 24 .686 4560 29.13 .696 9233 29.02 77 3557 28.94 .717 7585 28.86 25 1.686 6308 29.13 1.697 0974 29.02 1.707 5293 28.93 1.717 9317 28.86 26 .686 8055 29.13 .697 2716 29.02 .707 7029 28.93 .718 1048 28.86 27 .686 9803 29.12 .697 4457 29.02 .707 8765 28.93 .718 2780 28.86 28 .687 1550 29.12 .697 6198 29.02 .708 0501 28.93 .718 4511 28.86 29 .687 3297 29.12 .697 7939 29.02 .708 2237 28.93 .718 6242 28.85 30 1.687 5044 29.12 1.697 9680 29.02 1.708 3972 28.93 1.718 7974 28.85 31 .687 6791 29.12 .698 1421 29.01 .708 5708 28.93 .718 9705 28.85 32 .687 8538 29.11 .698 3162 29.01 .708 7444 28.92 .719 1436 28.85 33 .688 0285 29.11 .698 4902 29.01 .708 9179 28.92 .719 3167 28.85 34 .688 2032 29.11 .698 6643 29.01 .709 0914 28.92 .719 4898 28.85 35 1.688 3778 29.11 1.698 8383 29.01 1.709 2650 28.92 1.719 6629 28.85 36 .688 5525 29.1 1 .699 0124 29.01 .709 4385 28.92 .719 8360 28.85 37 .688 7271 ,29.10 .699 1864 29.00 .709 6120 28.92 .720 0090 28.85 38 .688 9017 29.10 .699 3604 29.00 709 7855 28.92 .720 1821 28.84 39 .689 .0764 29.10 .699 5345 29.00 .709 9590 28.92 .720 3552 28.84 4O 1.689 2510 29.10 1.699 7085 29.00 1.710 1325 28.91 1.720 5282 28.84 41 .689 4256 29.10 .699 8824 29.00 .710 3060 28.91 .720 7013 28.84 42 .689 6001 29.09 .700 0564 29.00 .710 4794 28.91 .720 8743 28.84 43 .689 7747 29.09 .700 2304 29.00 .710 6529 28.91 .721 0474 28.84 44 .689 9493 29.09 .700 4044 28.99 .710 8263 28.91 .721 2204 28.84 45 1.690 1238 29.09 1.700 5783 28.99 1.710 9998 28.91 1.721 3934 28.84 46 .690 2984 29.09 .700 7523 28.99 .711 1732 28.91 .721 5665 28.84 47 .690 4729 29.09 .700 9262 28.99 .711 3467 28.90 .721 7395 28.84 48 .690 6474 29.09 .701 1001 28.99 .711 5201 28.90 .721 9125 28.83 49 .690 8219 29.08 .701 2741 28.99 7" 6935 28.90 .722 0855 28.83 50 1.690 9964 29.08 1.701 4480 28.98 1.711 8669 28.90 1.722 2585 28.83 51 .691 1709 29.08 .701 6219 28.98 .712 0403 28.90 .722 4315 28.83 52 .691 3454 29.08 .701 7958 28.98 .712 2137 28.90 .722 6044 28.83 53 .691 5199 29.08 .701 9697 28.98 .712 3871 28.90 .722 7774 28.83 54 .691 6943 29.08 .702 1435 28.98 .712 5605 28.90 .722 9504 28.83 55 1.691 8688 29.07 1.702 3174 28.98 1.712 7339 28.90 1.723 1233 28.83 56 .692 0432 29.07 .702 4913 28.98 .712 9072 28.89 .723 2963 28.83 57 .692 2176 29.07 .702 6651 28.97 .713 0806 28.89 ,723 4693 28.82 58 .692 3920 29.07 .702 8389 28.97 7i3 2539 28.89 .723 6422 28.82 59 .692 5664 29.07 .703 0128 28.97 .713 4273 28.89 .723 8151 28.82 6O 1.692 7408 29.07 1.703 1866 23.97 1.713 6006 28.89 1.723 9881 28.82 581 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 64 65 66 67 logM. Diff. 1". log M. Diff. I". logM. Diff. 1". logM. Diff. 1". 0' 1.723 9881 28.82 1-734 3539 28.77 1.744 73 X 28.73 1.755 405 28.70 1 .724 1610 28.82 734 5265 28.77 744 8755 28-73 755 2127 28.70 2 .724 3339 28.82 734 6 99 J 28.77 745 479 28.73 755 3849 28.70 3 .724 5068 28.82 734 8718 28.77 .745 2202 28.73 755 557i 28.70 4 .724 6798 28.82 735 444 28.77 745 3926 28.73 755 7293 28.70 5 1.724 8527 28.82 1.735 2169 28.76 I -745 5 6 5 28.73 1-755 9oi5 28.70 6 .725 0256 28.82 735 3895 28.76 745 7373 28. 73 .756 0737 28,70 7 .725 1984 28.81 .735 5621 28.76 745 997 28.73 75 6 2459 28.70 8 .725 3713 28.81 735 7347 28.76 .746 0820 28.72 .756 4181 28.70 9 725 5442 28.81 735 973 28.76 .746 2544 28.72 756 593 28.70 10 1.725 7171 28.81 1.736 0798 28.76 1.746 4267 28.72 1.756 7625 28.70 11 .725 8900 28.81 .736 2524 28.76 .746 5991 28.72 75 6 9347 28.70 12 .726 0628 28.81 .736 4250 28.76 .746 7714 28.72 .757 1069 28.70 13 .726 2357 28.81 73 6 5975 28.76 .746 9437 28.72 757 2791 28.70 14 .726 4085 28.81 .736 7701 28.76 .747 1161 28.72 757 4513 28.70 15 1.726 5814 28.81 1.736 9426 28.76 1.747 2884 28.72 r -757 6235 28.70 16 .726 7542 28.81 737 H5 2 28.76 .747 4607 28.72 757 7957 28.70 17 .726 9270 28.81 737 2877 28.76 747 6 33 28.72 757 9 6 79 28.70 18 .727 0999 28.80 737 4602 28.76 747 8054 28.72 .758 1401 28.70 19 .727 2727 28.80 737 6328 28.75 747 9777 28.72 758 3123 28.70 20 1.727 4455 28.80 1.737 8053 28.75 1.748 1500 28.72 1.758 4844 28.70 21 .727 6183 28.80 737 9778 28.75 .748 3223 28.72 .758 6566 28.70 22 .727 7911 28.80 .738 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 oio 28.70 24 .728 1367 28.80 738 4953 28.75 .748 83-92 28.72 759 I 73 I 28.70 25 1.728 3095 28.80 1.738 6679 28.75 1.749 OI1 5 28.72 '759 3453 28.70 26 .728 4823 28.80 .738 8404 28.75 749 l8 3 8 28.72 759 5*75 28.70 27 .728 6551 28.80 .739 0129 28.75 749 35 61 28.72 759 68 97 28.70 28 .728 8279 28.80 739 l8 53 28.75 749 5 2 84 28.72 759 8618 28.69 29 .729 0006 28.79 739 3578 28.75 749 7007 28.71 .760 0340 28.69 30 1.729 1734 28.79 1-739 533 28.75 1.749 873 28.71 1.760 2062 28.69 31 .729 3461 28.79 .739 7028 28.75 .750 0453 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 74 477 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 28.74 i-75o 7344 28.71 1.761 0670 28.69 36 .730 2099 28.79 74 5 6 5 X 28.74 .750 9067 28.71 .761 2392 28.69 37 .730 3826 28.79 .740 7376 28.74 .751 0789 28.71 .761 4113 28.69 38 73 5553 28.79 .740 9101 28.74 .751 2512 28.71 .761 5835 28.69 39 .730 7280 28.79 .741 0825 28.74 751 4234 28.71 .761 7556 28.69 40 1.730 9007 28.78 1.741 2550 28.74 I -75 I 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 42 .731 2462 28.78 .741 5998 28.74 .751 9402 28.71 .762 2721 28.69 43 .731 4189 28.78 .741 7723 28.74 .752 1125 28.71 .762 4442 28.69 44 73 1 59*5 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 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 .732x2823 28.78 .742 6344 28.74 752 9737 28.71 .763 3050 28.69 49 .732 4549 28.78 .742 8068 28.74 753 H 6 28.71 .763 4771 28.69 50 1.732 6276 28.78 1.742 9792 28.74 1.753 3 lg 2 28.71 1.763 6493 28.69 51 .732 8002 28.78 743 J 5i6 28.73 753 494 28.71 .763 8214 28.69 52 .732 9729 28.77 743 3 2 40 28.73 753 662 7 28.71 .763 9936 28.69 53 733 I4S5 28.77 743 49 6 4 28.73 753 8349 28.71 .764 1657 28.69 54 733 3182 28.77 .743 6688 28.73 754 7i 28.70 764 3379 28.69 55 1.733 49o8 28.77 1.743 8412 28.73 1.754 1794 28.70 1.764 5100 28.69 56 733 6635 28.77 744 OI 3 6 28.73 754 35 l6 28.70 .764 6821 28.69 57 58 733 8361 734 0087 28.77 28.77 .744 1860 744 3584 28.73 28.73 754 5238 .754 6960 28.70 28.70 .764 8543 .765 0264 28.69 28.69 59 734 1813 28.77 744 53 8 28.73 .754 8682 28.70 .765 1985 28.69 6O '734 3539 28.77 '744 7031 28.73 1.755 45 28.70 1-765 3707 28.69 582 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 68 69 70 71 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1.765 3707 28.69 1.775 6 985 28.69 1.786 0284 28.70 1.796 3650 28.73 1 .765 5428 28.69 .775 8706 28.69 .786 2006 28.70 79 6 5374 28.73 2 .765 7150 28.69 .776 0427 28.69 .786 3728 28.70 .796 7097 28.73 3 .765 8871 28.69 .776 2149 28.69 .786 5450 28.70 .796 8821 28.73 4 .766 0592 28.69 .776 3870 28.69 .786 7172 28.70 797 0545 28.73 5 1.766 2314 28.69 i-776 559 1 28.69 1.786 8894 28.70 1.797 2268 28.73 6 .766 4035 28.69 77 6 73 T 3 28.69 .787 0617 28.70 797 399 2 28.73 7 .766 5756 28.69 .776 9034 28.69 787 2339 28.70 797 57i6 28.73 8 .766 7478 28.69 777 755 28.69 .787 4061 28.70 797 744 28.73 9 .766 9199 28.69 777 2 477 28.69 787 5783 28.70 797 9 l6 4 28.73 1O 1.767 0920 28.69 1.777 4*9 8 28.69 1.787 7506 28.70 1.798 0888 28.73 11 .767 2642 28.69 777 59 20 28.69 .787 9228 28.71 .798 2611 28.73 12 767 43 6 3 28.69 777 7641 28.69 .788 0950 28.71 798 4335 28.73 13 .767 6084 28.69 777 93 6 3 28.69 .788 2673 28.71 .798 6060 28.73 14 .767 7805 28.69 .778 1084 28.69 .788 4395 28.71 .798 7784 28.73 15 1.767 9527 28.69 1.778 2806 28.69 1.788 6117 28.71 1.798 9508 28.73 16 .768 1248 28.69 .778 4527 28.69 .788 7840 28.71 799 12 3 2 28.74 17 .768 2969 28.69 .778 6248 28.69 .788 9562 28.71 799 2 95 6 28.74 18 .768 4691 28.69 .778 7970 28.69 .789 1284 28.71 .799 4680 28.74 19 .768 6412 28.69 778 9 6 9' 28.69 .789 3007 28.71 799 6 44 28.74 20 1.768 8133 28.69 1.779 H'3 28.69 1.789 4730 28.71 1.799 8128 28.74 21 .768 9854 28.69 779 3'40 28.69 .789 6452 28.71 799 9853 28.74 22 .769 1576 28.69 779 4862 28.69 .789 8175 28.71 .800 1577 28.74 23 .769 3297 28.69 779 6 578 28.69 .789 9897 28.71 .800 3301 28.74 24 .769 5018 28.69 779 8299 28.69 .790 1620 28.71 .800 5026 28.74 25 1.769 6740 28.69 1.780 OO2I 28.69 1.790 3342 28.71 i. 800 6750 28.74 26 .769 8461 28.69 .780 I 74 2 28.69 .790 5065 28.71 .800 8475 28.74 27 .770 0182 28.69 .780 3464 28.69 .790 6788 28.71 .801 0199 28.74 28 .770 1903 28.69 .780 5185 28.69 79 8510 28.71 .801 1924 28.74 29 .770 3625 28.69 .780 6907 28.69 791 02 33 28.71 .801 3648 28.74 30 1.770 5346 28.69 1.780 8629 28.69 1.791 1956 28.71 1-801 5373 28.74 31 .770 7067 28.69 .781 0350 28.69 .791 3678 28.71 .801 7107 28.74 32 .770 8788 28.69 .781 2072 28.69 .791 5401 28.71 .801 8822 28.74 33 .771 0510 28.69 .781 3793 28.69 .791 7124 28.71 .802 0547 28.75 34 .771 2231 28.69 781 55'5 28.69 .791 8847 28.71 .802 2271 28.75 35 1.771 3952 28.69 1.781 7237 28.69 1.792 0570 28.71 1.802 3996 28.75 36 .771 5673 28.69 .781 8959 28.69 .792 2293 28.71 .802 5721 28.75 37 77i 7395 28.69 .782 0680 28.70 .792 4016 28.72 .802 7446 28.75 38 .771 9116 28.69 .782 2402 28.70 .792 5738 28.72 .802 9171 28.75 39 .772 0837 28.69 .782 4124 28.70 .792 7461 28.72 .803 0896 28.75 40 1.772 2559 28.69 1.782 5845 28.70 1.792 9184 28.72 1.803 2 ^ 21 28.75 41 .772 4280 28.69 .782 7567 28.70 793 97 28.72 803 4346 28.75 42 .772 6001 28.69 .782 9289 28.70 .793 2630 2.8.72 .803 6071 28.75 43 .772 7722 28.69 .783 ion 28.70 793 4354 28.72 .803 7796 28.75 44 .772 9444 28.69 .783 2732 28.70 793 6 77 28.72 .803 9521 28.75 45 1.773 "65 28.69 1-783 4454 28.70 1.793 7800 28.72 1.804 1246 28.75 46 .773 2886 28.69 .783 6176 28.70 793 95 2 3 28.72 .804 2971 28.75 47 .773 4607 28.69 .783 7898 28.70 .794 1246 28.72 .804 4697 28.75 48 773 6 3 2 9 28.69 .783 9620 28.70 .794 2969 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 977i 28.69 1.784 3064 28.70 1-794 6 4 l6 28.72 1.804 9873 28.76 51 .774 1493 28.69 .784 4786 28.70 794 8139 28.72 .805 1598 28.76 52 774 3 2I 4 28.69 .784 6508 28.70 .794 9862 28.72 805 3324 28.76 53 774 4935 28.69 .784 8230 28.70 795 i5 86 28.72 .805 5049 28.76 54 774 66 57 28.69 784 995 2 28.70 795 339 28.72 805 6775 28.76 55 1.774 8378 28.69 1.785 1674 28.70 1-795 533 28.72 1.805 8500 28.76 56 .775 0099 28.69 785 339 6 28.70 795 6 75 6 28.72 .806 0226 28.76 57 .775 1821 28.69 .785 5118 28.70 .795 8480 28.72 .806 1952 28.76 58 .775 3542 28.69 .785 6840 28.70 .796 0203 28.73 .806 3677 22.76 59 775 5 26 3 28.69 .785 8562 28.70 .796 1927 28.73 .806 5403 28.76 60 1-775 6 985 28.69 1.786 0284 28.70 1.796 3650 28.73 i. 806 7129 28.76 583 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 72 73 74 75 log It Diff. I". log M. Diff. I". log Mi Diff. I". log M. Diff. 1". 0' 1. 806 712,9 28.76 1.817 0765 28.81 1.827 4602 28.88 1.837 8686 28.95 1 .806 8855 28.76 .817 2494 28.81 .827 6335 28.88 838 0423 28.95 2 .807 0581 28.77 .817 4222 28.82 .827 8068 28.88 .838 2160 28.95 3 .807 2307 28.77 .817 5951 28.82 .827 9800 28.88 .838 3898 28.95 4 .807 4033 28.77 .817 7680 28.82 .828 1533 28.88 838 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 .8|o 1274 28.97 14 .809 1295 28.77 .819 4974 28.83 .829 8865 28.89 .840 3012 28.97 15 1.809 3021 28.78 1.819 6704 28.83 1.830 0599 28.89 1.840 4751 28.97 16 .809 4748 28.78 .819 8433 28.83 .830 2332 28.89 .840 6489 28.97 17 .809 6474 28.78 .820 0163 28.83 .830 4066 28.90 .840 8227 28.97 18 .809 8201 28.78 .820 1893 28.83 .830 5800 28.90 .840 9966 28.97 19 .809 9928 28.78 .820 3623 28.83 8 3 7533 28.90 .841 1704 28.98 20 1.810 1655 28.78 1.820 5353 28.83 1.830 9267 28.90 1.841 3443 28.98 21 .810 3381 28.78 .820 7083 28.83 .831 looi 28.90 .841 5182 28.98 22 .810 5108 28.78 .820 8813 28.84 83 1 2 735 28.90 .841 6921 28.98 23 .810 6835 28.78 .821 0543 28.84 .831 4470 28.90 .841 8659 28.98 24 .810 8562 28.78 .821 2273 28.84 .831 6204 28.90 .842 0398 28.98 25 1.811 0289 28'. 7 8 1.821 4003 28.84 1.831 7938 28.91 1.842 2138 28.98 26 .811 2016 28.78 .821 5734 28.84 .831 9672 28.91 .842 3877 28.99 27 .811 3743 28.78 .821 7464 28.84 .832 1407 28.91 .842 5616 28.99 28 .811 5470 28.79 .821 9194 28.84 .832 3141 28.91 .842 7355 28.99 29 .811 7197 28.79 .822 0925 28.84 .832 4876 28.91 .842 9095 28.99 30 1.811 8924 28.79 1.822 2656 28.84 1.832 6611 28.91 1.843 0834 28.99 31 .812 0652 28.79 .822 4386 28.84 .832 8345 28.91 .843 2574 28.99 32 .812 2379 28.79 .822 6117 28.85 .833 0080 28.92 .843 4313 29.00 33 .812 4106 28.79 .822 7848 28.85 .833 1815 28.92 843 6 53 29.00 34 .812 5834 28.79 .822 9578 28.85 8 33 355 28.92 843 7793 29.00 35 i. 812 7561 28.79 1.823 I 39 28.85 1.833 5^85 28.92 1-843 9533 29.00 36 .812 9289 28.79 .823 3040 28.85 .833 7020 28.92 .844 1273 29.00 37 .813 1016 28.79 .823 4771 28.85 833 8755 28.92 .844 3013 29.00 38 .813 2744 28.79 .823 6502 28.85 .834 0491 28.92 844 4753 29.00 39 .813 4472 28.79 .823 8233 28.85 .834 2226 28.92 .844 6494 29.01 40 1.813 6199 28.80 1.823 9965 28.85 1.834 3961 28.92 1.844 8234 29.01 41 .813 7927 28.80 .824 1696 28.85 .834 5697 28.93 .844 9974 29.01 42 .813 9655 28.80 .824 3427 28.86 .834 7432 28.93 845 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 45 1.814 4839 28.80 1.824 8622 28.86 1.835 2,640 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 29.02 51 .815 5208 28.81 .825 9012 28.87 .836 3056 28.94 .846 7384 29.02 52 .815 6936 28.81 .826 0744 28.87 .836 4792 28.94 .846 9125 29.03 53 54 .815 8664 .816 0393 28.81 28.81 .826 2476 .826 4208 28.87 28.87 .836 6529 .836 8265 28.94 28.94 .847 0867 .847 2609 29.03 29.03 55 1.816 2121 28.81 1.826 5940 28.87 1.837 OOO2 28.94 1.847 4350 29.03 56 57 .816 3850 .816 5578 28.81 28.81 .826 7673 .826 9405 28.87 28.87 .837 1739 8 37 3475 28.95 28.95 .847 6092 .847 7834 29.03 29.03 58 .816 7307 28.81 .827 1137 28.87 .837 5212 28.95 .847 9576 29.03 59 .816 9036 28.81 .827 2870 28.87 .837 6949 28.95 .848 1318 29.04 60 1.817 0765 28.81 1.827 4602 28.88 1.837 8686 28.95 1.848 3060 29.04 584 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 76 77 78 79 logM. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". O' 1.848 3060 29.04 1.858 7769 29.14 1.869 2857 29.25 1.879 8369 29-37 1 .848 4803 29.04 .858 9517 29.14 .869 4612 .880 0131 29.37 2 .848 6545 29.04 .859 1266 29.14 .869 6367 29.25 .880 1894 29-38 3 .848 8287 29.04 859 3 OI 4 29.14 .869 8122 29.25 .880 3656 29.38 4 .849 0030 29.04 .859 4763 29.15 .869 9878 29.26 .880 5419 29.38 5 1.849 1773 29.04 1.859 6512 29.15 1.870 1633 29.26 1. 880 7182 29.38 6 849 35*5 29.05 .859 8260 29.15 .870 3389 29.26 .880 8945 29.38 7 .849 5258 29.05 .860 0009 29.15 .870 5144 29.26 .881 0708 29.39 8 .849 7001 29.05 .860 1758 29.15 .870 6900 29.26 .881 2471 29-39 9 .849 8744 29.05 .860 3507 29.15 .870 8656 29.26 .881 4235 29-39 10 1.850 0487 29.05 i. 860 5256 29.15 1.871 0412 29.27 1.881 5998 29-39 11 .850 2231 29.05 .860 7006 29.16 .871 2168 29.27 .881 7762 29.39 12 .850 3974 29.06 .860 8755 29.16 .871 3924 29.27 .881 9526 29.40 13 .850 5717 29.06 .861 0505 29.16 .871 5681 29.27 .882 1290 29.40 14 .850 7461 29.06 .861 2254 29.16 871 7437 29.28 .882 3054 29.40 15 1.850 9204 29.06 1.861 4004 29.16 1.871 9194 29.28 1.882 4818 29.40 16 .851 0948 29.06 .861 5754 29.16 .872 0950 29.28 .882 6582 29.41 17 .851 2692 29.06 .861 7504 29.17 .872 2707 29.28 .882 8347 29-41 18 19 .851 4436 .851 6180 29.07 29.07 .861 9254 .862 1004 29.17 29.17 .872 4464 .872 6221 29.28 29.29 .883 0112 .883 l8 7 6 29.41 2941 20 1.851 7924 29.07 1.862 2754 29.17 1.872 7979 29.29 1.883 3641 2942 21 .851 9668 29.07 .862 4505 29.17 .872 9736 29.29 .883 5406 29.42 22 .852 1412 29.07 .862 6255 29.18 873 1493 29.29 883 7171 29.42 23 .852 3157 29.07 .862 8006 29.18 .873 3251 29.29 .883 8937 29.42 34 .852 4901 29.07 .862 9756 29.18 .873 5008 29.30 .884 0702 2942 25 1.852 6646 29.08 1.863 1507 29.18 1.873 6766 29.30 1.884 2468 29-43 26 .852 8391 29.08 .863 3258 29.18 .873 8524 29.30 .884 4233 29.43 27 853 OI 35 29.08 .863 5009 29.18 .874 0282 29.30 884 5999 29.43 28 .853 1880 29.08 .863 6760 29.19 .874 2041 29.30 .884 7765 29-43 29 .853 3625 29.08 .863 8512 29.19 874 3799 29.31 884 9531 29-44 30 I - 8 53 537 29.09 1.864 0263 29.19 1-874 5557 29.31 1.885 1297 29-44 31 853 7115 29.09 .864 2015 29.19 .874 7316 29.31 .885 3064 29.44 32 .853 8861 29.09 .864 3766 29.19 .874 9074 29.31 .885 4830 29.44 33 .854 0606 29.09 .864 5518 29.20 875 0833 29.31 .885 6597 29.45 34 .854 2351 29.09 .864 7270 29.20 875 2592 29.32 .885 8364 29-45 35 1.854 4097 29.09 1.864 9022 29.20 1.875 435 1 29.32 1.886 0131 29.45 36 .854 5843 29.10 .865 0774 29.20 .875 our 29.32 .886 1898 29-45 37 854 7588 29.10 .865 2526 29.20 .875 7870 29.32 .886 3665 29.45 38 854 9334 29.10 .865 4278 29.20 .875 9629 29.32 -886 5432 29.46 39 .855 1080 29.10 .865 6030 29.21 .876 1389 29-33 .886 7200 29.46 40 1.855 2826 29.10 1.865 7783 29.21 1.876 3148 29.33 1.886 8967 29.46 41 .855 4572 29.10 .865 9536 29.21 .876 4908 29-33 -887 0735 2946 42 855 6319 29.11 .866 1288 29.21 .876 6668 29-33 .887 2503 29-47 43 .855 8065 29.11 .866 3041 29.21 .876 8428 29-33 .887 4271 29-47 44 855 9811 29.11 .866 4794 29.22 .877 0188 29.34 .887 6039 29-47 45 1.856 1558 29.11 1.866 6547 29.22 1.877 1949 29-34 1.887 7807 29.47 46 856 3305 29.11 .866 8301 29.22 877 379 29-34 887 9576 29.48 47 856 5052 29.11 .867 0054 29.22 .877 5470 29-34 .888 1344 29.48 48 .856 6799 29.12 .867 1807 29.22 .877 7230 29.34 .888 3113 29.48 49 .856 8546 29.12 .867 3561 29.23 .877 8991 29.35 .888 4882 2948 50 1.857 0293 29.12 1.867 5314 29.23 1.878 0752 29-35 1.888 6651 29.48 51 .857 2040 29.12 .867 7068 29.23 .878 2513 29-35 .888 8420 29-49 52 857 3787 29.12 .867 8822 29.23 .878 4275 29.35 .889 0189 29-49 53 .857 5534 29.12 .868 0576 29-23 .878 6036 29-35 .889 1959 29.49 54 .857 7282 29.13 .868 2330 29.24 .878 7797 29.36 .889 3728 29-49 55 1.857 9030 29.13 1.868 4084 29.24 1.878 9559 29.36 1.889 5498 29.49 56 858 0777 29.13 .868 5839 29.24 .879 1321 29.36 .889 7268 29.50 57 .858 2525 29.13 .868 7593 29.24 .879 3082 29.36 .889 8038 29.50 58 .858 4273 29.13 .868 9348 29.24 -879 4844 29.36 .890 0808 29.50 59 .858 6021 29.13 .869 IIO2 29.25 .879 6606 29-37 .890 2578 29.51 60 1.858 7769 29.14 1.869 2857 29-25 1.879 8369 29.37 1.890 4349 29.51 585 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 1 80 81 82 83 / log M. Diff. I". log M. Diff. 1". log M. Diff. 1". log M. Diff. 1". 0' 1.890 4349 29.51 1.901 0841 29.66 I. 9 II 7893 29.82 1.922 5548 29.99 1 .890 6119 2951 .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 29.83 .923 0947 30.00 4 .891 1432 29.52 .901 7960 29.67 .912 5050 29.83 .923 2747 30.00 5 1.891 3203 29.52 1.901 9740 29.67 1.912 6840 29.83 1.923 4548 30.01 6 .891 4974 29.52 .902 1521 29.67 .912 8630 29.84 .923 6348 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 29.84 923 995 30.02 9 .892 0289 29-53 .902 6862 29.68 .913 4001 29.84 .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 29.69 913 75 8 3 29.85 924 5354 30.03 12 .892 5605 29.54 .903 2105 29.69 .913 9374 29.85 924 7155 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 1.893 922 29.54 I -93 755 29.70 1.914 4748 29.86 1.925 2561 30.04 16 17 .893 2695 .893 4467 29-55 29-55 .903 9332 .904 1114 29.70 29.70 .914 6540 9*4 8331 29.86 29.87 .925 4364 .925 6166 30.04 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 2O 1.893 9787 29.56 1.904 6461 29.71 1.915 3708 29.87 1.926 1575 30.05 21 .894 1560 29.56 .904 8243 29.71 915 55 01 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 1.894 8656 29.57 .905 5376 29.72 1.916 2673 29.89 1.927 0593 30.07 26 .895 0430 29.57 95 7i59 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 1.895 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 29.91 .928 1422 30.09 32 .896 1078 29.59 .906 7863 29.74 .917 5231 29.91 .928 3227 30.09 33 .896 2854 29.59 .906 9648 29.74 .917 7025 29.9! .928 5032 30.09 34 .896 4628 29-59 .907 1432 29-75 .917 8820 29.92 .928 6838 30.10 35 1.896 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 29-93 929 5869 30.11 40 .897 5284 29.61 .908 2144 29.76 1.918 9593 29-93 1.929 7676 30.12 41 42 .897 7060 .897 8837 29.61 29.61 .908 3930 .908 5716 29.77 29.77 .919 1389 .919 3185 29.94 29.94 929 9483 .930 1291 30.12 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 J -9i9 8575 29.95 1.930 6713 SO-H 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 93 1 33 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 .931 3946 30.14 50 51 52 53 54 '899 3055 .899 4833 -899 6611 .899 8389 .900 0168 29.63 29-63 29.64 29.64 29.64 .910 ooio .910 1798 910 3585 91 5373 .910 7161 29.79 29.79 29.80 29.80 29.80 1.920 7561 920 9359 .921 1157 .921 2956 921 4754 29.96 29.97 29.97 29.97 29.98 93 1 5755 .931 7564 93 1 9373 932 1183 .932 2992 3- I 5 S -^ 30.15 30.16 30.16 55 .900 1946 29.64 .910 8949 29.80 .921 6552 29.98 .932 4802 30.16 56 .900 3725 29.65 .911 0738 29.81 .921 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 59 .900 7283 .900 9062 29.65 29.66 .911 4315 .911 6104 29.81 29.82 .922 1949 .922 3748 29.99 29.99 933 0232 933 2043 30.17 30.18 60 .901 0841 29.66 .911 7893 29.82 .922 5548 29.99 933 3853 30.18 586 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 84 85 86 87 logM. Diff. I". log M. Diff. 1". logM. Diff. 1". log M. Diff. l". I -933 3853 30.18 1.944 2856 30.38 1.955 2602 30.59 1.966 3140 30.82 1 933 5 66 4 30.18 .944 4678 30.38 955 4438 30.60 .966 4990 30.82 2 933 7475 30.19 944 6 52 3-39 955 6274 30.60 .966 6839 30.83 3 933 9287 30.19 944 8 3*5 3-39 .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 1.934 2910 30.20 1.945 *972 30.40 1.956 1783 30.61 1.967 2389 30.84 6 .934 4722 30.20 945 379 6 30.40 .956 3619 30.61 .967 4240 30.84 7 934 6 533 30.20 945 5620 30.40 .956 5456 30.62 .967 6090 30.85 8 934 8346 30.21 945 7444 30.41 .956 7294 30.62 .967 7941 30.85 9 935 OI 5 8 30.21 945 9269 30.41 .956 9131 30.63 .967 9792 30.85 10 !-935 *97* 30.21 1.946 1094 30.41 1.957 0969 30.63 1.968 1644 30.86 11 935 37H 30.22 .946 2919 30.42 957 2807 30.63 .968 3496 30.86 12 935 5597 30.22 .946 4744 30.42 957 4 6 45 30.64 .968 5347 30-87 13 935 74 10 30.22 .946 6569 30.42 957 6483 30.64 .968 7200 30-87 14 935 9223 30.22 .946 8395 3-43 957 8 32a 30.64 .968 9052 30.87 15 1.936 1037 30.23 1.947 O22I 3-43 1.958 0160 30.65 1.969 0905 30.88 16 .936 2851 30.23 .947 2047 3-44 .958 1999 30.65 .969 2757 30.88 17 .936 4665 30.23 947 3 8 73 30.44 958 3839 30.66 .969 4610 30.89 18 .936 6479 30.24 947 5 6 99 30.44 958 5678 30.66 .969 6464 30.89 19 .936 8293 30.24 947 7526 30-45 .958 7518 30.66 .969 8317 30.89 20 1.937 0108 30.24 r -947 9353 30.45 1.958 9358 30.67 1.970 0171 30.90 21 937 '922 30.25 .948 1180 3-45 959 1198 30.67 .970 2025 30.90 22 937 3737 30.25 .948 3007 30.46 959 3038 30-67 97 3879 30-91 23 937 5553 30.25 948 4834 30.46 959 4879 30.68 970 5734 30.91 24 937 73 68 30.26 .948 6662 30.46 .959 6720 30.68 .970 7589 30.91 25 1.937 9184 30.26 1.948 8490 30-47 1.959 8561 30.69 1.970 9443 30.92 26 .938 0999 30.26 949 3 l8 3-47 .960 0402 30.69 .971 1299 30.92 27 .938 2815 30.27 .949 2146 3-47 .960 2243 30.69 971 3*54 30-93 28 .938 4632 30.27 949 3975 30.48 .960 4085 30-7 .971 5010 30-93 29 .938 6448 30.27 949 5804 30.48 .960 5927 30.70 .971 6866 3-93 30 1.938 8264 30.28 1-949 7 6 33 30.48 1.960 7769 30.70 1.971 8722 3-94 31 .939 0081 30.28 .949 9462 3-49 .960 9612 30.71 .972 0578 30-94 32 939 1898 30.28 .950 1291 3-49 .961 1454 30.71 .972 2435 3-95 33 939 3715 30.29 95 3'2i 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 1-939 735 30.29 1.950 6781 30.50 1.961 6983 30.72 1.972 8006 30.96 36 .939 9168 30.30 .950 8611 30-5I .961 8827 30-73 .972 9864 30.96 37 .940 0986 30.30 95 i 044i 30-5 1 .962 0671 3-73 .973 1722 3-97 38 .940 2804 30.30 .951 2272 3 -5i .962 2515 30-73 973 358o 3-97 39 .940 4623 30.31 .951 4103 30.52 .962 4359 30-74 973 543 8 30-97 40 1.940 6441 30.31 I-95 1 5934 30.52 1.962 6203 30.74 1.973 7297 30.98 41 .940 8260 30.31 .951 7766 30.52 .962 8048 30-75 973 9i5 6 30.98 42 .941 0079 30.32 95 1 9597 30-53 .962 9893 30.75 974 IOI 5 30.99 43 .941 1898 30.32 .952 1429 30-53 .963 1738 30.75 974 2874 30.99 44 94 1 3717 30.32 .952 3261 30-53 .963 3583 30.76 974 4734 30.99 45 I -94 I 5537 3-33 1.952 5093 3-54 1.963 5429 30.76 1.974 6 593 31.00 46 941 7357 3-33 .952 6925 30.54 .963 7275 3-77 974 8 454 31.00 47 .941 9177 3-34 .952 8758 30-55 .963 9121 30-77 975 3 J 4 31.01 48 .942 0997 3-34 953 59* 3-55 .964 0967 3-77 .975 2174 31.01 49 .942 2817 3-34 953 2424 30-55 .964 2814 30.78 975 4035 31.01 50 1.942 4638 3-35 i-953 4^57 30.56 1.964 4660 30.78 1.975 5896 31.02 51 .942 6459 3-35 953 609 1 30.56 .964 6507 30.78 975 7757 31.02 52 .942 8280 3-35 953 79 2 4 30.56 .964 8354 30.79 975 9619 31.03 53 .943 oioi 30.36 953 975 8 30.57 .965 O2O2 30-79 .976 1481 31.03 54 943 !9 2 3 30.36 954 159^ 3-57 .965 2050 30.80 976 3343 31.04 55 56 J -943 3744 943 55 66 30.36 3-37 J-954 34^7 954 5262 30-57 30.58 1.965 3 8 97 .965 5746 30.80 30.80 1.976 5205 .976 7067 31.04 31.04 57 943 73 8 8 3-37 954 709 6 30.58 965 7594 30.81 .976 8930 3i-o5 58 943 9211 3-37 954 893 1 30-59 .965 9442 30.81 977 0793 3 I -5 59 944 i33 30.38 955 7 66 30-59 .966 1291 30.81 .977 2656 31.06 60 1.944 2856 30.38 1.955 2602 30.59 1.966 3140 30.82 1.977 4520 31.06 587 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 88 89 90 91 V. logM. Diff. 1". log M. Diff. I". logM. Diff. 1". log M. Diff. 1". 0' 977 4520 31.06 1.988 6789 31-31 2.000 0000 3I-58 2.OI I 4203 31-87 1 2 -977 6383 .977 8247 31.06 31.07 .988 8668 .989 0548 31.32 3'-32 .000 1895 .000 3790 31-59 3^-59 .on 6115 .on 8027 31-88 3 .978 0112 31.07 .989 2427 3-33 .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 .978 3841 31.08 1.989 6187 3'-34 2.000 9478 31.60 2.012 3766 31.89 6 .978 5706 31.08 .989 8067 .001 1375 31.61 .OI2 5680 31.89 7 8 .978 7571 978 943 6 31.09 31.09 .989 9948 .990 1829 31-34 31-35 .001 3272 .001 5169 31.61 31.62 .012 7594 .012 9508 31.90 31.90 9 979 I 3 02 31.10 .990 3710 3!-35 .001 7066 31.62 .013 1422 3 I -9 I 10 979 3168 31.10 1.990 5591 31.36 2.001 8963 31.63 2.013 3337 3 I -9 I 11 .979 5034 31.11 99 7473 31.36 .OO2 o86l 3 I - 6 3 .013 5252 31.92 12 .979 6901 31.11 99 9355 3*-37 .002 2759 31.64- .013 7167 31.92 13 .979 8768 31.11 .991 1237 31-37 .OO2 4658 31.64 .013 9083 31-93 14 .980 0635 31.12 .991 3119 3^-38 .002 6557 31-65 .014 0999 31-93 15 .980 2502 31.12 1.991 5002 31.38 2.002 8456 31.65 2.014 2915 31.94 16 .980 4369 .991 6885 3I-38 .003 0355 31.66 .014 4831 3 J -94 17 .980 6237 31-13 .991 8768 .003 2254 31.66 .014 6748 31-95 18 .980 8105 31-13 .992 0651 31-39 .003 4154 31.67 .014 8665 31-95 19 .980 9973 31.14 .992 2535 31.40 .003 6054 31.67 .015 0582 31.96 20 .981 1842 31.14 1.992 4419 31.40 2.003 7955 31.68 2.015 2500 31.96 21 22 .981 3710 .981 5579 3i-i5 3 I - I 5 .992 6304 .992 8188 .003 9855 .004 1756 31.68 31.68 .015 4418 .015 6336 31-97 23 .981 7449 31.16 993 73 31.42 .004 3658 31.69 .015 8255 3I-98 24 .981 9318 31.16 993 X 95 8 31.42 4 5559 31.69 .016 0174 31.98 25 1.982 1188 31.16 i-993 3 8 43 31.42 2.004 7461 31.70 2.016 2093 31.99 26 .982 3058 3 I - I 7 993 5729 .004 9363 31.70 .016 4012 3'-99 27 .982 4928 31.17 993 7 6l 5 3M3 .005 1265 .016 5932 32.00 28 .982 6798 31.18 993 95 01 3M4 .005 3168 31.71 .016 7852 32.00 29 .982 8669 31.18 .994 1387 3M4 .005 5071 31.72 .016 9772 32.01 30 1.983 0540 31.18 1.994 3274 3M5 2.005 6974 31.72 2.017 1^93 32.01 31 .983 2411 31-19 994 5161 3M5 .005 8878 3J-73 .017 3614 32.02 32 -983 4283 31-19 .994 7048 31.46 .006 0781 31.73 017 5535 32.02 33 .983 6155 31.20 994 8 93 6 31.46 .006 2685 3 J -74 .017 7456 32.03 34 .983 8027 31.20 995 82 3 31.46 .006 4590 31-74- .017 9378 32.03 35 1.983 9899 31.21 1.995 2711 31.47 2.006 6494 31-75 2.018 1300 32.04 36 .984 1772 31.21 995 4 6o 3M7 .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 55 J 7 31.22 995 8377 31.48 .007 2210 31.76 .018 7068 32.05 39 .984 7391 31.22 .996 0266 3M9 .007 4116 31.77 .018 8992 32.06 40 1.984 9264 31.23 1.996 2155 3M9 2.OO7 6O22 31-77 2.019 0915 32.06 41 985 1138 31.23 .996 4045 3 x -5 .007 7928 31-77 .019 2839 32.07 42 .985 3012 31.24 996 5935 3 I -5 .007 9835 3I-78 .019.4763 32.07 43 .985 4886 31.24 .996 7825 .008 1742 3I-78 .019 6688 32.08 44 .985 6761 31.24 .996 9716 3i-5i .OO8 3649 3 J -79 .019 8613 32.08 45 1.985 8636 3 J - 2 5 1.997 1606 Si-5 1 2.OO8 5556 3'-79 2.020 0538 32.09 46 .986 0511 31-25 997 3497 .008 7464 31.80 .O2O 2463 32.09 47 .986 2386 31.26 997 5389 31.52 .OO8 9372 31.80 .020 4389 32.10 48 .986 4262 31.26 .997 7280 ^1.53 .009 I28o 31.81 .O2O 6315 32.10 49 .986 6138 31.27 997 9172 31-53 .009 3189 31.81 .020 8241 32.11 50 1.986 8014 31.27 1.998 1064 31-54- 2.009 598 31.82 2.021 Ol68 32.11 51 .986 9890 31.28 .998 2956 31-54- .009 7007 31.82 .O2I 2095 32.12 52 987 1767 31.28 998 4849 3*-55 .009 8917 31-83 .021 4022 32.12 53 .987 3644 31.28 .998 6742 .010 0826 31-83 .021 5949 32.13 54 .987 5521 31.29 .998 8635 3M 6 .010 2736 31.84 .021 7877 55 1.987 7398 31.29 1.999 0529 31.56 2.OIO 4647 31.84 2.021 9805 32.14 56 .987 9276 31.30 .999 2422 31.56 .010 6557 .022 1734 32.14 57 .988 1154 31.30 .999 4316 3i-57 .010 8468 31-85 .022 3662 32.15 58 .988 3032 3 I -3 I .999 6211 .on 0380 31.86 .022 5591 32-15 59 .988 4911 999 8105 3 I -5 8 .on 2291 31.86 .022 7521 32.16 60 1.988 6789 3i-3i 2.000 0000 31.58 2.01 I 4203 3i-*7 2.022 9450 32.16 588 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. ??. 92 93 94 95 C/t log M. Diff. I". logM. Diff. I". log- M. Diff. 1". 1 gM. Diff. I". O' 2.022 9450 32.16 2.034 5797 32.48 2.046 3296 32.80 2.058 2005 33-*5 1 .023 1380 32.17 034 7745 32.48 .046 5264 32.81 .058 3994 33-*5 2 .023 33 H 32.17 .034 9694 32.49 .046 7233 32.82 .058 5983 33.16 3 .023 5241 32.18 .035 1644 32-49 .046 9^202 32.82 .058 7973 33.16 4 .023 7172 32.18 35 3593 32.50 .047 II 7 2 32.83 .058 9963 33- 1 ? 5 2.023 9103 32.19 2-035 5543 32.50 2.047 3H 1 32.83 2-059 1953 33-i8 6 .024 1035 32.19 35 7494 32-51 .047 5111 32.84 59 3944 33-i8 7 .024 2967 32.20 35 9444 3 2 -5i .047 7082 32.84 59 5935 33-*9 8 .024 4899 32.20 .036 1395 32.52 .047 9053 32.85 .059 7927 33- J 9 9 .024 6831 32.21 036 3347 32.52 .048 1024 32.85 .059 9919 33-20 10 2.O24 8764 32.21 2.036 5298 32.53 2.048 2995 32.86 2.060 1911 33.21 11 .025 0697 32.22 .036 7250 32.53 ' .048 4967 32.87 .060 3904 33.21 12 .025 2630 32.22 .036 9202 32-54 .048 6939 32.87 .060 5897 33-22 13 .025 4564 32.23 .037 1155 32-54 .048 8912 32.88 .060 7890 33-22 14 .025 6498 32.23 .037 3108 32.55 .049 0884 32.88 .060 9884 33-23 15 2.025 8432 32.24 2.037 5061 32.55 2.049 2857 32.89 2.061 1878 33-24 16 .026 0367 32.24 .037 7015 32.56 .049 4831 32.89 .061 3872 33-24 17 .026 2301 32.25 .037 8969 32.57 .049 6805 32.90 .061 5867 33-25 18 .026 4236 32.26 .038 0923 32.57 .049 8879 32.90 .061 7862 33-25 19 .026 6172 32.26 .038 2877 32.58 .050 0753 32.91 .061 9857 33-26 20 2.026 8108 32.27 2.038 4832 32.58 2.050 2728 32.92 2.062 1853 33-27 21 .027 0044 32.27 .038 6787 32-59 .050 4703 32.92 .062 3849 33-27 22 .027 1980 32.28 .038 8743 32-59 .050 6679 32.93 .062 5846 33.28 23 .027 3917 32.28 .039 0699 32.60 .050 8655 32.93 .062 7842 33.28 24 .027 5854 32.29 .039 2655 32.61 .051 0631 32.94 .062 9840 33-29 25 2.027 779 I 32.29 2.039 4611 32.61 2.051 2608 32.95 2.063 1837 33-3 26 .027 9729 3*3 .039 6568 32.62 .051 4585 32.95 .063 3835 33-3 27 .028 1667 32.30 .039 8525 32.62 .051 6562 32.96 .063 5833 33-3 1 28 .028 3605 32.31 .040 0482 32.63 .051 8539 32.96 .063 7832 33-3 1 29 .028 5544 32.31 .040 2440 32.63 .052 0517 32.97 .063 9831 33-32 30 2.028 7483 32-32 2.040 4399 32.64 2.052 2496 32.97 2.064 I ^3 I 33-33 31 .028 9422 32.32 .040 6357 32.64 .052 4474 32.98 .064 3830 33-33 32 .029 1361 32-33 .040 8316 32.65 .052 6453 32.98 .064 5830 33-34 33 .029 3301 32-33 .041 0275 32.65 .052 8432 32.99 .064 7831 33-34 34 .029 5241 32.34 .041 2234 32.66 .053 0412 33- .064 9832 33-35 35 2.029 7182 32-34 2.041 4194 32.67. 2053 2392 33-0 2.065 J 833 33-36 36 .029 9123 32-35 .041 6154 32.67 053 4372 33.01 6 5 3834 33-3 6 37 .030 1064 3^-35 .041 8114 32.68 53 6 353 33-oi .065 5836 33 37 38 .030 3005 3 2 -3 6 .042 0075 32.68 53 8 334 33.02 .065 7839 33 37 39 .030 4947 32-36 .042 2036 32.69 .054 0315 33.03 .065 9841 33-38 40 2.030 6889 32.37 2.042 3998 32-69 2.054 2297 33.03 2.066 1844 33-39 41 .030 8831 32-37 .042 5960 32.70 .054 4279 33-04 .066 3847 33-39 42 .031 0774 32.38 .042 7922 32.70 .054 6262 33-4 .066 5851 33-4 43 .031 2717 32-39 .042 9834 32.71 .054 8244 33-05 .066 7855 33.40 44 .031 4660 32.39 .043 1847 32.71 .055 0227 33-05 .066 9860 33-41 45 2.031 6604 32.40 2.043 3810 32-72 2.055 221 I 33.06 2.067 *865 33-42 46 .031 8548 32.40 043 5773 32.73 55 4195 33-07 .067 3870 33-42 47 .032 0492 32.41 043 7737 32-73 .055 6179 33-07 .067 5875 33-43 48 .032 2437 32.41 .043 9701 32-74 .055 8163 33.08 .067 7881 33-43 49 .032 4382 32.42 .044 1665 32.74 .056 0148 33.08 .067 9887 33-44 5O 2.032 6327 32.42 2.044 3630 32.75 2.056 2133 33-09 2.068 1894 33-45 51 .032 8272 32-43 44 5595 32-75 .056 4119 33.10 .068 3901 33-45 52 .033 0218 32-43 .044 7561 32.76 .056 6105 33.10 .068 5908 33-46 53 .033 2164 32-44 .044 9526 32.76 .056 8091 33-n .068 7916 33-47 54 .033 4111 32.44 .045 1492 32.77 .057 0078 33-" .068 9924 33-47 55 2.033 6058 32.45 2-045 3459 32.78 2.057 2065 33-12 2.069 1933 33-48 56 .033 8005 32-45 .045 5426 32.78 .057 4052 33.12 .069 3942 33-48 57 033 995^ 32.46 45 7393 32-79 ,057 6040 33-13 .069 5951 33-49 58 .034 1900 32-47 .045 9360 32-79 .057 8028 33-14 .069 7960 33-5 59 .034 3848 32-47 .046 1328 32.80 .058 0016 33-H .069 9970 33-5 60 2-034 5797 32.48 2.046 3296 32.80 2.058 2005 33-15 2.070 1980 33-5 1 589 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. > 96 97 98 99 / loglC Diff. 1". logic. Diff. 1". log M. Diff. I". log M. Diff. I". 0' 2.070 1980 33-5 1 2.082 3282 33-88 2.094 5971 34.28 2.107 0109 34-69 1 .070 3991 33-51 .082 5316 33-89 .094 8028 34.29 .107 2190 34-70 2 .070 6002 33-52 .082 7349 33-9 .095 0085 34-29 .107 4272 34.70 3 .070 8014 33-53 .082 9383 33-9 .095 2143 34-3 107 6355 34-71 4 .071 0025 33-53 .083 1418 33-9 1 .095 4201 34- 3 .107 8437 34-72 5 2.071 2037 33-54 2.083 3453 33-92 2.095 6260 34- 3 i 2.108 0521 34-72 6 .071 4050 33-54 .083 5488 33-92 .095 8318 34-32 .108 2604 34-73 7 .071 6063 33-55 .083 7523 33-93 .096 0378 34-33 .108 4689 34-74 8 .071 8076 33-56 j J j 083 9559 33-94 .096 2438 34-33 .108 6773 34-75 9 .072 0090 33-5 6 .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 33-9 6 .097 0681 34-36 .109 5116 34-77 13 .072 8148 33-59 .084 9745 33-97 .097 2742 34-37 .109 7202 34.78 14 .073 0163 33-59 .085 1783 33.98 .097 4804 34-37 .109 9289 34-79 15 2.073 2179 33.60 2.085 3822 33-9 8 2.097 6867 34-38 2.IIO 1377 34-8o 16 .073 4195 33- 61 .085 5861 33-99 .097 8930 34-39 .no 3465 34-8o 17 .073 6212 33- 61 .085 7901 33-99 .098 0993 34-39 - 110 5553 34-8i 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 34-4 1 .no 9731 34-82 20 2.074 2264 33-63 2.086 4021 34.01 2.098 7186 34-41 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- 6 4 .086 8104 34.03 .099 1316 34-43 .III 6001 34.85 23 .074 8320 33- 6 5 .087 0146 34-03 99 3382 34-43 .III 8092 34-85 24 75 339 33.66 .087 2188 34-04 99 5449 34-44 .112 0184 34-86 25 2.075 2358 33-66 2.087 4231 34-05 2.099 7515 34-45 2. 112 2275 34.87 26 .075 4378 33-67 .087 6274 34.05 .099 9582 34-45 .112 4368 34-87 27 .075 6399 ]33- 6 7 .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 2.076 2462 33-69 2.088 4449 34.08 2.100 7855 34-48 2.II3 2741 34-9 31 .076 4484 33-7 .088 6494 34-09 .100 9924 34-49 ."3 4835 34-9 1 32 .076 6507 33-71 .088 8540 34-09 .101 1993 34-5 .113 6930 34-92 33 .076 8529 33-71 .089 0586 34.10 .101 4063 34-5 .113 9025 34-92 34 .077 0552 33-72 .089 2632 34-n .101 6134 34-5 1 .114 II2I 34-93 35 2.077 2575 33-73 2.089 4678 34-n 2.IOI 8204 34-52 2.114 3 2I 7 34-94 36 .077 4599 33-73 .089 6725 34.12 .IO2 0276 34-52 "4 53*3 34-95 37 .077 6623 33-74 .089 8772 34.12 .102 2347 34-53 .114 7410 34-95 38 .077 8647 33-74 .090 0820 34-13 .102 4419 34-54 .114 9508 34-96 39 .078 0672 33-75 .090 2868 34.14 .102 6492 34-54 .115 1605 34-97 10 2.078 2697 33-76 2.090 4917 34-15 2.102 8564 34-55 2.115 3704 34-97 41 .078 4723 33-76 .090 6966 34-15 .103 0638 34-56 .115. 5802 34-98 42 .078 6749 33-77 .090 9015 34.16 .103 2711 34.56 .115 7901 34-99 43 .078 8775 33-78 .091 1065 34-17 .103 4785 34-57 .116 oooi 35-0 44 .079 0802 33.78 .091 3115 34.17 .103 6860 34.58 .Il6 2101 35-oo 45 2.079 2829 33-79 2.091 5165 34.18 2.103 8935 34-59 2.1 1 6 4201 35-oi 46 .079 4857 33.80 .091 7216 34-19 .104 ioio 34-59 .Il6 6301 35-02 47 .079 6885 33.80 .091 9268 34-19 .104 3086 34.60 .Il6 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 2.080 2971 33.82 2.092 5424 34.21 2.104 93*6 34.62 2.II7 4710 35-05 51 .080 5000 33-83 .092 7477 34.22 i5 1393 34-63 .1 17 6813 35-5 52 .080 7030 33-83 .092 9530 34.22 .105 3471 34-63 .117 8916 35-o6 53 .080 9060 33-84 .093 1584 34-^3 .105 5549 34-64 .Il8 IO2O 35-07 54 .081 1091 33-85 093 3638 34-24 .105 7628 34-65 .Il8 3124 35.08 55 2.081 3122 33.8| 2.093 5692 34-*5 2.105 9707 34-66 2.118 5229 35.08 56 .081 5153 33-86 093 7747 34-25 .106 1786 34-66 .118 7334 35-09 57 .081 7185 33-87 .093 9803 34.26 .106 3866 34-67 .118 9440 35- 10 58 .081 9217 33-87 .094 1858 34-27 .106 5947 34.68 ,119 1546 | 35.10 59 .082 1249 33-88 .094 3914 34-27 .106 8027 34.68 119 3652 j 35- 11 6O 2.082 3282 33-88 2.094 5971 34.28 2.107 0109 34-69 2.119 5759 35-'2 590 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 100 101 102 103 logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". logM. Diff. I". O' 2.119 5759 35-12 2.132 2989 35-57 2.145 *866 36.03 2.158 2460 36.52 1 .119 7867 35-13 .132 5123 35-57 .145 4028 36.04 .158 4652 36.53 2 .119 9974 35-13 .132 7258 35-58 .145 6191 36.05 .158 6844 36.54 3 .120 2083 35-H 132 9393 35-59 H5 8354 36.06 .158 9036 36.55 4 .120 4191 35-15 133 1529 35-6o .146 0518 36.07 .159 1229 36.55 5 2.120 6301 35-i6 2.133 3665 35-6i 2.146 2682 36.07 2-159 3423 36.56 6 .I2O 8410 35-i6 .133 5802 35-6i .146 4847 36.08 .159 5617 36.57 7 .121 0520 35-17 !33 7939 35.62 .146 7012 36.09 .159 7811 36.58 8 .121 2630 35-i8 .134 0076 35-63 .146 9178 36.10 .160 0006 36.59 9 .121 4741 35-19 .134 2214 35-64 147 1344 36.11 .l6o 22O2 36.60 10 2. 121 6853 3S-*9 2-134 4352 35-64 2.147 3510 36.11 2.l6o 4398 36.60 11 .121 8965 35.20 .134 6491 35.65 147 5677 36.12 .l6o 6594 36.61 12 .122 1077 35-2i .134 8631 35-66 .147 7845 36.13 ,l6o 8791 36.62 13 .122 3190 35-2i .135 0770 35.67 .148 0013 36.14 .l6l 0989 36.63 14 .122 5303 35.22 .135 2910 35-67 .148 2182 36.15 .l6l 3187 36.64 15 2.122 7416 35-23 2-135 5051 35-68 2.148 4351 36.15 2.161 5385 36-65 16 .122 9530 35-24 .135 7192 35.69 .148 6520 36.16 .l6l 7584 36-65 17 .123 1644 35.24 135 9334 35-70 .148 8690 36.17 .l6l 9784 36.66 18 I2 3 3759 35-25 136 1476 35-71 .149 0861 36.18 .162 1984 36-67 19 .123 5875 35.26 .136 3619 35-71 .149 3032 36.19 .162 4185 36.68 20 2.123 799 35-27 2.136 5762 35-72 2.149 5203 36.19 2.162 6386 36.69 21 .124 0107 35-27 .136 7905 35-73 .149 7375 36.20 .162 8587 36.70 22 .124 2223 35.28 .137 0049 35-74 .149 9547 36.21 .163 0789 36.70 23 24 .124 4340 .124 6458 35-29 35-3 .137 2193 *37 4338 35-74 35-75 .150 1720 .150 3893 36.22 36-23 .163 2 99 2 .163 5195 36.71 36.72 25 2.124 8576 35-30 2.137 6484 35-76 2.150 6067 36-23 2.163 7398 36.73 26 .125 0694 35-31 .137 8630 35-77 .150 8242 36.24 .163 9602 36.74 27 .125 2813 35-32 .138 0776 35-77 .151 0417 36.25 .164 1807 36.74 28 I2 5 4933 35-33 .138 2922 35-78 .151 2592 36.26 .164 4OI2. 36.75 29 .125 7052 35-33 .138 5070 35-79 .151 4768 36.27 .164 62l8 36.76 30 2.125 9173 35-34 2.138 7217 35.80 2.151 6944 36.28 2.164 8424 36.77 31 .126 12,93 35-35 .138 93 6 5 35-8i .151 9121 36.28 .165 0630 36-78 32 .126 3414 35-35 .139 1514 35-8i .152 1298 36.29 .165 2837 36.79 33 .126 5536 35-3 6 .139 3663 35.82 .152 3476 36.30 .165 5045 36.80 34 .126 7658 35-37 .139 5813 35-83 .152 5654 36.31 .165 7253 36.81 35 2.126 9780 35-38 2-139 79 6 3 35-84 2.152 7833 36.32 2.165 9462 36.81 36 .127 1903 35-39 .140 0113 35-84 .153 0012 36.32 .l66 1671 36.82 37 .127 4027 35-39 .140 2264 35-85 .153 2I 9 2 36.33 .166 3881 36.83 38 .127 6151 35.40 .140 4415 35-86 153 4372 36.34 .166 6091 36-84 39 .127 8275 35-41 .140 6567 35.87 .153 6552 36.35 .166 8301 36.85 40 2.128 0400 35-42 2.140 8720 35-88 2-153 8734 36.35 2.167 0513 36.86 41 .128 2525 35-42 .141 0873 35-88 .154 0915 36.36 .167 2724 36-87 42 .128 4650 35-43 .141 3026 35.89 .154 397 36.37 .167 4936 36-87 43 .128 6776 35-44 .141 5180 35-9 .154 5280 36.38 .167 7149 36.88 44 .128 8903 35-45 Hi 7334 35-9 1 .154 7463 36.39 .167 9362 36.8 9 45 2.129 I0 3 35-45 2.141 9489 35-92 2.154 9647 36.40 2.168 1576 36.90 46 .129 3157 35-46 .142 1644 35-92 .155 1831 36.41 .168 3790 36.91 47 .129 5285 35-47 .142 3799 35-93 .155 4015 36.41 .l68 6005 36.92 48 .129 7414 35-48 .142 5955 35-94 .155 6200 36.42 .168 8220 36.93 49 .129 9542 35.48 .142 8112 35-95 .155 8386 36.43 .169 0436 36.93 50 2.130 1672 35-49 2.143 0269 35-96 2.156 0572 36.44 2.169 2652 36.94 51 52 .130 3801 .130 5931 35-5 35-51 .143 2427 .143 4585 35-96 35-97 .156 2759' .156 4946 36.45 36.46 .169 4869 .169 7087 36-95 36.96 53 .130 8062 35-51 .143 6743 35.98 .156 7133 36.46 .169 9304 36.97 54 .131 0193 35-52 .143 8902 35-99 .156 93 2I 36.47 .170 1523 36.98 55 2.131 2325 35-53 2.144 1062 36.00 2.157 1510 36-48 2.170 3742 36.99 56 H 1 4457 35-54 .144 3222 36.00 .157 3699 3 6 -49 .170 5961 36.99 57 .131 6589 35-54 .144 5382 36.01 -157 5889 36-5 .170 8181 37.00 58 .131 8722 35-55 .144 7543 36.02 .157 8079 36.50 .171 0401 37.01 59 .132 0855 35-5 6 .144 9704 36.03 .158 0269 36.51 .171 2622 37.02 60 2.132 2989 35-57 2.145 *866 36.03 2.158 2460 36.52 2.171 4844 37.03 591 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 104 105 106 107 logM. Diff. I". logM. Diff. I". log M. Diff. 1". logM. Diff. I". 0' 1 2 3 .171 4844 .171 7066 .171 9288 .172 1511 37.03 37.04 37-05 37.05 .184 9092 .185 1346 .185 3600 -l8 5 5855 37.56 37-57 37.57 37.58 .198 5282 .198 7568 .198 9856 .199 2144 38.11 38.12 38.13 38.14 .212 3493 .212 5814 .212 8136 .213 0458 38.68 38.69 38.70 38.71 4 .172 3735 37.06 .185 8110 37-59 .199 4432 38.14 .213 2781 38.72 5 .172 5959 37.07 .186 0366 37.60 .199 6721 38.15 .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 2078 38.76 9 .173 4860 .186 9395 37.64 .200 5882 38.19 .214 4404 38.77 10 .173 7087 37.12 .187 1653 37-65 .2OO 8174 38.20 2.214 673 38-78 11 .173 9314 37.12 .187 3912 37.66 .201 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-H .187 8432 37-67 .201 5053 38.23 215 3713 38.81 J _ rt 14 .174 5999 37-15 .188 0693 37-68 .201 7347 38.24 .215 6042 38.82 15 16 .174 8228 .175 0458 37.16 37-17 .188 2954 .188 5216 37.69 37-70 .201 9642 .202 1937 38.25 38.26 2.215 8371 .216 0701 38.83 38.84 17 r o o .175 2606 .188 7478 37-71 .202 4233 38.27 .216 3032 38.85 18 175 49*9 37.18 .188 9741 37-72 .202 6529 38.28 .216 5363 38.86 19 .175 7150 37-19 .189 2005 37-73 .202 8826 38.29 .216 7694 38.87 20 .175 9382 37-20 2.189 4269 37.74 2.203 1123 38.30 2.217 0027 38.88 21 .176 1615 37.21 .189 6533 37-75 .203 3421 38.31 .217 2360 38.89 22 .176 3848 37.22 .189 8798 37.76 .203 5720 38.31 217 4693 38.90 23 .176 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 2.2O4 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 37-8o .204 7 222 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 4.129 38.39 2.219 3382 38.98 31 .178 3968 37-3 .191 9209 37-84 .205 6433 38.40 .219 5721 3 8 -99 32 .178 6206 37.31 .192 1480 37-85 .205 8737 38.41 .219 8061 39.00 33 .178 8445 37-32 .192 3751 37.86 .206 1042 38-42 .220 0401 39.01 34 .179 0684 37-33 .192 6023 .206 3348 38.43 .220 2741 39-02 35 2.179 2924 37-33 2.192 8295 37-88 2.206 5654 38.44 2.22O 5082 39-03 36 .179 5164 37-34 .193 0568 37-88 .206 7961 38.45 .220 7424 39-4 37 .179 7405 37-35 .193 2841 37.89 .207 0268 38.46 .220 9767 39-5 38 .179 9646 37-36 .193 5115 37-9 .207 2575 38.47 .221 2110 39.06 39 .180 1888 37-37 .193 7389 37-91 .207 4884 38-48 .221 4453 39.07 40 2.180 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 39-09 42 .180 8617 37-4 .194 4216 37-94 .208 1812 38-5I .222 1488 39.10 43 .181 0861 37-4 1 .194 6493 37-95 .208 4123 38.52 .222 3834 39- 11 44 .181 3106 3741 .194 8770 37-96 .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-'3 46 .181 7597 37-43 -19S 3326 37.98 .209 1058 38.54 .223 0876 39- J 4 47 .181 9843 37-44 .195 5605 37-99 .209 3371 38.55 .223 3224 39-'5 48 .182 1089 37-45 .195 7885 38.00 .209 5685 38-56 223 5573 39.16 49 .182 4337 37.46 .196 0165 38.00 .209 7999 38.57 -223 7923 39- J 7 50 2.182 6584 37-47 2.196 2445 38.01 2.2IO 0314 38.58 2.224 0273 39.18 51 .182 8833 3748 .196 4726 38.02 .210 2629 38.59 .224 2624 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. 211 1896 38.63 2.225 2033 39-23 56 .184 0082 37-52 .197 6140 38-07 .211 4214 38-64 .225 4387 39-24 57 .184 2334 37-53 .197 8425 38.08 .211 6533 38.65 .225 6741 39-25 58 .184 4586 37-54 .198 0710 38.09 .211 8852 38.66 .225 9096 39.26 59 .184 6839 37-55 .198 2995 38.10 .212 1172 38.6 7 .226 1452 39.27 60 2.184 9092 37.56 2.198 5282 38.11 2.212 3493 38.68 2.226 3808 39.28 592 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 108 109 110 111 log M. Diff. 1". logM. Diff. 1". logM. Diff. I". logM. Diff. I". 0' 2.226 3808 39.28 2.240 6314 39.90 2.255 1099 4- 5 4 2.269 8255 41.21 1 .226 6165 39.29 .240 8708 39-91 255 3532 40.55 .270 0728 41.23 2 .226 8523 39.30 .241 1103 39-92 .255 5965 40.56 .270 3202 41.24 3 .227 0881 39-31 .241 3498 39-93 .255 8399 40.58 .270 5676 41.25 4 .227 3240 39-32 .241 5894 39-94 .256 0834 40.59 .270 8152 41.26 5 2.227 5599 39-33 2.241 8291 39-95 2.256 3270 40.60 2.271 0628 41.27 6 .227 7959 39-34 .242 0688 39-96 .256 5706 40.61 .271 3104 41.28 7 .228 0320 39-35 .242 3086 39-97 .256 8143 40.62 .271 5582 41.29 8 .228 2681 39-3 6 .242 5485 39.98 .257 0580 40.63 .271 8060 41.30 9 .228 5043 39-37 .242 7884 39-99 257 3 OI 9 40.64 .272 0538 41.32 10 2.228 7405 39.38 2.243 0284 40.00 2-257 5458 40.65 2.272 3018 4'-33 11 .228 9768 39-39 .243 2685 40.01 .257 7897 40.66 .272 5498 41.34 12 .229 2131 39.40 .243 5086 40.02 .258 0337 40.68 .272 7979 4'-35 13 .229 4496 39.41 .243 7488 40.03 .258 2778 40.69 .273 0460 41.36 14 .229 6861 39-4^ .243 9890 40.05 .258 5220 40.70 273 2942 41.38 15 2.229 9226 39-43 2..H4 2293 40.06 2.258 7662 40.71 2-273 5425 4*-39 16 .230 1592 39-44 .'.'.44 4697 40.07 .259 0105 40.72 273 799 41.40 17 .230 3959 39-45 .244 7101 40.08 259 2548 4-73 274 393 41.41 18 .230 6326 39-46 .244 9506 40.09 .259 4992 40.74 .274 2878 41.42 19 .230 8694 39-47 .245 1912 40.10 259 7437 40.75 -274 53 6 4 4 J -43 20 2.231 1063 39.48 2.245 43i8 40.11 2.259 9883 40.76 2.274 785 41.44 21 .231 3432 39-49 MS 6 725 40.12 .260 2329 40.78 .275 0337 41.46 22 .231 5802 39-5 -245 9*32 40.13 .260 4776 40.79 .275 2825 41.47 23 .231 8172 39-5 1 .246 1541 40.14 .260 7223 40.80 275 53'3 41.48 24 .232 0543 39-52 246 3949 40.15 .260 9671 40.81 .275 7802 41.49 25 2.232 2915 39-53 2.246 6359 40.16 2.26l 2120 40.82 2.276 0292 41.50 26 .232 5287 39-54 .246 8769 40.17 .26l 4570 40.83 .276 2783 41.51 27 .232 7660 39-55 .247 1180 40.18 .26l 7020 40.84 .276 5274 4i-53 28 29 .233 0033 .233 2407 39-5 6 39-57 .247 3591 .247 6003 40.19 40.21 .261 9471 .262 1922 40.85 40.86 .276 7766 .277 0258 4M4 41-55 30 2.233 4782 39-58 2.247 8416 40.22 2.262 4374 40.88 2.277 2752 41.56 31 233 7157 39-59 .248 0829 40.23 .262 6827 40.89 277 5246 4'-57 32 233 9533 39.60 .248 3243 40.24 .262 9281 40.90 .277 7740 41.58 33 .234 1910 39.61 .248 5658 40.25 .263 1735 40.91 .278 0236 41.60 34 .234 4287 39-63 .248 8073 40.26 .263 4190 40.92 .278 2732 41.61 35 2.234 6665 39- 6 4 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 5323 40.29 .264 1559 40.95 .279 0224 41.64 38 .235 3802 39-67 249 774i 40.30 .264 4016 40.96 .279 2723 41.65 39 .235 6183 39.68 .250 0159 40.31 .264 6474 40.98 .279 5223 41.67 40 2.235 8563 39-69 2.250 2578 40.32 2.264 8933 40.99 2.279 7723 41.68 41 .236 0945 39-7 .250 4998 4-34 .265 1393 41.00 .280 0224 41.69 42 .236 3327 39-7 1 .250 7419 4-35 265 3 8 53 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 4-37 .265 8776 41.03 .280 7731 41.72 45 2.237 0478 39-74 2.251 4684 40.38 2.266 1238 41. OA 2.281 0235 41.74 46 .237 2862 39-75 .251 7107 40.39 .266 3701 41.06 .281 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 4*-77 49 .238 OO2O 39.78 .252 4380 40.42 .267 1094 41.0 9 .282 0258 41.78 50 2.238 2407 39-79 2.252 6806 4-43 2.267 356 41.10 2.282 2765 41.80 51 .238 4795 39.80 .252 9232 40.44 .267 6026 41.11 .282 5273 41.81 52 .238 7284 39.81 253 l6 59 40.46 .267 8493 41.12 .282 7782 41.82 53 238 9573 39.82 .253 4087 40.47 .268 0961 41.13 .283 0291 41.83 54 .239 1962 39-83 -253 6515 40.48 .268 3430 41.15 .283 2801 41.84 55 2-239 4353 39-84 2.253 8944 40.49 2.268 5899 41.16 2.283 5712 41.85 56 .239 6744 39-86 -254 1374 40.50 .268 8369 41.17 -283 7824 41.87 57 .239 9235 39-87 .254 3804 40.51 .269 0839 4I.I8 .284 0336 41.88 58 .240 1528 39.88 254 6 235 40.52 .269 3310 41.19 .284 2849 41.89 59 .240 392,1 39.89 .254 8666 4-53 .269 5782 41.20 .284 5363 41.90 60 2.240 6314 39.90 2.255 1099 40.54 2.269 8255 41.21 2.284 7878 41.91 593 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 112 113 114 115 V. logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". 0' 1 2 3 4 .284 7878 .285 0393 .285 2909 285 5425 285 7943 41.91 4*-,93 41.94 41.95 41.96 .300 0067 .300 2626 .300 5186 .300 7746 .301 0307 42.64 42.65 42.67 42.68 42.69 .315 4927 .315 7531 ,316 0136 .316 2742 .316 5348 43-4 43-41 43-42 43-44 43-45 .331 2564 .331 5216 .331 7868 332 0521 332 3!75 44.18 44.20 44.21 44.22 44:24 5 .286 0461 41.97 .301 2869 42.70 .316 7956 43-46 .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*73 43-49 .333 1141 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 .317 8393 43-5 1 333 6456 44.31 10 .287 3062 42.03 .302 5689 42.76 2.318 1004 43-53 333 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 33 339 42.80 .318 8842 43.56 334 7096 44.36 14 .288 3155 42.08 .303 5958 42.81 .319 1456 43-58 334 9758 44-37 15 .288 5680 42.09 303 8528 42-83 2.319 4072 43-59 335 2421 44-39 16 .288 8206 42.10 .304 1098 42.84 .319 6687 43.60 335 5 84 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 .289 8317 42.15 .305 1385 42.89 2.320 7159 43.66 2.33 6 5747 44-45 21 .290 0847 42.16 35 3959 42.90 .320 9778 43-67 336 84x4 44-47 22 290 3377 42.18 35 6533 42.91 .321 2399 43-68 337 1-083 44-48 23 .290 5908 42.19 .305 9109 42.93 .321 5020 43-69 337 3752 44-49 24 .290 8440 42.20 .306 1685 42.94 .321 7642 43-7 337 6422 44- 5 i 25 .291 0972 42.21 2.306 4261 42-95 2.322 0265 43.72 2.337 9093 44-52 26 27 .291 3505 .291 6039 42.22 42.24 .306 6839 .306 9417 42.96 42.98 .322 2889 .322 5513 43-73 43-75 338 1765 .338 4437 44-53 44-55 28 .291 8574 42.25 .307 1996 42.99 322 8139 43-76 .338 7111 44-56 29 .292 1109 42.26 .307 4576 43.00 .323 0765 43-77 338 9785 44-58 30 .292 3645 42.27 2.307 7157 43.02 2.323 339 1 43-79 2.339 2460 44-59 31 .292 6182 42.29 37 973 s 43-3 .323 6019 43.80 339 5135 44-6o 32 .292 8719 42.30 .308 2320 43-04 .323 8647 43.81 339 7812 44.62 33 .293 1258 42-31 .308 4903 43-5 .324 1277 43.83 .340 0490 44.63 34 293 3797 42.32 .308 7486 43.07 324 397 43-84 .340 3168 44.64 35 2.293 6 336 42-33 2.309 0071 43.08 2.324 6537 43-85 2.340 5847 44-66 36 293 8877 42.35 .309 2656 43-09 .324 9169 43-87 .340 8527 44-67 37 .294 1418 42.36 .309 5242 43.10 .325 1801 43-88 .341 1207 44.69 38 294 396o 42.37 .309 7828 43.12 325 4434 43-89 .341 3889 44.70 39 294 6 503 42.38 .310 0416 43-13 325 7o68 43-91 .341 6571 44.71 40 2.294 9046 42.40 2.310 3004 43.14 2-325 973 43-92 2.341 9255 44-73 41 .295 1590 42.41 .310 5593 43- J 5 326 2339 43-93 342 1939 44-74 42 295 4135 42.42 .310 8182 43- 1 ? 326 4975 43-94 342 4623 44-75 43 .295 6680 42.43 .311 0773 43.18 .326 7612 43-96 342 739 44-77 44 .295 9227 42-44 3 11 33 6 4 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 2-343 2683 44-80 46 .296 4321 42.47 .311 8549 43-22 327 5528 44.00 343 537i 44-81 47 .296 6870 42.48 .312 1142 43-23 .327 8168 44.01 .343 8060 44.82 48 .296 9419 42.49 .312 3736 43-24 .328 0809 44.02 344 75 44.84 49 .297 1969 42.51 .312 6331 43.26 328 3451 44.04 344 344 44.85 50 2.297 4520 42.52 2.312 8927 43-27 2.328 6094 44.05 2-344 6132 44.86 51 .297 7071 42.53 313 '524 43.28 .328 8737 44.06 344 8824 44-88 52 297 9623 42.54 .313 4121 43-29 .329 1382 44.08 345 IS 1 ? 44.89 53 .298 2176 42.55 .313 6719 43-3 1 .329 4027 44.09 345 4211 44.91 54 .298 4730 .313 93x8 43-32 .329 6672 44.10 .345 6906 44-92 55 2.298 7284 42.58 2.314 1917 43-33 2.329 9319 44.12 2.345 9601 44-93 56 .298 9839 42.59 .314 4518 43-35 33 *9 6 7 44.13 .346 2298 44-95 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 34 6 7693 44-97 59 .299 7509 42.63 .315 2323 33 99H 44.17 347 0392 44-99 60 2.300 0067 42.64 2.315 4927 43-40 2.331 2564 44.18 2-347 3092 45.00 594 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 116 117 118 119 logM. Diff. I". logM. Diff. 1". log M. Diff. 1". logM. Diff. I". 0' 2.347 3092 45.00 2.363 6626 45.86 2.380 3290 46.74 2.397 3210 47.66 1 347 5792 45-2 3 6 3 9378 45.87 .380 6095 46.76 .397 6070 47.68 2 347 8494 45-3 .364 2131 45.88 .380 8901 46.77 397 8931 47-70 3 .348 1196 45.04 .364 4885 45.90 .381 1708 46.79 .398 1794 47.71 4 .348 3899 45.06 3 6 4 7639 45.91 .381 4515 46.80 398 4657 47-73 5 2.348 6603 45-07 2-3 6 5 394 45-93 2.381 7324 46.82 2.398 7521 47-74 G .348 9308 45-09 .3 6 5 3150 45-94 .382 0133 46.83 399 386 47.76 7 349 2014 45.10 365 5907 45.96 .382 2944 46.85 399 3252 47-77 8 349 4720 45'" .365 8665 45-97 382 5755 46.86 399 6lI 9 47-79 9 349 74*8 45-13 .366 1423 45-99 382 8567 46.88 399 8987 47.81 10 2.350 0136 45-14 2.366 4183 46.00 2.383 1380 46.89 2.400 1856 47.82 11 .350 2845 45.16 .366 6944 46.01 383 4194 46.91 40 4725 47.84 12 350 5554 45-17 .366 9705 46.03 .383 7009 46.92 .400 7596 47.85 13 .350 8265 45.18 .367 2467 46.04 .383 9825 46.94 .401 0468 47.87 14 .351 0977 45.20 .367 5230 46.06 .384 2642 46.95 .401 3340 47.89 15 2.351 3689 45.21 2.3 6 7 7994 46.07 2.384 5460 46.97 2.401 6214 47-9 16 .351 6402 45.23 .368 0759 46.09 .384 8278 46.98 .401 9088 47-92 17 .351 9116 45.24 368 3525 46.10 .385 1098 46.99 .402 1964 47-93 18 .352 1831 45-25 .368 6291 46.12 .385 39i8 47.01 .402 4840 47-95 19 352 4547 45.27 .368 9059 46.13 385 6739 47.03 .402 7718 47-97 20 2.352 7263 45.28 2.369 1827 46.15 2-385 9562 47.05 2.403 0596 47.98 21 .352 9981 45.30 .369 4596 46.16 .386 2385 47.06 43 3475 48.00 22 353 2699 45-3 * .369 7367 46.18 .386 5209 47-08 .403 6356 48.01 23 353 54i8 45-33 .370 0138 46.19 .386 8034 47-09 43 9237 48.03 24 353 8138 45-34 .370 2909 46.21 .387 0860 47.11 .404 2119 48.04 25 2.354 0859 45-35 2.370 5682 46.22 2.387 3687 47. i a 2.404 5002 48.06 2G 354 358i 45-37 .370 8456 46.24 387 6514 47.14 .404 7886 48.08 27 354 6303 45.38 .371 1230 46.25 387 9343 47.15 .405 0771 48.09 28 354 927 45-40 .371 4006 46.26 .388 2173 47-17 45 3657 48.11 29 355 i75i 45-4 1 .371 6782 46.28 .388 5003 47.18 45 6544 48.12 30 2-355 447 6 45-42 2-371 9559 46.29 2-388 7835 47.20 2.405 9432 48.14 31 355 7*02 45-44 .372 2337 46.31 .389 0667 47.21 .406 2321 48.16 32 355 99*8 45-45 372 5 116 46.32 .389 3500 47-23 .406 5211 48.17 33 .356 2656 45-47 .372 7896 46.34 389 6 335 47.24 .406 8102 48.19 34 35 6 5385 45.48 373 6 77 46.35 .389 9170 47.26 .407 0993 48.20 35 2.356 8114 45-5 2-373 3459 46.37 2.390 2006 47.28 2.407 3886 48.22 36 357 0844 45-5i -373 6241 46.38 .390 4843 47.29 .407 6780 48.24 37 357 3575 45-52 373 9024 46.40 .390 7681 47-31 .407 9674 48.25 38 357 6307 45-54 .374 1809 46.41 .391 0519 47-32 .408 2570 48.27 39 357 904 45-55 374 4594 46.43 391 3359 47-34 .408 5467 48.28 40 2-358 1773 45-57 2-374 738o 46.44 2.391 6200 47-35 2.408 8364 48.30 41 .358 4508 45.58 375 Ol6 7 46.46 392 9042 47-37 .409 1263 48-32 42 358 7M3 45.60 375 2955 46.47 .392 1884 47.38 .409 4162 48.33 43 358 9979 45.61 375 5744 46.49 .392 4728 47.40 .409 7063 48.35 44 359 2 7i6 45.62 375 8533 46.50 .392 7572 47.41 .409 9964 48.37 45 2-359 5454 45.64 2.376 1324 46.51 2.393 04*7 47-43 2.410 2866 48.38 46 359 8193 45.65 .376 4115 46.53 393 3264 47-45 .410 5770 48.40 47 .360 0933 45.67 .376 6908 46.55 .393 6111 47-46 .410 8674 48.41 48 .360 3673 45.68 .376 9701 46.56 393 8959 47.48 .411 1579 48.43 49 .360 6415 45.70 377 2495 46.58 .394 1808 47-49 .411 4486 48.45 50 2.360 9157 45-7 1 2.377 5290 46-59 2.394 4658 47-51 2.411 7393 48.46 51 .361 1900 45-72 .377 8086 46.60 394 759 47-52 .412 0301 48.48 52 .361 4644 45-74 .378 0883 46.62 395 3 61 47-54 .412 3210 48.49 53 .361 7389 45-75 .378 3681 46.64 395 32H 47-55 .412 6120 48.51 54 .362 0134 45-77 .378 6479 46.65 395 6067 47-57 .412 9031 48.53 55 2.362 2881 45.78 2-378 9279 46.67 2-395 8922 47-59 2.413 1944 48.54 56 .362 5628 45.80 .379 2079 46.68 .396 1778 47.60 .413 4857 48.56 57 .362 8376 45.81 379 4881 46.70 .396 4634 47.62 4'3 777i 48.58 58 .363 1126 45.82 .379 7683 46.71 39 6 7492 47- 6 3 .414 0686 48.59 59 .363 3876 45.84 .380 0486 46-73 397 0350 47-65 .414 3602 48.61 60 2.363 6626 45.86 2.380 3290 46.74 2.397 3210 47.66 2.414 6519 48.62 595 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 120 121 122 123 logM. Diflf. 1". logM, Dlff. I". logM. Diflf. 1". logM. Diff. 1". 0' 1 9 3 4 .414 6519 .414 9437 .415 2356 .415 5276 .415 8197 48.62 48.64 48.66 48.67 48.69 2.432 3356 .432 6334 43 2 93*3 -433 22 93 433 5 2 74 49.62 49 1t 49.66 49.68 49.69 .450 3868 .450 6908 .450 9950 45 1 2 99 2 .451 6036 50.67 50.68 50.70 50.72 50-74 .468 8205 .469 1311 .469 4418 .469 7526 .470 0634 51-75 51-77 5'-79 51.81 51.82 5 .416 1119 48.71 2-433 82 57 49.71 .451 9081 5-75 .470 3744 5'*i 6 .416 4042 48.72 .434 1240 49-73 .452 2127 50.77 .470 6856 51.86 7 .416 6965 48.74 434 4 22 4 49-74 .452 5174 5-79 .470 9968 51.88 8 .416 9890 48.76 .434 7209 49.76 .452 8222 50.81 .471 3081 51.90 9 .417 2816 48.77 435 OI 95 49.78 453 ' 2 7i 50.83 .471 6196 51.92 1O .417 5743 48.79 2.435 3 lg 2 49.80 453 43 21 5 -oi .471 9311 51-94 11 .417 8671 48.81 435 6l 7* 49.81 453 737 2 50.86 .472 2428 51-95 12 .418 1600 48.82 435 9160 49- 8 3 454 4 2 4 50.88 .472 5546 51-97 13 .418 4529 48.84 .436 2150 49- 8 5 454 3477 50.90 .472 8665 51.99 14 .418 7460 48.85 .436 5141 49.86 454 6 53 2 50.92 473 '7 8 5 52.01 15 .419 0392 48.87 2.436 8134 49.88 454 95 8 7 5-93 .473 4906 52.03 16 4*9 33 2 5 48.89 .437 1127 49.90 .455 2644 50.95 .473 8028 5 2 -5 17 .419 6258 48.90 .437 4122 49.92 455 57i 50-97 .474 1152 52.07 18 .419 9193 48.92 437 7"7 49-93 455 8 7 6 50.99 .474 4276 52.09 19 .420 2129 48.94 .438 0114 49-95 .456 1820 51.00 474 74 2 52.10 20 2.420 5066 48.95 2.438 3111 49-97 2.456 4881 51.02 . 2.475 0529 52.12 21 22 .420 8003 .421 0942 48.97 48.99 .438 6110 .438 9109 49.98 50.00 45 6 7943 .457 1006 51.04 51.06 475 3657 .475 6786 52.14 52.16 23 .421 3882 49.00 439 2II 50.02 457 4 7o 51.08 475 99 l6 52.18 24 .421 6822 49.02 439 5" 2 50.04 457 7135 51.09 .476 3047 52.20 25 2.421 9764 49.03 2.439 8lI 4 50.05 2.458 0201 51.11 2.476 6180 52.22 26 .422 2707 49.05 .440 1118 50.07 .458 3268 $i-?3 .476 9313 52.23 27 .422 5650 49.07 .440 4123 50.09 45 8 6 337 5z. 15 477 2 44 8 52.25 28 .422 8595 49.09 .440 7129 50.11 .458 9406 5i-i7 477 55 8 4 52.27 29 .423 1541 49.10 .441 0136 50.12 -459 2 477 51.18 .477 8721 52.29 30 2.423 4488 49.12 2 -44 J 3*43 50.14 2.459 554 8 51.20 2.478 1859 5 2 -3' 31 .423 7435 49.14 .441 6152 50.16 .459 8621 51.22 .478 4998 5 2 -33 32 .424 0384 49.15 .441 9162 50.18 .460 1695 51.24 .478 8138 5 2 -35 33 4 2 4 3334 49.17 .442 2173 50.19 .460 4770 51.26 .479 1280 5 2 -37 34 .424 6284 49.19 .442 5185 50.21 .460 7846 51.28 479 44 22 52.39 35 2.424 9236 49.20 2.442 8199 50.23 2.461 0923 51.29 2.479 7566 52.40 36 .425 2189 49.22 443 I2I 3 50.24 .461 4001 5i-3i .480 0711 52.42 37 .425 5142 49.24 .443 42 a 50.26 .461 7080 5 I -33 .480 3857 52.44 38 .425 8097 49.25 443 7 2 44 50.28 .462 0161 51-35 .480 7004 52.46 39 .426 1053 49.27 .444 0261 50.30 .462 3242 $i-f? .481 0152 52.48 40 2.426 4010 49.29 2.444 3 28 50.31 2.462 6325 51.38 2.481 3301 52.50 41 .426 6967 49-3 .444 6299 5-33 .462 9408 51.40 .481 6452 52.52 42 .426 9926 49-3 2 .444 9320 5-35 .463 2493 51.42 .481 9604 52.54 43 .427 2886 49-34 445 2 34i 5-37 4 6 3 5579 51.44 .482 2756 52.56 44 .427 5847 49-35 445 53 6 4 50.38 .463 8666 51.46 .482 5910 52.58 45 2.427 8808 49-37 2.445 8 3 8 7 50.40 2.464 1754 51.48 2.482 9065 52.59 46 .428 1771 49-39 .446 1412 50.42 .464 4843 51.49 .483 2222 52.61 47 .428 4735 49.40 .446 4437 50.44 464 7933 5*-$* 483 5379 52.63 48 .428 7700 49.42 .446 7464 50.45 .465 1024 5i-53 -483 8 537 52.65 49 .429 0665 49-44 447 0492 5-47 .465 4116 51-55 .484 1697 52.67 50 2.429 3632 49.46 2.447 35 2 ' 50.49 2.465 7210 5i-57 2.484 4858 52.69 51 .429 6600 49-47 447 6 55* 50.51 .466 0305 5*-59 .484 8020 52.71 52 .429 9569 49-49 .447 9582 5-53 .466 3400 51.60 .485 1183 5 2 -73 53 .430 2539 49-51 .448 2614 50.54 .466 6497 51.62 .485 4347 5 2 -75 54 .430 5510 49.52 .448 5647 50.56 .466 9595 51.64 4 8 5 75*3 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 -467 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 .431 7403 49-59 449 779 50.63 .468 1997 51.71 .487 0186 52.84 59 .432 0379 49.61 .450 0828 50.65 .468 5101 51-73 4 8 7 3357 52.86 60 2.432 3356 49.62 2.450 3868 50.67 2.468 8205 51-75 2.487 6529 52.88 596 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 124 125 126 127 logK, Diff. I". logM. Diff. I". logM. Diff. 1". log M. Diff. 1". O' 2.487 6529 52.88 2.506 9006 54.06 2.526 5813 55-29 2-546 7135 56.57 1 .487 9702 52.90 .507 2251 54.08 .526 9131 55-3 1 547 53 56.59 2 .488 2877 52.92 .507 5496 54.10 .527 2450 55-33 547 3926 56.61 3 .488 6053 52-94 .507 8742 54.12 527 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 2.489 2408 52.98 2.508 5239 54.16 2.528 2415 55-39 2.548 4122 56.68 6 .489 5587 53.00 .508 8489 54.18 528 5739 55-4i 548 7523 56.70 7 .489 8767 53-02 .509 1741 54-20 .528 9065 55-43 .549 0926 56.72 8 .490 1949 53-3 59 4993 54-22 .529 2391 55-45 549 433 56.74 9 .490 5132 53-05 .509 8247 54.24 529 5719 55.48 549 7735 56.76 1O 2.490 8315 53-07 2.510 1502 54.26 2.529 9048 55-5 2.550 1141 56.79 11 .491 1500 53-9 .510 4758 54.28 530 2379 55-52 55 4549 56.81 12 .491 4686 53-n .510 8016 54-3 .530 5710 55-54 55 7958 56.83 13 .491 7874 53-13 .511 1274 54-32 53 9043 55-56 .551 1369 56.85 14 .492 1063 53-15 5" 4534 54-34 .531 2378 55.58 .551 4781 56.8 7 15 2.492 4252 53-17 2.511 7795 54.36 2-531 5713 55-60 2.551 8194 56.90 16 492 7443 53-19 .512 1057 54-38 53 1 95 55-62 .552 1608 56.92 17 18 493 6 35 .493 3828 '53- 2 i 53-23 .512 4321 .512 7586 54-4 54.42 532 2388 532 5727 55.64 55- 6 7 .552 5024 .552 8441 56.96 19 493 7023 53-25 .513 0852 54-44 .532 9068 55-69 553 l8 59 56.98 20 2.494 0218 53-27 2.513 4119 54-46 2.533 2410 55-71 2-553 5279 57.01 21 494 3415 53-29 5^3 7387 54.48 533 5753 55-73 553 8700 57.03 22 .494 6613 53-31 .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 8 97i 57.10 25 2.495 6213 53-37 2-5I5 473 54.56 2-534 9138 55.81 2.555 2398 57-12 26 495 94i6 53-39 5!5 3747 54-58 535 2487 55-84 555 5825 57.14 27 .496 2619 53-41 515 7023 54.60 535 5838 55-86 555 9254 57.16 28 .496 5824 53-42 .516 0300 54-63 535 9190 55-88 .556 2685 57.i8 29 .496 9030 53-44 .516 3578 54-65 53 6 2543 55.90 .556 6116 57-21 30 2.497 2238 53.46 2.516 6857 54-67 2.536 5898 55-92 2-556 9549 57-23 31 .497 5446 53-48 .517 0138 54-69 536 9254 55-94 557 2984 57-25 32 497 ^656 53-50 .517 3420 54-7i 537 2611 55-96 557 6420 57*27 33 .498 1867 53-52 .517 6703 54-73 537 5970 55.98 557 9857 57.29 34 .498 5079 53-54 .517 9987 54-75 537 9329 56.01 55 8 3295 57-32 35 2.498 8292 53-5 6 2.518 3273 54-77 2.538 2690 56.03 2-558 6735 57-34 36 .499 1506 53.58 .518 6559 54-79 .538 6052 56.05 559 OI 7 6 57.36 37 .499 4721 53.60 .518 9847 54-8i .538 9416 56.07 559 3618 57-38 38 499 7938 53.62 5>9 3i37 54.83 539 2781 56.09 559 7062 57.41 39 .500 1156 53- 6 4 .519 6427 54.85 539 6 H7 56.11 .560 0507 57-43 40 2.500 4375 53.66 2.519 9719 54.87 2-539 9514 56.13 2.560 3953 57-45 41 5 7595 53-68 .520 3012 54.89 .540 2883 56.15 .560 7401 57-47 42 .501 0817 53.70 .520 6306 54-9 1 .540 6253 56.18 .561 0850 57.50 43 .501 4039 53-72 .520 9601 54-93 .540 9625 56.20 .561 4301 57.52 44 .501 7263 53-74 .521 2898 54-95 .541 2997 56.22 .561 7753 57-54 45 2.502 0488 53-7 6 2.521 6196 54-97 2.541 6371 56.24 2.562 1206 57.56 46 .502 3714 53-78 .521 9495 54-99 .541 9746 56.26 .562 4660 57-59 47 .502 6942 53.80 522 2795 55.02 542 3*23 56-29 .562 8116 57.6i 48 .503 0170 53.82 .522 6097 55.04 .542 6500 56-31 563 1574 57-63 49 .503 3400 53-84 .522 9400 55.06 .542 9880 56.33 .563 5032 57.65 50 2.503 6631 53.86 2.523 2704 55.08 2-543 3260 56-35 2.563 8492 57-68 51 .503 9863 53-88 .523 6009 55.10 543 6641 56-37 564 1953 57-7 52 .504 3096 53-9 .523 9316 55-12 .544 0024 56.39 .564 5416 57-72 53 .504 6331 53-92 .524 2624 55-H 544 349 56.42 .564 8880 57-74 54 .504 9567 53-94 524 5933 55.16 544 6 794 56.44 565 2345 57-77 55 2.505 2804 53-9 6 2.524 9243 55.18 2.545 0181 56.46 2-565 5812 57-79 56 .505 6042 53-98 525 2555 55.20 545 3569 56.48 .565 9280 57.81 57 .505 9282 54.00 .525 5867 55.22 545 6959 56.50 .566 2750 57.84 58 .506 2522 54-02 .525 9181 55-24 546 0350 56.52 .566 6221 57-86 59 .506 5763 54-04 526 2497 55.26 .546 3742 56-55 .566 9693 57.88 60 2.506 9006 54.06 2.526 5813 55-29 2-546 7135 5 6 -57 2.567 3166 57-9 597 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 128 129 130 131 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". 0' 1 2 3 4 567 3166 567 6641 568 0117 5 68 3595 .568 7074 57.90 57-93 57-95 57-97 57-99 588 4112 588 7670 589 1230 589 4792 589 8355 59-3 59-32 59-35 59-37 59-39 610 0188 6 10 3834 610 7481 611 1130 611 4781 60.75 60.78 60.80 60.83 60.85 .632 1622 .632 5360 .632 9099 .633 2839 633 6581 62.28 62.30 62.33 62.35 62.38 5 6 7 8 9 569 0554 .569 4036 569 75*9 .570 1004 .570 4490 58.02 58.04 58.06 58-09 58.11 590 1919 59 5485 590 9052 .591 2620 .591 6190 59.42 59-44 59-47 59-49 59-5 1 611 8433 .612 2086 .612 5741 .612 9397 .613 3055 60.88 60.90 60.93 60.95 60.98 .634 0325 634 47 .634 7817 635 ^65 635 5315 62.41 62.43 62.46 62.48 62.51 10 11 12 .570 7977 .571 1465 57 1 4955 58.13 58.15 58.18 .591 9762 592 3335 592 6909 59-54 59-56 59-58 .613 6715 .614 0376 .614 4038 6 1. oo 61.03 61.05 .635 9066 .636 2819 .636 6573 62.54 62.56 13 14 .571 8447 572 1939 58.20 58.22 .593 0485 593 4062 59.61 59-63 .614 7702 .615 1368 6 1. 08 61.10 637 329 .637 4 87 62.64 15 16 572 5434 .572 8929 58.25 58.27 593 7641 .594 1221 59.66 59-68 615 535 .615 8703 61.13 61.15 .637 7846 .638 1607 62.67 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 '970 59-75 .616 9718 61.23 .639 2899 62.77 20 21 22 574 2925 574 6427 574 993 1 58-36 58.38 58.41 595 5556 595 9*43 .596 2732 59.78 59.80 59.82 .617 3392 .617 7068 .6.18 0746 61.25 61.28 61.30 2.639 6666 .640 0435 .640 4205 62.80 62.82 62.85 23 24 575 3436 575 6943 58.43 58.45 .596 6322 596 99H 59-85 59.87 .618 4425 .618 8105 61.33 61.36 .640 7977 .641 1750 62.88 62.90 25 .576 0451 58.48 2-597 357 59-9 2.619 J 787 61.38 2.641 5525 62.93 26 .576 3960 58.50 597 7 102 59.92 .619 5471 61.41 .641 9302 62.96 27 J ' r J ' 576 747 i 58.52 .598 069? 59-95 .619 9156 61.44 .642 3080 62.98 28 29 577 0983 577 4496 58.55 58.57 598 4295 598 7894 59-97 59-99 .620 2843 .620 6531 61.46 61.48 .642 6860 .643 0641 63.01 63.04 30 2.577 8011 58.59 2-599 J 494 60.02 2.621 0220 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 32 578 545 58.64 599 8699 60.07 .621 7604 61.56 644 1995 63.12 33 .578 8564 58.66 .600 2304 60.09 .622 1298 61.58 644 5783 63.14 34 579 2085 58.69 .600 5910 60.12 .622 4994 61.61 .644 9572 63.17 35 2.579 5607 58-71 2.600 9518 60.14 2.622 8691 61.63 2-645 3363 63.19 36 579 9 I 3 58.73 .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 63-30 40 2.581 3237 58.83 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 8432 60.34 .625 8325 61.84 .648 3748 63.41 44 .582 8267 58.92 .604 2053 60.36 .626 2036 61.86 648 7553 63-44 45 2.583 0903 58-94 2.604 5675 60.38 2.626 5748 61.89 2.649 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 63.52 48 584 15*9 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 5 8 5 5694 59.11 .607 1073 60.56 .629 1780 62.07 .651 8053 63.65 53 585 9241 59-'3 .607 4707 60.58 .629 5505 62.09 .652 1873 63.68 54 .586 2790 59.16 607 8343 60.61 .629 92*31 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 63-73 56 .586 9891 59.20 .608 5618 60.66 .630 6689 62.17 653 3342 63.76 57 587 3444 59-23 .608 9258 60.68 .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 63-87 598 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 132 133 134 135 log M. Diff. I". log M. Diff. I". logM. Diff. 1". logM. Diff. 1". 0' 2.654 8657 63.87 2.678 1547 65-53 2.702 0562 67.27 2.726 5990 69.09 1 .655 2490 63.89 .678 5480 65.56 .702 4600 67.30 .727 0137 69.12 2 .655 6324 63.92 .678 9414 65-59 .702 8638 67.33 .727 4285 69.15 3 .656 0160 63-95 .679 3350 65.61 .703 2679 67.36 .727 8435 69.19 4 .656 3998 63-97 .679 7288 65.64 .703 6721 67.39 .728 2587 69.22 5 2.656 7837 64.00 2.68o 1227 65.67 2.704 0766 67.42 2.728 6741 69. 2C 6 .657 1678 64.03 .680 5168 65.70 .704 4812 67.45 .729 0897 6 9 .28 7 .657 5521 64.06 .680 9111 65-73 .704 8860 67.48 7*9 555 69.31 8 .657 9365 64.08 .681 3056 65.76 .705 2909 67.51 .729 9215 69-34 9 .658 3211 64.11 ,68l 7002 65.79 .705 6961 67-54 .730 3376 69.37 10 2.658 7058 64.14 2.682 0950 65.81 2.706 1014 67.57 2-73 7539 69.40 11 .659 0907 64.17 .682 4900 65.84 .706 5069 67.60 73 1 J 75 69.44 12 6 59 475 8 64,19 .682 8851 65.87 .706 9126 67.63 .731 5872 69.47 13 .659 8611 64.22 .683 2804 65.90 .707 3184 67.66 .732 0041 69.50 14 .660 2465 64.25 .683 6759 65-93 .707 7244 67.69 .732 4212 69-53 15 2.660 6320 64.28 2.684 7 l6 65.96 2.708 1307 67.72 2.732 8385 69.56 16 .661 0178 64.30 .684 4674 65-99 .708 5371 67.75 733 2 559 69-59 17 .661 4037 6 4-33 .684 8634 66. 01 .708 9436 67.78 733 6736 69.62 18 .661 7897 64.36 .685 2596 66.04 .709 3504 67.81 .734 0914 69.66 19 .662- 1760 64.38 .685 6559 66.07 79 7573 67.84 734 594 69.69 20 2.662 5623 64.41 2.686 0524 66.10 2.710 1645 67.87 2.734 9277 69.72 21 .662 9489 64.44 .686 4491 66.13 .710 5718 67.90 735 3461 69-75 22 663 3356 64.47 .686 8460 66.16 .710 9792 67-93 735 7647 69.78 23 .663 7225 64.49 .687 2430 66.19 .711 3869 67.96 .736 1835 69.81 24 .664 1096 64.52 .687 6402 66.22 .711 7947 67.99 .736 6025 69.85 25 2.664 4968 64-55 2.688 0376 66.25 2.712 2028 68.02 2.737 0216 69.88 26 .664 8842 64.57 .688 4352 66.27 .712 61 10 68.05 737 441 69.91 27 .665 2717 64.60 .688 8329 66.30 .713 0194 68.08 .737 8605 69.94 28 .665 6594 64.63 .689 2308 66.33 .713 4279 68.11 .738 2803 69.97 29 .666 0473 64.66 .689 6289 66.36 .713 8367 68.14 .738 7002 70.00 3O 2.666 4354 64.69 2.690 0272 66.39 2.714 2456 68. i 7 2.739 1203 70.04 31 .666 8236 64.72 .690 4256 66.42 .714 6547 68.20 739 54 6 70.07 32 .667 2I2O 64.74 .690 8242 66.45 .715 0640 68.23 -739 9612 7O.IO 33 .667 6005 64.77 .691 2230 66.48 7*5 4735 68.26 .740 3819 70.13 34 .667 9892 64.80 .691 6219 66.51 .715 8832 68.29 .740 8027 70.l6 35 2.668 3781 64.83 2.692 O2IO 66.54 2.716 2930 68.32 2.741 2238 7O.2O 36 .668 7672 64.86 .692 4203 66.56 .716 7031 68.35 .741 6451 70.23 37 .669 1564 64.88 .692 8198 66.59 7*7 H33 68.38 .742 0666 70.26 38 .669 5457 64.91 .693 2194 66.62 7 J 7 5*37 68.41 .742 4882 70.29 39 669 9353 64.94 .693 6193 66.65 .717 9342 68.44 .742 9101 70.32 40 2.670 3250 64.97 2.694 0193 66.68 2.718 3450 68.48 2 -743 33 21 70.36 41 .670 7149 65.00 .694 4194 66.71 .718 7560 68.51 743 7543 70-39 42 .671 1050 65.02 .694 8198 66.74 .719 1671 68.54 744 1768 70.42 43 .671 4952 65.05 .695 2203 66.77 .719 5784 68.57 .744 5994 70.45 44 .671 8856 65.08 .695 62IO 66.80 .719 9899 68.60 .745 0222 7 0. 4 8 45 2.672 2761 65.11 2.696 0219 66.83 2.720 4016 68.63 2 -745 445 2 70.52 46 .672 6668 6 5- I 3 .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 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 I39i 70.65 50 2.674 2314 65.25 2.698 0289 66.97 2.722 4628 68.78 2.747 5631 70.68 51 .674 6230 65.28 .698 4308 67.00 .722 8756 68. 81 747 9 8 73 70.71 52 .675 0147 65.30 .698 8330 67.03 .723 2885 68.84 .748 4116 70.74 53 .675 4066 65-33 .699 2353 67.06 .723 7017 68.88 .748 8362 70.78 54 .675 7987 65.36 .699 6377 67.09 .724 1150 68.91 .749 2609 70.81 55 2.676 1909 65-39 2.700 0404 67.12 2.724 5286 68.94 2.749 6859 70.84 56 .676 5833 65.42 .700 4432 67.15 .724 9423 68.97 .750 mo 70.87 57 .676 9759 65.44 .700 8462 67.18 .725 3562 69.00 75 5364 70.90 58 .677 3687 65-47 .701 2494 67.21 .725 7703 69.03 .750 9619 70.94 59 .677 7616 65.50 .701 6527 67.24 .726 1846 69.06 .751 3876 70.97 60 2.678 1547 65-53 2.702 0562 67.27 2.726 5990 69.09 2.751 8135 71.00 599 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 136 137 138 139 V, logM. Diff. 1". logM. Diff. I". logM. Diff. I". log M. Diff. 1". O' 2.751 8135 71.00 2.777 73 22 73.01 2.804 3 8 95 75-n 2.831 8224 77-3 2 1 .752 2396 71.03 .778 1703 73-4 .804 8403 75-14 .832 2864 77-35 2 / J J S .752 6659 71.07 .778 6087 73-7 .805 2912 75.18 .832 7506 77-39 3 753 9 2 5 71.10 .779 0472 73.11 .805 7424 75- 2 i .833 2I 5 I 77-43 4 753 5'9* 7LI3 779 4 8 59 73-H .806 1938 75- 2 5 .833 6798 77-47 5 2.753 946i 71.17 2.779 9249 73.18 2.8o6 6454 75.29 2.834 1447 77-5 6 754 373 2 71.20 .780 3641 73.21 .807 0973 75-3 2 .834 6098 77-54 7 .754 8004 71.23 .780 8034 73- 2 4 .807 5493 75-36 8 35 75 2 77-58 8 755 22 79 71.26 .781 2430 73.28 .808 0016 75-4 .835 5408 77.62 9 755 6 55 6 71.30 .781 6828 73-31 .808 4541 75-43 .836 0066 77.66 10 2.756 0835 71-33 2.782 1228 73-35 2.808 9068 75-47 2.836 4727 77-69 11 .756 5116 71.36 .782 5630 73-3 8 .809 3597 75-5 .836 9390 77-73 12 75 6 9399 71.40 .783 0034 73-4 2 .809 8128 75-54 8 37 455 77-77 13 14 757 3 68 3 757 797 7*-43 71.46 .783 4440 .783 8848 73-45 73-49 .810 2662 .810 7197 75-58 75.6i .837 8722 .838 3392 77.81 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 17 .759 0842 71.56 .785 2085 73-59 .812 0817 75-7 2 839 74H 77-96 18 759 5 J 37 71-59 .785 6502 73- 6 3 .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 21 2.760 3732 .760 8032 71.66 71.69 2.786 5341 .786 9764 73-7 73-73 2.813 4457 .813 9008 75.83 75-87 2.841 1458 .841 6144 78.08 78.11 22 .761 2335 71.73 .787 4189 73.76 .814 3561 75-9 .842 0832 78.15 23 .761 6639 71.76 .787 8615 73.80 .814 8117 75-94 .842 5522 78.19 24 .762 0940 71.79 .788 3044 73- 8 3 .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 7!-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 3128 78.50 32 .765 5470 72.06 .791 8552 74.11 .818 9214 76.27 .846 7839 78.54 33 .765 9795 72.09 .792 3000 74-15 .819 3792 76.31 847 2 553 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 !9 2 74.22 2.820 2953 76.38 2.848 1986 78.66 36 .767 2781 72.19 793 6356 74.25 .820 7537 76.42 .848 6707 78.69 37 .767 7113 72.23 794 8 *3 74.29 .821 2123 76.46 .849 1430 78.73 38 .768 1448 72.26 .794 5271 74-3 2 .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 2.769 0123 7 2 -33 2.795 4*94 74.40 2.822 5895 76.57 2.850 5612 78.85 41 .769 4464 72.36 795 86 59 74-43 .823 0491 76.60 851 0344 78.89 42 .769 8806 7 2 -39 .796 3126 74-47 .823 5088 76.64 .851 5079 78.93 43 .770 3151 7 2 -43 .796 7595 74.50 .823 9688 76.68 .851 9816 78.97 44 .770 7498 72.46 .797 2066 74-54 .824 4289 76.72 8 5 2 4555 79.01 45 2.771 1846 72.50 2 -797 6539 74-58 2.824 8894 76.75 2.852 9297 79.05 46 .771 6197 7 2 -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 79.20 50 2.773 3 62 i 72.67 2.799 8 93 8 74-75 2.827 1947 76.94 2.855 3040 79.24 51 773 79 82 72.70 .800 3424 74-79 .827 6565 76.98 855 7795 79.28 52 774 2 344 7 2 -73 .800 7912 74.82 .828 1185 77.01 856 2553 79.32 53 774 6 79 72.77 .801 2402 74.86 .828 5807 77.05 856 73*4 79-36 54 775 i77 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- J 3 2.857 6842 79-44 56 775 9 8l 7 72.87 .802 5886 74.96 .829 9686 77.16 .858 1609 79.48 57 .776 4190 72.90 .803 0385 75.00 .830 4317 77.20 .858 6379 79.52 58 776 8565 72.94 .803 4886 75.04 .830 8951 77.24 859 "5 1 79-56 59 .777 2942 72.97 .803 9390 75.08 .831 3586 77.28 .859 5926 79.60 60 2.777 73 22 73.01 2.804 3 8 95 75.11 2.831 8224 77-3* 2.860 0703 79.64 600 TABLE VI. For 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. I". logic Diff. 1". logM. Diff. 1". O' 2.860 0703 79.64 2.889 J 754 82.08 2.919 1831 84.65 2.950 1420 87-37 1 .860 5482 79 .68 .889 6680 82.12 .919 6911 84.70 .950 6664 87.41. 2 .861 0264 79.72 .890 1609 82.16 .920 1994 84.74 .951 1910 87.46 3 .861 5048 79.76 .890 6540 82.20 .920 7080 84.78 .951 7159 87.50 4 '.861 9835 79.80 .891 1473 82.25 .921 2169 84.83 .952 2411 87.55 5 2.862 4624 79.84 2.891 6409 82.29 2.921 7260 84.87 2.952 7665 87.60 6 .862 9415 79.88 .892 1348 82.33 .922 2353 84.92 953 2923 87.65 7 .863 4209 79.92 .892 6289 82.37 .922 7450 84.96 953 8183 87.69 8 .863 9005 79.96 .893 1233 82.41 .923 2549 85.01 954 344 6 87.74 9 .864 3803 80.00 .893 6179 82.46 .923 7650 85.05 954 8711 87.79 10 2.864 8604 80.04 2.894 1127 82.50 2.924 2755 85.10 2-955 398o 87.83 11 .865 3408 80.08 .894 6078 82.54 .924 7862 85.14 955 9251 87.88 12 .865 8213 80. 1 2 .895 1032 82.58 .925 2972 85.18 956 4525 87.93 13 .866 3021 80.16 .895 5989 82.63 .925 8084 85.23 .956 9802 87.97 14 .866 7832 80.20 .896 0948 82.67 .926 3199 85.27 957 5082 88.02 15 2.867 2645 80.24 2.896 5909 82.71 2.926 8317 85.32 2.958 0365 88.07 16 .867 7460 80.28 .897 0873 82.75 .927 3437 85.36 .958 5651 88.11 17 .868 2278 80.32 .897 5839 82.79 .927 8560 85.41 959 939 88.16 18 .868 7098 80.36 .898 0808 82.84 .928 3686 85-45 959 623 88.21 19 .869 1921 80.40 .898 5780 82.88 .928 8814 85.50 .960 1524 88.26 20 2.869 6 74 6 80.44 2.899 0754 82.92 2.929 3945 85.54 2.960 6821 88.30 21 .870 1573 80.48 .899 5730 82.96 .929 9079 85-59 .961 2120 88.35 22 .870 6403 80.52 .900 0709 83.01 .930 4216 85.63 .961 7423 88.40 23 .871 1235 80.56 .900 5691 83.05 .930 9355 85.68 .962 2728 88.45 24 .871 6070 80.60 .901 0675 83.09 .931 4497 85.72 .962 7036 88.49 25 2.872 0907 80.64 2.901 5662 83.13 2.931 9641 85-77 2.963 3347 ^'54 26 .872 5747 80.68 .902 0651 83.18 .932 4788 85.81 .963 8661 88.59 27 .873 0589 80.72 .902 5643 83.22 .932 9938 85.86 .964 3978 88.64 28 8 73 5433 80.76 .903 0638 83.26 933 59! 85.91 .964 9297 88.68 29 .874 0280 80.80 .903 5635 83-3I 934 247 8 5.95 .965 4620 88.73 30 2.874 5 I2 9 80.84 2.904 0635 83-35 2-934 5405 85.99 2.965 9945 88.78 31 .874 9981 80.88 .904 5637 83-39 935 565 86.04 .966 5273 88.83 32 875 4835 80.92 .905 0642 83-43 935 5729 86.08 .967 0604 88.87 33 .875 9692 80.96 .905 5649 83.48 .936 0895 86.13 .967 5938 88.92 34 .876 4551 8l.oi .906 0659 83.52 .936 6064 86.17 .968 1275 88.97 35 2.876 9413 81.05 2.906 5672 83.56 2.937 1236 86.22 2.968 6615 89.02 36 .877 4277 81.09 .907 0687 83.61 937 6410 86.26 .969 1957 89.07 37 8 77 9H3 81.13 .907 5704 83.65 .938 1587 86.31 .969 7303 89.12 38 .878 4012 81.17 .908 0725 83.69 .938 6767 86.35 .970 2651 89.17 39 .878 8883 81.21 .908 5748 83.74 939 195 86.40 .970 8002 89.21 40 2 -879 3757 81.25 2.909 0773 83.78 2-939 7135 86.45 2.971 3356 89.26 41 .879 8633 81.29 .909 5801 83.82 .940 2323 86.49 .971 8713 89-3' 42 .880 3512 81.33 .910 0832 83.87 94 75H 86.54 .972 4073 89.36 43 .880 8393 81.37 .910 5865 83.91 .941 2708 86.58 972 943 6 80.40 44 .881 3277 81.42 .911 0901 83-95 .941 7904 86.63 973 4801 89.45 45 2.881 8163 81.46 2.911 5940 83-99 2.942 3103 86.67 2.974 0170 89.50 46 .882 3052 81.50 .912 0981 84.04 .942 8305 86.72 974 554i 89.55 47 .882 7943 81.54 .912 6024 84.08 943 35 10 86.77 975 9i6 89.60 48 .883 2837 81.58 .913 1070 84.13 943 8717 86.81 975 6 293 89.65 49 883 7733 81.62 .913 6119 84.17 944 39 2 7 86.86 .976 1673 89.69 5O 2.884 2631 81.66 2.914 1171 84.22 2.944 9*4 86.90 2.976 7056 89.74 51 .884 7532 81.70 .914 6225 84.26 945 4355 86.95 977 2442 89.79 52 .885 2436 81.75 .915 1282 84.30 945 9574 87.00 977 7831 89.84 53 .885 7342 81.79 .915 6341 84-34 .946 4795 87.04 .978 3223 89.89 54 .886 2251 81.83 .916 1403 84.39 .947 0019 87.09 .978 8618 89.94 55 2.886 7162 81.87 2.916 6468 84.43 2.947 5245 87.13 2.979 4015 89.99 56 .887 2075 81.91 9 J 7 1535 84.48 948 0475 87.18 979 94 l6 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 84.56 949 942 87.27 .981 1226 90.13 59 .888 6831 82.04 .918 6753 84.61 .949 6180 87.32 .981 6636 90.18 6O 2.889 1754 82.08 2.919 1831 84.65 2.950 1420 87.37 2.982 1048 90.23 601 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 144 145 146 147 log M. Diff. I". logM. Diff. 1". log M. Diff. 1". logM. Diff. I". 0' 1 .982 1048 .982 6463 90.23 90.28 .015 1281 .015 6878 93.26 93-3 1 .049 2733 .049 8522 9647 96.52 .084 6070 .o8c 2064 f 99.87 99.92 2 .983 1882 9-33 .016 2478 93.36 .050 4315 96.58 .085 8061 99.98 3 4 983 733 .984 2727 00.38 90.43 .016 8082 .017 3688 93-4 2 93-47 .051 0112 .051 5911 96.69 .086 4062 .087 0066 100.04 100.10 5 .984 8154 90.48 .017 9298 93.52 .052 1714 96.74 .087 6073 100.16 6 7 985 3584 .985 9017 9-53 90.58 .018 4911 .019 0526 93-57 93.62 .052 7520 .053 3329 96.80 96.85 .088 2085 .088 8099 100.22 100.28 8 .986 4453 90.63 .019 6145 93-68 .053 9142 96.91 .089 4118 I00. 33 9 .986 9892 90.67 .020 1768 93-73 .054 4959 96.96 .090 0140 100.39 10 987 5334 90.72 .020 7393 93-78 .055 0778 97.01 .090 6165 100.45 11 -988 0779 90.77 .021 3021 93-83 .055 6601 97.07 .091 2194 100.51 12 .988 6227 90.82 .021 8653 93-89 .056 2427 97.I3 .091 8226 100.57 13 .989 1678 90.87 .022 4288 93-94 .056 8256 97.19 .092 4262 100.63 14 .989 7132 90.92 .022 9926 93-99 .057 4089 97.24 .093 0302 100.69 15 2.990 2589 90.97 .023 5567 94.04 .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 ioo. 81 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 19 992 4446 91.17 .025 8163 94.26 .060 3304 97.52 .096 0553 100.98 20 2.992 9918 91.22 .026 3820 94-31 .060 9157 97.58 ,.096 6614 101.04 21 993 5393 91.27 .026 9480 94-36 .061 5013 97.63 .097 2678 IOI.IO 22 .994 0871 91.32 .027 5143 94.41 .062 0873 97.69 .097 8746 101.16 23 24 994 6351 995 l8 35 9 I -37 91.42 .028 o8lO .028 6479 94-47 94.52 .062 6736 .063 2602 97-75 97.80 .098 4818 .099 0893 IOI.22 101.28 25 2.995 7322 91.47 [.029 2152 94-57 .063 8472 97.86 3.099 6972 101.34 26 .996 2812 91.52 .029 7828 .064 4345 97.91 .100 3054 101.40 27 .996 8305 9 J -57 .030 3507 94^68 .065 O222 97-97 .100 9140 101.46 28 997 3801 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 |.032 0564 94.84 ;.o66 7872 98.14 3-IO2 742O 101.64 31 999 37 91.77 .032 6256 94-89 .067 3762 98.20 .103 3520 101.70 32 999 5815 91.82 .033 1951 94-94 .067 9655 98.25 .103 9624 101.76 33 I.OQO 1326 91.87 .033 7650 95.00 .068 5552 98.31 .104 5732 101.82 34 .000 6840 91.93 34 335 1 95.05 .069 1453 98.37 .105 1843 101.88 35 3.001 2357 91.98 3.034 9056 95.11 3.069 7357 98.42 3.105 7958 101.94 36 .001 7877 92.03 .035 4764 95.16 .070 3264 98.48 .Io6 4076 IO2.OO 37 .002 3400 92.08 .036 0475 95.22 .070 9174 98.54 .107 0198 102.07 38 .002 8926 92.13 .036 6190 95.27 .071 5088 98.60 .107 6324 102.13 39 .003 4456 92.18 .037 1908 95-3* .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 IO2.25 41 .004 5523 92.28 3 8 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 I02.3 7 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 11 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. 6l 47 .007 8800 92.59 .041 7767 95-76 .076 8469 99.11 .113 1620 102.67 48 .008 4357 92.64 .042 3514 95-8i .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 5O 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 6,615 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 .oil 7763 92.95 .045 8064 96.14 .081 0181 99.52 117 4833 103.10 55 3.012 1342 93.00 3.046 3834 96.19 3.081 6154 99-57 3.118 1022 103.16 56 .012 8923 93-05 .046 9607 96.25 .082 2130 .Il8 7213 103.23 57 .013 4508 93.10 .047 5383 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 60 3.015 I28l 93.26 3.049 2733 96.47 3.084 6070 99.87 3-I2I 20l8 103.48 602 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 148 149 150 151 log M. Diff. 1". logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". 0' 3.121 2018 103.48 3.159 1367 107.31 3.198 4984 111.41 3.239 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 IJI -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 116.08 5 3.124 3107 103.79 3.162 3611 107.65 3.201 8459 111.76 3.242 8608 116.15 6 .124 9336 103.85 .163 0072 107.71 .202 5166 111.83 .243 5580 116.23 7 1*5 5569 103.91 .163 6536 107.78 .203 1878 111.90 .244 2556 Il6.70 8 .120 1805 103.97 .164 3005 107.85 .203 8594 111.97 .244 9536 116.38 9 .126 8045 104.04 .164 9478 107.91 .204 5315 112.04 .245 6521 116.45 10 3.127 4289 104.10 3- l6 5 5955 107.98 3.205 2040 112. II 3.246 3511 116.53 11 .128 0537 104.16 .166 2435 108.04 .205 8769 112. l8 .247 0505 116.61 12 .128 6789 104.22 .166 8920 108.11 .206 5502 112.26 .247 7503 116.68 13 .129 3044 104.29 .167 5409 108.18 .207 2239 "2-33 .248 4507 116.76 14 ,I2 9 9303 i4-35 .168 1901 108.25 .207 8981 1 1 2.40 .249 1515 116.84 15 3.130 5566 104.41 3.168 8398 108.31 3.208 5727 112.47 3.249 8527 116.91 16 .I 3 I 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 '33 0655 104.67 .171 4426 108.58 .211 2755 112.76 .252 6623 117.22 20 3- I 33 6 937 104.73 3.172 0942 108.65 3-2II 9522 112.83 3-*53 3 6 5 8 117.30 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 2 54 7743 117.45 23 135 5 8 5 104.92 .174 0517 108.85 .213 9851 113.05 .255 4792 "7-53 24 .136 2IO2 104.98 .174 7051 108.92 .214 6636 113.12 .256 1846 117.60 25 3.136 8403 105.05 3-175 3588 108.99 3- aI 5 3425 113.19 3.256 8905 117.68 26 .137 4708 105.11 .176 0129 109.06 .216 0219 113.26 .257 5968 117.76 27 .138 1016 105.17 .176 6674 109.12 .216 7017 H3-34 .258 3036 117.84 28 .138 7329 105.24 .177 3224 109.19 .217 3819 113.41 .259 0109 117.91 29 139 3645 105.30 .177 9777 109.26 .218 0626 113.48 .259 7180 117.99 30 3.139 9965 105.36 3.178 6335 109.33 3- 218 7437 "3-55 3.260 4268 118.07 31 .140 6289 i5-43 .179 2897 109.40 .219 4252 113.63 .261 1354 118.15 32 .141 2616 105.49 .179 9462 109.46 .220 1072 113.70 .261 8446 118.23 33 .141 8948 i5-55 .180 6032 i9-53 .220 7896 113.77 .262 5542 118.30 34 .142 5283 105.62 .181 2606 109.60 .221 4724 113.84 .263 2642 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 113.99 .264 6857 118.54 37 .144 4312 105.81 .183 2353 109.81 .223 5236 114.06 .265 3972 118.62 38 .145 0663 105.87 .183 8943 109.87 .224 2082 114.14 .266 1091 118.70 39 .145 7018 105.94 .184 5538 109.94 .224 8933 114.21 .266 8216 118.77 40 3.146 3376 106.00 3.185 2136 IIO.OI 3.225 5788 114.28 3- 26 7 5345 118.85 41 .146 9739 106.07 .185 8739 110.08 .226 2647 114.36 .268 2478 118.93 42 .147 6105 106.14 .186 5346 110.15 .226 9511 114.43 .268 9616 119.01 43 .148 2475 106.20 .187 1957 110.22 .227 6379 114.51 .269 6759 119.09 44 .148 8849 106.27 .187 8572 110.29 .228 3252 114.58 .270 3907 119.17 45 3.149 5227 106.33 3.188 5192 110.36 3.229 0129 114.65 3.271 1060 119.25 46 .150 1609 106.40 .189 1815 II0. 4 3 .229 7OIO ii4-73 .271 8217 "9-33 47 ^o 7995 106.46 .189 8443 110.50 .230 3896 114.80 .272 5379 119.41 48 '5 1 43 8 5 106.53 .190 5075 110.57 .231 0786 114.88 .273 2546 119.49 49 .152 0778 106.59 .191 1711 110.64 .231 7681 114.95 .273 9717 119.57 50 3.152 7176 106.66 3.191 8351 110.71 3.232 4581 115.03 3.274 6894 119.65 51 *53 3577 106.72 .192 4996 110-77 .233 1484 115.10 .275 4075 119.73 52 153 99 8 3 106.79 .193 1644 110.84 .233 8392 115.17 .276 1261 119.81 53 .154 6392 106.85 .193 8297 IlO.gi 234 535 115.25 .276 8452 119.89 54 .155 2805 106.92 .194 4954 IlO.gS .235 2222 115.32 .277 5647 119.97 55 3.155 9222 106.99 3.195 1615 II 1.05 3.235 9144 115.40 3.278 2848 120.05 56 .156 5643 107.05 .195 8281 III. 12 .236 6070 115.47 .279 0053 120.13 57 .157 2068 107.12 .196 4950 III.I9 .237 3001 "5-55 .279 7263 I2O.2I 58 .157 8497 107.18 .197 1624 111.26 .237 9936 115.62 .280 4477 120.29 59 .158 4930 107.25 .197 8302 111-34 .238 6876 115.70 .281 1697 120.37 60 3.159 1367 107.31 3.198 4984 III.4I 3.239 3820 115.77 3.281 8921 120.45 603 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 152 153 154 155 logM. Diff. 1". log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". O' 1 3.281 8921 .282 6151 120.45 120.53 3.326 1448 .326 8978 125.46 125.55 3.372 2684 373 53 8 i3- 8 5 130.94 ,.420 4064 .421 2266 136.66 I36 r'^ 2 .283 3385 120.61 .327 6513 125.63 373 8397 131.04 .422 0475 136.86 3 .284 0624 120.69 .328 4054 125.72 .374 6262 H 1 -^ .422 8690 136.96 4 .284 7868 120.77 .329 1600 125.81 375 4133 131.22 .423 6910 137.06 5 3.285 5116 120.85 3.329 9151 125.89 [.376 2009 131-32 3424 5137 137.16 6 .286 2370 120.93 .330 6707 125.98 .376 9890 131.41 425 337 137.26 7 .286 9628 121. OI .331 4268 126.07 377 7778 131.50 .426 1609 137-37 8 .287 6891 121. 10 .332 1835 126.16 .378 5 6 7! 131.60 .426 9854 137-47 9 .288 4160 I2I.I8 .332 9407 126.24 379 3570 131.69 .427 8105 137-57 10 3.289 1433 121.26 3.333 6984 126.33 3.380 1474 I3I i?> 3.428 6362 137.67 11 .289 8711 121.34 .334 4567 126.42 .380 9384 131.88 .429 4626 137-77 12 .290 5993 121.42 335 2154 126.51 .381 7300 131.98 .430 2895 137.88 13 .291 3281 121.50 335 9747 126.59 .382 5221 132.07 .431 1171 I37-98 14 .292 0574 121.59 .336 7346 126.68 .383 3148 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 132.35 433 6034 138.29 17 .294 2481 121.83 339 OI 72 126.95 385 6963 132.45 434 4334 138.39 18 .294 9794 121.91 339 7792 127.03 .386 4913 132-54 435 2641 138.49 19 .295 7111 122.00 .340 5417 127.12 .387 2869 132.64 436 0953 138.59 20 3.296 4433 122.08 3-341 3047 127.21 3.388 0830 132.73 3.436 9272 138.70 21 .297 1761 I22.I6 .342 0682 127.30 -388 8797 132.83 437 7597 138.80 22 .297 9093 122.24 .342 8323 127.39 .389 6770 132.93 438 59 2 8 138.90 23 .298 6430 122-33 343 59 6 9 127.48 39 4749 133.02 .439 4266 139.01 24 .299 3772 122.41 344 3620 127.57 39 1 2733 133.12 .440 2609 139.11 25 3.300 1119 122.49 3-345 I2 77 127.66 3.392 0723 133.22 3.441 0959 139.22 26 .300 8471 122.58 345 8939 127-75 .392 8719 !33-3i 441 9315 I39-32 27 .301 5828 122.66 .346 6606 127.84 .393 6720 '33-41 442 7677 13942 28 .302 3190 122.74 347 4279 127.93 394 4728 133-5 443 6046 '39-53 29 33 557 122.83 .348 1958 128.02 395 2741 133.60 444 442i 139.63 30 3.303 7929 122.91 3.348 9641 I28.II 3.396 0760 133.70 3.445 2802 1 39-74 31 .304 5306 , 122.99 349 733 128.19 .396 8785 133-79 .446 1189 139.84 32 .305 2688 123.08 35 5024 128.28 397 6815, I33-89 .446 9583 139-95 33 .306 0075 123.16 .351 2724 128.37 .398 4852 J33-99 .447 7983 140.05 34 .306 7468 123.24 .352 0429 128.46 399 2894 134.09 .448 6389 140.16 35 3.307 4865 123.33 3.352 8140 128.55 3.400 0942 I34-I9 3-449 4802 140.26 36 .308 2267 123.41 353 5 8 56 128.65 .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 i34 128.83 .402 5122 134.48 452 0077 i4 -57 39 .310 4504 123.66 355 937 128.92 .403 3193 134-57 452 8515 140.68 40 3.311 1926 123-75 3.356 6774 129.01 3.404 1270 134-67 3-453 6959 140.79 41 3" 9354 123.83 357 4517 129.10 404 9354 134.77 454 54 10 140.90 42 .312 6786 123.92 .358 2266 129.19 45 7443 134.87 455 3867 141.00 43 3i3 4224 124.00 .359 0020 129.28 .406 5538 134-97 .456 2330 141.11 44 .314 1667 124.09 359 778o 129.37 .407 3639 i35-7 .457 0800 141.21 45 3-3H 9'i5 124.17 3-3 6 5545 129.46 3.408 1746 135.16 3.457 9276 141.32 46 .315 6567 124.26 .361 3316 129.56 .408 9859 135.26 458 7759 141.43 47 .316 4025 124.34 .362 1092 129.65 .409 7977 I35-3 6 .459 6248 141.54 48 .317 1489 124.43 .362 8873 129.74 .410 6102 I35-46 .460 4743 141.64 49 3 J 7 8957 124.51 .363 6660 129.83 .411 4233 I35-56 .461 3245 141.75 50 3.318 6430 124.60 3.364 4453 129.92 3.412 2369 135.66 3.462 1753 141.86 51 .319 3909 124.68 .365 2251 I 30.01 .413 0512 135.76 .463 0268 141.97 52 .320 1392 124.77 .366 0055 130.11 .413 8660 135.86 .463 8789 142.07 53 .320 8881 124.86 .366 7864 130.20 .414 6815 135.96 464 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 3-3 68 3499 130.38 3.416 3142 136.16 3.466 4392 142.40 56 3*3 '379 125.11 .369 1325 130.48 .417 1314 136.26 467 2939 142.51 57 .323 8888 125.20 .369 9156 I30-57 .417 9492 136.36 .468 1492 142.61 58 .324 6403 125.29 .370 6993 130.66 .418 7677 136.46 .469 0052 142.72 59 3*5 39*3 125.37 .371 4836 130.76 .419 5867 136.56 .469 8619 142.83 60 3.326 1448 125.46 3.372 2684 130.85 3.420 4064 136.66 3.470 7192 142.94 604 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 156 157 158 159 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". log M. Diff. 1". 0' 3.470 7192 142.94 3-5 2 3 3 8 75 H9-75 3.578 6154 157.17 3.636 6 35I 165.28 1 .471 5772 HS-QS .524 2864 149.87 579 5588 157-3 .637 6272 165.42 2 .472 4358 143.16 .525 1860 149.99 .580 5030 157-43 .638 6202 165.56 3 473 2951 143.27 .526 0863 150.1 1 .581 4480 !57-5 6 .639 6140 165.71 4 -474 '55 I43-3 8 .526 9873 150.23 582 3937 157.69 .640 6087 165.85 5 3.475 0156 143.49 3.527 8890 1S-1S 3-583 343 157.82 3.641 6042 165.99 6 475 8 7 6 9 143.60 .528 7915 I 5-47 .584 2876 J 57-95 .642 6006 166.13 7 .476 7388 143.71 .529 6947 I50-59 585 2357 158.08 643 5978 166.28 8 477 6014 143.82 53 5985 150.71 .586 1846 158.21 6 44 5959 166.42 9 .478 4646 H3-93 531 53! 150.83 .587 1342 158-34 .645 5948 166.56 10 3-479 3285 144.04 3.532 4085 150.95 3.588 0847 158.47 3.646 5946 166.71 11 .480 1931 144.15 533 3'45 151.07 .589 0359 158.61 647 5953 166.85 12 .481 0583 144.26 -534 2213 151.19 .589 9880 158.74 .648 5968 166.99 13 .481 9242 H4-37 535 1*88 151-31 .590 9408 158.87 .649 5992 167.14 14 .482 7907 144.48 .536 0370 151-43 .591 8944 159.00 .650 6025 167.28 15 3.483 6579 '44-59 3-536 9459 I5I-55 3.592 8488 iSW 3.651 6066 167.42 16 .484 5258 144.70 537 8556 151.67 593 8040 159.26 .652 6116 l 6 7-57 17 4 8 5 3944 144.81 .538 7660 151.79 .594 7600 159.40 .653 6175 167.72 18 .486 2636 H4-93 539 6 77i 151.91 595 7 l6 7 !59-53 .654 6242 167.86 19 .487 1335 145.04 .540 5890 152.04 .596 6743 159.66 .655 6318 168.01 20 3.488 0040 145.15 3.541 5015 152.16 3-597 6 327 159-79 3.656 6403 168.15 21 .488 8752 145.26 .542 4148 152.28 598 5919 '59-93 657 6497 168.30 22 .489 7472 145-37 543 3 28 9 152.40 599 55i8 160.06 .658 6599 168.45 23 .490 6198 '45-49 544 2 436 152.52 .600 5126 160.19 .659 6710 168.59 24 .491 4930 145.60 545 i59i 152.65 .601 4742 160.33 .660 6830 168.74 25 3.492 3670 145.71 3.546 0754 152.77 3.602 4365 160.46 3.661 6959 168.89 26 493 2416 145.82 .546 9924 152.89 .603 3997 160.60 .662 7096 169.03 27 .494 1 1 68 145.94 .547 9101 153.01 .60, 3637 160.73 .663 7243 169.18 28 .494 9928 146.05 .548 8285 153-14 .605 3285 160.87 .664 7398 169.33 29 495 8695 146.16 549 7477 153.26 .606 2941 161.00 .665 7562 169.48 30 3.496 7468 146.28 3.550 6677 I53-38 3.607 2605 161.14 3.666 7735 169.62 31 .497 6248 146.39 .551 5883 !53-5i .608 2277 161.27 .667 7917 169.77 32 498 535 146.50 .552 5097 i53- 6 3 .609 1957 161.41 .668 8108 169.92 33 499 3828 146.62 553 43 J 9 J 53-75 .610 1646 161.54 .669 8308 170.07 34 .500 2629 146.73 554 3548 153.88 .611 1342 161.68 .670 8516 170.22 35 3.501 1436 146.85 3-555 ^785 154.00 3.612 1047 161.81 3.671 8734 170.37 36 .502 0250 146.96 .556 2029 I54-I3 .613 0760 161.95 .672 8961 170.52 37 .502 9071 147.08 .557 1280 I54-25 .614 0481 162.09 .673 9196 170.67 38 .503 7899 147.19 55 8 0539 !54-3 8 .615 O2IO 162.22 .674 9441 170.82 39 .504 6734 I47-3I .558 9806 154-5 .615 9948 162.36 675 9694 170.97 40 3-55 557 6 147.42 3-559 98o I54-63 3.616 9693 162.50 3.676 9957 171.12 41 .506 4425 H7-54 .560 8361 1 54-75 .617 9447 162.63 .678 0228 171.27 42 .507 3280 147.65 .561 7650 154.88 .618 9209 162.77 .679 0509 171.42 43 .508 2143 147.77 .562 6947 155.01 .619 8980 162.91 .680 0799 J 7i-57 44 .509 1012 147.88 .563 6251 !55-i3 .620 8758 163.05 .681 1098 171.72 45 3.509 9889 148.00 3.564 5562 155.26 3.621 8545 163.18 3.682 1406 171.87 46 .510 8772 148.11 .565 4882 I55-38 .622 8340 163.32 .683 1723 172.03 47 .511 7662 148.23 .566 4209 155-51 .623 8144 163.46 .684 2049 172.18 48 .512 6560 148.34 5 6 7 3543 155.64 .624 7956 163.60 .685 2384 172.33 49 513 5464 148.46 .568 2885 J 55-7 6 .625 7776 163.74 .686 2728 172.48 50 3-5H 4375 148.58 3.569 2235 155.89 3.626 7604 163.88 3.687 3082 172.64 51 52 515 3294 .516 2219 148.70 148.81 57 *592 57 1 957 156.02 156.15 .627 7441 .628 7287 164.02 164.16 .688 3445 .689 3817 172.79 172.94 53 .517 1151 148.93 .572 0330 156.27 .629 7140 164.30 .690 4198 173.10 54 .518 0090 149.05 .572 9710 156.40 .630 7002 164.44 .691 4588 173.25 55 3.518 9037 149.17 3-573 99 8 156.53 3.631 6873 164.58 3.692 4988 173.40 56 .519 7990 149.28 .574 8494 156.66 .632 6751 164.72 693 5397 I73-56 57 58 .520 6951 .521 5918 149.40 149.52 575 7897 .576 7308 156.79 156.92 .633 6638 634 6534 164.86 165.00 .694 5815 .695 6243 173.71 I73-87 59 .522 4893 149.64 577 6727 157.04 .635 6438 165.14 .696 6680 174.02 60 3-5*3 3875 149-75 3.578 6154 157.17 3.636 6351 165.28 3.697 7126 174.18 605 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 160 161 162 163 log M. Diff. 1". logM. Diff. 1". logM. Diff. 1". logM. Diff. I". 0' 1 2 3 4 .697 7126 .698 7581 .699 8046 .700 8520 .701 9003 174.18 174-34 174-49 174.65 174.80 3.762 1539 .763 2584 .764 3639 .765 4704 .766 5780 183.99 184.16 184.34 184.51 184.68 .830 3147 .831 4845 832 6554 .833 8275 .835 0008 194.87 195.06 195.25 195.44 195.64 .902 6107 93 8534 .905 0973 .906 3425 .907 5890 207.00 207.21 207.43 207.64 207.86 5 6 7 8 9 .702 9496 .703 9999 .705 0511 .706 1032 .707 1562 174.96 175.12 175.28 175-43 3.767 6867 .768 7963 .769 9070 .771 0187 772 13*5 184.86 185.03 185.20 185.38 185.55 3.836 1752 .837 3508 -838 5275 839 754 .840 8844 I95-83 196.02 196.22 196.41 196.60 .908 8368 .910 0859 .911 3363 .912 5880 .913 8410 208.08 208.29 208.51 208.72 208.94 10 11 12 13 .708 2102 .709 2652 .710 3211 .711 3780 175-75 I75-9I 176.07 176.22 3-773 2454 774 3 6 3 775 4762 77 6 5932 185.73 185.90 186.08 186.25 1.842 0646 .843 2460 .844 4286 .845 6123 196.80 196.99 197.19 I97-38 3-9*5 953 .916 3509 .917 6078 .918 8661 209.16 209.38 209.60 209.81 14 .712 4358 176.38 777 7"2 186.43 .846 7972 I97-58 .920 1256 210.03 15 .713 4946 176.54 3.778 8303 186.60 3-847 9833 197.78 3.921 3865 210.25 16 .714 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 .851 5486 198.37 925 177 210.92 19 717 7392 177.18 -783 3*74 187.31 .852 7394 198.57 .926 4432 211.14 20 .718 8028 J 77-34 3.784 4418 187.49 3-853 93H 198.76 3.927 7107 211.36 21 .719 8673 177.50 785 5672 187.67 .855 1245 198.96 .928 9795 211.58 22 .720 9328 177.66 .786 6938 187.85 .856 3189 199.16 .930 2497 211.81 23 .721 9993 177.83 .787 8214 188.03 857 5*45 199.36 .931 5212 212.03 24 .723 0668 178.00 .788 9501 188.21 .858 7112 199.56 .932 7940 212.25 25 3.724 1352 178.15 3.790 0799 188.39 3.859 9092 199.76 3.934 0682 212.48 26 .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 212.93 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 9i7i 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 73 1 6 4'3 179.29 .798 0186 189.65 .868 3287 201.17 943 02 54 214.06 33 .732 7176 179-45 799 i57i 189.83 869 5363 201.37 944 3 I0 5 214.29 34 733 794 s 179.62 .800 2966 190.01 .870 7452 201.58 945 59 6 9 214.52 35 3-734 873 179-78 3.801 4372 190.20 3-871 9552 201.78 3.946 8847 214.74 36 735 9522 J 79-95 .802 5790 190.38 .873 1665 201.98 .948 1738 214.97 37 .737 0324 180.11 .803 7218 190.56 .874 3791 202.19 .949 4644 215.20 38 738 "3 6 180.28 .804 8657 190.65 .875 5928 202.39 .950 7563 215.43 39 739 J 957 180.45 .806 0108 190.93 .876 8078 202.60 .952 0496 215.66 40 3.740 2789 180.61 3.807 1569 I9I.II 3.878 0240 202.8o 3-953 3443 216.90 41 180.78 .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 i8i.ii .810 6020 191.67 .881 6800 203.42 957 2366 216.59 44 .744 6216 181.28 8n 7525 191.86 .882 9012 203.63 958 53 6 9 216.82 45 3-745 7097 181.45 3.812 9042 192.04 3.884 1236 203.84 3-959 8385 217.06 46 .746 7989 181.61 .814 0570 192.23 885 3473 204.05 .961 1416 217.29 47 747 8891 181.78 .815 2110 192.41 .886 5722 204.26 .962 4460 217-53 48 49 .748 9803 .750 0725 181.95 182.12 .8l6 3660 .817 5222 192.60 192.79 -887 7983 .889 0257 204.46 204.67 .963 7519 .965 0592 217.76 218.00 50 3.751 1657 182.29 3.818 6795 192.98 3.890 2544 204.88 3.966 3678 2l2,v23 51 752 2599 182.46 .819 8379 193.16 .891 4843 205.09 .967 6779 218.47 52 753 3552 182.63 .820 9974 '93-35 892 7155 205.31 .968 9895 218.70 53 754 45H 182.80 .822 1581 J 93-54 .893 9480 205.52 .970 3024 218.94 54 755 5487 182.97 823 3199 193-73 .895 1817 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 7464 183.3! .825 6470 194.11 897 6529 206.15 974 2498 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 3-979 533 220.61 606 TABLE VI. For finding the True Anomaly or tlie Time from the Perihelion in a Parabolic Orbit. V. 164 165 166 167 log M. Diff. 1". logM. Diff. I". logM. Diff. 1". log M. Diff. 1". 0' 3-979 533 220.62 4.061 6673 236.01 4.149 7198 2 53-57 4- 2 44 5537 273.78 1 .980 8574 220.86 .063 0842 236.28 .151 2422 253.88 .246 1975 274.14 2 .982 1833 221. IO .064 5027 236.56 .152 7664 254.19 .247 8434 274.51 3 .983 5106 221.34 .065 9229 236.83 .154 2925 254.51 .249 4916 274.87 4 .984 8394 221.58 .067 3447 237.11 .155 8205 254.83 .251 1419 275.24 5 3.986 1696 221.83 4.068 7682 237.39 4-157 354 255.14 4.252 7944 275.60 6 9 8 7 5 OI 3 222.07 .070 1933 237.66 .158 8822 255.46 .254 4491 2 75-97 7 .988 8345 222.31 .071 6201 237-94 .160 4159 255.78 .256 1061 276.34 8 .990 1691 222.56 .073 0486 238.22 .161 9515 256.10 .257 7652 276.71 9 .991 5051 222.80 .074 4787 238.50 .163 4891 256.42 .259 4266 277.08 10 11 3.992 8427 994 1817 223.0 5 223.29 4.075 9106 .077 3441 238.78 239.06 4.165 0285 .166 5699 256.74 257.06 4.261 0902 .262 7560 2 77-45 277.82 12 995 5 222 22 3-54 .078 7792 *39-34 .168 1132 257.38 .264 4240 278.20 13 .996 8642 223.79 .080 2161 239.62 .169 6585 257.70 .266 0943 278.57 14 .998 2077 224.03 .081 6546 239.90 .171 2056 258.02 .267 7669 278.95 15 3-999 5527 224.28 4.083 0948 240. 1 8 4.172 7547 2 58-35 4.269 4417 279.32 16 4.000 8991 224.53 .084 5368 240.46 .174 3058 258.67 .271 1187 279.70 17 .002 2471 224.78 .085 9804 240.75 .175 8588 259.00 .272 7981 280.08 18 .003 5965 225.03 .087 4257 241.03 .177 4138 2 59-33 .274 4797 280.46 19 .004 9474 225.28 .088 8728 241.32 .178 9707 2 59- 6 5 .276 1635 280.84 20 4.006 2999 "5-53 4.090 3215 241.60 4.180 5296 259.98 4.277 8497 281.22 21 .007 6538 225.78 .091 7720 241.89 .182 0905 260.31 .279 5381 281.60 22 .009 0093 226.04 .093 2242 242.08 .183 6534 260.64 .281 2289 281.98 23 .010 3663 226.29 .094 6781 242.56 260.97 .282 9219 282.36 24 .on 7248 226.54 .096 1337 242.75 .186 7850 261.30 .284 6173 282.75 25 4.013 0848 226.79 4.097 5911 243.04 4.188 3538 261.63 4.286 3149 283.14 26 .014 4463 227.05 .099 0502 2 43-33 .189 9246 261.96 .288 0149 283.52 27 .015 8093 227.30 .100 5110 243.62 .191 4974 262.30 .289 7172 283.91 28 .017 1739 227.55 .101 9736 243.91 .193 0722 262.63 .291 4218 284.30 29 .018 5400 227.81 .103 4379 244.20 .194 6490 262.97 .293 1288 284.69 30 4.019 9077 228.06 4.104 9040 244.49 4.196 2278 263.30 4.294 8381 285.08 31 .021 2769 228.32 .106 3718 244.78 .197 8086 263.64 .296 5498 285.47 32 .022 6476 228.58 .107 8414 245.08 !99 39 J 5 263.98 .298 2638 285.87 33 .624 0199 228.84 .109 3127 245-37 .200 9764 264.32 .299 9802 286.26 34 25 3937 229.09 .no 7858 245.67 .202 5633 264.66 .301 6990 286.66 35 4.026 7691 229.35 4.112 2607 245.96 4.204 1523 265.00 4.303 4201 287.05 36 .028 1460 229.62 "3 7374 246.26 .205 7473 265. 3 .305 1436 287.45 37 .029 5245 229.88 .115 2158 246.55 .207 3363 265.68 .306 8695 287.85 38 .030 9045 230.14 ,116 6960 246.85 .208 9314 266.02 .308 5978 288.25 39 .032 2861 230.40 .118 1780 247.15 .210 5286 266.37 .310 3285 288.65 40 4.033 6693 230.66 4.119 6618 247.45 4-212 1278 266.71 4.312 0616 289.05 41 .035 0540 230.92 .121 1474 2 47-75 .213 7291 267.06 .313 7971 289.45 42 .036 4404 231.18 .122 6348 248.05 .215 3325 267.40 3'5 535 289.86 43 .037 8283 231.45 .124 1239 248.35 .216 9379 267.75 .317 2753 290.26 44 .039 2177 231.71 .125 6149 248.65 .218 5455 268.10 .319 0181 290.67 45 4.040 6088 231.97 4.127 1077 248.95 4.220 1551 268.44 4.320 7633 291.07 46 .042 0015 232.24 .128 6021 249.25 .221 7668 268.79 .322 5110 291.48 47 43 3957 232.51 ,130 0988 249.56 .223 3806 269.14 .324 2611 291.89 48 .044 7915 232.77 .131 5970 249.86 .224 9965 269.50 .326 0137 292.30 49 .046 1890 233.04 .133 0971 250.17 .226 6146 269.85 .327 7688 292.71 50 4.047 5880 233-3 1 4.134 5990 2 5-47 4.228 2347 270.20 4.329 5263 293.13 51 .048 9887 233-57 .136 1028 250.78 .229 8570 2 7-55 .331 2863 293-54 52 .050 3909 233.84 .137 6084 251.08 .231 4814 270.91 333 0487 293-95 53 .051 7948 234.11 .139 1158 2 5 I -39 .233 1079 271.27 334 8137 294-37 54 .053 2003 234.38 .140 6251 251.70 .234 7366 271.62 .336 5812 294.79 55 4.054 6074 234.65 4.142 1362 252.01 4.236 3674 271.98 4-33 8 35" 295.20 56 .056 0161 234.92 .143 6492 252.32 .238 0003 272.34 .340 1236 295.62 57 .057 4264 235-!9 .145 1641 252.63 2 39 6 354 272.70 .341 8986 296.04 58 .058 8384 235.46 .146 6808 252.94 .241 2727 273.06 -343 6762 296.47 59 .060 2520 235-73 .148 1994 253-25 .242 9121 273.42 345 4562 296.89 60 4.061 6673 236.01 4.149 7198 2 53-57 4- 2 44 5537 273.78 4-347 2 388 297.31 607 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 168 169 170 171 ; V* log M. Diff. 1". logM. Diff. I". logM. Diff. 1". log M. Diff. 1". 0' 1 4-347 2388 .349 0240 297.31 297.74 4-459 1242 .461 0761 325-07 4.581 9445 .584 0962 358.31 358.92 4-7*7 9835 .720 3790 398.87 399.62 2 .350 8117 298.16 .463 0311 326.08 .586 2516 359-53 .722 7790 400.38 3 4 .352 6019 354 3948 298.59 299.02 .464 9891 .466 9501 326.59 327.10 .588 4106 59 5734 360.76 .725 1835 .727 5926 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 I 6 .357 9882 299.88 .470 8814 328.12 594 9*0 362.00 732 4245 403.43 | 7 359 7888 300.31 472 8517 328.64 597 0838 362.62 734 8474 404.19 8 .361 5919 300.75 .474 8250 329-15 599 2615 363-25 737 2749 404.96 9 363 3977 301.18 .476 8015 329.67 .601 4428 363-88 739 7070 405.74 10 4.365 2061 301.62 4.478 7811 33-*9 4.603 6280 364.50 4.742 1438 406.52 11 .367 0171 302.05 480 7637 .605 8169 365^*4 744 5852 407.30 12 .368 8308 302.49 .482 7495 331.23 .608 0096 365-77 -747 3*4 408.08 13 .370 6470 302.93 .484 7385 33*-75 .610 2061 366.40 .749 4822 408.87 14 .372 4659 303.37 .486 7306 332.28 .612 4064 367.04 75* 9378 409.66 15 4.374 2875 303.81 4.488 7258 332-81 4.614 6106 367.68 4.754 3981 4*0-45 16 .376 1117 304.26 .490 7242 333-33 .616 8186 368.32 75 6 8632 411.24 17 377 9386 304.70 .492 7258 333-86 .619 0304 368.96 759 333 412.04 18 .379 7681 35-*5 .494 7306 334-4 .621 2461 369.61 .761 8077 412.84 19 r r .381 6003 305.59 49 6 7386 334-93 623 4657 370.26 .764 2872 20 4-383 4352 306.04 4.498 7498 335-46 4.625 6892 370.91 4.766 7715 414.46 21 .385 2728 306.49 .500 7642 336.00 .627 9166 37L56 .769 2606 415.27 22 387 **3* 306.94 .502 7818 336-54 .630 1480 372.21 77* 7547 416.08 23 .388 9561 3 7.39 .504 8026 337.08 ,632 3832 372.87 774 2536 416.90 24 .390 8019 .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-*9 4.779 2662 418.54 26 394 5 OI 5 308.76 .510 8847 338.7* .639 1127 374-86 .781 7799 4*9-37 27 39 6 3554 309.21 .512 9186 339.26 .641 3639 375-52 .784 2986 420.20 28 .398 2121 309.67 5*4 9558 339.80 .643 6190 376.19 .786 8222 421.03 29 .400 0715 3*o-*3. .516 9962 340-35 .645 8781 376.86 789 3509 421.86 30 4.401 9337 310.59 4.519 0400 340.9* 4.648 1413 377-53 4.791 8846 422.70 31 .403 7986 311.06 .521 0871 341.46 .650 4085 378.21 794 4233 423.54 32 .405 6663 311.52 .523 1376 342.02 .652 6798 378.89 .796 9671 424.39 33 .407 5368 3*i-99 .525 1913 342.57 6 54 9552 379-57 799 5160 425.24 34 .409 4102 3*2.45 .527 2484 343-*3 .657 2346 380.25 ,802 0700 426.09 35 [..411 2863 312.92 4.529 3089 343.69 4.659 5182 380.93 4.804 6291 426.95 36 .413 1652 3*3-39 -53* 3728 344.26 .661 8059 381.62 .807 1934 427.81 37 .415 0469 3*3-86 533 44 344.82 .664 0977 382.31 .809 7628 428.67 38 .416 9315 535 5106 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 3*5-28 4.539 6620 346-52 4.670 9980 384.39 4.817 5022 431.28 41 .422 6022 3*5-75 .541 7429 347.09 .673 3064 385.09 .820 0925 432-15 42 .424 4982 316.23 -543 8272 347.67 .675 6191 385.80 .822 6881 433.03 43 .426 3970 316.71 545 9*49 348.24 677 9360 386.50 .825 2889 433-9* 44 .428 2987 3*7-19 .548 0061 348.82 .680 2571 387-21 .827 8950 434.80 45 4.430 2031 3*7-67 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 * 2 34 436.59 47 .434 0212 318.64 554 35 350-56 .687 2460 389-34 835 7456 437.48 48 435 9345 3*9-13 556 4056 .689 5842 390.06 -838 3732 438-38 49 437 8507 319.61 55 s 5*43 35*-73 .691 9268 390.78 .841 0062 439.29 50 51 4-439 7698 .441 6919 320.10 320.59 4.560 6264 .562 7421 352.32 352-91 4.694 2736 .696 6248 391.50 392.23 t!# &6 440.20 441.11 52 .443 6169 321.08 .564 8614 353-50 .698 9803 392.96 .848 9380 442-03 53 445 5449 321-58 .566 9842 354-10 .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 322.57 4.571 2405 355-29 4.706 0733 395-*5 4.856 9193 444.80 56 .451 3466 323.06 573 374* 355-89 .708 4464 395-89 859 599 445-73 57 453 2865 323.56 575 5**3 356.49 .710 8240 396.63 .862 2680 446.66 58 455 2294 324.06 577 6521 357-10 .713 2060 397-38 .864 9508 447.60 59 457 J753 324.56 579 7965 357-70 7*5 5925 398-12 .867 6392 448.54 60 4.459 1242 325.07 4.581 9445 358.31 4-7*7 9835 398-87 4-870 3333 449-49 608 TABLE VI, For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 172 173 174 175 log M. Diff. V. log M. Diff. 1". logM. Diff. I". log M. Diff. 1". 4- 8 7 3333 449-49. 5.043 3285 5H47 5.243 3165 601.00 5.480 1373 722.00 1 .873 0331 450-44 .046 4191 5!5-7i .246 9276 602.69 484 4765 724.42 2 875 73 8 6 451-39 .049 5171 516.96 .250 5488 604.38 .488 8304 726.87 3 .878 4499 .052 6226 518.21 .254 1802 606.08 493 T 9 8 9 729-33 4 .881 1668 453-3 1 055 7356 51947 .257 8218 607.80 497 5823 731.80 5 4.883 8896 454.28 5.058 8562 520.73 5.261 4738 609.53 5.501 9806 734-3 6 .886 6182 455-25 .061 9843 522.00 .265 1361 611.26 .506 3939 736.81 7 .889 3526 456-23 .065 1202 523-28 .268 8089 613.00 .510 8223 739-33 8 .892 0929 457-20 .068 2637 524.56 272 4922 614.75 .515 2659 741.87 9 .894 8391 458.19 .071 4149 525.85 .276 1860 616.52 .519 7248 744.44 10 4.897 5912, 459- J 7 5.074 5738 527.14 5.279 8904 618.29 5.524 1992 747-02 : 11 .900 3492 460.16 .077 7406 528.44 .283 6055 620.08 .528 6890 749.61 ; 12 .903 1132 461.16 .080 9151 287 3313 621.87 533 J 946 752-23 13 .905 8831 462.16 .084 0976 531.06 .291 0680 623.67 537 7158 754-86 j 14 .908 6591 463.16 .087 2879 532.38 .294 8154 625.49 542 2529 757-51 15 4.911 4411 464.17 5.090 4862 533-71 5-298 5738 627.31 5.546 8060 760.18 16 .914 2291 465.18 .093 6924 535-04 .302 3432 629.15 55 1 375 1 762.87 17 .917 0233 466.20 .096 9067 536.38 .306 1237 631.00 555 9605 765-58 18 .919 8235 467.22 .100 1290 537-73 .309 9152 632-85 .560 5621 768.31 | 19 .922 6299 468.25 I0 3 3594 539.08 313 7179 .565 1802 771.05 20 4.925 4425 469.28 5.106 5980 540.44 5-3!7 53 J 9 636.60 5.569 8148 773.82 21 .928 2612 470-3 1 .109 8447 541.81 .321 3571 638.49 574 4661 776.61 22 .931 0862 471-35 .113 0997 543-^8 325 1938 640.39 579 J 34* 779-41 23 933 9 J 74 472.39 .116 3629 544^6 .329 0418 642.30 .583 8190 782.24 | 24 936 7549 473-44 .119 6344 545-95 332 9 OI 4 644.23 .588 5210 785.08 25 4-939 59 8 7 474-49 5.122 9143 547-34 5.336 7726 646.16 5.593 2401 787.95 26 .942 4489 475-55 .126 2026 548.74 .340 6554 648.11 597 9764 790.84 27 945 353 476.61 .129 4992 550.I5 344 5499 650.07 .602 7302 793-75 28 .948 1682 477-68 .132 8044 .348 4562 652.04 .607 5014 796.68 29 95 1 375 478.75 .136 1181 552-99 352 3744 654.02 .612 2903 799.63 30 4-953 9132 479. 8 3 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 481.99 .146 1106 557-3 .364 2007 660.04 .626 7642 808.62 33 .962 5793 483.08 .149 4588 558.75 .368 1671 662.07 .631 6250 811.66 34 .965 4811 484.18 .152 8157 560.21 372 1456 664.11 .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 486.38 159 5558 563.16 .380 1396 668.24 .646 3179 820.92 37 .974 2260 487.49 .162 9392 564.64 3 8 4 J 553 670.32 .651 2528 824.05 j 38 977 *543 488.61 .166 3315 566.13 .388 1834 672.41 .656 2065 827.21 39 .980 0893 489-73 .169 7328 567.63 .392 2242 674.52 .661 1793 8 3-39 40 4-983 0311 490.85 5.173 1431 569-13 5.396 2777 676.64 5.666 1713 833.60 41 9 8 5 9795 491.98 .176 5624 570.65 .400 3439 678.77 .671 1825 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 495-4 .186 8752 575-24 .412 6199 685.25 .686 3336 846.67 45 4.997 8418 496.55 5.190 3312 576.78 5.416 7379 687.44 5.691 4236 850.00 46 5.000 8246 497-71 .193 7966 578.34 .420 8692 689.64 696 5337 853.36 47 .003 8143 498.87 .197 2713 579.90 .425 0136 691.85 .701 6640 856.75 48 .006 8m 500.04 .200 7554 581.47 .429 1714 694.08 .706 8147 860.16 49 .009 8148 501.21 .204 2489 583-05 433 3427 696.33 .711 9860 863.60 50 5.012 8256 52-39 5.207 7520 584.64 5-437 5274 698.59 5.717 1779 867.06 51 .015 8435 503-57 .211 2646 586.23 .441 7258 700.86 .722 3908 870.56 52 .018 8685 504.76 .214 7868 587-84 445 9378 73- I 5 .727 6247 874.08 53 .021 9006 505.95 .218 3186 589.45 .450 1636 705-45 732 8798 877.63 54 .024 9399 .221 8602 591.07 .454 4032 707.77 .738 1563 881.21 55 5.027 9864 508.36 5.225 4116 592-7I 5.458 6568 710.10 5-743 4544 884.82 56 .031 0402 509.57 .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 717.19 759 4798 895.83 59 .040 2454 513.24 239 7156 599-32 475 8125 7I9-59 .764 8659 899.56 60 5.043 3285 5H-47 5.243 3165 601.00 5.480 1373 722.00 5.770 2745 903.31 609 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. V. 176 177 178 179 logM. Diff. I". logM. Diff. I". logM. Diff. 1". logM. Diff. 1". 0' 1 5.770 2745 775 7 58 93-3 907.1 6.144 6239 .151 8807 1205.3 I2I2.0 6.672 5724 .683 4709 1808. 8 1824.0 7.575 4640 597 359 6 3619 3680 2 .781 1599 910.9 159 1733 1218.8 .694 4613 I839-5 .619 6295 3744 i 3 .786 6370 914.8 .166 5070 1225.7 75 5454 1855.3 .642 2868 3809 ! 4 .792 1374 918.7 .173 8823 1232.7 .716 7248 1871.3 .665 3452 3877 j 5 5.797 6612 922.6 6.181 2997 1239.8 6.728 ooio 1887.5 7.688 8192 3948 6 .803 2086 926.6 .188 7597 1246.9 739 3758 1904.1 .712 7239 4021 7 .808 7798 930.6 .196 2628 1254.1 .750 8509 1921.0 737 0756 4097 8 .814 3751 934- 6 .203 8095 1261.4 .762 4279 1938.2 .761 8913 4176 9 .819 9946 938.6 .211 4002 1268.8 774 i9 1955-6 .787 1889 4257 JO 5.825 6386 942.7 6.219 354 1276.3 6.785 8958 1973-4 7.812 9876 4343 11 .831 3073 946.8 .226 7158 1283.8 797 794 1991.5 -839 3075 443 x 12 .837 0008 951.0 .234 4419 1291.5 .809 7946 2010.0 .866 1702 4524 13 .842 7195 955-2 .242 2142 1299.2 .821 9106 2028.8 .893 5986 4620 14 .848 4634 959-5 25 333 1307.1 8 34 !44 2048.0 .921 6170 4720 15 5.854 2329 963.7 6.257 8997 1315.0 6.846 4863 2067.5 7-95 2513 4825 16 .860 0282 968.0 .265 8139 1323.0 .858 9503 2087.3 7-979 5292 4935 17 .865 8495 972.4 .273 7766 1331.1 .871 5348 2IO7.6 8.009 4802 55 18 .871 6970 976.8 .281 7884 1339-4 .884 2422 2128.3 .040 1361 5*7 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 7011 5428 21 .889 3993 990.2 .306 1244 1364-7 .923 1261 2192.8 .136 6857 5568 22 8 95 3542 994.8 3H 33 8 7 '373-3 93 6 3498 2215.2 .170 5274 57H 23 .901 3365 999-4 .322 6052 1382.1 949 7093 2238.0 .205 2717 5869 24 .907 3465 1004.0 33 9247 1391.0 9 6 3 2073 2261.4 .240 9679 6032 25 5-9*3 3 8 45 1008.7 6.339 2977 1400.0 6.976 8466 2285.2 8.277 6700 6204 26 .919 4507 1013.4 347 7H-9 1409.1 6.990 6304 2309.6 315 436i 6387 27 9 2 5 5454 1018.1 .356 2072 1418.3 7.004 5616 2334-3 354 3298 6580 28 .931 6688 1022.9 .364 7451 1427.6 .018 6437 2359-7 394 4205 6786 29 937 8213 1027.8 373 3395 1437.1 .032 8796 2385-7 435 7842 7004 30 5.944 0030 1032.7 6.381 9910 1446.7 7.047 2729 2412.2 8.478 5044 7238 31 .950 2144 1037.6 .390 7005 1456.4 .061 8271 2439.4 .522 6731 7488 32 .956 4556 1042.6 399 4 6 87 1466.2 .076 5458 2467.1 .568 3920 7755 33 .962 7269 1047.7 .408 2965 1476.2 .091 4329 2495.4 615 7739 8042 34 .969 0287 1052.9 .417 1846 1486.4 .106 4921 2524.5 .664 9442 8352 35 5-975 3 6l 3 1058.0 6.426 1337 1496.7 7.121 7276 2554.2 8.716 0431 8686 36 .981 7249 1063.2 435 H49 1507.0 137 H34 2584.6 .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 6.007 495 8 1084.5 6.471 8275 1550.2 7.200 6993 2713.9 9.006 7690 10857 41 .014 0192 1089.9 .481 1620 1561.3 .217 0850 2748.3 73 5974 11429 42 .020 5756 1095.4 .490 5641 1572.6 .233 6796 2783-5 .144 0401 12064 43 .027 1652 IIOI.O .500 0346 1584.1 .250 4884 2819.7 .218 5102 12773 44 .033 7885 1106.7 .509 5746 1595.8 .267 5170 2856.8 297 49 6 3 13572 45 6.040 4457 1112.4 6.519 1850 1607.7 7.284 7712 2894.8 9.381 5820 14476 46 .047 1372 1118.1 .528 8669 1619.6 .302 2571 2934.1 .471 4711 15510 47 .053 8634 1123.9 .538 6216 1631.8 .319 9810 2974.2 .568 0247 16704 48 .060 6246 1129.8 .548 4499 1644.2 337 9494 3015.6 .672 31061 18096 49 .067 4212 "35-7 558 353 1656.8 .356 1692 30 S 8.I 785 6758 19741 50 6.074 2535 1141.7 6.568 3320 1669.6 7.374 6475 3IOI.7 9.909 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 1160.0 .598 7368 1708.9 .431 7097 3240.7 374 5584 3 IQ 23 54 .joi 9479 1166.3 .609 0317 1722.6 .451 2999 3289.9 575 3986 36197 55 6.108 9647 1172.6 6.619 4086 i73 6 -4 7.471 1892 3340.3 10.812 9421 4345 56 .116 0196 1179.0 .629 8689 i75 -3 .491 3870 3392.6 11.103 6719 57 .123 1131 1185.4 .640 4141 1764.5 .511 9029 3446.5 11.478 4880 58 .130 2455 1192.0 .651 0455 1779.0 .532 7472 3502.1 12.006 7617 59 .137 4173 1198.6 .661 7645 1793-8 553 935 3559-6 12.909 8516 60 6.144 6289 1205.3 6.672 5724 1808. 8 7.575 4640 3618.7 610 TABLE VII, For finding the True Anomaly in a Parabolic Orbit when v is nearly 180. w * Diff. w 4, Diff. w 1 Diff. O I / // it 1 / // it o / i a n 155 5 10 15 3 23-09 '9-74 16.43 3-35 3-3' 3.26 160 5 10 15 I 6.70 5-33 3-97 2.64 -33 165 10 20 30 o 15.85 14.98 14.16 17.78 0.87 0.82 0.78 20 25 9-95 6.77 3.22 3.18 20 25 i-33 0.04 31 .29 .26 40 50 * j'i'-' 12.63 11.91 0.75 0.72 0.69 155 30 35 40 3 3-63 o-54 2 57-49 3-09 3-5 160 30 35 40 o 58.78 57-54 .24 23 166 10 20 O 11.22 10.57 9-95 0.65 0.62 45 50 55 5448 51.51 48-58 3.01 2-97 2-93 2 80 45 50 55 55.11 53-93 52.77 .20 .18 .16 30 40 50 y 7J 9.36 8.80 8.26 0.59 0.56 o-54 z.oy .14 0.51 156 5 10 15 20 25 2 45.69 42.84 40.03 37-26 34-53 31-83 2.8 5 2.81 2-77 2-73 2.70 2.66 161 5 10 15 20 25 o 51.63 50.50 49.40 48.32 47.26 46.21 13 .10 .08 .06 .05 .02 167 10 20 30 40 50 o 7.75 6*\ 5.96 5-57 0.48 0.46 0.44 0.41 0.39 0.37 156 30 35 40 45 50 55 2 29.17 26.55 23-97 21-43 18.92 16.44 2.62 2.58 2-54 2-5 1 2.48 2.44 161 30 35 40 45 50 55 o 45.19 44.18 43-19 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 4.84 4.51 4.20 3-90 3.62 0.36 -33 0.31 0.30 0.28 0.26 157 5 2 14.00 11.59 2.41 162 5 o 39.41 78. ci 0.90 169 10 o 3.36 3.11 0.25 10 . 15 9.22 6.89 2.37 2-33 10 15 e- 37.62 0.89 0.87 20 30 2.88 2.66 0.23 0.22 20 25 2.3-1 2.31 2.27 2.23 20 25 35-9 35.06 0.85 0.84 0.82 40 50 2.46 2.27 O.2O O.ig 0.18 157 30 35 40 45 50 55 2 0.08 I 57.89 55-72 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 0.81 0.79 0.78 0.76 o-75 0.73 170 10 20 30 40 50 o 2.09 92 76 .62 .48 35 0.17 o.i 6 0.14 0.14 0.13 0.12 158 5 10 15 20 25 i 47-35 45-34 43-35 4i-39 39-47 37-57 2.01 .99 .96 .92 .90 .87 163 5 10 15 20 25 o 29.62 28.90 28.20 27.51 26.83 26.16 0.72 0.70 0.69 0.68 0.67 0.65 171 10 20 30 40 50 o .23 .12 .02 0.93 0.84 0.76 O.I I O.IC 0.09 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 .83 .81 78 .76 72 .70 163 30 35 40 45 50 55 o 25.51 24.88 24.25 23.64 23.04 22.45 0.63 0.63 0.61 0.60 0.59 0-57 172 10 20 30 40 50 o 0.68 0.61 0.55 0.49 0.44 -39 0.07 O.o6 0.06 0.05 0.05 0.04 159 5 10 15 i 25.10 2343 21.78 20.16 i 7 65 .62 164 5 10 15 o 21.88 21.31 20.76 2O.22 -57 0.55 0.54 173 10 20 30 o 0.35 0.31 0.27 0.24 0.04 0.04 0.03 20 25 18.57 17.00 59 57 55 20 25 19.69 I9.I8 -53 0.51 0.51 40 50 0.21 0.19 0.03 O.O2 O.O7 159 30 1 "545 164 30 o 18.67 174 o 0.16 35 40 45 50 55 13-94 12.44 10.97 9-53 8.10 .50 47 -44 43 35 40 45 50 55 18.17 17.69 17.21 16.75 16.29 0.50 0.48 0.48 0.46 0.46 175 176 177 178 179 0.07 0.02 O.OI 0.00 0.00 '0.05 o.oi O.OI o.oo .40 7 0.44 0.00 160 i 6.70 165 o 15.85 180 o o.oo 611 TABLE VIII. For finding the Time from the Perihelion in a Parabolic Orbit. B log & Diff. V log N Diff. C log N Diff. O / 1 / 30 1 0.025 5763 .025 5749 .025 5707 H 30 30 31 O.O2O 7913 .020 6368 .O2O 4802 '545 1566 60 30 61 0.008 8644 .008 6458 .008 4277 2186 2181 30 2 30 .025 5638 .025 5542 .025 5418 09 96 124 152 30 32 30 .020 3215 .O2O 1607 .019 9979 1608 1628 1649 30 62 30 .008 2103 .007 9934 .007 7774 2174 2169 2160 2I 53 3 30 4 30 0.025 5266 .025 5087 .025 4881 .025 4647 179 206 234 261 33 30 34 30 0.019 8330 .019 6662 .019 4974 .019 3267 1668 1688 1707 63 30 64 30 0.007 5621 .007 3477 .007 1343 .006 9220 2144 2134 2123 5 .025 4386 ,o n 35 .019 1540 1727 65 .006 7108 2112 30 .025 4097 209 316 30 .018 9795 1745 1765 30 .006 5008 2IOO 2086 6 30 7 30 8 30 0.025 378i 25 3437 .025 3066 .025 2668 .025 2243 .025 1791 344 398 425 452 480 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 2025 2008 1988 9 30 1O 0.025 'S 11 .025 0805 .025 0271 506 534 39 30 40 0.017 7072 .017 5186 .017 3283 1886 1903 69 30 70 0.005 0729 .004 8760 .004 68 i i 1969 1949 30 .024 9711 500 30 .017 1365 1918 30 .004 4884 1927 11 30 .024 9124 .024 8510 614 641 41 30 .016 9432 .016 7483 '933 1949 1963 71 30 .004 2980 .004 noo 1904 1880 1855 12 30 13 30 14 30 0.024 7869 .024 7201 .024 6507 .024 5786 .024 5039 .024 4266 668 694 721 747 III 42 30 43 30 44 30 0.016 5520 .016 3542 .016 1550 OI 5 9545 .015 7526 OI 5 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 1774 1745 1714 1682 15 30 16 30 0.024 3466 .024 2641 .024 1789 .024 0911 825 852 878 45 30 46 30 0.015 3450 .015 1394 .014 9326 .014 7247 TJ 2056 2068 2079 75 30 76 30 O.OO2 8698 .OO2 7049 .002 5433 .002 3854 ^49 1616 1579 17 30 .024 0008 .023 9079 93 929 47 30 .014 5157 .014 3057 2090 2100 77 30 .002 2311 .002 0806 1543 954 2110 1465 18 0.023 8125 48 0.014 0947 78 o.ooi 9341 30 19 30 20 30 .023 7145 .023 6140 .023 5109 .023 4054 .023 2973 1005 1031 10 55 1081 1 105 30 49 30 50 30 .013 8827 .013 6698 .013 4561 .013 2416 .013 0263 2I2O 2129 2137 2I 45 2153 2l6o 30 79 30 80 30 .001 7917 .001 6535 .001 5196 .001 3903 .001 2656 1424 1382 *339 1293 1247 1198 21 30 22 0.023 1868 .023 0738 .022 9584 1130 1154 51 30 52 0.012 8103 .012 5936 .012 3764 2167 2172 81 30 82 o.ooi 1458 .001 0309 .000 9211 1149 1098 30 23 30 .022 8405 .022 7202 .022 5975 1179 1203 1227 1251 30 53 30 .012 1585 .on 9402 .on 7215 2I 79 2183 2187 21 QI 30 83 30 .000 8l66 .000 7175 .000 6240 1045 991 935 876 24 30 25 30 26 30 0.022 4724 .022 3449 .022 2151 .022 0829 .021 9484 .021 8116 1275 1298 1322 '345 1368 1390 54 30 55 30 56 30 o.on 5024 .011 2829 .on 0632 .010 8432 .010 6231 .010 4029 ~ * 7 * 2195 2197 2200 22OI 2202 2202 84 30 85 30 86 30 o.ooo 5364 .000 4546 .000 3790 .000 3096 .000 2468 .000 1906 J 1 \J 818 756 694 628 562 4Q3 ' 27 30, 28 30 29 30 0.021 6726 .021 5312 .021 3876 .021 2418 .021 0938 .020 9436 1414 1458 1480 1502 57 30 58 30 59 30 o.oio 1827 .009 9625 .009 7424 .009 5225 .009 3028 .009 0834 2202 2201 2I 99 2197 2I 94 87 30 88 30 89 30 o.ooo 1413 .000 0990 .000 0639 .000 0363 .000 0163 .000 0041 T J J 423 351 2 7 6 200 122 | 30 0.020 7913 60 0.008 8644 2190 90 o.ooo oooo 4 1 612 TABLE VIII, For finding the Time from the Perihelion in a Parabolic Orbit. V log N' Diff. V log N' Diff. V log N' Diff. o t \ f o / 90 30 91 O.OOO 0000 9.999 9876 999 957 124 369 120 30 121 9.963 1069 .962 0074 .960 8971 10995 II 103 150 30 151 9.889 0321 .887 8738 .886 7259 "583 11479 30 92 30 999 8893 999 8039 -999 6944 6 14 854 1095 30 122 30 959 7764 .958 6454 957 54 6 1 1 207 11310 11408 11504 30 152 30 .885 5887 .884 4627 883 3481 11372 11260 11146 11026 93 9.999 5613 123 9.956 3542 153 9.882 2455 30 .999 4046 1507 30 955 J 945 11597 _ , /- o _ 30 .881 1552 10903 94 30 95 30 .999 2246 .999 0215 .998 7955 .998 5468 I oOO 2031 2260 2487 124 30 125 30 954 0258 .952 8483 .951 6624 .950 4684 I I OOy 11775 11859 11940 I20l8 154 30 155 30 .880 0775 .879 0129 .877 9616 .876 9242 10777 10646 10513 10374 10232 96 30 97 30 98 30 9.998 2757 .997 9824 .997 6669 997 3 2 97 996 9708 .996 5906 2 933 3155 3372 3589 3802 4015 126 30 127 30 128 30 9.949 2666 948 0573 .946 8408 .945 6174 944 3875 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 9470 9307 99 30 9.996 1891 .995 7666 4225 129 30 9.941 9092 .940 6615 12477 159 30 9.870 0792 .869 1652 9140 100 30 101 30 995 3234 994 8596 994 3755 993 8712 4841 543 5242 130 30 131 30 939 4 o8 5 .938 1506 .936 8881 935 6213 12530 12579 12625 12668 12707 160 30 161 30 .868 2683 .867 3886 .866 5266 .865 6827 8969 8797 8620 8439 8257 102 30 103 30 104 30 9-993 3470 .992 8031 .992 2397 .991 6570 991 0553 .990 4347 5439 5634 5827 6017 6206 6391 132 30 133 30 134 30 9.934 3506 933 0763 931 7987 .930 5183 9 2 9 2353 .927 9501 12743 12776 12804 12830 12852 12871 162 30 163 30 164 30 9.864 8570 .864 0500 .863 2620 .862 4932 .861 7439 .861 0145 8070 7880 7688 7493 7294 7092 105 30 106 30 107 30 9.989 7956 .989 1380 .988 4622 .987 7685 .987 0571 .986 3281 6576 6758 6937 7114 7290 7462 135 30 136 30 137 30 9.926 6630 9^5 3745 .924 0848 .922 7943 .921 5035 .920 2126 12885 12897 12905 12908 12909 12906 165 30 166 30 167 30 9.860 3053 .859 6164 .858 9482 .858 3010 857 6750 .857 0704 6889 6682 6472 6260 6046 5829 108 30 109 30 110 30 9.985 5819 .984 8186 .984 0385 .983 2418 .982 4288 .981 5996 7633 7801 7967 8130 8292 8451 138 30 139 30 140 30 9.918 9220 .917 6321 9i6 3433 .915 0559 .913 7703 .912 4870 12899 12888 12874 12856 12833 12808 168 30 169 30 170 30 9.856 4875 .855 9266 .855 3878 .854 8714 854 3775 8 53 9 6 5 5609 5388 5164 4939 4710 4481 111 30 112 9.980 7545 979 8938 979 OI 77 8607 8761 141 30 142 9.911 2062 .909 9283 .908 6538 12779 12745 171 30 172 9.853 4584 853 335 .852 6319 4249 4016 30 113 30 .978 1264 .977 2202 .976 2993 89*3 9062 9209 9353 30 143 30 .907 3 8 3I .906 1164 .904 8542 12707 12667 12622 I2 573 30 173 30 .852 2538 .851 8994 .851 5687 3544 337 3067 114 30 115 30 116 30 9-975 364 974 4H5 .973 4510 .972 4739 .971 4833 .970 4796 9495 9635 9771 9906 10037 10167 144 30 145 30 146 30 9.903 5969 .902 3449 .901 0985 .899 8582 .898 6243 897 397^ 12520 12464 12403 12339 12271 12198 174 30 175 30 176 30 9.851 2620 .850 9794 .850 7209 .850 4868 .850 2770 .850 0917 2826 2585 234 1 2098 1853 I l6o8 ; 117 30 118 30 119 30 9.969 4629 968 4337 967 39 20 .966 3382 j .965 2726 .964 1954 10292 10417 10538 10656 10772 10885 147 30 148 30 149 30 9.896 1774 .894 9652 .893 7610 .892 5652 .891 3782 .890 2004 I2I22 12042 11958 11870 II 77 8 ii6? 3 177 30 178 30 179 30 9.849 9309 ,849 7948 849 6833 .849 5966 849 5346 .849 4974 1361 III5 86 7 62O 37^ 124 120 9.963 1069 15O i 9.889 0321 180 9.849 4850 613 TABLE IX. For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. X A Diff. B Diff. c B' Diff. C' o 0.00 it o.ooo If o.ooo II 0.000 II 0.000 1 0.00 o.oo 0.000 o.ooo o.ooo o.ooo 2 O.OI O.OI 0.000 o.ooo 0.000 o.oco 3 0.05 0.04 o.ooo 0.000 o.ooo 0.000 4 0.12 0.07 0.000 o.ooo 0.000 o.ooo O. II 5 0.23 , o.ooo o.ooo 0.000 o.ooo 6 O.7Q 0. 1 O o.ooo 0.000 o.ooo o.ooo 7 r 0.62 0.23 0.000 o.ooo 0.000 0.000 8 0.93 0.31 o.ooo 0.000 o.ooo o.ooo 9 I33 0.40 0.000 o.ooo 0.000 o.ooo j o 0.49 1O 1.82 o.ooo 0.000 0.000 0.000 11 2.42 0.60 o.ooo o.ooo o.ooo o.ooo 12 3.14 0.72 0.000 0.000 0.000 0.000 13 3-99 0.85 o.ooo o.ooo o.ooo o.ooo 14 4-99 I.OO 1.14 0.00 1 0.000 O.OOI 0.000 15 6.13 O.OOI o.ooo O.OOI o.ooo 16 J 7.43 1.30 0.002 .001 0.000 O.OOI .000 o.ooo 17 8.90 1.47 O.OO2 .000 o.ooo O.OO2 .001 o.ooo 18 19 iQ-55 12.40 1.65 1.85 2.05 0.003 0.004 .001 .001 .001 0.000 o.ooo 0.002 0.003 .000 .001 .001 o.ooo 0.000 2O 21 14.45 16.70 2.25 0.005 0.006 .001 0.000 o.ooo 0.004 0.005 .001 o.ooo o.ooo 22 19.18 2.48 0.008 .002 0.000 0.006 .001 o.ooo 23 21.89 2.71 O.OIO .002 0.000 0.008 .002 o.ooo 24 24.83 2.94 0.012 .002 0.000 O.OIO .002 o.ooo 3.20 .002 .002 25 28.03 0.014 0.000 0.012 0.000 26 27 28 29 31.48 35.20 39.19 43-47 3-45 3-72 3-99 4.28 4-57 0.017 0.020 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 0.000 o.ooo 0.000 30 31 32 33 34 48.04 52.91 58.09 63-59 69.42 4.87 5.18 5-5 5.83 6.15 0.035 0.041 0.047 -55 0.064 .006 .006 .008 .009 .009 o.ooo 0.000 0.000 0.000 o.ooo 0.028 0.033 0.039 0.045 0.052 .005 .006 .006 .007 .008 o.ooo 0.000 o.ooo 0.000 0,000 35 36 75-57 82.07 6.50 0.073 0.084 .Oil 0.000 o.ooo 0.060 0.068 .008 o.ooo o.ooo 37 38 39 88.92 96.12 103.68 6.85 7.20 7-56 7-93 0.096 0.109 0.123 .012 .013 .014 .016 0.000 o.ooo 0.000 0.078 0.088 O.I 00 .010 .010 .012 .013 o.ooo 0.000 o.ooo 40 111.61 0.139 o.ooo 0.113 o.ooo 41 42 43 44 119.92 128.62 137.70 147.18 8.31 8.70 9.08 9.48 0.156 0-175 0.196 0.218 .017 .019 .021 .022 0.000 o.ooo 0.000 o.ooo 0.127 0.142 0.159 0.177 .014 .015 .017 .018 o.ooo 0.000 o.ooo o.ooo 9.87 .025 .020 45 46 47 48 49 !57-5 167.34 178.04 189.16 200.71 10.^9 10.70 II. 12 u-55 0.243 0.269 0.298 0.328 0.361 .026 .029 .030 033 0.000 o.ooo o.ooo o.ooo 0.000 0.197 0.219 0.242 0.267 0.294 .022 .023 .025 .027 0.000 o.ooo 0.000 o.ooo 0.000 1 1.90 .036 .029 50 51 212.69 225.10 12.41 -397 0.436 039 0.000 o.ooo 0.323 0.354 .031 0.000 o.ooo 52 53 54 2 37-95 251.25 265.01 12.85 13-3 13.76 14.20 0.477 0.521 0.567 .041 .044 .046 .050 O.OOI O.OOI O.OOI 0.388 0.424 0.462 .034 .036 .038 .040 0.000 o.ooo o.ooo 55 56 57 58 59 279.21 293.88 309.02 324.62 340.70 14.67 15.14 15.60 16.08 16.56 0.617 0.671 0.727 0.787 0.851 .054 .056 .060 .064 .068 O.OOI O.OO2 0.002 O.OO2 O.OO2 0.502 0.546 0.592 0.641 0-693 .044 .046 .049 .052 .056 0.000 O.OOI O.OOI O.OOI O.OOI 6O 357.26 0.919 0.003 0.749 0.002 614 TABLE IX, For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. X A Diff. B Diff. C B> Diff. C' o n II n 60 61 62 63 64 357-26 374-30 391.84 409.86 428.38 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 0.807 0.869 o-935 1.004 .058 .062 .066 .069 073 0.002 O.OO2 O.OO2 O.OO2 O.OO2 65 66 67 68 69 447.40 466.92 486.96 507-5I 528.58 19.52 20.04 20.55 21.07 21.59 1.318 1.411 1.510 1.613 1.721 93 .099 .103 .108 .114 0.004 0.005 0.005 0.006 0.006 1.077 1.154 1.235 1.321 1.411 .077 .O8l .086 .090 .094 0.003 0.003 0.003 0.004 0.004 7O 71 72 550.17 572.29 594-94 22.12 22.65 01 T 9 I-835 1.954 2.078 .119 .124 0.007 0.007 0.008 1.605 1.709 .100 .104 O.OO4 0.005 0.005 73 74 618.12 641-85 23*10 23-73 24.28 2.209 2-345 '*$ H3 0.009 0.009 1.819 1.934 . I I O .115 .121 O.OO6 O.OO6 75 666.13 2.488 O.OIO 2.055 T-rfi O.OO7 76 77 690.96 7 l6 -34 25-38 2.637 2-793 .149 O.OII O.OI2 2.181 2.314 .120 133 0.007 O.OO8 78 79 742-29 768.81 25-95 26.52 27.09 2.956 3-125 .169 .177 0.013 0.014 2-453 2-599 139 .146 '53 0.008 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 0.018 0.020 2.752 2.912 3-79 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 3I-32 31.96 32.61 4-303 4.529 4-764 5.008 5.262 .226 235 244 254 O.O2I 0.023 0.024 0.026 0.028 3-631 3-833 4.044 4.266 4.498 .202 .211 .222 .232 0.015 0.016 0.018 0.019 O.O2I 33.27 .265 243 90 1 100.08 5 527 0.030 4.741 0.023 91 92 1 134.02 1168.64 33-94 34-62 5.801 6.087 .274 .286 -?n8 0.032 0.034 4.996 5-263 .'267 O.O25 0.027 93 94 1203.95 1239.97 35-3 1 36.02 36-75 6.694 .29 39 322 0.036 0.038 5-544 5-838 294 39 O.O29 0.032 95 96 97 1276.72 1314.21 37-49 38-24 7.016 7-35 7.698 334 348 0.041 0.044 0.047 6.147 6.471 6.812 .324 -341 0.035 0.038 0.041 98 99 1391.46 1431.27 39.01 39.81 40.61 8.060 8-437 .362 377 392 0.050 0.053 7.171 7-549 359 .378 397 0.045 . 0.049 100 1471.88 20 6i 8.829 0.056 7.946 .206 0.053 30 101 30 102 30 1492.50 I5I3-33 I 534-3 8 '555-64 1577.12 20.83 21.05 21.26 21.48 21.70 9.032 9-238 9-449 9.664 9.883 .203 .206 .211 .215 .219 .225 0.058 0.060 0.062 0.064 0.066 8.152 Sift! 8.805 9.035 .212 .218 .223 .230 .236 0.055 0.058 0.060 0.063 0.066 103 1598.82 10.108 0.068 9.271 0.069 30 104 30 1620.75 1642.91 1665.30 21.93 22.16 22.39 10.337 10.570 10.809 .229 233 239 0.070 0.072 0.074 9-5I3 9.761 10.017 !256 0.072 0.075 0.078 105 30 1687.93 1710.80 22.63 22.87 23.12 11.053 11.302 244 249 .255 0.077 0.079 10.280 10.550 .203 .270 .278 O.o82 0.085 106 30 107 30 108 30 I733-92 1757.28 1780.90 1804.77 1828.90 1853-3 23.36 23.62 23.87 24-13 24.40 24.67 "557 1 1.817 12.083 I2 -354 12.632 12.916 .260 .266 .271 .278 .284 .291 0.082 0.084 0.087 0.090 0.093 o 096 10.828 ii. 1 14 1 1.408 1 1.71 1 12.022 12.343 .286 .294 33 .311 .321 33 0.089 0.093 0.098 0.102 O.IO7 O.I I 2 109 1877.97 13.207 0.099 12.673 O.II7 615 TABLE IX, For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity. X. A Diff. B Diff. C Diff. B' Diff. C" Diff. O 1 ,/ n n 1 109 30 no o 30 111 30 1877.97 1902.91 1928.13 1953.64 1979.44 2005.54 24.94 1 25.22 25-5 1 25.80 26.10 26.40 13.207 13-504 13.808 14.119 14.438 14.764 -297 .304 3 11 319 .326 -333 0.099 0.102 0.106 0.109 0.113 o.i 1 6 .003 .004 3 .004 003 .004 12.673 13.013 J 3-3 6 3 13-724 14.095 14.478 34 350 .361 -371 .383 39 6 0.117 O.I 22 0.128 0.134 0.141 0.148 .005 .006 .006 .0-07 .007 .007 112 30 113 30 2031.94 2058.64 2085.66 21 I-J.OO 26.70 27.02 15.097 15-439 15.789 16.148 342 35 359 O.I20 0.124 0.128 0.132 .004 .004 .004 14.874 15.282 15.702 16.135 .408 .420 433 0-155 O.l62 0.170 0.178 .007 .008 .008 114 30 2140.66 2 7-oo 2168.66 28 / 28.34 16.515 16.892 367 377 .386 0.137 0.142 .005 005 .005 16.583 I7-045 .448 .462 477 0.187 0.196 .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 30.54 17.278 17.674 18.080 18.496 18.924 19.363 396 .406 .416 .428 439 45 0.147 0.152 0.157 0.162 0.168 0.174 .005 .005 .005 .006 .006 .006 17.522 18.015 18.524 19.050 19-594 20.156 493 59 -526 544 .562 .582 O.2O6 0.2 1 6 0.227 0.239 0.251 0.264 .010 .Oil .012 .012 .013 .013 118 30 119 30 2374.60 2405.54 2436.88 2468.64 3-94 31-34 31.76 19.813 20.276 20.751 21.240 .463 475 -489 0.180 0.186 0.193 0.200 .006 .007 .007 20.738 21.339 21.962 22.606 .601 .623 644 0.277 0.291 0.306 O.'222 .014 .015 .Ol6 120 30 2500.83 2533-45 32.19 32.62 33-o6 21.742 22.258 .502 .516 53 1 0.207 O.2I4 .007 .007 .008 23.273 23.964 .667 .691 .716 , j ** o-339 -357 .017 .018 .019 121 30 122 2566.51 2600.03 2634.02 33-52 33-99 22.789 23.336 23.898 547 .562 0.222 0.230 0.239 .008 .009 24.680 25.422 26.191 .742 769 0.376 0.396 0.417 .020 .021 30 123 30 2668.49 2703.46 2738.93 34-47 34-97 35-47 35.98 24.477 25-073, 25.687' 579 596 .614 633 0.248 0.258 0.268 .009 .010 .010 .010 26.988 27.815 28.673 797 .827 .858 .891 T^ / -439 0.463 0.488 .022 .024 .025 .027 124 30 125 30 126 30 2774.91 2811.43 2848.50 2886.13 2963.12 36-52 37-07 37-63 38.20 39.41 26.320 26.973 27.646 28.341 29.057 29.797 .653 .673 6 95 .716 .740 .765 0.278 0.289 0.300 0.312 0.325 0.338 .Oil .Oil .012 013 .013 .014 29.564 30.489 32.448 33-485 34-5 6 3 .925 .961 .998 1.037 1.078 1. 122 o-544 -574 0.606 0.640 0.676 .029 .030 032 34 .036 .039 127 30 128 30 129 30 3002.53 3042.56 3083.23 3166.59 3209.31 40.03 40.67 41.34 42.02 42-72 43-45 30.562 31-351 32.167 33-011 -789 .816 844 874 .904 936 0.352 0.367 0.382 0.398 0.415 o-433 .015 .015 .Ol6 .017 .018 .019 35.685 36-852 38.067 39-33 1 40. 649 42.022 1.167 I.2I 5 1.264 I. 3 l8 1-373 1.430 0.715 o-757 0.800 0.846 0.896 o-949 .042 43 .046 53 .056 130 20 40 131 20 40 3252.76 3282.13 3341.90 3372.31 3403.09 29-37 29.72 30.05 30.41 30.78 35-725 36.367 37-025 38:389 39.097 .642 658 .674 .690 .708 -725 0.452 0.465 0-479 0-493 0.508 0.523! 1 .013 .014 .014 .015 .015 .Ol6 43-452 44-439 45-455 46.500 47-575 48.682 0.987 1.016 1.045 1.075 1.107 1.138 1.005 1.045 1.087 1.130 1.175 1.223 .040 .042 043 45 .048 .050 132 20 40 133 20 40 3434.23 3465-74 3497.63 3529.91 3562.60 3595.69 31-89 32.28 32.69 33-09 33-51 39.822 40.564 41.326 42.108 42.910 43-733 .742 .762 .782 .802 .823 843 o-539/ 0-555L 0.572; 0.590 0.609 0.629 .Ol6 ^.017 .018 .019 .O2O .O2O 49.820 50.992 52.199 53-442 54-723 56.042 1.172 1.207 1.243 1.281 1.273 1-3*5 !-379 1.436 1.495 1.558 .052 .054 .057 .059 .063 .o6c 134 20 40 135 20 40 136 3629.20 3663.13 3697-50 3732.31 3767.58 3803.31 3839.52 33-93 34-37 35-27 35-73 36-21 44-57 6 45.442 46.331 47-245 48.183 49.147. 50.138 .866 .889 .914 938 .964 .991 0.649 0.669 0.691 0.714 0.738 0763 0.788 .020 .022 .023 .024 .025 .025 57.401 58.802 60.247 61.736 63-273 64.857 66.491 1.401 1.445 1.489 J-537 1.584 1.634 1.623 1.692 1.764 1.839 1.917 2.000 2.087 j .069 .072 75 078 .083 .087 616 TABLE IX, For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity X A Diff. B Diff. c Diff. B' Diff. C' Diff. o / a n ,, n n 136 20 40 137 20 40 3839-52 3876.21 3913.41 3951.12 3989-35 4028.11 36.69 37.20 ! 7 *:ll 38.76 39-3 1 50.138 51.156 52.203 53.280 54-388 55-528 I.OlS 1.047 1.077 1.108 1.140 1.174 0.788 0.815 0.843 0.873 0.904 0.936 .027 .028 .030 .031 .032 33 66.491 68.178 69.920 71.718 73-575 75-493 1.687 1.742 1.798 1.857 1.918 1.982 2.087 2.178 2.274 2.375 2.480 2.591 .091 .096 .101 .105 .in .117 138 20 40 139 20 40 4067.42 4107.28 4147.72 4188.75 4230.38 4272.63 39.86 40.44 41.03 41,63 42.25 42.89 56.702 57-910 59-^54 60.436 61.757 63.119 1. 208 1.244 J.282 I. 3 2I 1.362 1.404 0.969 .004 .041 .079 .119 .161 .035 .037 .038 .040 .042 .044 77-475 79.523 81.641 83.830 86.094 88.436 2.048 2.118 2.189 2.264 2.342 2.424 2.708 2.831 2.960 3.096 3-239 3-39 .123 .129 .136 143 .151 .159 140 20 40 141 20 40 43I5-S2 4359.06 4403.26 4448.15 4493-73 4540.03 43-54 44.20 44.89 45-58 46.30 47.04 64.523 65.971 67.465 69.007 70-599 72.243 1.448 1.494 1.542 1.592 1.644 1.698 .205 .251 .299 35 .404 .460 .046 .048 .051 54 .056 .058 90.860 93-369 95-967 98.657 101.443 104.331 2.509 2.598 2.690 2.786 2.888 2.993 3-549 3-7I7 3-893 4.080 4.277 4.484 .168 .176 .187 .197 .207 .220 142 10 20 30 40 50 4587.07 4610.88 4634.88 4659.07 4683.46 4708.05 23.81 24.00 24,19 24-39 24-59 24.79 73-941 74.811 75-695 76.595 77-509 78.439 0.870 0.884 0.900 0.914 0.930 0.946 .518 549 .580 .612 .645 .679 .031 .031 .032 .033 34 35 107.324 108.861 110.427 112.022 113.646 II5.30I l ' 5 ll 1.566 1.595 1.624 I-655 1.685 4.704 4.819 4-936 5-57 5.181 5-309 .115 .117 .121 .124 .128 J3 1 143 10 20 30 40 50 4732.84 4757.84 4783.05 4808.46 4834.10 4859.95 25.00 25.21 25.41 25.64 25.85 26.07 79-385 80.347 81.325 82.321 83-333 84-363 0.962 0.978 0.996 1. 012 1.030 1.048 .714 $1 .823 .862 .901 035 37 037 -39 .039 .041 116.986 118.704 120.452 122.233 124.049 125.899 1.718 1.748 1.781 1.816 1.850 1.886 5.440 5.575 5-715 5.858 6.005 6.157 !35 .140 .143 .147 .152 .156 144 10 20 30 40 50 4886.02 4912.31 4938.83 4965.58 4992.56 5019.78 26.29 26.52 26.75 26.98 27.22 27.45 85.411 86.478 87.564 88.668 89.793 90.938 1.067 1.086 I.I04 I.I25 I.I45 1.165 .942 984 2.026 2.070 2.116 2.162 .042 .042 .044 .046 .046 127.785 129.707 131.666 133.663 135.698 137-774 1.922 1.959 1.997 2-035 2.076 2.116 6.313 6-473 6.639 6.809 6.984 7.165 .160 .166 .170 175 .181 .!86 145 10 20 30 40 50 5047.23 5 74-93 5102.88 5131.08 5'59-53 5188.24 27.70 27.95 28.20 28.45 28.71 28.97 92.103 93.290 94-498 95.729 96.982 98.259 1.187 1. 208 I.23I I - 2 53 1.277 1.300 2.210 2.259 2.309 2.361 2.414 2.469 .049 .050 .052 053 055 .057 139.890 142.048 144.249 146.494 148.784 I5I.I20 2.158 2.2OI 2.245 2.290 2.336 2.383 7-35 1 7-543 7-74 7-943 8 - I 53 8.369 .192 .197 .203 .210 .216 .223 146 10 20 30 40 50 5217.21 5246.45 5275-95 53 5-73 5335-79 5366.13 29.24 29.50 29.78 30.06 30-34 30.63 99-559 100.884 102.234 103.610 105.012 106.441 1-325 1.350 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 I 55-934 158-415 160.947 l6 3-53i 166.168 2.431 2.481 2.532 2.584 2.637 2.692 8.592 8.822 9.060 9-304 9-555 9.815 .230 .238 .244 .260 .268 147 10 20 30 40 50 5396.76 5427.67 5458.88 549-39 5522.20 5554-33 30.91 31.21 S'-S' 31-81 32.13 32.44 107.897 109.382 110.896 112.439 114.013 115.619 1.485 1.514 1-543 ''I 7 ! i. 606 1.637 2.900 2.969 3.040 3'"3 3.188 3.266 .069 .071 073 75 .078 .080 168.860 171.608 174.414 177.280 180.206 183.194 2.748 2.806 2.866 2.926 2.988 3.052 10.083 10 -359 10.645 10.940 11.244 11.558 .2 7 6 .286 -295 34 34 325 148 10 20 30 40 50 5586.77 5619.52 5652.60 5686.01 57I9-75 5753-83 32.75 33.08 33-41 33-74 34.08 34-43 117.256 118.926 120.631 122.370 124.144 125.955 1.670 1.705 1-739 1-774 i.8n 1.849 3.346 3.428 3-5I3 3.601 3.691 3-784 .082 .085 .088 .090 -093 .097 186.246 189.364 192.549 195.804 199.130 202.528 3.118 3.185 3-255 3.326 3.398 3-474 11.883 12.218 12.564 12.921 13.291 13.673 335 -346 357 -370 .382 -394 149 5788.26 127.804 3-88i 206.002 14.067 617 TABLE X, For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits. A Ellipse. Hyperbola. log B Diff. logC log I. Diff. log half II. Diff. log B Diff. log C log I. Diff. log half II. Diff. o.ooo o.ooo o.oo oooo 0.000 0000 4.23990 1.778 0000 0.000 0000 4.23982,4 1.771 .01 0007 7 .001 7432 .24286 0007 9.998 2688 .23686 .767 .02 0030 2 3 .003 4985 .24583 ^88 0030 37 99 6 5493 .23392 .762 .03 0067 37 C 1 .005 2659 .24885 794 0067 ^ x 994 8414 .23098 .758 .04 0120 11 .007 0457 .25190 799 0118 66 993 '45 .22807 753 O.OC 0188 g 0.008 8381 4- 2 5497 1.805 0184 81 9.991 4599 4.22518,! 1.748 .06 0272 QQ .010 6432 .25806 .811 0265 94 989 7859 .22230 743 .07 .08 .09 0371 0485 0615 77 114 I 3 .012 4613 .014 2924 .016 1367 .26116 .26427 .26741 .816 .821 .827 359 0468 0591 109 123 137 .988 1231 .986 4711 .984 8298 2I 943 .21659 .21376 739 734 73 0.10 0762 l62 0.017 9945 4.27057 I-833 0728 152 9.983 1992 4 .2I0 94n 1.725 .11 0924 .019 8659 .27376 839 0880 .981 5791 .20815 .720 .12 JI02 IQ4. .021 7511 .27697 .845 1045 178 979 9 6 94 .20537 .716 .14 1296 1507 211 22 7 .023 6503 .025 5637 .28020 .28344 .851 .857 1223 1416 / 193 206 978 3 6 99 .976 7805 .20260 .19986 .'706 0.15 .16 1734 1977 2 43 261 0.027 49 l 6 .029 4340 4.28670 .28999 1.863 .869 1622 1842 220 233 9.975 2011 973 6 3 l6 .19440" 1.700 .695 17 2238 277 .031 3913 2 933 J 875 2075 246 .972 0719 .19170 f2 .18 .19 2515 2809 / / 294 3" .033 3636 .035 3511 .29665 .30001 .882 .888 2321 2581 260 *73 .970 5218 .968 9813 .18901 18633 .685 .679 0.20 .21 .22 2 3 .24 3120 3448 3793 4156 4537 328 345 3 6 3 398 0.037 3542 .039 3730 .041 4077 .043 4585 .045 5259 4-3339 .30679 .31022 .31368 .31716 1.895 .901 .908 9*5 .922 2854 3140 3439 4076 286 299 312 3 2 5 338 9.967 4502 .965 9285 .964 4159 .962 9124 .961 4180 .18102 .17840 17579 1.672 .666 .66! .655 .649 0.25 .26 4935 416 A"iA 0.047 6099 .049 7109 4.32066 .32418 1.929 93 6 4414 4765 767 9-959 93 2 4 .958 4556 4.17061,, .16803 1.643 637 .27 5785 T JT .051 8290 .32773 943 5128 376 .956 9875 .16547 .631 .28 6237 471 .053 9646 .951 554 955 5 2g i .16292 .625 .29 6708 / * 488 .056 1179 3349 2 .958 5893 401 .954 0771 .16038 .618 O.JO 7196 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. 0.00 .01 .02 3 .04 o.ooooo .00992 .01969 .02930 .03877 992 977 961 947 931 0.00000 .01008 .02033 .03074 .04132 1008 1025 1041 1058 1077 O.2O .21 .22 2 3 .24 0.17266 .18008 .18740 .19462 .20174 742 73 2 722 712 704 0.23867 .25309 .26779 .28280 29813 1442 1470 1501 J 533 1564 a oi ::i .09 0.04808 .05726 .06630 .07521 .08398 918 904 8 9I 877 865 0.05209 .06303 .07417 .08550 .09702 1094 1114 "33 1152 1173 *U .27 .28 .29 0.20878 2I 573 .22258 .22935 .23604 695 685 677 669 661 0.31377 O.IO .11 .12 13 .14 0.09263 .10116 .10956 .11783 .12599 853 840 827 816 805 0.10875 .12069 .13285 .14522 .15782 1194 1216 1237 1260 1285 0.30 31 3 2 33 34 0.24265 .24917 .25561 .26198 .26826 652 644 637 628 . 621 c . ^ :S .19 0.13404 .14198 .14981 '5753 .16515 794 783 772 762 75 1 0.17067 .18375 .19709 .21068 .22454 1308 *334 1359 1386 1413 -35 .36 :ii 39 0.27447 .28061 .28668 .29268 .29860 6i 4 V 607.' 6 oo, a III'- 0.20 0.17266 0.23867 0.40 0.30446 T 618 TABLE XL For the Motion in a Parabolic Orbit. 1? logju. Diff. , log/x Diff. log/u. Diff. 0.000 o.ooo oooo 0.060 o.ooo 0652 0.120 o.ooo 2617 .OOI .000 0000 o .061 .000 0674 22 .121 .000 2661 44 .002 .OOO OOOI 1 .062 .000 0697 23 .122 .000 2705 44 .003 .000 0002 I .063 .000 0719 22 .123 .000 2750 45 .004 .000 0003 I I .064 .000 0742 23 24 .124 .000 2795 46 0.005 o.ooo 0004 0.065 o.ooo 0766 0.125 o.ooo 2841 .OO6 .000 0006 2 .066 .000 0790 24 .126 .000 2886 45 .007 .000 0009 3 .067 .000 0814 24 .127 .000 2933 47 i A . .008 .OOO OOI2 3 .068 .000 0838 24 .128 .000 2979 46 .009 .000 0015 3 3 .069 .000 0863 2 5 .129 .000 3026 47 ; 48 0.010 o.ooo 0018 0.070 o.ooo 0888 g 0.130 o.ooo 3074 .Oil .OOO OO22 4 .071 .000 0914 , .131 .000 3121 47 .012 .OOO OO26 4 .072 .000 0940 76 .132 .000 3169 4 .013 .000 0031 5 .073 .000 0966 2U 133 .000 3218 49 .014 .000 0035 4 6 .074 .000 0993 27 27 134 .000 3267 49 49 0.015 o.ooo 0041 0.075 0.000 1020 0.135 o.ooo 3316 .016 .000 0046 5 .076 .000 1047 27 28 .136 .000 3365 49 .017 .000 0052 .077 .000 1075 -0 137 .000 3415 5 .018 .000 0059 I .078 .000 1103 2o .138 .000 3466 5 1 .019 .000 0065 7 .079 .000 1132 29 29 '39 .000 3516 5 51 0.020 o.ooo 0072 0.080 o.ooo 1161 0.140 o.ooo 3567 .021 .000 0080 .081 .000 1190 29 .141 .000 3619 5 2 .022 .000 0088 .082 .000 1219 29 .142 .000 3671 5 2 .023 .000 0096 .083 .000 1249 3 .000 3723 5 2 .024 .000 0104 o 9 .084 .000 1280 3 1 3 1 .144 .000 3775 5 2 53 0.025 o.ooo 0113 0.085 o.ooo 1311 0.145 o.ooo 3828 .026 .000 0122 9 .086 .000 1342 3 1 .146 .000 3882 54 .027 .000 0132 IO .087 .000 1373 3 1 .147 .000 3935 53 .028 .000 0142 IO .088 .000 1405 3 2 .148 .000 3989 54 .029 .000 0152 I O II .089 .000 1437 3 2 33 .149 .000 4044 55 55 0.030 o.ooo 0163 0.090 o.ooo 1470 0.150 o.ooo 4099 .031 .000 0174 I I .091 .000 1502 3 .151 .000 4154 55 .032 .000 0185 I I .092 .000 1536 34 .152 .000 4209 55 33 .000 0197 12 93 .000 1569 33 153 .000 4265 5 34 .000 0209 I 2 .094 .000 1603 34 35 .154 .000 4322 56 0.035 .036 0.000 0222 .000 0235 13 0.095 .096 o.ooo 1638 .000 1673 35 0.155 .156 o.ooo 4378 .000 4435 3 037 .038 .000 0248 .000 0262 14 .097 .098 .000 1708 .000 1743 35 157 .158 .000 4493 .000 4551 c8 39 .000 0275 15 .099 .000 1779 36 .159 .000 4609 5 59 0.040 o.ooo 0290 O.IOO o.ooo 1815 o.i 60 o.ooo 4668 -0 .041 .000 0304 It .101 .000 1852 37 .161 .000 4726 5 .042 .000 0320 I .102 .000 1889 37 .162 .000 4786 OO 043 .000 0335 *5 .103 .000 1926 11 .163 .000 4846 f, .044 .000 0351 I O 16 .104 .000 1964 I* .164 .000 4906 60 0.045 .046 o.ooo 0367 .000 0383 16 0.105 .IO6 0.000 2002 .000 2040 38 0.165 .166 o.ooo 4966 .000 5027 61 fir .047 .000 0400 1 7 .107 .000 2079 39 .167 .000 5088 O I .048 .049 .000 0417 .000 0435 18 18 .108 .109 .000 21 I 8 .000 2158 39 40 4 .168 .169 .000 5150 .000 5212 62 62 0.050 o.ooo 0453 t o O.IIO o.ooo 2198 0.170 o.ooo 5274 .051 .000 0471 I o .III .000 2238 4 .171 .000 5337 23 .052 053 .054 .000 0490 .oor 0509 .000 0528 J 9 19 20 .112 .113 .114 .000 2279 .000 2320 .000 2361 U W M .172 173 .174 .000 5400 .000 5464 .000 5528 64 64 0.055 .056 o.ooo c 4.8 .000 ^68 20 O.II5 .1 1 6 o.ooo 2403 .000 2445 42 .176 o.ooo 5592 .000 5657 65 .057 .oor ^89 2 1 .117 .000 2487 42 .177 .000 5722 ,5 .058 .00 .,10 21 .118 .000 2530 43 .178 .000 5787 66 .059 .00 71 2 1 21 .119 .000 2573 43 44 .179 .000 5853 66 0.060 o.oo. 52 0.120 o.ooo 2617 0.180 o.ooo 5919 619 TABLE XI, For the Motion in a Parabolic Orbit. ,, log/n Diff. , log p Diff. i? log ft. Diff. o.i8o .181 .182 .183 .184 o.ooo 5919 .000 5986 .000 6053 .000 6120 .000 6l88 67 67 68 68 0.240 .241 .242 .243 .244 o.ooi 0603 .001 0693 .001 0784 .001 0875 .001 0966 90 9 1 9 1 91 92 0.300 .301 .302 33 34 o.ooi 6733 .001 6848 .001 6963 .001 7079 .001 7195 "5 "5 116 116 117 0.185 .186 o.ooo 6256 .000 6325 69 68 0.245 .246 o.ooi 1058 .001 1150 92 0.305 .306 o.ooi 7312 .001 7429 117 117 .187 .188 .189 .000 6393 .000 6463 .000 6532 Oo 7 69 7 .247 .248 .249 .001 1242 .001 1335 .001 1429 93 94 93 37 .308 .309 .001 7546 .001 7664 .001 7783 118 119 118 0.190 .191 .192 .193 .194 o.ooo 6602 .000 6673 .000 6744 .000 6815 .000 6887 72 72 0.250 .251 .252 .254 o.ooi 1522 .001 1617 .001 1711 .001 1806 .001 1901 95 94 95 95 96 0.310 .311 .312 313 3*4 o.ooi 7901 .001 8020 .001 8140 .001 8260 .001 8381 119 I2O 120 121 121 0.195 o.ooo 6959 0.255 o.ooi 1997 q6 0.315 o.ooi 8502 o r 121 .196 .197 ..198 .199 .000 7031 .000 7104 .000 7177 .000 7250 72 73 73 73 74 .256 .257 .258 .259 .001 2093 .001 2190 .001 2287 .001 2384 7 97 97 97 98 .316 3i7 .318 .319 .001 8623 .001 8745 .001 8867 .001 8989 122 122 122 124 O.2OO .201 .202 .203 .204 o.ooo 7324 .000 7399 .000 7473 .000 7548 .000 7624 75 74 g 76 0.260 .261 .262 .263 .264 o.ooi 2482 .001 2580 .001 2679 .001 2778 .001 2877 98 99 99 99 100 0.320 .321 .322 3*3 .324 o.ooi 9113 .001 9236 .001 9360 .001 9484 .001 9609 I2 3 I2 4 124 I2 5 125 O.2O5 .206 .207 .208 o.ooo 7700 .000 7776 .000 7853 .000 7930 76 77 77 0.265 .266 .267 .268 o.ooi 2977 .001 3077 .001 3178 .001 3279 IOO 101 101 I OZ 0.325 .326 3 2 7 .328 o.ooi 9734 .001 9860 .001 9986 .002 0113 126 126 I2 7 .209 .000 8007 78 .269 .001 3381 101 3 2 9 .002 0240 127 0.210 .211 o.ooo 8085 .000 8163 78 0.270 .271 o.ooi 3482 .001 3585 103 0.330 331 0.002 0367 .002 0495 !28 .212 .213 .214 .000 8242 .000 8321 .000 8400 79 79 .272 .273 .274 .001 3688 .001 3791 .001 3894 103 103 103 104 33* 333 334 .002 0624 .OO2 0752 .OO2 0882 "1 130 129 O.2I5 .216 o.ooo 8480 .000 8560 80 Ki 0.275 .276 o.ooi 3998 .001 4103 T CtA -336 0.002 1 01 1 .002 II4I 130 .217 .000 8641 O I Q T .277 .001 4207 104 I O6 337 .OO2 1272 ~ .218 .000 8722 O I 81 .278 .001 4313 338 .002 1403 T -5 T .219 .000 8803 O I 82 .279 .001 4418 106 339 .002 1534 1 3 f 132 0.220 o.ooo 8885 0_ 0.280 o.ooi 4524 0.340 0.002 l666 .221 .222 .000 8967 .000 9050 82 ll .281 .282 .001 4631 .001 4738 107 107 T O*7 341 342 .OO2 1799 .002 1931 '33 132 .223 .224 .000 9132 .000 9216 oZ> 84 84 .283 .284 .001 4845 .001 4953 i<_>7 1 08 tot 343 344 .OO2 2065 .002 2198 133 0.225 .226 o.ooo 9300 .000 9384 84 0.285 .286 o.ooi 5061 .001 5169 108 o-345 34 6 O.OO2 2333 .002 2467 134 .227 .228 .229 .000 9468 .000 9553 .000 9638 84 85 85 86 .287 .288 .289 .001 5278 .001 5388 .001 5497 109 IIO 109 in 347 348 349 .OO2 2602 .002 2738 .002 2874 I 35 136 136 136 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 no in 112 0.350 35 1 35* O.OO2 3OIO .002 3147 .OO2 3284 137 137 1 18 2 33 .000 9984 .001 0071 87 88 .294 .001 5941 .001 6053 112 112 353 354 .002 3422 .OO2 3560 * 3" 138 139 0.235 .236 o.ooi 0159 .001 0247 88 88 0.295 .296 o.ooi 6165 .001 6278 "3 0-355 356 O.OO2 3699 .002 3838 139 I 70 237 .238 .001 0335 .001 0424 89 .297 .298 .001 6391 .001 6505 II 4 357 358 .002 3977 .002 4117 1 57 140 .239 .001 0513 9 .299 .001 6619 114 114 359 .002 4258 141 0.240 o.ooi 0603 0.300 o.ooi 6733 0.360 0.002 4399 620 TABLE XI. For the Motion in a Parabolic Orbit. , log/a Diff. , * Diff. log /a Diff. 0.360 . 3 6i .362 .363 0.002 4399 .002 4540 .OO2 4682 .002 4824 141 142 142 0.420 .421 .422 .423 0.003 3720 .003 3890 .003 4061 .003 4232 170 171 171 0.480 .481 .482 483 0.004 4858 .004 5061 .004 5263 .004 5467 203 202 204 .364 .OO2 4967 J 43 .424 .003 4404 172 172 .484 .004 5670 203 205 o-3 6 5 0.002 5110 0.425 0.003 4576 0.485 0.004 5875 .366 .002 5254 144 .426 .003 4749 X 73 .486 .004 6080 205 .367 .368 .369 .002 5398 .002 5543 .002 5688 144 H5 '45 146 427 .428 .429 .003 4923 .003 5096 .003 5271 J 74 173 175 174 487 .488 489 .004 6285 .004 6492 .004 6698 205 207 206 208 0.370 0.002 5834 T ,g 0.430 0.003 5445 0.490 0.004 6906 371 372 .002 5980 .002 6126 140 146 43 i 432 .003 5621 .003 5797 1 7 7 6 nfi .491 .492 .004 7113 .004 7322 207 20 9 373 374 .002 6273 .002 6421 148 H7 433 434 .003 5973 .003 6150 170 177 177 493 494 .004 7531 .004 7740 2OC^ 20 9 211 0-375 .376 377 378 379 0.002 6568 .OO2 6717 .002 6866 .002 7015 .002 7165 149 149 149 150 150 o-435 .436 437 438 439 0.003 6327 .003 6505 .003 6683 .003 6862 .003 7042 I 7 8 I 7 8 I 79 180 1 80 o-495 49 6 497 498 499 0.004 7951 .004 8161 .004 8373 .004 8585 .004 8797 210 212 212 212 213 0.380 .381 382 .383 384 0.002 7315 .002 7466 .002 7617 .002 7769 .OO2 7921 152 152 0.440 .441 .442 443 444 0.003 7222 .003 7402 003 7583 .003 7765 .003 7947 180 181 i*a 182 183 0.500 5 1 52 53 54 0.004 9010 .005 1173 005 3397 .005 5681 .005 8029 2163 2224 2284 2348 ! 2412 1 0.385 .386 387 .388 389 O.OO2 8073 .002 8226 .OO2 8380 .002 8534 .OO2 8689 153 154- 154 155 155 o-445 .446 447 448 449 0.003 8130 .003 8313 .003 8496 .003 8680 .003 8865 183 183 184 185 185 o-55 .56 31 59 0.006 0441 .006 2919 .006 5464 .006 8079 .007 0765 2545 2686 2760 0.390 391 0.002 8844 .OO2 8999 155 T r n 0.450 45 1 0.003 95 .003 9236 186 186 0.60 .61 0.007 3525 .007 6361 2836 392 393 394 .002 9155 .OO2 9311 .002 9468 5 ? 156 157 158 452 453 454 .003 9422 .003 9609 .003 9797 187 188 187 .62 i 3 .64 .007 9274 .008 2268 .008 5345 2913 2994 | 3077 3163 o-395 .396 397 398 O.OO2 9626 .002 9734 .002 9942 .003 oioi 158 158 159 o-455 .456 457 458 0.003 9984 .004 0173 .004 0362 .004 0551 189 189 189 0.65 .66 .67 .68 0.008 8508 .009 1759 .009 5103 .009 8542 3251 3344 3439 399 .003 0260 160 459 .004 0741 190 191 .69 .010 2081 3539 3642 0.400 0.003 0420 rfin 0.460 0.004 0932 0.70 o.oio 5723 .401 .003 0580 1 DO ifil .461 .004 1123 191 .71 .010 9473 375 JtAfl .402 43 .404 .003 0741 .003 0903 .003 1064 I D I 162 161 163 .462 .463 .464 .004 1315 .004 1507 .004 1700 192 192 193 72 73 74 .on 3336 .on 7316 .012 1419 33 3980 4103 4233 0.405 .406 47 .408 .409 0.003 1227 .003 1389 003 1553 .003 1716 .003 1881 162 164 163 'Js 164 .466 .467 .468 .469 0.004 1893 .004 2087 .004 2281 .004 2476 .004 2672 194 194 196 0.75 .76 77 .78 79 0.012 5652 .013 0022 013 453 6 .013 9202 .014 4031 437 45*4 4666 4829 5002 0.410 .411 .412 .413 0.003 2045 .003 221 I .003 2376 .003 2543 166 i J 67 166 0.470 .471 472 473 0.004 2868 .004 3064 .004 3261 004 3459 196 197 198 198 0.80 .81 .82 i 3 0.014 9033 .015 4219 .015 9603 .016 5202 5186 5384 5599 .414 .003 2709 168 474 .004 3657 199 .84 .017 1033 6087 ; 0.415 .416 0.003 2877 .003 3044 167 169 0.475 .476 0.004 3856 .004 4055 199 2OO 0.85 .86 0.017 7120 .018 3486 6366 417 .418 .419 .003 3213 .003 3381 .003 3550 168 169 170 477 .478 479 .004 4255 .004 4456 .004 4657 201 2O I 201 .87 .88 8 9 .019 0165 .019 7195 .020 4629 7030 | 7434 ; 0.420 0.003 3720 0.480 0.004 4858 0.90 0.021 2529 621 TABLE XII, y z i' z 2' z 3 ' 'Z *' ? log m l log m 2 TOJ ?n 2 m z i , m, in 2 Blj o / 00 o.oooo 90 o 90 o 180 o 180 o 180 o O O 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 4 46 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 HI 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 ii 55 91 41 9 I 41 173 19 173 19 175 355 355 2 3' 6 .9686 9.9968 14 19 92 I 92 I 171 59 171 59 174 o 354 o 354 28 7 .7698 9-9957 1 6 42 9 2 22 92 22 170 38 170 38 172 59 353 i 353 32 8 .5981 9-9943 19 7 92 42 92 42 169 1 8 169 1 8 171 59 35 2 i 352 37! 9 4473 9.9928 21 32 93 3 93 3 167 57 167 57 170 58 35 1 2 35i 42| 10 .3130 9.9911 23 57 93 2 5 93 2 5 166 35 166 35 169 57 35 3 350 47 11 .1922 9.9892 26 23 93 46 93 46 165 14 165 H 1 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 5 6 ; 13 0.9821 9.9848 3 1 i? 94 31 94 3 1 162 29 162 29 166 51 347 8 348 i 14 0.8898 9.9823 33 46 94 53 94 53 161 7 161 7 165 48 346 ii 347 6 15 0.8045 9.9796 3 6 15 95 i? 95 i? 159 43 159 43 164 44 345 14 346 ii 16 0.7254 9.9767 38 46 95 4 95 40 158 20 158 20 163 40 344 17 345 l6 17 0.6518 9.9736 41 18 96 5 9 6 5 156 55 156 55 162 34 343 ai 344 21 18 0.5830 9.9702 43 5 1 96 30 96 30 155 3 155 3 161 27 342 27 343 2 7 19 0.5185 9.9667 46 26 96 56 96 56 154 4 154 4 1 60 19 34i 3 2 342 3 2 20 0.4581 9.9629 49 2 97 23 97 23 152- 37 15* 37 159 9 340 38 341 37: 21 0.4013 9.9588 51 41 97 5 97 5 151 10 151 10 157 58 339 45 340 43 22 0.3479 9-9545 54 22 98 19 98 19 149 41 149 41 i5 6 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 i 59 5i 99 20 99 20 146 40 146 40 154 ii 337 9 338 o 25 0.2053 9.9400 62 40 99 53 99 53 145 7 145 7 152 50 336 19 337 6 26 0.1631 9-9345 65 33 100 28 100 28 143 32 143 32 151 25 335 28 336 13 27 0.1232 9.9287 68 30 101 5 101 5 141 55 141 55 149 56 334 38 335 19 28 0.0857 9.9226 7i 33 101 45 101 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 r- 30 0.0170 9.9092 77 58 I0 3 I 3 103 I 3 136 46 136 46 144 55 332 12 33 2 39 31 9.9857 9.9019 81 23 104 4 104 4 134 S 6 134 5 6 142 59 33i 24 33 1 46 32 9.95 6 5 9.8940 85 o 105 i 105 i I3 2 - 59 *3 2 59 140 51 33 37 33 54 33 9.9292 9.8856 88 54 106 6 106 6 13 54 13 54 138 27 3 2 9 49 33 2 34 9.9040 9-8765 93 ii 107, 22 107 22 128 38 128 38 135 39 329 2 329 1 35 9.8808 9.8665 98 7 108 58 108 58 126 2 126 2 132 13 328 I 4 328 19 36 9.8600 9-8555 104 20 in 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 34 326 45 3^6 45 This table exhibits the limits of the roots of the equation sin (V C) = m sin 4 z', when there are four real roots. The quantities m t and m 2 are the limiting values of m w and the values of /, z 2 ', z a f , and z 4 ', corresponding to each of these, give the limits of the four real roots of the equation. 622 TABLE XII. Y z i z 2 2 3 z *' 5 log m l log m 2 m 2 m, i 1 M1 2 m x n, 2 + 00 00 0.0000 90 o 90 o 180 o 180 o 180 o 1 4.2976 9-9999 I I 20 I 20 89 4 89 40 177 37 1 80 55 181 o 2 3-395 9.9996 2 2 40 2 40 89 20 89 20 175 H 181 51 182 o 3 2.8675 9.9992 3 4 o 4 o 89 o 89 o 172 52 182 46 183 o; 4 2.4938 9.9986 4 o 5 20 5 20 88 40 88 40 170 28 183 42 184 o' 5 2.2044 9.9978 5 o 6 41 6 41 88 19 88 19 168 5 184 37 185 o' 6 1.9686 9.9968 6 o 8 i 8 i 87 59 87 59 165 4i 185 32 186 o 7 1.7698 9-9957 7 i 9 22 9 22 87 38 87 38 163 18 186 28 186 59 8 1.5981 9-9943 8 i 10 42 10 42 87 18 87 18 160 53 187 23 187 59; 9 1-4473 9.9928 9 2 12 3 12 3 86 57 86 57 158 28 188 18 188 58 10 1.3130 9.9911 10 3 13 25 13 25 86 35 86 35 156 3 189 13 189 57 11 1.1922 9.9892 ii 5 14 46 14 46 86 14 86 14 153 37 190 8 190 56 12 1.0824 9.9871 12 6 16 8 16 8 85 52 85 52 151 10 191 4 191 54 13 0.9821 9.9848 13 9 17 31 17 31 85 29 85 29 H8 43 191 59 192 52, 14 0.8898 9.9823 14 12 18 53 18 53 85 7 85 7 146 14 192 54 193 49 15 0.8045 9.9796 15 16 20 I 7 20 17 84 43 84 43 H3 45 193 49 194 46 16 0.7254 9.9767 1 6 20 21 40 21 40 84 20 84 20 141 14 194 44 195 43 ; 17 0.6518 9.9736 17 26 23 5 23 5 83 55 83 55 138 42 195 39 196 39 18 0.5830 9.9702 18 33 24 30 24 30 83 3 83 30 136 9 196 33 197 33 19 0.5185 9.9667 19 41 25 5 6 25 5 6 83 4 83 4 133 34 197 28 198 28 20 0.4581 9.9629 20 51 27 23 27 23 82 37 82 37 130 58 198 23 I 99 22 21 0.4013 9.9588 22 2 28 50 28 50 82 10 82 10 128 19 199 i7 200 15 22 0.3479 9-9545 23 15 30 19 3 19 81 41 81 41 125 38 200 II 201 07 23 0.2976 9-9499 24 31 3i 49 3i 49 81 ii 81 ii 122 55 201 6 2-O2 O 24 0.2501 9.945I 25 49 33 20 33 20 80 40 80 40 120 9 2O2 O 202 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 ^8 35 36 28 36 28 79 32 79 32 114 27 203 47 204 32 27 0.1232 9.9287 30 4 38 5 38 5 78 55 78 55 III 30 204 41 205 22 28 0.0857 9.9226 3i 38 39 45 39 45 78 15 78 15 108 27 205 35 206 II 29 0.0503 9.9161 33 18 4i 27 4i 27 77 33 77 33 105 19 206 28 206 59 30 0.0170 9.9092 35 5 43 13 43 13 ?6 47 76 47 102 3 207 21 207 48 31 9.9857 9.9019 37 i 45 4 45 4 75 5 6 75 5 6 98 37 208 14 208 36 32 9.95 6 5 9.8940 39 9 47 i 47 i 74 59 74 59 95 o 209 06 209 23 33 9.9292 9.8856 4i 33 49 6 49 6 73 54 73 54 91 6 209 58 210 II 34 9.9040 9-8765 44 21 5-1 22 51 22 72 38 72 38 86 49 210 50 210 58 35 9.8808 9.8665 47 47 53 58 53 58 71 2 71 2 81 53 211 41 211 46 36 9.8600 9- 8 555 S 2 3i 57 13 57 13 68 47 68 47 75 4o 212 32 212 33 +36 52.2 9.8443 9.8443 63 26 63 26 63 26 63 26 63 26 63 26 213 15 213 15 This table exhibits the limits of the roots of the equation sin (z' C) = m sin 4 z'. when there are four real roots. The quantities m x and w 2 are the limiting values of m , and the values of z/, z 2 ', z s ', and z/, corresponding to each of these, give the limits of the four real roots of the equation. 623 TABLE XIII. For finding the Katio of the Sector to the Triangle. 1 ** Diff. n log 3 Diff. > logs 2 Diff. 0.0000 .0001 .OOO2 .0003 .0004 0.000 0000 .000 0965 .000 1930 .000 2894 .000 3858 965 965 964 964 963 0.0060 .0061 .0062 .0063 .0064 0.005 7298 .005 8243 .005 9187 .OO6 OI3I .006 1075 945 944 944 944 944 O.OI2O .0121 .OI22 .0123 .OI24 o.on 3417 .on 4343 .on 5268 .on 6193 .on 7118 926 925 925 925 925 0.0005 .ooo6 .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 O.OO6 2OI9 .006 2962 .OO6 3905 .006 4847 .OO6 579O 943 943 942 943 942 0.0125 .OI26 .0127 .0128 .0129 o.on 8043 .on 8967 .011 9890 .012 0814 .012 1737 924 923 924 923 923 O.OOIO .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 .OO6 7673 .006 8614 .006 9555 .007 0496 941 941 941 941 940 0.0130 .0131 .0132 .0133 .0134 0.012 2660 .012 3583 .012 4505 .OI2 5427 .012 6348 923 922 922 921 921 0.0015 .0016 o.ooi 4438 .001 5398 960 0.0075 .0076 0.007 1436 .007 2376 94 n 4 o 0.0135 .0136 0.012 7269 .012 8190 921 .0017 .0018 .001 6357 .001 7316 959 959 .0077 .0078 .007 3316 .007 4255 940 939 .0137 .0138 .OI2 9III .013 0032 921 .0019 .001 8275 959 959 .0079 .007 5194 939 939 .0139 .013 0952 919 0.0020 .0021 .0022 .0023 .0024 o.ooi 9234 .002 0192 .002 1150 .002 2107 .002 3064 958 958 957 957 0.0080 .0081 .0082 .0083 .0084 0.007 6133 .007 7071 .007 8009 .007 8947 .007 9884 938 938 938 937 0.0140 .0141 .0142 .0143 .0144 0.013 l %7 1 .013 2791 .013 3710 .013 4629 OI 3 5547 920 919 919 918 nT o 957 937 Q I o 0.0025 .0026 .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 93 6 0.0145 .0146 .0147 .0148 .0149 0.013 6465 .013 7383 .013 8301 .013 9218 .014 0135 918 918 917 917 917 0.0030 .0031 .0032 .0033 .0034 0.002 8800 .002 9755 ^.003 0709 .003 1663 .003 2617 955 954 954 954 0.0090 .0091 .0092 .0093 .0094 0.008 5502 .008 6437 .008 7372 .008 8306 .008 9240 935 935 934 934 0.0150 .OI5I .0152 0153 .0154 0.014 IO 5 2 .014 1968 .014 2884 .014 3800 .014 4716 916 916 9^6 | ?>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 5631 .014 6546 9'S .0037 .0038 .0039 .003 5476 .003 6428 .003 7380 953 952 952 952 .0097 .0098 .0099 .009 2041 .009 2974 .009 3906 933 933 932 932 .0157 .0158 .0159 .014 7460 .014 8374 .014 9288 914 914 914 914 0.0040 0.003 8332 O.OIOO 0.009 4838 0.0160 0.015 0202 .0041 .003 9284 952 .0101 .009 5770 932 .0161 .015 III5 9*3 .0042 .004 0235 95 1 .0102 .009 6702 932 .0162 .015 2028 9*3 .0043 .004 i i 86 95 1 n c n .0103 .009 7633 93 1 .0163 .OI5 2941 913 .0044 .004 2136 9tju .0104 .009 8564 93 1 .0164 .015 3854 913 95 93 1 912 0.0045 .0046 0.004 3086 .004 4036 950 0.0105 .0106 0.009 9495 .010 0425 93 0.0165 .0166 0.015 4766 .015 5678 912 .0047 .0048 .004 4985 .004 5934 949 949 .0107 .0108 .010 1355 .010 2285 93 93 .0167 .0168 .015 6589 .015 7500 911 911 .0049 .004 6883 949 C\A n .0109 .010 3215 93 .0169 .015 8411 911 V4V 929 911 0.0050 .0051 .0052 0.004 7832 .004 8780 .004 9728 94 8 948 O.OIIO .0111 .0112 o.oio 4144 .010 5073 .010 6001 929 928 0.0170 .0171 .0172 0.015 9322 .Ol6 0232 .Ol6 1142 910 910 .0053 .0054 .005 0675 .005 1622 947 947 947 .0113 .0114 .010 6929 .010 7857 928 928 928 .0173 .0174 .Ol6 2052 .Ol6 2961 910 909 909 0.0055 .0056 .0057 .0058 .0059 0.005 2569 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 927 927 926 926 926 0.0175 .0176 .0177 .0178 .0179 0.016 3870 .016 4779 .016 5688 .016 6596 .016 7504 909 909 908 908 908 0.0060 0.005 7298 0.0120 o.on 3417 y 0.0180 0.016 8412 y TABLE XIII, For finding the Katio of the Sector to the Triangle. 1) Iogs2 Diff. n logs 2 Diff. 1? logs 2 Diff. 0.0180 .0181 0.016 8412 .016 9319 907 0.0240 .0241 0.022 2330 .022 3220 890 80_ 0.0300 .0301 0.027 5218 .027 6091 873 Q _,. .0182 .017 0226 907 .0242 .022 4109 09 S8n .0302 .027 6964 8 73 0_- .0183 .0184 .017 1133 .017 2039 97 906 906 .0243 .0244 .022 4998 .022 5887 o oy 889 88 9 .0303 .0304 .027 7836 .027 8708 872 872 872 0.0185 .0186 0.017 2945 .017 3851 906 0.0245 .0246 0.022 6776 .022 7664 888 0.0305 .0306 0.027 9580 .028 0452 872 o_ , .0187 .017 4757 900 .0247 .022 8552 .0307 .028 1323 871 o .0188 .017 5662 905 .0248 .022 9440 .0308 .028 2194 871 .0189 .017 6567 95 904 .0249 .023 0328 887 .0309 .028 3065 871 871 0.0190 .0191 .0192 0.017 747 * .017 8376 .017 9280 95 904 0.0250 .0251 .0252 0.023 I2I 5 .023 2102 .023 2988 887 886 QQ_ 0.0310 .0311 .0312 0.028 3936 .028 4806 .028 5676 870 870 0_ .0193 .0194 .018 0183 .018 1087 93 904 903 .0253 .0254 .023 3875 .023 4761 007 886 886 .0313 .0314 .028 6546 .028 7415 o7O 86 9 869 0.0195 .0196 0.018 1990 .018 2893 93 0.0255 .0256 0.023 5 6 47 .023 6532 885 885 0.0315 .0316 0.028 8284 .028 9153 86 9 0(.' .0197 .0198 .0199 .018 3796 .018 4698 .018 5600 903 902 9O2 901 .0257 .0258 .0259 .023 7417 .023 8302 .023 9187 885 885 884 .0317 .0318 .0319 .029 0022 .029 0890 .029 I 75 8 009 868 868 868 0.0200 .0201 .0202 .0203 .0204 0.018 6501 .018 7403 .018 8304 .018 9205 .019 0105 9O2 901 9OI 900 900 0.0260 .0261 .0262 .0263 .0264 0.024 007! .024 0956 .024 1839 .024 2723 .024 3606 885 883 884 883 883 0.0320 .0321 .0322 .0323 .0324 0.029 2626 .029 3494 .029 4361 .029 5228 .029 6095 868 867 867 ^67 866 O.O2O5 .0206 .0207 .0208 .0209 0.019 I00 5 .019 1905 .019 2805 .019 3704 .019 4603 90O 9OO 899 899 899 0.0265 .0266 .0267 .0268 .0269 0.024 4489 .024 5372 .024 6254 .024 7136 .024 8018 883 882 882 882 882 0.0325 .0326 .0327 .0328 0329 0.029 6961 .029 7827 .029 8693 .029 9559 .030 0424 866 866 866 865 866 0.0210 0.019 55 02 8nn 0.0270 0.024 8900 QQ T 0.0330 0.030 1290 Of.. .0211 .0212 .019 6401 .019 7299 99 898 8n8 .0271 .0272 .024 9781 .025 0662 oo I 88 1 QQ T .0331 0332 .030 2154 .030 3019 004 865 or~. .0213 .0214 .019 8197 .019 9094 oQ o III .0273 .0274 .025 1543 .025 2423 oo I 880 880 0333 334 .030 3883 .030 4747 864 864. 864 0.0215 .02l6 .0217 .0218 .0219 0.019 999 2 .020 0889 .020 1785 .O2O 2682 .020 3578 III 8 97 896 8 9 6 0.0275 .0276 .0277 .0278 .0279 0.025 3303 .025 4183 .025 5063 .025 5942 .025 6821 880 880 879 879 879 0.0335 .0336 0337 0338 .0339 0.030 5611 .030 6475 .030 7338 .030 8201 .030 9064 86A 86 3 863 863 862 O.O22O .0221 .O222 .0223 .0224 0.020 4474 .020 5369 .020 6264 .020 7159 .020 8054 895 8 95 8 95 8 95 894 0.0280 .0281 .0282 .0283 .0284 0.025 7700 .025 8579 025 9457 .026 0335 .026 1213 879 878 878 878 877 0.0340 .0341 .0342 0343 344 0.030 9926 .031 0788 .031 1650 .031 2512 03 1 3373 862 862 862 861 86 1 0.0225 .O226 .0227 .0228 .0229 O.O2O 8948 .020 9842 .021 0736 .021 1630 .021 2523 8 94 8 94 8 94 893 893 0.0285 .0286 .0287 .0288 .0289 0.026 2090 .026 2967 .026 3844 .026 4721 .026 5597 877 877 877 876 876 0.0345 .0346 347 .0348 .0349 0.031 4234 .031 5095 .031 5956 .031 6816 .031 7676 861 861 860 860 860 0.0230 .0231 .0232 .0233 .0234 0.021 3416 .021 4309 .021 5201 .021 6093 .O2I 6985 893- 892 892 892 891 0.0290 .0291 .0292 .0293 .0294 0.026 6473 .026 7349 .026 8224 .026 9099 .026 9974 876 875 875 875 875 0.0350 .0351 .0352 353 354 0.031 8536 .031 9396 .032 0255 .032 1114 .032 1973 860 859 859 859 858 0.0235 .0236 .0237 .0238 .0239 0.021 7876 .021 8768 .021 9659 .022 0549 .022 1440 892 8 9 i 890 8 9 i 890 0.0295 .0296 .0297 .0298 .0299 0.027 0849 .027 1723 .027 2597 .027 3471 .027 4345 874 874 874 874 873 0-0355 .0356 .0357 0358 0359 0.032 2831 .032 3689 .032 4547 .032 5405 .032 6262 858 858 858 857 858 0.0240 0.022 2330 0.0300 0.027 5218 0.0360 0.032 7120 40 (525 TABLE XIH. For finding the Eatio of the Sector to the Triangle. 1? 10g2 Diff i\ *., Diff. ^ Diff. SO sD so VO SO CO CO CO CO CO q q q q q 6 0.032 7120 .032 7976 .032 8833 .032 9689 .033 0546 8 5 6 857 856 857 855 0.060 .061 .062 .063 .064 0.052 5626 .053 3602 .054 1556 .054 9488 055 7397 7976 7954 7932 799 7888 0.120 .121 .122 .123 .124 0.096 8849 .097 5692 .098 2520 .098 9331 .099 6127 6843 6828 6811 6796 6780 so VO so so so CO CO CO CO CO q q q o q o 0.033 1401 .of3 2257 .033 3112 .033 3967 .033 4822 8 5 6 855 855 855 855 0.065 .066 .067 .068 .069 0.056 5285 057 3 J 5 .058 0994 .058 8817 .059 6618 7865 7844 7823 7801 7780 0.125 .126 .!2 7 .128 .129 o.ioo 2907 .100 9672 .101 6421 .102 3154 .102 9873 6765 6749 6 733 6719 6703 0.0370 .0371 .0372 0373 374 0-033 5677 033 6531 033 7385 .033 8239 .033 9092 854 854 854 853 854 0.070 .071 .072 .073 .074 0.060 4398 .061 2157 .061 9895 .062 7612 .063 5308 7759 7738 7717 7696 7676 0.130 .131 .132 133 .134 0.103 6576 .104 3264 .104 9936 .105 6594 .106 3237 6688 6672 6658 6643 6628 0.0375 .0376 0377 0.033 9946 .034 0799 .034 1651 853 852 8 et 0.075 .076 .077 0.064 2984 .065 0639 .065 8274 7655 7635 '.\$ 137 0.106 9865 .107 6478 .108 3076 6613 6598 .0378 379 .034 2504 34 335 6 8 53 852 852 .078 .079 .066 5888 .067 3482 7614 7594 7575 .138 .139 .108 9660 .109 6229 6569 6554 w cj tort- OO OO OO OO OO CO CO CO CO CO q o o o o d 0.034 4208 .034 5059 .034 5911 .034 6762 034 7613 851 852 851 851 851 0.080 .081 .082 .083 .084 0.068 1057 .068 8612 .069 6146 .070 3661 .071 1157 7555 7534 7515 7496 7476 0.140 .141 .142 143 .144 o.no 2783 .no 9323 .III 5849 .112 2360 .112 8857 6540 6526 6511 6497 648-2 0.0385 .0386 0387 .0388 .0389 0.034 8464 34 93 J 4 .035 0164 .035 1014 .035 1864 850 850 850 850 849 0.085 .086 .087 .088 .089 0.071 8633 .072 6090 .073 3527 .074 0945 .074 8345 7457 7437 7418 7400 7380 0.145 .146 .147 .148 .149 0.113 534 .114 1809 .114 8264 .115 4704 .116 1131 T^ J 6469 6 455 6440 6427 6413 0.0390 .0391 0.035 2713 035 3562 849 0.090 .091 o-075 5725 .076 3087 7362 0.150 .151 0.116 7544 "7 3943 6 399 r. of: .0392 .035 4411 88 .092 .077 0430 7343 .152 .118 0329 0300 0393 .0394 035 5259 .035 6108 849 848 93 94 77 7754 .078 5060 7324 7306 7288 '53 .154 .118 6701 .119 3059 6372 6358 6345 0.0395 .0396 .0397 .0398 .0399 0.035 6956 .035 7804 .035 8651 .035 9499 .036 0346 848 847 848 847 846 0.095 .096 .097 .098 .099 0.079 2348 .079 9617 .080 6868 .081 4101 .082 1316 7269 7251 7215 7197 0.155 .156 157 .158 .159 0.119 944 120 5735 .121 2053 .121 8357 .122 4649 6331 6318 6304 6292 6278 0.040 .041 .042 .043 .044 0.036 1192 .036 9646 .037 8075 .038 6478 .039 4856 8454 8429 8403 8378 8353 O.IOO .101 .102 .103 .104 0.082 8513 .083 5693 .084 2854 .084 9999 .085 7125 7180 7161 7H5 7126 7110 0.160 .161 .162 .,63 .164 0.123 927 .123 7192 .124 3444 .124 9682 .125 5908 6265 6252 6238 6226 6213 0.045 .046 .047 0.040 3209 .041 1537 .o 4I 9841 8328 ;34 0.105 .106 .107 0.086 4235 087 1327 .087 8401 7092 7074 0.165 .166 .167 0.126 2121 .126 8321 .127 4508 6200 6l8 7 .048 .042 8121 , o .108 .088 5459 7058 .168 .128 0683 6 Sl 5 .049 .043 6376 8255 8231 .109 .089 2500 7041 7023 .169 .128 6845 6l62 6149 0.050 .051 .052 053 .054 0.044 4607 .045 2814 .046 0997 .046 9157 .047 7294 8207 8183 8160 8i37 8113 O.IIO .HI .112 "3 .114 0.089 9523 .090 6530 .091 3520 .092 0494 .092 7451 7007 6990 6974 6957 6940 0.170 .171 .172 173 .174 0.129 2994 .129 9131 .130 5255 .131 1367 .131 7466 6137 6124 6112 6099 6087 0.055 .056 .057 .058 .059 0.048 5407 .049 3496 .050 1563 .050 9607 .051 7628 8089 8067 8044 8021 7998 0.115 .116 .117 .118 .119 -93 439 1 .094 1315 .094 8223 .095 5114 .096 1990 6924 6908 6891 6876 6859 0.175 .176 .177 .178 .179 0-132 3553 1 .132 9628 .133 5690 .134 1740 .134 7778 6075 6062 6050 6038 6026 0.060 0.052 5626 0.120 0.096 8849 0.180 0.135 3804 626 TABLE XIII, For finding the Katio of the Sector to the Triangle. 1? logs 2 Diff. * logs 2 Diff. I logs 2 Diff. o.i8o .181 .182 .183 .184 0.135 3804 .135 9818 .136 5821 .137 1811 .137 7789 6014 6003 5990 5978 5966 0.240 .241 .242 .243 .244 0.169 59* .170 0470 .170 5838 .171 1197 .171 6547 5378 5368 5359 535 534 0.300 .301 .302 33 .304 O.2OO 2285 .200 7157 .2OI 2O2I .201 6878 .202 1727 4872 4864 i 4857 i 4849 4842 0.185 .186 .187 0-138 3755 .138 9710 139 5653 5955 5943 0.245 .246 *47 0.172 1887 .172 7218 .173 2540 533i 53** 0.305 .306 37 0.202 6569 .203 1403 .203 6230 4834 48*7 A 9"> rt .188 .189 .140 1585 .140 7504 593* 5019 5908 .248 .249 173 7853 "^ .174 3156 5303 5*95 .308 39 .204 1050 .204 5862 4.0X0 4812 4805 0.190 .191 0.141 3412 .141 9309 5897 -oo r 0.250 .251 0.174 8451 .175 3736 5*85 0.310 .311 0.205 0667 .205 5464 4797 .192 193 .194 .142 5194 .143 1068 .143 6931 5 o 5 5874 5863 585! .252 *53 .254 .175 9013 .176 4280 .176 9538 5277 5*67 5*58 5*5 .312 3'4 .206 0254 .206 5037 .206 9813 479 4783 4776 4768 0.195 .196 .197 .198 .199 0.144 2782 .144 8622 .145 4450 .146 0268 .146 6074 5840 5828 5818 5806 5795 0.255 .256 .257 .258 .259 0.177 4788 .178 0029 .178 5261 .179 0484 .179 5698 5*4i 5*3* 5223 5214 5205 .316 .317 .318 .319 0.207 4581 .207 9342 .208 4096 .208 8843 .209 3582 4761 4754 4747 4739 4733 O.2OO .201 0.147 1869 .147 7653 5784 0.260 .261 o.i 80 0903 .180 6100 o o 0.320 .321 0.209 8315 .2IO 3040 47*5 .202 .203 .204 .148 3427 .148 9189 .149 4940 5774 5762 575 1 574 1 .262 .263 .264 .181 1288 .181 6467 .182 1638 5188 5 J 79 5171 5162 .322 3*3 3*4 .210 7759 .211 2470 .211 7174 4719 4711 4704 4697 0.205 0.150 0681 0.265 0.182 6800 0.325 0.212 1871 .206 .150 6411 573 .266 .183 1953 5153 .212 6562 leU .20 7 .151 2130 57 X 9 M/SQ .267 .183 7098 5H5 3*7 .213 1245 4083 .208 .209 .151 7838 '5* 3535 5708 5697 5687 .268 .269 .184 2235 .184 7363 5128 5120 .328 3*9 .213 59 2I .214 0591 4676 467 4662 0.210 .211 .212 .213 .214 0.152 9222 .153 4899 .154 0565 .154 6220 155 1865 5677 5666 5655 5645 5634 0.270 .271 .272 *73 .274 0.185 *483 .185 7594 .186 2696 .186 7791 .187 2877 5111 5102 5095 5086 5078 0.330 331 33* 333 334 0.214 5253 .214 9909 .215 4558 .215 9200 .216 3835 4656 4649 4642 4635 4629 0.215 .2l6 .217 .218 .219 0.155 7499 .156 3123 .156 8737 157 434 *57 9933 5624 5614 5 6 3 5593 5583 0.275 .276 .277 .278 .279 0.187 7955 .188 3024 .188 8085 .189 3138 .189 8183 5069 5061 553 545 537 o-335 336 337 338 339 0.2 1 6 8464 .217 3085 .217 7700 .218 2308 .218 6910 4621 4615 4608 4602 4595 0.220 .221 .222 .223 .224 0.158 5516 .159 1089 .159 6652 .160 2204 .160 7747 5573 55 6 3 555* 5543 553* 0.280 .281 .282 .283 .284 0.190 3220 .190 8249 .191 3269 .191 8281 .192 3286 5*9 5020 5012 5005 4996 0.340 .341 34* 343 344 0.219 I 55 .219 6093 .220 0675 .220 5250 .220 9818 4588 4582 4575 4568 4562 O.225 .226 .227 0.161 3279 .161 8802 .162 4315 55*3 5513 0.285 .286 .287 0.192 8282 .193 3271 .193 8251 4989 4980 o-345 34 6 347 0.221 4380 .221 8935 .222 3483 4555 4548 .228 .229 .162 9817 .163 5310 5502 5493 - . 6- .288 .289 .194 3224 .194 8188 4973 4964 348 349 .222 8025 ,223 2561 454* 4536 543 4957 45*9 0.230 .231 .232 *33 *34 0.164 793 .164 6267 .165 1730 .165 7184 .166 2628 5474 5463 5454 5444 5435 0.290 .291 .292 .294 0.195 3145 .195 8094 .196 3035 .196 7968 .197 2894 4949 4941 4933 4926 49 J 7 0.350 351 35* 353 354 0.223 7090 .224 1613 .224 6130 .225 0640 .225 5143 45*3 45 r 7 ! 4510 4503 4497 0.235 .236 *37 .238 *39 0.166 8063 .167 3488 .167 8903 .168 4309 .168 9705 54*5 5415 54o6 5396 5387 0.295 .296 .297 .298 .299 0.197 7811 .198 2721 .198 7624 | .199 2518 .199 7406 4910 4903 4894 4888 4879 o-355 356 357 .358 359 0.225 9640 .226 4131 .226 8615 .227 3093 .227 7565 4491 4484 4478 447* 4466 0.240 0.169 5092 0.300 0.200 2285 0.360 0.228 2031 TABLE XIII, For finding the Katio of the Sector to the Triangle. 17 logs* Diff. n log2 Diff. T) logss Diff. 0.360 .361 .362 .363 .364 0.228 2031 .228 6490 .229 0943 .229 5390 .229 9831 4459 4453 4447 4441 4434 0.420 .421 .422 423 .424 0-253 9'53 .254 3269 254 7379 .255 1484 255 5584 4116 4110 4105 4100 4095 0.480 . 4 8l .482 .483 .484 0.277 7272 .278 1096 .278 4916 278 8732 279 2543 3824 3820 3816 3811 3806 -3J>5 .366 367 .368 3 6 9 0.230 4265 .230 8694 .231 3116 .231 7532 .232 1942 4429 4422 4416 4410 4404 0.425 .426 .427 .428 .429 0.255 9679 256 3769 256 7853 .257 1932 .257 6006 4090 4084 4079 4074 4069 0.485 .486 .487 .488 489 0.279 6349 .280 0151 280 3949 280 7743 .281 1532 3802 3798 3794 3785 3784 0.370 .371 .372 0.232 6346 233 0743 233 5 J 35 4397 4392 . -6c 0.430 431 43 2 0.258 0075 258 4139 .258 8198 4064 4059 0.490 491 492 0.281 5316 .281 9096 .282 2872 3780 3776 373 374 .233 9521 .234 3900 4380 4379 4374 433 434 259 2252 .259 6300 454 4048 444 493 494 .282 6644 .283 0411 3772 3767 3762 o-375 376 377 378 379 0.234 8274 .235 2642 .235 7003 .236 1359 .236 5709 4368 43 61 4356 4350 -435 .436 437 .438 439 0.260 0344 .260 4382 .260 8415 .261 2444 .261 6467 4038 433 4029 4023 o-495 .496 497 498 499 0.283 4173 283 7932 .284 1686 .284 5436 .284 9181 3759 3754 3750 3745 4344 4019 3742 0.380 .381 .382 3^3 .384 0.237 0053 .237 4391 .237 8723 .238 3050 .238 7370 4338 433 2 4327 4320 0.440 .441 .442 443 444 0.262 0486 .262 4499 .262 8507 .263 2511 .263 6509 4013 4008 4004 3998 0.500 .501 .502 53 .504 0.285 2923 .285 6660 .286 0392 .286 4121 .286 7845 3737 3732 3729 3724 43*5 3994 3720 0.385 .386 387 388 0.239 I 685 239 5993 .240 0296 .240 4594 4308 433 4298 0.445 .446 447 .448 0.264 53 .264 4492 .264 8475 .265 2454 3989 3983 3979 0.505 .506 507 .508 0.287 ^65 .287 5281 .287 8992 .288 2700 3716 3711 3708 389 .240 8885 4291 4286 449 .265 6428 3974 39 6 9 509 .288 6403 3703 3 6 99 0.390 .391 392 393 394 0.241 3171 .241 7451 .242 1725 .242 5994 .243 0257 4280 4274 4269 4263 4 2 57 0.450 451 452 453 454 0.266 0397 .266 4362 .266 8321 .267 2276 .267 6226 3965 3959 3955 395 3945 0.510 .511 .512 513 .514 0.289 !O2 .289 3797 .289 7487 .290 1174 .290 4856 3 6 95 3690 3687 3682 3679 -395 396 397 .398 399 0.243 45 H .243 8766 .244 3012 .244 7252 .245 1487 4252 4246 4240 4235 4229 o-455 456 457 458 459 0.268 0171 .268 4111 .268 8046 269 1977 269 5903 3940 3935 393 1 3926 3921 o-5i5 .516 517 .518 .519 0.290 8535 .291 2209 .291 5879 .291 9545 .292 3207 3674 3670 3666 3662 3657 0.400 .401 .402 .403 .404 0.245 57i6 .245 9940 .246 4158 .246 8371 .247 2578 4224 4218 4213 4207 4201 0.460 .461 .462 463 .464 0.269 9824 .270 3741 .270 7652 .271 1559 .271 5462 39 r 7 39" 397 393 3898 0.520 .521 .522 523 524 0.292 6864 .293 0518 .293 4168 .293 7813 294 H55 3654 3650 3 6 45 3642 3637 0.405 .406 .407 .408 .409 0.247 6779 .248 0975 .248 5166 .248 9351 249 353 * 4196 4191 4185 4180 4174 $ .467 .468 469 0.271 9360 .272 3253 .272 7141 .273 1025 .273 4904 3893 3884 3879 3874 0.525 .526 .527 .528 529 0.294 5092 .294 8726 295 2355 .295 5981 .295 9602 3 6 34 3629 3626 3621 3618 0.410 .411 .412 .413 .414 0.249 7705 .250 1874 .250 6038 .251 0196 251 4349 4169 4164 4158 4153 4H7 0.470 .471 .472 473 474 0.273 8778 .274 2648 274 6513 275 0374 275 4230 3870 3865 3861 3856 3852 0.530 531 532 533 534 0.296 3220 .296 6833 297 0443 .297 4049 .297 7650 3613 3610 3606 3601 3598 0.415 .416 .417 .418 .419 0.251 8496 .252 2638 .252 6775 .253 0906 253 5032 4142 4^37 4131 4126 .121 o-475 476 477 478 479 0.275 8082 .276 1929 .276 5771 .276 9609 277 3443 3847 3842 3838 3834 3829 0-535 536 537 538 539 0.298 1248 .298 4842 .298 8432 .299 2018 .299 5600 3594 359 3586 3582 3578 0.420 0.253 9153 0.480 0.277 7272 0.540 0.299 9178 628 TABLE XIIL For finding the Ratio of the Sector to the Triangle. 1 .0,, Diff. * * Diff. * logs 2 Diff. 0.540 54 1 .542 543 544 0.299 9 J 78 .300 2752 .300 6323 .300 9890 .301 3452 3574 357i 3567 3562 3559 0.560 .561 .562 .563 .564 0.306 9938 37 3437 37 6931 .308 0422 .308 3910 3499 3494 349 1 3488 7484 0.580 .581 .582 583 .584 0.313 9215 .314 264! .314 6064 .314 9483 .315 2898 3426 3423 3419 34*5 3412 0-545 546 547 548 549 0.301 7011 .302 0566 .302 4117 .302 7664 .303 1208 3555 355 1 3547 3544 354 ^566 5 6 7 .568 .569 0.308 7394 .309 0874 39 435 .309 7823 .310 1292 3476 3473 3469 3466 0.585 .586 .587 .588 589 0.315 6310 .315 9719 .316 3124 .316 6525 .316 9923 3409 345 3401 3398 3395 0.550 55i 552 j -553 554 0.303 4748 .303 8284 .304 1816 34 5344 .304 8869 353 6 3532 3528 3525 352i 0.570 571 572 573 574 0.310 4758 .310 8220 .311 1678 .311 8584 3462 3458 3455 345 1 3447 0.590 59 1 592 593 594 0.317 3318 .317 6709 .318 0096 .318 3480 .318 6861 33 3 87 3384 3377 0-555 556 0.305 2390 .305 5907 35'7 o-575 576 0.312 2031 3 12 5475 3444 o-595 59 6 0.319 0238 .319 3612 3374 557 558 559 .7o<: 0420 j-> j l f .000 0249 23 .051 .000 1532 61 .000 1444 si .022 .023 .024 .000 0280 .000 0306 .000 0334 11 28 28 .000 0273 .000 0298 .000 0325 27 27 .052 .053 054 .000 1593 .000 1656 .000 1720 63 64 65 .000 1500 .000 1558 .000 1616 S u 58 58 59 0.025 .026 o.ooo 0362 .000 0392 30 o.ooo 0352 .000 0381 29 0.055 .056 o.ooo 1785 .000 1852 67 68 o.ooo 1675 .000 1736 61 62 .027 .028 .029 .000 0423 .000 0455 .000 0489 3 1 34 34 .000 0410 .000 0441 .000 0473 3 1 32 33 057 .058 .059 .000 1920 .000 1989 .000 2060 .000 1798 .000 1860 .000 1924 62 64 64 0.030 o.ooo 0523 o.ooo 0506 0.060 o.ooo 2131 o.ooo 1988 629 TABLE XIV. For finding the Ratio of the Sector to the Triangle. X | X Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.o6o .061 o.ooo 2131 .000 2204 73 o.ooo 1988 .000 2054 66 67 O.I 2O .121 o.ooo 8845 .000 8999 154 o.ooo 7698 .000 7822 If ft .062 .063 .004 .000 2278 .000 2354 .000 2431 77 78 .000 2121 .OOO 2189 .000 2257 u / 68 68 70 .122 .123 .124 .000 9154 .000 9311 .000 9469 157 158 .000 7948 .000 8074 .000 8202 I 20 126 128 128 0.065 .066 .067 .068 .069 o.ooo 2509 .000 2588 .000 2669 .000 2751 .000 2834 79 81 82 83 84 o.ooo 2327 .000 2398 .oco 2470 .000 2543 .000 2617 7 1 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 o.ooo 8330 .000 8459 .000 8590 .000 8721 .000 8853 129 '3 1 132 J 33 0.070 o.ooo 2918 o.ooo 2691 76 0.130 o.ooi 0447 168 o.ooo 8986 .071 .072 .000 3004 .000 3091 87 .000 2767 .000 2844 /** .132 .001 0615 .001 0784 169 171 .000 9120 .000 9255 J 34 35 073 .074 .000 3180 .000 3269 89 9 1 .000 2922 .000 3001 79 80 '34 .001 0955 .001 1128 1 1 A 173 173 .000 9390 .000 9527 137 138 0.075 .076 o.ooo 3360 .000 3453 93 o.ooo 3081 .000 3162 81 82 0*1:35 .136 o.ooi 1301 .001 1477 176 T T7 o.ooo 9665 .000 9803 138 .077 .078 .079 .000 3546 .000 3641 .000 3738 93 95 97 97 .000 3244 .000 3327 .000 3411 83 84 85 137 .138 ->39 .001 1654 .001 1832 .001 2012 177 180 181 .000 9943 .001 0083 .001 0224 I -|-O 140 141 142 0.080 .081 o.ooo 3835 .000 3934 99 o.ooo 3496 .000 3582 86 0.140 .141 o.ooi 2193 .001 2376 183 T Q A o.ooi 0366 .001 0509 143 .082 .000 4034 IOO .000 3669 88 .142 .001 2560 I 54 185 .001 0653 144 .083 .000 4136 .000 3757 80 .143 .001 2745 ,QQ .001 0798 J 45 .084 .000 4239 103 104 .000 3846 09 90 .144 .001 2933 loo 188 .001 0944 140 147 0.085 o.ooo 4343 o.ooo 3936 0.145 o.ooi 3121 o.ooi 1091 .086 .087 .000 4448 .000 4555 107 .000 4027 .000 4119 92 .146 .147 .001 3311 .001 3503 190 192 .001 1238 .001 1387 147 149 .088 .000 4663 108 .000 4212 93 .148 .001 3696 J 93 .001 1536 149 .089 .000 4773 I IO .000 4306 94 .149 .001 3891 J 95 .001 1686 150 iii 95 I 90 152 0.090 .091 .092 93 o.ooo 4884 .000 4996 .000 5109 .000 5224 112 "3 "5 o.ooo 4401 .000 4496 .000 4593 .000 4691 95 97 98 0.150 .151 .152 '53 o.ooi 4087 .001 4285 .001 4484 .001 4684 198 199 2OO o.ooi. 1838 .001 1990 .001 2143 .001 2296 153 153 .094 .000 5341 1 17 117 .000 4790 99 IOO .154 .001 4886 2O2 204 .001 2451 156 0.095 .096 .097 .098 .099 o.ooo 5458 .000 5577 .000 5697 .000 5819 .000 5942 119 120 122 123 124 o.ooo 4890 .000 4991 .000 5092 .000 5195 .000 5299 101 101 103 104 104 0.155 .156 ?57 .158 '59 o.ooi 5090 .001 5295 .001 5502 .001 5710 .001 5920 205 207 208 210 211 o.ooi 2607 .001 2763 .001 2921 .001 3079 .001 3238 156 158 158 J 59 160 O.I 00 .101 .102 .103 .104 o.ooo 6066 .000 6192 .000 6319 .000 6448 .000 6578 126 127 I2 9 130 o.ooo 5403 .000 5509 .000 5616 .000 5723 .000 5832 1 06 107 107 109 109 o.i 60 .161 .162 .163 .164 o.ooi 6131 .001 6344 .001 6559 .001 6775 .001 6992 213 216 217 219 o.ooi 3398 .001 3559 .001 3721 .001 3883 .001 4047 161 162 162 164 164 0.105 .IO6 .107 .108 o.ooo 6709 .000 6842 .000 6976 .000 7111 133 135 o.ooo 5941 .000 6052 .000 6163 .000 6275 III III 112 0.165 .166 .167 .168 o.ooi 7211 .001 7432 .001 7654 .001 7878 221 222 224 o.ooi 4211 .001 4377 .001 4543 .001 4710 166 166 167 .109 .000 7248 137 I 3 8 .000 6389 114 114 .169 .001 8103 225 227 .001 4878 168 169 O.I 10 .III .112 "3 .114 o.ooo 7386 .000 7526 .000 7667 .000 7809 .000 7953 140 I 4 I 142 I 44 H5 o.ooo 6503 .000 66l8 .000 6734 .000 6851 .000 6969 1 16 117 118 119 0.170 .171 .172 .173 .174 o.ooi 8330 .001 8558 .001 8788 .001 9020 .001 9253 228 230 232 2 33 234 o.ooi 5047 .001 5216 .001 5387 .001 5558 .001 5730 169 171 171 172 173 0.115 .116 .117 .118 .119 o.ooo 8098 .000 8245 .000 8393 .000 8542 .000 8693 r$ 149 152 o.ooo 7088 .000 7208 .000 7329 .000 7451 .000 7574 120 121 122 I2 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 174 I 7 7 l 176 178 0.120 o.ooo 8845 o.ooo 7698 0.180 0.002 0635 o.ooi 6782 630 TABLE XIV. For finding the Ratio of the Sector to the Triangle. X X * Ellipse. Diff. Hyperbola. Diff. Ellipse. Diff. Hyperbola. Diff. o.i8o .181 .182 .183 .184 0.002 0685 .002 0929 .002 1175 .002 1422 .OO2 1671 244 246 247 249 o.ooi 6782 .001 6960 .001 7139 .001 7319 .001 7500 178 179 180 181 T o T 0.240 .241 .242 .243 .244 0.003 8289 .003 8635 .003 8983 3 9333 .003 9685 346 348 35 35 2 0.002 8939 .002 9166 .002 9394 .002 9623 .OO2 9852 227 228 229 229 251 354 231 0.185 .186 .187 .188 0.002 1922 .OO2 2174 .002 2428 .OO2 2683 252 254 2 55 o.ooi 7681 .001 7864 .001 8047 .001 8231 183 183 184 T o - 0.245 .246 .247 .248 0.004 39 .004 0394 .004 0752 .004 mi 355 358 359 0.003 83 .003 0314 .003 0545 .003 0778 231 231 2 33 .189 .002 2941 258 258 .001 8416 1*6 .249 .004 1472 361 363 ,003 ion 2 33 2 34 0.190 .191 .192 .193 O.OO2 3199 .002 3460 .OO2 3722 .002 3985 261 262 263 266 o.ooi 8602 .001 8789 .001 8976 .001 9165 187 187 189 189 0.250 .251 .252 2 53 0.004 ^835 .004 2199 .004 2566 .004 2934 3 6 4 367 368 0.003 I2 45 .003 1480 .003 1716 .003 1952 235 236 236 .194 .OO2 4251 267 .001 9354 190 .254 .004 3305 372 .003 2189 *J7 238 0.195 .196 .197 .198 .199 O.OO2 4518 .002 4786 .002 5056 .002 5328 .002 5602 268 270 272 274 o.ooi 9544 .001 9735 .001 9926 .002 0119 .002 0312 191 191 '93 193 0.255 .256 .257 .258 .259 0.004 3 6 77 .004 4051 .004 4427 .004 4804 .004 5184 374 376 377 380 385. 0.003 2427 .003 2666 .003 2905 .003 3146 .003 3387 239 239 241 241 0.200 .201 .202 .203 .204 0.002 5877 .002 6154 .OO2 6433 .002 6713 .OO2 6995 277 279 280 282 283 0.002 0507 .002 0702 .002 0897 .002 1094 ,OO2 1292 195 197 198 198 0.260 .261 .262 .263 ,264 0.004 5566 .004 5949 .004 6334 .004 6721 .004 7111 383 385 387 39 39 1 0.003 3628 .003 3871 .003 4114 .003 4358 .003 4603 243 2 43 / 2 45 245 0.205 .206 .207 .208 .209 0.002 7278 .002 7564 .002 7851 .OO2 8139 .002 8429 286 287 288 290 293 0.002 1490 .002 1689 .002 1889 .OO2 2090 .002 2291 199 200 201 201 20 3 0.265 .266 .267 .268 .269 0.004 75 02 .004 7894 .004 8289 .004 8686 .004 9085 392 395 397 399 400 0.003 4848 .003 5094 .003 5341 .003 5589 .003 5838 246 247 248 249 249 0.210 .21 I .212 .213 .214 0.002 8722 .OO2 9015 .002 9311 OO2 9608 .002 9907 2 93 296 297 299 300 0.002 2494 .OO2 2697 .002 2901 .OO2 3106 .002 3311 20 3 204 205 205 207 0.270 .271 .272 .273 .274 0.004 9485 .004 9888 .005 0292 .005 0699 .005 1107 43 404 407 408 410 0.003 6087 .003 6337 .003 6587 .003 6839 .003 7091 250 250 252 252 253 O.2I5 .2l6 .217 .218 .219 0.003 O2O7 .003 0509 .003 0814 .003 III9 .003 1427 302 35 35 308 39 0.002 3518 .002 3725 .002 3932 .002 4142 .002 4352 20 7 207 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 254 2 54 25 I 256 257 0.220 .221 .222 0.003 J 73^ .003 2047 .003 2359 3 11 312 0.002 4562 .002 4774 .002 4986 212 212 0.280 .281 .282 0.005 3598 .005 4020 .005 4444 422 424 0.003 8620 .003 8877 .003 9135 257 258 .223 .224 .003 2674 .003 2990 3*5 316 3i8 .OO2 5199 .002 5412 213 213 2I 5 .283 .284 .005 4870 .005 5298 428 43 .003 9394 .003 9654 259 260 260 0.225 .226 0.003 338 .003 3627 3*9 0.002 5627 .OO2 5842 215 0.285 .286 0.005 57 2 8 .005 6160 432 0.003 9914 .004 0175 261 262 .227 .228 .229 .003 3949 .003 4272 .003 4597 322 3 2 3 3 2 5 3 2 7 .002 6058 .OO2 6275 .002 6493 217 218 .287 .288 .289 .005 6594 .005 7030 .005 7468 434 43 6 438 44 .004 0437 .004 0700 .004 0963 263' 263 264 0.230 .231 .232 2 33 2 34 0.003 49 2 4 .003 5252 .003 5582 .003 5914 .003 6248 328 33 33 2 334 33 6 O.OO2 6711 .002 6931 .OO2 7151 .002 7371 .002 7593 220 22O 220 222 223 0.290 .291 .292 2 93 .294 0.005 7908 .005 8350 .005 8795 .005 9241 .005 9689 442 445 446 448 450 0.004 1227 .004 1491 .004 1757 .004 2023 .004 2290 264 266 266 267 267 0.235 .236 2 37 .238 .239 0.003 ^584 .003 6921 .003 7260 .003 7601 .003 7944 337 339 343 345 0.002 7816 .OO2 8039 .002 8263 .OO2 8487 .002 8713 223 224 224 226 226 0.295 .296 .297 .298 .299 0.006 0139 .006 0591 .006 1045 .006 1502 .006 1960 452 454 457 458 461 0.004 2 557 .004 2826 .004 3095 .004 3364 .004 3635 269 269 269 271 271 0.240 0.003 8289 0.002 8939 0.300 0.006 2421 0.004 396 631 TABLE XV, For Elliptic Orbits of great eccentricity. c or 8 log B or log B ' Diff. log JV Diff. e or S log B or log BQ' Diff. log N Diff. o o 1 o.ooo oooo .000 oooo o.ooo oooo .000 0007 7 2 I 30 31 o.ooo 0066 .000 0075 9 II o.ooo 6400 .000 6836 43 6 45 2 .000 OOOO o .000 0028 16 32 .000 0086 II .000 7286 464 3 .000 oooo o .000 0064 J 4.0 33 .000 0097 12 .000 7750 479 4 .000 booo o o .000 0113 64 34 .000 0109 13 .000 8229 493 5 0.000 OOOO o.ooo 0177 78 35 0.000 0122 1C o.ooo 8722 508 6 .000 oooo Q .000 0255 / 92 36 .OOO 0137 16 .000 9230 5 2 3 7 .000 oooo .000 0347 y 107 37 .000 0153 18 .000 9753 537 8 .000 OOOO .000 0454 1 20 38 .000 0171 19 .001 0290 9 .000 0001 .000 0574 135 39 .000 0190 20 .001 0842 5 6 7 10 O.OOO OOOI Q o.ooo 0709 I4-Q 40 O.OOO O2IO 22 o.ooi 1409 5 8 ' 11 .000 OOOI J .000 0858 163 41 .000 0232 23 .001 1990 596 12 .000 0002 Q .000 I 02 I 78 42 .000 0255 26 .001 2586 611 13 .000 0002 J .000 1199 / 43 .000 0281 27 .001 3197 626 14 .000 0003 I .000 1390 206 44 .000 0308 29 .001 3823 640 15 16 o.ooo 0004 .000 0005 2, o.ooo 1596 .000 1816 220 45 46 o.ooo 0337 .000 0368 31 33 o.ooi 4463 .001 5118 o r> 655 670 17 .000 0007 .000 2051 248 47 .000 0401 36 .001 5788 685 18 19 .000 0009 .000 oon 2 2 .000 2299 .000 2562 277 48 49 .000 0437 .000 0475 3 8 40 .001 6473 .001 7173 700 7i5 2O 21 22 23 24 o.ooo 0013 .000 0016 .000 0019 .000 0023 .000 0027 3 3 4 4 5 o.ooo 2839 .000 3131 .000 3437 .000 3757 .000 4091 292 306 320 334 349 50 51 52 53 54 o.ooo 0515 .000 0558 .000 0604 .000 0652 .000 0703 a 48 51 54 o.ooi 7888 .001 86l8 .001 9362 .OO2 OI22 .002 0897 73 744 760 775 79 25 o.ooo 0032 o.ooo 4440 55 o.ooo 0757 58 O.OO2 1687 806 26 .000 0037 I .000 4803 778 56 .000 0815 60 .002 2493 820 27 28 .000 0043 .000 0050 7 7 .000 5181 .000 5573 J / 39* 407 57 58 .000 0875 .000 0939 64 68 .002 3313 .002 4149 836 851 29 .000 0057 / 9 .000 5980 T 420 59 .000 1007 71 .OO2 5OOO 866 30 o.ooo 0066 o.ooo 6400 60 o.ooo 1078 O.OO2 5866 TABLE XVI. For Hyperbolic Orbits. m orn log Q or log <$ log I. Diff. log half II. Diff. m orn log Q or log Q' log I. Diff. log half IK Diff. 0.00 o.ooo oooo _ 2.1149,, O.IO 9.998 7021 3.412-56,! 2.1046,1 .01 9-999 9 8 7 2.41 S97n 2.1146,! .11 .998 4308 3.45326,, 2.1025,1 .02 999 9479 2.1142,! .12 .998 1342 3.49028,1 2.1003,1 .03 .999 8828 2.89259,1 2. 1 1 37,1 13 .997 8123 3.52423,1 2.0978,1 .04 999 79 1 7 3.01741 2.1 130,1 .14 997 4 6 54 3-55547 2.0952,1 0.05 9.999 6746 3.1141171 2.1 I2I n O.I5 9.997 0936 3.58453,1 2.0923,1 .06 999 53^6 3.19290,! 2.IIIOft .l6 .996 6971 3.61154,1 2.0892,1 .07 .08 .999 3628 .999 1682 3.25940,! 3'3 l68 7n 2.1o82 n .18 .996 2760 995 8 35 3.63679,1 3.66048,! 2.o86o n 2.0826,, .09 .998 9479 3-3 6 745 2.1065,1 .19 995 3 6 8 3.68276,1 2.0790* O.IO 9.998 7021 3.41256,! 2.1046,1 0.20 9.994 8671 3.7037 8 , 2 - 75J " 632 TABLE XVIL For special Perturbations. . a', 2" For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. o.oooo 0.477 1213 1086 0.301 0300 869 0.477 1213 1086 0.301 0300 869 .0001 .0002 .0003 .0004 .477 0127 .476 9042 .476 7957 .476 6872 1085 1085 1085 1085 .300 9431 .300 8563 .300 7695 .300 6827 868 868 868 868 477 2299 477 3385 477 447 * 477 5558 1086 1086 1087 1087 .301 1169 .301 2037 .301 2906 .301 3776 868 869 870 869 0.0005 .0006 .0007 .0008 .0009 0.476 5787 .476 4702 .476 7618 476 2534 .476 1450 1085 1084 1084 1084 1083 0-30 5959 .300 5092 .300 4224 .300 3357 .300 2490 867 868 867 867 867 0.477 6645 477 7732 477 8819 .477 9906 .478 0994 1087 1087 1087 1088 1088 0.301 4645 3 01 55*5 .301 6384 .301 7254 .301 8124 870 869 870 870 i 871 1 0.0010 .0011 .0012 .0013 .0014 0.476 0367 475 9284 .475 8201 475 7u8 475 6035 1083 1083 1083 1083 1082 0.300 1623 .300 0756 .299 9889' .299 9023 .299 8157 867 867 866 866 866 0.478 2082 .478 3170 .478 4259 .478 5348 .478 6437 1088 1089 1089 1089 1089 0.301 8995 .301 9865 .302 0736 .302 1606 ..302 2477 870 871 870 871 871 0.0015 .0016 .0017 .0018 .0019 o-475 4953 475 3871 -475 2789 .475 1707 .475 0626 1082 1082 1082 1081 1081 0.299 7291 .299 6425 299 5559 .299 4693 .299 3828 866 866 866 865 865 0.478 7526 .478 8615 478 975 479 795 479 1885 1089 1090 1090 1090 1090 0.302 3348 .302 4220 .302 5091 .302 5963 .302 6835 872 871 872 872 872 O.OO2O .0021 .0022 .0023 .0024 -474 9545 474 8464 474 7383 .474 6303 474 52^3 1081 1081 1080 1080 1080 0.299 2963 .299 2098 .299 1233 .299 0368 .298 9504 865 865 865 864 865 0.479 2975 -479 4 6 5 479 5 J 5 6 479 62 47 479 733^ 1090 1091 1091 1091 1092 0.302 7707 .302 8579 .302 9451 .303 0324 .303 1196 872 872 873 872 873 0.0025 .0026 .OO27 .0028 .0029 0.474 4143 474 3 6 3 .474 1983 474 0904 473 9825 1080 1080 1079 1079 1079 0.298 8639 .298 7775 .298 6911 .298 6047 .298 5184 864 864 864 863 864 0.479 8 43 479 9522 .480 0614 .480 1706 .480 2798 1092 1092 1092 1092 1093 0.303 2069 .303 2942 303 3815 33 4 68 9 33 55 6 2 873 873 874 873 874 0.0030 .0031 .0032 .0033 .0034 0.473 8746 473 7667 .473 6589 473 55" 473 4433 1079 1078 1078 1078 1078 0.298 4320 .298 3457 .298 2594 .298 1731 .298 0868 863 863 863 863 863 0.480 3891 .480 4984 .480 6077 .480 7170 .480 8264 1093 1093 1093 1094 1094 0-303 6 43 6 .303 7310 .303 8184 .303 9058 33 9933 874 874 874 875 874 0.0035 .0036 .0037 .0038 .0039 o-473 3355 473 2278 473 1^0! 473 o'*4 .472 9047 1077 1077 1077 1077 1077 0.298 0005 .297 9143 .297 8280 .297 7418 .297 6556 862 863 862 862 861 0.480 9358 .481 0452 .481 1547 .481 2641 4 8 * 373 6 1094 1095 1094 1095 1095 0.304 0807 .304 1682 34 2557 34 3432 34 438 875 875 875 876 875 0.0040 .0041 .0042 .0043 .0044 0.472 7970 .472 6894 .472 5818 .472 4742 .472 3666 1076 1076 1076 1076 1075 0.297 5695 .297 4833 .297 3972 .297 3110 .297 2249 862 861 862 861 861 0.481 4831 .481 5926 .481 7022 .481 8118 .481 9214 1095 1096 1096 1096 1096 0.304 5183 .304 6059 .304 6935 .304 7811 .304 8687 876 876 876 876 876 0.0045 .0046 .0047 .0048 .0049 0.472 2591 .472 1516 .472 0441 .471 0366 .471 8292 1075 1075 1075 1074 1074 0.297 1388 .297 0528 .296. 9667 .296 8807 .296 7946 860 861 860 861 860 0.482 0310 .482 1407 .482 2504 .482 3601 .482 4698 1097 1097 1097 1097 1098 0.304 9563 35 44 .305 1317 .305 2194 .305 3071 877 877 877 877 877 0.0050 .0051 .0052 .0053 .0054 0.471 7218 .471 6144 .471 5070 .471 3996 .471 2923 1074 1074 1074 1073 1073 0.296 7086 .296 6226 .296 5367 .296 4507 .296 3648 860 859 860 859 860 0.482 5796 .482 6894 .482 7992 .482 9090 .483 0188 1098 1098 1098 1098 1099 0.305 3948 35 4825 .305 5703 .305 6581 35 7459 877 878 878 8-78 878 0.0055 .0056 .0057 .0058 .0059 .0060 0.471 1850 .471 0777 .470 9704 .470 8632 .470 7560 .470 6488 1073 1073 1072 1072 1072 0.296 2788 .296 1929 .296 1070 .296 O2I2 *95 9353 .295 8495 859 859 858 859 858 0.483 1287 .483 2386 .483 3485 .483 4584 .483 5684 .483 6784 1099 1099 1099 I 100 I 100 0.305 8337 305 9215 .306 0094 .306 0973 .306 1851 .306 2.730 878 879 879 878 879 ' 633 TABLE XVII. For special Perturbations. 9, 9*. 2" For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. 0.0060 .0061 .0062 .0063 .0064 0.470 6488 .470 5416 .470 4345 .470 3274 .470 2203 1072 1071 1071 1071 1071 0.295 8495 .295 7637 .295 6779 .295 5921 .295 5063 858 858 858 858 858 0.483 6784 .483 7884 .483 8984 .484 0085 .484 1186 1 100 I IOO IIOI IIOI IIOI 0.306 2730 .306 3610 .306 4489 .306 5369 .306 6248 880 879 880 879 880 0.0065 .0066 .0067 .0068 .0069 0.470 1132 .470 0062 .469 8992 .469 7922 .469 6852 1070 1070 1070 1070 1070 0.295 4 20 5 2 95 334 8 .295 3491 .295 1634 .295 0777 857 857 857 857 857 0.484 2287 .484 3388 .484 4490 .484 5592 .484 6694 IIOI IIO2 1102 1 102 1102 0.306 7128 .306 8009 .306 8889 .306 9769 .307 0650 881 880 880 88 1 881 0.0070 .0071 .0072 .0073 .0074 0.469 5782 .469 4713 .469 3644 .469 2575 .469 1506 o o o o o ON ON ON ON ON xoxoxovo xo 0.294 99^o .294 9064 .294 8208 .294 7351 .294 6495 856 856 857 856 855 0.484 7796 .484 8898 .485 oooi .485 1104 .485 2207 1102 1103 1103 1103 1104 0.307 1531 .307 2412 37 3*93 .307 4174 .307 5056 881 881 881 882 882 0.0075 .0076 .0077 .0078 .0079 0.469 0437 .468 9369 .468 8301 .468 7233 .468 6165 1068 1068 1068 1068 1067 0.294 5640 .294 4784 .294 3928 294 373 .294 2218 856 856 855 855 855 0.485 3311 .485 4415 485 5519 .485 6623 .485 7728 1104 1104 1104 1105 1105 0.307 5938 .307 6820 .307 7702 .307 8584 .307 9466 882 882 882 882 883 0.0080 .0081 .0082 .0083 .0084 0.468 5098 .468 4031 .468 2964 .468 1897 .468 0831 1067 1067 1067 I066 1066 0.294 1363 .294 0508 .293 9653 .293 8799 2 93 7945 855 855 854 854 855 0.485 8833 .485 9938 .486 1043 .486 2149 .486 3255 1105 IIO5 I 1 06 1 106 1106 0.308 0349 .308 1232 .308 2115 .308 2998 .308 3881 883 883 i 883 883 884 0.0085 .0086 .0087 .0088 .0089 0.467 9765 .467 8699 .467 7633 .467 6567 .467 5502 1066 1066 1066 1065 1065 0.293 709 .293 6236 293 5383 .293 4529 2 93 3 6 75 854 853 854 854 853 0.486 4361 .486 5467 .486 6573 .486 7680 .486 8787 1106 1106 1107 1107 1107 0.308 4765 .308 5648 .308 6532 .308 7416 .308 8301 883 884 884 885 884 0.0090 .0091 .0092 .0093 .0094 0.467 4437 .467 3372 .467 2307 . .467 1243 .467 0179 1065 1065 1064 1064 1064 0.293 2822 .293 1969 .293 1116 .293 0263 .292 9411 853 853 853 852 853 0.486 9894 .487 iooi .487 2109 487 3 2I 7 487 43 2 5 1107 1108 1108 1108 1108 0.308 9185 .309 0070 .309 0954 .309 1839 .309 2725 885 884 885 886 885 0.0095 .0096 .0097 .0098 .0099 0.466 9115 .466 8051 .466 6988 .466 5925 .466 4862 1064 1063 1063 1063 1063 0.292 8558 .292 7706 .292 6854 .292 6002 .292 5150 852 852 852 852 852 0.487 5433 .487 6542 .487 7651 .487 8760 .487 9869 1 109 1109 1109 1109 IIIO 0.309 3610 .309 4495 .309 5381 .309 6267 39 7153 885 886 886 886 886 O.OIOO .OIOI .0102 .0103 .0104 0.466 3799 .466 2736 .466 1674 .466 0612 .465 9550 1063 1062 1062 1062 1062 0.292 4298 .292 3447 .292 2595 .292 1744 .292 0893 8 5 I 852 851 851 850 0.488 0979 .488 2089 .488 3199 .488 4309 .488 5420 IIIO IIIO IIIO IIII II II 0.309 8039 .309 8926 .309 9812 .310 0699 .310 1586 887 886 887 887 887 0.0105 .OI06 .OIO7 .0108 .0109 0.465 8488 .465 7427 .465 6366 465 535 .465 4244 1061 1061 1061 1061 1061 0.292 0043 I .291 9192 .291 8341 .291 7491 .291 6641 851 851 850 850 850 0.488 6531 .488 7642 .488 8753 .488 9865 .489 0977 IIII IIII III2 III2 III2 0.310 2473 .310 3360 .310 4248 .310 5136 .310 6023 887 888 888 887 888 O.OIIO .0111 .0112 .0113 .0114 0.465 3183 .465 2123 .465 1063 .465 0003 .464 8943 1060 1060 1060 1060 1059 0.291 5791 .291 4941 .291 4092 .291 3242 .291 2393 850 849 850 849 849 0.489 2089 .489 3201 .489 4314 .489 5427 .489 6540 II 12 III 3 III3 I III 0.310 6911 .310 7800 .310 8688 .310 9577 .311 0465 889 888 889 888 88q O.OII5 .0116 .0117 .0118 .0119 .0120 0.464 7884 .464 6825 .464 5766 .464 4707 .464 3648 .464 2590 1059 1059 1059 1059 1058 0.291 1544 .291 069.5 .290 9846 .290 8997 .290 8149 .290 7300 849 849 849 848 849 0.489 7653 .489 8767 .489 9881 .490 0995 .490 2109 .490 3223 j III4 III4 III4 III4 0.311 1354 .311 2243 3 11 3*33 .311 4022 .311 4912 .311 5802 ooy 889 890 889 890 890 634 TABLE XVII, For special Perturbations. 9, ^ 2" For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. 0.0120 ,0121 .0122 .0123 .0124 0.464 2590 .464 1532 .464 0474 .463 9416 .463 8359 1058 1058 1058 1057 1057 0.290 7300 .290 6452 .290 5604 .290 4756 .290 3909 848 848 848 III 0.490 3223 49 4338 .490 5453 .490 6568 .490 7684 1115 III5 1115 1116 1116 0.311 5802 .311 6692 .311 7582 .311 8472 .311 9363 890 890 890 891 891 0.0125 .0126 .0127 .0128 .0129 0.463 7302 .463 6245 .463 5188 .463 4132 .463 3076 1057 1057 1056 1056 1056 0.290 3061 .290 2214 .290 1367 .290 0520 .289 9673 847 847 847 847 847 0.490 8800 .490 9916 .491 1032 .491 2149 .491 3266 1116 1 116 1117 1117 1117 0.312 0254 .312 1145 .312 2036 .312 2927 .312 3819 891 891 891 892 891 0.0130 .0131 .0132 .0133 OI 34 0.463 2020 .463 0964 .462 9908 .462 8853 .462 7798 1056 1056 1055 1055 1055 0.289 8826 .289 7980 .289 7134 .289 6287 .289 5441 846 846 847 846 845 0.491 4383 .491 5500 .491 6618 .491 7736 .491 8854 1117 1118 1118 1118 1118 0.312 4710 .312 5602 .312 6494 .312 7387 .312 8279 892 892 8 93 893 0.0135 .OT 36 .0137 .0138 .0139 0.462 6743 .462 5688 .462 4633 .462 3579 .462 2525 1055 1055 1054 1054 0.289 4596 .289 3750 .289 2904 .289 2059 .289 1214 846 846 845 845 845 0.491 9972 .492 1091 .492 22IO 492 3329 .492 4448 1119 1119 1119 1119 1119 0.313 9172 .313 0064 3*3 957 .313 1850 3 '3 2744 892 893 893 894 893 0.0140 .0141 .0142 .0143 .0144 0.462 1471 .462 0417 .461 9364 .461 8311 .461 7258 1054 1053 i53 1053 0.289 369 .288 9524 .288 8679 .288 7835 .288 6990 845 845 844 845 844 0.492 5567 .492 6687 .492 7807 .492 8927 .493 0047 1120 1120 1 1 20 II 20 II2I 0.313 3637 3!3 453 1 3 J 3 5425 .313 6319 .313 7213 894 894 894 894 895 0.0145 .0146 .0147 .0148 .0149 0.461 6205 .461 5153 .461 4101 .461 3049 .461 1997 1052 1052 1052 1052 1052 0.288 6146 .288 5302 .288 4458 .288 3615 .288 2771 844 844 843 844 843 0.493 1168 493 2289 493 34 10 493 4532 493 5 6 54 II2I II2I 1 1 22 1122 1122 0.313 8108 .313 9002 .313 9897 .314 0792 .314 1687 894 895 895 895 896 O.OI5O .0151 .0152 0153 .0154 0.461 0945 .460 9894 .460 8843 .460 7792 .460 6741 1051 1051 1051 1051 1051 0.288 1928 .288 1085 .288 0241 287 9399 287 8556 843 844 842 843 843 0.493 6776 493 7898 .493 9021 .494 0144 .494 1267 1 1 22 I12 3 II2 3 II2 3 1123 0.314 2583 3H 3478 3*4 4374 3'4 5270 .314 6166 895 896 896 896 896 0.0155 .0156 .0157 .0158 .0159 0.460 5690 .460 4640 .460 3590 .460 2540 .460 1490 1050 1050 1050 1050 1049 0.287 7713 .287 6871 .287 6029 .287 5187 287 4345 842 842 842 842 842 0.494 2390 494 35 J 4 494 4 6 3 8 .494 5762 .494 6886 1124 1124 1 124 1124 I 124 0.314 7062 3H 7959 .314 8855 3H 9752 .315 0649 897 896 897 897 897 0.0160 .0161 .0162 .0163 0.460 0441 459 9392 459 8343 459 7294 1049 1049 1049 0.287 3503 .287 2661 .287 1820 .287 0979 842 841 841 9,-. 0.494 8010 494 9 J 35 .495 0260 495 1385 1125 1125 1125 0.315 1546 .315 2444 315 334i 3 J 5 4239 898 897 898 0^0 .0164 459 6 245 1049 1048 .287 0138 841 841 .495 2510 II2| 1126 3'5 5137 o y o 898 0.0165 o-459 5197 1048 0.286 9297 a ' 0.495 3636 I I 2.6 0.315 6035 SQQ .0166 .0167 .0168 .0169 459 4H9 459 3 101 459 2053 .459 1006 1048 1048 1047 1047 .286 8456 o 41 .286 7615 |4 -86 5935 lH 495 4762 .495 5888 495 7015 .495 8142 1126 1127 1127 1127 .315 6934 .315 7832 .315 8731 .315 9630 yy 8 9 8 8 99 899 8 99 0.0170 .0171 .0172 .0173 .0174 0.458 9959 .458 8912 .458 7865 .458 6818 45 8 5772 1047 1047 1047 1046 1046 0.286 5095 .286 4255 .286 3415 .286 2575 .286 1736 840 840 840 839 8 4 0.495 9269 .496 0396 .496 1524 .496 2652 .496 3780 1127 1128 1128 1128 1128 0.316 0529 .316 1428 .316 2327 .316 3227 .316 4127 899 899 900 900 900 0.0175 .0176 .0177 .0178 .0179 .0180 0.458 4726 .458 3680 .458 2634 .458 1589 458 0544 457 9499 1046 1046 1045 I0 45 1045 0.286 0896 .286 0057 .285 9218 .285 8380 285 754i .285 6702 839 839 838 839 839 0.496 4908 .496 6037 .496 7166 .496 8295 .496 9424 497 0554 1129 1129 1129 1129 1130 0.316 5027 .316 5927 .316 6827 .316 7728 .316 8629 .316 9530 900 900 901 901 901 635 TABLE XVII. For special Perturbations. ft ', Q" For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. 0.0180 .0181 .0182 .0183 0-457 9499 457 8454 457 749 457 6365 1045 1045 1044 0.285 6702 .285 5864 .285 5026 .285 4188 838 838 838 838 0-497 554 .497 1684 497 2814 497 3944 1130 1130 "3 0.316 9530 .317 0431 .317 1332 .317 2234 901 901 902 .0184 457 53* 1 1044 1044 * 8 5 335 838 497 5075 1131 3*7 3'35 90 I 902 0.0185 .0186 .0187 .0188 .0189 0.457 4277 457 3*33 457 2189 .457 1146 457 0103 1044 1044 1043 1043 I0 43 0.285 *5 12 .285 1675 .285 0838 .285 oooo .284 9163 837 837 838 837 837 0.497 6206 497 7337 .497 8468 .497 9600 498 0732 1131 1132 1132 "3* 0.317 4037 3*7 4939 .317 5841 .317 6744 .317 7646 902 902 93 902 903 0.0190 .0191 .0192 .0193 .0194 0.456 9060 .456 8017 .456 6975 45f 5933 .456 4891 1043 1042 1042 1042 1042 0.284 8326 .284 7490 .284 6653 .284 5817 .284 4981 836 837 836 836 836 0.498 1864 .498 2996 .498 4129 .498 5262 .498 6395 "32 "33 "33 "33 11 33 0.317 8549 317 9452 3 l8 0355 .318 1259 .318 2162 93 93 904 93 904 0.0195 .0196 .0197 .0198 ,0199 0.456 3849 .456 2808 .456 1767 .456 0726 455 9 68 5 1041 1041 1041 1041 1041 0.284 4H5 .284 3309 .284 2473 284 1637 .284 0802 836 836 836 835 835 0.498 7528 .498 8662 498 9796 499 93 .499 2064 1134 "34 "34 "34 "35 0.318 3066 .318 3970 .318 4874 .318 5778 .318 5683 , 904 904 904 95 905 0.0200 .0201 .0202 .O2O3 .0204 0-455 8644 .455 7604 455 6 564 455 55*4 455 4484 1040 1040 1040 1040 1040 0.283 9967 .283 9132 .283 8297 .283 7462 .283 6627 835 835 f 35 835 834 0.499 3199 499 4334 499 54 6 9 .499 6604 499 774 "35 "35 "35 1136 1136 0.318 7588 .318 8492 .318 9398 .319 0303 .319 1208 904 906 905 905 906 0.0205 .O2O6 .0207 .0208 .0209 o-455 3444 .455 2405 455 1366 455 3 2 7 .454 9288 1039 1039 1039 1039 1039 0.283 5793 .283 4958 .283 4124 .283 3290 .283 2456 835 834 834 834 833 0.499 8876 .500 OOI2 .500 1149 .500 2286 .500 3423 1136 "37 "37 "37 "37 0.319 2114 .319 3020 .319 3926 .319 4832 3i9 5738 906 906 906 906 907 0.0210 .O2II .0212 .0213 .0214 0.454 8249 .454 7211 454 6 i73 454 5135 454 497 1038 1038 1038 1038 1037 0.283 1623 .283 0789 .282 9956 .282 9123 .282 8290 834 833 833 833 833 0.500 4560 .500 5697 .500 6835 .500 7973 .500 9111 "37 1138 1138 1138 "39 0.319 6645 3i9 7552 .319 8459 .319 9366 .320 0273 907 907 907 907 908 0.0215 .O2l6 .0217 .0218 .0219 0.454 3060 .454 2023 .454 0986 453 9949 453 8912 1037 1037 1037 1037 1036 0.282 7457 .282 6624 .282 5792 .282 4959 .282 4127 833 832 833 832 832 0.501 0250 .501 1389 .501 2528 .501 3667 .501 4807 "39 "39 "39 1140 1140 0.320 1181 .320 2088 .320 2996 .320 3904 .320 4813 907 908 908 909 908 0.0220 .0221 .0222 .0223 .0224 0-453 7876 .453 6840 453 5804 453 4768 453 3733 1036 1036 1036 i35 1035 0.282 3295 .282 2463 .282 1631 .282 0800 .281 9968 832 832 831 832 831 0.501 5947 .501 7087 .501 8227 .501 9368 .502 0509 1140 1140 1141 1141 1141 0.320 5721 .320 6630 .320 7539 .320 8448 .320 9357 909 909 909 909 909 0.0225 .0226 .O227 .0228 .0229 0.453 *6g8 453 1663 .453 0628 45* 9593 .452 8558 1035 1035 I0 35 I0 35 I0 34 0.281 9137 .281 8306 .281 7475 .281 6644 .281 5814 831 831 831 830 831 0.502 1650 .502 2791 52 3933 .502 5075 .502 6217 1141 1142 1142 1142 "43 0.321 0266 .321 1176 .321 2086 .321 2996 .321 3906 910 910 910 910 910 0.0230 .0231 .0232 *33 .0234 0.452 7524 45* 6 49 45* 5456 45* 44** 45* 3389 1034 I0 34 1034 1033 1033 0.281 4983 .281 4153 .281 3323 .281 2493 ' .281 1663 830 830 830 830 830 0.502 7360 .502 8503 .502 9646 .503 0789 .503 1932 "43 "43 "43 "43 "44 0.321 4816 .321 5727 .321 6637 .321 7548 .321 8460 | 9" 910 9" 912 QI I 0.0235 .0236 .0237 .0238 .0239 0.452 2356 45* 13*3 .452 0290 .451 9258 .451 8226 1033 1033 1032 1032 0.281 0^33 .281 0004 .280 9174 .280 8345 .280 7516 829 830 829 829 0.503 3076 .503 4220 503 53 6 4 .503 6508 .503 760 "44 "44 "44 "45 0.321 9371 .322 0282 9" 3*2 H94 HI .322 2106 9J2 .0240 .451 7194 1032 .280 6687 829 j j / j j 503 8798 "45 .322 3930 9 12 636 TABLE XVII. For special Perturbations. ~ For positive values of the Argument. For negative values of the Argument. log/ Diff. log/', log/" Diff. log/ Diff. log/', log/" Diff. 0.0240 .0241 0.451 7194 .451 6162 1032 0.280 6687 .280 5858 829 0,0 0.503 8798 .503 9943 "45 0.322 3930 .322 4843 9^3 .0242 .0243 .0244 45 i 5*3 45 1 4099 .451 3068 1031 1031 1031 .280 5030 .280 4201 .280 3373 829 828 828 .504 1089 .504 2235 .504 3381 1 146 1146 1146 1146 .322 5756 .322 6668 .322 7581 912 913 9 J 4 0.0245 .0246 .0247 .0248 .0249 0.451 2037 .451 1006 .450 9975 .450 8945 45 79 J 5 1031 1031 1030 1030 1030 0.280 2545 .280 1717 .280 0889 .280 0062 .279 9234 828 828 827 828 827 0.504 4527 .504 5674 .504 6821 .504 7968 .504 9115 "47 "47 "47 "47 1148 0.322 8495 .322 9408 .323 0322 .323 1236 .323 2150 914 914 914 914 0.0250 .0251 .0252 .0253 .0254 0.450 6885 45 5855 .450 4825 .450 3796 .450 2767 1030 1030 1029 1029 1029 0.279 8407 .279 7580 .279 6753 .279 5926 .279 5099 827 827 827 827 826 0.505 0263 .505 1411 55 2 559 55 3707 .505 4856 1148 1148 1148 "49 "49 0.323 3064 3*3 3978 3 2 3 4893 .323 5808 .323 6723 914 9'5 0.0255 .0256 .0257 .0258 .0259 0.450 1738 .450 0709 .449 9681 449 8653 449 7625 1029 1028 1028 1028 1028 0.279 4 2 73 .279 3446 .279 2620 .279 1794 .279 0968 827 826 826 826 825 0.505 6005 .505 7154 .505 8303 505 9453 .506 0603 "49 "49 1150 1150 1150 0.323 7638 3*3 8553 .323 9469 .324 0384 .324 1300 915 916 9'5 916 917 0.0260 .0261 .0262 .0263 .0264 0.449 6 597 449 55 6 9 449 4542 449 35 * 5 .449 2488 1028 1027 1027 1027 1027 0.279 OI 43 .278 9317 .278 8492 .278 7666 .278 6841 826 825 826 825 825 0.506 1753 .506 2903 .506 4054 .506 5205 .506 6356 1150 1151 1151 1152 0.324 2217 3 2 4 3 J 33 .324 4049 .324 4966 3M 5883 916 916 917 917 917 0.0265 .0266 .0267 .0268 .0269 0.449 X 46i 449 0435 .448 9409 .448 8383 448 7357 1026 1026 1026 1026 1026 0.278 6016 .278 5191 .278 4367 .278 3542 .278 2718 825 824 825 824 824 0.506 7508 .506 8660 .506 9813 .507 0965 .507 2117 1152 "53 1152 1152 "53 0.324 6800 .324 7717 .324 8635 3*4 9553 .325 0470 917 918 918 919 0.0270 .0271 .0272 .0273 .0274 0.448 6331 448 535 .448 4280 448 3*55 .448 2230 1026 1025 1025 1025 1025 0.278 1894 .278 1070 .278 0246 .277 9422 .277 8599 824 824 824 823 824 0.507 3270 .507 4423 57 5577 .507 6731 .507 7885 "53 "54 "54 "54 "54 0.325 1389 .325 2307 .325 3225 .325 4144 3 2 5 5 6 3 918 918 919 919 919 0.0275 .0276 .0277 .0278 .0279. 0.448 1205 .448 01 8 i 447 9 J 57 .447 8133 447 7109 1024 1024 1024 1024 1024 0.277 7775 .277 6952 .277 6129 .277 5306 .277 4483 823 823 823 823 822 0.507 9039 .508 0194 .508 1349 .508 2504 .508 3659 "55 "55 "55 "55 "55 0.325 5982 .325 6901 .325 7821 .325 8740 .325 9660 919 920 919 920 920 0.0280 .0281 .0282 .0283 .0284 0.447 6085 .447 5062 447 439 447 3 1 6 447 2993 1023 1023 1023 1023 1023 0.277 3661 .277 2838 .277 2016 .277 1194 .277 0372 823 822 822 822 822 0.508 4814 .508 5970 .508 7126 .508 8282 .508 9439 1156 1156 1156 "57 "57 0.326 0580 .326 1500 .326 2421 .326 3341 .326 4262 920 921 920 9 2! 921 0.0285 .0286 .0287 .0288 .0289 0.447 0970 .446 9948 .446 8926 .446 7904 .446 6882 1022 1022 IO22 1022 1021 0.276 9550 .276 8728 .276 7907 .276 7086 .276 6264 822 821 821 822 821 0*509 0596 509 1753 .509 2910 .509 4068 .509 5226 "57 "57 1158 1158 1158 0.326 5183 .326 6104 .326 7026 .326 7947 .326 8869 9 2I 922 9 2I 9 22 922 0.0290 .0291 .0292 .0293 .0294 0.446 5861 .446 4840 .446 3819 .446 2798 .446 1777 102 I 1 02 I 1021 1021 1021 0.276 5443 .276 4622 .276 3802 .276 2981 .276 2161 821 820 821 820 821 0.509 6384 59 7543 .509 8702 .509 9861 .510 1020 "59 "59 "59 "59 "59 0.326 9791 .327 0713 .327 1635 .327 2558 .327 3481 9 22 922 923 923 923 I 0.0295 .0296 0.446 0756 445 97 3 6 1020 I O2O 0.276 1340 .276 0520 820 820 0.510 2179 5 10 3339 1160 1 1 60 0.327 4404 3^7 53 2 7 923 .0297 .0298 .0299 .03-00 445 8716 .445 7696 445 6676 445 5657 JO2O 1020 1019 .275 9700 .275 8880 .275 8061 .275 7241 820 819 820 .510 4499 .510 5659 .510 6819 .510 7980 1 1 60 1160 1161 .327 6250 .327 7174 .327 8097 .327 9021 9*3 924 923 904 637 TABLE XVIII. Elements of the Orbits of Comets which have been observed. ardt ardt. ardt. Hind. // Burckh M . r- . -^ rH. ~ SW W PQfi "P So -g &i>'! 1 "IP ft S3 O C 3'rt S a 0*13 ' 1 tf Q rj-oo oo to O M cl t^so so H d cotoOi-OON-<$-ON>H sooo^l-'OcooOc* 00 ? O CoS ON^^. CO^ <$- C^ t^O^SO CO C< 00 SO OOOt^wi-l r-00 SO CO !>, Tj- CO !H ON ON r+- to r~- co to Tl-OO O SO Tj- to u-iso SO t^ ONQi-iOOSO OO coso co to SO t^OO SO OO M O SO Tj- t~- so oo to tooo ON ONOO I--, i-- t^oo ONOO ON rj-toONONt~~.ootot~~.ONt-^ooootot--t-~. ^> t~. r. c t>. ON ON ON oV ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON ON CO CO SO f-- OO >O ON o o o o o o o o q o q q q q q q q q q q q q q q q q q q ON q q q q q ON :; O O O O O OOOOO OOOOO OOOOO OOOOO OOOOO O O O M to rj- to ,. OOOOO HwONOco O O ^> O to tot^ooOso tonsoONto f^O'-'Ot^ TJ-C>ONONCO CO CO ^- CO to CO to d to w to M to CO Cl n M C( CO _ O t^ ^- o M ^ so K M ON r^ t^- oo co so to oo o so t^ cJ os t^ ^J" n -* to to t^ ci oo o co to ^- tOOQOO^O OOOOO OOOOO OOOOO OOOOO OOOOO 000^-to ^OOO-O sot^ONcoto OOOOO OoOMOt-- i-iONOtocJ toOOOOO ONSO so O t-- Co ^J-tO O COwtoCOCO tOTj-COCOTj- 1-ICOTj-COlHCOTJr tOCO wcJCOC* o cl N ONOO oo rj- oo O so O ^" to to t*~. co t*^ t^ co c^ l*^ oo cooo M ^ oo oo r^ to ON ON to cJ to ON QCOwOOtoto ONHONOtoOOcJCJOl-1 toOONnrJ-SOCOTj-SOO OOSOCOTj-t-t CT\C^tOrM Z O O O O O OOOOO OOOOO OOOOO OOOOO OOOOO OOOOO ^QtoO'OO t^-CT\t-~coco Otooooo ooOOO" t^c^Ococo OOt--cl^i - QtoONcJON to co TJ-CO lOcJco^-cortcl COT^- rt- CO>-ITJ-T}-P-IT}-TJ-C< to M M COOO SO CO t ONOO rt" ^-SO d ^ n CO rt SO ON IH HH IH SO OO OO CO O *H to t^ ^- ON ONOO O c* to t^ M oo M ^- tooo so so so to co co Tj- so ON O oo O to ^- to >H to O co H t^ cl d O COrtClCO COMCOCJCJ CCMCOlH c> Cl IHCCOCO NHrtCOiH (^elcOiHlH "OOOOO ONTj-inOto OOOONO OOOOt^- tOMOOco OO^OO too toso ON Tj* M cl CO Tj- Ci C< to xj* ^ M to CO ^- H ^" CO CO S O O O O O OO coso OO OOOsOc* M ONSO OON rt-ONOONCOSOtortOco into-J- 'i-sO cJrh to cJiHCOcl'^- MIH Tj-Mwcoto lowHcl^- tortOtoO sosOrhOco O O O -> t>- cot-^MOoo soOHONto WWIOONTJ- tnOrlHco ^f os cTcToT oo^t^^o^T-To* c co*'c^od^ H^Hjrco^trtTor ^'oTc^'io'cc' THC^rHC^ O1 COTH THC^COTHCOTHrH iH C to O 00 O HI VO vo HI r^VD ON HI O O O -* O rj- t~~- ON O to ONVO t^OO 00 -* O O t-- ON t^ t^ t^> vo O J tooo T- O t-- t^ f^ ON vo O ONOO oo t-^OO vo ON O ONVO to ONOO ON ONOO ON ON ON ON O- CN O ON ON ON ON O t^- ON ON ON ONOO ON ON ON ON ON ON ON O ON O ON OS ON O O O O O 00 00 VO ON O O O O O O O O O HI OS Tj- HI O O vo O t-. t>-00 O rj- ON ON ON O O ON t^ to ONVO 00 VO ON ON ON O 00000 O CO to O VO 000 * i co O T*- o O . t~^ vo ONOO oo t- O M O M tO O OO HI OO vo O O O O vo HI HI HI OO vo t^ O co O vo vovo O O O HI vo O vo O ^ o j; vo vo ON TJ- -sj-vo oo CO O HI CO OO OO vo vo T*- O 'vo O <* M VO ^ tToo" vo HI H CO C^ CO O t^ O 00 t^ VO HI 00 vo CO VO O w VO Tj- VOOO CO vo t^. HI co vo ^ HI vovo O ON ^ OO HI VO VO O oco 00 O vo rt- O O 00000 vo to HI vo O vo O ON O ONOO oo O O ON HI r^- vo O O O vo vo Tj- vo O T}- vo O O co vo ON HI o to O HI ON HI VO VO I s - O O to rj- Tj- 00 O CO VO rj- to w "* Tj- Tj- "fr * H o O covo vo to Tf ON t^OO oo 00 O ON ON to VO t-- vo 1--.VO vo O\ HI vo O ONOO HI vo O HI HI t> O k t; 00 VO O O VO H, H, fO v. vooo t^. ON TJ- HI VO HI VO vo oo r~-vo o vo r-, c-- OO O VOOO co O HI HI oo cooo CO CO 00 VO HI vo O O O vo HI tO CO CO VO VO tO O HI O vo !>. to t--. ^J- rj- to t-~ vo t^ vo O tO tO M t^ ON vo HI O Tj" Tj- VO tO VO O O M vo 00 O vo O OO co to o co O 000 vovo CO HI f^VO VO t^- HI t) O O O T$- vovo vo CO O VO VO HI vo vo TJ: ON rj- T^- CO vo CO cooo O ON HI to to t-^ t~- O vo O O ON O CO O CO rf- vo co vo to to O O vo O O to to o HI *> <* -*- rj- O 00 O TJ- CO HI OO ON ONOO VO CO CO CO - 00 S< QO t^ Tin CO OO CO OS OS OO CO t- CO O T-H T^ e3*'8 F '^'& (M lO O CO CO 00 00 O5 O5 OS bC> > "S S 1-- 00 00 C 00 O O O O i ( ^J V . 4) V H c s e a ST 3 c3 S 3 OQ 1-5 h, h-5 l-j CO OS t^- I>- OS CN (M CO CO CO 6 W059000'* THIOTH ^*-*-* * 5^i5,x r-l (51 W * i SSSSi TH .2 -g ^ s ,x ^r c zv % Hf^ Motion. CJ OJ O 'ill t- BD^J to O O O O g -S $S$ftp3 // Direct. // // Eetrograde. Direct. Eetrograde. Direct. // Eetrograde. ^ nJ &D "~ OJ ^i .5 * * "? A *i ^H 5: 5: S: =: S o5 o s s 0*^000 p Swp Eetrograde. Direct. Eetrograde. Direct. // w cP VO 00 VO rt rj- f^ ro IH T*- rj- ON co O 00 M CO CO ONVO Cl O\ VO OO OO OO OO rt el c* >$ ON vo M co M t-^ rf-vo OO covo VO oo r rt ^ rt IH VO 1-1 ONVO C* O ON rf- rj- ^- m ro t~^ ri- O r< ON ON ro t^, vo ro vo ro vo ON O oo oo ro O ONOO c< 4- VO O O O vo vo M ON r> O oo ONOO TJ- 6) M vo to M vo co OO ON IH IH -4- co rl * ri-00 C^ 00 vo o r- r^vo co r) ON rt * ^^?: vo r^. ri oo t^ OO VO M VO M ON T*- vn r) ^ f^ oo t-- vo O 00 O HI M * "* TO W t^ tH CO rt M M VO 00 CO -* rt IH I^ rj- VO t-00 Cl !* M O t-- co O vo r^CO M vo cs 5 vo t^ ON * el ^J- IO cl VO T}- ^ r}- M vo r--oo CO CO IH T}- M VO ON O O f- IH c vo rt H 00 H O O VO M VO VO VO rj- VOOO f> IH rt ro m 5-8 SE-* ON Tj- VO VO CO IH O IH co ON c* MM VO Tj- vooo 0-4-0 VO CO c< CO M IH vooo * Ti- VO IH 4 oo O oo vo O IH CO VO Cl M M O C* VO 4- o ^t- M vovo O cl C4 IH CO CO ON rt cooo co rj- M CO Cl 00 V0 Tj- ,1- co co >H co vo t M c* m t^ * vn t< IH CO CO M <4- t^- r-x co o CJ IH IH M VO vo cl OO fl C< IH IH ci ^ oo t-~ vo vovo vo co ON H CO IH i * M t--00 vo TJ- VO CO VO . CO t^ VO CO fj O O O O oo VO M Cl O t-^oo oo O -4- c* O l^ H vo rt vo vo O ON t^ r}- c* H c Ti- vo vo O O t^ M co rt- H ON O VO t^OO VO C* 3-vO co o O C? vo CO CO T}- IH ONOO 00 t^ d t-~- t^ r- o IH ON rj- ON r^ vooo d t~- co IH t~^ c) vo O to ON rj- rj- u-, VO CO O OO IH ro IH r^-VO 00 ^J- 10 Tj- LO O rj- O vo t^- t~- O IH (^ IH OO CO VO IH vo VO ONVO O T*- rj- CO M VO T$- vo e* co co O ON f--VO OO OO O ON vo vo o rj-vo co ON ON rj- M ON ON <* oo IH rJ oo vo O vo O vo O ON O O -M O co O ON vo O V> CO M IH O S 3 S ^Lo VOOO Tj- t^ Tf moo M *3- rt IH co cooo ON VO CO M VO ^J- CO IH IH CO O r^oo oo r-- r-^ rfS rj- o rh t^ ON ON IH t-^ co co IH H M O H 00 O ro IH rt PH 00 CO Tj- O vo C* M VO VO TJ- ON C< cl O * M M C* rt IH M T}- t^-OO O CO C^ IH b COOOrHCO * -^i $ i III rH^OQ^g (M CO CO HH t^ ^i ^ "tf -H rj< t^ t^ t^ 1>- I>- SSsfstf JS e* 9 o> o -8J ^^Or-sS 00 00 t^ 00 OS Ttl Tt< 1C IO IO 1>- CO GO r- 1 (N (M rH (M tH > ^ > J2 o rfi (N rH (M tH (M rj 'S bC > ^J PH o O O EH^O^ CO CO C5 CO CO CO 1^ t^ O5 CO )O 1C ^f r-t rH rH lllli rH C OS O5 CO (M _>^> ^ & O O rH T 1 CO CO CO CO CO 00 rH t^ 00 O 00 CM (M CO JH e *S rj ^ i^J'S Tfl IO O CO CO CO 00 CO CO CO 6 ; rH (N 09 ^ l& I- l I'. 1 1- i>.QCOiO l> t> t>- CN. OD IH 5 C ^ W5 QC GC OC QC QC 50 r>. oc Ct-O aOGCGCceci THIN 09HU9 CSCSOiCiCS Dt>.GCCiO CiCiCtCtO iH *1 C * 1* OOOOO 640 TABLE XVIII. Elements of the Orbits of Comets which have been observed. Computed by e ci c J ^ c 2 // Zach. Piazzi. Saron. If Arrest. Encke. Olbers. " // Burckhardt. Wahl. // Burckhardt. Olbers. Gauss. II., l.ifll Nicollet. Gerling. Bessel. Burckhardt. Pogson. | Motion. 1 1 Retrograde. // // // Direct. it Retrograde. // Direct. Retrograde. 1 // // Retrograde. Direct. Retrograde. // Direct. Retrograde. Direct. it Retrograde. // Direct. n it CM bd <4-oo vo vo O t-^ m d nvo d VO Os COVO m O oo oo O HI VO O vo rf oo in m co f~- Os ^ co t^vo O HI oo O t^ Os HI Osvo HI ^ HI os O ON O inoo r-- H CO M ^j- COOO r}-oo M in M C* OS H 00 OS m HI t~--vo oo t^ CO rj- HI 00 cooo O vo m r*- Tj-00 O 00 TJ- m t-^- Os os d os w co d ON t^- <* O O 00 O 00 vo * os m HI TJ-OO HI M O vo HI co ON rt r- -3- o O co in co M Os m Os O oo m oo Os m O m t-, m os l~- in M co os 00 CO H, d rj- co os in os O 00 00 w Os Os r-- os o HI oo Os d Os 1-. HI o Os O m d HI vo vo m osoo t~-~ d T+- rj- co n in *-oo oo oo vo oo O O vo oo Os O Os Os O ON O Os Os O ON O Os ON Os Os Os Os O O Os Os O Os Os Os ON O O Os Os O O 00 Os O HI CO Os oo 00 00 oo oo" O O O O OS HI c* oo oo t-- O oo oo vo m o in rj- ^ HI os oo t^ Os o O oo cooo d coo H, OS HI Tj- O 1-^ in m d *$ osoo in O O ON ON os vS Os CO O O Os O O M HI HI M -. to t*-. o *i" oo w r* m TJ- vo n 00 H rj- OsO Tj- HI VO c* v"> r--. i"-** ^" o O VO *J" H CO co o\ ON O -< O co Tj-vO vo O co -rl- c rfr CO CO CO Tj- CO rj- O CO Cl HI vo m TJ- Tj- d m oo d d CO CO CO C< covo r O co CO CO M Tt- co co in co in M M COVO rj- Os m HI HI co oo M d t-. f- M HI M in Os HI co HI d covo t^ co t^ co d invo T}- co M in d d co d Os m d HI HI Tt" CO O d oo <$ ^ d ^ VO tO^O C^ f*^ ro r$- c$ co ^ HI VO Tj- 00 d ON 3-0 CJ HI d -^- os O cJ vo t^ M o cJ HI co cl co ON d m cl O cooo O Osoo M co m co o oo in f^ O Osvo HI VO O vo co r-.oo HI TJ- m M w rj- w M r- m rj-vo O d HI co in oo O oo * Tj- vo vo d d t^ o O in in i>> vo HI HI CO M d CO O COOO C* co Osoo O ri- t^ Os H Os CO M C C> * CO CO HI Cl OsVO rj* O VO ON d ^- M f^ CO CO HI * HI d vo d co m cJ vo c> CO Cl CO H CO ^ O O co co d HI T}- Os in co HI d O d co co O vo ^oo d in co d ^ ON t- rh m t-^ m rj- co ^ Tj-OO ON rf Tj- t^> rt in o co Cl CO CO CO O S C 00 05 CO O5 O^ rH d "H^ rH rH 00 OO 00 00 00 OO 00 GO CO 00 GO OO OO QO OO CO 00 00 OO OO OOOOri SSS33 c* X O rH (5-1 CO -^ I 5l *51 ^1 ?1 M rH (M 0? rH 1 wwwwS 41 641 TABLE XVIIL Elements of the Orbits of Comets which have been observed. enber bart cke. Heili w > & ' 1 ti 1 T2 - 3*5 jj | 3 S.aj-'i -*. ||jw HW fc Q-^O > E.S .. PPnft o> 1 -4J fcJO -+J o o o 2;- cv ~H -g .a fttf ft Re Di Re Retrograde. Direct. Ret Dir n n Ret M M <3- i-i ^- r-^ ON covo c* H t^oo O HI co co in ON ro co co co cooo m vo ON m O "3- oo oo m d oo OHt^Ost^ t~- H N cooo mvo O to fl O ON in inoo ON ON t^ in ON O' ON O*N ON ON ONOO' cV ON ON O c< O NO HI cJ^oooooo t^rtMO^l- co r^ oo ^}~ vo ON co HH O in c^ inoo O ON ooOt-~-HON wvot^ ^-vo -3-OQ tnvo ^J- OomniMOO vot^-touoHi ^-OONT}-t-- in in t-x rt ^f rJ-coONinO t^-corOO Ocot-~.OON ONinoCNco c-JONTt-c^-ON O" ON ON o' ON ON ON o' ON o' ON ONOO' ON ON t>-oo ^ H m in co rj- r^ O 00 O vo 00 ON CO in "3- ONOO ONOO "^- ON m to tovo ON co HI in ON O in O\ oV oV ON ON ON ON ON O in in M co oo O m ON Tt- 00 toM m inmMt-^ O 1 ^ MO *}-C< Tj- OOOOHiON COTj- Tj-OO DO co oocom t--H W t-. ^}-co t~. OO^J-vOON rJVO rj-co oo in vo T!" vo ' HI ^ invo oo ON ^f* in HI r+. u-j oo ^}" ON O ^" ON T^- o ON ^J- <4" *n ooooot^^oooooo ON q o^ q' q q oo ON t^- q q q q q q ONOO q q oo q i>- q q q M HI O* HI* O* O' HI M O* M O* HI* W M* M* H* o' O" O' HI HI* M* HI HI HI Q* O" HI HI Q' O' HI HI HI O 1-- ON ONVO ONC^COOO ONCOCOCO ONCOCOCOON minONt-^co HI Tt- CO ^ Tl- M ON ON r^ Tj- Tj- M a rtcorJ-ONO M- T^-VO t\ Tt- MOO COVO 00 cooo HI rJ-roOcoco t^^J-c>r>vn rt i to M co cl co to M co H HI inMONt-xTj- co^t- ONVO rt rt cooo ON HI vo ^j- co ON * rtcocort-Hi M o ro M Tt- MincOM coclrtTj-M h ^-vo OO HI OO OO ON O HI l^% t^- HI ON HI ^- vo M co ON t-~-ONrlt--ON HiTj-Ort-co O t^-oo ONt^ int^mcotx. Ot^-cJOt^ vo co ON n M t~^ t^vo t--^ HI vo HI o HI comwcoON inmniMin O M t^ O w c^ d H oo ^ONrJOr' ONONOOOON moo oo ONVO co^j-inTj-in ^t^Ooomoo-^-OHiHi HI ro co vn ^* r< c^ ^" in co in HI c^ ^ M n in HI ^~ co in ^" in TJ~ to M in M vnc^cocoo ooOcJHi (N ^3s*J ?5*Jk3 j )ftl-5CcS <1oQft^* OOCOOCiOi Ci-H,,-, _,_,-,---. -.-.-.-_ CO CO CO CO CO CO 00 CO 00 00 CO CO CO CO 00 CO CO CO 00 00 00 CO CO 00 00 CO CO CO CO 00 CO CO CO CO CO 642 TABLE XVIIL Elements of the Orbits of Comets which have been observed. Computed by Encke. Westphalen. Encke. Peters and O. Struve. Plantamour. A ilJ-L - Tj- VO Tj- HI Tf vo vo rj- d d T}- VO VOVO T$- co vo co d d O vo t^ O M COOO VO CO Tj- VO VO HI OO ON ONVO vo M co co HI d ON ONOO O CO d rj- Tj-00 00 d d vo d co oo oo o oo oo co vo d d ON HI <$ co co M t^-oo ^ ON Ox HI VO VO vo d O HI vo T^- vO O \O d couovo covo f-~OO co ON Ox HI oo d O t^. O ON COVO d ^ ^- ONVO t-. ON d vo ON ON -4- ^J- a% 0\ OS CN -frvo 00 OXOO * O COOO 00 d oo HI vovo d_00 ON TJ- d T}- vo Tj- ONVO VO 00 rj- t> HI t^ O\ ro t* oo vo t^r^Hi vo d t-- HI o M rj- M cooo VO ON O HI Tj- CO *^ CO HI eT HI d CO d M CO CO CO CO M d ~ ~ d d M co co HI d co M co vo M d -d H, d vod d ir\ i-t Mm TJ- M d CO HI rj- vo CO k ** d co d HI HI t-x T*. t^ d vo O vo OxOO HI CO M HI HI CO d HI rf d d vo H t~~ CO HI CO d d co co co M ON d Oxvo 00 rj- rj- |>- Ox Cl CO HI cJ C CO Tf- HI d d f- ON H c< to ONVO d O HI ONOO <* HI HI d d oo vo t^-vo vo ON 1-- CO rt- HI OX * O t^ d co HI c d HI oo d d 't^ vo ON ON ON co^ vo vod ON t^ rh M d ONOO vo co vo t--- Ox ON ^t- ON ^ to CO HI HI r}- < oo d O O co vo d d vo vo ' -O ONOO HI co d ^J- VO CO ON M d ^ CO CO i-l LQ OS d O LO vO t i I IO Oi O C<1 OS 02 05^ COCO A J bb > o - ^ 3 O v 3 iO iO CO O O CO CO CO ^ "^ CO OO CO 00 OO B >'Z d ja PU P &4 1 (M C* CO CO CJ oo oo M <* M^oo rj-vr> oo oo r-- r- O roONrou-i OO roi-i *J- m N O O oo m TJ- t-^vo co t~- w ON t-^ rl t M I-H r~, O co T}- in c< ON w co ^ in O\ O* O' ON ON t-~.OO OO r* rl rJ-OO ro CsOO O t^- f< vn ro O rooo O n ooo a\\o t~^ vn o ^o w oo O oo t>. inoo r-- o q q oo O cf\ cf\ o\ o\ o\ o* O O cr\ co ON M TJ-VO vo O w O r--00 oo co m vo vo in OO OO OO ON c* OO -4- VO ON ON ^J" oo O vo M w w M rJ moo ON Tj- CO rf- rt r inoo O ON ON >-t in m w ONOO c< O Ot^ONONON t^OOOO VO M O O ON rj- Tt- m t->-vO M O 00 ONOO co rj- M ON t- covo m co r-- OvOO *O O <* ON ON ON ONOO moo rj- vo w co M ON co co t-. wONOOO ON co w OOOwO M WMMM 00000 00000 00000 M m i-i c4 m O IH c Tj- MtH coos coco ^ ^^ \- 1 rt co ON IH Tj- O M co ON IH O co t^ *+ in O t"^ OO OO t^ c*" CS McocomincoM-^Hc* r^mincoinco cl^cl r}- O c< n ONOO co TJ- rt- t^- mvo ON O O O I s * m t^ rj-oo O M Tj- Tj-VO ^ Tt- C C W Tj- H COOO rj- ONOO ON co II CO TJ-VO M comr-xcooN ONrtOoot-^ O vo covo t^oo ^J- c< I-" ro in VO rj- m ON M H M cJ co co M ON to in in H M f- co M cl rH (M $& Seeling. Oppolzer. Engelmann. // Frischauf. L! rH .

u J^ .2 a d ',' +$ ^3<| S^ Illll 1 fe > ife-jr HIS Motion. o5 "8 o 5 * "8 S "S M fc fi qj "S .-1 * *= ft Retrograde. Direct. n Retrograde. Direct. Retrograde. // // Direct. Retrograde. 1 nJ fcX) s3 o g * ft fe ft w w Direct. Retrograde. // 2 5; t ft O C< ** Tj- ON vo d HO VO OO rj- cooo OO ON t^ OO rj- vo TJ-VO vo VO M -rj- rj- rt ro r- H M 8O H vo f-, ^60 co O co r- ON HI O vo O OO t~-OO O | OO CO>O CO CJ O t^ !>. vo cl vo o r-. HI vo H. CO M Tj- r-vo IM pj ro (^ Cv C> CJv u-> ONOO O O cl Ov ON ON ON O oo ON t-^oo rt ON TJ-OO w c< t--oo oo M oo OO ONVO t-~ vo VOVO 00 TJ- M ON ONOO O T*- ON vovo O OO c ON vo ON rt M O O ON ON ON O O ON O O ON O\ Ox ON ON ON ON ON ON ON O ON ON ON O ON ON ON ON O 00 ON O VO c vo O CO M VO ON ON co c* co t^ VO NO t-^ ON <* vo O O ONOO vo 5- H 1^ ON O O O O ON co ? * co co ON vo oo O *J- co t^ co vo vo 00 00 -3- O ON ON O 00 oo O r-- c* ON vo ON O ON O ON O ON 1 O O O O O Tj- HI H t^. CO t^~ vo t^~ ON vo ON ON ON ON ON O O O 00 C* H) ONOO vo HI CO VO * vo vo r~- ON vo rj- ON vooo O HH HI O O O H, HI H. M HI O O HI O H, H. H. H M M M M M H, * $3-:r:: t^. VO VO Tj-00 CO rj- vo CO O HI oo co O to f c vo co covo co t co vo vn M * l-t VO t-^ ON t .vo c vr> r-~ VO ^1-00 VO H H M OOO rj- CO vr> oo r^vo oo vo VO M VO CO w O oo t^ t-^ t^ $ w oo HI C M t*- HI OO VO t VO 00 CO Tj- VO HI CO ^ H 00 ONOO O ro rj- vo ON O * H vo vo O covo c O c> ONOO O .vo O m rooo co vr> ui vn u-i m rn rt vo TJ- io LO rt vo vo Tj-vo rt VO H H ON ON CO M w r^vo H c> vo c> c> rl 1-1 *$ vo ro ^J- co CO CO VO HI 10 T^- CO VO HI VO VO O * vo rj- ON O t~- H vo co O HI CO HI CO tf ON t~^ Tj-oo < ^- vo vo H OO HI CO CO *$ ONOO o-i tj- O cl t^-. TJ- co M r M co vo t^ VAVO M rJ m u-i M u-> CO M CO M t* ON t^ ^- o 4- T*- ON t^ Cl CO NH H vo w co O co ON co O vo vo n 11 H HI ONOO 00 co O t^vo Or) HI t: * vo H, * *^" CO CO VO oo vo c> r- M OO HI HI VO O HI CO C! HI co O vovo HI O rt t^vo O r* ro HI ON c T*- O O ^ vo ro *s*\ t-o N H <* ^J- c vo t--oo oo ^J- M H -H VO e* c M NO t-- VO CO CO CO OO vo VO t^. t- VO H, M h vovo VO >~ ON O ON c^ co vo $ Tj- VO CO O H, t-^ t-- vovo M co ON moo M *f r}- l-^ vo ON Tj- U-) H t^ H -4- f< ON * 10 rj- O CO VO O ONVO 00 00 rt- ON w p M O vr, cJ vo <* O ON vo H VO rj- t^-VO Vi t^ ON w CO r-^ VO OO O vo>O O O t^OO ON ON O 00 vo co t^VO rj- vo ON Tj- Tj- M O CO r-~oo vo vo g vo vo ON vo ON O O o w r^vo o O ON ON t-~ * T*- d F; t^. t-v cooo rt d O\ vovo t^ O VO ON H rj- * O M 00 M M d O M oo * ON ()-. M co ON w r-^oo co O O O r* H h o c3 ^> PH O o^ O 'IS -^^Qq^ CO CO CO CO Tt< O CO CO CO CO 00 GO 00 00 GO a"s"f^ $* 8 8 g <1CPQ^ Tj< "* Tf Tfl lO CO CO CO CO CO 00 00.00 OO 00 rH rf 05 I- rH rH rH (M C -Q r~ -O a.* s r H-,fer-3fH co co r^- 1^ CD CO CO CO CO 00 OO 00 d fc Ot>-ae>CiO >*"** rH O 91 (M 91 91 91 r-KMM-^lO oo o oo 91 91 91 91 91 Oi>-aoctO O OOO 91 91 (N (N 91 !-iiee^> t* t> t>- 1>- 1 -!XCiO r- !>!> r> of 91 91 91 91 91 645 TABLE XIX, Elements of the Orbits of the Minor Planets. I ^gJg| 1,Jt 'L'Lt l**A 3 t 9 w! ^ rO 'C F 2 . 8 S rrt ^ ^ &4 O rZ C G ^ tn * 0^3 g g ^ 3 W2 O O OJ g?2 % a Sosos BWW33 oKeKtf o^wwo OKKcto KiK^^K Date of Discovery. . **E _J g *f* g O O O O rj< CO 00 00 00 CO ^~ ti^j C 'E "g P O P< PH t~ t~ b- OO OS Th -rC rji r- rf OO OO OO OO CO <5 CO ^ S l ~S O O O i I i 1 CO CO GO CO 00 lllll > 'C 'C o o o L, p^ (M - rt- r-- os COOS rt- OO 00 rt- rt- OS O co vn to O vn w O vo d Osoo w d d OO vO HI W d O O co vn d co d O Osvo Os co os vn Os vn co O C-* w VO OsvO 00 00 co oo oo r-^ d o l-v os co vno oo t-~ t-^ O vn Osoo oo vo vo HI t-~ vn i-^ O w d vn oo i-^ vn t^ cooo r~- rt- d rt- cooo co co vn w o vo d vn vn w t-^oo OO OO OO rt-OO VO rt- d rj- CO rt-0 d rh rt- OO cs d co t^ vo Os w co d OO OS Os CO w O vo vo vn rt- vn t-.oo vnoo oo vnvo oo t~-^ w OS rt" t^ t^ t^vo w t-^ vn l-~ t^OO OSOO OS d VO d rt- COVO OO VO CO Os Os O Osvo rt- rt- I s --- CO vn d Os vn vn d OS OSOO 00 00 Os d O O oo O w N to rh r-. os o os os CO rl- CO'O CO co w co co vn Os l-^OO vo Os Osvo vo csvn w oo vo vo r-> oo ost-^oo o co d w vn O co vn co o oo VO OO 1--OO VO oo vnoo co w 00 d C0r|- rt- w Os w l-^ d co co d d i-- VO vo OO Os rj- cod vn coco coj^vo w t^ Q- ^ d t-- vnvo O w d co co VO CO rt- vnOO co vn vn rt- rj- O vo d VO w vnrt- w co os t--- vn rt- rt- t-- COd vn rt- rl- d CO d oo os co t-^ rj- CO vn CO rl- vnd rt- OS O W HI O d O O co w d co w I*** f-^ d csoo rt-OO OO d OO rt- HI vn vn Os Os vn d co Os vn covo rt- r-- os t-~ vnvo co d vn co O OO t-. rt-VO vn co w HI vn OSOO rt- t- O COOO rt" w w rj- vn r|- vn rt- co t-~ Os d rt- rl- rt- d d rt- Osvo d oo oo coO O Os O O co d VO 1--OO vn rt- d d Osoo vn vn rt-VO w CO t--OO CO CO OS O O Os Osvo - ;; oo oo w vn os d cod ^ O d woo Os COrl- w O rt- co f^ vn co cooo O O vo OO covo Os rl- d vn CO rt- rt- vn vn vn co W 1--.OO t-- l-x w rl- CO w t-^ CO O t"-^ rt" co d co r}- rfOO VO OS w F-.VO vo rt- OS vn w r$- covo os d w co co rt- covnO M O OS 1--VO d vo r)- d vn co vn co cooo r|- rt- w rt" CO CO CO O O w rt- d d rt- co co d co w osvo d vn vn Osoo t^- w d rt- l-^ Osvo vo O d d vn d w vo vn w vnoo d d vo oo w co cooo vo d oo Os t-^ co t^ vn vo d 1-- w t-v d oo d vo co co vn Os d d Os vn os vnoo C2 5: l^s vnvo d 00 vn co d w O CO Os w vo vnrj-rt-w d _, O d O co w 00 f~- t-^O rt- t~- vnoo vn w w vn r|- CO Cs l-~ r-- w CO CO rt- w CO rt" 00 OSQ 00 VO co vn w vo oo w d w d f- w VO C0r|- d rt- vn d w l-^ rt- vn d d CO w r}- vn vn vn covo co d CO rt-OO Os t-^ Osvo m ro to H i^ CO (^ (S ^J- vnO wvo vn d vn w O w w w d d CO rt- w t--. vn O d vn d co rt" w O VO 1--VO rt- OO vo vo CO w Cs w os vn Os vn co rf-OO w O OS vn rt- rt- CO vn CO rf-vo OO rt- OS rt- vn O w COCO vn rl- rl-vo vn rt- w vn CO CO w oo vnoo w CO rt- rt- vn w O Osr)- vn co d oo vn d w co rt-00 OSVO w f>. w co w d rt-0 Ooo vn O vO O d oo rt- w vo vn rt- O VO VO 00 OS vo w t^ d O 5. OO vo w OO OO d w oo d os w vn vnvo O f>. 1-- w rt- w OO VO wvo CO vn co os w r-^ vo d vo Os os h ^ d w vo Osrt- d vn w w Q 00 d rt-Osvn vo cort- coQ d vn w vn w d w vn rt- osvn d d w co in vn VO rf- vn t-- OS d d vn d vn O vn O oo t*> vn Osoo os w rt- rt- t-^oo co O d vn vn vnvo oo d co vn vn d VO t-- d vo w d w w co o rt- d CO rt- CO rt- d O w w w d w r^ co CO CO HI w CO w w CO d w w CO w d 1-. O Osoo vn vn O vo f-~ O w os co d vn vo w o co d w oscor-xoo vo cooo coO O O t-- vn 1-- . ^ vn Osoo vn d CO r|- d CO O vo O d co O Os CO rt- d vn Os w os w W CO T|- CO VO rfr- t^ CO rt- ri- d CO rt- vnvo vo O M ri-w d O t- rt- r|- t-^ vn w t-^ d vn vn vn d co d vn t>.oo w d d rl- vn d co d r)- CO 00 rt- O CO OS t-^ w OSVO rt- co w d w co to cod d covo vn I--. Os oo vo co vn os Osvo O rt- d covo d co d vn f- O 00 w w t>-00 vnvo rt- OSOO O OS f~-00 vo rt- t^ d w OS CO rl-VO vn rt- O O O CO w CO w CO M d rh o r-. w d r)- t-~ rt- . o o o o o o o o o cs o o o o o o 10 o o >o o o o o o 00000 O O O O O g a rH CS CO 05 C*T-4 I-H O O I 1 O O OS OO O e^ co cs T-H r-* CO "rH O CO i 1 CN CN T-. i 1 o' io I-H o c-i ! i 1 C^ C CO TH CO O CC CO >O O O oo oo co co oo co to t- -* LO lllll co c co eo o uo CO CO CO O OO OO CO OO OO lllll t~ O CO iO CO CO O CO CO CS 00 CO OO OO 00 g O "ci s > "co m III /I H fq J^ H! OH H PH 2 D " B c ei hi 1 ILs'^ " 3 O "o . o ** 10 ot-oocso ^N W"*i Ot^QOOiQ p*e* W^ift ^OOCSO *-< N w ^ 646 TABLE XIX, Elements of the Orbits of the Minor Planets. . midt. idt. 'a g 'a a a a n . a a 'a 1 5 s * 5 i ,3 . C G J^ pfit M ^ yC| i3 -! . ,J3 ,C O "*~* *3 r3 j-H fi O ,^3 , ~ T* ^-! iS r* 03 *3 TO O O O M 01 ^ O M M O C> a; i 3 OoiO^a:^ coC? T OT 2 -S 2 2 So Sc2 2 ) 2 2 ! 3 2 2 Is 2 5 5 ? 2 c S 4 5s 5 St:3 ? 2 o 3 ,a ,a o ooooo o 3 o o o> ei o 3 o g ops^S ogciSS 303^0 Ci5 h-3 O O O O P-i AH O O p-ii-^OOfe H} O H? O co Ct3 (-3 i-H 1 O PH Of&OHH H PH j CO O tw *** o fe .2 o *3 T-H CO 's mil iO iO CO CO O co co co co co C<1 CO * O O OJ tM (M rH T-H C^ CO CO t- i t- co co co co co "* C> CJ O u <1 CO CO CCO !>. J>- ir- i- t- CO CO CO CO CO cc co co co co oo co co so co ft < 3 p< eu CC CO t% CO t t- C5 O O O CO CO CO CO CO OO O ONOO VO CO rj- M VO "^ i-i t VO O OO t^ Pl Plplt^-Plvo 'd-OvOVOvD fxoo 1--.O-4- co^OMr^ oo IH t^ pl vo pl t^ co -rj- rh CO pl co rt- vo ^t-OO rf-OO co Tj-TJ-TfTJ-cO r *" c P c T irr>T ?" 66666 66666 tj- Q c O vo vovo ON O pl ON pl OO co O vo ONOO Pl "^"^"^"^"1" o d o o 6 pl Tj- O M TJ- TJ-VO Pl ONOO cl M vo pl O O OvOrt-Od TJ-OMVOO MVOONt^-Pl M r^OO rj- rj- \O vo vo rj-vo ON 1-1 M vo vo M ON ONOO ON vo pl O O OO ONcoplcot>. dpl-^-4-ON ON-rJ-ddO rj- M M O M COMOOCOO rt-ONcoi>- T^-VO ONOO rh co TJ- ^f- co r-^ M ON CO COOO t-- ON t-~- d CO pi OO OOOOO OOOOO OOOOO OOOOO vo M co vo oo vo vo oo O oo d oo t-^ vovo < co -^- O vo co vo vovo ON I ON rj- vo vo vo ON O CO vo i vovorlOOr}- ONONONCO- rt- M TJ- t^ i-i ONVO d O ON t-^ d oo O. co t^oo t-- t^ O oo 4M VO cooo Tj- r~- ONO o> Tj- ON ON OO I*-* VOOO CO cooo ON O ^h vo ON d d rj- cooo O d M vo -^ M vo ON M o oo co i , vo vr, VO O VO -4" rj- f^ vo co ON t^ ONVO OO r-^ t~x ON r^ d ON d O O d O ^-vo co O M vo M ^- o vo O ON M ^t" ^ M ON O vo t^ cooo cot-~.t-~- oovodOt^ covo vo ON ^t- O oo co cooo ON T}- covo ON oo d ON cooo rj- ro ON ON vo oo vo l^ t^ ON O f--00 O M M vo d ON TJ- vo O VO d Ti-vo ON co VO ONOO vo OO ON t--vO OO d M l>- d vo covo 6 Pl t>. Ov co vo ri oo pi VO M pl COOO ON r^ vo M 10 co co co ON *- ON rj- VOOO vo OvvO OO CO vo vo oo vo oo' t-^- vo d oo O vo d co ^ to VO CO ONOO T}- ON t-x T^- covo to vo M M oo rj-oo vo -^i-oo co vo d vo O t^vo co M rj- ON ON t^ tvO vo ON M O co CO VO CO ONO 000 oo to co M oo M q vovo O Mvopi'^-Ti- ooqq ONOO &"^. pivovo vovocoOoo MONONdpi ^plvovort- picodt^vo dOOOO'OOO piMO\MoX OOOco COOO Vo t^-vO pi ON vo vo rj- ON O IH M pl VO MCOrJ-VOd CO d^-^1- VO^t-COTj-rJ- TJ- VOMM plMMVO MCOCOMCO r^ O ONOO i-- o oo covo O ^h vooo to i co d o t^ vo M oo vo vooo to oo d vo M to VO vo Pl CO CO vo f^OO M a, ON rt- O VOOO OO O CO M CO d to o O ri-vo ON vo vo ONOO ON t^- vovo OO O q M to d oo 6 pi oo t^. -4- Tj-VO vo ON O r}- O -4- O O CO CO M CO CO O COOO tO VO *^>3Q Of) ON vd 6 t~. 6 oo lONTJ-Md MONMQt-- vo co q pi t^- oV t> 4-OO M d * M ON VOM t^ Pl vo d oo vo M pi pi oo pi t>. M d VO Tj- CO d oo q M M O VO M O Tj- ON vo I 4- Pl VO M M pl VO M Tj-VO fl >H O vn ON rj- co ON VOOO OO OO w H i* H vo ON f- vo M rj- f^ O VOOO CO Pl CO CO Pl a <* S . ri s o o o o > o o o o cc oo icN|: -BS-sll S HS 1-5 1-5 O Hi H^HjH>Jpf*a HS H-3 h-5 iCCiC^COCO i^iGOOCCCO tftO'^OCD O^OOiOr-H O O CO T ' * i CO^Ov^OO O^OOOO COCOOOCC CDCOCOCOO COCOCOCCCO ioOODCCOO OOOOODOJcX) CCCOODOOQO QOCCCCCOCO COCOCOCOCO OOcX)OOCOCO CCOOOOCOCC zi g .S . . IceS -J|. a .* M M -2 c^pnce^ -^a ^' * 5 "35 * O '4- oV ON t^- t~- -4" O rj- ur, vo HI VO to rj- HI O ON vo N rj- to HI OOVOONO HlOVOWVO ONt^ONHtON ONNO tOOO CO VO ^NOONwdNO OON t^.00 N l^ ON O O 00 tOOO O O\ t^- CO N ^T" w w T^- vo CO T^- to HI HI pi p) ^J- CO N N T^Tj-ONNt^ cOwNt^-vo cooo OO N N tOtOpJp* M HI Tj- HI COHIP>HICO _ N t^- ON t> rj- COOO HtrJ-to COWOOONN u NOvo CO ONVON4-VO rj- tOOO CO N pi to to HI co HI HI co co co HI to ON co 00 CO PJ T}-CO N topj HI HI NO VO t^ l"^ NO NO O NO VO CO HI to vo CO CO p CO HI vooo vo HI O vo N vo vo COpl fl HI OvO tri(> 00 N ^vod 4-1 HI N VOTj- N M VO CO HI NO ON CO NO CO Tj- N Pj Tj- 4" N w ON O\ vr> OO -4-NO ON N VO t^ tr> to vo to N to to to ' O O O O O O O ur 10 o O O CO* -* rH CO T-H T-l I-H * co O to N oo pj O 00 N O N O tOOO t-> HI CO ? t^- to ON ONNO O HI oo t~s to HI to T}- N co N co O O N vo 8 NO *4- ONNO HI pj O O O HI to o o o o o o o o o o o o o o o o o o o o o o o o o o o o o +1 1 ++ 1 1 co t^NO O HI vo Ti- N 8O to r-- N O ONND NO OO O NO to f^ ON vo co r^coM WNO *$ toNO NO tooo NO NO oo O O HI ONrJ-voTj-O NOOOOOOO dddddddd < HI VO CONO to HI i oo 4- O N toco * vo to N to CN O r-- 1 5: Tj-OO NO HI tOOO Hi O to HI oo ONNO l^ pj vo HI N T{- Tj- Ot^,CO HI HI N O HI a * : ^ CO f \ ' i ONW N ON t OOVO N ON CON covo 1 1 1 1 cs N C t d t N f OO ' vo t St JN vo ON T}- tooo i cooo t^. to HI -I HI tO tO CO i- O NO vo ON t^ O N to VO 4- 00 00 HI N O ->. T^- ON HI t-^ CO HI HI h v. tj-oo HI N to t^ O - ON <* ONNO w N <* HI N HI tO * N HI O NO NO OO H4 CO ^ t^. co tosO ^t" ONNO O *" COVO tOCONwCO HI co O tooo ON w t-- N ^ to p* co HI rJ-OO ON N w ON f^ to HI CO HI ^ MONvoOtoOON ^ co co d vo covo tooo " rj- HI voN N to OtoONNtoOOOto CO CO p4 HI p4 ^ ONO HI ONO HI cot^to Epoch and Mean Equinox. Greenwich Mean Time. O a o o fS o CO CO rH 1 111 sill! 648 TABLE XXL Constants, &c, log Base of Naperian logarithms e 2.71828183 0.43429448 Modulus of the common logarithms . . A = 0.43429448 9.63778431 10 Kadius of a Circle in seconds r 206264.806 5.31442513 it i' a a minutes r 3437.7468 3.53627388 '/ // " a degrees r = 57.29578 1.75812263 Circumference of a Circle in seconds .... 1296000 6.11260500 // " '/ whenr=zl. . . . TT = 3.14159265 0.49714987 Sine of 1 second 0.000004848137 4.68557487 Equatorial horizontal parallax of the sun, according to Encke 8".57116 0.9330396 Length of the sidereal year, according to Hansen and Olufsen 365.2563582 days 2.56259778 Length of the tropical year, according to Hansen and Olufsen 365.2422008 // 2.56258095 This value of the length of the tropical year is for 1850.0. The annual variation is O.*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 // n // // it if it in se- conds of arc 3548.18761 3.55000657 Constant of Aberration, according to Struve 20".4451 // o Nutation, '// // Peters 9".2231 Mean Obliquity of the ecliptic for 1750 + t, according to Bessel .... 23 28' 18".00 0".48368< 0".00000272295* 2 Mean Obliquity of the ecliptic for 1800 + t, according to Struve and Peters . . 23 27' 54".22 0".4738< 0".0000014< 2 General Precession for the year 1750 + t, according to Bessel 50".21129 + 0".0002442966* // // n it n a Struve 50".22980 + 0".000226< MASSES OF THE PLANETS, THE MASS OF THE SUN BEING THE UNIT. Mercury ra = , Jupiter .... m = , 4865751 1047.879 1 1 Venus > Saturn . 390000 3501.6 Earth ' Uranus 649 EXPLANATION OF THE TABLES. TABLE I. contains the values of the angle of the vertical and of the logarithm of the earth's radius, with the geographical latitude as the argument. The adopted elements are those derived by Bessel. De- noting by p the radius of the earth, by

0.0000018 cos 4? + &c., p being expressed in parts of the equatorial radius as the unit. These quantities are required in the determination of the parallax of a heavenly body. The formulae for the parallax in right ascension and in declination are given in Art. 61. TABLE II. gives the intervals of sidereal time corresponding to given intervals of mean time. It is required for the conversion of mean solar into sidereal time. TABLE III. gives the intervals of mean time corresponding to given intervals of sidereal time. It is required for the conversion of sidereal into mean solar time. TABLE IV. furnishes the numbers required in converting hours, minutes, and seconds into decimals of a day. Thus, to convert 13/i 19m 43.5s into the decimal of a day, we find from the Table 13/i =0.5416667 19m =0.0131944 43s =0.0004977 0.5s = 0.0000058 Therefore 13/i 19m 43.5s = 0.5553646 651 652 THEORETICAL ASTRONOMY. The decimal corresponding to 0.5s is found from that for 5s by changing the place of the decimal point. TABLE V. serves to find, for any instant, the number of days from the beginning of the year. Thus, for 1863 Sept. 14, l&h 53m 37.2s, we have Sept. 0.0 = 243.00000 days from the beginning of the year. Ud 15h 53m 37.28= 14.66224 Kequired number of days = 257.66224 TABLE Vlt contains the values of M= 75 tan %v + 25 tan 3 ^v for values of v at intervals of one minute from to 180. For an ex- planation of its construction and use, see Articles 22, 27, 29, 41, and 72. In the case of parabolic motion the formulae are wherein log C 9.9601277. From these, by means of the Table, v may be found when t T is given, or t T when v is known. From v = 30 to v = 180 the Table contains the values of log M. TABLE VII., the construction of which is explained in Art. 23, serves to determine, in the case of parabolic motion, the true anomaly or the time from the perihelion when v approaches near to 180. The formulae are 8 /200 200 qi smw V-iF v w-f A, t T=-~->-^- 3 , * M C sin 3 10 w being taken in the second quadrant. The Table gives the values of A O with w as the argument. As an example, let it be required to find the true anomaly corresponding to the values t T= 22.5 days and log q = 7.902720. From these we derive log M =4.4582302. Table VI. gives for this value of log M, taking into account the second differences, v = 168 59' 32".49 ; but, using Table VII., we have w = 168 59' 29".ll, A O = 3".37, EXPLANATION OF THE TABLES. 653 and hence v = ti;-}-A = 168 59'32".48, the two results agreeing completely. TABLE VIII. serves to find the time from the perihelion in the case of parabolic motion. For an explanation of its construction and use, see Articles 24, 69, and 72. TABLE IX. is used in the determination of the true anomaly or the time from the perihelion in the case of orbits of great eccen- tricity. Its construction is fully explained in Art. 28, and its use in Art. 41. TABLE X. serves to find the value of v or of t T in the case of elliptic or hyperbolic orbits. The construction of this Table is ex- plained in Art. 29. The first part gives the values of log B and log C, with A as the argument, for the ellipse and the hyperbola. In the case of log C there are given also log I. Diff. and log half II. Diff.. expressed in units of the seventh decimal place, by means of which the interpolation is facilitated. Thus, if we denote by log (C) the value which the Table gives directly for the argument next less than the given value of A, and by &A the difference between this argument and the given value of J., expressed in units of the second decimal place, we have, for the required value, log C== log (0) + AJL X I- Diff. + A^l 2 X half II. Diff. For example, let it be required to find the value of log C correspond- ing to A = 0.02497944, and the process will be: (1) (2) Arg. 0.02, log ( (7) = 0.0034986 log I. Diff. = 4.24585 log half II.Diff. = 1.778 (1) = 8770.6 logAJ. =9.69718 2 log & A =9.394 &A= 0.497944, (2)= _ 14.8 3.94303 1.172 log = 0.0043771 The second part of the Table gives the values of A corresponding to given values of r. TABLE XI. serves to determine the chord of the orbit when the extreme radii-vectores and the time of describing the parabolic arc are given. For an explanation of the construction and use of this Table, see Articles 68, 72, and 117. 654 THEORETICAL ASTRONOMY. TABLE XII. exhibits the limits of the real roots of the equation sin (V C) = m sin 4 z'. The construction and use of this table are fully explained in Articles 84 and 93. TABLES XIII. and XI Y. are used in finding the ratio of the sector included by two radii-vectores to the triangle included by the same radii-vectores and the chord joining their extremities. For an explanation of the construction and use of these Tables, see Articles 88, 89, 93, and 101. TABLE XV. is used in the determination of the chord of the part of the orbit described in a given time in the case of very eccentric elliptic motion, and in the determination of the interval of time whenever the chord is known. For an explanation of its construc- tion and use, see Articles 116, 117, and 119. TABLE XYI. is used in finding the chord or the interval of time in the case of hyperbolic motion. See Articles 118 and 119 for an explanation of the use of the Table, and also the explanation of Table X. for an illustration of the use of the columns headed log I. Diff. and log half II. Diff. TABLE XVII. is used in the computation of special perturbations when the terms depending on the squares and higher powers of the masses are taken into account, For an explanation of its construc- tion and use, see Articles 157, 165, 166, 170, and 171. TABLE XVIII. contains the elements of the orbits of the comets which have been observed. These elements are: T, the time of peri- helion passage (mean time at Greenwich); TT, the longitude of the perihelion; &, the longitude of the ascending node; i, the inclina- tion of the orbit to the plane of the ecliptic; e, the eccentricity of the orbit; and q, the perihelion distance. The longitudes for Nos. 1, 2, 12, 16, 91, 92, 115, 127, 138, 155, 156, 159, 160, 162, 171, 173-175. 180, 181, 185, 191, 192, 195-199, 201, 203, 204, 207, 208, 212-215, 217-219, 221-228, 230, 233, 234, 237-248, 251-258, 261-267, 269-275, 277-279, are in each case measured from the mean equinox of the beginning of the year. In the case of Nos. 134, 146, 172, 182, 189, 190, 205, 231, 232, 236, 259, and 268, the longitudes are EXPLANATION OF THE TABLES. 655 measured from the mean equinox of the beginning of the next year. The longitudes for Nos. 19 and 27 are measured from the mean equinox of 1850.0; for No. 186, from the mean equinox of July 3; for No. 187, from the mean equinox of Nov. 9; for No. 200, from the mean equinox of July 1 ; for No. 202, from the mean equinox of Oct. 1 ; for No. 206, from the mean equinox of Oct. 7; for No. 211, from the mean equinox of 1848.0; for No. 216, from the mean equi- nox of Feb. 20 ; for No. 229, from the mean equinox of April 1 ; for No. 250, from the mean equinox of Oct. 1 ; and for No. 276, from the mean equinox of 1865 Oct. 4.0. Nos. 1, 2, 11, 12, 20, 23, 29, 41, 53, 80, and 177 give the elements for the successive appearances of Halley's comet; Nos. 104, 116, 126, 143, 149, 157, 167, 170, 176, 178, 183, 194, 210, 220, 235, 249, and 260, those for Encke's comet, the longitudes being measured from the mean equinox for the instant of the perihelion passage. Nos. 92, 127, 159, 172, 196, and 222 give the elements for the successive ap- pearances of Biela's comet; Nos. 187, 216, 250, and 276, those for Faye's comet; Nos. 197 and 238, those for Brorsen's comet; Nos. 217 and 243, those for D' Arrest's comet; and Nos. 145 and 245, those for Winnecke's comet. For epochs previous to 1583 the dates are given according to the old style. This Table is useful for identifying a comet which may appear with one previously observed, by means of a similarity of the ele- ments, its periodic character being otherwise unknown or at least un- certain. The elements given are those which appear to represent the observations most completely. For a collection of elements by vari- ous computers, and also for information in regard to the observations made and in regard to the place and manner of their publication, consult Carl's Repertorium der Cometen-Astronomie (Munich, 1864), or Galle's Cometen-Verzeichniss appended to the latest edition of Olbers's Methode die Balm eines Cometen zu berechnen. TABLE XIX. contains the elements of the orbits of the minor planets, derived chiefly from the Berliner Aslronomisches Jahrbuch fur 1868. The epoch is given in Berlin mean time; M denotes the mean anomaly, y> the angle of eccentricity, p the mean daily motion, and a the semi-transverse axis. The elements of Vesta, Iris^ Flora, Metis, Victoria, Eunomia, Melpomene, Lutetia, Proserpina, and Pomona are mean elements ; the others are osculating for the epoch. The date of the discovery of the planet, and the name of the dis- coverer, are also added. 656 THEOKETICAL ASTRONOMY. TABLE XX. contains the mean elements of the orbits of the major planets, together with the amount of their variations during a period of one hundred years. The epoch is expressed in Greenwich mean time, and L denotes the mean longitude of the planet. TABLE XXI. gives the values of the masses of the major planets, and also various constants which are used in astronomical calcula- tions. APPENDIX. A. Precession. If we adopt the values for the precession and for the variation of the position of the plane of the ecliptic given in Art. 40, and put M = 171 36' 10" + 39".79 (* 1750), the formulae for the annual precession in longitude (^) and latitude (/?) become, for the instant t, -- = 50".2113 + 0".0002443 (* 1750) -1- (0".4889 0".00000614 (* 1750)) cos (A M) tan ft (J) Ojf = (0".4889 0".00000614 (t 1750)) sin (A M). If we denote the planetary precession by a, the luni-solar preces- sion by l n and the obliquity of the fixed ecliptic, at the time 1750 + r, by e , we have, according to Bessel, ~ = 0".17926 0".0005320786 r, CLL ^L = 5Q".37572 0".000243589 r, at e fl = 23 28' 18".0 + 0".0000098423 T, and if we put dl f da . dl, the formulse for the annual precession in right ascension () and declination (d) become da, ... d8 / N - = m + n tan d sin a, =-- = n cos a, (2) at at 42 657 658 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".0000970204 (t 1750). To determine the precession during the interval t r t, we compute the annual variation for the instant J (t f -f t) and this variation mul- tiplied byt'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, A2, = _ 17".2405 sin ft + 0".2073 sin 2ft 0".2041 sin 2 C + 0".0677 sin ( < F) 1".2694 sin 20 + 0".1279 sin (0 r) 0".0213sin(0 + r), (3) Ae = + 9".2231 cos ft 0".0897 cos 2ft + 0".0886 cos 2([ + 0".5510 cos 20 + 0".0093 cos (0 -f r), for the year 1800, and A a = _ 17".2577 sin ft + 0".2073 sin 2ft 0".2041 sin 2C + 0".0677 sin ( r') 1".2695 sin 20 + 0".1275 sin (0 r) 0".0213 sin (0 + r), Ae = + 9".2240 cos ft 0".0896 cos 2ft + 0".08S5 cos 2< + 0".5507 cos 20 + 0".0092 cos (0 + r), for the year 1900. In these equations ft denotes the longitude of the ascending node of the moon's orbit, referred to the mean equinox, ( the true longitude of the moon, the true longitude of the sun, F the true longitude of the sun's perigee, and P the true longitude of the moon's perigee. The values of these quantities may be derived from the solar and lunar tables, and thus the required values of A^ and AS may be found. The equations give the corrections for the reduction from the mean equinox to the true equinox. To find the nutation in right ascension and in declination, if we consider only the terms of the first order, we have _ da, da iiii sr ^ cW ^ ds The values of AA and AS are found from the preceding equations, and for the differential coefficients we have APPENDIX. 659 da dd - = cos s -j- sm e tan o sin a, == cos a sin e, MA u/, da dd T- cos a tan o. -7- = sm a. Cfe C? 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 r , 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 A = mr -f- AA cos -f- (^ r ~r A ^ sin e) sin a tan d A cos a tan d, A<5 = (nr -j- AA sin e~) cos a -|- A sin a. Let us now put mr -f- AA cos /, nr -f- AA sin = g sin G, (7) A = g COS G, and the equations (6) become A =/+ 9 sin ( G + a ) tan ^ , as already given in Art. 40. The astronomical ephemerides give at intervals of a few days the values of the quantities/, y, and G- for the reduction of the place of the body from the mean equinox of the beginning of the year to the true equinox of the date; and, in order to obtain uniformity and accuracy, the beginning of the year is taken at the instant when the mean longitude of the sun is 280. When these tables are not avail- able, the values of/, g, and G may be found directly by means of the equations (7). The reduction from the true equinox of t' to the mean equinox of t will be obtained by changing the signs of the corrections. C. Aberration. The aberration in the case of the planets and comets may be considered in three different modes : 1. If we subtract from the observed time the interval occupied by 660 THEORETICAL ASTRONOMY. the light in passing to the earth, the result will be the time for which the true place is identical with the apparent place for the observed time. 2. If we compute the time occupied by light in traversing the distance between the body and the earth, and, by means of the rate of the variation of the geocentric spherical co-ordinates, compute the motion during this interval, we may derive the true place at the in- stant of observation. 3. We may consider the observed place corrected fdr the aberration of the fixed stars as the true place at the instant when the light was emitted, but as seen from the place of the earth at the instant of observation. The formula for the actual aberration of the fixed stars are AA = 20".4451 cos .(A Q) sec ft 0".3429 cos (A T) sec ft, A0 = + 20".4451 sin (A Q ) sin /? + 0".3429 sin (/I T) sin /5), ' in the case of the longitude and latitude, and Aa = 20".4451 (cos Q cos e cos a -f- sin O sin a) sec <5 0".3429 (cos F cos e cos a + g i n r sm a ) sec ^ Afl = -f- 20".4451 cos O (sin a sin d cos e cos d sin e) (10) - 20".4451 sin Q cos a sin d 4- 0".3429 cos F (sin a sin d cos e cos d sin e) -0".3429sinrcosasin(5, in the case of the right ascension and declination. In these formulae F denotes the longitude of the sun's perigee, and they give the cor- rections for the reduction from the true place to the apparent place. D. Intensity of Light. If we denote by r the distance of a planet or comet from the sun, by J its distance from the earth, and by C a constant quantity depending on the magnitude of the body and on its capacity for reflecting the light, the intensity of the light of the body as seen from the earth will be J =T&- (ID When the constant C is unknown, we may determine the relative brilliancy of the comet at different times by means of the formula APPENDIX. 661 In the case of the planets we adopt as the unit of the intensity of light the value of I when the planet is in opposition and both it and the earth are at their mean distances from the sun. Thus we obtain C=a?(a I) 2 , and hence ,.(-!). > Let us now denote by R the ratio of the intensities of the light for two consecutive stellar magnitudes ; then, if we denote by M the apparent stellar magnitude of the planet when 1= 1, and by m the magnitude for any value of J, we shall have R* I== W and hence By means of photometric determinations of the relative brilliancy of the stars, it has been found that R = 2.56, and hence we derive m=M 2.45 log J, (15) by means of which the apparent stellar magnitude of a planet may be determined, J being found by means of equation (13). The value of M must be determined for each planet by means of observed values of m. EXAMPLE. The value of M for Eurynome is 10.4; required the apparent stellar magnitude of the planet when log a = 0.38795, log r = 0.2956, and log J = 9.9952. The equation (13) gives log /== 0.5129, and from (15) we derive m = 10.4 1.3 = 9.1. For the values log r = 0.4338, log J- 0.2357, we obtain log I =9.7555 10, and m = 10.4 + 2.45 X 0.2445 = 11.0. 662 THEOKETICAL ASTKONOMY. E. Numerical Calculations. The extended numerical calculations required in many of the problems of Theoretical Astronomy, render it important that a judicious arrangement of the details should be effected. The beginner will not, in general, be able to effect such an arrangement at the outset ; and it would only confuse to attempt to give any specific directions. Familiarity with the formulae to be applied, and practice in the performance of calculations of this character, will speedily suggest those various devices of arrangement by which skillful computers expedite the mechanical part of the solution. There are, however, a few general suggestions which may be of service. Thus, it will always facilitate the calculation, when several values of a variable are to be computed, to arrange it so that the values of each function involved shall appear in the same verti- cal or horizontal column. The course of the differences will then indicate the existence of errors which might not otherwise be dis- covered until the greater part if not the entire calculation has been completed; and, besides, by carrying along the several parts simulta- neously the use of the logarithmic and other tables will be facilitated. Numbers which are to be frequently used may be written on slips of paper and applied wherever they may be required ; and by performing the addition or subtraction of two logarithms or of two numbers from left to right (which will be effected easily and certainly after a little practice), the sum or difference to be used as the argument in the tables may be retained in the memory, and thus the required number or arc may be written down directly. The number of the decimal figures of the logarithms to be used will depend on the character of the data as well as on the accuracy sought to be obtained, and the use of approximate formula? will be governed by the same considerations. Whenever the formulae furnish checks or tests of the accuracy of the numerical process, they should be applied ; and whenever these are not provided, the use of differences for the same purpose should not be overlooked. By proper attention to these suggestions, much time and labor will be saved. The agreement of the several proofs will beget confidence, relieve the mind from much anxiety, and thus greatly facilitate the progress of the work. THE END. PROSPECTUS. Messrs, J, B, LIPPIJTOOTT & CO,, Philadelphia, propose to publish by Subscription A TREATISE ON THE MOTIONS OF THE HEAVENLY BODIES REVOLVING AROUND THE SUN IN ACCORDANCE WITH THE LAW OF UNIVERSAL GRAVITATION: EMBRACING AN ANALYTICAL AND SYSTEMATIC DERIVATION OF THE FORMULA FOB THE CALCULATION OF THE GEOCENTRIC AND HELIOCENTRIC PLACES, FOR THE DETERMINATION OF THE ORBITS OF PLANETS AND COMETS AND THE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COMPUTATION OF SPECI*L PERTURBATIONS; TOGETHER WITH THE THEORY OF THE COMBI- NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. itl] Ifameriral (fampte an& ^wriliarg BY JAMES C. "WATSON, DIRECTOR OF THE OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IX THE UNIVERSITY OF MICHIGAN. One 8vo. Vol, of about 500 pages, similar in style to Ohauvenet's "MANUAL OP SPHERICAL AND PRACTICAL ASTRONOMY," Price $5, THERE has long existed a want for a volume which should contain, within convenient ' limits, a connected and systematic development of the formulae now generally used in calculations relating to the undisturbed motion of planets and comets, and in the computation of the special perturbations preparatory to the most complete determination of the elements of their orbits, together with such information in regard to the numer- ical applications as may have resulted from the combined ex- 2 PROSPECTUS. perience of astronomers up to the present time. To supply this want the work above announced has been prepared. Since the publication of the Mecanique Celeste and of Gauss' Theoria Motus, many elegant improvements have been intro- duced into the methods given in those works, by the researches of Hansen, Bessel, Encke, and others, and the special investi- gations of those who have labored on these problems are to be found in monographs, now with difficulty to be obtained, or are scattered through the pages of Transactions of learned Societies and in astronomical periodicals such as Zach's Mo- natliche Corresponded, Zeitschrifl fur Astronomic, Astronomische Nachrichten, Astronomical Journal, etc., etc. The work now an- nounced embodies all that is of practical value to be obtained from these different sources; and the author, besides having had much experience in the actual performance of the intricate numerical calculations involved, has read with care everything accessible which could aid in simplifying the different phases of the problems under consideration, or in giving to the formulae greater analytical elegance. The following is a brief synopsis of the Contents of the work: CHAPTER I. Investigation of the fundamental equations of motion and of the formulas for determining, from known elements, the heliocentric and geocentric places of a body, adapted to numerical computation for cases of any excentricity whatever. CHAPTER IL Investigation of the differential formulae which express the relation between the geocentric places of a heavenly body and the variation of the elements of its orbit. CHAPTER III. Determination of the orbits of comets moving in parabolas and the correction of approximate elements by the variation of the geocentric distance. CHAPTER IV. Determination, from three complete observations, of the elements of the orbit of a heavenly body, including the excentricity or form of the conic section. CHAPTER V. Determination of the elements of the orbit of a planet from four observed posi- tions, two of which are complete. PROSPECTUS. CHAPTER VI. Investigation of various formulae for correcting approximate orbits, with illus- trations of the applicability of the different methods. CHAPTER VII. Method of the least squares, theory of the combination of observations, and determination of the most probable system of elements from a series of observa- tions. CHAPTER VIII. Investigation of formulae for the computation of special perturbations. At the end of each chapter will be found a resume of the formulae required in the solution of the different problems, with numerical examples involving data derived from actual observations. The collection of auxiliary tables will be both complete and accurate. In the preparation of the work great care has been taken to secure a logical sequence in the development 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 mathemat- ical knowledge, desire to be informed of the means by which astronomers are enabled to arrive at so many grand results connected with the motions of the heavenly bodies, and by which the grandeur and sublimity of creation are unveiled. The Publishers will spare no pains in the general mechanical execution of the work, which will go promptly forward so soon as the number of subscriptions will warrant. J. B. UPPINOOTT & CO., Publishers, Philadelphia. PUBLICATION OF WATSON'S ASTRONOMY. One Royal Octavo Volume of or?r 65 O payes. DEAR SIR: We have the pleasure of informing you that Prof. Watson's "Theoretical Astronomy," which has been for a long time in course of preparation, is now ready for delivery to subscribers.- Owing to the great ex- pense attending its issue nearly double what was at first expected we are compelled to make the price to subscribers 7.50 instead of $5.00, as we at first con- templated. The retail price through the trade, and to those who have not subscribed in advance of publica- tion, will be $10.00. We have had the pleasure of receiving your name as a subscriber for /cop^/. ; please say how * / V 4/- shall be forwarded to you, And oblige, Faithfully yours, J. B. LIPPINCOTT & CO. PHILADELPHIA, March 2,5, 1868. 14 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRONOMY, MATHEMATICS - STATISTICS LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. rar LD 21-40m-5,'G5 (F4308slO)476 General Library University of California Berkeley